TPTP Problem File: ITP290^4.p
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- Solve Problem
%------------------------------------------------------------------------------
% File : ITP290^4 : TPTP v8.2.0. Released v8.0.0.
% Domain : Interactive Theorem Proving
% Problem : Sledgehammer problem VEBT_DelImperative 00505_031086
% 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 : 0095_VEBT_DelImperative_00505_031086 [Des22]
% Status : Theorem
% Rating : 1.00 v8.1.0
% Syntax : Number of formulae : 10158 (2975 unt; 608 typ; 0 def)
% Number of atoms : 30683 (9811 equ; 0 cnn)
% Maximal formula atoms : 71 ( 3 avg)
% Number of connectives : 204594 (2463 ~; 339 |;2386 &;185076 @)
% ( 0 <=>;14330 =>; 0 <=; 0 <~>)
% Maximal formula depth : 40 ( 7 avg)
% Number of types : 18 ( 17 usr)
% Number of type conns : 3461 (3461 >; 0 *; 0 +; 0 <<)
% Number of symbols : 594 ( 591 usr; 24 con; 0-7 aty)
% Number of variables : 30096 (2542 ^;26260 !; 855 ?;30096 :)
% ( 439 !>; 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 22:49:44.209
%------------------------------------------------------------------------------
% Could-be-implicit typings (31)
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_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_Enum_Ofinite__3,type,
finite_3: $tType ).
thf(ty_t_Enum_Ofinite__2,type,
finite_2: $tType ).
thf(ty_t_Enum_Ofinite__1,type,
finite_1: $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_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_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 (577)
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_Rings_Oring,type,
ring:
!>[A: $tType] : $o ).
thf(sy_cl_Groups_Oplus,type,
plus:
!>[A: $tType] : $o ).
thf(sy_cl_Groups_Ozero,type,
zero:
!>[A: $tType] : $o ).
thf(sy_cl_Num_Onumeral,type,
numeral:
!>[A: $tType] : $o ).
thf(sy_cl_Power_Opower,type,
power:
!>[A: $tType] : $o ).
thf(sy_cl_Fields_Ofield,type,
field:
!>[A: $tType] : $o ).
thf(sy_cl_Groups_Ominus,type,
minus:
!>[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_Oord,type,
ord:
!>[A: $tType] : $o ).
thf(sy_cl_Fields_Oinverse,type,
inverse:
!>[A: $tType] : $o ).
thf(sy_cl_Rings_Osemiring,type,
semiring:
!>[A: $tType] : $o ).
thf(sy_cl_Nat_Oring__char__0,type,
ring_char_0:
!>[A: $tType] : $o ).
thf(sy_cl_Num_Oneg__numeral,type,
neg_numeral:
!>[A: $tType] : $o ).
thf(sy_cl_Orderings_Oorder,type,
order:
!>[A: $tType] : $o ).
thf(sy_cl_Rings_Ocomm__ring,type,
comm_ring:
!>[A: $tType] : $o ).
thf(sy_cl_Rings_Omult__zero,type,
mult_zero:
!>[A: $tType] : $o ).
thf(sy_cl_Type__Length_Olen,type,
type_len:
!>[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_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_Type__Length_Olen0,type,
type_len0:
!>[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_Rings_Oidom__divide,type,
idom_divide:
!>[A: $tType] : $o ).
thf(sy_cl_Rings_Oidom__modulo,type,
idom_modulo:
!>[A: $tType] : $o ).
thf(sy_cl_Transcendental_Oln,type,
ln:
!>[A: $tType] : $o ).
thf(sy_cl_Cardinality_OCARD__1,type,
cARD_1:
!>[A: $tType] : $o ).
thf(sy_cl_Groups_Omonoid__mult,type,
monoid_mult:
!>[A: $tType] : $o ).
thf(sy_cl_Orderings_Olinorder,type,
linorder:
!>[A: $tType] : $o ).
thf(sy_cl_Orderings_Opreorder,type,
preorder:
!>[A: $tType] : $o ).
thf(sy_cl_Parity_Oring__parity,type,
ring_parity:
!>[A: $tType] : $o ).
thf(sy_cl_Rings_Oidom__abs__sgn,type,
idom_abs_sgn:
!>[A: $tType] : $o ).
thf(sy_cl_Rings_Oordered__ring,type,
ordered_ring:
!>[A: $tType] : $o ).
thf(sy_cl_Rings_Ozero__neq__one,type,
zero_neq_one:
!>[A: $tType] : $o ).
thf(sy_cl_Fields_Ofield__char__0,type,
field_char_0:
!>[A: $tType] : $o ).
thf(sy_cl_Groups_Oab__group__add,type,
ab_group_add:
!>[A: $tType] : $o ).
thf(sy_cl_Nat_Osemiring__char__0,type,
semiring_char_0:
!>[A: $tType] : $o ).
thf(sy_cl_Orderings_Oorder__bot,type,
order_bot:
!>[A: $tType] : $o ).
thf(sy_cl_Orderings_Oorder__top,type,
order_top:
!>[A: $tType] : $o ).
thf(sy_cl_Orderings_Owellorder,type,
wellorder:
!>[A: $tType] : $o ).
thf(sy_cl_Rings_Ocomm__semiring,type,
comm_semiring:
!>[A: $tType] : $o ).
thf(sy_cl_Rings_Ozero__less__one,type,
zero_less_one:
!>[A: $tType] : $o ).
thf(sy_cl_Fields_Odivision__ring,type,
division_ring:
!>[A: $tType] : $o ).
thf(sy_cl_Fields_Ofield__abs__sgn,type,
field_abs_sgn:
!>[A: $tType] : $o ).
thf(sy_cl_Groups_Osemigroup__add,type,
semigroup_add:
!>[A: $tType] : $o ).
thf(sy_cl_Num_Osemiring__numeral,type,
semiring_numeral:
!>[A: $tType] : $o ).
thf(sy_cl_Rings_Osemidom__divide,type,
semidom_divide:
!>[A: $tType] : $o ).
thf(sy_cl_Rings_Osemidom__modulo,type,
semidom_modulo:
!>[A: $tType] : $o ).
thf(sy_cl_Complete__Lattices_OInf,type,
complete_Inf:
!>[A: $tType] : $o ).
thf(sy_cl_Complete__Lattices_OSup,type,
complete_Sup:
!>[A: $tType] : $o ).
thf(sy_cl_Groups_Osemigroup__mult,type,
semigroup_mult:
!>[A: $tType] : $o ).
thf(sy_cl_Orderings_Odense__order,type,
dense_order:
!>[A: $tType] : $o ).
thf(sy_cl_Rings_Ocomm__semiring__0,type,
comm_semiring_0:
!>[A: $tType] : $o ).
thf(sy_cl_Rings_Ocomm__semiring__1,type,
comm_semiring_1:
!>[A: $tType] : $o ).
thf(sy_cl_Rings_Olinordered__idom,type,
linordered_idom:
!>[A: $tType] : $o ).
thf(sy_cl_Rings_Olinordered__ring,type,
linordered_ring:
!>[A: $tType] : $o ).
thf(sy_cl_Rings_Osemiring__modulo,type,
semiring_modulo:
!>[A: $tType] : $o ).
thf(sy_cl_Groups_Ocomm__monoid__add,type,
comm_monoid_add:
!>[A: $tType] : $o ).
thf(sy_cl_Parity_Osemiring__parity,type,
semiring_parity:
!>[A: $tType] : $o ).
thf(sy_cl_Rings_Oordered__ring__abs,type,
ordered_ring_abs:
!>[A: $tType] : $o ).
thf(sy_cl_Rings_Oordered__semiring,type,
ordered_semiring:
!>[A: $tType] : $o ).
thf(sy_cl_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_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_Least__significant__bit_Olsb,type,
least_6119777620449941438nt_lsb:
!>[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_Groups_Ocancel__semigroup__add,type,
cancel_semigroup_add:
!>[A: $tType] : $o ).
thf(sy_cl_Groups_Oordered__ab__group__add,type,
ordered_ab_group_add:
!>[A: $tType] : $o ).
thf(sy_cl_Rings_Olinordered__semiring__1,type,
linord6961819062388156250ring_1:
!>[A: $tType] : $o ).
thf(sy_cl_Rings_Oordered__comm__semiring,type,
ordere2520102378445227354miring:
!>[A: $tType] : $o ).
thf(sy_cl_Topological__Spaces_Ot1__space,type,
topological_t1_space:
!>[A: $tType] : $o ).
thf(sy_cl_Topological__Spaces_Ot2__space,type,
topological_t2_space:
!>[A: $tType] : $o ).
thf(sy_cl_Bit__Operations_Osemiring__bits,type,
bit_semiring_bits:
!>[A: $tType] : $o ).
thf(sy_cl_Limits_Otopological__group__add,type,
topolo1633459387980952147up_add:
!>[A: $tType] : $o ).
thf(sy_cl_Real__Vector__Spaces_Odist__norm,type,
real_V6936659425649961206t_norm:
!>[A: $tType] : $o ).
thf(sy_cl_Rings_Ocomm__semiring__1__cancel,type,
comm_s4317794764714335236cancel:
!>[A: $tType] : $o ).
thf(sy_cl_Rings_Olinordered__ring__strict,type,
linord4710134922213307826strict:
!>[A: $tType] : $o ).
thf(sy_cl_Groups_Ocancel__comm__monoid__add,type,
cancel1802427076303600483id_add:
!>[A: $tType] : $o ).
thf(sy_cl_Limits_Otopological__monoid__add,type,
topolo6943815403480290642id_add:
!>[A: $tType] : $o ).
thf(sy_cl_Real__Vector__Spaces_Oreal__field,type,
real_V7773925162809079976_field:
!>[A: $tType] : $o ).
thf(sy_cl_Rings_Oring__1__no__zero__divisors,type,
ring_15535105094025558882visors:
!>[A: $tType] : $o ).
thf(sy_cl_Groups_Ocancel__ab__semigroup__add,type,
cancel2418104881723323429up_add:
!>[A: $tType] : $o ).
thf(sy_cl_Groups_Olinordered__ab__group__add,type,
linord5086331880401160121up_add:
!>[A: $tType] : $o ).
thf(sy_cl_Groups_Oordered__comm__monoid__add,type,
ordere6911136660526730532id_add:
!>[A: $tType] : $o ).
thf(sy_cl_Limits_Otopological__monoid__mult,type,
topolo1898628316856586783d_mult:
!>[A: $tType] : $o ).
thf(sy_cl_Real__Vector__Spaces_Oreal__vector,type,
real_V4867850818363320053vector:
!>[A: $tType] : $o ).
thf(sy_cl_Archimedean__Field_Ofloor__ceiling,type,
archim2362893244070406136eiling:
!>[A: $tType] : $o ).
thf(sy_cl_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_Real__Vector__Spaces_Oreal__algebra,type,
real_V6157519004096292374lgebra:
!>[A: $tType] : $o ).
thf(sy_cl_Rings_Osemiring__no__zero__divisors,type,
semiri3467727345109120633visors:
!>[A: $tType] : $o ).
thf(sy_cl_Boolean__Algebras_Oboolean__algebra,type,
boolea8198339166811842893lgebra:
!>[A: $tType] : $o ).
thf(sy_cl_Rings_Olinordered__semiring__strict,type,
linord8928482502909563296strict:
!>[A: $tType] : $o ).
thf(sy_cl_Topological__Spaces_Operfect__space,type,
topolo8386298272705272623_space:
!>[A: $tType] : $o ).
thf(sy_cl_Topological__Spaces_Ouniform__space,type,
topolo7287701948861334536_space:
!>[A: $tType] : $o ).
thf(sy_cl_Euclidean__Division_Oeuclidean__ring,type,
euclid5891614535332579305n_ring:
!>[A: $tType] : $o ).
thf(sy_cl_Limits_Otopological__semigroup__mult,type,
topolo4211221413907600880p_mult:
!>[A: $tType] : $o ).
thf(sy_cl_Real__Vector__Spaces_Ocomplete__space,type,
real_V8037385150606011577_space:
!>[A: $tType] : $o ).
thf(sy_cl_Real__Vector__Spaces_Oreal__algebra__1,type,
real_V2191834092415804123ebra_1:
!>[A: $tType] : $o ).
thf(sy_cl_Rings_Olinordered__nonzero__semiring,type,
linord181362715937106298miring:
!>[A: $tType] : $o ).
thf(sy_cl_Rings_Osemiring__1__no__zero__divisors,type,
semiri2026040879449505780visors:
!>[A: $tType] : $o ).
thf(sy_cl_Topological__Spaces_Oorder__topology,type,
topolo2564578578187576103pology:
!>[A: $tType] : $o ).
thf(sy_cl_Bit__Operations_Oring__bit__operations,type,
bit_ri3973907225187159222ations:
!>[A: $tType] : $o ).
thf(sy_cl_Complete__Lattices_Ocomplete__lattice,type,
comple6319245703460814977attice:
!>[A: $tType] : $o ).
thf(sy_cl_Groups_Olinordered__ab__semigroup__add,type,
linord4140545234300271783up_add:
!>[A: $tType] : $o ).
thf(sy_cl_Limits_Otopological__comm__monoid__add,type,
topolo5987344860129210374id_add:
!>[A: $tType] : $o ).
thf(sy_cl_Orderings_Ounbounded__dense__linorder,type,
unboun7993243217541854897norder:
!>[A: $tType] : $o ).
thf(sy_cl_Rings_Olinordered__semiring__1__strict,type,
linord715952674999750819strict:
!>[A: $tType] : $o ).
thf(sy_cl_Archimedean__Field_Oarchimedean__field,type,
archim462609752435547400_field:
!>[A: $tType] : $o ).
thf(sy_cl_Complete__Lattices_Ocomplete__linorder,type,
comple5582772986160207858norder:
!>[A: $tType] : $o ).
thf(sy_cl_Limits_Otopological__comm__monoid__mult,type,
topolo4987421752381908075d_mult:
!>[A: $tType] : $o ).
thf(sy_cl_Real__Vector__Spaces_Oreal__div__algebra,type,
real_V5047593784448816457lgebra:
!>[A: $tType] : $o ).
thf(sy_cl_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_Odiscrete__topology,type,
topolo8865339358273720382pology:
!>[A: $tType] : $o ).
thf(sy_cl_Topological__Spaces_Olinorder__topology,type,
topolo1944317154257567458pology:
!>[A: $tType] : $o ).
thf(sy_cl_Topological__Spaces_Otopological__space,type,
topolo4958980785337419405_space:
!>[A: $tType] : $o ).
thf(sy_cl_Euclidean__Division_Oeuclidean__semiring,type,
euclid3725896446679973847miring:
!>[A: $tType] : $o ).
thf(sy_cl_Groups_Ocanonically__ordered__monoid__add,type,
canoni5634975068530333245id_add:
!>[A: $tType] : $o ).
thf(sy_cl_Groups_Oordered__cancel__comm__monoid__add,type,
ordere8940638589300402666id_add:
!>[A: $tType] : $o ).
thf(sy_cl_Groups_Ostrict__ordered__comm__monoid__add,type,
strict7427464778891057005id_add:
!>[A: $tType] : $o ).
thf(sy_cl_Real__Vector__Spaces_Oreal__normed__vector,type,
real_V822414075346904944vector:
!>[A: $tType] : $o ).
thf(sy_cl_Rings_Olinordered__comm__semiring__strict,type,
linord2810124833399127020strict:
!>[A: $tType] : $o ).
thf(sy_cl_Bit__Operations_Osemiring__bit__operations,type,
bit_se359711467146920520ations:
!>[A: $tType] : $o ).
thf(sy_cl_Groups_Oordered__ab__semigroup__add__imp__le,type,
ordere2412721322843649153imp_le:
!>[A: $tType] : $o ).
thf(sy_cl_Groups_Oordered__cancel__ab__semigroup__add,type,
ordere580206878836729694up_add:
!>[A: $tType] : $o ).
thf(sy_cl_Groups_Oordered__cancel__comm__monoid__diff,type,
ordere1170586879665033532d_diff:
!>[A: $tType] : $o ).
thf(sy_cl_Groups_Ostrict__ordered__ab__semigroup__add,type,
strict9044650504122735259up_add:
!>[A: $tType] : $o ).
thf(sy_cl_Real__Vector__Spaces_Oordered__real__vector,type,
real_V5355595471888546746vector:
!>[A: $tType] : $o ).
thf(sy_cl_Real__Vector__Spaces_Oreal__normed__algebra,type,
real_V4412858255891104859lgebra:
!>[A: $tType] : $o ).
thf(sy_cl_Rings_Osemiring__no__zero__divisors__cancel,type,
semiri6575147826004484403cancel:
!>[A: $tType] : $o ).
thf(sy_cl_Euclidean__Division_Oeuclidean__ring__cancel,type,
euclid8851590272496341667cancel:
!>[A: $tType] : $o ).
thf(sy_cl_Real__Vector__Spaces_Oreal__normed__algebra__1,type,
real_V2822296259951069270ebra_1:
!>[A: $tType] : $o ).
thf(sy_cl_Divides_Ounique__euclidean__semiring__numeral,type,
unique1627219031080169319umeral:
!>[A: $tType] : $o ).
thf(sy_cl_Real__Vector__Spaces_Oreal__normed__div__algebra,type,
real_V8999393235501362500lgebra:
!>[A: $tType] : $o ).
thf(sy_cl_Topological__Spaces_Ofirst__countable__topology,type,
topolo3112930676232923870pology:
!>[A: $tType] : $o ).
thf(sy_cl_Euclidean__Division_Oeuclidean__semiring__cancel,type,
euclid4440199948858584721cancel:
!>[A: $tType] : $o ).
thf(sy_cl_Euclidean__Division_Ounique__euclidean__semiring,type,
euclid3128863361964157862miring:
!>[A: $tType] : $o ).
thf(sy_cl_Topological__Spaces_Olinear__continuum__topology,type,
topolo8458572112393995274pology:
!>[A: $tType] : $o ).
thf(sy_cl_Groups_Oordered__ab__semigroup__monoid__add__imp__le,type,
ordere1937475149494474687imp_le:
!>[A: $tType] : $o ).
thf(sy_cl_Conditionally__Complete__Lattices_Olinear__continuum,type,
condit5016429287641298734tinuum:
!>[A: $tType] : $o ).
thf(sy_cl_Euclidean__Division_Ounique__euclidean__ring__with__nat,type,
euclid8789492081693882211th_nat:
!>[A: $tType] : $o ).
thf(sy_cl_Euclidean__Division_Ounique__euclidean__semiring__with__nat,type,
euclid5411537665997757685th_nat:
!>[A: $tType] : $o ).
thf(sy_cl_Countable__Complete__Lattices_Ocountable__complete__lattice,type,
counta3822494911875563373attice:
!>[A: $tType] : $o ).
thf(sy_cl_Semiring__Normalization_Ocomm__semiring__1__cancel__crossproduct,type,
semiri1453513574482234551roduct:
!>[A: $tType] : $o ).
thf(sy_cl_Bit__Operations_Ounique__euclidean__semiring__with__bit__operations,type,
bit_un5681908812861735899ations:
!>[A: $tType] : $o ).
thf(sy_cl_Conditionally__Complete__Lattices_Oconditionally__complete__lattice,type,
condit1219197933456340205attice:
!>[A: $tType] : $o ).
thf(sy_cl_Conditionally__Complete__Lattices_Oconditionally__complete__linorder,type,
condit6923001295902523014norder:
!>[A: $tType] : $o ).
thf(sy_c_Archimedean__Field_Oceiling,type,
archimedean_ceiling:
!>[A: $tType] : ( A > int ) ).
thf(sy_c_Archimedean__Field_Ofloor__ceiling__class_Ofloor,type,
archim6421214686448440834_floor:
!>[A: $tType] : ( A > int ) ).
thf(sy_c_Archimedean__Field_Ofrac,type,
archimedean_frac:
!>[A: $tType] : ( A > A ) ).
thf(sy_c_Archimedean__Field_Oround,type,
archimedean_round:
!>[A: $tType] : ( A > int ) ).
thf(sy_c_Array__Time_Ofreeze,type,
array_freeze:
!>[A: $tType] : ( ( array @ A ) > ( heap_Time_Heap @ ( list @ A ) ) ) ).
thf(sy_c_Array__Time_Olen,type,
array_len:
!>[A: $tType] : ( ( array @ A ) > ( heap_Time_Heap @ nat ) ) ).
thf(sy_c_Array__Time_Onew,type,
array_new:
!>[A: $tType] : ( nat > 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_Oof__list,type,
array_of_list:
!>[A: $tType] : ( ( list @ A ) > ( heap_Time_Heap @ ( array @ 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_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_Automation_OFI__QUERY,type,
fI_QUERY: assn > assn > assn > $o ).
thf(sy_c_Binomial_Obinomial,type,
binomial: nat > nat > nat ).
thf(sy_c_Binomial_Ogbinomial,type,
gbinomial:
!>[A: $tType] : ( A > nat > A ) ).
thf(sy_c_Bit__Operations_Oand__int__rel,type,
bit_and_int_rel: ( product_prod @ int @ int ) > ( product_prod @ int @ int ) > $o ).
thf(sy_c_Bit__Operations_Oconcat__bit,type,
bit_concat_bit: nat > int > int > int ).
thf(sy_c_Bit__Operations_Oor__not__num__neg,type,
bit_or_not_num_neg: num > num > num ).
thf(sy_c_Bit__Operations_Oor__not__num__neg__rel,type,
bit_or3848514188828904588eg_rel: ( product_prod @ num @ num ) > ( product_prod @ num @ num ) > $o ).
thf(sy_c_Bit__Operations_Oring__bit__operations__class_Onot,type,
bit_ri4277139882892585799ns_not:
!>[A: $tType] : ( A > A ) ).
thf(sy_c_Bit__Operations_Oring__bit__operations__class_Osigned__take__bit,type,
bit_ri4674362597316999326ke_bit:
!>[A: $tType] : ( nat > A > A ) ).
thf(sy_c_Bit__Operations_Osemiring__bit__operations__class_Oand,type,
bit_se5824344872417868541ns_and:
!>[A: $tType] : ( A > A > A ) ).
thf(sy_c_Bit__Operations_Osemiring__bit__operations__class_Odrop__bit,type,
bit_se4197421643247451524op_bit:
!>[A: $tType] : ( nat > A > A ) ).
thf(sy_c_Bit__Operations_Osemiring__bit__operations__class_Oflip__bit,type,
bit_se8732182000553998342ip_bit:
!>[A: $tType] : ( nat > A > A ) ).
thf(sy_c_Bit__Operations_Osemiring__bit__operations__class_Omask,type,
bit_se2239418461657761734s_mask:
!>[A: $tType] : ( nat > A ) ).
thf(sy_c_Bit__Operations_Osemiring__bit__operations__class_Oor,type,
bit_se1065995026697491101ons_or:
!>[A: $tType] : ( A > A > A ) ).
thf(sy_c_Bit__Operations_Osemiring__bit__operations__class_Opush__bit,type,
bit_se4730199178511100633sh_bit:
!>[A: $tType] : ( nat > A > A ) ).
thf(sy_c_Bit__Operations_Osemiring__bit__operations__class_Oset__bit,type,
bit_se5668285175392031749et_bit:
!>[A: $tType] : ( nat > A > A ) ).
thf(sy_c_Bit__Operations_Osemiring__bit__operations__class_Otake__bit,type,
bit_se2584673776208193580ke_bit:
!>[A: $tType] : ( nat > A > A ) ).
thf(sy_c_Bit__Operations_Osemiring__bit__operations__class_Ounset__bit,type,
bit_se2638667681897837118et_bit:
!>[A: $tType] : ( nat > A > A ) ).
thf(sy_c_Bit__Operations_Osemiring__bit__operations__class_Oxor,type,
bit_se5824344971392196577ns_xor:
!>[A: $tType] : ( A > A > A ) ).
thf(sy_c_Bit__Operations_Osemiring__bits__class_Obit,type,
bit_se5641148757651400278ts_bit:
!>[A: $tType] : ( A > nat > $o ) ).
thf(sy_c_Bit__Operations_Osemiring__bits__class_Opossible__bit,type,
bit_se6407376104438227557le_bit:
!>[A: $tType] : ( ( itself @ A ) > nat > $o ) ).
thf(sy_c_Bit__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_OBit__integer,type,
bits_Bit_integer: code_integer > $o > code_integer ).
thf(sy_c_Bits__Integer_Obin__last__integer,type,
bits_b8758750999018896077nteger: code_integer > $o ).
thf(sy_c_Bits__Integer_Obin__rest__integer,type,
bits_b2549910563261871055nteger: 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_Odup,type,
code_dup: code_integer > code_integer ).
thf(sy_c_Code__Numeral_Ointeger_Ointeger__of__int,type,
code_integer_of_int: int > code_integer ).
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_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_OArg,type,
arg: complex > real ).
thf(sy_c_Complex_Ocis,type,
cis: real > complex ).
thf(sy_c_Complex_Ocomplex_OComplex,type,
complex2: real > real > complex ).
thf(sy_c_Complex_Ocsqrt,type,
csqrt: complex > complex ).
thf(sy_c_Complex_Oimaginary__unit,type,
imaginary_unit: complex ).
thf(sy_c_Deriv_Odifferentiable,type,
differentiable:
!>[A: $tType,B: $tType] : ( ( A > B ) > ( filter @ A ) > $o ) ).
thf(sy_c_Deriv_Ohas__derivative,type,
has_derivative:
!>[A: $tType,B: $tType] : ( ( A > B ) > ( A > B ) > ( filter @ A ) > $o ) ).
thf(sy_c_Deriv_Ohas__field__derivative,type,
has_field_derivative:
!>[A: $tType] : ( ( A > A ) > A > ( filter @ A ) > $o ) ).
thf(sy_c_Divides_Oadjust__div,type,
adjust_div: ( product_prod @ int @ int ) > int ).
thf(sy_c_Divides_Odivmod__nat,type,
divmod_nat: nat > nat > ( product_prod @ nat @ nat ) ).
thf(sy_c_Divides_Oeucl__rel__int,type,
eucl_rel_int: int > int > ( product_prod @ int @ int ) > $o ).
thf(sy_c_Divides_Ounique__euclidean__semiring__numeral__class_Odivides__aux,type,
unique5940410009612947441es_aux:
!>[A: $tType] : ( ( product_prod @ A @ A ) > $o ) ).
thf(sy_c_Divides_Ounique__euclidean__semiring__numeral__class_Odivmod,type,
unique8689654367752047608divmod:
!>[A: $tType] : ( num > num > ( product_prod @ A @ A ) ) ).
thf(sy_c_Divides_Ounique__euclidean__semiring__numeral__class_Odivmod__step,type,
unique1321980374590559556d_step:
!>[A: $tType] : ( num > ( product_prod @ A @ A ) > ( product_prod @ A @ A ) ) ).
thf(sy_c_Factorial_Ocomm__semiring__1__class_Opochhammer,type,
comm_s3205402744901411588hammer:
!>[A: $tType] : ( A > nat > A ) ).
thf(sy_c_Factorial_Osemiring__char__0__class_Ofact,type,
semiring_char_0_fact:
!>[A: $tType] : ( nat > A ) ).
thf(sy_c_Fields_Oinverse__class_Oinverse,type,
inverse_inverse:
!>[A: $tType] : ( A > A ) ).
thf(sy_c_Filter_Oat__bot,type,
at_bot:
!>[A: $tType] : ( filter @ A ) ).
thf(sy_c_Filter_Oat__top,type,
at_top:
!>[A: $tType] : ( filter @ A ) ).
thf(sy_c_Filter_Oeventually,type,
eventually:
!>[A: $tType] : ( ( A > $o ) > ( filter @ A ) > $o ) ).
thf(sy_c_Filter_Ofilterlim,type,
filterlim:
!>[A: $tType,B: $tType] : ( ( A > B ) > ( filter @ B ) > ( filter @ A ) > $o ) ).
thf(sy_c_Finite__Set_Ocard,type,
finite_card:
!>[B: $tType] : ( ( set @ B ) > nat ) ).
thf(sy_c_Finite__Set_Ofinite,type,
finite_finite2:
!>[A: $tType] : ( ( set @ A ) > $o ) ).
thf(sy_c_Finite__Set_Ofold,type,
finite_fold:
!>[A: $tType,B: $tType] : ( ( A > B > B ) > B > ( set @ A ) > B ) ).
thf(sy_c_Fun_Obij__betw,type,
bij_betw:
!>[A: $tType,B: $tType] : ( ( A > B ) > ( set @ A ) > ( set @ B ) > $o ) ).
thf(sy_c_Fun_Oinj__on,type,
inj_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_Osemiring__1__class_Osemiring__char,type,
semiri4206861660011772517g_char:
!>[A: $tType] : ( ( itself @ A ) > nat ) ).
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__class_Osum,type,
groups7311177749621191930dd_sum:
!>[B: $tType,A: $tType] : ( ( B > A ) > ( set @ B ) > A ) ).
thf(sy_c_Groups__Big_Ocomm__monoid__add__class_Osum_H,type,
groups1027152243600224163dd_sum:
!>[C: $tType,A: $tType] : ( ( C > A ) > ( set @ C ) > A ) ).
thf(sy_c_Groups__Big_Ocomm__monoid__mult__class_Oprod,type,
groups7121269368397514597t_prod:
!>[B: $tType,A: $tType] : ( ( B > A ) > ( set @ B ) > A ) ).
thf(sy_c_Groups__List_Ocomm__semiring__0__class_Ohorner__sum,type,
groups4207007520872428315er_sum:
!>[B: $tType,A: $tType] : ( ( B > A ) > A > ( list @ B ) > A ) ).
thf(sy_c_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_Hash__Instances_Ohash__code__list,type,
hash_hash_code_list:
!>[A: $tType] : ( ( A > uint32 ) > ( list @ A ) > uint32 ) ).
thf(sy_c_Hash__Instances_Ohash__code__option,type,
hash_h1887023736457453652option:
!>[A: $tType] : ( ( A > uint32 ) > ( option @ A ) > uint32 ) ).
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_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__Big_Olinorder__class_OMax,type,
lattic643756798349783984er_Max:
!>[A: $tType] : ( ( set @ A ) > A ) ).
thf(sy_c_Least__significant__bit_Olsb__class_Olsb,type,
least_8051144512741203767sb_lsb:
!>[A: $tType] : ( A > $o ) ).
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_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_Ofoldr,type,
foldr:
!>[A: $tType,B: $tType] : ( ( A > B > B ) > ( list @ A ) > B > B ) ).
thf(sy_c_List_Olist_OCons,type,
cons:
!>[A: $tType] : ( A > ( list @ A ) > ( list @ A ) ) ).
thf(sy_c_List_Olist_Omap,type,
map:
!>[A: $tType,Aa: $tType] : ( ( A > Aa ) > ( list @ A ) > ( list @ Aa ) ) ).
thf(sy_c_List_Olist_Oset,type,
set2:
!>[A: $tType] : ( ( list @ A ) > ( set @ A ) ) ).
thf(sy_c_List_Olist__update,type,
list_update:
!>[A: $tType] : ( ( list @ A ) > nat > 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_Oproduct,type,
product:
!>[A: $tType,B: $tType] : ( ( list @ A ) > ( list @ B ) > ( list @ ( product_prod @ A @ B ) ) ) ).
thf(sy_c_List_Oproduct__lists,type,
product_lists:
!>[A: $tType] : ( ( list @ ( list @ A ) ) > ( list @ ( list @ A ) ) ) ).
thf(sy_c_List_Oreplicate,type,
replicate:
!>[A: $tType] : ( nat > A > ( list @ A ) ) ).
thf(sy_c_List_Osubseqs,type,
subseqs:
!>[A: $tType] : ( ( list @ A ) > ( list @ ( list @ A ) ) ) ).
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_Map_Oran,type,
ran:
!>[A: $tType,B: $tType] : ( ( A > ( option @ B ) ) > ( set @ B ) ) ).
thf(sy_c_Misc_Ofun__of__rel,type,
fun_of_rel:
!>[B: $tType,A: $tType] : ( ( set @ ( product_prod @ B @ A ) ) > B > A ) ).
thf(sy_c_Misc_Oinv__on,type,
inv_on:
!>[A: $tType,B: $tType] : ( ( A > B ) > ( set @ A ) > B > A ) ).
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_Oslice,type,
slice:
!>[A: $tType] : ( nat > nat > ( list @ A ) > ( list @ A ) ) ).
thf(sy_c_Most__significant__bit_Omsb__class_Omsb,type,
most_s684356279273892711sb_msb:
!>[A: $tType] : ( A > $o ) ).
thf(sy_c_Nat_OSuc,type,
suc: nat > nat ).
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_Oset__decode,type,
nat_set_decode: nat > ( set @ nat ) ).
thf(sy_c_Nat__Bijection_Oset__encode,type,
nat_set_encode: ( set @ nat ) > nat ).
thf(sy_c_Nat__Bijection_Otriangle,type,
nat_triangle: nat > nat ).
thf(sy_c_NthRoot_Oroot,type,
root: nat > real > real ).
thf(sy_c_NthRoot_Osqrt,type,
sqrt: real > real ).
thf(sy_c_Num_OBitM,type,
bitM: num > num ).
thf(sy_c_Num_Oinc,type,
inc: num > num ).
thf(sy_c_Num_Oneg__numeral__class_Odbl,type,
neg_numeral_dbl:
!>[A: $tType] : ( A > A ) ).
thf(sy_c_Num_Oneg__numeral__class_Odbl__dec,type,
neg_numeral_dbl_dec:
!>[A: $tType] : ( A > A ) ).
thf(sy_c_Num_Oneg__numeral__class_Odbl__inc,type,
neg_numeral_dbl_inc:
!>[A: $tType] : ( A > A ) ).
thf(sy_c_Num_Onum_OBit0,type,
bit0: num > num ).
thf(sy_c_Num_Onum_OBit1,type,
bit1: num > num ).
thf(sy_c_Num_Onum_OOne,type,
one2: num ).
thf(sy_c_Num_Onum_Osize__num,type,
size_num: num > nat ).
thf(sy_c_Num_Onumeral__class_Onumeral,type,
numeral_numeral:
!>[A: $tType] : ( num > A ) ).
thf(sy_c_Num_Opow,type,
pow: num > num > num ).
thf(sy_c_Num_Opred__numeral,type,
pred_numeral: num > nat ).
thf(sy_c_Numeral__Type_Obit1_OAbs__bit1,type,
numeral_Abs_bit1:
!>[A: $tType] : ( int > ( numeral_bit1 @ A ) ) ).
thf(sy_c_Option_Ooption_ONone,type,
none:
!>[A: $tType] : ( option @ A ) ).
thf(sy_c_Option_Ooption_OSome,type,
some:
!>[A: $tType] : ( A > ( option @ A ) ) ).
thf(sy_c_Option_Ooption_Osize__option,type,
size_option:
!>[A: $tType] : ( ( A > nat ) > ( option @ A ) > nat ) ).
thf(sy_c_Option_Ooption_Othe,type,
the2:
!>[A: $tType] : ( ( option @ A ) > A ) ).
thf(sy_c_Option_Othese,type,
these:
!>[A: $tType] : ( ( set @ ( option @ A ) ) > ( set @ A ) ) ).
thf(sy_c_Orderings_Obot__class_Obot,type,
bot_bot:
!>[A: $tType] : A ).
thf(sy_c_Orderings_Oord__class_Oless,type,
ord_less:
!>[A: $tType] : ( A > A > $o ) ).
thf(sy_c_Orderings_Oord__class_Oless__eq,type,
ord_less_eq:
!>[A: $tType] : ( A > A > $o ) ).
thf(sy_c_Orderings_Oord__class_Omax,type,
ord_max:
!>[A: $tType] : ( A > A > A ) ).
thf(sy_c_Orderings_Oord__class_Omin,type,
ord_min:
!>[A: $tType] : ( A > A > A ) ).
thf(sy_c_Orderings_Oorder__class_OGreatest,type,
order_Greatest:
!>[A: $tType] : ( ( A > $o ) > A ) ).
thf(sy_c_Orderings_Oorder__class_Oantimono,type,
order_antimono:
!>[A: $tType,B: $tType] : ( ( A > B ) > $o ) ).
thf(sy_c_Orderings_Otop__class_Otop,type,
top_top:
!>[A: $tType] : A ).
thf(sy_c_Power_Opower__class_Opower,type,
power_power:
!>[A: $tType] : ( A > nat > A ) ).
thf(sy_c_Product__Type_OPair,type,
product_Pair:
!>[A: $tType,B: $tType] : ( A > B > ( product_prod @ A @ B ) ) ).
thf(sy_c_Product__Type_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_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_Rat_OFrct,type,
frct: ( product_prod @ int @ int ) > rat ).
thf(sy_c_Rat_Onormalize,type,
normalize: ( product_prod @ int @ int ) > ( product_prod @ int @ int ) ).
thf(sy_c_Rat_Oof__int,type,
of_int: int > rat ).
thf(sy_c_Rat_Oquotient__of,type,
quotient_of: rat > ( 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_Odist__class_Odist,type,
real_V557655796197034286t_dist:
!>[A: $tType] : ( A > A > real ) ).
thf(sy_c_Real__Vector__Spaces_Onorm__class_Onorm,type,
real_V7770717601297561774m_norm:
!>[A: $tType] : ( A > real ) ).
thf(sy_c_Real__Vector__Spaces_Oof__real,type,
real_Vector_of_real:
!>[A: $tType] : ( real > A ) ).
thf(sy_c_Real__Vector__Spaces_OscaleR__class_OscaleR,type,
real_V8093663219630862766scaleR:
!>[A: $tType] : ( real > A > A ) ).
thf(sy_c_Refine__Imp__Hol_Orefines,type,
refine_Imp_refines:
!>[A: $tType] : ( ( heap_Time_Heap @ A ) > ( heap_Time_Heap @ 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_OCollect,type,
collect:
!>[A: $tType] : ( ( A > $o ) > ( set @ A ) ) ).
thf(sy_c_Set_Ofilter,type,
filter2:
!>[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_Othe__elem,type,
the_elem:
!>[A: $tType] : ( ( set @ A ) > A ) ).
thf(sy_c_Set__Interval_Ofold__atLeastAtMost__nat,type,
set_fo6178422350223883121st_nat:
!>[A: $tType] : ( ( nat > A > A ) > nat > nat > A > A ) ).
thf(sy_c_Set__Interval_Ofold__atLeastAtMost__nat__rel,type,
set_fo1817059534552279752at_rel:
!>[A: $tType] : ( ( product_prod @ ( nat > A > A ) @ ( product_prod @ nat @ ( product_prod @ nat @ A ) ) ) > ( product_prod @ ( nat > A > A ) @ ( product_prod @ nat @ ( product_prod @ nat @ A ) ) ) > $o ) ).
thf(sy_c_Set__Interval_Oord__class_OatLeast,type,
set_ord_atLeast:
!>[A: $tType] : ( A > ( set @ A ) ) ).
thf(sy_c_Set__Interval_Oord__class_OatLeastAtMost,type,
set_or1337092689740270186AtMost:
!>[A: $tType] : ( A > A > ( set @ A ) ) ).
thf(sy_c_Set__Interval_Oord__class_OatLeastLessThan,type,
set_or7035219750837199246ssThan:
!>[A: $tType] : ( A > A > ( set @ A ) ) ).
thf(sy_c_Set__Interval_Oord__class_OatMost,type,
set_ord_atMost:
!>[A: $tType] : ( A > ( set @ A ) ) ).
thf(sy_c_Set__Interval_Oord__class_OgreaterThan,type,
set_ord_greaterThan:
!>[A: $tType] : ( A > ( set @ A ) ) ).
thf(sy_c_Set__Interval_Oord__class_OgreaterThanAtMost,type,
set_or3652927894154168847AtMost:
!>[A: $tType] : ( A > A > ( set @ A ) ) ).
thf(sy_c_Set__Interval_Oord__class_OgreaterThanLessThan,type,
set_or5935395276787703475ssThan:
!>[A: $tType] : ( A > A > ( set @ A ) ) ).
thf(sy_c_Set__Interval_Oord__class_OlessThan,type,
set_ord_lessThan:
!>[A: $tType] : ( A > ( set @ A ) ) ).
thf(sy_c_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_Time__Reasoning_OTBOUND,type,
time_TBOUND:
!>[A: $tType] : ( ( heap_Time_Heap @ A ) > nat > $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_Ocontinuous,type,
topolo3448309680560233919inuous:
!>[A: $tType,B: $tType] : ( ( filter @ A ) > ( A > B ) > $o ) ).
thf(sy_c_Topological__Spaces_Ocontinuous__on,type,
topolo81223032696312382ous_on:
!>[A: $tType,B: $tType] : ( ( set @ A ) > ( A > B ) > $o ) ).
thf(sy_c_Topological__Spaces_Omonoseq,type,
topological_monoseq:
!>[A: $tType] : ( ( nat > A ) > $o ) ).
thf(sy_c_Topological__Spaces_Oopen__class_Oopen,type,
topolo1002775350975398744n_open:
!>[A: $tType] : ( ( set @ A ) > $o ) ).
thf(sy_c_Topological__Spaces_Ot2__space__class_OLim,type,
topolo3827282254853284352ce_Lim:
!>[F: $tType,A: $tType] : ( ( filter @ F ) > ( F > A ) > A ) ).
thf(sy_c_Topological__Spaces_Otopological__space__class_Oat__within,type,
topolo174197925503356063within:
!>[A: $tType] : ( A > ( set @ A ) > ( filter @ A ) ) ).
thf(sy_c_Topological__Spaces_Otopological__space__class_Onhds,type,
topolo7230453075368039082e_nhds:
!>[A: $tType] : ( A > ( filter @ A ) ) ).
thf(sy_c_Topological__Spaces_Ouniform__space__class_OCauchy,type,
topolo3814608138187158403Cauchy:
!>[A: $tType] : ( ( nat > A ) > $o ) ).
thf(sy_c_Topological__Spaces_Ouniform__space__class_Ototally__bounded,type,
topolo6688025880775521714ounded:
!>[A: $tType] : ( ( set @ A ) > $o ) ).
thf(sy_c_Transcendental_Oarccos,type,
arccos: real > real ).
thf(sy_c_Transcendental_Oarcosh,type,
arcosh:
!>[A: $tType] : ( A > A ) ).
thf(sy_c_Transcendental_Oarcsin,type,
arcsin: real > real ).
thf(sy_c_Transcendental_Oarctan,type,
arctan: real > real ).
thf(sy_c_Transcendental_Oarsinh,type,
arsinh:
!>[A: $tType] : ( A > A ) ).
thf(sy_c_Transcendental_Oartanh,type,
artanh:
!>[A: $tType] : ( A > A ) ).
thf(sy_c_Transcendental_Ocos,type,
cos:
!>[A: $tType] : ( A > A ) ).
thf(sy_c_Transcendental_Ocos__coeff,type,
cos_coeff: nat > real ).
thf(sy_c_Transcendental_Ocosh,type,
cosh:
!>[A: $tType] : ( A > A ) ).
thf(sy_c_Transcendental_Ocot,type,
cot:
!>[A: $tType] : ( A > A ) ).
thf(sy_c_Transcendental_Odiffs,type,
diffs:
!>[A: $tType] : ( ( nat > A ) > nat > A ) ).
thf(sy_c_Transcendental_Oexp,type,
exp:
!>[A: $tType] : ( A > A ) ).
thf(sy_c_Transcendental_Oln__class_Oln,type,
ln_ln:
!>[A: $tType] : ( A > A ) ).
thf(sy_c_Transcendental_Olog,type,
log: real > real > real ).
thf(sy_c_Transcendental_Opi,type,
pi: real ).
thf(sy_c_Transcendental_Opowr,type,
powr:
!>[A: $tType] : ( A > A > A ) ).
thf(sy_c_Transcendental_Osin,type,
sin:
!>[A: $tType] : ( A > A ) ).
thf(sy_c_Transcendental_Osin__coeff,type,
sin_coeff: nat > real ).
thf(sy_c_Transcendental_Osinh,type,
sinh:
!>[A: $tType] : ( A > A ) ).
thf(sy_c_Transcendental_Otan,type,
tan:
!>[A: $tType] : ( A > A ) ).
thf(sy_c_Transcendental_Otanh,type,
tanh:
!>[A: $tType] : ( A > A ) ).
thf(sy_c_Type__Length_Olen0__class_Olen__of,type,
type_len0_len_of:
!>[A: $tType] : ( ( itself @ A ) > 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,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_OminNulli__rel,type,
vEBT_V5740978063120863272li_rel: vEBT_VEBTi > vEBT_VEBTi > $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__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_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__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__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__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_Ooption__shift__rel,type,
vEBT_V459564278314245337ft_rel:
!>[A: $tType] : ( ( product_prod @ ( A > A > A ) @ ( product_prod @ ( option @ A ) @ ( option @ A ) ) ) > ( product_prod @ ( A > A > A ) @ ( product_prod @ ( option @ A ) @ ( option @ A ) ) ) > $o ) ).
thf(sy_c_VEBT__MinMax_OVEBT__internal_Opower,type,
vEBT_VEBT_power: ( option @ nat ) > ( option @ nat ) > ( option @ nat ) ).
thf(sy_c_VEBT__MinMax_Ovebt__maxt,type,
vEBT_vebt_maxt: vEBT_VEBT > ( option @ nat ) ).
thf(sy_c_VEBT__MinMax_Ovebt__maxt__rel,type,
vEBT_vebt_maxt_rel: vEBT_VEBT > vEBT_VEBT > $o ).
thf(sy_c_VEBT__MinMax_Ovebt__mint,type,
vEBT_vebt_mint: vEBT_VEBT > ( option @ nat ) ).
thf(sy_c_VEBT__MinMax_Ovebt__mint__rel,type,
vEBT_vebt_mint_rel: vEBT_VEBT > vEBT_VEBT > $o ).
thf(sy_c_VEBT__Pred_Ois__pred__in__set,type,
vEBT_is_pred_in_set: ( set @ nat ) > nat > nat > $o ).
thf(sy_c_VEBT__Pred_Ovebt__pred,type,
vEBT_vebt_pred: vEBT_VEBT > nat > ( option @ nat ) ).
thf(sy_c_VEBT__Pred_Ovebt__pred__rel,type,
vEBT_vebt_pred_rel: ( product_prod @ vEBT_VEBT @ nat ) > ( product_prod @ vEBT_VEBT @ nat ) > $o ).
thf(sy_c_VEBT__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_Wellfounded_Oaccp,type,
accp:
!>[A: $tType] : ( ( A > A > $o ) > A > $o ) ).
thf(sy_c_Wellfounded_Omeasure,type,
measure:
!>[A: $tType] : ( ( A > nat ) > ( set @ ( product_prod @ A @ A ) ) ) ).
thf(sy_c_Word_OWord,type,
word2:
!>[A: $tType] : ( int > ( word @ A ) ) ).
thf(sy_c_Word_Oeven__word,type,
even_word:
!>[A: $tType] : ( ( word @ A ) > $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_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__pred,type,
word_pred:
!>[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_fChoice,type,
fChoice:
!>[A: $tType] : ( ( A > $o ) > A ) ).
thf(sy_c_member,type,
member:
!>[A: $tType] : ( A > ( set @ A ) > $o ) ).
thf(sy_v_aktnode____,type,
aktnode: vEBT_VEBT ).
thf(sy_v_ma____,type,
ma: nat ).
thf(sy_v_mi____,type,
mi: nat ).
thf(sy_v_minew____,type,
minew: nat ).
thf(sy_v_na____,type,
na: nat ).
thf(sy_v_newnode____,type,
newnode: vEBT_VEBT ).
thf(sy_v_summary____,type,
summary: vEBT_VEBT ).
thf(sy_v_tia____,type,
tia: vEBT_VEBTi ).
thf(sy_v_treeList____,type,
treeList: list @ vEBT_VEBT ).
thf(sy_v_tree__is______,type,
tree_is: list @ vEBT_VEBTi ).
thf(sy_v_va____,type,
va: nat ).
thf(sy_v_x13______,type,
x13: array @ vEBT_VEBTi ).
thf(sy_v_x14______,type,
x14: vEBT_VEBTi ).
thf(sy_v_xa____,type,
xa: nat ).
thf(sy_v_xaa______,type,
xaa: vEBT_VEBT ).
thf(sy_v_xb______,type,
xb: vEBT_VEBTi ).
thf(sy_v_xba______,type,
xba: option @ nat ).
thf(sy_v_xc,type,
xc: vEBT_VEBTi ).
thf(sy_v_xnew____,type,
xnew: nat ).
% Relevant facts (8175)
thf(fact_0_groupy,axiom,
! [A2: assn,B2: assn,C2: assn,D: assn,X: assn] :
( ( entails @ ( times_times @ assn @ ( times_times @ assn @ A2 @ B2 ) @ ( times_times @ assn @ C2 @ D ) ) @ X )
=> ( entails @ ( times_times @ assn @ ( times_times @ assn @ ( times_times @ assn @ A2 @ B2 ) @ C2 ) @ D ) @ X ) ) ).
% groupy
thf(fact_1_midextr,axiom,
! [P: assn,Q: assn,Q2: assn,R: assn,X: assn] :
( ( entails @ ( times_times @ assn @ ( times_times @ assn @ ( times_times @ assn @ P @ Q ) @ Q2 ) @ R ) @ X )
=> ( entails @ ( times_times @ assn @ ( times_times @ assn @ ( times_times @ assn @ P @ R ) @ Q ) @ Q2 ) @ X ) ) ).
% midextr
thf(fact_2_swappa,axiom,
! [B2: assn,A2: assn,C2: assn,X: assn] :
( ( entails @ ( times_times @ assn @ ( times_times @ assn @ B2 @ A2 ) @ C2 ) @ X )
=> ( entails @ ( times_times @ assn @ ( times_times @ assn @ A2 @ B2 ) @ C2 ) @ X ) ) ).
% swappa
thf(fact_3_minNullmin,axiom,
! [T2: vEBT_VEBT] :
( ( vEBT_VEBT_minNull @ T2 )
=> ( ( vEBT_vebt_mint @ T2 )
= ( none @ nat ) ) ) ).
% minNullmin
thf(fact_4_minminNull,axiom,
! [T2: vEBT_VEBT] :
( ( ( vEBT_vebt_mint @ T2 )
= ( none @ nat ) )
=> ( vEBT_VEBT_minNull @ T2 ) ) ).
% minminNull
thf(fact_5_bit__split__inv,axiom,
! [X2: nat,D2: nat] :
( ( vEBT_VEBT_bit_concat @ ( vEBT_VEBT_high @ X2 @ D2 ) @ ( vEBT_VEBT_low @ X2 @ D2 ) @ D2 )
= X2 ) ).
% bit_split_inv
thf(fact_6_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_7_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_8_nth__list__update__eq,axiom,
! [A: $tType,I: nat,Xs: list @ A,X2: A] :
( ( ord_less @ nat @ I @ ( size_size @ ( list @ A ) @ Xs ) )
=> ( ( nth @ A @ ( list_update @ A @ Xs @ I @ X2 ) @ I )
= X2 ) ) ).
% nth_list_update_eq
thf(fact_9_nth__update__invalid,axiom,
! [A: $tType,I: nat,L: list @ A,J: nat,X2: A] :
( ~ ( ord_less @ nat @ I @ ( size_size @ ( list @ A ) @ L ) )
=> ( ( nth @ A @ ( list_update @ A @ L @ J @ X2 ) @ I )
= ( nth @ A @ L @ I ) ) ) ).
% nth_update_invalid
thf(fact_10_divide__less__eq__numeral1_I1_J,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [B3: A,W: num,A3: A] :
( ( ord_less @ A @ ( divide_divide @ A @ B3 @ ( numeral_numeral @ A @ W ) ) @ A3 )
= ( ord_less @ A @ B3 @ ( times_times @ A @ A3 @ ( numeral_numeral @ A @ W ) ) ) ) ) ).
% divide_less_eq_numeral1(1)
thf(fact_11_less__divide__eq__numeral1_I1_J,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [A3: A,B3: A,W: num] :
( ( ord_less @ A @ A3 @ ( divide_divide @ A @ B3 @ ( numeral_numeral @ A @ W ) ) )
= ( ord_less @ A @ ( times_times @ A @ A3 @ ( numeral_numeral @ A @ W ) ) @ B3 ) ) ) ).
% less_divide_eq_numeral1(1)
thf(fact_12_delt__out__of__range,axiom,
! [X2: nat,Mi: nat,Ma: nat,Deg: nat,TreeList: list @ vEBT_VEBT,Summary: vEBT_VEBT] :
( ( ( ord_less @ nat @ X2 @ Mi )
| ( ord_less @ nat @ Ma @ X2 ) )
=> ( ( 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 ) @ X2 )
= ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi @ Ma ) ) @ Deg @ TreeList @ Summary ) ) ) ) ).
% delt_out_of_range
thf(fact_13_list__update__id,axiom,
! [A: $tType,Xs: list @ A,I: nat] :
( ( list_update @ A @ Xs @ I @ ( nth @ A @ Xs @ I ) )
= Xs ) ).
% list_update_id
thf(fact_14_nth__list__update__neq,axiom,
! [A: $tType,I: nat,J: nat,Xs: list @ A,X2: A] :
( ( I != J )
=> ( ( nth @ A @ ( list_update @ A @ Xs @ I @ X2 ) @ J )
= ( nth @ A @ Xs @ J ) ) ) ).
% nth_list_update_neq
thf(fact_15_length__list__update,axiom,
! [A: $tType,Xs: list @ A,I: nat,X2: A] :
( ( size_size @ ( list @ A ) @ ( list_update @ A @ Xs @ I @ X2 ) )
= ( size_size @ ( list @ A ) @ Xs ) ) ).
% length_list_update
thf(fact_16_not__None__eq,axiom,
! [A: $tType,X2: option @ A] :
( ( X2
!= ( none @ A ) )
= ( ? [Y: A] :
( X2
= ( some @ A @ Y ) ) ) ) ).
% not_None_eq
thf(fact_17_not__Some__eq,axiom,
! [A: $tType,X2: option @ A] :
( ( ! [Y: A] :
( X2
!= ( some @ A @ Y ) ) )
= ( X2
= ( none @ A ) ) ) ).
% not_Some_eq
thf(fact_18_lessI,axiom,
! [N: nat] : ( ord_less @ nat @ N @ ( suc @ N ) ) ).
% lessI
thf(fact_19_max__in__set__def,axiom,
( vEBT_VEBT_max_in_set
= ( ^ [Xs2: set @ nat,X3: nat] :
( ( member @ nat @ X3 @ Xs2 )
& ! [Y: nat] :
( ( member @ nat @ Y @ Xs2 )
=> ( ord_less_eq @ nat @ Y @ X3 ) ) ) ) ) ).
% max_in_set_def
thf(fact_20_min__in__set__def,axiom,
( vEBT_VEBT_min_in_set
= ( ^ [Xs2: set @ nat,X3: nat] :
( ( member @ nat @ X3 @ Xs2 )
& ! [Y: nat] :
( ( member @ nat @ Y @ Xs2 )
=> ( ord_less_eq @ nat @ X3 @ Y ) ) ) ) ) ).
% min_in_set_def
thf(fact_21_numeral__eq__iff,axiom,
! [A: $tType] :
( ( semiring_char_0 @ A )
=> ! [M: num,N: num] :
( ( ( numeral_numeral @ A @ M )
= ( numeral_numeral @ A @ N ) )
= ( M = N ) ) ) ).
% numeral_eq_iff
thf(fact_22_old_Onat_Oinject,axiom,
! [Nat: nat,Nat2: nat] :
( ( ( suc @ Nat )
= ( suc @ Nat2 ) )
= ( Nat = Nat2 ) ) ).
% old.nat.inject
thf(fact_23_nat_Oinject,axiom,
! [X22: nat,Y2: nat] :
( ( ( suc @ X22 )
= ( suc @ Y2 ) )
= ( X22 = Y2 ) ) ).
% nat.inject
thf(fact_24_option_Oinject,axiom,
! [A: $tType,X22: A,Y2: A] :
( ( ( some @ A @ X22 )
= ( some @ A @ Y2 ) )
= ( X22 = Y2 ) ) ).
% option.inject
thf(fact_25_list__update__overwrite,axiom,
! [A: $tType,Xs: list @ A,I: nat,X2: A,Y3: A] :
( ( list_update @ A @ ( list_update @ A @ Xs @ I @ X2 ) @ I @ Y3 )
= ( list_update @ A @ Xs @ I @ Y3 ) ) ).
% list_update_overwrite
thf(fact_26_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_27_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_28_num__double,axiom,
! [N: num] :
( ( times_times @ num @ ( bit0 @ one2 ) @ N )
= ( bit0 @ N ) ) ).
% num_double
thf(fact_29_mult__numeral__left__semiring__numeral,axiom,
! [A: $tType] :
( ( semiring_numeral @ A )
=> ! [V: num,W: num,Z: A] :
( ( times_times @ A @ ( numeral_numeral @ A @ V ) @ ( times_times @ A @ ( numeral_numeral @ A @ W ) @ Z ) )
= ( times_times @ A @ ( numeral_numeral @ A @ ( times_times @ num @ V @ W ) ) @ Z ) ) ) ).
% mult_numeral_left_semiring_numeral
thf(fact_30_numeral__times__numeral,axiom,
! [A: $tType] :
( ( semiring_numeral @ A )
=> ! [M: num,N: num] :
( ( times_times @ A @ ( numeral_numeral @ A @ M ) @ ( numeral_numeral @ A @ N ) )
= ( numeral_numeral @ A @ ( times_times @ num @ M @ N ) ) ) ) ).
% numeral_times_numeral
thf(fact_31_del__single__cont,axiom,
! [X2: nat,Mi: nat,Ma: nat,Deg: nat,TreeList: list @ vEBT_VEBT,Summary: vEBT_VEBT] :
( ( ( X2 = Mi )
& ( X2 = 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 ) @ X2 )
= ( vEBT_Node @ ( none @ ( product_prod @ nat @ nat ) ) @ Deg @ TreeList @ Summary ) ) ) ) ).
% del_single_cont
thf(fact_32_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_33_Suc__mono,axiom,
! [M: nat,N: nat] :
( ( ord_less @ nat @ M @ N )
=> ( ord_less @ nat @ ( suc @ M ) @ ( suc @ N ) ) ) ).
% Suc_mono
thf(fact_34_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_35_not__Some__eq2,axiom,
! [B: $tType,A: $tType,V: option @ ( product_prod @ A @ B )] :
( ( ! [X3: A,Y: B] :
( V
!= ( some @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X3 @ Y ) ) ) )
= ( V
= ( none @ ( product_prod @ A @ B ) ) ) ) ).
% not_Some_eq2
thf(fact_36_lesseq__shift,axiom,
( ( ord_less_eq @ nat )
= ( ^ [X3: nat,Y: nat] : ( vEBT_VEBT_lesseq @ ( some @ nat @ X3 ) @ ( some @ nat @ Y ) ) ) ) ).
% lesseq_shift
thf(fact_37_list__update__beyond,axiom,
! [A: $tType,Xs: list @ A,I: nat,X2: A] :
( ( ord_less_eq @ nat @ ( size_size @ ( list @ A ) @ Xs ) @ I )
=> ( ( list_update @ A @ Xs @ I @ X2 )
= Xs ) ) ).
% list_update_beyond
thf(fact_38_le__divide__eq__numeral1_I1_J,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [A3: A,B3: A,W: num] :
( ( ord_less_eq @ A @ A3 @ ( divide_divide @ A @ B3 @ ( numeral_numeral @ A @ W ) ) )
= ( ord_less_eq @ A @ ( times_times @ A @ A3 @ ( numeral_numeral @ A @ W ) ) @ B3 ) ) ) ).
% le_divide_eq_numeral1(1)
thf(fact_39_divide__le__eq__numeral1_I1_J,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [B3: A,W: num,A3: A] :
( ( ord_less_eq @ A @ ( divide_divide @ A @ B3 @ ( numeral_numeral @ A @ W ) ) @ A3 )
= ( ord_less_eq @ A @ B3 @ ( times_times @ A @ A3 @ ( numeral_numeral @ A @ W ) ) ) ) ) ).
% divide_le_eq_numeral1(1)
thf(fact_40_lift__Suc__mono__le,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [F2: nat > A,N: nat,N2: nat] :
( ! [N3: nat] : ( ord_less_eq @ A @ ( F2 @ N3 ) @ ( F2 @ ( suc @ N3 ) ) )
=> ( ( ord_less_eq @ nat @ N @ N2 )
=> ( ord_less_eq @ A @ ( F2 @ N ) @ ( F2 @ N2 ) ) ) ) ) ).
% lift_Suc_mono_le
thf(fact_41_lift__Suc__antimono__le,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [F2: nat > A,N: nat,N2: nat] :
( ! [N3: nat] : ( ord_less_eq @ A @ ( F2 @ ( suc @ N3 ) ) @ ( F2 @ N3 ) )
=> ( ( ord_less_eq @ nat @ N @ N2 )
=> ( ord_less_eq @ A @ ( F2 @ N2 ) @ ( F2 @ N ) ) ) ) ) ).
% lift_Suc_antimono_le
thf(fact_42_le__cube,axiom,
! [M: nat] : ( ord_less_eq @ nat @ M @ ( times_times @ nat @ M @ ( times_times @ nat @ M @ M ) ) ) ).
% le_cube
thf(fact_43_mem__Collect__eq,axiom,
! [A: $tType,A3: A,P: A > $o] :
( ( member @ A @ A3 @ ( collect @ A @ P ) )
= ( P @ A3 ) ) ).
% mem_Collect_eq
thf(fact_44_Collect__mem__eq,axiom,
! [A: $tType,A2: set @ A] :
( ( collect @ A
@ ^ [X3: A] : ( member @ A @ X3 @ A2 ) )
= A2 ) ).
% Collect_mem_eq
thf(fact_45_Collect__cong,axiom,
! [A: $tType,P: A > $o,Q: A > $o] :
( ! [X4: A] :
( ( P @ X4 )
= ( Q @ X4 ) )
=> ( ( collect @ A @ P )
= ( collect @ A @ Q ) ) ) ).
% Collect_cong
thf(fact_46_HOL_Oext,axiom,
! [B: $tType,A: $tType,F2: A > B,G: A > B] :
( ! [X4: A] :
( ( F2 @ X4 )
= ( G @ X4 ) )
=> ( F2 = G ) ) ).
% HOL.ext
thf(fact_47_le__refl,axiom,
! [N: nat] : ( ord_less_eq @ nat @ N @ N ) ).
% le_refl
thf(fact_48_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_49_eq__imp__le,axiom,
! [M: nat,N: nat] :
( ( M = N )
=> ( ord_less_eq @ nat @ M @ N ) ) ).
% eq_imp_le
thf(fact_50_le__square,axiom,
! [M: nat] : ( ord_less_eq @ nat @ M @ ( times_times @ nat @ M @ M ) ) ).
% le_square
thf(fact_51_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_52_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_53_le__some__optE,axiom,
! [A: $tType] :
( ( preorder @ A )
=> ! [M: A,X2: option @ A] :
( ( ord_less_eq @ ( option @ A ) @ ( some @ A @ M ) @ X2 )
=> ~ ! [M2: A] :
( ( X2
= ( some @ A @ M2 ) )
=> ~ ( ord_less_eq @ A @ M @ M2 ) ) ) ) ).
% le_some_optE
thf(fact_54_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_55_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_56_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_57_ord__eq__le__eq__trans,axiom,
! [A: $tType] :
( ( ord @ A )
=> ! [A3: A,B3: A,C3: A,D2: A] :
( ( A3 = B3 )
=> ( ( ord_less_eq @ A @ B3 @ C3 )
=> ( ( C3 = D2 )
=> ( ord_less_eq @ A @ A3 @ D2 ) ) ) ) ) ).
% ord_eq_le_eq_trans
thf(fact_58_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_59_Nat_Oex__has__greatest__nat,axiom,
! [P: nat > $o,K: nat,B3: nat] :
( ( P @ K )
=> ( ! [Y4: nat] :
( ( P @ Y4 )
=> ( ord_less_eq @ nat @ Y4 @ B3 ) )
=> ? [X4: nat] :
( ( P @ X4 )
& ! [Y5: nat] :
( ( P @ Y5 )
=> ( ord_less_eq @ nat @ Y5 @ X4 ) ) ) ) ) ).
% Nat.ex_has_greatest_nat
thf(fact_60_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_61_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_62_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_63_transitive__stepwise__le,axiom,
! [M: nat,N: nat,R: nat > nat > $o] :
( ( ord_less_eq @ nat @ M @ N )
=> ( ! [X4: nat] : ( R @ X4 @ X4 )
=> ( ! [X4: nat,Y4: nat,Z2: nat] :
( ( R @ X4 @ Y4 )
=> ( ( R @ Y4 @ Z2 )
=> ( R @ X4 @ Z2 ) ) )
=> ( ! [N3: nat] : ( R @ N3 @ ( suc @ N3 ) )
=> ( R @ M @ N ) ) ) ) ) ).
% transitive_stepwise_le
thf(fact_64_nat__induct__at__least,axiom,
! [M: nat,N: nat,P: nat > $o] :
( ( ord_less_eq @ nat @ M @ N )
=> ( ( P @ M )
=> ( ! [N3: nat] :
( ( ord_less_eq @ nat @ M @ N3 )
=> ( ( P @ N3 )
=> ( P @ ( suc @ N3 ) ) ) )
=> ( P @ N ) ) ) ) ).
% nat_induct_at_least
thf(fact_65_full__nat__induct,axiom,
! [P: nat > $o,N: nat] :
( ! [N3: nat] :
( ! [M3: nat] :
( ( ord_less_eq @ nat @ ( suc @ M3 ) @ N3 )
=> ( P @ M3 ) )
=> ( P @ N3 ) )
=> ( P @ N ) ) ).
% full_nat_induct
thf(fact_66_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_67_Suc__n__not__le__n,axiom,
! [N: nat] :
~ ( ord_less_eq @ nat @ ( suc @ N ) @ N ) ).
% Suc_n_not_le_n
thf(fact_68_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_69_Suc__le__D,axiom,
! [N: nat,M4: nat] :
( ( ord_less_eq @ nat @ ( suc @ N ) @ M4 )
=> ? [M5: nat] :
( M4
= ( suc @ M5 ) ) ) ).
% Suc_le_D
thf(fact_70_le__SucI,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq @ nat @ M @ N )
=> ( ord_less_eq @ nat @ M @ ( suc @ N ) ) ) ).
% le_SucI
thf(fact_71_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_72_Suc__leD,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq @ nat @ ( suc @ M ) @ N )
=> ( ord_less_eq @ nat @ M @ N ) ) ).
% Suc_leD
thf(fact_73_less__mono__imp__le__mono,axiom,
! [F2: nat > nat,I: nat,J: nat] :
( ! [I2: nat,J2: nat] :
( ( ord_less @ nat @ I2 @ J2 )
=> ( ord_less @ nat @ ( F2 @ I2 ) @ ( F2 @ J2 ) ) )
=> ( ( ord_less_eq @ nat @ I @ J )
=> ( ord_less_eq @ nat @ ( F2 @ I ) @ ( F2 @ J ) ) ) ) ).
% less_mono_imp_le_mono
thf(fact_74_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_75_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_76_le__eq__less__or__eq,axiom,
( ( ord_less_eq @ nat )
= ( ^ [M6: nat,N4: nat] :
( ( ord_less @ nat @ M6 @ N4 )
| ( M6 = N4 ) ) ) ) ).
% le_eq_less_or_eq
thf(fact_77_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_78_nat__less__le,axiom,
( ( ord_less @ nat )
= ( ^ [M6: nat,N4: nat] :
( ( ord_less_eq @ nat @ M6 @ N4 )
& ( M6 != N4 ) ) ) ) ).
% nat_less_le
thf(fact_79_exists__leI,axiom,
! [N: nat,P: nat > $o] :
( ( ! [N5: nat] :
( ( ord_less @ nat @ N5 @ N )
=> ~ ( P @ N5 ) )
=> ( P @ N ) )
=> ? [N6: nat] :
( ( ord_less_eq @ nat @ N6 @ N )
& ( P @ N6 ) ) ) ).
% exists_leI
thf(fact_80_div__mult2__eq,axiom,
! [M: nat,N: nat,Q3: nat] :
( ( divide_divide @ nat @ M @ ( times_times @ nat @ N @ Q3 ) )
= ( divide_divide @ nat @ ( divide_divide @ nat @ M @ N ) @ Q3 ) ) ).
% div_mult2_eq
thf(fact_81_div__le__dividend,axiom,
! [M: nat,N: nat] : ( ord_less_eq @ nat @ ( divide_divide @ nat @ M @ N ) @ M ) ).
% div_le_dividend
thf(fact_82_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_83_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_84_div__mult2__numeral__eq,axiom,
! [A: $tType] :
( ( euclid5411537665997757685th_nat @ A )
=> ! [A3: A,K: num,L: num] :
( ( divide_divide @ A @ ( divide_divide @ A @ A3 @ ( numeral_numeral @ A @ K ) ) @ ( numeral_numeral @ A @ L ) )
= ( divide_divide @ A @ A3 @ ( numeral_numeral @ A @ ( times_times @ num @ K @ L ) ) ) ) ) ).
% div_mult2_numeral_eq
thf(fact_85_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_86_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_87_less__eq__Suc__le,axiom,
( ( ord_less @ nat )
= ( ^ [N4: nat] : ( ord_less_eq @ nat @ ( suc @ N4 ) ) ) ) ).
% less_eq_Suc_le
thf(fact_88_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_89_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_90_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_91_inc__induct,axiom,
! [I: nat,J: nat,P: nat > $o] :
( ( ord_less_eq @ nat @ I @ J )
=> ( ( P @ J )
=> ( ! [N3: nat] :
( ( ord_less_eq @ nat @ I @ N3 )
=> ( ( ord_less @ nat @ N3 @ J )
=> ( ( P @ ( suc @ N3 ) )
=> ( P @ N3 ) ) ) )
=> ( P @ I ) ) ) ) ).
% inc_induct
thf(fact_92_dec__induct,axiom,
! [I: nat,J: nat,P: nat > $o] :
( ( ord_less_eq @ nat @ I @ J )
=> ( ( P @ I )
=> ( ! [N3: nat] :
( ( ord_less_eq @ nat @ I @ N3 )
=> ( ( ord_less @ nat @ N3 @ J )
=> ( ( P @ N3 )
=> ( P @ ( suc @ N3 ) ) ) ) )
=> ( P @ J ) ) ) ) ).
% dec_induct
thf(fact_93_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_94_Suc__leI,axiom,
! [M: nat,N: nat] :
( ( ord_less @ nat @ M @ N )
=> ( ord_less_eq @ nat @ ( suc @ M ) @ N ) ) ).
% Suc_leI
thf(fact_95_nat__in__between__eq_I1_J,axiom,
! [A3: nat,B3: nat] :
( ( ( ord_less @ nat @ A3 @ B3 )
& ( ord_less_eq @ nat @ B3 @ ( suc @ A3 ) ) )
= ( B3
= ( suc @ A3 ) ) ) ).
% nat_in_between_eq(1)
thf(fact_96_nat__in__between__eq_I2_J,axiom,
! [A3: nat,B3: nat] :
( ( ( ord_less_eq @ nat @ A3 @ B3 )
& ( ord_less @ nat @ B3 @ ( suc @ A3 ) ) )
= ( B3 = A3 ) ) ).
% nat_in_between_eq(2)
thf(fact_97_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_98_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_99_measure__induct__rule,axiom,
! [B: $tType,A: $tType] :
( ( wellorder @ B )
=> ! [F2: A > B,P: A > $o,A3: A] :
( ! [X4: A] :
( ! [Y5: A] :
( ( ord_less @ B @ ( F2 @ Y5 ) @ ( F2 @ X4 ) )
=> ( P @ Y5 ) )
=> ( P @ X4 ) )
=> ( P @ A3 ) ) ) ).
% measure_induct_rule
thf(fact_100_measure__induct,axiom,
! [B: $tType,A: $tType] :
( ( wellorder @ B )
=> ! [F2: A > B,P: A > $o,A3: A] :
( ! [X4: A] :
( ! [Y5: A] :
( ( ord_less @ B @ ( F2 @ Y5 ) @ ( F2 @ X4 ) )
=> ( P @ Y5 ) )
=> ( P @ X4 ) )
=> ( P @ A3 ) ) ) ).
% measure_induct
thf(fact_101_bex2I,axiom,
! [A: $tType,B: $tType,A3: A,B3: B,S: set @ ( product_prod @ A @ B ),P: A > B > $o] :
( ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ A3 @ B3 ) @ S )
=> ( ( ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ A3 @ B3 ) @ S )
=> ( P @ A3 @ B3 ) )
=> ? [A4: A,B4: B] :
( ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ A4 @ B4 ) @ S )
& ( P @ A4 @ B4 ) ) ) ) ).
% bex2I
thf(fact_102_pairself_Ocases,axiom,
! [B: $tType,A: $tType,X2: product_prod @ ( A > B ) @ ( product_prod @ A @ A )] :
~ ! [F3: A > B,A4: A,B4: A] :
( X2
!= ( product_Pair @ ( A > B ) @ ( product_prod @ A @ A ) @ F3 @ ( product_Pair @ A @ A @ A4 @ B4 ) ) ) ).
% pairself.cases
thf(fact_103_n__not__Suc__n,axiom,
! [N: nat] :
( N
!= ( suc @ N ) ) ).
% n_not_Suc_n
thf(fact_104_Suc__inject,axiom,
! [X2: nat,Y3: nat] :
( ( ( suc @ X2 )
= ( suc @ Y3 ) )
=> ( X2 = Y3 ) ) ).
% Suc_inject
thf(fact_105_infinite__descent__measure,axiom,
! [A: $tType,P: A > $o,V2: A > nat,X2: A] :
( ! [X4: A] :
( ~ ( P @ X4 )
=> ? [Y5: A] :
( ( ord_less @ nat @ ( V2 @ Y5 ) @ ( V2 @ X4 ) )
& ~ ( P @ Y5 ) ) )
=> ( P @ X2 ) ) ).
% infinite_descent_measure
thf(fact_106_linorder__neqE__nat,axiom,
! [X2: nat,Y3: nat] :
( ( X2 != Y3 )
=> ( ~ ( ord_less @ nat @ X2 @ Y3 )
=> ( ord_less @ nat @ Y3 @ X2 ) ) ) ).
% linorder_neqE_nat
thf(fact_107_infinite__descent,axiom,
! [P: nat > $o,N: nat] :
( ! [N3: nat] :
( ~ ( P @ N3 )
=> ? [M3: nat] :
( ( ord_less @ nat @ M3 @ N3 )
& ~ ( P @ M3 ) ) )
=> ( P @ N ) ) ).
% infinite_descent
thf(fact_108_nat__less__induct,axiom,
! [P: nat > $o,N: nat] :
( ! [N3: nat] :
( ! [M3: nat] :
( ( ord_less @ nat @ M3 @ N3 )
=> ( P @ M3 ) )
=> ( P @ N3 ) )
=> ( P @ N ) ) ).
% nat_less_induct
thf(fact_109_less__irrefl__nat,axiom,
! [N: nat] :
~ ( ord_less @ nat @ N @ N ) ).
% less_irrefl_nat
thf(fact_110_less__not__refl3,axiom,
! [S2: nat,T2: nat] :
( ( ord_less @ nat @ S2 @ T2 )
=> ( S2 != T2 ) ) ).
% less_not_refl3
thf(fact_111_less__not__refl2,axiom,
! [N: nat,M: nat] :
( ( ord_less @ nat @ N @ M )
=> ( M != N ) ) ).
% less_not_refl2
thf(fact_112_less__not__refl,axiom,
! [N: nat] :
~ ( ord_less @ nat @ N @ N ) ).
% less_not_refl
thf(fact_113_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_114_size__neq__size__imp__neq,axiom,
! [A: $tType] :
( ( size @ A )
=> ! [X2: A,Y3: A] :
( ( ( size_size @ A @ X2 )
!= ( size_size @ A @ Y3 ) )
=> ( X2 != Y3 ) ) ) ).
% size_neq_size_imp_neq
thf(fact_115_neq__if__length__neq,axiom,
! [A: $tType,Xs: list @ A,Ys: list @ A] :
( ( ( size_size @ ( list @ A ) @ Xs )
!= ( size_size @ ( list @ A ) @ Ys ) )
=> ( Xs != Ys ) ) ).
% neq_if_length_neq
thf(fact_116_Ex__list__of__length,axiom,
! [A: $tType,N: nat] :
? [Xs3: list @ A] :
( ( size_size @ ( list @ A ) @ Xs3 )
= N ) ).
% Ex_list_of_length
thf(fact_117_list__update__swap,axiom,
! [A: $tType,I: nat,I3: nat,Xs: list @ A,X2: A,X5: A] :
( ( I != I3 )
=> ( ( list_update @ A @ ( list_update @ A @ Xs @ I @ X2 ) @ I3 @ X5 )
= ( list_update @ A @ ( list_update @ A @ Xs @ I3 @ X5 ) @ I @ X2 ) ) ) ).
% list_update_swap
thf(fact_118_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_119_strict__inc__induct,axiom,
! [I: nat,J: nat,P: nat > $o] :
( ( ord_less @ nat @ I @ J )
=> ( ! [I2: nat] :
( ( J
= ( suc @ I2 ) )
=> ( P @ I2 ) )
=> ( ! [I2: nat] :
( ( ord_less @ nat @ I2 @ J )
=> ( ( P @ ( suc @ I2 ) )
=> ( P @ I2 ) ) )
=> ( P @ I ) ) ) ) ).
% strict_inc_induct
thf(fact_120_less__Suc__induct,axiom,
! [I: nat,J: nat,P: nat > nat > $o] :
( ( ord_less @ nat @ I @ J )
=> ( ! [I2: nat] : ( P @ I2 @ ( suc @ I2 ) )
=> ( ! [I2: nat,J2: nat,K2: nat] :
( ( ord_less @ nat @ I2 @ J2 )
=> ( ( ord_less @ nat @ J2 @ K2 )
=> ( ( P @ I2 @ J2 )
=> ( ( P @ J2 @ K2 )
=> ( P @ I2 @ K2 ) ) ) ) )
=> ( P @ I @ J ) ) ) ) ).
% less_Suc_induct
thf(fact_121_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_122_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_123_less__antisym,axiom,
! [N: nat,M: nat] :
( ~ ( ord_less @ nat @ N @ M )
=> ( ( ord_less @ nat @ N @ ( suc @ M ) )
=> ( M = N ) ) ) ).
% less_antisym
thf(fact_124_Suc__less__eq2,axiom,
! [N: nat,M: nat] :
( ( ord_less @ nat @ ( suc @ N ) @ M )
= ( ? [M7: nat] :
( ( M
= ( suc @ M7 ) )
& ( ord_less @ nat @ N @ M7 ) ) ) ) ).
% Suc_less_eq2
thf(fact_125_Nat_OAll__less__Suc,axiom,
! [N: nat,P: nat > $o] :
( ( ! [I4: nat] :
( ( ord_less @ nat @ I4 @ ( suc @ N ) )
=> ( P @ I4 ) ) )
= ( ( P @ N )
& ! [I4: nat] :
( ( ord_less @ nat @ I4 @ N )
=> ( P @ I4 ) ) ) ) ).
% Nat.All_less_Suc
thf(fact_126_not__less__eq,axiom,
! [M: nat,N: nat] :
( ( ~ ( ord_less @ nat @ M @ N ) )
= ( ord_less @ nat @ N @ ( suc @ M ) ) ) ).
% not_less_eq
thf(fact_127_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_128_Ex__less__Suc,axiom,
! [N: nat,P: nat > $o] :
( ( ? [I4: nat] :
( ( ord_less @ nat @ I4 @ ( suc @ N ) )
& ( P @ I4 ) ) )
= ( ( P @ N )
| ? [I4: nat] :
( ( ord_less @ nat @ I4 @ N )
& ( P @ I4 ) ) ) ) ).
% Ex_less_Suc
thf(fact_129_less__SucI,axiom,
! [M: nat,N: nat] :
( ( ord_less @ nat @ M @ N )
=> ( ord_less @ nat @ M @ ( suc @ N ) ) ) ).
% less_SucI
thf(fact_130_less__SucE,axiom,
! [M: nat,N: nat] :
( ( ord_less @ nat @ M @ ( suc @ N ) )
=> ( ~ ( ord_less @ nat @ M @ N )
=> ( M = N ) ) ) ).
% less_SucE
thf(fact_131_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_132_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_133_Suc__lessD,axiom,
! [M: nat,N: nat] :
( ( ord_less @ nat @ ( suc @ M ) @ N )
=> ( ord_less @ nat @ M @ N ) ) ).
% Suc_lessD
thf(fact_134_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_135_length__induct,axiom,
! [A: $tType,P: ( list @ A ) > $o,Xs: list @ A] :
( ! [Xs3: list @ A] :
( ! [Ys2: list @ A] :
( ( ord_less @ nat @ ( size_size @ ( list @ A ) @ Ys2 ) @ ( size_size @ ( list @ A ) @ Xs3 ) )
=> ( P @ Ys2 ) )
=> ( P @ Xs3 ) )
=> ( P @ Xs ) ) ).
% length_induct
thf(fact_136_combine__options__cases,axiom,
! [A: $tType,B: $tType,X2: option @ A,P: ( option @ A ) > ( option @ B ) > $o,Y3: option @ B] :
( ( ( X2
= ( none @ A ) )
=> ( P @ X2 @ Y3 ) )
=> ( ( ( Y3
= ( none @ B ) )
=> ( P @ X2 @ Y3 ) )
=> ( ! [A4: A,B4: B] :
( ( X2
= ( some @ A @ A4 ) )
=> ( ( Y3
= ( some @ B @ B4 ) )
=> ( P @ X2 @ Y3 ) ) )
=> ( P @ X2 @ Y3 ) ) ) ) ).
% combine_options_cases
thf(fact_137_split__option__all,axiom,
! [A: $tType] :
( ( ^ [P2: ( option @ A ) > $o] :
! [X6: option @ A] : ( P2 @ X6 ) )
= ( ^ [P3: ( option @ A ) > $o] :
( ( P3 @ ( none @ A ) )
& ! [X3: A] : ( P3 @ ( some @ A @ X3 ) ) ) ) ) ).
% split_option_all
thf(fact_138_split__option__ex,axiom,
! [A: $tType] :
( ( ^ [P2: ( option @ A ) > $o] :
? [X6: option @ A] : ( P2 @ X6 ) )
= ( ^ [P3: ( option @ A ) > $o] :
( ( P3 @ ( none @ A ) )
| ? [X3: A] : ( P3 @ ( some @ A @ X3 ) ) ) ) ) ).
% split_option_ex
thf(fact_139_option_Oexhaust,axiom,
! [A: $tType,Y3: option @ A] :
( ( Y3
!= ( none @ A ) )
=> ~ ! [X23: A] :
( Y3
!= ( some @ A @ X23 ) ) ) ).
% option.exhaust
thf(fact_140_option_OdiscI,axiom,
! [A: $tType,Option: option @ A,X22: A] :
( ( Option
= ( some @ A @ X22 ) )
=> ( Option
!= ( none @ A ) ) ) ).
% option.discI
thf(fact_141_option_Odistinct_I1_J,axiom,
! [A: $tType,X22: A] :
( ( none @ A )
!= ( some @ A @ X22 ) ) ).
% option.distinct(1)
thf(fact_142_mult__numeral__1__right,axiom,
! [A: $tType] :
( ( semiring_numeral @ A )
=> ! [A3: A] :
( ( times_times @ A @ A3 @ ( numeral_numeral @ A @ one2 ) )
= A3 ) ) ).
% mult_numeral_1_right
thf(fact_143_mult__numeral__1,axiom,
! [A: $tType] :
( ( semiring_numeral @ A )
=> ! [A3: A] :
( ( times_times @ A @ ( numeral_numeral @ A @ one2 ) @ A3 )
= A3 ) ) ).
% mult_numeral_1
thf(fact_144_divide__numeral__1,axiom,
! [A: $tType] :
( ( field @ A )
=> ! [A3: A] :
( ( divide_divide @ A @ A3 @ ( numeral_numeral @ A @ one2 ) )
= A3 ) ) ).
% divide_numeral_1
thf(fact_145_lift__Suc__mono__less__iff,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [F2: nat > A,N: nat,M: nat] :
( ! [N3: nat] : ( ord_less @ A @ ( F2 @ N3 ) @ ( F2 @ ( suc @ N3 ) ) )
=> ( ( ord_less @ A @ ( F2 @ N ) @ ( F2 @ M ) )
= ( ord_less @ nat @ N @ M ) ) ) ) ).
% lift_Suc_mono_less_iff
thf(fact_146_lift__Suc__mono__less,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [F2: nat > A,N: nat,N2: nat] :
( ! [N3: nat] : ( ord_less @ A @ ( F2 @ N3 ) @ ( F2 @ ( suc @ N3 ) ) )
=> ( ( ord_less @ nat @ N @ N2 )
=> ( ord_less @ A @ ( F2 @ N ) @ ( F2 @ N2 ) ) ) ) ) ).
% lift_Suc_mono_less
thf(fact_147_obtain__list__from__elements,axiom,
! [A: $tType,N: nat,P: A > nat > $o] :
( ! [I2: nat] :
( ( ord_less @ nat @ I2 @ N )
=> ? [Li: A] : ( P @ Li @ I2 ) )
=> ~ ! [L2: list @ A] :
( ( ( size_size @ ( list @ A ) @ L2 )
= N )
=> ~ ! [I5: nat] :
( ( ord_less @ nat @ I5 @ N )
=> ( P @ ( nth @ A @ L2 @ I5 ) @ I5 ) ) ) ) ).
% obtain_list_from_elements
thf(fact_148_list__eq__iff__nth__eq,axiom,
! [A: $tType] :
( ( ^ [Y6: list @ A,Z3: list @ A] : Y6 = Z3 )
= ( ^ [Xs2: list @ A,Ys3: list @ A] :
( ( ( size_size @ ( list @ A ) @ Xs2 )
= ( size_size @ ( list @ A ) @ Ys3 ) )
& ! [I4: nat] :
( ( ord_less @ nat @ I4 @ ( size_size @ ( list @ A ) @ Xs2 ) )
=> ( ( nth @ A @ Xs2 @ I4 )
= ( nth @ A @ Ys3 @ I4 ) ) ) ) ) ) ).
% list_eq_iff_nth_eq
thf(fact_149_Skolem__list__nth,axiom,
! [A: $tType,K: nat,P: nat > A > $o] :
( ( ! [I4: nat] :
( ( ord_less @ nat @ I4 @ K )
=> ? [X7: A] : ( P @ I4 @ X7 ) ) )
= ( ? [Xs2: list @ A] :
( ( ( size_size @ ( list @ A ) @ Xs2 )
= K )
& ! [I4: nat] :
( ( ord_less @ nat @ I4 @ K )
=> ( P @ I4 @ ( nth @ A @ Xs2 @ I4 ) ) ) ) ) ) ).
% Skolem_list_nth
thf(fact_150_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_151_numeral__Bit0__div__2,axiom,
! [A: $tType] :
( ( euclid5411537665997757685th_nat @ A )
=> ! [N: num] :
( ( divide_divide @ A @ ( numeral_numeral @ A @ ( bit0 @ N ) ) @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) )
= ( numeral_numeral @ A @ N ) ) ) ).
% numeral_Bit0_div_2
thf(fact_152_list__update__same__conv,axiom,
! [A: $tType,I: nat,Xs: list @ A,X2: A] :
( ( ord_less @ nat @ I @ ( size_size @ ( list @ A ) @ Xs ) )
=> ( ( ( list_update @ A @ Xs @ I @ X2 )
= Xs )
= ( ( nth @ A @ Xs @ I )
= X2 ) ) ) ).
% list_update_same_conv
thf(fact_153_nth__list__update_H,axiom,
! [A: $tType,I: nat,J: nat,L: list @ A,X2: A] :
( ( ( ( I = J )
& ( ord_less @ nat @ I @ ( size_size @ ( list @ A ) @ L ) ) )
=> ( ( nth @ A @ ( list_update @ A @ L @ I @ X2 ) @ J )
= X2 ) )
& ( ~ ( ( I = J )
& ( ord_less @ nat @ I @ ( size_size @ ( list @ A ) @ L ) ) )
=> ( ( nth @ A @ ( list_update @ A @ L @ I @ X2 ) @ J )
= ( nth @ A @ L @ J ) ) ) ) ).
% nth_list_update'
thf(fact_154_nth__list__update,axiom,
! [A: $tType,I: nat,Xs: list @ A,J: nat,X2: A] :
( ( ord_less @ nat @ I @ ( size_size @ ( list @ A ) @ Xs ) )
=> ( ( ( I = J )
=> ( ( nth @ A @ ( list_update @ A @ Xs @ I @ X2 ) @ J )
= X2 ) )
& ( ( I != J )
=> ( ( nth @ A @ ( list_update @ A @ Xs @ I @ X2 ) @ J )
= ( nth @ A @ Xs @ J ) ) ) ) ) ).
% nth_list_update
thf(fact_155_highbound,axiom,
( ( ma != mi )
=> ( ( ord_less_eq @ nat @ xa @ ma )
=> ( ord_less @ nat @ ( vEBT_VEBT_high @ xa @ ( divide_divide @ nat @ na @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( size_size @ ( list @ vEBT_VEBT ) @ treeList ) ) ) ) ).
% highbound
thf(fact_156_xbound,axiom,
( ( ord_less_eq @ nat @ mi @ xa )
=> ( ( ord_less_eq @ nat @ xa @ ma )
=> ( ord_less_eq @ nat @ ( vEBT_VEBT_high @ xa @ ( divide_divide @ nat @ na @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( size_size @ ( list @ vEBT_VEBT ) @ treeList ) ) ) ) ).
% xbound
thf(fact_157_member__inv,axiom,
! [Mi: nat,Ma: nat,Deg: nat,TreeList: list @ vEBT_VEBT,Summary: vEBT_VEBT,X2: nat] :
( ( vEBT_vebt_member @ ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi @ Ma ) ) @ Deg @ TreeList @ Summary ) @ X2 )
=> ( ( ord_less_eq @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ Deg )
& ( ( X2 = Mi )
| ( X2 = Ma )
| ( ( ord_less @ nat @ X2 @ Ma )
& ( ord_less @ nat @ Mi @ X2 )
& ( ord_less @ nat @ ( vEBT_VEBT_high @ X2 @ ( 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 @ X2 @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( vEBT_VEBT_low @ X2 @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) ) ) ) ).
% member_inv
thf(fact_158_succ__list__to__short,axiom,
! [Deg: nat,Mi: nat,X2: 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 @ X2 )
=> ( ( ord_less_eq @ nat @ ( size_size @ ( list @ vEBT_VEBT ) @ TreeList ) @ ( vEBT_VEBT_high @ X2 @ ( 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 ) @ X2 )
= ( none @ nat ) ) ) ) ) ).
% succ_list_to_short
thf(fact_159_pred__list__to__short,axiom,
! [Deg: nat,X2: 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 @ X2 @ Ma )
=> ( ( ord_less_eq @ nat @ ( size_size @ ( list @ vEBT_VEBT ) @ TreeList ) @ ( vEBT_VEBT_high @ X2 @ ( 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 ) @ X2 )
= ( none @ nat ) ) ) ) ) ).
% pred_list_to_short
thf(fact_160_insert__simp__mima,axiom,
! [X2: nat,Mi: nat,Ma: nat,Deg: nat,TreeList: list @ vEBT_VEBT,Summary: vEBT_VEBT] :
( ( ( X2 = Mi )
| ( X2 = 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 ) @ X2 )
= ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi @ Ma ) ) @ Deg @ TreeList @ Summary ) ) ) ) ).
% insert_simp_mima
thf(fact_161_succ__min,axiom,
! [Deg: nat,X2: 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 @ X2 @ Mi )
=> ( ( vEBT_vebt_succ @ ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi @ Ma ) ) @ Deg @ TreeList @ Summary ) @ X2 )
= ( some @ nat @ Mi ) ) ) ) ).
% succ_min
thf(fact_162_pred__max,axiom,
! [Deg: nat,Ma: nat,X2: nat,Mi: nat,TreeList: list @ vEBT_VEBT,Summary: vEBT_VEBT] :
( ( ord_less_eq @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ Deg )
=> ( ( ord_less @ nat @ Ma @ X2 )
=> ( ( vEBT_vebt_pred @ ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi @ Ma ) ) @ Deg @ TreeList @ Summary ) @ X2 )
= ( some @ nat @ Ma ) ) ) ) ).
% pred_max
thf(fact_163_highboundn,axiom,
( ( ma != mi )
=> ( ( ord_less_eq @ nat @ xa @ ma )
=> ( ord_less @ nat @ ( vEBT_VEBT_high @ xnew @ ( divide_divide @ nat @ na @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( size_size @ ( list @ vEBT_VEBT ) @ treeList ) ) ) ) ).
% highboundn
thf(fact_164_less__option__None__Some__code,axiom,
! [A: $tType] :
( ( preorder @ A )
=> ! [X2: A] : ( ord_less @ ( option @ A ) @ ( none @ A ) @ ( some @ A @ X2 ) ) ) ).
% less_option_None_Some_code
thf(fact_165_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_166_less__option__Some,axiom,
! [A: $tType] :
( ( preorder @ A )
=> ! [X2: A,Y3: A] :
( ( ord_less @ ( option @ A ) @ ( some @ A @ X2 ) @ ( some @ A @ Y3 ) )
= ( ord_less @ A @ X2 @ Y3 ) ) ) ).
% less_option_Some
thf(fact_167_min__Null__member,axiom,
! [T2: vEBT_VEBT,X2: nat] :
( ( vEBT_VEBT_minNull @ T2 )
=> ~ ( vEBT_vebt_member @ T2 @ X2 ) ) ).
% min_Null_member
thf(fact_168_less__eq__option__None__code,axiom,
! [A: $tType] :
( ( preorder @ A )
=> ! [X2: option @ A] : ( ord_less_eq @ ( option @ A ) @ ( none @ A ) @ X2 ) ) ).
% less_eq_option_None_code
thf(fact_169_less__option__None,axiom,
! [A: $tType] :
( ( preorder @ A )
=> ! [X2: option @ A] :
~ ( ord_less @ ( option @ A ) @ X2 @ ( none @ A ) ) ) ).
% less_option_None
thf(fact_170_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_171_less__eq__option__Some,axiom,
! [A: $tType] :
( ( preorder @ A )
=> ! [X2: A,Y3: A] :
( ( ord_less_eq @ ( option @ A ) @ ( some @ A @ X2 ) @ ( some @ A @ Y3 ) )
= ( ord_less_eq @ A @ X2 @ Y3 ) ) ) ).
% less_eq_option_Some
thf(fact_172_less__eq__option__Some__None,axiom,
! [A: $tType] :
( ( preorder @ A )
=> ! [X2: A] :
~ ( ord_less_eq @ ( option @ A ) @ ( some @ A @ X2 ) @ ( none @ A ) ) ) ).
% less_eq_option_Some_None
thf(fact_173__C7_Oprems_C,axiom,
vEBT_invar_vebt @ ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ mi @ ma ) ) @ ( suc @ ( suc @ va ) ) @ treeList @ summary ) @ na ).
% "7.prems"
thf(fact_174_enat__less__induct,axiom,
! [P: extended_enat > $o,N: extended_enat] :
( ! [N3: extended_enat] :
( ! [M3: extended_enat] :
( ( ord_less @ extended_enat @ M3 @ N3 )
=> ( P @ M3 ) )
=> ( P @ N3 ) )
=> ( P @ N ) ) ).
% enat_less_induct
thf(fact_175_less__eq__option__None,axiom,
! [A: $tType] :
( ( preorder @ A )
=> ! [X2: option @ A] : ( ord_less_eq @ ( option @ A ) @ ( none @ A ) @ X2 ) ) ).
% less_eq_option_None
thf(fact_176_less__eq__option__None__is__None,axiom,
! [A: $tType] :
( ( preorder @ A )
=> ! [X2: option @ A] :
( ( ord_less_eq @ ( option @ A ) @ X2 @ ( none @ A ) )
=> ( X2
= ( none @ A ) ) ) ) ).
% less_eq_option_None_is_None
thf(fact_177_le__num__One__iff,axiom,
! [X2: num] :
( ( ord_less_eq @ num @ X2 @ one2 )
= ( X2 = one2 ) ) ).
% le_num_One_iff
thf(fact_178_less__option__None__Some,axiom,
! [A: $tType] :
( ( preorder @ A )
=> ! [X2: A] : ( ord_less @ ( option @ A ) @ ( none @ A ) @ ( some @ A @ X2 ) ) ) ).
% less_option_None_Some
thf(fact_179_less__option__None__is__Some,axiom,
! [A: $tType] :
( ( preorder @ A )
=> ! [X2: option @ A] :
( ( ord_less @ ( option @ A ) @ ( none @ A ) @ X2 )
=> ? [Z2: A] :
( X2
= ( some @ A @ Z2 ) ) ) ) ).
% less_option_None_is_Some
thf(fact_180_vebt__member_Osimps_I5_J,axiom,
! [Mi: nat,Ma: nat,Va: nat,TreeList: list @ vEBT_VEBT,Summary: vEBT_VEBT,X2: nat] :
( ( vEBT_vebt_member @ ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi @ Ma ) ) @ ( suc @ ( suc @ Va ) ) @ TreeList @ Summary ) @ X2 )
= ( ( X2 != Mi )
=> ( ( X2 != Ma )
=> ( ~ ( ord_less @ nat @ X2 @ Mi )
& ( ~ ( ord_less @ nat @ X2 @ Mi )
=> ( ~ ( ord_less @ nat @ Ma @ X2 )
& ( ~ ( ord_less @ nat @ Ma @ X2 )
=> ( ( ( ord_less @ nat @ ( vEBT_VEBT_high @ X2 @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( size_size @ ( list @ vEBT_VEBT ) @ TreeList ) )
=> ( vEBT_vebt_member @ ( nth @ vEBT_VEBT @ TreeList @ ( vEBT_VEBT_high @ X2 @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( vEBT_VEBT_low @ X2 @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) )
& ( ord_less @ nat @ ( vEBT_VEBT_high @ X2 @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( size_size @ ( list @ vEBT_VEBT ) @ TreeList ) ) ) ) ) ) ) ) ) ) ).
% vebt_member.simps(5)
thf(fact_181_insert__simp__excp,axiom,
! [Mi: nat,Deg: nat,TreeList: list @ vEBT_VEBT,X2: 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 @ X2 @ Mi )
=> ( ( ord_less_eq @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ Deg )
=> ( ( X2 != Ma )
=> ( ( vEBT_vebt_insert @ ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi @ Ma ) ) @ Deg @ TreeList @ Summary ) @ X2 )
= ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ X2 @ ( 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_182_insert__simp__norm,axiom,
! [X2: nat,Deg: nat,TreeList: list @ vEBT_VEBT,Mi: nat,Ma: nat,Summary: vEBT_VEBT] :
( ( ord_less @ nat @ ( vEBT_VEBT_high @ X2 @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( size_size @ ( list @ vEBT_VEBT ) @ TreeList ) )
=> ( ( ord_less @ nat @ Mi @ X2 )
=> ( ( ord_less_eq @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ Deg )
=> ( ( X2 != Ma )
=> ( ( vEBT_vebt_insert @ ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi @ Ma ) ) @ Deg @ TreeList @ Summary ) @ X2 )
= ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi @ ( ord_max @ nat @ X2 @ Ma ) ) ) @ Deg @ ( list_update @ vEBT_VEBT @ TreeList @ ( vEBT_VEBT_high @ X2 @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( vEBT_vebt_insert @ ( nth @ vEBT_VEBT @ TreeList @ ( vEBT_VEBT_high @ X2 @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( vEBT_VEBT_low @ X2 @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) @ ( if @ vEBT_VEBT @ ( vEBT_VEBT_minNull @ ( nth @ vEBT_VEBT @ TreeList @ ( vEBT_VEBT_high @ X2 @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) @ ( vEBT_vebt_insert @ Summary @ ( vEBT_VEBT_high @ X2 @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ Summary ) ) ) ) ) ) ) ).
% insert_simp_norm
thf(fact_183_semiring__norm_I69_J,axiom,
! [M: num] :
~ ( ord_less_eq @ num @ ( bit0 @ M ) @ one2 ) ).
% semiring_norm(69)
thf(fact_184_semiring__norm_I76_J,axiom,
! [N: num] : ( ord_less @ num @ one2 @ ( bit0 @ N ) ) ).
% semiring_norm(76)
thf(fact_185_mimaxprop,axiom,
( ( ord_less_eq @ nat @ mi @ ma )
& ( ord_less_eq @ nat @ ma @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ na ) ) ) ).
% mimaxprop
thf(fact_186_vebt__insert_Osimps_I4_J,axiom,
! [V: nat,TreeList: list @ vEBT_VEBT,Summary: vEBT_VEBT,X2: nat] :
( ( vEBT_vebt_insert @ ( vEBT_Node @ ( none @ ( product_prod @ nat @ nat ) ) @ ( suc @ ( suc @ V ) ) @ TreeList @ Summary ) @ X2 )
= ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ X2 @ X2 ) ) @ ( suc @ ( suc @ V ) ) @ TreeList @ Summary ) ) ).
% vebt_insert.simps(4)
thf(fact_187_semiring__norm_I68_J,axiom,
! [N: num] : ( ord_less_eq @ num @ one2 @ N ) ).
% semiring_norm(68)
thf(fact_188_semiring__norm_I75_J,axiom,
! [M: num] :
~ ( ord_less @ num @ M @ one2 ) ).
% semiring_norm(75)
thf(fact_189_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_190_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_191_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_192_delete__pres__valid,axiom,
! [T2: vEBT_VEBT,N: nat,X2: nat] :
( ( vEBT_invar_vebt @ T2 @ N )
=> ( vEBT_invar_vebt @ ( vEBT_vebt_delete @ T2 @ X2 ) @ N ) ) ).
% delete_pres_valid
thf(fact_193_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,S3: vEBT_VEBT] :
( Tree
= ( vEBT_Node @ Info2 @ ( suc @ ( suc @ N ) ) @ TreeList2 @ S3 ) ) ) ).
% deg_SUcn_Node
thf(fact_194_dele__member__cont__corr,axiom,
! [T2: vEBT_VEBT,N: nat,X2: nat,Y3: nat] :
( ( vEBT_invar_vebt @ T2 @ N )
=> ( ( vEBT_vebt_member @ ( vEBT_vebt_delete @ T2 @ X2 ) @ Y3 )
= ( ( X2 != Y3 )
& ( vEBT_vebt_member @ T2 @ Y3 ) ) ) ) ).
% dele_member_cont_corr
thf(fact_195_power__shift,axiom,
! [X2: nat,Y3: nat,Z: nat] :
( ( ( power_power @ nat @ X2 @ Y3 )
= Z )
= ( ( vEBT_VEBT_power @ ( some @ nat @ X2 ) @ ( some @ nat @ Y3 ) )
= ( some @ nat @ Z ) ) ) ).
% power_shift
thf(fact_196_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_197_semiring__norm_I87_J,axiom,
! [M: num,N: num] :
( ( ( bit0 @ M )
= ( bit0 @ N ) )
= ( M = N ) ) ).
% semiring_norm(87)
thf(fact_198_mint__corr__help,axiom,
! [T2: vEBT_VEBT,N: nat,Mini: nat,X2: nat] :
( ( vEBT_invar_vebt @ T2 @ N )
=> ( ( ( vEBT_vebt_mint @ T2 )
= ( some @ nat @ Mini ) )
=> ( ( vEBT_vebt_member @ T2 @ X2 )
=> ( ord_less_eq @ nat @ Mini @ X2 ) ) ) ) ).
% mint_corr_help
thf(fact_199_mulcomm,axiom,
! [I: nat,Va: nat] :
( ( times_times @ nat @ I @ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ Va @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) )
= ( times_times @ nat @ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ Va @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ I ) ) ).
% mulcomm
thf(fact_200_high__def,axiom,
( vEBT_VEBT_high
= ( ^ [X3: nat,N4: nat] : ( divide_divide @ nat @ X3 @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N4 ) ) ) ) ).
% high_def
thf(fact_201_member__bound,axiom,
! [Tree: vEBT_VEBT,X2: nat,N: nat] :
( ( vEBT_vebt_member @ Tree @ X2 )
=> ( ( vEBT_invar_vebt @ Tree @ N )
=> ( ord_less @ nat @ X2 @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N ) ) ) ) ).
% member_bound
thf(fact_202_geqmaxNone,axiom,
! [Mi: nat,Ma: nat,Deg: nat,TreeList: list @ vEBT_VEBT,Summary: vEBT_VEBT,N: nat,X2: 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 @ X2 )
=> ( ( vEBT_vebt_succ @ ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi @ Ma ) ) @ Deg @ TreeList @ Summary ) @ X2 )
= ( none @ nat ) ) ) ) ).
% geqmaxNone
thf(fact_203_semiring__norm_I83_J,axiom,
! [N: num] :
( one2
!= ( bit0 @ N ) ) ).
% semiring_norm(83)
thf(fact_204_semiring__norm_I85_J,axiom,
! [M: num] :
( ( bit0 @ M )
!= one2 ) ).
% semiring_norm(85)
thf(fact_205_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_206_helpypredd,axiom,
! [T2: vEBT_VEBT,N: nat,X2: nat,Y3: nat] :
( ( vEBT_invar_vebt @ T2 @ N )
=> ( ( ( vEBT_vebt_pred @ T2 @ X2 )
= ( some @ nat @ Y3 ) )
=> ( ord_less @ nat @ Y3 @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N ) ) ) ) ).
% helpypredd
thf(fact_207_helpyd,axiom,
! [T2: vEBT_VEBT,N: nat,X2: nat,Y3: nat] :
( ( vEBT_invar_vebt @ T2 @ N )
=> ( ( ( vEBT_vebt_succ @ T2 @ X2 )
= ( some @ nat @ Y3 ) )
=> ( ord_less @ nat @ Y3 @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N ) ) ) ) ).
% helpyd
thf(fact_208_post__member__pre__member,axiom,
! [T2: vEBT_VEBT,N: nat,X2: nat,Y3: nat] :
( ( vEBT_invar_vebt @ T2 @ N )
=> ( ( ord_less @ nat @ X2 @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N ) )
=> ( ( ord_less @ nat @ Y3 @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N ) )
=> ( ( vEBT_vebt_member @ ( vEBT_vebt_insert @ T2 @ X2 ) @ Y3 )
=> ( ( vEBT_vebt_member @ T2 @ Y3 )
| ( X2 = Y3 ) ) ) ) ) ) ).
% post_member_pre_member
thf(fact_209_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_210_semiring__norm_I13_J,axiom,
! [M: num,N: num] :
( ( times_times @ num @ ( bit0 @ M ) @ ( bit0 @ N ) )
= ( bit0 @ ( bit0 @ ( times_times @ num @ M @ N ) ) ) ) ).
% semiring_norm(13)
thf(fact_211_semiring__norm_I11_J,axiom,
! [M: num] :
( ( times_times @ num @ M @ one2 )
= M ) ).
% semiring_norm(11)
thf(fact_212_semiring__norm_I12_J,axiom,
! [N: num] :
( ( times_times @ num @ one2 @ N )
= N ) ).
% semiring_norm(12)
thf(fact_213_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_214_member__correct,axiom,
! [T2: vEBT_VEBT,N: nat,X2: nat] :
( ( vEBT_invar_vebt @ T2 @ N )
=> ( ( vEBT_vebt_member @ T2 @ X2 )
= ( member @ nat @ X2 @ ( vEBT_set_vebt @ T2 ) ) ) ) ).
% member_correct
thf(fact_215_max__number__of_I1_J,axiom,
! [A: $tType] :
( ( ( numeral @ A )
& ( ord @ A ) )
=> ! [U: num,V: num] :
( ( ( ord_less_eq @ A @ ( numeral_numeral @ A @ U ) @ ( numeral_numeral @ A @ V ) )
=> ( ( ord_max @ A @ ( numeral_numeral @ A @ U ) @ ( numeral_numeral @ A @ V ) )
= ( numeral_numeral @ A @ V ) ) )
& ( ~ ( ord_less_eq @ A @ ( numeral_numeral @ A @ U ) @ ( numeral_numeral @ A @ V ) )
=> ( ( ord_max @ A @ ( numeral_numeral @ A @ U ) @ ( numeral_numeral @ A @ V ) )
= ( numeral_numeral @ A @ U ) ) ) ) ) ).
% max_number_of(1)
thf(fact_216_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_217_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_218_VEBT__internal_OminNull_Osimps_I5_J,axiom,
! [Uz: product_prod @ nat @ nat,Va: nat,Vb: list @ vEBT_VEBT,Vc: vEBT_VEBT] :
~ ( vEBT_VEBT_minNull @ ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ Uz ) @ Va @ Vb @ Vc ) ) ).
% VEBT_internal.minNull.simps(5)
thf(fact_219_VEBT__internal_OminNull_Osimps_I4_J,axiom,
! [Uw: nat,Ux: list @ vEBT_VEBT,Uy: vEBT_VEBT] : ( vEBT_VEBT_minNull @ ( vEBT_Node @ ( none @ ( product_prod @ nat @ nat ) ) @ Uw @ Ux @ Uy ) ) ).
% VEBT_internal.minNull.simps(4)
thf(fact_220_vebt__member_Osimps_I2_J,axiom,
! [Uu: nat,Uv: list @ vEBT_VEBT,Uw: vEBT_VEBT,X2: nat] :
~ ( vEBT_vebt_member @ ( vEBT_Node @ ( none @ ( product_prod @ nat @ nat ) ) @ Uu @ Uv @ Uw ) @ X2 ) ).
% vebt_member.simps(2)
thf(fact_221_vebt__insert_Osimps_I5_J,axiom,
! [Mi: nat,Ma: nat,Va: nat,TreeList: list @ vEBT_VEBT,Summary: vEBT_VEBT,X2: nat] :
( ( vEBT_vebt_insert @ ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi @ Ma ) ) @ ( suc @ ( suc @ Va ) ) @ TreeList @ Summary ) @ X2 )
= ( if @ vEBT_VEBT
@ ( ( ord_less @ nat @ ( vEBT_VEBT_high @ ( if @ nat @ ( ord_less @ nat @ X2 @ Mi ) @ Mi @ X2 ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( size_size @ ( list @ vEBT_VEBT ) @ TreeList ) )
& ~ ( ( X2 = Mi )
| ( X2 = Ma ) ) )
@ ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ ( if @ nat @ ( ord_less @ nat @ X2 @ Mi ) @ X2 @ Mi ) @ ( ord_max @ nat @ ( if @ nat @ ( ord_less @ nat @ X2 @ Mi ) @ Mi @ X2 ) @ Ma ) ) ) @ ( suc @ ( suc @ Va ) ) @ ( list_update @ vEBT_VEBT @ TreeList @ ( vEBT_VEBT_high @ ( if @ nat @ ( ord_less @ nat @ X2 @ Mi ) @ Mi @ X2 ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( vEBT_vebt_insert @ ( nth @ vEBT_VEBT @ TreeList @ ( vEBT_VEBT_high @ ( if @ nat @ ( ord_less @ nat @ X2 @ Mi ) @ Mi @ X2 ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( vEBT_VEBT_low @ ( if @ nat @ ( ord_less @ nat @ X2 @ Mi ) @ Mi @ X2 ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) @ ( if @ vEBT_VEBT @ ( vEBT_VEBT_minNull @ ( nth @ vEBT_VEBT @ TreeList @ ( vEBT_VEBT_high @ ( if @ nat @ ( ord_less @ nat @ X2 @ Mi ) @ Mi @ X2 ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) @ ( vEBT_vebt_insert @ Summary @ ( vEBT_VEBT_high @ ( if @ nat @ ( ord_less @ nat @ X2 @ Mi ) @ Mi @ X2 ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ Summary ) )
@ ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi @ Ma ) ) @ ( suc @ ( suc @ Va ) ) @ TreeList @ Summary ) ) ) ).
% vebt_insert.simps(5)
thf(fact_222_mint__corr,axiom,
! [T2: vEBT_VEBT,N: nat,X2: nat] :
( ( vEBT_invar_vebt @ T2 @ N )
=> ( ( ( vEBT_vebt_mint @ T2 )
= ( some @ nat @ X2 ) )
=> ( vEBT_VEBT_min_in_set @ ( vEBT_VEBT_set_vebt @ T2 ) @ X2 ) ) ) ).
% mint_corr
thf(fact_223_mint__sound,axiom,
! [T2: vEBT_VEBT,N: nat,X2: nat] :
( ( vEBT_invar_vebt @ T2 @ N )
=> ( ( vEBT_VEBT_min_in_set @ ( vEBT_VEBT_set_vebt @ T2 ) @ X2 )
=> ( ( vEBT_vebt_mint @ T2 )
= ( some @ nat @ X2 ) ) ) ) ).
% mint_sound
thf(fact_224_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 )
& ? [M5: nat] :
( ( ( some @ nat @ M5 )
= ( vEBT_vebt_mint @ Summary ) )
& ( ord_less @ nat @ M5 @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( minus_minus @ nat @ N @ ( divide_divide @ nat @ N @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) ) ) ) ) ).
% mintlistlength
thf(fact_225_power__mult__numeral,axiom,
! [A: $tType] :
( ( monoid_mult @ A )
=> ! [A3: A,M: num,N: num] :
( ( power_power @ A @ ( power_power @ A @ A3 @ ( numeral_numeral @ nat @ M ) ) @ ( numeral_numeral @ nat @ N ) )
= ( power_power @ A @ A3 @ ( numeral_numeral @ nat @ ( times_times @ num @ M @ N ) ) ) ) ) ).
% power_mult_numeral
thf(fact_226_listlength,axiom,
( ( size_size @ ( list @ vEBT_VEBT ) @ treeList )
= ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( minus_minus @ nat @ na @ ( divide_divide @ nat @ na @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) ).
% listlength
thf(fact_227_two__pow__div__gt__le,axiom,
! [V: nat,N: nat,M: nat] :
( ( ord_less @ nat @ V @ ( 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_228_sumprop,axiom,
vEBT_invar_vebt @ summary @ ( minus_minus @ nat @ na @ ( divide_divide @ nat @ na @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ).
% sumprop
thf(fact_229_power__odd__eq,axiom,
! [A: $tType] :
( ( monoid_mult @ A )
=> ! [A3: A,N: nat] :
( ( power_power @ A @ A3 @ ( suc @ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N ) ) )
= ( times_times @ A @ A3 @ ( power_power @ A @ ( power_power @ A @ A3 @ N ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) ).
% power_odd_eq
thf(fact_230_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_231_both__member__options__ding,axiom,
! [Info: option @ ( product_prod @ nat @ nat ),Deg: nat,TreeList: list @ vEBT_VEBT,Summary: vEBT_VEBT,N: nat,X2: nat] :
( ( vEBT_invar_vebt @ ( vEBT_Node @ Info @ Deg @ TreeList @ Summary ) @ N )
=> ( ( ord_less @ nat @ X2 @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ Deg ) )
=> ( ( vEBT_V8194947554948674370ptions @ ( nth @ vEBT_VEBT @ TreeList @ ( vEBT_VEBT_high @ X2 @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( vEBT_VEBT_low @ X2 @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) )
=> ( vEBT_V8194947554948674370ptions @ ( vEBT_Node @ Info @ Deg @ TreeList @ Summary ) @ X2 ) ) ) ) ).
% both_member_options_ding
thf(fact_232_setprop,axiom,
! [T2: vEBT_VEBT] :
( ( member @ vEBT_VEBT @ T2 @ ( set2 @ vEBT_VEBT @ treeList ) )
=> ( vEBT_invar_vebt @ T2 @ ( divide_divide @ nat @ na @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ).
% setprop
thf(fact_233_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_234_set__vebt_H__def,axiom,
( vEBT_VEBT_set_vebt
= ( ^ [T3: vEBT_VEBT] : ( collect @ nat @ ( vEBT_vebt_member @ T3 ) ) ) ) ).
% set_vebt'_def
thf(fact_235_dele__bmo__cont__corr,axiom,
! [T2: vEBT_VEBT,N: nat,X2: nat,Y3: nat] :
( ( vEBT_invar_vebt @ T2 @ N )
=> ( ( vEBT_V8194947554948674370ptions @ ( vEBT_vebt_delete @ T2 @ X2 ) @ Y3 )
= ( ( X2 != Y3 )
& ( vEBT_V8194947554948674370ptions @ T2 @ Y3 ) ) ) ) ).
% dele_bmo_cont_corr
thf(fact_236_valid__member__both__member__options,axiom,
! [T2: vEBT_VEBT,N: nat,X2: nat] :
( ( vEBT_invar_vebt @ T2 @ N )
=> ( ( vEBT_V8194947554948674370ptions @ T2 @ X2 )
=> ( vEBT_vebt_member @ T2 @ X2 ) ) ) ).
% valid_member_both_member_options
thf(fact_237_both__member__options__equiv__member,axiom,
! [T2: vEBT_VEBT,N: nat,X2: nat] :
( ( vEBT_invar_vebt @ T2 @ N )
=> ( ( vEBT_V8194947554948674370ptions @ T2 @ X2 )
= ( vEBT_vebt_member @ T2 @ X2 ) ) ) ).
% both_member_options_equiv_member
thf(fact_238_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_239_inthall,axiom,
! [A: $tType,Xs: list @ A,P: A > $o,N: nat] :
( ! [X4: A] :
( ( member @ A @ X4 @ ( set2 @ A @ Xs ) )
=> ( P @ X4 ) )
=> ( ( ord_less @ nat @ N @ ( size_size @ ( list @ A ) @ Xs ) )
=> ( P @ ( nth @ A @ Xs @ N ) ) ) ) ).
% inthall
thf(fact_240_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 )
=> ( ! [X8: vEBT_VEBT] :
( ( member @ vEBT_VEBT @ X8 @ ( set2 @ vEBT_VEBT @ TreeList ) )
=> ~ ? [X_12: nat] : ( vEBT_V8194947554948674370ptions @ X8 @ X_12 ) )
& ~ ? [X_12: nat] : ( vEBT_V8194947554948674370ptions @ Summary @ X_12 ) ) ) ) ).
% mi_eq_ma_no_ch
thf(fact_241_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_242_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_243_valid__insert__both__member__options__add,axiom,
! [T2: vEBT_VEBT,N: nat,X2: nat] :
( ( vEBT_invar_vebt @ T2 @ N )
=> ( ( ord_less @ nat @ X2 @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N ) )
=> ( vEBT_V8194947554948674370ptions @ ( vEBT_vebt_insert @ T2 @ X2 ) @ X2 ) ) ) ).
% valid_insert_both_member_options_add
thf(fact_244_valid__insert__both__member__options__pres,axiom,
! [T2: vEBT_VEBT,N: nat,X2: nat,Y3: nat] :
( ( vEBT_invar_vebt @ T2 @ N )
=> ( ( ord_less @ nat @ X2 @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N ) )
=> ( ( ord_less @ nat @ Y3 @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N ) )
=> ( ( vEBT_V8194947554948674370ptions @ T2 @ X2 )
=> ( vEBT_V8194947554948674370ptions @ ( vEBT_vebt_insert @ T2 @ Y3 ) @ X2 ) ) ) ) ) ).
% valid_insert_both_member_options_pres
thf(fact_245_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_246_left__diff__distrib__numeral,axiom,
! [A: $tType] :
( ( ( numeral @ A )
& ( ring @ A ) )
=> ! [A3: A,B3: A,V: num] :
( ( times_times @ A @ ( minus_minus @ A @ A3 @ B3 ) @ ( numeral_numeral @ A @ V ) )
= ( minus_minus @ A @ ( times_times @ A @ A3 @ ( numeral_numeral @ A @ V ) ) @ ( times_times @ A @ B3 @ ( numeral_numeral @ A @ V ) ) ) ) ) ).
% left_diff_distrib_numeral
thf(fact_247_right__diff__distrib__numeral,axiom,
! [A: $tType] :
( ( ( numeral @ A )
& ( ring @ A ) )
=> ! [V: num,B3: A,C3: A] :
( ( times_times @ A @ ( numeral_numeral @ A @ V ) @ ( minus_minus @ A @ B3 @ C3 ) )
= ( minus_minus @ A @ ( times_times @ A @ ( numeral_numeral @ A @ V ) @ B3 ) @ ( times_times @ A @ ( numeral_numeral @ A @ V ) @ C3 ) ) ) ) ).
% right_diff_distrib_numeral
thf(fact_248_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_249_succ__member,axiom,
! [T2: vEBT_VEBT,X2: nat,Y3: nat] :
( ( vEBT_is_succ_in_set @ ( vEBT_VEBT_set_vebt @ T2 ) @ X2 @ Y3 )
= ( ( vEBT_vebt_member @ T2 @ Y3 )
& ( ord_less @ nat @ X2 @ Y3 )
& ! [Z4: nat] :
( ( ( vEBT_vebt_member @ T2 @ Z4 )
& ( ord_less @ nat @ X2 @ Z4 ) )
=> ( ord_less_eq @ nat @ Y3 @ Z4 ) ) ) ) ).
% succ_member
thf(fact_250_pred__member,axiom,
! [T2: vEBT_VEBT,X2: nat,Y3: nat] :
( ( vEBT_is_pred_in_set @ ( vEBT_VEBT_set_vebt @ T2 ) @ X2 @ Y3 )
= ( ( vEBT_vebt_member @ T2 @ Y3 )
& ( ord_less @ nat @ Y3 @ X2 )
& ! [Z4: nat] :
( ( ( vEBT_vebt_member @ T2 @ Z4 )
& ( ord_less @ nat @ Z4 @ X2 ) )
=> ( ord_less_eq @ nat @ Z4 @ Y3 ) ) ) ) ).
% pred_member
thf(fact_251_pred__corr,axiom,
! [T2: vEBT_VEBT,N: nat,X2: nat,Px: nat] :
( ( vEBT_invar_vebt @ T2 @ N )
=> ( ( ( vEBT_vebt_pred @ T2 @ X2 )
= ( some @ nat @ Px ) )
= ( vEBT_is_pred_in_set @ ( vEBT_VEBT_set_vebt @ T2 ) @ X2 @ Px ) ) ) ).
% pred_corr
thf(fact_252_succ__corr,axiom,
! [T2: vEBT_VEBT,N: nat,X2: nat,Sx: nat] :
( ( vEBT_invar_vebt @ T2 @ N )
=> ( ( ( vEBT_vebt_succ @ T2 @ X2 )
= ( some @ nat @ Sx ) )
= ( vEBT_is_succ_in_set @ ( vEBT_VEBT_set_vebt @ T2 ) @ X2 @ Sx ) ) ) ).
% succ_corr
thf(fact_253_pred__correct,axiom,
! [T2: vEBT_VEBT,N: nat,X2: nat,Sx: nat] :
( ( vEBT_invar_vebt @ T2 @ N )
=> ( ( ( vEBT_vebt_pred @ T2 @ X2 )
= ( some @ nat @ Sx ) )
= ( vEBT_is_pred_in_set @ ( vEBT_set_vebt @ T2 ) @ X2 @ Sx ) ) ) ).
% pred_correct
thf(fact_254_succ__correct,axiom,
! [T2: vEBT_VEBT,N: nat,X2: nat,Sx: nat] :
( ( vEBT_invar_vebt @ T2 @ N )
=> ( ( ( vEBT_vebt_succ @ T2 @ X2 )
= ( some @ nat @ Sx ) )
= ( vEBT_is_succ_in_set @ ( vEBT_set_vebt @ T2 ) @ X2 @ Sx ) ) ) ).
% succ_correct
thf(fact_255_local_Opower__def,axiom,
( vEBT_VEBT_power
= ( vEBT_V2048590022279873568_shift @ nat @ ( power_power @ nat ) ) ) ).
% local.power_def
thf(fact_256_subset__code_I1_J,axiom,
! [A: $tType,Xs: list @ A,B2: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ ( set2 @ A @ Xs ) @ B2 )
= ( ! [X3: A] :
( ( member @ A @ X3 @ ( set2 @ A @ Xs ) )
=> ( member @ A @ X3 @ B2 ) ) ) ) ).
% subset_code(1)
thf(fact_257_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_258_zero__induct__lemma,axiom,
! [P: nat > $o,K: nat,I: nat] :
( ( P @ K )
=> ( ! [N3: nat] :
( ( P @ ( suc @ N3 ) )
=> ( P @ N3 ) )
=> ( P @ ( minus_minus @ nat @ K @ I ) ) ) ) ).
% zero_induct_lemma
thf(fact_259_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_260_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_261_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_262_le__diff__iff_H,axiom,
! [A3: nat,C3: nat,B3: nat] :
( ( ord_less_eq @ nat @ A3 @ C3 )
=> ( ( ord_less_eq @ nat @ B3 @ C3 )
=> ( ( ord_less_eq @ nat @ ( minus_minus @ nat @ C3 @ A3 ) @ ( minus_minus @ nat @ C3 @ B3 ) )
= ( ord_less_eq @ nat @ B3 @ A3 ) ) ) ) ).
% le_diff_iff'
thf(fact_263_diff__le__self,axiom,
! [M: nat,N: nat] : ( ord_less_eq @ nat @ ( minus_minus @ nat @ M @ N ) @ M ) ).
% diff_le_self
thf(fact_264_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_265_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_266_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_267_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_268_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_269_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_270_set__update__subsetI,axiom,
! [A: $tType,Xs: list @ A,A2: set @ A,X2: A,I: nat] :
( ( ord_less_eq @ ( set @ A ) @ ( set2 @ A @ Xs ) @ A2 )
=> ( ( member @ A @ X2 @ A2 )
=> ( ord_less_eq @ ( set @ A ) @ ( set2 @ A @ ( list_update @ A @ Xs @ I @ X2 ) ) @ A2 ) ) ) ).
% set_update_subsetI
thf(fact_271_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_272_diff__less__Suc,axiom,
! [M: nat,N: nat] : ( ord_less @ nat @ ( minus_minus @ nat @ M @ N ) @ ( suc @ M ) ) ).
% diff_less_Suc
thf(fact_273_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_274_diff__less__mono,axiom,
! [A3: nat,B3: nat,C3: nat] :
( ( ord_less @ nat @ A3 @ B3 )
=> ( ( ord_less_eq @ nat @ C3 @ A3 )
=> ( ord_less @ nat @ ( minus_minus @ nat @ A3 @ C3 ) @ ( minus_minus @ nat @ B3 @ C3 ) ) ) ) ).
% diff_less_mono
thf(fact_275_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_276_power2__commute,axiom,
! [A: $tType] :
( ( comm_ring_1 @ A )
=> ! [X2: A,Y3: A] :
( ( power_power @ A @ ( minus_minus @ A @ X2 @ Y3 ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) )
= ( power_power @ A @ ( minus_minus @ A @ Y3 @ X2 ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ).
% power2_commute
thf(fact_277_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_278_list__ball__nth,axiom,
! [A: $tType,N: nat,Xs: list @ A,P: A > $o] :
( ( ord_less @ nat @ N @ ( size_size @ ( list @ A ) @ Xs ) )
=> ( ! [X4: A] :
( ( member @ A @ X4 @ ( set2 @ A @ Xs ) )
=> ( P @ X4 ) )
=> ( P @ ( nth @ A @ Xs @ N ) ) ) ) ).
% list_ball_nth
thf(fact_279_in__set__conv__nth,axiom,
! [A: $tType,X2: A,Xs: list @ A] :
( ( member @ A @ X2 @ ( set2 @ A @ Xs ) )
= ( ? [I4: nat] :
( ( ord_less @ nat @ I4 @ ( size_size @ ( list @ A ) @ Xs ) )
& ( ( nth @ A @ Xs @ I4 )
= X2 ) ) ) ) ).
% in_set_conv_nth
thf(fact_280_all__set__conv__nth,axiom,
! [A: $tType,L: list @ A,P: A > $o] :
( ( ! [X3: A] :
( ( member @ A @ X3 @ ( set2 @ A @ L ) )
=> ( P @ X3 ) ) )
= ( ! [I4: nat] :
( ( ord_less @ nat @ I4 @ ( size_size @ ( list @ A ) @ L ) )
=> ( P @ ( nth @ A @ L @ I4 ) ) ) ) ) ).
% all_set_conv_nth
thf(fact_281_all__nth__imp__all__set,axiom,
! [A: $tType,Xs: list @ A,P: A > $o,X2: A] :
( ! [I2: nat] :
( ( ord_less @ nat @ I2 @ ( size_size @ ( list @ A ) @ Xs ) )
=> ( P @ ( nth @ A @ Xs @ I2 ) ) )
=> ( ( member @ A @ X2 @ ( set2 @ A @ Xs ) )
=> ( P @ X2 ) ) ) ).
% all_nth_imp_all_set
thf(fact_282_all__set__conv__all__nth,axiom,
! [A: $tType,Xs: list @ A,P: A > $o] :
( ( ! [X3: A] :
( ( member @ A @ X3 @ ( set2 @ A @ Xs ) )
=> ( P @ X3 ) ) )
= ( ! [I4: nat] :
( ( ord_less @ nat @ I4 @ ( size_size @ ( list @ A ) @ Xs ) )
=> ( P @ ( nth @ A @ Xs @ I4 ) ) ) ) ) ).
% all_set_conv_all_nth
thf(fact_283_in__set__upd__eq,axiom,
! [A: $tType,I: nat,L: list @ A,X2: A,Y3: A] :
( ( ord_less @ nat @ I @ ( size_size @ ( list @ A ) @ L ) )
=> ( ( member @ A @ X2 @ ( set2 @ A @ ( list_update @ A @ L @ I @ Y3 ) ) )
= ( ( X2 = Y3 )
| ( ( member @ A @ X2 @ ( set2 @ A @ L ) )
& ! [Y: A] : ( member @ A @ X2 @ ( set2 @ A @ ( list_update @ A @ L @ I @ Y ) ) ) ) ) ) ) ).
% in_set_upd_eq
thf(fact_284_set__update__memI,axiom,
! [A: $tType,N: nat,Xs: list @ A,X2: A] :
( ( ord_less @ nat @ N @ ( size_size @ ( list @ A ) @ Xs ) )
=> ( member @ A @ X2 @ ( set2 @ A @ ( list_update @ A @ Xs @ N @ X2 ) ) ) ) ).
% set_update_memI
thf(fact_285_in__set__upd__cases,axiom,
! [A: $tType,X2: A,L: list @ A,I: nat,Y3: A] :
( ( member @ A @ X2 @ ( set2 @ A @ ( list_update @ A @ L @ I @ Y3 ) ) )
=> ( ( ( ord_less @ nat @ I @ ( size_size @ ( list @ A ) @ L ) )
=> ( X2 != Y3 ) )
=> ( member @ A @ X2 @ ( set2 @ A @ L ) ) ) ) ).
% in_set_upd_cases
thf(fact_286_in__set__upd__eq__aux,axiom,
! [A: $tType,I: nat,L: list @ A,X2: A,Y3: A] :
( ( ord_less @ nat @ I @ ( size_size @ ( list @ A ) @ L ) )
=> ( ( member @ A @ X2 @ ( set2 @ A @ ( list_update @ A @ L @ I @ Y3 ) ) )
= ( ( X2 = Y3 )
| ! [Y: A] : ( member @ A @ X2 @ ( set2 @ A @ ( list_update @ A @ L @ I @ Y ) ) ) ) ) ) ).
% in_set_upd_eq_aux
thf(fact_287_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_288_power__minus__is__div,axiom,
! [B3: nat,A3: nat] :
( ( ord_less_eq @ nat @ B3 @ A3 )
=> ( ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( minus_minus @ nat @ A3 @ B3 ) )
= ( divide_divide @ nat @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ A3 ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ B3 ) ) ) ) ).
% power_minus_is_div
thf(fact_289_less__two__pow__divD,axiom,
! [X2: nat,N: nat,M: nat] :
( ( ord_less @ nat @ X2 @ ( 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 @ X2 @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( minus_minus @ nat @ N @ M ) ) ) ) ) ).
% less_two_pow_divD
thf(fact_290_less__two__pow__divI,axiom,
! [X2: nat,N: nat,M: nat] :
( ( ord_less @ nat @ X2 @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( minus_minus @ nat @ N @ M ) ) )
=> ( ( ord_less_eq @ nat @ M @ N )
=> ( ord_less @ nat @ X2 @ ( 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_291_power__commutes,axiom,
! [A: $tType] :
( ( monoid_mult @ A )
=> ! [A3: A,N: nat] :
( ( times_times @ A @ ( power_power @ A @ A3 @ N ) @ A3 )
= ( times_times @ A @ A3 @ ( power_power @ A @ A3 @ N ) ) ) ) ).
% power_commutes
thf(fact_292_power__mult__distrib,axiom,
! [A: $tType] :
( ( comm_monoid_mult @ A )
=> ! [A3: A,B3: A,N: nat] :
( ( power_power @ A @ ( times_times @ A @ A3 @ B3 ) @ N )
= ( times_times @ A @ ( power_power @ A @ A3 @ N ) @ ( power_power @ A @ B3 @ N ) ) ) ) ).
% power_mult_distrib
thf(fact_293_power__commuting__commutes,axiom,
! [A: $tType] :
( ( monoid_mult @ A )
=> ! [X2: A,Y3: A,N: nat] :
( ( ( times_times @ A @ X2 @ Y3 )
= ( times_times @ A @ Y3 @ X2 ) )
=> ( ( times_times @ A @ ( power_power @ A @ X2 @ N ) @ Y3 )
= ( times_times @ A @ Y3 @ ( power_power @ A @ X2 @ N ) ) ) ) ) ).
% power_commuting_commutes
thf(fact_294_power__divide,axiom,
! [A: $tType] :
( ( field @ A )
=> ! [A3: A,B3: A,N: nat] :
( ( power_power @ A @ ( divide_divide @ A @ A3 @ B3 ) @ N )
= ( divide_divide @ A @ ( power_power @ A @ A3 @ N ) @ ( power_power @ A @ B3 @ N ) ) ) ) ).
% power_divide
thf(fact_295_power__mult,axiom,
! [A: $tType] :
( ( monoid_mult @ A )
=> ! [A3: A,M: nat,N: nat] :
( ( power_power @ A @ A3 @ ( times_times @ nat @ M @ N ) )
= ( power_power @ A @ ( power_power @ A @ A3 @ M ) @ N ) ) ) ).
% power_mult
thf(fact_296_power__Suc,axiom,
! [A: $tType] :
( ( power @ A )
=> ! [A3: A,N: nat] :
( ( power_power @ A @ A3 @ ( suc @ N ) )
= ( times_times @ A @ A3 @ ( power_power @ A @ A3 @ N ) ) ) ) ).
% power_Suc
thf(fact_297_power__Suc2,axiom,
! [A: $tType] :
( ( monoid_mult @ A )
=> ! [A3: A,N: nat] :
( ( power_power @ A @ A3 @ ( suc @ N ) )
= ( times_times @ A @ ( power_power @ A @ A3 @ N ) @ A3 ) ) ) ).
% power_Suc2
thf(fact_298_div__mult__le,axiom,
! [A3: nat,B3: nat] : ( ord_less_eq @ nat @ ( times_times @ nat @ ( divide_divide @ nat @ A3 @ B3 ) @ B3 ) @ A3 ) ).
% div_mult_le
thf(fact_299_power__numeral__even,axiom,
! [A: $tType] :
( ( monoid_mult @ A )
=> ! [Z: A,W: num] :
( ( power_power @ A @ Z @ ( numeral_numeral @ nat @ ( bit0 @ W ) ) )
= ( times_times @ A @ ( power_power @ A @ Z @ ( numeral_numeral @ nat @ W ) ) @ ( power_power @ A @ Z @ ( numeral_numeral @ nat @ W ) ) ) ) ) ).
% power_numeral_even
thf(fact_300_power4__eq__xxxx,axiom,
! [A: $tType] :
( ( monoid_mult @ A )
=> ! [X2: A] :
( ( power_power @ A @ X2 @ ( numeral_numeral @ nat @ ( bit0 @ ( bit0 @ one2 ) ) ) )
= ( times_times @ A @ ( times_times @ A @ ( times_times @ A @ X2 @ X2 ) @ X2 ) @ X2 ) ) ) ).
% power4_eq_xxxx
thf(fact_301_power2__eq__square,axiom,
! [A: $tType] :
( ( monoid_mult @ A )
=> ! [A3: A] :
( ( power_power @ A @ A3 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) )
= ( times_times @ A @ A3 @ A3 ) ) ) ).
% power2_eq_square
thf(fact_302_power__even__eq,axiom,
! [A: $tType] :
( ( monoid_mult @ A )
=> ! [A3: A,N: nat] :
( ( power_power @ A @ A3 @ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N ) )
= ( power_power @ A @ ( power_power @ A @ A3 @ N ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ).
% power_even_eq
thf(fact_303_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_304_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_305_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_306_in__children__def,axiom,
( vEBT_V5917875025757280293ildren
= ( ^ [N4: nat,TreeList3: list @ vEBT_VEBT,X3: nat] : ( vEBT_V8194947554948674370ptions @ ( nth @ vEBT_VEBT @ TreeList3 @ ( vEBT_VEBT_high @ X3 @ N4 ) ) @ ( vEBT_VEBT_low @ X3 @ N4 ) ) ) ) ).
% in_children_def
thf(fact_307_both__member__options__from__chilf__to__complete__tree,axiom,
! [X2: nat,Deg: nat,TreeList: list @ vEBT_VEBT,Mi: nat,Ma: nat,Summary: vEBT_VEBT] :
( ( ord_less @ nat @ ( vEBT_VEBT_high @ X2 @ ( 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 @ X2 @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( vEBT_VEBT_low @ X2 @ ( 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 ) @ X2 ) ) ) ) ).
% both_member_options_from_chilf_to_complete_tree
thf(fact_308_both__member__options__from__complete__tree__to__child,axiom,
! [Deg: nat,Mi: nat,Ma: nat,TreeList: list @ vEBT_VEBT,Summary: vEBT_VEBT,X2: 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 ) @ X2 )
=> ( ( vEBT_V8194947554948674370ptions @ ( nth @ vEBT_VEBT @ TreeList @ ( vEBT_VEBT_high @ X2 @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( vEBT_VEBT_low @ X2 @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) )
| ( X2 = Mi )
| ( X2 = Ma ) ) ) ) ).
% both_member_options_from_complete_tree_to_child
thf(fact_309_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_310_set__n__deg__not__0,axiom,
! [TreeList: list @ vEBT_VEBT,N: nat,M: nat] :
( ! [X4: vEBT_VEBT] :
( ( member @ vEBT_VEBT @ X4 @ ( set2 @ vEBT_VEBT @ TreeList ) )
=> ( vEBT_invar_vebt @ X4 @ 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_311_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_312_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_313_xndef,axiom,
( xnew
= ( plus_plus @ nat @ ( times_times @ nat @ ( the2 @ nat @ ( vEBT_vebt_mint @ summary ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ ( suc @ ( suc @ va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ treeList @ ( the2 @ nat @ ( vEBT_vebt_mint @ summary ) ) ) ) ) ) ) ).
% xndef
thf(fact_314_bit__concat__def,axiom,
( vEBT_VEBT_bit_concat
= ( ^ [H: nat,L3: nat,D3: nat] : ( plus_plus @ nat @ ( times_times @ nat @ H @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ D3 ) ) @ L3 ) ) ) ).
% bit_concat_def
thf(fact_315_low__inv,axiom,
! [X2: nat,N: nat,Y3: nat] :
( ( ord_less @ nat @ X2 @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N ) )
=> ( ( vEBT_VEBT_low @ ( plus_plus @ nat @ ( times_times @ nat @ Y3 @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N ) ) @ X2 ) @ N )
= X2 ) ) ).
% low_inv
thf(fact_316_high__inv,axiom,
! [X2: nat,N: nat,Y3: nat] :
( ( ord_less @ nat @ X2 @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N ) )
=> ( ( vEBT_VEBT_high @ ( plus_plus @ nat @ ( times_times @ nat @ Y3 @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N ) ) @ X2 ) @ N )
= Y3 ) ) ).
% high_inv
thf(fact_317_even__odd__cases,axiom,
! [X2: nat] :
( ! [N3: nat] :
( X2
!= ( plus_plus @ nat @ N3 @ N3 ) )
=> ~ ! [N3: nat] :
( X2
!= ( plus_plus @ nat @ N3 @ ( suc @ N3 ) ) ) ) ).
% even_odd_cases
thf(fact_318_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_319_pow__sum,axiom,
! [A3: nat,B3: nat] :
( ( divide_divide @ nat @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( plus_plus @ nat @ A3 @ B3 ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ A3 ) )
= ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ B3 ) ) ).
% pow_sum
thf(fact_320_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_321_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_322_numeral__plus__numeral,axiom,
! [A: $tType] :
( ( numeral @ A )
=> ! [M: num,N: num] :
( ( plus_plus @ A @ ( numeral_numeral @ A @ M ) @ ( numeral_numeral @ A @ N ) )
= ( numeral_numeral @ A @ ( plus_plus @ num @ M @ N ) ) ) ) ).
% numeral_plus_numeral
thf(fact_323_add__numeral__left,axiom,
! [A: $tType] :
( ( numeral @ A )
=> ! [V: num,W: num,Z: A] :
( ( plus_plus @ A @ ( numeral_numeral @ A @ V ) @ ( plus_plus @ A @ ( numeral_numeral @ A @ W ) @ Z ) )
= ( plus_plus @ A @ ( numeral_numeral @ A @ ( plus_plus @ num @ V @ W ) ) @ Z ) ) ) ).
% add_numeral_left
thf(fact_324_power__one,axiom,
! [A: $tType] :
( ( monoid_mult @ A )
=> ! [N: nat] :
( ( power_power @ A @ ( one_one @ A ) @ N )
= ( one_one @ A ) ) ) ).
% power_one
thf(fact_325_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_326_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_327_power__one__right,axiom,
! [A: $tType] :
( ( monoid_mult @ A )
=> ! [A3: A] :
( ( power_power @ A @ A3 @ ( one_one @ nat ) )
= A3 ) ) ).
% power_one_right
thf(fact_328_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_329_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_330_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_331_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_332_minewdef,axiom,
( minew
= ( plus_plus @ nat @ ( times_times @ nat @ ( the2 @ nat @ ( vEBT_vebt_mint @ summary ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ ( suc @ ( suc @ va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ treeList @ ( the2 @ nat @ ( vEBT_vebt_mint @ summary ) ) ) ) ) ) ) ).
% minewdef
thf(fact_333_one__eq__numeral__iff,axiom,
! [A: $tType] :
( ( semiring_char_0 @ A )
=> ! [N: num] :
( ( ( one_one @ A )
= ( numeral_numeral @ A @ N ) )
= ( one2 = N ) ) ) ).
% one_eq_numeral_iff
thf(fact_334_numeral__eq__one__iff,axiom,
! [A: $tType] :
( ( semiring_char_0 @ A )
=> ! [N: num] :
( ( ( numeral_numeral @ A @ N )
= ( one_one @ A ) )
= ( N = one2 ) ) ) ).
% numeral_eq_one_iff
thf(fact_335_distrib__right__numeral,axiom,
! [A: $tType] :
( ( ( numeral @ A )
& ( semiring @ A ) )
=> ! [A3: A,B3: A,V: num] :
( ( times_times @ A @ ( plus_plus @ A @ A3 @ B3 ) @ ( numeral_numeral @ A @ V ) )
= ( plus_plus @ A @ ( times_times @ A @ A3 @ ( numeral_numeral @ A @ V ) ) @ ( times_times @ A @ B3 @ ( numeral_numeral @ A @ V ) ) ) ) ) ).
% distrib_right_numeral
thf(fact_336_distrib__left__numeral,axiom,
! [A: $tType] :
( ( ( numeral @ A )
& ( semiring @ A ) )
=> ! [V: num,B3: A,C3: A] :
( ( times_times @ A @ ( numeral_numeral @ A @ V ) @ ( plus_plus @ A @ B3 @ C3 ) )
= ( plus_plus @ A @ ( times_times @ A @ ( numeral_numeral @ A @ V ) @ B3 ) @ ( times_times @ A @ ( numeral_numeral @ A @ V ) @ C3 ) ) ) ) ).
% distrib_left_numeral
thf(fact_337_power__inject__exp,axiom,
! [A: $tType] :
( ( linordered_semidom @ A )
=> ! [A3: A,M: nat,N: nat] :
( ( ord_less @ A @ ( one_one @ A ) @ A3 )
=> ( ( ( power_power @ A @ A3 @ M )
= ( power_power @ A @ A3 @ N ) )
= ( M = N ) ) ) ) ).
% power_inject_exp
thf(fact_338_diff__Suc__1,axiom,
! [N: nat] :
( ( minus_minus @ nat @ ( suc @ N ) @ ( one_one @ nat ) )
= N ) ).
% diff_Suc_1
thf(fact_339_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_340_max__0__1_I6_J,axiom,
! [A: $tType] :
( ( linord181362715937106298miring @ A )
=> ! [X2: num] :
( ( ord_max @ A @ ( numeral_numeral @ A @ X2 ) @ ( one_one @ A ) )
= ( numeral_numeral @ A @ X2 ) ) ) ).
% max_0_1(6)
thf(fact_341_max__0__1_I5_J,axiom,
! [A: $tType] :
( ( linord181362715937106298miring @ A )
=> ! [X2: num] :
( ( ord_max @ A @ ( one_one @ A ) @ ( numeral_numeral @ A @ X2 ) )
= ( numeral_numeral @ A @ X2 ) ) ) ).
% max_0_1(5)
thf(fact_342_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_343_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_344_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_345_option_Ocollapse,axiom,
! [A: $tType,Option: option @ A] :
( ( Option
!= ( none @ A ) )
=> ( ( some @ A @ ( the2 @ A @ Option ) )
= Option ) ) ).
% option.collapse
thf(fact_346_newnodedef,axiom,
( newnode
= ( vEBT_vebt_delete @ ( nth @ vEBT_VEBT @ treeList @ ( vEBT_VEBT_high @ ( plus_plus @ nat @ ( times_times @ nat @ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ va @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_mint @ summary ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ treeList @ ( the2 @ nat @ ( vEBT_vebt_mint @ summary ) ) ) ) ) ) @ ( suc @ ( divide_divide @ nat @ va @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) @ ( 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 @ ( suc @ ( suc @ va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ treeList @ ( the2 @ nat @ ( vEBT_vebt_mint @ summary ) ) ) ) ) ) @ ( divide_divide @ nat @ ( suc @ ( suc @ va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) ).
% newnodedef
thf(fact_347_one__plus__numeral,axiom,
! [A: $tType] :
( ( numeral @ A )
=> ! [N: num] :
( ( plus_plus @ A @ ( one_one @ A ) @ ( numeral_numeral @ A @ N ) )
= ( numeral_numeral @ A @ ( plus_plus @ num @ one2 @ N ) ) ) ) ).
% one_plus_numeral
thf(fact_348_numeral__plus__one,axiom,
! [A: $tType] :
( ( numeral @ A )
=> ! [N: num] :
( ( plus_plus @ A @ ( numeral_numeral @ A @ N ) @ ( one_one @ A ) )
= ( numeral_numeral @ A @ ( plus_plus @ num @ N @ one2 ) ) ) ) ).
% numeral_plus_one
thf(fact_349_aktnodedef,axiom,
( ( ma != mi )
=> ( ( ord_less_eq @ nat @ xa @ ma )
=> ( aktnode
= ( nth @ vEBT_VEBT @ treeList @ ( vEBT_VEBT_high @ ( plus_plus @ nat @ ( times_times @ nat @ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ va @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_mint @ summary ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ treeList @ ( the2 @ nat @ ( vEBT_vebt_mint @ summary ) ) ) ) ) ) @ ( suc @ ( divide_divide @ nat @ va @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) ) ) ) ).
% aktnodedef
thf(fact_350_power__strict__increasing__iff,axiom,
! [A: $tType] :
( ( linordered_semidom @ A )
=> ! [B3: A,X2: nat,Y3: nat] :
( ( ord_less @ A @ ( one_one @ A ) @ B3 )
=> ( ( ord_less @ A @ ( power_power @ A @ B3 @ X2 ) @ ( power_power @ A @ B3 @ Y3 ) )
= ( ord_less @ nat @ X2 @ Y3 ) ) ) ) ).
% power_strict_increasing_iff
thf(fact_351_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_352_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_353_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_354_nested__mint,axiom,
! [Mi: nat,Ma: nat,Deg: nat,TreeList: list @ vEBT_VEBT,Summary: vEBT_VEBT,N: nat,Va: nat] :
( ( vEBT_invar_vebt @ ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi @ Ma ) ) @ Deg @ TreeList @ Summary ) @ N )
=> ( ( N
= ( suc @ ( suc @ Va ) ) )
=> ( ~ ( ord_less @ nat @ Ma @ Mi )
=> ( ( Ma != Mi )
=> ( ord_less @ nat @ ( vEBT_VEBT_high @ ( plus_plus @ nat @ ( times_times @ nat @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ Va @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ ( suc @ ( divide_divide @ nat @ Va @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( size_size @ ( list @ vEBT_VEBT ) @ TreeList ) ) ) ) ) ) ).
% nested_mint
thf(fact_355_one__add__one,axiom,
! [A: $tType] :
( ( numeral @ A )
=> ( ( plus_plus @ A @ ( one_one @ A ) @ ( one_one @ A ) )
= ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) ) ).
% one_add_one
thf(fact_356_power__increasing__iff,axiom,
! [A: $tType] :
( ( linordered_semidom @ A )
=> ! [B3: A,X2: nat,Y3: nat] :
( ( ord_less @ A @ ( one_one @ A ) @ B3 )
=> ( ( ord_less_eq @ A @ ( power_power @ A @ B3 @ X2 ) @ ( power_power @ A @ B3 @ Y3 ) )
= ( ord_less_eq @ nat @ X2 @ Y3 ) ) ) ) ).
% power_increasing_iff
thf(fact_357_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_358_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_359_Suc__1,axiom,
( ( suc @ ( one_one @ nat ) )
= ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ).
% Suc_1
thf(fact_360_add__self__div__2,axiom,
! [M: nat] :
( ( divide_divide @ nat @ ( plus_plus @ nat @ M @ M ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) )
= M ) ).
% add_self_div_2
thf(fact_361_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_362_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_363__092_060open_062_092_060And_062thesis_O_A_I_092_060And_062aktnode_O_A_I_092_060lbrakk_062ma_A_092_060noteq_062_Ami_059_Ax_A_092_060le_062_Ama_092_060rbrakk_062_A_092_060Longrightarrow_062_Aaktnode_A_061_AtreeList_A_B_Ahigh_A_I2_A_K_A2_A_094_A_Iva_Adiv_A2_J_A_K_Athe_A_Ivebt__mint_Asummary_J_A_L_Athe_A_Ivebt__mint_A_ItreeList_A_B_Athe_A_Ivebt__mint_Asummary_J_J_J_J_A_ISuc_A_Iva_Adiv_A2_J_J_J_A_092_060Longrightarrow_062_Athesis_J_A_092_060Longrightarrow_062_Athesis_092_060close_062,axiom,
~ ! [Aktnode: vEBT_VEBT] :
~ ( ( ma != mi )
=> ( ( ord_less_eq @ nat @ xa @ ma )
=> ( Aktnode
= ( nth @ vEBT_VEBT @ treeList @ ( vEBT_VEBT_high @ ( plus_plus @ nat @ ( times_times @ nat @ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ va @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_mint @ summary ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ treeList @ ( the2 @ nat @ ( vEBT_vebt_mint @ summary ) ) ) ) ) ) @ ( suc @ ( divide_divide @ nat @ va @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) ) ) ) ).
% \<open>\<And>thesis. (\<And>aktnode. (\<lbrakk>ma \<noteq> mi; x \<le> ma\<rbrakk> \<Longrightarrow> aktnode = treeList ! high (2 * 2 ^ (va div 2) * the (vebt_mint summary) + the (vebt_mint (treeList ! the (vebt_mint summary)))) (Suc (va div 2))) \<Longrightarrow> thesis) \<Longrightarrow> thesis\<close>
thf(fact_364_is__num__normalize_I1_J,axiom,
! [A: $tType] :
( ( neg_numeral @ A )
=> ! [A3: A,B3: A,C3: A] :
( ( plus_plus @ A @ ( plus_plus @ A @ A3 @ B3 ) @ C3 )
= ( plus_plus @ A @ A3 @ ( plus_plus @ A @ B3 @ C3 ) ) ) ) ).
% is_num_normalize(1)
thf(fact_365_subset__Collect__conv,axiom,
! [A: $tType,S: set @ A,P: A > $o] :
( ( ord_less_eq @ ( set @ A ) @ S @ ( collect @ A @ P ) )
= ( ! [X3: A] :
( ( member @ A @ X3 @ S )
=> ( P @ X3 ) ) ) ) ).
% subset_Collect_conv
thf(fact_366_one__plus__numeral__commute,axiom,
! [A: $tType] :
( ( numeral @ A )
=> ! [X2: num] :
( ( plus_plus @ A @ ( one_one @ A ) @ ( numeral_numeral @ A @ X2 ) )
= ( plus_plus @ A @ ( numeral_numeral @ A @ X2 ) @ ( one_one @ A ) ) ) ) ).
% one_plus_numeral_commute
thf(fact_367_Suc__eq__plus1,axiom,
( suc
= ( ^ [N4: nat] : ( plus_plus @ nat @ N4 @ ( one_one @ nat ) ) ) ) ).
% Suc_eq_plus1
thf(fact_368_plus__1__eq__Suc,axiom,
( ( plus_plus @ nat @ ( one_one @ nat ) )
= suc ) ).
% plus_1_eq_Suc
thf(fact_369_Suc__eq__plus1__left,axiom,
( suc
= ( plus_plus @ nat @ ( one_one @ nat ) ) ) ).
% Suc_eq_plus1_left
thf(fact_370_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_371_add__Suc,axiom,
! [M: nat,N: nat] :
( ( plus_plus @ nat @ ( suc @ M ) @ N )
= ( suc @ ( plus_plus @ nat @ M @ N ) ) ) ).
% add_Suc
thf(fact_372_nat__arith_Osuc1,axiom,
! [A2: nat,K: nat,A3: nat] :
( ( A2
= ( plus_plus @ nat @ K @ A3 ) )
=> ( ( suc @ A2 )
= ( plus_plus @ nat @ K @ ( suc @ A3 ) ) ) ) ).
% nat_arith.suc1
thf(fact_373_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_374_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_375_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_376_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_377_not__add__less2,axiom,
! [J: nat,I: nat] :
~ ( ord_less @ nat @ ( plus_plus @ nat @ J @ I ) @ I ) ).
% not_add_less2
thf(fact_378_not__add__less1,axiom,
! [I: nat,J: nat] :
~ ( ord_less @ nat @ ( plus_plus @ nat @ I @ J ) @ I ) ).
% not_add_less1
thf(fact_379_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_380_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_381_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_382_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_383_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_384_le__add1,axiom,
! [N: nat,M: nat] : ( ord_less_eq @ nat @ N @ ( plus_plus @ nat @ N @ M ) ) ).
% le_add1
thf(fact_385_le__add2,axiom,
! [N: nat,M: nat] : ( ord_less_eq @ nat @ N @ ( plus_plus @ nat @ M @ N ) ) ).
% le_add2
thf(fact_386_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_387_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_388_le__Suc__ex,axiom,
! [K: nat,L: nat] :
( ( ord_less_eq @ nat @ K @ L )
=> ? [N3: nat] :
( L
= ( plus_plus @ nat @ K @ N3 ) ) ) ).
% le_Suc_ex
thf(fact_389_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_390_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_391_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_392_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_393_nat__le__iff__add,axiom,
( ( ord_less_eq @ nat )
= ( ^ [M6: nat,N4: nat] :
? [K3: nat] :
( N4
= ( plus_plus @ nat @ M6 @ K3 ) ) ) ) ).
% nat_le_iff_add
thf(fact_394_diff__add__inverse2,axiom,
! [M: nat,N: nat] :
( ( minus_minus @ nat @ ( plus_plus @ nat @ M @ N ) @ N )
= M ) ).
% diff_add_inverse2
thf(fact_395_diff__add__inverse,axiom,
! [N: nat,M: nat] :
( ( minus_minus @ nat @ ( plus_plus @ nat @ N @ M ) @ N )
= M ) ).
% diff_add_inverse
thf(fact_396_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_397_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_398_left__add__mult__distrib,axiom,
! [I: nat,U: nat,J: nat,K: nat] :
( ( plus_plus @ nat @ ( times_times @ nat @ I @ U ) @ ( plus_plus @ nat @ ( times_times @ nat @ J @ U ) @ K ) )
= ( plus_plus @ nat @ ( times_times @ nat @ ( plus_plus @ nat @ I @ J ) @ U ) @ K ) ) ).
% left_add_mult_distrib
thf(fact_399_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_400_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_401_nat__mult__1__right,axiom,
! [N: nat] :
( ( times_times @ nat @ N @ ( one_one @ nat ) )
= N ) ).
% nat_mult_1_right
thf(fact_402_nat__mult__1,axiom,
! [N: nat] :
( ( times_times @ nat @ ( one_one @ nat ) @ N )
= N ) ).
% nat_mult_1
thf(fact_403_option_Osel,axiom,
! [A: $tType,X22: A] :
( ( the2 @ A @ ( some @ A @ X22 ) )
= X22 ) ).
% option.sel
thf(fact_404_option_Oexpand,axiom,
! [A: $tType,Option: option @ A,Option2: option @ A] :
( ( ( Option
= ( none @ A ) )
= ( Option2
= ( none @ A ) ) )
=> ( ( ( Option
!= ( none @ A ) )
=> ( ( Option2
!= ( none @ A ) )
=> ( ( the2 @ A @ Option )
= ( the2 @ A @ Option2 ) ) ) )
=> ( Option = Option2 ) ) ) ).
% option.expand
thf(fact_405_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_406_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_407_nat__1__add__1,axiom,
( ( plus_plus @ nat @ ( one_one @ nat ) @ ( one_one @ nat ) )
= ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ).
% nat_1_add_1
thf(fact_408_numeral__Bit0,axiom,
! [A: $tType] :
( ( numeral @ A )
=> ! [N: num] :
( ( numeral_numeral @ A @ ( bit0 @ N ) )
= ( plus_plus @ A @ ( numeral_numeral @ A @ N ) @ ( numeral_numeral @ A @ N ) ) ) ) ).
% numeral_Bit0
thf(fact_409_power__add,axiom,
! [A: $tType] :
( ( monoid_mult @ A )
=> ! [A3: A,M: nat,N: nat] :
( ( power_power @ A @ A3 @ ( plus_plus @ nat @ M @ N ) )
= ( times_times @ A @ ( power_power @ A @ A3 @ M ) @ ( power_power @ A @ A3 @ N ) ) ) ) ).
% power_add
thf(fact_410_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_411_less__iff__Suc__add,axiom,
( ( ord_less @ nat )
= ( ^ [M6: nat,N4: nat] :
? [K3: nat] :
( N4
= ( suc @ ( plus_plus @ nat @ M6 @ K3 ) ) ) ) ) ).
% less_iff_Suc_add
thf(fact_412_less__add__Suc2,axiom,
! [I: nat,M: nat] : ( ord_less @ nat @ I @ ( suc @ ( plus_plus @ nat @ M @ I ) ) ) ).
% less_add_Suc2
thf(fact_413_less__add__Suc1,axiom,
! [I: nat,M: nat] : ( ord_less @ nat @ I @ ( suc @ ( plus_plus @ nat @ I @ M ) ) ) ).
% less_add_Suc1
thf(fact_414_less__natE,axiom,
! [M: nat,N: nat] :
( ( ord_less @ nat @ M @ N )
=> ~ ! [Q4: nat] :
( N
!= ( suc @ ( plus_plus @ nat @ M @ Q4 ) ) ) ) ).
% less_natE
thf(fact_415_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_416_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_417_mono__nat__linear__lb,axiom,
! [F2: nat > nat,M: nat,K: nat] :
( ! [M5: nat,N3: nat] :
( ( ord_less @ nat @ M5 @ N3 )
=> ( ord_less @ nat @ ( F2 @ M5 ) @ ( F2 @ N3 ) ) )
=> ( ord_less_eq @ nat @ ( plus_plus @ nat @ ( F2 @ M ) @ K ) @ ( F2 @ ( plus_plus @ nat @ M @ K ) ) ) ) ).
% mono_nat_linear_lb
thf(fact_418_numeral__One,axiom,
! [A: $tType] :
( ( numeral @ A )
=> ( ( numeral_numeral @ A @ one2 )
= ( one_one @ A ) ) ) ).
% numeral_One
thf(fact_419_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_420_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_421_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_422_one__le__power,axiom,
! [A: $tType] :
( ( linordered_semidom @ A )
=> ! [A3: A,N: nat] :
( ( ord_less_eq @ A @ ( one_one @ A ) @ A3 )
=> ( ord_less_eq @ A @ ( one_one @ A ) @ ( power_power @ A @ A3 @ N ) ) ) ) ).
% one_le_power
thf(fact_423_mlex__bound,axiom,
! [A3: nat,A2: nat,B3: nat,N7: nat] :
( ( ord_less @ nat @ A3 @ A2 )
=> ( ( ord_less @ nat @ B3 @ N7 )
=> ( ord_less @ nat @ ( plus_plus @ nat @ ( times_times @ nat @ A3 @ N7 ) @ B3 ) @ ( times_times @ nat @ A2 @ N7 ) ) ) ) ).
% mlex_bound
thf(fact_424_mlex__fst__decrI,axiom,
! [A3: nat,A5: nat,B3: nat,N7: nat,B5: nat] :
( ( ord_less @ nat @ A3 @ A5 )
=> ( ( ord_less @ nat @ B3 @ N7 )
=> ( ( ord_less @ nat @ B5 @ N7 )
=> ( ord_less @ nat @ ( plus_plus @ nat @ ( times_times @ nat @ A3 @ N7 ) @ B3 ) @ ( plus_plus @ nat @ ( times_times @ nat @ A5 @ N7 ) @ B5 ) ) ) ) ) ).
% mlex_fst_decrI
thf(fact_425_mlex__snd__decrI,axiom,
! [A3: nat,A5: nat,B3: nat,B5: nat,N7: nat] :
( ( A3 = A5 )
=> ( ( ord_less @ nat @ B3 @ B5 )
=> ( ord_less @ nat @ ( plus_plus @ nat @ ( times_times @ nat @ A3 @ N7 ) @ B3 ) @ ( plus_plus @ nat @ ( times_times @ nat @ A5 @ N7 ) @ B5 ) ) ) ) ).
% mlex_snd_decrI
thf(fact_426_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_427_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_428_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_429_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_430_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_431_left__right__inverse__power,axiom,
! [A: $tType] :
( ( monoid_mult @ A )
=> ! [X2: A,Y3: A,N: nat] :
( ( ( times_times @ A @ X2 @ Y3 )
= ( one_one @ A ) )
=> ( ( times_times @ A @ ( power_power @ A @ X2 @ N ) @ ( power_power @ A @ Y3 @ N ) )
= ( one_one @ A ) ) ) ) ).
% left_right_inverse_power
thf(fact_432_mlex__leI,axiom,
! [A3: nat,A5: nat,B3: nat,B5: nat,N7: nat] :
( ( ord_less_eq @ nat @ A3 @ A5 )
=> ( ( ord_less_eq @ nat @ B3 @ B5 )
=> ( ord_less_eq @ nat @ ( plus_plus @ nat @ ( times_times @ nat @ A3 @ N7 ) @ B3 ) @ ( plus_plus @ nat @ ( times_times @ nat @ A5 @ N7 ) @ B5 ) ) ) ) ).
% mlex_leI
thf(fact_433_power__one__over,axiom,
! [A: $tType] :
( ( division_ring @ A )
=> ! [A3: A,N: nat] :
( ( power_power @ A @ ( divide_divide @ A @ ( one_one @ A ) @ A3 ) @ N )
= ( divide_divide @ A @ ( one_one @ A ) @ ( power_power @ A @ A3 @ N ) ) ) ) ).
% power_one_over
thf(fact_434_numerals_I1_J,axiom,
( ( numeral_numeral @ nat @ one2 )
= ( one_one @ nat ) ) ).
% numerals(1)
thf(fact_435_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_436_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_437_option_Oexhaust__sel,axiom,
! [A: $tType,Option: option @ A] :
( ( Option
!= ( none @ A ) )
=> ( Option
= ( some @ A @ ( the2 @ A @ Option ) ) ) ) ).
% option.exhaust_sel
thf(fact_438_numeral__code_I2_J,axiom,
! [A: $tType] :
( ( numeral @ A )
=> ! [N: num] :
( ( numeral_numeral @ A @ ( bit0 @ N ) )
= ( plus_plus @ A @ ( numeral_numeral @ A @ N ) @ ( numeral_numeral @ A @ N ) ) ) ) ).
% numeral_code(2)
thf(fact_439_power__2__mult__step__le,axiom,
! [N2: nat,N: nat,K4: nat,K: nat] :
( ( ord_less_eq @ nat @ N2 @ N )
=> ( ( ord_less @ nat @ ( times_times @ nat @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N2 ) @ K4 ) @ ( 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 ) ) @ N2 ) @ ( plus_plus @ nat @ K4 @ ( one_one @ nat ) ) ) @ ( times_times @ nat @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N ) @ K ) ) ) ) ).
% power_2_mult_step_le
thf(fact_440_ex__power__ivl2,axiom,
! [B3: nat,K: nat] :
( ( ord_less_eq @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ B3 )
=> ( ( ord_less_eq @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ K )
=> ? [N3: nat] :
( ( ord_less @ nat @ ( power_power @ nat @ B3 @ N3 ) @ K )
& ( ord_less_eq @ nat @ K @ ( power_power @ nat @ B3 @ ( plus_plus @ nat @ N3 @ ( one_one @ nat ) ) ) ) ) ) ) ).
% ex_power_ivl2
thf(fact_441_ex__power__ivl1,axiom,
! [B3: nat,K: nat] :
( ( ord_less_eq @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ B3 )
=> ( ( ord_less_eq @ nat @ ( one_one @ nat ) @ K )
=> ? [N3: nat] :
( ( ord_less_eq @ nat @ ( power_power @ nat @ B3 @ N3 ) @ K )
& ( ord_less @ nat @ K @ ( power_power @ nat @ B3 @ ( plus_plus @ nat @ N3 @ ( one_one @ nat ) ) ) ) ) ) ) ).
% ex_power_ivl1
thf(fact_442_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_443_power__less__power__Suc,axiom,
! [A: $tType] :
( ( linordered_semidom @ A )
=> ! [A3: A,N: nat] :
( ( ord_less @ A @ ( one_one @ A ) @ A3 )
=> ( ord_less @ A @ ( power_power @ A @ A3 @ N ) @ ( times_times @ A @ A3 @ ( power_power @ A @ A3 @ N ) ) ) ) ) ).
% power_less_power_Suc
thf(fact_444_power__gt1__lemma,axiom,
! [A: $tType] :
( ( linordered_semidom @ A )
=> ! [A3: A,N: nat] :
( ( ord_less @ A @ ( one_one @ A ) @ A3 )
=> ( ord_less @ A @ ( one_one @ A ) @ ( times_times @ A @ A3 @ ( power_power @ A @ A3 @ N ) ) ) ) ) ).
% power_gt1_lemma
thf(fact_445_power__gt1,axiom,
! [A: $tType] :
( ( linordered_semidom @ A )
=> ! [A3: A,N: nat] :
( ( ord_less @ A @ ( one_one @ A ) @ A3 )
=> ( ord_less @ A @ ( one_one @ A ) @ ( power_power @ A @ A3 @ ( suc @ N ) ) ) ) ) ).
% power_gt1
thf(fact_446_power__strict__increasing,axiom,
! [A: $tType] :
( ( linordered_semidom @ A )
=> ! [N: nat,N7: nat,A3: A] :
( ( ord_less @ nat @ N @ N7 )
=> ( ( ord_less @ A @ ( one_one @ A ) @ A3 )
=> ( ord_less @ A @ ( power_power @ A @ A3 @ N ) @ ( power_power @ A @ A3 @ N7 ) ) ) ) ) ).
% power_strict_increasing
thf(fact_447_power__less__imp__less__exp,axiom,
! [A: $tType] :
( ( linordered_semidom @ A )
=> ! [A3: A,M: nat,N: nat] :
( ( ord_less @ A @ ( one_one @ A ) @ A3 )
=> ( ( ord_less @ A @ ( power_power @ A @ A3 @ M ) @ ( power_power @ A @ A3 @ N ) )
=> ( ord_less @ nat @ M @ N ) ) ) ) ).
% power_less_imp_less_exp
thf(fact_448_power__increasing,axiom,
! [A: $tType] :
( ( linordered_semidom @ A )
=> ! [N: nat,N7: nat,A3: A] :
( ( ord_less_eq @ nat @ N @ N7 )
=> ( ( ord_less_eq @ A @ ( one_one @ A ) @ A3 )
=> ( ord_less_eq @ A @ ( power_power @ A @ A3 @ N ) @ ( power_power @ A @ A3 @ N7 ) ) ) ) ) ).
% power_increasing
thf(fact_449_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_450_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_451_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_452_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_453_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_454_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_455_left__add__twice,axiom,
! [A: $tType] :
( ( semiring_numeral @ A )
=> ! [A3: A,B3: A] :
( ( plus_plus @ A @ A3 @ ( plus_plus @ A @ A3 @ B3 ) )
= ( plus_plus @ A @ ( times_times @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ A3 ) @ B3 ) ) ) ).
% left_add_twice
thf(fact_456_mult__2__right,axiom,
! [A: $tType] :
( ( semiring_numeral @ A )
=> ! [Z: A] :
( ( times_times @ A @ Z @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) )
= ( plus_plus @ A @ Z @ Z ) ) ) ).
% mult_2_right
thf(fact_457_mult__2,axiom,
! [A: $tType] :
( ( semiring_numeral @ A )
=> ! [Z: A] :
( ( times_times @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ Z )
= ( plus_plus @ A @ Z @ Z ) ) ) ).
% mult_2
thf(fact_458_one__power2,axiom,
! [A: $tType] :
( ( semiring_1 @ A )
=> ( ( power_power @ A @ ( one_one @ A ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) )
= ( one_one @ A ) ) ) ).
% one_power2
thf(fact_459_power__le__imp__le__exp,axiom,
! [A: $tType] :
( ( linordered_semidom @ A )
=> ! [A3: A,M: nat,N: nat] :
( ( ord_less @ A @ ( one_one @ A ) @ A3 )
=> ( ( ord_less_eq @ A @ ( power_power @ A @ A3 @ M ) @ ( power_power @ A @ A3 @ N ) )
=> ( ord_less_eq @ nat @ M @ N ) ) ) ) ).
% power_le_imp_le_exp
thf(fact_460_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_461_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_462_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_463_nat__add__offset__less,axiom,
! [Y3: nat,N: nat,X2: nat,M: nat,Sz: nat] :
( ( ord_less @ nat @ Y3 @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N ) )
=> ( ( ord_less @ nat @ X2 @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ M ) )
=> ( ( Sz
= ( plus_plus @ nat @ M @ N ) )
=> ( ord_less @ nat @ ( plus_plus @ nat @ ( times_times @ nat @ X2 @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N ) ) @ Y3 ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ Sz ) ) ) ) ) ).
% nat_add_offset_less
thf(fact_464_power2__sum,axiom,
! [A: $tType] :
( ( comm_semiring_1 @ A )
=> ! [X2: A,Y3: A] :
( ( power_power @ A @ ( plus_plus @ A @ X2 @ Y3 ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) )
= ( plus_plus @ A @ ( plus_plus @ A @ ( power_power @ A @ X2 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ ( power_power @ A @ Y3 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( times_times @ A @ ( times_times @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ X2 ) @ Y3 ) ) ) ) ).
% power2_sum
thf(fact_465_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_466_power2__diff,axiom,
! [A: $tType] :
( ( comm_ring_1 @ A )
=> ! [X2: A,Y3: A] :
( ( power_power @ A @ ( minus_minus @ A @ X2 @ Y3 ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) )
= ( minus_minus @ A @ ( plus_plus @ A @ ( power_power @ A @ X2 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ ( power_power @ A @ Y3 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( times_times @ A @ ( times_times @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ X2 ) @ Y3 ) ) ) ) ).
% power2_diff
thf(fact_467_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_468_tdeletemimi_H,axiom,
! [Deg: nat,Mi: nat,TreeList: list @ vEBT_VEBT,Summary: vEBT_VEBT,X2: 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 ) @ X2 ) @ ( one_one @ nat ) ) ) ).
% tdeletemimi'
thf(fact_469_invar__vebt_Ointros_I5_J,axiom,
! [TreeList: list @ vEBT_VEBT,N: nat,Summary: vEBT_VEBT,M: nat,Deg: nat,Mi: nat,Ma: nat] :
( ! [X4: vEBT_VEBT] :
( ( member @ vEBT_VEBT @ X4 @ ( set2 @ vEBT_VEBT @ TreeList ) )
=> ( vEBT_invar_vebt @ X4 @ 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 ) )
=> ( ( ? [X7: nat] : ( vEBT_V8194947554948674370ptions @ ( nth @ vEBT_VEBT @ TreeList @ I2 ) @ X7 ) )
= ( vEBT_V8194947554948674370ptions @ Summary @ I2 ) ) )
=> ( ( ( Mi = Ma )
=> ! [X4: vEBT_VEBT] :
( ( member @ vEBT_VEBT @ X4 @ ( set2 @ vEBT_VEBT @ TreeList ) )
=> ~ ? [X_1: nat] : ( vEBT_V8194947554948674370ptions @ X4 @ X_1 ) ) )
=> ( ( ord_less_eq @ nat @ Mi @ Ma )
=> ( ( ord_less @ nat @ Ma @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ Deg ) )
=> ( ( ( Mi != Ma )
=> ! [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 ) ) )
& ! [X4: nat] :
( ( ( ( vEBT_VEBT_high @ X4 @ N )
= I2 )
& ( vEBT_V8194947554948674370ptions @ ( nth @ vEBT_VEBT @ TreeList @ I2 ) @ ( vEBT_VEBT_low @ X4 @ N ) ) )
=> ( ( ord_less @ nat @ Mi @ X4 )
& ( ord_less_eq @ nat @ X4 @ Ma ) ) ) ) ) )
=> ( vEBT_invar_vebt @ ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi @ Ma ) ) @ Deg @ TreeList @ Summary ) @ Deg ) ) ) ) ) ) ) ) ) ) ) ).
% invar_vebt.intros(5)
thf(fact_470_invar__vebt_Ointros_I4_J,axiom,
! [TreeList: list @ vEBT_VEBT,N: nat,Summary: vEBT_VEBT,M: nat,Deg: nat,Mi: nat,Ma: nat] :
( ! [X4: vEBT_VEBT] :
( ( member @ vEBT_VEBT @ X4 @ ( set2 @ vEBT_VEBT @ TreeList ) )
=> ( vEBT_invar_vebt @ X4 @ 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 ) )
=> ( ( ? [X7: nat] : ( vEBT_V8194947554948674370ptions @ ( nth @ vEBT_VEBT @ TreeList @ I2 ) @ X7 ) )
= ( vEBT_V8194947554948674370ptions @ Summary @ I2 ) ) )
=> ( ( ( Mi = Ma )
=> ! [X4: vEBT_VEBT] :
( ( member @ vEBT_VEBT @ X4 @ ( set2 @ vEBT_VEBT @ TreeList ) )
=> ~ ? [X_1: nat] : ( vEBT_V8194947554948674370ptions @ X4 @ X_1 ) ) )
=> ( ( ord_less_eq @ nat @ Mi @ Ma )
=> ( ( ord_less @ nat @ Ma @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ Deg ) )
=> ( ( ( Mi != Ma )
=> ! [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 ) ) )
& ! [X4: nat] :
( ( ( ( vEBT_VEBT_high @ X4 @ N )
= I2 )
& ( vEBT_V8194947554948674370ptions @ ( nth @ vEBT_VEBT @ TreeList @ I2 ) @ ( vEBT_VEBT_low @ X4 @ N ) ) )
=> ( ( ord_less @ nat @ Mi @ X4 )
& ( ord_less_eq @ nat @ X4 @ Ma ) ) ) ) ) )
=> ( vEBT_invar_vebt @ ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi @ Ma ) ) @ Deg @ TreeList @ Summary ) @ Deg ) ) ) ) ) ) ) ) ) ) ) ).
% invar_vebt.intros(4)
thf(fact_471_minNull__delete__time__bound_H,axiom,
! [T2: vEBT_VEBT,N: nat,X2: nat] :
( ( vEBT_invar_vebt @ T2 @ N )
=> ( ( vEBT_VEBT_minNull @ ( vEBT_vebt_delete @ T2 @ X2 ) )
=> ( ord_less_eq @ nat @ ( vEBT_V1232361888498592333_e_t_e @ T2 @ X2 ) @ ( one_one @ nat ) ) ) ) ).
% minNull_delete_time_bound'
thf(fact_472_delete__bound__height_H,axiom,
! [T2: vEBT_VEBT,N: nat,X2: nat] :
( ( vEBT_invar_vebt @ T2 @ N )
=> ( ord_less_eq @ nat @ ( vEBT_V1232361888498592333_e_t_e @ T2 @ X2 ) @ ( plus_plus @ nat @ ( one_one @ nat ) @ ( vEBT_VEBT_height @ T2 ) ) ) ) ).
% delete_bound_height'
thf(fact_473_del__x__mia,axiom,
! [X2: nat,Mi: nat,Ma: nat,Deg: nat,TreeList: list @ vEBT_VEBT,Summary: vEBT_VEBT] :
( ( ( X2 = Mi )
& ( ord_less @ nat @ X2 @ 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 ) @ X2 )
= ( 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_474_del__x__mi__lets__in__minNull,axiom,
! [X2: 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] :
( ( ( X2 = Mi )
& ( ord_less @ nat @ X2 @ 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 ) @ X2 )
= ( 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_475_del__x__mi__lets__in,axiom,
! [X2: 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] :
( ( ( X2 = Mi )
& ( ord_less @ nat @ X2 @ 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 ) @ X2 )
= ( 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 ) @ X2 )
= ( 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_476_del__x__mi,axiom,
! [X2: nat,Mi: nat,Ma: nat,Deg: nat,Xn: nat,H2: nat,Summary: vEBT_VEBT,TreeList: list @ vEBT_VEBT,L: nat] :
( ( ( X2 = Mi )
& ( ord_less @ nat @ X2 @ 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 ) @ X2 )
= ( 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_477_del__in__range,axiom,
! [Mi: nat,X2: nat,Ma: nat,Deg: nat,TreeList: list @ vEBT_VEBT,Summary: vEBT_VEBT] :
( ( ( ord_less_eq @ nat @ Mi @ X2 )
& ( ord_less_eq @ nat @ X2 @ 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 ) @ X2 )
= ( if @ vEBT_VEBT @ ( ord_less @ nat @ ( vEBT_VEBT_high @ ( if @ nat @ ( X2 = 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 ) ) ) ) ) ) @ X2 ) @ ( 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 @ ( X2 = 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 ) ) ) ) ) ) @ X2 ) @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( vEBT_VEBT_low @ ( if @ nat @ ( X2 = 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 ) ) ) ) ) ) @ X2 ) @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) )
@ ( vEBT_Node
@ ( some @ ( product_prod @ nat @ nat )
@ ( product_Pair @ nat @ nat @ ( if @ nat @ ( X2 = Mi ) @ ( if @ nat @ ( X2 = 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 ) ) ) ) ) ) @ X2 ) @ Mi )
@ ( if @ nat
@ ( ( ( X2 = 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 ) )
& ( ( X2 != Mi )
=> ( X2 = Ma ) ) )
@ ( if @ nat
@ ( ( vEBT_vebt_maxt @ ( vEBT_vebt_delete @ Summary @ ( vEBT_VEBT_high @ ( if @ nat @ ( X2 = 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 ) ) ) ) ) ) @ X2 ) @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) )
= ( none @ nat ) )
@ ( if @ nat @ ( X2 = Mi ) @ ( if @ nat @ ( X2 = 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 ) ) ) ) ) ) @ X2 ) @ 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 @ ( X2 = 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 ) ) ) ) ) ) @ X2 ) @ ( 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 @ ( X2 = 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 ) ) ) ) ) ) @ X2 ) @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( vEBT_vebt_delete @ ( nth @ vEBT_VEBT @ TreeList @ ( vEBT_VEBT_high @ ( if @ nat @ ( X2 = 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 ) ) ) ) ) ) @ X2 ) @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( vEBT_VEBT_low @ ( if @ nat @ ( X2 = 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 ) ) ) ) ) ) @ X2 ) @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_maxt @ ( vEBT_vebt_delete @ Summary @ ( vEBT_VEBT_high @ ( if @ nat @ ( X2 = 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 ) ) ) ) ) ) @ X2 ) @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) ) ) ) ) ) )
@ Ma ) ) )
@ Deg
@ ( list_update @ vEBT_VEBT @ TreeList @ ( vEBT_VEBT_high @ ( if @ nat @ ( X2 = 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 ) ) ) ) ) ) @ X2 ) @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( vEBT_vebt_delete @ ( nth @ vEBT_VEBT @ TreeList @ ( vEBT_VEBT_high @ ( if @ nat @ ( X2 = 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 ) ) ) ) ) ) @ X2 ) @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( vEBT_VEBT_low @ ( if @ nat @ ( X2 = 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 ) ) ) ) ) ) @ X2 ) @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) )
@ ( vEBT_vebt_delete @ Summary @ ( vEBT_VEBT_high @ ( if @ nat @ ( X2 = 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 ) ) ) ) ) ) @ X2 ) @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) )
@ ( vEBT_Node
@ ( some @ ( product_prod @ nat @ nat )
@ ( product_Pair @ nat @ nat @ ( if @ nat @ ( X2 = Mi ) @ ( if @ nat @ ( X2 = 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 ) ) ) ) ) ) @ X2 ) @ Mi )
@ ( if @ nat
@ ( ( ( X2 = 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 ) )
& ( ( X2 != Mi )
=> ( X2 = Ma ) ) )
@ ( plus_plus @ nat @ ( times_times @ nat @ ( vEBT_VEBT_high @ ( if @ nat @ ( X2 = 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 ) ) ) ) ) ) @ X2 ) @ ( 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 @ ( X2 = 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 ) ) ) ) ) ) @ X2 ) @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( vEBT_vebt_delete @ ( nth @ vEBT_VEBT @ TreeList @ ( vEBT_VEBT_high @ ( if @ nat @ ( X2 = 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 ) ) ) ) ) ) @ X2 ) @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( vEBT_VEBT_low @ ( if @ nat @ ( X2 = 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 ) ) ) ) ) ) @ X2 ) @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) @ ( vEBT_VEBT_high @ ( if @ nat @ ( X2 = 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 ) ) ) ) ) ) @ X2 ) @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) ) )
@ Ma ) ) )
@ Deg
@ ( list_update @ vEBT_VEBT @ TreeList @ ( vEBT_VEBT_high @ ( if @ nat @ ( X2 = 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 ) ) ) ) ) ) @ X2 ) @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( vEBT_vebt_delete @ ( nth @ vEBT_VEBT @ TreeList @ ( vEBT_VEBT_high @ ( if @ nat @ ( X2 = 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 ) ) ) ) ) ) @ X2 ) @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( vEBT_VEBT_low @ ( if @ nat @ ( X2 = 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 ) ) ) ) ) ) @ X2 ) @ ( 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_478_del__x__not__mia,axiom,
! [Mi: nat,X2: nat,Ma: nat,Deg: nat,H2: nat,L: nat,TreeList: list @ vEBT_VEBT,Summary: vEBT_VEBT] :
( ( ( ord_less @ nat @ Mi @ X2 )
& ( ord_less_eq @ nat @ X2 @ Ma ) )
=> ( ( Mi != Ma )
=> ( ( ord_less_eq @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ Deg )
=> ( ( ( vEBT_VEBT_high @ X2 @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) )
= H2 )
=> ( ( ( vEBT_VEBT_low @ X2 @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) )
= L )
=> ( ( ord_less @ nat @ ( vEBT_VEBT_high @ X2 @ ( 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 ) @ X2 )
= ( 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 @ ( X2 = 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 @ ( X2 = 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_479_maxbmo,axiom,
! [T2: vEBT_VEBT,X2: nat] :
( ( ( vEBT_vebt_maxt @ T2 )
= ( some @ nat @ X2 ) )
=> ( vEBT_V8194947554948674370ptions @ T2 @ X2 ) ) ).
% maxbmo
thf(fact_480_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_481_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_482_maxt__corr__help,axiom,
! [T2: vEBT_VEBT,N: nat,Maxi: nat,X2: nat] :
( ( vEBT_invar_vebt @ T2 @ N )
=> ( ( ( vEBT_vebt_maxt @ T2 )
= ( some @ nat @ Maxi ) )
=> ( ( vEBT_vebt_member @ T2 @ X2 )
=> ( ord_less_eq @ nat @ X2 @ Maxi ) ) ) ) ).
% maxt_corr_help
thf(fact_483_maxt__corr,axiom,
! [T2: vEBT_VEBT,N: nat,X2: nat] :
( ( vEBT_invar_vebt @ T2 @ N )
=> ( ( ( vEBT_vebt_maxt @ T2 )
= ( some @ nat @ X2 ) )
=> ( vEBT_VEBT_max_in_set @ ( vEBT_VEBT_set_vebt @ T2 ) @ X2 ) ) ) ).
% maxt_corr
thf(fact_484_maxt__sound,axiom,
! [T2: vEBT_VEBT,N: nat,X2: nat] :
( ( vEBT_invar_vebt @ T2 @ N )
=> ( ( vEBT_VEBT_max_in_set @ ( vEBT_VEBT_set_vebt @ T2 ) @ X2 )
=> ( ( vEBT_vebt_maxt @ T2 )
= ( some @ nat @ X2 ) ) ) ) ).
% maxt_sound
thf(fact_485_semiring__norm_I6_J,axiom,
! [M: num,N: num] :
( ( plus_plus @ num @ ( bit0 @ M ) @ ( bit0 @ N ) )
= ( bit0 @ ( plus_plus @ num @ M @ N ) ) ) ).
% semiring_norm(6)
thf(fact_486_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_487_semiring__norm_I2_J,axiom,
( ( plus_plus @ num @ one2 @ one2 )
= ( bit0 @ one2 ) ) ).
% semiring_norm(2)
thf(fact_488_Suc__numeral,axiom,
! [N: num] :
( ( suc @ ( numeral_numeral @ nat @ N ) )
= ( numeral_numeral @ nat @ ( plus_plus @ num @ N @ one2 ) ) ) ).
% Suc_numeral
thf(fact_489_power__add__numeral,axiom,
! [A: $tType] :
( ( monoid_mult @ A )
=> ! [A3: A,M: num,N: num] :
( ( times_times @ A @ ( power_power @ A @ A3 @ ( numeral_numeral @ nat @ M ) ) @ ( power_power @ A @ A3 @ ( numeral_numeral @ nat @ N ) ) )
= ( power_power @ A @ A3 @ ( numeral_numeral @ nat @ ( plus_plus @ num @ M @ N ) ) ) ) ) ).
% power_add_numeral
thf(fact_490_power__add__numeral2,axiom,
! [A: $tType] :
( ( monoid_mult @ A )
=> ! [A3: A,M: num,N: num,B3: A] :
( ( times_times @ A @ ( power_power @ A @ A3 @ ( numeral_numeral @ nat @ M ) ) @ ( times_times @ A @ ( power_power @ A @ A3 @ ( numeral_numeral @ nat @ N ) ) @ B3 ) )
= ( times_times @ A @ ( power_power @ A @ A3 @ ( numeral_numeral @ nat @ ( plus_plus @ num @ M @ N ) ) ) @ B3 ) ) ) ).
% power_add_numeral2
thf(fact_491_del__x__not__mi__newnode__not__nil,axiom,
! [Mi: nat,X2: 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 @ X2 )
& ( ord_less_eq @ nat @ X2 @ Ma ) )
=> ( ( Mi != Ma )
=> ( ( ord_less_eq @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ Deg )
=> ( ( ( vEBT_VEBT_high @ X2 @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) )
= H2 )
=> ( ( ( vEBT_VEBT_low @ X2 @ ( 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 @ X2 @ ( 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 ) @ X2 )
= ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi @ ( if @ nat @ ( X2 = 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_492_del__x__mi__lets__in__not__minNull,axiom,
! [X2: 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] :
( ( ( X2 = Mi )
& ( ord_less @ nat @ X2 @ 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 ) @ X2 )
= ( 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_493_del__x__not__mi,axiom,
! [Mi: nat,X2: 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 @ X2 )
& ( ord_less_eq @ nat @ X2 @ Ma ) )
=> ( ( Mi != Ma )
=> ( ( ord_less_eq @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ Deg )
=> ( ( ( vEBT_VEBT_high @ X2 @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) )
= H2 )
=> ( ( ( vEBT_VEBT_low @ X2 @ ( 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 @ X2 @ ( 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 ) @ X2 )
= ( vEBT_Node
@ ( some @ ( product_prod @ nat @ nat )
@ ( product_Pair @ nat @ nat @ Mi
@ ( if @ nat @ ( X2 = 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 ) @ X2 )
= ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi @ ( if @ nat @ ( X2 = 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_494_del__x__not__mi__new__node__nil,axiom,
! [Mi: nat,X2: 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 @ X2 )
& ( ord_less_eq @ nat @ X2 @ Ma ) )
=> ( ( Mi != Ma )
=> ( ( ord_less_eq @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ Deg )
=> ( ( ( vEBT_VEBT_high @ X2 @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) )
= H2 )
=> ( ( ( vEBT_VEBT_low @ X2 @ ( 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 @ X2 @ ( 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 ) @ X2 )
= ( vEBT_Node
@ ( some @ ( product_prod @ nat @ nat )
@ ( product_Pair @ nat @ nat @ Mi
@ ( if @ nat @ ( X2 = 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_495_z1pdiv2,axiom,
! [B3: int] :
( ( divide_divide @ int @ ( plus_plus @ int @ ( times_times @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ B3 ) @ ( one_one @ int ) ) @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) )
= B3 ) ).
% z1pdiv2
thf(fact_496_add__One__commute,axiom,
! [N: num] :
( ( plus_plus @ num @ one2 @ N )
= ( plus_plus @ num @ N @ one2 ) ) ).
% add_One_commute
thf(fact_497_add__diff__assoc__enat,axiom,
! [Z: extended_enat,Y3: extended_enat,X2: extended_enat] :
( ( ord_less_eq @ extended_enat @ Z @ Y3 )
=> ( ( plus_plus @ extended_enat @ X2 @ ( minus_minus @ extended_enat @ Y3 @ Z ) )
= ( minus_minus @ extended_enat @ ( plus_plus @ extended_enat @ X2 @ Y3 ) @ Z ) ) ) ).
% add_diff_assoc_enat
thf(fact_498_Suc__nat__number__of__add,axiom,
! [V: num,N: nat] :
( ( suc @ ( plus_plus @ nat @ ( numeral_numeral @ nat @ V ) @ N ) )
= ( plus_plus @ nat @ ( numeral_numeral @ nat @ ( plus_plus @ num @ V @ one2 ) ) @ N ) ) ).
% Suc_nat_number_of_add
thf(fact_499_set__vebt__def,axiom,
( vEBT_set_vebt
= ( ^ [T3: vEBT_VEBT] : ( collect @ nat @ ( vEBT_V8194947554948674370ptions @ T3 ) ) ) ) ).
% set_vebt_def
thf(fact_500_invar__vebt_Ointros_I2_J,axiom,
! [TreeList: list @ vEBT_VEBT,N: nat,Summary: vEBT_VEBT,M: nat,Deg: nat] :
( ! [X4: vEBT_VEBT] :
( ( member @ vEBT_VEBT @ X4 @ ( set2 @ vEBT_VEBT @ TreeList ) )
=> ( vEBT_invar_vebt @ X4 @ 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 )
=> ( ! [X4: vEBT_VEBT] :
( ( member @ vEBT_VEBT @ X4 @ ( set2 @ vEBT_VEBT @ TreeList ) )
=> ~ ? [X_1: nat] : ( vEBT_V8194947554948674370ptions @ X4 @ X_1 ) )
=> ( vEBT_invar_vebt @ ( vEBT_Node @ ( none @ ( product_prod @ nat @ nat ) ) @ Deg @ TreeList @ Summary ) @ Deg ) ) ) ) ) ) ) ) ).
% invar_vebt.intros(2)
thf(fact_501_invar__vebt_Ointros_I3_J,axiom,
! [TreeList: list @ vEBT_VEBT,N: nat,Summary: vEBT_VEBT,M: nat,Deg: nat] :
( ! [X4: vEBT_VEBT] :
( ( member @ vEBT_VEBT @ X4 @ ( set2 @ vEBT_VEBT @ TreeList ) )
=> ( vEBT_invar_vebt @ X4 @ 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 )
=> ( ! [X4: vEBT_VEBT] :
( ( member @ vEBT_VEBT @ X4 @ ( set2 @ vEBT_VEBT @ TreeList ) )
=> ~ ? [X_1: nat] : ( vEBT_V8194947554948674370ptions @ X4 @ X_1 ) )
=> ( vEBT_invar_vebt @ ( vEBT_Node @ ( none @ ( product_prod @ nat @ nat ) ) @ Deg @ TreeList @ Summary ) @ Deg ) ) ) ) ) ) ) ) ).
% invar_vebt.intros(3)
thf(fact_502_vebt__delete_Osimps_I7_J,axiom,
! [X2: nat,Mi: nat,Ma: nat,Va: nat,TreeList: list @ vEBT_VEBT,Summary: vEBT_VEBT] :
( ( ( ( ord_less @ nat @ X2 @ Mi )
| ( ord_less @ nat @ Ma @ X2 ) )
=> ( ( vEBT_vebt_delete @ ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi @ Ma ) ) @ ( suc @ ( suc @ Va ) ) @ TreeList @ Summary ) @ X2 )
= ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi @ Ma ) ) @ ( suc @ ( suc @ Va ) ) @ TreeList @ Summary ) ) )
& ( ~ ( ( ord_less @ nat @ X2 @ Mi )
| ( ord_less @ nat @ Ma @ X2 ) )
=> ( ( ( ( X2 = Mi )
& ( X2 = Ma ) )
=> ( ( vEBT_vebt_delete @ ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi @ Ma ) ) @ ( suc @ ( suc @ Va ) ) @ TreeList @ Summary ) @ X2 )
= ( vEBT_Node @ ( none @ ( product_prod @ nat @ nat ) ) @ ( suc @ ( suc @ Va ) ) @ TreeList @ Summary ) ) )
& ( ~ ( ( X2 = Mi )
& ( X2 = Ma ) )
=> ( ( vEBT_vebt_delete @ ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi @ Ma ) ) @ ( suc @ ( suc @ Va ) ) @ TreeList @ Summary ) @ X2 )
= ( if @ vEBT_VEBT @ ( ord_less @ nat @ ( vEBT_VEBT_high @ ( if @ nat @ ( X2 = Mi ) @ ( plus_plus @ nat @ ( times_times @ nat @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ X2 ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( size_size @ ( list @ vEBT_VEBT ) @ TreeList ) )
@ ( if @ vEBT_VEBT @ ( vEBT_VEBT_minNull @ ( vEBT_vebt_delete @ ( nth @ vEBT_VEBT @ TreeList @ ( vEBT_VEBT_high @ ( if @ nat @ ( X2 = Mi ) @ ( plus_plus @ nat @ ( times_times @ nat @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ X2 ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( vEBT_VEBT_low @ ( if @ nat @ ( X2 = Mi ) @ ( plus_plus @ nat @ ( times_times @ nat @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ X2 ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) )
@ ( vEBT_Node
@ ( some @ ( product_prod @ nat @ nat )
@ ( product_Pair @ nat @ nat @ ( if @ nat @ ( X2 = Mi ) @ ( if @ nat @ ( X2 = Mi ) @ ( plus_plus @ nat @ ( times_times @ nat @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ X2 ) @ Mi )
@ ( if @ nat
@ ( ( ( X2 = Mi )
=> ( ( plus_plus @ nat @ ( times_times @ nat @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) )
= Ma ) )
& ( ( X2 != Mi )
=> ( X2 = Ma ) ) )
@ ( if @ nat
@ ( ( vEBT_vebt_maxt @ ( vEBT_vebt_delete @ Summary @ ( vEBT_VEBT_high @ ( if @ nat @ ( X2 = Mi ) @ ( plus_plus @ nat @ ( times_times @ nat @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ X2 ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) )
= ( none @ nat ) )
@ ( if @ nat @ ( X2 = Mi ) @ ( if @ nat @ ( X2 = Mi ) @ ( plus_plus @ nat @ ( times_times @ nat @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ X2 ) @ Mi )
@ ( plus_plus @ nat @ ( times_times @ nat @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_maxt @ ( vEBT_vebt_delete @ Summary @ ( vEBT_VEBT_high @ ( if @ nat @ ( X2 = Mi ) @ ( plus_plus @ nat @ ( times_times @ nat @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ X2 ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_maxt @ ( nth @ vEBT_VEBT @ ( list_update @ vEBT_VEBT @ TreeList @ ( vEBT_VEBT_high @ ( if @ nat @ ( X2 = Mi ) @ ( plus_plus @ nat @ ( times_times @ nat @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ X2 ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( vEBT_vebt_delete @ ( nth @ vEBT_VEBT @ TreeList @ ( vEBT_VEBT_high @ ( if @ nat @ ( X2 = Mi ) @ ( plus_plus @ nat @ ( times_times @ nat @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ X2 ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( vEBT_VEBT_low @ ( if @ nat @ ( X2 = Mi ) @ ( plus_plus @ nat @ ( times_times @ nat @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ X2 ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_maxt @ ( vEBT_vebt_delete @ Summary @ ( vEBT_VEBT_high @ ( if @ nat @ ( X2 = Mi ) @ ( plus_plus @ nat @ ( times_times @ nat @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ X2 ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) ) ) ) ) ) )
@ Ma ) ) )
@ ( suc @ ( suc @ Va ) )
@ ( list_update @ vEBT_VEBT @ TreeList @ ( vEBT_VEBT_high @ ( if @ nat @ ( X2 = Mi ) @ ( plus_plus @ nat @ ( times_times @ nat @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ X2 ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( vEBT_vebt_delete @ ( nth @ vEBT_VEBT @ TreeList @ ( vEBT_VEBT_high @ ( if @ nat @ ( X2 = Mi ) @ ( plus_plus @ nat @ ( times_times @ nat @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ X2 ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( vEBT_VEBT_low @ ( if @ nat @ ( X2 = Mi ) @ ( plus_plus @ nat @ ( times_times @ nat @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ X2 ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) )
@ ( vEBT_vebt_delete @ Summary @ ( vEBT_VEBT_high @ ( if @ nat @ ( X2 = Mi ) @ ( plus_plus @ nat @ ( times_times @ nat @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ X2 ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) )
@ ( vEBT_Node
@ ( some @ ( product_prod @ nat @ nat )
@ ( product_Pair @ nat @ nat @ ( if @ nat @ ( X2 = Mi ) @ ( if @ nat @ ( X2 = Mi ) @ ( plus_plus @ nat @ ( times_times @ nat @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ X2 ) @ Mi )
@ ( if @ nat
@ ( ( ( X2 = Mi )
=> ( ( plus_plus @ nat @ ( times_times @ nat @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) )
= Ma ) )
& ( ( X2 != Mi )
=> ( X2 = Ma ) ) )
@ ( plus_plus @ nat @ ( times_times @ nat @ ( vEBT_VEBT_high @ ( if @ nat @ ( X2 = Mi ) @ ( plus_plus @ nat @ ( times_times @ nat @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ X2 ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_maxt @ ( nth @ vEBT_VEBT @ ( list_update @ vEBT_VEBT @ TreeList @ ( vEBT_VEBT_high @ ( if @ nat @ ( X2 = Mi ) @ ( plus_plus @ nat @ ( times_times @ nat @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ X2 ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( vEBT_vebt_delete @ ( nth @ vEBT_VEBT @ TreeList @ ( vEBT_VEBT_high @ ( if @ nat @ ( X2 = Mi ) @ ( plus_plus @ nat @ ( times_times @ nat @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ X2 ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( vEBT_VEBT_low @ ( if @ nat @ ( X2 = Mi ) @ ( plus_plus @ nat @ ( times_times @ nat @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ X2 ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) @ ( vEBT_VEBT_high @ ( if @ nat @ ( X2 = Mi ) @ ( plus_plus @ nat @ ( times_times @ nat @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ X2 ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) ) )
@ Ma ) ) )
@ ( suc @ ( suc @ Va ) )
@ ( list_update @ vEBT_VEBT @ TreeList @ ( vEBT_VEBT_high @ ( if @ nat @ ( X2 = Mi ) @ ( plus_plus @ nat @ ( times_times @ nat @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ X2 ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( vEBT_vebt_delete @ ( nth @ vEBT_VEBT @ TreeList @ ( vEBT_VEBT_high @ ( if @ nat @ ( X2 = Mi ) @ ( plus_plus @ nat @ ( times_times @ nat @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ X2 ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( vEBT_VEBT_low @ ( if @ nat @ ( X2 = Mi ) @ ( plus_plus @ nat @ ( times_times @ nat @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ X2 ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) )
@ Summary ) )
@ ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi @ Ma ) ) @ ( suc @ ( suc @ Va ) ) @ TreeList @ Summary ) ) ) ) ) ) ) ).
% vebt_delete.simps(7)
thf(fact_503_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,
! [X2: nat,Mi: nat,Ma: nat,Va: nat,TreeList: list @ vEBT_VEBT,Summary: vEBT_VEBT] :
( ( ( ( ord_less @ nat @ X2 @ Mi )
| ( ord_less @ nat @ Ma @ X2 ) )
=> ( ( vEBT_V1232361888498592333_e_t_e @ ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi @ Ma ) ) @ ( suc @ ( suc @ Va ) ) @ TreeList @ Summary ) @ X2 )
= ( one_one @ nat ) ) )
& ( ~ ( ( ord_less @ nat @ X2 @ Mi )
| ( ord_less @ nat @ Ma @ X2 ) )
=> ( ( ( ( X2 = Mi )
& ( X2 = Ma ) )
=> ( ( vEBT_V1232361888498592333_e_t_e @ ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi @ Ma ) ) @ ( suc @ ( suc @ Va ) ) @ TreeList @ Summary ) @ X2 )
= ( one_one @ nat ) ) )
& ( ~ ( ( X2 = Mi )
& ( X2 = Ma ) )
=> ( ( vEBT_V1232361888498592333_e_t_e @ ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi @ Ma ) ) @ ( suc @ ( suc @ Va ) ) @ TreeList @ Summary ) @ X2 )
= ( if @ nat @ ( ord_less @ nat @ ( vEBT_VEBT_high @ ( if @ nat @ ( X2 = Mi ) @ ( plus_plus @ nat @ ( times_times @ nat @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ X2 ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( 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 @ ( X2 = Mi ) @ ( plus_plus @ nat @ ( times_times @ nat @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ X2 ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( vEBT_VEBT_low @ ( if @ nat @ ( X2 = Mi ) @ ( plus_plus @ nat @ ( times_times @ nat @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ X2 ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( if @ nat @ ( vEBT_VEBT_minNull @ ( vEBT_vebt_delete @ ( nth @ vEBT_VEBT @ TreeList @ ( vEBT_VEBT_high @ ( if @ nat @ ( X2 = Mi ) @ ( plus_plus @ nat @ ( times_times @ nat @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ X2 ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( vEBT_VEBT_low @ ( if @ nat @ ( X2 = Mi ) @ ( plus_plus @ nat @ ( times_times @ nat @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ X2 ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) @ ( vEBT_V1232361888498592333_e_t_e @ Summary @ ( vEBT_VEBT_high @ ( if @ nat @ ( X2 = Mi ) @ ( plus_plus @ nat @ ( times_times @ nat @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ X2 ) @ ( 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'.simps(7)
thf(fact_504_sum__squares__bound,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [X2: A,Y3: A] : ( ord_less_eq @ A @ ( times_times @ A @ ( times_times @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ X2 ) @ Y3 ) @ ( plus_plus @ A @ ( power_power @ A @ X2 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ ( power_power @ A @ Y3 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) ).
% sum_squares_bound
thf(fact_505_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_506__C7_OIH_C_I2_J,axiom,
! [Xa: nat,Xb: nat,Xc: nat,Xd: nat,Xe: vEBT_VEBT,Xf: list @ vEBT_VEBT,N: nat,Ti: vEBT_VEBTi] :
( ~ ( ( ord_less @ nat @ xa @ mi )
| ( ord_less @ nat @ ma @ xa ) )
=> ( ~ ( ( xa = mi )
& ( xa = ma ) )
=> ( ( ( ( xa = mi )
=> ( Xa
= ( plus_plus @ nat @ ( times_times @ nat @ ( the2 @ nat @ ( vEBT_vebt_mint @ summary ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ ( suc @ ( suc @ va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ treeList @ ( the2 @ nat @ ( vEBT_vebt_mint @ summary ) ) ) ) ) ) ) )
& ( ( xa != mi )
=> ( Xa = xa ) ) )
=> ( ( ( ( xa = mi )
=> ( Xb = Xa ) )
& ( ( xa != mi )
=> ( Xb = mi ) ) )
=> ( ( Xc
= ( vEBT_VEBT_low @ Xa @ ( divide_divide @ nat @ ( suc @ ( suc @ va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) )
=> ( ( Xd
= ( vEBT_VEBT_high @ Xa @ ( divide_divide @ nat @ ( suc @ ( suc @ va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) )
=> ( ( ord_less @ nat @ Xd @ ( size_size @ ( list @ vEBT_VEBT ) @ treeList ) )
=> ( ( Xe
= ( vEBT_vebt_delete @ ( nth @ vEBT_VEBT @ treeList @ Xd ) @ Xc ) )
=> ( ( Xf
= ( list_update @ vEBT_VEBT @ treeList @ Xd @ Xe ) )
=> ( ( vEBT_VEBT_minNull @ Xe )
=> ( ( vEBT_invar_vebt @ summary @ N )
=> ( hoare_hoare_triple @ vEBT_VEBTi @ ( vEBT_vebt_assn_raw @ summary @ Ti ) @ ( vEBT_V1365221501068881998eletei @ summary @ Ti @ Xd ) @ ( vEBT_vebt_assn_raw @ ( vEBT_vebt_delete @ summary @ Xd ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).
% "7.IH"(2)
thf(fact_507_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_508_le__add__diff__inverse,axiom,
! [A: $tType] :
( ( linordered_semidom @ A )
=> ! [B3: A,A3: A] :
( ( ord_less_eq @ A @ B3 @ A3 )
=> ( ( plus_plus @ A @ B3 @ ( minus_minus @ A @ A3 @ B3 ) )
= A3 ) ) ) ).
% le_add_diff_inverse
thf(fact_509_le__add__diff__inverse2,axiom,
! [A: $tType] :
( ( linordered_semidom @ A )
=> ! [B3: A,A3: A] :
( ( ord_less_eq @ A @ B3 @ A3 )
=> ( ( plus_plus @ A @ ( minus_minus @ A @ A3 @ B3 ) @ B3 )
= A3 ) ) ) ).
% le_add_diff_inverse2
thf(fact_510__C7_OIH_C_I1_J,axiom,
! [Xa: nat,Xb: nat,Xc: nat,Xd: nat,N: nat,Ti: vEBT_VEBTi] :
( ~ ( ( ord_less @ nat @ xa @ mi )
| ( ord_less @ nat @ ma @ xa ) )
=> ( ~ ( ( xa = mi )
& ( xa = ma ) )
=> ( ( ( ( xa = mi )
=> ( Xa
= ( plus_plus @ nat @ ( times_times @ nat @ ( the2 @ nat @ ( vEBT_vebt_mint @ summary ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ ( suc @ ( suc @ va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ treeList @ ( the2 @ nat @ ( vEBT_vebt_mint @ summary ) ) ) ) ) ) ) )
& ( ( xa != mi )
=> ( Xa = xa ) ) )
=> ( ( ( ( xa = mi )
=> ( Xb = Xa ) )
& ( ( xa != mi )
=> ( Xb = mi ) ) )
=> ( ( Xc
= ( vEBT_VEBT_low @ Xa @ ( divide_divide @ nat @ ( suc @ ( suc @ va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) )
=> ( ( Xd
= ( vEBT_VEBT_high @ Xa @ ( divide_divide @ nat @ ( suc @ ( suc @ va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) )
=> ( ( ord_less @ nat @ Xd @ ( size_size @ ( list @ vEBT_VEBT ) @ treeList ) )
=> ( ( vEBT_invar_vebt @ ( nth @ vEBT_VEBT @ treeList @ Xd ) @ N )
=> ( hoare_hoare_triple @ vEBT_VEBTi @ ( vEBT_vebt_assn_raw @ ( nth @ vEBT_VEBT @ treeList @ Xd ) @ Ti ) @ ( vEBT_V1365221501068881998eletei @ ( nth @ vEBT_VEBT @ treeList @ Xd ) @ Ti @ Xc ) @ ( vEBT_vebt_assn_raw @ ( vEBT_vebt_delete @ ( nth @ vEBT_VEBT @ treeList @ Xd ) @ Xc ) ) ) ) ) ) ) ) ) ) ) ).
% "7.IH"(1)
thf(fact_511_div__exp__eq,axiom,
! [A: $tType] :
( ( bit_semiring_bits @ A )
=> ! [A3: A,M: nat,N: nat] :
( ( divide_divide @ A @ ( divide_divide @ A @ A3 @ ( power_power @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ M ) ) @ ( power_power @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ N ) )
= ( divide_divide @ A @ A3 @ ( power_power @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ ( plus_plus @ nat @ M @ N ) ) ) ) ) ).
% div_exp_eq
thf(fact_512_bits__div__by__1,axiom,
! [A: $tType] :
( ( bit_semiring_bits @ A )
=> ! [A3: A] :
( ( divide_divide @ A @ A3 @ ( one_one @ A ) )
= A3 ) ) ).
% bits_div_by_1
thf(fact_513_div__by__1,axiom,
! [A: $tType] :
( ( semidom_divide @ A )
=> ! [A3: A] :
( ( divide_divide @ A @ A3 @ ( one_one @ A ) )
= A3 ) ) ).
% div_by_1
thf(fact_514_vebt__inserti_H__rf__abstr,axiom,
! [T2: vEBT_VEBT,Ti: vEBT_VEBTi,X2: nat] : ( hoare_hoare_triple @ vEBT_VEBTi @ ( vEBT_vebt_assn_raw @ T2 @ Ti ) @ ( vEBT_V3964819847710782039nserti @ T2 @ Ti @ X2 ) @ ( vEBT_vebt_assn_raw @ ( vEBT_vebt_insert @ T2 @ X2 ) ) ) ).
% vebt_inserti'_rf_abstr
thf(fact_515_linorder__neqE__linordered__idom,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [X2: A,Y3: A] :
( ( X2 != Y3 )
=> ( ~ ( ord_less @ A @ X2 @ Y3 )
=> ( ord_less @ A @ Y3 @ X2 ) ) ) ) ).
% linorder_neqE_linordered_idom
thf(fact_516_combine__common__factor,axiom,
! [A: $tType] :
( ( semiring @ A )
=> ! [A3: A,E: A,B3: A,C3: A] :
( ( plus_plus @ A @ ( times_times @ A @ A3 @ E ) @ ( plus_plus @ A @ ( times_times @ A @ B3 @ E ) @ C3 ) )
= ( plus_plus @ A @ ( times_times @ A @ ( plus_plus @ A @ A3 @ B3 ) @ E ) @ C3 ) ) ) ).
% combine_common_factor
thf(fact_517_distrib__right,axiom,
! [A: $tType] :
( ( semiring @ A )
=> ! [A3: A,B3: A,C3: A] :
( ( times_times @ A @ ( plus_plus @ A @ A3 @ B3 ) @ C3 )
= ( plus_plus @ A @ ( times_times @ A @ A3 @ C3 ) @ ( times_times @ A @ B3 @ C3 ) ) ) ) ).
% distrib_right
thf(fact_518_distrib__left,axiom,
! [A: $tType] :
( ( semiring @ A )
=> ! [A3: A,B3: A,C3: A] :
( ( times_times @ A @ A3 @ ( plus_plus @ A @ B3 @ C3 ) )
= ( plus_plus @ A @ ( times_times @ A @ A3 @ B3 ) @ ( times_times @ A @ A3 @ C3 ) ) ) ) ).
% distrib_left
thf(fact_519_comm__semiring__class_Odistrib,axiom,
! [A: $tType] :
( ( comm_semiring @ A )
=> ! [A3: A,B3: A,C3: A] :
( ( times_times @ A @ ( plus_plus @ A @ A3 @ B3 ) @ C3 )
= ( plus_plus @ A @ ( times_times @ A @ A3 @ C3 ) @ ( times_times @ A @ B3 @ C3 ) ) ) ) ).
% comm_semiring_class.distrib
thf(fact_520_ring__class_Oring__distribs_I1_J,axiom,
! [A: $tType] :
( ( ring @ A )
=> ! [A3: A,B3: A,C3: A] :
( ( times_times @ A @ A3 @ ( plus_plus @ A @ B3 @ C3 ) )
= ( plus_plus @ A @ ( times_times @ A @ A3 @ B3 ) @ ( times_times @ A @ A3 @ C3 ) ) ) ) ).
% ring_class.ring_distribs(1)
thf(fact_521_ring__class_Oring__distribs_I2_J,axiom,
! [A: $tType] :
( ( ring @ A )
=> ! [A3: A,B3: A,C3: A] :
( ( times_times @ A @ ( plus_plus @ A @ A3 @ B3 ) @ C3 )
= ( plus_plus @ A @ ( times_times @ A @ A3 @ C3 ) @ ( times_times @ A @ B3 @ C3 ) ) ) ) ).
% ring_class.ring_distribs(2)
thf(fact_522_right__diff__distrib_H,axiom,
! [A: $tType] :
( ( comm_s4317794764714335236cancel @ A )
=> ! [A3: A,B3: A,C3: A] :
( ( times_times @ A @ A3 @ ( minus_minus @ A @ B3 @ C3 ) )
= ( minus_minus @ A @ ( times_times @ A @ A3 @ B3 ) @ ( times_times @ A @ A3 @ C3 ) ) ) ) ).
% right_diff_distrib'
thf(fact_523_left__diff__distrib_H,axiom,
! [A: $tType] :
( ( comm_s4317794764714335236cancel @ A )
=> ! [B3: A,C3: A,A3: A] :
( ( times_times @ A @ ( minus_minus @ A @ B3 @ C3 ) @ A3 )
= ( minus_minus @ A @ ( times_times @ A @ B3 @ A3 ) @ ( times_times @ A @ C3 @ A3 ) ) ) ) ).
% left_diff_distrib'
thf(fact_524_right__diff__distrib,axiom,
! [A: $tType] :
( ( ring @ A )
=> ! [A3: A,B3: A,C3: A] :
( ( times_times @ A @ A3 @ ( minus_minus @ A @ B3 @ C3 ) )
= ( minus_minus @ A @ ( times_times @ A @ A3 @ B3 ) @ ( times_times @ A @ A3 @ C3 ) ) ) ) ).
% right_diff_distrib
thf(fact_525_left__diff__distrib,axiom,
! [A: $tType] :
( ( ring @ A )
=> ! [A3: A,B3: A,C3: A] :
( ( times_times @ A @ ( minus_minus @ A @ A3 @ B3 ) @ C3 )
= ( minus_minus @ A @ ( times_times @ A @ A3 @ C3 ) @ ( times_times @ A @ B3 @ C3 ) ) ) ) ).
% left_diff_distrib
thf(fact_526_lambda__one,axiom,
! [A: $tType] :
( ( monoid_mult @ A )
=> ( ( ^ [X3: A] : X3 )
= ( times_times @ A @ ( one_one @ A ) ) ) ) ).
% lambda_one
thf(fact_527_less__add__one,axiom,
! [A: $tType] :
( ( linordered_semidom @ A )
=> ! [A3: A] : ( ord_less @ A @ A3 @ ( plus_plus @ A @ A3 @ ( one_one @ A ) ) ) ) ).
% less_add_one
thf(fact_528_add__mono1,axiom,
! [A: $tType] :
( ( linord181362715937106298miring @ A )
=> ! [A3: A,B3: A] :
( ( ord_less @ A @ A3 @ B3 )
=> ( ord_less @ A @ ( plus_plus @ A @ A3 @ ( one_one @ A ) ) @ ( plus_plus @ A @ B3 @ ( one_one @ A ) ) ) ) ) ).
% add_mono1
thf(fact_529_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_530_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_531_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_532_linordered__semidom__class_Oadd__diff__inverse,axiom,
! [A: $tType] :
( ( linordered_semidom @ A )
=> ! [A3: A,B3: A] :
( ~ ( ord_less @ A @ A3 @ B3 )
=> ( ( plus_plus @ A @ B3 @ ( minus_minus @ A @ A3 @ B3 ) )
= A3 ) ) ) ).
% linordered_semidom_class.add_diff_inverse
thf(fact_533_four__x__squared,axiom,
! [X2: real] :
( ( times_times @ real @ ( numeral_numeral @ real @ ( bit0 @ ( bit0 @ one2 ) ) ) @ ( power_power @ real @ X2 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) )
= ( power_power @ real @ ( times_times @ real @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) @ X2 ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ).
% four_x_squared
thf(fact_534_L2__set__mult__ineq__lemma,axiom,
! [A3: real,C3: real,B3: real,D2: real] : ( ord_less_eq @ real @ ( times_times @ real @ ( times_times @ real @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) @ ( times_times @ real @ A3 @ C3 ) ) @ ( times_times @ real @ B3 @ D2 ) ) @ ( plus_plus @ real @ ( times_times @ real @ ( power_power @ real @ A3 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ ( power_power @ real @ D2 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( times_times @ real @ ( power_power @ real @ B3 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ ( power_power @ real @ C3 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) ).
% L2_set_mult_ineq_lemma
thf(fact_535_eq__add__iff1,axiom,
! [A: $tType] :
( ( ring @ A )
=> ! [A3: A,E: A,C3: A,B3: A,D2: A] :
( ( ( plus_plus @ A @ ( times_times @ A @ A3 @ E ) @ C3 )
= ( plus_plus @ A @ ( times_times @ A @ B3 @ E ) @ D2 ) )
= ( ( plus_plus @ A @ ( times_times @ A @ ( minus_minus @ A @ A3 @ B3 ) @ E ) @ C3 )
= D2 ) ) ) ).
% eq_add_iff1
thf(fact_536_eq__add__iff2,axiom,
! [A: $tType] :
( ( ring @ A )
=> ! [A3: A,E: A,C3: A,B3: A,D2: A] :
( ( ( plus_plus @ A @ ( times_times @ A @ A3 @ E ) @ C3 )
= ( plus_plus @ A @ ( times_times @ A @ B3 @ E ) @ D2 ) )
= ( C3
= ( plus_plus @ A @ ( times_times @ A @ ( minus_minus @ A @ B3 @ A3 ) @ E ) @ D2 ) ) ) ) ).
% eq_add_iff2
thf(fact_537_square__diff__square__factored,axiom,
! [A: $tType] :
( ( comm_ring @ A )
=> ! [X2: A,Y3: A] :
( ( minus_minus @ A @ ( times_times @ A @ X2 @ X2 ) @ ( times_times @ A @ Y3 @ Y3 ) )
= ( times_times @ A @ ( plus_plus @ A @ X2 @ Y3 ) @ ( minus_minus @ A @ X2 @ Y3 ) ) ) ) ).
% square_diff_square_factored
thf(fact_538_vebt__delete_Osimps_I4_J,axiom,
! [Deg: nat,TreeList: list @ vEBT_VEBT,Summary: vEBT_VEBT,Uu: nat] :
( ( vEBT_vebt_delete @ ( vEBT_Node @ ( none @ ( product_prod @ nat @ nat ) ) @ Deg @ TreeList @ Summary ) @ Uu )
= ( vEBT_Node @ ( none @ ( product_prod @ nat @ nat ) ) @ Deg @ TreeList @ Summary ) ) ).
% vebt_delete.simps(4)
thf(fact_539_ordered__ring__class_Ole__add__iff1,axiom,
! [A: $tType] :
( ( ordered_ring @ A )
=> ! [A3: A,E: A,C3: A,B3: A,D2: A] :
( ( ord_less_eq @ A @ ( plus_plus @ A @ ( times_times @ A @ A3 @ E ) @ C3 ) @ ( plus_plus @ A @ ( times_times @ A @ B3 @ E ) @ D2 ) )
= ( ord_less_eq @ A @ ( plus_plus @ A @ ( times_times @ A @ ( minus_minus @ A @ A3 @ B3 ) @ E ) @ C3 ) @ D2 ) ) ) ).
% ordered_ring_class.le_add_iff1
thf(fact_540_ordered__ring__class_Ole__add__iff2,axiom,
! [A: $tType] :
( ( ordered_ring @ A )
=> ! [A3: A,E: A,C3: A,B3: A,D2: A] :
( ( ord_less_eq @ A @ ( plus_plus @ A @ ( times_times @ A @ A3 @ E ) @ C3 ) @ ( plus_plus @ A @ ( times_times @ A @ B3 @ E ) @ D2 ) )
= ( ord_less_eq @ A @ C3 @ ( plus_plus @ A @ ( times_times @ A @ ( minus_minus @ A @ B3 @ A3 ) @ E ) @ D2 ) ) ) ) ).
% ordered_ring_class.le_add_iff2
thf(fact_541_less__add__iff2,axiom,
! [A: $tType] :
( ( ordered_ring @ A )
=> ! [A3: A,E: A,C3: A,B3: A,D2: A] :
( ( ord_less @ A @ ( plus_plus @ A @ ( times_times @ A @ A3 @ E ) @ C3 ) @ ( plus_plus @ A @ ( times_times @ A @ B3 @ E ) @ D2 ) )
= ( ord_less @ A @ C3 @ ( plus_plus @ A @ ( times_times @ A @ ( minus_minus @ A @ B3 @ A3 ) @ E ) @ D2 ) ) ) ) ).
% less_add_iff2
thf(fact_542_less__add__iff1,axiom,
! [A: $tType] :
( ( ordered_ring @ A )
=> ! [A3: A,E: A,C3: A,B3: A,D2: A] :
( ( ord_less @ A @ ( plus_plus @ A @ ( times_times @ A @ A3 @ E ) @ C3 ) @ ( plus_plus @ A @ ( times_times @ A @ B3 @ E ) @ D2 ) )
= ( ord_less @ A @ ( plus_plus @ A @ ( times_times @ A @ ( minus_minus @ A @ A3 @ B3 ) @ E ) @ C3 ) @ D2 ) ) ) ).
% less_add_iff1
thf(fact_543_square__diff__one__factored,axiom,
! [A: $tType] :
( ( ring_1 @ A )
=> ! [X2: A] :
( ( minus_minus @ A @ ( times_times @ A @ X2 @ X2 ) @ ( one_one @ A ) )
= ( times_times @ A @ ( plus_plus @ A @ X2 @ ( one_one @ A ) ) @ ( minus_minus @ A @ X2 @ ( one_one @ A ) ) ) ) ) ).
% square_diff_one_factored
thf(fact_544_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,Uu: nat] :
( ( vEBT_V1232361888498592333_e_t_e @ ( vEBT_Node @ ( none @ ( product_prod @ nat @ nat ) ) @ Deg @ TreeList @ Summary ) @ Uu )
= ( one_one @ nat ) ) ).
% VEBT_internal.T\<^sub>d\<^sub>e\<^sub>l\<^sub>e\<^sub>t\<^sub>e'.simps(4)
thf(fact_545__092_060open_062_092_060And_062x22_Ax21_O_Ati_A_061_ALeafi_Ax21_Ax22_A_092_060Longrightarrow_062_A_060vebt__assn__raw_A_INode_A_ISome_A_Imi_M_Ama_J_J_A_ISuc_A_ISuc_Ava_J_J_AtreeList_Asummary_J_Ati_062_Avebt__deletei_H_A_INode_A_ISome_A_Imi_M_Ama_J_J_A_ISuc_A_ISuc_Ava_J_J_AtreeList_Asummary_J_Ati_Ax_A_060vebt__assn__raw_A_Ivebt__delete_A_INode_A_ISome_A_Imi_M_Ama_J_J_A_ISuc_A_ISuc_Ava_J_J_AtreeList_Asummary_J_Ax_J_062_092_060close_062,axiom,
! [X21: $o,X222: $o] :
( ( tia
= ( vEBT_Leafi @ X21 @ X222 ) )
=> ( hoare_hoare_triple @ vEBT_VEBTi @ ( vEBT_vebt_assn_raw @ ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ mi @ ma ) ) @ ( suc @ ( suc @ va ) ) @ treeList @ summary ) @ tia ) @ ( vEBT_V1365221501068881998eletei @ ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ mi @ ma ) ) @ ( suc @ ( suc @ va ) ) @ treeList @ summary ) @ tia @ xa ) @ ( vEBT_vebt_assn_raw @ ( vEBT_vebt_delete @ ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ mi @ ma ) ) @ ( suc @ ( suc @ va ) ) @ treeList @ summary ) @ xa ) ) ) ) ).
% \<open>\<And>x22 x21. ti = Leafi x21 x22 \<Longrightarrow> <vebt_assn_raw (Node (Some (mi, ma)) (Suc (Suc va)) treeList summary) ti> vebt_deletei' (Node (Some (mi, ma)) (Suc (Suc va)) treeList summary) ti x <vebt_assn_raw (vebt_delete (Node (Some (mi, ma)) (Suc (Suc va)) treeList summary) x)>\<close>
thf(fact_546_real__average__minus__first,axiom,
! [A3: real,B3: real] :
( ( minus_minus @ real @ ( divide_divide @ real @ ( plus_plus @ real @ A3 @ B3 ) @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) @ A3 )
= ( divide_divide @ real @ ( minus_minus @ real @ B3 @ A3 ) @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) ) ).
% real_average_minus_first
thf(fact_547_real__average__minus__second,axiom,
! [B3: real,A3: real] :
( ( minus_minus @ real @ ( divide_divide @ real @ ( plus_plus @ real @ B3 @ A3 ) @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) @ A3 )
= ( divide_divide @ real @ ( minus_minus @ real @ B3 @ A3 ) @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) ) ).
% real_average_minus_second
thf(fact_548_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_549_zdiv__numeral__Bit0,axiom,
! [V: num,W: num] :
( ( divide_divide @ int @ ( numeral_numeral @ int @ ( bit0 @ V ) ) @ ( numeral_numeral @ int @ ( bit0 @ W ) ) )
= ( divide_divide @ int @ ( numeral_numeral @ int @ V ) @ ( numeral_numeral @ int @ W ) ) ) ).
% zdiv_numeral_Bit0
thf(fact_550_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_551_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,Va: nat,TreeList: list @ vEBT_VEBT,Summary: vEBT_VEBT,X2: nat] :
( ( vEBT_T_i_n_s_e_r_t2 @ ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi @ Ma ) ) @ ( suc @ ( suc @ Va ) ) @ TreeList @ Summary ) @ X2 )
= ( if @ nat
@ ( ( ord_less @ nat @ ( vEBT_VEBT_high @ ( if @ nat @ ( ord_less @ nat @ X2 @ Mi ) @ Mi @ X2 ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( size_size @ ( list @ vEBT_VEBT ) @ TreeList ) )
& ~ ( ( X2 = Mi )
| ( X2 = Ma ) ) )
@ ( plus_plus @ nat @ ( vEBT_T_i_n_s_e_r_t2 @ ( nth @ vEBT_VEBT @ TreeList @ ( vEBT_VEBT_high @ ( if @ nat @ ( ord_less @ nat @ X2 @ Mi ) @ Mi @ X2 ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( vEBT_VEBT_low @ ( if @ nat @ ( ord_less @ nat @ X2 @ Mi ) @ Mi @ X2 ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( if @ nat @ ( vEBT_VEBT_minNull @ ( nth @ vEBT_VEBT @ TreeList @ ( vEBT_VEBT_high @ ( if @ nat @ ( ord_less @ nat @ X2 @ Mi ) @ Mi @ X2 ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) @ ( vEBT_T_i_n_s_e_r_t2 @ Summary @ ( vEBT_VEBT_high @ ( if @ nat @ ( ord_less @ nat @ X2 @ Mi ) @ Mi @ X2 ) @ ( 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'.simps(5)
thf(fact_552_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_553_field__less__half__sum,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [X2: A,Y3: A] :
( ( ord_less @ A @ X2 @ Y3 )
=> ( ord_less @ A @ X2 @ ( divide_divide @ A @ ( plus_plus @ A @ X2 @ Y3 ) @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) ) ) ) ).
% field_less_half_sum
thf(fact_554_succ__empty,axiom,
! [T2: vEBT_VEBT,N: nat,X2: nat] :
( ( vEBT_invar_vebt @ T2 @ N )
=> ( ( ( vEBT_vebt_succ @ T2 @ X2 )
= ( none @ nat ) )
= ( ( collect @ nat
@ ^ [Y: nat] :
( ( vEBT_vebt_member @ T2 @ Y )
& ( ord_less @ nat @ X2 @ Y ) ) )
= ( bot_bot @ ( set @ nat ) ) ) ) ) ).
% succ_empty
thf(fact_555_buildup__gives__empty,axiom,
! [N: nat] :
( ( vEBT_VEBT_set_vebt @ ( vEBT_vebt_buildup @ N ) )
= ( bot_bot @ ( set @ nat ) ) ) ).
% buildup_gives_empty
thf(fact_556_semiring__norm_I90_J,axiom,
! [M: num,N: num] :
( ( ( bit1 @ M )
= ( bit1 @ N ) )
= ( M = N ) ) ).
% semiring_norm(90)
thf(fact_557_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_558_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_559_pred__empty,axiom,
! [T2: vEBT_VEBT,N: nat,X2: nat] :
( ( vEBT_invar_vebt @ T2 @ N )
=> ( ( ( vEBT_vebt_pred @ T2 @ X2 )
= ( none @ nat ) )
= ( ( collect @ nat
@ ^ [Y: nat] :
( ( vEBT_vebt_member @ T2 @ Y )
& ( ord_less @ nat @ Y @ X2 ) ) )
= ( bot_bot @ ( set @ nat ) ) ) ) ) ).
% pred_empty
thf(fact_560_semiring__norm_I89_J,axiom,
! [M: num,N: num] :
( ( bit1 @ M )
!= ( bit0 @ N ) ) ).
% semiring_norm(89)
thf(fact_561_semiring__norm_I88_J,axiom,
! [M: num,N: num] :
( ( bit0 @ M )
!= ( bit1 @ N ) ) ).
% semiring_norm(88)
thf(fact_562_semiring__norm_I86_J,axiom,
! [M: num] :
( ( bit1 @ M )
!= one2 ) ).
% semiring_norm(86)
thf(fact_563_semiring__norm_I84_J,axiom,
! [N: num] :
( one2
!= ( bit1 @ N ) ) ).
% semiring_norm(84)
thf(fact_564_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_565_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_566_semiring__norm_I7_J,axiom,
! [M: num,N: num] :
( ( plus_plus @ num @ ( bit0 @ M ) @ ( bit1 @ N ) )
= ( bit1 @ ( plus_plus @ num @ M @ N ) ) ) ).
% semiring_norm(7)
thf(fact_567_semiring__norm_I9_J,axiom,
! [M: num,N: num] :
( ( plus_plus @ num @ ( bit1 @ M ) @ ( bit0 @ N ) )
= ( bit1 @ ( plus_plus @ num @ M @ N ) ) ) ).
% semiring_norm(9)
thf(fact_568_semiring__norm_I14_J,axiom,
! [M: num,N: num] :
( ( times_times @ num @ ( bit0 @ M ) @ ( bit1 @ N ) )
= ( bit0 @ ( times_times @ num @ M @ ( bit1 @ N ) ) ) ) ).
% semiring_norm(14)
thf(fact_569_semiring__norm_I15_J,axiom,
! [M: num,N: num] :
( ( times_times @ num @ ( bit1 @ M ) @ ( bit0 @ N ) )
= ( bit0 @ ( times_times @ num @ ( bit1 @ M ) @ N ) ) ) ).
% semiring_norm(15)
thf(fact_570_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_571_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_572_semiring__norm_I70_J,axiom,
! [M: num] :
~ ( ord_less_eq @ num @ ( bit1 @ M ) @ one2 ) ).
% semiring_norm(70)
thf(fact_573_semiring__norm_I77_J,axiom,
! [N: num] : ( ord_less @ num @ one2 @ ( bit1 @ N ) ) ).
% semiring_norm(77)
thf(fact_574_zdiv__numeral__Bit1,axiom,
! [V: num,W: num] :
( ( divide_divide @ int @ ( numeral_numeral @ int @ ( bit1 @ V ) ) @ ( numeral_numeral @ int @ ( bit0 @ W ) ) )
= ( divide_divide @ int @ ( numeral_numeral @ int @ V ) @ ( numeral_numeral @ int @ W ) ) ) ).
% zdiv_numeral_Bit1
thf(fact_575_semiring__norm_I10_J,axiom,
! [M: num,N: num] :
( ( plus_plus @ num @ ( bit1 @ M ) @ ( bit1 @ N ) )
= ( bit0 @ ( plus_plus @ num @ ( plus_plus @ num @ M @ N ) @ one2 ) ) ) ).
% semiring_norm(10)
thf(fact_576_semiring__norm_I8_J,axiom,
! [M: num] :
( ( plus_plus @ num @ ( bit1 @ M ) @ one2 )
= ( bit0 @ ( plus_plus @ num @ M @ one2 ) ) ) ).
% semiring_norm(8)
thf(fact_577_semiring__norm_I5_J,axiom,
! [M: num] :
( ( plus_plus @ num @ ( bit0 @ M ) @ one2 )
= ( bit1 @ M ) ) ).
% semiring_norm(5)
thf(fact_578_semiring__norm_I4_J,axiom,
! [N: num] :
( ( plus_plus @ num @ one2 @ ( bit1 @ N ) )
= ( bit0 @ ( plus_plus @ num @ N @ one2 ) ) ) ).
% semiring_norm(4)
thf(fact_579_semiring__norm_I3_J,axiom,
! [N: num] :
( ( plus_plus @ num @ one2 @ ( bit0 @ N ) )
= ( bit1 @ N ) ) ).
% semiring_norm(3)
thf(fact_580_semiring__norm_I16_J,axiom,
! [M: num,N: num] :
( ( times_times @ num @ ( bit1 @ M ) @ ( bit1 @ N ) )
= ( bit1 @ ( plus_plus @ num @ ( plus_plus @ num @ M @ N ) @ ( bit0 @ ( times_times @ num @ M @ N ) ) ) ) ) ).
% semiring_norm(16)
thf(fact_581_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_582_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_583_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_584_Suc__div__eq__add3__div__numeral,axiom,
! [M: nat,V: num] :
( ( divide_divide @ nat @ ( suc @ ( suc @ ( suc @ M ) ) ) @ ( numeral_numeral @ nat @ V ) )
= ( divide_divide @ nat @ ( plus_plus @ nat @ ( numeral_numeral @ nat @ ( bit1 @ one2 ) ) @ M ) @ ( numeral_numeral @ nat @ V ) ) ) ).
% Suc_div_eq_add3_div_numeral
thf(fact_585_set__notEmptyE,axiom,
! [A: $tType,S: set @ A] :
( ( S
!= ( bot_bot @ ( set @ A ) ) )
=> ~ ! [X4: A] :
~ ( member @ A @ X4 @ S ) ) ).
% set_notEmptyE
thf(fact_586_memb__imp__not__empty,axiom,
! [A: $tType,X2: A,S: set @ A] :
( ( member @ A @ X2 @ S )
=> ( S
!= ( bot_bot @ ( set @ A ) ) ) ) ).
% memb_imp_not_empty
thf(fact_587_subset__minus__empty,axiom,
! [A: $tType,A2: set @ A,B2: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ A2 @ B2 )
=> ( ( minus_minus @ ( set @ A ) @ A2 @ B2 )
= ( bot_bot @ ( set @ A ) ) ) ) ).
% subset_minus_empty
thf(fact_588_num_Oexhaust,axiom,
! [Y3: num] :
( ( Y3 != one2 )
=> ( ! [X23: num] :
( Y3
!= ( bit0 @ X23 ) )
=> ~ ! [X32: num] :
( Y3
!= ( bit1 @ X32 ) ) ) ) ).
% num.exhaust
thf(fact_589_xor__num_Ocases,axiom,
! [X2: product_prod @ num @ num] :
( ( X2
!= ( product_Pair @ num @ num @ one2 @ one2 ) )
=> ( ! [N3: num] :
( X2
!= ( product_Pair @ num @ num @ one2 @ ( bit0 @ N3 ) ) )
=> ( ! [N3: num] :
( X2
!= ( product_Pair @ num @ num @ one2 @ ( bit1 @ N3 ) ) )
=> ( ! [M5: num] :
( X2
!= ( product_Pair @ num @ num @ ( bit0 @ M5 ) @ one2 ) )
=> ( ! [M5: num,N3: num] :
( X2
!= ( product_Pair @ num @ num @ ( bit0 @ M5 ) @ ( bit0 @ N3 ) ) )
=> ( ! [M5: num,N3: num] :
( X2
!= ( product_Pair @ num @ num @ ( bit0 @ M5 ) @ ( bit1 @ N3 ) ) )
=> ( ! [M5: num] :
( X2
!= ( product_Pair @ num @ num @ ( bit1 @ M5 ) @ one2 ) )
=> ( ! [M5: num,N3: num] :
( X2
!= ( product_Pair @ num @ num @ ( bit1 @ M5 ) @ ( bit0 @ N3 ) ) )
=> ~ ! [M5: num,N3: num] :
( X2
!= ( product_Pair @ num @ num @ ( bit1 @ M5 ) @ ( bit1 @ N3 ) ) ) ) ) ) ) ) ) ) ) ).
% xor_num.cases
thf(fact_590_numeral__Bit1,axiom,
! [A: $tType] :
( ( numeral @ A )
=> ! [N: num] :
( ( numeral_numeral @ A @ ( bit1 @ N ) )
= ( plus_plus @ A @ ( plus_plus @ A @ ( numeral_numeral @ A @ N ) @ ( numeral_numeral @ A @ N ) ) @ ( one_one @ A ) ) ) ) ).
% numeral_Bit1
thf(fact_591_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_592_less__eq__real__def,axiom,
( ( ord_less_eq @ real )
= ( ^ [X3: real,Y: real] :
( ( ord_less @ real @ X3 @ Y )
| ( X3 = Y ) ) ) ) ).
% less_eq_real_def
thf(fact_593_complete__real,axiom,
! [S: set @ real] :
( ? [X8: real] : ( member @ real @ X8 @ S )
=> ( ? [Z5: real] :
! [X4: real] :
( ( member @ real @ X4 @ S )
=> ( ord_less_eq @ real @ X4 @ Z5 ) )
=> ? [Y4: real] :
( ! [X8: real] :
( ( member @ real @ X8 @ S )
=> ( ord_less_eq @ real @ X8 @ Y4 ) )
& ! [Z5: real] :
( ! [X4: real] :
( ( member @ real @ X4 @ S )
=> ( ord_less_eq @ real @ X4 @ Z5 ) )
=> ( ord_less_eq @ real @ Y4 @ Z5 ) ) ) ) ) ).
% complete_real
thf(fact_594_numeral__code_I3_J,axiom,
! [A: $tType] :
( ( numeral @ A )
=> ! [N: num] :
( ( numeral_numeral @ A @ ( bit1 @ N ) )
= ( plus_plus @ A @ ( plus_plus @ A @ ( numeral_numeral @ A @ N ) @ ( numeral_numeral @ A @ N ) ) @ ( one_one @ A ) ) ) ) ).
% numeral_code(3)
thf(fact_595_power__numeral__odd,axiom,
! [A: $tType] :
( ( monoid_mult @ A )
=> ! [Z: A,W: num] :
( ( power_power @ A @ Z @ ( numeral_numeral @ nat @ ( bit1 @ W ) ) )
= ( times_times @ A @ ( times_times @ A @ Z @ ( power_power @ A @ Z @ ( numeral_numeral @ nat @ W ) ) ) @ ( power_power @ A @ Z @ ( numeral_numeral @ nat @ W ) ) ) ) ) ).
% power_numeral_odd
thf(fact_596_numeral__Bit1__div__2,axiom,
! [A: $tType] :
( ( euclid5411537665997757685th_nat @ A )
=> ! [N: num] :
( ( divide_divide @ A @ ( numeral_numeral @ A @ ( bit1 @ N ) ) @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) )
= ( numeral_numeral @ A @ N ) ) ) ).
% numeral_Bit1_div_2
thf(fact_597_power3__eq__cube,axiom,
! [A: $tType] :
( ( monoid_mult @ A )
=> ! [A3: A] :
( ( power_power @ A @ A3 @ ( numeral_numeral @ nat @ ( bit1 @ one2 ) ) )
= ( times_times @ A @ ( times_times @ A @ A3 @ A3 ) @ A3 ) ) ) ).
% power3_eq_cube
thf(fact_598_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_599_mult__commute__abs,axiom,
! [A: $tType] :
( ( ab_semigroup_mult @ A )
=> ! [C3: A] :
( ( ^ [X3: A] : ( times_times @ A @ X3 @ C3 ) )
= ( times_times @ A @ C3 ) ) ) ).
% mult_commute_abs
thf(fact_600_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_601_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,
! [V: nat,TreeList: list @ vEBT_VEBT,Summary: vEBT_VEBT,X2: nat] :
( ( vEBT_T_i_n_s_e_r_t2 @ ( vEBT_Node @ ( none @ ( product_prod @ nat @ nat ) ) @ ( suc @ ( suc @ V ) ) @ TreeList @ Summary ) @ X2 )
= ( one_one @ nat ) ) ).
% T\<^sub>i\<^sub>n\<^sub>s\<^sub>e\<^sub>r\<^sub>t'.simps(4)
thf(fact_602_insersimp_H,axiom,
! [T2: vEBT_VEBT,N: nat,Y3: nat] :
( ( vEBT_invar_vebt @ T2 @ N )
=> ( ~ ? [X_1: nat] : ( vEBT_V8194947554948674370ptions @ T2 @ X_1 )
=> ( ord_less_eq @ nat @ ( vEBT_T_i_n_s_e_r_t2 @ T2 @ Y3 ) @ ( one_one @ nat ) ) ) ) ).
% insersimp'
thf(fact_603_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_604_add__diff__add,axiom,
! [A: $tType] :
( ( ab_group_add @ A )
=> ! [A3: A,C3: A,B3: A,D2: A] :
( ( minus_minus @ A @ ( plus_plus @ A @ A3 @ C3 ) @ ( plus_plus @ A @ B3 @ D2 ) )
= ( plus_plus @ A @ ( minus_minus @ A @ A3 @ B3 ) @ ( minus_minus @ A @ C3 @ D2 ) ) ) ) ).
% add_diff_add
thf(fact_605_real__arch__pow,axiom,
! [X2: real,Y3: real] :
( ( ord_less @ real @ ( one_one @ real ) @ X2 )
=> ? [N3: nat] : ( ord_less @ real @ Y3 @ ( power_power @ real @ X2 @ N3 ) ) ) ).
% real_arch_pow
thf(fact_606_insert_H__bound__height,axiom,
! [T2: vEBT_VEBT,N: nat,X2: nat] :
( ( vEBT_invar_vebt @ T2 @ N )
=> ( ord_less_eq @ nat @ ( vEBT_T_i_n_s_e_r_t2 @ T2 @ X2 ) @ ( plus_plus @ nat @ ( one_one @ nat ) @ ( vEBT_VEBT_height @ T2 ) ) ) ) ).
% insert'_bound_height
thf(fact_607_small__powers__of__2,axiom,
! [X2: nat] :
( ( ord_less_eq @ nat @ ( numeral_numeral @ nat @ ( bit1 @ one2 ) ) @ X2 )
=> ( ord_less @ nat @ X2 @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( minus_minus @ nat @ X2 @ ( one_one @ nat ) ) ) ) ) ).
% small_powers_of_2
thf(fact_608_mult__diff__mult,axiom,
! [A: $tType] :
( ( ring @ A )
=> ! [X2: A,Y3: A,A3: A,B3: A] :
( ( minus_minus @ A @ ( times_times @ A @ X2 @ Y3 ) @ ( times_times @ A @ A3 @ B3 ) )
= ( plus_plus @ A @ ( times_times @ A @ X2 @ ( minus_minus @ A @ Y3 @ B3 ) ) @ ( times_times @ A @ ( minus_minus @ A @ X2 @ A3 ) @ B3 ) ) ) ) ).
% mult_diff_mult
thf(fact_609_discrete,axiom,
! [A: $tType] :
( ( unique1627219031080169319umeral @ A )
=> ( ( ord_less @ A )
= ( ^ [A6: A] : ( ord_less_eq @ A @ ( plus_plus @ A @ A6 @ ( one_one @ A ) ) ) ) ) ) ).
% discrete
thf(fact_610_field__sum__of__halves,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [X2: A] :
( ( plus_plus @ A @ ( divide_divide @ A @ X2 @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) @ ( divide_divide @ A @ X2 @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) )
= X2 ) ) ).
% field_sum_of_halves
thf(fact_611_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_612_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_613_delete__bound__height,axiom,
! [T2: vEBT_VEBT,N: nat,X2: nat] :
( ( vEBT_invar_vebt @ T2 @ N )
=> ( ord_less_eq @ nat @ ( vEBT_T_d_e_l_e_t_e @ T2 @ X2 ) @ ( 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_614_minNull__delete__time__bound,axiom,
! [T2: vEBT_VEBT,N: nat,X2: nat] :
( ( vEBT_invar_vebt @ T2 @ N )
=> ( ( vEBT_VEBT_minNull @ ( vEBT_vebt_delete @ T2 @ X2 ) )
=> ( ord_less_eq @ nat @ ( vEBT_T_d_e_l_e_t_e @ T2 @ X2 ) @ ( numeral_numeral @ nat @ ( bit1 @ ( bit0 @ ( bit0 @ one2 ) ) ) ) ) ) ) ).
% minNull_delete_time_bound
thf(fact_615_tdeletemimi,axiom,
! [Deg: nat,Mi: nat,TreeList: list @ vEBT_VEBT,Summary: vEBT_VEBT,X2: 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 ) @ X2 ) @ ( numeral_numeral @ nat @ ( bit1 @ ( bit0 @ ( bit0 @ one2 ) ) ) ) ) ) ).
% tdeletemimi
thf(fact_616_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_617_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_618_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_619_TBOUND__vebt__inserti,axiom,
! [T2: vEBT_VEBT,Ti: vEBT_VEBTi,X2: nat] : ( time_TBOUND @ vEBT_VEBTi @ ( vEBT_V3964819847710782039nserti @ T2 @ Ti @ X2 ) @ ( 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_620_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_621_space__space_H,axiom,
! [T2: vEBT_VEBT] : ( ord_less @ nat @ ( vEBT_VEBT_space @ T2 ) @ ( vEBT_VEBT_space2 @ T2 ) ) ).
% space_space'
thf(fact_622_TBOUND__vebt__buildupi,axiom,
! [N: nat] : ( time_TBOUND @ vEBT_VEBTi @ ( vEBT_V739175172307565963ildupi @ N ) @ ( vEBT_V441764108873111860ildupi @ N ) ) ).
% TBOUND_vebt_buildupi
thf(fact_623_TBOUND__minNull,axiom,
! [T2: vEBT_VEBT,Ti: vEBT_VEBTi,X2: nat] :
( ( vEBT_VEBT_minNull @ T2 )
=> ( time_TBOUND @ vEBT_VEBTi @ ( vEBT_V3964819847710782039nserti @ T2 @ Ti @ X2 ) @ ( one_one @ nat ) ) ) ).
% TBOUND_minNull
thf(fact_624_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,Uu: nat] :
( ( vEBT_T_d_e_l_e_t_e @ ( vEBT_Node @ ( none @ ( product_prod @ nat @ nat ) ) @ Deg @ TreeList @ Summary ) @ Uu )
= ( one_one @ nat ) ) ).
% T\<^sub>d\<^sub>e\<^sub>l\<^sub>e\<^sub>t\<^sub>e.simps(4)
thf(fact_625_TBOUND__replicate,axiom,
! [A: $tType,X2: heap_Time_Heap @ A,C3: nat,N: nat] :
( ( time_TBOUND @ A @ X2 @ C3 )
=> ( time_TBOUND @ ( list @ A ) @ ( vEBT_VEBT_replicatei @ A @ N @ X2 ) @ ( plus_plus @ nat @ ( one_one @ nat ) @ ( times_times @ nat @ ( plus_plus @ nat @ ( one_one @ nat ) @ C3 ) @ N ) ) ) ) ).
% TBOUND_replicate
thf(fact_626_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_627_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_628_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_629_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_630_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_631_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_632_zle__add1__eq__le,axiom,
! [W: int,Z: int] :
( ( ord_less @ int @ W @ ( plus_plus @ int @ Z @ ( one_one @ int ) ) )
= ( ord_less_eq @ int @ W @ Z ) ) ).
% zle_add1_eq_le
thf(fact_633_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,Va: nat,TreeList: list @ vEBT_VEBT,Summary: vEBT_VEBT,X2: nat] :
( ( vEBT_T_m_e_m_b_e_r @ ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi @ Ma ) ) @ ( suc @ ( suc @ Va ) ) @ TreeList @ Summary ) @ X2 )
= ( plus_plus @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( if @ nat @ ( X2 = Mi ) @ ( one_one @ nat ) @ ( plus_plus @ nat @ ( one_one @ nat ) @ ( if @ nat @ ( X2 = Ma ) @ ( one_one @ nat ) @ ( plus_plus @ nat @ ( one_one @ nat ) @ ( if @ nat @ ( ord_less @ nat @ X2 @ Mi ) @ ( one_one @ nat ) @ ( plus_plus @ nat @ ( one_one @ nat ) @ ( if @ nat @ ( ord_less @ nat @ Ma @ X2 ) @ ( one_one @ nat ) @ ( plus_plus @ nat @ ( numeral_numeral @ nat @ ( bit1 @ ( bit0 @ ( bit0 @ one2 ) ) ) ) @ ( if @ nat @ ( ord_less @ nat @ ( vEBT_VEBT_high @ X2 @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( 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 @ X2 @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( vEBT_VEBT_low @ X2 @ ( 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.simps(5)
thf(fact_634_Diff__eq__empty__iff,axiom,
! [A: $tType,A2: set @ A,B2: set @ A] :
( ( ( minus_minus @ ( set @ A ) @ A2 @ B2 )
= ( bot_bot @ ( set @ A ) ) )
= ( ord_less_eq @ ( set @ A ) @ A2 @ B2 ) ) ).
% Diff_eq_empty_iff
thf(fact_635_subsetI,axiom,
! [A: $tType,A2: set @ A,B2: set @ A] :
( ! [X4: A] :
( ( member @ A @ X4 @ A2 )
=> ( member @ A @ X4 @ B2 ) )
=> ( ord_less_eq @ ( set @ A ) @ A2 @ B2 ) ) ).
% subsetI
thf(fact_636_psubsetI,axiom,
! [A: $tType,A2: set @ A,B2: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ A2 @ B2 )
=> ( ( A2 != B2 )
=> ( ord_less @ ( set @ A ) @ A2 @ B2 ) ) ) ).
% psubsetI
thf(fact_637_subset__antisym,axiom,
! [A: $tType,A2: set @ A,B2: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ A2 @ B2 )
=> ( ( ord_less_eq @ ( set @ A ) @ B2 @ A2 )
=> ( A2 = B2 ) ) ) ).
% subset_antisym
thf(fact_638_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_639_DiffI,axiom,
! [A: $tType,C3: A,A2: set @ A,B2: set @ A] :
( ( member @ A @ C3 @ A2 )
=> ( ~ ( member @ A @ C3 @ B2 )
=> ( member @ A @ C3 @ ( minus_minus @ ( set @ A ) @ A2 @ B2 ) ) ) ) ).
% DiffI
thf(fact_640_Diff__iff,axiom,
! [A: $tType,C3: A,A2: set @ A,B2: set @ A] :
( ( member @ A @ C3 @ ( minus_minus @ ( set @ A ) @ A2 @ B2 ) )
= ( ( member @ A @ C3 @ A2 )
& ~ ( member @ A @ C3 @ B2 ) ) ) ).
% Diff_iff
thf(fact_641_Diff__idemp,axiom,
! [A: $tType,A2: set @ A,B2: set @ A] :
( ( minus_minus @ ( set @ A ) @ ( minus_minus @ ( set @ A ) @ A2 @ B2 ) @ B2 )
= ( minus_minus @ ( set @ A ) @ A2 @ B2 ) ) ).
% Diff_idemp
thf(fact_642_time__replicate,axiom,
! [A: $tType,X2: heap_Time_Heap @ A,C3: nat,N: nat,H2: heap_ext @ product_unit] :
( ! [H3: heap_ext @ product_unit] : ( ord_less_eq @ nat @ ( time_time @ A @ X2 @ H3 ) @ C3 )
=> ( ord_less_eq @ nat @ ( time_time @ ( list @ A ) @ ( vEBT_VEBT_replicatei @ A @ N @ X2 ) @ H2 ) @ ( plus_plus @ nat @ ( one_one @ nat ) @ ( times_times @ nat @ ( plus_plus @ nat @ ( one_one @ nat ) @ C3 ) @ N ) ) ) ) ).
% time_replicate
thf(fact_643_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_644_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_645_subset__empty,axiom,
! [A: $tType,A2: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ A2 @ ( bot_bot @ ( set @ A ) ) )
= ( A2
= ( bot_bot @ ( set @ A ) ) ) ) ).
% subset_empty
thf(fact_646_empty__subsetI,axiom,
! [A: $tType,A2: set @ A] : ( ord_less_eq @ ( set @ A ) @ ( bot_bot @ ( set @ A ) ) @ A2 ) ).
% empty_subsetI
thf(fact_647_Diff__empty,axiom,
! [A: $tType,A2: set @ A] :
( ( minus_minus @ ( set @ A ) @ A2 @ ( bot_bot @ ( set @ A ) ) )
= A2 ) ).
% Diff_empty
thf(fact_648_empty__Diff,axiom,
! [A: $tType,A2: set @ A] :
( ( minus_minus @ ( set @ A ) @ ( bot_bot @ ( set @ A ) ) @ A2 )
= ( bot_bot @ ( set @ A ) ) ) ).
% empty_Diff
thf(fact_649_Diff__cancel,axiom,
! [A: $tType,A2: set @ A] :
( ( minus_minus @ ( set @ A ) @ A2 @ A2 )
= ( bot_bot @ ( set @ A ) ) ) ).
% Diff_cancel
thf(fact_650_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_651_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_652_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_653_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_654_of__nat__numeral,axiom,
! [A: $tType] :
( ( semiring_1 @ A )
=> ! [N: num] :
( ( semiring_1_of_nat @ A @ ( numeral_numeral @ nat @ N ) )
= ( numeral_numeral @ A @ N ) ) ) ).
% of_nat_numeral
thf(fact_655_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_656_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_657_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_658_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_659_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_660_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_661_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_662_of__nat__power__eq__of__nat__cancel__iff,axiom,
! [A: $tType] :
( ( semiring_char_0 @ A )
=> ! [X2: nat,B3: nat,W: nat] :
( ( ( semiring_1_of_nat @ A @ X2 )
= ( power_power @ A @ ( semiring_1_of_nat @ A @ B3 ) @ W ) )
= ( X2
= ( power_power @ nat @ B3 @ W ) ) ) ) ).
% of_nat_power_eq_of_nat_cancel_iff
thf(fact_663_of__nat__eq__of__nat__power__cancel__iff,axiom,
! [A: $tType] :
( ( semiring_char_0 @ A )
=> ! [B3: nat,W: nat,X2: nat] :
( ( ( power_power @ A @ ( semiring_1_of_nat @ A @ B3 ) @ W )
= ( semiring_1_of_nat @ A @ X2 ) )
= ( ( power_power @ nat @ B3 @ W )
= X2 ) ) ) ).
% of_nat_eq_of_nat_power_cancel_iff
thf(fact_664_semiring__1__class_Oof__nat__power,axiom,
! [A: $tType] :
( ( semiring_1 @ A )
=> ! [M: nat,N: nat] :
( ( semiring_1_of_nat @ A @ ( power_power @ nat @ M @ N ) )
= ( power_power @ A @ ( semiring_1_of_nat @ A @ M ) @ N ) ) ) ).
% semiring_1_class.of_nat_power
thf(fact_665_zle__diff1__eq,axiom,
! [W: int,Z: int] :
( ( ord_less_eq @ int @ W @ ( minus_minus @ int @ Z @ ( one_one @ int ) ) )
= ( ord_less @ int @ W @ Z ) ) ).
% zle_diff1_eq
thf(fact_666_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_667_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_668_numeral__power__eq__of__nat__cancel__iff,axiom,
! [A: $tType] :
( ( semiring_char_0 @ A )
=> ! [X2: num,N: nat,Y3: nat] :
( ( ( power_power @ A @ ( numeral_numeral @ A @ X2 ) @ N )
= ( semiring_1_of_nat @ A @ Y3 ) )
= ( ( power_power @ nat @ ( numeral_numeral @ nat @ X2 ) @ N )
= Y3 ) ) ) ).
% numeral_power_eq_of_nat_cancel_iff
thf(fact_669_real__of__nat__eq__numeral__power__cancel__iff,axiom,
! [A: $tType] :
( ( semiring_char_0 @ A )
=> ! [Y3: nat,X2: num,N: nat] :
( ( ( semiring_1_of_nat @ A @ Y3 )
= ( power_power @ A @ ( numeral_numeral @ A @ X2 ) @ N ) )
= ( Y3
= ( power_power @ nat @ ( numeral_numeral @ nat @ X2 ) @ N ) ) ) ) ).
% real_of_nat_eq_numeral_power_cancel_iff
thf(fact_670_of__nat__less__of__nat__power__cancel__iff,axiom,
! [A: $tType] :
( ( linordered_semidom @ A )
=> ! [B3: nat,W: nat,X2: nat] :
( ( ord_less @ A @ ( power_power @ A @ ( semiring_1_of_nat @ A @ B3 ) @ W ) @ ( semiring_1_of_nat @ A @ X2 ) )
= ( ord_less @ nat @ ( power_power @ nat @ B3 @ W ) @ X2 ) ) ) ).
% of_nat_less_of_nat_power_cancel_iff
thf(fact_671_of__nat__power__less__of__nat__cancel__iff,axiom,
! [A: $tType] :
( ( linordered_semidom @ A )
=> ! [X2: nat,B3: nat,W: nat] :
( ( ord_less @ A @ ( semiring_1_of_nat @ A @ X2 ) @ ( power_power @ A @ ( semiring_1_of_nat @ A @ B3 ) @ W ) )
= ( ord_less @ nat @ X2 @ ( power_power @ nat @ B3 @ W ) ) ) ) ).
% of_nat_power_less_of_nat_cancel_iff
thf(fact_672_of__nat__power__le__of__nat__cancel__iff,axiom,
! [A: $tType] :
( ( linordered_semidom @ A )
=> ! [X2: nat,B3: nat,W: nat] :
( ( ord_less_eq @ A @ ( semiring_1_of_nat @ A @ X2 ) @ ( power_power @ A @ ( semiring_1_of_nat @ A @ B3 ) @ W ) )
= ( ord_less_eq @ nat @ X2 @ ( power_power @ nat @ B3 @ W ) ) ) ) ).
% of_nat_power_le_of_nat_cancel_iff
thf(fact_673_of__nat__le__of__nat__power__cancel__iff,axiom,
! [A: $tType] :
( ( linordered_semidom @ A )
=> ! [B3: nat,W: nat,X2: nat] :
( ( ord_less_eq @ A @ ( power_power @ A @ ( semiring_1_of_nat @ A @ B3 ) @ W ) @ ( semiring_1_of_nat @ A @ X2 ) )
= ( ord_less_eq @ nat @ ( power_power @ nat @ B3 @ W ) @ X2 ) ) ) ).
% of_nat_le_of_nat_power_cancel_iff
thf(fact_674_numeral__less__real__of__nat__iff,axiom,
! [W: num,N: nat] :
( ( ord_less @ real @ ( numeral_numeral @ real @ W ) @ ( semiring_1_of_nat @ real @ N ) )
= ( ord_less @ nat @ ( numeral_numeral @ nat @ W ) @ N ) ) ).
% numeral_less_real_of_nat_iff
thf(fact_675_real__of__nat__less__numeral__iff,axiom,
! [N: nat,W: num] :
( ( ord_less @ real @ ( semiring_1_of_nat @ real @ N ) @ ( numeral_numeral @ real @ W ) )
= ( ord_less @ nat @ N @ ( numeral_numeral @ nat @ W ) ) ) ).
% real_of_nat_less_numeral_iff
thf(fact_676_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_677_numeral__power__less__of__nat__cancel__iff,axiom,
! [A: $tType] :
( ( linordered_semidom @ A )
=> ! [I: num,N: nat,X2: nat] :
( ( ord_less @ A @ ( power_power @ A @ ( numeral_numeral @ A @ I ) @ N ) @ ( semiring_1_of_nat @ A @ X2 ) )
= ( ord_less @ nat @ ( power_power @ nat @ ( numeral_numeral @ nat @ I ) @ N ) @ X2 ) ) ) ).
% numeral_power_less_of_nat_cancel_iff
thf(fact_678_of__nat__less__numeral__power__cancel__iff,axiom,
! [A: $tType] :
( ( linordered_semidom @ A )
=> ! [X2: nat,I: num,N: nat] :
( ( ord_less @ A @ ( semiring_1_of_nat @ A @ X2 ) @ ( power_power @ A @ ( numeral_numeral @ A @ I ) @ N ) )
= ( ord_less @ nat @ X2 @ ( power_power @ nat @ ( numeral_numeral @ nat @ I ) @ N ) ) ) ) ).
% of_nat_less_numeral_power_cancel_iff
thf(fact_679_of__nat__le__numeral__power__cancel__iff,axiom,
! [A: $tType] :
( ( linordered_semidom @ A )
=> ! [X2: nat,I: num,N: nat] :
( ( ord_less_eq @ A @ ( semiring_1_of_nat @ A @ X2 ) @ ( power_power @ A @ ( numeral_numeral @ A @ I ) @ N ) )
= ( ord_less_eq @ nat @ X2 @ ( power_power @ nat @ ( numeral_numeral @ nat @ I ) @ N ) ) ) ) ).
% of_nat_le_numeral_power_cancel_iff
thf(fact_680_numeral__power__le__of__nat__cancel__iff,axiom,
! [A: $tType] :
( ( linordered_semidom @ A )
=> ! [I: num,N: nat,X2: nat] :
( ( ord_less_eq @ A @ ( power_power @ A @ ( numeral_numeral @ A @ I ) @ N ) @ ( semiring_1_of_nat @ A @ X2 ) )
= ( ord_less_eq @ nat @ ( power_power @ nat @ ( numeral_numeral @ nat @ I ) @ N ) @ X2 ) ) ) ).
% numeral_power_le_of_nat_cancel_iff
thf(fact_681_TBOUND__vebt__memberi,axiom,
! [T2: vEBT_VEBT,Ti: vEBT_VEBTi,X2: nat] : ( time_TBOUND @ $o @ ( vEBT_V854960066525838166emberi @ T2 @ Ti @ X2 ) @ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ ( bit0 @ one2 ) ) ) @ ( plus_plus @ nat @ ( one_one @ nat ) @ ( vEBT_VEBT_height @ T2 ) ) ) ) ).
% TBOUND_vebt_memberi
thf(fact_682_TBOUND__vebt__predi,axiom,
! [T2: vEBT_VEBT,Ti: vEBT_VEBTi,X2: nat] : ( time_TBOUND @ ( option @ nat ) @ ( vEBT_VEBT_vebt_predi @ T2 @ Ti @ X2 ) @ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit1 @ ( bit1 @ one2 ) ) ) @ ( plus_plus @ nat @ ( one_one @ nat ) @ ( vEBT_VEBT_height @ T2 ) ) ) ) ).
% TBOUND_vebt_predi
thf(fact_683_TBOUND__vebt__succi,axiom,
! [T2: vEBT_VEBT,Ti: vEBT_VEBTi,X2: nat] : ( time_TBOUND @ ( option @ nat ) @ ( vEBT_VEBT_vebt_succi @ T2 @ Ti @ X2 ) @ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit1 @ ( bit1 @ one2 ) ) ) @ ( plus_plus @ nat @ ( one_one @ nat ) @ ( vEBT_VEBT_height @ T2 ) ) ) ) ).
% TBOUND_vebt_succi
thf(fact_684_htt__vebt__inserti,axiom,
! [T2: vEBT_VEBT,Ti: vEBT_VEBTi,X2: nat] : ( time_htt @ vEBT_VEBTi @ ( vEBT_vebt_assn_raw @ T2 @ Ti ) @ ( vEBT_vebt_inserti @ Ti @ X2 ) @ ( vEBT_vebt_assn_raw @ ( vEBT_vebt_insert @ T2 @ X2 ) ) @ ( 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_685_mult__of__nat__commute,axiom,
! [A: $tType] :
( ( semiring_1 @ A )
=> ! [X2: nat,Y3: A] :
( ( times_times @ A @ ( semiring_1_of_nat @ A @ X2 ) @ Y3 )
= ( times_times @ A @ Y3 @ ( semiring_1_of_nat @ A @ X2 ) ) ) ) ).
% mult_of_nat_commute
thf(fact_686_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_687_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_688_div__mult2__eq_H,axiom,
! [A: $tType] :
( ( euclid5411537665997757685th_nat @ A )
=> ! [A3: A,M: nat,N: nat] :
( ( divide_divide @ A @ A3 @ ( times_times @ A @ ( semiring_1_of_nat @ A @ M ) @ ( semiring_1_of_nat @ A @ N ) ) )
= ( divide_divide @ A @ ( divide_divide @ A @ A3 @ ( semiring_1_of_nat @ A @ M ) ) @ ( semiring_1_of_nat @ A @ N ) ) ) ) ).
% div_mult2_eq'
thf(fact_689_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_690_unique__euclidean__semiring__with__nat__class_Oof__nat__div,axiom,
! [A: $tType] :
( ( euclid5411537665997757685th_nat @ A )
=> ! [M: nat,N: nat] :
( ( semiring_1_of_nat @ A @ ( divide_divide @ nat @ M @ N ) )
= ( divide_divide @ A @ ( semiring_1_of_nat @ A @ M ) @ ( semiring_1_of_nat @ A @ N ) ) ) ) ).
% unique_euclidean_semiring_with_nat_class.of_nat_div
thf(fact_691_of__nat__max,axiom,
! [A: $tType] :
( ( linord181362715937106298miring @ A )
=> ! [X2: nat,Y3: nat] :
( ( semiring_1_of_nat @ A @ ( ord_max @ nat @ X2 @ Y3 ) )
= ( ord_max @ A @ ( semiring_1_of_nat @ A @ X2 ) @ ( semiring_1_of_nat @ A @ Y3 ) ) ) ) ).
% of_nat_max
thf(fact_692_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_693_real__of__nat__div4,axiom,
! [N: nat,X2: nat] : ( ord_less_eq @ real @ ( semiring_1_of_nat @ real @ ( divide_divide @ nat @ N @ X2 ) ) @ ( divide_divide @ real @ ( semiring_1_of_nat @ real @ N ) @ ( semiring_1_of_nat @ real @ X2 ) ) ) ).
% real_of_nat_div4
thf(fact_694_in__mono,axiom,
! [A: $tType,A2: set @ A,B2: set @ A,X2: A] :
( ( ord_less_eq @ ( set @ A ) @ A2 @ B2 )
=> ( ( member @ A @ X2 @ A2 )
=> ( member @ A @ X2 @ B2 ) ) ) ).
% in_mono
thf(fact_695_subsetD,axiom,
! [A: $tType,A2: set @ A,B2: set @ A,C3: A] :
( ( ord_less_eq @ ( set @ A ) @ A2 @ B2 )
=> ( ( member @ A @ C3 @ A2 )
=> ( member @ A @ C3 @ B2 ) ) ) ).
% subsetD
thf(fact_696_psubsetE,axiom,
! [A: $tType,A2: set @ A,B2: set @ A] :
( ( ord_less @ ( set @ A ) @ A2 @ B2 )
=> ~ ( ( ord_less_eq @ ( set @ A ) @ A2 @ B2 )
=> ( ord_less_eq @ ( set @ A ) @ B2 @ A2 ) ) ) ).
% psubsetE
thf(fact_697_equalityE,axiom,
! [A: $tType,A2: set @ A,B2: set @ A] :
( ( A2 = B2 )
=> ~ ( ( ord_less_eq @ ( set @ A ) @ A2 @ B2 )
=> ~ ( ord_less_eq @ ( set @ A ) @ B2 @ A2 ) ) ) ).
% equalityE
thf(fact_698_subset__eq,axiom,
! [A: $tType] :
( ( ord_less_eq @ ( set @ A ) )
= ( ^ [A7: set @ A,B6: set @ A] :
! [X3: A] :
( ( member @ A @ X3 @ A7 )
=> ( member @ A @ X3 @ B6 ) ) ) ) ).
% subset_eq
thf(fact_699_equalityD1,axiom,
! [A: $tType,A2: set @ A,B2: set @ A] :
( ( A2 = B2 )
=> ( ord_less_eq @ ( set @ A ) @ A2 @ B2 ) ) ).
% equalityD1
thf(fact_700_Set_OequalityD2,axiom,
! [A: $tType,A2: set @ A,B2: set @ A] :
( ( A2 = B2 )
=> ( ord_less_eq @ ( set @ A ) @ B2 @ A2 ) ) ).
% Set.equalityD2
thf(fact_701_psubset__eq,axiom,
! [A: $tType] :
( ( ord_less @ ( set @ A ) )
= ( ^ [A7: set @ A,B6: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ A7 @ B6 )
& ( A7 != B6 ) ) ) ) ).
% psubset_eq
thf(fact_702_subset__iff,axiom,
! [A: $tType] :
( ( ord_less_eq @ ( set @ A ) )
= ( ^ [A7: set @ A,B6: set @ A] :
! [T3: A] :
( ( member @ A @ T3 @ A7 )
=> ( member @ A @ T3 @ B6 ) ) ) ) ).
% subset_iff
thf(fact_703_subset__refl,axiom,
! [A: $tType,A2: set @ A] : ( ord_less_eq @ ( set @ A ) @ A2 @ A2 ) ).
% subset_refl
thf(fact_704_Collect__mono,axiom,
! [A: $tType,P: A > $o,Q: A > $o] :
( ! [X4: A] :
( ( P @ X4 )
=> ( Q @ X4 ) )
=> ( ord_less_eq @ ( set @ A ) @ ( collect @ A @ P ) @ ( collect @ A @ Q ) ) ) ).
% Collect_mono
thf(fact_705_subset__trans,axiom,
! [A: $tType,A2: set @ A,B2: set @ A,C2: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ A2 @ B2 )
=> ( ( ord_less_eq @ ( set @ A ) @ B2 @ C2 )
=> ( ord_less_eq @ ( set @ A ) @ A2 @ C2 ) ) ) ).
% subset_trans
thf(fact_706_set__eq__subset,axiom,
! [A: $tType] :
( ( ^ [Y6: set @ A,Z3: set @ A] : Y6 = Z3 )
= ( ^ [A7: set @ A,B6: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ A7 @ B6 )
& ( ord_less_eq @ ( set @ A ) @ B6 @ A7 ) ) ) ) ).
% set_eq_subset
thf(fact_707_Collect__mono__iff,axiom,
! [A: $tType,P: A > $o,Q: A > $o] :
( ( ord_less_eq @ ( set @ A ) @ ( collect @ A @ P ) @ ( collect @ A @ Q ) )
= ( ! [X3: A] :
( ( P @ X3 )
=> ( Q @ X3 ) ) ) ) ).
% Collect_mono_iff
thf(fact_708_psubset__imp__subset,axiom,
! [A: $tType,A2: set @ A,B2: set @ A] :
( ( ord_less @ ( set @ A ) @ A2 @ B2 )
=> ( ord_less_eq @ ( set @ A ) @ A2 @ B2 ) ) ).
% psubset_imp_subset
thf(fact_709_psubset__subset__trans,axiom,
! [A: $tType,A2: set @ A,B2: set @ A,C2: set @ A] :
( ( ord_less @ ( set @ A ) @ A2 @ B2 )
=> ( ( ord_less_eq @ ( set @ A ) @ B2 @ C2 )
=> ( ord_less @ ( set @ A ) @ A2 @ C2 ) ) ) ).
% psubset_subset_trans
thf(fact_710_subset__not__subset__eq,axiom,
! [A: $tType] :
( ( ord_less @ ( set @ A ) )
= ( ^ [A7: set @ A,B6: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ A7 @ B6 )
& ~ ( ord_less_eq @ ( set @ A ) @ B6 @ A7 ) ) ) ) ).
% subset_not_subset_eq
thf(fact_711_subset__psubset__trans,axiom,
! [A: $tType,A2: set @ A,B2: set @ A,C2: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ A2 @ B2 )
=> ( ( ord_less @ ( set @ A ) @ B2 @ C2 )
=> ( ord_less @ ( set @ A ) @ A2 @ C2 ) ) ) ).
% subset_psubset_trans
thf(fact_712_subset__iff__psubset__eq,axiom,
! [A: $tType] :
( ( ord_less_eq @ ( set @ A ) )
= ( ^ [A7: set @ A,B6: set @ A] :
( ( ord_less @ ( set @ A ) @ A7 @ B6 )
| ( A7 = B6 ) ) ) ) ).
% subset_iff_psubset_eq
thf(fact_713_int__less__induct,axiom,
! [I: int,K: int,P: int > $o] :
( ( ord_less @ int @ I @ K )
=> ( ( P @ ( minus_minus @ int @ K @ ( one_one @ int ) ) )
=> ( ! [I2: int] :
( ( ord_less @ int @ I2 @ K )
=> ( ( P @ I2 )
=> ( P @ ( minus_minus @ int @ I2 @ ( one_one @ int ) ) ) ) )
=> ( P @ I ) ) ) ) ).
% int_less_induct
thf(fact_714_psubset__imp__ex__mem,axiom,
! [A: $tType,A2: set @ A,B2: set @ A] :
( ( ord_less @ ( set @ A ) @ A2 @ B2 )
=> ? [B4: A] : ( member @ A @ B4 @ ( minus_minus @ ( set @ A ) @ B2 @ A2 ) ) ) ).
% psubset_imp_ex_mem
thf(fact_715_DiffE,axiom,
! [A: $tType,C3: A,A2: set @ A,B2: set @ A] :
( ( member @ A @ C3 @ ( minus_minus @ ( set @ A ) @ A2 @ B2 ) )
=> ~ ( ( member @ A @ C3 @ A2 )
=> ( member @ A @ C3 @ B2 ) ) ) ).
% DiffE
thf(fact_716_DiffD1,axiom,
! [A: $tType,C3: A,A2: set @ A,B2: set @ A] :
( ( member @ A @ C3 @ ( minus_minus @ ( set @ A ) @ A2 @ B2 ) )
=> ( member @ A @ C3 @ A2 ) ) ).
% DiffD1
thf(fact_717_DiffD2,axiom,
! [A: $tType,C3: A,A2: set @ A,B2: set @ A] :
( ( member @ A @ C3 @ ( minus_minus @ ( set @ A ) @ A2 @ B2 ) )
=> ~ ( member @ A @ C3 @ B2 ) ) ).
% DiffD2
thf(fact_718_int__distrib_I3_J,axiom,
! [Z1: int,Z22: int,W: int] :
( ( times_times @ int @ ( minus_minus @ int @ Z1 @ Z22 ) @ W )
= ( minus_minus @ int @ ( times_times @ int @ Z1 @ W ) @ ( times_times @ int @ Z22 @ W ) ) ) ).
% int_distrib(3)
thf(fact_719_int__distrib_I4_J,axiom,
! [W: int,Z1: int,Z22: int] :
( ( times_times @ int @ W @ ( minus_minus @ int @ Z1 @ Z22 ) )
= ( minus_minus @ int @ ( times_times @ int @ W @ Z1 ) @ ( times_times @ int @ W @ Z22 ) ) ) ).
% int_distrib(4)
thf(fact_720_nat__less__real__le,axiom,
( ( ord_less @ nat )
= ( ^ [N4: nat,M6: nat] : ( ord_less_eq @ real @ ( plus_plus @ real @ ( semiring_1_of_nat @ real @ N4 ) @ ( one_one @ real ) ) @ ( semiring_1_of_nat @ real @ M6 ) ) ) ) ).
% nat_less_real_le
thf(fact_721_nat__le__real__less,axiom,
( ( ord_less_eq @ nat )
= ( ^ [N4: nat,M6: nat] : ( ord_less @ real @ ( semiring_1_of_nat @ real @ N4 ) @ ( plus_plus @ real @ ( semiring_1_of_nat @ real @ M6 ) @ ( one_one @ real ) ) ) ) ) ).
% nat_le_real_less
thf(fact_722_Set_Oempty__def,axiom,
! [A: $tType] :
( ( bot_bot @ ( set @ A ) )
= ( collect @ A
@ ^ [X3: A] : $false ) ) ).
% Set.empty_def
thf(fact_723_Collect__subset,axiom,
! [A: $tType,A2: set @ A,P: A > $o] :
( ord_less_eq @ ( set @ A )
@ ( collect @ A
@ ^ [X3: A] :
( ( member @ A @ X3 @ A2 )
& ( P @ X3 ) ) )
@ A2 ) ).
% Collect_subset
thf(fact_724_less__eq__set__def,axiom,
! [A: $tType] :
( ( ord_less_eq @ ( set @ A ) )
= ( ^ [A7: set @ A,B6: set @ A] :
( ord_less_eq @ ( A > $o )
@ ^ [X3: A] : ( member @ A @ X3 @ A7 )
@ ^ [X3: A] : ( member @ A @ X3 @ B6 ) ) ) ) ).
% less_eq_set_def
thf(fact_725_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_726_set__diff__eq,axiom,
! [A: $tType] :
( ( minus_minus @ ( set @ A ) )
= ( ^ [A7: set @ A,B6: set @ A] :
( collect @ A
@ ^ [X3: A] :
( ( member @ A @ X3 @ A7 )
& ~ ( member @ A @ X3 @ B6 ) ) ) ) ) ).
% set_diff_eq
thf(fact_727_minus__set__def,axiom,
! [A: $tType] :
( ( minus_minus @ ( set @ A ) )
= ( ^ [A7: set @ A,B6: set @ A] :
( collect @ A
@ ( minus_minus @ ( A > $o )
@ ^ [X3: A] : ( member @ A @ X3 @ A7 )
@ ^ [X3: A] : ( member @ A @ X3 @ B6 ) ) ) ) ) ).
% minus_set_def
thf(fact_728_real__of__nat__div3,axiom,
! [N: nat,X2: nat] : ( ord_less_eq @ real @ ( minus_minus @ real @ ( divide_divide @ real @ ( semiring_1_of_nat @ real @ N ) @ ( semiring_1_of_nat @ real @ X2 ) ) @ ( semiring_1_of_nat @ real @ ( divide_divide @ nat @ N @ X2 ) ) ) @ ( one_one @ real ) ) ).
% real_of_nat_div3
thf(fact_729_double__diff,axiom,
! [A: $tType,A2: set @ A,B2: set @ A,C2: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ A2 @ B2 )
=> ( ( ord_less_eq @ ( set @ A ) @ B2 @ C2 )
=> ( ( minus_minus @ ( set @ A ) @ B2 @ ( minus_minus @ ( set @ A ) @ C2 @ A2 ) )
= A2 ) ) ) ).
% double_diff
thf(fact_730_Diff__subset,axiom,
! [A: $tType,A2: set @ A,B2: set @ A] : ( ord_less_eq @ ( set @ A ) @ ( minus_minus @ ( set @ A ) @ A2 @ B2 ) @ A2 ) ).
% Diff_subset
thf(fact_731_Diff__mono,axiom,
! [A: $tType,A2: set @ A,C2: set @ A,D: set @ A,B2: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ A2 @ C2 )
=> ( ( ord_less_eq @ ( set @ A ) @ D @ B2 )
=> ( ord_less_eq @ ( set @ A ) @ ( minus_minus @ ( set @ A ) @ A2 @ B2 ) @ ( minus_minus @ ( set @ A ) @ C2 @ D ) ) ) ) ).
% Diff_mono
thf(fact_732_int__le__induct,axiom,
! [I: int,K: int,P: int > $o] :
( ( ord_less_eq @ int @ I @ K )
=> ( ( P @ K )
=> ( ! [I2: int] :
( ( ord_less_eq @ int @ I2 @ K )
=> ( ( P @ I2 )
=> ( P @ ( minus_minus @ int @ I2 @ ( one_one @ int ) ) ) ) )
=> ( P @ I ) ) ) ) ).
% int_le_induct
thf(fact_733_zless__add1__eq,axiom,
! [W: int,Z: int] :
( ( ord_less @ int @ W @ ( plus_plus @ int @ Z @ ( one_one @ int ) ) )
= ( ( ord_less @ int @ W @ Z )
| ( W = Z ) ) ) ).
% zless_add1_eq
thf(fact_734_int__gr__induct,axiom,
! [K: int,I: int,P: int > $o] :
( ( ord_less @ int @ K @ I )
=> ( ( P @ ( plus_plus @ int @ K @ ( one_one @ int ) ) )
=> ( ! [I2: int] :
( ( ord_less @ int @ K @ I2 )
=> ( ( P @ I2 )
=> ( P @ ( plus_plus @ int @ I2 @ ( one_one @ int ) ) ) ) )
=> ( P @ I ) ) ) ) ).
% int_gr_induct
thf(fact_735_int__distrib_I1_J,axiom,
! [Z1: int,Z22: int,W: int] :
( ( times_times @ int @ ( plus_plus @ int @ Z1 @ Z22 ) @ W )
= ( plus_plus @ int @ ( times_times @ int @ Z1 @ W ) @ ( times_times @ int @ Z22 @ W ) ) ) ).
% int_distrib(1)
thf(fact_736_int__distrib_I2_J,axiom,
! [W: int,Z1: int,Z22: int] :
( ( times_times @ int @ W @ ( plus_plus @ int @ Z1 @ Z22 ) )
= ( plus_plus @ int @ ( times_times @ int @ W @ Z1 ) @ ( times_times @ int @ W @ Z22 ) ) ) ).
% int_distrib(2)
thf(fact_737_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,
! [Uu: nat,Uv: list @ vEBT_VEBT,Uw: vEBT_VEBT,X2: nat] :
( ( vEBT_T_m_e_m_b_e_r @ ( vEBT_Node @ ( none @ ( product_prod @ nat @ nat ) ) @ Uu @ Uv @ Uw ) @ X2 )
= ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ).
% T\<^sub>m\<^sub>e\<^sub>m\<^sub>b\<^sub>e\<^sub>r.simps(2)
thf(fact_738_add1__zle__eq,axiom,
! [W: int,Z: int] :
( ( ord_less_eq @ int @ ( plus_plus @ int @ W @ ( one_one @ int ) ) @ Z )
= ( ord_less @ int @ W @ Z ) ) ).
% add1_zle_eq
thf(fact_739_int__ge__induct,axiom,
! [K: int,I: int,P: int > $o] :
( ( ord_less_eq @ int @ K @ I )
=> ( ( P @ K )
=> ( ! [I2: int] :
( ( ord_less_eq @ int @ K @ I2 )
=> ( ( P @ I2 )
=> ( P @ ( plus_plus @ int @ I2 @ ( one_one @ int ) ) ) ) )
=> ( P @ I ) ) ) ) ).
% int_ge_induct
thf(fact_740_zless__imp__add1__zle,axiom,
! [W: int,Z: int] :
( ( ord_less @ int @ W @ Z )
=> ( ord_less_eq @ int @ ( plus_plus @ int @ W @ ( one_one @ int ) ) @ Z ) ) ).
% zless_imp_add1_zle
thf(fact_741_int__induct,axiom,
! [P: int > $o,K: int,I: int] :
( ( P @ K )
=> ( ! [I2: int] :
( ( ord_less_eq @ int @ K @ I2 )
=> ( ( P @ I2 )
=> ( P @ ( plus_plus @ int @ I2 @ ( one_one @ int ) ) ) ) )
=> ( ! [I2: int] :
( ( ord_less_eq @ int @ I2 @ K )
=> ( ( P @ I2 )
=> ( P @ ( minus_minus @ int @ I2 @ ( one_one @ int ) ) ) ) )
=> ( P @ I ) ) ) ) ).
% int_induct
thf(fact_742_member__bound__height,axiom,
! [T2: vEBT_VEBT,N: nat,X2: nat] :
( ( vEBT_invar_vebt @ T2 @ N )
=> ( ord_less_eq @ nat @ ( vEBT_T_m_e_m_b_e_r @ T2 @ X2 ) @ ( 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_743_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_744_cnt__cnt__eq,axiom,
( vEBT_VEBT_cnt
= ( ^ [T3: vEBT_VEBT] : ( semiring_1_of_nat @ real @ ( vEBT_VEBT_cnt2 @ T3 ) ) ) ) ).
% cnt_cnt_eq
thf(fact_745_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_746_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_747_delete__bound__size__univ,axiom,
! [T2: vEBT_VEBT,N: nat,U: real,X2: 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 @ X2 ) ) @ ( 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_748_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_749_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_750_delete__bound__size__univ_H,axiom,
! [T2: vEBT_VEBT,N: nat,U: real,X2: 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 @ X2 ) ) @ ( 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_751_vebt__minti_Ocases,axiom,
! [X2: vEBT_VEBTi] :
( ! [A4: $o,B4: $o] :
( X2
!= ( vEBT_Leafi @ A4 @ B4 ) )
=> ( ! [Uu2: nat,Uv2: array @ vEBT_VEBTi,Uw2: vEBT_VEBTi] :
( X2
!= ( vEBT_Nodei @ ( none @ ( product_prod @ nat @ nat ) ) @ Uu2 @ Uv2 @ Uw2 ) )
=> ~ ! [Mi2: nat,Ma2: nat,Ux2: nat,Uy2: array @ vEBT_VEBTi,Uz2: vEBT_VEBTi] :
( X2
!= ( vEBT_Nodei @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi2 @ Ma2 ) ) @ Ux2 @ Uy2 @ Uz2 ) ) ) ) ).
% vebt_minti.cases
thf(fact_752_valid__0__not,axiom,
! [T2: vEBT_VEBT] :
~ ( vEBT_invar_vebt @ T2 @ ( zero_zero @ nat ) ) ).
% valid_0_not
thf(fact_753_valid__tree__deg__neq__0,axiom,
! [T2: vEBT_VEBT] :
~ ( vEBT_invar_vebt @ T2 @ ( zero_zero @ nat ) ) ).
% valid_tree_deg_neq_0
thf(fact_754_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_755_T__vebt__buildupi__gq__0,axiom,
! [N: nat] : ( ord_less @ nat @ ( zero_zero @ nat ) @ ( vEBT_V441764108873111860ildupi @ N ) ) ).
% T_vebt_buildupi_gq_0
thf(fact_756_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_757_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_758_double__eq__0__iff,axiom,
! [A: $tType] :
( ( linord5086331880401160121up_add @ A )
=> ! [A3: A] :
( ( ( plus_plus @ A @ A3 @ A3 )
= ( zero_zero @ A ) )
= ( A3
= ( zero_zero @ A ) ) ) ) ).
% double_eq_0_iff
thf(fact_759_mult__cancel__right,axiom,
! [A: $tType] :
( ( semiri6575147826004484403cancel @ A )
=> ! [A3: A,C3: A,B3: A] :
( ( ( times_times @ A @ A3 @ C3 )
= ( times_times @ A @ B3 @ C3 ) )
= ( ( C3
= ( zero_zero @ A ) )
| ( A3 = B3 ) ) ) ) ).
% mult_cancel_right
thf(fact_760_mult__cancel__left,axiom,
! [A: $tType] :
( ( semiri6575147826004484403cancel @ A )
=> ! [C3: A,A3: A,B3: A] :
( ( ( times_times @ A @ C3 @ A3 )
= ( times_times @ A @ C3 @ B3 ) )
= ( ( C3
= ( zero_zero @ A ) )
| ( A3 = B3 ) ) ) ) ).
% mult_cancel_left
thf(fact_761_mult__eq__0__iff,axiom,
! [A: $tType] :
( ( semiri3467727345109120633visors @ A )
=> ! [A3: A,B3: A] :
( ( ( times_times @ A @ A3 @ B3 )
= ( zero_zero @ A ) )
= ( ( A3
= ( zero_zero @ A ) )
| ( B3
= ( zero_zero @ A ) ) ) ) ) ).
% mult_eq_0_iff
thf(fact_762_mult__zero__right,axiom,
! [A: $tType] :
( ( mult_zero @ A )
=> ! [A3: A] :
( ( times_times @ A @ A3 @ ( zero_zero @ A ) )
= ( zero_zero @ A ) ) ) ).
% mult_zero_right
thf(fact_763_mult__zero__left,axiom,
! [A: $tType] :
( ( mult_zero @ A )
=> ! [A3: A] :
( ( times_times @ A @ ( zero_zero @ A ) @ A3 )
= ( zero_zero @ A ) ) ) ).
% mult_zero_left
thf(fact_764_bits__div__by__0,axiom,
! [A: $tType] :
( ( bit_semiring_bits @ A )
=> ! [A3: A] :
( ( divide_divide @ A @ A3 @ ( zero_zero @ A ) )
= ( zero_zero @ A ) ) ) ).
% bits_div_by_0
thf(fact_765_bits__div__0,axiom,
! [A: $tType] :
( ( bit_semiring_bits @ A )
=> ! [A3: A] :
( ( divide_divide @ A @ ( zero_zero @ A ) @ A3 )
= ( zero_zero @ A ) ) ) ).
% bits_div_0
thf(fact_766_div__by__0,axiom,
! [A: $tType] :
( ( semidom_divide @ A )
=> ! [A3: A] :
( ( divide_divide @ A @ A3 @ ( zero_zero @ A ) )
= ( zero_zero @ A ) ) ) ).
% div_by_0
thf(fact_767_div__0,axiom,
! [A: $tType] :
( ( semidom_divide @ A )
=> ! [A3: A] :
( ( divide_divide @ A @ ( zero_zero @ A ) @ A3 )
= ( zero_zero @ A ) ) ) ).
% div_0
thf(fact_768_less__nat__zero__code,axiom,
! [N: nat] :
~ ( ord_less @ nat @ N @ ( zero_zero @ nat ) ) ).
% less_nat_zero_code
thf(fact_769_neq0__conv,axiom,
! [N: nat] :
( ( N
!= ( zero_zero @ nat ) )
= ( ord_less @ nat @ ( zero_zero @ nat ) @ N ) ) ).
% neq0_conv
thf(fact_770_bot__nat__0_Onot__eq__extremum,axiom,
! [A3: nat] :
( ( A3
!= ( zero_zero @ nat ) )
= ( ord_less @ nat @ ( zero_zero @ nat ) @ A3 ) ) ).
% bot_nat_0.not_eq_extremum
thf(fact_771_Nat_Oadd__0__right,axiom,
! [M: nat] :
( ( plus_plus @ nat @ M @ ( zero_zero @ nat ) )
= M ) ).
% Nat.add_0_right
thf(fact_772_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_773_bot__nat__0_Oextremum,axiom,
! [A3: nat] : ( ord_less_eq @ nat @ ( zero_zero @ nat ) @ A3 ) ).
% bot_nat_0.extremum
thf(fact_774_le0,axiom,
! [N: nat] : ( ord_less_eq @ nat @ ( zero_zero @ nat ) @ N ) ).
% le0
thf(fact_775_int__eq__iff__numeral,axiom,
! [M: nat,V: num] :
( ( ( semiring_1_of_nat @ int @ M )
= ( numeral_numeral @ int @ V ) )
= ( M
= ( numeral_numeral @ nat @ V ) ) ) ).
% int_eq_iff_numeral
thf(fact_776_diff__self__eq__0,axiom,
! [M: nat] :
( ( minus_minus @ nat @ M @ M )
= ( zero_zero @ nat ) ) ).
% diff_self_eq_0
thf(fact_777_diff__0__eq__0,axiom,
! [N: nat] :
( ( minus_minus @ nat @ ( zero_zero @ nat ) @ N )
= ( zero_zero @ nat ) ) ).
% diff_0_eq_0
thf(fact_778_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_779_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_780_mult__0__right,axiom,
! [M: nat] :
( ( times_times @ nat @ M @ ( zero_zero @ nat ) )
= ( zero_zero @ nat ) ) ).
% mult_0_right
thf(fact_781_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_782_max__0R,axiom,
! [N: nat] :
( ( ord_max @ nat @ N @ ( zero_zero @ nat ) )
= N ) ).
% max_0R
thf(fact_783_max__0L,axiom,
! [N: nat] :
( ( ord_max @ nat @ ( zero_zero @ nat ) @ N )
= N ) ).
% max_0L
thf(fact_784_max__nat_Oright__neutral,axiom,
! [A3: nat] :
( ( ord_max @ nat @ A3 @ ( zero_zero @ nat ) )
= A3 ) ).
% max_nat.right_neutral
thf(fact_785_max__nat_Oneutr__eq__iff,axiom,
! [A3: nat,B3: nat] :
( ( ( zero_zero @ nat )
= ( ord_max @ nat @ A3 @ B3 ) )
= ( ( A3
= ( zero_zero @ nat ) )
& ( B3
= ( zero_zero @ nat ) ) ) ) ).
% max_nat.neutr_eq_iff
thf(fact_786_max__nat_Oleft__neutral,axiom,
! [A3: nat] :
( ( ord_max @ nat @ ( zero_zero @ nat ) @ A3 )
= A3 ) ).
% max_nat.left_neutral
thf(fact_787_max__nat_Oeq__neutr__iff,axiom,
! [A3: nat,B3: nat] :
( ( ( ord_max @ nat @ A3 @ B3 )
= ( zero_zero @ nat ) )
= ( ( A3
= ( zero_zero @ nat ) )
& ( B3
= ( zero_zero @ nat ) ) ) ) ).
% max_nat.eq_neutr_iff
thf(fact_788_zero__comp__diff__simps_I1_J,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [A3: A,B3: A] :
( ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( minus_minus @ A @ A3 @ B3 ) )
= ( ord_less_eq @ A @ B3 @ A3 ) ) ) ).
% zero_comp_diff_simps(1)
thf(fact_789_zero__comp__diff__simps_I2_J,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [A3: A,B3: A] :
( ( ord_less @ A @ ( zero_zero @ A ) @ ( minus_minus @ A @ A3 @ B3 ) )
= ( ord_less @ A @ B3 @ A3 ) ) ) ).
% zero_comp_diff_simps(2)
thf(fact_790_sum__squares__eq__zero__iff,axiom,
! [A: $tType] :
( ( linord4710134922213307826strict @ A )
=> ! [X2: A,Y3: A] :
( ( ( plus_plus @ A @ ( times_times @ A @ X2 @ X2 ) @ ( times_times @ A @ Y3 @ Y3 ) )
= ( zero_zero @ A ) )
= ( ( X2
= ( zero_zero @ A ) )
& ( Y3
= ( zero_zero @ A ) ) ) ) ) ).
% sum_squares_eq_zero_iff
thf(fact_791_mult__cancel__left1,axiom,
! [A: $tType] :
( ( ring_15535105094025558882visors @ A )
=> ! [C3: A,B3: A] :
( ( C3
= ( times_times @ A @ C3 @ B3 ) )
= ( ( C3
= ( zero_zero @ A ) )
| ( B3
= ( one_one @ A ) ) ) ) ) ).
% mult_cancel_left1
thf(fact_792_mult__cancel__left2,axiom,
! [A: $tType] :
( ( ring_15535105094025558882visors @ A )
=> ! [C3: A,A3: A] :
( ( ( times_times @ A @ C3 @ A3 )
= C3 )
= ( ( C3
= ( zero_zero @ A ) )
| ( A3
= ( one_one @ A ) ) ) ) ) ).
% mult_cancel_left2
thf(fact_793_mult__cancel__right1,axiom,
! [A: $tType] :
( ( ring_15535105094025558882visors @ A )
=> ! [C3: A,B3: A] :
( ( C3
= ( times_times @ A @ B3 @ C3 ) )
= ( ( C3
= ( zero_zero @ A ) )
| ( B3
= ( one_one @ A ) ) ) ) ) ).
% mult_cancel_right1
thf(fact_794_mult__cancel__right2,axiom,
! [A: $tType] :
( ( ring_15535105094025558882visors @ A )
=> ! [A3: A,C3: A] :
( ( ( times_times @ A @ A3 @ C3 )
= C3 )
= ( ( C3
= ( zero_zero @ A ) )
| ( A3
= ( one_one @ A ) ) ) ) ) ).
% mult_cancel_right2
thf(fact_795_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_796_div__self,axiom,
! [A: $tType] :
( ( semidom_divide @ A )
=> ! [A3: A] :
( ( A3
!= ( zero_zero @ A ) )
=> ( ( divide_divide @ A @ A3 @ A3 )
= ( one_one @ A ) ) ) ) ).
% div_self
thf(fact_797_nonzero__mult__div__cancel__right,axiom,
! [A: $tType] :
( ( semidom_divide @ A )
=> ! [B3: A,A3: A] :
( ( B3
!= ( zero_zero @ A ) )
=> ( ( divide_divide @ A @ ( times_times @ A @ A3 @ B3 ) @ B3 )
= A3 ) ) ) ).
% nonzero_mult_div_cancel_right
thf(fact_798_nonzero__mult__div__cancel__left,axiom,
! [A: $tType] :
( ( semidom_divide @ A )
=> ! [A3: A,B3: A] :
( ( A3
!= ( zero_zero @ A ) )
=> ( ( divide_divide @ A @ ( times_times @ A @ A3 @ B3 ) @ A3 )
= B3 ) ) ) ).
% nonzero_mult_div_cancel_left
thf(fact_799_div__mult__mult1__if,axiom,
! [A: $tType] :
( ( euclid4440199948858584721cancel @ A )
=> ! [C3: A,A3: A,B3: A] :
( ( ( C3
= ( zero_zero @ A ) )
=> ( ( divide_divide @ A @ ( times_times @ A @ C3 @ A3 ) @ ( times_times @ A @ C3 @ B3 ) )
= ( zero_zero @ A ) ) )
& ( ( C3
!= ( zero_zero @ A ) )
=> ( ( divide_divide @ A @ ( times_times @ A @ C3 @ A3 ) @ ( times_times @ A @ C3 @ B3 ) )
= ( divide_divide @ A @ A3 @ B3 ) ) ) ) ) ).
% div_mult_mult1_if
thf(fact_800_div__mult__mult2,axiom,
! [A: $tType] :
( ( euclid4440199948858584721cancel @ A )
=> ! [C3: A,A3: A,B3: A] :
( ( C3
!= ( zero_zero @ A ) )
=> ( ( divide_divide @ A @ ( times_times @ A @ A3 @ C3 ) @ ( times_times @ A @ B3 @ C3 ) )
= ( divide_divide @ A @ A3 @ B3 ) ) ) ) ).
% div_mult_mult2
thf(fact_801_div__mult__mult1,axiom,
! [A: $tType] :
( ( euclid4440199948858584721cancel @ A )
=> ! [C3: A,A3: A,B3: A] :
( ( C3
!= ( zero_zero @ A ) )
=> ( ( divide_divide @ A @ ( times_times @ A @ C3 @ A3 ) @ ( times_times @ A @ C3 @ B3 ) )
= ( divide_divide @ A @ A3 @ B3 ) ) ) ) ).
% div_mult_mult1
thf(fact_802_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_803_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_804_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_805_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_806_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_807_power__Suc0__right,axiom,
! [A: $tType] :
( ( monoid_mult @ A )
=> ! [A3: A] :
( ( power_power @ A @ A3 @ ( suc @ ( zero_zero @ nat ) ) )
= A3 ) ) ).
% power_Suc0_right
thf(fact_808_zero__less__Suc,axiom,
! [N: nat] : ( ord_less @ nat @ ( zero_zero @ nat ) @ ( suc @ N ) ) ).
% zero_less_Suc
thf(fact_809_less__Suc0,axiom,
! [N: nat] :
( ( ord_less @ nat @ N @ ( suc @ ( zero_zero @ nat ) ) )
= ( N
= ( zero_zero @ nat ) ) ) ).
% less_Suc0
thf(fact_810_div__by__Suc__0,axiom,
! [M: nat] :
( ( divide_divide @ nat @ M @ ( suc @ ( zero_zero @ nat ) ) )
= M ) ).
% div_by_Suc_0
thf(fact_811_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_812_less__one,axiom,
! [N: nat] :
( ( ord_less @ nat @ N @ ( one_one @ nat ) )
= ( N
= ( zero_zero @ nat ) ) ) ).
% less_one
thf(fact_813_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_814_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_815_div__less,axiom,
! [M: nat,N: nat] :
( ( ord_less @ nat @ M @ N )
=> ( ( divide_divide @ nat @ M @ N )
= ( zero_zero @ nat ) ) ) ).
% div_less
thf(fact_816_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_817_power__Suc__0,axiom,
! [N: nat] :
( ( power_power @ nat @ ( suc @ ( zero_zero @ nat ) ) @ N )
= ( suc @ ( zero_zero @ nat ) ) ) ).
% power_Suc_0
thf(fact_818_nat__power__eq__Suc__0__iff,axiom,
! [X2: nat,M: nat] :
( ( ( power_power @ nat @ X2 @ M )
= ( suc @ ( zero_zero @ nat ) ) )
= ( ( M
= ( zero_zero @ nat ) )
| ( X2
= ( suc @ ( zero_zero @ nat ) ) ) ) ) ).
% nat_power_eq_Suc_0_iff
thf(fact_819_max__0__1_I4_J,axiom,
! [A: $tType] :
( ( linord181362715937106298miring @ A )
=> ! [X2: num] :
( ( ord_max @ A @ ( numeral_numeral @ A @ X2 ) @ ( zero_zero @ A ) )
= ( numeral_numeral @ A @ X2 ) ) ) ).
% max_0_1(4)
thf(fact_820_max__0__1_I3_J,axiom,
! [A: $tType] :
( ( linord181362715937106298miring @ A )
=> ! [X2: num] :
( ( ord_max @ A @ ( zero_zero @ A ) @ ( numeral_numeral @ A @ X2 ) )
= ( numeral_numeral @ A @ X2 ) ) ) ).
% max_0_1(3)
thf(fact_821_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_822_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_823_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_824_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_825_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_826_nat__zero__less__power__iff,axiom,
! [X2: nat,N: nat] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ ( power_power @ nat @ X2 @ N ) )
= ( ( ord_less @ nat @ ( zero_zero @ nat ) @ X2 )
| ( N
= ( zero_zero @ nat ) ) ) ) ).
% nat_zero_less_power_iff
thf(fact_827_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_828_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_829_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_830_eq__divide__eq__numeral1_I1_J,axiom,
! [A: $tType] :
( ( division_ring @ A )
=> ! [A3: A,B3: A,W: num] :
( ( A3
= ( divide_divide @ A @ B3 @ ( numeral_numeral @ A @ W ) ) )
= ( ( ( ( numeral_numeral @ A @ W )
!= ( zero_zero @ A ) )
=> ( ( times_times @ A @ A3 @ ( numeral_numeral @ A @ W ) )
= B3 ) )
& ( ( ( numeral_numeral @ A @ W )
= ( zero_zero @ A ) )
=> ( A3
= ( zero_zero @ A ) ) ) ) ) ) ).
% eq_divide_eq_numeral1(1)
thf(fact_831_divide__eq__eq__numeral1_I1_J,axiom,
! [A: $tType] :
( ( division_ring @ A )
=> ! [B3: A,W: num,A3: A] :
( ( ( divide_divide @ A @ B3 @ ( numeral_numeral @ A @ W ) )
= A3 )
= ( ( ( ( numeral_numeral @ A @ W )
!= ( zero_zero @ A ) )
=> ( B3
= ( times_times @ A @ A3 @ ( numeral_numeral @ A @ W ) ) ) )
& ( ( ( numeral_numeral @ A @ W )
= ( zero_zero @ A ) )
=> ( A3
= ( zero_zero @ A ) ) ) ) ) ) ).
% divide_eq_eq_numeral1(1)
thf(fact_832_div__mult__self4,axiom,
! [A: $tType] :
( ( euclid4440199948858584721cancel @ A )
=> ! [B3: A,C3: A,A3: A] :
( ( B3
!= ( zero_zero @ A ) )
=> ( ( divide_divide @ A @ ( plus_plus @ A @ ( times_times @ A @ B3 @ C3 ) @ A3 ) @ B3 )
= ( plus_plus @ A @ C3 @ ( divide_divide @ A @ A3 @ B3 ) ) ) ) ) ).
% div_mult_self4
thf(fact_833_div__mult__self3,axiom,
! [A: $tType] :
( ( euclid4440199948858584721cancel @ A )
=> ! [B3: A,C3: A,A3: A] :
( ( B3
!= ( zero_zero @ A ) )
=> ( ( divide_divide @ A @ ( plus_plus @ A @ ( times_times @ A @ C3 @ B3 ) @ A3 ) @ B3 )
= ( plus_plus @ A @ C3 @ ( divide_divide @ A @ A3 @ B3 ) ) ) ) ) ).
% div_mult_self3
thf(fact_834_div__mult__self2,axiom,
! [A: $tType] :
( ( euclid4440199948858584721cancel @ A )
=> ! [B3: A,A3: A,C3: A] :
( ( B3
!= ( zero_zero @ A ) )
=> ( ( divide_divide @ A @ ( plus_plus @ A @ A3 @ ( times_times @ A @ B3 @ C3 ) ) @ B3 )
= ( plus_plus @ A @ C3 @ ( divide_divide @ A @ A3 @ B3 ) ) ) ) ) ).
% div_mult_self2
thf(fact_835_div__mult__self1,axiom,
! [A: $tType] :
( ( euclid4440199948858584721cancel @ A )
=> ! [B3: A,A3: A,C3: A] :
( ( B3
!= ( zero_zero @ A ) )
=> ( ( divide_divide @ A @ ( plus_plus @ A @ A3 @ ( times_times @ A @ C3 @ B3 ) ) @ B3 )
= ( plus_plus @ A @ C3 @ ( divide_divide @ A @ A3 @ B3 ) ) ) ) ) ).
% div_mult_self1
thf(fact_836_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_837_power__eq__0__iff,axiom,
! [A: $tType] :
( ( semiri2026040879449505780visors @ A )
=> ! [A3: A,N: nat] :
( ( ( power_power @ A @ A3 @ N )
= ( zero_zero @ A ) )
= ( ( A3
= ( zero_zero @ A ) )
& ( ord_less @ nat @ ( zero_zero @ nat ) @ N ) ) ) ) ).
% power_eq_0_iff
thf(fact_838_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_839_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_840_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_841_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_842_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_843_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_844_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_845_zero__eq__power2,axiom,
! [A: $tType] :
( ( semiri2026040879449505780visors @ A )
=> ! [A3: A] :
( ( ( power_power @ A @ A3 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) )
= ( zero_zero @ A ) )
= ( A3
= ( zero_zero @ A ) ) ) ) ).
% zero_eq_power2
thf(fact_846_power__strict__decreasing__iff,axiom,
! [A: $tType] :
( ( linordered_semidom @ A )
=> ! [B3: A,M: nat,N: nat] :
( ( ord_less @ A @ ( zero_zero @ A ) @ B3 )
=> ( ( ord_less @ A @ B3 @ ( one_one @ A ) )
=> ( ( ord_less @ A @ ( power_power @ A @ B3 @ M ) @ ( power_power @ A @ B3 @ N ) )
= ( ord_less @ nat @ N @ M ) ) ) ) ) ).
% power_strict_decreasing_iff
thf(fact_847_power__mono__iff,axiom,
! [A: $tType] :
( ( linordered_semidom @ A )
=> ! [A3: A,B3: A,N: nat] :
( ( ord_less_eq @ A @ ( zero_zero @ A ) @ A3 )
=> ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ B3 )
=> ( ( ord_less @ nat @ ( zero_zero @ nat ) @ N )
=> ( ( ord_less_eq @ A @ ( power_power @ A @ A3 @ N ) @ ( power_power @ A @ B3 @ N ) )
= ( ord_less_eq @ A @ A3 @ B3 ) ) ) ) ) ) ).
% power_mono_iff
thf(fact_848_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_849_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_850_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_851_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_852_power2__eq__iff__nonneg,axiom,
! [A: $tType] :
( ( linordered_semidom @ A )
=> ! [X2: A,Y3: A] :
( ( ord_less_eq @ A @ ( zero_zero @ A ) @ X2 )
=> ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ Y3 )
=> ( ( ( power_power @ A @ X2 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) )
= ( power_power @ A @ Y3 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) )
= ( X2 = Y3 ) ) ) ) ) ).
% power2_eq_iff_nonneg
thf(fact_853_power2__less__eq__zero__iff,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [A3: A] :
( ( ord_less_eq @ A @ ( power_power @ A @ A3 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ ( zero_zero @ A ) )
= ( A3
= ( zero_zero @ A ) ) ) ) ).
% power2_less_eq_zero_iff
thf(fact_854_zero__less__power2,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [A3: A] :
( ( ord_less @ A @ ( zero_zero @ A ) @ ( power_power @ A @ A3 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) )
= ( A3
!= ( zero_zero @ A ) ) ) ) ).
% zero_less_power2
thf(fact_855_sum__power2__eq__zero__iff,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [X2: A,Y3: A] :
( ( ( plus_plus @ A @ ( power_power @ A @ X2 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ ( power_power @ A @ Y3 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) )
= ( zero_zero @ A ) )
= ( ( X2
= ( zero_zero @ A ) )
& ( Y3
= ( zero_zero @ A ) ) ) ) ) ).
% sum_power2_eq_zero_iff
thf(fact_856_power__decreasing__iff,axiom,
! [A: $tType] :
( ( linordered_semidom @ A )
=> ! [B3: A,M: nat,N: nat] :
( ( ord_less @ A @ ( zero_zero @ A ) @ B3 )
=> ( ( ord_less @ A @ B3 @ ( one_one @ A ) )
=> ( ( ord_less_eq @ A @ ( power_power @ A @ B3 @ M ) @ ( power_power @ A @ B3 @ N ) )
= ( ord_less_eq @ nat @ N @ M ) ) ) ) ) ).
% power_decreasing_iff
thf(fact_857_of__nat__zero__less__power__iff,axiom,
! [A: $tType] :
( ( linordered_semidom @ A )
=> ! [X2: nat,N: nat] :
( ( ord_less @ A @ ( zero_zero @ A ) @ ( power_power @ A @ ( semiring_1_of_nat @ A @ X2 ) @ N ) )
= ( ( ord_less @ nat @ ( zero_zero @ nat ) @ X2 )
| ( N
= ( zero_zero @ nat ) ) ) ) ) ).
% of_nat_zero_less_power_iff
thf(fact_858_int__diff__cases,axiom,
! [Z: int] :
~ ! [M5: nat,N3: nat] :
( Z
!= ( minus_minus @ int @ ( semiring_1_of_nat @ int @ M5 ) @ ( semiring_1_of_nat @ int @ N3 ) ) ) ).
% int_diff_cases
thf(fact_859_option_Osize__neq,axiom,
! [A: $tType,X2: option @ A] :
( ( size_size @ ( option @ A ) @ X2 )
!= ( zero_zero @ nat ) ) ).
% option.size_neq
thf(fact_860_VEBT__internal_Oreplicatei_Ocases,axiom,
! [A: $tType,X2: product_prod @ nat @ ( heap_Time_Heap @ A )] :
( ! [X4: heap_Time_Heap @ A] :
( X2
!= ( product_Pair @ nat @ ( heap_Time_Heap @ A ) @ ( zero_zero @ nat ) @ X4 ) )
=> ~ ! [N3: nat,X4: heap_Time_Heap @ A] :
( X2
!= ( product_Pair @ nat @ ( heap_Time_Heap @ A ) @ ( suc @ N3 ) @ X4 ) ) ) ).
% VEBT_internal.replicatei.cases
thf(fact_861_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_862_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_863_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_864_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_865_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_866_zero__neq__numeral,axiom,
! [A: $tType] :
( ( semiring_char_0 @ A )
=> ! [N: num] :
( ( zero_zero @ A )
!= ( numeral_numeral @ A @ N ) ) ) ).
% zero_neq_numeral
thf(fact_867_zero__neq__one,axiom,
! [A: $tType] :
( ( zero_neq_one @ A )
=> ( ( zero_zero @ A )
!= ( one_one @ A ) ) ) ).
% zero_neq_one
thf(fact_868_mult__right__cancel,axiom,
! [A: $tType] :
( ( semiri6575147826004484403cancel @ A )
=> ! [C3: A,A3: A,B3: A] :
( ( C3
!= ( zero_zero @ A ) )
=> ( ( ( times_times @ A @ A3 @ C3 )
= ( times_times @ A @ B3 @ C3 ) )
= ( A3 = B3 ) ) ) ) ).
% mult_right_cancel
thf(fact_869_mult__left__cancel,axiom,
! [A: $tType] :
( ( semiri6575147826004484403cancel @ A )
=> ! [C3: A,A3: A,B3: A] :
( ( C3
!= ( zero_zero @ A ) )
=> ( ( ( times_times @ A @ C3 @ A3 )
= ( times_times @ A @ C3 @ B3 ) )
= ( A3 = B3 ) ) ) ) ).
% mult_left_cancel
thf(fact_870_no__zero__divisors,axiom,
! [A: $tType] :
( ( semiri3467727345109120633visors @ A )
=> ! [A3: A,B3: A] :
( ( A3
!= ( zero_zero @ A ) )
=> ( ( B3
!= ( zero_zero @ A ) )
=> ( ( times_times @ A @ A3 @ B3 )
!= ( zero_zero @ A ) ) ) ) ) ).
% no_zero_divisors
thf(fact_871_divisors__zero,axiom,
! [A: $tType] :
( ( semiri3467727345109120633visors @ A )
=> ! [A3: A,B3: A] :
( ( ( times_times @ A @ A3 @ B3 )
= ( zero_zero @ A ) )
=> ( ( A3
= ( zero_zero @ A ) )
| ( B3
= ( zero_zero @ A ) ) ) ) ) ).
% divisors_zero
thf(fact_872_mult__not__zero,axiom,
! [A: $tType] :
( ( mult_zero @ A )
=> ! [A3: A,B3: A] :
( ( ( times_times @ A @ A3 @ B3 )
!= ( zero_zero @ A ) )
=> ( ( A3
!= ( zero_zero @ A ) )
& ( B3
!= ( zero_zero @ A ) ) ) ) ) ).
% mult_not_zero
thf(fact_873_semiring__1__no__zero__divisors__class_Opower__not__zero,axiom,
! [A: $tType] :
( ( semiri2026040879449505780visors @ A )
=> ! [A3: A,N: nat] :
( ( A3
!= ( zero_zero @ A ) )
=> ( ( power_power @ A @ A3 @ N )
!= ( zero_zero @ A ) ) ) ) ).
% semiring_1_no_zero_divisors_class.power_not_zero
thf(fact_874_num_Osize_I4_J,axiom,
( ( size_size @ num @ one2 )
= ( zero_zero @ nat ) ) ).
% num.size(4)
thf(fact_875_not0__implies__Suc,axiom,
! [N: nat] :
( ( N
!= ( zero_zero @ nat ) )
=> ? [M5: nat] :
( N
= ( suc @ M5 ) ) ) ).
% not0_implies_Suc
thf(fact_876_Zero__not__Suc,axiom,
! [M: nat] :
( ( zero_zero @ nat )
!= ( suc @ M ) ) ).
% Zero_not_Suc
thf(fact_877_Zero__neq__Suc,axiom,
! [M: nat] :
( ( zero_zero @ nat )
!= ( suc @ M ) ) ).
% Zero_neq_Suc
thf(fact_878_Suc__neq__Zero,axiom,
! [M: nat] :
( ( suc @ M )
!= ( zero_zero @ nat ) ) ).
% Suc_neq_Zero
thf(fact_879_zero__induct,axiom,
! [P: nat > $o,K: nat] :
( ( P @ K )
=> ( ! [N3: nat] :
( ( P @ ( suc @ N3 ) )
=> ( P @ N3 ) )
=> ( P @ ( zero_zero @ nat ) ) ) ) ).
% zero_induct
thf(fact_880_diff__induct,axiom,
! [P: nat > nat > $o,M: nat,N: nat] :
( ! [X4: nat] : ( P @ X4 @ ( zero_zero @ nat ) )
=> ( ! [Y4: nat] : ( P @ ( zero_zero @ nat ) @ ( suc @ Y4 ) )
=> ( ! [X4: nat,Y4: nat] :
( ( P @ X4 @ Y4 )
=> ( P @ ( suc @ X4 ) @ ( suc @ Y4 ) ) )
=> ( P @ M @ N ) ) ) ) ).
% diff_induct
thf(fact_881_nat__induct,axiom,
! [P: nat > $o,N: nat] :
( ( P @ ( zero_zero @ nat ) )
=> ( ! [N3: nat] :
( ( P @ N3 )
=> ( P @ ( suc @ N3 ) ) )
=> ( P @ N ) ) ) ).
% nat_induct
thf(fact_882_old_Onat_Oexhaust,axiom,
! [Y3: nat] :
( ( Y3
!= ( zero_zero @ nat ) )
=> ~ ! [Nat3: nat] :
( Y3
!= ( suc @ Nat3 ) ) ) ).
% old.nat.exhaust
thf(fact_883_nat_OdiscI,axiom,
! [Nat: nat,X22: nat] :
( ( Nat
= ( suc @ X22 ) )
=> ( Nat
!= ( zero_zero @ nat ) ) ) ).
% nat.discI
thf(fact_884_old_Onat_Odistinct_I1_J,axiom,
! [Nat2: nat] :
( ( zero_zero @ nat )
!= ( suc @ Nat2 ) ) ).
% old.nat.distinct(1)
thf(fact_885_old_Onat_Odistinct_I2_J,axiom,
! [Nat4: nat] :
( ( suc @ Nat4 )
!= ( zero_zero @ nat ) ) ).
% old.nat.distinct(2)
thf(fact_886_nat_Odistinct_I1_J,axiom,
! [X22: nat] :
( ( zero_zero @ nat )
!= ( suc @ X22 ) ) ).
% nat.distinct(1)
thf(fact_887_infinite__descent0__measure,axiom,
! [A: $tType,V2: A > nat,P: A > $o,X2: A] :
( ! [X4: A] :
( ( ( V2 @ X4 )
= ( zero_zero @ nat ) )
=> ( P @ X4 ) )
=> ( ! [X4: A] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ ( V2 @ X4 ) )
=> ( ~ ( P @ X4 )
=> ? [Y5: A] :
( ( ord_less @ nat @ ( V2 @ Y5 ) @ ( V2 @ X4 ) )
& ~ ( P @ Y5 ) ) ) )
=> ( P @ X2 ) ) ) ).
% infinite_descent0_measure
thf(fact_888_infinite__descent0,axiom,
! [P: nat > $o,N: nat] :
( ( P @ ( zero_zero @ nat ) )
=> ( ! [N3: nat] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ N3 )
=> ( ~ ( P @ N3 )
=> ? [M3: nat] :
( ( ord_less @ nat @ M3 @ N3 )
& ~ ( P @ M3 ) ) ) )
=> ( P @ N ) ) ) ).
% infinite_descent0
thf(fact_889_gr__implies__not0,axiom,
! [M: nat,N: nat] :
( ( ord_less @ nat @ M @ N )
=> ( N
!= ( zero_zero @ nat ) ) ) ).
% gr_implies_not0
thf(fact_890_less__zeroE,axiom,
! [N: nat] :
~ ( ord_less @ nat @ N @ ( zero_zero @ nat ) ) ).
% less_zeroE
thf(fact_891_not__less0,axiom,
! [N: nat] :
~ ( ord_less @ nat @ N @ ( zero_zero @ nat ) ) ).
% not_less0
thf(fact_892_not__gr0,axiom,
! [N: nat] :
( ( ~ ( ord_less @ nat @ ( zero_zero @ nat ) @ N ) )
= ( N
= ( zero_zero @ nat ) ) ) ).
% not_gr0
thf(fact_893_gr0I,axiom,
! [N: nat] :
( ( N
!= ( zero_zero @ nat ) )
=> ( ord_less @ nat @ ( zero_zero @ nat ) @ N ) ) ).
% gr0I
thf(fact_894_bot__nat__0_Oextremum__strict,axiom,
! [A3: nat] :
~ ( ord_less @ nat @ A3 @ ( zero_zero @ nat ) ) ).
% bot_nat_0.extremum_strict
thf(fact_895_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_896_plus__nat_Oadd__0,axiom,
! [N: nat] :
( ( plus_plus @ nat @ ( zero_zero @ nat ) @ N )
= N ) ).
% plus_nat.add_0
thf(fact_897_less__eq__nat_Osimps_I1_J,axiom,
! [N: nat] : ( ord_less_eq @ nat @ ( zero_zero @ nat ) @ N ) ).
% less_eq_nat.simps(1)
thf(fact_898_bot__nat__0_Oextremum__unique,axiom,
! [A3: nat] :
( ( ord_less_eq @ nat @ A3 @ ( zero_zero @ nat ) )
= ( A3
= ( zero_zero @ nat ) ) ) ).
% bot_nat_0.extremum_unique
thf(fact_899_bot__nat__0_Oextremum__uniqueI,axiom,
! [A3: nat] :
( ( ord_less_eq @ nat @ A3 @ ( zero_zero @ nat ) )
=> ( A3
= ( zero_zero @ nat ) ) ) ).
% bot_nat_0.extremum_uniqueI
thf(fact_900_le__0__eq,axiom,
! [N: nat] :
( ( ord_less_eq @ nat @ N @ ( zero_zero @ nat ) )
= ( N
= ( zero_zero @ nat ) ) ) ).
% le_0_eq
thf(fact_901_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_902_minus__nat_Odiff__0,axiom,
! [M: nat] :
( ( minus_minus @ nat @ M @ ( zero_zero @ nat ) )
= M ) ).
% minus_nat.diff_0
thf(fact_903_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_904_mult__0,axiom,
! [N: nat] :
( ( times_times @ nat @ ( zero_zero @ nat ) @ N )
= ( zero_zero @ nat ) ) ).
% mult_0
thf(fact_905_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_906_power__eq__iff__eq__base,axiom,
! [A: $tType] :
( ( linordered_semidom @ A )
=> ! [N: nat,A3: A,B3: A] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ N )
=> ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ A3 )
=> ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ B3 )
=> ( ( ( power_power @ A @ A3 @ N )
= ( power_power @ A @ B3 @ N ) )
= ( A3 = B3 ) ) ) ) ) ) ).
% power_eq_iff_eq_base
thf(fact_907_power__eq__imp__eq__base,axiom,
! [A: $tType] :
( ( linordered_semidom @ A )
=> ! [A3: A,N: nat,B3: A] :
( ( ( power_power @ A @ A3 @ N )
= ( power_power @ A @ B3 @ N ) )
=> ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ A3 )
=> ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ B3 )
=> ( ( ord_less @ nat @ ( zero_zero @ nat ) @ N )
=> ( A3 = B3 ) ) ) ) ) ) ).
% power_eq_imp_eq_base
thf(fact_908_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_909_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_910_lambda__zero,axiom,
! [A: $tType] :
( ( mult_zero @ A )
=> ( ( ^ [H: A] : ( zero_zero @ A ) )
= ( times_times @ A @ ( zero_zero @ A ) ) ) ) ).
% lambda_zero
thf(fact_911_power__strict__mono,axiom,
! [A: $tType] :
( ( linordered_semidom @ A )
=> ! [A3: A,B3: A,N: nat] :
( ( ord_less @ A @ A3 @ B3 )
=> ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ A3 )
=> ( ( ord_less @ nat @ ( zero_zero @ nat ) @ N )
=> ( ord_less @ A @ ( power_power @ A @ A3 @ N ) @ ( power_power @ A @ B3 @ N ) ) ) ) ) ) ).
% power_strict_mono
thf(fact_912_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_913_zadd__int__left,axiom,
! [M: nat,N: nat,Z: int] :
( ( plus_plus @ int @ ( semiring_1_of_nat @ int @ M ) @ ( plus_plus @ int @ ( semiring_1_of_nat @ int @ N ) @ Z ) )
= ( plus_plus @ int @ ( semiring_1_of_nat @ int @ ( plus_plus @ nat @ M @ N ) ) @ Z ) ) ).
% zadd_int_left
thf(fact_914_zle__iff__zadd,axiom,
( ( ord_less_eq @ int )
= ( ^ [W2: int,Z4: int] :
? [N4: nat] :
( Z4
= ( plus_plus @ int @ W2 @ ( semiring_1_of_nat @ int @ N4 ) ) ) ) ) ).
% zle_iff_zadd
thf(fact_915_zdiv__int,axiom,
! [A3: nat,B3: nat] :
( ( semiring_1_of_nat @ int @ ( divide_divide @ nat @ A3 @ B3 ) )
= ( divide_divide @ int @ ( semiring_1_of_nat @ int @ A3 ) @ ( semiring_1_of_nat @ int @ B3 ) ) ) ).
% zdiv_int
thf(fact_916_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_917_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_918_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_919_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_920_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_921_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_922_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_923_mult__mono,axiom,
! [A: $tType] :
( ( ordered_semiring @ A )
=> ! [A3: A,B3: A,C3: A,D2: A] :
( ( ord_less_eq @ A @ A3 @ B3 )
=> ( ( ord_less_eq @ A @ C3 @ D2 )
=> ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ B3 )
=> ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ C3 )
=> ( ord_less_eq @ A @ ( times_times @ A @ A3 @ C3 ) @ ( times_times @ A @ B3 @ D2 ) ) ) ) ) ) ) ).
% mult_mono
thf(fact_924_mult__mono_H,axiom,
! [A: $tType] :
( ( ordered_semiring @ A )
=> ! [A3: A,B3: A,C3: A,D2: A] :
( ( ord_less_eq @ A @ A3 @ B3 )
=> ( ( ord_less_eq @ A @ C3 @ D2 )
=> ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ A3 )
=> ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ C3 )
=> ( ord_less_eq @ A @ ( times_times @ A @ A3 @ C3 ) @ ( times_times @ A @ B3 @ D2 ) ) ) ) ) ) ) ).
% mult_mono'
thf(fact_925_zero__le__square,axiom,
! [A: $tType] :
( ( linordered_ring @ A )
=> ! [A3: A] : ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( times_times @ A @ A3 @ A3 ) ) ) ).
% zero_le_square
thf(fact_926_split__mult__pos__le,axiom,
! [A: $tType] :
( ( ordered_ring @ A )
=> ! [A3: A,B3: A] :
( ( ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ A3 )
& ( ord_less_eq @ A @ ( zero_zero @ A ) @ B3 ) )
| ( ( ord_less_eq @ A @ A3 @ ( zero_zero @ A ) )
& ( ord_less_eq @ A @ B3 @ ( zero_zero @ A ) ) ) )
=> ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( times_times @ A @ A3 @ B3 ) ) ) ) ).
% split_mult_pos_le
thf(fact_927_mult__left__mono__neg,axiom,
! [A: $tType] :
( ( ordered_ring @ A )
=> ! [B3: A,A3: A,C3: A] :
( ( ord_less_eq @ A @ B3 @ A3 )
=> ( ( ord_less_eq @ A @ C3 @ ( zero_zero @ A ) )
=> ( ord_less_eq @ A @ ( times_times @ A @ C3 @ A3 ) @ ( times_times @ A @ C3 @ B3 ) ) ) ) ) ).
% mult_left_mono_neg
thf(fact_928_mult__nonpos__nonpos,axiom,
! [A: $tType] :
( ( ordered_ring @ A )
=> ! [A3: A,B3: A] :
( ( ord_less_eq @ A @ A3 @ ( zero_zero @ A ) )
=> ( ( ord_less_eq @ A @ B3 @ ( zero_zero @ A ) )
=> ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( times_times @ A @ A3 @ B3 ) ) ) ) ) ).
% mult_nonpos_nonpos
thf(fact_929_mult__left__mono,axiom,
! [A: $tType] :
( ( ordered_semiring @ A )
=> ! [A3: A,B3: A,C3: A] :
( ( ord_less_eq @ A @ A3 @ B3 )
=> ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ C3 )
=> ( ord_less_eq @ A @ ( times_times @ A @ C3 @ A3 ) @ ( times_times @ A @ C3 @ B3 ) ) ) ) ) ).
% mult_left_mono
thf(fact_930_mult__right__mono__neg,axiom,
! [A: $tType] :
( ( ordered_ring @ A )
=> ! [B3: A,A3: A,C3: A] :
( ( ord_less_eq @ A @ B3 @ A3 )
=> ( ( ord_less_eq @ A @ C3 @ ( zero_zero @ A ) )
=> ( ord_less_eq @ A @ ( times_times @ A @ A3 @ C3 ) @ ( times_times @ A @ B3 @ C3 ) ) ) ) ) ).
% mult_right_mono_neg
thf(fact_931_mult__right__mono,axiom,
! [A: $tType] :
( ( ordered_semiring @ A )
=> ! [A3: A,B3: A,C3: A] :
( ( ord_less_eq @ A @ A3 @ B3 )
=> ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ C3 )
=> ( ord_less_eq @ A @ ( times_times @ A @ A3 @ C3 ) @ ( times_times @ A @ B3 @ C3 ) ) ) ) ) ).
% mult_right_mono
thf(fact_932_mult__le__0__iff,axiom,
! [A: $tType] :
( ( linord4710134922213307826strict @ A )
=> ! [A3: A,B3: A] :
( ( ord_less_eq @ A @ ( times_times @ A @ A3 @ B3 ) @ ( zero_zero @ A ) )
= ( ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ A3 )
& ( ord_less_eq @ A @ B3 @ ( zero_zero @ A ) ) )
| ( ( ord_less_eq @ A @ A3 @ ( zero_zero @ A ) )
& ( ord_less_eq @ A @ ( zero_zero @ A ) @ B3 ) ) ) ) ) ).
% mult_le_0_iff
thf(fact_933_split__mult__neg__le,axiom,
! [A: $tType] :
( ( ordered_semiring_0 @ A )
=> ! [A3: A,B3: A] :
( ( ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ A3 )
& ( ord_less_eq @ A @ B3 @ ( zero_zero @ A ) ) )
| ( ( ord_less_eq @ A @ A3 @ ( zero_zero @ A ) )
& ( ord_less_eq @ A @ ( zero_zero @ A ) @ B3 ) ) )
=> ( ord_less_eq @ A @ ( times_times @ A @ A3 @ B3 ) @ ( zero_zero @ A ) ) ) ) ).
% split_mult_neg_le
thf(fact_934_mult__nonneg__nonneg,axiom,
! [A: $tType] :
( ( ordered_semiring_0 @ A )
=> ! [A3: A,B3: A] :
( ( ord_less_eq @ A @ ( zero_zero @ A ) @ A3 )
=> ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ B3 )
=> ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( times_times @ A @ A3 @ B3 ) ) ) ) ) ).
% mult_nonneg_nonneg
thf(fact_935_mult__nonneg__nonpos,axiom,
! [A: $tType] :
( ( ordered_semiring_0 @ A )
=> ! [A3: A,B3: A] :
( ( ord_less_eq @ A @ ( zero_zero @ A ) @ A3 )
=> ( ( ord_less_eq @ A @ B3 @ ( zero_zero @ A ) )
=> ( ord_less_eq @ A @ ( times_times @ A @ A3 @ B3 ) @ ( zero_zero @ A ) ) ) ) ) ).
% mult_nonneg_nonpos
thf(fact_936_mult__nonpos__nonneg,axiom,
! [A: $tType] :
( ( ordered_semiring_0 @ A )
=> ! [A3: A,B3: A] :
( ( ord_less_eq @ A @ A3 @ ( zero_zero @ A ) )
=> ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ B3 )
=> ( ord_less_eq @ A @ ( times_times @ A @ A3 @ B3 ) @ ( zero_zero @ A ) ) ) ) ) ).
% mult_nonpos_nonneg
thf(fact_937_mult__nonneg__nonpos2,axiom,
! [A: $tType] :
( ( ordered_semiring_0 @ A )
=> ! [A3: A,B3: A] :
( ( ord_less_eq @ A @ ( zero_zero @ A ) @ A3 )
=> ( ( ord_less_eq @ A @ B3 @ ( zero_zero @ A ) )
=> ( ord_less_eq @ A @ ( times_times @ A @ B3 @ A3 ) @ ( zero_zero @ A ) ) ) ) ) ).
% mult_nonneg_nonpos2
thf(fact_938_zero__le__mult__iff,axiom,
! [A: $tType] :
( ( linord4710134922213307826strict @ A )
=> ! [A3: A,B3: A] :
( ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( times_times @ A @ A3 @ B3 ) )
= ( ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ A3 )
& ( ord_less_eq @ A @ ( zero_zero @ A ) @ B3 ) )
| ( ( ord_less_eq @ A @ A3 @ ( zero_zero @ A ) )
& ( ord_less_eq @ A @ B3 @ ( zero_zero @ A ) ) ) ) ) ) ).
% zero_le_mult_iff
thf(fact_939_ordered__comm__semiring__class_Ocomm__mult__left__mono,axiom,
! [A: $tType] :
( ( ordere2520102378445227354miring @ A )
=> ! [A3: A,B3: A,C3: A] :
( ( ord_less_eq @ A @ A3 @ B3 )
=> ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ C3 )
=> ( ord_less_eq @ A @ ( times_times @ A @ C3 @ A3 ) @ ( times_times @ A @ C3 @ B3 ) ) ) ) ) ).
% ordered_comm_semiring_class.comm_mult_left_mono
thf(fact_940_add__less__zeroD,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [X2: A,Y3: A] :
( ( ord_less @ A @ ( plus_plus @ A @ X2 @ Y3 ) @ ( zero_zero @ A ) )
=> ( ( ord_less @ A @ X2 @ ( zero_zero @ A ) )
| ( ord_less @ A @ Y3 @ ( zero_zero @ A ) ) ) ) ) ).
% add_less_zeroD
thf(fact_941_zero__less__one,axiom,
! [A: $tType] :
( ( zero_less_one @ A )
=> ( ord_less @ A @ ( zero_zero @ A ) @ ( one_one @ A ) ) ) ).
% zero_less_one
thf(fact_942_not__one__less__zero,axiom,
! [A: $tType] :
( ( linord181362715937106298miring @ A )
=> ~ ( ord_less @ A @ ( one_one @ A ) @ ( zero_zero @ A ) ) ) ).
% not_one_less_zero
thf(fact_943_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_944_linordered__comm__semiring__strict__class_Ocomm__mult__strict__left__mono,axiom,
! [A: $tType] :
( ( linord2810124833399127020strict @ A )
=> ! [A3: A,B3: A,C3: A] :
( ( ord_less @ A @ A3 @ B3 )
=> ( ( ord_less @ A @ ( zero_zero @ A ) @ C3 )
=> ( ord_less @ A @ ( times_times @ A @ C3 @ A3 ) @ ( times_times @ A @ C3 @ B3 ) ) ) ) ) ).
% linordered_comm_semiring_strict_class.comm_mult_strict_left_mono
thf(fact_945_mult__less__cancel__right__disj,axiom,
! [A: $tType] :
( ( linord4710134922213307826strict @ A )
=> ! [A3: A,C3: A,B3: A] :
( ( ord_less @ A @ ( times_times @ A @ A3 @ C3 ) @ ( times_times @ A @ B3 @ C3 ) )
= ( ( ( ord_less @ A @ ( zero_zero @ A ) @ C3 )
& ( ord_less @ A @ A3 @ B3 ) )
| ( ( ord_less @ A @ C3 @ ( zero_zero @ A ) )
& ( ord_less @ A @ B3 @ A3 ) ) ) ) ) ).
% mult_less_cancel_right_disj
thf(fact_946_mult__strict__right__mono,axiom,
! [A: $tType] :
( ( linord8928482502909563296strict @ A )
=> ! [A3: A,B3: A,C3: A] :
( ( ord_less @ A @ A3 @ B3 )
=> ( ( ord_less @ A @ ( zero_zero @ A ) @ C3 )
=> ( ord_less @ A @ ( times_times @ A @ A3 @ C3 ) @ ( times_times @ A @ B3 @ C3 ) ) ) ) ) ).
% mult_strict_right_mono
thf(fact_947_mult__strict__right__mono__neg,axiom,
! [A: $tType] :
( ( linord4710134922213307826strict @ A )
=> ! [B3: A,A3: A,C3: A] :
( ( ord_less @ A @ B3 @ A3 )
=> ( ( ord_less @ A @ C3 @ ( zero_zero @ A ) )
=> ( ord_less @ A @ ( times_times @ A @ A3 @ C3 ) @ ( times_times @ A @ B3 @ C3 ) ) ) ) ) ).
% mult_strict_right_mono_neg
thf(fact_948_mult__less__cancel__left__disj,axiom,
! [A: $tType] :
( ( linord4710134922213307826strict @ A )
=> ! [C3: A,A3: A,B3: A] :
( ( ord_less @ A @ ( times_times @ A @ C3 @ A3 ) @ ( times_times @ A @ C3 @ B3 ) )
= ( ( ( ord_less @ A @ ( zero_zero @ A ) @ C3 )
& ( ord_less @ A @ A3 @ B3 ) )
| ( ( ord_less @ A @ C3 @ ( zero_zero @ A ) )
& ( ord_less @ A @ B3 @ A3 ) ) ) ) ) ).
% mult_less_cancel_left_disj
thf(fact_949_mult__strict__left__mono,axiom,
! [A: $tType] :
( ( linord8928482502909563296strict @ A )
=> ! [A3: A,B3: A,C3: A] :
( ( ord_less @ A @ A3 @ B3 )
=> ( ( ord_less @ A @ ( zero_zero @ A ) @ C3 )
=> ( ord_less @ A @ ( times_times @ A @ C3 @ A3 ) @ ( times_times @ A @ C3 @ B3 ) ) ) ) ) ).
% mult_strict_left_mono
thf(fact_950_mult__strict__left__mono__neg,axiom,
! [A: $tType] :
( ( linord4710134922213307826strict @ A )
=> ! [B3: A,A3: A,C3: A] :
( ( ord_less @ A @ B3 @ A3 )
=> ( ( ord_less @ A @ C3 @ ( zero_zero @ A ) )
=> ( ord_less @ A @ ( times_times @ A @ C3 @ A3 ) @ ( times_times @ A @ C3 @ B3 ) ) ) ) ) ).
% mult_strict_left_mono_neg
thf(fact_951_mult__less__cancel__left__pos,axiom,
! [A: $tType] :
( ( linord4710134922213307826strict @ A )
=> ! [C3: A,A3: A,B3: A] :
( ( ord_less @ A @ ( zero_zero @ A ) @ C3 )
=> ( ( ord_less @ A @ ( times_times @ A @ C3 @ A3 ) @ ( times_times @ A @ C3 @ B3 ) )
= ( ord_less @ A @ A3 @ B3 ) ) ) ) ).
% mult_less_cancel_left_pos
thf(fact_952_mult__less__cancel__left__neg,axiom,
! [A: $tType] :
( ( linord4710134922213307826strict @ A )
=> ! [C3: A,A3: A,B3: A] :
( ( ord_less @ A @ C3 @ ( zero_zero @ A ) )
=> ( ( ord_less @ A @ ( times_times @ A @ C3 @ A3 ) @ ( times_times @ A @ C3 @ B3 ) )
= ( ord_less @ A @ B3 @ A3 ) ) ) ) ).
% mult_less_cancel_left_neg
thf(fact_953_zero__less__mult__pos2,axiom,
! [A: $tType] :
( ( linord8928482502909563296strict @ A )
=> ! [B3: A,A3: A] :
( ( ord_less @ A @ ( zero_zero @ A ) @ ( times_times @ A @ B3 @ A3 ) )
=> ( ( ord_less @ A @ ( zero_zero @ A ) @ A3 )
=> ( ord_less @ A @ ( zero_zero @ A ) @ B3 ) ) ) ) ).
% zero_less_mult_pos2
thf(fact_954_zero__less__mult__pos,axiom,
! [A: $tType] :
( ( linord8928482502909563296strict @ A )
=> ! [A3: A,B3: A] :
( ( ord_less @ A @ ( zero_zero @ A ) @ ( times_times @ A @ A3 @ B3 ) )
=> ( ( ord_less @ A @ ( zero_zero @ A ) @ A3 )
=> ( ord_less @ A @ ( zero_zero @ A ) @ B3 ) ) ) ) ).
% zero_less_mult_pos
thf(fact_955_zero__less__mult__iff,axiom,
! [A: $tType] :
( ( linord4710134922213307826strict @ A )
=> ! [A3: A,B3: A] :
( ( ord_less @ A @ ( zero_zero @ A ) @ ( times_times @ A @ A3 @ B3 ) )
= ( ( ( ord_less @ A @ ( zero_zero @ A ) @ A3 )
& ( ord_less @ A @ ( zero_zero @ A ) @ B3 ) )
| ( ( ord_less @ A @ A3 @ ( zero_zero @ A ) )
& ( ord_less @ A @ B3 @ ( zero_zero @ A ) ) ) ) ) ) ).
% zero_less_mult_iff
thf(fact_956_mult__pos__neg2,axiom,
! [A: $tType] :
( ( linord8928482502909563296strict @ A )
=> ! [A3: A,B3: A] :
( ( ord_less @ A @ ( zero_zero @ A ) @ A3 )
=> ( ( ord_less @ A @ B3 @ ( zero_zero @ A ) )
=> ( ord_less @ A @ ( times_times @ A @ B3 @ A3 ) @ ( zero_zero @ A ) ) ) ) ) ).
% mult_pos_neg2
thf(fact_957_mult__pos__pos,axiom,
! [A: $tType] :
( ( linord8928482502909563296strict @ A )
=> ! [A3: A,B3: A] :
( ( ord_less @ A @ ( zero_zero @ A ) @ A3 )
=> ( ( ord_less @ A @ ( zero_zero @ A ) @ B3 )
=> ( ord_less @ A @ ( zero_zero @ A ) @ ( times_times @ A @ A3 @ B3 ) ) ) ) ) ).
% mult_pos_pos
thf(fact_958_mult__pos__neg,axiom,
! [A: $tType] :
( ( linord8928482502909563296strict @ A )
=> ! [A3: A,B3: A] :
( ( ord_less @ A @ ( zero_zero @ A ) @ A3 )
=> ( ( ord_less @ A @ B3 @ ( zero_zero @ A ) )
=> ( ord_less @ A @ ( times_times @ A @ A3 @ B3 ) @ ( zero_zero @ A ) ) ) ) ) ).
% mult_pos_neg
thf(fact_959_mult__neg__pos,axiom,
! [A: $tType] :
( ( linord8928482502909563296strict @ A )
=> ! [A3: A,B3: A] :
( ( ord_less @ A @ A3 @ ( zero_zero @ A ) )
=> ( ( ord_less @ A @ ( zero_zero @ A ) @ B3 )
=> ( ord_less @ A @ ( times_times @ A @ A3 @ B3 ) @ ( zero_zero @ A ) ) ) ) ) ).
% mult_neg_pos
thf(fact_960_mult__less__0__iff,axiom,
! [A: $tType] :
( ( linord4710134922213307826strict @ A )
=> ! [A3: A,B3: A] :
( ( ord_less @ A @ ( times_times @ A @ A3 @ B3 ) @ ( zero_zero @ A ) )
= ( ( ( ord_less @ A @ ( zero_zero @ A ) @ A3 )
& ( ord_less @ A @ B3 @ ( zero_zero @ A ) ) )
| ( ( ord_less @ A @ A3 @ ( zero_zero @ A ) )
& ( ord_less @ A @ ( zero_zero @ A ) @ B3 ) ) ) ) ) ).
% mult_less_0_iff
thf(fact_961_not__square__less__zero,axiom,
! [A: $tType] :
( ( linordered_ring @ A )
=> ! [A3: A] :
~ ( ord_less @ A @ ( times_times @ A @ A3 @ A3 ) @ ( zero_zero @ A ) ) ) ).
% not_square_less_zero
thf(fact_962_mult__neg__neg,axiom,
! [A: $tType] :
( ( linord4710134922213307826strict @ A )
=> ! [A3: A,B3: A] :
( ( ord_less @ A @ A3 @ ( zero_zero @ A ) )
=> ( ( ord_less @ A @ B3 @ ( zero_zero @ A ) )
=> ( ord_less @ A @ ( zero_zero @ A ) @ ( times_times @ A @ A3 @ B3 ) ) ) ) ) ).
% mult_neg_neg
thf(fact_963_power__mono,axiom,
! [A: $tType] :
( ( linordered_semidom @ A )
=> ! [A3: A,B3: A,N: nat] :
( ( ord_less_eq @ A @ A3 @ B3 )
=> ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ A3 )
=> ( ord_less_eq @ A @ ( power_power @ A @ A3 @ N ) @ ( power_power @ A @ B3 @ N ) ) ) ) ) ).
% power_mono
thf(fact_964_zero__le__power,axiom,
! [A: $tType] :
( ( linordered_semidom @ A )
=> ! [A3: A,N: nat] :
( ( ord_less_eq @ A @ ( zero_zero @ A ) @ A3 )
=> ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( power_power @ A @ A3 @ N ) ) ) ) ).
% zero_le_power
thf(fact_965_zero__less__power,axiom,
! [A: $tType] :
( ( linordered_semidom @ A )
=> ! [A3: A,N: nat] :
( ( ord_less @ A @ ( zero_zero @ A ) @ A3 )
=> ( ord_less @ A @ ( zero_zero @ A ) @ ( power_power @ A @ A3 @ N ) ) ) ) ).
% zero_less_power
thf(fact_966_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_967_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_968_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_969_power__0,axiom,
! [A: $tType] :
( ( power @ A )
=> ! [A3: A] :
( ( power_power @ A @ A3 @ ( zero_zero @ nat ) )
= ( one_one @ A ) ) ) ).
% power_0
thf(fact_970_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_971_gr0__implies__Suc,axiom,
! [N: nat] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ N )
=> ? [M5: nat] :
( N
= ( suc @ M5 ) ) ) ).
% gr0_implies_Suc
thf(fact_972_All__less__Suc2,axiom,
! [N: nat,P: nat > $o] :
( ( ! [I4: nat] :
( ( ord_less @ nat @ I4 @ ( suc @ N ) )
=> ( P @ I4 ) ) )
= ( ( P @ ( zero_zero @ nat ) )
& ! [I4: nat] :
( ( ord_less @ nat @ I4 @ N )
=> ( P @ ( suc @ I4 ) ) ) ) ) ).
% All_less_Suc2
thf(fact_973_gr0__conv__Suc,axiom,
! [N: nat] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ N )
= ( ? [M6: nat] :
( N
= ( suc @ M6 ) ) ) ) ).
% gr0_conv_Suc
thf(fact_974_Ex__less__Suc2,axiom,
! [N: nat,P: nat > $o] :
( ( ? [I4: nat] :
( ( ord_less @ nat @ I4 @ ( suc @ N ) )
& ( P @ I4 ) ) )
= ( ( P @ ( zero_zero @ nat ) )
| ? [I4: nat] :
( ( ord_less @ nat @ I4 @ N )
& ( P @ ( suc @ I4 ) ) ) ) ) ).
% Ex_less_Suc2
thf(fact_975_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_976_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_977_nat__compl__induct,axiom,
! [P: nat > $o,N: nat] :
( ( P @ ( zero_zero @ nat ) )
=> ( ! [N3: nat] :
( ! [Nn: nat] :
( ( ord_less_eq @ nat @ Nn @ N3 )
=> ( P @ Nn ) )
=> ( P @ ( suc @ N3 ) ) )
=> ( P @ N ) ) ) ).
% nat_compl_induct
thf(fact_978_nat__compl__induct_H,axiom,
! [P: nat > $o,N: nat] :
( ( P @ ( zero_zero @ nat ) )
=> ( ! [N3: nat] :
( ! [Nn: nat] :
( ( ord_less_eq @ nat @ Nn @ N3 )
=> ( P @ Nn ) )
=> ( P @ ( suc @ N3 ) ) )
=> ( P @ N ) ) ) ).
% nat_compl_induct'
thf(fact_979_option_Osize_I4_J,axiom,
! [A: $tType,X22: A] :
( ( size_size @ ( option @ A ) @ ( some @ A @ X22 ) )
= ( suc @ ( zero_zero @ nat ) ) ) ).
% option.size(4)
thf(fact_980_One__nat__def,axiom,
( ( one_one @ nat )
= ( suc @ ( zero_zero @ nat ) ) ) ).
% One_nat_def
thf(fact_981_option_Osize_I3_J,axiom,
! [A: $tType] :
( ( size_size @ ( option @ A ) @ ( none @ A ) )
= ( suc @ ( zero_zero @ nat ) ) ) ).
% option.size(3)
thf(fact_982_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_983_ex__least__nat__le,axiom,
! [P: nat > $o,N: nat] :
( ( P @ N )
=> ( ~ ( P @ ( zero_zero @ nat ) )
=> ? [K2: nat] :
( ( ord_less_eq @ nat @ K2 @ N )
& ! [I5: nat] :
( ( ord_less @ nat @ I5 @ K2 )
=> ~ ( P @ I5 ) )
& ( P @ K2 ) ) ) ) ).
% ex_least_nat_le
thf(fact_984_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_985_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_986_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_987_nat__geq__1__eq__neqz,axiom,
! [X2: nat] :
( ( ord_less_eq @ nat @ ( one_one @ nat ) @ X2 )
= ( X2
!= ( zero_zero @ nat ) ) ) ).
% nat_geq_1_eq_neqz
thf(fact_988_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_989_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_990_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_991_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_992_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_993_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_994_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_995_vebt__insert_Osimps_I2_J,axiom,
! [Info: option @ ( product_prod @ nat @ nat ),Ts: list @ vEBT_VEBT,S2: vEBT_VEBT,X2: nat] :
( ( vEBT_vebt_insert @ ( vEBT_Node @ Info @ ( zero_zero @ nat ) @ Ts @ S2 ) @ X2 )
= ( vEBT_Node @ Info @ ( zero_zero @ nat ) @ Ts @ S2 ) ) ).
% vebt_insert.simps(2)
thf(fact_996_zless__iff__Suc__zadd,axiom,
( ( ord_less @ int )
= ( ^ [W2: int,Z4: int] :
? [N4: nat] :
( Z4
= ( plus_plus @ int @ W2 @ ( semiring_1_of_nat @ int @ ( suc @ N4 ) ) ) ) ) ) ).
% zless_iff_Suc_zadd
thf(fact_997_mult__less__le__imp__less,axiom,
! [A: $tType] :
( ( linord8928482502909563296strict @ A )
=> ! [A3: A,B3: A,C3: A,D2: A] :
( ( ord_less @ A @ A3 @ B3 )
=> ( ( ord_less_eq @ A @ C3 @ D2 )
=> ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ A3 )
=> ( ( ord_less @ A @ ( zero_zero @ A ) @ C3 )
=> ( ord_less @ A @ ( times_times @ A @ A3 @ C3 ) @ ( times_times @ A @ B3 @ D2 ) ) ) ) ) ) ) ).
% mult_less_le_imp_less
thf(fact_998_mult__le__less__imp__less,axiom,
! [A: $tType] :
( ( linord8928482502909563296strict @ A )
=> ! [A3: A,B3: A,C3: A,D2: A] :
( ( ord_less_eq @ A @ A3 @ B3 )
=> ( ( ord_less @ A @ C3 @ D2 )
=> ( ( ord_less @ A @ ( zero_zero @ A ) @ A3 )
=> ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ C3 )
=> ( ord_less @ A @ ( times_times @ A @ A3 @ C3 ) @ ( times_times @ A @ B3 @ D2 ) ) ) ) ) ) ) ).
% mult_le_less_imp_less
thf(fact_999_mult__right__le__imp__le,axiom,
! [A: $tType] :
( ( linord8928482502909563296strict @ A )
=> ! [A3: A,C3: A,B3: A] :
( ( ord_less_eq @ A @ ( times_times @ A @ A3 @ C3 ) @ ( times_times @ A @ B3 @ C3 ) )
=> ( ( ord_less @ A @ ( zero_zero @ A ) @ C3 )
=> ( ord_less_eq @ A @ A3 @ B3 ) ) ) ) ).
% mult_right_le_imp_le
thf(fact_1000_mult__left__le__imp__le,axiom,
! [A: $tType] :
( ( linord8928482502909563296strict @ A )
=> ! [C3: A,A3: A,B3: A] :
( ( ord_less_eq @ A @ ( times_times @ A @ C3 @ A3 ) @ ( times_times @ A @ C3 @ B3 ) )
=> ( ( ord_less @ A @ ( zero_zero @ A ) @ C3 )
=> ( ord_less_eq @ A @ A3 @ B3 ) ) ) ) ).
% mult_left_le_imp_le
thf(fact_1001_mult__le__cancel__left__pos,axiom,
! [A: $tType] :
( ( linord4710134922213307826strict @ A )
=> ! [C3: A,A3: A,B3: A] :
( ( ord_less @ A @ ( zero_zero @ A ) @ C3 )
=> ( ( ord_less_eq @ A @ ( times_times @ A @ C3 @ A3 ) @ ( times_times @ A @ C3 @ B3 ) )
= ( ord_less_eq @ A @ A3 @ B3 ) ) ) ) ).
% mult_le_cancel_left_pos
thf(fact_1002_mult__le__cancel__left__neg,axiom,
! [A: $tType] :
( ( linord4710134922213307826strict @ A )
=> ! [C3: A,A3: A,B3: A] :
( ( ord_less @ A @ C3 @ ( zero_zero @ A ) )
=> ( ( ord_less_eq @ A @ ( times_times @ A @ C3 @ A3 ) @ ( times_times @ A @ C3 @ B3 ) )
= ( ord_less_eq @ A @ B3 @ A3 ) ) ) ) ).
% mult_le_cancel_left_neg
thf(fact_1003_mult__less__cancel__right,axiom,
! [A: $tType] :
( ( linord4710134922213307826strict @ A )
=> ! [A3: A,C3: A,B3: A] :
( ( ord_less @ A @ ( times_times @ A @ A3 @ C3 ) @ ( times_times @ A @ B3 @ C3 ) )
= ( ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ C3 )
=> ( ord_less @ A @ A3 @ B3 ) )
& ( ( ord_less_eq @ A @ C3 @ ( zero_zero @ A ) )
=> ( ord_less @ A @ B3 @ A3 ) ) ) ) ) ).
% mult_less_cancel_right
thf(fact_1004_mult__strict__mono_H,axiom,
! [A: $tType] :
( ( linord8928482502909563296strict @ A )
=> ! [A3: A,B3: A,C3: A,D2: A] :
( ( ord_less @ A @ A3 @ B3 )
=> ( ( ord_less @ A @ C3 @ D2 )
=> ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ A3 )
=> ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ C3 )
=> ( ord_less @ A @ ( times_times @ A @ A3 @ C3 ) @ ( times_times @ A @ B3 @ D2 ) ) ) ) ) ) ) ).
% mult_strict_mono'
thf(fact_1005_mult__right__less__imp__less,axiom,
! [A: $tType] :
( ( linordered_semiring @ A )
=> ! [A3: A,C3: A,B3: A] :
( ( ord_less @ A @ ( times_times @ A @ A3 @ C3 ) @ ( times_times @ A @ B3 @ C3 ) )
=> ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ C3 )
=> ( ord_less @ A @ A3 @ B3 ) ) ) ) ).
% mult_right_less_imp_less
thf(fact_1006_mult__less__cancel__left,axiom,
! [A: $tType] :
( ( linord4710134922213307826strict @ A )
=> ! [C3: A,A3: A,B3: A] :
( ( ord_less @ A @ ( times_times @ A @ C3 @ A3 ) @ ( times_times @ A @ C3 @ B3 ) )
= ( ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ C3 )
=> ( ord_less @ A @ A3 @ B3 ) )
& ( ( ord_less_eq @ A @ C3 @ ( zero_zero @ A ) )
=> ( ord_less @ A @ B3 @ A3 ) ) ) ) ) ).
% mult_less_cancel_left
thf(fact_1007_mult__strict__mono,axiom,
! [A: $tType] :
( ( linord8928482502909563296strict @ A )
=> ! [A3: A,B3: A,C3: A,D2: A] :
( ( ord_less @ A @ A3 @ B3 )
=> ( ( ord_less @ A @ C3 @ D2 )
=> ( ( ord_less @ A @ ( zero_zero @ A ) @ B3 )
=> ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ C3 )
=> ( ord_less @ A @ ( times_times @ A @ A3 @ C3 ) @ ( times_times @ A @ B3 @ D2 ) ) ) ) ) ) ) ).
% mult_strict_mono
thf(fact_1008_mult__left__less__imp__less,axiom,
! [A: $tType] :
( ( linordered_semiring @ A )
=> ! [C3: A,A3: A,B3: A] :
( ( ord_less @ A @ ( times_times @ A @ C3 @ A3 ) @ ( times_times @ A @ C3 @ B3 ) )
=> ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ C3 )
=> ( ord_less @ A @ A3 @ B3 ) ) ) ) ).
% mult_left_less_imp_less
thf(fact_1009_mult__le__cancel__right,axiom,
! [A: $tType] :
( ( linord4710134922213307826strict @ A )
=> ! [A3: A,C3: A,B3: A] :
( ( ord_less_eq @ A @ ( times_times @ A @ A3 @ C3 ) @ ( times_times @ A @ B3 @ C3 ) )
= ( ( ( ord_less @ A @ ( zero_zero @ A ) @ C3 )
=> ( ord_less_eq @ A @ A3 @ B3 ) )
& ( ( ord_less @ A @ C3 @ ( zero_zero @ A ) )
=> ( ord_less_eq @ A @ B3 @ A3 ) ) ) ) ) ).
% mult_le_cancel_right
thf(fact_1010_mult__le__cancel__left,axiom,
! [A: $tType] :
( ( linord4710134922213307826strict @ A )
=> ! [C3: A,A3: A,B3: A] :
( ( ord_less_eq @ A @ ( times_times @ A @ C3 @ A3 ) @ ( times_times @ A @ C3 @ B3 ) )
= ( ( ( ord_less @ A @ ( zero_zero @ A ) @ C3 )
=> ( ord_less_eq @ A @ A3 @ B3 ) )
& ( ( ord_less @ A @ C3 @ ( zero_zero @ A ) )
=> ( ord_less_eq @ A @ B3 @ A3 ) ) ) ) ) ).
% mult_le_cancel_left
thf(fact_1011_sum__squares__ge__zero,axiom,
! [A: $tType] :
( ( linordered_ring @ A )
=> ! [X2: A,Y3: A] : ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( plus_plus @ A @ ( times_times @ A @ X2 @ X2 ) @ ( times_times @ A @ Y3 @ Y3 ) ) ) ) ).
% sum_squares_ge_zero
thf(fact_1012_sum__squares__le__zero__iff,axiom,
! [A: $tType] :
( ( linord4710134922213307826strict @ A )
=> ! [X2: A,Y3: A] :
( ( ord_less_eq @ A @ ( plus_plus @ A @ ( times_times @ A @ X2 @ X2 ) @ ( times_times @ A @ Y3 @ Y3 ) ) @ ( zero_zero @ A ) )
= ( ( X2
= ( zero_zero @ A ) )
& ( Y3
= ( zero_zero @ A ) ) ) ) ) ).
% sum_squares_le_zero_iff
thf(fact_1013_mult__left__le__one__le,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [X2: A,Y3: A] :
( ( ord_less_eq @ A @ ( zero_zero @ A ) @ X2 )
=> ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ Y3 )
=> ( ( ord_less_eq @ A @ Y3 @ ( one_one @ A ) )
=> ( ord_less_eq @ A @ ( times_times @ A @ Y3 @ X2 ) @ X2 ) ) ) ) ) ).
% mult_left_le_one_le
thf(fact_1014_mult__right__le__one__le,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [X2: A,Y3: A] :
( ( ord_less_eq @ A @ ( zero_zero @ A ) @ X2 )
=> ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ Y3 )
=> ( ( ord_less_eq @ A @ Y3 @ ( one_one @ A ) )
=> ( ord_less_eq @ A @ ( times_times @ A @ X2 @ Y3 ) @ X2 ) ) ) ) ) ).
% mult_right_le_one_le
thf(fact_1015_mult__le__one,axiom,
! [A: $tType] :
( ( linord181362715937106298miring @ A )
=> ! [A3: A,B3: A] :
( ( ord_less_eq @ A @ A3 @ ( one_one @ A ) )
=> ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ B3 )
=> ( ( ord_less_eq @ A @ B3 @ ( one_one @ A ) )
=> ( ord_less_eq @ A @ ( times_times @ A @ A3 @ B3 ) @ ( one_one @ A ) ) ) ) ) ) ).
% mult_le_one
thf(fact_1016_mult__left__le,axiom,
! [A: $tType] :
( ( linord181362715937106298miring @ A )
=> ! [C3: A,A3: A] :
( ( ord_less_eq @ A @ C3 @ ( one_one @ A ) )
=> ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ A3 )
=> ( ord_less_eq @ A @ ( times_times @ A @ A3 @ C3 ) @ A3 ) ) ) ) ).
% mult_left_le
thf(fact_1017_div__positive,axiom,
! [A: $tType] :
( ( unique1627219031080169319umeral @ A )
=> ! [B3: A,A3: A] :
( ( ord_less @ A @ ( zero_zero @ A ) @ B3 )
=> ( ( ord_less_eq @ A @ B3 @ A3 )
=> ( ord_less @ A @ ( zero_zero @ A ) @ ( divide_divide @ A @ A3 @ B3 ) ) ) ) ) ).
% div_positive
thf(fact_1018_unique__euclidean__semiring__numeral__class_Odiv__less,axiom,
! [A: $tType] :
( ( unique1627219031080169319umeral @ A )
=> ! [A3: A,B3: A] :
( ( ord_less_eq @ A @ ( zero_zero @ A ) @ A3 )
=> ( ( ord_less @ A @ A3 @ B3 )
=> ( ( divide_divide @ A @ A3 @ B3 )
= ( zero_zero @ A ) ) ) ) ) ).
% unique_euclidean_semiring_numeral_class.div_less
thf(fact_1019_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_1020_not__sum__squares__lt__zero,axiom,
! [A: $tType] :
( ( linordered_ring @ A )
=> ! [X2: A,Y3: A] :
~ ( ord_less @ A @ ( plus_plus @ A @ ( times_times @ A @ X2 @ X2 ) @ ( times_times @ A @ Y3 @ Y3 ) ) @ ( zero_zero @ A ) ) ) ).
% not_sum_squares_lt_zero
thf(fact_1021_sum__squares__gt__zero__iff,axiom,
! [A: $tType] :
( ( linord4710134922213307826strict @ A )
=> ! [X2: A,Y3: A] :
( ( ord_less @ A @ ( zero_zero @ A ) @ ( plus_plus @ A @ ( times_times @ A @ X2 @ X2 ) @ ( times_times @ A @ Y3 @ Y3 ) ) )
= ( ( X2
!= ( zero_zero @ A ) )
| ( Y3
!= ( zero_zero @ A ) ) ) ) ) ).
% sum_squares_gt_zero_iff
thf(fact_1022_power__less__imp__less__base,axiom,
! [A: $tType] :
( ( linordered_semidom @ A )
=> ! [A3: A,N: nat,B3: A] :
( ( ord_less @ A @ ( power_power @ A @ A3 @ N ) @ ( power_power @ A @ B3 @ N ) )
=> ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ B3 )
=> ( ord_less @ A @ A3 @ B3 ) ) ) ) ).
% power_less_imp_less_base
thf(fact_1023_unique__euclidean__semiring__numeral__class_Odiv__mult2__eq,axiom,
! [A: $tType] :
( ( unique1627219031080169319umeral @ A )
=> ! [C3: A,A3: A,B3: A] :
( ( ord_less_eq @ A @ ( zero_zero @ A ) @ C3 )
=> ( ( divide_divide @ A @ A3 @ ( times_times @ A @ B3 @ C3 ) )
= ( divide_divide @ A @ ( divide_divide @ A @ A3 @ B3 ) @ C3 ) ) ) ) ).
% unique_euclidean_semiring_numeral_class.div_mult2_eq
thf(fact_1024_power__le__one,axiom,
! [A: $tType] :
( ( linordered_semidom @ A )
=> ! [A3: A,N: nat] :
( ( ord_less_eq @ A @ ( zero_zero @ A ) @ A3 )
=> ( ( ord_less_eq @ A @ A3 @ ( one_one @ A ) )
=> ( ord_less_eq @ A @ ( power_power @ A @ A3 @ N ) @ ( one_one @ A ) ) ) ) ) ).
% power_le_one
thf(fact_1025_eq__divide__eq__numeral_I1_J,axiom,
! [A: $tType] :
( ( division_ring @ A )
=> ! [W: num,B3: A,C3: A] :
( ( ( numeral_numeral @ A @ W )
= ( divide_divide @ A @ B3 @ C3 ) )
= ( ( ( C3
!= ( zero_zero @ A ) )
=> ( ( times_times @ A @ ( numeral_numeral @ A @ W ) @ C3 )
= B3 ) )
& ( ( C3
= ( zero_zero @ A ) )
=> ( ( numeral_numeral @ A @ W )
= ( zero_zero @ A ) ) ) ) ) ) ).
% eq_divide_eq_numeral(1)
thf(fact_1026_divide__eq__eq__numeral_I1_J,axiom,
! [A: $tType] :
( ( division_ring @ A )
=> ! [B3: A,C3: A,W: num] :
( ( ( divide_divide @ A @ B3 @ C3 )
= ( numeral_numeral @ A @ W ) )
= ( ( ( C3
!= ( zero_zero @ A ) )
=> ( B3
= ( times_times @ A @ ( numeral_numeral @ A @ W ) @ C3 ) ) )
& ( ( C3
= ( zero_zero @ A ) )
=> ( ( numeral_numeral @ A @ W )
= ( zero_zero @ A ) ) ) ) ) ) ).
% divide_eq_eq_numeral(1)
thf(fact_1027_power__inject__base,axiom,
! [A: $tType] :
( ( linordered_semidom @ A )
=> ! [A3: A,N: nat,B3: A] :
( ( ( power_power @ A @ A3 @ ( suc @ N ) )
= ( power_power @ A @ B3 @ ( suc @ N ) ) )
=> ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ A3 )
=> ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ B3 )
=> ( A3 = B3 ) ) ) ) ) ).
% power_inject_base
thf(fact_1028_power__le__imp__le__base,axiom,
! [A: $tType] :
( ( linordered_semidom @ A )
=> ! [A3: A,N: nat,B3: A] :
( ( ord_less_eq @ A @ ( power_power @ A @ A3 @ ( suc @ N ) ) @ ( power_power @ A @ B3 @ ( suc @ N ) ) )
=> ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ B3 )
=> ( ord_less_eq @ A @ A3 @ B3 ) ) ) ) ).
% power_le_imp_le_base
thf(fact_1029_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_1030_div__add__self2,axiom,
! [A: $tType] :
( ( euclid4440199948858584721cancel @ A )
=> ! [B3: A,A3: A] :
( ( B3
!= ( zero_zero @ A ) )
=> ( ( divide_divide @ A @ ( plus_plus @ A @ A3 @ B3 ) @ B3 )
= ( plus_plus @ A @ ( divide_divide @ A @ A3 @ B3 ) @ ( one_one @ A ) ) ) ) ) ).
% div_add_self2
thf(fact_1031_div__add__self1,axiom,
! [A: $tType] :
( ( euclid4440199948858584721cancel @ A )
=> ! [B3: A,A3: A] :
( ( B3
!= ( zero_zero @ A ) )
=> ( ( divide_divide @ A @ ( plus_plus @ A @ B3 @ A3 ) @ B3 )
= ( plus_plus @ A @ ( divide_divide @ A @ A3 @ B3 ) @ ( one_one @ A ) ) ) ) ) ).
% div_add_self1
thf(fact_1032_numeral__1__eq__Suc__0,axiom,
( ( numeral_numeral @ nat @ one2 )
= ( suc @ ( zero_zero @ nat ) ) ) ).
% numeral_1_eq_Suc_0
thf(fact_1033_num_Osize_I5_J,axiom,
! [X22: num] :
( ( size_size @ num @ ( bit0 @ X22 ) )
= ( plus_plus @ nat @ ( size_size @ num @ X22 ) @ ( suc @ ( zero_zero @ nat ) ) ) ) ).
% num.size(5)
thf(fact_1034_ex__least__nat__less,axiom,
! [P: nat > $o,N: nat] :
( ( P @ N )
=> ( ~ ( P @ ( zero_zero @ nat ) )
=> ? [K2: nat] :
( ( ord_less @ nat @ K2 @ N )
& ! [I5: nat] :
( ( ord_less_eq @ nat @ I5 @ K2 )
=> ~ ( P @ I5 ) )
& ( P @ ( suc @ K2 ) ) ) ) ) ).
% ex_least_nat_less
thf(fact_1035_nat__induct__non__zero,axiom,
! [N: nat,P: nat > $o] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ N )
=> ( ( P @ ( one_one @ nat ) )
=> ( ! [N3: nat] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ N3 )
=> ( ( P @ N3 )
=> ( P @ ( suc @ N3 ) ) ) )
=> ( P @ N ) ) ) ) ).
% nat_induct_non_zero
thf(fact_1036_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_1037_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_1038_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_1039_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_1040_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_1041_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_1042_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_1043_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_1044_nat__diff__split,axiom,
! [P: nat > $o,A3: nat,B3: nat] :
( ( P @ ( minus_minus @ nat @ A3 @ B3 ) )
= ( ( ( ord_less @ nat @ A3 @ B3 )
=> ( P @ ( zero_zero @ nat ) ) )
& ! [D3: nat] :
( ( A3
= ( plus_plus @ nat @ B3 @ D3 ) )
=> ( P @ D3 ) ) ) ) ).
% nat_diff_split
thf(fact_1045_nat__diff__split__asm,axiom,
! [P: nat > $o,A3: nat,B3: nat] :
( ( P @ ( minus_minus @ nat @ A3 @ B3 ) )
= ( ~ ( ( ( ord_less @ nat @ A3 @ B3 )
& ~ ( P @ ( zero_zero @ nat ) ) )
| ? [D3: nat] :
( ( A3
= ( plus_plus @ nat @ B3 @ D3 ) )
& ~ ( P @ D3 ) ) ) ) ) ).
% nat_diff_split_asm
thf(fact_1046_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_1047_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_1048_length__pos__if__in__set,axiom,
! [A: $tType,X2: A,Xs: list @ A] :
( ( member @ A @ X2 @ ( set2 @ A @ Xs ) )
=> ( ord_less @ nat @ ( zero_zero @ nat ) @ ( size_size @ ( list @ A ) @ Xs ) ) ) ).
% length_pos_if_in_set
thf(fact_1049_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_1050_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_1051_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_1052_td__gal__lt,axiom,
! [C3: nat,A3: nat,B3: nat] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ C3 )
=> ( ( ord_less @ nat @ A3 @ ( times_times @ nat @ B3 @ C3 ) )
= ( ord_less @ nat @ ( divide_divide @ nat @ A3 @ C3 ) @ B3 ) ) ) ).
% td_gal_lt
thf(fact_1053_vebt__insert_Osimps_I3_J,axiom,
! [Info: option @ ( product_prod @ nat @ nat ),Ts: list @ vEBT_VEBT,S2: vEBT_VEBT,X2: nat] :
( ( vEBT_vebt_insert @ ( vEBT_Node @ Info @ ( suc @ ( zero_zero @ nat ) ) @ Ts @ S2 ) @ X2 )
= ( vEBT_Node @ Info @ ( suc @ ( zero_zero @ nat ) ) @ Ts @ S2 ) ) ).
% vebt_insert.simps(3)
thf(fact_1054_vebt__member_Osimps_I3_J,axiom,
! [V: product_prod @ nat @ nat,Uy: list @ vEBT_VEBT,Uz: vEBT_VEBT,X2: nat] :
~ ( vEBT_vebt_member @ ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ V ) @ ( zero_zero @ nat ) @ Uy @ Uz ) @ X2 ) ).
% vebt_member.simps(3)
thf(fact_1055_T_092_060_094sub_062i_092_060_094sub_062n_092_060_094sub_062s_092_060_094sub_062e_092_060_094sub_062r_092_060_094sub_062t_H_Osimps_I2_J,axiom,
! [Info: option @ ( product_prod @ nat @ nat ),Ts: list @ vEBT_VEBT,S2: vEBT_VEBT,X2: nat] :
( ( vEBT_T_i_n_s_e_r_t2 @ ( vEBT_Node @ Info @ ( zero_zero @ nat ) @ Ts @ S2 ) @ X2 )
= ( one_one @ nat ) ) ).
% T\<^sub>i\<^sub>n\<^sub>s\<^sub>e\<^sub>r\<^sub>t'.simps(2)
thf(fact_1056_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_1057_mult__less__cancel__right2,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [A3: A,C3: A] :
( ( ord_less @ A @ ( times_times @ A @ A3 @ C3 ) @ C3 )
= ( ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ C3 )
=> ( ord_less @ A @ A3 @ ( one_one @ A ) ) )
& ( ( ord_less_eq @ A @ C3 @ ( zero_zero @ A ) )
=> ( ord_less @ A @ ( one_one @ A ) @ A3 ) ) ) ) ) ).
% mult_less_cancel_right2
thf(fact_1058_mult__less__cancel__right1,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [C3: A,B3: A] :
( ( ord_less @ A @ C3 @ ( times_times @ A @ B3 @ C3 ) )
= ( ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ C3 )
=> ( ord_less @ A @ ( one_one @ A ) @ B3 ) )
& ( ( ord_less_eq @ A @ C3 @ ( zero_zero @ A ) )
=> ( ord_less @ A @ B3 @ ( one_one @ A ) ) ) ) ) ) ).
% mult_less_cancel_right1
thf(fact_1059_mult__less__cancel__left2,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [C3: A,A3: A] :
( ( ord_less @ A @ ( times_times @ A @ C3 @ A3 ) @ C3 )
= ( ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ C3 )
=> ( ord_less @ A @ A3 @ ( one_one @ A ) ) )
& ( ( ord_less_eq @ A @ C3 @ ( zero_zero @ A ) )
=> ( ord_less @ A @ ( one_one @ A ) @ A3 ) ) ) ) ) ).
% mult_less_cancel_left2
thf(fact_1060_mult__less__cancel__left1,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [C3: A,B3: A] :
( ( ord_less @ A @ C3 @ ( times_times @ A @ C3 @ B3 ) )
= ( ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ C3 )
=> ( ord_less @ A @ ( one_one @ A ) @ B3 ) )
& ( ( ord_less_eq @ A @ C3 @ ( zero_zero @ A ) )
=> ( ord_less @ A @ B3 @ ( one_one @ A ) ) ) ) ) ) ).
% mult_less_cancel_left1
thf(fact_1061_mult__le__cancel__right2,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [A3: A,C3: A] :
( ( ord_less_eq @ A @ ( times_times @ A @ A3 @ C3 ) @ C3 )
= ( ( ( ord_less @ A @ ( zero_zero @ A ) @ C3 )
=> ( ord_less_eq @ A @ A3 @ ( one_one @ A ) ) )
& ( ( ord_less @ A @ C3 @ ( zero_zero @ A ) )
=> ( ord_less_eq @ A @ ( one_one @ A ) @ A3 ) ) ) ) ) ).
% mult_le_cancel_right2
thf(fact_1062_mult__le__cancel__right1,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [C3: A,B3: A] :
( ( ord_less_eq @ A @ C3 @ ( times_times @ A @ B3 @ C3 ) )
= ( ( ( ord_less @ A @ ( zero_zero @ A ) @ C3 )
=> ( ord_less_eq @ A @ ( one_one @ A ) @ B3 ) )
& ( ( ord_less @ A @ C3 @ ( zero_zero @ A ) )
=> ( ord_less_eq @ A @ B3 @ ( one_one @ A ) ) ) ) ) ) ).
% mult_le_cancel_right1
thf(fact_1063_mult__le__cancel__left2,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [C3: A,A3: A] :
( ( ord_less_eq @ A @ ( times_times @ A @ C3 @ A3 ) @ C3 )
= ( ( ( ord_less @ A @ ( zero_zero @ A ) @ C3 )
=> ( ord_less_eq @ A @ A3 @ ( one_one @ A ) ) )
& ( ( ord_less @ A @ C3 @ ( zero_zero @ A ) )
=> ( ord_less_eq @ A @ ( one_one @ A ) @ A3 ) ) ) ) ) ).
% mult_le_cancel_left2
thf(fact_1064_mult__le__cancel__left1,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [C3: A,B3: A] :
( ( ord_less_eq @ A @ C3 @ ( times_times @ A @ C3 @ B3 ) )
= ( ( ( ord_less @ A @ ( zero_zero @ A ) @ C3 )
=> ( ord_less_eq @ A @ ( one_one @ A ) @ B3 ) )
& ( ( ord_less @ A @ C3 @ ( zero_zero @ A ) )
=> ( ord_less_eq @ A @ B3 @ ( one_one @ A ) ) ) ) ) ) ).
% mult_le_cancel_left1
thf(fact_1065_convex__bound__le,axiom,
! [A: $tType] :
( ( linord6961819062388156250ring_1 @ A )
=> ! [X2: A,A3: A,Y3: A,U: A,V: A] :
( ( ord_less_eq @ A @ X2 @ A3 )
=> ( ( ord_less_eq @ A @ Y3 @ A3 )
=> ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ U )
=> ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ V )
=> ( ( ( plus_plus @ A @ U @ V )
= ( one_one @ A ) )
=> ( ord_less_eq @ A @ ( plus_plus @ A @ ( times_times @ A @ U @ X2 ) @ ( times_times @ A @ V @ Y3 ) ) @ A3 ) ) ) ) ) ) ) ).
% convex_bound_le
thf(fact_1066_less__divide__eq__numeral_I1_J,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [W: num,B3: A,C3: A] :
( ( ord_less @ A @ ( numeral_numeral @ A @ W ) @ ( divide_divide @ A @ B3 @ C3 ) )
= ( ( ( ord_less @ A @ ( zero_zero @ A ) @ C3 )
=> ( ord_less @ A @ ( times_times @ A @ ( numeral_numeral @ A @ W ) @ C3 ) @ B3 ) )
& ( ~ ( ord_less @ A @ ( zero_zero @ A ) @ C3 )
=> ( ( ( ord_less @ A @ C3 @ ( zero_zero @ A ) )
=> ( ord_less @ A @ B3 @ ( times_times @ A @ ( numeral_numeral @ A @ W ) @ C3 ) ) )
& ( ~ ( ord_less @ A @ C3 @ ( zero_zero @ A ) )
=> ( ord_less @ A @ ( numeral_numeral @ A @ W ) @ ( zero_zero @ A ) ) ) ) ) ) ) ) ).
% less_divide_eq_numeral(1)
thf(fact_1067_divide__less__eq__numeral_I1_J,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [B3: A,C3: A,W: num] :
( ( ord_less @ A @ ( divide_divide @ A @ B3 @ C3 ) @ ( numeral_numeral @ A @ W ) )
= ( ( ( ord_less @ A @ ( zero_zero @ A ) @ C3 )
=> ( ord_less @ A @ B3 @ ( times_times @ A @ ( numeral_numeral @ A @ W ) @ C3 ) ) )
& ( ~ ( ord_less @ A @ ( zero_zero @ A ) @ C3 )
=> ( ( ( ord_less @ A @ C3 @ ( zero_zero @ A ) )
=> ( ord_less @ A @ ( times_times @ A @ ( numeral_numeral @ A @ W ) @ C3 ) @ B3 ) )
& ( ~ ( ord_less @ A @ C3 @ ( zero_zero @ A ) )
=> ( ord_less @ A @ ( zero_zero @ A ) @ ( numeral_numeral @ A @ W ) ) ) ) ) ) ) ) ).
% divide_less_eq_numeral(1)
thf(fact_1068_power__Suc__less,axiom,
! [A: $tType] :
( ( linordered_semidom @ A )
=> ! [A3: A,N: nat] :
( ( ord_less @ A @ ( zero_zero @ A ) @ A3 )
=> ( ( ord_less @ A @ A3 @ ( one_one @ A ) )
=> ( ord_less @ A @ ( times_times @ A @ A3 @ ( power_power @ A @ A3 @ N ) ) @ ( power_power @ A @ A3 @ N ) ) ) ) ) ).
% power_Suc_less
thf(fact_1069_power__Suc__le__self,axiom,
! [A: $tType] :
( ( linordered_semidom @ A )
=> ! [A3: A,N: nat] :
( ( ord_less_eq @ A @ ( zero_zero @ A ) @ A3 )
=> ( ( ord_less_eq @ A @ A3 @ ( one_one @ A ) )
=> ( ord_less_eq @ A @ ( power_power @ A @ A3 @ ( suc @ N ) ) @ A3 ) ) ) ) ).
% power_Suc_le_self
thf(fact_1070_power__Suc__less__one,axiom,
! [A: $tType] :
( ( linordered_semidom @ A )
=> ! [A3: A,N: nat] :
( ( ord_less @ A @ ( zero_zero @ A ) @ A3 )
=> ( ( ord_less @ A @ A3 @ ( one_one @ A ) )
=> ( ord_less @ A @ ( power_power @ A @ A3 @ ( suc @ N ) ) @ ( one_one @ A ) ) ) ) ) ).
% power_Suc_less_one
thf(fact_1071_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_1072_power__strict__decreasing,axiom,
! [A: $tType] :
( ( linordered_semidom @ A )
=> ! [N: nat,N7: nat,A3: A] :
( ( ord_less @ nat @ N @ N7 )
=> ( ( ord_less @ A @ ( zero_zero @ A ) @ A3 )
=> ( ( ord_less @ A @ A3 @ ( one_one @ A ) )
=> ( ord_less @ A @ ( power_power @ A @ A3 @ N7 ) @ ( power_power @ A @ A3 @ N ) ) ) ) ) ) ).
% power_strict_decreasing
thf(fact_1073_power__decreasing,axiom,
! [A: $tType] :
( ( linordered_semidom @ A )
=> ! [N: nat,N7: nat,A3: A] :
( ( ord_less_eq @ nat @ N @ N7 )
=> ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ A3 )
=> ( ( ord_less_eq @ A @ A3 @ ( one_one @ A ) )
=> ( ord_less_eq @ A @ ( power_power @ A @ A3 @ N7 ) @ ( power_power @ A @ A3 @ N ) ) ) ) ) ) ).
% power_decreasing
thf(fact_1074_numeral__2__eq__2,axiom,
( ( numeral_numeral @ nat @ ( bit0 @ one2 ) )
= ( suc @ ( suc @ ( zero_zero @ nat ) ) ) ) ).
% numeral_2_eq_2
thf(fact_1075_self__le__power,axiom,
! [A: $tType] :
( ( linordered_semidom @ A )
=> ! [A3: A,N: nat] :
( ( ord_less_eq @ A @ ( one_one @ A ) @ A3 )
=> ( ( ord_less @ nat @ ( zero_zero @ nat ) @ N )
=> ( ord_less_eq @ A @ A3 @ ( power_power @ A @ A3 @ N ) ) ) ) ) ).
% self_le_power
thf(fact_1076_pos2,axiom,
ord_less @ nat @ ( zero_zero @ nat ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ).
% pos2
thf(fact_1077_power__diff,axiom,
! [A: $tType] :
( ( semidom_divide @ A )
=> ! [A3: A,N: nat,M: nat] :
( ( A3
!= ( zero_zero @ A ) )
=> ( ( ord_less_eq @ nat @ N @ M )
=> ( ( power_power @ A @ A3 @ ( minus_minus @ nat @ M @ N ) )
= ( divide_divide @ A @ ( power_power @ A @ A3 @ M ) @ ( power_power @ A @ A3 @ N ) ) ) ) ) ) ).
% power_diff
thf(fact_1078_one__less__power,axiom,
! [A: $tType] :
( ( linordered_semidom @ A )
=> ! [A3: A,N: nat] :
( ( ord_less @ A @ ( one_one @ A ) @ A3 )
=> ( ( ord_less @ nat @ ( zero_zero @ nat ) @ N )
=> ( ord_less @ A @ ( one_one @ A ) @ ( power_power @ A @ A3 @ N ) ) ) ) ) ).
% one_less_power
thf(fact_1079_numeral__3__eq__3,axiom,
( ( numeral_numeral @ nat @ ( bit1 @ one2 ) )
= ( suc @ ( suc @ ( suc @ ( zero_zero @ nat ) ) ) ) ) ).
% numeral_3_eq_3
thf(fact_1080_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_1081_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_1082_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_1083_div__if,axiom,
( ( divide_divide @ nat )
= ( ^ [M6: nat,N4: nat] :
( if @ nat
@ ( ( ord_less @ nat @ M6 @ N4 )
| ( N4
= ( zero_zero @ nat ) ) )
@ ( zero_zero @ nat )
@ ( suc @ ( divide_divide @ nat @ ( minus_minus @ nat @ M6 @ N4 ) @ N4 ) ) ) ) ) ).
% div_if
thf(fact_1084_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_1085_add__eq__if,axiom,
( ( plus_plus @ nat )
= ( ^ [M6: nat,N4: nat] :
( if @ nat
@ ( M6
= ( zero_zero @ nat ) )
@ N4
@ ( suc @ ( plus_plus @ nat @ ( minus_minus @ nat @ M6 @ ( one_one @ nat ) ) @ N4 ) ) ) ) ) ).
% add_eq_if
thf(fact_1086_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_1087_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_1088_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_1089_split__div,axiom,
! [P: nat > $o,M: nat,N: nat] :
( ( P @ ( divide_divide @ nat @ M @ N ) )
= ( ( ( N
= ( zero_zero @ nat ) )
=> ( P @ ( zero_zero @ nat ) ) )
& ( ( N
!= ( zero_zero @ nat ) )
=> ! [I4: nat,J3: nat] :
( ( ord_less @ nat @ J3 @ N )
=> ( ( M
= ( plus_plus @ nat @ ( times_times @ nat @ N @ I4 ) @ J3 ) )
=> ( P @ I4 ) ) ) ) ) ) ).
% split_div
thf(fact_1090_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_1091_td__gal,axiom,
! [C3: nat,B3: nat,A3: nat] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ C3 )
=> ( ( ord_less_eq @ nat @ ( times_times @ nat @ B3 @ C3 ) @ A3 )
= ( ord_less_eq @ nat @ B3 @ ( divide_divide @ nat @ A3 @ C3 ) ) ) ) ).
% td_gal
thf(fact_1092_mult__eq__if,axiom,
( ( times_times @ nat )
= ( ^ [M6: nat,N4: nat] :
( if @ nat
@ ( M6
= ( zero_zero @ nat ) )
@ ( zero_zero @ nat )
@ ( plus_plus @ nat @ N4 @ ( times_times @ nat @ ( minus_minus @ nat @ M6 @ ( one_one @ nat ) ) @ N4 ) ) ) ) ) ).
% mult_eq_if
thf(fact_1093_nat__mult__power__less__eq,axiom,
! [B3: nat,A3: nat,N: nat,M: nat] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ B3 )
=> ( ( ord_less @ nat @ ( times_times @ nat @ A3 @ ( power_power @ nat @ B3 @ N ) ) @ ( power_power @ nat @ B3 @ M ) )
= ( ord_less @ nat @ A3 @ ( power_power @ nat @ B3 @ ( minus_minus @ nat @ M @ N ) ) ) ) ) ).
% nat_mult_power_less_eq
thf(fact_1094_vebt__delete_Osimps_I5_J,axiom,
! [Mi: nat,Ma: nat,TrLst: list @ vEBT_VEBT,Smry: vEBT_VEBT,X2: nat] :
( ( vEBT_vebt_delete @ ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi @ Ma ) ) @ ( zero_zero @ nat ) @ TrLst @ Smry ) @ X2 )
= ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi @ Ma ) ) @ ( zero_zero @ nat ) @ TrLst @ Smry ) ) ).
% vebt_delete.simps(5)
thf(fact_1095_vebt__member_Osimps_I4_J,axiom,
! [V: product_prod @ nat @ nat,Vb: list @ vEBT_VEBT,Vc: vEBT_VEBT,X2: nat] :
~ ( vEBT_vebt_member @ ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ V ) @ ( suc @ ( zero_zero @ nat ) ) @ Vb @ Vc ) @ X2 ) ).
% vebt_member.simps(4)
thf(fact_1096_T_092_060_094sub_062i_092_060_094sub_062n_092_060_094sub_062s_092_060_094sub_062e_092_060_094sub_062r_092_060_094sub_062t_H_Osimps_I3_J,axiom,
! [Info: option @ ( product_prod @ nat @ nat ),Ts: list @ vEBT_VEBT,S2: vEBT_VEBT,X2: nat] :
( ( vEBT_T_i_n_s_e_r_t2 @ ( vEBT_Node @ Info @ ( suc @ ( zero_zero @ nat ) ) @ Ts @ S2 ) @ X2 )
= ( one_one @ nat ) ) ).
% T\<^sub>i\<^sub>n\<^sub>s\<^sub>e\<^sub>r\<^sub>t'.simps(3)
thf(fact_1097_convex__bound__lt,axiom,
! [A: $tType] :
( ( linord715952674999750819strict @ A )
=> ! [X2: A,A3: A,Y3: A,U: A,V: A] :
( ( ord_less @ A @ X2 @ A3 )
=> ( ( ord_less @ A @ Y3 @ A3 )
=> ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ U )
=> ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ V )
=> ( ( ( plus_plus @ A @ U @ V )
= ( one_one @ A ) )
=> ( ord_less @ A @ ( plus_plus @ A @ ( times_times @ A @ U @ X2 ) @ ( times_times @ A @ V @ Y3 ) ) @ A3 ) ) ) ) ) ) ) ).
% convex_bound_lt
thf(fact_1098_le__divide__eq__numeral_I1_J,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [W: num,B3: A,C3: A] :
( ( ord_less_eq @ A @ ( numeral_numeral @ A @ W ) @ ( divide_divide @ A @ B3 @ C3 ) )
= ( ( ( ord_less @ A @ ( zero_zero @ A ) @ C3 )
=> ( ord_less_eq @ A @ ( times_times @ A @ ( numeral_numeral @ A @ W ) @ C3 ) @ B3 ) )
& ( ~ ( ord_less @ A @ ( zero_zero @ A ) @ C3 )
=> ( ( ( ord_less @ A @ C3 @ ( zero_zero @ A ) )
=> ( ord_less_eq @ A @ B3 @ ( times_times @ A @ ( numeral_numeral @ A @ W ) @ C3 ) ) )
& ( ~ ( ord_less @ A @ C3 @ ( zero_zero @ A ) )
=> ( ord_less_eq @ A @ ( numeral_numeral @ A @ W ) @ ( zero_zero @ A ) ) ) ) ) ) ) ) ).
% le_divide_eq_numeral(1)
thf(fact_1099_divide__le__eq__numeral_I1_J,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [B3: A,C3: A,W: num] :
( ( ord_less_eq @ A @ ( divide_divide @ A @ B3 @ C3 ) @ ( numeral_numeral @ A @ W ) )
= ( ( ( ord_less @ A @ ( zero_zero @ A ) @ C3 )
=> ( ord_less_eq @ A @ B3 @ ( times_times @ A @ ( numeral_numeral @ A @ W ) @ C3 ) ) )
& ( ~ ( ord_less @ A @ ( zero_zero @ A ) @ C3 )
=> ( ( ( ord_less @ A @ C3 @ ( zero_zero @ A ) )
=> ( ord_less_eq @ A @ ( times_times @ A @ ( numeral_numeral @ A @ W ) @ C3 ) @ B3 ) )
& ( ~ ( ord_less @ A @ C3 @ ( zero_zero @ A ) )
=> ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( numeral_numeral @ A @ W ) ) ) ) ) ) ) ) ).
% divide_le_eq_numeral(1)
thf(fact_1100_half__gt__zero__iff,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [A3: A] :
( ( ord_less @ A @ ( zero_zero @ A ) @ ( divide_divide @ A @ A3 @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) )
= ( ord_less @ A @ ( zero_zero @ A ) @ A3 ) ) ) ).
% half_gt_zero_iff
thf(fact_1101_half__gt__zero,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [A3: A] :
( ( ord_less @ A @ ( zero_zero @ A ) @ A3 )
=> ( ord_less @ A @ ( zero_zero @ A ) @ ( divide_divide @ A @ A3 @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) ) ) ) ).
% half_gt_zero
thf(fact_1102_power2__le__imp__le,axiom,
! [A: $tType] :
( ( linordered_semidom @ A )
=> ! [X2: A,Y3: A] :
( ( ord_less_eq @ A @ ( power_power @ A @ X2 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ ( power_power @ A @ Y3 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) )
=> ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ Y3 )
=> ( ord_less_eq @ A @ X2 @ Y3 ) ) ) ) ).
% power2_le_imp_le
thf(fact_1103_power2__eq__imp__eq,axiom,
! [A: $tType] :
( ( linordered_semidom @ A )
=> ! [X2: A,Y3: A] :
( ( ( power_power @ A @ X2 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) )
= ( power_power @ A @ Y3 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) )
=> ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ X2 )
=> ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ Y3 )
=> ( X2 = Y3 ) ) ) ) ) ).
% power2_eq_imp_eq
thf(fact_1104_zero__le__power2,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [A3: A] : ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( power_power @ A @ A3 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ).
% zero_le_power2
thf(fact_1105_power2__less__0,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [A3: A] :
~ ( ord_less @ A @ ( power_power @ A @ A3 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ ( zero_zero @ A ) ) ) ).
% power2_less_0
thf(fact_1106_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_1107_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_1108_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_1109_power__diff__power__eq,axiom,
! [A: $tType] :
( ( euclid4440199948858584721cancel @ A )
=> ! [A3: A,N: nat,M: nat] :
( ( A3
!= ( zero_zero @ A ) )
=> ( ( ( ord_less_eq @ nat @ N @ M )
=> ( ( divide_divide @ A @ ( power_power @ A @ A3 @ M ) @ ( power_power @ A @ A3 @ N ) )
= ( power_power @ A @ A3 @ ( minus_minus @ nat @ M @ N ) ) ) )
& ( ~ ( ord_less_eq @ nat @ N @ M )
=> ( ( divide_divide @ A @ ( power_power @ A @ A3 @ M ) @ ( power_power @ A @ A3 @ N ) )
= ( divide_divide @ A @ ( one_one @ A ) @ ( power_power @ A @ A3 @ ( minus_minus @ nat @ N @ M ) ) ) ) ) ) ) ) ).
% power_diff_power_eq
thf(fact_1110_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_1111_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_1112_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_1113_power__eq__if,axiom,
! [A: $tType] :
( ( power @ A )
=> ( ( power_power @ A )
= ( ^ [P4: A,M6: nat] :
( if @ A
@ ( M6
= ( zero_zero @ nat ) )
@ ( one_one @ A )
@ ( times_times @ A @ P4 @ ( power_power @ A @ P4 @ ( minus_minus @ nat @ M6 @ ( one_one @ nat ) ) ) ) ) ) ) ) ).
% power_eq_if
thf(fact_1114_power__minus__mult,axiom,
! [A: $tType] :
( ( monoid_mult @ A )
=> ! [N: nat,A3: A] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ N )
=> ( ( times_times @ A @ ( power_power @ A @ A3 @ ( minus_minus @ nat @ N @ ( one_one @ nat ) ) ) @ A3 )
= ( power_power @ A @ A3 @ N ) ) ) ) ).
% power_minus_mult
thf(fact_1115_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_1116_split__div_H,axiom,
! [P: nat > $o,M: nat,N: nat] :
( ( P @ ( divide_divide @ nat @ M @ N ) )
= ( ( ( N
= ( zero_zero @ nat ) )
& ( P @ ( zero_zero @ nat ) ) )
| ? [Q5: nat] :
( ( ord_less_eq @ nat @ ( times_times @ nat @ N @ Q5 ) @ M )
& ( ord_less @ nat @ M @ ( times_times @ nat @ N @ ( suc @ Q5 ) ) )
& ( P @ Q5 ) ) ) ) ).
% split_div'
thf(fact_1117_power__sub,axiom,
! [N: nat,M: nat,A3: nat] :
( ( ord_less_eq @ nat @ N @ M )
=> ( ( ord_less @ nat @ ( zero_zero @ nat ) @ A3 )
=> ( ( power_power @ nat @ A3 @ ( minus_minus @ nat @ M @ N ) )
= ( divide_divide @ nat @ ( power_power @ nat @ A3 @ M ) @ ( power_power @ nat @ A3 @ N ) ) ) ) ) ).
% power_sub
thf(fact_1118_vebt__delete_Osimps_I6_J,axiom,
! [Mi: nat,Ma: nat,Tr: list @ vEBT_VEBT,Sm: vEBT_VEBT,X2: nat] :
( ( vEBT_vebt_delete @ ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi @ Ma ) ) @ ( suc @ ( zero_zero @ nat ) ) @ Tr @ Sm ) @ X2 )
= ( 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_1119_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,X2: 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 ) @ X2 )
= ( one_one @ nat ) ) ).
% T\<^sub>d\<^sub>e\<^sub>l\<^sub>e\<^sub>t\<^sub>e.simps(5)
thf(fact_1120_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,X2: nat] :
( ( vEBT_V1232361888498592333_e_t_e @ ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi @ Ma ) ) @ ( zero_zero @ nat ) @ TreeList @ Summary ) @ X2 )
= ( one_one @ nat ) ) ).
% VEBT_internal.T\<^sub>d\<^sub>e\<^sub>l\<^sub>e\<^sub>t\<^sub>e'.simps(5)
thf(fact_1121_power2__less__imp__less,axiom,
! [A: $tType] :
( ( linordered_semidom @ A )
=> ! [X2: A,Y3: A] :
( ( ord_less @ A @ ( power_power @ A @ X2 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ ( power_power @ A @ Y3 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) )
=> ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ Y3 )
=> ( ord_less @ A @ X2 @ Y3 ) ) ) ) ).
% power2_less_imp_less
thf(fact_1122_sum__power2__le__zero__iff,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [X2: A,Y3: A] :
( ( ord_less_eq @ A @ ( plus_plus @ A @ ( power_power @ A @ X2 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ ( power_power @ A @ Y3 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( zero_zero @ A ) )
= ( ( X2
= ( zero_zero @ A ) )
& ( Y3
= ( zero_zero @ A ) ) ) ) ) ).
% sum_power2_le_zero_iff
thf(fact_1123_sum__power2__ge__zero,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [X2: A,Y3: A] : ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( plus_plus @ A @ ( power_power @ A @ X2 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ ( power_power @ A @ Y3 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) ).
% sum_power2_ge_zero
thf(fact_1124_sum__power2__gt__zero__iff,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [X2: A,Y3: A] :
( ( ord_less @ A @ ( zero_zero @ A ) @ ( plus_plus @ A @ ( power_power @ A @ X2 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ ( power_power @ A @ Y3 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) )
= ( ( X2
!= ( zero_zero @ A ) )
| ( Y3
!= ( zero_zero @ A ) ) ) ) ) ).
% sum_power2_gt_zero_iff
thf(fact_1125_not__sum__power2__lt__zero,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [X2: A,Y3: A] :
~ ( ord_less @ A @ ( plus_plus @ A @ ( power_power @ A @ X2 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ ( power_power @ A @ Y3 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( zero_zero @ A ) ) ) ).
% not_sum_power2_lt_zero
thf(fact_1126_zero__le__even__power_H,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [A3: A,N: nat] : ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( power_power @ A @ A3 @ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N ) ) ) ) ).
% zero_le_even_power'
thf(fact_1127_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_1128_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_1129_nat__bit__induct,axiom,
! [P: nat > $o,N: nat] :
( ( P @ ( zero_zero @ nat ) )
=> ( ! [N3: nat] :
( ( P @ N3 )
=> ( ( ord_less @ nat @ ( zero_zero @ nat ) @ N3 )
=> ( P @ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N3 ) ) ) )
=> ( ! [N3: nat] :
( ( P @ N3 )
=> ( P @ ( suc @ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N3 ) ) ) )
=> ( P @ N ) ) ) ) ).
% nat_bit_induct
thf(fact_1130_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,
! [V: product_prod @ nat @ nat,Uy: list @ vEBT_VEBT,Uz: vEBT_VEBT,X2: nat] :
( ( vEBT_T_m_e_m_b_e_r @ ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ V ) @ ( zero_zero @ nat ) @ Uy @ Uz ) @ X2 )
= ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ).
% T\<^sub>m\<^sub>e\<^sub>m\<^sub>b\<^sub>e\<^sub>r.simps(3)
thf(fact_1131_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,X2: 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 ) @ X2 )
= ( one_one @ nat ) ) ).
% T\<^sub>d\<^sub>e\<^sub>l\<^sub>e\<^sub>t\<^sub>e.simps(6)
thf(fact_1132_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,X2: 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 ) @ X2 )
= ( one_one @ nat ) ) ).
% VEBT_internal.T\<^sub>d\<^sub>e\<^sub>l\<^sub>e\<^sub>t\<^sub>e'.simps(6)
thf(fact_1133_odd__0__le__power__imp__0__le,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [A3: A,N: nat] :
( ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( power_power @ A @ A3 @ ( suc @ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N ) ) ) )
=> ( ord_less_eq @ A @ ( zero_zero @ A ) @ A3 ) ) ) ).
% odd_0_le_power_imp_0_le
thf(fact_1134_odd__power__less__zero,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [A3: A,N: nat] :
( ( ord_less @ A @ A3 @ ( zero_zero @ A ) )
=> ( ord_less @ A @ ( power_power @ A @ A3 @ ( suc @ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N ) ) ) @ ( zero_zero @ A ) ) ) ) ).
% odd_power_less_zero
thf(fact_1135_VEBTi_Oexhaust,axiom,
! [Y3: vEBT_VEBTi] :
( ! [X112: option @ ( product_prod @ nat @ nat ),X122: nat,X132: array @ vEBT_VEBTi,X142: vEBT_VEBTi] :
( Y3
!= ( vEBT_Nodei @ X112 @ X122 @ X132 @ X142 ) )
=> ~ ! [X212: $o,X223: $o] :
( Y3
!= ( vEBT_Leafi @ X212 @ X223 ) ) ) ).
% VEBTi.exhaust
thf(fact_1136_VEBTi_Odistinct_I1_J,axiom,
! [X11: option @ ( product_prod @ nat @ nat ),X12: nat,X13: array @ vEBT_VEBTi,X14: vEBT_VEBTi,X213: $o,X224: $o] :
( ( vEBT_Nodei @ X11 @ X12 @ X13 @ X14 )
!= ( vEBT_Leafi @ X213 @ X224 ) ) ).
% VEBTi.distinct(1)
thf(fact_1137_VEBT__internal_Oexp__split__high__low_I1_J,axiom,
! [X2: nat,N: nat,M: nat] :
( ( ord_less @ nat @ X2 @ ( 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 @ X2 @ N ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ M ) ) ) ) ) ).
% VEBT_internal.exp_split_high_low(1)
thf(fact_1138_VEBT__internal_Oexp__split__high__low_I2_J,axiom,
! [X2: nat,N: nat,M: nat] :
( ( ord_less @ nat @ X2 @ ( 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 @ X2 @ N ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N ) ) ) ) ) ).
% VEBT_internal.exp_split_high_low(2)
thf(fact_1139_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,
! [V: product_prod @ nat @ nat,Vb: list @ vEBT_VEBT,Vc: vEBT_VEBT,X2: nat] :
( ( vEBT_T_m_e_m_b_e_r @ ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ V ) @ ( suc @ ( zero_zero @ nat ) ) @ Vb @ Vc ) @ X2 )
= ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ).
% T\<^sub>m\<^sub>e\<^sub>m\<^sub>b\<^sub>e\<^sub>r.simps(4)
thf(fact_1140_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_1141_arith__geo__mean,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [U: A,X2: A,Y3: A] :
( ( ( power_power @ A @ U @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) )
= ( times_times @ A @ X2 @ Y3 ) )
=> ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ X2 )
=> ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ Y3 )
=> ( ord_less_eq @ A @ U @ ( divide_divide @ A @ ( plus_plus @ A @ X2 @ Y3 ) @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) ) ) ) ) ) ).
% arith_geo_mean
thf(fact_1142_member__bound__size__univ,axiom,
! [T2: vEBT_VEBT,N: nat,U: real,X2: 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 @ X2 ) ) @ ( 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_1143_VEBT__internal_OminNulli_Ocases,axiom,
! [X2: vEBT_VEBTi] :
( ( X2
!= ( vEBT_Leafi @ $false @ $false ) )
=> ( ! [Uv2: $o] :
( X2
!= ( vEBT_Leafi @ $true @ Uv2 ) )
=> ( ! [Uu2: $o] :
( X2
!= ( vEBT_Leafi @ Uu2 @ $true ) )
=> ( ! [Uw2: nat,Ux2: array @ vEBT_VEBTi,Uy2: vEBT_VEBTi] :
( X2
!= ( vEBT_Nodei @ ( none @ ( product_prod @ nat @ nat ) ) @ Uw2 @ Ux2 @ Uy2 ) )
=> ~ ! [Uz2: product_prod @ nat @ nat,Va2: nat,Vb2: array @ vEBT_VEBTi,Vc2: vEBT_VEBTi] :
( X2
!= ( vEBT_Nodei @ ( some @ ( product_prod @ nat @ nat ) @ Uz2 ) @ Va2 @ Vb2 @ Vc2 ) ) ) ) ) ) ).
% VEBT_internal.minNulli.cases
thf(fact_1144_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_1145_le__divide__eq__1__pos,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [A3: A,B3: A] :
( ( ord_less @ A @ ( zero_zero @ A ) @ A3 )
=> ( ( ord_less_eq @ A @ ( one_one @ A ) @ ( divide_divide @ A @ B3 @ A3 ) )
= ( ord_less_eq @ A @ A3 @ B3 ) ) ) ) ).
% le_divide_eq_1_pos
thf(fact_1146_le__divide__eq__1__neg,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [A3: A,B3: A] :
( ( ord_less @ A @ A3 @ ( zero_zero @ A ) )
=> ( ( ord_less_eq @ A @ ( one_one @ A ) @ ( divide_divide @ A @ B3 @ A3 ) )
= ( ord_less_eq @ A @ B3 @ A3 ) ) ) ) ).
% le_divide_eq_1_neg
thf(fact_1147_divide__le__eq__1__pos,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [A3: A,B3: A] :
( ( ord_less @ A @ ( zero_zero @ A ) @ A3 )
=> ( ( ord_less_eq @ A @ ( divide_divide @ A @ B3 @ A3 ) @ ( one_one @ A ) )
= ( ord_less_eq @ A @ B3 @ A3 ) ) ) ) ).
% divide_le_eq_1_pos
thf(fact_1148_divide__le__eq__1__neg,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [A3: A,B3: A] :
( ( ord_less @ A @ A3 @ ( zero_zero @ A ) )
=> ( ( ord_less_eq @ A @ ( divide_divide @ A @ B3 @ A3 ) @ ( one_one @ A ) )
= ( ord_less_eq @ A @ A3 @ B3 ) ) ) ) ).
% divide_le_eq_1_neg
thf(fact_1149_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_1150_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_1151_TBOUND__minNulli,axiom,
! [T2: vEBT_VEBTi] : ( time_TBOUND @ $o @ ( vEBT_VEBT_minNulli @ T2 ) @ ( one_one @ nat ) ) ).
% TBOUND_minNulli
thf(fact_1152_nonzero__divide__mult__cancel__right,axiom,
! [A: $tType] :
( ( field @ A )
=> ! [B3: A,A3: A] :
( ( B3
!= ( zero_zero @ A ) )
=> ( ( divide_divide @ A @ B3 @ ( times_times @ A @ A3 @ B3 ) )
= ( divide_divide @ A @ ( one_one @ A ) @ A3 ) ) ) ) ).
% nonzero_divide_mult_cancel_right
thf(fact_1153_nonzero__divide__mult__cancel__left,axiom,
! [A: $tType] :
( ( field @ A )
=> ! [A3: A,B3: A] :
( ( A3
!= ( zero_zero @ A ) )
=> ( ( divide_divide @ A @ A3 @ ( times_times @ A @ A3 @ B3 ) )
= ( divide_divide @ A @ ( one_one @ A ) @ B3 ) ) ) ) ).
% nonzero_divide_mult_cancel_left
thf(fact_1154_cnt__non__neg,axiom,
! [T2: vEBT_VEBT] : ( ord_less_eq @ real @ ( zero_zero @ real ) @ ( vEBT_VEBT_cnt @ T2 ) ) ).
% cnt_non_neg
thf(fact_1155_Tb__Tb_H,axiom,
( vEBT_VEBT_Tb
= ( ^ [T3: nat] : ( semiring_1_of_nat @ int @ ( vEBT_VEBT_Tb2 @ T3 ) ) ) ) ).
% Tb_Tb'
thf(fact_1156_i0__less,axiom,
! [N: extended_enat] :
( ( ord_less @ extended_enat @ ( zero_zero @ extended_enat ) @ N )
= ( N
!= ( zero_zero @ extended_enat ) ) ) ).
% i0_less
thf(fact_1157_idiff__0,axiom,
! [N: extended_enat] :
( ( minus_minus @ extended_enat @ ( zero_zero @ extended_enat ) @ N )
= ( zero_zero @ extended_enat ) ) ).
% idiff_0
thf(fact_1158_idiff__0__right,axiom,
! [N: extended_enat] :
( ( minus_minus @ extended_enat @ N @ ( zero_zero @ extended_enat ) )
= N ) ).
% idiff_0_right
thf(fact_1159_divide__eq__0__iff,axiom,
! [A: $tType] :
( ( field @ A )
=> ! [A3: A,B3: A] :
( ( ( divide_divide @ A @ A3 @ B3 )
= ( zero_zero @ A ) )
= ( ( A3
= ( zero_zero @ A ) )
| ( B3
= ( zero_zero @ A ) ) ) ) ) ).
% divide_eq_0_iff
thf(fact_1160_divide__cancel__left,axiom,
! [A: $tType] :
( ( field @ A )
=> ! [C3: A,A3: A,B3: A] :
( ( ( divide_divide @ A @ C3 @ A3 )
= ( divide_divide @ A @ C3 @ B3 ) )
= ( ( C3
= ( zero_zero @ A ) )
| ( A3 = B3 ) ) ) ) ).
% divide_cancel_left
thf(fact_1161_divide__cancel__right,axiom,
! [A: $tType] :
( ( field @ A )
=> ! [A3: A,C3: A,B3: A] :
( ( ( divide_divide @ A @ A3 @ C3 )
= ( divide_divide @ A @ B3 @ C3 ) )
= ( ( C3
= ( zero_zero @ A ) )
| ( A3 = B3 ) ) ) ) ).
% divide_cancel_right
thf(fact_1162_division__ring__divide__zero,axiom,
! [A: $tType] :
( ( division_ring @ A )
=> ! [A3: A] :
( ( divide_divide @ A @ A3 @ ( zero_zero @ A ) )
= ( zero_zero @ A ) ) ) ).
% division_ring_divide_zero
thf(fact_1163_times__divide__eq__left,axiom,
! [A: $tType] :
( ( field @ A )
=> ! [B3: A,C3: A,A3: A] :
( ( times_times @ A @ ( divide_divide @ A @ B3 @ C3 ) @ A3 )
= ( divide_divide @ A @ ( times_times @ A @ B3 @ A3 ) @ C3 ) ) ) ).
% times_divide_eq_left
thf(fact_1164_divide__divide__eq__left,axiom,
! [A: $tType] :
( ( field @ A )
=> ! [A3: A,B3: A,C3: A] :
( ( divide_divide @ A @ ( divide_divide @ A @ A3 @ B3 ) @ C3 )
= ( divide_divide @ A @ A3 @ ( times_times @ A @ B3 @ C3 ) ) ) ) ).
% divide_divide_eq_left
thf(fact_1165_divide__divide__eq__right,axiom,
! [A: $tType] :
( ( field @ A )
=> ! [A3: A,B3: A,C3: A] :
( ( divide_divide @ A @ A3 @ ( divide_divide @ A @ B3 @ C3 ) )
= ( divide_divide @ A @ ( times_times @ A @ A3 @ C3 ) @ B3 ) ) ) ).
% divide_divide_eq_right
thf(fact_1166_times__divide__eq__right,axiom,
! [A: $tType] :
( ( division_ring @ A )
=> ! [A3: A,B3: A,C3: A] :
( ( times_times @ A @ A3 @ ( divide_divide @ A @ B3 @ C3 ) )
= ( divide_divide @ A @ ( times_times @ A @ A3 @ B3 ) @ C3 ) ) ) ).
% times_divide_eq_right
thf(fact_1167_log__one,axiom,
! [A3: real] :
( ( log @ A3 @ ( one_one @ real ) )
= ( zero_zero @ real ) ) ).
% log_one
thf(fact_1168_divide__eq__1__iff,axiom,
! [A: $tType] :
( ( field @ A )
=> ! [A3: A,B3: A] :
( ( ( divide_divide @ A @ A3 @ B3 )
= ( one_one @ A ) )
= ( ( B3
!= ( zero_zero @ A ) )
& ( A3 = B3 ) ) ) ) ).
% divide_eq_1_iff
thf(fact_1169_one__eq__divide__iff,axiom,
! [A: $tType] :
( ( field @ A )
=> ! [A3: A,B3: A] :
( ( ( one_one @ A )
= ( divide_divide @ A @ A3 @ B3 ) )
= ( ( B3
!= ( zero_zero @ A ) )
& ( A3 = B3 ) ) ) ) ).
% one_eq_divide_iff
thf(fact_1170_divide__self,axiom,
! [A: $tType] :
( ( division_ring @ A )
=> ! [A3: A] :
( ( A3
!= ( zero_zero @ A ) )
=> ( ( divide_divide @ A @ A3 @ A3 )
= ( one_one @ A ) ) ) ) ).
% divide_self
thf(fact_1171_divide__self__if,axiom,
! [A: $tType] :
( ( division_ring @ A )
=> ! [A3: A] :
( ( ( A3
= ( zero_zero @ A ) )
=> ( ( divide_divide @ A @ A3 @ A3 )
= ( zero_zero @ A ) ) )
& ( ( A3
!= ( zero_zero @ A ) )
=> ( ( divide_divide @ A @ A3 @ A3 )
= ( one_one @ A ) ) ) ) ) ).
% divide_self_if
thf(fact_1172_divide__eq__eq__1,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [B3: A,A3: A] :
( ( ( divide_divide @ A @ B3 @ A3 )
= ( one_one @ A ) )
= ( ( A3
!= ( zero_zero @ A ) )
& ( A3 = B3 ) ) ) ) ).
% divide_eq_eq_1
thf(fact_1173_eq__divide__eq__1,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [B3: A,A3: A] :
( ( ( one_one @ A )
= ( divide_divide @ A @ B3 @ A3 ) )
= ( ( A3
!= ( zero_zero @ A ) )
& ( A3 = B3 ) ) ) ) ).
% eq_divide_eq_1
thf(fact_1174_one__divide__eq__0__iff,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [A3: A] :
( ( ( divide_divide @ A @ ( one_one @ A ) @ A3 )
= ( zero_zero @ A ) )
= ( A3
= ( zero_zero @ A ) ) ) ) ).
% one_divide_eq_0_iff
thf(fact_1175_zero__eq__1__divide__iff,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [A3: A] :
( ( ( zero_zero @ A )
= ( divide_divide @ A @ ( one_one @ A ) @ A3 ) )
= ( A3
= ( zero_zero @ A ) ) ) ) ).
% zero_eq_1_divide_iff
thf(fact_1176_mult__divide__mult__cancel__left__if,axiom,
! [A: $tType] :
( ( field @ A )
=> ! [C3: A,A3: A,B3: A] :
( ( ( C3
= ( zero_zero @ A ) )
=> ( ( divide_divide @ A @ ( times_times @ A @ C3 @ A3 ) @ ( times_times @ A @ C3 @ B3 ) )
= ( zero_zero @ A ) ) )
& ( ( C3
!= ( zero_zero @ A ) )
=> ( ( divide_divide @ A @ ( times_times @ A @ C3 @ A3 ) @ ( times_times @ A @ C3 @ B3 ) )
= ( divide_divide @ A @ A3 @ B3 ) ) ) ) ) ).
% mult_divide_mult_cancel_left_if
thf(fact_1177_nonzero__mult__divide__mult__cancel__left,axiom,
! [A: $tType] :
( ( field @ A )
=> ! [C3: A,A3: A,B3: A] :
( ( C3
!= ( zero_zero @ A ) )
=> ( ( divide_divide @ A @ ( times_times @ A @ C3 @ A3 ) @ ( times_times @ A @ C3 @ B3 ) )
= ( divide_divide @ A @ A3 @ B3 ) ) ) ) ).
% nonzero_mult_divide_mult_cancel_left
thf(fact_1178_nonzero__mult__divide__mult__cancel__left2,axiom,
! [A: $tType] :
( ( field @ A )
=> ! [C3: A,A3: A,B3: A] :
( ( C3
!= ( zero_zero @ A ) )
=> ( ( divide_divide @ A @ ( times_times @ A @ C3 @ A3 ) @ ( times_times @ A @ B3 @ C3 ) )
= ( divide_divide @ A @ A3 @ B3 ) ) ) ) ).
% nonzero_mult_divide_mult_cancel_left2
thf(fact_1179_nonzero__mult__divide__mult__cancel__right,axiom,
! [A: $tType] :
( ( field @ A )
=> ! [C3: A,A3: A,B3: A] :
( ( C3
!= ( zero_zero @ A ) )
=> ( ( divide_divide @ A @ ( times_times @ A @ A3 @ C3 ) @ ( times_times @ A @ B3 @ C3 ) )
= ( divide_divide @ A @ A3 @ B3 ) ) ) ) ).
% nonzero_mult_divide_mult_cancel_right
thf(fact_1180_nonzero__mult__divide__mult__cancel__right2,axiom,
! [A: $tType] :
( ( field @ A )
=> ! [C3: A,A3: A,B3: A] :
( ( C3
!= ( zero_zero @ A ) )
=> ( ( divide_divide @ A @ ( times_times @ A @ A3 @ C3 ) @ ( times_times @ A @ C3 @ B3 ) )
= ( divide_divide @ A @ A3 @ B3 ) ) ) ) ).
% nonzero_mult_divide_mult_cancel_right2
thf(fact_1181_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_1182_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_1183_int__div__same__is__1,axiom,
! [A3: int,B3: int] :
( ( ord_less @ int @ ( zero_zero @ int ) @ A3 )
=> ( ( ( divide_divide @ int @ A3 @ B3 )
= A3 )
= ( B3
= ( one_one @ int ) ) ) ) ).
% int_div_same_is_1
thf(fact_1184_log__eq__one,axiom,
! [A3: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ A3 )
=> ( ( A3
!= ( one_one @ real ) )
=> ( ( log @ A3 @ A3 )
= ( one_one @ real ) ) ) ) ).
% log_eq_one
thf(fact_1185_log__less__cancel__iff,axiom,
! [A3: real,X2: real,Y3: real] :
( ( ord_less @ real @ ( one_one @ real ) @ A3 )
=> ( ( ord_less @ real @ ( zero_zero @ real ) @ X2 )
=> ( ( ord_less @ real @ ( zero_zero @ real ) @ Y3 )
=> ( ( ord_less @ real @ ( log @ A3 @ X2 ) @ ( log @ A3 @ Y3 ) )
= ( ord_less @ real @ X2 @ Y3 ) ) ) ) ) ).
% log_less_cancel_iff
thf(fact_1186_log__less__one__cancel__iff,axiom,
! [A3: real,X2: real] :
( ( ord_less @ real @ ( one_one @ real ) @ A3 )
=> ( ( ord_less @ real @ ( zero_zero @ real ) @ X2 )
=> ( ( ord_less @ real @ ( log @ A3 @ X2 ) @ ( one_one @ real ) )
= ( ord_less @ real @ X2 @ A3 ) ) ) ) ).
% log_less_one_cancel_iff
thf(fact_1187_one__less__log__cancel__iff,axiom,
! [A3: real,X2: real] :
( ( ord_less @ real @ ( one_one @ real ) @ A3 )
=> ( ( ord_less @ real @ ( zero_zero @ real ) @ X2 )
=> ( ( ord_less @ real @ ( one_one @ real ) @ ( log @ A3 @ X2 ) )
= ( ord_less @ real @ A3 @ X2 ) ) ) ) ).
% one_less_log_cancel_iff
thf(fact_1188_log__less__zero__cancel__iff,axiom,
! [A3: real,X2: real] :
( ( ord_less @ real @ ( one_one @ real ) @ A3 )
=> ( ( ord_less @ real @ ( zero_zero @ real ) @ X2 )
=> ( ( ord_less @ real @ ( log @ A3 @ X2 ) @ ( zero_zero @ real ) )
= ( ord_less @ real @ X2 @ ( one_one @ real ) ) ) ) ) ).
% log_less_zero_cancel_iff
thf(fact_1189_zero__less__log__cancel__iff,axiom,
! [A3: real,X2: real] :
( ( ord_less @ real @ ( one_one @ real ) @ A3 )
=> ( ( ord_less @ real @ ( zero_zero @ real ) @ X2 )
=> ( ( ord_less @ real @ ( zero_zero @ real ) @ ( log @ A3 @ X2 ) )
= ( ord_less @ real @ ( one_one @ real ) @ X2 ) ) ) ) ).
% zero_less_log_cancel_iff
thf(fact_1190_zero__le__divide__1__iff,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [A3: A] :
( ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( divide_divide @ A @ ( one_one @ A ) @ A3 ) )
= ( ord_less_eq @ A @ ( zero_zero @ A ) @ A3 ) ) ) ).
% zero_le_divide_1_iff
thf(fact_1191_divide__le__0__1__iff,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [A3: A] :
( ( ord_less_eq @ A @ ( divide_divide @ A @ ( one_one @ A ) @ A3 ) @ ( zero_zero @ A ) )
= ( ord_less_eq @ A @ A3 @ ( zero_zero @ A ) ) ) ) ).
% divide_le_0_1_iff
thf(fact_1192_divide__less__0__1__iff,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [A3: A] :
( ( ord_less @ A @ ( divide_divide @ A @ ( one_one @ A ) @ A3 ) @ ( zero_zero @ A ) )
= ( ord_less @ A @ A3 @ ( zero_zero @ A ) ) ) ) ).
% divide_less_0_1_iff
thf(fact_1193_divide__less__eq__1__neg,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [A3: A,B3: A] :
( ( ord_less @ A @ A3 @ ( zero_zero @ A ) )
=> ( ( ord_less @ A @ ( divide_divide @ A @ B3 @ A3 ) @ ( one_one @ A ) )
= ( ord_less @ A @ A3 @ B3 ) ) ) ) ).
% divide_less_eq_1_neg
thf(fact_1194_divide__less__eq__1__pos,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [A3: A,B3: A] :
( ( ord_less @ A @ ( zero_zero @ A ) @ A3 )
=> ( ( ord_less @ A @ ( divide_divide @ A @ B3 @ A3 ) @ ( one_one @ A ) )
= ( ord_less @ A @ B3 @ A3 ) ) ) ) ).
% divide_less_eq_1_pos
thf(fact_1195_less__divide__eq__1__neg,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [A3: A,B3: A] :
( ( ord_less @ A @ A3 @ ( zero_zero @ A ) )
=> ( ( ord_less @ A @ ( one_one @ A ) @ ( divide_divide @ A @ B3 @ A3 ) )
= ( ord_less @ A @ B3 @ A3 ) ) ) ) ).
% less_divide_eq_1_neg
thf(fact_1196_less__divide__eq__1__pos,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [A3: A,B3: A] :
( ( ord_less @ A @ ( zero_zero @ A ) @ A3 )
=> ( ( ord_less @ A @ ( one_one @ A ) @ ( divide_divide @ A @ B3 @ A3 ) )
= ( ord_less @ A @ A3 @ B3 ) ) ) ) ).
% less_divide_eq_1_pos
thf(fact_1197_zero__less__divide__1__iff,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [A3: A] :
( ( ord_less @ A @ ( zero_zero @ A ) @ ( divide_divide @ A @ ( one_one @ A ) @ A3 ) )
= ( ord_less @ A @ ( zero_zero @ A ) @ A3 ) ) ) ).
% zero_less_divide_1_iff
thf(fact_1198_log__le__cancel__iff,axiom,
! [A3: real,X2: real,Y3: real] :
( ( ord_less @ real @ ( one_one @ real ) @ A3 )
=> ( ( ord_less @ real @ ( zero_zero @ real ) @ X2 )
=> ( ( ord_less @ real @ ( zero_zero @ real ) @ Y3 )
=> ( ( ord_less_eq @ real @ ( log @ A3 @ X2 ) @ ( log @ A3 @ Y3 ) )
= ( ord_less_eq @ real @ X2 @ Y3 ) ) ) ) ) ).
% log_le_cancel_iff
thf(fact_1199_log__le__one__cancel__iff,axiom,
! [A3: real,X2: real] :
( ( ord_less @ real @ ( one_one @ real ) @ A3 )
=> ( ( ord_less @ real @ ( zero_zero @ real ) @ X2 )
=> ( ( ord_less_eq @ real @ ( log @ A3 @ X2 ) @ ( one_one @ real ) )
= ( ord_less_eq @ real @ X2 @ A3 ) ) ) ) ).
% log_le_one_cancel_iff
thf(fact_1200_one__le__log__cancel__iff,axiom,
! [A3: real,X2: real] :
( ( ord_less @ real @ ( one_one @ real ) @ A3 )
=> ( ( ord_less @ real @ ( zero_zero @ real ) @ X2 )
=> ( ( ord_less_eq @ real @ ( one_one @ real ) @ ( log @ A3 @ X2 ) )
= ( ord_less_eq @ real @ A3 @ X2 ) ) ) ) ).
% one_le_log_cancel_iff
thf(fact_1201_log__le__zero__cancel__iff,axiom,
! [A3: real,X2: real] :
( ( ord_less @ real @ ( one_one @ real ) @ A3 )
=> ( ( ord_less @ real @ ( zero_zero @ real ) @ X2 )
=> ( ( ord_less_eq @ real @ ( log @ A3 @ X2 ) @ ( zero_zero @ real ) )
= ( ord_less_eq @ real @ X2 @ ( one_one @ real ) ) ) ) ) ).
% log_le_zero_cancel_iff
thf(fact_1202_zero__le__log__cancel__iff,axiom,
! [A3: real,X2: real] :
( ( ord_less @ real @ ( one_one @ real ) @ A3 )
=> ( ( ord_less @ real @ ( zero_zero @ real ) @ X2 )
=> ( ( ord_less_eq @ real @ ( zero_zero @ real ) @ ( log @ A3 @ X2 ) )
= ( ord_less_eq @ real @ ( one_one @ real ) @ X2 ) ) ) ) ).
% zero_le_log_cancel_iff
thf(fact_1203_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_1204_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_1205_log__pow__cancel,axiom,
! [A3: real,B3: nat] :
( ( ord_less @ real @ ( zero_zero @ real ) @ A3 )
=> ( ( A3
!= ( one_one @ real ) )
=> ( ( log @ A3 @ ( power_power @ real @ A3 @ B3 ) )
= ( semiring_1_of_nat @ real @ B3 ) ) ) ) ).
% log_pow_cancel
thf(fact_1206_less__eq__int__code_I1_J,axiom,
ord_less_eq @ int @ ( zero_zero @ int ) @ ( zero_zero @ int ) ).
% less_eq_int_code(1)
thf(fact_1207_zero__le__imp__eq__int,axiom,
! [K: int] :
( ( ord_less_eq @ int @ ( zero_zero @ int ) @ K )
=> ? [N3: nat] :
( K
= ( semiring_1_of_nat @ int @ N3 ) ) ) ).
% zero_le_imp_eq_int
thf(fact_1208_nonneg__int__cases,axiom,
! [K: int] :
( ( ord_less_eq @ int @ ( zero_zero @ int ) @ K )
=> ~ ! [N3: nat] :
( K
!= ( semiring_1_of_nat @ int @ N3 ) ) ) ).
% nonneg_int_cases
thf(fact_1209_plus__int__code_I1_J,axiom,
! [K: int] :
( ( plus_plus @ int @ K @ ( zero_zero @ int ) )
= K ) ).
% plus_int_code(1)
thf(fact_1210_plus__int__code_I2_J,axiom,
! [L: int] :
( ( plus_plus @ int @ ( zero_zero @ int ) @ L )
= L ) ).
% plus_int_code(2)
thf(fact_1211_minus__int__code_I1_J,axiom,
! [K: int] :
( ( minus_minus @ int @ K @ ( zero_zero @ int ) )
= K ) ).
% minus_int_code(1)
thf(fact_1212_not__iless0,axiom,
! [N: extended_enat] :
~ ( ord_less @ extended_enat @ N @ ( zero_zero @ extended_enat ) ) ).
% not_iless0
thf(fact_1213_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_1214_iadd__is__0,axiom,
! [M: extended_enat,N: extended_enat] :
( ( ( plus_plus @ extended_enat @ M @ N )
= ( zero_zero @ extended_enat ) )
= ( ( M
= ( zero_zero @ extended_enat ) )
& ( N
= ( zero_zero @ extended_enat ) ) ) ) ).
% iadd_is_0
thf(fact_1215_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_1216_i0__lb,axiom,
! [N: extended_enat] : ( ord_less_eq @ extended_enat @ ( zero_zero @ extended_enat ) @ N ) ).
% i0_lb
thf(fact_1217_odd__nonzero,axiom,
! [Z: int] :
( ( plus_plus @ int @ ( plus_plus @ int @ ( one_one @ int ) @ Z ) @ Z )
!= ( zero_zero @ int ) ) ).
% odd_nonzero
thf(fact_1218_linordered__field__no__lb,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [X8: A] :
? [Y4: A] : ( ord_less @ A @ Y4 @ X8 ) ) ).
% linordered_field_no_lb
thf(fact_1219_linordered__field__no__ub,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [X8: A] :
? [X_1: A] : ( ord_less @ A @ X8 @ X_1 ) ) ).
% linordered_field_no_ub
thf(fact_1220_log__base__change,axiom,
! [A3: real,B3: real,X2: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ A3 )
=> ( ( A3
!= ( one_one @ real ) )
=> ( ( log @ B3 @ X2 )
= ( divide_divide @ real @ ( log @ A3 @ X2 ) @ ( log @ A3 @ B3 ) ) ) ) ) ).
% log_base_change
thf(fact_1221_pos__int__cases,axiom,
! [K: int] :
( ( ord_less @ int @ ( zero_zero @ int ) @ K )
=> ~ ! [N3: nat] :
( ( K
= ( semiring_1_of_nat @ int @ N3 ) )
=> ~ ( ord_less @ nat @ ( zero_zero @ nat ) @ N3 ) ) ) ).
% pos_int_cases
thf(fact_1222_zero__less__imp__eq__int,axiom,
! [K: int] :
( ( ord_less @ int @ ( zero_zero @ int ) @ K )
=> ? [N3: nat] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ N3 )
& ( K
= ( semiring_1_of_nat @ int @ N3 ) ) ) ) ).
% zero_less_imp_eq_int
thf(fact_1223_realpow__pos__nth2,axiom,
! [A3: real,N: nat] :
( ( ord_less @ real @ ( zero_zero @ real ) @ A3 )
=> ? [R3: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ R3 )
& ( ( power_power @ real @ R3 @ ( suc @ N ) )
= A3 ) ) ) ).
% realpow_pos_nth2
thf(fact_1224_real__arch__pow__inv,axiom,
! [Y3: real,X2: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ Y3 )
=> ( ( ord_less @ real @ X2 @ ( one_one @ real ) )
=> ? [N3: nat] : ( ord_less @ real @ ( power_power @ real @ X2 @ N3 ) @ Y3 ) ) ) ).
% real_arch_pow_inv
thf(fact_1225_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_1226_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_1227_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_1228_nonneg1__imp__zdiv__pos__iff,axiom,
! [A3: int,B3: int] :
( ( ord_less_eq @ int @ ( zero_zero @ int ) @ A3 )
=> ( ( ord_less @ int @ ( zero_zero @ int ) @ ( divide_divide @ int @ A3 @ B3 ) )
= ( ( ord_less_eq @ int @ B3 @ A3 )
& ( ord_less @ int @ ( zero_zero @ int ) @ B3 ) ) ) ) ).
% nonneg1_imp_zdiv_pos_iff
thf(fact_1229_pos__imp__zdiv__nonneg__iff,axiom,
! [B3: int,A3: int] :
( ( ord_less @ int @ ( zero_zero @ int ) @ B3 )
=> ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ ( divide_divide @ int @ A3 @ B3 ) )
= ( ord_less_eq @ int @ ( zero_zero @ int ) @ A3 ) ) ) ).
% pos_imp_zdiv_nonneg_iff
thf(fact_1230_neg__imp__zdiv__nonneg__iff,axiom,
! [B3: int,A3: int] :
( ( ord_less @ int @ B3 @ ( zero_zero @ int ) )
=> ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ ( divide_divide @ int @ A3 @ B3 ) )
= ( ord_less_eq @ int @ A3 @ ( zero_zero @ int ) ) ) ) ).
% neg_imp_zdiv_nonneg_iff
thf(fact_1231_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_1232_div__nonpos__pos__le0,axiom,
! [A3: int,B3: int] :
( ( ord_less_eq @ int @ A3 @ ( zero_zero @ int ) )
=> ( ( ord_less @ int @ ( zero_zero @ int ) @ B3 )
=> ( ord_less_eq @ int @ ( divide_divide @ int @ A3 @ B3 ) @ ( zero_zero @ int ) ) ) ) ).
% div_nonpos_pos_le0
thf(fact_1233_div__nonneg__neg__le0,axiom,
! [A3: int,B3: int] :
( ( ord_less_eq @ int @ ( zero_zero @ int ) @ A3 )
=> ( ( ord_less @ int @ B3 @ ( zero_zero @ int ) )
=> ( ord_less_eq @ int @ ( divide_divide @ int @ A3 @ B3 ) @ ( zero_zero @ int ) ) ) ) ).
% div_nonneg_neg_le0
thf(fact_1234_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_1235_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_1236_zdiv__mono2__neg,axiom,
! [A3: int,B5: int,B3: int] :
( ( ord_less @ int @ A3 @ ( zero_zero @ int ) )
=> ( ( ord_less @ int @ ( zero_zero @ int ) @ B5 )
=> ( ( ord_less_eq @ int @ B5 @ B3 )
=> ( ord_less_eq @ int @ ( divide_divide @ int @ A3 @ B5 ) @ ( divide_divide @ int @ A3 @ B3 ) ) ) ) ) ).
% zdiv_mono2_neg
thf(fact_1237_zdiv__mono1__neg,axiom,
! [A3: int,A5: int,B3: int] :
( ( ord_less_eq @ int @ A3 @ A5 )
=> ( ( ord_less @ int @ B3 @ ( zero_zero @ int ) )
=> ( ord_less_eq @ int @ ( divide_divide @ int @ A5 @ B3 ) @ ( divide_divide @ int @ A3 @ B3 ) ) ) ) ).
% zdiv_mono1_neg
thf(fact_1238_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_1239_zdiv__mono2,axiom,
! [A3: int,B5: int,B3: int] :
( ( ord_less_eq @ int @ ( zero_zero @ int ) @ A3 )
=> ( ( ord_less @ int @ ( zero_zero @ int ) @ B5 )
=> ( ( ord_less_eq @ int @ B5 @ B3 )
=> ( ord_less_eq @ int @ ( divide_divide @ int @ A3 @ B3 ) @ ( divide_divide @ int @ A3 @ B5 ) ) ) ) ) ).
% zdiv_mono2
thf(fact_1240_zdiv__mono1,axiom,
! [A3: int,A5: int,B3: int] :
( ( ord_less_eq @ int @ A3 @ A5 )
=> ( ( ord_less @ int @ ( zero_zero @ int ) @ B3 )
=> ( ord_less_eq @ int @ ( divide_divide @ int @ A3 @ B3 ) @ ( divide_divide @ int @ A5 @ B3 ) ) ) ) ).
% zdiv_mono1
thf(fact_1241_zdiv__le__dividend,axiom,
! [A3: int,B3: int] :
( ( ord_less_eq @ int @ ( zero_zero @ int ) @ A3 )
=> ( ( ord_less @ int @ ( zero_zero @ int ) @ B3 )
=> ( ord_less_eq @ int @ ( divide_divide @ int @ A3 @ B3 ) @ A3 ) ) ) ).
% zdiv_le_dividend
thf(fact_1242_int__div__less__self,axiom,
! [X2: int,K: int] :
( ( ord_less @ int @ ( zero_zero @ int ) @ X2 )
=> ( ( ord_less @ int @ ( one_one @ int ) @ K )
=> ( ord_less @ int @ ( divide_divide @ int @ X2 @ K ) @ X2 ) ) ) ).
% int_div_less_self
thf(fact_1243_zdiv__zmult2__eq,axiom,
! [C3: int,A3: int,B3: int] :
( ( ord_less_eq @ int @ ( zero_zero @ int ) @ C3 )
=> ( ( divide_divide @ int @ A3 @ ( times_times @ int @ B3 @ C3 ) )
= ( divide_divide @ int @ ( divide_divide @ int @ A3 @ B3 ) @ C3 ) ) ) ).
% zdiv_zmult2_eq
thf(fact_1244_zdiv__mult__self,axiom,
! [M: int,A3: int,N: int] :
( ( M
!= ( zero_zero @ int ) )
=> ( ( divide_divide @ int @ ( plus_plus @ int @ A3 @ ( times_times @ int @ M @ N ) ) @ M )
= ( plus_plus @ int @ ( divide_divide @ int @ A3 @ M ) @ N ) ) ) ).
% zdiv_mult_self
thf(fact_1245_log__mult,axiom,
! [A3: real,X2: real,Y3: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ A3 )
=> ( ( A3
!= ( one_one @ real ) )
=> ( ( ord_less @ real @ ( zero_zero @ real ) @ X2 )
=> ( ( ord_less @ real @ ( zero_zero @ real ) @ Y3 )
=> ( ( log @ A3 @ ( times_times @ real @ X2 @ Y3 ) )
= ( plus_plus @ real @ ( log @ A3 @ X2 ) @ ( log @ A3 @ Y3 ) ) ) ) ) ) ) ).
% log_mult
thf(fact_1246_log__divide,axiom,
! [A3: real,X2: real,Y3: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ A3 )
=> ( ( A3
!= ( one_one @ real ) )
=> ( ( ord_less @ real @ ( zero_zero @ real ) @ X2 )
=> ( ( ord_less @ real @ ( zero_zero @ real ) @ Y3 )
=> ( ( log @ A3 @ ( divide_divide @ real @ X2 @ Y3 ) )
= ( minus_minus @ real @ ( log @ A3 @ X2 ) @ ( log @ A3 @ Y3 ) ) ) ) ) ) ) ).
% log_divide
thf(fact_1247_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_1248_realpow__pos__nth__unique,axiom,
! [N: nat,A3: real] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ N )
=> ( ( ord_less @ real @ ( zero_zero @ real ) @ A3 )
=> ? [X4: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ X4 )
& ( ( power_power @ real @ X4 @ N )
= A3 )
& ! [Y5: real] :
( ( ( ord_less @ real @ ( zero_zero @ real ) @ Y5 )
& ( ( power_power @ real @ Y5 @ N )
= A3 ) )
=> ( Y5 = X4 ) ) ) ) ) ).
% realpow_pos_nth_unique
thf(fact_1249_realpow__pos__nth,axiom,
! [N: nat,A3: real] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ N )
=> ( ( ord_less @ real @ ( zero_zero @ real ) @ A3 )
=> ? [R3: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ R3 )
& ( ( power_power @ real @ R3 @ N )
= A3 ) ) ) ) ).
% realpow_pos_nth
thf(fact_1250_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_1251_q__pos__lemma,axiom,
! [B5: int,Q6: int,R4: int] :
( ( ord_less_eq @ int @ ( zero_zero @ int ) @ ( plus_plus @ int @ ( times_times @ int @ B5 @ Q6 ) @ R4 ) )
=> ( ( ord_less @ int @ R4 @ B5 )
=> ( ( ord_less @ int @ ( zero_zero @ int ) @ B5 )
=> ( ord_less_eq @ int @ ( zero_zero @ int ) @ Q6 ) ) ) ) ).
% q_pos_lemma
thf(fact_1252_zdiv__mono2__lemma,axiom,
! [B3: int,Q3: int,R2: int,B5: int,Q6: int,R4: int] :
( ( ( plus_plus @ int @ ( times_times @ int @ B3 @ Q3 ) @ R2 )
= ( plus_plus @ int @ ( times_times @ int @ B5 @ Q6 ) @ R4 ) )
=> ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ ( plus_plus @ int @ ( times_times @ int @ B5 @ Q6 ) @ R4 ) )
=> ( ( ord_less @ int @ R4 @ B5 )
=> ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ R2 )
=> ( ( ord_less @ int @ ( zero_zero @ int ) @ B5 )
=> ( ( ord_less_eq @ int @ B5 @ B3 )
=> ( ord_less_eq @ int @ Q3 @ Q6 ) ) ) ) ) ) ) ).
% zdiv_mono2_lemma
thf(fact_1253_zdiv__mono2__neg__lemma,axiom,
! [B3: int,Q3: int,R2: int,B5: int,Q6: int,R4: int] :
( ( ( plus_plus @ int @ ( times_times @ int @ B3 @ Q3 ) @ R2 )
= ( plus_plus @ int @ ( times_times @ int @ B5 @ Q6 ) @ R4 ) )
=> ( ( ord_less @ int @ ( plus_plus @ int @ ( times_times @ int @ B5 @ Q6 ) @ R4 ) @ ( zero_zero @ int ) )
=> ( ( ord_less @ int @ R2 @ B3 )
=> ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ R4 )
=> ( ( ord_less @ int @ ( zero_zero @ int ) @ B5 )
=> ( ( ord_less_eq @ int @ B5 @ B3 )
=> ( ord_less_eq @ int @ Q6 @ Q3 ) ) ) ) ) ) ) ).
% zdiv_mono2_neg_lemma
thf(fact_1254_unique__quotient__lemma,axiom,
! [B3: int,Q6: int,R4: int,Q3: int,R2: int] :
( ( ord_less_eq @ int @ ( plus_plus @ int @ ( times_times @ int @ B3 @ Q6 ) @ R4 ) @ ( plus_plus @ int @ ( times_times @ int @ B3 @ Q3 ) @ R2 ) )
=> ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ R4 )
=> ( ( ord_less @ int @ R4 @ B3 )
=> ( ( ord_less @ int @ R2 @ B3 )
=> ( ord_less_eq @ int @ Q6 @ Q3 ) ) ) ) ) ).
% unique_quotient_lemma
thf(fact_1255_unique__quotient__lemma__neg,axiom,
! [B3: int,Q6: int,R4: int,Q3: int,R2: int] :
( ( ord_less_eq @ int @ ( plus_plus @ int @ ( times_times @ int @ B3 @ Q6 ) @ R4 ) @ ( plus_plus @ int @ ( times_times @ int @ B3 @ Q3 ) @ R2 ) )
=> ( ( ord_less_eq @ int @ R2 @ ( zero_zero @ int ) )
=> ( ( ord_less @ int @ B3 @ R2 )
=> ( ( ord_less @ int @ B3 @ R4 )
=> ( ord_less_eq @ int @ Q3 @ Q6 ) ) ) ) ) ).
% unique_quotient_lemma_neg
thf(fact_1256_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_1257_real__archimedian__rdiv__eq__0,axiom,
! [X2: real,C3: real] :
( ( ord_less_eq @ real @ ( zero_zero @ real ) @ X2 )
=> ( ( ord_less_eq @ real @ ( zero_zero @ real ) @ C3 )
=> ( ! [M5: nat] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ M5 )
=> ( ord_less_eq @ real @ ( times_times @ real @ ( semiring_1_of_nat @ real @ M5 ) @ X2 ) @ C3 ) )
=> ( X2
= ( zero_zero @ real ) ) ) ) ) ).
% real_archimedian_rdiv_eq_0
thf(fact_1258_split__zdiv,axiom,
! [P: int > $o,N: int,K: int] :
( ( P @ ( divide_divide @ int @ N @ K ) )
= ( ( ( K
= ( zero_zero @ int ) )
=> ( P @ ( zero_zero @ int ) ) )
& ( ( ord_less @ int @ ( zero_zero @ int ) @ K )
=> ! [I4: int,J3: int] :
( ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ J3 )
& ( ord_less @ int @ J3 @ K )
& ( N
= ( plus_plus @ int @ ( times_times @ int @ K @ I4 ) @ J3 ) ) )
=> ( P @ I4 ) ) )
& ( ( ord_less @ int @ K @ ( zero_zero @ int ) )
=> ! [I4: int,J3: int] :
( ( ( ord_less @ int @ K @ J3 )
& ( ord_less_eq @ int @ J3 @ ( zero_zero @ int ) )
& ( N
= ( plus_plus @ int @ ( times_times @ int @ K @ I4 ) @ J3 ) ) )
=> ( P @ I4 ) ) ) ) ) ).
% split_zdiv
thf(fact_1259_int__div__neg__eq,axiom,
! [A3: int,B3: int,Q3: int,R2: int] :
( ( A3
= ( plus_plus @ int @ ( times_times @ int @ B3 @ Q3 ) @ R2 ) )
=> ( ( ord_less_eq @ int @ R2 @ ( zero_zero @ int ) )
=> ( ( ord_less @ int @ B3 @ R2 )
=> ( ( divide_divide @ int @ A3 @ B3 )
= Q3 ) ) ) ) ).
% int_div_neg_eq
thf(fact_1260_int__div__pos__eq,axiom,
! [A3: int,B3: int,Q3: int,R2: int] :
( ( A3
= ( plus_plus @ int @ ( times_times @ int @ B3 @ Q3 ) @ R2 ) )
=> ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ R2 )
=> ( ( ord_less @ int @ R2 @ B3 )
=> ( ( divide_divide @ int @ A3 @ B3 )
= Q3 ) ) ) ) ).
% int_div_pos_eq
thf(fact_1261_real__of__nat__div2,axiom,
! [N: nat,X2: 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 @ X2 ) ) @ ( semiring_1_of_nat @ real @ ( divide_divide @ nat @ N @ X2 ) ) ) ) ).
% real_of_nat_div2
thf(fact_1262_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_1263_axxdiv2,axiom,
! [X2: int] :
( ( ( divide_divide @ int @ ( plus_plus @ int @ ( plus_plus @ int @ ( one_one @ int ) @ X2 ) @ X2 ) @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) )
= X2 )
& ( ( divide_divide @ int @ ( plus_plus @ int @ ( plus_plus @ int @ ( zero_zero @ int ) @ X2 ) @ X2 ) @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) )
= X2 ) ) ).
% axxdiv2
thf(fact_1264_add__divide__distrib,axiom,
! [A: $tType] :
( ( division_ring @ A )
=> ! [A3: A,B3: A,C3: A] :
( ( divide_divide @ A @ ( plus_plus @ A @ A3 @ B3 ) @ C3 )
= ( plus_plus @ A @ ( divide_divide @ A @ A3 @ C3 ) @ ( divide_divide @ A @ B3 @ C3 ) ) ) ) ).
% add_divide_distrib
thf(fact_1265_times__divide__times__eq,axiom,
! [A: $tType] :
( ( field @ A )
=> ! [X2: A,Y3: A,Z: A,W: A] :
( ( times_times @ A @ ( divide_divide @ A @ X2 @ Y3 ) @ ( divide_divide @ A @ Z @ W ) )
= ( divide_divide @ A @ ( times_times @ A @ X2 @ Z ) @ ( times_times @ A @ Y3 @ W ) ) ) ) ).
% times_divide_times_eq
thf(fact_1266_divide__divide__times__eq,axiom,
! [A: $tType] :
( ( field @ A )
=> ! [X2: A,Y3: A,Z: A,W: A] :
( ( divide_divide @ A @ ( divide_divide @ A @ X2 @ Y3 ) @ ( divide_divide @ A @ Z @ W ) )
= ( divide_divide @ A @ ( times_times @ A @ X2 @ W ) @ ( times_times @ A @ Y3 @ Z ) ) ) ) ).
% divide_divide_times_eq
thf(fact_1267_divide__divide__eq__left_H,axiom,
! [A: $tType] :
( ( division_ring @ A )
=> ! [A3: A,B3: A,C3: A] :
( ( divide_divide @ A @ ( divide_divide @ A @ A3 @ B3 ) @ C3 )
= ( divide_divide @ A @ A3 @ ( times_times @ A @ C3 @ B3 ) ) ) ) ).
% divide_divide_eq_left'
thf(fact_1268_diff__divide__distrib,axiom,
! [A: $tType] :
( ( division_ring @ A )
=> ! [A3: A,B3: A,C3: A] :
( ( divide_divide @ A @ ( minus_minus @ A @ A3 @ B3 ) @ C3 )
= ( minus_minus @ A @ ( divide_divide @ A @ A3 @ C3 ) @ ( divide_divide @ A @ B3 @ C3 ) ) ) ) ).
% diff_divide_distrib
thf(fact_1269_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_1270_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_1271_neg__zdiv__mult__2,axiom,
! [A3: int,B3: int] :
( ( ord_less_eq @ int @ A3 @ ( zero_zero @ int ) )
=> ( ( divide_divide @ int @ ( plus_plus @ int @ ( one_one @ int ) @ ( times_times @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ B3 ) ) @ ( times_times @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ A3 ) )
= ( divide_divide @ int @ ( plus_plus @ int @ B3 @ ( one_one @ int ) ) @ A3 ) ) ) ).
% neg_zdiv_mult_2
thf(fact_1272_pos__zdiv__mult__2,axiom,
! [A3: int,B3: int] :
( ( ord_less_eq @ int @ ( zero_zero @ int ) @ A3 )
=> ( ( divide_divide @ int @ ( plus_plus @ int @ ( one_one @ int ) @ ( times_times @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ B3 ) ) @ ( times_times @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ A3 ) )
= ( divide_divide @ int @ B3 @ A3 ) ) ) ).
% pos_zdiv_mult_2
thf(fact_1273_divide__right__mono__neg,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [A3: A,B3: A,C3: A] :
( ( ord_less_eq @ A @ A3 @ B3 )
=> ( ( ord_less_eq @ A @ C3 @ ( zero_zero @ A ) )
=> ( ord_less_eq @ A @ ( divide_divide @ A @ B3 @ C3 ) @ ( divide_divide @ A @ A3 @ C3 ) ) ) ) ) ).
% divide_right_mono_neg
thf(fact_1274_divide__nonpos__nonpos,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [X2: A,Y3: A] :
( ( ord_less_eq @ A @ X2 @ ( zero_zero @ A ) )
=> ( ( ord_less_eq @ A @ Y3 @ ( zero_zero @ A ) )
=> ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( divide_divide @ A @ X2 @ Y3 ) ) ) ) ) ).
% divide_nonpos_nonpos
thf(fact_1275_divide__nonpos__nonneg,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [X2: A,Y3: A] :
( ( ord_less_eq @ A @ X2 @ ( zero_zero @ A ) )
=> ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ Y3 )
=> ( ord_less_eq @ A @ ( divide_divide @ A @ X2 @ Y3 ) @ ( zero_zero @ A ) ) ) ) ) ).
% divide_nonpos_nonneg
thf(fact_1276_divide__nonneg__nonpos,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [X2: A,Y3: A] :
( ( ord_less_eq @ A @ ( zero_zero @ A ) @ X2 )
=> ( ( ord_less_eq @ A @ Y3 @ ( zero_zero @ A ) )
=> ( ord_less_eq @ A @ ( divide_divide @ A @ X2 @ Y3 ) @ ( zero_zero @ A ) ) ) ) ) ).
% divide_nonneg_nonpos
thf(fact_1277_divide__nonneg__nonneg,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [X2: A,Y3: A] :
( ( ord_less_eq @ A @ ( zero_zero @ A ) @ X2 )
=> ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ Y3 )
=> ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( divide_divide @ A @ X2 @ Y3 ) ) ) ) ) ).
% divide_nonneg_nonneg
thf(fact_1278_zero__le__divide__iff,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [A3: A,B3: A] :
( ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( divide_divide @ A @ A3 @ B3 ) )
= ( ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ A3 )
& ( ord_less_eq @ A @ ( zero_zero @ A ) @ B3 ) )
| ( ( ord_less_eq @ A @ A3 @ ( zero_zero @ A ) )
& ( ord_less_eq @ A @ B3 @ ( zero_zero @ A ) ) ) ) ) ) ).
% zero_le_divide_iff
thf(fact_1279_divide__right__mono,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [A3: A,B3: A,C3: A] :
( ( ord_less_eq @ A @ A3 @ B3 )
=> ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ C3 )
=> ( ord_less_eq @ A @ ( divide_divide @ A @ A3 @ C3 ) @ ( divide_divide @ A @ B3 @ C3 ) ) ) ) ) ).
% divide_right_mono
thf(fact_1280_divide__le__0__iff,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [A3: A,B3: A] :
( ( ord_less_eq @ A @ ( divide_divide @ A @ A3 @ B3 ) @ ( zero_zero @ A ) )
= ( ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ A3 )
& ( ord_less_eq @ A @ B3 @ ( zero_zero @ A ) ) )
| ( ( ord_less_eq @ A @ A3 @ ( zero_zero @ A ) )
& ( ord_less_eq @ A @ ( zero_zero @ A ) @ B3 ) ) ) ) ) ).
% divide_le_0_iff
thf(fact_1281_divide__neg__neg,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [X2: A,Y3: A] :
( ( ord_less @ A @ X2 @ ( zero_zero @ A ) )
=> ( ( ord_less @ A @ Y3 @ ( zero_zero @ A ) )
=> ( ord_less @ A @ ( zero_zero @ A ) @ ( divide_divide @ A @ X2 @ Y3 ) ) ) ) ) ).
% divide_neg_neg
thf(fact_1282_divide__neg__pos,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [X2: A,Y3: A] :
( ( ord_less @ A @ X2 @ ( zero_zero @ A ) )
=> ( ( ord_less @ A @ ( zero_zero @ A ) @ Y3 )
=> ( ord_less @ A @ ( divide_divide @ A @ X2 @ Y3 ) @ ( zero_zero @ A ) ) ) ) ) ).
% divide_neg_pos
thf(fact_1283_divide__pos__neg,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [X2: A,Y3: A] :
( ( ord_less @ A @ ( zero_zero @ A ) @ X2 )
=> ( ( ord_less @ A @ Y3 @ ( zero_zero @ A ) )
=> ( ord_less @ A @ ( divide_divide @ A @ X2 @ Y3 ) @ ( zero_zero @ A ) ) ) ) ) ).
% divide_pos_neg
thf(fact_1284_divide__pos__pos,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [X2: A,Y3: A] :
( ( ord_less @ A @ ( zero_zero @ A ) @ X2 )
=> ( ( ord_less @ A @ ( zero_zero @ A ) @ Y3 )
=> ( ord_less @ A @ ( zero_zero @ A ) @ ( divide_divide @ A @ X2 @ Y3 ) ) ) ) ) ).
% divide_pos_pos
thf(fact_1285_divide__less__0__iff,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [A3: A,B3: A] :
( ( ord_less @ A @ ( divide_divide @ A @ A3 @ B3 ) @ ( zero_zero @ A ) )
= ( ( ( ord_less @ A @ ( zero_zero @ A ) @ A3 )
& ( ord_less @ A @ B3 @ ( zero_zero @ A ) ) )
| ( ( ord_less @ A @ A3 @ ( zero_zero @ A ) )
& ( ord_less @ A @ ( zero_zero @ A ) @ B3 ) ) ) ) ) ).
% divide_less_0_iff
thf(fact_1286_divide__less__cancel,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [A3: A,C3: A,B3: A] :
( ( ord_less @ A @ ( divide_divide @ A @ A3 @ C3 ) @ ( divide_divide @ A @ B3 @ C3 ) )
= ( ( ( ord_less @ A @ ( zero_zero @ A ) @ C3 )
=> ( ord_less @ A @ A3 @ B3 ) )
& ( ( ord_less @ A @ C3 @ ( zero_zero @ A ) )
=> ( ord_less @ A @ B3 @ A3 ) )
& ( C3
!= ( zero_zero @ A ) ) ) ) ) ).
% divide_less_cancel
thf(fact_1287_zero__less__divide__iff,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [A3: A,B3: A] :
( ( ord_less @ A @ ( zero_zero @ A ) @ ( divide_divide @ A @ A3 @ B3 ) )
= ( ( ( ord_less @ A @ ( zero_zero @ A ) @ A3 )
& ( ord_less @ A @ ( zero_zero @ A ) @ B3 ) )
| ( ( ord_less @ A @ A3 @ ( zero_zero @ A ) )
& ( ord_less @ A @ B3 @ ( zero_zero @ A ) ) ) ) ) ) ).
% zero_less_divide_iff
thf(fact_1288_divide__strict__right__mono,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [A3: A,B3: A,C3: A] :
( ( ord_less @ A @ A3 @ B3 )
=> ( ( ord_less @ A @ ( zero_zero @ A ) @ C3 )
=> ( ord_less @ A @ ( divide_divide @ A @ A3 @ C3 ) @ ( divide_divide @ A @ B3 @ C3 ) ) ) ) ) ).
% divide_strict_right_mono
thf(fact_1289_divide__strict__right__mono__neg,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [B3: A,A3: A,C3: A] :
( ( ord_less @ A @ B3 @ A3 )
=> ( ( ord_less @ A @ C3 @ ( zero_zero @ A ) )
=> ( ord_less @ A @ ( divide_divide @ A @ A3 @ C3 ) @ ( divide_divide @ A @ B3 @ C3 ) ) ) ) ) ).
% divide_strict_right_mono_neg
thf(fact_1290_right__inverse__eq,axiom,
! [A: $tType] :
( ( division_ring @ A )
=> ! [B3: A,A3: A] :
( ( B3
!= ( zero_zero @ A ) )
=> ( ( ( divide_divide @ A @ A3 @ B3 )
= ( one_one @ A ) )
= ( A3 = B3 ) ) ) ) ).
% right_inverse_eq
thf(fact_1291_frac__eq__eq,axiom,
! [A: $tType] :
( ( field @ A )
=> ! [Y3: A,Z: A,X2: A,W: A] :
( ( Y3
!= ( zero_zero @ A ) )
=> ( ( Z
!= ( zero_zero @ A ) )
=> ( ( ( divide_divide @ A @ X2 @ Y3 )
= ( divide_divide @ A @ W @ Z ) )
= ( ( times_times @ A @ X2 @ Z )
= ( times_times @ A @ W @ Y3 ) ) ) ) ) ) ).
% frac_eq_eq
thf(fact_1292_divide__eq__eq,axiom,
! [A: $tType] :
( ( division_ring @ A )
=> ! [B3: A,C3: A,A3: A] :
( ( ( divide_divide @ A @ B3 @ C3 )
= A3 )
= ( ( ( C3
!= ( zero_zero @ A ) )
=> ( B3
= ( times_times @ A @ A3 @ C3 ) ) )
& ( ( C3
= ( zero_zero @ A ) )
=> ( A3
= ( zero_zero @ A ) ) ) ) ) ) ).
% divide_eq_eq
thf(fact_1293_eq__divide__eq,axiom,
! [A: $tType] :
( ( division_ring @ A )
=> ! [A3: A,B3: A,C3: A] :
( ( A3
= ( divide_divide @ A @ B3 @ C3 ) )
= ( ( ( C3
!= ( zero_zero @ A ) )
=> ( ( times_times @ A @ A3 @ C3 )
= B3 ) )
& ( ( C3
= ( zero_zero @ A ) )
=> ( A3
= ( zero_zero @ A ) ) ) ) ) ) ).
% eq_divide_eq
thf(fact_1294_divide__eq__imp,axiom,
! [A: $tType] :
( ( division_ring @ A )
=> ! [C3: A,B3: A,A3: A] :
( ( C3
!= ( zero_zero @ A ) )
=> ( ( B3
= ( times_times @ A @ A3 @ C3 ) )
=> ( ( divide_divide @ A @ B3 @ C3 )
= A3 ) ) ) ) ).
% divide_eq_imp
thf(fact_1295_eq__divide__imp,axiom,
! [A: $tType] :
( ( division_ring @ A )
=> ! [C3: A,A3: A,B3: A] :
( ( C3
!= ( zero_zero @ A ) )
=> ( ( ( times_times @ A @ A3 @ C3 )
= B3 )
=> ( A3
= ( divide_divide @ A @ B3 @ C3 ) ) ) ) ) ).
% eq_divide_imp
thf(fact_1296_nonzero__divide__eq__eq,axiom,
! [A: $tType] :
( ( division_ring @ A )
=> ! [C3: A,B3: A,A3: A] :
( ( C3
!= ( zero_zero @ A ) )
=> ( ( ( divide_divide @ A @ B3 @ C3 )
= A3 )
= ( B3
= ( times_times @ A @ A3 @ C3 ) ) ) ) ) ).
% nonzero_divide_eq_eq
thf(fact_1297_nonzero__eq__divide__eq,axiom,
! [A: $tType] :
( ( division_ring @ A )
=> ! [C3: A,A3: A,B3: A] :
( ( C3
!= ( zero_zero @ A ) )
=> ( ( A3
= ( divide_divide @ A @ B3 @ C3 ) )
= ( ( times_times @ A @ A3 @ C3 )
= B3 ) ) ) ) ).
% nonzero_eq_divide_eq
thf(fact_1298_field__le__epsilon,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [X2: A,Y3: A] :
( ! [E2: A] :
( ( ord_less @ A @ ( zero_zero @ A ) @ E2 )
=> ( ord_less_eq @ A @ X2 @ ( plus_plus @ A @ Y3 @ E2 ) ) )
=> ( ord_less_eq @ A @ X2 @ Y3 ) ) ) ).
% field_le_epsilon
thf(fact_1299_frac__le,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [Y3: A,X2: A,W: A,Z: A] :
( ( ord_less_eq @ A @ ( zero_zero @ A ) @ Y3 )
=> ( ( ord_less_eq @ A @ X2 @ Y3 )
=> ( ( ord_less @ A @ ( zero_zero @ A ) @ W )
=> ( ( ord_less_eq @ A @ W @ Z )
=> ( ord_less_eq @ A @ ( divide_divide @ A @ X2 @ Z ) @ ( divide_divide @ A @ Y3 @ W ) ) ) ) ) ) ) ).
% frac_le
thf(fact_1300_frac__less,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [X2: A,Y3: A,W: A,Z: A] :
( ( ord_less_eq @ A @ ( zero_zero @ A ) @ X2 )
=> ( ( ord_less @ A @ X2 @ Y3 )
=> ( ( ord_less @ A @ ( zero_zero @ A ) @ W )
=> ( ( ord_less_eq @ A @ W @ Z )
=> ( ord_less @ A @ ( divide_divide @ A @ X2 @ Z ) @ ( divide_divide @ A @ Y3 @ W ) ) ) ) ) ) ) ).
% frac_less
thf(fact_1301_frac__less2,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [X2: A,Y3: A,W: A,Z: A] :
( ( ord_less @ A @ ( zero_zero @ A ) @ X2 )
=> ( ( ord_less_eq @ A @ X2 @ Y3 )
=> ( ( ord_less @ A @ ( zero_zero @ A ) @ W )
=> ( ( ord_less @ A @ W @ Z )
=> ( ord_less @ A @ ( divide_divide @ A @ X2 @ Z ) @ ( divide_divide @ A @ Y3 @ W ) ) ) ) ) ) ) ).
% frac_less2
thf(fact_1302_divide__le__cancel,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [A3: A,C3: A,B3: A] :
( ( ord_less_eq @ A @ ( divide_divide @ A @ A3 @ C3 ) @ ( divide_divide @ A @ B3 @ C3 ) )
= ( ( ( ord_less @ A @ ( zero_zero @ A ) @ C3 )
=> ( ord_less_eq @ A @ A3 @ B3 ) )
& ( ( ord_less @ A @ C3 @ ( zero_zero @ A ) )
=> ( ord_less_eq @ A @ B3 @ A3 ) ) ) ) ) ).
% divide_le_cancel
thf(fact_1303_divide__nonneg__neg,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [X2: A,Y3: A] :
( ( ord_less_eq @ A @ ( zero_zero @ A ) @ X2 )
=> ( ( ord_less @ A @ Y3 @ ( zero_zero @ A ) )
=> ( ord_less_eq @ A @ ( divide_divide @ A @ X2 @ Y3 ) @ ( zero_zero @ A ) ) ) ) ) ).
% divide_nonneg_neg
thf(fact_1304_divide__nonneg__pos,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [X2: A,Y3: A] :
( ( ord_less_eq @ A @ ( zero_zero @ A ) @ X2 )
=> ( ( ord_less @ A @ ( zero_zero @ A ) @ Y3 )
=> ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( divide_divide @ A @ X2 @ Y3 ) ) ) ) ) ).
% divide_nonneg_pos
thf(fact_1305_divide__nonpos__neg,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [X2: A,Y3: A] :
( ( ord_less_eq @ A @ X2 @ ( zero_zero @ A ) )
=> ( ( ord_less @ A @ Y3 @ ( zero_zero @ A ) )
=> ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( divide_divide @ A @ X2 @ Y3 ) ) ) ) ) ).
% divide_nonpos_neg
thf(fact_1306_divide__nonpos__pos,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [X2: A,Y3: A] :
( ( ord_less_eq @ A @ X2 @ ( zero_zero @ A ) )
=> ( ( ord_less @ A @ ( zero_zero @ A ) @ Y3 )
=> ( ord_less_eq @ A @ ( divide_divide @ A @ X2 @ Y3 ) @ ( zero_zero @ A ) ) ) ) ) ).
% divide_nonpos_pos
thf(fact_1307_divide__less__eq__1,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [B3: A,A3: A] :
( ( ord_less @ A @ ( divide_divide @ A @ B3 @ A3 ) @ ( one_one @ A ) )
= ( ( ( ord_less @ A @ ( zero_zero @ A ) @ A3 )
& ( ord_less @ A @ B3 @ A3 ) )
| ( ( ord_less @ A @ A3 @ ( zero_zero @ A ) )
& ( ord_less @ A @ A3 @ B3 ) )
| ( A3
= ( zero_zero @ A ) ) ) ) ) ).
% divide_less_eq_1
thf(fact_1308_less__divide__eq__1,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [B3: A,A3: A] :
( ( ord_less @ A @ ( one_one @ A ) @ ( divide_divide @ A @ B3 @ A3 ) )
= ( ( ( ord_less @ A @ ( zero_zero @ A ) @ A3 )
& ( ord_less @ A @ A3 @ B3 ) )
| ( ( ord_less @ A @ A3 @ ( zero_zero @ A ) )
& ( ord_less @ A @ B3 @ A3 ) ) ) ) ) ).
% less_divide_eq_1
thf(fact_1309_divide__less__eq,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [B3: A,C3: A,A3: A] :
( ( ord_less @ A @ ( divide_divide @ A @ B3 @ C3 ) @ A3 )
= ( ( ( ord_less @ A @ ( zero_zero @ A ) @ C3 )
=> ( ord_less @ A @ B3 @ ( times_times @ A @ A3 @ C3 ) ) )
& ( ~ ( ord_less @ A @ ( zero_zero @ A ) @ C3 )
=> ( ( ( ord_less @ A @ C3 @ ( zero_zero @ A ) )
=> ( ord_less @ A @ ( times_times @ A @ A3 @ C3 ) @ B3 ) )
& ( ~ ( ord_less @ A @ C3 @ ( zero_zero @ A ) )
=> ( ord_less @ A @ ( zero_zero @ A ) @ A3 ) ) ) ) ) ) ) ).
% divide_less_eq
thf(fact_1310_less__divide__eq,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [A3: A,B3: A,C3: A] :
( ( ord_less @ A @ A3 @ ( divide_divide @ A @ B3 @ C3 ) )
= ( ( ( ord_less @ A @ ( zero_zero @ A ) @ C3 )
=> ( ord_less @ A @ ( times_times @ A @ A3 @ C3 ) @ B3 ) )
& ( ~ ( ord_less @ A @ ( zero_zero @ A ) @ C3 )
=> ( ( ( ord_less @ A @ C3 @ ( zero_zero @ A ) )
=> ( ord_less @ A @ B3 @ ( times_times @ A @ A3 @ C3 ) ) )
& ( ~ ( ord_less @ A @ C3 @ ( zero_zero @ A ) )
=> ( ord_less @ A @ A3 @ ( zero_zero @ A ) ) ) ) ) ) ) ) ).
% less_divide_eq
thf(fact_1311_neg__divide__less__eq,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [C3: A,B3: A,A3: A] :
( ( ord_less @ A @ C3 @ ( zero_zero @ A ) )
=> ( ( ord_less @ A @ ( divide_divide @ A @ B3 @ C3 ) @ A3 )
= ( ord_less @ A @ ( times_times @ A @ A3 @ C3 ) @ B3 ) ) ) ) ).
% neg_divide_less_eq
thf(fact_1312_neg__less__divide__eq,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [C3: A,A3: A,B3: A] :
( ( ord_less @ A @ C3 @ ( zero_zero @ A ) )
=> ( ( ord_less @ A @ A3 @ ( divide_divide @ A @ B3 @ C3 ) )
= ( ord_less @ A @ B3 @ ( times_times @ A @ A3 @ C3 ) ) ) ) ) ).
% neg_less_divide_eq
thf(fact_1313_pos__divide__less__eq,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [C3: A,B3: A,A3: A] :
( ( ord_less @ A @ ( zero_zero @ A ) @ C3 )
=> ( ( ord_less @ A @ ( divide_divide @ A @ B3 @ C3 ) @ A3 )
= ( ord_less @ A @ B3 @ ( times_times @ A @ A3 @ C3 ) ) ) ) ) ).
% pos_divide_less_eq
thf(fact_1314_pos__less__divide__eq,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [C3: A,A3: A,B3: A] :
( ( ord_less @ A @ ( zero_zero @ A ) @ C3 )
=> ( ( ord_less @ A @ A3 @ ( divide_divide @ A @ B3 @ C3 ) )
= ( ord_less @ A @ ( times_times @ A @ A3 @ C3 ) @ B3 ) ) ) ) ).
% pos_less_divide_eq
thf(fact_1315_mult__imp__div__pos__less,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [Y3: A,X2: A,Z: A] :
( ( ord_less @ A @ ( zero_zero @ A ) @ Y3 )
=> ( ( ord_less @ A @ X2 @ ( times_times @ A @ Z @ Y3 ) )
=> ( ord_less @ A @ ( divide_divide @ A @ X2 @ Y3 ) @ Z ) ) ) ) ).
% mult_imp_div_pos_less
thf(fact_1316_mult__imp__less__div__pos,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [Y3: A,Z: A,X2: A] :
( ( ord_less @ A @ ( zero_zero @ A ) @ Y3 )
=> ( ( ord_less @ A @ ( times_times @ A @ Z @ Y3 ) @ X2 )
=> ( ord_less @ A @ Z @ ( divide_divide @ A @ X2 @ Y3 ) ) ) ) ) ).
% mult_imp_less_div_pos
thf(fact_1317_divide__strict__left__mono,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [B3: A,A3: A,C3: A] :
( ( ord_less @ A @ B3 @ A3 )
=> ( ( ord_less @ A @ ( zero_zero @ A ) @ C3 )
=> ( ( ord_less @ A @ ( zero_zero @ A ) @ ( times_times @ A @ A3 @ B3 ) )
=> ( ord_less @ A @ ( divide_divide @ A @ C3 @ A3 ) @ ( divide_divide @ A @ C3 @ B3 ) ) ) ) ) ) ).
% divide_strict_left_mono
thf(fact_1318_divide__strict__left__mono__neg,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [A3: A,B3: A,C3: A] :
( ( ord_less @ A @ A3 @ B3 )
=> ( ( ord_less @ A @ C3 @ ( zero_zero @ A ) )
=> ( ( ord_less @ A @ ( zero_zero @ A ) @ ( times_times @ A @ A3 @ B3 ) )
=> ( ord_less @ A @ ( divide_divide @ A @ C3 @ A3 ) @ ( divide_divide @ A @ C3 @ B3 ) ) ) ) ) ) ).
% divide_strict_left_mono_neg
thf(fact_1319_add__divide__eq__if__simps_I2_J,axiom,
! [A: $tType] :
( ( division_ring @ A )
=> ! [Z: A,A3: A,B3: A] :
( ( ( Z
= ( zero_zero @ A ) )
=> ( ( plus_plus @ A @ ( divide_divide @ A @ A3 @ Z ) @ B3 )
= B3 ) )
& ( ( Z
!= ( zero_zero @ A ) )
=> ( ( plus_plus @ A @ ( divide_divide @ A @ A3 @ Z ) @ B3 )
= ( divide_divide @ A @ ( plus_plus @ A @ A3 @ ( times_times @ A @ B3 @ Z ) ) @ Z ) ) ) ) ) ).
% add_divide_eq_if_simps(2)
thf(fact_1320_add__divide__eq__if__simps_I1_J,axiom,
! [A: $tType] :
( ( division_ring @ A )
=> ! [Z: A,A3: A,B3: A] :
( ( ( Z
= ( zero_zero @ A ) )
=> ( ( plus_plus @ A @ A3 @ ( divide_divide @ A @ B3 @ Z ) )
= A3 ) )
& ( ( Z
!= ( zero_zero @ A ) )
=> ( ( plus_plus @ A @ A3 @ ( divide_divide @ A @ B3 @ Z ) )
= ( divide_divide @ A @ ( plus_plus @ A @ ( times_times @ A @ A3 @ Z ) @ B3 ) @ Z ) ) ) ) ) ).
% add_divide_eq_if_simps(1)
thf(fact_1321_add__frac__eq,axiom,
! [A: $tType] :
( ( field @ A )
=> ! [Y3: A,Z: A,X2: A,W: A] :
( ( Y3
!= ( zero_zero @ A ) )
=> ( ( Z
!= ( zero_zero @ A ) )
=> ( ( plus_plus @ A @ ( divide_divide @ A @ X2 @ Y3 ) @ ( divide_divide @ A @ W @ Z ) )
= ( divide_divide @ A @ ( plus_plus @ A @ ( times_times @ A @ X2 @ Z ) @ ( times_times @ A @ W @ Y3 ) ) @ ( times_times @ A @ Y3 @ Z ) ) ) ) ) ) ).
% add_frac_eq
thf(fact_1322_add__frac__num,axiom,
! [A: $tType] :
( ( field @ A )
=> ! [Y3: A,X2: A,Z: A] :
( ( Y3
!= ( zero_zero @ A ) )
=> ( ( plus_plus @ A @ ( divide_divide @ A @ X2 @ Y3 ) @ Z )
= ( divide_divide @ A @ ( plus_plus @ A @ X2 @ ( times_times @ A @ Z @ Y3 ) ) @ Y3 ) ) ) ) ).
% add_frac_num
thf(fact_1323_add__num__frac,axiom,
! [A: $tType] :
( ( field @ A )
=> ! [Y3: A,Z: A,X2: A] :
( ( Y3
!= ( zero_zero @ A ) )
=> ( ( plus_plus @ A @ Z @ ( divide_divide @ A @ X2 @ Y3 ) )
= ( divide_divide @ A @ ( plus_plus @ A @ X2 @ ( times_times @ A @ Z @ Y3 ) ) @ Y3 ) ) ) ) ).
% add_num_frac
thf(fact_1324_add__divide__eq__iff,axiom,
! [A: $tType] :
( ( division_ring @ A )
=> ! [Z: A,X2: A,Y3: A] :
( ( Z
!= ( zero_zero @ A ) )
=> ( ( plus_plus @ A @ X2 @ ( divide_divide @ A @ Y3 @ Z ) )
= ( divide_divide @ A @ ( plus_plus @ A @ ( times_times @ A @ X2 @ Z ) @ Y3 ) @ Z ) ) ) ) ).
% add_divide_eq_iff
thf(fact_1325_divide__add__eq__iff,axiom,
! [A: $tType] :
( ( division_ring @ A )
=> ! [Z: A,X2: A,Y3: A] :
( ( Z
!= ( zero_zero @ A ) )
=> ( ( plus_plus @ A @ ( divide_divide @ A @ X2 @ Z ) @ Y3 )
= ( divide_divide @ A @ ( plus_plus @ A @ X2 @ ( times_times @ A @ Y3 @ Z ) ) @ Z ) ) ) ) ).
% divide_add_eq_iff
thf(fact_1326_gt__half__sum,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [A3: A,B3: A] :
( ( ord_less @ A @ A3 @ B3 )
=> ( ord_less @ A @ ( divide_divide @ A @ ( plus_plus @ A @ A3 @ B3 ) @ ( plus_plus @ A @ ( one_one @ A ) @ ( one_one @ A ) ) ) @ B3 ) ) ) ).
% gt_half_sum
thf(fact_1327_less__half__sum,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [A3: A,B3: A] :
( ( ord_less @ A @ A3 @ B3 )
=> ( ord_less @ A @ A3 @ ( divide_divide @ A @ ( plus_plus @ A @ A3 @ B3 ) @ ( plus_plus @ A @ ( one_one @ A ) @ ( one_one @ A ) ) ) ) ) ) ).
% less_half_sum
thf(fact_1328_add__divide__eq__if__simps_I4_J,axiom,
! [A: $tType] :
( ( division_ring @ A )
=> ! [Z: A,A3: A,B3: A] :
( ( ( Z
= ( zero_zero @ A ) )
=> ( ( minus_minus @ A @ A3 @ ( divide_divide @ A @ B3 @ Z ) )
= A3 ) )
& ( ( Z
!= ( zero_zero @ A ) )
=> ( ( minus_minus @ A @ A3 @ ( divide_divide @ A @ B3 @ Z ) )
= ( divide_divide @ A @ ( minus_minus @ A @ ( times_times @ A @ A3 @ Z ) @ B3 ) @ Z ) ) ) ) ) ).
% add_divide_eq_if_simps(4)
thf(fact_1329_diff__frac__eq,axiom,
! [A: $tType] :
( ( field @ A )
=> ! [Y3: A,Z: A,X2: A,W: A] :
( ( Y3
!= ( zero_zero @ A ) )
=> ( ( Z
!= ( zero_zero @ A ) )
=> ( ( minus_minus @ A @ ( divide_divide @ A @ X2 @ Y3 ) @ ( divide_divide @ A @ W @ Z ) )
= ( divide_divide @ A @ ( minus_minus @ A @ ( times_times @ A @ X2 @ Z ) @ ( times_times @ A @ W @ Y3 ) ) @ ( times_times @ A @ Y3 @ Z ) ) ) ) ) ) ).
% diff_frac_eq
thf(fact_1330_diff__divide__eq__iff,axiom,
! [A: $tType] :
( ( division_ring @ A )
=> ! [Z: A,X2: A,Y3: A] :
( ( Z
!= ( zero_zero @ A ) )
=> ( ( minus_minus @ A @ X2 @ ( divide_divide @ A @ Y3 @ Z ) )
= ( divide_divide @ A @ ( minus_minus @ A @ ( times_times @ A @ X2 @ Z ) @ Y3 ) @ Z ) ) ) ) ).
% diff_divide_eq_iff
thf(fact_1331_divide__diff__eq__iff,axiom,
! [A: $tType] :
( ( division_ring @ A )
=> ! [Z: A,X2: A,Y3: A] :
( ( Z
!= ( zero_zero @ A ) )
=> ( ( minus_minus @ A @ ( divide_divide @ A @ X2 @ Z ) @ Y3 )
= ( divide_divide @ A @ ( minus_minus @ A @ X2 @ ( times_times @ A @ Y3 @ Z ) ) @ Z ) ) ) ) ).
% divide_diff_eq_iff
thf(fact_1332_field__le__mult__one__interval,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [X2: A,Y3: A] :
( ! [Z2: A] :
( ( ord_less @ A @ ( zero_zero @ A ) @ Z2 )
=> ( ( ord_less @ A @ Z2 @ ( one_one @ A ) )
=> ( ord_less_eq @ A @ ( times_times @ A @ Z2 @ X2 ) @ Y3 ) ) )
=> ( ord_less_eq @ A @ X2 @ Y3 ) ) ) ).
% field_le_mult_one_interval
thf(fact_1333_divide__le__eq__1,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [B3: A,A3: A] :
( ( ord_less_eq @ A @ ( divide_divide @ A @ B3 @ A3 ) @ ( one_one @ A ) )
= ( ( ( ord_less @ A @ ( zero_zero @ A ) @ A3 )
& ( ord_less_eq @ A @ B3 @ A3 ) )
| ( ( ord_less @ A @ A3 @ ( zero_zero @ A ) )
& ( ord_less_eq @ A @ A3 @ B3 ) )
| ( A3
= ( zero_zero @ A ) ) ) ) ) ).
% divide_le_eq_1
thf(fact_1334_le__divide__eq__1,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [B3: A,A3: A] :
( ( ord_less_eq @ A @ ( one_one @ A ) @ ( divide_divide @ A @ B3 @ A3 ) )
= ( ( ( ord_less @ A @ ( zero_zero @ A ) @ A3 )
& ( ord_less_eq @ A @ A3 @ B3 ) )
| ( ( ord_less @ A @ A3 @ ( zero_zero @ A ) )
& ( ord_less_eq @ A @ B3 @ A3 ) ) ) ) ) ).
% le_divide_eq_1
thf(fact_1335_divide__le__eq,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [B3: A,C3: A,A3: A] :
( ( ord_less_eq @ A @ ( divide_divide @ A @ B3 @ C3 ) @ A3 )
= ( ( ( ord_less @ A @ ( zero_zero @ A ) @ C3 )
=> ( ord_less_eq @ A @ B3 @ ( times_times @ A @ A3 @ C3 ) ) )
& ( ~ ( ord_less @ A @ ( zero_zero @ A ) @ C3 )
=> ( ( ( ord_less @ A @ C3 @ ( zero_zero @ A ) )
=> ( ord_less_eq @ A @ ( times_times @ A @ A3 @ C3 ) @ B3 ) )
& ( ~ ( ord_less @ A @ C3 @ ( zero_zero @ A ) )
=> ( ord_less_eq @ A @ ( zero_zero @ A ) @ A3 ) ) ) ) ) ) ) ).
% divide_le_eq
thf(fact_1336_le__divide__eq,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [A3: A,B3: A,C3: A] :
( ( ord_less_eq @ A @ A3 @ ( divide_divide @ A @ B3 @ C3 ) )
= ( ( ( ord_less @ A @ ( zero_zero @ A ) @ C3 )
=> ( ord_less_eq @ A @ ( times_times @ A @ A3 @ C3 ) @ B3 ) )
& ( ~ ( ord_less @ A @ ( zero_zero @ A ) @ C3 )
=> ( ( ( ord_less @ A @ C3 @ ( zero_zero @ A ) )
=> ( ord_less_eq @ A @ B3 @ ( times_times @ A @ A3 @ C3 ) ) )
& ( ~ ( ord_less @ A @ C3 @ ( zero_zero @ A ) )
=> ( ord_less_eq @ A @ A3 @ ( zero_zero @ A ) ) ) ) ) ) ) ) ).
% le_divide_eq
thf(fact_1337_divide__left__mono,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [B3: A,A3: A,C3: A] :
( ( ord_less_eq @ A @ B3 @ A3 )
=> ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ C3 )
=> ( ( ord_less @ A @ ( zero_zero @ A ) @ ( times_times @ A @ A3 @ B3 ) )
=> ( ord_less_eq @ A @ ( divide_divide @ A @ C3 @ A3 ) @ ( divide_divide @ A @ C3 @ B3 ) ) ) ) ) ) ).
% divide_left_mono
thf(fact_1338_neg__divide__le__eq,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [C3: A,B3: A,A3: A] :
( ( ord_less @ A @ C3 @ ( zero_zero @ A ) )
=> ( ( ord_less_eq @ A @ ( divide_divide @ A @ B3 @ C3 ) @ A3 )
= ( ord_less_eq @ A @ ( times_times @ A @ A3 @ C3 ) @ B3 ) ) ) ) ).
% neg_divide_le_eq
thf(fact_1339_neg__le__divide__eq,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [C3: A,A3: A,B3: A] :
( ( ord_less @ A @ C3 @ ( zero_zero @ A ) )
=> ( ( ord_less_eq @ A @ A3 @ ( divide_divide @ A @ B3 @ C3 ) )
= ( ord_less_eq @ A @ B3 @ ( times_times @ A @ A3 @ C3 ) ) ) ) ) ).
% neg_le_divide_eq
thf(fact_1340_pos__divide__le__eq,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [C3: A,B3: A,A3: A] :
( ( ord_less @ A @ ( zero_zero @ A ) @ C3 )
=> ( ( ord_less_eq @ A @ ( divide_divide @ A @ B3 @ C3 ) @ A3 )
= ( ord_less_eq @ A @ B3 @ ( times_times @ A @ A3 @ C3 ) ) ) ) ) ).
% pos_divide_le_eq
thf(fact_1341_pos__le__divide__eq,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [C3: A,A3: A,B3: A] :
( ( ord_less @ A @ ( zero_zero @ A ) @ C3 )
=> ( ( ord_less_eq @ A @ A3 @ ( divide_divide @ A @ B3 @ C3 ) )
= ( ord_less_eq @ A @ ( times_times @ A @ A3 @ C3 ) @ B3 ) ) ) ) ).
% pos_le_divide_eq
thf(fact_1342_mult__imp__div__pos__le,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [Y3: A,X2: A,Z: A] :
( ( ord_less @ A @ ( zero_zero @ A ) @ Y3 )
=> ( ( ord_less_eq @ A @ X2 @ ( times_times @ A @ Z @ Y3 ) )
=> ( ord_less_eq @ A @ ( divide_divide @ A @ X2 @ Y3 ) @ Z ) ) ) ) ).
% mult_imp_div_pos_le
thf(fact_1343_mult__imp__le__div__pos,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [Y3: A,Z: A,X2: A] :
( ( ord_less @ A @ ( zero_zero @ A ) @ Y3 )
=> ( ( ord_less_eq @ A @ ( times_times @ A @ Z @ Y3 ) @ X2 )
=> ( ord_less_eq @ A @ Z @ ( divide_divide @ A @ X2 @ Y3 ) ) ) ) ) ).
% mult_imp_le_div_pos
thf(fact_1344_divide__left__mono__neg,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [A3: A,B3: A,C3: A] :
( ( ord_less_eq @ A @ A3 @ B3 )
=> ( ( ord_less_eq @ A @ C3 @ ( zero_zero @ A ) )
=> ( ( ord_less @ A @ ( zero_zero @ A ) @ ( times_times @ A @ A3 @ B3 ) )
=> ( ord_less_eq @ A @ ( divide_divide @ A @ C3 @ A3 ) @ ( divide_divide @ A @ C3 @ B3 ) ) ) ) ) ) ).
% divide_left_mono_neg
thf(fact_1345_frac__le__eq,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [Y3: A,Z: A,X2: A,W: A] :
( ( Y3
!= ( zero_zero @ A ) )
=> ( ( Z
!= ( zero_zero @ A ) )
=> ( ( ord_less_eq @ A @ ( divide_divide @ A @ X2 @ Y3 ) @ ( divide_divide @ A @ W @ Z ) )
= ( ord_less_eq @ A @ ( divide_divide @ A @ ( minus_minus @ A @ ( times_times @ A @ X2 @ Z ) @ ( times_times @ A @ W @ Y3 ) ) @ ( times_times @ A @ Y3 @ Z ) ) @ ( zero_zero @ A ) ) ) ) ) ) ).
% frac_le_eq
thf(fact_1346_frac__less__eq,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [Y3: A,Z: A,X2: A,W: A] :
( ( Y3
!= ( zero_zero @ A ) )
=> ( ( Z
!= ( zero_zero @ A ) )
=> ( ( ord_less @ A @ ( divide_divide @ A @ X2 @ Y3 ) @ ( divide_divide @ A @ W @ Z ) )
= ( ord_less @ A @ ( divide_divide @ A @ ( minus_minus @ A @ ( times_times @ A @ X2 @ Z ) @ ( times_times @ A @ W @ Y3 ) ) @ ( times_times @ A @ Y3 @ Z ) ) @ ( zero_zero @ A ) ) ) ) ) ) ).
% frac_less_eq
thf(fact_1347_less__log__of__power,axiom,
! [B3: real,N: nat,M: real] :
( ( ord_less @ real @ ( power_power @ real @ B3 @ N ) @ M )
=> ( ( ord_less @ real @ ( one_one @ real ) @ B3 )
=> ( ord_less @ real @ ( semiring_1_of_nat @ real @ N ) @ ( log @ B3 @ M ) ) ) ) ).
% less_log_of_power
thf(fact_1348_log__of__power__eq,axiom,
! [M: nat,B3: real,N: nat] :
( ( ( semiring_1_of_nat @ real @ M )
= ( power_power @ real @ B3 @ N ) )
=> ( ( ord_less @ real @ ( one_one @ real ) @ B3 )
=> ( ( semiring_1_of_nat @ real @ N )
= ( log @ B3 @ ( semiring_1_of_nat @ real @ M ) ) ) ) ) ).
% log_of_power_eq
thf(fact_1349_scaling__mono,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [U: A,V: A,R2: A,S2: A] :
( ( ord_less_eq @ A @ U @ V )
=> ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ R2 )
=> ( ( ord_less_eq @ A @ R2 @ S2 )
=> ( ord_less_eq @ A @ ( plus_plus @ A @ U @ ( divide_divide @ A @ ( times_times @ A @ R2 @ ( minus_minus @ A @ V @ U ) ) @ S2 ) ) @ V ) ) ) ) ) ).
% scaling_mono
thf(fact_1350_le__log__of__power,axiom,
! [B3: real,N: nat,M: real] :
( ( ord_less_eq @ real @ ( power_power @ real @ B3 @ N ) @ M )
=> ( ( ord_less @ real @ ( one_one @ real ) @ B3 )
=> ( ord_less_eq @ real @ ( semiring_1_of_nat @ real @ N ) @ ( log @ B3 @ M ) ) ) ) ).
% le_log_of_power
thf(fact_1351_log2__of__power__eq,axiom,
! [M: nat,N: nat] :
( ( M
= ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N ) )
=> ( ( semiring_1_of_nat @ real @ N )
= ( log @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) @ ( semiring_1_of_nat @ real @ M ) ) ) ) ).
% log2_of_power_eq
thf(fact_1352_log__of__power__less,axiom,
! [M: nat,B3: real,N: nat] :
( ( ord_less @ real @ ( semiring_1_of_nat @ real @ M ) @ ( power_power @ real @ B3 @ N ) )
=> ( ( ord_less @ real @ ( one_one @ real ) @ B3 )
=> ( ( ord_less @ nat @ ( zero_zero @ nat ) @ M )
=> ( ord_less @ real @ ( log @ B3 @ ( semiring_1_of_nat @ real @ M ) ) @ ( semiring_1_of_nat @ real @ N ) ) ) ) ) ).
% log_of_power_less
thf(fact_1353_log__of__power__le,axiom,
! [M: nat,B3: real,N: nat] :
( ( ord_less_eq @ real @ ( semiring_1_of_nat @ real @ M ) @ ( power_power @ real @ B3 @ N ) )
=> ( ( ord_less @ real @ ( one_one @ real ) @ B3 )
=> ( ( ord_less @ nat @ ( zero_zero @ nat ) @ M )
=> ( ord_less_eq @ real @ ( log @ B3 @ ( semiring_1_of_nat @ real @ M ) ) @ ( semiring_1_of_nat @ real @ N ) ) ) ) ) ).
% log_of_power_le
thf(fact_1354_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_1355_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_1356_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_1357_vebt__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_buildupi_rule
thf(fact_1358_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,Va: nat,TreeList: list @ vEBT_VEBT,Summary: vEBT_VEBT,X2: nat] :
( ( vEBT_T_m_e_m_b_e_r2 @ ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi @ Ma ) ) @ ( suc @ ( suc @ Va ) ) @ TreeList @ Summary ) @ X2 )
= ( plus_plus @ nat @ ( one_one @ nat )
@ ( if @ nat @ ( X2 = Mi ) @ ( zero_zero @ nat )
@ ( if @ nat @ ( X2 = Ma ) @ ( zero_zero @ nat )
@ ( if @ nat @ ( ord_less @ nat @ X2 @ Mi ) @ ( zero_zero @ nat )
@ ( if @ nat @ ( ord_less @ nat @ Ma @ X2 ) @ ( zero_zero @ nat )
@ ( if @ nat
@ ( ( ord_less @ nat @ Mi @ X2 )
& ( ord_less @ nat @ X2 @ Ma ) )
@ ( if @ nat @ ( ord_less @ nat @ ( vEBT_VEBT_high @ X2 @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( 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 @ X2 @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( vEBT_VEBT_low @ X2 @ ( 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'.simps(5)
thf(fact_1359_diff__add__zero,axiom,
! [A: $tType] :
( ( comm_monoid_diff @ A )
=> ! [A3: A,B3: A] :
( ( minus_minus @ A @ A3 @ ( plus_plus @ A @ A3 @ B3 ) )
= ( zero_zero @ A ) ) ) ).
% diff_add_zero
thf(fact_1360_diff__gt__0__iff__gt,axiom,
! [A: $tType] :
( ( ordered_ab_group_add @ A )
=> ! [A3: A,B3: A] :
( ( ord_less @ A @ ( zero_zero @ A ) @ ( minus_minus @ A @ A3 @ B3 ) )
= ( ord_less @ A @ B3 @ A3 ) ) ) ).
% diff_gt_0_iff_gt
thf(fact_1361_diff__ge__0__iff__ge,axiom,
! [A: $tType] :
( ( ordered_ab_group_add @ A )
=> ! [A3: A,B3: A] :
( ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( minus_minus @ A @ A3 @ B3 ) )
= ( ord_less_eq @ A @ B3 @ A3 ) ) ) ).
% diff_ge_0_iff_ge
thf(fact_1362_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,
! [X2: vEBT_VEBT,Xa: nat,Y3: nat] :
( ( ( vEBT_V1232361888498592333_e_t_e @ X2 @ Xa )
= Y3 )
=> ( ( ? [A4: $o,B4: $o] :
( X2
= ( vEBT_Leaf @ A4 @ B4 ) )
=> ( ( Xa
= ( zero_zero @ nat ) )
=> ( Y3
!= ( one_one @ nat ) ) ) )
=> ( ( ? [A4: $o,B4: $o] :
( X2
= ( vEBT_Leaf @ A4 @ B4 ) )
=> ( ( Xa
= ( suc @ ( zero_zero @ nat ) ) )
=> ( Y3
!= ( one_one @ nat ) ) ) )
=> ( ( ? [A4: $o,B4: $o] :
( X2
= ( vEBT_Leaf @ A4 @ B4 ) )
=> ( ? [N3: nat] :
( Xa
= ( suc @ ( suc @ N3 ) ) )
=> ( Y3
!= ( one_one @ nat ) ) ) )
=> ( ( ? [Deg2: nat,TreeList2: list @ vEBT_VEBT,Summary2: vEBT_VEBT] :
( X2
= ( vEBT_Node @ ( none @ ( product_prod @ nat @ nat ) ) @ Deg2 @ TreeList2 @ Summary2 ) )
=> ( Y3
!= ( one_one @ nat ) ) )
=> ( ( ? [Mi2: nat,Ma2: nat,TreeList2: list @ vEBT_VEBT,Summary2: vEBT_VEBT] :
( X2
= ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi2 @ Ma2 ) ) @ ( zero_zero @ nat ) @ TreeList2 @ Summary2 ) )
=> ( Y3
!= ( one_one @ nat ) ) )
=> ( ( ? [Mi2: nat,Ma2: nat,TreeList2: list @ vEBT_VEBT,Summary2: vEBT_VEBT] :
( X2
= ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( zero_zero @ nat ) ) @ TreeList2 @ Summary2 ) )
=> ( Y3
!= ( one_one @ nat ) ) )
=> ~ ! [Mi2: nat,Ma2: nat,Va3: nat,TreeList2: list @ vEBT_VEBT,Summary2: vEBT_VEBT] :
( ( X2
= ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( suc @ Va3 ) ) @ TreeList2 @ Summary2 ) )
=> ~ ( ( ( ( ord_less @ nat @ Xa @ Mi2 )
| ( ord_less @ nat @ Ma2 @ Xa ) )
=> ( Y3
= ( one_one @ nat ) ) )
& ( ~ ( ( ord_less @ nat @ Xa @ Mi2 )
| ( ord_less @ nat @ Ma2 @ Xa ) )
=> ( ( ( ( Xa = Mi2 )
& ( Xa = Ma2 ) )
=> ( Y3
= ( one_one @ nat ) ) )
& ( ~ ( ( Xa = Mi2 )
& ( Xa = Ma2 ) )
=> ( Y3
= ( if @ nat @ ( ord_less @ nat @ ( vEBT_VEBT_high @ ( if @ nat @ ( Xa = Mi2 ) @ ( plus_plus @ nat @ ( times_times @ nat @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary2 ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList2 @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary2 ) ) ) ) ) ) @ Xa ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( size_size @ ( list @ vEBT_VEBT ) @ TreeList2 ) ) @ ( plus_plus @ nat @ ( vEBT_V1232361888498592333_e_t_e @ ( nth @ vEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ ( if @ nat @ ( Xa = Mi2 ) @ ( plus_plus @ nat @ ( times_times @ nat @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary2 ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList2 @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary2 ) ) ) ) ) ) @ Xa ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( vEBT_VEBT_low @ ( if @ nat @ ( Xa = Mi2 ) @ ( plus_plus @ nat @ ( times_times @ nat @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary2 ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList2 @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary2 ) ) ) ) ) ) @ Xa ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( if @ nat @ ( vEBT_VEBT_minNull @ ( vEBT_vebt_delete @ ( nth @ vEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ ( if @ nat @ ( Xa = Mi2 ) @ ( plus_plus @ nat @ ( times_times @ nat @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary2 ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList2 @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary2 ) ) ) ) ) ) @ Xa ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( vEBT_VEBT_low @ ( if @ nat @ ( Xa = Mi2 ) @ ( plus_plus @ nat @ ( times_times @ nat @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary2 ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList2 @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary2 ) ) ) ) ) ) @ Xa ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) @ ( vEBT_V1232361888498592333_e_t_e @ Summary2 @ ( vEBT_VEBT_high @ ( if @ nat @ ( Xa = Mi2 ) @ ( plus_plus @ nat @ ( times_times @ nat @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary2 ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList2 @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary2 ) ) ) ) ) ) @ Xa ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va3 ) ) @ ( 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_1363_zero__less__double__add__iff__zero__less__single__add,axiom,
! [A: $tType] :
( ( linord5086331880401160121up_add @ A )
=> ! [A3: A] :
( ( ord_less @ A @ ( zero_zero @ A ) @ ( plus_plus @ A @ A3 @ A3 ) )
= ( ord_less @ A @ ( zero_zero @ A ) @ A3 ) ) ) ).
% zero_less_double_add_iff_zero_less_single_add
thf(fact_1364_double__add__less__zero__iff__single__add__less__zero,axiom,
! [A: $tType] :
( ( linord5086331880401160121up_add @ A )
=> ! [A3: A] :
( ( ord_less @ A @ ( plus_plus @ A @ A3 @ A3 ) @ ( zero_zero @ A ) )
= ( ord_less @ A @ A3 @ ( zero_zero @ A ) ) ) ) ).
% double_add_less_zero_iff_single_add_less_zero
thf(fact_1365_Leaf__0__not,axiom,
! [A3: $o,B3: $o] :
~ ( vEBT_invar_vebt @ ( vEBT_Leaf @ A3 @ B3 ) @ ( zero_zero @ nat ) ) ).
% Leaf_0_not
thf(fact_1366_deg1Leaf,axiom,
! [T2: vEBT_VEBT] :
( ( vEBT_invar_vebt @ T2 @ ( one_one @ nat ) )
= ( ? [A6: $o,B7: $o] :
( T2
= ( vEBT_Leaf @ A6 @ B7 ) ) ) ) ).
% deg1Leaf
thf(fact_1367_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_1368_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_1369_Tbuildupi__buildupi_H,axiom,
! [N: nat] :
( ( semiring_1_of_nat @ int @ ( vEBT_V441764108873111860ildupi @ N ) )
= ( vEBT_V9176841429113362141ildupi @ N ) ) ).
% Tbuildupi_buildupi'
thf(fact_1370_add__left__cancel,axiom,
! [A: $tType] :
( ( cancel_semigroup_add @ A )
=> ! [A3: A,B3: A,C3: A] :
( ( ( plus_plus @ A @ A3 @ B3 )
= ( plus_plus @ A @ A3 @ C3 ) )
= ( B3 = C3 ) ) ) ).
% add_left_cancel
thf(fact_1371_add__right__cancel,axiom,
! [A: $tType] :
( ( cancel_semigroup_add @ A )
=> ! [B3: A,A3: A,C3: A] :
( ( ( plus_plus @ A @ B3 @ A3 )
= ( plus_plus @ A @ C3 @ A3 ) )
= ( B3 = C3 ) ) ) ).
% add_right_cancel
thf(fact_1372_max__enat__simps_I2_J,axiom,
! [Q3: extended_enat] :
( ( ord_max @ extended_enat @ Q3 @ ( zero_zero @ extended_enat ) )
= Q3 ) ).
% max_enat_simps(2)
thf(fact_1373_max__enat__simps_I3_J,axiom,
! [Q3: extended_enat] :
( ( ord_max @ extended_enat @ ( zero_zero @ extended_enat ) @ Q3 )
= Q3 ) ).
% max_enat_simps(3)
thf(fact_1374_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_1375_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_1376_add_Oright__neutral,axiom,
! [A: $tType] :
( ( monoid_add @ A )
=> ! [A3: A] :
( ( plus_plus @ A @ A3 @ ( zero_zero @ A ) )
= A3 ) ) ).
% add.right_neutral
thf(fact_1377_double__zero__sym,axiom,
! [A: $tType] :
( ( linord5086331880401160121up_add @ A )
=> ! [A3: A] :
( ( ( zero_zero @ A )
= ( plus_plus @ A @ A3 @ A3 ) )
= ( A3
= ( zero_zero @ A ) ) ) ) ).
% double_zero_sym
thf(fact_1378_add__cancel__left__left,axiom,
! [A: $tType] :
( ( cancel1802427076303600483id_add @ A )
=> ! [B3: A,A3: A] :
( ( ( plus_plus @ A @ B3 @ A3 )
= A3 )
= ( B3
= ( zero_zero @ A ) ) ) ) ).
% add_cancel_left_left
thf(fact_1379_add__cancel__left__right,axiom,
! [A: $tType] :
( ( cancel1802427076303600483id_add @ A )
=> ! [A3: A,B3: A] :
( ( ( plus_plus @ A @ A3 @ B3 )
= A3 )
= ( B3
= ( zero_zero @ A ) ) ) ) ).
% add_cancel_left_right
thf(fact_1380_add__cancel__right__left,axiom,
! [A: $tType] :
( ( cancel1802427076303600483id_add @ A )
=> ! [A3: A,B3: A] :
( ( A3
= ( plus_plus @ A @ B3 @ A3 ) )
= ( B3
= ( zero_zero @ A ) ) ) ) ).
% add_cancel_right_left
thf(fact_1381_add__cancel__right__right,axiom,
! [A: $tType] :
( ( cancel1802427076303600483id_add @ A )
=> ! [A3: A,B3: A] :
( ( A3
= ( plus_plus @ A @ A3 @ B3 ) )
= ( B3
= ( zero_zero @ A ) ) ) ) ).
% add_cancel_right_right
thf(fact_1382_add__eq__0__iff__both__eq__0,axiom,
! [A: $tType] :
( ( canoni5634975068530333245id_add @ A )
=> ! [X2: A,Y3: A] :
( ( ( plus_plus @ A @ X2 @ Y3 )
= ( zero_zero @ A ) )
= ( ( X2
= ( zero_zero @ A ) )
& ( Y3
= ( zero_zero @ A ) ) ) ) ) ).
% add_eq_0_iff_both_eq_0
thf(fact_1383_zero__eq__add__iff__both__eq__0,axiom,
! [A: $tType] :
( ( canoni5634975068530333245id_add @ A )
=> ! [X2: A,Y3: A] :
( ( ( zero_zero @ A )
= ( plus_plus @ A @ X2 @ Y3 ) )
= ( ( X2
= ( zero_zero @ A ) )
& ( Y3
= ( zero_zero @ A ) ) ) ) ) ).
% zero_eq_add_iff_both_eq_0
thf(fact_1384_add__0,axiom,
! [A: $tType] :
( ( monoid_add @ A )
=> ! [A3: A] :
( ( plus_plus @ A @ ( zero_zero @ A ) @ A3 )
= A3 ) ) ).
% add_0
thf(fact_1385_add__le__cancel__left,axiom,
! [A: $tType] :
( ( ordere2412721322843649153imp_le @ A )
=> ! [C3: A,A3: A,B3: A] :
( ( ord_less_eq @ A @ ( plus_plus @ A @ C3 @ A3 ) @ ( plus_plus @ A @ C3 @ B3 ) )
= ( ord_less_eq @ A @ A3 @ B3 ) ) ) ).
% add_le_cancel_left
thf(fact_1386_add__le__cancel__right,axiom,
! [A: $tType] :
( ( ordere2412721322843649153imp_le @ A )
=> ! [A3: A,C3: A,B3: A] :
( ( ord_less_eq @ A @ ( plus_plus @ A @ A3 @ C3 ) @ ( plus_plus @ A @ B3 @ C3 ) )
= ( ord_less_eq @ A @ A3 @ B3 ) ) ) ).
% add_le_cancel_right
thf(fact_1387_add__less__cancel__right,axiom,
! [A: $tType] :
( ( ordere2412721322843649153imp_le @ A )
=> ! [A3: A,C3: A,B3: A] :
( ( ord_less @ A @ ( plus_plus @ A @ A3 @ C3 ) @ ( plus_plus @ A @ B3 @ C3 ) )
= ( ord_less @ A @ A3 @ B3 ) ) ) ).
% add_less_cancel_right
thf(fact_1388_add__less__cancel__left,axiom,
! [A: $tType] :
( ( ordere2412721322843649153imp_le @ A )
=> ! [C3: A,A3: A,B3: A] :
( ( ord_less @ A @ ( plus_plus @ A @ C3 @ A3 ) @ ( plus_plus @ A @ C3 @ B3 ) )
= ( ord_less @ A @ A3 @ B3 ) ) ) ).
% add_less_cancel_left
thf(fact_1389_diff__self,axiom,
! [A: $tType] :
( ( group_add @ A )
=> ! [A3: A] :
( ( minus_minus @ A @ A3 @ A3 )
= ( zero_zero @ A ) ) ) ).
% diff_self
thf(fact_1390_diff__0__right,axiom,
! [A: $tType] :
( ( group_add @ A )
=> ! [A3: A] :
( ( minus_minus @ A @ A3 @ ( zero_zero @ A ) )
= A3 ) ) ).
% diff_0_right
thf(fact_1391_zero__diff,axiom,
! [A: $tType] :
( ( comm_monoid_diff @ A )
=> ! [A3: A] :
( ( minus_minus @ A @ ( zero_zero @ A ) @ A3 )
= ( zero_zero @ A ) ) ) ).
% zero_diff
thf(fact_1392_diff__zero,axiom,
! [A: $tType] :
( ( cancel1802427076303600483id_add @ A )
=> ! [A3: A] :
( ( minus_minus @ A @ A3 @ ( zero_zero @ A ) )
= A3 ) ) ).
% diff_zero
thf(fact_1393_cancel__comm__monoid__add__class_Odiff__cancel,axiom,
! [A: $tType] :
( ( cancel1802427076303600483id_add @ A )
=> ! [A3: A] :
( ( minus_minus @ A @ A3 @ A3 )
= ( zero_zero @ A ) ) ) ).
% cancel_comm_monoid_add_class.diff_cancel
thf(fact_1394_mult__1,axiom,
! [A: $tType] :
( ( monoid_mult @ A )
=> ! [A3: A] :
( ( times_times @ A @ ( one_one @ A ) @ A3 )
= A3 ) ) ).
% mult_1
thf(fact_1395_mult_Oright__neutral,axiom,
! [A: $tType] :
( ( monoid_mult @ A )
=> ! [A3: A] :
( ( times_times @ A @ A3 @ ( one_one @ A ) )
= A3 ) ) ).
% mult.right_neutral
thf(fact_1396_add__diff__cancel,axiom,
! [A: $tType] :
( ( group_add @ A )
=> ! [A3: A,B3: A] :
( ( minus_minus @ A @ ( plus_plus @ A @ A3 @ B3 ) @ B3 )
= A3 ) ) ).
% add_diff_cancel
thf(fact_1397_diff__add__cancel,axiom,
! [A: $tType] :
( ( group_add @ A )
=> ! [A3: A,B3: A] :
( ( plus_plus @ A @ ( minus_minus @ A @ A3 @ B3 ) @ B3 )
= A3 ) ) ).
% diff_add_cancel
thf(fact_1398_add__diff__cancel__left,axiom,
! [A: $tType] :
( ( cancel2418104881723323429up_add @ A )
=> ! [C3: A,A3: A,B3: A] :
( ( minus_minus @ A @ ( plus_plus @ A @ C3 @ A3 ) @ ( plus_plus @ A @ C3 @ B3 ) )
= ( minus_minus @ A @ A3 @ B3 ) ) ) ).
% add_diff_cancel_left
thf(fact_1399_add__diff__cancel__left_H,axiom,
! [A: $tType] :
( ( cancel2418104881723323429up_add @ A )
=> ! [A3: A,B3: A] :
( ( minus_minus @ A @ ( plus_plus @ A @ A3 @ B3 ) @ A3 )
= B3 ) ) ).
% add_diff_cancel_left'
thf(fact_1400_add__diff__cancel__right,axiom,
! [A: $tType] :
( ( cancel2418104881723323429up_add @ A )
=> ! [A3: A,C3: A,B3: A] :
( ( minus_minus @ A @ ( plus_plus @ A @ A3 @ C3 ) @ ( plus_plus @ A @ B3 @ C3 ) )
= ( minus_minus @ A @ A3 @ B3 ) ) ) ).
% add_diff_cancel_right
thf(fact_1401_add__diff__cancel__right_H,axiom,
! [A: $tType] :
( ( cancel2418104881723323429up_add @ A )
=> ! [A3: A,B3: A] :
( ( minus_minus @ A @ ( plus_plus @ A @ A3 @ B3 ) @ B3 )
= A3 ) ) ).
% add_diff_cancel_right'
thf(fact_1402_zero__le__double__add__iff__zero__le__single__add,axiom,
! [A: $tType] :
( ( linord5086331880401160121up_add @ A )
=> ! [A3: A] :
( ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( plus_plus @ A @ A3 @ A3 ) )
= ( ord_less_eq @ A @ ( zero_zero @ A ) @ A3 ) ) ) ).
% zero_le_double_add_iff_zero_le_single_add
thf(fact_1403_double__add__le__zero__iff__single__add__le__zero,axiom,
! [A: $tType] :
( ( linord5086331880401160121up_add @ A )
=> ! [A3: A] :
( ( ord_less_eq @ A @ ( plus_plus @ A @ A3 @ A3 ) @ ( zero_zero @ A ) )
= ( ord_less_eq @ A @ A3 @ ( zero_zero @ A ) ) ) ) ).
% double_add_le_zero_iff_single_add_le_zero
thf(fact_1404_le__add__same__cancel2,axiom,
! [A: $tType] :
( ( ordere1937475149494474687imp_le @ A )
=> ! [A3: A,B3: A] :
( ( ord_less_eq @ A @ A3 @ ( plus_plus @ A @ B3 @ A3 ) )
= ( ord_less_eq @ A @ ( zero_zero @ A ) @ B3 ) ) ) ).
% le_add_same_cancel2
thf(fact_1405_le__add__same__cancel1,axiom,
! [A: $tType] :
( ( ordere1937475149494474687imp_le @ A )
=> ! [A3: A,B3: A] :
( ( ord_less_eq @ A @ A3 @ ( plus_plus @ A @ A3 @ B3 ) )
= ( ord_less_eq @ A @ ( zero_zero @ A ) @ B3 ) ) ) ).
% le_add_same_cancel1
thf(fact_1406_add__le__same__cancel2,axiom,
! [A: $tType] :
( ( ordere1937475149494474687imp_le @ A )
=> ! [A3: A,B3: A] :
( ( ord_less_eq @ A @ ( plus_plus @ A @ A3 @ B3 ) @ B3 )
= ( ord_less_eq @ A @ A3 @ ( zero_zero @ A ) ) ) ) ).
% add_le_same_cancel2
thf(fact_1407_add__le__same__cancel1,axiom,
! [A: $tType] :
( ( ordere1937475149494474687imp_le @ A )
=> ! [B3: A,A3: A] :
( ( ord_less_eq @ A @ ( plus_plus @ A @ B3 @ A3 ) @ B3 )
= ( ord_less_eq @ A @ A3 @ ( zero_zero @ A ) ) ) ) ).
% add_le_same_cancel1
thf(fact_1408_add__less__same__cancel1,axiom,
! [A: $tType] :
( ( ordere1937475149494474687imp_le @ A )
=> ! [B3: A,A3: A] :
( ( ord_less @ A @ ( plus_plus @ A @ B3 @ A3 ) @ B3 )
= ( ord_less @ A @ A3 @ ( zero_zero @ A ) ) ) ) ).
% add_less_same_cancel1
thf(fact_1409_add__less__same__cancel2,axiom,
! [A: $tType] :
( ( ordere1937475149494474687imp_le @ A )
=> ! [A3: A,B3: A] :
( ( ord_less @ A @ ( plus_plus @ A @ A3 @ B3 ) @ B3 )
= ( ord_less @ A @ A3 @ ( zero_zero @ A ) ) ) ) ).
% add_less_same_cancel2
thf(fact_1410_less__add__same__cancel1,axiom,
! [A: $tType] :
( ( ordere1937475149494474687imp_le @ A )
=> ! [A3: A,B3: A] :
( ( ord_less @ A @ A3 @ ( plus_plus @ A @ A3 @ B3 ) )
= ( ord_less @ A @ ( zero_zero @ A ) @ B3 ) ) ) ).
% less_add_same_cancel1
thf(fact_1411_less__add__same__cancel2,axiom,
! [A: $tType] :
( ( ordere1937475149494474687imp_le @ A )
=> ! [A3: A,B3: A] :
( ( ord_less @ A @ A3 @ ( plus_plus @ A @ B3 @ A3 ) )
= ( ord_less @ A @ ( zero_zero @ A ) @ B3 ) ) ) ).
% less_add_same_cancel2
thf(fact_1412_imult__is__0,axiom,
! [M: extended_enat,N: extended_enat] :
( ( ( times_times @ extended_enat @ M @ N )
= ( zero_zero @ extended_enat ) )
= ( ( M
= ( zero_zero @ extended_enat ) )
| ( N
= ( zero_zero @ extended_enat ) ) ) ) ).
% imult_is_0
thf(fact_1413_zero__one__enat__neq_I1_J,axiom,
( ( zero_zero @ extended_enat )
!= ( one_one @ extended_enat ) ) ).
% zero_one_enat_neq(1)
thf(fact_1414_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,
! [A3: $o,B3: $o,X2: nat] :
( ( vEBT_T_m_e_m_b_e_r2 @ ( vEBT_Leaf @ A3 @ B3 ) @ X2 )
= ( one_one @ nat ) ) ).
% T\<^sub>m\<^sub>e\<^sub>m\<^sub>b\<^sub>e\<^sub>r'.simps(1)
thf(fact_1415_vebt__assn__raw_Osimps_I1_J,axiom,
! [A3: $o,B3: $o,Ai: $o,Bi: $o] :
( ( vEBT_vebt_assn_raw @ ( vEBT_Leaf @ A3 @ B3 ) @ ( vEBT_Leafi @ Ai @ Bi ) )
= ( pure_assn
@ ( ( Ai = A3 )
& ( Bi = B3 ) ) ) ) ).
% vebt_assn_raw.simps(1)
thf(fact_1416_VEBT_Oexhaust,axiom,
! [Y3: vEBT_VEBT] :
( ! [X112: option @ ( product_prod @ nat @ nat ),X122: nat,X132: list @ vEBT_VEBT,X142: vEBT_VEBT] :
( Y3
!= ( vEBT_Node @ X112 @ X122 @ X132 @ X142 ) )
=> ~ ! [X212: $o,X223: $o] :
( Y3
!= ( vEBT_Leaf @ X212 @ X223 ) ) ) ).
% VEBT.exhaust
thf(fact_1417_VEBT_Odistinct_I1_J,axiom,
! [X11: option @ ( product_prod @ nat @ nat ),X12: nat,X13: list @ vEBT_VEBT,X14: vEBT_VEBT,X213: $o,X224: $o] :
( ( vEBT_Node @ X11 @ X12 @ X13 @ X14 )
!= ( vEBT_Leaf @ X213 @ X224 ) ) ).
% VEBT.distinct(1)
thf(fact_1418_VEBT__internal_Ovalid_H_Ocases,axiom,
! [X2: product_prod @ vEBT_VEBT @ nat] :
( ! [Uu2: $o,Uv2: $o,D4: nat] :
( X2
!= ( product_Pair @ vEBT_VEBT @ nat @ ( vEBT_Leaf @ Uu2 @ Uv2 ) @ D4 ) )
=> ~ ! [Mima: option @ ( product_prod @ nat @ nat ),Deg2: nat,TreeList2: list @ vEBT_VEBT,Summary2: vEBT_VEBT,Deg3: nat] :
( X2
!= ( product_Pair @ vEBT_VEBT @ nat @ ( vEBT_Node @ Mima @ Deg2 @ TreeList2 @ Summary2 ) @ Deg3 ) ) ) ).
% VEBT_internal.valid'.cases
thf(fact_1419_VEBT__internal_OminNull_Osimps_I1_J,axiom,
vEBT_VEBT_minNull @ ( vEBT_Leaf @ $false @ $false ) ).
% VEBT_internal.minNull.simps(1)
thf(fact_1420_VEBT__internal_OminNull_Osimps_I2_J,axiom,
! [Uv: $o] :
~ ( vEBT_VEBT_minNull @ ( vEBT_Leaf @ $true @ Uv ) ) ).
% VEBT_internal.minNull.simps(2)
thf(fact_1421_VEBT__internal_OminNull_Osimps_I3_J,axiom,
! [Uu: $o] :
~ ( vEBT_VEBT_minNull @ ( vEBT_Leaf @ Uu @ $true ) ) ).
% VEBT_internal.minNull.simps(3)
thf(fact_1422_vebt__delete_Osimps_I3_J,axiom,
! [A3: $o,B3: $o,N: nat] :
( ( vEBT_vebt_delete @ ( vEBT_Leaf @ A3 @ B3 ) @ ( suc @ ( suc @ N ) ) )
= ( vEBT_Leaf @ A3 @ B3 ) ) ).
% vebt_delete.simps(3)
thf(fact_1423_vebt__delete_Osimps_I1_J,axiom,
! [A3: $o,B3: $o] :
( ( vEBT_vebt_delete @ ( vEBT_Leaf @ A3 @ B3 ) @ ( zero_zero @ nat ) )
= ( vEBT_Leaf @ $false @ B3 ) ) ).
% vebt_delete.simps(1)
thf(fact_1424_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,
! [A3: $o,B3: $o,X2: nat] :
( ( vEBT_T_i_n_s_e_r_t2 @ ( vEBT_Leaf @ A3 @ B3 ) @ X2 )
= ( one_one @ nat ) ) ).
% T\<^sub>i\<^sub>n\<^sub>s\<^sub>e\<^sub>r\<^sub>t'.simps(1)
thf(fact_1425_VEBT__internal_Onaive__member_Ocases,axiom,
! [X2: product_prod @ vEBT_VEBT @ nat] :
( ! [A4: $o,B4: $o,X4: nat] :
( X2
!= ( product_Pair @ vEBT_VEBT @ nat @ ( vEBT_Leaf @ A4 @ B4 ) @ X4 ) )
=> ( ! [Uu2: option @ ( product_prod @ nat @ nat ),Uv2: list @ vEBT_VEBT,Uw2: vEBT_VEBT,Ux2: nat] :
( X2
!= ( product_Pair @ vEBT_VEBT @ nat @ ( vEBT_Node @ Uu2 @ ( zero_zero @ nat ) @ Uv2 @ Uw2 ) @ Ux2 ) )
=> ~ ! [Uy2: option @ ( product_prod @ nat @ nat ),V3: nat,TreeList2: list @ vEBT_VEBT,S3: vEBT_VEBT,X4: nat] :
( X2
!= ( product_Pair @ vEBT_VEBT @ nat @ ( vEBT_Node @ Uy2 @ ( suc @ V3 ) @ TreeList2 @ S3 ) @ X4 ) ) ) ) ).
% VEBT_internal.naive_member.cases
thf(fact_1426_invar__vebt_Ointros_I1_J,axiom,
! [A3: $o,B3: $o] : ( vEBT_invar_vebt @ ( vEBT_Leaf @ A3 @ B3 ) @ ( suc @ ( zero_zero @ nat ) ) ) ).
% invar_vebt.intros(1)
thf(fact_1427_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_1428_vebt__delete_Osimps_I2_J,axiom,
! [A3: $o,B3: $o] :
( ( vEBT_vebt_delete @ ( vEBT_Leaf @ A3 @ B3 ) @ ( suc @ ( zero_zero @ nat ) ) )
= ( vEBT_Leaf @ A3 @ $false ) ) ).
% vebt_delete.simps(2)
thf(fact_1429_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,
! [X2: vEBT_VEBT] :
( ( X2
!= ( vEBT_Leaf @ $false @ $false ) )
=> ( ! [Uv2: $o] :
( X2
!= ( vEBT_Leaf @ $true @ Uv2 ) )
=> ( ! [Uu2: $o] :
( X2
!= ( vEBT_Leaf @ Uu2 @ $true ) )
=> ( ! [Uw2: nat,Ux2: list @ vEBT_VEBT,Uy2: vEBT_VEBT] :
( X2
!= ( vEBT_Node @ ( none @ ( product_prod @ nat @ nat ) ) @ Uw2 @ Ux2 @ Uy2 ) )
=> ~ ! [Uz2: product_prod @ nat @ nat,Va2: nat,Vb2: list @ vEBT_VEBT,Vc2: vEBT_VEBT] :
( X2
!= ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ Uz2 ) @ Va2 @ Vb2 @ Vc2 ) ) ) ) ) ) ).
% T\<^sub>m\<^sub>i\<^sub>n\<^sub>N\<^sub>u\<^sub>l\<^sub>l.cases
thf(fact_1430_vebt__member_Osimps_I1_J,axiom,
! [A3: $o,B3: $o,X2: nat] :
( ( vEBT_vebt_member @ ( vEBT_Leaf @ A3 @ B3 ) @ X2 )
= ( ( ( X2
= ( zero_zero @ nat ) )
=> A3 )
& ( ( X2
!= ( zero_zero @ nat ) )
=> ( ( ( X2
= ( one_one @ nat ) )
=> B3 )
& ( X2
= ( one_one @ nat ) ) ) ) ) ) ).
% vebt_member.simps(1)
thf(fact_1431_vebt__buildup_Osimps_I2_J,axiom,
( ( vEBT_vebt_buildup @ ( suc @ ( zero_zero @ nat ) ) )
= ( vEBT_Leaf @ $false @ $false ) ) ).
% vebt_buildup.simps(2)
thf(fact_1432_VEBT__internal_OminNull_Oelims_I3_J,axiom,
! [X2: vEBT_VEBT] :
( ~ ( vEBT_VEBT_minNull @ X2 )
=> ( ! [Uv2: $o] :
( X2
!= ( vEBT_Leaf @ $true @ Uv2 ) )
=> ( ! [Uu2: $o] :
( X2
!= ( vEBT_Leaf @ Uu2 @ $true ) )
=> ~ ! [Uz2: product_prod @ nat @ nat,Va2: nat,Vb2: list @ vEBT_VEBT,Vc2: vEBT_VEBT] :
( X2
!= ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ Uz2 ) @ Va2 @ Vb2 @ Vc2 ) ) ) ) ) ).
% VEBT_internal.minNull.elims(3)
thf(fact_1433_vebt__assn__raw_Ocases,axiom,
! [X2: product_prod @ vEBT_VEBT @ vEBT_VEBTi] :
( ! [A4: $o,B4: $o,Ai2: $o,Bi2: $o] :
( X2
!= ( product_Pair @ vEBT_VEBT @ vEBT_VEBTi @ ( vEBT_Leaf @ A4 @ B4 ) @ ( vEBT_Leafi @ Ai2 @ Bi2 ) ) )
=> ( ! [Mmo: option @ ( product_prod @ nat @ nat ),Deg2: nat,Tree_list: list @ vEBT_VEBT,Summary2: vEBT_VEBT,Mmoi: option @ ( product_prod @ nat @ nat ),Degi: nat,Tree_array: array @ vEBT_VEBTi,Summaryi: vEBT_VEBTi] :
( X2
!= ( product_Pair @ vEBT_VEBT @ vEBT_VEBTi @ ( vEBT_Node @ Mmo @ Deg2 @ Tree_list @ Summary2 ) @ ( vEBT_Nodei @ Mmoi @ Degi @ Tree_array @ Summaryi ) ) )
=> ( ! [V3: option @ ( product_prod @ nat @ nat ),Va3: nat,Vb3: list @ vEBT_VEBT,Vc3: vEBT_VEBT,Vd: $o,Ve: $o] :
( X2
!= ( product_Pair @ vEBT_VEBT @ vEBT_VEBTi @ ( vEBT_Node @ V3 @ Va3 @ Vb3 @ Vc3 ) @ ( vEBT_Leafi @ Vd @ Ve ) ) )
=> ~ ! [Vd: $o,Ve: $o,V3: option @ ( product_prod @ nat @ nat ),Va3: nat,Vb3: array @ vEBT_VEBTi,Vc3: vEBT_VEBTi] :
( X2
!= ( product_Pair @ vEBT_VEBT @ vEBT_VEBTi @ ( vEBT_Leaf @ Vd @ Ve ) @ ( vEBT_Nodei @ V3 @ Va3 @ Vb3 @ Vc3 ) ) ) ) ) ) ).
% vebt_assn_raw.cases
thf(fact_1434_vebt__insert_Osimps_I1_J,axiom,
! [X2: nat,A3: $o,B3: $o] :
( ( ( X2
= ( zero_zero @ nat ) )
=> ( ( vEBT_vebt_insert @ ( vEBT_Leaf @ A3 @ B3 ) @ X2 )
= ( vEBT_Leaf @ $true @ B3 ) ) )
& ( ( X2
!= ( zero_zero @ nat ) )
=> ( ( ( X2
= ( one_one @ nat ) )
=> ( ( vEBT_vebt_insert @ ( vEBT_Leaf @ A3 @ B3 ) @ X2 )
= ( vEBT_Leaf @ A3 @ $true ) ) )
& ( ( X2
!= ( one_one @ nat ) )
=> ( ( vEBT_vebt_insert @ ( vEBT_Leaf @ A3 @ B3 ) @ X2 )
= ( vEBT_Leaf @ A3 @ B3 ) ) ) ) ) ) ).
% vebt_insert.simps(1)
thf(fact_1435_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_1436_group__cancel_Oadd1,axiom,
! [A: $tType] :
( ( comm_monoid_add @ A )
=> ! [A2: A,K: A,A3: A,B3: A] :
( ( A2
= ( plus_plus @ A @ K @ A3 ) )
=> ( ( plus_plus @ A @ A2 @ B3 )
= ( plus_plus @ A @ K @ ( plus_plus @ A @ A3 @ B3 ) ) ) ) ) ).
% group_cancel.add1
thf(fact_1437_group__cancel_Oadd2,axiom,
! [A: $tType] :
( ( comm_monoid_add @ A )
=> ! [B2: A,K: A,B3: A,A3: A] :
( ( B2
= ( plus_plus @ A @ K @ B3 ) )
=> ( ( plus_plus @ A @ A3 @ B2 )
= ( plus_plus @ A @ K @ ( plus_plus @ A @ A3 @ B3 ) ) ) ) ) ).
% group_cancel.add2
thf(fact_1438_add_Oassoc,axiom,
! [A: $tType] :
( ( semigroup_add @ A )
=> ! [A3: A,B3: A,C3: A] :
( ( plus_plus @ A @ ( plus_plus @ A @ A3 @ B3 ) @ C3 )
= ( plus_plus @ A @ A3 @ ( plus_plus @ A @ B3 @ C3 ) ) ) ) ).
% add.assoc
thf(fact_1439_add_Oleft__cancel,axiom,
! [A: $tType] :
( ( group_add @ A )
=> ! [A3: A,B3: A,C3: A] :
( ( ( plus_plus @ A @ A3 @ B3 )
= ( plus_plus @ A @ A3 @ C3 ) )
= ( B3 = C3 ) ) ) ).
% add.left_cancel
thf(fact_1440_add_Oright__cancel,axiom,
! [A: $tType] :
( ( group_add @ A )
=> ! [B3: A,A3: A,C3: A] :
( ( ( plus_plus @ A @ B3 @ A3 )
= ( plus_plus @ A @ C3 @ A3 ) )
= ( B3 = C3 ) ) ) ).
% add.right_cancel
thf(fact_1441_ab__semigroup__add__class_Oadd_Ocommute,axiom,
! [A: $tType] :
( ( ab_semigroup_add @ A )
=> ( ( plus_plus @ A )
= ( ^ [A6: A,B7: A] : ( plus_plus @ A @ B7 @ A6 ) ) ) ) ).
% ab_semigroup_add_class.add.commute
thf(fact_1442_ab__semigroup__add__class_Oadd_Oleft__commute,axiom,
! [A: $tType] :
( ( ab_semigroup_add @ A )
=> ! [B3: A,A3: A,C3: A] :
( ( plus_plus @ A @ B3 @ ( plus_plus @ A @ A3 @ C3 ) )
= ( plus_plus @ A @ A3 @ ( plus_plus @ A @ B3 @ C3 ) ) ) ) ).
% ab_semigroup_add_class.add.left_commute
thf(fact_1443_add__left__imp__eq,axiom,
! [A: $tType] :
( ( cancel_semigroup_add @ A )
=> ! [A3: A,B3: A,C3: A] :
( ( ( plus_plus @ A @ A3 @ B3 )
= ( plus_plus @ A @ A3 @ C3 ) )
=> ( B3 = C3 ) ) ) ).
% add_left_imp_eq
thf(fact_1444_add__right__imp__eq,axiom,
! [A: $tType] :
( ( cancel_semigroup_add @ A )
=> ! [B3: A,A3: A,C3: A] :
( ( ( plus_plus @ A @ B3 @ A3 )
= ( plus_plus @ A @ C3 @ A3 ) )
=> ( B3 = C3 ) ) ) ).
% add_right_imp_eq
thf(fact_1445_one__reorient,axiom,
! [A: $tType] :
( ( one @ A )
=> ! [X2: A] :
( ( ( one_one @ A )
= X2 )
= ( X2
= ( one_one @ A ) ) ) ) ).
% one_reorient
thf(fact_1446_ab__semigroup__mult__class_Omult_Oleft__commute,axiom,
! [A: $tType] :
( ( ab_semigroup_mult @ A )
=> ! [B3: A,A3: A,C3: A] :
( ( times_times @ A @ B3 @ ( times_times @ A @ A3 @ C3 ) )
= ( times_times @ A @ A3 @ ( times_times @ A @ B3 @ C3 ) ) ) ) ).
% ab_semigroup_mult_class.mult.left_commute
thf(fact_1447_ab__semigroup__mult__class_Omult_Ocommute,axiom,
! [A: $tType] :
( ( ab_semigroup_mult @ A )
=> ( ( times_times @ A )
= ( ^ [A6: A,B7: A] : ( times_times @ A @ B7 @ A6 ) ) ) ) ).
% ab_semigroup_mult_class.mult.commute
thf(fact_1448_mult_Oassoc,axiom,
! [A: $tType] :
( ( semigroup_mult @ A )
=> ! [A3: A,B3: A,C3: A] :
( ( times_times @ A @ ( times_times @ A @ A3 @ B3 ) @ C3 )
= ( times_times @ A @ A3 @ ( times_times @ A @ B3 @ C3 ) ) ) ) ).
% mult.assoc
thf(fact_1449_VEBT__internal_OminNull_Oelims_I2_J,axiom,
! [X2: vEBT_VEBT] :
( ( vEBT_VEBT_minNull @ X2 )
=> ( ( X2
!= ( vEBT_Leaf @ $false @ $false ) )
=> ~ ! [Uw2: nat,Ux2: list @ vEBT_VEBT,Uy2: vEBT_VEBT] :
( X2
!= ( vEBT_Node @ ( none @ ( product_prod @ nat @ nat ) ) @ Uw2 @ Ux2 @ Uy2 ) ) ) ) ).
% VEBT_internal.minNull.elims(2)
thf(fact_1450_diff__eq__diff__eq,axiom,
! [A: $tType] :
( ( group_add @ A )
=> ! [A3: A,B3: A,C3: A,D2: A] :
( ( ( minus_minus @ A @ A3 @ B3 )
= ( minus_minus @ A @ C3 @ D2 ) )
=> ( ( A3 = B3 )
= ( C3 = D2 ) ) ) ) ).
% diff_eq_diff_eq
thf(fact_1451_cancel__ab__semigroup__add__class_Odiff__right__commute,axiom,
! [A: $tType] :
( ( cancel2418104881723323429up_add @ A )
=> ! [A3: A,C3: A,B3: A] :
( ( minus_minus @ A @ ( minus_minus @ A @ A3 @ C3 ) @ B3 )
= ( minus_minus @ A @ ( minus_minus @ A @ A3 @ B3 ) @ C3 ) ) ) ).
% cancel_ab_semigroup_add_class.diff_right_commute
thf(fact_1452_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,
! [A3: $o,B3: $o,N: nat] :
( ( vEBT_T_d_e_l_e_t_e @ ( vEBT_Leaf @ A3 @ B3 ) @ ( suc @ ( suc @ N ) ) )
= ( one_one @ nat ) ) ).
% T\<^sub>d\<^sub>e\<^sub>l\<^sub>e\<^sub>t\<^sub>e.simps(3)
thf(fact_1453_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,
! [A3: $o,B3: $o,N: nat] :
( ( vEBT_V1232361888498592333_e_t_e @ ( vEBT_Leaf @ A3 @ B3 ) @ ( 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_1454_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,
! [A3: $o,B3: $o] :
( ( vEBT_T_d_e_l_e_t_e @ ( vEBT_Leaf @ A3 @ B3 ) @ ( zero_zero @ nat ) )
= ( one_one @ nat ) ) ).
% T\<^sub>d\<^sub>e\<^sub>l\<^sub>e\<^sub>t\<^sub>e.simps(1)
thf(fact_1455_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,
! [A3: $o,B3: $o] :
( ( vEBT_V1232361888498592333_e_t_e @ ( vEBT_Leaf @ A3 @ B3 ) @ ( 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_1456_T_092_060_094sub_062m_092_060_094sub_062i_092_060_094sub_062n_092_060_094sub_062t_Ocases,axiom,
! [X2: vEBT_VEBT] :
( ! [A4: $o,B4: $o] :
( X2
!= ( vEBT_Leaf @ A4 @ B4 ) )
=> ( ! [Uu2: nat,Uv2: list @ vEBT_VEBT,Uw2: vEBT_VEBT] :
( X2
!= ( vEBT_Node @ ( none @ ( product_prod @ nat @ nat ) ) @ Uu2 @ Uv2 @ Uw2 ) )
=> ~ ! [Mi2: nat,Ma2: nat,Ux2: nat,Uy2: list @ vEBT_VEBT,Uz2: vEBT_VEBT] :
( X2
!= ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi2 @ Ma2 ) ) @ Ux2 @ Uy2 @ Uz2 ) ) ) ) ).
% T\<^sub>m\<^sub>i\<^sub>n\<^sub>t.cases
thf(fact_1457_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,
! [A3: $o,B3: $o] :
( ( vEBT_T_d_e_l_e_t_e @ ( vEBT_Leaf @ A3 @ B3 ) @ ( suc @ ( zero_zero @ nat ) ) )
= ( one_one @ nat ) ) ).
% T\<^sub>d\<^sub>e\<^sub>l\<^sub>e\<^sub>t\<^sub>e.simps(2)
thf(fact_1458_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,
! [A3: $o,B3: $o] :
( ( vEBT_V1232361888498592333_e_t_e @ ( vEBT_Leaf @ A3 @ B3 ) @ ( 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_1459_VEBT__internal_OminNull_Oelims_I1_J,axiom,
! [X2: vEBT_VEBT,Y3: $o] :
( ( ( vEBT_VEBT_minNull @ X2 )
= Y3 )
=> ( ( ( X2
= ( vEBT_Leaf @ $false @ $false ) )
=> ~ Y3 )
=> ( ( ? [Uv2: $o] :
( X2
= ( vEBT_Leaf @ $true @ Uv2 ) )
=> Y3 )
=> ( ( ? [Uu2: $o] :
( X2
= ( vEBT_Leaf @ Uu2 @ $true ) )
=> Y3 )
=> ( ( ? [Uw2: nat,Ux2: list @ vEBT_VEBT,Uy2: vEBT_VEBT] :
( X2
= ( vEBT_Node @ ( none @ ( product_prod @ nat @ nat ) ) @ Uw2 @ Ux2 @ Uy2 ) )
=> ~ Y3 )
=> ~ ( ? [Uz2: product_prod @ nat @ nat,Va2: nat,Vb2: list @ vEBT_VEBT,Vc2: vEBT_VEBT] :
( X2
= ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ Uz2 ) @ Va2 @ Vb2 @ Vc2 ) )
=> Y3 ) ) ) ) ) ) ).
% VEBT_internal.minNull.elims(1)
thf(fact_1460_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_1461_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,
! [Uu: nat,Uv: list @ vEBT_VEBT,Uw: vEBT_VEBT,X2: nat] :
( ( vEBT_T_m_e_m_b_e_r2 @ ( vEBT_Node @ ( none @ ( product_prod @ nat @ nat ) ) @ Uu @ Uv @ Uw ) @ X2 )
= ( one_one @ nat ) ) ).
% T\<^sub>m\<^sub>e\<^sub>m\<^sub>b\<^sub>e\<^sub>r'.simps(2)
thf(fact_1462_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,
! [X2: product_prod @ vEBT_VEBT @ nat] :
( ! [A4: $o,B4: $o] :
( X2
!= ( product_Pair @ vEBT_VEBT @ nat @ ( vEBT_Leaf @ A4 @ B4 ) @ ( zero_zero @ nat ) ) )
=> ( ! [A4: $o,B4: $o] :
( X2
!= ( product_Pair @ vEBT_VEBT @ nat @ ( vEBT_Leaf @ A4 @ B4 ) @ ( suc @ ( zero_zero @ nat ) ) ) )
=> ( ! [A4: $o,B4: $o,N3: nat] :
( X2
!= ( product_Pair @ vEBT_VEBT @ nat @ ( vEBT_Leaf @ A4 @ B4 ) @ ( suc @ ( suc @ N3 ) ) ) )
=> ( ! [Deg2: nat,TreeList2: list @ vEBT_VEBT,Summary2: vEBT_VEBT,Uu2: nat] :
( X2
!= ( product_Pair @ vEBT_VEBT @ nat @ ( vEBT_Node @ ( none @ ( product_prod @ nat @ nat ) ) @ Deg2 @ TreeList2 @ Summary2 ) @ Uu2 ) )
=> ( ! [Mi2: nat,Ma2: nat,TreeList2: list @ vEBT_VEBT,Summary2: vEBT_VEBT,X4: nat] :
( X2
!= ( product_Pair @ vEBT_VEBT @ nat @ ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi2 @ Ma2 ) ) @ ( zero_zero @ nat ) @ TreeList2 @ Summary2 ) @ X4 ) )
=> ( ! [Mi2: nat,Ma2: nat,TreeList2: list @ vEBT_VEBT,Summary2: vEBT_VEBT,X4: nat] :
( X2
!= ( product_Pair @ vEBT_VEBT @ nat @ ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( zero_zero @ nat ) ) @ TreeList2 @ Summary2 ) @ X4 ) )
=> ~ ! [Mi2: nat,Ma2: nat,Va3: nat,TreeList2: list @ vEBT_VEBT,Summary2: vEBT_VEBT,X4: nat] :
( X2
!= ( product_Pair @ vEBT_VEBT @ nat @ ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( suc @ Va3 ) ) @ TreeList2 @ Summary2 ) @ X4 ) ) ) ) ) ) ) ) ).
% VEBT_internal.T\<^sub>d\<^sub>e\<^sub>l\<^sub>e\<^sub>t\<^sub>e'.cases
thf(fact_1463_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,
! [X2: product_prod @ vEBT_VEBT @ nat] :
( ! [A4: $o,B4: $o,X4: nat] :
( X2
!= ( product_Pair @ vEBT_VEBT @ nat @ ( vEBT_Leaf @ A4 @ B4 ) @ X4 ) )
=> ( ! [Uu2: nat,Uv2: list @ vEBT_VEBT,Uw2: vEBT_VEBT,X4: nat] :
( X2
!= ( product_Pair @ vEBT_VEBT @ nat @ ( vEBT_Node @ ( none @ ( product_prod @ nat @ nat ) ) @ Uu2 @ Uv2 @ Uw2 ) @ X4 ) )
=> ( ! [V3: product_prod @ nat @ nat,Uy2: list @ vEBT_VEBT,Uz2: vEBT_VEBT,X4: nat] :
( X2
!= ( product_Pair @ vEBT_VEBT @ nat @ ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ V3 ) @ ( zero_zero @ nat ) @ Uy2 @ Uz2 ) @ X4 ) )
=> ( ! [V3: product_prod @ nat @ nat,Vb2: list @ vEBT_VEBT,Vc2: vEBT_VEBT,X4: nat] :
( X2
!= ( product_Pair @ vEBT_VEBT @ nat @ ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ V3 ) @ ( suc @ ( zero_zero @ nat ) ) @ Vb2 @ Vc2 ) @ X4 ) )
=> ~ ! [Mi2: nat,Ma2: nat,Va3: nat,TreeList2: list @ vEBT_VEBT,Summary2: vEBT_VEBT,X4: nat] :
( X2
!= ( product_Pair @ vEBT_VEBT @ nat @ ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( suc @ Va3 ) ) @ TreeList2 @ Summary2 ) @ X4 ) ) ) ) ) ) ).
% T\<^sub>m\<^sub>e\<^sub>m\<^sub>b\<^sub>e\<^sub>r'.cases
thf(fact_1464_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,
! [X2: product_prod @ vEBT_VEBT @ nat] :
( ! [A4: $o,B4: $o,X4: nat] :
( X2
!= ( product_Pair @ vEBT_VEBT @ nat @ ( vEBT_Leaf @ A4 @ B4 ) @ X4 ) )
=> ( ! [Info2: option @ ( product_prod @ nat @ nat ),Ts2: list @ vEBT_VEBT,S3: vEBT_VEBT,X4: nat] :
( X2
!= ( product_Pair @ vEBT_VEBT @ nat @ ( vEBT_Node @ Info2 @ ( zero_zero @ nat ) @ Ts2 @ S3 ) @ X4 ) )
=> ( ! [Info2: option @ ( product_prod @ nat @ nat ),Ts2: list @ vEBT_VEBT,S3: vEBT_VEBT,X4: nat] :
( X2
!= ( product_Pair @ vEBT_VEBT @ nat @ ( vEBT_Node @ Info2 @ ( suc @ ( zero_zero @ nat ) ) @ Ts2 @ S3 ) @ X4 ) )
=> ( ! [V3: nat,TreeList2: list @ vEBT_VEBT,Summary2: vEBT_VEBT,X4: nat] :
( X2
!= ( product_Pair @ vEBT_VEBT @ nat @ ( vEBT_Node @ ( none @ ( product_prod @ nat @ nat ) ) @ ( suc @ ( suc @ V3 ) ) @ TreeList2 @ Summary2 ) @ X4 ) )
=> ~ ! [Mi2: nat,Ma2: nat,Va3: nat,TreeList2: list @ vEBT_VEBT,Summary2: vEBT_VEBT,X4: nat] :
( X2
!= ( product_Pair @ vEBT_VEBT @ nat @ ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( suc @ Va3 ) ) @ TreeList2 @ Summary2 ) @ X4 ) ) ) ) ) ) ).
% T\<^sub>i\<^sub>n\<^sub>s\<^sub>e\<^sub>r\<^sub>t'.cases
thf(fact_1465_T_092_060_094sub_062s_092_060_094sub_062u_092_060_094sub_062c_092_060_094sub_062c_H_Ocases,axiom,
! [X2: product_prod @ vEBT_VEBT @ nat] :
( ! [Uu2: $o,B4: $o] :
( X2
!= ( product_Pair @ vEBT_VEBT @ nat @ ( vEBT_Leaf @ Uu2 @ B4 ) @ ( zero_zero @ nat ) ) )
=> ( ! [Uv2: $o,Uw2: $o,N3: nat] :
( X2
!= ( product_Pair @ vEBT_VEBT @ nat @ ( vEBT_Leaf @ Uv2 @ Uw2 ) @ ( suc @ N3 ) ) )
=> ( ! [Ux2: nat,Uy2: list @ vEBT_VEBT,Uz2: vEBT_VEBT,Va2: nat] :
( X2
!= ( product_Pair @ vEBT_VEBT @ nat @ ( vEBT_Node @ ( none @ ( product_prod @ nat @ nat ) ) @ Ux2 @ Uy2 @ Uz2 ) @ Va2 ) )
=> ( ! [V3: product_prod @ nat @ nat,Vc2: list @ vEBT_VEBT,Vd2: vEBT_VEBT,Ve2: nat] :
( X2
!= ( product_Pair @ vEBT_VEBT @ nat @ ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ V3 ) @ ( zero_zero @ nat ) @ Vc2 @ Vd2 ) @ Ve2 ) )
=> ( ! [V3: product_prod @ nat @ nat,Vg: list @ vEBT_VEBT,Vh: vEBT_VEBT,Vi: nat] :
( X2
!= ( product_Pair @ vEBT_VEBT @ nat @ ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ V3 ) @ ( suc @ ( zero_zero @ nat ) ) @ Vg @ Vh ) @ Vi ) )
=> ~ ! [Mi2: nat,Ma2: nat,Va3: nat,TreeList2: list @ vEBT_VEBT,Summary2: vEBT_VEBT,X4: nat] :
( X2
!= ( product_Pair @ vEBT_VEBT @ nat @ ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( suc @ Va3 ) ) @ TreeList2 @ Summary2 ) @ X4 ) ) ) ) ) ) ) ).
% T\<^sub>s\<^sub>u\<^sub>c\<^sub>c'.cases
thf(fact_1466_T_092_060_094sub_062p_092_060_094sub_062r_092_060_094sub_062e_092_060_094sub_062d_H_Ocases,axiom,
! [X2: product_prod @ vEBT_VEBT @ nat] :
( ! [Uu2: $o,Uv2: $o] :
( X2
!= ( product_Pair @ vEBT_VEBT @ nat @ ( vEBT_Leaf @ Uu2 @ Uv2 ) @ ( zero_zero @ nat ) ) )
=> ( ! [A4: $o,Uw2: $o] :
( X2
!= ( product_Pair @ vEBT_VEBT @ nat @ ( vEBT_Leaf @ A4 @ Uw2 ) @ ( suc @ ( zero_zero @ nat ) ) ) )
=> ( ! [A4: $o,B4: $o,Va3: nat] :
( X2
!= ( product_Pair @ vEBT_VEBT @ nat @ ( vEBT_Leaf @ A4 @ B4 ) @ ( suc @ ( suc @ Va3 ) ) ) )
=> ( ! [Uy2: nat,Uz2: list @ vEBT_VEBT,Va2: vEBT_VEBT,Vb2: nat] :
( X2
!= ( product_Pair @ vEBT_VEBT @ nat @ ( vEBT_Node @ ( none @ ( product_prod @ nat @ nat ) ) @ Uy2 @ Uz2 @ Va2 ) @ Vb2 ) )
=> ( ! [V3: product_prod @ nat @ nat,Vd2: list @ vEBT_VEBT,Ve2: vEBT_VEBT,Vf: nat] :
( X2
!= ( product_Pair @ vEBT_VEBT @ nat @ ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ V3 ) @ ( zero_zero @ nat ) @ Vd2 @ Ve2 ) @ Vf ) )
=> ( ! [V3: product_prod @ nat @ nat,Vh: list @ vEBT_VEBT,Vi: vEBT_VEBT,Vj: nat] :
( X2
!= ( product_Pair @ vEBT_VEBT @ nat @ ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ V3 ) @ ( suc @ ( zero_zero @ nat ) ) @ Vh @ Vi ) @ Vj ) )
=> ~ ! [Mi2: nat,Ma2: nat,Va3: nat,TreeList2: list @ vEBT_VEBT,Summary2: vEBT_VEBT,X4: nat] :
( X2
!= ( product_Pair @ vEBT_VEBT @ nat @ ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( suc @ Va3 ) ) @ TreeList2 @ Summary2 ) @ X4 ) ) ) ) ) ) ) ) ).
% T\<^sub>p\<^sub>r\<^sub>e\<^sub>d'.cases
thf(fact_1467_VEBT__internal_Omembermima_Ocases,axiom,
! [X2: product_prod @ vEBT_VEBT @ nat] :
( ! [Uu2: $o,Uv2: $o,Uw2: nat] :
( X2
!= ( product_Pair @ vEBT_VEBT @ nat @ ( vEBT_Leaf @ Uu2 @ Uv2 ) @ Uw2 ) )
=> ( ! [Ux2: list @ vEBT_VEBT,Uy2: vEBT_VEBT,Uz2: nat] :
( X2
!= ( product_Pair @ vEBT_VEBT @ nat @ ( vEBT_Node @ ( none @ ( product_prod @ nat @ nat ) ) @ ( zero_zero @ nat ) @ Ux2 @ Uy2 ) @ Uz2 ) )
=> ( ! [Mi2: nat,Ma2: nat,Va2: list @ vEBT_VEBT,Vb2: vEBT_VEBT,X4: nat] :
( X2
!= ( product_Pair @ vEBT_VEBT @ nat @ ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi2 @ Ma2 ) ) @ ( zero_zero @ nat ) @ Va2 @ Vb2 ) @ X4 ) )
=> ( ! [Mi2: nat,Ma2: nat,V3: nat,TreeList2: list @ vEBT_VEBT,Vc2: vEBT_VEBT,X4: nat] :
( X2
!= ( product_Pair @ vEBT_VEBT @ nat @ ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi2 @ Ma2 ) ) @ ( suc @ V3 ) @ TreeList2 @ Vc2 ) @ X4 ) )
=> ~ ! [V3: nat,TreeList2: list @ vEBT_VEBT,Vd2: vEBT_VEBT,X4: nat] :
( X2
!= ( product_Pair @ vEBT_VEBT @ nat @ ( vEBT_Node @ ( none @ ( product_prod @ nat @ nat ) ) @ ( suc @ V3 ) @ TreeList2 @ Vd2 ) @ X4 ) ) ) ) ) ) ).
% VEBT_internal.membermima.cases
thf(fact_1468_zero__le,axiom,
! [A: $tType] :
( ( canoni5634975068530333245id_add @ A )
=> ! [X2: A] : ( ord_less_eq @ A @ ( zero_zero @ A ) @ X2 ) ) ).
% zero_le
thf(fact_1469_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_1470_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_1471_not__less__zero,axiom,
! [A: $tType] :
( ( canoni5634975068530333245id_add @ A )
=> ! [N: A] :
~ ( ord_less @ A @ N @ ( zero_zero @ A ) ) ) ).
% not_less_zero
thf(fact_1472_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_1473_comm__monoid__add__class_Oadd__0,axiom,
! [A: $tType] :
( ( comm_monoid_add @ A )
=> ! [A3: A] :
( ( plus_plus @ A @ ( zero_zero @ A ) @ A3 )
= A3 ) ) ).
% comm_monoid_add_class.add_0
thf(fact_1474_add_Ocomm__neutral,axiom,
! [A: $tType] :
( ( comm_monoid_add @ A )
=> ! [A3: A] :
( ( plus_plus @ A @ A3 @ ( zero_zero @ A ) )
= A3 ) ) ).
% add.comm_neutral
thf(fact_1475_add_Ogroup__left__neutral,axiom,
! [A: $tType] :
( ( group_add @ A )
=> ! [A3: A] :
( ( plus_plus @ A @ ( zero_zero @ A ) @ A3 )
= A3 ) ) ).
% add.group_left_neutral
thf(fact_1476_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_1477_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_1478_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_1479_add__mono,axiom,
! [A: $tType] :
( ( ordere6658533253407199908up_add @ A )
=> ! [A3: A,B3: A,C3: A,D2: A] :
( ( ord_less_eq @ A @ A3 @ B3 )
=> ( ( ord_less_eq @ A @ C3 @ D2 )
=> ( ord_less_eq @ A @ ( plus_plus @ A @ A3 @ C3 ) @ ( plus_plus @ A @ B3 @ D2 ) ) ) ) ) ).
% add_mono
thf(fact_1480_add__left__mono,axiom,
! [A: $tType] :
( ( ordere6658533253407199908up_add @ A )
=> ! [A3: A,B3: A,C3: A] :
( ( ord_less_eq @ A @ A3 @ B3 )
=> ( ord_less_eq @ A @ ( plus_plus @ A @ C3 @ A3 ) @ ( plus_plus @ A @ C3 @ B3 ) ) ) ) ).
% add_left_mono
thf(fact_1481_less__eqE,axiom,
! [A: $tType] :
( ( canoni5634975068530333245id_add @ A )
=> ! [A3: A,B3: A] :
( ( ord_less_eq @ A @ A3 @ B3 )
=> ~ ! [C4: A] :
( B3
!= ( plus_plus @ A @ A3 @ C4 ) ) ) ) ).
% less_eqE
thf(fact_1482_add__right__mono,axiom,
! [A: $tType] :
( ( ordere6658533253407199908up_add @ A )
=> ! [A3: A,B3: A,C3: A] :
( ( ord_less_eq @ A @ A3 @ B3 )
=> ( ord_less_eq @ A @ ( plus_plus @ A @ A3 @ C3 ) @ ( plus_plus @ A @ B3 @ C3 ) ) ) ) ).
% add_right_mono
thf(fact_1483_le__iff__add,axiom,
! [A: $tType] :
( ( canoni5634975068530333245id_add @ A )
=> ( ( ord_less_eq @ A )
= ( ^ [A6: A,B7: A] :
? [C5: A] :
( B7
= ( plus_plus @ A @ A6 @ C5 ) ) ) ) ) ).
% le_iff_add
thf(fact_1484_add__le__imp__le__left,axiom,
! [A: $tType] :
( ( ordere2412721322843649153imp_le @ A )
=> ! [C3: A,A3: A,B3: A] :
( ( ord_less_eq @ A @ ( plus_plus @ A @ C3 @ A3 ) @ ( plus_plus @ A @ C3 @ B3 ) )
=> ( ord_less_eq @ A @ A3 @ B3 ) ) ) ).
% add_le_imp_le_left
thf(fact_1485_add__le__imp__le__right,axiom,
! [A: $tType] :
( ( ordere2412721322843649153imp_le @ A )
=> ! [A3: A,C3: A,B3: A] :
( ( ord_less_eq @ A @ ( plus_plus @ A @ A3 @ C3 ) @ ( plus_plus @ A @ B3 @ C3 ) )
=> ( ord_less_eq @ A @ A3 @ B3 ) ) ) ).
% add_le_imp_le_right
thf(fact_1486_add__less__imp__less__right,axiom,
! [A: $tType] :
( ( ordere2412721322843649153imp_le @ A )
=> ! [A3: A,C3: A,B3: A] :
( ( ord_less @ A @ ( plus_plus @ A @ A3 @ C3 ) @ ( plus_plus @ A @ B3 @ C3 ) )
=> ( ord_less @ A @ A3 @ B3 ) ) ) ).
% add_less_imp_less_right
thf(fact_1487_add__less__imp__less__left,axiom,
! [A: $tType] :
( ( ordere2412721322843649153imp_le @ A )
=> ! [C3: A,A3: A,B3: A] :
( ( ord_less @ A @ ( plus_plus @ A @ C3 @ A3 ) @ ( plus_plus @ A @ C3 @ B3 ) )
=> ( ord_less @ A @ A3 @ B3 ) ) ) ).
% add_less_imp_less_left
thf(fact_1488_add__strict__right__mono,axiom,
! [A: $tType] :
( ( ordere580206878836729694up_add @ A )
=> ! [A3: A,B3: A,C3: A] :
( ( ord_less @ A @ A3 @ B3 )
=> ( ord_less @ A @ ( plus_plus @ A @ A3 @ C3 ) @ ( plus_plus @ A @ B3 @ C3 ) ) ) ) ).
% add_strict_right_mono
thf(fact_1489_add__strict__left__mono,axiom,
! [A: $tType] :
( ( ordere580206878836729694up_add @ A )
=> ! [A3: A,B3: A,C3: A] :
( ( ord_less @ A @ A3 @ B3 )
=> ( ord_less @ A @ ( plus_plus @ A @ C3 @ A3 ) @ ( plus_plus @ A @ C3 @ B3 ) ) ) ) ).
% add_strict_left_mono
thf(fact_1490_add__strict__mono,axiom,
! [A: $tType] :
( ( strict9044650504122735259up_add @ A )
=> ! [A3: A,B3: A,C3: A,D2: A] :
( ( ord_less @ A @ A3 @ B3 )
=> ( ( ord_less @ A @ C3 @ D2 )
=> ( ord_less @ A @ ( plus_plus @ A @ A3 @ C3 ) @ ( plus_plus @ A @ B3 @ D2 ) ) ) ) ) ).
% add_strict_mono
thf(fact_1491_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_1492_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_1493_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_1494_eq__iff__diff__eq__0,axiom,
! [A: $tType] :
( ( group_add @ A )
=> ( ( ^ [Y6: A,Z3: A] : Y6 = Z3 )
= ( ^ [A6: A,B7: A] :
( ( minus_minus @ A @ A6 @ B7 )
= ( zero_zero @ A ) ) ) ) ) ).
% eq_iff_diff_eq_0
thf(fact_1495_diff__mono,axiom,
! [A: $tType] :
( ( ordered_ab_group_add @ A )
=> ! [A3: A,B3: A,D2: A,C3: A] :
( ( ord_less_eq @ A @ A3 @ B3 )
=> ( ( ord_less_eq @ A @ D2 @ C3 )
=> ( ord_less_eq @ A @ ( minus_minus @ A @ A3 @ C3 ) @ ( minus_minus @ A @ B3 @ D2 ) ) ) ) ) ).
% diff_mono
thf(fact_1496_diff__left__mono,axiom,
! [A: $tType] :
( ( ordered_ab_group_add @ A )
=> ! [B3: A,A3: A,C3: A] :
( ( ord_less_eq @ A @ B3 @ A3 )
=> ( ord_less_eq @ A @ ( minus_minus @ A @ C3 @ A3 ) @ ( minus_minus @ A @ C3 @ B3 ) ) ) ) ).
% diff_left_mono
thf(fact_1497_diff__right__mono,axiom,
! [A: $tType] :
( ( ordered_ab_group_add @ A )
=> ! [A3: A,B3: A,C3: A] :
( ( ord_less_eq @ A @ A3 @ B3 )
=> ( ord_less_eq @ A @ ( minus_minus @ A @ A3 @ C3 ) @ ( minus_minus @ A @ B3 @ C3 ) ) ) ) ).
% diff_right_mono
thf(fact_1498_diff__eq__diff__less__eq,axiom,
! [A: $tType] :
( ( ordered_ab_group_add @ A )
=> ! [A3: A,B3: A,C3: A,D2: A] :
( ( ( minus_minus @ A @ A3 @ B3 )
= ( minus_minus @ A @ C3 @ D2 ) )
=> ( ( ord_less_eq @ A @ A3 @ B3 )
= ( ord_less_eq @ A @ C3 @ D2 ) ) ) ) ).
% diff_eq_diff_less_eq
thf(fact_1499_diff__strict__right__mono,axiom,
! [A: $tType] :
( ( ordered_ab_group_add @ A )
=> ! [A3: A,B3: A,C3: A] :
( ( ord_less @ A @ A3 @ B3 )
=> ( ord_less @ A @ ( minus_minus @ A @ A3 @ C3 ) @ ( minus_minus @ A @ B3 @ C3 ) ) ) ) ).
% diff_strict_right_mono
thf(fact_1500_diff__strict__left__mono,axiom,
! [A: $tType] :
( ( ordered_ab_group_add @ A )
=> ! [B3: A,A3: A,C3: A] :
( ( ord_less @ A @ B3 @ A3 )
=> ( ord_less @ A @ ( minus_minus @ A @ C3 @ A3 ) @ ( minus_minus @ A @ C3 @ B3 ) ) ) ) ).
% diff_strict_left_mono
thf(fact_1501_diff__eq__diff__less,axiom,
! [A: $tType] :
( ( ordered_ab_group_add @ A )
=> ! [A3: A,B3: A,C3: A,D2: A] :
( ( ( minus_minus @ A @ A3 @ B3 )
= ( minus_minus @ A @ C3 @ D2 ) )
=> ( ( ord_less @ A @ A3 @ B3 )
= ( ord_less @ A @ C3 @ D2 ) ) ) ) ).
% diff_eq_diff_less
thf(fact_1502_diff__strict__mono,axiom,
! [A: $tType] :
( ( ordered_ab_group_add @ A )
=> ! [A3: A,B3: A,D2: A,C3: A] :
( ( ord_less @ A @ A3 @ B3 )
=> ( ( ord_less @ A @ D2 @ C3 )
=> ( ord_less @ A @ ( minus_minus @ A @ A3 @ C3 ) @ ( minus_minus @ A @ B3 @ D2 ) ) ) ) ) ).
% diff_strict_mono
thf(fact_1503_mult_Ocomm__neutral,axiom,
! [A: $tType] :
( ( comm_monoid_mult @ A )
=> ! [A3: A] :
( ( times_times @ A @ A3 @ ( one_one @ A ) )
= A3 ) ) ).
% mult.comm_neutral
thf(fact_1504_comm__monoid__mult__class_Omult__1,axiom,
! [A: $tType] :
( ( comm_monoid_mult @ A )
=> ! [A3: A] :
( ( times_times @ A @ ( one_one @ A ) @ A3 )
= A3 ) ) ).
% comm_monoid_mult_class.mult_1
thf(fact_1505_group__cancel_Osub1,axiom,
! [A: $tType] :
( ( ab_group_add @ A )
=> ! [A2: A,K: A,A3: A,B3: A] :
( ( A2
= ( plus_plus @ A @ K @ A3 ) )
=> ( ( minus_minus @ A @ A2 @ B3 )
= ( plus_plus @ A @ K @ ( minus_minus @ A @ A3 @ B3 ) ) ) ) ) ).
% group_cancel.sub1
thf(fact_1506_diff__eq__eq,axiom,
! [A: $tType] :
( ( group_add @ A )
=> ! [A3: A,B3: A,C3: A] :
( ( ( minus_minus @ A @ A3 @ B3 )
= C3 )
= ( A3
= ( plus_plus @ A @ C3 @ B3 ) ) ) ) ).
% diff_eq_eq
thf(fact_1507_eq__diff__eq,axiom,
! [A: $tType] :
( ( group_add @ A )
=> ! [A3: A,C3: A,B3: A] :
( ( A3
= ( minus_minus @ A @ C3 @ B3 ) )
= ( ( plus_plus @ A @ A3 @ B3 )
= C3 ) ) ) ).
% eq_diff_eq
thf(fact_1508_add__diff__eq,axiom,
! [A: $tType] :
( ( group_add @ A )
=> ! [A3: A,B3: A,C3: A] :
( ( plus_plus @ A @ A3 @ ( minus_minus @ A @ B3 @ C3 ) )
= ( minus_minus @ A @ ( plus_plus @ A @ A3 @ B3 ) @ C3 ) ) ) ).
% add_diff_eq
thf(fact_1509_diff__diff__eq2,axiom,
! [A: $tType] :
( ( group_add @ A )
=> ! [A3: A,B3: A,C3: A] :
( ( minus_minus @ A @ A3 @ ( minus_minus @ A @ B3 @ C3 ) )
= ( minus_minus @ A @ ( plus_plus @ A @ A3 @ C3 ) @ B3 ) ) ) ).
% diff_diff_eq2
thf(fact_1510_diff__add__eq,axiom,
! [A: $tType] :
( ( ab_group_add @ A )
=> ! [A3: A,B3: A,C3: A] :
( ( plus_plus @ A @ ( minus_minus @ A @ A3 @ B3 ) @ C3 )
= ( minus_minus @ A @ ( plus_plus @ A @ A3 @ C3 ) @ B3 ) ) ) ).
% diff_add_eq
thf(fact_1511_diff__add__eq__diff__diff__swap,axiom,
! [A: $tType] :
( ( group_add @ A )
=> ! [A3: A,B3: A,C3: A] :
( ( minus_minus @ A @ A3 @ ( plus_plus @ A @ B3 @ C3 ) )
= ( minus_minus @ A @ ( minus_minus @ A @ A3 @ C3 ) @ B3 ) ) ) ).
% diff_add_eq_diff_diff_swap
thf(fact_1512_add__implies__diff,axiom,
! [A: $tType] :
( ( cancel1802427076303600483id_add @ A )
=> ! [C3: A,B3: A,A3: A] :
( ( ( plus_plus @ A @ C3 @ B3 )
= A3 )
=> ( C3
= ( minus_minus @ A @ A3 @ B3 ) ) ) ) ).
% add_implies_diff
thf(fact_1513_diff__diff__eq,axiom,
! [A: $tType] :
( ( cancel2418104881723323429up_add @ A )
=> ! [A3: A,B3: A,C3: A] :
( ( minus_minus @ A @ ( minus_minus @ A @ A3 @ B3 ) @ C3 )
= ( minus_minus @ A @ A3 @ ( plus_plus @ A @ B3 @ C3 ) ) ) ) ).
% diff_diff_eq
thf(fact_1514_max__add__distrib__left,axiom,
! [A: $tType] :
( ( ordere2412721322843649153imp_le @ A )
=> ! [X2: A,Y3: A,Z: A] :
( ( plus_plus @ A @ ( ord_max @ A @ X2 @ Y3 ) @ Z )
= ( ord_max @ A @ ( plus_plus @ A @ X2 @ Z ) @ ( plus_plus @ A @ Y3 @ Z ) ) ) ) ).
% max_add_distrib_left
thf(fact_1515_max__add__distrib__right,axiom,
! [A: $tType] :
( ( ordere2412721322843649153imp_le @ A )
=> ! [X2: A,Y3: A,Z: A] :
( ( plus_plus @ A @ X2 @ ( ord_max @ A @ Y3 @ Z ) )
= ( ord_max @ A @ ( plus_plus @ A @ X2 @ Y3 ) @ ( plus_plus @ A @ X2 @ Z ) ) ) ) ).
% max_add_distrib_right
thf(fact_1516_max__diff__distrib__left,axiom,
! [A: $tType] :
( ( ordered_ab_group_add @ A )
=> ! [X2: A,Y3: A,Z: A] :
( ( minus_minus @ A @ ( ord_max @ A @ X2 @ Y3 ) @ Z )
= ( ord_max @ A @ ( minus_minus @ A @ X2 @ Z ) @ ( minus_minus @ A @ Y3 @ Z ) ) ) ) ).
% max_diff_distrib_left
thf(fact_1517_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,
! [A3: $o,B3: $o,X2: nat] :
( ( vEBT_T_m_e_m_b_e_r @ ( vEBT_Leaf @ A3 @ B3 ) @ X2 )
= ( plus_plus @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) )
@ ( if @ nat
@ ( X2
= ( 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_1518_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,
! [V: product_prod @ nat @ nat,Uy: list @ vEBT_VEBT,Uz: vEBT_VEBT,X2: nat] :
( ( vEBT_T_m_e_m_b_e_r2 @ ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ V ) @ ( zero_zero @ nat ) @ Uy @ Uz ) @ X2 )
= ( one_one @ nat ) ) ).
% T\<^sub>m\<^sub>e\<^sub>m\<^sub>b\<^sub>e\<^sub>r'.simps(3)
thf(fact_1519_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,
! [V: product_prod @ nat @ nat,Vb: list @ vEBT_VEBT,Vc: vEBT_VEBT,X2: nat] :
( ( vEBT_T_m_e_m_b_e_r2 @ ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ V ) @ ( suc @ ( zero_zero @ nat ) ) @ Vb @ Vc ) @ X2 )
= ( one_one @ nat ) ) ).
% T\<^sub>m\<^sub>e\<^sub>m\<^sub>b\<^sub>e\<^sub>r'.simps(4)
thf(fact_1520_member__bound__height_H,axiom,
! [T2: vEBT_VEBT,N: nat,X2: nat] :
( ( vEBT_invar_vebt @ T2 @ N )
=> ( ord_less_eq @ nat @ ( vEBT_T_m_e_m_b_e_r2 @ T2 @ X2 ) @ ( plus_plus @ nat @ ( one_one @ nat ) @ ( vEBT_VEBT_height @ T2 ) ) ) ) ).
% member_bound_height'
thf(fact_1521_add__nonpos__eq__0__iff,axiom,
! [A: $tType] :
( ( ordere6911136660526730532id_add @ A )
=> ! [X2: A,Y3: A] :
( ( ord_less_eq @ A @ X2 @ ( zero_zero @ A ) )
=> ( ( ord_less_eq @ A @ Y3 @ ( zero_zero @ A ) )
=> ( ( ( plus_plus @ A @ X2 @ Y3 )
= ( zero_zero @ A ) )
= ( ( X2
= ( zero_zero @ A ) )
& ( Y3
= ( zero_zero @ A ) ) ) ) ) ) ) ).
% add_nonpos_eq_0_iff
thf(fact_1522_add__nonneg__eq__0__iff,axiom,
! [A: $tType] :
( ( ordere6911136660526730532id_add @ A )
=> ! [X2: A,Y3: A] :
( ( ord_less_eq @ A @ ( zero_zero @ A ) @ X2 )
=> ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ Y3 )
=> ( ( ( plus_plus @ A @ X2 @ Y3 )
= ( zero_zero @ A ) )
= ( ( X2
= ( zero_zero @ A ) )
& ( Y3
= ( zero_zero @ A ) ) ) ) ) ) ) ).
% add_nonneg_eq_0_iff
thf(fact_1523_add__nonpos__nonpos,axiom,
! [A: $tType] :
( ( ordere6911136660526730532id_add @ A )
=> ! [A3: A,B3: A] :
( ( ord_less_eq @ A @ A3 @ ( zero_zero @ A ) )
=> ( ( ord_less_eq @ A @ B3 @ ( zero_zero @ A ) )
=> ( ord_less_eq @ A @ ( plus_plus @ A @ A3 @ B3 ) @ ( zero_zero @ A ) ) ) ) ) ).
% add_nonpos_nonpos
thf(fact_1524_add__nonneg__nonneg,axiom,
! [A: $tType] :
( ( ordere6911136660526730532id_add @ A )
=> ! [A3: A,B3: A] :
( ( ord_less_eq @ A @ ( zero_zero @ A ) @ A3 )
=> ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ B3 )
=> ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( plus_plus @ A @ A3 @ B3 ) ) ) ) ) ).
% add_nonneg_nonneg
thf(fact_1525_add__increasing2,axiom,
! [A: $tType] :
( ( ordere6911136660526730532id_add @ A )
=> ! [C3: A,B3: A,A3: A] :
( ( ord_less_eq @ A @ ( zero_zero @ A ) @ C3 )
=> ( ( ord_less_eq @ A @ B3 @ A3 )
=> ( ord_less_eq @ A @ B3 @ ( plus_plus @ A @ A3 @ C3 ) ) ) ) ) ).
% add_increasing2
thf(fact_1526_add__decreasing2,axiom,
! [A: $tType] :
( ( ordere6911136660526730532id_add @ A )
=> ! [C3: A,A3: A,B3: A] :
( ( ord_less_eq @ A @ C3 @ ( zero_zero @ A ) )
=> ( ( ord_less_eq @ A @ A3 @ B3 )
=> ( ord_less_eq @ A @ ( plus_plus @ A @ A3 @ C3 ) @ B3 ) ) ) ) ).
% add_decreasing2
thf(fact_1527_add__increasing,axiom,
! [A: $tType] :
( ( ordere6911136660526730532id_add @ A )
=> ! [A3: A,B3: A,C3: A] :
( ( ord_less_eq @ A @ ( zero_zero @ A ) @ A3 )
=> ( ( ord_less_eq @ A @ B3 @ C3 )
=> ( ord_less_eq @ A @ B3 @ ( plus_plus @ A @ A3 @ C3 ) ) ) ) ) ).
% add_increasing
thf(fact_1528_add__decreasing,axiom,
! [A: $tType] :
( ( ordere6911136660526730532id_add @ A )
=> ! [A3: A,C3: A,B3: A] :
( ( ord_less_eq @ A @ A3 @ ( zero_zero @ A ) )
=> ( ( ord_less_eq @ A @ C3 @ B3 )
=> ( ord_less_eq @ A @ ( plus_plus @ A @ A3 @ C3 ) @ B3 ) ) ) ) ).
% add_decreasing
thf(fact_1529_add__neg__neg,axiom,
! [A: $tType] :
( ( ordere6911136660526730532id_add @ A )
=> ! [A3: A,B3: A] :
( ( ord_less @ A @ A3 @ ( zero_zero @ A ) )
=> ( ( ord_less @ A @ B3 @ ( zero_zero @ A ) )
=> ( ord_less @ A @ ( plus_plus @ A @ A3 @ B3 ) @ ( zero_zero @ A ) ) ) ) ) ).
% add_neg_neg
thf(fact_1530_add__pos__pos,axiom,
! [A: $tType] :
( ( ordere6911136660526730532id_add @ A )
=> ! [A3: A,B3: A] :
( ( ord_less @ A @ ( zero_zero @ A ) @ A3 )
=> ( ( ord_less @ A @ ( zero_zero @ A ) @ B3 )
=> ( ord_less @ A @ ( zero_zero @ A ) @ ( plus_plus @ A @ A3 @ B3 ) ) ) ) ) ).
% add_pos_pos
thf(fact_1531_canonically__ordered__monoid__add__class_OlessE,axiom,
! [A: $tType] :
( ( canoni5634975068530333245id_add @ A )
=> ! [A3: A,B3: A] :
( ( ord_less @ A @ A3 @ B3 )
=> ~ ! [C4: A] :
( ( B3
= ( plus_plus @ A @ A3 @ C4 ) )
=> ( C4
= ( zero_zero @ A ) ) ) ) ) ).
% canonically_ordered_monoid_add_class.lessE
thf(fact_1532_pos__add__strict,axiom,
! [A: $tType] :
( ( strict7427464778891057005id_add @ A )
=> ! [A3: A,B3: A,C3: A] :
( ( ord_less @ A @ ( zero_zero @ A ) @ A3 )
=> ( ( ord_less @ A @ B3 @ C3 )
=> ( ord_less @ A @ B3 @ ( plus_plus @ A @ A3 @ C3 ) ) ) ) ) ).
% pos_add_strict
thf(fact_1533_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_1534_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_1535_add__le__less__mono,axiom,
! [A: $tType] :
( ( ordere580206878836729694up_add @ A )
=> ! [A3: A,B3: A,C3: A,D2: A] :
( ( ord_less_eq @ A @ A3 @ B3 )
=> ( ( ord_less @ A @ C3 @ D2 )
=> ( ord_less @ A @ ( plus_plus @ A @ A3 @ C3 ) @ ( plus_plus @ A @ B3 @ D2 ) ) ) ) ) ).
% add_le_less_mono
thf(fact_1536_add__less__le__mono,axiom,
! [A: $tType] :
( ( ordere580206878836729694up_add @ A )
=> ! [A3: A,B3: A,C3: A,D2: A] :
( ( ord_less @ A @ A3 @ B3 )
=> ( ( ord_less_eq @ A @ C3 @ D2 )
=> ( ord_less @ A @ ( plus_plus @ A @ A3 @ C3 ) @ ( plus_plus @ A @ B3 @ D2 ) ) ) ) ) ).
% add_less_le_mono
thf(fact_1537_le__iff__diff__le__0,axiom,
! [A: $tType] :
( ( ordered_ab_group_add @ A )
=> ( ( ord_less_eq @ A )
= ( ^ [A6: A,B7: A] : ( ord_less_eq @ A @ ( minus_minus @ A @ A6 @ B7 ) @ ( zero_zero @ A ) ) ) ) ) ).
% le_iff_diff_le_0
thf(fact_1538_less__iff__diff__less__0,axiom,
! [A: $tType] :
( ( ordered_ab_group_add @ A )
=> ( ( ord_less @ A )
= ( ^ [A6: A,B7: A] : ( ord_less @ A @ ( minus_minus @ A @ A6 @ B7 ) @ ( zero_zero @ A ) ) ) ) ) ).
% less_iff_diff_less_0
thf(fact_1539_ordered__cancel__comm__monoid__diff__class_Ole__imp__diff__is__add,axiom,
! [A: $tType] :
( ( ordere1170586879665033532d_diff @ A )
=> ! [A3: A,B3: A,C3: A] :
( ( ord_less_eq @ A @ A3 @ B3 )
=> ( ( ord_less_eq @ A @ A3 @ B3 )
=> ( ( ( minus_minus @ A @ B3 @ A3 )
= C3 )
= ( B3
= ( plus_plus @ A @ C3 @ A3 ) ) ) ) ) ) ).
% ordered_cancel_comm_monoid_diff_class.le_imp_diff_is_add
thf(fact_1540_ordered__cancel__comm__monoid__diff__class_Oadd__diff__inverse,axiom,
! [A: $tType] :
( ( ordere1170586879665033532d_diff @ A )
=> ! [A3: A,B3: A] :
( ( ord_less_eq @ A @ A3 @ B3 )
=> ( ( plus_plus @ A @ A3 @ ( minus_minus @ A @ B3 @ A3 ) )
= B3 ) ) ) ).
% ordered_cancel_comm_monoid_diff_class.add_diff_inverse
thf(fact_1541_ordered__cancel__comm__monoid__diff__class_Odiff__diff__right,axiom,
! [A: $tType] :
( ( ordere1170586879665033532d_diff @ A )
=> ! [A3: A,B3: A,C3: A] :
( ( ord_less_eq @ A @ A3 @ B3 )
=> ( ( minus_minus @ A @ C3 @ ( minus_minus @ A @ B3 @ A3 ) )
= ( minus_minus @ A @ ( plus_plus @ A @ C3 @ A3 ) @ B3 ) ) ) ) ).
% ordered_cancel_comm_monoid_diff_class.diff_diff_right
thf(fact_1542_ordered__cancel__comm__monoid__diff__class_Odiff__add__assoc2,axiom,
! [A: $tType] :
( ( ordere1170586879665033532d_diff @ A )
=> ! [A3: A,B3: A,C3: A] :
( ( ord_less_eq @ A @ A3 @ B3 )
=> ( ( minus_minus @ A @ ( plus_plus @ A @ B3 @ C3 ) @ A3 )
= ( plus_plus @ A @ ( minus_minus @ A @ B3 @ A3 ) @ C3 ) ) ) ) ).
% ordered_cancel_comm_monoid_diff_class.diff_add_assoc2
thf(fact_1543_ordered__cancel__comm__monoid__diff__class_Oadd__diff__assoc2,axiom,
! [A: $tType] :
( ( ordere1170586879665033532d_diff @ A )
=> ! [A3: A,B3: A,C3: A] :
( ( ord_less_eq @ A @ A3 @ B3 )
=> ( ( plus_plus @ A @ ( minus_minus @ A @ B3 @ A3 ) @ C3 )
= ( minus_minus @ A @ ( plus_plus @ A @ B3 @ C3 ) @ A3 ) ) ) ) ).
% ordered_cancel_comm_monoid_diff_class.add_diff_assoc2
thf(fact_1544_ordered__cancel__comm__monoid__diff__class_Odiff__add__assoc,axiom,
! [A: $tType] :
( ( ordere1170586879665033532d_diff @ A )
=> ! [A3: A,B3: A,C3: A] :
( ( ord_less_eq @ A @ A3 @ B3 )
=> ( ( minus_minus @ A @ ( plus_plus @ A @ C3 @ B3 ) @ A3 )
= ( plus_plus @ A @ C3 @ ( minus_minus @ A @ B3 @ A3 ) ) ) ) ) ).
% ordered_cancel_comm_monoid_diff_class.diff_add_assoc
thf(fact_1545_ordered__cancel__comm__monoid__diff__class_Oadd__diff__assoc,axiom,
! [A: $tType] :
( ( ordere1170586879665033532d_diff @ A )
=> ! [A3: A,B3: A,C3: A] :
( ( ord_less_eq @ A @ A3 @ B3 )
=> ( ( plus_plus @ A @ C3 @ ( minus_minus @ A @ B3 @ A3 ) )
= ( minus_minus @ A @ ( plus_plus @ A @ C3 @ B3 ) @ A3 ) ) ) ) ).
% ordered_cancel_comm_monoid_diff_class.add_diff_assoc
thf(fact_1546_ordered__cancel__comm__monoid__diff__class_Ole__diff__conv2,axiom,
! [A: $tType] :
( ( ordere1170586879665033532d_diff @ A )
=> ! [A3: A,B3: A,C3: A] :
( ( ord_less_eq @ A @ A3 @ B3 )
=> ( ( ord_less_eq @ A @ C3 @ ( minus_minus @ A @ B3 @ A3 ) )
= ( ord_less_eq @ A @ ( plus_plus @ A @ C3 @ A3 ) @ B3 ) ) ) ) ).
% ordered_cancel_comm_monoid_diff_class.le_diff_conv2
thf(fact_1547_le__add__diff,axiom,
! [A: $tType] :
( ( ordere1170586879665033532d_diff @ A )
=> ! [A3: A,B3: A,C3: A] :
( ( ord_less_eq @ A @ A3 @ B3 )
=> ( ord_less_eq @ A @ C3 @ ( minus_minus @ A @ ( plus_plus @ A @ B3 @ C3 ) @ A3 ) ) ) ) ).
% le_add_diff
thf(fact_1548_ordered__cancel__comm__monoid__diff__class_Odiff__add,axiom,
! [A: $tType] :
( ( ordere1170586879665033532d_diff @ A )
=> ! [A3: A,B3: A] :
( ( ord_less_eq @ A @ A3 @ B3 )
=> ( ( plus_plus @ A @ ( minus_minus @ A @ B3 @ A3 ) @ A3 )
= B3 ) ) ) ).
% ordered_cancel_comm_monoid_diff_class.diff_add
thf(fact_1549_le__diff__eq,axiom,
! [A: $tType] :
( ( ordered_ab_group_add @ A )
=> ! [A3: A,C3: A,B3: A] :
( ( ord_less_eq @ A @ A3 @ ( minus_minus @ A @ C3 @ B3 ) )
= ( ord_less_eq @ A @ ( plus_plus @ A @ A3 @ B3 ) @ C3 ) ) ) ).
% le_diff_eq
thf(fact_1550_diff__le__eq,axiom,
! [A: $tType] :
( ( ordered_ab_group_add @ A )
=> ! [A3: A,B3: A,C3: A] :
( ( ord_less_eq @ A @ ( minus_minus @ A @ A3 @ B3 ) @ C3 )
= ( ord_less_eq @ A @ A3 @ ( plus_plus @ A @ C3 @ B3 ) ) ) ) ).
% diff_le_eq
thf(fact_1551_diff__less__eq,axiom,
! [A: $tType] :
( ( ordered_ab_group_add @ A )
=> ! [A3: A,B3: A,C3: A] :
( ( ord_less @ A @ ( minus_minus @ A @ A3 @ B3 ) @ C3 )
= ( ord_less @ A @ A3 @ ( plus_plus @ A @ C3 @ B3 ) ) ) ) ).
% diff_less_eq
thf(fact_1552_less__diff__eq,axiom,
! [A: $tType] :
( ( ordered_ab_group_add @ A )
=> ! [A3: A,C3: A,B3: A] :
( ( ord_less @ A @ A3 @ ( minus_minus @ A @ C3 @ B3 ) )
= ( ord_less @ A @ ( plus_plus @ A @ A3 @ B3 ) @ C3 ) ) ) ).
% less_diff_eq
thf(fact_1553_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,
! [X2: vEBT_VEBT,Xa: nat,Y3: nat] :
( ( ( vEBT_T_m_e_m_b_e_r2 @ X2 @ Xa )
= Y3 )
=> ( ( ? [A4: $o,B4: $o] :
( X2
= ( vEBT_Leaf @ A4 @ B4 ) )
=> ( Y3
!= ( one_one @ nat ) ) )
=> ( ( ? [Uu2: nat,Uv2: list @ vEBT_VEBT,Uw2: vEBT_VEBT] :
( X2
= ( vEBT_Node @ ( none @ ( product_prod @ nat @ nat ) ) @ Uu2 @ Uv2 @ Uw2 ) )
=> ( Y3
!= ( one_one @ nat ) ) )
=> ( ( ? [V3: product_prod @ nat @ nat,Uy2: list @ vEBT_VEBT,Uz2: vEBT_VEBT] :
( X2
= ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ V3 ) @ ( zero_zero @ nat ) @ Uy2 @ Uz2 ) )
=> ( Y3
!= ( one_one @ nat ) ) )
=> ( ( ? [V3: product_prod @ nat @ nat,Vb2: list @ vEBT_VEBT,Vc2: vEBT_VEBT] :
( X2
= ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ V3 ) @ ( suc @ ( zero_zero @ nat ) ) @ Vb2 @ Vc2 ) )
=> ( Y3
!= ( one_one @ nat ) ) )
=> ~ ! [Mi2: nat,Ma2: nat,Va3: nat,TreeList2: list @ vEBT_VEBT] :
( ? [Summary2: vEBT_VEBT] :
( X2
= ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( suc @ Va3 ) ) @ TreeList2 @ Summary2 ) )
=> ( Y3
!= ( plus_plus @ nat @ ( one_one @ nat )
@ ( if @ nat @ ( Xa = Mi2 ) @ ( zero_zero @ nat )
@ ( if @ nat @ ( Xa = Ma2 ) @ ( zero_zero @ nat )
@ ( if @ nat @ ( ord_less @ nat @ Xa @ Mi2 ) @ ( zero_zero @ nat )
@ ( if @ nat @ ( ord_less @ nat @ Ma2 @ Xa ) @ ( zero_zero @ nat )
@ ( if @ nat
@ ( ( ord_less @ nat @ Mi2 @ Xa )
& ( ord_less @ nat @ Xa @ Ma2 ) )
@ ( if @ nat @ ( ord_less @ nat @ ( vEBT_VEBT_high @ Xa @ ( divide_divide @ nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( size_size @ ( list @ vEBT_VEBT ) @ TreeList2 ) ) @ ( vEBT_T_m_e_m_b_e_r2 @ ( nth @ vEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ Xa @ ( divide_divide @ nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( vEBT_VEBT_low @ Xa @ ( divide_divide @ nat @ ( suc @ ( suc @ Va3 ) ) @ ( 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_1554_add__neg__nonpos,axiom,
! [A: $tType] :
( ( ordere6911136660526730532id_add @ A )
=> ! [A3: A,B3: A] :
( ( ord_less @ A @ A3 @ ( zero_zero @ A ) )
=> ( ( ord_less_eq @ A @ B3 @ ( zero_zero @ A ) )
=> ( ord_less @ A @ ( plus_plus @ A @ A3 @ B3 ) @ ( zero_zero @ A ) ) ) ) ) ).
% add_neg_nonpos
thf(fact_1555_add__nonneg__pos,axiom,
! [A: $tType] :
( ( ordere6911136660526730532id_add @ A )
=> ! [A3: A,B3: A] :
( ( ord_less_eq @ A @ ( zero_zero @ A ) @ A3 )
=> ( ( ord_less @ A @ ( zero_zero @ A ) @ B3 )
=> ( ord_less @ A @ ( zero_zero @ A ) @ ( plus_plus @ A @ A3 @ B3 ) ) ) ) ) ).
% add_nonneg_pos
thf(fact_1556_add__nonpos__neg,axiom,
! [A: $tType] :
( ( ordere6911136660526730532id_add @ A )
=> ! [A3: A,B3: A] :
( ( ord_less_eq @ A @ A3 @ ( zero_zero @ A ) )
=> ( ( ord_less @ A @ B3 @ ( zero_zero @ A ) )
=> ( ord_less @ A @ ( plus_plus @ A @ A3 @ B3 ) @ ( zero_zero @ A ) ) ) ) ) ).
% add_nonpos_neg
thf(fact_1557_add__pos__nonneg,axiom,
! [A: $tType] :
( ( ordere6911136660526730532id_add @ A )
=> ! [A3: A,B3: A] :
( ( ord_less @ A @ ( zero_zero @ A ) @ A3 )
=> ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ B3 )
=> ( ord_less @ A @ ( zero_zero @ A ) @ ( plus_plus @ A @ A3 @ B3 ) ) ) ) ) ).
% add_pos_nonneg
thf(fact_1558_add__strict__increasing,axiom,
! [A: $tType] :
( ( ordere8940638589300402666id_add @ A )
=> ! [A3: A,B3: A,C3: A] :
( ( ord_less @ A @ ( zero_zero @ A ) @ A3 )
=> ( ( ord_less_eq @ A @ B3 @ C3 )
=> ( ord_less @ A @ B3 @ ( plus_plus @ A @ A3 @ C3 ) ) ) ) ) ).
% add_strict_increasing
thf(fact_1559_add__strict__increasing2,axiom,
! [A: $tType] :
( ( ordere8940638589300402666id_add @ A )
=> ! [A3: A,B3: A,C3: A] :
( ( ord_less_eq @ A @ ( zero_zero @ A ) @ A3 )
=> ( ( ord_less @ A @ B3 @ C3 )
=> ( ord_less @ A @ B3 @ ( plus_plus @ A @ A3 @ C3 ) ) ) ) ) ).
% add_strict_increasing2
thf(fact_1560_vebt__member_Oelims_I2_J,axiom,
! [X2: vEBT_VEBT,Xa: nat] :
( ( vEBT_vebt_member @ X2 @ Xa )
=> ( ! [A4: $o,B4: $o] :
( ( X2
= ( vEBT_Leaf @ A4 @ B4 ) )
=> ~ ( ( ( Xa
= ( zero_zero @ nat ) )
=> A4 )
& ( ( Xa
!= ( zero_zero @ nat ) )
=> ( ( ( Xa
= ( one_one @ nat ) )
=> B4 )
& ( Xa
= ( one_one @ nat ) ) ) ) ) )
=> ~ ! [Mi2: nat,Ma2: nat,Va3: nat,TreeList2: list @ vEBT_VEBT] :
( ? [Summary2: vEBT_VEBT] :
( X2
= ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( suc @ Va3 ) ) @ TreeList2 @ Summary2 ) )
=> ~ ( ( Xa != Mi2 )
=> ( ( Xa != Ma2 )
=> ( ~ ( ord_less @ nat @ Xa @ Mi2 )
& ( ~ ( ord_less @ nat @ Xa @ Mi2 )
=> ( ~ ( ord_less @ nat @ Ma2 @ Xa )
& ( ~ ( ord_less @ nat @ Ma2 @ Xa )
=> ( ( ( ord_less @ nat @ ( vEBT_VEBT_high @ Xa @ ( divide_divide @ nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( size_size @ ( list @ vEBT_VEBT ) @ TreeList2 ) )
=> ( vEBT_vebt_member @ ( nth @ vEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ Xa @ ( divide_divide @ nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( vEBT_VEBT_low @ Xa @ ( divide_divide @ nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) )
& ( ord_less @ nat @ ( vEBT_VEBT_high @ Xa @ ( divide_divide @ nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( size_size @ ( list @ vEBT_VEBT ) @ TreeList2 ) ) ) ) ) ) ) ) ) ) ) ) ).
% vebt_member.elims(2)
thf(fact_1561_vebt__member_Oelims_I1_J,axiom,
! [X2: vEBT_VEBT,Xa: nat,Y3: $o] :
( ( ( vEBT_vebt_member @ X2 @ Xa )
= Y3 )
=> ( ! [A4: $o,B4: $o] :
( ( X2
= ( vEBT_Leaf @ A4 @ B4 ) )
=> ( Y3
= ( ~ ( ( ( Xa
= ( zero_zero @ nat ) )
=> A4 )
& ( ( Xa
!= ( zero_zero @ nat ) )
=> ( ( ( Xa
= ( one_one @ nat ) )
=> B4 )
& ( Xa
= ( one_one @ nat ) ) ) ) ) ) ) )
=> ( ( ? [Uu2: nat,Uv2: list @ vEBT_VEBT,Uw2: vEBT_VEBT] :
( X2
= ( vEBT_Node @ ( none @ ( product_prod @ nat @ nat ) ) @ Uu2 @ Uv2 @ Uw2 ) )
=> Y3 )
=> ( ( ? [V3: product_prod @ nat @ nat,Uy2: list @ vEBT_VEBT,Uz2: vEBT_VEBT] :
( X2
= ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ V3 ) @ ( zero_zero @ nat ) @ Uy2 @ Uz2 ) )
=> Y3 )
=> ( ( ? [V3: product_prod @ nat @ nat,Vb2: list @ vEBT_VEBT,Vc2: vEBT_VEBT] :
( X2
= ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ V3 ) @ ( suc @ ( zero_zero @ nat ) ) @ Vb2 @ Vc2 ) )
=> Y3 )
=> ~ ! [Mi2: nat,Ma2: nat,Va3: nat,TreeList2: list @ vEBT_VEBT] :
( ? [Summary2: vEBT_VEBT] :
( X2
= ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( suc @ Va3 ) ) @ TreeList2 @ Summary2 ) )
=> ( Y3
= ( ~ ( ( Xa != Mi2 )
=> ( ( Xa != Ma2 )
=> ( ~ ( ord_less @ nat @ Xa @ Mi2 )
& ( ~ ( ord_less @ nat @ Xa @ Mi2 )
=> ( ~ ( ord_less @ nat @ Ma2 @ Xa )
& ( ~ ( ord_less @ nat @ Ma2 @ Xa )
=> ( ( ( ord_less @ nat @ ( vEBT_VEBT_high @ Xa @ ( divide_divide @ nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( size_size @ ( list @ vEBT_VEBT ) @ TreeList2 ) )
=> ( vEBT_vebt_member @ ( nth @ vEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ Xa @ ( divide_divide @ nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( vEBT_VEBT_low @ Xa @ ( divide_divide @ nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) )
& ( ord_less @ nat @ ( vEBT_VEBT_high @ Xa @ ( divide_divide @ nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( size_size @ ( list @ vEBT_VEBT ) @ TreeList2 ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).
% vebt_member.elims(1)
thf(fact_1562_vebt__member_Oelims_I3_J,axiom,
! [X2: vEBT_VEBT,Xa: nat] :
( ~ ( vEBT_vebt_member @ X2 @ Xa )
=> ( ! [A4: $o,B4: $o] :
( ( X2
= ( vEBT_Leaf @ A4 @ B4 ) )
=> ( ( ( Xa
= ( zero_zero @ nat ) )
=> A4 )
& ( ( Xa
!= ( zero_zero @ nat ) )
=> ( ( ( Xa
= ( one_one @ nat ) )
=> B4 )
& ( Xa
= ( one_one @ nat ) ) ) ) ) )
=> ( ! [Uu2: nat,Uv2: list @ vEBT_VEBT,Uw2: vEBT_VEBT] :
( X2
!= ( vEBT_Node @ ( none @ ( product_prod @ nat @ nat ) ) @ Uu2 @ Uv2 @ Uw2 ) )
=> ( ! [V3: product_prod @ nat @ nat,Uy2: list @ vEBT_VEBT,Uz2: vEBT_VEBT] :
( X2
!= ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ V3 ) @ ( zero_zero @ nat ) @ Uy2 @ Uz2 ) )
=> ( ! [V3: product_prod @ nat @ nat,Vb2: list @ vEBT_VEBT,Vc2: vEBT_VEBT] :
( X2
!= ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ V3 ) @ ( suc @ ( zero_zero @ nat ) ) @ Vb2 @ Vc2 ) )
=> ~ ! [Mi2: nat,Ma2: nat,Va3: nat,TreeList2: list @ vEBT_VEBT] :
( ? [Summary2: vEBT_VEBT] :
( X2
= ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( suc @ Va3 ) ) @ TreeList2 @ Summary2 ) )
=> ( ( Xa != Mi2 )
=> ( ( Xa != Ma2 )
=> ( ~ ( ord_less @ nat @ Xa @ Mi2 )
& ( ~ ( ord_less @ nat @ Xa @ Mi2 )
=> ( ~ ( ord_less @ nat @ Ma2 @ Xa )
& ( ~ ( ord_less @ nat @ Ma2 @ Xa )
=> ( ( ( ord_less @ nat @ ( vEBT_VEBT_high @ Xa @ ( divide_divide @ nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( size_size @ ( list @ vEBT_VEBT ) @ TreeList2 ) )
=> ( vEBT_vebt_member @ ( nth @ vEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ Xa @ ( divide_divide @ nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( vEBT_VEBT_low @ Xa @ ( divide_divide @ nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) )
& ( ord_less @ nat @ ( vEBT_VEBT_high @ Xa @ ( divide_divide @ nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( size_size @ ( list @ vEBT_VEBT ) @ TreeList2 ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).
% vebt_member.elims(3)
thf(fact_1563_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,
! [X2: vEBT_VEBT,Xa: nat,Y3: nat] :
( ( ( vEBT_T_m_e_m_b_e_r @ X2 @ Xa )
= Y3 )
=> ( ( ? [A4: $o,B4: $o] :
( X2
= ( vEBT_Leaf @ A4 @ B4 ) )
=> ( Y3
!= ( plus_plus @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) )
@ ( if @ nat
@ ( Xa
= ( zero_zero @ nat ) )
@ ( one_one @ nat )
@ ( plus_plus @ nat @ ( one_one @ nat ) @ ( one_one @ nat ) ) ) ) ) )
=> ( ( ? [Uu2: nat,Uv2: list @ vEBT_VEBT,Uw2: vEBT_VEBT] :
( X2
= ( vEBT_Node @ ( none @ ( product_prod @ nat @ nat ) ) @ Uu2 @ Uv2 @ Uw2 ) )
=> ( Y3
!= ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) )
=> ( ( ? [V3: product_prod @ nat @ nat,Uy2: list @ vEBT_VEBT,Uz2: vEBT_VEBT] :
( X2
= ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ V3 ) @ ( zero_zero @ nat ) @ Uy2 @ Uz2 ) )
=> ( Y3
!= ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) )
=> ( ( ? [V3: product_prod @ nat @ nat,Vb2: list @ vEBT_VEBT,Vc2: vEBT_VEBT] :
( X2
= ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ V3 ) @ ( suc @ ( zero_zero @ nat ) ) @ Vb2 @ Vc2 ) )
=> ( Y3
!= ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) )
=> ~ ! [Mi2: nat,Ma2: nat,Va3: nat,TreeList2: list @ vEBT_VEBT] :
( ? [Summary2: vEBT_VEBT] :
( X2
= ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( suc @ Va3 ) ) @ TreeList2 @ Summary2 ) )
=> ( Y3
!= ( plus_plus @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( if @ nat @ ( Xa = Mi2 ) @ ( one_one @ nat ) @ ( plus_plus @ nat @ ( one_one @ nat ) @ ( if @ nat @ ( Xa = Ma2 ) @ ( one_one @ nat ) @ ( plus_plus @ nat @ ( one_one @ nat ) @ ( if @ nat @ ( ord_less @ nat @ Xa @ Mi2 ) @ ( one_one @ nat ) @ ( plus_plus @ nat @ ( one_one @ nat ) @ ( if @ nat @ ( ord_less @ nat @ Ma2 @ Xa ) @ ( one_one @ nat ) @ ( plus_plus @ nat @ ( numeral_numeral @ nat @ ( bit1 @ ( bit0 @ ( bit0 @ one2 ) ) ) ) @ ( if @ nat @ ( ord_less @ nat @ ( vEBT_VEBT_high @ Xa @ ( divide_divide @ nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( size_size @ ( list @ vEBT_VEBT ) @ TreeList2 ) ) @ ( plus_plus @ nat @ ( one_one @ nat ) @ ( vEBT_T_m_e_m_b_e_r @ ( nth @ vEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ Xa @ ( divide_divide @ nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( vEBT_VEBT_low @ Xa @ ( divide_divide @ nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) @ ( one_one @ nat ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).
% T\<^sub>m\<^sub>e\<^sub>m\<^sub>b\<^sub>e\<^sub>r.elims
thf(fact_1564_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,
! [X2: vEBT_VEBT,Xa: nat,Y3: nat] :
( ( ( vEBT_T_i_n_s_e_r_t2 @ X2 @ Xa )
= Y3 )
=> ( ( ? [A4: $o,B4: $o] :
( X2
= ( vEBT_Leaf @ A4 @ B4 ) )
=> ( Y3
!= ( one_one @ nat ) ) )
=> ( ( ? [Info2: option @ ( product_prod @ nat @ nat ),Ts2: list @ vEBT_VEBT,S3: vEBT_VEBT] :
( X2
= ( vEBT_Node @ Info2 @ ( zero_zero @ nat ) @ Ts2 @ S3 ) )
=> ( Y3
!= ( one_one @ nat ) ) )
=> ( ( ? [Info2: option @ ( product_prod @ nat @ nat ),Ts2: list @ vEBT_VEBT,S3: vEBT_VEBT] :
( X2
= ( vEBT_Node @ Info2 @ ( suc @ ( zero_zero @ nat ) ) @ Ts2 @ S3 ) )
=> ( Y3
!= ( one_one @ nat ) ) )
=> ( ( ? [V3: nat,TreeList2: list @ vEBT_VEBT,Summary2: vEBT_VEBT] :
( X2
= ( vEBT_Node @ ( none @ ( product_prod @ nat @ nat ) ) @ ( suc @ ( suc @ V3 ) ) @ TreeList2 @ Summary2 ) )
=> ( Y3
!= ( one_one @ nat ) ) )
=> ~ ! [Mi2: nat,Ma2: nat,Va3: nat,TreeList2: list @ vEBT_VEBT,Summary2: vEBT_VEBT] :
( ( X2
= ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( suc @ Va3 ) ) @ TreeList2 @ Summary2 ) )
=> ( Y3
!= ( if @ nat
@ ( ( ord_less @ nat @ ( vEBT_VEBT_high @ ( if @ nat @ ( ord_less @ nat @ Xa @ Mi2 ) @ Mi2 @ Xa ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( size_size @ ( list @ vEBT_VEBT ) @ TreeList2 ) )
& ~ ( ( Xa = Mi2 )
| ( Xa = Ma2 ) ) )
@ ( plus_plus @ nat @ ( vEBT_T_i_n_s_e_r_t2 @ ( nth @ vEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ ( if @ nat @ ( ord_less @ nat @ Xa @ Mi2 ) @ Mi2 @ Xa ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( vEBT_VEBT_low @ ( if @ nat @ ( ord_less @ nat @ Xa @ Mi2 ) @ Mi2 @ Xa ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( if @ nat @ ( vEBT_VEBT_minNull @ ( nth @ vEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ ( if @ nat @ ( ord_less @ nat @ Xa @ Mi2 ) @ Mi2 @ Xa ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) @ ( vEBT_T_i_n_s_e_r_t2 @ Summary2 @ ( vEBT_VEBT_high @ ( if @ nat @ ( ord_less @ nat @ Xa @ Mi2 ) @ Mi2 @ Xa ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va3 ) ) @ ( 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_1565_invar__vebt_Osimps,axiom,
( vEBT_invar_vebt
= ( ^ [A1: vEBT_VEBT,A22: nat] :
( ( ? [A6: $o,B7: $o] :
( A1
= ( vEBT_Leaf @ A6 @ B7 ) )
& ( A22
= ( suc @ ( zero_zero @ nat ) ) ) )
| ? [TreeList3: list @ vEBT_VEBT,N4: nat,Summary3: vEBT_VEBT] :
( ( A1
= ( vEBT_Node @ ( none @ ( product_prod @ nat @ nat ) ) @ A22 @ TreeList3 @ Summary3 ) )
& ! [X3: vEBT_VEBT] :
( ( member @ vEBT_VEBT @ X3 @ ( set2 @ vEBT_VEBT @ TreeList3 ) )
=> ( vEBT_invar_vebt @ X3 @ N4 ) )
& ( vEBT_invar_vebt @ Summary3 @ N4 )
& ( ( size_size @ ( list @ vEBT_VEBT ) @ TreeList3 )
= ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N4 ) )
& ( A22
= ( plus_plus @ nat @ N4 @ N4 ) )
& ~ ? [X7: nat] : ( vEBT_V8194947554948674370ptions @ Summary3 @ X7 )
& ! [X3: vEBT_VEBT] :
( ( member @ vEBT_VEBT @ X3 @ ( set2 @ vEBT_VEBT @ TreeList3 ) )
=> ~ ? [X7: nat] : ( vEBT_V8194947554948674370ptions @ X3 @ X7 ) ) )
| ? [TreeList3: list @ vEBT_VEBT,N4: nat,Summary3: vEBT_VEBT] :
( ( A1
= ( vEBT_Node @ ( none @ ( product_prod @ nat @ nat ) ) @ A22 @ TreeList3 @ Summary3 ) )
& ! [X3: vEBT_VEBT] :
( ( member @ vEBT_VEBT @ X3 @ ( set2 @ vEBT_VEBT @ TreeList3 ) )
=> ( vEBT_invar_vebt @ X3 @ N4 ) )
& ( vEBT_invar_vebt @ Summary3 @ ( suc @ N4 ) )
& ( ( size_size @ ( list @ vEBT_VEBT ) @ TreeList3 )
= ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( suc @ N4 ) ) )
& ( A22
= ( plus_plus @ nat @ N4 @ ( suc @ N4 ) ) )
& ~ ? [X7: nat] : ( vEBT_V8194947554948674370ptions @ Summary3 @ X7 )
& ! [X3: vEBT_VEBT] :
( ( member @ vEBT_VEBT @ X3 @ ( set2 @ vEBT_VEBT @ TreeList3 ) )
=> ~ ? [X7: nat] : ( vEBT_V8194947554948674370ptions @ X3 @ X7 ) ) )
| ? [TreeList3: list @ vEBT_VEBT,N4: nat,Summary3: vEBT_VEBT,Mi3: nat,Ma3: nat] :
( ( A1
= ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi3 @ Ma3 ) ) @ A22 @ TreeList3 @ Summary3 ) )
& ! [X3: vEBT_VEBT] :
( ( member @ vEBT_VEBT @ X3 @ ( set2 @ vEBT_VEBT @ TreeList3 ) )
=> ( vEBT_invar_vebt @ X3 @ N4 ) )
& ( vEBT_invar_vebt @ Summary3 @ N4 )
& ( ( size_size @ ( list @ vEBT_VEBT ) @ TreeList3 )
= ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N4 ) )
& ( A22
= ( plus_plus @ nat @ N4 @ N4 ) )
& ! [I4: nat] :
( ( ord_less @ nat @ I4 @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N4 ) )
=> ( ( ? [X7: nat] : ( vEBT_V8194947554948674370ptions @ ( nth @ vEBT_VEBT @ TreeList3 @ I4 ) @ X7 ) )
= ( vEBT_V8194947554948674370ptions @ Summary3 @ I4 ) ) )
& ( ( Mi3 = Ma3 )
=> ! [X3: vEBT_VEBT] :
( ( member @ vEBT_VEBT @ X3 @ ( set2 @ vEBT_VEBT @ TreeList3 ) )
=> ~ ? [X7: nat] : ( vEBT_V8194947554948674370ptions @ X3 @ X7 ) ) )
& ( ord_less_eq @ nat @ Mi3 @ Ma3 )
& ( ord_less @ nat @ Ma3 @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ A22 ) )
& ( ( Mi3 != Ma3 )
=> ! [I4: nat] :
( ( ord_less @ nat @ I4 @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N4 ) )
=> ( ( ( ( vEBT_VEBT_high @ Ma3 @ N4 )
= I4 )
=> ( vEBT_V8194947554948674370ptions @ ( nth @ vEBT_VEBT @ TreeList3 @ I4 ) @ ( vEBT_VEBT_low @ Ma3 @ N4 ) ) )
& ! [X3: nat] :
( ( ( ( vEBT_VEBT_high @ X3 @ N4 )
= I4 )
& ( vEBT_V8194947554948674370ptions @ ( nth @ vEBT_VEBT @ TreeList3 @ I4 ) @ ( vEBT_VEBT_low @ X3 @ N4 ) ) )
=> ( ( ord_less @ nat @ Mi3 @ X3 )
& ( ord_less_eq @ nat @ X3 @ Ma3 ) ) ) ) ) ) )
| ? [TreeList3: list @ vEBT_VEBT,N4: nat,Summary3: vEBT_VEBT,Mi3: nat,Ma3: nat] :
( ( A1
= ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi3 @ Ma3 ) ) @ A22 @ TreeList3 @ Summary3 ) )
& ! [X3: vEBT_VEBT] :
( ( member @ vEBT_VEBT @ X3 @ ( set2 @ vEBT_VEBT @ TreeList3 ) )
=> ( vEBT_invar_vebt @ X3 @ N4 ) )
& ( vEBT_invar_vebt @ Summary3 @ ( suc @ N4 ) )
& ( ( size_size @ ( list @ vEBT_VEBT ) @ TreeList3 )
= ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( suc @ N4 ) ) )
& ( A22
= ( plus_plus @ nat @ N4 @ ( suc @ N4 ) ) )
& ! [I4: nat] :
( ( ord_less @ nat @ I4 @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( suc @ N4 ) ) )
=> ( ( ? [X7: nat] : ( vEBT_V8194947554948674370ptions @ ( nth @ vEBT_VEBT @ TreeList3 @ I4 ) @ X7 ) )
= ( vEBT_V8194947554948674370ptions @ Summary3 @ I4 ) ) )
& ( ( Mi3 = Ma3 )
=> ! [X3: vEBT_VEBT] :
( ( member @ vEBT_VEBT @ X3 @ ( set2 @ vEBT_VEBT @ TreeList3 ) )
=> ~ ? [X7: nat] : ( vEBT_V8194947554948674370ptions @ X3 @ X7 ) ) )
& ( ord_less_eq @ nat @ Mi3 @ Ma3 )
& ( ord_less @ nat @ Ma3 @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ A22 ) )
& ( ( Mi3 != Ma3 )
=> ! [I4: nat] :
( ( ord_less @ nat @ I4 @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( suc @ N4 ) ) )
=> ( ( ( ( vEBT_VEBT_high @ Ma3 @ N4 )
= I4 )
=> ( vEBT_V8194947554948674370ptions @ ( nth @ vEBT_VEBT @ TreeList3 @ I4 ) @ ( vEBT_VEBT_low @ Ma3 @ N4 ) ) )
& ! [X3: nat] :
( ( ( ( vEBT_VEBT_high @ X3 @ N4 )
= I4 )
& ( vEBT_V8194947554948674370ptions @ ( nth @ vEBT_VEBT @ TreeList3 @ I4 ) @ ( vEBT_VEBT_low @ X3 @ N4 ) ) )
=> ( ( ord_less @ nat @ Mi3 @ X3 )
& ( ord_less_eq @ nat @ X3 @ Ma3 ) ) ) ) ) ) ) ) ) ) ).
% invar_vebt.simps
thf(fact_1566_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,N3: nat,Summary2: vEBT_VEBT,M5: nat,Deg2: nat] :
( ( A12
= ( vEBT_Node @ ( none @ ( product_prod @ nat @ nat ) ) @ Deg2 @ TreeList2 @ Summary2 ) )
=> ( ( A23 = Deg2 )
=> ( ! [X8: vEBT_VEBT] :
( ( member @ vEBT_VEBT @ X8 @ ( set2 @ vEBT_VEBT @ TreeList2 ) )
=> ( vEBT_invar_vebt @ X8 @ N3 ) )
=> ( ( vEBT_invar_vebt @ Summary2 @ M5 )
=> ( ( ( size_size @ ( list @ vEBT_VEBT ) @ TreeList2 )
= ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ M5 ) )
=> ( ( M5 = N3 )
=> ( ( Deg2
= ( plus_plus @ nat @ N3 @ M5 ) )
=> ( ~ ? [X_12: nat] : ( vEBT_V8194947554948674370ptions @ Summary2 @ X_12 )
=> ~ ! [X8: vEBT_VEBT] :
( ( member @ vEBT_VEBT @ X8 @ ( set2 @ vEBT_VEBT @ TreeList2 ) )
=> ~ ? [X_12: nat] : ( vEBT_V8194947554948674370ptions @ X8 @ X_12 ) ) ) ) ) ) ) ) ) )
=> ( ! [TreeList2: list @ vEBT_VEBT,N3: nat,Summary2: vEBT_VEBT,M5: nat,Deg2: nat] :
( ( A12
= ( vEBT_Node @ ( none @ ( product_prod @ nat @ nat ) ) @ Deg2 @ TreeList2 @ Summary2 ) )
=> ( ( A23 = Deg2 )
=> ( ! [X8: vEBT_VEBT] :
( ( member @ vEBT_VEBT @ X8 @ ( set2 @ vEBT_VEBT @ TreeList2 ) )
=> ( vEBT_invar_vebt @ X8 @ N3 ) )
=> ( ( vEBT_invar_vebt @ Summary2 @ M5 )
=> ( ( ( size_size @ ( list @ vEBT_VEBT ) @ TreeList2 )
= ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ M5 ) )
=> ( ( M5
= ( suc @ N3 ) )
=> ( ( Deg2
= ( plus_plus @ nat @ N3 @ M5 ) )
=> ( ~ ? [X_12: nat] : ( vEBT_V8194947554948674370ptions @ Summary2 @ X_12 )
=> ~ ! [X8: vEBT_VEBT] :
( ( member @ vEBT_VEBT @ X8 @ ( set2 @ vEBT_VEBT @ TreeList2 ) )
=> ~ ? [X_12: nat] : ( vEBT_V8194947554948674370ptions @ X8 @ X_12 ) ) ) ) ) ) ) ) ) )
=> ( ! [TreeList2: list @ vEBT_VEBT,N3: nat,Summary2: vEBT_VEBT,M5: 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 )
=> ( ! [X8: vEBT_VEBT] :
( ( member @ vEBT_VEBT @ X8 @ ( set2 @ vEBT_VEBT @ TreeList2 ) )
=> ( vEBT_invar_vebt @ X8 @ N3 ) )
=> ( ( vEBT_invar_vebt @ Summary2 @ M5 )
=> ( ( ( size_size @ ( list @ vEBT_VEBT ) @ TreeList2 )
= ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ M5 ) )
=> ( ( M5 = N3 )
=> ( ( Deg2
= ( plus_plus @ nat @ N3 @ M5 ) )
=> ( ! [I5: nat] :
( ( ord_less @ nat @ I5 @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ M5 ) )
=> ( ( ? [X7: nat] : ( vEBT_V8194947554948674370ptions @ ( nth @ vEBT_VEBT @ TreeList2 @ I5 ) @ X7 ) )
= ( vEBT_V8194947554948674370ptions @ Summary2 @ I5 ) ) )
=> ( ( ( Mi2 = Ma2 )
=> ! [X8: vEBT_VEBT] :
( ( member @ vEBT_VEBT @ X8 @ ( set2 @ vEBT_VEBT @ TreeList2 ) )
=> ~ ? [X_12: nat] : ( vEBT_V8194947554948674370ptions @ X8 @ X_12 ) ) )
=> ( ( ord_less_eq @ nat @ Mi2 @ Ma2 )
=> ( ( ord_less @ nat @ Ma2 @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ Deg2 ) )
=> ~ ( ( Mi2 != Ma2 )
=> ! [I5: nat] :
( ( ord_less @ nat @ I5 @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ M5 ) )
=> ( ( ( ( vEBT_VEBT_high @ Ma2 @ N3 )
= I5 )
=> ( vEBT_V8194947554948674370ptions @ ( nth @ vEBT_VEBT @ TreeList2 @ I5 ) @ ( vEBT_VEBT_low @ Ma2 @ N3 ) ) )
& ! [X8: nat] :
( ( ( ( vEBT_VEBT_high @ X8 @ N3 )
= I5 )
& ( vEBT_V8194947554948674370ptions @ ( nth @ vEBT_VEBT @ TreeList2 @ I5 ) @ ( vEBT_VEBT_low @ X8 @ N3 ) ) )
=> ( ( ord_less @ nat @ Mi2 @ X8 )
& ( ord_less_eq @ nat @ X8 @ Ma2 ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) )
=> ~ ! [TreeList2: list @ vEBT_VEBT,N3: nat,Summary2: vEBT_VEBT,M5: 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 )
=> ( ! [X8: vEBT_VEBT] :
( ( member @ vEBT_VEBT @ X8 @ ( set2 @ vEBT_VEBT @ TreeList2 ) )
=> ( vEBT_invar_vebt @ X8 @ N3 ) )
=> ( ( vEBT_invar_vebt @ Summary2 @ M5 )
=> ( ( ( size_size @ ( list @ vEBT_VEBT ) @ TreeList2 )
= ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ M5 ) )
=> ( ( M5
= ( suc @ N3 ) )
=> ( ( Deg2
= ( plus_plus @ nat @ N3 @ M5 ) )
=> ( ! [I5: nat] :
( ( ord_less @ nat @ I5 @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ M5 ) )
=> ( ( ? [X7: nat] : ( vEBT_V8194947554948674370ptions @ ( nth @ vEBT_VEBT @ TreeList2 @ I5 ) @ X7 ) )
= ( vEBT_V8194947554948674370ptions @ Summary2 @ I5 ) ) )
=> ( ( ( Mi2 = Ma2 )
=> ! [X8: vEBT_VEBT] :
( ( member @ vEBT_VEBT @ X8 @ ( set2 @ vEBT_VEBT @ TreeList2 ) )
=> ~ ? [X_12: nat] : ( vEBT_V8194947554948674370ptions @ X8 @ X_12 ) ) )
=> ( ( ord_less_eq @ nat @ Mi2 @ Ma2 )
=> ( ( ord_less @ nat @ Ma2 @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ Deg2 ) )
=> ~ ( ( Mi2 != Ma2 )
=> ! [I5: nat] :
( ( ord_less @ nat @ I5 @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ M5 ) )
=> ( ( ( ( vEBT_VEBT_high @ Ma2 @ N3 )
= I5 )
=> ( vEBT_V8194947554948674370ptions @ ( nth @ vEBT_VEBT @ TreeList2 @ I5 ) @ ( vEBT_VEBT_low @ Ma2 @ N3 ) ) )
& ! [X8: nat] :
( ( ( ( vEBT_VEBT_high @ X8 @ N3 )
= I5 )
& ( vEBT_V8194947554948674370ptions @ ( nth @ vEBT_VEBT @ TreeList2 @ I5 ) @ ( vEBT_VEBT_low @ X8 @ N3 ) ) )
=> ( ( ord_less @ nat @ Mi2 @ X8 )
& ( ord_less_eq @ nat @ X8 @ Ma2 ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).
% invar_vebt.cases
thf(fact_1567_vebt__insert_Oelims,axiom,
! [X2: vEBT_VEBT,Xa: nat,Y3: vEBT_VEBT] :
( ( ( vEBT_vebt_insert @ X2 @ Xa )
= Y3 )
=> ( ! [A4: $o,B4: $o] :
( ( X2
= ( vEBT_Leaf @ A4 @ B4 ) )
=> ~ ( ( ( Xa
= ( zero_zero @ nat ) )
=> ( Y3
= ( vEBT_Leaf @ $true @ B4 ) ) )
& ( ( Xa
!= ( zero_zero @ nat ) )
=> ( ( ( Xa
= ( one_one @ nat ) )
=> ( Y3
= ( vEBT_Leaf @ A4 @ $true ) ) )
& ( ( Xa
!= ( one_one @ nat ) )
=> ( Y3
= ( vEBT_Leaf @ A4 @ B4 ) ) ) ) ) ) )
=> ( ! [Info2: option @ ( product_prod @ nat @ nat ),Ts2: list @ vEBT_VEBT,S3: vEBT_VEBT] :
( ( X2
= ( vEBT_Node @ Info2 @ ( zero_zero @ nat ) @ Ts2 @ S3 ) )
=> ( Y3
!= ( vEBT_Node @ Info2 @ ( zero_zero @ nat ) @ Ts2 @ S3 ) ) )
=> ( ! [Info2: option @ ( product_prod @ nat @ nat ),Ts2: list @ vEBT_VEBT,S3: vEBT_VEBT] :
( ( X2
= ( vEBT_Node @ Info2 @ ( suc @ ( zero_zero @ nat ) ) @ Ts2 @ S3 ) )
=> ( Y3
!= ( vEBT_Node @ Info2 @ ( suc @ ( zero_zero @ nat ) ) @ Ts2 @ S3 ) ) )
=> ( ! [V3: nat,TreeList2: list @ vEBT_VEBT,Summary2: vEBT_VEBT] :
( ( X2
= ( vEBT_Node @ ( none @ ( product_prod @ nat @ nat ) ) @ ( suc @ ( suc @ V3 ) ) @ TreeList2 @ Summary2 ) )
=> ( Y3
!= ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Xa @ Xa ) ) @ ( suc @ ( suc @ V3 ) ) @ TreeList2 @ Summary2 ) ) )
=> ~ ! [Mi2: nat,Ma2: nat,Va3: nat,TreeList2: list @ vEBT_VEBT,Summary2: vEBT_VEBT] :
( ( X2
= ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( suc @ Va3 ) ) @ TreeList2 @ Summary2 ) )
=> ( Y3
!= ( if @ vEBT_VEBT
@ ( ( ord_less @ nat @ ( vEBT_VEBT_high @ ( if @ nat @ ( ord_less @ nat @ Xa @ Mi2 ) @ Mi2 @ Xa ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( size_size @ ( list @ vEBT_VEBT ) @ TreeList2 ) )
& ~ ( ( Xa = Mi2 )
| ( Xa = Ma2 ) ) )
@ ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ ( if @ nat @ ( ord_less @ nat @ Xa @ Mi2 ) @ Xa @ Mi2 ) @ ( ord_max @ nat @ ( if @ nat @ ( ord_less @ nat @ Xa @ Mi2 ) @ Mi2 @ Xa ) @ Ma2 ) ) ) @ ( suc @ ( suc @ Va3 ) ) @ ( list_update @ vEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ ( if @ nat @ ( ord_less @ nat @ Xa @ Mi2 ) @ Mi2 @ Xa ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( vEBT_vebt_insert @ ( nth @ vEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ ( if @ nat @ ( ord_less @ nat @ Xa @ Mi2 ) @ Mi2 @ Xa ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( vEBT_VEBT_low @ ( if @ nat @ ( ord_less @ nat @ Xa @ Mi2 ) @ Mi2 @ Xa ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) @ ( if @ vEBT_VEBT @ ( vEBT_VEBT_minNull @ ( nth @ vEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ ( if @ nat @ ( ord_less @ nat @ Xa @ Mi2 ) @ Mi2 @ Xa ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) @ ( vEBT_vebt_insert @ Summary2 @ ( vEBT_VEBT_high @ ( if @ nat @ ( ord_less @ nat @ Xa @ Mi2 ) @ Mi2 @ Xa ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ Summary2 ) )
@ ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( suc @ Va3 ) ) @ TreeList2 @ Summary2 ) ) ) ) ) ) ) ) ) ).
% vebt_insert.elims
thf(fact_1568_vebt__delete_Oelims,axiom,
! [X2: vEBT_VEBT,Xa: nat,Y3: vEBT_VEBT] :
( ( ( vEBT_vebt_delete @ X2 @ Xa )
= Y3 )
=> ( ! [A4: $o,B4: $o] :
( ( X2
= ( vEBT_Leaf @ A4 @ B4 ) )
=> ( ( Xa
= ( zero_zero @ nat ) )
=> ( Y3
!= ( vEBT_Leaf @ $false @ B4 ) ) ) )
=> ( ! [A4: $o] :
( ? [B4: $o] :
( X2
= ( vEBT_Leaf @ A4 @ B4 ) )
=> ( ( Xa
= ( suc @ ( zero_zero @ nat ) ) )
=> ( Y3
!= ( vEBT_Leaf @ A4 @ $false ) ) ) )
=> ( ! [A4: $o,B4: $o] :
( ( X2
= ( vEBT_Leaf @ A4 @ B4 ) )
=> ( ? [N3: nat] :
( Xa
= ( suc @ ( suc @ N3 ) ) )
=> ( Y3
!= ( vEBT_Leaf @ A4 @ B4 ) ) ) )
=> ( ! [Deg2: nat,TreeList2: list @ vEBT_VEBT,Summary2: vEBT_VEBT] :
( ( X2
= ( vEBT_Node @ ( none @ ( product_prod @ nat @ nat ) ) @ Deg2 @ TreeList2 @ Summary2 ) )
=> ( Y3
!= ( vEBT_Node @ ( none @ ( product_prod @ nat @ nat ) ) @ Deg2 @ TreeList2 @ Summary2 ) ) )
=> ( ! [Mi2: nat,Ma2: nat,TrLst2: list @ vEBT_VEBT,Smry2: vEBT_VEBT] :
( ( X2
= ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi2 @ Ma2 ) ) @ ( zero_zero @ nat ) @ TrLst2 @ Smry2 ) )
=> ( Y3
!= ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi2 @ Ma2 ) ) @ ( zero_zero @ nat ) @ TrLst2 @ Smry2 ) ) )
=> ( ! [Mi2: nat,Ma2: nat,Tr2: list @ vEBT_VEBT,Sm2: vEBT_VEBT] :
( ( X2
= ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( zero_zero @ nat ) ) @ Tr2 @ Sm2 ) )
=> ( Y3
!= ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( zero_zero @ nat ) ) @ Tr2 @ Sm2 ) ) )
=> ~ ! [Mi2: nat,Ma2: nat,Va3: nat,TreeList2: list @ vEBT_VEBT,Summary2: vEBT_VEBT] :
( ( X2
= ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( suc @ Va3 ) ) @ TreeList2 @ Summary2 ) )
=> ~ ( ( ( ( ord_less @ nat @ Xa @ Mi2 )
| ( ord_less @ nat @ Ma2 @ Xa ) )
=> ( Y3
= ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( suc @ Va3 ) ) @ TreeList2 @ Summary2 ) ) )
& ( ~ ( ( ord_less @ nat @ Xa @ Mi2 )
| ( ord_less @ nat @ Ma2 @ Xa ) )
=> ( ( ( ( Xa = Mi2 )
& ( Xa = Ma2 ) )
=> ( Y3
= ( vEBT_Node @ ( none @ ( product_prod @ nat @ nat ) ) @ ( suc @ ( suc @ Va3 ) ) @ TreeList2 @ Summary2 ) ) )
& ( ~ ( ( Xa = Mi2 )
& ( Xa = Ma2 ) )
=> ( Y3
= ( if @ vEBT_VEBT @ ( ord_less @ nat @ ( vEBT_VEBT_high @ ( if @ nat @ ( Xa = Mi2 ) @ ( plus_plus @ nat @ ( times_times @ nat @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary2 ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList2 @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary2 ) ) ) ) ) ) @ Xa ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( size_size @ ( list @ vEBT_VEBT ) @ TreeList2 ) )
@ ( if @ vEBT_VEBT @ ( vEBT_VEBT_minNull @ ( vEBT_vebt_delete @ ( nth @ vEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ ( if @ nat @ ( Xa = Mi2 ) @ ( plus_plus @ nat @ ( times_times @ nat @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary2 ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList2 @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary2 ) ) ) ) ) ) @ Xa ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( vEBT_VEBT_low @ ( if @ nat @ ( Xa = Mi2 ) @ ( plus_plus @ nat @ ( times_times @ nat @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary2 ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList2 @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary2 ) ) ) ) ) ) @ Xa ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) )
@ ( vEBT_Node
@ ( some @ ( product_prod @ nat @ nat )
@ ( product_Pair @ nat @ nat @ ( if @ nat @ ( Xa = Mi2 ) @ ( if @ nat @ ( Xa = Mi2 ) @ ( plus_plus @ nat @ ( times_times @ nat @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary2 ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList2 @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary2 ) ) ) ) ) ) @ Xa ) @ Mi2 )
@ ( if @ nat
@ ( ( ( Xa = Mi2 )
=> ( ( plus_plus @ nat @ ( times_times @ nat @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary2 ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList2 @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary2 ) ) ) ) ) )
= Ma2 ) )
& ( ( Xa != Mi2 )
=> ( Xa = Ma2 ) ) )
@ ( if @ nat
@ ( ( vEBT_vebt_maxt @ ( vEBT_vebt_delete @ Summary2 @ ( vEBT_VEBT_high @ ( if @ nat @ ( Xa = Mi2 ) @ ( plus_plus @ nat @ ( times_times @ nat @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary2 ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList2 @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary2 ) ) ) ) ) ) @ Xa ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) )
= ( none @ nat ) )
@ ( if @ nat @ ( Xa = Mi2 ) @ ( if @ nat @ ( Xa = Mi2 ) @ ( plus_plus @ nat @ ( times_times @ nat @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary2 ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList2 @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary2 ) ) ) ) ) ) @ Xa ) @ Mi2 )
@ ( plus_plus @ nat @ ( times_times @ nat @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_maxt @ ( vEBT_vebt_delete @ Summary2 @ ( vEBT_VEBT_high @ ( if @ nat @ ( Xa = Mi2 ) @ ( plus_plus @ nat @ ( times_times @ nat @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary2 ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList2 @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary2 ) ) ) ) ) ) @ Xa ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_maxt @ ( nth @ vEBT_VEBT @ ( list_update @ vEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ ( if @ nat @ ( Xa = Mi2 ) @ ( plus_plus @ nat @ ( times_times @ nat @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary2 ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList2 @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary2 ) ) ) ) ) ) @ Xa ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( vEBT_vebt_delete @ ( nth @ vEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ ( if @ nat @ ( Xa = Mi2 ) @ ( plus_plus @ nat @ ( times_times @ nat @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary2 ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList2 @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary2 ) ) ) ) ) ) @ Xa ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( vEBT_VEBT_low @ ( if @ nat @ ( Xa = Mi2 ) @ ( plus_plus @ nat @ ( times_times @ nat @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary2 ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList2 @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary2 ) ) ) ) ) ) @ Xa ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_maxt @ ( vEBT_vebt_delete @ Summary2 @ ( vEBT_VEBT_high @ ( if @ nat @ ( Xa = Mi2 ) @ ( plus_plus @ nat @ ( times_times @ nat @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary2 ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList2 @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary2 ) ) ) ) ) ) @ Xa ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) ) ) ) ) ) )
@ Ma2 ) ) )
@ ( suc @ ( suc @ Va3 ) )
@ ( list_update @ vEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ ( if @ nat @ ( Xa = Mi2 ) @ ( plus_plus @ nat @ ( times_times @ nat @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary2 ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList2 @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary2 ) ) ) ) ) ) @ Xa ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( vEBT_vebt_delete @ ( nth @ vEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ ( if @ nat @ ( Xa = Mi2 ) @ ( plus_plus @ nat @ ( times_times @ nat @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary2 ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList2 @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary2 ) ) ) ) ) ) @ Xa ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( vEBT_VEBT_low @ ( if @ nat @ ( Xa = Mi2 ) @ ( plus_plus @ nat @ ( times_times @ nat @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary2 ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList2 @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary2 ) ) ) ) ) ) @ Xa ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) )
@ ( vEBT_vebt_delete @ Summary2 @ ( vEBT_VEBT_high @ ( if @ nat @ ( Xa = Mi2 ) @ ( plus_plus @ nat @ ( times_times @ nat @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary2 ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList2 @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary2 ) ) ) ) ) ) @ Xa ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) )
@ ( vEBT_Node
@ ( some @ ( product_prod @ nat @ nat )
@ ( product_Pair @ nat @ nat @ ( if @ nat @ ( Xa = Mi2 ) @ ( if @ nat @ ( Xa = Mi2 ) @ ( plus_plus @ nat @ ( times_times @ nat @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary2 ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList2 @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary2 ) ) ) ) ) ) @ Xa ) @ Mi2 )
@ ( if @ nat
@ ( ( ( Xa = Mi2 )
=> ( ( plus_plus @ nat @ ( times_times @ nat @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary2 ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList2 @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary2 ) ) ) ) ) )
= Ma2 ) )
& ( ( Xa != Mi2 )
=> ( Xa = Ma2 ) ) )
@ ( plus_plus @ nat @ ( times_times @ nat @ ( vEBT_VEBT_high @ ( if @ nat @ ( Xa = Mi2 ) @ ( plus_plus @ nat @ ( times_times @ nat @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary2 ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList2 @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary2 ) ) ) ) ) ) @ Xa ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_maxt @ ( nth @ vEBT_VEBT @ ( list_update @ vEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ ( if @ nat @ ( Xa = Mi2 ) @ ( plus_plus @ nat @ ( times_times @ nat @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary2 ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList2 @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary2 ) ) ) ) ) ) @ Xa ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( vEBT_vebt_delete @ ( nth @ vEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ ( if @ nat @ ( Xa = Mi2 ) @ ( plus_plus @ nat @ ( times_times @ nat @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary2 ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList2 @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary2 ) ) ) ) ) ) @ Xa ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( vEBT_VEBT_low @ ( if @ nat @ ( Xa = Mi2 ) @ ( plus_plus @ nat @ ( times_times @ nat @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary2 ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList2 @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary2 ) ) ) ) ) ) @ Xa ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) @ ( vEBT_VEBT_high @ ( if @ nat @ ( Xa = Mi2 ) @ ( plus_plus @ nat @ ( times_times @ nat @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary2 ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList2 @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary2 ) ) ) ) ) ) @ Xa ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) ) )
@ Ma2 ) ) )
@ ( suc @ ( suc @ Va3 ) )
@ ( list_update @ vEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ ( if @ nat @ ( Xa = Mi2 ) @ ( plus_plus @ nat @ ( times_times @ nat @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary2 ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList2 @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary2 ) ) ) ) ) ) @ Xa ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( vEBT_vebt_delete @ ( nth @ vEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ ( if @ nat @ ( Xa = Mi2 ) @ ( plus_plus @ nat @ ( times_times @ nat @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary2 ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList2 @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary2 ) ) ) ) ) ) @ Xa ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( vEBT_VEBT_low @ ( if @ nat @ ( Xa = Mi2 ) @ ( plus_plus @ nat @ ( times_times @ nat @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary2 ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList2 @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary2 ) ) ) ) ) ) @ Xa ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) )
@ Summary2 ) )
@ ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( suc @ Va3 ) ) @ TreeList2 @ Summary2 ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).
% vebt_delete.elims
thf(fact_1569_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_1570_vebt__maxt_Oelims,axiom,
! [X2: vEBT_VEBT,Y3: option @ nat] :
( ( ( vEBT_vebt_maxt @ X2 )
= Y3 )
=> ( ! [A4: $o,B4: $o] :
( ( X2
= ( vEBT_Leaf @ A4 @ B4 ) )
=> ~ ( ( B4
=> ( Y3
= ( some @ nat @ ( one_one @ nat ) ) ) )
& ( ~ B4
=> ( ( A4
=> ( Y3
= ( some @ nat @ ( zero_zero @ nat ) ) ) )
& ( ~ A4
=> ( Y3
= ( none @ nat ) ) ) ) ) ) )
=> ( ( ? [Uu2: nat,Uv2: list @ vEBT_VEBT,Uw2: vEBT_VEBT] :
( X2
= ( vEBT_Node @ ( none @ ( product_prod @ nat @ nat ) ) @ Uu2 @ Uv2 @ Uw2 ) )
=> ( Y3
!= ( none @ nat ) ) )
=> ~ ! [Mi2: nat,Ma2: nat] :
( ? [Ux2: nat,Uy2: list @ vEBT_VEBT,Uz2: vEBT_VEBT] :
( X2
= ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi2 @ Ma2 ) ) @ Ux2 @ Uy2 @ Uz2 ) )
=> ( Y3
!= ( some @ nat @ Ma2 ) ) ) ) ) ) ).
% vebt_maxt.elims
thf(fact_1571_vebt__mint_Oelims,axiom,
! [X2: vEBT_VEBT,Y3: option @ nat] :
( ( ( vEBT_vebt_mint @ X2 )
= Y3 )
=> ( ! [A4: $o,B4: $o] :
( ( X2
= ( vEBT_Leaf @ A4 @ B4 ) )
=> ~ ( ( A4
=> ( Y3
= ( some @ nat @ ( zero_zero @ nat ) ) ) )
& ( ~ A4
=> ( ( B4
=> ( Y3
= ( some @ nat @ ( one_one @ nat ) ) ) )
& ( ~ B4
=> ( Y3
= ( none @ nat ) ) ) ) ) ) )
=> ( ( ? [Uu2: nat,Uv2: list @ vEBT_VEBT,Uw2: vEBT_VEBT] :
( X2
= ( vEBT_Node @ ( none @ ( product_prod @ nat @ nat ) ) @ Uu2 @ Uv2 @ Uw2 ) )
=> ( Y3
!= ( none @ nat ) ) )
=> ~ ! [Mi2: nat] :
( ? [Ma2: nat,Ux2: nat,Uy2: list @ vEBT_VEBT,Uz2: vEBT_VEBT] :
( X2
= ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi2 @ Ma2 ) ) @ Ux2 @ Uy2 @ Uz2 ) )
=> ( Y3
!= ( some @ nat @ Mi2 ) ) ) ) ) ) ).
% vebt_mint.elims
thf(fact_1572_norm__pre__pure__iff__sng,axiom,
! [A: $tType,B3: $o,F2: heap_Time_Heap @ A,Q: A > assn] :
( ( hoare_hoare_triple @ A @ ( pure_assn @ B3 ) @ F2 @ Q )
= ( B3
=> ( hoare_hoare_triple @ A @ ( one_one @ assn ) @ F2 @ Q ) ) ) ).
% norm_pre_pure_iff_sng
thf(fact_1573_ent__pure__pre__iff__sng,axiom,
! [B3: $o,Q: assn] :
( ( entails @ ( pure_assn @ B3 ) @ Q )
= ( B3
=> ( entails @ ( one_one @ assn ) @ Q ) ) ) ).
% ent_pure_pre_iff_sng
thf(fact_1574_norm__pre__pure__iff,axiom,
! [A: $tType,P: assn,B3: $o,F2: heap_Time_Heap @ A,Q: A > assn] :
( ( hoare_hoare_triple @ A @ ( times_times @ assn @ P @ ( pure_assn @ B3 ) ) @ F2 @ Q )
= ( B3
=> ( hoare_hoare_triple @ A @ P @ F2 @ Q ) ) ) ).
% norm_pre_pure_iff
thf(fact_1575_ent__pure__pre__iff,axiom,
! [P: assn,B3: $o,Q: assn] :
( ( entails @ ( times_times @ assn @ P @ ( pure_assn @ B3 ) ) @ Q )
= ( B3
=> ( entails @ P @ Q ) ) ) ).
% ent_pure_pre_iff
thf(fact_1576_vebt__pred_Osimps_I3_J,axiom,
! [B3: $o,A3: $o,Va: nat] :
( ( B3
=> ( ( vEBT_vebt_pred @ ( vEBT_Leaf @ A3 @ B3 ) @ ( suc @ ( suc @ Va ) ) )
= ( some @ nat @ ( one_one @ nat ) ) ) )
& ( ~ B3
=> ( ( A3
=> ( ( vEBT_vebt_pred @ ( vEBT_Leaf @ A3 @ B3 ) @ ( suc @ ( suc @ Va ) ) )
= ( some @ nat @ ( zero_zero @ nat ) ) ) )
& ( ~ A3
=> ( ( vEBT_vebt_pred @ ( vEBT_Leaf @ A3 @ B3 ) @ ( suc @ ( suc @ Va ) ) )
= ( none @ nat ) ) ) ) ) ) ).
% vebt_pred.simps(3)
thf(fact_1577_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_1578_vebt__memberi_H__rf__abstr,axiom,
! [T2: vEBT_VEBT,Ti: vEBT_VEBTi,X2: nat] :
( hoare_hoare_triple @ $o @ ( vEBT_vebt_assn_raw @ T2 @ Ti ) @ ( vEBT_V854960066525838166emberi @ T2 @ Ti @ X2 )
@ ^ [R5: $o] :
( times_times @ assn @ ( vEBT_vebt_assn_raw @ T2 @ Ti )
@ ( pure_assn
@ ( R5
= ( vEBT_vebt_member @ T2 @ X2 ) ) ) ) ) ).
% vebt_memberi'_rf_abstr
thf(fact_1579_vebt__pred_H__rf__abstr,axiom,
! [T2: vEBT_VEBT,N: nat,Ti: vEBT_VEBTi,X2: nat] :
( ( vEBT_invar_vebt @ T2 @ N )
=> ( hoare_hoare_triple @ ( option @ nat ) @ ( vEBT_vebt_assn_raw @ T2 @ Ti ) @ ( vEBT_VEBT_vebt_predi @ T2 @ Ti @ X2 )
@ ^ [R5: option @ nat] :
( times_times @ assn @ ( vEBT_vebt_assn_raw @ T2 @ Ti )
@ ( pure_assn
@ ( R5
= ( vEBT_vebt_pred @ T2 @ X2 ) ) ) ) ) ) ).
% vebt_pred'_rf_abstr
thf(fact_1580_vebt__succi_H__rf__abstr,axiom,
! [T2: vEBT_VEBT,N: nat,Ti: vEBT_VEBTi,X2: nat] :
( ( vEBT_invar_vebt @ T2 @ N )
=> ( hoare_hoare_triple @ ( option @ nat ) @ ( vEBT_vebt_assn_raw @ T2 @ Ti ) @ ( vEBT_VEBT_vebt_succi @ T2 @ Ti @ X2 )
@ ^ [R5: option @ nat] :
( times_times @ assn @ ( vEBT_vebt_assn_raw @ T2 @ Ti )
@ ( pure_assn
@ ( R5
= ( vEBT_vebt_succ @ T2 @ X2 ) ) ) ) ) ) ).
% vebt_succi'_rf_abstr
thf(fact_1581_merge__pure__star,axiom,
! [A3: $o,B3: $o] :
( ( times_times @ assn @ ( pure_assn @ A3 ) @ ( pure_assn @ B3 ) )
= ( pure_assn
@ ( A3
& B3 ) ) ) ).
% merge_pure_star
thf(fact_1582_pure__true,axiom,
( ( pure_assn @ $true )
= ( one_one @ assn ) ) ).
% pure_true
thf(fact_1583_pure__assn__eq__emp__iff,axiom,
! [P: $o] :
( ( ( pure_assn @ P )
= ( one_one @ assn ) )
= P ) ).
% pure_assn_eq_emp_iff
thf(fact_1584_htt__vebt__memberi,axiom,
! [T2: vEBT_VEBT,Ti: vEBT_VEBTi,X2: nat] :
( time_htt @ $o @ ( vEBT_vebt_assn_raw @ T2 @ Ti ) @ ( vEBT_vebt_memberi @ Ti @ X2 )
@ ^ [R5: $o] :
( times_times @ assn @ ( vEBT_vebt_assn_raw @ T2 @ Ti )
@ ( pure_assn
@ ( R5
= ( vEBT_vebt_member @ T2 @ X2 ) ) ) )
@ ( 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_1585_assn__times__comm,axiom,
( ( times_times @ assn )
= ( ^ [P3: assn,Q7: assn] : ( times_times @ assn @ Q7 @ P3 ) ) ) ).
% assn_times_comm
thf(fact_1586_assn__times__assoc,axiom,
! [P: assn,Q: assn,R: assn] :
( ( times_times @ assn @ ( times_times @ assn @ P @ Q ) @ R )
= ( times_times @ assn @ P @ ( times_times @ assn @ Q @ R ) ) ) ).
% assn_times_assoc
thf(fact_1587_ent__iffI,axiom,
! [A2: assn,B2: assn] :
( ( entails @ A2 @ B2 )
=> ( ( entails @ B2 @ A2 )
=> ( A2 = B2 ) ) ) ).
% ent_iffI
thf(fact_1588_ent__refl,axiom,
! [P: assn] : ( entails @ P @ P ) ).
% ent_refl
thf(fact_1589_ent__trans,axiom,
! [P: assn,Q: assn,R: assn] :
( ( entails @ P @ Q )
=> ( ( entails @ Q @ R )
=> ( entails @ P @ R ) ) ) ).
% ent_trans
thf(fact_1590_VEBT__internal_Ooption__shift_Ocases,axiom,
! [A: $tType,X2: product_prod @ ( A > A > A ) @ ( product_prod @ ( option @ A ) @ ( option @ A ) )] :
( ! [Uu2: A > A > A,Uv2: option @ A] :
( X2
!= ( product_Pair @ ( A > A > A ) @ ( product_prod @ ( option @ A ) @ ( option @ A ) ) @ Uu2 @ ( product_Pair @ ( option @ A ) @ ( option @ A ) @ ( none @ A ) @ Uv2 ) ) )
=> ( ! [Uw2: A > A > A,V3: A] :
( X2
!= ( product_Pair @ ( A > A > A ) @ ( product_prod @ ( option @ A ) @ ( option @ A ) ) @ Uw2 @ ( product_Pair @ ( option @ A ) @ ( option @ A ) @ ( some @ A @ V3 ) @ ( none @ A ) ) ) )
=> ~ ! [F3: A > A > A,A4: A,B4: A] :
( X2
!= ( product_Pair @ ( A > A > A ) @ ( product_prod @ ( option @ A ) @ ( option @ A ) ) @ F3 @ ( product_Pair @ ( option @ A ) @ ( option @ A ) @ ( some @ A @ A4 ) @ ( some @ A @ B4 ) ) ) ) ) ) ).
% VEBT_internal.option_shift.cases
thf(fact_1591_VEBT__internal_Ooption__comp__shift_Ocases,axiom,
! [A: $tType,X2: product_prod @ ( A > A > $o ) @ ( product_prod @ ( option @ A ) @ ( option @ A ) )] :
( ! [Uu2: A > A > $o,Uv2: option @ A] :
( X2
!= ( product_Pair @ ( A > A > $o ) @ ( product_prod @ ( option @ A ) @ ( option @ A ) ) @ Uu2 @ ( product_Pair @ ( option @ A ) @ ( option @ A ) @ ( none @ A ) @ Uv2 ) ) )
=> ( ! [Uw2: A > A > $o,V3: A] :
( X2
!= ( product_Pair @ ( A > A > $o ) @ ( product_prod @ ( option @ A ) @ ( option @ A ) ) @ Uw2 @ ( product_Pair @ ( option @ A ) @ ( option @ A ) @ ( some @ A @ V3 ) @ ( none @ A ) ) ) )
=> ~ ! [F3: A > A > $o,X4: A,Y4: A] :
( X2
!= ( product_Pair @ ( A > A > $o ) @ ( product_prod @ ( option @ A ) @ ( option @ A ) ) @ F3 @ ( product_Pair @ ( option @ A ) @ ( option @ A ) @ ( some @ A @ X4 ) @ ( some @ A @ Y4 ) ) ) ) ) ) ).
% VEBT_internal.option_comp_shift.cases
thf(fact_1592_ent__star__mono,axiom,
! [P: assn,P5: assn,Q: assn,Q2: assn] :
( ( entails @ P @ P5 )
=> ( ( entails @ Q @ Q2 )
=> ( entails @ ( times_times @ assn @ P @ Q ) @ ( times_times @ assn @ P5 @ Q2 ) ) ) ) ).
% ent_star_mono
thf(fact_1593_cons__rule,axiom,
! [A: $tType,P: assn,P5: assn,Q: A > assn,Q2: A > assn,C3: heap_Time_Heap @ A] :
( ( entails @ P @ P5 )
=> ( ! [X4: A] : ( entails @ ( Q @ X4 ) @ ( Q2 @ X4 ) )
=> ( ( hoare_hoare_triple @ A @ P5 @ C3 @ Q )
=> ( hoare_hoare_triple @ A @ P @ C3 @ Q2 ) ) ) ) ).
% cons_rule
thf(fact_1594_cons__post__rule,axiom,
! [A: $tType,P: assn,C3: heap_Time_Heap @ A,Q: A > assn,Q2: A > assn] :
( ( hoare_hoare_triple @ A @ P @ C3 @ Q )
=> ( ! [X4: A] : ( entails @ ( Q @ X4 ) @ ( Q2 @ X4 ) )
=> ( hoare_hoare_triple @ A @ P @ C3 @ Q2 ) ) ) ).
% cons_post_rule
thf(fact_1595_assn__one__left,axiom,
! [P: assn] :
( ( times_times @ assn @ ( one_one @ assn ) @ P )
= P ) ).
% assn_one_left
thf(fact_1596_VEBT__internal_Ooption__shift_Osimps_I3_J,axiom,
! [A: $tType,F2: A > A > A,A3: A,B3: A] :
( ( vEBT_V2048590022279873568_shift @ A @ F2 @ ( some @ A @ A3 ) @ ( some @ A @ B3 ) )
= ( some @ A @ ( F2 @ A3 @ B3 ) ) ) ).
% VEBT_internal.option_shift.simps(3)
thf(fact_1597_VEBT__internal_Ooption__shift_Osimps_I1_J,axiom,
! [A: $tType,Uu: A > A > A,Uv: option @ A] :
( ( vEBT_V2048590022279873568_shift @ A @ Uu @ ( none @ A ) @ Uv )
= ( none @ A ) ) ).
% VEBT_internal.option_shift.simps(1)
thf(fact_1598_frame__rule,axiom,
! [A: $tType,P: assn,C3: heap_Time_Heap @ A,Q: A > assn,R: assn] :
( ( hoare_hoare_triple @ A @ P @ C3 @ Q )
=> ( hoare_hoare_triple @ A @ ( times_times @ assn @ P @ R ) @ C3
@ ^ [X3: A] : ( times_times @ assn @ ( Q @ X3 ) @ R ) ) ) ).
% frame_rule
thf(fact_1599_cons__pre__rule,axiom,
! [A: $tType,P: assn,P5: assn,C3: heap_Time_Heap @ A,Q: A > assn] :
( ( entails @ P @ P5 )
=> ( ( hoare_hoare_triple @ A @ P5 @ C3 @ Q )
=> ( hoare_hoare_triple @ A @ P @ C3 @ Q ) ) ) ).
% cons_pre_rule
thf(fact_1600_norm__pre__pure__rule1,axiom,
! [A: $tType,B3: $o,P: assn,F2: heap_Time_Heap @ A,Q: A > assn] :
( ( B3
=> ( hoare_hoare_triple @ A @ P @ F2 @ Q ) )
=> ( hoare_hoare_triple @ A @ ( times_times @ assn @ P @ ( pure_assn @ B3 ) ) @ F2 @ Q ) ) ).
% norm_pre_pure_rule1
thf(fact_1601_norm__pre__pure__rule2,axiom,
! [A: $tType,B3: $o,F2: heap_Time_Heap @ A,Q: A > assn] :
( ( B3
=> ( hoare_hoare_triple @ A @ ( one_one @ assn ) @ F2 @ Q ) )
=> ( hoare_hoare_triple @ A @ ( pure_assn @ B3 ) @ F2 @ Q ) ) ).
% norm_pre_pure_rule2
thf(fact_1602_VEBT__internal_Ooption__shift_Osimps_I2_J,axiom,
! [A: $tType,Uw: A > A > A,V: A] :
( ( vEBT_V2048590022279873568_shift @ A @ Uw @ ( some @ A @ V ) @ ( none @ A ) )
= ( none @ A ) ) ).
% VEBT_internal.option_shift.simps(2)
thf(fact_1603_VEBT__internal_Ooption__shift_Oelims,axiom,
! [A: $tType,X2: A > A > A,Xa: option @ A,Xb: option @ A,Y3: option @ A] :
( ( ( vEBT_V2048590022279873568_shift @ A @ X2 @ Xa @ Xb )
= Y3 )
=> ( ( ( Xa
= ( none @ A ) )
=> ( Y3
!= ( none @ A ) ) )
=> ( ( ? [V3: A] :
( Xa
= ( some @ A @ V3 ) )
=> ( ( Xb
= ( none @ A ) )
=> ( Y3
!= ( none @ A ) ) ) )
=> ~ ! [A4: A] :
( ( Xa
= ( some @ A @ A4 ) )
=> ! [B4: A] :
( ( Xb
= ( some @ A @ B4 ) )
=> ( Y3
!= ( some @ A @ ( X2 @ A4 @ B4 ) ) ) ) ) ) ) ) ).
% VEBT_internal.option_shift.elims
thf(fact_1604_is__pred__in__set__def,axiom,
( vEBT_is_pred_in_set
= ( ^ [Xs2: set @ nat,X3: nat,Y: nat] :
( ( member @ nat @ Y @ Xs2 )
& ( ord_less @ nat @ Y @ X3 )
& ! [Z4: nat] :
( ( member @ nat @ Z4 @ Xs2 )
=> ( ( ord_less @ nat @ Z4 @ X3 )
=> ( ord_less_eq @ nat @ Z4 @ Y ) ) ) ) ) ) ).
% is_pred_in_set_def
thf(fact_1605_vebt__mint_Osimps_I2_J,axiom,
! [Uu: nat,Uv: list @ vEBT_VEBT,Uw: vEBT_VEBT] :
( ( vEBT_vebt_mint @ ( vEBT_Node @ ( none @ ( product_prod @ nat @ nat ) ) @ Uu @ Uv @ Uw ) )
= ( none @ nat ) ) ).
% vebt_mint.simps(2)
thf(fact_1606_vebt__maxt_Osimps_I2_J,axiom,
! [Uu: nat,Uv: list @ vEBT_VEBT,Uw: vEBT_VEBT] :
( ( vEBT_vebt_maxt @ ( vEBT_Node @ ( none @ ( product_prod @ nat @ nat ) ) @ Uu @ Uv @ Uw ) )
= ( none @ nat ) ) ).
% vebt_maxt.simps(2)
thf(fact_1607_vebt__pred_Osimps_I1_J,axiom,
! [Uu: $o,Uv: $o] :
( ( vEBT_vebt_pred @ ( vEBT_Leaf @ Uu @ Uv ) @ ( zero_zero @ nat ) )
= ( none @ nat ) ) ).
% vebt_pred.simps(1)
thf(fact_1608_vebt__pred_Osimps_I4_J,axiom,
! [Uy: nat,Uz: list @ vEBT_VEBT,Va: vEBT_VEBT,Vb: nat] :
( ( vEBT_vebt_pred @ ( vEBT_Node @ ( none @ ( product_prod @ nat @ nat ) ) @ Uy @ Uz @ Va ) @ Vb )
= ( none @ nat ) ) ).
% vebt_pred.simps(4)
thf(fact_1609_vebt__mint_Osimps_I3_J,axiom,
! [Mi: nat,Ma: nat,Ux: nat,Uy: list @ vEBT_VEBT,Uz: vEBT_VEBT] :
( ( vEBT_vebt_mint @ ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi @ Ma ) ) @ Ux @ Uy @ Uz ) )
= ( some @ nat @ Mi ) ) ).
% vebt_mint.simps(3)
thf(fact_1610_vebt__maxt_Osimps_I3_J,axiom,
! [Mi: nat,Ma: nat,Ux: nat,Uy: list @ vEBT_VEBT,Uz: vEBT_VEBT] :
( ( vEBT_vebt_maxt @ ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi @ Ma ) ) @ Ux @ Uy @ Uz ) )
= ( some @ nat @ Ma ) ) ).
% vebt_maxt.simps(3)
thf(fact_1611_vebt__pred_Osimps_I5_J,axiom,
! [V: product_prod @ nat @ nat,Vd3: list @ vEBT_VEBT,Ve3: vEBT_VEBT,Vf2: nat] :
( ( vEBT_vebt_pred @ ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ V ) @ ( zero_zero @ nat ) @ Vd3 @ Ve3 ) @ Vf2 )
= ( none @ nat ) ) ).
% vebt_pred.simps(5)
thf(fact_1612_vebt__mint_Osimps_I1_J,axiom,
! [A3: $o,B3: $o] :
( ( A3
=> ( ( vEBT_vebt_mint @ ( vEBT_Leaf @ A3 @ B3 ) )
= ( some @ nat @ ( zero_zero @ nat ) ) ) )
& ( ~ A3
=> ( ( B3
=> ( ( vEBT_vebt_mint @ ( vEBT_Leaf @ A3 @ B3 ) )
= ( some @ nat @ ( one_one @ nat ) ) ) )
& ( ~ B3
=> ( ( vEBT_vebt_mint @ ( vEBT_Leaf @ A3 @ B3 ) )
= ( none @ nat ) ) ) ) ) ) ).
% vebt_mint.simps(1)
thf(fact_1613_vebt__maxt_Osimps_I1_J,axiom,
! [B3: $o,A3: $o] :
( ( B3
=> ( ( vEBT_vebt_maxt @ ( vEBT_Leaf @ A3 @ B3 ) )
= ( some @ nat @ ( one_one @ nat ) ) ) )
& ( ~ B3
=> ( ( A3
=> ( ( vEBT_vebt_maxt @ ( vEBT_Leaf @ A3 @ B3 ) )
= ( some @ nat @ ( zero_zero @ nat ) ) ) )
& ( ~ A3
=> ( ( vEBT_vebt_maxt @ ( vEBT_Leaf @ A3 @ B3 ) )
= ( none @ nat ) ) ) ) ) ) ).
% vebt_maxt.simps(1)
thf(fact_1614_vebt__pred_Osimps_I6_J,axiom,
! [V: product_prod @ nat @ nat,Vh2: list @ vEBT_VEBT,Vi2: vEBT_VEBT,Vj2: nat] :
( ( vEBT_vebt_pred @ ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ V ) @ ( suc @ ( zero_zero @ nat ) ) @ Vh2 @ Vi2 ) @ Vj2 )
= ( none @ nat ) ) ).
% vebt_pred.simps(6)
thf(fact_1615_vebt__pred_Osimps_I2_J,axiom,
! [A3: $o,Uw: $o] :
( ( A3
=> ( ( vEBT_vebt_pred @ ( vEBT_Leaf @ A3 @ Uw ) @ ( suc @ ( zero_zero @ nat ) ) )
= ( some @ nat @ ( zero_zero @ nat ) ) ) )
& ( ~ A3
=> ( ( vEBT_vebt_pred @ ( vEBT_Leaf @ A3 @ Uw ) @ ( suc @ ( zero_zero @ nat ) ) )
= ( none @ nat ) ) ) ) ).
% vebt_pred.simps(2)
thf(fact_1616_linear__plus__1__le__power,axiom,
! [X2: real,N: nat] :
( ( ord_less_eq @ real @ ( zero_zero @ real ) @ X2 )
=> ( ord_less_eq @ real @ ( plus_plus @ real @ ( times_times @ real @ ( semiring_1_of_nat @ real @ N ) @ X2 ) @ ( one_one @ real ) ) @ ( power_power @ real @ ( plus_plus @ real @ X2 @ ( one_one @ real ) ) @ N ) ) ) ).
% linear_plus_1_le_power
thf(fact_1617_vebt__maxtilist,axiom,
! [I: nat,Ts: list @ vEBT_VEBT,Tsi: list @ vEBT_VEBTi] :
( ( ord_less @ nat @ I @ ( size_size @ ( list @ vEBT_VEBT ) @ Ts ) )
=> ( hoare_hoare_triple @ ( option @ nat ) @ ( vEBT_List_list_assn @ vEBT_VEBT @ vEBT_VEBTi @ vEBT_vebt_assn_raw @ Ts @ Tsi ) @ ( vEBT_vebt_maxti @ ( nth @ vEBT_VEBTi @ Tsi @ I ) )
@ ^ [R5: option @ nat] :
( times_times @ assn
@ ( pure_assn
@ ( R5
= ( vEBT_vebt_maxt @ ( nth @ vEBT_VEBT @ Ts @ I ) ) ) )
@ ( vEBT_List_list_assn @ vEBT_VEBT @ vEBT_VEBTi @ vEBT_vebt_assn_raw @ Ts @ Tsi ) ) ) ) ).
% vebt_maxtilist
thf(fact_1618_vebt__mintilist,axiom,
! [I: nat,Ts: list @ vEBT_VEBT,Tsi: list @ vEBT_VEBTi] :
( ( ord_less @ nat @ I @ ( size_size @ ( list @ vEBT_VEBT ) @ Ts ) )
=> ( hoare_hoare_triple @ ( option @ nat ) @ ( vEBT_List_list_assn @ vEBT_VEBT @ vEBT_VEBTi @ vEBT_vebt_assn_raw @ Ts @ Tsi ) @ ( vEBT_vebt_minti @ ( nth @ vEBT_VEBTi @ Tsi @ I ) )
@ ^ [R5: option @ nat] :
( times_times @ assn
@ ( pure_assn
@ ( R5
= ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ Ts @ I ) ) ) )
@ ( vEBT_List_list_assn @ vEBT_VEBT @ vEBT_VEBTi @ vEBT_vebt_assn_raw @ Ts @ Tsi ) ) ) ) ).
% vebt_mintilist
thf(fact_1619_minus__apply,axiom,
! [B: $tType,A: $tType] :
( ( minus @ B )
=> ( ( minus_minus @ ( A > B ) )
= ( ^ [A7: A > B,B6: A > B,X3: A] : ( minus_minus @ B @ ( A7 @ X3 ) @ ( B6 @ X3 ) ) ) ) ) ).
% minus_apply
thf(fact_1620_arcosh__1,axiom,
! [A: $tType] :
( ( ln @ A )
=> ( ( arcosh @ A @ ( one_one @ A ) )
= ( zero_zero @ A ) ) ) ).
% arcosh_1
thf(fact_1621_less__shift,axiom,
( ( ord_less @ nat )
= ( ^ [X3: nat,Y: nat] : ( vEBT_VEBT_less @ ( some @ nat @ X3 ) @ ( some @ nat @ Y ) ) ) ) ).
% less_shift
thf(fact_1622_greater__shift,axiom,
( ( ord_less @ nat )
= ( ^ [Y: nat,X3: nat] : ( vEBT_VEBT_greater @ ( some @ nat @ X3 ) @ ( some @ nat @ Y ) ) ) ) ).
% greater_shift
thf(fact_1623_TBOUND__vebt__maxti,axiom,
! [T2: vEBT_VEBTi] : ( time_TBOUND @ ( option @ nat ) @ ( vEBT_vebt_maxti @ T2 ) @ ( one_one @ nat ) ) ).
% TBOUND_vebt_maxti
thf(fact_1624_TBOUND__vebt__minti,axiom,
! [T2: vEBT_VEBTi] : ( time_TBOUND @ ( option @ nat ) @ ( vEBT_vebt_minti @ T2 ) @ ( one_one @ nat ) ) ).
% TBOUND_vebt_minti
thf(fact_1625_max_Oidem,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A3: A] :
( ( ord_max @ A @ A3 @ A3 )
= A3 ) ) ).
% max.idem
thf(fact_1626_max_Oleft__idem,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A3: A,B3: A] :
( ( ord_max @ A @ A3 @ ( ord_max @ A @ A3 @ B3 ) )
= ( ord_max @ A @ A3 @ B3 ) ) ) ).
% max.left_idem
thf(fact_1627_max_Oright__idem,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A3: A,B3: A] :
( ( ord_max @ A @ ( ord_max @ A @ A3 @ B3 ) @ B3 )
= ( ord_max @ A @ A3 @ B3 ) ) ) ).
% max.right_idem
thf(fact_1628_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_1629_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_1630_max_Oabsorb1,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [B3: A,A3: A] :
( ( ord_less_eq @ A @ B3 @ A3 )
=> ( ( ord_max @ A @ A3 @ B3 )
= A3 ) ) ) ).
% max.absorb1
thf(fact_1631_max_Oabsorb2,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A3: A,B3: A] :
( ( ord_less_eq @ A @ A3 @ B3 )
=> ( ( ord_max @ A @ A3 @ B3 )
= B3 ) ) ) ).
% max.absorb2
thf(fact_1632_max_Obounded__iff,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [B3: A,C3: A,A3: A] :
( ( ord_less_eq @ A @ ( ord_max @ A @ B3 @ C3 ) @ A3 )
= ( ( ord_less_eq @ A @ B3 @ A3 )
& ( ord_less_eq @ A @ C3 @ A3 ) ) ) ) ).
% max.bounded_iff
thf(fact_1633_max__less__iff__conj,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [X2: A,Y3: A,Z: A] :
( ( ord_less @ A @ ( ord_max @ A @ X2 @ Y3 ) @ Z )
= ( ( ord_less @ A @ X2 @ Z )
& ( ord_less @ A @ Y3 @ Z ) ) ) ) ).
% max_less_iff_conj
thf(fact_1634_max_Oabsorb4,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A3: A,B3: A] :
( ( ord_less @ A @ A3 @ B3 )
=> ( ( ord_max @ A @ A3 @ B3 )
= B3 ) ) ) ).
% max.absorb4
thf(fact_1635_max_Oabsorb3,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [B3: A,A3: A] :
( ( ord_less @ A @ B3 @ A3 )
=> ( ( ord_max @ A @ A3 @ B3 )
= A3 ) ) ) ).
% max.absorb3
thf(fact_1636_max_Oassoc,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A3: A,B3: A,C3: A] :
( ( ord_max @ A @ ( ord_max @ A @ A3 @ B3 ) @ C3 )
= ( ord_max @ A @ A3 @ ( ord_max @ A @ B3 @ C3 ) ) ) ) ).
% max.assoc
thf(fact_1637_max_Ocommute,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ( ( ord_max @ A )
= ( ^ [A6: A,B7: A] : ( ord_max @ A @ B7 @ A6 ) ) ) ) ).
% max.commute
thf(fact_1638_max_Oleft__commute,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [B3: A,A3: A,C3: A] :
( ( ord_max @ A @ B3 @ ( ord_max @ A @ A3 @ C3 ) )
= ( ord_max @ A @ A3 @ ( ord_max @ A @ B3 @ C3 ) ) ) ) ).
% max.left_commute
thf(fact_1639_max_Omono,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [C3: A,A3: A,D2: A,B3: A] :
( ( ord_less_eq @ A @ C3 @ A3 )
=> ( ( ord_less_eq @ A @ D2 @ B3 )
=> ( ord_less_eq @ A @ ( ord_max @ A @ C3 @ D2 ) @ ( ord_max @ A @ A3 @ B3 ) ) ) ) ) ).
% max.mono
thf(fact_1640_max_OorderE,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [B3: A,A3: A] :
( ( ord_less_eq @ A @ B3 @ A3 )
=> ( A3
= ( ord_max @ A @ A3 @ B3 ) ) ) ) ).
% max.orderE
thf(fact_1641_max_OorderI,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A3: A,B3: A] :
( ( A3
= ( ord_max @ A @ A3 @ B3 ) )
=> ( ord_less_eq @ A @ B3 @ A3 ) ) ) ).
% max.orderI
thf(fact_1642_max_OboundedE,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [B3: A,C3: A,A3: A] :
( ( ord_less_eq @ A @ ( ord_max @ A @ B3 @ C3 ) @ A3 )
=> ~ ( ( ord_less_eq @ A @ B3 @ A3 )
=> ~ ( ord_less_eq @ A @ C3 @ A3 ) ) ) ) ).
% max.boundedE
thf(fact_1643_max_OboundedI,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [B3: A,A3: A,C3: A] :
( ( ord_less_eq @ A @ B3 @ A3 )
=> ( ( ord_less_eq @ A @ C3 @ A3 )
=> ( ord_less_eq @ A @ ( ord_max @ A @ B3 @ C3 ) @ A3 ) ) ) ) ).
% max.boundedI
thf(fact_1644_max_Oorder__iff,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ( ( ord_less_eq @ A )
= ( ^ [B7: A,A6: A] :
( A6
= ( ord_max @ A @ A6 @ B7 ) ) ) ) ) ).
% max.order_iff
thf(fact_1645_max_Ocobounded1,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A3: A,B3: A] : ( ord_less_eq @ A @ A3 @ ( ord_max @ A @ A3 @ B3 ) ) ) ).
% max.cobounded1
thf(fact_1646_max_Ocobounded2,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [B3: A,A3: A] : ( ord_less_eq @ A @ B3 @ ( ord_max @ A @ A3 @ B3 ) ) ) ).
% max.cobounded2
thf(fact_1647_le__max__iff__disj,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [Z: A,X2: A,Y3: A] :
( ( ord_less_eq @ A @ Z @ ( ord_max @ A @ X2 @ Y3 ) )
= ( ( ord_less_eq @ A @ Z @ X2 )
| ( ord_less_eq @ A @ Z @ Y3 ) ) ) ) ).
% le_max_iff_disj
thf(fact_1648_max_Oabsorb__iff1,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ( ( ord_less_eq @ A )
= ( ^ [B7: A,A6: A] :
( ( ord_max @ A @ A6 @ B7 )
= A6 ) ) ) ) ).
% max.absorb_iff1
thf(fact_1649_max_Oabsorb__iff2,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ( ( ord_less_eq @ A )
= ( ^ [A6: A,B7: A] :
( ( ord_max @ A @ A6 @ B7 )
= B7 ) ) ) ) ).
% max.absorb_iff2
thf(fact_1650_max_OcoboundedI1,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [C3: A,A3: A,B3: A] :
( ( ord_less_eq @ A @ C3 @ A3 )
=> ( ord_less_eq @ A @ C3 @ ( ord_max @ A @ A3 @ B3 ) ) ) ) ).
% max.coboundedI1
thf(fact_1651_max_OcoboundedI2,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [C3: A,B3: A,A3: A] :
( ( ord_less_eq @ A @ C3 @ B3 )
=> ( ord_less_eq @ A @ C3 @ ( ord_max @ A @ A3 @ B3 ) ) ) ) ).
% max.coboundedI2
thf(fact_1652_max_Ostrict__coboundedI2,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [C3: A,B3: A,A3: A] :
( ( ord_less @ A @ C3 @ B3 )
=> ( ord_less @ A @ C3 @ ( ord_max @ A @ A3 @ B3 ) ) ) ) ).
% max.strict_coboundedI2
thf(fact_1653_max_Ostrict__coboundedI1,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [C3: A,A3: A,B3: A] :
( ( ord_less @ A @ C3 @ A3 )
=> ( ord_less @ A @ C3 @ ( ord_max @ A @ A3 @ B3 ) ) ) ) ).
% max.strict_coboundedI1
thf(fact_1654_max_Ostrict__order__iff,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ( ( ord_less @ A )
= ( ^ [B7: A,A6: A] :
( ( A6
= ( ord_max @ A @ A6 @ B7 ) )
& ( A6 != B7 ) ) ) ) ) ).
% max.strict_order_iff
thf(fact_1655_max_Ostrict__boundedE,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [B3: A,C3: A,A3: A] :
( ( ord_less @ A @ ( ord_max @ A @ B3 @ C3 ) @ A3 )
=> ~ ( ( ord_less @ A @ B3 @ A3 )
=> ~ ( ord_less @ A @ C3 @ A3 ) ) ) ) ).
% max.strict_boundedE
thf(fact_1656_less__max__iff__disj,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [Z: A,X2: A,Y3: A] :
( ( ord_less @ A @ Z @ ( ord_max @ A @ X2 @ Y3 ) )
= ( ( ord_less @ A @ Z @ X2 )
| ( ord_less @ A @ Z @ Y3 ) ) ) ) ).
% less_max_iff_disj
thf(fact_1657_fun__diff__def,axiom,
! [B: $tType,A: $tType] :
( ( minus @ B )
=> ( ( minus_minus @ ( A > B ) )
= ( ^ [A7: A > B,B6: A > B,X3: A] : ( minus_minus @ B @ ( A7 @ X3 ) @ ( B6 @ X3 ) ) ) ) ) ).
% fun_diff_def
thf(fact_1658_Bolzano,axiom,
! [A3: real,B3: real,P: real > real > $o] :
( ( ord_less_eq @ real @ A3 @ B3 )
=> ( ! [A4: real,B4: real,C4: real] :
( ( P @ A4 @ B4 )
=> ( ( P @ B4 @ C4 )
=> ( ( ord_less_eq @ real @ A4 @ B4 )
=> ( ( ord_less_eq @ real @ B4 @ C4 )
=> ( P @ A4 @ C4 ) ) ) ) )
=> ( ! [X4: real] :
( ( ord_less_eq @ real @ A3 @ X4 )
=> ( ( ord_less_eq @ real @ X4 @ B3 )
=> ? [D5: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ D5 )
& ! [A4: real,B4: real] :
( ( ( ord_less_eq @ real @ A4 @ X4 )
& ( ord_less_eq @ real @ X4 @ B4 )
& ( ord_less @ real @ ( minus_minus @ real @ B4 @ A4 ) @ D5 ) )
=> ( P @ A4 @ B4 ) ) ) ) )
=> ( P @ A3 @ B3 ) ) ) ) ).
% Bolzano
thf(fact_1659_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_1660_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_1661_T_092_060_094sub_062s_092_060_094sub_062u_092_060_094sub_062c_092_060_094sub_062c_H_Oelims,axiom,
! [X2: vEBT_VEBT,Xa: nat,Y3: nat] :
( ( ( vEBT_T_s_u_c_c2 @ X2 @ Xa )
= Y3 )
=> ( ( ? [Uu2: $o,B4: $o] :
( X2
= ( vEBT_Leaf @ Uu2 @ B4 ) )
=> ( ( Xa
= ( zero_zero @ nat ) )
=> ( Y3
!= ( one_one @ nat ) ) ) )
=> ( ( ? [Uv2: $o,Uw2: $o] :
( X2
= ( vEBT_Leaf @ Uv2 @ Uw2 ) )
=> ( ? [N3: nat] :
( Xa
= ( suc @ N3 ) )
=> ( Y3
!= ( one_one @ nat ) ) ) )
=> ( ( ? [Ux2: nat,Uy2: list @ vEBT_VEBT,Uz2: vEBT_VEBT] :
( X2
= ( vEBT_Node @ ( none @ ( product_prod @ nat @ nat ) ) @ Ux2 @ Uy2 @ Uz2 ) )
=> ( Y3
!= ( one_one @ nat ) ) )
=> ( ( ? [V3: product_prod @ nat @ nat,Vc2: list @ vEBT_VEBT,Vd2: vEBT_VEBT] :
( X2
= ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ V3 ) @ ( zero_zero @ nat ) @ Vc2 @ Vd2 ) )
=> ( Y3
!= ( one_one @ nat ) ) )
=> ( ( ? [V3: product_prod @ nat @ nat,Vg: list @ vEBT_VEBT,Vh: vEBT_VEBT] :
( X2
= ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ V3 ) @ ( suc @ ( zero_zero @ nat ) ) @ Vg @ Vh ) )
=> ( Y3
!= ( one_one @ nat ) ) )
=> ~ ! [Mi2: nat,Ma2: nat,Va3: nat,TreeList2: list @ vEBT_VEBT,Summary2: vEBT_VEBT] :
( ( X2
= ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( suc @ Va3 ) ) @ TreeList2 @ Summary2 ) )
=> ~ ( ( ( ord_less @ nat @ Xa @ Mi2 )
=> ( Y3
= ( one_one @ nat ) ) )
& ( ~ ( ord_less @ nat @ Xa @ Mi2 )
=> ( Y3
= ( if @ nat @ ( ord_less @ nat @ ( vEBT_VEBT_high @ Xa @ ( divide_divide @ nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( size_size @ ( list @ vEBT_VEBT ) @ TreeList2 ) )
@ ( if @ nat
@ ( ( ( vEBT_vebt_maxt @ ( nth @ vEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ Xa @ ( divide_divide @ nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) )
!= ( none @ nat ) )
& ( vEBT_VEBT_less @ ( some @ nat @ ( vEBT_VEBT_low @ Xa @ ( divide_divide @ nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( vEBT_vebt_maxt @ ( nth @ vEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ Xa @ ( divide_divide @ nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) ) )
@ ( plus_plus @ nat @ ( one_one @ nat ) @ ( vEBT_T_s_u_c_c2 @ ( nth @ vEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ Xa @ ( divide_divide @ nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( vEBT_VEBT_low @ Xa @ ( divide_divide @ nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) )
@ ( plus_plus @ nat @ ( vEBT_T_s_u_c_c2 @ Summary2 @ ( vEBT_VEBT_high @ Xa @ ( divide_divide @ nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( one_one @ nat ) ) )
@ ( one_one @ nat ) ) ) ) ) ) ) ) ) ) ) ) ).
% T\<^sub>s\<^sub>u\<^sub>c\<^sub>c'.elims
thf(fact_1662_T_092_060_094sub_062p_092_060_094sub_062r_092_060_094sub_062e_092_060_094sub_062d_H_Oelims,axiom,
! [X2: vEBT_VEBT,Xa: nat,Y3: nat] :
( ( ( vEBT_T_p_r_e_d2 @ X2 @ Xa )
= Y3 )
=> ( ( ? [Uu2: $o,Uv2: $o] :
( X2
= ( vEBT_Leaf @ Uu2 @ Uv2 ) )
=> ( ( Xa
= ( zero_zero @ nat ) )
=> ( Y3
!= ( one_one @ nat ) ) ) )
=> ( ( ? [A4: $o,Uw2: $o] :
( X2
= ( vEBT_Leaf @ A4 @ Uw2 ) )
=> ( ( Xa
= ( suc @ ( zero_zero @ nat ) ) )
=> ( Y3
!= ( one_one @ nat ) ) ) )
=> ( ( ? [A4: $o,B4: $o] :
( X2
= ( vEBT_Leaf @ A4 @ B4 ) )
=> ( ? [Va3: nat] :
( Xa
= ( suc @ ( suc @ Va3 ) ) )
=> ( Y3
!= ( one_one @ nat ) ) ) )
=> ( ( ? [Uy2: nat,Uz2: list @ vEBT_VEBT,Va2: vEBT_VEBT] :
( X2
= ( vEBT_Node @ ( none @ ( product_prod @ nat @ nat ) ) @ Uy2 @ Uz2 @ Va2 ) )
=> ( Y3
!= ( one_one @ nat ) ) )
=> ( ( ? [V3: product_prod @ nat @ nat,Vd2: list @ vEBT_VEBT,Ve2: vEBT_VEBT] :
( X2
= ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ V3 ) @ ( zero_zero @ nat ) @ Vd2 @ Ve2 ) )
=> ( Y3
!= ( one_one @ nat ) ) )
=> ( ( ? [V3: product_prod @ nat @ nat,Vh: list @ vEBT_VEBT,Vi: vEBT_VEBT] :
( X2
= ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ V3 ) @ ( suc @ ( zero_zero @ nat ) ) @ Vh @ Vi ) )
=> ( Y3
!= ( one_one @ nat ) ) )
=> ~ ! [Mi2: nat,Ma2: nat,Va3: nat,TreeList2: list @ vEBT_VEBT,Summary2: vEBT_VEBT] :
( ( X2
= ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( suc @ Va3 ) ) @ TreeList2 @ Summary2 ) )
=> ~ ( ( ( ord_less @ nat @ Ma2 @ Xa )
=> ( Y3
= ( one_one @ nat ) ) )
& ( ~ ( ord_less @ nat @ Ma2 @ Xa )
=> ( Y3
= ( if @ nat @ ( ord_less @ nat @ ( vEBT_VEBT_high @ Xa @ ( divide_divide @ nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( size_size @ ( list @ vEBT_VEBT ) @ TreeList2 ) )
@ ( if @ nat
@ ( ( ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ Xa @ ( divide_divide @ nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) )
!= ( none @ nat ) )
& ( vEBT_VEBT_greater @ ( some @ nat @ ( vEBT_VEBT_low @ Xa @ ( divide_divide @ nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ Xa @ ( divide_divide @ nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) ) )
@ ( plus_plus @ nat @ ( one_one @ nat ) @ ( vEBT_T_p_r_e_d2 @ ( nth @ vEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ Xa @ ( divide_divide @ nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( vEBT_VEBT_low @ Xa @ ( divide_divide @ nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) )
@ ( plus_plus @ nat @ ( vEBT_T_p_r_e_d2 @ Summary2 @ ( vEBT_VEBT_high @ Xa @ ( divide_divide @ nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( one_one @ nat ) ) )
@ ( one_one @ nat ) ) ) ) ) ) ) ) ) ) ) ) ) ).
% T\<^sub>p\<^sub>r\<^sub>e\<^sub>d'.elims
thf(fact_1663_T_092_060_094sub_062s_092_060_094sub_062u_092_060_094sub_062c_092_060_094sub_062c_H_Osimps_I6_J,axiom,
! [X2: nat,Mi: nat,Ma: nat,Va: nat,TreeList: list @ vEBT_VEBT,Summary: vEBT_VEBT] :
( ( ( ord_less @ nat @ X2 @ Mi )
=> ( ( vEBT_T_s_u_c_c2 @ ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi @ Ma ) ) @ ( suc @ ( suc @ Va ) ) @ TreeList @ Summary ) @ X2 )
= ( one_one @ nat ) ) )
& ( ~ ( ord_less @ nat @ X2 @ Mi )
=> ( ( vEBT_T_s_u_c_c2 @ ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi @ Ma ) ) @ ( suc @ ( suc @ Va ) ) @ TreeList @ Summary ) @ X2 )
= ( if @ nat @ ( ord_less @ nat @ ( vEBT_VEBT_high @ X2 @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( size_size @ ( list @ vEBT_VEBT ) @ TreeList ) )
@ ( if @ nat
@ ( ( ( vEBT_vebt_maxt @ ( nth @ vEBT_VEBT @ TreeList @ ( vEBT_VEBT_high @ X2 @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) )
!= ( none @ nat ) )
& ( vEBT_VEBT_less @ ( some @ nat @ ( vEBT_VEBT_low @ X2 @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( vEBT_vebt_maxt @ ( nth @ vEBT_VEBT @ TreeList @ ( vEBT_VEBT_high @ X2 @ ( 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 @ TreeList @ ( vEBT_VEBT_high @ X2 @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( vEBT_VEBT_low @ X2 @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) )
@ ( plus_plus @ nat @ ( vEBT_T_s_u_c_c2 @ Summary @ ( vEBT_VEBT_high @ X2 @ ( 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'.simps(6)
thf(fact_1664_T_092_060_094sub_062p_092_060_094sub_062r_092_060_094sub_062e_092_060_094sub_062d_H_Osimps_I7_J,axiom,
! [Ma: nat,X2: nat,Mi: nat,Va: nat,TreeList: list @ vEBT_VEBT,Summary: vEBT_VEBT] :
( ( ( ord_less @ nat @ Ma @ X2 )
=> ( ( vEBT_T_p_r_e_d2 @ ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi @ Ma ) ) @ ( suc @ ( suc @ Va ) ) @ TreeList @ Summary ) @ X2 )
= ( one_one @ nat ) ) )
& ( ~ ( ord_less @ nat @ Ma @ X2 )
=> ( ( vEBT_T_p_r_e_d2 @ ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi @ Ma ) ) @ ( suc @ ( suc @ Va ) ) @ TreeList @ Summary ) @ X2 )
= ( if @ nat @ ( ord_less @ nat @ ( vEBT_VEBT_high @ X2 @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( size_size @ ( list @ vEBT_VEBT ) @ TreeList ) )
@ ( if @ nat
@ ( ( ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList @ ( vEBT_VEBT_high @ X2 @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) )
!= ( none @ nat ) )
& ( vEBT_VEBT_greater @ ( some @ nat @ ( vEBT_VEBT_low @ X2 @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList @ ( vEBT_VEBT_high @ X2 @ ( 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 @ TreeList @ ( vEBT_VEBT_high @ X2 @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( vEBT_VEBT_low @ X2 @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) )
@ ( plus_plus @ nat @ ( vEBT_T_p_r_e_d2 @ Summary @ ( vEBT_VEBT_high @ X2 @ ( 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'.simps(7)
thf(fact_1665_vebt__succ_Osimps_I1_J,axiom,
! [B3: $o,Uu: $o] :
( ( B3
=> ( ( vEBT_vebt_succ @ ( vEBT_Leaf @ Uu @ B3 ) @ ( zero_zero @ nat ) )
= ( some @ nat @ ( one_one @ nat ) ) ) )
& ( ~ B3
=> ( ( vEBT_vebt_succ @ ( vEBT_Leaf @ Uu @ B3 ) @ ( zero_zero @ nat ) )
= ( none @ nat ) ) ) ) ).
% vebt_succ.simps(1)
thf(fact_1666_T_092_060_094sub_062p_092_060_094sub_062r_092_060_094sub_062e_092_060_094sub_062d_H_Osimps_I3_J,axiom,
! [A3: $o,B3: $o,Va: nat] :
( ( vEBT_T_p_r_e_d2 @ ( vEBT_Leaf @ A3 @ B3 ) @ ( suc @ ( suc @ Va ) ) )
= ( one_one @ nat ) ) ).
% T\<^sub>p\<^sub>r\<^sub>e\<^sub>d'.simps(3)
thf(fact_1667_T_092_060_094sub_062p_092_060_094sub_062r_092_060_094sub_062e_092_060_094sub_062d_H_Osimps_I1_J,axiom,
! [Uu: $o,Uv: $o] :
( ( vEBT_T_p_r_e_d2 @ ( vEBT_Leaf @ Uu @ Uv ) @ ( zero_zero @ nat ) )
= ( one_one @ nat ) ) ).
% T\<^sub>p\<^sub>r\<^sub>e\<^sub>d'.simps(1)
thf(fact_1668_T_092_060_094sub_062s_092_060_094sub_062u_092_060_094sub_062c_092_060_094sub_062c_H_Osimps_I2_J,axiom,
! [Uv: $o,Uw: $o,N: nat] :
( ( vEBT_T_s_u_c_c2 @ ( vEBT_Leaf @ Uv @ Uw ) @ ( suc @ N ) )
= ( one_one @ nat ) ) ).
% T\<^sub>s\<^sub>u\<^sub>c\<^sub>c'.simps(2)
thf(fact_1669_T_092_060_094sub_062p_092_060_094sub_062r_092_060_094sub_062e_092_060_094sub_062d_H_Osimps_I4_J,axiom,
! [Uy: nat,Uz: list @ vEBT_VEBT,Va: vEBT_VEBT,Vb: nat] :
( ( vEBT_T_p_r_e_d2 @ ( vEBT_Node @ ( none @ ( product_prod @ nat @ nat ) ) @ Uy @ Uz @ Va ) @ Vb )
= ( one_one @ nat ) ) ).
% T\<^sub>p\<^sub>r\<^sub>e\<^sub>d'.simps(4)
thf(fact_1670_T_092_060_094sub_062s_092_060_094sub_062u_092_060_094sub_062c_092_060_094sub_062c_H_Osimps_I1_J,axiom,
! [Uu: $o,B3: $o] :
( ( vEBT_T_s_u_c_c2 @ ( vEBT_Leaf @ Uu @ B3 ) @ ( zero_zero @ nat ) )
= ( one_one @ nat ) ) ).
% T\<^sub>s\<^sub>u\<^sub>c\<^sub>c'.simps(1)
thf(fact_1671_T_092_060_094sub_062s_092_060_094sub_062u_092_060_094sub_062c_092_060_094sub_062c_H_Osimps_I3_J,axiom,
! [Ux: nat,Uy: list @ vEBT_VEBT,Uz: vEBT_VEBT,Va: nat] :
( ( vEBT_T_s_u_c_c2 @ ( vEBT_Node @ ( none @ ( product_prod @ nat @ nat ) ) @ Ux @ Uy @ Uz ) @ Va )
= ( one_one @ nat ) ) ).
% T\<^sub>s\<^sub>u\<^sub>c\<^sub>c'.simps(3)
thf(fact_1672_T_092_060_094sub_062p_092_060_094sub_062r_092_060_094sub_062e_092_060_094sub_062d_H_Osimps_I2_J,axiom,
! [A3: $o,Uw: $o] :
( ( vEBT_T_p_r_e_d2 @ ( vEBT_Leaf @ A3 @ Uw ) @ ( suc @ ( zero_zero @ nat ) ) )
= ( one_one @ nat ) ) ).
% T\<^sub>p\<^sub>r\<^sub>e\<^sub>d'.simps(2)
thf(fact_1673_T_092_060_094sub_062p_092_060_094sub_062r_092_060_094sub_062e_092_060_094sub_062d_H_Osimps_I5_J,axiom,
! [V: product_prod @ nat @ nat,Vd3: list @ vEBT_VEBT,Ve3: vEBT_VEBT,Vf2: nat] :
( ( vEBT_T_p_r_e_d2 @ ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ V ) @ ( zero_zero @ nat ) @ Vd3 @ Ve3 ) @ Vf2 )
= ( one_one @ nat ) ) ).
% T\<^sub>p\<^sub>r\<^sub>e\<^sub>d'.simps(5)
thf(fact_1674_T_092_060_094sub_062s_092_060_094sub_062u_092_060_094sub_062c_092_060_094sub_062c_H_Osimps_I4_J,axiom,
! [V: product_prod @ nat @ nat,Vc: list @ vEBT_VEBT,Vd3: vEBT_VEBT,Ve3: nat] :
( ( vEBT_T_s_u_c_c2 @ ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ V ) @ ( zero_zero @ nat ) @ Vc @ Vd3 ) @ Ve3 )
= ( one_one @ nat ) ) ).
% T\<^sub>s\<^sub>u\<^sub>c\<^sub>c'.simps(4)
thf(fact_1675_T_092_060_094sub_062p_092_060_094sub_062r_092_060_094sub_062e_092_060_094sub_062d_H_Osimps_I6_J,axiom,
! [V: 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 ) @ V ) @ ( suc @ ( zero_zero @ nat ) ) @ Vh2 @ Vi2 ) @ Vj2 )
= ( one_one @ nat ) ) ).
% T\<^sub>p\<^sub>r\<^sub>e\<^sub>d'.simps(6)
thf(fact_1676_T_092_060_094sub_062s_092_060_094sub_062u_092_060_094sub_062c_092_060_094sub_062c_H_Osimps_I5_J,axiom,
! [V: 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 ) @ V ) @ ( suc @ ( zero_zero @ nat ) ) @ Vg2 @ Vh2 ) @ Vi2 )
= ( one_one @ nat ) ) ).
% T\<^sub>s\<^sub>u\<^sub>c\<^sub>c'.simps(5)
thf(fact_1677_pred__bound__height_H,axiom,
! [T2: vEBT_VEBT,N: nat,X2: nat] :
( ( vEBT_invar_vebt @ T2 @ N )
=> ( ord_less_eq @ nat @ ( vEBT_T_p_r_e_d2 @ T2 @ X2 ) @ ( plus_plus @ nat @ ( one_one @ nat ) @ ( vEBT_VEBT_height @ T2 ) ) ) ) ).
% pred_bound_height'
thf(fact_1678_succ_H__bound__height,axiom,
! [T2: vEBT_VEBT,N: nat,X2: nat] :
( ( vEBT_invar_vebt @ T2 @ N )
=> ( ord_less_eq @ nat @ ( vEBT_T_s_u_c_c2 @ T2 @ X2 ) @ ( plus_plus @ nat @ ( one_one @ nat ) @ ( vEBT_VEBT_height @ T2 ) ) ) ) ).
% succ'_bound_height
thf(fact_1679_is__succ__in__set__def,axiom,
( vEBT_is_succ_in_set
= ( ^ [Xs2: set @ nat,X3: nat,Y: nat] :
( ( member @ nat @ Y @ Xs2 )
& ( ord_less @ nat @ X3 @ Y )
& ! [Z4: nat] :
( ( member @ nat @ Z4 @ Xs2 )
=> ( ( ord_less @ nat @ X3 @ Z4 )
=> ( ord_less_eq @ nat @ Y @ Z4 ) ) ) ) ) ) ).
% is_succ_in_set_def
thf(fact_1680_pred__bound__size__univ_H,axiom,
! [T2: vEBT_VEBT,N: nat,U: real,X2: 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 @ X2 ) ) @ ( 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_1681_succ__bound__size__univ_H,axiom,
! [T2: vEBT_VEBT,N: nat,U: real,X2: 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 @ X2 ) ) @ ( 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_1682_vebt__succ_Osimps_I2_J,axiom,
! [Uv: $o,Uw: $o,N: nat] :
( ( vEBT_vebt_succ @ ( vEBT_Leaf @ Uv @ Uw ) @ ( suc @ N ) )
= ( none @ nat ) ) ).
% vebt_succ.simps(2)
thf(fact_1683_vebt__succ_Osimps_I3_J,axiom,
! [Ux: nat,Uy: list @ vEBT_VEBT,Uz: vEBT_VEBT,Va: nat] :
( ( vEBT_vebt_succ @ ( vEBT_Node @ ( none @ ( product_prod @ nat @ nat ) ) @ Ux @ Uy @ Uz ) @ Va )
= ( none @ nat ) ) ).
% vebt_succ.simps(3)
thf(fact_1684_vebt__succ_Osimps_I4_J,axiom,
! [V: product_prod @ nat @ nat,Vc: list @ vEBT_VEBT,Vd3: vEBT_VEBT,Ve3: nat] :
( ( vEBT_vebt_succ @ ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ V ) @ ( zero_zero @ nat ) @ Vc @ Vd3 ) @ Ve3 )
= ( none @ nat ) ) ).
% vebt_succ.simps(4)
thf(fact_1685_vebt__succ_Osimps_I5_J,axiom,
! [V: product_prod @ nat @ nat,Vg2: list @ vEBT_VEBT,Vh2: vEBT_VEBT,Vi2: nat] :
( ( vEBT_vebt_succ @ ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ V ) @ ( suc @ ( zero_zero @ nat ) ) @ Vg2 @ Vh2 ) @ Vi2 )
= ( none @ nat ) ) ).
% vebt_succ.simps(5)
thf(fact_1686_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_1687_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_1688_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_1689_zdiff__int__split,axiom,
! [P: int > $o,X2: nat,Y3: nat] :
( ( P @ ( semiring_1_of_nat @ int @ ( minus_minus @ nat @ X2 @ Y3 ) ) )
= ( ( ( ord_less_eq @ nat @ Y3 @ X2 )
=> ( P @ ( minus_minus @ int @ ( semiring_1_of_nat @ int @ X2 ) @ ( semiring_1_of_nat @ int @ Y3 ) ) ) )
& ( ( ord_less @ nat @ X2 @ Y3 )
=> ( P @ ( zero_zero @ int ) ) ) ) ) ).
% zdiff_int_split
thf(fact_1690_heaphelp,axiom,
! [A: $tType,Xa: 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 @ Xa @ 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 @ Xa @ Xb ) ) ) )
@ H2 )
=> ( rep_assn @ ( vEBT_vebt_assn_raw @ ( vEBT_Node @ ( none @ ( product_prod @ nat @ nat ) ) @ N @ TreeList @ Summary ) @ Xc ) @ H2 ) ) ).
% heaphelp
thf(fact_1691_nat__induct2,axiom,
! [P: nat > $o,N: nat] :
( ( P @ ( zero_zero @ nat ) )
=> ( ( P @ ( one_one @ nat ) )
=> ( ! [N3: nat] :
( ( P @ N3 )
=> ( P @ ( plus_plus @ nat @ N3 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) )
=> ( P @ N ) ) ) ) ).
% nat_induct2
thf(fact_1692_VEBT__internal_Omembermima_Oelims_I3_J,axiom,
! [X2: vEBT_VEBT,Xa: nat] :
( ~ ( vEBT_VEBT_membermima @ X2 @ Xa )
=> ( ! [Uu2: $o,Uv2: $o] :
( X2
!= ( vEBT_Leaf @ Uu2 @ Uv2 ) )
=> ( ! [Ux2: list @ vEBT_VEBT,Uy2: vEBT_VEBT] :
( X2
!= ( vEBT_Node @ ( none @ ( product_prod @ nat @ nat ) ) @ ( zero_zero @ nat ) @ Ux2 @ Uy2 ) )
=> ( ! [Mi2: nat,Ma2: nat] :
( ? [Va2: list @ vEBT_VEBT,Vb2: vEBT_VEBT] :
( X2
= ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi2 @ Ma2 ) ) @ ( zero_zero @ nat ) @ Va2 @ Vb2 ) )
=> ( ( Xa = Mi2 )
| ( Xa = Ma2 ) ) )
=> ( ! [Mi2: nat,Ma2: nat,V3: nat,TreeList2: list @ vEBT_VEBT] :
( ? [Vc2: vEBT_VEBT] :
( X2
= ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi2 @ Ma2 ) ) @ ( suc @ V3 ) @ TreeList2 @ Vc2 ) )
=> ( ( Xa = Mi2 )
| ( Xa = Ma2 )
| ( ( ( ord_less @ nat @ ( vEBT_VEBT_high @ Xa @ ( divide_divide @ nat @ ( suc @ V3 ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( size_size @ ( list @ vEBT_VEBT ) @ TreeList2 ) )
=> ( vEBT_VEBT_membermima @ ( nth @ vEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ Xa @ ( divide_divide @ nat @ ( suc @ V3 ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( vEBT_VEBT_low @ Xa @ ( divide_divide @ nat @ ( suc @ V3 ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) )
& ( ord_less @ nat @ ( vEBT_VEBT_high @ Xa @ ( divide_divide @ nat @ ( suc @ V3 ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( size_size @ ( list @ vEBT_VEBT ) @ TreeList2 ) ) ) ) )
=> ~ ! [V3: nat,TreeList2: list @ vEBT_VEBT] :
( ? [Vd2: vEBT_VEBT] :
( X2
= ( vEBT_Node @ ( none @ ( product_prod @ nat @ nat ) ) @ ( suc @ V3 ) @ TreeList2 @ Vd2 ) )
=> ( ( ( ord_less @ nat @ ( vEBT_VEBT_high @ Xa @ ( divide_divide @ nat @ ( suc @ V3 ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( size_size @ ( list @ vEBT_VEBT ) @ TreeList2 ) )
=> ( vEBT_VEBT_membermima @ ( nth @ vEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ Xa @ ( divide_divide @ nat @ ( suc @ V3 ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( vEBT_VEBT_low @ Xa @ ( divide_divide @ nat @ ( suc @ V3 ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) )
& ( ord_less @ nat @ ( vEBT_VEBT_high @ Xa @ ( divide_divide @ nat @ ( suc @ V3 ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( size_size @ ( list @ vEBT_VEBT ) @ TreeList2 ) ) ) ) ) ) ) ) ) ).
% VEBT_internal.membermima.elims(3)
thf(fact_1693_VEBT__internal_Omembermima_Oelims_I1_J,axiom,
! [X2: vEBT_VEBT,Xa: nat,Y3: $o] :
( ( ( vEBT_VEBT_membermima @ X2 @ Xa )
= Y3 )
=> ( ( ? [Uu2: $o,Uv2: $o] :
( X2
= ( vEBT_Leaf @ Uu2 @ Uv2 ) )
=> Y3 )
=> ( ( ? [Ux2: list @ vEBT_VEBT,Uy2: vEBT_VEBT] :
( X2
= ( vEBT_Node @ ( none @ ( product_prod @ nat @ nat ) ) @ ( zero_zero @ nat ) @ Ux2 @ Uy2 ) )
=> Y3 )
=> ( ! [Mi2: nat,Ma2: nat] :
( ? [Va2: list @ vEBT_VEBT,Vb2: vEBT_VEBT] :
( X2
= ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi2 @ Ma2 ) ) @ ( zero_zero @ nat ) @ Va2 @ Vb2 ) )
=> ( Y3
= ( ~ ( ( Xa = Mi2 )
| ( Xa = Ma2 ) ) ) ) )
=> ( ! [Mi2: nat,Ma2: nat,V3: nat,TreeList2: list @ vEBT_VEBT] :
( ? [Vc2: vEBT_VEBT] :
( X2
= ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi2 @ Ma2 ) ) @ ( suc @ V3 ) @ TreeList2 @ Vc2 ) )
=> ( Y3
= ( ~ ( ( Xa = Mi2 )
| ( Xa = Ma2 )
| ( ( ( ord_less @ nat @ ( vEBT_VEBT_high @ Xa @ ( divide_divide @ nat @ ( suc @ V3 ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( size_size @ ( list @ vEBT_VEBT ) @ TreeList2 ) )
=> ( vEBT_VEBT_membermima @ ( nth @ vEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ Xa @ ( divide_divide @ nat @ ( suc @ V3 ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( vEBT_VEBT_low @ Xa @ ( divide_divide @ nat @ ( suc @ V3 ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) )
& ( ord_less @ nat @ ( vEBT_VEBT_high @ Xa @ ( divide_divide @ nat @ ( suc @ V3 ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( size_size @ ( list @ vEBT_VEBT ) @ TreeList2 ) ) ) ) ) ) )
=> ~ ! [V3: nat,TreeList2: list @ vEBT_VEBT] :
( ? [Vd2: vEBT_VEBT] :
( X2
= ( vEBT_Node @ ( none @ ( product_prod @ nat @ nat ) ) @ ( suc @ V3 ) @ TreeList2 @ Vd2 ) )
=> ( Y3
= ( ~ ( ( ( ord_less @ nat @ ( vEBT_VEBT_high @ Xa @ ( divide_divide @ nat @ ( suc @ V3 ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( size_size @ ( list @ vEBT_VEBT ) @ TreeList2 ) )
=> ( vEBT_VEBT_membermima @ ( nth @ vEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ Xa @ ( divide_divide @ nat @ ( suc @ V3 ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( vEBT_VEBT_low @ Xa @ ( divide_divide @ nat @ ( suc @ V3 ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) )
& ( ord_less @ nat @ ( vEBT_VEBT_high @ Xa @ ( divide_divide @ nat @ ( suc @ V3 ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( size_size @ ( list @ vEBT_VEBT ) @ TreeList2 ) ) ) ) ) ) ) ) ) ) ) ).
% VEBT_internal.membermima.elims(1)
thf(fact_1694_buildup__nothing__in__min__max,axiom,
! [N: nat,X2: nat] :
~ ( vEBT_VEBT_membermima @ ( vEBT_vebt_buildup @ N ) @ X2 ) ).
% buildup_nothing_in_min_max
thf(fact_1695_mod__h__bot__iff_I5_J,axiom,
! [P: assn,Q: assn,H2: heap_ext @ product_unit] :
( ( rep_assn @ ( times_times @ assn @ P @ Q ) @ ( product_Pair @ ( heap_ext @ product_unit ) @ ( set @ nat ) @ H2 @ ( bot_bot @ ( set @ nat ) ) ) )
= ( ( rep_assn @ P @ ( product_Pair @ ( heap_ext @ product_unit ) @ ( set @ nat ) @ H2 @ ( bot_bot @ ( set @ nat ) ) ) )
& ( rep_assn @ Q @ ( product_Pair @ ( heap_ext @ product_unit ) @ ( set @ nat ) @ H2 @ ( bot_bot @ ( set @ nat ) ) ) ) ) ) ).
% mod_h_bot_iff(5)
thf(fact_1696_mod__pure__star__dist,axiom,
! [P: assn,B3: $o,H2: product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat )] :
( ( rep_assn @ ( times_times @ assn @ P @ ( pure_assn @ B3 ) ) @ H2 )
= ( ( rep_assn @ P @ H2 )
& B3 ) ) ).
% mod_pure_star_dist
thf(fact_1697_mod__h__bot__iff_I4_J,axiom,
! [B: $tType] :
( ( heap @ B )
=> ! [Q3: array @ B,Y3: list @ B,H2: heap_ext @ product_unit] :
~ ( rep_assn @ ( snga_assn @ B @ Q3 @ Y3 ) @ ( product_Pair @ ( heap_ext @ product_unit ) @ ( set @ nat ) @ H2 @ ( bot_bot @ ( set @ nat ) ) ) ) ) ).
% mod_h_bot_iff(4)
thf(fact_1698_ent__pure__post__iff,axiom,
! [P: assn,Q: assn,B3: $o] :
( ( entails @ P @ ( times_times @ assn @ Q @ ( pure_assn @ B3 ) ) )
= ( ! [H: product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat )] :
( ( rep_assn @ P @ H )
=> B3 )
& ( entails @ P @ Q ) ) ) ).
% ent_pure_post_iff
thf(fact_1699_ent__pure__post__iff__sng,axiom,
! [P: assn,B3: $o] :
( ( entails @ P @ ( pure_assn @ B3 ) )
= ( ! [H: product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat )] :
( ( rep_assn @ P @ H )
=> B3 )
& ( entails @ P @ ( one_one @ assn ) ) ) ) ).
% ent_pure_post_iff_sng
thf(fact_1700_mod__starE,axiom,
! [A3: assn,B3: assn,H2: product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat )] :
( ( rep_assn @ ( times_times @ assn @ A3 @ B3 ) @ H2 )
=> ~ ( ? [X_1: product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat )] : ( rep_assn @ A3 @ X_1 )
=> ! [H_2: product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat )] :
~ ( rep_assn @ B3 @ H_2 ) ) ) ).
% mod_starE
thf(fact_1701_mod__starD,axiom,
! [A2: assn,B2: assn,H2: product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat )] :
( ( rep_assn @ ( times_times @ assn @ A2 @ B2 ) @ H2 )
=> ? [H1: product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ),H22: product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat )] :
( ( rep_assn @ A2 @ H1 )
& ( rep_assn @ B2 @ H22 ) ) ) ).
% mod_starD
thf(fact_1702_ent__fwd,axiom,
! [P: assn,H2: product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ),Q: assn] :
( ( rep_assn @ P @ H2 )
=> ( ( entails @ P @ Q )
=> ( rep_assn @ Q @ H2 ) ) ) ).
% ent_fwd
thf(fact_1703_entailsD,axiom,
! [P: assn,Q: assn,H2: product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat )] :
( ( entails @ P @ Q )
=> ( ( rep_assn @ P @ H2 )
=> ( rep_assn @ Q @ H2 ) ) ) ).
% entailsD
thf(fact_1704_entailsI,axiom,
! [P: assn,Q: assn] :
( ! [H3: product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat )] :
( ( rep_assn @ P @ H3 )
=> ( rep_assn @ Q @ H3 ) )
=> ( entails @ P @ Q ) ) ).
% entailsI
thf(fact_1705_entails__def,axiom,
( entails
= ( ^ [P3: assn,Q7: assn] :
! [H: product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat )] :
( ( rep_assn @ P3 @ H )
=> ( rep_assn @ Q7 @ H ) ) ) ) ).
% entails_def
thf(fact_1706_ex__nat__less,axiom,
! [N: nat,P: nat > $o] :
( ( ? [M6: nat] :
( ( ord_less_eq @ nat @ M6 @ N )
& ( P @ M6 ) ) )
= ( ? [X3: nat] :
( ( member @ nat @ X3 @ ( set_or1337092689740270186AtMost @ nat @ ( zero_zero @ nat ) @ N ) )
& ( P @ X3 ) ) ) ) ).
% ex_nat_less
thf(fact_1707_all__nat__less,axiom,
! [N: nat,P: nat > $o] :
( ( ! [M6: nat] :
( ( ord_less_eq @ nat @ M6 @ N )
=> ( P @ M6 ) ) )
= ( ! [X3: nat] :
( ( member @ nat @ X3 @ ( set_or1337092689740270186AtMost @ nat @ ( zero_zero @ nat ) @ N ) )
=> ( P @ X3 ) ) ) ) ).
% all_nat_less
thf(fact_1708_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_1709_pinf_I1_J,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [P: A > $o,P5: A > $o,Q: A > $o,Q2: A > $o] :
( ? [Z5: A] :
! [X4: A] :
( ( ord_less @ A @ Z5 @ X4 )
=> ( ( P @ X4 )
= ( P5 @ X4 ) ) )
=> ( ? [Z5: A] :
! [X4: A] :
( ( ord_less @ A @ Z5 @ X4 )
=> ( ( Q @ X4 )
= ( Q2 @ X4 ) ) )
=> ? [Z2: A] :
! [X8: A] :
( ( ord_less @ A @ Z2 @ X8 )
=> ( ( ( P @ X8 )
& ( Q @ X8 ) )
= ( ( P5 @ X8 )
& ( Q2 @ X8 ) ) ) ) ) ) ) ).
% pinf(1)
thf(fact_1710_pinf_I2_J,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [P: A > $o,P5: A > $o,Q: A > $o,Q2: A > $o] :
( ? [Z5: A] :
! [X4: A] :
( ( ord_less @ A @ Z5 @ X4 )
=> ( ( P @ X4 )
= ( P5 @ X4 ) ) )
=> ( ? [Z5: A] :
! [X4: A] :
( ( ord_less @ A @ Z5 @ X4 )
=> ( ( Q @ X4 )
= ( Q2 @ X4 ) ) )
=> ? [Z2: A] :
! [X8: A] :
( ( ord_less @ A @ Z2 @ X8 )
=> ( ( ( P @ X8 )
| ( Q @ X8 ) )
= ( ( P5 @ X8 )
| ( Q2 @ X8 ) ) ) ) ) ) ) ).
% pinf(2)
thf(fact_1711_pinf_I3_J,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [T2: A] :
? [Z2: A] :
! [X8: A] :
( ( ord_less @ A @ Z2 @ X8 )
=> ( X8 != T2 ) ) ) ).
% pinf(3)
thf(fact_1712_pinf_I4_J,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [T2: A] :
? [Z2: A] :
! [X8: A] :
( ( ord_less @ A @ Z2 @ X8 )
=> ( X8 != T2 ) ) ) ).
% pinf(4)
thf(fact_1713_pinf_I5_J,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [T2: A] :
? [Z2: A] :
! [X8: A] :
( ( ord_less @ A @ Z2 @ X8 )
=> ~ ( ord_less @ A @ X8 @ T2 ) ) ) ).
% pinf(5)
thf(fact_1714_pinf_I7_J,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [T2: A] :
? [Z2: A] :
! [X8: A] :
( ( ord_less @ A @ Z2 @ X8 )
=> ( ord_less @ A @ T2 @ X8 ) ) ) ).
% pinf(7)
thf(fact_1715_pinf_I11_J,axiom,
! [C: $tType,D6: $tType] :
( ( ord @ C )
=> ! [F4: D6] :
? [Z2: C] :
! [X8: C] :
( ( ord_less @ C @ Z2 @ X8 )
=> ( F4 = F4 ) ) ) ).
% pinf(11)
thf(fact_1716_minf_I1_J,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [P: A > $o,P5: A > $o,Q: A > $o,Q2: A > $o] :
( ? [Z5: A] :
! [X4: A] :
( ( ord_less @ A @ X4 @ Z5 )
=> ( ( P @ X4 )
= ( P5 @ X4 ) ) )
=> ( ? [Z5: A] :
! [X4: A] :
( ( ord_less @ A @ X4 @ Z5 )
=> ( ( Q @ X4 )
= ( Q2 @ X4 ) ) )
=> ? [Z2: A] :
! [X8: A] :
( ( ord_less @ A @ X8 @ Z2 )
=> ( ( ( P @ X8 )
& ( Q @ X8 ) )
= ( ( P5 @ X8 )
& ( Q2 @ X8 ) ) ) ) ) ) ) ).
% minf(1)
thf(fact_1717_minf_I2_J,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [P: A > $o,P5: A > $o,Q: A > $o,Q2: A > $o] :
( ? [Z5: A] :
! [X4: A] :
( ( ord_less @ A @ X4 @ Z5 )
=> ( ( P @ X4 )
= ( P5 @ X4 ) ) )
=> ( ? [Z5: A] :
! [X4: A] :
( ( ord_less @ A @ X4 @ Z5 )
=> ( ( Q @ X4 )
= ( Q2 @ X4 ) ) )
=> ? [Z2: A] :
! [X8: A] :
( ( ord_less @ A @ X8 @ Z2 )
=> ( ( ( P @ X8 )
| ( Q @ X8 ) )
= ( ( P5 @ X8 )
| ( Q2 @ X8 ) ) ) ) ) ) ) ).
% minf(2)
thf(fact_1718_minf_I3_J,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [T2: A] :
? [Z2: A] :
! [X8: A] :
( ( ord_less @ A @ X8 @ Z2 )
=> ( X8 != T2 ) ) ) ).
% minf(3)
thf(fact_1719_minf_I4_J,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [T2: A] :
? [Z2: A] :
! [X8: A] :
( ( ord_less @ A @ X8 @ Z2 )
=> ( X8 != T2 ) ) ) ).
% minf(4)
thf(fact_1720_minf_I5_J,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [T2: A] :
? [Z2: A] :
! [X8: A] :
( ( ord_less @ A @ X8 @ Z2 )
=> ( ord_less @ A @ X8 @ T2 ) ) ) ).
% minf(5)
thf(fact_1721_minf_I7_J,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [T2: A] :
? [Z2: A] :
! [X8: A] :
( ( ord_less @ A @ X8 @ Z2 )
=> ~ ( ord_less @ A @ T2 @ X8 ) ) ) ).
% minf(7)
thf(fact_1722_minf_I11_J,axiom,
! [C: $tType,D6: $tType] :
( ( ord @ C )
=> ! [F4: D6] :
? [Z2: C] :
! [X8: C] :
( ( ord_less @ C @ X8 @ Z2 )
=> ( F4 = F4 ) ) ) ).
% minf(11)
thf(fact_1723_VEBT__internal_Omembermima_Osimps_I2_J,axiom,
! [Ux: list @ vEBT_VEBT,Uy: vEBT_VEBT,Uz: nat] :
~ ( vEBT_VEBT_membermima @ ( vEBT_Node @ ( none @ ( product_prod @ nat @ nat ) ) @ ( zero_zero @ nat ) @ Ux @ Uy ) @ Uz ) ).
% VEBT_internal.membermima.simps(2)
thf(fact_1724_VEBT__internal_Omembermima_Osimps_I3_J,axiom,
! [Mi: nat,Ma: nat,Va: list @ vEBT_VEBT,Vb: vEBT_VEBT,X2: nat] :
( ( vEBT_VEBT_membermima @ ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi @ Ma ) ) @ ( zero_zero @ nat ) @ Va @ Vb ) @ X2 )
= ( ( X2 = Mi )
| ( X2 = Ma ) ) ) ).
% VEBT_internal.membermima.simps(3)
thf(fact_1725_pinf_I6_J,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [T2: A] :
? [Z2: A] :
! [X8: A] :
( ( ord_less @ A @ Z2 @ X8 )
=> ~ ( ord_less_eq @ A @ X8 @ T2 ) ) ) ).
% pinf(6)
thf(fact_1726_pinf_I8_J,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [T2: A] :
? [Z2: A] :
! [X8: A] :
( ( ord_less @ A @ Z2 @ X8 )
=> ( ord_less_eq @ A @ T2 @ X8 ) ) ) ).
% pinf(8)
thf(fact_1727_minf_I6_J,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [T2: A] :
? [Z2: A] :
! [X8: A] :
( ( ord_less @ A @ X8 @ Z2 )
=> ( ord_less_eq @ A @ X8 @ T2 ) ) ) ).
% minf(6)
thf(fact_1728_minf_I8_J,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [T2: A] :
? [Z2: A] :
! [X8: A] :
( ( ord_less @ A @ X8 @ Z2 )
=> ~ ( ord_less_eq @ A @ T2 @ X8 ) ) ) ).
% minf(8)
thf(fact_1729_inf__period_I1_J,axiom,
! [A: $tType] :
( ( ( comm_ring @ A )
& ( dvd @ A ) )
=> ! [P: A > $o,D: A,Q: A > $o] :
( ! [X4: A,K2: A] :
( ( P @ X4 )
= ( P @ ( minus_minus @ A @ X4 @ ( times_times @ A @ K2 @ D ) ) ) )
=> ( ! [X4: A,K2: A] :
( ( Q @ X4 )
= ( Q @ ( minus_minus @ A @ X4 @ ( times_times @ A @ K2 @ D ) ) ) )
=> ! [X8: A,K5: A] :
( ( ( P @ X8 )
& ( Q @ X8 ) )
= ( ( P @ ( minus_minus @ A @ X8 @ ( times_times @ A @ K5 @ D ) ) )
& ( Q @ ( minus_minus @ A @ X8 @ ( times_times @ A @ K5 @ D ) ) ) ) ) ) ) ) ).
% inf_period(1)
thf(fact_1730_inf__period_I2_J,axiom,
! [A: $tType] :
( ( ( comm_ring @ A )
& ( dvd @ A ) )
=> ! [P: A > $o,D: A,Q: A > $o] :
( ! [X4: A,K2: A] :
( ( P @ X4 )
= ( P @ ( minus_minus @ A @ X4 @ ( times_times @ A @ K2 @ D ) ) ) )
=> ( ! [X4: A,K2: A] :
( ( Q @ X4 )
= ( Q @ ( minus_minus @ A @ X4 @ ( times_times @ A @ K2 @ D ) ) ) )
=> ! [X8: A,K5: A] :
( ( ( P @ X8 )
| ( Q @ X8 ) )
= ( ( P @ ( minus_minus @ A @ X8 @ ( times_times @ A @ K5 @ D ) ) )
| ( Q @ ( minus_minus @ A @ X8 @ ( times_times @ A @ K5 @ D ) ) ) ) ) ) ) ) ).
% inf_period(2)
thf(fact_1731_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,
! [X2: nat] :
( ( X2
!= ( zero_zero @ nat ) )
=> ( ( X2
!= ( suc @ ( zero_zero @ nat ) ) )
=> ~ ! [Va3: nat] :
( X2
!= ( suc @ ( suc @ Va3 ) ) ) ) ) ).
% VEBT_internal.T\<^sub>b\<^sub>u\<^sub>i\<^sub>l\<^sub>d\<^sub>u\<^sub>p.cases
thf(fact_1732_conj__le__cong,axiom,
! [X2: int,X5: int,P: $o,P5: $o] :
( ( X2 = X5 )
=> ( ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ X5 )
=> ( P = P5 ) )
=> ( ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ X2 )
& P )
= ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ X5 )
& P5 ) ) ) ) ).
% conj_le_cong
thf(fact_1733_imp__le__cong,axiom,
! [X2: int,X5: int,P: $o,P5: $o] :
( ( X2 = X5 )
=> ( ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ X5 )
=> ( P = P5 ) )
=> ( ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ X2 )
=> P )
= ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ X5 )
=> P5 ) ) ) ) ).
% imp_le_cong
thf(fact_1734_VEBT__internal_Ocnt_Osimps_I1_J,axiom,
! [A3: $o,B3: $o] :
( ( vEBT_VEBT_cnt @ ( vEBT_Leaf @ A3 @ B3 ) )
= ( one_one @ real ) ) ).
% VEBT_internal.cnt.simps(1)
thf(fact_1735_VEBT__internal_Ocnt_H_Osimps_I1_J,axiom,
! [A3: $o,B3: $o] :
( ( vEBT_VEBT_cnt2 @ ( vEBT_Leaf @ A3 @ B3 ) )
= ( one_one @ nat ) ) ).
% VEBT_internal.cnt'.simps(1)
thf(fact_1736_plusinfinity,axiom,
! [D2: int,P5: int > $o,P: int > $o] :
( ( ord_less @ int @ ( zero_zero @ int ) @ D2 )
=> ( ! [X4: int,K2: int] :
( ( P5 @ X4 )
= ( P5 @ ( minus_minus @ int @ X4 @ ( times_times @ int @ K2 @ D2 ) ) ) )
=> ( ? [Z5: int] :
! [X4: int] :
( ( ord_less @ int @ Z5 @ X4 )
=> ( ( P @ X4 )
= ( P5 @ X4 ) ) )
=> ( ? [X_12: int] : ( P5 @ X_12 )
=> ? [X_1: int] : ( P @ X_1 ) ) ) ) ) ).
% plusinfinity
thf(fact_1737_minusinfinity,axiom,
! [D2: int,P1: int > $o,P: int > $o] :
( ( ord_less @ int @ ( zero_zero @ int ) @ D2 )
=> ( ! [X4: int,K2: int] :
( ( P1 @ X4 )
= ( P1 @ ( minus_minus @ int @ X4 @ ( times_times @ int @ K2 @ D2 ) ) ) )
=> ( ? [Z5: int] :
! [X4: int] :
( ( ord_less @ int @ X4 @ Z5 )
=> ( ( P @ X4 )
= ( P1 @ X4 ) ) )
=> ( ? [X_12: int] : ( P1 @ X_12 )
=> ? [X_1: int] : ( P @ X_1 ) ) ) ) ) ).
% minusinfinity
thf(fact_1738_VEBT__internal_Omembermima_Osimps_I5_J,axiom,
! [V: nat,TreeList: list @ vEBT_VEBT,Vd3: vEBT_VEBT,X2: nat] :
( ( vEBT_VEBT_membermima @ ( vEBT_Node @ ( none @ ( product_prod @ nat @ nat ) ) @ ( suc @ V ) @ TreeList @ Vd3 ) @ X2 )
= ( ( ( ord_less @ nat @ ( vEBT_VEBT_high @ X2 @ ( divide_divide @ nat @ ( suc @ V ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( size_size @ ( list @ vEBT_VEBT ) @ TreeList ) )
=> ( vEBT_VEBT_membermima @ ( nth @ vEBT_VEBT @ TreeList @ ( vEBT_VEBT_high @ X2 @ ( divide_divide @ nat @ ( suc @ V ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( vEBT_VEBT_low @ X2 @ ( divide_divide @ nat @ ( suc @ V ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) )
& ( ord_less @ nat @ ( vEBT_VEBT_high @ X2 @ ( divide_divide @ nat @ ( suc @ V ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( size_size @ ( list @ vEBT_VEBT ) @ TreeList ) ) ) ) ).
% VEBT_internal.membermima.simps(5)
thf(fact_1739_VEBT__internal_Omembermima_Osimps_I4_J,axiom,
! [Mi: nat,Ma: nat,V: nat,TreeList: list @ vEBT_VEBT,Vc: vEBT_VEBT,X2: nat] :
( ( vEBT_VEBT_membermima @ ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi @ Ma ) ) @ ( suc @ V ) @ TreeList @ Vc ) @ X2 )
= ( ( X2 = Mi )
| ( X2 = Ma )
| ( ( ( ord_less @ nat @ ( vEBT_VEBT_high @ X2 @ ( divide_divide @ nat @ ( suc @ V ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( size_size @ ( list @ vEBT_VEBT ) @ TreeList ) )
=> ( vEBT_VEBT_membermima @ ( nth @ vEBT_VEBT @ TreeList @ ( vEBT_VEBT_high @ X2 @ ( divide_divide @ nat @ ( suc @ V ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( vEBT_VEBT_low @ X2 @ ( divide_divide @ nat @ ( suc @ V ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) )
& ( ord_less @ nat @ ( vEBT_VEBT_high @ X2 @ ( divide_divide @ nat @ ( suc @ V ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( size_size @ ( list @ vEBT_VEBT ) @ TreeList ) ) ) ) ) ).
% VEBT_internal.membermima.simps(4)
thf(fact_1740_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_1741_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_1742_incr__mult__lemma,axiom,
! [D2: int,P: int > $o,K: int] :
( ( ord_less @ int @ ( zero_zero @ int ) @ D2 )
=> ( ! [X4: int] :
( ( P @ X4 )
=> ( P @ ( plus_plus @ int @ X4 @ D2 ) ) )
=> ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ K )
=> ! [X8: int] :
( ( P @ X8 )
=> ( P @ ( plus_plus @ int @ X8 @ ( times_times @ int @ K @ D2 ) ) ) ) ) ) ) ).
% incr_mult_lemma
thf(fact_1743_decr__mult__lemma,axiom,
! [D2: int,P: int > $o,K: int] :
( ( ord_less @ int @ ( zero_zero @ int ) @ D2 )
=> ( ! [X4: int] :
( ( P @ X4 )
=> ( P @ ( minus_minus @ int @ X4 @ D2 ) ) )
=> ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ K )
=> ! [X8: int] :
( ( P @ X8 )
=> ( P @ ( minus_minus @ int @ X8 @ ( times_times @ int @ K @ D2 ) ) ) ) ) ) ) ).
% decr_mult_lemma
thf(fact_1744_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_1745_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_1746_VEBT__internal_Ospace_H_Osimps_I1_J,axiom,
! [A3: $o,B3: $o] :
( ( vEBT_VEBT_space2 @ ( vEBT_Leaf @ A3 @ B3 ) )
= ( numeral_numeral @ nat @ ( bit0 @ ( bit0 @ one2 ) ) ) ) ).
% VEBT_internal.space'.simps(1)
thf(fact_1747_VEBT__internal_Ospace_Osimps_I1_J,axiom,
! [A3: $o,B3: $o] :
( ( vEBT_VEBT_space @ ( vEBT_Leaf @ A3 @ B3 ) )
= ( numeral_numeral @ nat @ ( bit1 @ one2 ) ) ) ).
% VEBT_internal.space.simps(1)
thf(fact_1748_VEBT__internal_Omembermima_Oelims_I2_J,axiom,
! [X2: vEBT_VEBT,Xa: nat] :
( ( vEBT_VEBT_membermima @ X2 @ Xa )
=> ( ! [Mi2: nat,Ma2: nat] :
( ? [Va2: list @ vEBT_VEBT,Vb2: vEBT_VEBT] :
( X2
= ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi2 @ Ma2 ) ) @ ( zero_zero @ nat ) @ Va2 @ Vb2 ) )
=> ~ ( ( Xa = Mi2 )
| ( Xa = Ma2 ) ) )
=> ( ! [Mi2: nat,Ma2: nat,V3: nat,TreeList2: list @ vEBT_VEBT] :
( ? [Vc2: vEBT_VEBT] :
( X2
= ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi2 @ Ma2 ) ) @ ( suc @ V3 ) @ TreeList2 @ Vc2 ) )
=> ~ ( ( Xa = Mi2 )
| ( Xa = Ma2 )
| ( ( ( ord_less @ nat @ ( vEBT_VEBT_high @ Xa @ ( divide_divide @ nat @ ( suc @ V3 ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( size_size @ ( list @ vEBT_VEBT ) @ TreeList2 ) )
=> ( vEBT_VEBT_membermima @ ( nth @ vEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ Xa @ ( divide_divide @ nat @ ( suc @ V3 ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( vEBT_VEBT_low @ Xa @ ( divide_divide @ nat @ ( suc @ V3 ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) )
& ( ord_less @ nat @ ( vEBT_VEBT_high @ Xa @ ( divide_divide @ nat @ ( suc @ V3 ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( size_size @ ( list @ vEBT_VEBT ) @ TreeList2 ) ) ) ) )
=> ~ ! [V3: nat,TreeList2: list @ vEBT_VEBT] :
( ? [Vd2: vEBT_VEBT] :
( X2
= ( vEBT_Node @ ( none @ ( product_prod @ nat @ nat ) ) @ ( suc @ V3 ) @ TreeList2 @ Vd2 ) )
=> ~ ( ( ( ord_less @ nat @ ( vEBT_VEBT_high @ Xa @ ( divide_divide @ nat @ ( suc @ V3 ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( size_size @ ( list @ vEBT_VEBT ) @ TreeList2 ) )
=> ( vEBT_VEBT_membermima @ ( nth @ vEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ Xa @ ( divide_divide @ nat @ ( suc @ V3 ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( vEBT_VEBT_low @ Xa @ ( divide_divide @ nat @ ( suc @ V3 ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) )
& ( ord_less @ nat @ ( vEBT_VEBT_high @ Xa @ ( divide_divide @ nat @ ( suc @ V3 ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( size_size @ ( list @ vEBT_VEBT ) @ TreeList2 ) ) ) ) ) ) ) ).
% VEBT_internal.membermima.elims(2)
thf(fact_1749_member__valid__both__member__options,axiom,
! [Tree: vEBT_VEBT,N: nat,X2: nat] :
( ( vEBT_invar_vebt @ Tree @ N )
=> ( ( vEBT_vebt_member @ Tree @ X2 )
=> ( ( vEBT_V5719532721284313246member @ Tree @ X2 )
| ( vEBT_VEBT_membermima @ Tree @ X2 ) ) ) ) ).
% member_valid_both_member_options
thf(fact_1750_atLeastatMost__subset__iff,axiom,
! [A: $tType] :
( ( preorder @ A )
=> ! [A3: A,B3: A,C3: A,D2: A] :
( ( ord_less_eq @ ( set @ A ) @ ( set_or1337092689740270186AtMost @ A @ A3 @ B3 ) @ ( set_or1337092689740270186AtMost @ A @ C3 @ D2 ) )
= ( ~ ( ord_less_eq @ A @ A3 @ B3 )
| ( ( ord_less_eq @ A @ C3 @ A3 )
& ( ord_less_eq @ A @ B3 @ D2 ) ) ) ) ) ).
% atLeastatMost_subset_iff
thf(fact_1751_atLeastatMost__empty,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [B3: A,A3: A] :
( ( ord_less @ A @ B3 @ A3 )
=> ( ( set_or1337092689740270186AtMost @ A @ A3 @ B3 )
= ( bot_bot @ ( set @ A ) ) ) ) ) ).
% atLeastatMost_empty
thf(fact_1752_atLeastatMost__empty__iff2,axiom,
! [A: $tType] :
( ( preorder @ A )
=> ! [A3: A,B3: A] :
( ( ( bot_bot @ ( set @ A ) )
= ( set_or1337092689740270186AtMost @ A @ A3 @ B3 ) )
= ( ~ ( ord_less_eq @ A @ A3 @ B3 ) ) ) ) ).
% atLeastatMost_empty_iff2
thf(fact_1753_atLeastatMost__empty__iff,axiom,
! [A: $tType] :
( ( preorder @ A )
=> ! [A3: A,B3: A] :
( ( ( set_or1337092689740270186AtMost @ A @ A3 @ B3 )
= ( bot_bot @ ( set @ A ) ) )
= ( ~ ( ord_less_eq @ A @ A3 @ B3 ) ) ) ) ).
% atLeastatMost_empty_iff
thf(fact_1754_both__member__options__def,axiom,
( vEBT_V8194947554948674370ptions
= ( ^ [T3: vEBT_VEBT,X3: nat] :
( ( vEBT_V5719532721284313246member @ T3 @ X3 )
| ( vEBT_VEBT_membermima @ T3 @ X3 ) ) ) ) ).
% both_member_options_def
thf(fact_1755_nat__approx__posE,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [E: A] :
( ( ord_less @ A @ ( zero_zero @ A ) @ E )
=> ~ ! [N3: nat] :
~ ( ord_less @ A @ ( divide_divide @ A @ ( one_one @ A ) @ ( semiring_1_of_nat @ A @ ( suc @ N3 ) ) ) @ E ) ) ) ).
% nat_approx_posE
thf(fact_1756_buildup__nothing__in__leaf,axiom,
! [N: nat,X2: nat] :
~ ( vEBT_V5719532721284313246member @ ( vEBT_vebt_buildup @ N ) @ X2 ) ).
% buildup_nothing_in_leaf
thf(fact_1757_Icc__eq__Icc,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [L: A,H2: A,L4: A,H4: A] :
( ( ( set_or1337092689740270186AtMost @ A @ L @ H2 )
= ( set_or1337092689740270186AtMost @ A @ L4 @ H4 ) )
= ( ( ( L = L4 )
& ( H2 = H4 ) )
| ( ~ ( ord_less_eq @ A @ L @ H2 )
& ~ ( ord_less_eq @ A @ L4 @ H4 ) ) ) ) ) ).
% Icc_eq_Icc
thf(fact_1758_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_1759_aset_I2_J,axiom,
! [D: int,A2: set @ int,P: int > $o,Q: int > $o] :
( ! [X4: int] :
( ! [Xa2: int] :
( ( member @ int @ Xa2 @ ( set_or1337092689740270186AtMost @ int @ ( one_one @ int ) @ D ) )
=> ! [Xb2: int] :
( ( member @ int @ Xb2 @ A2 )
=> ( X4
!= ( minus_minus @ int @ Xb2 @ Xa2 ) ) ) )
=> ( ( P @ X4 )
=> ( P @ ( plus_plus @ int @ X4 @ D ) ) ) )
=> ( ! [X4: int] :
( ! [Xa2: int] :
( ( member @ int @ Xa2 @ ( set_or1337092689740270186AtMost @ int @ ( one_one @ int ) @ D ) )
=> ! [Xb2: int] :
( ( member @ int @ Xb2 @ A2 )
=> ( X4
!= ( minus_minus @ int @ Xb2 @ Xa2 ) ) ) )
=> ( ( Q @ X4 )
=> ( Q @ ( plus_plus @ int @ X4 @ D ) ) ) )
=> ! [X8: int] :
( ! [Xa3: int] :
( ( member @ int @ Xa3 @ ( set_or1337092689740270186AtMost @ int @ ( one_one @ int ) @ D ) )
=> ! [Xb3: int] :
( ( member @ int @ Xb3 @ A2 )
=> ( X8
!= ( minus_minus @ int @ Xb3 @ Xa3 ) ) ) )
=> ( ( ( P @ X8 )
| ( Q @ X8 ) )
=> ( ( P @ ( plus_plus @ int @ X8 @ D ) )
| ( Q @ ( plus_plus @ int @ X8 @ D ) ) ) ) ) ) ) ).
% aset(2)
thf(fact_1760_aset_I1_J,axiom,
! [D: int,A2: set @ int,P: int > $o,Q: int > $o] :
( ! [X4: int] :
( ! [Xa2: int] :
( ( member @ int @ Xa2 @ ( set_or1337092689740270186AtMost @ int @ ( one_one @ int ) @ D ) )
=> ! [Xb2: int] :
( ( member @ int @ Xb2 @ A2 )
=> ( X4
!= ( minus_minus @ int @ Xb2 @ Xa2 ) ) ) )
=> ( ( P @ X4 )
=> ( P @ ( plus_plus @ int @ X4 @ D ) ) ) )
=> ( ! [X4: int] :
( ! [Xa2: int] :
( ( member @ int @ Xa2 @ ( set_or1337092689740270186AtMost @ int @ ( one_one @ int ) @ D ) )
=> ! [Xb2: int] :
( ( member @ int @ Xb2 @ A2 )
=> ( X4
!= ( minus_minus @ int @ Xb2 @ Xa2 ) ) ) )
=> ( ( Q @ X4 )
=> ( Q @ ( plus_plus @ int @ X4 @ D ) ) ) )
=> ! [X8: int] :
( ! [Xa3: int] :
( ( member @ int @ Xa3 @ ( set_or1337092689740270186AtMost @ int @ ( one_one @ int ) @ D ) )
=> ! [Xb3: int] :
( ( member @ int @ Xb3 @ A2 )
=> ( X8
!= ( minus_minus @ int @ Xb3 @ Xa3 ) ) ) )
=> ( ( ( P @ X8 )
& ( Q @ X8 ) )
=> ( ( P @ ( plus_plus @ int @ X8 @ D ) )
& ( Q @ ( plus_plus @ int @ X8 @ D ) ) ) ) ) ) ) ).
% aset(1)
thf(fact_1761_bset_I2_J,axiom,
! [D: int,B2: set @ int,P: int > $o,Q: int > $o] :
( ! [X4: int] :
( ! [Xa2: int] :
( ( member @ int @ Xa2 @ ( set_or1337092689740270186AtMost @ int @ ( one_one @ int ) @ D ) )
=> ! [Xb2: int] :
( ( member @ int @ Xb2 @ B2 )
=> ( X4
!= ( plus_plus @ int @ Xb2 @ Xa2 ) ) ) )
=> ( ( P @ X4 )
=> ( P @ ( minus_minus @ int @ X4 @ D ) ) ) )
=> ( ! [X4: int] :
( ! [Xa2: int] :
( ( member @ int @ Xa2 @ ( set_or1337092689740270186AtMost @ int @ ( one_one @ int ) @ D ) )
=> ! [Xb2: int] :
( ( member @ int @ Xb2 @ B2 )
=> ( X4
!= ( plus_plus @ int @ Xb2 @ Xa2 ) ) ) )
=> ( ( Q @ X4 )
=> ( Q @ ( minus_minus @ int @ X4 @ D ) ) ) )
=> ! [X8: int] :
( ! [Xa3: int] :
( ( member @ int @ Xa3 @ ( set_or1337092689740270186AtMost @ int @ ( one_one @ int ) @ D ) )
=> ! [Xb3: int] :
( ( member @ int @ Xb3 @ B2 )
=> ( X8
!= ( plus_plus @ int @ Xb3 @ Xa3 ) ) ) )
=> ( ( ( P @ X8 )
| ( Q @ X8 ) )
=> ( ( P @ ( minus_minus @ int @ X8 @ D ) )
| ( Q @ ( minus_minus @ int @ X8 @ D ) ) ) ) ) ) ) ).
% bset(2)
thf(fact_1762_bset_I1_J,axiom,
! [D: int,B2: set @ int,P: int > $o,Q: int > $o] :
( ! [X4: int] :
( ! [Xa2: int] :
( ( member @ int @ Xa2 @ ( set_or1337092689740270186AtMost @ int @ ( one_one @ int ) @ D ) )
=> ! [Xb2: int] :
( ( member @ int @ Xb2 @ B2 )
=> ( X4
!= ( plus_plus @ int @ Xb2 @ Xa2 ) ) ) )
=> ( ( P @ X4 )
=> ( P @ ( minus_minus @ int @ X4 @ D ) ) ) )
=> ( ! [X4: int] :
( ! [Xa2: int] :
( ( member @ int @ Xa2 @ ( set_or1337092689740270186AtMost @ int @ ( one_one @ int ) @ D ) )
=> ! [Xb2: int] :
( ( member @ int @ Xb2 @ B2 )
=> ( X4
!= ( plus_plus @ int @ Xb2 @ Xa2 ) ) ) )
=> ( ( Q @ X4 )
=> ( Q @ ( minus_minus @ int @ X4 @ D ) ) ) )
=> ! [X8: int] :
( ! [Xa3: int] :
( ( member @ int @ Xa3 @ ( set_or1337092689740270186AtMost @ int @ ( one_one @ int ) @ D ) )
=> ! [Xb3: int] :
( ( member @ int @ Xb3 @ B2 )
=> ( X8
!= ( plus_plus @ int @ Xb3 @ Xa3 ) ) ) )
=> ( ( ( P @ X8 )
& ( Q @ X8 ) )
=> ( ( P @ ( minus_minus @ int @ X8 @ D ) )
& ( Q @ ( minus_minus @ int @ X8 @ D ) ) ) ) ) ) ) ).
% bset(1)
thf(fact_1763_VEBT__internal_Onaive__member_Osimps_I2_J,axiom,
! [Uu: option @ ( product_prod @ nat @ nat ),Uv: list @ vEBT_VEBT,Uw: vEBT_VEBT,Ux: nat] :
~ ( vEBT_V5719532721284313246member @ ( vEBT_Node @ Uu @ ( zero_zero @ nat ) @ Uv @ Uw ) @ Ux ) ).
% VEBT_internal.naive_member.simps(2)
thf(fact_1764_bounded__Max__nat,axiom,
! [P: nat > $o,X2: nat,M8: nat] :
( ( P @ X2 )
=> ( ! [X4: nat] :
( ( P @ X4 )
=> ( ord_less_eq @ nat @ X4 @ M8 ) )
=> ~ ! [M5: nat] :
( ( P @ M5 )
=> ~ ! [X8: nat] :
( ( P @ X8 )
=> ( ord_less_eq @ nat @ X8 @ M5 ) ) ) ) ) ).
% bounded_Max_nat
thf(fact_1765_fold__atLeastAtMost__nat_Ocases,axiom,
! [A: $tType,X2: product_prod @ ( nat > A > A ) @ ( product_prod @ nat @ ( product_prod @ nat @ A ) )] :
~ ! [F3: nat > A > A,A4: nat,B4: nat,Acc: A] :
( X2
!= ( product_Pair @ ( nat > A > A ) @ ( product_prod @ nat @ ( product_prod @ nat @ A ) ) @ F3 @ ( product_Pair @ nat @ ( product_prod @ nat @ A ) @ A4 @ ( product_Pair @ nat @ A @ B4 @ Acc ) ) ) ) ).
% fold_atLeastAtMost_nat.cases
thf(fact_1766_VEBT__internal_Onaive__member_Osimps_I1_J,axiom,
! [A3: $o,B3: $o,X2: nat] :
( ( vEBT_V5719532721284313246member @ ( vEBT_Leaf @ A3 @ B3 ) @ X2 )
= ( ( ( X2
= ( zero_zero @ nat ) )
=> A3 )
& ( ( X2
!= ( zero_zero @ nat ) )
=> ( ( ( X2
= ( one_one @ nat ) )
=> B3 )
& ( X2
= ( one_one @ nat ) ) ) ) ) ) ).
% VEBT_internal.naive_member.simps(1)
thf(fact_1767_aset_I7_J,axiom,
! [D: int,A2: set @ int,T2: int] :
( ( ord_less @ int @ ( zero_zero @ int ) @ D )
=> ! [X8: int] :
( ! [Xa3: int] :
( ( member @ int @ Xa3 @ ( set_or1337092689740270186AtMost @ int @ ( one_one @ int ) @ D ) )
=> ! [Xb3: int] :
( ( member @ int @ Xb3 @ A2 )
=> ( X8
!= ( minus_minus @ int @ Xb3 @ Xa3 ) ) ) )
=> ( ( ord_less @ int @ T2 @ X8 )
=> ( ord_less @ int @ T2 @ ( plus_plus @ int @ X8 @ D ) ) ) ) ) ).
% aset(7)
thf(fact_1768_aset_I5_J,axiom,
! [D: int,T2: int,A2: set @ int] :
( ( ord_less @ int @ ( zero_zero @ int ) @ D )
=> ( ( member @ int @ T2 @ A2 )
=> ! [X8: int] :
( ! [Xa3: int] :
( ( member @ int @ Xa3 @ ( set_or1337092689740270186AtMost @ int @ ( one_one @ int ) @ D ) )
=> ! [Xb3: int] :
( ( member @ int @ Xb3 @ A2 )
=> ( X8
!= ( minus_minus @ int @ Xb3 @ Xa3 ) ) ) )
=> ( ( ord_less @ int @ X8 @ T2 )
=> ( ord_less @ int @ ( plus_plus @ int @ X8 @ D ) @ T2 ) ) ) ) ) ).
% aset(5)
thf(fact_1769_aset_I4_J,axiom,
! [D: int,T2: int,A2: set @ int] :
( ( ord_less @ int @ ( zero_zero @ int ) @ D )
=> ( ( member @ int @ T2 @ A2 )
=> ! [X8: int] :
( ! [Xa3: int] :
( ( member @ int @ Xa3 @ ( set_or1337092689740270186AtMost @ int @ ( one_one @ int ) @ D ) )
=> ! [Xb3: int] :
( ( member @ int @ Xb3 @ A2 )
=> ( X8
!= ( minus_minus @ int @ Xb3 @ Xa3 ) ) ) )
=> ( ( X8 != T2 )
=> ( ( plus_plus @ int @ X8 @ D )
!= T2 ) ) ) ) ) ).
% aset(4)
thf(fact_1770_aset_I3_J,axiom,
! [D: int,T2: int,A2: set @ int] :
( ( ord_less @ int @ ( zero_zero @ int ) @ D )
=> ( ( member @ int @ ( plus_plus @ int @ T2 @ ( one_one @ int ) ) @ A2 )
=> ! [X8: int] :
( ! [Xa3: int] :
( ( member @ int @ Xa3 @ ( set_or1337092689740270186AtMost @ int @ ( one_one @ int ) @ D ) )
=> ! [Xb3: int] :
( ( member @ int @ Xb3 @ A2 )
=> ( X8
!= ( minus_minus @ int @ Xb3 @ Xa3 ) ) ) )
=> ( ( X8 = T2 )
=> ( ( plus_plus @ int @ X8 @ D )
= T2 ) ) ) ) ) ).
% aset(3)
thf(fact_1771_bset_I7_J,axiom,
! [D: int,T2: int,B2: set @ int] :
( ( ord_less @ int @ ( zero_zero @ int ) @ D )
=> ( ( member @ int @ T2 @ B2 )
=> ! [X8: int] :
( ! [Xa3: int] :
( ( member @ int @ Xa3 @ ( set_or1337092689740270186AtMost @ int @ ( one_one @ int ) @ D ) )
=> ! [Xb3: int] :
( ( member @ int @ Xb3 @ B2 )
=> ( X8
!= ( plus_plus @ int @ Xb3 @ Xa3 ) ) ) )
=> ( ( ord_less @ int @ T2 @ X8 )
=> ( ord_less @ int @ T2 @ ( minus_minus @ int @ X8 @ D ) ) ) ) ) ) ).
% bset(7)
thf(fact_1772_bset_I5_J,axiom,
! [D: int,B2: set @ int,T2: int] :
( ( ord_less @ int @ ( zero_zero @ int ) @ D )
=> ! [X8: int] :
( ! [Xa3: int] :
( ( member @ int @ Xa3 @ ( set_or1337092689740270186AtMost @ int @ ( one_one @ int ) @ D ) )
=> ! [Xb3: int] :
( ( member @ int @ Xb3 @ B2 )
=> ( X8
!= ( plus_plus @ int @ Xb3 @ Xa3 ) ) ) )
=> ( ( ord_less @ int @ X8 @ T2 )
=> ( ord_less @ int @ ( minus_minus @ int @ X8 @ D ) @ T2 ) ) ) ) ).
% bset(5)
thf(fact_1773_bset_I4_J,axiom,
! [D: int,T2: int,B2: set @ int] :
( ( ord_less @ int @ ( zero_zero @ int ) @ D )
=> ( ( member @ int @ T2 @ B2 )
=> ! [X8: int] :
( ! [Xa3: int] :
( ( member @ int @ Xa3 @ ( set_or1337092689740270186AtMost @ int @ ( one_one @ int ) @ D ) )
=> ! [Xb3: int] :
( ( member @ int @ Xb3 @ B2 )
=> ( X8
!= ( plus_plus @ int @ Xb3 @ Xa3 ) ) ) )
=> ( ( X8 != T2 )
=> ( ( minus_minus @ int @ X8 @ D )
!= T2 ) ) ) ) ) ).
% bset(4)
thf(fact_1774_bset_I3_J,axiom,
! [D: int,T2: int,B2: set @ int] :
( ( ord_less @ int @ ( zero_zero @ int ) @ D )
=> ( ( member @ int @ ( minus_minus @ int @ T2 @ ( one_one @ int ) ) @ B2 )
=> ! [X8: int] :
( ! [Xa3: int] :
( ( member @ int @ Xa3 @ ( set_or1337092689740270186AtMost @ int @ ( one_one @ int ) @ D ) )
=> ! [Xb3: int] :
( ( member @ int @ Xb3 @ B2 )
=> ( X8
!= ( plus_plus @ int @ Xb3 @ Xa3 ) ) ) )
=> ( ( X8 = T2 )
=> ( ( minus_minus @ int @ X8 @ D )
= T2 ) ) ) ) ) ).
% bset(3)
thf(fact_1775_periodic__finite__ex,axiom,
! [D2: int,P: int > $o] :
( ( ord_less @ int @ ( zero_zero @ int ) @ D2 )
=> ( ! [X4: int,K2: int] :
( ( P @ X4 )
= ( P @ ( minus_minus @ int @ X4 @ ( times_times @ int @ K2 @ D2 ) ) ) )
=> ( ( ? [X7: int] : ( P @ X7 ) )
= ( ? [X3: int] :
( ( member @ int @ X3 @ ( set_or1337092689740270186AtMost @ int @ ( one_one @ int ) @ D2 ) )
& ( P @ X3 ) ) ) ) ) ) ).
% periodic_finite_ex
thf(fact_1776_bset_I6_J,axiom,
! [D: int,B2: set @ int,T2: int] :
( ( ord_less @ int @ ( zero_zero @ int ) @ D )
=> ! [X8: int] :
( ! [Xa3: int] :
( ( member @ int @ Xa3 @ ( set_or1337092689740270186AtMost @ int @ ( one_one @ int ) @ D ) )
=> ! [Xb3: int] :
( ( member @ int @ Xb3 @ B2 )
=> ( X8
!= ( plus_plus @ int @ Xb3 @ Xa3 ) ) ) )
=> ( ( ord_less_eq @ int @ X8 @ T2 )
=> ( ord_less_eq @ int @ ( minus_minus @ int @ X8 @ D ) @ T2 ) ) ) ) ).
% bset(6)
thf(fact_1777_bset_I8_J,axiom,
! [D: int,T2: int,B2: set @ int] :
( ( ord_less @ int @ ( zero_zero @ int ) @ D )
=> ( ( member @ int @ ( minus_minus @ int @ T2 @ ( one_one @ int ) ) @ B2 )
=> ! [X8: int] :
( ! [Xa3: int] :
( ( member @ int @ Xa3 @ ( set_or1337092689740270186AtMost @ int @ ( one_one @ int ) @ D ) )
=> ! [Xb3: int] :
( ( member @ int @ Xb3 @ B2 )
=> ( X8
!= ( plus_plus @ int @ Xb3 @ Xa3 ) ) ) )
=> ( ( ord_less_eq @ int @ T2 @ X8 )
=> ( ord_less_eq @ int @ T2 @ ( minus_minus @ int @ X8 @ D ) ) ) ) ) ) ).
% bset(8)
thf(fact_1778_aset_I6_J,axiom,
! [D: int,T2: int,A2: set @ int] :
( ( ord_less @ int @ ( zero_zero @ int ) @ D )
=> ( ( member @ int @ ( plus_plus @ int @ T2 @ ( one_one @ int ) ) @ A2 )
=> ! [X8: int] :
( ! [Xa3: int] :
( ( member @ int @ Xa3 @ ( set_or1337092689740270186AtMost @ int @ ( one_one @ int ) @ D ) )
=> ! [Xb3: int] :
( ( member @ int @ Xb3 @ A2 )
=> ( X8
!= ( minus_minus @ int @ Xb3 @ Xa3 ) ) ) )
=> ( ( ord_less_eq @ int @ X8 @ T2 )
=> ( ord_less_eq @ int @ ( plus_plus @ int @ X8 @ D ) @ T2 ) ) ) ) ) ).
% aset(6)
thf(fact_1779_aset_I8_J,axiom,
! [D: int,A2: set @ int,T2: int] :
( ( ord_less @ int @ ( zero_zero @ int ) @ D )
=> ! [X8: int] :
( ! [Xa3: int] :
( ( member @ int @ Xa3 @ ( set_or1337092689740270186AtMost @ int @ ( one_one @ int ) @ D ) )
=> ! [Xb3: int] :
( ( member @ int @ Xb3 @ A2 )
=> ( X8
!= ( minus_minus @ int @ Xb3 @ Xa3 ) ) ) )
=> ( ( ord_less_eq @ int @ T2 @ X8 )
=> ( ord_less_eq @ int @ T2 @ ( plus_plus @ int @ X8 @ D ) ) ) ) ) ).
% aset(8)
thf(fact_1780_cpmi,axiom,
! [D: int,P: int > $o,P5: int > $o,B2: set @ int] :
( ( ord_less @ int @ ( zero_zero @ int ) @ D )
=> ( ? [Z5: int] :
! [X4: int] :
( ( ord_less @ int @ X4 @ Z5 )
=> ( ( P @ X4 )
= ( P5 @ X4 ) ) )
=> ( ! [X4: int] :
( ! [Xa2: int] :
( ( member @ int @ Xa2 @ ( set_or1337092689740270186AtMost @ int @ ( one_one @ int ) @ D ) )
=> ! [Xb2: int] :
( ( member @ int @ Xb2 @ B2 )
=> ( X4
!= ( plus_plus @ int @ Xb2 @ Xa2 ) ) ) )
=> ( ( P @ X4 )
=> ( P @ ( minus_minus @ int @ X4 @ D ) ) ) )
=> ( ! [X4: int,K2: int] :
( ( P5 @ X4 )
= ( P5 @ ( minus_minus @ int @ X4 @ ( times_times @ int @ K2 @ D ) ) ) )
=> ( ( ? [X7: int] : ( P @ X7 ) )
= ( ? [X3: int] :
( ( member @ int @ X3 @ ( set_or1337092689740270186AtMost @ int @ ( one_one @ int ) @ D ) )
& ( P5 @ X3 ) )
| ? [X3: int] :
( ( member @ int @ X3 @ ( set_or1337092689740270186AtMost @ int @ ( one_one @ int ) @ D ) )
& ? [Y: int] :
( ( member @ int @ Y @ B2 )
& ( P @ ( plus_plus @ int @ Y @ X3 ) ) ) ) ) ) ) ) ) ) ).
% cpmi
thf(fact_1781_cppi,axiom,
! [D: int,P: int > $o,P5: int > $o,A2: set @ int] :
( ( ord_less @ int @ ( zero_zero @ int ) @ D )
=> ( ? [Z5: int] :
! [X4: int] :
( ( ord_less @ int @ Z5 @ X4 )
=> ( ( P @ X4 )
= ( P5 @ X4 ) ) )
=> ( ! [X4: int] :
( ! [Xa2: int] :
( ( member @ int @ Xa2 @ ( set_or1337092689740270186AtMost @ int @ ( one_one @ int ) @ D ) )
=> ! [Xb2: int] :
( ( member @ int @ Xb2 @ A2 )
=> ( X4
!= ( minus_minus @ int @ Xb2 @ Xa2 ) ) ) )
=> ( ( P @ X4 )
=> ( P @ ( plus_plus @ int @ X4 @ D ) ) ) )
=> ( ! [X4: int,K2: int] :
( ( P5 @ X4 )
= ( P5 @ ( minus_minus @ int @ X4 @ ( times_times @ int @ K2 @ D ) ) ) )
=> ( ( ? [X7: int] : ( P @ X7 ) )
= ( ? [X3: int] :
( ( member @ int @ X3 @ ( set_or1337092689740270186AtMost @ int @ ( one_one @ int ) @ D ) )
& ( P5 @ X3 ) )
| ? [X3: int] :
( ( member @ int @ X3 @ ( set_or1337092689740270186AtMost @ int @ ( one_one @ int ) @ D ) )
& ? [Y: int] :
( ( member @ int @ Y @ A2 )
& ( P @ ( minus_minus @ int @ Y @ X3 ) ) ) ) ) ) ) ) ) ) ).
% cppi
thf(fact_1782_exists__least__lemma,axiom,
! [P: nat > $o] :
( ~ ( P @ ( zero_zero @ nat ) )
=> ( ? [X_12: nat] : ( P @ X_12 )
=> ? [N3: nat] :
( ~ ( P @ N3 )
& ( P @ ( suc @ N3 ) ) ) ) ) ).
% exists_least_lemma
thf(fact_1783_real__arch__simple,axiom,
! [A: $tType] :
( ( archim462609752435547400_field @ A )
=> ! [X2: A] :
? [N3: nat] : ( ord_less_eq @ A @ X2 @ ( semiring_1_of_nat @ A @ N3 ) ) ) ).
% real_arch_simple
thf(fact_1784_reals__Archimedean2,axiom,
! [A: $tType] :
( ( archim462609752435547400_field @ A )
=> ! [X2: A] :
? [N3: nat] : ( ord_less @ A @ X2 @ ( semiring_1_of_nat @ A @ N3 ) ) ) ).
% reals_Archimedean2
thf(fact_1785_ex__less__of__nat__mult,axiom,
! [A: $tType] :
( ( archim462609752435547400_field @ A )
=> ! [X2: A,Y3: A] :
( ( ord_less @ A @ ( zero_zero @ A ) @ X2 )
=> ? [N3: nat] : ( ord_less @ A @ Y3 @ ( times_times @ A @ ( semiring_1_of_nat @ A @ N3 ) @ X2 ) ) ) ) ).
% ex_less_of_nat_mult
thf(fact_1786_atLeastatMost__psubset__iff,axiom,
! [A: $tType] :
( ( preorder @ A )
=> ! [A3: A,B3: A,C3: A,D2: A] :
( ( ord_less @ ( set @ A ) @ ( set_or1337092689740270186AtMost @ A @ A3 @ B3 ) @ ( set_or1337092689740270186AtMost @ A @ C3 @ D2 ) )
= ( ( ~ ( ord_less_eq @ A @ A3 @ B3 )
| ( ( ord_less_eq @ A @ C3 @ A3 )
& ( ord_less_eq @ A @ B3 @ D2 )
& ( ( ord_less @ A @ C3 @ A3 )
| ( ord_less @ A @ B3 @ D2 ) ) ) )
& ( ord_less_eq @ A @ C3 @ D2 ) ) ) ) ).
% atLeastatMost_psubset_iff
thf(fact_1787_VEBT__internal_Onaive__member_Osimps_I3_J,axiom,
! [Uy: option @ ( product_prod @ nat @ nat ),V: nat,TreeList: list @ vEBT_VEBT,S2: vEBT_VEBT,X2: nat] :
( ( vEBT_V5719532721284313246member @ ( vEBT_Node @ Uy @ ( suc @ V ) @ TreeList @ S2 ) @ X2 )
= ( ( ( ord_less @ nat @ ( vEBT_VEBT_high @ X2 @ ( divide_divide @ nat @ ( suc @ V ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( size_size @ ( list @ vEBT_VEBT ) @ TreeList ) )
=> ( vEBT_V5719532721284313246member @ ( nth @ vEBT_VEBT @ TreeList @ ( vEBT_VEBT_high @ X2 @ ( divide_divide @ nat @ ( suc @ V ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( vEBT_VEBT_low @ X2 @ ( divide_divide @ nat @ ( suc @ V ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) )
& ( ord_less @ nat @ ( vEBT_VEBT_high @ X2 @ ( divide_divide @ nat @ ( suc @ V ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( size_size @ ( list @ vEBT_VEBT ) @ TreeList ) ) ) ) ).
% VEBT_internal.naive_member.simps(3)
thf(fact_1788_VEBT__internal_Onaive__member_Oelims_I3_J,axiom,
! [X2: vEBT_VEBT,Xa: nat] :
( ~ ( vEBT_V5719532721284313246member @ X2 @ Xa )
=> ( ! [A4: $o,B4: $o] :
( ( X2
= ( vEBT_Leaf @ A4 @ B4 ) )
=> ( ( ( Xa
= ( zero_zero @ nat ) )
=> A4 )
& ( ( Xa
!= ( zero_zero @ nat ) )
=> ( ( ( Xa
= ( one_one @ nat ) )
=> B4 )
& ( Xa
= ( one_one @ nat ) ) ) ) ) )
=> ( ! [Uu2: option @ ( product_prod @ nat @ nat ),Uv2: list @ vEBT_VEBT,Uw2: vEBT_VEBT] :
( X2
!= ( vEBT_Node @ Uu2 @ ( zero_zero @ nat ) @ Uv2 @ Uw2 ) )
=> ~ ! [Uy2: option @ ( product_prod @ nat @ nat ),V3: nat,TreeList2: list @ vEBT_VEBT] :
( ? [S3: vEBT_VEBT] :
( X2
= ( vEBT_Node @ Uy2 @ ( suc @ V3 ) @ TreeList2 @ S3 ) )
=> ( ( ( ord_less @ nat @ ( vEBT_VEBT_high @ Xa @ ( divide_divide @ nat @ ( suc @ V3 ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( size_size @ ( list @ vEBT_VEBT ) @ TreeList2 ) )
=> ( vEBT_V5719532721284313246member @ ( nth @ vEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ Xa @ ( divide_divide @ nat @ ( suc @ V3 ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( vEBT_VEBT_low @ Xa @ ( divide_divide @ nat @ ( suc @ V3 ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) )
& ( ord_less @ nat @ ( vEBT_VEBT_high @ Xa @ ( divide_divide @ nat @ ( suc @ V3 ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( size_size @ ( list @ vEBT_VEBT ) @ TreeList2 ) ) ) ) ) ) ) ).
% VEBT_internal.naive_member.elims(3)
thf(fact_1789_VEBT__internal_Onaive__member_Oelims_I2_J,axiom,
! [X2: vEBT_VEBT,Xa: nat] :
( ( vEBT_V5719532721284313246member @ X2 @ Xa )
=> ( ! [A4: $o,B4: $o] :
( ( X2
= ( vEBT_Leaf @ A4 @ B4 ) )
=> ~ ( ( ( Xa
= ( zero_zero @ nat ) )
=> A4 )
& ( ( Xa
!= ( zero_zero @ nat ) )
=> ( ( ( Xa
= ( one_one @ nat ) )
=> B4 )
& ( Xa
= ( one_one @ nat ) ) ) ) ) )
=> ~ ! [Uy2: option @ ( product_prod @ nat @ nat ),V3: nat,TreeList2: list @ vEBT_VEBT] :
( ? [S3: vEBT_VEBT] :
( X2
= ( vEBT_Node @ Uy2 @ ( suc @ V3 ) @ TreeList2 @ S3 ) )
=> ~ ( ( ( ord_less @ nat @ ( vEBT_VEBT_high @ Xa @ ( divide_divide @ nat @ ( suc @ V3 ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( size_size @ ( list @ vEBT_VEBT ) @ TreeList2 ) )
=> ( vEBT_V5719532721284313246member @ ( nth @ vEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ Xa @ ( divide_divide @ nat @ ( suc @ V3 ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( vEBT_VEBT_low @ Xa @ ( divide_divide @ nat @ ( suc @ V3 ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) )
& ( ord_less @ nat @ ( vEBT_VEBT_high @ Xa @ ( divide_divide @ nat @ ( suc @ V3 ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( size_size @ ( list @ vEBT_VEBT ) @ TreeList2 ) ) ) ) ) ) ).
% VEBT_internal.naive_member.elims(2)
thf(fact_1790_VEBT__internal_Onaive__member_Oelims_I1_J,axiom,
! [X2: vEBT_VEBT,Xa: nat,Y3: $o] :
( ( ( vEBT_V5719532721284313246member @ X2 @ Xa )
= Y3 )
=> ( ! [A4: $o,B4: $o] :
( ( X2
= ( vEBT_Leaf @ A4 @ B4 ) )
=> ( Y3
= ( ~ ( ( ( Xa
= ( zero_zero @ nat ) )
=> A4 )
& ( ( Xa
!= ( zero_zero @ nat ) )
=> ( ( ( Xa
= ( one_one @ nat ) )
=> B4 )
& ( Xa
= ( one_one @ nat ) ) ) ) ) ) ) )
=> ( ( ? [Uu2: option @ ( product_prod @ nat @ nat ),Uv2: list @ vEBT_VEBT,Uw2: vEBT_VEBT] :
( X2
= ( vEBT_Node @ Uu2 @ ( zero_zero @ nat ) @ Uv2 @ Uw2 ) )
=> Y3 )
=> ~ ! [Uy2: option @ ( product_prod @ nat @ nat ),V3: nat,TreeList2: list @ vEBT_VEBT] :
( ? [S3: vEBT_VEBT] :
( X2
= ( vEBT_Node @ Uy2 @ ( suc @ V3 ) @ TreeList2 @ S3 ) )
=> ( Y3
= ( ~ ( ( ( ord_less @ nat @ ( vEBT_VEBT_high @ Xa @ ( divide_divide @ nat @ ( suc @ V3 ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( size_size @ ( list @ vEBT_VEBT ) @ TreeList2 ) )
=> ( vEBT_V5719532721284313246member @ ( nth @ vEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ Xa @ ( divide_divide @ nat @ ( suc @ V3 ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( vEBT_VEBT_low @ Xa @ ( divide_divide @ nat @ ( suc @ V3 ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) )
& ( ord_less @ nat @ ( vEBT_VEBT_high @ Xa @ ( divide_divide @ nat @ ( suc @ V3 ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( size_size @ ( list @ vEBT_VEBT ) @ TreeList2 ) ) ) ) ) ) ) ) ) ).
% VEBT_internal.naive_member.elims(1)
thf(fact_1791_htt__def,axiom,
! [A: $tType] :
( ( time_htt @ A )
= ( ^ [P3: assn,C5: heap_Time_Heap @ A,Q7: A > assn,T3: nat] :
( ( hoare_hoare_triple @ A @ P3 @ C5 @ Q7 )
& ! [H: heap_ext @ product_unit,As: set @ nat] :
( ( rep_assn @ P3 @ ( product_Pair @ ( heap_ext @ product_unit ) @ ( set @ nat ) @ H @ As ) )
=> ( ord_less_eq @ nat @ ( time_time @ A @ C5 @ H ) @ T3 ) ) ) ) ) ).
% htt_def
thf(fact_1792_httI,axiom,
! [A: $tType,P: assn,C3: heap_Time_Heap @ A,Q: A > assn,T2: nat] :
( ( hoare_hoare_triple @ A @ P @ C3 @ Q )
=> ( ! [H3: heap_ext @ product_unit,As2: set @ nat] :
( ( rep_assn @ P @ ( product_Pair @ ( heap_ext @ product_unit ) @ ( set @ nat ) @ H3 @ As2 ) )
=> ( ord_less_eq @ nat @ ( time_time @ A @ C3 @ H3 ) @ T2 ) )
=> ( time_htt @ A @ P @ C3 @ Q @ T2 ) ) ) ).
% httI
thf(fact_1793_int__ops_I6_J,axiom,
! [A3: nat,B3: nat] :
( ( ( ord_less @ int @ ( semiring_1_of_nat @ int @ A3 ) @ ( semiring_1_of_nat @ int @ B3 ) )
=> ( ( semiring_1_of_nat @ int @ ( minus_minus @ nat @ A3 @ B3 ) )
= ( zero_zero @ int ) ) )
& ( ~ ( ord_less @ int @ ( semiring_1_of_nat @ int @ A3 ) @ ( semiring_1_of_nat @ int @ B3 ) )
=> ( ( semiring_1_of_nat @ int @ ( minus_minus @ nat @ A3 @ B3 ) )
= ( minus_minus @ int @ ( semiring_1_of_nat @ int @ A3 ) @ ( semiring_1_of_nat @ int @ B3 ) ) ) ) ) ).
% int_ops(6)
thf(fact_1794_max__bot2,axiom,
! [A: $tType] :
( ( order_bot @ A )
=> ! [X2: A] :
( ( ord_max @ A @ X2 @ ( bot_bot @ A ) )
= X2 ) ) ).
% max_bot2
thf(fact_1795_max__bot,axiom,
! [A: $tType] :
( ( order_bot @ A )
=> ! [X2: A] :
( ( ord_max @ A @ ( bot_bot @ A ) @ X2 )
= X2 ) ) ).
% max_bot
thf(fact_1796_pos__mult__pos__ge,axiom,
! [X2: int,N: int] :
( ( ord_less @ int @ ( zero_zero @ int ) @ X2 )
=> ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ N )
=> ( ord_less_eq @ int @ ( times_times @ int @ N @ ( one_one @ int ) ) @ ( times_times @ int @ N @ X2 ) ) ) ) ).
% pos_mult_pos_ge
thf(fact_1797_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_1798_order__refl,axiom,
! [A: $tType] :
( ( preorder @ A )
=> ! [X2: A] : ( ord_less_eq @ A @ X2 @ X2 ) ) ).
% order_refl
thf(fact_1799_dual__order_Orefl,axiom,
! [A: $tType] :
( ( preorder @ A )
=> ! [A3: A] : ( ord_less_eq @ A @ A3 @ A3 ) ) ).
% dual_order.refl
thf(fact_1800_verit__eq__simplify_I8_J,axiom,
! [X22: num,Y2: num] :
( ( ( bit0 @ X22 )
= ( bit0 @ Y2 ) )
= ( X22 = Y2 ) ) ).
% verit_eq_simplify(8)
thf(fact_1801_star__false__right,axiom,
! [P: assn] :
( ( times_times @ assn @ P @ ( bot_bot @ assn ) )
= ( bot_bot @ assn ) ) ).
% star_false_right
thf(fact_1802_star__false__left,axiom,
! [P: assn] :
( ( times_times @ assn @ ( bot_bot @ assn ) @ P )
= ( bot_bot @ assn ) ) ).
% star_false_left
thf(fact_1803_assn__basic__inequalities_I3_J,axiom,
( ( bot_bot @ assn )
!= ( one_one @ assn ) ) ).
% assn_basic_inequalities(3)
thf(fact_1804_predicate1I,axiom,
! [A: $tType,P: A > $o,Q: A > $o] :
( ! [X4: A] :
( ( P @ X4 )
=> ( Q @ X4 ) )
=> ( ord_less_eq @ ( A > $o ) @ P @ Q ) ) ).
% predicate1I
thf(fact_1805_ent__false__iff,axiom,
! [P: assn] :
( ( entails @ P @ ( bot_bot @ assn ) )
= ( ! [H: product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat )] :
~ ( rep_assn @ P @ H ) ) ) ).
% ent_false_iff
thf(fact_1806_snga__same__false,axiom,
! [A: $tType] :
( ( heap @ A )
=> ! [P6: array @ A,X2: list @ A,Y3: list @ A] :
( ( times_times @ assn @ ( snga_assn @ A @ P6 @ X2 ) @ ( snga_assn @ A @ P6 @ Y3 ) )
= ( bot_bot @ assn ) ) ) ).
% snga_same_false
thf(fact_1807_ceiling__numeral,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [V: num] :
( ( archimedean_ceiling @ A @ ( numeral_numeral @ A @ V ) )
= ( numeral_numeral @ int @ V ) ) ) ).
% ceiling_numeral
thf(fact_1808_ceiling__one,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ( ( archimedean_ceiling @ A @ ( one_one @ A ) )
= ( one_one @ int ) ) ) ).
% ceiling_one
thf(fact_1809_ceiling__le__zero,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [X2: A] :
( ( ord_less_eq @ int @ ( archimedean_ceiling @ A @ X2 ) @ ( zero_zero @ int ) )
= ( ord_less_eq @ A @ X2 @ ( zero_zero @ A ) ) ) ) ).
% ceiling_le_zero
thf(fact_1810_ceiling__le__numeral,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [X2: A,V: num] :
( ( ord_less_eq @ int @ ( archimedean_ceiling @ A @ X2 ) @ ( numeral_numeral @ int @ V ) )
= ( ord_less_eq @ A @ X2 @ ( numeral_numeral @ A @ V ) ) ) ) ).
% ceiling_le_numeral
thf(fact_1811_zero__less__ceiling,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [X2: A] :
( ( ord_less @ int @ ( zero_zero @ int ) @ ( archimedean_ceiling @ A @ X2 ) )
= ( ord_less @ A @ ( zero_zero @ A ) @ X2 ) ) ) ).
% zero_less_ceiling
thf(fact_1812_numeral__less__ceiling,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [V: num,X2: A] :
( ( ord_less @ int @ ( numeral_numeral @ int @ V ) @ ( archimedean_ceiling @ A @ X2 ) )
= ( ord_less @ A @ ( numeral_numeral @ A @ V ) @ X2 ) ) ) ).
% numeral_less_ceiling
thf(fact_1813_ceiling__less__one,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [X2: A] :
( ( ord_less @ int @ ( archimedean_ceiling @ A @ X2 ) @ ( one_one @ int ) )
= ( ord_less_eq @ A @ X2 @ ( zero_zero @ A ) ) ) ) ).
% ceiling_less_one
thf(fact_1814_one__le__ceiling,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [X2: A] :
( ( ord_less_eq @ int @ ( one_one @ int ) @ ( archimedean_ceiling @ A @ X2 ) )
= ( ord_less @ A @ ( zero_zero @ A ) @ X2 ) ) ) ).
% one_le_ceiling
thf(fact_1815_ceiling__add__numeral,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [X2: A,V: num] :
( ( archimedean_ceiling @ A @ ( plus_plus @ A @ X2 @ ( numeral_numeral @ A @ V ) ) )
= ( plus_plus @ int @ ( archimedean_ceiling @ A @ X2 ) @ ( numeral_numeral @ int @ V ) ) ) ) ).
% ceiling_add_numeral
thf(fact_1816_ceiling__le__one,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [X2: A] :
( ( ord_less_eq @ int @ ( archimedean_ceiling @ A @ X2 ) @ ( one_one @ int ) )
= ( ord_less_eq @ A @ X2 @ ( one_one @ A ) ) ) ) ).
% ceiling_le_one
thf(fact_1817_one__less__ceiling,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [X2: A] :
( ( ord_less @ int @ ( one_one @ int ) @ ( archimedean_ceiling @ A @ X2 ) )
= ( ord_less @ A @ ( one_one @ A ) @ X2 ) ) ) ).
% one_less_ceiling
thf(fact_1818_ceiling__add__one,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [X2: A] :
( ( archimedean_ceiling @ A @ ( plus_plus @ A @ X2 @ ( one_one @ A ) ) )
= ( plus_plus @ int @ ( archimedean_ceiling @ A @ X2 ) @ ( one_one @ int ) ) ) ) ).
% ceiling_add_one
thf(fact_1819_ceiling__diff__numeral,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [X2: A,V: num] :
( ( archimedean_ceiling @ A @ ( minus_minus @ A @ X2 @ ( numeral_numeral @ A @ V ) ) )
= ( minus_minus @ int @ ( archimedean_ceiling @ A @ X2 ) @ ( numeral_numeral @ int @ V ) ) ) ) ).
% ceiling_diff_numeral
thf(fact_1820_ceiling__numeral__power,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [X2: num,N: nat] :
( ( archimedean_ceiling @ A @ ( power_power @ A @ ( numeral_numeral @ A @ X2 ) @ N ) )
= ( power_power @ int @ ( numeral_numeral @ int @ X2 ) @ N ) ) ) ).
% ceiling_numeral_power
thf(fact_1821_ceiling__diff__one,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [X2: A] :
( ( archimedean_ceiling @ A @ ( minus_minus @ A @ X2 @ ( one_one @ A ) ) )
= ( minus_minus @ int @ ( archimedean_ceiling @ A @ X2 ) @ ( one_one @ int ) ) ) ) ).
% ceiling_diff_one
thf(fact_1822_ceiling__less__numeral,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [X2: A,V: num] :
( ( ord_less @ int @ ( archimedean_ceiling @ A @ X2 ) @ ( numeral_numeral @ int @ V ) )
= ( ord_less_eq @ A @ X2 @ ( minus_minus @ A @ ( numeral_numeral @ A @ V ) @ ( one_one @ A ) ) ) ) ) ).
% ceiling_less_numeral
thf(fact_1823_numeral__le__ceiling,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [V: num,X2: A] :
( ( ord_less_eq @ int @ ( numeral_numeral @ int @ V ) @ ( archimedean_ceiling @ A @ X2 ) )
= ( ord_less @ A @ ( minus_minus @ A @ ( numeral_numeral @ A @ V ) @ ( one_one @ A ) ) @ X2 ) ) ) ).
% numeral_le_ceiling
thf(fact_1824_bot__nat__def,axiom,
( ( bot_bot @ nat )
= ( zero_zero @ nat ) ) ).
% bot_nat_def
thf(fact_1825_bot__option__def,axiom,
! [A: $tType] :
( ( order @ A )
=> ( ( bot_bot @ ( option @ A ) )
= ( none @ A ) ) ) ).
% bot_option_def
thf(fact_1826_ent__false,axiom,
! [P: assn] : ( entails @ ( bot_bot @ assn ) @ P ) ).
% ent_false
thf(fact_1827_bot__enat__def,axiom,
( ( bot_bot @ extended_enat )
= ( zero_zero @ extended_enat ) ) ).
% bot_enat_def
thf(fact_1828_ceiling__mono,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [Y3: A,X2: A] :
( ( ord_less_eq @ A @ Y3 @ X2 )
=> ( ord_less_eq @ int @ ( archimedean_ceiling @ A @ Y3 ) @ ( archimedean_ceiling @ A @ X2 ) ) ) ) ).
% ceiling_mono
thf(fact_1829_ceiling__less__cancel,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [X2: A,Y3: A] :
( ( ord_less @ int @ ( archimedean_ceiling @ A @ X2 ) @ ( archimedean_ceiling @ A @ Y3 ) )
=> ( ord_less @ A @ X2 @ Y3 ) ) ) ).
% ceiling_less_cancel
thf(fact_1830_ceiling__add__le,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [X2: A,Y3: A] : ( ord_less_eq @ int @ ( archimedean_ceiling @ A @ ( plus_plus @ A @ X2 @ Y3 ) ) @ ( plus_plus @ int @ ( archimedean_ceiling @ A @ X2 ) @ ( archimedean_ceiling @ A @ Y3 ) ) ) ) ).
% ceiling_add_le
thf(fact_1831_verit__comp__simplify1_I2_J,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [A3: A] : ( ord_less_eq @ A @ A3 @ A3 ) ) ).
% verit_comp_simplify1(2)
thf(fact_1832_nle__le,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A3: A,B3: A] :
( ( ~ ( ord_less_eq @ A @ A3 @ B3 ) )
= ( ( ord_less_eq @ A @ B3 @ A3 )
& ( B3 != A3 ) ) ) ) ).
% nle_le
thf(fact_1833_le__cases3,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [X2: A,Y3: A,Z: A] :
( ( ( ord_less_eq @ A @ X2 @ Y3 )
=> ~ ( ord_less_eq @ A @ Y3 @ Z ) )
=> ( ( ( ord_less_eq @ A @ Y3 @ X2 )
=> ~ ( ord_less_eq @ A @ X2 @ Z ) )
=> ( ( ( ord_less_eq @ A @ X2 @ Z )
=> ~ ( ord_less_eq @ A @ Z @ Y3 ) )
=> ( ( ( ord_less_eq @ A @ Z @ Y3 )
=> ~ ( ord_less_eq @ A @ Y3 @ X2 ) )
=> ( ( ( ord_less_eq @ A @ Y3 @ Z )
=> ~ ( ord_less_eq @ A @ Z @ X2 ) )
=> ~ ( ( ord_less_eq @ A @ Z @ X2 )
=> ~ ( ord_less_eq @ A @ X2 @ Y3 ) ) ) ) ) ) ) ) ).
% le_cases3
thf(fact_1834_order__class_Oorder__eq__iff,axiom,
! [A: $tType] :
( ( order @ A )
=> ( ( ^ [Y6: A,Z3: A] : Y6 = Z3 )
= ( ^ [X3: A,Y: A] :
( ( ord_less_eq @ A @ X3 @ Y )
& ( ord_less_eq @ A @ Y @ X3 ) ) ) ) ) ).
% order_class.order_eq_iff
thf(fact_1835_ord__eq__le__trans,axiom,
! [A: $tType] :
( ( ord @ A )
=> ! [A3: A,B3: A,C3: A] :
( ( A3 = B3 )
=> ( ( ord_less_eq @ A @ B3 @ C3 )
=> ( ord_less_eq @ A @ A3 @ C3 ) ) ) ) ).
% ord_eq_le_trans
thf(fact_1836_ord__le__eq__trans,axiom,
! [A: $tType] :
( ( ord @ A )
=> ! [A3: A,B3: A,C3: A] :
( ( ord_less_eq @ A @ A3 @ B3 )
=> ( ( B3 = C3 )
=> ( ord_less_eq @ A @ A3 @ C3 ) ) ) ) ).
% ord_le_eq_trans
thf(fact_1837_order__antisym,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [X2: A,Y3: A] :
( ( ord_less_eq @ A @ X2 @ Y3 )
=> ( ( ord_less_eq @ A @ Y3 @ X2 )
=> ( X2 = Y3 ) ) ) ) ).
% order_antisym
thf(fact_1838_order_Otrans,axiom,
! [A: $tType] :
( ( preorder @ A )
=> ! [A3: A,B3: A,C3: A] :
( ( ord_less_eq @ A @ A3 @ B3 )
=> ( ( ord_less_eq @ A @ B3 @ C3 )
=> ( ord_less_eq @ A @ A3 @ C3 ) ) ) ) ).
% order.trans
thf(fact_1839_order__trans,axiom,
! [A: $tType] :
( ( preorder @ A )
=> ! [X2: A,Y3: A,Z: A] :
( ( ord_less_eq @ A @ X2 @ Y3 )
=> ( ( ord_less_eq @ A @ Y3 @ Z )
=> ( ord_less_eq @ A @ X2 @ Z ) ) ) ) ).
% order_trans
thf(fact_1840_linorder__wlog,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [P: A > A > $o,A3: A,B3: A] :
( ! [A4: A,B4: A] :
( ( ord_less_eq @ A @ A4 @ B4 )
=> ( P @ A4 @ B4 ) )
=> ( ! [A4: A,B4: A] :
( ( P @ B4 @ A4 )
=> ( P @ A4 @ B4 ) )
=> ( P @ A3 @ B3 ) ) ) ) ).
% linorder_wlog
thf(fact_1841_dual__order_Oeq__iff,axiom,
! [A: $tType] :
( ( order @ A )
=> ( ( ^ [Y6: A,Z3: A] : Y6 = Z3 )
= ( ^ [A6: A,B7: A] :
( ( ord_less_eq @ A @ B7 @ A6 )
& ( ord_less_eq @ A @ A6 @ B7 ) ) ) ) ) ).
% dual_order.eq_iff
thf(fact_1842_dual__order_Oantisym,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [B3: A,A3: A] :
( ( ord_less_eq @ A @ B3 @ A3 )
=> ( ( ord_less_eq @ A @ A3 @ B3 )
=> ( A3 = B3 ) ) ) ) ).
% dual_order.antisym
thf(fact_1843_dual__order_Otrans,axiom,
! [A: $tType] :
( ( preorder @ A )
=> ! [B3: A,A3: A,C3: A] :
( ( ord_less_eq @ A @ B3 @ A3 )
=> ( ( ord_less_eq @ A @ C3 @ B3 )
=> ( ord_less_eq @ A @ C3 @ A3 ) ) ) ) ).
% dual_order.trans
thf(fact_1844_antisym,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [A3: A,B3: A] :
( ( ord_less_eq @ A @ A3 @ B3 )
=> ( ( ord_less_eq @ A @ B3 @ A3 )
=> ( A3 = B3 ) ) ) ) ).
% antisym
thf(fact_1845_Orderings_Oorder__eq__iff,axiom,
! [A: $tType] :
( ( order @ A )
=> ( ( ^ [Y6: A,Z3: A] : Y6 = Z3 )
= ( ^ [A6: A,B7: A] :
( ( ord_less_eq @ A @ A6 @ B7 )
& ( ord_less_eq @ A @ B7 @ A6 ) ) ) ) ) ).
% Orderings.order_eq_iff
thf(fact_1846_order__subst1,axiom,
! [A: $tType,B: $tType] :
( ( ( order @ B )
& ( order @ A ) )
=> ! [A3: A,F2: B > A,B3: B,C3: B] :
( ( ord_less_eq @ A @ A3 @ ( F2 @ B3 ) )
=> ( ( ord_less_eq @ B @ B3 @ C3 )
=> ( ! [X4: B,Y4: B] :
( ( ord_less_eq @ B @ X4 @ Y4 )
=> ( ord_less_eq @ A @ ( F2 @ X4 ) @ ( F2 @ Y4 ) ) )
=> ( ord_less_eq @ A @ A3 @ ( F2 @ C3 ) ) ) ) ) ) ).
% order_subst1
thf(fact_1847_order__subst2,axiom,
! [A: $tType,C: $tType] :
( ( ( order @ C )
& ( order @ A ) )
=> ! [A3: A,B3: A,F2: A > C,C3: C] :
( ( ord_less_eq @ A @ A3 @ B3 )
=> ( ( ord_less_eq @ C @ ( F2 @ B3 ) @ C3 )
=> ( ! [X4: A,Y4: A] :
( ( ord_less_eq @ A @ X4 @ Y4 )
=> ( ord_less_eq @ C @ ( F2 @ X4 ) @ ( F2 @ Y4 ) ) )
=> ( ord_less_eq @ C @ ( F2 @ A3 ) @ C3 ) ) ) ) ) ).
% order_subst2
thf(fact_1848_order__eq__refl,axiom,
! [A: $tType] :
( ( preorder @ A )
=> ! [X2: A,Y3: A] :
( ( X2 = Y3 )
=> ( ord_less_eq @ A @ X2 @ Y3 ) ) ) ).
% order_eq_refl
thf(fact_1849_verit__la__disequality,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A3: A,B3: A] :
( ( A3 = B3 )
| ~ ( ord_less_eq @ A @ A3 @ B3 )
| ~ ( ord_less_eq @ A @ B3 @ A3 ) ) ) ).
% verit_la_disequality
thf(fact_1850_linorder__linear,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [X2: A,Y3: A] :
( ( ord_less_eq @ A @ X2 @ Y3 )
| ( ord_less_eq @ A @ Y3 @ X2 ) ) ) ).
% linorder_linear
thf(fact_1851_ord__eq__le__subst,axiom,
! [A: $tType,B: $tType] :
( ( ( ord @ B )
& ( ord @ A ) )
=> ! [A3: A,F2: B > A,B3: B,C3: B] :
( ( A3
= ( F2 @ B3 ) )
=> ( ( ord_less_eq @ B @ B3 @ C3 )
=> ( ! [X4: B,Y4: B] :
( ( ord_less_eq @ B @ X4 @ Y4 )
=> ( ord_less_eq @ A @ ( F2 @ X4 ) @ ( F2 @ Y4 ) ) )
=> ( ord_less_eq @ A @ A3 @ ( F2 @ C3 ) ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_1852_ord__le__eq__subst,axiom,
! [A: $tType,B: $tType] :
( ( ( ord @ B )
& ( ord @ A ) )
=> ! [A3: A,B3: A,F2: A > B,C3: B] :
( ( ord_less_eq @ A @ A3 @ B3 )
=> ( ( ( F2 @ B3 )
= C3 )
=> ( ! [X4: A,Y4: A] :
( ( ord_less_eq @ A @ X4 @ Y4 )
=> ( ord_less_eq @ B @ ( F2 @ X4 ) @ ( F2 @ Y4 ) ) )
=> ( ord_less_eq @ B @ ( F2 @ A3 ) @ C3 ) ) ) ) ) ).
% ord_le_eq_subst
thf(fact_1853_linorder__le__cases,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [X2: A,Y3: A] :
( ~ ( ord_less_eq @ A @ X2 @ Y3 )
=> ( ord_less_eq @ A @ Y3 @ X2 ) ) ) ).
% linorder_le_cases
thf(fact_1854_order__antisym__conv,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [Y3: A,X2: A] :
( ( ord_less_eq @ A @ Y3 @ X2 )
=> ( ( ord_less_eq @ A @ X2 @ Y3 )
= ( X2 = Y3 ) ) ) ) ).
% order_antisym_conv
thf(fact_1855_order__less__imp__not__less,axiom,
! [A: $tType] :
( ( preorder @ A )
=> ! [X2: A,Y3: A] :
( ( ord_less @ A @ X2 @ Y3 )
=> ~ ( ord_less @ A @ Y3 @ X2 ) ) ) ).
% order_less_imp_not_less
thf(fact_1856_order__less__imp__not__eq2,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [X2: A,Y3: A] :
( ( ord_less @ A @ X2 @ Y3 )
=> ( Y3 != X2 ) ) ) ).
% order_less_imp_not_eq2
thf(fact_1857_order__less__imp__not__eq,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [X2: A,Y3: A] :
( ( ord_less @ A @ X2 @ Y3 )
=> ( X2 != Y3 ) ) ) ).
% order_less_imp_not_eq
thf(fact_1858_linorder__less__linear,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [X2: A,Y3: A] :
( ( ord_less @ A @ X2 @ Y3 )
| ( X2 = Y3 )
| ( ord_less @ A @ Y3 @ X2 ) ) ) ).
% linorder_less_linear
thf(fact_1859_order__less__imp__triv,axiom,
! [A: $tType] :
( ( preorder @ A )
=> ! [X2: A,Y3: A,P: $o] :
( ( ord_less @ A @ X2 @ Y3 )
=> ( ( ord_less @ A @ Y3 @ X2 )
=> P ) ) ) ).
% order_less_imp_triv
thf(fact_1860_order__less__not__sym,axiom,
! [A: $tType] :
( ( preorder @ A )
=> ! [X2: A,Y3: A] :
( ( ord_less @ A @ X2 @ Y3 )
=> ~ ( ord_less @ A @ Y3 @ X2 ) ) ) ).
% order_less_not_sym
thf(fact_1861_order__less__subst2,axiom,
! [A: $tType,C: $tType] :
( ( ( order @ C )
& ( order @ A ) )
=> ! [A3: A,B3: A,F2: A > C,C3: C] :
( ( ord_less @ A @ A3 @ B3 )
=> ( ( ord_less @ C @ ( F2 @ B3 ) @ C3 )
=> ( ! [X4: A,Y4: A] :
( ( ord_less @ A @ X4 @ Y4 )
=> ( ord_less @ C @ ( F2 @ X4 ) @ ( F2 @ Y4 ) ) )
=> ( ord_less @ C @ ( F2 @ A3 ) @ C3 ) ) ) ) ) ).
% order_less_subst2
thf(fact_1862_order__less__subst1,axiom,
! [A: $tType,B: $tType] :
( ( ( order @ B )
& ( order @ A ) )
=> ! [A3: A,F2: B > A,B3: B,C3: B] :
( ( ord_less @ A @ A3 @ ( F2 @ B3 ) )
=> ( ( ord_less @ B @ B3 @ C3 )
=> ( ! [X4: B,Y4: B] :
( ( ord_less @ B @ X4 @ Y4 )
=> ( ord_less @ A @ ( F2 @ X4 ) @ ( F2 @ Y4 ) ) )
=> ( ord_less @ A @ A3 @ ( F2 @ C3 ) ) ) ) ) ) ).
% order_less_subst1
thf(fact_1863_order__less__irrefl,axiom,
! [A: $tType] :
( ( preorder @ A )
=> ! [X2: A] :
~ ( ord_less @ A @ X2 @ X2 ) ) ).
% order_less_irrefl
thf(fact_1864_ord__less__eq__subst,axiom,
! [A: $tType,B: $tType] :
( ( ( ord @ B )
& ( ord @ A ) )
=> ! [A3: A,B3: A,F2: A > B,C3: B] :
( ( ord_less @ A @ A3 @ B3 )
=> ( ( ( F2 @ B3 )
= C3 )
=> ( ! [X4: A,Y4: A] :
( ( ord_less @ A @ X4 @ Y4 )
=> ( ord_less @ B @ ( F2 @ X4 ) @ ( F2 @ Y4 ) ) )
=> ( ord_less @ B @ ( F2 @ A3 ) @ C3 ) ) ) ) ) ).
% ord_less_eq_subst
thf(fact_1865_ord__eq__less__subst,axiom,
! [A: $tType,B: $tType] :
( ( ( ord @ B )
& ( ord @ A ) )
=> ! [A3: A,F2: B > A,B3: B,C3: B] :
( ( A3
= ( F2 @ B3 ) )
=> ( ( ord_less @ B @ B3 @ C3 )
=> ( ! [X4: B,Y4: B] :
( ( ord_less @ B @ X4 @ Y4 )
=> ( ord_less @ A @ ( F2 @ X4 ) @ ( F2 @ Y4 ) ) )
=> ( ord_less @ A @ A3 @ ( F2 @ C3 ) ) ) ) ) ) ).
% ord_eq_less_subst
thf(fact_1866_order__less__trans,axiom,
! [A: $tType] :
( ( preorder @ A )
=> ! [X2: A,Y3: A,Z: A] :
( ( ord_less @ A @ X2 @ Y3 )
=> ( ( ord_less @ A @ Y3 @ Z )
=> ( ord_less @ A @ X2 @ Z ) ) ) ) ).
% order_less_trans
thf(fact_1867_order__less__asym_H,axiom,
! [A: $tType] :
( ( preorder @ A )
=> ! [A3: A,B3: A] :
( ( ord_less @ A @ A3 @ B3 )
=> ~ ( ord_less @ A @ B3 @ A3 ) ) ) ).
% order_less_asym'
thf(fact_1868_linorder__neq__iff,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [X2: A,Y3: A] :
( ( X2 != Y3 )
= ( ( ord_less @ A @ X2 @ Y3 )
| ( ord_less @ A @ Y3 @ X2 ) ) ) ) ).
% linorder_neq_iff
thf(fact_1869_order__less__asym,axiom,
! [A: $tType] :
( ( preorder @ A )
=> ! [X2: A,Y3: A] :
( ( ord_less @ A @ X2 @ Y3 )
=> ~ ( ord_less @ A @ Y3 @ X2 ) ) ) ).
% order_less_asym
thf(fact_1870_linorder__neqE,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [X2: A,Y3: A] :
( ( X2 != Y3 )
=> ( ~ ( ord_less @ A @ X2 @ Y3 )
=> ( ord_less @ A @ Y3 @ X2 ) ) ) ) ).
% linorder_neqE
thf(fact_1871_dual__order_Ostrict__implies__not__eq,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [B3: A,A3: A] :
( ( ord_less @ A @ B3 @ A3 )
=> ( A3 != B3 ) ) ) ).
% dual_order.strict_implies_not_eq
thf(fact_1872_order_Ostrict__implies__not__eq,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [A3: A,B3: A] :
( ( ord_less @ A @ A3 @ B3 )
=> ( A3 != B3 ) ) ) ).
% order.strict_implies_not_eq
thf(fact_1873_dual__order_Ostrict__trans,axiom,
! [A: $tType] :
( ( preorder @ A )
=> ! [B3: A,A3: A,C3: A] :
( ( ord_less @ A @ B3 @ A3 )
=> ( ( ord_less @ A @ C3 @ B3 )
=> ( ord_less @ A @ C3 @ A3 ) ) ) ) ).
% dual_order.strict_trans
thf(fact_1874_not__less__iff__gr__or__eq,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [X2: A,Y3: A] :
( ( ~ ( ord_less @ A @ X2 @ Y3 ) )
= ( ( ord_less @ A @ Y3 @ X2 )
| ( X2 = Y3 ) ) ) ) ).
% not_less_iff_gr_or_eq
thf(fact_1875_order_Ostrict__trans,axiom,
! [A: $tType] :
( ( preorder @ A )
=> ! [A3: A,B3: A,C3: A] :
( ( ord_less @ A @ A3 @ B3 )
=> ( ( ord_less @ A @ B3 @ C3 )
=> ( ord_less @ A @ A3 @ C3 ) ) ) ) ).
% order.strict_trans
thf(fact_1876_linorder__less__wlog,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [P: A > A > $o,A3: A,B3: A] :
( ! [A4: A,B4: A] :
( ( ord_less @ A @ A4 @ B4 )
=> ( P @ A4 @ B4 ) )
=> ( ! [A4: A] : ( P @ A4 @ A4 )
=> ( ! [A4: A,B4: A] :
( ( P @ B4 @ A4 )
=> ( P @ A4 @ B4 ) )
=> ( P @ A3 @ B3 ) ) ) ) ) ).
% linorder_less_wlog
thf(fact_1877_exists__least__iff,axiom,
! [A: $tType] :
( ( wellorder @ A )
=> ( ( ^ [P2: A > $o] :
? [X6: A] : ( P2 @ X6 ) )
= ( ^ [P3: A > $o] :
? [N4: A] :
( ( P3 @ N4 )
& ! [M6: A] :
( ( ord_less @ A @ M6 @ N4 )
=> ~ ( P3 @ M6 ) ) ) ) ) ) ).
% exists_least_iff
thf(fact_1878_dual__order_Oirrefl,axiom,
! [A: $tType] :
( ( preorder @ A )
=> ! [A3: A] :
~ ( ord_less @ A @ A3 @ A3 ) ) ).
% dual_order.irrefl
thf(fact_1879_dual__order_Oasym,axiom,
! [A: $tType] :
( ( preorder @ A )
=> ! [B3: A,A3: A] :
( ( ord_less @ A @ B3 @ A3 )
=> ~ ( ord_less @ A @ A3 @ B3 ) ) ) ).
% dual_order.asym
thf(fact_1880_linorder__cases,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [X2: A,Y3: A] :
( ~ ( ord_less @ A @ X2 @ Y3 )
=> ( ( X2 != Y3 )
=> ( ord_less @ A @ Y3 @ X2 ) ) ) ) ).
% linorder_cases
thf(fact_1881_antisym__conv3,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [Y3: A,X2: A] :
( ~ ( ord_less @ A @ Y3 @ X2 )
=> ( ( ~ ( ord_less @ A @ X2 @ Y3 ) )
= ( X2 = Y3 ) ) ) ) ).
% antisym_conv3
thf(fact_1882_less__induct,axiom,
! [A: $tType] :
( ( wellorder @ A )
=> ! [P: A > $o,A3: A] :
( ! [X4: A] :
( ! [Y5: A] :
( ( ord_less @ A @ Y5 @ X4 )
=> ( P @ Y5 ) )
=> ( P @ X4 ) )
=> ( P @ A3 ) ) ) ).
% less_induct
thf(fact_1883_ord__less__eq__trans,axiom,
! [A: $tType] :
( ( ord @ A )
=> ! [A3: A,B3: A,C3: A] :
( ( ord_less @ A @ A3 @ B3 )
=> ( ( B3 = C3 )
=> ( ord_less @ A @ A3 @ C3 ) ) ) ) ).
% ord_less_eq_trans
thf(fact_1884_ord__eq__less__trans,axiom,
! [A: $tType] :
( ( ord @ A )
=> ! [A3: A,B3: A,C3: A] :
( ( A3 = B3 )
=> ( ( ord_less @ A @ B3 @ C3 )
=> ( ord_less @ A @ A3 @ C3 ) ) ) ) ).
% ord_eq_less_trans
thf(fact_1885_order_Oasym,axiom,
! [A: $tType] :
( ( preorder @ A )
=> ! [A3: A,B3: A] :
( ( ord_less @ A @ A3 @ B3 )
=> ~ ( ord_less @ A @ B3 @ A3 ) ) ) ).
% order.asym
thf(fact_1886_less__imp__neq,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [X2: A,Y3: A] :
( ( ord_less @ A @ X2 @ Y3 )
=> ( X2 != Y3 ) ) ) ).
% less_imp_neq
thf(fact_1887_dense,axiom,
! [A: $tType] :
( ( dense_order @ A )
=> ! [X2: A,Y3: A] :
( ( ord_less @ A @ X2 @ Y3 )
=> ? [Z2: A] :
( ( ord_less @ A @ X2 @ Z2 )
& ( ord_less @ A @ Z2 @ Y3 ) ) ) ) ).
% dense
thf(fact_1888_gt__ex,axiom,
! [A: $tType] :
( ( no_top @ A )
=> ! [X2: A] :
? [X_1: A] : ( ord_less @ A @ X2 @ X_1 ) ) ).
% gt_ex
thf(fact_1889_lt__ex,axiom,
! [A: $tType] :
( ( no_bot @ A )
=> ! [X2: A] :
? [Y4: A] : ( ord_less @ A @ Y4 @ X2 ) ) ).
% lt_ex
thf(fact_1890_verit__comp__simplify1_I1_J,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [A3: A] :
~ ( ord_less @ A @ A3 @ A3 ) ) ).
% verit_comp_simplify1(1)
thf(fact_1891_vebt__assn__raw_Osimps_I4_J,axiom,
! [Vd3: $o,Ve3: $o,V: option @ ( product_prod @ nat @ nat ),Va: nat,Vb: array @ vEBT_VEBTi,Vc: vEBT_VEBTi] :
( ( vEBT_vebt_assn_raw @ ( vEBT_Leaf @ Vd3 @ Ve3 ) @ ( vEBT_Nodei @ V @ Va @ Vb @ Vc ) )
= ( bot_bot @ assn ) ) ).
% vebt_assn_raw.simps(4)
thf(fact_1892_vebt__assn__raw_Osimps_I3_J,axiom,
! [V: option @ ( product_prod @ nat @ nat ),Va: nat,Vb: list @ vEBT_VEBT,Vc: vEBT_VEBT,Vd3: $o,Ve3: $o] :
( ( vEBT_vebt_assn_raw @ ( vEBT_Node @ V @ Va @ Vb @ Vc ) @ ( vEBT_Leafi @ Vd3 @ Ve3 ) )
= ( bot_bot @ assn ) ) ).
% vebt_assn_raw.simps(3)
thf(fact_1893_verit__la__generic,axiom,
! [A3: int,X2: int] :
( ( ord_less_eq @ int @ A3 @ X2 )
| ( A3 = X2 )
| ( ord_less_eq @ int @ X2 @ A3 ) ) ).
% verit_la_generic
thf(fact_1894_less__fun__def,axiom,
! [B: $tType,A: $tType] :
( ( ord @ B )
=> ( ( ord_less @ ( A > B ) )
= ( ^ [F5: A > B,G2: A > B] :
( ( ord_less_eq @ ( A > B ) @ F5 @ G2 )
& ~ ( ord_less_eq @ ( A > B ) @ G2 @ F5 ) ) ) ) ) ).
% less_fun_def
thf(fact_1895_predicate1D,axiom,
! [A: $tType,P: A > $o,Q: A > $o,X2: A] :
( ( ord_less_eq @ ( A > $o ) @ P @ Q )
=> ( ( P @ X2 )
=> ( Q @ X2 ) ) ) ).
% predicate1D
thf(fact_1896_rev__predicate1D,axiom,
! [A: $tType,P: A > $o,X2: A,Q: A > $o] :
( ( P @ X2 )
=> ( ( ord_less_eq @ ( A > $o ) @ P @ Q )
=> ( Q @ X2 ) ) ) ).
% rev_predicate1D
thf(fact_1897_mult__ceiling__le,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [A3: A,B3: A] :
( ( ord_less_eq @ A @ ( zero_zero @ A ) @ A3 )
=> ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ B3 )
=> ( ord_less_eq @ int @ ( archimedean_ceiling @ A @ ( times_times @ A @ A3 @ B3 ) ) @ ( times_times @ int @ ( archimedean_ceiling @ A @ A3 ) @ ( archimedean_ceiling @ A @ B3 ) ) ) ) ) ) ).
% mult_ceiling_le
thf(fact_1898_order__le__imp__less__or__eq,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [X2: A,Y3: A] :
( ( ord_less_eq @ A @ X2 @ Y3 )
=> ( ( ord_less @ A @ X2 @ Y3 )
| ( X2 = Y3 ) ) ) ) ).
% order_le_imp_less_or_eq
thf(fact_1899_linorder__le__less__linear,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [X2: A,Y3: A] :
( ( ord_less_eq @ A @ X2 @ Y3 )
| ( ord_less @ A @ Y3 @ X2 ) ) ) ).
% linorder_le_less_linear
thf(fact_1900_order__less__le__subst2,axiom,
! [A: $tType,C: $tType] :
( ( ( order @ C )
& ( order @ A ) )
=> ! [A3: A,B3: A,F2: A > C,C3: C] :
( ( ord_less @ A @ A3 @ B3 )
=> ( ( ord_less_eq @ C @ ( F2 @ B3 ) @ C3 )
=> ( ! [X4: A,Y4: A] :
( ( ord_less @ A @ X4 @ Y4 )
=> ( ord_less @ C @ ( F2 @ X4 ) @ ( F2 @ Y4 ) ) )
=> ( ord_less @ C @ ( F2 @ A3 ) @ C3 ) ) ) ) ) ).
% order_less_le_subst2
thf(fact_1901_order__less__le__subst1,axiom,
! [A: $tType,B: $tType] :
( ( ( order @ B )
& ( order @ A ) )
=> ! [A3: A,F2: B > A,B3: B,C3: B] :
( ( ord_less @ A @ A3 @ ( F2 @ B3 ) )
=> ( ( ord_less_eq @ B @ B3 @ C3 )
=> ( ! [X4: B,Y4: B] :
( ( ord_less_eq @ B @ X4 @ Y4 )
=> ( ord_less_eq @ A @ ( F2 @ X4 ) @ ( F2 @ Y4 ) ) )
=> ( ord_less @ A @ A3 @ ( F2 @ C3 ) ) ) ) ) ) ).
% order_less_le_subst1
thf(fact_1902_order__le__less__subst2,axiom,
! [A: $tType,C: $tType] :
( ( ( order @ C )
& ( order @ A ) )
=> ! [A3: A,B3: A,F2: A > C,C3: C] :
( ( ord_less_eq @ A @ A3 @ B3 )
=> ( ( ord_less @ C @ ( F2 @ B3 ) @ C3 )
=> ( ! [X4: A,Y4: A] :
( ( ord_less_eq @ A @ X4 @ Y4 )
=> ( ord_less_eq @ C @ ( F2 @ X4 ) @ ( F2 @ Y4 ) ) )
=> ( ord_less @ C @ ( F2 @ A3 ) @ C3 ) ) ) ) ) ).
% order_le_less_subst2
thf(fact_1903_order__le__less__subst1,axiom,
! [A: $tType,B: $tType] :
( ( ( order @ B )
& ( order @ A ) )
=> ! [A3: A,F2: B > A,B3: B,C3: B] :
( ( ord_less_eq @ A @ A3 @ ( F2 @ B3 ) )
=> ( ( ord_less @ B @ B3 @ C3 )
=> ( ! [X4: B,Y4: B] :
( ( ord_less @ B @ X4 @ Y4 )
=> ( ord_less @ A @ ( F2 @ X4 ) @ ( F2 @ Y4 ) ) )
=> ( ord_less @ A @ A3 @ ( F2 @ C3 ) ) ) ) ) ) ).
% order_le_less_subst1
thf(fact_1904_order__less__le__trans,axiom,
! [A: $tType] :
( ( preorder @ A )
=> ! [X2: A,Y3: A,Z: A] :
( ( ord_less @ A @ X2 @ Y3 )
=> ( ( ord_less_eq @ A @ Y3 @ Z )
=> ( ord_less @ A @ X2 @ Z ) ) ) ) ).
% order_less_le_trans
thf(fact_1905_order__le__less__trans,axiom,
! [A: $tType] :
( ( preorder @ A )
=> ! [X2: A,Y3: A,Z: A] :
( ( ord_less_eq @ A @ X2 @ Y3 )
=> ( ( ord_less @ A @ Y3 @ Z )
=> ( ord_less @ A @ X2 @ Z ) ) ) ) ).
% order_le_less_trans
thf(fact_1906_order__neq__le__trans,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [A3: A,B3: A] :
( ( A3 != B3 )
=> ( ( ord_less_eq @ A @ A3 @ B3 )
=> ( ord_less @ A @ A3 @ B3 ) ) ) ) ).
% order_neq_le_trans
thf(fact_1907_order__le__neq__trans,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [A3: A,B3: A] :
( ( ord_less_eq @ A @ A3 @ B3 )
=> ( ( A3 != B3 )
=> ( ord_less @ A @ A3 @ B3 ) ) ) ) ).
% order_le_neq_trans
thf(fact_1908_order__less__imp__le,axiom,
! [A: $tType] :
( ( preorder @ A )
=> ! [X2: A,Y3: A] :
( ( ord_less @ A @ X2 @ Y3 )
=> ( ord_less_eq @ A @ X2 @ Y3 ) ) ) ).
% order_less_imp_le
thf(fact_1909_linorder__not__less,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [X2: A,Y3: A] :
( ( ~ ( ord_less @ A @ X2 @ Y3 ) )
= ( ord_less_eq @ A @ Y3 @ X2 ) ) ) ).
% linorder_not_less
thf(fact_1910_linorder__not__le,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [X2: A,Y3: A] :
( ( ~ ( ord_less_eq @ A @ X2 @ Y3 ) )
= ( ord_less @ A @ Y3 @ X2 ) ) ) ).
% linorder_not_le
thf(fact_1911_order__less__le,axiom,
! [A: $tType] :
( ( order @ A )
=> ( ( ord_less @ A )
= ( ^ [X3: A,Y: A] :
( ( ord_less_eq @ A @ X3 @ Y )
& ( X3 != Y ) ) ) ) ) ).
% order_less_le
thf(fact_1912_order__le__less,axiom,
! [A: $tType] :
( ( order @ A )
=> ( ( ord_less_eq @ A )
= ( ^ [X3: A,Y: A] :
( ( ord_less @ A @ X3 @ Y )
| ( X3 = Y ) ) ) ) ) ).
% order_le_less
thf(fact_1913_dual__order_Ostrict__implies__order,axiom,
! [A: $tType] :
( ( preorder @ A )
=> ! [B3: A,A3: A] :
( ( ord_less @ A @ B3 @ A3 )
=> ( ord_less_eq @ A @ B3 @ A3 ) ) ) ).
% dual_order.strict_implies_order
thf(fact_1914_order_Ostrict__implies__order,axiom,
! [A: $tType] :
( ( preorder @ A )
=> ! [A3: A,B3: A] :
( ( ord_less @ A @ A3 @ B3 )
=> ( ord_less_eq @ A @ A3 @ B3 ) ) ) ).
% order.strict_implies_order
thf(fact_1915_dual__order_Ostrict__iff__not,axiom,
! [A: $tType] :
( ( preorder @ A )
=> ( ( ord_less @ A )
= ( ^ [B7: A,A6: A] :
( ( ord_less_eq @ A @ B7 @ A6 )
& ~ ( ord_less_eq @ A @ A6 @ B7 ) ) ) ) ) ).
% dual_order.strict_iff_not
thf(fact_1916_dual__order_Ostrict__trans2,axiom,
! [A: $tType] :
( ( preorder @ A )
=> ! [B3: A,A3: A,C3: A] :
( ( ord_less @ A @ B3 @ A3 )
=> ( ( ord_less_eq @ A @ C3 @ B3 )
=> ( ord_less @ A @ C3 @ A3 ) ) ) ) ).
% dual_order.strict_trans2
thf(fact_1917_dual__order_Ostrict__trans1,axiom,
! [A: $tType] :
( ( preorder @ A )
=> ! [B3: A,A3: A,C3: A] :
( ( ord_less_eq @ A @ B3 @ A3 )
=> ( ( ord_less @ A @ C3 @ B3 )
=> ( ord_less @ A @ C3 @ A3 ) ) ) ) ).
% dual_order.strict_trans1
thf(fact_1918_dual__order_Ostrict__iff__order,axiom,
! [A: $tType] :
( ( order @ A )
=> ( ( ord_less @ A )
= ( ^ [B7: A,A6: A] :
( ( ord_less_eq @ A @ B7 @ A6 )
& ( A6 != B7 ) ) ) ) ) ).
% dual_order.strict_iff_order
thf(fact_1919_dual__order_Oorder__iff__strict,axiom,
! [A: $tType] :
( ( order @ A )
=> ( ( ord_less_eq @ A )
= ( ^ [B7: A,A6: A] :
( ( ord_less @ A @ B7 @ A6 )
| ( A6 = B7 ) ) ) ) ) ).
% dual_order.order_iff_strict
thf(fact_1920_dense__le__bounded,axiom,
! [A: $tType] :
( ( dense_linorder @ A )
=> ! [X2: A,Y3: A,Z: A] :
( ( ord_less @ A @ X2 @ Y3 )
=> ( ! [W3: A] :
( ( ord_less @ A @ X2 @ W3 )
=> ( ( ord_less @ A @ W3 @ Y3 )
=> ( ord_less_eq @ A @ W3 @ Z ) ) )
=> ( ord_less_eq @ A @ Y3 @ Z ) ) ) ) ).
% dense_le_bounded
thf(fact_1921_dense__ge__bounded,axiom,
! [A: $tType] :
( ( dense_linorder @ A )
=> ! [Z: A,X2: A,Y3: A] :
( ( ord_less @ A @ Z @ X2 )
=> ( ! [W3: A] :
( ( ord_less @ A @ Z @ W3 )
=> ( ( ord_less @ A @ W3 @ X2 )
=> ( ord_less_eq @ A @ Y3 @ W3 ) ) )
=> ( ord_less_eq @ A @ Y3 @ Z ) ) ) ) ).
% dense_ge_bounded
thf(fact_1922_order_Ostrict__iff__not,axiom,
! [A: $tType] :
( ( preorder @ A )
=> ( ( ord_less @ A )
= ( ^ [A6: A,B7: A] :
( ( ord_less_eq @ A @ A6 @ B7 )
& ~ ( ord_less_eq @ A @ B7 @ A6 ) ) ) ) ) ).
% order.strict_iff_not
thf(fact_1923_order_Ostrict__trans2,axiom,
! [A: $tType] :
( ( preorder @ A )
=> ! [A3: A,B3: A,C3: A] :
( ( ord_less @ A @ A3 @ B3 )
=> ( ( ord_less_eq @ A @ B3 @ C3 )
=> ( ord_less @ A @ A3 @ C3 ) ) ) ) ).
% order.strict_trans2
thf(fact_1924_order_Ostrict__trans1,axiom,
! [A: $tType] :
( ( preorder @ A )
=> ! [A3: A,B3: A,C3: A] :
( ( ord_less_eq @ A @ A3 @ B3 )
=> ( ( ord_less @ A @ B3 @ C3 )
=> ( ord_less @ A @ A3 @ C3 ) ) ) ) ).
% order.strict_trans1
thf(fact_1925_order_Ostrict__iff__order,axiom,
! [A: $tType] :
( ( order @ A )
=> ( ( ord_less @ A )
= ( ^ [A6: A,B7: A] :
( ( ord_less_eq @ A @ A6 @ B7 )
& ( A6 != B7 ) ) ) ) ) ).
% order.strict_iff_order
thf(fact_1926_order_Oorder__iff__strict,axiom,
! [A: $tType] :
( ( order @ A )
=> ( ( ord_less_eq @ A )
= ( ^ [A6: A,B7: A] :
( ( ord_less @ A @ A6 @ B7 )
| ( A6 = B7 ) ) ) ) ) ).
% order.order_iff_strict
thf(fact_1927_not__le__imp__less,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [Y3: A,X2: A] :
( ~ ( ord_less_eq @ A @ Y3 @ X2 )
=> ( ord_less @ A @ X2 @ Y3 ) ) ) ).
% not_le_imp_less
thf(fact_1928_less__le__not__le,axiom,
! [A: $tType] :
( ( preorder @ A )
=> ( ( ord_less @ A )
= ( ^ [X3: A,Y: A] :
( ( ord_less_eq @ A @ X3 @ Y )
& ~ ( ord_less_eq @ A @ Y @ X3 ) ) ) ) ) ).
% less_le_not_le
thf(fact_1929_dense__le,axiom,
! [A: $tType] :
( ( dense_linorder @ A )
=> ! [Y3: A,Z: A] :
( ! [X4: A] :
( ( ord_less @ A @ X4 @ Y3 )
=> ( ord_less_eq @ A @ X4 @ Z ) )
=> ( ord_less_eq @ A @ Y3 @ Z ) ) ) ).
% dense_le
thf(fact_1930_dense__ge,axiom,
! [A: $tType] :
( ( dense_linorder @ A )
=> ! [Z: A,Y3: A] :
( ! [X4: A] :
( ( ord_less @ A @ Z @ X4 )
=> ( ord_less_eq @ A @ Y3 @ X4 ) )
=> ( ord_less_eq @ A @ Y3 @ Z ) ) ) ).
% dense_ge
thf(fact_1931_antisym__conv2,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [X2: A,Y3: A] :
( ( ord_less_eq @ A @ X2 @ Y3 )
=> ( ( ~ ( ord_less @ A @ X2 @ Y3 ) )
= ( X2 = Y3 ) ) ) ) ).
% antisym_conv2
thf(fact_1932_antisym__conv1,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [X2: A,Y3: A] :
( ~ ( ord_less @ A @ X2 @ Y3 )
=> ( ( ord_less_eq @ A @ X2 @ Y3 )
= ( X2 = Y3 ) ) ) ) ).
% antisym_conv1
thf(fact_1933_nless__le,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [A3: A,B3: A] :
( ( ~ ( ord_less @ A @ A3 @ B3 ) )
= ( ~ ( ord_less_eq @ A @ A3 @ B3 )
| ( A3 = B3 ) ) ) ) ).
% nless_le
thf(fact_1934_leI,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [X2: A,Y3: A] :
( ~ ( ord_less @ A @ X2 @ Y3 )
=> ( ord_less_eq @ A @ Y3 @ X2 ) ) ) ).
% leI
thf(fact_1935_leD,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [Y3: A,X2: A] :
( ( ord_less_eq @ A @ Y3 @ X2 )
=> ~ ( ord_less @ A @ X2 @ Y3 ) ) ) ).
% leD
thf(fact_1936_verit__comp__simplify1_I3_J,axiom,
! [B: $tType] :
( ( linorder @ B )
=> ! [B5: B,A5: B] :
( ( ~ ( ord_less_eq @ B @ B5 @ A5 ) )
= ( ord_less @ B @ A5 @ B5 ) ) ) ).
% verit_comp_simplify1(3)
thf(fact_1937_verit__sum__simplify,axiom,
! [A: $tType] :
( ( cancel1802427076303600483id_add @ A )
=> ! [A3: A] :
( ( plus_plus @ A @ A3 @ ( zero_zero @ A ) )
= A3 ) ) ).
% verit_sum_simplify
thf(fact_1938_verit__eq__simplify_I10_J,axiom,
! [X22: num] :
( one2
!= ( bit0 @ X22 ) ) ).
% verit_eq_simplify(10)
thf(fact_1939_bot_Oextremum,axiom,
! [A: $tType] :
( ( order_bot @ A )
=> ! [A3: A] : ( ord_less_eq @ A @ ( bot_bot @ A ) @ A3 ) ) ).
% bot.extremum
thf(fact_1940_bot_Oextremum__unique,axiom,
! [A: $tType] :
( ( order_bot @ A )
=> ! [A3: A] :
( ( ord_less_eq @ A @ A3 @ ( bot_bot @ A ) )
= ( A3
= ( bot_bot @ A ) ) ) ) ).
% bot.extremum_unique
thf(fact_1941_bot_Oextremum__uniqueI,axiom,
! [A: $tType] :
( ( order_bot @ A )
=> ! [A3: A] :
( ( ord_less_eq @ A @ A3 @ ( bot_bot @ A ) )
=> ( A3
= ( bot_bot @ A ) ) ) ) ).
% bot.extremum_uniqueI
thf(fact_1942_bot_Onot__eq__extremum,axiom,
! [A: $tType] :
( ( order_bot @ A )
=> ! [A3: A] :
( ( A3
!= ( bot_bot @ A ) )
= ( ord_less @ A @ ( bot_bot @ A ) @ A3 ) ) ) ).
% bot.not_eq_extremum
thf(fact_1943_bot_Oextremum__strict,axiom,
! [A: $tType] :
( ( order_bot @ A )
=> ! [A3: A] :
~ ( ord_less @ A @ A3 @ ( bot_bot @ A ) ) ) ).
% bot.extremum_strict
thf(fact_1944_verit__eq__simplify_I14_J,axiom,
! [X22: num,X33: num] :
( ( bit0 @ X22 )
!= ( bit1 @ X33 ) ) ).
% verit_eq_simplify(14)
thf(fact_1945_verit__eq__simplify_I12_J,axiom,
! [X33: num] :
( one2
!= ( bit1 @ X33 ) ) ).
% verit_eq_simplify(12)
thf(fact_1946_max__def,axiom,
! [A: $tType] :
( ( ord @ A )
=> ( ( ord_max @ A )
= ( ^ [A6: A,B7: A] : ( if @ A @ ( ord_less_eq @ A @ A6 @ B7 ) @ B7 @ A6 ) ) ) ) ).
% max_def
thf(fact_1947_max__absorb1,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [Y3: A,X2: A] :
( ( ord_less_eq @ A @ Y3 @ X2 )
=> ( ( ord_max @ A @ X2 @ Y3 )
= X2 ) ) ) ).
% max_absorb1
thf(fact_1948_max__absorb2,axiom,
! [A: $tType] :
( ( ord @ A )
=> ! [X2: A,Y3: A] :
( ( ord_less_eq @ A @ X2 @ Y3 )
=> ( ( ord_max @ A @ X2 @ Y3 )
= Y3 ) ) ) ).
% max_absorb2
thf(fact_1949_le__funD,axiom,
! [B: $tType,A: $tType] :
( ( ord @ B )
=> ! [F2: A > B,G: A > B,X2: A] :
( ( ord_less_eq @ ( A > B ) @ F2 @ G )
=> ( ord_less_eq @ B @ ( F2 @ X2 ) @ ( G @ X2 ) ) ) ) ).
% le_funD
thf(fact_1950_le__funE,axiom,
! [B: $tType,A: $tType] :
( ( ord @ B )
=> ! [F2: A > B,G: A > B,X2: A] :
( ( ord_less_eq @ ( A > B ) @ F2 @ G )
=> ( ord_less_eq @ B @ ( F2 @ X2 ) @ ( G @ X2 ) ) ) ) ).
% le_funE
thf(fact_1951_le__funI,axiom,
! [B: $tType,A: $tType] :
( ( ord @ B )
=> ! [F2: A > B,G: A > B] :
( ! [X4: A] : ( ord_less_eq @ B @ ( F2 @ X4 ) @ ( G @ X4 ) )
=> ( ord_less_eq @ ( A > B ) @ F2 @ G ) ) ) ).
% le_funI
thf(fact_1952_le__fun__def,axiom,
! [B: $tType,A: $tType] :
( ( ord @ B )
=> ( ( ord_less_eq @ ( A > B ) )
= ( ^ [F5: A > B,G2: A > B] :
! [X3: A] : ( ord_less_eq @ B @ ( F5 @ X3 ) @ ( G2 @ X3 ) ) ) ) ) ).
% le_fun_def
thf(fact_1953_TBOUND__mono,axiom,
! [A: $tType,C3: heap_Time_Heap @ A,T2: nat,T4: nat] :
( ( time_TBOUND @ A @ C3 @ T2 )
=> ( ( ord_less_eq @ nat @ T2 @ T4 )
=> ( time_TBOUND @ A @ C3 @ T4 ) ) ) ).
% TBOUND_mono
thf(fact_1954_TBOUND__if__max,axiom,
! [A: $tType,P: $o,M: heap_Time_Heap @ A,Bm: nat,N: heap_Time_Heap @ A,Bn: nat] :
( ( P
=> ( time_TBOUND @ A @ M @ Bm ) )
=> ( ( ~ P
=> ( time_TBOUND @ A @ N @ Bn ) )
=> ( time_TBOUND @ A @ ( if @ ( heap_Time_Heap @ A ) @ P @ M @ N ) @ ( ord_max @ nat @ Bm @ Bn ) ) ) ) ).
% TBOUND_if_max
thf(fact_1955_max__def__raw,axiom,
! [A: $tType] :
( ( ord @ A )
=> ( ( ord_max @ A )
= ( ^ [A6: A,B7: A] : ( if @ A @ ( ord_less_eq @ A @ A6 @ B7 ) @ B7 @ A6 ) ) ) ) ).
% max_def_raw
thf(fact_1956_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_1957_int__ops_I3_J,axiom,
! [N: num] :
( ( semiring_1_of_nat @ int @ ( numeral_numeral @ nat @ N ) )
= ( numeral_numeral @ int @ N ) ) ).
% int_ops(3)
thf(fact_1958_nat__int__comparison_I2_J,axiom,
( ( ord_less @ nat )
= ( ^ [A6: nat,B7: nat] : ( ord_less @ int @ ( semiring_1_of_nat @ int @ A6 ) @ ( semiring_1_of_nat @ int @ B7 ) ) ) ) ).
% nat_int_comparison(2)
thf(fact_1959_nat__int__comparison_I3_J,axiom,
( ( ord_less_eq @ nat )
= ( ^ [A6: nat,B7: nat] : ( ord_less_eq @ int @ ( semiring_1_of_nat @ int @ A6 ) @ ( semiring_1_of_nat @ int @ B7 ) ) ) ) ).
% nat_int_comparison(3)
thf(fact_1960_int__ops_I2_J,axiom,
( ( semiring_1_of_nat @ int @ ( one_one @ nat ) )
= ( one_one @ int ) ) ).
% int_ops(2)
thf(fact_1961_int__plus,axiom,
! [N: nat,M: nat] :
( ( semiring_1_of_nat @ int @ ( plus_plus @ nat @ N @ M ) )
= ( plus_plus @ int @ ( semiring_1_of_nat @ int @ N ) @ ( semiring_1_of_nat @ int @ M ) ) ) ).
% int_plus
thf(fact_1962_int__ops_I5_J,axiom,
! [A3: nat,B3: nat] :
( ( semiring_1_of_nat @ int @ ( plus_plus @ nat @ A3 @ B3 ) )
= ( plus_plus @ int @ ( semiring_1_of_nat @ int @ A3 ) @ ( semiring_1_of_nat @ int @ B3 ) ) ) ).
% int_ops(5)
thf(fact_1963_int__ops_I7_J,axiom,
! [A3: nat,B3: nat] :
( ( semiring_1_of_nat @ int @ ( times_times @ nat @ A3 @ B3 ) )
= ( times_times @ int @ ( semiring_1_of_nat @ int @ A3 ) @ ( semiring_1_of_nat @ int @ B3 ) ) ) ).
% int_ops(7)
thf(fact_1964_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_1965_TBOUNDI,axiom,
! [A: $tType,M: heap_Time_Heap @ A,T2: nat] :
( ! [H3: heap_ext @ product_unit] : ( ord_less_eq @ nat @ ( time_time @ A @ M @ H3 ) @ T2 )
=> ( time_TBOUND @ A @ M @ T2 ) ) ).
% TBOUNDI
thf(fact_1966_TBOUND__def,axiom,
! [A: $tType] :
( ( time_TBOUND @ A )
= ( ^ [M6: heap_Time_Heap @ A,T3: nat] :
! [H: heap_ext @ product_unit] : ( ord_less_eq @ nat @ ( time_time @ A @ M6 @ H ) @ T3 ) ) ) ).
% TBOUND_def
thf(fact_1967_ceiling__log__nat__eq__if,axiom,
! [B3: nat,N: nat,K: nat] :
( ( ord_less @ nat @ ( power_power @ nat @ B3 @ N ) @ K )
=> ( ( ord_less_eq @ nat @ K @ ( power_power @ nat @ B3 @ ( plus_plus @ nat @ N @ ( one_one @ nat ) ) ) )
=> ( ( ord_less_eq @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ B3 )
=> ( ( archimedean_ceiling @ real @ ( log @ ( semiring_1_of_nat @ real @ B3 ) @ ( 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_1968_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_1969_norm__pre__pure__iff__htt_H,axiom,
! [A: $tType,B3: $o,P: assn,F2: heap_Time_Heap @ A,Q: A > assn,T2: nat] :
( ( time_htt @ A @ ( times_times @ assn @ ( pure_assn @ B3 ) @ P ) @ F2 @ Q @ T2 )
= ( B3
=> ( time_htt @ A @ P @ F2 @ Q @ T2 ) ) ) ).
% norm_pre_pure_iff_htt'
thf(fact_1970_norm__pre__pure__iff__htt,axiom,
! [A: $tType,P: assn,B3: $o,F2: heap_Time_Heap @ A,Q: A > assn,T2: nat] :
( ( time_htt @ A @ ( times_times @ assn @ P @ ( pure_assn @ B3 ) ) @ F2 @ Q @ T2 )
= ( B3
=> ( time_htt @ A @ P @ F2 @ Q @ T2 ) ) ) ).
% norm_pre_pure_iff_htt
thf(fact_1971_nat__less__as__int,axiom,
( ( ord_less @ nat )
= ( ^ [A6: nat,B7: nat] : ( ord_less @ int @ ( semiring_1_of_nat @ int @ A6 ) @ ( semiring_1_of_nat @ int @ B7 ) ) ) ) ).
% nat_less_as_int
thf(fact_1972_nat__leq__as__int,axiom,
( ( ord_less_eq @ nat )
= ( ^ [A6: nat,B7: nat] : ( ord_less_eq @ int @ ( semiring_1_of_nat @ int @ A6 ) @ ( semiring_1_of_nat @ int @ B7 ) ) ) ) ).
% nat_leq_as_int
thf(fact_1973_ceiling__log__nat__eq__powr__iff,axiom,
! [B3: nat,K: nat,N: nat] :
( ( ord_less_eq @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ B3 )
=> ( ( ord_less @ nat @ ( zero_zero @ nat ) @ K )
=> ( ( ( archimedean_ceiling @ real @ ( log @ ( semiring_1_of_nat @ real @ B3 ) @ ( semiring_1_of_nat @ real @ K ) ) )
= ( plus_plus @ int @ ( semiring_1_of_nat @ int @ N ) @ ( one_one @ int ) ) )
= ( ( ord_less @ nat @ ( power_power @ nat @ B3 @ N ) @ K )
& ( ord_less_eq @ nat @ K @ ( power_power @ nat @ B3 @ ( plus_plus @ nat @ N @ ( one_one @ nat ) ) ) ) ) ) ) ) ).
% ceiling_log_nat_eq_powr_iff
thf(fact_1974_htt__cons__rule,axiom,
! [A: $tType,P5: assn,C3: heap_Time_Heap @ A,Q2: A > assn,T4: nat,P: assn,Q: A > assn,T2: nat] :
( ( time_htt @ A @ P5 @ C3 @ Q2 @ T4 )
=> ( ( entails @ P @ P5 )
=> ( ! [X4: A] : ( entails @ ( Q2 @ X4 ) @ ( Q @ X4 ) )
=> ( ( ord_less_eq @ nat @ T4 @ T2 )
=> ( time_htt @ A @ P @ C3 @ Q @ T2 ) ) ) ) ) ).
% htt_cons_rule
thf(fact_1975_word__unat__power,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [N: nat] :
( ( power_power @ ( word @ A ) @ ( numeral_numeral @ ( word @ A ) @ ( bit0 @ one2 ) ) @ N )
= ( semiring_1_of_nat @ ( word @ A ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N ) ) ) ) ).
% word_unat_power
thf(fact_1976_word__less__two__pow__divD,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [X2: word @ A,N: nat,M: nat] :
( ( ord_less @ ( word @ A ) @ X2 @ ( 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 ) @ X2 @ ( power_power @ ( word @ A ) @ ( numeral_numeral @ ( word @ A ) @ ( bit0 @ one2 ) ) @ ( minus_minus @ nat @ N @ M ) ) ) ) ) ) ).
% word_less_two_pow_divD
thf(fact_1977_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_1978_int__ops_I4_J,axiom,
! [A3: nat] :
( ( semiring_1_of_nat @ int @ ( suc @ A3 ) )
= ( plus_plus @ int @ ( semiring_1_of_nat @ int @ A3 ) @ ( one_one @ int ) ) ) ).
% int_ops(4)
thf(fact_1979_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_1980_htt__vebt__memberi__invar__vebt,axiom,
! [T2: vEBT_VEBT,N: nat,Ti: vEBT_VEBTi,X2: nat] :
( ( vEBT_invar_vebt @ T2 @ N )
=> ( time_htt @ $o @ ( vEBT_vebt_assn_raw @ T2 @ Ti ) @ ( vEBT_vebt_memberi @ Ti @ X2 )
@ ^ [R5: $o] :
( times_times @ assn @ ( vEBT_vebt_assn_raw @ T2 @ Ti )
@ ( pure_assn
@ ( R5
= ( vEBT_vebt_member @ T2 @ X2 ) ) ) )
@ ( 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_1981_htt__vebt__inserti__invar__vebt,axiom,
! [T2: vEBT_VEBT,N: nat,Ti: vEBT_VEBTi,X2: nat] :
( ( vEBT_invar_vebt @ T2 @ N )
=> ( time_htt @ vEBT_VEBTi @ ( vEBT_vebt_assn_raw @ T2 @ Ti ) @ ( vEBT_vebt_inserti @ Ti @ X2 ) @ ( vEBT_vebt_assn_raw @ ( vEBT_vebt_insert @ T2 @ X2 ) ) @ ( 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_1982_htt__vebt__predi,axiom,
! [T2: vEBT_VEBT,N: nat,Ti: vEBT_VEBTi,X2: nat] :
( ( vEBT_invar_vebt @ T2 @ N )
=> ( time_htt @ ( option @ nat ) @ ( vEBT_vebt_assn_raw @ T2 @ Ti ) @ ( vEBT_vebt_predi @ Ti @ X2 )
@ ^ [R5: option @ nat] :
( times_times @ assn @ ( vEBT_vebt_assn_raw @ T2 @ Ti )
@ ( pure_assn
@ ( R5
= ( vEBT_vebt_pred @ T2 @ X2 ) ) ) )
@ ( 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_1983_htt__vebt__succi,axiom,
! [T2: vEBT_VEBT,N: nat,Ti: vEBT_VEBTi,X2: nat] :
( ( vEBT_invar_vebt @ T2 @ N )
=> ( time_htt @ ( option @ nat ) @ ( vEBT_vebt_assn_raw @ T2 @ Ti ) @ ( vEBT_vebt_succi @ Ti @ X2 )
@ ^ [R5: option @ nat] :
( times_times @ assn @ ( vEBT_vebt_assn_raw @ T2 @ Ti )
@ ( pure_assn
@ ( R5
= ( vEBT_vebt_succ @ T2 @ X2 ) ) ) )
@ ( 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_1984_log__ceil__idem,axiom,
! [X2: real] :
( ( ord_less_eq @ real @ ( one_one @ real ) @ X2 )
=> ( ( archimedean_ceiling @ real @ ( log @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) @ X2 ) )
= ( archimedean_ceiling @ real @ ( log @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) @ ( ring_1_of_int @ real @ ( archimedean_ceiling @ real @ X2 ) ) ) ) ) ) ).
% log_ceil_idem
thf(fact_1985_word__size__gt__0,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [W: word @ A] : ( ord_less @ nat @ ( zero_zero @ nat ) @ ( size_size @ ( word @ A ) @ W ) ) ) ).
% word_size_gt_0
thf(fact_1986_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,
! [X2: vEBT_VEBT,Xa: nat,Y3: nat] :
( ( ( vEBT_V1232361888498592333_e_t_e @ X2 @ Xa )
= Y3 )
=> ( ( accp @ ( product_prod @ vEBT_VEBT @ nat ) @ vEBT_V6368547301243506412_e_rel @ ( product_Pair @ vEBT_VEBT @ nat @ X2 @ Xa ) )
=> ( ! [A4: $o,B4: $o] :
( ( X2
= ( vEBT_Leaf @ A4 @ B4 ) )
=> ( ( Xa
= ( zero_zero @ nat ) )
=> ( ( Y3
= ( 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] :
( ( X2
= ( vEBT_Leaf @ A4 @ B4 ) )
=> ( ( Xa
= ( suc @ ( zero_zero @ nat ) ) )
=> ( ( Y3
= ( 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] :
( ( X2
= ( vEBT_Leaf @ A4 @ B4 ) )
=> ! [N3: nat] :
( ( Xa
= ( suc @ ( suc @ N3 ) ) )
=> ( ( Y3
= ( one_one @ nat ) )
=> ~ ( accp @ ( product_prod @ vEBT_VEBT @ nat ) @ vEBT_V6368547301243506412_e_rel @ ( product_Pair @ vEBT_VEBT @ nat @ ( vEBT_Leaf @ A4 @ B4 ) @ ( suc @ ( suc @ N3 ) ) ) ) ) ) )
=> ( ! [Deg2: nat,TreeList2: list @ vEBT_VEBT,Summary2: vEBT_VEBT] :
( ( X2
= ( vEBT_Node @ ( none @ ( product_prod @ nat @ nat ) ) @ Deg2 @ TreeList2 @ Summary2 ) )
=> ( ( Y3
= ( one_one @ nat ) )
=> ~ ( accp @ ( product_prod @ vEBT_VEBT @ nat ) @ vEBT_V6368547301243506412_e_rel @ ( product_Pair @ vEBT_VEBT @ nat @ ( vEBT_Node @ ( none @ ( product_prod @ nat @ nat ) ) @ Deg2 @ TreeList2 @ Summary2 ) @ Xa ) ) ) )
=> ( ! [Mi2: nat,Ma2: nat,TreeList2: list @ vEBT_VEBT,Summary2: vEBT_VEBT] :
( ( X2
= ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi2 @ Ma2 ) ) @ ( zero_zero @ nat ) @ TreeList2 @ Summary2 ) )
=> ( ( Y3
= ( one_one @ nat ) )
=> ~ ( accp @ ( product_prod @ vEBT_VEBT @ nat ) @ vEBT_V6368547301243506412_e_rel @ ( product_Pair @ vEBT_VEBT @ nat @ ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi2 @ Ma2 ) ) @ ( zero_zero @ nat ) @ TreeList2 @ Summary2 ) @ Xa ) ) ) )
=> ( ! [Mi2: nat,Ma2: nat,TreeList2: list @ vEBT_VEBT,Summary2: vEBT_VEBT] :
( ( X2
= ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( zero_zero @ nat ) ) @ TreeList2 @ Summary2 ) )
=> ( ( Y3
= ( one_one @ nat ) )
=> ~ ( accp @ ( product_prod @ vEBT_VEBT @ nat ) @ vEBT_V6368547301243506412_e_rel @ ( product_Pair @ vEBT_VEBT @ nat @ ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( zero_zero @ nat ) ) @ TreeList2 @ Summary2 ) @ Xa ) ) ) )
=> ~ ! [Mi2: nat,Ma2: nat,Va3: nat,TreeList2: list @ vEBT_VEBT,Summary2: vEBT_VEBT] :
( ( X2
= ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( suc @ Va3 ) ) @ TreeList2 @ Summary2 ) )
=> ( ( ( ( ( ord_less @ nat @ Xa @ Mi2 )
| ( ord_less @ nat @ Ma2 @ Xa ) )
=> ( Y3
= ( one_one @ nat ) ) )
& ( ~ ( ( ord_less @ nat @ Xa @ Mi2 )
| ( ord_less @ nat @ Ma2 @ Xa ) )
=> ( ( ( ( Xa = Mi2 )
& ( Xa = Ma2 ) )
=> ( Y3
= ( one_one @ nat ) ) )
& ( ~ ( ( Xa = Mi2 )
& ( Xa = Ma2 ) )
=> ( Y3
= ( if @ nat @ ( ord_less @ nat @ ( vEBT_VEBT_high @ ( if @ nat @ ( Xa = Mi2 ) @ ( plus_plus @ nat @ ( times_times @ nat @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary2 ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList2 @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary2 ) ) ) ) ) ) @ Xa ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( size_size @ ( list @ vEBT_VEBT ) @ TreeList2 ) ) @ ( plus_plus @ nat @ ( vEBT_V1232361888498592333_e_t_e @ ( nth @ vEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ ( if @ nat @ ( Xa = Mi2 ) @ ( plus_plus @ nat @ ( times_times @ nat @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary2 ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList2 @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary2 ) ) ) ) ) ) @ Xa ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( vEBT_VEBT_low @ ( if @ nat @ ( Xa = Mi2 ) @ ( plus_plus @ nat @ ( times_times @ nat @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary2 ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList2 @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary2 ) ) ) ) ) ) @ Xa ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( if @ nat @ ( vEBT_VEBT_minNull @ ( vEBT_vebt_delete @ ( nth @ vEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ ( if @ nat @ ( Xa = Mi2 ) @ ( plus_plus @ nat @ ( times_times @ nat @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary2 ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList2 @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary2 ) ) ) ) ) ) @ Xa ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( vEBT_VEBT_low @ ( if @ nat @ ( Xa = Mi2 ) @ ( plus_plus @ nat @ ( times_times @ nat @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary2 ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList2 @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary2 ) ) ) ) ) ) @ Xa ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) @ ( vEBT_V1232361888498592333_e_t_e @ Summary2 @ ( vEBT_VEBT_high @ ( if @ nat @ ( Xa = Mi2 ) @ ( plus_plus @ nat @ ( times_times @ nat @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary2 ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList2 @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary2 ) ) ) ) ) ) @ Xa ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va3 ) ) @ ( 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 @ Va3 ) ) @ TreeList2 @ Summary2 ) @ Xa ) ) ) ) ) ) ) ) ) ) ) ) ).
% VEBT_internal.T\<^sub>d\<^sub>e\<^sub>l\<^sub>e\<^sub>t\<^sub>e'.pelims
thf(fact_1987_word__le__sub1__numberof,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [W: num] :
( ( ( numeral_numeral @ ( word @ A ) @ W )
!= ( zero_zero @ ( word @ A ) ) )
=> ( ( ord_less_eq @ ( word @ A ) @ ( one_one @ ( word @ A ) ) @ ( numeral_numeral @ ( word @ A ) @ W ) )
= ( ord_less_eq @ ( word @ A ) @ ( zero_zero @ ( word @ A ) ) @ ( minus_minus @ ( word @ A ) @ ( numeral_numeral @ ( word @ A ) @ W ) @ ( one_one @ ( word @ A ) ) ) ) ) ) ) ).
% word_le_sub1_numberof
thf(fact_1988_word__less__1,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [X2: word @ A] :
( ( ord_less @ ( word @ A ) @ X2 @ ( one_one @ ( word @ A ) ) )
= ( X2
= ( zero_zero @ ( word @ A ) ) ) ) ) ).
% word_less_1
thf(fact_1989_of__int__numeral,axiom,
! [A: $tType] :
( ( ring_1 @ A )
=> ! [K: num] :
( ( ring_1_of_int @ A @ ( numeral_numeral @ int @ K ) )
= ( numeral_numeral @ A @ K ) ) ) ).
% of_int_numeral
thf(fact_1990_of__int__eq__numeral__iff,axiom,
! [A: $tType] :
( ( ring_char_0 @ A )
=> ! [Z: int,N: num] :
( ( ( ring_1_of_int @ A @ Z )
= ( numeral_numeral @ A @ N ) )
= ( Z
= ( numeral_numeral @ int @ N ) ) ) ) ).
% of_int_eq_numeral_iff
thf(fact_1991_of__int__le__iff,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [W: int,Z: int] :
( ( ord_less_eq @ A @ ( ring_1_of_int @ A @ W ) @ ( ring_1_of_int @ A @ Z ) )
= ( ord_less_eq @ int @ W @ Z ) ) ) ).
% of_int_le_iff
thf(fact_1992_of__int__less__iff,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [W: int,Z: int] :
( ( ord_less @ A @ ( ring_1_of_int @ A @ W ) @ ( ring_1_of_int @ A @ Z ) )
= ( ord_less @ int @ W @ Z ) ) ) ).
% of_int_less_iff
thf(fact_1993_of__int__eq__1__iff,axiom,
! [A: $tType] :
( ( ring_char_0 @ A )
=> ! [Z: int] :
( ( ( ring_1_of_int @ A @ Z )
= ( one_one @ A ) )
= ( Z
= ( one_one @ int ) ) ) ) ).
% of_int_eq_1_iff
thf(fact_1994_of__int__1,axiom,
! [A: $tType] :
( ( ring_1 @ A )
=> ( ( ring_1_of_int @ A @ ( one_one @ int ) )
= ( one_one @ A ) ) ) ).
% of_int_1
thf(fact_1995_of__int__add,axiom,
! [A: $tType] :
( ( ring_1 @ A )
=> ! [W: int,Z: int] :
( ( ring_1_of_int @ A @ ( plus_plus @ int @ W @ Z ) )
= ( plus_plus @ A @ ( ring_1_of_int @ A @ W ) @ ( ring_1_of_int @ A @ Z ) ) ) ) ).
% of_int_add
thf(fact_1996_of__int__mult,axiom,
! [A: $tType] :
( ( ring_1 @ A )
=> ! [W: int,Z: int] :
( ( ring_1_of_int @ A @ ( times_times @ int @ W @ Z ) )
= ( times_times @ A @ ( ring_1_of_int @ A @ W ) @ ( ring_1_of_int @ A @ Z ) ) ) ) ).
% of_int_mult
thf(fact_1997_nat__numeral,axiom,
! [K: num] :
( ( nat2 @ ( numeral_numeral @ int @ K ) )
= ( numeral_numeral @ nat @ K ) ) ).
% nat_numeral
thf(fact_1998_of__int__diff,axiom,
! [A: $tType] :
( ( ring_1 @ A )
=> ! [W: int,Z: int] :
( ( ring_1_of_int @ A @ ( minus_minus @ int @ W @ Z ) )
= ( minus_minus @ A @ ( ring_1_of_int @ A @ W ) @ ( ring_1_of_int @ A @ Z ) ) ) ) ).
% of_int_diff
thf(fact_1999_of__int__power__eq__of__int__cancel__iff,axiom,
! [A: $tType] :
( ( ring_char_0 @ A )
=> ! [X2: int,B3: int,W: nat] :
( ( ( ring_1_of_int @ A @ X2 )
= ( power_power @ A @ ( ring_1_of_int @ A @ B3 ) @ W ) )
= ( X2
= ( power_power @ int @ B3 @ W ) ) ) ) ).
% of_int_power_eq_of_int_cancel_iff
thf(fact_2000_of__int__eq__of__int__power__cancel__iff,axiom,
! [A: $tType] :
( ( ring_char_0 @ A )
=> ! [B3: int,W: nat,X2: int] :
( ( ( power_power @ A @ ( ring_1_of_int @ A @ B3 ) @ W )
= ( ring_1_of_int @ A @ X2 ) )
= ( ( power_power @ int @ B3 @ W )
= X2 ) ) ) ).
% of_int_eq_of_int_power_cancel_iff
thf(fact_2001_of__int__power,axiom,
! [A: $tType] :
( ( ring_1 @ A )
=> ! [Z: int,N: nat] :
( ( ring_1_of_int @ A @ ( power_power @ int @ Z @ N ) )
= ( power_power @ A @ ( ring_1_of_int @ A @ Z ) @ N ) ) ) ).
% of_int_power
thf(fact_2002_word__less__sub1__numberof,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [W: num] :
( ( ( numeral_numeral @ ( word @ A ) @ W )
!= ( zero_zero @ ( word @ A ) ) )
=> ( ( ord_less @ ( word @ A ) @ ( one_one @ ( word @ A ) ) @ ( numeral_numeral @ ( word @ A ) @ W ) )
= ( ord_less @ ( word @ A ) @ ( zero_zero @ ( word @ A ) ) @ ( minus_minus @ ( word @ A ) @ ( numeral_numeral @ ( word @ A ) @ W ) @ ( one_one @ ( word @ A ) ) ) ) ) ) ) ).
% word_less_sub1_numberof
thf(fact_2003_nat__1,axiom,
( ( nat2 @ ( one_one @ int ) )
= ( suc @ ( zero_zero @ nat ) ) ) ).
% nat_1
thf(fact_2004_nat__le__0,axiom,
! [Z: int] :
( ( ord_less_eq @ int @ Z @ ( zero_zero @ int ) )
=> ( ( nat2 @ Z )
= ( zero_zero @ nat ) ) ) ).
% nat_le_0
thf(fact_2005_nat__0__iff,axiom,
! [I: int] :
( ( ( nat2 @ I )
= ( zero_zero @ nat ) )
= ( ord_less_eq @ int @ I @ ( zero_zero @ int ) ) ) ).
% nat_0_iff
thf(fact_2006_zless__nat__conj,axiom,
! [W: int,Z: int] :
( ( ord_less @ nat @ ( nat2 @ W ) @ ( nat2 @ Z ) )
= ( ( ord_less @ int @ ( zero_zero @ int ) @ Z )
& ( ord_less @ int @ W @ Z ) ) ) ).
% zless_nat_conj
thf(fact_2007_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_2008_ceiling__add__of__int,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [X2: A,Z: int] :
( ( archimedean_ceiling @ A @ ( plus_plus @ A @ X2 @ ( ring_1_of_int @ A @ Z ) ) )
= ( plus_plus @ int @ ( archimedean_ceiling @ A @ X2 ) @ Z ) ) ) ).
% ceiling_add_of_int
thf(fact_2009_ceiling__diff__of__int,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [X2: A,Z: int] :
( ( archimedean_ceiling @ A @ ( minus_minus @ A @ X2 @ ( ring_1_of_int @ A @ Z ) ) )
= ( minus_minus @ int @ ( archimedean_ceiling @ A @ X2 ) @ Z ) ) ) ).
% ceiling_diff_of_int
thf(fact_2010_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_2011_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_2012_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_2013_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_2014_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_2015_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_2016_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_2017_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_2018_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_2019_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_2020_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_2021_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_2022_of__int__eq__numeral__power__cancel__iff,axiom,
! [A: $tType] :
( ( ring_char_0 @ A )
=> ! [Y3: int,X2: num,N: nat] :
( ( ( ring_1_of_int @ A @ Y3 )
= ( power_power @ A @ ( numeral_numeral @ A @ X2 ) @ N ) )
= ( Y3
= ( power_power @ int @ ( numeral_numeral @ int @ X2 ) @ N ) ) ) ) ).
% of_int_eq_numeral_power_cancel_iff
thf(fact_2023_numeral__power__eq__of__int__cancel__iff,axiom,
! [A: $tType] :
( ( ring_char_0 @ A )
=> ! [X2: num,N: nat,Y3: int] :
( ( ( power_power @ A @ ( numeral_numeral @ A @ X2 ) @ N )
= ( ring_1_of_int @ A @ Y3 ) )
= ( ( power_power @ int @ ( numeral_numeral @ int @ X2 ) @ N )
= Y3 ) ) ) ).
% numeral_power_eq_of_int_cancel_iff
thf(fact_2024_of__int__power__le__of__int__cancel__iff,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [X2: int,B3: int,W: nat] :
( ( ord_less_eq @ A @ ( ring_1_of_int @ A @ X2 ) @ ( power_power @ A @ ( ring_1_of_int @ A @ B3 ) @ W ) )
= ( ord_less_eq @ int @ X2 @ ( power_power @ int @ B3 @ W ) ) ) ) ).
% of_int_power_le_of_int_cancel_iff
thf(fact_2025_of__int__le__of__int__power__cancel__iff,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [B3: int,W: nat,X2: int] :
( ( ord_less_eq @ A @ ( power_power @ A @ ( ring_1_of_int @ A @ B3 ) @ W ) @ ( ring_1_of_int @ A @ X2 ) )
= ( ord_less_eq @ int @ ( power_power @ int @ B3 @ W ) @ X2 ) ) ) ).
% of_int_le_of_int_power_cancel_iff
thf(fact_2026_of__int__power__less__of__int__cancel__iff,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [X2: int,B3: int,W: nat] :
( ( ord_less @ A @ ( ring_1_of_int @ A @ X2 ) @ ( power_power @ A @ ( ring_1_of_int @ A @ B3 ) @ W ) )
= ( ord_less @ int @ X2 @ ( power_power @ int @ B3 @ W ) ) ) ) ).
% of_int_power_less_of_int_cancel_iff
thf(fact_2027_of__int__less__of__int__power__cancel__iff,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [B3: int,W: nat,X2: int] :
( ( ord_less @ A @ ( power_power @ A @ ( ring_1_of_int @ A @ B3 ) @ W ) @ ( ring_1_of_int @ A @ X2 ) )
= ( ord_less @ int @ ( power_power @ int @ B3 @ W ) @ X2 ) ) ) ).
% of_int_less_of_int_power_cancel_iff
thf(fact_2028_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_2029_diff__nat__numeral,axiom,
! [V: num,V4: num] :
( ( minus_minus @ nat @ ( numeral_numeral @ nat @ V ) @ ( numeral_numeral @ nat @ V4 ) )
= ( nat2 @ ( minus_minus @ int @ ( numeral_numeral @ int @ V ) @ ( numeral_numeral @ int @ V4 ) ) ) ) ).
% diff_nat_numeral
thf(fact_2030_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_2031_numeral__power__eq__nat__cancel__iff,axiom,
! [X2: num,N: nat,Y3: int] :
( ( ( power_power @ nat @ ( numeral_numeral @ nat @ X2 ) @ N )
= ( nat2 @ Y3 ) )
= ( ( power_power @ int @ ( numeral_numeral @ int @ X2 ) @ N )
= Y3 ) ) ).
% numeral_power_eq_nat_cancel_iff
thf(fact_2032_nat__eq__numeral__power__cancel__iff,axiom,
! [Y3: int,X2: num,N: nat] :
( ( ( nat2 @ Y3 )
= ( power_power @ nat @ ( numeral_numeral @ nat @ X2 ) @ N ) )
= ( Y3
= ( power_power @ int @ ( numeral_numeral @ int @ X2 ) @ N ) ) ) ).
% nat_eq_numeral_power_cancel_iff
thf(fact_2033_nat__ceiling__le__eq,axiom,
! [X2: real,A3: nat] :
( ( ord_less_eq @ nat @ ( nat2 @ ( archimedean_ceiling @ real @ X2 ) ) @ A3 )
= ( ord_less_eq @ real @ X2 @ ( semiring_1_of_nat @ real @ A3 ) ) ) ).
% nat_ceiling_le_eq
thf(fact_2034_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_2035_numeral__power__le__of__int__cancel__iff,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [X2: num,N: nat,A3: int] :
( ( ord_less_eq @ A @ ( power_power @ A @ ( numeral_numeral @ A @ X2 ) @ N ) @ ( ring_1_of_int @ A @ A3 ) )
= ( ord_less_eq @ int @ ( power_power @ int @ ( numeral_numeral @ int @ X2 ) @ N ) @ A3 ) ) ) ).
% numeral_power_le_of_int_cancel_iff
thf(fact_2036_of__int__le__numeral__power__cancel__iff,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [A3: int,X2: num,N: nat] :
( ( ord_less_eq @ A @ ( ring_1_of_int @ A @ A3 ) @ ( power_power @ A @ ( numeral_numeral @ A @ X2 ) @ N ) )
= ( ord_less_eq @ int @ A3 @ ( power_power @ int @ ( numeral_numeral @ int @ X2 ) @ N ) ) ) ) ).
% of_int_le_numeral_power_cancel_iff
thf(fact_2037_of__int__less__numeral__power__cancel__iff,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [A3: int,X2: num,N: nat] :
( ( ord_less @ A @ ( ring_1_of_int @ A @ A3 ) @ ( power_power @ A @ ( numeral_numeral @ A @ X2 ) @ N ) )
= ( ord_less @ int @ A3 @ ( power_power @ int @ ( numeral_numeral @ int @ X2 ) @ N ) ) ) ) ).
% of_int_less_numeral_power_cancel_iff
thf(fact_2038_numeral__power__less__of__int__cancel__iff,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [X2: num,N: nat,A3: int] :
( ( ord_less @ A @ ( power_power @ A @ ( numeral_numeral @ A @ X2 ) @ N ) @ ( ring_1_of_int @ A @ A3 ) )
= ( ord_less @ int @ ( power_power @ int @ ( numeral_numeral @ int @ X2 ) @ N ) @ A3 ) ) ) ).
% numeral_power_less_of_int_cancel_iff
thf(fact_2039_nat__numeral__diff__1,axiom,
! [V: num] :
( ( minus_minus @ nat @ ( numeral_numeral @ nat @ V ) @ ( one_one @ nat ) )
= ( nat2 @ ( minus_minus @ int @ ( numeral_numeral @ int @ V ) @ ( one_one @ int ) ) ) ) ).
% nat_numeral_diff_1
thf(fact_2040_numeral__power__less__nat__cancel__iff,axiom,
! [X2: num,N: nat,A3: int] :
( ( ord_less @ nat @ ( power_power @ nat @ ( numeral_numeral @ nat @ X2 ) @ N ) @ ( nat2 @ A3 ) )
= ( ord_less @ int @ ( power_power @ int @ ( numeral_numeral @ int @ X2 ) @ N ) @ A3 ) ) ).
% numeral_power_less_nat_cancel_iff
thf(fact_2041_nat__less__numeral__power__cancel__iff,axiom,
! [A3: int,X2: num,N: nat] :
( ( ord_less @ nat @ ( nat2 @ A3 ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ X2 ) @ N ) )
= ( ord_less @ int @ A3 @ ( power_power @ int @ ( numeral_numeral @ int @ X2 ) @ N ) ) ) ).
% nat_less_numeral_power_cancel_iff
thf(fact_2042_nat__le__numeral__power__cancel__iff,axiom,
! [A3: int,X2: num,N: nat] :
( ( ord_less_eq @ nat @ ( nat2 @ A3 ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ X2 ) @ N ) )
= ( ord_less_eq @ int @ A3 @ ( power_power @ int @ ( numeral_numeral @ int @ X2 ) @ N ) ) ) ).
% nat_le_numeral_power_cancel_iff
thf(fact_2043_numeral__power__le__nat__cancel__iff,axiom,
! [X2: num,N: nat,A3: int] :
( ( ord_less_eq @ nat @ ( power_power @ nat @ ( numeral_numeral @ nat @ X2 ) @ N ) @ ( nat2 @ A3 ) )
= ( ord_less_eq @ int @ ( power_power @ int @ ( numeral_numeral @ int @ X2 ) @ N ) @ A3 ) ) ).
% numeral_power_le_nat_cancel_iff
thf(fact_2044_word__subset__less,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [X2: word @ A,R2: word @ A,Y3: word @ A,S2: word @ A] :
( ( ord_less_eq @ ( set @ ( word @ A ) ) @ ( set_or1337092689740270186AtMost @ ( word @ A ) @ X2 @ ( minus_minus @ ( word @ A ) @ ( plus_plus @ ( word @ A ) @ X2 @ R2 ) @ ( one_one @ ( word @ A ) ) ) ) @ ( set_or1337092689740270186AtMost @ ( word @ A ) @ Y3 @ ( minus_minus @ ( word @ A ) @ ( plus_plus @ ( word @ A ) @ Y3 @ S2 ) @ ( one_one @ ( word @ A ) ) ) ) )
=> ( ( ord_less_eq @ ( word @ A ) @ X2 @ ( minus_minus @ ( word @ A ) @ ( plus_plus @ ( word @ A ) @ X2 @ R2 ) @ ( one_one @ ( word @ A ) ) ) )
=> ( ( ord_less_eq @ ( word @ A ) @ Y3 @ ( minus_minus @ ( word @ A ) @ ( plus_plus @ ( word @ A ) @ Y3 @ S2 ) @ ( one_one @ ( word @ A ) ) ) )
=> ( ( S2
!= ( zero_zero @ ( word @ A ) ) )
=> ( ord_less_eq @ ( word @ A ) @ R2 @ S2 ) ) ) ) ) ) ).
% word_subset_less
thf(fact_2045_word__overflow,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [X2: word @ A] :
( ( ord_less @ ( word @ A ) @ X2 @ ( plus_plus @ ( word @ A ) @ X2 @ ( one_one @ ( word @ A ) ) ) )
| ( ( plus_plus @ ( word @ A ) @ X2 @ ( one_one @ ( word @ A ) ) )
= ( zero_zero @ ( word @ A ) ) ) ) ) ).
% word_overflow
thf(fact_2046_word__gr0__conv__Suc,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [M: word @ A] :
( ( ord_less @ ( word @ A ) @ ( zero_zero @ ( word @ A ) ) @ M )
=> ? [N3: word @ A] :
( M
= ( plus_plus @ ( word @ A ) @ N3 @ ( one_one @ ( word @ A ) ) ) ) ) ) ).
% word_gr0_conv_Suc
thf(fact_2047_less__is__non__zero__p1,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [A3: word @ A,K: word @ A] :
( ( ord_less @ ( word @ A ) @ A3 @ K )
=> ( ( plus_plus @ ( word @ A ) @ A3 @ ( one_one @ ( word @ A ) ) )
!= ( zero_zero @ ( word @ A ) ) ) ) ) ).
% less_is_non_zero_p1
thf(fact_2048_div__less__dividend__word,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [X2: word @ A,N: word @ A] :
( ( X2
!= ( zero_zero @ ( word @ A ) ) )
=> ( ( N
!= ( one_one @ ( word @ A ) ) )
=> ( ord_less @ ( word @ A ) @ ( divide_divide @ ( word @ A ) @ X2 @ N ) @ X2 ) ) ) ) ).
% div_less_dividend_word
thf(fact_2049_word__div__1,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [N: word @ A] :
( ( divide_divide @ ( word @ A ) @ N @ ( one_one @ ( word @ A ) ) )
= N ) ) ).
% word_div_1
thf(fact_2050_word__less__cases,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [X2: word @ A,Y3: word @ A] :
( ( ord_less @ ( word @ A ) @ X2 @ Y3 )
=> ( ( X2
= ( minus_minus @ ( word @ A ) @ Y3 @ ( one_one @ ( word @ A ) ) ) )
| ( ord_less @ ( word @ A ) @ X2 @ ( minus_minus @ ( word @ A ) @ Y3 @ ( one_one @ ( word @ A ) ) ) ) ) ) ) ).
% word_less_cases
thf(fact_2051_gt0__iff__gem1,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [X2: word @ A] :
( ( ord_less @ ( word @ A ) @ ( zero_zero @ ( word @ A ) ) @ X2 )
= ( ord_less @ ( word @ A ) @ ( minus_minus @ ( word @ A ) @ X2 @ ( one_one @ ( word @ A ) ) ) @ X2 ) ) ) ).
% gt0_iff_gem1
thf(fact_2052_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_2053_le__m1__iff__lt,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [X2: word @ A,Y3: word @ A] :
( ( ord_less @ ( word @ A ) @ ( zero_zero @ ( word @ A ) ) @ X2 )
= ( ( ord_less_eq @ ( word @ A ) @ Y3 @ ( minus_minus @ ( word @ A ) @ X2 @ ( one_one @ ( word @ A ) ) ) )
= ( ord_less @ ( word @ A ) @ Y3 @ X2 ) ) ) ) ).
% le_m1_iff_lt
thf(fact_2054_word__div__sub,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [Y3: word @ A,X2: word @ A] :
( ( ord_less_eq @ ( word @ A ) @ Y3 @ X2 )
=> ( ( ord_less @ ( word @ A ) @ ( zero_zero @ ( word @ A ) ) @ Y3 )
=> ( ( divide_divide @ ( word @ A ) @ ( minus_minus @ ( word @ A ) @ X2 @ Y3 ) @ Y3 )
= ( minus_minus @ ( word @ A ) @ ( divide_divide @ ( word @ A ) @ X2 @ Y3 ) @ ( one_one @ ( word @ A ) ) ) ) ) ) ) ).
% word_div_sub
thf(fact_2055_word__leq__le__minus__one,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [X2: word @ A,Y3: word @ A] :
( ( ord_less_eq @ ( word @ A ) @ X2 @ Y3 )
=> ( ( X2
!= ( zero_zero @ ( word @ A ) ) )
=> ( ord_less @ ( word @ A ) @ ( minus_minus @ ( word @ A ) @ X2 @ ( one_one @ ( word @ A ) ) ) @ Y3 ) ) ) ) ).
% word_leq_le_minus_one
thf(fact_2056_word__leq__minus__one__le,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [Y3: word @ A,X2: word @ A] :
( ( Y3
!= ( zero_zero @ ( word @ A ) ) )
=> ( ( ord_less_eq @ ( word @ A ) @ X2 @ ( minus_minus @ ( word @ A ) @ Y3 @ ( one_one @ ( word @ A ) ) ) )
=> ( ord_less @ ( word @ A ) @ X2 @ Y3 ) ) ) ) ).
% word_leq_minus_one_le
thf(fact_2057_word__sub__plus__one__nonzero,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [N2: word @ A,N: word @ A] :
( ( ord_less_eq @ ( word @ A ) @ N2 @ N )
=> ( ( N2
!= ( zero_zero @ ( word @ A ) ) )
=> ( ( plus_plus @ ( word @ A ) @ ( minus_minus @ ( word @ A ) @ N @ N2 ) @ ( one_one @ ( word @ A ) ) )
!= ( zero_zero @ ( word @ A ) ) ) ) ) ) ).
% word_sub_plus_one_nonzero
thf(fact_2058_word__must__wrap,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [X2: word @ A,N: word @ A] :
( ( ord_less_eq @ ( word @ A ) @ X2 @ ( minus_minus @ ( word @ A ) @ N @ ( one_one @ ( word @ A ) ) ) )
=> ( ( ord_less_eq @ ( word @ A ) @ N @ X2 )
=> ( N
= ( zero_zero @ ( word @ A ) ) ) ) ) ) ).
% word_must_wrap
thf(fact_2059_word__sub__1__le,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [X2: word @ A] :
( ( X2
!= ( zero_zero @ ( word @ A ) ) )
=> ( ord_less_eq @ ( word @ A ) @ ( minus_minus @ ( word @ A ) @ X2 @ ( one_one @ ( word @ A ) ) ) @ X2 ) ) ) ).
% word_sub_1_le
thf(fact_2060_word__le__minus__one__leq,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [X2: word @ A,Y3: word @ A] :
( ( ord_less @ ( word @ A ) @ X2 @ Y3 )
=> ( ord_less_eq @ ( word @ A ) @ X2 @ ( minus_minus @ ( word @ A ) @ Y3 @ ( one_one @ ( word @ A ) ) ) ) ) ) ).
% word_le_minus_one_leq
thf(fact_2061_word__minus__one__le__leq,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [X2: word @ A,Y3: word @ A] :
( ( ord_less @ ( word @ A ) @ ( minus_minus @ ( word @ A ) @ X2 @ ( one_one @ ( word @ A ) ) ) @ Y3 )
=> ( ord_less_eq @ ( word @ A ) @ X2 @ Y3 ) ) ) ).
% word_minus_one_le_leq
thf(fact_2062_le__step__down__word__2,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [X2: word @ A,Y3: word @ A] :
( ( ord_less_eq @ ( word @ A ) @ X2 @ Y3 )
=> ( ( X2 != Y3 )
=> ( ord_less_eq @ ( word @ A ) @ X2 @ ( minus_minus @ ( word @ A ) @ Y3 @ ( one_one @ ( word @ A ) ) ) ) ) ) ) ).
% le_step_down_word_2
thf(fact_2063_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_2064_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_2065_word__diff__ls_H_H_I1_J,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [Y3: word @ A,X2: word @ A,W: word @ A] :
( ( ord_less @ ( word @ A ) @ ( minus_minus @ ( word @ A ) @ Y3 @ X2 ) @ ( minus_minus @ ( word @ A ) @ ( plus_plus @ ( word @ A ) @ W @ X2 ) @ X2 ) )
=> ( ( ord_less_eq @ ( word @ A ) @ X2 @ ( plus_plus @ ( word @ A ) @ W @ X2 ) )
=> ( ord_less @ ( word @ A ) @ Y3 @ ( plus_plus @ ( word @ A ) @ W @ X2 ) ) ) ) ) ).
% word_diff_ls''(1)
thf(fact_2066_word__diff__ls_H_H_I2_J,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [Y3: word @ A,W: word @ A,X2: word @ A] :
( ( ord_less @ ( word @ A ) @ Y3 @ ( plus_plus @ ( word @ A ) @ W @ X2 ) )
=> ( ( ord_less_eq @ ( word @ A ) @ X2 @ Y3 )
=> ( ord_less @ ( word @ A ) @ ( minus_minus @ ( word @ A ) @ Y3 @ X2 ) @ ( minus_minus @ ( word @ A ) @ ( plus_plus @ ( word @ A ) @ W @ X2 ) @ X2 ) ) ) ) ) ).
% word_diff_ls''(2)
thf(fact_2067_word__l__diffs_H_I1_J,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [W: word @ A,X2: word @ A,Z: word @ A] :
( ( ord_less @ ( word @ A ) @ ( minus_minus @ ( word @ A ) @ ( plus_plus @ ( word @ A ) @ W @ X2 ) @ X2 ) @ ( minus_minus @ ( word @ A ) @ Z @ X2 ) )
=> ( ( ord_less_eq @ ( word @ A ) @ X2 @ Z )
=> ( ord_less @ ( word @ A ) @ ( plus_plus @ ( word @ A ) @ W @ X2 ) @ Z ) ) ) ) ).
% word_l_diffs'(1)
thf(fact_2068_word__l__diffs_H_I2_J,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [W: word @ A,X2: word @ A,Z: word @ A] :
( ( ord_less @ ( word @ A ) @ ( plus_plus @ ( word @ A ) @ W @ X2 ) @ Z )
=> ( ( ord_less_eq @ ( word @ A ) @ X2 @ ( plus_plus @ ( word @ A ) @ W @ X2 ) )
=> ( ord_less @ ( word @ A ) @ ( minus_minus @ ( word @ A ) @ ( plus_plus @ ( word @ A ) @ W @ X2 ) @ X2 ) @ ( minus_minus @ ( word @ A ) @ Z @ X2 ) ) ) ) ) ).
% word_l_diffs'(2)
thf(fact_2069_word__diff__ls_H_I1_J,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [Y3: word @ A,X2: word @ A,W: word @ A] :
( ( ord_less @ ( word @ A ) @ ( minus_minus @ ( word @ A ) @ Y3 @ X2 ) @ W )
=> ( ( ord_less_eq @ ( word @ A ) @ X2 @ ( plus_plus @ ( word @ A ) @ W @ X2 ) )
=> ( ord_less @ ( word @ A ) @ Y3 @ ( plus_plus @ ( word @ A ) @ W @ X2 ) ) ) ) ) ).
% word_diff_ls'(1)
thf(fact_2070_word__diff__ls_H_I2_J,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [Y3: word @ A,W: word @ A,X2: word @ A] :
( ( ord_less @ ( word @ A ) @ Y3 @ ( plus_plus @ ( word @ A ) @ W @ X2 ) )
=> ( ( ord_less_eq @ ( word @ A ) @ X2 @ Y3 )
=> ( ord_less @ ( word @ A ) @ ( minus_minus @ ( word @ A ) @ Y3 @ X2 ) @ W ) ) ) ) ).
% word_diff_ls'(2)
thf(fact_2071_More__Word_Oword__l__diffs_I1_J,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [W: word @ A,Z: word @ A,X2: word @ A] :
( ( ord_less @ ( word @ A ) @ W @ ( minus_minus @ ( word @ A ) @ Z @ X2 ) )
=> ( ( ord_less_eq @ ( word @ A ) @ X2 @ Z )
=> ( ord_less @ ( word @ A ) @ ( plus_plus @ ( word @ A ) @ W @ X2 ) @ Z ) ) ) ) ).
% More_Word.word_l_diffs(1)
thf(fact_2072_More__Word_Oword__l__diffs_I2_J,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [W: word @ A,X2: word @ A,Z: word @ A] :
( ( ord_less @ ( word @ A ) @ ( plus_plus @ ( word @ A ) @ W @ X2 ) @ Z )
=> ( ( ord_less_eq @ ( word @ A ) @ X2 @ ( plus_plus @ ( word @ A ) @ W @ X2 ) )
=> ( ord_less @ ( word @ A ) @ W @ ( minus_minus @ ( word @ A ) @ Z @ X2 ) ) ) ) ) ).
% More_Word.word_l_diffs(2)
thf(fact_2073_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_2074_word__less__nowrapI_H,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [X2: word @ A,Z: word @ A,K: word @ A] :
( ( ord_less_eq @ ( word @ A ) @ X2 @ ( 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 ) @ X2 @ ( plus_plus @ ( word @ A ) @ X2 @ K ) ) ) ) ) ) ).
% word_less_nowrapI'
thf(fact_2075_div__to__mult__word__lt,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [X2: word @ A,Y3: word @ A,Z: word @ A] :
( ( ord_less_eq @ ( word @ A ) @ X2 @ ( divide_divide @ ( word @ A ) @ Y3 @ Z ) )
=> ( ord_less_eq @ ( word @ A ) @ ( times_times @ ( word @ A ) @ X2 @ Z ) @ Y3 ) ) ) ).
% div_to_mult_word_lt
thf(fact_2076_word__plus__strict__mono__right,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [Y3: word @ A,Z: word @ A,X2: word @ A] :
( ( ord_less @ ( word @ A ) @ Y3 @ Z )
=> ( ( ord_less_eq @ ( word @ A ) @ X2 @ ( plus_plus @ ( word @ A ) @ X2 @ Z ) )
=> ( ord_less @ ( word @ A ) @ ( plus_plus @ ( word @ A ) @ X2 @ Y3 ) @ ( plus_plus @ ( word @ A ) @ X2 @ Z ) ) ) ) ) ).
% word_plus_strict_mono_right
thf(fact_2077_word__le__plus,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [A3: word @ A,B3: word @ A,C3: word @ A] :
( ( ord_less @ ( word @ A ) @ A3 @ ( plus_plus @ ( word @ A ) @ A3 @ B3 ) )
=> ( ( ord_less @ ( word @ A ) @ C3 @ B3 )
=> ( ord_less_eq @ ( word @ A ) @ A3 @ ( plus_plus @ ( word @ A ) @ A3 @ C3 ) ) ) ) ) ).
% word_le_plus
thf(fact_2078_word__not__le,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [X2: word @ A,Y3: word @ A] :
( ( ~ ( ord_less_eq @ ( word @ A ) @ X2 @ Y3 ) )
= ( ord_less @ ( word @ A ) @ Y3 @ X2 ) ) ) ).
% word_not_le
thf(fact_2079_word__le__not__less,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ( ( ord_less_eq @ ( word @ A ) )
= ( ^ [B7: word @ A,A6: word @ A] :
~ ( ord_less @ ( word @ A ) @ A6 @ B7 ) ) ) ) ).
% word_le_not_less
thf(fact_2080_word__le__add,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [X2: word @ A,Y3: word @ A] :
( ( ord_less_eq @ ( word @ A ) @ X2 @ Y3 )
=> ? [N3: nat] :
( Y3
= ( plus_plus @ ( word @ A ) @ X2 @ ( semiring_1_of_nat @ ( word @ A ) @ N3 ) ) ) ) ) ).
% word_le_add
thf(fact_2081_word__1__0,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [A3: word @ A,B3: word @ A,X2: nat] :
( ( ord_less_eq @ ( word @ A ) @ ( plus_plus @ ( word @ A ) @ A3 @ ( one_one @ ( word @ A ) ) ) @ B3 )
=> ( ( ord_less @ ( word @ A ) @ A3 @ ( semiring_1_of_nat @ ( word @ A ) @ X2 ) )
=> ( ord_less @ ( word @ A ) @ A3 @ B3 ) ) ) ) ).
% word_1_0
thf(fact_2082_word__le__plus__1,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [Y3: word @ A,N: word @ A,A3: word @ A] :
( ( ord_less @ ( word @ A ) @ Y3 @ ( plus_plus @ ( word @ A ) @ Y3 @ N ) )
=> ( ( ord_less @ ( word @ A ) @ A3 @ N )
=> ( ord_less_eq @ ( word @ A ) @ ( plus_plus @ ( word @ A ) @ Y3 @ A3 ) @ ( plus_plus @ ( word @ A ) @ ( plus_plus @ ( word @ A ) @ Y3 @ A3 ) @ ( one_one @ ( word @ A ) ) ) ) ) ) ) ).
% word_le_plus_1
thf(fact_2083_plus__one__helper,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [X2: word @ A,N: word @ A] :
( ( ord_less @ ( word @ A ) @ X2 @ ( plus_plus @ ( word @ A ) @ N @ ( one_one @ ( word @ A ) ) ) )
=> ( ord_less_eq @ ( word @ A ) @ X2 @ N ) ) ) ).
% plus_one_helper
thf(fact_2084_word__plus__one__nonzero,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [X2: word @ A,Y3: word @ A] :
( ( ord_less_eq @ ( word @ A ) @ X2 @ ( plus_plus @ ( word @ A ) @ X2 @ Y3 ) )
=> ( ( Y3
!= ( zero_zero @ ( word @ A ) ) )
=> ( ( plus_plus @ ( word @ A ) @ X2 @ ( one_one @ ( word @ A ) ) )
!= ( zero_zero @ ( word @ A ) ) ) ) ) ) ).
% word_plus_one_nonzero
thf(fact_2085_lt1__neq0,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [X2: word @ A] :
( ( ord_less_eq @ ( word @ A ) @ ( one_one @ ( word @ A ) ) @ X2 )
= ( X2
!= ( zero_zero @ ( word @ A ) ) ) ) ) ).
% lt1_neq0
thf(fact_2086_plus__one__helper2,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [X2: word @ A,N: word @ A] :
( ( ord_less_eq @ ( word @ A ) @ X2 @ N )
=> ( ( ( plus_plus @ ( word @ A ) @ N @ ( one_one @ ( word @ A ) ) )
!= ( zero_zero @ ( word @ A ) ) )
=> ( ord_less @ ( word @ A ) @ X2 @ ( plus_plus @ ( word @ A ) @ N @ ( one_one @ ( word @ A ) ) ) ) ) ) ) ).
% plus_one_helper2
thf(fact_2087_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_2088_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_2089_word__le__less__eq,axiom,
! [Z6: $tType] :
( ( type_len @ Z6 )
=> ( ( ord_less_eq @ ( word @ Z6 ) )
= ( ^ [X3: word @ Z6,Y: word @ Z6] :
( ( X3 = Y )
| ( ord_less @ ( word @ Z6 ) @ X3 @ Y ) ) ) ) ) ).
% word_le_less_eq
thf(fact_2090_plus__le__left__cancel__nowrap,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [X2: word @ A,Y7: word @ A,Y3: word @ A] :
( ( ord_less_eq @ ( word @ A ) @ X2 @ ( plus_plus @ ( word @ A ) @ X2 @ Y7 ) )
=> ( ( ord_less_eq @ ( word @ A ) @ X2 @ ( plus_plus @ ( word @ A ) @ X2 @ Y3 ) )
=> ( ( ord_less @ ( word @ A ) @ ( plus_plus @ ( word @ A ) @ X2 @ Y7 ) @ ( plus_plus @ ( word @ A ) @ X2 @ Y3 ) )
= ( ord_less @ ( word @ A ) @ Y7 @ Y3 ) ) ) ) ) ).
% plus_le_left_cancel_nowrap
thf(fact_2091_word__div__mult__le,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [A3: word @ A,B3: word @ A] : ( ord_less_eq @ ( word @ A ) @ ( times_times @ ( word @ A ) @ ( divide_divide @ ( word @ A ) @ A3 @ B3 ) @ B3 ) @ A3 ) ) ).
% word_div_mult_le
thf(fact_2092_div__le__mult,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [I: word @ A,K: word @ A,X2: word @ A] :
( ( ord_less_eq @ ( word @ A ) @ I @ ( divide_divide @ ( word @ A ) @ K @ X2 ) )
=> ( ( ord_less @ ( word @ A ) @ ( zero_zero @ ( word @ A ) ) @ X2 )
=> ( ord_less_eq @ ( word @ A ) @ ( times_times @ ( word @ A ) @ I @ X2 ) @ K ) ) ) ) ).
% div_le_mult
thf(fact_2093_word__less__nowrapI,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [X2: word @ A,Z: word @ A,K: word @ A] :
( ( ord_less @ ( word @ A ) @ X2 @ ( 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 ) @ X2 @ ( plus_plus @ ( word @ A ) @ X2 @ K ) ) ) ) ) ) ).
% word_less_nowrapI
thf(fact_2094_plus__minus__not__NULL,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [X2: word @ A,Ab: word @ A,C3: word @ A] :
( ( ord_less @ ( word @ A ) @ X2 @ ( minus_minus @ ( word @ A ) @ Ab @ C3 ) )
=> ( ( ord_less_eq @ ( word @ A ) @ C3 @ Ab )
=> ( ( C3
!= ( zero_zero @ ( word @ A ) ) )
=> ( ( plus_plus @ ( word @ A ) @ X2 @ C3 )
!= ( zero_zero @ ( word @ A ) ) ) ) ) ) ) ).
% plus_minus_not_NULL
thf(fact_2095_sub__wrap,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [X2: word @ A,Z: word @ A] :
( ( ord_less_eq @ ( word @ A ) @ X2 @ ( minus_minus @ ( word @ A ) @ X2 @ Z ) )
= ( ( Z
= ( zero_zero @ ( word @ A ) ) )
| ( ord_less @ ( word @ A ) @ X2 @ Z ) ) ) ) ).
% sub_wrap
thf(fact_2096_word__diff__ls_I2_J,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [Y3: word @ A,W: word @ A,Xa: word @ A,X2: word @ A] :
( ( ord_less @ ( word @ A ) @ Y3 @ ( plus_plus @ ( word @ A ) @ W @ Xa ) )
=> ( ( ord_less_eq @ ( word @ A ) @ X2 @ Y3 )
=> ( ord_less @ ( word @ A ) @ ( minus_minus @ ( word @ A ) @ Y3 @ X2 ) @ ( minus_minus @ ( word @ A ) @ ( plus_plus @ ( word @ A ) @ W @ Xa ) @ X2 ) ) ) ) ) ).
% word_diff_ls(2)
thf(fact_2097_word__diff__ls_I1_J,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [Y3: word @ A,X2: word @ A,W: word @ A,Xa: word @ A] :
( ( ord_less @ ( word @ A ) @ ( minus_minus @ ( word @ A ) @ Y3 @ X2 ) @ ( minus_minus @ ( word @ A ) @ ( plus_plus @ ( word @ A ) @ W @ Xa ) @ X2 ) )
=> ( ( ord_less_eq @ ( word @ A ) @ X2 @ ( plus_plus @ ( word @ A ) @ W @ Xa ) )
=> ( ord_less @ ( word @ A ) @ Y3 @ ( plus_plus @ ( word @ A ) @ W @ Xa ) ) ) ) ) ).
% word_diff_ls(1)
thf(fact_2098_Word_Oword__l__diffs_I2_J,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [W: word @ A,Xa: word @ A,Z: word @ A,X2: word @ A] :
( ( ord_less @ ( word @ A ) @ ( plus_plus @ ( word @ A ) @ W @ Xa ) @ Z )
=> ( ( ord_less_eq @ ( word @ A ) @ X2 @ ( plus_plus @ ( word @ A ) @ W @ Xa ) )
=> ( ord_less @ ( word @ A ) @ ( minus_minus @ ( word @ A ) @ ( plus_plus @ ( word @ A ) @ W @ Xa ) @ X2 ) @ ( minus_minus @ ( word @ A ) @ Z @ X2 ) ) ) ) ) ).
% Word.word_l_diffs(2)
thf(fact_2099_Word_Oword__l__diffs_I1_J,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [W: word @ A,Xa: word @ A,X2: word @ A,Z: word @ A] :
( ( ord_less @ ( word @ A ) @ ( minus_minus @ ( word @ A ) @ ( plus_plus @ ( word @ A ) @ W @ Xa ) @ X2 ) @ ( minus_minus @ ( word @ A ) @ Z @ X2 ) )
=> ( ( ord_less_eq @ ( word @ A ) @ X2 @ Z )
=> ( ord_less @ ( word @ A ) @ ( plus_plus @ ( word @ A ) @ W @ Xa ) @ Z ) ) ) ) ).
% Word.word_l_diffs(1)
thf(fact_2100_word__plus__mcs_I2_J,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [V: word @ A,Xb: word @ A,W: word @ A,Xa: word @ A,X2: word @ A] :
( ( ord_less @ ( word @ A ) @ ( plus_plus @ ( word @ A ) @ V @ Xb ) @ ( plus_plus @ ( word @ A ) @ W @ Xa ) )
=> ( ( ord_less_eq @ ( word @ A ) @ X2 @ ( plus_plus @ ( word @ A ) @ V @ Xb ) )
=> ( ord_less @ ( word @ A ) @ ( minus_minus @ ( word @ A ) @ ( plus_plus @ ( word @ A ) @ V @ Xb ) @ X2 ) @ ( minus_minus @ ( word @ A ) @ ( plus_plus @ ( word @ A ) @ W @ Xa ) @ X2 ) ) ) ) ) ).
% word_plus_mcs(2)
thf(fact_2101_word__plus__mcs_I1_J,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [V: word @ A,Xb: word @ A,X2: word @ A,W: word @ A,Xa: word @ A] :
( ( ord_less @ ( word @ A ) @ ( minus_minus @ ( word @ A ) @ ( plus_plus @ ( word @ A ) @ V @ Xb ) @ X2 ) @ ( minus_minus @ ( word @ A ) @ ( plus_plus @ ( word @ A ) @ W @ Xa ) @ X2 ) )
=> ( ( ord_less_eq @ ( word @ A ) @ X2 @ ( plus_plus @ ( word @ A ) @ W @ Xa ) )
=> ( ord_less @ ( word @ A ) @ ( plus_plus @ ( word @ A ) @ V @ Xb ) @ ( plus_plus @ ( word @ A ) @ W @ Xa ) ) ) ) ) ).
% word_plus_mcs(1)
thf(fact_2102_word__less__add__right,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [X2: word @ A,Y3: word @ A,Z: word @ A] :
( ( ord_less @ ( word @ A ) @ X2 @ ( minus_minus @ ( word @ A ) @ Y3 @ Z ) )
=> ( ( ord_less_eq @ ( word @ A ) @ Z @ Y3 )
=> ( ord_less @ ( word @ A ) @ ( plus_plus @ ( word @ A ) @ X2 @ Z ) @ Y3 ) ) ) ) ).
% word_less_add_right
thf(fact_2103_word__less__sub__right,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [X2: word @ A,Y3: word @ A,Z: word @ A] :
( ( ord_less @ ( word @ A ) @ X2 @ ( plus_plus @ ( word @ A ) @ Y3 @ Z ) )
=> ( ( ord_less_eq @ ( word @ A ) @ Y3 @ X2 )
=> ( ord_less @ ( word @ A ) @ ( minus_minus @ ( word @ A ) @ X2 @ Y3 ) @ Z ) ) ) ) ).
% word_less_sub_right
thf(fact_2104_plus__minus__no__overflow,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [X2: word @ A,Ab: word @ A,C3: word @ A] :
( ( ord_less @ ( word @ A ) @ X2 @ ( minus_minus @ ( word @ A ) @ Ab @ C3 ) )
=> ( ( ord_less_eq @ ( word @ A ) @ C3 @ Ab )
=> ( ord_less_eq @ ( word @ A ) @ X2 @ ( plus_plus @ ( word @ A ) @ X2 @ C3 ) ) ) ) ) ).
% plus_minus_no_overflow
thf(fact_2105_word__less__minus__mono__left,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [Y3: word @ A,Z: word @ A,X2: word @ A] :
( ( ord_less @ ( word @ A ) @ Y3 @ Z )
=> ( ( ord_less_eq @ ( word @ A ) @ X2 @ Y3 )
=> ( ord_less @ ( word @ A ) @ ( minus_minus @ ( word @ A ) @ Y3 @ X2 ) @ ( minus_minus @ ( word @ A ) @ Z @ X2 ) ) ) ) ) ).
% word_less_minus_mono_left
thf(fact_2106_word__less__minus__cancel,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [Y3: word @ A,X2: word @ A,Z: word @ A] :
( ( ord_less @ ( word @ A ) @ ( minus_minus @ ( word @ A ) @ Y3 @ X2 ) @ ( minus_minus @ ( word @ A ) @ Z @ X2 ) )
=> ( ( ord_less_eq @ ( word @ A ) @ X2 @ Z )
=> ( ord_less @ ( word @ A ) @ Y3 @ Z ) ) ) ) ).
% word_less_minus_cancel
thf(fact_2107_word__le__sub1,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [X2: word @ A] :
( ( X2
!= ( zero_zero @ ( word @ A ) ) )
=> ( ( ord_less_eq @ ( word @ A ) @ ( one_one @ ( word @ A ) ) @ X2 )
= ( ord_less_eq @ ( word @ A ) @ ( zero_zero @ ( word @ A ) ) @ ( minus_minus @ ( word @ A ) @ X2 @ ( one_one @ ( word @ A ) ) ) ) ) ) ) ).
% word_le_sub1
thf(fact_2108_word__less__sub1,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [X2: word @ A] :
( ( X2
!= ( zero_zero @ ( word @ A ) ) )
=> ( ( ord_less @ ( word @ A ) @ ( one_one @ ( word @ A ) ) @ X2 )
= ( ord_less @ ( word @ A ) @ ( zero_zero @ ( word @ A ) ) @ ( minus_minus @ ( word @ A ) @ X2 @ ( one_one @ ( word @ A ) ) ) ) ) ) ) ).
% word_less_sub1
thf(fact_2109_sub__wrap__lt,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [X2: word @ A,Z: word @ A] :
( ( ord_less @ ( word @ A ) @ X2 @ ( minus_minus @ ( word @ A ) @ X2 @ Z ) )
= ( ord_less @ ( word @ A ) @ X2 @ Z ) ) ) ).
% sub_wrap_lt
thf(fact_2110_word__sub__less__iff,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [X2: word @ A,Y3: word @ A] :
( ( ord_less @ ( word @ A ) @ X2 @ ( minus_minus @ ( word @ A ) @ X2 @ Y3 ) )
= ( ord_less @ ( word @ A ) @ X2 @ Y3 ) ) ) ).
% word_sub_less_iff
thf(fact_2111_word__less__minus__mono,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [A3: word @ A,C3: word @ A,D2: word @ A,B3: word @ A] :
( ( ord_less @ ( word @ A ) @ A3 @ C3 )
=> ( ( ord_less @ ( word @ A ) @ D2 @ B3 )
=> ( ( ord_less @ ( word @ A ) @ ( minus_minus @ ( word @ A ) @ A3 @ B3 ) @ A3 )
=> ( ( ord_less @ ( word @ A ) @ ( minus_minus @ ( word @ A ) @ C3 @ D2 ) @ C3 )
=> ( ord_less @ ( word @ A ) @ ( minus_minus @ ( word @ A ) @ A3 @ B3 ) @ ( minus_minus @ ( word @ A ) @ C3 @ D2 ) ) ) ) ) ) ) ).
% word_less_minus_mono
thf(fact_2112_plus__le__left__cancel__wrap,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [X2: word @ A,Y7: word @ A,Y3: word @ A] :
( ( ord_less @ ( word @ A ) @ ( plus_plus @ ( word @ A ) @ X2 @ Y7 ) @ X2 )
=> ( ( ord_less @ ( word @ A ) @ ( plus_plus @ ( word @ A ) @ X2 @ Y3 ) @ X2 )
=> ( ( ord_less @ ( word @ A ) @ ( plus_plus @ ( word @ A ) @ X2 @ Y7 ) @ ( plus_plus @ ( word @ A ) @ X2 @ Y3 ) )
= ( ord_less @ ( word @ A ) @ Y7 @ Y3 ) ) ) ) ) ).
% plus_le_left_cancel_wrap
thf(fact_2113_word__induct,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [P: ( word @ A ) > $o,M: word @ A] :
( ( P @ ( zero_zero @ ( word @ A ) ) )
=> ( ! [N3: word @ A] :
( ( P @ N3 )
=> ( P @ ( plus_plus @ ( word @ A ) @ ( one_one @ ( word @ A ) ) @ N3 ) ) )
=> ( P @ M ) ) ) ) ).
% word_induct
thf(fact_2114_word__induct2,axiom,
! [B: $tType] :
( ( type_len @ B )
=> ! [P: ( word @ B ) > $o,N: word @ B] :
( ( P @ ( zero_zero @ ( word @ B ) ) )
=> ( ! [N3: word @ B] :
( ( ( plus_plus @ ( word @ B ) @ ( one_one @ ( word @ B ) ) @ N3 )
!= ( zero_zero @ ( word @ B ) ) )
=> ( ( P @ N3 )
=> ( P @ ( plus_plus @ ( word @ B ) @ ( one_one @ ( word @ B ) ) @ N3 ) ) ) )
=> ( P @ N ) ) ) ) ).
% word_induct2
thf(fact_2115_div__word__self,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [W: word @ A] :
( ( W
!= ( zero_zero @ ( word @ A ) ) )
=> ( ( divide_divide @ ( word @ A ) @ W @ W )
= ( one_one @ ( word @ A ) ) ) ) ) ).
% div_word_self
thf(fact_2116_word__induct__less,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [P: ( word @ A ) > $o,M: word @ A] :
( ( P @ ( zero_zero @ ( word @ A ) ) )
=> ( ! [N3: word @ A] :
( ( ord_less @ ( word @ A ) @ N3 @ M )
=> ( ( P @ N3 )
=> ( P @ ( plus_plus @ ( word @ A ) @ ( one_one @ ( word @ A ) ) @ N3 ) ) ) )
=> ( P @ M ) ) ) ) ).
% word_induct_less
thf(fact_2117_ex__le__of__int,axiom,
! [A: $tType] :
( ( archim462609752435547400_field @ A )
=> ! [X2: A] :
? [Z2: int] : ( ord_less_eq @ A @ X2 @ ( ring_1_of_int @ A @ Z2 ) ) ) ).
% ex_le_of_int
thf(fact_2118_ex__of__int__less,axiom,
! [A: $tType] :
( ( archim462609752435547400_field @ A )
=> ! [X2: A] :
? [Z2: int] : ( ord_less @ A @ ( ring_1_of_int @ A @ Z2 ) @ X2 ) ) ).
% ex_of_int_less
thf(fact_2119_ex__less__of__int,axiom,
! [A: $tType] :
( ( archim462609752435547400_field @ A )
=> ! [X2: A] :
? [Z2: int] : ( ord_less @ A @ X2 @ ( ring_1_of_int @ A @ Z2 ) ) ) ).
% ex_less_of_int
thf(fact_2120_mult__of__int__commute,axiom,
! [A: $tType] :
( ( ring_1 @ A )
=> ! [X2: int,Y3: A] :
( ( times_times @ A @ ( ring_1_of_int @ A @ X2 ) @ Y3 )
= ( times_times @ A @ Y3 @ ( ring_1_of_int @ A @ X2 ) ) ) ) ).
% mult_of_int_commute
thf(fact_2121_of__int__max,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [X2: int,Y3: int] :
( ( ring_1_of_int @ A @ ( ord_max @ int @ X2 @ Y3 ) )
= ( ord_max @ A @ ( ring_1_of_int @ A @ X2 ) @ ( ring_1_of_int @ A @ Y3 ) ) ) ) ).
% of_int_max
thf(fact_2122_nat__numeral__as__int,axiom,
( ( numeral_numeral @ nat )
= ( ^ [I4: num] : ( nat2 @ ( numeral_numeral @ int @ I4 ) ) ) ) ).
% nat_numeral_as_int
thf(fact_2123_nat__mono,axiom,
! [X2: int,Y3: int] :
( ( ord_less_eq @ int @ X2 @ Y3 )
=> ( ord_less_eq @ nat @ ( nat2 @ X2 ) @ ( nat2 @ Y3 ) ) ) ).
% nat_mono
thf(fact_2124_nat__one__as__int,axiom,
( ( one_one @ nat )
= ( nat2 @ ( one_one @ int ) ) ) ).
% nat_one_as_int
thf(fact_2125_ex__nat,axiom,
( ( ^ [P2: nat > $o] :
? [X6: nat] : ( P2 @ X6 ) )
= ( ^ [P3: nat > $o] :
? [X3: int] :
( ( ord_less_eq @ int @ ( zero_zero @ int ) @ X3 )
& ( P3 @ ( nat2 @ X3 ) ) ) ) ) ).
% ex_nat
thf(fact_2126_all__nat,axiom,
( ( ^ [P2: nat > $o] :
! [X6: nat] : ( P2 @ X6 ) )
= ( ^ [P3: nat > $o] :
! [X3: int] :
( ( ord_less_eq @ int @ ( zero_zero @ int ) @ X3 )
=> ( P3 @ ( nat2 @ X3 ) ) ) ) ) ).
% all_nat
thf(fact_2127_eq__nat__nat__iff,axiom,
! [Z: int,Z7: int] :
( ( ord_less_eq @ int @ ( zero_zero @ int ) @ Z )
=> ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ Z7 )
=> ( ( ( nat2 @ Z )
= ( nat2 @ Z7 ) )
= ( Z = Z7 ) ) ) ) ).
% eq_nat_nat_iff
thf(fact_2128_le__of__int__ceiling,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [X2: A] : ( ord_less_eq @ A @ X2 @ ( ring_1_of_int @ A @ ( archimedean_ceiling @ A @ X2 ) ) ) ) ).
% le_of_int_ceiling
thf(fact_2129_nat__mono__iff,axiom,
! [Z: int,W: int] :
( ( ord_less @ int @ ( zero_zero @ int ) @ Z )
=> ( ( ord_less @ nat @ ( nat2 @ W ) @ ( nat2 @ Z ) )
= ( ord_less @ int @ W @ Z ) ) ) ).
% nat_mono_iff
thf(fact_2130_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_2131_nat__le__iff,axiom,
! [X2: int,N: nat] :
( ( ord_less_eq @ nat @ ( nat2 @ X2 ) @ N )
= ( ord_less_eq @ int @ X2 @ ( semiring_1_of_nat @ int @ N ) ) ) ).
% nat_le_iff
thf(fact_2132_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_2133_nat__int__add,axiom,
! [A3: nat,B3: nat] :
( ( nat2 @ ( plus_plus @ int @ ( semiring_1_of_nat @ int @ A3 ) @ ( semiring_1_of_nat @ int @ B3 ) ) )
= ( plus_plus @ nat @ A3 @ B3 ) ) ).
% nat_int_add
thf(fact_2134_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_2135_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_2136_ceiling__le__iff,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [X2: A,Z: int] :
( ( ord_less_eq @ int @ ( archimedean_ceiling @ A @ X2 ) @ Z )
= ( ord_less_eq @ A @ X2 @ ( ring_1_of_int @ A @ Z ) ) ) ) ).
% ceiling_le_iff
thf(fact_2137_ceiling__le,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [X2: A,A3: int] :
( ( ord_less_eq @ A @ X2 @ ( ring_1_of_int @ A @ A3 ) )
=> ( ord_less_eq @ int @ ( archimedean_ceiling @ A @ X2 ) @ A3 ) ) ) ).
% ceiling_le
thf(fact_2138_less__ceiling__iff,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [Z: int,X2: A] :
( ( ord_less @ int @ Z @ ( archimedean_ceiling @ A @ X2 ) )
= ( ord_less @ A @ ( ring_1_of_int @ A @ Z ) @ X2 ) ) ) ).
% less_ceiling_iff
thf(fact_2139_int__minus,axiom,
! [N: nat,M: nat] :
( ( semiring_1_of_nat @ int @ ( minus_minus @ nat @ N @ M ) )
= ( semiring_1_of_nat @ int @ ( nat2 @ ( minus_minus @ int @ ( semiring_1_of_nat @ int @ N ) @ ( semiring_1_of_nat @ int @ M ) ) ) ) ) ).
% int_minus
thf(fact_2140_real__of__int__div4,axiom,
! [N: int,X2: int] : ( ord_less_eq @ real @ ( ring_1_of_int @ real @ ( divide_divide @ int @ N @ X2 ) ) @ ( divide_divide @ real @ ( ring_1_of_int @ real @ N ) @ ( ring_1_of_int @ real @ X2 ) ) ) ).
% real_of_int_div4
thf(fact_2141_le__minus,axiom,
! [Aa: $tType,A: $tType] :
( ( ( type_len @ A )
& ( order @ Aa ) )
=> ! [Y3: Aa,X2: Aa,A3: word @ A,C3: word @ A,B3: word @ A] :
( ( ord_less_eq @ Aa @ Y3 @ X2 )
=> ( ( ord_less_eq @ ( word @ A ) @ ( plus_plus @ ( word @ A ) @ A3 @ C3 ) @ B3 )
=> ( ( ord_less_eq @ ( word @ A ) @ A3 @ ( plus_plus @ ( word @ A ) @ A3 @ C3 ) )
=> ( ord_less_eq @ ( word @ A ) @ C3 @ ( minus_minus @ ( word @ A ) @ B3 @ A3 ) ) ) ) ) ) ).
% le_minus
thf(fact_2142_real__nat__ceiling__ge,axiom,
! [X2: real] : ( ord_less_eq @ real @ X2 @ ( semiring_1_of_nat @ real @ ( nat2 @ ( archimedean_ceiling @ real @ X2 ) ) ) ) ).
% real_nat_ceiling_ge
thf(fact_2143_nat__plus__as__int,axiom,
( ( plus_plus @ nat )
= ( ^ [A6: nat,B7: nat] : ( nat2 @ ( plus_plus @ int @ ( semiring_1_of_nat @ int @ A6 ) @ ( semiring_1_of_nat @ int @ B7 ) ) ) ) ) ).
% nat_plus_as_int
thf(fact_2144_Abs__fnat__hom__mult,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [A3: nat,B3: nat] :
( ( times_times @ ( word @ A ) @ ( semiring_1_of_nat @ ( word @ A ) @ A3 ) @ ( semiring_1_of_nat @ ( word @ A ) @ B3 ) )
= ( semiring_1_of_nat @ ( word @ A ) @ ( times_times @ nat @ A3 @ B3 ) ) ) ) ).
% Abs_fnat_hom_mult
thf(fact_2145_nat__times__as__int,axiom,
( ( times_times @ nat )
= ( ^ [A6: nat,B7: nat] : ( nat2 @ ( times_times @ int @ ( semiring_1_of_nat @ int @ A6 ) @ ( semiring_1_of_nat @ int @ B7 ) ) ) ) ) ).
% nat_times_as_int
thf(fact_2146_nat__minus__as__int,axiom,
( ( minus_minus @ nat )
= ( ^ [A6: nat,B7: nat] : ( nat2 @ ( minus_minus @ int @ ( semiring_1_of_nat @ int @ A6 ) @ ( semiring_1_of_nat @ int @ B7 ) ) ) ) ) ).
% nat_minus_as_int
thf(fact_2147_nat__div__as__int,axiom,
( ( divide_divide @ nat )
= ( ^ [A6: nat,B7: nat] : ( nat2 @ ( divide_divide @ int @ ( semiring_1_of_nat @ int @ A6 ) @ ( semiring_1_of_nat @ int @ B7 ) ) ) ) ) ).
% nat_div_as_int
thf(fact_2148_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_2149_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_2150_floor__exists,axiom,
! [A: $tType] :
( ( archim462609752435547400_field @ A )
=> ! [X2: A] :
? [Z2: int] :
( ( ord_less_eq @ A @ ( ring_1_of_int @ A @ Z2 ) @ X2 )
& ( ord_less @ A @ X2 @ ( ring_1_of_int @ A @ ( plus_plus @ int @ Z2 @ ( one_one @ int ) ) ) ) ) ) ).
% floor_exists
thf(fact_2151_floor__exists1,axiom,
! [A: $tType] :
( ( archim462609752435547400_field @ A )
=> ! [X2: A] :
? [X4: int] :
( ( ord_less_eq @ A @ ( ring_1_of_int @ A @ X4 ) @ X2 )
& ( ord_less @ A @ X2 @ ( ring_1_of_int @ A @ ( plus_plus @ int @ X4 @ ( one_one @ int ) ) ) )
& ! [Y5: int] :
( ( ( ord_less_eq @ A @ ( ring_1_of_int @ A @ Y5 ) @ X2 )
& ( ord_less @ A @ X2 @ ( ring_1_of_int @ A @ ( plus_plus @ int @ Y5 @ ( one_one @ int ) ) ) ) )
=> ( Y5 = X4 ) ) ) ) ).
% floor_exists1
thf(fact_2152_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_2153_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_2154_of__nat__less__of__int__iff,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [N: nat,X2: int] :
( ( ord_less @ A @ ( semiring_1_of_nat @ A @ N ) @ ( ring_1_of_int @ A @ X2 ) )
= ( ord_less @ int @ ( semiring_1_of_nat @ int @ N ) @ X2 ) ) ) ).
% of_nat_less_of_int_iff
thf(fact_2155_nat__less__eq__zless,axiom,
! [W: int,Z: int] :
( ( ord_less_eq @ int @ ( zero_zero @ int ) @ W )
=> ( ( ord_less @ nat @ ( nat2 @ W ) @ ( nat2 @ Z ) )
= ( ord_less @ int @ W @ Z ) ) ) ).
% nat_less_eq_zless
thf(fact_2156_nat__eq__iff,axiom,
! [W: int,M: nat] :
( ( ( nat2 @ W )
= M )
= ( ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ W )
=> ( W
= ( semiring_1_of_nat @ int @ M ) ) )
& ( ~ ( ord_less_eq @ int @ ( zero_zero @ int ) @ W )
=> ( M
= ( zero_zero @ nat ) ) ) ) ) ).
% nat_eq_iff
thf(fact_2157_nat__eq__iff2,axiom,
! [M: nat,W: int] :
( ( M
= ( nat2 @ W ) )
= ( ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ W )
=> ( W
= ( semiring_1_of_nat @ int @ M ) ) )
& ( ~ ( ord_less_eq @ int @ ( zero_zero @ int ) @ W )
=> ( M
= ( zero_zero @ nat ) ) ) ) ) ).
% nat_eq_iff2
thf(fact_2158_nat__le__eq__zle,axiom,
! [W: int,Z: int] :
( ( ( ord_less @ int @ ( zero_zero @ int ) @ W )
| ( ord_less_eq @ int @ ( zero_zero @ int ) @ Z ) )
=> ( ( ord_less_eq @ nat @ ( nat2 @ W ) @ ( nat2 @ Z ) )
= ( ord_less_eq @ int @ W @ Z ) ) ) ).
% nat_le_eq_zle
thf(fact_2159_nat__add__distrib,axiom,
! [Z: int,Z7: int] :
( ( ord_less_eq @ int @ ( zero_zero @ int ) @ Z )
=> ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ Z7 )
=> ( ( nat2 @ ( plus_plus @ int @ Z @ Z7 ) )
= ( plus_plus @ nat @ ( nat2 @ Z ) @ ( nat2 @ Z7 ) ) ) ) ) ).
% nat_add_distrib
thf(fact_2160_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_2161_Suc__as__int,axiom,
( suc
= ( ^ [A6: nat] : ( nat2 @ ( plus_plus @ int @ ( semiring_1_of_nat @ int @ A6 ) @ ( one_one @ int ) ) ) ) ) ).
% Suc_as_int
thf(fact_2162_nat__mult__distrib,axiom,
! [Z: int,Z7: int] :
( ( ord_less_eq @ int @ ( zero_zero @ int ) @ Z )
=> ( ( nat2 @ ( times_times @ int @ Z @ Z7 ) )
= ( times_times @ nat @ ( nat2 @ Z ) @ ( nat2 @ Z7 ) ) ) ) ).
% nat_mult_distrib
thf(fact_2163_nat__diff__distrib_H,axiom,
! [X2: int,Y3: int] :
( ( ord_less_eq @ int @ ( zero_zero @ int ) @ X2 )
=> ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ Y3 )
=> ( ( nat2 @ ( minus_minus @ int @ X2 @ Y3 ) )
= ( minus_minus @ nat @ ( nat2 @ X2 ) @ ( nat2 @ Y3 ) ) ) ) ) ).
% nat_diff_distrib'
thf(fact_2164_nat__diff__distrib,axiom,
! [Z7: int,Z: int] :
( ( ord_less_eq @ int @ ( zero_zero @ int ) @ Z7 )
=> ( ( ord_less_eq @ int @ Z7 @ Z )
=> ( ( nat2 @ ( minus_minus @ int @ Z @ Z7 ) )
= ( minus_minus @ nat @ ( nat2 @ Z ) @ ( nat2 @ Z7 ) ) ) ) ) ).
% nat_diff_distrib
thf(fact_2165_nat__div__distrib,axiom,
! [X2: int,Y3: int] :
( ( ord_less_eq @ int @ ( zero_zero @ int ) @ X2 )
=> ( ( nat2 @ ( divide_divide @ int @ X2 @ Y3 ) )
= ( divide_divide @ nat @ ( nat2 @ X2 ) @ ( nat2 @ Y3 ) ) ) ) ).
% nat_div_distrib
thf(fact_2166_nat__div__distrib_H,axiom,
! [Y3: int,X2: int] :
( ( ord_less_eq @ int @ ( zero_zero @ int ) @ Y3 )
=> ( ( nat2 @ ( divide_divide @ int @ X2 @ Y3 ) )
= ( divide_divide @ nat @ ( nat2 @ X2 ) @ ( nat2 @ Y3 ) ) ) ) ).
% nat_div_distrib'
thf(fact_2167_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_2168_int__le__real__less,axiom,
( ( ord_less_eq @ int )
= ( ^ [N4: int,M6: int] : ( ord_less @ real @ ( ring_1_of_int @ real @ N4 ) @ ( plus_plus @ real @ ( ring_1_of_int @ real @ M6 ) @ ( one_one @ real ) ) ) ) ) ).
% int_le_real_less
thf(fact_2169_int__less__real__le,axiom,
( ( ord_less @ int )
= ( ^ [N4: int,M6: int] : ( ord_less_eq @ real @ ( plus_plus @ real @ ( ring_1_of_int @ real @ N4 ) @ ( one_one @ real ) ) @ ( ring_1_of_int @ real @ M6 ) ) ) ) ).
% int_less_real_le
thf(fact_2170_nat__2,axiom,
( ( nat2 @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) )
= ( suc @ ( suc @ ( zero_zero @ nat ) ) ) ) ).
% nat_2
thf(fact_2171_ceiling__split,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [P: int > $o,T2: A] :
( ( P @ ( archimedean_ceiling @ A @ T2 ) )
= ( ! [I4: int] :
( ( ( ord_less @ A @ ( minus_minus @ A @ ( ring_1_of_int @ A @ I4 ) @ ( one_one @ A ) ) @ T2 )
& ( ord_less_eq @ A @ T2 @ ( ring_1_of_int @ A @ I4 ) ) )
=> ( P @ I4 ) ) ) ) ) ).
% ceiling_split
thf(fact_2172_ceiling__eq__iff,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [X2: A,A3: int] :
( ( ( archimedean_ceiling @ A @ X2 )
= A3 )
= ( ( ord_less @ A @ ( minus_minus @ A @ ( ring_1_of_int @ A @ A3 ) @ ( one_one @ A ) ) @ X2 )
& ( ord_less_eq @ A @ X2 @ ( ring_1_of_int @ A @ A3 ) ) ) ) ) ).
% ceiling_eq_iff
thf(fact_2173_ceiling__unique,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [Z: int,X2: A] :
( ( ord_less @ A @ ( minus_minus @ A @ ( ring_1_of_int @ A @ Z ) @ ( one_one @ A ) ) @ X2 )
=> ( ( ord_less_eq @ A @ X2 @ ( ring_1_of_int @ A @ Z ) )
=> ( ( archimedean_ceiling @ A @ X2 )
= Z ) ) ) ) ).
% ceiling_unique
thf(fact_2174_ceiling__correct,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [X2: A] :
( ( ord_less @ A @ ( minus_minus @ A @ ( ring_1_of_int @ A @ ( archimedean_ceiling @ A @ X2 ) ) @ ( one_one @ A ) ) @ X2 )
& ( ord_less_eq @ A @ X2 @ ( ring_1_of_int @ A @ ( archimedean_ceiling @ A @ X2 ) ) ) ) ) ).
% ceiling_correct
thf(fact_2175_ceiling__less__iff,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [X2: A,Z: int] :
( ( ord_less @ int @ ( archimedean_ceiling @ A @ X2 ) @ Z )
= ( ord_less_eq @ A @ X2 @ ( minus_minus @ A @ ( ring_1_of_int @ A @ Z ) @ ( one_one @ A ) ) ) ) ) ).
% ceiling_less_iff
thf(fact_2176_le__ceiling__iff,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [Z: int,X2: A] :
( ( ord_less_eq @ int @ Z @ ( archimedean_ceiling @ A @ X2 ) )
= ( ord_less @ A @ ( minus_minus @ A @ ( ring_1_of_int @ A @ Z ) @ ( one_one @ A ) ) @ X2 ) ) ) ).
% le_ceiling_iff
thf(fact_2177_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_2178_nat__less__iff,axiom,
! [W: int,M: nat] :
( ( ord_less_eq @ int @ ( zero_zero @ int ) @ W )
=> ( ( ord_less @ nat @ ( nat2 @ W ) @ M )
= ( ord_less @ int @ W @ ( semiring_1_of_nat @ int @ M ) ) ) ) ).
% nat_less_iff
thf(fact_2179_real__of__int__div2,axiom,
! [N: int,X2: 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 @ X2 ) ) @ ( ring_1_of_int @ real @ ( divide_divide @ int @ N @ X2 ) ) ) ) ).
% real_of_int_div2
thf(fact_2180_real__of__int__div3,axiom,
! [N: int,X2: int] : ( ord_less_eq @ real @ ( minus_minus @ real @ ( divide_divide @ real @ ( ring_1_of_int @ real @ N ) @ ( ring_1_of_int @ real @ X2 ) ) @ ( ring_1_of_int @ real @ ( divide_divide @ int @ N @ X2 ) ) ) @ ( one_one @ real ) ) ).
% real_of_int_div3
thf(fact_2181_diff__nat__eq__if,axiom,
! [Z7: int,Z: int] :
( ( ( ord_less @ int @ Z7 @ ( zero_zero @ int ) )
=> ( ( minus_minus @ nat @ ( nat2 @ Z ) @ ( nat2 @ Z7 ) )
= ( nat2 @ Z ) ) )
& ( ~ ( ord_less @ int @ Z7 @ ( zero_zero @ int ) )
=> ( ( minus_minus @ nat @ ( nat2 @ Z ) @ ( nat2 @ Z7 ) )
= ( if @ nat @ ( ord_less @ int @ ( minus_minus @ int @ Z @ Z7 ) @ ( zero_zero @ int ) ) @ ( zero_zero @ nat ) @ ( nat2 @ ( minus_minus @ int @ Z @ Z7 ) ) ) ) ) ) ).
% diff_nat_eq_if
thf(fact_2182_ceiling__divide__upper,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [Q3: A,P6: A] :
( ( ord_less @ A @ ( zero_zero @ A ) @ Q3 )
=> ( ord_less_eq @ A @ P6 @ ( times_times @ A @ ( ring_1_of_int @ A @ ( archimedean_ceiling @ A @ ( divide_divide @ A @ P6 @ Q3 ) ) ) @ Q3 ) ) ) ) ).
% ceiling_divide_upper
thf(fact_2183_ceiling__divide__lower,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [Q3: A,P6: A] :
( ( ord_less @ A @ ( zero_zero @ A ) @ Q3 )
=> ( ord_less @ A @ ( times_times @ A @ ( minus_minus @ A @ ( ring_1_of_int @ A @ ( archimedean_ceiling @ A @ ( divide_divide @ A @ P6 @ Q3 ) ) ) @ ( one_one @ A ) ) @ Q3 ) @ P6 ) ) ) ).
% ceiling_divide_lower
thf(fact_2184_ceiling__eq,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [N: int,X2: A] :
( ( ord_less @ A @ ( ring_1_of_int @ A @ N ) @ X2 )
=> ( ( ord_less_eq @ A @ X2 @ ( plus_plus @ A @ ( ring_1_of_int @ A @ N ) @ ( one_one @ A ) ) )
=> ( ( archimedean_ceiling @ A @ X2 )
= ( plus_plus @ int @ N @ ( one_one @ int ) ) ) ) ) ) ).
% ceiling_eq
thf(fact_2185_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_2186_Abs__fnat__hom__add,axiom,
! [A: $tType] :
( ( semiring_1 @ A )
=> ! [A3: nat,B3: nat] :
( ( plus_plus @ A @ ( semiring_1_of_nat @ A @ A3 ) @ ( semiring_1_of_nat @ A @ B3 ) )
= ( semiring_1_of_nat @ A @ ( plus_plus @ nat @ A3 @ B3 ) ) ) ) ).
% Abs_fnat_hom_add
thf(fact_2187_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_2188_vebt__delete_Opelims,axiom,
! [X2: vEBT_VEBT,Xa: nat,Y3: vEBT_VEBT] :
( ( ( vEBT_vebt_delete @ X2 @ Xa )
= Y3 )
=> ( ( accp @ ( product_prod @ vEBT_VEBT @ nat ) @ vEBT_vebt_delete_rel @ ( product_Pair @ vEBT_VEBT @ nat @ X2 @ Xa ) )
=> ( ! [A4: $o,B4: $o] :
( ( X2
= ( vEBT_Leaf @ A4 @ B4 ) )
=> ( ( Xa
= ( zero_zero @ nat ) )
=> ( ( Y3
= ( 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] :
( ( X2
= ( vEBT_Leaf @ A4 @ B4 ) )
=> ( ( Xa
= ( suc @ ( zero_zero @ nat ) ) )
=> ( ( Y3
= ( 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] :
( ( X2
= ( vEBT_Leaf @ A4 @ B4 ) )
=> ! [N3: nat] :
( ( Xa
= ( suc @ ( suc @ N3 ) ) )
=> ( ( Y3
= ( 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 @ N3 ) ) ) ) ) ) )
=> ( ! [Deg2: nat,TreeList2: list @ vEBT_VEBT,Summary2: vEBT_VEBT] :
( ( X2
= ( vEBT_Node @ ( none @ ( product_prod @ nat @ nat ) ) @ Deg2 @ TreeList2 @ Summary2 ) )
=> ( ( Y3
= ( vEBT_Node @ ( none @ ( product_prod @ nat @ nat ) ) @ Deg2 @ TreeList2 @ Summary2 ) )
=> ~ ( accp @ ( product_prod @ vEBT_VEBT @ nat ) @ vEBT_vebt_delete_rel @ ( product_Pair @ vEBT_VEBT @ nat @ ( vEBT_Node @ ( none @ ( product_prod @ nat @ nat ) ) @ Deg2 @ TreeList2 @ Summary2 ) @ Xa ) ) ) )
=> ( ! [Mi2: nat,Ma2: nat,TrLst2: list @ vEBT_VEBT,Smry2: vEBT_VEBT] :
( ( X2
= ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi2 @ Ma2 ) ) @ ( zero_zero @ nat ) @ TrLst2 @ Smry2 ) )
=> ( ( Y3
= ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi2 @ Ma2 ) ) @ ( zero_zero @ nat ) @ TrLst2 @ Smry2 ) )
=> ~ ( accp @ ( product_prod @ vEBT_VEBT @ nat ) @ vEBT_vebt_delete_rel @ ( product_Pair @ vEBT_VEBT @ nat @ ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi2 @ Ma2 ) ) @ ( zero_zero @ nat ) @ TrLst2 @ Smry2 ) @ Xa ) ) ) )
=> ( ! [Mi2: nat,Ma2: nat,Tr2: list @ vEBT_VEBT,Sm2: vEBT_VEBT] :
( ( X2
= ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( zero_zero @ nat ) ) @ Tr2 @ Sm2 ) )
=> ( ( Y3
= ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( zero_zero @ nat ) ) @ Tr2 @ Sm2 ) )
=> ~ ( accp @ ( product_prod @ vEBT_VEBT @ nat ) @ vEBT_vebt_delete_rel @ ( product_Pair @ vEBT_VEBT @ nat @ ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( zero_zero @ nat ) ) @ Tr2 @ Sm2 ) @ Xa ) ) ) )
=> ~ ! [Mi2: nat,Ma2: nat,Va3: nat,TreeList2: list @ vEBT_VEBT,Summary2: vEBT_VEBT] :
( ( X2
= ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( suc @ Va3 ) ) @ TreeList2 @ Summary2 ) )
=> ( ( ( ( ( ord_less @ nat @ Xa @ Mi2 )
| ( ord_less @ nat @ Ma2 @ Xa ) )
=> ( Y3
= ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( suc @ Va3 ) ) @ TreeList2 @ Summary2 ) ) )
& ( ~ ( ( ord_less @ nat @ Xa @ Mi2 )
| ( ord_less @ nat @ Ma2 @ Xa ) )
=> ( ( ( ( Xa = Mi2 )
& ( Xa = Ma2 ) )
=> ( Y3
= ( vEBT_Node @ ( none @ ( product_prod @ nat @ nat ) ) @ ( suc @ ( suc @ Va3 ) ) @ TreeList2 @ Summary2 ) ) )
& ( ~ ( ( Xa = Mi2 )
& ( Xa = Ma2 ) )
=> ( Y3
= ( if @ vEBT_VEBT @ ( ord_less @ nat @ ( vEBT_VEBT_high @ ( if @ nat @ ( Xa = Mi2 ) @ ( plus_plus @ nat @ ( times_times @ nat @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary2 ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList2 @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary2 ) ) ) ) ) ) @ Xa ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( size_size @ ( list @ vEBT_VEBT ) @ TreeList2 ) )
@ ( if @ vEBT_VEBT @ ( vEBT_VEBT_minNull @ ( vEBT_vebt_delete @ ( nth @ vEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ ( if @ nat @ ( Xa = Mi2 ) @ ( plus_plus @ nat @ ( times_times @ nat @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary2 ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList2 @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary2 ) ) ) ) ) ) @ Xa ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( vEBT_VEBT_low @ ( if @ nat @ ( Xa = Mi2 ) @ ( plus_plus @ nat @ ( times_times @ nat @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary2 ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList2 @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary2 ) ) ) ) ) ) @ Xa ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) )
@ ( vEBT_Node
@ ( some @ ( product_prod @ nat @ nat )
@ ( product_Pair @ nat @ nat @ ( if @ nat @ ( Xa = Mi2 ) @ ( if @ nat @ ( Xa = Mi2 ) @ ( plus_plus @ nat @ ( times_times @ nat @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary2 ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList2 @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary2 ) ) ) ) ) ) @ Xa ) @ Mi2 )
@ ( if @ nat
@ ( ( ( Xa = Mi2 )
=> ( ( plus_plus @ nat @ ( times_times @ nat @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary2 ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList2 @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary2 ) ) ) ) ) )
= Ma2 ) )
& ( ( Xa != Mi2 )
=> ( Xa = Ma2 ) ) )
@ ( if @ nat
@ ( ( vEBT_vebt_maxt @ ( vEBT_vebt_delete @ Summary2 @ ( vEBT_VEBT_high @ ( if @ nat @ ( Xa = Mi2 ) @ ( plus_plus @ nat @ ( times_times @ nat @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary2 ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList2 @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary2 ) ) ) ) ) ) @ Xa ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) )
= ( none @ nat ) )
@ ( if @ nat @ ( Xa = Mi2 ) @ ( if @ nat @ ( Xa = Mi2 ) @ ( plus_plus @ nat @ ( times_times @ nat @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary2 ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList2 @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary2 ) ) ) ) ) ) @ Xa ) @ Mi2 )
@ ( plus_plus @ nat @ ( times_times @ nat @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_maxt @ ( vEBT_vebt_delete @ Summary2 @ ( vEBT_VEBT_high @ ( if @ nat @ ( Xa = Mi2 ) @ ( plus_plus @ nat @ ( times_times @ nat @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary2 ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList2 @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary2 ) ) ) ) ) ) @ Xa ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_maxt @ ( nth @ vEBT_VEBT @ ( list_update @ vEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ ( if @ nat @ ( Xa = Mi2 ) @ ( plus_plus @ nat @ ( times_times @ nat @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary2 ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList2 @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary2 ) ) ) ) ) ) @ Xa ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( vEBT_vebt_delete @ ( nth @ vEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ ( if @ nat @ ( Xa = Mi2 ) @ ( plus_plus @ nat @ ( times_times @ nat @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary2 ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList2 @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary2 ) ) ) ) ) ) @ Xa ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( vEBT_VEBT_low @ ( if @ nat @ ( Xa = Mi2 ) @ ( plus_plus @ nat @ ( times_times @ nat @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary2 ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList2 @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary2 ) ) ) ) ) ) @ Xa ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_maxt @ ( vEBT_vebt_delete @ Summary2 @ ( vEBT_VEBT_high @ ( if @ nat @ ( Xa = Mi2 ) @ ( plus_plus @ nat @ ( times_times @ nat @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary2 ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList2 @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary2 ) ) ) ) ) ) @ Xa ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) ) ) ) ) ) )
@ Ma2 ) ) )
@ ( suc @ ( suc @ Va3 ) )
@ ( list_update @ vEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ ( if @ nat @ ( Xa = Mi2 ) @ ( plus_plus @ nat @ ( times_times @ nat @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary2 ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList2 @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary2 ) ) ) ) ) ) @ Xa ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( vEBT_vebt_delete @ ( nth @ vEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ ( if @ nat @ ( Xa = Mi2 ) @ ( plus_plus @ nat @ ( times_times @ nat @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary2 ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList2 @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary2 ) ) ) ) ) ) @ Xa ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( vEBT_VEBT_low @ ( if @ nat @ ( Xa = Mi2 ) @ ( plus_plus @ nat @ ( times_times @ nat @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary2 ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList2 @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary2 ) ) ) ) ) ) @ Xa ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) )
@ ( vEBT_vebt_delete @ Summary2 @ ( vEBT_VEBT_high @ ( if @ nat @ ( Xa = Mi2 ) @ ( plus_plus @ nat @ ( times_times @ nat @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary2 ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList2 @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary2 ) ) ) ) ) ) @ Xa ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) )
@ ( vEBT_Node
@ ( some @ ( product_prod @ nat @ nat )
@ ( product_Pair @ nat @ nat @ ( if @ nat @ ( Xa = Mi2 ) @ ( if @ nat @ ( Xa = Mi2 ) @ ( plus_plus @ nat @ ( times_times @ nat @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary2 ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList2 @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary2 ) ) ) ) ) ) @ Xa ) @ Mi2 )
@ ( if @ nat
@ ( ( ( Xa = Mi2 )
=> ( ( plus_plus @ nat @ ( times_times @ nat @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary2 ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList2 @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary2 ) ) ) ) ) )
= Ma2 ) )
& ( ( Xa != Mi2 )
=> ( Xa = Ma2 ) ) )
@ ( plus_plus @ nat @ ( times_times @ nat @ ( vEBT_VEBT_high @ ( if @ nat @ ( Xa = Mi2 ) @ ( plus_plus @ nat @ ( times_times @ nat @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary2 ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList2 @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary2 ) ) ) ) ) ) @ Xa ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_maxt @ ( nth @ vEBT_VEBT @ ( list_update @ vEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ ( if @ nat @ ( Xa = Mi2 ) @ ( plus_plus @ nat @ ( times_times @ nat @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary2 ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList2 @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary2 ) ) ) ) ) ) @ Xa ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( vEBT_vebt_delete @ ( nth @ vEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ ( if @ nat @ ( Xa = Mi2 ) @ ( plus_plus @ nat @ ( times_times @ nat @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary2 ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList2 @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary2 ) ) ) ) ) ) @ Xa ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( vEBT_VEBT_low @ ( if @ nat @ ( Xa = Mi2 ) @ ( plus_plus @ nat @ ( times_times @ nat @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary2 ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList2 @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary2 ) ) ) ) ) ) @ Xa ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) @ ( vEBT_VEBT_high @ ( if @ nat @ ( Xa = Mi2 ) @ ( plus_plus @ nat @ ( times_times @ nat @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary2 ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList2 @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary2 ) ) ) ) ) ) @ Xa ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) ) )
@ Ma2 ) ) )
@ ( suc @ ( suc @ Va3 ) )
@ ( list_update @ vEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ ( if @ nat @ ( Xa = Mi2 ) @ ( plus_plus @ nat @ ( times_times @ nat @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary2 ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList2 @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary2 ) ) ) ) ) ) @ Xa ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( vEBT_vebt_delete @ ( nth @ vEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ ( if @ nat @ ( Xa = Mi2 ) @ ( plus_plus @ nat @ ( times_times @ nat @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary2 ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList2 @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary2 ) ) ) ) ) ) @ Xa ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( vEBT_VEBT_low @ ( if @ nat @ ( Xa = Mi2 ) @ ( plus_plus @ nat @ ( times_times @ nat @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary2 ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList2 @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary2 ) ) ) ) ) ) @ Xa ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) )
@ Summary2 ) )
@ ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( suc @ Va3 ) ) @ 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 @ Va3 ) ) @ TreeList2 @ Summary2 ) @ Xa ) ) ) ) ) ) ) ) ) ) ) ) ).
% vebt_delete.pelims
thf(fact_2189_T_092_060_094sub_062s_092_060_094sub_062u_092_060_094sub_062c_092_060_094sub_062c_H_Opelims,axiom,
! [X2: vEBT_VEBT,Xa: nat,Y3: nat] :
( ( ( vEBT_T_s_u_c_c2 @ X2 @ Xa )
= Y3 )
=> ( ( accp @ ( product_prod @ vEBT_VEBT @ nat ) @ vEBT_T_s_u_c_c_rel @ ( product_Pair @ vEBT_VEBT @ nat @ X2 @ Xa ) )
=> ( ! [Uu2: $o,B4: $o] :
( ( X2
= ( vEBT_Leaf @ Uu2 @ B4 ) )
=> ( ( Xa
= ( zero_zero @ nat ) )
=> ( ( Y3
= ( one_one @ nat ) )
=> ~ ( accp @ ( product_prod @ vEBT_VEBT @ nat ) @ vEBT_T_s_u_c_c_rel @ ( product_Pair @ vEBT_VEBT @ nat @ ( vEBT_Leaf @ Uu2 @ B4 ) @ ( zero_zero @ nat ) ) ) ) ) )
=> ( ! [Uv2: $o,Uw2: $o] :
( ( X2
= ( vEBT_Leaf @ Uv2 @ Uw2 ) )
=> ! [N3: nat] :
( ( Xa
= ( suc @ N3 ) )
=> ( ( Y3
= ( one_one @ nat ) )
=> ~ ( accp @ ( product_prod @ vEBT_VEBT @ nat ) @ vEBT_T_s_u_c_c_rel @ ( product_Pair @ vEBT_VEBT @ nat @ ( vEBT_Leaf @ Uv2 @ Uw2 ) @ ( suc @ N3 ) ) ) ) ) )
=> ( ! [Ux2: nat,Uy2: list @ vEBT_VEBT,Uz2: vEBT_VEBT] :
( ( X2
= ( vEBT_Node @ ( none @ ( product_prod @ nat @ nat ) ) @ Ux2 @ Uy2 @ Uz2 ) )
=> ( ( Y3
= ( one_one @ nat ) )
=> ~ ( accp @ ( product_prod @ vEBT_VEBT @ nat ) @ vEBT_T_s_u_c_c_rel @ ( product_Pair @ vEBT_VEBT @ nat @ ( vEBT_Node @ ( none @ ( product_prod @ nat @ nat ) ) @ Ux2 @ Uy2 @ Uz2 ) @ Xa ) ) ) )
=> ( ! [V3: product_prod @ nat @ nat,Vc2: list @ vEBT_VEBT,Vd2: vEBT_VEBT] :
( ( X2
= ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ V3 ) @ ( zero_zero @ nat ) @ Vc2 @ Vd2 ) )
=> ( ( Y3
= ( one_one @ nat ) )
=> ~ ( accp @ ( product_prod @ vEBT_VEBT @ nat ) @ vEBT_T_s_u_c_c_rel @ ( product_Pair @ vEBT_VEBT @ nat @ ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ V3 ) @ ( zero_zero @ nat ) @ Vc2 @ Vd2 ) @ Xa ) ) ) )
=> ( ! [V3: product_prod @ nat @ nat,Vg: list @ vEBT_VEBT,Vh: vEBT_VEBT] :
( ( X2
= ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ V3 ) @ ( suc @ ( zero_zero @ nat ) ) @ Vg @ Vh ) )
=> ( ( Y3
= ( one_one @ nat ) )
=> ~ ( accp @ ( product_prod @ vEBT_VEBT @ nat ) @ vEBT_T_s_u_c_c_rel @ ( product_Pair @ vEBT_VEBT @ nat @ ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ V3 ) @ ( suc @ ( zero_zero @ nat ) ) @ Vg @ Vh ) @ Xa ) ) ) )
=> ~ ! [Mi2: nat,Ma2: nat,Va3: nat,TreeList2: list @ vEBT_VEBT,Summary2: vEBT_VEBT] :
( ( X2
= ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( suc @ Va3 ) ) @ TreeList2 @ Summary2 ) )
=> ( ( ( ( ord_less @ nat @ Xa @ Mi2 )
=> ( Y3
= ( one_one @ nat ) ) )
& ( ~ ( ord_less @ nat @ Xa @ Mi2 )
=> ( Y3
= ( if @ nat @ ( ord_less @ nat @ ( vEBT_VEBT_high @ Xa @ ( divide_divide @ nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( size_size @ ( list @ vEBT_VEBT ) @ TreeList2 ) )
@ ( if @ nat
@ ( ( ( vEBT_vebt_maxt @ ( nth @ vEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ Xa @ ( divide_divide @ nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) )
!= ( none @ nat ) )
& ( vEBT_VEBT_less @ ( some @ nat @ ( vEBT_VEBT_low @ Xa @ ( divide_divide @ nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( vEBT_vebt_maxt @ ( nth @ vEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ Xa @ ( divide_divide @ nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) ) )
@ ( plus_plus @ nat @ ( one_one @ nat ) @ ( vEBT_T_s_u_c_c2 @ ( nth @ vEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ Xa @ ( divide_divide @ nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( vEBT_VEBT_low @ Xa @ ( divide_divide @ nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) )
@ ( plus_plus @ nat @ ( vEBT_T_s_u_c_c2 @ Summary2 @ ( vEBT_VEBT_high @ Xa @ ( divide_divide @ nat @ ( suc @ ( suc @ Va3 ) ) @ ( 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 @ Va3 ) ) @ TreeList2 @ Summary2 ) @ Xa ) ) ) ) ) ) ) ) ) ) ) ).
% T\<^sub>s\<^sub>u\<^sub>c\<^sub>c'.pelims
thf(fact_2190_vebt__insert_Opelims,axiom,
! [X2: vEBT_VEBT,Xa: nat,Y3: vEBT_VEBT] :
( ( ( vEBT_vebt_insert @ X2 @ Xa )
= Y3 )
=> ( ( accp @ ( product_prod @ vEBT_VEBT @ nat ) @ vEBT_vebt_insert_rel @ ( product_Pair @ vEBT_VEBT @ nat @ X2 @ Xa ) )
=> ( ! [A4: $o,B4: $o] :
( ( X2
= ( vEBT_Leaf @ A4 @ B4 ) )
=> ( ( ( ( Xa
= ( zero_zero @ nat ) )
=> ( Y3
= ( vEBT_Leaf @ $true @ B4 ) ) )
& ( ( Xa
!= ( zero_zero @ nat ) )
=> ( ( ( Xa
= ( one_one @ nat ) )
=> ( Y3
= ( vEBT_Leaf @ A4 @ $true ) ) )
& ( ( Xa
!= ( one_one @ nat ) )
=> ( Y3
= ( vEBT_Leaf @ A4 @ B4 ) ) ) ) ) )
=> ~ ( accp @ ( product_prod @ vEBT_VEBT @ nat ) @ vEBT_vebt_insert_rel @ ( product_Pair @ vEBT_VEBT @ nat @ ( vEBT_Leaf @ A4 @ B4 ) @ Xa ) ) ) )
=> ( ! [Info2: option @ ( product_prod @ nat @ nat ),Ts2: list @ vEBT_VEBT,S3: vEBT_VEBT] :
( ( X2
= ( vEBT_Node @ Info2 @ ( zero_zero @ nat ) @ Ts2 @ S3 ) )
=> ( ( Y3
= ( vEBT_Node @ Info2 @ ( zero_zero @ nat ) @ Ts2 @ S3 ) )
=> ~ ( accp @ ( product_prod @ vEBT_VEBT @ nat ) @ vEBT_vebt_insert_rel @ ( product_Pair @ vEBT_VEBT @ nat @ ( vEBT_Node @ Info2 @ ( zero_zero @ nat ) @ Ts2 @ S3 ) @ Xa ) ) ) )
=> ( ! [Info2: option @ ( product_prod @ nat @ nat ),Ts2: list @ vEBT_VEBT,S3: vEBT_VEBT] :
( ( X2
= ( vEBT_Node @ Info2 @ ( suc @ ( zero_zero @ nat ) ) @ Ts2 @ S3 ) )
=> ( ( Y3
= ( vEBT_Node @ Info2 @ ( suc @ ( zero_zero @ nat ) ) @ Ts2 @ S3 ) )
=> ~ ( accp @ ( product_prod @ vEBT_VEBT @ nat ) @ vEBT_vebt_insert_rel @ ( product_Pair @ vEBT_VEBT @ nat @ ( vEBT_Node @ Info2 @ ( suc @ ( zero_zero @ nat ) ) @ Ts2 @ S3 ) @ Xa ) ) ) )
=> ( ! [V3: nat,TreeList2: list @ vEBT_VEBT,Summary2: vEBT_VEBT] :
( ( X2
= ( vEBT_Node @ ( none @ ( product_prod @ nat @ nat ) ) @ ( suc @ ( suc @ V3 ) ) @ TreeList2 @ Summary2 ) )
=> ( ( Y3
= ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Xa @ Xa ) ) @ ( suc @ ( suc @ V3 ) ) @ 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 @ V3 ) ) @ TreeList2 @ Summary2 ) @ Xa ) ) ) )
=> ~ ! [Mi2: nat,Ma2: nat,Va3: nat,TreeList2: list @ vEBT_VEBT,Summary2: vEBT_VEBT] :
( ( X2
= ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( suc @ Va3 ) ) @ TreeList2 @ Summary2 ) )
=> ( ( Y3
= ( if @ vEBT_VEBT
@ ( ( ord_less @ nat @ ( vEBT_VEBT_high @ ( if @ nat @ ( ord_less @ nat @ Xa @ Mi2 ) @ Mi2 @ Xa ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( size_size @ ( list @ vEBT_VEBT ) @ TreeList2 ) )
& ~ ( ( Xa = Mi2 )
| ( Xa = Ma2 ) ) )
@ ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ ( if @ nat @ ( ord_less @ nat @ Xa @ Mi2 ) @ Xa @ Mi2 ) @ ( ord_max @ nat @ ( if @ nat @ ( ord_less @ nat @ Xa @ Mi2 ) @ Mi2 @ Xa ) @ Ma2 ) ) ) @ ( suc @ ( suc @ Va3 ) ) @ ( list_update @ vEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ ( if @ nat @ ( ord_less @ nat @ Xa @ Mi2 ) @ Mi2 @ Xa ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( vEBT_vebt_insert @ ( nth @ vEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ ( if @ nat @ ( ord_less @ nat @ Xa @ Mi2 ) @ Mi2 @ Xa ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( vEBT_VEBT_low @ ( if @ nat @ ( ord_less @ nat @ Xa @ Mi2 ) @ Mi2 @ Xa ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) @ ( if @ vEBT_VEBT @ ( vEBT_VEBT_minNull @ ( nth @ vEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ ( if @ nat @ ( ord_less @ nat @ Xa @ Mi2 ) @ Mi2 @ Xa ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) @ ( vEBT_vebt_insert @ Summary2 @ ( vEBT_VEBT_high @ ( if @ nat @ ( ord_less @ nat @ Xa @ Mi2 ) @ Mi2 @ Xa ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ Summary2 ) )
@ ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( suc @ Va3 ) ) @ 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 @ Va3 ) ) @ TreeList2 @ Summary2 ) @ Xa ) ) ) ) ) ) ) ) ) ) ).
% vebt_insert.pelims
thf(fact_2191_T_092_060_094sub_062p_092_060_094sub_062r_092_060_094sub_062e_092_060_094sub_062d_H_Opelims,axiom,
! [X2: vEBT_VEBT,Xa: nat,Y3: nat] :
( ( ( vEBT_T_p_r_e_d2 @ X2 @ Xa )
= Y3 )
=> ( ( accp @ ( product_prod @ vEBT_VEBT @ nat ) @ vEBT_T_p_r_e_d_rel @ ( product_Pair @ vEBT_VEBT @ nat @ X2 @ Xa ) )
=> ( ! [Uu2: $o,Uv2: $o] :
( ( X2
= ( vEBT_Leaf @ Uu2 @ Uv2 ) )
=> ( ( Xa
= ( zero_zero @ nat ) )
=> ( ( Y3
= ( one_one @ nat ) )
=> ~ ( accp @ ( product_prod @ vEBT_VEBT @ nat ) @ vEBT_T_p_r_e_d_rel @ ( product_Pair @ vEBT_VEBT @ nat @ ( vEBT_Leaf @ Uu2 @ Uv2 ) @ ( zero_zero @ nat ) ) ) ) ) )
=> ( ! [A4: $o,Uw2: $o] :
( ( X2
= ( vEBT_Leaf @ A4 @ Uw2 ) )
=> ( ( Xa
= ( suc @ ( zero_zero @ nat ) ) )
=> ( ( Y3
= ( one_one @ nat ) )
=> ~ ( accp @ ( product_prod @ vEBT_VEBT @ nat ) @ vEBT_T_p_r_e_d_rel @ ( product_Pair @ vEBT_VEBT @ nat @ ( vEBT_Leaf @ A4 @ Uw2 ) @ ( suc @ ( zero_zero @ nat ) ) ) ) ) ) )
=> ( ! [A4: $o,B4: $o] :
( ( X2
= ( vEBT_Leaf @ A4 @ B4 ) )
=> ! [Va3: nat] :
( ( Xa
= ( suc @ ( suc @ Va3 ) ) )
=> ( ( Y3
= ( 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 @ Va3 ) ) ) ) ) ) )
=> ( ! [Uy2: nat,Uz2: list @ vEBT_VEBT,Va2: vEBT_VEBT] :
( ( X2
= ( vEBT_Node @ ( none @ ( product_prod @ nat @ nat ) ) @ Uy2 @ Uz2 @ Va2 ) )
=> ( ( Y3
= ( one_one @ nat ) )
=> ~ ( accp @ ( product_prod @ vEBT_VEBT @ nat ) @ vEBT_T_p_r_e_d_rel @ ( product_Pair @ vEBT_VEBT @ nat @ ( vEBT_Node @ ( none @ ( product_prod @ nat @ nat ) ) @ Uy2 @ Uz2 @ Va2 ) @ Xa ) ) ) )
=> ( ! [V3: product_prod @ nat @ nat,Vd2: list @ vEBT_VEBT,Ve2: vEBT_VEBT] :
( ( X2
= ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ V3 ) @ ( zero_zero @ nat ) @ Vd2 @ Ve2 ) )
=> ( ( Y3
= ( one_one @ nat ) )
=> ~ ( accp @ ( product_prod @ vEBT_VEBT @ nat ) @ vEBT_T_p_r_e_d_rel @ ( product_Pair @ vEBT_VEBT @ nat @ ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ V3 ) @ ( zero_zero @ nat ) @ Vd2 @ Ve2 ) @ Xa ) ) ) )
=> ( ! [V3: product_prod @ nat @ nat,Vh: list @ vEBT_VEBT,Vi: vEBT_VEBT] :
( ( X2
= ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ V3 ) @ ( suc @ ( zero_zero @ nat ) ) @ Vh @ Vi ) )
=> ( ( Y3
= ( one_one @ nat ) )
=> ~ ( accp @ ( product_prod @ vEBT_VEBT @ nat ) @ vEBT_T_p_r_e_d_rel @ ( product_Pair @ vEBT_VEBT @ nat @ ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ V3 ) @ ( suc @ ( zero_zero @ nat ) ) @ Vh @ Vi ) @ Xa ) ) ) )
=> ~ ! [Mi2: nat,Ma2: nat,Va3: nat,TreeList2: list @ vEBT_VEBT,Summary2: vEBT_VEBT] :
( ( X2
= ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( suc @ Va3 ) ) @ TreeList2 @ Summary2 ) )
=> ( ( ( ( ord_less @ nat @ Ma2 @ Xa )
=> ( Y3
= ( one_one @ nat ) ) )
& ( ~ ( ord_less @ nat @ Ma2 @ Xa )
=> ( Y3
= ( if @ nat @ ( ord_less @ nat @ ( vEBT_VEBT_high @ Xa @ ( divide_divide @ nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( size_size @ ( list @ vEBT_VEBT ) @ TreeList2 ) )
@ ( if @ nat
@ ( ( ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ Xa @ ( divide_divide @ nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) )
!= ( none @ nat ) )
& ( vEBT_VEBT_greater @ ( some @ nat @ ( vEBT_VEBT_low @ Xa @ ( divide_divide @ nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ Xa @ ( divide_divide @ nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) ) )
@ ( plus_plus @ nat @ ( one_one @ nat ) @ ( vEBT_T_p_r_e_d2 @ ( nth @ vEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ Xa @ ( divide_divide @ nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( vEBT_VEBT_low @ Xa @ ( divide_divide @ nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) )
@ ( plus_plus @ nat @ ( vEBT_T_p_r_e_d2 @ Summary2 @ ( vEBT_VEBT_high @ Xa @ ( divide_divide @ nat @ ( suc @ ( suc @ Va3 ) ) @ ( 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 @ Va3 ) ) @ TreeList2 @ Summary2 ) @ Xa ) ) ) ) ) ) ) ) ) ) ) ) ).
% T\<^sub>p\<^sub>r\<^sub>e\<^sub>d'.pelims
thf(fact_2192_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,
! [X2: vEBT_VEBT,Xa: nat,Y3: nat] :
( ( ( vEBT_T_i_n_s_e_r_t2 @ X2 @ Xa )
= Y3 )
=> ( ( accp @ ( product_prod @ vEBT_VEBT @ nat ) @ vEBT_T5076183648494686801_t_rel @ ( product_Pair @ vEBT_VEBT @ nat @ X2 @ Xa ) )
=> ( ! [A4: $o,B4: $o] :
( ( X2
= ( vEBT_Leaf @ A4 @ B4 ) )
=> ( ( Y3
= ( one_one @ nat ) )
=> ~ ( accp @ ( product_prod @ vEBT_VEBT @ nat ) @ vEBT_T5076183648494686801_t_rel @ ( product_Pair @ vEBT_VEBT @ nat @ ( vEBT_Leaf @ A4 @ B4 ) @ Xa ) ) ) )
=> ( ! [Info2: option @ ( product_prod @ nat @ nat ),Ts2: list @ vEBT_VEBT,S3: vEBT_VEBT] :
( ( X2
= ( vEBT_Node @ Info2 @ ( zero_zero @ nat ) @ Ts2 @ S3 ) )
=> ( ( Y3
= ( one_one @ nat ) )
=> ~ ( accp @ ( product_prod @ vEBT_VEBT @ nat ) @ vEBT_T5076183648494686801_t_rel @ ( product_Pair @ vEBT_VEBT @ nat @ ( vEBT_Node @ Info2 @ ( zero_zero @ nat ) @ Ts2 @ S3 ) @ Xa ) ) ) )
=> ( ! [Info2: option @ ( product_prod @ nat @ nat ),Ts2: list @ vEBT_VEBT,S3: vEBT_VEBT] :
( ( X2
= ( vEBT_Node @ Info2 @ ( suc @ ( zero_zero @ nat ) ) @ Ts2 @ S3 ) )
=> ( ( Y3
= ( one_one @ nat ) )
=> ~ ( accp @ ( product_prod @ vEBT_VEBT @ nat ) @ vEBT_T5076183648494686801_t_rel @ ( product_Pair @ vEBT_VEBT @ nat @ ( vEBT_Node @ Info2 @ ( suc @ ( zero_zero @ nat ) ) @ Ts2 @ S3 ) @ Xa ) ) ) )
=> ( ! [V3: nat,TreeList2: list @ vEBT_VEBT,Summary2: vEBT_VEBT] :
( ( X2
= ( vEBT_Node @ ( none @ ( product_prod @ nat @ nat ) ) @ ( suc @ ( suc @ V3 ) ) @ TreeList2 @ Summary2 ) )
=> ( ( Y3
= ( one_one @ nat ) )
=> ~ ( accp @ ( product_prod @ vEBT_VEBT @ nat ) @ vEBT_T5076183648494686801_t_rel @ ( product_Pair @ vEBT_VEBT @ nat @ ( vEBT_Node @ ( none @ ( product_prod @ nat @ nat ) ) @ ( suc @ ( suc @ V3 ) ) @ TreeList2 @ Summary2 ) @ Xa ) ) ) )
=> ~ ! [Mi2: nat,Ma2: nat,Va3: nat,TreeList2: list @ vEBT_VEBT,Summary2: vEBT_VEBT] :
( ( X2
= ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( suc @ Va3 ) ) @ TreeList2 @ Summary2 ) )
=> ( ( Y3
= ( if @ nat
@ ( ( ord_less @ nat @ ( vEBT_VEBT_high @ ( if @ nat @ ( ord_less @ nat @ Xa @ Mi2 ) @ Mi2 @ Xa ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( size_size @ ( list @ vEBT_VEBT ) @ TreeList2 ) )
& ~ ( ( Xa = Mi2 )
| ( Xa = Ma2 ) ) )
@ ( plus_plus @ nat @ ( vEBT_T_i_n_s_e_r_t2 @ ( nth @ vEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ ( if @ nat @ ( ord_less @ nat @ Xa @ Mi2 ) @ Mi2 @ Xa ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( vEBT_VEBT_low @ ( if @ nat @ ( ord_less @ nat @ Xa @ Mi2 ) @ Mi2 @ Xa ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( if @ nat @ ( vEBT_VEBT_minNull @ ( nth @ vEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ ( if @ nat @ ( ord_less @ nat @ Xa @ Mi2 ) @ Mi2 @ Xa ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) @ ( vEBT_T_i_n_s_e_r_t2 @ Summary2 @ ( vEBT_VEBT_high @ ( if @ nat @ ( ord_less @ nat @ Xa @ Mi2 ) @ Mi2 @ Xa ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va3 ) ) @ ( 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 @ Va3 ) ) @ TreeList2 @ Summary2 ) @ Xa ) ) ) ) ) ) ) ) ) ) ).
% T\<^sub>i\<^sub>n\<^sub>s\<^sub>e\<^sub>r\<^sub>t'.pelims
thf(fact_2193_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,
! [X2: vEBT_VEBT,Xa: nat,Y3: nat] :
( ( ( vEBT_T_m_e_m_b_e_r @ X2 @ Xa )
= Y3 )
=> ( ( accp @ ( product_prod @ vEBT_VEBT @ nat ) @ vEBT_T5837161174952499735_r_rel @ ( product_Pair @ vEBT_VEBT @ nat @ X2 @ Xa ) )
=> ( ! [A4: $o,B4: $o] :
( ( X2
= ( vEBT_Leaf @ A4 @ B4 ) )
=> ( ( Y3
= ( plus_plus @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) )
@ ( if @ nat
@ ( Xa
= ( zero_zero @ nat ) )
@ ( one_one @ nat )
@ ( plus_plus @ nat @ ( one_one @ nat ) @ ( one_one @ nat ) ) ) ) )
=> ~ ( accp @ ( product_prod @ vEBT_VEBT @ nat ) @ vEBT_T5837161174952499735_r_rel @ ( product_Pair @ vEBT_VEBT @ nat @ ( vEBT_Leaf @ A4 @ B4 ) @ Xa ) ) ) )
=> ( ! [Uu2: nat,Uv2: list @ vEBT_VEBT,Uw2: vEBT_VEBT] :
( ( X2
= ( vEBT_Node @ ( none @ ( product_prod @ nat @ nat ) ) @ Uu2 @ Uv2 @ Uw2 ) )
=> ( ( Y3
= ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) )
=> ~ ( accp @ ( product_prod @ vEBT_VEBT @ nat ) @ vEBT_T5837161174952499735_r_rel @ ( product_Pair @ vEBT_VEBT @ nat @ ( vEBT_Node @ ( none @ ( product_prod @ nat @ nat ) ) @ Uu2 @ Uv2 @ Uw2 ) @ Xa ) ) ) )
=> ( ! [V3: product_prod @ nat @ nat,Uy2: list @ vEBT_VEBT,Uz2: vEBT_VEBT] :
( ( X2
= ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ V3 ) @ ( zero_zero @ nat ) @ Uy2 @ Uz2 ) )
=> ( ( Y3
= ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) )
=> ~ ( accp @ ( product_prod @ vEBT_VEBT @ nat ) @ vEBT_T5837161174952499735_r_rel @ ( product_Pair @ vEBT_VEBT @ nat @ ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ V3 ) @ ( zero_zero @ nat ) @ Uy2 @ Uz2 ) @ Xa ) ) ) )
=> ( ! [V3: product_prod @ nat @ nat,Vb2: list @ vEBT_VEBT,Vc2: vEBT_VEBT] :
( ( X2
= ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ V3 ) @ ( suc @ ( zero_zero @ nat ) ) @ Vb2 @ Vc2 ) )
=> ( ( Y3
= ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) )
=> ~ ( accp @ ( product_prod @ vEBT_VEBT @ nat ) @ vEBT_T5837161174952499735_r_rel @ ( product_Pair @ vEBT_VEBT @ nat @ ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ V3 ) @ ( suc @ ( zero_zero @ nat ) ) @ Vb2 @ Vc2 ) @ Xa ) ) ) )
=> ~ ! [Mi2: nat,Ma2: nat,Va3: nat,TreeList2: list @ vEBT_VEBT,Summary2: vEBT_VEBT] :
( ( X2
= ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( suc @ Va3 ) ) @ TreeList2 @ Summary2 ) )
=> ( ( Y3
= ( plus_plus @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( if @ nat @ ( Xa = Mi2 ) @ ( one_one @ nat ) @ ( plus_plus @ nat @ ( one_one @ nat ) @ ( if @ nat @ ( Xa = Ma2 ) @ ( one_one @ nat ) @ ( plus_plus @ nat @ ( one_one @ nat ) @ ( if @ nat @ ( ord_less @ nat @ Xa @ Mi2 ) @ ( one_one @ nat ) @ ( plus_plus @ nat @ ( one_one @ nat ) @ ( if @ nat @ ( ord_less @ nat @ Ma2 @ Xa ) @ ( one_one @ nat ) @ ( plus_plus @ nat @ ( numeral_numeral @ nat @ ( bit1 @ ( bit0 @ ( bit0 @ one2 ) ) ) ) @ ( if @ nat @ ( ord_less @ nat @ ( vEBT_VEBT_high @ Xa @ ( divide_divide @ nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( size_size @ ( list @ vEBT_VEBT ) @ TreeList2 ) ) @ ( plus_plus @ nat @ ( one_one @ nat ) @ ( vEBT_T_m_e_m_b_e_r @ ( nth @ vEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ Xa @ ( divide_divide @ nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( vEBT_VEBT_low @ Xa @ ( divide_divide @ nat @ ( suc @ ( suc @ Va3 ) ) @ ( 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 @ Va3 ) ) @ TreeList2 @ Summary2 ) @ Xa ) ) ) ) ) ) ) ) ) ) ).
% T\<^sub>m\<^sub>e\<^sub>m\<^sub>b\<^sub>e\<^sub>r.pelims
thf(fact_2194_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,
! [X2: vEBT_VEBT,Xa: nat,Y3: nat] :
( ( ( vEBT_T_m_e_m_b_e_r2 @ X2 @ Xa )
= Y3 )
=> ( ( accp @ ( product_prod @ vEBT_VEBT @ nat ) @ vEBT_T8099345112685741742_r_rel @ ( product_Pair @ vEBT_VEBT @ nat @ X2 @ Xa ) )
=> ( ! [A4: $o,B4: $o] :
( ( X2
= ( vEBT_Leaf @ A4 @ B4 ) )
=> ( ( Y3
= ( one_one @ nat ) )
=> ~ ( accp @ ( product_prod @ vEBT_VEBT @ nat ) @ vEBT_T8099345112685741742_r_rel @ ( product_Pair @ vEBT_VEBT @ nat @ ( vEBT_Leaf @ A4 @ B4 ) @ Xa ) ) ) )
=> ( ! [Uu2: nat,Uv2: list @ vEBT_VEBT,Uw2: vEBT_VEBT] :
( ( X2
= ( vEBT_Node @ ( none @ ( product_prod @ nat @ nat ) ) @ Uu2 @ Uv2 @ Uw2 ) )
=> ( ( Y3
= ( one_one @ nat ) )
=> ~ ( accp @ ( product_prod @ vEBT_VEBT @ nat ) @ vEBT_T8099345112685741742_r_rel @ ( product_Pair @ vEBT_VEBT @ nat @ ( vEBT_Node @ ( none @ ( product_prod @ nat @ nat ) ) @ Uu2 @ Uv2 @ Uw2 ) @ Xa ) ) ) )
=> ( ! [V3: product_prod @ nat @ nat,Uy2: list @ vEBT_VEBT,Uz2: vEBT_VEBT] :
( ( X2
= ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ V3 ) @ ( zero_zero @ nat ) @ Uy2 @ Uz2 ) )
=> ( ( Y3
= ( one_one @ nat ) )
=> ~ ( accp @ ( product_prod @ vEBT_VEBT @ nat ) @ vEBT_T8099345112685741742_r_rel @ ( product_Pair @ vEBT_VEBT @ nat @ ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ V3 ) @ ( zero_zero @ nat ) @ Uy2 @ Uz2 ) @ Xa ) ) ) )
=> ( ! [V3: product_prod @ nat @ nat,Vb2: list @ vEBT_VEBT,Vc2: vEBT_VEBT] :
( ( X2
= ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ V3 ) @ ( suc @ ( zero_zero @ nat ) ) @ Vb2 @ Vc2 ) )
=> ( ( Y3
= ( one_one @ nat ) )
=> ~ ( accp @ ( product_prod @ vEBT_VEBT @ nat ) @ vEBT_T8099345112685741742_r_rel @ ( product_Pair @ vEBT_VEBT @ nat @ ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ V3 ) @ ( suc @ ( zero_zero @ nat ) ) @ Vb2 @ Vc2 ) @ Xa ) ) ) )
=> ~ ! [Mi2: nat,Ma2: nat,Va3: nat,TreeList2: list @ vEBT_VEBT,Summary2: vEBT_VEBT] :
( ( X2
= ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( suc @ Va3 ) ) @ TreeList2 @ Summary2 ) )
=> ( ( Y3
= ( plus_plus @ nat @ ( one_one @ nat )
@ ( if @ nat @ ( Xa = Mi2 ) @ ( zero_zero @ nat )
@ ( if @ nat @ ( Xa = Ma2 ) @ ( zero_zero @ nat )
@ ( if @ nat @ ( ord_less @ nat @ Xa @ Mi2 ) @ ( zero_zero @ nat )
@ ( if @ nat @ ( ord_less @ nat @ Ma2 @ Xa ) @ ( zero_zero @ nat )
@ ( if @ nat
@ ( ( ord_less @ nat @ Mi2 @ Xa )
& ( ord_less @ nat @ Xa @ Ma2 ) )
@ ( if @ nat @ ( ord_less @ nat @ ( vEBT_VEBT_high @ Xa @ ( divide_divide @ nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( size_size @ ( list @ vEBT_VEBT ) @ TreeList2 ) ) @ ( vEBT_T_m_e_m_b_e_r2 @ ( nth @ vEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ Xa @ ( divide_divide @ nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( vEBT_VEBT_low @ Xa @ ( divide_divide @ nat @ ( suc @ ( suc @ Va3 ) ) @ ( 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 @ Va3 ) ) @ TreeList2 @ Summary2 ) @ Xa ) ) ) ) ) ) ) ) ) ) ).
% T\<^sub>m\<^sub>e\<^sub>m\<^sub>b\<^sub>e\<^sub>r'.pelims
thf(fact_2195_word__of__int__1,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ( ( ring_1_of_int @ ( word @ A ) @ ( one_one @ int ) )
= ( one_one @ ( word @ A ) ) ) ) ).
% word_of_int_1
thf(fact_2196_word__le__0__iff,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [X2: word @ A] :
( ( ord_less_eq @ ( word @ A ) @ X2 @ ( zero_zero @ ( word @ A ) ) )
= ( X2
= ( zero_zero @ ( word @ A ) ) ) ) ) ).
% word_le_0_iff
thf(fact_2197_word__coorder_Oextremum__unique,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [A3: word @ A] :
( ( ord_less_eq @ ( word @ A ) @ A3 @ ( zero_zero @ ( word @ A ) ) )
= ( A3
= ( zero_zero @ ( word @ A ) ) ) ) ) ).
% word_coorder.extremum_unique
thf(fact_2198_range__subset__card,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [A3: word @ A,B3: word @ A,C3: word @ A,D2: word @ A] :
( ( ord_less_eq @ ( set @ ( word @ A ) ) @ ( set_or1337092689740270186AtMost @ ( word @ A ) @ A3 @ B3 ) @ ( set_or1337092689740270186AtMost @ ( word @ A ) @ C3 @ D2 ) )
=> ( ( ord_less_eq @ ( word @ A ) @ A3 @ B3 )
=> ( ( ord_less_eq @ ( word @ A ) @ C3 @ D2 )
& ( ord_less_eq @ ( word @ A ) @ ( minus_minus @ ( word @ A ) @ B3 @ A3 ) @ ( minus_minus @ ( word @ A ) @ D2 @ C3 ) ) ) ) ) ) ).
% range_subset_card
thf(fact_2199_More__Word_Oword__l__diffs_I4_J,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [W: word @ A,X2: word @ A,Z: word @ A] :
( ( ord_less_eq @ ( word @ A ) @ ( plus_plus @ ( word @ A ) @ W @ X2 ) @ Z )
=> ( ( ord_less_eq @ ( word @ A ) @ X2 @ ( plus_plus @ ( word @ A ) @ W @ X2 ) )
=> ( ord_less_eq @ ( word @ A ) @ W @ ( minus_minus @ ( word @ A ) @ Z @ X2 ) ) ) ) ) ).
% More_Word.word_l_diffs(4)
thf(fact_2200_More__Word_Oword__l__diffs_I3_J,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [W: word @ A,Z: word @ A,X2: word @ A] :
( ( ord_less_eq @ ( word @ A ) @ W @ ( minus_minus @ ( word @ A ) @ Z @ X2 ) )
=> ( ( ord_less_eq @ ( word @ A ) @ X2 @ Z )
=> ( ord_less_eq @ ( word @ A ) @ ( plus_plus @ ( word @ A ) @ W @ X2 ) @ Z ) ) ) ) ).
% More_Word.word_l_diffs(3)
thf(fact_2201_word__diff__ls_H_I4_J,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [Y3: word @ A,W: word @ A,X2: word @ A] :
( ( ord_less_eq @ ( word @ A ) @ Y3 @ ( plus_plus @ ( word @ A ) @ W @ X2 ) )
=> ( ( ord_less_eq @ ( word @ A ) @ X2 @ Y3 )
=> ( ord_less_eq @ ( word @ A ) @ ( minus_minus @ ( word @ A ) @ Y3 @ X2 ) @ W ) ) ) ) ).
% word_diff_ls'(4)
thf(fact_2202_word__diff__ls_H_I3_J,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [Y3: word @ A,X2: word @ A,W: word @ A] :
( ( ord_less_eq @ ( word @ A ) @ ( minus_minus @ ( word @ A ) @ Y3 @ X2 ) @ W )
=> ( ( ord_less_eq @ ( word @ A ) @ X2 @ ( plus_plus @ ( word @ A ) @ W @ X2 ) )
=> ( ord_less_eq @ ( word @ A ) @ Y3 @ ( plus_plus @ ( word @ A ) @ W @ X2 ) ) ) ) ) ).
% word_diff_ls'(3)
thf(fact_2203_word__l__diffs_H_I4_J,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [W: word @ A,X2: word @ A,Z: word @ A] :
( ( ord_less_eq @ ( word @ A ) @ ( plus_plus @ ( word @ A ) @ W @ X2 ) @ Z )
=> ( ( ord_less_eq @ ( word @ A ) @ X2 @ ( plus_plus @ ( word @ A ) @ W @ X2 ) )
=> ( ord_less_eq @ ( word @ A ) @ ( minus_minus @ ( word @ A ) @ ( plus_plus @ ( word @ A ) @ W @ X2 ) @ X2 ) @ ( minus_minus @ ( word @ A ) @ Z @ X2 ) ) ) ) ) ).
% word_l_diffs'(4)
thf(fact_2204_word__l__diffs_H_I3_J,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [W: word @ A,X2: word @ A,Z: word @ A] :
( ( ord_less_eq @ ( word @ A ) @ ( minus_minus @ ( word @ A ) @ ( plus_plus @ ( word @ A ) @ W @ X2 ) @ X2 ) @ ( minus_minus @ ( word @ A ) @ Z @ X2 ) )
=> ( ( ord_less_eq @ ( word @ A ) @ X2 @ Z )
=> ( ord_less_eq @ ( word @ A ) @ ( plus_plus @ ( word @ A ) @ W @ X2 ) @ Z ) ) ) ) ).
% word_l_diffs'(3)
thf(fact_2205_word__diff__ls_H_H_I4_J,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [Y3: word @ A,W: word @ A,X2: word @ A] :
( ( ord_less_eq @ ( word @ A ) @ Y3 @ ( plus_plus @ ( word @ A ) @ W @ X2 ) )
=> ( ( ord_less_eq @ ( word @ A ) @ X2 @ Y3 )
=> ( ord_less_eq @ ( word @ A ) @ ( minus_minus @ ( word @ A ) @ Y3 @ X2 ) @ ( minus_minus @ ( word @ A ) @ ( plus_plus @ ( word @ A ) @ W @ X2 ) @ X2 ) ) ) ) ) ).
% word_diff_ls''(4)
thf(fact_2206_word__diff__ls_H_H_I3_J,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [Y3: word @ A,X2: word @ A,W: word @ A] :
( ( ord_less_eq @ ( word @ A ) @ ( minus_minus @ ( word @ A ) @ Y3 @ X2 ) @ ( minus_minus @ ( word @ A ) @ ( plus_plus @ ( word @ A ) @ W @ X2 ) @ X2 ) )
=> ( ( ord_less_eq @ ( word @ A ) @ X2 @ ( plus_plus @ ( word @ A ) @ W @ X2 ) )
=> ( ord_less_eq @ ( word @ A ) @ Y3 @ ( plus_plus @ ( word @ A ) @ W @ X2 ) ) ) ) ) ).
% word_diff_ls''(3)
thf(fact_2207_word__le__minus__mono__right,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [Z: word @ A,Y3: word @ A,X2: word @ A] :
( ( ord_less_eq @ ( word @ A ) @ Z @ Y3 )
=> ( ( ord_less_eq @ ( word @ A ) @ Y3 @ X2 )
=> ( ( ord_less_eq @ ( word @ A ) @ Z @ X2 )
=> ( ord_less_eq @ ( word @ A ) @ ( minus_minus @ ( word @ A ) @ X2 @ Y3 ) @ ( minus_minus @ ( word @ A ) @ X2 @ Z ) ) ) ) ) ) ).
% word_le_minus_mono_right
thf(fact_2208_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_2209_neq__0__no__wrap,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [X2: word @ A,Y3: word @ A] :
( ( ord_less_eq @ ( word @ A ) @ X2 @ ( plus_plus @ ( word @ A ) @ X2 @ Y3 ) )
=> ( ( X2
!= ( zero_zero @ ( word @ A ) ) )
=> ( ( plus_plus @ ( word @ A ) @ X2 @ Y3 )
!= ( zero_zero @ ( word @ A ) ) ) ) ) ) ).
% neq_0_no_wrap
thf(fact_2210_word__plus__mono__left,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [Y3: word @ A,Z: word @ A,X2: word @ A] :
( ( ord_less_eq @ ( word @ A ) @ Y3 @ Z )
=> ( ( ord_less_eq @ ( word @ A ) @ X2 @ ( plus_plus @ ( word @ A ) @ X2 @ Z ) )
=> ( ord_less_eq @ ( word @ A ) @ ( plus_plus @ ( word @ A ) @ Y3 @ X2 ) @ ( plus_plus @ ( word @ A ) @ Z @ X2 ) ) ) ) ) ).
% word_plus_mono_left
thf(fact_2211_word__add__increasing,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [P6: word @ A,W: word @ A,X2: word @ A] :
( ( ord_less_eq @ ( word @ A ) @ ( plus_plus @ ( word @ A ) @ P6 @ W ) @ X2 )
=> ( ( ord_less_eq @ ( word @ A ) @ P6 @ ( plus_plus @ ( word @ A ) @ P6 @ W ) )
=> ( ord_less_eq @ ( word @ A ) @ P6 @ X2 ) ) ) ) ).
% word_add_increasing
thf(fact_2212_word__plus__mcs__4_H,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [X2: word @ A,V: word @ A,W: word @ A] :
( ( ord_less_eq @ ( word @ A ) @ ( plus_plus @ ( word @ A ) @ X2 @ V ) @ ( plus_plus @ ( word @ A ) @ X2 @ W ) )
=> ( ( ord_less_eq @ ( word @ A ) @ X2 @ ( plus_plus @ ( word @ A ) @ X2 @ V ) )
=> ( ord_less_eq @ ( word @ A ) @ V @ W ) ) ) ) ).
% word_plus_mcs_4'
thf(fact_2213_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_2214_word__plus__mcs__4,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [V: word @ A,X2: word @ A,W: word @ A] :
( ( ord_less_eq @ ( word @ A ) @ ( plus_plus @ ( word @ A ) @ V @ X2 ) @ ( plus_plus @ ( word @ A ) @ W @ X2 ) )
=> ( ( ord_less_eq @ ( word @ A ) @ X2 @ ( plus_plus @ ( word @ A ) @ V @ X2 ) )
=> ( ord_less_eq @ ( word @ A ) @ V @ W ) ) ) ) ).
% word_plus_mcs_4
thf(fact_2215_word__plus__mcs__3,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [V: word @ A,W: word @ A,X2: word @ A] :
( ( ord_less_eq @ ( word @ A ) @ V @ W )
=> ( ( ord_less_eq @ ( word @ A ) @ X2 @ ( plus_plus @ ( word @ A ) @ W @ X2 ) )
=> ( ord_less_eq @ ( word @ A ) @ ( plus_plus @ ( word @ A ) @ V @ X2 ) @ ( plus_plus @ ( word @ A ) @ W @ X2 ) ) ) ) ) ).
% word_plus_mcs_3
thf(fact_2216_word__sub__mono2,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [A3: word @ A,B3: word @ A,C3: word @ A,D2: word @ A] :
( ( ord_less_eq @ ( word @ A ) @ ( plus_plus @ ( word @ A ) @ A3 @ B3 ) @ ( plus_plus @ ( word @ A ) @ C3 @ D2 ) )
=> ( ( ord_less_eq @ ( word @ A ) @ C3 @ A3 )
=> ( ( ord_less_eq @ ( word @ A ) @ B3 @ ( plus_plus @ ( word @ A ) @ A3 @ B3 ) )
=> ( ( ord_less_eq @ ( word @ A ) @ D2 @ ( plus_plus @ ( word @ A ) @ C3 @ D2 ) )
=> ( ord_less_eq @ ( word @ A ) @ B3 @ D2 ) ) ) ) ) ) ).
% word_sub_mono2
thf(fact_2217_word__random,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [P6: word @ A,X5: word @ A,X2: word @ A] :
( ( ord_less_eq @ ( word @ A ) @ P6 @ ( plus_plus @ ( word @ A ) @ P6 @ X5 ) )
=> ( ( ord_less_eq @ ( word @ A ) @ X2 @ X5 )
=> ( ord_less_eq @ ( word @ A ) @ P6 @ ( plus_plus @ ( word @ A ) @ P6 @ X2 ) ) ) ) ) ).
% word_random
thf(fact_2218_constraint__expand,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [X2: word @ A,Lower: word @ A,Upper: word @ A] :
( ( member @ ( word @ A ) @ X2
@ ( collect @ ( word @ A )
@ ^ [Y: word @ A] :
( ( ord_less_eq @ ( word @ A ) @ Lower @ Y )
& ( ord_less_eq @ ( word @ A ) @ Y @ Upper ) ) ) )
= ( ( ord_less_eq @ ( word @ A ) @ Lower @ X2 )
& ( ord_less_eq @ ( word @ A ) @ X2 @ Upper ) ) ) ) ).
% constraint_expand
thf(fact_2219_olen__add__eqv,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [Y3: word @ A,X2: word @ A] :
( ( ord_less_eq @ ( word @ A ) @ Y3 @ ( plus_plus @ ( word @ A ) @ Y3 @ X2 ) )
= ( ord_less_eq @ ( word @ A ) @ X2 @ ( plus_plus @ ( word @ A ) @ Y3 @ X2 ) ) ) ) ).
% olen_add_eqv
thf(fact_2220_le__no__overflow,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [X2: word @ A,B3: word @ A,A3: word @ A] :
( ( ord_less_eq @ ( word @ A ) @ X2 @ B3 )
=> ( ( ord_less_eq @ ( word @ A ) @ A3 @ ( plus_plus @ ( word @ A ) @ A3 @ B3 ) )
=> ( ord_less_eq @ ( word @ A ) @ X2 @ ( plus_plus @ ( word @ A ) @ A3 @ B3 ) ) ) ) ) ).
% le_no_overflow
thf(fact_2221_word__le__plus__either,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [X2: word @ A,Y3: word @ A,Z: word @ A] :
( ( ( ord_less_eq @ ( word @ A ) @ X2 @ Y3 )
| ( ord_less_eq @ ( word @ A ) @ X2 @ Z ) )
=> ( ( ord_less_eq @ ( word @ A ) @ Y3 @ ( plus_plus @ ( word @ A ) @ Y3 @ Z ) )
=> ( ord_less_eq @ ( word @ A ) @ X2 @ ( plus_plus @ ( word @ A ) @ Y3 @ Z ) ) ) ) ) ).
% word_le_plus_either
thf(fact_2222_word__plus__mono__right,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [Y3: word @ A,Z: word @ A,X2: word @ A] :
( ( ord_less_eq @ ( word @ A ) @ Y3 @ Z )
=> ( ( ord_less_eq @ ( word @ A ) @ X2 @ ( plus_plus @ ( word @ A ) @ X2 @ Z ) )
=> ( ord_less_eq @ ( word @ A ) @ ( plus_plus @ ( word @ A ) @ X2 @ Y3 ) @ ( plus_plus @ ( word @ A ) @ X2 @ Z ) ) ) ) ) ).
% word_plus_mono_right
thf(fact_2223_word__plus__mono__right2,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [A3: word @ A,B3: word @ A,C3: word @ A] :
( ( ord_less_eq @ ( word @ A ) @ A3 @ ( plus_plus @ ( word @ A ) @ A3 @ B3 ) )
=> ( ( ord_less_eq @ ( word @ A ) @ C3 @ B3 )
=> ( ord_less_eq @ ( word @ A ) @ A3 @ ( plus_plus @ ( word @ A ) @ A3 @ C3 ) ) ) ) ) ).
% word_plus_mono_right2
thf(fact_2224_word__zero__le,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [Y3: word @ A] : ( ord_less_eq @ ( word @ A ) @ ( zero_zero @ ( word @ A ) ) @ Y3 ) ) ).
% word_zero_le
thf(fact_2225_word__coorder_Oextremum__uniqueI,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [A3: word @ A] :
( ( ord_less_eq @ ( word @ A ) @ A3 @ ( zero_zero @ ( word @ A ) ) )
=> ( A3
= ( zero_zero @ ( word @ A ) ) ) ) ) ).
% word_coorder.extremum_uniqueI
thf(fact_2226_word__coorder_Oextremum,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [A3: word @ A] : ( ord_less_eq @ ( word @ A ) @ ( zero_zero @ ( word @ A ) ) @ A3 ) ) ).
% word_coorder.extremum
thf(fact_2227_plus__minus__not__NULL__ab,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [X2: word @ A,Ab: word @ A,C3: word @ A] :
( ( ord_less_eq @ ( word @ A ) @ X2 @ ( minus_minus @ ( word @ A ) @ Ab @ C3 ) )
=> ( ( ord_less_eq @ ( word @ A ) @ C3 @ Ab )
=> ( ( C3
!= ( zero_zero @ ( word @ A ) ) )
=> ( ( plus_plus @ ( word @ A ) @ X2 @ C3 )
!= ( zero_zero @ ( word @ A ) ) ) ) ) ) ) ).
% plus_minus_not_NULL_ab
thf(fact_2228_word__sub__le,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [Y3: word @ A,X2: word @ A] :
( ( ord_less_eq @ ( word @ A ) @ Y3 @ X2 )
=> ( ord_less_eq @ ( word @ A ) @ ( minus_minus @ ( word @ A ) @ X2 @ Y3 ) @ X2 ) ) ) ).
% word_sub_le
thf(fact_2229_word__sub__le__iff,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [X2: word @ A,Y3: word @ A] :
( ( ord_less_eq @ ( word @ A ) @ ( minus_minus @ ( word @ A ) @ X2 @ Y3 ) @ X2 )
= ( ord_less_eq @ ( word @ A ) @ Y3 @ X2 ) ) ) ).
% word_sub_le_iff
thf(fact_2230_word__le__minus__mono,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [A3: word @ A,C3: word @ A,D2: word @ A,B3: word @ A] :
( ( ord_less_eq @ ( word @ A ) @ A3 @ C3 )
=> ( ( ord_less_eq @ ( word @ A ) @ D2 @ B3 )
=> ( ( ord_less_eq @ ( word @ A ) @ ( minus_minus @ ( word @ A ) @ A3 @ B3 ) @ A3 )
=> ( ( ord_less_eq @ ( word @ A ) @ ( minus_minus @ ( word @ A ) @ C3 @ D2 ) @ C3 )
=> ( ord_less_eq @ ( word @ A ) @ ( minus_minus @ ( word @ A ) @ A3 @ B3 ) @ ( minus_minus @ ( word @ A ) @ C3 @ D2 ) ) ) ) ) ) ) ).
% word_le_minus_mono
thf(fact_2231_word__le__minus__cancel,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [Y3: word @ A,X2: word @ A,Z: word @ A] :
( ( ord_less_eq @ ( word @ A ) @ ( minus_minus @ ( word @ A ) @ Y3 @ X2 ) @ ( minus_minus @ ( word @ A ) @ Z @ X2 ) )
=> ( ( ord_less_eq @ ( word @ A ) @ X2 @ Z )
=> ( ord_less_eq @ ( word @ A ) @ Y3 @ Z ) ) ) ) ).
% word_le_minus_cancel
thf(fact_2232_word__le__minus__mono__left,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [Y3: word @ A,Z: word @ A,X2: word @ A] :
( ( ord_less_eq @ ( word @ A ) @ Y3 @ Z )
=> ( ( ord_less_eq @ ( word @ A ) @ X2 @ Y3 )
=> ( ord_less_eq @ ( word @ A ) @ ( minus_minus @ ( word @ A ) @ Y3 @ X2 ) @ ( minus_minus @ ( word @ A ) @ Z @ X2 ) ) ) ) ) ).
% word_le_minus_mono_left
thf(fact_2233_plus__minus__no__overflow__ab,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [X2: word @ A,Ab: word @ A,C3: word @ A] :
( ( ord_less_eq @ ( word @ A ) @ X2 @ ( minus_minus @ ( word @ A ) @ Ab @ C3 ) )
=> ( ( ord_less_eq @ ( word @ A ) @ C3 @ Ab )
=> ( ord_less_eq @ ( word @ A ) @ X2 @ ( plus_plus @ ( word @ A ) @ X2 @ C3 ) ) ) ) ) ).
% plus_minus_no_overflow_ab
thf(fact_2234_le__minus_H,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [A3: word @ A,C3: word @ A,B3: word @ A] :
( ( ord_less_eq @ ( word @ A ) @ ( plus_plus @ ( word @ A ) @ A3 @ C3 ) @ B3 )
=> ( ( ord_less_eq @ ( word @ A ) @ A3 @ ( plus_plus @ ( word @ A ) @ A3 @ C3 ) )
=> ( ord_less_eq @ ( word @ A ) @ C3 @ ( minus_minus @ ( word @ A ) @ B3 @ A3 ) ) ) ) ) ).
% le_minus'
thf(fact_2235_le__plus_H,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [A3: word @ A,B3: word @ A,C3: word @ A] :
( ( ord_less_eq @ ( word @ A ) @ A3 @ B3 )
=> ( ( ord_less_eq @ ( word @ A ) @ C3 @ ( minus_minus @ ( word @ A ) @ B3 @ A3 ) )
=> ( ord_less_eq @ ( word @ A ) @ ( plus_plus @ ( word @ A ) @ A3 @ C3 ) @ B3 ) ) ) ) ).
% le_plus'
thf(fact_2236_le__plus,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [C3: word @ A,B3: word @ A,A3: word @ A] :
( ( ord_less_eq @ ( word @ A ) @ C3 @ ( minus_minus @ ( word @ A ) @ B3 @ A3 ) )
=> ( ( ord_less_eq @ ( word @ A ) @ A3 @ B3 )
=> ( ord_less_eq @ ( word @ A ) @ ( plus_plus @ ( word @ A ) @ A3 @ C3 ) @ B3 ) ) ) ) ).
% le_plus
thf(fact_2237_word__plus__mcs_I3_J,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [V: word @ A,Xb: word @ A,X2: word @ A,W: word @ A,Xa: word @ A] :
( ( ord_less_eq @ ( word @ A ) @ ( minus_minus @ ( word @ A ) @ ( plus_plus @ ( word @ A ) @ V @ Xb ) @ X2 ) @ ( minus_minus @ ( word @ A ) @ ( plus_plus @ ( word @ A ) @ W @ Xa ) @ X2 ) )
=> ( ( ord_less_eq @ ( word @ A ) @ X2 @ ( plus_plus @ ( word @ A ) @ W @ Xa ) )
=> ( ord_less_eq @ ( word @ A ) @ ( plus_plus @ ( word @ A ) @ V @ Xb ) @ ( plus_plus @ ( word @ A ) @ W @ Xa ) ) ) ) ) ).
% word_plus_mcs(3)
thf(fact_2238_word__plus__mcs_I4_J,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [V: word @ A,Xb: word @ A,W: word @ A,Xa: word @ A,X2: word @ A] :
( ( ord_less_eq @ ( word @ A ) @ ( plus_plus @ ( word @ A ) @ V @ Xb ) @ ( plus_plus @ ( word @ A ) @ W @ Xa ) )
=> ( ( ord_less_eq @ ( word @ A ) @ X2 @ ( plus_plus @ ( word @ A ) @ V @ Xb ) )
=> ( ord_less_eq @ ( word @ A ) @ ( minus_minus @ ( word @ A ) @ ( plus_plus @ ( word @ A ) @ V @ Xb ) @ X2 ) @ ( minus_minus @ ( word @ A ) @ ( plus_plus @ ( word @ A ) @ W @ Xa ) @ X2 ) ) ) ) ) ).
% word_plus_mcs(4)
thf(fact_2239_Word_Oword__l__diffs_I3_J,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [W: word @ A,Xa: word @ A,X2: word @ A,Z: word @ A] :
( ( ord_less_eq @ ( word @ A ) @ ( minus_minus @ ( word @ A ) @ ( plus_plus @ ( word @ A ) @ W @ Xa ) @ X2 ) @ ( minus_minus @ ( word @ A ) @ Z @ X2 ) )
=> ( ( ord_less_eq @ ( word @ A ) @ X2 @ Z )
=> ( ord_less_eq @ ( word @ A ) @ ( plus_plus @ ( word @ A ) @ W @ Xa ) @ Z ) ) ) ) ).
% Word.word_l_diffs(3)
thf(fact_2240_Word_Oword__l__diffs_I4_J,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [W: word @ A,Xa: word @ A,Z: word @ A,X2: word @ A] :
( ( ord_less_eq @ ( word @ A ) @ ( plus_plus @ ( word @ A ) @ W @ Xa ) @ Z )
=> ( ( ord_less_eq @ ( word @ A ) @ X2 @ ( plus_plus @ ( word @ A ) @ W @ Xa ) )
=> ( ord_less_eq @ ( word @ A ) @ ( minus_minus @ ( word @ A ) @ ( plus_plus @ ( word @ A ) @ W @ Xa ) @ X2 ) @ ( minus_minus @ ( word @ A ) @ Z @ X2 ) ) ) ) ) ).
% Word.word_l_diffs(4)
thf(fact_2241_word__diff__ls_I3_J,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [Y3: word @ A,X2: word @ A,W: word @ A,Xa: word @ A] :
( ( ord_less_eq @ ( word @ A ) @ ( minus_minus @ ( word @ A ) @ Y3 @ X2 ) @ ( minus_minus @ ( word @ A ) @ ( plus_plus @ ( word @ A ) @ W @ Xa ) @ X2 ) )
=> ( ( ord_less_eq @ ( word @ A ) @ X2 @ ( plus_plus @ ( word @ A ) @ W @ Xa ) )
=> ( ord_less_eq @ ( word @ A ) @ Y3 @ ( plus_plus @ ( word @ A ) @ W @ Xa ) ) ) ) ) ).
% word_diff_ls(3)
thf(fact_2242_word__diff__ls_I4_J,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [Y3: word @ A,W: word @ A,Xa: word @ A,X2: word @ A] :
( ( ord_less_eq @ ( word @ A ) @ Y3 @ ( plus_plus @ ( word @ A ) @ W @ Xa ) )
=> ( ( ord_less_eq @ ( word @ A ) @ X2 @ Y3 )
=> ( ord_less_eq @ ( word @ A ) @ ( minus_minus @ ( word @ A ) @ Y3 @ X2 ) @ ( minus_minus @ ( word @ A ) @ ( plus_plus @ ( word @ A ) @ W @ Xa ) @ X2 ) ) ) ) ) ).
% word_diff_ls(4)
thf(fact_2243_wi__hom__add,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [A3: int,B3: int] :
( ( plus_plus @ ( word @ A ) @ ( ring_1_of_int @ ( word @ A ) @ A3 ) @ ( ring_1_of_int @ ( word @ A ) @ B3 ) )
= ( ring_1_of_int @ ( word @ A ) @ ( plus_plus @ int @ A3 @ B3 ) ) ) ) ).
% wi_hom_add
thf(fact_2244_wi__hom__sub,axiom,
! [B: $tType] :
( ( type_len @ B )
=> ! [A3: int,B3: int] :
( ( minus_minus @ ( word @ B ) @ ( ring_1_of_int @ ( word @ B ) @ A3 ) @ ( ring_1_of_int @ ( word @ B ) @ B3 ) )
= ( ring_1_of_int @ ( word @ B ) @ ( minus_minus @ int @ A3 @ B3 ) ) ) ) ).
% wi_hom_sub
thf(fact_2245_word__of__int__nat,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [X2: int] :
( ( ord_less_eq @ int @ ( zero_zero @ int ) @ X2 )
=> ( ( ring_1_of_int @ ( word @ A ) @ X2 )
= ( semiring_1_of_nat @ ( word @ A ) @ ( nat2 @ X2 ) ) ) ) ) ).
% word_of_int_nat
thf(fact_2246_word__of__int__2p,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [N: nat] :
( ( ring_1_of_int @ ( word @ A ) @ ( power_power @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ N ) )
= ( power_power @ ( word @ A ) @ ( numeral_numeral @ ( word @ A ) @ ( bit0 @ one2 ) ) @ N ) ) ) ).
% word_of_int_2p
thf(fact_2247_vebt__member_Opelims_I1_J,axiom,
! [X2: vEBT_VEBT,Xa: nat,Y3: $o] :
( ( ( vEBT_vebt_member @ X2 @ Xa )
= Y3 )
=> ( ( accp @ ( product_prod @ vEBT_VEBT @ nat ) @ vEBT_vebt_member_rel @ ( product_Pair @ vEBT_VEBT @ nat @ X2 @ Xa ) )
=> ( ! [A4: $o,B4: $o] :
( ( X2
= ( vEBT_Leaf @ A4 @ B4 ) )
=> ( ( Y3
= ( ( ( Xa
= ( zero_zero @ nat ) )
=> A4 )
& ( ( Xa
!= ( zero_zero @ nat ) )
=> ( ( ( Xa
= ( one_one @ nat ) )
=> B4 )
& ( Xa
= ( one_one @ nat ) ) ) ) ) )
=> ~ ( accp @ ( product_prod @ vEBT_VEBT @ nat ) @ vEBT_vebt_member_rel @ ( product_Pair @ vEBT_VEBT @ nat @ ( vEBT_Leaf @ A4 @ B4 ) @ Xa ) ) ) )
=> ( ! [Uu2: nat,Uv2: list @ vEBT_VEBT,Uw2: vEBT_VEBT] :
( ( X2
= ( vEBT_Node @ ( none @ ( product_prod @ nat @ nat ) ) @ Uu2 @ Uv2 @ Uw2 ) )
=> ( ~ Y3
=> ~ ( accp @ ( product_prod @ vEBT_VEBT @ nat ) @ vEBT_vebt_member_rel @ ( product_Pair @ vEBT_VEBT @ nat @ ( vEBT_Node @ ( none @ ( product_prod @ nat @ nat ) ) @ Uu2 @ Uv2 @ Uw2 ) @ Xa ) ) ) )
=> ( ! [V3: product_prod @ nat @ nat,Uy2: list @ vEBT_VEBT,Uz2: vEBT_VEBT] :
( ( X2
= ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ V3 ) @ ( zero_zero @ nat ) @ Uy2 @ Uz2 ) )
=> ( ~ Y3
=> ~ ( accp @ ( product_prod @ vEBT_VEBT @ nat ) @ vEBT_vebt_member_rel @ ( product_Pair @ vEBT_VEBT @ nat @ ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ V3 ) @ ( zero_zero @ nat ) @ Uy2 @ Uz2 ) @ Xa ) ) ) )
=> ( ! [V3: product_prod @ nat @ nat,Vb2: list @ vEBT_VEBT,Vc2: vEBT_VEBT] :
( ( X2
= ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ V3 ) @ ( suc @ ( zero_zero @ nat ) ) @ Vb2 @ Vc2 ) )
=> ( ~ Y3
=> ~ ( accp @ ( product_prod @ vEBT_VEBT @ nat ) @ vEBT_vebt_member_rel @ ( product_Pair @ vEBT_VEBT @ nat @ ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ V3 ) @ ( suc @ ( zero_zero @ nat ) ) @ Vb2 @ Vc2 ) @ Xa ) ) ) )
=> ~ ! [Mi2: nat,Ma2: nat,Va3: nat,TreeList2: list @ vEBT_VEBT,Summary2: vEBT_VEBT] :
( ( X2
= ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( suc @ Va3 ) ) @ TreeList2 @ Summary2 ) )
=> ( ( Y3
= ( ( Xa != Mi2 )
=> ( ( Xa != Ma2 )
=> ( ~ ( ord_less @ nat @ Xa @ Mi2 )
& ( ~ ( ord_less @ nat @ Xa @ Mi2 )
=> ( ~ ( ord_less @ nat @ Ma2 @ Xa )
& ( ~ ( ord_less @ nat @ Ma2 @ Xa )
=> ( ( ( ord_less @ nat @ ( vEBT_VEBT_high @ Xa @ ( divide_divide @ nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( size_size @ ( list @ vEBT_VEBT ) @ TreeList2 ) )
=> ( vEBT_vebt_member @ ( nth @ vEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ Xa @ ( divide_divide @ nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( vEBT_VEBT_low @ Xa @ ( divide_divide @ nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) )
& ( ord_less @ nat @ ( vEBT_VEBT_high @ Xa @ ( divide_divide @ nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( size_size @ ( list @ vEBT_VEBT ) @ TreeList2 ) ) ) ) ) ) ) ) ) )
=> ~ ( accp @ ( product_prod @ vEBT_VEBT @ nat ) @ vEBT_vebt_member_rel @ ( product_Pair @ vEBT_VEBT @ nat @ ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( suc @ Va3 ) ) @ TreeList2 @ Summary2 ) @ Xa ) ) ) ) ) ) ) ) ) ) ).
% vebt_member.pelims(1)
thf(fact_2248_vebt__member_Opelims_I3_J,axiom,
! [X2: vEBT_VEBT,Xa: nat] :
( ~ ( vEBT_vebt_member @ X2 @ Xa )
=> ( ( accp @ ( product_prod @ vEBT_VEBT @ nat ) @ vEBT_vebt_member_rel @ ( product_Pair @ vEBT_VEBT @ nat @ X2 @ Xa ) )
=> ( ! [A4: $o,B4: $o] :
( ( X2
= ( vEBT_Leaf @ A4 @ B4 ) )
=> ( ( accp @ ( product_prod @ vEBT_VEBT @ nat ) @ vEBT_vebt_member_rel @ ( product_Pair @ vEBT_VEBT @ nat @ ( vEBT_Leaf @ A4 @ B4 ) @ Xa ) )
=> ( ( ( Xa
= ( zero_zero @ nat ) )
=> A4 )
& ( ( Xa
!= ( zero_zero @ nat ) )
=> ( ( ( Xa
= ( one_one @ nat ) )
=> B4 )
& ( Xa
= ( one_one @ nat ) ) ) ) ) ) )
=> ( ! [Uu2: nat,Uv2: list @ vEBT_VEBT,Uw2: vEBT_VEBT] :
( ( X2
= ( vEBT_Node @ ( none @ ( product_prod @ nat @ nat ) ) @ Uu2 @ Uv2 @ Uw2 ) )
=> ~ ( accp @ ( product_prod @ vEBT_VEBT @ nat ) @ vEBT_vebt_member_rel @ ( product_Pair @ vEBT_VEBT @ nat @ ( vEBT_Node @ ( none @ ( product_prod @ nat @ nat ) ) @ Uu2 @ Uv2 @ Uw2 ) @ Xa ) ) )
=> ( ! [V3: product_prod @ nat @ nat,Uy2: list @ vEBT_VEBT,Uz2: vEBT_VEBT] :
( ( X2
= ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ V3 ) @ ( zero_zero @ nat ) @ Uy2 @ Uz2 ) )
=> ~ ( accp @ ( product_prod @ vEBT_VEBT @ nat ) @ vEBT_vebt_member_rel @ ( product_Pair @ vEBT_VEBT @ nat @ ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ V3 ) @ ( zero_zero @ nat ) @ Uy2 @ Uz2 ) @ Xa ) ) )
=> ( ! [V3: product_prod @ nat @ nat,Vb2: list @ vEBT_VEBT,Vc2: vEBT_VEBT] :
( ( X2
= ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ V3 ) @ ( 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 ) @ V3 ) @ ( suc @ ( zero_zero @ nat ) ) @ Vb2 @ Vc2 ) @ Xa ) ) )
=> ~ ! [Mi2: nat,Ma2: nat,Va3: nat,TreeList2: list @ vEBT_VEBT,Summary2: vEBT_VEBT] :
( ( X2
= ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( suc @ Va3 ) ) @ 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 @ Va3 ) ) @ TreeList2 @ Summary2 ) @ Xa ) )
=> ( ( Xa != Mi2 )
=> ( ( Xa != Ma2 )
=> ( ~ ( ord_less @ nat @ Xa @ Mi2 )
& ( ~ ( ord_less @ nat @ Xa @ Mi2 )
=> ( ~ ( ord_less @ nat @ Ma2 @ Xa )
& ( ~ ( ord_less @ nat @ Ma2 @ Xa )
=> ( ( ( ord_less @ nat @ ( vEBT_VEBT_high @ Xa @ ( divide_divide @ nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( size_size @ ( list @ vEBT_VEBT ) @ TreeList2 ) )
=> ( vEBT_vebt_member @ ( nth @ vEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ Xa @ ( divide_divide @ nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( vEBT_VEBT_low @ Xa @ ( divide_divide @ nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) )
& ( ord_less @ nat @ ( vEBT_VEBT_high @ Xa @ ( divide_divide @ nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( size_size @ ( list @ vEBT_VEBT ) @ TreeList2 ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).
% vebt_member.pelims(3)
thf(fact_2249_VEBT__internal_Onaive__member_Opelims_I1_J,axiom,
! [X2: vEBT_VEBT,Xa: nat,Y3: $o] :
( ( ( vEBT_V5719532721284313246member @ X2 @ Xa )
= Y3 )
=> ( ( accp @ ( product_prod @ vEBT_VEBT @ nat ) @ vEBT_V5765760719290551771er_rel @ ( product_Pair @ vEBT_VEBT @ nat @ X2 @ Xa ) )
=> ( ! [A4: $o,B4: $o] :
( ( X2
= ( vEBT_Leaf @ A4 @ B4 ) )
=> ( ( Y3
= ( ( ( Xa
= ( zero_zero @ nat ) )
=> A4 )
& ( ( Xa
!= ( zero_zero @ nat ) )
=> ( ( ( Xa
= ( one_one @ nat ) )
=> B4 )
& ( Xa
= ( one_one @ nat ) ) ) ) ) )
=> ~ ( accp @ ( product_prod @ vEBT_VEBT @ nat ) @ vEBT_V5765760719290551771er_rel @ ( product_Pair @ vEBT_VEBT @ nat @ ( vEBT_Leaf @ A4 @ B4 ) @ Xa ) ) ) )
=> ( ! [Uu2: option @ ( product_prod @ nat @ nat ),Uv2: list @ vEBT_VEBT,Uw2: vEBT_VEBT] :
( ( X2
= ( vEBT_Node @ Uu2 @ ( zero_zero @ nat ) @ Uv2 @ Uw2 ) )
=> ( ~ Y3
=> ~ ( accp @ ( product_prod @ vEBT_VEBT @ nat ) @ vEBT_V5765760719290551771er_rel @ ( product_Pair @ vEBT_VEBT @ nat @ ( vEBT_Node @ Uu2 @ ( zero_zero @ nat ) @ Uv2 @ Uw2 ) @ Xa ) ) ) )
=> ~ ! [Uy2: option @ ( product_prod @ nat @ nat ),V3: nat,TreeList2: list @ vEBT_VEBT,S3: vEBT_VEBT] :
( ( X2
= ( vEBT_Node @ Uy2 @ ( suc @ V3 ) @ TreeList2 @ S3 ) )
=> ( ( Y3
= ( ( ( ord_less @ nat @ ( vEBT_VEBT_high @ Xa @ ( divide_divide @ nat @ ( suc @ V3 ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( size_size @ ( list @ vEBT_VEBT ) @ TreeList2 ) )
=> ( vEBT_V5719532721284313246member @ ( nth @ vEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ Xa @ ( divide_divide @ nat @ ( suc @ V3 ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( vEBT_VEBT_low @ Xa @ ( divide_divide @ nat @ ( suc @ V3 ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) )
& ( ord_less @ nat @ ( vEBT_VEBT_high @ Xa @ ( divide_divide @ nat @ ( suc @ V3 ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( size_size @ ( list @ vEBT_VEBT ) @ TreeList2 ) ) ) )
=> ~ ( accp @ ( product_prod @ vEBT_VEBT @ nat ) @ vEBT_V5765760719290551771er_rel @ ( product_Pair @ vEBT_VEBT @ nat @ ( vEBT_Node @ Uy2 @ ( suc @ V3 ) @ TreeList2 @ S3 ) @ Xa ) ) ) ) ) ) ) ) ).
% VEBT_internal.naive_member.pelims(1)
thf(fact_2250_VEBT__internal_Onaive__member_Opelims_I2_J,axiom,
! [X2: vEBT_VEBT,Xa: nat] :
( ( vEBT_V5719532721284313246member @ X2 @ Xa )
=> ( ( accp @ ( product_prod @ vEBT_VEBT @ nat ) @ vEBT_V5765760719290551771er_rel @ ( product_Pair @ vEBT_VEBT @ nat @ X2 @ Xa ) )
=> ( ! [A4: $o,B4: $o] :
( ( X2
= ( vEBT_Leaf @ A4 @ B4 ) )
=> ( ( accp @ ( product_prod @ vEBT_VEBT @ nat ) @ vEBT_V5765760719290551771er_rel @ ( product_Pair @ vEBT_VEBT @ nat @ ( vEBT_Leaf @ A4 @ B4 ) @ Xa ) )
=> ~ ( ( ( Xa
= ( zero_zero @ nat ) )
=> A4 )
& ( ( Xa
!= ( zero_zero @ nat ) )
=> ( ( ( Xa
= ( one_one @ nat ) )
=> B4 )
& ( Xa
= ( one_one @ nat ) ) ) ) ) ) )
=> ~ ! [Uy2: option @ ( product_prod @ nat @ nat ),V3: nat,TreeList2: list @ vEBT_VEBT,S3: vEBT_VEBT] :
( ( X2
= ( vEBT_Node @ Uy2 @ ( suc @ V3 ) @ TreeList2 @ S3 ) )
=> ( ( accp @ ( product_prod @ vEBT_VEBT @ nat ) @ vEBT_V5765760719290551771er_rel @ ( product_Pair @ vEBT_VEBT @ nat @ ( vEBT_Node @ Uy2 @ ( suc @ V3 ) @ TreeList2 @ S3 ) @ Xa ) )
=> ~ ( ( ( ord_less @ nat @ ( vEBT_VEBT_high @ Xa @ ( divide_divide @ nat @ ( suc @ V3 ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( size_size @ ( list @ vEBT_VEBT ) @ TreeList2 ) )
=> ( vEBT_V5719532721284313246member @ ( nth @ vEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ Xa @ ( divide_divide @ nat @ ( suc @ V3 ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( vEBT_VEBT_low @ Xa @ ( divide_divide @ nat @ ( suc @ V3 ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) )
& ( ord_less @ nat @ ( vEBT_VEBT_high @ Xa @ ( divide_divide @ nat @ ( suc @ V3 ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( size_size @ ( list @ vEBT_VEBT ) @ TreeList2 ) ) ) ) ) ) ) ) ).
% VEBT_internal.naive_member.pelims(2)
thf(fact_2251_VEBT__internal_Onaive__member_Opelims_I3_J,axiom,
! [X2: vEBT_VEBT,Xa: nat] :
( ~ ( vEBT_V5719532721284313246member @ X2 @ Xa )
=> ( ( accp @ ( product_prod @ vEBT_VEBT @ nat ) @ vEBT_V5765760719290551771er_rel @ ( product_Pair @ vEBT_VEBT @ nat @ X2 @ Xa ) )
=> ( ! [A4: $o,B4: $o] :
( ( X2
= ( vEBT_Leaf @ A4 @ B4 ) )
=> ( ( accp @ ( product_prod @ vEBT_VEBT @ nat ) @ vEBT_V5765760719290551771er_rel @ ( product_Pair @ vEBT_VEBT @ nat @ ( vEBT_Leaf @ A4 @ B4 ) @ Xa ) )
=> ( ( ( Xa
= ( zero_zero @ nat ) )
=> A4 )
& ( ( Xa
!= ( zero_zero @ nat ) )
=> ( ( ( Xa
= ( one_one @ nat ) )
=> B4 )
& ( Xa
= ( one_one @ nat ) ) ) ) ) ) )
=> ( ! [Uu2: option @ ( product_prod @ nat @ nat ),Uv2: list @ vEBT_VEBT,Uw2: vEBT_VEBT] :
( ( X2
= ( vEBT_Node @ Uu2 @ ( zero_zero @ nat ) @ Uv2 @ Uw2 ) )
=> ~ ( accp @ ( product_prod @ vEBT_VEBT @ nat ) @ vEBT_V5765760719290551771er_rel @ ( product_Pair @ vEBT_VEBT @ nat @ ( vEBT_Node @ Uu2 @ ( zero_zero @ nat ) @ Uv2 @ Uw2 ) @ Xa ) ) )
=> ~ ! [Uy2: option @ ( product_prod @ nat @ nat ),V3: nat,TreeList2: list @ vEBT_VEBT,S3: vEBT_VEBT] :
( ( X2
= ( vEBT_Node @ Uy2 @ ( suc @ V3 ) @ TreeList2 @ S3 ) )
=> ( ( accp @ ( product_prod @ vEBT_VEBT @ nat ) @ vEBT_V5765760719290551771er_rel @ ( product_Pair @ vEBT_VEBT @ nat @ ( vEBT_Node @ Uy2 @ ( suc @ V3 ) @ TreeList2 @ S3 ) @ Xa ) )
=> ( ( ( ord_less @ nat @ ( vEBT_VEBT_high @ Xa @ ( divide_divide @ nat @ ( suc @ V3 ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( size_size @ ( list @ vEBT_VEBT ) @ TreeList2 ) )
=> ( vEBT_V5719532721284313246member @ ( nth @ vEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ Xa @ ( divide_divide @ nat @ ( suc @ V3 ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( vEBT_VEBT_low @ Xa @ ( divide_divide @ nat @ ( suc @ V3 ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) )
& ( ord_less @ nat @ ( vEBT_VEBT_high @ Xa @ ( divide_divide @ nat @ ( suc @ V3 ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( size_size @ ( list @ vEBT_VEBT ) @ TreeList2 ) ) ) ) ) ) ) ) ) ).
% VEBT_internal.naive_member.pelims(3)
thf(fact_2252_vebt__member_Opelims_I2_J,axiom,
! [X2: vEBT_VEBT,Xa: nat] :
( ( vEBT_vebt_member @ X2 @ Xa )
=> ( ( accp @ ( product_prod @ vEBT_VEBT @ nat ) @ vEBT_vebt_member_rel @ ( product_Pair @ vEBT_VEBT @ nat @ X2 @ Xa ) )
=> ( ! [A4: $o,B4: $o] :
( ( X2
= ( vEBT_Leaf @ A4 @ B4 ) )
=> ( ( accp @ ( product_prod @ vEBT_VEBT @ nat ) @ vEBT_vebt_member_rel @ ( product_Pair @ vEBT_VEBT @ nat @ ( vEBT_Leaf @ A4 @ B4 ) @ Xa ) )
=> ~ ( ( ( Xa
= ( zero_zero @ nat ) )
=> A4 )
& ( ( Xa
!= ( zero_zero @ nat ) )
=> ( ( ( Xa
= ( one_one @ nat ) )
=> B4 )
& ( Xa
= ( one_one @ nat ) ) ) ) ) ) )
=> ~ ! [Mi2: nat,Ma2: nat,Va3: nat,TreeList2: list @ vEBT_VEBT,Summary2: vEBT_VEBT] :
( ( X2
= ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( suc @ Va3 ) ) @ 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 @ Va3 ) ) @ TreeList2 @ Summary2 ) @ Xa ) )
=> ~ ( ( Xa != Mi2 )
=> ( ( Xa != Ma2 )
=> ( ~ ( ord_less @ nat @ Xa @ Mi2 )
& ( ~ ( ord_less @ nat @ Xa @ Mi2 )
=> ( ~ ( ord_less @ nat @ Ma2 @ Xa )
& ( ~ ( ord_less @ nat @ Ma2 @ Xa )
=> ( ( ( ord_less @ nat @ ( vEBT_VEBT_high @ Xa @ ( divide_divide @ nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( size_size @ ( list @ vEBT_VEBT ) @ TreeList2 ) )
=> ( vEBT_vebt_member @ ( nth @ vEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ Xa @ ( divide_divide @ nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( vEBT_VEBT_low @ Xa @ ( divide_divide @ nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) )
& ( ord_less @ nat @ ( vEBT_VEBT_high @ Xa @ ( divide_divide @ nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( size_size @ ( list @ vEBT_VEBT ) @ TreeList2 ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).
% vebt_member.pelims(2)
thf(fact_2253_VEBT__internal_Omembermima_Opelims_I3_J,axiom,
! [X2: vEBT_VEBT,Xa: nat] :
( ~ ( vEBT_VEBT_membermima @ X2 @ Xa )
=> ( ( accp @ ( product_prod @ vEBT_VEBT @ nat ) @ vEBT_V4351362008482014158ma_rel @ ( product_Pair @ vEBT_VEBT @ nat @ X2 @ Xa ) )
=> ( ! [Uu2: $o,Uv2: $o] :
( ( X2
= ( vEBT_Leaf @ Uu2 @ Uv2 ) )
=> ~ ( accp @ ( product_prod @ vEBT_VEBT @ nat ) @ vEBT_V4351362008482014158ma_rel @ ( product_Pair @ vEBT_VEBT @ nat @ ( vEBT_Leaf @ Uu2 @ Uv2 ) @ Xa ) ) )
=> ( ! [Ux2: list @ vEBT_VEBT,Uy2: vEBT_VEBT] :
( ( X2
= ( vEBT_Node @ ( none @ ( product_prod @ nat @ nat ) ) @ ( zero_zero @ nat ) @ Ux2 @ Uy2 ) )
=> ~ ( accp @ ( product_prod @ vEBT_VEBT @ nat ) @ vEBT_V4351362008482014158ma_rel @ ( product_Pair @ vEBT_VEBT @ nat @ ( vEBT_Node @ ( none @ ( product_prod @ nat @ nat ) ) @ ( zero_zero @ nat ) @ Ux2 @ Uy2 ) @ Xa ) ) )
=> ( ! [Mi2: nat,Ma2: nat,Va2: list @ vEBT_VEBT,Vb2: vEBT_VEBT] :
( ( X2
= ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi2 @ Ma2 ) ) @ ( zero_zero @ nat ) @ Va2 @ Vb2 ) )
=> ( ( accp @ ( product_prod @ vEBT_VEBT @ nat ) @ vEBT_V4351362008482014158ma_rel @ ( product_Pair @ vEBT_VEBT @ nat @ ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi2 @ Ma2 ) ) @ ( zero_zero @ nat ) @ Va2 @ Vb2 ) @ Xa ) )
=> ( ( Xa = Mi2 )
| ( Xa = Ma2 ) ) ) )
=> ( ! [Mi2: nat,Ma2: nat,V3: nat,TreeList2: list @ vEBT_VEBT,Vc2: vEBT_VEBT] :
( ( X2
= ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi2 @ Ma2 ) ) @ ( suc @ V3 ) @ 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 @ V3 ) @ TreeList2 @ Vc2 ) @ Xa ) )
=> ( ( Xa = Mi2 )
| ( Xa = Ma2 )
| ( ( ( ord_less @ nat @ ( vEBT_VEBT_high @ Xa @ ( divide_divide @ nat @ ( suc @ V3 ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( size_size @ ( list @ vEBT_VEBT ) @ TreeList2 ) )
=> ( vEBT_VEBT_membermima @ ( nth @ vEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ Xa @ ( divide_divide @ nat @ ( suc @ V3 ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( vEBT_VEBT_low @ Xa @ ( divide_divide @ nat @ ( suc @ V3 ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) )
& ( ord_less @ nat @ ( vEBT_VEBT_high @ Xa @ ( divide_divide @ nat @ ( suc @ V3 ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( size_size @ ( list @ vEBT_VEBT ) @ TreeList2 ) ) ) ) ) )
=> ~ ! [V3: nat,TreeList2: list @ vEBT_VEBT,Vd2: vEBT_VEBT] :
( ( X2
= ( vEBT_Node @ ( none @ ( product_prod @ nat @ nat ) ) @ ( suc @ V3 ) @ TreeList2 @ Vd2 ) )
=> ( ( accp @ ( product_prod @ vEBT_VEBT @ nat ) @ vEBT_V4351362008482014158ma_rel @ ( product_Pair @ vEBT_VEBT @ nat @ ( vEBT_Node @ ( none @ ( product_prod @ nat @ nat ) ) @ ( suc @ V3 ) @ TreeList2 @ Vd2 ) @ Xa ) )
=> ( ( ( ord_less @ nat @ ( vEBT_VEBT_high @ Xa @ ( divide_divide @ nat @ ( suc @ V3 ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( size_size @ ( list @ vEBT_VEBT ) @ TreeList2 ) )
=> ( vEBT_VEBT_membermima @ ( nth @ vEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ Xa @ ( divide_divide @ nat @ ( suc @ V3 ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( vEBT_VEBT_low @ Xa @ ( divide_divide @ nat @ ( suc @ V3 ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) )
& ( ord_less @ nat @ ( vEBT_VEBT_high @ Xa @ ( divide_divide @ nat @ ( suc @ V3 ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( size_size @ ( list @ vEBT_VEBT ) @ TreeList2 ) ) ) ) ) ) ) ) ) ) ) ).
% VEBT_internal.membermima.pelims(3)
thf(fact_2254_VEBT__internal_Omembermima_Opelims_I1_J,axiom,
! [X2: vEBT_VEBT,Xa: nat,Y3: $o] :
( ( ( vEBT_VEBT_membermima @ X2 @ Xa )
= Y3 )
=> ( ( accp @ ( product_prod @ vEBT_VEBT @ nat ) @ vEBT_V4351362008482014158ma_rel @ ( product_Pair @ vEBT_VEBT @ nat @ X2 @ Xa ) )
=> ( ! [Uu2: $o,Uv2: $o] :
( ( X2
= ( vEBT_Leaf @ Uu2 @ Uv2 ) )
=> ( ~ Y3
=> ~ ( accp @ ( product_prod @ vEBT_VEBT @ nat ) @ vEBT_V4351362008482014158ma_rel @ ( product_Pair @ vEBT_VEBT @ nat @ ( vEBT_Leaf @ Uu2 @ Uv2 ) @ Xa ) ) ) )
=> ( ! [Ux2: list @ vEBT_VEBT,Uy2: vEBT_VEBT] :
( ( X2
= ( vEBT_Node @ ( none @ ( product_prod @ nat @ nat ) ) @ ( zero_zero @ nat ) @ Ux2 @ Uy2 ) )
=> ( ~ Y3
=> ~ ( accp @ ( product_prod @ vEBT_VEBT @ nat ) @ vEBT_V4351362008482014158ma_rel @ ( product_Pair @ vEBT_VEBT @ nat @ ( vEBT_Node @ ( none @ ( product_prod @ nat @ nat ) ) @ ( zero_zero @ nat ) @ Ux2 @ Uy2 ) @ Xa ) ) ) )
=> ( ! [Mi2: nat,Ma2: nat,Va2: list @ vEBT_VEBT,Vb2: vEBT_VEBT] :
( ( X2
= ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi2 @ Ma2 ) ) @ ( zero_zero @ nat ) @ Va2 @ Vb2 ) )
=> ( ( Y3
= ( ( Xa = Mi2 )
| ( Xa = Ma2 ) ) )
=> ~ ( accp @ ( product_prod @ vEBT_VEBT @ nat ) @ vEBT_V4351362008482014158ma_rel @ ( product_Pair @ vEBT_VEBT @ nat @ ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi2 @ Ma2 ) ) @ ( zero_zero @ nat ) @ Va2 @ Vb2 ) @ Xa ) ) ) )
=> ( ! [Mi2: nat,Ma2: nat,V3: nat,TreeList2: list @ vEBT_VEBT,Vc2: vEBT_VEBT] :
( ( X2
= ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi2 @ Ma2 ) ) @ ( suc @ V3 ) @ TreeList2 @ Vc2 ) )
=> ( ( Y3
= ( ( Xa = Mi2 )
| ( Xa = Ma2 )
| ( ( ( ord_less @ nat @ ( vEBT_VEBT_high @ Xa @ ( divide_divide @ nat @ ( suc @ V3 ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( size_size @ ( list @ vEBT_VEBT ) @ TreeList2 ) )
=> ( vEBT_VEBT_membermima @ ( nth @ vEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ Xa @ ( divide_divide @ nat @ ( suc @ V3 ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( vEBT_VEBT_low @ Xa @ ( divide_divide @ nat @ ( suc @ V3 ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) )
& ( ord_less @ nat @ ( vEBT_VEBT_high @ Xa @ ( divide_divide @ nat @ ( suc @ V3 ) @ ( 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 @ V3 ) @ TreeList2 @ Vc2 ) @ Xa ) ) ) )
=> ~ ! [V3: nat,TreeList2: list @ vEBT_VEBT,Vd2: vEBT_VEBT] :
( ( X2
= ( vEBT_Node @ ( none @ ( product_prod @ nat @ nat ) ) @ ( suc @ V3 ) @ TreeList2 @ Vd2 ) )
=> ( ( Y3
= ( ( ( ord_less @ nat @ ( vEBT_VEBT_high @ Xa @ ( divide_divide @ nat @ ( suc @ V3 ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( size_size @ ( list @ vEBT_VEBT ) @ TreeList2 ) )
=> ( vEBT_VEBT_membermima @ ( nth @ vEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ Xa @ ( divide_divide @ nat @ ( suc @ V3 ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( vEBT_VEBT_low @ Xa @ ( divide_divide @ nat @ ( suc @ V3 ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) )
& ( ord_less @ nat @ ( vEBT_VEBT_high @ Xa @ ( divide_divide @ nat @ ( suc @ V3 ) @ ( 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 @ V3 ) @ TreeList2 @ Vd2 ) @ Xa ) ) ) ) ) ) ) ) ) ) ).
% VEBT_internal.membermima.pelims(1)
thf(fact_2255_VEBT__internal_Omembermima_Opelims_I2_J,axiom,
! [X2: vEBT_VEBT,Xa: nat] :
( ( vEBT_VEBT_membermima @ X2 @ Xa )
=> ( ( accp @ ( product_prod @ vEBT_VEBT @ nat ) @ vEBT_V4351362008482014158ma_rel @ ( product_Pair @ vEBT_VEBT @ nat @ X2 @ Xa ) )
=> ( ! [Mi2: nat,Ma2: nat,Va2: list @ vEBT_VEBT,Vb2: vEBT_VEBT] :
( ( X2
= ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi2 @ Ma2 ) ) @ ( zero_zero @ nat ) @ Va2 @ Vb2 ) )
=> ( ( accp @ ( product_prod @ vEBT_VEBT @ nat ) @ vEBT_V4351362008482014158ma_rel @ ( product_Pair @ vEBT_VEBT @ nat @ ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi2 @ Ma2 ) ) @ ( zero_zero @ nat ) @ Va2 @ Vb2 ) @ Xa ) )
=> ~ ( ( Xa = Mi2 )
| ( Xa = Ma2 ) ) ) )
=> ( ! [Mi2: nat,Ma2: nat,V3: nat,TreeList2: list @ vEBT_VEBT,Vc2: vEBT_VEBT] :
( ( X2
= ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi2 @ Ma2 ) ) @ ( suc @ V3 ) @ 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 @ V3 ) @ TreeList2 @ Vc2 ) @ Xa ) )
=> ~ ( ( Xa = Mi2 )
| ( Xa = Ma2 )
| ( ( ( ord_less @ nat @ ( vEBT_VEBT_high @ Xa @ ( divide_divide @ nat @ ( suc @ V3 ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( size_size @ ( list @ vEBT_VEBT ) @ TreeList2 ) )
=> ( vEBT_VEBT_membermima @ ( nth @ vEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ Xa @ ( divide_divide @ nat @ ( suc @ V3 ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( vEBT_VEBT_low @ Xa @ ( divide_divide @ nat @ ( suc @ V3 ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) )
& ( ord_less @ nat @ ( vEBT_VEBT_high @ Xa @ ( divide_divide @ nat @ ( suc @ V3 ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( size_size @ ( list @ vEBT_VEBT ) @ TreeList2 ) ) ) ) ) )
=> ~ ! [V3: nat,TreeList2: list @ vEBT_VEBT,Vd2: vEBT_VEBT] :
( ( X2
= ( vEBT_Node @ ( none @ ( product_prod @ nat @ nat ) ) @ ( suc @ V3 ) @ TreeList2 @ Vd2 ) )
=> ( ( accp @ ( product_prod @ vEBT_VEBT @ nat ) @ vEBT_V4351362008482014158ma_rel @ ( product_Pair @ vEBT_VEBT @ nat @ ( vEBT_Node @ ( none @ ( product_prod @ nat @ nat ) ) @ ( suc @ V3 ) @ TreeList2 @ Vd2 ) @ Xa ) )
=> ~ ( ( ( ord_less @ nat @ ( vEBT_VEBT_high @ Xa @ ( divide_divide @ nat @ ( suc @ V3 ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( size_size @ ( list @ vEBT_VEBT ) @ TreeList2 ) )
=> ( vEBT_VEBT_membermima @ ( nth @ vEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ Xa @ ( divide_divide @ nat @ ( suc @ V3 ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( vEBT_VEBT_low @ Xa @ ( divide_divide @ nat @ ( suc @ V3 ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) )
& ( ord_less @ nat @ ( vEBT_VEBT_high @ Xa @ ( divide_divide @ nat @ ( suc @ V3 ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( size_size @ ( list @ vEBT_VEBT ) @ TreeList2 ) ) ) ) ) ) ) ) ) ).
% VEBT_internal.membermima.pelims(2)
thf(fact_2256_set__bit__0,axiom,
! [A: $tType] :
( ( bit_se359711467146920520ations @ A )
=> ! [A3: A] :
( ( bit_se5668285175392031749et_bit @ A @ ( zero_zero @ nat ) @ A3 )
= ( plus_plus @ A @ ( one_one @ A ) @ ( times_times @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ ( divide_divide @ A @ A3 @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) ) ) ) ) ).
% set_bit_0
thf(fact_2257_pred__less__length__list,axiom,
! [Deg: nat,X2: 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 @ X2 @ Ma )
=> ( ( ord_less @ nat @ ( vEBT_VEBT_high @ X2 @ ( 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 ) @ X2 )
= ( if @ ( option @ nat )
@ ( ( ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList @ ( vEBT_VEBT_high @ X2 @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) )
!= ( none @ nat ) )
& ( vEBT_VEBT_greater @ ( some @ nat @ ( vEBT_VEBT_low @ X2 @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList @ ( vEBT_VEBT_high @ X2 @ ( 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 @ X2 @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) @ ( vEBT_vebt_pred @ ( nth @ vEBT_VEBT @ TreeList @ ( vEBT_VEBT_high @ X2 @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( vEBT_VEBT_low @ X2 @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) )
@ ( if @ ( option @ nat )
@ ( ( vEBT_vebt_pred @ Summary @ ( vEBT_VEBT_high @ X2 @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) )
= ( none @ nat ) )
@ ( if @ ( option @ nat ) @ ( ord_less @ nat @ Mi @ X2 ) @ ( 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 @ X2 @ ( 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 @ X2 @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).
% pred_less_length_list
thf(fact_2258_pred__lesseq__max,axiom,
! [Deg: nat,X2: 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 @ X2 @ Ma )
=> ( ( vEBT_vebt_pred @ ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi @ Ma ) ) @ Deg @ TreeList @ Summary ) @ X2 )
= ( if @ ( option @ nat ) @ ( ord_less @ nat @ ( vEBT_VEBT_high @ X2 @ ( 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 @ X2 @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) )
!= ( none @ nat ) )
& ( vEBT_VEBT_greater @ ( some @ nat @ ( vEBT_VEBT_low @ X2 @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList @ ( vEBT_VEBT_high @ X2 @ ( 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 @ X2 @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) @ ( vEBT_vebt_pred @ ( nth @ vEBT_VEBT @ TreeList @ ( vEBT_VEBT_high @ X2 @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( vEBT_VEBT_low @ X2 @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) )
@ ( if @ ( option @ nat )
@ ( ( vEBT_vebt_pred @ Summary @ ( vEBT_VEBT_high @ X2 @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) )
= ( none @ nat ) )
@ ( if @ ( option @ nat ) @ ( ord_less @ nat @ Mi @ X2 ) @ ( 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 @ X2 @ ( 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 @ X2 @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) ) ) ) ) )
@ ( none @ nat ) ) ) ) ) ).
% pred_lesseq_max
thf(fact_2259_add__shift,axiom,
! [X2: nat,Y3: nat,Z: nat] :
( ( ( plus_plus @ nat @ X2 @ Y3 )
= Z )
= ( ( vEBT_VEBT_add @ ( some @ nat @ X2 ) @ ( some @ nat @ Y3 ) )
= ( some @ nat @ Z ) ) ) ).
% add_shift
thf(fact_2260_mul__shift,axiom,
! [X2: nat,Y3: nat,Z: nat] :
( ( ( times_times @ nat @ X2 @ Y3 )
= Z )
= ( ( vEBT_VEBT_mul @ ( some @ nat @ X2 ) @ ( some @ nat @ Y3 ) )
= ( some @ nat @ Z ) ) ) ).
% mul_shift
thf(fact_2261_add__def,axiom,
( vEBT_VEBT_add
= ( vEBT_V2048590022279873568_shift @ nat @ ( plus_plus @ nat ) ) ) ).
% add_def
thf(fact_2262_mul__def,axiom,
( vEBT_VEBT_mul
= ( vEBT_V2048590022279873568_shift @ nat @ ( times_times @ nat ) ) ) ).
% mul_def
thf(fact_2263_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_2264_succ__less__length__list,axiom,
! [Deg: nat,Mi: nat,X2: 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 @ X2 )
=> ( ( ord_less @ nat @ ( vEBT_VEBT_high @ X2 @ ( 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 ) @ X2 )
= ( if @ ( option @ nat )
@ ( ( ( vEBT_vebt_maxt @ ( nth @ vEBT_VEBT @ TreeList @ ( vEBT_VEBT_high @ X2 @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) )
!= ( none @ nat ) )
& ( vEBT_VEBT_less @ ( some @ nat @ ( vEBT_VEBT_low @ X2 @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( vEBT_vebt_maxt @ ( nth @ vEBT_VEBT @ TreeList @ ( vEBT_VEBT_high @ X2 @ ( 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 @ X2 @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) @ ( vEBT_vebt_succ @ ( nth @ vEBT_VEBT @ TreeList @ ( vEBT_VEBT_high @ X2 @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( vEBT_VEBT_low @ X2 @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) )
@ ( if @ ( option @ nat )
@ ( ( vEBT_vebt_succ @ Summary @ ( vEBT_VEBT_high @ X2 @ ( 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 @ X2 @ ( 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 @ X2 @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).
% succ_less_length_list
thf(fact_2265_succ__greatereq__min,axiom,
! [Deg: nat,Mi: nat,X2: 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 @ X2 )
=> ( ( vEBT_vebt_succ @ ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi @ Ma ) ) @ Deg @ TreeList @ Summary ) @ X2 )
= ( if @ ( option @ nat ) @ ( ord_less @ nat @ ( vEBT_VEBT_high @ X2 @ ( 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 @ X2 @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) )
!= ( none @ nat ) )
& ( vEBT_VEBT_less @ ( some @ nat @ ( vEBT_VEBT_low @ X2 @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( vEBT_vebt_maxt @ ( nth @ vEBT_VEBT @ TreeList @ ( vEBT_VEBT_high @ X2 @ ( 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 @ X2 @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) @ ( vEBT_vebt_succ @ ( nth @ vEBT_VEBT @ TreeList @ ( vEBT_VEBT_high @ X2 @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( vEBT_VEBT_low @ X2 @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) )
@ ( if @ ( option @ nat )
@ ( ( vEBT_vebt_succ @ Summary @ ( vEBT_VEBT_high @ X2 @ ( 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 @ X2 @ ( 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 @ X2 @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) ) ) ) ) )
@ ( none @ nat ) ) ) ) ) ).
% succ_greatereq_min
thf(fact_2266_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_2267_vebt__pred_Osimps_I7_J,axiom,
! [Ma: nat,X2: nat,Mi: nat,Va: nat,TreeList: list @ vEBT_VEBT,Summary: vEBT_VEBT] :
( ( ( ord_less @ nat @ Ma @ X2 )
=> ( ( vEBT_vebt_pred @ ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi @ Ma ) ) @ ( suc @ ( suc @ Va ) ) @ TreeList @ Summary ) @ X2 )
= ( some @ nat @ Ma ) ) )
& ( ~ ( ord_less @ nat @ Ma @ X2 )
=> ( ( vEBT_vebt_pred @ ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi @ Ma ) ) @ ( suc @ ( suc @ Va ) ) @ TreeList @ Summary ) @ X2 )
= ( if @ ( option @ nat ) @ ( ord_less @ nat @ ( vEBT_VEBT_high @ X2 @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( size_size @ ( list @ vEBT_VEBT ) @ TreeList ) )
@ ( if @ ( option @ nat )
@ ( ( ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList @ ( vEBT_VEBT_high @ X2 @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) )
!= ( none @ nat ) )
& ( vEBT_VEBT_greater @ ( some @ nat @ ( vEBT_VEBT_low @ X2 @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList @ ( vEBT_VEBT_high @ X2 @ ( 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 @ X2 @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) @ ( vEBT_vebt_pred @ ( nth @ vEBT_VEBT @ TreeList @ ( vEBT_VEBT_high @ X2 @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( vEBT_VEBT_low @ X2 @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) )
@ ( if @ ( option @ nat )
@ ( ( vEBT_vebt_pred @ Summary @ ( vEBT_VEBT_high @ X2 @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) )
= ( none @ nat ) )
@ ( if @ ( option @ nat ) @ ( ord_less @ nat @ Mi @ X2 ) @ ( some @ nat @ Mi ) @ ( none @ nat ) )
@ ( vEBT_VEBT_add @ ( vEBT_VEBT_mul @ ( some @ nat @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( vEBT_vebt_pred @ Summary @ ( vEBT_VEBT_high @ X2 @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) @ ( vEBT_vebt_maxt @ ( nth @ vEBT_VEBT @ TreeList @ ( the2 @ nat @ ( vEBT_vebt_pred @ Summary @ ( vEBT_VEBT_high @ X2 @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) ) ) ) ) )
@ ( none @ nat ) ) ) ) ) ).
% vebt_pred.simps(7)
thf(fact_2268_vebt__succ_Osimps_I6_J,axiom,
! [X2: nat,Mi: nat,Ma: nat,Va: nat,TreeList: list @ vEBT_VEBT,Summary: vEBT_VEBT] :
( ( ( ord_less @ nat @ X2 @ Mi )
=> ( ( vEBT_vebt_succ @ ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi @ Ma ) ) @ ( suc @ ( suc @ Va ) ) @ TreeList @ Summary ) @ X2 )
= ( some @ nat @ Mi ) ) )
& ( ~ ( ord_less @ nat @ X2 @ Mi )
=> ( ( vEBT_vebt_succ @ ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi @ Ma ) ) @ ( suc @ ( suc @ Va ) ) @ TreeList @ Summary ) @ X2 )
= ( if @ ( option @ nat ) @ ( ord_less @ nat @ ( vEBT_VEBT_high @ X2 @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( size_size @ ( list @ vEBT_VEBT ) @ TreeList ) )
@ ( if @ ( option @ nat )
@ ( ( ( vEBT_vebt_maxt @ ( nth @ vEBT_VEBT @ TreeList @ ( vEBT_VEBT_high @ X2 @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) )
!= ( none @ nat ) )
& ( vEBT_VEBT_less @ ( some @ nat @ ( vEBT_VEBT_low @ X2 @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( vEBT_vebt_maxt @ ( nth @ vEBT_VEBT @ TreeList @ ( vEBT_VEBT_high @ X2 @ ( 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 @ X2 @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) @ ( vEBT_vebt_succ @ ( nth @ vEBT_VEBT @ TreeList @ ( vEBT_VEBT_high @ X2 @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( vEBT_VEBT_low @ X2 @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) )
@ ( if @ ( option @ nat )
@ ( ( vEBT_vebt_succ @ Summary @ ( vEBT_VEBT_high @ X2 @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) )
= ( none @ nat ) )
@ ( none @ nat )
@ ( vEBT_VEBT_add @ ( vEBT_VEBT_mul @ ( some @ nat @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( vEBT_vebt_succ @ Summary @ ( vEBT_VEBT_high @ X2 @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) @ ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList @ ( the2 @ nat @ ( vEBT_vebt_succ @ Summary @ ( vEBT_VEBT_high @ X2 @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) ) ) ) ) )
@ ( none @ nat ) ) ) ) ) ).
% vebt_succ.simps(6)
thf(fact_2269_vebt__pred_Oelims,axiom,
! [X2: vEBT_VEBT,Xa: nat,Y3: option @ nat] :
( ( ( vEBT_vebt_pred @ X2 @ Xa )
= Y3 )
=> ( ( ? [Uu2: $o,Uv2: $o] :
( X2
= ( vEBT_Leaf @ Uu2 @ Uv2 ) )
=> ( ( Xa
= ( zero_zero @ nat ) )
=> ( Y3
!= ( none @ nat ) ) ) )
=> ( ! [A4: $o] :
( ? [Uw2: $o] :
( X2
= ( vEBT_Leaf @ A4 @ Uw2 ) )
=> ( ( Xa
= ( suc @ ( zero_zero @ nat ) ) )
=> ~ ( ( A4
=> ( Y3
= ( some @ nat @ ( zero_zero @ nat ) ) ) )
& ( ~ A4
=> ( Y3
= ( none @ nat ) ) ) ) ) )
=> ( ! [A4: $o,B4: $o] :
( ( X2
= ( vEBT_Leaf @ A4 @ B4 ) )
=> ( ? [Va3: nat] :
( Xa
= ( suc @ ( suc @ Va3 ) ) )
=> ~ ( ( B4
=> ( Y3
= ( some @ nat @ ( one_one @ nat ) ) ) )
& ( ~ B4
=> ( ( A4
=> ( Y3
= ( some @ nat @ ( zero_zero @ nat ) ) ) )
& ( ~ A4
=> ( Y3
= ( none @ nat ) ) ) ) ) ) ) )
=> ( ( ? [Uy2: nat,Uz2: list @ vEBT_VEBT,Va2: vEBT_VEBT] :
( X2
= ( vEBT_Node @ ( none @ ( product_prod @ nat @ nat ) ) @ Uy2 @ Uz2 @ Va2 ) )
=> ( Y3
!= ( none @ nat ) ) )
=> ( ( ? [V3: product_prod @ nat @ nat,Vd2: list @ vEBT_VEBT,Ve2: vEBT_VEBT] :
( X2
= ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ V3 ) @ ( zero_zero @ nat ) @ Vd2 @ Ve2 ) )
=> ( Y3
!= ( none @ nat ) ) )
=> ( ( ? [V3: product_prod @ nat @ nat,Vh: list @ vEBT_VEBT,Vi: vEBT_VEBT] :
( X2
= ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ V3 ) @ ( suc @ ( zero_zero @ nat ) ) @ Vh @ Vi ) )
=> ( Y3
!= ( none @ nat ) ) )
=> ~ ! [Mi2: nat,Ma2: nat,Va3: nat,TreeList2: list @ vEBT_VEBT,Summary2: vEBT_VEBT] :
( ( X2
= ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( suc @ Va3 ) ) @ TreeList2 @ Summary2 ) )
=> ~ ( ( ( ord_less @ nat @ Ma2 @ Xa )
=> ( Y3
= ( some @ nat @ Ma2 ) ) )
& ( ~ ( ord_less @ nat @ Ma2 @ Xa )
=> ( Y3
= ( if @ ( option @ nat ) @ ( ord_less @ nat @ ( vEBT_VEBT_high @ Xa @ ( divide_divide @ nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( size_size @ ( list @ vEBT_VEBT ) @ TreeList2 ) )
@ ( if @ ( option @ nat )
@ ( ( ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ Xa @ ( divide_divide @ nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) )
!= ( none @ nat ) )
& ( vEBT_VEBT_greater @ ( some @ nat @ ( vEBT_VEBT_low @ Xa @ ( divide_divide @ nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ Xa @ ( divide_divide @ nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) ) )
@ ( vEBT_VEBT_add @ ( vEBT_VEBT_mul @ ( some @ nat @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( some @ nat @ ( vEBT_VEBT_high @ Xa @ ( divide_divide @ nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) @ ( vEBT_vebt_pred @ ( nth @ vEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ Xa @ ( divide_divide @ nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( vEBT_VEBT_low @ Xa @ ( divide_divide @ nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) )
@ ( if @ ( option @ nat )
@ ( ( vEBT_vebt_pred @ Summary2 @ ( vEBT_VEBT_high @ Xa @ ( divide_divide @ nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) )
= ( none @ nat ) )
@ ( if @ ( option @ nat ) @ ( ord_less @ nat @ Mi2 @ Xa ) @ ( some @ nat @ Mi2 ) @ ( none @ nat ) )
@ ( vEBT_VEBT_add @ ( vEBT_VEBT_mul @ ( some @ nat @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( vEBT_vebt_pred @ Summary2 @ ( vEBT_VEBT_high @ Xa @ ( divide_divide @ nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) @ ( vEBT_vebt_maxt @ ( nth @ vEBT_VEBT @ TreeList2 @ ( the2 @ nat @ ( vEBT_vebt_pred @ Summary2 @ ( vEBT_VEBT_high @ Xa @ ( divide_divide @ nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) ) ) ) ) )
@ ( none @ nat ) ) ) ) ) ) ) ) ) ) ) ) ) ).
% vebt_pred.elims
thf(fact_2270_vebt__succ_Oelims,axiom,
! [X2: vEBT_VEBT,Xa: nat,Y3: option @ nat] :
( ( ( vEBT_vebt_succ @ X2 @ Xa )
= Y3 )
=> ( ! [Uu2: $o,B4: $o] :
( ( X2
= ( vEBT_Leaf @ Uu2 @ B4 ) )
=> ( ( Xa
= ( zero_zero @ nat ) )
=> ~ ( ( B4
=> ( Y3
= ( some @ nat @ ( one_one @ nat ) ) ) )
& ( ~ B4
=> ( Y3
= ( none @ nat ) ) ) ) ) )
=> ( ( ? [Uv2: $o,Uw2: $o] :
( X2
= ( vEBT_Leaf @ Uv2 @ Uw2 ) )
=> ( ? [N3: nat] :
( Xa
= ( suc @ N3 ) )
=> ( Y3
!= ( none @ nat ) ) ) )
=> ( ( ? [Ux2: nat,Uy2: list @ vEBT_VEBT,Uz2: vEBT_VEBT] :
( X2
= ( vEBT_Node @ ( none @ ( product_prod @ nat @ nat ) ) @ Ux2 @ Uy2 @ Uz2 ) )
=> ( Y3
!= ( none @ nat ) ) )
=> ( ( ? [V3: product_prod @ nat @ nat,Vc2: list @ vEBT_VEBT,Vd2: vEBT_VEBT] :
( X2
= ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ V3 ) @ ( zero_zero @ nat ) @ Vc2 @ Vd2 ) )
=> ( Y3
!= ( none @ nat ) ) )
=> ( ( ? [V3: product_prod @ nat @ nat,Vg: list @ vEBT_VEBT,Vh: vEBT_VEBT] :
( X2
= ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ V3 ) @ ( suc @ ( zero_zero @ nat ) ) @ Vg @ Vh ) )
=> ( Y3
!= ( none @ nat ) ) )
=> ~ ! [Mi2: nat,Ma2: nat,Va3: nat,TreeList2: list @ vEBT_VEBT,Summary2: vEBT_VEBT] :
( ( X2
= ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( suc @ Va3 ) ) @ TreeList2 @ Summary2 ) )
=> ~ ( ( ( ord_less @ nat @ Xa @ Mi2 )
=> ( Y3
= ( some @ nat @ Mi2 ) ) )
& ( ~ ( ord_less @ nat @ Xa @ Mi2 )
=> ( Y3
= ( if @ ( option @ nat ) @ ( ord_less @ nat @ ( vEBT_VEBT_high @ Xa @ ( divide_divide @ nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( size_size @ ( list @ vEBT_VEBT ) @ TreeList2 ) )
@ ( if @ ( option @ nat )
@ ( ( ( vEBT_vebt_maxt @ ( nth @ vEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ Xa @ ( divide_divide @ nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) )
!= ( none @ nat ) )
& ( vEBT_VEBT_less @ ( some @ nat @ ( vEBT_VEBT_low @ Xa @ ( divide_divide @ nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( vEBT_vebt_maxt @ ( nth @ vEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ Xa @ ( divide_divide @ nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) ) )
@ ( vEBT_VEBT_add @ ( vEBT_VEBT_mul @ ( some @ nat @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( some @ nat @ ( vEBT_VEBT_high @ Xa @ ( divide_divide @ nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) @ ( vEBT_vebt_succ @ ( nth @ vEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ Xa @ ( divide_divide @ nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( vEBT_VEBT_low @ Xa @ ( divide_divide @ nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) )
@ ( if @ ( option @ nat )
@ ( ( vEBT_vebt_succ @ Summary2 @ ( vEBT_VEBT_high @ Xa @ ( divide_divide @ nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) )
= ( none @ nat ) )
@ ( none @ nat )
@ ( vEBT_VEBT_add @ ( vEBT_VEBT_mul @ ( some @ nat @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( vEBT_vebt_succ @ Summary2 @ ( vEBT_VEBT_high @ Xa @ ( divide_divide @ nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) @ ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList2 @ ( the2 @ nat @ ( vEBT_vebt_succ @ Summary2 @ ( vEBT_VEBT_high @ Xa @ ( divide_divide @ nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) ) ) ) ) )
@ ( none @ nat ) ) ) ) ) ) ) ) ) ) ) ) ).
% vebt_succ.elims
thf(fact_2271_vebt__succ_Opelims,axiom,
! [X2: vEBT_VEBT,Xa: nat,Y3: option @ nat] :
( ( ( vEBT_vebt_succ @ X2 @ Xa )
= Y3 )
=> ( ( accp @ ( product_prod @ vEBT_VEBT @ nat ) @ vEBT_vebt_succ_rel @ ( product_Pair @ vEBT_VEBT @ nat @ X2 @ Xa ) )
=> ( ! [Uu2: $o,B4: $o] :
( ( X2
= ( vEBT_Leaf @ Uu2 @ B4 ) )
=> ( ( Xa
= ( zero_zero @ nat ) )
=> ( ( ( B4
=> ( Y3
= ( some @ nat @ ( one_one @ nat ) ) ) )
& ( ~ B4
=> ( Y3
= ( none @ nat ) ) ) )
=> ~ ( accp @ ( product_prod @ vEBT_VEBT @ nat ) @ vEBT_vebt_succ_rel @ ( product_Pair @ vEBT_VEBT @ nat @ ( vEBT_Leaf @ Uu2 @ B4 ) @ ( zero_zero @ nat ) ) ) ) ) )
=> ( ! [Uv2: $o,Uw2: $o] :
( ( X2
= ( vEBT_Leaf @ Uv2 @ Uw2 ) )
=> ! [N3: nat] :
( ( Xa
= ( suc @ N3 ) )
=> ( ( Y3
= ( none @ nat ) )
=> ~ ( accp @ ( product_prod @ vEBT_VEBT @ nat ) @ vEBT_vebt_succ_rel @ ( product_Pair @ vEBT_VEBT @ nat @ ( vEBT_Leaf @ Uv2 @ Uw2 ) @ ( suc @ N3 ) ) ) ) ) )
=> ( ! [Ux2: nat,Uy2: list @ vEBT_VEBT,Uz2: vEBT_VEBT] :
( ( X2
= ( vEBT_Node @ ( none @ ( product_prod @ nat @ nat ) ) @ Ux2 @ Uy2 @ Uz2 ) )
=> ( ( Y3
= ( none @ nat ) )
=> ~ ( accp @ ( product_prod @ vEBT_VEBT @ nat ) @ vEBT_vebt_succ_rel @ ( product_Pair @ vEBT_VEBT @ nat @ ( vEBT_Node @ ( none @ ( product_prod @ nat @ nat ) ) @ Ux2 @ Uy2 @ Uz2 ) @ Xa ) ) ) )
=> ( ! [V3: product_prod @ nat @ nat,Vc2: list @ vEBT_VEBT,Vd2: vEBT_VEBT] :
( ( X2
= ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ V3 ) @ ( zero_zero @ nat ) @ Vc2 @ Vd2 ) )
=> ( ( Y3
= ( none @ nat ) )
=> ~ ( accp @ ( product_prod @ vEBT_VEBT @ nat ) @ vEBT_vebt_succ_rel @ ( product_Pair @ vEBT_VEBT @ nat @ ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ V3 ) @ ( zero_zero @ nat ) @ Vc2 @ Vd2 ) @ Xa ) ) ) )
=> ( ! [V3: product_prod @ nat @ nat,Vg: list @ vEBT_VEBT,Vh: vEBT_VEBT] :
( ( X2
= ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ V3 ) @ ( suc @ ( zero_zero @ nat ) ) @ Vg @ Vh ) )
=> ( ( Y3
= ( none @ nat ) )
=> ~ ( accp @ ( product_prod @ vEBT_VEBT @ nat ) @ vEBT_vebt_succ_rel @ ( product_Pair @ vEBT_VEBT @ nat @ ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ V3 ) @ ( suc @ ( zero_zero @ nat ) ) @ Vg @ Vh ) @ Xa ) ) ) )
=> ~ ! [Mi2: nat,Ma2: nat,Va3: nat,TreeList2: list @ vEBT_VEBT,Summary2: vEBT_VEBT] :
( ( X2
= ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( suc @ Va3 ) ) @ TreeList2 @ Summary2 ) )
=> ( ( ( ( ord_less @ nat @ Xa @ Mi2 )
=> ( Y3
= ( some @ nat @ Mi2 ) ) )
& ( ~ ( ord_less @ nat @ Xa @ Mi2 )
=> ( Y3
= ( if @ ( option @ nat ) @ ( ord_less @ nat @ ( vEBT_VEBT_high @ Xa @ ( divide_divide @ nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( size_size @ ( list @ vEBT_VEBT ) @ TreeList2 ) )
@ ( if @ ( option @ nat )
@ ( ( ( vEBT_vebt_maxt @ ( nth @ vEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ Xa @ ( divide_divide @ nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) )
!= ( none @ nat ) )
& ( vEBT_VEBT_less @ ( some @ nat @ ( vEBT_VEBT_low @ Xa @ ( divide_divide @ nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( vEBT_vebt_maxt @ ( nth @ vEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ Xa @ ( divide_divide @ nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) ) )
@ ( vEBT_VEBT_add @ ( vEBT_VEBT_mul @ ( some @ nat @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( some @ nat @ ( vEBT_VEBT_high @ Xa @ ( divide_divide @ nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) @ ( vEBT_vebt_succ @ ( nth @ vEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ Xa @ ( divide_divide @ nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( vEBT_VEBT_low @ Xa @ ( divide_divide @ nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) )
@ ( if @ ( option @ nat )
@ ( ( vEBT_vebt_succ @ Summary2 @ ( vEBT_VEBT_high @ Xa @ ( divide_divide @ nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) )
= ( none @ nat ) )
@ ( none @ nat )
@ ( vEBT_VEBT_add @ ( vEBT_VEBT_mul @ ( some @ nat @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( vEBT_vebt_succ @ Summary2 @ ( vEBT_VEBT_high @ Xa @ ( divide_divide @ nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) @ ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList2 @ ( the2 @ nat @ ( vEBT_vebt_succ @ Summary2 @ ( vEBT_VEBT_high @ Xa @ ( divide_divide @ nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) ) ) ) ) )
@ ( none @ nat ) ) ) ) )
=> ~ ( accp @ ( product_prod @ vEBT_VEBT @ nat ) @ vEBT_vebt_succ_rel @ ( product_Pair @ vEBT_VEBT @ nat @ ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( suc @ Va3 ) ) @ TreeList2 @ Summary2 ) @ Xa ) ) ) ) ) ) ) ) ) ) ) ).
% vebt_succ.pelims
thf(fact_2272_vebt__pred_Opelims,axiom,
! [X2: vEBT_VEBT,Xa: nat,Y3: option @ nat] :
( ( ( vEBT_vebt_pred @ X2 @ Xa )
= Y3 )
=> ( ( accp @ ( product_prod @ vEBT_VEBT @ nat ) @ vEBT_vebt_pred_rel @ ( product_Pair @ vEBT_VEBT @ nat @ X2 @ Xa ) )
=> ( ! [Uu2: $o,Uv2: $o] :
( ( X2
= ( vEBT_Leaf @ Uu2 @ Uv2 ) )
=> ( ( Xa
= ( zero_zero @ nat ) )
=> ( ( Y3
= ( none @ nat ) )
=> ~ ( accp @ ( product_prod @ vEBT_VEBT @ nat ) @ vEBT_vebt_pred_rel @ ( product_Pair @ vEBT_VEBT @ nat @ ( vEBT_Leaf @ Uu2 @ Uv2 ) @ ( zero_zero @ nat ) ) ) ) ) )
=> ( ! [A4: $o,Uw2: $o] :
( ( X2
= ( vEBT_Leaf @ A4 @ Uw2 ) )
=> ( ( Xa
= ( suc @ ( zero_zero @ nat ) ) )
=> ( ( ( A4
=> ( Y3
= ( some @ nat @ ( zero_zero @ nat ) ) ) )
& ( ~ A4
=> ( Y3
= ( none @ nat ) ) ) )
=> ~ ( accp @ ( product_prod @ vEBT_VEBT @ nat ) @ vEBT_vebt_pred_rel @ ( product_Pair @ vEBT_VEBT @ nat @ ( vEBT_Leaf @ A4 @ Uw2 ) @ ( suc @ ( zero_zero @ nat ) ) ) ) ) ) )
=> ( ! [A4: $o,B4: $o] :
( ( X2
= ( vEBT_Leaf @ A4 @ B4 ) )
=> ! [Va3: nat] :
( ( Xa
= ( suc @ ( suc @ Va3 ) ) )
=> ( ( ( B4
=> ( Y3
= ( some @ nat @ ( one_one @ nat ) ) ) )
& ( ~ B4
=> ( ( A4
=> ( Y3
= ( some @ nat @ ( zero_zero @ nat ) ) ) )
& ( ~ A4
=> ( Y3
= ( none @ nat ) ) ) ) ) )
=> ~ ( accp @ ( product_prod @ vEBT_VEBT @ nat ) @ vEBT_vebt_pred_rel @ ( product_Pair @ vEBT_VEBT @ nat @ ( vEBT_Leaf @ A4 @ B4 ) @ ( suc @ ( suc @ Va3 ) ) ) ) ) ) )
=> ( ! [Uy2: nat,Uz2: list @ vEBT_VEBT,Va2: vEBT_VEBT] :
( ( X2
= ( vEBT_Node @ ( none @ ( product_prod @ nat @ nat ) ) @ Uy2 @ Uz2 @ Va2 ) )
=> ( ( Y3
= ( none @ nat ) )
=> ~ ( accp @ ( product_prod @ vEBT_VEBT @ nat ) @ vEBT_vebt_pred_rel @ ( product_Pair @ vEBT_VEBT @ nat @ ( vEBT_Node @ ( none @ ( product_prod @ nat @ nat ) ) @ Uy2 @ Uz2 @ Va2 ) @ Xa ) ) ) )
=> ( ! [V3: product_prod @ nat @ nat,Vd2: list @ vEBT_VEBT,Ve2: vEBT_VEBT] :
( ( X2
= ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ V3 ) @ ( zero_zero @ nat ) @ Vd2 @ Ve2 ) )
=> ( ( Y3
= ( none @ nat ) )
=> ~ ( accp @ ( product_prod @ vEBT_VEBT @ nat ) @ vEBT_vebt_pred_rel @ ( product_Pair @ vEBT_VEBT @ nat @ ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ V3 ) @ ( zero_zero @ nat ) @ Vd2 @ Ve2 ) @ Xa ) ) ) )
=> ( ! [V3: product_prod @ nat @ nat,Vh: list @ vEBT_VEBT,Vi: vEBT_VEBT] :
( ( X2
= ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ V3 ) @ ( suc @ ( zero_zero @ nat ) ) @ Vh @ Vi ) )
=> ( ( Y3
= ( none @ nat ) )
=> ~ ( accp @ ( product_prod @ vEBT_VEBT @ nat ) @ vEBT_vebt_pred_rel @ ( product_Pair @ vEBT_VEBT @ nat @ ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ V3 ) @ ( suc @ ( zero_zero @ nat ) ) @ Vh @ Vi ) @ Xa ) ) ) )
=> ~ ! [Mi2: nat,Ma2: nat,Va3: nat,TreeList2: list @ vEBT_VEBT,Summary2: vEBT_VEBT] :
( ( X2
= ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( suc @ Va3 ) ) @ TreeList2 @ Summary2 ) )
=> ( ( ( ( ord_less @ nat @ Ma2 @ Xa )
=> ( Y3
= ( some @ nat @ Ma2 ) ) )
& ( ~ ( ord_less @ nat @ Ma2 @ Xa )
=> ( Y3
= ( if @ ( option @ nat ) @ ( ord_less @ nat @ ( vEBT_VEBT_high @ Xa @ ( divide_divide @ nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( size_size @ ( list @ vEBT_VEBT ) @ TreeList2 ) )
@ ( if @ ( option @ nat )
@ ( ( ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ Xa @ ( divide_divide @ nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) )
!= ( none @ nat ) )
& ( vEBT_VEBT_greater @ ( some @ nat @ ( vEBT_VEBT_low @ Xa @ ( divide_divide @ nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ Xa @ ( divide_divide @ nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) ) )
@ ( vEBT_VEBT_add @ ( vEBT_VEBT_mul @ ( some @ nat @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( some @ nat @ ( vEBT_VEBT_high @ Xa @ ( divide_divide @ nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) @ ( vEBT_vebt_pred @ ( nth @ vEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ Xa @ ( divide_divide @ nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( vEBT_VEBT_low @ Xa @ ( divide_divide @ nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) )
@ ( if @ ( option @ nat )
@ ( ( vEBT_vebt_pred @ Summary2 @ ( vEBT_VEBT_high @ Xa @ ( divide_divide @ nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) )
= ( none @ nat ) )
@ ( if @ ( option @ nat ) @ ( ord_less @ nat @ Mi2 @ Xa ) @ ( some @ nat @ Mi2 ) @ ( none @ nat ) )
@ ( vEBT_VEBT_add @ ( vEBT_VEBT_mul @ ( some @ nat @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( vEBT_vebt_pred @ Summary2 @ ( vEBT_VEBT_high @ Xa @ ( divide_divide @ nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) @ ( vEBT_vebt_maxt @ ( nth @ vEBT_VEBT @ TreeList2 @ ( the2 @ nat @ ( vEBT_vebt_pred @ Summary2 @ ( vEBT_VEBT_high @ Xa @ ( divide_divide @ nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) ) ) ) ) )
@ ( none @ nat ) ) ) ) )
=> ~ ( accp @ ( product_prod @ vEBT_VEBT @ nat ) @ vEBT_vebt_pred_rel @ ( product_Pair @ vEBT_VEBT @ nat @ ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( suc @ Va3 ) ) @ TreeList2 @ Summary2 ) @ Xa ) ) ) ) ) ) ) ) ) ) ) ) ).
% vebt_pred.pelims
thf(fact_2273_succ__bound__size__univ,axiom,
! [T2: vEBT_VEBT,N: nat,U: real,X2: 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 @ X2 ) ) @ ( 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_2274_insert__bound__size__univ,axiom,
! [T2: vEBT_VEBT,N: nat,U: real,X2: 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 @ X2 ) ) @ ( 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_2275_pred__bound__size__univ,axiom,
! [T2: vEBT_VEBT,N: nat,U: real,X2: 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 @ X2 ) ) @ ( 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_2276_delete__correct,axiom,
! [T2: vEBT_VEBT,N: nat,X2: nat] :
( ( vEBT_invar_vebt @ T2 @ N )
=> ( ( vEBT_VEBT_set_vebt @ ( vEBT_vebt_delete @ T2 @ X2 ) )
= ( minus_minus @ ( set @ nat ) @ ( vEBT_set_vebt @ T2 ) @ ( insert @ nat @ X2 @ ( bot_bot @ ( set @ nat ) ) ) ) ) ) ).
% delete_correct
thf(fact_2277_delete__correct_H,axiom,
! [T2: vEBT_VEBT,N: nat,X2: nat] :
( ( vEBT_invar_vebt @ T2 @ N )
=> ( ( vEBT_VEBT_set_vebt @ ( vEBT_vebt_delete @ T2 @ X2 ) )
= ( minus_minus @ ( set @ nat ) @ ( vEBT_VEBT_set_vebt @ T2 ) @ ( insert @ nat @ X2 @ ( bot_bot @ ( set @ nat ) ) ) ) ) ) ).
% delete_correct'
thf(fact_2278_insert__subset,axiom,
! [A: $tType,X2: A,A2: set @ A,B2: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ ( insert @ A @ X2 @ A2 ) @ B2 )
= ( ( member @ A @ X2 @ B2 )
& ( ord_less_eq @ ( set @ A ) @ A2 @ B2 ) ) ) ).
% insert_subset
thf(fact_2279_Diff__insert0,axiom,
! [A: $tType,X2: A,A2: set @ A,B2: set @ A] :
( ~ ( member @ A @ X2 @ A2 )
=> ( ( minus_minus @ ( set @ A ) @ A2 @ ( insert @ A @ X2 @ B2 ) )
= ( minus_minus @ ( set @ A ) @ A2 @ B2 ) ) ) ).
% Diff_insert0
thf(fact_2280_insert__Diff1,axiom,
! [A: $tType,X2: A,B2: set @ A,A2: set @ A] :
( ( member @ A @ X2 @ B2 )
=> ( ( minus_minus @ ( set @ A ) @ ( insert @ A @ X2 @ A2 ) @ B2 )
= ( minus_minus @ ( set @ A ) @ A2 @ B2 ) ) ) ).
% insert_Diff1
thf(fact_2281_singleton__conv2,axiom,
! [A: $tType,A3: A] :
( ( collect @ A
@ ( ^ [Y6: A,Z3: A] : Y6 = Z3
@ A3 ) )
= ( insert @ A @ A3 @ ( bot_bot @ ( set @ A ) ) ) ) ).
% singleton_conv2
thf(fact_2282_singleton__conv,axiom,
! [A: $tType,A3: A] :
( ( collect @ A
@ ^ [X3: A] : X3 = A3 )
= ( insert @ A @ A3 @ ( bot_bot @ ( set @ A ) ) ) ) ).
% singleton_conv
thf(fact_2283_singleton__insert__inj__eq,axiom,
! [A: $tType,B3: A,A3: A,A2: set @ A] :
( ( ( insert @ A @ B3 @ ( bot_bot @ ( set @ A ) ) )
= ( insert @ A @ A3 @ A2 ) )
= ( ( A3 = B3 )
& ( ord_less_eq @ ( set @ A ) @ A2 @ ( insert @ A @ B3 @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ).
% singleton_insert_inj_eq
thf(fact_2284_singleton__insert__inj__eq_H,axiom,
! [A: $tType,A3: A,A2: set @ A,B3: A] :
( ( ( insert @ A @ A3 @ A2 )
= ( insert @ A @ B3 @ ( bot_bot @ ( set @ A ) ) ) )
= ( ( A3 = B3 )
& ( ord_less_eq @ ( set @ A ) @ A2 @ ( insert @ A @ B3 @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ).
% singleton_insert_inj_eq'
thf(fact_2285_insert__Diff__single,axiom,
! [A: $tType,A3: A,A2: set @ A] :
( ( insert @ A @ A3 @ ( minus_minus @ ( set @ A ) @ A2 @ ( insert @ A @ A3 @ ( bot_bot @ ( set @ A ) ) ) ) )
= ( insert @ A @ A3 @ A2 ) ) ).
% insert_Diff_single
thf(fact_2286_insert__compr,axiom,
! [A: $tType] :
( ( insert @ A )
= ( ^ [A6: A,B6: set @ A] :
( collect @ A
@ ^ [X3: A] :
( ( X3 = A6 )
| ( member @ A @ X3 @ B6 ) ) ) ) ) ).
% insert_compr
thf(fact_2287_insert__Collect,axiom,
! [A: $tType,A3: A,P: A > $o] :
( ( insert @ A @ A3 @ ( collect @ A @ P ) )
= ( collect @ A
@ ^ [U2: A] :
( ( U2 != A3 )
=> ( P @ U2 ) ) ) ) ).
% insert_Collect
thf(fact_2288_insert__mono,axiom,
! [A: $tType,C2: set @ A,D: set @ A,A3: A] :
( ( ord_less_eq @ ( set @ A ) @ C2 @ D )
=> ( ord_less_eq @ ( set @ A ) @ ( insert @ A @ A3 @ C2 ) @ ( insert @ A @ A3 @ D ) ) ) ).
% insert_mono
thf(fact_2289_subset__insert,axiom,
! [A: $tType,X2: A,A2: set @ A,B2: set @ A] :
( ~ ( member @ A @ X2 @ A2 )
=> ( ( ord_less_eq @ ( set @ A ) @ A2 @ ( insert @ A @ X2 @ B2 ) )
= ( ord_less_eq @ ( set @ A ) @ A2 @ B2 ) ) ) ).
% subset_insert
thf(fact_2290_subset__insertI,axiom,
! [A: $tType,B2: set @ A,A3: A] : ( ord_less_eq @ ( set @ A ) @ B2 @ ( insert @ A @ A3 @ B2 ) ) ).
% subset_insertI
thf(fact_2291_subset__insertI2,axiom,
! [A: $tType,A2: set @ A,B2: set @ A,B3: A] :
( ( ord_less_eq @ ( set @ A ) @ A2 @ B2 )
=> ( ord_less_eq @ ( set @ A ) @ A2 @ ( insert @ A @ B3 @ B2 ) ) ) ).
% subset_insertI2
thf(fact_2292_insert__Diff__if,axiom,
! [A: $tType,X2: A,B2: set @ A,A2: set @ A] :
( ( ( member @ A @ X2 @ B2 )
=> ( ( minus_minus @ ( set @ A ) @ ( insert @ A @ X2 @ A2 ) @ B2 )
= ( minus_minus @ ( set @ A ) @ A2 @ B2 ) ) )
& ( ~ ( member @ A @ X2 @ B2 )
=> ( ( minus_minus @ ( set @ A ) @ ( insert @ A @ X2 @ A2 ) @ B2 )
= ( insert @ A @ X2 @ ( minus_minus @ ( set @ A ) @ A2 @ B2 ) ) ) ) ) ).
% insert_Diff_if
thf(fact_2293_Collect__conv__if2,axiom,
! [A: $tType,P: A > $o,A3: A] :
( ( ( P @ A3 )
=> ( ( collect @ A
@ ^ [X3: A] :
( ( A3 = X3 )
& ( P @ X3 ) ) )
= ( insert @ A @ A3 @ ( bot_bot @ ( set @ A ) ) ) ) )
& ( ~ ( P @ A3 )
=> ( ( collect @ A
@ ^ [X3: A] :
( ( A3 = X3 )
& ( P @ X3 ) ) )
= ( bot_bot @ ( set @ A ) ) ) ) ) ).
% Collect_conv_if2
thf(fact_2294_Collect__conv__if,axiom,
! [A: $tType,P: A > $o,A3: A] :
( ( ( P @ A3 )
=> ( ( collect @ A
@ ^ [X3: A] :
( ( X3 = A3 )
& ( P @ X3 ) ) )
= ( insert @ A @ A3 @ ( bot_bot @ ( set @ A ) ) ) ) )
& ( ~ ( P @ A3 )
=> ( ( collect @ A
@ ^ [X3: A] :
( ( X3 = A3 )
& ( P @ X3 ) ) )
= ( bot_bot @ ( set @ A ) ) ) ) ) ).
% Collect_conv_if
thf(fact_2295_subset__singletonD,axiom,
! [A: $tType,A2: set @ A,X2: A] :
( ( ord_less_eq @ ( set @ A ) @ A2 @ ( insert @ A @ X2 @ ( bot_bot @ ( set @ A ) ) ) )
=> ( ( A2
= ( bot_bot @ ( set @ A ) ) )
| ( A2
= ( insert @ A @ X2 @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ).
% subset_singletonD
thf(fact_2296_subset__singleton__iff,axiom,
! [A: $tType,X: set @ A,A3: A] :
( ( ord_less_eq @ ( set @ A ) @ X @ ( insert @ A @ A3 @ ( bot_bot @ ( set @ A ) ) ) )
= ( ( X
= ( bot_bot @ ( set @ A ) ) )
| ( X
= ( insert @ A @ A3 @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ).
% subset_singleton_iff
thf(fact_2297_Diff__insert,axiom,
! [A: $tType,A2: set @ A,A3: A,B2: set @ A] :
( ( minus_minus @ ( set @ A ) @ A2 @ ( insert @ A @ A3 @ B2 ) )
= ( minus_minus @ ( set @ A ) @ ( minus_minus @ ( set @ A ) @ A2 @ B2 ) @ ( insert @ A @ A3 @ ( bot_bot @ ( set @ A ) ) ) ) ) ).
% Diff_insert
thf(fact_2298_insert__Diff,axiom,
! [A: $tType,A3: A,A2: set @ A] :
( ( member @ A @ A3 @ A2 )
=> ( ( insert @ A @ A3 @ ( minus_minus @ ( set @ A ) @ A2 @ ( insert @ A @ A3 @ ( bot_bot @ ( set @ A ) ) ) ) )
= A2 ) ) ).
% insert_Diff
thf(fact_2299_Diff__insert2,axiom,
! [A: $tType,A2: set @ A,A3: A,B2: set @ A] :
( ( minus_minus @ ( set @ A ) @ A2 @ ( insert @ A @ A3 @ B2 ) )
= ( minus_minus @ ( set @ A ) @ ( minus_minus @ ( set @ A ) @ A2 @ ( insert @ A @ A3 @ ( bot_bot @ ( set @ A ) ) ) ) @ B2 ) ) ).
% Diff_insert2
thf(fact_2300_Diff__insert__absorb,axiom,
! [A: $tType,X2: A,A2: set @ A] :
( ~ ( member @ A @ X2 @ A2 )
=> ( ( minus_minus @ ( set @ A ) @ ( insert @ A @ X2 @ A2 ) @ ( insert @ A @ X2 @ ( bot_bot @ ( set @ A ) ) ) )
= A2 ) ) ).
% Diff_insert_absorb
thf(fact_2301_set__minus__singleton__eq,axiom,
! [A: $tType,X2: A,X: set @ A] :
( ~ ( member @ A @ X2 @ X )
=> ( ( minus_minus @ ( set @ A ) @ X @ ( insert @ A @ X2 @ ( bot_bot @ ( set @ A ) ) ) )
= X ) ) ).
% set_minus_singleton_eq
thf(fact_2302_insert__minus__eq,axiom,
! [A: $tType,X2: A,Y3: A,A2: set @ A] :
( ( X2 != Y3 )
=> ( ( minus_minus @ ( set @ A ) @ ( insert @ A @ X2 @ A2 ) @ ( insert @ A @ Y3 @ ( bot_bot @ ( set @ A ) ) ) )
= ( insert @ A @ X2 @ ( minus_minus @ ( set @ A ) @ A2 @ ( insert @ A @ Y3 @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ) ).
% insert_minus_eq
thf(fact_2303_subset__Diff__insert,axiom,
! [A: $tType,A2: set @ A,B2: set @ A,X2: A,C2: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ A2 @ ( minus_minus @ ( set @ A ) @ B2 @ ( insert @ A @ X2 @ C2 ) ) )
= ( ( ord_less_eq @ ( set @ A ) @ A2 @ ( minus_minus @ ( set @ A ) @ B2 @ C2 ) )
& ~ ( member @ A @ X2 @ A2 ) ) ) ).
% subset_Diff_insert
thf(fact_2304_Diff__single__insert,axiom,
! [A: $tType,A2: set @ A,X2: A,B2: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ ( minus_minus @ ( set @ A ) @ A2 @ ( insert @ A @ X2 @ ( bot_bot @ ( set @ A ) ) ) ) @ B2 )
=> ( ord_less_eq @ ( set @ A ) @ A2 @ ( insert @ A @ X2 @ B2 ) ) ) ).
% Diff_single_insert
thf(fact_2305_subset__insert__iff,axiom,
! [A: $tType,A2: set @ A,X2: A,B2: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ A2 @ ( insert @ A @ X2 @ B2 ) )
= ( ( ( member @ A @ X2 @ A2 )
=> ( ord_less_eq @ ( set @ A ) @ ( minus_minus @ ( set @ A ) @ A2 @ ( insert @ A @ X2 @ ( bot_bot @ ( set @ A ) ) ) ) @ B2 ) )
& ( ~ ( member @ A @ X2 @ A2 )
=> ( ord_less_eq @ ( set @ A ) @ A2 @ B2 ) ) ) ) ).
% subset_insert_iff
thf(fact_2306_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_2307_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_2308_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_2309_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_2310_remove__subset,axiom,
! [A: $tType,X2: A,S: set @ A] :
( ( member @ A @ X2 @ S )
=> ( ord_less @ ( set @ A ) @ ( minus_minus @ ( set @ A ) @ S @ ( insert @ A @ X2 @ ( bot_bot @ ( set @ A ) ) ) ) @ S ) ) ).
% remove_subset
thf(fact_2311_set__update__subset__insert,axiom,
! [A: $tType,Xs: list @ A,I: nat,X2: A] : ( ord_less_eq @ ( set @ A ) @ ( set2 @ A @ ( list_update @ A @ Xs @ I @ X2 ) ) @ ( insert @ A @ X2 @ ( set2 @ A @ Xs ) ) ) ).
% set_update_subset_insert
thf(fact_2312_psubset__insert__iff,axiom,
! [A: $tType,A2: set @ A,X2: A,B2: set @ A] :
( ( ord_less @ ( set @ A ) @ A2 @ ( insert @ A @ X2 @ B2 ) )
= ( ( ( member @ A @ X2 @ B2 )
=> ( ord_less @ ( set @ A ) @ A2 @ B2 ) )
& ( ~ ( member @ A @ X2 @ B2 )
=> ( ( ( member @ A @ X2 @ A2 )
=> ( ord_less @ ( set @ A ) @ ( minus_minus @ ( set @ A ) @ A2 @ ( insert @ A @ X2 @ ( bot_bot @ ( set @ A ) ) ) ) @ B2 ) )
& ( ~ ( member @ A @ X2 @ A2 )
=> ( ord_less_eq @ ( set @ A ) @ A2 @ B2 ) ) ) ) ) ) ).
% psubset_insert_iff
thf(fact_2313_T_092_060_094sub_062i_092_060_094sub_062n_092_060_094sub_062s_092_060_094sub_062e_092_060_094sub_062r_092_060_094sub_062t_Osimps_I2_J,axiom,
! [Info: option @ ( product_prod @ nat @ nat ),Ts: list @ vEBT_VEBT,S2: vEBT_VEBT,X2: nat] :
( ( vEBT_T_i_n_s_e_r_t @ ( vEBT_Node @ Info @ ( zero_zero @ nat ) @ Ts @ S2 ) @ X2 )
= ( one_one @ nat ) ) ).
% T\<^sub>i\<^sub>n\<^sub>s\<^sub>e\<^sub>r\<^sub>t.simps(2)
thf(fact_2314_T_092_060_094sub_062p_092_060_094sub_062r_092_060_094sub_062e_092_060_094sub_062d_Osimps_I1_J,axiom,
! [Uu: $o,Uv: $o] :
( ( vEBT_T_p_r_e_d @ ( vEBT_Leaf @ Uu @ Uv ) @ ( zero_zero @ nat ) )
= ( one_one @ nat ) ) ).
% T\<^sub>p\<^sub>r\<^sub>e\<^sub>d.simps(1)
thf(fact_2315_T_092_060_094sub_062s_092_060_094sub_062u_092_060_094sub_062c_092_060_094sub_062c_Osimps_I2_J,axiom,
! [Uv: $o,Uw: $o,N: nat] :
( ( vEBT_T_s_u_c_c @ ( vEBT_Leaf @ Uv @ Uw ) @ ( suc @ N ) )
= ( one_one @ nat ) ) ).
% T\<^sub>s\<^sub>u\<^sub>c\<^sub>c.simps(2)
thf(fact_2316_T_092_060_094sub_062p_092_060_094sub_062r_092_060_094sub_062e_092_060_094sub_062d_Osimps_I4_J,axiom,
! [Uy: nat,Uz: list @ vEBT_VEBT,Va: vEBT_VEBT,Vb: nat] :
( ( vEBT_T_p_r_e_d @ ( vEBT_Node @ ( none @ ( product_prod @ nat @ nat ) ) @ Uy @ Uz @ Va ) @ Vb )
= ( one_one @ nat ) ) ).
% T\<^sub>p\<^sub>r\<^sub>e\<^sub>d.simps(4)
thf(fact_2317_T_092_060_094sub_062s_092_060_094sub_062u_092_060_094sub_062c_092_060_094sub_062c_Osimps_I3_J,axiom,
! [Ux: nat,Uy: list @ vEBT_VEBT,Uz: vEBT_VEBT,Va: nat] :
( ( vEBT_T_s_u_c_c @ ( vEBT_Node @ ( none @ ( product_prod @ nat @ nat ) ) @ Ux @ Uy @ Uz ) @ Va )
= ( one_one @ nat ) ) ).
% T\<^sub>s\<^sub>u\<^sub>c\<^sub>c.simps(3)
thf(fact_2318_insert__swap__set__eq,axiom,
! [A: $tType,I: nat,L: list @ A,X2: A] :
( ( ord_less @ nat @ I @ ( size_size @ ( list @ A ) @ L ) )
=> ( ( insert @ A @ ( nth @ A @ L @ I ) @ ( set2 @ A @ ( list_update @ A @ L @ I @ X2 ) ) )
= ( insert @ A @ X2 @ ( set2 @ A @ L ) ) ) ) ).
% insert_swap_set_eq
thf(fact_2319_T_092_060_094sub_062i_092_060_094sub_062n_092_060_094sub_062s_092_060_094sub_062e_092_060_094sub_062r_092_060_094sub_062t_Osimps_I3_J,axiom,
! [Info: option @ ( product_prod @ nat @ nat ),Ts: list @ vEBT_VEBT,S2: vEBT_VEBT,X2: nat] :
( ( vEBT_T_i_n_s_e_r_t @ ( vEBT_Node @ Info @ ( suc @ ( zero_zero @ nat ) ) @ Ts @ S2 ) @ X2 )
= ( one_one @ nat ) ) ).
% T\<^sub>i\<^sub>n\<^sub>s\<^sub>e\<^sub>r\<^sub>t.simps(3)
thf(fact_2320_T_092_060_094sub_062p_092_060_094sub_062r_092_060_094sub_062e_092_060_094sub_062d_Osimps_I3_J,axiom,
! [A3: $o,B3: $o,Va: nat] :
( ( vEBT_T_p_r_e_d @ ( vEBT_Leaf @ A3 @ B3 ) @ ( suc @ ( suc @ Va ) ) )
= ( plus_plus @ nat @ ( one_one @ nat ) @ ( if @ nat @ B3 @ ( 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_2321_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,
! [A3: $o,B3: $o,X2: nat] :
( ( vEBT_T_i_n_s_e_r_t @ ( vEBT_Leaf @ A3 @ B3 ) @ X2 )
= ( plus_plus @ nat @ ( one_one @ nat )
@ ( if @ nat
@ ( X2
= ( 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_2322_T_092_060_094sub_062p_092_060_094sub_062r_092_060_094sub_062e_092_060_094sub_062d_Osimps_I5_J,axiom,
! [V: product_prod @ nat @ nat,Vd3: list @ vEBT_VEBT,Ve3: vEBT_VEBT,Vf2: nat] :
( ( vEBT_T_p_r_e_d @ ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ V ) @ ( zero_zero @ nat ) @ Vd3 @ Ve3 ) @ Vf2 )
= ( one_one @ nat ) ) ).
% T\<^sub>p\<^sub>r\<^sub>e\<^sub>d.simps(5)
thf(fact_2323_T_092_060_094sub_062s_092_060_094sub_062u_092_060_094sub_062c_092_060_094sub_062c_Osimps_I1_J,axiom,
! [Uu: $o,B3: $o] :
( ( vEBT_T_s_u_c_c @ ( vEBT_Leaf @ Uu @ B3 ) @ ( 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_2324_T_092_060_094sub_062s_092_060_094sub_062u_092_060_094sub_062c_092_060_094sub_062c_Osimps_I4_J,axiom,
! [V: product_prod @ nat @ nat,Vc: list @ vEBT_VEBT,Vd3: vEBT_VEBT,Ve3: nat] :
( ( vEBT_T_s_u_c_c @ ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ V ) @ ( zero_zero @ nat ) @ Vc @ Vd3 ) @ Ve3 )
= ( one_one @ nat ) ) ).
% T\<^sub>s\<^sub>u\<^sub>c\<^sub>c.simps(4)
thf(fact_2325_T_092_060_094sub_062p_092_060_094sub_062r_092_060_094sub_062e_092_060_094sub_062d_Osimps_I2_J,axiom,
! [A3: $o,Uw: $o] :
( ( vEBT_T_p_r_e_d @ ( vEBT_Leaf @ A3 @ Uw ) @ ( 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_2326_T_092_060_094sub_062p_092_060_094sub_062r_092_060_094sub_062e_092_060_094sub_062d_Osimps_I6_J,axiom,
! [V: 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 ) @ V ) @ ( suc @ ( zero_zero @ nat ) ) @ Vh2 @ Vi2 ) @ Vj2 )
= ( one_one @ nat ) ) ).
% T\<^sub>p\<^sub>r\<^sub>e\<^sub>d.simps(6)
thf(fact_2327_T_092_060_094sub_062s_092_060_094sub_062u_092_060_094sub_062c_092_060_094sub_062c_Osimps_I5_J,axiom,
! [V: 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 ) @ V ) @ ( suc @ ( zero_zero @ nat ) ) @ Vg2 @ Vh2 ) @ Vi2 )
= ( one_one @ nat ) ) ).
% T\<^sub>s\<^sub>u\<^sub>c\<^sub>c.simps(5)
thf(fact_2328_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,
! [V: nat,TreeList: list @ vEBT_VEBT,Summary: vEBT_VEBT,X2: nat] :
( ( vEBT_T_i_n_s_e_r_t @ ( vEBT_Node @ ( none @ ( product_prod @ nat @ nat ) ) @ ( suc @ ( suc @ V ) ) @ TreeList @ Summary ) @ X2 )
= ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ).
% T\<^sub>i\<^sub>n\<^sub>s\<^sub>e\<^sub>r\<^sub>t.simps(4)
thf(fact_2329_insersimp,axiom,
! [T2: vEBT_VEBT,N: nat,Y3: nat] :
( ( vEBT_invar_vebt @ T2 @ N )
=> ( ~ ? [X_1: nat] : ( vEBT_V8194947554948674370ptions @ T2 @ X_1 )
=> ( ord_less_eq @ nat @ ( vEBT_T_i_n_s_e_r_t @ T2 @ Y3 ) @ ( numeral_numeral @ nat @ ( bit1 @ one2 ) ) ) ) ) ).
% insersimp
thf(fact_2330_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_2331_insert__bound__height,axiom,
! [T2: vEBT_VEBT,N: nat,X2: nat] :
( ( vEBT_invar_vebt @ T2 @ N )
=> ( ord_less_eq @ nat @ ( vEBT_T_i_n_s_e_r_t @ T2 @ X2 ) @ ( 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_2332_pred__bound__height,axiom,
! [T2: vEBT_VEBT,N: nat,X2: nat] :
( ( vEBT_invar_vebt @ T2 @ N )
=> ( ord_less_eq @ nat @ ( vEBT_T_p_r_e_d @ T2 @ X2 ) @ ( 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_2333_succ__bound__height,axiom,
! [T2: vEBT_VEBT,N: nat,X2: nat] :
( ( vEBT_invar_vebt @ T2 @ N )
=> ( ord_less_eq @ nat @ ( vEBT_T_s_u_c_c @ T2 @ X2 ) @ ( 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_2334_listI__assn__wrap__insert,axiom,
! [E3: $tType,P: assn,Uu: vEBT_VEBT,Uua: nat,Xi: vEBT_VEBTi,I6: set @ nat,I: nat,Xs: list @ vEBT_VEBT,Xsi: list @ vEBT_VEBTi,F4: assn,C3: heap_Time_Heap @ E3,Q: E3 > assn] :
( ( entails @ P @ ( times_times @ assn @ ( times_times @ assn @ ( vEBT_vebt_assn_raw @ ( vEBT_vebt_insert @ Uu @ 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 ) ) @ F4 ) )
=> ( ( ord_less @ nat @ I @ ( size_size @ ( list @ vEBT_VEBT ) @ Xs ) )
=> ( ( member @ nat @ I @ I6 )
=> ( ( hoare_hoare_triple @ E3 @ ( 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 @ Uu @ Uua ) ) @ ( list_update @ vEBT_VEBTi @ Xsi @ I @ Xi ) ) @ F4 ) @ C3 @ Q )
=> ( hoare_hoare_triple @ E3 @ P @ C3 @ Q ) ) ) ) ) ).
% listI_assn_wrap_insert
thf(fact_2335_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,
! [X2: vEBT_VEBT,Xa: nat,Y3: nat] :
( ( ( vEBT_T_i_n_s_e_r_t @ X2 @ Xa )
= Y3 )
=> ( ( ? [A4: $o,B4: $o] :
( X2
= ( vEBT_Leaf @ A4 @ B4 ) )
=> ( Y3
!= ( plus_plus @ nat @ ( one_one @ nat )
@ ( if @ nat
@ ( Xa
= ( zero_zero @ nat ) )
@ ( one_one @ nat )
@ ( plus_plus @ nat @ ( one_one @ nat ) @ ( one_one @ nat ) ) ) ) ) )
=> ( ( ? [Info2: option @ ( product_prod @ nat @ nat ),Ts2: list @ vEBT_VEBT,S3: vEBT_VEBT] :
( X2
= ( vEBT_Node @ Info2 @ ( zero_zero @ nat ) @ Ts2 @ S3 ) )
=> ( Y3
!= ( one_one @ nat ) ) )
=> ( ( ? [Info2: option @ ( product_prod @ nat @ nat ),Ts2: list @ vEBT_VEBT,S3: vEBT_VEBT] :
( X2
= ( vEBT_Node @ Info2 @ ( suc @ ( zero_zero @ nat ) ) @ Ts2 @ S3 ) )
=> ( Y3
!= ( one_one @ nat ) ) )
=> ( ( ? [V3: nat,TreeList2: list @ vEBT_VEBT,Summary2: vEBT_VEBT] :
( X2
= ( vEBT_Node @ ( none @ ( product_prod @ nat @ nat ) ) @ ( suc @ ( suc @ V3 ) ) @ TreeList2 @ Summary2 ) )
=> ( Y3
!= ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) )
=> ~ ! [Mi2: nat,Ma2: nat,Va3: nat,TreeList2: list @ vEBT_VEBT,Summary2: vEBT_VEBT] :
( ( X2
= ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( suc @ Va3 ) ) @ TreeList2 @ Summary2 ) )
=> ( Y3
!= ( plus_plus @ nat @ ( numeral_numeral @ nat @ ( bit1 @ ( bit1 @ ( bit0 @ ( bit0 @ one2 ) ) ) ) )
@ ( if @ nat
@ ( ( ord_less @ nat @ ( vEBT_VEBT_high @ ( if @ nat @ ( ord_less @ nat @ Xa @ Mi2 ) @ Mi2 @ Xa ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( size_size @ ( list @ vEBT_VEBT ) @ TreeList2 ) )
& ~ ( ( Xa = Mi2 )
| ( Xa = Ma2 ) ) )
@ ( plus_plus @ nat @ ( plus_plus @ nat @ ( vEBT_T_i_n_s_e_r_t @ ( nth @ vEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ ( if @ nat @ ( ord_less @ nat @ Xa @ Mi2 ) @ Mi2 @ Xa ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( vEBT_VEBT_low @ ( if @ nat @ ( ord_less @ nat @ Xa @ Mi2 ) @ Mi2 @ Xa ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( vEBT_T_m_i_n_N_u_l_l @ ( nth @ vEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ ( if @ nat @ ( ord_less @ nat @ Xa @ Mi2 ) @ Mi2 @ Xa ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) ) @ ( if @ nat @ ( vEBT_VEBT_minNull @ ( nth @ vEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ ( if @ nat @ ( ord_less @ nat @ Xa @ Mi2 ) @ Mi2 @ Xa ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) @ ( vEBT_T_i_n_s_e_r_t @ Summary2 @ ( vEBT_VEBT_high @ ( if @ nat @ ( ord_less @ nat @ Xa @ Mi2 ) @ Mi2 @ Xa ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va3 ) ) @ ( 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_2336_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,Va: nat,TreeList: list @ vEBT_VEBT,Summary: vEBT_VEBT,X2: nat] :
( ( vEBT_T_i_n_s_e_r_t @ ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi @ Ma ) ) @ ( suc @ ( suc @ Va ) ) @ TreeList @ Summary ) @ X2 )
= ( plus_plus @ nat @ ( numeral_numeral @ nat @ ( bit1 @ ( bit1 @ ( bit0 @ ( bit0 @ one2 ) ) ) ) )
@ ( if @ nat
@ ( ( ord_less @ nat @ ( vEBT_VEBT_high @ ( if @ nat @ ( ord_less @ nat @ X2 @ Mi ) @ Mi @ X2 ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( size_size @ ( list @ vEBT_VEBT ) @ TreeList ) )
& ~ ( ( X2 = Mi )
| ( X2 = 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 @ X2 @ Mi ) @ Mi @ X2 ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( vEBT_VEBT_low @ ( if @ nat @ ( ord_less @ nat @ X2 @ Mi ) @ Mi @ X2 ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( 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 @ X2 @ Mi ) @ Mi @ X2 ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) ) @ ( if @ nat @ ( vEBT_VEBT_minNull @ ( nth @ vEBT_VEBT @ TreeList @ ( vEBT_VEBT_high @ ( if @ nat @ ( ord_less @ nat @ X2 @ Mi ) @ Mi @ X2 ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) @ ( vEBT_T_i_n_s_e_r_t @ Summary @ ( vEBT_VEBT_high @ ( if @ nat @ ( ord_less @ nat @ X2 @ Mi ) @ Mi @ X2 ) @ ( 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.simps(5)
thf(fact_2337_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,
! [X2: vEBT_VEBT,Xa: nat,Y3: nat] :
( ( ( vEBT_T_i_n_s_e_r_t @ X2 @ Xa )
= Y3 )
=> ( ( accp @ ( product_prod @ vEBT_VEBT @ nat ) @ vEBT_T9217963907923527482_t_rel @ ( product_Pair @ vEBT_VEBT @ nat @ X2 @ Xa ) )
=> ( ! [A4: $o,B4: $o] :
( ( X2
= ( vEBT_Leaf @ A4 @ B4 ) )
=> ( ( Y3
= ( plus_plus @ nat @ ( one_one @ nat )
@ ( if @ nat
@ ( Xa
= ( zero_zero @ nat ) )
@ ( one_one @ nat )
@ ( plus_plus @ nat @ ( one_one @ nat ) @ ( one_one @ nat ) ) ) ) )
=> ~ ( accp @ ( product_prod @ vEBT_VEBT @ nat ) @ vEBT_T9217963907923527482_t_rel @ ( product_Pair @ vEBT_VEBT @ nat @ ( vEBT_Leaf @ A4 @ B4 ) @ Xa ) ) ) )
=> ( ! [Info2: option @ ( product_prod @ nat @ nat ),Ts2: list @ vEBT_VEBT,S3: vEBT_VEBT] :
( ( X2
= ( vEBT_Node @ Info2 @ ( zero_zero @ nat ) @ Ts2 @ S3 ) )
=> ( ( Y3
= ( one_one @ nat ) )
=> ~ ( accp @ ( product_prod @ vEBT_VEBT @ nat ) @ vEBT_T9217963907923527482_t_rel @ ( product_Pair @ vEBT_VEBT @ nat @ ( vEBT_Node @ Info2 @ ( zero_zero @ nat ) @ Ts2 @ S3 ) @ Xa ) ) ) )
=> ( ! [Info2: option @ ( product_prod @ nat @ nat ),Ts2: list @ vEBT_VEBT,S3: vEBT_VEBT] :
( ( X2
= ( vEBT_Node @ Info2 @ ( suc @ ( zero_zero @ nat ) ) @ Ts2 @ S3 ) )
=> ( ( Y3
= ( one_one @ nat ) )
=> ~ ( accp @ ( product_prod @ vEBT_VEBT @ nat ) @ vEBT_T9217963907923527482_t_rel @ ( product_Pair @ vEBT_VEBT @ nat @ ( vEBT_Node @ Info2 @ ( suc @ ( zero_zero @ nat ) ) @ Ts2 @ S3 ) @ Xa ) ) ) )
=> ( ! [V3: nat,TreeList2: list @ vEBT_VEBT,Summary2: vEBT_VEBT] :
( ( X2
= ( vEBT_Node @ ( none @ ( product_prod @ nat @ nat ) ) @ ( suc @ ( suc @ V3 ) ) @ TreeList2 @ Summary2 ) )
=> ( ( Y3
= ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) )
=> ~ ( accp @ ( product_prod @ vEBT_VEBT @ nat ) @ vEBT_T9217963907923527482_t_rel @ ( product_Pair @ vEBT_VEBT @ nat @ ( vEBT_Node @ ( none @ ( product_prod @ nat @ nat ) ) @ ( suc @ ( suc @ V3 ) ) @ TreeList2 @ Summary2 ) @ Xa ) ) ) )
=> ~ ! [Mi2: nat,Ma2: nat,Va3: nat,TreeList2: list @ vEBT_VEBT,Summary2: vEBT_VEBT] :
( ( X2
= ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( suc @ Va3 ) ) @ TreeList2 @ Summary2 ) )
=> ( ( Y3
= ( plus_plus @ nat @ ( numeral_numeral @ nat @ ( bit1 @ ( bit1 @ ( bit0 @ ( bit0 @ one2 ) ) ) ) )
@ ( if @ nat
@ ( ( ord_less @ nat @ ( vEBT_VEBT_high @ ( if @ nat @ ( ord_less @ nat @ Xa @ Mi2 ) @ Mi2 @ Xa ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( size_size @ ( list @ vEBT_VEBT ) @ TreeList2 ) )
& ~ ( ( Xa = Mi2 )
| ( Xa = Ma2 ) ) )
@ ( plus_plus @ nat @ ( plus_plus @ nat @ ( vEBT_T_i_n_s_e_r_t @ ( nth @ vEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ ( if @ nat @ ( ord_less @ nat @ Xa @ Mi2 ) @ Mi2 @ Xa ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( vEBT_VEBT_low @ ( if @ nat @ ( ord_less @ nat @ Xa @ Mi2 ) @ Mi2 @ Xa ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( vEBT_T_m_i_n_N_u_l_l @ ( nth @ vEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ ( if @ nat @ ( ord_less @ nat @ Xa @ Mi2 ) @ Mi2 @ Xa ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) ) @ ( if @ nat @ ( vEBT_VEBT_minNull @ ( nth @ vEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ ( if @ nat @ ( ord_less @ nat @ Xa @ Mi2 ) @ Mi2 @ Xa ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) @ ( vEBT_T_i_n_s_e_r_t @ Summary2 @ ( vEBT_VEBT_high @ ( if @ nat @ ( ord_less @ nat @ Xa @ Mi2 ) @ Mi2 @ Xa ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va3 ) ) @ ( 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 @ Va3 ) ) @ TreeList2 @ Summary2 ) @ Xa ) ) ) ) ) ) ) ) ) ) ).
% T\<^sub>i\<^sub>n\<^sub>s\<^sub>e\<^sub>r\<^sub>t.pelims
thf(fact_2338_T_092_060_094sub_062s_092_060_094sub_062u_092_060_094sub_062c_092_060_094sub_062c_Oelims,axiom,
! [X2: vEBT_VEBT,Xa: nat,Y3: nat] :
( ( ( vEBT_T_s_u_c_c @ X2 @ Xa )
= Y3 )
=> ( ( ? [Uu2: $o,B4: $o] :
( X2
= ( vEBT_Leaf @ Uu2 @ B4 ) )
=> ( ( Xa
= ( zero_zero @ nat ) )
=> ( Y3
!= ( plus_plus @ nat @ ( one_one @ nat ) @ ( one_one @ nat ) ) ) ) )
=> ( ( ? [Uv2: $o,Uw2: $o] :
( X2
= ( vEBT_Leaf @ Uv2 @ Uw2 ) )
=> ( ? [N3: nat] :
( Xa
= ( suc @ N3 ) )
=> ( Y3
!= ( one_one @ nat ) ) ) )
=> ( ( ? [Ux2: nat,Uy2: list @ vEBT_VEBT,Uz2: vEBT_VEBT] :
( X2
= ( vEBT_Node @ ( none @ ( product_prod @ nat @ nat ) ) @ Ux2 @ Uy2 @ Uz2 ) )
=> ( Y3
!= ( one_one @ nat ) ) )
=> ( ( ? [V3: product_prod @ nat @ nat,Vc2: list @ vEBT_VEBT,Vd2: vEBT_VEBT] :
( X2
= ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ V3 ) @ ( zero_zero @ nat ) @ Vc2 @ Vd2 ) )
=> ( Y3
!= ( one_one @ nat ) ) )
=> ( ( ? [V3: product_prod @ nat @ nat,Vg: list @ vEBT_VEBT,Vh: vEBT_VEBT] :
( X2
= ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ V3 ) @ ( suc @ ( zero_zero @ nat ) ) @ Vg @ Vh ) )
=> ( Y3
!= ( one_one @ nat ) ) )
=> ~ ! [Mi2: nat,Ma2: nat,Va3: nat,TreeList2: list @ vEBT_VEBT,Summary2: vEBT_VEBT] :
( ( X2
= ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( suc @ Va3 ) ) @ TreeList2 @ Summary2 ) )
=> ( Y3
!= ( plus_plus @ nat @ ( one_one @ nat )
@ ( if @ nat @ ( ord_less @ nat @ Xa @ Mi2 ) @ ( one_one @ nat )
@ ( plus_plus @ nat @ ( numeral_numeral @ nat @ ( bit0 @ ( bit1 @ ( bit0 @ one2 ) ) ) )
@ ( if @ nat @ ( ord_less @ nat @ ( vEBT_VEBT_high @ Xa @ ( divide_divide @ nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( size_size @ ( list @ vEBT_VEBT ) @ TreeList2 ) )
@ ( plus_plus @ nat @ ( plus_plus @ nat @ ( one_one @ nat ) @ ( vEBT_T_m_a_x_t @ ( nth @ vEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ Xa @ ( divide_divide @ nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) )
@ ( plus_plus @ nat @ ( numeral_numeral @ nat @ ( bit1 @ one2 ) )
@ ( if @ nat
@ ( ( ( vEBT_vebt_maxt @ ( nth @ vEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ Xa @ ( divide_divide @ nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) )
!= ( none @ nat ) )
& ( vEBT_VEBT_less @ ( some @ nat @ ( vEBT_VEBT_low @ Xa @ ( divide_divide @ nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( vEBT_vebt_maxt @ ( nth @ vEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ Xa @ ( divide_divide @ nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) ) )
@ ( plus_plus @ nat @ ( numeral_numeral @ nat @ ( bit0 @ ( bit0 @ one2 ) ) ) @ ( vEBT_T_s_u_c_c @ ( nth @ vEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ Xa @ ( divide_divide @ nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( vEBT_VEBT_low @ Xa @ ( divide_divide @ nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) )
@ ( plus_plus @ nat @ ( plus_plus @ nat @ ( plus_plus @ nat @ ( one_one @ nat ) @ ( vEBT_T_s_u_c_c @ Summary2 @ ( vEBT_VEBT_high @ Xa @ ( divide_divide @ nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) @ ( one_one @ nat ) )
@ ( if @ nat
@ ( ( vEBT_vebt_succ @ Summary2 @ ( vEBT_VEBT_high @ Xa @ ( divide_divide @ nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) )
= ( none @ nat ) )
@ ( one_one @ nat )
@ ( plus_plus @ nat @ ( numeral_numeral @ nat @ ( bit0 @ ( bit0 @ one2 ) ) ) @ ( vEBT_T_m_i_n_t @ ( nth @ vEBT_VEBT @ TreeList2 @ ( the2 @ nat @ ( vEBT_vebt_succ @ Summary2 @ ( vEBT_VEBT_high @ Xa @ ( divide_divide @ nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) ) ) ) ) ) ) ) )
@ ( one_one @ nat ) ) ) ) ) ) ) ) ) ) ) ) ) ).
% T\<^sub>s\<^sub>u\<^sub>c\<^sub>c.elims
thf(fact_2339_T_092_060_094sub_062p_092_060_094sub_062r_092_060_094sub_062e_092_060_094sub_062d_Oelims,axiom,
! [X2: vEBT_VEBT,Xa: nat,Y3: nat] :
( ( ( vEBT_T_p_r_e_d @ X2 @ Xa )
= Y3 )
=> ( ( ? [Uu2: $o,Uv2: $o] :
( X2
= ( vEBT_Leaf @ Uu2 @ Uv2 ) )
=> ( ( Xa
= ( zero_zero @ nat ) )
=> ( Y3
!= ( one_one @ nat ) ) ) )
=> ( ( ? [A4: $o,Uw2: $o] :
( X2
= ( vEBT_Leaf @ A4 @ Uw2 ) )
=> ( ( Xa
= ( suc @ ( zero_zero @ nat ) ) )
=> ( Y3
!= ( plus_plus @ nat @ ( one_one @ nat ) @ ( one_one @ nat ) ) ) ) )
=> ( ! [A4: $o,B4: $o] :
( ( X2
= ( vEBT_Leaf @ A4 @ B4 ) )
=> ( ? [Va3: nat] :
( Xa
= ( suc @ ( suc @ Va3 ) ) )
=> ( Y3
!= ( plus_plus @ nat @ ( one_one @ nat ) @ ( if @ nat @ B4 @ ( one_one @ nat ) @ ( plus_plus @ nat @ ( one_one @ nat ) @ ( one_one @ nat ) ) ) ) ) ) )
=> ( ( ? [Uy2: nat,Uz2: list @ vEBT_VEBT,Va2: vEBT_VEBT] :
( X2
= ( vEBT_Node @ ( none @ ( product_prod @ nat @ nat ) ) @ Uy2 @ Uz2 @ Va2 ) )
=> ( Y3
!= ( one_one @ nat ) ) )
=> ( ( ? [V3: product_prod @ nat @ nat,Vd2: list @ vEBT_VEBT,Ve2: vEBT_VEBT] :
( X2
= ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ V3 ) @ ( zero_zero @ nat ) @ Vd2 @ Ve2 ) )
=> ( Y3
!= ( one_one @ nat ) ) )
=> ( ( ? [V3: product_prod @ nat @ nat,Vh: list @ vEBT_VEBT,Vi: vEBT_VEBT] :
( X2
= ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ V3 ) @ ( suc @ ( zero_zero @ nat ) ) @ Vh @ Vi ) )
=> ( Y3
!= ( one_one @ nat ) ) )
=> ~ ! [Mi2: nat,Ma2: nat,Va3: nat,TreeList2: list @ vEBT_VEBT,Summary2: vEBT_VEBT] :
( ( X2
= ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( suc @ Va3 ) ) @ TreeList2 @ Summary2 ) )
=> ( Y3
!= ( plus_plus @ nat @ ( one_one @ nat )
@ ( if @ nat @ ( ord_less @ nat @ Ma2 @ Xa ) @ ( one_one @ nat )
@ ( plus_plus @ nat @ ( plus_plus @ nat @ ( numeral_numeral @ nat @ ( bit0 @ ( bit1 @ ( bit0 @ one2 ) ) ) ) @ ( one_one @ nat ) )
@ ( if @ nat @ ( ord_less @ nat @ ( vEBT_VEBT_high @ Xa @ ( divide_divide @ nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( size_size @ ( list @ vEBT_VEBT ) @ TreeList2 ) )
@ ( plus_plus @ nat @ ( plus_plus @ nat @ ( plus_plus @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( vEBT_T_m_i_n_t @ ( nth @ vEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ Xa @ ( divide_divide @ nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) ) @ ( numeral_numeral @ nat @ ( bit1 @ one2 ) ) )
@ ( if @ nat
@ ( ( ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ Xa @ ( divide_divide @ nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) )
!= ( none @ nat ) )
& ( vEBT_VEBT_greater @ ( some @ nat @ ( vEBT_VEBT_low @ Xa @ ( divide_divide @ nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ Xa @ ( divide_divide @ nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) ) )
@ ( plus_plus @ nat @ ( numeral_numeral @ nat @ ( bit0 @ ( bit0 @ one2 ) ) ) @ ( vEBT_T_p_r_e_d @ ( nth @ vEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ Xa @ ( divide_divide @ nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( vEBT_VEBT_low @ Xa @ ( divide_divide @ nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) )
@ ( plus_plus @ nat @ ( plus_plus @ nat @ ( plus_plus @ nat @ ( one_one @ nat ) @ ( vEBT_T_p_r_e_d @ Summary2 @ ( vEBT_VEBT_high @ Xa @ ( divide_divide @ nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) @ ( one_one @ nat ) )
@ ( if @ nat
@ ( ( vEBT_vebt_pred @ Summary2 @ ( vEBT_VEBT_high @ Xa @ ( divide_divide @ nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) )
= ( none @ nat ) )
@ ( plus_plus @ nat @ ( one_one @ nat ) @ ( one_one @ nat ) )
@ ( plus_plus @ nat @ ( numeral_numeral @ nat @ ( bit0 @ ( bit0 @ one2 ) ) ) @ ( vEBT_T_m_a_x_t @ ( nth @ vEBT_VEBT @ TreeList2 @ ( the2 @ nat @ ( vEBT_vebt_pred @ Summary2 @ ( vEBT_VEBT_high @ Xa @ ( divide_divide @ nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) ) ) ) ) ) ) )
@ ( one_one @ nat ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).
% T\<^sub>p\<^sub>r\<^sub>e\<^sub>d.elims
thf(fact_2340_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_2341_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_2342_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,
! [Uv: $o] :
( ( vEBT_T_m_i_n_N_u_l_l @ ( vEBT_Leaf @ $true @ Uv ) )
= ( one_one @ nat ) ) ).
% T\<^sub>m\<^sub>i\<^sub>n\<^sub>N\<^sub>u\<^sub>l\<^sub>l.simps(2)
thf(fact_2343_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,
! [Uu: $o] :
( ( vEBT_T_m_i_n_N_u_l_l @ ( vEBT_Leaf @ Uu @ $true ) )
= ( one_one @ nat ) ) ).
% T\<^sub>m\<^sub>i\<^sub>n\<^sub>N\<^sub>u\<^sub>l\<^sub>l.simps(3)
thf(fact_2344_T_092_060_094sub_062m_092_060_094sub_062a_092_060_094sub_062x_092_060_094sub_062t_Osimps_I1_J,axiom,
! [A3: $o,B3: $o] :
( ( vEBT_T_m_a_x_t @ ( vEBT_Leaf @ A3 @ B3 ) )
= ( plus_plus @ nat @ ( one_one @ nat ) @ ( if @ nat @ B3 @ ( 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_2345_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_2346_simp__from__to,axiom,
( ( set_or1337092689740270186AtMost @ int )
= ( ^ [I4: int,J3: int] : ( if @ ( set @ int ) @ ( ord_less @ int @ J3 @ I4 ) @ ( bot_bot @ ( set @ int ) ) @ ( insert @ int @ I4 @ ( set_or1337092689740270186AtMost @ int @ ( plus_plus @ int @ I4 @ ( one_one @ int ) ) @ J3 ) ) ) ) ) ).
% simp_from_to
thf(fact_2347_T_092_060_094sub_062m_092_060_094sub_062a_092_060_094sub_062x_092_060_094sub_062t_Osimps_I2_J,axiom,
! [Uu: nat,Uv: list @ vEBT_VEBT,Uw: vEBT_VEBT] :
( ( vEBT_T_m_a_x_t @ ( vEBT_Node @ ( none @ ( product_prod @ nat @ nat ) ) @ Uu @ Uv @ Uw ) )
= ( one_one @ nat ) ) ).
% T\<^sub>m\<^sub>a\<^sub>x\<^sub>t.simps(2)
thf(fact_2348_T_092_060_094sub_062m_092_060_094sub_062i_092_060_094sub_062n_092_060_094sub_062t_Osimps_I2_J,axiom,
! [Uu: nat,Uv: list @ vEBT_VEBT,Uw: vEBT_VEBT] :
( ( vEBT_T_m_i_n_t @ ( vEBT_Node @ ( none @ ( product_prod @ nat @ nat ) ) @ Uu @ Uv @ Uw ) )
= ( one_one @ nat ) ) ).
% T\<^sub>m\<^sub>i\<^sub>n\<^sub>t.simps(2)
thf(fact_2349_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,Va: 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 ) @ Va @ Vb @ Vc ) )
= ( one_one @ nat ) ) ).
% T\<^sub>m\<^sub>i\<^sub>n\<^sub>N\<^sub>u\<^sub>l\<^sub>l.simps(5)
thf(fact_2350_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,
! [Uw: nat,Ux: list @ vEBT_VEBT,Uy: vEBT_VEBT] :
( ( vEBT_T_m_i_n_N_u_l_l @ ( vEBT_Node @ ( none @ ( product_prod @ nat @ nat ) ) @ Uw @ Ux @ Uy ) )
= ( one_one @ nat ) ) ).
% T\<^sub>m\<^sub>i\<^sub>n\<^sub>N\<^sub>u\<^sub>l\<^sub>l.simps(4)
thf(fact_2351_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_2352_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,Ux: nat,Uy: 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 ) ) @ Ux @ Uy @ Uz ) )
= ( one_one @ nat ) ) ).
% T\<^sub>m\<^sub>a\<^sub>x\<^sub>t.simps(3)
thf(fact_2353_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_2354_T_092_060_094sub_062m_092_060_094sub_062i_092_060_094sub_062n_092_060_094sub_062t_Osimps_I1_J,axiom,
! [A3: $o,B3: $o] :
( ( vEBT_T_m_i_n_t @ ( vEBT_Leaf @ A3 @ B3 ) )
= ( plus_plus @ nat @ ( one_one @ nat ) @ ( if @ nat @ A3 @ ( 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_2355_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,Ux: nat,Uy: 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 ) ) @ Ux @ Uy @ Uz ) )
= ( one_one @ nat ) ) ).
% T\<^sub>m\<^sub>i\<^sub>n\<^sub>t.simps(3)
thf(fact_2356_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,
! [X2: vEBT_VEBT,Y3: nat] :
( ( ( vEBT_T_m_i_n_N_u_l_l @ X2 )
= Y3 )
=> ( ( ( X2
= ( vEBT_Leaf @ $false @ $false ) )
=> ( Y3
!= ( one_one @ nat ) ) )
=> ( ( ? [Uv2: $o] :
( X2
= ( vEBT_Leaf @ $true @ Uv2 ) )
=> ( Y3
!= ( one_one @ nat ) ) )
=> ( ( ? [Uu2: $o] :
( X2
= ( vEBT_Leaf @ Uu2 @ $true ) )
=> ( Y3
!= ( one_one @ nat ) ) )
=> ( ( ? [Uw2: nat,Ux2: list @ vEBT_VEBT,Uy2: vEBT_VEBT] :
( X2
= ( vEBT_Node @ ( none @ ( product_prod @ nat @ nat ) ) @ Uw2 @ Ux2 @ Uy2 ) )
=> ( Y3
!= ( one_one @ nat ) ) )
=> ~ ( ? [Uz2: product_prod @ nat @ nat,Va2: nat,Vb2: list @ vEBT_VEBT,Vc2: vEBT_VEBT] :
( X2
= ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ Uz2 ) @ Va2 @ Vb2 @ Vc2 ) )
=> ( Y3
!= ( one_one @ nat ) ) ) ) ) ) ) ) ).
% T\<^sub>m\<^sub>i\<^sub>n\<^sub>N\<^sub>u\<^sub>l\<^sub>l.elims
thf(fact_2357_T_092_060_094sub_062m_092_060_094sub_062a_092_060_094sub_062x_092_060_094sub_062t_Oelims,axiom,
! [X2: vEBT_VEBT,Y3: nat] :
( ( ( vEBT_T_m_a_x_t @ X2 )
= Y3 )
=> ( ! [A4: $o,B4: $o] :
( ( X2
= ( vEBT_Leaf @ A4 @ B4 ) )
=> ( Y3
!= ( plus_plus @ nat @ ( one_one @ nat ) @ ( if @ nat @ B4 @ ( one_one @ nat ) @ ( plus_plus @ nat @ ( one_one @ nat ) @ ( one_one @ nat ) ) ) ) ) )
=> ( ( ? [Uu2: nat,Uv2: list @ vEBT_VEBT,Uw2: vEBT_VEBT] :
( X2
= ( vEBT_Node @ ( none @ ( product_prod @ nat @ nat ) ) @ Uu2 @ Uv2 @ Uw2 ) )
=> ( Y3
!= ( one_one @ nat ) ) )
=> ~ ( ? [Mi2: nat,Ma2: nat,Ux2: nat,Uy2: list @ vEBT_VEBT,Uz2: vEBT_VEBT] :
( X2
= ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi2 @ Ma2 ) ) @ Ux2 @ Uy2 @ Uz2 ) )
=> ( Y3
!= ( one_one @ nat ) ) ) ) ) ) ).
% T\<^sub>m\<^sub>a\<^sub>x\<^sub>t.elims
thf(fact_2358_T_092_060_094sub_062m_092_060_094sub_062i_092_060_094sub_062n_092_060_094sub_062t_Oelims,axiom,
! [X2: vEBT_VEBT,Y3: nat] :
( ( ( vEBT_T_m_i_n_t @ X2 )
= Y3 )
=> ( ! [A4: $o] :
( ? [B4: $o] :
( X2
= ( vEBT_Leaf @ A4 @ B4 ) )
=> ( Y3
!= ( plus_plus @ nat @ ( one_one @ nat ) @ ( if @ nat @ A4 @ ( zero_zero @ nat ) @ ( plus_plus @ nat @ ( one_one @ nat ) @ ( one_one @ nat ) ) ) ) ) )
=> ( ( ? [Uu2: nat,Uv2: list @ vEBT_VEBT,Uw2: vEBT_VEBT] :
( X2
= ( vEBT_Node @ ( none @ ( product_prod @ nat @ nat ) ) @ Uu2 @ Uv2 @ Uw2 ) )
=> ( Y3
!= ( one_one @ nat ) ) )
=> ~ ( ? [Mi2: nat,Ma2: nat,Ux2: nat,Uy2: list @ vEBT_VEBT,Uz2: vEBT_VEBT] :
( X2
= ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi2 @ Ma2 ) ) @ Ux2 @ Uy2 @ Uz2 ) )
=> ( Y3
!= ( one_one @ nat ) ) ) ) ) ) ).
% T\<^sub>m\<^sub>i\<^sub>n\<^sub>t.elims
thf(fact_2359_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,Va: nat,TreeList: list @ vEBT_VEBT,Summary: vEBT_VEBT,X2: nat] :
( ( vEBT_T_d_e_l_e_t_e @ ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi @ Ma ) ) @ ( suc @ ( suc @ Va ) ) @ TreeList @ Summary ) @ X2 )
= ( plus_plus @ nat @ ( numeral_numeral @ nat @ ( bit1 @ one2 ) )
@ ( if @ nat
@ ( ( ord_less @ nat @ X2 @ Mi )
| ( ord_less @ nat @ Ma @ X2 ) )
@ ( one_one @ nat )
@ ( plus_plus @ nat @ ( numeral_numeral @ nat @ ( bit1 @ one2 ) )
@ ( if @ nat
@ ( ( X2 = Mi )
& ( X2 = Ma ) )
@ ( numeral_numeral @ nat @ ( bit1 @ one2 ) )
@ ( plus_plus @ nat @ ( plus_plus @ nat @ ( plus_plus @ nat @ ( numeral_numeral @ nat @ ( bit1 @ ( bit0 @ ( bit1 @ one2 ) ) ) ) @ ( if @ nat @ ( X2 = 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 @ ( X2 = Mi ) @ ( plus_plus @ nat @ ( times_times @ nat @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ X2 ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( 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 @ ( X2 = Mi ) @ ( plus_plus @ nat @ ( times_times @ nat @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ X2 ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( vEBT_VEBT_low @ ( if @ nat @ ( X2 = Mi ) @ ( plus_plus @ nat @ ( times_times @ nat @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ X2 ) @ ( 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 @ TreeList @ ( vEBT_VEBT_high @ ( if @ nat @ ( X2 = Mi ) @ ( plus_plus @ nat @ ( times_times @ nat @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ X2 ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( vEBT_VEBT_low @ ( if @ nat @ ( X2 = Mi ) @ ( plus_plus @ nat @ ( times_times @ nat @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ X2 ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) )
@ ( if @ nat @ ( vEBT_VEBT_minNull @ ( vEBT_vebt_delete @ ( nth @ vEBT_VEBT @ TreeList @ ( vEBT_VEBT_high @ ( if @ nat @ ( X2 = Mi ) @ ( plus_plus @ nat @ ( times_times @ nat @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ X2 ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( vEBT_VEBT_low @ ( if @ nat @ ( X2 = Mi ) @ ( plus_plus @ nat @ ( times_times @ nat @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ X2 ) @ ( 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 @ Summary @ ( vEBT_VEBT_high @ ( if @ nat @ ( X2 = Mi ) @ ( plus_plus @ nat @ ( times_times @ nat @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ X2 ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) )
@ ( plus_plus @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) )
@ ( if @ nat
@ ( ( ( X2 = Mi )
=> ( ( plus_plus @ nat @ ( times_times @ nat @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) )
= Ma ) )
& ( ( X2 != Mi )
=> ( X2 = 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 @ ( X2 = Mi ) @ ( plus_plus @ nat @ ( times_times @ nat @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ X2 ) @ ( 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 @ Summary @ ( vEBT_VEBT_high @ ( if @ nat @ ( X2 = Mi ) @ ( plus_plus @ nat @ ( times_times @ nat @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ X2 ) @ ( 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 @ TreeList @ ( vEBT_VEBT_high @ ( if @ nat @ ( X2 = Mi ) @ ( plus_plus @ nat @ ( times_times @ nat @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ X2 ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( vEBT_vebt_delete @ ( nth @ vEBT_VEBT @ TreeList @ ( vEBT_VEBT_high @ ( if @ nat @ ( X2 = Mi ) @ ( plus_plus @ nat @ ( times_times @ nat @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ X2 ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( vEBT_VEBT_low @ ( if @ nat @ ( X2 = Mi ) @ ( plus_plus @ nat @ ( times_times @ nat @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ X2 ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_maxt @ ( vEBT_vebt_delete @ Summary @ ( vEBT_VEBT_high @ ( if @ nat @ ( X2 = Mi ) @ ( plus_plus @ nat @ ( times_times @ nat @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ X2 ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) ) ) ) ) ) ) )
@ ( one_one @ nat ) ) ) )
@ ( plus_plus @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) )
@ ( if @ nat
@ ( ( ( X2 = Mi )
=> ( ( plus_plus @ nat @ ( times_times @ nat @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) )
= Ma ) )
& ( ( X2 != Mi )
=> ( X2 = 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 @ ( X2 = Mi ) @ ( plus_plus @ nat @ ( times_times @ nat @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ X2 ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( vEBT_vebt_delete @ ( nth @ vEBT_VEBT @ TreeList @ ( vEBT_VEBT_high @ ( if @ nat @ ( X2 = Mi ) @ ( plus_plus @ nat @ ( times_times @ nat @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ X2 ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( vEBT_VEBT_low @ ( if @ nat @ ( X2 = Mi ) @ ( plus_plus @ nat @ ( times_times @ nat @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ X2 ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) @ ( vEBT_VEBT_high @ ( if @ nat @ ( X2 = Mi ) @ ( plus_plus @ nat @ ( times_times @ nat @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ X2 ) @ ( 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.simps(7)
thf(fact_2360_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,
! [X2: vEBT_VEBT,Xa: nat,Y3: nat] :
( ( ( vEBT_T_d_e_l_e_t_e @ X2 @ Xa )
= Y3 )
=> ( ( ? [A4: $o,B4: $o] :
( X2
= ( vEBT_Leaf @ A4 @ B4 ) )
=> ( ( Xa
= ( zero_zero @ nat ) )
=> ( Y3
!= ( one_one @ nat ) ) ) )
=> ( ( ? [A4: $o,B4: $o] :
( X2
= ( vEBT_Leaf @ A4 @ B4 ) )
=> ( ( Xa
= ( suc @ ( zero_zero @ nat ) ) )
=> ( Y3
!= ( one_one @ nat ) ) ) )
=> ( ( ? [A4: $o,B4: $o] :
( X2
= ( vEBT_Leaf @ A4 @ B4 ) )
=> ( ? [N3: nat] :
( Xa
= ( suc @ ( suc @ N3 ) ) )
=> ( Y3
!= ( one_one @ nat ) ) ) )
=> ( ( ? [Deg2: nat,TreeList2: list @ vEBT_VEBT,Summary2: vEBT_VEBT] :
( X2
= ( vEBT_Node @ ( none @ ( product_prod @ nat @ nat ) ) @ Deg2 @ TreeList2 @ Summary2 ) )
=> ( Y3
!= ( one_one @ nat ) ) )
=> ( ( ? [Mi2: nat,Ma2: nat,TreeList2: list @ vEBT_VEBT,Summary2: vEBT_VEBT] :
( X2
= ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi2 @ Ma2 ) ) @ ( zero_zero @ nat ) @ TreeList2 @ Summary2 ) )
=> ( Y3
!= ( one_one @ nat ) ) )
=> ( ( ? [Mi2: nat,Ma2: nat,TreeList2: list @ vEBT_VEBT,Summary2: vEBT_VEBT] :
( X2
= ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( zero_zero @ nat ) ) @ TreeList2 @ Summary2 ) )
=> ( Y3
!= ( one_one @ nat ) ) )
=> ~ ! [Mi2: nat,Ma2: nat,Va3: nat,TreeList2: list @ vEBT_VEBT,Summary2: vEBT_VEBT] :
( ( X2
= ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( suc @ Va3 ) ) @ TreeList2 @ Summary2 ) )
=> ( Y3
!= ( plus_plus @ nat @ ( numeral_numeral @ nat @ ( bit1 @ one2 ) )
@ ( if @ nat
@ ( ( ord_less @ nat @ Xa @ Mi2 )
| ( ord_less @ nat @ Ma2 @ Xa ) )
@ ( one_one @ nat )
@ ( plus_plus @ nat @ ( numeral_numeral @ nat @ ( bit1 @ one2 ) )
@ ( if @ nat
@ ( ( Xa = Mi2 )
& ( Xa = Ma2 ) )
@ ( numeral_numeral @ nat @ ( bit1 @ one2 ) )
@ ( plus_plus @ nat @ ( plus_plus @ nat @ ( plus_plus @ nat @ ( numeral_numeral @ nat @ ( bit1 @ ( bit0 @ ( bit1 @ one2 ) ) ) ) @ ( if @ nat @ ( Xa = Mi2 ) @ ( plus_plus @ nat @ ( plus_plus @ nat @ ( vEBT_T_m_i_n_t @ Summary2 ) @ ( vEBT_T_m_i_n_t @ ( nth @ vEBT_VEBT @ TreeList2 @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary2 ) ) ) ) ) @ ( numeral_numeral @ nat @ ( bit1 @ ( bit1 @ one2 ) ) ) ) @ ( one_one @ nat ) ) ) @ ( one_one @ nat ) )
@ ( if @ nat @ ( ord_less @ nat @ ( vEBT_VEBT_high @ ( if @ nat @ ( Xa = Mi2 ) @ ( plus_plus @ nat @ ( times_times @ nat @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary2 ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList2 @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary2 ) ) ) ) ) ) @ Xa ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( size_size @ ( list @ vEBT_VEBT ) @ TreeList2 ) )
@ ( plus_plus @ nat @ ( plus_plus @ nat @ ( numeral_numeral @ nat @ ( bit0 @ ( bit0 @ one2 ) ) ) @ ( vEBT_T_d_e_l_e_t_e @ ( nth @ vEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ ( if @ nat @ ( Xa = Mi2 ) @ ( plus_plus @ nat @ ( times_times @ nat @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary2 ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList2 @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary2 ) ) ) ) ) ) @ Xa ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( vEBT_VEBT_low @ ( if @ nat @ ( Xa = Mi2 ) @ ( plus_plus @ nat @ ( times_times @ nat @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary2 ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList2 @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary2 ) ) ) ) ) ) @ Xa ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) )
@ ( plus_plus @ nat @ ( plus_plus @ nat @ ( one_one @ nat ) @ ( vEBT_T_m_i_n_N_u_l_l @ ( vEBT_vebt_delete @ ( nth @ vEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ ( if @ nat @ ( Xa = Mi2 ) @ ( plus_plus @ nat @ ( times_times @ nat @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary2 ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList2 @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary2 ) ) ) ) ) ) @ Xa ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( vEBT_VEBT_low @ ( if @ nat @ ( Xa = Mi2 ) @ ( plus_plus @ nat @ ( times_times @ nat @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary2 ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList2 @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary2 ) ) ) ) ) ) @ Xa ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) )
@ ( if @ nat @ ( vEBT_VEBT_minNull @ ( vEBT_vebt_delete @ ( nth @ vEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ ( if @ nat @ ( Xa = Mi2 ) @ ( plus_plus @ nat @ ( times_times @ nat @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary2 ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList2 @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary2 ) ) ) ) ) ) @ Xa ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( vEBT_VEBT_low @ ( if @ nat @ ( Xa = Mi2 ) @ ( plus_plus @ nat @ ( times_times @ nat @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary2 ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList2 @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary2 ) ) ) ) ) ) @ Xa ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) )
@ ( plus_plus @ nat @ ( plus_plus @ nat @ ( one_one @ nat ) @ ( vEBT_T_d_e_l_e_t_e @ Summary2 @ ( vEBT_VEBT_high @ ( if @ nat @ ( Xa = Mi2 ) @ ( plus_plus @ nat @ ( times_times @ nat @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary2 ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList2 @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary2 ) ) ) ) ) ) @ Xa ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) )
@ ( plus_plus @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) )
@ ( if @ nat
@ ( ( ( Xa = Mi2 )
=> ( ( plus_plus @ nat @ ( times_times @ nat @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary2 ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList2 @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary2 ) ) ) ) ) )
= Ma2 ) )
& ( ( Xa != Mi2 )
=> ( Xa = Ma2 ) ) )
@ ( plus_plus @ nat @ ( plus_plus @ nat @ ( one_one @ nat ) @ ( vEBT_T_m_a_x_t @ ( vEBT_vebt_delete @ Summary2 @ ( vEBT_VEBT_high @ ( if @ nat @ ( Xa = Mi2 ) @ ( plus_plus @ nat @ ( times_times @ nat @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary2 ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList2 @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary2 ) ) ) ) ) ) @ Xa ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) )
@ ( plus_plus @ nat @ ( one_one @ nat )
@ ( if @ nat
@ ( ( vEBT_vebt_maxt @ ( vEBT_vebt_delete @ Summary2 @ ( vEBT_VEBT_high @ ( if @ nat @ ( Xa = Mi2 ) @ ( plus_plus @ nat @ ( times_times @ nat @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary2 ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList2 @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary2 ) ) ) ) ) ) @ Xa ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) )
= ( none @ nat ) )
@ ( one_one @ nat )
@ ( plus_plus @ nat @ ( numeral_numeral @ nat @ ( bit0 @ ( bit0 @ ( bit0 @ one2 ) ) ) ) @ ( vEBT_T_m_a_x_t @ ( nth @ vEBT_VEBT @ ( list_update @ vEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ ( if @ nat @ ( Xa = Mi2 ) @ ( plus_plus @ nat @ ( times_times @ nat @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary2 ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList2 @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary2 ) ) ) ) ) ) @ Xa ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( vEBT_vebt_delete @ ( nth @ vEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ ( if @ nat @ ( Xa = Mi2 ) @ ( plus_plus @ nat @ ( times_times @ nat @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary2 ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList2 @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary2 ) ) ) ) ) ) @ Xa ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( vEBT_VEBT_low @ ( if @ nat @ ( Xa = Mi2 ) @ ( plus_plus @ nat @ ( times_times @ nat @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary2 ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList2 @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary2 ) ) ) ) ) ) @ Xa ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_maxt @ ( vEBT_vebt_delete @ Summary2 @ ( vEBT_VEBT_high @ ( if @ nat @ ( Xa = Mi2 ) @ ( plus_plus @ nat @ ( times_times @ nat @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary2 ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList2 @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary2 ) ) ) ) ) ) @ Xa ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) ) ) ) ) ) ) )
@ ( one_one @ nat ) ) ) )
@ ( plus_plus @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) )
@ ( if @ nat
@ ( ( ( Xa = Mi2 )
=> ( ( plus_plus @ nat @ ( times_times @ nat @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary2 ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList2 @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary2 ) ) ) ) ) )
= Ma2 ) )
& ( ( Xa != Mi2 )
=> ( Xa = Ma2 ) ) )
@ ( plus_plus @ nat @ ( numeral_numeral @ nat @ ( bit0 @ ( bit1 @ one2 ) ) ) @ ( vEBT_T_m_a_x_t @ ( nth @ vEBT_VEBT @ ( list_update @ vEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ ( if @ nat @ ( Xa = Mi2 ) @ ( plus_plus @ nat @ ( times_times @ nat @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary2 ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList2 @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary2 ) ) ) ) ) ) @ Xa ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( vEBT_vebt_delete @ ( nth @ vEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ ( if @ nat @ ( Xa = Mi2 ) @ ( plus_plus @ nat @ ( times_times @ nat @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary2 ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList2 @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary2 ) ) ) ) ) ) @ Xa ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( vEBT_VEBT_low @ ( if @ nat @ ( Xa = Mi2 ) @ ( plus_plus @ nat @ ( times_times @ nat @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary2 ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList2 @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary2 ) ) ) ) ) ) @ Xa ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) @ ( vEBT_VEBT_high @ ( if @ nat @ ( Xa = Mi2 ) @ ( plus_plus @ nat @ ( times_times @ nat @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary2 ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList2 @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary2 ) ) ) ) ) ) @ Xa ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va3 ) ) @ ( 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_2361_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,Va: nat,TreeList: list @ vEBT_VEBT,Summary: vEBT_VEBT,X2: nat] :
( ( vEBT_T_p_r_e_d @ ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi @ Ma ) ) @ ( suc @ ( suc @ Va ) ) @ TreeList @ Summary ) @ X2 )
= ( plus_plus @ nat @ ( one_one @ nat )
@ ( if @ nat @ ( ord_less @ nat @ Ma @ X2 ) @ ( 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 @ X2 @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( 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 @ X2 @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) ) @ ( numeral_numeral @ nat @ ( bit1 @ one2 ) ) )
@ ( if @ nat
@ ( ( ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList @ ( vEBT_VEBT_high @ X2 @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) )
!= ( none @ nat ) )
& ( vEBT_VEBT_greater @ ( some @ nat @ ( vEBT_VEBT_low @ X2 @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList @ ( vEBT_VEBT_high @ X2 @ ( 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 @ TreeList @ ( vEBT_VEBT_high @ X2 @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( vEBT_VEBT_low @ X2 @ ( 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 @ Summary @ ( vEBT_VEBT_high @ X2 @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) @ ( one_one @ nat ) )
@ ( if @ nat
@ ( ( vEBT_vebt_pred @ Summary @ ( vEBT_VEBT_high @ X2 @ ( 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 @ TreeList @ ( the2 @ nat @ ( vEBT_vebt_pred @ Summary @ ( vEBT_VEBT_high @ X2 @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) ) ) ) ) ) ) )
@ ( one_one @ nat ) ) ) ) ) ) ).
% T\<^sub>p\<^sub>r\<^sub>e\<^sub>d.simps(7)
thf(fact_2362_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,Va: nat,TreeList: list @ vEBT_VEBT,Summary: vEBT_VEBT,X2: nat] :
( ( vEBT_T_s_u_c_c @ ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi @ Ma ) ) @ ( suc @ ( suc @ Va ) ) @ TreeList @ Summary ) @ X2 )
= ( plus_plus @ nat @ ( one_one @ nat )
@ ( if @ nat @ ( ord_less @ nat @ X2 @ Mi ) @ ( one_one @ nat )
@ ( plus_plus @ nat @ ( numeral_numeral @ nat @ ( bit0 @ ( bit1 @ ( bit0 @ one2 ) ) ) )
@ ( if @ nat @ ( ord_less @ nat @ ( vEBT_VEBT_high @ X2 @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( 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 @ X2 @ ( 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 @ TreeList @ ( vEBT_VEBT_high @ X2 @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) )
!= ( none @ nat ) )
& ( vEBT_VEBT_less @ ( some @ nat @ ( vEBT_VEBT_low @ X2 @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( vEBT_vebt_maxt @ ( nth @ vEBT_VEBT @ TreeList @ ( vEBT_VEBT_high @ X2 @ ( 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 @ TreeList @ ( vEBT_VEBT_high @ X2 @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( vEBT_VEBT_low @ X2 @ ( 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 @ Summary @ ( vEBT_VEBT_high @ X2 @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) @ ( one_one @ nat ) )
@ ( if @ nat
@ ( ( vEBT_vebt_succ @ Summary @ ( vEBT_VEBT_high @ X2 @ ( 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 @ TreeList @ ( the2 @ nat @ ( vEBT_vebt_succ @ Summary @ ( vEBT_VEBT_high @ X2 @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) ) ) ) ) ) ) ) )
@ ( one_one @ nat ) ) ) ) ) ) ).
% T\<^sub>s\<^sub>u\<^sub>c\<^sub>c.simps(6)
thf(fact_2363_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,
! [X2: vEBT_VEBT,Xa: nat,Y3: nat] :
( ( ( vEBT_T_d_e_l_e_t_e @ X2 @ Xa )
= Y3 )
=> ( ( accp @ ( product_prod @ vEBT_VEBT @ nat ) @ vEBT_T8441311223069195367_e_rel @ ( product_Pair @ vEBT_VEBT @ nat @ X2 @ Xa ) )
=> ( ! [A4: $o,B4: $o] :
( ( X2
= ( vEBT_Leaf @ A4 @ B4 ) )
=> ( ( Xa
= ( zero_zero @ nat ) )
=> ( ( Y3
= ( one_one @ nat ) )
=> ~ ( accp @ ( product_prod @ vEBT_VEBT @ nat ) @ vEBT_T8441311223069195367_e_rel @ ( product_Pair @ vEBT_VEBT @ nat @ ( vEBT_Leaf @ A4 @ B4 ) @ ( zero_zero @ nat ) ) ) ) ) )
=> ( ! [A4: $o,B4: $o] :
( ( X2
= ( vEBT_Leaf @ A4 @ B4 ) )
=> ( ( Xa
= ( suc @ ( zero_zero @ nat ) ) )
=> ( ( Y3
= ( one_one @ nat ) )
=> ~ ( accp @ ( product_prod @ vEBT_VEBT @ nat ) @ vEBT_T8441311223069195367_e_rel @ ( product_Pair @ vEBT_VEBT @ nat @ ( vEBT_Leaf @ A4 @ B4 ) @ ( suc @ ( zero_zero @ nat ) ) ) ) ) ) )
=> ( ! [A4: $o,B4: $o] :
( ( X2
= ( vEBT_Leaf @ A4 @ B4 ) )
=> ! [N3: nat] :
( ( Xa
= ( suc @ ( suc @ N3 ) ) )
=> ( ( Y3
= ( one_one @ nat ) )
=> ~ ( accp @ ( product_prod @ vEBT_VEBT @ nat ) @ vEBT_T8441311223069195367_e_rel @ ( product_Pair @ vEBT_VEBT @ nat @ ( vEBT_Leaf @ A4 @ B4 ) @ ( suc @ ( suc @ N3 ) ) ) ) ) ) )
=> ( ! [Deg2: nat,TreeList2: list @ vEBT_VEBT,Summary2: vEBT_VEBT] :
( ( X2
= ( vEBT_Node @ ( none @ ( product_prod @ nat @ nat ) ) @ Deg2 @ TreeList2 @ Summary2 ) )
=> ( ( Y3
= ( one_one @ nat ) )
=> ~ ( accp @ ( product_prod @ vEBT_VEBT @ nat ) @ vEBT_T8441311223069195367_e_rel @ ( product_Pair @ vEBT_VEBT @ nat @ ( vEBT_Node @ ( none @ ( product_prod @ nat @ nat ) ) @ Deg2 @ TreeList2 @ Summary2 ) @ Xa ) ) ) )
=> ( ! [Mi2: nat,Ma2: nat,TreeList2: list @ vEBT_VEBT,Summary2: vEBT_VEBT] :
( ( X2
= ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi2 @ Ma2 ) ) @ ( zero_zero @ nat ) @ TreeList2 @ Summary2 ) )
=> ( ( Y3
= ( one_one @ nat ) )
=> ~ ( accp @ ( product_prod @ vEBT_VEBT @ nat ) @ vEBT_T8441311223069195367_e_rel @ ( product_Pair @ vEBT_VEBT @ nat @ ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi2 @ Ma2 ) ) @ ( zero_zero @ nat ) @ TreeList2 @ Summary2 ) @ Xa ) ) ) )
=> ( ! [Mi2: nat,Ma2: nat,TreeList2: list @ vEBT_VEBT,Summary2: vEBT_VEBT] :
( ( X2
= ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( zero_zero @ nat ) ) @ TreeList2 @ Summary2 ) )
=> ( ( Y3
= ( one_one @ nat ) )
=> ~ ( accp @ ( product_prod @ vEBT_VEBT @ nat ) @ vEBT_T8441311223069195367_e_rel @ ( product_Pair @ vEBT_VEBT @ nat @ ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( zero_zero @ nat ) ) @ TreeList2 @ Summary2 ) @ Xa ) ) ) )
=> ~ ! [Mi2: nat,Ma2: nat,Va3: nat,TreeList2: list @ vEBT_VEBT,Summary2: vEBT_VEBT] :
( ( X2
= ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( suc @ Va3 ) ) @ TreeList2 @ Summary2 ) )
=> ( ( Y3
= ( plus_plus @ nat @ ( numeral_numeral @ nat @ ( bit1 @ one2 ) )
@ ( if @ nat
@ ( ( ord_less @ nat @ Xa @ Mi2 )
| ( ord_less @ nat @ Ma2 @ Xa ) )
@ ( one_one @ nat )
@ ( plus_plus @ nat @ ( numeral_numeral @ nat @ ( bit1 @ one2 ) )
@ ( if @ nat
@ ( ( Xa = Mi2 )
& ( Xa = Ma2 ) )
@ ( numeral_numeral @ nat @ ( bit1 @ one2 ) )
@ ( plus_plus @ nat @ ( plus_plus @ nat @ ( plus_plus @ nat @ ( numeral_numeral @ nat @ ( bit1 @ ( bit0 @ ( bit1 @ one2 ) ) ) ) @ ( if @ nat @ ( Xa = Mi2 ) @ ( plus_plus @ nat @ ( plus_plus @ nat @ ( vEBT_T_m_i_n_t @ Summary2 ) @ ( vEBT_T_m_i_n_t @ ( nth @ vEBT_VEBT @ TreeList2 @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary2 ) ) ) ) ) @ ( numeral_numeral @ nat @ ( bit1 @ ( bit1 @ one2 ) ) ) ) @ ( one_one @ nat ) ) ) @ ( one_one @ nat ) )
@ ( if @ nat @ ( ord_less @ nat @ ( vEBT_VEBT_high @ ( if @ nat @ ( Xa = Mi2 ) @ ( plus_plus @ nat @ ( times_times @ nat @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary2 ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList2 @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary2 ) ) ) ) ) ) @ Xa ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( size_size @ ( list @ vEBT_VEBT ) @ TreeList2 ) )
@ ( plus_plus @ nat @ ( plus_plus @ nat @ ( numeral_numeral @ nat @ ( bit0 @ ( bit0 @ one2 ) ) ) @ ( vEBT_T_d_e_l_e_t_e @ ( nth @ vEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ ( if @ nat @ ( Xa = Mi2 ) @ ( plus_plus @ nat @ ( times_times @ nat @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary2 ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList2 @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary2 ) ) ) ) ) ) @ Xa ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( vEBT_VEBT_low @ ( if @ nat @ ( Xa = Mi2 ) @ ( plus_plus @ nat @ ( times_times @ nat @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary2 ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList2 @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary2 ) ) ) ) ) ) @ Xa ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) )
@ ( plus_plus @ nat @ ( plus_plus @ nat @ ( one_one @ nat ) @ ( vEBT_T_m_i_n_N_u_l_l @ ( vEBT_vebt_delete @ ( nth @ vEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ ( if @ nat @ ( Xa = Mi2 ) @ ( plus_plus @ nat @ ( times_times @ nat @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary2 ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList2 @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary2 ) ) ) ) ) ) @ Xa ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( vEBT_VEBT_low @ ( if @ nat @ ( Xa = Mi2 ) @ ( plus_plus @ nat @ ( times_times @ nat @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary2 ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList2 @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary2 ) ) ) ) ) ) @ Xa ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) )
@ ( if @ nat @ ( vEBT_VEBT_minNull @ ( vEBT_vebt_delete @ ( nth @ vEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ ( if @ nat @ ( Xa = Mi2 ) @ ( plus_plus @ nat @ ( times_times @ nat @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary2 ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList2 @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary2 ) ) ) ) ) ) @ Xa ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( vEBT_VEBT_low @ ( if @ nat @ ( Xa = Mi2 ) @ ( plus_plus @ nat @ ( times_times @ nat @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary2 ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList2 @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary2 ) ) ) ) ) ) @ Xa ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) )
@ ( plus_plus @ nat @ ( plus_plus @ nat @ ( one_one @ nat ) @ ( vEBT_T_d_e_l_e_t_e @ Summary2 @ ( vEBT_VEBT_high @ ( if @ nat @ ( Xa = Mi2 ) @ ( plus_plus @ nat @ ( times_times @ nat @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary2 ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList2 @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary2 ) ) ) ) ) ) @ Xa ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) )
@ ( plus_plus @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) )
@ ( if @ nat
@ ( ( ( Xa = Mi2 )
=> ( ( plus_plus @ nat @ ( times_times @ nat @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary2 ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList2 @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary2 ) ) ) ) ) )
= Ma2 ) )
& ( ( Xa != Mi2 )
=> ( Xa = Ma2 ) ) )
@ ( plus_plus @ nat @ ( plus_plus @ nat @ ( one_one @ nat ) @ ( vEBT_T_m_a_x_t @ ( vEBT_vebt_delete @ Summary2 @ ( vEBT_VEBT_high @ ( if @ nat @ ( Xa = Mi2 ) @ ( plus_plus @ nat @ ( times_times @ nat @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary2 ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList2 @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary2 ) ) ) ) ) ) @ Xa ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) )
@ ( plus_plus @ nat @ ( one_one @ nat )
@ ( if @ nat
@ ( ( vEBT_vebt_maxt @ ( vEBT_vebt_delete @ Summary2 @ ( vEBT_VEBT_high @ ( if @ nat @ ( Xa = Mi2 ) @ ( plus_plus @ nat @ ( times_times @ nat @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary2 ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList2 @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary2 ) ) ) ) ) ) @ Xa ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) )
= ( none @ nat ) )
@ ( one_one @ nat )
@ ( plus_plus @ nat @ ( numeral_numeral @ nat @ ( bit0 @ ( bit0 @ ( bit0 @ one2 ) ) ) ) @ ( vEBT_T_m_a_x_t @ ( nth @ vEBT_VEBT @ ( list_update @ vEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ ( if @ nat @ ( Xa = Mi2 ) @ ( plus_plus @ nat @ ( times_times @ nat @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary2 ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList2 @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary2 ) ) ) ) ) ) @ Xa ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( vEBT_vebt_delete @ ( nth @ vEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ ( if @ nat @ ( Xa = Mi2 ) @ ( plus_plus @ nat @ ( times_times @ nat @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary2 ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList2 @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary2 ) ) ) ) ) ) @ Xa ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( vEBT_VEBT_low @ ( if @ nat @ ( Xa = Mi2 ) @ ( plus_plus @ nat @ ( times_times @ nat @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary2 ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList2 @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary2 ) ) ) ) ) ) @ Xa ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_maxt @ ( vEBT_vebt_delete @ Summary2 @ ( vEBT_VEBT_high @ ( if @ nat @ ( Xa = Mi2 ) @ ( plus_plus @ nat @ ( times_times @ nat @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary2 ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList2 @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary2 ) ) ) ) ) ) @ Xa ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) ) ) ) ) ) ) )
@ ( one_one @ nat ) ) ) )
@ ( plus_plus @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) )
@ ( if @ nat
@ ( ( ( Xa = Mi2 )
=> ( ( plus_plus @ nat @ ( times_times @ nat @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary2 ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList2 @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary2 ) ) ) ) ) )
= Ma2 ) )
& ( ( Xa != Mi2 )
=> ( Xa = Ma2 ) ) )
@ ( plus_plus @ nat @ ( numeral_numeral @ nat @ ( bit0 @ ( bit1 @ one2 ) ) ) @ ( vEBT_T_m_a_x_t @ ( nth @ vEBT_VEBT @ ( list_update @ vEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ ( if @ nat @ ( Xa = Mi2 ) @ ( plus_plus @ nat @ ( times_times @ nat @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary2 ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList2 @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary2 ) ) ) ) ) ) @ Xa ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( vEBT_vebt_delete @ ( nth @ vEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ ( if @ nat @ ( Xa = Mi2 ) @ ( plus_plus @ nat @ ( times_times @ nat @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary2 ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList2 @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary2 ) ) ) ) ) ) @ Xa ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( vEBT_VEBT_low @ ( if @ nat @ ( Xa = Mi2 ) @ ( plus_plus @ nat @ ( times_times @ nat @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary2 ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList2 @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary2 ) ) ) ) ) ) @ Xa ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) @ ( vEBT_VEBT_high @ ( if @ nat @ ( Xa = Mi2 ) @ ( plus_plus @ nat @ ( times_times @ nat @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary2 ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList2 @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary2 ) ) ) ) ) ) @ Xa ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va3 ) ) @ ( 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 @ Va3 ) ) @ TreeList2 @ Summary2 ) @ Xa ) ) ) ) ) ) ) ) ) ) ) ) ).
% T\<^sub>d\<^sub>e\<^sub>l\<^sub>e\<^sub>t\<^sub>e.pelims
thf(fact_2364_T_092_060_094sub_062s_092_060_094sub_062u_092_060_094sub_062c_092_060_094sub_062c_Opelims,axiom,
! [X2: vEBT_VEBT,Xa: nat,Y3: nat] :
( ( ( vEBT_T_s_u_c_c @ X2 @ Xa )
= Y3 )
=> ( ( accp @ ( product_prod @ vEBT_VEBT @ nat ) @ vEBT_T_s_u_c_c_rel2 @ ( product_Pair @ vEBT_VEBT @ nat @ X2 @ Xa ) )
=> ( ! [Uu2: $o,B4: $o] :
( ( X2
= ( vEBT_Leaf @ Uu2 @ B4 ) )
=> ( ( Xa
= ( zero_zero @ nat ) )
=> ( ( Y3
= ( plus_plus @ nat @ ( one_one @ nat ) @ ( one_one @ nat ) ) )
=> ~ ( accp @ ( product_prod @ vEBT_VEBT @ nat ) @ vEBT_T_s_u_c_c_rel2 @ ( product_Pair @ vEBT_VEBT @ nat @ ( vEBT_Leaf @ Uu2 @ B4 ) @ ( zero_zero @ nat ) ) ) ) ) )
=> ( ! [Uv2: $o,Uw2: $o] :
( ( X2
= ( vEBT_Leaf @ Uv2 @ Uw2 ) )
=> ! [N3: nat] :
( ( Xa
= ( suc @ N3 ) )
=> ( ( Y3
= ( one_one @ nat ) )
=> ~ ( accp @ ( product_prod @ vEBT_VEBT @ nat ) @ vEBT_T_s_u_c_c_rel2 @ ( product_Pair @ vEBT_VEBT @ nat @ ( vEBT_Leaf @ Uv2 @ Uw2 ) @ ( suc @ N3 ) ) ) ) ) )
=> ( ! [Ux2: nat,Uy2: list @ vEBT_VEBT,Uz2: vEBT_VEBT] :
( ( X2
= ( vEBT_Node @ ( none @ ( product_prod @ nat @ nat ) ) @ Ux2 @ Uy2 @ Uz2 ) )
=> ( ( Y3
= ( one_one @ nat ) )
=> ~ ( accp @ ( product_prod @ vEBT_VEBT @ nat ) @ vEBT_T_s_u_c_c_rel2 @ ( product_Pair @ vEBT_VEBT @ nat @ ( vEBT_Node @ ( none @ ( product_prod @ nat @ nat ) ) @ Ux2 @ Uy2 @ Uz2 ) @ Xa ) ) ) )
=> ( ! [V3: product_prod @ nat @ nat,Vc2: list @ vEBT_VEBT,Vd2: vEBT_VEBT] :
( ( X2
= ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ V3 ) @ ( zero_zero @ nat ) @ Vc2 @ Vd2 ) )
=> ( ( Y3
= ( one_one @ nat ) )
=> ~ ( accp @ ( product_prod @ vEBT_VEBT @ nat ) @ vEBT_T_s_u_c_c_rel2 @ ( product_Pair @ vEBT_VEBT @ nat @ ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ V3 ) @ ( zero_zero @ nat ) @ Vc2 @ Vd2 ) @ Xa ) ) ) )
=> ( ! [V3: product_prod @ nat @ nat,Vg: list @ vEBT_VEBT,Vh: vEBT_VEBT] :
( ( X2
= ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ V3 ) @ ( suc @ ( zero_zero @ nat ) ) @ Vg @ Vh ) )
=> ( ( Y3
= ( one_one @ nat ) )
=> ~ ( accp @ ( product_prod @ vEBT_VEBT @ nat ) @ vEBT_T_s_u_c_c_rel2 @ ( product_Pair @ vEBT_VEBT @ nat @ ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ V3 ) @ ( suc @ ( zero_zero @ nat ) ) @ Vg @ Vh ) @ Xa ) ) ) )
=> ~ ! [Mi2: nat,Ma2: nat,Va3: nat,TreeList2: list @ vEBT_VEBT,Summary2: vEBT_VEBT] :
( ( X2
= ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( suc @ Va3 ) ) @ TreeList2 @ Summary2 ) )
=> ( ( Y3
= ( plus_plus @ nat @ ( one_one @ nat )
@ ( if @ nat @ ( ord_less @ nat @ Xa @ Mi2 ) @ ( one_one @ nat )
@ ( plus_plus @ nat @ ( numeral_numeral @ nat @ ( bit0 @ ( bit1 @ ( bit0 @ one2 ) ) ) )
@ ( if @ nat @ ( ord_less @ nat @ ( vEBT_VEBT_high @ Xa @ ( divide_divide @ nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( size_size @ ( list @ vEBT_VEBT ) @ TreeList2 ) )
@ ( plus_plus @ nat @ ( plus_plus @ nat @ ( one_one @ nat ) @ ( vEBT_T_m_a_x_t @ ( nth @ vEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ Xa @ ( divide_divide @ nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) )
@ ( plus_plus @ nat @ ( numeral_numeral @ nat @ ( bit1 @ one2 ) )
@ ( if @ nat
@ ( ( ( vEBT_vebt_maxt @ ( nth @ vEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ Xa @ ( divide_divide @ nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) )
!= ( none @ nat ) )
& ( vEBT_VEBT_less @ ( some @ nat @ ( vEBT_VEBT_low @ Xa @ ( divide_divide @ nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( vEBT_vebt_maxt @ ( nth @ vEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ Xa @ ( divide_divide @ nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) ) )
@ ( plus_plus @ nat @ ( numeral_numeral @ nat @ ( bit0 @ ( bit0 @ one2 ) ) ) @ ( vEBT_T_s_u_c_c @ ( nth @ vEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ Xa @ ( divide_divide @ nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( vEBT_VEBT_low @ Xa @ ( divide_divide @ nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) )
@ ( plus_plus @ nat @ ( plus_plus @ nat @ ( plus_plus @ nat @ ( one_one @ nat ) @ ( vEBT_T_s_u_c_c @ Summary2 @ ( vEBT_VEBT_high @ Xa @ ( divide_divide @ nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) @ ( one_one @ nat ) )
@ ( if @ nat
@ ( ( vEBT_vebt_succ @ Summary2 @ ( vEBT_VEBT_high @ Xa @ ( divide_divide @ nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) )
= ( none @ nat ) )
@ ( one_one @ nat )
@ ( plus_plus @ nat @ ( numeral_numeral @ nat @ ( bit0 @ ( bit0 @ one2 ) ) ) @ ( vEBT_T_m_i_n_t @ ( nth @ vEBT_VEBT @ TreeList2 @ ( the2 @ nat @ ( vEBT_vebt_succ @ Summary2 @ ( vEBT_VEBT_high @ Xa @ ( divide_divide @ nat @ ( suc @ ( suc @ Va3 ) ) @ ( 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 @ Va3 ) ) @ TreeList2 @ Summary2 ) @ Xa ) ) ) ) ) ) ) ) ) ) ) ).
% T\<^sub>s\<^sub>u\<^sub>c\<^sub>c.pelims
thf(fact_2365_T_092_060_094sub_062p_092_060_094sub_062r_092_060_094sub_062e_092_060_094sub_062d_Opelims,axiom,
! [X2: vEBT_VEBT,Xa: nat,Y3: nat] :
( ( ( vEBT_T_p_r_e_d @ X2 @ Xa )
= Y3 )
=> ( ( accp @ ( product_prod @ vEBT_VEBT @ nat ) @ vEBT_T_p_r_e_d_rel2 @ ( product_Pair @ vEBT_VEBT @ nat @ X2 @ Xa ) )
=> ( ! [Uu2: $o,Uv2: $o] :
( ( X2
= ( vEBT_Leaf @ Uu2 @ Uv2 ) )
=> ( ( Xa
= ( zero_zero @ nat ) )
=> ( ( Y3
= ( one_one @ nat ) )
=> ~ ( accp @ ( product_prod @ vEBT_VEBT @ nat ) @ vEBT_T_p_r_e_d_rel2 @ ( product_Pair @ vEBT_VEBT @ nat @ ( vEBT_Leaf @ Uu2 @ Uv2 ) @ ( zero_zero @ nat ) ) ) ) ) )
=> ( ! [A4: $o,Uw2: $o] :
( ( X2
= ( vEBT_Leaf @ A4 @ Uw2 ) )
=> ( ( Xa
= ( suc @ ( zero_zero @ nat ) ) )
=> ( ( Y3
= ( plus_plus @ nat @ ( one_one @ nat ) @ ( one_one @ nat ) ) )
=> ~ ( accp @ ( product_prod @ vEBT_VEBT @ nat ) @ vEBT_T_p_r_e_d_rel2 @ ( product_Pair @ vEBT_VEBT @ nat @ ( vEBT_Leaf @ A4 @ Uw2 ) @ ( suc @ ( zero_zero @ nat ) ) ) ) ) ) )
=> ( ! [A4: $o,B4: $o] :
( ( X2
= ( vEBT_Leaf @ A4 @ B4 ) )
=> ! [Va3: nat] :
( ( Xa
= ( suc @ ( suc @ Va3 ) ) )
=> ( ( Y3
= ( 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 @ Va3 ) ) ) ) ) ) )
=> ( ! [Uy2: nat,Uz2: list @ vEBT_VEBT,Va2: vEBT_VEBT] :
( ( X2
= ( vEBT_Node @ ( none @ ( product_prod @ nat @ nat ) ) @ Uy2 @ Uz2 @ Va2 ) )
=> ( ( Y3
= ( one_one @ nat ) )
=> ~ ( accp @ ( product_prod @ vEBT_VEBT @ nat ) @ vEBT_T_p_r_e_d_rel2 @ ( product_Pair @ vEBT_VEBT @ nat @ ( vEBT_Node @ ( none @ ( product_prod @ nat @ nat ) ) @ Uy2 @ Uz2 @ Va2 ) @ Xa ) ) ) )
=> ( ! [V3: product_prod @ nat @ nat,Vd2: list @ vEBT_VEBT,Ve2: vEBT_VEBT] :
( ( X2
= ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ V3 ) @ ( zero_zero @ nat ) @ Vd2 @ Ve2 ) )
=> ( ( Y3
= ( one_one @ nat ) )
=> ~ ( accp @ ( product_prod @ vEBT_VEBT @ nat ) @ vEBT_T_p_r_e_d_rel2 @ ( product_Pair @ vEBT_VEBT @ nat @ ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ V3 ) @ ( zero_zero @ nat ) @ Vd2 @ Ve2 ) @ Xa ) ) ) )
=> ( ! [V3: product_prod @ nat @ nat,Vh: list @ vEBT_VEBT,Vi: vEBT_VEBT] :
( ( X2
= ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ V3 ) @ ( suc @ ( zero_zero @ nat ) ) @ Vh @ Vi ) )
=> ( ( Y3
= ( one_one @ nat ) )
=> ~ ( accp @ ( product_prod @ vEBT_VEBT @ nat ) @ vEBT_T_p_r_e_d_rel2 @ ( product_Pair @ vEBT_VEBT @ nat @ ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ V3 ) @ ( suc @ ( zero_zero @ nat ) ) @ Vh @ Vi ) @ Xa ) ) ) )
=> ~ ! [Mi2: nat,Ma2: nat,Va3: nat,TreeList2: list @ vEBT_VEBT,Summary2: vEBT_VEBT] :
( ( X2
= ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( suc @ Va3 ) ) @ TreeList2 @ Summary2 ) )
=> ( ( Y3
= ( plus_plus @ nat @ ( one_one @ nat )
@ ( if @ nat @ ( ord_less @ nat @ Ma2 @ Xa ) @ ( one_one @ nat )
@ ( plus_plus @ nat @ ( plus_plus @ nat @ ( numeral_numeral @ nat @ ( bit0 @ ( bit1 @ ( bit0 @ one2 ) ) ) ) @ ( one_one @ nat ) )
@ ( if @ nat @ ( ord_less @ nat @ ( vEBT_VEBT_high @ Xa @ ( divide_divide @ nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( size_size @ ( list @ vEBT_VEBT ) @ TreeList2 ) )
@ ( plus_plus @ nat @ ( plus_plus @ nat @ ( plus_plus @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( vEBT_T_m_i_n_t @ ( nth @ vEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ Xa @ ( divide_divide @ nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) ) @ ( numeral_numeral @ nat @ ( bit1 @ one2 ) ) )
@ ( if @ nat
@ ( ( ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ Xa @ ( divide_divide @ nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) )
!= ( none @ nat ) )
& ( vEBT_VEBT_greater @ ( some @ nat @ ( vEBT_VEBT_low @ Xa @ ( divide_divide @ nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ Xa @ ( divide_divide @ nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) ) )
@ ( plus_plus @ nat @ ( numeral_numeral @ nat @ ( bit0 @ ( bit0 @ one2 ) ) ) @ ( vEBT_T_p_r_e_d @ ( nth @ vEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ Xa @ ( divide_divide @ nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( vEBT_VEBT_low @ Xa @ ( divide_divide @ nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) )
@ ( plus_plus @ nat @ ( plus_plus @ nat @ ( plus_plus @ nat @ ( one_one @ nat ) @ ( vEBT_T_p_r_e_d @ Summary2 @ ( vEBT_VEBT_high @ Xa @ ( divide_divide @ nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) @ ( one_one @ nat ) )
@ ( if @ nat
@ ( ( vEBT_vebt_pred @ Summary2 @ ( vEBT_VEBT_high @ Xa @ ( divide_divide @ nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) )
= ( none @ nat ) )
@ ( plus_plus @ nat @ ( one_one @ nat ) @ ( one_one @ nat ) )
@ ( plus_plus @ nat @ ( numeral_numeral @ nat @ ( bit0 @ ( bit0 @ one2 ) ) ) @ ( vEBT_T_m_a_x_t @ ( nth @ vEBT_VEBT @ TreeList2 @ ( the2 @ nat @ ( vEBT_vebt_pred @ Summary2 @ ( vEBT_VEBT_high @ Xa @ ( divide_divide @ nat @ ( suc @ ( suc @ Va3 ) ) @ ( 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 @ Va3 ) ) @ TreeList2 @ Summary2 ) @ Xa ) ) ) ) ) ) ) ) ) ) ) ) ).
% T\<^sub>p\<^sub>r\<^sub>e\<^sub>d.pelims
thf(fact_2366_big__assn__simp_H,axiom,
! [H2: nat,TreeList: list @ vEBT_VEBT,Xaa: vEBT_VEBT,L: nat,X2: 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 @ X2 )
@ ( 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 @ X2 ) ) @ ( 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 @ X2 ) ) @ ( 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 @ X2 ) ) ) ) ) ) ).
% big_assn_simp'
thf(fact_2367_big__assn__simp,axiom,
! [H2: nat,TreeList: list @ vEBT_VEBT,L: nat,X2: 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 ) @ X2 )
@ ( 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 @ X2 ) ) @ ( 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 @ X2 ) ) @ ( 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 @ X2 ) ) ) ) ) ).
% big_assn_simp
thf(fact_2368_tcd,axiom,
! [A: $tType,I: nat,TreeList: list @ vEBT_VEBT,TreeList4: list @ A,Y3: vEBT_VEBT,X2: 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 ) @ TreeList4 ) )
=> ( entails @ ( times_times @ assn @ ( times_times @ assn @ ( times_times @ assn @ ( vEBT_vebt_assn_raw @ Y3 @ X2 ) @ ( snga_assn @ vEBT_VEBTi @ X13 @ ( list_update @ vEBT_VEBTi @ Tree_is @ I @ X2 ) ) ) @ ( 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 @ Y3 ) @ ( list_update @ vEBT_VEBTi @ Tree_is @ I @ X2 ) ) ) @ ( times_times @ assn @ ( times_times @ assn @ ( snga_assn @ vEBT_VEBTi @ X13 @ ( list_update @ vEBT_VEBTi @ Tree_is @ I @ X2 ) ) @ ( vEBT_vebt_assn_raw @ Summary @ X14 ) ) @ ( vEBT_List_list_assn @ vEBT_VEBT @ vEBT_VEBTi @ vEBT_vebt_assn_raw @ ( list_update @ vEBT_VEBT @ TreeList @ I @ Y3 ) @ ( list_update @ vEBT_VEBTi @ Tree_is @ I @ X2 ) ) ) ) ) ) ).
% tcd
thf(fact_2369_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_2370_atLeastLessThan__empty,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [B3: A,A3: A] :
( ( ord_less_eq @ A @ B3 @ A3 )
=> ( ( set_or7035219750837199246ssThan @ A @ A3 @ B3 )
= ( bot_bot @ ( set @ A ) ) ) ) ) ).
% atLeastLessThan_empty
thf(fact_2371_atLeastLessThan__empty__iff2,axiom,
! [A: $tType] :
( ( preorder @ A )
=> ! [A3: A,B3: A] :
( ( ( bot_bot @ ( set @ A ) )
= ( set_or7035219750837199246ssThan @ A @ A3 @ B3 ) )
= ( ~ ( ord_less @ A @ A3 @ B3 ) ) ) ) ).
% atLeastLessThan_empty_iff2
thf(fact_2372_atLeastLessThan__empty__iff,axiom,
! [A: $tType] :
( ( preorder @ A )
=> ! [A3: A,B3: A] :
( ( ( set_or7035219750837199246ssThan @ A @ A3 @ B3 )
= ( bot_bot @ ( set @ A ) ) )
= ( ~ ( ord_less @ A @ A3 @ B3 ) ) ) ) ).
% atLeastLessThan_empty_iff
thf(fact_2373_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_2374_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_2375_local_Oext,axiom,
! [Y3: nat,TreeList: list @ vEBT_VEBT,X13: array @ vEBT_VEBTi,Tree_is: list @ vEBT_VEBTi,Summary: vEBT_VEBT,X14: vEBT_VEBTi] :
( ( ord_less @ nat @ Y3 @ ( 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 @ Y3 ) @ ( nth @ vEBT_VEBTi @ Tree_is @ Y3 ) ) @ ( 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 @ Y3 @ ( 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 @ Y3 @ ( bot_bot @ ( set @ nat ) ) ) ) @ vEBT_vebt_assn_raw @ TreeList @ Tree_is ) ) @ ( vEBT_vebt_assn_raw @ ( nth @ vEBT_VEBT @ TreeList @ Y3 ) @ ( nth @ vEBT_VEBTi @ Tree_is @ Y3 ) ) ) ) ) ).
% local.ext
thf(fact_2376_atLeastLessThan__singleton,axiom,
! [M: nat] :
( ( set_or7035219750837199246ssThan @ nat @ M @ ( suc @ M ) )
= ( insert @ nat @ M @ ( bot_bot @ ( set @ nat ) ) ) ) ).
% atLeastLessThan_singleton
thf(fact_2377_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_2378_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_2379_txe,axiom,
! [Y3: nat,TreeList: list @ vEBT_VEBT,Tree_is: list @ vEBT_VEBTi,X13: array @ vEBT_VEBTi,Summary: vEBT_VEBT,X14: vEBT_VEBTi] :
( ( ord_less @ nat @ Y3 @ ( size_size @ ( list @ vEBT_VEBT ) @ TreeList ) )
=> ( entails @ ( times_times @ assn @ ( times_times @ assn @ ( times_times @ assn @ ( vEBT_vebt_assn_raw @ ( nth @ vEBT_VEBT @ TreeList @ Y3 ) @ ( nth @ vEBT_VEBTi @ Tree_is @ Y3 ) ) @ ( 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 @ Y3 @ ( 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_2380_atLeastLessThan__eq__iff,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A3: A,B3: A,C3: A,D2: A] :
( ( ord_less @ A @ A3 @ B3 )
=> ( ( ord_less @ A @ C3 @ D2 )
=> ( ( ( set_or7035219750837199246ssThan @ A @ A3 @ B3 )
= ( set_or7035219750837199246ssThan @ A @ C3 @ D2 ) )
= ( ( A3 = C3 )
& ( B3 = D2 ) ) ) ) ) ) ).
% atLeastLessThan_eq_iff
thf(fact_2381_atLeastLessThan__inj_I1_J,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A3: A,B3: A,C3: A,D2: A] :
( ( ( set_or7035219750837199246ssThan @ A @ A3 @ B3 )
= ( set_or7035219750837199246ssThan @ A @ C3 @ D2 ) )
=> ( ( ord_less @ A @ A3 @ B3 )
=> ( ( ord_less @ A @ C3 @ D2 )
=> ( A3 = C3 ) ) ) ) ) ).
% atLeastLessThan_inj(1)
thf(fact_2382_atLeastLessThan__inj_I2_J,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A3: A,B3: A,C3: A,D2: A] :
( ( ( set_or7035219750837199246ssThan @ A @ A3 @ B3 )
= ( set_or7035219750837199246ssThan @ A @ C3 @ D2 ) )
=> ( ( ord_less @ A @ A3 @ B3 )
=> ( ( ord_less @ A @ C3 @ D2 )
=> ( B3 = D2 ) ) ) ) ) ).
% atLeastLessThan_inj(2)
thf(fact_2383_atLeastLessThan__subset__iff,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A3: A,B3: A,C3: A,D2: A] :
( ( ord_less_eq @ ( set @ A ) @ ( set_or7035219750837199246ssThan @ A @ A3 @ B3 ) @ ( set_or7035219750837199246ssThan @ A @ C3 @ D2 ) )
=> ( ( ord_less_eq @ A @ B3 @ A3 )
| ( ( ord_less_eq @ A @ C3 @ A3 )
& ( ord_less_eq @ A @ B3 @ D2 ) ) ) ) ) ).
% atLeastLessThan_subset_iff
thf(fact_2384_all__nat__less__eq,axiom,
! [N: nat,P: nat > $o] :
( ( ! [M6: nat] :
( ( ord_less @ nat @ M6 @ N )
=> ( P @ M6 ) ) )
= ( ! [X3: nat] :
( ( member @ nat @ X3 @ ( set_or7035219750837199246ssThan @ nat @ ( zero_zero @ nat ) @ N ) )
=> ( P @ X3 ) ) ) ) ).
% all_nat_less_eq
thf(fact_2385_ex__nat__less__eq,axiom,
! [N: nat,P: nat > $o] :
( ( ? [M6: nat] :
( ( ord_less @ nat @ M6 @ N )
& ( P @ M6 ) ) )
= ( ? [X3: nat] :
( ( member @ nat @ X3 @ ( set_or7035219750837199246ssThan @ nat @ ( zero_zero @ nat ) @ N ) )
& ( P @ X3 ) ) ) ) ).
% ex_nat_less_eq
thf(fact_2386_atLeastLessThanSuc__atLeastAtMost,axiom,
! [L: nat,U: nat] :
( ( set_or7035219750837199246ssThan @ nat @ L @ ( suc @ U ) )
= ( set_or1337092689740270186AtMost @ nat @ L @ U ) ) ).
% atLeastLessThanSuc_atLeastAtMost
thf(fact_2387_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_2388_atLeastLessThan__subseteq__atLeastAtMost__iff,axiom,
! [A: $tType] :
( ( dense_linorder @ A )
=> ! [A3: A,B3: A,C3: A,D2: A] :
( ( ord_less_eq @ ( set @ A ) @ ( set_or7035219750837199246ssThan @ A @ A3 @ B3 ) @ ( set_or1337092689740270186AtMost @ A @ C3 @ D2 ) )
= ( ( ord_less @ A @ A3 @ B3 )
=> ( ( ord_less_eq @ A @ C3 @ A3 )
& ( ord_less_eq @ A @ B3 @ D2 ) ) ) ) ) ).
% atLeastLessThan_subseteq_atLeastAtMost_iff
thf(fact_2389_atLeastAtMost__subseteq__atLeastLessThan__iff,axiom,
! [A: $tType] :
( ( preorder @ A )
=> ! [A3: A,B3: A,C3: A,D2: A] :
( ( ord_less_eq @ ( set @ A ) @ ( set_or1337092689740270186AtMost @ A @ A3 @ B3 ) @ ( set_or7035219750837199246ssThan @ A @ C3 @ D2 ) )
= ( ( ord_less_eq @ A @ A3 @ B3 )
=> ( ( ord_less_eq @ A @ C3 @ A3 )
& ( ord_less @ A @ B3 @ D2 ) ) ) ) ) ).
% atLeastAtMost_subseteq_atLeastLessThan_iff
thf(fact_2390_atLeastLessThan__eq__atLeastAtMost__diff,axiom,
! [A: $tType] :
( ( order @ A )
=> ( ( set_or7035219750837199246ssThan @ A )
= ( ^ [A6: A,B7: A] : ( minus_minus @ ( set @ A ) @ ( set_or1337092689740270186AtMost @ A @ A6 @ B7 ) @ ( insert @ A @ B7 @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ) ).
% atLeastLessThan_eq_atLeastAtMost_diff
thf(fact_2391_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_2392_rule__at__index,axiom,
! [A: $tType,B: $tType,C: $tType,P: assn,A2: A > B > assn,Xs: list @ A,Xsi: list @ B,F4: assn,I: nat,C3: heap_Time_Heap @ C,Q2: C > assn,F6: C > assn] :
( ( entails @ P @ ( times_times @ assn @ ( vEBT_List_list_assn @ A @ B @ A2 @ Xs @ Xsi ) @ F4 ) )
=> ( ( ord_less @ nat @ I @ ( size_size @ ( list @ A ) @ Xs ) )
=> ( ( hoare_hoare_triple @ C @ ( times_times @ assn @ ( times_times @ assn @ ( A2 @ ( 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 ) ) ) ) @ A2 @ Xs @ Xsi ) ) @ F4 ) @ C3 @ Q2 )
=> ( ! [R3: C] : ( entails @ ( Q2 @ R3 ) @ ( times_times @ assn @ ( times_times @ assn @ ( A2 @ ( 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 ) ) ) ) @ A2 @ Xs @ Xsi ) ) @ ( F6 @ R3 ) ) )
=> ( hoare_hoare_triple @ C @ P @ C3
@ ^ [R5: C] : ( times_times @ assn @ ( vEBT_List_list_assn @ A @ B @ A2 @ Xs @ Xsi ) @ ( F6 @ R5 ) ) ) ) ) ) ) ).
% rule_at_index
thf(fact_2393_listI__assn__reinsert__upd_H,axiom,
! [C: $tType,D6: $tType,E3: $tType,P: assn,A2: C > D6 > assn,X2: C,Xi: D6,I6: set @ nat,I: nat,Xs: list @ C,Xsi: list @ D6,F4: assn,C3: heap_Time_Heap @ E3,Q: E3 > assn] :
( ( entails @ P @ ( times_times @ assn @ ( times_times @ assn @ ( A2 @ X2 @ Xi ) @ ( vEBT_List_listI_assn @ C @ D6 @ ( minus_minus @ ( set @ nat ) @ I6 @ ( insert @ nat @ I @ ( bot_bot @ ( set @ nat ) ) ) ) @ A2 @ Xs @ Xsi ) ) @ F4 ) )
=> ( ( ord_less @ nat @ I @ ( size_size @ ( list @ C ) @ Xs ) )
=> ( ( member @ nat @ I @ I6 )
=> ( ( hoare_hoare_triple @ E3 @ ( times_times @ assn @ ( vEBT_List_listI_assn @ C @ D6 @ I6 @ A2 @ ( list_update @ C @ Xs @ I @ X2 ) @ ( list_update @ D6 @ Xsi @ I @ Xi ) ) @ F4 ) @ C3 @ Q )
=> ( hoare_hoare_triple @ E3 @ P @ C3 @ Q ) ) ) ) ) ).
% listI_assn_reinsert_upd'
thf(fact_2394_listI__assn__reinsert_H,axiom,
! [A: $tType,B: $tType,C: $tType,P: assn,A2: A > B > assn,Xs: list @ A,I: nat,Xsi: list @ B,I6: set @ nat,F4: assn,C3: heap_Time_Heap @ C,Q: C > assn] :
( ( entails @ P @ ( times_times @ assn @ ( times_times @ assn @ ( A2 @ ( 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 ) ) ) ) @ A2 @ Xs @ Xsi ) ) @ F4 ) )
=> ( ( 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 @ A2 @ Xs @ Xsi ) @ F4 ) @ C3 @ Q )
=> ( hoare_hoare_triple @ C @ P @ C3 @ Q ) ) ) ) ) ).
% listI_assn_reinsert'
thf(fact_2395_listI__assn__reinsert__upd,axiom,
! [D6: $tType,C: $tType,P: assn,A2: C > D6 > assn,X2: C,Xi: D6,I6: set @ nat,I: nat,Xs: list @ C,Xsi: list @ D6,F4: assn,Q: assn] :
( ( entails @ P @ ( times_times @ assn @ ( times_times @ assn @ ( A2 @ X2 @ Xi ) @ ( vEBT_List_listI_assn @ C @ D6 @ ( minus_minus @ ( set @ nat ) @ I6 @ ( insert @ nat @ I @ ( bot_bot @ ( set @ nat ) ) ) ) @ A2 @ Xs @ Xsi ) ) @ F4 ) )
=> ( ( ord_less @ nat @ I @ ( size_size @ ( list @ C ) @ Xs ) )
=> ( ( member @ nat @ I @ I6 )
=> ( ( entails @ ( times_times @ assn @ ( vEBT_List_listI_assn @ C @ D6 @ I6 @ A2 @ ( list_update @ C @ Xs @ I @ X2 ) @ ( list_update @ D6 @ Xsi @ I @ Xi ) ) @ F4 ) @ Q )
=> ( entails @ P @ Q ) ) ) ) ) ).
% listI_assn_reinsert_upd
thf(fact_2396_listI__assn__reinsert,axiom,
! [B: $tType,A: $tType,P: assn,A2: A > B > assn,Xs: list @ A,I: nat,Xsi: list @ B,I6: set @ nat,F4: assn,Q: assn] :
( ( entails @ P @ ( times_times @ assn @ ( times_times @ assn @ ( A2 @ ( 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 ) ) ) ) @ A2 @ Xs @ Xsi ) ) @ F4 ) )
=> ( ( ord_less @ nat @ I @ ( size_size @ ( list @ A ) @ Xs ) )
=> ( ( member @ nat @ I @ I6 )
=> ( ( entails @ ( times_times @ assn @ ( vEBT_List_listI_assn @ A @ B @ I6 @ A2 @ Xs @ Xsi ) @ F4 ) @ Q )
=> ( entails @ P @ Q ) ) ) ) ) ).
% listI_assn_reinsert
thf(fact_2397_listI__assn__extract,axiom,
! [A: $tType,B: $tType,I: nat,I6: set @ nat,Xs: list @ A,A2: 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 @ A2 @ Xs @ Xsi )
= ( times_times @ assn @ ( A2 @ ( 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 ) ) ) ) @ A2 @ Xs @ Xsi ) ) ) ) ) ).
% listI_assn_extract
thf(fact_2398_word__to__1__set,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ( ( set_or7035219750837199246ssThan @ ( word @ A ) @ ( zero_zero @ ( word @ A ) ) @ ( one_one @ ( word @ A ) ) )
= ( insert @ ( word @ A ) @ ( zero_zero @ ( word @ A ) ) @ ( bot_bot @ ( set @ ( word @ A ) ) ) ) ) ) ).
% word_to_1_set
thf(fact_2399_atLeastLessThanPlusOne__atLeastAtMost__int,axiom,
! [L: int,U: int] :
( ( set_or7035219750837199246ssThan @ int @ L @ ( plus_plus @ int @ U @ ( one_one @ int ) ) )
= ( set_or1337092689740270186AtMost @ int @ L @ U ) ) ).
% atLeastLessThanPlusOne_atLeastAtMost_int
thf(fact_2400_word__range__minus__1,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [B3: word @ A,A3: word @ A] :
( ( B3
!= ( zero_zero @ ( word @ A ) ) )
=> ( ( set_or1337092689740270186AtMost @ ( word @ A ) @ A3 @ ( minus_minus @ ( word @ A ) @ B3 @ ( one_one @ ( word @ A ) ) ) )
= ( set_or7035219750837199246ssThan @ ( word @ A ) @ A3 @ B3 ) ) ) ) ).
% word_range_minus_1
thf(fact_2401_list__assn__cong,axiom,
! [A: $tType,B: $tType,Xs: list @ A,Xs4: list @ A,Xsi: list @ B,Xsi2: list @ B,A2: A > B > assn,A8: A > B > assn] :
( ( Xs = Xs4 )
=> ( ( Xsi = Xsi2 )
=> ( ! [X4: A,Xi2: B] :
( ( member @ A @ X4 @ ( set2 @ A @ Xs4 ) )
=> ( ( member @ B @ Xi2 @ ( set2 @ B @ Xsi2 ) )
=> ( ( A2 @ X4 @ Xi2 )
= ( A8 @ X4 @ Xi2 ) ) ) )
=> ( ( vEBT_List_list_assn @ A @ B @ A2 @ Xs @ Xsi )
= ( vEBT_List_list_assn @ A @ B @ A8 @ Xs4 @ Xsi2 ) ) ) ) ) ).
% list_assn_cong
thf(fact_2402_list__assn__mono,axiom,
! [A: $tType,B: $tType,P: A > B > assn,P5: A > B > assn,L: list @ A,L4: list @ B] :
( ! [X4: A,X9: B] : ( entails @ ( P @ X4 @ X9 ) @ ( P5 @ X4 @ X9 ) )
=> ( entails @ ( vEBT_List_list_assn @ A @ B @ P @ L @ L4 ) @ ( vEBT_List_list_assn @ A @ B @ P5 @ L @ L4 ) ) ) ).
% list_assn_mono
thf(fact_2403_extract__pre__list__assn__lengthD,axiom,
! [B: $tType,A: $tType,A2: 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 @ A2 @ Xs @ Xsi ) @ H2 )
=> ( ( size_size @ ( list @ B ) @ Xsi )
= ( size_size @ ( list @ A ) @ Xs ) ) ) ).
% extract_pre_list_assn_lengthD
thf(fact_2404_list__assn__aux__ineq__len,axiom,
! [B: $tType,A: $tType,L: list @ A,Li2: list @ B,A2: A > B > assn] :
( ( ( size_size @ ( list @ A ) @ L )
!= ( size_size @ ( list @ B ) @ Li2 ) )
=> ( ( vEBT_List_list_assn @ A @ B @ A2 @ L @ Li2 )
= ( bot_bot @ assn ) ) ) ).
% list_assn_aux_ineq_len
thf(fact_2405_listI__assn__weak__cong,axiom,
! [A: $tType,B: $tType,I6: set @ nat,I7: set @ nat,A2: A > B > assn,A8: A > B > assn,Xs: list @ A,Xs4: list @ A,Xsi: list @ B,Xsi2: list @ B] :
( ( I6 = I7 )
=> ( ( A2 = A8 )
=> ( ( ( 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 @ A2 @ Xs @ Xsi )
= ( vEBT_List_listI_assn @ A @ B @ I7 @ A8 @ Xs4 @ Xsi2 ) ) ) ) ) ) ) ).
% listI_assn_weak_cong
thf(fact_2406_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,A2: A > B > assn,A8: 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 ) )
& ( ( A2 @ ( nth @ A @ Xs @ I2 ) @ ( nth @ B @ Xsi @ I2 ) )
= ( A8 @ ( nth @ A @ Xs4 @ I2 ) @ ( nth @ B @ Xsi2 @ I2 ) ) ) ) ) ) )
=> ( ( vEBT_List_listI_assn @ A @ B @ I6 @ A2 @ Xs @ Xsi )
= ( vEBT_List_listI_assn @ A @ B @ I7 @ A8 @ Xs4 @ Xsi2 ) ) ) ) ) ) ).
% listI_assn_cong
thf(fact_2407_subst__not__in,axiom,
! [A: $tType,B: $tType,I: nat,I6: set @ nat,Xs: list @ A,A2: A > B > assn,X1: A,Xsi: list @ B,X22: B] :
( ~ ( member @ nat @ I @ I6 )
=> ( ( ord_less @ nat @ I @ ( size_size @ ( list @ A ) @ Xs ) )
=> ( ( vEBT_List_listI_assn @ A @ B @ I6 @ A2 @ ( list_update @ A @ Xs @ I @ X1 ) @ ( list_update @ B @ Xsi @ I @ X22 ) )
= ( vEBT_List_listI_assn @ A @ B @ I6 @ A2 @ Xs @ Xsi ) ) ) ) ).
% subst_not_in
thf(fact_2408_listI__assn__conv,axiom,
! [A: $tType,B: $tType,N: nat,Xs: list @ A,A2: 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 ) @ A2 @ Xs @ Xsi )
= ( vEBT_List_list_assn @ A @ B @ A2 @ Xs @ Xsi ) ) ) ).
% listI_assn_conv
thf(fact_2409_list__assn__conv__idx,axiom,
! [B: $tType,A: $tType] :
( ( vEBT_List_list_assn @ A @ B )
= ( ^ [A7: A > B > assn,Xs2: list @ A] : ( vEBT_List_listI_assn @ A @ B @ ( set_or7035219750837199246ssThan @ nat @ ( zero_zero @ nat ) @ ( size_size @ ( list @ A ) @ Xs2 ) ) @ A7 @ Xs2 ) ) ) ).
% list_assn_conv_idx
thf(fact_2410_listI__assn__insert,axiom,
! [A: $tType,B: $tType,I: nat,I6: set @ nat,Xs: list @ A,A2: 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 ) @ A2 @ Xs @ Xsi )
= ( times_times @ assn @ ( A2 @ ( nth @ A @ Xs @ I ) @ ( nth @ B @ Xsi @ I ) ) @ ( vEBT_List_listI_assn @ A @ B @ I6 @ A2 @ Xs @ Xsi ) ) ) ) ) ).
% listI_assn_insert
thf(fact_2411_listI__assn__conv_H,axiom,
! [B: $tType,A: $tType,N: nat,Xs: list @ A,A2: A > B > assn,Xsi: list @ B,F4: assn] :
( ( N
= ( size_size @ ( list @ A ) @ Xs ) )
=> ( ( times_times @ assn @ ( vEBT_List_listI_assn @ A @ B @ ( set_or7035219750837199246ssThan @ nat @ ( zero_zero @ nat ) @ N ) @ A2 @ Xs @ Xsi ) @ F4 )
= ( times_times @ assn @ ( vEBT_List_list_assn @ A @ B @ A2 @ Xs @ Xsi ) @ F4 ) ) ) ).
% listI_assn_conv'
thf(fact_2412_listI__assn__subst,axiom,
! [A: $tType,B: $tType,I: nat,I6: set @ nat,Xs: list @ A,A2: A > B > assn,X1: A,Xsi: list @ B,X22: B] :
( ~ ( member @ nat @ I @ I6 )
=> ( ( ord_less @ nat @ I @ ( size_size @ ( list @ A ) @ Xs ) )
=> ( ( vEBT_List_listI_assn @ A @ B @ ( insert @ nat @ I @ I6 ) @ A2 @ ( list_update @ A @ Xs @ I @ X1 ) @ ( list_update @ B @ Xsi @ I @ X22 ) )
= ( times_times @ assn @ ( A2 @ X1 @ X22 ) @ ( vEBT_List_listI_assn @ A @ B @ I6 @ A2 @ Xs @ Xsi ) ) ) ) ) ).
% listI_assn_subst
thf(fact_2413_lemma__termdiff3,axiom,
! [A: $tType] :
( ( real_V3459762299906320749_field @ A )
=> ! [H2: A,Z: A,K6: real,N: nat] :
( ( H2
!= ( zero_zero @ A ) )
=> ( ( ord_less_eq @ real @ ( real_V7770717601297561774m_norm @ A @ Z ) @ K6 )
=> ( ( ord_less_eq @ real @ ( real_V7770717601297561774m_norm @ A @ ( plus_plus @ A @ Z @ H2 ) ) @ K6 )
=> ( ord_less_eq @ real @ ( real_V7770717601297561774m_norm @ A @ ( minus_minus @ A @ ( divide_divide @ A @ ( minus_minus @ A @ ( power_power @ A @ ( plus_plus @ A @ 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 @ K6 @ ( minus_minus @ nat @ N @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( real_V7770717601297561774m_norm @ A @ H2 ) ) ) ) ) ) ) ).
% lemma_termdiff3
thf(fact_2414_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_2415_rel__of__empty,axiom,
! [B: $tType,A: $tType,P: ( product_prod @ A @ B ) > $o] :
( ( rel_of @ A @ B
@ ^ [X3: A] : ( none @ B )
@ P )
= ( bot_bot @ ( set @ ( product_prod @ A @ B ) ) ) ) ).
% rel_of_empty
thf(fact_2416_foldr__same,axiom,
! [Xs: list @ real,Y3: real] :
( ! [X4: real,Y4: real] :
( ( member @ real @ X4 @ ( set2 @ real @ Xs ) )
=> ( ( member @ real @ Y4 @ ( set2 @ real @ Xs ) )
=> ( X4 = Y4 ) ) )
=> ( ! [X4: real] :
( ( member @ real @ X4 @ ( set2 @ real @ Xs ) )
=> ( X4 = Y3 ) )
=> ( ( foldr @ real @ real @ ( plus_plus @ real ) @ Xs @ ( zero_zero @ real ) )
= ( times_times @ real @ ( semiring_1_of_nat @ real @ ( size_size @ ( list @ real ) @ Xs ) ) @ Y3 ) ) ) ) ).
% foldr_same
thf(fact_2417_lowi__hT,axiom,
! [X2: nat,N: nat] :
( time_htt @ nat @ ( one_one @ assn ) @ ( vEBT_VEBT_lowi @ X2 @ N )
@ ^ [R5: nat] :
( pure_assn
@ ( R5
= ( vEBT_VEBT_low @ X2 @ N ) ) )
@ ( one_one @ nat ) ) ).
% lowi_hT
thf(fact_2418_highi__hT,axiom,
! [X2: nat,N: nat] :
( time_htt @ nat @ ( one_one @ assn ) @ ( vEBT_VEBT_highi @ X2 @ N )
@ ^ [R5: nat] :
( pure_assn
@ ( R5
= ( vEBT_VEBT_high @ X2 @ N ) ) )
@ ( one_one @ nat ) ) ).
% highi_hT
thf(fact_2419_TBOUND__lowi,axiom,
! [X2: nat,N: nat] : ( time_TBOUND @ nat @ ( vEBT_VEBT_lowi @ X2 @ N ) @ ( one_one @ nat ) ) ).
% TBOUND_lowi
thf(fact_2420_TBOUND__highi,axiom,
! [X2: nat,N: nat] : ( time_TBOUND @ nat @ ( vEBT_VEBT_highi @ X2 @ N ) @ ( one_one @ nat ) ) ).
% TBOUND_highi
thf(fact_2421_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_2422_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_2423_highi__h,axiom,
! [X2: nat,N: nat] :
( hoare_hoare_triple @ nat @ ( one_one @ assn ) @ ( vEBT_VEBT_highi @ X2 @ N )
@ ^ [R5: nat] :
( pure_assn
@ ( R5
= ( vEBT_VEBT_high @ X2 @ N ) ) ) ) ).
% highi_h
thf(fact_2424_lowi__h,axiom,
! [X2: nat,N: nat] :
( hoare_hoare_triple @ nat @ ( one_one @ assn ) @ ( vEBT_VEBT_lowi @ X2 @ N )
@ ^ [R5: nat] :
( pure_assn
@ ( R5
= ( vEBT_VEBT_low @ X2 @ N ) ) ) ) ).
% lowi_h
thf(fact_2425_foldr__same__int,axiom,
! [Xs: list @ nat,Y3: nat] :
( ! [X4: nat,Y4: nat] :
( ( member @ nat @ X4 @ ( set2 @ nat @ Xs ) )
=> ( ( member @ nat @ Y4 @ ( set2 @ nat @ Xs ) )
=> ( X4 = Y4 ) ) )
=> ( ! [X4: nat] :
( ( member @ nat @ X4 @ ( set2 @ nat @ Xs ) )
=> ( X4 = Y3 ) )
=> ( ( foldr @ nat @ nat @ ( plus_plus @ nat ) @ Xs @ ( zero_zero @ nat ) )
= ( times_times @ nat @ ( size_size @ ( list @ nat ) @ Xs ) @ Y3 ) ) ) ) ).
% foldr_same_int
thf(fact_2426_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_2427_foldr__length,axiom,
! [A: $tType,L: list @ A] :
( ( foldr @ A @ nat
@ ^ [X3: A] : suc
@ L
@ ( zero_zero @ nat ) )
= ( size_size @ ( list @ A ) @ L ) ) ).
% foldr_length
thf(fact_2428_foldr__cong,axiom,
! [B: $tType,A: $tType,A3: A,B3: A,L: list @ B,K: list @ B,F2: B > A > A,G: B > A > A] :
( ( A3 = B3 )
=> ( ( L = K )
=> ( ! [A4: A,X4: B] :
( ( member @ B @ X4 @ ( set2 @ B @ L ) )
=> ( ( F2 @ X4 @ A4 )
= ( G @ X4 @ A4 ) ) )
=> ( ( foldr @ B @ A @ F2 @ L @ A3 )
= ( foldr @ B @ A @ G @ K @ B3 ) ) ) ) ) ).
% foldr_cong
thf(fact_2429_foldr__length__aux,axiom,
! [A: $tType,L: list @ A,A3: nat] :
( ( foldr @ A @ nat
@ ^ [X3: A] : suc
@ L
@ A3 )
= ( plus_plus @ nat @ A3 @ ( size_size @ ( list @ A ) @ L ) ) ) ).
% foldr_length_aux
thf(fact_2430_lemma__NBseq__def2,axiom,
! [A: $tType,B: $tType] :
( ( real_V822414075346904944vector @ B )
=> ! [X: A > B] :
( ( ? [K7: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ K7 )
& ! [N4: A] : ( ord_less_eq @ real @ ( real_V7770717601297561774m_norm @ B @ ( X @ N4 ) ) @ K7 ) ) )
= ( ? [N8: nat] :
! [N4: A] : ( ord_less @ real @ ( real_V7770717601297561774m_norm @ B @ ( X @ N4 ) ) @ ( semiring_1_of_nat @ real @ ( suc @ N8 ) ) ) ) ) ) ).
% lemma_NBseq_def2
thf(fact_2431_lemma__NBseq__def,axiom,
! [A: $tType,B: $tType] :
( ( real_V822414075346904944vector @ B )
=> ! [X: A > B] :
( ( ? [K7: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ K7 )
& ! [N4: A] : ( ord_less_eq @ real @ ( real_V7770717601297561774m_norm @ B @ ( X @ N4 ) ) @ K7 ) ) )
= ( ? [N8: nat] :
! [N4: A] : ( ord_less_eq @ real @ ( real_V7770717601297561774m_norm @ B @ ( X @ N4 ) ) @ ( semiring_1_of_nat @ real @ ( suc @ N8 ) ) ) ) ) ) ).
% lemma_NBseq_def
thf(fact_2432_norm__divide__numeral,axiom,
! [A: $tType] :
( ( real_V3459762299906320749_field @ A )
=> ! [A3: A,W: num] :
( ( real_V7770717601297561774m_norm @ A @ ( divide_divide @ A @ A3 @ ( numeral_numeral @ A @ W ) ) )
= ( divide_divide @ real @ ( real_V7770717601297561774m_norm @ A @ A3 ) @ ( numeral_numeral @ real @ W ) ) ) ) ).
% norm_divide_numeral
thf(fact_2433_norm__mult__numeral1,axiom,
! [A: $tType] :
( ( real_V3459762299906320749_field @ A )
=> ! [W: num,A3: A] :
( ( real_V7770717601297561774m_norm @ A @ ( times_times @ A @ ( numeral_numeral @ A @ W ) @ A3 ) )
= ( times_times @ real @ ( numeral_numeral @ real @ W ) @ ( real_V7770717601297561774m_norm @ A @ A3 ) ) ) ) ).
% norm_mult_numeral1
thf(fact_2434_norm__mult__numeral2,axiom,
! [A: $tType] :
( ( real_V3459762299906320749_field @ A )
=> ! [A3: A,W: num] :
( ( real_V7770717601297561774m_norm @ A @ ( times_times @ A @ A3 @ ( numeral_numeral @ A @ W ) ) )
= ( times_times @ real @ ( real_V7770717601297561774m_norm @ A @ A3 ) @ ( numeral_numeral @ real @ W ) ) ) ) ).
% norm_mult_numeral2
thf(fact_2435_norm__le__zero__iff,axiom,
! [A: $tType] :
( ( real_V822414075346904944vector @ A )
=> ! [X2: A] :
( ( ord_less_eq @ real @ ( real_V7770717601297561774m_norm @ A @ X2 ) @ ( zero_zero @ real ) )
= ( X2
= ( zero_zero @ A ) ) ) ) ).
% norm_le_zero_iff
thf(fact_2436_norm__numeral,axiom,
! [A: $tType] :
( ( real_V2822296259951069270ebra_1 @ A )
=> ! [W: num] :
( ( real_V7770717601297561774m_norm @ A @ ( numeral_numeral @ A @ W ) )
= ( numeral_numeral @ real @ W ) ) ) ).
% norm_numeral
thf(fact_2437_norm__one,axiom,
! [A: $tType] :
( ( real_V2822296259951069270ebra_1 @ A )
=> ( ( real_V7770717601297561774m_norm @ A @ ( one_one @ A ) )
= ( one_one @ real ) ) ) ).
% norm_one
thf(fact_2438_norm__minus__commute,axiom,
! [A: $tType] :
( ( real_V822414075346904944vector @ A )
=> ! [A3: A,B3: A] :
( ( real_V7770717601297561774m_norm @ A @ ( minus_minus @ A @ A3 @ B3 ) )
= ( real_V7770717601297561774m_norm @ A @ ( minus_minus @ A @ B3 @ A3 ) ) ) ) ).
% norm_minus_commute
thf(fact_2439_norm__mult,axiom,
! [A: $tType] :
( ( real_V8999393235501362500lgebra @ A )
=> ! [X2: A,Y3: A] :
( ( real_V7770717601297561774m_norm @ A @ ( times_times @ A @ X2 @ Y3 ) )
= ( times_times @ real @ ( real_V7770717601297561774m_norm @ A @ X2 ) @ ( real_V7770717601297561774m_norm @ A @ Y3 ) ) ) ) ).
% norm_mult
thf(fact_2440_norm__ge__zero,axiom,
! [A: $tType] :
( ( real_V822414075346904944vector @ A )
=> ! [X2: A] : ( ord_less_eq @ real @ ( zero_zero @ real ) @ ( real_V7770717601297561774m_norm @ A @ X2 ) ) ) ).
% norm_ge_zero
thf(fact_2441_norm__divide,axiom,
! [A: $tType] :
( ( real_V3459762299906320749_field @ A )
=> ! [A3: A,B3: A] :
( ( real_V7770717601297561774m_norm @ A @ ( divide_divide @ A @ A3 @ B3 ) )
= ( divide_divide @ real @ ( real_V7770717601297561774m_norm @ A @ A3 ) @ ( real_V7770717601297561774m_norm @ A @ B3 ) ) ) ) ).
% norm_divide
thf(fact_2442_norm__power,axiom,
! [A: $tType] :
( ( real_V8999393235501362500lgebra @ A )
=> ! [X2: A,N: nat] :
( ( real_V7770717601297561774m_norm @ A @ ( power_power @ A @ X2 @ N ) )
= ( power_power @ real @ ( real_V7770717601297561774m_norm @ A @ X2 ) @ N ) ) ) ).
% norm_power
thf(fact_2443_power__eq__imp__eq__norm,axiom,
! [A: $tType] :
( ( real_V8999393235501362500lgebra @ A )
=> ! [W: A,N: nat,Z: A] :
( ( ( power_power @ A @ W @ N )
= ( power_power @ A @ Z @ N ) )
=> ( ( ord_less @ nat @ ( zero_zero @ nat ) @ N )
=> ( ( real_V7770717601297561774m_norm @ A @ W )
= ( real_V7770717601297561774m_norm @ A @ Z ) ) ) ) ) ).
% power_eq_imp_eq_norm
thf(fact_2444_nonzero__norm__divide,axiom,
! [A: $tType] :
( ( real_V3459762299906320749_field @ A )
=> ! [B3: A,A3: A] :
( ( B3
!= ( zero_zero @ A ) )
=> ( ( real_V7770717601297561774m_norm @ A @ ( divide_divide @ A @ A3 @ B3 ) )
= ( divide_divide @ real @ ( real_V7770717601297561774m_norm @ A @ A3 ) @ ( real_V7770717601297561774m_norm @ A @ B3 ) ) ) ) ) ).
% nonzero_norm_divide
thf(fact_2445_norm__mult__less,axiom,
! [A: $tType] :
( ( real_V4412858255891104859lgebra @ A )
=> ! [X2: A,R2: real,Y3: A,S2: real] :
( ( ord_less @ real @ ( real_V7770717601297561774m_norm @ A @ X2 ) @ R2 )
=> ( ( ord_less @ real @ ( real_V7770717601297561774m_norm @ A @ Y3 ) @ S2 )
=> ( ord_less @ real @ ( real_V7770717601297561774m_norm @ A @ ( times_times @ A @ X2 @ Y3 ) ) @ ( times_times @ real @ R2 @ S2 ) ) ) ) ) ).
% norm_mult_less
thf(fact_2446_norm__mult__ineq,axiom,
! [A: $tType] :
( ( real_V4412858255891104859lgebra @ A )
=> ! [X2: A,Y3: A] : ( ord_less_eq @ real @ ( real_V7770717601297561774m_norm @ A @ ( times_times @ A @ X2 @ Y3 ) ) @ ( times_times @ real @ ( real_V7770717601297561774m_norm @ A @ X2 ) @ ( real_V7770717601297561774m_norm @ A @ Y3 ) ) ) ) ).
% norm_mult_ineq
thf(fact_2447_norm__add__less,axiom,
! [A: $tType] :
( ( real_V822414075346904944vector @ A )
=> ! [X2: A,R2: real,Y3: A,S2: real] :
( ( ord_less @ real @ ( real_V7770717601297561774m_norm @ A @ X2 ) @ R2 )
=> ( ( ord_less @ real @ ( real_V7770717601297561774m_norm @ A @ Y3 ) @ S2 )
=> ( ord_less @ real @ ( real_V7770717601297561774m_norm @ A @ ( plus_plus @ A @ X2 @ Y3 ) ) @ ( plus_plus @ real @ R2 @ S2 ) ) ) ) ) ).
% norm_add_less
thf(fact_2448_norm__triangle__lt,axiom,
! [A: $tType] :
( ( real_V822414075346904944vector @ A )
=> ! [X2: A,Y3: A,E: real] :
( ( ord_less @ real @ ( plus_plus @ real @ ( real_V7770717601297561774m_norm @ A @ X2 ) @ ( real_V7770717601297561774m_norm @ A @ Y3 ) ) @ E )
=> ( ord_less @ real @ ( real_V7770717601297561774m_norm @ A @ ( plus_plus @ A @ X2 @ Y3 ) ) @ E ) ) ) ).
% norm_triangle_lt
thf(fact_2449_norm__triangle__le,axiom,
! [A: $tType] :
( ( real_V822414075346904944vector @ A )
=> ! [X2: A,Y3: A,E: real] :
( ( ord_less_eq @ real @ ( plus_plus @ real @ ( real_V7770717601297561774m_norm @ A @ X2 ) @ ( real_V7770717601297561774m_norm @ A @ Y3 ) ) @ E )
=> ( ord_less_eq @ real @ ( real_V7770717601297561774m_norm @ A @ ( plus_plus @ A @ X2 @ Y3 ) ) @ E ) ) ) ).
% norm_triangle_le
thf(fact_2450_norm__triangle__mono,axiom,
! [A: $tType] :
( ( real_V822414075346904944vector @ A )
=> ! [A3: A,R2: real,B3: A,S2: real] :
( ( ord_less_eq @ real @ ( real_V7770717601297561774m_norm @ A @ A3 ) @ R2 )
=> ( ( ord_less_eq @ real @ ( real_V7770717601297561774m_norm @ A @ B3 ) @ S2 )
=> ( ord_less_eq @ real @ ( real_V7770717601297561774m_norm @ A @ ( plus_plus @ A @ A3 @ B3 ) ) @ ( plus_plus @ real @ R2 @ S2 ) ) ) ) ) ).
% norm_triangle_mono
thf(fact_2451_norm__triangle__ineq,axiom,
! [A: $tType] :
( ( real_V822414075346904944vector @ A )
=> ! [X2: A,Y3: A] : ( ord_less_eq @ real @ ( real_V7770717601297561774m_norm @ A @ ( plus_plus @ A @ X2 @ Y3 ) ) @ ( plus_plus @ real @ ( real_V7770717601297561774m_norm @ A @ X2 ) @ ( real_V7770717601297561774m_norm @ A @ Y3 ) ) ) ) ).
% norm_triangle_ineq
thf(fact_2452_norm__add__leD,axiom,
! [A: $tType] :
( ( real_V822414075346904944vector @ A )
=> ! [A3: A,B3: A,C3: real] :
( ( ord_less_eq @ real @ ( real_V7770717601297561774m_norm @ A @ ( plus_plus @ A @ A3 @ B3 ) ) @ C3 )
=> ( ord_less_eq @ real @ ( real_V7770717601297561774m_norm @ A @ B3 ) @ ( plus_plus @ real @ ( real_V7770717601297561774m_norm @ A @ A3 ) @ C3 ) ) ) ) ).
% norm_add_leD
thf(fact_2453_norm__diff__triangle__less,axiom,
! [A: $tType] :
( ( real_V822414075346904944vector @ A )
=> ! [X2: A,Y3: A,E1: real,Z: A,E22: real] :
( ( ord_less @ real @ ( real_V7770717601297561774m_norm @ A @ ( minus_minus @ A @ X2 @ Y3 ) ) @ E1 )
=> ( ( ord_less @ real @ ( real_V7770717601297561774m_norm @ A @ ( minus_minus @ A @ Y3 @ Z ) ) @ E22 )
=> ( ord_less @ real @ ( real_V7770717601297561774m_norm @ A @ ( minus_minus @ A @ X2 @ Z ) ) @ ( plus_plus @ real @ E1 @ E22 ) ) ) ) ) ).
% norm_diff_triangle_less
thf(fact_2454_norm__power__ineq,axiom,
! [A: $tType] :
( ( real_V2822296259951069270ebra_1 @ A )
=> ! [X2: A,N: nat] : ( ord_less_eq @ real @ ( real_V7770717601297561774m_norm @ A @ ( power_power @ A @ X2 @ N ) ) @ ( power_power @ real @ ( real_V7770717601297561774m_norm @ A @ X2 ) @ N ) ) ) ).
% norm_power_ineq
thf(fact_2455_norm__triangle__le__diff,axiom,
! [A: $tType] :
( ( real_V822414075346904944vector @ A )
=> ! [X2: A,Y3: A,E: real] :
( ( ord_less_eq @ real @ ( plus_plus @ real @ ( real_V7770717601297561774m_norm @ A @ X2 ) @ ( real_V7770717601297561774m_norm @ A @ Y3 ) ) @ E )
=> ( ord_less_eq @ real @ ( real_V7770717601297561774m_norm @ A @ ( minus_minus @ A @ X2 @ Y3 ) ) @ E ) ) ) ).
% norm_triangle_le_diff
thf(fact_2456_norm__diff__triangle__le,axiom,
! [A: $tType] :
( ( real_V822414075346904944vector @ A )
=> ! [X2: A,Y3: A,E1: real,Z: A,E22: real] :
( ( ord_less_eq @ real @ ( real_V7770717601297561774m_norm @ A @ ( minus_minus @ A @ X2 @ Y3 ) ) @ E1 )
=> ( ( ord_less_eq @ real @ ( real_V7770717601297561774m_norm @ A @ ( minus_minus @ A @ Y3 @ Z ) ) @ E22 )
=> ( ord_less_eq @ real @ ( real_V7770717601297561774m_norm @ A @ ( minus_minus @ A @ X2 @ Z ) ) @ ( plus_plus @ real @ E1 @ E22 ) ) ) ) ) ).
% norm_diff_triangle_le
thf(fact_2457_norm__triangle__ineq4,axiom,
! [A: $tType] :
( ( real_V822414075346904944vector @ A )
=> ! [A3: A,B3: A] : ( ord_less_eq @ real @ ( real_V7770717601297561774m_norm @ A @ ( minus_minus @ A @ A3 @ B3 ) ) @ ( plus_plus @ real @ ( real_V7770717601297561774m_norm @ A @ A3 ) @ ( real_V7770717601297561774m_norm @ A @ B3 ) ) ) ) ).
% norm_triangle_ineq4
thf(fact_2458_norm__triangle__sub,axiom,
! [A: $tType] :
( ( real_V822414075346904944vector @ A )
=> ! [X2: A,Y3: A] : ( ord_less_eq @ real @ ( real_V7770717601297561774m_norm @ A @ X2 ) @ ( plus_plus @ real @ ( real_V7770717601297561774m_norm @ A @ Y3 ) @ ( real_V7770717601297561774m_norm @ A @ ( minus_minus @ A @ X2 @ Y3 ) ) ) ) ) ).
% norm_triangle_sub
thf(fact_2459_norm__diff__ineq,axiom,
! [A: $tType] :
( ( real_V822414075346904944vector @ A )
=> ! [A3: A,B3: A] : ( ord_less_eq @ real @ ( minus_minus @ real @ ( real_V7770717601297561774m_norm @ A @ A3 ) @ ( real_V7770717601297561774m_norm @ A @ B3 ) ) @ ( real_V7770717601297561774m_norm @ A @ ( plus_plus @ A @ A3 @ B3 ) ) ) ) ).
% norm_diff_ineq
thf(fact_2460_norm__triangle__ineq2,axiom,
! [A: $tType] :
( ( real_V822414075346904944vector @ A )
=> ! [A3: A,B3: A] : ( ord_less_eq @ real @ ( minus_minus @ real @ ( real_V7770717601297561774m_norm @ A @ A3 ) @ ( real_V7770717601297561774m_norm @ A @ B3 ) ) @ ( real_V7770717601297561774m_norm @ A @ ( minus_minus @ A @ A3 @ B3 ) ) ) ) ).
% norm_triangle_ineq2
thf(fact_2461_power__eq__1__iff,axiom,
! [A: $tType] :
( ( real_V8999393235501362500lgebra @ A )
=> ! [W: A,N: nat] :
( ( ( power_power @ A @ W @ N )
= ( one_one @ A ) )
=> ( ( ( real_V7770717601297561774m_norm @ A @ W )
= ( one_one @ real ) )
| ( N
= ( zero_zero @ nat ) ) ) ) ) ).
% power_eq_1_iff
thf(fact_2462_norm__diff__triangle__ineq,axiom,
! [A: $tType] :
( ( real_V822414075346904944vector @ A )
=> ! [A3: A,B3: A,C3: A,D2: A] : ( ord_less_eq @ real @ ( real_V7770717601297561774m_norm @ A @ ( minus_minus @ A @ ( plus_plus @ A @ A3 @ B3 ) @ ( plus_plus @ A @ C3 @ D2 ) ) ) @ ( plus_plus @ real @ ( real_V7770717601297561774m_norm @ A @ ( minus_minus @ A @ A3 @ C3 ) ) @ ( real_V7770717601297561774m_norm @ A @ ( minus_minus @ A @ B3 @ D2 ) ) ) ) ) ).
% norm_diff_triangle_ineq
thf(fact_2463_square__norm__one,axiom,
! [A: $tType] :
( ( real_V8999393235501362500lgebra @ A )
=> ! [X2: A] :
( ( ( power_power @ A @ X2 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) )
= ( one_one @ A ) )
=> ( ( real_V7770717601297561774m_norm @ A @ X2 )
= ( one_one @ real ) ) ) ) ).
% square_norm_one
thf(fact_2464_norm__power__diff,axiom,
! [A: $tType] :
( ( ( comm_monoid_mult @ A )
& ( real_V2822296259951069270ebra_1 @ A ) )
=> ! [Z: A,W: A,M: nat] :
( ( ord_less_eq @ real @ ( real_V7770717601297561774m_norm @ A @ Z ) @ ( one_one @ real ) )
=> ( ( ord_less_eq @ real @ ( real_V7770717601297561774m_norm @ A @ W ) @ ( one_one @ real ) )
=> ( ord_less_eq @ real @ ( real_V7770717601297561774m_norm @ A @ ( minus_minus @ A @ ( power_power @ A @ Z @ M ) @ ( power_power @ A @ W @ M ) ) ) @ ( times_times @ real @ ( semiring_1_of_nat @ real @ M ) @ ( real_V7770717601297561774m_norm @ A @ ( minus_minus @ A @ Z @ W ) ) ) ) ) ) ) ).
% norm_power_diff
thf(fact_2465_list__every__elemnt__bound__sum__bound__real,axiom,
! [A: $tType,Xs: list @ A,F2: A > real,Bound: real,I: real] :
( ! [X4: A] :
( ( member @ A @ X4 @ ( set2 @ A @ Xs ) )
=> ( ord_less_eq @ real @ ( F2 @ X4 ) @ Bound ) )
=> ( ord_less_eq @ real @ ( foldr @ real @ real @ ( plus_plus @ real ) @ ( map @ A @ real @ F2 @ Xs ) @ I ) @ ( plus_plus @ real @ ( times_times @ real @ ( semiring_1_of_nat @ real @ ( size_size @ ( list @ A ) @ Xs ) ) @ Bound ) @ I ) ) ) ).
% list_every_elemnt_bound_sum_bound_real
thf(fact_2466_f__g__map__foldr__bound,axiom,
! [A: $tType,Xs: list @ A,F2: A > real,C3: real,G: A > real,D2: real] :
( ! [X4: A] :
( ( member @ A @ X4 @ ( set2 @ A @ Xs ) )
=> ( ord_less_eq @ real @ ( F2 @ X4 ) @ ( times_times @ real @ C3 @ ( G @ X4 ) ) ) )
=> ( ord_less_eq @ real @ ( foldr @ real @ real @ ( plus_plus @ real ) @ ( map @ A @ real @ F2 @ Xs ) @ D2 ) @ ( plus_plus @ real @ ( times_times @ real @ C3 @ ( foldr @ real @ real @ ( plus_plus @ real ) @ ( map @ A @ real @ G @ Xs ) @ ( zero_zero @ real ) ) ) @ D2 ) ) ) ).
% f_g_map_foldr_bound
thf(fact_2467_list__every__elemnt__bound__sum__bound,axiom,
! [A: $tType,Xs: list @ A,F2: A > nat,Bound: nat,I: nat] :
( ! [X4: A] :
( ( member @ A @ X4 @ ( set2 @ A @ Xs ) )
=> ( ord_less_eq @ nat @ ( F2 @ X4 ) @ Bound ) )
=> ( ord_less_eq @ nat @ ( foldr @ nat @ nat @ ( plus_plus @ nat ) @ ( map @ A @ nat @ F2 @ Xs ) @ I ) @ ( plus_plus @ nat @ ( times_times @ nat @ ( size_size @ ( list @ A ) @ Xs ) @ Bound ) @ I ) ) ) ).
% list_every_elemnt_bound_sum_bound
thf(fact_2468_vebt__memberi__refines,axiom,
! [Ti: vEBT_VEBTi,X2: nat,T2: vEBT_VEBT] : ( refine_Imp_refines @ $o @ ( vEBT_vebt_memberi @ Ti @ X2 ) @ ( vEBT_V854960066525838166emberi @ T2 @ Ti @ X2 ) ) ).
% vebt_memberi_refines
thf(fact_2469_Suc__if__eq,axiom,
! [A: $tType,F2: nat > A,H2: nat > A,G: A,N: nat] :
( ! [N3: nat] :
( ( F2 @ ( suc @ N3 ) )
= ( H2 @ N3 ) )
=> ( ( ( F2 @ ( zero_zero @ nat ) )
= G )
=> ( ( ( N
= ( zero_zero @ nat ) )
=> ( ( F2 @ N )
= G ) )
& ( ( N
!= ( zero_zero @ nat ) )
=> ( ( F2 @ N )
= ( H2 @ ( minus_minus @ nat @ N @ ( one_one @ nat ) ) ) ) ) ) ) ) ).
% Suc_if_eq
thf(fact_2470_refines__replicate,axiom,
! [A: $tType,F2: heap_Time_Heap @ A,F7: heap_Time_Heap @ A,N: nat] :
( ( refine_Imp_refines @ A @ F2 @ F7 )
=> ( refine_Imp_refines @ ( list @ A ) @ ( vEBT_VEBT_replicatei @ A @ N @ F2 ) @ ( vEBT_VEBT_replicatei @ A @ N @ F7 ) ) ) ).
% refines_replicate
thf(fact_2471_map__ident,axiom,
! [A: $tType] :
( ( map @ A @ A
@ ^ [X3: A] : X3 )
= ( ^ [Xs2: list @ A] : Xs2 ) ) ).
% map_ident
thf(fact_2472_listsum__bound,axiom,
! [A: $tType,Xs: list @ A,F2: A > real,Y3: real] :
( ! [X4: A] :
( ( member @ A @ X4 @ ( set2 @ A @ Xs ) )
=> ( ord_less_eq @ real @ ( zero_zero @ real ) @ ( F2 @ X4 ) ) )
=> ( ord_less_eq @ real @ Y3 @ ( foldr @ real @ real @ ( plus_plus @ real ) @ ( map @ A @ real @ F2 @ Xs ) @ Y3 ) ) ) ).
% listsum_bound
thf(fact_2473_length__map,axiom,
! [A: $tType,B: $tType,F2: B > A,Xs: list @ B] :
( ( size_size @ ( list @ A ) @ ( map @ B @ A @ F2 @ Xs ) )
= ( size_size @ ( list @ B ) @ Xs ) ) ).
% length_map
thf(fact_2474_map__eq__conv,axiom,
! [A: $tType,B: $tType,F2: B > A,Xs: list @ B,G: B > A] :
( ( ( map @ B @ A @ F2 @ Xs )
= ( map @ B @ A @ G @ Xs ) )
= ( ! [X3: B] :
( ( member @ B @ X3 @ ( set2 @ B @ Xs ) )
=> ( ( F2 @ X3 )
= ( G @ X3 ) ) ) ) ) ).
% map_eq_conv
thf(fact_2475_real__nat__list,axiom,
! [A: $tType,F2: A > nat,Xs: list @ A,C3: nat] :
( ( semiring_1_of_nat @ real @ ( foldr @ nat @ nat @ ( plus_plus @ nat ) @ ( map @ A @ nat @ F2 @ Xs ) @ C3 ) )
= ( foldr @ real @ real @ ( plus_plus @ real )
@ ( map @ A @ real
@ ^ [X3: A] : ( semiring_1_of_nat @ real @ ( F2 @ X3 ) )
@ Xs )
@ ( semiring_1_of_nat @ real @ C3 ) ) ) ).
% real_nat_list
thf(fact_2476_nth__map,axiom,
! [B: $tType,A: $tType,N: nat,Xs: list @ A,F2: A > B] :
( ( ord_less @ nat @ N @ ( size_size @ ( list @ A ) @ Xs ) )
=> ( ( nth @ B @ ( map @ A @ B @ F2 @ Xs ) @ N )
= ( F2 @ ( nth @ A @ Xs @ N ) ) ) ) ).
% nth_map
thf(fact_2477_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_2478_list_Omap__ident,axiom,
! [A: $tType,T2: list @ A] :
( ( map @ A @ A
@ ^ [X3: A] : X3
@ T2 )
= T2 ) ).
% list.map_ident
thf(fact_2479_map__eq__imp__length__eq,axiom,
! [A: $tType,B: $tType,C: $tType,F2: B > A,Xs: list @ B,G: C > A,Ys: list @ C] :
( ( ( map @ B @ A @ F2 @ Xs )
= ( map @ C @ A @ G @ Ys ) )
=> ( ( size_size @ ( list @ B ) @ Xs )
= ( size_size @ ( list @ C ) @ Ys ) ) ) ).
% map_eq_imp_length_eq
thf(fact_2480_list_Omap__cong,axiom,
! [B: $tType,A: $tType,X2: list @ A,Ya: list @ A,F2: A > B,G: A > B] :
( ( X2 = Ya )
=> ( ! [Z2: A] :
( ( member @ A @ Z2 @ ( set2 @ A @ Ya ) )
=> ( ( F2 @ Z2 )
= ( G @ Z2 ) ) )
=> ( ( map @ A @ B @ F2 @ X2 )
= ( map @ A @ B @ G @ Ya ) ) ) ) ).
% list.map_cong
thf(fact_2481_list_Omap__cong0,axiom,
! [B: $tType,A: $tType,X2: list @ A,F2: A > B,G: A > B] :
( ! [Z2: A] :
( ( member @ A @ Z2 @ ( set2 @ A @ X2 ) )
=> ( ( F2 @ Z2 )
= ( G @ Z2 ) ) )
=> ( ( map @ A @ B @ F2 @ X2 )
= ( map @ A @ B @ G @ X2 ) ) ) ).
% list.map_cong0
thf(fact_2482_list_Oinj__map__strong,axiom,
! [B: $tType,A: $tType,X2: list @ A,Xa: list @ A,F2: A > B,Fa: A > B] :
( ! [Z2: A,Za: A] :
( ( member @ A @ Z2 @ ( set2 @ A @ X2 ) )
=> ( ( member @ A @ Za @ ( set2 @ A @ Xa ) )
=> ( ( ( F2 @ Z2 )
= ( Fa @ Za ) )
=> ( Z2 = Za ) ) ) )
=> ( ( ( map @ A @ B @ F2 @ X2 )
= ( map @ A @ B @ Fa @ Xa ) )
=> ( X2 = Xa ) ) ) ).
% list.inj_map_strong
thf(fact_2483_map__ext,axiom,
! [B: $tType,A: $tType,Xs: list @ A,F2: A > B,G: A > B] :
( ! [X4: A] :
( ( member @ A @ X4 @ ( set2 @ A @ Xs ) )
=> ( ( F2 @ X4 )
= ( G @ X4 ) ) )
=> ( ( map @ A @ B @ F2 @ Xs )
= ( map @ A @ B @ G @ Xs ) ) ) ).
% map_ext
thf(fact_2484_map__idI,axiom,
! [A: $tType,Xs: list @ A,F2: A > A] :
( ! [X4: A] :
( ( member @ A @ X4 @ ( set2 @ A @ Xs ) )
=> ( ( F2 @ X4 )
= X4 ) )
=> ( ( map @ A @ A @ F2 @ Xs )
= Xs ) ) ).
% map_idI
thf(fact_2485_map__cong,axiom,
! [B: $tType,A: $tType,Xs: list @ A,Ys: list @ A,F2: A > B,G: A > B] :
( ( Xs = Ys )
=> ( ! [X4: A] :
( ( member @ A @ X4 @ ( set2 @ A @ Ys ) )
=> ( ( F2 @ X4 )
= ( G @ X4 ) ) )
=> ( ( map @ A @ B @ F2 @ Xs )
= ( map @ A @ B @ G @ Ys ) ) ) ) ).
% map_cong
thf(fact_2486_ex__map__conv,axiom,
! [A: $tType,B: $tType,Ys: list @ B,F2: A > B] :
( ( ? [Xs2: list @ A] :
( Ys
= ( map @ A @ B @ F2 @ Xs2 ) ) )
= ( ! [X3: B] :
( ( member @ B @ X3 @ ( set2 @ B @ Ys ) )
=> ? [Y: A] :
( X3
= ( F2 @ Y ) ) ) ) ) ).
% ex_map_conv
thf(fact_2487_map__eq__nth__eq,axiom,
! [A: $tType,B: $tType,F2: B > A,L: list @ B,L4: list @ B,I: nat] :
( ( ( map @ B @ A @ F2 @ L )
= ( map @ B @ A @ F2 @ L4 ) )
=> ( ( F2 @ ( nth @ B @ L @ I ) )
= ( F2 @ ( nth @ B @ L4 @ I ) ) ) ) ).
% map_eq_nth_eq
thf(fact_2488_map__update,axiom,
! [A: $tType,B: $tType,F2: B > A,Xs: list @ B,K: nat,Y3: B] :
( ( map @ B @ A @ F2 @ ( list_update @ B @ Xs @ K @ Y3 ) )
= ( list_update @ A @ ( map @ B @ A @ F2 @ Xs ) @ K @ ( F2 @ Y3 ) ) ) ).
% map_update
thf(fact_2489_map__upd__eq,axiom,
! [B: $tType,A: $tType,I: nat,L: list @ A,F2: A > B,X2: A] :
( ( ( ord_less @ nat @ I @ ( size_size @ ( list @ A ) @ L ) )
=> ( ( F2 @ ( nth @ A @ L @ I ) )
= ( F2 @ X2 ) ) )
=> ( ( map @ A @ B @ F2 @ ( list_update @ A @ L @ I @ X2 ) )
= ( map @ A @ B @ F2 @ L ) ) ) ).
% map_upd_eq
thf(fact_2490_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_2491_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_2492_VEBT__internal_Ocnt_H_Oelims,axiom,
! [X2: vEBT_VEBT,Y3: nat] :
( ( ( vEBT_VEBT_cnt2 @ X2 )
= Y3 )
=> ( ( ? [A4: $o,B4: $o] :
( X2
= ( vEBT_Leaf @ A4 @ B4 ) )
=> ( Y3
!= ( one_one @ nat ) ) )
=> ~ ! [Info2: option @ ( product_prod @ nat @ nat ),Deg2: nat,TreeList2: list @ vEBT_VEBT,Summary2: vEBT_VEBT] :
( ( X2
= ( vEBT_Node @ Info2 @ Deg2 @ TreeList2 @ Summary2 ) )
=> ( Y3
!= ( 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_2493_VEBT__internal_Ocnt_Oelims,axiom,
! [X2: vEBT_VEBT,Y3: real] :
( ( ( vEBT_VEBT_cnt @ X2 )
= Y3 )
=> ( ( ? [A4: $o,B4: $o] :
( X2
= ( vEBT_Leaf @ A4 @ B4 ) )
=> ( Y3
!= ( one_one @ real ) ) )
=> ~ ! [Info2: option @ ( product_prod @ nat @ nat ),Deg2: nat,TreeList2: list @ vEBT_VEBT,Summary2: vEBT_VEBT] :
( ( X2
= ( vEBT_Node @ Info2 @ Deg2 @ TreeList2 @ Summary2 ) )
=> ( Y3
!= ( 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_2494_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_2495_VEBT__internal_Ospace_H_Oelims,axiom,
! [X2: vEBT_VEBT,Y3: nat] :
( ( ( vEBT_VEBT_space2 @ X2 )
= Y3 )
=> ( ( ? [A4: $o,B4: $o] :
( X2
= ( vEBT_Leaf @ A4 @ B4 ) )
=> ( Y3
!= ( numeral_numeral @ nat @ ( bit0 @ ( bit0 @ one2 ) ) ) ) )
=> ~ ! [Info2: option @ ( product_prod @ nat @ nat ),Deg2: nat,TreeList2: list @ vEBT_VEBT,Summary2: vEBT_VEBT] :
( ( X2
= ( vEBT_Node @ Info2 @ Deg2 @ TreeList2 @ Summary2 ) )
=> ( Y3
!= ( 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_2496_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_2497_VEBT__internal_Ospace_Oelims,axiom,
! [X2: vEBT_VEBT,Y3: nat] :
( ( ( vEBT_VEBT_space @ X2 )
= Y3 )
=> ( ( ? [A4: $o,B4: $o] :
( X2
= ( vEBT_Leaf @ A4 @ B4 ) )
=> ( Y3
!= ( numeral_numeral @ nat @ ( bit1 @ one2 ) ) ) )
=> ~ ! [Info2: option @ ( product_prod @ nat @ nat ),Deg2: nat,TreeList2: list @ vEBT_VEBT,Summary2: vEBT_VEBT] :
( ( X2
= ( vEBT_Node @ Info2 @ Deg2 @ TreeList2 @ Summary2 ) )
=> ( Y3
!= ( 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_2498_round__unique,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [X2: A,Y3: int] :
( ( ord_less @ A @ ( minus_minus @ A @ X2 @ ( divide_divide @ A @ ( one_one @ A ) @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) ) @ ( ring_1_of_int @ A @ Y3 ) )
=> ( ( ord_less_eq @ A @ ( ring_1_of_int @ A @ Y3 ) @ ( plus_plus @ A @ X2 @ ( divide_divide @ A @ ( one_one @ A ) @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) ) )
=> ( ( archimedean_round @ A @ X2 )
= Y3 ) ) ) ) ).
% round_unique
thf(fact_2499_mult__le__cancel__iff1,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [Z: A,X2: A,Y3: A] :
( ( ord_less @ A @ ( zero_zero @ A ) @ Z )
=> ( ( ord_less_eq @ A @ ( times_times @ A @ X2 @ Z ) @ ( times_times @ A @ Y3 @ Z ) )
= ( ord_less_eq @ A @ X2 @ Y3 ) ) ) ) ).
% mult_le_cancel_iff1
thf(fact_2500_mult__le__cancel__iff2,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [Z: A,X2: A,Y3: A] :
( ( ord_less @ A @ ( zero_zero @ A ) @ Z )
=> ( ( ord_less_eq @ A @ ( times_times @ A @ Z @ X2 ) @ ( times_times @ A @ Z @ Y3 ) )
= ( ord_less_eq @ A @ X2 @ Y3 ) ) ) ) ).
% mult_le_cancel_iff2
thf(fact_2501_vebt__succi__refines,axiom,
! [Ti: vEBT_VEBTi,X2: nat,T2: vEBT_VEBT] : ( refine_Imp_refines @ ( option @ nat ) @ ( vEBT_vebt_succi @ Ti @ X2 ) @ ( vEBT_VEBT_vebt_succi @ T2 @ Ti @ X2 ) ) ).
% vebt_succi_refines
thf(fact_2502_vebt__predi__refines,axiom,
! [Ti: vEBT_VEBTi,X2: nat,T2: vEBT_VEBT] : ( refine_Imp_refines @ ( option @ nat ) @ ( vEBT_vebt_predi @ Ti @ X2 ) @ ( vEBT_VEBT_vebt_predi @ T2 @ Ti @ X2 ) ) ).
% vebt_predi_refines
thf(fact_2503_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_2504_vebt__buildupi__refines,axiom,
! [N: nat] : ( refine_Imp_refines @ vEBT_VEBTi @ ( vEBT_vebt_buildupi @ N ) @ ( vEBT_V739175172307565963ildupi @ N ) ) ).
% vebt_buildupi_refines
thf(fact_2505_vebt__inserti__refines,axiom,
! [Ti: vEBT_VEBTi,X2: nat,T2: vEBT_VEBT] : ( refine_Imp_refines @ vEBT_VEBTi @ ( vEBT_vebt_inserti @ Ti @ X2 ) @ ( vEBT_V3964819847710782039nserti @ T2 @ Ti @ X2 ) ) ).
% vebt_inserti_refines
thf(fact_2506_round__numeral,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [N: num] :
( ( archimedean_round @ A @ ( numeral_numeral @ A @ N ) )
= ( numeral_numeral @ int @ N ) ) ) ).
% round_numeral
thf(fact_2507_round__1,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ( ( archimedean_round @ A @ ( one_one @ A ) )
= ( one_one @ int ) ) ) ).
% round_1
thf(fact_2508_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_2509_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_2510_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_2511_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_2512_TBOUND__VEBT__case,axiom,
! [A: $tType,Ti: vEBT_VEBTi,F2: $o > $o > ( heap_Time_Heap @ A ),Bnd: $o > $o > nat,F7: ( 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 @ ( F2 @ 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 @ ( F7 @ Info2 @ Deg2 @ TreeArray @ Summary2 ) @ ( Bnd2 @ Info2 @ Deg2 @ TreeArray @ Summary2 ) ) )
=> ( time_TBOUND @ A @ ( vEBT_case_VEBTi @ ( heap_Time_Heap @ A ) @ F7 @ F2 @ Ti ) @ ( vEBT_case_VEBTi @ nat @ Bnd2 @ Bnd @ Ti ) ) ) ) ).
% TBOUND_VEBT_case
thf(fact_2513_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
@ ^ [X15: option @ ( product_prod @ nat @ nat ),X24: nat,X34: array @ vEBT_VEBTi,X42: vEBT_VEBTi] : ( H2 @ ( F1 @ X15 @ X24 @ X34 @ X42 ) )
@ ^ [X15: $o,X24: $o] : ( H2 @ ( F22 @ X15 @ X24 ) )
@ VEBTi ) ) ).
% VEBTi.case_distrib
thf(fact_2514_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_2515_round__mono,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [X2: A,Y3: A] :
( ( ord_less_eq @ A @ X2 @ Y3 )
=> ( ord_less_eq @ int @ ( archimedean_round @ A @ X2 ) @ ( archimedean_round @ A @ Y3 ) ) ) ) ).
% round_mono
thf(fact_2516_ceiling__ge__round,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [X2: A] : ( ord_less_eq @ int @ ( archimedean_round @ A @ X2 ) @ ( archimedean_ceiling @ A @ X2 ) ) ) ).
% ceiling_ge_round
thf(fact_2517_divmod__divmod__step,axiom,
! [A: $tType] :
( ( unique1627219031080169319umeral @ A )
=> ( ( unique8689654367752047608divmod @ A )
= ( ^ [M6: num,N4: num] : ( if @ ( product_prod @ A @ A ) @ ( ord_less @ num @ M6 @ N4 ) @ ( product_Pair @ A @ A @ ( zero_zero @ A ) @ ( numeral_numeral @ A @ M6 ) ) @ ( unique1321980374590559556d_step @ A @ N4 @ ( unique8689654367752047608divmod @ A @ M6 @ ( bit0 @ N4 ) ) ) ) ) ) ) ).
% divmod_divmod_step
thf(fact_2518_mult__less__iff1,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [Z: A,X2: A,Y3: A] :
( ( ord_less @ A @ ( zero_zero @ A ) @ Z )
=> ( ( ord_less @ A @ ( times_times @ A @ X2 @ Z ) @ ( times_times @ A @ Y3 @ Z ) )
= ( ord_less @ A @ X2 @ Y3 ) ) ) ) ).
% mult_less_iff1
thf(fact_2519_of__int__round__le,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [X2: A] : ( ord_less_eq @ A @ ( ring_1_of_int @ A @ ( archimedean_round @ A @ X2 ) ) @ ( plus_plus @ A @ X2 @ ( divide_divide @ A @ ( one_one @ A ) @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) ) ) ) ).
% of_int_round_le
thf(fact_2520_of__int__round__ge,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [X2: A] : ( ord_less_eq @ A @ ( minus_minus @ A @ X2 @ ( divide_divide @ A @ ( one_one @ A ) @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) ) @ ( ring_1_of_int @ A @ ( archimedean_round @ A @ X2 ) ) ) ) ).
% of_int_round_ge
thf(fact_2521_of__int__round__gt,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [X2: A] : ( ord_less @ A @ ( minus_minus @ A @ X2 @ ( divide_divide @ A @ ( one_one @ A ) @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) ) @ ( ring_1_of_int @ A @ ( archimedean_round @ A @ X2 ) ) ) ) ).
% of_int_round_gt
thf(fact_2522_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_2523_VEBT__internal_OT__vebt__buildupi_H_Oelims,axiom,
! [X2: nat,Y3: int] :
( ( ( vEBT_V9176841429113362141ildupi @ X2 )
= Y3 )
=> ( ( ( X2
= ( zero_zero @ nat ) )
=> ( Y3
!= ( one_one @ int ) ) )
=> ( ( ( X2
= ( suc @ ( zero_zero @ nat ) ) )
=> ( Y3
!= ( one_one @ int ) ) )
=> ~ ! [N3: nat] :
( ( X2
= ( suc @ ( suc @ N3 ) ) )
=> ~ ( ( ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N3 )
=> ( Y3
= ( plus_plus @ int @ ( numeral_numeral @ int @ ( bit1 @ one2 ) ) @ ( plus_plus @ int @ ( vEBT_V9176841429113362141ildupi @ ( suc @ ( divide_divide @ nat @ N3 @ ( 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 @ N3 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( times_times @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ ( times_times @ int @ ( vEBT_V9176841429113362141ildupi @ ( suc @ ( divide_divide @ nat @ N3 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( power_power @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ N3 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) ) ) ) ) )
& ( ~ ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N3 )
=> ( Y3
= ( plus_plus @ int @ ( numeral_numeral @ int @ ( bit1 @ one2 ) ) @ ( plus_plus @ int @ ( vEBT_V9176841429113362141ildupi @ ( suc @ ( suc @ ( divide_divide @ nat @ N3 @ ( 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 @ N3 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( times_times @ int @ ( numeral_numeral @ int @ ( bit0 @ ( bit0 @ one2 ) ) ) @ ( times_times @ int @ ( vEBT_V9176841429113362141ildupi @ ( suc @ ( divide_divide @ nat @ N3 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( power_power @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ N3 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).
% VEBT_internal.T_vebt_buildupi'.elims
thf(fact_2524_highi__def,axiom,
( vEBT_VEBT_highi
= ( ^ [X3: nat,N4: nat] : ( heap_Time_return @ nat @ ( divide_divide @ nat @ X3 @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N4 ) ) ) ) ) ).
% highi_def
thf(fact_2525_nth__rule,axiom,
! [A: $tType] :
( ( heap @ A )
=> ! [I: nat,Xs: list @ A,A3: array @ A] :
( ( ord_less @ nat @ I @ ( size_size @ ( list @ A ) @ Xs ) )
=> ( hoare_hoare_triple @ A @ ( snga_assn @ A @ A3 @ Xs ) @ ( array_nth @ A @ A3 @ I )
@ ^ [R5: A] :
( times_times @ assn @ ( snga_assn @ A @ A3 @ Xs )
@ ( pure_assn
@ ( R5
= ( nth @ A @ Xs @ I ) ) ) ) ) ) ) ).
% nth_rule
thf(fact_2526_unset__bit__0,axiom,
! [A: $tType] :
( ( bit_se359711467146920520ations @ A )
=> ! [A3: A] :
( ( bit_se2638667681897837118et_bit @ A @ ( zero_zero @ nat ) @ A3 )
= ( times_times @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ ( divide_divide @ A @ A3 @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) ) ) ) ).
% unset_bit_0
thf(fact_2527_fi__rule,axiom,
! [A: $tType,P: assn,C3: heap_Time_Heap @ A,Q: A > assn,Ps: assn,F4: assn] :
( ( hoare_hoare_triple @ A @ P @ C3 @ Q )
=> ( ( entails @ Ps @ ( times_times @ assn @ P @ F4 ) )
=> ( hoare_hoare_triple @ A @ Ps @ C3
@ ^ [X3: A] : ( times_times @ assn @ ( Q @ X3 ) @ F4 ) ) ) ) ).
% fi_rule
thf(fact_2528_highsimp,axiom,
! [X2: nat,N: nat] :
( ( heap_Time_return @ nat @ ( vEBT_VEBT_high @ X2 @ N ) )
= ( vEBT_VEBT_highi @ X2 @ N ) ) ).
% highsimp
thf(fact_2529_lowsimp,axiom,
! [X2: nat,N: nat] :
( ( heap_Time_return @ nat @ ( vEBT_VEBT_low @ X2 @ N ) )
= ( vEBT_VEBT_lowi @ X2 @ N ) ) ).
% lowsimp
thf(fact_2530_dvd__add__triv__right__iff,axiom,
! [A: $tType] :
( ( comm_s4317794764714335236cancel @ A )
=> ! [A3: A,B3: A] :
( ( dvd_dvd @ A @ A3 @ ( plus_plus @ A @ B3 @ A3 ) )
= ( dvd_dvd @ A @ A3 @ B3 ) ) ) ).
% dvd_add_triv_right_iff
thf(fact_2531_dvd__add__triv__left__iff,axiom,
! [A: $tType] :
( ( comm_s4317794764714335236cancel @ A )
=> ! [A3: A,B3: A] :
( ( dvd_dvd @ A @ A3 @ ( plus_plus @ A @ A3 @ B3 ) )
= ( dvd_dvd @ A @ A3 @ B3 ) ) ) ).
% dvd_add_triv_left_iff
thf(fact_2532_div__dvd__div,axiom,
! [A: $tType] :
( ( algebraic_semidom @ A )
=> ! [A3: A,B3: A,C3: A] :
( ( dvd_dvd @ A @ A3 @ B3 )
=> ( ( dvd_dvd @ A @ A3 @ C3 )
=> ( ( dvd_dvd @ A @ ( divide_divide @ A @ B3 @ A3 ) @ ( divide_divide @ A @ C3 @ A3 ) )
= ( dvd_dvd @ A @ B3 @ C3 ) ) ) ) ) ).
% div_dvd_div
thf(fact_2533_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_2534_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_2535_dvd__times__right__cancel__iff,axiom,
! [A: $tType] :
( ( algebraic_semidom @ A )
=> ! [A3: A,B3: A,C3: A] :
( ( A3
!= ( zero_zero @ A ) )
=> ( ( dvd_dvd @ A @ ( times_times @ A @ B3 @ A3 ) @ ( times_times @ A @ C3 @ A3 ) )
= ( dvd_dvd @ A @ B3 @ C3 ) ) ) ) ).
% dvd_times_right_cancel_iff
thf(fact_2536_dvd__times__left__cancel__iff,axiom,
! [A: $tType] :
( ( algebraic_semidom @ A )
=> ! [A3: A,B3: A,C3: A] :
( ( A3
!= ( zero_zero @ A ) )
=> ( ( dvd_dvd @ A @ ( times_times @ A @ A3 @ B3 ) @ ( times_times @ A @ A3 @ C3 ) )
= ( dvd_dvd @ A @ B3 @ C3 ) ) ) ) ).
% dvd_times_left_cancel_iff
thf(fact_2537_dvd__mult__cancel__right,axiom,
! [A: $tType] :
( ( idom @ A )
=> ! [A3: A,C3: A,B3: A] :
( ( dvd_dvd @ A @ ( times_times @ A @ A3 @ C3 ) @ ( times_times @ A @ B3 @ C3 ) )
= ( ( C3
= ( zero_zero @ A ) )
| ( dvd_dvd @ A @ A3 @ B3 ) ) ) ) ).
% dvd_mult_cancel_right
thf(fact_2538_dvd__mult__cancel__left,axiom,
! [A: $tType] :
( ( idom @ A )
=> ! [C3: A,A3: A,B3: A] :
( ( dvd_dvd @ A @ ( times_times @ A @ C3 @ A3 ) @ ( times_times @ A @ C3 @ B3 ) )
= ( ( C3
= ( zero_zero @ A ) )
| ( dvd_dvd @ A @ A3 @ B3 ) ) ) ) ).
% dvd_mult_cancel_left
thf(fact_2539_dvd__add__times__triv__right__iff,axiom,
! [A: $tType] :
( ( comm_s4317794764714335236cancel @ A )
=> ! [A3: A,B3: A,C3: A] :
( ( dvd_dvd @ A @ A3 @ ( plus_plus @ A @ B3 @ ( times_times @ A @ C3 @ A3 ) ) )
= ( dvd_dvd @ A @ A3 @ B3 ) ) ) ).
% dvd_add_times_triv_right_iff
thf(fact_2540_dvd__add__times__triv__left__iff,axiom,
! [A: $tType] :
( ( comm_s4317794764714335236cancel @ A )
=> ! [A3: A,C3: A,B3: A] :
( ( dvd_dvd @ A @ A3 @ ( plus_plus @ A @ ( times_times @ A @ C3 @ A3 ) @ B3 ) )
= ( dvd_dvd @ A @ A3 @ B3 ) ) ) ).
% dvd_add_times_triv_left_iff
thf(fact_2541_unit__prod,axiom,
! [A: $tType] :
( ( algebraic_semidom @ A )
=> ! [A3: A,B3: A] :
( ( dvd_dvd @ A @ A3 @ ( one_one @ A ) )
=> ( ( dvd_dvd @ A @ B3 @ ( one_one @ A ) )
=> ( dvd_dvd @ A @ ( times_times @ A @ A3 @ B3 ) @ ( one_one @ A ) ) ) ) ) ).
% unit_prod
thf(fact_2542_div__add,axiom,
! [A: $tType] :
( ( algebraic_semidom @ A )
=> ! [C3: A,A3: A,B3: A] :
( ( dvd_dvd @ A @ C3 @ A3 )
=> ( ( dvd_dvd @ A @ C3 @ B3 )
=> ( ( divide_divide @ A @ ( plus_plus @ A @ A3 @ B3 ) @ C3 )
= ( plus_plus @ A @ ( divide_divide @ A @ A3 @ C3 ) @ ( divide_divide @ A @ B3 @ C3 ) ) ) ) ) ) ).
% div_add
thf(fact_2543_unit__div,axiom,
! [A: $tType] :
( ( algebraic_semidom @ A )
=> ! [A3: A,B3: A] :
( ( dvd_dvd @ A @ A3 @ ( one_one @ A ) )
=> ( ( dvd_dvd @ A @ B3 @ ( one_one @ A ) )
=> ( dvd_dvd @ A @ ( divide_divide @ A @ A3 @ B3 ) @ ( one_one @ A ) ) ) ) ) ).
% unit_div
thf(fact_2544_unit__div__1__unit,axiom,
! [A: $tType] :
( ( algebraic_semidom @ A )
=> ! [A3: A] :
( ( dvd_dvd @ A @ A3 @ ( one_one @ A ) )
=> ( dvd_dvd @ A @ ( divide_divide @ A @ ( one_one @ A ) @ A3 ) @ ( one_one @ A ) ) ) ) ).
% unit_div_1_unit
thf(fact_2545_unit__div__1__div__1,axiom,
! [A: $tType] :
( ( algebraic_semidom @ A )
=> ! [A3: A] :
( ( dvd_dvd @ A @ A3 @ ( one_one @ A ) )
=> ( ( divide_divide @ A @ ( one_one @ A ) @ ( divide_divide @ A @ ( one_one @ A ) @ A3 ) )
= A3 ) ) ) ).
% unit_div_1_div_1
thf(fact_2546_dvd__div__mult__self,axiom,
! [A: $tType] :
( ( algebraic_semidom @ A )
=> ! [A3: A,B3: A] :
( ( dvd_dvd @ A @ A3 @ B3 )
=> ( ( times_times @ A @ ( divide_divide @ A @ B3 @ A3 ) @ A3 )
= B3 ) ) ) ).
% dvd_div_mult_self
thf(fact_2547_dvd__mult__div__cancel,axiom,
! [A: $tType] :
( ( algebraic_semidom @ A )
=> ! [A3: A,B3: A] :
( ( dvd_dvd @ A @ A3 @ B3 )
=> ( ( times_times @ A @ A3 @ ( divide_divide @ A @ B3 @ A3 ) )
= B3 ) ) ) ).
% dvd_mult_div_cancel
thf(fact_2548_div__diff,axiom,
! [A: $tType] :
( ( idom_modulo @ A )
=> ! [C3: A,A3: A,B3: A] :
( ( dvd_dvd @ A @ C3 @ A3 )
=> ( ( dvd_dvd @ A @ C3 @ B3 )
=> ( ( divide_divide @ A @ ( minus_minus @ A @ A3 @ B3 ) @ C3 )
= ( minus_minus @ A @ ( divide_divide @ A @ A3 @ C3 ) @ ( divide_divide @ A @ B3 @ C3 ) ) ) ) ) ) ).
% div_diff
thf(fact_2549_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_2550_dvd__1__left,axiom,
! [K: nat] : ( dvd_dvd @ nat @ ( suc @ ( zero_zero @ nat ) ) @ K ) ).
% dvd_1_left
thf(fact_2551_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_2552_unit__mult__div__div,axiom,
! [A: $tType] :
( ( algebraic_semidom @ A )
=> ! [A3: A,B3: A] :
( ( dvd_dvd @ A @ A3 @ ( one_one @ A ) )
=> ( ( times_times @ A @ B3 @ ( divide_divide @ A @ ( one_one @ A ) @ A3 ) )
= ( divide_divide @ A @ B3 @ A3 ) ) ) ) ).
% unit_mult_div_div
thf(fact_2553_unit__div__mult__self,axiom,
! [A: $tType] :
( ( algebraic_semidom @ A )
=> ! [A3: A,B3: A] :
( ( dvd_dvd @ A @ A3 @ ( one_one @ A ) )
=> ( ( times_times @ A @ ( divide_divide @ A @ B3 @ A3 ) @ A3 )
= B3 ) ) ) ).
% unit_div_mult_self
thf(fact_2554_odd__add,axiom,
! [A: $tType] :
( ( semiring_parity @ A )
=> ! [A3: A,B3: A] :
( ( ~ ( dvd_dvd @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ ( plus_plus @ A @ A3 @ B3 ) ) )
= ( ( ~ ( dvd_dvd @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ A3 ) )
!= ( ~ ( dvd_dvd @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ B3 ) ) ) ) ) ).
% odd_add
thf(fact_2555_even__add,axiom,
! [A: $tType] :
( ( semiring_parity @ A )
=> ! [A3: A,B3: A] :
( ( dvd_dvd @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ ( plus_plus @ A @ A3 @ B3 ) )
= ( ( dvd_dvd @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ A3 )
= ( dvd_dvd @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ B3 ) ) ) ) ).
% even_add
thf(fact_2556_even__mult__iff,axiom,
! [A: $tType] :
( ( semiring_parity @ A )
=> ! [A3: A,B3: A] :
( ( dvd_dvd @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ ( times_times @ A @ A3 @ B3 ) )
= ( ( dvd_dvd @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ A3 )
| ( dvd_dvd @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ B3 ) ) ) ) ).
% even_mult_iff
thf(fact_2557_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_2558_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_2559_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_2560_even__plus__one__iff,axiom,
! [A: $tType] :
( ( semiring_parity @ A )
=> ! [A3: A] :
( ( dvd_dvd @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ ( plus_plus @ A @ A3 @ ( one_one @ A ) ) )
= ( ~ ( dvd_dvd @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ A3 ) ) ) ) ).
% even_plus_one_iff
thf(fact_2561_even__diff,axiom,
! [A: $tType] :
( ( ring_parity @ A )
=> ! [A3: A,B3: A] :
( ( dvd_dvd @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ ( minus_minus @ A @ A3 @ B3 ) )
= ( dvd_dvd @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ ( plus_plus @ A @ A3 @ B3 ) ) ) ) ).
% even_diff
thf(fact_2562_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_2563_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_2564_even__succ__div__2,axiom,
! [A: $tType] :
( ( bit_semiring_bits @ A )
=> ! [A3: A] :
( ( dvd_dvd @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ A3 )
=> ( ( divide_divide @ A @ ( plus_plus @ A @ ( one_one @ A ) @ A3 ) @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) )
= ( divide_divide @ A @ A3 @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) ) ) ) ).
% even_succ_div_2
thf(fact_2565_odd__succ__div__two,axiom,
! [A: $tType] :
( ( euclid5411537665997757685th_nat @ A )
=> ! [A3: A] :
( ~ ( dvd_dvd @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ A3 )
=> ( ( divide_divide @ A @ ( plus_plus @ A @ A3 @ ( one_one @ A ) ) @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) )
= ( plus_plus @ A @ ( divide_divide @ A @ A3 @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) @ ( one_one @ A ) ) ) ) ) ).
% odd_succ_div_two
thf(fact_2566_even__succ__div__two,axiom,
! [A: $tType] :
( ( euclid5411537665997757685th_nat @ A )
=> ! [A3: A] :
( ( dvd_dvd @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ A3 )
=> ( ( divide_divide @ A @ ( plus_plus @ A @ A3 @ ( one_one @ A ) ) @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) )
= ( divide_divide @ A @ A3 @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) ) ) ) ).
% even_succ_div_two
thf(fact_2567_even__power,axiom,
! [A: $tType] :
( ( semiring_parity @ A )
=> ! [A3: A,N: nat] :
( ( dvd_dvd @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ ( power_power @ A @ A3 @ N ) )
= ( ( dvd_dvd @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ A3 )
& ( ord_less @ nat @ ( zero_zero @ nat ) @ N ) ) ) ) ).
% even_power
thf(fact_2568_zero__le__power__eq__numeral,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [A3: A,W: num] :
( ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( power_power @ A @ A3 @ ( numeral_numeral @ nat @ W ) ) )
= ( ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( numeral_numeral @ nat @ W ) )
| ( ~ ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( numeral_numeral @ nat @ W ) )
& ( ord_less_eq @ A @ ( zero_zero @ A ) @ A3 ) ) ) ) ) ).
% zero_le_power_eq_numeral
thf(fact_2569_power__less__zero__eq,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [A3: A,N: nat] :
( ( ord_less @ A @ ( power_power @ A @ A3 @ N ) @ ( zero_zero @ A ) )
= ( ~ ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N )
& ( ord_less @ A @ A3 @ ( zero_zero @ A ) ) ) ) ) ).
% power_less_zero_eq
thf(fact_2570_power__less__zero__eq__numeral,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [A3: A,W: num] :
( ( ord_less @ A @ ( power_power @ A @ A3 @ ( numeral_numeral @ nat @ W ) ) @ ( zero_zero @ A ) )
= ( ~ ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( numeral_numeral @ nat @ W ) )
& ( ord_less @ A @ A3 @ ( zero_zero @ A ) ) ) ) ) ).
% power_less_zero_eq_numeral
thf(fact_2571_even__of__nat,axiom,
! [A: $tType] :
( ( semiring_parity @ A )
=> ! [N: nat] :
( ( dvd_dvd @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ ( semiring_1_of_nat @ A @ N ) )
= ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N ) ) ) ).
% even_of_nat
thf(fact_2572_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_2573_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_2574_odd__two__times__div__two__succ,axiom,
! [A: $tType] :
( ( euclid5411537665997757685th_nat @ A )
=> ! [A3: A] :
( ~ ( dvd_dvd @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ A3 )
=> ( ( plus_plus @ A @ ( times_times @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ ( divide_divide @ A @ A3 @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) ) @ ( one_one @ A ) )
= A3 ) ) ) ).
% odd_two_times_div_two_succ
thf(fact_2575_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_2576_zero__less__power__eq__numeral,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [A3: A,W: num] :
( ( ord_less @ A @ ( zero_zero @ A ) @ ( power_power @ A @ A3 @ ( numeral_numeral @ nat @ W ) ) )
= ( ( ( numeral_numeral @ nat @ W )
= ( zero_zero @ nat ) )
| ( ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( numeral_numeral @ nat @ W ) )
& ( A3
!= ( zero_zero @ A ) ) )
| ( ~ ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( numeral_numeral @ nat @ W ) )
& ( ord_less @ A @ ( zero_zero @ A ) @ A3 ) ) ) ) ) ).
% zero_less_power_eq_numeral
thf(fact_2577_odd__two__times__div__two__nat,axiom,
! [N: nat] :
( ~ ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N )
=> ( ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ N @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) )
= ( minus_minus @ nat @ N @ ( one_one @ nat ) ) ) ) ).
% odd_two_times_div_two_nat
thf(fact_2578_power__le__zero__eq__numeral,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [A3: A,W: num] :
( ( ord_less_eq @ A @ ( power_power @ A @ A3 @ ( numeral_numeral @ nat @ W ) ) @ ( zero_zero @ A ) )
= ( ( ord_less @ nat @ ( zero_zero @ nat ) @ ( numeral_numeral @ nat @ W ) )
& ( ( ~ ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( numeral_numeral @ nat @ W ) )
& ( ord_less_eq @ A @ A3 @ ( zero_zero @ A ) ) )
| ( ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( numeral_numeral @ nat @ W ) )
& ( A3
= ( zero_zero @ A ) ) ) ) ) ) ) ).
% power_le_zero_eq_numeral
thf(fact_2579_even__succ__div__exp,axiom,
! [A: $tType] :
( ( bit_semiring_bits @ A )
=> ! [A3: A,N: nat] :
( ( dvd_dvd @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ A3 )
=> ( ( ord_less @ nat @ ( zero_zero @ nat ) @ N )
=> ( ( divide_divide @ A @ ( plus_plus @ A @ ( one_one @ A ) @ A3 ) @ ( power_power @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ N ) )
= ( divide_divide @ A @ A3 @ ( power_power @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ N ) ) ) ) ) ) ).
% even_succ_div_exp
thf(fact_2580_dvd__antisym,axiom,
! [M: nat,N: nat] :
( ( dvd_dvd @ nat @ M @ N )
=> ( ( dvd_dvd @ nat @ N @ M )
=> ( M = N ) ) ) ).
% dvd_antisym
thf(fact_2581_of__nat__dvd__iff,axiom,
! [A: $tType] :
( ( euclid5411537665997757685th_nat @ A )
=> ! [M: nat,N: nat] :
( ( dvd_dvd @ A @ ( semiring_1_of_nat @ A @ M ) @ ( semiring_1_of_nat @ A @ N ) )
= ( dvd_dvd @ nat @ M @ N ) ) ) ).
% of_nat_dvd_iff
thf(fact_2582_dvd__add__right__iff,axiom,
! [A: $tType] :
( ( comm_s4317794764714335236cancel @ A )
=> ! [A3: A,B3: A,C3: A] :
( ( dvd_dvd @ A @ A3 @ B3 )
=> ( ( dvd_dvd @ A @ A3 @ ( plus_plus @ A @ B3 @ C3 ) )
= ( dvd_dvd @ A @ A3 @ C3 ) ) ) ) ).
% dvd_add_right_iff
thf(fact_2583_dvd__add__left__iff,axiom,
! [A: $tType] :
( ( comm_s4317794764714335236cancel @ A )
=> ! [A3: A,C3: A,B3: A] :
( ( dvd_dvd @ A @ A3 @ C3 )
=> ( ( dvd_dvd @ A @ A3 @ ( plus_plus @ A @ B3 @ C3 ) )
= ( dvd_dvd @ A @ A3 @ B3 ) ) ) ) ).
% dvd_add_left_iff
thf(fact_2584_dvd__add,axiom,
! [A: $tType] :
( ( comm_semiring_1 @ A )
=> ! [A3: A,B3: A,C3: A] :
( ( dvd_dvd @ A @ A3 @ B3 )
=> ( ( dvd_dvd @ A @ A3 @ C3 )
=> ( dvd_dvd @ A @ A3 @ ( plus_plus @ A @ B3 @ C3 ) ) ) ) ) ).
% dvd_add
thf(fact_2585_dvd__unit__imp__unit,axiom,
! [A: $tType] :
( ( algebraic_semidom @ A )
=> ! [A3: A,B3: A] :
( ( dvd_dvd @ A @ A3 @ B3 )
=> ( ( dvd_dvd @ A @ B3 @ ( one_one @ A ) )
=> ( dvd_dvd @ A @ A3 @ ( one_one @ A ) ) ) ) ) ).
% dvd_unit_imp_unit
thf(fact_2586_unit__imp__dvd,axiom,
! [A: $tType] :
( ( algebraic_semidom @ A )
=> ! [B3: A,A3: A] :
( ( dvd_dvd @ A @ B3 @ ( one_one @ A ) )
=> ( dvd_dvd @ A @ B3 @ A3 ) ) ) ).
% unit_imp_dvd
thf(fact_2587_one__dvd,axiom,
! [A: $tType] :
( ( comm_monoid_mult @ A )
=> ! [A3: A] : ( dvd_dvd @ A @ ( one_one @ A ) @ A3 ) ) ).
% one_dvd
thf(fact_2588_dvdE,axiom,
! [A: $tType] :
( ( dvd @ A )
=> ! [B3: A,A3: A] :
( ( dvd_dvd @ A @ B3 @ A3 )
=> ~ ! [K2: A] :
( A3
!= ( times_times @ A @ B3 @ K2 ) ) ) ) ).
% dvdE
thf(fact_2589_dvdI,axiom,
! [A: $tType] :
( ( dvd @ A )
=> ! [A3: A,B3: A,K: A] :
( ( A3
= ( times_times @ A @ B3 @ K ) )
=> ( dvd_dvd @ A @ B3 @ A3 ) ) ) ).
% dvdI
thf(fact_2590_dvd__def,axiom,
! [A: $tType] :
( ( dvd @ A )
=> ( ( dvd_dvd @ A )
= ( ^ [B7: A,A6: A] :
? [K3: A] :
( A6
= ( times_times @ A @ B7 @ K3 ) ) ) ) ) ).
% dvd_def
thf(fact_2591_dvd__mult,axiom,
! [A: $tType] :
( ( comm_monoid_mult @ A )
=> ! [A3: A,C3: A,B3: A] :
( ( dvd_dvd @ A @ A3 @ C3 )
=> ( dvd_dvd @ A @ A3 @ ( times_times @ A @ B3 @ C3 ) ) ) ) ).
% dvd_mult
thf(fact_2592_dvd__mult2,axiom,
! [A: $tType] :
( ( comm_monoid_mult @ A )
=> ! [A3: A,B3: A,C3: A] :
( ( dvd_dvd @ A @ A3 @ B3 )
=> ( dvd_dvd @ A @ A3 @ ( times_times @ A @ B3 @ C3 ) ) ) ) ).
% dvd_mult2
thf(fact_2593_dvd__mult__left,axiom,
! [A: $tType] :
( ( comm_monoid_mult @ A )
=> ! [A3: A,B3: A,C3: A] :
( ( dvd_dvd @ A @ ( times_times @ A @ A3 @ B3 ) @ C3 )
=> ( dvd_dvd @ A @ A3 @ C3 ) ) ) ).
% dvd_mult_left
thf(fact_2594_dvd__triv__left,axiom,
! [A: $tType] :
( ( comm_monoid_mult @ A )
=> ! [A3: A,B3: A] : ( dvd_dvd @ A @ A3 @ ( times_times @ A @ A3 @ B3 ) ) ) ).
% dvd_triv_left
thf(fact_2595_mult__dvd__mono,axiom,
! [A: $tType] :
( ( comm_monoid_mult @ A )
=> ! [A3: A,B3: A,C3: A,D2: A] :
( ( dvd_dvd @ A @ A3 @ B3 )
=> ( ( dvd_dvd @ A @ C3 @ D2 )
=> ( dvd_dvd @ A @ ( times_times @ A @ A3 @ C3 ) @ ( times_times @ A @ B3 @ D2 ) ) ) ) ) ).
% mult_dvd_mono
thf(fact_2596_dvd__mult__right,axiom,
! [A: $tType] :
( ( comm_monoid_mult @ A )
=> ! [A3: A,B3: A,C3: A] :
( ( dvd_dvd @ A @ ( times_times @ A @ A3 @ B3 ) @ C3 )
=> ( dvd_dvd @ A @ B3 @ C3 ) ) ) ).
% dvd_mult_right
thf(fact_2597_dvd__triv__right,axiom,
! [A: $tType] :
( ( comm_monoid_mult @ A )
=> ! [A3: A,B3: A] : ( dvd_dvd @ A @ A3 @ ( times_times @ A @ B3 @ A3 ) ) ) ).
% dvd_triv_right
thf(fact_2598_dvd__diff__commute,axiom,
! [A: $tType] :
( ( euclid5891614535332579305n_ring @ A )
=> ! [A3: A,C3: A,B3: A] :
( ( dvd_dvd @ A @ A3 @ ( minus_minus @ A @ C3 @ B3 ) )
= ( dvd_dvd @ A @ A3 @ ( minus_minus @ A @ B3 @ C3 ) ) ) ) ).
% dvd_diff_commute
thf(fact_2599_dvd__diff,axiom,
! [A: $tType] :
( ( comm_ring_1 @ A )
=> ! [X2: A,Y3: A,Z: A] :
( ( dvd_dvd @ A @ X2 @ Y3 )
=> ( ( dvd_dvd @ A @ X2 @ Z )
=> ( dvd_dvd @ A @ X2 @ ( minus_minus @ A @ Y3 @ Z ) ) ) ) ) ).
% dvd_diff
thf(fact_2600_dvd__div__eq__iff,axiom,
! [A: $tType] :
( ( semidom_divide @ A )
=> ! [C3: A,A3: A,B3: A] :
( ( dvd_dvd @ A @ C3 @ A3 )
=> ( ( dvd_dvd @ A @ C3 @ B3 )
=> ( ( ( divide_divide @ A @ A3 @ C3 )
= ( divide_divide @ A @ B3 @ C3 ) )
= ( A3 = B3 ) ) ) ) ) ).
% dvd_div_eq_iff
thf(fact_2601_dvd__div__eq__cancel,axiom,
! [A: $tType] :
( ( semidom_divide @ A )
=> ! [A3: A,C3: A,B3: A] :
( ( ( divide_divide @ A @ A3 @ C3 )
= ( divide_divide @ A @ B3 @ C3 ) )
=> ( ( dvd_dvd @ A @ C3 @ A3 )
=> ( ( dvd_dvd @ A @ C3 @ B3 )
=> ( A3 = B3 ) ) ) ) ) ).
% dvd_div_eq_cancel
thf(fact_2602_div__div__div__same,axiom,
! [A: $tType] :
( ( algebraic_semidom @ A )
=> ! [D2: A,B3: A,A3: A] :
( ( dvd_dvd @ A @ D2 @ B3 )
=> ( ( dvd_dvd @ A @ B3 @ A3 )
=> ( ( divide_divide @ A @ ( divide_divide @ A @ A3 @ D2 ) @ ( divide_divide @ A @ B3 @ D2 ) )
= ( divide_divide @ A @ A3 @ B3 ) ) ) ) ) ).
% div_div_div_same
thf(fact_2603_dvd__power__same,axiom,
! [A: $tType] :
( ( comm_semiring_1 @ A )
=> ! [X2: A,Y3: A,N: nat] :
( ( dvd_dvd @ A @ X2 @ Y3 )
=> ( dvd_dvd @ A @ ( power_power @ A @ X2 @ N ) @ ( power_power @ A @ Y3 @ N ) ) ) ) ).
% dvd_power_same
thf(fact_2604_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_2605_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_2606_subset__divisors__dvd,axiom,
! [A: $tType] :
( ( comm_monoid_mult @ A )
=> ! [A3: A,B3: A] :
( ( ord_less_eq @ ( set @ A )
@ ( collect @ A
@ ^ [C5: A] : ( dvd_dvd @ A @ C5 @ A3 ) )
@ ( collect @ A
@ ^ [C5: A] : ( dvd_dvd @ A @ C5 @ B3 ) ) )
= ( dvd_dvd @ A @ A3 @ B3 ) ) ) ).
% subset_divisors_dvd
thf(fact_2607_strict__subset__divisors__dvd,axiom,
! [A: $tType] :
( ( comm_monoid_mult @ A )
=> ! [A3: A,B3: A] :
( ( ord_less @ ( set @ A )
@ ( collect @ A
@ ^ [C5: A] : ( dvd_dvd @ A @ C5 @ A3 ) )
@ ( collect @ A
@ ^ [C5: A] : ( dvd_dvd @ A @ C5 @ B3 ) ) )
= ( ( dvd_dvd @ A @ A3 @ B3 )
& ~ ( dvd_dvd @ A @ B3 @ A3 ) ) ) ) ).
% strict_subset_divisors_dvd
thf(fact_2608_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_2609_pinf_I9_J,axiom,
! [B: $tType] :
( ( ( plus @ B )
& ( linorder @ B )
& ( dvd @ B ) )
=> ! [D2: B,S2: B] :
? [Z2: B] :
! [X8: B] :
( ( ord_less @ B @ Z2 @ X8 )
=> ( ( dvd_dvd @ B @ D2 @ ( plus_plus @ B @ X8 @ S2 ) )
= ( dvd_dvd @ B @ D2 @ ( plus_plus @ B @ X8 @ S2 ) ) ) ) ) ).
% pinf(9)
thf(fact_2610_pinf_I10_J,axiom,
! [B: $tType] :
( ( ( plus @ B )
& ( linorder @ B )
& ( dvd @ B ) )
=> ! [D2: B,S2: B] :
? [Z2: B] :
! [X8: B] :
( ( ord_less @ B @ Z2 @ X8 )
=> ( ( ~ ( dvd_dvd @ B @ D2 @ ( plus_plus @ B @ X8 @ S2 ) ) )
= ( ~ ( dvd_dvd @ B @ D2 @ ( plus_plus @ B @ X8 @ S2 ) ) ) ) ) ) ).
% pinf(10)
thf(fact_2611_minf_I9_J,axiom,
! [B: $tType] :
( ( ( plus @ B )
& ( linorder @ B )
& ( dvd @ B ) )
=> ! [D2: B,S2: B] :
? [Z2: B] :
! [X8: B] :
( ( ord_less @ B @ X8 @ Z2 )
=> ( ( dvd_dvd @ B @ D2 @ ( plus_plus @ B @ X8 @ S2 ) )
= ( dvd_dvd @ B @ D2 @ ( plus_plus @ B @ X8 @ S2 ) ) ) ) ) ).
% minf(9)
thf(fact_2612_minf_I10_J,axiom,
! [B: $tType] :
( ( ( plus @ B )
& ( linorder @ B )
& ( dvd @ B ) )
=> ! [D2: B,S2: B] :
? [Z2: B] :
! [X8: B] :
( ( ord_less @ B @ X8 @ Z2 )
=> ( ( ~ ( dvd_dvd @ B @ D2 @ ( plus_plus @ B @ X8 @ S2 ) ) )
= ( ~ ( dvd_dvd @ B @ D2 @ ( plus_plus @ B @ X8 @ S2 ) ) ) ) ) ) ).
% minf(10)
thf(fact_2613_dvd__div__eq__0__iff,axiom,
! [A: $tType] :
( ( semidom_divide @ A )
=> ! [B3: A,A3: A] :
( ( dvd_dvd @ A @ B3 @ A3 )
=> ( ( ( divide_divide @ A @ A3 @ B3 )
= ( zero_zero @ A ) )
= ( A3
= ( zero_zero @ A ) ) ) ) ) ).
% dvd_div_eq_0_iff
thf(fact_2614_unit__mult__right__cancel,axiom,
! [A: $tType] :
( ( algebraic_semidom @ A )
=> ! [A3: A,B3: A,C3: A] :
( ( dvd_dvd @ A @ A3 @ ( one_one @ A ) )
=> ( ( ( times_times @ A @ B3 @ A3 )
= ( times_times @ A @ C3 @ A3 ) )
= ( B3 = C3 ) ) ) ) ).
% unit_mult_right_cancel
thf(fact_2615_unit__mult__left__cancel,axiom,
! [A: $tType] :
( ( algebraic_semidom @ A )
=> ! [A3: A,B3: A,C3: A] :
( ( dvd_dvd @ A @ A3 @ ( one_one @ A ) )
=> ( ( ( times_times @ A @ A3 @ B3 )
= ( times_times @ A @ A3 @ C3 ) )
= ( B3 = C3 ) ) ) ) ).
% unit_mult_left_cancel
thf(fact_2616_mult__unit__dvd__iff_H,axiom,
! [A: $tType] :
( ( algebraic_semidom @ A )
=> ! [A3: A,B3: A,C3: A] :
( ( dvd_dvd @ A @ A3 @ ( one_one @ A ) )
=> ( ( dvd_dvd @ A @ ( times_times @ A @ A3 @ B3 ) @ C3 )
= ( dvd_dvd @ A @ B3 @ C3 ) ) ) ) ).
% mult_unit_dvd_iff'
thf(fact_2617_dvd__mult__unit__iff_H,axiom,
! [A: $tType] :
( ( algebraic_semidom @ A )
=> ! [B3: A,A3: A,C3: A] :
( ( dvd_dvd @ A @ B3 @ ( one_one @ A ) )
=> ( ( dvd_dvd @ A @ A3 @ ( times_times @ A @ B3 @ C3 ) )
= ( dvd_dvd @ A @ A3 @ C3 ) ) ) ) ).
% dvd_mult_unit_iff'
thf(fact_2618_mult__unit__dvd__iff,axiom,
! [A: $tType] :
( ( algebraic_semidom @ A )
=> ! [B3: A,A3: A,C3: A] :
( ( dvd_dvd @ A @ B3 @ ( one_one @ A ) )
=> ( ( dvd_dvd @ A @ ( times_times @ A @ A3 @ B3 ) @ C3 )
= ( dvd_dvd @ A @ A3 @ C3 ) ) ) ) ).
% mult_unit_dvd_iff
thf(fact_2619_dvd__mult__unit__iff,axiom,
! [A: $tType] :
( ( algebraic_semidom @ A )
=> ! [B3: A,A3: A,C3: A] :
( ( dvd_dvd @ A @ B3 @ ( one_one @ A ) )
=> ( ( dvd_dvd @ A @ A3 @ ( times_times @ A @ C3 @ B3 ) )
= ( dvd_dvd @ A @ A3 @ C3 ) ) ) ) ).
% dvd_mult_unit_iff
thf(fact_2620_is__unit__mult__iff,axiom,
! [A: $tType] :
( ( algebraic_semidom @ A )
=> ! [A3: A,B3: A] :
( ( dvd_dvd @ A @ ( times_times @ A @ A3 @ B3 ) @ ( one_one @ A ) )
= ( ( dvd_dvd @ A @ A3 @ ( one_one @ A ) )
& ( dvd_dvd @ A @ B3 @ ( one_one @ A ) ) ) ) ) ).
% is_unit_mult_iff
thf(fact_2621_div__plus__div__distrib__dvd__right,axiom,
! [A: $tType] :
( ( euclid4440199948858584721cancel @ A )
=> ! [C3: A,B3: A,A3: A] :
( ( dvd_dvd @ A @ C3 @ B3 )
=> ( ( divide_divide @ A @ ( plus_plus @ A @ A3 @ B3 ) @ C3 )
= ( plus_plus @ A @ ( divide_divide @ A @ A3 @ C3 ) @ ( divide_divide @ A @ B3 @ C3 ) ) ) ) ) ).
% div_plus_div_distrib_dvd_right
thf(fact_2622_div__plus__div__distrib__dvd__left,axiom,
! [A: $tType] :
( ( euclid4440199948858584721cancel @ A )
=> ! [C3: A,A3: A,B3: A] :
( ( dvd_dvd @ A @ C3 @ A3 )
=> ( ( divide_divide @ A @ ( plus_plus @ A @ A3 @ B3 ) @ C3 )
= ( plus_plus @ A @ ( divide_divide @ A @ A3 @ C3 ) @ ( divide_divide @ A @ B3 @ C3 ) ) ) ) ) ).
% div_plus_div_distrib_dvd_left
thf(fact_2623_unit__div__cancel,axiom,
! [A: $tType] :
( ( algebraic_semidom @ A )
=> ! [A3: A,B3: A,C3: A] :
( ( dvd_dvd @ A @ A3 @ ( one_one @ A ) )
=> ( ( ( divide_divide @ A @ B3 @ A3 )
= ( divide_divide @ A @ C3 @ A3 ) )
= ( B3 = C3 ) ) ) ) ).
% unit_div_cancel
thf(fact_2624_div__unit__dvd__iff,axiom,
! [A: $tType] :
( ( algebraic_semidom @ A )
=> ! [B3: A,A3: A,C3: A] :
( ( dvd_dvd @ A @ B3 @ ( one_one @ A ) )
=> ( ( dvd_dvd @ A @ ( divide_divide @ A @ A3 @ B3 ) @ C3 )
= ( dvd_dvd @ A @ A3 @ C3 ) ) ) ) ).
% div_unit_dvd_iff
thf(fact_2625_dvd__div__unit__iff,axiom,
! [A: $tType] :
( ( algebraic_semidom @ A )
=> ! [B3: A,A3: A,C3: A] :
( ( dvd_dvd @ A @ B3 @ ( one_one @ A ) )
=> ( ( dvd_dvd @ A @ A3 @ ( divide_divide @ A @ C3 @ B3 ) )
= ( dvd_dvd @ A @ A3 @ C3 ) ) ) ) ).
% dvd_div_unit_iff
thf(fact_2626_dvd__div__mult,axiom,
! [A: $tType] :
( ( algebraic_semidom @ A )
=> ! [C3: A,B3: A,A3: A] :
( ( dvd_dvd @ A @ C3 @ B3 )
=> ( ( times_times @ A @ ( divide_divide @ A @ B3 @ C3 ) @ A3 )
= ( divide_divide @ A @ ( times_times @ A @ B3 @ A3 ) @ C3 ) ) ) ) ).
% dvd_div_mult
thf(fact_2627_div__mult__swap,axiom,
! [A: $tType] :
( ( algebraic_semidom @ A )
=> ! [C3: A,B3: A,A3: A] :
( ( dvd_dvd @ A @ C3 @ B3 )
=> ( ( times_times @ A @ A3 @ ( divide_divide @ A @ B3 @ C3 ) )
= ( divide_divide @ A @ ( times_times @ A @ A3 @ B3 ) @ C3 ) ) ) ) ).
% div_mult_swap
thf(fact_2628_div__div__eq__right,axiom,
! [A: $tType] :
( ( algebraic_semidom @ A )
=> ! [C3: A,B3: A,A3: A] :
( ( dvd_dvd @ A @ C3 @ B3 )
=> ( ( dvd_dvd @ A @ B3 @ A3 )
=> ( ( divide_divide @ A @ A3 @ ( divide_divide @ A @ B3 @ C3 ) )
= ( times_times @ A @ ( divide_divide @ A @ A3 @ B3 ) @ C3 ) ) ) ) ) ).
% div_div_eq_right
thf(fact_2629_dvd__div__mult2__eq,axiom,
! [A: $tType] :
( ( algebraic_semidom @ A )
=> ! [B3: A,C3: A,A3: A] :
( ( dvd_dvd @ A @ ( times_times @ A @ B3 @ C3 ) @ A3 )
=> ( ( divide_divide @ A @ A3 @ ( times_times @ A @ B3 @ C3 ) )
= ( divide_divide @ A @ ( divide_divide @ A @ A3 @ B3 ) @ C3 ) ) ) ) ).
% dvd_div_mult2_eq
thf(fact_2630_dvd__mult__imp__div,axiom,
! [A: $tType] :
( ( algebraic_semidom @ A )
=> ! [A3: A,C3: A,B3: A] :
( ( dvd_dvd @ A @ ( times_times @ A @ A3 @ C3 ) @ B3 )
=> ( dvd_dvd @ A @ A3 @ ( divide_divide @ A @ B3 @ C3 ) ) ) ) ).
% dvd_mult_imp_div
thf(fact_2631_div__mult__div__if__dvd,axiom,
! [A: $tType] :
( ( algebraic_semidom @ A )
=> ! [B3: A,A3: A,D2: A,C3: A] :
( ( dvd_dvd @ A @ B3 @ A3 )
=> ( ( dvd_dvd @ A @ D2 @ C3 )
=> ( ( times_times @ A @ ( divide_divide @ A @ A3 @ B3 ) @ ( divide_divide @ A @ C3 @ D2 ) )
= ( divide_divide @ A @ ( times_times @ A @ A3 @ C3 ) @ ( times_times @ A @ B3 @ D2 ) ) ) ) ) ) ).
% div_mult_div_if_dvd
thf(fact_2632_div__power,axiom,
! [A: $tType] :
( ( algebraic_semidom @ A )
=> ! [B3: A,A3: A,N: nat] :
( ( dvd_dvd @ A @ B3 @ A3 )
=> ( ( power_power @ A @ ( divide_divide @ A @ A3 @ B3 ) @ N )
= ( divide_divide @ A @ ( power_power @ A @ A3 @ N ) @ ( power_power @ A @ B3 @ N ) ) ) ) ) ).
% div_power
thf(fact_2633_le__imp__power__dvd,axiom,
! [A: $tType] :
( ( comm_semiring_1 @ A )
=> ! [M: nat,N: nat,A3: A] :
( ( ord_less_eq @ nat @ M @ N )
=> ( dvd_dvd @ A @ ( power_power @ A @ A3 @ M ) @ ( power_power @ A @ A3 @ N ) ) ) ) ).
% le_imp_power_dvd
thf(fact_2634_power__le__dvd,axiom,
! [A: $tType] :
( ( comm_semiring_1 @ A )
=> ! [A3: A,N: nat,B3: A,M: nat] :
( ( dvd_dvd @ A @ ( power_power @ A @ A3 @ N ) @ B3 )
=> ( ( ord_less_eq @ nat @ M @ N )
=> ( dvd_dvd @ A @ ( power_power @ A @ A3 @ M ) @ B3 ) ) ) ) ).
% power_le_dvd
thf(fact_2635_dvd__power__le,axiom,
! [A: $tType] :
( ( comm_semiring_1 @ A )
=> ! [X2: A,Y3: A,N: nat,M: nat] :
( ( dvd_dvd @ A @ X2 @ Y3 )
=> ( ( ord_less_eq @ nat @ N @ M )
=> ( dvd_dvd @ A @ ( power_power @ A @ X2 @ N ) @ ( power_power @ A @ Y3 @ M ) ) ) ) ) ).
% dvd_power_le
thf(fact_2636_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_2637_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_2638_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_2639_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_2640_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_2641_return__cons__rule,axiom,
! [A: $tType,P: assn,Q: A > assn,X2: A] :
( ( entails @ P @ ( Q @ X2 ) )
=> ( hoare_hoare_triple @ A @ P @ ( heap_Time_return @ A @ X2 ) @ Q ) ) ).
% return_cons_rule
thf(fact_2642_even__unset__bit__iff,axiom,
! [A: $tType] :
( ( bit_se359711467146920520ations @ A )
=> ! [M: nat,A3: A] :
( ( dvd_dvd @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ ( bit_se2638667681897837118et_bit @ A @ M @ A3 ) )
= ( ( dvd_dvd @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ A3 )
| ( M
= ( zero_zero @ nat ) ) ) ) ) ).
% even_unset_bit_iff
thf(fact_2643_even__of__int__iff,axiom,
! [A: $tType] :
( ( bit_ri3973907225187159222ations @ A )
=> ! [K: int] :
( ( dvd_dvd @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ ( ring_1_of_int @ A @ K ) )
= ( dvd_dvd @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ K ) ) ) ).
% even_of_int_iff
thf(fact_2644_even__numeral,axiom,
! [A: $tType] :
( ( semiring_parity @ A )
=> ! [N: num] : ( dvd_dvd @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ ( numeral_numeral @ A @ ( bit0 @ N ) ) ) ) ).
% even_numeral
thf(fact_2645_unity__coeff__ex,axiom,
! [A: $tType] :
( ( ( dvd @ A )
& ( semiring_0 @ A ) )
=> ! [P: A > $o,L: A] :
( ( ? [X3: A] : ( P @ ( times_times @ A @ L @ X3 ) ) )
= ( ? [X3: A] :
( ( dvd_dvd @ A @ L @ ( plus_plus @ A @ X3 @ ( zero_zero @ A ) ) )
& ( P @ X3 ) ) ) ) ) ).
% unity_coeff_ex
thf(fact_2646_unit__dvdE,axiom,
! [A: $tType] :
( ( algebraic_semidom @ A )
=> ! [A3: A,B3: A] :
( ( dvd_dvd @ A @ A3 @ ( one_one @ A ) )
=> ~ ( ( A3
!= ( zero_zero @ A ) )
=> ! [C4: A] :
( B3
!= ( times_times @ A @ A3 @ C4 ) ) ) ) ) ).
% unit_dvdE
thf(fact_2647_unit__div__eq__0__iff,axiom,
! [A: $tType] :
( ( algebraic_semidom @ A )
=> ! [B3: A,A3: A] :
( ( dvd_dvd @ A @ B3 @ ( one_one @ A ) )
=> ( ( ( divide_divide @ A @ A3 @ B3 )
= ( zero_zero @ A ) )
= ( A3
= ( zero_zero @ A ) ) ) ) ) ).
% unit_div_eq_0_iff
thf(fact_2648_dvd__div__eq__mult,axiom,
! [A: $tType] :
( ( algebraic_semidom @ A )
=> ! [A3: A,B3: A,C3: A] :
( ( A3
!= ( zero_zero @ A ) )
=> ( ( dvd_dvd @ A @ A3 @ B3 )
=> ( ( ( divide_divide @ A @ B3 @ A3 )
= C3 )
= ( B3
= ( times_times @ A @ C3 @ A3 ) ) ) ) ) ) ).
% dvd_div_eq_mult
thf(fact_2649_div__dvd__iff__mult,axiom,
! [A: $tType] :
( ( algebraic_semidom @ A )
=> ! [B3: A,A3: A,C3: A] :
( ( B3
!= ( zero_zero @ A ) )
=> ( ( dvd_dvd @ A @ B3 @ A3 )
=> ( ( dvd_dvd @ A @ ( divide_divide @ A @ A3 @ B3 ) @ C3 )
= ( dvd_dvd @ A @ A3 @ ( times_times @ A @ C3 @ B3 ) ) ) ) ) ) ).
% div_dvd_iff_mult
thf(fact_2650_dvd__div__iff__mult,axiom,
! [A: $tType] :
( ( algebraic_semidom @ A )
=> ! [C3: A,B3: A,A3: A] :
( ( C3
!= ( zero_zero @ A ) )
=> ( ( dvd_dvd @ A @ C3 @ B3 )
=> ( ( dvd_dvd @ A @ A3 @ ( divide_divide @ A @ B3 @ C3 ) )
= ( dvd_dvd @ A @ ( times_times @ A @ A3 @ C3 ) @ B3 ) ) ) ) ) ).
% dvd_div_iff_mult
thf(fact_2651_dvd__div__div__eq__mult,axiom,
! [A: $tType] :
( ( algebraic_semidom @ A )
=> ! [A3: A,C3: A,B3: A,D2: A] :
( ( A3
!= ( zero_zero @ A ) )
=> ( ( C3
!= ( zero_zero @ A ) )
=> ( ( dvd_dvd @ A @ A3 @ B3 )
=> ( ( dvd_dvd @ A @ C3 @ D2 )
=> ( ( ( divide_divide @ A @ B3 @ A3 )
= ( divide_divide @ A @ D2 @ C3 ) )
= ( ( times_times @ A @ B3 @ C3 )
= ( times_times @ A @ A3 @ D2 ) ) ) ) ) ) ) ) ).
% dvd_div_div_eq_mult
thf(fact_2652_inf__period_I3_J,axiom,
! [A: $tType] :
( ( ( comm_ring @ A )
& ( dvd @ A ) )
=> ! [D2: A,D: A,T2: A] :
( ( dvd_dvd @ A @ D2 @ D )
=> ! [X8: A,K5: A] :
( ( dvd_dvd @ A @ D2 @ ( plus_plus @ A @ X8 @ T2 ) )
= ( dvd_dvd @ A @ D2 @ ( plus_plus @ A @ ( minus_minus @ A @ X8 @ ( times_times @ A @ K5 @ D ) ) @ T2 ) ) ) ) ) ).
% inf_period(3)
thf(fact_2653_inf__period_I4_J,axiom,
! [A: $tType] :
( ( ( comm_ring @ A )
& ( dvd @ A ) )
=> ! [D2: A,D: A,T2: A] :
( ( dvd_dvd @ A @ D2 @ D )
=> ! [X8: A,K5: A] :
( ( ~ ( dvd_dvd @ A @ D2 @ ( plus_plus @ A @ X8 @ T2 ) ) )
= ( ~ ( dvd_dvd @ A @ D2 @ ( plus_plus @ A @ ( minus_minus @ A @ X8 @ ( times_times @ A @ K5 @ D ) ) @ T2 ) ) ) ) ) ) ).
% inf_period(4)
thf(fact_2654_unit__eq__div1,axiom,
! [A: $tType] :
( ( algebraic_semidom @ A )
=> ! [B3: A,A3: A,C3: A] :
( ( dvd_dvd @ A @ B3 @ ( one_one @ A ) )
=> ( ( ( divide_divide @ A @ A3 @ B3 )
= C3 )
= ( A3
= ( times_times @ A @ C3 @ B3 ) ) ) ) ) ).
% unit_eq_div1
thf(fact_2655_unit__eq__div2,axiom,
! [A: $tType] :
( ( algebraic_semidom @ A )
=> ! [B3: A,A3: A,C3: A] :
( ( dvd_dvd @ A @ B3 @ ( one_one @ A ) )
=> ( ( A3
= ( divide_divide @ A @ C3 @ B3 ) )
= ( ( times_times @ A @ A3 @ B3 )
= C3 ) ) ) ) ).
% unit_eq_div2
thf(fact_2656_div__mult__unit2,axiom,
! [A: $tType] :
( ( algebraic_semidom @ A )
=> ! [C3: A,B3: A,A3: A] :
( ( dvd_dvd @ A @ C3 @ ( one_one @ A ) )
=> ( ( dvd_dvd @ A @ B3 @ A3 )
=> ( ( divide_divide @ A @ A3 @ ( times_times @ A @ B3 @ C3 ) )
= ( divide_divide @ A @ ( divide_divide @ A @ A3 @ B3 ) @ C3 ) ) ) ) ) ).
% div_mult_unit2
thf(fact_2657_unit__div__commute,axiom,
! [A: $tType] :
( ( algebraic_semidom @ A )
=> ! [B3: A,A3: A,C3: A] :
( ( dvd_dvd @ A @ B3 @ ( one_one @ A ) )
=> ( ( times_times @ A @ ( divide_divide @ A @ A3 @ B3 ) @ C3 )
= ( divide_divide @ A @ ( times_times @ A @ A3 @ C3 ) @ B3 ) ) ) ) ).
% unit_div_commute
thf(fact_2658_unit__div__mult__swap,axiom,
! [A: $tType] :
( ( algebraic_semidom @ A )
=> ! [C3: A,A3: A,B3: A] :
( ( dvd_dvd @ A @ C3 @ ( one_one @ A ) )
=> ( ( times_times @ A @ A3 @ ( divide_divide @ A @ B3 @ C3 ) )
= ( divide_divide @ A @ ( times_times @ A @ A3 @ B3 ) @ C3 ) ) ) ) ).
% unit_div_mult_swap
thf(fact_2659_is__unit__div__mult2__eq,axiom,
! [A: $tType] :
( ( algebraic_semidom @ A )
=> ! [B3: A,C3: A,A3: A] :
( ( dvd_dvd @ A @ B3 @ ( one_one @ A ) )
=> ( ( dvd_dvd @ A @ C3 @ ( one_one @ A ) )
=> ( ( divide_divide @ A @ A3 @ ( times_times @ A @ B3 @ C3 ) )
= ( divide_divide @ A @ ( divide_divide @ A @ A3 @ B3 ) @ C3 ) ) ) ) ) ).
% is_unit_div_mult2_eq
thf(fact_2660_is__unit__power__iff,axiom,
! [A: $tType] :
( ( algebraic_semidom @ A )
=> ! [A3: A,N: nat] :
( ( dvd_dvd @ A @ ( power_power @ A @ A3 @ N ) @ ( one_one @ A ) )
= ( ( dvd_dvd @ A @ A3 @ ( one_one @ A ) )
| ( N
= ( zero_zero @ nat ) ) ) ) ) ).
% is_unit_power_iff
thf(fact_2661_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_2662_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_2663_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_2664_TBOUND__return,axiom,
! [A: $tType,X2: A] : ( time_TBOUND @ A @ ( heap_Time_return @ A @ X2 ) @ ( one_one @ nat ) ) ).
% TBOUND_return
thf(fact_2665_real__of__nat__div,axiom,
! [D2: nat,N: nat] :
( ( dvd_dvd @ nat @ D2 @ N )
=> ( ( semiring_1_of_nat @ real @ ( divide_divide @ nat @ N @ D2 ) )
= ( divide_divide @ real @ ( semiring_1_of_nat @ real @ N ) @ ( semiring_1_of_nat @ real @ D2 ) ) ) ) ).
% real_of_nat_div
thf(fact_2666_time__return,axiom,
! [A: $tType,X2: A,H2: heap_ext @ product_unit] :
( ( time_time @ A @ ( heap_Time_return @ A @ X2 ) @ H2 )
= ( one_one @ nat ) ) ).
% time_return
thf(fact_2667_assn__aci_I10_J,axiom,
! [A3: assn,B3: assn,C3: assn] :
( ( times_times @ assn @ ( times_times @ assn @ A3 @ B3 ) @ C3 )
= ( times_times @ assn @ ( times_times @ assn @ A3 @ C3 ) @ B3 ) ) ).
% assn_aci(10)
thf(fact_2668_star__aci_I3_J,axiom,
! [A3: assn,B3: assn,C3: assn] :
( ( times_times @ assn @ A3 @ ( times_times @ assn @ B3 @ C3 ) )
= ( times_times @ assn @ B3 @ ( times_times @ assn @ A3 @ C3 ) ) ) ).
% star_aci(3)
thf(fact_2669_star__aci_I2_J,axiom,
( ( times_times @ assn )
= ( ^ [A6: assn,B7: assn] : ( times_times @ assn @ B7 @ A6 ) ) ) ).
% star_aci(2)
thf(fact_2670_star__assoc,axiom,
! [A3: assn,B3: assn,C3: assn] :
( ( times_times @ assn @ ( times_times @ assn @ A3 @ B3 ) @ C3 )
= ( times_times @ assn @ A3 @ ( times_times @ assn @ B3 @ C3 ) ) ) ).
% star_assoc
thf(fact_2671_is__entails,axiom,
! [P: assn,Q: assn] :
( ( entails @ P @ Q )
=> ( entails @ P @ Q ) ) ).
% is_entails
thf(fact_2672_TBOUND__nth,axiom,
! [A: $tType] :
( ( heap @ A )
=> ! [Xs: array @ A,I: nat] : ( time_TBOUND @ A @ ( array_nth @ A @ Xs @ I ) @ ( one_one @ nat ) ) ) ).
% TBOUND_nth
thf(fact_2673_even__zero,axiom,
! [A: $tType] :
( ( semiring_parity @ A )
=> ( dvd_dvd @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ ( zero_zero @ A ) ) ) ).
% even_zero
thf(fact_2674_odd__even__add,axiom,
! [A: $tType] :
( ( semiring_parity @ A )
=> ! [A3: A,B3: A] :
( ~ ( dvd_dvd @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ A3 )
=> ( ~ ( dvd_dvd @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ B3 )
=> ( dvd_dvd @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ ( plus_plus @ A @ A3 @ B3 ) ) ) ) ) ).
% odd_even_add
thf(fact_2675_odd__one,axiom,
! [A: $tType] :
( ( semiring_parity @ A )
=> ~ ( dvd_dvd @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ ( one_one @ A ) ) ) ).
% odd_one
thf(fact_2676_evenE,axiom,
! [A: $tType] :
( ( semiring_parity @ A )
=> ! [A3: A] :
( ( dvd_dvd @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ A3 )
=> ~ ! [B4: A] :
( A3
!= ( times_times @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ B4 ) ) ) ) ).
% evenE
thf(fact_2677_bit__eq__rec,axiom,
! [A: $tType] :
( ( bit_semiring_bits @ A )
=> ( ( ^ [Y6: A,Z3: A] : Y6 = Z3 )
= ( ^ [A6: A,B7: A] :
( ( ( dvd_dvd @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ A6 )
= ( dvd_dvd @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ B7 ) )
& ( ( divide_divide @ A @ A6 @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) )
= ( divide_divide @ A @ B7 @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) ) ) ) ) ) ).
% bit_eq_rec
thf(fact_2678_is__unit__div__mult__cancel__right,axiom,
! [A: $tType] :
( ( algebraic_semidom @ A )
=> ! [A3: A,B3: A] :
( ( A3
!= ( zero_zero @ A ) )
=> ( ( dvd_dvd @ A @ B3 @ ( one_one @ A ) )
=> ( ( divide_divide @ A @ A3 @ ( times_times @ A @ B3 @ A3 ) )
= ( divide_divide @ A @ ( one_one @ A ) @ B3 ) ) ) ) ) ).
% is_unit_div_mult_cancel_right
thf(fact_2679_is__unit__div__mult__cancel__left,axiom,
! [A: $tType] :
( ( algebraic_semidom @ A )
=> ! [A3: A,B3: A] :
( ( A3
!= ( zero_zero @ A ) )
=> ( ( dvd_dvd @ A @ B3 @ ( one_one @ A ) )
=> ( ( divide_divide @ A @ A3 @ ( times_times @ A @ A3 @ B3 ) )
= ( divide_divide @ A @ ( one_one @ A ) @ B3 ) ) ) ) ) ).
% is_unit_div_mult_cancel_left
thf(fact_2680_is__unitE,axiom,
! [A: $tType] :
( ( algebraic_semidom @ A )
=> ! [A3: A,C3: A] :
( ( dvd_dvd @ A @ A3 @ ( one_one @ A ) )
=> ~ ( ( A3
!= ( zero_zero @ A ) )
=> ! [B4: A] :
( ( B4
!= ( zero_zero @ A ) )
=> ( ( dvd_dvd @ A @ B4 @ ( one_one @ A ) )
=> ( ( ( divide_divide @ A @ ( one_one @ A ) @ A3 )
= B4 )
=> ( ( ( divide_divide @ A @ ( one_one @ A ) @ B4 )
= A3 )
=> ( ( ( times_times @ A @ A3 @ B4 )
= ( one_one @ A ) )
=> ( ( divide_divide @ A @ C3 @ A3 )
!= ( times_times @ A @ C3 @ B4 ) ) ) ) ) ) ) ) ) ) ).
% is_unitE
thf(fact_2681_odd__numeral,axiom,
! [A: $tType] :
( ( semiring_parity @ A )
=> ! [N: num] :
~ ( dvd_dvd @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ ( numeral_numeral @ A @ ( bit1 @ N ) ) ) ) ).
% odd_numeral
thf(fact_2682_dvd__power__iff,axiom,
! [A: $tType] :
( ( algebraic_semidom @ A )
=> ! [X2: A,M: nat,N: nat] :
( ( X2
!= ( zero_zero @ A ) )
=> ( ( dvd_dvd @ A @ ( power_power @ A @ X2 @ M ) @ ( power_power @ A @ X2 @ N ) )
= ( ( dvd_dvd @ A @ X2 @ ( one_one @ A ) )
| ( ord_less_eq @ nat @ M @ N ) ) ) ) ) ).
% dvd_power_iff
thf(fact_2683_dvd__power,axiom,
! [A: $tType] :
( ( comm_semiring_1 @ A )
=> ! [N: nat,X2: A] :
( ( ( ord_less @ nat @ ( zero_zero @ nat ) @ N )
| ( X2
= ( one_one @ A ) ) )
=> ( dvd_dvd @ A @ X2 @ ( power_power @ A @ X2 @ N ) ) ) ) ).
% dvd_power
thf(fact_2684_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_2685_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_2686_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_2687_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_2688_even__two__times__div__two,axiom,
! [A: $tType] :
( ( euclid5411537665997757685th_nat @ A )
=> ! [A3: A] :
( ( dvd_dvd @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ A3 )
=> ( ( times_times @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ ( divide_divide @ A @ A3 @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) )
= A3 ) ) ) ).
% even_two_times_div_two
thf(fact_2689_power__mono__odd,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [N: nat,A3: A,B3: A] :
( ~ ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N )
=> ( ( ord_less_eq @ A @ A3 @ B3 )
=> ( ord_less_eq @ A @ ( power_power @ A @ A3 @ N ) @ ( power_power @ A @ B3 @ N ) ) ) ) ) ).
% power_mono_odd
thf(fact_2690_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_2691_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_2692_even__set__bit__iff,axiom,
! [A: $tType] :
( ( bit_se359711467146920520ations @ A )
=> ! [M: nat,A3: A] :
( ( dvd_dvd @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ ( bit_se5668285175392031749et_bit @ A @ M @ A3 ) )
= ( ( dvd_dvd @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ A3 )
& ( M
!= ( zero_zero @ nat ) ) ) ) ) ).
% even_set_bit_iff
thf(fact_2693_return__sp__rule,axiom,
! [A: $tType,P: assn,X2: A] :
( hoare_hoare_triple @ A @ P @ ( heap_Time_return @ A @ X2 )
@ ^ [R5: A] : ( times_times @ assn @ P @ ( pure_assn @ ( R5 = X2 ) ) ) ) ).
% return_sp_rule
thf(fact_2694_oddE,axiom,
! [A: $tType] :
( ( semiring_parity @ A )
=> ! [A3: A] :
( ~ ( dvd_dvd @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ A3 )
=> ~ ! [B4: A] :
( A3
!= ( plus_plus @ A @ ( times_times @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ B4 ) @ ( one_one @ A ) ) ) ) ) ).
% oddE
thf(fact_2695_zero__le__power__eq,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [A3: A,N: nat] :
( ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( power_power @ A @ A3 @ N ) )
= ( ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N )
| ( ~ ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N )
& ( ord_less_eq @ A @ ( zero_zero @ A ) @ A3 ) ) ) ) ) ).
% zero_le_power_eq
thf(fact_2696_zero__le__odd__power,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [N: nat,A3: A] :
( ~ ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N )
=> ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( power_power @ A @ A3 @ N ) )
= ( ord_less_eq @ A @ ( zero_zero @ A ) @ A3 ) ) ) ) ).
% zero_le_odd_power
thf(fact_2697_zero__le__even__power,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [N: nat,A3: A] :
( ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N )
=> ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( power_power @ A @ A3 @ N ) ) ) ) ).
% zero_le_even_power
thf(fact_2698_zero__less__power__eq,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [A3: A,N: nat] :
( ( ord_less @ A @ ( zero_zero @ A ) @ ( power_power @ A @ A3 @ N ) )
= ( ( N
= ( zero_zero @ nat ) )
| ( ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N )
& ( A3
!= ( zero_zero @ A ) ) )
| ( ~ ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N )
& ( ord_less @ A @ ( zero_zero @ A ) @ A3 ) ) ) ) ) ).
% zero_less_power_eq
thf(fact_2699_fr__rot,axiom,
! [A2: assn,B2: assn,C2: assn] :
( ( entails @ ( times_times @ assn @ A2 @ B2 ) @ C2 )
=> ( entails @ ( times_times @ assn @ B2 @ A2 ) @ C2 ) ) ).
% fr_rot
thf(fact_2700_fr__refl,axiom,
! [A2: assn,B2: assn,C2: assn] :
( ( entails @ A2 @ B2 )
=> ( entails @ ( times_times @ assn @ A2 @ C2 ) @ ( times_times @ assn @ B2 @ C2 ) ) ) ).
% fr_refl
thf(fact_2701_fr__rot__rhs,axiom,
! [A2: assn,B2: assn,C2: assn] :
( ( entails @ A2 @ ( times_times @ assn @ B2 @ C2 ) )
=> ( entails @ A2 @ ( times_times @ assn @ C2 @ B2 ) ) ) ).
% fr_rot_rhs
thf(fact_2702_ent__frame__fwd,axiom,
! [P: assn,R: assn,Ps: assn,F4: assn,Q: assn] :
( ( entails @ P @ R )
=> ( ( entails @ Ps @ ( times_times @ assn @ P @ F4 ) )
=> ( ( entails @ ( times_times @ assn @ R @ F4 ) @ Q )
=> ( entails @ Ps @ Q ) ) ) ) ).
% ent_frame_fwd
thf(fact_2703_norm__assertion__simps_I2_J,axiom,
! [A3: assn] :
( ( times_times @ assn @ A3 @ ( one_one @ assn ) )
= A3 ) ).
% norm_assertion_simps(2)
thf(fact_2704_norm__assertion__simps_I1_J,axiom,
! [A3: assn] :
( ( times_times @ assn @ ( one_one @ assn ) @ A3 )
= A3 ) ).
% norm_assertion_simps(1)
thf(fact_2705_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_2706_power__le__zero__eq,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [A3: A,N: nat] :
( ( ord_less_eq @ A @ ( power_power @ A @ A3 @ N ) @ ( zero_zero @ A ) )
= ( ( ord_less @ nat @ ( zero_zero @ nat ) @ N )
& ( ( ~ ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N )
& ( ord_less_eq @ A @ A3 @ ( zero_zero @ A ) ) )
| ( ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N )
& ( A3
= ( zero_zero @ A ) ) ) ) ) ) ) ).
% power_le_zero_eq
thf(fact_2707_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_2708_frame__rule__left,axiom,
! [A: $tType,P: assn,C3: heap_Time_Heap @ A,Q: A > assn,R: assn] :
( ( hoare_hoare_triple @ A @ P @ C3 @ Q )
=> ( hoare_hoare_triple @ A @ ( times_times @ assn @ R @ P ) @ C3
@ ^ [X3: A] : ( times_times @ assn @ R @ ( Q @ X3 ) ) ) ) ).
% frame_rule_left
thf(fact_2709_Bernoulli__inequality__even,axiom,
! [N: nat,X2: 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 ) @ X2 ) ) @ ( power_power @ real @ ( plus_plus @ real @ ( one_one @ real ) @ X2 ) @ N ) ) ) ).
% Bernoulli_inequality_even
thf(fact_2710_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_2711_even__mult__exp__div__exp__iff,axiom,
! [A: $tType] :
( ( bit_semiring_bits @ A )
=> ! [A3: A,M: nat,N: nat] :
( ( dvd_dvd @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ ( divide_divide @ A @ ( times_times @ A @ A3 @ ( power_power @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ 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 @ A3 @ ( power_power @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ ( minus_minus @ nat @ N @ M ) ) ) ) ) ) ) ) ).
% even_mult_exp_div_exp_iff
thf(fact_2712_VEBT__internal_OT__vebt__buildupi_Oelims,axiom,
! [X2: nat,Y3: nat] :
( ( ( vEBT_V441764108873111860ildupi @ X2 )
= Y3 )
=> ( ( ( X2
= ( zero_zero @ nat ) )
=> ( Y3
!= ( suc @ ( zero_zero @ nat ) ) ) )
=> ( ( ( X2
= ( suc @ ( zero_zero @ nat ) ) )
=> ( Y3
!= ( suc @ ( zero_zero @ nat ) ) ) )
=> ~ ! [N3: nat] :
( ( X2
= ( suc @ ( suc @ N3 ) ) )
=> ~ ( ( ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N3 )
=> ( Y3
= ( suc @ ( suc @ ( suc @ ( plus_plus @ nat @ ( vEBT_V441764108873111860ildupi @ ( suc @ ( divide_divide @ nat @ N3 @ ( 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 @ N3 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( times_times @ nat @ ( vEBT_V441764108873111860ildupi @ ( suc @ ( divide_divide @ nat @ N3 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ N3 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) ) ) ) ) ) ) )
& ( ~ ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N3 )
=> ( Y3
= ( suc @ ( suc @ ( suc @ ( plus_plus @ nat @ ( vEBT_V441764108873111860ildupi @ ( suc @ ( suc @ ( divide_divide @ nat @ N3 @ ( 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 @ N3 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ ( bit0 @ one2 ) ) ) @ ( times_times @ nat @ ( vEBT_V441764108873111860ildupi @ ( suc @ ( divide_divide @ nat @ N3 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ N3 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).
% VEBT_internal.T_vebt_buildupi.elims
thf(fact_2713_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_2714_VEBT__internal_OTb_H_Oelims,axiom,
! [X2: nat,Y3: nat] :
( ( ( vEBT_VEBT_Tb2 @ X2 )
= Y3 )
=> ( ( ( X2
= ( zero_zero @ nat ) )
=> ( Y3
!= ( numeral_numeral @ nat @ ( bit1 @ one2 ) ) ) )
=> ( ( ( X2
= ( suc @ ( zero_zero @ nat ) ) )
=> ( Y3
!= ( numeral_numeral @ nat @ ( bit1 @ one2 ) ) ) )
=> ~ ! [N3: nat] :
( ( X2
= ( suc @ ( suc @ N3 ) ) )
=> ~ ( ( ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N3 )
=> ( Y3
= ( plus_plus @ nat @ ( plus_plus @ nat @ ( numeral_numeral @ nat @ ( bit1 @ ( bit0 @ one2 ) ) ) @ ( vEBT_VEBT_Tb2 @ ( suc @ ( divide_divide @ nat @ N3 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) @ ( times_times @ nat @ ( vEBT_VEBT_Tb2 @ ( suc @ ( divide_divide @ nat @ N3 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( suc @ ( divide_divide @ nat @ N3 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) ) ) )
& ( ~ ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N3 )
=> ( Y3
= ( plus_plus @ nat @ ( plus_plus @ nat @ ( numeral_numeral @ nat @ ( bit1 @ ( bit0 @ one2 ) ) ) @ ( vEBT_VEBT_Tb2 @ ( suc @ ( suc @ ( divide_divide @ nat @ N3 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) ) @ ( times_times @ nat @ ( vEBT_VEBT_Tb2 @ ( suc @ ( divide_divide @ nat @ N3 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( suc @ ( suc @ ( divide_divide @ nat @ N3 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).
% VEBT_internal.Tb'.elims
thf(fact_2715_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_2716_mod__frame__fwd,axiom,
! [Ps: assn,H2: product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ),P: assn,R: assn,F4: assn] :
( ( rep_assn @ Ps @ H2 )
=> ( ( entails @ P @ R )
=> ( ( entails @ Ps @ ( times_times @ assn @ P @ F4 ) )
=> ( rep_assn @ ( times_times @ assn @ R @ F4 ) @ H2 ) ) ) ) ).
% mod_frame_fwd
thf(fact_2717_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_2718_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,
! [Va: nat] :
( ( ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( suc @ ( suc @ Va ) ) )
=> ( ( vEBT_V8346862874174094_d_u_p @ ( suc @ ( suc @ Va ) ) )
= ( 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 ) ) )
=> ( ( vEBT_V8346862874174094_d_u_p @ ( suc @ ( suc @ Va ) ) )
= ( 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.simps(3)
thf(fact_2719_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,
! [Va: nat] :
( ( ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( suc @ ( suc @ Va ) ) )
=> ( ( vEBT_V8646137997579335489_i_l_d @ ( suc @ ( suc @ Va ) ) )
= ( 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 ) ) )
=> ( ( vEBT_V8646137997579335489_i_l_d @ ( suc @ ( suc @ Va ) ) )
= ( 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.simps(3)
thf(fact_2720_VEBT__internal_OTb_Oelims,axiom,
! [X2: nat,Y3: int] :
( ( ( vEBT_VEBT_Tb @ X2 )
= Y3 )
=> ( ( ( X2
= ( zero_zero @ nat ) )
=> ( Y3
!= ( numeral_numeral @ int @ ( bit1 @ one2 ) ) ) )
=> ( ( ( X2
= ( suc @ ( zero_zero @ nat ) ) )
=> ( Y3
!= ( numeral_numeral @ int @ ( bit1 @ one2 ) ) ) )
=> ~ ! [N3: nat] :
( ( X2
= ( suc @ ( suc @ N3 ) ) )
=> ~ ( ( ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N3 )
=> ( Y3
= ( plus_plus @ int @ ( plus_plus @ int @ ( numeral_numeral @ int @ ( bit1 @ ( bit0 @ one2 ) ) ) @ ( vEBT_VEBT_Tb @ ( suc @ ( divide_divide @ nat @ N3 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) @ ( times_times @ int @ ( vEBT_VEBT_Tb @ ( suc @ ( divide_divide @ nat @ N3 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( power_power @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ ( suc @ ( divide_divide @ nat @ N3 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) ) ) )
& ( ~ ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N3 )
=> ( Y3
= ( plus_plus @ int @ ( plus_plus @ int @ ( numeral_numeral @ int @ ( bit1 @ ( bit0 @ one2 ) ) ) @ ( vEBT_VEBT_Tb @ ( suc @ ( suc @ ( divide_divide @ nat @ N3 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) ) @ ( times_times @ int @ ( vEBT_VEBT_Tb @ ( suc @ ( divide_divide @ nat @ N3 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( power_power @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ ( suc @ ( suc @ ( divide_divide @ nat @ N3 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).
% VEBT_internal.Tb.elims
thf(fact_2721_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,
! [X2: nat,Y3: nat] :
( ( ( vEBT_V8346862874174094_d_u_p @ X2 )
= Y3 )
=> ( ( ( X2
= ( zero_zero @ nat ) )
=> ( Y3
!= ( numeral_numeral @ nat @ ( bit1 @ one2 ) ) ) )
=> ( ( ( X2
= ( suc @ ( zero_zero @ nat ) ) )
=> ( Y3
!= ( numeral_numeral @ nat @ ( bit1 @ one2 ) ) ) )
=> ~ ! [Va3: nat] :
( ( X2
= ( suc @ ( suc @ Va3 ) ) )
=> ~ ( ( ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( suc @ ( suc @ Va3 ) ) )
=> ( Y3
= ( 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 @ Va3 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( times_times @ nat @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( plus_plus @ nat @ ( vEBT_V8346862874174094_d_u_p @ ( divide_divide @ nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( one_one @ nat ) ) ) ) ) ) )
& ( ~ ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( suc @ ( suc @ Va3 ) ) )
=> ( Y3
= ( plus_plus @ nat @ ( plus_plus @ nat @ ( numeral_numeral @ nat @ ( bit1 @ ( bit1 @ ( bit0 @ one2 ) ) ) ) @ ( vEBT_V8346862874174094_d_u_p @ ( suc @ ( divide_divide @ nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) @ ( times_times @ nat @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( suc @ ( divide_divide @ nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( plus_plus @ nat @ ( vEBT_V8346862874174094_d_u_p @ ( divide_divide @ nat @ ( suc @ ( suc @ Va3 ) ) @ ( 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_2722_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,
! [X2: nat,Y3: nat] :
( ( ( vEBT_V8646137997579335489_i_l_d @ X2 )
= Y3 )
=> ( ( ( X2
= ( zero_zero @ nat ) )
=> ( Y3
!= ( numeral_numeral @ nat @ ( bit0 @ ( bit0 @ one2 ) ) ) ) )
=> ( ( ( X2
= ( suc @ ( zero_zero @ nat ) ) )
=> ( Y3
!= ( numeral_numeral @ nat @ ( bit0 @ ( bit0 @ one2 ) ) ) ) )
=> ~ ! [Va3: nat] :
( ( X2
= ( suc @ ( suc @ Va3 ) ) )
=> ~ ( ( ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( suc @ ( suc @ Va3 ) ) )
=> ( Y3
= ( 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 @ Va3 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( times_times @ nat @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( vEBT_V8646137997579335489_i_l_d @ ( divide_divide @ nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) ) ) )
& ( ~ ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( suc @ ( suc @ Va3 ) ) )
=> ( Y3
= ( plus_plus @ nat @ ( plus_plus @ nat @ ( numeral_numeral @ nat @ ( bit0 @ ( bit0 @ ( bit1 @ one2 ) ) ) ) @ ( vEBT_V8646137997579335489_i_l_d @ ( suc @ ( divide_divide @ nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) @ ( times_times @ nat @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( suc @ ( divide_divide @ nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( vEBT_V8646137997579335489_i_l_d @ ( divide_divide @ nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) ) ) ) ) ) ) ) ).
% VEBT_internal.T\<^sub>b\<^sub>u\<^sub>i\<^sub>l\<^sub>d.elims
thf(fact_2723_pow__divides__pow__iff,axiom,
! [A: $tType] :
( ( semiring_gcd @ A )
=> ! [N: nat,A3: A,B3: A] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ N )
=> ( ( dvd_dvd @ A @ ( power_power @ A @ A3 @ N ) @ ( power_power @ A @ B3 @ N ) )
= ( dvd_dvd @ A @ A3 @ B3 ) ) ) ) ).
% pow_divides_pow_iff
thf(fact_2724_div2__even__ext__nat,axiom,
! [X2: nat,Y3: nat] :
( ( ( divide_divide @ nat @ X2 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) )
= ( divide_divide @ nat @ Y3 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) )
=> ( ( ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ X2 )
= ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ Y3 ) )
=> ( X2 = Y3 ) ) ) ).
% div2_even_ext_nat
thf(fact_2725_lowi__def,axiom,
( vEBT_VEBT_lowi
= ( ^ [X3: nat,N4: nat] : ( heap_Time_return @ nat @ ( modulo_modulo @ nat @ X3 @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N4 ) ) ) ) ) ).
% lowi_def
thf(fact_2726_bezout__add__strong__nat,axiom,
! [A3: nat,B3: nat] :
( ( A3
!= ( zero_zero @ nat ) )
=> ? [D4: nat,X4: nat,Y4: nat] :
( ( dvd_dvd @ nat @ D4 @ A3 )
& ( dvd_dvd @ nat @ D4 @ B3 )
& ( ( times_times @ nat @ A3 @ X4 )
= ( plus_plus @ nat @ ( times_times @ nat @ B3 @ Y4 ) @ D4 ) ) ) ) ).
% bezout_add_strong_nat
thf(fact_2727_low__def,axiom,
( vEBT_VEBT_low
= ( ^ [X3: nat,N4: nat] : ( modulo_modulo @ nat @ X3 @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N4 ) ) ) ) ).
% low_def
thf(fact_2728_neg__eucl__rel__int__mult__2,axiom,
! [B3: int,A3: int,Q3: int,R2: int] :
( ( ord_less_eq @ int @ B3 @ ( zero_zero @ int ) )
=> ( ( eucl_rel_int @ ( plus_plus @ int @ A3 @ ( one_one @ int ) ) @ B3 @ ( product_Pair @ int @ int @ Q3 @ R2 ) )
=> ( eucl_rel_int @ ( plus_plus @ int @ ( one_one @ int ) @ ( times_times @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ A3 ) ) @ ( times_times @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ B3 ) @ ( 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_2729_mod__mod__trivial,axiom,
! [A: $tType] :
( ( euclid4440199948858584721cancel @ A )
=> ! [A3: A,B3: A] :
( ( modulo_modulo @ A @ ( modulo_modulo @ A @ A3 @ B3 ) @ B3 )
= ( modulo_modulo @ A @ A3 @ B3 ) ) ) ).
% mod_mod_trivial
thf(fact_2730_mod__add__self2,axiom,
! [A: $tType] :
( ( euclid4440199948858584721cancel @ A )
=> ! [A3: A,B3: A] :
( ( modulo_modulo @ A @ ( plus_plus @ A @ A3 @ B3 ) @ B3 )
= ( modulo_modulo @ A @ A3 @ B3 ) ) ) ).
% mod_add_self2
thf(fact_2731_mod__add__self1,axiom,
! [A: $tType] :
( ( euclid4440199948858584721cancel @ A )
=> ! [B3: A,A3: A] :
( ( modulo_modulo @ A @ ( plus_plus @ A @ B3 @ A3 ) @ B3 )
= ( modulo_modulo @ A @ A3 @ B3 ) ) ) ).
% mod_add_self1
thf(fact_2732_minus__mod__self2,axiom,
! [A: $tType] :
( ( euclid8851590272496341667cancel @ A )
=> ! [A3: A,B3: A] :
( ( modulo_modulo @ A @ ( minus_minus @ A @ A3 @ B3 ) @ B3 )
= ( modulo_modulo @ A @ A3 @ B3 ) ) ) ).
% minus_mod_self2
thf(fact_2733_mod__less,axiom,
! [M: nat,N: nat] :
( ( ord_less @ nat @ M @ N )
=> ( ( modulo_modulo @ nat @ M @ N )
= M ) ) ).
% mod_less
thf(fact_2734_nat__mod__eq_H,axiom,
! [A3: nat,N: nat] :
( ( ord_less @ nat @ A3 @ N )
=> ( ( modulo_modulo @ nat @ A3 @ N )
= A3 ) ) ).
% nat_mod_eq'
thf(fact_2735_mod__by__1,axiom,
! [A: $tType] :
( ( semidom_modulo @ A )
=> ! [A3: A] :
( ( modulo_modulo @ A @ A3 @ ( one_one @ A ) )
= ( zero_zero @ A ) ) ) ).
% mod_by_1
thf(fact_2736_bits__mod__by__1,axiom,
! [A: $tType] :
( ( bit_semiring_bits @ A )
=> ! [A3: A] :
( ( modulo_modulo @ A @ A3 @ ( one_one @ A ) )
= ( zero_zero @ A ) ) ) ).
% bits_mod_by_1
thf(fact_2737_mod__mult__self1__is__0,axiom,
! [A: $tType] :
( ( euclid4440199948858584721cancel @ A )
=> ! [B3: A,A3: A] :
( ( modulo_modulo @ A @ ( times_times @ A @ B3 @ A3 ) @ B3 )
= ( zero_zero @ A ) ) ) ).
% mod_mult_self1_is_0
thf(fact_2738_mod__mult__self2__is__0,axiom,
! [A: $tType] :
( ( euclid4440199948858584721cancel @ A )
=> ! [A3: A,B3: A] :
( ( modulo_modulo @ A @ ( times_times @ A @ A3 @ B3 ) @ B3 )
= ( zero_zero @ A ) ) ) ).
% mod_mult_self2_is_0
thf(fact_2739_bits__mod__div__trivial,axiom,
! [A: $tType] :
( ( bit_semiring_bits @ A )
=> ! [A3: A,B3: A] :
( ( divide_divide @ A @ ( modulo_modulo @ A @ A3 @ B3 ) @ B3 )
= ( zero_zero @ A ) ) ) ).
% bits_mod_div_trivial
thf(fact_2740_mod__div__trivial,axiom,
! [A: $tType] :
( ( euclid4440199948858584721cancel @ A )
=> ! [A3: A,B3: A] :
( ( divide_divide @ A @ ( modulo_modulo @ A @ A3 @ B3 ) @ B3 )
= ( zero_zero @ A ) ) ) ).
% mod_div_trivial
thf(fact_2741_mod__mult__self1,axiom,
! [A: $tType] :
( ( euclid4440199948858584721cancel @ A )
=> ! [A3: A,C3: A,B3: A] :
( ( modulo_modulo @ A @ ( plus_plus @ A @ A3 @ ( times_times @ A @ C3 @ B3 ) ) @ B3 )
= ( modulo_modulo @ A @ A3 @ B3 ) ) ) ).
% mod_mult_self1
thf(fact_2742_mod__mult__self2,axiom,
! [A: $tType] :
( ( euclid4440199948858584721cancel @ A )
=> ! [A3: A,B3: A,C3: A] :
( ( modulo_modulo @ A @ ( plus_plus @ A @ A3 @ ( times_times @ A @ B3 @ C3 ) ) @ B3 )
= ( modulo_modulo @ A @ A3 @ B3 ) ) ) ).
% mod_mult_self2
thf(fact_2743_mod__mult__self3,axiom,
! [A: $tType] :
( ( euclid4440199948858584721cancel @ A )
=> ! [C3: A,B3: A,A3: A] :
( ( modulo_modulo @ A @ ( plus_plus @ A @ ( times_times @ A @ C3 @ B3 ) @ A3 ) @ B3 )
= ( modulo_modulo @ A @ A3 @ B3 ) ) ) ).
% mod_mult_self3
thf(fact_2744_mod__mult__self4,axiom,
! [A: $tType] :
( ( euclid4440199948858584721cancel @ A )
=> ! [B3: A,C3: A,A3: A] :
( ( modulo_modulo @ A @ ( plus_plus @ A @ ( times_times @ A @ B3 @ C3 ) @ A3 ) @ B3 )
= ( modulo_modulo @ A @ A3 @ B3 ) ) ) ).
% mod_mult_self4
thf(fact_2745_mod__by__Suc__0,axiom,
! [M: nat] :
( ( modulo_modulo @ nat @ M @ ( suc @ ( zero_zero @ nat ) ) )
= ( zero_zero @ nat ) ) ).
% mod_by_Suc_0
thf(fact_2746_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_2747_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_2748_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_2749_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_2750_bits__one__mod__two__eq__one,axiom,
! [A: $tType] :
( ( bit_semiring_bits @ A )
=> ( ( modulo_modulo @ A @ ( one_one @ A ) @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) )
= ( one_one @ A ) ) ) ).
% bits_one_mod_two_eq_one
thf(fact_2751_one__mod__two__eq__one,axiom,
! [A: $tType] :
( ( euclid5411537665997757685th_nat @ A )
=> ( ( modulo_modulo @ A @ ( one_one @ A ) @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) )
= ( one_one @ A ) ) ) ).
% one_mod_two_eq_one
thf(fact_2752_even__mod__2__iff,axiom,
! [A: $tType] :
( ( semiring_parity @ A )
=> ! [A3: A] :
( ( dvd_dvd @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ ( modulo_modulo @ A @ A3 @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) )
= ( dvd_dvd @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ A3 ) ) ) ).
% even_mod_2_iff
thf(fact_2753_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_2754_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_2755_not__mod__2__eq__1__eq__0,axiom,
! [A: $tType] :
( ( semiring_parity @ A )
=> ! [A3: A] :
( ( ( modulo_modulo @ A @ A3 @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) )
!= ( one_one @ A ) )
= ( ( modulo_modulo @ A @ A3 @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) )
= ( zero_zero @ A ) ) ) ) ).
% not_mod_2_eq_1_eq_0
thf(fact_2756_not__mod__2__eq__0__eq__1,axiom,
! [A: $tType] :
( ( semiring_parity @ A )
=> ! [A3: A] :
( ( ( modulo_modulo @ A @ A3 @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) )
!= ( zero_zero @ A ) )
= ( ( modulo_modulo @ A @ A3 @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) )
= ( one_one @ A ) ) ) ) ).
% not_mod_2_eq_0_eq_1
thf(fact_2757_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_2758_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_2759_Suc__mod__eq__add3__mod__numeral,axiom,
! [M: nat,V: num] :
( ( modulo_modulo @ nat @ ( suc @ ( suc @ ( suc @ M ) ) ) @ ( numeral_numeral @ nat @ V ) )
= ( modulo_modulo @ nat @ ( plus_plus @ nat @ ( numeral_numeral @ nat @ ( bit1 @ one2 ) ) @ M ) @ ( numeral_numeral @ nat @ V ) ) ) ).
% Suc_mod_eq_add3_mod_numeral
thf(fact_2760_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_2761_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_2762_even__succ__mod__exp,axiom,
! [A: $tType] :
( ( bit_semiring_bits @ A )
=> ! [A3: A,N: nat] :
( ( dvd_dvd @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ A3 )
=> ( ( ord_less @ nat @ ( zero_zero @ nat ) @ N )
=> ( ( modulo_modulo @ A @ ( plus_plus @ A @ ( one_one @ A ) @ A3 ) @ ( power_power @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ N ) )
= ( plus_plus @ A @ ( one_one @ A ) @ ( modulo_modulo @ A @ A3 @ ( power_power @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ N ) ) ) ) ) ) ) ).
% even_succ_mod_exp
thf(fact_2763_of__nat__mod,axiom,
! [A: $tType] :
( ( euclid5411537665997757685th_nat @ A )
=> ! [M: nat,N: nat] :
( ( semiring_1_of_nat @ A @ ( modulo_modulo @ nat @ M @ N ) )
= ( modulo_modulo @ A @ ( semiring_1_of_nat @ A @ M ) @ ( semiring_1_of_nat @ A @ N ) ) ) ) ).
% of_nat_mod
thf(fact_2764_mod__add__right__eq,axiom,
! [A: $tType] :
( ( euclid4440199948858584721cancel @ A )
=> ! [A3: A,B3: A,C3: A] :
( ( modulo_modulo @ A @ ( plus_plus @ A @ A3 @ ( modulo_modulo @ A @ B3 @ C3 ) ) @ C3 )
= ( modulo_modulo @ A @ ( plus_plus @ A @ A3 @ B3 ) @ C3 ) ) ) ).
% mod_add_right_eq
thf(fact_2765_mod__add__left__eq,axiom,
! [A: $tType] :
( ( euclid4440199948858584721cancel @ A )
=> ! [A3: A,C3: A,B3: A] :
( ( modulo_modulo @ A @ ( plus_plus @ A @ ( modulo_modulo @ A @ A3 @ C3 ) @ B3 ) @ C3 )
= ( modulo_modulo @ A @ ( plus_plus @ A @ A3 @ B3 ) @ C3 ) ) ) ).
% mod_add_left_eq
thf(fact_2766_mod__add__cong,axiom,
! [A: $tType] :
( ( euclid4440199948858584721cancel @ A )
=> ! [A3: A,C3: A,A5: A,B3: A,B5: A] :
( ( ( modulo_modulo @ A @ A3 @ C3 )
= ( modulo_modulo @ A @ A5 @ C3 ) )
=> ( ( ( modulo_modulo @ A @ B3 @ C3 )
= ( modulo_modulo @ A @ B5 @ C3 ) )
=> ( ( modulo_modulo @ A @ ( plus_plus @ A @ A3 @ B3 ) @ C3 )
= ( modulo_modulo @ A @ ( plus_plus @ A @ A5 @ B5 ) @ C3 ) ) ) ) ) ).
% mod_add_cong
thf(fact_2767_mod__add__eq,axiom,
! [A: $tType] :
( ( euclid4440199948858584721cancel @ A )
=> ! [A3: A,C3: A,B3: A] :
( ( modulo_modulo @ A @ ( plus_plus @ A @ ( modulo_modulo @ A @ A3 @ C3 ) @ ( modulo_modulo @ A @ B3 @ C3 ) ) @ C3 )
= ( modulo_modulo @ A @ ( plus_plus @ A @ A3 @ B3 ) @ C3 ) ) ) ).
% mod_add_eq
thf(fact_2768_mod__mult__right__eq,axiom,
! [A: $tType] :
( ( euclid4440199948858584721cancel @ A )
=> ! [A3: A,B3: A,C3: A] :
( ( modulo_modulo @ A @ ( times_times @ A @ A3 @ ( modulo_modulo @ A @ B3 @ C3 ) ) @ C3 )
= ( modulo_modulo @ A @ ( times_times @ A @ A3 @ B3 ) @ C3 ) ) ) ).
% mod_mult_right_eq
thf(fact_2769_mod__mult__left__eq,axiom,
! [A: $tType] :
( ( euclid4440199948858584721cancel @ A )
=> ! [A3: A,C3: A,B3: A] :
( ( modulo_modulo @ A @ ( times_times @ A @ ( modulo_modulo @ A @ A3 @ C3 ) @ B3 ) @ C3 )
= ( modulo_modulo @ A @ ( times_times @ A @ A3 @ B3 ) @ C3 ) ) ) ).
% mod_mult_left_eq
thf(fact_2770_mult__mod__right,axiom,
! [A: $tType] :
( ( euclid4440199948858584721cancel @ A )
=> ! [C3: A,A3: A,B3: A] :
( ( times_times @ A @ C3 @ ( modulo_modulo @ A @ A3 @ B3 ) )
= ( modulo_modulo @ A @ ( times_times @ A @ C3 @ A3 ) @ ( times_times @ A @ C3 @ B3 ) ) ) ) ).
% mult_mod_right
thf(fact_2771_mod__mult__mult2,axiom,
! [A: $tType] :
( ( euclid4440199948858584721cancel @ A )
=> ! [A3: A,C3: A,B3: A] :
( ( modulo_modulo @ A @ ( times_times @ A @ A3 @ C3 ) @ ( times_times @ A @ B3 @ C3 ) )
= ( times_times @ A @ ( modulo_modulo @ A @ A3 @ B3 ) @ C3 ) ) ) ).
% mod_mult_mult2
thf(fact_2772_mod__mult__cong,axiom,
! [A: $tType] :
( ( euclid4440199948858584721cancel @ A )
=> ! [A3: A,C3: A,A5: A,B3: A,B5: A] :
( ( ( modulo_modulo @ A @ A3 @ C3 )
= ( modulo_modulo @ A @ A5 @ C3 ) )
=> ( ( ( modulo_modulo @ A @ B3 @ C3 )
= ( modulo_modulo @ A @ B5 @ C3 ) )
=> ( ( modulo_modulo @ A @ ( times_times @ A @ A3 @ B3 ) @ C3 )
= ( modulo_modulo @ A @ ( times_times @ A @ A5 @ B5 ) @ C3 ) ) ) ) ) ).
% mod_mult_cong
thf(fact_2773_mod__mult__eq,axiom,
! [A: $tType] :
( ( euclid4440199948858584721cancel @ A )
=> ! [A3: A,C3: A,B3: A] :
( ( modulo_modulo @ A @ ( times_times @ A @ ( modulo_modulo @ A @ A3 @ C3 ) @ ( modulo_modulo @ A @ B3 @ C3 ) ) @ C3 )
= ( modulo_modulo @ A @ ( times_times @ A @ A3 @ B3 ) @ C3 ) ) ) ).
% mod_mult_eq
thf(fact_2774_mod__diff__right__eq,axiom,
! [A: $tType] :
( ( euclid8851590272496341667cancel @ A )
=> ! [A3: A,B3: A,C3: A] :
( ( modulo_modulo @ A @ ( minus_minus @ A @ A3 @ ( modulo_modulo @ A @ B3 @ C3 ) ) @ C3 )
= ( modulo_modulo @ A @ ( minus_minus @ A @ A3 @ B3 ) @ C3 ) ) ) ).
% mod_diff_right_eq
thf(fact_2775_mod__diff__left__eq,axiom,
! [A: $tType] :
( ( euclid8851590272496341667cancel @ A )
=> ! [A3: A,C3: A,B3: A] :
( ( modulo_modulo @ A @ ( minus_minus @ A @ ( modulo_modulo @ A @ A3 @ C3 ) @ B3 ) @ C3 )
= ( modulo_modulo @ A @ ( minus_minus @ A @ A3 @ B3 ) @ C3 ) ) ) ).
% mod_diff_left_eq
thf(fact_2776_mod__diff__cong,axiom,
! [A: $tType] :
( ( euclid8851590272496341667cancel @ A )
=> ! [A3: A,C3: A,A5: A,B3: A,B5: A] :
( ( ( modulo_modulo @ A @ A3 @ C3 )
= ( modulo_modulo @ A @ A5 @ C3 ) )
=> ( ( ( modulo_modulo @ A @ B3 @ C3 )
= ( modulo_modulo @ A @ B5 @ C3 ) )
=> ( ( modulo_modulo @ A @ ( minus_minus @ A @ A3 @ B3 ) @ C3 )
= ( modulo_modulo @ A @ ( minus_minus @ A @ A5 @ B5 ) @ C3 ) ) ) ) ) ).
% mod_diff_cong
thf(fact_2777_mod__diff__eq,axiom,
! [A: $tType] :
( ( euclid8851590272496341667cancel @ A )
=> ! [A3: A,C3: A,B3: A] :
( ( modulo_modulo @ A @ ( minus_minus @ A @ ( modulo_modulo @ A @ A3 @ C3 ) @ ( modulo_modulo @ A @ B3 @ C3 ) ) @ C3 )
= ( modulo_modulo @ A @ ( minus_minus @ A @ A3 @ B3 ) @ C3 ) ) ) ).
% mod_diff_eq
thf(fact_2778_power__mod,axiom,
! [A: $tType] :
( ( euclid4440199948858584721cancel @ A )
=> ! [A3: A,B3: A,N: nat] :
( ( modulo_modulo @ A @ ( power_power @ A @ ( modulo_modulo @ A @ A3 @ B3 ) @ N ) @ B3 )
= ( modulo_modulo @ A @ ( power_power @ A @ A3 @ N ) @ B3 ) ) ) ).
% power_mod
thf(fact_2779_mod__mod__cancel,axiom,
! [A: $tType] :
( ( euclid4440199948858584721cancel @ A )
=> ! [C3: A,B3: A,A3: A] :
( ( dvd_dvd @ A @ C3 @ B3 )
=> ( ( modulo_modulo @ A @ ( modulo_modulo @ A @ A3 @ B3 ) @ C3 )
= ( modulo_modulo @ A @ A3 @ C3 ) ) ) ) ).
% mod_mod_cancel
thf(fact_2780_dvd__mod,axiom,
! [A: $tType] :
( ( euclid4440199948858584721cancel @ A )
=> ! [K: A,M: A,N: A] :
( ( dvd_dvd @ A @ K @ M )
=> ( ( dvd_dvd @ A @ K @ N )
=> ( dvd_dvd @ A @ K @ ( modulo_modulo @ A @ M @ N ) ) ) ) ) ).
% dvd_mod
thf(fact_2781_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_2782_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_2783_nat__mod__eq,axiom,
! [B3: nat,N: nat,A3: nat] :
( ( ord_less @ nat @ B3 @ N )
=> ( ( ( modulo_modulo @ nat @ A3 @ N )
= ( modulo_modulo @ nat @ B3 @ N ) )
=> ( ( modulo_modulo @ nat @ A3 @ N )
= B3 ) ) ) ).
% nat_mod_eq
thf(fact_2784_mod__plus__right,axiom,
! [A3: nat,X2: nat,M: nat,B3: nat] :
( ( ( modulo_modulo @ nat @ ( plus_plus @ nat @ A3 @ X2 ) @ M )
= ( modulo_modulo @ nat @ ( plus_plus @ nat @ B3 @ X2 ) @ M ) )
= ( ( modulo_modulo @ nat @ A3 @ M )
= ( modulo_modulo @ nat @ B3 @ M ) ) ) ).
% mod_plus_right
thf(fact_2785_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_2786_zdvd__zdiffD,axiom,
! [K: int,M: int,N: int] :
( ( dvd_dvd @ int @ K @ ( minus_minus @ int @ M @ N ) )
=> ( ( dvd_dvd @ int @ K @ N )
=> ( dvd_dvd @ int @ K @ M ) ) ) ).
% zdvd_zdiffD
thf(fact_2787_gcd__nat__induct,axiom,
! [P: nat > nat > $o,M: nat,N: nat] :
( ! [M5: nat] : ( P @ M5 @ ( zero_zero @ nat ) )
=> ( ! [M5: nat,N3: nat] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ N3 )
=> ( ( P @ N3 @ ( modulo_modulo @ nat @ M5 @ N3 ) )
=> ( P @ M5 @ N3 ) ) )
=> ( P @ M @ N ) ) ) ).
% gcd_nat_induct
thf(fact_2788_unique__euclidean__semiring__numeral__class_Omod__less__eq__dividend,axiom,
! [A: $tType] :
( ( unique1627219031080169319umeral @ A )
=> ! [A3: A,B3: A] :
( ( ord_less_eq @ A @ ( zero_zero @ A ) @ A3 )
=> ( ord_less_eq @ A @ ( modulo_modulo @ A @ A3 @ B3 ) @ A3 ) ) ) ).
% unique_euclidean_semiring_numeral_class.mod_less_eq_dividend
thf(fact_2789_unique__euclidean__semiring__numeral__class_Opos__mod__bound,axiom,
! [A: $tType] :
( ( unique1627219031080169319umeral @ A )
=> ! [B3: A,A3: A] :
( ( ord_less @ A @ ( zero_zero @ A ) @ B3 )
=> ( ord_less @ A @ ( modulo_modulo @ A @ A3 @ B3 ) @ B3 ) ) ) ).
% unique_euclidean_semiring_numeral_class.pos_mod_bound
thf(fact_2790_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_2791_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_2792_mod__eq__self__iff__div__eq__0,axiom,
! [A: $tType] :
( ( euclid3725896446679973847miring @ A )
=> ! [A3: A,B3: A] :
( ( ( modulo_modulo @ A @ A3 @ B3 )
= A3 )
= ( ( divide_divide @ A @ A3 @ B3 )
= ( zero_zero @ A ) ) ) ) ).
% mod_eq_self_iff_div_eq_0
thf(fact_2793_mod__eqE,axiom,
! [A: $tType] :
( ( euclid8851590272496341667cancel @ A )
=> ! [A3: A,C3: A,B3: A] :
( ( ( modulo_modulo @ A @ A3 @ C3 )
= ( modulo_modulo @ A @ B3 @ C3 ) )
=> ~ ! [D4: A] :
( B3
!= ( plus_plus @ A @ A3 @ ( times_times @ A @ C3 @ D4 ) ) ) ) ) ).
% mod_eqE
thf(fact_2794_div__add1__eq,axiom,
! [A: $tType] :
( ( euclid3128863361964157862miring @ A )
=> ! [A3: A,B3: A,C3: A] :
( ( divide_divide @ A @ ( plus_plus @ A @ A3 @ B3 ) @ C3 )
= ( plus_plus @ A @ ( plus_plus @ A @ ( divide_divide @ A @ A3 @ C3 ) @ ( divide_divide @ A @ B3 @ C3 ) ) @ ( divide_divide @ A @ ( plus_plus @ A @ ( modulo_modulo @ A @ A3 @ C3 ) @ ( modulo_modulo @ A @ B3 @ C3 ) ) @ C3 ) ) ) ) ).
% div_add1_eq
thf(fact_2795_mod__eq__dvd__iff,axiom,
! [A: $tType] :
( ( euclid8851590272496341667cancel @ A )
=> ! [A3: A,C3: A,B3: A] :
( ( ( modulo_modulo @ A @ A3 @ C3 )
= ( modulo_modulo @ A @ B3 @ C3 ) )
= ( dvd_dvd @ A @ C3 @ ( minus_minus @ A @ A3 @ B3 ) ) ) ) ).
% mod_eq_dvd_iff
thf(fact_2796_dvd__minus__mod,axiom,
! [A: $tType] :
( ( semidom_modulo @ A )
=> ! [B3: A,A3: A] : ( dvd_dvd @ A @ B3 @ ( minus_minus @ A @ A3 @ ( modulo_modulo @ A @ A3 @ B3 ) ) ) ) ).
% dvd_minus_mod
thf(fact_2797_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_2798_mod__induct,axiom,
! [P: nat > $o,N: nat,P6: nat,M: nat] :
( ( P @ N )
=> ( ( ord_less @ nat @ N @ P6 )
=> ( ( ord_less @ nat @ M @ P6 )
=> ( ! [N3: nat] :
( ( ord_less @ nat @ N3 @ P6 )
=> ( ( P @ N3 )
=> ( P @ ( modulo_modulo @ nat @ ( suc @ N3 ) @ P6 ) ) ) )
=> ( P @ M ) ) ) ) ) ).
% mod_induct
thf(fact_2799_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_2800_nat__mod__lem,axiom,
! [N: nat,B3: nat] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ N )
=> ( ( ord_less @ nat @ B3 @ N )
= ( ( modulo_modulo @ nat @ B3 @ N )
= B3 ) ) ) ).
% nat_mod_lem
thf(fact_2801_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_2802_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_2803_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_2804_mod__if,axiom,
( ( modulo_modulo @ nat )
= ( ^ [M6: nat,N4: nat] : ( if @ nat @ ( ord_less @ nat @ M6 @ N4 ) @ M6 @ ( modulo_modulo @ nat @ ( minus_minus @ nat @ M6 @ N4 ) @ N4 ) ) ) ) ).
% mod_if
thf(fact_2805_mod__nat__sub,axiom,
! [X2: nat,Z: nat,Y3: nat] :
( ( ord_less @ nat @ X2 @ Z )
=> ( ( modulo_modulo @ nat @ ( minus_minus @ nat @ X2 @ Y3 ) @ Z )
= ( minus_minus @ nat @ X2 @ Y3 ) ) ) ).
% mod_nat_sub
thf(fact_2806_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_2807_mod__eq__0D,axiom,
! [M: nat,D2: nat] :
( ( ( modulo_modulo @ nat @ M @ D2 )
= ( zero_zero @ nat ) )
=> ? [Q4: nat] :
( M
= ( times_times @ nat @ D2 @ Q4 ) ) ) ).
% mod_eq_0D
thf(fact_2808_nat__minus__mod__plus__right,axiom,
! [N: nat,X2: nat,M: nat] :
( ( modulo_modulo @ nat @ ( minus_minus @ nat @ ( plus_plus @ nat @ N @ X2 ) @ ( modulo_modulo @ nat @ N @ M ) ) @ M )
= ( modulo_modulo @ nat @ X2 @ M ) ) ).
% nat_minus_mod_plus_right
thf(fact_2809_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_2810_msrevs_I2_J,axiom,
! [K: nat,N: nat,M: nat] :
( ( modulo_modulo @ nat @ ( plus_plus @ nat @ ( times_times @ nat @ K @ N ) @ M ) @ N )
= ( modulo_modulo @ nat @ M @ N ) ) ).
% msrevs(2)
thf(fact_2811_nat__mod__eq__iff,axiom,
! [X2: nat,N: nat,Y3: nat] :
( ( ( modulo_modulo @ nat @ X2 @ N )
= ( modulo_modulo @ nat @ Y3 @ N ) )
= ( ? [Q1: nat,Q22: nat] :
( ( plus_plus @ nat @ X2 @ ( times_times @ nat @ N @ Q1 ) )
= ( plus_plus @ nat @ Y3 @ ( times_times @ nat @ N @ Q22 ) ) ) ) ) ).
% nat_mod_eq_iff
thf(fact_2812_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_2813_zdvd__period,axiom,
! [A3: int,D2: int,X2: int,T2: int,C3: int] :
( ( dvd_dvd @ int @ A3 @ D2 )
=> ( ( dvd_dvd @ int @ A3 @ ( plus_plus @ int @ X2 @ T2 ) )
= ( dvd_dvd @ int @ A3 @ ( plus_plus @ int @ ( plus_plus @ int @ X2 @ ( times_times @ int @ C3 @ D2 ) ) @ T2 ) ) ) ) ).
% zdvd_period
thf(fact_2814_zdvd__reduce,axiom,
! [K: int,N: int,M: int] :
( ( dvd_dvd @ int @ K @ ( plus_plus @ int @ N @ ( times_times @ int @ K @ M ) ) )
= ( dvd_dvd @ int @ K @ N ) ) ).
% zdvd_reduce
thf(fact_2815_unique__euclidean__semiring__numeral__class_Omod__less,axiom,
! [A: $tType] :
( ( unique1627219031080169319umeral @ A )
=> ! [A3: A,B3: A] :
( ( ord_less_eq @ A @ ( zero_zero @ A ) @ A3 )
=> ( ( ord_less @ A @ A3 @ B3 )
=> ( ( modulo_modulo @ A @ A3 @ B3 )
= A3 ) ) ) ) ).
% unique_euclidean_semiring_numeral_class.mod_less
thf(fact_2816_unique__euclidean__semiring__numeral__class_Opos__mod__sign,axiom,
! [A: $tType] :
( ( unique1627219031080169319umeral @ A )
=> ! [B3: A,A3: A] :
( ( ord_less @ A @ ( zero_zero @ A ) @ B3 )
=> ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( modulo_modulo @ A @ A3 @ B3 ) ) ) ) ).
% unique_euclidean_semiring_numeral_class.pos_mod_sign
thf(fact_2817_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_2818_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_2819_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_2820_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_2821_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_2822_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_2823_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_2824_mult__div__mod__eq,axiom,
! [A: $tType] :
( ( semiring_modulo @ A )
=> ! [B3: A,A3: A] :
( ( plus_plus @ A @ ( times_times @ A @ B3 @ ( divide_divide @ A @ A3 @ B3 ) ) @ ( modulo_modulo @ A @ A3 @ B3 ) )
= A3 ) ) ).
% mult_div_mod_eq
thf(fact_2825_mod__mult__div__eq,axiom,
! [A: $tType] :
( ( semiring_modulo @ A )
=> ! [A3: A,B3: A] :
( ( plus_plus @ A @ ( modulo_modulo @ A @ A3 @ B3 ) @ ( times_times @ A @ B3 @ ( divide_divide @ A @ A3 @ B3 ) ) )
= A3 ) ) ).
% mod_mult_div_eq
thf(fact_2826_mod__div__mult__eq,axiom,
! [A: $tType] :
( ( semiring_modulo @ A )
=> ! [A3: A,B3: A] :
( ( plus_plus @ A @ ( modulo_modulo @ A @ A3 @ B3 ) @ ( times_times @ A @ ( divide_divide @ A @ A3 @ B3 ) @ B3 ) )
= A3 ) ) ).
% mod_div_mult_eq
thf(fact_2827_div__mult__mod__eq,axiom,
! [A: $tType] :
( ( semiring_modulo @ A )
=> ! [A3: A,B3: A] :
( ( plus_plus @ A @ ( times_times @ A @ ( divide_divide @ A @ A3 @ B3 ) @ B3 ) @ ( modulo_modulo @ A @ A3 @ B3 ) )
= A3 ) ) ).
% div_mult_mod_eq
thf(fact_2828_mod__div__decomp,axiom,
! [A: $tType] :
( ( semiring_modulo @ A )
=> ! [A3: A,B3: A] :
( A3
= ( plus_plus @ A @ ( times_times @ A @ ( divide_divide @ A @ A3 @ B3 ) @ B3 ) @ ( modulo_modulo @ A @ A3 @ B3 ) ) ) ) ).
% mod_div_decomp
thf(fact_2829_cancel__div__mod__rules_I1_J,axiom,
! [A: $tType] :
( ( semidom_modulo @ A )
=> ! [A3: A,B3: A,C3: A] :
( ( plus_plus @ A @ ( plus_plus @ A @ ( times_times @ A @ ( divide_divide @ A @ A3 @ B3 ) @ B3 ) @ ( modulo_modulo @ A @ A3 @ B3 ) ) @ C3 )
= ( plus_plus @ A @ A3 @ C3 ) ) ) ).
% cancel_div_mod_rules(1)
thf(fact_2830_cancel__div__mod__rules_I2_J,axiom,
! [A: $tType] :
( ( semidom_modulo @ A )
=> ! [B3: A,A3: A,C3: A] :
( ( plus_plus @ A @ ( plus_plus @ A @ ( times_times @ A @ B3 @ ( divide_divide @ A @ A3 @ B3 ) ) @ ( modulo_modulo @ A @ A3 @ B3 ) ) @ C3 )
= ( plus_plus @ A @ A3 @ C3 ) ) ) ).
% cancel_div_mod_rules(2)
thf(fact_2831_div__mult1__eq,axiom,
! [A: $tType] :
( ( euclid3128863361964157862miring @ A )
=> ! [A3: A,B3: A,C3: A] :
( ( divide_divide @ A @ ( times_times @ A @ A3 @ B3 ) @ C3 )
= ( plus_plus @ A @ ( times_times @ A @ A3 @ ( divide_divide @ A @ B3 @ C3 ) ) @ ( divide_divide @ A @ ( times_times @ A @ A3 @ ( modulo_modulo @ A @ B3 @ C3 ) ) @ C3 ) ) ) ) ).
% div_mult1_eq
thf(fact_2832_minus__mult__div__eq__mod,axiom,
! [A: $tType] :
( ( semiring_modulo @ A )
=> ! [A3: A,B3: A] :
( ( minus_minus @ A @ A3 @ ( times_times @ A @ B3 @ ( divide_divide @ A @ A3 @ B3 ) ) )
= ( modulo_modulo @ A @ A3 @ B3 ) ) ) ).
% minus_mult_div_eq_mod
thf(fact_2833_minus__mod__eq__mult__div,axiom,
! [A: $tType] :
( ( semiring_modulo @ A )
=> ! [A3: A,B3: A] :
( ( minus_minus @ A @ A3 @ ( modulo_modulo @ A @ A3 @ B3 ) )
= ( times_times @ A @ B3 @ ( divide_divide @ A @ A3 @ B3 ) ) ) ) ).
% minus_mod_eq_mult_div
thf(fact_2834_minus__mod__eq__div__mult,axiom,
! [A: $tType] :
( ( semiring_modulo @ A )
=> ! [A3: A,B3: A] :
( ( minus_minus @ A @ A3 @ ( modulo_modulo @ A @ A3 @ B3 ) )
= ( times_times @ A @ ( divide_divide @ A @ A3 @ B3 ) @ B3 ) ) ) ).
% minus_mod_eq_div_mult
thf(fact_2835_minus__div__mult__eq__mod,axiom,
! [A: $tType] :
( ( semiring_modulo @ A )
=> ! [A3: A,B3: A] :
( ( minus_minus @ A @ A3 @ ( times_times @ A @ ( divide_divide @ A @ A3 @ B3 ) @ B3 ) )
= ( modulo_modulo @ A @ A3 @ B3 ) ) ) ).
% minus_div_mult_eq_mod
thf(fact_2836_zmde,axiom,
! [A: $tType] :
( ( ( group_add @ A )
& ( semiring_modulo @ A ) )
=> ! [B3: A,A3: A] :
( ( times_times @ A @ B3 @ ( divide_divide @ A @ A3 @ B3 ) )
= ( minus_minus @ A @ A3 @ ( modulo_modulo @ A @ A3 @ B3 ) ) ) ) ).
% zmde
thf(fact_2837_unit__imp__mod__eq__0,axiom,
! [A: $tType] :
( ( euclid3725896446679973847miring @ A )
=> ! [B3: A,A3: A] :
( ( dvd_dvd @ A @ B3 @ ( one_one @ A ) )
=> ( ( modulo_modulo @ A @ A3 @ B3 )
= ( zero_zero @ A ) ) ) ) ).
% unit_imp_mod_eq_0
thf(fact_2838_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_2839_div__less__mono,axiom,
! [A2: nat,B2: nat,N: nat] :
( ( ord_less @ nat @ A2 @ B2 )
=> ( ( ord_less @ nat @ ( zero_zero @ nat ) @ N )
=> ( ( ( modulo_modulo @ nat @ A2 @ N )
= ( zero_zero @ nat ) )
=> ( ( ( modulo_modulo @ nat @ B2 @ N )
= ( zero_zero @ nat ) )
=> ( ord_less @ nat @ ( divide_divide @ nat @ A2 @ N ) @ ( divide_divide @ nat @ B2 @ N ) ) ) ) ) ) ).
% div_less_mono
thf(fact_2840_mod__nat__add,axiom,
! [X2: nat,Z: nat,Y3: nat] :
( ( ord_less @ nat @ X2 @ Z )
=> ( ( ord_less @ nat @ Y3 @ Z )
=> ( ( ( ord_less @ nat @ ( plus_plus @ nat @ X2 @ Y3 ) @ Z )
=> ( ( modulo_modulo @ nat @ ( plus_plus @ nat @ X2 @ Y3 ) @ Z )
= ( plus_plus @ nat @ X2 @ Y3 ) ) )
& ( ~ ( ord_less @ nat @ ( plus_plus @ nat @ X2 @ Y3 ) @ Z )
=> ( ( modulo_modulo @ nat @ ( plus_plus @ nat @ X2 @ Y3 ) @ Z )
= ( minus_minus @ nat @ ( plus_plus @ nat @ X2 @ Y3 ) @ Z ) ) ) ) ) ) ).
% mod_nat_add
thf(fact_2841_nat__mod__eq__lemma,axiom,
! [X2: nat,N: nat,Y3: nat] :
( ( ( modulo_modulo @ nat @ X2 @ N )
= ( modulo_modulo @ nat @ Y3 @ N ) )
=> ( ( ord_less_eq @ nat @ Y3 @ X2 )
=> ? [Q4: nat] :
( X2
= ( plus_plus @ nat @ Y3 @ ( times_times @ nat @ N @ Q4 ) ) ) ) ) ).
% nat_mod_eq_lemma
thf(fact_2842_mod__eq__nat2E,axiom,
! [M: nat,Q3: nat,N: nat] :
( ( ( modulo_modulo @ nat @ M @ Q3 )
= ( modulo_modulo @ nat @ N @ Q3 ) )
=> ( ( ord_less_eq @ nat @ M @ N )
=> ~ ! [S3: nat] :
( N
!= ( plus_plus @ nat @ M @ ( times_times @ nat @ Q3 @ S3 ) ) ) ) ) ).
% mod_eq_nat2E
thf(fact_2843_mod__eq__nat1E,axiom,
! [M: nat,Q3: nat,N: nat] :
( ( ( modulo_modulo @ nat @ M @ Q3 )
= ( modulo_modulo @ nat @ N @ Q3 ) )
=> ( ( ord_less_eq @ nat @ N @ M )
=> ~ ! [S3: nat] :
( M
!= ( plus_plus @ nat @ N @ ( times_times @ nat @ Q3 @ S3 ) ) ) ) ) ).
% mod_eq_nat1E
thf(fact_2844_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_2845_divmod_H__nat__def,axiom,
( ( unique8689654367752047608divmod @ nat )
= ( ^ [M6: num,N4: num] : ( product_Pair @ nat @ nat @ ( divide_divide @ nat @ ( numeral_numeral @ nat @ M6 ) @ ( numeral_numeral @ nat @ N4 ) ) @ ( modulo_modulo @ nat @ ( numeral_numeral @ nat @ M6 ) @ ( numeral_numeral @ nat @ N4 ) ) ) ) ) ).
% divmod'_nat_def
thf(fact_2846_div__mod__decomp,axiom,
! [A2: nat,N: nat] :
( A2
= ( plus_plus @ nat @ ( times_times @ nat @ ( divide_divide @ nat @ A2 @ N ) @ N ) @ ( modulo_modulo @ nat @ A2 @ N ) ) ) ).
% div_mod_decomp
thf(fact_2847_mod__mult2__eq,axiom,
! [M: nat,N: nat,Q3: nat] :
( ( modulo_modulo @ nat @ M @ ( times_times @ nat @ N @ Q3 ) )
= ( plus_plus @ nat @ ( times_times @ nat @ N @ ( modulo_modulo @ nat @ ( divide_divide @ nat @ M @ N ) @ Q3 ) ) @ ( modulo_modulo @ nat @ M @ N ) ) ) ).
% mod_mult2_eq
thf(fact_2848_modulo__nat__def,axiom,
( ( modulo_modulo @ nat )
= ( ^ [M6: nat,N4: nat] : ( minus_minus @ nat @ M6 @ ( times_times @ nat @ ( divide_divide @ nat @ M6 @ N4 ) @ N4 ) ) ) ) ).
% modulo_nat_def
thf(fact_2849_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_2850_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_2851_VEBT__internal_OminNulli_Osimps_I5_J,axiom,
! [Uz: product_prod @ nat @ nat,Va: nat,Vb: array @ vEBT_VEBTi,Vc: vEBT_VEBTi] :
( ( vEBT_VEBT_minNulli @ ( vEBT_Nodei @ ( some @ ( product_prod @ nat @ nat ) @ Uz ) @ Va @ Vb @ Vc ) )
= ( heap_Time_return @ $o @ $false ) ) ).
% VEBT_internal.minNulli.simps(5)
thf(fact_2852_VEBT__internal_OminNulli_Osimps_I4_J,axiom,
! [Uw: nat,Ux: array @ vEBT_VEBTi,Uy: vEBT_VEBTi] :
( ( vEBT_VEBT_minNulli @ ( vEBT_Nodei @ ( none @ ( product_prod @ nat @ nat ) ) @ Uw @ Ux @ Uy ) )
= ( heap_Time_return @ $o @ $true ) ) ).
% VEBT_internal.minNulli.simps(4)
thf(fact_2853_prod__decode__aux_Ocases,axiom,
! [X2: product_prod @ nat @ nat] :
~ ! [K2: nat,M5: nat] :
( X2
!= ( product_Pair @ nat @ nat @ K2 @ M5 ) ) ).
% prod_decode_aux.cases
thf(fact_2854_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_2855_mod__mult2__eq_H,axiom,
! [A: $tType] :
( ( euclid5411537665997757685th_nat @ A )
=> ! [A3: A,M: nat,N: nat] :
( ( modulo_modulo @ A @ A3 @ ( times_times @ A @ ( semiring_1_of_nat @ A @ M ) @ ( semiring_1_of_nat @ A @ N ) ) )
= ( plus_plus @ A @ ( times_times @ A @ ( semiring_1_of_nat @ A @ M ) @ ( modulo_modulo @ A @ ( divide_divide @ A @ A3 @ ( semiring_1_of_nat @ A @ M ) ) @ ( semiring_1_of_nat @ A @ N ) ) ) @ ( modulo_modulo @ A @ A3 @ ( semiring_1_of_nat @ A @ M ) ) ) ) ) ).
% mod_mult2_eq'
thf(fact_2856_even__even__mod__4__iff,axiom,
! [N: nat] :
( ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N )
= ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( modulo_modulo @ nat @ N @ ( numeral_numeral @ nat @ ( bit0 @ ( bit0 @ one2 ) ) ) ) ) ) ).
% even_even_mod_4_iff
thf(fact_2857_field__char__0__class_Oof__nat__div,axiom,
! [A: $tType] :
( ( field_char_0 @ A )
=> ! [M: nat,N: nat] :
( ( semiring_1_of_nat @ A @ ( divide_divide @ nat @ M @ N ) )
= ( divide_divide @ A @ ( minus_minus @ A @ ( semiring_1_of_nat @ A @ M ) @ ( semiring_1_of_nat @ A @ ( modulo_modulo @ nat @ M @ N ) ) ) @ ( semiring_1_of_nat @ A @ N ) ) ) ) ).
% field_char_0_class.of_nat_div
thf(fact_2858_split__mod,axiom,
! [P: nat > $o,M: nat,N: nat] :
( ( P @ ( modulo_modulo @ nat @ M @ N ) )
= ( ( ( N
= ( zero_zero @ nat ) )
=> ( P @ M ) )
& ( ( N
!= ( zero_zero @ nat ) )
=> ! [I4: nat,J3: nat] :
( ( ord_less @ nat @ J3 @ N )
=> ( ( M
= ( plus_plus @ nat @ ( times_times @ nat @ N @ I4 ) @ J3 ) )
=> ( P @ J3 ) ) ) ) ) ) ).
% split_mod
thf(fact_2859_mod__lemma,axiom,
! [C3: nat,R2: nat,B3: nat,Q3: nat] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ C3 )
=> ( ( ord_less @ nat @ R2 @ B3 )
=> ( ord_less @ nat @ ( plus_plus @ nat @ ( times_times @ nat @ B3 @ ( modulo_modulo @ nat @ Q3 @ C3 ) ) @ R2 ) @ ( times_times @ nat @ B3 @ C3 ) ) ) ) ).
% mod_lemma
thf(fact_2860_divmod__def,axiom,
! [A: $tType] :
( ( unique1627219031080169319umeral @ A )
=> ( ( unique8689654367752047608divmod @ A )
= ( ^ [M6: num,N4: num] : ( product_Pair @ A @ A @ ( divide_divide @ A @ ( numeral_numeral @ A @ M6 ) @ ( numeral_numeral @ A @ N4 ) ) @ ( modulo_modulo @ A @ ( numeral_numeral @ A @ M6 ) @ ( numeral_numeral @ A @ N4 ) ) ) ) ) ) ).
% divmod_def
thf(fact_2861_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_2862_diff__mod__le,axiom,
! [A3: nat,D2: nat,B3: nat] :
( ( ord_less @ nat @ A3 @ D2 )
=> ( ( dvd_dvd @ nat @ B3 @ D2 )
=> ( ord_less_eq @ nat @ ( minus_minus @ nat @ A3 @ ( modulo_modulo @ nat @ A3 @ B3 ) ) @ ( minus_minus @ nat @ D2 @ B3 ) ) ) ) ).
% diff_mod_le
thf(fact_2863_VEBT__internal_Ovebt__buildupi_H_Osimps_I2_J,axiom,
( ( vEBT_V739175172307565963ildupi @ ( suc @ ( zero_zero @ nat ) ) )
= ( heap_Time_return @ vEBT_VEBTi @ ( vEBT_Leafi @ $false @ $false ) ) ) ).
% VEBT_internal.vebt_buildupi'.simps(2)
thf(fact_2864_vebt__buildupi_Osimps_I2_J,axiom,
( ( vEBT_vebt_buildupi @ ( suc @ ( zero_zero @ nat ) ) )
= ( heap_Time_return @ vEBT_VEBTi @ ( vEBT_Leafi @ $false @ $false ) ) ) ).
% vebt_buildupi.simps(2)
thf(fact_2865_real__of__nat__div__aux,axiom,
! [X2: nat,D2: nat] :
( ( divide_divide @ real @ ( semiring_1_of_nat @ real @ X2 ) @ ( semiring_1_of_nat @ real @ D2 ) )
= ( plus_plus @ real @ ( semiring_1_of_nat @ real @ ( divide_divide @ nat @ X2 @ D2 ) ) @ ( divide_divide @ real @ ( semiring_1_of_nat @ real @ ( modulo_modulo @ nat @ X2 @ D2 ) ) @ ( semiring_1_of_nat @ real @ D2 ) ) ) ) ).
% real_of_nat_div_aux
thf(fact_2866_vebt__minti_Osimps_I2_J,axiom,
! [Uu: nat,Uv: array @ vEBT_VEBTi,Uw: vEBT_VEBTi] :
( ( vEBT_vebt_minti @ ( vEBT_Nodei @ ( none @ ( product_prod @ nat @ nat ) ) @ Uu @ Uv @ Uw ) )
= ( heap_Time_return @ ( option @ nat ) @ ( none @ nat ) ) ) ).
% vebt_minti.simps(2)
thf(fact_2867_vebt__maxti_Osimps_I2_J,axiom,
! [Uu: nat,Uv: array @ vEBT_VEBTi,Uw: vEBT_VEBTi] :
( ( vEBT_vebt_maxti @ ( vEBT_Nodei @ ( none @ ( product_prod @ nat @ nat ) ) @ Uu @ Uv @ Uw ) )
= ( heap_Time_return @ ( option @ nat ) @ ( none @ nat ) ) ) ).
% vebt_maxti.simps(2)
thf(fact_2868_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_2869_bset_I9_J,axiom,
! [D2: int,D: int,B2: set @ int,T2: int] :
( ( dvd_dvd @ int @ D2 @ D )
=> ! [X8: int] :
( ! [Xa3: int] :
( ( member @ int @ Xa3 @ ( set_or1337092689740270186AtMost @ int @ ( one_one @ int ) @ D ) )
=> ! [Xb3: int] :
( ( member @ int @ Xb3 @ B2 )
=> ( X8
!= ( plus_plus @ int @ Xb3 @ Xa3 ) ) ) )
=> ( ( dvd_dvd @ int @ D2 @ ( plus_plus @ int @ X8 @ T2 ) )
=> ( dvd_dvd @ int @ D2 @ ( plus_plus @ int @ ( minus_minus @ int @ X8 @ D ) @ T2 ) ) ) ) ) ).
% bset(9)
thf(fact_2870_bset_I10_J,axiom,
! [D2: int,D: int,B2: set @ int,T2: int] :
( ( dvd_dvd @ int @ D2 @ D )
=> ! [X8: int] :
( ! [Xa3: int] :
( ( member @ int @ Xa3 @ ( set_or1337092689740270186AtMost @ int @ ( one_one @ int ) @ D ) )
=> ! [Xb3: int] :
( ( member @ int @ Xb3 @ B2 )
=> ( X8
!= ( plus_plus @ int @ Xb3 @ Xa3 ) ) ) )
=> ( ~ ( dvd_dvd @ int @ D2 @ ( plus_plus @ int @ X8 @ T2 ) )
=> ~ ( dvd_dvd @ int @ D2 @ ( plus_plus @ int @ ( minus_minus @ int @ X8 @ D ) @ T2 ) ) ) ) ) ).
% bset(10)
thf(fact_2871_aset_I9_J,axiom,
! [D2: int,D: int,A2: set @ int,T2: int] :
( ( dvd_dvd @ int @ D2 @ D )
=> ! [X8: int] :
( ! [Xa3: int] :
( ( member @ int @ Xa3 @ ( set_or1337092689740270186AtMost @ int @ ( one_one @ int ) @ D ) )
=> ! [Xb3: int] :
( ( member @ int @ Xb3 @ A2 )
=> ( X8
!= ( minus_minus @ int @ Xb3 @ Xa3 ) ) ) )
=> ( ( dvd_dvd @ int @ D2 @ ( plus_plus @ int @ X8 @ T2 ) )
=> ( dvd_dvd @ int @ D2 @ ( plus_plus @ int @ ( plus_plus @ int @ X8 @ D ) @ T2 ) ) ) ) ) ).
% aset(9)
thf(fact_2872_aset_I10_J,axiom,
! [D2: int,D: int,A2: set @ int,T2: int] :
( ( dvd_dvd @ int @ D2 @ D )
=> ! [X8: int] :
( ! [Xa3: int] :
( ( member @ int @ Xa3 @ ( set_or1337092689740270186AtMost @ int @ ( one_one @ int ) @ D ) )
=> ! [Xb3: int] :
( ( member @ int @ Xb3 @ A2 )
=> ( X8
!= ( minus_minus @ int @ Xb3 @ Xa3 ) ) ) )
=> ( ~ ( dvd_dvd @ int @ D2 @ ( plus_plus @ int @ X8 @ T2 ) )
=> ~ ( dvd_dvd @ int @ D2 @ ( plus_plus @ int @ ( plus_plus @ int @ X8 @ D ) @ T2 ) ) ) ) ) ).
% aset(10)
thf(fact_2873_unique__euclidean__semiring__numeral__class_Omod__mult2__eq,axiom,
! [A: $tType] :
( ( unique1627219031080169319umeral @ A )
=> ! [C3: A,A3: A,B3: A] :
( ( ord_less_eq @ A @ ( zero_zero @ A ) @ C3 )
=> ( ( modulo_modulo @ A @ A3 @ ( times_times @ A @ B3 @ C3 ) )
= ( plus_plus @ A @ ( times_times @ A @ B3 @ ( modulo_modulo @ A @ ( divide_divide @ A @ A3 @ B3 ) @ C3 ) ) @ ( modulo_modulo @ A @ A3 @ B3 ) ) ) ) ) ).
% unique_euclidean_semiring_numeral_class.mod_mult2_eq
thf(fact_2874_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_2875_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_2876_even__iff__mod__2__eq__zero,axiom,
! [A: $tType] :
( ( semiring_parity @ A )
=> ! [A3: A] :
( ( dvd_dvd @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ A3 )
= ( ( modulo_modulo @ A @ A3 @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) )
= ( zero_zero @ A ) ) ) ) ).
% even_iff_mod_2_eq_zero
thf(fact_2877_odd__iff__mod__2__eq__one,axiom,
! [A: $tType] :
( ( semiring_parity @ A )
=> ! [A3: A] :
( ( ~ ( dvd_dvd @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ A3 ) )
= ( ( modulo_modulo @ A @ A3 @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) )
= ( one_one @ A ) ) ) ) ).
% odd_iff_mod_2_eq_one
thf(fact_2878_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_2879_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_2880_even__diff__iff,axiom,
! [K: int,L: int] :
( ( dvd_dvd @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ ( minus_minus @ int @ K @ L ) )
= ( dvd_dvd @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ ( plus_plus @ int @ K @ L ) ) ) ).
% even_diff_iff
thf(fact_2881_VEBT__internal_OminNulli_Oelims,axiom,
! [X2: vEBT_VEBTi,Y3: heap_Time_Heap @ $o] :
( ( ( vEBT_VEBT_minNulli @ X2 )
= Y3 )
=> ( ( ( X2
= ( vEBT_Leafi @ $false @ $false ) )
=> ( Y3
!= ( heap_Time_return @ $o @ $true ) ) )
=> ( ( ? [Uv2: $o] :
( X2
= ( vEBT_Leafi @ $true @ Uv2 ) )
=> ( Y3
!= ( heap_Time_return @ $o @ $false ) ) )
=> ( ( ? [Uu2: $o] :
( X2
= ( vEBT_Leafi @ Uu2 @ $true ) )
=> ( Y3
!= ( heap_Time_return @ $o @ $false ) ) )
=> ( ( ? [Uw2: nat,Ux2: array @ vEBT_VEBTi,Uy2: vEBT_VEBTi] :
( X2
= ( vEBT_Nodei @ ( none @ ( product_prod @ nat @ nat ) ) @ Uw2 @ Ux2 @ Uy2 ) )
=> ( Y3
!= ( heap_Time_return @ $o @ $true ) ) )
=> ~ ( ? [Uz2: product_prod @ nat @ nat,Va2: nat,Vb2: array @ vEBT_VEBTi,Vc2: vEBT_VEBTi] :
( X2
= ( vEBT_Nodei @ ( some @ ( product_prod @ nat @ nat ) @ Uz2 ) @ Va2 @ Vb2 @ Vc2 ) )
=> ( Y3
!= ( heap_Time_return @ $o @ $false ) ) ) ) ) ) ) ) ).
% VEBT_internal.minNulli.elims
thf(fact_2882_vebt__minti_Osimps_I3_J,axiom,
! [Mi: nat,Ma: nat,Ux: nat,Uy: array @ vEBT_VEBTi,Uz: vEBT_VEBTi] :
( ( vEBT_vebt_minti @ ( vEBT_Nodei @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi @ Ma ) ) @ Ux @ Uy @ Uz ) )
= ( heap_Time_return @ ( option @ nat ) @ ( some @ nat @ Mi ) ) ) ).
% vebt_minti.simps(3)
thf(fact_2883_vebt__maxti_Osimps_I3_J,axiom,
! [Mi: nat,Ma: nat,Ux: nat,Uy: array @ vEBT_VEBTi,Uz: vEBT_VEBTi] :
( ( vEBT_vebt_maxti @ ( vEBT_Nodei @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi @ Ma ) ) @ Ux @ Uy @ Uz ) )
= ( heap_Time_return @ ( option @ nat ) @ ( some @ nat @ Ma ) ) ) ).
% vebt_maxti.simps(3)
thf(fact_2884_divmod__digit__0_I2_J,axiom,
! [A: $tType] :
( ( unique1627219031080169319umeral @ A )
=> ! [B3: A,A3: A] :
( ( ord_less @ A @ ( zero_zero @ A ) @ B3 )
=> ( ( ord_less @ A @ ( modulo_modulo @ A @ A3 @ ( times_times @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ B3 ) ) @ B3 )
=> ( ( modulo_modulo @ A @ A3 @ ( times_times @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ B3 ) )
= ( modulo_modulo @ A @ A3 @ B3 ) ) ) ) ) ).
% divmod_digit_0(2)
thf(fact_2885_bits__stable__imp__add__self,axiom,
! [A: $tType] :
( ( bit_semiring_bits @ A )
=> ! [A3: A] :
( ( ( divide_divide @ A @ A3 @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) )
= A3 )
=> ( ( plus_plus @ A @ A3 @ ( modulo_modulo @ A @ A3 @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) )
= ( zero_zero @ A ) ) ) ) ).
% bits_stable_imp_add_self
thf(fact_2886_parity__cases,axiom,
! [A: $tType] :
( ( semiring_parity @ A )
=> ! [A3: A] :
( ( ( dvd_dvd @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ A3 )
=> ( ( modulo_modulo @ A @ A3 @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) )
!= ( zero_zero @ A ) ) )
=> ~ ( ~ ( dvd_dvd @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ A3 )
=> ( ( modulo_modulo @ A @ A3 @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) )
!= ( one_one @ A ) ) ) ) ) ).
% parity_cases
thf(fact_2887_mod2__eq__if,axiom,
! [A: $tType] :
( ( semiring_parity @ A )
=> ! [A3: A] :
( ( ( dvd_dvd @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ A3 )
=> ( ( modulo_modulo @ A @ A3 @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) )
= ( zero_zero @ A ) ) )
& ( ~ ( dvd_dvd @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ A3 )
=> ( ( modulo_modulo @ A @ A3 @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) )
= ( one_one @ A ) ) ) ) ) ).
% mod2_eq_if
thf(fact_2888_div__exp__mod__exp__eq,axiom,
! [A: $tType] :
( ( bit_semiring_bits @ A )
=> ! [A3: A,N: nat,M: nat] :
( ( modulo_modulo @ A @ ( divide_divide @ A @ A3 @ ( power_power @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ N ) ) @ ( power_power @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ M ) )
= ( divide_divide @ A @ ( modulo_modulo @ A @ A3 @ ( power_power @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ ( plus_plus @ nat @ N @ M ) ) ) @ ( power_power @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ N ) ) ) ) ).
% div_exp_mod_exp_eq
thf(fact_2889_power__mod__div,axiom,
! [X2: nat,N: nat,M: nat] :
( ( divide_divide @ nat @ ( modulo_modulo @ nat @ X2 @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ M ) )
= ( modulo_modulo @ nat @ ( divide_divide @ nat @ X2 @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ M ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( minus_minus @ nat @ N @ M ) ) ) ) ).
% power_mod_div
thf(fact_2890_verit__le__mono__div,axiom,
! [A2: nat,B2: nat,N: nat] :
( ( ord_less @ nat @ A2 @ B2 )
=> ( ( ord_less @ nat @ ( zero_zero @ nat ) @ N )
=> ( ord_less_eq @ nat
@ ( plus_plus @ nat @ ( divide_divide @ nat @ A2 @ N )
@ ( if @ nat
@ ( ( modulo_modulo @ nat @ B2 @ N )
= ( zero_zero @ nat ) )
@ ( one_one @ nat )
@ ( zero_zero @ nat ) ) )
@ ( divide_divide @ nat @ B2 @ N ) ) ) ) ).
% verit_le_mono_div
thf(fact_2891_vebt__maxti_Osimps_I1_J,axiom,
! [B3: $o,A3: $o] :
( ( B3
=> ( ( vEBT_vebt_maxti @ ( vEBT_Leafi @ A3 @ B3 ) )
= ( heap_Time_return @ ( option @ nat ) @ ( some @ nat @ ( one_one @ nat ) ) ) ) )
& ( ~ B3
=> ( ( A3
=> ( ( vEBT_vebt_maxti @ ( vEBT_Leafi @ A3 @ B3 ) )
= ( heap_Time_return @ ( option @ nat ) @ ( some @ nat @ ( zero_zero @ nat ) ) ) ) )
& ( ~ A3
=> ( ( vEBT_vebt_maxti @ ( vEBT_Leafi @ A3 @ B3 ) )
= ( heap_Time_return @ ( option @ nat ) @ ( none @ nat ) ) ) ) ) ) ) ).
% vebt_maxti.simps(1)
thf(fact_2892_vebt__minti_Osimps_I1_J,axiom,
! [A3: $o,B3: $o] :
( ( A3
=> ( ( vEBT_vebt_minti @ ( vEBT_Leafi @ A3 @ B3 ) )
= ( heap_Time_return @ ( option @ nat ) @ ( some @ nat @ ( zero_zero @ nat ) ) ) ) )
& ( ~ A3
=> ( ( B3
=> ( ( vEBT_vebt_minti @ ( vEBT_Leafi @ A3 @ B3 ) )
= ( heap_Time_return @ ( option @ nat ) @ ( some @ nat @ ( one_one @ nat ) ) ) ) )
& ( ~ B3
=> ( ( vEBT_vebt_minti @ ( vEBT_Leafi @ A3 @ B3 ) )
= ( heap_Time_return @ ( option @ nat ) @ ( none @ nat ) ) ) ) ) ) ) ).
% vebt_minti.simps(1)
thf(fact_2893_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_2894_divmod__digit__0_I1_J,axiom,
! [A: $tType] :
( ( unique1627219031080169319umeral @ A )
=> ! [B3: A,A3: A] :
( ( ord_less @ A @ ( zero_zero @ A ) @ B3 )
=> ( ( ord_less @ A @ ( modulo_modulo @ A @ A3 @ ( times_times @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ B3 ) ) @ B3 )
=> ( ( times_times @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ ( divide_divide @ A @ A3 @ ( times_times @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ B3 ) ) )
= ( divide_divide @ A @ A3 @ B3 ) ) ) ) ) ).
% divmod_digit_0(1)
thf(fact_2895_mult__exp__mod__exp__eq,axiom,
! [A: $tType] :
( ( bit_semiring_bits @ A )
=> ! [M: nat,N: nat,A3: A] :
( ( ord_less_eq @ nat @ M @ N )
=> ( ( modulo_modulo @ A @ ( times_times @ A @ A3 @ ( power_power @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ M ) ) @ ( power_power @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ N ) )
= ( times_times @ A @ ( modulo_modulo @ A @ A3 @ ( power_power @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ ( minus_minus @ nat @ N @ M ) ) ) @ ( power_power @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ M ) ) ) ) ) ).
% mult_exp_mod_exp_eq
thf(fact_2896_list__decode_Ocases,axiom,
! [X2: nat] :
( ( X2
!= ( zero_zero @ nat ) )
=> ~ ! [N3: nat] :
( X2
!= ( suc @ N3 ) ) ) ).
% list_decode.cases
thf(fact_2897_dvd__productE,axiom,
! [A: $tType] :
( ( semiring_gcd @ A )
=> ! [P6: A,A3: A,B3: A] :
( ( dvd_dvd @ A @ P6 @ ( times_times @ A @ A3 @ B3 ) )
=> ~ ! [X4: A,Y4: A] :
( ( P6
= ( times_times @ A @ X4 @ Y4 ) )
=> ( ( dvd_dvd @ A @ X4 @ A3 )
=> ~ ( dvd_dvd @ A @ Y4 @ B3 ) ) ) ) ) ).
% dvd_productE
thf(fact_2898_division__decomp,axiom,
! [A: $tType] :
( ( semiring_gcd @ A )
=> ! [A3: A,B3: A,C3: A] :
( ( dvd_dvd @ A @ A3 @ ( times_times @ A @ B3 @ C3 ) )
=> ? [B8: A,C6: A] :
( ( A3
= ( times_times @ A @ B8 @ C6 ) )
& ( dvd_dvd @ A @ B8 @ B3 )
& ( dvd_dvd @ A @ C6 @ C3 ) ) ) ) ).
% division_decomp
thf(fact_2899_Euclid__induct,axiom,
! [P: nat > nat > $o,A3: nat,B3: nat] :
( ! [A4: nat,B4: nat] :
( ( P @ A4 @ B4 )
= ( P @ B4 @ A4 ) )
=> ( ! [A4: nat] : ( P @ A4 @ ( zero_zero @ nat ) )
=> ( ! [A4: nat,B4: nat] :
( ( P @ A4 @ B4 )
=> ( P @ A4 @ ( plus_plus @ nat @ A4 @ B4 ) ) )
=> ( P @ A3 @ B3 ) ) ) ) ).
% Euclid_induct
thf(fact_2900_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_2901_mod__double__modulus,axiom,
! [A: $tType] :
( ( unique1627219031080169319umeral @ A )
=> ! [M: A,X2: A] :
( ( ord_less @ A @ ( zero_zero @ A ) @ M )
=> ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ X2 )
=> ( ( ( modulo_modulo @ A @ X2 @ ( times_times @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ M ) )
= ( modulo_modulo @ A @ X2 @ M ) )
| ( ( modulo_modulo @ A @ X2 @ ( times_times @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ M ) )
= ( plus_plus @ A @ ( modulo_modulo @ A @ X2 @ M ) @ M ) ) ) ) ) ) ).
% mod_double_modulus
thf(fact_2902_divmod__digit__1_I2_J,axiom,
! [A: $tType] :
( ( unique1627219031080169319umeral @ A )
=> ! [A3: A,B3: A] :
( ( ord_less_eq @ A @ ( zero_zero @ A ) @ A3 )
=> ( ( ord_less @ A @ ( zero_zero @ A ) @ B3 )
=> ( ( ord_less_eq @ A @ B3 @ ( modulo_modulo @ A @ A3 @ ( times_times @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ B3 ) ) )
=> ( ( minus_minus @ A @ ( modulo_modulo @ A @ A3 @ ( times_times @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ B3 ) ) @ B3 )
= ( modulo_modulo @ A @ A3 @ B3 ) ) ) ) ) ) ).
% divmod_digit_1(2)
thf(fact_2903_unset__bit__Suc,axiom,
! [A: $tType] :
( ( bit_se359711467146920520ations @ A )
=> ! [N: nat,A3: A] :
( ( bit_se2638667681897837118et_bit @ A @ ( suc @ N ) @ A3 )
= ( plus_plus @ A @ ( modulo_modulo @ A @ A3 @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) @ ( times_times @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ ( bit_se2638667681897837118et_bit @ A @ N @ ( divide_divide @ A @ A3 @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) ) ) ) ) ) ).
% unset_bit_Suc
thf(fact_2904_set__bit__Suc,axiom,
! [A: $tType] :
( ( bit_se359711467146920520ations @ A )
=> ! [N: nat,A3: A] :
( ( bit_se5668285175392031749et_bit @ A @ ( suc @ N ) @ A3 )
= ( plus_plus @ A @ ( modulo_modulo @ A @ A3 @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) @ ( times_times @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ ( bit_se5668285175392031749et_bit @ A @ N @ ( divide_divide @ A @ A3 @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) ) ) ) ) ) ).
% set_bit_Suc
thf(fact_2905_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_2906_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_2907_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_2908_divmod__digit__1_I1_J,axiom,
! [A: $tType] :
( ( unique1627219031080169319umeral @ A )
=> ! [A3: A,B3: A] :
( ( ord_less_eq @ A @ ( zero_zero @ A ) @ A3 )
=> ( ( ord_less @ A @ ( zero_zero @ A ) @ B3 )
=> ( ( ord_less_eq @ A @ B3 @ ( modulo_modulo @ A @ A3 @ ( times_times @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ B3 ) ) )
=> ( ( plus_plus @ A @ ( times_times @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ ( divide_divide @ A @ A3 @ ( times_times @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ B3 ) ) ) @ ( one_one @ A ) )
= ( divide_divide @ A @ A3 @ B3 ) ) ) ) ) ) ).
% divmod_digit_1(1)
thf(fact_2909_vebt__minti_Oelims,axiom,
! [X2: vEBT_VEBTi,Y3: heap_Time_Heap @ ( option @ nat )] :
( ( ( vEBT_vebt_minti @ X2 )
= Y3 )
=> ( ! [A4: $o,B4: $o] :
( ( X2
= ( vEBT_Leafi @ A4 @ B4 ) )
=> ~ ( ( A4
=> ( Y3
= ( heap_Time_return @ ( option @ nat ) @ ( some @ nat @ ( zero_zero @ nat ) ) ) ) )
& ( ~ A4
=> ( ( B4
=> ( Y3
= ( heap_Time_return @ ( option @ nat ) @ ( some @ nat @ ( one_one @ nat ) ) ) ) )
& ( ~ B4
=> ( Y3
= ( heap_Time_return @ ( option @ nat ) @ ( none @ nat ) ) ) ) ) ) ) )
=> ( ( ? [Uu2: nat,Uv2: array @ vEBT_VEBTi,Uw2: vEBT_VEBTi] :
( X2
= ( vEBT_Nodei @ ( none @ ( product_prod @ nat @ nat ) ) @ Uu2 @ Uv2 @ Uw2 ) )
=> ( Y3
!= ( heap_Time_return @ ( option @ nat ) @ ( none @ nat ) ) ) )
=> ~ ! [Mi2: nat] :
( ? [Ma2: nat,Ux2: nat,Uy2: array @ vEBT_VEBTi,Uz2: vEBT_VEBTi] :
( X2
= ( vEBT_Nodei @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi2 @ Ma2 ) ) @ Ux2 @ Uy2 @ Uz2 ) )
=> ( Y3
!= ( heap_Time_return @ ( option @ nat ) @ ( some @ nat @ Mi2 ) ) ) ) ) ) ) ).
% vebt_minti.elims
thf(fact_2910_vebt__maxti_Oelims,axiom,
! [X2: vEBT_VEBTi,Y3: heap_Time_Heap @ ( option @ nat )] :
( ( ( vEBT_vebt_maxti @ X2 )
= Y3 )
=> ( ! [A4: $o,B4: $o] :
( ( X2
= ( vEBT_Leafi @ A4 @ B4 ) )
=> ~ ( ( B4
=> ( Y3
= ( heap_Time_return @ ( option @ nat ) @ ( some @ nat @ ( one_one @ nat ) ) ) ) )
& ( ~ B4
=> ( ( A4
=> ( Y3
= ( heap_Time_return @ ( option @ nat ) @ ( some @ nat @ ( zero_zero @ nat ) ) ) ) )
& ( ~ A4
=> ( Y3
= ( heap_Time_return @ ( option @ nat ) @ ( none @ nat ) ) ) ) ) ) ) )
=> ( ( ? [Uu2: nat,Uv2: array @ vEBT_VEBTi,Uw2: vEBT_VEBTi] :
( X2
= ( vEBT_Nodei @ ( none @ ( product_prod @ nat @ nat ) ) @ Uu2 @ Uv2 @ Uw2 ) )
=> ( Y3
!= ( heap_Time_return @ ( option @ nat ) @ ( none @ nat ) ) ) )
=> ~ ! [Mi2: nat,Ma2: nat] :
( ? [Ux2: nat,Uy2: array @ vEBT_VEBTi,Uz2: vEBT_VEBTi] :
( X2
= ( vEBT_Nodei @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi2 @ Ma2 ) ) @ Ux2 @ Uy2 @ Uz2 ) )
=> ( Y3
!= ( heap_Time_return @ ( option @ nat ) @ ( some @ nat @ Ma2 ) ) ) ) ) ) ) ).
% vebt_maxti.elims
thf(fact_2911_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_2912_bezout__add__nat,axiom,
! [A3: nat,B3: nat] :
? [D4: nat,X4: nat,Y4: nat] :
( ( dvd_dvd @ nat @ D4 @ A3 )
& ( dvd_dvd @ nat @ D4 @ B3 )
& ( ( ( times_times @ nat @ A3 @ X4 )
= ( plus_plus @ nat @ ( times_times @ nat @ B3 @ Y4 ) @ D4 ) )
| ( ( times_times @ nat @ B3 @ X4 )
= ( plus_plus @ nat @ ( times_times @ nat @ A3 @ Y4 ) @ D4 ) ) ) ) ).
% bezout_add_nat
thf(fact_2913_bezout__lemma__nat,axiom,
! [D2: nat,A3: nat,B3: nat,X2: nat,Y3: nat] :
( ( dvd_dvd @ nat @ D2 @ A3 )
=> ( ( dvd_dvd @ nat @ D2 @ B3 )
=> ( ( ( ( times_times @ nat @ A3 @ X2 )
= ( plus_plus @ nat @ ( times_times @ nat @ B3 @ Y3 ) @ D2 ) )
| ( ( times_times @ nat @ B3 @ X2 )
= ( plus_plus @ nat @ ( times_times @ nat @ A3 @ Y3 ) @ D2 ) ) )
=> ? [X4: nat,Y4: nat] :
( ( dvd_dvd @ nat @ D2 @ A3 )
& ( dvd_dvd @ nat @ D2 @ ( plus_plus @ nat @ A3 @ B3 ) )
& ( ( ( times_times @ nat @ A3 @ X4 )
= ( plus_plus @ nat @ ( times_times @ nat @ ( plus_plus @ nat @ A3 @ B3 ) @ Y4 ) @ D2 ) )
| ( ( times_times @ nat @ ( plus_plus @ nat @ A3 @ B3 ) @ X4 )
= ( plus_plus @ nat @ ( times_times @ nat @ A3 @ Y4 ) @ D2 ) ) ) ) ) ) ) ).
% bezout_lemma_nat
thf(fact_2914_bezout1__nat,axiom,
! [A3: nat,B3: nat] :
? [D4: nat,X4: nat,Y4: nat] :
( ( dvd_dvd @ nat @ D4 @ A3 )
& ( dvd_dvd @ nat @ D4 @ B3 )
& ( ( ( minus_minus @ nat @ ( times_times @ nat @ A3 @ X4 ) @ ( times_times @ nat @ B3 @ Y4 ) )
= D4 )
| ( ( minus_minus @ nat @ ( times_times @ nat @ B3 @ X4 ) @ ( times_times @ nat @ A3 @ Y4 ) )
= D4 ) ) ) ).
% bezout1_nat
thf(fact_2915_pos__eucl__rel__int__mult__2,axiom,
! [B3: int,A3: int,Q3: int,R2: int] :
( ( ord_less_eq @ int @ ( zero_zero @ int ) @ B3 )
=> ( ( eucl_rel_int @ A3 @ B3 @ ( product_Pair @ int @ int @ Q3 @ R2 ) )
=> ( eucl_rel_int @ ( plus_plus @ int @ ( one_one @ int ) @ ( times_times @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ A3 ) ) @ ( times_times @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ B3 ) @ ( 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_2916_div__half__nat,axiom,
! [Y3: nat,X2: nat] :
( ( Y3
!= ( zero_zero @ nat ) )
=> ( ( product_Pair @ nat @ nat @ ( divide_divide @ nat @ X2 @ Y3 ) @ ( modulo_modulo @ nat @ X2 @ Y3 ) )
= ( if @ ( product_prod @ nat @ nat ) @ ( ord_less_eq @ nat @ Y3 @ ( minus_minus @ nat @ X2 @ ( times_times @ nat @ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ ( divide_divide @ nat @ X2 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ Y3 ) ) @ Y3 ) ) ) @ ( product_Pair @ nat @ nat @ ( plus_plus @ nat @ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ ( divide_divide @ nat @ X2 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ Y3 ) ) @ ( one_one @ nat ) ) @ ( minus_minus @ nat @ ( minus_minus @ nat @ X2 @ ( times_times @ nat @ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ ( divide_divide @ nat @ X2 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ Y3 ) ) @ Y3 ) ) @ Y3 ) ) @ ( product_Pair @ nat @ nat @ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ ( divide_divide @ nat @ X2 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ Y3 ) ) @ ( minus_minus @ nat @ X2 @ ( times_times @ nat @ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ ( divide_divide @ nat @ X2 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ Y3 ) ) @ Y3 ) ) ) ) ) ) ).
% div_half_nat
thf(fact_2917_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_2918_flip__bit__Suc,axiom,
! [A: $tType] :
( ( bit_se359711467146920520ations @ A )
=> ! [N: nat,A3: A] :
( ( bit_se8732182000553998342ip_bit @ A @ ( suc @ N ) @ A3 )
= ( plus_plus @ A @ ( modulo_modulo @ A @ A3 @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) @ ( times_times @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ ( bit_se8732182000553998342ip_bit @ A @ N @ ( divide_divide @ A @ A3 @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) ) ) ) ) ) ).
% flip_bit_Suc
thf(fact_2919_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_2920_vebt__assn__raw_Oelims,axiom,
! [X2: vEBT_VEBT,Xa: vEBT_VEBTi,Y3: assn] :
( ( ( vEBT_vebt_assn_raw @ X2 @ Xa )
= Y3 )
=> ( ! [A4: $o,B4: $o] :
( ( X2
= ( vEBT_Leaf @ A4 @ B4 ) )
=> ! [Ai2: $o,Bi2: $o] :
( ( Xa
= ( vEBT_Leafi @ Ai2 @ Bi2 ) )
=> ( Y3
!= ( pure_assn
@ ( ( Ai2 = A4 )
& ( Bi2 = B4 ) ) ) ) ) )
=> ( ! [Mmo: option @ ( product_prod @ nat @ nat ),Deg2: nat,Tree_list: list @ vEBT_VEBT,Summary2: vEBT_VEBT] :
( ( X2
= ( vEBT_Node @ Mmo @ Deg2 @ Tree_list @ Summary2 ) )
=> ! [Mmoi: option @ ( product_prod @ nat @ nat ),Degi: nat,Tree_array: array @ vEBT_VEBTi,Summaryi: vEBT_VEBTi] :
( ( Xa
= ( vEBT_Nodei @ Mmoi @ Degi @ Tree_array @ Summaryi ) )
=> ( Y3
!= ( times_times @ assn
@ ( times_times @ assn
@ ( pure_assn
@ ( ( Mmoi = Mmo )
& ( Degi = Deg2 ) ) )
@ ( vEBT_vebt_assn_raw @ Summary2 @ 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 ) ) ) ) ) ) )
=> ( ( ? [V3: option @ ( product_prod @ nat @ nat ),Va3: nat,Vb3: list @ vEBT_VEBT,Vc3: vEBT_VEBT] :
( X2
= ( vEBT_Node @ V3 @ Va3 @ Vb3 @ Vc3 ) )
=> ( ? [Vd: $o,Ve: $o] :
( Xa
= ( vEBT_Leafi @ Vd @ Ve ) )
=> ( Y3
!= ( bot_bot @ assn ) ) ) )
=> ~ ( ? [Vd: $o,Ve: $o] :
( X2
= ( vEBT_Leaf @ Vd @ Ve ) )
=> ( ? [V3: option @ ( product_prod @ nat @ nat ),Va3: nat,Vb3: array @ vEBT_VEBTi,Vc3: vEBT_VEBTi] :
( Xa
= ( vEBT_Nodei @ V3 @ Va3 @ Vb3 @ Vc3 ) )
=> ( Y3
!= ( bot_bot @ assn ) ) ) ) ) ) ) ) ).
% vebt_assn_raw.elims
thf(fact_2921_even__word__def,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ( ( even_word @ A )
= ( dvd_dvd @ ( word @ A ) @ ( numeral_numeral @ ( word @ A ) @ ( bit0 @ one2 ) ) ) ) ) ).
% even_word_def
thf(fact_2922_ex__assn__const,axiom,
! [A: $tType,C3: assn] :
( ( ex_assn @ A
@ ^ [X3: A] : C3 )
= C3 ) ).
% ex_assn_const
thf(fact_2923_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_2924_norm__assertion__simps_I17_J,axiom,
! [B: $tType,R: assn,Q: B > assn] :
( ( times_times @ assn @ R @ ( ex_assn @ B @ Q ) )
= ( ex_assn @ B
@ ^ [X3: B] : ( times_times @ assn @ R @ ( Q @ X3 ) ) ) ) ).
% norm_assertion_simps(17)
thf(fact_2925_norm__assertion__simps_I16_J,axiom,
! [A: $tType,Q: A > assn,R: assn] :
( ( times_times @ assn @ ( ex_assn @ A @ Q ) @ R )
= ( ex_assn @ A
@ ^ [X3: A] : ( times_times @ assn @ ( Q @ X3 ) @ R ) ) ) ).
% norm_assertion_simps(16)
thf(fact_2926_triv__exI,axiom,
! [A: $tType,Q: A > assn,X2: A] : ( entails @ ( Q @ X2 ) @ ( ex_assn @ A @ Q ) ) ).
% triv_exI
thf(fact_2927_mod__ex__dist,axiom,
! [A: $tType,P: A > assn,H2: product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat )] :
( ( rep_assn @ ( ex_assn @ A @ P ) @ H2 )
= ( ? [X3: A] : ( rep_assn @ ( P @ X3 ) @ H2 ) ) ) ).
% mod_ex_dist
thf(fact_2928_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_2929_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_2930_length__product,axiom,
! [A: $tType,B: $tType,Xs: list @ A,Ys: list @ B] :
( ( size_size @ ( list @ ( product_prod @ A @ B ) ) @ ( product @ A @ B @ Xs @ Ys ) )
= ( times_times @ nat @ ( size_size @ ( list @ A ) @ Xs ) @ ( size_size @ ( list @ B ) @ Ys ) ) ) ).
% length_product
thf(fact_2931_zmod__numeral__Bit0,axiom,
! [V: num,W: num] :
( ( modulo_modulo @ int @ ( numeral_numeral @ int @ ( bit0 @ V ) ) @ ( numeral_numeral @ int @ ( bit0 @ W ) ) )
= ( times_times @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ ( modulo_modulo @ int @ ( numeral_numeral @ int @ V ) @ ( numeral_numeral @ int @ W ) ) ) ) ).
% zmod_numeral_Bit0
thf(fact_2932_mod__h__bot__iff_I8_J,axiom,
! [C: $tType,R: C > assn,H2: heap_ext @ product_unit] :
( ( rep_assn @ ( ex_assn @ C @ R ) @ ( product_Pair @ ( heap_ext @ product_unit ) @ ( set @ nat ) @ H2 @ ( bot_bot @ ( set @ nat ) ) ) )
= ( ? [X3: C] : ( rep_assn @ ( R @ X3 ) @ ( product_Pair @ ( heap_ext @ product_unit ) @ ( set @ nat ) @ H2 @ ( bot_bot @ ( set @ nat ) ) ) ) ) ) ).
% mod_h_bot_iff(8)
thf(fact_2933_one__mod__exp__eq__one,axiom,
! [N: nat] :
( ( modulo_modulo @ int @ ( one_one @ int ) @ ( times_times @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ ( power_power @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ N ) ) )
= ( one_one @ int ) ) ).
% one_mod_exp_eq_one
thf(fact_2934_zmod__numeral__Bit1,axiom,
! [V: num,W: num] :
( ( modulo_modulo @ int @ ( numeral_numeral @ int @ ( bit1 @ V ) ) @ ( numeral_numeral @ int @ ( bit0 @ W ) ) )
= ( plus_plus @ int @ ( times_times @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ ( modulo_modulo @ int @ ( numeral_numeral @ int @ V ) @ ( numeral_numeral @ int @ W ) ) ) @ ( one_one @ int ) ) ) ).
% zmod_numeral_Bit1
thf(fact_2935_mod__plus__cong,axiom,
! [B3: int,B5: int,X2: int,X5: int,Y3: int,Y7: int,Z7: int] :
( ( B3 = B5 )
=> ( ( ( modulo_modulo @ int @ X2 @ B5 )
= ( modulo_modulo @ int @ X5 @ B5 ) )
=> ( ( ( modulo_modulo @ int @ Y3 @ B5 )
= ( modulo_modulo @ int @ Y7 @ B5 ) )
=> ( ( ( plus_plus @ int @ X5 @ Y7 )
= Z7 )
=> ( ( modulo_modulo @ int @ ( plus_plus @ int @ X2 @ Y3 ) @ B3 )
= ( modulo_modulo @ int @ Z7 @ B5 ) ) ) ) ) ) ).
% mod_plus_cong
thf(fact_2936_zmod__helper,axiom,
! [N: int,M: int,K: int,A3: int] :
( ( ( modulo_modulo @ int @ N @ M )
= K )
=> ( ( modulo_modulo @ int @ ( plus_plus @ int @ N @ A3 ) @ M )
= ( modulo_modulo @ int @ ( plus_plus @ int @ K @ A3 ) @ M ) ) ) ).
% zmod_helper
thf(fact_2937_Word_Omod__minus__cong,axiom,
! [B3: int,B5: int,X2: int,X5: int,Y3: int,Y7: int,Z7: int] :
( ( B3 = B5 )
=> ( ( ( modulo_modulo @ int @ X2 @ B5 )
= ( modulo_modulo @ int @ X5 @ B5 ) )
=> ( ( ( modulo_modulo @ int @ Y3 @ B5 )
= ( modulo_modulo @ int @ Y7 @ B5 ) )
=> ( ( ( minus_minus @ int @ X5 @ Y7 )
= Z7 )
=> ( ( modulo_modulo @ int @ ( minus_minus @ int @ X2 @ Y3 ) @ B3 )
= ( modulo_modulo @ int @ Z7 @ B5 ) ) ) ) ) ) ).
% Word.mod_minus_cong
thf(fact_2938_ex__distrib__star,axiom,
! [A: $tType,P: A > assn,Q: assn] :
( ( ex_assn @ A
@ ^ [X3: A] : ( times_times @ assn @ ( P @ X3 ) @ Q ) )
= ( times_times @ assn @ ( ex_assn @ A @ P ) @ Q ) ) ).
% ex_distrib_star
thf(fact_2939_enorm__exI_H,axiom,
! [A: $tType,Z8: A > $o,P: assn,Q: A > assn] :
( ! [X4: A] :
( ( Z8 @ X4 )
=> ( entails @ P @ ( Q @ X4 ) ) )
=> ( ? [X_12: A] : ( Z8 @ X_12 )
=> ( entails @ P @ ( ex_assn @ A @ Q ) ) ) ) ).
% enorm_exI'
thf(fact_2940_ent__ex__preI,axiom,
! [A: $tType,P: A > assn,Q: assn] :
( ! [X4: A] : ( entails @ ( P @ X4 ) @ Q )
=> ( entails @ ( ex_assn @ A @ P ) @ Q ) ) ).
% ent_ex_preI
thf(fact_2941_ent__ex__postI,axiom,
! [A: $tType,P: assn,Q: A > assn,X2: A] :
( ( entails @ P @ ( Q @ X2 ) )
=> ( entails @ P @ ( ex_assn @ A @ Q ) ) ) ).
% ent_ex_postI
thf(fact_2942_norm__pre__ex__rule,axiom,
! [A: $tType,B: $tType,P: A > assn,F2: heap_Time_Heap @ B,Q: B > assn] :
( ! [X4: A] : ( hoare_hoare_triple @ B @ ( P @ X4 ) @ F2 @ Q )
=> ( hoare_hoare_triple @ B @ ( ex_assn @ A @ P ) @ F2 @ Q ) ) ).
% norm_pre_ex_rule
thf(fact_2943_post__exI__rule,axiom,
! [B: $tType,A: $tType,P: assn,C3: heap_Time_Heap @ A,Q: A > B > assn,X2: B] :
( ( hoare_hoare_triple @ A @ P @ C3
@ ^ [R5: A] : ( Q @ R5 @ X2 ) )
=> ( hoare_hoare_triple @ A @ P @ C3
@ ^ [R5: A] : ( ex_assn @ B @ ( Q @ R5 ) ) ) ) ).
% post_exI_rule
thf(fact_2944_mod__exE,axiom,
! [A: $tType,P: A > assn,H2: product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat )] :
( ( rep_assn @ ( ex_assn @ A @ P ) @ H2 )
=> ~ ! [X4: A] :
~ ( rep_assn @ ( P @ X4 ) @ H2 ) ) ).
% mod_exE
thf(fact_2945_mod__exI,axiom,
! [A: $tType,P: A > assn,H2: product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat )] :
( ? [X8: A] : ( rep_assn @ ( P @ X8 ) @ H2 )
=> ( rep_assn @ ( ex_assn @ A @ P ) @ H2 ) ) ).
% mod_exI
thf(fact_2946_ex__one__point__gen,axiom,
! [A: $tType,P: A > assn,V: A] :
( ! [H3: product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ),X4: A] :
( ( rep_assn @ ( P @ X4 ) @ H3 )
=> ( X4 = V ) )
=> ( ( ex_assn @ A @ P )
= ( P @ V ) ) ) ).
% ex_one_point_gen
thf(fact_2947_norm__post__ex__rule__htt,axiom,
! [A: $tType,B: $tType,P: assn,F2: heap_Time_Heap @ A,Q: B > A > assn,X2: B,T2: nat] :
( ( time_htt @ A @ P @ F2 @ ( Q @ X2 ) @ T2 )
=> ( time_htt @ A @ P @ F2
@ ^ [R5: A] :
( ex_assn @ B
@ ^ [X3: B] : ( Q @ X3 @ R5 ) )
@ T2 ) ) ).
% norm_post_ex_rule_htt
thf(fact_2948_norm__pre__ex__rule__htt,axiom,
! [B: $tType,A: $tType,P: A > assn,F2: heap_Time_Heap @ B,Q: B > assn,T2: nat] :
( ! [X4: A] : ( time_htt @ B @ ( P @ X4 ) @ F2 @ Q @ T2 )
=> ( time_htt @ B @ ( ex_assn @ A @ P ) @ F2 @ Q @ T2 ) ) ).
% norm_pre_ex_rule_htt
thf(fact_2949_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_2950_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_2951_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_2952_int__mod__ge,axiom,
! [A3: int,N: int] :
( ( ord_less @ int @ A3 @ N )
=> ( ( ord_less @ int @ ( zero_zero @ int ) @ N )
=> ( ord_less_eq @ int @ A3 @ ( modulo_modulo @ int @ A3 @ N ) ) ) ) ).
% int_mod_ge
thf(fact_2953_neg__mod__conj,axiom,
! [B3: int,A3: int] :
( ( ord_less @ int @ B3 @ ( zero_zero @ int ) )
=> ( ( ord_less_eq @ int @ ( modulo_modulo @ int @ A3 @ B3 ) @ ( zero_zero @ int ) )
& ( ord_less @ int @ B3 @ ( modulo_modulo @ int @ A3 @ B3 ) ) ) ) ).
% neg_mod_conj
thf(fact_2954_pos__mod__conj,axiom,
! [B3: int,A3: int] :
( ( ord_less @ int @ ( zero_zero @ int ) @ B3 )
=> ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ ( modulo_modulo @ int @ A3 @ B3 ) )
& ( ord_less @ int @ ( modulo_modulo @ int @ A3 @ B3 ) @ B3 ) ) ) ).
% pos_mod_conj
thf(fact_2955_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_2956_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_2957_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_2958_int__mod__lem,axiom,
! [N: int,B3: int] :
( ( ord_less @ int @ ( zero_zero @ int ) @ N )
=> ( ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ B3 )
& ( ord_less @ int @ B3 @ N ) )
= ( ( modulo_modulo @ int @ B3 @ N )
= B3 ) ) ) ).
% int_mod_lem
thf(fact_2959_int__mod__eq,axiom,
! [B3: int,N: int,A3: int] :
( ( ord_less_eq @ int @ ( zero_zero @ int ) @ B3 )
=> ( ( ord_less @ int @ B3 @ N )
=> ( ( ( modulo_modulo @ int @ A3 @ N )
= ( modulo_modulo @ int @ B3 @ N ) )
=> ( ( modulo_modulo @ int @ A3 @ N )
= B3 ) ) ) ) ).
% int_mod_eq
thf(fact_2960_int__mod__le_H,axiom,
! [B3: int,N: int] :
( ( ord_less_eq @ int @ ( zero_zero @ int ) @ ( minus_minus @ int @ B3 @ N ) )
=> ( ord_less_eq @ int @ ( modulo_modulo @ int @ B3 @ N ) @ ( minus_minus @ int @ B3 @ N ) ) ) ).
% int_mod_le'
thf(fact_2961_nonneg__mod__div,axiom,
! [A3: int,B3: int] :
( ( ord_less_eq @ int @ ( zero_zero @ int ) @ A3 )
=> ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ B3 )
=> ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ ( modulo_modulo @ int @ A3 @ B3 ) )
& ( ord_less_eq @ int @ ( zero_zero @ int ) @ ( divide_divide @ int @ A3 @ B3 ) ) ) ) ) ).
% nonneg_mod_div
thf(fact_2962_div__mod__decomp__int,axiom,
! [A2: int,N: int] :
( A2
= ( plus_plus @ int @ ( times_times @ int @ ( divide_divide @ int @ A2 @ N ) @ N ) @ ( modulo_modulo @ int @ A2 @ N ) ) ) ).
% div_mod_decomp_int
thf(fact_2963_mod__div__equality__div__eq,axiom,
! [A3: int,B3: int] :
( ( times_times @ int @ ( divide_divide @ int @ A3 @ B3 ) @ B3 )
= ( minus_minus @ int @ A3 @ ( modulo_modulo @ int @ A3 @ B3 ) ) ) ).
% mod_div_equality_div_eq
thf(fact_2964_word__mod__div__equality,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [N: word @ A,B3: word @ A] :
( ( plus_plus @ ( word @ A ) @ ( times_times @ ( word @ A ) @ ( divide_divide @ ( word @ A ) @ N @ B3 ) @ B3 ) @ ( modulo_modulo @ ( word @ A ) @ N @ B3 ) )
= N ) ) ).
% word_mod_div_equality
thf(fact_2965_nat__mod__as__int,axiom,
( ( modulo_modulo @ nat )
= ( ^ [A6: nat,B7: nat] : ( nat2 @ ( modulo_modulo @ int @ ( semiring_1_of_nat @ int @ A6 ) @ ( semiring_1_of_nat @ int @ B7 ) ) ) ) ) ).
% nat_mod_as_int
thf(fact_2966_pos__mod__bound2,axiom,
! [A3: int] : ( ord_less @ int @ ( modulo_modulo @ int @ A3 @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) ) @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) ) ).
% pos_mod_bound2
thf(fact_2967_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_2968_int__mod__ge_H,axiom,
! [B3: int,N: int] :
( ( ord_less @ int @ B3 @ ( zero_zero @ int ) )
=> ( ( ord_less @ int @ ( zero_zero @ int ) @ N )
=> ( ord_less_eq @ int @ ( plus_plus @ int @ B3 @ N ) @ ( modulo_modulo @ int @ B3 @ N ) ) ) ) ).
% int_mod_ge'
thf(fact_2969_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_2970_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_2971_nat__mod__distrib,axiom,
! [X2: int,Y3: int] :
( ( ord_less_eq @ int @ ( zero_zero @ int ) @ X2 )
=> ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ Y3 )
=> ( ( nat2 @ ( modulo_modulo @ int @ X2 @ Y3 ) )
= ( modulo_modulo @ nat @ ( nat2 @ X2 ) @ ( nat2 @ Y3 ) ) ) ) ) ).
% nat_mod_distrib
thf(fact_2972_real__of__int__div__aux,axiom,
! [X2: int,D2: int] :
( ( divide_divide @ real @ ( ring_1_of_int @ real @ X2 ) @ ( ring_1_of_int @ real @ D2 ) )
= ( plus_plus @ real @ ( ring_1_of_int @ real @ ( divide_divide @ int @ X2 @ D2 ) ) @ ( divide_divide @ real @ ( ring_1_of_int @ real @ ( modulo_modulo @ int @ X2 @ D2 ) ) @ ( ring_1_of_int @ real @ D2 ) ) ) ) ).
% real_of_int_div_aux
thf(fact_2973_pos__mod__sign2,axiom,
! [A3: int] : ( ord_less_eq @ int @ ( zero_zero @ int ) @ ( modulo_modulo @ int @ A3 @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) ) ) ).
% pos_mod_sign2
thf(fact_2974_nmod2,axiom,
! [N: int] :
( ( ( modulo_modulo @ int @ N @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) )
= ( zero_zero @ int ) )
| ( ( modulo_modulo @ int @ N @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) )
= ( one_one @ int ) ) ) ).
% nmod2
thf(fact_2975_mod__2__neq__1__eq__eq__0,axiom,
! [K: int] :
( ( ( modulo_modulo @ int @ K @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) )
!= ( one_one @ int ) )
= ( ( modulo_modulo @ int @ K @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) )
= ( zero_zero @ int ) ) ) ).
% mod_2_neq_1_eq_eq_0
thf(fact_2976_mod__exp__less__eq__exp,axiom,
! [A3: int,N: nat] : ( ord_less @ int @ ( modulo_modulo @ int @ A3 @ ( 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_2977_mod__power__lem,axiom,
! [A3: int,M: nat,N: nat] :
( ( ord_less @ int @ ( one_one @ int ) @ A3 )
=> ( ( ( ord_less_eq @ nat @ M @ N )
=> ( ( modulo_modulo @ int @ ( power_power @ int @ A3 @ N ) @ ( power_power @ int @ A3 @ M ) )
= ( zero_zero @ int ) ) )
& ( ~ ( ord_less_eq @ nat @ M @ N )
=> ( ( modulo_modulo @ int @ ( power_power @ int @ A3 @ N ) @ ( power_power @ int @ A3 @ M ) )
= ( power_power @ int @ A3 @ N ) ) ) ) ) ).
% mod_power_lem
thf(fact_2978_split__zmod,axiom,
! [P: int > $o,N: int,K: int] :
( ( P @ ( modulo_modulo @ int @ N @ K ) )
= ( ( ( K
= ( zero_zero @ int ) )
=> ( P @ N ) )
& ( ( ord_less @ int @ ( zero_zero @ int ) @ K )
=> ! [I4: int,J3: int] :
( ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ J3 )
& ( ord_less @ int @ J3 @ K )
& ( N
= ( plus_plus @ int @ ( times_times @ int @ K @ I4 ) @ J3 ) ) )
=> ( P @ J3 ) ) )
& ( ( ord_less @ int @ K @ ( zero_zero @ int ) )
=> ! [I4: int,J3: int] :
( ( ( ord_less @ int @ K @ J3 )
& ( ord_less_eq @ int @ J3 @ ( zero_zero @ int ) )
& ( N
= ( plus_plus @ int @ ( times_times @ int @ K @ I4 ) @ J3 ) ) )
=> ( P @ J3 ) ) ) ) ) ).
% split_zmod
thf(fact_2979_int__mod__neg__eq,axiom,
! [A3: int,B3: int,Q3: int,R2: int] :
( ( A3
= ( plus_plus @ int @ ( times_times @ int @ B3 @ Q3 ) @ R2 ) )
=> ( ( ord_less_eq @ int @ R2 @ ( zero_zero @ int ) )
=> ( ( ord_less @ int @ B3 @ R2 )
=> ( ( modulo_modulo @ int @ A3 @ B3 )
= R2 ) ) ) ) ).
% int_mod_neg_eq
thf(fact_2980_int__mod__pos__eq,axiom,
! [A3: int,B3: int,Q3: int,R2: int] :
( ( A3
= ( plus_plus @ int @ ( times_times @ int @ B3 @ Q3 ) @ R2 ) )
=> ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ R2 )
=> ( ( ord_less @ int @ R2 @ B3 )
=> ( ( modulo_modulo @ int @ A3 @ B3 )
= R2 ) ) ) ) ).
% int_mod_pos_eq
thf(fact_2981_mod__add__if__z,axiom,
! [X2: int,Z: int,Y3: int] :
( ( ord_less @ int @ X2 @ Z )
=> ( ( ord_less @ int @ Y3 @ Z )
=> ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ Y3 )
=> ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ X2 )
=> ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ Z )
=> ( ( ( ord_less @ int @ ( plus_plus @ int @ X2 @ Y3 ) @ Z )
=> ( ( modulo_modulo @ int @ ( plus_plus @ int @ X2 @ Y3 ) @ Z )
= ( plus_plus @ int @ X2 @ Y3 ) ) )
& ( ~ ( ord_less @ int @ ( plus_plus @ int @ X2 @ Y3 ) @ Z )
=> ( ( modulo_modulo @ int @ ( plus_plus @ int @ X2 @ Y3 ) @ Z )
= ( minus_minus @ int @ ( plus_plus @ int @ X2 @ Y3 ) @ Z ) ) ) ) ) ) ) ) ) ).
% mod_add_if_z
thf(fact_2982_mod__sub__if__z,axiom,
! [X2: int,Z: int,Y3: int] :
( ( ord_less @ int @ X2 @ Z )
=> ( ( ord_less @ int @ Y3 @ Z )
=> ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ Y3 )
=> ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ X2 )
=> ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ Z )
=> ( ( ( ord_less_eq @ int @ Y3 @ X2 )
=> ( ( modulo_modulo @ int @ ( minus_minus @ int @ X2 @ Y3 ) @ Z )
= ( minus_minus @ int @ X2 @ Y3 ) ) )
& ( ~ ( ord_less_eq @ int @ Y3 @ X2 )
=> ( ( modulo_modulo @ int @ ( minus_minus @ int @ X2 @ Y3 ) @ Z )
= ( plus_plus @ int @ ( minus_minus @ int @ X2 @ Y3 ) @ Z ) ) ) ) ) ) ) ) ) ).
% mod_sub_if_z
thf(fact_2983_zmod__zmult2__eq,axiom,
! [C3: int,A3: int,B3: int] :
( ( ord_less_eq @ int @ ( zero_zero @ int ) @ C3 )
=> ( ( modulo_modulo @ int @ A3 @ ( times_times @ int @ B3 @ C3 ) )
= ( plus_plus @ int @ ( times_times @ int @ B3 @ ( modulo_modulo @ int @ ( divide_divide @ int @ A3 @ B3 ) @ C3 ) ) @ ( modulo_modulo @ int @ A3 @ B3 ) ) ) ) ).
% zmod_zmult2_eq
thf(fact_2984_axxmod2,axiom,
! [X2: int] :
( ( ( modulo_modulo @ int @ ( plus_plus @ int @ ( plus_plus @ int @ ( one_one @ int ) @ X2 ) @ X2 ) @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) )
= ( one_one @ int ) )
& ( ( modulo_modulo @ int @ ( plus_plus @ int @ ( plus_plus @ int @ ( zero_zero @ int ) @ X2 ) @ X2 ) @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) )
= ( zero_zero @ int ) ) ) ).
% axxmod2
thf(fact_2985_z1pmod2,axiom,
! [B3: int] :
( ( modulo_modulo @ int @ ( plus_plus @ int @ ( times_times @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ B3 ) @ ( one_one @ int ) ) @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) )
= ( one_one @ int ) ) ).
% z1pmod2
thf(fact_2986_verit__le__mono__div__int,axiom,
! [A2: int,B2: int,N: int] :
( ( ord_less @ int @ A2 @ B2 )
=> ( ( ord_less @ int @ ( zero_zero @ int ) @ N )
=> ( ord_less_eq @ int
@ ( plus_plus @ int @ ( divide_divide @ int @ A2 @ N )
@ ( if @ int
@ ( ( modulo_modulo @ int @ B2 @ N )
= ( zero_zero @ int ) )
@ ( one_one @ int )
@ ( zero_zero @ int ) ) )
@ ( divide_divide @ int @ B2 @ N ) ) ) ) ).
% verit_le_mono_div_int
thf(fact_2987_split__neg__lemma,axiom,
! [K: int,P: int > int > $o,N: int] :
( ( ord_less @ int @ K @ ( zero_zero @ int ) )
=> ( ( P @ ( divide_divide @ int @ N @ K ) @ ( modulo_modulo @ int @ N @ K ) )
= ( ! [I4: int,J3: int] :
( ( ( ord_less @ int @ K @ J3 )
& ( ord_less_eq @ int @ J3 @ ( zero_zero @ int ) )
& ( N
= ( plus_plus @ int @ ( times_times @ int @ K @ I4 ) @ J3 ) ) )
=> ( P @ I4 @ J3 ) ) ) ) ) ).
% split_neg_lemma
thf(fact_2988_split__pos__lemma,axiom,
! [K: int,P: int > int > $o,N: int] :
( ( ord_less @ int @ ( zero_zero @ int ) @ K )
=> ( ( P @ ( divide_divide @ int @ N @ K ) @ ( modulo_modulo @ int @ N @ K ) )
= ( ! [I4: int,J3: int] :
( ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ J3 )
& ( ord_less @ int @ J3 @ K )
& ( N
= ( plus_plus @ int @ ( times_times @ int @ K @ I4 ) @ J3 ) ) )
=> ( P @ I4 @ J3 ) ) ) ) ) ).
% split_pos_lemma
thf(fact_2989_p1mod22k,axiom,
! [B3: int,N: nat] :
( ( modulo_modulo @ int @ ( plus_plus @ int @ ( times_times @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ B3 ) @ ( one_one @ int ) ) @ ( times_times @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ ( power_power @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ N ) ) )
= ( plus_plus @ int @ ( times_times @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ ( modulo_modulo @ int @ B3 @ ( power_power @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ N ) ) ) @ ( one_one @ int ) ) ) ).
% p1mod22k
thf(fact_2990_p1mod22k_H,axiom,
! [B3: int,N: nat] :
( ( modulo_modulo @ int @ ( plus_plus @ int @ ( one_one @ int ) @ ( times_times @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ B3 ) ) @ ( times_times @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ ( power_power @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ N ) ) )
= ( plus_plus @ int @ ( one_one @ int ) @ ( times_times @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ ( modulo_modulo @ int @ B3 @ ( power_power @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ N ) ) ) ) ) ).
% p1mod22k'
thf(fact_2991_eq__diff__eq_H,axiom,
! [X2: real,Y3: real,Z: real] :
( ( X2
= ( minus_minus @ real @ Y3 @ Z ) )
= ( Y3
= ( plus_plus @ real @ X2 @ Z ) ) ) ).
% eq_diff_eq'
thf(fact_2992_pos__zmod__mult__2,axiom,
! [A3: int,B3: int] :
( ( ord_less_eq @ int @ ( zero_zero @ int ) @ A3 )
=> ( ( modulo_modulo @ int @ ( plus_plus @ int @ ( one_one @ int ) @ ( times_times @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ B3 ) ) @ ( times_times @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ A3 ) )
= ( plus_plus @ int @ ( one_one @ int ) @ ( times_times @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ ( modulo_modulo @ int @ B3 @ A3 ) ) ) ) ) ).
% pos_zmod_mult_2
thf(fact_2993_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_2994_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_2995_even__flip__bit__iff,axiom,
! [A: $tType] :
( ( bit_se359711467146920520ations @ A )
=> ! [M: nat,A3: A] :
( ( dvd_dvd @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ ( bit_se8732182000553998342ip_bit @ A @ M @ A3 ) )
= ( ( dvd_dvd @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ A3 )
!= ( M
= ( zero_zero @ nat ) ) ) ) ) ).
% even_flip_bit_iff
thf(fact_2996_sb__inc__lem,axiom,
! [A3: int,K: nat] :
( ( ord_less @ int @ ( plus_plus @ int @ A3 @ ( power_power @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ K ) ) @ ( zero_zero @ int ) )
=> ( ord_less_eq @ int @ ( plus_plus @ int @ ( plus_plus @ int @ A3 @ ( 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 @ A3 @ ( power_power @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ K ) ) @ ( power_power @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ ( suc @ K ) ) ) ) ) ).
% sb_inc_lem
thf(fact_2997_vebt__assn__raw_Osimps_I2_J,axiom,
! [Mmo2: option @ ( product_prod @ nat @ nat ),Deg: nat,Tree_list2: list @ vEBT_VEBT,Summary: vEBT_VEBT,Mmoi2: option @ ( product_prod @ nat @ nat ),Degi2: nat,Tree_array2: array @ vEBT_VEBTi,Summaryi2: vEBT_VEBTi] :
( ( vEBT_vebt_assn_raw @ ( vEBT_Node @ Mmo2 @ Deg @ Tree_list2 @ Summary ) @ ( vEBT_Nodei @ Mmoi2 @ Degi2 @ Tree_array2 @ Summaryi2 ) )
= ( times_times @ assn
@ ( times_times @ assn
@ ( pure_assn
@ ( ( Mmoi2 = Mmo2 )
& ( Degi2 = Deg ) ) )
@ ( vEBT_vebt_assn_raw @ Summary @ 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 ) ) ) ) ) ).
% vebt_assn_raw.simps(2)
thf(fact_2998_neg__zmod__mult__2,axiom,
! [A3: int,B3: int] :
( ( ord_less_eq @ int @ A3 @ ( zero_zero @ int ) )
=> ( ( modulo_modulo @ int @ ( plus_plus @ int @ ( one_one @ int ) @ ( times_times @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ B3 ) ) @ ( times_times @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ A3 ) )
= ( minus_minus @ int @ ( times_times @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ ( modulo_modulo @ int @ ( plus_plus @ int @ B3 @ ( one_one @ int ) ) @ A3 ) ) @ ( one_one @ int ) ) ) ) ).
% neg_zmod_mult_2
thf(fact_2999_triangle__def,axiom,
( nat_triangle
= ( ^ [N4: nat] : ( divide_divide @ nat @ ( times_times @ nat @ N4 @ ( suc @ N4 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ).
% triangle_def
thf(fact_3000_obtain__set__pred,axiom,
! [Z: nat,X2: nat,A2: set @ nat] :
( ( ord_less @ nat @ Z @ X2 )
=> ( ( vEBT_VEBT_min_in_set @ A2 @ Z )
=> ( ( finite_finite2 @ nat @ A2 )
=> ? [X_1: nat] : ( vEBT_is_pred_in_set @ A2 @ X2 @ X_1 ) ) ) ) ).
% obtain_set_pred
thf(fact_3001_obtain__set__succ,axiom,
! [X2: nat,Z: nat,A2: set @ nat,B2: set @ nat] :
( ( ord_less @ nat @ X2 @ Z )
=> ( ( vEBT_VEBT_max_in_set @ A2 @ Z )
=> ( ( finite_finite2 @ nat @ B2 )
=> ( ( A2 = B2 )
=> ? [X_1: nat] : ( vEBT_is_succ_in_set @ A2 @ X2 @ X_1 ) ) ) ) ) ).
% obtain_set_succ
thf(fact_3002_vebt__buildup_Oelims,axiom,
! [X2: nat,Y3: vEBT_VEBT] :
( ( ( vEBT_vebt_buildup @ X2 )
= Y3 )
=> ( ( ( X2
= ( zero_zero @ nat ) )
=> ( Y3
!= ( vEBT_Leaf @ $false @ $false ) ) )
=> ( ( ( X2
= ( suc @ ( zero_zero @ nat ) ) )
=> ( Y3
!= ( vEBT_Leaf @ $false @ $false ) ) )
=> ~ ! [Va3: nat] :
( ( X2
= ( suc @ ( suc @ Va3 ) ) )
=> ~ ( ( ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( suc @ ( suc @ Va3 ) ) )
=> ( Y3
= ( vEBT_Node @ ( none @ ( product_prod @ nat @ nat ) ) @ ( suc @ ( suc @ Va3 ) ) @ ( replicate @ vEBT_VEBT @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( vEBT_vebt_buildup @ ( divide_divide @ nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( vEBT_vebt_buildup @ ( divide_divide @ nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) )
& ( ~ ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( suc @ ( suc @ Va3 ) ) )
=> ( Y3
= ( vEBT_Node @ ( none @ ( product_prod @ nat @ nat ) ) @ ( suc @ ( suc @ Va3 ) ) @ ( replicate @ vEBT_VEBT @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( suc @ ( divide_divide @ nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( vEBT_vebt_buildup @ ( divide_divide @ nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( vEBT_vebt_buildup @ ( suc @ ( divide_divide @ nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) ) ) ) ) ) ) ) ).
% vebt_buildup.elims
thf(fact_3003_ceiling__log__eq__powr__iff,axiom,
! [X2: real,B3: real,K: nat] :
( ( ord_less @ real @ ( zero_zero @ real ) @ X2 )
=> ( ( ord_less @ real @ ( one_one @ real ) @ B3 )
=> ( ( ( archimedean_ceiling @ real @ ( log @ B3 @ X2 ) )
= ( plus_plus @ int @ ( semiring_1_of_nat @ int @ K ) @ ( one_one @ int ) ) )
= ( ( ord_less @ real @ ( powr @ real @ B3 @ ( semiring_1_of_nat @ real @ K ) ) @ X2 )
& ( ord_less_eq @ real @ X2 @ ( powr @ real @ B3 @ ( semiring_1_of_nat @ real @ ( plus_plus @ nat @ K @ ( one_one @ nat ) ) ) ) ) ) ) ) ) ).
% ceiling_log_eq_powr_iff
thf(fact_3004_intind,axiom,
! [A: $tType,I: nat,N: nat,P: A > $o,X2: A] :
( ( ord_less @ nat @ I @ N )
=> ( ( P @ X2 )
=> ( P @ ( nth @ A @ ( replicate @ A @ N @ X2 ) @ I ) ) ) ) ).
% intind
thf(fact_3005_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_3006_succ__none__empty,axiom,
! [Xs: set @ nat,A3: nat] :
( ~ ? [X_1: nat] : ( vEBT_is_succ_in_set @ Xs @ A3 @ X_1 )
=> ( ( finite_finite2 @ nat @ Xs )
=> ~ ? [X8: nat] :
( ( member @ nat @ X8 @ Xs )
& ( ord_less @ nat @ A3 @ X8 ) ) ) ) ).
% succ_none_empty
thf(fact_3007_pred__none__empty,axiom,
! [Xs: set @ nat,A3: nat] :
( ~ ? [X_1: nat] : ( vEBT_is_pred_in_set @ Xs @ A3 @ X_1 )
=> ( ( finite_finite2 @ nat @ Xs )
=> ~ ? [X8: nat] :
( ( member @ nat @ X8 @ Xs )
& ( ord_less @ nat @ X8 @ A3 ) ) ) ) ).
% pred_none_empty
thf(fact_3008_repli__cons__repl,axiom,
! [B: $tType,A: $tType,Q: assn,X2: heap_Time_Heap @ A,A2: B > A > assn,Y3: B,N: nat] :
( ( hoare_hoare_triple @ A @ Q @ X2
@ ^ [R5: A] : ( times_times @ assn @ Q @ ( A2 @ Y3 @ R5 ) ) )
=> ( hoare_hoare_triple @ ( list @ A ) @ Q @ ( vEBT_VEBT_replicatei @ A @ N @ X2 )
@ ^ [R5: list @ A] : ( times_times @ assn @ Q @ ( vEBT_List_list_assn @ B @ A @ A2 @ ( replicate @ B @ N @ Y3 ) @ R5 ) ) ) ) ).
% repli_cons_repl
thf(fact_3009_repli__emp,axiom,
! [A: $tType,B: $tType,X2: heap_Time_Heap @ A,A2: B > A > assn,Y3: B,N: nat] :
( ( hoare_hoare_triple @ A @ ( one_one @ assn ) @ X2 @ ( A2 @ Y3 ) )
=> ( hoare_hoare_triple @ ( list @ A ) @ ( one_one @ assn ) @ ( vEBT_VEBT_replicatei @ A @ N @ X2 ) @ ( vEBT_List_list_assn @ B @ A @ A2 @ ( replicate @ B @ N @ Y3 ) ) ) ) ).
% repli_emp
thf(fact_3010_List_Ofinite__set,axiom,
! [A: $tType,Xs: list @ A] : ( finite_finite2 @ A @ ( set2 @ A @ Xs ) ) ).
% List.finite_set
thf(fact_3011_powr__one__eq__one,axiom,
! [A: $tType] :
( ( ln @ A )
=> ! [A3: A] :
( ( powr @ A @ ( one_one @ A ) @ A3 )
= ( one_one @ A ) ) ) ).
% powr_one_eq_one
thf(fact_3012_replicate__eq__replicate,axiom,
! [A: $tType,M: nat,X2: A,N: nat,Y3: A] :
( ( ( replicate @ A @ M @ X2 )
= ( replicate @ A @ N @ Y3 ) )
= ( ( M = N )
& ( ( M
!= ( zero_zero @ nat ) )
=> ( X2 = Y3 ) ) ) ) ).
% replicate_eq_replicate
thf(fact_3013_length__replicate,axiom,
! [A: $tType,N: nat,X2: A] :
( ( size_size @ ( list @ A ) @ ( replicate @ A @ N @ X2 ) )
= N ) ).
% length_replicate
thf(fact_3014_map__replicate,axiom,
! [A: $tType,B: $tType,F2: B > A,N: nat,X2: B] :
( ( map @ B @ A @ F2 @ ( replicate @ B @ N @ X2 ) )
= ( replicate @ A @ N @ ( F2 @ X2 ) ) ) ).
% map_replicate
thf(fact_3015_infinite__Icc__iff,axiom,
! [A: $tType] :
( ( dense_linorder @ A )
=> ! [A3: A,B3: A] :
( ( ~ ( finite_finite2 @ A @ ( set_or1337092689740270186AtMost @ A @ A3 @ B3 ) ) )
= ( ord_less @ A @ A3 @ B3 ) ) ) ).
% infinite_Icc_iff
thf(fact_3016_infinite__Ico__iff,axiom,
! [A: $tType] :
( ( dense_linorder @ A )
=> ! [A3: A,B3: A] :
( ( ~ ( finite_finite2 @ A @ ( set_or7035219750837199246ssThan @ A @ A3 @ B3 ) ) )
= ( ord_less @ A @ A3 @ B3 ) ) ) ).
% infinite_Ico_iff
thf(fact_3017_powr__zero__eq__one,axiom,
! [A: $tType] :
( ( ln @ A )
=> ! [X2: A] :
( ( ( X2
= ( zero_zero @ A ) )
=> ( ( powr @ A @ X2 @ ( zero_zero @ A ) )
= ( zero_zero @ A ) ) )
& ( ( X2
!= ( zero_zero @ A ) )
=> ( ( powr @ A @ X2 @ ( zero_zero @ A ) )
= ( one_one @ A ) ) ) ) ) ).
% powr_zero_eq_one
thf(fact_3018_Ball__set__replicate,axiom,
! [A: $tType,N: nat,A3: A,P: A > $o] :
( ( ! [X3: A] :
( ( member @ A @ X3 @ ( set2 @ A @ ( replicate @ A @ N @ A3 ) ) )
=> ( P @ X3 ) ) )
= ( ( P @ A3 )
| ( N
= ( zero_zero @ nat ) ) ) ) ).
% Ball_set_replicate
thf(fact_3019_Bex__set__replicate,axiom,
! [A: $tType,N: nat,A3: A,P: A > $o] :
( ( ? [X3: A] :
( ( member @ A @ X3 @ ( set2 @ A @ ( replicate @ A @ N @ A3 ) ) )
& ( P @ X3 ) ) )
= ( ( P @ A3 )
& ( N
!= ( zero_zero @ nat ) ) ) ) ).
% Bex_set_replicate
thf(fact_3020_in__set__replicate,axiom,
! [A: $tType,X2: A,N: nat,Y3: A] :
( ( member @ A @ X2 @ ( set2 @ A @ ( replicate @ A @ N @ Y3 ) ) )
= ( ( X2 = Y3 )
& ( N
!= ( zero_zero @ nat ) ) ) ) ).
% in_set_replicate
thf(fact_3021_nth__replicate,axiom,
! [A: $tType,I: nat,N: nat,X2: A] :
( ( ord_less @ nat @ I @ N )
=> ( ( nth @ A @ ( replicate @ A @ N @ X2 ) @ I )
= X2 ) ) ).
% nth_replicate
thf(fact_3022_powr__nonneg__iff,axiom,
! [A3: real,X2: real] :
( ( ord_less_eq @ real @ ( powr @ real @ A3 @ X2 ) @ ( zero_zero @ real ) )
= ( A3
= ( zero_zero @ real ) ) ) ).
% powr_nonneg_iff
thf(fact_3023_powr__less__cancel__iff,axiom,
! [X2: real,A3: real,B3: real] :
( ( ord_less @ real @ ( one_one @ real ) @ X2 )
=> ( ( ord_less @ real @ ( powr @ real @ X2 @ A3 ) @ ( powr @ real @ X2 @ B3 ) )
= ( ord_less @ real @ A3 @ B3 ) ) ) ).
% powr_less_cancel_iff
thf(fact_3024_triangle__Suc,axiom,
! [N: nat] :
( ( nat_triangle @ ( suc @ N ) )
= ( plus_plus @ nat @ ( nat_triangle @ N ) @ ( suc @ N ) ) ) ).
% triangle_Suc
thf(fact_3025_powr__eq__one__iff,axiom,
! [A3: real,X2: real] :
( ( ord_less @ real @ ( one_one @ real ) @ A3 )
=> ( ( ( powr @ real @ A3 @ X2 )
= ( one_one @ real ) )
= ( X2
= ( zero_zero @ real ) ) ) ) ).
% powr_eq_one_iff
thf(fact_3026_powr__one__gt__zero__iff,axiom,
! [X2: real] :
( ( ( powr @ real @ X2 @ ( one_one @ real ) )
= X2 )
= ( ord_less_eq @ real @ ( zero_zero @ real ) @ X2 ) ) ).
% powr_one_gt_zero_iff
thf(fact_3027_powr__one,axiom,
! [X2: real] :
( ( ord_less_eq @ real @ ( zero_zero @ real ) @ X2 )
=> ( ( powr @ real @ X2 @ ( one_one @ real ) )
= X2 ) ) ).
% powr_one
thf(fact_3028_powr__le__cancel__iff,axiom,
! [X2: real,A3: real,B3: real] :
( ( ord_less @ real @ ( one_one @ real ) @ X2 )
=> ( ( ord_less_eq @ real @ ( powr @ real @ X2 @ A3 ) @ ( powr @ real @ X2 @ B3 ) )
= ( ord_less_eq @ real @ A3 @ B3 ) ) ) ).
% powr_le_cancel_iff
thf(fact_3029_numeral__powr__numeral__real,axiom,
! [M: num,N: num] :
( ( powr @ real @ ( numeral_numeral @ real @ M ) @ ( numeral_numeral @ real @ N ) )
= ( power_power @ real @ ( numeral_numeral @ real @ M ) @ ( numeral_numeral @ nat @ N ) ) ) ).
% numeral_powr_numeral_real
thf(fact_3030_set__replicate,axiom,
! [A: $tType,N: nat,X2: A] :
( ( N
!= ( zero_zero @ nat ) )
=> ( ( set2 @ A @ ( replicate @ A @ N @ X2 ) )
= ( insert @ A @ X2 @ ( bot_bot @ ( set @ A ) ) ) ) ) ).
% set_replicate
thf(fact_3031_powr__log__cancel,axiom,
! [A3: real,X2: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ A3 )
=> ( ( A3
!= ( one_one @ real ) )
=> ( ( ord_less @ real @ ( zero_zero @ real ) @ X2 )
=> ( ( powr @ real @ A3 @ ( log @ A3 @ X2 ) )
= X2 ) ) ) ) ).
% powr_log_cancel
thf(fact_3032_log__powr__cancel,axiom,
! [A3: real,Y3: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ A3 )
=> ( ( A3
!= ( one_one @ real ) )
=> ( ( log @ A3 @ ( powr @ real @ A3 @ Y3 ) )
= Y3 ) ) ) ).
% log_powr_cancel
thf(fact_3033_powr__numeral,axiom,
! [X2: real,N: num] :
( ( ord_less_eq @ real @ ( zero_zero @ real ) @ X2 )
=> ( ( powr @ real @ X2 @ ( numeral_numeral @ real @ N ) )
= ( power_power @ real @ X2 @ ( numeral_numeral @ nat @ N ) ) ) ) ).
% powr_numeral
thf(fact_3034_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_3035_finite__if__eq__beyond__finite,axiom,
! [A: $tType,S: set @ A,S4: set @ A] :
( ( finite_finite2 @ A @ S )
=> ( finite_finite2 @ ( set @ A )
@ ( collect @ ( set @ A )
@ ^ [S5: set @ A] :
( ( minus_minus @ ( set @ A ) @ S5 @ S )
= ( minus_minus @ ( set @ A ) @ S4 @ S ) ) ) ) ) ).
% finite_if_eq_beyond_finite
thf(fact_3036_bounded__nat__set__is__finite,axiom,
! [N7: set @ nat,N: nat] :
( ! [X4: nat] :
( ( member @ nat @ X4 @ N7 )
=> ( ord_less @ nat @ X4 @ N ) )
=> ( finite_finite2 @ nat @ N7 ) ) ).
% bounded_nat_set_is_finite
thf(fact_3037_finite__nat__set__iff__bounded,axiom,
( ( finite_finite2 @ nat )
= ( ^ [N8: set @ nat] :
? [M6: nat] :
! [X3: nat] :
( ( member @ nat @ X3 @ N8 )
=> ( ord_less @ nat @ X3 @ M6 ) ) ) ) ).
% finite_nat_set_iff_bounded
thf(fact_3038_finite__nat__set__iff__bounded__le,axiom,
( ( finite_finite2 @ nat )
= ( ^ [N8: set @ nat] :
? [M6: nat] :
! [X3: nat] :
( ( member @ nat @ X3 @ N8 )
=> ( ord_less_eq @ nat @ X3 @ M6 ) ) ) ) ).
% finite_nat_set_iff_bounded_le
thf(fact_3039_finite__list,axiom,
! [A: $tType,A2: set @ A] :
( ( finite_finite2 @ A @ A2 )
=> ? [Xs3: list @ A] :
( ( set2 @ A @ Xs3 )
= A2 ) ) ).
% finite_list
thf(fact_3040_finite__M__bounded__by__nat,axiom,
! [P: nat > $o,I: nat] :
( finite_finite2 @ nat
@ ( collect @ nat
@ ^ [K3: nat] :
( ( P @ K3 )
& ( ord_less @ nat @ K3 @ I ) ) ) ) ).
% finite_M_bounded_by_nat
thf(fact_3041_finite__less__ub,axiom,
! [F2: nat > nat,U: nat] :
( ! [N3: nat] : ( ord_less_eq @ nat @ N3 @ ( F2 @ N3 ) )
=> ( finite_finite2 @ nat
@ ( collect @ nat
@ ^ [N4: nat] : ( ord_less_eq @ nat @ ( F2 @ N4 ) @ U ) ) ) ) ).
% finite_less_ub
thf(fact_3042_finite__lists__length__eq,axiom,
! [A: $tType,A2: set @ A,N: nat] :
( ( finite_finite2 @ A @ A2 )
=> ( finite_finite2 @ ( list @ A )
@ ( collect @ ( list @ A )
@ ^ [Xs2: list @ A] :
( ( ord_less_eq @ ( set @ A ) @ ( set2 @ A @ Xs2 ) @ A2 )
& ( ( size_size @ ( list @ A ) @ Xs2 )
= N ) ) ) ) ) ).
% finite_lists_length_eq
thf(fact_3043_powr__ge__pzero,axiom,
! [X2: real,Y3: real] : ( ord_less_eq @ real @ ( zero_zero @ real ) @ ( powr @ real @ X2 @ Y3 ) ) ).
% powr_ge_pzero
thf(fact_3044_powr__mono2,axiom,
! [A3: real,X2: real,Y3: real] :
( ( ord_less_eq @ real @ ( zero_zero @ real ) @ A3 )
=> ( ( ord_less_eq @ real @ ( zero_zero @ real ) @ X2 )
=> ( ( ord_less_eq @ real @ X2 @ Y3 )
=> ( ord_less_eq @ real @ ( powr @ real @ X2 @ A3 ) @ ( powr @ real @ Y3 @ A3 ) ) ) ) ) ).
% powr_mono2
thf(fact_3045_powr__less__mono,axiom,
! [A3: real,B3: real,X2: real] :
( ( ord_less @ real @ A3 @ B3 )
=> ( ( ord_less @ real @ ( one_one @ real ) @ X2 )
=> ( ord_less @ real @ ( powr @ real @ X2 @ A3 ) @ ( powr @ real @ X2 @ B3 ) ) ) ) ).
% powr_less_mono
thf(fact_3046_powr__less__cancel,axiom,
! [X2: real,A3: real,B3: real] :
( ( ord_less @ real @ ( powr @ real @ X2 @ A3 ) @ ( powr @ real @ X2 @ B3 ) )
=> ( ( ord_less @ real @ ( one_one @ real ) @ X2 )
=> ( ord_less @ real @ A3 @ B3 ) ) ) ).
% powr_less_cancel
thf(fact_3047_powr__mono,axiom,
! [A3: real,B3: real,X2: real] :
( ( ord_less_eq @ real @ A3 @ B3 )
=> ( ( ord_less_eq @ real @ ( one_one @ real ) @ X2 )
=> ( ord_less_eq @ real @ ( powr @ real @ X2 @ A3 ) @ ( powr @ real @ X2 @ B3 ) ) ) ) ).
% powr_mono
thf(fact_3048_replicate__eqI,axiom,
! [A: $tType,Xs: list @ A,N: nat,X2: A] :
( ( ( size_size @ ( list @ A ) @ Xs )
= N )
=> ( ! [Y4: A] :
( ( member @ A @ Y4 @ ( set2 @ A @ Xs ) )
=> ( Y4 = X2 ) )
=> ( Xs
= ( replicate @ A @ N @ X2 ) ) ) ) ).
% replicate_eqI
thf(fact_3049_replicate__length__same,axiom,
! [A: $tType,Xs: list @ A,X2: A] :
( ! [X4: A] :
( ( member @ A @ X4 @ ( set2 @ A @ Xs ) )
=> ( X4 = X2 ) )
=> ( ( replicate @ A @ ( size_size @ ( list @ A ) @ Xs ) @ X2 )
= Xs ) ) ).
% replicate_length_same
thf(fact_3050_infinite__Icc,axiom,
! [A: $tType] :
( ( dense_linorder @ A )
=> ! [A3: A,B3: A] :
( ( ord_less @ A @ A3 @ B3 )
=> ~ ( finite_finite2 @ A @ ( set_or1337092689740270186AtMost @ A @ A3 @ B3 ) ) ) ) ).
% infinite_Icc
thf(fact_3051_infinite__Ico,axiom,
! [A: $tType] :
( ( dense_linorder @ A )
=> ! [A3: A,B3: A] :
( ( ord_less @ A @ A3 @ B3 )
=> ~ ( finite_finite2 @ A @ ( set_or7035219750837199246ssThan @ A @ A3 @ B3 ) ) ) ) ).
% infinite_Ico
thf(fact_3052_finite__lists__length__le,axiom,
! [A: $tType,A2: set @ A,N: nat] :
( ( finite_finite2 @ A @ A2 )
=> ( finite_finite2 @ ( list @ A )
@ ( collect @ ( list @ A )
@ ^ [Xs2: list @ A] :
( ( ord_less_eq @ ( set @ A ) @ ( set2 @ A @ Xs2 ) @ A2 )
& ( ord_less_eq @ nat @ ( size_size @ ( list @ A ) @ Xs2 ) @ N ) ) ) ) ) ).
% finite_lists_length_le
thf(fact_3053_map__replicate__const,axiom,
! [B: $tType,A: $tType,K: A,Lst: list @ B] :
( ( map @ B @ A
@ ^ [X3: B] : K
@ Lst )
= ( replicate @ A @ ( size_size @ ( list @ B ) @ Lst ) @ K ) ) ).
% map_replicate_const
thf(fact_3054_powr__mono2_H,axiom,
! [A3: real,X2: real,Y3: real] :
( ( ord_less_eq @ real @ A3 @ ( zero_zero @ real ) )
=> ( ( ord_less @ real @ ( zero_zero @ real ) @ X2 )
=> ( ( ord_less_eq @ real @ X2 @ Y3 )
=> ( ord_less_eq @ real @ ( powr @ real @ Y3 @ A3 ) @ ( powr @ real @ X2 @ A3 ) ) ) ) ) ).
% powr_mono2'
thf(fact_3055_powr__less__mono2,axiom,
! [A3: real,X2: real,Y3: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ A3 )
=> ( ( ord_less_eq @ real @ ( zero_zero @ real ) @ X2 )
=> ( ( ord_less @ real @ X2 @ Y3 )
=> ( ord_less @ real @ ( powr @ real @ X2 @ A3 ) @ ( powr @ real @ Y3 @ A3 ) ) ) ) ) ).
% powr_less_mono2
thf(fact_3056_gr__one__powr,axiom,
! [X2: real,Y3: real] :
( ( ord_less @ real @ ( one_one @ real ) @ X2 )
=> ( ( ord_less @ real @ ( zero_zero @ real ) @ Y3 )
=> ( ord_less @ real @ ( one_one @ real ) @ ( powr @ real @ X2 @ Y3 ) ) ) ) ).
% gr_one_powr
thf(fact_3057_powr__inj,axiom,
! [A3: real,X2: real,Y3: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ A3 )
=> ( ( A3
!= ( one_one @ real ) )
=> ( ( ( powr @ real @ A3 @ X2 )
= ( powr @ real @ A3 @ Y3 ) )
= ( X2 = Y3 ) ) ) ) ).
% powr_inj
thf(fact_3058_ge__one__powr__ge__zero,axiom,
! [X2: real,A3: real] :
( ( ord_less_eq @ real @ ( one_one @ real ) @ X2 )
=> ( ( ord_less_eq @ real @ ( zero_zero @ real ) @ A3 )
=> ( ord_less_eq @ real @ ( one_one @ real ) @ ( powr @ real @ X2 @ A3 ) ) ) ) ).
% ge_one_powr_ge_zero
thf(fact_3059_powr__mono__both,axiom,
! [A3: real,B3: real,X2: real,Y3: real] :
( ( ord_less_eq @ real @ ( zero_zero @ real ) @ A3 )
=> ( ( ord_less_eq @ real @ A3 @ B3 )
=> ( ( ord_less_eq @ real @ ( one_one @ real ) @ X2 )
=> ( ( ord_less_eq @ real @ X2 @ Y3 )
=> ( ord_less_eq @ real @ ( powr @ real @ X2 @ A3 ) @ ( powr @ real @ Y3 @ B3 ) ) ) ) ) ) ).
% powr_mono_both
thf(fact_3060_powr__le1,axiom,
! [A3: real,X2: real] :
( ( ord_less_eq @ real @ ( zero_zero @ real ) @ A3 )
=> ( ( ord_less_eq @ real @ ( zero_zero @ real ) @ X2 )
=> ( ( ord_less_eq @ real @ X2 @ ( one_one @ real ) )
=> ( ord_less_eq @ real @ ( powr @ real @ X2 @ A3 ) @ ( one_one @ real ) ) ) ) ) ).
% powr_le1
thf(fact_3061_powr__divide,axiom,
! [X2: real,Y3: real,A3: real] :
( ( ord_less_eq @ real @ ( zero_zero @ real ) @ X2 )
=> ( ( ord_less_eq @ real @ ( zero_zero @ real ) @ Y3 )
=> ( ( powr @ real @ ( divide_divide @ real @ X2 @ Y3 ) @ A3 )
= ( divide_divide @ real @ ( powr @ real @ X2 @ A3 ) @ ( powr @ real @ Y3 @ A3 ) ) ) ) ) ).
% powr_divide
thf(fact_3062_powr__mult,axiom,
! [X2: real,Y3: real,A3: real] :
( ( ord_less_eq @ real @ ( zero_zero @ real ) @ X2 )
=> ( ( ord_less_eq @ real @ ( zero_zero @ real ) @ Y3 )
=> ( ( powr @ real @ ( times_times @ real @ X2 @ Y3 ) @ A3 )
= ( times_times @ real @ ( powr @ real @ X2 @ A3 ) @ ( powr @ real @ Y3 @ A3 ) ) ) ) ) ).
% powr_mult
thf(fact_3063_powr__add,axiom,
! [A: $tType] :
( ( ( real_V3459762299906320749_field @ A )
& ( ln @ A ) )
=> ! [X2: A,A3: A,B3: A] :
( ( powr @ A @ X2 @ ( plus_plus @ A @ A3 @ B3 ) )
= ( times_times @ A @ ( powr @ A @ X2 @ A3 ) @ ( powr @ A @ X2 @ B3 ) ) ) ) ).
% powr_add
thf(fact_3064_powr__diff,axiom,
! [A: $tType] :
( ( ( real_V3459762299906320749_field @ A )
& ( ln @ A ) )
=> ! [W: A,Z1: A,Z22: A] :
( ( powr @ A @ W @ ( minus_minus @ A @ Z1 @ Z22 ) )
= ( divide_divide @ A @ ( powr @ A @ W @ Z1 ) @ ( powr @ A @ W @ Z22 ) ) ) ) ).
% powr_diff
thf(fact_3065_finite__divisors__nat,axiom,
! [M: nat] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ M )
=> ( finite_finite2 @ nat
@ ( collect @ nat
@ ^ [D3: nat] : ( dvd_dvd @ nat @ D3 @ M ) ) ) ) ).
% finite_divisors_nat
thf(fact_3066_subset__eq__atLeast0__lessThan__finite,axiom,
! [N7: set @ nat,N: nat] :
( ( ord_less_eq @ ( set @ nat ) @ N7 @ ( set_or7035219750837199246ssThan @ nat @ ( zero_zero @ nat ) @ N ) )
=> ( finite_finite2 @ nat @ N7 ) ) ).
% subset_eq_atLeast0_lessThan_finite
thf(fact_3067_subset__eq__atLeast0__atMost__finite,axiom,
! [N7: set @ nat,N: nat] :
( ( ord_less_eq @ ( set @ nat ) @ N7 @ ( set_or1337092689740270186AtMost @ nat @ ( zero_zero @ nat ) @ N ) )
=> ( finite_finite2 @ nat @ N7 ) ) ).
% subset_eq_atLeast0_atMost_finite
thf(fact_3068_set__replicate__Suc,axiom,
! [A: $tType,N: nat,X2: A] :
( ( set2 @ A @ ( replicate @ A @ ( suc @ N ) @ X2 ) )
= ( insert @ A @ X2 @ ( bot_bot @ ( set @ A ) ) ) ) ).
% set_replicate_Suc
thf(fact_3069_set__replicate__conv__if,axiom,
! [A: $tType,N: nat,X2: A] :
( ( ( N
= ( zero_zero @ nat ) )
=> ( ( set2 @ A @ ( replicate @ A @ N @ X2 ) )
= ( bot_bot @ ( set @ A ) ) ) )
& ( ( N
!= ( zero_zero @ nat ) )
=> ( ( set2 @ A @ ( replicate @ A @ N @ X2 ) )
= ( insert @ A @ X2 @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ).
% set_replicate_conv_if
thf(fact_3070_powr__less__iff,axiom,
! [B3: real,X2: real,Y3: real] :
( ( ord_less @ real @ ( one_one @ real ) @ B3 )
=> ( ( ord_less @ real @ ( zero_zero @ real ) @ X2 )
=> ( ( ord_less @ real @ ( powr @ real @ B3 @ Y3 ) @ X2 )
= ( ord_less @ real @ Y3 @ ( log @ B3 @ X2 ) ) ) ) ) ).
% powr_less_iff
thf(fact_3071_less__powr__iff,axiom,
! [B3: real,X2: real,Y3: real] :
( ( ord_less @ real @ ( one_one @ real ) @ B3 )
=> ( ( ord_less @ real @ ( zero_zero @ real ) @ X2 )
=> ( ( ord_less @ real @ X2 @ ( powr @ real @ B3 @ Y3 ) )
= ( ord_less @ real @ ( log @ B3 @ X2 ) @ Y3 ) ) ) ) ).
% less_powr_iff
thf(fact_3072_log__less__iff,axiom,
! [B3: real,X2: real,Y3: real] :
( ( ord_less @ real @ ( one_one @ real ) @ B3 )
=> ( ( ord_less @ real @ ( zero_zero @ real ) @ X2 )
=> ( ( ord_less @ real @ ( log @ B3 @ X2 ) @ Y3 )
= ( ord_less @ real @ X2 @ ( powr @ real @ B3 @ Y3 ) ) ) ) ) ).
% log_less_iff
thf(fact_3073_less__log__iff,axiom,
! [B3: real,X2: real,Y3: real] :
( ( ord_less @ real @ ( one_one @ real ) @ B3 )
=> ( ( ord_less @ real @ ( zero_zero @ real ) @ X2 )
=> ( ( ord_less @ real @ Y3 @ ( log @ B3 @ X2 ) )
= ( ord_less @ real @ ( powr @ real @ B3 @ Y3 ) @ X2 ) ) ) ) ).
% less_log_iff
thf(fact_3074_finite__roots__unity,axiom,
! [A: $tType] :
( ( ( real_V8999393235501362500lgebra @ A )
& ( idom @ A ) )
=> ! [N: nat] :
( ( ord_less_eq @ nat @ ( one_one @ nat ) @ N )
=> ( finite_finite2 @ A
@ ( collect @ A
@ ^ [Z4: A] :
( ( power_power @ A @ Z4 @ N )
= ( one_one @ A ) ) ) ) ) ) ).
% finite_roots_unity
thf(fact_3075_powr__mult__base,axiom,
! [X2: real,Y3: real] :
( ( ord_less_eq @ real @ ( zero_zero @ real ) @ X2 )
=> ( ( times_times @ real @ X2 @ ( powr @ real @ X2 @ Y3 ) )
= ( powr @ real @ X2 @ ( plus_plus @ real @ ( one_one @ real ) @ Y3 ) ) ) ) ).
% powr_mult_base
thf(fact_3076_powr__le__iff,axiom,
! [B3: real,X2: real,Y3: real] :
( ( ord_less @ real @ ( one_one @ real ) @ B3 )
=> ( ( ord_less @ real @ ( zero_zero @ real ) @ X2 )
=> ( ( ord_less_eq @ real @ ( powr @ real @ B3 @ Y3 ) @ X2 )
= ( ord_less_eq @ real @ Y3 @ ( log @ B3 @ X2 ) ) ) ) ) ).
% powr_le_iff
thf(fact_3077_le__powr__iff,axiom,
! [B3: real,X2: real,Y3: real] :
( ( ord_less @ real @ ( one_one @ real ) @ B3 )
=> ( ( ord_less @ real @ ( zero_zero @ real ) @ X2 )
=> ( ( ord_less_eq @ real @ X2 @ ( powr @ real @ B3 @ Y3 ) )
= ( ord_less_eq @ real @ ( log @ B3 @ X2 ) @ Y3 ) ) ) ) ).
% le_powr_iff
thf(fact_3078_log__le__iff,axiom,
! [B3: real,X2: real,Y3: real] :
( ( ord_less @ real @ ( one_one @ real ) @ B3 )
=> ( ( ord_less @ real @ ( zero_zero @ real ) @ X2 )
=> ( ( ord_less_eq @ real @ ( log @ B3 @ X2 ) @ Y3 )
= ( ord_less_eq @ real @ X2 @ ( powr @ real @ B3 @ Y3 ) ) ) ) ) ).
% log_le_iff
thf(fact_3079_le__log__iff,axiom,
! [B3: real,X2: real,Y3: real] :
( ( ord_less @ real @ ( one_one @ real ) @ B3 )
=> ( ( ord_less @ real @ ( zero_zero @ real ) @ X2 )
=> ( ( ord_less_eq @ real @ Y3 @ ( log @ B3 @ X2 ) )
= ( ord_less_eq @ real @ ( powr @ real @ B3 @ Y3 ) @ X2 ) ) ) ) ).
% le_log_iff
thf(fact_3080_log__add__eq__powr,axiom,
! [B3: real,X2: real,Y3: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ B3 )
=> ( ( B3
!= ( one_one @ real ) )
=> ( ( ord_less @ real @ ( zero_zero @ real ) @ X2 )
=> ( ( plus_plus @ real @ ( log @ B3 @ X2 ) @ Y3 )
= ( log @ B3 @ ( times_times @ real @ X2 @ ( powr @ real @ B3 @ Y3 ) ) ) ) ) ) ) ).
% log_add_eq_powr
thf(fact_3081_add__log__eq__powr,axiom,
! [B3: real,X2: real,Y3: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ B3 )
=> ( ( B3
!= ( one_one @ real ) )
=> ( ( ord_less @ real @ ( zero_zero @ real ) @ X2 )
=> ( ( plus_plus @ real @ Y3 @ ( log @ B3 @ X2 ) )
= ( log @ B3 @ ( times_times @ real @ ( powr @ real @ B3 @ Y3 ) @ X2 ) ) ) ) ) ) ).
% add_log_eq_powr
thf(fact_3082_minus__log__eq__powr,axiom,
! [B3: real,X2: real,Y3: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ B3 )
=> ( ( B3
!= ( one_one @ real ) )
=> ( ( ord_less @ real @ ( zero_zero @ real ) @ X2 )
=> ( ( minus_minus @ real @ Y3 @ ( log @ B3 @ X2 ) )
= ( log @ B3 @ ( divide_divide @ real @ ( powr @ real @ B3 @ Y3 ) @ X2 ) ) ) ) ) ) ).
% minus_log_eq_powr
thf(fact_3083_vebt__buildup_Osimps_I3_J,axiom,
! [Va: nat] :
( ( ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( suc @ ( suc @ Va ) ) )
=> ( ( vEBT_vebt_buildup @ ( suc @ ( suc @ Va ) ) )
= ( vEBT_Node @ ( none @ ( product_prod @ nat @ nat ) ) @ ( suc @ ( suc @ Va ) ) @ ( replicate @ vEBT_VEBT @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( vEBT_vebt_buildup @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( vEBT_vebt_buildup @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) )
& ( ~ ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( suc @ ( suc @ Va ) ) )
=> ( ( vEBT_vebt_buildup @ ( suc @ ( suc @ Va ) ) )
= ( vEBT_Node @ ( none @ ( product_prod @ nat @ nat ) ) @ ( suc @ ( suc @ Va ) ) @ ( replicate @ vEBT_VEBT @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( suc @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( vEBT_vebt_buildup @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( vEBT_vebt_buildup @ ( suc @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) ) ) ) ).
% vebt_buildup.simps(3)
thf(fact_3084_finite__Diff__insert,axiom,
! [A: $tType,A2: set @ A,A3: A,B2: set @ A] :
( ( finite_finite2 @ A @ ( minus_minus @ ( set @ A ) @ A2 @ ( insert @ A @ A3 @ B2 ) ) )
= ( finite_finite2 @ A @ ( minus_minus @ ( set @ A ) @ A2 @ B2 ) ) ) ).
% finite_Diff_insert
thf(fact_3085_finite__Collect__le__nat,axiom,
! [K: nat] :
( finite_finite2 @ nat
@ ( collect @ nat
@ ^ [N4: nat] : ( ord_less_eq @ nat @ N4 @ K ) ) ) ).
% finite_Collect_le_nat
thf(fact_3086_finite__Collect__less__nat,axiom,
! [K: nat] :
( finite_finite2 @ nat
@ ( collect @ nat
@ ^ [N4: nat] : ( ord_less @ nat @ N4 @ K ) ) ) ).
% finite_Collect_less_nat
thf(fact_3087_finite__Collect__subsets,axiom,
! [A: $tType,A2: set @ A] :
( ( finite_finite2 @ A @ A2 )
=> ( finite_finite2 @ ( set @ A )
@ ( collect @ ( set @ A )
@ ^ [B6: set @ A] : ( ord_less_eq @ ( set @ A ) @ B6 @ A2 ) ) ) ) ).
% finite_Collect_subsets
thf(fact_3088_finite__induct__select,axiom,
! [A: $tType,S: set @ A,P: ( set @ A ) > $o] :
( ( finite_finite2 @ A @ S )
=> ( ( P @ ( bot_bot @ ( set @ A ) ) )
=> ( ! [T5: set @ A] :
( ( ord_less @ ( set @ A ) @ T5 @ S )
=> ( ( P @ T5 )
=> ? [X8: A] :
( ( member @ A @ X8 @ ( minus_minus @ ( set @ A ) @ S @ T5 ) )
& ( P @ ( insert @ A @ X8 @ T5 ) ) ) ) )
=> ( P @ S ) ) ) ) ).
% finite_induct_select
thf(fact_3089_finite__Collect__disjI,axiom,
! [A: $tType,P: A > $o,Q: A > $o] :
( ( finite_finite2 @ A
@ ( collect @ A
@ ^ [X3: A] :
( ( P @ X3 )
| ( Q @ X3 ) ) ) )
= ( ( finite_finite2 @ A @ ( collect @ A @ P ) )
& ( finite_finite2 @ A @ ( collect @ A @ Q ) ) ) ) ).
% finite_Collect_disjI
thf(fact_3090_finite__Collect__conjI,axiom,
! [A: $tType,P: A > $o,Q: A > $o] :
( ( ( finite_finite2 @ A @ ( collect @ A @ P ) )
| ( finite_finite2 @ A @ ( collect @ A @ Q ) ) )
=> ( finite_finite2 @ A
@ ( collect @ A
@ ^ [X3: A] :
( ( P @ X3 )
& ( Q @ X3 ) ) ) ) ) ).
% finite_Collect_conjI
thf(fact_3091_finite__interval__int1,axiom,
! [A3: int,B3: int] :
( finite_finite2 @ int
@ ( collect @ int
@ ^ [I4: int] :
( ( ord_less_eq @ int @ A3 @ I4 )
& ( ord_less_eq @ int @ I4 @ B3 ) ) ) ) ).
% finite_interval_int1
thf(fact_3092_finite__interval__int4,axiom,
! [A3: int,B3: int] :
( finite_finite2 @ int
@ ( collect @ int
@ ^ [I4: int] :
( ( ord_less @ int @ A3 @ I4 )
& ( ord_less @ int @ I4 @ B3 ) ) ) ) ).
% finite_interval_int4
thf(fact_3093_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_3094_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_3095_finite__Diff,axiom,
! [A: $tType,A2: set @ A,B2: set @ A] :
( ( finite_finite2 @ A @ A2 )
=> ( finite_finite2 @ A @ ( minus_minus @ ( set @ A ) @ A2 @ B2 ) ) ) ).
% finite_Diff
thf(fact_3096_finite__Diff2,axiom,
! [A: $tType,B2: set @ A,A2: set @ A] :
( ( finite_finite2 @ A @ B2 )
=> ( ( finite_finite2 @ A @ ( minus_minus @ ( set @ A ) @ A2 @ B2 ) )
= ( finite_finite2 @ A @ A2 ) ) ) ).
% finite_Diff2
thf(fact_3097_finite__interval__int2,axiom,
! [A3: int,B3: int] :
( finite_finite2 @ int
@ ( collect @ int
@ ^ [I4: int] :
( ( ord_less_eq @ int @ A3 @ I4 )
& ( ord_less @ int @ I4 @ B3 ) ) ) ) ).
% finite_interval_int2
thf(fact_3098_finite__interval__int3,axiom,
! [A3: int,B3: int] :
( finite_finite2 @ int
@ ( collect @ int
@ ^ [I4: int] :
( ( ord_less @ int @ A3 @ I4 )
& ( ord_less_eq @ int @ I4 @ B3 ) ) ) ) ).
% finite_interval_int3
thf(fact_3099_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_3100_finite__maxlen,axiom,
! [A: $tType,M8: set @ ( list @ A )] :
( ( finite_finite2 @ ( list @ A ) @ M8 )
=> ? [N3: nat] :
! [X8: list @ A] :
( ( member @ ( list @ A ) @ X8 @ M8 )
=> ( ord_less @ nat @ ( size_size @ ( list @ A ) @ X8 ) @ N3 ) ) ) ).
% finite_maxlen
thf(fact_3101_finite__divisors__int,axiom,
! [I: int] :
( ( I
!= ( zero_zero @ int ) )
=> ( finite_finite2 @ int
@ ( collect @ int
@ ^ [D3: int] : ( dvd_dvd @ int @ D3 @ I ) ) ) ) ).
% finite_divisors_int
thf(fact_3102_pigeonhole__infinite__rel,axiom,
! [B: $tType,A: $tType,A2: set @ A,B2: set @ B,R: A > B > $o] :
( ~ ( finite_finite2 @ A @ A2 )
=> ( ( finite_finite2 @ B @ B2 )
=> ( ! [X4: A] :
( ( member @ A @ X4 @ A2 )
=> ? [Xa2: B] :
( ( member @ B @ Xa2 @ B2 )
& ( R @ X4 @ Xa2 ) ) )
=> ? [X4: B] :
( ( member @ B @ X4 @ B2 )
& ~ ( finite_finite2 @ A
@ ( collect @ A
@ ^ [A6: A] :
( ( member @ A @ A6 @ A2 )
& ( R @ A6 @ X4 ) ) ) ) ) ) ) ) ).
% pigeonhole_infinite_rel
thf(fact_3103_not__finite__existsD,axiom,
! [A: $tType,P: A > $o] :
( ~ ( finite_finite2 @ A @ ( collect @ A @ P ) )
=> ? [X_1: A] : ( P @ X_1 ) ) ).
% not_finite_existsD
thf(fact_3104_finite__has__minimal2,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [A2: set @ A,A3: A] :
( ( finite_finite2 @ A @ A2 )
=> ( ( member @ A @ A3 @ A2 )
=> ? [X4: A] :
( ( member @ A @ X4 @ A2 )
& ( ord_less_eq @ A @ X4 @ A3 )
& ! [Xa2: A] :
( ( member @ A @ Xa2 @ A2 )
=> ( ( ord_less_eq @ A @ Xa2 @ X4 )
=> ( X4 = Xa2 ) ) ) ) ) ) ) ).
% finite_has_minimal2
thf(fact_3105_finite__has__maximal2,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [A2: set @ A,A3: A] :
( ( finite_finite2 @ A @ A2 )
=> ( ( member @ A @ A3 @ A2 )
=> ? [X4: A] :
( ( member @ A @ X4 @ A2 )
& ( ord_less_eq @ A @ A3 @ X4 )
& ! [Xa2: A] :
( ( member @ A @ Xa2 @ A2 )
=> ( ( ord_less_eq @ A @ X4 @ Xa2 )
=> ( X4 = Xa2 ) ) ) ) ) ) ) ).
% finite_has_maximal2
thf(fact_3106_finite__subset,axiom,
! [A: $tType,A2: set @ A,B2: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ A2 @ B2 )
=> ( ( finite_finite2 @ A @ B2 )
=> ( finite_finite2 @ A @ A2 ) ) ) ).
% finite_subset
thf(fact_3107_infinite__super,axiom,
! [A: $tType,S: set @ A,T6: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ S @ T6 )
=> ( ~ ( finite_finite2 @ A @ S )
=> ~ ( finite_finite2 @ A @ T6 ) ) ) ).
% infinite_super
thf(fact_3108_rev__finite__subset,axiom,
! [A: $tType,B2: set @ A,A2: set @ A] :
( ( finite_finite2 @ A @ B2 )
=> ( ( ord_less_eq @ ( set @ A ) @ A2 @ B2 )
=> ( finite_finite2 @ A @ A2 ) ) ) ).
% rev_finite_subset
thf(fact_3109_Diff__infinite__finite,axiom,
! [A: $tType,T6: set @ A,S: set @ A] :
( ( finite_finite2 @ A @ T6 )
=> ( ~ ( finite_finite2 @ A @ S )
=> ~ ( finite_finite2 @ A @ ( minus_minus @ ( set @ A ) @ S @ T6 ) ) ) ) ).
% Diff_infinite_finite
thf(fact_3110_finite__has__maximal,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [A2: set @ A] :
( ( finite_finite2 @ A @ A2 )
=> ( ( A2
!= ( bot_bot @ ( set @ A ) ) )
=> ? [X4: A] :
( ( member @ A @ X4 @ A2 )
& ! [Xa2: A] :
( ( member @ A @ Xa2 @ A2 )
=> ( ( ord_less_eq @ A @ X4 @ Xa2 )
=> ( X4 = Xa2 ) ) ) ) ) ) ) ).
% finite_has_maximal
thf(fact_3111_finite__has__minimal,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [A2: set @ A] :
( ( finite_finite2 @ A @ A2 )
=> ( ( A2
!= ( bot_bot @ ( set @ A ) ) )
=> ? [X4: A] :
( ( member @ A @ X4 @ A2 )
& ! [Xa2: A] :
( ( member @ A @ Xa2 @ A2 )
=> ( ( ord_less_eq @ A @ Xa2 @ X4 )
=> ( X4 = Xa2 ) ) ) ) ) ) ) ).
% finite_has_minimal
thf(fact_3112_finite__subset__induct_H,axiom,
! [A: $tType,F4: set @ A,A2: set @ A,P: ( set @ A ) > $o] :
( ( finite_finite2 @ A @ F4 )
=> ( ( ord_less_eq @ ( set @ A ) @ F4 @ A2 )
=> ( ( P @ ( bot_bot @ ( set @ A ) ) )
=> ( ! [A4: A,F8: set @ A] :
( ( finite_finite2 @ A @ F8 )
=> ( ( member @ A @ A4 @ A2 )
=> ( ( ord_less_eq @ ( set @ A ) @ F8 @ A2 )
=> ( ~ ( member @ A @ A4 @ F8 )
=> ( ( P @ F8 )
=> ( P @ ( insert @ A @ A4 @ F8 ) ) ) ) ) ) )
=> ( P @ F4 ) ) ) ) ) ).
% finite_subset_induct'
thf(fact_3113_finite__subset__induct,axiom,
! [A: $tType,F4: set @ A,A2: set @ A,P: ( set @ A ) > $o] :
( ( finite_finite2 @ A @ F4 )
=> ( ( ord_less_eq @ ( set @ A ) @ F4 @ A2 )
=> ( ( P @ ( bot_bot @ ( set @ A ) ) )
=> ( ! [A4: A,F8: set @ A] :
( ( finite_finite2 @ A @ F8 )
=> ( ( member @ A @ A4 @ A2 )
=> ( ~ ( member @ A @ A4 @ F8 )
=> ( ( P @ F8 )
=> ( P @ ( insert @ A @ A4 @ F8 ) ) ) ) ) )
=> ( P @ F4 ) ) ) ) ) ).
% finite_subset_induct
thf(fact_3114_infinite__remove,axiom,
! [A: $tType,S: set @ A,A3: A] :
( ~ ( finite_finite2 @ A @ S )
=> ~ ( finite_finite2 @ A @ ( minus_minus @ ( set @ A ) @ S @ ( insert @ A @ A3 @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ).
% infinite_remove
thf(fact_3115_infinite__coinduct,axiom,
! [A: $tType,X: ( set @ A ) > $o,A2: set @ A] :
( ( X @ A2 )
=> ( ! [A9: set @ A] :
( ( X @ A9 )
=> ? [X8: A] :
( ( member @ A @ X8 @ A9 )
& ( ( X @ ( minus_minus @ ( set @ A ) @ A9 @ ( insert @ A @ X8 @ ( bot_bot @ ( set @ A ) ) ) ) )
| ~ ( finite_finite2 @ A @ ( minus_minus @ ( set @ A ) @ A9 @ ( insert @ A @ X8 @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ) )
=> ~ ( finite_finite2 @ A @ A2 ) ) ) ).
% infinite_coinduct
thf(fact_3116_finite__empty__induct,axiom,
! [A: $tType,A2: set @ A,P: ( set @ A ) > $o] :
( ( finite_finite2 @ A @ A2 )
=> ( ( P @ A2 )
=> ( ! [A4: A,A9: set @ A] :
( ( finite_finite2 @ A @ A9 )
=> ( ( member @ A @ A4 @ A9 )
=> ( ( P @ A9 )
=> ( P @ ( minus_minus @ ( set @ A ) @ A9 @ ( insert @ A @ A4 @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ) )
=> ( P @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ).
% finite_empty_induct
thf(fact_3117_finite__remove__induct,axiom,
! [A: $tType,B2: set @ A,P: ( set @ A ) > $o] :
( ( finite_finite2 @ A @ B2 )
=> ( ( P @ ( bot_bot @ ( set @ A ) ) )
=> ( ! [A9: set @ A] :
( ( finite_finite2 @ A @ A9 )
=> ( ( A9
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( ord_less_eq @ ( set @ A ) @ A9 @ B2 )
=> ( ! [X8: A] :
( ( member @ A @ X8 @ A9 )
=> ( P @ ( minus_minus @ ( set @ A ) @ A9 @ ( insert @ A @ X8 @ ( bot_bot @ ( set @ A ) ) ) ) ) )
=> ( P @ A9 ) ) ) ) )
=> ( P @ B2 ) ) ) ) ).
% finite_remove_induct
thf(fact_3118_remove__induct,axiom,
! [A: $tType,P: ( set @ A ) > $o,B2: set @ A] :
( ( P @ ( bot_bot @ ( set @ A ) ) )
=> ( ( ~ ( finite_finite2 @ A @ B2 )
=> ( P @ B2 ) )
=> ( ! [A9: set @ A] :
( ( finite_finite2 @ A @ A9 )
=> ( ( A9
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( ord_less_eq @ ( set @ A ) @ A9 @ B2 )
=> ( ! [X8: A] :
( ( member @ A @ X8 @ A9 )
=> ( P @ ( minus_minus @ ( set @ A ) @ A9 @ ( insert @ A @ X8 @ ( bot_bot @ ( set @ A ) ) ) ) ) )
=> ( P @ A9 ) ) ) ) )
=> ( P @ B2 ) ) ) ) ).
% remove_induct
thf(fact_3119_finite__nth__roots,axiom,
! [N: nat,C3: complex] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ N )
=> ( finite_finite2 @ complex
@ ( collect @ complex
@ ^ [Z4: complex] :
( ( power_power @ complex @ Z4 @ N )
= C3 ) ) ) ) ).
% finite_nth_roots
thf(fact_3120_set__encode__insert,axiom,
! [A2: set @ nat,N: nat] :
( ( finite_finite2 @ nat @ A2 )
=> ( ~ ( member @ nat @ N @ A2 )
=> ( ( nat_set_encode @ ( insert @ nat @ N @ A2 ) )
= ( plus_plus @ nat @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N ) @ ( nat_set_encode @ A2 ) ) ) ) ) ).
% set_encode_insert
thf(fact_3121_diff__preserves__multiset,axiom,
! [A: $tType,M8: A > nat,N7: A > nat] :
( ( finite_finite2 @ A
@ ( collect @ A
@ ^ [X3: A] : ( ord_less @ nat @ ( zero_zero @ nat ) @ ( M8 @ X3 ) ) ) )
=> ( finite_finite2 @ A
@ ( collect @ A
@ ^ [X3: A] : ( ord_less @ nat @ ( zero_zero @ nat ) @ ( minus_minus @ nat @ ( M8 @ X3 ) @ ( N7 @ X3 ) ) ) ) ) ) ).
% diff_preserves_multiset
thf(fact_3122_add__mset__in__multiset,axiom,
! [A: $tType,M8: A > nat,A3: A] :
( ( finite_finite2 @ A
@ ( collect @ A
@ ^ [X3: A] : ( ord_less @ nat @ ( zero_zero @ nat ) @ ( M8 @ X3 ) ) ) )
=> ( finite_finite2 @ A
@ ( collect @ A
@ ^ [X3: A] : ( ord_less @ nat @ ( zero_zero @ nat ) @ ( if @ nat @ ( X3 = A3 ) @ ( suc @ ( M8 @ X3 ) ) @ ( M8 @ X3 ) ) ) ) ) ) ).
% add_mset_in_multiset
thf(fact_3123_finite__linorder__max__induct,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A2: set @ A,P: ( set @ A ) > $o] :
( ( finite_finite2 @ A @ A2 )
=> ( ( P @ ( bot_bot @ ( set @ A ) ) )
=> ( ! [B4: A,A9: set @ A] :
( ( finite_finite2 @ A @ A9 )
=> ( ! [X8: A] :
( ( member @ A @ X8 @ A9 )
=> ( ord_less @ A @ X8 @ B4 ) )
=> ( ( P @ A9 )
=> ( P @ ( insert @ A @ B4 @ A9 ) ) ) ) )
=> ( P @ A2 ) ) ) ) ) ).
% finite_linorder_max_induct
thf(fact_3124_size__union,axiom,
! [A: $tType,M8: multiset @ A,N7: multiset @ A] :
( ( size_size @ ( multiset @ A ) @ ( plus_plus @ ( multiset @ A ) @ M8 @ N7 ) )
= ( plus_plus @ nat @ ( size_size @ ( multiset @ A ) @ M8 ) @ ( size_size @ ( multiset @ A ) @ N7 ) ) ) ).
% size_union
thf(fact_3125_ex__has__least__nat,axiom,
! [A: $tType,P: A > $o,K: A,M: A > nat] :
( ( P @ K )
=> ? [X4: A] :
( ( P @ X4 )
& ! [Y5: A] :
( ( P @ Y5 )
=> ( ord_less_eq @ nat @ ( M @ X4 ) @ ( M @ Y5 ) ) ) ) ) ).
% ex_has_least_nat
thf(fact_3126_nonempty__has__size,axiom,
! [A: $tType,S: multiset @ A] :
( ( S
!= ( zero_zero @ ( multiset @ A ) ) )
= ( ord_less @ nat @ ( zero_zero @ nat ) @ ( size_size @ ( multiset @ A ) @ S ) ) ) ).
% nonempty_has_size
thf(fact_3127_Lattices__Big_Oex__has__greatest__nat,axiom,
! [A: $tType,P: A > $o,K: A,F2: A > nat,B3: nat] :
( ( P @ K )
=> ( ! [Y4: A] :
( ( P @ Y4 )
=> ( ord_less @ nat @ ( F2 @ Y4 ) @ B3 ) )
=> ? [X4: A] :
( ( P @ X4 )
& ! [Y5: A] :
( ( P @ Y5 )
=> ( ord_less_eq @ nat @ ( F2 @ Y5 ) @ ( F2 @ X4 ) ) ) ) ) ) ).
% Lattices_Big.ex_has_greatest_nat
thf(fact_3128_diff__size__le__size__Diff,axiom,
! [A: $tType,M8: multiset @ A,M9: multiset @ A] : ( ord_less_eq @ nat @ ( minus_minus @ nat @ ( size_size @ ( multiset @ A ) @ M8 ) @ ( size_size @ ( multiset @ A ) @ M9 ) ) @ ( size_size @ ( multiset @ A ) @ ( minus_minus @ ( multiset @ A ) @ M8 @ M9 ) ) ) ).
% diff_size_le_size_Diff
thf(fact_3129_complex__mod__triangle__ineq2,axiom,
! [B3: complex,A3: complex] : ( ord_less_eq @ real @ ( minus_minus @ real @ ( real_V7770717601297561774m_norm @ complex @ ( plus_plus @ complex @ B3 @ A3 ) ) @ ( real_V7770717601297561774m_norm @ complex @ B3 ) ) @ ( real_V7770717601297561774m_norm @ complex @ A3 ) ) ).
% complex_mod_triangle_ineq2
thf(fact_3130_even__set__encode__iff,axiom,
! [A2: set @ nat] :
( ( finite_finite2 @ nat @ A2 )
=> ( ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( nat_set_encode @ A2 ) )
= ( ~ ( member @ nat @ ( zero_zero @ nat ) @ A2 ) ) ) ) ).
% even_set_encode_iff
thf(fact_3131_infinite__growing,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [X: set @ A] :
( ( X
!= ( bot_bot @ ( set @ A ) ) )
=> ( ! [X4: A] :
( ( member @ A @ X4 @ X )
=> ? [Xa2: A] :
( ( member @ A @ Xa2 @ X )
& ( ord_less @ A @ X4 @ Xa2 ) ) )
=> ~ ( finite_finite2 @ A @ X ) ) ) ) ).
% infinite_growing
thf(fact_3132_ex__min__if__finite,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [S: set @ A] :
( ( finite_finite2 @ A @ S )
=> ( ( S
!= ( bot_bot @ ( set @ A ) ) )
=> ? [X4: A] :
( ( member @ A @ X4 @ S )
& ~ ? [Xa2: A] :
( ( member @ A @ Xa2 @ S )
& ( ord_less @ A @ Xa2 @ X4 ) ) ) ) ) ) ).
% ex_min_if_finite
thf(fact_3133_ex__has__greatest__nat__lemma,axiom,
! [A: $tType,P: A > $o,K: A,F2: A > nat,N: nat] :
( ( P @ K )
=> ( ! [X4: A] :
( ( P @ X4 )
=> ? [Y5: A] :
( ( P @ Y5 )
& ~ ( ord_less_eq @ nat @ ( F2 @ Y5 ) @ ( F2 @ X4 ) ) ) )
=> ? [Y4: A] :
( ( P @ Y4 )
& ~ ( ord_less @ nat @ ( F2 @ Y4 ) @ ( plus_plus @ nat @ ( F2 @ K ) @ N ) ) ) ) ) ).
% ex_has_greatest_nat_lemma
thf(fact_3134_filter__preserves__multiset,axiom,
! [A: $tType,M8: A > nat,P: A > $o] :
( ( finite_finite2 @ A
@ ( collect @ A
@ ^ [X3: A] : ( ord_less @ nat @ ( zero_zero @ nat ) @ ( M8 @ X3 ) ) ) )
=> ( finite_finite2 @ A
@ ( collect @ A
@ ^ [X3: A] : ( ord_less @ nat @ ( zero_zero @ nat ) @ ( if @ nat @ ( P @ X3 ) @ ( M8 @ X3 ) @ ( zero_zero @ nat ) ) ) ) ) ) ).
% filter_preserves_multiset
thf(fact_3135_finite__ranking__induct,axiom,
! [A: $tType,B: $tType] :
( ( linorder @ A )
=> ! [S: set @ B,P: ( set @ B ) > $o,F2: B > A] :
( ( finite_finite2 @ B @ S )
=> ( ( P @ ( bot_bot @ ( set @ B ) ) )
=> ( ! [X4: B,S6: set @ B] :
( ( finite_finite2 @ B @ S6 )
=> ( ! [Y5: B] :
( ( member @ B @ Y5 @ S6 )
=> ( ord_less_eq @ A @ ( F2 @ Y5 ) @ ( F2 @ X4 ) ) )
=> ( ( P @ S6 )
=> ( P @ ( insert @ B @ X4 @ S6 ) ) ) ) )
=> ( P @ S ) ) ) ) ) ).
% finite_ranking_induct
thf(fact_3136_finite__linorder__min__induct,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A2: set @ A,P: ( set @ A ) > $o] :
( ( finite_finite2 @ A @ A2 )
=> ( ( P @ ( bot_bot @ ( set @ A ) ) )
=> ( ! [B4: A,A9: set @ A] :
( ( finite_finite2 @ A @ A9 )
=> ( ! [X8: A] :
( ( member @ A @ X8 @ A9 )
=> ( ord_less @ A @ B4 @ X8 ) )
=> ( ( P @ A9 )
=> ( P @ ( insert @ A @ B4 @ A9 ) ) ) ) )
=> ( P @ A2 ) ) ) ) ) ).
% finite_linorder_min_induct
thf(fact_3137_freeze__rule,axiom,
! [A: $tType] :
( ( heap @ A )
=> ! [A3: array @ A,Xs: list @ A] :
( hoare_hoare_triple @ ( list @ A ) @ ( snga_assn @ A @ A3 @ Xs ) @ ( array_freeze @ A @ A3 )
@ ^ [R5: list @ A] : ( times_times @ assn @ ( snga_assn @ A @ A3 @ Xs ) @ ( pure_assn @ ( R5 = Xs ) ) ) ) ) ).
% freeze_rule
thf(fact_3138_prod_Ofinite__Collect__op,axiom,
! [A: $tType,B: $tType] :
( ( comm_monoid_mult @ A )
=> ! [I6: set @ B,X2: B > A,Y3: B > A] :
( ( finite_finite2 @ B
@ ( collect @ B
@ ^ [I4: B] :
( ( member @ B @ I4 @ I6 )
& ( ( X2 @ I4 )
!= ( one_one @ A ) ) ) ) )
=> ( ( finite_finite2 @ B
@ ( collect @ B
@ ^ [I4: B] :
( ( member @ B @ I4 @ I6 )
& ( ( Y3 @ I4 )
!= ( one_one @ A ) ) ) ) )
=> ( finite_finite2 @ B
@ ( collect @ B
@ ^ [I4: B] :
( ( member @ B @ I4 @ I6 )
& ( ( times_times @ A @ ( X2 @ I4 ) @ ( Y3 @ I4 ) )
!= ( one_one @ A ) ) ) ) ) ) ) ) ).
% prod.finite_Collect_op
thf(fact_3139_sum_Ofinite__Collect__op,axiom,
! [A: $tType,B: $tType] :
( ( comm_monoid_add @ A )
=> ! [I6: set @ B,X2: B > A,Y3: B > A] :
( ( finite_finite2 @ B
@ ( collect @ B
@ ^ [I4: B] :
( ( member @ B @ I4 @ I6 )
& ( ( X2 @ I4 )
!= ( zero_zero @ A ) ) ) ) )
=> ( ( finite_finite2 @ B
@ ( collect @ B
@ ^ [I4: B] :
( ( member @ B @ I4 @ I6 )
& ( ( Y3 @ I4 )
!= ( zero_zero @ A ) ) ) ) )
=> ( finite_finite2 @ B
@ ( collect @ B
@ ^ [I4: B] :
( ( member @ B @ I4 @ I6 )
& ( ( plus_plus @ A @ ( X2 @ I4 ) @ ( Y3 @ I4 ) )
!= ( zero_zero @ A ) ) ) ) ) ) ) ) ).
% sum.finite_Collect_op
thf(fact_3140_sum__gp,axiom,
! [A: $tType] :
( ( ( division_ring @ A )
& ( comm_ring @ A ) )
=> ! [N: nat,M: nat,X2: A] :
( ( ( ord_less @ nat @ N @ M )
=> ( ( groups7311177749621191930dd_sum @ nat @ A @ ( power_power @ A @ X2 ) @ ( set_or1337092689740270186AtMost @ nat @ M @ N ) )
= ( zero_zero @ A ) ) )
& ( ~ ( ord_less @ nat @ N @ M )
=> ( ( ( X2
= ( one_one @ A ) )
=> ( ( groups7311177749621191930dd_sum @ nat @ A @ ( power_power @ A @ X2 ) @ ( set_or1337092689740270186AtMost @ nat @ M @ N ) )
= ( semiring_1_of_nat @ A @ ( minus_minus @ nat @ ( plus_plus @ nat @ N @ ( one_one @ nat ) ) @ M ) ) ) )
& ( ( X2
!= ( one_one @ A ) )
=> ( ( groups7311177749621191930dd_sum @ nat @ A @ ( power_power @ A @ X2 ) @ ( set_or1337092689740270186AtMost @ nat @ M @ N ) )
= ( divide_divide @ A @ ( minus_minus @ A @ ( power_power @ A @ X2 @ M ) @ ( power_power @ A @ X2 @ ( suc @ N ) ) ) @ ( minus_minus @ A @ ( one_one @ A ) @ X2 ) ) ) ) ) ) ) ) ).
% sum_gp
thf(fact_3141_subset__mset_Oadd__eq__0__iff__both__eq__0,axiom,
! [A: $tType,X2: multiset @ A,Y3: multiset @ A] :
( ( ( plus_plus @ ( multiset @ A ) @ X2 @ Y3 )
= ( zero_zero @ ( multiset @ A ) ) )
= ( ( X2
= ( zero_zero @ ( multiset @ A ) ) )
& ( Y3
= ( zero_zero @ ( multiset @ A ) ) ) ) ) ).
% subset_mset.add_eq_0_iff_both_eq_0
thf(fact_3142_subset__mset_Ozero__eq__add__iff__both__eq__0,axiom,
! [A: $tType,X2: multiset @ A,Y3: multiset @ A] :
( ( ( zero_zero @ ( multiset @ A ) )
= ( plus_plus @ ( multiset @ A ) @ X2 @ Y3 ) )
= ( ( X2
= ( zero_zero @ ( multiset @ A ) ) )
& ( Y3
= ( zero_zero @ ( multiset @ A ) ) ) ) ) ).
% subset_mset.zero_eq_add_iff_both_eq_0
thf(fact_3143_empty__eq__union,axiom,
! [A: $tType,M8: multiset @ A,N7: multiset @ A] :
( ( ( zero_zero @ ( multiset @ A ) )
= ( plus_plus @ ( multiset @ A ) @ M8 @ N7 ) )
= ( ( M8
= ( zero_zero @ ( multiset @ A ) ) )
& ( N7
= ( zero_zero @ ( multiset @ A ) ) ) ) ) ).
% empty_eq_union
thf(fact_3144_union__eq__empty,axiom,
! [A: $tType,M8: multiset @ A,N7: multiset @ A] :
( ( ( plus_plus @ ( multiset @ A ) @ M8 @ N7 )
= ( zero_zero @ ( multiset @ A ) ) )
= ( ( M8
= ( zero_zero @ ( multiset @ A ) ) )
& ( N7
= ( zero_zero @ ( multiset @ A ) ) ) ) ) ).
% union_eq_empty
thf(fact_3145_diff__diff__add__mset,axiom,
! [A: $tType,M8: multiset @ A,N7: multiset @ A,P: multiset @ A] :
( ( minus_minus @ ( multiset @ A ) @ ( minus_minus @ ( multiset @ A ) @ M8 @ N7 ) @ P )
= ( minus_minus @ ( multiset @ A ) @ M8 @ ( plus_plus @ ( multiset @ A ) @ N7 @ P ) ) ) ).
% diff_diff_add_mset
thf(fact_3146_sum_Oneutral__const,axiom,
! [B: $tType,A: $tType] :
( ( comm_monoid_add @ A )
=> ! [A2: set @ B] :
( ( groups7311177749621191930dd_sum @ B @ A
@ ^ [Uu3: B] : ( zero_zero @ A )
@ A2 )
= ( zero_zero @ A ) ) ) ).
% sum.neutral_const
thf(fact_3147_of__nat__sum,axiom,
! [A: $tType,B: $tType] :
( ( semiring_1 @ A )
=> ! [F2: B > nat,A2: set @ B] :
( ( semiring_1_of_nat @ A @ ( groups7311177749621191930dd_sum @ B @ nat @ F2 @ A2 ) )
= ( groups7311177749621191930dd_sum @ B @ A
@ ^ [X3: B] : ( semiring_1_of_nat @ A @ ( F2 @ X3 ) )
@ A2 ) ) ) ).
% of_nat_sum
thf(fact_3148_of__int__sum,axiom,
! [A: $tType,B: $tType] :
( ( ring_1 @ A )
=> ! [F2: B > int,A2: set @ B] :
( ( ring_1_of_int @ A @ ( groups7311177749621191930dd_sum @ B @ int @ F2 @ A2 ) )
= ( groups7311177749621191930dd_sum @ B @ A
@ ^ [X3: B] : ( ring_1_of_int @ A @ ( F2 @ X3 ) )
@ A2 ) ) ) ).
% of_int_sum
thf(fact_3149_sum_Odelta_H,axiom,
! [B: $tType,A: $tType] :
( ( comm_monoid_add @ A )
=> ! [S: set @ B,A3: B,B3: B > A] :
( ( finite_finite2 @ B @ S )
=> ( ( ( member @ B @ A3 @ S )
=> ( ( groups7311177749621191930dd_sum @ B @ A
@ ^ [K3: B] : ( if @ A @ ( A3 = K3 ) @ ( B3 @ K3 ) @ ( zero_zero @ A ) )
@ S )
= ( B3 @ A3 ) ) )
& ( ~ ( member @ B @ A3 @ S )
=> ( ( groups7311177749621191930dd_sum @ B @ A
@ ^ [K3: B] : ( if @ A @ ( A3 = K3 ) @ ( B3 @ K3 ) @ ( zero_zero @ A ) )
@ S )
= ( zero_zero @ A ) ) ) ) ) ) ).
% sum.delta'
thf(fact_3150_sum_Odelta,axiom,
! [B: $tType,A: $tType] :
( ( comm_monoid_add @ A )
=> ! [S: set @ B,A3: B,B3: B > A] :
( ( finite_finite2 @ B @ S )
=> ( ( ( member @ B @ A3 @ S )
=> ( ( groups7311177749621191930dd_sum @ B @ A
@ ^ [K3: B] : ( if @ A @ ( K3 = A3 ) @ ( B3 @ K3 ) @ ( zero_zero @ A ) )
@ S )
= ( B3 @ A3 ) ) )
& ( ~ ( member @ B @ A3 @ S )
=> ( ( groups7311177749621191930dd_sum @ B @ A
@ ^ [K3: B] : ( if @ A @ ( K3 = A3 ) @ ( B3 @ K3 ) @ ( zero_zero @ A ) )
@ S )
= ( zero_zero @ A ) ) ) ) ) ) ).
% sum.delta
thf(fact_3151_sum_Oinsert,axiom,
! [A: $tType,B: $tType] :
( ( comm_monoid_add @ A )
=> ! [A2: set @ B,X2: B,G: B > A] :
( ( finite_finite2 @ B @ A2 )
=> ( ~ ( member @ B @ X2 @ A2 )
=> ( ( groups7311177749621191930dd_sum @ B @ A @ G @ ( insert @ B @ X2 @ A2 ) )
= ( plus_plus @ A @ ( G @ X2 ) @ ( groups7311177749621191930dd_sum @ B @ A @ G @ A2 ) ) ) ) ) ) ).
% sum.insert
thf(fact_3152_sum_Oop__ivl__Suc,axiom,
! [A: $tType] :
( ( comm_monoid_add @ A )
=> ! [N: nat,M: nat,G: nat > A] :
( ( ( ord_less @ nat @ N @ M )
=> ( ( groups7311177749621191930dd_sum @ nat @ A @ G @ ( set_or7035219750837199246ssThan @ nat @ M @ ( suc @ N ) ) )
= ( zero_zero @ A ) ) )
& ( ~ ( ord_less @ nat @ N @ M )
=> ( ( groups7311177749621191930dd_sum @ nat @ A @ G @ ( set_or7035219750837199246ssThan @ nat @ M @ ( suc @ N ) ) )
= ( plus_plus @ A @ ( groups7311177749621191930dd_sum @ nat @ A @ G @ ( set_or7035219750837199246ssThan @ nat @ M @ N ) ) @ ( G @ N ) ) ) ) ) ) ).
% sum.op_ivl_Suc
thf(fact_3153_sum_Ocl__ivl__Suc,axiom,
! [A: $tType] :
( ( comm_monoid_add @ A )
=> ! [N: nat,M: nat,G: nat > A] :
( ( ( ord_less @ nat @ ( suc @ N ) @ M )
=> ( ( groups7311177749621191930dd_sum @ nat @ A @ G @ ( set_or1337092689740270186AtMost @ nat @ M @ ( suc @ N ) ) )
= ( zero_zero @ A ) ) )
& ( ~ ( ord_less @ nat @ ( suc @ N ) @ M )
=> ( ( groups7311177749621191930dd_sum @ nat @ A @ G @ ( set_or1337092689740270186AtMost @ nat @ M @ ( suc @ N ) ) )
= ( plus_plus @ A @ ( groups7311177749621191930dd_sum @ nat @ A @ G @ ( set_or1337092689740270186AtMost @ nat @ M @ N ) ) @ ( G @ ( suc @ N ) ) ) ) ) ) ) ).
% sum.cl_ivl_Suc
thf(fact_3154_sum__zero__power,axiom,
! [A: $tType] :
( ( division_ring @ A )
=> ! [A2: set @ nat,C3: nat > A] :
( ( ( ( finite_finite2 @ nat @ A2 )
& ( member @ nat @ ( zero_zero @ nat ) @ A2 ) )
=> ( ( groups7311177749621191930dd_sum @ nat @ A
@ ^ [I4: nat] : ( times_times @ A @ ( C3 @ I4 ) @ ( power_power @ A @ ( zero_zero @ A ) @ I4 ) )
@ A2 )
= ( C3 @ ( zero_zero @ nat ) ) ) )
& ( ~ ( ( finite_finite2 @ nat @ A2 )
& ( member @ nat @ ( zero_zero @ nat ) @ A2 ) )
=> ( ( groups7311177749621191930dd_sum @ nat @ A
@ ^ [I4: nat] : ( times_times @ A @ ( C3 @ I4 ) @ ( power_power @ A @ ( zero_zero @ A ) @ I4 ) )
@ A2 )
= ( zero_zero @ A ) ) ) ) ) ).
% sum_zero_power
thf(fact_3155_sum__zero__power_H,axiom,
! [A: $tType] :
( ( field @ A )
=> ! [A2: set @ nat,C3: nat > A,D2: nat > A] :
( ( ( ( finite_finite2 @ nat @ A2 )
& ( member @ nat @ ( zero_zero @ nat ) @ A2 ) )
=> ( ( groups7311177749621191930dd_sum @ nat @ A
@ ^ [I4: nat] : ( divide_divide @ A @ ( times_times @ A @ ( C3 @ I4 ) @ ( power_power @ A @ ( zero_zero @ A ) @ I4 ) ) @ ( D2 @ I4 ) )
@ A2 )
= ( divide_divide @ A @ ( C3 @ ( zero_zero @ nat ) ) @ ( D2 @ ( zero_zero @ nat ) ) ) ) )
& ( ~ ( ( finite_finite2 @ nat @ A2 )
& ( member @ nat @ ( zero_zero @ nat ) @ A2 ) )
=> ( ( groups7311177749621191930dd_sum @ nat @ A
@ ^ [I4: nat] : ( divide_divide @ A @ ( times_times @ A @ ( C3 @ I4 ) @ ( power_power @ A @ ( zero_zero @ A ) @ I4 ) ) @ ( D2 @ I4 ) )
@ A2 )
= ( zero_zero @ A ) ) ) ) ) ).
% sum_zero_power'
thf(fact_3156_Multiset_Odiff__add,axiom,
! [A: $tType,M8: multiset @ A,N7: multiset @ A,Q: multiset @ A] :
( ( minus_minus @ ( multiset @ A ) @ M8 @ ( plus_plus @ ( multiset @ A ) @ N7 @ Q ) )
= ( minus_minus @ ( multiset @ A ) @ ( minus_minus @ ( multiset @ A ) @ M8 @ N7 ) @ Q ) ) ).
% Multiset.diff_add
thf(fact_3157_diff__union__cancelL,axiom,
! [A: $tType,N7: multiset @ A,M8: multiset @ A] :
( ( minus_minus @ ( multiset @ A ) @ ( plus_plus @ ( multiset @ A ) @ N7 @ M8 ) @ N7 )
= M8 ) ).
% diff_union_cancelL
thf(fact_3158_diff__union__cancelR,axiom,
! [A: $tType,M8: multiset @ A,N7: multiset @ A] :
( ( minus_minus @ ( multiset @ A ) @ ( plus_plus @ ( multiset @ A ) @ M8 @ N7 ) @ N7 )
= M8 ) ).
% diff_union_cancelR
thf(fact_3159_sum_Oswap__restrict,axiom,
! [A: $tType,B: $tType,C: $tType] :
( ( comm_monoid_add @ A )
=> ! [A2: set @ B,B2: set @ C,G: B > C > A,R: B > C > $o] :
( ( finite_finite2 @ B @ A2 )
=> ( ( finite_finite2 @ C @ B2 )
=> ( ( groups7311177749621191930dd_sum @ B @ A
@ ^ [X3: B] :
( groups7311177749621191930dd_sum @ C @ A @ ( G @ X3 )
@ ( collect @ C
@ ^ [Y: C] :
( ( member @ C @ Y @ B2 )
& ( R @ X3 @ Y ) ) ) )
@ A2 )
= ( groups7311177749621191930dd_sum @ C @ A
@ ^ [Y: C] :
( groups7311177749621191930dd_sum @ B @ A
@ ^ [X3: B] : ( G @ X3 @ Y )
@ ( collect @ B
@ ^ [X3: B] :
( ( member @ B @ X3 @ A2 )
& ( R @ X3 @ Y ) ) ) )
@ B2 ) ) ) ) ) ).
% sum.swap_restrict
thf(fact_3160_empty__neutral_I2_J,axiom,
! [A: $tType,X2: multiset @ A] :
( ( plus_plus @ ( multiset @ A ) @ X2 @ ( zero_zero @ ( multiset @ A ) ) )
= X2 ) ).
% empty_neutral(2)
thf(fact_3161_empty__neutral_I1_J,axiom,
! [A: $tType,X2: multiset @ A] :
( ( plus_plus @ ( multiset @ A ) @ ( zero_zero @ ( multiset @ A ) ) @ X2 )
= X2 ) ).
% empty_neutral(1)
thf(fact_3162_union__less__mono,axiom,
! [A: $tType] :
( ( preorder @ A )
=> ! [A2: multiset @ A,C2: multiset @ A,B2: multiset @ A,D: multiset @ A] :
( ( ord_less @ ( multiset @ A ) @ A2 @ C2 )
=> ( ( ord_less @ ( multiset @ A ) @ B2 @ D )
=> ( ord_less @ ( multiset @ A ) @ ( plus_plus @ ( multiset @ A ) @ A2 @ B2 ) @ ( plus_plus @ ( multiset @ A ) @ C2 @ D ) ) ) ) ) ).
% union_less_mono
thf(fact_3163_union__le__mono2,axiom,
! [A: $tType] :
( ( preorder @ A )
=> ! [B2: multiset @ A,D: multiset @ A,C2: multiset @ A] :
( ( ord_less @ ( multiset @ A ) @ B2 @ D )
=> ( ord_less @ ( multiset @ A ) @ ( plus_plus @ ( multiset @ A ) @ C2 @ B2 ) @ ( plus_plus @ ( multiset @ A ) @ C2 @ D ) ) ) ) ).
% union_le_mono2
thf(fact_3164_union__le__mono1,axiom,
! [A: $tType] :
( ( preorder @ A )
=> ! [B2: multiset @ A,D: multiset @ A,C2: multiset @ A] :
( ( ord_less @ ( multiset @ A ) @ B2 @ D )
=> ( ord_less @ ( multiset @ A ) @ ( plus_plus @ ( multiset @ A ) @ B2 @ C2 ) @ ( plus_plus @ ( multiset @ A ) @ D @ C2 ) ) ) ) ).
% union_le_mono1
thf(fact_3165_union__assoc,axiom,
! [A: $tType,M8: multiset @ A,N7: multiset @ A,K6: multiset @ A] :
( ( plus_plus @ ( multiset @ A ) @ ( plus_plus @ ( multiset @ A ) @ M8 @ N7 ) @ K6 )
= ( plus_plus @ ( multiset @ A ) @ M8 @ ( plus_plus @ ( multiset @ A ) @ N7 @ K6 ) ) ) ).
% union_assoc
thf(fact_3166_union__lcomm,axiom,
! [A: $tType,M8: multiset @ A,N7: multiset @ A,K6: multiset @ A] :
( ( plus_plus @ ( multiset @ A ) @ M8 @ ( plus_plus @ ( multiset @ A ) @ N7 @ K6 ) )
= ( plus_plus @ ( multiset @ A ) @ N7 @ ( plus_plus @ ( multiset @ A ) @ M8 @ K6 ) ) ) ).
% union_lcomm
thf(fact_3167_union__commute,axiom,
! [A: $tType] :
( ( plus_plus @ ( multiset @ A ) )
= ( ^ [M10: multiset @ A,N8: multiset @ A] : ( plus_plus @ ( multiset @ A ) @ N8 @ M10 ) ) ) ).
% union_commute
thf(fact_3168_union__left__cancel,axiom,
! [A: $tType,K6: multiset @ A,M8: multiset @ A,N7: multiset @ A] :
( ( ( plus_plus @ ( multiset @ A ) @ K6 @ M8 )
= ( plus_plus @ ( multiset @ A ) @ K6 @ N7 ) )
= ( M8 = N7 ) ) ).
% union_left_cancel
thf(fact_3169_union__right__cancel,axiom,
! [A: $tType,M8: multiset @ A,K6: multiset @ A,N7: multiset @ A] :
( ( ( plus_plus @ ( multiset @ A ) @ M8 @ K6 )
= ( plus_plus @ ( multiset @ A ) @ N7 @ K6 ) )
= ( M8 = N7 ) ) ).
% union_right_cancel
thf(fact_3170_multi__union__self__other__eq,axiom,
! [A: $tType,A2: multiset @ A,X: multiset @ A,Y8: multiset @ A] :
( ( ( plus_plus @ ( multiset @ A ) @ A2 @ X )
= ( plus_plus @ ( multiset @ A ) @ A2 @ Y8 ) )
=> ( X = Y8 ) ) ).
% multi_union_self_other_eq
thf(fact_3171_Multiset_Odiff__right__commute,axiom,
! [A: $tType,M8: multiset @ A,N7: multiset @ A,Q: multiset @ A] :
( ( minus_minus @ ( multiset @ A ) @ ( minus_minus @ ( multiset @ A ) @ M8 @ N7 ) @ Q )
= ( minus_minus @ ( multiset @ A ) @ ( minus_minus @ ( multiset @ A ) @ M8 @ Q ) @ N7 ) ) ).
% Multiset.diff_right_commute
thf(fact_3172_Multiset_Odiff__cancel,axiom,
! [A: $tType,A2: multiset @ A] :
( ( minus_minus @ ( multiset @ A ) @ A2 @ A2 )
= ( zero_zero @ ( multiset @ A ) ) ) ).
% Multiset.diff_cancel
thf(fact_3173_diff__empty,axiom,
! [A: $tType,M8: multiset @ A] :
( ( ( minus_minus @ ( multiset @ A ) @ M8 @ ( zero_zero @ ( multiset @ A ) ) )
= M8 )
& ( ( minus_minus @ ( multiset @ A ) @ ( zero_zero @ ( multiset @ A ) ) @ M8 )
= ( zero_zero @ ( multiset @ A ) ) ) ) ).
% diff_empty
thf(fact_3174_sum__subtractf,axiom,
! [A: $tType,B: $tType] :
( ( ab_group_add @ A )
=> ! [F2: B > A,G: B > A,A2: set @ B] :
( ( groups7311177749621191930dd_sum @ B @ A
@ ^ [X3: B] : ( minus_minus @ A @ ( F2 @ X3 ) @ ( G @ X3 ) )
@ A2 )
= ( minus_minus @ A @ ( groups7311177749621191930dd_sum @ B @ A @ F2 @ A2 ) @ ( groups7311177749621191930dd_sum @ B @ A @ G @ A2 ) ) ) ) ).
% sum_subtractf
thf(fact_3175_sum__norm__le,axiom,
! [A: $tType,B: $tType] :
( ( real_V822414075346904944vector @ A )
=> ! [S: set @ B,F2: B > A,G: B > real] :
( ! [X4: B] :
( ( member @ B @ X4 @ S )
=> ( ord_less_eq @ real @ ( real_V7770717601297561774m_norm @ A @ ( F2 @ X4 ) ) @ ( G @ X4 ) ) )
=> ( ord_less_eq @ real @ ( real_V7770717601297561774m_norm @ A @ ( groups7311177749621191930dd_sum @ B @ A @ F2 @ S ) ) @ ( groups7311177749621191930dd_sum @ B @ real @ G @ S ) ) ) ) ).
% sum_norm_le
thf(fact_3176_sum__mono,axiom,
! [A: $tType,B: $tType] :
( ( ordere6911136660526730532id_add @ A )
=> ! [K6: set @ B,F2: B > A,G: B > A] :
( ! [I2: B] :
( ( member @ B @ I2 @ K6 )
=> ( ord_less_eq @ A @ ( F2 @ I2 ) @ ( G @ I2 ) ) )
=> ( ord_less_eq @ A @ ( groups7311177749621191930dd_sum @ B @ A @ F2 @ K6 ) @ ( groups7311177749621191930dd_sum @ B @ A @ G @ K6 ) ) ) ) ).
% sum_mono
thf(fact_3177_sum__distrib__left,axiom,
! [A: $tType,B: $tType] :
( ( semiring_0 @ A )
=> ! [R2: A,F2: B > A,A2: set @ B] :
( ( times_times @ A @ R2 @ ( groups7311177749621191930dd_sum @ B @ A @ F2 @ A2 ) )
= ( groups7311177749621191930dd_sum @ B @ A
@ ^ [N4: B] : ( times_times @ A @ R2 @ ( F2 @ N4 ) )
@ A2 ) ) ) ).
% sum_distrib_left
thf(fact_3178_sum__distrib__right,axiom,
! [A: $tType,B: $tType] :
( ( semiring_0 @ A )
=> ! [F2: B > A,A2: set @ B,R2: A] :
( ( times_times @ A @ ( groups7311177749621191930dd_sum @ B @ A @ F2 @ A2 ) @ R2 )
= ( groups7311177749621191930dd_sum @ B @ A
@ ^ [N4: B] : ( times_times @ A @ ( F2 @ N4 ) @ R2 )
@ A2 ) ) ) ).
% sum_distrib_right
thf(fact_3179_sum__product,axiom,
! [B: $tType,C: $tType,A: $tType] :
( ( semiring_0 @ B )
=> ! [F2: A > B,A2: set @ A,G: C > B,B2: set @ C] :
( ( times_times @ B @ ( groups7311177749621191930dd_sum @ A @ B @ F2 @ A2 ) @ ( groups7311177749621191930dd_sum @ C @ B @ G @ B2 ) )
= ( groups7311177749621191930dd_sum @ A @ B
@ ^ [I4: A] :
( groups7311177749621191930dd_sum @ C @ B
@ ^ [J3: C] : ( times_times @ B @ ( F2 @ I4 ) @ ( G @ J3 ) )
@ B2 )
@ A2 ) ) ) ).
% sum_product
thf(fact_3180_sum_Oswap,axiom,
! [A: $tType,B: $tType,C: $tType] :
( ( comm_monoid_add @ A )
=> ! [G: B > C > A,B2: set @ C,A2: set @ B] :
( ( groups7311177749621191930dd_sum @ B @ A
@ ^ [I4: B] : ( groups7311177749621191930dd_sum @ C @ A @ ( G @ I4 ) @ B2 )
@ A2 )
= ( groups7311177749621191930dd_sum @ C @ A
@ ^ [J3: C] :
( groups7311177749621191930dd_sum @ B @ A
@ ^ [I4: B] : ( G @ I4 @ J3 )
@ A2 )
@ B2 ) ) ) ).
% sum.swap
thf(fact_3181_sum__divide__distrib,axiom,
! [A: $tType,B: $tType] :
( ( field @ A )
=> ! [F2: B > A,A2: set @ B,R2: A] :
( ( divide_divide @ A @ ( groups7311177749621191930dd_sum @ B @ A @ F2 @ A2 ) @ R2 )
= ( groups7311177749621191930dd_sum @ B @ A
@ ^ [N4: B] : ( divide_divide @ A @ ( F2 @ N4 ) @ R2 )
@ A2 ) ) ) ).
% sum_divide_distrib
thf(fact_3182_norm__sum,axiom,
! [A: $tType,B: $tType] :
( ( real_V822414075346904944vector @ A )
=> ! [F2: B > A,A2: set @ B] :
( ord_less_eq @ real @ ( real_V7770717601297561774m_norm @ A @ ( groups7311177749621191930dd_sum @ B @ A @ F2 @ A2 ) )
@ ( groups7311177749621191930dd_sum @ B @ real
@ ^ [I4: B] : ( real_V7770717601297561774m_norm @ A @ ( F2 @ I4 ) )
@ A2 ) ) ) ).
% norm_sum
thf(fact_3183_sum_Odistrib,axiom,
! [A: $tType,B: $tType] :
( ( comm_monoid_add @ A )
=> ! [G: B > A,H2: B > A,A2: set @ B] :
( ( groups7311177749621191930dd_sum @ B @ A
@ ^ [X3: B] : ( plus_plus @ A @ ( G @ X3 ) @ ( H2 @ X3 ) )
@ A2 )
= ( plus_plus @ A @ ( groups7311177749621191930dd_sum @ B @ A @ G @ A2 ) @ ( groups7311177749621191930dd_sum @ B @ A @ H2 @ A2 ) ) ) ) ).
% sum.distrib
thf(fact_3184_mset__distrib,axiom,
! [A: $tType,A2: multiset @ A,B2: multiset @ A,M8: multiset @ A,N7: multiset @ A] :
( ( ( plus_plus @ ( multiset @ A ) @ A2 @ B2 )
= ( plus_plus @ ( multiset @ A ) @ M8 @ N7 ) )
=> ~ ! [Am: multiset @ A,An: multiset @ A] :
( ( A2
= ( plus_plus @ ( multiset @ A ) @ Am @ An ) )
=> ! [Bm2: multiset @ A,Bn2: multiset @ A] :
( ( B2
= ( plus_plus @ ( multiset @ A ) @ Bm2 @ Bn2 ) )
=> ( ( M8
= ( plus_plus @ ( multiset @ A ) @ Am @ Bm2 ) )
=> ( N7
!= ( plus_plus @ ( multiset @ A ) @ An @ Bn2 ) ) ) ) ) ) ).
% mset_distrib
thf(fact_3185_union__diff__assoc,axiom,
! [A: $tType,C2: multiset @ A,B2: multiset @ A,A2: multiset @ A] :
( ( ( minus_minus @ ( multiset @ A ) @ C2 @ B2 )
= ( zero_zero @ ( multiset @ A ) ) )
=> ( ( minus_minus @ ( multiset @ A ) @ ( plus_plus @ ( multiset @ A ) @ A2 @ B2 ) @ C2 )
= ( plus_plus @ ( multiset @ A ) @ A2 @ ( minus_minus @ ( multiset @ A ) @ B2 @ C2 ) ) ) ) ).
% union_diff_assoc
thf(fact_3186_sum__nonpos,axiom,
! [B: $tType,A: $tType] :
( ( ordere6911136660526730532id_add @ A )
=> ! [A2: set @ B,F2: B > A] :
( ! [X4: B] :
( ( member @ B @ X4 @ A2 )
=> ( ord_less_eq @ A @ ( F2 @ X4 ) @ ( zero_zero @ A ) ) )
=> ( ord_less_eq @ A @ ( groups7311177749621191930dd_sum @ B @ A @ F2 @ A2 ) @ ( zero_zero @ A ) ) ) ) ).
% sum_nonpos
thf(fact_3187_sum__nonneg,axiom,
! [A: $tType,B: $tType] :
( ( ordere6911136660526730532id_add @ A )
=> ! [A2: set @ B,F2: B > A] :
( ! [X4: B] :
( ( member @ B @ X4 @ A2 )
=> ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( F2 @ X4 ) ) )
=> ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( groups7311177749621191930dd_sum @ B @ A @ F2 @ A2 ) ) ) ) ).
% sum_nonneg
thf(fact_3188_sum__mono__inv,axiom,
! [A: $tType,I8: $tType] :
( ( ordere8940638589300402666id_add @ A )
=> ! [F2: I8 > A,I6: set @ I8,G: I8 > A,I: I8] :
( ( ( groups7311177749621191930dd_sum @ I8 @ A @ F2 @ I6 )
= ( groups7311177749621191930dd_sum @ I8 @ A @ G @ I6 ) )
=> ( ! [I2: I8] :
( ( member @ I8 @ I2 @ I6 )
=> ( ord_less_eq @ A @ ( F2 @ I2 ) @ ( G @ I2 ) ) )
=> ( ( member @ I8 @ I @ I6 )
=> ( ( finite_finite2 @ I8 @ I6 )
=> ( ( F2 @ I )
= ( G @ I ) ) ) ) ) ) ) ).
% sum_mono_inv
thf(fact_3189_sum__cong__Suc,axiom,
! [A: $tType] :
( ( comm_monoid_add @ A )
=> ! [A2: set @ nat,F2: nat > A,G: nat > A] :
( ~ ( member @ nat @ ( zero_zero @ nat ) @ A2 )
=> ( ! [X4: nat] :
( ( member @ nat @ ( suc @ X4 ) @ A2 )
=> ( ( F2 @ ( suc @ X4 ) )
= ( G @ ( suc @ X4 ) ) ) )
=> ( ( groups7311177749621191930dd_sum @ nat @ A @ F2 @ A2 )
= ( groups7311177749621191930dd_sum @ nat @ A @ G @ A2 ) ) ) ) ) ).
% sum_cong_Suc
thf(fact_3190_mod__sum__eq,axiom,
! [B: $tType,A: $tType] :
( ( euclid4440199948858584721cancel @ A )
=> ! [F2: B > A,A3: A,A2: set @ B] :
( ( modulo_modulo @ A
@ ( groups7311177749621191930dd_sum @ B @ A
@ ^ [I4: B] : ( modulo_modulo @ A @ ( F2 @ I4 ) @ A3 )
@ A2 )
@ A3 )
= ( modulo_modulo @ A @ ( groups7311177749621191930dd_sum @ B @ A @ F2 @ A2 ) @ A3 ) ) ) ).
% mod_sum_eq
thf(fact_3191_sum_Ointer__filter,axiom,
! [A: $tType,B: $tType] :
( ( comm_monoid_add @ A )
=> ! [A2: set @ B,G: B > A,P: B > $o] :
( ( finite_finite2 @ B @ A2 )
=> ( ( groups7311177749621191930dd_sum @ B @ A @ G
@ ( collect @ B
@ ^ [X3: B] :
( ( member @ B @ X3 @ A2 )
& ( P @ X3 ) ) ) )
= ( groups7311177749621191930dd_sum @ B @ A
@ ^ [X3: B] : ( if @ A @ ( P @ X3 ) @ ( G @ X3 ) @ ( zero_zero @ A ) )
@ A2 ) ) ) ) ).
% sum.inter_filter
thf(fact_3192_sum__le__included,axiom,
! [B: $tType,A: $tType,C: $tType] :
( ( ordere6911136660526730532id_add @ A )
=> ! [S2: set @ B,T2: set @ C,G: C > A,I: C > B,F2: B > A] :
( ( finite_finite2 @ B @ S2 )
=> ( ( finite_finite2 @ C @ T2 )
=> ( ! [X4: C] :
( ( member @ C @ X4 @ T2 )
=> ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( G @ X4 ) ) )
=> ( ! [X4: B] :
( ( member @ B @ X4 @ S2 )
=> ? [Xa2: C] :
( ( member @ C @ Xa2 @ T2 )
& ( ( I @ Xa2 )
= X4 )
& ( ord_less_eq @ A @ ( F2 @ X4 ) @ ( G @ Xa2 ) ) ) )
=> ( ord_less_eq @ A @ ( groups7311177749621191930dd_sum @ B @ A @ F2 @ S2 ) @ ( groups7311177749621191930dd_sum @ C @ A @ G @ T2 ) ) ) ) ) ) ) ).
% sum_le_included
thf(fact_3193_sum__nonneg__eq__0__iff,axiom,
! [A: $tType,B: $tType] :
( ( ordere6911136660526730532id_add @ A )
=> ! [A2: set @ B,F2: B > A] :
( ( finite_finite2 @ B @ A2 )
=> ( ! [X4: B] :
( ( member @ B @ X4 @ A2 )
=> ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( F2 @ X4 ) ) )
=> ( ( ( groups7311177749621191930dd_sum @ B @ A @ F2 @ A2 )
= ( zero_zero @ A ) )
= ( ! [X3: B] :
( ( member @ B @ X3 @ A2 )
=> ( ( F2 @ X3 )
= ( zero_zero @ A ) ) ) ) ) ) ) ) ).
% sum_nonneg_eq_0_iff
thf(fact_3194_sum__strict__mono__ex1,axiom,
! [A: $tType,I8: $tType] :
( ( ordere8940638589300402666id_add @ A )
=> ! [A2: set @ I8,F2: I8 > A,G: I8 > A] :
( ( finite_finite2 @ I8 @ A2 )
=> ( ! [X4: I8] :
( ( member @ I8 @ X4 @ A2 )
=> ( ord_less_eq @ A @ ( F2 @ X4 ) @ ( G @ X4 ) ) )
=> ( ? [X8: I8] :
( ( member @ I8 @ X8 @ A2 )
& ( ord_less @ A @ ( F2 @ X8 ) @ ( G @ X8 ) ) )
=> ( ord_less @ A @ ( groups7311177749621191930dd_sum @ I8 @ A @ F2 @ A2 ) @ ( groups7311177749621191930dd_sum @ I8 @ A @ G @ A2 ) ) ) ) ) ) ).
% sum_strict_mono_ex1
thf(fact_3195_sum_Orelated,axiom,
! [A: $tType,B: $tType] :
( ( comm_monoid_add @ A )
=> ! [R: A > A > $o,S: set @ B,H2: B > A,G: B > A] :
( ( R @ ( zero_zero @ A ) @ ( zero_zero @ A ) )
=> ( ! [X16: A,Y1: A,X23: A,Y22: A] :
( ( ( R @ X16 @ X23 )
& ( R @ Y1 @ Y22 ) )
=> ( R @ ( plus_plus @ A @ X16 @ Y1 ) @ ( plus_plus @ A @ X23 @ Y22 ) ) )
=> ( ( finite_finite2 @ B @ S )
=> ( ! [X4: B] :
( ( member @ B @ X4 @ S )
=> ( R @ ( H2 @ X4 ) @ ( G @ X4 ) ) )
=> ( R @ ( groups7311177749621191930dd_sum @ B @ A @ H2 @ S ) @ ( groups7311177749621191930dd_sum @ B @ A @ G @ S ) ) ) ) ) ) ) ).
% sum.related
thf(fact_3196_sum__strict__mono,axiom,
! [A: $tType,B: $tType] :
( ( strict7427464778891057005id_add @ A )
=> ! [A2: set @ B,F2: B > A,G: B > A] :
( ( finite_finite2 @ B @ A2 )
=> ( ( A2
!= ( bot_bot @ ( set @ B ) ) )
=> ( ! [X4: B] :
( ( member @ B @ X4 @ A2 )
=> ( ord_less @ A @ ( F2 @ X4 ) @ ( G @ X4 ) ) )
=> ( ord_less @ A @ ( groups7311177749621191930dd_sum @ B @ A @ F2 @ A2 ) @ ( groups7311177749621191930dd_sum @ B @ A @ G @ A2 ) ) ) ) ) ) ).
% sum_strict_mono
thf(fact_3197_sum_Oinsert__if,axiom,
! [A: $tType,B: $tType] :
( ( comm_monoid_add @ A )
=> ! [A2: set @ B,X2: B,G: B > A] :
( ( finite_finite2 @ B @ A2 )
=> ( ( ( member @ B @ X2 @ A2 )
=> ( ( groups7311177749621191930dd_sum @ B @ A @ G @ ( insert @ B @ X2 @ A2 ) )
= ( groups7311177749621191930dd_sum @ B @ A @ G @ A2 ) ) )
& ( ~ ( member @ B @ X2 @ A2 )
=> ( ( groups7311177749621191930dd_sum @ B @ A @ G @ ( insert @ B @ X2 @ A2 ) )
= ( plus_plus @ A @ ( G @ X2 ) @ ( groups7311177749621191930dd_sum @ B @ A @ G @ A2 ) ) ) ) ) ) ) ).
% sum.insert_if
thf(fact_3198_sum_Oreindex__bij__witness__not__neutral,axiom,
! [B: $tType,A: $tType,C: $tType] :
( ( comm_monoid_add @ A )
=> ! [S7: set @ B,T7: set @ C,S: set @ B,I: C > B,J: B > C,T6: set @ C,G: B > A,H2: C > A] :
( ( finite_finite2 @ B @ S7 )
=> ( ( finite_finite2 @ C @ T7 )
=> ( ! [A4: B] :
( ( member @ B @ A4 @ ( minus_minus @ ( set @ B ) @ S @ S7 ) )
=> ( ( I @ ( J @ A4 ) )
= A4 ) )
=> ( ! [A4: B] :
( ( member @ B @ A4 @ ( minus_minus @ ( set @ B ) @ S @ S7 ) )
=> ( member @ C @ ( J @ A4 ) @ ( minus_minus @ ( set @ C ) @ T6 @ T7 ) ) )
=> ( ! [B4: C] :
( ( member @ C @ B4 @ ( minus_minus @ ( set @ C ) @ T6 @ T7 ) )
=> ( ( J @ ( I @ B4 ) )
= B4 ) )
=> ( ! [B4: C] :
( ( member @ C @ B4 @ ( minus_minus @ ( set @ C ) @ T6 @ T7 ) )
=> ( member @ B @ ( I @ B4 ) @ ( minus_minus @ ( set @ B ) @ S @ S7 ) ) )
=> ( ! [A4: B] :
( ( member @ B @ A4 @ S7 )
=> ( ( G @ A4 )
= ( zero_zero @ A ) ) )
=> ( ! [B4: C] :
( ( member @ C @ B4 @ T7 )
=> ( ( H2 @ B4 )
= ( zero_zero @ A ) ) )
=> ( ! [A4: B] :
( ( member @ B @ A4 @ S )
=> ( ( H2 @ ( J @ A4 ) )
= ( G @ A4 ) ) )
=> ( ( groups7311177749621191930dd_sum @ B @ A @ G @ S )
= ( groups7311177749621191930dd_sum @ C @ A @ H2 @ T6 ) ) ) ) ) ) ) ) ) ) ) ) ).
% sum.reindex_bij_witness_not_neutral
thf(fact_3199_sum__nonneg__0,axiom,
! [B: $tType,A: $tType] :
( ( ordere6911136660526730532id_add @ A )
=> ! [S2: set @ B,F2: B > A,I: B] :
( ( finite_finite2 @ B @ S2 )
=> ( ! [I2: B] :
( ( member @ B @ I2 @ S2 )
=> ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( F2 @ I2 ) ) )
=> ( ( ( groups7311177749621191930dd_sum @ B @ A @ F2 @ S2 )
= ( zero_zero @ A ) )
=> ( ( member @ B @ I @ S2 )
=> ( ( F2 @ I )
= ( zero_zero @ A ) ) ) ) ) ) ) ).
% sum_nonneg_0
thf(fact_3200_sum__nonneg__leq__bound,axiom,
! [B: $tType,A: $tType] :
( ( ordere6911136660526730532id_add @ A )
=> ! [S2: set @ B,F2: B > A,B2: A,I: B] :
( ( finite_finite2 @ B @ S2 )
=> ( ! [I2: B] :
( ( member @ B @ I2 @ S2 )
=> ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( F2 @ I2 ) ) )
=> ( ( ( groups7311177749621191930dd_sum @ B @ A @ F2 @ S2 )
= B2 )
=> ( ( member @ B @ I @ S2 )
=> ( ord_less_eq @ A @ ( F2 @ I ) @ B2 ) ) ) ) ) ) ).
% sum_nonneg_leq_bound
thf(fact_3201_sum_Osetdiff__irrelevant,axiom,
! [A: $tType,B: $tType] :
( ( comm_monoid_add @ A )
=> ! [A2: set @ B,G: B > A] :
( ( finite_finite2 @ B @ A2 )
=> ( ( groups7311177749621191930dd_sum @ B @ A @ G
@ ( minus_minus @ ( set @ B ) @ A2
@ ( collect @ B
@ ^ [X3: B] :
( ( G @ X3 )
= ( zero_zero @ A ) ) ) ) )
= ( groups7311177749621191930dd_sum @ B @ A @ G @ A2 ) ) ) ) ).
% sum.setdiff_irrelevant
thf(fact_3202_sum_Oshift__bounds__Suc__ivl,axiom,
! [A: $tType] :
( ( comm_monoid_add @ A )
=> ! [G: nat > A,M: nat,N: nat] :
( ( groups7311177749621191930dd_sum @ nat @ A @ G @ ( set_or7035219750837199246ssThan @ nat @ ( suc @ M ) @ ( suc @ N ) ) )
= ( groups7311177749621191930dd_sum @ nat @ A
@ ^ [I4: nat] : ( G @ ( suc @ I4 ) )
@ ( set_or7035219750837199246ssThan @ nat @ M @ N ) ) ) ) ).
% sum.shift_bounds_Suc_ivl
thf(fact_3203_sum_Oshift__bounds__cl__Suc__ivl,axiom,
! [A: $tType] :
( ( comm_monoid_add @ A )
=> ! [G: nat > A,M: nat,N: nat] :
( ( groups7311177749621191930dd_sum @ nat @ A @ G @ ( set_or1337092689740270186AtMost @ nat @ ( suc @ M ) @ ( suc @ N ) ) )
= ( groups7311177749621191930dd_sum @ nat @ A
@ ^ [I4: nat] : ( G @ ( suc @ I4 ) )
@ ( set_or1337092689740270186AtMost @ nat @ M @ N ) ) ) ) ).
% sum.shift_bounds_cl_Suc_ivl
thf(fact_3204_sum_Oshift__bounds__nat__ivl,axiom,
! [A: $tType] :
( ( comm_monoid_add @ A )
=> ! [G: nat > A,M: nat,K: nat,N: nat] :
( ( groups7311177749621191930dd_sum @ nat @ A @ G @ ( set_or7035219750837199246ssThan @ nat @ ( plus_plus @ nat @ M @ K ) @ ( plus_plus @ nat @ N @ K ) ) )
= ( groups7311177749621191930dd_sum @ nat @ A
@ ^ [I4: nat] : ( G @ ( plus_plus @ nat @ I4 @ K ) )
@ ( set_or7035219750837199246ssThan @ nat @ M @ N ) ) ) ) ).
% sum.shift_bounds_nat_ivl
thf(fact_3205_sum_Oshift__bounds__cl__nat__ivl,axiom,
! [A: $tType] :
( ( comm_monoid_add @ A )
=> ! [G: nat > A,M: nat,K: nat,N: nat] :
( ( groups7311177749621191930dd_sum @ nat @ A @ G @ ( set_or1337092689740270186AtMost @ nat @ ( plus_plus @ nat @ M @ K ) @ ( plus_plus @ nat @ N @ K ) ) )
= ( groups7311177749621191930dd_sum @ nat @ A
@ ^ [I4: nat] : ( G @ ( plus_plus @ nat @ I4 @ K ) )
@ ( set_or1337092689740270186AtMost @ nat @ M @ N ) ) ) ) ).
% sum.shift_bounds_cl_nat_ivl
thf(fact_3206_sum__pos2,axiom,
! [A: $tType,B: $tType] :
( ( ordere6911136660526730532id_add @ A )
=> ! [I6: set @ B,I: B,F2: B > A] :
( ( finite_finite2 @ B @ I6 )
=> ( ( member @ B @ I @ I6 )
=> ( ( ord_less @ A @ ( zero_zero @ A ) @ ( F2 @ I ) )
=> ( ! [I2: B] :
( ( member @ B @ I2 @ I6 )
=> ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( F2 @ I2 ) ) )
=> ( ord_less @ A @ ( zero_zero @ A ) @ ( groups7311177749621191930dd_sum @ B @ A @ F2 @ I6 ) ) ) ) ) ) ) ).
% sum_pos2
thf(fact_3207_sum__pos,axiom,
! [A: $tType,B: $tType] :
( ( ordere6911136660526730532id_add @ A )
=> ! [I6: set @ B,F2: B > A] :
( ( finite_finite2 @ B @ I6 )
=> ( ( I6
!= ( bot_bot @ ( set @ B ) ) )
=> ( ! [I2: B] :
( ( member @ B @ I2 @ I6 )
=> ( ord_less @ A @ ( zero_zero @ A ) @ ( F2 @ I2 ) ) )
=> ( ord_less @ A @ ( zero_zero @ A ) @ ( groups7311177749621191930dd_sum @ B @ A @ F2 @ I6 ) ) ) ) ) ) ).
% sum_pos
thf(fact_3208_sum_Osame__carrier,axiom,
! [A: $tType,B: $tType] :
( ( comm_monoid_add @ A )
=> ! [C2: set @ B,A2: set @ B,B2: set @ B,G: B > A,H2: B > A] :
( ( finite_finite2 @ B @ C2 )
=> ( ( ord_less_eq @ ( set @ B ) @ A2 @ C2 )
=> ( ( ord_less_eq @ ( set @ B ) @ B2 @ C2 )
=> ( ! [A4: B] :
( ( member @ B @ A4 @ ( minus_minus @ ( set @ B ) @ C2 @ A2 ) )
=> ( ( G @ A4 )
= ( zero_zero @ A ) ) )
=> ( ! [B4: B] :
( ( member @ B @ B4 @ ( minus_minus @ ( set @ B ) @ C2 @ B2 ) )
=> ( ( H2 @ B4 )
= ( zero_zero @ A ) ) )
=> ( ( ( groups7311177749621191930dd_sum @ B @ A @ G @ A2 )
= ( groups7311177749621191930dd_sum @ B @ A @ H2 @ B2 ) )
= ( ( groups7311177749621191930dd_sum @ B @ A @ G @ C2 )
= ( groups7311177749621191930dd_sum @ B @ A @ H2 @ C2 ) ) ) ) ) ) ) ) ) ).
% sum.same_carrier
thf(fact_3209_sum_Osame__carrierI,axiom,
! [A: $tType,B: $tType] :
( ( comm_monoid_add @ A )
=> ! [C2: set @ B,A2: set @ B,B2: set @ B,G: B > A,H2: B > A] :
( ( finite_finite2 @ B @ C2 )
=> ( ( ord_less_eq @ ( set @ B ) @ A2 @ C2 )
=> ( ( ord_less_eq @ ( set @ B ) @ B2 @ C2 )
=> ( ! [A4: B] :
( ( member @ B @ A4 @ ( minus_minus @ ( set @ B ) @ C2 @ A2 ) )
=> ( ( G @ A4 )
= ( zero_zero @ A ) ) )
=> ( ! [B4: B] :
( ( member @ B @ B4 @ ( minus_minus @ ( set @ B ) @ C2 @ B2 ) )
=> ( ( H2 @ B4 )
= ( zero_zero @ A ) ) )
=> ( ( ( groups7311177749621191930dd_sum @ B @ A @ G @ C2 )
= ( groups7311177749621191930dd_sum @ B @ A @ H2 @ C2 ) )
=> ( ( groups7311177749621191930dd_sum @ B @ A @ G @ A2 )
= ( groups7311177749621191930dd_sum @ B @ A @ H2 @ B2 ) ) ) ) ) ) ) ) ) ).
% sum.same_carrierI
thf(fact_3210_sum_Omono__neutral__left,axiom,
! [A: $tType,B: $tType] :
( ( comm_monoid_add @ A )
=> ! [T6: set @ B,S: set @ B,G: B > A] :
( ( finite_finite2 @ B @ T6 )
=> ( ( ord_less_eq @ ( set @ B ) @ S @ T6 )
=> ( ! [X4: B] :
( ( member @ B @ X4 @ ( minus_minus @ ( set @ B ) @ T6 @ S ) )
=> ( ( G @ X4 )
= ( zero_zero @ A ) ) )
=> ( ( groups7311177749621191930dd_sum @ B @ A @ G @ S )
= ( groups7311177749621191930dd_sum @ B @ A @ G @ T6 ) ) ) ) ) ) ).
% sum.mono_neutral_left
thf(fact_3211_sum_Omono__neutral__right,axiom,
! [A: $tType,B: $tType] :
( ( comm_monoid_add @ A )
=> ! [T6: set @ B,S: set @ B,G: B > A] :
( ( finite_finite2 @ B @ T6 )
=> ( ( ord_less_eq @ ( set @ B ) @ S @ T6 )
=> ( ! [X4: B] :
( ( member @ B @ X4 @ ( minus_minus @ ( set @ B ) @ T6 @ S ) )
=> ( ( G @ X4 )
= ( zero_zero @ A ) ) )
=> ( ( groups7311177749621191930dd_sum @ B @ A @ G @ T6 )
= ( groups7311177749621191930dd_sum @ B @ A @ G @ S ) ) ) ) ) ) ).
% sum.mono_neutral_right
thf(fact_3212_sum_Omono__neutral__cong__left,axiom,
! [A: $tType,B: $tType] :
( ( comm_monoid_add @ A )
=> ! [T6: set @ B,S: set @ B,H2: B > A,G: B > A] :
( ( finite_finite2 @ B @ T6 )
=> ( ( ord_less_eq @ ( set @ B ) @ S @ T6 )
=> ( ! [X4: B] :
( ( member @ B @ X4 @ ( minus_minus @ ( set @ B ) @ T6 @ S ) )
=> ( ( H2 @ X4 )
= ( zero_zero @ A ) ) )
=> ( ! [X4: B] :
( ( member @ B @ X4 @ S )
=> ( ( G @ X4 )
= ( H2 @ X4 ) ) )
=> ( ( groups7311177749621191930dd_sum @ B @ A @ G @ S )
= ( groups7311177749621191930dd_sum @ B @ A @ H2 @ T6 ) ) ) ) ) ) ) ).
% sum.mono_neutral_cong_left
thf(fact_3213_sum_Omono__neutral__cong__right,axiom,
! [A: $tType,B: $tType] :
( ( comm_monoid_add @ A )
=> ! [T6: set @ B,S: set @ B,G: B > A,H2: B > A] :
( ( finite_finite2 @ B @ T6 )
=> ( ( ord_less_eq @ ( set @ B ) @ S @ T6 )
=> ( ! [X4: B] :
( ( member @ B @ X4 @ ( minus_minus @ ( set @ B ) @ T6 @ S ) )
=> ( ( G @ X4 )
= ( zero_zero @ A ) ) )
=> ( ! [X4: B] :
( ( member @ B @ X4 @ S )
=> ( ( G @ X4 )
= ( H2 @ X4 ) ) )
=> ( ( groups7311177749621191930dd_sum @ B @ A @ G @ T6 )
= ( groups7311177749621191930dd_sum @ B @ A @ H2 @ S ) ) ) ) ) ) ) ).
% sum.mono_neutral_cong_right
thf(fact_3214_sum_Osubset__diff,axiom,
! [A: $tType,B: $tType] :
( ( comm_monoid_add @ A )
=> ! [B2: set @ B,A2: set @ B,G: B > A] :
( ( ord_less_eq @ ( set @ B ) @ B2 @ A2 )
=> ( ( finite_finite2 @ B @ A2 )
=> ( ( groups7311177749621191930dd_sum @ B @ A @ G @ A2 )
= ( plus_plus @ A @ ( groups7311177749621191930dd_sum @ B @ A @ G @ ( minus_minus @ ( set @ B ) @ A2 @ B2 ) ) @ ( groups7311177749621191930dd_sum @ B @ A @ G @ B2 ) ) ) ) ) ) ).
% sum.subset_diff
thf(fact_3215_sum__diff,axiom,
! [A: $tType,B: $tType] :
( ( ab_group_add @ A )
=> ! [A2: set @ B,B2: set @ B,F2: B > A] :
( ( finite_finite2 @ B @ A2 )
=> ( ( ord_less_eq @ ( set @ B ) @ B2 @ A2 )
=> ( ( groups7311177749621191930dd_sum @ B @ A @ F2 @ ( minus_minus @ ( set @ B ) @ A2 @ B2 ) )
= ( minus_minus @ A @ ( groups7311177749621191930dd_sum @ B @ A @ F2 @ A2 ) @ ( groups7311177749621191930dd_sum @ B @ A @ F2 @ B2 ) ) ) ) ) ) ).
% sum_diff
thf(fact_3216_sum_Oivl__cong,axiom,
! [A: $tType,B: $tType] :
( ( ( ord @ B )
& ( comm_monoid_add @ A ) )
=> ! [A3: B,C3: B,B3: B,D2: B,G: B > A,H2: B > A] :
( ( A3 = C3 )
=> ( ( B3 = D2 )
=> ( ! [X4: B] :
( ( ord_less_eq @ B @ C3 @ X4 )
=> ( ( ord_less @ B @ X4 @ D2 )
=> ( ( G @ X4 )
= ( H2 @ X4 ) ) ) )
=> ( ( groups7311177749621191930dd_sum @ B @ A @ G @ ( set_or7035219750837199246ssThan @ B @ A3 @ B3 ) )
= ( groups7311177749621191930dd_sum @ B @ A @ H2 @ ( set_or7035219750837199246ssThan @ B @ C3 @ D2 ) ) ) ) ) ) ) ).
% sum.ivl_cong
thf(fact_3217_sum_OatLeastLessThan__concat,axiom,
! [A: $tType] :
( ( comm_monoid_add @ A )
=> ! [M: nat,N: nat,P6: nat,G: nat > A] :
( ( ord_less_eq @ nat @ M @ N )
=> ( ( ord_less_eq @ nat @ N @ P6 )
=> ( ( plus_plus @ A @ ( groups7311177749621191930dd_sum @ nat @ A @ G @ ( set_or7035219750837199246ssThan @ nat @ M @ N ) ) @ ( groups7311177749621191930dd_sum @ nat @ A @ G @ ( set_or7035219750837199246ssThan @ nat @ N @ P6 ) ) )
= ( groups7311177749621191930dd_sum @ nat @ A @ G @ ( set_or7035219750837199246ssThan @ nat @ M @ P6 ) ) ) ) ) ) ).
% sum.atLeastLessThan_concat
thf(fact_3218_sum__diff__nat__ivl,axiom,
! [A: $tType] :
( ( ab_group_add @ A )
=> ! [M: nat,N: nat,P6: nat,F2: nat > A] :
( ( ord_less_eq @ nat @ M @ N )
=> ( ( ord_less_eq @ nat @ N @ P6 )
=> ( ( minus_minus @ A @ ( groups7311177749621191930dd_sum @ nat @ A @ F2 @ ( set_or7035219750837199246ssThan @ nat @ M @ P6 ) ) @ ( groups7311177749621191930dd_sum @ nat @ A @ F2 @ ( set_or7035219750837199246ssThan @ nat @ M @ N ) ) )
= ( groups7311177749621191930dd_sum @ nat @ A @ F2 @ ( set_or7035219750837199246ssThan @ nat @ N @ P6 ) ) ) ) ) ) ).
% sum_diff_nat_ivl
thf(fact_3219_sum__power__add,axiom,
! [A: $tType] :
( ( ( monoid_mult @ A )
& ( comm_ring @ A ) )
=> ! [X2: A,M: nat,I6: set @ nat] :
( ( groups7311177749621191930dd_sum @ nat @ A
@ ^ [I4: nat] : ( power_power @ A @ X2 @ ( plus_plus @ nat @ M @ I4 ) )
@ I6 )
= ( times_times @ A @ ( power_power @ A @ X2 @ M ) @ ( groups7311177749621191930dd_sum @ nat @ A @ ( power_power @ A @ X2 ) @ I6 ) ) ) ) ).
% sum_power_add
thf(fact_3220_sum__mono2,axiom,
! [A: $tType,B: $tType] :
( ( ordere6911136660526730532id_add @ A )
=> ! [B2: set @ B,A2: set @ B,F2: B > A] :
( ( finite_finite2 @ B @ B2 )
=> ( ( ord_less_eq @ ( set @ B ) @ A2 @ B2 )
=> ( ! [B4: B] :
( ( member @ B @ B4 @ ( minus_minus @ ( set @ B ) @ B2 @ A2 ) )
=> ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( F2 @ B4 ) ) )
=> ( ord_less_eq @ A @ ( groups7311177749621191930dd_sum @ B @ A @ F2 @ A2 ) @ ( groups7311177749621191930dd_sum @ B @ A @ F2 @ B2 ) ) ) ) ) ) ).
% sum_mono2
thf(fact_3221_sum_OatLeastAtMost__rev,axiom,
! [A: $tType] :
( ( comm_monoid_add @ A )
=> ! [G: nat > A,N: nat,M: nat] :
( ( groups7311177749621191930dd_sum @ nat @ A @ G @ ( set_or1337092689740270186AtMost @ nat @ N @ M ) )
= ( groups7311177749621191930dd_sum @ nat @ A
@ ^ [I4: nat] : ( G @ ( minus_minus @ nat @ ( plus_plus @ nat @ M @ N ) @ I4 ) )
@ ( set_or1337092689740270186AtMost @ nat @ N @ M ) ) ) ) ).
% sum.atLeastAtMost_rev
thf(fact_3222_sum_Oinsert__remove,axiom,
! [A: $tType,B: $tType] :
( ( comm_monoid_add @ A )
=> ! [A2: set @ B,G: B > A,X2: B] :
( ( finite_finite2 @ B @ A2 )
=> ( ( groups7311177749621191930dd_sum @ B @ A @ G @ ( insert @ B @ X2 @ A2 ) )
= ( plus_plus @ A @ ( G @ X2 ) @ ( groups7311177749621191930dd_sum @ B @ A @ G @ ( minus_minus @ ( set @ B ) @ A2 @ ( insert @ B @ X2 @ ( bot_bot @ ( set @ B ) ) ) ) ) ) ) ) ) ).
% sum.insert_remove
thf(fact_3223_sum_Oremove,axiom,
! [A: $tType,B: $tType] :
( ( comm_monoid_add @ A )
=> ! [A2: set @ B,X2: B,G: B > A] :
( ( finite_finite2 @ B @ A2 )
=> ( ( member @ B @ X2 @ A2 )
=> ( ( groups7311177749621191930dd_sum @ B @ A @ G @ A2 )
= ( plus_plus @ A @ ( G @ X2 ) @ ( groups7311177749621191930dd_sum @ B @ A @ G @ ( minus_minus @ ( set @ B ) @ A2 @ ( insert @ B @ X2 @ ( bot_bot @ ( set @ B ) ) ) ) ) ) ) ) ) ) ).
% sum.remove
thf(fact_3224_sum__diff1,axiom,
! [A: $tType,B: $tType] :
( ( ab_group_add @ A )
=> ! [A2: set @ B,A3: B,F2: B > A] :
( ( finite_finite2 @ B @ A2 )
=> ( ( ( member @ B @ A3 @ A2 )
=> ( ( groups7311177749621191930dd_sum @ B @ A @ F2 @ ( minus_minus @ ( set @ B ) @ A2 @ ( insert @ B @ A3 @ ( bot_bot @ ( set @ B ) ) ) ) )
= ( minus_minus @ A @ ( groups7311177749621191930dd_sum @ B @ A @ F2 @ A2 ) @ ( F2 @ A3 ) ) ) )
& ( ~ ( member @ B @ A3 @ A2 )
=> ( ( groups7311177749621191930dd_sum @ B @ A @ F2 @ ( minus_minus @ ( set @ B ) @ A2 @ ( insert @ B @ A3 @ ( bot_bot @ ( set @ B ) ) ) ) )
= ( groups7311177749621191930dd_sum @ B @ A @ F2 @ A2 ) ) ) ) ) ) ).
% sum_diff1
thf(fact_3225_sum_Odelta__remove,axiom,
! [A: $tType,B: $tType] :
( ( comm_monoid_add @ A )
=> ! [S: set @ B,A3: B,B3: B > A,C3: B > A] :
( ( finite_finite2 @ B @ S )
=> ( ( ( member @ B @ A3 @ S )
=> ( ( groups7311177749621191930dd_sum @ B @ A
@ ^ [K3: B] : ( if @ A @ ( K3 = A3 ) @ ( B3 @ K3 ) @ ( C3 @ K3 ) )
@ S )
= ( plus_plus @ A @ ( B3 @ A3 ) @ ( groups7311177749621191930dd_sum @ B @ A @ C3 @ ( minus_minus @ ( set @ B ) @ S @ ( insert @ B @ A3 @ ( bot_bot @ ( set @ B ) ) ) ) ) ) ) )
& ( ~ ( member @ B @ A3 @ S )
=> ( ( groups7311177749621191930dd_sum @ B @ A
@ ^ [K3: B] : ( if @ A @ ( K3 = A3 ) @ ( B3 @ K3 ) @ ( C3 @ K3 ) )
@ S )
= ( groups7311177749621191930dd_sum @ B @ A @ C3 @ ( minus_minus @ ( set @ B ) @ S @ ( insert @ B @ A3 @ ( bot_bot @ ( set @ B ) ) ) ) ) ) ) ) ) ) ).
% sum.delta_remove
thf(fact_3226_sum__shift__lb__Suc0__0__upt,axiom,
! [A: $tType] :
( ( comm_monoid_add @ A )
=> ! [F2: nat > A,K: nat] :
( ( ( F2 @ ( zero_zero @ nat ) )
= ( zero_zero @ A ) )
=> ( ( groups7311177749621191930dd_sum @ nat @ A @ F2 @ ( set_or7035219750837199246ssThan @ nat @ ( suc @ ( zero_zero @ nat ) ) @ K ) )
= ( groups7311177749621191930dd_sum @ nat @ A @ F2 @ ( set_or7035219750837199246ssThan @ nat @ ( zero_zero @ nat ) @ K ) ) ) ) ) ).
% sum_shift_lb_Suc0_0_upt
thf(fact_3227_sum__shift__lb__Suc0__0,axiom,
! [A: $tType] :
( ( comm_monoid_add @ A )
=> ! [F2: nat > A,K: nat] :
( ( ( F2 @ ( zero_zero @ nat ) )
= ( zero_zero @ A ) )
=> ( ( groups7311177749621191930dd_sum @ nat @ A @ F2 @ ( set_or1337092689740270186AtMost @ nat @ ( suc @ ( zero_zero @ nat ) ) @ K ) )
= ( groups7311177749621191930dd_sum @ nat @ A @ F2 @ ( set_or1337092689740270186AtMost @ nat @ ( zero_zero @ nat ) @ K ) ) ) ) ) ).
% sum_shift_lb_Suc0_0
thf(fact_3228_sum_OatLeast0__lessThan__Suc,axiom,
! [A: $tType] :
( ( comm_monoid_add @ A )
=> ! [G: nat > A,N: nat] :
( ( groups7311177749621191930dd_sum @ nat @ A @ G @ ( set_or7035219750837199246ssThan @ nat @ ( zero_zero @ nat ) @ ( suc @ N ) ) )
= ( plus_plus @ A @ ( groups7311177749621191930dd_sum @ nat @ A @ G @ ( set_or7035219750837199246ssThan @ nat @ ( zero_zero @ nat ) @ N ) ) @ ( G @ N ) ) ) ) ).
% sum.atLeast0_lessThan_Suc
thf(fact_3229_sum_OatLeast__Suc__lessThan,axiom,
! [A: $tType] :
( ( comm_monoid_add @ A )
=> ! [M: nat,N: nat,G: nat > A] :
( ( ord_less @ nat @ M @ N )
=> ( ( groups7311177749621191930dd_sum @ nat @ A @ G @ ( set_or7035219750837199246ssThan @ nat @ M @ N ) )
= ( plus_plus @ A @ ( G @ M ) @ ( groups7311177749621191930dd_sum @ nat @ A @ G @ ( set_or7035219750837199246ssThan @ nat @ ( suc @ M ) @ N ) ) ) ) ) ) ).
% sum.atLeast_Suc_lessThan
thf(fact_3230_sum_OatLeast0__atMost__Suc,axiom,
! [A: $tType] :
( ( comm_monoid_add @ A )
=> ! [G: nat > A,N: nat] :
( ( groups7311177749621191930dd_sum @ nat @ A @ G @ ( set_or1337092689740270186AtMost @ nat @ ( zero_zero @ nat ) @ ( suc @ N ) ) )
= ( plus_plus @ A @ ( groups7311177749621191930dd_sum @ nat @ A @ G @ ( set_or1337092689740270186AtMost @ nat @ ( zero_zero @ nat ) @ N ) ) @ ( G @ ( suc @ N ) ) ) ) ) ).
% sum.atLeast0_atMost_Suc
thf(fact_3231_sum_OatLeastLessThan__Suc,axiom,
! [A: $tType] :
( ( comm_monoid_add @ A )
=> ! [A3: nat,B3: nat,G: nat > A] :
( ( ord_less_eq @ nat @ A3 @ B3 )
=> ( ( groups7311177749621191930dd_sum @ nat @ A @ G @ ( set_or7035219750837199246ssThan @ nat @ A3 @ ( suc @ B3 ) ) )
= ( plus_plus @ A @ ( groups7311177749621191930dd_sum @ nat @ A @ G @ ( set_or7035219750837199246ssThan @ nat @ A3 @ B3 ) ) @ ( G @ B3 ) ) ) ) ) ).
% sum.atLeastLessThan_Suc
thf(fact_3232_sum_OatLeast__Suc__atMost,axiom,
! [A: $tType] :
( ( comm_monoid_add @ A )
=> ! [M: nat,N: nat,G: nat > A] :
( ( ord_less_eq @ nat @ M @ N )
=> ( ( groups7311177749621191930dd_sum @ nat @ A @ G @ ( set_or1337092689740270186AtMost @ nat @ M @ N ) )
= ( plus_plus @ A @ ( G @ M ) @ ( groups7311177749621191930dd_sum @ nat @ A @ G @ ( set_or1337092689740270186AtMost @ nat @ ( suc @ M ) @ N ) ) ) ) ) ) ).
% sum.atLeast_Suc_atMost
thf(fact_3233_sum_Onat__ivl__Suc_H,axiom,
! [A: $tType] :
( ( comm_monoid_add @ A )
=> ! [M: nat,N: nat,G: nat > A] :
( ( ord_less_eq @ nat @ M @ ( suc @ N ) )
=> ( ( groups7311177749621191930dd_sum @ nat @ A @ G @ ( set_or1337092689740270186AtMost @ nat @ M @ ( suc @ N ) ) )
= ( plus_plus @ A @ ( G @ ( suc @ N ) ) @ ( groups7311177749621191930dd_sum @ nat @ A @ G @ ( set_or1337092689740270186AtMost @ nat @ M @ N ) ) ) ) ) ) ).
% sum.nat_ivl_Suc'
thf(fact_3234_sum_Olast__plus,axiom,
! [A: $tType] :
( ( comm_monoid_add @ A )
=> ! [M: nat,N: nat,G: nat > A] :
( ( ord_less_eq @ nat @ M @ N )
=> ( ( groups7311177749621191930dd_sum @ nat @ A @ G @ ( set_or1337092689740270186AtMost @ nat @ M @ N ) )
= ( plus_plus @ A @ ( G @ N ) @ ( groups7311177749621191930dd_sum @ nat @ A @ G @ ( set_or7035219750837199246ssThan @ nat @ M @ N ) ) ) ) ) ) ).
% sum.last_plus
thf(fact_3235_sum__strict__mono2,axiom,
! [B: $tType,A: $tType] :
( ( ordere8940638589300402666id_add @ B )
=> ! [B2: set @ A,A2: set @ A,B3: A,F2: A > B] :
( ( finite_finite2 @ A @ B2 )
=> ( ( ord_less_eq @ ( set @ A ) @ A2 @ B2 )
=> ( ( member @ A @ B3 @ ( minus_minus @ ( set @ A ) @ B2 @ A2 ) )
=> ( ( ord_less @ B @ ( zero_zero @ B ) @ ( F2 @ B3 ) )
=> ( ! [X4: A] :
( ( member @ A @ X4 @ B2 )
=> ( ord_less_eq @ B @ ( zero_zero @ B ) @ ( F2 @ X4 ) ) )
=> ( ord_less @ B @ ( groups7311177749621191930dd_sum @ A @ B @ F2 @ A2 ) @ ( groups7311177749621191930dd_sum @ A @ B @ F2 @ B2 ) ) ) ) ) ) ) ) ).
% sum_strict_mono2
thf(fact_3236_member__le__sum,axiom,
! [B: $tType,C: $tType] :
( ( ( ordere6911136660526730532id_add @ B )
& ( semiring_1 @ B ) )
=> ! [I: C,A2: set @ C,F2: C > B] :
( ( member @ C @ I @ A2 )
=> ( ! [X4: C] :
( ( member @ C @ X4 @ ( minus_minus @ ( set @ C ) @ A2 @ ( insert @ C @ I @ ( bot_bot @ ( set @ C ) ) ) ) )
=> ( ord_less_eq @ B @ ( zero_zero @ B ) @ ( F2 @ X4 ) ) )
=> ( ( finite_finite2 @ C @ A2 )
=> ( ord_less_eq @ B @ ( F2 @ I ) @ ( groups7311177749621191930dd_sum @ C @ B @ F2 @ A2 ) ) ) ) ) ) ).
% member_le_sum
thf(fact_3237_sum_OSuc__reindex__ivl,axiom,
! [A: $tType] :
( ( comm_monoid_add @ A )
=> ! [M: nat,N: nat,G: nat > A] :
( ( ord_less_eq @ nat @ M @ N )
=> ( ( plus_plus @ A @ ( groups7311177749621191930dd_sum @ nat @ A @ G @ ( set_or1337092689740270186AtMost @ nat @ M @ N ) ) @ ( G @ ( suc @ N ) ) )
= ( plus_plus @ A @ ( G @ M )
@ ( groups7311177749621191930dd_sum @ nat @ A
@ ^ [I4: nat] : ( G @ ( suc @ I4 ) )
@ ( set_or1337092689740270186AtMost @ nat @ M @ N ) ) ) ) ) ) ).
% sum.Suc_reindex_ivl
thf(fact_3238_sum__Suc__diff_H,axiom,
! [A: $tType] :
( ( ab_group_add @ A )
=> ! [M: nat,N: nat,F2: nat > A] :
( ( ord_less_eq @ nat @ M @ N )
=> ( ( groups7311177749621191930dd_sum @ nat @ A
@ ^ [I4: nat] : ( minus_minus @ A @ ( F2 @ ( suc @ I4 ) ) @ ( F2 @ I4 ) )
@ ( set_or7035219750837199246ssThan @ nat @ M @ N ) )
= ( minus_minus @ A @ ( F2 @ N ) @ ( F2 @ M ) ) ) ) ) ).
% sum_Suc_diff'
thf(fact_3239_sum__Suc__diff,axiom,
! [A: $tType] :
( ( ab_group_add @ A )
=> ! [M: nat,N: nat,F2: nat > A] :
( ( ord_less_eq @ nat @ M @ ( suc @ N ) )
=> ( ( groups7311177749621191930dd_sum @ nat @ A
@ ^ [I4: nat] : ( minus_minus @ A @ ( F2 @ ( suc @ I4 ) ) @ ( F2 @ I4 ) )
@ ( set_or1337092689740270186AtMost @ nat @ M @ N ) )
= ( minus_minus @ A @ ( F2 @ ( suc @ N ) ) @ ( F2 @ M ) ) ) ) ) ).
% sum_Suc_diff
thf(fact_3240_sum_OatLeastLessThan__rev,axiom,
! [A: $tType] :
( ( comm_monoid_add @ A )
=> ! [G: nat > A,N: nat,M: nat] :
( ( groups7311177749621191930dd_sum @ nat @ A @ G @ ( set_or7035219750837199246ssThan @ nat @ N @ M ) )
= ( groups7311177749621191930dd_sum @ nat @ A
@ ^ [I4: nat] : ( G @ ( minus_minus @ nat @ ( plus_plus @ nat @ M @ N ) @ ( suc @ I4 ) ) )
@ ( set_or7035219750837199246ssThan @ nat @ N @ M ) ) ) ) ).
% sum.atLeastLessThan_rev
thf(fact_3241_sum_Onested__swap,axiom,
! [A: $tType] :
( ( comm_monoid_add @ A )
=> ! [A3: nat > nat > A,N: nat] :
( ( groups7311177749621191930dd_sum @ nat @ A
@ ^ [I4: nat] : ( groups7311177749621191930dd_sum @ nat @ A @ ( A3 @ I4 ) @ ( set_or7035219750837199246ssThan @ nat @ ( zero_zero @ nat ) @ I4 ) )
@ ( set_or1337092689740270186AtMost @ nat @ ( zero_zero @ nat ) @ N ) )
= ( groups7311177749621191930dd_sum @ nat @ A
@ ^ [J3: nat] :
( groups7311177749621191930dd_sum @ nat @ A
@ ^ [I4: nat] : ( A3 @ I4 @ J3 )
@ ( set_or1337092689740270186AtMost @ nat @ ( suc @ J3 ) @ N ) )
@ ( set_or7035219750837199246ssThan @ nat @ ( zero_zero @ nat ) @ N ) ) ) ) ).
% sum.nested_swap
thf(fact_3242_sum_Oub__add__nat,axiom,
! [A: $tType] :
( ( comm_monoid_add @ A )
=> ! [M: nat,N: nat,G: nat > A,P6: nat] :
( ( ord_less_eq @ nat @ M @ ( plus_plus @ nat @ N @ ( one_one @ nat ) ) )
=> ( ( groups7311177749621191930dd_sum @ nat @ A @ G @ ( set_or1337092689740270186AtMost @ nat @ M @ ( plus_plus @ nat @ N @ P6 ) ) )
= ( plus_plus @ A @ ( groups7311177749621191930dd_sum @ nat @ A @ G @ ( set_or1337092689740270186AtMost @ nat @ M @ N ) ) @ ( groups7311177749621191930dd_sum @ nat @ A @ G @ ( set_or1337092689740270186AtMost @ nat @ ( plus_plus @ nat @ N @ ( one_one @ nat ) ) @ ( plus_plus @ nat @ N @ P6 ) ) ) ) ) ) ) ).
% sum.ub_add_nat
thf(fact_3243_sum_Ohead__if,axiom,
! [A: $tType] :
( ( comm_monoid_add @ A )
=> ! [N: nat,M: nat,G: nat > A] :
( ( ( ord_less @ nat @ N @ M )
=> ( ( groups7311177749621191930dd_sum @ nat @ A @ G @ ( set_or1337092689740270186AtMost @ nat @ M @ N ) )
= ( zero_zero @ A ) ) )
& ( ~ ( ord_less @ nat @ N @ M )
=> ( ( groups7311177749621191930dd_sum @ nat @ A @ G @ ( set_or1337092689740270186AtMost @ nat @ M @ N ) )
= ( plus_plus @ A @ ( groups7311177749621191930dd_sum @ nat @ A @ G @ ( set_or7035219750837199246ssThan @ nat @ M @ N ) ) @ ( G @ N ) ) ) ) ) ) ).
% sum.head_if
thf(fact_3244_set__encode__def,axiom,
( nat_set_encode
= ( groups7311177749621191930dd_sum @ nat @ nat @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ).
% set_encode_def
thf(fact_3245_sum_OatLeastLessThan__rev__at__least__Suc__atMost,axiom,
! [A: $tType] :
( ( comm_monoid_add @ A )
=> ! [G: nat > A,N: nat,M: nat] :
( ( groups7311177749621191930dd_sum @ nat @ A @ G @ ( set_or7035219750837199246ssThan @ nat @ N @ M ) )
= ( groups7311177749621191930dd_sum @ nat @ A
@ ^ [I4: nat] : ( G @ ( minus_minus @ nat @ ( plus_plus @ nat @ M @ N ) @ I4 ) )
@ ( set_or1337092689740270186AtMost @ nat @ ( suc @ N ) @ M ) ) ) ) ).
% sum.atLeastLessThan_rev_at_least_Suc_atMost
thf(fact_3246_sum__natinterval__diff,axiom,
! [A: $tType] :
( ( ab_group_add @ A )
=> ! [M: nat,N: nat,F2: nat > A] :
( ( ( ord_less_eq @ nat @ M @ N )
=> ( ( groups7311177749621191930dd_sum @ nat @ A
@ ^ [K3: nat] : ( minus_minus @ A @ ( F2 @ K3 ) @ ( F2 @ ( plus_plus @ nat @ K3 @ ( one_one @ nat ) ) ) )
@ ( set_or1337092689740270186AtMost @ nat @ M @ N ) )
= ( minus_minus @ A @ ( F2 @ M ) @ ( F2 @ ( plus_plus @ nat @ N @ ( one_one @ nat ) ) ) ) ) )
& ( ~ ( ord_less_eq @ nat @ M @ N )
=> ( ( groups7311177749621191930dd_sum @ nat @ A
@ ^ [K3: nat] : ( minus_minus @ A @ ( F2 @ K3 ) @ ( F2 @ ( plus_plus @ nat @ K3 @ ( one_one @ nat ) ) ) )
@ ( set_or1337092689740270186AtMost @ nat @ M @ N ) )
= ( zero_zero @ A ) ) ) ) ) ).
% sum_natinterval_diff
thf(fact_3247_sum__telescope_H_H,axiom,
! [A: $tType] :
( ( ab_group_add @ A )
=> ! [M: nat,N: nat,F2: nat > A] :
( ( ord_less_eq @ nat @ M @ N )
=> ( ( groups7311177749621191930dd_sum @ nat @ A
@ ^ [K3: nat] : ( minus_minus @ A @ ( F2 @ K3 ) @ ( F2 @ ( minus_minus @ nat @ K3 @ ( one_one @ nat ) ) ) )
@ ( set_or1337092689740270186AtMost @ nat @ ( suc @ M ) @ N ) )
= ( minus_minus @ A @ ( F2 @ N ) @ ( F2 @ M ) ) ) ) ) ).
% sum_telescope''
thf(fact_3248_mask__eq__sum__exp,axiom,
! [A: $tType] :
( ( semiring_parity @ A )
=> ! [N: nat] :
( ( minus_minus @ A @ ( power_power @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ N ) @ ( one_one @ A ) )
= ( groups7311177749621191930dd_sum @ nat @ A @ ( power_power @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) )
@ ( collect @ nat
@ ^ [Q5: nat] : ( ord_less @ nat @ Q5 @ N ) ) ) ) ) ).
% mask_eq_sum_exp
thf(fact_3249_sum_Oin__pairs,axiom,
! [A: $tType] :
( ( comm_monoid_add @ A )
=> ! [G: nat > A,M: nat,N: nat] :
( ( groups7311177749621191930dd_sum @ nat @ A @ G @ ( set_or1337092689740270186AtMost @ nat @ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ M ) @ ( suc @ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N ) ) ) )
= ( groups7311177749621191930dd_sum @ nat @ A
@ ^ [I4: nat] : ( plus_plus @ A @ ( G @ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ I4 ) ) @ ( G @ ( suc @ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ I4 ) ) ) )
@ ( set_or1337092689740270186AtMost @ nat @ M @ N ) ) ) ) ).
% sum.in_pairs
thf(fact_3250_sum__gp__multiplied,axiom,
! [A: $tType] :
( ( ( monoid_mult @ A )
& ( comm_ring @ A ) )
=> ! [M: nat,N: nat,X2: A] :
( ( ord_less_eq @ nat @ M @ N )
=> ( ( times_times @ A @ ( minus_minus @ A @ ( one_one @ A ) @ X2 ) @ ( groups7311177749621191930dd_sum @ nat @ A @ ( power_power @ A @ X2 ) @ ( set_or1337092689740270186AtMost @ nat @ M @ N ) ) )
= ( minus_minus @ A @ ( power_power @ A @ X2 @ M ) @ ( power_power @ A @ X2 @ ( suc @ N ) ) ) ) ) ) ).
% sum_gp_multiplied
thf(fact_3251_mask__eq__sum__exp__nat,axiom,
! [N: nat] :
( ( minus_minus @ nat @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N ) @ ( suc @ ( zero_zero @ nat ) ) )
= ( groups7311177749621191930dd_sum @ nat @ nat @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) )
@ ( collect @ nat
@ ^ [Q5: nat] : ( ord_less @ nat @ Q5 @ N ) ) ) ) ).
% mask_eq_sum_exp_nat
thf(fact_3252_gauss__sum__nat,axiom,
! [N: nat] :
( ( groups7311177749621191930dd_sum @ nat @ nat
@ ^ [X3: nat] : X3
@ ( 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_3253_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_3254_Sum__Ico__nat,axiom,
! [M: nat,N: nat] :
( ( groups7311177749621191930dd_sum @ nat @ nat
@ ^ [X3: nat] : X3
@ ( set_or7035219750837199246ssThan @ nat @ M @ N ) )
= ( divide_divide @ nat @ ( minus_minus @ nat @ ( times_times @ nat @ N @ ( minus_minus @ nat @ N @ ( one_one @ nat ) ) ) @ ( times_times @ nat @ M @ ( minus_minus @ nat @ M @ ( one_one @ nat ) ) ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ).
% Sum_Ico_nat
thf(fact_3255_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_3256_double__arith__series,axiom,
! [A: $tType] :
( ( comm_semiring_1 @ A )
=> ! [A3: A,D2: A,N: nat] :
( ( times_times @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) )
@ ( groups7311177749621191930dd_sum @ nat @ A
@ ^ [I4: nat] : ( plus_plus @ A @ A3 @ ( times_times @ A @ ( semiring_1_of_nat @ A @ I4 ) @ 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 ) ) @ A3 ) @ ( times_times @ A @ ( semiring_1_of_nat @ A @ N ) @ D2 ) ) ) ) ) ).
% double_arith_series
thf(fact_3257_arith__series__nat,axiom,
! [A3: nat,D2: nat,N: nat] :
( ( groups7311177749621191930dd_sum @ nat @ nat
@ ^ [I4: nat] : ( plus_plus @ nat @ A3 @ ( times_times @ nat @ I4 @ 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 ) ) @ A3 ) @ ( times_times @ nat @ N @ D2 ) ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ).
% arith_series_nat
thf(fact_3258_Sum__Icc__nat,axiom,
! [M: nat,N: nat] :
( ( groups7311177749621191930dd_sum @ nat @ nat
@ ^ [X3: nat] : X3
@ ( set_or1337092689740270186AtMost @ nat @ M @ N ) )
= ( divide_divide @ nat @ ( minus_minus @ nat @ ( times_times @ nat @ N @ ( plus_plus @ nat @ N @ ( one_one @ nat ) ) ) @ ( times_times @ nat @ M @ ( minus_minus @ nat @ M @ ( one_one @ nat ) ) ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ).
% Sum_Icc_nat
thf(fact_3259_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_3260_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_3261_arith__series,axiom,
! [A: $tType] :
( ( euclid5411537665997757685th_nat @ A )
=> ! [A3: A,D2: A,N: nat] :
( ( groups7311177749621191930dd_sum @ nat @ A
@ ^ [I4: nat] : ( plus_plus @ A @ A3 @ ( times_times @ A @ ( semiring_1_of_nat @ A @ I4 ) @ 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 ) ) @ A3 ) @ ( times_times @ A @ ( semiring_1_of_nat @ A @ N ) @ D2 ) ) ) @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) ) ) ).
% arith_series
thf(fact_3262_sum__gp__offset,axiom,
! [A: $tType] :
( ( ( division_ring @ A )
& ( comm_ring @ A ) )
=> ! [X2: A,M: nat,N: nat] :
( ( ( X2
= ( one_one @ A ) )
=> ( ( groups7311177749621191930dd_sum @ nat @ A @ ( power_power @ A @ X2 ) @ ( set_or1337092689740270186AtMost @ nat @ M @ ( plus_plus @ nat @ M @ N ) ) )
= ( plus_plus @ A @ ( semiring_1_of_nat @ A @ N ) @ ( one_one @ A ) ) ) )
& ( ( X2
!= ( one_one @ A ) )
=> ( ( groups7311177749621191930dd_sum @ nat @ A @ ( power_power @ A @ X2 ) @ ( set_or1337092689740270186AtMost @ nat @ M @ ( plus_plus @ nat @ M @ N ) ) )
= ( divide_divide @ A @ ( times_times @ A @ ( power_power @ A @ X2 @ M ) @ ( minus_minus @ A @ ( one_one @ A ) @ ( power_power @ A @ X2 @ ( suc @ N ) ) ) ) @ ( minus_minus @ A @ ( one_one @ A ) @ X2 ) ) ) ) ) ) ).
% sum_gp_offset
thf(fact_3263_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_3264_Chebyshev__sum__upper,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [N: nat,A3: nat > A,B3: nat > A] :
( ! [I2: nat,J2: nat] :
( ( ord_less_eq @ nat @ I2 @ J2 )
=> ( ( ord_less @ nat @ J2 @ N )
=> ( ord_less_eq @ A @ ( A3 @ I2 ) @ ( A3 @ J2 ) ) ) )
=> ( ! [I2: nat,J2: nat] :
( ( ord_less_eq @ nat @ I2 @ J2 )
=> ( ( ord_less @ nat @ J2 @ N )
=> ( ord_less_eq @ A @ ( B3 @ J2 ) @ ( B3 @ I2 ) ) ) )
=> ( ord_less_eq @ A
@ ( times_times @ A @ ( semiring_1_of_nat @ A @ N )
@ ( groups7311177749621191930dd_sum @ nat @ A
@ ^ [K3: nat] : ( times_times @ A @ ( A3 @ K3 ) @ ( B3 @ K3 ) )
@ ( set_or7035219750837199246ssThan @ nat @ ( zero_zero @ nat ) @ N ) ) )
@ ( times_times @ A @ ( groups7311177749621191930dd_sum @ nat @ A @ A3 @ ( set_or7035219750837199246ssThan @ nat @ ( zero_zero @ nat ) @ N ) ) @ ( groups7311177749621191930dd_sum @ nat @ A @ B3 @ ( set_or7035219750837199246ssThan @ nat @ ( zero_zero @ nat ) @ N ) ) ) ) ) ) ) ).
% Chebyshev_sum_upper
thf(fact_3265_Chebyshev__sum__upper__nat,axiom,
! [N: nat,A3: nat > nat,B3: nat > nat] :
( ! [I2: nat,J2: nat] :
( ( ord_less_eq @ nat @ I2 @ J2 )
=> ( ( ord_less @ nat @ J2 @ N )
=> ( ord_less_eq @ nat @ ( A3 @ I2 ) @ ( A3 @ J2 ) ) ) )
=> ( ! [I2: nat,J2: nat] :
( ( ord_less_eq @ nat @ I2 @ J2 )
=> ( ( ord_less @ nat @ J2 @ N )
=> ( ord_less_eq @ nat @ ( B3 @ J2 ) @ ( B3 @ I2 ) ) ) )
=> ( ord_less_eq @ nat
@ ( times_times @ nat @ N
@ ( groups7311177749621191930dd_sum @ nat @ nat
@ ^ [I4: nat] : ( times_times @ nat @ ( A3 @ I4 ) @ ( B3 @ I4 ) )
@ ( set_or7035219750837199246ssThan @ nat @ ( zero_zero @ nat ) @ N ) ) )
@ ( times_times @ nat @ ( groups7311177749621191930dd_sum @ nat @ nat @ A3 @ ( set_or7035219750837199246ssThan @ nat @ ( zero_zero @ nat ) @ N ) ) @ ( groups7311177749621191930dd_sum @ nat @ nat @ B3 @ ( set_or7035219750837199246ssThan @ nat @ ( zero_zero @ nat ) @ N ) ) ) ) ) ) ).
% Chebyshev_sum_upper_nat
thf(fact_3266_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
@ ^ [P4: nat] :
( groups7311177749621191930dd_sum @ nat @ A
@ ^ [Q5: nat] : ( times_times @ A @ ( power_power @ A @ ( plus_plus @ A @ Z @ H2 ) @ Q5 ) @ ( power_power @ A @ Z @ ( minus_minus @ nat @ ( minus_minus @ nat @ N @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ Q5 ) ) )
@ ( set_ord_lessThan @ nat @ ( minus_minus @ nat @ ( minus_minus @ nat @ N @ ( suc @ ( zero_zero @ nat ) ) ) @ P4 ) ) )
@ ( set_ord_lessThan @ nat @ ( minus_minus @ nat @ N @ ( suc @ ( zero_zero @ nat ) ) ) ) ) ) ) ) ) ).
% lemma_termdiff2
thf(fact_3267_length__rule,axiom,
! [A: $tType] :
( ( heap @ A )
=> ! [A3: array @ A,Xs: list @ A] :
( hoare_hoare_triple @ nat @ ( snga_assn @ A @ A3 @ Xs ) @ ( array_len @ A @ A3 )
@ ^ [R5: nat] :
( times_times @ assn @ ( snga_assn @ A @ A3 @ Xs )
@ ( pure_assn
@ ( R5
= ( size_size @ ( list @ A ) @ Xs ) ) ) ) ) ) ).
% length_rule
thf(fact_3268_dbl__inc__simps_I3_J,axiom,
! [A: $tType] :
( ( neg_numeral @ A )
=> ( ( neg_numeral_dbl_inc @ A @ ( one_one @ A ) )
= ( numeral_numeral @ A @ ( bit1 @ one2 ) ) ) ) ).
% dbl_inc_simps(3)
thf(fact_3269_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_3270_lessThan__subset__iff,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [X2: A,Y3: A] :
( ( ord_less_eq @ ( set @ A ) @ ( set_ord_lessThan @ A @ X2 ) @ ( set_ord_lessThan @ A @ Y3 ) )
= ( ord_less_eq @ A @ X2 @ Y3 ) ) ) ).
% lessThan_subset_iff
thf(fact_3271_lessThan__minus__lessThan,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [N: A,M: A] :
( ( minus_minus @ ( set @ A ) @ ( set_ord_lessThan @ A @ N ) @ ( set_ord_lessThan @ A @ M ) )
= ( set_or7035219750837199246ssThan @ A @ M @ N ) ) ) ).
% lessThan_minus_lessThan
thf(fact_3272_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_3273_dbl__inc__simps_I5_J,axiom,
! [A: $tType] :
( ( neg_numeral @ A )
=> ! [K: num] :
( ( neg_numeral_dbl_inc @ A @ ( numeral_numeral @ A @ K ) )
= ( numeral_numeral @ A @ ( bit1 @ K ) ) ) ) ).
% dbl_inc_simps(5)
thf(fact_3274_sum_OlessThan__Suc,axiom,
! [A: $tType] :
( ( comm_monoid_add @ A )
=> ! [G: nat > A,N: nat] :
( ( groups7311177749621191930dd_sum @ nat @ A @ G @ ( set_ord_lessThan @ nat @ ( suc @ N ) ) )
= ( plus_plus @ A @ ( groups7311177749621191930dd_sum @ nat @ A @ G @ ( set_ord_lessThan @ nat @ N ) ) @ ( G @ N ) ) ) ) ).
% sum.lessThan_Suc
thf(fact_3275_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_3276_sum__diff__distrib,axiom,
! [A: $tType] :
( ( ord @ A )
=> ! [Q: A > nat,P: A > nat,N: A] :
( ! [X4: A] : ( ord_less_eq @ nat @ ( Q @ X4 ) @ ( P @ X4 ) )
=> ( ( minus_minus @ nat @ ( groups7311177749621191930dd_sum @ A @ nat @ P @ ( set_ord_lessThan @ A @ N ) ) @ ( groups7311177749621191930dd_sum @ A @ nat @ Q @ ( set_ord_lessThan @ A @ N ) ) )
= ( groups7311177749621191930dd_sum @ A @ nat
@ ^ [X3: A] : ( minus_minus @ nat @ ( P @ X3 ) @ ( Q @ X3 ) )
@ ( set_ord_lessThan @ A @ N ) ) ) ) ) ).
% sum_diff_distrib
thf(fact_3277_int__sum,axiom,
! [B: $tType,F2: B > nat,A2: set @ B] :
( ( semiring_1_of_nat @ int @ ( groups7311177749621191930dd_sum @ B @ nat @ F2 @ A2 ) )
= ( groups7311177749621191930dd_sum @ B @ int
@ ^ [X3: B] : ( semiring_1_of_nat @ int @ ( F2 @ X3 ) )
@ A2 ) ) ).
% int_sum
thf(fact_3278_less__eq__multiset__def,axiom,
! [A: $tType] :
( ( preorder @ A )
=> ( ( ord_less_eq @ ( multiset @ A ) )
= ( ^ [M10: multiset @ A,N8: multiset @ A] :
( ( ord_less @ ( multiset @ A ) @ M10 @ N8 )
| ( M10 = N8 ) ) ) ) ) ).
% less_eq_multiset_def
thf(fact_3279_lessThan__def,axiom,
! [A: $tType] :
( ( ord @ A )
=> ( ( set_ord_lessThan @ A )
= ( ^ [U2: A] :
( collect @ A
@ ^ [X3: A] : ( ord_less @ A @ X3 @ U2 ) ) ) ) ) ).
% lessThan_def
thf(fact_3280_time__array__len,axiom,
! [A: $tType] :
( ( heap @ A )
=> ! [P6: array @ A,H2: heap_ext @ product_unit] :
( ( time_time @ nat @ ( array_len @ A @ P6 ) @ H2 )
= ( one_one @ nat ) ) ) ).
% time_array_len
thf(fact_3281_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_3282_lessThan__Suc,axiom,
! [K: nat] :
( ( set_ord_lessThan @ nat @ ( suc @ K ) )
= ( insert @ nat @ K @ ( set_ord_lessThan @ nat @ K ) ) ) ).
% lessThan_Suc
thf(fact_3283_sum__subtractf__nat,axiom,
! [A: $tType,A2: set @ A,G: A > nat,F2: A > nat] :
( ! [X4: A] :
( ( member @ A @ X4 @ A2 )
=> ( ord_less_eq @ nat @ ( G @ X4 ) @ ( F2 @ X4 ) ) )
=> ( ( groups7311177749621191930dd_sum @ A @ nat
@ ^ [X3: A] : ( minus_minus @ nat @ ( F2 @ X3 ) @ ( G @ X3 ) )
@ A2 )
= ( minus_minus @ nat @ ( groups7311177749621191930dd_sum @ A @ nat @ F2 @ A2 ) @ ( groups7311177749621191930dd_sum @ A @ nat @ G @ A2 ) ) ) ) ).
% sum_subtractf_nat
thf(fact_3284_sum__SucD,axiom,
! [A: $tType,F2: A > nat,A2: set @ A,N: nat] :
( ( ( groups7311177749621191930dd_sum @ A @ nat @ F2 @ A2 )
= ( suc @ N ) )
=> ? [X4: A] :
( ( member @ A @ X4 @ A2 )
& ( ord_less @ nat @ ( zero_zero @ nat ) @ ( F2 @ X4 ) ) ) ) ).
% sum_SucD
thf(fact_3285_sum__eq__Suc0__iff,axiom,
! [A: $tType,A2: set @ A,F2: A > nat] :
( ( finite_finite2 @ A @ A2 )
=> ( ( ( groups7311177749621191930dd_sum @ A @ nat @ F2 @ A2 )
= ( suc @ ( zero_zero @ nat ) ) )
= ( ? [X3: A] :
( ( member @ A @ X3 @ A2 )
& ( ( F2 @ X3 )
= ( suc @ ( zero_zero @ nat ) ) )
& ! [Y: A] :
( ( member @ A @ Y @ A2 )
=> ( ( X3 != Y )
=> ( ( F2 @ Y )
= ( zero_zero @ nat ) ) ) ) ) ) ) ) ).
% sum_eq_Suc0_iff
thf(fact_3286_sum__eq__1__iff,axiom,
! [A: $tType,A2: set @ A,F2: A > nat] :
( ( finite_finite2 @ A @ A2 )
=> ( ( ( groups7311177749621191930dd_sum @ A @ nat @ F2 @ A2 )
= ( one_one @ nat ) )
= ( ? [X3: A] :
( ( member @ A @ X3 @ A2 )
& ( ( F2 @ X3 )
= ( one_one @ nat ) )
& ! [Y: A] :
( ( member @ A @ Y @ A2 )
=> ( ( X3 != Y )
=> ( ( F2 @ Y )
= ( zero_zero @ nat ) ) ) ) ) ) ) ) ).
% sum_eq_1_iff
thf(fact_3287_sum_Onat__diff__reindex,axiom,
! [A: $tType] :
( ( comm_monoid_add @ A )
=> ! [G: nat > A,N: nat] :
( ( groups7311177749621191930dd_sum @ nat @ A
@ ^ [I4: nat] : ( G @ ( minus_minus @ nat @ N @ ( suc @ I4 ) ) )
@ ( set_ord_lessThan @ nat @ N ) )
= ( groups7311177749621191930dd_sum @ nat @ A @ G @ ( set_ord_lessThan @ nat @ N ) ) ) ) ).
% sum.nat_diff_reindex
thf(fact_3288_sum__nth__roots,axiom,
! [N: nat,C3: complex] :
( ( ord_less @ nat @ ( one_one @ nat ) @ N )
=> ( ( groups7311177749621191930dd_sum @ complex @ complex
@ ^ [X3: complex] : X3
@ ( collect @ complex
@ ^ [Z4: complex] :
( ( power_power @ complex @ Z4 @ N )
= C3 ) ) )
= ( zero_zero @ complex ) ) ) ).
% sum_nth_roots
thf(fact_3289_sum__roots__unity,axiom,
! [N: nat] :
( ( ord_less @ nat @ ( one_one @ nat ) @ N )
=> ( ( groups7311177749621191930dd_sum @ complex @ complex
@ ^ [X3: complex] : X3
@ ( collect @ complex
@ ^ [Z4: complex] :
( ( power_power @ complex @ Z4 @ N )
= ( one_one @ complex ) ) ) )
= ( zero_zero @ complex ) ) ) ).
% sum_roots_unity
thf(fact_3290_sum__diff__nat,axiom,
! [A: $tType,B2: set @ A,A2: set @ A,F2: A > nat] :
( ( finite_finite2 @ A @ B2 )
=> ( ( ord_less_eq @ ( set @ A ) @ B2 @ A2 )
=> ( ( groups7311177749621191930dd_sum @ A @ nat @ F2 @ ( minus_minus @ ( set @ A ) @ A2 @ B2 ) )
= ( minus_minus @ nat @ ( groups7311177749621191930dd_sum @ A @ nat @ F2 @ A2 ) @ ( groups7311177749621191930dd_sum @ A @ nat @ F2 @ B2 ) ) ) ) ) ).
% sum_diff_nat
thf(fact_3291_sum__diff1__nat,axiom,
! [A: $tType,A3: A,A2: set @ A,F2: A > nat] :
( ( ( member @ A @ A3 @ A2 )
=> ( ( groups7311177749621191930dd_sum @ A @ nat @ F2 @ ( minus_minus @ ( set @ A ) @ A2 @ ( insert @ A @ A3 @ ( bot_bot @ ( set @ A ) ) ) ) )
= ( minus_minus @ nat @ ( groups7311177749621191930dd_sum @ A @ nat @ F2 @ A2 ) @ ( F2 @ A3 ) ) ) )
& ( ~ ( member @ A @ A3 @ A2 )
=> ( ( groups7311177749621191930dd_sum @ A @ nat @ F2 @ ( minus_minus @ ( set @ A ) @ A2 @ ( insert @ A @ A3 @ ( bot_bot @ ( set @ A ) ) ) ) )
= ( groups7311177749621191930dd_sum @ A @ nat @ F2 @ A2 ) ) ) ) ).
% sum_diff1_nat
thf(fact_3292_dbl__inc__def,axiom,
! [A: $tType] :
( ( neg_numeral @ A )
=> ( ( neg_numeral_dbl_inc @ A )
= ( ^ [X3: A] : ( plus_plus @ A @ ( plus_plus @ A @ X3 @ X3 ) @ ( one_one @ A ) ) ) ) ) ).
% dbl_inc_def
thf(fact_3293_sum_OlessThan__Suc__shift,axiom,
! [A: $tType] :
( ( comm_monoid_add @ A )
=> ! [G: nat > A,N: nat] :
( ( groups7311177749621191930dd_sum @ nat @ A @ G @ ( set_ord_lessThan @ nat @ ( suc @ N ) ) )
= ( plus_plus @ A @ ( G @ ( zero_zero @ nat ) )
@ ( groups7311177749621191930dd_sum @ nat @ A
@ ^ [I4: nat] : ( G @ ( suc @ I4 ) )
@ ( set_ord_lessThan @ nat @ N ) ) ) ) ) ).
% sum.lessThan_Suc_shift
thf(fact_3294_sum__lessThan__telescope_H,axiom,
! [A: $tType] :
( ( ab_group_add @ A )
=> ! [F2: nat > A,M: nat] :
( ( groups7311177749621191930dd_sum @ nat @ A
@ ^ [N4: nat] : ( minus_minus @ A @ ( F2 @ N4 ) @ ( F2 @ ( suc @ N4 ) ) )
@ ( set_ord_lessThan @ nat @ M ) )
= ( minus_minus @ A @ ( F2 @ ( zero_zero @ nat ) ) @ ( F2 @ M ) ) ) ) ).
% sum_lessThan_telescope'
thf(fact_3295_sum__lessThan__telescope,axiom,
! [A: $tType] :
( ( ab_group_add @ A )
=> ! [F2: nat > A,M: nat] :
( ( groups7311177749621191930dd_sum @ nat @ A
@ ^ [N4: nat] : ( minus_minus @ A @ ( F2 @ ( suc @ N4 ) ) @ ( F2 @ N4 ) )
@ ( set_ord_lessThan @ nat @ M ) )
= ( minus_minus @ A @ ( F2 @ M ) @ ( F2 @ ( zero_zero @ nat ) ) ) ) ) ).
% sum_lessThan_telescope
thf(fact_3296_sumr__diff__mult__const2,axiom,
! [A: $tType] :
( ( ring_1 @ A )
=> ! [F2: nat > A,N: nat,R2: A] :
( ( minus_minus @ A @ ( groups7311177749621191930dd_sum @ nat @ A @ F2 @ ( set_ord_lessThan @ nat @ N ) ) @ ( times_times @ A @ ( semiring_1_of_nat @ A @ N ) @ R2 ) )
= ( groups7311177749621191930dd_sum @ nat @ A
@ ^ [I4: nat] : ( minus_minus @ A @ ( F2 @ I4 ) @ R2 )
@ ( set_ord_lessThan @ nat @ N ) ) ) ) ).
% sumr_diff_mult_const2
thf(fact_3297_sum_OatLeast1__atMost__eq,axiom,
! [A: $tType] :
( ( comm_monoid_add @ A )
=> ! [G: nat > A,N: nat] :
( ( groups7311177749621191930dd_sum @ nat @ A @ G @ ( set_or1337092689740270186AtMost @ nat @ ( suc @ ( zero_zero @ nat ) ) @ N ) )
= ( groups7311177749621191930dd_sum @ nat @ A
@ ^ [K3: nat] : ( G @ ( suc @ K3 ) )
@ ( set_ord_lessThan @ nat @ N ) ) ) ) ).
% sum.atLeast1_atMost_eq
thf(fact_3298_sum_Onat__group,axiom,
! [A: $tType] :
( ( comm_monoid_add @ A )
=> ! [G: nat > A,K: nat,N: nat] :
( ( groups7311177749621191930dd_sum @ nat @ A
@ ^ [M6: nat] : ( groups7311177749621191930dd_sum @ nat @ A @ G @ ( set_or7035219750837199246ssThan @ nat @ ( times_times @ nat @ M6 @ K ) @ ( plus_plus @ nat @ ( times_times @ nat @ M6 @ K ) @ K ) ) )
@ ( set_ord_lessThan @ nat @ N ) )
= ( groups7311177749621191930dd_sum @ nat @ A @ G @ ( set_ord_lessThan @ nat @ ( times_times @ nat @ N @ K ) ) ) ) ) ).
% sum.nat_group
thf(fact_3299_one__diff__power__eq,axiom,
! [A: $tType] :
( ( ( monoid_mult @ A )
& ( comm_ring @ A ) )
=> ! [X2: A,N: nat] :
( ( minus_minus @ A @ ( one_one @ A ) @ ( power_power @ A @ X2 @ N ) )
= ( times_times @ A @ ( minus_minus @ A @ ( one_one @ A ) @ X2 ) @ ( groups7311177749621191930dd_sum @ nat @ A @ ( power_power @ A @ X2 ) @ ( set_ord_lessThan @ nat @ N ) ) ) ) ) ).
% one_diff_power_eq
thf(fact_3300_power__diff__1__eq,axiom,
! [A: $tType] :
( ( ( monoid_mult @ A )
& ( comm_ring @ A ) )
=> ! [X2: A,N: nat] :
( ( minus_minus @ A @ ( power_power @ A @ X2 @ N ) @ ( one_one @ A ) )
= ( times_times @ A @ ( minus_minus @ A @ X2 @ ( one_one @ A ) ) @ ( groups7311177749621191930dd_sum @ nat @ A @ ( power_power @ A @ X2 ) @ ( set_ord_lessThan @ nat @ N ) ) ) ) ) ).
% power_diff_1_eq
thf(fact_3301_geometric__sum,axiom,
! [A: $tType] :
( ( field @ A )
=> ! [X2: A,N: nat] :
( ( X2
!= ( one_one @ A ) )
=> ( ( groups7311177749621191930dd_sum @ nat @ A @ ( power_power @ A @ X2 ) @ ( set_ord_lessThan @ nat @ N ) )
= ( divide_divide @ A @ ( minus_minus @ A @ ( power_power @ A @ X2 @ N ) @ ( one_one @ A ) ) @ ( minus_minus @ A @ X2 @ ( one_one @ A ) ) ) ) ) ) ).
% geometric_sum
thf(fact_3302_TBOUND__len,axiom,
! [A: $tType] :
( ( heap @ A )
=> ! [Xs: array @ A] : ( time_TBOUND @ nat @ ( array_len @ A @ Xs ) @ ( one_one @ nat ) ) ) ).
% TBOUND_len
thf(fact_3303_lemma__termdiff1,axiom,
! [A: $tType] :
( ( ( monoid_mult @ A )
& ( comm_ring @ A ) )
=> ! [Z: A,H2: A,M: nat] :
( ( groups7311177749621191930dd_sum @ nat @ A
@ ^ [P4: nat] : ( minus_minus @ A @ ( times_times @ A @ ( power_power @ A @ ( plus_plus @ A @ Z @ H2 ) @ ( minus_minus @ nat @ M @ P4 ) ) @ ( power_power @ A @ Z @ P4 ) ) @ ( power_power @ A @ Z @ M ) )
@ ( set_ord_lessThan @ nat @ M ) )
= ( groups7311177749621191930dd_sum @ nat @ A
@ ^ [P4: nat] : ( times_times @ A @ ( power_power @ A @ Z @ P4 ) @ ( minus_minus @ A @ ( power_power @ A @ ( plus_plus @ A @ Z @ H2 ) @ ( minus_minus @ nat @ M @ P4 ) ) @ ( power_power @ A @ Z @ ( minus_minus @ nat @ M @ P4 ) ) ) )
@ ( set_ord_lessThan @ nat @ M ) ) ) ) ).
% lemma_termdiff1
thf(fact_3304_sum__gp__strict,axiom,
! [A: $tType] :
( ( ( division_ring @ A )
& ( comm_ring @ A ) )
=> ! [X2: A,N: nat] :
( ( ( X2
= ( one_one @ A ) )
=> ( ( groups7311177749621191930dd_sum @ nat @ A @ ( power_power @ A @ X2 ) @ ( set_ord_lessThan @ nat @ N ) )
= ( semiring_1_of_nat @ A @ N ) ) )
& ( ( X2
!= ( one_one @ A ) )
=> ( ( groups7311177749621191930dd_sum @ nat @ A @ ( power_power @ A @ X2 ) @ ( set_ord_lessThan @ nat @ N ) )
= ( divide_divide @ A @ ( minus_minus @ A @ ( one_one @ A ) @ ( power_power @ A @ X2 @ N ) ) @ ( minus_minus @ A @ ( one_one @ A ) @ X2 ) ) ) ) ) ) ).
% sum_gp_strict
thf(fact_3305_power__diff__sumr2,axiom,
! [A: $tType] :
( ( ( monoid_mult @ A )
& ( comm_ring @ A ) )
=> ! [X2: A,N: nat,Y3: A] :
( ( minus_minus @ A @ ( power_power @ A @ X2 @ N ) @ ( power_power @ A @ Y3 @ N ) )
= ( times_times @ A @ ( minus_minus @ A @ X2 @ Y3 )
@ ( groups7311177749621191930dd_sum @ nat @ A
@ ^ [I4: nat] : ( times_times @ A @ ( power_power @ A @ Y3 @ ( minus_minus @ nat @ N @ ( suc @ I4 ) ) ) @ ( power_power @ A @ X2 @ I4 ) )
@ ( set_ord_lessThan @ nat @ N ) ) ) ) ) ).
% power_diff_sumr2
thf(fact_3306_diff__power__eq__sum,axiom,
! [A: $tType] :
( ( ( monoid_mult @ A )
& ( comm_ring @ A ) )
=> ! [X2: A,N: nat,Y3: A] :
( ( minus_minus @ A @ ( power_power @ A @ X2 @ ( suc @ N ) ) @ ( power_power @ A @ Y3 @ ( suc @ N ) ) )
= ( times_times @ A @ ( minus_minus @ A @ X2 @ Y3 )
@ ( groups7311177749621191930dd_sum @ nat @ A
@ ^ [P4: nat] : ( times_times @ A @ ( power_power @ A @ X2 @ P4 ) @ ( power_power @ A @ Y3 @ ( minus_minus @ nat @ N @ P4 ) ) )
@ ( set_ord_lessThan @ nat @ ( suc @ N ) ) ) ) ) ) ).
% diff_power_eq_sum
thf(fact_3307_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_3308_real__sum__nat__ivl__bounded2,axiom,
! [A: $tType] :
( ( linordered_semidom @ A )
=> ! [N: nat,F2: nat > A,K6: A,K: nat] :
( ! [P7: nat] :
( ( ord_less @ nat @ P7 @ N )
=> ( ord_less_eq @ A @ ( F2 @ P7 ) @ K6 ) )
=> ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ K6 )
=> ( ord_less_eq @ A @ ( groups7311177749621191930dd_sum @ nat @ A @ F2 @ ( set_ord_lessThan @ nat @ ( minus_minus @ nat @ N @ K ) ) ) @ ( times_times @ A @ ( semiring_1_of_nat @ A @ N ) @ K6 ) ) ) ) ) ).
% real_sum_nat_ivl_bounded2
thf(fact_3309_one__diff__power__eq_H,axiom,
! [A: $tType] :
( ( ( monoid_mult @ A )
& ( comm_ring @ A ) )
=> ! [X2: A,N: nat] :
( ( minus_minus @ A @ ( one_one @ A ) @ ( power_power @ A @ X2 @ N ) )
= ( times_times @ A @ ( minus_minus @ A @ ( one_one @ A ) @ X2 )
@ ( groups7311177749621191930dd_sum @ nat @ A
@ ^ [I4: nat] : ( power_power @ A @ X2 @ ( minus_minus @ nat @ N @ ( suc @ I4 ) ) )
@ ( set_ord_lessThan @ nat @ N ) ) ) ) ) ).
% one_diff_power_eq'
thf(fact_3310_sum__split__even__odd,axiom,
! [F2: nat > real,G: nat > real,N: nat] :
( ( groups7311177749621191930dd_sum @ nat @ real
@ ^ [I4: nat] : ( if @ real @ ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ I4 ) @ ( F2 @ I4 ) @ ( G @ I4 ) )
@ ( set_ord_lessThan @ nat @ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N ) ) )
= ( plus_plus @ real
@ ( groups7311177749621191930dd_sum @ nat @ real
@ ^ [I4: nat] : ( F2 @ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ I4 ) )
@ ( set_ord_lessThan @ nat @ N ) )
@ ( groups7311177749621191930dd_sum @ nat @ real
@ ^ [I4: nat] : ( G @ ( plus_plus @ nat @ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ I4 ) @ ( one_one @ nat ) ) )
@ ( set_ord_lessThan @ nat @ N ) ) ) ) ).
% sum_split_even_odd
thf(fact_3311_Sum__Icc__int,axiom,
! [M: int,N: int] :
( ( ord_less_eq @ int @ M @ N )
=> ( ( groups7311177749621191930dd_sum @ int @ int
@ ^ [X3: int] : X3
@ ( 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_3312_sum__bounds__lt__plus1,axiom,
! [A: $tType] :
( ( comm_monoid_add @ A )
=> ! [F2: nat > A,Mm: nat] :
( ( groups7311177749621191930dd_sum @ nat @ A
@ ^ [K3: nat] : ( F2 @ ( suc @ K3 ) )
@ ( set_ord_lessThan @ nat @ Mm ) )
= ( groups7311177749621191930dd_sum @ nat @ A @ F2 @ ( set_or1337092689740270186AtMost @ nat @ ( one_one @ nat ) @ Mm ) ) ) ) ).
% sum_bounds_lt_plus1
thf(fact_3313_length__corresp,axiom,
! [B: $tType,A: $tType] :
( ( heap @ A )
=> ! [Tree_array2: array @ A,Tree_is: list @ B] :
( ( ( ex_assn @ ( list @ A ) @ ( snga_assn @ A @ Tree_array2 ) )
= ( top_top @ assn ) )
=> ( ( heap_Time_return @ nat @ ( size_size @ ( list @ B ) @ Tree_is ) )
= ( array_len @ A @ Tree_array2 ) ) ) ) ).
% length_corresp
thf(fact_3314_sumr__cos__zero__one,axiom,
! [N: nat] :
( ( groups7311177749621191930dd_sum @ nat @ real
@ ^ [M6: nat] : ( times_times @ real @ ( cos_coeff @ M6 ) @ ( power_power @ real @ ( zero_zero @ real ) @ M6 ) )
@ ( set_ord_lessThan @ nat @ ( suc @ N ) ) )
= ( one_one @ real ) ) ).
% sumr_cos_zero_one
thf(fact_3315_finite__nat__bounded,axiom,
! [S: set @ nat] :
( ( finite_finite2 @ nat @ S )
=> ? [K2: nat] : ( ord_less_eq @ ( set @ nat ) @ S @ ( set_ord_lessThan @ nat @ K2 ) ) ) ).
% finite_nat_bounded
thf(fact_3316_finite__nat__iff__bounded,axiom,
( ( finite_finite2 @ nat )
= ( ^ [S8: set @ nat] :
? [K3: nat] : ( ord_less_eq @ ( set @ nat ) @ S8 @ ( set_ord_lessThan @ nat @ K3 ) ) ) ) ).
% finite_nat_iff_bounded
thf(fact_3317_max__top,axiom,
! [A: $tType] :
( ( order_top @ A )
=> ! [X2: A] :
( ( ord_max @ A @ ( top_top @ A ) @ X2 )
= ( top_top @ A ) ) ) ).
% max_top
thf(fact_3318_max__top2,axiom,
! [A: $tType] :
( ( order_top @ A )
=> ! [X2: A] :
( ( ord_max @ A @ X2 @ ( top_top @ A ) )
= ( top_top @ A ) ) ) ).
% max_top2
thf(fact_3319_merge__true__star,axiom,
( ( times_times @ assn @ ( top_top @ assn ) @ ( top_top @ assn ) )
= ( top_top @ assn ) ) ).
% merge_true_star
thf(fact_3320_assn__basic__inequalities_I1_J,axiom,
( ( top_top @ assn )
!= ( one_one @ assn ) ) ).
% assn_basic_inequalities(1)
thf(fact_3321_cos__coeff__0,axiom,
( ( cos_coeff @ ( zero_zero @ nat ) )
= ( one_one @ real ) ) ).
% cos_coeff_0
thf(fact_3322_top_Oextremum__strict,axiom,
! [A: $tType] :
( ( order_top @ A )
=> ! [A3: A] :
~ ( ord_less @ A @ ( top_top @ A ) @ A3 ) ) ).
% top.extremum_strict
thf(fact_3323_top_Onot__eq__extremum,axiom,
! [A: $tType] :
( ( order_top @ A )
=> ! [A3: A] :
( ( A3
!= ( top_top @ A ) )
= ( ord_less @ A @ A3 @ ( top_top @ A ) ) ) ) ).
% top.not_eq_extremum
thf(fact_3324_merge__true__star__ctx,axiom,
! [P: assn] :
( ( times_times @ assn @ ( top_top @ assn ) @ ( times_times @ assn @ ( top_top @ assn ) @ P ) )
= ( times_times @ assn @ ( top_top @ assn ) @ P ) ) ).
% merge_true_star_ctx
thf(fact_3325_top__option__def,axiom,
! [A: $tType] :
( ( order_top @ A )
=> ( ( top_top @ ( option @ A ) )
= ( some @ A @ ( top_top @ A ) ) ) ) ).
% top_option_def
thf(fact_3326_top__greatest,axiom,
! [A: $tType] :
( ( order_top @ A )
=> ! [A3: A] : ( ord_less_eq @ A @ A3 @ ( top_top @ A ) ) ) ).
% top_greatest
thf(fact_3327_top_Oextremum__unique,axiom,
! [A: $tType] :
( ( order_top @ A )
=> ! [A3: A] :
( ( ord_less_eq @ A @ ( top_top @ A ) @ A3 )
= ( A3
= ( top_top @ A ) ) ) ) ).
% top.extremum_unique
thf(fact_3328_top_Oextremum__uniqueI,axiom,
! [A: $tType] :
( ( order_top @ A )
=> ! [A3: A] :
( ( ord_less_eq @ A @ ( top_top @ A ) @ A3 )
=> ( A3
= ( top_top @ A ) ) ) ) ).
% top.extremum_uniqueI
thf(fact_3329_ent__true,axiom,
! [P: assn] : ( entails @ P @ ( top_top @ assn ) ) ).
% ent_true
thf(fact_3330_ent__true__drop_I2_J,axiom,
! [P: assn,Q: assn] :
( ( entails @ P @ Q )
=> ( entails @ P @ ( times_times @ assn @ Q @ ( top_top @ assn ) ) ) ) ).
% ent_true_drop(2)
thf(fact_3331_ent__true__drop_I1_J,axiom,
! [P: assn,Q: assn,R: assn] :
( ( entails @ P @ ( times_times @ assn @ Q @ ( top_top @ assn ) ) )
=> ( entails @ ( times_times @ assn @ P @ R ) @ ( times_times @ assn @ Q @ ( top_top @ assn ) ) ) ) ).
% ent_true_drop(1)
thf(fact_3332_ent__refl__true,axiom,
! [A2: assn] : ( entails @ A2 @ ( times_times @ assn @ A2 @ ( top_top @ assn ) ) ) ).
% ent_refl_true
thf(fact_3333_ent__star__mono__true,axiom,
! [A2: assn,A8: assn,B2: assn,B9: assn] :
( ( entails @ A2 @ ( times_times @ assn @ A8 @ ( top_top @ assn ) ) )
=> ( ( entails @ B2 @ ( times_times @ assn @ B9 @ ( top_top @ assn ) ) )
=> ( entails @ ( times_times @ assn @ ( times_times @ assn @ A2 @ B2 ) @ ( top_top @ assn ) ) @ ( times_times @ assn @ ( times_times @ assn @ A8 @ B9 ) @ ( top_top @ assn ) ) ) ) ) ).
% ent_star_mono_true
thf(fact_3334_mod__star__trueE,axiom,
! [P: assn,H2: product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat )] :
( ( rep_assn @ ( times_times @ assn @ P @ ( top_top @ assn ) ) @ H2 )
=> ~ ! [H5: product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat )] :
~ ( rep_assn @ P @ H5 ) ) ).
% mod_star_trueE
thf(fact_3335_mod__star__trueI,axiom,
! [P: assn,H2: product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat )] :
( ( rep_assn @ P @ H2 )
=> ( rep_assn @ ( times_times @ assn @ P @ ( top_top @ assn ) ) @ H2 ) ) ).
% mod_star_trueI
thf(fact_3336_unbounded__k__infinite,axiom,
! [K: nat,S: set @ nat] :
( ! [M5: nat] :
( ( ord_less @ nat @ K @ M5 )
=> ? [N9: nat] :
( ( ord_less @ nat @ M5 @ N9 )
& ( member @ nat @ N9 @ S ) ) )
=> ~ ( finite_finite2 @ nat @ S ) ) ).
% unbounded_k_infinite
thf(fact_3337_infinite__nat__iff__unbounded,axiom,
! [S: set @ nat] :
( ( ~ ( finite_finite2 @ nat @ S ) )
= ( ! [M6: nat] :
? [N4: nat] :
( ( ord_less @ nat @ M6 @ N4 )
& ( member @ nat @ N4 @ S ) ) ) ) ).
% infinite_nat_iff_unbounded
thf(fact_3338_infinite__nat__iff__unbounded__le,axiom,
! [S: set @ nat] :
( ( ~ ( finite_finite2 @ nat @ S ) )
= ( ! [M6: nat] :
? [N4: nat] :
( ( ord_less_eq @ nat @ M6 @ N4 )
& ( member @ nat @ N4 @ S ) ) ) ) ).
% infinite_nat_iff_unbounded_le
thf(fact_3339_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
@ ^ [N4: nat] : ( times_times @ A @ ( semiring_1_of_nat @ A @ ( suc @ N4 ) ) @ ( power_power @ A @ Z @ N4 ) )
@ ( 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_3340_vebt__assn__raw_Opelims,axiom,
! [X2: vEBT_VEBT,Xa: vEBT_VEBTi,Y3: assn] :
( ( ( vEBT_vebt_assn_raw @ X2 @ Xa )
= Y3 )
=> ( ( accp @ ( product_prod @ vEBT_VEBT @ vEBT_VEBTi ) @ vEBT_v8524038756793281170aw_rel @ ( product_Pair @ vEBT_VEBT @ vEBT_VEBTi @ X2 @ Xa ) )
=> ( ! [A4: $o,B4: $o] :
( ( X2
= ( vEBT_Leaf @ A4 @ B4 ) )
=> ! [Ai2: $o,Bi2: $o] :
( ( Xa
= ( vEBT_Leafi @ Ai2 @ Bi2 ) )
=> ( ( Y3
= ( pure_assn
@ ( ( Ai2 = A4 )
& ( Bi2 = B4 ) ) ) )
=> ~ ( accp @ ( product_prod @ vEBT_VEBT @ vEBT_VEBTi ) @ vEBT_v8524038756793281170aw_rel @ ( product_Pair @ vEBT_VEBT @ vEBT_VEBTi @ ( vEBT_Leaf @ A4 @ B4 ) @ ( vEBT_Leafi @ Ai2 @ Bi2 ) ) ) ) ) )
=> ( ! [Mmo: option @ ( product_prod @ nat @ nat ),Deg2: nat,Tree_list: list @ vEBT_VEBT,Summary2: vEBT_VEBT] :
( ( X2
= ( vEBT_Node @ Mmo @ Deg2 @ Tree_list @ Summary2 ) )
=> ! [Mmoi: option @ ( product_prod @ nat @ nat ),Degi: nat,Tree_array: array @ vEBT_VEBTi,Summaryi: vEBT_VEBTi] :
( ( Xa
= ( vEBT_Nodei @ Mmoi @ Degi @ Tree_array @ Summaryi ) )
=> ( ( Y3
= ( times_times @ assn
@ ( times_times @ assn
@ ( pure_assn
@ ( ( Mmoi = Mmo )
& ( Degi = Deg2 ) ) )
@ ( vEBT_vebt_assn_raw @ Summary2 @ 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 ) ) ) ) )
=> ~ ( accp @ ( product_prod @ vEBT_VEBT @ vEBT_VEBTi ) @ vEBT_v8524038756793281170aw_rel @ ( product_Pair @ vEBT_VEBT @ vEBT_VEBTi @ ( vEBT_Node @ Mmo @ Deg2 @ Tree_list @ Summary2 ) @ ( vEBT_Nodei @ Mmoi @ Degi @ Tree_array @ Summaryi ) ) ) ) ) )
=> ( ! [V3: option @ ( product_prod @ nat @ nat ),Va3: nat,Vb3: list @ vEBT_VEBT,Vc3: vEBT_VEBT] :
( ( X2
= ( vEBT_Node @ V3 @ Va3 @ Vb3 @ Vc3 ) )
=> ! [Vd: $o,Ve: $o] :
( ( Xa
= ( vEBT_Leafi @ Vd @ Ve ) )
=> ( ( Y3
= ( bot_bot @ assn ) )
=> ~ ( accp @ ( product_prod @ vEBT_VEBT @ vEBT_VEBTi ) @ vEBT_v8524038756793281170aw_rel @ ( product_Pair @ vEBT_VEBT @ vEBT_VEBTi @ ( vEBT_Node @ V3 @ Va3 @ Vb3 @ Vc3 ) @ ( vEBT_Leafi @ Vd @ Ve ) ) ) ) ) )
=> ~ ! [Vd: $o,Ve: $o] :
( ( X2
= ( vEBT_Leaf @ Vd @ Ve ) )
=> ! [V3: option @ ( product_prod @ nat @ nat ),Va3: nat,Vb3: array @ vEBT_VEBTi,Vc3: vEBT_VEBTi] :
( ( Xa
= ( vEBT_Nodei @ V3 @ Va3 @ Vb3 @ Vc3 ) )
=> ( ( Y3
= ( bot_bot @ assn ) )
=> ~ ( accp @ ( product_prod @ vEBT_VEBT @ vEBT_VEBTi ) @ vEBT_v8524038756793281170aw_rel @ ( product_Pair @ vEBT_VEBT @ vEBT_VEBTi @ ( vEBT_Leaf @ Vd @ Ve ) @ ( vEBT_Nodei @ V3 @ Va3 @ Vb3 @ Vc3 ) ) ) ) ) ) ) ) ) ) ) ).
% vebt_assn_raw.pelims
thf(fact_3341_VEBT__internal_Oheight_Osimps_I1_J,axiom,
! [A3: $o,B3: $o] :
( ( vEBT_VEBT_height @ ( vEBT_Leaf @ A3 @ B3 ) )
= ( zero_zero @ nat ) ) ).
% VEBT_internal.height.simps(1)
thf(fact_3342_pochhammer__double,axiom,
! [A: $tType] :
( ( field_char_0 @ A )
=> ! [Z: A,N: nat] :
( ( comm_s3205402744901411588hammer @ A @ ( times_times @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ Z ) @ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N ) )
= ( times_times @ A @ ( times_times @ A @ ( semiring_1_of_nat @ A @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N ) ) ) @ ( comm_s3205402744901411588hammer @ A @ Z @ N ) ) @ ( comm_s3205402744901411588hammer @ A @ ( plus_plus @ A @ Z @ ( divide_divide @ A @ ( one_one @ A ) @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) ) @ N ) ) ) ) ).
% pochhammer_double
thf(fact_3343_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_3344_Collect__const,axiom,
! [A: $tType,P: $o] :
( ( P
=> ( ( collect @ A
@ ^ [S5: A] : P )
= ( top_top @ ( set @ A ) ) ) )
& ( ~ P
=> ( ( collect @ A
@ ^ [S5: A] : P )
= ( bot_bot @ ( set @ A ) ) ) ) ) ).
% Collect_const
thf(fact_3345_finite__Collect__not,axiom,
! [A: $tType,P: A > $o] :
( ( finite_finite2 @ A @ ( collect @ A @ P ) )
=> ( ( finite_finite2 @ A
@ ( collect @ A
@ ^ [X3: A] :
~ ( P @ X3 ) ) )
= ( finite_finite2 @ A @ ( top_top @ ( set @ A ) ) ) ) ) ).
% finite_Collect_not
thf(fact_3346_Diff__UNIV,axiom,
! [A: $tType,A2: set @ A] :
( ( minus_minus @ ( set @ A ) @ A2 @ ( top_top @ ( set @ A ) ) )
= ( bot_bot @ ( set @ A ) ) ) ).
% Diff_UNIV
thf(fact_3347_pochhammer__1,axiom,
! [A: $tType] :
( ( comm_semiring_1 @ A )
=> ! [A3: A] :
( ( comm_s3205402744901411588hammer @ A @ A3 @ ( one_one @ nat ) )
= A3 ) ) ).
% pochhammer_1
thf(fact_3348_pochhammer__0,axiom,
! [A: $tType] :
( ( comm_semiring_1 @ A )
=> ! [A3: A] :
( ( comm_s3205402744901411588hammer @ A @ A3 @ ( zero_zero @ nat ) )
= ( one_one @ A ) ) ) ).
% pochhammer_0
thf(fact_3349_pochhammer__Suc0,axiom,
! [A: $tType] :
( ( comm_semiring_1 @ A )
=> ! [A3: A] :
( ( comm_s3205402744901411588hammer @ A @ A3 @ ( suc @ ( zero_zero @ nat ) ) )
= A3 ) ) ).
% pochhammer_Suc0
thf(fact_3350_powser__sums__zero__iff,axiom,
! [A: $tType] :
( ( real_V8999393235501362500lgebra @ A )
=> ! [A3: nat > A,X2: A] :
( ( sums @ A
@ ^ [N4: nat] : ( times_times @ A @ ( A3 @ N4 ) @ ( power_power @ A @ ( zero_zero @ A ) @ N4 ) )
@ X2 )
= ( ( A3 @ ( zero_zero @ nat ) )
= X2 ) ) ) ).
% powser_sums_zero_iff
thf(fact_3351_subset__UNIV,axiom,
! [A: $tType,A2: set @ A] : ( ord_less_eq @ ( set @ A ) @ A2 @ ( top_top @ ( set @ A ) ) ) ).
% subset_UNIV
thf(fact_3352_infinite__UNIV__listI,axiom,
! [A: $tType] :
~ ( finite_finite2 @ ( list @ A ) @ ( top_top @ ( set @ ( list @ A ) ) ) ) ).
% infinite_UNIV_listI
thf(fact_3353_pochhammer__pos,axiom,
! [A: $tType] :
( ( linordered_semidom @ A )
=> ! [X2: A,N: nat] :
( ( ord_less @ A @ ( zero_zero @ A ) @ X2 )
=> ( ord_less @ A @ ( zero_zero @ A ) @ ( comm_s3205402744901411588hammer @ A @ X2 @ N ) ) ) ) ).
% pochhammer_pos
thf(fact_3354_pochhammer__eq__0__mono,axiom,
! [A: $tType] :
( ( field_char_0 @ A )
=> ! [A3: A,N: nat,M: nat] :
( ( ( comm_s3205402744901411588hammer @ A @ A3 @ N )
= ( zero_zero @ A ) )
=> ( ( ord_less_eq @ nat @ N @ M )
=> ( ( comm_s3205402744901411588hammer @ A @ A3 @ M )
= ( zero_zero @ A ) ) ) ) ) ).
% pochhammer_eq_0_mono
thf(fact_3355_pochhammer__neq__0__mono,axiom,
! [A: $tType] :
( ( field_char_0 @ A )
=> ! [A3: A,M: nat,N: nat] :
( ( ( comm_s3205402744901411588hammer @ A @ A3 @ M )
!= ( zero_zero @ A ) )
=> ( ( ord_less_eq @ nat @ N @ M )
=> ( ( comm_s3205402744901411588hammer @ A @ A3 @ N )
!= ( zero_zero @ A ) ) ) ) ) ).
% pochhammer_neq_0_mono
thf(fact_3356_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_3357_pochhammer__nonneg,axiom,
! [A: $tType] :
( ( linordered_semidom @ A )
=> ! [X2: A,N: nat] :
( ( ord_less @ A @ ( zero_zero @ A ) @ X2 )
=> ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( comm_s3205402744901411588hammer @ A @ X2 @ N ) ) ) ) ).
% pochhammer_nonneg
thf(fact_3358_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_3359_powser__sums__zero,axiom,
! [A: $tType] :
( ( real_V8999393235501362500lgebra @ A )
=> ! [A3: nat > A] :
( sums @ A
@ ^ [N4: nat] : ( times_times @ A @ ( A3 @ N4 ) @ ( power_power @ A @ ( zero_zero @ A ) @ N4 ) )
@ ( A3 @ ( zero_zero @ nat ) ) ) ) ).
% powser_sums_zero
thf(fact_3360_pochhammer__rec,axiom,
! [A: $tType] :
( ( comm_semiring_1 @ A )
=> ! [A3: A,N: nat] :
( ( comm_s3205402744901411588hammer @ A @ A3 @ ( suc @ N ) )
= ( times_times @ A @ A3 @ ( comm_s3205402744901411588hammer @ A @ ( plus_plus @ A @ A3 @ ( one_one @ A ) ) @ N ) ) ) ) ).
% pochhammer_rec
thf(fact_3361_pochhammer__Suc,axiom,
! [A: $tType] :
( ( comm_semiring_1 @ A )
=> ! [A3: A,N: nat] :
( ( comm_s3205402744901411588hammer @ A @ A3 @ ( suc @ N ) )
= ( times_times @ A @ ( comm_s3205402744901411588hammer @ A @ A3 @ N ) @ ( plus_plus @ A @ A3 @ ( semiring_1_of_nat @ A @ N ) ) ) ) ) ).
% pochhammer_Suc
thf(fact_3362_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_3363_pochhammer__product_H,axiom,
! [A: $tType] :
( ( comm_semiring_1 @ A )
=> ! [Z: A,N: nat,M: nat] :
( ( comm_s3205402744901411588hammer @ A @ Z @ ( plus_plus @ nat @ N @ M ) )
= ( times_times @ A @ ( comm_s3205402744901411588hammer @ A @ Z @ N ) @ ( comm_s3205402744901411588hammer @ A @ ( plus_plus @ A @ Z @ ( semiring_1_of_nat @ A @ N ) ) @ M ) ) ) ) ).
% pochhammer_product'
thf(fact_3364_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_3365_VEBT__internal_Oheight_Ocases,axiom,
! [X2: vEBT_VEBT] :
( ! [A4: $o,B4: $o] :
( X2
!= ( vEBT_Leaf @ A4 @ B4 ) )
=> ~ ! [Uu2: option @ ( product_prod @ nat @ nat ),Deg2: nat,TreeList2: list @ vEBT_VEBT,Summary2: vEBT_VEBT] :
( X2
!= ( vEBT_Node @ Uu2 @ Deg2 @ TreeList2 @ Summary2 ) ) ) ).
% VEBT_internal.height.cases
thf(fact_3366_sums__if_H,axiom,
! [G: nat > real,X2: real] :
( ( sums @ real @ G @ X2 )
=> ( sums @ real
@ ^ [N4: nat] : ( if @ real @ ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N4 ) @ ( zero_zero @ real ) @ ( G @ ( divide_divide @ nat @ ( minus_minus @ nat @ N4 @ ( one_one @ nat ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) )
@ X2 ) ) ).
% sums_if'
thf(fact_3367_sums__if,axiom,
! [G: nat > real,X2: real,F2: nat > real,Y3: real] :
( ( sums @ real @ G @ X2 )
=> ( ( sums @ real @ F2 @ Y3 )
=> ( sums @ real
@ ^ [N4: nat] : ( if @ real @ ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N4 ) @ ( F2 @ ( divide_divide @ nat @ N4 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( G @ ( divide_divide @ nat @ ( minus_minus @ nat @ N4 @ ( one_one @ nat ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) )
@ ( plus_plus @ real @ X2 @ Y3 ) ) ) ) ).
% sums_if
thf(fact_3368_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_3369_sums__group,axiom,
! [A: $tType] :
( ( ( comm_monoid_add @ A )
& ( topolo4958980785337419405_space @ A ) )
=> ! [F2: nat > A,S2: A,K: nat] :
( ( sums @ A @ F2 @ S2 )
=> ( ( ord_less @ nat @ ( zero_zero @ nat ) @ K )
=> ( sums @ A
@ ^ [N4: nat] : ( groups7311177749621191930dd_sum @ nat @ A @ F2 @ ( set_or7035219750837199246ssThan @ nat @ ( times_times @ nat @ N4 @ K ) @ ( plus_plus @ nat @ ( times_times @ nat @ N4 @ K ) @ K ) ) )
@ S2 ) ) ) ) ).
% sums_group
thf(fact_3370_power__half__series,axiom,
( sums @ real
@ ^ [N4: nat] : ( power_power @ real @ ( divide_divide @ real @ ( one_one @ real ) @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) @ ( suc @ N4 ) )
@ ( one_one @ real ) ) ).
% power_half_series
thf(fact_3371_sums__zero,axiom,
! [A: $tType] :
( ( ( comm_monoid_add @ A )
& ( topolo4958980785337419405_space @ A ) )
=> ( sums @ A
@ ^ [N4: nat] : ( zero_zero @ A )
@ ( zero_zero @ A ) ) ) ).
% sums_zero
thf(fact_3372_sums__If__finite__set_H,axiom,
! [A: $tType] :
( ( ( topolo1287966508704411220up_add @ A )
& ( topological_t2_space @ A ) )
=> ! [G: nat > A,S: A,A2: set @ nat,S7: A,F2: nat > A] :
( ( sums @ A @ G @ S )
=> ( ( finite_finite2 @ nat @ A2 )
=> ( ( S7
= ( plus_plus @ A @ S
@ ( groups7311177749621191930dd_sum @ nat @ A
@ ^ [N4: nat] : ( minus_minus @ A @ ( F2 @ N4 ) @ ( G @ N4 ) )
@ A2 ) ) )
=> ( sums @ A
@ ^ [N4: nat] : ( if @ A @ ( member @ nat @ N4 @ A2 ) @ ( F2 @ N4 ) @ ( G @ N4 ) )
@ S7 ) ) ) ) ) ).
% sums_If_finite_set'
thf(fact_3373_UNIV__def,axiom,
! [A: $tType] :
( ( top_top @ ( set @ A ) )
= ( collect @ A
@ ^ [X3: A] : $true ) ) ).
% UNIV_def
thf(fact_3374_sums__le,axiom,
! [A: $tType] :
( ( ( ordere6911136660526730532id_add @ A )
& ( topolo1944317154257567458pology @ A ) )
=> ! [F2: nat > A,G: nat > A,S2: A,T2: A] :
( ! [N3: nat] : ( ord_less_eq @ A @ ( F2 @ N3 ) @ ( G @ N3 ) )
=> ( ( sums @ A @ F2 @ S2 )
=> ( ( sums @ A @ G @ T2 )
=> ( ord_less_eq @ A @ S2 @ T2 ) ) ) ) ) ).
% sums_le
thf(fact_3375_sums__single,axiom,
! [A: $tType] :
( ( ( comm_monoid_add @ A )
& ( topolo4958980785337419405_space @ A ) )
=> ! [I: nat,F2: nat > A] :
( sums @ A
@ ^ [R5: nat] : ( if @ A @ ( R5 = I ) @ ( F2 @ R5 ) @ ( zero_zero @ A ) )
@ ( F2 @ I ) ) ) ).
% sums_single
thf(fact_3376_sums__add,axiom,
! [A: $tType] :
( ( ( topolo5987344860129210374id_add @ A )
& ( topological_t2_space @ A ) )
=> ! [F2: nat > A,A3: A,G: nat > A,B3: A] :
( ( sums @ A @ F2 @ A3 )
=> ( ( sums @ A @ G @ B3 )
=> ( sums @ A
@ ^ [N4: nat] : ( plus_plus @ A @ ( F2 @ N4 ) @ ( G @ N4 ) )
@ ( plus_plus @ A @ A3 @ B3 ) ) ) ) ) ).
% sums_add
thf(fact_3377_sums__mult,axiom,
! [A: $tType] :
( ( real_V4412858255891104859lgebra @ A )
=> ! [F2: nat > A,A3: A,C3: A] :
( ( sums @ A @ F2 @ A3 )
=> ( sums @ A
@ ^ [N4: nat] : ( times_times @ A @ C3 @ ( F2 @ N4 ) )
@ ( times_times @ A @ C3 @ A3 ) ) ) ) ).
% sums_mult
thf(fact_3378_sums__mult2,axiom,
! [A: $tType] :
( ( real_V4412858255891104859lgebra @ A )
=> ! [F2: nat > A,A3: A,C3: A] :
( ( sums @ A @ F2 @ A3 )
=> ( sums @ A
@ ^ [N4: nat] : ( times_times @ A @ ( F2 @ N4 ) @ C3 )
@ ( times_times @ A @ A3 @ C3 ) ) ) ) ).
% sums_mult2
thf(fact_3379_sums__diff,axiom,
! [A: $tType] :
( ( real_V822414075346904944vector @ A )
=> ! [F2: nat > A,A3: A,G: nat > A,B3: A] :
( ( sums @ A @ F2 @ A3 )
=> ( ( sums @ A @ G @ B3 )
=> ( sums @ A
@ ^ [N4: nat] : ( minus_minus @ A @ ( F2 @ N4 ) @ ( G @ N4 ) )
@ ( minus_minus @ A @ A3 @ B3 ) ) ) ) ) ).
% sums_diff
thf(fact_3380_sums__divide,axiom,
! [A: $tType] :
( ( real_V3459762299906320749_field @ A )
=> ! [F2: nat > A,A3: A,C3: A] :
( ( sums @ A @ F2 @ A3 )
=> ( sums @ A
@ ^ [N4: nat] : ( divide_divide @ A @ ( F2 @ N4 ) @ C3 )
@ ( divide_divide @ A @ A3 @ C3 ) ) ) ) ).
% sums_divide
thf(fact_3381_sums__sum,axiom,
! [A: $tType,I8: $tType] :
( ( ( topolo5987344860129210374id_add @ A )
& ( topological_t2_space @ A ) )
=> ! [I6: set @ I8,F2: I8 > nat > A,X2: I8 > A] :
( ! [I2: I8] :
( ( member @ I8 @ I2 @ I6 )
=> ( sums @ A @ ( F2 @ I2 ) @ ( X2 @ I2 ) ) )
=> ( sums @ A
@ ^ [N4: nat] :
( groups7311177749621191930dd_sum @ I8 @ A
@ ^ [I4: I8] : ( F2 @ I4 @ N4 )
@ I6 )
@ ( groups7311177749621191930dd_sum @ I8 @ A @ X2 @ I6 ) ) ) ) ).
% sums_sum
thf(fact_3382_sums__mult__iff,axiom,
! [A: $tType] :
( ( ( field @ A )
& ( real_V4412858255891104859lgebra @ A ) )
=> ! [C3: A,F2: nat > A,D2: A] :
( ( C3
!= ( zero_zero @ A ) )
=> ( ( sums @ A
@ ^ [N4: nat] : ( times_times @ A @ C3 @ ( F2 @ N4 ) )
@ ( times_times @ A @ C3 @ D2 ) )
= ( sums @ A @ F2 @ D2 ) ) ) ) ).
% sums_mult_iff
thf(fact_3383_sums__mult2__iff,axiom,
! [A: $tType] :
( ( ( field @ A )
& ( real_V4412858255891104859lgebra @ A ) )
=> ! [C3: A,F2: nat > A,D2: A] :
( ( C3
!= ( zero_zero @ A ) )
=> ( ( sums @ A
@ ^ [N4: nat] : ( times_times @ A @ ( F2 @ N4 ) @ C3 )
@ ( times_times @ A @ D2 @ C3 ) )
= ( sums @ A @ F2 @ D2 ) ) ) ) ).
% sums_mult2_iff
thf(fact_3384_sums__mult__D,axiom,
! [A: $tType] :
( ( real_V3459762299906320749_field @ A )
=> ! [C3: A,F2: nat > A,A3: A] :
( ( sums @ A
@ ^ [N4: nat] : ( times_times @ A @ C3 @ ( F2 @ N4 ) )
@ A3 )
=> ( ( C3
!= ( zero_zero @ A ) )
=> ( sums @ A @ F2 @ ( divide_divide @ A @ A3 @ C3 ) ) ) ) ) ).
% sums_mult_D
thf(fact_3385_sums__Suc__imp,axiom,
! [A: $tType] :
( ( real_V822414075346904944vector @ A )
=> ! [F2: nat > A,S2: A] :
( ( ( F2 @ ( zero_zero @ nat ) )
= ( zero_zero @ A ) )
=> ( ( sums @ A
@ ^ [N4: nat] : ( F2 @ ( suc @ N4 ) )
@ S2 )
=> ( sums @ A @ F2 @ S2 ) ) ) ) ).
% sums_Suc_imp
thf(fact_3386_sums__Suc__iff,axiom,
! [A: $tType] :
( ( real_V822414075346904944vector @ A )
=> ! [F2: nat > A,S2: A] :
( ( sums @ A
@ ^ [N4: nat] : ( F2 @ ( suc @ N4 ) )
@ S2 )
= ( sums @ A @ F2 @ ( plus_plus @ A @ S2 @ ( F2 @ ( zero_zero @ nat ) ) ) ) ) ) ).
% sums_Suc_iff
thf(fact_3387_sums__Suc,axiom,
! [A: $tType] :
( ( ( topolo5987344860129210374id_add @ A )
& ( topological_t2_space @ A ) )
=> ! [F2: nat > A,L: A] :
( ( sums @ A
@ ^ [N4: nat] : ( F2 @ ( suc @ N4 ) )
@ L )
=> ( sums @ A @ F2 @ ( plus_plus @ A @ L @ ( F2 @ ( zero_zero @ nat ) ) ) ) ) ) ).
% sums_Suc
thf(fact_3388_sums__zero__iff__shift,axiom,
! [A: $tType] :
( ( real_V822414075346904944vector @ A )
=> ! [N: nat,F2: nat > A,S2: A] :
( ! [I2: nat] :
( ( ord_less @ nat @ I2 @ N )
=> ( ( F2 @ I2 )
= ( zero_zero @ A ) ) )
=> ( ( sums @ A
@ ^ [I4: nat] : ( F2 @ ( plus_plus @ nat @ I4 @ N ) )
@ S2 )
= ( sums @ A @ F2 @ S2 ) ) ) ) ).
% sums_zero_iff_shift
thf(fact_3389_sums__finite,axiom,
! [A: $tType] :
( ( ( comm_monoid_add @ A )
& ( topolo4958980785337419405_space @ A ) )
=> ! [N7: set @ nat,F2: nat > A] :
( ( finite_finite2 @ nat @ N7 )
=> ( ! [N3: nat] :
( ~ ( member @ nat @ N3 @ N7 )
=> ( ( F2 @ N3 )
= ( zero_zero @ A ) ) )
=> ( sums @ A @ F2 @ ( groups7311177749621191930dd_sum @ nat @ A @ F2 @ N7 ) ) ) ) ) ).
% sums_finite
thf(fact_3390_sums__If__finite,axiom,
! [A: $tType] :
( ( ( comm_monoid_add @ A )
& ( topolo4958980785337419405_space @ A ) )
=> ! [P: nat > $o,F2: nat > A] :
( ( finite_finite2 @ nat @ ( collect @ nat @ P ) )
=> ( sums @ A
@ ^ [R5: nat] : ( if @ A @ ( P @ R5 ) @ ( F2 @ R5 ) @ ( zero_zero @ A ) )
@ ( groups7311177749621191930dd_sum @ nat @ A @ F2 @ ( collect @ nat @ P ) ) ) ) ) ).
% sums_If_finite
thf(fact_3391_sums__If__finite__set,axiom,
! [A: $tType] :
( ( ( comm_monoid_add @ A )
& ( topolo4958980785337419405_space @ A ) )
=> ! [A2: set @ nat,F2: nat > A] :
( ( finite_finite2 @ nat @ A2 )
=> ( sums @ A
@ ^ [R5: nat] : ( if @ A @ ( member @ nat @ R5 @ A2 ) @ ( F2 @ R5 ) @ ( zero_zero @ A ) )
@ ( groups7311177749621191930dd_sum @ nat @ A @ F2 @ A2 ) ) ) ) ).
% sums_If_finite_set
thf(fact_3392_powser__sums__if,axiom,
! [A: $tType] :
( ( ( ring_1 @ A )
& ( topolo4958980785337419405_space @ A ) )
=> ! [M: nat,Z: A] :
( sums @ A
@ ^ [N4: nat] : ( times_times @ A @ ( if @ A @ ( N4 = M ) @ ( one_one @ A ) @ ( zero_zero @ A ) ) @ ( power_power @ A @ Z @ N4 ) )
@ ( power_power @ A @ Z @ M ) ) ) ).
% powser_sums_if
thf(fact_3393_sums__iff__shift,axiom,
! [A: $tType] :
( ( real_V822414075346904944vector @ A )
=> ! [F2: nat > A,N: nat,S2: A] :
( ( sums @ A
@ ^ [I4: nat] : ( F2 @ ( plus_plus @ nat @ I4 @ N ) )
@ S2 )
= ( sums @ A @ F2 @ ( plus_plus @ A @ S2 @ ( groups7311177749621191930dd_sum @ nat @ A @ F2 @ ( set_ord_lessThan @ nat @ N ) ) ) ) ) ) ).
% sums_iff_shift
thf(fact_3394_sums__split__initial__segment,axiom,
! [A: $tType] :
( ( real_V822414075346904944vector @ A )
=> ! [F2: nat > A,S2: A,N: nat] :
( ( sums @ A @ F2 @ S2 )
=> ( sums @ A
@ ^ [I4: nat] : ( F2 @ ( plus_plus @ nat @ I4 @ N ) )
@ ( minus_minus @ A @ S2 @ ( groups7311177749621191930dd_sum @ nat @ A @ F2 @ ( set_ord_lessThan @ nat @ N ) ) ) ) ) ) ).
% sums_split_initial_segment
thf(fact_3395_sums__iff__shift_H,axiom,
! [A: $tType] :
( ( real_V822414075346904944vector @ A )
=> ! [F2: nat > A,N: nat,S2: A] :
( ( sums @ A
@ ^ [I4: nat] : ( F2 @ ( plus_plus @ nat @ I4 @ N ) )
@ ( minus_minus @ A @ S2 @ ( groups7311177749621191930dd_sum @ nat @ A @ F2 @ ( set_ord_lessThan @ nat @ N ) ) ) )
= ( sums @ A @ F2 @ S2 ) ) ) ).
% sums_iff_shift'
thf(fact_3396_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_3397_pochhammer__code,axiom,
! [A: $tType] :
( ( comm_semiring_1 @ A )
=> ( ( comm_s3205402744901411588hammer @ A )
= ( ^ [A6: A,N4: nat] :
( if @ A
@ ( N4
= ( zero_zero @ nat ) )
@ ( one_one @ A )
@ ( set_fo6178422350223883121st_nat @ A
@ ^ [O: nat] : ( times_times @ A @ ( plus_plus @ A @ A6 @ ( semiring_1_of_nat @ A @ O ) ) )
@ ( zero_zero @ nat )
@ ( minus_minus @ nat @ N4 @ ( one_one @ nat ) )
@ ( one_one @ A ) ) ) ) ) ) ).
% pochhammer_code
thf(fact_3398_of__nat__code,axiom,
! [A: $tType] :
( ( semiring_1 @ A )
=> ( ( semiring_1_of_nat @ A )
= ( ^ [N4: nat] :
( semiri8178284476397505188at_aux @ A
@ ^ [I4: A] : ( plus_plus @ A @ I4 @ ( one_one @ A ) )
@ N4
@ ( zero_zero @ A ) ) ) ) ) ).
% of_nat_code
thf(fact_3399_floor__log__nat__eq__powr__iff,axiom,
! [B3: nat,K: nat,N: nat] :
( ( ord_less_eq @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ B3 )
=> ( ( ord_less @ nat @ ( zero_zero @ nat ) @ K )
=> ( ( ( archim6421214686448440834_floor @ real @ ( log @ ( semiring_1_of_nat @ real @ B3 ) @ ( semiring_1_of_nat @ real @ K ) ) )
= ( semiring_1_of_nat @ int @ N ) )
= ( ( ord_less_eq @ nat @ ( power_power @ nat @ B3 @ N ) @ K )
& ( ord_less @ nat @ K @ ( power_power @ nat @ B3 @ ( plus_plus @ nat @ N @ ( one_one @ nat ) ) ) ) ) ) ) ) ).
% floor_log_nat_eq_powr_iff
thf(fact_3400_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_3401_gbinomial__1,axiom,
! [A: $tType] :
( ( ( semiring_char_0 @ A )
& ( semidom_divide @ A ) )
=> ! [A3: A] :
( ( gbinomial @ A @ A3 @ ( one_one @ nat ) )
= A3 ) ) ).
% gbinomial_1
thf(fact_3402_prod_Oneutral__const,axiom,
! [B: $tType,A: $tType] :
( ( comm_monoid_mult @ A )
=> ! [A2: set @ B] :
( ( groups7121269368397514597t_prod @ B @ A
@ ^ [Uu3: B] : ( one_one @ A )
@ A2 )
= ( one_one @ A ) ) ) ).
% prod.neutral_const
thf(fact_3403_of__nat__prod,axiom,
! [A: $tType,B: $tType] :
( ( comm_semiring_1 @ A )
=> ! [F2: B > nat,A2: set @ B] :
( ( semiring_1_of_nat @ A @ ( groups7121269368397514597t_prod @ B @ nat @ F2 @ A2 ) )
= ( groups7121269368397514597t_prod @ B @ A
@ ^ [X3: B] : ( semiring_1_of_nat @ A @ ( F2 @ X3 ) )
@ A2 ) ) ) ).
% of_nat_prod
thf(fact_3404_of__int__prod,axiom,
! [A: $tType,B: $tType] :
( ( comm_ring_1 @ A )
=> ! [F2: B > int,A2: set @ B] :
( ( ring_1_of_int @ A @ ( groups7121269368397514597t_prod @ B @ int @ F2 @ A2 ) )
= ( groups7121269368397514597t_prod @ B @ A
@ ^ [X3: B] : ( ring_1_of_int @ A @ ( F2 @ X3 ) )
@ A2 ) ) ) ).
% of_int_prod
thf(fact_3405_prod_Oempty,axiom,
! [B: $tType,A: $tType] :
( ( comm_monoid_mult @ A )
=> ! [G: B > A] :
( ( groups7121269368397514597t_prod @ B @ A @ G @ ( bot_bot @ ( set @ B ) ) )
= ( one_one @ A ) ) ) ).
% prod.empty
thf(fact_3406_prod_Oinfinite,axiom,
! [B: $tType,A: $tType] :
( ( comm_monoid_mult @ A )
=> ! [A2: set @ B,G: B > A] :
( ~ ( finite_finite2 @ B @ A2 )
=> ( ( groups7121269368397514597t_prod @ B @ A @ G @ A2 )
= ( one_one @ A ) ) ) ) ).
% prod.infinite
thf(fact_3407_floor__numeral,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [V: num] :
( ( archim6421214686448440834_floor @ A @ ( numeral_numeral @ A @ V ) )
= ( numeral_numeral @ int @ V ) ) ) ).
% floor_numeral
thf(fact_3408_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_3409_gbinomial__0_I1_J,axiom,
! [A: $tType] :
( ( ( semiring_char_0 @ A )
& ( semidom_divide @ A ) )
=> ! [A3: A] :
( ( gbinomial @ A @ A3 @ ( zero_zero @ nat ) )
= ( one_one @ A ) ) ) ).
% gbinomial_0(1)
thf(fact_3410_floor__one,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ( ( archim6421214686448440834_floor @ A @ ( one_one @ A ) )
= ( one_one @ int ) ) ) ).
% floor_one
thf(fact_3411_gbinomial__Suc0,axiom,
! [A: $tType] :
( ( ( semiring_char_0 @ A )
& ( semidom_divide @ A ) )
=> ! [A3: A] :
( ( gbinomial @ A @ A3 @ ( suc @ ( zero_zero @ nat ) ) )
= A3 ) ) ).
% gbinomial_Suc0
thf(fact_3412_prod_Odelta,axiom,
! [B: $tType,A: $tType] :
( ( comm_monoid_mult @ A )
=> ! [S: set @ B,A3: B,B3: B > A] :
( ( finite_finite2 @ B @ S )
=> ( ( ( member @ B @ A3 @ S )
=> ( ( groups7121269368397514597t_prod @ B @ A
@ ^ [K3: B] : ( if @ A @ ( K3 = A3 ) @ ( B3 @ K3 ) @ ( one_one @ A ) )
@ S )
= ( B3 @ A3 ) ) )
& ( ~ ( member @ B @ A3 @ S )
=> ( ( groups7121269368397514597t_prod @ B @ A
@ ^ [K3: B] : ( if @ A @ ( K3 = A3 ) @ ( B3 @ K3 ) @ ( one_one @ A ) )
@ S )
= ( one_one @ A ) ) ) ) ) ) ).
% prod.delta
thf(fact_3413_prod_Odelta_H,axiom,
! [B: $tType,A: $tType] :
( ( comm_monoid_mult @ A )
=> ! [S: set @ B,A3: B,B3: B > A] :
( ( finite_finite2 @ B @ S )
=> ( ( ( member @ B @ A3 @ S )
=> ( ( groups7121269368397514597t_prod @ B @ A
@ ^ [K3: B] : ( if @ A @ ( A3 = K3 ) @ ( B3 @ K3 ) @ ( one_one @ A ) )
@ S )
= ( B3 @ A3 ) ) )
& ( ~ ( member @ B @ A3 @ S )
=> ( ( groups7121269368397514597t_prod @ B @ A
@ ^ [K3: B] : ( if @ A @ ( A3 = K3 ) @ ( B3 @ K3 ) @ ( one_one @ A ) )
@ S )
= ( one_one @ A ) ) ) ) ) ) ).
% prod.delta'
thf(fact_3414_prod_Oinsert,axiom,
! [A: $tType,B: $tType] :
( ( comm_monoid_mult @ A )
=> ! [A2: set @ B,X2: B,G: B > A] :
( ( finite_finite2 @ B @ A2 )
=> ( ~ ( member @ B @ X2 @ A2 )
=> ( ( groups7121269368397514597t_prod @ B @ A @ G @ ( insert @ B @ X2 @ A2 ) )
= ( times_times @ A @ ( G @ X2 ) @ ( groups7121269368397514597t_prod @ B @ A @ G @ A2 ) ) ) ) ) ) ).
% prod.insert
thf(fact_3415_prod_OlessThan__Suc,axiom,
! [A: $tType] :
( ( comm_monoid_mult @ A )
=> ! [G: nat > A,N: nat] :
( ( groups7121269368397514597t_prod @ nat @ A @ G @ ( set_ord_lessThan @ nat @ ( suc @ N ) ) )
= ( times_times @ A @ ( groups7121269368397514597t_prod @ nat @ A @ G @ ( set_ord_lessThan @ nat @ N ) ) @ ( G @ N ) ) ) ) ).
% prod.lessThan_Suc
thf(fact_3416_floor__diff__of__int,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [X2: A,Z: int] :
( ( archim6421214686448440834_floor @ A @ ( minus_minus @ A @ X2 @ ( ring_1_of_int @ A @ Z ) ) )
= ( minus_minus @ int @ ( archim6421214686448440834_floor @ A @ X2 ) @ Z ) ) ) ).
% floor_diff_of_int
thf(fact_3417_zero__le__floor,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [X2: A] :
( ( ord_less_eq @ int @ ( zero_zero @ int ) @ ( archim6421214686448440834_floor @ A @ X2 ) )
= ( ord_less_eq @ A @ ( zero_zero @ A ) @ X2 ) ) ) ).
% zero_le_floor
thf(fact_3418_numeral__le__floor,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [V: num,X2: A] :
( ( ord_less_eq @ int @ ( numeral_numeral @ int @ V ) @ ( archim6421214686448440834_floor @ A @ X2 ) )
= ( ord_less_eq @ A @ ( numeral_numeral @ A @ V ) @ X2 ) ) ) ).
% numeral_le_floor
thf(fact_3419_floor__less__zero,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [X2: A] :
( ( ord_less @ int @ ( archim6421214686448440834_floor @ A @ X2 ) @ ( zero_zero @ int ) )
= ( ord_less @ A @ X2 @ ( zero_zero @ A ) ) ) ) ).
% floor_less_zero
thf(fact_3420_floor__less__numeral,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [X2: A,V: num] :
( ( ord_less @ int @ ( archim6421214686448440834_floor @ A @ X2 ) @ ( numeral_numeral @ int @ V ) )
= ( ord_less @ A @ X2 @ ( numeral_numeral @ A @ V ) ) ) ) ).
% floor_less_numeral
thf(fact_3421_zero__less__floor,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [X2: A] :
( ( ord_less @ int @ ( zero_zero @ int ) @ ( archim6421214686448440834_floor @ A @ X2 ) )
= ( ord_less_eq @ A @ ( one_one @ A ) @ X2 ) ) ) ).
% zero_less_floor
thf(fact_3422_floor__le__zero,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [X2: A] :
( ( ord_less_eq @ int @ ( archim6421214686448440834_floor @ A @ X2 ) @ ( zero_zero @ int ) )
= ( ord_less @ A @ X2 @ ( one_one @ A ) ) ) ) ).
% floor_le_zero
thf(fact_3423_one__le__floor,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [X2: A] :
( ( ord_less_eq @ int @ ( one_one @ int ) @ ( archim6421214686448440834_floor @ A @ X2 ) )
= ( ord_less_eq @ A @ ( one_one @ A ) @ X2 ) ) ) ).
% one_le_floor
thf(fact_3424_floor__less__one,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [X2: A] :
( ( ord_less @ int @ ( archim6421214686448440834_floor @ A @ X2 ) @ ( one_one @ int ) )
= ( ord_less @ A @ X2 @ ( one_one @ A ) ) ) ) ).
% floor_less_one
thf(fact_3425_floor__diff__numeral,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [X2: A,V: num] :
( ( archim6421214686448440834_floor @ A @ ( minus_minus @ A @ X2 @ ( numeral_numeral @ A @ V ) ) )
= ( minus_minus @ int @ ( archim6421214686448440834_floor @ A @ X2 ) @ ( numeral_numeral @ int @ V ) ) ) ) ).
% floor_diff_numeral
thf(fact_3426_floor__numeral__power,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [X2: num,N: nat] :
( ( archim6421214686448440834_floor @ A @ ( power_power @ A @ ( numeral_numeral @ A @ X2 ) @ N ) )
= ( power_power @ int @ ( numeral_numeral @ int @ X2 ) @ N ) ) ) ).
% floor_numeral_power
thf(fact_3427_floor__diff__one,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [X2: A] :
( ( archim6421214686448440834_floor @ A @ ( minus_minus @ A @ X2 @ ( one_one @ A ) ) )
= ( minus_minus @ int @ ( archim6421214686448440834_floor @ A @ X2 ) @ ( one_one @ int ) ) ) ) ).
% floor_diff_one
thf(fact_3428_prod_Oop__ivl__Suc,axiom,
! [A: $tType] :
( ( comm_monoid_mult @ A )
=> ! [N: nat,M: nat,G: nat > A] :
( ( ( ord_less @ nat @ N @ M )
=> ( ( groups7121269368397514597t_prod @ nat @ A @ G @ ( set_or7035219750837199246ssThan @ nat @ M @ ( suc @ N ) ) )
= ( one_one @ A ) ) )
& ( ~ ( ord_less @ nat @ N @ M )
=> ( ( groups7121269368397514597t_prod @ nat @ A @ G @ ( set_or7035219750837199246ssThan @ nat @ M @ ( suc @ N ) ) )
= ( times_times @ A @ ( groups7121269368397514597t_prod @ nat @ A @ G @ ( set_or7035219750837199246ssThan @ nat @ M @ N ) ) @ ( G @ N ) ) ) ) ) ) ).
% prod.op_ivl_Suc
thf(fact_3429_prod_Ocl__ivl__Suc,axiom,
! [A: $tType] :
( ( comm_monoid_mult @ A )
=> ! [N: nat,M: nat,G: nat > A] :
( ( ( ord_less @ nat @ ( suc @ N ) @ M )
=> ( ( groups7121269368397514597t_prod @ nat @ A @ G @ ( set_or1337092689740270186AtMost @ nat @ M @ ( suc @ N ) ) )
= ( one_one @ A ) ) )
& ( ~ ( ord_less @ nat @ ( suc @ N ) @ M )
=> ( ( groups7121269368397514597t_prod @ nat @ A @ G @ ( set_or1337092689740270186AtMost @ nat @ M @ ( suc @ N ) ) )
= ( times_times @ A @ ( groups7121269368397514597t_prod @ nat @ A @ G @ ( set_or1337092689740270186AtMost @ nat @ M @ N ) ) @ ( G @ ( suc @ N ) ) ) ) ) ) ) ).
% prod.cl_ivl_Suc
thf(fact_3430_numeral__less__floor,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [V: num,X2: A] :
( ( ord_less @ int @ ( numeral_numeral @ int @ V ) @ ( archim6421214686448440834_floor @ A @ X2 ) )
= ( ord_less_eq @ A @ ( plus_plus @ A @ ( numeral_numeral @ A @ V ) @ ( one_one @ A ) ) @ X2 ) ) ) ).
% numeral_less_floor
thf(fact_3431_floor__le__numeral,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [X2: A,V: num] :
( ( ord_less_eq @ int @ ( archim6421214686448440834_floor @ A @ X2 ) @ ( numeral_numeral @ int @ V ) )
= ( ord_less @ A @ X2 @ ( plus_plus @ A @ ( numeral_numeral @ A @ V ) @ ( one_one @ A ) ) ) ) ) ).
% floor_le_numeral
thf(fact_3432_one__less__floor,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [X2: A] :
( ( ord_less @ int @ ( one_one @ int ) @ ( archim6421214686448440834_floor @ A @ X2 ) )
= ( ord_less_eq @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ X2 ) ) ) ).
% one_less_floor
thf(fact_3433_floor__le__one,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [X2: A] :
( ( ord_less_eq @ int @ ( archim6421214686448440834_floor @ A @ X2 ) @ ( one_one @ int ) )
= ( ord_less @ A @ X2 @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) ) ) ).
% floor_le_one
thf(fact_3434_floor__one__divide__eq__div__numeral,axiom,
! [B3: num] :
( ( archim6421214686448440834_floor @ real @ ( divide_divide @ real @ ( one_one @ real ) @ ( numeral_numeral @ real @ B3 ) ) )
= ( divide_divide @ int @ ( one_one @ int ) @ ( numeral_numeral @ int @ B3 ) ) ) ).
% floor_one_divide_eq_div_numeral
thf(fact_3435_norm__prod__le,axiom,
! [A: $tType,B: $tType] :
( ( ( comm_monoid_mult @ A )
& ( real_V2822296259951069270ebra_1 @ A ) )
=> ! [F2: B > A,A2: set @ B] :
( ord_less_eq @ real @ ( real_V7770717601297561774m_norm @ A @ ( groups7121269368397514597t_prod @ B @ A @ F2 @ A2 ) )
@ ( groups7121269368397514597t_prod @ B @ real
@ ^ [A6: B] : ( real_V7770717601297561774m_norm @ A @ ( F2 @ A6 ) )
@ A2 ) ) ) ).
% norm_prod_le
thf(fact_3436_prod_Oneutral,axiom,
! [B: $tType,A: $tType] :
( ( comm_monoid_mult @ A )
=> ! [A2: set @ B,G: B > A] :
( ! [X4: B] :
( ( member @ B @ X4 @ A2 )
=> ( ( G @ X4 )
= ( one_one @ A ) ) )
=> ( ( groups7121269368397514597t_prod @ B @ A @ G @ A2 )
= ( one_one @ A ) ) ) ) ).
% prod.neutral
thf(fact_3437_prod_Onot__neutral__contains__not__neutral,axiom,
! [B: $tType,A: $tType] :
( ( comm_monoid_mult @ A )
=> ! [G: B > A,A2: set @ B] :
( ( ( groups7121269368397514597t_prod @ B @ A @ G @ A2 )
!= ( one_one @ A ) )
=> ~ ! [A4: B] :
( ( member @ B @ A4 @ A2 )
=> ( ( G @ A4 )
= ( one_one @ A ) ) ) ) ) ).
% prod.not_neutral_contains_not_neutral
thf(fact_3438_prod__norm,axiom,
! [B: $tType,A: $tType] :
( ( ( real_V8999393235501362500lgebra @ B )
& ( comm_semiring_1 @ B ) )
=> ! [F2: A > B,A2: set @ A] :
( ( groups7121269368397514597t_prod @ A @ real
@ ^ [X3: A] : ( real_V7770717601297561774m_norm @ B @ ( F2 @ X3 ) )
@ A2 )
= ( real_V7770717601297561774m_norm @ B @ ( groups7121269368397514597t_prod @ A @ B @ F2 @ A2 ) ) ) ) ).
% prod_norm
thf(fact_3439_prod_Oswap,axiom,
! [A: $tType,B: $tType,C: $tType] :
( ( comm_monoid_mult @ A )
=> ! [G: B > C > A,B2: set @ C,A2: set @ B] :
( ( groups7121269368397514597t_prod @ B @ A
@ ^ [I4: B] : ( groups7121269368397514597t_prod @ C @ A @ ( G @ I4 ) @ B2 )
@ A2 )
= ( groups7121269368397514597t_prod @ C @ A
@ ^ [J3: C] :
( groups7121269368397514597t_prod @ B @ A
@ ^ [I4: B] : ( G @ I4 @ J3 )
@ A2 )
@ B2 ) ) ) ).
% prod.swap
thf(fact_3440_prod_Odistrib,axiom,
! [A: $tType,B: $tType] :
( ( comm_monoid_mult @ A )
=> ! [G: B > A,H2: B > A,A2: set @ B] :
( ( groups7121269368397514597t_prod @ B @ A
@ ^ [X3: B] : ( times_times @ A @ ( G @ X3 ) @ ( H2 @ X3 ) )
@ A2 )
= ( times_times @ A @ ( groups7121269368397514597t_prod @ B @ A @ G @ A2 ) @ ( groups7121269368397514597t_prod @ B @ A @ H2 @ A2 ) ) ) ) ).
% prod.distrib
thf(fact_3441_prod__dividef,axiom,
! [A: $tType,B: $tType] :
( ( field @ A )
=> ! [F2: B > A,G: B > A,A2: set @ B] :
( ( groups7121269368397514597t_prod @ B @ A
@ ^ [X3: B] : ( divide_divide @ A @ ( F2 @ X3 ) @ ( G @ X3 ) )
@ A2 )
= ( divide_divide @ A @ ( groups7121269368397514597t_prod @ B @ A @ F2 @ A2 ) @ ( groups7121269368397514597t_prod @ B @ A @ G @ A2 ) ) ) ) ).
% prod_dividef
thf(fact_3442_prod__power__distrib,axiom,
! [B: $tType,A: $tType] :
( ( comm_semiring_1 @ B )
=> ! [F2: A > B,A2: set @ A,N: nat] :
( ( power_power @ B @ ( groups7121269368397514597t_prod @ A @ B @ F2 @ A2 ) @ N )
= ( groups7121269368397514597t_prod @ A @ B
@ ^ [X3: A] : ( power_power @ B @ ( F2 @ X3 ) @ N )
@ A2 ) ) ) ).
% prod_power_distrib
thf(fact_3443_prod_Oswap__restrict,axiom,
! [A: $tType,B: $tType,C: $tType] :
( ( comm_monoid_mult @ A )
=> ! [A2: set @ B,B2: set @ C,G: B > C > A,R: B > C > $o] :
( ( finite_finite2 @ B @ A2 )
=> ( ( finite_finite2 @ C @ B2 )
=> ( ( groups7121269368397514597t_prod @ B @ A
@ ^ [X3: B] :
( groups7121269368397514597t_prod @ C @ A @ ( G @ X3 )
@ ( collect @ C
@ ^ [Y: C] :
( ( member @ C @ Y @ B2 )
& ( R @ X3 @ Y ) ) ) )
@ A2 )
= ( groups7121269368397514597t_prod @ C @ A
@ ^ [Y: C] :
( groups7121269368397514597t_prod @ B @ A
@ ^ [X3: B] : ( G @ X3 @ Y )
@ ( collect @ B
@ ^ [X3: B] :
( ( member @ B @ X3 @ A2 )
& ( R @ X3 @ Y ) ) ) )
@ B2 ) ) ) ) ) ).
% prod.swap_restrict
thf(fact_3444_mod__prod__eq,axiom,
! [B: $tType,A: $tType] :
( ( euclid4440199948858584721cancel @ A )
=> ! [F2: B > A,A3: A,A2: set @ B] :
( ( modulo_modulo @ A
@ ( groups7121269368397514597t_prod @ B @ A
@ ^ [I4: B] : ( modulo_modulo @ A @ ( F2 @ I4 ) @ A3 )
@ A2 )
@ A3 )
= ( modulo_modulo @ A @ ( groups7121269368397514597t_prod @ B @ A @ F2 @ A2 ) @ A3 ) ) ) ).
% mod_prod_eq
thf(fact_3445_prod__nonneg,axiom,
! [A: $tType,B: $tType] :
( ( linordered_semidom @ A )
=> ! [A2: set @ B,F2: B > A] :
( ! [X4: B] :
( ( member @ B @ X4 @ A2 )
=> ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( F2 @ X4 ) ) )
=> ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( groups7121269368397514597t_prod @ B @ A @ F2 @ A2 ) ) ) ) ).
% prod_nonneg
thf(fact_3446_prod__mono,axiom,
! [A: $tType,B: $tType] :
( ( linordered_semidom @ A )
=> ! [A2: set @ B,F2: B > A,G: B > A] :
( ! [I2: B] :
( ( member @ B @ I2 @ A2 )
=> ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( F2 @ I2 ) )
& ( ord_less_eq @ A @ ( F2 @ I2 ) @ ( G @ I2 ) ) ) )
=> ( ord_less_eq @ A @ ( groups7121269368397514597t_prod @ B @ A @ F2 @ A2 ) @ ( groups7121269368397514597t_prod @ B @ A @ G @ A2 ) ) ) ) ).
% prod_mono
thf(fact_3447_prod__pos,axiom,
! [A: $tType,B: $tType] :
( ( linordered_semidom @ A )
=> ! [A2: set @ B,F2: B > A] :
( ! [X4: B] :
( ( member @ B @ X4 @ A2 )
=> ( ord_less @ A @ ( zero_zero @ A ) @ ( F2 @ X4 ) ) )
=> ( ord_less @ A @ ( zero_zero @ A ) @ ( groups7121269368397514597t_prod @ B @ A @ F2 @ A2 ) ) ) ) ).
% prod_pos
thf(fact_3448_prod__ge__1,axiom,
! [A: $tType,B: $tType] :
( ( linord181362715937106298miring @ A )
=> ! [A2: set @ B,F2: B > A] :
( ! [X4: B] :
( ( member @ B @ X4 @ A2 )
=> ( ord_less_eq @ A @ ( one_one @ A ) @ ( F2 @ X4 ) ) )
=> ( ord_less_eq @ A @ ( one_one @ A ) @ ( groups7121269368397514597t_prod @ B @ A @ F2 @ A2 ) ) ) ) ).
% prod_ge_1
thf(fact_3449_of__int__floor__le,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [X2: A] : ( ord_less_eq @ A @ ( ring_1_of_int @ A @ ( archim6421214686448440834_floor @ A @ X2 ) ) @ X2 ) ) ).
% of_int_floor_le
thf(fact_3450_floor__mono,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [X2: A,Y3: A] :
( ( ord_less_eq @ A @ X2 @ Y3 )
=> ( ord_less_eq @ int @ ( archim6421214686448440834_floor @ A @ X2 ) @ ( archim6421214686448440834_floor @ A @ Y3 ) ) ) ) ).
% floor_mono
thf(fact_3451_floor__less__cancel,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [X2: A,Y3: A] :
( ( ord_less @ int @ ( archim6421214686448440834_floor @ A @ X2 ) @ ( archim6421214686448440834_floor @ A @ Y3 ) )
=> ( ord_less @ A @ X2 @ Y3 ) ) ) ).
% floor_less_cancel
thf(fact_3452_prod__atLeastAtMost__code,axiom,
! [A: $tType] :
( ( comm_monoid_mult @ A )
=> ! [F2: nat > A,A3: nat,B3: nat] :
( ( groups7121269368397514597t_prod @ nat @ A @ F2 @ ( set_or1337092689740270186AtMost @ nat @ A3 @ B3 ) )
= ( set_fo6178422350223883121st_nat @ A
@ ^ [A6: nat] : ( times_times @ A @ ( F2 @ A6 ) )
@ A3
@ B3
@ ( one_one @ A ) ) ) ) ).
% prod_atLeastAtMost_code
thf(fact_3453_floor__le__ceiling,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [X2: A] : ( ord_less_eq @ int @ ( archim6421214686448440834_floor @ A @ X2 ) @ ( archimedean_ceiling @ A @ X2 ) ) ) ).
% floor_le_ceiling
thf(fact_3454_floor__le__round,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [X2: A] : ( ord_less_eq @ int @ ( archim6421214686448440834_floor @ A @ X2 ) @ ( archimedean_round @ A @ X2 ) ) ) ).
% floor_le_round
thf(fact_3455_prod_Ointer__filter,axiom,
! [A: $tType,B: $tType] :
( ( comm_monoid_mult @ A )
=> ! [A2: set @ B,G: B > A,P: B > $o] :
( ( finite_finite2 @ B @ A2 )
=> ( ( groups7121269368397514597t_prod @ B @ A @ G
@ ( collect @ B
@ ^ [X3: B] :
( ( member @ B @ X3 @ A2 )
& ( P @ X3 ) ) ) )
= ( groups7121269368397514597t_prod @ B @ A
@ ^ [X3: B] : ( if @ A @ ( P @ X3 ) @ ( G @ X3 ) @ ( one_one @ A ) )
@ A2 ) ) ) ) ).
% prod.inter_filter
thf(fact_3456_prod_Oshift__bounds__Suc__ivl,axiom,
! [A: $tType] :
( ( comm_monoid_mult @ A )
=> ! [G: nat > A,M: nat,N: nat] :
( ( groups7121269368397514597t_prod @ nat @ A @ G @ ( set_or7035219750837199246ssThan @ nat @ ( suc @ M ) @ ( suc @ N ) ) )
= ( groups7121269368397514597t_prod @ nat @ A
@ ^ [I4: nat] : ( G @ ( suc @ I4 ) )
@ ( set_or7035219750837199246ssThan @ nat @ M @ N ) ) ) ) ).
% prod.shift_bounds_Suc_ivl
thf(fact_3457_prod_Oshift__bounds__cl__Suc__ivl,axiom,
! [A: $tType] :
( ( comm_monoid_mult @ A )
=> ! [G: nat > A,M: nat,N: nat] :
( ( groups7121269368397514597t_prod @ nat @ A @ G @ ( set_or1337092689740270186AtMost @ nat @ ( suc @ M ) @ ( suc @ N ) ) )
= ( groups7121269368397514597t_prod @ nat @ A
@ ^ [I4: nat] : ( G @ ( suc @ I4 ) )
@ ( set_or1337092689740270186AtMost @ nat @ M @ N ) ) ) ) ).
% prod.shift_bounds_cl_Suc_ivl
thf(fact_3458_power__sum,axiom,
! [A: $tType,B: $tType] :
( ( comm_monoid_mult @ A )
=> ! [C3: A,F2: B > nat,A2: set @ B] :
( ( power_power @ A @ C3 @ ( groups7311177749621191930dd_sum @ B @ nat @ F2 @ A2 ) )
= ( groups7121269368397514597t_prod @ B @ A
@ ^ [A6: B] : ( power_power @ A @ C3 @ ( F2 @ A6 ) )
@ A2 ) ) ) ).
% power_sum
thf(fact_3459_prod_Oshift__bounds__nat__ivl,axiom,
! [A: $tType] :
( ( comm_monoid_mult @ A )
=> ! [G: nat > A,M: nat,K: nat,N: nat] :
( ( groups7121269368397514597t_prod @ nat @ A @ G @ ( set_or7035219750837199246ssThan @ nat @ ( plus_plus @ nat @ M @ K ) @ ( plus_plus @ nat @ N @ K ) ) )
= ( groups7121269368397514597t_prod @ nat @ A
@ ^ [I4: nat] : ( G @ ( plus_plus @ nat @ I4 @ K ) )
@ ( set_or7035219750837199246ssThan @ nat @ M @ N ) ) ) ) ).
% prod.shift_bounds_nat_ivl
thf(fact_3460_prod_Oshift__bounds__cl__nat__ivl,axiom,
! [A: $tType] :
( ( comm_monoid_mult @ A )
=> ! [G: nat > A,M: nat,K: nat,N: nat] :
( ( groups7121269368397514597t_prod @ nat @ A @ G @ ( set_or1337092689740270186AtMost @ nat @ ( plus_plus @ nat @ M @ K ) @ ( plus_plus @ nat @ N @ K ) ) )
= ( groups7121269368397514597t_prod @ nat @ A
@ ^ [I4: nat] : ( G @ ( plus_plus @ nat @ I4 @ K ) )
@ ( set_or1337092689740270186AtMost @ nat @ M @ N ) ) ) ) ).
% prod.shift_bounds_cl_nat_ivl
thf(fact_3461_prod__le__1,axiom,
! [B: $tType,A: $tType] :
( ( linord181362715937106298miring @ A )
=> ! [A2: set @ B,F2: B > A] :
( ! [X4: B] :
( ( member @ B @ X4 @ A2 )
=> ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( F2 @ X4 ) )
& ( ord_less_eq @ A @ ( F2 @ X4 ) @ ( one_one @ A ) ) ) )
=> ( ord_less_eq @ A @ ( groups7121269368397514597t_prod @ B @ A @ F2 @ A2 ) @ ( one_one @ A ) ) ) ) ).
% prod_le_1
thf(fact_3462_prod_Orelated,axiom,
! [A: $tType,B: $tType] :
( ( comm_monoid_mult @ A )
=> ! [R: A > A > $o,S: set @ B,H2: B > A,G: B > A] :
( ( R @ ( one_one @ A ) @ ( one_one @ A ) )
=> ( ! [X16: A,Y1: A,X23: A,Y22: A] :
( ( ( R @ X16 @ X23 )
& ( R @ Y1 @ Y22 ) )
=> ( R @ ( times_times @ A @ X16 @ Y1 ) @ ( times_times @ A @ X23 @ Y22 ) ) )
=> ( ( finite_finite2 @ B @ S )
=> ( ! [X4: B] :
( ( member @ B @ X4 @ S )
=> ( R @ ( H2 @ X4 ) @ ( G @ X4 ) ) )
=> ( R @ ( groups7121269368397514597t_prod @ B @ A @ H2 @ S ) @ ( groups7121269368397514597t_prod @ B @ A @ G @ S ) ) ) ) ) ) ) ).
% prod.related
thf(fact_3463_prod_Oivl__cong,axiom,
! [A: $tType,B: $tType] :
( ( ( ord @ B )
& ( comm_monoid_mult @ A ) )
=> ! [A3: B,C3: B,B3: B,D2: B,G: B > A,H2: B > A] :
( ( A3 = C3 )
=> ( ( B3 = D2 )
=> ( ! [X4: B] :
( ( ord_less_eq @ B @ C3 @ X4 )
=> ( ( ord_less @ B @ X4 @ D2 )
=> ( ( G @ X4 )
= ( H2 @ X4 ) ) ) )
=> ( ( groups7121269368397514597t_prod @ B @ A @ G @ ( set_or7035219750837199246ssThan @ B @ A3 @ B3 ) )
= ( groups7121269368397514597t_prod @ B @ A @ H2 @ ( set_or7035219750837199246ssThan @ B @ C3 @ D2 ) ) ) ) ) ) ) ).
% prod.ivl_cong
thf(fact_3464_prod_Oinsert__if,axiom,
! [A: $tType,B: $tType] :
( ( comm_monoid_mult @ A )
=> ! [A2: set @ B,X2: B,G: B > A] :
( ( finite_finite2 @ B @ A2 )
=> ( ( ( member @ B @ X2 @ A2 )
=> ( ( groups7121269368397514597t_prod @ B @ A @ G @ ( insert @ B @ X2 @ A2 ) )
= ( groups7121269368397514597t_prod @ B @ A @ G @ A2 ) ) )
& ( ~ ( member @ B @ X2 @ A2 )
=> ( ( groups7121269368397514597t_prod @ B @ A @ G @ ( insert @ B @ X2 @ A2 ) )
= ( times_times @ A @ ( G @ X2 ) @ ( groups7121269368397514597t_prod @ B @ A @ G @ A2 ) ) ) ) ) ) ) ).
% prod.insert_if
thf(fact_3465_prod_Oreindex__bij__witness__not__neutral,axiom,
! [B: $tType,A: $tType,C: $tType] :
( ( comm_monoid_mult @ A )
=> ! [S7: set @ B,T7: set @ C,S: set @ B,I: C > B,J: B > C,T6: set @ C,G: B > A,H2: C > A] :
( ( finite_finite2 @ B @ S7 )
=> ( ( finite_finite2 @ C @ T7 )
=> ( ! [A4: B] :
( ( member @ B @ A4 @ ( minus_minus @ ( set @ B ) @ S @ S7 ) )
=> ( ( I @ ( J @ A4 ) )
= A4 ) )
=> ( ! [A4: B] :
( ( member @ B @ A4 @ ( minus_minus @ ( set @ B ) @ S @ S7 ) )
=> ( member @ C @ ( J @ A4 ) @ ( minus_minus @ ( set @ C ) @ T6 @ T7 ) ) )
=> ( ! [B4: C] :
( ( member @ C @ B4 @ ( minus_minus @ ( set @ C ) @ T6 @ T7 ) )
=> ( ( J @ ( I @ B4 ) )
= B4 ) )
=> ( ! [B4: C] :
( ( member @ C @ B4 @ ( minus_minus @ ( set @ C ) @ T6 @ T7 ) )
=> ( member @ B @ ( I @ B4 ) @ ( minus_minus @ ( set @ B ) @ S @ S7 ) ) )
=> ( ! [A4: B] :
( ( member @ B @ A4 @ S7 )
=> ( ( G @ A4 )
= ( one_one @ A ) ) )
=> ( ! [B4: C] :
( ( member @ C @ B4 @ T7 )
=> ( ( H2 @ B4 )
= ( one_one @ A ) ) )
=> ( ! [A4: B] :
( ( member @ B @ A4 @ S )
=> ( ( H2 @ ( J @ A4 ) )
= ( G @ A4 ) ) )
=> ( ( groups7121269368397514597t_prod @ B @ A @ G @ S )
= ( groups7121269368397514597t_prod @ C @ A @ H2 @ T6 ) ) ) ) ) ) ) ) ) ) ) ) ).
% prod.reindex_bij_witness_not_neutral
thf(fact_3466_prod__dvd__prod__subset2,axiom,
! [A: $tType,B: $tType] :
( ( comm_semiring_1 @ A )
=> ! [B2: set @ B,A2: set @ B,F2: B > A,G: B > A] :
( ( finite_finite2 @ B @ B2 )
=> ( ( ord_less_eq @ ( set @ B ) @ A2 @ B2 )
=> ( ! [A4: B] :
( ( member @ B @ A4 @ A2 )
=> ( dvd_dvd @ A @ ( F2 @ A4 ) @ ( G @ A4 ) ) )
=> ( dvd_dvd @ A @ ( groups7121269368397514597t_prod @ B @ A @ F2 @ A2 ) @ ( groups7121269368397514597t_prod @ B @ A @ G @ B2 ) ) ) ) ) ) ).
% prod_dvd_prod_subset2
thf(fact_3467_prod__dvd__prod__subset,axiom,
! [A: $tType,B: $tType] :
( ( comm_monoid_mult @ A )
=> ! [B2: set @ B,A2: set @ B,F2: B > A] :
( ( finite_finite2 @ B @ B2 )
=> ( ( ord_less_eq @ ( set @ B ) @ A2 @ B2 )
=> ( dvd_dvd @ A @ ( groups7121269368397514597t_prod @ B @ A @ F2 @ A2 ) @ ( groups7121269368397514597t_prod @ B @ A @ F2 @ B2 ) ) ) ) ) ).
% prod_dvd_prod_subset
thf(fact_3468_le__floor__iff,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [Z: int,X2: A] :
( ( ord_less_eq @ int @ Z @ ( archim6421214686448440834_floor @ A @ X2 ) )
= ( ord_less_eq @ A @ ( ring_1_of_int @ A @ Z ) @ X2 ) ) ) ).
% le_floor_iff
thf(fact_3469_floor__less__iff,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [X2: A,Z: int] :
( ( ord_less @ int @ ( archim6421214686448440834_floor @ A @ X2 ) @ Z )
= ( ord_less @ A @ X2 @ ( ring_1_of_int @ A @ Z ) ) ) ) ).
% floor_less_iff
thf(fact_3470_int__add__floor,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [Z: int,X2: A] :
( ( plus_plus @ int @ Z @ ( archim6421214686448440834_floor @ A @ X2 ) )
= ( archim6421214686448440834_floor @ A @ ( plus_plus @ A @ ( ring_1_of_int @ A @ Z ) @ X2 ) ) ) ) ).
% int_add_floor
thf(fact_3471_floor__add__int,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [X2: A,Z: int] :
( ( plus_plus @ int @ ( archim6421214686448440834_floor @ A @ X2 ) @ Z )
= ( archim6421214686448440834_floor @ A @ ( plus_plus @ A @ X2 @ ( ring_1_of_int @ A @ Z ) ) ) ) ) ).
% floor_add_int
thf(fact_3472_le__floor__add,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [X2: A,Y3: A] : ( ord_less_eq @ int @ ( plus_plus @ int @ ( archim6421214686448440834_floor @ A @ X2 ) @ ( archim6421214686448440834_floor @ A @ Y3 ) ) @ ( archim6421214686448440834_floor @ A @ ( plus_plus @ A @ X2 @ Y3 ) ) ) ) ).
% le_floor_add
thf(fact_3473_floor__power,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [X2: A,N: nat] :
( ( X2
= ( ring_1_of_int @ A @ ( archim6421214686448440834_floor @ A @ X2 ) ) )
=> ( ( archim6421214686448440834_floor @ A @ ( power_power @ A @ X2 @ N ) )
= ( power_power @ int @ ( archim6421214686448440834_floor @ A @ X2 ) @ N ) ) ) ) ).
% floor_power
thf(fact_3474_floor__divide__of__int__eq,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [K: int,L: int] :
( ( archim6421214686448440834_floor @ A @ ( divide_divide @ A @ ( ring_1_of_int @ A @ K ) @ ( ring_1_of_int @ A @ L ) ) )
= ( divide_divide @ int @ K @ L ) ) ) ).
% floor_divide_of_int_eq
thf(fact_3475_prod_OatLeastLessThan__concat,axiom,
! [A: $tType] :
( ( comm_monoid_mult @ A )
=> ! [M: nat,N: nat,P6: nat,G: nat > A] :
( ( ord_less_eq @ nat @ M @ N )
=> ( ( ord_less_eq @ nat @ N @ P6 )
=> ( ( times_times @ A @ ( groups7121269368397514597t_prod @ nat @ A @ G @ ( set_or7035219750837199246ssThan @ nat @ M @ N ) ) @ ( groups7121269368397514597t_prod @ nat @ A @ G @ ( set_or7035219750837199246ssThan @ nat @ N @ P6 ) ) )
= ( groups7121269368397514597t_prod @ nat @ A @ G @ ( set_or7035219750837199246ssThan @ nat @ M @ P6 ) ) ) ) ) ) ).
% prod.atLeastLessThan_concat
thf(fact_3476_gbinomial__altdef__of__nat,axiom,
! [A: $tType] :
( ( field_char_0 @ A )
=> ( ( gbinomial @ A )
= ( ^ [A6: A,K3: nat] :
( groups7121269368397514597t_prod @ nat @ A
@ ^ [I4: nat] : ( divide_divide @ A @ ( minus_minus @ A @ A6 @ ( semiring_1_of_nat @ A @ I4 ) ) @ ( semiring_1_of_nat @ A @ ( minus_minus @ nat @ K3 @ I4 ) ) )
@ ( set_or7035219750837199246ssThan @ nat @ ( zero_zero @ nat ) @ K3 ) ) ) ) ) ).
% gbinomial_altdef_of_nat
thf(fact_3477_gbinomial__Suc__Suc,axiom,
! [A: $tType] :
( ( field_char_0 @ A )
=> ! [A3: A,K: nat] :
( ( gbinomial @ A @ ( plus_plus @ A @ A3 @ ( one_one @ A ) ) @ ( suc @ K ) )
= ( plus_plus @ A @ ( gbinomial @ A @ A3 @ K ) @ ( gbinomial @ A @ A3 @ ( suc @ K ) ) ) ) ) ).
% gbinomial_Suc_Suc
thf(fact_3478_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_3479_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_3480_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_3481_prod_Osetdiff__irrelevant,axiom,
! [A: $tType,B: $tType] :
( ( comm_monoid_mult @ A )
=> ! [A2: set @ B,G: B > A] :
( ( finite_finite2 @ B @ A2 )
=> ( ( groups7121269368397514597t_prod @ B @ A @ G
@ ( minus_minus @ ( set @ B ) @ A2
@ ( collect @ B
@ ^ [X3: B] :
( ( G @ X3 )
= ( one_one @ A ) ) ) ) )
= ( groups7121269368397514597t_prod @ B @ A @ G @ A2 ) ) ) ) ).
% prod.setdiff_irrelevant
thf(fact_3482_prod_Onat__diff__reindex,axiom,
! [A: $tType] :
( ( comm_monoid_mult @ A )
=> ! [G: nat > A,N: nat] :
( ( groups7121269368397514597t_prod @ nat @ A
@ ^ [I4: nat] : ( G @ ( minus_minus @ nat @ N @ ( suc @ I4 ) ) )
@ ( set_ord_lessThan @ nat @ N ) )
= ( groups7121269368397514597t_prod @ nat @ A @ G @ ( set_ord_lessThan @ nat @ N ) ) ) ) ).
% prod.nat_diff_reindex
thf(fact_3483_prod_OatLeastAtMost__rev,axiom,
! [A: $tType] :
( ( comm_monoid_mult @ A )
=> ! [G: nat > A,N: nat,M: nat] :
( ( groups7121269368397514597t_prod @ nat @ A @ G @ ( set_or1337092689740270186AtMost @ nat @ N @ M ) )
= ( groups7121269368397514597t_prod @ nat @ A
@ ^ [I4: nat] : ( G @ ( minus_minus @ nat @ ( plus_plus @ nat @ M @ N ) @ I4 ) )
@ ( set_or1337092689740270186AtMost @ nat @ N @ M ) ) ) ) ).
% prod.atLeastAtMost_rev
thf(fact_3484_less__1__prod2,axiom,
! [B: $tType,A: $tType] :
( ( linordered_idom @ B )
=> ! [I6: set @ A,I: A,F2: A > B] :
( ( finite_finite2 @ A @ I6 )
=> ( ( member @ A @ I @ I6 )
=> ( ( ord_less @ B @ ( one_one @ B ) @ ( F2 @ I ) )
=> ( ! [I2: A] :
( ( member @ A @ I2 @ I6 )
=> ( ord_less_eq @ B @ ( one_one @ B ) @ ( F2 @ I2 ) ) )
=> ( ord_less @ B @ ( one_one @ B ) @ ( groups7121269368397514597t_prod @ A @ B @ F2 @ I6 ) ) ) ) ) ) ) ).
% less_1_prod2
thf(fact_3485_less__1__prod,axiom,
! [B: $tType,A: $tType] :
( ( linordered_idom @ B )
=> ! [I6: set @ A,F2: A > B] :
( ( finite_finite2 @ A @ I6 )
=> ( ( I6
!= ( bot_bot @ ( set @ A ) ) )
=> ( ! [I2: A] :
( ( member @ A @ I2 @ I6 )
=> ( ord_less @ B @ ( one_one @ B ) @ ( F2 @ I2 ) ) )
=> ( ord_less @ B @ ( one_one @ B ) @ ( groups7121269368397514597t_prod @ A @ B @ F2 @ I6 ) ) ) ) ) ) ).
% less_1_prod
thf(fact_3486_prod_Omono__neutral__cong__right,axiom,
! [A: $tType,B: $tType] :
( ( comm_monoid_mult @ A )
=> ! [T6: set @ B,S: set @ B,G: B > A,H2: B > A] :
( ( finite_finite2 @ B @ T6 )
=> ( ( ord_less_eq @ ( set @ B ) @ S @ T6 )
=> ( ! [X4: B] :
( ( member @ B @ X4 @ ( minus_minus @ ( set @ B ) @ T6 @ S ) )
=> ( ( G @ X4 )
= ( one_one @ A ) ) )
=> ( ! [X4: B] :
( ( member @ B @ X4 @ S )
=> ( ( G @ X4 )
= ( H2 @ X4 ) ) )
=> ( ( groups7121269368397514597t_prod @ B @ A @ G @ T6 )
= ( groups7121269368397514597t_prod @ B @ A @ H2 @ S ) ) ) ) ) ) ) ).
% prod.mono_neutral_cong_right
thf(fact_3487_prod_Omono__neutral__cong__left,axiom,
! [A: $tType,B: $tType] :
( ( comm_monoid_mult @ A )
=> ! [T6: set @ B,S: set @ B,H2: B > A,G: B > A] :
( ( finite_finite2 @ B @ T6 )
=> ( ( ord_less_eq @ ( set @ B ) @ S @ T6 )
=> ( ! [X4: B] :
( ( member @ B @ X4 @ ( minus_minus @ ( set @ B ) @ T6 @ S ) )
=> ( ( H2 @ X4 )
= ( one_one @ A ) ) )
=> ( ! [X4: B] :
( ( member @ B @ X4 @ S )
=> ( ( G @ X4 )
= ( H2 @ X4 ) ) )
=> ( ( groups7121269368397514597t_prod @ B @ A @ G @ S )
= ( groups7121269368397514597t_prod @ B @ A @ H2 @ T6 ) ) ) ) ) ) ) ).
% prod.mono_neutral_cong_left
thf(fact_3488_prod_Omono__neutral__right,axiom,
! [A: $tType,B: $tType] :
( ( comm_monoid_mult @ A )
=> ! [T6: set @ B,S: set @ B,G: B > A] :
( ( finite_finite2 @ B @ T6 )
=> ( ( ord_less_eq @ ( set @ B ) @ S @ T6 )
=> ( ! [X4: B] :
( ( member @ B @ X4 @ ( minus_minus @ ( set @ B ) @ T6 @ S ) )
=> ( ( G @ X4 )
= ( one_one @ A ) ) )
=> ( ( groups7121269368397514597t_prod @ B @ A @ G @ T6 )
= ( groups7121269368397514597t_prod @ B @ A @ G @ S ) ) ) ) ) ) ).
% prod.mono_neutral_right
thf(fact_3489_prod_Omono__neutral__left,axiom,
! [A: $tType,B: $tType] :
( ( comm_monoid_mult @ A )
=> ! [T6: set @ B,S: set @ B,G: B > A] :
( ( finite_finite2 @ B @ T6 )
=> ( ( ord_less_eq @ ( set @ B ) @ S @ T6 )
=> ( ! [X4: B] :
( ( member @ B @ X4 @ ( minus_minus @ ( set @ B ) @ T6 @ S ) )
=> ( ( G @ X4 )
= ( one_one @ A ) ) )
=> ( ( groups7121269368397514597t_prod @ B @ A @ G @ S )
= ( groups7121269368397514597t_prod @ B @ A @ G @ T6 ) ) ) ) ) ) ).
% prod.mono_neutral_left
thf(fact_3490_prod_Osame__carrierI,axiom,
! [A: $tType,B: $tType] :
( ( comm_monoid_mult @ A )
=> ! [C2: set @ B,A2: set @ B,B2: set @ B,G: B > A,H2: B > A] :
( ( finite_finite2 @ B @ C2 )
=> ( ( ord_less_eq @ ( set @ B ) @ A2 @ C2 )
=> ( ( ord_less_eq @ ( set @ B ) @ B2 @ C2 )
=> ( ! [A4: B] :
( ( member @ B @ A4 @ ( minus_minus @ ( set @ B ) @ C2 @ A2 ) )
=> ( ( G @ A4 )
= ( one_one @ A ) ) )
=> ( ! [B4: B] :
( ( member @ B @ B4 @ ( minus_minus @ ( set @ B ) @ C2 @ B2 ) )
=> ( ( H2 @ B4 )
= ( one_one @ A ) ) )
=> ( ( ( groups7121269368397514597t_prod @ B @ A @ G @ C2 )
= ( groups7121269368397514597t_prod @ B @ A @ H2 @ C2 ) )
=> ( ( groups7121269368397514597t_prod @ B @ A @ G @ A2 )
= ( groups7121269368397514597t_prod @ B @ A @ H2 @ B2 ) ) ) ) ) ) ) ) ) ).
% prod.same_carrierI
thf(fact_3491_prod_Osame__carrier,axiom,
! [A: $tType,B: $tType] :
( ( comm_monoid_mult @ A )
=> ! [C2: set @ B,A2: set @ B,B2: set @ B,G: B > A,H2: B > A] :
( ( finite_finite2 @ B @ C2 )
=> ( ( ord_less_eq @ ( set @ B ) @ A2 @ C2 )
=> ( ( ord_less_eq @ ( set @ B ) @ B2 @ C2 )
=> ( ! [A4: B] :
( ( member @ B @ A4 @ ( minus_minus @ ( set @ B ) @ C2 @ A2 ) )
=> ( ( G @ A4 )
= ( one_one @ A ) ) )
=> ( ! [B4: B] :
( ( member @ B @ B4 @ ( minus_minus @ ( set @ B ) @ C2 @ B2 ) )
=> ( ( H2 @ B4 )
= ( one_one @ A ) ) )
=> ( ( ( groups7121269368397514597t_prod @ B @ A @ G @ A2 )
= ( groups7121269368397514597t_prod @ B @ A @ H2 @ B2 ) )
= ( ( groups7121269368397514597t_prod @ B @ A @ G @ C2 )
= ( groups7121269368397514597t_prod @ B @ A @ H2 @ C2 ) ) ) ) ) ) ) ) ) ).
% prod.same_carrier
thf(fact_3492_prod_Osubset__diff,axiom,
! [A: $tType,B: $tType] :
( ( comm_monoid_mult @ A )
=> ! [B2: set @ B,A2: set @ B,G: B > A] :
( ( ord_less_eq @ ( set @ B ) @ B2 @ A2 )
=> ( ( finite_finite2 @ B @ A2 )
=> ( ( groups7121269368397514597t_prod @ B @ A @ G @ A2 )
= ( times_times @ A @ ( groups7121269368397514597t_prod @ B @ A @ G @ ( minus_minus @ ( set @ B ) @ A2 @ B2 ) ) @ ( groups7121269368397514597t_prod @ B @ A @ G @ B2 ) ) ) ) ) ) ).
% prod.subset_diff
thf(fact_3493_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_3494_one__add__floor,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [X2: A] :
( ( plus_plus @ int @ ( archim6421214686448440834_floor @ A @ X2 ) @ ( one_one @ int ) )
= ( archim6421214686448440834_floor @ A @ ( plus_plus @ A @ X2 @ ( one_one @ A ) ) ) ) ) ).
% one_add_floor
thf(fact_3495_floor__divide__of__nat__eq,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [M: nat,N: nat] :
( ( archim6421214686448440834_floor @ A @ ( divide_divide @ A @ ( semiring_1_of_nat @ A @ M ) @ ( semiring_1_of_nat @ A @ N ) ) )
= ( semiring_1_of_nat @ int @ ( divide_divide @ nat @ M @ N ) ) ) ) ).
% floor_divide_of_nat_eq
thf(fact_3496_le__mult__nat__floor,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [A3: A,B3: A] : ( ord_less_eq @ nat @ ( times_times @ nat @ ( nat2 @ ( archim6421214686448440834_floor @ A @ A3 ) ) @ ( nat2 @ ( archim6421214686448440834_floor @ A @ B3 ) ) ) @ ( nat2 @ ( archim6421214686448440834_floor @ A @ ( times_times @ A @ A3 @ B3 ) ) ) ) ) ).
% le_mult_nat_floor
thf(fact_3497_prod_OatLeast0__lessThan__Suc,axiom,
! [A: $tType] :
( ( comm_monoid_mult @ A )
=> ! [G: nat > A,N: nat] :
( ( groups7121269368397514597t_prod @ nat @ A @ G @ ( set_or7035219750837199246ssThan @ nat @ ( zero_zero @ nat ) @ ( suc @ N ) ) )
= ( times_times @ A @ ( groups7121269368397514597t_prod @ nat @ A @ G @ ( set_or7035219750837199246ssThan @ nat @ ( zero_zero @ nat ) @ N ) ) @ ( G @ N ) ) ) ) ).
% prod.atLeast0_lessThan_Suc
thf(fact_3498_prod_OatLeast__Suc__lessThan,axiom,
! [A: $tType] :
( ( comm_monoid_mult @ A )
=> ! [M: nat,N: nat,G: nat > A] :
( ( ord_less @ nat @ M @ N )
=> ( ( groups7121269368397514597t_prod @ nat @ A @ G @ ( set_or7035219750837199246ssThan @ nat @ M @ N ) )
= ( times_times @ A @ ( G @ M ) @ ( groups7121269368397514597t_prod @ nat @ A @ G @ ( set_or7035219750837199246ssThan @ nat @ ( suc @ M ) @ N ) ) ) ) ) ) ).
% prod.atLeast_Suc_lessThan
thf(fact_3499_prod_OatLeast0__atMost__Suc,axiom,
! [A: $tType] :
( ( comm_monoid_mult @ A )
=> ! [G: nat > A,N: nat] :
( ( groups7121269368397514597t_prod @ nat @ A @ G @ ( set_or1337092689740270186AtMost @ nat @ ( zero_zero @ nat ) @ ( suc @ N ) ) )
= ( times_times @ A @ ( groups7121269368397514597t_prod @ nat @ A @ G @ ( set_or1337092689740270186AtMost @ nat @ ( zero_zero @ nat ) @ N ) ) @ ( G @ ( suc @ N ) ) ) ) ) ).
% prod.atLeast0_atMost_Suc
thf(fact_3500_prod_OatLeastLessThan__Suc,axiom,
! [A: $tType] :
( ( comm_monoid_mult @ A )
=> ! [A3: nat,B3: nat,G: nat > A] :
( ( ord_less_eq @ nat @ A3 @ B3 )
=> ( ( groups7121269368397514597t_prod @ nat @ A @ G @ ( set_or7035219750837199246ssThan @ nat @ A3 @ ( suc @ B3 ) ) )
= ( times_times @ A @ ( groups7121269368397514597t_prod @ nat @ A @ G @ ( set_or7035219750837199246ssThan @ nat @ A3 @ B3 ) ) @ ( G @ B3 ) ) ) ) ) ).
% prod.atLeastLessThan_Suc
thf(fact_3501_prod_OatLeast__Suc__atMost,axiom,
! [A: $tType] :
( ( comm_monoid_mult @ A )
=> ! [M: nat,N: nat,G: nat > A] :
( ( ord_less_eq @ nat @ M @ N )
=> ( ( groups7121269368397514597t_prod @ nat @ A @ G @ ( set_or1337092689740270186AtMost @ nat @ M @ N ) )
= ( times_times @ A @ ( G @ M ) @ ( groups7121269368397514597t_prod @ nat @ A @ G @ ( set_or1337092689740270186AtMost @ nat @ ( suc @ M ) @ N ) ) ) ) ) ) ).
% prod.atLeast_Suc_atMost
thf(fact_3502_prod_Onat__ivl__Suc_H,axiom,
! [A: $tType] :
( ( comm_monoid_mult @ A )
=> ! [M: nat,N: nat,G: nat > A] :
( ( ord_less_eq @ nat @ M @ ( suc @ N ) )
=> ( ( groups7121269368397514597t_prod @ nat @ A @ G @ ( set_or1337092689740270186AtMost @ nat @ M @ ( suc @ N ) ) )
= ( times_times @ A @ ( G @ ( suc @ N ) ) @ ( groups7121269368397514597t_prod @ nat @ A @ G @ ( set_or1337092689740270186AtMost @ nat @ M @ N ) ) ) ) ) ) ).
% prod.nat_ivl_Suc'
thf(fact_3503_nat__floor__neg,axiom,
! [X2: real] :
( ( ord_less_eq @ real @ X2 @ ( zero_zero @ real ) )
=> ( ( nat2 @ ( archim6421214686448440834_floor @ real @ X2 ) )
= ( zero_zero @ nat ) ) ) ).
% nat_floor_neg
thf(fact_3504_prod_Olast__plus,axiom,
! [A: $tType] :
( ( comm_monoid_mult @ A )
=> ! [M: nat,N: nat,G: nat > A] :
( ( ord_less_eq @ nat @ M @ N )
=> ( ( groups7121269368397514597t_prod @ nat @ A @ G @ ( set_or1337092689740270186AtMost @ nat @ M @ N ) )
= ( times_times @ A @ ( G @ N ) @ ( groups7121269368397514597t_prod @ nat @ A @ G @ ( set_or7035219750837199246ssThan @ nat @ M @ N ) ) ) ) ) ) ).
% prod.last_plus
thf(fact_3505_floor__eq3,axiom,
! [N: nat,X2: real] :
( ( ord_less @ real @ ( semiring_1_of_nat @ real @ N ) @ X2 )
=> ( ( ord_less @ real @ X2 @ ( semiring_1_of_nat @ real @ ( suc @ N ) ) )
=> ( ( nat2 @ ( archim6421214686448440834_floor @ real @ X2 ) )
= N ) ) ) ).
% floor_eq3
thf(fact_3506_ceiling__altdef,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ( ( archimedean_ceiling @ A )
= ( ^ [X3: A] :
( if @ int
@ ( X3
= ( ring_1_of_int @ A @ ( archim6421214686448440834_floor @ A @ X3 ) ) )
@ ( archim6421214686448440834_floor @ A @ X3 )
@ ( plus_plus @ int @ ( archim6421214686448440834_floor @ A @ X3 ) @ ( one_one @ int ) ) ) ) ) ) ).
% ceiling_altdef
thf(fact_3507_le__nat__floor,axiom,
! [X2: nat,A3: real] :
( ( ord_less_eq @ real @ ( semiring_1_of_nat @ real @ X2 ) @ A3 )
=> ( ord_less_eq @ nat @ X2 @ ( nat2 @ ( archim6421214686448440834_floor @ real @ A3 ) ) ) ) ).
% le_nat_floor
thf(fact_3508_gbinomial__addition__formula,axiom,
! [A: $tType] :
( ( field_char_0 @ A )
=> ! [A3: A,K: nat] :
( ( gbinomial @ A @ A3 @ ( suc @ K ) )
= ( plus_plus @ A @ ( gbinomial @ A @ ( minus_minus @ A @ A3 @ ( one_one @ A ) ) @ ( suc @ K ) ) @ ( gbinomial @ A @ ( minus_minus @ A @ A3 @ ( one_one @ A ) ) @ K ) ) ) ) ).
% gbinomial_addition_formula
thf(fact_3509_ceiling__diff__floor__le__1,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [X2: A] : ( ord_less_eq @ int @ ( minus_minus @ int @ ( archimedean_ceiling @ A @ X2 ) @ ( archim6421214686448440834_floor @ A @ X2 ) ) @ ( one_one @ int ) ) ) ).
% ceiling_diff_floor_le_1
thf(fact_3510_gbinomial__mult__1,axiom,
! [A: $tType] :
( ( field_char_0 @ A )
=> ! [A3: A,K: nat] :
( ( times_times @ A @ A3 @ ( gbinomial @ A @ A3 @ K ) )
= ( plus_plus @ A @ ( times_times @ A @ ( semiring_1_of_nat @ A @ K ) @ ( gbinomial @ A @ A3 @ K ) ) @ ( times_times @ A @ ( semiring_1_of_nat @ A @ ( suc @ K ) ) @ ( gbinomial @ A @ A3 @ ( suc @ K ) ) ) ) ) ) ).
% gbinomial_mult_1
thf(fact_3511_gbinomial__mult__1_H,axiom,
! [A: $tType] :
( ( field_char_0 @ A )
=> ! [A3: A,K: nat] :
( ( times_times @ A @ ( gbinomial @ A @ A3 @ K ) @ A3 )
= ( plus_plus @ A @ ( times_times @ A @ ( semiring_1_of_nat @ A @ K ) @ ( gbinomial @ A @ A3 @ K ) ) @ ( times_times @ A @ ( semiring_1_of_nat @ A @ ( suc @ K ) ) @ ( gbinomial @ A @ A3 @ ( suc @ K ) ) ) ) ) ) ).
% gbinomial_mult_1'
thf(fact_3512_gbinomial__absorb__comp,axiom,
! [A: $tType] :
( ( field_char_0 @ A )
=> ! [A3: A,K: nat] :
( ( times_times @ A @ ( minus_minus @ A @ A3 @ ( semiring_1_of_nat @ A @ K ) ) @ ( gbinomial @ A @ A3 @ K ) )
= ( times_times @ A @ A3 @ ( gbinomial @ A @ ( minus_minus @ A @ A3 @ ( one_one @ A ) ) @ K ) ) ) ) ).
% gbinomial_absorb_comp
thf(fact_3513_gbinomial__ge__n__over__k__pow__k,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [K: nat,A3: A] :
( ( ord_less_eq @ A @ ( semiring_1_of_nat @ A @ K ) @ A3 )
=> ( ord_less_eq @ A @ ( power_power @ A @ ( divide_divide @ A @ A3 @ ( semiring_1_of_nat @ A @ K ) ) @ K ) @ ( gbinomial @ A @ A3 @ K ) ) ) ) ).
% gbinomial_ge_n_over_k_pow_k
thf(fact_3514_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_3515_floor__eq,axiom,
! [N: int,X2: real] :
( ( ord_less @ real @ ( ring_1_of_int @ real @ N ) @ X2 )
=> ( ( ord_less @ real @ X2 @ ( plus_plus @ real @ ( ring_1_of_int @ real @ N ) @ ( one_one @ real ) ) )
=> ( ( archim6421214686448440834_floor @ real @ X2 )
= N ) ) ) ).
% floor_eq
thf(fact_3516_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_3517_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_3518_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_3519_prod_OlessThan__Suc__shift,axiom,
! [A: $tType] :
( ( comm_monoid_mult @ A )
=> ! [G: nat > A,N: nat] :
( ( groups7121269368397514597t_prod @ nat @ A @ G @ ( set_ord_lessThan @ nat @ ( suc @ N ) ) )
= ( times_times @ A @ ( G @ ( zero_zero @ nat ) )
@ ( groups7121269368397514597t_prod @ nat @ A
@ ^ [I4: nat] : ( G @ ( suc @ I4 ) )
@ ( set_ord_lessThan @ nat @ N ) ) ) ) ) ).
% prod.lessThan_Suc_shift
thf(fact_3520_prod_OSuc__reindex__ivl,axiom,
! [A: $tType] :
( ( comm_monoid_mult @ A )
=> ! [M: nat,N: nat,G: nat > A] :
( ( ord_less_eq @ nat @ M @ N )
=> ( ( times_times @ A @ ( groups7121269368397514597t_prod @ nat @ A @ G @ ( set_or1337092689740270186AtMost @ nat @ M @ N ) ) @ ( G @ ( suc @ N ) ) )
= ( times_times @ A @ ( G @ M )
@ ( groups7121269368397514597t_prod @ nat @ A
@ ^ [I4: nat] : ( G @ ( suc @ I4 ) )
@ ( set_or1337092689740270186AtMost @ nat @ M @ N ) ) ) ) ) ) ).
% prod.Suc_reindex_ivl
thf(fact_3521_prod_OatLeastLessThan__rev,axiom,
! [A: $tType] :
( ( comm_monoid_mult @ A )
=> ! [G: nat > A,N: nat,M: nat] :
( ( groups7121269368397514597t_prod @ nat @ A @ G @ ( set_or7035219750837199246ssThan @ nat @ N @ M ) )
= ( groups7121269368397514597t_prod @ nat @ A
@ ^ [I4: nat] : ( G @ ( minus_minus @ nat @ ( plus_plus @ nat @ M @ N ) @ ( suc @ I4 ) ) )
@ ( set_or7035219750837199246ssThan @ nat @ N @ M ) ) ) ) ).
% prod.atLeastLessThan_rev
thf(fact_3522_prod_Onested__swap,axiom,
! [A: $tType] :
( ( comm_monoid_mult @ A )
=> ! [A3: nat > nat > A,N: nat] :
( ( groups7121269368397514597t_prod @ nat @ A
@ ^ [I4: nat] : ( groups7121269368397514597t_prod @ nat @ A @ ( A3 @ I4 ) @ ( set_or7035219750837199246ssThan @ nat @ ( zero_zero @ nat ) @ I4 ) )
@ ( set_or1337092689740270186AtMost @ nat @ ( zero_zero @ nat ) @ N ) )
= ( groups7121269368397514597t_prod @ nat @ A
@ ^ [J3: nat] :
( groups7121269368397514597t_prod @ nat @ A
@ ^ [I4: nat] : ( A3 @ I4 @ J3 )
@ ( set_or1337092689740270186AtMost @ nat @ ( suc @ J3 ) @ N ) )
@ ( set_or7035219750837199246ssThan @ nat @ ( zero_zero @ nat ) @ N ) ) ) ) ).
% prod.nested_swap
thf(fact_3523_prod_OatLeast1__atMost__eq,axiom,
! [A: $tType] :
( ( comm_monoid_mult @ A )
=> ! [G: nat > A,N: nat] :
( ( groups7121269368397514597t_prod @ nat @ A @ G @ ( set_or1337092689740270186AtMost @ nat @ ( suc @ ( zero_zero @ nat ) ) @ N ) )
= ( groups7121269368397514597t_prod @ nat @ A
@ ^ [K3: nat] : ( G @ ( suc @ K3 ) )
@ ( set_ord_lessThan @ nat @ N ) ) ) ) ).
% prod.atLeast1_atMost_eq
thf(fact_3524_prod_Onat__group,axiom,
! [A: $tType] :
( ( comm_monoid_mult @ A )
=> ! [G: nat > A,K: nat,N: nat] :
( ( groups7121269368397514597t_prod @ nat @ A
@ ^ [M6: nat] : ( groups7121269368397514597t_prod @ nat @ A @ G @ ( set_or7035219750837199246ssThan @ nat @ ( times_times @ nat @ M6 @ K ) @ ( plus_plus @ nat @ ( times_times @ nat @ M6 @ K ) @ K ) ) )
@ ( set_ord_lessThan @ nat @ N ) )
= ( groups7121269368397514597t_prod @ nat @ A @ G @ ( set_ord_lessThan @ nat @ ( times_times @ nat @ N @ K ) ) ) ) ) ).
% prod.nat_group
thf(fact_3525_prod__mono__strict,axiom,
! [A: $tType,B: $tType] :
( ( linordered_semidom @ A )
=> ! [A2: set @ B,F2: B > A,G: B > A] :
( ( finite_finite2 @ B @ A2 )
=> ( ! [I2: B] :
( ( member @ B @ I2 @ A2 )
=> ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( F2 @ I2 ) )
& ( ord_less @ A @ ( F2 @ I2 ) @ ( G @ I2 ) ) ) )
=> ( ( A2
!= ( bot_bot @ ( set @ B ) ) )
=> ( ord_less @ A @ ( groups7121269368397514597t_prod @ B @ A @ F2 @ A2 ) @ ( groups7121269368397514597t_prod @ B @ A @ G @ A2 ) ) ) ) ) ) ).
% prod_mono_strict
thf(fact_3526_even__prod__iff,axiom,
! [A: $tType,B: $tType] :
( ( semiring_parity @ A )
=> ! [A2: set @ B,F2: B > A] :
( ( finite_finite2 @ B @ A2 )
=> ( ( dvd_dvd @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ ( groups7121269368397514597t_prod @ B @ A @ F2 @ A2 ) )
= ( ? [X3: B] :
( ( member @ B @ X3 @ A2 )
& ( dvd_dvd @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ ( F2 @ X3 ) ) ) ) ) ) ) ).
% even_prod_iff
thf(fact_3527_floor__unique,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [Z: int,X2: A] :
( ( ord_less_eq @ A @ ( ring_1_of_int @ A @ Z ) @ X2 )
=> ( ( ord_less @ A @ X2 @ ( plus_plus @ A @ ( ring_1_of_int @ A @ Z ) @ ( one_one @ A ) ) )
=> ( ( archim6421214686448440834_floor @ A @ X2 )
= Z ) ) ) ) ).
% floor_unique
thf(fact_3528_floor__eq__iff,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [X2: A,A3: int] :
( ( ( archim6421214686448440834_floor @ A @ X2 )
= A3 )
= ( ( ord_less_eq @ A @ ( ring_1_of_int @ A @ A3 ) @ X2 )
& ( ord_less @ A @ X2 @ ( plus_plus @ A @ ( ring_1_of_int @ A @ A3 ) @ ( one_one @ A ) ) ) ) ) ) ).
% floor_eq_iff
thf(fact_3529_floor__split,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [P: int > $o,T2: A] :
( ( P @ ( archim6421214686448440834_floor @ A @ T2 ) )
= ( ! [I4: int] :
( ( ( ord_less_eq @ A @ ( ring_1_of_int @ A @ I4 ) @ T2 )
& ( ord_less @ A @ T2 @ ( plus_plus @ A @ ( ring_1_of_int @ A @ I4 ) @ ( one_one @ A ) ) ) )
=> ( P @ I4 ) ) ) ) ) ).
% floor_split
thf(fact_3530_prod_Oinsert__remove,axiom,
! [A: $tType,B: $tType] :
( ( comm_monoid_mult @ A )
=> ! [A2: set @ B,G: B > A,X2: B] :
( ( finite_finite2 @ B @ A2 )
=> ( ( groups7121269368397514597t_prod @ B @ A @ G @ ( insert @ B @ X2 @ A2 ) )
= ( times_times @ A @ ( G @ X2 ) @ ( groups7121269368397514597t_prod @ B @ A @ G @ ( minus_minus @ ( set @ B ) @ A2 @ ( insert @ B @ X2 @ ( bot_bot @ ( set @ B ) ) ) ) ) ) ) ) ) ).
% prod.insert_remove
thf(fact_3531_prod_Oremove,axiom,
! [A: $tType,B: $tType] :
( ( comm_monoid_mult @ A )
=> ! [A2: set @ B,X2: B,G: B > A] :
( ( finite_finite2 @ B @ A2 )
=> ( ( member @ B @ X2 @ A2 )
=> ( ( groups7121269368397514597t_prod @ B @ A @ G @ A2 )
= ( times_times @ A @ ( G @ X2 ) @ ( groups7121269368397514597t_prod @ B @ A @ G @ ( minus_minus @ ( set @ B ) @ A2 @ ( insert @ B @ X2 @ ( bot_bot @ ( set @ B ) ) ) ) ) ) ) ) ) ) ).
% prod.remove
thf(fact_3532_less__floor__iff,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [Z: int,X2: A] :
( ( ord_less @ int @ Z @ ( archim6421214686448440834_floor @ A @ X2 ) )
= ( ord_less_eq @ A @ ( plus_plus @ A @ ( ring_1_of_int @ A @ Z ) @ ( one_one @ A ) ) @ X2 ) ) ) ).
% less_floor_iff
thf(fact_3533_floor__le__iff,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [X2: A,Z: int] :
( ( ord_less_eq @ int @ ( archim6421214686448440834_floor @ A @ X2 ) @ Z )
= ( ord_less @ A @ X2 @ ( plus_plus @ A @ ( ring_1_of_int @ A @ Z ) @ ( one_one @ A ) ) ) ) ) ).
% floor_le_iff
thf(fact_3534_le__mult__floor,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [A3: A,B3: A] :
( ( ord_less_eq @ A @ ( zero_zero @ A ) @ A3 )
=> ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ B3 )
=> ( ord_less_eq @ int @ ( times_times @ int @ ( archim6421214686448440834_floor @ A @ A3 ) @ ( archim6421214686448440834_floor @ A @ B3 ) ) @ ( archim6421214686448440834_floor @ A @ ( times_times @ A @ A3 @ B3 ) ) ) ) ) ) ).
% le_mult_floor
thf(fact_3535_floor__correct,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [X2: A] :
( ( ord_less_eq @ A @ ( ring_1_of_int @ A @ ( archim6421214686448440834_floor @ A @ X2 ) ) @ X2 )
& ( ord_less @ A @ X2 @ ( ring_1_of_int @ A @ ( plus_plus @ int @ ( archim6421214686448440834_floor @ A @ X2 ) @ ( one_one @ int ) ) ) ) ) ) ).
% floor_correct
thf(fact_3536_prod_Oub__add__nat,axiom,
! [A: $tType] :
( ( comm_monoid_mult @ A )
=> ! [M: nat,N: nat,G: nat > A,P6: nat] :
( ( ord_less_eq @ nat @ M @ ( plus_plus @ nat @ N @ ( one_one @ nat ) ) )
=> ( ( groups7121269368397514597t_prod @ nat @ A @ G @ ( set_or1337092689740270186AtMost @ nat @ M @ ( plus_plus @ nat @ N @ P6 ) ) )
= ( times_times @ A @ ( groups7121269368397514597t_prod @ nat @ A @ G @ ( set_or1337092689740270186AtMost @ nat @ M @ N ) ) @ ( groups7121269368397514597t_prod @ nat @ A @ G @ ( set_or1337092689740270186AtMost @ nat @ ( plus_plus @ nat @ N @ ( one_one @ nat ) ) @ ( plus_plus @ nat @ N @ P6 ) ) ) ) ) ) ) ).
% prod.ub_add_nat
thf(fact_3537_prod_Ohead__if,axiom,
! [A: $tType] :
( ( comm_monoid_mult @ A )
=> ! [N: nat,M: nat,G: nat > A] :
( ( ( ord_less @ nat @ N @ M )
=> ( ( groups7121269368397514597t_prod @ nat @ A @ G @ ( set_or1337092689740270186AtMost @ nat @ M @ N ) )
= ( one_one @ A ) ) )
& ( ~ ( ord_less @ nat @ N @ M )
=> ( ( groups7121269368397514597t_prod @ nat @ A @ G @ ( set_or1337092689740270186AtMost @ nat @ M @ N ) )
= ( times_times @ A @ ( groups7121269368397514597t_prod @ nat @ A @ G @ ( set_or7035219750837199246ssThan @ nat @ M @ N ) ) @ ( G @ N ) ) ) ) ) ) ).
% prod.head_if
thf(fact_3538_floor__eq4,axiom,
! [N: nat,X2: real] :
( ( ord_less_eq @ real @ ( semiring_1_of_nat @ real @ N ) @ X2 )
=> ( ( ord_less @ real @ X2 @ ( semiring_1_of_nat @ real @ ( suc @ N ) ) )
=> ( ( nat2 @ ( archim6421214686448440834_floor @ real @ X2 ) )
= N ) ) ) ).
% floor_eq4
thf(fact_3539_fold__atLeastAtMost__nat_Oelims,axiom,
! [A: $tType,X2: nat > A > A,Xa: nat,Xb: nat,Xc: A,Y3: A] :
( ( ( set_fo6178422350223883121st_nat @ A @ X2 @ Xa @ Xb @ Xc )
= Y3 )
=> ( ( ( ord_less @ nat @ Xb @ Xa )
=> ( Y3 = Xc ) )
& ( ~ ( ord_less @ nat @ Xb @ Xa )
=> ( Y3
= ( set_fo6178422350223883121st_nat @ A @ X2 @ ( plus_plus @ nat @ Xa @ ( one_one @ nat ) ) @ Xb @ ( X2 @ Xa @ Xc ) ) ) ) ) ) ).
% fold_atLeastAtMost_nat.elims
thf(fact_3540_fold__atLeastAtMost__nat_Osimps,axiom,
! [A: $tType] :
( ( set_fo6178422350223883121st_nat @ A )
= ( ^ [F5: nat > A > A,A6: nat,B7: nat,Acc2: A] : ( if @ A @ ( ord_less @ nat @ B7 @ A6 ) @ Acc2 @ ( set_fo6178422350223883121st_nat @ A @ F5 @ ( plus_plus @ nat @ A6 @ ( one_one @ nat ) ) @ B7 @ ( F5 @ A6 @ Acc2 ) ) ) ) ) ).
% fold_atLeastAtMost_nat.simps
thf(fact_3541_Suc__times__gbinomial,axiom,
! [A: $tType] :
( ( field_char_0 @ A )
=> ! [K: nat,A3: A] :
( ( times_times @ A @ ( semiring_1_of_nat @ A @ ( suc @ K ) ) @ ( gbinomial @ A @ ( plus_plus @ A @ A3 @ ( one_one @ A ) ) @ ( suc @ K ) ) )
= ( times_times @ A @ ( plus_plus @ A @ A3 @ ( one_one @ A ) ) @ ( gbinomial @ A @ A3 @ K ) ) ) ) ).
% Suc_times_gbinomial
thf(fact_3542_gbinomial__absorption,axiom,
! [A: $tType] :
( ( field_char_0 @ A )
=> ! [K: nat,A3: A] :
( ( times_times @ A @ ( semiring_1_of_nat @ A @ ( suc @ K ) ) @ ( gbinomial @ A @ A3 @ ( suc @ K ) ) )
= ( times_times @ A @ A3 @ ( gbinomial @ A @ ( minus_minus @ A @ A3 @ ( one_one @ A ) ) @ K ) ) ) ) ).
% gbinomial_absorption
thf(fact_3543_floor__eq2,axiom,
! [N: int,X2: real] :
( ( ord_less_eq @ real @ ( ring_1_of_int @ real @ N ) @ X2 )
=> ( ( ord_less @ real @ X2 @ ( plus_plus @ real @ ( ring_1_of_int @ real @ N ) @ ( one_one @ real ) ) )
=> ( ( archim6421214686448440834_floor @ real @ X2 )
= N ) ) ) ).
% floor_eq2
thf(fact_3544_prod_Odelta__remove,axiom,
! [A: $tType,B: $tType] :
( ( comm_monoid_mult @ A )
=> ! [S: set @ B,A3: B,B3: B > A,C3: B > A] :
( ( finite_finite2 @ B @ S )
=> ( ( ( member @ B @ A3 @ S )
=> ( ( groups7121269368397514597t_prod @ B @ A
@ ^ [K3: B] : ( if @ A @ ( K3 = A3 ) @ ( B3 @ K3 ) @ ( C3 @ K3 ) )
@ S )
= ( times_times @ A @ ( B3 @ A3 ) @ ( groups7121269368397514597t_prod @ B @ A @ C3 @ ( minus_minus @ ( set @ B ) @ S @ ( insert @ B @ A3 @ ( bot_bot @ ( set @ B ) ) ) ) ) ) ) )
& ( ~ ( member @ B @ A3 @ S )
=> ( ( groups7121269368397514597t_prod @ B @ A
@ ^ [K3: B] : ( if @ A @ ( K3 = A3 ) @ ( B3 @ K3 ) @ ( C3 @ K3 ) )
@ S )
= ( groups7121269368397514597t_prod @ B @ A @ C3 @ ( minus_minus @ ( set @ B ) @ S @ ( insert @ B @ A3 @ ( bot_bot @ ( set @ B ) ) ) ) ) ) ) ) ) ) ).
% prod.delta_remove
thf(fact_3545_floor__divide__real__eq__div,axiom,
! [B3: int,A3: real] :
( ( ord_less_eq @ int @ ( zero_zero @ int ) @ B3 )
=> ( ( archim6421214686448440834_floor @ real @ ( divide_divide @ real @ A3 @ ( ring_1_of_int @ real @ B3 ) ) )
= ( divide_divide @ int @ ( archim6421214686448440834_floor @ real @ A3 ) @ B3 ) ) ) ).
% floor_divide_real_eq_div
thf(fact_3546_gbinomial__trinomial__revision,axiom,
! [A: $tType] :
( ( field_char_0 @ A )
=> ! [K: nat,M: nat,A3: A] :
( ( ord_less_eq @ nat @ K @ M )
=> ( ( times_times @ A @ ( gbinomial @ A @ A3 @ M ) @ ( gbinomial @ A @ ( semiring_1_of_nat @ A @ M ) @ K ) )
= ( times_times @ A @ ( gbinomial @ A @ A3 @ K ) @ ( gbinomial @ A @ ( minus_minus @ A @ A3 @ ( semiring_1_of_nat @ A @ K ) ) @ ( minus_minus @ nat @ M @ K ) ) ) ) ) ) ).
% gbinomial_trinomial_revision
thf(fact_3547_norm__prod__diff,axiom,
! [A: $tType,I8: $tType] :
( ( ( comm_monoid_mult @ A )
& ( real_V2822296259951069270ebra_1 @ A ) )
=> ! [I6: set @ I8,Z: I8 > A,W: I8 > A] :
( ! [I2: I8] :
( ( member @ I8 @ I2 @ I6 )
=> ( ord_less_eq @ real @ ( real_V7770717601297561774m_norm @ A @ ( Z @ I2 ) ) @ ( one_one @ real ) ) )
=> ( ! [I2: I8] :
( ( member @ I8 @ I2 @ I6 )
=> ( ord_less_eq @ real @ ( real_V7770717601297561774m_norm @ A @ ( W @ I2 ) ) @ ( one_one @ real ) ) )
=> ( ord_less_eq @ real @ ( real_V7770717601297561774m_norm @ A @ ( minus_minus @ A @ ( groups7121269368397514597t_prod @ I8 @ A @ Z @ I6 ) @ ( groups7121269368397514597t_prod @ I8 @ A @ W @ I6 ) ) )
@ ( groups7311177749621191930dd_sum @ I8 @ real
@ ^ [I4: I8] : ( real_V7770717601297561774m_norm @ A @ ( minus_minus @ A @ ( Z @ I4 ) @ ( W @ I4 ) ) )
@ I6 ) ) ) ) ) ).
% norm_prod_diff
thf(fact_3548_prod_OatLeastLessThan__rev__at__least__Suc__atMost,axiom,
! [A: $tType] :
( ( comm_monoid_mult @ A )
=> ! [G: nat > A,N: nat,M: nat] :
( ( groups7121269368397514597t_prod @ nat @ A @ G @ ( set_or7035219750837199246ssThan @ nat @ N @ M ) )
= ( groups7121269368397514597t_prod @ nat @ A
@ ^ [I4: nat] : ( G @ ( minus_minus @ nat @ ( plus_plus @ nat @ M @ N ) @ I4 ) )
@ ( set_or1337092689740270186AtMost @ nat @ ( suc @ N ) @ M ) ) ) ) ).
% prod.atLeastLessThan_rev_at_least_Suc_atMost
thf(fact_3549_pochhammer__prod,axiom,
! [A: $tType] :
( ( comm_semiring_1 @ A )
=> ( ( comm_s3205402744901411588hammer @ A )
= ( ^ [A6: A,N4: nat] :
( groups7121269368397514597t_prod @ nat @ A
@ ^ [I4: nat] : ( plus_plus @ A @ A6 @ ( semiring_1_of_nat @ A @ I4 ) )
@ ( set_or7035219750837199246ssThan @ nat @ ( zero_zero @ nat ) @ N4 ) ) ) ) ) ).
% pochhammer_prod
thf(fact_3550_floor__divide__lower,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [Q3: A,P6: A] :
( ( ord_less @ A @ ( zero_zero @ A ) @ Q3 )
=> ( ord_less_eq @ A @ ( times_times @ A @ ( ring_1_of_int @ A @ ( archim6421214686448440834_floor @ A @ ( divide_divide @ A @ P6 @ Q3 ) ) ) @ Q3 ) @ P6 ) ) ) ).
% floor_divide_lower
thf(fact_3551_prod__mono2,axiom,
! [B: $tType,A: $tType] :
( ( linordered_idom @ B )
=> ! [B2: set @ A,A2: set @ A,F2: A > B] :
( ( finite_finite2 @ A @ B2 )
=> ( ( ord_less_eq @ ( set @ A ) @ A2 @ B2 )
=> ( ! [B4: A] :
( ( member @ A @ B4 @ ( minus_minus @ ( set @ A ) @ B2 @ A2 ) )
=> ( ord_less_eq @ B @ ( one_one @ B ) @ ( F2 @ B4 ) ) )
=> ( ! [A4: A] :
( ( member @ A @ A4 @ A2 )
=> ( ord_less_eq @ B @ ( zero_zero @ B ) @ ( F2 @ A4 ) ) )
=> ( ord_less_eq @ B @ ( groups7121269368397514597t_prod @ A @ B @ F2 @ A2 ) @ ( groups7121269368397514597t_prod @ A @ B @ F2 @ B2 ) ) ) ) ) ) ) ).
% prod_mono2
thf(fact_3552_prod__diff1,axiom,
! [A: $tType,B: $tType] :
( ( semidom_divide @ A )
=> ! [A2: set @ B,F2: B > A,A3: B] :
( ( finite_finite2 @ B @ A2 )
=> ( ( ( F2 @ A3 )
!= ( zero_zero @ A ) )
=> ( ( ( member @ B @ A3 @ A2 )
=> ( ( groups7121269368397514597t_prod @ B @ A @ F2 @ ( minus_minus @ ( set @ B ) @ A2 @ ( insert @ B @ A3 @ ( bot_bot @ ( set @ B ) ) ) ) )
= ( divide_divide @ A @ ( groups7121269368397514597t_prod @ B @ A @ F2 @ A2 ) @ ( F2 @ A3 ) ) ) )
& ( ~ ( member @ B @ A3 @ A2 )
=> ( ( groups7121269368397514597t_prod @ B @ A @ F2 @ ( minus_minus @ ( set @ B ) @ A2 @ ( insert @ B @ A3 @ ( bot_bot @ ( set @ B ) ) ) ) )
= ( groups7121269368397514597t_prod @ B @ A @ F2 @ A2 ) ) ) ) ) ) ) ).
% prod_diff1
thf(fact_3553_gbinomial__factors,axiom,
! [A: $tType] :
( ( field_char_0 @ A )
=> ! [A3: A,K: nat] :
( ( gbinomial @ A @ ( plus_plus @ A @ A3 @ ( one_one @ A ) ) @ ( suc @ K ) )
= ( times_times @ A @ ( divide_divide @ A @ ( plus_plus @ A @ A3 @ ( one_one @ A ) ) @ ( semiring_1_of_nat @ A @ ( suc @ K ) ) ) @ ( gbinomial @ A @ A3 @ K ) ) ) ) ).
% gbinomial_factors
thf(fact_3554_gbinomial__rec,axiom,
! [A: $tType] :
( ( field_char_0 @ A )
=> ! [A3: A,K: nat] :
( ( gbinomial @ A @ ( plus_plus @ A @ A3 @ ( one_one @ A ) ) @ ( suc @ K ) )
= ( times_times @ A @ ( gbinomial @ A @ A3 @ K ) @ ( divide_divide @ A @ ( plus_plus @ A @ A3 @ ( one_one @ A ) ) @ ( semiring_1_of_nat @ A @ ( suc @ K ) ) ) ) ) ) ).
% gbinomial_rec
thf(fact_3555_pochhammer__Suc__prod,axiom,
! [A: $tType] :
( ( comm_semiring_1 @ A )
=> ! [A3: A,N: nat] :
( ( comm_s3205402744901411588hammer @ A @ A3 @ ( suc @ N ) )
= ( groups7121269368397514597t_prod @ nat @ A
@ ^ [I4: nat] : ( plus_plus @ A @ A3 @ ( semiring_1_of_nat @ A @ I4 ) )
@ ( set_or1337092689740270186AtMost @ nat @ ( zero_zero @ nat ) @ N ) ) ) ) ).
% pochhammer_Suc_prod
thf(fact_3556_pochhammer__prod__rev,axiom,
! [A: $tType] :
( ( comm_semiring_1 @ A )
=> ( ( comm_s3205402744901411588hammer @ A )
= ( ^ [A6: A,N4: nat] :
( groups7121269368397514597t_prod @ nat @ A
@ ^ [I4: nat] : ( plus_plus @ A @ A6 @ ( semiring_1_of_nat @ A @ ( minus_minus @ nat @ N4 @ I4 ) ) )
@ ( set_or1337092689740270186AtMost @ nat @ ( one_one @ nat ) @ N4 ) ) ) ) ) ).
% pochhammer_prod_rev
thf(fact_3557_floor__divide__upper,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [Q3: A,P6: A] :
( ( ord_less @ A @ ( zero_zero @ A ) @ Q3 )
=> ( ord_less @ A @ P6 @ ( times_times @ A @ ( plus_plus @ A @ ( ring_1_of_int @ A @ ( archim6421214686448440834_floor @ A @ ( divide_divide @ A @ P6 @ Q3 ) ) ) @ ( one_one @ A ) ) @ Q3 ) ) ) ) ).
% floor_divide_upper
thf(fact_3558_round__def,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ( ( archimedean_round @ A )
= ( ^ [X3: A] : ( archim6421214686448440834_floor @ A @ ( plus_plus @ A @ X3 @ ( divide_divide @ A @ ( one_one @ A ) @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) ) ) ) ) ) ).
% round_def
thf(fact_3559_prod_Oin__pairs,axiom,
! [A: $tType] :
( ( comm_monoid_mult @ A )
=> ! [G: nat > A,M: nat,N: nat] :
( ( groups7121269368397514597t_prod @ nat @ A @ G @ ( set_or1337092689740270186AtMost @ nat @ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ M ) @ ( suc @ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N ) ) ) )
= ( groups7121269368397514597t_prod @ nat @ A
@ ^ [I4: nat] : ( times_times @ A @ ( G @ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ I4 ) ) @ ( G @ ( suc @ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ I4 ) ) ) )
@ ( set_or1337092689740270186AtMost @ nat @ M @ N ) ) ) ) ).
% prod.in_pairs
thf(fact_3560_gbinomial__reduce__nat,axiom,
! [A: $tType] :
( ( field_char_0 @ A )
=> ! [K: nat,A3: A] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ K )
=> ( ( gbinomial @ A @ A3 @ K )
= ( plus_plus @ A @ ( gbinomial @ A @ ( minus_minus @ A @ A3 @ ( one_one @ A ) ) @ ( minus_minus @ nat @ K @ ( one_one @ nat ) ) ) @ ( gbinomial @ A @ ( minus_minus @ A @ A3 @ ( one_one @ A ) ) @ K ) ) ) ) ) ).
% gbinomial_reduce_nat
thf(fact_3561_sum__atLeastAtMost__code,axiom,
! [A: $tType] :
( ( comm_monoid_add @ A )
=> ! [F2: nat > A,A3: nat,B3: nat] :
( ( groups7311177749621191930dd_sum @ nat @ A @ F2 @ ( set_or1337092689740270186AtMost @ nat @ A3 @ B3 ) )
= ( set_fo6178422350223883121st_nat @ A
@ ^ [A6: nat] : ( plus_plus @ A @ ( F2 @ A6 ) )
@ A3
@ B3
@ ( zero_zero @ A ) ) ) ) ).
% sum_atLeastAtMost_code
thf(fact_3562_pochhammer__Suc__prod__rev,axiom,
! [A: $tType] :
( ( comm_semiring_1 @ A )
=> ! [A3: A,N: nat] :
( ( comm_s3205402744901411588hammer @ A @ A3 @ ( suc @ N ) )
= ( groups7121269368397514597t_prod @ nat @ A
@ ^ [I4: nat] : ( plus_plus @ A @ A3 @ ( semiring_1_of_nat @ A @ ( minus_minus @ nat @ N @ I4 ) ) )
@ ( set_or1337092689740270186AtMost @ nat @ ( zero_zero @ nat ) @ N ) ) ) ) ).
% pochhammer_Suc_prod_rev
thf(fact_3563_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_3564_gbinomial__absorption_H,axiom,
! [A: $tType] :
( ( field_char_0 @ A )
=> ! [K: nat,A3: A] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ K )
=> ( ( gbinomial @ A @ A3 @ K )
= ( times_times @ A @ ( divide_divide @ A @ A3 @ ( semiring_1_of_nat @ A @ K ) ) @ ( gbinomial @ A @ ( minus_minus @ A @ A3 @ ( one_one @ A ) ) @ ( minus_minus @ nat @ K @ ( one_one @ nat ) ) ) ) ) ) ) ).
% gbinomial_absorption'
thf(fact_3565_floor__log__eq__powr__iff,axiom,
! [X2: real,B3: real,K: int] :
( ( ord_less @ real @ ( zero_zero @ real ) @ X2 )
=> ( ( ord_less @ real @ ( one_one @ real ) @ B3 )
=> ( ( ( archim6421214686448440834_floor @ real @ ( log @ B3 @ X2 ) )
= K )
= ( ( ord_less_eq @ real @ ( powr @ real @ B3 @ ( ring_1_of_int @ real @ K ) ) @ X2 )
& ( ord_less @ real @ X2 @ ( powr @ real @ B3 @ ( ring_1_of_int @ real @ ( plus_plus @ int @ K @ ( one_one @ int ) ) ) ) ) ) ) ) ) ).
% floor_log_eq_powr_iff
thf(fact_3566_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_3567_floor__log__nat__eq__if,axiom,
! [B3: nat,N: nat,K: nat] :
( ( ord_less_eq @ nat @ ( power_power @ nat @ B3 @ N ) @ K )
=> ( ( ord_less @ nat @ K @ ( power_power @ nat @ B3 @ ( plus_plus @ nat @ N @ ( one_one @ nat ) ) ) )
=> ( ( ord_less_eq @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ B3 )
=> ( ( archim6421214686448440834_floor @ real @ ( log @ ( semiring_1_of_nat @ real @ B3 ) @ ( semiring_1_of_nat @ real @ K ) ) )
= ( semiring_1_of_nat @ int @ N ) ) ) ) ) ).
% floor_log_nat_eq_if
thf(fact_3568_gbinomial__partial__row__sum,axiom,
! [A: $tType] :
( ( field_char_0 @ A )
=> ! [A3: A,M: nat] :
( ( groups7311177749621191930dd_sum @ nat @ A
@ ^ [K3: nat] : ( times_times @ A @ ( gbinomial @ A @ A3 @ K3 ) @ ( minus_minus @ A @ ( divide_divide @ A @ A3 @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) @ ( semiring_1_of_nat @ A @ K3 ) ) )
@ ( set_ord_atMost @ nat @ M ) )
= ( times_times @ A @ ( divide_divide @ A @ ( plus_plus @ A @ ( semiring_1_of_nat @ A @ M ) @ ( one_one @ A ) ) @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) @ ( gbinomial @ A @ A3 @ ( plus_plus @ nat @ M @ ( one_one @ nat ) ) ) ) ) ) ).
% gbinomial_partial_row_sum
thf(fact_3569_round__altdef,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ( ( archimedean_round @ A )
= ( ^ [X3: A] : ( if @ int @ ( ord_less_eq @ A @ ( divide_divide @ A @ ( one_one @ A ) @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) @ ( archimedean_frac @ A @ X3 ) ) @ ( archimedean_ceiling @ A @ X3 ) @ ( archim6421214686448440834_floor @ A @ X3 ) ) ) ) ) ).
% round_altdef
thf(fact_3570_gbinomial__r__part__sum,axiom,
! [A: $tType] :
( ( field_char_0 @ A )
=> ! [M: nat] :
( ( groups7311177749621191930dd_sum @ nat @ A @ ( gbinomial @ A @ ( plus_plus @ A @ ( times_times @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ ( semiring_1_of_nat @ A @ M ) ) @ ( one_one @ A ) ) ) @ ( set_ord_atMost @ nat @ M ) )
= ( power_power @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ M ) ) ) ) ).
% gbinomial_r_part_sum
thf(fact_3571_prod__induct7,axiom,
! [G3: $tType,F: $tType,E3: $tType,D6: $tType,C: $tType,B: $tType,A: $tType,P: ( product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D6 @ ( product_prod @ E3 @ ( product_prod @ F @ G3 ) ) ) ) ) ) > $o,X2: product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D6 @ ( product_prod @ E3 @ ( product_prod @ F @ G3 ) ) ) ) )] :
( ! [A4: A,B4: B,C4: C,D4: D6,E2: E3,F3: F,G4: G3] : ( P @ ( product_Pair @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D6 @ ( product_prod @ E3 @ ( product_prod @ F @ G3 ) ) ) ) ) @ A4 @ ( product_Pair @ B @ ( product_prod @ C @ ( product_prod @ D6 @ ( product_prod @ E3 @ ( product_prod @ F @ G3 ) ) ) ) @ B4 @ ( product_Pair @ C @ ( product_prod @ D6 @ ( product_prod @ E3 @ ( product_prod @ F @ G3 ) ) ) @ C4 @ ( product_Pair @ D6 @ ( product_prod @ E3 @ ( product_prod @ F @ G3 ) ) @ D4 @ ( product_Pair @ E3 @ ( product_prod @ F @ G3 ) @ E2 @ ( product_Pair @ F @ G3 @ F3 @ G4 ) ) ) ) ) ) )
=> ( P @ X2 ) ) ).
% prod_induct7
thf(fact_3572_prod__induct6,axiom,
! [F: $tType,E3: $tType,D6: $tType,C: $tType,B: $tType,A: $tType,P: ( product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D6 @ ( product_prod @ E3 @ F ) ) ) ) ) > $o,X2: product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D6 @ ( product_prod @ E3 @ F ) ) ) )] :
( ! [A4: A,B4: B,C4: C,D4: D6,E2: E3,F3: F] : ( P @ ( product_Pair @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D6 @ ( product_prod @ E3 @ F ) ) ) ) @ A4 @ ( product_Pair @ B @ ( product_prod @ C @ ( product_prod @ D6 @ ( product_prod @ E3 @ F ) ) ) @ B4 @ ( product_Pair @ C @ ( product_prod @ D6 @ ( product_prod @ E3 @ F ) ) @ C4 @ ( product_Pair @ D6 @ ( product_prod @ E3 @ F ) @ D4 @ ( product_Pair @ E3 @ F @ E2 @ F3 ) ) ) ) ) )
=> ( P @ X2 ) ) ).
% prod_induct6
thf(fact_3573_prod_Oinject,axiom,
! [A: $tType,B: $tType,X1: A,X22: B,Y15: A,Y2: B] :
( ( ( product_Pair @ A @ B @ X1 @ X22 )
= ( product_Pair @ A @ B @ Y15 @ Y2 ) )
= ( ( X1 = Y15 )
& ( X22 = Y2 ) ) ) ).
% prod.inject
thf(fact_3574_old_Oprod_Oinject,axiom,
! [A: $tType,B: $tType,A3: A,B3: B,A5: A,B5: B] :
( ( ( product_Pair @ A @ B @ A3 @ B3 )
= ( product_Pair @ A @ B @ A5 @ B5 ) )
= ( ( A3 = A5 )
& ( B3 = B5 ) ) ) ).
% old.prod.inject
thf(fact_3575_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_3576_atMost__subset__iff,axiom,
! [A: $tType] :
( ( preorder @ A )
=> ! [X2: A,Y3: A] :
( ( ord_less_eq @ ( set @ A ) @ ( set_ord_atMost @ A @ X2 ) @ ( set_ord_atMost @ A @ Y3 ) )
= ( ord_less_eq @ A @ X2 @ Y3 ) ) ) ).
% atMost_subset_iff
thf(fact_3577_prod__eq__1__iff,axiom,
! [A: $tType,A2: set @ A,F2: A > nat] :
( ( finite_finite2 @ A @ A2 )
=> ( ( ( groups7121269368397514597t_prod @ A @ nat @ F2 @ A2 )
= ( one_one @ nat ) )
= ( ! [X3: A] :
( ( member @ A @ X3 @ A2 )
=> ( ( F2 @ X3 )
= ( one_one @ nat ) ) ) ) ) ) ).
% prod_eq_1_iff
thf(fact_3578_Icc__subset__Iic__iff,axiom,
! [A: $tType] :
( ( preorder @ A )
=> ! [L: A,H2: A,H4: A] :
( ( ord_less_eq @ ( set @ A ) @ ( set_or1337092689740270186AtMost @ A @ L @ H2 ) @ ( set_ord_atMost @ A @ H4 ) )
= ( ~ ( ord_less_eq @ A @ L @ H2 )
| ( ord_less_eq @ A @ H2 @ H4 ) ) ) ) ).
% Icc_subset_Iic_iff
thf(fact_3579_sum_OatMost__Suc,axiom,
! [A: $tType] :
( ( comm_monoid_add @ A )
=> ! [G: nat > A,N: nat] :
( ( groups7311177749621191930dd_sum @ nat @ A @ G @ ( set_ord_atMost @ nat @ ( suc @ N ) ) )
= ( plus_plus @ A @ ( groups7311177749621191930dd_sum @ nat @ A @ G @ ( set_ord_atMost @ nat @ N ) ) @ ( G @ ( suc @ N ) ) ) ) ) ).
% sum.atMost_Suc
thf(fact_3580_prod_OatMost__Suc,axiom,
! [A: $tType] :
( ( comm_monoid_mult @ A )
=> ! [G: nat > A,N: nat] :
( ( groups7121269368397514597t_prod @ nat @ A @ G @ ( set_ord_atMost @ nat @ ( suc @ N ) ) )
= ( times_times @ A @ ( groups7121269368397514597t_prod @ nat @ A @ G @ ( set_ord_atMost @ nat @ N ) ) @ ( G @ ( suc @ N ) ) ) ) ) ).
% prod.atMost_Suc
thf(fact_3581_prod__pos__nat__iff,axiom,
! [A: $tType,A2: set @ A,F2: A > nat] :
( ( finite_finite2 @ A @ A2 )
=> ( ( ord_less @ nat @ ( zero_zero @ nat ) @ ( groups7121269368397514597t_prod @ A @ nat @ F2 @ A2 ) )
= ( ! [X3: A] :
( ( member @ A @ X3 @ A2 )
=> ( ord_less @ nat @ ( zero_zero @ nat ) @ ( F2 @ X3 ) ) ) ) ) ) ).
% prod_pos_nat_iff
thf(fact_3582_int__prod,axiom,
! [B: $tType,F2: B > nat,A2: set @ B] :
( ( semiring_1_of_nat @ int @ ( groups7121269368397514597t_prod @ B @ nat @ F2 @ A2 ) )
= ( groups7121269368397514597t_prod @ B @ int
@ ^ [X3: B] : ( semiring_1_of_nat @ int @ ( F2 @ X3 ) )
@ A2 ) ) ).
% int_prod
thf(fact_3583_atMost__def,axiom,
! [A: $tType] :
( ( ord @ A )
=> ( ( set_ord_atMost @ A )
= ( ^ [U2: A] :
( collect @ A
@ ^ [X3: A] : ( ord_less_eq @ A @ X3 @ U2 ) ) ) ) ) ).
% atMost_def
thf(fact_3584_lessThan__Suc__atMost,axiom,
! [K: nat] :
( ( set_ord_lessThan @ nat @ ( suc @ K ) )
= ( set_ord_atMost @ nat @ K ) ) ).
% lessThan_Suc_atMost
thf(fact_3585_atMost__Suc,axiom,
! [K: nat] :
( ( set_ord_atMost @ nat @ ( suc @ K ) )
= ( insert @ nat @ ( suc @ K ) @ ( set_ord_atMost @ nat @ K ) ) ) ).
% atMost_Suc
thf(fact_3586_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_3587_not__Iic__le__Icc,axiom,
! [A: $tType] :
( ( no_bot @ A )
=> ! [H2: A,L4: A,H4: A] :
~ ( ord_less_eq @ ( set @ A ) @ ( set_ord_atMost @ A @ H2 ) @ ( set_or1337092689740270186AtMost @ A @ L4 @ H4 ) ) ) ).
% not_Iic_le_Icc
thf(fact_3588_finite__nat__iff__bounded__le,axiom,
( ( finite_finite2 @ nat )
= ( ^ [S8: set @ nat] :
? [K3: nat] : ( ord_less_eq @ ( set @ nat ) @ S8 @ ( set_ord_atMost @ nat @ K3 ) ) ) ) ).
% finite_nat_iff_bounded_le
thf(fact_3589_frac__ge__0,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [X2: A] : ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( archimedean_frac @ A @ X2 ) ) ) ).
% frac_ge_0
thf(fact_3590_frac__lt__1,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [X2: A] : ( ord_less @ A @ ( archimedean_frac @ A @ X2 ) @ ( one_one @ A ) ) ) ).
% frac_lt_1
thf(fact_3591_frac__1__eq,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [X2: A] :
( ( archimedean_frac @ A @ ( plus_plus @ A @ X2 @ ( one_one @ A ) ) )
= ( archimedean_frac @ A @ X2 ) ) ) ).
% frac_1_eq
thf(fact_3592_old_Oprod_Oexhaust,axiom,
! [A: $tType,B: $tType,Y3: product_prod @ A @ B] :
~ ! [A4: A,B4: B] :
( Y3
!= ( product_Pair @ A @ B @ A4 @ B4 ) ) ).
% old.prod.exhaust
thf(fact_3593_surj__pair,axiom,
! [A: $tType,B: $tType,P6: product_prod @ A @ B] :
? [X4: A,Y4: B] :
( P6
= ( product_Pair @ A @ B @ X4 @ Y4 ) ) ).
% surj_pair
thf(fact_3594_prod__cases,axiom,
! [B: $tType,A: $tType,P: ( product_prod @ A @ B ) > $o,P6: product_prod @ A @ B] :
( ! [A4: A,B4: B] : ( P @ ( product_Pair @ A @ B @ A4 @ B4 ) )
=> ( P @ P6 ) ) ).
% prod_cases
thf(fact_3595_Pair__inject,axiom,
! [A: $tType,B: $tType,A3: A,B3: B,A5: A,B5: B] :
( ( ( product_Pair @ A @ B @ A3 @ B3 )
= ( product_Pair @ A @ B @ A5 @ B5 ) )
=> ~ ( ( A3 = A5 )
=> ( B3 != B5 ) ) ) ).
% Pair_inject
thf(fact_3596_Iic__subset__Iio__iff,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A3: A,B3: A] :
( ( ord_less_eq @ ( set @ A ) @ ( set_ord_atMost @ A @ A3 ) @ ( set_ord_lessThan @ A @ B3 ) )
= ( ord_less @ A @ A3 @ B3 ) ) ) ).
% Iic_subset_Iio_iff
thf(fact_3597_prod__int__eq,axiom,
! [I: nat,J: nat] :
( ( groups7121269368397514597t_prod @ nat @ int @ ( semiring_1_of_nat @ int ) @ ( set_or1337092689740270186AtMost @ nat @ I @ J ) )
= ( groups7121269368397514597t_prod @ int @ int
@ ^ [X3: int] : X3
@ ( set_or1337092689740270186AtMost @ int @ ( semiring_1_of_nat @ int @ I ) @ ( semiring_1_of_nat @ int @ J ) ) ) ) ).
% prod_int_eq
thf(fact_3598_frac__def,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ( ( archimedean_frac @ A )
= ( ^ [X3: A] : ( minus_minus @ A @ X3 @ ( ring_1_of_int @ A @ ( archim6421214686448440834_floor @ A @ X3 ) ) ) ) ) ) ).
% frac_def
thf(fact_3599_sum_OatMost__Suc__shift,axiom,
! [A: $tType] :
( ( comm_monoid_add @ A )
=> ! [G: nat > A,N: nat] :
( ( groups7311177749621191930dd_sum @ nat @ A @ G @ ( set_ord_atMost @ nat @ ( suc @ N ) ) )
= ( plus_plus @ A @ ( G @ ( zero_zero @ nat ) )
@ ( groups7311177749621191930dd_sum @ nat @ A
@ ^ [I4: nat] : ( G @ ( suc @ I4 ) )
@ ( set_ord_atMost @ nat @ N ) ) ) ) ) ).
% sum.atMost_Suc_shift
thf(fact_3600_sum__telescope,axiom,
! [A: $tType] :
( ( ab_group_add @ A )
=> ! [F2: nat > A,I: nat] :
( ( groups7311177749621191930dd_sum @ nat @ A
@ ^ [I4: nat] : ( minus_minus @ A @ ( F2 @ I4 ) @ ( F2 @ ( suc @ I4 ) ) )
@ ( set_ord_atMost @ nat @ I ) )
= ( minus_minus @ A @ ( F2 @ ( zero_zero @ nat ) ) @ ( F2 @ ( suc @ I ) ) ) ) ) ).
% sum_telescope
thf(fact_3601_polyfun__eq__coeffs,axiom,
! [A: $tType] :
( ( ( real_V8999393235501362500lgebra @ A )
& ( idom @ A ) )
=> ! [C3: nat > A,N: nat,D2: nat > A] :
( ( ! [X3: A] :
( ( groups7311177749621191930dd_sum @ nat @ A
@ ^ [I4: nat] : ( times_times @ A @ ( C3 @ I4 ) @ ( power_power @ A @ X3 @ I4 ) )
@ ( set_ord_atMost @ nat @ N ) )
= ( groups7311177749621191930dd_sum @ nat @ A
@ ^ [I4: nat] : ( times_times @ A @ ( D2 @ I4 ) @ ( power_power @ A @ X3 @ I4 ) )
@ ( set_ord_atMost @ nat @ N ) ) ) )
= ( ! [I4: nat] :
( ( ord_less_eq @ nat @ I4 @ N )
=> ( ( C3 @ I4 )
= ( D2 @ I4 ) ) ) ) ) ) ).
% polyfun_eq_coeffs
thf(fact_3602_prod_OatMost__Suc__shift,axiom,
! [A: $tType] :
( ( comm_monoid_mult @ A )
=> ! [G: nat > A,N: nat] :
( ( groups7121269368397514597t_prod @ nat @ A @ G @ ( set_ord_atMost @ nat @ ( suc @ N ) ) )
= ( times_times @ A @ ( G @ ( zero_zero @ nat ) )
@ ( groups7121269368397514597t_prod @ nat @ A
@ ^ [I4: nat] : ( G @ ( suc @ I4 ) )
@ ( set_ord_atMost @ nat @ N ) ) ) ) ) ).
% prod.atMost_Suc_shift
thf(fact_3603_prod__int__plus__eq,axiom,
! [I: nat,J: nat] :
( ( groups7121269368397514597t_prod @ nat @ int @ ( semiring_1_of_nat @ int ) @ ( set_or1337092689740270186AtMost @ nat @ I @ ( plus_plus @ nat @ I @ J ) ) )
= ( groups7121269368397514597t_prod @ int @ int
@ ^ [X3: int] : X3
@ ( set_or1337092689740270186AtMost @ int @ ( semiring_1_of_nat @ int @ I ) @ ( semiring_1_of_nat @ int @ ( plus_plus @ nat @ I @ J ) ) ) ) ) ).
% prod_int_plus_eq
thf(fact_3604_sum_Onested__swap_H,axiom,
! [A: $tType] :
( ( comm_monoid_add @ A )
=> ! [A3: nat > nat > A,N: nat] :
( ( groups7311177749621191930dd_sum @ nat @ A
@ ^ [I4: nat] : ( groups7311177749621191930dd_sum @ nat @ A @ ( A3 @ I4 ) @ ( set_ord_lessThan @ nat @ I4 ) )
@ ( set_ord_atMost @ nat @ N ) )
= ( groups7311177749621191930dd_sum @ nat @ A
@ ^ [J3: nat] :
( groups7311177749621191930dd_sum @ nat @ A
@ ^ [I4: nat] : ( A3 @ I4 @ J3 )
@ ( set_or1337092689740270186AtMost @ nat @ ( suc @ J3 ) @ N ) )
@ ( set_ord_lessThan @ nat @ N ) ) ) ) ).
% sum.nested_swap'
thf(fact_3605_prod_Onested__swap_H,axiom,
! [A: $tType] :
( ( comm_monoid_mult @ A )
=> ! [A3: nat > nat > A,N: nat] :
( ( groups7121269368397514597t_prod @ nat @ A
@ ^ [I4: nat] : ( groups7121269368397514597t_prod @ nat @ A @ ( A3 @ I4 ) @ ( set_ord_lessThan @ nat @ I4 ) )
@ ( set_ord_atMost @ nat @ N ) )
= ( groups7121269368397514597t_prod @ nat @ A
@ ^ [J3: nat] :
( groups7121269368397514597t_prod @ nat @ A
@ ^ [I4: nat] : ( A3 @ I4 @ J3 )
@ ( set_or1337092689740270186AtMost @ nat @ ( suc @ J3 ) @ N ) )
@ ( set_ord_lessThan @ nat @ N ) ) ) ) ).
% prod.nested_swap'
thf(fact_3606_frac__eq,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [X2: A] :
( ( ( archimedean_frac @ A @ X2 )
= X2 )
= ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ X2 )
& ( ord_less @ A @ X2 @ ( one_one @ A ) ) ) ) ) ).
% frac_eq
thf(fact_3607_frac__add,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [X2: A,Y3: A] :
( ( ( ord_less @ A @ ( plus_plus @ A @ ( archimedean_frac @ A @ X2 ) @ ( archimedean_frac @ A @ Y3 ) ) @ ( one_one @ A ) )
=> ( ( archimedean_frac @ A @ ( plus_plus @ A @ X2 @ Y3 ) )
= ( plus_plus @ A @ ( archimedean_frac @ A @ X2 ) @ ( archimedean_frac @ A @ Y3 ) ) ) )
& ( ~ ( ord_less @ A @ ( plus_plus @ A @ ( archimedean_frac @ A @ X2 ) @ ( archimedean_frac @ A @ Y3 ) ) @ ( one_one @ A ) )
=> ( ( archimedean_frac @ A @ ( plus_plus @ A @ X2 @ Y3 ) )
= ( minus_minus @ A @ ( plus_plus @ A @ ( archimedean_frac @ A @ X2 ) @ ( archimedean_frac @ A @ Y3 ) ) @ ( one_one @ A ) ) ) ) ) ) ).
% frac_add
thf(fact_3608_zero__polynom__imp__zero__coeffs,axiom,
! [A: $tType] :
( ( ( ab_semigroup_mult @ A )
& ( real_V8999393235501362500lgebra @ A ) )
=> ! [C3: nat > A,N: nat,K: nat] :
( ! [W3: A] :
( ( groups7311177749621191930dd_sum @ nat @ A
@ ^ [I4: nat] : ( times_times @ A @ ( C3 @ I4 ) @ ( power_power @ A @ W3 @ I4 ) )
@ ( 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_3609_polyfun__eq__0,axiom,
! [A: $tType] :
( ( ( real_V8999393235501362500lgebra @ A )
& ( idom @ A ) )
=> ! [C3: nat > A,N: nat] :
( ( ! [X3: A] :
( ( groups7311177749621191930dd_sum @ nat @ A
@ ^ [I4: nat] : ( times_times @ A @ ( C3 @ I4 ) @ ( power_power @ A @ X3 @ I4 ) )
@ ( set_ord_atMost @ nat @ N ) )
= ( zero_zero @ A ) ) )
= ( ! [I4: nat] :
( ( ord_less_eq @ nat @ I4 @ N )
=> ( ( C3 @ I4 )
= ( zero_zero @ A ) ) ) ) ) ) ).
% polyfun_eq_0
thf(fact_3610_sum_OatMost__shift,axiom,
! [A: $tType] :
( ( comm_monoid_add @ A )
=> ! [G: nat > A,N: nat] :
( ( groups7311177749621191930dd_sum @ nat @ A @ G @ ( set_ord_atMost @ nat @ N ) )
= ( plus_plus @ A @ ( G @ ( zero_zero @ nat ) )
@ ( groups7311177749621191930dd_sum @ nat @ A
@ ^ [I4: nat] : ( G @ ( suc @ I4 ) )
@ ( set_ord_lessThan @ nat @ N ) ) ) ) ) ).
% sum.atMost_shift
thf(fact_3611_sum__up__index__split,axiom,
! [A: $tType] :
( ( comm_monoid_add @ A )
=> ! [F2: nat > A,M: nat,N: nat] :
( ( groups7311177749621191930dd_sum @ nat @ A @ F2 @ ( set_ord_atMost @ nat @ ( plus_plus @ nat @ M @ N ) ) )
= ( plus_plus @ A @ ( groups7311177749621191930dd_sum @ nat @ A @ F2 @ ( set_ord_atMost @ nat @ M ) ) @ ( groups7311177749621191930dd_sum @ nat @ A @ F2 @ ( set_or1337092689740270186AtMost @ nat @ ( suc @ M ) @ ( plus_plus @ nat @ M @ N ) ) ) ) ) ) ).
% sum_up_index_split
thf(fact_3612_prod_OatMost__shift,axiom,
! [A: $tType] :
( ( comm_monoid_mult @ A )
=> ! [G: nat > A,N: nat] :
( ( groups7121269368397514597t_prod @ nat @ A @ G @ ( set_ord_atMost @ nat @ N ) )
= ( times_times @ A @ ( G @ ( zero_zero @ nat ) )
@ ( groups7121269368397514597t_prod @ nat @ A
@ ^ [I4: nat] : ( G @ ( suc @ I4 ) )
@ ( set_ord_lessThan @ nat @ N ) ) ) ) ) ).
% prod.atMost_shift
thf(fact_3613_gbinomial__parallel__sum,axiom,
! [A: $tType] :
( ( field_char_0 @ A )
=> ! [A3: A,N: nat] :
( ( groups7311177749621191930dd_sum @ nat @ A
@ ^ [K3: nat] : ( gbinomial @ A @ ( plus_plus @ A @ A3 @ ( semiring_1_of_nat @ A @ K3 ) ) @ K3 )
@ ( set_ord_atMost @ nat @ N ) )
= ( gbinomial @ A @ ( plus_plus @ A @ ( plus_plus @ A @ A3 @ ( semiring_1_of_nat @ A @ N ) ) @ ( one_one @ A ) ) @ N ) ) ) ).
% gbinomial_parallel_sum
thf(fact_3614_sum__gp__basic,axiom,
! [A: $tType] :
( ( ( monoid_mult @ A )
& ( comm_ring @ A ) )
=> ! [X2: A,N: nat] :
( ( times_times @ A @ ( minus_minus @ A @ ( one_one @ A ) @ X2 ) @ ( groups7311177749621191930dd_sum @ nat @ A @ ( power_power @ A @ X2 ) @ ( set_ord_atMost @ nat @ N ) ) )
= ( minus_minus @ A @ ( one_one @ A ) @ ( power_power @ A @ X2 @ ( suc @ N ) ) ) ) ) ).
% sum_gp_basic
thf(fact_3615_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
@ ^ [Z4: A] :
( ( groups7311177749621191930dd_sum @ nat @ A
@ ^ [I4: nat] : ( times_times @ A @ ( C3 @ I4 ) @ ( power_power @ A @ Z4 @ I4 ) )
@ ( set_ord_atMost @ nat @ N ) )
= ( zero_zero @ A ) ) ) ) ) ) ) ).
% polyfun_roots_finite
thf(fact_3616_polyfun__finite__roots,axiom,
! [A: $tType] :
( ( ( real_V8999393235501362500lgebra @ A )
& ( idom @ A ) )
=> ! [C3: nat > A,N: nat] :
( ( finite_finite2 @ A
@ ( collect @ A
@ ^ [X3: A] :
( ( groups7311177749621191930dd_sum @ nat @ A
@ ^ [I4: nat] : ( times_times @ A @ ( C3 @ I4 ) @ ( power_power @ A @ X3 @ I4 ) )
@ ( set_ord_atMost @ nat @ N ) )
= ( zero_zero @ A ) ) ) )
= ( ? [I4: nat] :
( ( ord_less_eq @ nat @ I4 @ N )
& ( ( C3 @ I4 )
!= ( zero_zero @ A ) ) ) ) ) ) ).
% polyfun_finite_roots
thf(fact_3617_polyfun__linear__factor__root,axiom,
! [A: $tType] :
( ( idom @ A )
=> ! [C3: nat > A,A3: A,N: nat] :
( ( ( groups7311177749621191930dd_sum @ nat @ A
@ ^ [I4: nat] : ( times_times @ A @ ( C3 @ I4 ) @ ( power_power @ A @ A3 @ I4 ) )
@ ( set_ord_atMost @ nat @ N ) )
= ( zero_zero @ A ) )
=> ~ ! [B4: nat > A] :
~ ! [Z5: A] :
( ( groups7311177749621191930dd_sum @ nat @ A
@ ^ [I4: nat] : ( times_times @ A @ ( C3 @ I4 ) @ ( power_power @ A @ Z5 @ I4 ) )
@ ( set_ord_atMost @ nat @ N ) )
= ( times_times @ A @ ( minus_minus @ A @ Z5 @ A3 )
@ ( groups7311177749621191930dd_sum @ nat @ A
@ ^ [I4: nat] : ( times_times @ A @ ( B4 @ I4 ) @ ( power_power @ A @ Z5 @ I4 ) )
@ ( set_ord_lessThan @ nat @ N ) ) ) ) ) ) ).
% polyfun_linear_factor_root
thf(fact_3618_polyfun__linear__factor,axiom,
! [A: $tType] :
( ( idom @ A )
=> ! [C3: nat > A,N: nat,A3: A] :
? [B4: nat > A] :
! [Z5: A] :
( ( groups7311177749621191930dd_sum @ nat @ A
@ ^ [I4: nat] : ( times_times @ A @ ( C3 @ I4 ) @ ( power_power @ A @ Z5 @ I4 ) )
@ ( set_ord_atMost @ nat @ N ) )
= ( plus_plus @ A
@ ( times_times @ A @ ( minus_minus @ A @ Z5 @ A3 )
@ ( groups7311177749621191930dd_sum @ nat @ A
@ ^ [I4: nat] : ( times_times @ A @ ( B4 @ I4 ) @ ( power_power @ A @ Z5 @ I4 ) )
@ ( set_ord_lessThan @ nat @ N ) ) )
@ ( groups7311177749621191930dd_sum @ nat @ A
@ ^ [I4: nat] : ( times_times @ A @ ( C3 @ I4 ) @ ( power_power @ A @ A3 @ I4 ) )
@ ( set_ord_atMost @ nat @ N ) ) ) ) ) ).
% polyfun_linear_factor
thf(fact_3619_sum__power__shift,axiom,
! [A: $tType] :
( ( ( monoid_mult @ A )
& ( comm_ring @ A ) )
=> ! [M: nat,N: nat,X2: A] :
( ( ord_less_eq @ nat @ M @ N )
=> ( ( groups7311177749621191930dd_sum @ nat @ A @ ( power_power @ A @ X2 ) @ ( set_or1337092689740270186AtMost @ nat @ M @ N ) )
= ( times_times @ A @ ( power_power @ A @ X2 @ M ) @ ( groups7311177749621191930dd_sum @ nat @ A @ ( power_power @ A @ X2 ) @ ( set_ord_atMost @ nat @ ( minus_minus @ nat @ N @ M ) ) ) ) ) ) ) ).
% sum_power_shift
thf(fact_3620_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_3621_sum_Oin__pairs__0,axiom,
! [A: $tType] :
( ( comm_monoid_add @ A )
=> ! [G: nat > A,N: nat] :
( ( groups7311177749621191930dd_sum @ nat @ A @ G @ ( set_ord_atMost @ nat @ ( suc @ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N ) ) ) )
= ( groups7311177749621191930dd_sum @ nat @ A
@ ^ [I4: nat] : ( plus_plus @ A @ ( G @ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ I4 ) ) @ ( G @ ( suc @ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ I4 ) ) ) )
@ ( set_ord_atMost @ nat @ N ) ) ) ) ).
% sum.in_pairs_0
thf(fact_3622_polynomial__product,axiom,
! [A: $tType] :
( ( idom @ A )
=> ! [M: nat,A3: nat > A,N: nat,B3: nat > A,X2: A] :
( ! [I2: nat] :
( ( ord_less @ nat @ M @ I2 )
=> ( ( A3 @ I2 )
= ( zero_zero @ A ) ) )
=> ( ! [J2: nat] :
( ( ord_less @ nat @ N @ J2 )
=> ( ( B3 @ J2 )
= ( zero_zero @ A ) ) )
=> ( ( times_times @ A
@ ( groups7311177749621191930dd_sum @ nat @ A
@ ^ [I4: nat] : ( times_times @ A @ ( A3 @ I4 ) @ ( power_power @ A @ X2 @ I4 ) )
@ ( set_ord_atMost @ nat @ M ) )
@ ( groups7311177749621191930dd_sum @ nat @ A
@ ^ [J3: nat] : ( times_times @ A @ ( B3 @ J3 ) @ ( power_power @ A @ X2 @ J3 ) )
@ ( set_ord_atMost @ nat @ N ) ) )
= ( groups7311177749621191930dd_sum @ nat @ A
@ ^ [R5: nat] :
( times_times @ A
@ ( groups7311177749621191930dd_sum @ nat @ A
@ ^ [K3: nat] : ( times_times @ A @ ( A3 @ K3 ) @ ( B3 @ ( minus_minus @ nat @ R5 @ K3 ) ) )
@ ( set_ord_atMost @ nat @ R5 ) )
@ ( power_power @ A @ X2 @ R5 ) )
@ ( set_ord_atMost @ nat @ ( plus_plus @ nat @ M @ N ) ) ) ) ) ) ) ).
% polynomial_product
thf(fact_3623_prod_Oin__pairs__0,axiom,
! [A: $tType] :
( ( comm_monoid_mult @ A )
=> ! [G: nat > A,N: nat] :
( ( groups7121269368397514597t_prod @ nat @ A @ G @ ( set_ord_atMost @ nat @ ( suc @ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N ) ) ) )
= ( groups7121269368397514597t_prod @ nat @ A
@ ^ [I4: nat] : ( times_times @ A @ ( G @ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ I4 ) ) @ ( G @ ( suc @ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ I4 ) ) ) )
@ ( set_ord_atMost @ nat @ N ) ) ) ) ).
% prod.in_pairs_0
thf(fact_3624_prod__cases3,axiom,
! [A: $tType,B: $tType,C: $tType,Y3: product_prod @ A @ ( product_prod @ B @ C )] :
~ ! [A4: A,B4: B,C4: C] :
( Y3
!= ( product_Pair @ A @ ( product_prod @ B @ C ) @ A4 @ ( product_Pair @ B @ C @ B4 @ C4 ) ) ) ).
% prod_cases3
thf(fact_3625_prod__induct3,axiom,
! [C: $tType,B: $tType,A: $tType,P: ( product_prod @ A @ ( product_prod @ B @ C ) ) > $o,X2: product_prod @ A @ ( product_prod @ B @ C )] :
( ! [A4: A,B4: B,C4: C] : ( P @ ( product_Pair @ A @ ( product_prod @ B @ C ) @ A4 @ ( product_Pair @ B @ C @ B4 @ C4 ) ) )
=> ( P @ X2 ) ) ).
% prod_induct3
thf(fact_3626_polyfun__eq__const,axiom,
! [A: $tType] :
( ( ( real_V8999393235501362500lgebra @ A )
& ( idom @ A ) )
=> ! [C3: nat > A,N: nat,K: A] :
( ( ! [X3: A] :
( ( groups7311177749621191930dd_sum @ nat @ A
@ ^ [I4: nat] : ( times_times @ A @ ( C3 @ I4 ) @ ( power_power @ A @ X3 @ I4 ) )
@ ( set_ord_atMost @ nat @ N ) )
= K ) )
= ( ( ( C3 @ ( zero_zero @ nat ) )
= K )
& ! [X3: nat] :
( ( member @ nat @ X3 @ ( set_or1337092689740270186AtMost @ nat @ ( one_one @ nat ) @ N ) )
=> ( ( C3 @ X3 )
= ( zero_zero @ A ) ) ) ) ) ) ).
% polyfun_eq_const
thf(fact_3627_polynomial__product__nat,axiom,
! [M: nat,A3: nat > nat,N: nat,B3: nat > nat,X2: nat] :
( ! [I2: nat] :
( ( ord_less @ nat @ M @ I2 )
=> ( ( A3 @ I2 )
= ( zero_zero @ nat ) ) )
=> ( ! [J2: nat] :
( ( ord_less @ nat @ N @ J2 )
=> ( ( B3 @ J2 )
= ( zero_zero @ nat ) ) )
=> ( ( times_times @ nat
@ ( groups7311177749621191930dd_sum @ nat @ nat
@ ^ [I4: nat] : ( times_times @ nat @ ( A3 @ I4 ) @ ( power_power @ nat @ X2 @ I4 ) )
@ ( set_ord_atMost @ nat @ M ) )
@ ( groups7311177749621191930dd_sum @ nat @ nat
@ ^ [J3: nat] : ( times_times @ nat @ ( B3 @ J3 ) @ ( power_power @ nat @ X2 @ J3 ) )
@ ( set_ord_atMost @ nat @ N ) ) )
= ( groups7311177749621191930dd_sum @ nat @ nat
@ ^ [R5: nat] :
( times_times @ nat
@ ( groups7311177749621191930dd_sum @ nat @ nat
@ ^ [K3: nat] : ( times_times @ nat @ ( A3 @ K3 ) @ ( B3 @ ( minus_minus @ nat @ R5 @ K3 ) ) )
@ ( set_ord_atMost @ nat @ R5 ) )
@ ( power_power @ nat @ X2 @ R5 ) )
@ ( set_ord_atMost @ nat @ ( plus_plus @ nat @ M @ N ) ) ) ) ) ) ).
% polynomial_product_nat
thf(fact_3628_floor__add,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [X2: A,Y3: A] :
( ( ( ord_less @ A @ ( plus_plus @ A @ ( archimedean_frac @ A @ X2 ) @ ( archimedean_frac @ A @ Y3 ) ) @ ( one_one @ A ) )
=> ( ( archim6421214686448440834_floor @ A @ ( plus_plus @ A @ X2 @ Y3 ) )
= ( plus_plus @ int @ ( archim6421214686448440834_floor @ A @ X2 ) @ ( archim6421214686448440834_floor @ A @ Y3 ) ) ) )
& ( ~ ( ord_less @ A @ ( plus_plus @ A @ ( archimedean_frac @ A @ X2 ) @ ( archimedean_frac @ A @ Y3 ) ) @ ( one_one @ A ) )
=> ( ( archim6421214686448440834_floor @ A @ ( plus_plus @ A @ X2 @ Y3 ) )
= ( plus_plus @ int @ ( plus_plus @ int @ ( archim6421214686448440834_floor @ A @ X2 ) @ ( archim6421214686448440834_floor @ A @ Y3 ) ) @ ( one_one @ int ) ) ) ) ) ) ).
% floor_add
thf(fact_3629_sum_Ozero__middle,axiom,
! [A: $tType] :
( ( comm_monoid_add @ A )
=> ! [P6: nat,K: nat,G: nat > A,H2: nat > A] :
( ( ord_less_eq @ nat @ ( one_one @ nat ) @ P6 )
=> ( ( ord_less_eq @ nat @ K @ P6 )
=> ( ( groups7311177749621191930dd_sum @ nat @ A
@ ^ [J3: nat] : ( if @ A @ ( ord_less @ nat @ J3 @ K ) @ ( G @ J3 ) @ ( if @ A @ ( J3 = K ) @ ( zero_zero @ A ) @ ( H2 @ ( minus_minus @ nat @ J3 @ ( suc @ ( zero_zero @ nat ) ) ) ) ) )
@ ( set_ord_atMost @ nat @ P6 ) )
= ( groups7311177749621191930dd_sum @ nat @ A
@ ^ [J3: nat] : ( if @ A @ ( ord_less @ nat @ J3 @ K ) @ ( G @ J3 ) @ ( H2 @ J3 ) )
@ ( set_ord_atMost @ nat @ ( minus_minus @ nat @ P6 @ ( suc @ ( zero_zero @ nat ) ) ) ) ) ) ) ) ) ).
% sum.zero_middle
thf(fact_3630_prod_Ozero__middle,axiom,
! [A: $tType] :
( ( comm_monoid_mult @ A )
=> ! [P6: nat,K: nat,G: nat > A,H2: nat > A] :
( ( ord_less_eq @ nat @ ( one_one @ nat ) @ P6 )
=> ( ( ord_less_eq @ nat @ K @ P6 )
=> ( ( groups7121269368397514597t_prod @ nat @ A
@ ^ [J3: nat] : ( if @ A @ ( ord_less @ nat @ J3 @ K ) @ ( G @ J3 ) @ ( if @ A @ ( J3 = K ) @ ( one_one @ A ) @ ( H2 @ ( minus_minus @ nat @ J3 @ ( suc @ ( zero_zero @ nat ) ) ) ) ) )
@ ( set_ord_atMost @ nat @ P6 ) )
= ( groups7121269368397514597t_prod @ nat @ A
@ ^ [J3: nat] : ( if @ A @ ( ord_less @ nat @ J3 @ K ) @ ( G @ J3 ) @ ( H2 @ J3 ) )
@ ( set_ord_atMost @ nat @ ( minus_minus @ nat @ P6 @ ( suc @ ( zero_zero @ nat ) ) ) ) ) ) ) ) ) ).
% prod.zero_middle
thf(fact_3631_sum__gp0,axiom,
! [A: $tType] :
( ( ( division_ring @ A )
& ( comm_ring @ A ) )
=> ! [X2: A,N: nat] :
( ( ( X2
= ( one_one @ A ) )
=> ( ( groups7311177749621191930dd_sum @ nat @ A @ ( power_power @ A @ X2 ) @ ( set_ord_atMost @ nat @ N ) )
= ( semiring_1_of_nat @ A @ ( plus_plus @ nat @ N @ ( one_one @ nat ) ) ) ) )
& ( ( X2
!= ( one_one @ A ) )
=> ( ( groups7311177749621191930dd_sum @ nat @ A @ ( power_power @ A @ X2 ) @ ( set_ord_atMost @ nat @ N ) )
= ( divide_divide @ A @ ( minus_minus @ A @ ( one_one @ A ) @ ( power_power @ A @ X2 @ ( suc @ N ) ) ) @ ( minus_minus @ A @ ( one_one @ A ) @ X2 ) ) ) ) ) ) ).
% sum_gp0
thf(fact_3632_gbinomial__sum__nat__pow2,axiom,
! [A: $tType] :
( ( field_char_0 @ A )
=> ! [M: nat] :
( ( groups7311177749621191930dd_sum @ nat @ A
@ ^ [K3: nat] : ( divide_divide @ A @ ( gbinomial @ A @ ( semiring_1_of_nat @ A @ ( plus_plus @ nat @ M @ K3 ) ) @ K3 ) @ ( power_power @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ K3 ) )
@ ( set_ord_atMost @ nat @ M ) )
= ( power_power @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ M ) ) ) ).
% gbinomial_sum_nat_pow2
thf(fact_3633_gbinomial__partial__sum__poly__xpos,axiom,
! [A: $tType] :
( ( field_char_0 @ A )
=> ! [M: nat,A3: A,X2: A,Y3: A] :
( ( groups7311177749621191930dd_sum @ nat @ A
@ ^ [K3: nat] : ( times_times @ A @ ( times_times @ A @ ( gbinomial @ A @ ( plus_plus @ A @ ( semiring_1_of_nat @ A @ M ) @ A3 ) @ K3 ) @ ( power_power @ A @ X2 @ K3 ) ) @ ( power_power @ A @ Y3 @ ( minus_minus @ nat @ M @ K3 ) ) )
@ ( set_ord_atMost @ nat @ M ) )
= ( groups7311177749621191930dd_sum @ nat @ A
@ ^ [K3: nat] : ( times_times @ A @ ( times_times @ A @ ( gbinomial @ A @ ( minus_minus @ A @ ( plus_plus @ A @ ( semiring_1_of_nat @ A @ K3 ) @ A3 ) @ ( one_one @ A ) ) @ K3 ) @ ( power_power @ A @ X2 @ K3 ) ) @ ( power_power @ A @ ( plus_plus @ A @ X2 @ Y3 ) @ ( minus_minus @ nat @ M @ K3 ) ) )
@ ( set_ord_atMost @ nat @ M ) ) ) ) ).
% gbinomial_partial_sum_poly_xpos
thf(fact_3634_polyfun__diff__alt,axiom,
! [A: $tType] :
( ( idom @ A )
=> ! [N: nat,A3: nat > A,X2: A,Y3: A] :
( ( ord_less_eq @ nat @ ( one_one @ nat ) @ N )
=> ( ( minus_minus @ A
@ ( groups7311177749621191930dd_sum @ nat @ A
@ ^ [I4: nat] : ( times_times @ A @ ( A3 @ I4 ) @ ( power_power @ A @ X2 @ I4 ) )
@ ( set_ord_atMost @ nat @ N ) )
@ ( groups7311177749621191930dd_sum @ nat @ A
@ ^ [I4: nat] : ( times_times @ A @ ( A3 @ I4 ) @ ( power_power @ A @ Y3 @ I4 ) )
@ ( set_ord_atMost @ nat @ N ) ) )
= ( times_times @ A @ ( minus_minus @ A @ X2 @ Y3 )
@ ( groups7311177749621191930dd_sum @ nat @ A
@ ^ [J3: nat] :
( groups7311177749621191930dd_sum @ nat @ A
@ ^ [K3: nat] : ( times_times @ A @ ( times_times @ A @ ( A3 @ ( plus_plus @ nat @ ( plus_plus @ nat @ J3 @ K3 ) @ ( one_one @ nat ) ) ) @ ( power_power @ A @ Y3 @ K3 ) ) @ ( power_power @ A @ X2 @ J3 ) )
@ ( set_ord_lessThan @ nat @ ( minus_minus @ nat @ N @ J3 ) ) )
@ ( set_ord_lessThan @ nat @ N ) ) ) ) ) ) ).
% polyfun_diff_alt
thf(fact_3635_polyfun__extremal__lemma,axiom,
! [A: $tType] :
( ( real_V8999393235501362500lgebra @ A )
=> ! [E: real,C3: nat > A,N: nat] :
( ( ord_less @ real @ ( zero_zero @ real ) @ E )
=> ? [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
@ ^ [I4: nat] : ( times_times @ A @ ( C3 @ I4 ) @ ( power_power @ A @ Z5 @ I4 ) )
@ ( set_ord_atMost @ nat @ N ) ) )
@ ( times_times @ real @ E @ ( power_power @ real @ ( real_V7770717601297561774m_norm @ A @ Z5 ) @ ( suc @ N ) ) ) ) ) ) ) ).
% polyfun_extremal_lemma
thf(fact_3636_polyfun__diff,axiom,
! [A: $tType] :
( ( idom @ A )
=> ! [N: nat,A3: nat > A,X2: A,Y3: A] :
( ( ord_less_eq @ nat @ ( one_one @ nat ) @ N )
=> ( ( minus_minus @ A
@ ( groups7311177749621191930dd_sum @ nat @ A
@ ^ [I4: nat] : ( times_times @ A @ ( A3 @ I4 ) @ ( power_power @ A @ X2 @ I4 ) )
@ ( set_ord_atMost @ nat @ N ) )
@ ( groups7311177749621191930dd_sum @ nat @ A
@ ^ [I4: nat] : ( times_times @ A @ ( A3 @ I4 ) @ ( power_power @ A @ Y3 @ I4 ) )
@ ( set_ord_atMost @ nat @ N ) ) )
= ( times_times @ A @ ( minus_minus @ A @ X2 @ Y3 )
@ ( groups7311177749621191930dd_sum @ nat @ A
@ ^ [J3: nat] :
( times_times @ A
@ ( groups7311177749621191930dd_sum @ nat @ A
@ ^ [I4: nat] : ( times_times @ A @ ( A3 @ I4 ) @ ( power_power @ A @ Y3 @ ( minus_minus @ nat @ ( minus_minus @ nat @ I4 @ J3 ) @ ( one_one @ nat ) ) ) )
@ ( set_or1337092689740270186AtMost @ nat @ ( suc @ J3 ) @ N ) )
@ ( power_power @ A @ X2 @ J3 ) )
@ ( set_ord_lessThan @ nat @ N ) ) ) ) ) ) ).
% polyfun_diff
thf(fact_3637_prod__cases4,axiom,
! [A: $tType,B: $tType,C: $tType,D6: $tType,Y3: product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ D6 ) )] :
~ ! [A4: A,B4: B,C4: C,D4: D6] :
( Y3
!= ( product_Pair @ A @ ( product_prod @ B @ ( product_prod @ C @ D6 ) ) @ A4 @ ( product_Pair @ B @ ( product_prod @ C @ D6 ) @ B4 @ ( product_Pair @ C @ D6 @ C4 @ D4 ) ) ) ) ).
% prod_cases4
thf(fact_3638_prod__cases5,axiom,
! [A: $tType,B: $tType,C: $tType,D6: $tType,E3: $tType,Y3: product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D6 @ E3 ) ) )] :
~ ! [A4: A,B4: B,C4: C,D4: D6,E2: E3] :
( Y3
!= ( product_Pair @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D6 @ E3 ) ) ) @ A4 @ ( product_Pair @ B @ ( product_prod @ C @ ( product_prod @ D6 @ E3 ) ) @ B4 @ ( product_Pair @ C @ ( product_prod @ D6 @ E3 ) @ C4 @ ( product_Pair @ D6 @ E3 @ D4 @ E2 ) ) ) ) ) ).
% prod_cases5
thf(fact_3639_prod__cases6,axiom,
! [A: $tType,B: $tType,C: $tType,D6: $tType,E3: $tType,F: $tType,Y3: product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D6 @ ( product_prod @ E3 @ F ) ) ) )] :
~ ! [A4: A,B4: B,C4: C,D4: D6,E2: E3,F3: F] :
( Y3
!= ( product_Pair @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D6 @ ( product_prod @ E3 @ F ) ) ) ) @ A4 @ ( product_Pair @ B @ ( product_prod @ C @ ( product_prod @ D6 @ ( product_prod @ E3 @ F ) ) ) @ B4 @ ( product_Pair @ C @ ( product_prod @ D6 @ ( product_prod @ E3 @ F ) ) @ C4 @ ( product_Pair @ D6 @ ( product_prod @ E3 @ F ) @ D4 @ ( product_Pair @ E3 @ F @ E2 @ F3 ) ) ) ) ) ) ).
% prod_cases6
thf(fact_3640_prod__cases7,axiom,
! [A: $tType,B: $tType,C: $tType,D6: $tType,E3: $tType,F: $tType,G3: $tType,Y3: product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D6 @ ( product_prod @ E3 @ ( product_prod @ F @ G3 ) ) ) ) )] :
~ ! [A4: A,B4: B,C4: C,D4: D6,E2: E3,F3: F,G4: G3] :
( Y3
!= ( product_Pair @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D6 @ ( product_prod @ E3 @ ( product_prod @ F @ G3 ) ) ) ) ) @ A4 @ ( product_Pair @ B @ ( product_prod @ C @ ( product_prod @ D6 @ ( product_prod @ E3 @ ( product_prod @ F @ G3 ) ) ) ) @ B4 @ ( product_Pair @ C @ ( product_prod @ D6 @ ( product_prod @ E3 @ ( product_prod @ F @ G3 ) ) ) @ C4 @ ( product_Pair @ D6 @ ( product_prod @ E3 @ ( product_prod @ F @ G3 ) ) @ D4 @ ( product_Pair @ E3 @ ( product_prod @ F @ G3 ) @ E2 @ ( product_Pair @ F @ G3 @ F3 @ G4 ) ) ) ) ) ) ) ).
% prod_cases7
thf(fact_3641_prod__induct4,axiom,
! [D6: $tType,C: $tType,B: $tType,A: $tType,P: ( product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ D6 ) ) ) > $o,X2: product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ D6 ) )] :
( ! [A4: A,B4: B,C4: C,D4: D6] : ( P @ ( product_Pair @ A @ ( product_prod @ B @ ( product_prod @ C @ D6 ) ) @ A4 @ ( product_Pair @ B @ ( product_prod @ C @ D6 ) @ B4 @ ( product_Pair @ C @ D6 @ C4 @ D4 ) ) ) )
=> ( P @ X2 ) ) ).
% prod_induct4
thf(fact_3642_prod__induct5,axiom,
! [E3: $tType,D6: $tType,C: $tType,B: $tType,A: $tType,P: ( product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D6 @ E3 ) ) ) ) > $o,X2: product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D6 @ E3 ) ) )] :
( ! [A4: A,B4: B,C4: C,D4: D6,E2: E3] : ( P @ ( product_Pair @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D6 @ E3 ) ) ) @ A4 @ ( product_Pair @ B @ ( product_prod @ C @ ( product_prod @ D6 @ E3 ) ) @ B4 @ ( product_Pair @ C @ ( product_prod @ D6 @ E3 ) @ C4 @ ( product_Pair @ D6 @ E3 @ D4 @ E2 ) ) ) ) )
=> ( P @ X2 ) ) ).
% prod_induct5
thf(fact_3643_choose__odd__sum,axiom,
! [A: $tType] :
( ( comm_ring_1 @ A )
=> ! [N: nat] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ N )
=> ( ( times_times @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) )
@ ( groups7311177749621191930dd_sum @ nat @ A
@ ^ [I4: nat] :
( if @ A
@ ~ ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ I4 )
@ ( semiring_1_of_nat @ A @ ( binomial @ N @ I4 ) )
@ ( zero_zero @ A ) )
@ ( set_ord_atMost @ nat @ N ) ) )
= ( power_power @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ N ) ) ) ) ).
% choose_odd_sum
thf(fact_3644_choose__even__sum,axiom,
! [A: $tType] :
( ( comm_ring_1 @ A )
=> ! [N: nat] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ N )
=> ( ( times_times @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) )
@ ( groups7311177749621191930dd_sum @ nat @ A
@ ^ [I4: nat] : ( if @ A @ ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ I4 ) @ ( semiring_1_of_nat @ A @ ( binomial @ N @ I4 ) ) @ ( zero_zero @ A ) )
@ ( set_ord_atMost @ nat @ N ) ) )
= ( power_power @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ N ) ) ) ) ).
% choose_even_sum
thf(fact_3645_old_Oprod_Orec,axiom,
! [A: $tType,T: $tType,B: $tType,F1: A > B > T,A3: A,B3: B] :
( ( product_rec_prod @ A @ B @ T @ F1 @ ( product_Pair @ A @ B @ A3 @ B3 ) )
= ( F1 @ A3 @ B3 ) ) ).
% old.prod.rec
thf(fact_3646_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_3647_binomial__Suc__n,axiom,
! [N: nat] :
( ( binomial @ ( suc @ N ) @ N )
= ( suc @ N ) ) ).
% binomial_Suc_n
thf(fact_3648_binomial__n__n,axiom,
! [N: nat] :
( ( binomial @ N @ N )
= ( one_one @ nat ) ) ).
% binomial_n_n
thf(fact_3649_binomial__1,axiom,
! [N: nat] :
( ( binomial @ N @ ( suc @ ( zero_zero @ nat ) ) )
= N ) ).
% binomial_1
thf(fact_3650_binomial__0__Suc,axiom,
! [K: nat] :
( ( binomial @ ( zero_zero @ nat ) @ ( suc @ K ) )
= ( zero_zero @ nat ) ) ).
% binomial_0_Suc
thf(fact_3651_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_3652_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_3653_binomial__n__0,axiom,
! [N: nat] :
( ( binomial @ N @ ( zero_zero @ nat ) )
= ( one_one @ nat ) ) ).
% binomial_n_0
thf(fact_3654_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_3655_choose__one,axiom,
! [N: nat] :
( ( binomial @ N @ ( one_one @ nat ) )
= N ) ).
% choose_one
thf(fact_3656_binomial__eq__0,axiom,
! [N: nat,K: nat] :
( ( ord_less @ nat @ N @ K )
=> ( ( binomial @ N @ K )
= ( zero_zero @ nat ) ) ) ).
% binomial_eq_0
thf(fact_3657_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_3658_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_3659_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_3660_choose__mult__lemma,axiom,
! [M: nat,R2: nat,K: nat] :
( ( times_times @ nat @ ( binomial @ ( plus_plus @ nat @ ( plus_plus @ nat @ M @ R2 ) @ K ) @ ( plus_plus @ nat @ M @ K ) ) @ ( binomial @ ( plus_plus @ nat @ M @ K ) @ K ) )
= ( times_times @ nat @ ( binomial @ ( plus_plus @ nat @ ( plus_plus @ nat @ M @ R2 ) @ K ) @ K ) @ ( binomial @ ( plus_plus @ nat @ M @ R2 ) @ M ) ) ) ).
% choose_mult_lemma
thf(fact_3661_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_3662_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_3663_Suc__times__binomial__add,axiom,
! [A3: nat,B3: nat] :
( ( times_times @ nat @ ( suc @ A3 ) @ ( binomial @ ( suc @ ( plus_plus @ nat @ A3 @ B3 ) ) @ ( suc @ A3 ) ) )
= ( times_times @ nat @ ( suc @ B3 ) @ ( binomial @ ( suc @ ( plus_plus @ nat @ A3 @ B3 ) ) @ A3 ) ) ) ).
% Suc_times_binomial_add
thf(fact_3664_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_3665_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_3666_binomial__absorb__comp,axiom,
! [N: nat,K: nat] :
( ( times_times @ nat @ ( minus_minus @ nat @ N @ K ) @ ( binomial @ N @ K ) )
= ( times_times @ nat @ N @ ( binomial @ ( minus_minus @ nat @ N @ ( one_one @ nat ) ) @ K ) ) ) ).
% binomial_absorb_comp
thf(fact_3667_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_3668_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_3669_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_3670_choose__rising__sum_I1_J,axiom,
! [N: nat,M: nat] :
( ( groups7311177749621191930dd_sum @ nat @ nat
@ ^ [J3: nat] : ( binomial @ ( plus_plus @ nat @ N @ J3 ) @ N )
@ ( set_ord_atMost @ nat @ M ) )
= ( binomial @ ( plus_plus @ nat @ ( plus_plus @ nat @ N @ M ) @ ( one_one @ nat ) ) @ ( plus_plus @ nat @ N @ ( one_one @ nat ) ) ) ) ).
% choose_rising_sum(1)
thf(fact_3671_choose__rising__sum_I2_J,axiom,
! [N: nat,M: nat] :
( ( groups7311177749621191930dd_sum @ nat @ nat
@ ^ [J3: nat] : ( binomial @ ( plus_plus @ nat @ N @ J3 ) @ N )
@ ( set_ord_atMost @ nat @ M ) )
= ( binomial @ ( plus_plus @ nat @ ( plus_plus @ nat @ N @ M ) @ ( one_one @ nat ) ) @ M ) ) ).
% choose_rising_sum(2)
thf(fact_3672_binomial__antimono,axiom,
! [K: nat,K4: nat,N: nat] :
( ( ord_less_eq @ nat @ K @ K4 )
=> ( ( ord_less_eq @ nat @ ( divide_divide @ nat @ N @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ K )
=> ( ( ord_less_eq @ nat @ K4 @ N )
=> ( ord_less_eq @ nat @ ( binomial @ N @ K4 ) @ ( binomial @ N @ K ) ) ) ) ) ).
% binomial_antimono
thf(fact_3673_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_3674_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_3675_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_3676_binomial__mono,axiom,
! [K: nat,K4: nat,N: nat] :
( ( ord_less_eq @ nat @ K @ K4 )
=> ( ( ord_less_eq @ nat @ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ K4 ) @ N )
=> ( ord_less_eq @ nat @ ( binomial @ N @ K ) @ ( binomial @ N @ K4 ) ) ) ) ).
% binomial_mono
thf(fact_3677_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_3678_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_3679_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_3680_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_3681_vandermonde,axiom,
! [M: nat,N: nat,R2: nat] :
( ( groups7311177749621191930dd_sum @ nat @ nat
@ ^ [K3: nat] : ( times_times @ nat @ ( binomial @ M @ K3 ) @ ( binomial @ N @ ( minus_minus @ nat @ R2 @ K3 ) ) )
@ ( set_ord_atMost @ nat @ R2 ) )
= ( binomial @ ( plus_plus @ nat @ M @ N ) @ R2 ) ) ).
% vandermonde
thf(fact_3682_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_3683_binomial__strict__antimono,axiom,
! [K: nat,K4: nat,N: nat] :
( ( ord_less @ nat @ K @ K4 )
=> ( ( ord_less_eq @ nat @ N @ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ K ) )
=> ( ( ord_less_eq @ nat @ K4 @ N )
=> ( ord_less @ nat @ ( binomial @ N @ K4 ) @ ( binomial @ N @ K ) ) ) ) ) ).
% binomial_strict_antimono
thf(fact_3684_binomial__strict__mono,axiom,
! [K: nat,K4: nat,N: nat] :
( ( ord_less @ nat @ K @ K4 )
=> ( ( ord_less_eq @ nat @ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ K4 ) @ N )
=> ( ord_less @ nat @ ( binomial @ N @ K ) @ ( binomial @ N @ K4 ) ) ) ) ).
% binomial_strict_mono
thf(fact_3685_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_3686_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_3687_choose__row__sum,axiom,
! [N: nat] :
( ( groups7311177749621191930dd_sum @ nat @ nat @ ( binomial @ N ) @ ( set_ord_atMost @ nat @ N ) )
= ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N ) ) ).
% choose_row_sum
thf(fact_3688_binomial,axiom,
! [A3: nat,B3: nat,N: nat] :
( ( power_power @ nat @ ( plus_plus @ nat @ A3 @ B3 ) @ N )
= ( groups7311177749621191930dd_sum @ nat @ nat
@ ^ [K3: nat] : ( times_times @ nat @ ( times_times @ nat @ ( semiring_1_of_nat @ nat @ ( binomial @ N @ K3 ) ) @ ( power_power @ nat @ A3 @ K3 ) ) @ ( power_power @ nat @ B3 @ ( minus_minus @ nat @ N @ K3 ) ) )
@ ( set_ord_atMost @ nat @ N ) ) ) ).
% binomial
thf(fact_3689_choose__two,axiom,
! [N: nat] :
( ( binomial @ N @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) )
= ( divide_divide @ nat @ ( times_times @ nat @ N @ ( minus_minus @ nat @ N @ ( one_one @ nat ) ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ).
% choose_two
thf(fact_3690_binomial__ring,axiom,
! [A: $tType] :
( ( comm_semiring_1 @ A )
=> ! [A3: A,B3: A,N: nat] :
( ( power_power @ A @ ( plus_plus @ A @ A3 @ B3 ) @ N )
= ( groups7311177749621191930dd_sum @ nat @ A
@ ^ [K3: nat] : ( times_times @ A @ ( times_times @ A @ ( semiring_1_of_nat @ A @ ( binomial @ N @ K3 ) ) @ ( power_power @ A @ A3 @ K3 ) ) @ ( power_power @ A @ B3 @ ( minus_minus @ nat @ N @ K3 ) ) )
@ ( set_ord_atMost @ nat @ N ) ) ) ) ).
% binomial_ring
thf(fact_3691_binomial__altdef__of__nat,axiom,
! [A: $tType] :
( ( field_char_0 @ A )
=> ! [K: nat,N: nat] :
( ( ord_less_eq @ nat @ K @ N )
=> ( ( semiring_1_of_nat @ A @ ( binomial @ N @ K ) )
= ( groups7121269368397514597t_prod @ nat @ A
@ ^ [I4: nat] : ( divide_divide @ A @ ( semiring_1_of_nat @ A @ ( minus_minus @ nat @ N @ I4 ) ) @ ( semiring_1_of_nat @ A @ ( minus_minus @ nat @ K @ I4 ) ) )
@ ( set_or7035219750837199246ssThan @ nat @ ( zero_zero @ nat ) @ K ) ) ) ) ) ).
% binomial_altdef_of_nat
thf(fact_3692_pochhammer__binomial__sum,axiom,
! [A: $tType] :
( ( comm_ring_1 @ A )
=> ! [A3: A,B3: A,N: nat] :
( ( comm_s3205402744901411588hammer @ A @ ( plus_plus @ A @ A3 @ B3 ) @ N )
= ( groups7311177749621191930dd_sum @ nat @ A
@ ^ [K3: nat] : ( times_times @ A @ ( times_times @ A @ ( semiring_1_of_nat @ A @ ( binomial @ N @ K3 ) ) @ ( comm_s3205402744901411588hammer @ A @ A3 @ K3 ) ) @ ( comm_s3205402744901411588hammer @ A @ B3 @ ( minus_minus @ nat @ N @ K3 ) ) )
@ ( set_ord_atMost @ nat @ N ) ) ) ) ).
% pochhammer_binomial_sum
thf(fact_3693_choose__square__sum,axiom,
! [N: nat] :
( ( groups7311177749621191930dd_sum @ nat @ nat
@ ^ [K3: nat] : ( power_power @ nat @ ( binomial @ N @ K3 ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) )
@ ( set_ord_atMost @ nat @ N ) )
= ( binomial @ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N ) @ N ) ) ).
% choose_square_sum
thf(fact_3694_binomial__r__part__sum,axiom,
! [M: nat] :
( ( groups7311177749621191930dd_sum @ nat @ nat @ ( binomial @ ( plus_plus @ nat @ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ M ) @ ( one_one @ nat ) ) ) @ ( set_ord_atMost @ nat @ M ) )
= ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ M ) ) ) ).
% binomial_r_part_sum
thf(fact_3695_choose__linear__sum,axiom,
! [N: nat] :
( ( groups7311177749621191930dd_sum @ nat @ nat
@ ^ [I4: nat] : ( times_times @ nat @ I4 @ ( binomial @ N @ I4 ) )
@ ( set_ord_atMost @ nat @ N ) )
= ( times_times @ nat @ N @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( minus_minus @ nat @ N @ ( one_one @ nat ) ) ) ) ) ).
% choose_linear_sum
thf(fact_3696_upd__rule,axiom,
! [A: $tType] :
( ( heap @ A )
=> ! [I: nat,Xs: list @ A,A3: array @ A,X2: A] :
( ( ord_less @ nat @ I @ ( size_size @ ( list @ A ) @ Xs ) )
=> ( hoare_hoare_triple @ ( array @ A ) @ ( snga_assn @ A @ A3 @ Xs ) @ ( array_upd @ A @ I @ X2 @ A3 )
@ ^ [R5: array @ A] : ( times_times @ assn @ ( snga_assn @ A @ A3 @ ( list_update @ A @ Xs @ I @ X2 ) ) @ ( pure_assn @ ( R5 = A3 ) ) ) ) ) ) ).
% upd_rule
thf(fact_3697_gbinomial__code,axiom,
! [A: $tType] :
( ( field_char_0 @ A )
=> ( ( gbinomial @ A )
= ( ^ [A6: A,K3: nat] :
( if @ A
@ ( K3
= ( zero_zero @ nat ) )
@ ( one_one @ A )
@ ( divide_divide @ A
@ ( set_fo6178422350223883121st_nat @ A
@ ^ [L3: nat] : ( times_times @ A @ ( minus_minus @ A @ A6 @ ( semiring_1_of_nat @ A @ L3 ) ) )
@ ( zero_zero @ nat )
@ ( minus_minus @ nat @ K3 @ ( one_one @ nat ) )
@ ( one_one @ A ) )
@ ( semiring_char_0_fact @ A @ K3 ) ) ) ) ) ) ).
% gbinomial_code
thf(fact_3698_powr__int,axiom,
! [X2: real,I: int] :
( ( ord_less @ real @ ( zero_zero @ real ) @ X2 )
=> ( ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ I )
=> ( ( powr @ real @ X2 @ ( ring_1_of_int @ real @ I ) )
= ( power_power @ real @ X2 @ ( nat2 @ I ) ) ) )
& ( ~ ( ord_less_eq @ int @ ( zero_zero @ int ) @ I )
=> ( ( powr @ real @ X2 @ ( ring_1_of_int @ real @ I ) )
= ( divide_divide @ real @ ( one_one @ real ) @ ( power_power @ real @ X2 @ ( nat2 @ ( uminus_uminus @ int @ I ) ) ) ) ) ) ) ) ).
% powr_int
thf(fact_3699_of__nat__id,axiom,
( ( semiring_1_of_nat @ nat )
= ( ^ [N4: nat] : N4 ) ) ).
% of_nat_id
thf(fact_3700_neg__le__iff__le,axiom,
! [A: $tType] :
( ( ordered_ab_group_add @ A )
=> ! [B3: A,A3: A] :
( ( ord_less_eq @ A @ ( uminus_uminus @ A @ B3 ) @ ( uminus_uminus @ A @ A3 ) )
= ( ord_less_eq @ A @ A3 @ B3 ) ) ) ).
% neg_le_iff_le
thf(fact_3701_neg__less__iff__less,axiom,
! [A: $tType] :
( ( ordered_ab_group_add @ A )
=> ! [B3: A,A3: A] :
( ( ord_less @ A @ ( uminus_uminus @ A @ B3 ) @ ( uminus_uminus @ A @ A3 ) )
= ( ord_less @ A @ A3 @ B3 ) ) ) ).
% neg_less_iff_less
thf(fact_3702_neg__numeral__eq__iff,axiom,
! [A: $tType] :
( ( ring_char_0 @ A )
=> ! [M: num,N: num] :
( ( ( uminus_uminus @ A @ ( numeral_numeral @ A @ M ) )
= ( uminus_uminus @ A @ ( numeral_numeral @ A @ N ) ) )
= ( M = N ) ) ) ).
% neg_numeral_eq_iff
thf(fact_3703_minus__add__distrib,axiom,
! [A: $tType] :
( ( ab_group_add @ A )
=> ! [A3: A,B3: A] :
( ( uminus_uminus @ A @ ( plus_plus @ A @ A3 @ B3 ) )
= ( plus_plus @ A @ ( uminus_uminus @ A @ A3 ) @ ( uminus_uminus @ A @ B3 ) ) ) ) ).
% minus_add_distrib
thf(fact_3704_minus__add__cancel,axiom,
! [A: $tType] :
( ( group_add @ A )
=> ! [A3: A,B3: A] :
( ( plus_plus @ A @ ( uminus_uminus @ A @ A3 ) @ ( plus_plus @ A @ A3 @ B3 ) )
= B3 ) ) ).
% minus_add_cancel
thf(fact_3705_add__minus__cancel,axiom,
! [A: $tType] :
( ( group_add @ A )
=> ! [A3: A,B3: A] :
( ( plus_plus @ A @ A3 @ ( plus_plus @ A @ ( uminus_uminus @ A @ A3 ) @ B3 ) )
= B3 ) ) ).
% add_minus_cancel
thf(fact_3706_mult__minus__left,axiom,
! [A: $tType] :
( ( ring @ A )
=> ! [A3: A,B3: A] :
( ( times_times @ A @ ( uminus_uminus @ A @ A3 ) @ B3 )
= ( uminus_uminus @ A @ ( times_times @ A @ A3 @ B3 ) ) ) ) ).
% mult_minus_left
thf(fact_3707_minus__mult__minus,axiom,
! [A: $tType] :
( ( ring @ A )
=> ! [A3: A,B3: A] :
( ( times_times @ A @ ( uminus_uminus @ A @ A3 ) @ ( uminus_uminus @ A @ B3 ) )
= ( times_times @ A @ A3 @ B3 ) ) ) ).
% minus_mult_minus
thf(fact_3708_mult__minus__right,axiom,
! [A: $tType] :
( ( ring @ A )
=> ! [A3: A,B3: A] :
( ( times_times @ A @ A3 @ ( uminus_uminus @ A @ B3 ) )
= ( uminus_uminus @ A @ ( times_times @ A @ A3 @ B3 ) ) ) ) ).
% mult_minus_right
thf(fact_3709_minus__diff__eq,axiom,
! [A: $tType] :
( ( group_add @ A )
=> ! [A3: A,B3: A] :
( ( uminus_uminus @ A @ ( minus_minus @ A @ A3 @ B3 ) )
= ( minus_minus @ A @ B3 @ A3 ) ) ) ).
% minus_diff_eq
thf(fact_3710_div__minus__minus,axiom,
! [A: $tType] :
( ( euclid8851590272496341667cancel @ A )
=> ! [A3: A,B3: A] :
( ( divide_divide @ A @ ( uminus_uminus @ A @ A3 ) @ ( uminus_uminus @ A @ B3 ) )
= ( divide_divide @ A @ A3 @ B3 ) ) ) ).
% div_minus_minus
thf(fact_3711_mod__minus__minus,axiom,
! [A: $tType] :
( ( euclid8851590272496341667cancel @ A )
=> ! [A3: A,B3: A] :
( ( modulo_modulo @ A @ ( uminus_uminus @ A @ A3 ) @ ( uminus_uminus @ A @ B3 ) )
= ( uminus_uminus @ A @ ( modulo_modulo @ A @ A3 @ B3 ) ) ) ) ).
% mod_minus_minus
thf(fact_3712_neg__less__eq__nonneg,axiom,
! [A: $tType] :
( ( linord5086331880401160121up_add @ A )
=> ! [A3: A] :
( ( ord_less_eq @ A @ ( uminus_uminus @ A @ A3 ) @ A3 )
= ( ord_less_eq @ A @ ( zero_zero @ A ) @ A3 ) ) ) ).
% neg_less_eq_nonneg
thf(fact_3713_less__eq__neg__nonpos,axiom,
! [A: $tType] :
( ( linord5086331880401160121up_add @ A )
=> ! [A3: A] :
( ( ord_less_eq @ A @ A3 @ ( uminus_uminus @ A @ A3 ) )
= ( ord_less_eq @ A @ A3 @ ( zero_zero @ A ) ) ) ) ).
% less_eq_neg_nonpos
thf(fact_3714_neg__le__0__iff__le,axiom,
! [A: $tType] :
( ( ordered_ab_group_add @ A )
=> ! [A3: A] :
( ( ord_less_eq @ A @ ( uminus_uminus @ A @ A3 ) @ ( zero_zero @ A ) )
= ( ord_less_eq @ A @ ( zero_zero @ A ) @ A3 ) ) ) ).
% neg_le_0_iff_le
thf(fact_3715_neg__0__le__iff__le,axiom,
! [A: $tType] :
( ( ordered_ab_group_add @ A )
=> ! [A3: A] :
( ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( uminus_uminus @ A @ A3 ) )
= ( ord_less_eq @ A @ A3 @ ( zero_zero @ A ) ) ) ) ).
% neg_0_le_iff_le
thf(fact_3716_neg__less__0__iff__less,axiom,
! [A: $tType] :
( ( ordered_ab_group_add @ A )
=> ! [A3: A] :
( ( ord_less @ A @ ( uminus_uminus @ A @ A3 ) @ ( zero_zero @ A ) )
= ( ord_less @ A @ ( zero_zero @ A ) @ A3 ) ) ) ).
% neg_less_0_iff_less
thf(fact_3717_neg__0__less__iff__less,axiom,
! [A: $tType] :
( ( ordered_ab_group_add @ A )
=> ! [A3: A] :
( ( ord_less @ A @ ( zero_zero @ A ) @ ( uminus_uminus @ A @ A3 ) )
= ( ord_less @ A @ A3 @ ( zero_zero @ A ) ) ) ) ).
% neg_0_less_iff_less
thf(fact_3718_neg__less__pos,axiom,
! [A: $tType] :
( ( linord5086331880401160121up_add @ A )
=> ! [A3: A] :
( ( ord_less @ A @ ( uminus_uminus @ A @ A3 ) @ A3 )
= ( ord_less @ A @ ( zero_zero @ A ) @ A3 ) ) ) ).
% neg_less_pos
thf(fact_3719_less__neg__neg,axiom,
! [A: $tType] :
( ( linord5086331880401160121up_add @ A )
=> ! [A3: A] :
( ( ord_less @ A @ A3 @ ( uminus_uminus @ A @ A3 ) )
= ( ord_less @ A @ A3 @ ( zero_zero @ A ) ) ) ) ).
% less_neg_neg
thf(fact_3720_ab__left__minus,axiom,
! [A: $tType] :
( ( group_add @ A )
=> ! [A3: A] :
( ( plus_plus @ A @ ( uminus_uminus @ A @ A3 ) @ A3 )
= ( zero_zero @ A ) ) ) ).
% ab_left_minus
thf(fact_3721_add_Oright__inverse,axiom,
! [A: $tType] :
( ( group_add @ A )
=> ! [A3: A] :
( ( plus_plus @ A @ A3 @ ( uminus_uminus @ A @ A3 ) )
= ( zero_zero @ A ) ) ) ).
% add.right_inverse
thf(fact_3722_add__neg__numeral__simps_I3_J,axiom,
! [A: $tType] :
( ( neg_numeral @ A )
=> ! [M: num,N: num] :
( ( plus_plus @ A @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ M ) ) @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ N ) ) )
= ( uminus_uminus @ A @ ( plus_plus @ A @ ( numeral_numeral @ A @ M ) @ ( numeral_numeral @ A @ N ) ) ) ) ) ).
% add_neg_numeral_simps(3)
thf(fact_3723_diff__0,axiom,
! [A: $tType] :
( ( group_add @ A )
=> ! [A3: A] :
( ( minus_minus @ A @ ( zero_zero @ A ) @ A3 )
= ( uminus_uminus @ A @ A3 ) ) ) ).
% diff_0
thf(fact_3724_verit__minus__simplify_I3_J,axiom,
! [B: $tType] :
( ( group_add @ B )
=> ! [B3: B] :
( ( minus_minus @ B @ ( zero_zero @ B ) @ B3 )
= ( uminus_uminus @ B @ B3 ) ) ) ).
% verit_minus_simplify(3)
thf(fact_3725_mult__minus1__right,axiom,
! [A: $tType] :
( ( ring_1 @ A )
=> ! [Z: A] :
( ( times_times @ A @ Z @ ( uminus_uminus @ A @ ( one_one @ A ) ) )
= ( uminus_uminus @ A @ Z ) ) ) ).
% mult_minus1_right
thf(fact_3726_mult__minus1,axiom,
! [A: $tType] :
( ( ring_1 @ A )
=> ! [Z: A] :
( ( times_times @ A @ ( uminus_uminus @ A @ ( one_one @ A ) ) @ Z )
= ( uminus_uminus @ A @ Z ) ) ) ).
% mult_minus1
thf(fact_3727_diff__minus__eq__add,axiom,
! [A: $tType] :
( ( group_add @ A )
=> ! [A3: A,B3: A] :
( ( minus_minus @ A @ A3 @ ( uminus_uminus @ A @ B3 ) )
= ( plus_plus @ A @ A3 @ B3 ) ) ) ).
% diff_minus_eq_add
thf(fact_3728_uminus__add__conv__diff,axiom,
! [A: $tType] :
( ( ab_group_add @ A )
=> ! [A3: A,B3: A] :
( ( plus_plus @ A @ ( uminus_uminus @ A @ A3 ) @ B3 )
= ( minus_minus @ A @ B3 @ A3 ) ) ) ).
% uminus_add_conv_diff
thf(fact_3729_divide__minus1,axiom,
! [A: $tType] :
( ( field @ A )
=> ! [X2: A] :
( ( divide_divide @ A @ X2 @ ( uminus_uminus @ A @ ( one_one @ A ) ) )
= ( uminus_uminus @ A @ X2 ) ) ) ).
% divide_minus1
thf(fact_3730_div__minus1__right,axiom,
! [A: $tType] :
( ( euclid8851590272496341667cancel @ A )
=> ! [A3: A] :
( ( divide_divide @ A @ A3 @ ( uminus_uminus @ A @ ( one_one @ A ) ) )
= ( uminus_uminus @ A @ A3 ) ) ) ).
% div_minus1_right
thf(fact_3731_minus__mod__self1,axiom,
! [A: $tType] :
( ( euclid8851590272496341667cancel @ A )
=> ! [B3: A,A3: A] :
( ( modulo_modulo @ A @ ( minus_minus @ A @ B3 @ A3 ) @ B3 )
= ( modulo_modulo @ A @ ( uminus_uminus @ A @ A3 ) @ B3 ) ) ) ).
% minus_mod_self1
thf(fact_3732_fact__0,axiom,
! [A: $tType] :
( ( semiring_char_0 @ A )
=> ( ( semiring_char_0_fact @ A @ ( zero_zero @ nat ) )
= ( one_one @ A ) ) ) ).
% fact_0
thf(fact_3733_fact__1,axiom,
! [A: $tType] :
( ( semiring_char_0 @ A )
=> ( ( semiring_char_0_fact @ A @ ( one_one @ nat ) )
= ( one_one @ A ) ) ) ).
% fact_1
thf(fact_3734_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_3735_dbl__inc__simps_I4_J,axiom,
! [A: $tType] :
( ( neg_numeral @ A )
=> ( ( neg_numeral_dbl_inc @ A @ ( uminus_uminus @ A @ ( one_one @ A ) ) )
= ( uminus_uminus @ A @ ( one_one @ A ) ) ) ) ).
% dbl_inc_simps(4)
thf(fact_3736_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_3737_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_3738_neg__one__eq__numeral__iff,axiom,
! [A: $tType] :
( ( ring_char_0 @ A )
=> ! [N: num] :
( ( ( uminus_uminus @ A @ ( one_one @ A ) )
= ( uminus_uminus @ A @ ( numeral_numeral @ A @ N ) ) )
= ( N = one2 ) ) ) ).
% neg_one_eq_numeral_iff
thf(fact_3739_numeral__eq__neg__one__iff,axiom,
! [A: $tType] :
( ( ring_char_0 @ A )
=> ! [N: num] :
( ( ( uminus_uminus @ A @ ( numeral_numeral @ A @ N ) )
= ( uminus_uminus @ A @ ( one_one @ A ) ) )
= ( N = one2 ) ) ) ).
% numeral_eq_neg_one_iff
thf(fact_3740_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_3741_left__minus__one__mult__self,axiom,
! [A: $tType] :
( ( comm_ring_1 @ A )
=> ! [N: nat,A3: A] :
( ( times_times @ A @ ( power_power @ A @ ( uminus_uminus @ A @ ( one_one @ A ) ) @ N ) @ ( times_times @ A @ ( power_power @ A @ ( uminus_uminus @ A @ ( one_one @ A ) ) @ N ) @ A3 ) )
= A3 ) ) ).
% left_minus_one_mult_self
thf(fact_3742_minus__one__mult__self,axiom,
! [A: $tType] :
( ( comm_ring_1 @ A )
=> ! [N: nat] :
( ( times_times @ A @ ( power_power @ A @ ( uminus_uminus @ A @ ( one_one @ A ) ) @ N ) @ ( power_power @ A @ ( uminus_uminus @ A @ ( one_one @ A ) ) @ N ) )
= ( one_one @ A ) ) ) ).
% minus_one_mult_self
thf(fact_3743_mod__minus1__right,axiom,
! [A: $tType] :
( ( euclid8851590272496341667cancel @ A )
=> ! [A3: A] :
( ( modulo_modulo @ A @ A3 @ ( uminus_uminus @ A @ ( one_one @ A ) ) )
= ( zero_zero @ A ) ) ) ).
% mod_minus1_right
thf(fact_3744_max__number__of_I2_J,axiom,
! [A: $tType] :
( ( ( uminus @ A )
& ( numeral @ A )
& ( ord @ A ) )
=> ! [U: num,V: num] :
( ( ( ord_less_eq @ A @ ( numeral_numeral @ A @ U ) @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ V ) ) )
=> ( ( ord_max @ A @ ( numeral_numeral @ A @ U ) @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ V ) ) )
= ( uminus_uminus @ A @ ( numeral_numeral @ A @ V ) ) ) )
& ( ~ ( ord_less_eq @ A @ ( numeral_numeral @ A @ U ) @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ V ) ) )
=> ( ( ord_max @ A @ ( numeral_numeral @ A @ U ) @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ V ) ) )
= ( numeral_numeral @ A @ U ) ) ) ) ) ).
% max_number_of(2)
thf(fact_3745_max__number__of_I3_J,axiom,
! [A: $tType] :
( ( ( uminus @ A )
& ( numeral @ A )
& ( ord @ A ) )
=> ! [U: num,V: num] :
( ( ( ord_less_eq @ A @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ U ) ) @ ( numeral_numeral @ A @ V ) )
=> ( ( ord_max @ A @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ U ) ) @ ( numeral_numeral @ A @ V ) )
= ( numeral_numeral @ A @ V ) ) )
& ( ~ ( ord_less_eq @ A @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ U ) ) @ ( numeral_numeral @ A @ V ) )
=> ( ( ord_max @ A @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ U ) ) @ ( numeral_numeral @ A @ V ) )
= ( uminus_uminus @ A @ ( numeral_numeral @ A @ U ) ) ) ) ) ) ).
% max_number_of(3)
thf(fact_3746_max__number__of_I4_J,axiom,
! [A: $tType] :
( ( ( uminus @ A )
& ( numeral @ A )
& ( ord @ A ) )
=> ! [U: num,V: num] :
( ( ( ord_less_eq @ A @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ U ) ) @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ V ) ) )
=> ( ( ord_max @ A @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ U ) ) @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ V ) ) )
= ( uminus_uminus @ A @ ( numeral_numeral @ A @ V ) ) ) )
& ( ~ ( ord_less_eq @ A @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ U ) ) @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ V ) ) )
=> ( ( ord_max @ A @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ U ) ) @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ V ) ) )
= ( uminus_uminus @ A @ ( numeral_numeral @ A @ U ) ) ) ) ) ) ).
% max_number_of(4)
thf(fact_3747_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_3748_norm__neg__numeral,axiom,
! [A: $tType] :
( ( real_V2822296259951069270ebra_1 @ A )
=> ! [W: num] :
( ( real_V7770717601297561774m_norm @ A @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ W ) ) )
= ( numeral_numeral @ real @ W ) ) ) ).
% norm_neg_numeral
thf(fact_3749_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_3750_semiring__norm_I168_J,axiom,
! [A: $tType] :
( ( neg_numeral @ A )
=> ! [V: num,W: num,Y3: A] :
( ( plus_plus @ A @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ V ) ) @ ( plus_plus @ A @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ W ) ) @ Y3 ) )
= ( plus_plus @ A @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ ( plus_plus @ num @ V @ W ) ) ) @ Y3 ) ) ) ).
% semiring_norm(168)
thf(fact_3751_diff__numeral__simps_I2_J,axiom,
! [A: $tType] :
( ( neg_numeral @ A )
=> ! [M: num,N: num] :
( ( minus_minus @ A @ ( numeral_numeral @ A @ M ) @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ N ) ) )
= ( numeral_numeral @ A @ ( plus_plus @ num @ M @ N ) ) ) ) ).
% diff_numeral_simps(2)
thf(fact_3752_diff__numeral__simps_I3_J,axiom,
! [A: $tType] :
( ( neg_numeral @ A )
=> ! [M: num,N: num] :
( ( minus_minus @ A @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ M ) ) @ ( numeral_numeral @ A @ N ) )
= ( uminus_uminus @ A @ ( numeral_numeral @ A @ ( plus_plus @ num @ M @ N ) ) ) ) ) ).
% diff_numeral_simps(3)
thf(fact_3753_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_3754_mult__neg__numeral__simps_I3_J,axiom,
! [A: $tType] :
( ( ring_1 @ A )
=> ! [M: num,N: num] :
( ( times_times @ A @ ( numeral_numeral @ A @ M ) @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ N ) ) )
= ( uminus_uminus @ A @ ( numeral_numeral @ A @ ( times_times @ num @ M @ N ) ) ) ) ) ).
% mult_neg_numeral_simps(3)
thf(fact_3755_mult__neg__numeral__simps_I2_J,axiom,
! [A: $tType] :
( ( ring_1 @ A )
=> ! [M: num,N: num] :
( ( times_times @ A @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ M ) ) @ ( numeral_numeral @ A @ N ) )
= ( uminus_uminus @ A @ ( numeral_numeral @ A @ ( times_times @ num @ M @ N ) ) ) ) ) ).
% mult_neg_numeral_simps(2)
thf(fact_3756_mult__neg__numeral__simps_I1_J,axiom,
! [A: $tType] :
( ( ring_1 @ A )
=> ! [M: num,N: num] :
( ( times_times @ A @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ M ) ) @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ N ) ) )
= ( numeral_numeral @ A @ ( times_times @ num @ M @ N ) ) ) ) ).
% mult_neg_numeral_simps(1)
thf(fact_3757_semiring__norm_I172_J,axiom,
! [A: $tType] :
( ( ring_1 @ A )
=> ! [V: num,W: num,Y3: A] :
( ( times_times @ A @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ V ) ) @ ( times_times @ A @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ W ) ) @ Y3 ) )
= ( times_times @ A @ ( numeral_numeral @ A @ ( times_times @ num @ V @ W ) ) @ Y3 ) ) ) ).
% semiring_norm(172)
thf(fact_3758_semiring__norm_I171_J,axiom,
! [A: $tType] :
( ( ring_1 @ A )
=> ! [V: num,W: num,Y3: A] :
( ( times_times @ A @ ( numeral_numeral @ A @ V ) @ ( times_times @ A @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ W ) ) @ Y3 ) )
= ( times_times @ A @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ ( times_times @ num @ V @ W ) ) ) @ Y3 ) ) ) ).
% semiring_norm(171)
thf(fact_3759_semiring__norm_I170_J,axiom,
! [A: $tType] :
( ( ring_1 @ A )
=> ! [V: num,W: num,Y3: A] :
( ( times_times @ A @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ V ) ) @ ( times_times @ A @ ( numeral_numeral @ A @ W ) @ Y3 ) )
= ( times_times @ A @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ ( times_times @ num @ V @ W ) ) ) @ Y3 ) ) ) ).
% semiring_norm(170)
thf(fact_3760_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_3761_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_3762_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_3763_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_3764_divide__eq__eq__numeral1_I2_J,axiom,
! [A: $tType] :
( ( division_ring @ A )
=> ! [B3: A,W: num,A3: A] :
( ( ( divide_divide @ A @ B3 @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ W ) ) )
= A3 )
= ( ( ( ( uminus_uminus @ A @ ( numeral_numeral @ A @ W ) )
!= ( zero_zero @ A ) )
=> ( B3
= ( times_times @ A @ A3 @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ W ) ) ) ) )
& ( ( ( uminus_uminus @ A @ ( numeral_numeral @ A @ W ) )
= ( zero_zero @ A ) )
=> ( A3
= ( zero_zero @ A ) ) ) ) ) ) ).
% divide_eq_eq_numeral1(2)
thf(fact_3765_eq__divide__eq__numeral1_I2_J,axiom,
! [A: $tType] :
( ( division_ring @ A )
=> ! [A3: A,B3: A,W: num] :
( ( A3
= ( divide_divide @ A @ B3 @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ W ) ) ) )
= ( ( ( ( uminus_uminus @ A @ ( numeral_numeral @ A @ W ) )
!= ( zero_zero @ A ) )
=> ( ( times_times @ A @ A3 @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ W ) ) )
= B3 ) )
& ( ( ( uminus_uminus @ A @ ( numeral_numeral @ A @ W ) )
= ( zero_zero @ A ) )
=> ( A3
= ( zero_zero @ A ) ) ) ) ) ) ).
% eq_divide_eq_numeral1(2)
thf(fact_3766_divide__le__eq__numeral1_I2_J,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [B3: A,W: num,A3: A] :
( ( ord_less_eq @ A @ ( divide_divide @ A @ B3 @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ W ) ) ) @ A3 )
= ( ord_less_eq @ A @ ( times_times @ A @ A3 @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ W ) ) ) @ B3 ) ) ) ).
% divide_le_eq_numeral1(2)
thf(fact_3767_le__divide__eq__numeral1_I2_J,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [A3: A,B3: A,W: num] :
( ( ord_less_eq @ A @ A3 @ ( divide_divide @ A @ B3 @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ W ) ) ) )
= ( ord_less_eq @ A @ B3 @ ( times_times @ A @ A3 @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ W ) ) ) ) ) ) ).
% le_divide_eq_numeral1(2)
thf(fact_3768_divide__less__eq__numeral1_I2_J,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [B3: A,W: num,A3: A] :
( ( ord_less @ A @ ( divide_divide @ A @ B3 @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ W ) ) ) @ A3 )
= ( ord_less @ A @ ( times_times @ A @ A3 @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ W ) ) ) @ B3 ) ) ) ).
% divide_less_eq_numeral1(2)
thf(fact_3769_less__divide__eq__numeral1_I2_J,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [A3: A,B3: A,W: num] :
( ( ord_less @ A @ A3 @ ( divide_divide @ A @ B3 @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ W ) ) ) )
= ( ord_less @ A @ B3 @ ( times_times @ A @ A3 @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ W ) ) ) ) ) ) ).
% less_divide_eq_numeral1(2)
thf(fact_3770_power2__minus,axiom,
! [A: $tType] :
( ( ring_1 @ A )
=> ! [A3: A] :
( ( power_power @ A @ ( uminus_uminus @ A @ A3 ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) )
= ( power_power @ A @ A3 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ).
% power2_minus
thf(fact_3771_fact__2,axiom,
! [A: $tType] :
( ( semiring_char_0 @ A )
=> ( ( semiring_char_0_fact @ A @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) )
= ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) ) ).
% fact_2
thf(fact_3772_floor__neg__numeral,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [V: num] :
( ( archim6421214686448440834_floor @ A @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ V ) ) )
= ( uminus_uminus @ int @ ( numeral_numeral @ int @ V ) ) ) ) ).
% floor_neg_numeral
thf(fact_3773_ceiling__neg__numeral,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [V: num] :
( ( archimedean_ceiling @ A @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ V ) ) )
= ( uminus_uminus @ int @ ( numeral_numeral @ int @ V ) ) ) ) ).
% ceiling_neg_numeral
thf(fact_3774_int__div__minus__is__minus1,axiom,
! [A3: int,B3: int] :
( ( ord_less @ int @ A3 @ ( zero_zero @ int ) )
=> ( ( ( divide_divide @ int @ A3 @ B3 )
= ( uminus_uminus @ int @ A3 ) )
= ( B3
= ( uminus_uminus @ int @ ( one_one @ int ) ) ) ) ) ).
% int_div_minus_is_minus1
thf(fact_3775_round__neg__numeral,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [N: num] :
( ( archimedean_round @ A @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ N ) ) )
= ( uminus_uminus @ int @ ( numeral_numeral @ int @ N ) ) ) ) ).
% round_neg_numeral
thf(fact_3776_add__neg__numeral__special_I9_J,axiom,
! [A: $tType] :
( ( neg_numeral @ A )
=> ( ( plus_plus @ A @ ( uminus_uminus @ A @ ( one_one @ A ) ) @ ( uminus_uminus @ A @ ( one_one @ A ) ) )
= ( uminus_uminus @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) ) ) ).
% add_neg_numeral_special(9)
thf(fact_3777_diff__numeral__special_I11_J,axiom,
! [A: $tType] :
( ( neg_numeral @ A )
=> ( ( minus_minus @ A @ ( one_one @ A ) @ ( uminus_uminus @ A @ ( one_one @ A ) ) )
= ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) ) ).
% diff_numeral_special(11)
thf(fact_3778_diff__numeral__special_I10_J,axiom,
! [A: $tType] :
( ( neg_numeral @ A )
=> ( ( minus_minus @ A @ ( uminus_uminus @ A @ ( one_one @ A ) ) @ ( one_one @ A ) )
= ( uminus_uminus @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) ) ) ).
% diff_numeral_special(10)
thf(fact_3779_minus__1__div__2__eq,axiom,
! [A: $tType] :
( ( euclid8789492081693882211th_nat @ A )
=> ( ( divide_divide @ A @ ( uminus_uminus @ A @ ( one_one @ A ) ) @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) )
= ( uminus_uminus @ A @ ( one_one @ A ) ) ) ) ).
% minus_1_div_2_eq
thf(fact_3780_bits__minus__1__mod__2__eq,axiom,
! [A: $tType] :
( ( bit_ri3973907225187159222ations @ A )
=> ( ( modulo_modulo @ A @ ( uminus_uminus @ A @ ( one_one @ A ) ) @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) )
= ( one_one @ A ) ) ) ).
% bits_minus_1_mod_2_eq
thf(fact_3781_minus__1__mod__2__eq,axiom,
! [A: $tType] :
( ( euclid8789492081693882211th_nat @ A )
=> ( ( modulo_modulo @ A @ ( uminus_uminus @ A @ ( one_one @ A ) ) @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) )
= ( one_one @ A ) ) ) ).
% minus_1_mod_2_eq
thf(fact_3782_Power_Oring__1__class_Opower__minus__even,axiom,
! [A: $tType] :
( ( ring_1 @ A )
=> ! [A3: A,N: nat] :
( ( power_power @ A @ ( uminus_uminus @ A @ A3 ) @ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N ) )
= ( power_power @ A @ A3 @ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N ) ) ) ) ).
% Power.ring_1_class.power_minus_even
thf(fact_3783_power__minus__odd,axiom,
! [A: $tType] :
( ( ring_1 @ A )
=> ! [N: nat,A3: A] :
( ~ ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N )
=> ( ( power_power @ A @ ( uminus_uminus @ A @ A3 ) @ N )
= ( uminus_uminus @ A @ ( power_power @ A @ A3 @ N ) ) ) ) ) ).
% power_minus_odd
thf(fact_3784_Parity_Oring__1__class_Opower__minus__even,axiom,
! [A: $tType] :
( ( ring_1 @ A )
=> ! [N: nat,A3: A] :
( ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N )
=> ( ( power_power @ A @ ( uminus_uminus @ A @ A3 ) @ N )
= ( power_power @ A @ A3 @ N ) ) ) ) ).
% Parity.ring_1_class.power_minus_even
thf(fact_3785_diff__numeral__special_I4_J,axiom,
! [A: $tType] :
( ( neg_numeral @ A )
=> ! [M: num] :
( ( minus_minus @ A @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ M ) ) @ ( one_one @ A ) )
= ( uminus_uminus @ A @ ( numeral_numeral @ A @ ( plus_plus @ num @ M @ one2 ) ) ) ) ) ).
% diff_numeral_special(4)
thf(fact_3786_diff__numeral__special_I3_J,axiom,
! [A: $tType] :
( ( neg_numeral @ A )
=> ! [N: num] :
( ( minus_minus @ A @ ( one_one @ A ) @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ N ) ) )
= ( numeral_numeral @ A @ ( plus_plus @ num @ one2 @ N ) ) ) ) ).
% diff_numeral_special(3)
thf(fact_3787_ceiling__less__zero,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [X2: A] :
( ( ord_less @ int @ ( archimedean_ceiling @ A @ X2 ) @ ( zero_zero @ int ) )
= ( ord_less_eq @ A @ X2 @ ( uminus_uminus @ A @ ( one_one @ A ) ) ) ) ) ).
% ceiling_less_zero
thf(fact_3788_zero__le__ceiling,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [X2: A] :
( ( ord_less_eq @ int @ ( zero_zero @ int ) @ ( archimedean_ceiling @ A @ X2 ) )
= ( ord_less @ A @ ( uminus_uminus @ A @ ( one_one @ A ) ) @ X2 ) ) ) ).
% zero_le_ceiling
thf(fact_3789_power__minus1__even,axiom,
! [A: $tType] :
( ( ring_1 @ A )
=> ! [N: nat] :
( ( power_power @ A @ ( uminus_uminus @ A @ ( one_one @ A ) ) @ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N ) )
= ( one_one @ A ) ) ) ).
% power_minus1_even
thf(fact_3790_neg__one__odd__power,axiom,
! [A: $tType] :
( ( ring_1 @ A )
=> ! [N: nat] :
( ~ ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N )
=> ( ( power_power @ A @ ( uminus_uminus @ A @ ( one_one @ A ) ) @ N )
= ( uminus_uminus @ A @ ( one_one @ A ) ) ) ) ) ).
% neg_one_odd_power
thf(fact_3791_neg__one__even__power,axiom,
! [A: $tType] :
( ( ring_1 @ A )
=> ! [N: nat] :
( ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N )
=> ( ( power_power @ A @ ( uminus_uminus @ A @ ( one_one @ A ) ) @ N )
= ( one_one @ A ) ) ) ) ).
% neg_one_even_power
thf(fact_3792_neg__numeral__power__eq__of__int__cancel__iff,axiom,
! [A: $tType] :
( ( ring_char_0 @ A )
=> ! [X2: num,N: nat,Y3: int] :
( ( ( power_power @ A @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ X2 ) ) @ N )
= ( ring_1_of_int @ A @ Y3 ) )
= ( ( power_power @ int @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ X2 ) ) @ N )
= Y3 ) ) ) ).
% neg_numeral_power_eq_of_int_cancel_iff
thf(fact_3793_of__int__eq__neg__numeral__power__cancel__iff,axiom,
! [A: $tType] :
( ( ring_char_0 @ A )
=> ! [Y3: int,X2: num,N: nat] :
( ( ( ring_1_of_int @ A @ Y3 )
= ( power_power @ A @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ X2 ) ) @ N ) )
= ( Y3
= ( power_power @ int @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ X2 ) ) @ N ) ) ) ) ).
% of_int_eq_neg_numeral_power_cancel_iff
thf(fact_3794_neg__numeral__le__floor,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [V: num,X2: A] :
( ( ord_less_eq @ int @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ V ) ) @ ( archim6421214686448440834_floor @ A @ X2 ) )
= ( ord_less_eq @ A @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ V ) ) @ X2 ) ) ) ).
% neg_numeral_le_floor
thf(fact_3795_floor__less__neg__numeral,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [X2: A,V: num] :
( ( ord_less @ int @ ( archim6421214686448440834_floor @ A @ X2 ) @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ V ) ) )
= ( ord_less @ A @ X2 @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ V ) ) ) ) ) ).
% floor_less_neg_numeral
thf(fact_3796_ceiling__le__neg__numeral,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [X2: A,V: num] :
( ( ord_less_eq @ int @ ( archimedean_ceiling @ A @ X2 ) @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ V ) ) )
= ( ord_less_eq @ A @ X2 @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ V ) ) ) ) ) ).
% ceiling_le_neg_numeral
thf(fact_3797_neg__numeral__less__ceiling,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [V: num,X2: A] :
( ( ord_less @ int @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ V ) ) @ ( archimedean_ceiling @ A @ X2 ) )
= ( ord_less @ A @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ V ) ) @ X2 ) ) ) ).
% neg_numeral_less_ceiling
thf(fact_3798_neg__numeral__less__floor,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [V: num,X2: A] :
( ( ord_less @ int @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ V ) ) @ ( archim6421214686448440834_floor @ A @ X2 ) )
= ( ord_less_eq @ A @ ( plus_plus @ A @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ V ) ) @ ( one_one @ A ) ) @ X2 ) ) ) ).
% neg_numeral_less_floor
thf(fact_3799_floor__le__neg__numeral,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [X2: A,V: num] :
( ( ord_less_eq @ int @ ( archim6421214686448440834_floor @ A @ X2 ) @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ V ) ) )
= ( ord_less @ A @ X2 @ ( plus_plus @ A @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ V ) ) @ ( one_one @ A ) ) ) ) ) ).
% floor_le_neg_numeral
thf(fact_3800_ceiling__less__neg__numeral,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [X2: A,V: num] :
( ( ord_less @ int @ ( archimedean_ceiling @ A @ X2 ) @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ V ) ) )
= ( ord_less_eq @ A @ X2 @ ( minus_minus @ A @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ V ) ) @ ( one_one @ A ) ) ) ) ) ).
% ceiling_less_neg_numeral
thf(fact_3801_neg__numeral__le__ceiling,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [V: num,X2: A] :
( ( ord_less_eq @ int @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ V ) ) @ ( archimedean_ceiling @ A @ X2 ) )
= ( ord_less @ A @ ( minus_minus @ A @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ V ) ) @ ( one_one @ A ) ) @ X2 ) ) ) ).
% neg_numeral_le_ceiling
thf(fact_3802_of__int__le__neg__numeral__power__cancel__iff,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [A3: int,X2: num,N: nat] :
( ( ord_less_eq @ A @ ( ring_1_of_int @ A @ A3 ) @ ( power_power @ A @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ X2 ) ) @ N ) )
= ( ord_less_eq @ int @ A3 @ ( power_power @ int @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ X2 ) ) @ N ) ) ) ) ).
% of_int_le_neg_numeral_power_cancel_iff
thf(fact_3803_neg__numeral__power__le__of__int__cancel__iff,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [X2: num,N: nat,A3: int] :
( ( ord_less_eq @ A @ ( power_power @ A @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ X2 ) ) @ N ) @ ( ring_1_of_int @ A @ A3 ) )
= ( ord_less_eq @ int @ ( power_power @ int @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ X2 ) ) @ N ) @ A3 ) ) ) ).
% neg_numeral_power_le_of_int_cancel_iff
thf(fact_3804_of__int__less__neg__numeral__power__cancel__iff,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [A3: int,X2: num,N: nat] :
( ( ord_less @ A @ ( ring_1_of_int @ A @ A3 ) @ ( power_power @ A @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ X2 ) ) @ N ) )
= ( ord_less @ int @ A3 @ ( power_power @ int @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ X2 ) ) @ N ) ) ) ) ).
% of_int_less_neg_numeral_power_cancel_iff
thf(fact_3805_neg__numeral__power__less__of__int__cancel__iff,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [X2: num,N: nat,A3: int] :
( ( ord_less @ A @ ( power_power @ A @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ X2 ) ) @ N ) @ ( ring_1_of_int @ A @ A3 ) )
= ( ord_less @ int @ ( power_power @ int @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ X2 ) ) @ N ) @ A3 ) ) ) ).
% neg_numeral_power_less_of_int_cancel_iff
thf(fact_3806_minus__less__iff,axiom,
! [A: $tType] :
( ( ordered_ab_group_add @ A )
=> ! [A3: A,B3: A] :
( ( ord_less @ A @ ( uminus_uminus @ A @ A3 ) @ B3 )
= ( ord_less @ A @ ( uminus_uminus @ A @ B3 ) @ A3 ) ) ) ).
% minus_less_iff
thf(fact_3807_less__minus__iff,axiom,
! [A: $tType] :
( ( ordered_ab_group_add @ A )
=> ! [A3: A,B3: A] :
( ( ord_less @ A @ A3 @ ( uminus_uminus @ A @ B3 ) )
= ( ord_less @ A @ B3 @ ( uminus_uminus @ A @ A3 ) ) ) ) ).
% less_minus_iff
thf(fact_3808_verit__negate__coefficient_I2_J,axiom,
! [A: $tType] :
( ( ordered_ab_group_add @ A )
=> ! [A3: A,B3: A] :
( ( ord_less @ A @ A3 @ B3 )
=> ( ord_less @ A @ ( uminus_uminus @ A @ B3 ) @ ( uminus_uminus @ A @ A3 ) ) ) ) ).
% verit_negate_coefficient(2)
thf(fact_3809_mod__minus__right,axiom,
! [A: $tType] :
( ( euclid8851590272496341667cancel @ A )
=> ! [A3: A,B3: A] :
( ( modulo_modulo @ A @ A3 @ ( uminus_uminus @ A @ B3 ) )
= ( uminus_uminus @ A @ ( modulo_modulo @ A @ ( uminus_uminus @ A @ A3 ) @ B3 ) ) ) ) ).
% mod_minus_right
thf(fact_3810_euclidean__ring__cancel__class_Omod__minus__cong,axiom,
! [A: $tType] :
( ( euclid8851590272496341667cancel @ A )
=> ! [A3: A,B3: A,A5: A] :
( ( ( modulo_modulo @ A @ A3 @ B3 )
= ( modulo_modulo @ A @ A5 @ B3 ) )
=> ( ( modulo_modulo @ A @ ( uminus_uminus @ A @ A3 ) @ B3 )
= ( modulo_modulo @ A @ ( uminus_uminus @ A @ A5 ) @ B3 ) ) ) ) ).
% euclidean_ring_cancel_class.mod_minus_cong
thf(fact_3811_mod__minus__eq,axiom,
! [A: $tType] :
( ( euclid8851590272496341667cancel @ A )
=> ! [A3: A,B3: A] :
( ( modulo_modulo @ A @ ( uminus_uminus @ A @ ( modulo_modulo @ A @ A3 @ B3 ) ) @ B3 )
= ( modulo_modulo @ A @ ( uminus_uminus @ A @ A3 ) @ B3 ) ) ) ).
% mod_minus_eq
thf(fact_3812_numeral__neq__neg__numeral,axiom,
! [A: $tType] :
( ( ring_char_0 @ A )
=> ! [M: num,N: num] :
( ( numeral_numeral @ A @ M )
!= ( uminus_uminus @ A @ ( numeral_numeral @ A @ N ) ) ) ) ).
% numeral_neq_neg_numeral
thf(fact_3813_neg__numeral__neq__numeral,axiom,
! [A: $tType] :
( ( ring_char_0 @ A )
=> ! [M: num,N: num] :
( ( uminus_uminus @ A @ ( numeral_numeral @ A @ M ) )
!= ( numeral_numeral @ A @ N ) ) ) ).
% neg_numeral_neq_numeral
thf(fact_3814_div__minus__right,axiom,
! [A: $tType] :
( ( euclid8851590272496341667cancel @ A )
=> ! [A3: A,B3: A] :
( ( divide_divide @ A @ A3 @ ( uminus_uminus @ A @ B3 ) )
= ( divide_divide @ A @ ( uminus_uminus @ A @ A3 ) @ B3 ) ) ) ).
% div_minus_right
thf(fact_3815_minus__divide__left,axiom,
! [A: $tType] :
( ( division_ring @ A )
=> ! [A3: A,B3: A] :
( ( uminus_uminus @ A @ ( divide_divide @ A @ A3 @ B3 ) )
= ( divide_divide @ A @ ( uminus_uminus @ A @ A3 ) @ B3 ) ) ) ).
% minus_divide_left
thf(fact_3816_minus__divide__divide,axiom,
! [A: $tType] :
( ( field @ A )
=> ! [A3: A,B3: A] :
( ( divide_divide @ A @ ( uminus_uminus @ A @ A3 ) @ ( uminus_uminus @ A @ B3 ) )
= ( divide_divide @ A @ A3 @ B3 ) ) ) ).
% minus_divide_divide
thf(fact_3817_minus__divide__right,axiom,
! [A: $tType] :
( ( field @ A )
=> ! [A3: A,B3: A] :
( ( uminus_uminus @ A @ ( divide_divide @ A @ A3 @ B3 ) )
= ( divide_divide @ A @ A3 @ ( uminus_uminus @ A @ B3 ) ) ) ) ).
% minus_divide_right
thf(fact_3818_minus__mult__commute,axiom,
! [A: $tType] :
( ( ring @ A )
=> ! [A3: A,B3: A] :
( ( times_times @ A @ ( uminus_uminus @ A @ A3 ) @ B3 )
= ( times_times @ A @ A3 @ ( uminus_uminus @ A @ B3 ) ) ) ) ).
% minus_mult_commute
thf(fact_3819_square__eq__iff,axiom,
! [A: $tType] :
( ( idom @ A )
=> ! [A3: A,B3: A] :
( ( ( times_times @ A @ A3 @ A3 )
= ( times_times @ A @ B3 @ B3 ) )
= ( ( A3 = B3 )
| ( A3
= ( uminus_uminus @ A @ B3 ) ) ) ) ) ).
% square_eq_iff
thf(fact_3820_is__num__normalize_I8_J,axiom,
! [A: $tType] :
( ( neg_numeral @ A )
=> ! [A3: A,B3: A] :
( ( uminus_uminus @ A @ ( plus_plus @ A @ A3 @ B3 ) )
= ( plus_plus @ A @ ( uminus_uminus @ A @ B3 ) @ ( uminus_uminus @ A @ A3 ) ) ) ) ).
% is_num_normalize(8)
thf(fact_3821_group__cancel_Oneg1,axiom,
! [A: $tType] :
( ( ab_group_add @ A )
=> ! [A2: A,K: A,A3: A] :
( ( A2
= ( plus_plus @ A @ K @ A3 ) )
=> ( ( uminus_uminus @ A @ A2 )
= ( plus_plus @ A @ ( uminus_uminus @ A @ K ) @ ( uminus_uminus @ A @ A3 ) ) ) ) ) ).
% group_cancel.neg1
thf(fact_3822_add_Oinverse__distrib__swap,axiom,
! [A: $tType] :
( ( group_add @ A )
=> ! [A3: A,B3: A] :
( ( uminus_uminus @ A @ ( plus_plus @ A @ A3 @ B3 ) )
= ( plus_plus @ A @ ( uminus_uminus @ A @ B3 ) @ ( uminus_uminus @ A @ A3 ) ) ) ) ).
% add.inverse_distrib_swap
thf(fact_3823_minus__diff__minus,axiom,
! [A: $tType] :
( ( ab_group_add @ A )
=> ! [A3: A,B3: A] :
( ( minus_minus @ A @ ( uminus_uminus @ A @ A3 ) @ ( uminus_uminus @ A @ B3 ) )
= ( uminus_uminus @ A @ ( minus_minus @ A @ A3 @ B3 ) ) ) ) ).
% minus_diff_minus
thf(fact_3824_minus__diff__commute,axiom,
! [A: $tType] :
( ( ab_group_add @ A )
=> ! [B3: A,A3: A] :
( ( minus_minus @ A @ ( uminus_uminus @ A @ B3 ) @ A3 )
= ( minus_minus @ A @ ( uminus_uminus @ A @ A3 ) @ B3 ) ) ) ).
% minus_diff_commute
thf(fact_3825_one__neq__neg__one,axiom,
! [A: $tType] :
( ( ring_char_0 @ A )
=> ( ( one_one @ A )
!= ( uminus_uminus @ A @ ( one_one @ A ) ) ) ) ).
% one_neq_neg_one
thf(fact_3826_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_3827_fact__ge__self,axiom,
! [N: nat] : ( ord_less_eq @ nat @ N @ ( semiring_char_0_fact @ nat @ N ) ) ).
% fact_ge_self
thf(fact_3828_le__minus__iff,axiom,
! [A: $tType] :
( ( ordered_ab_group_add @ A )
=> ! [A3: A,B3: A] :
( ( ord_less_eq @ A @ A3 @ ( uminus_uminus @ A @ B3 ) )
= ( ord_less_eq @ A @ B3 @ ( uminus_uminus @ A @ A3 ) ) ) ) ).
% le_minus_iff
thf(fact_3829_minus__le__iff,axiom,
! [A: $tType] :
( ( ordered_ab_group_add @ A )
=> ! [A3: A,B3: A] :
( ( ord_less_eq @ A @ ( uminus_uminus @ A @ A3 ) @ B3 )
= ( ord_less_eq @ A @ ( uminus_uminus @ A @ B3 ) @ A3 ) ) ) ).
% minus_le_iff
thf(fact_3830_le__imp__neg__le,axiom,
! [A: $tType] :
( ( ordered_ab_group_add @ A )
=> ! [A3: A,B3: A] :
( ( ord_less_eq @ A @ A3 @ B3 )
=> ( ord_less_eq @ A @ ( uminus_uminus @ A @ B3 ) @ ( uminus_uminus @ A @ A3 ) ) ) ) ).
% le_imp_neg_le
thf(fact_3831_of__int__neg__numeral,axiom,
! [A: $tType] :
( ( ring_1 @ A )
=> ! [K: num] :
( ( ring_1_of_int @ A @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ K ) ) )
= ( uminus_uminus @ A @ ( numeral_numeral @ A @ K ) ) ) ) ).
% of_int_neg_numeral
thf(fact_3832_sum__negf,axiom,
! [A: $tType,B: $tType] :
( ( ab_group_add @ A )
=> ! [F2: B > A,A2: set @ B] :
( ( groups7311177749621191930dd_sum @ B @ A
@ ^ [X3: B] : ( uminus_uminus @ A @ ( F2 @ X3 ) )
@ A2 )
= ( uminus_uminus @ A @ ( groups7311177749621191930dd_sum @ B @ A @ F2 @ A2 ) ) ) ) ).
% sum_negf
thf(fact_3833_sums__minus,axiom,
! [A: $tType] :
( ( real_V822414075346904944vector @ A )
=> ! [F2: nat > A,A3: A] :
( ( sums @ A @ F2 @ A3 )
=> ( sums @ A
@ ^ [N4: nat] : ( uminus_uminus @ A @ ( F2 @ N4 ) )
@ ( uminus_uminus @ A @ A3 ) ) ) ) ).
% sums_minus
thf(fact_3834_TBOUND__upd,axiom,
! [A: $tType] :
( ( heap @ A )
=> ! [Xs: nat,I: A,X2: array @ A] : ( time_TBOUND @ ( array @ A ) @ ( array_upd @ A @ Xs @ I @ X2 ) @ ( one_one @ nat ) ) ) ).
% TBOUND_upd
thf(fact_3835_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_3836_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_3837_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_3838_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_3839_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_3840_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_3841_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_3842_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_3843_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_3844_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_3845_neg__eq__iff__add__eq__0,axiom,
! [A: $tType] :
( ( group_add @ A )
=> ! [A3: A,B3: A] :
( ( ( uminus_uminus @ A @ A3 )
= B3 )
= ( ( plus_plus @ A @ A3 @ B3 )
= ( zero_zero @ A ) ) ) ) ).
% neg_eq_iff_add_eq_0
thf(fact_3846_eq__neg__iff__add__eq__0,axiom,
! [A: $tType] :
( ( group_add @ A )
=> ! [A3: A,B3: A] :
( ( A3
= ( uminus_uminus @ A @ B3 ) )
= ( ( plus_plus @ A @ A3 @ B3 )
= ( zero_zero @ A ) ) ) ) ).
% eq_neg_iff_add_eq_0
thf(fact_3847_add_Oinverse__unique,axiom,
! [A: $tType] :
( ( group_add @ A )
=> ! [A3: A,B3: A] :
( ( ( plus_plus @ A @ A3 @ B3 )
= ( zero_zero @ A ) )
=> ( ( uminus_uminus @ A @ A3 )
= B3 ) ) ) ).
% add.inverse_unique
thf(fact_3848_ab__group__add__class_Oab__left__minus,axiom,
! [A: $tType] :
( ( ab_group_add @ A )
=> ! [A3: A] :
( ( plus_plus @ A @ ( uminus_uminus @ A @ A3 ) @ A3 )
= ( zero_zero @ A ) ) ) ).
% ab_group_add_class.ab_left_minus
thf(fact_3849_add__eq__0__iff,axiom,
! [A: $tType] :
( ( group_add @ A )
=> ! [A3: A,B3: A] :
( ( ( plus_plus @ A @ A3 @ B3 )
= ( zero_zero @ A ) )
= ( B3
= ( uminus_uminus @ A @ A3 ) ) ) ) ).
% add_eq_0_iff
thf(fact_3850_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_3851_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_3852_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_3853_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_3854_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_3855_one__neq__neg__numeral,axiom,
! [A: $tType] :
( ( ring_char_0 @ A )
=> ! [N: num] :
( ( one_one @ A )
!= ( uminus_uminus @ A @ ( numeral_numeral @ A @ N ) ) ) ) ).
% one_neq_neg_numeral
thf(fact_3856_numeral__neq__neg__one,axiom,
! [A: $tType] :
( ( ring_char_0 @ A )
=> ! [N: num] :
( ( numeral_numeral @ A @ N )
!= ( uminus_uminus @ A @ ( one_one @ A ) ) ) ) ).
% numeral_neq_neg_one
thf(fact_3857_numeral__times__minus__swap,axiom,
! [A: $tType] :
( ( comm_ring_1 @ A )
=> ! [W: num,X2: A] :
( ( times_times @ A @ ( numeral_numeral @ A @ W ) @ ( uminus_uminus @ A @ X2 ) )
= ( times_times @ A @ X2 @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ W ) ) ) ) ) ).
% numeral_times_minus_swap
thf(fact_3858_nonzero__minus__divide__right,axiom,
! [A: $tType] :
( ( division_ring @ A )
=> ! [B3: A,A3: A] :
( ( B3
!= ( zero_zero @ A ) )
=> ( ( uminus_uminus @ A @ ( divide_divide @ A @ A3 @ B3 ) )
= ( divide_divide @ A @ A3 @ ( uminus_uminus @ A @ B3 ) ) ) ) ) ).
% nonzero_minus_divide_right
thf(fact_3859_nonzero__minus__divide__divide,axiom,
! [A: $tType] :
( ( division_ring @ A )
=> ! [B3: A,A3: A] :
( ( B3
!= ( zero_zero @ A ) )
=> ( ( divide_divide @ A @ ( uminus_uminus @ A @ A3 ) @ ( uminus_uminus @ A @ B3 ) )
= ( divide_divide @ A @ A3 @ B3 ) ) ) ) ).
% nonzero_minus_divide_divide
thf(fact_3860_square__eq__1__iff,axiom,
! [A: $tType] :
( ( ring_15535105094025558882visors @ A )
=> ! [X2: A] :
( ( ( times_times @ A @ X2 @ X2 )
= ( one_one @ A ) )
= ( ( X2
= ( one_one @ A ) )
| ( X2
= ( uminus_uminus @ A @ ( one_one @ A ) ) ) ) ) ) ).
% square_eq_1_iff
thf(fact_3861_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_3862_group__cancel_Osub2,axiom,
! [A: $tType] :
( ( ab_group_add @ A )
=> ! [B2: A,K: A,B3: A,A3: A] :
( ( B2
= ( plus_plus @ A @ K @ B3 ) )
=> ( ( minus_minus @ A @ A3 @ B2 )
= ( plus_plus @ A @ ( uminus_uminus @ A @ K ) @ ( minus_minus @ A @ A3 @ B3 ) ) ) ) ) ).
% group_cancel.sub2
thf(fact_3863_diff__conv__add__uminus,axiom,
! [A: $tType] :
( ( group_add @ A )
=> ( ( minus_minus @ A )
= ( ^ [A6: A,B7: A] : ( plus_plus @ A @ A6 @ ( uminus_uminus @ A @ B7 ) ) ) ) ) ).
% diff_conv_add_uminus
thf(fact_3864_ab__group__add__class_Oab__diff__conv__add__uminus,axiom,
! [A: $tType] :
( ( ab_group_add @ A )
=> ( ( minus_minus @ A )
= ( ^ [A6: A,B7: A] : ( plus_plus @ A @ A6 @ ( uminus_uminus @ A @ B7 ) ) ) ) ) ).
% ab_group_add_class.ab_diff_conv_add_uminus
thf(fact_3865_dvd__neg__div,axiom,
! [A: $tType] :
( ( idom_divide @ A )
=> ! [B3: A,A3: A] :
( ( dvd_dvd @ A @ B3 @ A3 )
=> ( ( divide_divide @ A @ ( uminus_uminus @ A @ A3 ) @ B3 )
= ( uminus_uminus @ A @ ( divide_divide @ A @ A3 @ B3 ) ) ) ) ) ).
% dvd_neg_div
thf(fact_3866_dvd__div__neg,axiom,
! [A: $tType] :
( ( idom_divide @ A )
=> ! [B3: A,A3: A] :
( ( dvd_dvd @ A @ B3 @ A3 )
=> ( ( divide_divide @ A @ A3 @ ( uminus_uminus @ A @ B3 ) )
= ( uminus_uminus @ A @ ( divide_divide @ A @ A3 @ B3 ) ) ) ) ) ).
% dvd_div_neg
thf(fact_3867_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_3868_int__cases,axiom,
! [Z: int] :
( ! [N3: nat] :
( Z
!= ( semiring_1_of_nat @ int @ N3 ) )
=> ~ ! [N3: nat] :
( Z
!= ( uminus_uminus @ int @ ( semiring_1_of_nat @ int @ ( suc @ N3 ) ) ) ) ) ).
% int_cases
thf(fact_3869_int__of__nat__induct,axiom,
! [P: int > $o,Z: int] :
( ! [N3: nat] : ( P @ ( semiring_1_of_nat @ int @ N3 ) )
=> ( ! [N3: nat] : ( P @ ( uminus_uminus @ int @ ( semiring_1_of_nat @ int @ ( suc @ N3 ) ) ) )
=> ( P @ Z ) ) ) ).
% int_of_nat_induct
thf(fact_3870_pos__zmult__eq__1__iff__lemma,axiom,
! [M: int,N: int] :
( ( ( times_times @ int @ M @ N )
= ( one_one @ int ) )
=> ( ( M
= ( one_one @ int ) )
| ( M
= ( uminus_uminus @ int @ ( one_one @ int ) ) ) ) ) ).
% pos_zmult_eq_1_iff_lemma
thf(fact_3871_zmult__eq__1__iff,axiom,
! [M: int,N: int] :
( ( ( times_times @ int @ M @ N )
= ( one_one @ int ) )
= ( ( ( M
= ( one_one @ int ) )
& ( N
= ( one_one @ int ) ) )
| ( ( M
= ( uminus_uminus @ int @ ( one_one @ int ) ) )
& ( N
= ( uminus_uminus @ int @ ( one_one @ int ) ) ) ) ) ) ).
% zmult_eq_1_iff
thf(fact_3872_minus__int__code_I2_J,axiom,
! [L: int] :
( ( minus_minus @ int @ ( zero_zero @ int ) @ L )
= ( uminus_uminus @ int @ L ) ) ).
% minus_int_code(2)
thf(fact_3873_pochhammer__fact,axiom,
! [A: $tType] :
( ( ( semiring_char_0 @ A )
& ( comm_semiring_1 @ A ) )
=> ( ( semiring_char_0_fact @ A )
= ( comm_s3205402744901411588hammer @ A @ ( one_one @ A ) ) ) ) ).
% pochhammer_fact
thf(fact_3874_pochhammer__same,axiom,
! [A: $tType] :
( ( ( semiring_char_0 @ A )
& ( comm_ring_1 @ A )
& ( semiri3467727345109120633visors @ A ) )
=> ! [N: nat] :
( ( comm_s3205402744901411588hammer @ A @ ( uminus_uminus @ A @ ( semiring_1_of_nat @ A @ N ) ) @ N )
= ( times_times @ A @ ( power_power @ A @ ( uminus_uminus @ A @ ( one_one @ A ) ) @ N ) @ ( semiring_char_0_fact @ A @ N ) ) ) ) ).
% pochhammer_same
thf(fact_3875_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_3876_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_3877_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_3878_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_3879_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_3880_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_3881_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_3882_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_3883_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_3884_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_3885_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_3886_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_3887_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_3888_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_3889_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_3890_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_3891_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_3892_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_3893_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_3894_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_3895_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_3896_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_3897_uminus__numeral__One,axiom,
! [A: $tType] :
( ( neg_numeral @ A )
=> ( ( uminus_uminus @ A @ ( numeral_numeral @ A @ one2 ) )
= ( uminus_uminus @ A @ ( one_one @ A ) ) ) ) ).
% uminus_numeral_One
thf(fact_3898_gbinomial__pochhammer,axiom,
! [A: $tType] :
( ( field_char_0 @ A )
=> ( ( gbinomial @ A )
= ( ^ [A6: A,K3: nat] : ( divide_divide @ A @ ( times_times @ A @ ( power_power @ A @ ( uminus_uminus @ A @ ( one_one @ A ) ) @ K3 ) @ ( comm_s3205402744901411588hammer @ A @ ( uminus_uminus @ A @ A6 ) @ K3 ) ) @ ( semiring_char_0_fact @ A @ K3 ) ) ) ) ) ).
% gbinomial_pochhammer
thf(fact_3899_mult__1s__ring__1_I2_J,axiom,
! [A: $tType] :
( ( ring_1 @ A )
=> ! [B3: A] :
( ( times_times @ A @ B3 @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ one2 ) ) )
= ( uminus_uminus @ A @ B3 ) ) ) ).
% mult_1s_ring_1(2)
thf(fact_3900_mult__1s__ring__1_I1_J,axiom,
! [A: $tType] :
( ( ring_1 @ A )
=> ! [B3: A] :
( ( times_times @ A @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ one2 ) ) @ B3 )
= ( uminus_uminus @ A @ B3 ) ) ) ).
% mult_1s_ring_1(1)
thf(fact_3901_divide__eq__minus__1__iff,axiom,
! [A: $tType] :
( ( field @ A )
=> ! [A3: A,B3: A] :
( ( ( divide_divide @ A @ A3 @ B3 )
= ( uminus_uminus @ A @ ( one_one @ A ) ) )
= ( ( B3
!= ( zero_zero @ A ) )
& ( A3
= ( uminus_uminus @ A @ B3 ) ) ) ) ) ).
% divide_eq_minus_1_iff
thf(fact_3902_eq__minus__divide__eq,axiom,
! [A: $tType] :
( ( division_ring @ A )
=> ! [A3: A,B3: A,C3: A] :
( ( A3
= ( uminus_uminus @ A @ ( divide_divide @ A @ B3 @ C3 ) ) )
= ( ( ( C3
!= ( zero_zero @ A ) )
=> ( ( times_times @ A @ A3 @ C3 )
= ( uminus_uminus @ A @ B3 ) ) )
& ( ( C3
= ( zero_zero @ A ) )
=> ( A3
= ( zero_zero @ A ) ) ) ) ) ) ).
% eq_minus_divide_eq
thf(fact_3903_minus__divide__eq__eq,axiom,
! [A: $tType] :
( ( division_ring @ A )
=> ! [B3: A,C3: A,A3: A] :
( ( ( uminus_uminus @ A @ ( divide_divide @ A @ B3 @ C3 ) )
= A3 )
= ( ( ( C3
!= ( zero_zero @ A ) )
=> ( ( uminus_uminus @ A @ B3 )
= ( times_times @ A @ A3 @ C3 ) ) )
& ( ( C3
= ( zero_zero @ A ) )
=> ( A3
= ( zero_zero @ A ) ) ) ) ) ) ).
% minus_divide_eq_eq
thf(fact_3904_nonzero__neg__divide__eq__eq,axiom,
! [A: $tType] :
( ( division_ring @ A )
=> ! [B3: A,A3: A,C3: A] :
( ( B3
!= ( zero_zero @ A ) )
=> ( ( ( uminus_uminus @ A @ ( divide_divide @ A @ A3 @ B3 ) )
= C3 )
= ( ( uminus_uminus @ A @ A3 )
= ( times_times @ A @ C3 @ B3 ) ) ) ) ) ).
% nonzero_neg_divide_eq_eq
thf(fact_3905_nonzero__neg__divide__eq__eq2,axiom,
! [A: $tType] :
( ( division_ring @ A )
=> ! [B3: A,C3: A,A3: A] :
( ( B3
!= ( zero_zero @ A ) )
=> ( ( C3
= ( uminus_uminus @ A @ ( divide_divide @ A @ A3 @ B3 ) ) )
= ( ( times_times @ A @ C3 @ B3 )
= ( uminus_uminus @ A @ A3 ) ) ) ) ) ).
% nonzero_neg_divide_eq_eq2
thf(fact_3906_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_3907_power__minus,axiom,
! [A: $tType] :
( ( ring_1 @ A )
=> ! [A3: A,N: nat] :
( ( power_power @ A @ ( uminus_uminus @ A @ A3 ) @ N )
= ( times_times @ A @ ( power_power @ A @ ( uminus_uminus @ A @ ( one_one @ A ) ) @ N ) @ ( power_power @ A @ A3 @ N ) ) ) ) ).
% power_minus
thf(fact_3908_power__minus__Bit0,axiom,
! [A: $tType] :
( ( ring_1 @ A )
=> ! [X2: A,K: num] :
( ( power_power @ A @ ( uminus_uminus @ A @ X2 ) @ ( numeral_numeral @ nat @ ( bit0 @ K ) ) )
= ( power_power @ A @ X2 @ ( numeral_numeral @ nat @ ( bit0 @ K ) ) ) ) ) ).
% power_minus_Bit0
thf(fact_3909_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_3910_fact__fact__dvd__fact,axiom,
! [A: $tType] :
( ( linordered_semidom @ A )
=> ! [K: nat,N: nat] : ( dvd_dvd @ A @ ( times_times @ A @ ( semiring_char_0_fact @ A @ K ) @ ( semiring_char_0_fact @ A @ N ) ) @ ( semiring_char_0_fact @ A @ ( plus_plus @ nat @ K @ N ) ) ) ) ).
% fact_fact_dvd_fact
thf(fact_3911_power__minus__Bit1,axiom,
! [A: $tType] :
( ( ring_1 @ A )
=> ! [X2: A,K: num] :
( ( power_power @ A @ ( uminus_uminus @ A @ X2 ) @ ( numeral_numeral @ nat @ ( bit1 @ K ) ) )
= ( uminus_uminus @ A @ ( power_power @ A @ X2 @ ( numeral_numeral @ nat @ ( bit1 @ K ) ) ) ) ) ) ).
% power_minus_Bit1
thf(fact_3912_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_3913_norm__uminus__minus,axiom,
! [A: $tType] :
( ( real_V822414075346904944vector @ A )
=> ! [X2: A,Y3: A] :
( ( real_V7770717601297561774m_norm @ A @ ( minus_minus @ A @ ( uminus_uminus @ A @ X2 ) @ Y3 ) )
= ( real_V7770717601297561774m_norm @ A @ ( plus_plus @ A @ X2 @ Y3 ) ) ) ) ).
% norm_uminus_minus
thf(fact_3914_powr__minus__divide,axiom,
! [A: $tType] :
( ( ( real_V3459762299906320749_field @ A )
& ( ln @ A ) )
=> ! [X2: A,A3: A] :
( ( powr @ A @ X2 @ ( uminus_uminus @ A @ A3 ) )
= ( divide_divide @ A @ ( one_one @ A ) @ ( powr @ A @ X2 @ A3 ) ) ) ) ).
% powr_minus_divide
thf(fact_3915_int__cases4,axiom,
! [M: int] :
( ! [N3: nat] :
( M
!= ( semiring_1_of_nat @ int @ N3 ) )
=> ~ ! [N3: nat] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ N3 )
=> ( M
!= ( uminus_uminus @ int @ ( semiring_1_of_nat @ int @ N3 ) ) ) ) ) ).
% int_cases4
thf(fact_3916_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_3917_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_3918_nonpos__int__cases,axiom,
! [K: int] :
( ( ord_less_eq @ int @ K @ ( zero_zero @ int ) )
=> ~ ! [N3: nat] :
( K
!= ( uminus_uminus @ int @ ( semiring_1_of_nat @ int @ N3 ) ) ) ) ).
% nonpos_int_cases
thf(fact_3919_zmod__zminus2__eq__if,axiom,
! [A3: int,B3: int] :
( ( ( ( modulo_modulo @ int @ A3 @ B3 )
= ( zero_zero @ int ) )
=> ( ( modulo_modulo @ int @ A3 @ ( uminus_uminus @ int @ B3 ) )
= ( zero_zero @ int ) ) )
& ( ( ( modulo_modulo @ int @ A3 @ B3 )
!= ( zero_zero @ int ) )
=> ( ( modulo_modulo @ int @ A3 @ ( uminus_uminus @ int @ B3 ) )
= ( minus_minus @ int @ ( modulo_modulo @ int @ A3 @ B3 ) @ B3 ) ) ) ) ).
% zmod_zminus2_eq_if
thf(fact_3920_zmod__zminus1__eq__if,axiom,
! [A3: int,B3: int] :
( ( ( ( modulo_modulo @ int @ A3 @ B3 )
= ( zero_zero @ int ) )
=> ( ( modulo_modulo @ int @ ( uminus_uminus @ int @ A3 ) @ B3 )
= ( zero_zero @ int ) ) )
& ( ( ( modulo_modulo @ int @ A3 @ B3 )
!= ( zero_zero @ int ) )
=> ( ( modulo_modulo @ int @ ( uminus_uminus @ int @ A3 ) @ B3 )
= ( minus_minus @ int @ B3 @ ( modulo_modulo @ int @ A3 @ B3 ) ) ) ) ) ).
% zmod_zminus1_eq_if
thf(fact_3921_fact__prod,axiom,
! [A: $tType] :
( ( semiring_char_0 @ A )
=> ( ( semiring_char_0_fact @ A )
= ( ^ [N4: nat] :
( semiring_1_of_nat @ A
@ ( groups7121269368397514597t_prod @ nat @ nat
@ ^ [X3: nat] : X3
@ ( set_or1337092689740270186AtMost @ nat @ ( one_one @ nat ) @ N4 ) ) ) ) ) ) ).
% fact_prod
thf(fact_3922_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_3923_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_3924_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_3925_less__minus__divide__eq,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [A3: A,B3: A,C3: A] :
( ( ord_less @ A @ A3 @ ( uminus_uminus @ A @ ( divide_divide @ A @ B3 @ C3 ) ) )
= ( ( ( ord_less @ A @ ( zero_zero @ A ) @ C3 )
=> ( ord_less @ A @ ( times_times @ A @ A3 @ C3 ) @ ( uminus_uminus @ A @ B3 ) ) )
& ( ~ ( ord_less @ A @ ( zero_zero @ A ) @ C3 )
=> ( ( ( ord_less @ A @ C3 @ ( zero_zero @ A ) )
=> ( ord_less @ A @ ( uminus_uminus @ A @ B3 ) @ ( times_times @ A @ A3 @ C3 ) ) )
& ( ~ ( ord_less @ A @ C3 @ ( zero_zero @ A ) )
=> ( ord_less @ A @ A3 @ ( zero_zero @ A ) ) ) ) ) ) ) ) ).
% less_minus_divide_eq
thf(fact_3926_minus__divide__less__eq,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [B3: A,C3: A,A3: A] :
( ( ord_less @ A @ ( uminus_uminus @ A @ ( divide_divide @ A @ B3 @ C3 ) ) @ A3 )
= ( ( ( ord_less @ A @ ( zero_zero @ A ) @ C3 )
=> ( ord_less @ A @ ( uminus_uminus @ A @ B3 ) @ ( times_times @ A @ A3 @ C3 ) ) )
& ( ~ ( ord_less @ A @ ( zero_zero @ A ) @ C3 )
=> ( ( ( ord_less @ A @ C3 @ ( zero_zero @ A ) )
=> ( ord_less @ A @ ( times_times @ A @ A3 @ C3 ) @ ( uminus_uminus @ A @ B3 ) ) )
& ( ~ ( ord_less @ A @ C3 @ ( zero_zero @ A ) )
=> ( ord_less @ A @ ( zero_zero @ A ) @ A3 ) ) ) ) ) ) ) ).
% minus_divide_less_eq
thf(fact_3927_neg__less__minus__divide__eq,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [C3: A,A3: A,B3: A] :
( ( ord_less @ A @ C3 @ ( zero_zero @ A ) )
=> ( ( ord_less @ A @ A3 @ ( uminus_uminus @ A @ ( divide_divide @ A @ B3 @ C3 ) ) )
= ( ord_less @ A @ ( uminus_uminus @ A @ B3 ) @ ( times_times @ A @ A3 @ C3 ) ) ) ) ) ).
% neg_less_minus_divide_eq
thf(fact_3928_neg__minus__divide__less__eq,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [C3: A,B3: A,A3: A] :
( ( ord_less @ A @ C3 @ ( zero_zero @ A ) )
=> ( ( ord_less @ A @ ( uminus_uminus @ A @ ( divide_divide @ A @ B3 @ C3 ) ) @ A3 )
= ( ord_less @ A @ ( times_times @ A @ A3 @ C3 ) @ ( uminus_uminus @ A @ B3 ) ) ) ) ) ).
% neg_minus_divide_less_eq
thf(fact_3929_pos__less__minus__divide__eq,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [C3: A,A3: A,B3: A] :
( ( ord_less @ A @ ( zero_zero @ A ) @ C3 )
=> ( ( ord_less @ A @ A3 @ ( uminus_uminus @ A @ ( divide_divide @ A @ B3 @ C3 ) ) )
= ( ord_less @ A @ ( times_times @ A @ A3 @ C3 ) @ ( uminus_uminus @ A @ B3 ) ) ) ) ) ).
% pos_less_minus_divide_eq
thf(fact_3930_pos__minus__divide__less__eq,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [C3: A,B3: A,A3: A] :
( ( ord_less @ A @ ( zero_zero @ A ) @ C3 )
=> ( ( ord_less @ A @ ( uminus_uminus @ A @ ( divide_divide @ A @ B3 @ C3 ) ) @ A3 )
= ( ord_less @ A @ ( uminus_uminus @ A @ B3 ) @ ( times_times @ A @ A3 @ C3 ) ) ) ) ) ).
% pos_minus_divide_less_eq
thf(fact_3931_divide__eq__eq__numeral_I2_J,axiom,
! [A: $tType] :
( ( division_ring @ A )
=> ! [B3: A,C3: A,W: num] :
( ( ( divide_divide @ A @ B3 @ C3 )
= ( uminus_uminus @ A @ ( numeral_numeral @ A @ W ) ) )
= ( ( ( C3
!= ( zero_zero @ A ) )
=> ( B3
= ( times_times @ A @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ W ) ) @ C3 ) ) )
& ( ( C3
= ( zero_zero @ A ) )
=> ( ( uminus_uminus @ A @ ( numeral_numeral @ A @ W ) )
= ( zero_zero @ A ) ) ) ) ) ) ).
% divide_eq_eq_numeral(2)
thf(fact_3932_eq__divide__eq__numeral_I2_J,axiom,
! [A: $tType] :
( ( division_ring @ A )
=> ! [W: num,B3: A,C3: A] :
( ( ( uminus_uminus @ A @ ( numeral_numeral @ A @ W ) )
= ( divide_divide @ A @ B3 @ C3 ) )
= ( ( ( C3
!= ( zero_zero @ A ) )
=> ( ( times_times @ A @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ W ) ) @ C3 )
= B3 ) )
& ( ( C3
= ( zero_zero @ A ) )
=> ( ( uminus_uminus @ A @ ( numeral_numeral @ A @ W ) )
= ( zero_zero @ A ) ) ) ) ) ) ).
% eq_divide_eq_numeral(2)
thf(fact_3933_minus__divide__add__eq__iff,axiom,
! [A: $tType] :
( ( division_ring @ A )
=> ! [Z: A,X2: A,Y3: A] :
( ( Z
!= ( zero_zero @ A ) )
=> ( ( plus_plus @ A @ ( uminus_uminus @ A @ ( divide_divide @ A @ X2 @ Z ) ) @ Y3 )
= ( divide_divide @ A @ ( plus_plus @ A @ ( uminus_uminus @ A @ X2 ) @ ( times_times @ A @ Y3 @ Z ) ) @ Z ) ) ) ) ).
% minus_divide_add_eq_iff
thf(fact_3934_add__divide__eq__if__simps_I3_J,axiom,
! [A: $tType] :
( ( division_ring @ A )
=> ! [Z: A,A3: A,B3: A] :
( ( ( Z
= ( zero_zero @ A ) )
=> ( ( plus_plus @ A @ ( uminus_uminus @ A @ ( divide_divide @ A @ A3 @ Z ) ) @ B3 )
= B3 ) )
& ( ( Z
!= ( zero_zero @ A ) )
=> ( ( plus_plus @ A @ ( uminus_uminus @ A @ ( divide_divide @ A @ A3 @ Z ) ) @ B3 )
= ( divide_divide @ A @ ( plus_plus @ A @ ( uminus_uminus @ A @ A3 ) @ ( times_times @ A @ B3 @ Z ) ) @ Z ) ) ) ) ) ).
% add_divide_eq_if_simps(3)
thf(fact_3935_minus__divide__diff__eq__iff,axiom,
! [A: $tType] :
( ( division_ring @ A )
=> ! [Z: A,X2: A,Y3: A] :
( ( Z
!= ( zero_zero @ A ) )
=> ( ( minus_minus @ A @ ( uminus_uminus @ A @ ( divide_divide @ A @ X2 @ Z ) ) @ Y3 )
= ( divide_divide @ A @ ( minus_minus @ A @ ( uminus_uminus @ A @ X2 ) @ ( times_times @ A @ Y3 @ Z ) ) @ Z ) ) ) ) ).
% minus_divide_diff_eq_iff
thf(fact_3936_add__divide__eq__if__simps_I5_J,axiom,
! [A: $tType] :
( ( division_ring @ A )
=> ! [Z: A,A3: A,B3: A] :
( ( ( Z
= ( zero_zero @ A ) )
=> ( ( minus_minus @ A @ ( divide_divide @ A @ A3 @ Z ) @ B3 )
= ( uminus_uminus @ A @ B3 ) ) )
& ( ( Z
!= ( zero_zero @ A ) )
=> ( ( minus_minus @ A @ ( divide_divide @ A @ A3 @ Z ) @ B3 )
= ( divide_divide @ A @ ( minus_minus @ A @ A3 @ ( times_times @ A @ B3 @ Z ) ) @ Z ) ) ) ) ) ).
% add_divide_eq_if_simps(5)
thf(fact_3937_add__divide__eq__if__simps_I6_J,axiom,
! [A: $tType] :
( ( division_ring @ A )
=> ! [Z: A,A3: A,B3: A] :
( ( ( Z
= ( zero_zero @ A ) )
=> ( ( minus_minus @ A @ ( uminus_uminus @ A @ ( divide_divide @ A @ A3 @ Z ) ) @ B3 )
= ( uminus_uminus @ A @ B3 ) ) )
& ( ( Z
!= ( zero_zero @ A ) )
=> ( ( minus_minus @ A @ ( uminus_uminus @ A @ ( divide_divide @ A @ A3 @ Z ) ) @ B3 )
= ( divide_divide @ A @ ( minus_minus @ A @ ( uminus_uminus @ A @ A3 ) @ ( times_times @ A @ B3 @ Z ) ) @ Z ) ) ) ) ) ).
% add_divide_eq_if_simps(6)
thf(fact_3938_even__minus,axiom,
! [A: $tType] :
( ( ring_parity @ A )
=> ! [A3: A] :
( ( dvd_dvd @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ ( uminus_uminus @ A @ A3 ) )
= ( dvd_dvd @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ A3 ) ) ) ).
% even_minus
thf(fact_3939_power2__eq__iff,axiom,
! [A: $tType] :
( ( idom @ A )
=> ! [X2: A,Y3: A] :
( ( ( power_power @ A @ X2 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) )
= ( power_power @ A @ Y3 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) )
= ( ( X2 = Y3 )
| ( X2
= ( uminus_uminus @ A @ Y3 ) ) ) ) ) ).
% power2_eq_iff
thf(fact_3940_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_3941_fact__prod__Suc,axiom,
! [A: $tType] :
( ( semiring_char_0 @ A )
=> ( ( semiring_char_0_fact @ A )
= ( ^ [N4: nat] : ( semiring_1_of_nat @ A @ ( groups7121269368397514597t_prod @ nat @ nat @ suc @ ( set_or7035219750837199246ssThan @ nat @ ( zero_zero @ nat ) @ N4 ) ) ) ) ) ) ).
% fact_prod_Suc
thf(fact_3942_int__cases3,axiom,
! [K: int] :
( ( K
!= ( zero_zero @ int ) )
=> ( ! [N3: nat] :
( ( K
= ( semiring_1_of_nat @ int @ N3 ) )
=> ~ ( ord_less @ nat @ ( zero_zero @ nat ) @ N3 ) )
=> ~ ! [N3: nat] :
( ( K
= ( uminus_uminus @ int @ ( semiring_1_of_nat @ int @ N3 ) ) )
=> ~ ( ord_less @ nat @ ( zero_zero @ nat ) @ N3 ) ) ) ) ).
% int_cases3
thf(fact_3943_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
@ ^ [X3: nat] : X3
@ ( set_or1337092689740270186AtMost @ nat @ ( suc @ N ) @ M ) ) ) ) ) ).
% fact_eq_fact_times
thf(fact_3944_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_3945_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_3946_pochhammer__eq__0__iff,axiom,
! [A: $tType] :
( ( field_char_0 @ A )
=> ! [A3: A,N: nat] :
( ( ( comm_s3205402744901411588hammer @ A @ A3 @ N )
= ( zero_zero @ A ) )
= ( ? [K3: nat] :
( ( ord_less @ nat @ K3 @ N )
& ( A3
= ( uminus_uminus @ A @ ( semiring_1_of_nat @ A @ K3 ) ) ) ) ) ) ) ).
% pochhammer_eq_0_iff
thf(fact_3947_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_3948_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_3949_negD,axiom,
! [X2: int] :
( ( ord_less @ int @ X2 @ ( zero_zero @ int ) )
=> ? [N3: nat] :
( X2
= ( uminus_uminus @ int @ ( semiring_1_of_nat @ int @ ( suc @ N3 ) ) ) ) ) ).
% negD
thf(fact_3950_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_3951_verit__less__mono__div__int2,axiom,
! [A2: int,B2: int,N: int] :
( ( ord_less_eq @ int @ A2 @ B2 )
=> ( ( ord_less @ int @ ( zero_zero @ int ) @ ( uminus_uminus @ int @ N ) )
=> ( ord_less_eq @ int @ ( divide_divide @ int @ B2 @ N ) @ ( divide_divide @ int @ A2 @ N ) ) ) ) ).
% verit_less_mono_div_int2
thf(fact_3952_div__eq__minus1,axiom,
! [B3: int] :
( ( ord_less @ int @ ( zero_zero @ int ) @ B3 )
=> ( ( divide_divide @ int @ ( uminus_uminus @ int @ ( one_one @ int ) ) @ B3 )
= ( uminus_uminus @ int @ ( one_one @ int ) ) ) ) ).
% div_eq_minus1
thf(fact_3953_ceiling__divide__eq__div,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [A3: int,B3: int] :
( ( archimedean_ceiling @ A @ ( divide_divide @ A @ ( ring_1_of_int @ A @ A3 ) @ ( ring_1_of_int @ A @ B3 ) ) )
= ( uminus_uminus @ int @ ( divide_divide @ int @ ( uminus_uminus @ int @ A3 ) @ B3 ) ) ) ) ).
% ceiling_divide_eq_div
thf(fact_3954_fact__prod__rev,axiom,
! [A: $tType] :
( ( semiring_char_0 @ A )
=> ( ( semiring_char_0_fact @ A )
= ( ^ [N4: nat] : ( semiring_1_of_nat @ A @ ( groups7121269368397514597t_prod @ nat @ nat @ ( minus_minus @ nat @ N4 ) @ ( set_or7035219750837199246ssThan @ nat @ ( zero_zero @ nat ) @ N4 ) ) ) ) ) ) ).
% fact_prod_rev
thf(fact_3955_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_3956_le__minus__divide__eq,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [A3: A,B3: A,C3: A] :
( ( ord_less_eq @ A @ A3 @ ( uminus_uminus @ A @ ( divide_divide @ A @ B3 @ C3 ) ) )
= ( ( ( ord_less @ A @ ( zero_zero @ A ) @ C3 )
=> ( ord_less_eq @ A @ ( times_times @ A @ A3 @ C3 ) @ ( uminus_uminus @ A @ B3 ) ) )
& ( ~ ( ord_less @ A @ ( zero_zero @ A ) @ C3 )
=> ( ( ( ord_less @ A @ C3 @ ( zero_zero @ A ) )
=> ( ord_less_eq @ A @ ( uminus_uminus @ A @ B3 ) @ ( times_times @ A @ A3 @ C3 ) ) )
& ( ~ ( ord_less @ A @ C3 @ ( zero_zero @ A ) )
=> ( ord_less_eq @ A @ A3 @ ( zero_zero @ A ) ) ) ) ) ) ) ) ).
% le_minus_divide_eq
thf(fact_3957_minus__divide__le__eq,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [B3: A,C3: A,A3: A] :
( ( ord_less_eq @ A @ ( uminus_uminus @ A @ ( divide_divide @ A @ B3 @ C3 ) ) @ A3 )
= ( ( ( ord_less @ A @ ( zero_zero @ A ) @ C3 )
=> ( ord_less_eq @ A @ ( uminus_uminus @ A @ B3 ) @ ( times_times @ A @ A3 @ C3 ) ) )
& ( ~ ( ord_less @ A @ ( zero_zero @ A ) @ C3 )
=> ( ( ( ord_less @ A @ C3 @ ( zero_zero @ A ) )
=> ( ord_less_eq @ A @ ( times_times @ A @ A3 @ C3 ) @ ( uminus_uminus @ A @ B3 ) ) )
& ( ~ ( ord_less @ A @ C3 @ ( zero_zero @ A ) )
=> ( ord_less_eq @ A @ ( zero_zero @ A ) @ A3 ) ) ) ) ) ) ) ).
% minus_divide_le_eq
thf(fact_3958_neg__le__minus__divide__eq,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [C3: A,A3: A,B3: A] :
( ( ord_less @ A @ C3 @ ( zero_zero @ A ) )
=> ( ( ord_less_eq @ A @ A3 @ ( uminus_uminus @ A @ ( divide_divide @ A @ B3 @ C3 ) ) )
= ( ord_less_eq @ A @ ( uminus_uminus @ A @ B3 ) @ ( times_times @ A @ A3 @ C3 ) ) ) ) ) ).
% neg_le_minus_divide_eq
thf(fact_3959_neg__minus__divide__le__eq,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [C3: A,B3: A,A3: A] :
( ( ord_less @ A @ C3 @ ( zero_zero @ A ) )
=> ( ( ord_less_eq @ A @ ( uminus_uminus @ A @ ( divide_divide @ A @ B3 @ C3 ) ) @ A3 )
= ( ord_less_eq @ A @ ( times_times @ A @ A3 @ C3 ) @ ( uminus_uminus @ A @ B3 ) ) ) ) ) ).
% neg_minus_divide_le_eq
thf(fact_3960_pos__le__minus__divide__eq,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [C3: A,A3: A,B3: A] :
( ( ord_less @ A @ ( zero_zero @ A ) @ C3 )
=> ( ( ord_less_eq @ A @ A3 @ ( uminus_uminus @ A @ ( divide_divide @ A @ B3 @ C3 ) ) )
= ( ord_less_eq @ A @ ( times_times @ A @ A3 @ C3 ) @ ( uminus_uminus @ A @ B3 ) ) ) ) ) ).
% pos_le_minus_divide_eq
thf(fact_3961_pos__minus__divide__le__eq,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [C3: A,B3: A,A3: A] :
( ( ord_less @ A @ ( zero_zero @ A ) @ C3 )
=> ( ( ord_less_eq @ A @ ( uminus_uminus @ A @ ( divide_divide @ A @ B3 @ C3 ) ) @ A3 )
= ( ord_less_eq @ A @ ( uminus_uminus @ A @ B3 ) @ ( times_times @ A @ A3 @ C3 ) ) ) ) ) ).
% pos_minus_divide_le_eq
thf(fact_3962_divide__less__eq__numeral_I2_J,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [B3: A,C3: A,W: num] :
( ( ord_less @ A @ ( divide_divide @ A @ B3 @ C3 ) @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ W ) ) )
= ( ( ( ord_less @ A @ ( zero_zero @ A ) @ C3 )
=> ( ord_less @ A @ B3 @ ( times_times @ A @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ W ) ) @ C3 ) ) )
& ( ~ ( ord_less @ A @ ( zero_zero @ A ) @ C3 )
=> ( ( ( ord_less @ A @ C3 @ ( zero_zero @ A ) )
=> ( ord_less @ A @ ( times_times @ A @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ W ) ) @ C3 ) @ B3 ) )
& ( ~ ( ord_less @ A @ C3 @ ( zero_zero @ A ) )
=> ( ord_less @ A @ ( zero_zero @ A ) @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ W ) ) ) ) ) ) ) ) ) ).
% divide_less_eq_numeral(2)
thf(fact_3963_less__divide__eq__numeral_I2_J,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [W: num,B3: A,C3: A] :
( ( ord_less @ A @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ W ) ) @ ( divide_divide @ A @ B3 @ C3 ) )
= ( ( ( ord_less @ A @ ( zero_zero @ A ) @ C3 )
=> ( ord_less @ A @ ( times_times @ A @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ W ) ) @ C3 ) @ B3 ) )
& ( ~ ( ord_less @ A @ ( zero_zero @ A ) @ C3 )
=> ( ( ( ord_less @ A @ C3 @ ( zero_zero @ A ) )
=> ( ord_less @ A @ B3 @ ( times_times @ A @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ W ) ) @ C3 ) ) )
& ( ~ ( ord_less @ A @ C3 @ ( zero_zero @ A ) )
=> ( ord_less @ A @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ W ) ) @ ( zero_zero @ A ) ) ) ) ) ) ) ) ).
% less_divide_eq_numeral(2)
thf(fact_3964_power2__eq__1__iff,axiom,
! [A: $tType] :
( ( ring_15535105094025558882visors @ A )
=> ! [A3: A] :
( ( ( power_power @ A @ A3 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) )
= ( one_one @ A ) )
= ( ( A3
= ( one_one @ A ) )
| ( A3
= ( uminus_uminus @ A @ ( one_one @ A ) ) ) ) ) ) ).
% power2_eq_1_iff
thf(fact_3965_uminus__power__if,axiom,
! [A: $tType] :
( ( ring_1 @ A )
=> ! [N: nat,A3: A] :
( ( ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N )
=> ( ( power_power @ A @ ( uminus_uminus @ A @ A3 ) @ N )
= ( power_power @ A @ A3 @ N ) ) )
& ( ~ ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N )
=> ( ( power_power @ A @ ( uminus_uminus @ A @ A3 ) @ N )
= ( uminus_uminus @ A @ ( power_power @ A @ A3 @ N ) ) ) ) ) ) ).
% uminus_power_if
thf(fact_3966_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_3967_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
@ ^ [X3: nat] : X3
@ ( set_or1337092689740270186AtMost @ nat @ ( plus_plus @ nat @ N @ ( one_one @ nat ) ) @ M ) ) ) ) ).
% fact_div_fact
thf(fact_3968_neg__int__cases,axiom,
! [K: int] :
( ( ord_less @ int @ K @ ( zero_zero @ int ) )
=> ~ ! [N3: nat] :
( ( K
= ( uminus_uminus @ int @ ( semiring_1_of_nat @ int @ N3 ) ) )
=> ~ ( ord_less @ nat @ ( zero_zero @ nat ) @ N3 ) ) ) ).
% neg_int_cases
thf(fact_3969_nat__mult__distrib__neg,axiom,
! [Z: int,Z7: int] :
( ( ord_less_eq @ int @ Z @ ( zero_zero @ int ) )
=> ( ( nat2 @ ( times_times @ int @ Z @ Z7 ) )
= ( times_times @ nat @ ( nat2 @ ( uminus_uminus @ int @ Z ) ) @ ( nat2 @ ( uminus_uminus @ int @ Z7 ) ) ) ) ) ).
% nat_mult_distrib_neg
thf(fact_3970_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_3971_zmod__minus1,axiom,
! [B3: int] :
( ( ord_less @ int @ ( zero_zero @ int ) @ B3 )
=> ( ( modulo_modulo @ int @ ( uminus_uminus @ int @ ( one_one @ int ) ) @ B3 )
= ( minus_minus @ int @ B3 @ ( one_one @ int ) ) ) ) ).
% zmod_minus1
thf(fact_3972_zdiv__zminus1__eq__if,axiom,
! [B3: int,A3: int] :
( ( B3
!= ( zero_zero @ int ) )
=> ( ( ( ( modulo_modulo @ int @ A3 @ B3 )
= ( zero_zero @ int ) )
=> ( ( divide_divide @ int @ ( uminus_uminus @ int @ A3 ) @ B3 )
= ( uminus_uminus @ int @ ( divide_divide @ int @ A3 @ B3 ) ) ) )
& ( ( ( modulo_modulo @ int @ A3 @ B3 )
!= ( zero_zero @ int ) )
=> ( ( divide_divide @ int @ ( uminus_uminus @ int @ A3 ) @ B3 )
= ( minus_minus @ int @ ( uminus_uminus @ int @ ( divide_divide @ int @ A3 @ B3 ) ) @ ( one_one @ int ) ) ) ) ) ) ).
% zdiv_zminus1_eq_if
thf(fact_3973_zdiv__zminus2__eq__if,axiom,
! [B3: int,A3: int] :
( ( B3
!= ( zero_zero @ int ) )
=> ( ( ( ( modulo_modulo @ int @ A3 @ B3 )
= ( zero_zero @ int ) )
=> ( ( divide_divide @ int @ A3 @ ( uminus_uminus @ int @ B3 ) )
= ( uminus_uminus @ int @ ( divide_divide @ int @ A3 @ B3 ) ) ) )
& ( ( ( modulo_modulo @ int @ A3 @ B3 )
!= ( zero_zero @ int ) )
=> ( ( divide_divide @ int @ A3 @ ( uminus_uminus @ int @ B3 ) )
= ( minus_minus @ int @ ( uminus_uminus @ int @ ( divide_divide @ int @ A3 @ B3 ) ) @ ( one_one @ int ) ) ) ) ) ) ).
% zdiv_zminus2_eq_if
thf(fact_3974_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_3975_zminus1__lemma,axiom,
! [A3: int,B3: int,Q3: int,R2: int] :
( ( eucl_rel_int @ A3 @ B3 @ ( product_Pair @ int @ int @ Q3 @ R2 ) )
=> ( ( B3
!= ( zero_zero @ int ) )
=> ( eucl_rel_int @ ( uminus_uminus @ int @ A3 ) @ B3
@ ( 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 @ B3 @ R2 ) ) ) ) ) ) ).
% zminus1_lemma
thf(fact_3976_divide__le__eq__numeral_I2_J,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [B3: A,C3: A,W: num] :
( ( ord_less_eq @ A @ ( divide_divide @ A @ B3 @ C3 ) @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ W ) ) )
= ( ( ( ord_less @ A @ ( zero_zero @ A ) @ C3 )
=> ( ord_less_eq @ A @ B3 @ ( times_times @ A @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ W ) ) @ C3 ) ) )
& ( ~ ( ord_less @ A @ ( zero_zero @ A ) @ C3 )
=> ( ( ( ord_less @ A @ C3 @ ( zero_zero @ A ) )
=> ( ord_less_eq @ A @ ( times_times @ A @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ W ) ) @ C3 ) @ B3 ) )
& ( ~ ( ord_less @ A @ C3 @ ( zero_zero @ A ) )
=> ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ W ) ) ) ) ) ) ) ) ) ).
% divide_le_eq_numeral(2)
thf(fact_3977_le__divide__eq__numeral_I2_J,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [W: num,B3: A,C3: A] :
( ( ord_less_eq @ A @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ W ) ) @ ( divide_divide @ A @ B3 @ C3 ) )
= ( ( ( ord_less @ A @ ( zero_zero @ A ) @ C3 )
=> ( ord_less_eq @ A @ ( times_times @ A @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ W ) ) @ C3 ) @ B3 ) )
& ( ~ ( ord_less @ A @ ( zero_zero @ A ) @ C3 )
=> ( ( ( ord_less @ A @ C3 @ ( zero_zero @ A ) )
=> ( ord_less_eq @ A @ B3 @ ( times_times @ A @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ W ) ) @ C3 ) ) )
& ( ~ ( ord_less @ A @ C3 @ ( zero_zero @ A ) )
=> ( ord_less_eq @ A @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ W ) ) @ ( zero_zero @ A ) ) ) ) ) ) ) ) ).
% le_divide_eq_numeral(2)
thf(fact_3978_square__le__1,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [X2: A] :
( ( ord_less_eq @ A @ ( uminus_uminus @ A @ ( one_one @ A ) ) @ X2 )
=> ( ( ord_less_eq @ A @ X2 @ ( one_one @ A ) )
=> ( ord_less_eq @ A @ ( power_power @ A @ X2 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ ( one_one @ A ) ) ) ) ) ).
% square_le_1
thf(fact_3979_minus__power__mult__self,axiom,
! [A: $tType] :
( ( comm_ring_1 @ A )
=> ! [A3: A,N: nat] :
( ( times_times @ A @ ( power_power @ A @ ( uminus_uminus @ A @ A3 ) @ N ) @ ( power_power @ A @ ( uminus_uminus @ A @ A3 ) @ N ) )
= ( power_power @ A @ A3 @ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N ) ) ) ) ).
% minus_power_mult_self
thf(fact_3980_fact__code,axiom,
! [A: $tType] :
( ( semiring_char_0 @ A )
=> ( ( semiring_char_0_fact @ A )
= ( ^ [N4: nat] : ( semiring_1_of_nat @ A @ ( set_fo6178422350223883121st_nat @ nat @ ( times_times @ nat ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N4 @ ( one_one @ nat ) ) ) ) ) ) ).
% fact_code
thf(fact_3981_fact__num__eq__if,axiom,
! [A: $tType] :
( ( semiring_char_0 @ A )
=> ( ( semiring_char_0_fact @ A )
= ( ^ [M6: nat] :
( if @ A
@ ( M6
= ( zero_zero @ nat ) )
@ ( one_one @ A )
@ ( times_times @ A @ ( semiring_1_of_nat @ A @ M6 ) @ ( semiring_char_0_fact @ A @ ( minus_minus @ nat @ M6 @ ( one_one @ nat ) ) ) ) ) ) ) ) ).
% fact_num_eq_if
thf(fact_3982_minus__one__power__iff,axiom,
! [A: $tType] :
( ( ring_1 @ A )
=> ! [N: nat] :
( ( ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N )
=> ( ( power_power @ A @ ( uminus_uminus @ A @ ( one_one @ A ) ) @ N )
= ( one_one @ A ) ) )
& ( ~ ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N )
=> ( ( power_power @ A @ ( uminus_uminus @ A @ ( one_one @ A ) ) @ N )
= ( uminus_uminus @ A @ ( one_one @ A ) ) ) ) ) ) ).
% minus_one_power_iff
thf(fact_3983_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_3984_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_3985_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_3986_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_3987_pochhammer__absorb__comp,axiom,
! [A: $tType] :
( ( comm_ring_1 @ A )
=> ! [R2: A,K: nat] :
( ( times_times @ A @ ( minus_minus @ A @ R2 @ ( semiring_1_of_nat @ A @ K ) ) @ ( comm_s3205402744901411588hammer @ A @ ( uminus_uminus @ A @ R2 ) @ K ) )
= ( times_times @ A @ R2 @ ( comm_s3205402744901411588hammer @ A @ ( plus_plus @ A @ ( uminus_uminus @ A @ R2 ) @ ( one_one @ A ) ) @ K ) ) ) ) ).
% pochhammer_absorb_comp
thf(fact_3988_gbinomial__index__swap,axiom,
! [A: $tType] :
( ( field_char_0 @ A )
=> ! [K: nat,N: nat] :
( ( times_times @ A @ ( power_power @ A @ ( uminus_uminus @ A @ ( one_one @ A ) ) @ K ) @ ( gbinomial @ A @ ( minus_minus @ A @ ( uminus_uminus @ A @ ( semiring_1_of_nat @ A @ N ) ) @ ( one_one @ A ) ) @ K ) )
= ( times_times @ A @ ( power_power @ A @ ( uminus_uminus @ A @ ( one_one @ A ) ) @ N ) @ ( gbinomial @ A @ ( minus_minus @ A @ ( uminus_uminus @ A @ ( semiring_1_of_nat @ A @ K ) ) @ ( one_one @ A ) ) @ N ) ) ) ) ).
% gbinomial_index_swap
thf(fact_3989_gbinomial__negated__upper,axiom,
! [A: $tType] :
( ( field_char_0 @ A )
=> ( ( gbinomial @ A )
= ( ^ [A6: A,K3: nat] : ( times_times @ A @ ( power_power @ A @ ( uminus_uminus @ A @ ( one_one @ A ) ) @ K3 ) @ ( gbinomial @ A @ ( minus_minus @ A @ ( minus_minus @ A @ ( semiring_1_of_nat @ A @ K3 ) @ A6 ) @ ( one_one @ A ) ) @ K3 ) ) ) ) ) ).
% gbinomial_negated_upper
thf(fact_3990_minus__1__div__exp__eq__int,axiom,
! [N: nat] :
( ( divide_divide @ int @ ( uminus_uminus @ int @ ( one_one @ int ) ) @ ( power_power @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ N ) )
= ( uminus_uminus @ int @ ( one_one @ int ) ) ) ).
% minus_1_div_exp_eq_int
thf(fact_3991_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_3992_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_3993_gbinomial__minus,axiom,
! [A: $tType] :
( ( field_char_0 @ A )
=> ! [A3: A,K: nat] :
( ( gbinomial @ A @ ( uminus_uminus @ A @ A3 ) @ K )
= ( times_times @ A @ ( power_power @ A @ ( uminus_uminus @ A @ ( one_one @ A ) ) @ K ) @ ( gbinomial @ A @ ( minus_minus @ A @ ( plus_plus @ A @ A3 @ ( semiring_1_of_nat @ A @ K ) ) @ ( one_one @ A ) ) @ K ) ) ) ) ).
% gbinomial_minus
thf(fact_3994_pochhammer__minus_H,axiom,
! [A: $tType] :
( ( comm_ring_1 @ A )
=> ! [B3: A,K: nat] :
( ( comm_s3205402744901411588hammer @ A @ ( plus_plus @ A @ ( minus_minus @ A @ B3 @ ( semiring_1_of_nat @ A @ K ) ) @ ( one_one @ A ) ) @ K )
= ( times_times @ A @ ( power_power @ A @ ( uminus_uminus @ A @ ( one_one @ A ) ) @ K ) @ ( comm_s3205402744901411588hammer @ A @ ( uminus_uminus @ A @ B3 ) @ K ) ) ) ) ).
% pochhammer_minus'
thf(fact_3995_pochhammer__minus,axiom,
! [A: $tType] :
( ( comm_ring_1 @ A )
=> ! [B3: A,K: nat] :
( ( comm_s3205402744901411588hammer @ A @ ( uminus_uminus @ A @ B3 ) @ K )
= ( times_times @ A @ ( power_power @ A @ ( uminus_uminus @ A @ ( one_one @ A ) ) @ K ) @ ( comm_s3205402744901411588hammer @ A @ ( plus_plus @ A @ ( minus_minus @ A @ B3 @ ( semiring_1_of_nat @ A @ K ) ) @ ( one_one @ A ) ) @ K ) ) ) ) ).
% pochhammer_minus
thf(fact_3996_gbinomial__pochhammer_H,axiom,
! [A: $tType] :
( ( field_char_0 @ A )
=> ( ( gbinomial @ A )
= ( ^ [A6: A,K3: nat] : ( divide_divide @ A @ ( comm_s3205402744901411588hammer @ A @ ( plus_plus @ A @ ( minus_minus @ A @ A6 @ ( semiring_1_of_nat @ A @ K3 ) ) @ ( one_one @ A ) ) @ K3 ) @ ( semiring_char_0_fact @ A @ K3 ) ) ) ) ) ).
% gbinomial_pochhammer'
thf(fact_3997_int__bit__induct,axiom,
! [P: int > $o,K: int] :
( ( P @ ( zero_zero @ int ) )
=> ( ( P @ ( uminus_uminus @ int @ ( one_one @ int ) ) )
=> ( ! [K2: int] :
( ( P @ K2 )
=> ( ( K2
!= ( zero_zero @ int ) )
=> ( P @ ( times_times @ int @ K2 @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) ) ) ) )
=> ( ! [K2: int] :
( ( P @ K2 )
=> ( ( K2
!= ( uminus_uminus @ int @ ( one_one @ int ) ) )
=> ( P @ ( plus_plus @ int @ ( one_one @ int ) @ ( times_times @ int @ K2 @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) ) ) ) ) )
=> ( P @ K ) ) ) ) ) ).
% int_bit_induct
thf(fact_3998_m1mod2k,axiom,
! [N: nat] :
( ( modulo_modulo @ int @ ( uminus_uminus @ int @ ( one_one @ int ) ) @ ( power_power @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ N ) )
= ( minus_minus @ int @ ( power_power @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ N ) @ ( one_one @ int ) ) ) ).
% m1mod2k
thf(fact_3999_gbinomial__sum__lower__neg,axiom,
! [A: $tType] :
( ( field_char_0 @ A )
=> ! [A3: A,M: nat] :
( ( groups7311177749621191930dd_sum @ nat @ A
@ ^ [K3: nat] : ( times_times @ A @ ( gbinomial @ A @ A3 @ K3 ) @ ( power_power @ A @ ( uminus_uminus @ A @ ( one_one @ A ) ) @ K3 ) )
@ ( set_ord_atMost @ nat @ M ) )
= ( times_times @ A @ ( power_power @ A @ ( uminus_uminus @ A @ ( one_one @ A ) ) @ M ) @ ( gbinomial @ A @ ( minus_minus @ A @ A3 @ ( one_one @ A ) ) @ M ) ) ) ) ).
% gbinomial_sum_lower_neg
thf(fact_4000_gbinomial__mult__fact,axiom,
! [A: $tType] :
( ( field_char_0 @ A )
=> ! [K: nat,A3: A] :
( ( times_times @ A @ ( semiring_char_0_fact @ A @ K ) @ ( gbinomial @ A @ A3 @ K ) )
= ( groups7121269368397514597t_prod @ nat @ A
@ ^ [I4: nat] : ( minus_minus @ A @ A3 @ ( semiring_1_of_nat @ A @ I4 ) )
@ ( set_or7035219750837199246ssThan @ nat @ ( zero_zero @ nat ) @ K ) ) ) ) ).
% gbinomial_mult_fact
thf(fact_4001_gbinomial__mult__fact_H,axiom,
! [A: $tType] :
( ( field_char_0 @ A )
=> ! [A3: A,K: nat] :
( ( times_times @ A @ ( gbinomial @ A @ A3 @ K ) @ ( semiring_char_0_fact @ A @ K ) )
= ( groups7121269368397514597t_prod @ nat @ A
@ ^ [I4: nat] : ( minus_minus @ A @ A3 @ ( semiring_1_of_nat @ A @ I4 ) )
@ ( set_or7035219750837199246ssThan @ nat @ ( zero_zero @ nat ) @ K ) ) ) ) ).
% gbinomial_mult_fact'
thf(fact_4002_gbinomial__prod__rev,axiom,
! [A: $tType] :
( ( ( semiring_char_0 @ A )
& ( semidom_divide @ A ) )
=> ( ( gbinomial @ A )
= ( ^ [A6: A,K3: nat] :
( divide_divide @ A
@ ( groups7121269368397514597t_prod @ nat @ A
@ ^ [I4: nat] : ( minus_minus @ A @ A6 @ ( semiring_1_of_nat @ A @ I4 ) )
@ ( set_or7035219750837199246ssThan @ nat @ ( zero_zero @ nat ) @ K3 ) )
@ ( semiring_char_0_fact @ A @ K3 ) ) ) ) ) ).
% gbinomial_prod_rev
thf(fact_4003_Maclaurin__zero,axiom,
! [A: $tType] :
( ( zero @ A )
=> ! [X2: real,N: nat,Diff: nat > A > real] :
( ( X2
= ( zero_zero @ real ) )
=> ( ( N
!= ( zero_zero @ nat ) )
=> ( ( groups7311177749621191930dd_sum @ nat @ real
@ ^ [M6: nat] : ( times_times @ real @ ( divide_divide @ real @ ( Diff @ M6 @ ( zero_zero @ A ) ) @ ( semiring_char_0_fact @ real @ M6 ) ) @ ( power_power @ real @ X2 @ M6 ) )
@ ( set_ord_lessThan @ nat @ N ) )
= ( Diff @ ( zero_zero @ nat ) @ ( zero_zero @ A ) ) ) ) ) ) ).
% Maclaurin_zero
thf(fact_4004_Maclaurin__lemma,axiom,
! [H2: real,F2: real > real,J: nat > real,N: nat] :
( ( ord_less @ real @ ( zero_zero @ real ) @ H2 )
=> ? [B10: real] :
( ( F2 @ H2 )
= ( plus_plus @ real
@ ( groups7311177749621191930dd_sum @ nat @ real
@ ^ [M6: nat] : ( times_times @ real @ ( divide_divide @ real @ ( J @ M6 ) @ ( semiring_char_0_fact @ real @ M6 ) ) @ ( power_power @ real @ H2 @ M6 ) )
@ ( set_ord_lessThan @ nat @ N ) )
@ ( times_times @ real @ B10 @ ( divide_divide @ real @ ( power_power @ real @ H2 @ N ) @ ( semiring_char_0_fact @ real @ N ) ) ) ) ) ) ).
% Maclaurin_lemma
thf(fact_4005_sb__dec__lem_H,axiom,
! [K: nat,A3: int] :
( ( ord_less_eq @ int @ ( power_power @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ K ) @ A3 )
=> ( ord_less_eq @ int @ ( modulo_modulo @ int @ ( plus_plus @ int @ A3 @ ( 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 ) ) @ A3 ) ) ) ).
% sb_dec_lem'
thf(fact_4006_m1mod22k,axiom,
! [N: nat] :
( ( modulo_modulo @ int @ ( uminus_uminus @ int @ ( one_one @ int ) ) @ ( times_times @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ ( power_power @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ N ) ) )
= ( minus_minus @ int @ ( times_times @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ ( power_power @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ N ) ) @ ( one_one @ int ) ) ) ).
% m1mod22k
thf(fact_4007_gbinomial__Suc,axiom,
! [A: $tType] :
( ( ( semiring_char_0 @ A )
& ( semidom_divide @ A ) )
=> ! [A3: A,K: nat] :
( ( gbinomial @ A @ A3 @ ( suc @ K ) )
= ( divide_divide @ A
@ ( groups7121269368397514597t_prod @ nat @ A
@ ^ [I4: nat] : ( minus_minus @ A @ A3 @ ( semiring_1_of_nat @ A @ I4 ) )
@ ( set_or1337092689740270186AtMost @ nat @ ( zero_zero @ nat ) @ K ) )
@ ( semiring_char_0_fact @ A @ ( suc @ K ) ) ) ) ) ).
% gbinomial_Suc
thf(fact_4008_gbinomial__partial__sum__poly,axiom,
! [A: $tType] :
( ( field_char_0 @ A )
=> ! [M: nat,A3: A,X2: A,Y3: A] :
( ( groups7311177749621191930dd_sum @ nat @ A
@ ^ [K3: nat] : ( times_times @ A @ ( times_times @ A @ ( gbinomial @ A @ ( plus_plus @ A @ ( semiring_1_of_nat @ A @ M ) @ A3 ) @ K3 ) @ ( power_power @ A @ X2 @ K3 ) ) @ ( power_power @ A @ Y3 @ ( minus_minus @ nat @ M @ K3 ) ) )
@ ( set_ord_atMost @ nat @ M ) )
= ( groups7311177749621191930dd_sum @ nat @ A
@ ^ [K3: nat] : ( times_times @ A @ ( times_times @ A @ ( gbinomial @ A @ ( uminus_uminus @ A @ A3 ) @ K3 ) @ ( power_power @ A @ ( uminus_uminus @ A @ X2 ) @ K3 ) ) @ ( power_power @ A @ ( plus_plus @ A @ X2 @ Y3 ) @ ( minus_minus @ nat @ M @ K3 ) ) )
@ ( set_ord_atMost @ nat @ M ) ) ) ) ).
% gbinomial_partial_sum_poly
thf(fact_4009_sb__inc__lem_H,axiom,
! [A3: int,K: nat] :
( ( ord_less @ int @ A3 @ ( uminus_uminus @ int @ ( power_power @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ K ) ) )
=> ( ord_less_eq @ int @ ( plus_plus @ int @ ( plus_plus @ int @ A3 @ ( 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 @ A3 @ ( power_power @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ K ) ) @ ( power_power @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ ( suc @ K ) ) ) ) ) ).
% sb_inc_lem'
thf(fact_4010_sb__dec__lem,axiom,
! [K: nat,A3: int] :
( ( ord_less_eq @ int @ ( zero_zero @ int ) @ ( plus_plus @ int @ ( uminus_uminus @ int @ ( power_power @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ K ) ) @ A3 ) )
=> ( ord_less_eq @ int @ ( modulo_modulo @ int @ ( plus_plus @ int @ A3 @ ( 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 ) ) @ A3 ) ) ) ).
% sb_dec_lem
thf(fact_4011_root__polyfun,axiom,
! [A: $tType] :
( ( idom @ A )
=> ! [N: nat,Z: A,A3: A] :
( ( ord_less_eq @ nat @ ( one_one @ nat ) @ N )
=> ( ( ( power_power @ A @ Z @ N )
= A3 )
= ( ( groups7311177749621191930dd_sum @ nat @ A
@ ^ [I4: nat] :
( times_times @ A
@ ( if @ A
@ ( I4
= ( zero_zero @ nat ) )
@ ( uminus_uminus @ A @ A3 )
@ ( if @ A @ ( I4 = N ) @ ( one_one @ A ) @ ( zero_zero @ A ) ) )
@ ( power_power @ A @ Z @ I4 ) )
@ ( set_ord_atMost @ nat @ N ) )
= ( zero_zero @ A ) ) ) ) ) ).
% root_polyfun
thf(fact_4012_choose__alternating__linear__sum,axiom,
! [A: $tType] :
( ( comm_ring_1 @ A )
=> ! [N: nat] :
( ( N
!= ( one_one @ nat ) )
=> ( ( groups7311177749621191930dd_sum @ nat @ A
@ ^ [I4: nat] : ( times_times @ A @ ( times_times @ A @ ( power_power @ A @ ( uminus_uminus @ A @ ( one_one @ A ) ) @ I4 ) @ ( semiring_1_of_nat @ A @ I4 ) ) @ ( semiring_1_of_nat @ A @ ( binomial @ N @ I4 ) ) )
@ ( set_ord_atMost @ nat @ N ) )
= ( zero_zero @ A ) ) ) ) ).
% choose_alternating_linear_sum
thf(fact_4013_fact__double,axiom,
! [A: $tType] :
( ( field_char_0 @ A )
=> ! [N: nat] :
( ( semiring_char_0_fact @ A @ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N ) )
= ( times_times @ A @ ( times_times @ A @ ( power_power @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N ) ) @ ( comm_s3205402744901411588hammer @ A @ ( divide_divide @ A @ ( one_one @ A ) @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) @ N ) ) @ ( semiring_char_0_fact @ A @ N ) ) ) ) ).
% fact_double
thf(fact_4014_binomial__code,axiom,
( binomial
= ( ^ [N4: nat,K3: nat] : ( if @ nat @ ( ord_less @ nat @ N4 @ K3 ) @ ( zero_zero @ nat ) @ ( if @ nat @ ( ord_less @ nat @ N4 @ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ K3 ) ) @ ( binomial @ N4 @ ( minus_minus @ nat @ N4 @ K3 ) ) @ ( divide_divide @ nat @ ( set_fo6178422350223883121st_nat @ nat @ ( times_times @ nat ) @ ( plus_plus @ nat @ ( minus_minus @ nat @ N4 @ K3 ) @ ( one_one @ nat ) ) @ N4 @ ( one_one @ nat ) ) @ ( semiring_char_0_fact @ nat @ K3 ) ) ) ) ) ) ).
% binomial_code
thf(fact_4015_choose__alternating__sum,axiom,
! [A: $tType] :
( ( comm_ring_1 @ A )
=> ! [N: nat] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ N )
=> ( ( groups7311177749621191930dd_sum @ nat @ A
@ ^ [I4: nat] : ( times_times @ A @ ( power_power @ A @ ( uminus_uminus @ A @ ( one_one @ A ) ) @ I4 ) @ ( semiring_1_of_nat @ A @ ( binomial @ N @ I4 ) ) )
@ ( set_ord_atMost @ nat @ N ) )
= ( zero_zero @ A ) ) ) ) ).
% choose_alternating_sum
thf(fact_4016_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_4017_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_4018_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_4019_compl__less__compl__iff,axiom,
! [A: $tType] :
( ( boolea8198339166811842893lgebra @ A )
=> ! [X2: A,Y3: A] :
( ( ord_less @ A @ ( uminus_uminus @ A @ X2 ) @ ( uminus_uminus @ A @ Y3 ) )
= ( ord_less @ A @ Y3 @ X2 ) ) ) ).
% compl_less_compl_iff
thf(fact_4020_Compl__subset__Compl__iff,axiom,
! [A: $tType,A2: set @ A,B2: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ ( uminus_uminus @ ( set @ A ) @ A2 ) @ ( uminus_uminus @ ( set @ A ) @ B2 ) )
= ( ord_less_eq @ ( set @ A ) @ B2 @ A2 ) ) ).
% Compl_subset_Compl_iff
thf(fact_4021_Compl__anti__mono,axiom,
! [A: $tType,A2: set @ A,B2: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ A2 @ B2 )
=> ( ord_less_eq @ ( set @ A ) @ ( uminus_uminus @ ( set @ A ) @ B2 ) @ ( uminus_uminus @ ( set @ A ) @ A2 ) ) ) ).
% Compl_anti_mono
thf(fact_4022_compl__le__compl__iff,axiom,
! [A: $tType] :
( ( boolea8198339166811842893lgebra @ A )
=> ! [X2: A,Y3: A] :
( ( ord_less_eq @ A @ ( uminus_uminus @ A @ X2 ) @ ( uminus_uminus @ A @ Y3 ) )
= ( ord_less_eq @ A @ Y3 @ X2 ) ) ) ).
% compl_le_compl_iff
thf(fact_4023_real__add__minus__iff,axiom,
! [X2: real,A3: real] :
( ( ( plus_plus @ real @ X2 @ ( uminus_uminus @ real @ A3 ) )
= ( zero_zero @ real ) )
= ( X2 = A3 ) ) ).
% real_add_minus_iff
thf(fact_4024_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_4025_word__n1__ge,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [Y3: word @ A] : ( ord_less_eq @ ( word @ A ) @ Y3 @ ( uminus_uminus @ ( word @ A ) @ ( one_one @ ( word @ A ) ) ) ) ) ).
% word_n1_ge
thf(fact_4026_word__order_Oextremum__unique,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [A3: word @ A] :
( ( ord_less_eq @ ( word @ A ) @ ( uminus_uminus @ ( word @ A ) @ ( one_one @ ( word @ A ) ) ) @ A3 )
= ( A3
= ( uminus_uminus @ ( word @ A ) @ ( one_one @ ( word @ A ) ) ) ) ) ) ).
% word_order.extremum_unique
thf(fact_4027_word__order_Oextremum,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [A3: word @ A] : ( ord_less_eq @ ( word @ A ) @ A3 @ ( uminus_uminus @ ( word @ A ) @ ( one_one @ ( word @ A ) ) ) ) ) ).
% word_order.extremum
thf(fact_4028_word__minus__one__le,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [X2: word @ A] :
( ( ord_less_eq @ ( word @ A ) @ ( uminus_uminus @ ( word @ A ) @ ( one_one @ ( word @ A ) ) ) @ X2 )
= ( X2
= ( uminus_uminus @ ( word @ A ) @ ( one_one @ ( word @ A ) ) ) ) ) ) ).
% word_minus_one_le
thf(fact_4029_max__word__less__eq__iff,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [W: word @ A] :
( ( ord_less_eq @ ( word @ A ) @ ( uminus_uminus @ ( word @ A ) @ ( one_one @ ( word @ A ) ) ) @ W )
= ( W
= ( uminus_uminus @ ( word @ A ) @ ( one_one @ ( word @ A ) ) ) ) ) ) ).
% max_word_less_eq_iff
thf(fact_4030_subset__Compl__singleton,axiom,
! [A: $tType,A2: set @ A,B3: A] :
( ( ord_less_eq @ ( set @ A ) @ A2 @ ( uminus_uminus @ ( set @ A ) @ ( insert @ A @ B3 @ ( bot_bot @ ( set @ A ) ) ) ) )
= ( ~ ( member @ A @ B3 @ A2 ) ) ) ).
% subset_Compl_singleton
thf(fact_4031_word__of__int__neg__1,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ( ( ring_1_of_int @ ( word @ A ) @ ( uminus_uminus @ int @ ( one_one @ int ) ) )
= ( uminus_uminus @ ( word @ A ) @ ( one_one @ ( word @ A ) ) ) ) ) ).
% word_of_int_neg_1
thf(fact_4032_floor__minus__one__divide__eq__div__numeral,axiom,
! [B3: num] :
( ( archim6421214686448440834_floor @ real @ ( uminus_uminus @ real @ ( divide_divide @ real @ ( one_one @ real ) @ ( numeral_numeral @ real @ B3 ) ) ) )
= ( divide_divide @ int @ ( uminus_uminus @ int @ ( one_one @ int ) ) @ ( numeral_numeral @ int @ B3 ) ) ) ).
% floor_minus_one_divide_eq_div_numeral
thf(fact_4033_subset__Compl__self__eq,axiom,
! [A: $tType,A2: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ A2 @ ( uminus_uminus @ ( set @ A ) @ A2 ) )
= ( A2
= ( bot_bot @ ( set @ A ) ) ) ) ).
% subset_Compl_self_eq
thf(fact_4034_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_4035_real__minus__mult__self__le,axiom,
! [U: real,X2: real] : ( ord_less_eq @ real @ ( uminus_uminus @ real @ ( times_times @ real @ U @ U ) ) @ ( times_times @ real @ X2 @ X2 ) ) ).
% real_minus_mult_self_le
thf(fact_4036_word__order_Oextremum__uniqueI,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [A3: word @ A] :
( ( ord_less_eq @ ( word @ A ) @ ( uminus_uminus @ ( word @ A ) @ ( one_one @ ( word @ A ) ) ) @ A3 )
=> ( A3
= ( uminus_uminus @ ( word @ A ) @ ( one_one @ ( word @ A ) ) ) ) ) ) ).
% word_order.extremum_uniqueI
thf(fact_4037_max__word__not__0,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ( ( uminus_uminus @ ( word @ A ) @ ( one_one @ ( word @ A ) ) )
!= ( zero_zero @ ( word @ A ) ) ) ) ).
% max_word_not_0
thf(fact_4038_word__lessThan__Suc__atMost,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [K: word @ A] :
( ( K
!= ( uminus_uminus @ ( word @ A ) @ ( one_one @ ( word @ A ) ) ) )
=> ( ( set_ord_lessThan @ ( word @ A ) @ ( plus_plus @ ( word @ A ) @ K @ ( one_one @ ( word @ A ) ) ) )
= ( set_ord_atMost @ ( word @ A ) @ K ) ) ) ) ).
% word_lessThan_Suc_atMost
thf(fact_4039_word__order_Onot__eq__extremum,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [A3: word @ A] :
( ( A3
!= ( uminus_uminus @ ( word @ A ) @ ( one_one @ ( word @ A ) ) ) )
= ( ord_less @ ( word @ A ) @ A3 @ ( uminus_uminus @ ( word @ A ) @ ( one_one @ ( word @ A ) ) ) ) ) ) ).
% word_order.not_eq_extremum
thf(fact_4040_word__order_Oextremum__strict,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [A3: word @ A] :
~ ( ord_less @ ( word @ A ) @ ( uminus_uminus @ ( word @ A ) @ ( one_one @ ( word @ A ) ) ) @ A3 ) ) ).
% word_order.extremum_strict
thf(fact_4041_word__not__simps_I3_J,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [Y3: word @ A] :
~ ( ord_less @ ( word @ A ) @ ( uminus_uminus @ ( word @ A ) @ ( one_one @ ( word @ A ) ) ) @ Y3 ) ) ).
% word_not_simps(3)
thf(fact_4042_max__word__not__less,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [X2: word @ A] :
~ ( ord_less @ ( word @ A ) @ ( uminus_uminus @ ( word @ A ) @ ( one_one @ ( word @ A ) ) ) @ X2 ) ) ).
% max_word_not_less
thf(fact_4043_minus__real__def,axiom,
( ( minus_minus @ real )
= ( ^ [X3: real,Y: real] : ( plus_plus @ real @ X3 @ ( uminus_uminus @ real @ Y ) ) ) ) ).
% minus_real_def
thf(fact_4044_complex__mod__minus__le__complex__mod,axiom,
! [X2: complex] : ( ord_less_eq @ real @ ( uminus_uminus @ real @ ( real_V7770717601297561774m_norm @ complex @ X2 ) ) @ ( real_V7770717601297561774m_norm @ complex @ X2 ) ) ).
% complex_mod_minus_le_complex_mod
thf(fact_4045_Compl__insert,axiom,
! [A: $tType,X2: A,A2: set @ A] :
( ( uminus_uminus @ ( set @ A ) @ ( insert @ A @ X2 @ A2 ) )
= ( minus_minus @ ( set @ A ) @ ( uminus_uminus @ ( set @ A ) @ A2 ) @ ( insert @ A @ X2 @ ( bot_bot @ ( set @ A ) ) ) ) ) ).
% Compl_insert
thf(fact_4046_real__add__less__0__iff,axiom,
! [X2: real,Y3: real] :
( ( ord_less @ real @ ( plus_plus @ real @ X2 @ Y3 ) @ ( zero_zero @ real ) )
= ( ord_less @ real @ Y3 @ ( uminus_uminus @ real @ X2 ) ) ) ).
% real_add_less_0_iff
thf(fact_4047_real__0__less__add__iff,axiom,
! [X2: real,Y3: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ ( plus_plus @ real @ X2 @ Y3 ) )
= ( ord_less @ real @ ( uminus_uminus @ real @ X2 ) @ Y3 ) ) ).
% real_0_less_add_iff
thf(fact_4048_real__0__le__add__iff,axiom,
! [X2: real,Y3: real] :
( ( ord_less_eq @ real @ ( zero_zero @ real ) @ ( plus_plus @ real @ X2 @ Y3 ) )
= ( ord_less_eq @ real @ ( uminus_uminus @ real @ X2 ) @ Y3 ) ) ).
% real_0_le_add_iff
thf(fact_4049_real__add__le__0__iff,axiom,
! [X2: real,Y3: real] :
( ( ord_less_eq @ real @ ( plus_plus @ real @ X2 @ Y3 ) @ ( zero_zero @ real ) )
= ( ord_less_eq @ real @ Y3 @ ( uminus_uminus @ real @ X2 ) ) ) ).
% real_add_le_0_iff
thf(fact_4050_plus__1__less,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [X2: word @ A] :
( ( ord_less_eq @ ( word @ A ) @ ( plus_plus @ ( word @ A ) @ X2 @ ( one_one @ ( word @ A ) ) ) @ X2 )
= ( X2
= ( uminus_uminus @ ( word @ A ) @ ( one_one @ ( word @ A ) ) ) ) ) ) ).
% plus_1_less
thf(fact_4051_overflow__plus__one__self,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [P6: word @ A] :
( ( ord_less_eq @ ( word @ A ) @ ( plus_plus @ ( word @ A ) @ ( one_one @ ( word @ A ) ) @ P6 ) @ P6 )
= ( P6
= ( uminus_uminus @ ( word @ A ) @ ( one_one @ ( word @ A ) ) ) ) ) ) ).
% overflow_plus_one_self
thf(fact_4052_max__word__wrap,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [X2: word @ A] :
( ( ( plus_plus @ ( word @ A ) @ X2 @ ( one_one @ ( word @ A ) ) )
= ( zero_zero @ ( word @ A ) ) )
=> ( X2
= ( uminus_uminus @ ( word @ A ) @ ( one_one @ ( word @ A ) ) ) ) ) ) ).
% max_word_wrap
thf(fact_4053_word__add__no__overflow,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [X2: word @ A] :
( ( ord_less @ ( word @ A ) @ X2 @ ( uminus_uminus @ ( word @ A ) @ ( one_one @ ( word @ A ) ) ) )
=> ( ord_less @ ( word @ A ) @ X2 @ ( plus_plus @ ( word @ A ) @ X2 @ ( one_one @ ( word @ A ) ) ) ) ) ) ).
% word_add_no_overflow
thf(fact_4054_less__x__plus__1,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [X2: word @ A,Y3: word @ A] :
( ( X2
!= ( uminus_uminus @ ( word @ A ) @ ( one_one @ ( word @ A ) ) ) )
=> ( ( ord_less @ ( word @ A ) @ Y3 @ ( plus_plus @ ( word @ A ) @ X2 @ ( one_one @ ( word @ A ) ) ) )
= ( ( ord_less @ ( word @ A ) @ Y3 @ X2 )
| ( Y3 = X2 ) ) ) ) ) ).
% less_x_plus_1
thf(fact_4055_no__plus__overflow__neg,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [X2: word @ A,Y3: word @ A] :
( ( ord_less @ ( word @ A ) @ X2 @ ( uminus_uminus @ ( word @ A ) @ Y3 ) )
=> ( ord_less_eq @ ( word @ A ) @ X2 @ ( plus_plus @ ( word @ A ) @ X2 @ Y3 ) ) ) ) ).
% no_plus_overflow_neg
thf(fact_4056_word__le__make__less,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [Y3: word @ A,X2: word @ A] :
( ( Y3
!= ( uminus_uminus @ ( word @ A ) @ ( one_one @ ( word @ A ) ) ) )
=> ( ( ord_less_eq @ ( word @ A ) @ X2 @ Y3 )
= ( ord_less @ ( word @ A ) @ X2 @ ( plus_plus @ ( word @ A ) @ Y3 @ ( one_one @ ( word @ A ) ) ) ) ) ) ) ).
% word_le_make_less
thf(fact_4057_word__Suc__leq,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [K: word @ A,X2: word @ A] :
( ( K
!= ( uminus_uminus @ ( word @ A ) @ ( one_one @ ( word @ A ) ) ) )
=> ( ( ord_less @ ( word @ A ) @ X2 @ ( plus_plus @ ( word @ A ) @ K @ ( one_one @ ( word @ A ) ) ) )
= ( ord_less_eq @ ( word @ A ) @ X2 @ K ) ) ) ) ).
% word_Suc_leq
thf(fact_4058_word__Suc__le,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [X2: word @ A,K: word @ A] :
( ( X2
!= ( uminus_uminus @ ( word @ A ) @ ( one_one @ ( word @ A ) ) ) )
=> ( ( ord_less_eq @ ( word @ A ) @ ( plus_plus @ ( word @ A ) @ X2 @ ( one_one @ ( word @ A ) ) ) @ K )
= ( ord_less @ ( word @ A ) @ X2 @ K ) ) ) ) ).
% word_Suc_le
thf(fact_4059_realpow__square__minus__le,axiom,
! [U: real,X2: real] : ( ord_less_eq @ real @ ( uminus_uminus @ real @ ( power_power @ real @ U @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( power_power @ real @ X2 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ).
% realpow_square_minus_le
thf(fact_4060_word__atLeastLessThan__Suc__atLeastAtMost,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [U: word @ A,L: word @ A] :
( ( U
!= ( uminus_uminus @ ( word @ A ) @ ( one_one @ ( word @ A ) ) ) )
=> ( ( set_or7035219750837199246ssThan @ ( word @ A ) @ L @ ( plus_plus @ ( word @ A ) @ U @ ( one_one @ ( word @ A ) ) ) )
= ( set_or1337092689740270186AtMost @ ( word @ A ) @ L @ U ) ) ) ) ).
% word_atLeastLessThan_Suc_atLeastAtMost
thf(fact_4061_powr__neg__one,axiom,
! [X2: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ X2 )
=> ( ( powr @ real @ X2 @ ( uminus_uminus @ real @ ( one_one @ real ) ) )
= ( divide_divide @ real @ ( one_one @ real ) @ X2 ) ) ) ).
% powr_neg_one
thf(fact_4062_Bernoulli__inequality,axiom,
! [X2: real,N: nat] :
( ( ord_less_eq @ real @ ( uminus_uminus @ real @ ( one_one @ real ) ) @ X2 )
=> ( ord_less_eq @ real @ ( plus_plus @ real @ ( one_one @ real ) @ ( times_times @ real @ ( semiring_1_of_nat @ real @ N ) @ X2 ) ) @ ( power_power @ real @ ( plus_plus @ real @ ( one_one @ real ) @ X2 ) @ N ) ) ) ).
% Bernoulli_inequality
thf(fact_4063_compl__le__swap2,axiom,
! [A: $tType] :
( ( boolea8198339166811842893lgebra @ A )
=> ! [Y3: A,X2: A] :
( ( ord_less_eq @ A @ ( uminus_uminus @ A @ Y3 ) @ X2 )
=> ( ord_less_eq @ A @ ( uminus_uminus @ A @ X2 ) @ Y3 ) ) ) ).
% compl_le_swap2
thf(fact_4064_compl__le__swap1,axiom,
! [A: $tType] :
( ( boolea8198339166811842893lgebra @ A )
=> ! [Y3: A,X2: A] :
( ( ord_less_eq @ A @ Y3 @ ( uminus_uminus @ A @ X2 ) )
=> ( ord_less_eq @ A @ X2 @ ( uminus_uminus @ A @ Y3 ) ) ) ) ).
% compl_le_swap1
thf(fact_4065_compl__mono,axiom,
! [A: $tType] :
( ( boolea8198339166811842893lgebra @ A )
=> ! [X2: A,Y3: A] :
( ( ord_less_eq @ A @ X2 @ Y3 )
=> ( ord_less_eq @ A @ ( uminus_uminus @ A @ Y3 ) @ ( uminus_uminus @ A @ X2 ) ) ) ) ).
% compl_mono
thf(fact_4066_compl__less__swap2,axiom,
! [A: $tType] :
( ( boolea8198339166811842893lgebra @ A )
=> ! [Y3: A,X2: A] :
( ( ord_less @ A @ ( uminus_uminus @ A @ Y3 ) @ X2 )
=> ( ord_less @ A @ ( uminus_uminus @ A @ X2 ) @ Y3 ) ) ) ).
% compl_less_swap2
thf(fact_4067_compl__less__swap1,axiom,
! [A: $tType] :
( ( boolea8198339166811842893lgebra @ A )
=> ! [Y3: A,X2: A] :
( ( ord_less @ A @ Y3 @ ( uminus_uminus @ A @ X2 ) )
=> ( ord_less @ A @ X2 @ ( uminus_uminus @ A @ Y3 ) ) ) ) ).
% compl_less_swap1
thf(fact_4068_log__minus__eq__powr,axiom,
! [B3: real,X2: real,Y3: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ B3 )
=> ( ( B3
!= ( one_one @ real ) )
=> ( ( ord_less @ real @ ( zero_zero @ real ) @ X2 )
=> ( ( minus_minus @ real @ ( log @ B3 @ X2 ) @ Y3 )
= ( log @ B3 @ ( times_times @ real @ X2 @ ( powr @ real @ B3 @ ( uminus_uminus @ real @ Y3 ) ) ) ) ) ) ) ) ).
% log_minus_eq_powr
thf(fact_4069_powr__neg__numeral,axiom,
! [X2: real,N: num] :
( ( ord_less @ real @ ( zero_zero @ real ) @ X2 )
=> ( ( powr @ real @ X2 @ ( uminus_uminus @ real @ ( numeral_numeral @ real @ N ) ) )
= ( divide_divide @ real @ ( one_one @ real ) @ ( power_power @ real @ X2 @ ( numeral_numeral @ nat @ N ) ) ) ) ) ).
% powr_neg_numeral
thf(fact_4070_diff__shunt__var,axiom,
! [A: $tType] :
( ( boolea8198339166811842893lgebra @ A )
=> ! [X2: A,Y3: A] :
( ( ( minus_minus @ A @ X2 @ Y3 )
= ( bot_bot @ A ) )
= ( ord_less_eq @ A @ X2 @ Y3 ) ) ) ).
% diff_shunt_var
thf(fact_4071_cos__coeff__def,axiom,
( cos_coeff
= ( ^ [N4: nat] : ( if @ real @ ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N4 ) @ ( divide_divide @ real @ ( power_power @ real @ ( uminus_uminus @ real @ ( one_one @ real ) ) @ ( divide_divide @ nat @ N4 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( semiring_char_0_fact @ real @ N4 ) ) @ ( zero_zero @ real ) ) ) ) ).
% cos_coeff_def
thf(fact_4072_sin__coeff__def,axiom,
( sin_coeff
= ( ^ [N4: nat] : ( if @ real @ ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N4 ) @ ( zero_zero @ real ) @ ( divide_divide @ real @ ( power_power @ real @ ( uminus_uminus @ real @ ( one_one @ real ) ) @ ( divide_divide @ nat @ ( minus_minus @ nat @ N4 @ ( suc @ ( zero_zero @ nat ) ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( semiring_char_0_fact @ real @ N4 ) ) ) ) ) ).
% sin_coeff_def
thf(fact_4073_dbl__dec__simps_I4_J,axiom,
! [A: $tType] :
( ( neg_numeral @ A )
=> ( ( neg_numeral_dbl_dec @ A @ ( uminus_uminus @ A @ ( one_one @ A ) ) )
= ( uminus_uminus @ A @ ( numeral_numeral @ A @ ( bit1 @ one2 ) ) ) ) ) ).
% dbl_dec_simps(4)
thf(fact_4074_signed__take__bit__rec,axiom,
! [A: $tType] :
( ( bit_ri3973907225187159222ations @ A )
=> ( ( bit_ri4674362597316999326ke_bit @ A )
= ( ^ [N4: nat,A6: A] :
( if @ A
@ ( N4
= ( zero_zero @ nat ) )
@ ( uminus_uminus @ A @ ( modulo_modulo @ A @ A6 @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) )
@ ( plus_plus @ A @ ( modulo_modulo @ A @ A6 @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) @ ( times_times @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ ( bit_ri4674362597316999326ke_bit @ A @ ( minus_minus @ nat @ N4 @ ( one_one @ nat ) ) @ ( divide_divide @ A @ A6 @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) ) ) ) ) ) ) ) ).
% signed_take_bit_rec
thf(fact_4075_dbl__simps_I4_J,axiom,
! [A: $tType] :
( ( neg_numeral @ A )
=> ( ( neg_numeral_dbl @ A @ ( uminus_uminus @ A @ ( one_one @ A ) ) )
= ( uminus_uminus @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) ) ) ).
% dbl_simps(4)
thf(fact_4076_dbl__simps_I2_J,axiom,
! [A: $tType] :
( ( neg_numeral @ A )
=> ( ( neg_numeral_dbl @ A @ ( zero_zero @ A ) )
= ( zero_zero @ A ) ) ) ).
% dbl_simps(2)
thf(fact_4077_dbl__dec__simps_I3_J,axiom,
! [A: $tType] :
( ( neg_numeral @ A )
=> ( ( neg_numeral_dbl_dec @ A @ ( one_one @ A ) )
= ( one_one @ A ) ) ) ).
% dbl_dec_simps(3)
thf(fact_4078_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_4079_signed__take__bit__numeral__of__1,axiom,
! [A: $tType] :
( ( bit_ri3973907225187159222ations @ A )
=> ! [K: num] :
( ( bit_ri4674362597316999326ke_bit @ A @ ( numeral_numeral @ nat @ K ) @ ( one_one @ A ) )
= ( one_one @ A ) ) ) ).
% signed_take_bit_numeral_of_1
thf(fact_4080_signed__take__bit__of__minus__1,axiom,
! [A: $tType] :
( ( bit_ri3973907225187159222ations @ A )
=> ! [N: nat] :
( ( bit_ri4674362597316999326ke_bit @ A @ N @ ( uminus_uminus @ A @ ( one_one @ A ) ) )
= ( uminus_uminus @ A @ ( one_one @ A ) ) ) ) ).
% signed_take_bit_of_minus_1
thf(fact_4081_dbl__simps_I5_J,axiom,
! [A: $tType] :
( ( neg_numeral @ A )
=> ! [K: num] :
( ( neg_numeral_dbl @ A @ ( numeral_numeral @ A @ K ) )
= ( numeral_numeral @ A @ ( bit0 @ K ) ) ) ) ).
% dbl_simps(5)
thf(fact_4082_dbl__simps_I1_J,axiom,
! [A: $tType] :
( ( neg_numeral @ A )
=> ! [K: num] :
( ( neg_numeral_dbl @ A @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ K ) ) )
= ( uminus_uminus @ A @ ( neg_numeral_dbl @ A @ ( numeral_numeral @ A @ K ) ) ) ) ) ).
% dbl_simps(1)
thf(fact_4083_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_4084_dbl__inc__simps_I1_J,axiom,
! [A: $tType] :
( ( neg_numeral @ A )
=> ! [K: num] :
( ( neg_numeral_dbl_inc @ A @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ K ) ) )
= ( uminus_uminus @ A @ ( neg_numeral_dbl_dec @ A @ ( numeral_numeral @ A @ K ) ) ) ) ) ).
% dbl_inc_simps(1)
thf(fact_4085_dbl__dec__simps_I1_J,axiom,
! [A: $tType] :
( ( neg_numeral @ A )
=> ! [K: num] :
( ( neg_numeral_dbl_dec @ A @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ K ) ) )
= ( uminus_uminus @ A @ ( neg_numeral_dbl_inc @ A @ ( numeral_numeral @ A @ K ) ) ) ) ) ).
% dbl_dec_simps(1)
thf(fact_4086_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_4087_dbl__simps_I3_J,axiom,
! [A: $tType] :
( ( neg_numeral @ A )
=> ( ( neg_numeral_dbl @ A @ ( one_one @ A ) )
= ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) ) ).
% dbl_simps(3)
thf(fact_4088_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_4089_signed__take__bit__0,axiom,
! [A: $tType] :
( ( bit_ri3973907225187159222ations @ A )
=> ! [A3: A] :
( ( bit_ri4674362597316999326ke_bit @ A @ ( zero_zero @ nat ) @ A3 )
= ( uminus_uminus @ A @ ( modulo_modulo @ A @ A3 @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) ) ) ) ).
% signed_take_bit_0
thf(fact_4090_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_4091_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_4092_uminus__set__def,axiom,
! [A: $tType] :
( ( uminus_uminus @ ( set @ A ) )
= ( ^ [A7: set @ A] :
( collect @ A
@ ( uminus_uminus @ ( A > $o )
@ ^ [X3: A] : ( member @ A @ X3 @ A7 ) ) ) ) ) ).
% uminus_set_def
thf(fact_4093_Compl__eq,axiom,
! [A: $tType] :
( ( uminus_uminus @ ( set @ A ) )
= ( ^ [A7: set @ A] :
( collect @ A
@ ^ [X3: A] :
~ ( member @ A @ X3 @ A7 ) ) ) ) ).
% Compl_eq
thf(fact_4094_Collect__neg__eq,axiom,
! [A: $tType,P: A > $o] :
( ( collect @ A
@ ^ [X3: A] :
~ ( P @ X3 ) )
= ( uminus_uminus @ ( set @ A ) @ ( collect @ A @ P ) ) ) ).
% Collect_neg_eq
thf(fact_4095_signed__take__bit__add,axiom,
! [N: nat,K: int,L: int] :
( ( bit_ri4674362597316999326ke_bit @ int @ N @ ( plus_plus @ int @ ( bit_ri4674362597316999326ke_bit @ int @ N @ K ) @ ( bit_ri4674362597316999326ke_bit @ int @ N @ L ) ) )
= ( bit_ri4674362597316999326ke_bit @ int @ N @ ( plus_plus @ int @ K @ L ) ) ) ).
% signed_take_bit_add
thf(fact_4096_signed__take__bit__diff,axiom,
! [N: nat,K: int,L: int] :
( ( bit_ri4674362597316999326ke_bit @ int @ N @ ( minus_minus @ int @ ( bit_ri4674362597316999326ke_bit @ int @ N @ K ) @ ( bit_ri4674362597316999326ke_bit @ int @ N @ L ) ) )
= ( bit_ri4674362597316999326ke_bit @ int @ N @ ( minus_minus @ int @ K @ L ) ) ) ).
% signed_take_bit_diff
thf(fact_4097_dbl__def,axiom,
! [A: $tType] :
( ( neg_numeral @ A )
=> ( ( neg_numeral_dbl @ A )
= ( ^ [X3: A] : ( plus_plus @ A @ X3 @ X3 ) ) ) ) ).
% dbl_def
thf(fact_4098_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_4099_even__signed__take__bit__iff,axiom,
! [A: $tType] :
( ( bit_ri3973907225187159222ations @ A )
=> ! [M: nat,A3: A] :
( ( dvd_dvd @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ ( bit_ri4674362597316999326ke_bit @ A @ M @ A3 ) )
= ( dvd_dvd @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ A3 ) ) ) ).
% even_signed_take_bit_iff
thf(fact_4100_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_4101_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_4102_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_4103_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_4104_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_4105_dbl__dec__def,axiom,
! [A: $tType] :
( ( neg_numeral @ A )
=> ( ( neg_numeral_dbl_dec @ A )
= ( ^ [X3: A] : ( minus_minus @ A @ ( plus_plus @ A @ X3 @ X3 ) @ ( one_one @ A ) ) ) ) ) ).
% dbl_dec_def
thf(fact_4106_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_4107_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_4108_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_4109_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_4110_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_4111_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_4112_signed__take__bit__Suc,axiom,
! [A: $tType] :
( ( bit_ri3973907225187159222ations @ A )
=> ! [N: nat,A3: A] :
( ( bit_ri4674362597316999326ke_bit @ A @ ( suc @ N ) @ A3 )
= ( plus_plus @ A @ ( modulo_modulo @ A @ A3 @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) @ ( times_times @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ ( bit_ri4674362597316999326ke_bit @ A @ N @ ( divide_divide @ A @ A3 @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) ) ) ) ) ) ).
% signed_take_bit_Suc
thf(fact_4113_sin__paired,axiom,
! [X2: real] :
( sums @ real
@ ^ [N4: nat] : ( times_times @ real @ ( divide_divide @ real @ ( power_power @ real @ ( uminus_uminus @ real @ ( one_one @ real ) ) @ N4 ) @ ( semiring_char_0_fact @ real @ ( plus_plus @ nat @ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N4 ) @ ( one_one @ nat ) ) ) ) @ ( power_power @ real @ X2 @ ( plus_plus @ nat @ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N4 ) @ ( one_one @ nat ) ) ) )
@ ( sin @ real @ X2 ) ) ).
% sin_paired
thf(fact_4114_signed__take__bit__numeral__minus__bit1,axiom,
! [L: num,K: num] :
( ( bit_ri4674362597316999326ke_bit @ int @ ( numeral_numeral @ nat @ L ) @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ ( bit1 @ K ) ) ) )
= ( plus_plus @ int @ ( times_times @ int @ ( bit_ri4674362597316999326ke_bit @ int @ ( pred_numeral @ L ) @ ( minus_minus @ int @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ K ) ) @ ( one_one @ int ) ) ) @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) ) @ ( one_one @ int ) ) ) ).
% signed_take_bit_numeral_minus_bit1
thf(fact_4115_cos__paired,axiom,
! [X2: real] :
( sums @ real
@ ^ [N4: nat] : ( times_times @ real @ ( divide_divide @ real @ ( power_power @ real @ ( uminus_uminus @ real @ ( one_one @ real ) ) @ N4 ) @ ( semiring_char_0_fact @ real @ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N4 ) ) ) @ ( power_power @ real @ X2 @ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N4 ) ) )
@ ( cos @ real @ X2 ) ) ).
% cos_paired
thf(fact_4116_or__int__unfold,axiom,
( ( bit_se1065995026697491101ons_or @ int )
= ( ^ [K3: int,L3: int] :
( if @ int
@ ( ( K3
= ( uminus_uminus @ int @ ( one_one @ int ) ) )
| ( L3
= ( uminus_uminus @ int @ ( one_one @ int ) ) ) )
@ ( uminus_uminus @ int @ ( one_one @ int ) )
@ ( if @ int
@ ( K3
= ( zero_zero @ int ) )
@ L3
@ ( if @ int
@ ( L3
= ( zero_zero @ int ) )
@ K3
@ ( plus_plus @ int @ ( ord_max @ int @ ( modulo_modulo @ int @ K3 @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) ) @ ( modulo_modulo @ int @ L3 @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) ) ) @ ( times_times @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ ( bit_se1065995026697491101ons_or @ int @ ( divide_divide @ int @ K3 @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) ) @ ( divide_divide @ int @ L3 @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) ) ) ) ) ) ) ) ) ) ).
% or_int_unfold
thf(fact_4117_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_4118_bit_Odisj__one__right,axiom,
! [A: $tType] :
( ( bit_ri3973907225187159222ations @ A )
=> ! [X2: A] :
( ( bit_se1065995026697491101ons_or @ A @ X2 @ ( uminus_uminus @ A @ ( one_one @ A ) ) )
= ( uminus_uminus @ A @ ( one_one @ A ) ) ) ) ).
% bit.disj_one_right
thf(fact_4119_bit_Odisj__one__left,axiom,
! [A: $tType] :
( ( bit_ri3973907225187159222ations @ A )
=> ! [X2: A] :
( ( bit_se1065995026697491101ons_or @ A @ ( uminus_uminus @ A @ ( one_one @ A ) ) @ X2 )
= ( uminus_uminus @ A @ ( one_one @ A ) ) ) ) ).
% bit.disj_one_left
thf(fact_4120_pred__numeral__simps_I1_J,axiom,
( ( pred_numeral @ one2 )
= ( zero_zero @ nat ) ) ).
% pred_numeral_simps(1)
thf(fact_4121_eq__numeral__Suc,axiom,
! [K: num,N: nat] :
( ( ( numeral_numeral @ nat @ K )
= ( suc @ N ) )
= ( ( pred_numeral @ K )
= N ) ) ).
% eq_numeral_Suc
thf(fact_4122_Suc__eq__numeral,axiom,
! [N: nat,K: num] :
( ( ( suc @ N )
= ( numeral_numeral @ nat @ K ) )
= ( N
= ( pred_numeral @ K ) ) ) ).
% Suc_eq_numeral
thf(fact_4123_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_4124_or__numerals_I8_J,axiom,
! [A: $tType] :
( ( bit_un5681908812861735899ations @ A )
=> ! [X2: num] :
( ( bit_se1065995026697491101ons_or @ A @ ( numeral_numeral @ A @ ( bit1 @ X2 ) ) @ ( one_one @ A ) )
= ( numeral_numeral @ A @ ( bit1 @ X2 ) ) ) ) ).
% or_numerals(8)
thf(fact_4125_or__numerals_I2_J,axiom,
! [A: $tType] :
( ( bit_un5681908812861735899ations @ A )
=> ! [Y3: num] :
( ( bit_se1065995026697491101ons_or @ A @ ( one_one @ A ) @ ( numeral_numeral @ A @ ( bit1 @ Y3 ) ) )
= ( numeral_numeral @ A @ ( bit1 @ Y3 ) ) ) ) ).
% or_numerals(2)
thf(fact_4126_pred__numeral__simps_I3_J,axiom,
! [K: num] :
( ( pred_numeral @ ( bit1 @ K ) )
= ( numeral_numeral @ nat @ ( bit0 @ K ) ) ) ).
% pred_numeral_simps(3)
thf(fact_4127_less__numeral__Suc,axiom,
! [K: num,N: nat] :
( ( ord_less @ nat @ ( numeral_numeral @ nat @ K ) @ ( suc @ N ) )
= ( ord_less @ nat @ ( pred_numeral @ K ) @ N ) ) ).
% less_numeral_Suc
thf(fact_4128_less__Suc__numeral,axiom,
! [N: nat,K: num] :
( ( ord_less @ nat @ ( suc @ N ) @ ( numeral_numeral @ nat @ K ) )
= ( ord_less @ nat @ N @ ( pred_numeral @ K ) ) ) ).
% less_Suc_numeral
thf(fact_4129_le__numeral__Suc,axiom,
! [K: num,N: nat] :
( ( ord_less_eq @ nat @ ( numeral_numeral @ nat @ K ) @ ( suc @ N ) )
= ( ord_less_eq @ nat @ ( pred_numeral @ K ) @ N ) ) ).
% le_numeral_Suc
thf(fact_4130_le__Suc__numeral,axiom,
! [N: nat,K: num] :
( ( ord_less_eq @ nat @ ( suc @ N ) @ ( numeral_numeral @ nat @ K ) )
= ( ord_less_eq @ nat @ N @ ( pred_numeral @ K ) ) ) ).
% le_Suc_numeral
thf(fact_4131_diff__numeral__Suc,axiom,
! [K: num,N: nat] :
( ( minus_minus @ nat @ ( numeral_numeral @ nat @ K ) @ ( suc @ N ) )
= ( minus_minus @ nat @ ( pred_numeral @ K ) @ N ) ) ).
% diff_numeral_Suc
thf(fact_4132_diff__Suc__numeral,axiom,
! [N: nat,K: num] :
( ( minus_minus @ nat @ ( suc @ N ) @ ( numeral_numeral @ nat @ K ) )
= ( minus_minus @ nat @ N @ ( pred_numeral @ K ) ) ) ).
% diff_Suc_numeral
thf(fact_4133_max__numeral__Suc,axiom,
! [K: num,N: nat] :
( ( ord_max @ nat @ ( numeral_numeral @ nat @ K ) @ ( suc @ N ) )
= ( suc @ ( ord_max @ nat @ ( pred_numeral @ K ) @ N ) ) ) ).
% max_numeral_Suc
thf(fact_4134_max__Suc__numeral,axiom,
! [N: nat,K: num] :
( ( ord_max @ nat @ ( suc @ N ) @ ( numeral_numeral @ nat @ K ) )
= ( suc @ ( ord_max @ nat @ N @ ( pred_numeral @ K ) ) ) ) ).
% max_Suc_numeral
thf(fact_4135_or__numerals_I3_J,axiom,
! [A: $tType] :
( ( bit_un5681908812861735899ations @ A )
=> ! [X2: num,Y3: num] :
( ( bit_se1065995026697491101ons_or @ A @ ( numeral_numeral @ A @ ( bit0 @ X2 ) ) @ ( numeral_numeral @ A @ ( bit0 @ Y3 ) ) )
= ( times_times @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ ( bit_se1065995026697491101ons_or @ A @ ( numeral_numeral @ A @ X2 ) @ ( numeral_numeral @ A @ Y3 ) ) ) ) ) ).
% or_numerals(3)
thf(fact_4136_or__numerals_I5_J,axiom,
! [A: $tType] :
( ( bit_un5681908812861735899ations @ A )
=> ! [X2: num] :
( ( bit_se1065995026697491101ons_or @ A @ ( numeral_numeral @ A @ ( bit0 @ X2 ) ) @ ( one_one @ A ) )
= ( numeral_numeral @ A @ ( bit1 @ X2 ) ) ) ) ).
% or_numerals(5)
thf(fact_4137_or__numerals_I1_J,axiom,
! [A: $tType] :
( ( bit_un5681908812861735899ations @ A )
=> ! [Y3: num] :
( ( bit_se1065995026697491101ons_or @ A @ ( one_one @ A ) @ ( numeral_numeral @ A @ ( bit0 @ Y3 ) ) )
= ( numeral_numeral @ A @ ( bit1 @ Y3 ) ) ) ) ).
% or_numerals(1)
thf(fact_4138_sin__cos__squared__add3,axiom,
! [A: $tType] :
( ( ( real_Vector_banach @ A )
& ( real_V3459762299906320749_field @ A ) )
=> ! [X2: A] :
( ( plus_plus @ A @ ( times_times @ A @ ( cos @ A @ X2 ) @ ( cos @ A @ X2 ) ) @ ( times_times @ A @ ( sin @ A @ X2 ) @ ( sin @ A @ X2 ) ) )
= ( one_one @ A ) ) ) ).
% sin_cos_squared_add3
thf(fact_4139_or__minus__numerals_I6_J,axiom,
! [N: num] :
( ( bit_se1065995026697491101ons_or @ int @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ ( bit1 @ N ) ) ) @ ( one_one @ int ) )
= ( uminus_uminus @ int @ ( numeral_numeral @ int @ ( bit1 @ N ) ) ) ) ).
% or_minus_numerals(6)
thf(fact_4140_or__minus__numerals_I2_J,axiom,
! [N: num] :
( ( bit_se1065995026697491101ons_or @ int @ ( one_one @ int ) @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ ( bit1 @ N ) ) ) )
= ( uminus_uminus @ int @ ( numeral_numeral @ int @ ( bit1 @ N ) ) ) ) ).
% or_minus_numerals(2)
thf(fact_4141_signed__take__bit__numeral__bit0,axiom,
! [L: num,K: num] :
( ( bit_ri4674362597316999326ke_bit @ int @ ( numeral_numeral @ nat @ L ) @ ( numeral_numeral @ int @ ( bit0 @ K ) ) )
= ( times_times @ int @ ( bit_ri4674362597316999326ke_bit @ int @ ( pred_numeral @ L ) @ ( numeral_numeral @ int @ K ) ) @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) ) ) ).
% signed_take_bit_numeral_bit0
thf(fact_4142_or__numerals_I4_J,axiom,
! [A: $tType] :
( ( bit_un5681908812861735899ations @ A )
=> ! [X2: num,Y3: num] :
( ( bit_se1065995026697491101ons_or @ A @ ( numeral_numeral @ A @ ( bit0 @ X2 ) ) @ ( numeral_numeral @ A @ ( bit1 @ Y3 ) ) )
= ( plus_plus @ A @ ( one_one @ A ) @ ( times_times @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ ( bit_se1065995026697491101ons_or @ A @ ( numeral_numeral @ A @ X2 ) @ ( numeral_numeral @ A @ Y3 ) ) ) ) ) ) ).
% or_numerals(4)
thf(fact_4143_or__numerals_I6_J,axiom,
! [A: $tType] :
( ( bit_un5681908812861735899ations @ A )
=> ! [X2: num,Y3: num] :
( ( bit_se1065995026697491101ons_or @ A @ ( numeral_numeral @ A @ ( bit1 @ X2 ) ) @ ( numeral_numeral @ A @ ( bit0 @ Y3 ) ) )
= ( plus_plus @ A @ ( one_one @ A ) @ ( times_times @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ ( bit_se1065995026697491101ons_or @ A @ ( numeral_numeral @ A @ X2 ) @ ( numeral_numeral @ A @ Y3 ) ) ) ) ) ) ).
% or_numerals(6)
thf(fact_4144_or__numerals_I7_J,axiom,
! [A: $tType] :
( ( bit_un5681908812861735899ations @ A )
=> ! [X2: num,Y3: num] :
( ( bit_se1065995026697491101ons_or @ A @ ( numeral_numeral @ A @ ( bit1 @ X2 ) ) @ ( numeral_numeral @ A @ ( bit1 @ Y3 ) ) )
= ( plus_plus @ A @ ( one_one @ A ) @ ( times_times @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ ( bit_se1065995026697491101ons_or @ A @ ( numeral_numeral @ A @ X2 ) @ ( numeral_numeral @ A @ Y3 ) ) ) ) ) ) ).
% or_numerals(7)
thf(fact_4145_sin__cos__squared__add,axiom,
! [A: $tType] :
( ( ( real_Vector_banach @ A )
& ( real_V3459762299906320749_field @ A ) )
=> ! [X2: A] :
( ( plus_plus @ A @ ( power_power @ A @ ( sin @ A @ X2 ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ ( power_power @ A @ ( cos @ A @ X2 ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) )
= ( one_one @ A ) ) ) ).
% sin_cos_squared_add
thf(fact_4146_sin__cos__squared__add2,axiom,
! [A: $tType] :
( ( ( real_Vector_banach @ A )
& ( real_V3459762299906320749_field @ A ) )
=> ! [X2: A] :
( ( plus_plus @ A @ ( power_power @ A @ ( cos @ A @ X2 ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ ( power_power @ A @ ( sin @ A @ X2 ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) )
= ( one_one @ A ) ) ) ).
% sin_cos_squared_add2
thf(fact_4147_signed__take__bit__numeral__minus__bit0,axiom,
! [L: num,K: num] :
( ( bit_ri4674362597316999326ke_bit @ int @ ( numeral_numeral @ nat @ L ) @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ ( bit0 @ K ) ) ) )
= ( times_times @ int @ ( bit_ri4674362597316999326ke_bit @ int @ ( pred_numeral @ L ) @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ K ) ) ) @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) ) ) ).
% signed_take_bit_numeral_minus_bit0
thf(fact_4148_signed__take__bit__numeral__bit1,axiom,
! [L: num,K: num] :
( ( bit_ri4674362597316999326ke_bit @ int @ ( numeral_numeral @ nat @ L ) @ ( numeral_numeral @ int @ ( bit1 @ K ) ) )
= ( plus_plus @ int @ ( times_times @ int @ ( bit_ri4674362597316999326ke_bit @ int @ ( pred_numeral @ L ) @ ( numeral_numeral @ int @ K ) ) @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) ) @ ( one_one @ int ) ) ) ).
% signed_take_bit_numeral_bit1
thf(fact_4149_sin__zero__norm__cos__one,axiom,
! [A: $tType] :
( ( ( real_Vector_banach @ A )
& ( real_V3459762299906320749_field @ A ) )
=> ! [X2: A] :
( ( ( sin @ A @ X2 )
= ( zero_zero @ A ) )
=> ( ( real_V7770717601297561774m_norm @ A @ ( cos @ A @ X2 ) )
= ( one_one @ real ) ) ) ) ).
% sin_zero_norm_cos_one
thf(fact_4150_cos__add,axiom,
! [A: $tType] :
( ( ( real_Vector_banach @ A )
& ( real_V3459762299906320749_field @ A ) )
=> ! [X2: A,Y3: A] :
( ( cos @ A @ ( plus_plus @ A @ X2 @ Y3 ) )
= ( minus_minus @ A @ ( times_times @ A @ ( cos @ A @ X2 ) @ ( cos @ A @ Y3 ) ) @ ( times_times @ A @ ( sin @ A @ X2 ) @ ( sin @ A @ Y3 ) ) ) ) ) ).
% cos_add
thf(fact_4151_cos__diff,axiom,
! [A: $tType] :
( ( ( real_Vector_banach @ A )
& ( real_V3459762299906320749_field @ A ) )
=> ! [X2: A,Y3: A] :
( ( cos @ A @ ( minus_minus @ A @ X2 @ Y3 ) )
= ( plus_plus @ A @ ( times_times @ A @ ( cos @ A @ X2 ) @ ( cos @ A @ Y3 ) ) @ ( times_times @ A @ ( sin @ A @ X2 ) @ ( sin @ A @ Y3 ) ) ) ) ) ).
% cos_diff
thf(fact_4152_cos__one__sin__zero,axiom,
! [A: $tType] :
( ( ( real_Vector_banach @ A )
& ( real_V3459762299906320749_field @ A ) )
=> ! [X2: A] :
( ( ( cos @ A @ X2 )
= ( one_one @ A ) )
=> ( ( sin @ A @ X2 )
= ( zero_zero @ A ) ) ) ) ).
% cos_one_sin_zero
thf(fact_4153_sin__diff,axiom,
! [A: $tType] :
( ( ( real_Vector_banach @ A )
& ( real_V3459762299906320749_field @ A ) )
=> ! [X2: A,Y3: A] :
( ( sin @ A @ ( minus_minus @ A @ X2 @ Y3 ) )
= ( minus_minus @ A @ ( times_times @ A @ ( sin @ A @ X2 ) @ ( cos @ A @ Y3 ) ) @ ( times_times @ A @ ( cos @ A @ X2 ) @ ( sin @ A @ Y3 ) ) ) ) ) ).
% sin_diff
thf(fact_4154_sin__add,axiom,
! [A: $tType] :
( ( ( real_Vector_banach @ A )
& ( real_V3459762299906320749_field @ A ) )
=> ! [X2: A,Y3: A] :
( ( sin @ A @ ( plus_plus @ A @ X2 @ Y3 ) )
= ( plus_plus @ A @ ( times_times @ A @ ( sin @ A @ X2 ) @ ( cos @ A @ Y3 ) ) @ ( times_times @ A @ ( cos @ A @ X2 ) @ ( sin @ A @ Y3 ) ) ) ) ) ).
% sin_add
thf(fact_4155_sin__double,axiom,
! [A: $tType] :
( ( ( real_Vector_banach @ A )
& ( real_V3459762299906320749_field @ A ) )
=> ! [X2: A] :
( ( sin @ A @ ( times_times @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ X2 ) )
= ( times_times @ A @ ( times_times @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ ( sin @ A @ X2 ) ) @ ( cos @ A @ X2 ) ) ) ) ).
% sin_double
thf(fact_4156_sin__x__le__x,axiom,
! [X2: real] :
( ( ord_less_eq @ real @ ( zero_zero @ real ) @ X2 )
=> ( ord_less_eq @ real @ ( sin @ real @ X2 ) @ X2 ) ) ).
% sin_x_le_x
thf(fact_4157_OR__lower,axiom,
! [X2: int,Y3: int] :
( ( ord_less_eq @ int @ ( zero_zero @ int ) @ X2 )
=> ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ Y3 )
=> ( ord_less_eq @ int @ ( zero_zero @ int ) @ ( bit_se1065995026697491101ons_or @ int @ X2 @ Y3 ) ) ) ) ).
% OR_lower
thf(fact_4158_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_4159_sin__le__one,axiom,
! [X2: real] : ( ord_less_eq @ real @ ( sin @ real @ X2 ) @ ( one_one @ real ) ) ).
% sin_le_one
thf(fact_4160_cos__le__one,axiom,
! [X2: real] : ( ord_less_eq @ real @ ( cos @ real @ X2 ) @ ( one_one @ real ) ) ).
% cos_le_one
thf(fact_4161_numeral__eq__Suc,axiom,
( ( numeral_numeral @ nat )
= ( ^ [K3: num] : ( suc @ ( pred_numeral @ K3 ) ) ) ) ).
% numeral_eq_Suc
thf(fact_4162_cos__squared__eq,axiom,
! [A: $tType] :
( ( ( real_Vector_banach @ A )
& ( real_V3459762299906320749_field @ A ) )
=> ! [X2: A] :
( ( power_power @ A @ ( cos @ A @ X2 ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) )
= ( minus_minus @ A @ ( one_one @ A ) @ ( power_power @ A @ ( sin @ A @ X2 ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) ).
% cos_squared_eq
thf(fact_4163_sin__squared__eq,axiom,
! [A: $tType] :
( ( ( real_Vector_banach @ A )
& ( real_V3459762299906320749_field @ A ) )
=> ! [X2: A] :
( ( power_power @ A @ ( sin @ A @ X2 ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) )
= ( minus_minus @ A @ ( one_one @ A ) @ ( power_power @ A @ ( cos @ A @ X2 ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) ).
% sin_squared_eq
thf(fact_4164_sin__x__ge__neg__x,axiom,
! [X2: real] :
( ( ord_less_eq @ real @ ( zero_zero @ real ) @ X2 )
=> ( ord_less_eq @ real @ ( uminus_uminus @ real @ X2 ) @ ( sin @ real @ X2 ) ) ) ).
% sin_x_ge_neg_x
thf(fact_4165_sin__ge__minus__one,axiom,
! [X2: real] : ( ord_less_eq @ real @ ( uminus_uminus @ real @ ( one_one @ real ) ) @ ( sin @ real @ X2 ) ) ).
% sin_ge_minus_one
thf(fact_4166_cos__ge__minus__one,axiom,
! [X2: real] : ( ord_less_eq @ real @ ( uminus_uminus @ real @ ( one_one @ real ) ) @ ( cos @ real @ X2 ) ) ).
% cos_ge_minus_one
thf(fact_4167_sin__times__sin,axiom,
! [A: $tType] :
( ( ( real_Vector_banach @ A )
& ( real_V3459762299906320749_field @ A ) )
=> ! [W: A,Z: A] :
( ( times_times @ A @ ( sin @ A @ W ) @ ( sin @ A @ Z ) )
= ( divide_divide @ A @ ( minus_minus @ A @ ( cos @ A @ ( minus_minus @ A @ W @ Z ) ) @ ( cos @ A @ ( plus_plus @ A @ W @ Z ) ) ) @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) ) ) ).
% sin_times_sin
thf(fact_4168_sin__times__cos,axiom,
! [A: $tType] :
( ( ( real_Vector_banach @ A )
& ( real_V3459762299906320749_field @ A ) )
=> ! [W: A,Z: A] :
( ( times_times @ A @ ( sin @ A @ W ) @ ( cos @ A @ Z ) )
= ( divide_divide @ A @ ( plus_plus @ A @ ( sin @ A @ ( plus_plus @ A @ W @ Z ) ) @ ( sin @ A @ ( minus_minus @ A @ W @ Z ) ) ) @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) ) ) ).
% sin_times_cos
thf(fact_4169_cos__times__sin,axiom,
! [A: $tType] :
( ( ( real_Vector_banach @ A )
& ( real_V3459762299906320749_field @ A ) )
=> ! [W: A,Z: A] :
( ( times_times @ A @ ( cos @ A @ W ) @ ( sin @ A @ Z ) )
= ( divide_divide @ A @ ( minus_minus @ A @ ( sin @ A @ ( plus_plus @ A @ W @ Z ) ) @ ( sin @ A @ ( minus_minus @ A @ W @ Z ) ) ) @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) ) ) ).
% cos_times_sin
thf(fact_4170_sin__plus__sin,axiom,
! [A: $tType] :
( ( ( real_Vector_banach @ A )
& ( real_V3459762299906320749_field @ A ) )
=> ! [W: A,Z: A] :
( ( plus_plus @ A @ ( sin @ A @ W ) @ ( sin @ A @ Z ) )
= ( times_times @ A @ ( times_times @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ ( sin @ A @ ( divide_divide @ A @ ( plus_plus @ A @ W @ Z ) @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) ) ) @ ( cos @ A @ ( divide_divide @ A @ ( minus_minus @ A @ W @ Z ) @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) ) ) ) ) ).
% sin_plus_sin
thf(fact_4171_sin__diff__sin,axiom,
! [A: $tType] :
( ( ( real_Vector_banach @ A )
& ( real_V3459762299906320749_field @ A ) )
=> ! [W: A,Z: A] :
( ( minus_minus @ A @ ( sin @ A @ W ) @ ( sin @ A @ Z ) )
= ( times_times @ A @ ( times_times @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ ( sin @ A @ ( divide_divide @ A @ ( minus_minus @ A @ W @ Z ) @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) ) ) @ ( cos @ A @ ( divide_divide @ A @ ( plus_plus @ A @ W @ Z ) @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) ) ) ) ) ).
% sin_diff_sin
thf(fact_4172_cos__diff__cos,axiom,
! [A: $tType] :
( ( ( real_Vector_banach @ A )
& ( real_V3459762299906320749_field @ A ) )
=> ! [W: A,Z: A] :
( ( minus_minus @ A @ ( cos @ A @ W ) @ ( cos @ A @ Z ) )
= ( times_times @ A @ ( times_times @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ ( sin @ A @ ( divide_divide @ A @ ( plus_plus @ A @ W @ Z ) @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) ) ) @ ( sin @ A @ ( divide_divide @ A @ ( minus_minus @ A @ Z @ W ) @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) ) ) ) ) ).
% cos_diff_cos
thf(fact_4173_cos__double,axiom,
! [A: $tType] :
( ( ( real_Vector_banach @ A )
& ( real_V3459762299906320749_field @ A ) )
=> ! [X2: A] :
( ( cos @ A @ ( times_times @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ X2 ) )
= ( minus_minus @ A @ ( power_power @ A @ ( cos @ A @ X2 ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ ( power_power @ A @ ( sin @ A @ X2 ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) ).
% cos_double
thf(fact_4174_pred__numeral__def,axiom,
( pred_numeral
= ( ^ [K3: num] : ( minus_minus @ nat @ ( numeral_numeral @ nat @ K3 ) @ ( one_one @ nat ) ) ) ) ).
% pred_numeral_def
thf(fact_4175_lessThan__nat__numeral,axiom,
! [K: num] :
( ( set_ord_lessThan @ nat @ ( numeral_numeral @ nat @ K ) )
= ( insert @ nat @ ( pred_numeral @ K ) @ ( set_ord_lessThan @ nat @ ( pred_numeral @ K ) ) ) ) ).
% lessThan_nat_numeral
thf(fact_4176_atMost__nat__numeral,axiom,
! [K: num] :
( ( set_ord_atMost @ nat @ ( numeral_numeral @ nat @ K ) )
= ( insert @ nat @ ( numeral_numeral @ nat @ K ) @ ( set_ord_atMost @ nat @ ( pred_numeral @ K ) ) ) ) ).
% atMost_nat_numeral
thf(fact_4177_cos__double__sin,axiom,
! [A: $tType] :
( ( ( real_Vector_banach @ A )
& ( real_V3459762299906320749_field @ A ) )
=> ! [W: A] :
( ( cos @ A @ ( times_times @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ W ) )
= ( minus_minus @ A @ ( one_one @ A ) @ ( times_times @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ ( power_power @ A @ ( sin @ A @ W ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) ) ).
% cos_double_sin
thf(fact_4178_even__or__iff,axiom,
! [A: $tType] :
( ( bit_se359711467146920520ations @ A )
=> ! [A3: A,B3: A] :
( ( dvd_dvd @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ ( bit_se1065995026697491101ons_or @ A @ A3 @ B3 ) )
= ( ( dvd_dvd @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ A3 )
& ( dvd_dvd @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ B3 ) ) ) ) ).
% even_or_iff
thf(fact_4179_cos__two__neq__zero,axiom,
( ( cos @ real @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) )
!= ( zero_zero @ real ) ) ).
% cos_two_neq_zero
thf(fact_4180_fact__numeral,axiom,
! [A: $tType] :
( ( semiring_char_0 @ A )
=> ! [K: num] :
( ( semiring_char_0_fact @ A @ ( numeral_numeral @ nat @ K ) )
= ( times_times @ A @ ( numeral_numeral @ A @ K ) @ ( semiring_char_0_fact @ A @ ( pred_numeral @ K ) ) ) ) ) ).
% fact_numeral
thf(fact_4181_sin__gt__zero__02,axiom,
! [X2: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ X2 )
=> ( ( ord_less @ real @ X2 @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) )
=> ( ord_less @ real @ ( zero_zero @ real ) @ ( sin @ real @ X2 ) ) ) ) ).
% sin_gt_zero_02
thf(fact_4182_cos__two__less__zero,axiom,
ord_less @ real @ ( cos @ real @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) @ ( zero_zero @ real ) ).
% cos_two_less_zero
thf(fact_4183_cos__is__zero,axiom,
? [X4: real] :
( ( ord_less_eq @ real @ ( zero_zero @ real ) @ X4 )
& ( ord_less_eq @ real @ X4 @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) )
& ( ( cos @ real @ X4 )
= ( zero_zero @ real ) )
& ! [Y5: real] :
( ( ( ord_less_eq @ real @ ( zero_zero @ real ) @ Y5 )
& ( ord_less_eq @ real @ Y5 @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) )
& ( ( cos @ real @ Y5 )
= ( zero_zero @ real ) ) )
=> ( Y5 = X4 ) ) ) ).
% cos_is_zero
thf(fact_4184_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_4185_atLeastLessThan__nat__numeral,axiom,
! [M: nat,K: num] :
( ( ( ord_less_eq @ nat @ M @ ( pred_numeral @ K ) )
=> ( ( set_or7035219750837199246ssThan @ nat @ M @ ( numeral_numeral @ nat @ K ) )
= ( insert @ nat @ ( pred_numeral @ K ) @ ( set_or7035219750837199246ssThan @ nat @ M @ ( pred_numeral @ K ) ) ) ) )
& ( ~ ( ord_less_eq @ nat @ M @ ( pred_numeral @ K ) )
=> ( ( set_or7035219750837199246ssThan @ nat @ M @ ( numeral_numeral @ nat @ K ) )
= ( bot_bot @ ( set @ nat ) ) ) ) ) ).
% atLeastLessThan_nat_numeral
thf(fact_4186_cos__times__cos,axiom,
! [A: $tType] :
( ( ( real_Vector_banach @ A )
& ( real_V3459762299906320749_field @ A ) )
=> ! [W: A,Z: A] :
( ( times_times @ A @ ( cos @ A @ W ) @ ( cos @ A @ Z ) )
= ( divide_divide @ A @ ( plus_plus @ A @ ( cos @ A @ ( minus_minus @ A @ W @ Z ) ) @ ( cos @ A @ ( plus_plus @ A @ W @ Z ) ) ) @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) ) ) ).
% cos_times_cos
thf(fact_4187_cos__plus__cos,axiom,
! [A: $tType] :
( ( ( real_Vector_banach @ A )
& ( real_V3459762299906320749_field @ A ) )
=> ! [W: A,Z: A] :
( ( plus_plus @ A @ ( cos @ A @ W ) @ ( cos @ A @ Z ) )
= ( times_times @ A @ ( times_times @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ ( cos @ A @ ( divide_divide @ A @ ( plus_plus @ A @ W @ Z ) @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) ) ) @ ( cos @ A @ ( divide_divide @ A @ ( minus_minus @ A @ W @ Z ) @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) ) ) ) ) ).
% cos_plus_cos
thf(fact_4188_OR__upper,axiom,
! [X2: int,N: nat,Y3: int] :
( ( ord_less_eq @ int @ ( zero_zero @ int ) @ X2 )
=> ( ( ord_less @ int @ X2 @ ( power_power @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ N ) )
=> ( ( ord_less @ int @ Y3 @ ( power_power @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ N ) )
=> ( ord_less @ int @ ( bit_se1065995026697491101ons_or @ int @ X2 @ Y3 ) @ ( power_power @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ N ) ) ) ) ) ).
% OR_upper
thf(fact_4189_cos__double__less__one,axiom,
! [X2: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ X2 )
=> ( ( ord_less @ real @ X2 @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) )
=> ( ord_less @ real @ ( cos @ real @ ( times_times @ real @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) @ X2 ) ) @ ( one_one @ real ) ) ) ) ).
% cos_double_less_one
thf(fact_4190_cos__double__cos,axiom,
! [A: $tType] :
( ( ( real_Vector_banach @ A )
& ( real_V3459762299906320749_field @ A ) )
=> ! [W: A] :
( ( cos @ A @ ( times_times @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ W ) )
= ( minus_minus @ A @ ( times_times @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ ( power_power @ A @ ( cos @ A @ W ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( one_one @ A ) ) ) ) ).
% cos_double_cos
thf(fact_4191_cos__treble__cos,axiom,
! [A: $tType] :
( ( ( real_Vector_banach @ A )
& ( real_V3459762299906320749_field @ A ) )
=> ! [X2: A] :
( ( cos @ A @ ( times_times @ A @ ( numeral_numeral @ A @ ( bit1 @ one2 ) ) @ X2 ) )
= ( minus_minus @ A @ ( times_times @ A @ ( numeral_numeral @ A @ ( bit0 @ ( bit0 @ one2 ) ) ) @ ( power_power @ A @ ( cos @ A @ X2 ) @ ( numeral_numeral @ nat @ ( bit1 @ one2 ) ) ) ) @ ( times_times @ A @ ( numeral_numeral @ A @ ( bit1 @ one2 ) ) @ ( cos @ A @ X2 ) ) ) ) ) ).
% cos_treble_cos
thf(fact_4192_sin__x__sin__y,axiom,
! [A: $tType] :
( ( ( real_Vector_banach @ A )
& ( real_V3459762299906320749_field @ A ) )
=> ! [X2: A,Y3: A] :
( sums @ A
@ ^ [P4: nat] :
( groups7311177749621191930dd_sum @ nat @ A
@ ^ [N4: nat] :
( if @ A
@ ( ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ P4 )
& ~ ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N4 ) )
@ ( 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 @ P4 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( semiring_1_of_nat @ int @ ( binomial @ P4 @ N4 ) ) ) ) @ ( semiring_char_0_fact @ real @ P4 ) ) ) @ ( power_power @ A @ X2 @ N4 ) ) @ ( power_power @ A @ Y3 @ ( minus_minus @ nat @ P4 @ N4 ) ) )
@ ( zero_zero @ A ) )
@ ( set_ord_atMost @ nat @ P4 ) )
@ ( times_times @ A @ ( sin @ A @ X2 ) @ ( sin @ A @ Y3 ) ) ) ) ).
% sin_x_sin_y
thf(fact_4193_sums__cos__x__plus__y,axiom,
! [A: $tType] :
( ( ( real_Vector_banach @ A )
& ( real_V3459762299906320749_field @ A ) )
=> ! [X2: A,Y3: A] :
( sums @ A
@ ^ [P4: nat] :
( groups7311177749621191930dd_sum @ nat @ A
@ ^ [N4: nat] : ( if @ A @ ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ P4 ) @ ( 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 @ P4 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( semiring_1_of_nat @ int @ ( binomial @ P4 @ N4 ) ) ) ) @ ( semiring_char_0_fact @ real @ P4 ) ) @ ( power_power @ A @ X2 @ N4 ) ) @ ( power_power @ A @ Y3 @ ( minus_minus @ nat @ P4 @ N4 ) ) ) @ ( zero_zero @ A ) )
@ ( set_ord_atMost @ nat @ P4 ) )
@ ( cos @ A @ ( plus_plus @ A @ X2 @ Y3 ) ) ) ) ).
% sums_cos_x_plus_y
thf(fact_4194_cos__x__cos__y,axiom,
! [A: $tType] :
( ( ( real_Vector_banach @ A )
& ( real_V3459762299906320749_field @ A ) )
=> ! [X2: A,Y3: A] :
( sums @ A
@ ^ [P4: nat] :
( groups7311177749621191930dd_sum @ nat @ A
@ ^ [N4: nat] :
( if @ A
@ ( ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ P4 )
& ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N4 ) )
@ ( 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 @ P4 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( semiring_1_of_nat @ int @ ( binomial @ P4 @ N4 ) ) ) ) @ ( semiring_char_0_fact @ real @ P4 ) ) @ ( power_power @ A @ X2 @ N4 ) ) @ ( power_power @ A @ Y3 @ ( minus_minus @ nat @ P4 @ N4 ) ) )
@ ( zero_zero @ A ) )
@ ( set_ord_atMost @ nat @ P4 ) )
@ ( times_times @ A @ ( cos @ A @ X2 ) @ ( cos @ A @ Y3 ) ) ) ) ).
% cos_x_cos_y
thf(fact_4195_Maclaurin__cos__expansion2,axiom,
! [X2: real,N: nat] :
( ( ord_less @ real @ ( zero_zero @ real ) @ X2 )
=> ( ( ord_less @ nat @ ( zero_zero @ nat ) @ N )
=> ? [T8: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ T8 )
& ( ord_less @ real @ T8 @ X2 )
& ( ( cos @ real @ X2 )
= ( plus_plus @ real
@ ( groups7311177749621191930dd_sum @ nat @ real
@ ^ [M6: nat] : ( times_times @ real @ ( cos_coeff @ M6 ) @ ( power_power @ real @ X2 @ M6 ) )
@ ( set_ord_lessThan @ nat @ N ) )
@ ( times_times @ real @ ( divide_divide @ real @ ( cos @ real @ ( plus_plus @ real @ T8 @ ( 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 @ X2 @ N ) ) ) ) ) ) ) ).
% Maclaurin_cos_expansion2
thf(fact_4196_mult__scaleR__right,axiom,
! [A: $tType] :
( ( real_V6157519004096292374lgebra @ A )
=> ! [X2: A,A3: real,Y3: A] :
( ( times_times @ A @ X2 @ ( real_V8093663219630862766scaleR @ A @ A3 @ Y3 ) )
= ( real_V8093663219630862766scaleR @ A @ A3 @ ( times_times @ A @ X2 @ Y3 ) ) ) ) ).
% mult_scaleR_right
thf(fact_4197_mult__scaleR__left,axiom,
! [A: $tType] :
( ( real_V6157519004096292374lgebra @ A )
=> ! [A3: real,X2: A,Y3: A] :
( ( times_times @ A @ ( real_V8093663219630862766scaleR @ A @ A3 @ X2 ) @ Y3 )
= ( real_V8093663219630862766scaleR @ A @ A3 @ ( times_times @ A @ X2 @ Y3 ) ) ) ) ).
% mult_scaleR_left
thf(fact_4198_scaleR__one,axiom,
! [A: $tType] :
( ( real_V4867850818363320053vector @ A )
=> ! [X2: A] :
( ( real_V8093663219630862766scaleR @ A @ ( one_one @ real ) @ X2 )
= X2 ) ) ).
% scaleR_one
thf(fact_4199_word__or__max,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [X2: word @ A] :
( ( bit_se1065995026697491101ons_or @ ( word @ A ) @ X2 @ ( uminus_uminus @ ( word @ A ) @ ( one_one @ ( word @ A ) ) ) )
= ( uminus_uminus @ ( word @ A ) @ ( one_one @ ( word @ A ) ) ) ) ) ).
% word_or_max
thf(fact_4200_word__bitwise__m1__simps_I4_J,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [X2: word @ A] :
( ( bit_se1065995026697491101ons_or @ ( word @ A ) @ ( uminus_uminus @ ( word @ A ) @ ( one_one @ ( word @ A ) ) ) @ X2 )
= ( uminus_uminus @ ( word @ A ) @ ( one_one @ ( word @ A ) ) ) ) ) ).
% word_bitwise_m1_simps(4)
thf(fact_4201_scaleR__eq__iff,axiom,
! [A: $tType] :
( ( real_V4867850818363320053vector @ A )
=> ! [B3: A,U: real,A3: A] :
( ( ( plus_plus @ A @ B3 @ ( real_V8093663219630862766scaleR @ A @ U @ A3 ) )
= ( plus_plus @ A @ A3 @ ( real_V8093663219630862766scaleR @ A @ U @ B3 ) ) )
= ( ( A3 = B3 )
| ( U
= ( one_one @ real ) ) ) ) ) ).
% scaleR_eq_iff
thf(fact_4202_scaleR__power,axiom,
! [A: $tType] :
( ( real_V2822296259951069270ebra_1 @ A )
=> ! [X2: real,Y3: A,N: nat] :
( ( power_power @ A @ ( real_V8093663219630862766scaleR @ A @ X2 @ Y3 ) @ N )
= ( real_V8093663219630862766scaleR @ A @ ( power_power @ real @ X2 @ N ) @ ( power_power @ A @ Y3 @ N ) ) ) ) ).
% scaleR_power
thf(fact_4203_sin__pi__minus,axiom,
! [X2: real] :
( ( sin @ real @ ( minus_minus @ real @ pi @ X2 ) )
= ( sin @ real @ X2 ) ) ).
% sin_pi_minus
thf(fact_4204_or__nat__numerals_I2_J,axiom,
! [Y3: num] :
( ( bit_se1065995026697491101ons_or @ nat @ ( suc @ ( zero_zero @ nat ) ) @ ( numeral_numeral @ nat @ ( bit1 @ Y3 ) ) )
= ( numeral_numeral @ nat @ ( bit1 @ Y3 ) ) ) ).
% or_nat_numerals(2)
thf(fact_4205_or__nat__numerals_I4_J,axiom,
! [X2: num] :
( ( bit_se1065995026697491101ons_or @ nat @ ( numeral_numeral @ nat @ ( bit1 @ X2 ) ) @ ( suc @ ( zero_zero @ nat ) ) )
= ( numeral_numeral @ nat @ ( bit1 @ X2 ) ) ) ).
% or_nat_numerals(4)
thf(fact_4206_scaleR__minus1__left,axiom,
! [A: $tType] :
( ( real_V4867850818363320053vector @ A )
=> ! [X2: A] :
( ( real_V8093663219630862766scaleR @ A @ ( uminus_uminus @ real @ ( one_one @ real ) ) @ X2 )
= ( uminus_uminus @ A @ X2 ) ) ) ).
% scaleR_minus1_left
thf(fact_4207_scaleR__collapse,axiom,
! [A: $tType] :
( ( real_V4867850818363320053vector @ A )
=> ! [U: real,A3: A] :
( ( plus_plus @ A @ ( real_V8093663219630862766scaleR @ A @ ( minus_minus @ real @ ( one_one @ real ) @ U ) @ A3 ) @ ( real_V8093663219630862766scaleR @ A @ U @ A3 ) )
= A3 ) ) ).
% scaleR_collapse
thf(fact_4208_cos__pi,axiom,
( ( cos @ real @ pi )
= ( uminus_uminus @ real @ ( one_one @ real ) ) ) ).
% cos_pi
thf(fact_4209_cos__periodic__pi,axiom,
! [X2: real] :
( ( cos @ real @ ( plus_plus @ real @ X2 @ pi ) )
= ( uminus_uminus @ real @ ( cos @ real @ X2 ) ) ) ).
% cos_periodic_pi
thf(fact_4210_cos__periodic__pi2,axiom,
! [X2: real] :
( ( cos @ real @ ( plus_plus @ real @ pi @ X2 ) )
= ( uminus_uminus @ real @ ( cos @ real @ X2 ) ) ) ).
% cos_periodic_pi2
thf(fact_4211_sin__periodic__pi,axiom,
! [X2: real] :
( ( sin @ real @ ( plus_plus @ real @ X2 @ pi ) )
= ( uminus_uminus @ real @ ( sin @ real @ X2 ) ) ) ).
% sin_periodic_pi
thf(fact_4212_sin__periodic__pi2,axiom,
! [X2: real] :
( ( sin @ real @ ( plus_plus @ real @ pi @ X2 ) )
= ( uminus_uminus @ real @ ( sin @ real @ X2 ) ) ) ).
% sin_periodic_pi2
thf(fact_4213_cos__pi__minus,axiom,
! [X2: real] :
( ( cos @ real @ ( minus_minus @ real @ pi @ X2 ) )
= ( uminus_uminus @ real @ ( cos @ real @ X2 ) ) ) ).
% cos_pi_minus
thf(fact_4214_cos__minus__pi,axiom,
! [X2: real] :
( ( cos @ real @ ( minus_minus @ real @ X2 @ pi ) )
= ( uminus_uminus @ real @ ( cos @ real @ X2 ) ) ) ).
% cos_minus_pi
thf(fact_4215_sin__minus__pi,axiom,
! [X2: real] :
( ( sin @ real @ ( minus_minus @ real @ X2 @ pi ) )
= ( uminus_uminus @ real @ ( sin @ real @ X2 ) ) ) ).
% sin_minus_pi
thf(fact_4216_or__nat__numerals_I3_J,axiom,
! [X2: num] :
( ( bit_se1065995026697491101ons_or @ nat @ ( numeral_numeral @ nat @ ( bit0 @ X2 ) ) @ ( suc @ ( zero_zero @ nat ) ) )
= ( numeral_numeral @ nat @ ( bit1 @ X2 ) ) ) ).
% or_nat_numerals(3)
thf(fact_4217_or__nat__numerals_I1_J,axiom,
! [Y3: num] :
( ( bit_se1065995026697491101ons_or @ nat @ ( suc @ ( zero_zero @ nat ) ) @ ( numeral_numeral @ nat @ ( bit0 @ Y3 ) ) )
= ( numeral_numeral @ nat @ ( bit1 @ Y3 ) ) ) ).
% or_nat_numerals(1)
thf(fact_4218_scaleR__times,axiom,
! [A: $tType] :
( ( real_V2191834092415804123ebra_1 @ A )
=> ! [U: num,W: num,A3: A] :
( ( real_V8093663219630862766scaleR @ A @ ( numeral_numeral @ real @ U ) @ ( times_times @ A @ ( numeral_numeral @ A @ W ) @ A3 ) )
= ( real_V8093663219630862766scaleR @ A @ ( times_times @ real @ ( numeral_numeral @ real @ U ) @ ( numeral_numeral @ real @ W ) ) @ A3 ) ) ) ).
% scaleR_times
thf(fact_4219_inverse__scaleR__times,axiom,
! [A: $tType] :
( ( real_V2191834092415804123ebra_1 @ A )
=> ! [V: num,W: num,A3: A] :
( ( real_V8093663219630862766scaleR @ A @ ( divide_divide @ real @ ( one_one @ real ) @ ( numeral_numeral @ real @ V ) ) @ ( times_times @ A @ ( numeral_numeral @ A @ W ) @ A3 ) )
= ( real_V8093663219630862766scaleR @ A @ ( divide_divide @ real @ ( numeral_numeral @ real @ W ) @ ( numeral_numeral @ real @ V ) ) @ A3 ) ) ) ).
% inverse_scaleR_times
thf(fact_4220_fraction__scaleR__times,axiom,
! [A: $tType] :
( ( real_V2191834092415804123ebra_1 @ A )
=> ! [U: num,V: num,W: num,A3: A] :
( ( real_V8093663219630862766scaleR @ A @ ( divide_divide @ real @ ( numeral_numeral @ real @ U ) @ ( numeral_numeral @ real @ V ) ) @ ( times_times @ A @ ( numeral_numeral @ A @ W ) @ A3 ) )
= ( real_V8093663219630862766scaleR @ A @ ( divide_divide @ real @ ( times_times @ real @ ( numeral_numeral @ real @ U ) @ ( numeral_numeral @ real @ W ) ) @ ( numeral_numeral @ real @ V ) ) @ A3 ) ) ) ).
% fraction_scaleR_times
thf(fact_4221_word__bitwise__1__simps_I8_J,axiom,
! [H6: $tType] :
( ( type_len @ H6 )
=> ! [A3: num] :
( ( bit_se1065995026697491101ons_or @ ( word @ H6 ) @ ( numeral_numeral @ ( word @ H6 ) @ A3 ) @ ( one_one @ ( word @ H6 ) ) )
= ( ring_1_of_int @ ( word @ H6 ) @ ( bit_se1065995026697491101ons_or @ int @ ( numeral_numeral @ int @ A3 ) @ ( one_one @ int ) ) ) ) ) ).
% word_bitwise_1_simps(8)
thf(fact_4222_word__bitwise__1__simps_I6_J,axiom,
! [F: $tType] :
( ( type_len @ F )
=> ! [B3: num] :
( ( bit_se1065995026697491101ons_or @ ( word @ F ) @ ( one_one @ ( word @ F ) ) @ ( numeral_numeral @ ( word @ F ) @ B3 ) )
= ( ring_1_of_int @ ( word @ F ) @ ( bit_se1065995026697491101ons_or @ int @ ( one_one @ int ) @ ( numeral_numeral @ int @ B3 ) ) ) ) ) ).
% word_bitwise_1_simps(6)
thf(fact_4223_scaleR__half__double,axiom,
! [A: $tType] :
( ( real_V4867850818363320053vector @ A )
=> ! [A3: A] :
( ( real_V8093663219630862766scaleR @ A @ ( divide_divide @ real @ ( one_one @ real ) @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) @ ( plus_plus @ A @ A3 @ A3 ) )
= A3 ) ) ).
% scaleR_half_double
thf(fact_4224_cos__pi__half,axiom,
( ( cos @ real @ ( divide_divide @ real @ pi @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) )
= ( zero_zero @ real ) ) ).
% cos_pi_half
thf(fact_4225_sin__two__pi,axiom,
( ( sin @ real @ ( times_times @ real @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) @ pi ) )
= ( zero_zero @ real ) ) ).
% sin_two_pi
thf(fact_4226_cos__two__pi,axiom,
( ( cos @ real @ ( times_times @ real @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) @ pi ) )
= ( one_one @ real ) ) ).
% cos_two_pi
thf(fact_4227_sin__pi__half,axiom,
( ( sin @ real @ ( divide_divide @ real @ pi @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) )
= ( one_one @ real ) ) ).
% sin_pi_half
thf(fact_4228_cos__periodic,axiom,
! [X2: real] :
( ( cos @ real @ ( plus_plus @ real @ X2 @ ( times_times @ real @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) @ pi ) ) )
= ( cos @ real @ X2 ) ) ).
% cos_periodic
thf(fact_4229_sin__periodic,axiom,
! [X2: real] :
( ( sin @ real @ ( plus_plus @ real @ X2 @ ( times_times @ real @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) @ pi ) ) )
= ( sin @ real @ X2 ) ) ).
% sin_periodic
thf(fact_4230_cos__2pi__minus,axiom,
! [X2: real] :
( ( cos @ real @ ( minus_minus @ real @ ( times_times @ real @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) @ pi ) @ X2 ) )
= ( cos @ real @ X2 ) ) ).
% cos_2pi_minus
thf(fact_4231_cos__npi,axiom,
! [N: nat] :
( ( cos @ real @ ( times_times @ real @ ( semiring_1_of_nat @ real @ N ) @ pi ) )
= ( power_power @ real @ ( uminus_uminus @ real @ ( one_one @ real ) ) @ N ) ) ).
% cos_npi
thf(fact_4232_cos__npi2,axiom,
! [N: nat] :
( ( cos @ real @ ( times_times @ real @ pi @ ( semiring_1_of_nat @ real @ N ) ) )
= ( power_power @ real @ ( uminus_uminus @ real @ ( one_one @ real ) ) @ N ) ) ).
% cos_npi2
thf(fact_4233_sin__2npi,axiom,
! [N: nat] :
( ( sin @ real @ ( times_times @ real @ ( times_times @ real @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) @ ( semiring_1_of_nat @ real @ N ) ) @ pi ) )
= ( zero_zero @ real ) ) ).
% sin_2npi
thf(fact_4234_cos__2npi,axiom,
! [N: nat] :
( ( cos @ real @ ( times_times @ real @ ( times_times @ real @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) @ ( semiring_1_of_nat @ real @ N ) ) @ pi ) )
= ( one_one @ real ) ) ).
% cos_2npi
thf(fact_4235_sin__2pi__minus,axiom,
! [X2: real] :
( ( sin @ real @ ( minus_minus @ real @ ( times_times @ real @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) @ pi ) @ X2 ) )
= ( uminus_uminus @ real @ ( sin @ real @ X2 ) ) ) ).
% sin_2pi_minus
thf(fact_4236_sin__int__2pin,axiom,
! [N: int] :
( ( sin @ real @ ( times_times @ real @ ( times_times @ real @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) @ pi ) @ ( ring_1_of_int @ real @ N ) ) )
= ( zero_zero @ real ) ) ).
% sin_int_2pin
thf(fact_4237_cos__int__2pin,axiom,
! [N: int] :
( ( cos @ real @ ( times_times @ real @ ( times_times @ real @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) @ pi ) @ ( ring_1_of_int @ real @ N ) ) )
= ( one_one @ real ) ) ).
% cos_int_2pin
thf(fact_4238_word__bitwise__1__simps_I7_J,axiom,
! [G3: $tType] :
( ( type_len @ G3 )
=> ! [B3: num] :
( ( bit_se1065995026697491101ons_or @ ( word @ G3 ) @ ( one_one @ ( word @ G3 ) ) @ ( uminus_uminus @ ( word @ G3 ) @ ( numeral_numeral @ ( word @ G3 ) @ B3 ) ) )
= ( ring_1_of_int @ ( word @ G3 ) @ ( bit_se1065995026697491101ons_or @ int @ ( one_one @ int ) @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ B3 ) ) ) ) ) ) ).
% word_bitwise_1_simps(7)
thf(fact_4239_word__bitwise__1__simps_I9_J,axiom,
! [I8: $tType] :
( ( type_len @ I8 )
=> ! [A3: num] :
( ( bit_se1065995026697491101ons_or @ ( word @ I8 ) @ ( uminus_uminus @ ( word @ I8 ) @ ( numeral_numeral @ ( word @ I8 ) @ A3 ) ) @ ( one_one @ ( word @ I8 ) ) )
= ( ring_1_of_int @ ( word @ I8 ) @ ( bit_se1065995026697491101ons_or @ int @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ A3 ) ) @ ( one_one @ int ) ) ) ) ) ).
% word_bitwise_1_simps(9)
thf(fact_4240_cos__3over2__pi,axiom,
( ( cos @ real @ ( times_times @ real @ ( divide_divide @ real @ ( numeral_numeral @ real @ ( bit1 @ one2 ) ) @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) @ pi ) )
= ( zero_zero @ real ) ) ).
% cos_3over2_pi
thf(fact_4241_sin__3over2__pi,axiom,
( ( sin @ real @ ( times_times @ real @ ( divide_divide @ real @ ( numeral_numeral @ real @ ( bit1 @ one2 ) ) @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) @ pi ) )
= ( uminus_uminus @ real @ ( one_one @ real ) ) ) ).
% sin_3over2_pi
thf(fact_4242_cos__npi__int,axiom,
! [N: int] :
( ( ( dvd_dvd @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ N )
=> ( ( cos @ real @ ( times_times @ real @ pi @ ( ring_1_of_int @ real @ N ) ) )
= ( one_one @ real ) ) )
& ( ~ ( dvd_dvd @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ N )
=> ( ( cos @ real @ ( times_times @ real @ pi @ ( ring_1_of_int @ real @ N ) ) )
= ( uminus_uminus @ real @ ( one_one @ real ) ) ) ) ) ).
% cos_npi_int
thf(fact_4243_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_4244_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_4245_le__word__or2,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [X2: word @ A,Y3: word @ A] : ( ord_less_eq @ ( word @ A ) @ X2 @ ( bit_se1065995026697491101ons_or @ ( word @ A ) @ X2 @ Y3 ) ) ) ).
% le_word_or2
thf(fact_4246_le__word__or1,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [C3: word @ A,Y3: word @ A] : ( ord_less_eq @ ( word @ A ) @ C3 @ ( bit_se1065995026697491101ons_or @ ( word @ A ) @ Y3 @ C3 ) ) ) ).
% le_word_or1
thf(fact_4247_scaleR__right__distrib,axiom,
! [A: $tType] :
( ( real_V4867850818363320053vector @ A )
=> ! [A3: real,X2: A,Y3: A] :
( ( real_V8093663219630862766scaleR @ A @ A3 @ ( plus_plus @ A @ X2 @ Y3 ) )
= ( plus_plus @ A @ ( real_V8093663219630862766scaleR @ A @ A3 @ X2 ) @ ( real_V8093663219630862766scaleR @ A @ A3 @ Y3 ) ) ) ) ).
% scaleR_right_distrib
thf(fact_4248_scaleR__right__diff__distrib,axiom,
! [A: $tType] :
( ( real_V4867850818363320053vector @ A )
=> ! [A3: real,X2: A,Y3: A] :
( ( real_V8093663219630862766scaleR @ A @ A3 @ ( minus_minus @ A @ X2 @ Y3 ) )
= ( minus_minus @ A @ ( real_V8093663219630862766scaleR @ A @ A3 @ X2 ) @ ( real_V8093663219630862766scaleR @ A @ A3 @ Y3 ) ) ) ) ).
% scaleR_right_diff_distrib
thf(fact_4249_scaleR__sum__right,axiom,
! [A: $tType,C: $tType] :
( ( real_V4867850818363320053vector @ A )
=> ! [A3: real,F2: C > A,A2: set @ C] :
( ( real_V8093663219630862766scaleR @ A @ A3 @ ( groups7311177749621191930dd_sum @ C @ A @ F2 @ A2 ) )
= ( groups7311177749621191930dd_sum @ C @ A
@ ^ [X3: C] : ( real_V8093663219630862766scaleR @ A @ A3 @ ( F2 @ X3 ) )
@ A2 ) ) ) ).
% scaleR_sum_right
thf(fact_4250_scaleR__right_Osum,axiom,
! [A: $tType,C: $tType] :
( ( real_V4867850818363320053vector @ A )
=> ! [A3: real,G: C > A,A2: set @ C] :
( ( real_V8093663219630862766scaleR @ A @ A3 @ ( groups7311177749621191930dd_sum @ C @ A @ G @ A2 ) )
= ( groups7311177749621191930dd_sum @ C @ A
@ ^ [X3: C] : ( real_V8093663219630862766scaleR @ A @ A3 @ ( G @ X3 ) )
@ A2 ) ) ) ).
% scaleR_right.sum
thf(fact_4251_sums__scaleR__right,axiom,
! [B: $tType] :
( ( real_V822414075346904944vector @ B )
=> ! [X: nat > B,A3: B,R2: real] :
( ( sums @ B @ X @ A3 )
=> ( sums @ B
@ ^ [N4: nat] : ( real_V8093663219630862766scaleR @ B @ R2 @ ( X @ N4 ) )
@ ( real_V8093663219630862766scaleR @ B @ R2 @ A3 ) ) ) ) ).
% sums_scaleR_right
thf(fact_4252_scaleR__left__distrib,axiom,
! [A: $tType] :
( ( real_V4867850818363320053vector @ A )
=> ! [A3: real,B3: real,X2: A] :
( ( real_V8093663219630862766scaleR @ A @ ( plus_plus @ real @ A3 @ B3 ) @ X2 )
= ( plus_plus @ A @ ( real_V8093663219630862766scaleR @ A @ A3 @ X2 ) @ ( real_V8093663219630862766scaleR @ A @ B3 @ X2 ) ) ) ) ).
% scaleR_left_distrib
thf(fact_4253_scaleR__left_Oadd,axiom,
! [A: $tType] :
( ( real_V4867850818363320053vector @ A )
=> ! [X2: real,Y3: real,Xa: A] :
( ( real_V8093663219630862766scaleR @ A @ ( plus_plus @ real @ X2 @ Y3 ) @ Xa )
= ( plus_plus @ A @ ( real_V8093663219630862766scaleR @ A @ X2 @ Xa ) @ ( real_V8093663219630862766scaleR @ A @ Y3 @ Xa ) ) ) ) ).
% scaleR_left.add
thf(fact_4254_scaleR__left__diff__distrib,axiom,
! [A: $tType] :
( ( real_V4867850818363320053vector @ A )
=> ! [A3: real,B3: real,X2: A] :
( ( real_V8093663219630862766scaleR @ A @ ( minus_minus @ real @ A3 @ B3 ) @ X2 )
= ( minus_minus @ A @ ( real_V8093663219630862766scaleR @ A @ A3 @ X2 ) @ ( real_V8093663219630862766scaleR @ A @ B3 @ X2 ) ) ) ) ).
% scaleR_left_diff_distrib
thf(fact_4255_scaleR__left_Odiff,axiom,
! [A: $tType] :
( ( real_V4867850818363320053vector @ A )
=> ! [X2: real,Y3: real,Xa: A] :
( ( real_V8093663219630862766scaleR @ A @ ( minus_minus @ real @ X2 @ Y3 ) @ Xa )
= ( minus_minus @ A @ ( real_V8093663219630862766scaleR @ A @ X2 @ Xa ) @ ( real_V8093663219630862766scaleR @ A @ Y3 @ Xa ) ) ) ) ).
% scaleR_left.diff
thf(fact_4256_pi__ge__zero,axiom,
ord_less_eq @ real @ ( zero_zero @ real ) @ pi ).
% pi_ge_zero
thf(fact_4257_scaleR__left_Osum,axiom,
! [A: $tType,C: $tType] :
( ( real_V4867850818363320053vector @ A )
=> ! [G: C > real,A2: set @ C,X2: A] :
( ( real_V8093663219630862766scaleR @ A @ ( groups7311177749621191930dd_sum @ C @ real @ G @ A2 ) @ X2 )
= ( groups7311177749621191930dd_sum @ C @ A
@ ^ [X3: C] : ( real_V8093663219630862766scaleR @ A @ ( G @ X3 ) @ X2 )
@ A2 ) ) ) ).
% scaleR_left.sum
thf(fact_4258_scaleR__sum__left,axiom,
! [A: $tType,C: $tType] :
( ( real_V4867850818363320053vector @ A )
=> ! [F2: C > real,A2: set @ C,X2: A] :
( ( real_V8093663219630862766scaleR @ A @ ( groups7311177749621191930dd_sum @ C @ real @ F2 @ A2 ) @ X2 )
= ( groups7311177749621191930dd_sum @ C @ A
@ ^ [A6: C] : ( real_V8093663219630862766scaleR @ A @ ( F2 @ A6 ) @ X2 )
@ A2 ) ) ) ).
% scaleR_sum_left
thf(fact_4259_sums__scaleR__left,axiom,
! [B: $tType] :
( ( real_V822414075346904944vector @ B )
=> ! [X: nat > real,A3: real,X2: B] :
( ( sums @ real @ X @ A3 )
=> ( sums @ B
@ ^ [N4: nat] : ( real_V8093663219630862766scaleR @ B @ ( X @ N4 ) @ X2 )
@ ( real_V8093663219630862766scaleR @ B @ A3 @ X2 ) ) ) ) ).
% sums_scaleR_left
thf(fact_4260_scaleR__right__mono__neg,axiom,
! [A: $tType] :
( ( real_V5355595471888546746vector @ A )
=> ! [B3: real,A3: real,C3: A] :
( ( ord_less_eq @ real @ B3 @ A3 )
=> ( ( ord_less_eq @ A @ C3 @ ( zero_zero @ A ) )
=> ( ord_less_eq @ A @ ( real_V8093663219630862766scaleR @ A @ A3 @ C3 ) @ ( real_V8093663219630862766scaleR @ A @ B3 @ C3 ) ) ) ) ) ).
% scaleR_right_mono_neg
thf(fact_4261_scaleR__right__mono,axiom,
! [A: $tType] :
( ( real_V5355595471888546746vector @ A )
=> ! [A3: real,B3: real,X2: A] :
( ( ord_less_eq @ real @ A3 @ B3 )
=> ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ X2 )
=> ( ord_less_eq @ A @ ( real_V8093663219630862766scaleR @ A @ A3 @ X2 ) @ ( real_V8093663219630862766scaleR @ A @ B3 @ X2 ) ) ) ) ) ).
% scaleR_right_mono
thf(fact_4262_scaleR__le__cancel__left__pos,axiom,
! [A: $tType] :
( ( real_V5355595471888546746vector @ A )
=> ! [C3: real,A3: A,B3: A] :
( ( ord_less @ real @ ( zero_zero @ real ) @ C3 )
=> ( ( ord_less_eq @ A @ ( real_V8093663219630862766scaleR @ A @ C3 @ A3 ) @ ( real_V8093663219630862766scaleR @ A @ C3 @ B3 ) )
= ( ord_less_eq @ A @ A3 @ B3 ) ) ) ) ).
% scaleR_le_cancel_left_pos
thf(fact_4263_scaleR__le__cancel__left__neg,axiom,
! [A: $tType] :
( ( real_V5355595471888546746vector @ A )
=> ! [C3: real,A3: A,B3: A] :
( ( ord_less @ real @ C3 @ ( zero_zero @ real ) )
=> ( ( ord_less_eq @ A @ ( real_V8093663219630862766scaleR @ A @ C3 @ A3 ) @ ( real_V8093663219630862766scaleR @ A @ C3 @ B3 ) )
= ( ord_less_eq @ A @ B3 @ A3 ) ) ) ) ).
% scaleR_le_cancel_left_neg
thf(fact_4264_scaleR__le__cancel__left,axiom,
! [A: $tType] :
( ( real_V5355595471888546746vector @ A )
=> ! [C3: real,A3: A,B3: A] :
( ( ord_less_eq @ A @ ( real_V8093663219630862766scaleR @ A @ C3 @ A3 ) @ ( real_V8093663219630862766scaleR @ A @ C3 @ B3 ) )
= ( ( ( ord_less @ real @ ( zero_zero @ real ) @ C3 )
=> ( ord_less_eq @ A @ A3 @ B3 ) )
& ( ( ord_less @ real @ C3 @ ( zero_zero @ real ) )
=> ( ord_less_eq @ A @ B3 @ A3 ) ) ) ) ) ).
% scaleR_le_cancel_left
thf(fact_4265_scaleR__left__mono__neg,axiom,
! [A: $tType] :
( ( real_V5355595471888546746vector @ A )
=> ! [B3: A,A3: A,C3: real] :
( ( ord_less_eq @ A @ B3 @ A3 )
=> ( ( ord_less_eq @ real @ C3 @ ( zero_zero @ real ) )
=> ( ord_less_eq @ A @ ( real_V8093663219630862766scaleR @ A @ C3 @ A3 ) @ ( real_V8093663219630862766scaleR @ A @ C3 @ B3 ) ) ) ) ) ).
% scaleR_left_mono_neg
thf(fact_4266_scaleR__left__mono,axiom,
! [A: $tType] :
( ( real_V5355595471888546746vector @ A )
=> ! [X2: A,Y3: A,A3: real] :
( ( ord_less_eq @ A @ X2 @ Y3 )
=> ( ( ord_less_eq @ real @ ( zero_zero @ real ) @ A3 )
=> ( ord_less_eq @ A @ ( real_V8093663219630862766scaleR @ A @ A3 @ X2 ) @ ( real_V8093663219630862766scaleR @ A @ A3 @ Y3 ) ) ) ) ) ).
% scaleR_left_mono
thf(fact_4267_Real__Vector__Spaces_Ole__add__iff2,axiom,
! [A: $tType] :
( ( real_V5355595471888546746vector @ A )
=> ! [A3: real,E: A,C3: A,B3: real,D2: A] :
( ( ord_less_eq @ A @ ( plus_plus @ A @ ( real_V8093663219630862766scaleR @ A @ A3 @ E ) @ C3 ) @ ( plus_plus @ A @ ( real_V8093663219630862766scaleR @ A @ B3 @ E ) @ D2 ) )
= ( ord_less_eq @ A @ C3 @ ( plus_plus @ A @ ( real_V8093663219630862766scaleR @ A @ ( minus_minus @ real @ B3 @ A3 ) @ E ) @ D2 ) ) ) ) ).
% Real_Vector_Spaces.le_add_iff2
thf(fact_4268_Real__Vector__Spaces_Ole__add__iff1,axiom,
! [A: $tType] :
( ( real_V5355595471888546746vector @ A )
=> ! [A3: real,E: A,C3: A,B3: real,D2: A] :
( ( ord_less_eq @ A @ ( plus_plus @ A @ ( real_V8093663219630862766scaleR @ A @ A3 @ E ) @ C3 ) @ ( plus_plus @ A @ ( real_V8093663219630862766scaleR @ A @ B3 @ E ) @ D2 ) )
= ( ord_less_eq @ A @ ( plus_plus @ A @ ( real_V8093663219630862766scaleR @ A @ ( minus_minus @ real @ A3 @ B3 ) @ E ) @ C3 ) @ D2 ) ) ) ).
% Real_Vector_Spaces.le_add_iff1
thf(fact_4269_cos__monotone__0__pi__le,axiom,
! [Y3: real,X2: real] :
( ( ord_less_eq @ real @ ( zero_zero @ real ) @ Y3 )
=> ( ( ord_less_eq @ real @ Y3 @ X2 )
=> ( ( ord_less_eq @ real @ X2 @ pi )
=> ( ord_less_eq @ real @ ( cos @ real @ X2 ) @ ( cos @ real @ Y3 ) ) ) ) ) ).
% cos_monotone_0_pi_le
thf(fact_4270_cos__mono__le__eq,axiom,
! [X2: real,Y3: real] :
( ( ord_less_eq @ real @ ( zero_zero @ real ) @ X2 )
=> ( ( ord_less_eq @ real @ X2 @ pi )
=> ( ( ord_less_eq @ real @ ( zero_zero @ real ) @ Y3 )
=> ( ( ord_less_eq @ real @ Y3 @ pi )
=> ( ( ord_less_eq @ real @ ( cos @ real @ X2 ) @ ( cos @ real @ Y3 ) )
= ( ord_less_eq @ real @ Y3 @ X2 ) ) ) ) ) ) ).
% cos_mono_le_eq
thf(fact_4271_cos__inj__pi,axiom,
! [X2: real,Y3: real] :
( ( ord_less_eq @ real @ ( zero_zero @ real ) @ X2 )
=> ( ( ord_less_eq @ real @ X2 @ pi )
=> ( ( ord_less_eq @ real @ ( zero_zero @ real ) @ Y3 )
=> ( ( ord_less_eq @ real @ Y3 @ pi )
=> ( ( ( cos @ real @ X2 )
= ( cos @ real @ Y3 ) )
=> ( X2 = Y3 ) ) ) ) ) ) ).
% cos_inj_pi
thf(fact_4272_sin__ge__zero,axiom,
! [X2: real] :
( ( ord_less_eq @ real @ ( zero_zero @ real ) @ X2 )
=> ( ( ord_less_eq @ real @ X2 @ pi )
=> ( ord_less_eq @ real @ ( zero_zero @ real ) @ ( sin @ real @ X2 ) ) ) ) ).
% sin_ge_zero
thf(fact_4273_zero__le__scaleR__iff,axiom,
! [A: $tType] :
( ( real_V5355595471888546746vector @ A )
=> ! [A3: real,B3: A] :
( ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( real_V8093663219630862766scaleR @ A @ A3 @ B3 ) )
= ( ( ( ord_less @ real @ ( zero_zero @ real ) @ A3 )
& ( ord_less_eq @ A @ ( zero_zero @ A ) @ B3 ) )
| ( ( ord_less @ real @ A3 @ ( zero_zero @ real ) )
& ( ord_less_eq @ A @ B3 @ ( zero_zero @ A ) ) )
| ( A3
= ( zero_zero @ real ) ) ) ) ) ).
% zero_le_scaleR_iff
thf(fact_4274_scaleR__le__0__iff,axiom,
! [A: $tType] :
( ( real_V5355595471888546746vector @ A )
=> ! [A3: real,B3: A] :
( ( ord_less_eq @ A @ ( real_V8093663219630862766scaleR @ A @ A3 @ B3 ) @ ( zero_zero @ A ) )
= ( ( ( ord_less @ real @ ( zero_zero @ real ) @ A3 )
& ( ord_less_eq @ A @ B3 @ ( zero_zero @ A ) ) )
| ( ( ord_less @ real @ A3 @ ( zero_zero @ real ) )
& ( ord_less_eq @ A @ ( zero_zero @ A ) @ B3 ) )
| ( A3
= ( zero_zero @ real ) ) ) ) ) ).
% scaleR_le_0_iff
thf(fact_4275_scaleR__nonpos__nonpos,axiom,
! [A: $tType] :
( ( real_V5355595471888546746vector @ A )
=> ! [A3: real,B3: A] :
( ( ord_less_eq @ real @ A3 @ ( zero_zero @ real ) )
=> ( ( ord_less_eq @ A @ B3 @ ( zero_zero @ A ) )
=> ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( real_V8093663219630862766scaleR @ A @ A3 @ B3 ) ) ) ) ) ).
% scaleR_nonpos_nonpos
thf(fact_4276_scaleR__nonpos__nonneg,axiom,
! [A: $tType] :
( ( real_V5355595471888546746vector @ A )
=> ! [A3: real,X2: A] :
( ( ord_less_eq @ real @ A3 @ ( zero_zero @ real ) )
=> ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ X2 )
=> ( ord_less_eq @ A @ ( real_V8093663219630862766scaleR @ A @ A3 @ X2 ) @ ( zero_zero @ A ) ) ) ) ) ).
% scaleR_nonpos_nonneg
thf(fact_4277_scaleR__nonneg__nonpos,axiom,
! [A: $tType] :
( ( real_V5355595471888546746vector @ A )
=> ! [A3: real,X2: A] :
( ( ord_less_eq @ real @ ( zero_zero @ real ) @ A3 )
=> ( ( ord_less_eq @ A @ X2 @ ( zero_zero @ A ) )
=> ( ord_less_eq @ A @ ( real_V8093663219630862766scaleR @ A @ A3 @ X2 ) @ ( zero_zero @ A ) ) ) ) ) ).
% scaleR_nonneg_nonpos
thf(fact_4278_scaleR__nonneg__nonneg,axiom,
! [A: $tType] :
( ( real_V5355595471888546746vector @ A )
=> ! [A3: real,X2: A] :
( ( ord_less_eq @ real @ ( zero_zero @ real ) @ A3 )
=> ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ X2 )
=> ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( real_V8093663219630862766scaleR @ A @ A3 @ X2 ) ) ) ) ) ).
% scaleR_nonneg_nonneg
thf(fact_4279_split__scaleR__pos__le,axiom,
! [A: $tType] :
( ( real_V5355595471888546746vector @ A )
=> ! [A3: real,B3: A] :
( ( ( ( ord_less_eq @ real @ ( zero_zero @ real ) @ A3 )
& ( ord_less_eq @ A @ ( zero_zero @ A ) @ B3 ) )
| ( ( ord_less_eq @ real @ A3 @ ( zero_zero @ real ) )
& ( ord_less_eq @ A @ B3 @ ( zero_zero @ A ) ) ) )
=> ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( real_V8093663219630862766scaleR @ A @ A3 @ B3 ) ) ) ) ).
% split_scaleR_pos_le
thf(fact_4280_split__scaleR__neg__le,axiom,
! [A: $tType] :
( ( real_V5355595471888546746vector @ A )
=> ! [A3: real,X2: A] :
( ( ( ( ord_less_eq @ real @ ( zero_zero @ real ) @ A3 )
& ( ord_less_eq @ A @ X2 @ ( zero_zero @ A ) ) )
| ( ( ord_less_eq @ real @ A3 @ ( zero_zero @ real ) )
& ( ord_less_eq @ A @ ( zero_zero @ A ) @ X2 ) ) )
=> ( ord_less_eq @ A @ ( real_V8093663219630862766scaleR @ A @ A3 @ X2 ) @ ( zero_zero @ A ) ) ) ) ).
% split_scaleR_neg_le
thf(fact_4281_scaleR__mono_H,axiom,
! [A: $tType] :
( ( real_V5355595471888546746vector @ A )
=> ! [A3: real,B3: real,C3: A,D2: A] :
( ( ord_less_eq @ real @ A3 @ B3 )
=> ( ( ord_less_eq @ A @ C3 @ D2 )
=> ( ( ord_less_eq @ real @ ( zero_zero @ real ) @ A3 )
=> ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ C3 )
=> ( ord_less_eq @ A @ ( real_V8093663219630862766scaleR @ A @ A3 @ C3 ) @ ( real_V8093663219630862766scaleR @ A @ B3 @ D2 ) ) ) ) ) ) ) ).
% scaleR_mono'
thf(fact_4282_scaleR__mono,axiom,
! [A: $tType] :
( ( real_V5355595471888546746vector @ A )
=> ! [A3: real,B3: real,X2: A,Y3: A] :
( ( ord_less_eq @ real @ A3 @ B3 )
=> ( ( ord_less_eq @ A @ X2 @ Y3 )
=> ( ( ord_less_eq @ real @ ( zero_zero @ real ) @ B3 )
=> ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ X2 )
=> ( ord_less_eq @ A @ ( real_V8093663219630862766scaleR @ A @ A3 @ X2 ) @ ( real_V8093663219630862766scaleR @ A @ B3 @ Y3 ) ) ) ) ) ) ) ).
% scaleR_mono
thf(fact_4283_scaleR__2,axiom,
! [A: $tType] :
( ( real_V4867850818363320053vector @ A )
=> ! [X2: A] :
( ( real_V8093663219630862766scaleR @ A @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) @ X2 )
= ( plus_plus @ A @ X2 @ X2 ) ) ) ).
% scaleR_2
thf(fact_4284_scaleR__left__le__one__le,axiom,
! [A: $tType] :
( ( real_V5355595471888546746vector @ A )
=> ! [X2: A,A3: real] :
( ( ord_less_eq @ A @ ( zero_zero @ A ) @ X2 )
=> ( ( ord_less_eq @ real @ A3 @ ( one_one @ real ) )
=> ( ord_less_eq @ A @ ( real_V8093663219630862766scaleR @ A @ A3 @ X2 ) @ X2 ) ) ) ) ).
% scaleR_left_le_one_le
thf(fact_4285_pi__less__4,axiom,
ord_less @ real @ pi @ ( numeral_numeral @ real @ ( bit0 @ ( bit0 @ one2 ) ) ) ).
% pi_less_4
thf(fact_4286_pi__ge__two,axiom,
ord_less_eq @ real @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) @ pi ).
% pi_ge_two
thf(fact_4287_pi__half__neq__two,axiom,
( ( divide_divide @ real @ pi @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) )
!= ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) ).
% pi_half_neq_two
thf(fact_4288_cos__monotone__0__pi,axiom,
! [Y3: real,X2: real] :
( ( ord_less_eq @ real @ ( zero_zero @ real ) @ Y3 )
=> ( ( ord_less @ real @ Y3 @ X2 )
=> ( ( ord_less_eq @ real @ X2 @ pi )
=> ( ord_less @ real @ ( cos @ real @ X2 ) @ ( cos @ real @ Y3 ) ) ) ) ) ).
% cos_monotone_0_pi
thf(fact_4289_cos__mono__less__eq,axiom,
! [X2: real,Y3: real] :
( ( ord_less_eq @ real @ ( zero_zero @ real ) @ X2 )
=> ( ( ord_less_eq @ real @ X2 @ pi )
=> ( ( ord_less_eq @ real @ ( zero_zero @ real ) @ Y3 )
=> ( ( ord_less_eq @ real @ Y3 @ pi )
=> ( ( ord_less @ real @ ( cos @ real @ X2 ) @ ( cos @ real @ Y3 ) )
= ( ord_less @ real @ Y3 @ X2 ) ) ) ) ) ) ).
% cos_mono_less_eq
thf(fact_4290_cos__monotone__minus__pi__0_H,axiom,
! [Y3: real,X2: real] :
( ( ord_less_eq @ real @ ( uminus_uminus @ real @ pi ) @ Y3 )
=> ( ( ord_less_eq @ real @ Y3 @ X2 )
=> ( ( ord_less_eq @ real @ X2 @ ( zero_zero @ real ) )
=> ( ord_less_eq @ real @ ( cos @ real @ Y3 ) @ ( cos @ real @ X2 ) ) ) ) ) ).
% cos_monotone_minus_pi_0'
thf(fact_4291_sin__converges,axiom,
! [A: $tType] :
( ( ( real_Vector_banach @ A )
& ( real_V2822296259951069270ebra_1 @ A ) )
=> ! [X2: A] :
( sums @ A
@ ^ [N4: nat] : ( real_V8093663219630862766scaleR @ A @ ( sin_coeff @ N4 ) @ ( power_power @ A @ X2 @ N4 ) )
@ ( sin @ A @ X2 ) ) ) ).
% sin_converges
thf(fact_4292_cos__converges,axiom,
! [A: $tType] :
( ( ( real_Vector_banach @ A )
& ( real_V2822296259951069270ebra_1 @ A ) )
=> ! [X2: A] :
( sums @ A
@ ^ [N4: nat] : ( real_V8093663219630862766scaleR @ A @ ( cos_coeff @ N4 ) @ ( power_power @ A @ X2 @ N4 ) )
@ ( cos @ A @ X2 ) ) ) ).
% cos_converges
thf(fact_4293_pi__half__neq__zero,axiom,
( ( divide_divide @ real @ pi @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) )
!= ( zero_zero @ real ) ) ).
% pi_half_neq_zero
thf(fact_4294_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_4295_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_4296_cos__monotone__minus__pi__0,axiom,
! [Y3: real,X2: real] :
( ( ord_less_eq @ real @ ( uminus_uminus @ real @ pi ) @ Y3 )
=> ( ( ord_less @ real @ Y3 @ X2 )
=> ( ( ord_less_eq @ real @ X2 @ ( zero_zero @ real ) )
=> ( ord_less @ real @ ( cos @ real @ Y3 ) @ ( cos @ real @ X2 ) ) ) ) ) ).
% cos_monotone_minus_pi_0
thf(fact_4297_cos__total,axiom,
! [Y3: real] :
( ( ord_less_eq @ real @ ( uminus_uminus @ real @ ( one_one @ real ) ) @ Y3 )
=> ( ( ord_less_eq @ real @ Y3 @ ( one_one @ real ) )
=> ? [X4: real] :
( ( ord_less_eq @ real @ ( zero_zero @ real ) @ X4 )
& ( ord_less_eq @ real @ X4 @ pi )
& ( ( cos @ real @ X4 )
= Y3 )
& ! [Y5: real] :
( ( ( ord_less_eq @ real @ ( zero_zero @ real ) @ Y5 )
& ( ord_less_eq @ real @ Y5 @ pi )
& ( ( cos @ real @ Y5 )
= Y3 ) )
=> ( Y5 = X4 ) ) ) ) ) ).
% cos_total
thf(fact_4298_sincos__principal__value,axiom,
! [X2: real] :
? [Y4: real] :
( ( ord_less @ real @ ( uminus_uminus @ real @ pi ) @ Y4 )
& ( ord_less_eq @ real @ Y4 @ pi )
& ( ( sin @ real @ Y4 )
= ( sin @ real @ X2 ) )
& ( ( cos @ real @ Y4 )
= ( cos @ real @ X2 ) ) ) ).
% sincos_principal_value
thf(fact_4299_sin__minus__converges,axiom,
! [A: $tType] :
( ( ( real_Vector_banach @ A )
& ( real_V2822296259951069270ebra_1 @ A ) )
=> ! [X2: A] :
( sums @ A
@ ^ [N4: nat] : ( uminus_uminus @ A @ ( real_V8093663219630862766scaleR @ A @ ( sin_coeff @ N4 ) @ ( power_power @ A @ ( uminus_uminus @ A @ X2 ) @ N4 ) ) )
@ ( sin @ A @ X2 ) ) ) ).
% sin_minus_converges
thf(fact_4300_cos__minus__converges,axiom,
! [A: $tType] :
( ( ( real_Vector_banach @ A )
& ( real_V2822296259951069270ebra_1 @ A ) )
=> ! [X2: A] :
( sums @ A
@ ^ [N4: nat] : ( real_V8093663219630862766scaleR @ A @ ( cos_coeff @ N4 ) @ ( power_power @ A @ ( uminus_uminus @ A @ X2 ) @ N4 ) )
@ ( cos @ A @ X2 ) ) ) ).
% cos_minus_converges
thf(fact_4301_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_4302_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_4303_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_4304_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_4305_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_4306_cos__gt__zero,axiom,
! [X2: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ X2 )
=> ( ( ord_less @ real @ X2 @ ( divide_divide @ real @ pi @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) )
=> ( ord_less @ real @ ( zero_zero @ real ) @ ( cos @ real @ X2 ) ) ) ) ).
% cos_gt_zero
thf(fact_4307_sin__gt__zero2,axiom,
! [X2: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ X2 )
=> ( ( ord_less @ real @ X2 @ ( divide_divide @ real @ pi @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) )
=> ( ord_less @ real @ ( zero_zero @ real ) @ ( sin @ real @ X2 ) ) ) ) ).
% sin_gt_zero2
thf(fact_4308_sin__lt__zero,axiom,
! [X2: real] :
( ( ord_less @ real @ pi @ X2 )
=> ( ( ord_less @ real @ X2 @ ( times_times @ real @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) @ pi ) )
=> ( ord_less @ real @ ( sin @ real @ X2 ) @ ( zero_zero @ real ) ) ) ) ).
% sin_lt_zero
thf(fact_4309_cos__60,axiom,
( ( cos @ real @ ( divide_divide @ real @ pi @ ( numeral_numeral @ real @ ( bit1 @ one2 ) ) ) )
= ( divide_divide @ real @ ( one_one @ real ) @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) ) ).
% cos_60
thf(fact_4310_sin__30,axiom,
( ( sin @ real @ ( divide_divide @ real @ pi @ ( numeral_numeral @ real @ ( bit0 @ ( bit1 @ one2 ) ) ) ) )
= ( divide_divide @ real @ ( one_one @ real ) @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) ) ).
% sin_30
thf(fact_4311_sin__monotone__2pi__le,axiom,
! [Y3: real,X2: real] :
( ( ord_less_eq @ real @ ( uminus_uminus @ real @ ( divide_divide @ real @ pi @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) ) @ Y3 )
=> ( ( ord_less_eq @ real @ Y3 @ X2 )
=> ( ( ord_less_eq @ real @ X2 @ ( divide_divide @ real @ pi @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) )
=> ( ord_less_eq @ real @ ( sin @ real @ Y3 ) @ ( sin @ real @ X2 ) ) ) ) ) ).
% sin_monotone_2pi_le
thf(fact_4312_sin__mono__le__eq,axiom,
! [X2: real,Y3: real] :
( ( ord_less_eq @ real @ ( uminus_uminus @ real @ ( divide_divide @ real @ pi @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) ) @ X2 )
=> ( ( ord_less_eq @ real @ X2 @ ( divide_divide @ real @ pi @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) )
=> ( ( ord_less_eq @ real @ ( uminus_uminus @ real @ ( divide_divide @ real @ pi @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) ) @ Y3 )
=> ( ( ord_less_eq @ real @ Y3 @ ( divide_divide @ real @ pi @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) )
=> ( ( ord_less_eq @ real @ ( sin @ real @ X2 ) @ ( sin @ real @ Y3 ) )
= ( ord_less_eq @ real @ X2 @ Y3 ) ) ) ) ) ) ).
% sin_mono_le_eq
thf(fact_4313_sin__inj__pi,axiom,
! [X2: real,Y3: real] :
( ( ord_less_eq @ real @ ( uminus_uminus @ real @ ( divide_divide @ real @ pi @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) ) @ X2 )
=> ( ( ord_less_eq @ real @ X2 @ ( divide_divide @ real @ pi @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) )
=> ( ( ord_less_eq @ real @ ( uminus_uminus @ real @ ( divide_divide @ real @ pi @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) ) @ Y3 )
=> ( ( ord_less_eq @ real @ Y3 @ ( divide_divide @ real @ pi @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) )
=> ( ( ( sin @ real @ X2 )
= ( sin @ real @ Y3 ) )
=> ( X2 = Y3 ) ) ) ) ) ) ).
% sin_inj_pi
thf(fact_4314_cos__one__2pi__int,axiom,
! [X2: real] :
( ( ( cos @ real @ X2 )
= ( one_one @ real ) )
= ( ? [X3: int] :
( X2
= ( times_times @ real @ ( times_times @ real @ ( ring_1_of_int @ real @ X3 ) @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) @ pi ) ) ) ) ).
% cos_one_2pi_int
thf(fact_4315_sin__le__zero,axiom,
! [X2: real] :
( ( ord_less_eq @ real @ pi @ X2 )
=> ( ( ord_less @ real @ X2 @ ( times_times @ real @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) @ pi ) )
=> ( ord_less_eq @ real @ ( sin @ real @ X2 ) @ ( zero_zero @ real ) ) ) ) ).
% sin_le_zero
thf(fact_4316_cos__gt__zero__pi,axiom,
! [X2: real] :
( ( ord_less @ real @ ( uminus_uminus @ real @ ( divide_divide @ real @ pi @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) ) @ X2 )
=> ( ( ord_less @ real @ X2 @ ( divide_divide @ real @ pi @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) )
=> ( ord_less @ real @ ( zero_zero @ real ) @ ( cos @ real @ X2 ) ) ) ) ).
% cos_gt_zero_pi
thf(fact_4317_sin__less__zero,axiom,
! [X2: real] :
( ( ord_less @ real @ ( divide_divide @ real @ ( uminus_uminus @ real @ pi ) @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) @ X2 )
=> ( ( ord_less @ real @ X2 @ ( zero_zero @ real ) )
=> ( ord_less @ real @ ( sin @ real @ X2 ) @ ( zero_zero @ real ) ) ) ) ).
% sin_less_zero
thf(fact_4318_cos__ge__zero,axiom,
! [X2: real] :
( ( ord_less_eq @ real @ ( uminus_uminus @ real @ ( divide_divide @ real @ pi @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) ) @ X2 )
=> ( ( ord_less_eq @ real @ X2 @ ( divide_divide @ real @ pi @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) )
=> ( ord_less_eq @ real @ ( zero_zero @ real ) @ ( cos @ real @ X2 ) ) ) ) ).
% cos_ge_zero
thf(fact_4319_sin__monotone__2pi,axiom,
! [Y3: real,X2: real] :
( ( ord_less_eq @ real @ ( uminus_uminus @ real @ ( divide_divide @ real @ pi @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) ) @ Y3 )
=> ( ( ord_less @ real @ Y3 @ X2 )
=> ( ( ord_less_eq @ real @ X2 @ ( divide_divide @ real @ pi @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) )
=> ( ord_less @ real @ ( sin @ real @ Y3 ) @ ( sin @ real @ X2 ) ) ) ) ) ).
% sin_monotone_2pi
thf(fact_4320_sin__mono__less__eq,axiom,
! [X2: real,Y3: real] :
( ( ord_less_eq @ real @ ( uminus_uminus @ real @ ( divide_divide @ real @ pi @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) ) @ X2 )
=> ( ( ord_less_eq @ real @ X2 @ ( divide_divide @ real @ pi @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) )
=> ( ( ord_less_eq @ real @ ( uminus_uminus @ real @ ( divide_divide @ real @ pi @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) ) @ Y3 )
=> ( ( ord_less_eq @ real @ Y3 @ ( divide_divide @ real @ pi @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) )
=> ( ( ord_less @ real @ ( sin @ real @ X2 ) @ ( sin @ real @ Y3 ) )
= ( ord_less @ real @ X2 @ Y3 ) ) ) ) ) ) ).
% sin_mono_less_eq
thf(fact_4321_sin__total,axiom,
! [Y3: real] :
( ( ord_less_eq @ real @ ( uminus_uminus @ real @ ( one_one @ real ) ) @ Y3 )
=> ( ( ord_less_eq @ real @ Y3 @ ( one_one @ real ) )
=> ? [X4: real] :
( ( ord_less_eq @ real @ ( uminus_uminus @ real @ ( divide_divide @ real @ pi @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) ) @ X4 )
& ( ord_less_eq @ real @ X4 @ ( divide_divide @ real @ pi @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) )
& ( ( sin @ real @ X4 )
= Y3 )
& ! [Y5: real] :
( ( ( ord_less_eq @ real @ ( uminus_uminus @ real @ ( divide_divide @ real @ pi @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) ) @ Y5 )
& ( ord_less_eq @ real @ Y5 @ ( divide_divide @ real @ pi @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) )
& ( ( sin @ real @ Y5 )
= Y3 ) )
=> ( Y5 = X4 ) ) ) ) ) ).
% sin_total
thf(fact_4322_or__nat__unfold,axiom,
( ( bit_se1065995026697491101ons_or @ nat )
= ( ^ [M6: nat,N4: nat] :
( if @ nat
@ ( M6
= ( zero_zero @ nat ) )
@ N4
@ ( if @ nat
@ ( N4
= ( zero_zero @ nat ) )
@ M6
@ ( plus_plus @ nat @ ( ord_max @ nat @ ( modulo_modulo @ nat @ M6 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ ( modulo_modulo @ nat @ N4 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( bit_se1065995026697491101ons_or @ nat @ ( divide_divide @ nat @ M6 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ ( divide_divide @ nat @ N4 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) ) ) ) ) ).
% or_nat_unfold
thf(fact_4323_cos__one__2pi,axiom,
! [X2: real] :
( ( ( cos @ real @ X2 )
= ( one_one @ real ) )
= ( ? [X3: nat] :
( X2
= ( times_times @ real @ ( times_times @ real @ ( semiring_1_of_nat @ real @ X3 ) @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) @ pi ) )
| ? [X3: nat] :
( X2
= ( uminus_uminus @ real @ ( times_times @ real @ ( times_times @ real @ ( semiring_1_of_nat @ real @ X3 ) @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) @ pi ) ) ) ) ) ).
% cos_one_2pi
thf(fact_4324_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_4325_sincos__total__pi,axiom,
! [Y3: real,X2: real] :
( ( ord_less_eq @ real @ ( zero_zero @ real ) @ Y3 )
=> ( ( ( plus_plus @ real @ ( power_power @ real @ X2 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ ( power_power @ real @ Y3 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) )
= ( one_one @ real ) )
=> ? [T8: real] :
( ( ord_less_eq @ real @ ( zero_zero @ real ) @ T8 )
& ( ord_less_eq @ real @ T8 @ pi )
& ( X2
= ( cos @ real @ T8 ) )
& ( Y3
= ( sin @ real @ T8 ) ) ) ) ) ).
% sincos_total_pi
thf(fact_4326_sin__expansion__lemma,axiom,
! [X2: real,M: nat] :
( ( sin @ real @ ( plus_plus @ real @ X2 @ ( divide_divide @ real @ ( times_times @ real @ ( semiring_1_of_nat @ real @ ( suc @ M ) ) @ pi ) @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) ) )
= ( cos @ real @ ( plus_plus @ real @ X2 @ ( divide_divide @ real @ ( times_times @ real @ ( semiring_1_of_nat @ real @ M ) @ pi ) @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) ) ) ) ).
% sin_expansion_lemma
thf(fact_4327_cos__zero__iff__int,axiom,
! [X2: real] :
( ( ( cos @ real @ X2 )
= ( zero_zero @ real ) )
= ( ? [I4: int] :
( ~ ( dvd_dvd @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ I4 )
& ( X2
= ( times_times @ real @ ( ring_1_of_int @ real @ I4 ) @ ( divide_divide @ real @ pi @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) ) ) ) ) ) ).
% cos_zero_iff_int
thf(fact_4328_sin__zero__iff__int,axiom,
! [X2: real] :
( ( ( sin @ real @ X2 )
= ( zero_zero @ real ) )
= ( ? [I4: int] :
( ( dvd_dvd @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ I4 )
& ( X2
= ( times_times @ real @ ( ring_1_of_int @ real @ I4 ) @ ( divide_divide @ real @ pi @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) ) ) ) ) ) ).
% sin_zero_iff_int
thf(fact_4329_cos__zero__lemma,axiom,
! [X2: real] :
( ( ord_less_eq @ real @ ( zero_zero @ real ) @ X2 )
=> ( ( ( cos @ real @ X2 )
= ( zero_zero @ real ) )
=> ? [N3: nat] :
( ~ ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N3 )
& ( X2
= ( times_times @ real @ ( semiring_1_of_nat @ real @ N3 ) @ ( divide_divide @ real @ pi @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) ) ) ) ) ) ).
% cos_zero_lemma
thf(fact_4330_sin__zero__lemma,axiom,
! [X2: real] :
( ( ord_less_eq @ real @ ( zero_zero @ real ) @ X2 )
=> ( ( ( sin @ real @ X2 )
= ( zero_zero @ real ) )
=> ? [N3: nat] :
( ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N3 )
& ( X2
= ( times_times @ real @ ( semiring_1_of_nat @ real @ N3 ) @ ( divide_divide @ real @ pi @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) ) ) ) ) ) ).
% sin_zero_lemma
thf(fact_4331_cos__zero__iff,axiom,
! [X2: real] :
( ( ( cos @ real @ X2 )
= ( zero_zero @ real ) )
= ( ? [N4: nat] :
( ~ ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N4 )
& ( X2
= ( times_times @ real @ ( semiring_1_of_nat @ real @ N4 ) @ ( divide_divide @ real @ pi @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) ) ) )
| ? [N4: nat] :
( ~ ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N4 )
& ( X2
= ( uminus_uminus @ real @ ( times_times @ real @ ( semiring_1_of_nat @ real @ N4 ) @ ( divide_divide @ real @ pi @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) ) ) ) ) ) ) ).
% cos_zero_iff
thf(fact_4332_sin__zero__iff,axiom,
! [X2: real] :
( ( ( sin @ real @ X2 )
= ( zero_zero @ real ) )
= ( ? [N4: nat] :
( ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N4 )
& ( X2
= ( times_times @ real @ ( semiring_1_of_nat @ real @ N4 ) @ ( divide_divide @ real @ pi @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) ) ) )
| ? [N4: nat] :
( ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N4 )
& ( X2
= ( uminus_uminus @ real @ ( times_times @ real @ ( semiring_1_of_nat @ real @ N4 ) @ ( divide_divide @ real @ pi @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) ) ) ) ) ) ) ).
% sin_zero_iff
thf(fact_4333_cos__expansion__lemma,axiom,
! [X2: real,M: nat] :
( ( cos @ real @ ( plus_plus @ real @ X2 @ ( 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 @ X2 @ ( divide_divide @ real @ ( times_times @ real @ ( semiring_1_of_nat @ real @ M ) @ pi ) @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) ) ) ) ) ).
% cos_expansion_lemma
thf(fact_4334_sincos__total__pi__half,axiom,
! [X2: real,Y3: real] :
( ( ord_less_eq @ real @ ( zero_zero @ real ) @ X2 )
=> ( ( ord_less_eq @ real @ ( zero_zero @ real ) @ Y3 )
=> ( ( ( plus_plus @ real @ ( power_power @ real @ X2 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ ( power_power @ real @ Y3 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) )
= ( one_one @ real ) )
=> ? [T8: real] :
( ( ord_less_eq @ real @ ( zero_zero @ real ) @ T8 )
& ( ord_less_eq @ real @ T8 @ ( divide_divide @ real @ pi @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) )
& ( X2
= ( cos @ real @ T8 ) )
& ( Y3
= ( sin @ real @ T8 ) ) ) ) ) ) ).
% sincos_total_pi_half
thf(fact_4335_sincos__total__2pi__le,axiom,
! [X2: real,Y3: real] :
( ( ( plus_plus @ real @ ( power_power @ real @ X2 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ ( power_power @ real @ Y3 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) )
= ( one_one @ real ) )
=> ? [T8: real] :
( ( ord_less_eq @ real @ ( zero_zero @ real ) @ T8 )
& ( ord_less_eq @ real @ T8 @ ( times_times @ real @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) @ pi ) )
& ( X2
= ( cos @ real @ T8 ) )
& ( Y3
= ( sin @ real @ T8 ) ) ) ) ).
% sincos_total_2pi_le
thf(fact_4336_sincos__total__2pi,axiom,
! [X2: real,Y3: real] :
( ( ( plus_plus @ real @ ( power_power @ real @ X2 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ ( power_power @ real @ Y3 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) )
= ( one_one @ real ) )
=> ~ ! [T8: real] :
( ( ord_less_eq @ real @ ( zero_zero @ real ) @ T8 )
=> ( ( ord_less @ real @ T8 @ ( times_times @ real @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) @ pi ) )
=> ( ( X2
= ( cos @ real @ T8 ) )
=> ( Y3
!= ( sin @ real @ T8 ) ) ) ) ) ) ).
% sincos_total_2pi
thf(fact_4337_Maclaurin__sin__expansion,axiom,
! [X2: real,N: nat] :
? [T8: real] :
( ( sin @ real @ X2 )
= ( plus_plus @ real
@ ( groups7311177749621191930dd_sum @ nat @ real
@ ^ [M6: nat] : ( times_times @ real @ ( sin_coeff @ M6 ) @ ( power_power @ real @ X2 @ M6 ) )
@ ( set_ord_lessThan @ nat @ N ) )
@ ( times_times @ real @ ( divide_divide @ real @ ( sin @ real @ ( plus_plus @ real @ T8 @ ( 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 @ X2 @ N ) ) ) ) ).
% Maclaurin_sin_expansion
thf(fact_4338_Maclaurin__sin__expansion4,axiom,
! [X2: real,N: nat] :
( ( ord_less @ real @ ( zero_zero @ real ) @ X2 )
=> ? [T8: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ T8 )
& ( ord_less_eq @ real @ T8 @ X2 )
& ( ( sin @ real @ X2 )
= ( plus_plus @ real
@ ( groups7311177749621191930dd_sum @ nat @ real
@ ^ [M6: nat] : ( times_times @ real @ ( sin_coeff @ M6 ) @ ( power_power @ real @ X2 @ M6 ) )
@ ( set_ord_lessThan @ nat @ N ) )
@ ( times_times @ real @ ( divide_divide @ real @ ( sin @ real @ ( plus_plus @ real @ T8 @ ( 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 @ X2 @ N ) ) ) ) ) ) ).
% Maclaurin_sin_expansion4
thf(fact_4339_Maclaurin__sin__expansion3,axiom,
! [N: nat,X2: real] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ N )
=> ( ( ord_less @ real @ ( zero_zero @ real ) @ X2 )
=> ? [T8: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ T8 )
& ( ord_less @ real @ T8 @ X2 )
& ( ( sin @ real @ X2 )
= ( plus_plus @ real
@ ( groups7311177749621191930dd_sum @ nat @ real
@ ^ [M6: nat] : ( times_times @ real @ ( sin_coeff @ M6 ) @ ( power_power @ real @ X2 @ M6 ) )
@ ( set_ord_lessThan @ nat @ N ) )
@ ( times_times @ real @ ( divide_divide @ real @ ( sin @ real @ ( plus_plus @ real @ T8 @ ( 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 @ X2 @ N ) ) ) ) ) ) ) ).
% Maclaurin_sin_expansion3
thf(fact_4340_Maclaurin__minus__cos__expansion,axiom,
! [N: nat,X2: real] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ N )
=> ( ( ord_less @ real @ X2 @ ( zero_zero @ real ) )
=> ? [T8: real] :
( ( ord_less @ real @ X2 @ T8 )
& ( ord_less @ real @ T8 @ ( zero_zero @ real ) )
& ( ( cos @ real @ X2 )
= ( plus_plus @ real
@ ( groups7311177749621191930dd_sum @ nat @ real
@ ^ [M6: nat] : ( times_times @ real @ ( cos_coeff @ M6 ) @ ( power_power @ real @ X2 @ M6 ) )
@ ( set_ord_lessThan @ nat @ N ) )
@ ( times_times @ real @ ( divide_divide @ real @ ( cos @ real @ ( plus_plus @ real @ T8 @ ( 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 @ X2 @ N ) ) ) ) ) ) ) ).
% Maclaurin_minus_cos_expansion
thf(fact_4341_Maclaurin__cos__expansion,axiom,
! [X2: real,N: nat] :
? [T8: real] :
( ( ord_less_eq @ real @ ( abs_abs @ real @ T8 ) @ ( abs_abs @ real @ X2 ) )
& ( ( cos @ real @ X2 )
= ( plus_plus @ real
@ ( groups7311177749621191930dd_sum @ nat @ real
@ ^ [M6: nat] : ( times_times @ real @ ( cos_coeff @ M6 ) @ ( power_power @ real @ X2 @ M6 ) )
@ ( set_ord_lessThan @ nat @ N ) )
@ ( times_times @ real @ ( divide_divide @ real @ ( cos @ real @ ( plus_plus @ real @ T8 @ ( 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 @ X2 @ N ) ) ) ) ) ).
% Maclaurin_cos_expansion
thf(fact_4342_Maclaurin__sin__expansion2,axiom,
! [X2: real,N: nat] :
? [T8: real] :
( ( ord_less_eq @ real @ ( abs_abs @ real @ T8 ) @ ( abs_abs @ real @ X2 ) )
& ( ( sin @ real @ X2 )
= ( plus_plus @ real
@ ( groups7311177749621191930dd_sum @ nat @ real
@ ^ [M6: nat] : ( times_times @ real @ ( sin_coeff @ M6 ) @ ( power_power @ real @ X2 @ M6 ) )
@ ( set_ord_lessThan @ nat @ N ) )
@ ( times_times @ real @ ( divide_divide @ real @ ( sin @ real @ ( plus_plus @ real @ T8 @ ( 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 @ X2 @ N ) ) ) ) ) ).
% Maclaurin_sin_expansion2
thf(fact_4343_pi__series,axiom,
( ( divide_divide @ real @ pi @ ( numeral_numeral @ real @ ( bit0 @ ( bit0 @ one2 ) ) ) )
= ( suminf @ real
@ ^ [K3: nat] : ( divide_divide @ real @ ( times_times @ real @ ( power_power @ real @ ( uminus_uminus @ real @ ( one_one @ real ) ) @ K3 ) @ ( one_one @ real ) ) @ ( semiring_1_of_nat @ real @ ( plus_plus @ nat @ ( times_times @ nat @ K3 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ ( one_one @ nat ) ) ) ) ) ) ).
% pi_series
thf(fact_4344_tan__double,axiom,
! [A: $tType] :
( ( ( real_Vector_banach @ A )
& ( real_V3459762299906320749_field @ A ) )
=> ! [X2: A] :
( ( ( cos @ A @ X2 )
!= ( zero_zero @ A ) )
=> ( ( ( cos @ A @ ( times_times @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ X2 ) )
!= ( zero_zero @ A ) )
=> ( ( tan @ A @ ( times_times @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ X2 ) )
= ( divide_divide @ A @ ( times_times @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ ( tan @ A @ X2 ) ) @ ( minus_minus @ A @ ( one_one @ A ) @ ( power_power @ A @ ( tan @ A @ X2 ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) ) ) ) ).
% tan_double
thf(fact_4345_abs__numeral,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [N: num] :
( ( abs_abs @ A @ ( numeral_numeral @ A @ N ) )
= ( numeral_numeral @ A @ N ) ) ) ).
% abs_numeral
thf(fact_4346_abs__add__abs,axiom,
! [A: $tType] :
( ( ordere166539214618696060dd_abs @ A )
=> ! [A3: A,B3: A] :
( ( abs_abs @ A @ ( plus_plus @ A @ ( abs_abs @ A @ A3 ) @ ( abs_abs @ A @ B3 ) ) )
= ( plus_plus @ A @ ( abs_abs @ A @ A3 ) @ ( abs_abs @ A @ B3 ) ) ) ) ).
% abs_add_abs
thf(fact_4347_abs__1,axiom,
! [A: $tType] :
( ( idom_abs_sgn @ A )
=> ( ( abs_abs @ A @ ( one_one @ A ) )
= ( one_one @ A ) ) ) ).
% abs_1
thf(fact_4348_abs__mult__self__eq,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [A3: A] :
( ( times_times @ A @ ( abs_abs @ A @ A3 ) @ ( abs_abs @ A @ A3 ) )
= ( times_times @ A @ A3 @ A3 ) ) ) ).
% abs_mult_self_eq
thf(fact_4349_abs__divide,axiom,
! [A: $tType] :
( ( field_abs_sgn @ A )
=> ! [A3: A,B3: A] :
( ( abs_abs @ A @ ( divide_divide @ A @ A3 @ B3 ) )
= ( divide_divide @ A @ ( abs_abs @ A @ A3 ) @ ( abs_abs @ A @ B3 ) ) ) ) ).
% abs_divide
thf(fact_4350_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_4351_tan__periodic__pi,axiom,
! [X2: real] :
( ( tan @ real @ ( plus_plus @ real @ X2 @ pi ) )
= ( tan @ real @ X2 ) ) ).
% tan_periodic_pi
thf(fact_4352_abs__sum__abs,axiom,
! [B: $tType,A: $tType] :
( ( ordere166539214618696060dd_abs @ B )
=> ! [F2: A > B,A2: set @ A] :
( ( abs_abs @ B
@ ( groups7311177749621191930dd_sum @ A @ B
@ ^ [A6: A] : ( abs_abs @ B @ ( F2 @ A6 ) )
@ A2 ) )
= ( groups7311177749621191930dd_sum @ A @ B
@ ^ [A6: A] : ( abs_abs @ B @ ( F2 @ A6 ) )
@ A2 ) ) ) ).
% abs_sum_abs
thf(fact_4353_suminf__zero,axiom,
! [A: $tType] :
( ( ( comm_monoid_add @ A )
& ( topological_t2_space @ A ) )
=> ( ( suminf @ A
@ ^ [N4: nat] : ( zero_zero @ A ) )
= ( zero_zero @ A ) ) ) ).
% suminf_zero
thf(fact_4354_abs__of__nonneg,axiom,
! [A: $tType] :
( ( ordere166539214618696060dd_abs @ A )
=> ! [A3: A] :
( ( ord_less_eq @ A @ ( zero_zero @ A ) @ A3 )
=> ( ( abs_abs @ A @ A3 )
= A3 ) ) ) ).
% abs_of_nonneg
thf(fact_4355_abs__le__self__iff,axiom,
! [A: $tType] :
( ( ordere166539214618696060dd_abs @ A )
=> ! [A3: A] :
( ( ord_less_eq @ A @ ( abs_abs @ A @ A3 ) @ A3 )
= ( ord_less_eq @ A @ ( zero_zero @ A ) @ A3 ) ) ) ).
% abs_le_self_iff
thf(fact_4356_abs__le__zero__iff,axiom,
! [A: $tType] :
( ( ordere166539214618696060dd_abs @ A )
=> ! [A3: A] :
( ( ord_less_eq @ A @ ( abs_abs @ A @ A3 ) @ ( zero_zero @ A ) )
= ( A3
= ( zero_zero @ A ) ) ) ) ).
% abs_le_zero_iff
thf(fact_4357_zero__less__abs__iff,axiom,
! [A: $tType] :
( ( ordere166539214618696060dd_abs @ A )
=> ! [A3: A] :
( ( ord_less @ A @ ( zero_zero @ A ) @ ( abs_abs @ A @ A3 ) )
= ( A3
!= ( zero_zero @ A ) ) ) ) ).
% zero_less_abs_iff
thf(fact_4358_abs__neg__numeral,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [N: num] :
( ( abs_abs @ A @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ N ) ) )
= ( numeral_numeral @ A @ N ) ) ) ).
% abs_neg_numeral
thf(fact_4359_abs__neg__one,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ( ( abs_abs @ A @ ( uminus_uminus @ A @ ( one_one @ A ) ) )
= ( one_one @ A ) ) ) ).
% abs_neg_one
thf(fact_4360_abs__power__minus,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [A3: A,N: nat] :
( ( abs_abs @ A @ ( power_power @ A @ ( uminus_uminus @ A @ A3 ) @ N ) )
= ( abs_abs @ A @ ( power_power @ A @ A3 @ N ) ) ) ) ).
% abs_power_minus
thf(fact_4361_sum__abs,axiom,
! [B: $tType,A: $tType] :
( ( ordere166539214618696060dd_abs @ B )
=> ! [F2: A > B,A2: set @ A] :
( ord_less_eq @ B @ ( abs_abs @ B @ ( groups7311177749621191930dd_sum @ A @ B @ F2 @ A2 ) )
@ ( groups7311177749621191930dd_sum @ A @ B
@ ^ [I4: A] : ( abs_abs @ B @ ( F2 @ I4 ) )
@ A2 ) ) ) ).
% sum_abs
thf(fact_4362_divide__le__0__abs__iff,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [A3: A,B3: A] :
( ( ord_less_eq @ A @ ( divide_divide @ A @ A3 @ ( abs_abs @ A @ B3 ) ) @ ( zero_zero @ A ) )
= ( ( ord_less_eq @ A @ A3 @ ( zero_zero @ A ) )
| ( B3
= ( zero_zero @ A ) ) ) ) ) ).
% divide_le_0_abs_iff
thf(fact_4363_zero__le__divide__abs__iff,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [A3: A,B3: A] :
( ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( divide_divide @ A @ A3 @ ( abs_abs @ A @ B3 ) ) )
= ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ A3 )
| ( B3
= ( zero_zero @ A ) ) ) ) ) ).
% zero_le_divide_abs_iff
thf(fact_4364_abs__of__nonpos,axiom,
! [A: $tType] :
( ( ordere166539214618696060dd_abs @ A )
=> ! [A3: A] :
( ( ord_less_eq @ A @ A3 @ ( zero_zero @ A ) )
=> ( ( abs_abs @ A @ A3 )
= ( uminus_uminus @ A @ A3 ) ) ) ) ).
% abs_of_nonpos
thf(fact_4365_tan__periodic__n,axiom,
! [X2: real,N: num] :
( ( tan @ real @ ( plus_plus @ real @ X2 @ ( times_times @ real @ ( numeral_numeral @ real @ N ) @ pi ) ) )
= ( tan @ real @ X2 ) ) ).
% tan_periodic_n
thf(fact_4366_tan__periodic__nat,axiom,
! [X2: real,N: nat] :
( ( tan @ real @ ( plus_plus @ real @ X2 @ ( times_times @ real @ ( semiring_1_of_nat @ real @ N ) @ pi ) ) )
= ( tan @ real @ X2 ) ) ).
% tan_periodic_nat
thf(fact_4367_tan__periodic__int,axiom,
! [X2: real,I: int] :
( ( tan @ real @ ( plus_plus @ real @ X2 @ ( times_times @ real @ ( ring_1_of_int @ real @ I ) @ pi ) ) )
= ( tan @ real @ X2 ) ) ).
% tan_periodic_int
thf(fact_4368_sum__abs__ge__zero,axiom,
! [B: $tType,A: $tType] :
( ( ordere166539214618696060dd_abs @ B )
=> ! [F2: A > B,A2: set @ A] :
( ord_less_eq @ B @ ( zero_zero @ B )
@ ( groups7311177749621191930dd_sum @ A @ B
@ ^ [I4: A] : ( abs_abs @ B @ ( F2 @ I4 ) )
@ A2 ) ) ) ).
% sum_abs_ge_zero
thf(fact_4369_zero__less__power__abs__iff,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [A3: A,N: nat] :
( ( ord_less @ A @ ( zero_zero @ A ) @ ( power_power @ A @ ( abs_abs @ A @ A3 ) @ N ) )
= ( ( A3
!= ( zero_zero @ A ) )
| ( N
= ( zero_zero @ nat ) ) ) ) ) ).
% zero_less_power_abs_iff
thf(fact_4370_power2__abs,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [A3: A] :
( ( power_power @ A @ ( abs_abs @ A @ A3 ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) )
= ( power_power @ A @ A3 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ).
% power2_abs
thf(fact_4371_abs__power2,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [A3: A] :
( ( abs_abs @ A @ ( power_power @ A @ A3 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) )
= ( power_power @ A @ A3 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ).
% abs_power2
thf(fact_4372_powser__zero,axiom,
! [A: $tType] :
( ( real_V2822296259951069270ebra_1 @ A )
=> ! [F2: nat > A] :
( ( suminf @ A
@ ^ [N4: nat] : ( times_times @ A @ ( F2 @ N4 ) @ ( power_power @ A @ ( zero_zero @ A ) @ N4 ) ) )
= ( F2 @ ( zero_zero @ nat ) ) ) ) ).
% powser_zero
thf(fact_4373_power__even__abs__numeral,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [W: num,A3: A] :
( ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( numeral_numeral @ nat @ W ) )
=> ( ( power_power @ A @ ( abs_abs @ A @ A3 ) @ ( numeral_numeral @ nat @ W ) )
= ( power_power @ A @ A3 @ ( numeral_numeral @ nat @ W ) ) ) ) ) ).
% power_even_abs_numeral
thf(fact_4374_tan__periodic,axiom,
! [X2: real] :
( ( tan @ real @ ( plus_plus @ real @ X2 @ ( times_times @ real @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) @ pi ) ) )
= ( tan @ real @ X2 ) ) ).
% tan_periodic
thf(fact_4375_square__powr__half,axiom,
! [X2: real] :
( ( powr @ real @ ( power_power @ real @ X2 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ ( divide_divide @ real @ ( one_one @ real ) @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) )
= ( abs_abs @ real @ X2 ) ) ).
% square_powr_half
thf(fact_4376_abs__one,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ( ( abs_abs @ A @ ( one_one @ A ) )
= ( one_one @ A ) ) ) ).
% abs_one
thf(fact_4377_abs__le__D1,axiom,
! [A: $tType] :
( ( ordere166539214618696060dd_abs @ A )
=> ! [A3: A,B3: A] :
( ( ord_less_eq @ A @ ( abs_abs @ A @ A3 ) @ B3 )
=> ( ord_less_eq @ A @ A3 @ B3 ) ) ) ).
% abs_le_D1
thf(fact_4378_abs__ge__self,axiom,
! [A: $tType] :
( ( ordere166539214618696060dd_abs @ A )
=> ! [A3: A] : ( ord_less_eq @ A @ A3 @ ( abs_abs @ A @ A3 ) ) ) ).
% abs_ge_self
thf(fact_4379_abs__minus__commute,axiom,
! [A: $tType] :
( ( ordere166539214618696060dd_abs @ A )
=> ! [A3: A,B3: A] :
( ( abs_abs @ A @ ( minus_minus @ A @ A3 @ B3 ) )
= ( abs_abs @ A @ ( minus_minus @ A @ B3 @ A3 ) ) ) ) ).
% abs_minus_commute
thf(fact_4380_abs__mult,axiom,
! [A: $tType] :
( ( idom_abs_sgn @ A )
=> ! [A3: A,B3: A] :
( ( abs_abs @ A @ ( times_times @ A @ A3 @ B3 ) )
= ( times_times @ A @ ( abs_abs @ A @ A3 ) @ ( abs_abs @ A @ B3 ) ) ) ) ).
% abs_mult
thf(fact_4381_power__abs,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [A3: A,N: nat] :
( ( abs_abs @ A @ ( power_power @ A @ A3 @ N ) )
= ( power_power @ A @ ( abs_abs @ A @ A3 ) @ N ) ) ) ).
% power_abs
thf(fact_4382_abs__prod,axiom,
! [A: $tType,B: $tType] :
( ( linordered_field @ A )
=> ! [F2: B > A,A2: set @ B] :
( ( abs_abs @ A @ ( groups7121269368397514597t_prod @ B @ A @ F2 @ A2 ) )
= ( groups7121269368397514597t_prod @ B @ A
@ ^ [X3: B] : ( abs_abs @ A @ ( F2 @ X3 ) )
@ A2 ) ) ) ).
% abs_prod
thf(fact_4383_abs__ge__zero,axiom,
! [A: $tType] :
( ( ordere166539214618696060dd_abs @ A )
=> ! [A3: A] : ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( abs_abs @ A @ A3 ) ) ) ).
% abs_ge_zero
thf(fact_4384_abs__of__pos,axiom,
! [A: $tType] :
( ( ordere166539214618696060dd_abs @ A )
=> ! [A3: A] :
( ( ord_less @ A @ ( zero_zero @ A ) @ A3 )
=> ( ( abs_abs @ A @ A3 )
= A3 ) ) ) ).
% abs_of_pos
thf(fact_4385_abs__not__less__zero,axiom,
! [A: $tType] :
( ( ordere166539214618696060dd_abs @ A )
=> ! [A3: A] :
~ ( ord_less @ A @ ( abs_abs @ A @ A3 ) @ ( zero_zero @ A ) ) ) ).
% abs_not_less_zero
thf(fact_4386_abs__triangle__ineq,axiom,
! [A: $tType] :
( ( ordere166539214618696060dd_abs @ A )
=> ! [A3: A,B3: A] : ( ord_less_eq @ A @ ( abs_abs @ A @ ( plus_plus @ A @ A3 @ B3 ) ) @ ( plus_plus @ A @ ( abs_abs @ A @ A3 ) @ ( abs_abs @ A @ B3 ) ) ) ) ).
% abs_triangle_ineq
thf(fact_4387_abs__mult__less,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [A3: A,C3: A,B3: A,D2: A] :
( ( ord_less @ A @ ( abs_abs @ A @ A3 ) @ C3 )
=> ( ( ord_less @ A @ ( abs_abs @ A @ B3 ) @ D2 )
=> ( ord_less @ A @ ( times_times @ A @ ( abs_abs @ A @ A3 ) @ ( abs_abs @ A @ B3 ) ) @ ( times_times @ A @ C3 @ D2 ) ) ) ) ) ).
% abs_mult_less
thf(fact_4388_abs__triangle__ineq2__sym,axiom,
! [A: $tType] :
( ( ordere166539214618696060dd_abs @ A )
=> ! [A3: A,B3: A] : ( ord_less_eq @ A @ ( minus_minus @ A @ ( abs_abs @ A @ A3 ) @ ( abs_abs @ A @ B3 ) ) @ ( abs_abs @ A @ ( minus_minus @ A @ B3 @ A3 ) ) ) ) ).
% abs_triangle_ineq2_sym
thf(fact_4389_abs__triangle__ineq3,axiom,
! [A: $tType] :
( ( ordere166539214618696060dd_abs @ A )
=> ! [A3: A,B3: A] : ( ord_less_eq @ A @ ( abs_abs @ A @ ( minus_minus @ A @ ( abs_abs @ A @ A3 ) @ ( abs_abs @ A @ B3 ) ) ) @ ( abs_abs @ A @ ( minus_minus @ A @ A3 @ B3 ) ) ) ) ).
% abs_triangle_ineq3
thf(fact_4390_abs__triangle__ineq2,axiom,
! [A: $tType] :
( ( ordere166539214618696060dd_abs @ A )
=> ! [A3: A,B3: A] : ( ord_less_eq @ A @ ( minus_minus @ A @ ( abs_abs @ A @ A3 ) @ ( abs_abs @ A @ B3 ) ) @ ( abs_abs @ A @ ( minus_minus @ A @ A3 @ B3 ) ) ) ) ).
% abs_triangle_ineq2
thf(fact_4391_nonzero__abs__divide,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [B3: A,A3: A] :
( ( B3
!= ( zero_zero @ A ) )
=> ( ( abs_abs @ A @ ( divide_divide @ A @ A3 @ B3 ) )
= ( divide_divide @ A @ ( abs_abs @ A @ A3 ) @ ( abs_abs @ A @ B3 ) ) ) ) ) ).
% nonzero_abs_divide
thf(fact_4392_abs__leI,axiom,
! [A: $tType] :
( ( ordere166539214618696060dd_abs @ A )
=> ! [A3: A,B3: A] :
( ( ord_less_eq @ A @ A3 @ B3 )
=> ( ( ord_less_eq @ A @ ( uminus_uminus @ A @ A3 ) @ B3 )
=> ( ord_less_eq @ A @ ( abs_abs @ A @ A3 ) @ B3 ) ) ) ) ).
% abs_leI
thf(fact_4393_abs__le__D2,axiom,
! [A: $tType] :
( ( ordere166539214618696060dd_abs @ A )
=> ! [A3: A,B3: A] :
( ( ord_less_eq @ A @ ( abs_abs @ A @ A3 ) @ B3 )
=> ( ord_less_eq @ A @ ( uminus_uminus @ A @ A3 ) @ B3 ) ) ) ).
% abs_le_D2
thf(fact_4394_abs__le__iff,axiom,
! [A: $tType] :
( ( ordere166539214618696060dd_abs @ A )
=> ! [A3: A,B3: A] :
( ( ord_less_eq @ A @ ( abs_abs @ A @ A3 ) @ B3 )
= ( ( ord_less_eq @ A @ A3 @ B3 )
& ( ord_less_eq @ A @ ( uminus_uminus @ A @ A3 ) @ B3 ) ) ) ) ).
% abs_le_iff
thf(fact_4395_abs__ge__minus__self,axiom,
! [A: $tType] :
( ( ordere166539214618696060dd_abs @ A )
=> ! [A3: A] : ( ord_less_eq @ A @ ( uminus_uminus @ A @ A3 ) @ ( abs_abs @ A @ A3 ) ) ) ).
% abs_ge_minus_self
thf(fact_4396_abs__less__iff,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [A3: A,B3: A] :
( ( ord_less @ A @ ( abs_abs @ A @ A3 ) @ B3 )
= ( ( ord_less @ A @ A3 @ B3 )
& ( ord_less @ A @ ( uminus_uminus @ A @ A3 ) @ B3 ) ) ) ) ).
% abs_less_iff
thf(fact_4397_abs__sin__x__le__abs__x,axiom,
! [X2: real] : ( ord_less_eq @ real @ ( abs_abs @ real @ ( sin @ real @ X2 ) ) @ ( abs_abs @ real @ X2 ) ) ).
% abs_sin_x_le_abs_x
thf(fact_4398_dense__eq0__I,axiom,
! [A: $tType] :
( ( ( ordere166539214618696060dd_abs @ A )
& ( dense_linorder @ A ) )
=> ! [X2: A] :
( ! [E2: A] :
( ( ord_less @ A @ ( zero_zero @ A ) @ E2 )
=> ( ord_less_eq @ A @ ( abs_abs @ A @ X2 ) @ E2 ) )
=> ( X2
= ( zero_zero @ A ) ) ) ) ).
% dense_eq0_I
thf(fact_4399_abs__mult__pos,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [X2: A,Y3: A] :
( ( ord_less_eq @ A @ ( zero_zero @ A ) @ X2 )
=> ( ( times_times @ A @ ( abs_abs @ A @ Y3 ) @ X2 )
= ( abs_abs @ A @ ( times_times @ A @ Y3 @ X2 ) ) ) ) ) ).
% abs_mult_pos
thf(fact_4400_abs__eq__mult,axiom,
! [A: $tType] :
( ( ordered_ring_abs @ A )
=> ! [A3: A,B3: A] :
( ( ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ A3 )
| ( ord_less_eq @ A @ A3 @ ( zero_zero @ A ) ) )
& ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ B3 )
| ( ord_less_eq @ A @ B3 @ ( zero_zero @ A ) ) ) )
=> ( ( abs_abs @ A @ ( times_times @ A @ A3 @ B3 ) )
= ( times_times @ A @ ( abs_abs @ A @ A3 ) @ ( abs_abs @ A @ B3 ) ) ) ) ) ).
% abs_eq_mult
thf(fact_4401_abs__div__pos,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [Y3: A,X2: A] :
( ( ord_less @ A @ ( zero_zero @ A ) @ Y3 )
=> ( ( divide_divide @ A @ ( abs_abs @ A @ X2 ) @ Y3 )
= ( abs_abs @ A @ ( divide_divide @ A @ X2 @ Y3 ) ) ) ) ) ).
% abs_div_pos
thf(fact_4402_zero__le__power__abs,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [A3: A,N: nat] : ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( power_power @ A @ ( abs_abs @ A @ A3 ) @ N ) ) ) ).
% zero_le_power_abs
thf(fact_4403_abs__minus__le__zero,axiom,
! [A: $tType] :
( ( ordere166539214618696060dd_abs @ A )
=> ! [A3: A] : ( ord_less_eq @ A @ ( uminus_uminus @ A @ ( abs_abs @ A @ A3 ) ) @ ( zero_zero @ A ) ) ) ).
% abs_minus_le_zero
thf(fact_4404_abs__eq__iff_H,axiom,
! [A: $tType] :
( ( linordered_ring @ A )
=> ! [A3: A,B3: A] :
( ( ( abs_abs @ A @ A3 )
= B3 )
= ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ B3 )
& ( ( A3 = B3 )
| ( A3
= ( uminus_uminus @ A @ B3 ) ) ) ) ) ) ).
% abs_eq_iff'
thf(fact_4405_eq__abs__iff_H,axiom,
! [A: $tType] :
( ( linordered_ring @ A )
=> ! [A3: A,B3: A] :
( ( A3
= ( abs_abs @ A @ B3 ) )
= ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ A3 )
& ( ( B3 = A3 )
| ( B3
= ( uminus_uminus @ A @ A3 ) ) ) ) ) ) ).
% eq_abs_iff'
thf(fact_4406_abs__if,axiom,
! [A: $tType] :
( ( abs_if @ A )
=> ( ( abs_abs @ A )
= ( ^ [A6: A] : ( if @ A @ ( ord_less @ A @ A6 @ ( zero_zero @ A ) ) @ ( uminus_uminus @ A @ A6 ) @ A6 ) ) ) ) ).
% abs_if
thf(fact_4407_abs__of__neg,axiom,
! [A: $tType] :
( ( ordere166539214618696060dd_abs @ A )
=> ! [A3: A] :
( ( ord_less @ A @ A3 @ ( zero_zero @ A ) )
=> ( ( abs_abs @ A @ A3 )
= ( uminus_uminus @ A @ A3 ) ) ) ) ).
% abs_of_neg
thf(fact_4408_abs__if__raw,axiom,
! [A: $tType] :
( ( abs_if @ A )
=> ( ( abs_abs @ A )
= ( ^ [A6: A] : ( if @ A @ ( ord_less @ A @ A6 @ ( zero_zero @ A ) ) @ ( uminus_uminus @ A @ A6 ) @ A6 ) ) ) ) ).
% abs_if_raw
thf(fact_4409_abs__diff__triangle__ineq,axiom,
! [A: $tType] :
( ( ordere166539214618696060dd_abs @ A )
=> ! [A3: A,B3: A,C3: A,D2: A] : ( ord_less_eq @ A @ ( abs_abs @ A @ ( minus_minus @ A @ ( plus_plus @ A @ A3 @ B3 ) @ ( plus_plus @ A @ C3 @ D2 ) ) ) @ ( plus_plus @ A @ ( abs_abs @ A @ ( minus_minus @ A @ A3 @ C3 ) ) @ ( abs_abs @ A @ ( minus_minus @ A @ B3 @ D2 ) ) ) ) ) ).
% abs_diff_triangle_ineq
thf(fact_4410_abs__triangle__ineq4,axiom,
! [A: $tType] :
( ( ordere166539214618696060dd_abs @ A )
=> ! [A3: A,B3: A] : ( ord_less_eq @ A @ ( abs_abs @ A @ ( minus_minus @ A @ A3 @ B3 ) ) @ ( plus_plus @ A @ ( abs_abs @ A @ A3 ) @ ( abs_abs @ A @ B3 ) ) ) ) ).
% abs_triangle_ineq4
thf(fact_4411_abs__diff__le__iff,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [X2: A,A3: A,R2: A] :
( ( ord_less_eq @ A @ ( abs_abs @ A @ ( minus_minus @ A @ X2 @ A3 ) ) @ R2 )
= ( ( ord_less_eq @ A @ ( minus_minus @ A @ A3 @ R2 ) @ X2 )
& ( ord_less_eq @ A @ X2 @ ( plus_plus @ A @ A3 @ R2 ) ) ) ) ) ).
% abs_diff_le_iff
thf(fact_4412_abs__diff__less__iff,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [X2: A,A3: A,R2: A] :
( ( ord_less @ A @ ( abs_abs @ A @ ( minus_minus @ A @ X2 @ A3 ) ) @ R2 )
= ( ( ord_less @ A @ ( minus_minus @ A @ A3 @ R2 ) @ X2 )
& ( ord_less @ A @ X2 @ ( plus_plus @ A @ A3 @ R2 ) ) ) ) ) ).
% abs_diff_less_iff
thf(fact_4413_lemma__interval__lt,axiom,
! [A3: real,X2: real,B3: real] :
( ( ord_less @ real @ A3 @ X2 )
=> ( ( ord_less @ real @ X2 @ B3 )
=> ? [D4: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ D4 )
& ! [Y5: real] :
( ( ord_less @ real @ ( abs_abs @ real @ ( minus_minus @ real @ X2 @ Y5 ) ) @ D4 )
=> ( ( ord_less @ real @ A3 @ Y5 )
& ( ord_less @ real @ Y5 @ B3 ) ) ) ) ) ) ).
% lemma_interval_lt
thf(fact_4414_abs__cos__le__one,axiom,
! [X2: real] : ( ord_less_eq @ real @ ( abs_abs @ real @ ( cos @ real @ X2 ) ) @ ( one_one @ real ) ) ).
% abs_cos_le_one
thf(fact_4415_abs__sin__le__one,axiom,
! [X2: real] : ( ord_less_eq @ real @ ( abs_abs @ real @ ( sin @ real @ X2 ) ) @ ( one_one @ real ) ) ).
% abs_sin_le_one
thf(fact_4416_sin__bound__lemma,axiom,
! [X2: real,Y3: real,U: real,V: real] :
( ( X2 = Y3 )
=> ( ( ord_less_eq @ real @ ( abs_abs @ real @ U ) @ V )
=> ( ord_less_eq @ real @ ( abs_abs @ real @ ( minus_minus @ real @ ( plus_plus @ real @ X2 @ U ) @ Y3 ) ) @ V ) ) ) ).
% sin_bound_lemma
thf(fact_4417_suminf__finite,axiom,
! [A: $tType] :
( ( ( comm_monoid_add @ A )
& ( topological_t2_space @ A ) )
=> ! [N7: set @ nat,F2: nat > A] :
( ( finite_finite2 @ nat @ N7 )
=> ( ! [N3: nat] :
( ~ ( member @ nat @ N3 @ N7 )
=> ( ( F2 @ N3 )
= ( zero_zero @ A ) ) )
=> ( ( suminf @ A @ F2 )
= ( groups7311177749621191930dd_sum @ nat @ A @ F2 @ N7 ) ) ) ) ) ).
% suminf_finite
thf(fact_4418_tan__bound__pi2,axiom,
! [X2: real] :
( ( ord_less @ real @ ( abs_abs @ real @ X2 ) @ ( divide_divide @ real @ pi @ ( numeral_numeral @ real @ ( bit0 @ ( bit0 @ one2 ) ) ) ) )
=> ( ord_less @ real @ ( abs_abs @ real @ ( tan @ real @ X2 ) ) @ ( one_one @ real ) ) ) ).
% tan_bound_pi2
thf(fact_4419_abs__add__one__gt__zero,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [X2: A] : ( ord_less @ A @ ( zero_zero @ A ) @ ( plus_plus @ A @ ( one_one @ A ) @ ( abs_abs @ A @ X2 ) ) ) ) ).
% abs_add_one_gt_zero
thf(fact_4420_tan__total__pi4,axiom,
! [X2: real] :
( ( ord_less @ real @ ( abs_abs @ real @ X2 ) @ ( one_one @ real ) )
=> ? [Z2: real] :
( ( ord_less @ real @ ( uminus_uminus @ real @ ( divide_divide @ real @ pi @ ( numeral_numeral @ real @ ( bit0 @ ( bit0 @ one2 ) ) ) ) ) @ Z2 )
& ( ord_less @ real @ Z2 @ ( divide_divide @ real @ pi @ ( numeral_numeral @ real @ ( bit0 @ ( bit0 @ one2 ) ) ) ) )
& ( ( tan @ real @ Z2 )
= X2 ) ) ) ).
% tan_total_pi4
thf(fact_4421_of__int__leD,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [N: int,X2: A] :
( ( ord_less_eq @ A @ ( abs_abs @ A @ ( ring_1_of_int @ A @ N ) ) @ X2 )
=> ( ( N
= ( zero_zero @ int ) )
| ( ord_less_eq @ A @ ( one_one @ A ) @ X2 ) ) ) ) ).
% of_int_leD
thf(fact_4422_of__int__lessD,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [N: int,X2: A] :
( ( ord_less @ A @ ( abs_abs @ A @ ( ring_1_of_int @ A @ N ) ) @ X2 )
=> ( ( N
= ( zero_zero @ int ) )
| ( ord_less @ A @ ( one_one @ A ) @ X2 ) ) ) ) ).
% of_int_lessD
thf(fact_4423_tan__def,axiom,
! [A: $tType] :
( ( ( real_Vector_banach @ A )
& ( real_V3459762299906320749_field @ A ) )
=> ( ( tan @ A )
= ( ^ [X3: A] : ( divide_divide @ A @ ( sin @ A @ X3 ) @ ( cos @ A @ X3 ) ) ) ) ) ).
% tan_def
thf(fact_4424_lemma__interval,axiom,
! [A3: real,X2: real,B3: real] :
( ( ord_less @ real @ A3 @ X2 )
=> ( ( ord_less @ real @ X2 @ B3 )
=> ? [D4: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ D4 )
& ! [Y5: real] :
( ( ord_less @ real @ ( abs_abs @ real @ ( minus_minus @ real @ X2 @ Y5 ) ) @ D4 )
=> ( ( ord_less_eq @ real @ A3 @ Y5 )
& ( ord_less_eq @ real @ Y5 @ B3 ) ) ) ) ) ) ).
% lemma_interval
thf(fact_4425_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_4426_norm__triangle__ineq3,axiom,
! [A: $tType] :
( ( real_V822414075346904944vector @ A )
=> ! [A3: A,B3: A] : ( ord_less_eq @ real @ ( abs_abs @ real @ ( minus_minus @ real @ ( real_V7770717601297561774m_norm @ A @ A3 ) @ ( real_V7770717601297561774m_norm @ A @ B3 ) ) ) @ ( real_V7770717601297561774m_norm @ A @ ( minus_minus @ A @ A3 @ B3 ) ) ) ) ).
% norm_triangle_ineq3
thf(fact_4427_sin__zero__abs__cos__one,axiom,
! [X2: real] :
( ( ( sin @ real @ X2 )
= ( zero_zero @ real ) )
=> ( ( abs_abs @ real @ ( cos @ real @ X2 ) )
= ( one_one @ real ) ) ) ).
% sin_zero_abs_cos_one
thf(fact_4428_sin__def,axiom,
! [A: $tType] :
( ( ( real_Vector_banach @ A )
& ( real_V2822296259951069270ebra_1 @ A ) )
=> ( ( sin @ A )
= ( ^ [X3: A] :
( suminf @ A
@ ^ [N4: nat] : ( real_V8093663219630862766scaleR @ A @ ( sin_coeff @ N4 ) @ ( power_power @ A @ X3 @ N4 ) ) ) ) ) ) ).
% sin_def
thf(fact_4429_cos__def,axiom,
! [A: $tType] :
( ( ( real_Vector_banach @ A )
& ( real_V2822296259951069270ebra_1 @ A ) )
=> ( ( cos @ A )
= ( ^ [X3: A] :
( suminf @ A
@ ^ [N4: nat] : ( real_V8093663219630862766scaleR @ A @ ( cos_coeff @ N4 ) @ ( power_power @ A @ X3 @ N4 ) ) ) ) ) ) ).
% cos_def
thf(fact_4430_abs__le__square__iff,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [X2: A,Y3: A] :
( ( ord_less_eq @ A @ ( abs_abs @ A @ X2 ) @ ( abs_abs @ A @ Y3 ) )
= ( ord_less_eq @ A @ ( power_power @ A @ X2 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ ( power_power @ A @ Y3 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) ).
% abs_le_square_iff
thf(fact_4431_abs__square__eq__1,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [X2: A] :
( ( ( power_power @ A @ X2 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) )
= ( one_one @ A ) )
= ( ( abs_abs @ A @ X2 )
= ( one_one @ A ) ) ) ) ).
% abs_square_eq_1
thf(fact_4432_power__even__abs,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [N: nat,A3: A] :
( ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N )
=> ( ( power_power @ A @ ( abs_abs @ A @ A3 ) @ N )
= ( power_power @ A @ A3 @ N ) ) ) ) ).
% power_even_abs
thf(fact_4433_abs__sqrt__wlog,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [P: A > A > $o,X2: A] :
( ! [X4: A] :
( ( ord_less_eq @ A @ ( zero_zero @ A ) @ X4 )
=> ( P @ X4 @ ( power_power @ A @ X4 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) )
=> ( P @ ( abs_abs @ A @ X2 ) @ ( power_power @ A @ X2 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) ).
% abs_sqrt_wlog
thf(fact_4434_power2__le__iff__abs__le,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [Y3: A,X2: A] :
( ( ord_less_eq @ A @ ( zero_zero @ A ) @ Y3 )
=> ( ( ord_less_eq @ A @ ( power_power @ A @ X2 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ ( power_power @ A @ Y3 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) )
= ( ord_less_eq @ A @ ( abs_abs @ A @ X2 ) @ Y3 ) ) ) ) ).
% power2_le_iff_abs_le
thf(fact_4435_abs__square__le__1,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [X2: A] :
( ( ord_less_eq @ A @ ( power_power @ A @ X2 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ ( one_one @ A ) )
= ( ord_less_eq @ A @ ( abs_abs @ A @ X2 ) @ ( one_one @ A ) ) ) ) ).
% abs_square_le_1
thf(fact_4436_abs__square__less__1,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [X2: A] :
( ( ord_less @ A @ ( power_power @ A @ X2 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ ( one_one @ A ) )
= ( ord_less @ A @ ( abs_abs @ A @ X2 ) @ ( one_one @ A ) ) ) ) ).
% abs_square_less_1
thf(fact_4437_power__mono__even,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [N: nat,A3: A,B3: A] :
( ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N )
=> ( ( ord_less_eq @ A @ ( abs_abs @ A @ A3 ) @ ( abs_abs @ A @ B3 ) )
=> ( ord_less_eq @ A @ ( power_power @ A @ A3 @ N ) @ ( power_power @ A @ B3 @ N ) ) ) ) ) ).
% power_mono_even
thf(fact_4438_tan__45,axiom,
( ( tan @ real @ ( divide_divide @ real @ pi @ ( numeral_numeral @ real @ ( bit0 @ ( bit0 @ one2 ) ) ) ) )
= ( one_one @ real ) ) ).
% tan_45
thf(fact_4439_convex__sum__bound__le,axiom,
! [A: $tType,B: $tType] :
( ( linordered_idom @ B )
=> ! [I6: set @ A,X2: A > B,A3: A > B,B3: B,Delta: B] :
( ! [I2: A] :
( ( member @ A @ I2 @ I6 )
=> ( ord_less_eq @ B @ ( zero_zero @ B ) @ ( X2 @ I2 ) ) )
=> ( ( ( groups7311177749621191930dd_sum @ A @ B @ X2 @ I6 )
= ( one_one @ B ) )
=> ( ! [I2: A] :
( ( member @ A @ I2 @ I6 )
=> ( ord_less_eq @ B @ ( abs_abs @ B @ ( minus_minus @ B @ ( A3 @ I2 ) @ B3 ) ) @ Delta ) )
=> ( ord_less_eq @ B
@ ( abs_abs @ B
@ ( minus_minus @ B
@ ( groups7311177749621191930dd_sum @ A @ B
@ ^ [I4: A] : ( times_times @ B @ ( A3 @ I4 ) @ ( X2 @ I4 ) )
@ I6 )
@ B3 ) )
@ Delta ) ) ) ) ) ).
% convex_sum_bound_le
thf(fact_4440_sin__cos__le1,axiom,
! [X2: real,Y3: real] : ( ord_less_eq @ real @ ( abs_abs @ real @ ( plus_plus @ real @ ( times_times @ real @ ( sin @ real @ X2 ) @ ( sin @ real @ Y3 ) ) @ ( times_times @ real @ ( cos @ real @ X2 ) @ ( cos @ real @ Y3 ) ) ) ) @ ( one_one @ real ) ) ).
% sin_cos_le1
thf(fact_4441_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_4442_lemma__tan__total,axiom,
! [Y3: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ Y3 )
=> ? [X4: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ X4 )
& ( ord_less @ real @ X4 @ ( divide_divide @ real @ pi @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) )
& ( ord_less @ real @ Y3 @ ( tan @ real @ X4 ) ) ) ) ).
% lemma_tan_total
thf(fact_4443_tan__gt__zero,axiom,
! [X2: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ X2 )
=> ( ( ord_less @ real @ X2 @ ( divide_divide @ real @ pi @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) )
=> ( ord_less @ real @ ( zero_zero @ real ) @ ( tan @ real @ X2 ) ) ) ) ).
% tan_gt_zero
thf(fact_4444_tan__total,axiom,
! [Y3: real] :
? [X4: real] :
( ( ord_less @ real @ ( uminus_uminus @ real @ ( divide_divide @ real @ pi @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) ) @ X4 )
& ( ord_less @ real @ X4 @ ( divide_divide @ real @ pi @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) )
& ( ( tan @ real @ X4 )
= Y3 )
& ! [Y5: real] :
( ( ( ord_less @ real @ ( uminus_uminus @ real @ ( divide_divide @ real @ pi @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) ) @ Y5 )
& ( ord_less @ real @ Y5 @ ( divide_divide @ real @ pi @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) )
& ( ( tan @ real @ Y5 )
= Y3 ) )
=> ( Y5 = X4 ) ) ) ).
% tan_total
thf(fact_4445_tan__monotone,axiom,
! [Y3: real,X2: real] :
( ( ord_less @ real @ ( uminus_uminus @ real @ ( divide_divide @ real @ pi @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) ) @ Y3 )
=> ( ( ord_less @ real @ Y3 @ X2 )
=> ( ( ord_less @ real @ X2 @ ( divide_divide @ real @ pi @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) )
=> ( ord_less @ real @ ( tan @ real @ Y3 ) @ ( tan @ real @ X2 ) ) ) ) ) ).
% tan_monotone
thf(fact_4446_tan__monotone_H,axiom,
! [Y3: real,X2: real] :
( ( ord_less @ real @ ( uminus_uminus @ real @ ( divide_divide @ real @ pi @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) ) @ Y3 )
=> ( ( ord_less @ real @ Y3 @ ( divide_divide @ real @ pi @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) )
=> ( ( ord_less @ real @ ( uminus_uminus @ real @ ( divide_divide @ real @ pi @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) ) @ X2 )
=> ( ( ord_less @ real @ X2 @ ( divide_divide @ real @ pi @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) )
=> ( ( ord_less @ real @ Y3 @ X2 )
= ( ord_less @ real @ ( tan @ real @ Y3 ) @ ( tan @ real @ X2 ) ) ) ) ) ) ) ).
% tan_monotone'
thf(fact_4447_tan__mono__lt__eq,axiom,
! [X2: real,Y3: real] :
( ( ord_less @ real @ ( uminus_uminus @ real @ ( divide_divide @ real @ pi @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) ) @ X2 )
=> ( ( ord_less @ real @ X2 @ ( divide_divide @ real @ pi @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) )
=> ( ( ord_less @ real @ ( uminus_uminus @ real @ ( divide_divide @ real @ pi @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) ) @ Y3 )
=> ( ( ord_less @ real @ Y3 @ ( divide_divide @ real @ pi @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) )
=> ( ( ord_less @ real @ ( tan @ real @ X2 ) @ ( tan @ real @ Y3 ) )
= ( ord_less @ real @ X2 @ Y3 ) ) ) ) ) ) ).
% tan_mono_lt_eq
thf(fact_4448_lemma__tan__total1,axiom,
! [Y3: real] :
? [X4: real] :
( ( ord_less @ real @ ( uminus_uminus @ real @ ( divide_divide @ real @ pi @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) ) @ X4 )
& ( ord_less @ real @ X4 @ ( divide_divide @ real @ pi @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) )
& ( ( tan @ real @ X4 )
= Y3 ) ) ).
% lemma_tan_total1
thf(fact_4449_tan__minus__45,axiom,
( ( tan @ real @ ( uminus_uminus @ real @ ( divide_divide @ real @ pi @ ( numeral_numeral @ real @ ( bit0 @ ( bit0 @ one2 ) ) ) ) ) )
= ( uminus_uminus @ real @ ( one_one @ real ) ) ) ).
% tan_minus_45
thf(fact_4450_tan__inverse,axiom,
! [Y3: real] :
( ( divide_divide @ real @ ( one_one @ real ) @ ( tan @ real @ Y3 ) )
= ( tan @ real @ ( minus_minus @ real @ ( divide_divide @ real @ pi @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) @ Y3 ) ) ) ).
% tan_inverse
thf(fact_4451_add__tan__eq,axiom,
! [A: $tType] :
( ( ( real_Vector_banach @ A )
& ( real_V3459762299906320749_field @ A ) )
=> ! [X2: A,Y3: A] :
( ( ( cos @ A @ X2 )
!= ( zero_zero @ A ) )
=> ( ( ( cos @ A @ Y3 )
!= ( zero_zero @ A ) )
=> ( ( plus_plus @ A @ ( tan @ A @ X2 ) @ ( tan @ A @ Y3 ) )
= ( divide_divide @ A @ ( sin @ A @ ( plus_plus @ A @ X2 @ Y3 ) ) @ ( times_times @ A @ ( cos @ A @ X2 ) @ ( cos @ A @ Y3 ) ) ) ) ) ) ) ).
% add_tan_eq
thf(fact_4452_tan__total__pos,axiom,
! [Y3: real] :
( ( ord_less_eq @ real @ ( zero_zero @ real ) @ Y3 )
=> ? [X4: real] :
( ( ord_less_eq @ real @ ( zero_zero @ real ) @ X4 )
& ( ord_less @ real @ X4 @ ( divide_divide @ real @ pi @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) )
& ( ( tan @ real @ X4 )
= Y3 ) ) ) ).
% tan_total_pos
thf(fact_4453_tan__pos__pi2__le,axiom,
! [X2: real] :
( ( ord_less_eq @ real @ ( zero_zero @ real ) @ X2 )
=> ( ( ord_less @ real @ X2 @ ( divide_divide @ real @ pi @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) )
=> ( ord_less_eq @ real @ ( zero_zero @ real ) @ ( tan @ real @ X2 ) ) ) ) ).
% tan_pos_pi2_le
thf(fact_4454_tan__less__zero,axiom,
! [X2: real] :
( ( ord_less @ real @ ( divide_divide @ real @ ( uminus_uminus @ real @ pi ) @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) @ X2 )
=> ( ( ord_less @ real @ X2 @ ( zero_zero @ real ) )
=> ( ord_less @ real @ ( tan @ real @ X2 ) @ ( zero_zero @ real ) ) ) ) ).
% tan_less_zero
thf(fact_4455_tan__mono__le,axiom,
! [X2: real,Y3: real] :
( ( ord_less @ real @ ( uminus_uminus @ real @ ( divide_divide @ real @ pi @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) ) @ X2 )
=> ( ( ord_less_eq @ real @ X2 @ Y3 )
=> ( ( ord_less @ real @ Y3 @ ( divide_divide @ real @ pi @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) )
=> ( ord_less_eq @ real @ ( tan @ real @ X2 ) @ ( tan @ real @ Y3 ) ) ) ) ) ).
% tan_mono_le
thf(fact_4456_tan__mono__le__eq,axiom,
! [X2: real,Y3: real] :
( ( ord_less @ real @ ( uminus_uminus @ real @ ( divide_divide @ real @ pi @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) ) @ X2 )
=> ( ( ord_less @ real @ X2 @ ( divide_divide @ real @ pi @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) )
=> ( ( ord_less @ real @ ( uminus_uminus @ real @ ( divide_divide @ real @ pi @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) ) @ Y3 )
=> ( ( ord_less @ real @ Y3 @ ( divide_divide @ real @ pi @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) )
=> ( ( ord_less_eq @ real @ ( tan @ real @ X2 ) @ ( tan @ real @ Y3 ) )
= ( ord_less_eq @ real @ X2 @ Y3 ) ) ) ) ) ) ).
% tan_mono_le_eq
thf(fact_4457_lemma__tan__add1,axiom,
! [A: $tType] :
( ( ( real_Vector_banach @ A )
& ( real_V3459762299906320749_field @ A ) )
=> ! [X2: A,Y3: A] :
( ( ( cos @ A @ X2 )
!= ( zero_zero @ A ) )
=> ( ( ( cos @ A @ Y3 )
!= ( zero_zero @ A ) )
=> ( ( minus_minus @ A @ ( one_one @ A ) @ ( times_times @ A @ ( tan @ A @ X2 ) @ ( tan @ A @ Y3 ) ) )
= ( divide_divide @ A @ ( cos @ A @ ( plus_plus @ A @ X2 @ Y3 ) ) @ ( times_times @ A @ ( cos @ A @ X2 ) @ ( cos @ A @ Y3 ) ) ) ) ) ) ) ).
% lemma_tan_add1
thf(fact_4458_tan__diff,axiom,
! [A: $tType] :
( ( ( real_Vector_banach @ A )
& ( real_V3459762299906320749_field @ A ) )
=> ! [X2: A,Y3: A] :
( ( ( cos @ A @ X2 )
!= ( zero_zero @ A ) )
=> ( ( ( cos @ A @ Y3 )
!= ( zero_zero @ A ) )
=> ( ( ( cos @ A @ ( minus_minus @ A @ X2 @ Y3 ) )
!= ( zero_zero @ A ) )
=> ( ( tan @ A @ ( minus_minus @ A @ X2 @ Y3 ) )
= ( divide_divide @ A @ ( minus_minus @ A @ ( tan @ A @ X2 ) @ ( tan @ A @ Y3 ) ) @ ( plus_plus @ A @ ( one_one @ A ) @ ( times_times @ A @ ( tan @ A @ X2 ) @ ( tan @ A @ Y3 ) ) ) ) ) ) ) ) ) ).
% tan_diff
thf(fact_4459_tan__add,axiom,
! [A: $tType] :
( ( ( real_Vector_banach @ A )
& ( real_V3459762299906320749_field @ A ) )
=> ! [X2: A,Y3: A] :
( ( ( cos @ A @ X2 )
!= ( zero_zero @ A ) )
=> ( ( ( cos @ A @ Y3 )
!= ( zero_zero @ A ) )
=> ( ( ( cos @ A @ ( plus_plus @ A @ X2 @ Y3 ) )
!= ( zero_zero @ A ) )
=> ( ( tan @ A @ ( plus_plus @ A @ X2 @ Y3 ) )
= ( divide_divide @ A @ ( plus_plus @ A @ ( tan @ A @ X2 ) @ ( tan @ A @ Y3 ) ) @ ( minus_minus @ A @ ( one_one @ A ) @ ( times_times @ A @ ( tan @ A @ X2 ) @ ( tan @ A @ Y3 ) ) ) ) ) ) ) ) ) ).
% tan_add
thf(fact_4460_of__int__round__abs__le,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [X2: A] : ( ord_less_eq @ A @ ( abs_abs @ A @ ( minus_minus @ A @ ( ring_1_of_int @ A @ ( archimedean_round @ A @ X2 ) ) @ X2 ) ) @ ( divide_divide @ A @ ( one_one @ A ) @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) ) ) ).
% of_int_round_abs_le
thf(fact_4461_round__unique_H,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [X2: A,N: int] :
( ( ord_less @ A @ ( abs_abs @ A @ ( minus_minus @ A @ X2 @ ( ring_1_of_int @ A @ N ) ) ) @ ( divide_divide @ A @ ( one_one @ A ) @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) )
=> ( ( archimedean_round @ A @ X2 )
= N ) ) ) ).
% round_unique'
thf(fact_4462_tan__half,axiom,
! [A: $tType] :
( ( ( real_Vector_banach @ A )
& ( real_V3459762299906320749_field @ A ) )
=> ( ( tan @ A )
= ( ^ [X3: A] : ( divide_divide @ A @ ( sin @ A @ ( times_times @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ X3 ) ) @ ( plus_plus @ A @ ( cos @ A @ ( times_times @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ X3 ) ) @ ( one_one @ A ) ) ) ) ) ) ).
% tan_half
thf(fact_4463_monoseq__arctan__series,axiom,
! [X2: real] :
( ( ord_less_eq @ real @ ( abs_abs @ real @ X2 ) @ ( one_one @ real ) )
=> ( topological_monoseq @ real
@ ^ [N4: nat] : ( times_times @ real @ ( divide_divide @ real @ ( one_one @ real ) @ ( semiring_1_of_nat @ real @ ( plus_plus @ nat @ ( times_times @ nat @ N4 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ ( one_one @ nat ) ) ) ) @ ( power_power @ real @ X2 @ ( plus_plus @ nat @ ( times_times @ nat @ N4 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ ( one_one @ nat ) ) ) ) ) ) ).
% monoseq_arctan_series
thf(fact_4464_arctan__series,axiom,
! [X2: real] :
( ( ord_less_eq @ real @ ( abs_abs @ real @ X2 ) @ ( one_one @ real ) )
=> ( ( arctan @ X2 )
= ( 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 @ X2 @ ( plus_plus @ nat @ ( times_times @ nat @ K3 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ ( one_one @ nat ) ) ) ) ) ) ) ) ).
% arctan_series
thf(fact_4465_ln__series,axiom,
! [X2: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ X2 )
=> ( ( ord_less @ real @ X2 @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) )
=> ( ( ln_ln @ real @ X2 )
= ( suminf @ real
@ ^ [N4: nat] : ( times_times @ real @ ( times_times @ real @ ( power_power @ real @ ( uminus_uminus @ real @ ( one_one @ real ) ) @ N4 ) @ ( divide_divide @ real @ ( one_one @ real ) @ ( semiring_1_of_nat @ real @ ( plus_plus @ nat @ N4 @ ( one_one @ nat ) ) ) ) ) @ ( power_power @ real @ ( minus_minus @ real @ X2 @ ( one_one @ real ) ) @ ( suc @ N4 ) ) ) ) ) ) ) ).
% ln_series
thf(fact_4466_summable__arctan__series,axiom,
! [X2: real] :
( ( ord_less_eq @ real @ ( abs_abs @ real @ X2 ) @ ( 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 @ X2 @ ( plus_plus @ nat @ ( times_times @ nat @ K3 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ ( one_one @ nat ) ) ) ) ) ) ) ).
% summable_arctan_series
thf(fact_4467_zdvd1__eq,axiom,
! [X2: int] :
( ( dvd_dvd @ int @ X2 @ ( one_one @ int ) )
= ( ( abs_abs @ int @ X2 )
= ( one_one @ int ) ) ) ).
% zdvd1_eq
thf(fact_4468_summable__zero,axiom,
! [A: $tType] :
( ( ( comm_monoid_add @ A )
& ( topolo4958980785337419405_space @ A ) )
=> ( summable @ A
@ ^ [N4: nat] : ( zero_zero @ A ) ) ) ).
% summable_zero
thf(fact_4469_summable__single,axiom,
! [A: $tType] :
( ( ( comm_monoid_add @ A )
& ( topolo4958980785337419405_space @ A ) )
=> ! [I: nat,F2: nat > A] :
( summable @ A
@ ^ [R5: nat] : ( if @ A @ ( R5 = I ) @ ( F2 @ R5 ) @ ( zero_zero @ A ) ) ) ) ).
% summable_single
thf(fact_4470_summable__iff__shift,axiom,
! [A: $tType] :
( ( real_V822414075346904944vector @ A )
=> ! [F2: nat > A,K: nat] :
( ( summable @ A
@ ^ [N4: nat] : ( F2 @ ( plus_plus @ nat @ N4 @ K ) ) )
= ( summable @ A @ F2 ) ) ) ).
% summable_iff_shift
thf(fact_4471_ln__one,axiom,
! [A: $tType] :
( ( ln @ A )
=> ( ( ln_ln @ A @ ( one_one @ A ) )
= ( zero_zero @ A ) ) ) ).
% ln_one
thf(fact_4472_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_4473_arctan__le__zero__iff,axiom,
! [X2: real] :
( ( ord_less_eq @ real @ ( arctan @ X2 ) @ ( zero_zero @ real ) )
= ( ord_less_eq @ real @ X2 @ ( zero_zero @ real ) ) ) ).
% arctan_le_zero_iff
thf(fact_4474_zero__le__arctan__iff,axiom,
! [X2: real] :
( ( ord_less_eq @ real @ ( zero_zero @ real ) @ ( arctan @ X2 ) )
= ( ord_less_eq @ real @ ( zero_zero @ real ) @ X2 ) ) ).
% zero_le_arctan_iff
thf(fact_4475_summable__cmult__iff,axiom,
! [A: $tType] :
( ( real_V3459762299906320749_field @ A )
=> ! [C3: A,F2: nat > A] :
( ( summable @ A
@ ^ [N4: nat] : ( times_times @ A @ C3 @ ( F2 @ N4 ) ) )
= ( ( C3
= ( zero_zero @ A ) )
| ( summable @ A @ F2 ) ) ) ) ).
% summable_cmult_iff
thf(fact_4476_summable__divide__iff,axiom,
! [A: $tType] :
( ( real_V3459762299906320749_field @ A )
=> ! [F2: nat > A,C3: A] :
( ( summable @ A
@ ^ [N4: nat] : ( divide_divide @ A @ ( F2 @ N4 ) @ C3 ) )
= ( ( C3
= ( zero_zero @ A ) )
| ( summable @ A @ F2 ) ) ) ) ).
% summable_divide_iff
thf(fact_4477_summable__If__finite__set,axiom,
! [A: $tType] :
( ( ( comm_monoid_add @ A )
& ( topolo4958980785337419405_space @ A ) )
=> ! [A2: set @ nat,F2: nat > A] :
( ( finite_finite2 @ nat @ A2 )
=> ( summable @ A
@ ^ [R5: nat] : ( if @ A @ ( member @ nat @ R5 @ A2 ) @ ( F2 @ R5 ) @ ( zero_zero @ A ) ) ) ) ) ).
% summable_If_finite_set
thf(fact_4478_summable__If__finite,axiom,
! [A: $tType] :
( ( ( comm_monoid_add @ A )
& ( topolo4958980785337419405_space @ A ) )
=> ! [P: nat > $o,F2: nat > A] :
( ( finite_finite2 @ nat @ ( collect @ nat @ P ) )
=> ( summable @ A
@ ^ [R5: nat] : ( if @ A @ ( P @ R5 ) @ ( F2 @ R5 ) @ ( zero_zero @ A ) ) ) ) ) ).
% summable_If_finite
thf(fact_4479_ln__le__cancel__iff,axiom,
! [X2: real,Y3: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ X2 )
=> ( ( ord_less @ real @ ( zero_zero @ real ) @ Y3 )
=> ( ( ord_less_eq @ real @ ( ln_ln @ real @ X2 ) @ ( ln_ln @ real @ Y3 ) )
= ( ord_less_eq @ real @ X2 @ Y3 ) ) ) ) ).
% ln_le_cancel_iff
thf(fact_4480_ln__eq__zero__iff,axiom,
! [X2: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ X2 )
=> ( ( ( ln_ln @ real @ X2 )
= ( zero_zero @ real ) )
= ( X2
= ( one_one @ real ) ) ) ) ).
% ln_eq_zero_iff
thf(fact_4481_ln__gt__zero__iff,axiom,
! [X2: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ X2 )
=> ( ( ord_less @ real @ ( zero_zero @ real ) @ ( ln_ln @ real @ X2 ) )
= ( ord_less @ real @ ( one_one @ real ) @ X2 ) ) ) ).
% ln_gt_zero_iff
thf(fact_4482_ln__less__zero__iff,axiom,
! [X2: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ X2 )
=> ( ( ord_less @ real @ ( ln_ln @ real @ X2 ) @ ( zero_zero @ real ) )
= ( ord_less @ real @ X2 @ ( one_one @ real ) ) ) ) ).
% ln_less_zero_iff
thf(fact_4483_ln__ge__zero__iff,axiom,
! [X2: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ X2 )
=> ( ( ord_less_eq @ real @ ( zero_zero @ real ) @ ( ln_ln @ real @ X2 ) )
= ( ord_less_eq @ real @ ( one_one @ real ) @ X2 ) ) ) ).
% ln_ge_zero_iff
thf(fact_4484_ln__le__zero__iff,axiom,
! [X2: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ X2 )
=> ( ( ord_less_eq @ real @ ( ln_ln @ real @ X2 ) @ ( zero_zero @ real ) )
= ( ord_less_eq @ real @ X2 @ ( one_one @ real ) ) ) ) ).
% ln_le_zero_iff
thf(fact_4485_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_4486_summable__rabs__cancel,axiom,
! [F2: nat > real] :
( ( summable @ real
@ ^ [N4: nat] : ( abs_abs @ real @ ( F2 @ N4 ) ) )
=> ( summable @ real @ F2 ) ) ).
% summable_rabs_cancel
thf(fact_4487_summable__minus,axiom,
! [A: $tType] :
( ( real_V822414075346904944vector @ A )
=> ! [F2: nat > A] :
( ( summable @ A @ F2 )
=> ( summable @ A
@ ^ [N4: nat] : ( uminus_uminus @ A @ ( F2 @ N4 ) ) ) ) ) ).
% summable_minus
thf(fact_4488_summable__minus__iff,axiom,
! [A: $tType] :
( ( real_V822414075346904944vector @ A )
=> ! [F2: nat > A] :
( ( summable @ A
@ ^ [N4: nat] : ( uminus_uminus @ A @ ( F2 @ N4 ) ) )
= ( summable @ A @ F2 ) ) ) ).
% summable_minus_iff
thf(fact_4489_summable__ignore__initial__segment,axiom,
! [A: $tType] :
( ( real_V822414075346904944vector @ A )
=> ! [F2: nat > A,K: nat] :
( ( summable @ A @ F2 )
=> ( summable @ A
@ ^ [N4: nat] : ( F2 @ ( plus_plus @ nat @ N4 @ K ) ) ) ) ) ).
% summable_ignore_initial_segment
thf(fact_4490_arctan__monotone_H,axiom,
! [X2: real,Y3: real] :
( ( ord_less_eq @ real @ X2 @ Y3 )
=> ( ord_less_eq @ real @ ( arctan @ X2 ) @ ( arctan @ Y3 ) ) ) ).
% arctan_monotone'
thf(fact_4491_arctan__le__iff,axiom,
! [X2: real,Y3: real] :
( ( ord_less_eq @ real @ ( arctan @ X2 ) @ ( arctan @ Y3 ) )
= ( ord_less_eq @ real @ X2 @ Y3 ) ) ).
% arctan_le_iff
thf(fact_4492_summable__norm__cancel,axiom,
! [A: $tType] :
( ( real_Vector_banach @ A )
=> ! [F2: nat > A] :
( ( summable @ real
@ ^ [N4: nat] : ( real_V7770717601297561774m_norm @ A @ ( F2 @ N4 ) ) )
=> ( summable @ A @ F2 ) ) ) ).
% summable_norm_cancel
thf(fact_4493_summable__divide,axiom,
! [A: $tType] :
( ( real_V3459762299906320749_field @ A )
=> ! [F2: nat > A,C3: A] :
( ( summable @ A @ F2 )
=> ( summable @ A
@ ^ [N4: nat] : ( divide_divide @ A @ ( F2 @ N4 ) @ C3 ) ) ) ) ).
% summable_divide
thf(fact_4494_summable__diff,axiom,
! [A: $tType] :
( ( real_V822414075346904944vector @ A )
=> ! [F2: nat > A,G: nat > A] :
( ( summable @ A @ F2 )
=> ( ( summable @ A @ G )
=> ( summable @ A
@ ^ [N4: nat] : ( minus_minus @ A @ ( F2 @ N4 ) @ ( G @ N4 ) ) ) ) ) ) ).
% summable_diff
thf(fact_4495_summable__Suc__iff,axiom,
! [A: $tType] :
( ( real_V822414075346904944vector @ A )
=> ! [F2: nat > A] :
( ( summable @ A
@ ^ [N4: nat] : ( F2 @ ( suc @ N4 ) ) )
= ( summable @ A @ F2 ) ) ) ).
% summable_Suc_iff
thf(fact_4496_summable__mult2,axiom,
! [A: $tType] :
( ( real_V4412858255891104859lgebra @ A )
=> ! [F2: nat > A,C3: A] :
( ( summable @ A @ F2 )
=> ( summable @ A
@ ^ [N4: nat] : ( times_times @ A @ ( F2 @ N4 ) @ C3 ) ) ) ) ).
% summable_mult2
thf(fact_4497_summable__mult,axiom,
! [A: $tType] :
( ( real_V4412858255891104859lgebra @ A )
=> ! [F2: nat > A,C3: A] :
( ( summable @ A @ F2 )
=> ( summable @ A
@ ^ [N4: nat] : ( times_times @ A @ C3 @ ( F2 @ N4 ) ) ) ) ) ).
% summable_mult
thf(fact_4498_summable__add,axiom,
! [A: $tType] :
( ( ( topolo5987344860129210374id_add @ A )
& ( topological_t2_space @ A ) )
=> ! [F2: nat > A,G: nat > A] :
( ( summable @ A @ F2 )
=> ( ( summable @ A @ G )
=> ( summable @ A
@ ^ [N4: nat] : ( plus_plus @ A @ ( F2 @ N4 ) @ ( G @ N4 ) ) ) ) ) ) ).
% summable_add
thf(fact_4499_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_4500_summable__comparison__test_H,axiom,
! [A: $tType] :
( ( real_Vector_banach @ A )
=> ! [G: nat > real,N7: nat,F2: nat > A] :
( ( summable @ real @ G )
=> ( ! [N3: nat] :
( ( ord_less_eq @ nat @ N7 @ N3 )
=> ( ord_less_eq @ real @ ( real_V7770717601297561774m_norm @ A @ ( F2 @ N3 ) ) @ ( G @ N3 ) ) )
=> ( summable @ A @ F2 ) ) ) ) ).
% summable_comparison_test'
thf(fact_4501_summable__comparison__test,axiom,
! [A: $tType] :
( ( real_Vector_banach @ A )
=> ! [F2: nat > A,G: nat > real] :
( ? [N10: nat] :
! [N3: nat] :
( ( ord_less_eq @ nat @ N10 @ N3 )
=> ( ord_less_eq @ real @ ( real_V7770717601297561774m_norm @ A @ ( F2 @ N3 ) ) @ ( G @ N3 ) ) )
=> ( ( summable @ real @ G )
=> ( summable @ A @ F2 ) ) ) ) ).
% summable_comparison_test
thf(fact_4502_summable__sum,axiom,
! [I8: $tType,A: $tType] :
( ( ( topolo5987344860129210374id_add @ A )
& ( topological_t2_space @ A ) )
=> ! [I6: set @ I8,F2: I8 > nat > A] :
( ! [I2: I8] :
( ( member @ I8 @ I2 @ I6 )
=> ( summable @ A @ ( F2 @ I2 ) ) )
=> ( summable @ A
@ ^ [N4: nat] :
( groups7311177749621191930dd_sum @ I8 @ A
@ ^ [I4: I8] : ( F2 @ I4 @ N4 )
@ I6 ) ) ) ) ).
% summable_sum
thf(fact_4503_summable__scaleR__right,axiom,
! [B: $tType] :
( ( real_V822414075346904944vector @ B )
=> ! [X: nat > B,R2: real] :
( ( summable @ B @ X )
=> ( summable @ B
@ ^ [N4: nat] : ( real_V8093663219630862766scaleR @ B @ R2 @ ( X @ N4 ) ) ) ) ) ).
% summable_scaleR_right
thf(fact_4504_summable__scaleR__left,axiom,
! [B: $tType] :
( ( real_V822414075346904944vector @ B )
=> ! [X: nat > real,X2: B] :
( ( summable @ real @ X )
=> ( summable @ B
@ ^ [N4: nat] : ( real_V8093663219630862766scaleR @ B @ ( X @ N4 ) @ X2 ) ) ) ) ).
% summable_scaleR_left
thf(fact_4505_powser__insidea,axiom,
! [A: $tType] :
( ( real_V8999393235501362500lgebra @ A )
=> ! [F2: nat > A,X2: A,Z: A] :
( ( summable @ A
@ ^ [N4: nat] : ( times_times @ A @ ( F2 @ N4 ) @ ( power_power @ A @ X2 @ N4 ) ) )
=> ( ( ord_less @ real @ ( real_V7770717601297561774m_norm @ A @ Z ) @ ( real_V7770717601297561774m_norm @ A @ X2 ) )
=> ( summable @ real
@ ^ [N4: nat] : ( real_V7770717601297561774m_norm @ A @ ( times_times @ A @ ( F2 @ N4 ) @ ( power_power @ A @ Z @ N4 ) ) ) ) ) ) ) ).
% powser_insidea
thf(fact_4506_suminf__le,axiom,
! [A: $tType] :
( ( ( ordere6911136660526730532id_add @ A )
& ( topolo1944317154257567458pology @ A ) )
=> ! [F2: nat > A,G: nat > A] :
( ! [N3: nat] : ( ord_less_eq @ A @ ( F2 @ N3 ) @ ( G @ N3 ) )
=> ( ( summable @ A @ F2 )
=> ( ( summable @ A @ G )
=> ( ord_less_eq @ A @ ( suminf @ A @ F2 ) @ ( suminf @ A @ G ) ) ) ) ) ) ).
% suminf_le
thf(fact_4507_abs__zmult__eq__1,axiom,
! [M: int,N: int] :
( ( ( abs_abs @ int @ ( times_times @ int @ M @ N ) )
= ( one_one @ int ) )
=> ( ( abs_abs @ int @ M )
= ( one_one @ int ) ) ) ).
% abs_zmult_eq_1
thf(fact_4508_summable__mult__D,axiom,
! [A: $tType] :
( ( real_V3459762299906320749_field @ A )
=> ! [C3: A,F2: nat > A] :
( ( summable @ A
@ ^ [N4: nat] : ( times_times @ A @ C3 @ ( F2 @ N4 ) ) )
=> ( ( C3
!= ( zero_zero @ A ) )
=> ( summable @ A @ F2 ) ) ) ) ).
% summable_mult_D
thf(fact_4509_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_4510_suminf__add,axiom,
! [A: $tType] :
( ( ( topolo5987344860129210374id_add @ A )
& ( topological_t2_space @ A ) )
=> ! [F2: nat > A,G: nat > A] :
( ( summable @ A @ F2 )
=> ( ( summable @ A @ G )
=> ( ( plus_plus @ A @ ( suminf @ A @ F2 ) @ ( suminf @ A @ G ) )
= ( suminf @ A
@ ^ [N4: nat] : ( plus_plus @ A @ ( F2 @ N4 ) @ ( G @ N4 ) ) ) ) ) ) ) ).
% suminf_add
thf(fact_4511_suminf__mult2,axiom,
! [A: $tType] :
( ( real_V4412858255891104859lgebra @ A )
=> ! [F2: nat > A,C3: A] :
( ( summable @ A @ F2 )
=> ( ( times_times @ A @ ( suminf @ A @ F2 ) @ C3 )
= ( suminf @ A
@ ^ [N4: nat] : ( times_times @ A @ ( F2 @ N4 ) @ C3 ) ) ) ) ) ).
% suminf_mult2
thf(fact_4512_suminf__mult,axiom,
! [A: $tType] :
( ( real_V4412858255891104859lgebra @ A )
=> ! [F2: nat > A,C3: A] :
( ( summable @ A @ F2 )
=> ( ( suminf @ A
@ ^ [N4: nat] : ( times_times @ A @ C3 @ ( F2 @ N4 ) ) )
= ( times_times @ A @ C3 @ ( suminf @ A @ F2 ) ) ) ) ) ).
% suminf_mult
thf(fact_4513_suminf__diff,axiom,
! [A: $tType] :
( ( real_V822414075346904944vector @ A )
=> ! [F2: nat > A,G: nat > A] :
( ( summable @ A @ F2 )
=> ( ( summable @ A @ G )
=> ( ( minus_minus @ A @ ( suminf @ A @ F2 ) @ ( suminf @ A @ G ) )
= ( suminf @ A
@ ^ [N4: nat] : ( minus_minus @ A @ ( F2 @ N4 ) @ ( G @ N4 ) ) ) ) ) ) ) ).
% suminf_diff
thf(fact_4514_suminf__divide,axiom,
! [A: $tType] :
( ( real_V3459762299906320749_field @ A )
=> ! [F2: nat > A,C3: A] :
( ( summable @ A @ F2 )
=> ( ( suminf @ A
@ ^ [N4: nat] : ( divide_divide @ A @ ( F2 @ N4 ) @ C3 ) )
= ( divide_divide @ A @ ( suminf @ A @ F2 ) @ C3 ) ) ) ) ).
% suminf_divide
thf(fact_4515_suminf__minus,axiom,
! [A: $tType] :
( ( real_V822414075346904944vector @ A )
=> ! [F2: nat > A] :
( ( summable @ A @ F2 )
=> ( ( suminf @ A
@ ^ [N4: nat] : ( uminus_uminus @ A @ ( F2 @ N4 ) ) )
= ( uminus_uminus @ A @ ( suminf @ A @ F2 ) ) ) ) ) ).
% suminf_minus
thf(fact_4516_suminf__sum,axiom,
! [A: $tType,I8: $tType] :
( ( ( topolo5987344860129210374id_add @ A )
& ( topological_t2_space @ A ) )
=> ! [I6: set @ I8,F2: I8 > nat > A] :
( ! [I2: I8] :
( ( member @ I8 @ I2 @ I6 )
=> ( summable @ A @ ( F2 @ I2 ) ) )
=> ( ( suminf @ A
@ ^ [N4: nat] :
( groups7311177749621191930dd_sum @ I8 @ A
@ ^ [I4: I8] : ( F2 @ I4 @ N4 )
@ I6 ) )
= ( groups7311177749621191930dd_sum @ I8 @ A
@ ^ [I4: I8] : ( suminf @ A @ ( F2 @ I4 ) )
@ I6 ) ) ) ) ).
% suminf_sum
thf(fact_4517_suminf__scaleR__right,axiom,
! [B: $tType] :
( ( real_V822414075346904944vector @ B )
=> ! [X: nat > B,R2: real] :
( ( summable @ B @ X )
=> ( ( real_V8093663219630862766scaleR @ B @ R2 @ ( suminf @ B @ X ) )
= ( suminf @ B
@ ^ [N4: nat] : ( real_V8093663219630862766scaleR @ B @ R2 @ ( X @ N4 ) ) ) ) ) ) ).
% suminf_scaleR_right
thf(fact_4518_summable__Cauchy__product,axiom,
! [A: $tType] :
( ( ( real_Vector_banach @ A )
& ( real_V4412858255891104859lgebra @ A ) )
=> ! [A3: nat > A,B3: nat > A] :
( ( summable @ real
@ ^ [K3: nat] : ( real_V7770717601297561774m_norm @ A @ ( A3 @ K3 ) ) )
=> ( ( summable @ real
@ ^ [K3: nat] : ( real_V7770717601297561774m_norm @ A @ ( B3 @ K3 ) ) )
=> ( summable @ A
@ ^ [K3: nat] :
( groups7311177749621191930dd_sum @ nat @ A
@ ^ [I4: nat] : ( times_times @ A @ ( A3 @ I4 ) @ ( B3 @ ( minus_minus @ nat @ K3 @ I4 ) ) )
@ ( set_ord_atMost @ nat @ K3 ) ) ) ) ) ) ).
% summable_Cauchy_product
thf(fact_4519_infinite__int__iff__unbounded__le,axiom,
! [S: set @ int] :
( ( ~ ( finite_finite2 @ int @ S ) )
= ( ! [M6: int] :
? [N4: int] :
( ( ord_less_eq @ int @ M6 @ ( abs_abs @ int @ N4 ) )
& ( member @ int @ N4 @ S ) ) ) ) ).
% infinite_int_iff_unbounded_le
thf(fact_4520_suminf__nonneg,axiom,
! [A: $tType] :
( ( ( ordere6911136660526730532id_add @ A )
& ( topolo1944317154257567458pology @ A ) )
=> ! [F2: nat > A] :
( ( summable @ A @ F2 )
=> ( ! [N3: nat] : ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( F2 @ N3 ) )
=> ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( suminf @ A @ F2 ) ) ) ) ) ).
% suminf_nonneg
thf(fact_4521_suminf__eq__zero__iff,axiom,
! [A: $tType] :
( ( ( ordere6911136660526730532id_add @ A )
& ( topolo1944317154257567458pology @ A ) )
=> ! [F2: nat > A] :
( ( summable @ A @ F2 )
=> ( ! [N3: nat] : ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( F2 @ N3 ) )
=> ( ( ( suminf @ A @ F2 )
= ( zero_zero @ A ) )
= ( ! [N4: nat] :
( ( F2 @ N4 )
= ( zero_zero @ A ) ) ) ) ) ) ) ).
% suminf_eq_zero_iff
thf(fact_4522_suminf__pos,axiom,
! [A: $tType] :
( ( ( ordere6911136660526730532id_add @ A )
& ( topolo1944317154257567458pology @ A ) )
=> ! [F2: nat > A] :
( ( summable @ A @ F2 )
=> ( ! [N3: nat] : ( ord_less @ A @ ( zero_zero @ A ) @ ( F2 @ N3 ) )
=> ( ord_less @ A @ ( zero_zero @ A ) @ ( suminf @ A @ F2 ) ) ) ) ) ).
% suminf_pos
thf(fact_4523_ln__bound,axiom,
! [X2: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ X2 )
=> ( ord_less_eq @ real @ ( ln_ln @ real @ X2 ) @ X2 ) ) ).
% ln_bound
thf(fact_4524_ln__gt__zero,axiom,
! [X2: real] :
( ( ord_less @ real @ ( one_one @ real ) @ X2 )
=> ( ord_less @ real @ ( zero_zero @ real ) @ ( ln_ln @ real @ X2 ) ) ) ).
% ln_gt_zero
thf(fact_4525_ln__less__zero,axiom,
! [X2: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ X2 )
=> ( ( ord_less @ real @ X2 @ ( one_one @ real ) )
=> ( ord_less @ real @ ( ln_ln @ real @ X2 ) @ ( zero_zero @ real ) ) ) ) ).
% ln_less_zero
thf(fact_4526_ln__gt__zero__imp__gt__one,axiom,
! [X2: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ ( ln_ln @ real @ X2 ) )
=> ( ( ord_less @ real @ ( zero_zero @ real ) @ X2 )
=> ( ord_less @ real @ ( one_one @ real ) @ X2 ) ) ) ).
% ln_gt_zero_imp_gt_one
thf(fact_4527_ln__ge__zero,axiom,
! [X2: real] :
( ( ord_less_eq @ real @ ( one_one @ real ) @ X2 )
=> ( ord_less_eq @ real @ ( zero_zero @ real ) @ ( ln_ln @ real @ X2 ) ) ) ).
% ln_ge_zero
thf(fact_4528_summable__zero__power_H,axiom,
! [A: $tType] :
( ( ( ring_1 @ A )
& ( topolo4958980785337419405_space @ A ) )
=> ! [F2: nat > A] :
( summable @ A
@ ^ [N4: nat] : ( times_times @ A @ ( F2 @ N4 ) @ ( power_power @ A @ ( zero_zero @ A ) @ N4 ) ) ) ) ).
% summable_zero_power'
thf(fact_4529_summable__0__powser,axiom,
! [A: $tType] :
( ( real_V8999393235501362500lgebra @ A )
=> ! [F2: nat > A] :
( summable @ A
@ ^ [N4: nat] : ( times_times @ A @ ( F2 @ N4 ) @ ( power_power @ A @ ( zero_zero @ A ) @ N4 ) ) ) ) ).
% summable_0_powser
thf(fact_4530_powser__split__head_I3_J,axiom,
! [A: $tType] :
( ( ( real_Vector_banach @ A )
& ( real_V8999393235501362500lgebra @ A ) )
=> ! [F2: nat > A,Z: A] :
( ( summable @ A
@ ^ [N4: nat] : ( times_times @ A @ ( F2 @ N4 ) @ ( power_power @ A @ Z @ N4 ) ) )
=> ( summable @ A
@ ^ [N4: nat] : ( times_times @ A @ ( F2 @ ( suc @ N4 ) ) @ ( power_power @ A @ Z @ N4 ) ) ) ) ) ).
% powser_split_head(3)
thf(fact_4531_summable__powser__split__head,axiom,
! [A: $tType] :
( ( real_V8999393235501362500lgebra @ A )
=> ! [F2: nat > A,Z: A] :
( ( summable @ A
@ ^ [N4: nat] : ( times_times @ A @ ( F2 @ ( suc @ N4 ) ) @ ( power_power @ A @ Z @ N4 ) ) )
= ( summable @ A
@ ^ [N4: nat] : ( times_times @ A @ ( F2 @ N4 ) @ ( power_power @ A @ Z @ N4 ) ) ) ) ) ).
% summable_powser_split_head
thf(fact_4532_summable__powser__ignore__initial__segment,axiom,
! [A: $tType] :
( ( real_V8999393235501362500lgebra @ A )
=> ! [F2: nat > A,M: nat,Z: A] :
( ( summable @ A
@ ^ [N4: nat] : ( times_times @ A @ ( F2 @ ( plus_plus @ nat @ N4 @ M ) ) @ ( power_power @ A @ Z @ N4 ) ) )
= ( summable @ A
@ ^ [N4: nat] : ( times_times @ A @ ( F2 @ N4 ) @ ( power_power @ A @ Z @ N4 ) ) ) ) ) ).
% summable_powser_ignore_initial_segment
thf(fact_4533_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_4534_nat__abs__mult__distrib,axiom,
! [W: int,Z: int] :
( ( nat2 @ ( abs_abs @ int @ ( times_times @ int @ W @ Z ) ) )
= ( times_times @ nat @ ( nat2 @ ( abs_abs @ int @ W ) ) @ ( nat2 @ ( abs_abs @ int @ Z ) ) ) ) ).
% nat_abs_mult_distrib
thf(fact_4535_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_4536_summable__norm__comparison__test,axiom,
! [A: $tType] :
( ( real_V822414075346904944vector @ A )
=> ! [F2: nat > A,G: nat > real] :
( ? [N10: nat] :
! [N3: nat] :
( ( ord_less_eq @ nat @ N10 @ N3 )
=> ( ord_less_eq @ real @ ( real_V7770717601297561774m_norm @ A @ ( F2 @ N3 ) ) @ ( G @ N3 ) ) )
=> ( ( summable @ real @ G )
=> ( summable @ real
@ ^ [N4: nat] : ( real_V7770717601297561774m_norm @ A @ ( F2 @ N4 ) ) ) ) ) ) ).
% summable_norm_comparison_test
thf(fact_4537_summable__rabs__comparison__test,axiom,
! [F2: nat > real,G: nat > real] :
( ? [N10: nat] :
! [N3: nat] :
( ( ord_less_eq @ nat @ N10 @ N3 )
=> ( ord_less_eq @ real @ ( abs_abs @ real @ ( F2 @ N3 ) ) @ ( G @ N3 ) ) )
=> ( ( summable @ real @ G )
=> ( summable @ real
@ ^ [N4: nat] : ( abs_abs @ real @ ( F2 @ N4 ) ) ) ) ) ).
% summable_rabs_comparison_test
thf(fact_4538_summable__rabs,axiom,
! [F2: nat > real] :
( ( summable @ real
@ ^ [N4: nat] : ( abs_abs @ real @ ( F2 @ N4 ) ) )
=> ( ord_less_eq @ real @ ( abs_abs @ real @ ( suminf @ real @ F2 ) )
@ ( suminf @ real
@ ^ [N4: nat] : ( abs_abs @ real @ ( F2 @ N4 ) ) ) ) ) ).
% summable_rabs
thf(fact_4539_suminf__scaleR__left,axiom,
! [B: $tType] :
( ( real_V822414075346904944vector @ B )
=> ! [X: nat > real,X2: B] :
( ( summable @ real @ X )
=> ( ( real_V8093663219630862766scaleR @ B @ ( suminf @ real @ X ) @ X2 )
= ( suminf @ B
@ ^ [N4: nat] : ( real_V8093663219630862766scaleR @ B @ ( X @ N4 ) @ X2 ) ) ) ) ) ).
% suminf_scaleR_left
thf(fact_4540_suminf__pos__iff,axiom,
! [A: $tType] :
( ( ( ordere6911136660526730532id_add @ A )
& ( topolo1944317154257567458pology @ A ) )
=> ! [F2: nat > A] :
( ( summable @ A @ F2 )
=> ( ! [N3: nat] : ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( F2 @ N3 ) )
=> ( ( ord_less @ A @ ( zero_zero @ A ) @ ( suminf @ A @ F2 ) )
= ( ? [I4: nat] : ( ord_less @ A @ ( zero_zero @ A ) @ ( F2 @ I4 ) ) ) ) ) ) ) ).
% suminf_pos_iff
thf(fact_4541_suminf__pos2,axiom,
! [A: $tType] :
( ( ( ordere6911136660526730532id_add @ A )
& ( topolo1944317154257567458pology @ A ) )
=> ! [F2: nat > A,I: nat] :
( ( summable @ A @ F2 )
=> ( ! [N3: nat] : ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( F2 @ N3 ) )
=> ( ( ord_less @ A @ ( zero_zero @ A ) @ ( F2 @ I ) )
=> ( ord_less @ A @ ( zero_zero @ A ) @ ( suminf @ A @ F2 ) ) ) ) ) ) ).
% suminf_pos2
thf(fact_4542_ln__ge__zero__imp__ge__one,axiom,
! [X2: real] :
( ( ord_less_eq @ real @ ( zero_zero @ real ) @ ( ln_ln @ real @ X2 ) )
=> ( ( ord_less @ real @ ( zero_zero @ real ) @ X2 )
=> ( ord_less_eq @ real @ ( one_one @ real ) @ X2 ) ) ) ).
% ln_ge_zero_imp_ge_one
thf(fact_4543_suminf__le__const,axiom,
! [A: $tType] :
( ( ( ordere6911136660526730532id_add @ A )
& ( topolo1944317154257567458pology @ A ) )
=> ! [F2: nat > A,X2: A] :
( ( summable @ A @ F2 )
=> ( ! [N3: nat] : ( ord_less_eq @ A @ ( groups7311177749621191930dd_sum @ nat @ A @ F2 @ ( set_ord_lessThan @ nat @ N3 ) ) @ X2 )
=> ( ord_less_eq @ A @ ( suminf @ A @ F2 ) @ X2 ) ) ) ) ).
% suminf_le_const
thf(fact_4544_ln__add__one__self__le__self,axiom,
! [X2: real] :
( ( ord_less_eq @ real @ ( zero_zero @ real ) @ X2 )
=> ( ord_less_eq @ real @ ( ln_ln @ real @ ( plus_plus @ real @ ( one_one @ real ) @ X2 ) ) @ X2 ) ) ).
% ln_add_one_self_le_self
thf(fact_4545_ln__mult,axiom,
! [X2: real,Y3: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ X2 )
=> ( ( ord_less @ real @ ( zero_zero @ real ) @ Y3 )
=> ( ( ln_ln @ real @ ( times_times @ real @ X2 @ Y3 ) )
= ( plus_plus @ real @ ( ln_ln @ real @ X2 ) @ ( ln_ln @ real @ Y3 ) ) ) ) ) ).
% ln_mult
thf(fact_4546_ln__eq__minus__one,axiom,
! [X2: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ X2 )
=> ( ( ( ln_ln @ real @ X2 )
= ( minus_minus @ real @ X2 @ ( one_one @ real ) ) )
=> ( X2
= ( one_one @ real ) ) ) ) ).
% ln_eq_minus_one
thf(fact_4547_ln__div,axiom,
! [X2: real,Y3: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ X2 )
=> ( ( ord_less @ real @ ( zero_zero @ real ) @ Y3 )
=> ( ( ln_ln @ real @ ( divide_divide @ real @ X2 @ Y3 ) )
= ( minus_minus @ real @ ( ln_ln @ real @ X2 ) @ ( ln_ln @ real @ Y3 ) ) ) ) ) ).
% ln_div
thf(fact_4548_powser__inside,axiom,
! [A: $tType] :
( ( ( real_Vector_banach @ A )
& ( real_V8999393235501362500lgebra @ A ) )
=> ! [F2: nat > A,X2: A,Z: A] :
( ( summable @ A
@ ^ [N4: nat] : ( times_times @ A @ ( F2 @ N4 ) @ ( power_power @ A @ X2 @ N4 ) ) )
=> ( ( ord_less @ real @ ( real_V7770717601297561774m_norm @ A @ Z ) @ ( real_V7770717601297561774m_norm @ A @ X2 ) )
=> ( summable @ A
@ ^ [N4: nat] : ( times_times @ A @ ( F2 @ N4 ) @ ( power_power @ A @ Z @ N4 ) ) ) ) ) ) ).
% powser_inside
thf(fact_4549_suminf__split__head,axiom,
! [A: $tType] :
( ( real_V822414075346904944vector @ A )
=> ! [F2: nat > A] :
( ( summable @ A @ F2 )
=> ( ( suminf @ A
@ ^ [N4: nat] : ( F2 @ ( suc @ N4 ) ) )
= ( minus_minus @ A @ ( suminf @ A @ F2 ) @ ( F2 @ ( zero_zero @ nat ) ) ) ) ) ) ).
% suminf_split_head
thf(fact_4550_summableI__nonneg__bounded,axiom,
! [A: $tType] :
( ( ( condit6923001295902523014norder @ A )
& ( ordere6911136660526730532id_add @ A )
& ( topolo1944317154257567458pology @ A ) )
=> ! [F2: nat > A,X2: A] :
( ! [N3: nat] : ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( F2 @ N3 ) )
=> ( ! [N3: nat] : ( ord_less_eq @ A @ ( groups7311177749621191930dd_sum @ nat @ A @ F2 @ ( set_ord_lessThan @ nat @ N3 ) ) @ X2 )
=> ( summable @ A @ F2 ) ) ) ) ).
% summableI_nonneg_bounded
thf(fact_4551_bounded__imp__summable,axiom,
! [A: $tType] :
( ( ( condit6923001295902523014norder @ A )
& ( linord2810124833399127020strict @ A )
& ( topolo1944317154257567458pology @ A ) )
=> ! [A3: nat > A,B2: A] :
( ! [N3: nat] : ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( A3 @ N3 ) )
=> ( ! [N3: nat] : ( ord_less_eq @ A @ ( groups7311177749621191930dd_sum @ nat @ A @ A3 @ ( set_ord_atMost @ nat @ N3 ) ) @ B2 )
=> ( summable @ A @ A3 ) ) ) ) ).
% bounded_imp_summable
thf(fact_4552_complete__algebra__summable__geometric,axiom,
! [A: $tType] :
( ( ( real_Vector_banach @ A )
& ( real_V2822296259951069270ebra_1 @ A ) )
=> ! [X2: A] :
( ( ord_less @ real @ ( real_V7770717601297561774m_norm @ A @ X2 ) @ ( one_one @ real ) )
=> ( summable @ A @ ( power_power @ A @ X2 ) ) ) ) ).
% complete_algebra_summable_geometric
thf(fact_4553_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_4554_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_4555_zdvd__mult__cancel1,axiom,
! [M: int,N: int] :
( ( M
!= ( zero_zero @ int ) )
=> ( ( dvd_dvd @ int @ ( times_times @ int @ M @ N ) @ M )
= ( ( abs_abs @ int @ N )
= ( one_one @ int ) ) ) ) ).
% zdvd_mult_cancel1
thf(fact_4556_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_4557_summable__norm,axiom,
! [A: $tType] :
( ( real_Vector_banach @ A )
=> ! [F2: nat > A] :
( ( summable @ real
@ ^ [N4: nat] : ( real_V7770717601297561774m_norm @ A @ ( F2 @ N4 ) ) )
=> ( ord_less_eq @ real @ ( real_V7770717601297561774m_norm @ A @ ( suminf @ A @ F2 ) )
@ ( suminf @ real
@ ^ [N4: nat] : ( real_V7770717601297561774m_norm @ A @ ( F2 @ N4 ) ) ) ) ) ) ).
% summable_norm
thf(fact_4558_summable__norm__sin,axiom,
! [A: $tType] :
( ( ( real_Vector_banach @ A )
& ( real_V2822296259951069270ebra_1 @ A ) )
=> ! [X2: A] :
( summable @ real
@ ^ [N4: nat] : ( real_V7770717601297561774m_norm @ A @ ( real_V8093663219630862766scaleR @ A @ ( sin_coeff @ N4 ) @ ( power_power @ A @ X2 @ N4 ) ) ) ) ) ).
% summable_norm_sin
thf(fact_4559_summable__norm__cos,axiom,
! [A: $tType] :
( ( ( real_Vector_banach @ A )
& ( real_V2822296259951069270ebra_1 @ A ) )
=> ! [X2: A] :
( summable @ real
@ ^ [N4: nat] : ( real_V7770717601297561774m_norm @ A @ ( real_V8093663219630862766scaleR @ A @ ( cos_coeff @ N4 ) @ ( power_power @ A @ X2 @ N4 ) ) ) ) ) ).
% summable_norm_cos
thf(fact_4560_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_4561_sum__le__suminf,axiom,
! [A: $tType] :
( ( ( ordere6911136660526730532id_add @ A )
& ( topolo1944317154257567458pology @ A ) )
=> ! [F2: nat > A,I6: set @ nat] :
( ( summable @ A @ F2 )
=> ( ( finite_finite2 @ nat @ I6 )
=> ( ! [N3: nat] :
( ( member @ nat @ N3 @ ( uminus_uminus @ ( set @ nat ) @ I6 ) )
=> ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( F2 @ N3 ) ) )
=> ( ord_less_eq @ A @ ( groups7311177749621191930dd_sum @ nat @ A @ F2 @ I6 ) @ ( suminf @ A @ F2 ) ) ) ) ) ) ).
% sum_le_suminf
thf(fact_4562_ln__le__minus__one,axiom,
! [X2: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ X2 )
=> ( ord_less_eq @ real @ ( ln_ln @ real @ X2 ) @ ( minus_minus @ real @ X2 @ ( one_one @ real ) ) ) ) ).
% ln_le_minus_one
thf(fact_4563_ln__diff__le,axiom,
! [X2: real,Y3: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ X2 )
=> ( ( ord_less @ real @ ( zero_zero @ real ) @ Y3 )
=> ( ord_less_eq @ real @ ( minus_minus @ real @ ( ln_ln @ real @ X2 ) @ ( ln_ln @ real @ Y3 ) ) @ ( divide_divide @ real @ ( minus_minus @ real @ X2 @ Y3 ) @ Y3 ) ) ) ) ).
% ln_diff_le
thf(fact_4564_ln__add__one__self__le__self2,axiom,
! [X2: real] :
( ( ord_less @ real @ ( uminus_uminus @ real @ ( one_one @ real ) ) @ X2 )
=> ( ord_less_eq @ real @ ( ln_ln @ real @ ( plus_plus @ real @ ( one_one @ real ) @ X2 ) ) @ X2 ) ) ).
% ln_add_one_self_le_self2
thf(fact_4565_even__add__abs__iff,axiom,
! [K: int,L: int] :
( ( dvd_dvd @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ ( plus_plus @ int @ K @ ( abs_abs @ int @ L ) ) )
= ( dvd_dvd @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ ( plus_plus @ int @ K @ L ) ) ) ).
% even_add_abs_iff
thf(fact_4566_even__abs__add__iff,axiom,
! [K: int,L: int] :
( ( dvd_dvd @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ ( plus_plus @ int @ ( abs_abs @ int @ K ) @ L ) )
= ( dvd_dvd @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ ( plus_plus @ int @ K @ L ) ) ) ).
% even_abs_add_iff
thf(fact_4567_suminf__split__initial__segment,axiom,
! [A: $tType] :
( ( real_V822414075346904944vector @ A )
=> ! [F2: nat > A,K: nat] :
( ( summable @ A @ F2 )
=> ( ( suminf @ A @ F2 )
= ( plus_plus @ A
@ ( suminf @ A
@ ^ [N4: nat] : ( F2 @ ( plus_plus @ nat @ N4 @ K ) ) )
@ ( groups7311177749621191930dd_sum @ nat @ A @ F2 @ ( set_ord_lessThan @ nat @ K ) ) ) ) ) ) ).
% suminf_split_initial_segment
thf(fact_4568_suminf__minus__initial__segment,axiom,
! [A: $tType] :
( ( real_V822414075346904944vector @ A )
=> ! [F2: nat > A,K: nat] :
( ( summable @ A @ F2 )
=> ( ( suminf @ A
@ ^ [N4: nat] : ( F2 @ ( plus_plus @ nat @ N4 @ K ) ) )
= ( minus_minus @ A @ ( suminf @ A @ F2 ) @ ( groups7311177749621191930dd_sum @ nat @ A @ F2 @ ( set_ord_lessThan @ nat @ K ) ) ) ) ) ) ).
% suminf_minus_initial_segment
thf(fact_4569_nat__abs__int__diff,axiom,
! [A3: nat,B3: nat] :
( ( ( ord_less_eq @ nat @ A3 @ B3 )
=> ( ( nat2 @ ( abs_abs @ int @ ( minus_minus @ int @ ( semiring_1_of_nat @ int @ A3 ) @ ( semiring_1_of_nat @ int @ B3 ) ) ) )
= ( minus_minus @ nat @ B3 @ A3 ) ) )
& ( ~ ( ord_less_eq @ nat @ A3 @ B3 )
=> ( ( nat2 @ ( abs_abs @ int @ ( minus_minus @ int @ ( semiring_1_of_nat @ int @ A3 ) @ ( semiring_1_of_nat @ int @ B3 ) ) ) )
= ( minus_minus @ nat @ A3 @ B3 ) ) ) ) ).
% nat_abs_int_diff
thf(fact_4570_sum__less__suminf,axiom,
! [A: $tType] :
( ( ( ordere8940638589300402666id_add @ A )
& ( topolo1944317154257567458pology @ A ) )
=> ! [F2: nat > A,N: nat] :
( ( summable @ A @ F2 )
=> ( ! [M5: nat] :
( ( ord_less_eq @ nat @ N @ M5 )
=> ( ord_less @ A @ ( zero_zero @ A ) @ ( F2 @ M5 ) ) )
=> ( ord_less @ A @ ( groups7311177749621191930dd_sum @ nat @ A @ F2 @ ( set_ord_lessThan @ nat @ N ) ) @ ( suminf @ A @ F2 ) ) ) ) ) ).
% sum_less_suminf
thf(fact_4571_summable__Cauchy,axiom,
! [A: $tType] :
( ( real_Vector_banach @ A )
=> ( ( summable @ A )
= ( ^ [F5: nat > A] :
! [E4: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ E4 )
=> ? [N8: nat] :
! [M6: nat] :
( ( ord_less_eq @ nat @ N8 @ M6 )
=> ! [N4: nat] : ( ord_less @ real @ ( real_V7770717601297561774m_norm @ A @ ( groups7311177749621191930dd_sum @ nat @ A @ F5 @ ( set_or7035219750837199246ssThan @ nat @ M6 @ N4 ) ) ) @ E4 ) ) ) ) ) ) ).
% summable_Cauchy
thf(fact_4572_powser__split__head_I1_J,axiom,
! [A: $tType] :
( ( ( real_Vector_banach @ A )
& ( real_V8999393235501362500lgebra @ A ) )
=> ! [F2: nat > A,Z: A] :
( ( summable @ A
@ ^ [N4: nat] : ( times_times @ A @ ( F2 @ N4 ) @ ( power_power @ A @ Z @ N4 ) ) )
=> ( ( suminf @ A
@ ^ [N4: nat] : ( times_times @ A @ ( F2 @ N4 ) @ ( power_power @ A @ Z @ N4 ) ) )
= ( plus_plus @ A @ ( F2 @ ( zero_zero @ nat ) )
@ ( times_times @ A
@ ( suminf @ A
@ ^ [N4: nat] : ( times_times @ A @ ( F2 @ ( suc @ N4 ) ) @ ( power_power @ A @ Z @ N4 ) ) )
@ Z ) ) ) ) ) ).
% powser_split_head(1)
thf(fact_4573_powser__split__head_I2_J,axiom,
! [A: $tType] :
( ( ( real_Vector_banach @ A )
& ( real_V8999393235501362500lgebra @ A ) )
=> ! [F2: nat > A,Z: A] :
( ( summable @ A
@ ^ [N4: nat] : ( times_times @ A @ ( F2 @ N4 ) @ ( power_power @ A @ Z @ N4 ) ) )
=> ( ( times_times @ A
@ ( suminf @ A
@ ^ [N4: nat] : ( times_times @ A @ ( F2 @ ( suc @ N4 ) ) @ ( power_power @ A @ Z @ N4 ) ) )
@ Z )
= ( minus_minus @ A
@ ( suminf @ A
@ ^ [N4: nat] : ( times_times @ A @ ( F2 @ N4 ) @ ( power_power @ A @ Z @ N4 ) ) )
@ ( F2 @ ( zero_zero @ nat ) ) ) ) ) ) ).
% powser_split_head(2)
thf(fact_4574_ln__one__minus__pos__upper__bound,axiom,
! [X2: real] :
( ( ord_less_eq @ real @ ( zero_zero @ real ) @ X2 )
=> ( ( ord_less @ real @ X2 @ ( one_one @ real ) )
=> ( ord_less_eq @ real @ ( ln_ln @ real @ ( minus_minus @ real @ ( one_one @ real ) @ X2 ) ) @ ( uminus_uminus @ real @ X2 ) ) ) ) ).
% ln_one_minus_pos_upper_bound
thf(fact_4575_ln__powr__bound,axiom,
! [X2: real,A3: real] :
( ( ord_less_eq @ real @ ( one_one @ real ) @ X2 )
=> ( ( ord_less @ real @ ( zero_zero @ real ) @ A3 )
=> ( ord_less_eq @ real @ ( ln_ln @ real @ X2 ) @ ( divide_divide @ real @ ( powr @ real @ X2 @ A3 ) @ A3 ) ) ) ) ).
% ln_powr_bound
thf(fact_4576_ln__powr__bound2,axiom,
! [X2: real,A3: real] :
( ( ord_less @ real @ ( one_one @ real ) @ X2 )
=> ( ( ord_less @ real @ ( zero_zero @ real ) @ A3 )
=> ( ord_less_eq @ real @ ( powr @ real @ ( ln_ln @ real @ X2 ) @ A3 ) @ ( times_times @ real @ ( powr @ real @ A3 @ A3 ) @ X2 ) ) ) ) ).
% ln_powr_bound2
thf(fact_4577_ln__prod,axiom,
! [A: $tType,I6: set @ A,F2: A > real] :
( ( finite_finite2 @ A @ I6 )
=> ( ! [I2: A] :
( ( member @ A @ I2 @ I6 )
=> ( ord_less @ real @ ( zero_zero @ real ) @ ( F2 @ I2 ) ) )
=> ( ( ln_ln @ real @ ( groups7121269368397514597t_prod @ A @ real @ F2 @ I6 ) )
= ( groups7311177749621191930dd_sum @ A @ real
@ ^ [X3: A] : ( ln_ln @ real @ ( F2 @ X3 ) )
@ I6 ) ) ) ) ).
% ln_prod
thf(fact_4578_log__eq__div__ln__mult__log,axiom,
! [A3: real,B3: real,X2: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ A3 )
=> ( ( A3
!= ( one_one @ real ) )
=> ( ( ord_less @ real @ ( zero_zero @ real ) @ B3 )
=> ( ( B3
!= ( one_one @ real ) )
=> ( ( ord_less @ real @ ( zero_zero @ real ) @ X2 )
=> ( ( log @ A3 @ X2 )
= ( times_times @ real @ ( divide_divide @ real @ ( ln_ln @ real @ B3 ) @ ( ln_ln @ real @ A3 ) ) @ ( log @ B3 @ X2 ) ) ) ) ) ) ) ) ).
% log_eq_div_ln_mult_log
thf(fact_4579_monoseq__realpow,axiom,
! [X2: real] :
( ( ord_less_eq @ real @ ( zero_zero @ real ) @ X2 )
=> ( ( ord_less_eq @ real @ X2 @ ( one_one @ real ) )
=> ( topological_monoseq @ real @ ( power_power @ real @ X2 ) ) ) ) ).
% monoseq_realpow
thf(fact_4580_suminf__exist__split,axiom,
! [A: $tType] :
( ( real_V822414075346904944vector @ A )
=> ! [R2: real,F2: nat > A] :
( ( ord_less @ real @ ( zero_zero @ real ) @ R2 )
=> ( ( summable @ A @ F2 )
=> ? [N11: nat] :
! [N9: nat] :
( ( ord_less_eq @ nat @ N11 @ N9 )
=> ( ord_less @ real
@ ( real_V7770717601297561774m_norm @ A
@ ( suminf @ A
@ ^ [I4: nat] : ( F2 @ ( plus_plus @ nat @ I4 @ N9 ) ) ) )
@ R2 ) ) ) ) ) ).
% suminf_exist_split
thf(fact_4581_summable__partial__sum__bound,axiom,
! [A: $tType] :
( ( real_Vector_banach @ A )
=> ! [F2: nat > A,E: real] :
( ( summable @ A @ F2 )
=> ( ( ord_less @ real @ ( zero_zero @ real ) @ E )
=> ~ ! [N11: nat] :
~ ! [M3: nat] :
( ( ord_less_eq @ nat @ N11 @ M3 )
=> ! [N9: nat] : ( ord_less @ real @ ( real_V7770717601297561774m_norm @ A @ ( groups7311177749621191930dd_sum @ nat @ A @ F2 @ ( set_or1337092689740270186AtMost @ nat @ M3 @ N9 ) ) ) @ E ) ) ) ) ) ).
% summable_partial_sum_bound
thf(fact_4582_summable__power__series,axiom,
! [F2: nat > real,Z: real] :
( ! [I2: nat] : ( ord_less_eq @ real @ ( F2 @ I2 ) @ ( one_one @ real ) )
=> ( ! [I2: nat] : ( ord_less_eq @ real @ ( zero_zero @ real ) @ ( F2 @ I2 ) )
=> ( ( ord_less_eq @ real @ ( zero_zero @ real ) @ Z )
=> ( ( ord_less @ real @ Z @ ( one_one @ real ) )
=> ( summable @ real
@ ^ [I4: nat] : ( times_times @ real @ ( F2 @ I4 ) @ ( power_power @ real @ Z @ I4 ) ) ) ) ) ) ) ).
% summable_power_series
thf(fact_4583_Abel__lemma,axiom,
! [A: $tType] :
( ( real_V822414075346904944vector @ A )
=> ! [R2: real,R0: real,A3: nat > A,M8: real] :
( ( ord_less_eq @ real @ ( zero_zero @ real ) @ R2 )
=> ( ( ord_less @ real @ R2 @ R0 )
=> ( ! [N3: nat] : ( ord_less_eq @ real @ ( times_times @ real @ ( real_V7770717601297561774m_norm @ A @ ( A3 @ N3 ) ) @ ( power_power @ real @ R0 @ N3 ) ) @ M8 )
=> ( summable @ real
@ ^ [N4: nat] : ( times_times @ real @ ( real_V7770717601297561774m_norm @ A @ ( A3 @ N4 ) ) @ ( power_power @ real @ R2 @ N4 ) ) ) ) ) ) ) ).
% Abel_lemma
thf(fact_4584_nat__intermed__int__val,axiom,
! [M: nat,N: nat,F2: nat > int,K: int] :
( ! [I2: nat] :
( ( ( ord_less_eq @ nat @ M @ I2 )
& ( ord_less @ nat @ I2 @ N ) )
=> ( ord_less_eq @ int @ ( abs_abs @ int @ ( minus_minus @ int @ ( F2 @ ( suc @ I2 ) ) @ ( F2 @ I2 ) ) ) @ ( one_one @ int ) ) )
=> ( ( ord_less_eq @ nat @ M @ N )
=> ( ( ord_less_eq @ int @ ( F2 @ M ) @ K )
=> ( ( ord_less_eq @ int @ K @ ( F2 @ N ) )
=> ? [I2: nat] :
( ( ord_less_eq @ nat @ M @ I2 )
& ( ord_less_eq @ nat @ I2 @ N )
& ( ( F2 @ I2 )
= K ) ) ) ) ) ) ).
% nat_intermed_int_val
thf(fact_4585_Cauchy__product,axiom,
! [A: $tType] :
( ( ( real_Vector_banach @ A )
& ( real_V4412858255891104859lgebra @ A ) )
=> ! [A3: nat > A,B3: nat > A] :
( ( summable @ real
@ ^ [K3: nat] : ( real_V7770717601297561774m_norm @ A @ ( A3 @ K3 ) ) )
=> ( ( summable @ real
@ ^ [K3: nat] : ( real_V7770717601297561774m_norm @ A @ ( B3 @ K3 ) ) )
=> ( ( times_times @ A @ ( suminf @ A @ A3 ) @ ( suminf @ A @ B3 ) )
= ( suminf @ A
@ ^ [K3: nat] :
( groups7311177749621191930dd_sum @ nat @ A
@ ^ [I4: nat] : ( times_times @ A @ ( A3 @ I4 ) @ ( B3 @ ( minus_minus @ nat @ K3 @ I4 ) ) )
@ ( set_ord_atMost @ nat @ K3 ) ) ) ) ) ) ) ).
% Cauchy_product
thf(fact_4586_decr__lemma,axiom,
! [D2: int,X2: int,Z: int] :
( ( ord_less @ int @ ( zero_zero @ int ) @ D2 )
=> ( ord_less @ int @ ( minus_minus @ int @ X2 @ ( times_times @ int @ ( plus_plus @ int @ ( abs_abs @ int @ ( minus_minus @ int @ X2 @ Z ) ) @ ( one_one @ int ) ) @ D2 ) ) @ Z ) ) ).
% decr_lemma
thf(fact_4587_incr__lemma,axiom,
! [D2: int,Z: int,X2: int] :
( ( ord_less @ int @ ( zero_zero @ int ) @ D2 )
=> ( ord_less @ int @ Z @ ( plus_plus @ int @ X2 @ ( times_times @ int @ ( plus_plus @ int @ ( abs_abs @ int @ ( minus_minus @ int @ X2 @ Z ) ) @ ( one_one @ int ) ) @ D2 ) ) ) ) ).
% incr_lemma
thf(fact_4588_summable__ratio__test,axiom,
! [A: $tType] :
( ( real_Vector_banach @ A )
=> ! [C3: real,N7: nat,F2: nat > A] :
( ( ord_less @ real @ C3 @ ( one_one @ real ) )
=> ( ! [N3: nat] :
( ( ord_less_eq @ nat @ N7 @ N3 )
=> ( ord_less_eq @ real @ ( real_V7770717601297561774m_norm @ A @ ( F2 @ ( suc @ N3 ) ) ) @ ( times_times @ real @ C3 @ ( real_V7770717601297561774m_norm @ A @ ( F2 @ N3 ) ) ) ) )
=> ( summable @ A @ F2 ) ) ) ) ).
% summable_ratio_test
thf(fact_4589_sum__less__suminf2,axiom,
! [A: $tType] :
( ( ( ordere8940638589300402666id_add @ A )
& ( topolo1944317154257567458pology @ A ) )
=> ! [F2: nat > A,N: nat,I: nat] :
( ( summable @ A @ F2 )
=> ( ! [M5: nat] :
( ( ord_less_eq @ nat @ N @ M5 )
=> ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( F2 @ M5 ) ) )
=> ( ( ord_less_eq @ nat @ N @ I )
=> ( ( ord_less @ A @ ( zero_zero @ A ) @ ( F2 @ I ) )
=> ( ord_less @ A @ ( groups7311177749621191930dd_sum @ nat @ A @ F2 @ ( set_ord_lessThan @ nat @ N ) ) @ ( suminf @ A @ F2 ) ) ) ) ) ) ) ).
% sum_less_suminf2
thf(fact_4590_arctan__ubound,axiom,
! [Y3: real] : ( ord_less @ real @ ( arctan @ Y3 ) @ ( divide_divide @ real @ pi @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) ) ).
% arctan_ubound
thf(fact_4591_arctan__one,axiom,
( ( arctan @ ( one_one @ real ) )
= ( divide_divide @ real @ pi @ ( numeral_numeral @ real @ ( bit0 @ ( bit0 @ one2 ) ) ) ) ) ).
% arctan_one
thf(fact_4592_nat__ivt__aux,axiom,
! [N: nat,F2: nat > int,K: int] :
( ! [I2: nat] :
( ( ord_less @ nat @ I2 @ N )
=> ( ord_less_eq @ int @ ( abs_abs @ int @ ( minus_minus @ int @ ( F2 @ ( suc @ I2 ) ) @ ( F2 @ I2 ) ) ) @ ( one_one @ int ) ) )
=> ( ( ord_less_eq @ int @ ( F2 @ ( zero_zero @ nat ) ) @ K )
=> ( ( ord_less_eq @ int @ K @ ( F2 @ N ) )
=> ? [I2: nat] :
( ( ord_less_eq @ nat @ I2 @ N )
& ( ( F2 @ I2 )
= K ) ) ) ) ) ).
% nat_ivt_aux
thf(fact_4593_Cauchy__product__sums,axiom,
! [A: $tType] :
( ( ( real_Vector_banach @ A )
& ( real_V4412858255891104859lgebra @ A ) )
=> ! [A3: nat > A,B3: nat > A] :
( ( summable @ real
@ ^ [K3: nat] : ( real_V7770717601297561774m_norm @ A @ ( A3 @ K3 ) ) )
=> ( ( summable @ real
@ ^ [K3: nat] : ( real_V7770717601297561774m_norm @ A @ ( B3 @ K3 ) ) )
=> ( sums @ A
@ ^ [K3: nat] :
( groups7311177749621191930dd_sum @ nat @ A
@ ^ [I4: nat] : ( times_times @ A @ ( A3 @ I4 ) @ ( B3 @ ( minus_minus @ nat @ K3 @ I4 ) ) )
@ ( set_ord_atMost @ nat @ K3 ) )
@ ( times_times @ A @ ( suminf @ A @ A3 ) @ ( suminf @ A @ B3 ) ) ) ) ) ) ).
% Cauchy_product_sums
thf(fact_4594_arctan__lbound,axiom,
! [Y3: real] : ( ord_less @ real @ ( uminus_uminus @ real @ ( divide_divide @ real @ pi @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) ) @ ( arctan @ Y3 ) ) ).
% arctan_lbound
thf(fact_4595_arctan__bounded,axiom,
! [Y3: real] :
( ( ord_less @ real @ ( uminus_uminus @ real @ ( divide_divide @ real @ pi @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) ) @ ( arctan @ Y3 ) )
& ( ord_less @ real @ ( arctan @ Y3 ) @ ( divide_divide @ real @ pi @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) ) ) ).
% arctan_bounded
thf(fact_4596_arctan,axiom,
! [Y3: real] :
( ( ord_less @ real @ ( uminus_uminus @ real @ ( divide_divide @ real @ pi @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) ) @ ( arctan @ Y3 ) )
& ( ord_less @ real @ ( arctan @ Y3 ) @ ( divide_divide @ real @ pi @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) )
& ( ( tan @ real @ ( arctan @ Y3 ) )
= Y3 ) ) ).
% arctan
thf(fact_4597_arctan__tan,axiom,
! [X2: real] :
( ( ord_less @ real @ ( uminus_uminus @ real @ ( divide_divide @ real @ pi @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) ) @ X2 )
=> ( ( ord_less @ real @ X2 @ ( divide_divide @ real @ pi @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) )
=> ( ( arctan @ ( tan @ real @ X2 ) )
= X2 ) ) ) ).
% arctan_tan
thf(fact_4598_arctan__unique,axiom,
! [X2: real,Y3: real] :
( ( ord_less @ real @ ( uminus_uminus @ real @ ( divide_divide @ real @ pi @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) ) @ X2 )
=> ( ( ord_less @ real @ X2 @ ( divide_divide @ real @ pi @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) )
=> ( ( ( tan @ real @ X2 )
= Y3 )
=> ( ( arctan @ Y3 )
= X2 ) ) ) ) ).
% arctan_unique
thf(fact_4599_nat0__intermed__int__val,axiom,
! [N: nat,F2: nat > int,K: int] :
( ! [I2: nat] :
( ( ord_less @ nat @ I2 @ N )
=> ( ord_less_eq @ int @ ( abs_abs @ int @ ( minus_minus @ int @ ( F2 @ ( plus_plus @ nat @ I2 @ ( one_one @ nat ) ) ) @ ( F2 @ I2 ) ) ) @ ( one_one @ int ) ) )
=> ( ( ord_less_eq @ int @ ( F2 @ ( zero_zero @ nat ) ) @ K )
=> ( ( ord_less_eq @ int @ K @ ( F2 @ N ) )
=> ? [I2: nat] :
( ( ord_less_eq @ nat @ I2 @ N )
& ( ( F2 @ I2 )
= K ) ) ) ) ) ).
% nat0_intermed_int_val
thf(fact_4600_ln__one__plus__pos__lower__bound,axiom,
! [X2: real] :
( ( ord_less_eq @ real @ ( zero_zero @ real ) @ X2 )
=> ( ( ord_less_eq @ real @ X2 @ ( one_one @ real ) )
=> ( ord_less_eq @ real @ ( minus_minus @ real @ X2 @ ( power_power @ real @ X2 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( ln_ln @ real @ ( plus_plus @ real @ ( one_one @ real ) @ X2 ) ) ) ) ) ).
% ln_one_plus_pos_lower_bound
thf(fact_4601_arctan__add,axiom,
! [X2: real,Y3: real] :
( ( ord_less_eq @ real @ ( abs_abs @ real @ X2 ) @ ( one_one @ real ) )
=> ( ( ord_less @ real @ ( abs_abs @ real @ Y3 ) @ ( one_one @ real ) )
=> ( ( plus_plus @ real @ ( arctan @ X2 ) @ ( arctan @ Y3 ) )
= ( arctan @ ( divide_divide @ real @ ( plus_plus @ real @ X2 @ Y3 ) @ ( minus_minus @ real @ ( one_one @ real ) @ ( times_times @ real @ X2 @ Y3 ) ) ) ) ) ) ) ).
% arctan_add
thf(fact_4602_abs__ln__one__plus__x__minus__x__bound__nonneg,axiom,
! [X2: real] :
( ( ord_less_eq @ real @ ( zero_zero @ real ) @ X2 )
=> ( ( ord_less_eq @ real @ X2 @ ( one_one @ real ) )
=> ( ord_less_eq @ real @ ( abs_abs @ real @ ( minus_minus @ real @ ( ln_ln @ real @ ( plus_plus @ real @ ( one_one @ real ) @ X2 ) ) @ X2 ) ) @ ( power_power @ real @ X2 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) ).
% abs_ln_one_plus_x_minus_x_bound_nonneg
thf(fact_4603_machin__Euler,axiom,
( ( plus_plus @ real @ ( times_times @ real @ ( numeral_numeral @ real @ ( bit1 @ ( bit0 @ one2 ) ) ) @ ( arctan @ ( divide_divide @ real @ ( one_one @ real ) @ ( numeral_numeral @ real @ ( bit1 @ ( bit1 @ one2 ) ) ) ) ) ) @ ( times_times @ real @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) @ ( arctan @ ( divide_divide @ real @ ( numeral_numeral @ real @ ( bit1 @ one2 ) ) @ ( numeral_numeral @ real @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit0 @ ( bit0 @ one2 ) ) ) ) ) ) ) ) ) ) )
= ( divide_divide @ real @ pi @ ( numeral_numeral @ real @ ( bit0 @ ( bit0 @ one2 ) ) ) ) ) ).
% machin_Euler
thf(fact_4604_machin,axiom,
( ( divide_divide @ real @ pi @ ( numeral_numeral @ real @ ( bit0 @ ( bit0 @ one2 ) ) ) )
= ( minus_minus @ real @ ( times_times @ real @ ( numeral_numeral @ real @ ( bit0 @ ( bit0 @ one2 ) ) ) @ ( arctan @ ( divide_divide @ real @ ( one_one @ real ) @ ( numeral_numeral @ real @ ( bit1 @ ( bit0 @ one2 ) ) ) ) ) ) @ ( arctan @ ( divide_divide @ real @ ( one_one @ real ) @ ( numeral_numeral @ real @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit0 @ ( bit1 @ ( bit1 @ one2 ) ) ) ) ) ) ) ) ) ) ) ) ).
% machin
thf(fact_4605_sum__pos__lt__pair,axiom,
! [F2: nat > real,K: nat] :
( ( summable @ real @ F2 )
=> ( ! [D4: nat] : ( ord_less @ real @ ( zero_zero @ real ) @ ( plus_plus @ real @ ( F2 @ ( plus_plus @ nat @ K @ ( times_times @ nat @ ( suc @ ( suc @ ( zero_zero @ nat ) ) ) @ D4 ) ) ) @ ( F2 @ ( plus_plus @ nat @ K @ ( plus_plus @ nat @ ( times_times @ nat @ ( suc @ ( suc @ ( zero_zero @ nat ) ) ) @ D4 ) @ ( one_one @ nat ) ) ) ) ) )
=> ( ord_less @ real @ ( groups7311177749621191930dd_sum @ nat @ real @ F2 @ ( set_ord_lessThan @ nat @ K ) ) @ ( suminf @ real @ F2 ) ) ) ) ).
% sum_pos_lt_pair
thf(fact_4606_ln__one__minus__pos__lower__bound,axiom,
! [X2: real] :
( ( ord_less_eq @ real @ ( zero_zero @ real ) @ X2 )
=> ( ( ord_less_eq @ real @ X2 @ ( divide_divide @ real @ ( one_one @ real ) @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) )
=> ( ord_less_eq @ real @ ( minus_minus @ real @ ( uminus_uminus @ real @ X2 ) @ ( times_times @ real @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) @ ( power_power @ real @ X2 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( ln_ln @ real @ ( minus_minus @ real @ ( one_one @ real ) @ X2 ) ) ) ) ) ).
% ln_one_minus_pos_lower_bound
thf(fact_4607_abs__ln__one__plus__x__minus__x__bound,axiom,
! [X2: real] :
( ( ord_less_eq @ real @ ( abs_abs @ real @ X2 ) @ ( 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 ) @ X2 ) ) @ X2 ) ) @ ( times_times @ real @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) @ ( power_power @ real @ X2 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) ).
% abs_ln_one_plus_x_minus_x_bound
thf(fact_4608_arctan__double,axiom,
! [X2: real] :
( ( ord_less @ real @ ( abs_abs @ real @ X2 ) @ ( one_one @ real ) )
=> ( ( times_times @ real @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) @ ( arctan @ X2 ) )
= ( arctan @ ( divide_divide @ real @ ( times_times @ real @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) @ X2 ) @ ( minus_minus @ real @ ( one_one @ real ) @ ( power_power @ real @ X2 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) ) ) ).
% arctan_double
thf(fact_4609_abs__ln__one__plus__x__minus__x__bound__nonpos,axiom,
! [X2: real] :
( ( ord_less_eq @ real @ ( uminus_uminus @ real @ ( divide_divide @ real @ ( one_one @ real ) @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) ) @ X2 )
=> ( ( ord_less_eq @ real @ X2 @ ( zero_zero @ real ) )
=> ( ord_less_eq @ real @ ( abs_abs @ real @ ( minus_minus @ real @ ( ln_ln @ real @ ( plus_plus @ real @ ( one_one @ real ) @ X2 ) ) @ X2 ) ) @ ( times_times @ real @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) @ ( power_power @ real @ X2 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) ) ).
% abs_ln_one_plus_x_minus_x_bound_nonpos
thf(fact_4610_artanh__def,axiom,
! [A: $tType] :
( ( ( real_V3459762299906320749_field @ A )
& ( ln @ A ) )
=> ( ( artanh @ A )
= ( ^ [X3: A] : ( divide_divide @ A @ ( ln_ln @ A @ ( divide_divide @ A @ ( plus_plus @ A @ ( one_one @ A ) @ X3 ) @ ( minus_minus @ A @ ( one_one @ A ) @ X3 ) ) ) @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) ) ) ) ).
% artanh_def
thf(fact_4611_diffs__equiv,axiom,
! [A: $tType] :
( ( ( real_V822414075346904944vector @ A )
& ( ring_1 @ A ) )
=> ! [C3: nat > A,X2: A] :
( ( summable @ A
@ ^ [N4: nat] : ( times_times @ A @ ( diffs @ A @ C3 @ N4 ) @ ( power_power @ A @ X2 @ N4 ) ) )
=> ( sums @ A
@ ^ [N4: nat] : ( times_times @ A @ ( times_times @ A @ ( semiring_1_of_nat @ A @ N4 ) @ ( C3 @ N4 ) ) @ ( power_power @ A @ X2 @ ( minus_minus @ nat @ N4 @ ( suc @ ( zero_zero @ nat ) ) ) ) )
@ ( suminf @ A
@ ^ [N4: nat] : ( times_times @ A @ ( diffs @ A @ C3 @ N4 ) @ ( power_power @ A @ X2 @ N4 ) ) ) ) ) ) ).
% diffs_equiv
thf(fact_4612_tanh__ln__real,axiom,
! [X2: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ X2 )
=> ( ( tanh @ real @ ( ln_ln @ real @ X2 ) )
= ( divide_divide @ real @ ( minus_minus @ real @ ( power_power @ real @ X2 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ ( one_one @ real ) ) @ ( plus_plus @ real @ ( power_power @ real @ X2 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ ( one_one @ real ) ) ) ) ) ).
% tanh_ln_real
thf(fact_4613_sin__tan,axiom,
! [X2: real] :
( ( ord_less @ real @ ( abs_abs @ real @ X2 ) @ ( divide_divide @ real @ pi @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) )
=> ( ( sin @ real @ X2 )
= ( divide_divide @ real @ ( tan @ real @ X2 ) @ ( sqrt @ ( plus_plus @ real @ ( one_one @ real ) @ ( power_power @ real @ ( tan @ real @ X2 ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) ) ) ).
% sin_tan
thf(fact_4614_real__sqrt__le__iff,axiom,
! [X2: real,Y3: real] :
( ( ord_less_eq @ real @ ( sqrt @ X2 ) @ ( sqrt @ Y3 ) )
= ( ord_less_eq @ real @ X2 @ Y3 ) ) ).
% real_sqrt_le_iff
thf(fact_4615_real__sqrt__eq__1__iff,axiom,
! [X2: real] :
( ( ( sqrt @ X2 )
= ( one_one @ real ) )
= ( X2
= ( one_one @ real ) ) ) ).
% real_sqrt_eq_1_iff
thf(fact_4616_real__sqrt__one,axiom,
( ( sqrt @ ( one_one @ real ) )
= ( one_one @ real ) ) ).
% real_sqrt_one
thf(fact_4617_tanh__real__le__iff,axiom,
! [X2: real,Y3: real] :
( ( ord_less_eq @ real @ ( tanh @ real @ X2 ) @ ( tanh @ real @ Y3 ) )
= ( ord_less_eq @ real @ X2 @ Y3 ) ) ).
% tanh_real_le_iff
thf(fact_4618_real__sqrt__ge__0__iff,axiom,
! [Y3: real] :
( ( ord_less_eq @ real @ ( zero_zero @ real ) @ ( sqrt @ Y3 ) )
= ( ord_less_eq @ real @ ( zero_zero @ real ) @ Y3 ) ) ).
% real_sqrt_ge_0_iff
thf(fact_4619_real__sqrt__le__0__iff,axiom,
! [X2: real] :
( ( ord_less_eq @ real @ ( sqrt @ X2 ) @ ( zero_zero @ real ) )
= ( ord_less_eq @ real @ X2 @ ( zero_zero @ real ) ) ) ).
% real_sqrt_le_0_iff
thf(fact_4620_real__sqrt__gt__1__iff,axiom,
! [Y3: real] :
( ( ord_less @ real @ ( one_one @ real ) @ ( sqrt @ Y3 ) )
= ( ord_less @ real @ ( one_one @ real ) @ Y3 ) ) ).
% real_sqrt_gt_1_iff
thf(fact_4621_real__sqrt__lt__1__iff,axiom,
! [X2: real] :
( ( ord_less @ real @ ( sqrt @ X2 ) @ ( one_one @ real ) )
= ( ord_less @ real @ X2 @ ( one_one @ real ) ) ) ).
% real_sqrt_lt_1_iff
thf(fact_4622_real__sqrt__le__1__iff,axiom,
! [X2: real] :
( ( ord_less_eq @ real @ ( sqrt @ X2 ) @ ( one_one @ real ) )
= ( ord_less_eq @ real @ X2 @ ( one_one @ real ) ) ) ).
% real_sqrt_le_1_iff
thf(fact_4623_real__sqrt__ge__1__iff,axiom,
! [Y3: real] :
( ( ord_less_eq @ real @ ( one_one @ real ) @ ( sqrt @ Y3 ) )
= ( ord_less_eq @ real @ ( one_one @ real ) @ Y3 ) ) ).
% real_sqrt_ge_1_iff
thf(fact_4624_tanh__real__nonpos__iff,axiom,
! [X2: real] :
( ( ord_less_eq @ real @ ( tanh @ real @ X2 ) @ ( zero_zero @ real ) )
= ( ord_less_eq @ real @ X2 @ ( zero_zero @ real ) ) ) ).
% tanh_real_nonpos_iff
thf(fact_4625_tanh__real__nonneg__iff,axiom,
! [X2: real] :
( ( ord_less_eq @ real @ ( zero_zero @ real ) @ ( tanh @ real @ X2 ) )
= ( ord_less_eq @ real @ ( zero_zero @ real ) @ X2 ) ) ).
% tanh_real_nonneg_iff
thf(fact_4626_real__sqrt__four,axiom,
( ( sqrt @ ( numeral_numeral @ real @ ( bit0 @ ( bit0 @ one2 ) ) ) )
= ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) ).
% real_sqrt_four
thf(fact_4627_artanh__minus__real,axiom,
! [X2: real] :
( ( ord_less @ real @ ( abs_abs @ real @ X2 ) @ ( one_one @ real ) )
=> ( ( artanh @ real @ ( uminus_uminus @ real @ X2 ) )
= ( uminus_uminus @ real @ ( artanh @ real @ X2 ) ) ) ) ).
% artanh_minus_real
thf(fact_4628_real__sqrt__abs,axiom,
! [X2: real] :
( ( sqrt @ ( power_power @ real @ X2 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) )
= ( abs_abs @ real @ X2 ) ) ).
% real_sqrt_abs
thf(fact_4629_real__sqrt__pow2__iff,axiom,
! [X2: real] :
( ( ( power_power @ real @ ( sqrt @ X2 ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) )
= X2 )
= ( ord_less_eq @ real @ ( zero_zero @ real ) @ X2 ) ) ).
% real_sqrt_pow2_iff
thf(fact_4630_real__sqrt__pow2,axiom,
! [X2: real] :
( ( ord_less_eq @ real @ ( zero_zero @ real ) @ X2 )
=> ( ( power_power @ real @ ( sqrt @ X2 ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) )
= X2 ) ) ).
% real_sqrt_pow2
thf(fact_4631_real__sqrt__sum__squares__mult__squared__eq,axiom,
! [X2: real,Y3: real,Xa: real,Ya: real] :
( ( power_power @ real @ ( sqrt @ ( times_times @ real @ ( plus_plus @ real @ ( power_power @ real @ X2 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ ( power_power @ real @ Y3 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( plus_plus @ real @ ( power_power @ real @ Xa @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ ( power_power @ real @ Ya @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) )
= ( times_times @ real @ ( plus_plus @ real @ ( power_power @ real @ X2 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ ( power_power @ real @ Y3 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( plus_plus @ real @ ( power_power @ real @ Xa @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ ( power_power @ real @ Ya @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) ).
% real_sqrt_sum_squares_mult_squared_eq
thf(fact_4632_real__sqrt__le__mono,axiom,
! [X2: real,Y3: real] :
( ( ord_less_eq @ real @ X2 @ Y3 )
=> ( ord_less_eq @ real @ ( sqrt @ X2 ) @ ( sqrt @ Y3 ) ) ) ).
% real_sqrt_le_mono
thf(fact_4633_real__sqrt__ge__zero,axiom,
! [X2: real] :
( ( ord_less_eq @ real @ ( zero_zero @ real ) @ X2 )
=> ( ord_less_eq @ real @ ( zero_zero @ real ) @ ( sqrt @ X2 ) ) ) ).
% real_sqrt_ge_zero
thf(fact_4634_real__sqrt__eq__zero__cancel,axiom,
! [X2: real] :
( ( ord_less_eq @ real @ ( zero_zero @ real ) @ X2 )
=> ( ( ( sqrt @ X2 )
= ( zero_zero @ real ) )
=> ( X2
= ( zero_zero @ real ) ) ) ) ).
% real_sqrt_eq_zero_cancel
thf(fact_4635_real__sqrt__ge__one,axiom,
! [X2: real] :
( ( ord_less_eq @ real @ ( one_one @ real ) @ X2 )
=> ( ord_less_eq @ real @ ( one_one @ real ) @ ( sqrt @ X2 ) ) ) ).
% real_sqrt_ge_one
thf(fact_4636_diffs__minus,axiom,
! [A: $tType] :
( ( ring_1 @ A )
=> ! [C3: nat > A] :
( ( diffs @ A
@ ^ [N4: nat] : ( uminus_uminus @ A @ ( C3 @ N4 ) ) )
= ( ^ [N4: nat] : ( uminus_uminus @ A @ ( diffs @ A @ C3 @ N4 ) ) ) ) ) ).
% diffs_minus
thf(fact_4637_tanh__real__lt__1,axiom,
! [X2: real] : ( ord_less @ real @ ( tanh @ real @ X2 ) @ ( one_one @ real ) ) ).
% tanh_real_lt_1
thf(fact_4638_real__div__sqrt,axiom,
! [X2: real] :
( ( ord_less_eq @ real @ ( zero_zero @ real ) @ X2 )
=> ( ( divide_divide @ real @ X2 @ ( sqrt @ X2 ) )
= ( sqrt @ X2 ) ) ) ).
% real_div_sqrt
thf(fact_4639_sqrt__add__le__add__sqrt,axiom,
! [X2: real,Y3: real] :
( ( ord_less_eq @ real @ ( zero_zero @ real ) @ X2 )
=> ( ( ord_less_eq @ real @ ( zero_zero @ real ) @ Y3 )
=> ( ord_less_eq @ real @ ( sqrt @ ( plus_plus @ real @ X2 @ Y3 ) ) @ ( plus_plus @ real @ ( sqrt @ X2 ) @ ( sqrt @ Y3 ) ) ) ) ) ).
% sqrt_add_le_add_sqrt
thf(fact_4640_le__real__sqrt__sumsq,axiom,
! [X2: real,Y3: real] : ( ord_less_eq @ real @ X2 @ ( sqrt @ ( plus_plus @ real @ ( times_times @ real @ X2 @ X2 ) @ ( times_times @ real @ Y3 @ Y3 ) ) ) ) ).
% le_real_sqrt_sumsq
thf(fact_4641_tanh__real__gt__neg1,axiom,
! [X2: real] : ( ord_less @ real @ ( uminus_uminus @ real @ ( one_one @ real ) ) @ ( tanh @ real @ X2 ) ) ).
% tanh_real_gt_neg1
thf(fact_4642_sqrt2__less__2,axiom,
ord_less @ real @ ( sqrt @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ).
% sqrt2_less_2
thf(fact_4643_diffs__def,axiom,
! [A: $tType] :
( ( ring_1 @ A )
=> ( ( diffs @ A )
= ( ^ [C5: nat > A,N4: nat] : ( times_times @ A @ ( semiring_1_of_nat @ A @ ( suc @ N4 ) ) @ ( C5 @ ( suc @ N4 ) ) ) ) ) ) ).
% diffs_def
thf(fact_4644_real__less__rsqrt,axiom,
! [X2: real,Y3: real] :
( ( ord_less @ real @ ( power_power @ real @ X2 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ Y3 )
=> ( ord_less @ real @ X2 @ ( sqrt @ Y3 ) ) ) ).
% real_less_rsqrt
thf(fact_4645_real__le__rsqrt,axiom,
! [X2: real,Y3: real] :
( ( ord_less_eq @ real @ ( power_power @ real @ X2 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ Y3 )
=> ( ord_less_eq @ real @ X2 @ ( sqrt @ Y3 ) ) ) ).
% real_le_rsqrt
thf(fact_4646_sqrt__le__D,axiom,
! [X2: real,Y3: real] :
( ( ord_less_eq @ real @ ( sqrt @ X2 ) @ Y3 )
=> ( ord_less_eq @ real @ X2 @ ( power_power @ real @ Y3 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ).
% sqrt_le_D
thf(fact_4647_termdiff__converges__all,axiom,
! [A: $tType] :
( ( ( real_Vector_banach @ A )
& ( real_V3459762299906320749_field @ A ) )
=> ! [C3: nat > A,X2: A] :
( ! [X4: A] :
( summable @ A
@ ^ [N4: nat] : ( times_times @ A @ ( C3 @ N4 ) @ ( power_power @ A @ X4 @ N4 ) ) )
=> ( summable @ A
@ ^ [N4: nat] : ( times_times @ A @ ( diffs @ A @ C3 @ N4 ) @ ( power_power @ A @ X2 @ N4 ) ) ) ) ) ).
% termdiff_converges_all
thf(fact_4648_real__sqrt__unique,axiom,
! [Y3: real,X2: real] :
( ( ( power_power @ real @ Y3 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) )
= X2 )
=> ( ( ord_less_eq @ real @ ( zero_zero @ real ) @ Y3 )
=> ( ( sqrt @ X2 )
= Y3 ) ) ) ).
% real_sqrt_unique
thf(fact_4649_real__le__lsqrt,axiom,
! [X2: real,Y3: real] :
( ( ord_less_eq @ real @ ( zero_zero @ real ) @ X2 )
=> ( ( ord_less_eq @ real @ ( zero_zero @ real ) @ Y3 )
=> ( ( ord_less_eq @ real @ X2 @ ( power_power @ real @ Y3 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) )
=> ( ord_less_eq @ real @ ( sqrt @ X2 ) @ Y3 ) ) ) ) ).
% real_le_lsqrt
thf(fact_4650_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_4651_real__sqrt__sum__squares__eq__cancel2,axiom,
! [X2: real,Y3: real] :
( ( ( sqrt @ ( plus_plus @ real @ ( power_power @ real @ X2 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ ( power_power @ real @ Y3 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) )
= Y3 )
=> ( X2
= ( zero_zero @ real ) ) ) ).
% real_sqrt_sum_squares_eq_cancel2
thf(fact_4652_real__sqrt__sum__squares__eq__cancel,axiom,
! [X2: real,Y3: real] :
( ( ( sqrt @ ( plus_plus @ real @ ( power_power @ real @ X2 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ ( power_power @ real @ Y3 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) )
= X2 )
=> ( Y3
= ( zero_zero @ real ) ) ) ).
% real_sqrt_sum_squares_eq_cancel
thf(fact_4653_real__sqrt__sum__squares__triangle__ineq,axiom,
! [A3: real,C3: real,B3: real,D2: real] : ( ord_less_eq @ real @ ( sqrt @ ( plus_plus @ real @ ( power_power @ real @ ( plus_plus @ real @ A3 @ C3 ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ ( power_power @ real @ ( plus_plus @ real @ B3 @ D2 ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( plus_plus @ real @ ( sqrt @ ( plus_plus @ real @ ( power_power @ real @ A3 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ ( power_power @ real @ B3 @ ( 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_4654_real__sqrt__sum__squares__ge2,axiom,
! [Y3: real,X2: real] : ( ord_less_eq @ real @ Y3 @ ( sqrt @ ( plus_plus @ real @ ( power_power @ real @ X2 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ ( power_power @ real @ Y3 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) ).
% real_sqrt_sum_squares_ge2
thf(fact_4655_real__sqrt__sum__squares__ge1,axiom,
! [X2: real,Y3: real] : ( ord_less_eq @ real @ X2 @ ( sqrt @ ( plus_plus @ real @ ( power_power @ real @ X2 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ ( power_power @ real @ Y3 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) ).
% real_sqrt_sum_squares_ge1
thf(fact_4656_sqrt__ge__absD,axiom,
! [X2: real,Y3: real] :
( ( ord_less_eq @ real @ ( abs_abs @ real @ X2 ) @ ( sqrt @ Y3 ) )
=> ( ord_less_eq @ real @ ( power_power @ real @ X2 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ Y3 ) ) ).
% sqrt_ge_absD
thf(fact_4657_cos__45,axiom,
( ( cos @ real @ ( divide_divide @ real @ pi @ ( numeral_numeral @ real @ ( bit0 @ ( bit0 @ one2 ) ) ) ) )
= ( divide_divide @ real @ ( sqrt @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) ) ).
% cos_45
thf(fact_4658_sin__45,axiom,
( ( sin @ real @ ( divide_divide @ real @ pi @ ( numeral_numeral @ real @ ( bit0 @ ( bit0 @ one2 ) ) ) ) )
= ( divide_divide @ real @ ( sqrt @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) ) ).
% sin_45
thf(fact_4659_tan__60,axiom,
( ( tan @ real @ ( divide_divide @ real @ pi @ ( numeral_numeral @ real @ ( bit1 @ one2 ) ) ) )
= ( sqrt @ ( numeral_numeral @ real @ ( bit1 @ one2 ) ) ) ) ).
% tan_60
thf(fact_4660_real__less__lsqrt,axiom,
! [X2: real,Y3: real] :
( ( ord_less_eq @ real @ ( zero_zero @ real ) @ X2 )
=> ( ( ord_less_eq @ real @ ( zero_zero @ real ) @ Y3 )
=> ( ( ord_less @ real @ X2 @ ( power_power @ real @ Y3 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) )
=> ( ord_less @ real @ ( sqrt @ X2 ) @ Y3 ) ) ) ) ).
% real_less_lsqrt
thf(fact_4661_sqrt__sum__squares__le__sum,axiom,
! [X2: real,Y3: real] :
( ( ord_less_eq @ real @ ( zero_zero @ real ) @ X2 )
=> ( ( ord_less_eq @ real @ ( zero_zero @ real ) @ Y3 )
=> ( ord_less_eq @ real @ ( sqrt @ ( plus_plus @ real @ ( power_power @ real @ X2 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ ( power_power @ real @ Y3 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( plus_plus @ real @ X2 @ Y3 ) ) ) ) ).
% sqrt_sum_squares_le_sum
thf(fact_4662_sqrt__even__pow2,axiom,
! [N: nat] :
( ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N )
=> ( ( sqrt @ ( power_power @ real @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) @ N ) )
= ( power_power @ real @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ N @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) ).
% sqrt_even_pow2
thf(fact_4663_sqrt__sum__squares__le__sum__abs,axiom,
! [X2: real,Y3: real] : ( ord_less_eq @ real @ ( sqrt @ ( plus_plus @ real @ ( power_power @ real @ X2 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ ( power_power @ real @ Y3 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( plus_plus @ real @ ( abs_abs @ real @ X2 ) @ ( abs_abs @ real @ Y3 ) ) ) ).
% sqrt_sum_squares_le_sum_abs
thf(fact_4664_real__sqrt__ge__abs2,axiom,
! [Y3: real,X2: real] : ( ord_less_eq @ real @ ( abs_abs @ real @ Y3 ) @ ( sqrt @ ( plus_plus @ real @ ( power_power @ real @ X2 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ ( power_power @ real @ Y3 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) ).
% real_sqrt_ge_abs2
thf(fact_4665_real__sqrt__ge__abs1,axiom,
! [X2: real,Y3: real] : ( ord_less_eq @ real @ ( abs_abs @ real @ X2 ) @ ( sqrt @ ( plus_plus @ real @ ( power_power @ real @ X2 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ ( power_power @ real @ Y3 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) ).
% real_sqrt_ge_abs1
thf(fact_4666_ln__sqrt,axiom,
! [X2: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ X2 )
=> ( ( ln_ln @ real @ ( sqrt @ X2 ) )
= ( divide_divide @ real @ ( ln_ln @ real @ X2 ) @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) ) ) ).
% ln_sqrt
thf(fact_4667_cos__30,axiom,
( ( cos @ real @ ( divide_divide @ real @ pi @ ( numeral_numeral @ real @ ( bit0 @ ( bit1 @ one2 ) ) ) ) )
= ( divide_divide @ real @ ( sqrt @ ( numeral_numeral @ real @ ( bit1 @ one2 ) ) ) @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) ) ).
% cos_30
thf(fact_4668_sin__60,axiom,
( ( sin @ real @ ( divide_divide @ real @ pi @ ( numeral_numeral @ real @ ( bit1 @ one2 ) ) ) )
= ( divide_divide @ real @ ( sqrt @ ( numeral_numeral @ real @ ( bit1 @ one2 ) ) ) @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) ) ).
% sin_60
thf(fact_4669_termdiff__converges,axiom,
! [A: $tType] :
( ( ( real_Vector_banach @ A )
& ( real_V3459762299906320749_field @ A ) )
=> ! [X2: A,K6: real,C3: nat > A] :
( ( ord_less @ real @ ( real_V7770717601297561774m_norm @ A @ X2 ) @ K6 )
=> ( ! [X4: A] :
( ( ord_less @ real @ ( real_V7770717601297561774m_norm @ A @ X4 ) @ K6 )
=> ( summable @ A
@ ^ [N4: nat] : ( times_times @ A @ ( C3 @ N4 ) @ ( power_power @ A @ X4 @ N4 ) ) ) )
=> ( summable @ A
@ ^ [N4: nat] : ( times_times @ A @ ( diffs @ A @ C3 @ N4 ) @ ( power_power @ A @ X2 @ N4 ) ) ) ) ) ) ).
% termdiff_converges
thf(fact_4670_real__sqrt__power__even,axiom,
! [N: nat,X2: real] :
( ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N )
=> ( ( ord_less_eq @ real @ ( zero_zero @ real ) @ X2 )
=> ( ( power_power @ real @ ( sqrt @ X2 ) @ N )
= ( power_power @ real @ X2 @ ( divide_divide @ nat @ N @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) ) ).
% real_sqrt_power_even
thf(fact_4671_arsinh__real__aux,axiom,
! [X2: real] : ( ord_less @ real @ ( zero_zero @ real ) @ ( plus_plus @ real @ X2 @ ( sqrt @ ( plus_plus @ real @ ( power_power @ real @ X2 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ ( one_one @ real ) ) ) ) ) ).
% arsinh_real_aux
thf(fact_4672_real__sqrt__sum__squares__mult__ge__zero,axiom,
! [X2: real,Y3: real,Xa: real,Ya: real] : ( ord_less_eq @ real @ ( zero_zero @ real ) @ ( sqrt @ ( times_times @ real @ ( plus_plus @ real @ ( power_power @ real @ X2 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ ( power_power @ real @ Y3 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( plus_plus @ real @ ( power_power @ real @ Xa @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ ( power_power @ real @ Ya @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) ) ).
% real_sqrt_sum_squares_mult_ge_zero
thf(fact_4673_arith__geo__mean__sqrt,axiom,
! [X2: real,Y3: real] :
( ( ord_less_eq @ real @ ( zero_zero @ real ) @ X2 )
=> ( ( ord_less_eq @ real @ ( zero_zero @ real ) @ Y3 )
=> ( ord_less_eq @ real @ ( sqrt @ ( times_times @ real @ X2 @ Y3 ) ) @ ( divide_divide @ real @ ( plus_plus @ real @ X2 @ Y3 ) @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) ) ) ) ).
% arith_geo_mean_sqrt
thf(fact_4674_powr__half__sqrt,axiom,
! [X2: real] :
( ( ord_less_eq @ real @ ( zero_zero @ real ) @ X2 )
=> ( ( powr @ real @ X2 @ ( divide_divide @ real @ ( one_one @ real ) @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) )
= ( sqrt @ X2 ) ) ) ).
% powr_half_sqrt
thf(fact_4675_tan__30,axiom,
( ( tan @ real @ ( divide_divide @ real @ pi @ ( numeral_numeral @ real @ ( bit0 @ ( bit1 @ one2 ) ) ) ) )
= ( divide_divide @ real @ ( one_one @ real ) @ ( sqrt @ ( numeral_numeral @ real @ ( bit1 @ one2 ) ) ) ) ) ).
% tan_30
thf(fact_4676_cos__x__y__le__one,axiom,
! [X2: real,Y3: real] : ( ord_less_eq @ real @ ( abs_abs @ real @ ( divide_divide @ real @ X2 @ ( sqrt @ ( plus_plus @ real @ ( power_power @ real @ X2 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ ( power_power @ real @ Y3 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) ) @ ( one_one @ real ) ) ).
% cos_x_y_le_one
thf(fact_4677_real__sqrt__sum__squares__less,axiom,
! [X2: real,U: real,Y3: real] :
( ( ord_less @ real @ ( abs_abs @ real @ X2 ) @ ( divide_divide @ real @ U @ ( sqrt @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) ) )
=> ( ( ord_less @ real @ ( abs_abs @ real @ Y3 ) @ ( divide_divide @ real @ U @ ( sqrt @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) ) )
=> ( ord_less @ real @ ( sqrt @ ( plus_plus @ real @ ( power_power @ real @ X2 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ ( power_power @ real @ Y3 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ U ) ) ) ).
% real_sqrt_sum_squares_less
thf(fact_4678_arcosh__real__def,axiom,
! [X2: real] :
( ( ord_less_eq @ real @ ( one_one @ real ) @ X2 )
=> ( ( arcosh @ real @ X2 )
= ( ln_ln @ real @ ( plus_plus @ real @ X2 @ ( sqrt @ ( minus_minus @ real @ ( power_power @ real @ X2 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ ( one_one @ real ) ) ) ) ) ) ) ).
% arcosh_real_def
thf(fact_4679_cos__arctan,axiom,
! [X2: real] :
( ( cos @ real @ ( arctan @ X2 ) )
= ( divide_divide @ real @ ( one_one @ real ) @ ( sqrt @ ( plus_plus @ real @ ( one_one @ real ) @ ( power_power @ real @ X2 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) ) ).
% cos_arctan
thf(fact_4680_sin__arctan,axiom,
! [X2: real] :
( ( sin @ real @ ( arctan @ X2 ) )
= ( divide_divide @ real @ X2 @ ( sqrt @ ( plus_plus @ real @ ( one_one @ real ) @ ( power_power @ real @ X2 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) ) ).
% sin_arctan
thf(fact_4681_sqrt__sum__squares__half__less,axiom,
! [X2: real,U: real,Y3: real] :
( ( ord_less @ real @ X2 @ ( divide_divide @ real @ U @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) )
=> ( ( ord_less @ real @ Y3 @ ( divide_divide @ real @ U @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) )
=> ( ( ord_less_eq @ real @ ( zero_zero @ real ) @ X2 )
=> ( ( ord_less_eq @ real @ ( zero_zero @ real ) @ Y3 )
=> ( ord_less @ real @ ( sqrt @ ( plus_plus @ real @ ( power_power @ real @ X2 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ ( power_power @ real @ Y3 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ U ) ) ) ) ) ).
% sqrt_sum_squares_half_less
thf(fact_4682_sin__cos__sqrt,axiom,
! [X2: real] :
( ( ord_less_eq @ real @ ( zero_zero @ real ) @ ( sin @ real @ X2 ) )
=> ( ( sin @ real @ X2 )
= ( sqrt @ ( minus_minus @ real @ ( one_one @ real ) @ ( power_power @ real @ ( cos @ real @ X2 ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) ) ).
% sin_cos_sqrt
thf(fact_4683_arctan__half,axiom,
( arctan
= ( ^ [X3: real] : ( times_times @ real @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) @ ( arctan @ ( divide_divide @ real @ X3 @ ( plus_plus @ real @ ( one_one @ real ) @ ( sqrt @ ( plus_plus @ real @ ( one_one @ real ) @ ( power_power @ real @ X3 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) ) ) ) ) ) ).
% arctan_half
thf(fact_4684_cos__tan,axiom,
! [X2: real] :
( ( ord_less @ real @ ( abs_abs @ real @ X2 ) @ ( divide_divide @ real @ pi @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) )
=> ( ( cos @ real @ X2 )
= ( divide_divide @ real @ ( one_one @ real ) @ ( sqrt @ ( plus_plus @ real @ ( one_one @ real ) @ ( power_power @ real @ ( tan @ real @ X2 ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) ) ) ).
% cos_tan
thf(fact_4685_arsinh__real__def,axiom,
( ( arsinh @ real )
= ( ^ [X3: real] : ( ln_ln @ real @ ( plus_plus @ real @ X3 @ ( sqrt @ ( plus_plus @ real @ ( power_power @ real @ X3 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ ( one_one @ real ) ) ) ) ) ) ) ).
% arsinh_real_def
thf(fact_4686_arcosh__def,axiom,
! [A: $tType] :
( ( ln @ A )
=> ( ( arcosh @ A )
= ( ^ [X3: A] : ( ln_ln @ A @ ( plus_plus @ A @ X3 @ ( powr @ A @ ( minus_minus @ A @ ( power_power @ A @ X3 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ ( one_one @ A ) ) @ ( real_Vector_of_real @ A @ ( divide_divide @ real @ ( one_one @ real ) @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) ) ) ) ) ) ) ) ).
% arcosh_def
thf(fact_4687_complex__unimodular__polar,axiom,
! [Z: complex] :
( ( ( real_V7770717601297561774m_norm @ complex @ Z )
= ( one_one @ real ) )
=> ~ ! [T8: real] :
( ( ord_less_eq @ real @ ( zero_zero @ real ) @ T8 )
=> ( ( ord_less @ real @ T8 @ ( times_times @ real @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) @ pi ) )
=> ( Z
!= ( complex2 @ ( cos @ real @ T8 ) @ ( sin @ real @ T8 ) ) ) ) ) ) ).
% complex_unimodular_polar
thf(fact_4688_summable__complex__of__real,axiom,
! [F2: nat > real] :
( ( summable @ complex
@ ^ [N4: nat] : ( real_Vector_of_real @ complex @ ( F2 @ N4 ) ) )
= ( summable @ real @ F2 ) ) ).
% summable_complex_of_real
thf(fact_4689_of__real__eq__1__iff,axiom,
! [A: $tType] :
( ( real_V2191834092415804123ebra_1 @ A )
=> ! [X2: real] :
( ( ( real_Vector_of_real @ A @ X2 )
= ( one_one @ A ) )
= ( X2
= ( one_one @ real ) ) ) ) ).
% of_real_eq_1_iff
thf(fact_4690_of__real__1,axiom,
! [A: $tType] :
( ( real_V2191834092415804123ebra_1 @ A )
=> ( ( real_Vector_of_real @ A @ ( one_one @ real ) )
= ( one_one @ A ) ) ) ).
% of_real_1
thf(fact_4691_of__real__numeral,axiom,
! [A: $tType] :
( ( real_V2191834092415804123ebra_1 @ A )
=> ! [W: num] :
( ( real_Vector_of_real @ A @ ( numeral_numeral @ real @ W ) )
= ( numeral_numeral @ A @ W ) ) ) ).
% of_real_numeral
thf(fact_4692_of__real__mult,axiom,
! [A: $tType] :
( ( real_V2191834092415804123ebra_1 @ A )
=> ! [X2: real,Y3: real] :
( ( real_Vector_of_real @ A @ ( times_times @ real @ X2 @ Y3 ) )
= ( times_times @ A @ ( real_Vector_of_real @ A @ X2 ) @ ( real_Vector_of_real @ A @ Y3 ) ) ) ) ).
% of_real_mult
thf(fact_4693_of__real__divide,axiom,
! [A: $tType] :
( ( real_V5047593784448816457lgebra @ A )
=> ! [X2: real,Y3: real] :
( ( real_Vector_of_real @ A @ ( divide_divide @ real @ X2 @ Y3 ) )
= ( divide_divide @ A @ ( real_Vector_of_real @ A @ X2 ) @ ( real_Vector_of_real @ A @ Y3 ) ) ) ) ).
% of_real_divide
thf(fact_4694_of__real__add,axiom,
! [A: $tType] :
( ( real_V2191834092415804123ebra_1 @ A )
=> ! [X2: real,Y3: real] :
( ( real_Vector_of_real @ A @ ( plus_plus @ real @ X2 @ Y3 ) )
= ( plus_plus @ A @ ( real_Vector_of_real @ A @ X2 ) @ ( real_Vector_of_real @ A @ Y3 ) ) ) ) ).
% of_real_add
thf(fact_4695_of__real__power,axiom,
! [A: $tType] :
( ( real_V2191834092415804123ebra_1 @ A )
=> ! [X2: real,N: nat] :
( ( real_Vector_of_real @ A @ ( power_power @ real @ X2 @ N ) )
= ( power_power @ A @ ( real_Vector_of_real @ A @ X2 ) @ N ) ) ) ).
% of_real_power
thf(fact_4696_of__real__diff,axiom,
! [A: $tType] :
( ( real_V2191834092415804123ebra_1 @ A )
=> ! [X2: real,Y3: real] :
( ( real_Vector_of_real @ A @ ( minus_minus @ real @ X2 @ Y3 ) )
= ( minus_minus @ A @ ( real_Vector_of_real @ A @ X2 ) @ ( real_Vector_of_real @ A @ Y3 ) ) ) ) ).
% of_real_diff
thf(fact_4697_of__real__sum,axiom,
! [A: $tType,B: $tType] :
( ( real_V2191834092415804123ebra_1 @ A )
=> ! [F2: B > real,S2: set @ B] :
( ( real_Vector_of_real @ A @ ( groups7311177749621191930dd_sum @ B @ real @ F2 @ S2 ) )
= ( groups7311177749621191930dd_sum @ B @ A
@ ^ [X3: B] : ( real_Vector_of_real @ A @ ( F2 @ X3 ) )
@ S2 ) ) ) ).
% of_real_sum
thf(fact_4698_of__real__prod,axiom,
! [A: $tType,B: $tType] :
( ( ( comm_monoid_mult @ A )
& ( real_V2191834092415804123ebra_1 @ A ) )
=> ! [F2: B > real,S2: set @ B] :
( ( real_Vector_of_real @ A @ ( groups7121269368397514597t_prod @ B @ real @ F2 @ S2 ) )
= ( groups7121269368397514597t_prod @ B @ A
@ ^ [X3: B] : ( real_Vector_of_real @ A @ ( F2 @ X3 ) )
@ S2 ) ) ) ).
% of_real_prod
thf(fact_4699_of__real__neg__numeral,axiom,
! [A: $tType] :
( ( real_V2191834092415804123ebra_1 @ A )
=> ! [W: num] :
( ( real_Vector_of_real @ A @ ( uminus_uminus @ real @ ( numeral_numeral @ real @ W ) ) )
= ( uminus_uminus @ A @ ( numeral_numeral @ A @ W ) ) ) ) ).
% of_real_neg_numeral
thf(fact_4700_cos__of__real__pi,axiom,
! [A: $tType] :
( ( ( real_Vector_banach @ A )
& ( real_V2822296259951069270ebra_1 @ A ) )
=> ( ( cos @ A @ ( real_Vector_of_real @ A @ pi ) )
= ( uminus_uminus @ A @ ( one_one @ A ) ) ) ) ).
% cos_of_real_pi
thf(fact_4701_norm__cos__sin,axiom,
! [T2: real] :
( ( real_V7770717601297561774m_norm @ complex @ ( complex2 @ ( cos @ real @ T2 ) @ ( sin @ real @ T2 ) ) )
= ( one_one @ real ) ) ).
% norm_cos_sin
thf(fact_4702_norm__of__real__add1,axiom,
! [A: $tType] :
( ( real_V8999393235501362500lgebra @ A )
=> ! [X2: real] :
( ( real_V7770717601297561774m_norm @ A @ ( plus_plus @ A @ ( real_Vector_of_real @ A @ X2 ) @ ( one_one @ A ) ) )
= ( abs_abs @ real @ ( plus_plus @ real @ X2 @ ( one_one @ real ) ) ) ) ) ).
% norm_of_real_add1
thf(fact_4703_norm__of__real__addn,axiom,
! [A: $tType] :
( ( real_V8999393235501362500lgebra @ A )
=> ! [X2: real,B3: num] :
( ( real_V7770717601297561774m_norm @ A @ ( plus_plus @ A @ ( real_Vector_of_real @ A @ X2 ) @ ( numeral_numeral @ A @ B3 ) ) )
= ( abs_abs @ real @ ( plus_plus @ real @ X2 @ ( numeral_numeral @ real @ B3 ) ) ) ) ) ).
% norm_of_real_addn
thf(fact_4704_cos__of__real__pi__half,axiom,
! [A: $tType] :
( ( ( real_Vector_banach @ A )
& ( real_V7773925162809079976_field @ A )
& ( real_V2822296259951069270ebra_1 @ A ) )
=> ( ( cos @ A @ ( divide_divide @ A @ ( real_Vector_of_real @ A @ pi ) @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) )
= ( zero_zero @ A ) ) ) ).
% cos_of_real_pi_half
thf(fact_4705_sin__of__real__pi__half,axiom,
! [A: $tType] :
( ( ( real_Vector_banach @ A )
& ( real_V7773925162809079976_field @ A )
& ( real_V2822296259951069270ebra_1 @ A ) )
=> ( ( sin @ A @ ( divide_divide @ A @ ( real_Vector_of_real @ A @ pi ) @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) )
= ( one_one @ A ) ) ) ).
% sin_of_real_pi_half
thf(fact_4706_diffs__of__real,axiom,
! [A: $tType] :
( ( real_V2191834092415804123ebra_1 @ A )
=> ! [F2: nat > real] :
( ( diffs @ A
@ ^ [N4: nat] : ( real_Vector_of_real @ A @ ( F2 @ N4 ) ) )
= ( ^ [N4: nat] : ( real_Vector_of_real @ A @ ( diffs @ real @ F2 @ N4 ) ) ) ) ) ).
% diffs_of_real
thf(fact_4707_complex__diff,axiom,
! [A3: real,B3: real,C3: real,D2: real] :
( ( minus_minus @ complex @ ( complex2 @ A3 @ B3 ) @ ( complex2 @ C3 @ D2 ) )
= ( complex2 @ ( minus_minus @ real @ A3 @ C3 ) @ ( minus_minus @ real @ B3 @ D2 ) ) ) ).
% complex_diff
thf(fact_4708_complex__of__real__add__Complex,axiom,
! [R2: real,X2: real,Y3: real] :
( ( plus_plus @ complex @ ( real_Vector_of_real @ complex @ R2 ) @ ( complex2 @ X2 @ Y3 ) )
= ( complex2 @ ( plus_plus @ real @ R2 @ X2 ) @ Y3 ) ) ).
% complex_of_real_add_Complex
thf(fact_4709_Complex__add__complex__of__real,axiom,
! [X2: real,Y3: real,R2: real] :
( ( plus_plus @ complex @ ( complex2 @ X2 @ Y3 ) @ ( real_Vector_of_real @ complex @ R2 ) )
= ( complex2 @ ( plus_plus @ real @ X2 @ R2 ) @ Y3 ) ) ).
% Complex_add_complex_of_real
thf(fact_4710_of__real__def,axiom,
! [A: $tType] :
( ( real_V2191834092415804123ebra_1 @ A )
=> ( ( real_Vector_of_real @ A )
= ( ^ [R5: real] : ( real_V8093663219630862766scaleR @ A @ R5 @ ( one_one @ A ) ) ) ) ) ).
% of_real_def
thf(fact_4711_scaleR__conv__of__real,axiom,
! [A: $tType] :
( ( real_V2191834092415804123ebra_1 @ A )
=> ( ( real_V8093663219630862766scaleR @ A )
= ( ^ [R5: real] : ( times_times @ A @ ( real_Vector_of_real @ A @ R5 ) ) ) ) ) ).
% scaleR_conv_of_real
thf(fact_4712_complex__add,axiom,
! [A3: real,B3: real,C3: real,D2: real] :
( ( plus_plus @ complex @ ( complex2 @ A3 @ B3 ) @ ( complex2 @ C3 @ D2 ) )
= ( complex2 @ ( plus_plus @ real @ A3 @ C3 ) @ ( plus_plus @ real @ B3 @ D2 ) ) ) ).
% complex_add
thf(fact_4713_summable__of__real,axiom,
! [A: $tType] :
( ( ( real_V2191834092415804123ebra_1 @ A )
& ( real_V822414075346904944vector @ A ) )
=> ! [X: nat > real] :
( ( summable @ real @ X )
=> ( summable @ A
@ ^ [N4: nat] : ( real_Vector_of_real @ A @ ( X @ N4 ) ) ) ) ) ).
% summable_of_real
thf(fact_4714_sums__of__real,axiom,
! [A: $tType] :
( ( ( real_V2191834092415804123ebra_1 @ A )
& ( real_V822414075346904944vector @ A ) )
=> ! [X: nat > real,A3: real] :
( ( sums @ real @ X @ A3 )
=> ( sums @ A
@ ^ [N4: nat] : ( real_Vector_of_real @ A @ ( X @ N4 ) )
@ ( real_Vector_of_real @ A @ A3 ) ) ) ) ).
% sums_of_real
thf(fact_4715_sums__of__real__iff,axiom,
! [A: $tType] :
( ( real_V8999393235501362500lgebra @ A )
=> ! [F2: nat > real,C3: real] :
( ( sums @ A
@ ^ [N4: nat] : ( real_Vector_of_real @ A @ ( F2 @ N4 ) )
@ ( real_Vector_of_real @ A @ C3 ) )
= ( sums @ real @ F2 @ C3 ) ) ) ).
% sums_of_real_iff
thf(fact_4716_nonzero__of__real__divide,axiom,
! [A: $tType] :
( ( real_V7773925162809079976_field @ A )
=> ! [Y3: real,X2: real] :
( ( Y3
!= ( zero_zero @ real ) )
=> ( ( real_Vector_of_real @ A @ ( divide_divide @ real @ X2 @ Y3 ) )
= ( divide_divide @ A @ ( real_Vector_of_real @ A @ X2 ) @ ( real_Vector_of_real @ A @ Y3 ) ) ) ) ) ).
% nonzero_of_real_divide
thf(fact_4717_complex__mult,axiom,
! [A3: real,B3: real,C3: real,D2: real] :
( ( times_times @ complex @ ( complex2 @ A3 @ B3 ) @ ( complex2 @ C3 @ D2 ) )
= ( complex2 @ ( minus_minus @ real @ ( times_times @ real @ A3 @ C3 ) @ ( times_times @ real @ B3 @ D2 ) ) @ ( plus_plus @ real @ ( times_times @ real @ A3 @ D2 ) @ ( times_times @ real @ B3 @ C3 ) ) ) ) ).
% complex_mult
thf(fact_4718_Complex__eq__1,axiom,
! [A3: real,B3: real] :
( ( ( complex2 @ A3 @ B3 )
= ( one_one @ complex ) )
= ( ( A3
= ( one_one @ real ) )
& ( B3
= ( zero_zero @ real ) ) ) ) ).
% Complex_eq_1
thf(fact_4719_one__complex_Ocode,axiom,
( ( one_one @ complex )
= ( complex2 @ ( one_one @ real ) @ ( zero_zero @ real ) ) ) ).
% one_complex.code
thf(fact_4720_suminf__of__real,axiom,
! [A: $tType] :
( ( ( real_V2191834092415804123ebra_1 @ A )
& ( real_V822414075346904944vector @ A ) )
=> ! [X: nat > real] :
( ( summable @ real @ X )
=> ( ( real_Vector_of_real @ A @ ( suminf @ real @ X ) )
= ( suminf @ A
@ ^ [N4: nat] : ( real_Vector_of_real @ A @ ( X @ N4 ) ) ) ) ) ) ).
% suminf_of_real
thf(fact_4721_norm__less__p1,axiom,
! [A: $tType] :
( ( real_V2822296259951069270ebra_1 @ A )
=> ! [X2: A] : ( ord_less @ real @ ( real_V7770717601297561774m_norm @ A @ X2 ) @ ( real_V7770717601297561774m_norm @ A @ ( plus_plus @ A @ ( real_Vector_of_real @ A @ ( real_V7770717601297561774m_norm @ A @ X2 ) ) @ ( one_one @ A ) ) ) ) ) ).
% norm_less_p1
thf(fact_4722_Complex__sum_H,axiom,
! [A: $tType,F2: A > real,S2: set @ A] :
( ( groups7311177749621191930dd_sum @ A @ complex
@ ^ [X3: A] : ( complex2 @ ( F2 @ X3 ) @ ( zero_zero @ real ) )
@ S2 )
= ( complex2 @ ( groups7311177749621191930dd_sum @ A @ real @ F2 @ S2 ) @ ( zero_zero @ real ) ) ) ).
% Complex_sum'
thf(fact_4723_Complex__eq__neg__1,axiom,
! [A3: real,B3: real] :
( ( ( complex2 @ A3 @ B3 )
= ( uminus_uminus @ complex @ ( one_one @ complex ) ) )
= ( ( A3
= ( uminus_uminus @ real @ ( one_one @ real ) ) )
& ( B3
= ( zero_zero @ real ) ) ) ) ).
% Complex_eq_neg_1
thf(fact_4724_norm__of__real__diff,axiom,
! [A: $tType] :
( ( real_V2822296259951069270ebra_1 @ A )
=> ! [B3: real,A3: real] : ( ord_less_eq @ real @ ( real_V7770717601297561774m_norm @ A @ ( minus_minus @ A @ ( real_Vector_of_real @ A @ B3 ) @ ( real_Vector_of_real @ A @ A3 ) ) ) @ ( abs_abs @ real @ ( minus_minus @ real @ B3 @ A3 ) ) ) ) ).
% norm_of_real_diff
thf(fact_4725_cos__int__times__real,axiom,
! [A: $tType] :
( ( ( real_Vector_banach @ A )
& ( real_V2822296259951069270ebra_1 @ A ) )
=> ! [M: int,X2: real] :
( ( cos @ A @ ( times_times @ A @ ( ring_1_of_int @ A @ M ) @ ( real_Vector_of_real @ A @ X2 ) ) )
= ( real_Vector_of_real @ A @ ( cos @ real @ ( times_times @ real @ ( ring_1_of_int @ real @ M ) @ X2 ) ) ) ) ) ).
% cos_int_times_real
thf(fact_4726_sin__int__times__real,axiom,
! [A: $tType] :
( ( ( real_Vector_banach @ A )
& ( real_V2822296259951069270ebra_1 @ A ) )
=> ! [M: int,X2: real] :
( ( sin @ A @ ( times_times @ A @ ( ring_1_of_int @ A @ M ) @ ( real_Vector_of_real @ A @ X2 ) ) )
= ( real_Vector_of_real @ A @ ( sin @ real @ ( times_times @ real @ ( ring_1_of_int @ real @ M ) @ X2 ) ) ) ) ) ).
% sin_int_times_real
thf(fact_4727_complex__norm,axiom,
! [X2: real,Y3: real] :
( ( real_V7770717601297561774m_norm @ complex @ ( complex2 @ X2 @ Y3 ) )
= ( sqrt @ ( plus_plus @ real @ ( power_power @ real @ X2 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ ( power_power @ real @ Y3 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) ).
% complex_norm
thf(fact_4728_sin__cos__eq,axiom,
! [A: $tType] :
( ( ( real_Vector_banach @ A )
& ( real_V3459762299906320749_field @ A ) )
=> ( ( sin @ A )
= ( ^ [X3: A] : ( cos @ A @ ( minus_minus @ A @ ( divide_divide @ A @ ( real_Vector_of_real @ A @ pi ) @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) @ X3 ) ) ) ) ) ).
% sin_cos_eq
thf(fact_4729_cos__sin__eq,axiom,
! [A: $tType] :
( ( ( real_Vector_banach @ A )
& ( real_V3459762299906320749_field @ A ) )
=> ( ( cos @ A )
= ( ^ [X3: A] : ( sin @ A @ ( minus_minus @ A @ ( divide_divide @ A @ ( real_Vector_of_real @ A @ pi ) @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) @ X3 ) ) ) ) ) ).
% cos_sin_eq
thf(fact_4730_arsinh__def,axiom,
! [A: $tType] :
( ( ln @ A )
=> ( ( arsinh @ A )
= ( ^ [X3: A] : ( ln_ln @ A @ ( plus_plus @ A @ X3 @ ( powr @ A @ ( plus_plus @ A @ ( power_power @ A @ X3 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ ( one_one @ A ) ) @ ( real_Vector_of_real @ A @ ( divide_divide @ real @ ( one_one @ real ) @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) ) ) ) ) ) ) ) ).
% arsinh_def
thf(fact_4731_minus__sin__cos__eq,axiom,
! [A: $tType] :
( ( ( real_Vector_banach @ A )
& ( real_V3459762299906320749_field @ A ) )
=> ! [X2: A] :
( ( uminus_uminus @ A @ ( sin @ A @ X2 ) )
= ( cos @ A @ ( plus_plus @ A @ X2 @ ( divide_divide @ A @ ( real_Vector_of_real @ A @ pi ) @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) ) ) ) ) ).
% minus_sin_cos_eq
thf(fact_4732_Maclaurin__exp__lt,axiom,
! [X2: real,N: nat] :
( ( X2
!= ( zero_zero @ real ) )
=> ( ( ord_less @ nat @ ( zero_zero @ nat ) @ N )
=> ? [T8: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ ( abs_abs @ real @ T8 ) )
& ( ord_less @ real @ ( abs_abs @ real @ T8 ) @ ( abs_abs @ real @ X2 ) )
& ( ( exp @ real @ X2 )
= ( plus_plus @ real
@ ( groups7311177749621191930dd_sum @ nat @ real
@ ^ [M6: nat] : ( divide_divide @ real @ ( power_power @ real @ X2 @ M6 ) @ ( semiring_char_0_fact @ real @ M6 ) )
@ ( set_ord_lessThan @ nat @ N ) )
@ ( times_times @ real @ ( divide_divide @ real @ ( exp @ real @ T8 ) @ ( semiring_char_0_fact @ real @ N ) ) @ ( power_power @ real @ X2 @ N ) ) ) ) ) ) ) ).
% Maclaurin_exp_lt
thf(fact_4733_cos__arcsin,axiom,
! [X2: real] :
( ( ord_less_eq @ real @ ( uminus_uminus @ real @ ( one_one @ real ) ) @ X2 )
=> ( ( ord_less_eq @ real @ X2 @ ( one_one @ real ) )
=> ( ( cos @ real @ ( arcsin @ X2 ) )
= ( sqrt @ ( minus_minus @ real @ ( one_one @ real ) @ ( power_power @ real @ X2 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) ) ) ).
% cos_arcsin
thf(fact_4734_sin__arccos__abs,axiom,
! [Y3: real] :
( ( ord_less_eq @ real @ ( abs_abs @ real @ Y3 ) @ ( one_one @ real ) )
=> ( ( sin @ real @ ( arccos @ Y3 ) )
= ( sqrt @ ( minus_minus @ real @ ( one_one @ real ) @ ( power_power @ real @ Y3 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) ) ).
% sin_arccos_abs
thf(fact_4735_exp__le__cancel__iff,axiom,
! [X2: real,Y3: real] :
( ( ord_less_eq @ real @ ( exp @ real @ X2 ) @ ( exp @ real @ Y3 ) )
= ( ord_less_eq @ real @ X2 @ Y3 ) ) ).
% exp_le_cancel_iff
thf(fact_4736_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_4737_exp__eq__one__iff,axiom,
! [X2: real] :
( ( ( exp @ real @ X2 )
= ( one_one @ real ) )
= ( X2
= ( zero_zero @ real ) ) ) ).
% exp_eq_one_iff
thf(fact_4738_arccos__1,axiom,
( ( arccos @ ( one_one @ real ) )
= ( zero_zero @ real ) ) ).
% arccos_1
thf(fact_4739_one__less__exp__iff,axiom,
! [X2: real] :
( ( ord_less @ real @ ( one_one @ real ) @ ( exp @ real @ X2 ) )
= ( ord_less @ real @ ( zero_zero @ real ) @ X2 ) ) ).
% one_less_exp_iff
thf(fact_4740_exp__less__one__iff,axiom,
! [X2: real] :
( ( ord_less @ real @ ( exp @ real @ X2 ) @ ( one_one @ real ) )
= ( ord_less @ real @ X2 @ ( zero_zero @ real ) ) ) ).
% exp_less_one_iff
thf(fact_4741_one__le__exp__iff,axiom,
! [X2: real] :
( ( ord_less_eq @ real @ ( one_one @ real ) @ ( exp @ real @ X2 ) )
= ( ord_less_eq @ real @ ( zero_zero @ real ) @ X2 ) ) ).
% one_le_exp_iff
thf(fact_4742_exp__le__one__iff,axiom,
! [X2: real] :
( ( ord_less_eq @ real @ ( exp @ real @ X2 ) @ ( one_one @ real ) )
= ( ord_less_eq @ real @ X2 @ ( zero_zero @ real ) ) ) ).
% exp_le_one_iff
thf(fact_4743_arccos__minus__1,axiom,
( ( arccos @ ( uminus_uminus @ real @ ( one_one @ real ) ) )
= pi ) ).
% arccos_minus_1
thf(fact_4744_cos__arccos,axiom,
! [Y3: real] :
( ( ord_less_eq @ real @ ( uminus_uminus @ real @ ( one_one @ real ) ) @ Y3 )
=> ( ( ord_less_eq @ real @ Y3 @ ( one_one @ real ) )
=> ( ( cos @ real @ ( arccos @ Y3 ) )
= Y3 ) ) ) ).
% cos_arccos
thf(fact_4745_sin__arcsin,axiom,
! [Y3: real] :
( ( ord_less_eq @ real @ ( uminus_uminus @ real @ ( one_one @ real ) ) @ Y3 )
=> ( ( ord_less_eq @ real @ Y3 @ ( one_one @ real ) )
=> ( ( sin @ real @ ( arcsin @ Y3 ) )
= Y3 ) ) ) ).
% sin_arcsin
thf(fact_4746_arccos__0,axiom,
( ( arccos @ ( zero_zero @ real ) )
= ( divide_divide @ real @ pi @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) ) ).
% arccos_0
thf(fact_4747_arcsin__1,axiom,
( ( arcsin @ ( one_one @ real ) )
= ( divide_divide @ real @ pi @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) ) ).
% arcsin_1
thf(fact_4748_arcsin__minus__1,axiom,
( ( arcsin @ ( uminus_uminus @ real @ ( one_one @ real ) ) )
= ( uminus_uminus @ real @ ( divide_divide @ real @ pi @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) ) ) ).
% arcsin_minus_1
thf(fact_4749_norm__exp,axiom,
! [A: $tType] :
( ( ( real_Vector_banach @ A )
& ( real_V2822296259951069270ebra_1 @ A ) )
=> ! [X2: A] : ( ord_less_eq @ real @ ( real_V7770717601297561774m_norm @ A @ ( exp @ A @ X2 ) ) @ ( exp @ real @ ( real_V7770717601297561774m_norm @ A @ X2 ) ) ) ) ).
% norm_exp
thf(fact_4750_exp__times__arg__commute,axiom,
! [A: $tType] :
( ( ( real_Vector_banach @ A )
& ( real_V2822296259951069270ebra_1 @ A ) )
=> ! [A2: A] :
( ( times_times @ A @ ( exp @ A @ A2 ) @ A2 )
= ( times_times @ A @ A2 @ ( exp @ A @ A2 ) ) ) ) ).
% exp_times_arg_commute
thf(fact_4751_not__exp__le__zero,axiom,
! [X2: real] :
~ ( ord_less_eq @ real @ ( exp @ real @ X2 ) @ ( zero_zero @ real ) ) ).
% not_exp_le_zero
thf(fact_4752_exp__ge__zero,axiom,
! [X2: real] : ( ord_less_eq @ real @ ( zero_zero @ real ) @ ( exp @ real @ X2 ) ) ).
% exp_ge_zero
thf(fact_4753_mult__exp__exp,axiom,
! [A: $tType] :
( ( ( real_Vector_banach @ A )
& ( real_V3459762299906320749_field @ A ) )
=> ! [X2: A,Y3: A] :
( ( times_times @ A @ ( exp @ A @ X2 ) @ ( exp @ A @ Y3 ) )
= ( exp @ A @ ( plus_plus @ A @ X2 @ Y3 ) ) ) ) ).
% mult_exp_exp
thf(fact_4754_exp__add__commuting,axiom,
! [A: $tType] :
( ( ( real_Vector_banach @ A )
& ( real_V2822296259951069270ebra_1 @ A ) )
=> ! [X2: A,Y3: A] :
( ( ( times_times @ A @ X2 @ Y3 )
= ( times_times @ A @ Y3 @ X2 ) )
=> ( ( exp @ A @ ( plus_plus @ A @ X2 @ Y3 ) )
= ( times_times @ A @ ( exp @ A @ X2 ) @ ( exp @ A @ Y3 ) ) ) ) ) ).
% exp_add_commuting
thf(fact_4755_exp__diff,axiom,
! [A: $tType] :
( ( ( real_Vector_banach @ A )
& ( real_V3459762299906320749_field @ A ) )
=> ! [X2: A,Y3: A] :
( ( exp @ A @ ( minus_minus @ A @ X2 @ Y3 ) )
= ( divide_divide @ A @ ( exp @ A @ X2 ) @ ( exp @ A @ Y3 ) ) ) ) ).
% exp_diff
thf(fact_4756_exp__gt__one,axiom,
! [X2: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ X2 )
=> ( ord_less @ real @ ( one_one @ real ) @ ( exp @ real @ X2 ) ) ) ).
% exp_gt_one
thf(fact_4757_exp__ge__add__one__self,axiom,
! [X2: real] : ( ord_less_eq @ real @ ( plus_plus @ real @ ( one_one @ real ) @ X2 ) @ ( exp @ real @ X2 ) ) ).
% exp_ge_add_one_self
thf(fact_4758_exp__minus__inverse,axiom,
! [A: $tType] :
( ( ( real_Vector_banach @ A )
& ( real_V2822296259951069270ebra_1 @ A ) )
=> ! [X2: A] :
( ( times_times @ A @ ( exp @ A @ X2 ) @ ( exp @ A @ ( uminus_uminus @ A @ X2 ) ) )
= ( one_one @ A ) ) ) ).
% exp_minus_inverse
thf(fact_4759_exp__of__nat__mult,axiom,
! [A: $tType] :
( ( ( real_Vector_banach @ A )
& ( real_V3459762299906320749_field @ A ) )
=> ! [N: nat,X2: A] :
( ( exp @ A @ ( times_times @ A @ ( semiring_1_of_nat @ A @ N ) @ X2 ) )
= ( power_power @ A @ ( exp @ A @ X2 ) @ N ) ) ) ).
% exp_of_nat_mult
thf(fact_4760_exp__of__nat2__mult,axiom,
! [A: $tType] :
( ( ( real_Vector_banach @ A )
& ( real_V3459762299906320749_field @ A ) )
=> ! [X2: A,N: nat] :
( ( exp @ A @ ( times_times @ A @ X2 @ ( semiring_1_of_nat @ A @ N ) ) )
= ( power_power @ A @ ( exp @ A @ X2 ) @ N ) ) ) ).
% exp_of_nat2_mult
thf(fact_4761_log__ln,axiom,
( ( ln_ln @ real )
= ( log @ ( exp @ real @ ( one_one @ real ) ) ) ) ).
% log_ln
thf(fact_4762_exp__sum,axiom,
! [B: $tType,A: $tType] :
( ( ( comm_monoid_mult @ B )
& ( real_Vector_banach @ B )
& ( real_V2822296259951069270ebra_1 @ B ) )
=> ! [I6: set @ A,F2: A > B] :
( ( finite_finite2 @ A @ I6 )
=> ( ( exp @ B @ ( groups7311177749621191930dd_sum @ A @ B @ F2 @ I6 ) )
= ( groups7121269368397514597t_prod @ A @ B
@ ^ [X3: A] : ( exp @ B @ ( F2 @ X3 ) )
@ I6 ) ) ) ) ).
% exp_sum
thf(fact_4763_arccos__le__arccos,axiom,
! [X2: real,Y3: real] :
( ( ord_less_eq @ real @ ( uminus_uminus @ real @ ( one_one @ real ) ) @ X2 )
=> ( ( ord_less_eq @ real @ X2 @ Y3 )
=> ( ( ord_less_eq @ real @ Y3 @ ( one_one @ real ) )
=> ( ord_less_eq @ real @ ( arccos @ Y3 ) @ ( arccos @ X2 ) ) ) ) ) ).
% arccos_le_arccos
thf(fact_4764_arccos__eq__iff,axiom,
! [X2: real,Y3: real] :
( ( ( ord_less_eq @ real @ ( abs_abs @ real @ X2 ) @ ( one_one @ real ) )
& ( ord_less_eq @ real @ ( abs_abs @ real @ Y3 ) @ ( one_one @ real ) ) )
=> ( ( ( arccos @ X2 )
= ( arccos @ Y3 ) )
= ( X2 = Y3 ) ) ) ).
% arccos_eq_iff
thf(fact_4765_arccos__le__mono,axiom,
! [X2: real,Y3: real] :
( ( ord_less_eq @ real @ ( abs_abs @ real @ X2 ) @ ( one_one @ real ) )
=> ( ( ord_less_eq @ real @ ( abs_abs @ real @ Y3 ) @ ( one_one @ real ) )
=> ( ( ord_less_eq @ real @ ( arccos @ X2 ) @ ( arccos @ Y3 ) )
= ( ord_less_eq @ real @ Y3 @ X2 ) ) ) ) ).
% arccos_le_mono
thf(fact_4766_arcsin__le__arcsin,axiom,
! [X2: real,Y3: real] :
( ( ord_less_eq @ real @ ( uminus_uminus @ real @ ( one_one @ real ) ) @ X2 )
=> ( ( ord_less_eq @ real @ X2 @ Y3 )
=> ( ( ord_less_eq @ real @ Y3 @ ( one_one @ real ) )
=> ( ord_less_eq @ real @ ( arcsin @ X2 ) @ ( arcsin @ Y3 ) ) ) ) ) ).
% arcsin_le_arcsin
thf(fact_4767_arcsin__minus,axiom,
! [X2: real] :
( ( ord_less_eq @ real @ ( uminus_uminus @ real @ ( one_one @ real ) ) @ X2 )
=> ( ( ord_less_eq @ real @ X2 @ ( one_one @ real ) )
=> ( ( arcsin @ ( uminus_uminus @ real @ X2 ) )
= ( uminus_uminus @ real @ ( arcsin @ X2 ) ) ) ) ) ).
% arcsin_minus
thf(fact_4768_arcsin__eq__iff,axiom,
! [X2: real,Y3: real] :
( ( ord_less_eq @ real @ ( abs_abs @ real @ X2 ) @ ( one_one @ real ) )
=> ( ( ord_less_eq @ real @ ( abs_abs @ real @ Y3 ) @ ( one_one @ real ) )
=> ( ( ( arcsin @ X2 )
= ( arcsin @ Y3 ) )
= ( X2 = Y3 ) ) ) ) ).
% arcsin_eq_iff
thf(fact_4769_arcsin__le__mono,axiom,
! [X2: real,Y3: real] :
( ( ord_less_eq @ real @ ( abs_abs @ real @ X2 ) @ ( one_one @ real ) )
=> ( ( ord_less_eq @ real @ ( abs_abs @ real @ Y3 ) @ ( one_one @ real ) )
=> ( ( ord_less_eq @ real @ ( arcsin @ X2 ) @ ( arcsin @ Y3 ) )
= ( ord_less_eq @ real @ X2 @ Y3 ) ) ) ) ).
% arcsin_le_mono
thf(fact_4770_exp__ge__add__one__self__aux,axiom,
! [X2: real] :
( ( ord_less_eq @ real @ ( zero_zero @ real ) @ X2 )
=> ( ord_less_eq @ real @ ( plus_plus @ real @ ( one_one @ real ) @ X2 ) @ ( exp @ real @ X2 ) ) ) ).
% exp_ge_add_one_self_aux
thf(fact_4771_lemma__exp__total,axiom,
! [Y3: real] :
( ( ord_less_eq @ real @ ( one_one @ real ) @ Y3 )
=> ? [X4: real] :
( ( ord_less_eq @ real @ ( zero_zero @ real ) @ X4 )
& ( ord_less_eq @ real @ X4 @ ( minus_minus @ real @ Y3 @ ( one_one @ real ) ) )
& ( ( exp @ real @ X4 )
= Y3 ) ) ) ).
% lemma_exp_total
thf(fact_4772_ln__ge__iff,axiom,
! [X2: real,Y3: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ X2 )
=> ( ( ord_less_eq @ real @ Y3 @ ( ln_ln @ real @ X2 ) )
= ( ord_less_eq @ real @ ( exp @ real @ Y3 ) @ X2 ) ) ) ).
% ln_ge_iff
thf(fact_4773_ln__x__over__x__mono,axiom,
! [X2: real,Y3: real] :
( ( ord_less_eq @ real @ ( exp @ real @ ( one_one @ real ) ) @ X2 )
=> ( ( ord_less_eq @ real @ X2 @ Y3 )
=> ( ord_less_eq @ real @ ( divide_divide @ real @ ( ln_ln @ real @ Y3 ) @ Y3 ) @ ( divide_divide @ real @ ( ln_ln @ real @ X2 ) @ X2 ) ) ) ) ).
% ln_x_over_x_mono
thf(fact_4774_powr__def,axiom,
! [A: $tType] :
( ( ln @ A )
=> ( ( powr @ A )
= ( ^ [X3: A,A6: A] :
( if @ A
@ ( X3
= ( zero_zero @ A ) )
@ ( zero_zero @ A )
@ ( exp @ A @ ( times_times @ A @ A6 @ ( ln_ln @ A @ X3 ) ) ) ) ) ) ) ).
% powr_def
thf(fact_4775_arccos__lbound,axiom,
! [Y3: real] :
( ( ord_less_eq @ real @ ( uminus_uminus @ real @ ( one_one @ real ) ) @ Y3 )
=> ( ( ord_less_eq @ real @ Y3 @ ( one_one @ real ) )
=> ( ord_less_eq @ real @ ( zero_zero @ real ) @ ( arccos @ Y3 ) ) ) ) ).
% arccos_lbound
thf(fact_4776_arccos__less__arccos,axiom,
! [X2: real,Y3: real] :
( ( ord_less_eq @ real @ ( uminus_uminus @ real @ ( one_one @ real ) ) @ X2 )
=> ( ( ord_less @ real @ X2 @ Y3 )
=> ( ( ord_less_eq @ real @ Y3 @ ( one_one @ real ) )
=> ( ord_less @ real @ ( arccos @ Y3 ) @ ( arccos @ X2 ) ) ) ) ) ).
% arccos_less_arccos
thf(fact_4777_arccos__less__mono,axiom,
! [X2: real,Y3: real] :
( ( ord_less_eq @ real @ ( abs_abs @ real @ X2 ) @ ( one_one @ real ) )
=> ( ( ord_less_eq @ real @ ( abs_abs @ real @ Y3 ) @ ( one_one @ real ) )
=> ( ( ord_less @ real @ ( arccos @ X2 ) @ ( arccos @ Y3 ) )
= ( ord_less @ real @ Y3 @ X2 ) ) ) ) ).
% arccos_less_mono
thf(fact_4778_exp__le,axiom,
ord_less_eq @ real @ ( exp @ real @ ( one_one @ real ) ) @ ( numeral_numeral @ real @ ( bit1 @ one2 ) ) ).
% exp_le
thf(fact_4779_arccos__ubound,axiom,
! [Y3: real] :
( ( ord_less_eq @ real @ ( uminus_uminus @ real @ ( one_one @ real ) ) @ Y3 )
=> ( ( ord_less_eq @ real @ Y3 @ ( one_one @ real ) )
=> ( ord_less_eq @ real @ ( arccos @ Y3 ) @ pi ) ) ) ).
% arccos_ubound
thf(fact_4780_arccos__cos,axiom,
! [X2: real] :
( ( ord_less_eq @ real @ ( zero_zero @ real ) @ X2 )
=> ( ( ord_less_eq @ real @ X2 @ pi )
=> ( ( arccos @ ( cos @ real @ X2 ) )
= X2 ) ) ) ).
% arccos_cos
thf(fact_4781_arcsin__less__arcsin,axiom,
! [X2: real,Y3: real] :
( ( ord_less_eq @ real @ ( uminus_uminus @ real @ ( one_one @ real ) ) @ X2 )
=> ( ( ord_less @ real @ X2 @ Y3 )
=> ( ( ord_less_eq @ real @ Y3 @ ( one_one @ real ) )
=> ( ord_less @ real @ ( arcsin @ X2 ) @ ( arcsin @ Y3 ) ) ) ) ) ).
% arcsin_less_arcsin
thf(fact_4782_arcsin__less__mono,axiom,
! [X2: real,Y3: real] :
( ( ord_less_eq @ real @ ( abs_abs @ real @ X2 ) @ ( one_one @ real ) )
=> ( ( ord_less_eq @ real @ ( abs_abs @ real @ Y3 ) @ ( one_one @ real ) )
=> ( ( ord_less @ real @ ( arcsin @ X2 ) @ ( arcsin @ Y3 ) )
= ( ord_less @ real @ X2 @ Y3 ) ) ) ) ).
% arcsin_less_mono
thf(fact_4783_cos__arccos__abs,axiom,
! [Y3: real] :
( ( ord_less_eq @ real @ ( abs_abs @ real @ Y3 ) @ ( one_one @ real ) )
=> ( ( cos @ real @ ( arccos @ Y3 ) )
= Y3 ) ) ).
% cos_arccos_abs
thf(fact_4784_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_4785_exp__divide__power__eq,axiom,
! [A: $tType] :
( ( ( real_Vector_banach @ A )
& ( real_V3459762299906320749_field @ A ) )
=> ! [N: nat,X2: A] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ N )
=> ( ( power_power @ A @ ( exp @ A @ ( divide_divide @ A @ X2 @ ( semiring_1_of_nat @ A @ N ) ) ) @ N )
= ( exp @ A @ X2 ) ) ) ) ).
% exp_divide_power_eq
thf(fact_4786_tanh__altdef,axiom,
! [A: $tType] :
( ( ( real_Vector_banach @ A )
& ( real_V3459762299906320749_field @ A ) )
=> ( ( tanh @ A )
= ( ^ [X3: A] : ( divide_divide @ A @ ( minus_minus @ A @ ( exp @ A @ X3 ) @ ( exp @ A @ ( uminus_uminus @ A @ X3 ) ) ) @ ( plus_plus @ A @ ( exp @ A @ X3 ) @ ( exp @ A @ ( uminus_uminus @ A @ X3 ) ) ) ) ) ) ) ).
% tanh_altdef
thf(fact_4787_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_4788_arccos__lt__bounded,axiom,
! [Y3: real] :
( ( ord_less @ real @ ( uminus_uminus @ real @ ( one_one @ real ) ) @ Y3 )
=> ( ( ord_less @ real @ Y3 @ ( one_one @ real ) )
=> ( ( ord_less @ real @ ( zero_zero @ real ) @ ( arccos @ Y3 ) )
& ( ord_less @ real @ ( arccos @ Y3 ) @ pi ) ) ) ) ).
% arccos_lt_bounded
thf(fact_4789_exp__double,axiom,
! [A: $tType] :
( ( ( real_Vector_banach @ A )
& ( real_V2822296259951069270ebra_1 @ A ) )
=> ! [Z: A] :
( ( exp @ A @ ( times_times @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ Z ) )
= ( power_power @ A @ ( exp @ A @ Z ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ).
% exp_double
thf(fact_4790_arccos__bounded,axiom,
! [Y3: real] :
( ( ord_less_eq @ real @ ( uminus_uminus @ real @ ( one_one @ real ) ) @ Y3 )
=> ( ( ord_less_eq @ real @ Y3 @ ( one_one @ real ) )
=> ( ( ord_less_eq @ real @ ( zero_zero @ real ) @ ( arccos @ Y3 ) )
& ( ord_less_eq @ real @ ( arccos @ Y3 ) @ pi ) ) ) ) ).
% arccos_bounded
thf(fact_4791_sin__arccos__nonzero,axiom,
! [X2: real] :
( ( ord_less @ real @ ( uminus_uminus @ real @ ( one_one @ real ) ) @ X2 )
=> ( ( ord_less @ real @ X2 @ ( one_one @ real ) )
=> ( ( sin @ real @ ( arccos @ X2 ) )
!= ( zero_zero @ real ) ) ) ) ).
% sin_arccos_nonzero
thf(fact_4792_arccos__cos2,axiom,
! [X2: real] :
( ( ord_less_eq @ real @ X2 @ ( zero_zero @ real ) )
=> ( ( ord_less_eq @ real @ ( uminus_uminus @ real @ pi ) @ X2 )
=> ( ( arccos @ ( cos @ real @ X2 ) )
= ( uminus_uminus @ real @ X2 ) ) ) ) ).
% arccos_cos2
thf(fact_4793_arccos__minus,axiom,
! [X2: real] :
( ( ord_less_eq @ real @ ( uminus_uminus @ real @ ( one_one @ real ) ) @ X2 )
=> ( ( ord_less_eq @ real @ X2 @ ( one_one @ real ) )
=> ( ( arccos @ ( uminus_uminus @ real @ X2 ) )
= ( minus_minus @ real @ pi @ ( arccos @ X2 ) ) ) ) ) ).
% arccos_minus
thf(fact_4794_cos__arcsin__nonzero,axiom,
! [X2: real] :
( ( ord_less @ real @ ( uminus_uminus @ real @ ( one_one @ real ) ) @ X2 )
=> ( ( ord_less @ real @ X2 @ ( one_one @ real ) )
=> ( ( cos @ real @ ( arcsin @ X2 ) )
!= ( zero_zero @ real ) ) ) ) ).
% cos_arcsin_nonzero
thf(fact_4795_arccos,axiom,
! [Y3: real] :
( ( ord_less_eq @ real @ ( uminus_uminus @ real @ ( one_one @ real ) ) @ Y3 )
=> ( ( ord_less_eq @ real @ Y3 @ ( one_one @ real ) )
=> ( ( ord_less_eq @ real @ ( zero_zero @ real ) @ ( arccos @ Y3 ) )
& ( ord_less_eq @ real @ ( arccos @ Y3 ) @ pi )
& ( ( cos @ real @ ( arccos @ Y3 ) )
= Y3 ) ) ) ) ).
% arccos
thf(fact_4796_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_4797_arccos__minus__abs,axiom,
! [X2: real] :
( ( ord_less_eq @ real @ ( abs_abs @ real @ X2 ) @ ( one_one @ real ) )
=> ( ( arccos @ ( uminus_uminus @ real @ X2 ) )
= ( minus_minus @ real @ pi @ ( arccos @ X2 ) ) ) ) ).
% arccos_minus_abs
thf(fact_4798_exp__bound,axiom,
! [X2: real] :
( ( ord_less_eq @ real @ ( zero_zero @ real ) @ X2 )
=> ( ( ord_less_eq @ real @ X2 @ ( one_one @ real ) )
=> ( ord_less_eq @ real @ ( exp @ real @ X2 ) @ ( plus_plus @ real @ ( plus_plus @ real @ ( one_one @ real ) @ X2 ) @ ( power_power @ real @ X2 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) ) ).
% exp_bound
thf(fact_4799_real__exp__bound__lemma,axiom,
! [X2: real] :
( ( ord_less_eq @ real @ ( zero_zero @ real ) @ X2 )
=> ( ( ord_less_eq @ real @ X2 @ ( divide_divide @ real @ ( one_one @ real ) @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) )
=> ( ord_less_eq @ real @ ( exp @ real @ X2 ) @ ( plus_plus @ real @ ( one_one @ real ) @ ( times_times @ real @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) @ X2 ) ) ) ) ) ).
% real_exp_bound_lemma
thf(fact_4800_exp__ge__one__plus__x__over__n__power__n,axiom,
! [N: nat,X2: real] :
( ( ord_less_eq @ real @ ( uminus_uminus @ real @ ( semiring_1_of_nat @ real @ N ) ) @ X2 )
=> ( ( ord_less @ nat @ ( zero_zero @ nat ) @ N )
=> ( ord_less_eq @ real @ ( power_power @ real @ ( plus_plus @ real @ ( one_one @ real ) @ ( divide_divide @ real @ X2 @ ( semiring_1_of_nat @ real @ N ) ) ) @ N ) @ ( exp @ real @ X2 ) ) ) ) ).
% exp_ge_one_plus_x_over_n_power_n
thf(fact_4801_exp__ge__one__minus__x__over__n__power__n,axiom,
! [X2: real,N: nat] :
( ( ord_less_eq @ real @ X2 @ ( 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 @ X2 @ ( semiring_1_of_nat @ real @ N ) ) ) @ N ) @ ( exp @ real @ ( uminus_uminus @ real @ X2 ) ) ) ) ) ).
% exp_ge_one_minus_x_over_n_power_n
thf(fact_4802_arccos__le__pi2,axiom,
! [Y3: real] :
( ( ord_less_eq @ real @ ( zero_zero @ real ) @ Y3 )
=> ( ( ord_less_eq @ real @ Y3 @ ( one_one @ real ) )
=> ( ord_less_eq @ real @ ( arccos @ Y3 ) @ ( divide_divide @ real @ pi @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) ) ) ) ).
% arccos_le_pi2
thf(fact_4803_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_4804_arcsin__lt__bounded,axiom,
! [Y3: real] :
( ( ord_less @ real @ ( uminus_uminus @ real @ ( one_one @ real ) ) @ Y3 )
=> ( ( ord_less @ real @ Y3 @ ( one_one @ real ) )
=> ( ( ord_less @ real @ ( uminus_uminus @ real @ ( divide_divide @ real @ pi @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) ) @ ( arcsin @ Y3 ) )
& ( ord_less @ real @ ( arcsin @ Y3 ) @ ( divide_divide @ real @ pi @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) ) ) ) ) ).
% arcsin_lt_bounded
thf(fact_4805_arcsin__bounded,axiom,
! [Y3: real] :
( ( ord_less_eq @ real @ ( uminus_uminus @ real @ ( one_one @ real ) ) @ Y3 )
=> ( ( ord_less_eq @ real @ Y3 @ ( one_one @ real ) )
=> ( ( ord_less_eq @ real @ ( uminus_uminus @ real @ ( divide_divide @ real @ pi @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) ) @ ( arcsin @ Y3 ) )
& ( ord_less_eq @ real @ ( arcsin @ Y3 ) @ ( divide_divide @ real @ pi @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) ) ) ) ) ).
% arcsin_bounded
thf(fact_4806_arcsin__ubound,axiom,
! [Y3: real] :
( ( ord_less_eq @ real @ ( uminus_uminus @ real @ ( one_one @ real ) ) @ Y3 )
=> ( ( ord_less_eq @ real @ Y3 @ ( one_one @ real ) )
=> ( ord_less_eq @ real @ ( arcsin @ Y3 ) @ ( divide_divide @ real @ pi @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) ) ) ) ).
% arcsin_ubound
thf(fact_4807_arcsin__lbound,axiom,
! [Y3: real] :
( ( ord_less_eq @ real @ ( uminus_uminus @ real @ ( one_one @ real ) ) @ Y3 )
=> ( ( ord_less_eq @ real @ Y3 @ ( one_one @ real ) )
=> ( ord_less_eq @ real @ ( uminus_uminus @ real @ ( divide_divide @ real @ pi @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) ) @ ( arcsin @ Y3 ) ) ) ) ).
% arcsin_lbound
thf(fact_4808_Maclaurin__exp__le,axiom,
! [X2: real,N: nat] :
? [T8: real] :
( ( ord_less_eq @ real @ ( abs_abs @ real @ T8 ) @ ( abs_abs @ real @ X2 ) )
& ( ( exp @ real @ X2 )
= ( plus_plus @ real
@ ( groups7311177749621191930dd_sum @ nat @ real
@ ^ [M6: nat] : ( divide_divide @ real @ ( power_power @ real @ X2 @ M6 ) @ ( semiring_char_0_fact @ real @ M6 ) )
@ ( set_ord_lessThan @ nat @ N ) )
@ ( times_times @ real @ ( divide_divide @ real @ ( exp @ real @ T8 ) @ ( semiring_char_0_fact @ real @ N ) ) @ ( power_power @ real @ X2 @ N ) ) ) ) ) ).
% Maclaurin_exp_le
thf(fact_4809_exp__lower__Taylor__quadratic,axiom,
! [X2: real] :
( ( ord_less_eq @ real @ ( zero_zero @ real ) @ X2 )
=> ( ord_less_eq @ real @ ( plus_plus @ real @ ( plus_plus @ real @ ( one_one @ real ) @ X2 ) @ ( divide_divide @ real @ ( power_power @ real @ X2 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) ) @ ( exp @ real @ X2 ) ) ) ).
% exp_lower_Taylor_quadratic
thf(fact_4810_arcsin__sin,axiom,
! [X2: real] :
( ( ord_less_eq @ real @ ( uminus_uminus @ real @ ( divide_divide @ real @ pi @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) ) @ X2 )
=> ( ( ord_less_eq @ real @ X2 @ ( divide_divide @ real @ pi @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) )
=> ( ( arcsin @ ( sin @ real @ X2 ) )
= X2 ) ) ) ).
% arcsin_sin
thf(fact_4811_log__base__10__eq2,axiom,
! [X2: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ X2 )
=> ( ( log @ ( numeral_numeral @ real @ ( bit0 @ ( bit1 @ ( bit0 @ one2 ) ) ) ) @ X2 )
= ( times_times @ real @ ( log @ ( numeral_numeral @ real @ ( bit0 @ ( bit1 @ ( bit0 @ one2 ) ) ) ) @ ( exp @ real @ ( one_one @ real ) ) ) @ ( ln_ln @ real @ X2 ) ) ) ) ).
% log_base_10_eq2
thf(fact_4812_tanh__real__altdef,axiom,
( ( tanh @ real )
= ( ^ [X3: real] : ( divide_divide @ real @ ( minus_minus @ real @ ( one_one @ real ) @ ( exp @ real @ ( times_times @ real @ ( uminus_uminus @ real @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) @ X3 ) ) ) @ ( plus_plus @ real @ ( one_one @ real ) @ ( exp @ real @ ( times_times @ real @ ( uminus_uminus @ real @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) @ X3 ) ) ) ) ) ) ).
% tanh_real_altdef
thf(fact_4813_log__base__10__eq1,axiom,
! [X2: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ X2 )
=> ( ( log @ ( numeral_numeral @ real @ ( bit0 @ ( bit1 @ ( bit0 @ one2 ) ) ) ) @ X2 )
= ( 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 @ X2 ) ) ) ) ).
% log_base_10_eq1
thf(fact_4814_le__arcsin__iff,axiom,
! [X2: real,Y3: real] :
( ( ord_less_eq @ real @ ( uminus_uminus @ real @ ( one_one @ real ) ) @ X2 )
=> ( ( ord_less_eq @ real @ X2 @ ( one_one @ real ) )
=> ( ( ord_less_eq @ real @ ( divide_divide @ real @ ( uminus_uminus @ real @ pi ) @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) @ Y3 )
=> ( ( ord_less_eq @ real @ Y3 @ ( divide_divide @ real @ pi @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) )
=> ( ( ord_less_eq @ real @ Y3 @ ( arcsin @ X2 ) )
= ( ord_less_eq @ real @ ( sin @ real @ Y3 ) @ X2 ) ) ) ) ) ) ).
% le_arcsin_iff
thf(fact_4815_arcsin__le__iff,axiom,
! [X2: real,Y3: real] :
( ( ord_less_eq @ real @ ( uminus_uminus @ real @ ( one_one @ real ) ) @ X2 )
=> ( ( ord_less_eq @ real @ X2 @ ( one_one @ real ) )
=> ( ( ord_less_eq @ real @ ( divide_divide @ real @ ( uminus_uminus @ real @ pi ) @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) @ Y3 )
=> ( ( ord_less_eq @ real @ Y3 @ ( divide_divide @ real @ pi @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) )
=> ( ( ord_less_eq @ real @ ( arcsin @ X2 ) @ Y3 )
= ( ord_less_eq @ real @ X2 @ ( sin @ real @ Y3 ) ) ) ) ) ) ) ).
% arcsin_le_iff
thf(fact_4816_arcsin__pi,axiom,
! [Y3: real] :
( ( ord_less_eq @ real @ ( uminus_uminus @ real @ ( one_one @ real ) ) @ Y3 )
=> ( ( ord_less_eq @ real @ Y3 @ ( one_one @ real ) )
=> ( ( ord_less_eq @ real @ ( uminus_uminus @ real @ ( divide_divide @ real @ pi @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) ) @ ( arcsin @ Y3 ) )
& ( ord_less_eq @ real @ ( arcsin @ Y3 ) @ pi )
& ( ( sin @ real @ ( arcsin @ Y3 ) )
= Y3 ) ) ) ) ).
% arcsin_pi
thf(fact_4817_arcsin,axiom,
! [Y3: real] :
( ( ord_less_eq @ real @ ( uminus_uminus @ real @ ( one_one @ real ) ) @ Y3 )
=> ( ( ord_less_eq @ real @ Y3 @ ( one_one @ real ) )
=> ( ( ord_less_eq @ real @ ( uminus_uminus @ real @ ( divide_divide @ real @ pi @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) ) @ ( arcsin @ Y3 ) )
& ( ord_less_eq @ real @ ( arcsin @ Y3 ) @ ( divide_divide @ real @ pi @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) )
& ( ( sin @ real @ ( arcsin @ Y3 ) )
= Y3 ) ) ) ) ).
% arcsin
thf(fact_4818_arccos__cos__eq__abs__2pi,axiom,
! [Theta: real] :
~ ! [K2: int] :
( ( arccos @ ( cos @ real @ Theta ) )
!= ( abs_abs @ real @ ( minus_minus @ real @ Theta @ ( times_times @ real @ ( ring_1_of_int @ real @ K2 ) @ ( times_times @ real @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) @ pi ) ) ) ) ) ).
% arccos_cos_eq_abs_2pi
thf(fact_4819_sin__arccos,axiom,
! [X2: real] :
( ( ord_less_eq @ real @ ( uminus_uminus @ real @ ( one_one @ real ) ) @ X2 )
=> ( ( ord_less_eq @ real @ X2 @ ( one_one @ real ) )
=> ( ( sin @ real @ ( arccos @ X2 ) )
= ( sqrt @ ( minus_minus @ real @ ( one_one @ real ) @ ( power_power @ real @ X2 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) ) ) ).
% sin_arccos
thf(fact_4820_monoI1,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [X: nat > A] :
( ! [M5: nat,N3: nat] :
( ( ord_less_eq @ nat @ M5 @ N3 )
=> ( ord_less_eq @ A @ ( X @ M5 ) @ ( X @ N3 ) ) )
=> ( topological_monoseq @ A @ X ) ) ) ).
% monoI1
thf(fact_4821_monoI2,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [X: nat > A] :
( ! [M5: nat,N3: nat] :
( ( ord_less_eq @ nat @ M5 @ N3 )
=> ( ord_less_eq @ A @ ( X @ N3 ) @ ( X @ M5 ) ) )
=> ( topological_monoseq @ A @ X ) ) ) ).
% monoI2
thf(fact_4822_monoseq__def,axiom,
! [A: $tType] :
( ( order @ A )
=> ( ( topological_monoseq @ A )
= ( ^ [X7: nat > A] :
( ! [M6: nat,N4: nat] :
( ( ord_less_eq @ nat @ M6 @ N4 )
=> ( ord_less_eq @ A @ ( X7 @ M6 ) @ ( X7 @ N4 ) ) )
| ! [M6: nat,N4: nat] :
( ( ord_less_eq @ nat @ M6 @ N4 )
=> ( ord_less_eq @ A @ ( X7 @ N4 ) @ ( X7 @ M6 ) ) ) ) ) ) ) ).
% monoseq_def
thf(fact_4823_monoseq__minus,axiom,
! [A: $tType] :
( ( ordered_ab_group_add @ A )
=> ! [A3: nat > A] :
( ( topological_monoseq @ A @ A3 )
=> ( topological_monoseq @ A
@ ^ [N4: nat] : ( uminus_uminus @ A @ ( A3 @ N4 ) ) ) ) ) ).
% monoseq_minus
thf(fact_4824_monoseq__Suc,axiom,
! [A: $tType] :
( ( order @ A )
=> ( ( topological_monoseq @ A )
= ( ^ [X7: nat > A] :
( ! [N4: nat] : ( ord_less_eq @ A @ ( X7 @ N4 ) @ ( X7 @ ( suc @ N4 ) ) )
| ! [N4: nat] : ( ord_less_eq @ A @ ( X7 @ ( suc @ N4 ) ) @ ( X7 @ N4 ) ) ) ) ) ) ).
% monoseq_Suc
thf(fact_4825_mono__SucI2,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [X: nat > A] :
( ! [N3: nat] : ( ord_less_eq @ A @ ( X @ ( suc @ N3 ) ) @ ( X @ N3 ) )
=> ( topological_monoseq @ A @ X ) ) ) ).
% mono_SucI2
thf(fact_4826_mono__SucI1,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [X: nat > A] :
( ! [N3: nat] : ( ord_less_eq @ A @ ( X @ N3 ) @ ( X @ ( suc @ N3 ) ) )
=> ( topological_monoseq @ A @ X ) ) ) ).
% mono_SucI1
thf(fact_4827_exp__first__two__terms,axiom,
! [A: $tType] :
( ( ( real_Vector_banach @ A )
& ( real_V2822296259951069270ebra_1 @ A ) )
=> ( ( exp @ A )
= ( ^ [X3: A] :
( plus_plus @ A @ ( plus_plus @ A @ ( one_one @ A ) @ X3 )
@ ( suminf @ A
@ ^ [N4: nat] : ( real_V8093663219630862766scaleR @ A @ ( inverse_inverse @ real @ ( semiring_char_0_fact @ real @ ( plus_plus @ nat @ N4 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( power_power @ A @ X3 @ ( plus_plus @ nat @ N4 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) ) ) ) ) ).
% exp_first_two_terms
thf(fact_4828_Maclaurin__sin__bound,axiom,
! [X2: real,N: nat] :
( ord_less_eq @ real
@ ( abs_abs @ real
@ ( minus_minus @ real @ ( sin @ real @ X2 )
@ ( groups7311177749621191930dd_sum @ nat @ real
@ ^ [M6: nat] : ( times_times @ real @ ( sin_coeff @ M6 ) @ ( power_power @ real @ X2 @ M6 ) )
@ ( set_ord_lessThan @ nat @ N ) ) ) )
@ ( times_times @ real @ ( inverse_inverse @ real @ ( semiring_char_0_fact @ real @ N ) ) @ ( power_power @ real @ ( abs_abs @ real @ X2 ) @ N ) ) ) ).
% Maclaurin_sin_bound
thf(fact_4829_cot__less__zero,axiom,
! [X2: real] :
( ( ord_less @ real @ ( divide_divide @ real @ ( uminus_uminus @ real @ pi ) @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) @ X2 )
=> ( ( ord_less @ real @ X2 @ ( zero_zero @ real ) )
=> ( ord_less @ real @ ( cot @ real @ X2 ) @ ( zero_zero @ real ) ) ) ) ).
% cot_less_zero
thf(fact_4830_inverse__1,axiom,
! [A: $tType] :
( ( division_ring @ A )
=> ( ( inverse_inverse @ A @ ( one_one @ A ) )
= ( one_one @ A ) ) ) ).
% inverse_1
thf(fact_4831_inverse__eq__1__iff,axiom,
! [A: $tType] :
( ( field @ A )
=> ! [X2: A] :
( ( ( inverse_inverse @ A @ X2 )
= ( one_one @ A ) )
= ( X2
= ( one_one @ A ) ) ) ) ).
% inverse_eq_1_iff
thf(fact_4832_inverse__mult__distrib,axiom,
! [A: $tType] :
( ( field @ A )
=> ! [A3: A,B3: A] :
( ( inverse_inverse @ A @ ( times_times @ A @ A3 @ B3 ) )
= ( times_times @ A @ ( inverse_inverse @ A @ A3 ) @ ( inverse_inverse @ A @ B3 ) ) ) ) ).
% inverse_mult_distrib
thf(fact_4833_inverse__divide,axiom,
! [A: $tType] :
( ( field @ A )
=> ! [A3: A,B3: A] :
( ( inverse_inverse @ A @ ( divide_divide @ A @ A3 @ B3 ) )
= ( divide_divide @ A @ B3 @ A3 ) ) ) ).
% inverse_divide
thf(fact_4834_inverse__nonnegative__iff__nonnegative,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [A3: A] :
( ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( inverse_inverse @ A @ A3 ) )
= ( ord_less_eq @ A @ ( zero_zero @ A ) @ A3 ) ) ) ).
% inverse_nonnegative_iff_nonnegative
thf(fact_4835_inverse__nonpositive__iff__nonpositive,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [A3: A] :
( ( ord_less_eq @ A @ ( inverse_inverse @ A @ A3 ) @ ( zero_zero @ A ) )
= ( ord_less_eq @ A @ A3 @ ( zero_zero @ A ) ) ) ) ).
% inverse_nonpositive_iff_nonpositive
thf(fact_4836_inverse__less__iff__less,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [A3: A,B3: A] :
( ( ord_less @ A @ ( zero_zero @ A ) @ A3 )
=> ( ( ord_less @ A @ ( zero_zero @ A ) @ B3 )
=> ( ( ord_less @ A @ ( inverse_inverse @ A @ A3 ) @ ( inverse_inverse @ A @ B3 ) )
= ( ord_less @ A @ B3 @ A3 ) ) ) ) ) ).
% inverse_less_iff_less
thf(fact_4837_inverse__less__iff__less__neg,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [A3: A,B3: A] :
( ( ord_less @ A @ A3 @ ( zero_zero @ A ) )
=> ( ( ord_less @ A @ B3 @ ( zero_zero @ A ) )
=> ( ( ord_less @ A @ ( inverse_inverse @ A @ A3 ) @ ( inverse_inverse @ A @ B3 ) )
= ( ord_less @ A @ B3 @ A3 ) ) ) ) ) ).
% inverse_less_iff_less_neg
thf(fact_4838_inverse__negative__iff__negative,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [A3: A] :
( ( ord_less @ A @ ( inverse_inverse @ A @ A3 ) @ ( zero_zero @ A ) )
= ( ord_less @ A @ A3 @ ( zero_zero @ A ) ) ) ) ).
% inverse_negative_iff_negative
thf(fact_4839_inverse__positive__iff__positive,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [A3: A] :
( ( ord_less @ A @ ( zero_zero @ A ) @ ( inverse_inverse @ A @ A3 ) )
= ( ord_less @ A @ ( zero_zero @ A ) @ A3 ) ) ) ).
% inverse_positive_iff_positive
thf(fact_4840_inverse__le__iff__le,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [A3: A,B3: A] :
( ( ord_less @ A @ ( zero_zero @ A ) @ A3 )
=> ( ( ord_less @ A @ ( zero_zero @ A ) @ B3 )
=> ( ( ord_less_eq @ A @ ( inverse_inverse @ A @ A3 ) @ ( inverse_inverse @ A @ B3 ) )
= ( ord_less_eq @ A @ B3 @ A3 ) ) ) ) ) ).
% inverse_le_iff_le
thf(fact_4841_inverse__le__iff__le__neg,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [A3: A,B3: A] :
( ( ord_less @ A @ A3 @ ( zero_zero @ A ) )
=> ( ( ord_less @ A @ B3 @ ( zero_zero @ A ) )
=> ( ( ord_less_eq @ A @ ( inverse_inverse @ A @ A3 ) @ ( inverse_inverse @ A @ B3 ) )
= ( ord_less_eq @ A @ B3 @ A3 ) ) ) ) ) ).
% inverse_le_iff_le_neg
thf(fact_4842_left__inverse,axiom,
! [A: $tType] :
( ( division_ring @ A )
=> ! [A3: A] :
( ( A3
!= ( zero_zero @ A ) )
=> ( ( times_times @ A @ ( inverse_inverse @ A @ A3 ) @ A3 )
= ( one_one @ A ) ) ) ) ).
% left_inverse
thf(fact_4843_right__inverse,axiom,
! [A: $tType] :
( ( division_ring @ A )
=> ! [A3: A] :
( ( A3
!= ( zero_zero @ A ) )
=> ( ( times_times @ A @ A3 @ ( inverse_inverse @ A @ A3 ) )
= ( one_one @ A ) ) ) ) ).
% right_inverse
thf(fact_4844_inverse__eq__divide__numeral,axiom,
! [A: $tType] :
( ( division_ring @ A )
=> ! [W: num] :
( ( inverse_inverse @ A @ ( numeral_numeral @ A @ W ) )
= ( divide_divide @ A @ ( one_one @ A ) @ ( numeral_numeral @ A @ W ) ) ) ) ).
% inverse_eq_divide_numeral
thf(fact_4845_inverse__eq__divide__neg__numeral,axiom,
! [A: $tType] :
( ( division_ring @ A )
=> ! [W: num] :
( ( inverse_inverse @ A @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ W ) ) )
= ( divide_divide @ A @ ( one_one @ A ) @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ W ) ) ) ) ) ).
% inverse_eq_divide_neg_numeral
thf(fact_4846_cot__periodic,axiom,
! [X2: real] :
( ( cot @ real @ ( plus_plus @ real @ X2 @ ( times_times @ real @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) @ pi ) ) )
= ( cot @ real @ X2 ) ) ).
% cot_periodic
thf(fact_4847_power__inverse,axiom,
! [A: $tType] :
( ( division_ring @ A )
=> ! [A3: A,N: nat] :
( ( power_power @ A @ ( inverse_inverse @ A @ A3 ) @ N )
= ( inverse_inverse @ A @ ( power_power @ A @ A3 @ N ) ) ) ) ).
% power_inverse
thf(fact_4848_mult__commute__imp__mult__inverse__commute,axiom,
! [A: $tType] :
( ( division_ring @ A )
=> ! [Y3: A,X2: A] :
( ( ( times_times @ A @ Y3 @ X2 )
= ( times_times @ A @ X2 @ Y3 ) )
=> ( ( times_times @ A @ ( inverse_inverse @ A @ Y3 ) @ X2 )
= ( times_times @ A @ X2 @ ( inverse_inverse @ A @ Y3 ) ) ) ) ) ).
% mult_commute_imp_mult_inverse_commute
thf(fact_4849_norm__inverse__le__norm,axiom,
! [A: $tType] :
( ( real_V8999393235501362500lgebra @ A )
=> ! [R2: real,X2: A] :
( ( ord_less_eq @ real @ R2 @ ( real_V7770717601297561774m_norm @ A @ X2 ) )
=> ( ( ord_less @ real @ ( zero_zero @ real ) @ R2 )
=> ( ord_less_eq @ real @ ( real_V7770717601297561774m_norm @ A @ ( inverse_inverse @ A @ X2 ) ) @ ( inverse_inverse @ real @ R2 ) ) ) ) ) ).
% norm_inverse_le_norm
thf(fact_4850_inverse__less__imp__less,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [A3: A,B3: A] :
( ( ord_less @ A @ ( inverse_inverse @ A @ A3 ) @ ( inverse_inverse @ A @ B3 ) )
=> ( ( ord_less @ A @ ( zero_zero @ A ) @ A3 )
=> ( ord_less @ A @ B3 @ A3 ) ) ) ) ).
% inverse_less_imp_less
thf(fact_4851_less__imp__inverse__less,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [A3: A,B3: A] :
( ( ord_less @ A @ A3 @ B3 )
=> ( ( ord_less @ A @ ( zero_zero @ A ) @ A3 )
=> ( ord_less @ A @ ( inverse_inverse @ A @ B3 ) @ ( inverse_inverse @ A @ A3 ) ) ) ) ) ).
% less_imp_inverse_less
thf(fact_4852_inverse__less__imp__less__neg,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [A3: A,B3: A] :
( ( ord_less @ A @ ( inverse_inverse @ A @ A3 ) @ ( inverse_inverse @ A @ B3 ) )
=> ( ( ord_less @ A @ B3 @ ( zero_zero @ A ) )
=> ( ord_less @ A @ B3 @ A3 ) ) ) ) ).
% inverse_less_imp_less_neg
thf(fact_4853_less__imp__inverse__less__neg,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [A3: A,B3: A] :
( ( ord_less @ A @ A3 @ B3 )
=> ( ( ord_less @ A @ B3 @ ( zero_zero @ A ) )
=> ( ord_less @ A @ ( inverse_inverse @ A @ B3 ) @ ( inverse_inverse @ A @ A3 ) ) ) ) ) ).
% less_imp_inverse_less_neg
thf(fact_4854_inverse__negative__imp__negative,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [A3: A] :
( ( ord_less @ A @ ( inverse_inverse @ A @ A3 ) @ ( zero_zero @ A ) )
=> ( ( A3
!= ( zero_zero @ A ) )
=> ( ord_less @ A @ A3 @ ( zero_zero @ A ) ) ) ) ) ).
% inverse_negative_imp_negative
thf(fact_4855_inverse__positive__imp__positive,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [A3: A] :
( ( ord_less @ A @ ( zero_zero @ A ) @ ( inverse_inverse @ A @ A3 ) )
=> ( ( A3
!= ( zero_zero @ A ) )
=> ( ord_less @ A @ ( zero_zero @ A ) @ A3 ) ) ) ) ).
% inverse_positive_imp_positive
thf(fact_4856_negative__imp__inverse__negative,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [A3: A] :
( ( ord_less @ A @ A3 @ ( zero_zero @ A ) )
=> ( ord_less @ A @ ( inverse_inverse @ A @ A3 ) @ ( zero_zero @ A ) ) ) ) ).
% negative_imp_inverse_negative
thf(fact_4857_positive__imp__inverse__positive,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [A3: A] :
( ( ord_less @ A @ ( zero_zero @ A ) @ A3 )
=> ( ord_less @ A @ ( zero_zero @ A ) @ ( inverse_inverse @ A @ A3 ) ) ) ) ).
% positive_imp_inverse_positive
thf(fact_4858_nonzero__inverse__mult__distrib,axiom,
! [A: $tType] :
( ( division_ring @ A )
=> ! [A3: A,B3: A] :
( ( A3
!= ( zero_zero @ A ) )
=> ( ( B3
!= ( zero_zero @ A ) )
=> ( ( inverse_inverse @ A @ ( times_times @ A @ A3 @ B3 ) )
= ( times_times @ A @ ( inverse_inverse @ A @ B3 ) @ ( inverse_inverse @ A @ A3 ) ) ) ) ) ) ).
% nonzero_inverse_mult_distrib
thf(fact_4859_inverse__numeral__1,axiom,
! [A: $tType] :
( ( division_ring @ A )
=> ( ( inverse_inverse @ A @ ( numeral_numeral @ A @ one2 ) )
= ( numeral_numeral @ A @ one2 ) ) ) ).
% inverse_numeral_1
thf(fact_4860_inverse__unique,axiom,
! [A: $tType] :
( ( division_ring @ A )
=> ! [A3: A,B3: A] :
( ( ( times_times @ A @ A3 @ B3 )
= ( one_one @ A ) )
=> ( ( inverse_inverse @ A @ A3 )
= B3 ) ) ) ).
% inverse_unique
thf(fact_4861_inverse__eq__divide,axiom,
! [A: $tType] :
( ( division_ring @ A )
=> ( ( inverse_inverse @ A )
= ( divide_divide @ A @ ( one_one @ A ) ) ) ) ).
% inverse_eq_divide
thf(fact_4862_divide__inverse__commute,axiom,
! [A: $tType] :
( ( field @ A )
=> ( ( divide_divide @ A )
= ( ^ [A6: A,B7: A] : ( times_times @ A @ ( inverse_inverse @ A @ B7 ) @ A6 ) ) ) ) ).
% divide_inverse_commute
thf(fact_4863_divide__inverse,axiom,
! [A: $tType] :
( ( division_ring @ A )
=> ( ( divide_divide @ A )
= ( ^ [A6: A,B7: A] : ( times_times @ A @ A6 @ ( inverse_inverse @ A @ B7 ) ) ) ) ) ).
% divide_inverse
thf(fact_4864_field__class_Ofield__divide__inverse,axiom,
! [A: $tType] :
( ( field @ A )
=> ( ( divide_divide @ A )
= ( ^ [A6: A,B7: A] : ( times_times @ A @ A6 @ ( inverse_inverse @ A @ B7 ) ) ) ) ) ).
% field_class.field_divide_inverse
thf(fact_4865_power__mult__inverse__distrib,axiom,
! [A: $tType] :
( ( division_ring @ A )
=> ! [X2: A,M: nat] :
( ( times_times @ A @ ( power_power @ A @ X2 @ M ) @ ( inverse_inverse @ A @ X2 ) )
= ( times_times @ A @ ( inverse_inverse @ A @ X2 ) @ ( power_power @ A @ X2 @ M ) ) ) ) ).
% power_mult_inverse_distrib
thf(fact_4866_power__mult__power__inverse__commute,axiom,
! [A: $tType] :
( ( division_ring @ A )
=> ! [X2: A,M: nat,N: nat] :
( ( times_times @ A @ ( power_power @ A @ X2 @ M ) @ ( power_power @ A @ ( inverse_inverse @ A @ X2 ) @ N ) )
= ( times_times @ A @ ( power_power @ A @ ( inverse_inverse @ A @ X2 ) @ N ) @ ( power_power @ A @ X2 @ M ) ) ) ) ).
% power_mult_power_inverse_commute
thf(fact_4867_mult__inverse__of__nat__commute,axiom,
! [A: $tType] :
( ( division_ring @ A )
=> ! [Xa: nat,X2: A] :
( ( times_times @ A @ ( inverse_inverse @ A @ ( semiring_1_of_nat @ A @ Xa ) ) @ X2 )
= ( times_times @ A @ X2 @ ( inverse_inverse @ A @ ( semiring_1_of_nat @ A @ Xa ) ) ) ) ) ).
% mult_inverse_of_nat_commute
thf(fact_4868_mult__inverse__of__int__commute,axiom,
! [A: $tType] :
( ( division_ring @ A )
=> ! [Xa: int,X2: A] :
( ( times_times @ A @ ( inverse_inverse @ A @ ( ring_1_of_int @ A @ Xa ) ) @ X2 )
= ( times_times @ A @ X2 @ ( inverse_inverse @ A @ ( ring_1_of_int @ A @ Xa ) ) ) ) ) ).
% mult_inverse_of_int_commute
thf(fact_4869_exp__fdiffs,axiom,
! [A: $tType] :
( ( ( real_Vector_banach @ A )
& ( real_V3459762299906320749_field @ A ) )
=> ( ( diffs @ A
@ ^ [N4: nat] : ( inverse_inverse @ A @ ( semiring_char_0_fact @ A @ N4 ) ) )
= ( ^ [N4: nat] : ( inverse_inverse @ A @ ( semiring_char_0_fact @ A @ N4 ) ) ) ) ) ).
% exp_fdiffs
thf(fact_4870_inverse__le__imp__le,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [A3: A,B3: A] :
( ( ord_less_eq @ A @ ( inverse_inverse @ A @ A3 ) @ ( inverse_inverse @ A @ B3 ) )
=> ( ( ord_less @ A @ ( zero_zero @ A ) @ A3 )
=> ( ord_less_eq @ A @ B3 @ A3 ) ) ) ) ).
% inverse_le_imp_le
thf(fact_4871_le__imp__inverse__le,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [A3: A,B3: A] :
( ( ord_less_eq @ A @ A3 @ B3 )
=> ( ( ord_less @ A @ ( zero_zero @ A ) @ A3 )
=> ( ord_less_eq @ A @ ( inverse_inverse @ A @ B3 ) @ ( inverse_inverse @ A @ A3 ) ) ) ) ) ).
% le_imp_inverse_le
thf(fact_4872_inverse__le__imp__le__neg,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [A3: A,B3: A] :
( ( ord_less_eq @ A @ ( inverse_inverse @ A @ A3 ) @ ( inverse_inverse @ A @ B3 ) )
=> ( ( ord_less @ A @ B3 @ ( zero_zero @ A ) )
=> ( ord_less_eq @ A @ B3 @ A3 ) ) ) ) ).
% inverse_le_imp_le_neg
thf(fact_4873_le__imp__inverse__le__neg,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [A3: A,B3: A] :
( ( ord_less_eq @ A @ A3 @ B3 )
=> ( ( ord_less @ A @ B3 @ ( zero_zero @ A ) )
=> ( ord_less_eq @ A @ ( inverse_inverse @ A @ B3 ) @ ( inverse_inverse @ A @ A3 ) ) ) ) ) ).
% le_imp_inverse_le_neg
thf(fact_4874_inverse__le__1__iff,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [X2: A] :
( ( ord_less_eq @ A @ ( inverse_inverse @ A @ X2 ) @ ( one_one @ A ) )
= ( ( ord_less_eq @ A @ X2 @ ( zero_zero @ A ) )
| ( ord_less_eq @ A @ ( one_one @ A ) @ X2 ) ) ) ) ).
% inverse_le_1_iff
thf(fact_4875_one__less__inverse,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [A3: A] :
( ( ord_less @ A @ ( zero_zero @ A ) @ A3 )
=> ( ( ord_less @ A @ A3 @ ( one_one @ A ) )
=> ( ord_less @ A @ ( one_one @ A ) @ ( inverse_inverse @ A @ A3 ) ) ) ) ) ).
% one_less_inverse
thf(fact_4876_one__less__inverse__iff,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [X2: A] :
( ( ord_less @ A @ ( one_one @ A ) @ ( inverse_inverse @ A @ X2 ) )
= ( ( ord_less @ A @ ( zero_zero @ A ) @ X2 )
& ( ord_less @ A @ X2 @ ( one_one @ A ) ) ) ) ) ).
% one_less_inverse_iff
thf(fact_4877_inverse__add,axiom,
! [A: $tType] :
( ( field @ A )
=> ! [A3: A,B3: A] :
( ( A3
!= ( zero_zero @ A ) )
=> ( ( B3
!= ( zero_zero @ A ) )
=> ( ( plus_plus @ A @ ( inverse_inverse @ A @ A3 ) @ ( inverse_inverse @ A @ B3 ) )
= ( times_times @ A @ ( times_times @ A @ ( plus_plus @ A @ A3 @ B3 ) @ ( inverse_inverse @ A @ A3 ) ) @ ( inverse_inverse @ A @ B3 ) ) ) ) ) ) ).
% inverse_add
thf(fact_4878_division__ring__inverse__add,axiom,
! [A: $tType] :
( ( division_ring @ A )
=> ! [A3: A,B3: A] :
( ( A3
!= ( zero_zero @ A ) )
=> ( ( B3
!= ( zero_zero @ A ) )
=> ( ( plus_plus @ A @ ( inverse_inverse @ A @ A3 ) @ ( inverse_inverse @ A @ B3 ) )
= ( times_times @ A @ ( times_times @ A @ ( inverse_inverse @ A @ A3 ) @ ( plus_plus @ A @ A3 @ B3 ) ) @ ( inverse_inverse @ A @ B3 ) ) ) ) ) ) ).
% division_ring_inverse_add
thf(fact_4879_field__class_Ofield__inverse,axiom,
! [A: $tType] :
( ( field @ A )
=> ! [A3: A] :
( ( A3
!= ( zero_zero @ A ) )
=> ( ( times_times @ A @ ( inverse_inverse @ A @ A3 ) @ A3 )
= ( one_one @ A ) ) ) ) ).
% field_class.field_inverse
thf(fact_4880_division__ring__inverse__diff,axiom,
! [A: $tType] :
( ( division_ring @ A )
=> ! [A3: A,B3: A] :
( ( A3
!= ( zero_zero @ A ) )
=> ( ( B3
!= ( zero_zero @ A ) )
=> ( ( minus_minus @ A @ ( inverse_inverse @ A @ A3 ) @ ( inverse_inverse @ A @ B3 ) )
= ( times_times @ A @ ( times_times @ A @ ( inverse_inverse @ A @ A3 ) @ ( minus_minus @ A @ B3 @ A3 ) ) @ ( inverse_inverse @ A @ B3 ) ) ) ) ) ) ).
% division_ring_inverse_diff
thf(fact_4881_nonzero__inverse__eq__divide,axiom,
! [A: $tType] :
( ( division_ring @ A )
=> ! [A3: A] :
( ( A3
!= ( zero_zero @ A ) )
=> ( ( inverse_inverse @ A @ A3 )
= ( divide_divide @ A @ ( one_one @ A ) @ A3 ) ) ) ) ).
% nonzero_inverse_eq_divide
thf(fact_4882_inverse__powr,axiom,
! [Y3: real,A3: real] :
( ( ord_less_eq @ real @ ( zero_zero @ real ) @ Y3 )
=> ( ( powr @ real @ ( inverse_inverse @ real @ Y3 ) @ A3 )
= ( inverse_inverse @ real @ ( powr @ real @ Y3 @ A3 ) ) ) ) ).
% inverse_powr
thf(fact_4883_one__le__inverse,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [A3: A] :
( ( ord_less @ A @ ( zero_zero @ A ) @ A3 )
=> ( ( ord_less_eq @ A @ A3 @ ( one_one @ A ) )
=> ( ord_less_eq @ A @ ( one_one @ A ) @ ( inverse_inverse @ A @ A3 ) ) ) ) ) ).
% one_le_inverse
thf(fact_4884_inverse__less__1__iff,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [X2: A] :
( ( ord_less @ A @ ( inverse_inverse @ A @ X2 ) @ ( one_one @ A ) )
= ( ( ord_less_eq @ A @ X2 @ ( zero_zero @ A ) )
| ( ord_less @ A @ ( one_one @ A ) @ X2 ) ) ) ) ).
% inverse_less_1_iff
thf(fact_4885_one__le__inverse__iff,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [X2: A] :
( ( ord_less_eq @ A @ ( one_one @ A ) @ ( inverse_inverse @ A @ X2 ) )
= ( ( ord_less @ A @ ( zero_zero @ A ) @ X2 )
& ( ord_less_eq @ A @ X2 @ ( one_one @ A ) ) ) ) ) ).
% one_le_inverse_iff
thf(fact_4886_inverse__less__iff,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [A3: A,B3: A] :
( ( ord_less @ A @ ( inverse_inverse @ A @ A3 ) @ ( inverse_inverse @ A @ B3 ) )
= ( ( ( ord_less @ A @ ( zero_zero @ A ) @ ( times_times @ A @ A3 @ B3 ) )
=> ( ord_less @ A @ B3 @ A3 ) )
& ( ( ord_less_eq @ A @ ( times_times @ A @ A3 @ B3 ) @ ( zero_zero @ A ) )
=> ( ord_less @ A @ A3 @ B3 ) ) ) ) ) ).
% inverse_less_iff
thf(fact_4887_inverse__le__iff,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [A3: A,B3: A] :
( ( ord_less_eq @ A @ ( inverse_inverse @ A @ A3 ) @ ( inverse_inverse @ A @ B3 ) )
= ( ( ( ord_less @ A @ ( zero_zero @ A ) @ ( times_times @ A @ A3 @ B3 ) )
=> ( ord_less_eq @ A @ B3 @ A3 ) )
& ( ( ord_less_eq @ A @ ( times_times @ A @ A3 @ B3 ) @ ( zero_zero @ A ) )
=> ( ord_less_eq @ A @ A3 @ B3 ) ) ) ) ) ).
% inverse_le_iff
thf(fact_4888_inverse__diff__inverse,axiom,
! [A: $tType] :
( ( division_ring @ A )
=> ! [A3: A,B3: A] :
( ( A3
!= ( zero_zero @ A ) )
=> ( ( B3
!= ( zero_zero @ A ) )
=> ( ( minus_minus @ A @ ( inverse_inverse @ A @ A3 ) @ ( inverse_inverse @ A @ B3 ) )
= ( uminus_uminus @ A @ ( times_times @ A @ ( times_times @ A @ ( inverse_inverse @ A @ A3 ) @ ( minus_minus @ A @ A3 @ B3 ) ) @ ( inverse_inverse @ A @ B3 ) ) ) ) ) ) ) ).
% inverse_diff_inverse
thf(fact_4889_reals__Archimedean,axiom,
! [A: $tType] :
( ( archim462609752435547400_field @ A )
=> ! [X2: A] :
( ( ord_less @ A @ ( zero_zero @ A ) @ X2 )
=> ? [N3: nat] : ( ord_less @ A @ ( inverse_inverse @ A @ ( semiring_1_of_nat @ A @ ( suc @ N3 ) ) ) @ X2 ) ) ) ).
% reals_Archimedean
thf(fact_4890_real__vector__eq__affinity,axiom,
! [A: $tType] :
( ( real_V4867850818363320053vector @ A )
=> ! [M: real,Y3: A,X2: A,C3: A] :
( ( M
!= ( zero_zero @ real ) )
=> ( ( Y3
= ( plus_plus @ A @ ( real_V8093663219630862766scaleR @ A @ M @ X2 ) @ C3 ) )
= ( ( minus_minus @ A @ ( real_V8093663219630862766scaleR @ A @ ( inverse_inverse @ real @ M ) @ Y3 ) @ ( real_V8093663219630862766scaleR @ A @ ( inverse_inverse @ real @ M ) @ C3 ) )
= X2 ) ) ) ) ).
% real_vector_eq_affinity
thf(fact_4891_real__vector__affinity__eq,axiom,
! [A: $tType] :
( ( real_V4867850818363320053vector @ A )
=> ! [M: real,X2: A,C3: A,Y3: A] :
( ( M
!= ( zero_zero @ real ) )
=> ( ( ( plus_plus @ A @ ( real_V8093663219630862766scaleR @ A @ M @ X2 ) @ C3 )
= Y3 )
= ( X2
= ( minus_minus @ A @ ( real_V8093663219630862766scaleR @ A @ ( inverse_inverse @ real @ M ) @ Y3 ) @ ( real_V8093663219630862766scaleR @ A @ ( inverse_inverse @ real @ M ) @ C3 ) ) ) ) ) ) ).
% real_vector_affinity_eq
thf(fact_4892_pos__divideR__le__eq,axiom,
! [A: $tType] :
( ( real_V5355595471888546746vector @ A )
=> ! [C3: real,B3: A,A3: A] :
( ( ord_less @ real @ ( zero_zero @ real ) @ C3 )
=> ( ( ord_less_eq @ A @ ( real_V8093663219630862766scaleR @ A @ ( inverse_inverse @ real @ C3 ) @ B3 ) @ A3 )
= ( ord_less_eq @ A @ B3 @ ( real_V8093663219630862766scaleR @ A @ C3 @ A3 ) ) ) ) ) ).
% pos_divideR_le_eq
thf(fact_4893_pos__le__divideR__eq,axiom,
! [A: $tType] :
( ( real_V5355595471888546746vector @ A )
=> ! [C3: real,A3: A,B3: A] :
( ( ord_less @ real @ ( zero_zero @ real ) @ C3 )
=> ( ( ord_less_eq @ A @ A3 @ ( real_V8093663219630862766scaleR @ A @ ( inverse_inverse @ real @ C3 ) @ B3 ) )
= ( ord_less_eq @ A @ ( real_V8093663219630862766scaleR @ A @ C3 @ A3 ) @ B3 ) ) ) ) ).
% pos_le_divideR_eq
thf(fact_4894_neg__divideR__le__eq,axiom,
! [A: $tType] :
( ( real_V5355595471888546746vector @ A )
=> ! [C3: real,B3: A,A3: A] :
( ( ord_less @ real @ C3 @ ( zero_zero @ real ) )
=> ( ( ord_less_eq @ A @ ( real_V8093663219630862766scaleR @ A @ ( inverse_inverse @ real @ C3 ) @ B3 ) @ A3 )
= ( ord_less_eq @ A @ ( real_V8093663219630862766scaleR @ A @ C3 @ A3 ) @ B3 ) ) ) ) ).
% neg_divideR_le_eq
thf(fact_4895_neg__le__divideR__eq,axiom,
! [A: $tType] :
( ( real_V5355595471888546746vector @ A )
=> ! [C3: real,A3: A,B3: A] :
( ( ord_less @ real @ C3 @ ( zero_zero @ real ) )
=> ( ( ord_less_eq @ A @ A3 @ ( real_V8093663219630862766scaleR @ A @ ( inverse_inverse @ real @ C3 ) @ B3 ) )
= ( ord_less_eq @ A @ B3 @ ( real_V8093663219630862766scaleR @ A @ C3 @ A3 ) ) ) ) ) ).
% neg_le_divideR_eq
thf(fact_4896_pos__divideR__less__eq,axiom,
! [A: $tType] :
( ( real_V5355595471888546746vector @ A )
=> ! [C3: real,B3: A,A3: A] :
( ( ord_less @ real @ ( zero_zero @ real ) @ C3 )
=> ( ( ord_less @ A @ ( real_V8093663219630862766scaleR @ A @ ( inverse_inverse @ real @ C3 ) @ B3 ) @ A3 )
= ( ord_less @ A @ B3 @ ( real_V8093663219630862766scaleR @ A @ C3 @ A3 ) ) ) ) ) ).
% pos_divideR_less_eq
thf(fact_4897_pos__less__divideR__eq,axiom,
! [A: $tType] :
( ( real_V5355595471888546746vector @ A )
=> ! [C3: real,A3: A,B3: A] :
( ( ord_less @ real @ ( zero_zero @ real ) @ C3 )
=> ( ( ord_less @ A @ A3 @ ( real_V8093663219630862766scaleR @ A @ ( inverse_inverse @ real @ C3 ) @ B3 ) )
= ( ord_less @ A @ ( real_V8093663219630862766scaleR @ A @ C3 @ A3 ) @ B3 ) ) ) ) ).
% pos_less_divideR_eq
thf(fact_4898_neg__divideR__less__eq,axiom,
! [A: $tType] :
( ( real_V5355595471888546746vector @ A )
=> ! [C3: real,B3: A,A3: A] :
( ( ord_less @ real @ C3 @ ( zero_zero @ real ) )
=> ( ( ord_less @ A @ ( real_V8093663219630862766scaleR @ A @ ( inverse_inverse @ real @ C3 ) @ B3 ) @ A3 )
= ( ord_less @ A @ ( real_V8093663219630862766scaleR @ A @ C3 @ A3 ) @ B3 ) ) ) ) ).
% neg_divideR_less_eq
thf(fact_4899_neg__less__divideR__eq,axiom,
! [A: $tType] :
( ( real_V5355595471888546746vector @ A )
=> ! [C3: real,A3: A,B3: A] :
( ( ord_less @ real @ C3 @ ( zero_zero @ real ) )
=> ( ( ord_less @ A @ A3 @ ( real_V8093663219630862766scaleR @ A @ ( inverse_inverse @ real @ C3 ) @ B3 ) )
= ( ord_less @ A @ B3 @ ( real_V8093663219630862766scaleR @ A @ C3 @ A3 ) ) ) ) ) ).
% neg_less_divideR_eq
thf(fact_4900_forall__pos__mono__1,axiom,
! [P: real > $o,E: real] :
( ! [D4: real,E2: real] :
( ( ord_less @ real @ D4 @ E2 )
=> ( ( P @ D4 )
=> ( P @ E2 ) ) )
=> ( ! [N3: nat] : ( P @ ( inverse_inverse @ real @ ( semiring_1_of_nat @ real @ ( suc @ N3 ) ) ) )
=> ( ( ord_less @ real @ ( zero_zero @ real ) @ E )
=> ( P @ E ) ) ) ) ).
% forall_pos_mono_1
thf(fact_4901_sqrt__divide__self__eq,axiom,
! [X2: real] :
( ( ord_less_eq @ real @ ( zero_zero @ real ) @ X2 )
=> ( ( divide_divide @ real @ ( sqrt @ X2 ) @ X2 )
= ( inverse_inverse @ real @ ( sqrt @ X2 ) ) ) ) ).
% sqrt_divide_self_eq
thf(fact_4902_summable__exp,axiom,
! [A: $tType] :
( ( ( real_Vector_banach @ A )
& ( real_V3459762299906320749_field @ A ) )
=> ! [X2: A] :
( summable @ A
@ ^ [N4: nat] : ( times_times @ A @ ( inverse_inverse @ A @ ( semiring_char_0_fact @ A @ N4 ) ) @ ( power_power @ A @ X2 @ N4 ) ) ) ) ).
% summable_exp
thf(fact_4903_summable__exp__generic,axiom,
! [A: $tType] :
( ( ( real_Vector_banach @ A )
& ( real_V2822296259951069270ebra_1 @ A ) )
=> ! [X2: A] :
( summable @ A
@ ^ [N4: nat] : ( real_V8093663219630862766scaleR @ A @ ( inverse_inverse @ real @ ( semiring_char_0_fact @ real @ N4 ) ) @ ( power_power @ A @ X2 @ N4 ) ) ) ) ).
% summable_exp_generic
thf(fact_4904_ex__inverse__of__nat__less,axiom,
! [A: $tType] :
( ( archim462609752435547400_field @ A )
=> ! [X2: A] :
( ( ord_less @ A @ ( zero_zero @ A ) @ X2 )
=> ? [N3: nat] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ N3 )
& ( ord_less @ A @ ( inverse_inverse @ A @ ( semiring_1_of_nat @ A @ N3 ) ) @ X2 ) ) ) ) ).
% ex_inverse_of_nat_less
thf(fact_4905_power__diff__conv__inverse,axiom,
! [A: $tType] :
( ( division_ring @ A )
=> ! [X2: A,M: nat,N: nat] :
( ( X2
!= ( zero_zero @ A ) )
=> ( ( ord_less_eq @ nat @ M @ N )
=> ( ( power_power @ A @ X2 @ ( minus_minus @ nat @ N @ M ) )
= ( times_times @ A @ ( power_power @ A @ X2 @ N ) @ ( power_power @ A @ ( inverse_inverse @ A @ X2 ) @ M ) ) ) ) ) ) ).
% power_diff_conv_inverse
thf(fact_4906_cot__def,axiom,
! [A: $tType] :
( ( ( real_Vector_banach @ A )
& ( real_V3459762299906320749_field @ A ) )
=> ( ( cot @ A )
= ( ^ [X3: A] : ( divide_divide @ A @ ( cos @ A @ X3 ) @ ( sin @ A @ X3 ) ) ) ) ) ).
% cot_def
thf(fact_4907_neg__minus__divideR__le__eq,axiom,
! [A: $tType] :
( ( real_V5355595471888546746vector @ A )
=> ! [C3: real,B3: A,A3: A] :
( ( ord_less @ real @ C3 @ ( zero_zero @ real ) )
=> ( ( ord_less_eq @ A @ ( uminus_uminus @ A @ ( real_V8093663219630862766scaleR @ A @ ( inverse_inverse @ real @ C3 ) @ B3 ) ) @ A3 )
= ( ord_less_eq @ A @ ( real_V8093663219630862766scaleR @ A @ C3 @ A3 ) @ ( uminus_uminus @ A @ B3 ) ) ) ) ) ).
% neg_minus_divideR_le_eq
thf(fact_4908_neg__le__minus__divideR__eq,axiom,
! [A: $tType] :
( ( real_V5355595471888546746vector @ A )
=> ! [C3: real,A3: A,B3: A] :
( ( ord_less @ real @ C3 @ ( zero_zero @ real ) )
=> ( ( ord_less_eq @ A @ A3 @ ( uminus_uminus @ A @ ( real_V8093663219630862766scaleR @ A @ ( inverse_inverse @ real @ C3 ) @ B3 ) ) )
= ( ord_less_eq @ A @ ( uminus_uminus @ A @ B3 ) @ ( real_V8093663219630862766scaleR @ A @ C3 @ A3 ) ) ) ) ) ).
% neg_le_minus_divideR_eq
thf(fact_4909_pos__minus__divideR__le__eq,axiom,
! [A: $tType] :
( ( real_V5355595471888546746vector @ A )
=> ! [C3: real,B3: A,A3: A] :
( ( ord_less @ real @ ( zero_zero @ real ) @ C3 )
=> ( ( ord_less_eq @ A @ ( uminus_uminus @ A @ ( real_V8093663219630862766scaleR @ A @ ( inverse_inverse @ real @ C3 ) @ B3 ) ) @ A3 )
= ( ord_less_eq @ A @ ( uminus_uminus @ A @ B3 ) @ ( real_V8093663219630862766scaleR @ A @ C3 @ A3 ) ) ) ) ) ).
% pos_minus_divideR_le_eq
thf(fact_4910_pos__le__minus__divideR__eq,axiom,
! [A: $tType] :
( ( real_V5355595471888546746vector @ A )
=> ! [C3: real,A3: A,B3: A] :
( ( ord_less @ real @ ( zero_zero @ real ) @ C3 )
=> ( ( ord_less_eq @ A @ A3 @ ( uminus_uminus @ A @ ( real_V8093663219630862766scaleR @ A @ ( inverse_inverse @ real @ C3 ) @ B3 ) ) )
= ( ord_less_eq @ A @ ( real_V8093663219630862766scaleR @ A @ C3 @ A3 ) @ ( uminus_uminus @ A @ B3 ) ) ) ) ) ).
% pos_le_minus_divideR_eq
thf(fact_4911_pos__less__minus__divideR__eq,axiom,
! [A: $tType] :
( ( real_V5355595471888546746vector @ A )
=> ! [C3: real,A3: A,B3: A] :
( ( ord_less @ real @ ( zero_zero @ real ) @ C3 )
=> ( ( ord_less @ A @ A3 @ ( uminus_uminus @ A @ ( real_V8093663219630862766scaleR @ A @ ( inverse_inverse @ real @ C3 ) @ B3 ) ) )
= ( ord_less @ A @ ( real_V8093663219630862766scaleR @ A @ C3 @ A3 ) @ ( uminus_uminus @ A @ B3 ) ) ) ) ) ).
% pos_less_minus_divideR_eq
thf(fact_4912_pos__minus__divideR__less__eq,axiom,
! [A: $tType] :
( ( real_V5355595471888546746vector @ A )
=> ! [C3: real,B3: A,A3: A] :
( ( ord_less @ real @ ( zero_zero @ real ) @ C3 )
=> ( ( ord_less @ A @ ( uminus_uminus @ A @ ( real_V8093663219630862766scaleR @ A @ ( inverse_inverse @ real @ C3 ) @ B3 ) ) @ A3 )
= ( ord_less @ A @ ( uminus_uminus @ A @ B3 ) @ ( real_V8093663219630862766scaleR @ A @ C3 @ A3 ) ) ) ) ) ).
% pos_minus_divideR_less_eq
thf(fact_4913_neg__less__minus__divideR__eq,axiom,
! [A: $tType] :
( ( real_V5355595471888546746vector @ A )
=> ! [C3: real,A3: A,B3: A] :
( ( ord_less @ real @ C3 @ ( zero_zero @ real ) )
=> ( ( ord_less @ A @ A3 @ ( uminus_uminus @ A @ ( real_V8093663219630862766scaleR @ A @ ( inverse_inverse @ real @ C3 ) @ B3 ) ) )
= ( ord_less @ A @ ( uminus_uminus @ A @ B3 ) @ ( real_V8093663219630862766scaleR @ A @ C3 @ A3 ) ) ) ) ) ).
% neg_less_minus_divideR_eq
thf(fact_4914_neg__minus__divideR__less__eq,axiom,
! [A: $tType] :
( ( real_V5355595471888546746vector @ A )
=> ! [C3: real,B3: A,A3: A] :
( ( ord_less @ real @ C3 @ ( zero_zero @ real ) )
=> ( ( ord_less @ A @ ( uminus_uminus @ A @ ( real_V8093663219630862766scaleR @ A @ ( inverse_inverse @ real @ C3 ) @ B3 ) ) @ A3 )
= ( ord_less @ A @ ( real_V8093663219630862766scaleR @ A @ C3 @ A3 ) @ ( uminus_uminus @ A @ B3 ) ) ) ) ) ).
% neg_minus_divideR_less_eq
thf(fact_4915_log__inverse,axiom,
! [A3: real,X2: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ A3 )
=> ( ( A3
!= ( one_one @ real ) )
=> ( ( ord_less @ real @ ( zero_zero @ real ) @ X2 )
=> ( ( log @ A3 @ ( inverse_inverse @ real @ X2 ) )
= ( uminus_uminus @ real @ ( log @ A3 @ X2 ) ) ) ) ) ) ).
% log_inverse
thf(fact_4916_exp__converges,axiom,
! [A: $tType] :
( ( ( real_Vector_banach @ A )
& ( real_V2822296259951069270ebra_1 @ A ) )
=> ! [X2: A] :
( sums @ A
@ ^ [N4: nat] : ( real_V8093663219630862766scaleR @ A @ ( inverse_inverse @ real @ ( semiring_char_0_fact @ real @ N4 ) ) @ ( power_power @ A @ X2 @ N4 ) )
@ ( exp @ A @ X2 ) ) ) ).
% exp_converges
thf(fact_4917_exp__def,axiom,
! [A: $tType] :
( ( ( real_Vector_banach @ A )
& ( real_V2822296259951069270ebra_1 @ A ) )
=> ( ( exp @ A )
= ( ^ [X3: A] :
( suminf @ A
@ ^ [N4: nat] : ( real_V8093663219630862766scaleR @ A @ ( inverse_inverse @ real @ ( semiring_char_0_fact @ real @ N4 ) ) @ ( power_power @ A @ X3 @ N4 ) ) ) ) ) ) ).
% exp_def
thf(fact_4918_summable__norm__exp,axiom,
! [A: $tType] :
( ( ( real_Vector_banach @ A )
& ( real_V2822296259951069270ebra_1 @ A ) )
=> ! [X2: A] :
( summable @ real
@ ^ [N4: nat] : ( real_V7770717601297561774m_norm @ A @ ( real_V8093663219630862766scaleR @ A @ ( inverse_inverse @ real @ ( semiring_char_0_fact @ real @ N4 ) ) @ ( power_power @ A @ X2 @ N4 ) ) ) ) ) ).
% summable_norm_exp
thf(fact_4919_exp__plus__inverse__exp,axiom,
! [X2: real] : ( ord_less_eq @ real @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) @ ( plus_plus @ real @ ( exp @ real @ X2 ) @ ( inverse_inverse @ real @ ( exp @ real @ X2 ) ) ) ) ).
% exp_plus_inverse_exp
thf(fact_4920_plus__inverse__ge__2,axiom,
! [X2: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ X2 )
=> ( ord_less_eq @ real @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) @ ( plus_plus @ real @ X2 @ ( inverse_inverse @ real @ X2 ) ) ) ) ).
% plus_inverse_ge_2
thf(fact_4921_real__inv__sqrt__pow2,axiom,
! [X2: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ X2 )
=> ( ( power_power @ real @ ( inverse_inverse @ real @ ( sqrt @ X2 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) )
= ( inverse_inverse @ real @ X2 ) ) ) ).
% real_inv_sqrt_pow2
thf(fact_4922_tan__cot,axiom,
! [X2: real] :
( ( tan @ real @ ( minus_minus @ real @ ( divide_divide @ real @ pi @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) @ X2 ) )
= ( inverse_inverse @ real @ ( tan @ real @ X2 ) ) ) ).
% tan_cot
thf(fact_4923_real__le__x__sinh,axiom,
! [X2: real] :
( ( ord_less_eq @ real @ ( zero_zero @ real ) @ X2 )
=> ( ord_less_eq @ real @ X2 @ ( divide_divide @ real @ ( minus_minus @ real @ ( exp @ real @ X2 ) @ ( inverse_inverse @ real @ ( exp @ real @ X2 ) ) ) @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) ) ) ).
% real_le_x_sinh
thf(fact_4924_real__le__abs__sinh,axiom,
! [X2: real] : ( ord_less_eq @ real @ ( abs_abs @ real @ X2 ) @ ( abs_abs @ real @ ( divide_divide @ real @ ( minus_minus @ real @ ( exp @ real @ X2 ) @ ( inverse_inverse @ real @ ( exp @ real @ X2 ) ) ) @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) ) ) ).
% real_le_abs_sinh
thf(fact_4925_exp__series__add__commuting,axiom,
! [A: $tType] :
( ( ( real_Vector_banach @ A )
& ( real_V2822296259951069270ebra_1 @ A ) )
=> ! [X2: A,Y3: A,N: nat] :
( ( ( times_times @ A @ X2 @ Y3 )
= ( times_times @ A @ Y3 @ X2 ) )
=> ( ( real_V8093663219630862766scaleR @ A @ ( inverse_inverse @ real @ ( semiring_char_0_fact @ real @ N ) ) @ ( power_power @ A @ ( plus_plus @ A @ X2 @ Y3 ) @ N ) )
= ( groups7311177749621191930dd_sum @ nat @ A
@ ^ [I4: nat] : ( times_times @ A @ ( real_V8093663219630862766scaleR @ A @ ( inverse_inverse @ real @ ( semiring_char_0_fact @ real @ I4 ) ) @ ( power_power @ A @ X2 @ I4 ) ) @ ( real_V8093663219630862766scaleR @ A @ ( inverse_inverse @ real @ ( semiring_char_0_fact @ real @ ( minus_minus @ nat @ N @ I4 ) ) ) @ ( power_power @ A @ Y3 @ ( minus_minus @ nat @ N @ I4 ) ) ) )
@ ( set_ord_atMost @ nat @ N ) ) ) ) ) ).
% exp_series_add_commuting
thf(fact_4926_exp__first__term,axiom,
! [A: $tType] :
( ( ( real_Vector_banach @ A )
& ( real_V2822296259951069270ebra_1 @ A ) )
=> ( ( exp @ A )
= ( ^ [X3: A] :
( plus_plus @ A @ ( one_one @ A )
@ ( suminf @ A
@ ^ [N4: nat] : ( real_V8093663219630862766scaleR @ A @ ( inverse_inverse @ real @ ( semiring_char_0_fact @ real @ ( suc @ N4 ) ) ) @ ( power_power @ A @ X3 @ ( suc @ N4 ) ) ) ) ) ) ) ) ).
% exp_first_term
thf(fact_4927_tan__sec,axiom,
! [A: $tType] :
( ( ( real_Vector_banach @ A )
& ( real_V3459762299906320749_field @ A ) )
=> ! [X2: A] :
( ( ( cos @ A @ X2 )
!= ( zero_zero @ A ) )
=> ( ( plus_plus @ A @ ( one_one @ A ) @ ( power_power @ A @ ( tan @ A @ X2 ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) )
= ( power_power @ A @ ( inverse_inverse @ A @ ( cos @ A @ X2 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) ).
% tan_sec
thf(fact_4928_powr__real__of__int,axiom,
! [X2: real,N: int] :
( ( ord_less @ real @ ( zero_zero @ real ) @ X2 )
=> ( ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ N )
=> ( ( powr @ real @ X2 @ ( ring_1_of_int @ real @ N ) )
= ( power_power @ real @ X2 @ ( nat2 @ N ) ) ) )
& ( ~ ( ord_less_eq @ int @ ( zero_zero @ int ) @ N )
=> ( ( powr @ real @ X2 @ ( ring_1_of_int @ real @ N ) )
= ( inverse_inverse @ real @ ( power_power @ real @ X2 @ ( nat2 @ ( uminus_uminus @ int @ N ) ) ) ) ) ) ) ) ).
% powr_real_of_int
thf(fact_4929_cot__gt__zero,axiom,
! [X2: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ X2 )
=> ( ( ord_less @ real @ X2 @ ( divide_divide @ real @ pi @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) )
=> ( ord_less @ real @ ( zero_zero @ real ) @ ( cot @ real @ X2 ) ) ) ) ).
% cot_gt_zero
thf(fact_4930_exp__first__terms,axiom,
! [A: $tType] :
( ( ( real_Vector_banach @ A )
& ( real_V2822296259951069270ebra_1 @ A ) )
=> ! [K: nat] :
( ( exp @ A )
= ( ^ [X3: A] :
( plus_plus @ A
@ ( groups7311177749621191930dd_sum @ nat @ A
@ ^ [N4: nat] : ( real_V8093663219630862766scaleR @ A @ ( inverse_inverse @ real @ ( semiring_char_0_fact @ real @ N4 ) ) @ ( power_power @ A @ X3 @ N4 ) )
@ ( set_ord_lessThan @ nat @ K ) )
@ ( suminf @ A
@ ^ [N4: nat] : ( real_V8093663219630862766scaleR @ A @ ( inverse_inverse @ real @ ( semiring_char_0_fact @ real @ ( plus_plus @ nat @ N4 @ K ) ) ) @ ( power_power @ A @ X3 @ ( plus_plus @ nat @ N4 @ K ) ) ) ) ) ) ) ) ).
% exp_first_terms
thf(fact_4931_tan__cot_H,axiom,
! [X2: real] :
( ( tan @ real @ ( minus_minus @ real @ ( divide_divide @ real @ pi @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) @ X2 ) )
= ( cot @ real @ X2 ) ) ).
% tan_cot'
thf(fact_4932_sinh__converges,axiom,
! [A: $tType] :
( ( ( real_Vector_banach @ A )
& ( real_V2822296259951069270ebra_1 @ A ) )
=> ! [X2: A] :
( sums @ A
@ ^ [N4: nat] : ( if @ A @ ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N4 ) @ ( zero_zero @ A ) @ ( real_V8093663219630862766scaleR @ A @ ( inverse_inverse @ real @ ( semiring_char_0_fact @ real @ N4 ) ) @ ( power_power @ A @ X2 @ N4 ) ) )
@ ( sinh @ A @ X2 ) ) ) ).
% sinh_converges
thf(fact_4933_cosh__converges,axiom,
! [A: $tType] :
( ( ( real_Vector_banach @ A )
& ( real_V2822296259951069270ebra_1 @ A ) )
=> ! [X2: A] :
( sums @ A
@ ^ [N4: nat] : ( if @ A @ ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N4 ) @ ( real_V8093663219630862766scaleR @ A @ ( inverse_inverse @ real @ ( semiring_char_0_fact @ real @ N4 ) ) @ ( power_power @ A @ X2 @ N4 ) ) @ ( zero_zero @ A ) )
@ ( cosh @ A @ X2 ) ) ) ).
% cosh_converges
thf(fact_4934_sint__range__size,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [W: word @ A] :
( ( ord_less_eq @ int @ ( uminus_uminus @ int @ ( power_power @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ ( minus_minus @ nat @ ( size_size @ ( word @ A ) @ W ) @ ( suc @ ( zero_zero @ nat ) ) ) ) ) @ ( ring_1_signed @ A @ int @ W ) )
& ( ord_less @ int @ ( ring_1_signed @ A @ int @ W ) @ ( power_power @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ ( minus_minus @ nat @ ( size_size @ ( word @ A ) @ W ) @ ( suc @ ( zero_zero @ nat ) ) ) ) ) ) ) ).
% sint_range_size
thf(fact_4935_sinh__real__le__iff,axiom,
! [X2: real,Y3: real] :
( ( ord_less_eq @ real @ ( sinh @ real @ X2 ) @ ( sinh @ real @ Y3 ) )
= ( ord_less_eq @ real @ X2 @ Y3 ) ) ).
% sinh_real_le_iff
thf(fact_4936_sinh__real__nonneg__iff,axiom,
! [X2: real] :
( ( ord_less_eq @ real @ ( zero_zero @ real ) @ ( sinh @ real @ X2 ) )
= ( ord_less_eq @ real @ ( zero_zero @ real ) @ X2 ) ) ).
% sinh_real_nonneg_iff
thf(fact_4937_sinh__real__nonpos__iff,axiom,
! [X2: real] :
( ( ord_less_eq @ real @ ( sinh @ real @ X2 ) @ ( zero_zero @ real ) )
= ( ord_less_eq @ real @ X2 @ ( zero_zero @ real ) ) ) ).
% sinh_real_nonpos_iff
thf(fact_4938_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_4939_signed__minus__1,axiom,
! [B: $tType,A: $tType] :
( ( ( ring_1 @ A )
& ( type_len @ B ) )
=> ( ( ring_1_signed @ B @ A @ ( uminus_uminus @ ( word @ B ) @ ( one_one @ ( word @ B ) ) ) )
= ( uminus_uminus @ A @ ( one_one @ A ) ) ) ) ).
% signed_minus_1
thf(fact_4940_sint__minus1,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [X2: word @ A] :
( ( ( ring_1_signed @ A @ int @ X2 )
= ( uminus_uminus @ int @ ( one_one @ int ) ) )
= ( X2
= ( uminus_uminus @ ( word @ A ) @ ( one_one @ ( word @ A ) ) ) ) ) ) ).
% sint_minus1
thf(fact_4941_sinh__diff,axiom,
! [A: $tType] :
( ( ( real_Vector_banach @ A )
& ( real_V3459762299906320749_field @ A ) )
=> ! [X2: A,Y3: A] :
( ( sinh @ A @ ( minus_minus @ A @ X2 @ Y3 ) )
= ( minus_minus @ A @ ( times_times @ A @ ( sinh @ A @ X2 ) @ ( cosh @ A @ Y3 ) ) @ ( times_times @ A @ ( cosh @ A @ X2 ) @ ( sinh @ A @ Y3 ) ) ) ) ) ).
% sinh_diff
thf(fact_4942_cosh__diff,axiom,
! [A: $tType] :
( ( ( real_Vector_banach @ A )
& ( real_V3459762299906320749_field @ A ) )
=> ! [X2: A,Y3: A] :
( ( cosh @ A @ ( minus_minus @ A @ X2 @ Y3 ) )
= ( minus_minus @ A @ ( times_times @ A @ ( cosh @ A @ X2 ) @ ( cosh @ A @ Y3 ) ) @ ( times_times @ A @ ( sinh @ A @ X2 ) @ ( sinh @ A @ Y3 ) ) ) ) ) ).
% cosh_diff
thf(fact_4943_sinh__add,axiom,
! [A: $tType] :
( ( ( real_Vector_banach @ A )
& ( real_V3459762299906320749_field @ A ) )
=> ! [X2: A,Y3: A] :
( ( sinh @ A @ ( plus_plus @ A @ X2 @ Y3 ) )
= ( plus_plus @ A @ ( times_times @ A @ ( sinh @ A @ X2 ) @ ( cosh @ A @ Y3 ) ) @ ( times_times @ A @ ( cosh @ A @ X2 ) @ ( sinh @ A @ Y3 ) ) ) ) ) ).
% sinh_add
thf(fact_4944_cosh__add,axiom,
! [A: $tType] :
( ( ( real_Vector_banach @ A )
& ( real_V3459762299906320749_field @ A ) )
=> ! [X2: A,Y3: A] :
( ( cosh @ A @ ( plus_plus @ A @ X2 @ Y3 ) )
= ( plus_plus @ A @ ( times_times @ A @ ( cosh @ A @ X2 ) @ ( cosh @ A @ Y3 ) ) @ ( times_times @ A @ ( sinh @ A @ X2 ) @ ( sinh @ A @ Y3 ) ) ) ) ) ).
% cosh_add
thf(fact_4945_sinh__le__cosh__real,axiom,
! [X2: real] : ( ord_less_eq @ real @ ( sinh @ real @ X2 ) @ ( cosh @ real @ X2 ) ) ).
% sinh_le_cosh_real
thf(fact_4946_tanh__def,axiom,
! [A: $tType] :
( ( ( real_Vector_banach @ A )
& ( real_V3459762299906320749_field @ A ) )
=> ( ( tanh @ A )
= ( ^ [X3: A] : ( divide_divide @ A @ ( sinh @ A @ X3 ) @ ( cosh @ A @ X3 ) ) ) ) ) ).
% tanh_def
thf(fact_4947_sinh__minus__cosh,axiom,
! [A: $tType] :
( ( ( real_Vector_banach @ A )
& ( real_V2822296259951069270ebra_1 @ A ) )
=> ! [X2: A] :
( ( minus_minus @ A @ ( sinh @ A @ X2 ) @ ( cosh @ A @ X2 ) )
= ( uminus_uminus @ A @ ( exp @ A @ ( uminus_uminus @ A @ X2 ) ) ) ) ) ).
% sinh_minus_cosh
thf(fact_4948_cosh__minus__sinh,axiom,
! [A: $tType] :
( ( ( real_Vector_banach @ A )
& ( real_V2822296259951069270ebra_1 @ A ) )
=> ! [X2: A] :
( ( minus_minus @ A @ ( cosh @ A @ X2 ) @ ( sinh @ A @ X2 ) )
= ( exp @ A @ ( uminus_uminus @ A @ X2 ) ) ) ) ).
% cosh_minus_sinh
thf(fact_4949_sinh__plus__cosh,axiom,
! [A: $tType] :
( ( ( real_Vector_banach @ A )
& ( real_V2822296259951069270ebra_1 @ A ) )
=> ! [X2: A] :
( ( plus_plus @ A @ ( sinh @ A @ X2 ) @ ( cosh @ A @ X2 ) )
= ( exp @ A @ X2 ) ) ) ).
% sinh_plus_cosh
thf(fact_4950_cosh__plus__sinh,axiom,
! [A: $tType] :
( ( ( real_Vector_banach @ A )
& ( real_V2822296259951069270ebra_1 @ A ) )
=> ! [X2: A] :
( ( plus_plus @ A @ ( cosh @ A @ X2 ) @ ( sinh @ A @ X2 ) )
= ( exp @ A @ X2 ) ) ) ).
% cosh_plus_sinh
thf(fact_4951_cosh__real__nonneg,axiom,
! [X2: real] : ( ord_less_eq @ real @ ( zero_zero @ real ) @ ( cosh @ real @ X2 ) ) ).
% cosh_real_nonneg
thf(fact_4952_cosh__real__nonneg__le__iff,axiom,
! [X2: real,Y3: real] :
( ( ord_less_eq @ real @ ( zero_zero @ real ) @ X2 )
=> ( ( ord_less_eq @ real @ ( zero_zero @ real ) @ Y3 )
=> ( ( ord_less_eq @ real @ ( cosh @ real @ X2 ) @ ( cosh @ real @ Y3 ) )
= ( ord_less_eq @ real @ X2 @ Y3 ) ) ) ) ).
% cosh_real_nonneg_le_iff
thf(fact_4953_cosh__real__nonpos__le__iff,axiom,
! [X2: real,Y3: real] :
( ( ord_less_eq @ real @ X2 @ ( zero_zero @ real ) )
=> ( ( ord_less_eq @ real @ Y3 @ ( zero_zero @ real ) )
=> ( ( ord_less_eq @ real @ ( cosh @ real @ X2 ) @ ( cosh @ real @ Y3 ) )
= ( ord_less_eq @ real @ Y3 @ X2 ) ) ) ) ).
% cosh_real_nonpos_le_iff
thf(fact_4954_sinh__double,axiom,
! [A: $tType] :
( ( ( real_Vector_banach @ A )
& ( real_V3459762299906320749_field @ A ) )
=> ! [X2: A] :
( ( sinh @ A @ ( times_times @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ X2 ) )
= ( times_times @ A @ ( times_times @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ ( sinh @ A @ X2 ) ) @ ( cosh @ A @ X2 ) ) ) ) ).
% sinh_double
thf(fact_4955_cosh__real__ge__1,axiom,
! [X2: real] : ( ord_less_eq @ real @ ( one_one @ real ) @ ( cosh @ real @ X2 ) ) ).
% cosh_real_ge_1
thf(fact_4956_cosh__real__nonpos__less__iff,axiom,
! [X2: real,Y3: real] :
( ( ord_less_eq @ real @ X2 @ ( zero_zero @ real ) )
=> ( ( ord_less_eq @ real @ Y3 @ ( zero_zero @ real ) )
=> ( ( ord_less @ real @ ( cosh @ real @ X2 ) @ ( cosh @ real @ Y3 ) )
= ( ord_less @ real @ Y3 @ X2 ) ) ) ) ).
% cosh_real_nonpos_less_iff
thf(fact_4957_cosh__real__nonneg__less__iff,axiom,
! [X2: real,Y3: real] :
( ( ord_less_eq @ real @ ( zero_zero @ real ) @ X2 )
=> ( ( ord_less_eq @ real @ ( zero_zero @ real ) @ Y3 )
=> ( ( ord_less @ real @ ( cosh @ real @ X2 ) @ ( cosh @ real @ Y3 ) )
= ( ord_less @ real @ X2 @ Y3 ) ) ) ) ).
% cosh_real_nonneg_less_iff
thf(fact_4958_cosh__real__strict__mono,axiom,
! [X2: real,Y3: real] :
( ( ord_less_eq @ real @ ( zero_zero @ real ) @ X2 )
=> ( ( ord_less @ real @ X2 @ Y3 )
=> ( ord_less @ real @ ( cosh @ real @ X2 ) @ ( cosh @ real @ Y3 ) ) ) ) ).
% cosh_real_strict_mono
thf(fact_4959_cosh__square__eq,axiom,
! [A: $tType] :
( ( ( real_Vector_banach @ A )
& ( real_V3459762299906320749_field @ A ) )
=> ! [X2: A] :
( ( power_power @ A @ ( cosh @ A @ X2 ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) )
= ( plus_plus @ A @ ( power_power @ A @ ( sinh @ A @ X2 ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ ( one_one @ A ) ) ) ) ).
% cosh_square_eq
thf(fact_4960_sinh__square__eq,axiom,
! [A: $tType] :
( ( ( real_Vector_banach @ A )
& ( real_V3459762299906320749_field @ A ) )
=> ! [X2: A] :
( ( power_power @ A @ ( sinh @ A @ X2 ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) )
= ( minus_minus @ A @ ( power_power @ A @ ( cosh @ A @ X2 ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ ( one_one @ A ) ) ) ) ).
% sinh_square_eq
thf(fact_4961_hyperbolic__pythagoras,axiom,
! [A: $tType] :
( ( ( real_Vector_banach @ A )
& ( real_V3459762299906320749_field @ A ) )
=> ! [X2: A] :
( ( minus_minus @ A @ ( power_power @ A @ ( cosh @ A @ X2 ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ ( power_power @ A @ ( sinh @ A @ X2 ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) )
= ( one_one @ A ) ) ) ).
% hyperbolic_pythagoras
thf(fact_4962_arcosh__cosh__real,axiom,
! [X2: real] :
( ( ord_less_eq @ real @ ( zero_zero @ real ) @ X2 )
=> ( ( arcosh @ real @ ( cosh @ real @ X2 ) )
= X2 ) ) ).
% arcosh_cosh_real
thf(fact_4963_cosh__double,axiom,
! [A: $tType] :
( ( ( real_Vector_banach @ A )
& ( real_V3459762299906320749_field @ A ) )
=> ! [X2: A] :
( ( cosh @ A @ ( times_times @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ X2 ) )
= ( plus_plus @ A @ ( power_power @ A @ ( cosh @ A @ X2 ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ ( power_power @ A @ ( sinh @ A @ X2 ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) ).
% cosh_double
thf(fact_4964_sint__n1,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ( ( ring_1_signed @ A @ int @ ( uminus_uminus @ ( word @ A ) @ ( one_one @ ( word @ A ) ) ) )
= ( uminus_uminus @ int @ ( one_one @ int ) ) ) ) ).
% sint_n1
thf(fact_4965_tanh__add,axiom,
! [A: $tType] :
( ( ( real_Vector_banach @ A )
& ( real_V3459762299906320749_field @ A ) )
=> ! [X2: A,Y3: A] :
( ( ( cosh @ A @ X2 )
!= ( zero_zero @ A ) )
=> ( ( ( cosh @ A @ Y3 )
!= ( zero_zero @ A ) )
=> ( ( tanh @ A @ ( plus_plus @ A @ X2 @ Y3 ) )
= ( divide_divide @ A @ ( plus_plus @ A @ ( tanh @ A @ X2 ) @ ( tanh @ A @ Y3 ) ) @ ( plus_plus @ A @ ( one_one @ A ) @ ( times_times @ A @ ( tanh @ A @ X2 ) @ ( tanh @ A @ Y3 ) ) ) ) ) ) ) ) ).
% tanh_add
thf(fact_4966_sinh__zero__iff,axiom,
! [A: $tType] :
( ( ( real_Vector_banach @ A )
& ( real_V3459762299906320749_field @ A ) )
=> ! [X2: A] :
( ( ( sinh @ A @ X2 )
= ( zero_zero @ A ) )
= ( member @ A @ ( exp @ A @ X2 ) @ ( insert @ A @ ( one_one @ A ) @ ( insert @ A @ ( uminus_uminus @ A @ ( one_one @ A ) ) @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ) ).
% sinh_zero_iff
thf(fact_4967_cosh__field__def,axiom,
! [A: $tType] :
( ( ( real_Vector_banach @ A )
& ( real_V3459762299906320749_field @ A ) )
=> ( ( cosh @ A )
= ( ^ [Z4: A] : ( divide_divide @ A @ ( plus_plus @ A @ ( exp @ A @ Z4 ) @ ( exp @ A @ ( uminus_uminus @ A @ Z4 ) ) ) @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) ) ) ) ).
% cosh_field_def
thf(fact_4968_complex__inverse,axiom,
! [A3: real,B3: real] :
( ( inverse_inverse @ complex @ ( complex2 @ A3 @ B3 ) )
= ( complex2 @ ( divide_divide @ real @ A3 @ ( plus_plus @ real @ ( power_power @ real @ A3 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ ( power_power @ real @ B3 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( divide_divide @ real @ ( uminus_uminus @ real @ B3 ) @ ( plus_plus @ real @ ( power_power @ real @ A3 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ ( power_power @ real @ B3 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) ) ).
% complex_inverse
thf(fact_4969_sinh__field__def,axiom,
! [A: $tType] :
( ( ( real_Vector_banach @ A )
& ( real_V3459762299906320749_field @ A ) )
=> ( ( sinh @ A )
= ( ^ [Z4: A] : ( divide_divide @ A @ ( minus_minus @ A @ ( exp @ A @ Z4 ) @ ( exp @ A @ ( uminus_uminus @ A @ Z4 ) ) ) @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) ) ) ) ).
% sinh_field_def
thf(fact_4970_cosh__zero__iff,axiom,
! [A: $tType] :
( ( ( real_Vector_banach @ A )
& ( real_V3459762299906320749_field @ A ) )
=> ! [X2: A] :
( ( ( cosh @ A @ X2 )
= ( zero_zero @ A ) )
= ( ( power_power @ A @ ( exp @ A @ X2 ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) )
= ( uminus_uminus @ A @ ( one_one @ A ) ) ) ) ) ).
% cosh_zero_iff
thf(fact_4971_sint__above__size,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [W: word @ A,X2: int] :
( ( ord_less_eq @ int @ ( power_power @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ ( minus_minus @ nat @ ( size_size @ ( word @ A ) @ W ) @ ( one_one @ nat ) ) ) @ X2 )
=> ( ord_less @ int @ ( ring_1_signed @ A @ int @ W ) @ X2 ) ) ) ).
% sint_above_size
thf(fact_4972_sint__below__size,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [X2: int,W: word @ A] :
( ( ord_less_eq @ int @ X2 @ ( uminus_uminus @ int @ ( power_power @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ ( minus_minus @ nat @ ( size_size @ ( word @ A ) @ W ) @ ( one_one @ nat ) ) ) ) )
=> ( ord_less_eq @ int @ X2 @ ( ring_1_signed @ A @ int @ W ) ) ) ) ).
% sint_below_size
thf(fact_4973_cosh__def,axiom,
! [A: $tType] :
( ( ( real_Vector_banach @ A )
& ( real_V2822296259951069270ebra_1 @ A ) )
=> ( ( cosh @ A )
= ( ^ [X3: A] : ( real_V8093663219630862766scaleR @ A @ ( inverse_inverse @ real @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) @ ( plus_plus @ A @ ( exp @ A @ X3 ) @ ( exp @ A @ ( uminus_uminus @ A @ X3 ) ) ) ) ) ) ) ).
% cosh_def
thf(fact_4974_cosh__ln__real,axiom,
! [X2: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ X2 )
=> ( ( cosh @ real @ ( ln_ln @ real @ X2 ) )
= ( divide_divide @ real @ ( plus_plus @ real @ X2 @ ( inverse_inverse @ real @ X2 ) ) @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) ) ) ).
% cosh_ln_real
thf(fact_4975_sinh__def,axiom,
! [A: $tType] :
( ( ( real_Vector_banach @ A )
& ( real_V2822296259951069270ebra_1 @ A ) )
=> ( ( sinh @ A )
= ( ^ [X3: A] : ( real_V8093663219630862766scaleR @ A @ ( inverse_inverse @ real @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) @ ( minus_minus @ A @ ( exp @ A @ X3 ) @ ( exp @ A @ ( uminus_uminus @ A @ X3 ) ) ) ) ) ) ) ).
% sinh_def
thf(fact_4976_sinh__ln__real,axiom,
! [X2: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ X2 )
=> ( ( sinh @ real @ ( ln_ln @ real @ X2 ) )
= ( divide_divide @ real @ ( minus_minus @ real @ X2 @ ( inverse_inverse @ real @ X2 ) ) @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) ) ) ).
% sinh_ln_real
thf(fact_4977_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_4978_i__even__power,axiom,
! [N: nat] :
( ( power_power @ complex @ imaginary_unit @ ( times_times @ nat @ N @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) )
= ( power_power @ complex @ ( uminus_uminus @ complex @ ( one_one @ complex ) ) @ N ) ) ).
% i_even_power
thf(fact_4979_option_Osize__gen_I2_J,axiom,
! [A: $tType,X2: A > nat,X22: A] :
( ( size_option @ A @ X2 @ ( some @ A @ X22 ) )
= ( plus_plus @ nat @ ( X2 @ X22 ) @ ( suc @ ( zero_zero @ nat ) ) ) ) ).
% option.size_gen(2)
thf(fact_4980_norm__ii,axiom,
( ( real_V7770717601297561774m_norm @ complex @ imaginary_unit )
= ( one_one @ real ) ) ).
% norm_ii
thf(fact_4981_dbl__dec__simps_I5_J,axiom,
! [A: $tType] :
( ( neg_numeral @ A )
=> ! [K: num] :
( ( neg_numeral_dbl_dec @ A @ ( numeral_numeral @ A @ K ) )
= ( numeral_numeral @ A @ ( bitM @ K ) ) ) ) ).
% dbl_dec_simps(5)
thf(fact_4982_pred__numeral__simps_I2_J,axiom,
! [K: num] :
( ( pred_numeral @ ( bit0 @ K ) )
= ( numeral_numeral @ nat @ ( bitM @ K ) ) ) ).
% pred_numeral_simps(2)
thf(fact_4983_i__squared,axiom,
( ( times_times @ complex @ imaginary_unit @ imaginary_unit )
= ( uminus_uminus @ complex @ ( one_one @ complex ) ) ) ).
% i_squared
thf(fact_4984_exp__pi__i,axiom,
( ( exp @ complex @ ( times_times @ complex @ ( real_Vector_of_real @ complex @ pi ) @ imaginary_unit ) )
= ( uminus_uminus @ complex @ ( one_one @ complex ) ) ) ).
% exp_pi_i
thf(fact_4985_exp__pi__i_H,axiom,
( ( exp @ complex @ ( times_times @ complex @ imaginary_unit @ ( real_Vector_of_real @ complex @ pi ) ) )
= ( uminus_uminus @ complex @ ( one_one @ complex ) ) ) ).
% exp_pi_i'
thf(fact_4986_exp__two__pi__i,axiom,
( ( exp @ complex @ ( times_times @ complex @ ( times_times @ complex @ ( numeral_numeral @ complex @ ( bit0 @ one2 ) ) @ ( real_Vector_of_real @ complex @ pi ) ) @ imaginary_unit ) )
= ( one_one @ complex ) ) ).
% exp_two_pi_i
thf(fact_4987_exp__two__pi__i_H,axiom,
( ( exp @ complex @ ( times_times @ complex @ imaginary_unit @ ( times_times @ complex @ ( real_Vector_of_real @ complex @ pi ) @ ( numeral_numeral @ complex @ ( bit0 @ one2 ) ) ) ) )
= ( one_one @ complex ) ) ).
% exp_two_pi_i'
thf(fact_4988_power2__i,axiom,
( ( power_power @ complex @ imaginary_unit @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) )
= ( uminus_uminus @ complex @ ( one_one @ complex ) ) ) ).
% power2_i
thf(fact_4989_semiring__norm_I26_J,axiom,
( ( bitM @ one2 )
= one2 ) ).
% semiring_norm(26)
thf(fact_4990_complex__i__not__one,axiom,
( imaginary_unit
!= ( one_one @ complex ) ) ).
% complex_i_not_one
thf(fact_4991_scast__n1,axiom,
! [B: $tType,A: $tType] :
( ( ( type_len @ A )
& ( type_len @ B ) )
=> ( ( ring_1_signed @ B @ ( word @ A ) @ ( uminus_uminus @ ( word @ B ) @ ( one_one @ ( word @ B ) ) ) )
= ( uminus_uminus @ ( word @ A ) @ ( one_one @ ( word @ A ) ) ) ) ) ).
% scast_n1
thf(fact_4992_semiring__norm_I28_J,axiom,
! [N: num] :
( ( bitM @ ( bit1 @ N ) )
= ( bit1 @ ( bit0 @ N ) ) ) ).
% semiring_norm(28)
thf(fact_4993_semiring__norm_I27_J,axiom,
! [N: num] :
( ( bitM @ ( bit0 @ N ) )
= ( bit1 @ ( bitM @ N ) ) ) ).
% semiring_norm(27)
thf(fact_4994_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_4995_one__plus__BitM,axiom,
! [N: num] :
( ( plus_plus @ num @ one2 @ ( bitM @ N ) )
= ( bit0 @ N ) ) ).
% one_plus_BitM
thf(fact_4996_BitM__plus__one,axiom,
! [N: num] :
( ( plus_plus @ num @ ( bitM @ N ) @ one2 )
= ( bit0 @ N ) ) ).
% BitM_plus_one
thf(fact_4997_Complex__eq__i,axiom,
! [X2: real,Y3: real] :
( ( ( complex2 @ X2 @ Y3 )
= imaginary_unit )
= ( ( X2
= ( zero_zero @ real ) )
& ( Y3
= ( one_one @ real ) ) ) ) ).
% Complex_eq_i
thf(fact_4998_imaginary__unit_Ocode,axiom,
( imaginary_unit
= ( complex2 @ ( zero_zero @ real ) @ ( one_one @ real ) ) ) ).
% imaginary_unit.code
thf(fact_4999_numeral__BitM,axiom,
! [A: $tType] :
( ( neg_numeral @ A )
=> ! [N: num] :
( ( numeral_numeral @ A @ ( bitM @ N ) )
= ( minus_minus @ A @ ( numeral_numeral @ A @ ( bit0 @ N ) ) @ ( one_one @ A ) ) ) ) ).
% numeral_BitM
thf(fact_5000_odd__numeral__BitM,axiom,
! [A: $tType] :
( ( semiring_parity @ A )
=> ! [W: num] :
~ ( dvd_dvd @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ ( numeral_numeral @ A @ ( bitM @ W ) ) ) ) ).
% odd_numeral_BitM
thf(fact_5001_Complex__eq,axiom,
( complex2
= ( ^ [A6: real,B7: real] : ( plus_plus @ complex @ ( real_Vector_of_real @ complex @ A6 ) @ ( times_times @ complex @ imaginary_unit @ ( real_Vector_of_real @ complex @ B7 ) ) ) ) ) ).
% Complex_eq
thf(fact_5002_complex__split__polar,axiom,
! [Z: complex] :
? [R3: real,A4: real] :
( Z
= ( times_times @ complex @ ( real_Vector_of_real @ complex @ R3 ) @ ( plus_plus @ complex @ ( real_Vector_of_real @ complex @ ( cos @ real @ A4 ) ) @ ( times_times @ complex @ imaginary_unit @ ( real_Vector_of_real @ complex @ ( sin @ real @ A4 ) ) ) ) ) ) ).
% complex_split_polar
thf(fact_5003_option_Osize__gen_I1_J,axiom,
! [A: $tType,X2: A > nat] :
( ( size_option @ A @ X2 @ ( none @ A ) )
= ( suc @ ( zero_zero @ nat ) ) ) ).
% option.size_gen(1)
thf(fact_5004_cmod__unit__one,axiom,
! [A3: real] :
( ( real_V7770717601297561774m_norm @ complex @ ( plus_plus @ complex @ ( real_Vector_of_real @ complex @ ( cos @ real @ A3 ) ) @ ( times_times @ complex @ imaginary_unit @ ( real_Vector_of_real @ complex @ ( sin @ real @ A3 ) ) ) ) )
= ( one_one @ real ) ) ).
% cmod_unit_one
thf(fact_5005_cmod__complex__polar,axiom,
! [R2: real,A3: real] :
( ( real_V7770717601297561774m_norm @ complex @ ( times_times @ complex @ ( real_Vector_of_real @ complex @ R2 ) @ ( plus_plus @ complex @ ( real_Vector_of_real @ complex @ ( cos @ real @ A3 ) ) @ ( times_times @ complex @ imaginary_unit @ ( real_Vector_of_real @ complex @ ( sin @ real @ A3 ) ) ) ) ) )
= ( abs_abs @ real @ R2 ) ) ).
% cmod_complex_polar
thf(fact_5006_Arg__minus__ii,axiom,
( ( arg @ ( uminus_uminus @ complex @ imaginary_unit ) )
= ( divide_divide @ real @ ( uminus_uminus @ real @ pi ) @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) ) ).
% Arg_minus_ii
thf(fact_5007_csqrt__ii,axiom,
( ( csqrt @ imaginary_unit )
= ( divide_divide @ complex @ ( plus_plus @ complex @ ( one_one @ complex ) @ imaginary_unit ) @ ( real_Vector_of_real @ complex @ ( sqrt @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) ) ) ) ).
% csqrt_ii
thf(fact_5008_Arg__ii,axiom,
( ( arg @ imaginary_unit )
= ( divide_divide @ real @ pi @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) ) ).
% Arg_ii
thf(fact_5009_csqrt__eq__1,axiom,
! [Z: complex] :
( ( ( csqrt @ Z )
= ( one_one @ complex ) )
= ( Z
= ( one_one @ complex ) ) ) ).
% csqrt_eq_1
thf(fact_5010_csqrt__1,axiom,
( ( csqrt @ ( one_one @ complex ) )
= ( one_one @ complex ) ) ).
% csqrt_1
thf(fact_5011_power2__csqrt,axiom,
! [Z: complex] :
( ( power_power @ complex @ ( csqrt @ Z ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) )
= Z ) ).
% power2_csqrt
thf(fact_5012_of__real__sqrt,axiom,
! [X2: real] :
( ( ord_less_eq @ real @ ( zero_zero @ real ) @ X2 )
=> ( ( real_Vector_of_real @ complex @ ( sqrt @ X2 ) )
= ( csqrt @ ( real_Vector_of_real @ complex @ X2 ) ) ) ) ).
% of_real_sqrt
thf(fact_5013_Arg__bounded,axiom,
! [Z: complex] :
( ( ord_less @ real @ ( uminus_uminus @ real @ pi ) @ ( arg @ Z ) )
& ( ord_less_eq @ real @ ( arg @ Z ) @ pi ) ) ).
% Arg_bounded
thf(fact_5014_cis__minus__pi__half,axiom,
( ( cis @ ( uminus_uminus @ real @ ( divide_divide @ real @ pi @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) ) )
= ( uminus_uminus @ complex @ imaginary_unit ) ) ).
% cis_minus_pi_half
thf(fact_5015_or__minus__numerals_I5_J,axiom,
! [N: num] :
( ( bit_se1065995026697491101ons_or @ int @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ ( bit0 @ N ) ) ) @ ( one_one @ int ) )
= ( uminus_uminus @ int @ ( numeral_numeral @ int @ ( bit_or_not_num_neg @ one2 @ ( bitM @ N ) ) ) ) ) ).
% or_minus_numerals(5)
thf(fact_5016_or__minus__numerals_I1_J,axiom,
! [N: num] :
( ( bit_se1065995026697491101ons_or @ int @ ( one_one @ int ) @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ ( bit0 @ N ) ) ) )
= ( uminus_uminus @ int @ ( numeral_numeral @ int @ ( bit_or_not_num_neg @ one2 @ ( bitM @ N ) ) ) ) ) ).
% or_minus_numerals(1)
thf(fact_5017_norm__cis,axiom,
! [A3: real] :
( ( real_V7770717601297561774m_norm @ complex @ ( cis @ A3 ) )
= ( one_one @ real ) ) ).
% norm_cis
thf(fact_5018_cis__zero,axiom,
( ( cis @ ( zero_zero @ real ) )
= ( one_one @ complex ) ) ).
% cis_zero
thf(fact_5019_cis__pi,axiom,
( ( cis @ pi )
= ( uminus_uminus @ complex @ ( one_one @ complex ) ) ) ).
% cis_pi
thf(fact_5020_or__minus__numerals_I8_J,axiom,
! [N: num,M: num] :
( ( bit_se1065995026697491101ons_or @ int @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ ( bit1 @ N ) ) ) @ ( numeral_numeral @ int @ M ) )
= ( uminus_uminus @ int @ ( numeral_numeral @ int @ ( bit_or_not_num_neg @ M @ ( bit0 @ N ) ) ) ) ) ).
% or_minus_numerals(8)
thf(fact_5021_or__minus__numerals_I4_J,axiom,
! [M: num,N: num] :
( ( bit_se1065995026697491101ons_or @ int @ ( numeral_numeral @ int @ M ) @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ ( bit1 @ N ) ) ) )
= ( uminus_uminus @ int @ ( numeral_numeral @ int @ ( bit_or_not_num_neg @ M @ ( bit0 @ N ) ) ) ) ) ).
% or_minus_numerals(4)
thf(fact_5022_or__minus__numerals_I3_J,axiom,
! [M: num,N: num] :
( ( bit_se1065995026697491101ons_or @ int @ ( numeral_numeral @ int @ M ) @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ ( bit0 @ N ) ) ) )
= ( uminus_uminus @ int @ ( numeral_numeral @ int @ ( bit_or_not_num_neg @ M @ ( bitM @ N ) ) ) ) ) ).
% or_minus_numerals(3)
thf(fact_5023_or__minus__numerals_I7_J,axiom,
! [N: num,M: num] :
( ( bit_se1065995026697491101ons_or @ int @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ ( bit0 @ N ) ) ) @ ( numeral_numeral @ int @ M ) )
= ( uminus_uminus @ int @ ( numeral_numeral @ int @ ( bit_or_not_num_neg @ M @ ( bitM @ N ) ) ) ) ) ).
% or_minus_numerals(7)
thf(fact_5024_cis__pi__half,axiom,
( ( cis @ ( divide_divide @ real @ pi @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) )
= imaginary_unit ) ).
% cis_pi_half
thf(fact_5025_cis__2pi,axiom,
( ( cis @ ( times_times @ real @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) @ pi ) )
= ( one_one @ complex ) ) ).
% cis_2pi
thf(fact_5026_or__not__num__neg_Osimps_I1_J,axiom,
( ( bit_or_not_num_neg @ one2 @ one2 )
= one2 ) ).
% or_not_num_neg.simps(1)
thf(fact_5027_or__not__num__neg_Osimps_I4_J,axiom,
! [N: num] :
( ( bit_or_not_num_neg @ ( bit0 @ N ) @ one2 )
= ( bit0 @ one2 ) ) ).
% or_not_num_neg.simps(4)
thf(fact_5028_or__not__num__neg_Osimps_I6_J,axiom,
! [N: num,M: num] :
( ( bit_or_not_num_neg @ ( bit0 @ N ) @ ( bit1 @ M ) )
= ( bit0 @ ( bit_or_not_num_neg @ N @ M ) ) ) ).
% or_not_num_neg.simps(6)
thf(fact_5029_or__not__num__neg_Osimps_I7_J,axiom,
! [N: num] :
( ( bit_or_not_num_neg @ ( bit1 @ N ) @ one2 )
= one2 ) ).
% or_not_num_neg.simps(7)
thf(fact_5030_or__not__num__neg_Osimps_I3_J,axiom,
! [M: num] :
( ( bit_or_not_num_neg @ one2 @ ( bit1 @ M ) )
= ( bit1 @ M ) ) ).
% or_not_num_neg.simps(3)
thf(fact_5031_or__not__num__neg_Osimps_I5_J,axiom,
! [N: num,M: num] :
( ( bit_or_not_num_neg @ ( bit0 @ N ) @ ( bit0 @ M ) )
= ( bitM @ ( bit_or_not_num_neg @ N @ M ) ) ) ).
% or_not_num_neg.simps(5)
thf(fact_5032_cis__mult,axiom,
! [A3: real,B3: real] :
( ( times_times @ complex @ ( cis @ A3 ) @ ( cis @ B3 ) )
= ( cis @ ( plus_plus @ real @ A3 @ B3 ) ) ) ).
% cis_mult
thf(fact_5033_cis__divide,axiom,
! [A3: real,B3: real] :
( ( divide_divide @ complex @ ( cis @ A3 ) @ ( cis @ B3 ) )
= ( cis @ ( minus_minus @ real @ A3 @ B3 ) ) ) ).
% cis_divide
thf(fact_5034_or__not__num__neg_Osimps_I2_J,axiom,
! [M: num] :
( ( bit_or_not_num_neg @ one2 @ ( bit0 @ M ) )
= ( bit1 @ M ) ) ).
% or_not_num_neg.simps(2)
thf(fact_5035_or__not__num__neg_Osimps_I8_J,axiom,
! [N: num,M: num] :
( ( bit_or_not_num_neg @ ( bit1 @ N ) @ ( bit0 @ M ) )
= ( bitM @ ( bit_or_not_num_neg @ N @ M ) ) ) ).
% or_not_num_neg.simps(8)
thf(fact_5036_or__not__num__neg_Oelims,axiom,
! [X2: num,Xa: num,Y3: num] :
( ( ( bit_or_not_num_neg @ X2 @ Xa )
= Y3 )
=> ( ( ( X2 = one2 )
=> ( ( Xa = one2 )
=> ( Y3 != one2 ) ) )
=> ( ( ( X2 = one2 )
=> ! [M5: num] :
( ( Xa
= ( bit0 @ M5 ) )
=> ( Y3
!= ( bit1 @ M5 ) ) ) )
=> ( ( ( X2 = one2 )
=> ! [M5: num] :
( ( Xa
= ( bit1 @ M5 ) )
=> ( Y3
!= ( bit1 @ M5 ) ) ) )
=> ( ( ? [N3: num] :
( X2
= ( bit0 @ N3 ) )
=> ( ( Xa = one2 )
=> ( Y3
!= ( bit0 @ one2 ) ) ) )
=> ( ! [N3: num] :
( ( X2
= ( bit0 @ N3 ) )
=> ! [M5: num] :
( ( Xa
= ( bit0 @ M5 ) )
=> ( Y3
!= ( bitM @ ( bit_or_not_num_neg @ N3 @ M5 ) ) ) ) )
=> ( ! [N3: num] :
( ( X2
= ( bit0 @ N3 ) )
=> ! [M5: num] :
( ( Xa
= ( bit1 @ M5 ) )
=> ( Y3
!= ( bit0 @ ( bit_or_not_num_neg @ N3 @ M5 ) ) ) ) )
=> ( ( ? [N3: num] :
( X2
= ( bit1 @ N3 ) )
=> ( ( Xa = one2 )
=> ( Y3 != one2 ) ) )
=> ( ! [N3: num] :
( ( X2
= ( bit1 @ N3 ) )
=> ! [M5: num] :
( ( Xa
= ( bit0 @ M5 ) )
=> ( Y3
!= ( bitM @ ( bit_or_not_num_neg @ N3 @ M5 ) ) ) ) )
=> ~ ! [N3: num] :
( ( X2
= ( bit1 @ N3 ) )
=> ! [M5: num] :
( ( Xa
= ( bit1 @ M5 ) )
=> ( Y3
!= ( bitM @ ( bit_or_not_num_neg @ N3 @ M5 ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).
% or_not_num_neg.elims
thf(fact_5037_bij__betw__roots__unity,axiom,
! [N: nat] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ N )
=> ( bij_betw @ nat @ complex
@ ^ [K3: nat] : ( cis @ ( divide_divide @ real @ ( times_times @ real @ ( times_times @ real @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) @ pi ) @ ( semiring_1_of_nat @ real @ K3 ) ) @ ( semiring_1_of_nat @ real @ N ) ) )
@ ( set_ord_lessThan @ nat @ N )
@ ( collect @ complex
@ ^ [Z4: complex] :
( ( power_power @ complex @ Z4 @ N )
= ( one_one @ complex ) ) ) ) ) ).
% bij_betw_roots_unity
thf(fact_5038_set__decode__0,axiom,
! [X2: nat] :
( ( member @ nat @ ( zero_zero @ nat ) @ ( nat_set_decode @ X2 ) )
= ( ~ ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ X2 ) ) ) ).
% set_decode_0
thf(fact_5039_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_5040_set__decode__Suc,axiom,
! [N: nat,X2: nat] :
( ( member @ nat @ ( suc @ N ) @ ( nat_set_decode @ X2 ) )
= ( member @ nat @ N @ ( nat_set_decode @ ( divide_divide @ nat @ X2 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) ).
% set_decode_Suc
thf(fact_5041_sum_Oreindex__bij__betw,axiom,
! [B: $tType,A: $tType,C: $tType] :
( ( comm_monoid_add @ A )
=> ! [H2: B > C,S: set @ B,T6: set @ C,G: C > A] :
( ( bij_betw @ B @ C @ H2 @ S @ T6 )
=> ( ( groups7311177749621191930dd_sum @ B @ A
@ ^ [X3: B] : ( G @ ( H2 @ X3 ) )
@ S )
= ( groups7311177749621191930dd_sum @ C @ A @ G @ T6 ) ) ) ) ).
% sum.reindex_bij_betw
thf(fact_5042_prod_Oreindex__bij__betw,axiom,
! [B: $tType,A: $tType,C: $tType] :
( ( comm_monoid_mult @ A )
=> ! [H2: B > C,S: set @ B,T6: set @ C,G: C > A] :
( ( bij_betw @ B @ C @ H2 @ S @ T6 )
=> ( ( groups7121269368397514597t_prod @ B @ A
@ ^ [X3: B] : ( G @ ( H2 @ X3 ) )
@ S )
= ( groups7121269368397514597t_prod @ C @ A @ G @ T6 ) ) ) ) ).
% prod.reindex_bij_betw
thf(fact_5043_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_5044_sum_Oreindex__bij__betw__not__neutral,axiom,
! [B: $tType,A: $tType,C: $tType] :
( ( comm_monoid_add @ A )
=> ! [S7: set @ B,T7: set @ C,H2: B > C,S: set @ B,T6: set @ C,G: C > A] :
( ( finite_finite2 @ B @ S7 )
=> ( ( finite_finite2 @ C @ T7 )
=> ( ( bij_betw @ B @ C @ H2 @ ( minus_minus @ ( set @ B ) @ S @ S7 ) @ ( minus_minus @ ( set @ C ) @ T6 @ T7 ) )
=> ( ! [A4: B] :
( ( member @ B @ A4 @ S7 )
=> ( ( G @ ( H2 @ A4 ) )
= ( zero_zero @ A ) ) )
=> ( ! [B4: C] :
( ( member @ C @ B4 @ T7 )
=> ( ( G @ B4 )
= ( zero_zero @ A ) ) )
=> ( ( groups7311177749621191930dd_sum @ B @ A
@ ^ [X3: B] : ( G @ ( H2 @ X3 ) )
@ S )
= ( groups7311177749621191930dd_sum @ C @ A @ G @ T6 ) ) ) ) ) ) ) ) ).
% sum.reindex_bij_betw_not_neutral
thf(fact_5045_prod_Oreindex__bij__betw__not__neutral,axiom,
! [B: $tType,A: $tType,C: $tType] :
( ( comm_monoid_mult @ A )
=> ! [S7: set @ B,T7: set @ C,H2: B > C,S: set @ B,T6: set @ C,G: C > A] :
( ( finite_finite2 @ B @ S7 )
=> ( ( finite_finite2 @ C @ T7 )
=> ( ( bij_betw @ B @ C @ H2 @ ( minus_minus @ ( set @ B ) @ S @ S7 ) @ ( minus_minus @ ( set @ C ) @ T6 @ T7 ) )
=> ( ! [A4: B] :
( ( member @ B @ A4 @ S7 )
=> ( ( G @ ( H2 @ A4 ) )
= ( one_one @ A ) ) )
=> ( ! [B4: C] :
( ( member @ C @ B4 @ T7 )
=> ( ( G @ B4 )
= ( one_one @ A ) ) )
=> ( ( groups7121269368397514597t_prod @ B @ A
@ ^ [X3: B] : ( G @ ( H2 @ X3 ) )
@ S )
= ( groups7121269368397514597t_prod @ C @ A @ G @ T6 ) ) ) ) ) ) ) ) ).
% prod.reindex_bij_betw_not_neutral
thf(fact_5046_set__decode__plus__power__2,axiom,
! [N: nat,Z: nat] :
( ~ ( member @ nat @ N @ ( nat_set_decode @ Z ) )
=> ( ( nat_set_decode @ ( plus_plus @ nat @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N ) @ Z ) )
= ( insert @ nat @ N @ ( nat_set_decode @ Z ) ) ) ) ).
% set_decode_plus_power_2
thf(fact_5047_set__decode__def,axiom,
( nat_set_decode
= ( ^ [X3: nat] :
( collect @ nat
@ ^ [N4: nat] :
~ ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ X3 @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N4 ) ) ) ) ) ) ).
% set_decode_def
thf(fact_5048_infinite__imp__bij__betw,axiom,
! [A: $tType,A2: set @ A,A3: A] :
( ~ ( finite_finite2 @ A @ A2 )
=> ? [H3: A > A] : ( bij_betw @ A @ A @ H3 @ A2 @ ( minus_minus @ ( set @ A ) @ A2 @ ( insert @ A @ A3 @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ).
% infinite_imp_bij_betw
thf(fact_5049_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_5050_flip__bit__0,axiom,
! [A: $tType] :
( ( bit_se359711467146920520ations @ A )
=> ! [A3: A] :
( ( bit_se8732182000553998342ip_bit @ A @ ( zero_zero @ nat ) @ A3 )
= ( plus_plus @ A @ ( zero_neq_one_of_bool @ A @ ( dvd_dvd @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ A3 ) ) @ ( times_times @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ ( divide_divide @ A @ A3 @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) ) ) ) ) ).
% flip_bit_0
thf(fact_5051_of__bool__less__eq__iff,axiom,
! [A: $tType] :
( ( linord181362715937106298miring @ A )
=> ! [P: $o,Q: $o] :
( ( ord_less_eq @ A @ ( zero_neq_one_of_bool @ A @ P ) @ ( zero_neq_one_of_bool @ A @ Q ) )
= ( P
=> Q ) ) ) ).
% of_bool_less_eq_iff
thf(fact_5052_of__bool__less__iff,axiom,
! [A: $tType] :
( ( linord181362715937106298miring @ A )
=> ! [P: $o,Q: $o] :
( ( ord_less @ A @ ( zero_neq_one_of_bool @ A @ P ) @ ( zero_neq_one_of_bool @ A @ Q ) )
= ( ~ P
& Q ) ) ) ).
% of_bool_less_iff
thf(fact_5053_of__bool__eq__1__iff,axiom,
! [A: $tType] :
( ( zero_neq_one @ A )
=> ! [P: $o] :
( ( ( zero_neq_one_of_bool @ A @ P )
= ( one_one @ A ) )
= P ) ) ).
% of_bool_eq_1_iff
thf(fact_5054_of__bool__eq_I2_J,axiom,
! [A: $tType] :
( ( zero_neq_one @ A )
=> ( ( zero_neq_one_of_bool @ A @ $true )
= ( one_one @ A ) ) ) ).
% of_bool_eq(2)
thf(fact_5055_of__nat__of__bool,axiom,
! [A: $tType] :
( ( semiring_1 @ A )
=> ! [P: $o] :
( ( semiring_1_of_nat @ A @ ( zero_neq_one_of_bool @ nat @ P ) )
= ( zero_neq_one_of_bool @ A @ P ) ) ) ).
% of_nat_of_bool
thf(fact_5056_of__bool__or__iff,axiom,
! [A: $tType] :
( ( linordered_semidom @ A )
=> ! [P: $o,Q: $o] :
( ( zero_neq_one_of_bool @ A
@ ( P
| Q ) )
= ( ord_max @ A @ ( zero_neq_one_of_bool @ A @ P ) @ ( zero_neq_one_of_bool @ A @ Q ) ) ) ) ).
% of_bool_or_iff
thf(fact_5057_div__word__one,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [W: word @ A] :
( ( divide_divide @ ( word @ A ) @ ( one_one @ ( word @ A ) ) @ W )
= ( zero_neq_one_of_bool @ ( word @ A )
@ ( W
= ( one_one @ ( word @ A ) ) ) ) ) ) ).
% div_word_one
thf(fact_5058_zero__less__of__bool__iff,axiom,
! [A: $tType] :
( ( linordered_semidom @ A )
=> ! [P: $o] :
( ( ord_less @ A @ ( zero_zero @ A ) @ ( zero_neq_one_of_bool @ A @ P ) )
= P ) ) ).
% zero_less_of_bool_iff
thf(fact_5059_of__bool__less__one__iff,axiom,
! [A: $tType] :
( ( linordered_semidom @ A )
=> ! [P: $o] :
( ( ord_less @ A @ ( zero_neq_one_of_bool @ A @ P ) @ ( one_one @ A ) )
= ~ P ) ) ).
% of_bool_less_one_iff
thf(fact_5060_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_5061_of__bool__not__iff,axiom,
! [A: $tType] :
( ( ring_1 @ A )
=> ! [P: $o] :
( ( zero_neq_one_of_bool @ A @ ~ P )
= ( minus_minus @ A @ ( one_one @ A ) @ ( zero_neq_one_of_bool @ A @ P ) ) ) ) ).
% of_bool_not_iff
thf(fact_5062_div__word__by__minus__1__eq,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [W: word @ A] :
( ( divide_divide @ ( word @ A ) @ W @ ( uminus_uminus @ ( word @ A ) @ ( one_one @ ( word @ A ) ) ) )
= ( zero_neq_one_of_bool @ ( word @ A )
@ ( W
= ( uminus_uminus @ ( word @ A ) @ ( one_one @ ( word @ A ) ) ) ) ) ) ) ).
% div_word_by_minus_1_eq
thf(fact_5063_mod__word__one,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [W: word @ A] :
( ( modulo_modulo @ ( word @ A ) @ ( one_one @ ( word @ A ) ) @ W )
= ( minus_minus @ ( word @ A ) @ ( one_one @ ( word @ A ) )
@ ( times_times @ ( word @ A ) @ W
@ ( zero_neq_one_of_bool @ ( word @ A )
@ ( W
= ( one_one @ ( word @ A ) ) ) ) ) ) ) ) ).
% mod_word_one
thf(fact_5064_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_5065_odd__of__bool__self,axiom,
! [A: $tType] :
( ( semiring_parity @ A )
=> ! [P6: $o] :
( ( ~ ( dvd_dvd @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ ( zero_neq_one_of_bool @ A @ P6 ) ) )
= P6 ) ) ).
% odd_of_bool_self
thf(fact_5066_mod__word__by__minus__1__eq,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [W: word @ A] :
( ( modulo_modulo @ ( word @ A ) @ W @ ( uminus_uminus @ ( word @ A ) @ ( one_one @ ( word @ A ) ) ) )
= ( times_times @ ( word @ A ) @ W @ ( zero_neq_one_of_bool @ ( word @ A ) @ ( ord_less @ ( word @ A ) @ W @ ( uminus_uminus @ ( word @ A ) @ ( one_one @ ( word @ A ) ) ) ) ) ) ) ) ).
% mod_word_by_minus_1_eq
thf(fact_5067_of__bool__half__eq__0,axiom,
! [A: $tType] :
( ( euclid5411537665997757685th_nat @ A )
=> ! [B3: $o] :
( ( divide_divide @ A @ ( zero_neq_one_of_bool @ A @ B3 ) @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) )
= ( zero_zero @ A ) ) ) ).
% of_bool_half_eq_0
thf(fact_5068_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_5069_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_5070_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_5071_of__bool__conj,axiom,
! [A: $tType] :
( ( semiring_1 @ A )
=> ! [P: $o,Q: $o] :
( ( zero_neq_one_of_bool @ A
@ ( P
& Q ) )
= ( times_times @ A @ ( zero_neq_one_of_bool @ A @ P ) @ ( zero_neq_one_of_bool @ A @ Q ) ) ) ) ).
% of_bool_conj
thf(fact_5072_zero__less__eq__of__bool,axiom,
! [A: $tType] :
( ( linordered_semidom @ A )
=> ! [P: $o] : ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( zero_neq_one_of_bool @ A @ P ) ) ) ).
% zero_less_eq_of_bool
thf(fact_5073_of__bool__def,axiom,
! [A: $tType] :
( ( zero_neq_one @ A )
=> ( ( zero_neq_one_of_bool @ A )
= ( ^ [P4: $o] : ( if @ A @ P4 @ ( one_one @ A ) @ ( zero_zero @ A ) ) ) ) ) ).
% of_bool_def
thf(fact_5074_split__of__bool,axiom,
! [A: $tType] :
( ( zero_neq_one @ A )
=> ! [P: A > $o,P6: $o] :
( ( P @ ( zero_neq_one_of_bool @ A @ P6 ) )
= ( ( P6
=> ( P @ ( one_one @ A ) ) )
& ( ~ P6
=> ( P @ ( zero_zero @ A ) ) ) ) ) ) ).
% split_of_bool
thf(fact_5075_split__of__bool__asm,axiom,
! [A: $tType] :
( ( zero_neq_one @ A )
=> ! [P: A > $o,P6: $o] :
( ( P @ ( zero_neq_one_of_bool @ A @ P6 ) )
= ( ~ ( ( P6
& ~ ( P @ ( one_one @ A ) ) )
| ( ~ P6
& ~ ( P @ ( zero_zero @ A ) ) ) ) ) ) ) ).
% split_of_bool_asm
thf(fact_5076_of__bool__less__eq__one,axiom,
! [A: $tType] :
( ( linordered_semidom @ A )
=> ! [P: $o] : ( ord_less_eq @ A @ ( zero_neq_one_of_bool @ A @ P ) @ ( one_one @ A ) ) ) ).
% of_bool_less_eq_one
thf(fact_5077_of__bool__odd__eq__mod__2,axiom,
! [A: $tType] :
( ( semiring_parity @ A )
=> ! [A3: A] :
( ( zero_neq_one_of_bool @ A
@ ~ ( dvd_dvd @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ A3 ) )
= ( modulo_modulo @ A @ A3 @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) ) ) ).
% of_bool_odd_eq_mod_2
thf(fact_5078_bits__induct,axiom,
! [A: $tType] :
( ( bit_semiring_bits @ A )
=> ! [P: A > $o,A3: A] :
( ! [A4: A] :
( ( ( divide_divide @ A @ A4 @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) )
= A4 )
=> ( P @ A4 ) )
=> ( ! [A4: A,B4: $o] :
( ( P @ A4 )
=> ( ( ( divide_divide @ A @ ( plus_plus @ A @ ( zero_neq_one_of_bool @ A @ B4 ) @ ( times_times @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ A4 ) ) @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) )
= A4 )
=> ( P @ ( plus_plus @ A @ ( zero_neq_one_of_bool @ A @ B4 ) @ ( times_times @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ A4 ) ) ) ) )
=> ( P @ A3 ) ) ) ) ).
% bits_induct
thf(fact_5079_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_5080_one__or__eq,axiom,
! [A: $tType] :
( ( bit_se359711467146920520ations @ A )
=> ! [A3: A] :
( ( bit_se1065995026697491101ons_or @ A @ ( one_one @ A ) @ A3 )
= ( plus_plus @ A @ A3 @ ( zero_neq_one_of_bool @ A @ ( dvd_dvd @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ A3 ) ) ) ) ) ).
% one_or_eq
thf(fact_5081_or__one__eq,axiom,
! [A: $tType] :
( ( bit_se359711467146920520ations @ A )
=> ! [A3: A] :
( ( bit_se1065995026697491101ons_or @ A @ A3 @ ( one_one @ A ) )
= ( plus_plus @ A @ A3 @ ( zero_neq_one_of_bool @ A @ ( dvd_dvd @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ A3 ) ) ) ) ) ).
% or_one_eq
thf(fact_5082_dependent__nat__choice,axiom,
! [A: $tType,P: nat > A > $o,Q: nat > A > A > $o] :
( ? [X_12: A] : ( P @ ( zero_zero @ nat ) @ X_12 )
=> ( ! [X4: A,N3: nat] :
( ( P @ N3 @ X4 )
=> ? [Y5: A] :
( ( P @ ( suc @ N3 ) @ Y5 )
& ( Q @ N3 @ X4 @ Y5 ) ) )
=> ? [F3: nat > A] :
! [N9: nat] :
( ( P @ N9 @ ( F3 @ N9 ) )
& ( Q @ N9 @ ( F3 @ N9 ) @ ( F3 @ ( suc @ N9 ) ) ) ) ) ) ).
% dependent_nat_choice
thf(fact_5083_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_5084_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_5085_or__nat__rec,axiom,
( ( bit_se1065995026697491101ons_or @ nat )
= ( ^ [M6: nat,N4: nat] :
( plus_plus @ nat
@ ( zero_neq_one_of_bool @ nat
@ ( ~ ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ M6 )
| ~ ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N4 ) ) )
@ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( bit_se1065995026697491101ons_or @ nat @ ( divide_divide @ nat @ M6 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ ( divide_divide @ nat @ N4 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) ) ) ).
% or_nat_rec
thf(fact_5086_or__int__rec,axiom,
( ( bit_se1065995026697491101ons_or @ int )
= ( ^ [K3: int,L3: int] :
( plus_plus @ int
@ ( zero_neq_one_of_bool @ int
@ ( ~ ( dvd_dvd @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ K3 )
| ~ ( dvd_dvd @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ L3 ) ) )
@ ( times_times @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ ( bit_se1065995026697491101ons_or @ int @ ( divide_divide @ int @ K3 @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) ) @ ( divide_divide @ int @ L3 @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) ) ) ) ) ) ) ).
% or_int_rec
thf(fact_5087_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_5088_bij__betw__nth__root__unity,axiom,
! [C3: complex,N: nat] :
( ( C3
!= ( zero_zero @ complex ) )
=> ( ( ord_less @ nat @ ( zero_zero @ nat ) @ N )
=> ( bij_betw @ complex @ complex @ ( times_times @ complex @ ( times_times @ complex @ ( real_Vector_of_real @ complex @ ( root @ N @ ( real_V7770717601297561774m_norm @ complex @ C3 ) ) ) @ ( cis @ ( divide_divide @ real @ ( arg @ C3 ) @ ( semiring_1_of_nat @ real @ N ) ) ) ) )
@ ( collect @ complex
@ ^ [Z4: complex] :
( ( power_power @ complex @ Z4 @ N )
= ( one_one @ complex ) ) )
@ ( collect @ complex
@ ^ [Z4: complex] :
( ( power_power @ complex @ Z4 @ N )
= C3 ) ) ) ) ) ).
% bij_betw_nth_root_unity
thf(fact_5089_and__int_Oelims,axiom,
! [X2: int,Xa: int,Y3: int] :
( ( ( bit_se5824344872417868541ns_and @ int @ X2 @ Xa )
= Y3 )
=> ( ( ( ( member @ int @ X2 @ ( insert @ int @ ( zero_zero @ int ) @ ( insert @ int @ ( uminus_uminus @ int @ ( one_one @ int ) ) @ ( bot_bot @ ( set @ int ) ) ) ) )
& ( member @ int @ Xa @ ( insert @ int @ ( zero_zero @ int ) @ ( insert @ int @ ( uminus_uminus @ int @ ( one_one @ int ) ) @ ( bot_bot @ ( set @ int ) ) ) ) ) )
=> ( Y3
= ( uminus_uminus @ int
@ ( zero_neq_one_of_bool @ int
@ ( ~ ( dvd_dvd @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ X2 )
& ~ ( dvd_dvd @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ Xa ) ) ) ) ) )
& ( ~ ( ( member @ int @ X2 @ ( insert @ int @ ( zero_zero @ int ) @ ( insert @ int @ ( uminus_uminus @ int @ ( one_one @ int ) ) @ ( bot_bot @ ( set @ int ) ) ) ) )
& ( member @ int @ Xa @ ( insert @ int @ ( zero_zero @ int ) @ ( insert @ int @ ( uminus_uminus @ int @ ( one_one @ int ) ) @ ( bot_bot @ ( set @ int ) ) ) ) ) )
=> ( Y3
= ( plus_plus @ int
@ ( zero_neq_one_of_bool @ int
@ ( ~ ( dvd_dvd @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ X2 )
& ~ ( dvd_dvd @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ Xa ) ) )
@ ( times_times @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ ( bit_se5824344872417868541ns_and @ int @ ( divide_divide @ int @ X2 @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) ) @ ( divide_divide @ int @ Xa @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) ) ) ) ) ) ) ) ) ).
% and_int.elims
thf(fact_5090_and__int_Osimps,axiom,
( ( bit_se5824344872417868541ns_and @ int )
= ( ^ [K3: int,L3: int] :
( if @ int
@ ( ( member @ int @ K3 @ ( insert @ int @ ( zero_zero @ int ) @ ( insert @ int @ ( uminus_uminus @ int @ ( one_one @ int ) ) @ ( bot_bot @ ( set @ int ) ) ) ) )
& ( member @ int @ L3 @ ( insert @ int @ ( zero_zero @ int ) @ ( insert @ int @ ( uminus_uminus @ int @ ( one_one @ int ) ) @ ( bot_bot @ ( set @ int ) ) ) ) ) )
@ ( uminus_uminus @ int
@ ( zero_neq_one_of_bool @ int
@ ( ~ ( dvd_dvd @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ K3 )
& ~ ( dvd_dvd @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ L3 ) ) ) )
@ ( plus_plus @ int
@ ( zero_neq_one_of_bool @ int
@ ( ~ ( dvd_dvd @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ K3 )
& ~ ( dvd_dvd @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ L3 ) ) )
@ ( times_times @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ ( bit_se5824344872417868541ns_and @ int @ ( divide_divide @ int @ K3 @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) ) @ ( divide_divide @ int @ L3 @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) ) ) ) ) ) ) ) ).
% and_int.simps
thf(fact_5091_and_Oleft__neutral,axiom,
! [A: $tType] :
( ( bit_ri3973907225187159222ations @ A )
=> ! [A3: A] :
( ( bit_se5824344872417868541ns_and @ A @ ( uminus_uminus @ A @ ( one_one @ A ) ) @ A3 )
= A3 ) ) ).
% and.left_neutral
thf(fact_5092_and_Oright__neutral,axiom,
! [A: $tType] :
( ( bit_ri3973907225187159222ations @ A )
=> ! [A3: A] :
( ( bit_se5824344872417868541ns_and @ A @ A3 @ ( uminus_uminus @ A @ ( one_one @ A ) ) )
= A3 ) ) ).
% and.right_neutral
thf(fact_5093_bit_Oconj__one__right,axiom,
! [A: $tType] :
( ( bit_ri3973907225187159222ations @ A )
=> ! [X2: A] :
( ( bit_se5824344872417868541ns_and @ A @ X2 @ ( uminus_uminus @ A @ ( one_one @ A ) ) )
= X2 ) ) ).
% bit.conj_one_right
thf(fact_5094_real__root__Suc__0,axiom,
! [X2: real] :
( ( root @ ( suc @ ( zero_zero @ nat ) ) @ X2 )
= X2 ) ).
% real_root_Suc_0
thf(fact_5095_real__root__eq__iff,axiom,
! [N: nat,X2: real,Y3: real] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ N )
=> ( ( ( root @ N @ X2 )
= ( root @ N @ Y3 ) )
= ( X2 = Y3 ) ) ) ).
% real_root_eq_iff
thf(fact_5096_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_5097_and__numerals_I2_J,axiom,
! [A: $tType] :
( ( bit_un5681908812861735899ations @ A )
=> ! [Y3: num] :
( ( bit_se5824344872417868541ns_and @ A @ ( one_one @ A ) @ ( numeral_numeral @ A @ ( bit1 @ Y3 ) ) )
= ( one_one @ A ) ) ) ).
% and_numerals(2)
thf(fact_5098_and__numerals_I8_J,axiom,
! [A: $tType] :
( ( bit_un5681908812861735899ations @ A )
=> ! [X2: num] :
( ( bit_se5824344872417868541ns_and @ A @ ( numeral_numeral @ A @ ( bit1 @ X2 ) ) @ ( one_one @ A ) )
= ( one_one @ A ) ) ) ).
% and_numerals(8)
thf(fact_5099_real__root__eq__0__iff,axiom,
! [N: nat,X2: real] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ N )
=> ( ( ( root @ N @ X2 )
= ( zero_zero @ real ) )
= ( X2
= ( zero_zero @ real ) ) ) ) ).
% real_root_eq_0_iff
thf(fact_5100_real__root__less__iff,axiom,
! [N: nat,X2: real,Y3: real] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ N )
=> ( ( ord_less @ real @ ( root @ N @ X2 ) @ ( root @ N @ Y3 ) )
= ( ord_less @ real @ X2 @ Y3 ) ) ) ).
% real_root_less_iff
thf(fact_5101_real__root__le__iff,axiom,
! [N: nat,X2: real,Y3: real] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ N )
=> ( ( ord_less_eq @ real @ ( root @ N @ X2 ) @ ( root @ N @ Y3 ) )
= ( ord_less_eq @ real @ X2 @ Y3 ) ) ) ).
% real_root_le_iff
thf(fact_5102_real__root__eq__1__iff,axiom,
! [N: nat,X2: real] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ N )
=> ( ( ( root @ N @ X2 )
= ( one_one @ real ) )
= ( X2
= ( one_one @ real ) ) ) ) ).
% real_root_eq_1_iff
thf(fact_5103_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_5104_and__numerals_I1_J,axiom,
! [A: $tType] :
( ( bit_un5681908812861735899ations @ A )
=> ! [Y3: num] :
( ( bit_se5824344872417868541ns_and @ A @ ( one_one @ A ) @ ( numeral_numeral @ A @ ( bit0 @ Y3 ) ) )
= ( zero_zero @ A ) ) ) ).
% and_numerals(1)
thf(fact_5105_and__numerals_I5_J,axiom,
! [A: $tType] :
( ( bit_un5681908812861735899ations @ A )
=> ! [X2: num] :
( ( bit_se5824344872417868541ns_and @ A @ ( numeral_numeral @ A @ ( bit0 @ X2 ) ) @ ( one_one @ A ) )
= ( zero_zero @ A ) ) ) ).
% and_numerals(5)
thf(fact_5106_and__numerals_I3_J,axiom,
! [A: $tType] :
( ( bit_un5681908812861735899ations @ A )
=> ! [X2: num,Y3: num] :
( ( bit_se5824344872417868541ns_and @ A @ ( numeral_numeral @ A @ ( bit0 @ X2 ) ) @ ( numeral_numeral @ A @ ( bit0 @ Y3 ) ) )
= ( times_times @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ ( bit_se5824344872417868541ns_and @ A @ ( numeral_numeral @ A @ X2 ) @ ( numeral_numeral @ A @ Y3 ) ) ) ) ) ).
% and_numerals(3)
thf(fact_5107_real__root__gt__0__iff,axiom,
! [N: nat,Y3: real] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ N )
=> ( ( ord_less @ real @ ( zero_zero @ real ) @ ( root @ N @ Y3 ) )
= ( ord_less @ real @ ( zero_zero @ real ) @ Y3 ) ) ) ).
% real_root_gt_0_iff
thf(fact_5108_real__root__lt__0__iff,axiom,
! [N: nat,X2: real] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ N )
=> ( ( ord_less @ real @ ( root @ N @ X2 ) @ ( zero_zero @ real ) )
= ( ord_less @ real @ X2 @ ( zero_zero @ real ) ) ) ) ).
% real_root_lt_0_iff
thf(fact_5109_real__root__le__0__iff,axiom,
! [N: nat,X2: real] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ N )
=> ( ( ord_less_eq @ real @ ( root @ N @ X2 ) @ ( zero_zero @ real ) )
= ( ord_less_eq @ real @ X2 @ ( zero_zero @ real ) ) ) ) ).
% real_root_le_0_iff
thf(fact_5110_real__root__ge__0__iff,axiom,
! [N: nat,Y3: real] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ N )
=> ( ( ord_less_eq @ real @ ( zero_zero @ real ) @ ( root @ N @ Y3 ) )
= ( ord_less_eq @ real @ ( zero_zero @ real ) @ Y3 ) ) ) ).
% real_root_ge_0_iff
thf(fact_5111_real__root__lt__1__iff,axiom,
! [N: nat,X2: real] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ N )
=> ( ( ord_less @ real @ ( root @ N @ X2 ) @ ( one_one @ real ) )
= ( ord_less @ real @ X2 @ ( one_one @ real ) ) ) ) ).
% real_root_lt_1_iff
thf(fact_5112_real__root__gt__1__iff,axiom,
! [N: nat,Y3: real] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ N )
=> ( ( ord_less @ real @ ( one_one @ real ) @ ( root @ N @ Y3 ) )
= ( ord_less @ real @ ( one_one @ real ) @ Y3 ) ) ) ).
% real_root_gt_1_iff
thf(fact_5113_real__root__ge__1__iff,axiom,
! [N: nat,Y3: real] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ N )
=> ( ( ord_less_eq @ real @ ( one_one @ real ) @ ( root @ N @ Y3 ) )
= ( ord_less_eq @ real @ ( one_one @ real ) @ Y3 ) ) ) ).
% real_root_ge_1_iff
thf(fact_5114_real__root__le__1__iff,axiom,
! [N: nat,X2: real] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ N )
=> ( ( ord_less_eq @ real @ ( root @ N @ X2 ) @ ( one_one @ real ) )
= ( ord_less_eq @ real @ X2 @ ( one_one @ real ) ) ) ) ).
% real_root_le_1_iff
thf(fact_5115_and__minus__numerals_I2_J,axiom,
! [N: num] :
( ( bit_se5824344872417868541ns_and @ int @ ( one_one @ int ) @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ ( bit1 @ N ) ) ) )
= ( one_one @ int ) ) ).
% and_minus_numerals(2)
thf(fact_5116_and__minus__numerals_I6_J,axiom,
! [N: num] :
( ( bit_se5824344872417868541ns_and @ int @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ ( bit1 @ N ) ) ) @ ( one_one @ int ) )
= ( one_one @ int ) ) ).
% and_minus_numerals(6)
thf(fact_5117_and__numerals_I6_J,axiom,
! [A: $tType] :
( ( bit_un5681908812861735899ations @ A )
=> ! [X2: num,Y3: num] :
( ( bit_se5824344872417868541ns_and @ A @ ( numeral_numeral @ A @ ( bit1 @ X2 ) ) @ ( numeral_numeral @ A @ ( bit0 @ Y3 ) ) )
= ( times_times @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ ( bit_se5824344872417868541ns_and @ A @ ( numeral_numeral @ A @ X2 ) @ ( numeral_numeral @ A @ Y3 ) ) ) ) ) ).
% and_numerals(6)
thf(fact_5118_and__numerals_I4_J,axiom,
! [A: $tType] :
( ( bit_un5681908812861735899ations @ A )
=> ! [X2: num,Y3: num] :
( ( bit_se5824344872417868541ns_and @ A @ ( numeral_numeral @ A @ ( bit0 @ X2 ) ) @ ( numeral_numeral @ A @ ( bit1 @ Y3 ) ) )
= ( times_times @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ ( bit_se5824344872417868541ns_and @ A @ ( numeral_numeral @ A @ X2 ) @ ( numeral_numeral @ A @ Y3 ) ) ) ) ) ).
% and_numerals(4)
thf(fact_5119_real__root__pow__pos2,axiom,
! [N: nat,X2: real] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ N )
=> ( ( ord_less_eq @ real @ ( zero_zero @ real ) @ X2 )
=> ( ( power_power @ real @ ( root @ N @ X2 ) @ N )
= X2 ) ) ) ).
% real_root_pow_pos2
thf(fact_5120_and__minus__numerals_I1_J,axiom,
! [N: num] :
( ( bit_se5824344872417868541ns_and @ int @ ( one_one @ int ) @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ ( bit0 @ N ) ) ) )
= ( zero_zero @ int ) ) ).
% and_minus_numerals(1)
thf(fact_5121_and__minus__numerals_I5_J,axiom,
! [N: num] :
( ( bit_se5824344872417868541ns_and @ int @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ ( bit0 @ N ) ) ) @ ( one_one @ int ) )
= ( zero_zero @ int ) ) ).
% and_minus_numerals(5)
thf(fact_5122_and__numerals_I7_J,axiom,
! [A: $tType] :
( ( bit_un5681908812861735899ations @ A )
=> ! [X2: num,Y3: num] :
( ( bit_se5824344872417868541ns_and @ A @ ( numeral_numeral @ A @ ( bit1 @ X2 ) ) @ ( numeral_numeral @ A @ ( bit1 @ Y3 ) ) )
= ( plus_plus @ A @ ( one_one @ A ) @ ( times_times @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ ( bit_se5824344872417868541ns_and @ A @ ( numeral_numeral @ A @ X2 ) @ ( numeral_numeral @ A @ Y3 ) ) ) ) ) ) ).
% and_numerals(7)
thf(fact_5123_and__eq__minus__1__iff,axiom,
! [A: $tType] :
( ( bit_ri3973907225187159222ations @ A )
=> ! [A3: A,B3: A] :
( ( ( bit_se5824344872417868541ns_and @ A @ A3 @ B3 )
= ( uminus_uminus @ A @ ( one_one @ A ) ) )
= ( ( A3
= ( uminus_uminus @ A @ ( one_one @ A ) ) )
& ( B3
= ( uminus_uminus @ A @ ( one_one @ A ) ) ) ) ) ) ).
% and_eq_minus_1_iff
thf(fact_5124_real__root__mult__exp,axiom,
! [M: nat,N: nat,X2: real] :
( ( root @ ( times_times @ nat @ M @ N ) @ X2 )
= ( root @ M @ ( root @ N @ X2 ) ) ) ).
% real_root_mult_exp
thf(fact_5125_real__root__pos__pos__le,axiom,
! [X2: real,N: nat] :
( ( ord_less_eq @ real @ ( zero_zero @ real ) @ X2 )
=> ( ord_less_eq @ real @ ( zero_zero @ real ) @ ( root @ N @ X2 ) ) ) ).
% real_root_pos_pos_le
thf(fact_5126_AND__upper2_H,axiom,
! [Y3: int,Z: int,X2: int] :
( ( ord_less_eq @ int @ ( zero_zero @ int ) @ Y3 )
=> ( ( ord_less_eq @ int @ Y3 @ Z )
=> ( ord_less_eq @ int @ ( bit_se5824344872417868541ns_and @ int @ X2 @ Y3 ) @ Z ) ) ) ).
% AND_upper2'
thf(fact_5127_AND__upper1_H,axiom,
! [Y3: int,Z: int,Ya: int] :
( ( ord_less_eq @ int @ ( zero_zero @ int ) @ Y3 )
=> ( ( ord_less_eq @ int @ Y3 @ Z )
=> ( ord_less_eq @ int @ ( bit_se5824344872417868541ns_and @ int @ Y3 @ Ya ) @ Z ) ) ) ).
% AND_upper1'
thf(fact_5128_AND__upper2,axiom,
! [Y3: int,X2: int] :
( ( ord_less_eq @ int @ ( zero_zero @ int ) @ Y3 )
=> ( ord_less_eq @ int @ ( bit_se5824344872417868541ns_and @ int @ X2 @ Y3 ) @ Y3 ) ) ).
% AND_upper2
thf(fact_5129_AND__upper1,axiom,
! [X2: int,Y3: int] :
( ( ord_less_eq @ int @ ( zero_zero @ int ) @ X2 )
=> ( ord_less_eq @ int @ ( bit_se5824344872417868541ns_and @ int @ X2 @ Y3 ) @ X2 ) ) ).
% AND_upper1
thf(fact_5130_AND__lower,axiom,
! [X2: int,Y3: int] :
( ( ord_less_eq @ int @ ( zero_zero @ int ) @ X2 )
=> ( ord_less_eq @ int @ ( zero_zero @ int ) @ ( bit_se5824344872417868541ns_and @ int @ X2 @ Y3 ) ) ) ).
% AND_lower
thf(fact_5131_plus__and__or,axiom,
! [X2: int,Y3: int] :
( ( plus_plus @ int @ ( bit_se5824344872417868541ns_and @ int @ X2 @ Y3 ) @ ( bit_se1065995026697491101ons_or @ int @ X2 @ Y3 ) )
= ( plus_plus @ int @ X2 @ Y3 ) ) ).
% plus_and_or
thf(fact_5132_real__root__less__mono,axiom,
! [N: nat,X2: real,Y3: real] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ N )
=> ( ( ord_less @ real @ X2 @ Y3 )
=> ( ord_less @ real @ ( root @ N @ X2 ) @ ( root @ N @ Y3 ) ) ) ) ).
% real_root_less_mono
thf(fact_5133_real__root__le__mono,axiom,
! [N: nat,X2: real,Y3: real] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ N )
=> ( ( ord_less_eq @ real @ X2 @ Y3 )
=> ( ord_less_eq @ real @ ( root @ N @ X2 ) @ ( root @ N @ Y3 ) ) ) ) ).
% real_root_le_mono
thf(fact_5134_real__root__power,axiom,
! [N: nat,X2: real,K: nat] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ N )
=> ( ( root @ N @ ( power_power @ real @ X2 @ K ) )
= ( power_power @ real @ ( root @ N @ X2 ) @ K ) ) ) ).
% real_root_power
thf(fact_5135_real__root__abs,axiom,
! [N: nat,X2: real] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ N )
=> ( ( root @ N @ ( abs_abs @ real @ X2 ) )
= ( abs_abs @ real @ ( root @ N @ X2 ) ) ) ) ).
% real_root_abs
thf(fact_5136_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_5137_AND__upper1_H_H,axiom,
! [Y3: int,Z: int,Ya: int] :
( ( ord_less_eq @ int @ ( zero_zero @ int ) @ Y3 )
=> ( ( ord_less @ int @ Y3 @ Z )
=> ( ord_less @ int @ ( bit_se5824344872417868541ns_and @ int @ Y3 @ Ya ) @ Z ) ) ) ).
% AND_upper1''
thf(fact_5138_AND__upper2_H_H,axiom,
! [Y3: int,Z: int,X2: int] :
( ( ord_less_eq @ int @ ( zero_zero @ int ) @ Y3 )
=> ( ( ord_less @ int @ Y3 @ Z )
=> ( ord_less @ int @ ( bit_se5824344872417868541ns_and @ int @ X2 @ Y3 ) @ Z ) ) ) ).
% AND_upper2''
thf(fact_5139_even__and__iff,axiom,
! [A: $tType] :
( ( bit_se359711467146920520ations @ A )
=> ! [A3: A,B3: A] :
( ( dvd_dvd @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ ( bit_se5824344872417868541ns_and @ A @ A3 @ B3 ) )
= ( ( dvd_dvd @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ A3 )
| ( dvd_dvd @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ B3 ) ) ) ) ).
% even_and_iff
thf(fact_5140_bit_Ocomplement__unique,axiom,
! [A: $tType] :
( ( bit_ri3973907225187159222ations @ A )
=> ! [A3: A,X2: A,Y3: A] :
( ( ( bit_se5824344872417868541ns_and @ A @ A3 @ X2 )
= ( zero_zero @ A ) )
=> ( ( ( bit_se1065995026697491101ons_or @ A @ A3 @ X2 )
= ( uminus_uminus @ A @ ( one_one @ A ) ) )
=> ( ( ( bit_se5824344872417868541ns_and @ A @ A3 @ Y3 )
= ( zero_zero @ A ) )
=> ( ( ( bit_se1065995026697491101ons_or @ A @ A3 @ Y3 )
= ( uminus_uminus @ A @ ( one_one @ A ) ) )
=> ( X2 = Y3 ) ) ) ) ) ) ).
% bit.complement_unique
thf(fact_5141_real__root__gt__zero,axiom,
! [N: nat,X2: real] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ N )
=> ( ( ord_less @ real @ ( zero_zero @ real ) @ X2 )
=> ( ord_less @ real @ ( zero_zero @ real ) @ ( root @ N @ X2 ) ) ) ) ).
% real_root_gt_zero
thf(fact_5142_real__root__strict__decreasing,axiom,
! [N: nat,N7: nat,X2: real] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ N )
=> ( ( ord_less @ nat @ N @ N7 )
=> ( ( ord_less @ real @ ( one_one @ real ) @ X2 )
=> ( ord_less @ real @ ( root @ N7 @ X2 ) @ ( root @ N @ X2 ) ) ) ) ) ).
% real_root_strict_decreasing
thf(fact_5143_sqrt__def,axiom,
( sqrt
= ( root @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ).
% sqrt_def
thf(fact_5144_root__abs__power,axiom,
! [N: nat,Y3: real] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ N )
=> ( ( abs_abs @ real @ ( root @ N @ ( power_power @ real @ Y3 @ N ) ) )
= ( abs_abs @ real @ Y3 ) ) ) ).
% root_abs_power
thf(fact_5145_even__and__iff__int,axiom,
! [K: int,L: int] :
( ( dvd_dvd @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ ( bit_se5824344872417868541ns_and @ int @ K @ L ) )
= ( ( dvd_dvd @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ K )
| ( dvd_dvd @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ L ) ) ) ).
% even_and_iff_int
thf(fact_5146_and__one__eq,axiom,
! [A: $tType] :
( ( bit_se359711467146920520ations @ A )
=> ! [A3: A] :
( ( bit_se5824344872417868541ns_and @ A @ A3 @ ( one_one @ A ) )
= ( modulo_modulo @ A @ A3 @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) ) ) ).
% and_one_eq
thf(fact_5147_one__and__eq,axiom,
! [A: $tType] :
( ( bit_se359711467146920520ations @ A )
=> ! [A3: A] :
( ( bit_se5824344872417868541ns_and @ A @ ( one_one @ A ) @ A3 )
= ( modulo_modulo @ A @ A3 @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) ) ) ).
% one_and_eq
thf(fact_5148_real__root__pos__pos,axiom,
! [N: nat,X2: real] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ N )
=> ( ( ord_less @ real @ ( zero_zero @ real ) @ X2 )
=> ( ord_less_eq @ real @ ( zero_zero @ real ) @ ( root @ N @ X2 ) ) ) ) ).
% real_root_pos_pos
thf(fact_5149_real__root__strict__increasing,axiom,
! [N: nat,N7: nat,X2: real] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ N )
=> ( ( ord_less @ nat @ N @ N7 )
=> ( ( ord_less @ real @ ( zero_zero @ real ) @ X2 )
=> ( ( ord_less @ real @ X2 @ ( one_one @ real ) )
=> ( ord_less @ real @ ( root @ N @ X2 ) @ ( root @ N7 @ X2 ) ) ) ) ) ) ).
% real_root_strict_increasing
thf(fact_5150_real__root__decreasing,axiom,
! [N: nat,N7: nat,X2: real] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ N )
=> ( ( ord_less_eq @ nat @ N @ N7 )
=> ( ( ord_less_eq @ real @ ( one_one @ real ) @ X2 )
=> ( ord_less_eq @ real @ ( root @ N7 @ X2 ) @ ( root @ N @ X2 ) ) ) ) ) ).
% real_root_decreasing
thf(fact_5151_odd__real__root__pow,axiom,
! [N: nat,X2: real] :
( ~ ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N )
=> ( ( power_power @ real @ ( root @ N @ X2 ) @ N )
= X2 ) ) ).
% odd_real_root_pow
thf(fact_5152_odd__real__root__unique,axiom,
! [N: nat,Y3: real,X2: real] :
( ~ ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N )
=> ( ( ( power_power @ real @ Y3 @ N )
= X2 )
=> ( ( root @ N @ X2 )
= Y3 ) ) ) ).
% odd_real_root_unique
thf(fact_5153_odd__real__root__power__cancel,axiom,
! [N: nat,X2: real] :
( ~ ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N )
=> ( ( root @ N @ ( power_power @ real @ X2 @ N ) )
= X2 ) ) ).
% odd_real_root_power_cancel
thf(fact_5154_real__root__pow__pos,axiom,
! [N: nat,X2: real] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ N )
=> ( ( ord_less @ real @ ( zero_zero @ real ) @ X2 )
=> ( ( power_power @ real @ ( root @ N @ X2 ) @ N )
= X2 ) ) ) ).
% real_root_pow_pos
thf(fact_5155_real__root__pos__unique,axiom,
! [N: nat,Y3: real,X2: real] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ N )
=> ( ( ord_less_eq @ real @ ( zero_zero @ real ) @ Y3 )
=> ( ( ( power_power @ real @ Y3 @ N )
= X2 )
=> ( ( root @ N @ X2 )
= Y3 ) ) ) ) ).
% real_root_pos_unique
thf(fact_5156_real__root__power__cancel,axiom,
! [N: nat,X2: real] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ N )
=> ( ( ord_less_eq @ real @ ( zero_zero @ real ) @ X2 )
=> ( ( root @ N @ ( power_power @ real @ X2 @ N ) )
= X2 ) ) ) ).
% real_root_power_cancel
thf(fact_5157_real__root__increasing,axiom,
! [N: nat,N7: nat,X2: real] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ N )
=> ( ( ord_less_eq @ nat @ N @ N7 )
=> ( ( ord_less_eq @ real @ ( zero_zero @ real ) @ X2 )
=> ( ( ord_less_eq @ real @ X2 @ ( one_one @ real ) )
=> ( ord_less_eq @ real @ ( root @ N @ X2 ) @ ( root @ N7 @ X2 ) ) ) ) ) ) ).
% real_root_increasing
thf(fact_5158_log__root,axiom,
! [N: nat,A3: real,B3: real] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ N )
=> ( ( ord_less @ real @ ( zero_zero @ real ) @ A3 )
=> ( ( log @ B3 @ ( root @ N @ A3 ) )
= ( divide_divide @ real @ ( log @ B3 @ A3 ) @ ( semiring_1_of_nat @ real @ N ) ) ) ) ) ).
% log_root
thf(fact_5159_log__base__root,axiom,
! [N: nat,B3: real,X2: real] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ N )
=> ( ( ord_less @ real @ ( zero_zero @ real ) @ B3 )
=> ( ( log @ ( root @ N @ B3 ) @ X2 )
= ( times_times @ real @ ( semiring_1_of_nat @ real @ N ) @ ( log @ B3 @ X2 ) ) ) ) ) ).
% log_base_root
thf(fact_5160_ln__root,axiom,
! [N: nat,B3: real] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ N )
=> ( ( ord_less @ real @ ( zero_zero @ real ) @ B3 )
=> ( ( ln_ln @ real @ ( root @ N @ B3 ) )
= ( divide_divide @ real @ ( ln_ln @ real @ B3 ) @ ( semiring_1_of_nat @ real @ N ) ) ) ) ) ).
% ln_root
thf(fact_5161_and__int__rec,axiom,
( ( bit_se5824344872417868541ns_and @ int )
= ( ^ [K3: int,L3: int] :
( plus_plus @ int
@ ( zero_neq_one_of_bool @ int
@ ( ~ ( dvd_dvd @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ K3 )
& ~ ( dvd_dvd @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ L3 ) ) )
@ ( times_times @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ ( bit_se5824344872417868541ns_and @ int @ ( divide_divide @ int @ K3 @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) ) @ ( divide_divide @ int @ L3 @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) ) ) ) ) ) ) ).
% and_int_rec
thf(fact_5162_root__powr__inverse,axiom,
! [N: nat,X2: real] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ N )
=> ( ( ord_less @ real @ ( zero_zero @ real ) @ X2 )
=> ( ( root @ N @ X2 )
= ( powr @ real @ X2 @ ( divide_divide @ real @ ( one_one @ real ) @ ( semiring_1_of_nat @ real @ N ) ) ) ) ) ) ).
% root_powr_inverse
thf(fact_5163_and__int__unfold,axiom,
( ( bit_se5824344872417868541ns_and @ int )
= ( ^ [K3: int,L3: int] :
( if @ int
@ ( ( K3
= ( zero_zero @ int ) )
| ( L3
= ( zero_zero @ int ) ) )
@ ( zero_zero @ int )
@ ( if @ int
@ ( K3
= ( uminus_uminus @ int @ ( one_one @ int ) ) )
@ L3
@ ( if @ int
@ ( L3
= ( uminus_uminus @ int @ ( one_one @ int ) ) )
@ K3
@ ( plus_plus @ int @ ( times_times @ int @ ( modulo_modulo @ int @ K3 @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) ) @ ( modulo_modulo @ int @ L3 @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) ) ) @ ( times_times @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ ( bit_se5824344872417868541ns_and @ int @ ( divide_divide @ int @ K3 @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) ) @ ( divide_divide @ int @ L3 @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) ) ) ) ) ) ) ) ) ) ).
% and_int_unfold
thf(fact_5164_and__int_Opsimps,axiom,
! [K: int,L: int] :
( ( accp @ ( product_prod @ int @ int ) @ bit_and_int_rel @ ( product_Pair @ int @ int @ K @ L ) )
=> ( ( ( ( member @ int @ K @ ( insert @ int @ ( zero_zero @ int ) @ ( insert @ int @ ( uminus_uminus @ int @ ( one_one @ int ) ) @ ( bot_bot @ ( set @ int ) ) ) ) )
& ( member @ int @ L @ ( insert @ int @ ( zero_zero @ int ) @ ( insert @ int @ ( uminus_uminus @ int @ ( one_one @ int ) ) @ ( bot_bot @ ( set @ int ) ) ) ) ) )
=> ( ( bit_se5824344872417868541ns_and @ int @ K @ L )
= ( uminus_uminus @ int
@ ( zero_neq_one_of_bool @ int
@ ( ~ ( dvd_dvd @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ K )
& ~ ( dvd_dvd @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ L ) ) ) ) ) )
& ( ~ ( ( member @ int @ K @ ( insert @ int @ ( zero_zero @ int ) @ ( insert @ int @ ( uminus_uminus @ int @ ( one_one @ int ) ) @ ( bot_bot @ ( set @ int ) ) ) ) )
& ( member @ int @ L @ ( insert @ int @ ( zero_zero @ int ) @ ( insert @ int @ ( uminus_uminus @ int @ ( one_one @ int ) ) @ ( bot_bot @ ( set @ int ) ) ) ) ) )
=> ( ( bit_se5824344872417868541ns_and @ int @ K @ L )
= ( plus_plus @ int
@ ( zero_neq_one_of_bool @ int
@ ( ~ ( dvd_dvd @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ K )
& ~ ( dvd_dvd @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ L ) ) )
@ ( times_times @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ ( bit_se5824344872417868541ns_and @ int @ ( divide_divide @ int @ K @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) ) @ ( divide_divide @ int @ L @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) ) ) ) ) ) ) ) ) ).
% and_int.psimps
thf(fact_5165_and__int_Opelims,axiom,
! [X2: int,Xa: int,Y3: int] :
( ( ( bit_se5824344872417868541ns_and @ int @ X2 @ Xa )
= Y3 )
=> ( ( accp @ ( product_prod @ int @ int ) @ bit_and_int_rel @ ( product_Pair @ int @ int @ X2 @ Xa ) )
=> ~ ( ( ( ( ( member @ int @ X2 @ ( insert @ int @ ( zero_zero @ int ) @ ( insert @ int @ ( uminus_uminus @ int @ ( one_one @ int ) ) @ ( bot_bot @ ( set @ int ) ) ) ) )
& ( member @ int @ Xa @ ( insert @ int @ ( zero_zero @ int ) @ ( insert @ int @ ( uminus_uminus @ int @ ( one_one @ int ) ) @ ( bot_bot @ ( set @ int ) ) ) ) ) )
=> ( Y3
= ( uminus_uminus @ int
@ ( zero_neq_one_of_bool @ int
@ ( ~ ( dvd_dvd @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ X2 )
& ~ ( dvd_dvd @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ Xa ) ) ) ) ) )
& ( ~ ( ( member @ int @ X2 @ ( insert @ int @ ( zero_zero @ int ) @ ( insert @ int @ ( uminus_uminus @ int @ ( one_one @ int ) ) @ ( bot_bot @ ( set @ int ) ) ) ) )
& ( member @ int @ Xa @ ( insert @ int @ ( zero_zero @ int ) @ ( insert @ int @ ( uminus_uminus @ int @ ( one_one @ int ) ) @ ( bot_bot @ ( set @ int ) ) ) ) ) )
=> ( Y3
= ( plus_plus @ int
@ ( zero_neq_one_of_bool @ int
@ ( ~ ( dvd_dvd @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ X2 )
& ~ ( dvd_dvd @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ Xa ) ) )
@ ( times_times @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ ( bit_se5824344872417868541ns_and @ int @ ( divide_divide @ int @ X2 @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) ) @ ( divide_divide @ int @ Xa @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) ) ) ) ) ) ) )
=> ~ ( accp @ ( product_prod @ int @ int ) @ bit_and_int_rel @ ( product_Pair @ int @ int @ X2 @ Xa ) ) ) ) ) ).
% and_int.pelims
thf(fact_5166_forall__finite_I3_J,axiom,
! [X2: nat,P: nat > $o] :
( ( ! [I4: nat] :
( ( ord_less @ nat @ I4 @ ( suc @ ( suc @ X2 ) ) )
=> ( P @ I4 ) ) )
= ( ( P @ ( zero_zero @ nat ) )
& ! [I4: nat] :
( ( ord_less @ nat @ I4 @ ( suc @ X2 ) )
=> ( P @ ( suc @ I4 ) ) ) ) ) ).
% forall_finite(3)
thf(fact_5167_word__bitwise__m1__simps_I2_J,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [X2: word @ A] :
( ( bit_se5824344872417868541ns_and @ ( word @ A ) @ ( uminus_uminus @ ( word @ A ) @ ( one_one @ ( word @ A ) ) ) @ X2 )
= X2 ) ) ).
% word_bitwise_m1_simps(2)
thf(fact_5168_word__and__max,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [X2: word @ A] :
( ( bit_se5824344872417868541ns_and @ ( word @ A ) @ X2 @ ( uminus_uminus @ ( word @ A ) @ ( one_one @ ( word @ A ) ) ) )
= X2 ) ) ).
% word_and_max
thf(fact_5169_word__plus__and__or,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [X2: word @ A,Y3: word @ A] :
( ( plus_plus @ ( word @ A ) @ ( bit_se5824344872417868541ns_and @ ( word @ A ) @ X2 @ Y3 ) @ ( bit_se1065995026697491101ons_or @ ( word @ A ) @ X2 @ Y3 ) )
= ( plus_plus @ ( word @ A ) @ X2 @ Y3 ) ) ) ).
% word_plus_and_or
thf(fact_5170_and__nat__numerals_I1_J,axiom,
! [Y3: num] :
( ( bit_se5824344872417868541ns_and @ nat @ ( suc @ ( zero_zero @ nat ) ) @ ( numeral_numeral @ nat @ ( bit0 @ Y3 ) ) )
= ( zero_zero @ nat ) ) ).
% and_nat_numerals(1)
thf(fact_5171_and__nat__numerals_I3_J,axiom,
! [X2: num] :
( ( bit_se5824344872417868541ns_and @ nat @ ( numeral_numeral @ nat @ ( bit0 @ X2 ) ) @ ( suc @ ( zero_zero @ nat ) ) )
= ( zero_zero @ nat ) ) ).
% and_nat_numerals(3)
thf(fact_5172_and__nat__numerals_I4_J,axiom,
! [X2: num] :
( ( bit_se5824344872417868541ns_and @ nat @ ( numeral_numeral @ nat @ ( bit1 @ X2 ) ) @ ( suc @ ( zero_zero @ nat ) ) )
= ( one_one @ nat ) ) ).
% and_nat_numerals(4)
thf(fact_5173_and__nat__numerals_I2_J,axiom,
! [Y3: num] :
( ( bit_se5824344872417868541ns_and @ nat @ ( suc @ ( zero_zero @ nat ) ) @ ( numeral_numeral @ nat @ ( bit1 @ Y3 ) ) )
= ( one_one @ nat ) ) ).
% and_nat_numerals(2)
thf(fact_5174_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_5175_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_5176_word__bitwise__1__simps_I2_J,axiom,
! [B: $tType] :
( ( type_len @ B )
=> ! [B3: num] :
( ( bit_se5824344872417868541ns_and @ ( word @ B ) @ ( one_one @ ( word @ B ) ) @ ( numeral_numeral @ ( word @ B ) @ B3 ) )
= ( ring_1_of_int @ ( word @ B ) @ ( bit_se5824344872417868541ns_and @ int @ ( one_one @ int ) @ ( numeral_numeral @ int @ B3 ) ) ) ) ) ).
% word_bitwise_1_simps(2)
thf(fact_5177_word__bitwise__1__simps_I4_J,axiom,
! [D6: $tType] :
( ( type_len @ D6 )
=> ! [A3: num] :
( ( bit_se5824344872417868541ns_and @ ( word @ D6 ) @ ( numeral_numeral @ ( word @ D6 ) @ A3 ) @ ( one_one @ ( word @ D6 ) ) )
= ( ring_1_of_int @ ( word @ D6 ) @ ( bit_se5824344872417868541ns_and @ int @ ( numeral_numeral @ int @ A3 ) @ ( one_one @ int ) ) ) ) ) ).
% word_bitwise_1_simps(4)
thf(fact_5178_word__bitwise__1__simps_I5_J,axiom,
! [E3: $tType] :
( ( type_len @ E3 )
=> ! [A3: num] :
( ( bit_se5824344872417868541ns_and @ ( word @ E3 ) @ ( uminus_uminus @ ( word @ E3 ) @ ( numeral_numeral @ ( word @ E3 ) @ A3 ) ) @ ( one_one @ ( word @ E3 ) ) )
= ( ring_1_of_int @ ( word @ E3 ) @ ( bit_se5824344872417868541ns_and @ int @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ A3 ) ) @ ( one_one @ int ) ) ) ) ) ).
% word_bitwise_1_simps(5)
thf(fact_5179_word__bitwise__1__simps_I3_J,axiom,
! [C: $tType] :
( ( type_len @ C )
=> ! [B3: num] :
( ( bit_se5824344872417868541ns_and @ ( word @ C ) @ ( one_one @ ( word @ C ) ) @ ( uminus_uminus @ ( word @ C ) @ ( numeral_numeral @ ( word @ C ) @ B3 ) ) )
= ( ring_1_of_int @ ( word @ C ) @ ( bit_se5824344872417868541ns_and @ int @ ( one_one @ int ) @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ B3 ) ) ) ) ) ) ).
% word_bitwise_1_simps(3)
thf(fact_5180_word__and__le1,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [Y3: word @ A,A3: word @ A] : ( ord_less_eq @ ( word @ A ) @ ( bit_se5824344872417868541ns_and @ ( word @ A ) @ Y3 @ A3 ) @ A3 ) ) ).
% word_and_le1
thf(fact_5181_word__and__le2,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [A3: word @ A,Y3: word @ A] : ( ord_less_eq @ ( word @ A ) @ ( bit_se5824344872417868541ns_and @ ( word @ A ) @ A3 @ Y3 ) @ A3 ) ) ).
% word_and_le2
thf(fact_5182_word__and__max__word,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [X2: word @ A,A3: word @ A] :
( ( X2
= ( uminus_uminus @ ( word @ A ) @ ( one_one @ ( word @ A ) ) ) )
=> ( ( bit_se5824344872417868541ns_and @ ( word @ A ) @ A3 @ X2 )
= A3 ) ) ) ).
% word_and_max_word
thf(fact_5183_word__plus__and__or__coroll,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [X2: word @ A,Y3: word @ A] :
( ( ( bit_se5824344872417868541ns_and @ ( word @ A ) @ X2 @ Y3 )
= ( zero_zero @ ( word @ A ) ) )
=> ( ( plus_plus @ ( word @ A ) @ X2 @ Y3 )
= ( bit_se1065995026697491101ons_or @ ( word @ A ) @ X2 @ Y3 ) ) ) ) ).
% word_plus_and_or_coroll
thf(fact_5184_even__word__iff,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ( ( even_word @ A )
= ( ^ [A6: word @ A] :
( ( bit_se5824344872417868541ns_and @ ( word @ A ) @ A6 @ ( one_one @ ( word @ A ) ) )
= ( zero_zero @ ( word @ A ) ) ) ) ) ) ).
% even_word_iff
thf(fact_5185_forall__finite_I1_J,axiom,
! [P: nat > $o,I5: nat] :
( ( ord_less @ nat @ I5 @ ( zero_zero @ nat ) )
=> ( P @ I5 ) ) ).
% forall_finite(1)
thf(fact_5186_and__nat__unfold,axiom,
( ( bit_se5824344872417868541ns_and @ nat )
= ( ^ [M6: nat,N4: nat] :
( if @ nat
@ ( ( M6
= ( zero_zero @ nat ) )
| ( N4
= ( zero_zero @ nat ) ) )
@ ( zero_zero @ nat )
@ ( plus_plus @ nat @ ( times_times @ nat @ ( modulo_modulo @ nat @ M6 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ ( modulo_modulo @ nat @ N4 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( bit_se5824344872417868541ns_and @ nat @ ( divide_divide @ nat @ M6 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ ( divide_divide @ nat @ N4 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) ) ) ) ).
% and_nat_unfold
thf(fact_5187_and__nat__rec,axiom,
( ( bit_se5824344872417868541ns_and @ nat )
= ( ^ [M6: nat,N4: nat] :
( plus_plus @ nat
@ ( zero_neq_one_of_bool @ nat
@ ( ~ ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ M6 )
& ~ ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N4 ) ) )
@ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( bit_se5824344872417868541ns_and @ nat @ ( divide_divide @ nat @ M6 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ ( divide_divide @ nat @ N4 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) ) ) ).
% and_nat_rec
thf(fact_5188_and__int_Opinduct,axiom,
! [A0: int,A12: int,P: int > int > $o] :
( ( accp @ ( product_prod @ int @ int ) @ bit_and_int_rel @ ( product_Pair @ int @ int @ A0 @ A12 ) )
=> ( ! [K2: int,L2: int] :
( ( accp @ ( product_prod @ int @ int ) @ bit_and_int_rel @ ( product_Pair @ int @ int @ K2 @ L2 ) )
=> ( ( ~ ( ( member @ int @ K2 @ ( insert @ int @ ( zero_zero @ int ) @ ( insert @ int @ ( uminus_uminus @ int @ ( one_one @ int ) ) @ ( bot_bot @ ( set @ int ) ) ) ) )
& ( member @ int @ L2 @ ( insert @ int @ ( zero_zero @ int ) @ ( insert @ int @ ( uminus_uminus @ int @ ( one_one @ int ) ) @ ( bot_bot @ ( set @ int ) ) ) ) ) )
=> ( P @ ( divide_divide @ int @ K2 @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) ) @ ( divide_divide @ int @ L2 @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) ) ) )
=> ( P @ K2 @ L2 ) ) )
=> ( P @ A0 @ A12 ) ) ) ).
% and_int.pinduct
thf(fact_5189_Comparator__Generator_OAll__less__Suc,axiom,
! [X2: nat,P: nat > $o] :
( ( ! [I4: nat] :
( ( ord_less @ nat @ I4 @ ( suc @ X2 ) )
=> ( P @ I4 ) ) )
= ( ( P @ ( zero_zero @ nat ) )
& ! [I4: nat] :
( ( ord_less @ nat @ I4 @ X2 )
=> ( P @ ( suc @ I4 ) ) ) ) ) ).
% Comparator_Generator.All_less_Suc
thf(fact_5190_forall__finite_I2_J,axiom,
! [P: nat > $o] :
( ( ! [I4: nat] :
( ( ord_less @ nat @ I4 @ ( suc @ ( zero_zero @ nat ) ) )
=> ( P @ I4 ) ) )
= ( P @ ( zero_zero @ nat ) ) ) ).
% forall_finite(2)
thf(fact_5191_upto_Opinduct,axiom,
! [A0: int,A12: int,P: int > int > $o] :
( ( accp @ ( product_prod @ int @ int ) @ upto_rel @ ( product_Pair @ int @ int @ A0 @ A12 ) )
=> ( ! [I2: int,J2: int] :
( ( accp @ ( product_prod @ int @ int ) @ upto_rel @ ( product_Pair @ int @ int @ I2 @ J2 ) )
=> ( ( ( ord_less_eq @ int @ I2 @ J2 )
=> ( P @ ( plus_plus @ int @ I2 @ ( one_one @ int ) ) @ J2 ) )
=> ( P @ I2 @ J2 ) ) )
=> ( P @ A0 @ A12 ) ) ) ).
% upto.pinduct
thf(fact_5192_take__bit__word__Bit1__eq,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [N: num,M: num] :
( ( bit_se2584673776208193580ke_bit @ ( word @ A ) @ ( numeral_numeral @ nat @ N ) @ ( numeral_numeral @ ( word @ A ) @ ( bit1 @ M ) ) )
= ( plus_plus @ ( word @ A ) @ ( one_one @ ( word @ A ) ) @ ( times_times @ ( word @ A ) @ ( numeral_numeral @ ( word @ A ) @ ( bit0 @ one2 ) ) @ ( bit_se2584673776208193580ke_bit @ ( word @ A ) @ ( pred_numeral @ N ) @ ( numeral_numeral @ ( word @ A ) @ M ) ) ) ) ) ) ).
% take_bit_word_Bit1_eq
thf(fact_5193_uint32_Osize__eq,axiom,
( ( size_size @ uint32 )
= ( ^ [P4: uint32] : ( numeral_numeral @ nat @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ one2 ) ) ) ) ) ) ) ) ).
% uint32.size_eq
thf(fact_5194_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_5195_take__bit__numeral__1,axiom,
! [A: $tType] :
( ( bit_un5681908812861735899ations @ A )
=> ! [L: num] :
( ( bit_se2584673776208193580ke_bit @ A @ ( numeral_numeral @ nat @ L ) @ ( one_one @ A ) )
= ( one_one @ A ) ) ) ).
% take_bit_numeral_1
thf(fact_5196_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_5197_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_5198_even__take__bit__eq,axiom,
! [A: $tType] :
( ( bit_se359711467146920520ations @ A )
=> ! [N: nat,A3: A] :
( ( dvd_dvd @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ ( bit_se2584673776208193580ke_bit @ A @ N @ A3 ) )
= ( ( N
= ( zero_zero @ nat ) )
| ( dvd_dvd @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ A3 ) ) ) ) ).
% even_take_bit_eq
thf(fact_5199_take__bit__Suc__0,axiom,
! [A: $tType] :
( ( bit_se359711467146920520ations @ A )
=> ! [A3: A] :
( ( bit_se2584673776208193580ke_bit @ A @ ( suc @ ( zero_zero @ nat ) ) @ A3 )
= ( modulo_modulo @ A @ A3 @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) ) ) ).
% take_bit_Suc_0
thf(fact_5200_take__bit__word__Bit0__eq,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [N: num,M: num] :
( ( bit_se2584673776208193580ke_bit @ ( word @ A ) @ ( numeral_numeral @ nat @ N ) @ ( numeral_numeral @ ( word @ A ) @ ( bit0 @ M ) ) )
= ( times_times @ ( word @ A ) @ ( numeral_numeral @ ( word @ A ) @ ( bit0 @ one2 ) ) @ ( bit_se2584673776208193580ke_bit @ ( word @ A ) @ ( pred_numeral @ N ) @ ( numeral_numeral @ ( word @ A ) @ M ) ) ) ) ) ).
% take_bit_word_Bit0_eq
thf(fact_5201_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_5202_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_5203_take__bit__word__minus__Bit0__eq,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [N: num,M: num] :
( ( bit_se2584673776208193580ke_bit @ ( word @ A ) @ ( numeral_numeral @ nat @ N ) @ ( uminus_uminus @ ( word @ A ) @ ( numeral_numeral @ ( word @ A ) @ ( bit0 @ M ) ) ) )
= ( times_times @ ( word @ A ) @ ( numeral_numeral @ ( word @ A ) @ ( bit0 @ one2 ) ) @ ( bit_se2584673776208193580ke_bit @ ( word @ A ) @ ( pred_numeral @ N ) @ ( uminus_uminus @ ( word @ A ) @ ( numeral_numeral @ ( word @ A ) @ M ) ) ) ) ) ) ).
% take_bit_word_minus_Bit0_eq
thf(fact_5204_take__bit__add,axiom,
! [A: $tType] :
( ( bit_un5681908812861735899ations @ A )
=> ! [N: nat,A3: A,B3: A] :
( ( bit_se2584673776208193580ke_bit @ A @ N @ ( plus_plus @ A @ ( bit_se2584673776208193580ke_bit @ A @ N @ A3 ) @ ( bit_se2584673776208193580ke_bit @ A @ N @ B3 ) ) )
= ( bit_se2584673776208193580ke_bit @ A @ N @ ( plus_plus @ A @ A3 @ B3 ) ) ) ) ).
% take_bit_add
thf(fact_5205_take__bit__tightened,axiom,
! [A: $tType] :
( ( bit_se359711467146920520ations @ A )
=> ! [N: nat,A3: A,B3: A,M: nat] :
( ( ( bit_se2584673776208193580ke_bit @ A @ N @ A3 )
= ( bit_se2584673776208193580ke_bit @ A @ N @ B3 ) )
=> ( ( ord_less_eq @ nat @ M @ N )
=> ( ( bit_se2584673776208193580ke_bit @ A @ M @ A3 )
= ( bit_se2584673776208193580ke_bit @ A @ M @ B3 ) ) ) ) ) ).
% take_bit_tightened
thf(fact_5206_signed__take__bit__eq__iff__take__bit__eq,axiom,
! [A: $tType] :
( ( bit_ri3973907225187159222ations @ A )
=> ! [N: nat,A3: A,B3: A] :
( ( ( bit_ri4674362597316999326ke_bit @ A @ N @ A3 )
= ( bit_ri4674362597316999326ke_bit @ A @ N @ B3 ) )
= ( ( bit_se2584673776208193580ke_bit @ A @ ( suc @ N ) @ A3 )
= ( bit_se2584673776208193580ke_bit @ A @ ( suc @ N ) @ B3 ) ) ) ) ).
% signed_take_bit_eq_iff_take_bit_eq
thf(fact_5207_signed__take__bit__take__bit,axiom,
! [A: $tType] :
( ( bit_ri3973907225187159222ations @ A )
=> ! [M: nat,N: nat,A3: A] :
( ( bit_ri4674362597316999326ke_bit @ A @ M @ ( bit_se2584673776208193580ke_bit @ A @ N @ A3 ) )
= ( if @ ( A > A ) @ ( ord_less_eq @ nat @ N @ M ) @ ( bit_se2584673776208193580ke_bit @ A @ N ) @ ( bit_ri4674362597316999326ke_bit @ A @ M ) @ A3 ) ) ) ).
% signed_take_bit_take_bit
thf(fact_5208_take__bit__unset__bit__eq,axiom,
! [A: $tType] :
( ( bit_se359711467146920520ations @ A )
=> ! [N: nat,M: nat,A3: A] :
( ( ( ord_less_eq @ nat @ N @ M )
=> ( ( bit_se2584673776208193580ke_bit @ A @ N @ ( bit_se2638667681897837118et_bit @ A @ M @ A3 ) )
= ( bit_se2584673776208193580ke_bit @ A @ N @ A3 ) ) )
& ( ~ ( ord_less_eq @ nat @ N @ M )
=> ( ( bit_se2584673776208193580ke_bit @ A @ N @ ( bit_se2638667681897837118et_bit @ A @ M @ A3 ) )
= ( bit_se2638667681897837118et_bit @ A @ M @ ( bit_se2584673776208193580ke_bit @ A @ N @ A3 ) ) ) ) ) ) ).
% take_bit_unset_bit_eq
thf(fact_5209_take__bit__set__bit__eq,axiom,
! [A: $tType] :
( ( bit_se359711467146920520ations @ A )
=> ! [N: nat,M: nat,A3: A] :
( ( ( ord_less_eq @ nat @ N @ M )
=> ( ( bit_se2584673776208193580ke_bit @ A @ N @ ( bit_se5668285175392031749et_bit @ A @ M @ A3 ) )
= ( bit_se2584673776208193580ke_bit @ A @ N @ A3 ) ) )
& ( ~ ( ord_less_eq @ nat @ N @ M )
=> ( ( bit_se2584673776208193580ke_bit @ A @ N @ ( bit_se5668285175392031749et_bit @ A @ M @ A3 ) )
= ( bit_se5668285175392031749et_bit @ A @ M @ ( bit_se2584673776208193580ke_bit @ A @ N @ A3 ) ) ) ) ) ) ).
% take_bit_set_bit_eq
thf(fact_5210_take__bit__flip__bit__eq,axiom,
! [A: $tType] :
( ( bit_se359711467146920520ations @ A )
=> ! [N: nat,M: nat,A3: A] :
( ( ( ord_less_eq @ nat @ N @ M )
=> ( ( bit_se2584673776208193580ke_bit @ A @ N @ ( bit_se8732182000553998342ip_bit @ A @ M @ A3 ) )
= ( bit_se2584673776208193580ke_bit @ A @ N @ A3 ) ) )
& ( ~ ( ord_less_eq @ nat @ N @ M )
=> ( ( bit_se2584673776208193580ke_bit @ A @ N @ ( bit_se8732182000553998342ip_bit @ A @ M @ A3 ) )
= ( bit_se8732182000553998342ip_bit @ A @ M @ ( bit_se2584673776208193580ke_bit @ A @ N @ A3 ) ) ) ) ) ) ).
% take_bit_flip_bit_eq
thf(fact_5211_take__bit__signed__take__bit,axiom,
! [A: $tType] :
( ( bit_ri3973907225187159222ations @ A )
=> ! [M: nat,N: nat,A3: A] :
( ( ord_less_eq @ nat @ M @ ( suc @ N ) )
=> ( ( bit_se2584673776208193580ke_bit @ A @ M @ ( bit_ri4674362597316999326ke_bit @ A @ N @ A3 ) )
= ( bit_se2584673776208193580ke_bit @ A @ M @ A3 ) ) ) ) ).
% take_bit_signed_take_bit
thf(fact_5212_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_5213_take__bit__eq__mod,axiom,
! [A: $tType] :
( ( bit_se359711467146920520ations @ A )
=> ( ( bit_se2584673776208193580ke_bit @ A )
= ( ^ [N4: nat,A6: A] : ( modulo_modulo @ A @ A6 @ ( power_power @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ N4 ) ) ) ) ) ).
% take_bit_eq_mod
thf(fact_5214_take__bit__eq__0__iff,axiom,
! [A: $tType] :
( ( bit_se359711467146920520ations @ A )
=> ! [N: nat,A3: A] :
( ( ( bit_se2584673776208193580ke_bit @ A @ N @ A3 )
= ( zero_zero @ A ) )
= ( dvd_dvd @ A @ ( power_power @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ N ) @ A3 ) ) ) ).
% take_bit_eq_0_iff
thf(fact_5215_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_5216_take__bit__numeral__bit0,axiom,
! [A: $tType] :
( ( bit_un5681908812861735899ations @ A )
=> ! [L: num,K: num] :
( ( bit_se2584673776208193580ke_bit @ A @ ( numeral_numeral @ nat @ L ) @ ( numeral_numeral @ A @ ( bit0 @ K ) ) )
= ( times_times @ A @ ( bit_se2584673776208193580ke_bit @ A @ ( pred_numeral @ L ) @ ( numeral_numeral @ A @ K ) ) @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) ) ) ).
% take_bit_numeral_bit0
thf(fact_5217_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_5218_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_5219_take__bit__numeral__minus__1__eq,axiom,
! [A: $tType] :
( ( bit_ri3973907225187159222ations @ A )
=> ! [K: num] :
( ( bit_se2584673776208193580ke_bit @ A @ ( numeral_numeral @ nat @ K ) @ ( uminus_uminus @ A @ ( one_one @ A ) ) )
= ( minus_minus @ A @ ( power_power @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ ( numeral_numeral @ nat @ K ) ) @ ( one_one @ A ) ) ) ) ).
% take_bit_numeral_minus_1_eq
thf(fact_5220_take__bit__Suc,axiom,
! [A: $tType] :
( ( bit_se359711467146920520ations @ A )
=> ! [N: nat,A3: A] :
( ( bit_se2584673776208193580ke_bit @ A @ ( suc @ N ) @ A3 )
= ( plus_plus @ A @ ( times_times @ A @ ( bit_se2584673776208193580ke_bit @ A @ N @ ( divide_divide @ A @ A3 @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) ) @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) @ ( modulo_modulo @ A @ A3 @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) ) ) ) ).
% take_bit_Suc
thf(fact_5221_stable__imp__take__bit__eq,axiom,
! [A: $tType] :
( ( bit_se359711467146920520ations @ A )
=> ! [A3: A,N: nat] :
( ( ( divide_divide @ A @ A3 @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) )
= A3 )
=> ( ( ( dvd_dvd @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ A3 )
=> ( ( bit_se2584673776208193580ke_bit @ A @ N @ A3 )
= ( zero_zero @ A ) ) )
& ( ~ ( dvd_dvd @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ A3 )
=> ( ( bit_se2584673776208193580ke_bit @ A @ N @ A3 )
= ( minus_minus @ A @ ( power_power @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ N ) @ ( one_one @ A ) ) ) ) ) ) ) ).
% stable_imp_take_bit_eq
thf(fact_5222_take__bit__numeral__bit1,axiom,
! [A: $tType] :
( ( bit_un5681908812861735899ations @ A )
=> ! [L: num,K: num] :
( ( bit_se2584673776208193580ke_bit @ A @ ( numeral_numeral @ nat @ L ) @ ( numeral_numeral @ A @ ( bit1 @ K ) ) )
= ( plus_plus @ A @ ( times_times @ A @ ( bit_se2584673776208193580ke_bit @ A @ ( pred_numeral @ L ) @ ( numeral_numeral @ A @ K ) ) @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) @ ( one_one @ A ) ) ) ) ).
% take_bit_numeral_bit1
thf(fact_5223_take__bit__rec,axiom,
! [A: $tType] :
( ( bit_se359711467146920520ations @ A )
=> ( ( bit_se2584673776208193580ke_bit @ A )
= ( ^ [N4: nat,A6: A] :
( if @ A
@ ( N4
= ( zero_zero @ nat ) )
@ ( zero_zero @ A )
@ ( plus_plus @ A @ ( times_times @ A @ ( bit_se2584673776208193580ke_bit @ A @ ( minus_minus @ nat @ N4 @ ( one_one @ nat ) ) @ ( divide_divide @ A @ A6 @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) ) @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) @ ( modulo_modulo @ A @ A6 @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) ) ) ) ) ) ).
% take_bit_rec
thf(fact_5224_take__bit__word__minus__Bit1__eq,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [N: num,M: num] :
( ( bit_se2584673776208193580ke_bit @ ( word @ A ) @ ( numeral_numeral @ nat @ N ) @ ( uminus_uminus @ ( word @ A ) @ ( numeral_numeral @ ( word @ A ) @ ( bit1 @ M ) ) ) )
= ( plus_plus @ ( word @ A ) @ ( one_one @ ( word @ A ) ) @ ( times_times @ ( word @ A ) @ ( numeral_numeral @ ( word @ A ) @ ( bit0 @ one2 ) ) @ ( bit_se2584673776208193580ke_bit @ ( word @ A ) @ ( pred_numeral @ N ) @ ( uminus_uminus @ ( word @ A ) @ ( numeral_numeral @ ( word @ A ) @ ( inc @ M ) ) ) ) ) ) ) ) ).
% take_bit_word_minus_Bit1_eq
thf(fact_5225_and__mask__arith_H,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [N: nat,W: word @ A] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ N )
=> ( ( bit_se5824344872417868541ns_and @ ( word @ A ) @ W @ ( bit_se2239418461657761734s_mask @ ( word @ A ) @ N ) )
= ( divide_divide @ ( word @ A ) @ ( times_times @ ( word @ A ) @ W @ ( power_power @ ( word @ A ) @ ( numeral_numeral @ ( word @ A ) @ ( bit0 @ one2 ) ) @ ( minus_minus @ nat @ ( size_size @ ( word @ A ) @ W ) @ N ) ) ) @ ( power_power @ ( word @ A ) @ ( numeral_numeral @ ( word @ A ) @ ( bit0 @ one2 ) ) @ ( minus_minus @ nat @ ( size_size @ ( word @ A ) @ W ) @ N ) ) ) ) ) ) ).
% and_mask_arith'
thf(fact_5226_divmod__algorithm__code_I6_J,axiom,
! [A: $tType] :
( ( unique1627219031080169319umeral @ A )
=> ! [M: num,N: num] :
( ( unique8689654367752047608divmod @ A @ ( bit1 @ M ) @ ( bit0 @ N ) )
= ( product_case_prod @ A @ A @ ( product_prod @ A @ A )
@ ^ [Q5: A,R5: A] : ( product_Pair @ A @ A @ Q5 @ ( plus_plus @ A @ ( times_times @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ R5 ) @ ( one_one @ A ) ) )
@ ( unique8689654367752047608divmod @ A @ M @ N ) ) ) ) ).
% divmod_algorithm_code(6)
thf(fact_5227_case__prod__conv,axiom,
! [B: $tType,A: $tType,C: $tType,F2: B > C > A,A3: B,B3: C] :
( ( product_case_prod @ B @ C @ A @ F2 @ ( product_Pair @ B @ C @ A3 @ B3 ) )
= ( F2 @ A3 @ B3 ) ) ).
% case_prod_conv
thf(fact_5228_pred__numeral__inc,axiom,
! [K: num] :
( ( pred_numeral @ ( inc @ K ) )
= ( numeral_numeral @ nat @ K ) ) ).
% pred_numeral_inc
thf(fact_5229_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_5230_take__bit__minus__one__eq__mask,axiom,
! [A: $tType] :
( ( bit_ri3973907225187159222ations @ A )
=> ! [N: nat] :
( ( bit_se2584673776208193580ke_bit @ A @ N @ ( uminus_uminus @ A @ ( one_one @ A ) ) )
= ( bit_se2239418461657761734s_mask @ A @ N ) ) ) ).
% take_bit_minus_one_eq_mask
thf(fact_5231_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_5232_add__neg__numeral__special_I6_J,axiom,
! [A: $tType] :
( ( neg_numeral @ A )
=> ! [M: num] :
( ( plus_plus @ A @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ M ) ) @ ( uminus_uminus @ A @ ( one_one @ A ) ) )
= ( uminus_uminus @ A @ ( numeral_numeral @ A @ ( inc @ M ) ) ) ) ) ).
% add_neg_numeral_special(6)
thf(fact_5233_add__neg__numeral__special_I5_J,axiom,
! [A: $tType] :
( ( neg_numeral @ A )
=> ! [N: num] :
( ( plus_plus @ A @ ( uminus_uminus @ A @ ( one_one @ A ) ) @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ N ) ) )
= ( uminus_uminus @ A @ ( numeral_numeral @ A @ ( inc @ N ) ) ) ) ) ).
% add_neg_numeral_special(5)
thf(fact_5234_diff__numeral__special_I6_J,axiom,
! [A: $tType] :
( ( neg_numeral @ A )
=> ! [M: num] :
( ( minus_minus @ A @ ( numeral_numeral @ A @ M ) @ ( uminus_uminus @ A @ ( one_one @ A ) ) )
= ( numeral_numeral @ A @ ( inc @ M ) ) ) ) ).
% diff_numeral_special(6)
thf(fact_5235_diff__numeral__special_I5_J,axiom,
! [A: $tType] :
( ( neg_numeral @ A )
=> ! [N: num] :
( ( minus_minus @ A @ ( uminus_uminus @ A @ ( one_one @ A ) ) @ ( numeral_numeral @ A @ N ) )
= ( uminus_uminus @ A @ ( numeral_numeral @ A @ ( inc @ N ) ) ) ) ) ).
% diff_numeral_special(5)
thf(fact_5236_divmod__algorithm__code_I5_J,axiom,
! [A: $tType] :
( ( unique1627219031080169319umeral @ A )
=> ! [M: num,N: num] :
( ( unique8689654367752047608divmod @ A @ ( bit0 @ M ) @ ( bit0 @ N ) )
= ( product_case_prod @ A @ A @ ( product_prod @ A @ A )
@ ^ [Q5: A,R5: A] : ( product_Pair @ A @ A @ Q5 @ ( times_times @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ R5 ) )
@ ( unique8689654367752047608divmod @ A @ M @ N ) ) ) ) ).
% divmod_algorithm_code(5)
thf(fact_5237_take__bit__diff,axiom,
! [N: nat,K: int,L: int] :
( ( bit_se2584673776208193580ke_bit @ int @ N @ ( minus_minus @ int @ ( bit_se2584673776208193580ke_bit @ int @ N @ K ) @ ( bit_se2584673776208193580ke_bit @ int @ N @ L ) ) )
= ( bit_se2584673776208193580ke_bit @ int @ N @ ( minus_minus @ int @ K @ L ) ) ) ).
% take_bit_diff
thf(fact_5238_mask__bin,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ( ( bit_se2239418461657761734s_mask @ ( word @ A ) )
= ( ^ [N4: nat] : ( ring_1_of_int @ ( word @ A ) @ ( bit_se2584673776208193580ke_bit @ int @ N4 @ ( uminus_uminus @ int @ ( one_one @ int ) ) ) ) ) ) ) ).
% mask_bin
thf(fact_5239_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_5240_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_5241_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_5242_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_5243_case__prodE2,axiom,
! [B: $tType,A: $tType,C: $tType,Q: A > $o,P: B > C > A,Z: product_prod @ B @ C] :
( ( Q @ ( product_case_prod @ B @ C @ A @ P @ Z ) )
=> ~ ! [X4: B,Y4: C] :
( ( Z
= ( product_Pair @ B @ C @ X4 @ Y4 ) )
=> ~ ( Q @ ( P @ X4 @ Y4 ) ) ) ) ).
% case_prodE2
thf(fact_5244_case__prod__eta,axiom,
! [C: $tType,B: $tType,A: $tType,F2: ( product_prod @ A @ B ) > C] :
( ( product_case_prod @ A @ B @ C
@ ^ [X3: A,Y: B] : ( F2 @ ( product_Pair @ A @ B @ X3 @ Y ) ) )
= F2 ) ).
% case_prod_eta
thf(fact_5245_cond__case__prod__eta,axiom,
! [C: $tType,B: $tType,A: $tType,F2: A > B > C,G: ( product_prod @ A @ B ) > C] :
( ! [X4: A,Y4: B] :
( ( F2 @ X4 @ Y4 )
= ( G @ ( product_Pair @ A @ B @ X4 @ Y4 ) ) )
=> ( ( product_case_prod @ A @ B @ C @ F2 )
= G ) ) ).
% cond_case_prod_eta
thf(fact_5246_num__induct,axiom,
! [P: num > $o,X2: num] :
( ( P @ one2 )
=> ( ! [X4: num] :
( ( P @ X4 )
=> ( P @ ( inc @ X4 ) ) )
=> ( P @ X2 ) ) ) ).
% num_induct
thf(fact_5247_old_Oprod_Ocase,axiom,
! [A: $tType,C: $tType,B: $tType,F2: A > B > C,X1: A,X22: B] :
( ( product_case_prod @ A @ B @ C @ F2 @ ( product_Pair @ A @ B @ X1 @ X22 ) )
= ( F2 @ X1 @ X22 ) ) ).
% old.prod.case
thf(fact_5248_prod_Ocase__distrib,axiom,
! [C: $tType,D6: $tType,B: $tType,A: $tType,H2: C > D6,F2: A > B > C,Prod: product_prod @ A @ B] :
( ( H2 @ ( product_case_prod @ A @ B @ C @ F2 @ Prod ) )
= ( product_case_prod @ A @ B @ D6
@ ^ [X15: A,X24: B] : ( H2 @ ( F2 @ X15 @ X24 ) )
@ Prod ) ) ).
% prod.case_distrib
thf(fact_5249_ex__mask__1,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ? [X4: nat] :
( ( bit_se2239418461657761734s_mask @ ( word @ A ) @ X4 )
= ( one_one @ ( word @ A ) ) ) ) ).
% ex_mask_1
thf(fact_5250_nested__case__prod__simp,axiom,
! [A: $tType,D6: $tType,C: $tType,B: $tType] :
( ( product_case_prod @ B @ C @ ( D6 > A ) )
= ( ^ [F5: B > C > D6 > A,X3: product_prod @ B @ C,Y: D6] :
( product_case_prod @ B @ C @ A
@ ^ [A6: B,B7: C] : ( F5 @ A6 @ B7 @ Y )
@ X3 ) ) ) ).
% nested_case_prod_simp
thf(fact_5251_add__inc,axiom,
! [X2: num,Y3: num] :
( ( plus_plus @ num @ X2 @ ( inc @ Y3 ) )
= ( inc @ ( plus_plus @ num @ X2 @ Y3 ) ) ) ).
% add_inc
thf(fact_5252_mask__twice2,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [N: nat,M: nat,X2: word @ A] :
( ( ord_less_eq @ nat @ N @ M )
=> ( ( bit_se5824344872417868541ns_and @ ( word @ A ) @ ( bit_se5824344872417868541ns_and @ ( word @ A ) @ X2 @ ( bit_se2239418461657761734s_mask @ ( word @ A ) @ M ) ) @ ( bit_se2239418461657761734s_mask @ ( word @ A ) @ N ) )
= ( bit_se5824344872417868541ns_and @ ( word @ A ) @ X2 @ ( bit_se2239418461657761734s_mask @ ( word @ A ) @ N ) ) ) ) ) ).
% mask_twice2
thf(fact_5253_le__mask__imp__and__mask,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [X2: word @ A,N: nat] :
( ( ord_less_eq @ ( word @ A ) @ X2 @ ( bit_se2239418461657761734s_mask @ ( word @ A ) @ N ) )
=> ( ( bit_se5824344872417868541ns_and @ ( word @ A ) @ X2 @ ( bit_se2239418461657761734s_mask @ ( word @ A ) @ N ) )
= X2 ) ) ) ).
% le_mask_imp_and_mask
thf(fact_5254_and__mask__eq__iff__le__mask,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [W: word @ A,N: nat] :
( ( ( bit_se5824344872417868541ns_and @ ( word @ A ) @ W @ ( bit_se2239418461657761734s_mask @ ( word @ A ) @ N ) )
= W )
= ( ord_less_eq @ ( word @ A ) @ W @ ( bit_se2239418461657761734s_mask @ ( word @ A ) @ N ) ) ) ) ).
% and_mask_eq_iff_le_mask
thf(fact_5255_mask__eqs_I7_J,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [A3: word @ A,N: nat,B3: word @ A] :
( ( bit_se5824344872417868541ns_and @ ( word @ A ) @ ( plus_plus @ ( word @ A ) @ ( bit_se5824344872417868541ns_and @ ( word @ A ) @ A3 @ ( bit_se2239418461657761734s_mask @ ( word @ A ) @ N ) ) @ ( bit_se5824344872417868541ns_and @ ( word @ A ) @ B3 @ ( bit_se2239418461657761734s_mask @ ( word @ A ) @ N ) ) ) @ ( bit_se2239418461657761734s_mask @ ( word @ A ) @ N ) )
= ( bit_se5824344872417868541ns_and @ ( word @ A ) @ ( plus_plus @ ( word @ A ) @ A3 @ B3 ) @ ( bit_se2239418461657761734s_mask @ ( word @ A ) @ N ) ) ) ) ).
% mask_eqs(7)
thf(fact_5256_mask__eqs_I2_J,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [A3: word @ A,B3: word @ A,N: nat] :
( ( bit_se5824344872417868541ns_and @ ( word @ A ) @ ( plus_plus @ ( word @ A ) @ A3 @ ( bit_se5824344872417868541ns_and @ ( word @ A ) @ B3 @ ( bit_se2239418461657761734s_mask @ ( word @ A ) @ N ) ) ) @ ( bit_se2239418461657761734s_mask @ ( word @ A ) @ N ) )
= ( bit_se5824344872417868541ns_and @ ( word @ A ) @ ( plus_plus @ ( word @ A ) @ A3 @ B3 ) @ ( bit_se2239418461657761734s_mask @ ( word @ A ) @ N ) ) ) ) ).
% mask_eqs(2)
thf(fact_5257_mask__eqs_I1_J,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [A3: word @ A,N: nat,B3: word @ A] :
( ( bit_se5824344872417868541ns_and @ ( word @ A ) @ ( plus_plus @ ( word @ A ) @ ( bit_se5824344872417868541ns_and @ ( word @ A ) @ A3 @ ( bit_se2239418461657761734s_mask @ ( word @ A ) @ N ) ) @ B3 ) @ ( bit_se2239418461657761734s_mask @ ( word @ A ) @ N ) )
= ( bit_se5824344872417868541ns_and @ ( word @ A ) @ ( plus_plus @ ( word @ A ) @ A3 @ B3 ) @ ( bit_se2239418461657761734s_mask @ ( word @ A ) @ N ) ) ) ) ).
% mask_eqs(1)
thf(fact_5258_mask__eqs_I8_J,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [A3: word @ A,N: nat,B3: word @ A] :
( ( bit_se5824344872417868541ns_and @ ( word @ A ) @ ( minus_minus @ ( word @ A ) @ ( bit_se5824344872417868541ns_and @ ( word @ A ) @ A3 @ ( bit_se2239418461657761734s_mask @ ( word @ A ) @ N ) ) @ ( bit_se5824344872417868541ns_and @ ( word @ A ) @ B3 @ ( bit_se2239418461657761734s_mask @ ( word @ A ) @ N ) ) ) @ ( bit_se2239418461657761734s_mask @ ( word @ A ) @ N ) )
= ( bit_se5824344872417868541ns_and @ ( word @ A ) @ ( minus_minus @ ( word @ A ) @ A3 @ B3 ) @ ( bit_se2239418461657761734s_mask @ ( word @ A ) @ N ) ) ) ) ).
% mask_eqs(8)
thf(fact_5259_mask__eqs_I4_J,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [A3: word @ A,B3: word @ A,N: nat] :
( ( bit_se5824344872417868541ns_and @ ( word @ A ) @ ( minus_minus @ ( word @ A ) @ A3 @ ( bit_se5824344872417868541ns_and @ ( word @ A ) @ B3 @ ( bit_se2239418461657761734s_mask @ ( word @ A ) @ N ) ) ) @ ( bit_se2239418461657761734s_mask @ ( word @ A ) @ N ) )
= ( bit_se5824344872417868541ns_and @ ( word @ A ) @ ( minus_minus @ ( word @ A ) @ A3 @ B3 ) @ ( bit_se2239418461657761734s_mask @ ( word @ A ) @ N ) ) ) ) ).
% mask_eqs(4)
thf(fact_5260_mask__eqs_I3_J,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [A3: word @ A,N: nat,B3: word @ A] :
( ( bit_se5824344872417868541ns_and @ ( word @ A ) @ ( minus_minus @ ( word @ A ) @ ( bit_se5824344872417868541ns_and @ ( word @ A ) @ A3 @ ( bit_se2239418461657761734s_mask @ ( word @ A ) @ N ) ) @ B3 ) @ ( bit_se2239418461657761734s_mask @ ( word @ A ) @ N ) )
= ( bit_se5824344872417868541ns_and @ ( word @ A ) @ ( minus_minus @ ( word @ A ) @ A3 @ B3 ) @ ( bit_se2239418461657761734s_mask @ ( word @ A ) @ N ) ) ) ) ).
% mask_eqs(3)
thf(fact_5261_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_5262_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_5263_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_5264_less__eq__mask__iff__take__bit__eq__self,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [W: word @ A,N: nat] :
( ( ord_less_eq @ ( word @ A ) @ W @ ( bit_se2239418461657761734s_mask @ ( word @ A ) @ N ) )
= ( ( bit_se2584673776208193580ke_bit @ ( word @ A ) @ N @ W )
= W ) ) ) ).
% less_eq_mask_iff_take_bit_eq_self
thf(fact_5265_mask__1,axiom,
! [A: $tType] :
( ( bit_se359711467146920520ations @ A )
=> ( ( bit_se2239418461657761734s_mask @ A @ ( one_one @ nat ) )
= ( one_one @ A ) ) ) ).
% mask_1
thf(fact_5266_inc_Osimps_I1_J,axiom,
( ( inc @ one2 )
= ( bit0 @ one2 ) ) ).
% inc.simps(1)
thf(fact_5267_inc_Osimps_I3_J,axiom,
! [X2: num] :
( ( inc @ ( bit1 @ X2 ) )
= ( bit0 @ ( inc @ X2 ) ) ) ).
% inc.simps(3)
thf(fact_5268_inc_Osimps_I2_J,axiom,
! [X2: num] :
( ( inc @ ( bit0 @ X2 ) )
= ( bit1 @ X2 ) ) ).
% inc.simps(2)
thf(fact_5269_add__One,axiom,
! [X2: num] :
( ( plus_plus @ num @ X2 @ one2 )
= ( inc @ X2 ) ) ).
% add_One
thf(fact_5270_inc__BitM__eq,axiom,
! [N: num] :
( ( inc @ ( bitM @ N ) )
= ( bit0 @ N ) ) ).
% inc_BitM_eq
thf(fact_5271_BitM__inc__eq,axiom,
! [N: num] :
( ( bitM @ ( inc @ N ) )
= ( bit1 @ N ) ) ).
% BitM_inc_eq
thf(fact_5272_mult__inc,axiom,
! [X2: num,Y3: num] :
( ( times_times @ num @ X2 @ ( inc @ Y3 ) )
= ( plus_plus @ num @ ( times_times @ num @ X2 @ Y3 ) @ X2 ) ) ).
% mult_inc
thf(fact_5273_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_5274_take__bit__decr__eq,axiom,
! [N: nat,K: int] :
( ( ( bit_se2584673776208193580ke_bit @ int @ N @ K )
!= ( zero_zero @ int ) )
=> ( ( bit_se2584673776208193580ke_bit @ int @ N @ ( minus_minus @ int @ K @ ( one_one @ int ) ) )
= ( minus_minus @ int @ ( bit_se2584673776208193580ke_bit @ int @ N @ K ) @ ( one_one @ int ) ) ) ) ).
% take_bit_decr_eq
thf(fact_5275_numeral__inc,axiom,
! [A: $tType] :
( ( numeral @ A )
=> ! [X2: num] :
( ( numeral_numeral @ A @ ( inc @ X2 ) )
= ( plus_plus @ A @ ( numeral_numeral @ A @ X2 ) @ ( one_one @ A ) ) ) ) ).
% numeral_inc
thf(fact_5276_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_5277_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_5278_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_5279_take__bit__nat__def,axiom,
( ( bit_se2584673776208193580ke_bit @ nat )
= ( ^ [N4: nat,M6: nat] : ( modulo_modulo @ nat @ M6 @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N4 ) ) ) ) ).
% take_bit_nat_def
thf(fact_5280_word__1FF__is__mask,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ( ( numeral_numeral @ ( word @ A ) @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ one2 ) ) ) ) ) ) ) ) )
= ( bit_se2239418461657761734s_mask @ ( word @ A ) @ ( numeral_numeral @ nat @ ( bit1 @ ( bit0 @ ( bit0 @ one2 ) ) ) ) ) ) ) ).
% word_1FF_is_mask
thf(fact_5281_word__FF__is__mask,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ( ( numeral_numeral @ ( word @ A ) @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ one2 ) ) ) ) ) ) ) )
= ( bit_se2239418461657761734s_mask @ ( word @ A ) @ ( numeral_numeral @ nat @ ( bit0 @ ( bit0 @ ( bit0 @ one2 ) ) ) ) ) ) ) ).
% word_FF_is_mask
thf(fact_5282_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_5283_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_5284_take__bit__int__def,axiom,
( ( bit_se2584673776208193580ke_bit @ int )
= ( ^ [N4: nat,K3: int] : ( modulo_modulo @ int @ K3 @ ( power_power @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ N4 ) ) ) ) ).
% take_bit_int_def
thf(fact_5285_take__bit__numeral__minus__bit1,axiom,
! [L: num,K: num] :
( ( bit_se2584673776208193580ke_bit @ int @ ( numeral_numeral @ nat @ L ) @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ ( bit1 @ K ) ) ) )
= ( plus_plus @ int @ ( times_times @ int @ ( bit_se2584673776208193580ke_bit @ int @ ( pred_numeral @ L ) @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ ( inc @ K ) ) ) ) @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) ) @ ( one_one @ int ) ) ) ).
% take_bit_numeral_minus_bit1
thf(fact_5286_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_5287_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_5288_mask__plus__1,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [N: nat] :
( ( plus_plus @ ( word @ A ) @ ( bit_se2239418461657761734s_mask @ ( word @ A ) @ N ) @ ( one_one @ ( word @ A ) ) )
= ( power_power @ ( word @ A ) @ ( numeral_numeral @ ( word @ A ) @ ( bit0 @ one2 ) ) @ N ) ) ) ).
% mask_plus_1
thf(fact_5289_less__mask__eq,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [X2: word @ A,N: nat] :
( ( ord_less @ ( word @ A ) @ X2 @ ( power_power @ ( word @ A ) @ ( numeral_numeral @ ( word @ A ) @ ( bit0 @ one2 ) ) @ N ) )
=> ( ( bit_se5824344872417868541ns_and @ ( word @ A ) @ X2 @ ( bit_se2239418461657761734s_mask @ ( word @ A ) @ N ) )
= X2 ) ) ) ).
% less_mask_eq
thf(fact_5290_mask__eq__decr__exp,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ( ( bit_se2239418461657761734s_mask @ ( word @ A ) )
= ( ^ [N4: nat] : ( minus_minus @ ( word @ A ) @ ( power_power @ ( word @ A ) @ ( numeral_numeral @ ( word @ A ) @ ( bit0 @ one2 ) ) @ N4 ) @ ( one_one @ ( word @ A ) ) ) ) ) ) ).
% mask_eq_decr_exp
thf(fact_5291_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_5292_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_5293_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_5294_mask__eq__exp__minus__1,axiom,
! [A: $tType] :
( ( bit_se359711467146920520ations @ A )
=> ( ( bit_se2239418461657761734s_mask @ A )
= ( ^ [N4: nat] : ( minus_minus @ A @ ( power_power @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ N4 ) @ ( one_one @ A ) ) ) ) ) ).
% mask_eq_exp_minus_1
thf(fact_5295_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_5296_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_5297_is__aligned__AND__less__0,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [U: word @ A,N: nat,V: word @ A] :
( ( ( bit_se5824344872417868541ns_and @ ( word @ A ) @ U @ ( bit_se2239418461657761734s_mask @ ( word @ A ) @ N ) )
= ( zero_zero @ ( word @ A ) ) )
=> ( ( ord_less @ ( word @ A ) @ V @ ( power_power @ ( word @ A ) @ ( numeral_numeral @ ( word @ A ) @ ( bit0 @ one2 ) ) @ N ) )
=> ( ( bit_se5824344872417868541ns_and @ ( word @ A ) @ U @ V )
= ( zero_zero @ ( word @ A ) ) ) ) ) ) ).
% is_aligned_AND_less_0
thf(fact_5298_add__mask__fold,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [X2: word @ A,N: nat] :
( ( minus_minus @ ( word @ A ) @ ( plus_plus @ ( word @ A ) @ X2 @ ( power_power @ ( word @ A ) @ ( numeral_numeral @ ( word @ A ) @ ( bit0 @ one2 ) ) @ N ) ) @ ( one_one @ ( word @ A ) ) )
= ( plus_plus @ ( word @ A ) @ X2 @ ( bit_se2239418461657761734s_mask @ ( word @ A ) @ N ) ) ) ) ).
% add_mask_fold
thf(fact_5299_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_5300_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_5301_take__bit__incr__eq,axiom,
! [N: nat,K: int] :
( ( ( bit_se2584673776208193580ke_bit @ int @ N @ K )
!= ( minus_minus @ int @ ( power_power @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ N ) @ ( one_one @ int ) ) )
=> ( ( bit_se2584673776208193580ke_bit @ int @ N @ ( plus_plus @ int @ K @ ( one_one @ int ) ) )
= ( plus_plus @ int @ ( one_one @ int ) @ ( bit_se2584673776208193580ke_bit @ int @ N @ K ) ) ) ) ).
% take_bit_incr_eq
thf(fact_5302_take__bit__numeral__minus__bit0,axiom,
! [L: num,K: num] :
( ( bit_se2584673776208193580ke_bit @ int @ ( numeral_numeral @ nat @ L ) @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ ( bit0 @ K ) ) ) )
= ( times_times @ int @ ( bit_se2584673776208193580ke_bit @ int @ ( pred_numeral @ L ) @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ K ) ) ) @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) ) ) ).
% take_bit_numeral_minus_bit0
thf(fact_5303_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_5304_mask__eq__iff__w2p,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [N: nat,W: word @ A] :
( ( ord_less @ nat @ N @ ( size_size @ ( word @ A ) @ W ) )
=> ( ( ( bit_se5824344872417868541ns_and @ ( word @ A ) @ W @ ( bit_se2239418461657761734s_mask @ ( word @ A ) @ N ) )
= W )
= ( ord_less @ ( word @ A ) @ W @ ( power_power @ ( word @ A ) @ ( numeral_numeral @ ( word @ A ) @ ( bit0 @ one2 ) ) @ N ) ) ) ) ) ).
% mask_eq_iff_w2p
thf(fact_5305_and__mask__less__size,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [N: nat,X2: word @ A] :
( ( ord_less @ nat @ N @ ( size_size @ ( word @ A ) @ X2 ) )
=> ( ord_less @ ( word @ A ) @ ( bit_se5824344872417868541ns_and @ ( word @ A ) @ X2 @ ( bit_se2239418461657761734s_mask @ ( word @ A ) @ N ) ) @ ( power_power @ ( word @ A ) @ ( numeral_numeral @ ( word @ A ) @ ( bit0 @ one2 ) ) @ N ) ) ) ) ).
% and_mask_less_size
thf(fact_5306_word__and__mask__le__2pm1,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [W: word @ A,N: nat] : ( ord_less_eq @ ( word @ A ) @ ( bit_se5824344872417868541ns_and @ ( word @ A ) @ W @ ( 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_5307_word__mod__2p__is__mask,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [N: nat,X2: 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 ) @ X2 @ ( power_power @ ( word @ A ) @ ( numeral_numeral @ ( word @ A ) @ ( bit0 @ one2 ) ) @ N ) )
= ( bit_se5824344872417868541ns_and @ ( word @ A ) @ X2 @ ( bit_se2239418461657761734s_mask @ ( word @ A ) @ N ) ) ) ) ) ).
% word_mod_2p_is_mask
thf(fact_5308_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_5309_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_5310_divmod__step__nat__def,axiom,
( ( unique1321980374590559556d_step @ nat )
= ( ^ [L3: num] :
( product_case_prod @ nat @ nat @ ( product_prod @ nat @ nat )
@ ^ [Q5: nat,R5: nat] : ( if @ ( product_prod @ nat @ nat ) @ ( ord_less_eq @ nat @ ( numeral_numeral @ nat @ L3 ) @ R5 ) @ ( product_Pair @ nat @ nat @ ( plus_plus @ nat @ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ Q5 ) @ ( one_one @ nat ) ) @ ( minus_minus @ nat @ R5 @ ( numeral_numeral @ nat @ L3 ) ) ) @ ( product_Pair @ nat @ nat @ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ Q5 ) @ R5 ) ) ) ) ) ).
% divmod_step_nat_def
thf(fact_5311_signed__take__bit__eq__take__bit__shift,axiom,
( ( bit_ri4674362597316999326ke_bit @ int )
= ( ^ [N4: nat,K3: int] : ( minus_minus @ int @ ( bit_se2584673776208193580ke_bit @ int @ ( suc @ N4 ) @ ( plus_plus @ int @ K3 @ ( power_power @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ N4 ) ) ) @ ( power_power @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ N4 ) ) ) ) ).
% signed_take_bit_eq_take_bit_shift
thf(fact_5312_divmod__step__int__def,axiom,
( ( unique1321980374590559556d_step @ int )
= ( ^ [L3: num] :
( product_case_prod @ int @ int @ ( product_prod @ int @ int )
@ ^ [Q5: int,R5: int] : ( if @ ( product_prod @ int @ int ) @ ( ord_less_eq @ int @ ( numeral_numeral @ int @ L3 ) @ R5 ) @ ( product_Pair @ int @ int @ ( plus_plus @ int @ ( times_times @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ Q5 ) @ ( one_one @ int ) ) @ ( minus_minus @ int @ R5 @ ( numeral_numeral @ int @ L3 ) ) ) @ ( product_Pair @ int @ int @ ( times_times @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ Q5 ) @ R5 ) ) ) ) ) ).
% divmod_step_int_def
thf(fact_5313_and__mask__arith,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [W: word @ A,N: nat] :
( ( bit_se5824344872417868541ns_and @ ( word @ A ) @ W @ ( bit_se2239418461657761734s_mask @ ( word @ A ) @ N ) )
= ( divide_divide @ ( word @ A ) @ ( times_times @ ( word @ A ) @ W @ ( power_power @ ( word @ A ) @ ( numeral_numeral @ ( word @ A ) @ ( bit0 @ one2 ) ) @ ( minus_minus @ nat @ ( size_size @ ( word @ A ) @ W ) @ N ) ) ) @ ( power_power @ ( word @ A ) @ ( numeral_numeral @ ( word @ A ) @ ( bit0 @ one2 ) ) @ ( minus_minus @ nat @ ( size_size @ ( word @ A ) @ W ) @ N ) ) ) ) ) ).
% and_mask_arith
thf(fact_5314_mask__numeral,axiom,
! [A: $tType] :
( ( bit_se359711467146920520ations @ A )
=> ! [N: num] :
( ( bit_se2239418461657761734s_mask @ A @ ( numeral_numeral @ nat @ N ) )
= ( plus_plus @ A @ ( one_one @ A ) @ ( times_times @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ ( bit_se2239418461657761734s_mask @ A @ ( pred_numeral @ N ) ) ) ) ) ) ).
% mask_numeral
thf(fact_5315_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_5316_divmod__step__def,axiom,
! [A: $tType] :
( ( unique1627219031080169319umeral @ A )
=> ( ( unique1321980374590559556d_step @ A )
= ( ^ [L3: num] :
( product_case_prod @ A @ A @ ( product_prod @ A @ A )
@ ^ [Q5: A,R5: A] : ( if @ ( product_prod @ A @ A ) @ ( ord_less_eq @ A @ ( numeral_numeral @ A @ L3 ) @ R5 ) @ ( product_Pair @ A @ A @ ( plus_plus @ A @ ( times_times @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ Q5 ) @ ( one_one @ A ) ) @ ( minus_minus @ A @ R5 @ ( numeral_numeral @ A @ L3 ) ) ) @ ( product_Pair @ A @ A @ ( times_times @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ Q5 ) @ R5 ) ) ) ) ) ) ).
% divmod_step_def
thf(fact_5317_divmod__step__integer__def,axiom,
( ( unique1321980374590559556d_step @ code_integer )
= ( ^ [L3: num] :
( product_case_prod @ code_integer @ code_integer @ ( product_prod @ code_integer @ code_integer )
@ ^ [Q5: code_integer,R5: code_integer] : ( if @ ( product_prod @ code_integer @ code_integer ) @ ( ord_less_eq @ code_integer @ ( numeral_numeral @ code_integer @ L3 ) @ R5 ) @ ( product_Pair @ code_integer @ code_integer @ ( plus_plus @ code_integer @ ( times_times @ code_integer @ ( numeral_numeral @ code_integer @ ( bit0 @ one2 ) ) @ Q5 ) @ ( one_one @ code_integer ) ) @ ( minus_minus @ code_integer @ R5 @ ( numeral_numeral @ code_integer @ L3 ) ) ) @ ( product_Pair @ code_integer @ code_integer @ ( times_times @ code_integer @ ( numeral_numeral @ code_integer @ ( bit0 @ one2 ) ) @ Q5 ) @ R5 ) ) ) ) ) ).
% divmod_step_integer_def
thf(fact_5318_divmod__nat__if,axiom,
( divmod_nat
= ( ^ [M6: nat,N4: nat] :
( if @ ( product_prod @ nat @ nat )
@ ( ( N4
= ( zero_zero @ nat ) )
| ( ord_less @ nat @ M6 @ N4 ) )
@ ( product_Pair @ nat @ nat @ ( zero_zero @ nat ) @ M6 )
@ ( product_case_prod @ nat @ nat @ ( product_prod @ nat @ nat )
@ ^ [Q5: nat] : ( product_Pair @ nat @ nat @ ( suc @ Q5 ) )
@ ( divmod_nat @ ( minus_minus @ nat @ M6 @ N4 ) @ N4 ) ) ) ) ) ).
% divmod_nat_if
thf(fact_5319_add__scale__eq__noteq,axiom,
! [A: $tType] :
( ( semiri1453513574482234551roduct @ A )
=> ! [R2: A,A3: A,B3: A,C3: A,D2: A] :
( ( R2
!= ( zero_zero @ A ) )
=> ( ( ( A3 = B3 )
& ( C3 != D2 ) )
=> ( ( plus_plus @ A @ A3 @ ( times_times @ A @ R2 @ C3 ) )
!= ( plus_plus @ A @ B3 @ ( times_times @ A @ R2 @ D2 ) ) ) ) ) ) ).
% add_scale_eq_noteq
thf(fact_5320_case__prodI,axiom,
! [A: $tType,B: $tType,F2: A > B > $o,A3: A,B3: B] :
( ( F2 @ A3 @ B3 )
=> ( product_case_prod @ A @ B @ $o @ F2 @ ( product_Pair @ A @ B @ A3 @ B3 ) ) ) ).
% case_prodI
thf(fact_5321_case__prodI2,axiom,
! [B: $tType,A: $tType,P6: product_prod @ A @ B,C3: A > B > $o] :
( ! [A4: A,B4: B] :
( ( P6
= ( product_Pair @ A @ B @ A4 @ B4 ) )
=> ( C3 @ A4 @ B4 ) )
=> ( product_case_prod @ A @ B @ $o @ C3 @ P6 ) ) ).
% case_prodI2
thf(fact_5322_case__prodI2_H,axiom,
! [A: $tType,B: $tType,C: $tType,P6: product_prod @ A @ B,C3: A > B > C > $o,X2: C] :
( ! [A4: A,B4: B] :
( ( ( product_Pair @ A @ B @ A4 @ B4 )
= P6 )
=> ( C3 @ A4 @ B4 @ X2 ) )
=> ( product_case_prod @ A @ B @ ( C > $o ) @ C3 @ P6 @ X2 ) ) ).
% case_prodI2'
thf(fact_5323_mem__case__prodI,axiom,
! [A: $tType,B: $tType,C: $tType,Z: A,C3: B > C > ( set @ A ),A3: B,B3: C] :
( ( member @ A @ Z @ ( C3 @ A3 @ B3 ) )
=> ( member @ A @ Z @ ( product_case_prod @ B @ C @ ( set @ A ) @ C3 @ ( product_Pair @ B @ C @ A3 @ B3 ) ) ) ) ).
% mem_case_prodI
thf(fact_5324_mem__case__prodI2,axiom,
! [C: $tType,B: $tType,A: $tType,P6: product_prod @ A @ B,Z: C,C3: A > B > ( set @ C )] :
( ! [A4: A,B4: B] :
( ( P6
= ( product_Pair @ A @ B @ A4 @ B4 ) )
=> ( member @ C @ Z @ ( C3 @ A4 @ B4 ) ) )
=> ( member @ C @ Z @ ( product_case_prod @ A @ B @ ( set @ C ) @ C3 @ P6 ) ) ) ).
% mem_case_prodI2
thf(fact_5325_Collect__const__case__prod,axiom,
! [B: $tType,A: $tType,P: $o] :
( ( P
=> ( ( collect @ ( product_prod @ A @ B )
@ ( product_case_prod @ A @ B @ $o
@ ^ [A6: A,B7: B] : P ) )
= ( top_top @ ( set @ ( product_prod @ A @ B ) ) ) ) )
& ( ~ P
=> ( ( collect @ ( product_prod @ A @ B )
@ ( product_case_prod @ A @ B @ $o
@ ^ [A6: A,B7: B] : P ) )
= ( bot_bot @ ( set @ ( product_prod @ A @ B ) ) ) ) ) ) ).
% Collect_const_case_prod
thf(fact_5326_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_5327_case__prodE,axiom,
! [A: $tType,B: $tType,C3: A > B > $o,P6: product_prod @ A @ B] :
( ( product_case_prod @ A @ B @ $o @ C3 @ P6 )
=> ~ ! [X4: A,Y4: B] :
( ( P6
= ( product_Pair @ A @ B @ X4 @ Y4 ) )
=> ~ ( C3 @ X4 @ Y4 ) ) ) ).
% case_prodE
thf(fact_5328_case__prodD,axiom,
! [A: $tType,B: $tType,F2: A > B > $o,A3: A,B3: B] :
( ( product_case_prod @ A @ B @ $o @ F2 @ ( product_Pair @ A @ B @ A3 @ B3 ) )
=> ( F2 @ A3 @ B3 ) ) ).
% case_prodD
thf(fact_5329_mem__case__prodE,axiom,
! [B: $tType,A: $tType,C: $tType,Z: A,C3: B > C > ( set @ A ),P6: product_prod @ B @ C] :
( ( member @ A @ Z @ ( product_case_prod @ B @ C @ ( set @ A ) @ C3 @ P6 ) )
=> ~ ! [X4: B,Y4: C] :
( ( P6
= ( product_Pair @ B @ C @ X4 @ Y4 ) )
=> ~ ( member @ A @ Z @ ( C3 @ X4 @ Y4 ) ) ) ) ).
% mem_case_prodE
thf(fact_5330_less__eq__mask,axiom,
! [N: nat] : ( ord_less_eq @ nat @ N @ ( bit_se2239418461657761734s_mask @ nat @ N ) ) ).
% less_eq_mask
thf(fact_5331_case__prodD_H,axiom,
! [B: $tType,A: $tType,C: $tType,R: A > B > C > $o,A3: A,B3: B,C3: C] :
( ( product_case_prod @ A @ B @ ( C > $o ) @ R @ ( product_Pair @ A @ B @ A3 @ B3 ) @ C3 )
=> ( R @ A3 @ B3 @ C3 ) ) ).
% case_prodD'
thf(fact_5332_case__prodE_H,axiom,
! [B: $tType,A: $tType,C: $tType,C3: A > B > C > $o,P6: product_prod @ A @ B,Z: C] :
( ( product_case_prod @ A @ B @ ( C > $o ) @ C3 @ P6 @ Z )
=> ~ ! [X4: A,Y4: B] :
( ( P6
= ( product_Pair @ A @ B @ X4 @ Y4 ) )
=> ~ ( C3 @ X4 @ Y4 @ Z ) ) ) ).
% case_prodE'
thf(fact_5333_TBOUND__prod__case,axiom,
! [C: $tType,B: $tType,A: $tType,T2: product_prod @ A @ B,F2: A > B > ( heap_Time_Heap @ C ),Bnd: A > B > nat] :
( ! [A4: A,B4: B] :
( ( T2
= ( product_Pair @ A @ B @ A4 @ B4 ) )
=> ( time_TBOUND @ C @ ( F2 @ A4 @ B4 ) @ ( Bnd @ A4 @ B4 ) ) )
=> ( time_TBOUND @ C @ ( product_case_prod @ A @ B @ ( heap_Time_Heap @ C ) @ F2 @ T2 ) @ ( product_case_prod @ A @ B @ nat @ Bnd @ T2 ) ) ) ).
% TBOUND_prod_case
thf(fact_5334_mask__nonnegative__int,axiom,
! [N: nat] : ( ord_less_eq @ int @ ( zero_zero @ int ) @ ( bit_se2239418461657761734s_mask @ int @ N ) ) ).
% mask_nonnegative_int
thf(fact_5335_sum_Otriangle__reindex__eq,axiom,
! [A: $tType] :
( ( comm_monoid_add @ A )
=> ! [G: nat > nat > A,N: nat] :
( ( groups7311177749621191930dd_sum @ ( product_prod @ nat @ nat ) @ A @ ( product_case_prod @ nat @ nat @ A @ G )
@ ( collect @ ( product_prod @ nat @ nat )
@ ( product_case_prod @ nat @ nat @ $o
@ ^ [I4: nat,J3: nat] : ( ord_less_eq @ nat @ ( plus_plus @ nat @ I4 @ J3 ) @ N ) ) ) )
= ( groups7311177749621191930dd_sum @ nat @ A
@ ^ [K3: nat] :
( groups7311177749621191930dd_sum @ nat @ A
@ ^ [I4: nat] : ( G @ I4 @ ( minus_minus @ nat @ K3 @ I4 ) )
@ ( set_ord_atMost @ nat @ K3 ) )
@ ( set_ord_atMost @ nat @ N ) ) ) ) ).
% sum.triangle_reindex_eq
thf(fact_5336_case__prod__rule,axiom,
! [A: $tType,B: $tType,C: $tType,X2: product_prod @ A @ B,P: assn,F2: A > B > ( heap_Time_Heap @ C ),Q: C > assn] :
( ! [A4: A,B4: B] :
( ( X2
= ( product_Pair @ A @ B @ A4 @ B4 ) )
=> ( hoare_hoare_triple @ C @ P @ ( F2 @ A4 @ B4 ) @ Q ) )
=> ( hoare_hoare_triple @ C @ P @ ( product_case_prod @ A @ B @ ( heap_Time_Heap @ C ) @ F2 @ X2 ) @ Q ) ) ).
% case_prod_rule
thf(fact_5337_prod_Otriangle__reindex__eq,axiom,
! [A: $tType] :
( ( comm_monoid_mult @ A )
=> ! [G: nat > nat > A,N: nat] :
( ( groups7121269368397514597t_prod @ ( product_prod @ nat @ nat ) @ A @ ( product_case_prod @ nat @ nat @ A @ G )
@ ( collect @ ( product_prod @ nat @ nat )
@ ( product_case_prod @ nat @ nat @ $o
@ ^ [I4: nat,J3: nat] : ( ord_less_eq @ nat @ ( plus_plus @ nat @ I4 @ J3 ) @ N ) ) ) )
= ( groups7121269368397514597t_prod @ nat @ A
@ ^ [K3: nat] :
( groups7121269368397514597t_prod @ nat @ A
@ ^ [I4: nat] : ( G @ I4 @ ( minus_minus @ nat @ K3 @ I4 ) )
@ ( set_ord_atMost @ nat @ K3 ) )
@ ( set_ord_atMost @ nat @ N ) ) ) ) ).
% prod.triangle_reindex_eq
thf(fact_5338_sum_Otriangle__reindex,axiom,
! [A: $tType] :
( ( comm_monoid_add @ A )
=> ! [G: nat > nat > A,N: nat] :
( ( groups7311177749621191930dd_sum @ ( product_prod @ nat @ nat ) @ A @ ( product_case_prod @ nat @ nat @ A @ G )
@ ( collect @ ( product_prod @ nat @ nat )
@ ( product_case_prod @ nat @ nat @ $o
@ ^ [I4: nat,J3: nat] : ( ord_less @ nat @ ( plus_plus @ nat @ I4 @ J3 ) @ N ) ) ) )
= ( groups7311177749621191930dd_sum @ nat @ A
@ ^ [K3: nat] :
( groups7311177749621191930dd_sum @ nat @ A
@ ^ [I4: nat] : ( G @ I4 @ ( minus_minus @ nat @ K3 @ I4 ) )
@ ( set_ord_atMost @ nat @ K3 ) )
@ ( set_ord_lessThan @ nat @ N ) ) ) ) ).
% sum.triangle_reindex
thf(fact_5339_prod_Otriangle__reindex,axiom,
! [A: $tType] :
( ( comm_monoid_mult @ A )
=> ! [G: nat > nat > A,N: nat] :
( ( groups7121269368397514597t_prod @ ( product_prod @ nat @ nat ) @ A @ ( product_case_prod @ nat @ nat @ A @ G )
@ ( collect @ ( product_prod @ nat @ nat )
@ ( product_case_prod @ nat @ nat @ $o
@ ^ [I4: nat,J3: nat] : ( ord_less @ nat @ ( plus_plus @ nat @ I4 @ J3 ) @ N ) ) ) )
= ( groups7121269368397514597t_prod @ nat @ A
@ ^ [K3: nat] :
( groups7121269368397514597t_prod @ nat @ A
@ ^ [I4: nat] : ( G @ I4 @ ( minus_minus @ nat @ K3 @ I4 ) )
@ ( set_ord_atMost @ nat @ K3 ) )
@ ( set_ord_lessThan @ nat @ N ) ) ) ) ).
% prod.triangle_reindex
thf(fact_5340_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_5341_take__bit__eq__mask__iff,axiom,
! [N: nat,K: int] :
( ( ( bit_se2584673776208193580ke_bit @ int @ N @ K )
= ( bit_se2239418461657761734s_mask @ int @ N ) )
= ( ( bit_se2584673776208193580ke_bit @ int @ N @ ( plus_plus @ int @ K @ ( one_one @ int ) ) )
= ( zero_zero @ int ) ) ) ).
% take_bit_eq_mask_iff
thf(fact_5342_one__natural_Orsp,axiom,
( ( one_one @ nat )
= ( one_one @ nat ) ) ).
% one_natural.rsp
thf(fact_5343_one__integer_Orsp,axiom,
( ( one_one @ int )
= ( one_one @ int ) ) ).
% one_integer.rsp
thf(fact_5344_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_5345_rel__of__def,axiom,
! [B: $tType,A: $tType] :
( ( rel_of @ A @ B )
= ( ^ [M6: A > ( option @ B ),P3: ( product_prod @ A @ B ) > $o] :
( collect @ ( product_prod @ A @ B )
@ ( product_case_prod @ A @ B @ $o
@ ^ [K3: A,V5: B] :
( ( ( M6 @ K3 )
= ( some @ B @ V5 ) )
& ( P3 @ ( product_Pair @ A @ B @ K3 @ V5 ) ) ) ) ) ) ) ).
% rel_of_def
thf(fact_5346_Divides_Oadjust__div__def,axiom,
( adjust_div
= ( product_case_prod @ int @ int @ int
@ ^ [Q5: int,R5: int] :
( plus_plus @ int @ Q5
@ ( zero_neq_one_of_bool @ int
@ ( R5
!= ( zero_zero @ int ) ) ) ) ) ) ).
% Divides.adjust_div_def
thf(fact_5347_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_5348_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_5349_mask__half__int,axiom,
! [N: nat] :
( ( divide_divide @ int @ ( bit_se2239418461657761734s_mask @ int @ N ) @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) )
= ( bit_se2239418461657761734s_mask @ int @ ( minus_minus @ nat @ N @ ( one_one @ nat ) ) ) ) ).
% mask_half_int
thf(fact_5350_mask__int__def,axiom,
( ( bit_se2239418461657761734s_mask @ int )
= ( ^ [N4: nat] : ( minus_minus @ int @ ( power_power @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ N4 ) @ ( one_one @ int ) ) ) ) ).
% mask_int_def
thf(fact_5351_mask__nat__def,axiom,
( ( bit_se2239418461657761734s_mask @ nat )
= ( ^ [N4: nat] : ( minus_minus @ nat @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N4 ) @ ( one_one @ nat ) ) ) ) ).
% mask_nat_def
thf(fact_5352_add__0__iff,axiom,
! [A: $tType] :
( ( semiri1453513574482234551roduct @ A )
=> ! [B3: A,A3: A] :
( ( B3
= ( plus_plus @ A @ B3 @ A3 ) )
= ( A3
= ( zero_zero @ A ) ) ) ) ).
% add_0_iff
thf(fact_5353_divmod__nat__def,axiom,
( divmod_nat
= ( ^ [M6: nat,N4: nat] : ( product_Pair @ nat @ nat @ ( divide_divide @ nat @ M6 @ N4 ) @ ( modulo_modulo @ nat @ M6 @ N4 ) ) ) ) ).
% divmod_nat_def
thf(fact_5354_crossproduct__eq,axiom,
! [A: $tType] :
( ( semiri1453513574482234551roduct @ A )
=> ! [W: A,Y3: A,X2: A,Z: A] :
( ( ( plus_plus @ A @ ( times_times @ A @ W @ Y3 ) @ ( times_times @ A @ X2 @ Z ) )
= ( plus_plus @ A @ ( times_times @ A @ W @ Z ) @ ( times_times @ A @ X2 @ Y3 ) ) )
= ( ( W = X2 )
| ( Y3 = Z ) ) ) ) ).
% crossproduct_eq
thf(fact_5355_crossproduct__noteq,axiom,
! [A: $tType] :
( ( semiri1453513574482234551roduct @ A )
=> ! [A3: A,B3: A,C3: A,D2: A] :
( ( ( A3 != B3 )
& ( C3 != D2 ) )
= ( ( plus_plus @ A @ ( times_times @ A @ A3 @ C3 ) @ ( times_times @ A @ B3 @ D2 ) )
!= ( plus_plus @ A @ ( times_times @ A @ A3 @ D2 ) @ ( times_times @ A @ B3 @ C3 ) ) ) ) ) ).
% crossproduct_noteq
thf(fact_5356_take__bit__eq__mask__iff__exp__dvd,axiom,
! [N: nat,K: int] :
( ( ( bit_se2584673776208193580ke_bit @ int @ N @ K )
= ( bit_se2239418461657761734s_mask @ int @ N ) )
= ( dvd_dvd @ int @ ( power_power @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ N ) @ ( plus_plus @ int @ K @ ( one_one @ int ) ) ) ) ).
% take_bit_eq_mask_iff_exp_dvd
thf(fact_5357_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_5358_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_5359_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_5360_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_5361_of__nat__code__if,axiom,
! [A: $tType] :
( ( semiring_1 @ A )
=> ( ( semiring_1_of_nat @ A )
= ( ^ [N4: nat] :
( if @ A
@ ( N4
= ( zero_zero @ nat ) )
@ ( zero_zero @ A )
@ ( product_case_prod @ nat @ nat @ A
@ ^ [M6: nat,Q5: nat] :
( if @ A
@ ( Q5
= ( zero_zero @ nat ) )
@ ( times_times @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ ( semiring_1_of_nat @ A @ M6 ) )
@ ( plus_plus @ A @ ( times_times @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ ( semiring_1_of_nat @ A @ M6 ) ) @ ( one_one @ A ) ) )
@ ( divmod_nat @ N4 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) ) ) ).
% of_nat_code_if
thf(fact_5362_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_5363_int__ge__less__than__def,axiom,
( int_ge_less_than
= ( ^ [D3: int] :
( collect @ ( product_prod @ int @ int )
@ ( product_case_prod @ int @ int @ $o
@ ^ [Z9: int,Z4: int] :
( ( ord_less_eq @ int @ D3 @ Z9 )
& ( ord_less @ int @ Z9 @ Z4 ) ) ) ) ) ) ).
% int_ge_less_than_def
thf(fact_5364_split__part,axiom,
! [B: $tType,A: $tType,P: $o,Q: A > B > $o] :
( ( product_case_prod @ A @ B @ $o
@ ^ [A6: A,B7: B] :
( P
& ( Q @ A6 @ B7 ) ) )
= ( ^ [Ab2: product_prod @ A @ B] :
( P
& ( product_case_prod @ A @ B @ $o @ Q @ Ab2 ) ) ) ) ).
% split_part
thf(fact_5365_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_5366_Collect__case__prod__mono,axiom,
! [B: $tType,A: $tType,A2: A > B > $o,B2: A > B > $o] :
( ( ord_less_eq @ ( A > B > $o ) @ A2 @ B2 )
=> ( ord_less_eq @ ( set @ ( product_prod @ A @ B ) ) @ ( collect @ ( product_prod @ A @ B ) @ ( product_case_prod @ A @ B @ $o @ A2 ) ) @ ( collect @ ( product_prod @ A @ B ) @ ( product_case_prod @ A @ B @ $o @ B2 ) ) ) ) ).
% Collect_case_prod_mono
thf(fact_5367_one__integer__def,axiom,
( ( one_one @ code_integer )
= ( code_integer_of_int @ ( one_one @ int ) ) ) ).
% one_integer_def
thf(fact_5368_less__eq__integer_Oabs__eq,axiom,
! [Xa: int,X2: int] :
( ( ord_less_eq @ code_integer @ ( code_integer_of_int @ Xa ) @ ( code_integer_of_int @ X2 ) )
= ( ord_less_eq @ int @ Xa @ X2 ) ) ).
% less_eq_integer.abs_eq
thf(fact_5369_plus__integer_Oabs__eq,axiom,
! [Xa: int,X2: int] :
( ( plus_plus @ code_integer @ ( code_integer_of_int @ Xa ) @ ( code_integer_of_int @ X2 ) )
= ( code_integer_of_int @ ( plus_plus @ int @ Xa @ X2 ) ) ) ).
% plus_integer.abs_eq
thf(fact_5370_minus__integer_Oabs__eq,axiom,
! [Xa: int,X2: int] :
( ( minus_minus @ code_integer @ ( code_integer_of_int @ Xa ) @ ( code_integer_of_int @ X2 ) )
= ( code_integer_of_int @ ( minus_minus @ int @ Xa @ X2 ) ) ) ).
% minus_integer.abs_eq
thf(fact_5371_int__ge__less__than2__def,axiom,
( int_ge_less_than2
= ( ^ [D3: int] :
( collect @ ( product_prod @ int @ int )
@ ( product_case_prod @ int @ int @ $o
@ ^ [Z9: int,Z4: int] :
( ( ord_less_eq @ int @ D3 @ Z4 )
& ( ord_less @ int @ Z9 @ Z4 ) ) ) ) ) ) ).
% int_ge_less_than2_def
thf(fact_5372_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_5373_listI__assn__def,axiom,
! [A: $tType,B: $tType] :
( ( vEBT_List_listI_assn @ A @ B )
= ( ^ [I9: set @ nat,A7: 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 ) @ I9 @ ( set_or7035219750837199246ssThan @ nat @ ( zero_zero @ nat ) @ ( size_size @ ( list @ A ) @ Xs2 ) ) ) ) )
@ ( finite_fold @ nat @ assn
@ ^ [I4: nat,A6: assn] : ( times_times @ assn @ A6 @ ( A7 @ ( nth @ A @ Xs2 @ I4 ) @ ( nth @ B @ Xsi3 @ I4 ) ) )
@ ( one_one @ assn )
@ I9 ) ) ) ) ).
% listI_assn_def
thf(fact_5374_predicate2I,axiom,
! [B: $tType,A: $tType,P: A > B > $o,Q: A > B > $o] :
( ! [X4: A,Y4: B] :
( ( P @ X4 @ Y4 )
=> ( Q @ X4 @ Y4 ) )
=> ( ord_less_eq @ ( A > B > $o ) @ P @ Q ) ) ).
% predicate2I
thf(fact_5375_slice__complete,axiom,
! [A: $tType,Xs: list @ A] :
( ( slice @ A @ ( zero_zero @ nat ) @ ( size_size @ ( list @ A ) @ Xs ) @ Xs )
= Xs ) ).
% slice_complete
thf(fact_5376_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_5377_predicate2D,axiom,
! [A: $tType,B: $tType,P: A > B > $o,Q: A > B > $o,X2: A,Y3: B] :
( ( ord_less_eq @ ( A > B > $o ) @ P @ Q )
=> ( ( P @ X2 @ Y3 )
=> ( Q @ X2 @ Y3 ) ) ) ).
% predicate2D
thf(fact_5378_rev__predicate2D,axiom,
! [A: $tType,B: $tType,P: A > B > $o,X2: A,Y3: B,Q: A > B > $o] :
( ( P @ X2 @ Y3 )
=> ( ( ord_less_eq @ ( A > B > $o ) @ P @ Q )
=> ( Q @ X2 @ Y3 ) ) ) ).
% rev_predicate2D
thf(fact_5379_pred__subset__eq2,axiom,
! [B: $tType,A: $tType,R: set @ ( product_prod @ A @ B ),S: set @ ( product_prod @ A @ B )] :
( ( ord_less_eq @ ( A > B > $o )
@ ^ [X3: A,Y: B] : ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X3 @ Y ) @ R )
@ ^ [X3: A,Y: B] : ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X3 @ Y ) @ S ) )
= ( ord_less_eq @ ( set @ ( product_prod @ A @ B ) ) @ R @ S ) ) ).
% pred_subset_eq2
thf(fact_5380_accp__subset,axiom,
! [A: $tType,R1: A > A > $o,R22: A > A > $o] :
( ( ord_less_eq @ ( A > A > $o ) @ R1 @ R22 )
=> ( ord_less_eq @ ( A > $o ) @ ( accp @ A @ R22 ) @ ( accp @ A @ R1 ) ) ) ).
% accp_subset
thf(fact_5381_word__2p__lem,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [N: nat,W: word @ A] :
( ( ord_less @ nat @ N @ ( size_size @ ( word @ A ) @ W ) )
=> ( ( ord_less @ ( word @ A ) @ W @ ( power_power @ ( word @ A ) @ ( numeral_numeral @ ( word @ A ) @ ( bit0 @ one2 ) ) @ N ) )
= ( ord_less @ int @ ( semiring_1_unsigned @ A @ int @ W ) @ ( power_power @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ N ) ) ) ) ) ).
% word_2p_lem
thf(fact_5382_unsigned__1,axiom,
! [B: $tType,A: $tType] :
( ( ( semiring_1 @ A )
& ( type_len @ B ) )
=> ( ( semiring_1_unsigned @ B @ A @ ( one_one @ ( word @ B ) ) )
= ( one_one @ A ) ) ) ).
% unsigned_1
thf(fact_5383_uint__nonnegative,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [W: word @ A] : ( ord_less_eq @ int @ ( zero_zero @ int ) @ ( semiring_1_unsigned @ A @ int @ W ) ) ) ).
% uint_nonnegative
thf(fact_5384_uint__le__0__iff,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [X2: word @ A] :
( ( ord_less_eq @ int @ ( semiring_1_unsigned @ A @ int @ X2 ) @ ( zero_zero @ int ) )
= ( ( semiring_1_unsigned @ A @ int @ X2 )
= ( zero_zero @ int ) ) ) ) ).
% uint_le_0_iff
thf(fact_5385_uint__ge__0,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [X2: word @ A] : ( ord_less_eq @ int @ ( zero_zero @ int ) @ ( semiring_1_unsigned @ A @ int @ X2 ) ) ) ).
% uint_ge_0
thf(fact_5386_word__le__no,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [A3: num,B3: num] :
( ( ord_less_eq @ ( word @ A ) @ ( numeral_numeral @ ( word @ A ) @ A3 ) @ ( numeral_numeral @ ( word @ A ) @ B3 ) )
= ( ord_less_eq @ int @ ( semiring_1_unsigned @ A @ int @ ( numeral_numeral @ ( word @ A ) @ A3 ) ) @ ( semiring_1_unsigned @ A @ int @ ( numeral_numeral @ ( word @ A ) @ B3 ) ) ) ) ) ).
% word_le_no
thf(fact_5387_uint__1__eq,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ( ( semiring_1_unsigned @ A @ int @ ( one_one @ ( word @ A ) ) )
= ( one_one @ int ) ) ) ).
% uint_1_eq
thf(fact_5388_word__le__def,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ( ( ord_less_eq @ ( word @ A ) )
= ( ^ [A6: word @ A,B7: word @ A] : ( ord_less_eq @ int @ ( semiring_1_unsigned @ A @ int @ A6 ) @ ( semiring_1_unsigned @ A @ int @ B7 ) ) ) ) ) ).
% word_le_def
thf(fact_5389_unsigned__greater__eq,axiom,
! [A: $tType,B: $tType] :
( ( ( type_len @ B )
& ( unique1627219031080169319umeral @ A ) )
=> ! [W: word @ B] : ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( semiring_1_unsigned @ B @ A @ W ) ) ) ).
% unsigned_greater_eq
thf(fact_5390_word__less__eq__iff__unsigned,axiom,
! [A: $tType,B: $tType] :
( ( ( type_len @ B )
& ( linordered_semidom @ A ) )
=> ( ( ord_less_eq @ ( word @ B ) )
= ( ^ [A6: word @ B,B7: word @ B] : ( ord_less_eq @ A @ ( semiring_1_unsigned @ B @ A @ A6 ) @ ( semiring_1_unsigned @ B @ A @ B7 ) ) ) ) ) ).
% word_less_eq_iff_unsigned
thf(fact_5391_word__less__iff__unsigned,axiom,
! [A: $tType,B: $tType] :
( ( ( type_len @ B )
& ( linordered_semidom @ A ) )
=> ( ( ord_less @ ( word @ B ) )
= ( ^ [A6: word @ B,B7: word @ B] : ( ord_less @ A @ ( semiring_1_unsigned @ B @ A @ A6 ) @ ( semiring_1_unsigned @ B @ A @ B7 ) ) ) ) ) ).
% word_less_iff_unsigned
thf(fact_5392_uint__add__ge0,axiom,
! [Aa: $tType,A: $tType] :
( ( ( type_len @ A )
& ( type_len @ Aa ) )
=> ! [Xa: word @ Aa,X2: word @ A] : ( ord_less_eq @ int @ ( zero_zero @ int ) @ ( plus_plus @ int @ ( semiring_1_unsigned @ Aa @ int @ Xa ) @ ( semiring_1_unsigned @ A @ int @ X2 ) ) ) ) ).
% uint_add_ge0
thf(fact_5393_uint__mult__ge0,axiom,
! [Aa: $tType,A: $tType] :
( ( ( type_len @ A )
& ( type_len @ Aa ) )
=> ! [Xa: word @ Aa,X2: word @ A] : ( ord_less_eq @ int @ ( zero_zero @ int ) @ ( times_times @ int @ ( semiring_1_unsigned @ Aa @ int @ Xa ) @ ( semiring_1_unsigned @ A @ int @ X2 ) ) ) ) ).
% uint_mult_ge0
thf(fact_5394_nat__uint__less__helper,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [Y3: word @ A,Z: nat,X2: word @ A] :
( ( ( nat2 @ ( semiring_1_unsigned @ A @ int @ Y3 ) )
= Z )
=> ( ( ord_less @ ( word @ A ) @ X2 @ Y3 )
=> ( ord_less @ nat @ ( nat2 @ ( semiring_1_unsigned @ A @ int @ X2 ) ) @ Z ) ) ) ) ).
% nat_uint_less_helper
thf(fact_5395_uint__add__le,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [X2: word @ A,Y3: word @ A] : ( ord_less_eq @ int @ ( semiring_1_unsigned @ A @ int @ ( plus_plus @ ( word @ A ) @ X2 @ Y3 ) ) @ ( plus_plus @ int @ ( semiring_1_unsigned @ A @ int @ X2 ) @ ( semiring_1_unsigned @ A @ int @ Y3 ) ) ) ) ).
% uint_add_le
thf(fact_5396_uint__plus__simple__iff,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [X2: word @ A,Y3: word @ A] :
( ( ord_less_eq @ ( word @ A ) @ X2 @ ( plus_plus @ ( word @ A ) @ X2 @ Y3 ) )
= ( ( semiring_1_unsigned @ A @ int @ ( plus_plus @ ( word @ A ) @ X2 @ Y3 ) )
= ( plus_plus @ int @ ( semiring_1_unsigned @ A @ int @ X2 ) @ ( semiring_1_unsigned @ A @ int @ Y3 ) ) ) ) ) ).
% uint_plus_simple_iff
thf(fact_5397_uint__plus__simple,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [X2: word @ A,Y3: word @ A] :
( ( ord_less_eq @ ( word @ A ) @ X2 @ ( plus_plus @ ( word @ A ) @ X2 @ Y3 ) )
=> ( ( semiring_1_unsigned @ A @ int @ ( plus_plus @ ( word @ A ) @ X2 @ Y3 ) )
= ( plus_plus @ int @ ( semiring_1_unsigned @ A @ int @ X2 ) @ ( semiring_1_unsigned @ A @ int @ Y3 ) ) ) ) ) ).
% uint_plus_simple
thf(fact_5398_no__ulen__sub,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [X2: word @ A,Y3: word @ A] :
( ( ord_less_eq @ ( word @ A ) @ ( minus_minus @ ( word @ A ) @ X2 @ Y3 ) @ X2 )
= ( ord_less_eq @ int @ ( semiring_1_unsigned @ A @ int @ Y3 ) @ ( semiring_1_unsigned @ A @ int @ X2 ) ) ) ) ).
% no_ulen_sub
thf(fact_5399_word__add__def,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ( ( plus_plus @ ( word @ A ) )
= ( ^ [A6: word @ A,B7: word @ A] : ( ring_1_of_int @ ( word @ A ) @ ( plus_plus @ int @ ( semiring_1_unsigned @ A @ int @ A6 ) @ ( semiring_1_unsigned @ A @ int @ B7 ) ) ) ) ) ) ).
% word_add_def
thf(fact_5400_uint__sub__ge,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [X2: word @ A,Y3: word @ A] : ( ord_less_eq @ int @ ( minus_minus @ int @ ( semiring_1_unsigned @ A @ int @ X2 ) @ ( semiring_1_unsigned @ A @ int @ Y3 ) ) @ ( semiring_1_unsigned @ A @ int @ ( minus_minus @ ( word @ A ) @ X2 @ Y3 ) ) ) ) ).
% uint_sub_ge
thf(fact_5401_uint__sub__lem,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [Y3: word @ A,X2: word @ A] :
( ( ord_less_eq @ int @ ( semiring_1_unsigned @ A @ int @ Y3 ) @ ( semiring_1_unsigned @ A @ int @ X2 ) )
= ( ( semiring_1_unsigned @ A @ int @ ( minus_minus @ ( word @ A ) @ X2 @ Y3 ) )
= ( minus_minus @ int @ ( semiring_1_unsigned @ A @ int @ X2 ) @ ( semiring_1_unsigned @ A @ int @ Y3 ) ) ) ) ) ).
% uint_sub_lem
thf(fact_5402_uint__minus__simple__iff,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [X2: word @ A,Y3: word @ A] :
( ( ord_less_eq @ ( word @ A ) @ ( minus_minus @ ( word @ A ) @ X2 @ Y3 ) @ X2 )
= ( ( semiring_1_unsigned @ A @ int @ ( minus_minus @ ( word @ A ) @ X2 @ Y3 ) )
= ( minus_minus @ int @ ( semiring_1_unsigned @ A @ int @ X2 ) @ ( semiring_1_unsigned @ A @ int @ Y3 ) ) ) ) ) ).
% uint_minus_simple_iff
thf(fact_5403_uint__minus__simple__alt,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ( ( ord_less_eq @ ( word @ A ) )
= ( ^ [Y: word @ A,X3: word @ A] :
( ( semiring_1_unsigned @ A @ int @ ( minus_minus @ ( word @ A ) @ X3 @ Y ) )
= ( minus_minus @ int @ ( semiring_1_unsigned @ A @ int @ X3 ) @ ( semiring_1_unsigned @ A @ int @ Y ) ) ) ) ) ) ).
% uint_minus_simple_alt
thf(fact_5404_word__sub__wi,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ( ( minus_minus @ ( word @ A ) )
= ( ^ [A6: word @ A,B7: word @ A] : ( ring_1_of_int @ ( word @ A ) @ ( minus_minus @ int @ ( semiring_1_unsigned @ A @ int @ A6 ) @ ( semiring_1_unsigned @ A @ int @ B7 ) ) ) ) ) ) ).
% word_sub_wi
thf(fact_5405_udvd__incr__lem0,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [Up: int,Uq: int,N: int,K6: word @ A,N2: int] :
( ( ord_less @ int @ Up @ Uq )
=> ( ( Up
= ( times_times @ int @ N @ ( semiring_1_unsigned @ A @ int @ K6 ) ) )
=> ( ( Uq
= ( times_times @ int @ N2 @ ( semiring_1_unsigned @ A @ int @ K6 ) ) )
=> ( ord_less_eq @ int @ ( plus_plus @ int @ Up @ ( semiring_1_unsigned @ A @ int @ K6 ) ) @ Uq ) ) ) ) ) ).
% udvd_incr_lem0
thf(fact_5406_udvd__incr__lem,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [Up: int,Uq: int,Ua: int,N: int,K6: word @ A,N2: int] :
( ( ord_less @ int @ Up @ Uq )
=> ( ( Up
= ( plus_plus @ int @ Ua @ ( times_times @ int @ N @ ( semiring_1_unsigned @ A @ int @ K6 ) ) ) )
=> ( ( Uq
= ( plus_plus @ int @ Ua @ ( times_times @ int @ N2 @ ( semiring_1_unsigned @ A @ int @ K6 ) ) ) )
=> ( ord_less_eq @ int @ ( plus_plus @ int @ Up @ ( semiring_1_unsigned @ A @ int @ K6 ) ) @ Uq ) ) ) ) ) ).
% udvd_incr_lem
thf(fact_5407_udvd__incr0,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [P6: word @ A,Q3: word @ A,N: int,K6: word @ A,N2: int] :
( ( ord_less @ ( word @ A ) @ P6 @ Q3 )
=> ( ( ( semiring_1_unsigned @ A @ int @ P6 )
= ( times_times @ int @ N @ ( semiring_1_unsigned @ A @ int @ K6 ) ) )
=> ( ( ( semiring_1_unsigned @ A @ int @ Q3 )
= ( times_times @ int @ N2 @ ( semiring_1_unsigned @ A @ int @ K6 ) ) )
=> ( ord_less_eq @ ( word @ A ) @ ( plus_plus @ ( word @ A ) @ P6 @ K6 ) @ Q3 ) ) ) ) ) ).
% udvd_incr0
thf(fact_5408_udvd__decr0,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [P6: word @ A,Q3: word @ A,N: int,K6: word @ A,N2: int] :
( ( ord_less @ ( word @ A ) @ P6 @ Q3 )
=> ( ( ( semiring_1_unsigned @ A @ int @ P6 )
= ( times_times @ int @ N @ ( semiring_1_unsigned @ A @ int @ K6 ) ) )
=> ( ( ( semiring_1_unsigned @ A @ int @ Q3 )
= ( times_times @ int @ N2 @ ( semiring_1_unsigned @ A @ int @ K6 ) ) )
=> ( ( ( semiring_1_unsigned @ A @ int @ Q3 )
= ( times_times @ int @ N2 @ ( semiring_1_unsigned @ A @ int @ K6 ) ) )
=> ( ord_less_eq @ ( word @ A ) @ P6 @ ( minus_minus @ ( word @ A ) @ Q3 @ K6 ) ) ) ) ) ) ) ).
% udvd_decr0
thf(fact_5409_pred__equals__eq2,axiom,
! [B: $tType,A: $tType,R: set @ ( product_prod @ A @ B ),S: set @ ( product_prod @ A @ B )] :
( ( ( ^ [X3: A,Y: B] : ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X3 @ Y ) @ R ) )
= ( ^ [X3: A,Y: B] : ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X3 @ Y ) @ S ) ) )
= ( R = S ) ) ).
% pred_equals_eq2
thf(fact_5410_udvd__incr_H,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [P6: word @ A,Q3: word @ A,Ua: int,N: int,K6: word @ A,N2: int] :
( ( ord_less @ ( word @ A ) @ P6 @ Q3 )
=> ( ( ( semiring_1_unsigned @ A @ int @ P6 )
= ( plus_plus @ int @ Ua @ ( times_times @ int @ N @ ( semiring_1_unsigned @ A @ int @ K6 ) ) ) )
=> ( ( ( semiring_1_unsigned @ A @ int @ Q3 )
= ( plus_plus @ int @ Ua @ ( times_times @ int @ N2 @ ( semiring_1_unsigned @ A @ int @ K6 ) ) ) )
=> ( ord_less_eq @ ( word @ A ) @ ( plus_plus @ ( word @ A ) @ P6 @ K6 ) @ Q3 ) ) ) ) ) ).
% udvd_incr'
thf(fact_5411_udvd__decr_H,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [P6: word @ A,Q3: word @ A,Ua: int,N: int,K6: word @ A,N2: int] :
( ( ord_less @ ( word @ A ) @ P6 @ Q3 )
=> ( ( ( semiring_1_unsigned @ A @ int @ P6 )
= ( plus_plus @ int @ Ua @ ( times_times @ int @ N @ ( semiring_1_unsigned @ A @ int @ K6 ) ) ) )
=> ( ( ( semiring_1_unsigned @ A @ int @ Q3 )
= ( plus_plus @ int @ Ua @ ( times_times @ int @ N2 @ ( semiring_1_unsigned @ A @ int @ K6 ) ) ) )
=> ( ( ( semiring_1_unsigned @ A @ int @ Q3 )
= ( plus_plus @ int @ Ua @ ( times_times @ int @ N2 @ ( semiring_1_unsigned @ A @ int @ K6 ) ) ) )
=> ( ord_less_eq @ ( word @ A ) @ P6 @ ( minus_minus @ ( word @ A ) @ Q3 @ K6 ) ) ) ) ) ) ) ).
% udvd_decr'
thf(fact_5412_and__mask__lt__2p,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [W: word @ A,N: nat] : ( ord_less @ int @ ( semiring_1_unsigned @ A @ int @ ( bit_se5824344872417868541ns_and @ ( word @ A ) @ W @ ( bit_se2239418461657761734s_mask @ ( word @ A ) @ N ) ) ) @ ( power_power @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ N ) ) ) ).
% and_mask_lt_2p
thf(fact_5413_mask__eq__iff,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [W: word @ A,N: nat] :
( ( ( bit_se5824344872417868541ns_and @ ( word @ A ) @ W @ ( bit_se2239418461657761734s_mask @ ( word @ A ) @ N ) )
= W )
= ( ord_less @ int @ ( semiring_1_unsigned @ A @ int @ W ) @ ( power_power @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ N ) ) ) ) ).
% mask_eq_iff
thf(fact_5414_subrelI,axiom,
! [B: $tType,A: $tType,R2: set @ ( product_prod @ A @ B ),S2: set @ ( product_prod @ A @ B )] :
( ! [X4: A,Y4: B] :
( ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X4 @ Y4 ) @ R2 )
=> ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X4 @ Y4 ) @ S2 ) )
=> ( ord_less_eq @ ( set @ ( product_prod @ A @ B ) ) @ R2 @ S2 ) ) ).
% subrelI
thf(fact_5415_accp__subset__induct,axiom,
! [A: $tType,D: A > $o,R: A > A > $o,X2: A,P: A > $o] :
( ( ord_less_eq @ ( A > $o ) @ D @ ( accp @ A @ R ) )
=> ( ! [X4: A,Z2: A] :
( ( D @ X4 )
=> ( ( R @ Z2 @ X4 )
=> ( D @ Z2 ) ) )
=> ( ( D @ X2 )
=> ( ! [X4: A] :
( ( D @ X4 )
=> ( ! [Z5: A] :
( ( R @ Z5 @ X4 )
=> ( P @ Z5 ) )
=> ( P @ X4 ) ) )
=> ( P @ X2 ) ) ) ) ) ).
% accp_subset_induct
thf(fact_5416_uint__range__size,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [W: word @ A] :
( ( ord_less_eq @ int @ ( zero_zero @ int ) @ ( semiring_1_unsigned @ A @ int @ W ) )
& ( ord_less @ int @ ( semiring_1_unsigned @ A @ int @ W ) @ ( power_power @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ ( size_size @ ( word @ A ) @ W ) ) ) ) ) ).
% uint_range_size
thf(fact_5417_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_5418_and__mask__mod__2p,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [W: word @ A,N: nat] :
( ( bit_se5824344872417868541ns_and @ ( word @ A ) @ W @ ( bit_se2239418461657761734s_mask @ ( word @ A ) @ N ) )
= ( ring_1_of_int @ ( word @ A ) @ ( modulo_modulo @ int @ ( semiring_1_unsigned @ A @ int @ W ) @ ( power_power @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ N ) ) ) ) ) ).
% and_mask_mod_2p
thf(fact_5419_and__mask__dvd,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [N: nat,W: word @ A] :
( ( dvd_dvd @ int @ ( power_power @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ N ) @ ( semiring_1_unsigned @ A @ int @ W ) )
= ( ( bit_se5824344872417868541ns_and @ ( word @ A ) @ W @ ( bit_se2239418461657761734s_mask @ ( word @ A ) @ N ) )
= ( zero_zero @ ( word @ A ) ) ) ) ) ).
% and_mask_dvd
thf(fact_5420_bot__empty__eq2,axiom,
! [B: $tType,A: $tType] :
( ( bot_bot @ ( A > B > $o ) )
= ( ^ [X3: A,Y: B] : ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X3 @ Y ) @ ( bot_bot @ ( set @ ( product_prod @ A @ B ) ) ) ) ) ) ).
% bot_empty_eq2
thf(fact_5421_top__empty__eq2,axiom,
! [B: $tType,A: $tType] :
( ( top_top @ ( A > B > $o ) )
= ( ^ [X3: A,Y: B] : ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X3 @ Y ) @ ( top_top @ ( set @ ( product_prod @ A @ B ) ) ) ) ) ) ).
% top_empty_eq2
thf(fact_5422_no__plus__overflow__uint__size,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [X2: word @ A,Y3: word @ A] :
( ( ord_less_eq @ ( word @ A ) @ X2 @ ( plus_plus @ ( word @ A ) @ X2 @ Y3 ) )
= ( ord_less @ int @ ( plus_plus @ int @ ( semiring_1_unsigned @ A @ int @ X2 ) @ ( semiring_1_unsigned @ A @ int @ Y3 ) ) @ ( power_power @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ ( size_size @ ( word @ A ) @ X2 ) ) ) ) ) ).
% no_plus_overflow_uint_size
thf(fact_5423_uint__plus__if__size,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [X2: word @ A,Y3: word @ A] :
( ( ( ord_less @ int @ ( plus_plus @ int @ ( semiring_1_unsigned @ A @ int @ X2 ) @ ( semiring_1_unsigned @ A @ int @ Y3 ) ) @ ( power_power @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ ( size_size @ ( word @ A ) @ X2 ) ) )
=> ( ( semiring_1_unsigned @ A @ int @ ( plus_plus @ ( word @ A ) @ X2 @ Y3 ) )
= ( plus_plus @ int @ ( semiring_1_unsigned @ A @ int @ X2 ) @ ( semiring_1_unsigned @ A @ int @ Y3 ) ) ) )
& ( ~ ( ord_less @ int @ ( plus_plus @ int @ ( semiring_1_unsigned @ A @ int @ X2 ) @ ( semiring_1_unsigned @ A @ int @ Y3 ) ) @ ( power_power @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ ( size_size @ ( word @ A ) @ X2 ) ) )
=> ( ( semiring_1_unsigned @ A @ int @ ( plus_plus @ ( word @ A ) @ X2 @ Y3 ) )
= ( minus_minus @ int @ ( plus_plus @ int @ ( semiring_1_unsigned @ A @ int @ X2 ) @ ( semiring_1_unsigned @ A @ int @ Y3 ) ) @ ( power_power @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ ( size_size @ ( word @ A ) @ X2 ) ) ) ) ) ) ) ).
% uint_plus_if_size
thf(fact_5424_pred__subset__eq,axiom,
! [A: $tType,R: set @ A,S: set @ A] :
( ( ord_less_eq @ ( A > $o )
@ ^ [X3: A] : ( member @ A @ X3 @ R )
@ ^ [X3: A] : ( member @ A @ X3 @ S ) )
= ( ord_less_eq @ ( set @ A ) @ R @ S ) ) ).
% pred_subset_eq
thf(fact_5425_uint__sub__if__size,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [Y3: word @ A,X2: word @ A] :
( ( ( ord_less_eq @ int @ ( semiring_1_unsigned @ A @ int @ Y3 ) @ ( semiring_1_unsigned @ A @ int @ X2 ) )
=> ( ( semiring_1_unsigned @ A @ int @ ( minus_minus @ ( word @ A ) @ X2 @ Y3 ) )
= ( minus_minus @ int @ ( semiring_1_unsigned @ A @ int @ X2 ) @ ( semiring_1_unsigned @ A @ int @ Y3 ) ) ) )
& ( ~ ( ord_less_eq @ int @ ( semiring_1_unsigned @ A @ int @ Y3 ) @ ( semiring_1_unsigned @ A @ int @ X2 ) )
=> ( ( semiring_1_unsigned @ A @ int @ ( minus_minus @ ( word @ A ) @ X2 @ Y3 ) )
= ( plus_plus @ int @ ( minus_minus @ int @ ( semiring_1_unsigned @ A @ int @ X2 ) @ ( semiring_1_unsigned @ A @ int @ Y3 ) ) @ ( power_power @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ ( size_size @ ( word @ A ) @ X2 ) ) ) ) ) ) ) ).
% uint_sub_if_size
thf(fact_5426_VEBT__internal_Ospace_Opelims,axiom,
! [X2: vEBT_VEBT,Y3: nat] :
( ( ( vEBT_VEBT_space @ X2 )
= Y3 )
=> ( ( accp @ vEBT_VEBT @ vEBT_VEBT_space_rel2 @ X2 )
=> ( ! [A4: $o,B4: $o] :
( ( X2
= ( vEBT_Leaf @ A4 @ B4 ) )
=> ( ( Y3
= ( 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] :
( ( X2
= ( vEBT_Node @ Info2 @ Deg2 @ TreeList2 @ Summary2 ) )
=> ( ( Y3
= ( 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_5427_VEBT__internal_OT__vebt__buildupi_H_Opelims,axiom,
! [X2: nat,Y3: int] :
( ( ( vEBT_V9176841429113362141ildupi @ X2 )
= Y3 )
=> ( ( accp @ nat @ vEBT_V3352910403632780892pi_rel @ X2 )
=> ( ( ( X2
= ( zero_zero @ nat ) )
=> ( ( Y3
= ( one_one @ int ) )
=> ~ ( accp @ nat @ vEBT_V3352910403632780892pi_rel @ ( zero_zero @ nat ) ) ) )
=> ( ( ( X2
= ( suc @ ( zero_zero @ nat ) ) )
=> ( ( Y3
= ( one_one @ int ) )
=> ~ ( accp @ nat @ vEBT_V3352910403632780892pi_rel @ ( suc @ ( zero_zero @ nat ) ) ) ) )
=> ~ ! [N3: nat] :
( ( X2
= ( suc @ ( suc @ N3 ) ) )
=> ( ( ( ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N3 )
=> ( Y3
= ( plus_plus @ int @ ( numeral_numeral @ int @ ( bit1 @ one2 ) ) @ ( plus_plus @ int @ ( vEBT_V9176841429113362141ildupi @ ( suc @ ( divide_divide @ nat @ N3 @ ( 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 @ N3 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( times_times @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ ( times_times @ int @ ( vEBT_V9176841429113362141ildupi @ ( suc @ ( divide_divide @ nat @ N3 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( power_power @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ N3 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) ) ) ) ) )
& ( ~ ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N3 )
=> ( Y3
= ( plus_plus @ int @ ( numeral_numeral @ int @ ( bit1 @ one2 ) ) @ ( plus_plus @ int @ ( vEBT_V9176841429113362141ildupi @ ( suc @ ( suc @ ( divide_divide @ nat @ N3 @ ( 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 @ N3 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( times_times @ int @ ( numeral_numeral @ int @ ( bit0 @ ( bit0 @ one2 ) ) ) @ ( times_times @ int @ ( vEBT_V9176841429113362141ildupi @ ( suc @ ( divide_divide @ nat @ N3 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( power_power @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ N3 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) ) ) ) ) ) )
=> ~ ( accp @ nat @ vEBT_V3352910403632780892pi_rel @ ( suc @ ( suc @ N3 ) ) ) ) ) ) ) ) ) ).
% VEBT_internal.T_vebt_buildupi'.pelims
thf(fact_5428_vebt__buildup_Opelims,axiom,
! [X2: nat,Y3: vEBT_VEBT] :
( ( ( vEBT_vebt_buildup @ X2 )
= Y3 )
=> ( ( accp @ nat @ vEBT_v4011308405150292612up_rel @ X2 )
=> ( ( ( X2
= ( zero_zero @ nat ) )
=> ( ( Y3
= ( vEBT_Leaf @ $false @ $false ) )
=> ~ ( accp @ nat @ vEBT_v4011308405150292612up_rel @ ( zero_zero @ nat ) ) ) )
=> ( ( ( X2
= ( suc @ ( zero_zero @ nat ) ) )
=> ( ( Y3
= ( vEBT_Leaf @ $false @ $false ) )
=> ~ ( accp @ nat @ vEBT_v4011308405150292612up_rel @ ( suc @ ( zero_zero @ nat ) ) ) ) )
=> ~ ! [Va3: nat] :
( ( X2
= ( suc @ ( suc @ Va3 ) ) )
=> ( ( ( ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( suc @ ( suc @ Va3 ) ) )
=> ( Y3
= ( vEBT_Node @ ( none @ ( product_prod @ nat @ nat ) ) @ ( suc @ ( suc @ Va3 ) ) @ ( replicate @ vEBT_VEBT @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( vEBT_vebt_buildup @ ( divide_divide @ nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( vEBT_vebt_buildup @ ( divide_divide @ nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) )
& ( ~ ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( suc @ ( suc @ Va3 ) ) )
=> ( Y3
= ( vEBT_Node @ ( none @ ( product_prod @ nat @ nat ) ) @ ( suc @ ( suc @ Va3 ) ) @ ( replicate @ vEBT_VEBT @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( suc @ ( divide_divide @ nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( vEBT_vebt_buildup @ ( divide_divide @ nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( vEBT_vebt_buildup @ ( suc @ ( divide_divide @ nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) ) ) )
=> ~ ( accp @ nat @ vEBT_v4011308405150292612up_rel @ ( suc @ ( suc @ Va3 ) ) ) ) ) ) ) ) ) ).
% vebt_buildup.pelims
thf(fact_5429_Suc__unat__minus__one,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [X2: word @ A] :
( ( X2
!= ( zero_zero @ ( word @ A ) ) )
=> ( ( suc @ ( semiring_1_unsigned @ A @ nat @ ( minus_minus @ ( word @ A ) @ X2 @ ( one_one @ ( word @ A ) ) ) ) )
= ( semiring_1_unsigned @ A @ nat @ X2 ) ) ) ) ).
% Suc_unat_minus_one
thf(fact_5430_le__ucast__ucast__le,axiom,
! [A: $tType,B: $tType] :
( ( ( type_len @ B )
& ( type_len @ A ) )
=> ! [X2: word @ A,Y3: word @ B] :
( ( ord_less_eq @ ( word @ A ) @ X2 @ ( semiring_1_unsigned @ B @ ( word @ A ) @ Y3 ) )
=> ( ord_less_eq @ ( word @ B ) @ ( semiring_1_unsigned @ A @ ( word @ B ) @ X2 ) @ Y3 ) ) ) ).
% le_ucast_ucast_le
thf(fact_5431_ucast__1,axiom,
! [A: $tType,B: $tType] :
( ( ( type_len @ B )
& ( type_len @ A ) )
=> ( ( semiring_1_unsigned @ A @ ( word @ B ) @ ( one_one @ ( word @ A ) ) )
= ( one_one @ ( word @ B ) ) ) ) ).
% ucast_1
thf(fact_5432_word__unat__Rep__inject1,axiom,
! [B: $tType,A: $tType] :
( ( ( type_len @ A )
& ( type_len @ B ) )
=> ! [X2: word @ A] :
( ( ( semiring_1_unsigned @ A @ nat @ X2 )
= ( semiring_1_unsigned @ B @ nat @ ( one_one @ ( word @ B ) ) ) )
= ( X2
= ( one_one @ ( word @ A ) ) ) ) ) ).
% word_unat_Rep_inject1
thf(fact_5433_unat__mono,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [A3: word @ A,B3: word @ A] :
( ( ord_less @ ( word @ A ) @ A3 @ B3 )
=> ( ord_less @ nat @ ( semiring_1_unsigned @ A @ nat @ A3 ) @ ( semiring_1_unsigned @ A @ nat @ B3 ) ) ) ) ).
% unat_mono
thf(fact_5434_word__less__nat__alt,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ( ( ord_less @ ( word @ A ) )
= ( ^ [A6: word @ A,B7: word @ A] : ( ord_less @ nat @ ( semiring_1_unsigned @ A @ nat @ A6 ) @ ( semiring_1_unsigned @ A @ nat @ B7 ) ) ) ) ) ).
% word_less_nat_alt
thf(fact_5435_word__le__nat__alt,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ( ( ord_less_eq @ ( word @ A ) )
= ( ^ [A6: word @ A,B7: word @ A] : ( ord_less_eq @ nat @ ( semiring_1_unsigned @ A @ nat @ A6 ) @ ( semiring_1_unsigned @ A @ nat @ B7 ) ) ) ) ) ).
% word_le_nat_alt
thf(fact_5436_unat__1,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ( ( semiring_1_unsigned @ A @ nat @ ( one_one @ ( word @ A ) ) )
= ( one_one @ nat ) ) ) ).
% unat_1
thf(fact_5437_word__unat__and__lt,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [X2: word @ A,N: nat,Y3: word @ A] :
( ( ( ord_less @ nat @ ( semiring_1_unsigned @ A @ nat @ X2 ) @ N )
| ( ord_less @ nat @ ( semiring_1_unsigned @ A @ nat @ Y3 ) @ N ) )
=> ( ord_less @ nat @ ( semiring_1_unsigned @ A @ nat @ ( bit_se5824344872417868541ns_and @ ( word @ A ) @ X2 @ Y3 ) ) @ N ) ) ) ).
% word_unat_and_lt
thf(fact_5438_le__unat__uoi,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [Y3: nat,Z: word @ A] :
( ( ord_less_eq @ nat @ Y3 @ ( semiring_1_unsigned @ A @ nat @ Z ) )
=> ( ( semiring_1_unsigned @ A @ nat @ ( semiring_1_of_nat @ ( word @ A ) @ Y3 ) )
= Y3 ) ) ) ).
% le_unat_uoi
thf(fact_5439_uno__simps_I2_J,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [Z: word @ A,N: nat] :
( ( semiring_1_unsigned @ A @ nat @ ( semiring_1_of_nat @ ( word @ A ) @ ( divide_divide @ nat @ ( semiring_1_unsigned @ A @ nat @ Z ) @ N ) ) )
= ( divide_divide @ nat @ ( semiring_1_unsigned @ A @ nat @ Z ) @ N ) ) ) ).
% uno_simps(2)
thf(fact_5440_unat__div,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [X2: word @ A,Y3: word @ A] :
( ( semiring_1_unsigned @ A @ nat @ ( divide_divide @ ( word @ A ) @ X2 @ Y3 ) )
= ( divide_divide @ nat @ ( semiring_1_unsigned @ A @ nat @ X2 ) @ ( semiring_1_unsigned @ A @ nat @ Y3 ) ) ) ) ).
% unat_div
thf(fact_5441_unat__div__distrib,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [V: word @ A,W: word @ A] :
( ( semiring_1_unsigned @ A @ nat @ ( divide_divide @ ( word @ A ) @ V @ W ) )
= ( divide_divide @ nat @ ( semiring_1_unsigned @ A @ nat @ V ) @ ( semiring_1_unsigned @ A @ nat @ W ) ) ) ) ).
% unat_div_distrib
thf(fact_5442_max__lt,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [A3: word @ A,B3: word @ A,C3: word @ A] :
( ( semiring_1_unsigned @ A @ nat @ ( divide_divide @ ( word @ A ) @ ( ord_max @ ( word @ A ) @ A3 @ B3 ) @ C3 ) )
= ( divide_divide @ nat @ ( semiring_1_unsigned @ A @ nat @ ( ord_max @ ( word @ A ) @ A3 @ B3 ) ) @ ( semiring_1_unsigned @ A @ nat @ C3 ) ) ) ) ).
% max_lt
thf(fact_5443_unat__eq__1,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [X2: word @ A] :
( ( ( semiring_1_unsigned @ A @ nat @ X2 )
= ( suc @ ( zero_zero @ nat ) ) )
= ( X2
= ( one_one @ ( word @ A ) ) ) ) ) ).
% unat_eq_1
thf(fact_5444_unat__gt__0,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [X2: word @ A] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ ( semiring_1_unsigned @ A @ nat @ X2 ) )
= ( X2
!= ( zero_zero @ ( word @ A ) ) ) ) ) ).
% unat_gt_0
thf(fact_5445_un__ui__le,axiom,
! [A: $tType,B: $tType] :
( ( ( type_len @ B )
& ( type_len @ A ) )
=> ! [A3: word @ A,B3: word @ B] :
( ( ord_less_eq @ nat @ ( semiring_1_unsigned @ A @ nat @ A3 ) @ ( semiring_1_unsigned @ B @ nat @ B3 ) )
= ( ord_less_eq @ int @ ( semiring_1_unsigned @ A @ int @ A3 ) @ ( semiring_1_unsigned @ B @ int @ B3 ) ) ) ) ).
% un_ui_le
thf(fact_5446_unat__plus__simple,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [X2: word @ A,Y3: word @ A] :
( ( ord_less_eq @ ( word @ A ) @ X2 @ ( plus_plus @ ( word @ A ) @ X2 @ Y3 ) )
= ( ( semiring_1_unsigned @ A @ nat @ ( plus_plus @ ( word @ A ) @ X2 @ Y3 ) )
= ( plus_plus @ nat @ ( semiring_1_unsigned @ A @ nat @ X2 ) @ ( semiring_1_unsigned @ A @ nat @ Y3 ) ) ) ) ) ).
% unat_plus_simple
thf(fact_5447_unat__sub,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [B3: word @ A,A3: word @ A] :
( ( ord_less_eq @ ( word @ A ) @ B3 @ A3 )
=> ( ( semiring_1_unsigned @ A @ nat @ ( minus_minus @ ( word @ A ) @ A3 @ B3 ) )
= ( minus_minus @ nat @ ( semiring_1_unsigned @ A @ nat @ A3 ) @ ( semiring_1_unsigned @ A @ nat @ B3 ) ) ) ) ) ).
% unat_sub
thf(fact_5448_unat__less__helper,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [X2: word @ A,N: nat] :
( ( ord_less @ ( word @ A ) @ X2 @ ( semiring_1_of_nat @ ( word @ A ) @ N ) )
=> ( ord_less @ nat @ ( semiring_1_unsigned @ A @ nat @ X2 ) @ N ) ) ) ).
% unat_less_helper
thf(fact_5449_word__of__nat__less,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [N: nat,X2: word @ A] :
( ( ord_less @ nat @ N @ ( semiring_1_unsigned @ A @ nat @ X2 ) )
=> ( ord_less @ ( word @ A ) @ ( semiring_1_of_nat @ ( word @ A ) @ N ) @ X2 ) ) ) ).
% word_of_nat_less
thf(fact_5450_word__of__nat__le,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [N: nat,X2: word @ A] :
( ( ord_less_eq @ nat @ N @ ( semiring_1_unsigned @ A @ nat @ X2 ) )
=> ( ord_less_eq @ ( word @ A ) @ ( semiring_1_of_nat @ ( word @ A ) @ N ) @ X2 ) ) ) ).
% word_of_nat_le
thf(fact_5451_word__unat__less__le,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [A3: word @ A,B3: nat] :
( ( ord_less_eq @ ( word @ A ) @ A3 @ ( semiring_1_of_nat @ ( word @ A ) @ B3 ) )
=> ( ord_less_eq @ nat @ ( semiring_1_unsigned @ A @ nat @ A3 ) @ B3 ) ) ) ).
% word_unat_less_le
thf(fact_5452_word__arith__nat__add,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ( ( plus_plus @ ( word @ A ) )
= ( ^ [A6: word @ A,B7: word @ A] : ( semiring_1_of_nat @ ( word @ A ) @ ( plus_plus @ nat @ ( semiring_1_unsigned @ A @ nat @ A6 ) @ ( semiring_1_unsigned @ A @ nat @ B7 ) ) ) ) ) ) ).
% word_arith_nat_add
thf(fact_5453_word__arith__nat__mult,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ( ( times_times @ ( word @ A ) )
= ( ^ [A6: word @ A,B7: word @ A] : ( semiring_1_of_nat @ ( word @ A ) @ ( times_times @ nat @ ( semiring_1_unsigned @ A @ nat @ A6 ) @ ( semiring_1_unsigned @ A @ nat @ B7 ) ) ) ) ) ) ).
% word_arith_nat_mult
thf(fact_5454_word__arith__nat__div,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ( ( divide_divide @ ( word @ A ) )
= ( ^ [A6: word @ A,B7: word @ A] : ( semiring_1_of_nat @ ( word @ A ) @ ( divide_divide @ nat @ ( semiring_1_unsigned @ A @ nat @ A6 ) @ ( semiring_1_unsigned @ A @ nat @ B7 ) ) ) ) ) ) ).
% word_arith_nat_div
thf(fact_5455_unat__1__0,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [X2: word @ A] :
( ( ord_less_eq @ ( word @ A ) @ ( one_one @ ( word @ A ) ) @ X2 )
= ( ord_less @ nat @ ( zero_zero @ nat ) @ ( semiring_1_unsigned @ A @ nat @ X2 ) ) ) ) ).
% unat_1_0
thf(fact_5456_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_5457_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_5458_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_5459_Suc__unat__diff__1,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [X2: word @ A] :
( ( ord_less_eq @ ( word @ A ) @ ( one_one @ ( word @ A ) ) @ X2 )
=> ( ( suc @ ( semiring_1_unsigned @ A @ nat @ ( minus_minus @ ( word @ A ) @ X2 @ ( one_one @ ( word @ A ) ) ) ) )
= ( semiring_1_unsigned @ A @ nat @ X2 ) ) ) ) ).
% Suc_unat_diff_1
thf(fact_5460_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_5461_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_5462_measure__unat,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [P6: word @ A] :
( ( P6
!= ( zero_zero @ ( word @ A ) ) )
=> ( ord_less @ nat @ ( semiring_1_unsigned @ A @ nat @ ( minus_minus @ ( word @ A ) @ P6 @ ( one_one @ ( word @ A ) ) ) ) @ ( semiring_1_unsigned @ A @ nat @ P6 ) ) ) ) ).
% measure_unat
thf(fact_5463_word__overflow__unat,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [X2: word @ A] :
( ( ( semiring_1_unsigned @ A @ nat @ ( plus_plus @ ( word @ A ) @ X2 @ ( one_one @ ( word @ A ) ) ) )
= ( plus_plus @ nat @ ( semiring_1_unsigned @ A @ nat @ X2 ) @ ( one_one @ nat ) ) )
| ( ( plus_plus @ ( word @ A ) @ X2 @ ( one_one @ ( word @ A ) ) )
= ( zero_zero @ ( word @ A ) ) ) ) ) ).
% word_overflow_unat
thf(fact_5464_unat__minus__one,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [W: word @ A] :
( ( W
!= ( zero_zero @ ( word @ A ) ) )
=> ( ( semiring_1_unsigned @ A @ nat @ ( minus_minus @ ( word @ A ) @ W @ ( one_one @ ( word @ A ) ) ) )
= ( minus_minus @ nat @ ( semiring_1_unsigned @ A @ nat @ W ) @ ( one_one @ nat ) ) ) ) ) ).
% unat_minus_one
thf(fact_5465_lt__plus__1__le__word,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [N: nat,MaxBound: word @ A,X2: word @ A] :
( ( ord_less @ nat @ N @ ( semiring_1_unsigned @ A @ nat @ MaxBound ) )
=> ( ( ord_less @ ( word @ A ) @ X2 @ ( plus_plus @ ( word @ A ) @ ( one_one @ ( word @ A ) ) @ ( semiring_1_of_nat @ ( word @ A ) @ N ) ) )
= ( ord_less_eq @ ( word @ A ) @ X2 @ ( semiring_1_of_nat @ ( word @ A ) @ N ) ) ) ) ) ).
% lt_plus_1_le_word
thf(fact_5466_even__word__imp__odd__next,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [X2: word @ A] :
( ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( semiring_1_unsigned @ A @ nat @ X2 ) )
=> ( ( ( plus_plus @ ( word @ A ) @ X2 @ ( one_one @ ( word @ A ) ) )
= ( zero_zero @ ( word @ A ) ) )
| ~ ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( semiring_1_unsigned @ A @ nat @ ( plus_plus @ ( word @ A ) @ X2 @ ( one_one @ ( word @ A ) ) ) ) ) ) ) ) ).
% even_word_imp_odd_next
thf(fact_5467_odd__word__imp__even__next,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [X2: word @ A] :
( ~ ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( semiring_1_unsigned @ A @ nat @ X2 ) )
=> ( ( ( plus_plus @ ( word @ A ) @ X2 @ ( one_one @ ( word @ A ) ) )
= ( zero_zero @ ( word @ A ) ) )
| ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( semiring_1_unsigned @ A @ nat @ ( plus_plus @ ( word @ A ) @ X2 @ ( one_one @ ( word @ A ) ) ) ) ) ) ) ) ).
% odd_word_imp_even_next
thf(fact_5468_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_5469_and__mask__dvd__nat,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [N: nat,W: word @ A] :
( ( dvd_dvd @ nat @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N ) @ ( semiring_1_unsigned @ A @ nat @ W ) )
= ( ( bit_se5824344872417868541ns_and @ ( word @ A ) @ W @ ( bit_se2239418461657761734s_mask @ ( word @ A ) @ N ) )
= ( zero_zero @ ( word @ A ) ) ) ) ) ).
% and_mask_dvd_nat
thf(fact_5470_of__nat__eq__size,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [N: nat,W: word @ A] :
( ( ( semiring_1_of_nat @ ( word @ A ) @ N )
= W )
= ( ? [Q5: nat] :
( N
= ( plus_plus @ nat @ ( semiring_1_unsigned @ A @ nat @ W ) @ ( times_times @ nat @ Q5 @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( size_size @ ( word @ A ) @ W ) ) ) ) ) ) ) ) ).
% of_nat_eq_size
thf(fact_5471_unat__plus__if__size,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [X2: word @ A,Y3: word @ A] :
( ( ( ord_less @ nat @ ( plus_plus @ nat @ ( semiring_1_unsigned @ A @ nat @ X2 ) @ ( semiring_1_unsigned @ A @ nat @ Y3 ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( size_size @ ( word @ A ) @ X2 ) ) )
=> ( ( semiring_1_unsigned @ A @ nat @ ( plus_plus @ ( word @ A ) @ X2 @ Y3 ) )
= ( plus_plus @ nat @ ( semiring_1_unsigned @ A @ nat @ X2 ) @ ( semiring_1_unsigned @ A @ nat @ Y3 ) ) ) )
& ( ~ ( ord_less @ nat @ ( plus_plus @ nat @ ( semiring_1_unsigned @ A @ nat @ X2 ) @ ( semiring_1_unsigned @ A @ nat @ Y3 ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( size_size @ ( word @ A ) @ X2 ) ) )
=> ( ( semiring_1_unsigned @ A @ nat @ ( plus_plus @ ( word @ A ) @ X2 @ Y3 ) )
= ( minus_minus @ nat @ ( plus_plus @ nat @ ( semiring_1_unsigned @ A @ nat @ X2 ) @ ( semiring_1_unsigned @ A @ nat @ Y3 ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( size_size @ ( word @ A ) @ X2 ) ) ) ) ) ) ) ).
% unat_plus_if_size
thf(fact_5472_unat__sub__if__size,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [Y3: word @ A,X2: word @ A] :
( ( ( ord_less_eq @ nat @ ( semiring_1_unsigned @ A @ nat @ Y3 ) @ ( semiring_1_unsigned @ A @ nat @ X2 ) )
=> ( ( semiring_1_unsigned @ A @ nat @ ( minus_minus @ ( word @ A ) @ X2 @ Y3 ) )
= ( minus_minus @ nat @ ( semiring_1_unsigned @ A @ nat @ X2 ) @ ( semiring_1_unsigned @ A @ nat @ Y3 ) ) ) )
& ( ~ ( ord_less_eq @ nat @ ( semiring_1_unsigned @ A @ nat @ Y3 ) @ ( semiring_1_unsigned @ A @ nat @ X2 ) )
=> ( ( semiring_1_unsigned @ A @ nat @ ( minus_minus @ ( word @ A ) @ X2 @ Y3 ) )
= ( minus_minus @ nat @ ( plus_plus @ nat @ ( semiring_1_unsigned @ A @ nat @ X2 ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( size_size @ ( word @ A ) @ X2 ) ) ) @ ( semiring_1_unsigned @ A @ nat @ Y3 ) ) ) ) ) ) ).
% unat_sub_if_size
thf(fact_5473_no__plus__overflow__unat__size,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [X2: word @ A,Y3: word @ A] :
( ( ord_less_eq @ ( word @ A ) @ X2 @ ( plus_plus @ ( word @ A ) @ X2 @ Y3 ) )
= ( ord_less @ nat @ ( plus_plus @ nat @ ( semiring_1_unsigned @ A @ nat @ X2 ) @ ( semiring_1_unsigned @ A @ nat @ Y3 ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( size_size @ ( word @ A ) @ X2 ) ) ) ) ) ).
% no_plus_overflow_unat_size
thf(fact_5474_word__unat__mask__lt,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [M: nat,W: word @ A] :
( ( ord_less_eq @ nat @ M @ ( size_size @ ( word @ A ) @ W ) )
=> ( ord_less @ nat @ ( semiring_1_unsigned @ A @ nat @ ( bit_se5824344872417868541ns_and @ ( word @ A ) @ W @ ( bit_se2239418461657761734s_mask @ ( word @ A ) @ M ) ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ M ) ) ) ) ).
% word_unat_mask_lt
thf(fact_5475_VEBT__internal_Ospace_H_Opelims,axiom,
! [X2: vEBT_VEBT,Y3: nat] :
( ( ( vEBT_VEBT_space2 @ X2 )
= Y3 )
=> ( ( accp @ vEBT_VEBT @ vEBT_VEBT_space_rel @ X2 )
=> ( ! [A4: $o,B4: $o] :
( ( X2
= ( vEBT_Leaf @ A4 @ B4 ) )
=> ( ( Y3
= ( 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] :
( ( X2
= ( vEBT_Node @ Info2 @ Deg2 @ TreeList2 @ Summary2 ) )
=> ( ( Y3
= ( 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_5476_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,
! [X2: nat,Y3: nat] :
( ( ( vEBT_V8646137997579335489_i_l_d @ X2 )
= Y3 )
=> ( ( accp @ nat @ vEBT_V5144397997797733112_d_rel @ X2 )
=> ( ( ( X2
= ( zero_zero @ nat ) )
=> ( ( Y3
= ( numeral_numeral @ nat @ ( bit0 @ ( bit0 @ one2 ) ) ) )
=> ~ ( accp @ nat @ vEBT_V5144397997797733112_d_rel @ ( zero_zero @ nat ) ) ) )
=> ( ( ( X2
= ( suc @ ( zero_zero @ nat ) ) )
=> ( ( Y3
= ( numeral_numeral @ nat @ ( bit0 @ ( bit0 @ one2 ) ) ) )
=> ~ ( accp @ nat @ vEBT_V5144397997797733112_d_rel @ ( suc @ ( zero_zero @ nat ) ) ) ) )
=> ~ ! [Va3: nat] :
( ( X2
= ( suc @ ( suc @ Va3 ) ) )
=> ( ( ( ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( suc @ ( suc @ Va3 ) ) )
=> ( Y3
= ( 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 @ Va3 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( times_times @ nat @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( vEBT_V8646137997579335489_i_l_d @ ( divide_divide @ nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) ) ) )
& ( ~ ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( suc @ ( suc @ Va3 ) ) )
=> ( Y3
= ( plus_plus @ nat @ ( plus_plus @ nat @ ( numeral_numeral @ nat @ ( bit0 @ ( bit0 @ ( bit1 @ one2 ) ) ) ) @ ( vEBT_V8646137997579335489_i_l_d @ ( suc @ ( divide_divide @ nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) @ ( times_times @ nat @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( suc @ ( divide_divide @ nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( vEBT_V8646137997579335489_i_l_d @ ( divide_divide @ nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) ) ) )
=> ~ ( accp @ nat @ vEBT_V5144397997797733112_d_rel @ ( suc @ ( suc @ Va3 ) ) ) ) ) ) ) ) ) ).
% VEBT_internal.T\<^sub>b\<^sub>u\<^sub>i\<^sub>l\<^sub>d.pelims
thf(fact_5477_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,
! [X2: nat,Y3: nat] :
( ( ( vEBT_V8346862874174094_d_u_p @ X2 )
= Y3 )
=> ( ( accp @ nat @ vEBT_V1247956027447740395_p_rel @ X2 )
=> ( ( ( X2
= ( zero_zero @ nat ) )
=> ( ( Y3
= ( numeral_numeral @ nat @ ( bit1 @ one2 ) ) )
=> ~ ( accp @ nat @ vEBT_V1247956027447740395_p_rel @ ( zero_zero @ nat ) ) ) )
=> ( ( ( X2
= ( suc @ ( zero_zero @ nat ) ) )
=> ( ( Y3
= ( numeral_numeral @ nat @ ( bit1 @ one2 ) ) )
=> ~ ( accp @ nat @ vEBT_V1247956027447740395_p_rel @ ( suc @ ( zero_zero @ nat ) ) ) ) )
=> ~ ! [Va3: nat] :
( ( X2
= ( suc @ ( suc @ Va3 ) ) )
=> ( ( ( ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( suc @ ( suc @ Va3 ) ) )
=> ( Y3
= ( 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 @ Va3 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( times_times @ nat @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( plus_plus @ nat @ ( vEBT_V8346862874174094_d_u_p @ ( divide_divide @ nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( one_one @ nat ) ) ) ) ) ) )
& ( ~ ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( suc @ ( suc @ Va3 ) ) )
=> ( Y3
= ( plus_plus @ nat @ ( plus_plus @ nat @ ( numeral_numeral @ nat @ ( bit1 @ ( bit1 @ ( bit0 @ one2 ) ) ) ) @ ( vEBT_V8346862874174094_d_u_p @ ( suc @ ( divide_divide @ nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) @ ( times_times @ nat @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( suc @ ( divide_divide @ nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( plus_plus @ nat @ ( vEBT_V8346862874174094_d_u_p @ ( divide_divide @ nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( one_one @ nat ) ) ) ) ) ) )
=> ~ ( accp @ nat @ vEBT_V1247956027447740395_p_rel @ ( suc @ ( suc @ Va3 ) ) ) ) ) ) ) ) ) ).
% VEBT_internal.T\<^sub>b\<^sub>u\<^sub>i\<^sub>l\<^sub>d\<^sub>u\<^sub>p.pelims
thf(fact_5478_VEBT__internal_OTb_Opelims,axiom,
! [X2: nat,Y3: int] :
( ( ( vEBT_VEBT_Tb @ X2 )
= Y3 )
=> ( ( accp @ nat @ vEBT_VEBT_Tb_rel2 @ X2 )
=> ( ( ( X2
= ( zero_zero @ nat ) )
=> ( ( Y3
= ( numeral_numeral @ int @ ( bit1 @ one2 ) ) )
=> ~ ( accp @ nat @ vEBT_VEBT_Tb_rel2 @ ( zero_zero @ nat ) ) ) )
=> ( ( ( X2
= ( suc @ ( zero_zero @ nat ) ) )
=> ( ( Y3
= ( numeral_numeral @ int @ ( bit1 @ one2 ) ) )
=> ~ ( accp @ nat @ vEBT_VEBT_Tb_rel2 @ ( suc @ ( zero_zero @ nat ) ) ) ) )
=> ~ ! [N3: nat] :
( ( X2
= ( suc @ ( suc @ N3 ) ) )
=> ( ( ( ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N3 )
=> ( Y3
= ( plus_plus @ int @ ( plus_plus @ int @ ( numeral_numeral @ int @ ( bit1 @ ( bit0 @ one2 ) ) ) @ ( vEBT_VEBT_Tb @ ( suc @ ( divide_divide @ nat @ N3 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) @ ( times_times @ int @ ( vEBT_VEBT_Tb @ ( suc @ ( divide_divide @ nat @ N3 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( power_power @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ ( suc @ ( divide_divide @ nat @ N3 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) ) ) )
& ( ~ ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N3 )
=> ( Y3
= ( plus_plus @ int @ ( plus_plus @ int @ ( numeral_numeral @ int @ ( bit1 @ ( bit0 @ one2 ) ) ) @ ( vEBT_VEBT_Tb @ ( suc @ ( suc @ ( divide_divide @ nat @ N3 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) ) @ ( times_times @ int @ ( vEBT_VEBT_Tb @ ( suc @ ( divide_divide @ nat @ N3 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( power_power @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ ( suc @ ( suc @ ( divide_divide @ nat @ N3 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) ) ) ) ) )
=> ~ ( accp @ nat @ vEBT_VEBT_Tb_rel2 @ ( suc @ ( suc @ N3 ) ) ) ) ) ) ) ) ) ).
% VEBT_internal.Tb.pelims
thf(fact_5479_VEBT__internal_OTb_H_Opelims,axiom,
! [X2: nat,Y3: nat] :
( ( ( vEBT_VEBT_Tb2 @ X2 )
= Y3 )
=> ( ( accp @ nat @ vEBT_VEBT_Tb_rel @ X2 )
=> ( ( ( X2
= ( zero_zero @ nat ) )
=> ( ( Y3
= ( numeral_numeral @ nat @ ( bit1 @ one2 ) ) )
=> ~ ( accp @ nat @ vEBT_VEBT_Tb_rel @ ( zero_zero @ nat ) ) ) )
=> ( ( ( X2
= ( suc @ ( zero_zero @ nat ) ) )
=> ( ( Y3
= ( numeral_numeral @ nat @ ( bit1 @ one2 ) ) )
=> ~ ( accp @ nat @ vEBT_VEBT_Tb_rel @ ( suc @ ( zero_zero @ nat ) ) ) ) )
=> ~ ! [N3: nat] :
( ( X2
= ( suc @ ( suc @ N3 ) ) )
=> ( ( ( ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N3 )
=> ( Y3
= ( plus_plus @ nat @ ( plus_plus @ nat @ ( numeral_numeral @ nat @ ( bit1 @ ( bit0 @ one2 ) ) ) @ ( vEBT_VEBT_Tb2 @ ( suc @ ( divide_divide @ nat @ N3 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) @ ( times_times @ nat @ ( vEBT_VEBT_Tb2 @ ( suc @ ( divide_divide @ nat @ N3 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( suc @ ( divide_divide @ nat @ N3 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) ) ) )
& ( ~ ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N3 )
=> ( Y3
= ( plus_plus @ nat @ ( plus_plus @ nat @ ( numeral_numeral @ nat @ ( bit1 @ ( bit0 @ one2 ) ) ) @ ( vEBT_VEBT_Tb2 @ ( suc @ ( suc @ ( divide_divide @ nat @ N3 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) ) @ ( times_times @ nat @ ( vEBT_VEBT_Tb2 @ ( suc @ ( divide_divide @ nat @ N3 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( suc @ ( suc @ ( divide_divide @ nat @ N3 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) ) ) ) ) )
=> ~ ( accp @ nat @ vEBT_VEBT_Tb_rel @ ( suc @ ( suc @ N3 ) ) ) ) ) ) ) ) ) ).
% VEBT_internal.Tb'.pelims
thf(fact_5480_VEBT__internal_Ocnt_Opelims,axiom,
! [X2: vEBT_VEBT,Y3: real] :
( ( ( vEBT_VEBT_cnt @ X2 )
= Y3 )
=> ( ( accp @ vEBT_VEBT @ vEBT_VEBT_cnt_rel2 @ X2 )
=> ( ! [A4: $o,B4: $o] :
( ( X2
= ( vEBT_Leaf @ A4 @ B4 ) )
=> ( ( Y3
= ( 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] :
( ( X2
= ( vEBT_Node @ Info2 @ Deg2 @ TreeList2 @ Summary2 ) )
=> ( ( Y3
= ( 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_5481_VEBT__internal_OT__vebt__buildupi_Opelims,axiom,
! [X2: nat,Y3: nat] :
( ( ( vEBT_V441764108873111860ildupi @ X2 )
= Y3 )
=> ( ( accp @ nat @ vEBT_V2957053500504383685pi_rel @ X2 )
=> ( ( ( X2
= ( zero_zero @ nat ) )
=> ( ( Y3
= ( suc @ ( zero_zero @ nat ) ) )
=> ~ ( accp @ nat @ vEBT_V2957053500504383685pi_rel @ ( zero_zero @ nat ) ) ) )
=> ( ( ( X2
= ( suc @ ( zero_zero @ nat ) ) )
=> ( ( Y3
= ( suc @ ( zero_zero @ nat ) ) )
=> ~ ( accp @ nat @ vEBT_V2957053500504383685pi_rel @ ( suc @ ( zero_zero @ nat ) ) ) ) )
=> ~ ! [N3: nat] :
( ( X2
= ( suc @ ( suc @ N3 ) ) )
=> ( ( ( ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N3 )
=> ( Y3
= ( suc @ ( suc @ ( suc @ ( plus_plus @ nat @ ( vEBT_V441764108873111860ildupi @ ( suc @ ( divide_divide @ nat @ N3 @ ( 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 @ N3 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( times_times @ nat @ ( vEBT_V441764108873111860ildupi @ ( suc @ ( divide_divide @ nat @ N3 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ N3 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) ) ) ) ) ) ) )
& ( ~ ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N3 )
=> ( Y3
= ( suc @ ( suc @ ( suc @ ( plus_plus @ nat @ ( vEBT_V441764108873111860ildupi @ ( suc @ ( suc @ ( divide_divide @ nat @ N3 @ ( 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 @ N3 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ ( bit0 @ one2 ) ) ) @ ( times_times @ nat @ ( vEBT_V441764108873111860ildupi @ ( suc @ ( divide_divide @ nat @ N3 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ N3 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) ) ) ) ) ) ) ) )
=> ~ ( accp @ nat @ vEBT_V2957053500504383685pi_rel @ ( suc @ ( suc @ N3 ) ) ) ) ) ) ) ) ) ).
% VEBT_internal.T_vebt_buildupi.pelims
thf(fact_5482_VEBT__internal_Ocnt_H_Opelims,axiom,
! [X2: vEBT_VEBT,Y3: nat] :
( ( ( vEBT_VEBT_cnt2 @ X2 )
= Y3 )
=> ( ( accp @ vEBT_VEBT @ vEBT_VEBT_cnt_rel @ X2 )
=> ( ! [A4: $o,B4: $o] :
( ( X2
= ( vEBT_Leaf @ A4 @ B4 ) )
=> ( ( Y3
= ( 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] :
( ( X2
= ( vEBT_Node @ Info2 @ Deg2 @ TreeList2 @ Summary2 ) )
=> ( ( Y3
= ( 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_5483_vebt__minti_Opelims,axiom,
! [X2: vEBT_VEBTi,Y3: heap_Time_Heap @ ( option @ nat )] :
( ( ( vEBT_vebt_minti @ X2 )
= Y3 )
=> ( ( accp @ vEBT_VEBTi @ vEBT_vebt_minti_rel @ X2 )
=> ( ! [A4: $o,B4: $o] :
( ( X2
= ( vEBT_Leafi @ A4 @ B4 ) )
=> ( ( ( A4
=> ( Y3
= ( heap_Time_return @ ( option @ nat ) @ ( some @ nat @ ( zero_zero @ nat ) ) ) ) )
& ( ~ A4
=> ( ( B4
=> ( Y3
= ( heap_Time_return @ ( option @ nat ) @ ( some @ nat @ ( one_one @ nat ) ) ) ) )
& ( ~ B4
=> ( Y3
= ( heap_Time_return @ ( option @ nat ) @ ( none @ nat ) ) ) ) ) ) )
=> ~ ( accp @ vEBT_VEBTi @ vEBT_vebt_minti_rel @ ( vEBT_Leafi @ A4 @ B4 ) ) ) )
=> ( ! [Uu2: nat,Uv2: array @ vEBT_VEBTi,Uw2: vEBT_VEBTi] :
( ( X2
= ( vEBT_Nodei @ ( none @ ( product_prod @ nat @ nat ) ) @ Uu2 @ Uv2 @ Uw2 ) )
=> ( ( Y3
= ( heap_Time_return @ ( option @ nat ) @ ( none @ nat ) ) )
=> ~ ( accp @ vEBT_VEBTi @ vEBT_vebt_minti_rel @ ( vEBT_Nodei @ ( none @ ( product_prod @ nat @ nat ) ) @ Uu2 @ Uv2 @ Uw2 ) ) ) )
=> ~ ! [Mi2: nat,Ma2: nat,Ux2: nat,Uy2: array @ vEBT_VEBTi,Uz2: vEBT_VEBTi] :
( ( X2
= ( vEBT_Nodei @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi2 @ Ma2 ) ) @ Ux2 @ Uy2 @ Uz2 ) )
=> ( ( Y3
= ( 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 ) ) @ Ux2 @ Uy2 @ Uz2 ) ) ) ) ) ) ) ) ).
% vebt_minti.pelims
thf(fact_5484_vebt__maxti_Opelims,axiom,
! [X2: vEBT_VEBTi,Y3: heap_Time_Heap @ ( option @ nat )] :
( ( ( vEBT_vebt_maxti @ X2 )
= Y3 )
=> ( ( accp @ vEBT_VEBTi @ vEBT_vebt_maxti_rel @ X2 )
=> ( ! [A4: $o,B4: $o] :
( ( X2
= ( vEBT_Leafi @ A4 @ B4 ) )
=> ( ( ( B4
=> ( Y3
= ( heap_Time_return @ ( option @ nat ) @ ( some @ nat @ ( one_one @ nat ) ) ) ) )
& ( ~ B4
=> ( ( A4
=> ( Y3
= ( heap_Time_return @ ( option @ nat ) @ ( some @ nat @ ( zero_zero @ nat ) ) ) ) )
& ( ~ A4
=> ( Y3
= ( heap_Time_return @ ( option @ nat ) @ ( none @ nat ) ) ) ) ) ) )
=> ~ ( accp @ vEBT_VEBTi @ vEBT_vebt_maxti_rel @ ( vEBT_Leafi @ A4 @ B4 ) ) ) )
=> ( ! [Uu2: nat,Uv2: array @ vEBT_VEBTi,Uw2: vEBT_VEBTi] :
( ( X2
= ( vEBT_Nodei @ ( none @ ( product_prod @ nat @ nat ) ) @ Uu2 @ Uv2 @ Uw2 ) )
=> ( ( Y3
= ( heap_Time_return @ ( option @ nat ) @ ( none @ nat ) ) )
=> ~ ( accp @ vEBT_VEBTi @ vEBT_vebt_maxti_rel @ ( vEBT_Nodei @ ( none @ ( product_prod @ nat @ nat ) ) @ Uu2 @ Uv2 @ Uw2 ) ) ) )
=> ~ ! [Mi2: nat,Ma2: nat,Ux2: nat,Uy2: array @ vEBT_VEBTi,Uz2: vEBT_VEBTi] :
( ( X2
= ( vEBT_Nodei @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi2 @ Ma2 ) ) @ Ux2 @ Uy2 @ Uz2 ) )
=> ( ( Y3
= ( 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 ) ) @ Ux2 @ Uy2 @ Uz2 ) ) ) ) ) ) ) ) ).
% vebt_maxti.pelims
thf(fact_5485_vebt__maxt_Opelims,axiom,
! [X2: vEBT_VEBT,Y3: option @ nat] :
( ( ( vEBT_vebt_maxt @ X2 )
= Y3 )
=> ( ( accp @ vEBT_VEBT @ vEBT_vebt_maxt_rel @ X2 )
=> ( ! [A4: $o,B4: $o] :
( ( X2
= ( vEBT_Leaf @ A4 @ B4 ) )
=> ( ( ( B4
=> ( Y3
= ( some @ nat @ ( one_one @ nat ) ) ) )
& ( ~ B4
=> ( ( A4
=> ( Y3
= ( some @ nat @ ( zero_zero @ nat ) ) ) )
& ( ~ A4
=> ( Y3
= ( none @ nat ) ) ) ) ) )
=> ~ ( accp @ vEBT_VEBT @ vEBT_vebt_maxt_rel @ ( vEBT_Leaf @ A4 @ B4 ) ) ) )
=> ( ! [Uu2: nat,Uv2: list @ vEBT_VEBT,Uw2: vEBT_VEBT] :
( ( X2
= ( vEBT_Node @ ( none @ ( product_prod @ nat @ nat ) ) @ Uu2 @ Uv2 @ Uw2 ) )
=> ( ( Y3
= ( none @ nat ) )
=> ~ ( accp @ vEBT_VEBT @ vEBT_vebt_maxt_rel @ ( vEBT_Node @ ( none @ ( product_prod @ nat @ nat ) ) @ Uu2 @ Uv2 @ Uw2 ) ) ) )
=> ~ ! [Mi2: nat,Ma2: nat,Ux2: nat,Uy2: list @ vEBT_VEBT,Uz2: vEBT_VEBT] :
( ( X2
= ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi2 @ Ma2 ) ) @ Ux2 @ Uy2 @ Uz2 ) )
=> ( ( Y3
= ( some @ nat @ Ma2 ) )
=> ~ ( accp @ vEBT_VEBT @ vEBT_vebt_maxt_rel @ ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi2 @ Ma2 ) ) @ Ux2 @ Uy2 @ Uz2 ) ) ) ) ) ) ) ) ).
% vebt_maxt.pelims
thf(fact_5486_vebt__mint_Opelims,axiom,
! [X2: vEBT_VEBT,Y3: option @ nat] :
( ( ( vEBT_vebt_mint @ X2 )
= Y3 )
=> ( ( accp @ vEBT_VEBT @ vEBT_vebt_mint_rel @ X2 )
=> ( ! [A4: $o,B4: $o] :
( ( X2
= ( vEBT_Leaf @ A4 @ B4 ) )
=> ( ( ( A4
=> ( Y3
= ( some @ nat @ ( zero_zero @ nat ) ) ) )
& ( ~ A4
=> ( ( B4
=> ( Y3
= ( some @ nat @ ( one_one @ nat ) ) ) )
& ( ~ B4
=> ( Y3
= ( none @ nat ) ) ) ) ) )
=> ~ ( accp @ vEBT_VEBT @ vEBT_vebt_mint_rel @ ( vEBT_Leaf @ A4 @ B4 ) ) ) )
=> ( ! [Uu2: nat,Uv2: list @ vEBT_VEBT,Uw2: vEBT_VEBT] :
( ( X2
= ( vEBT_Node @ ( none @ ( product_prod @ nat @ nat ) ) @ Uu2 @ Uv2 @ Uw2 ) )
=> ( ( Y3
= ( none @ nat ) )
=> ~ ( accp @ vEBT_VEBT @ vEBT_vebt_mint_rel @ ( vEBT_Node @ ( none @ ( product_prod @ nat @ nat ) ) @ Uu2 @ Uv2 @ Uw2 ) ) ) )
=> ~ ! [Mi2: nat,Ma2: nat,Ux2: nat,Uy2: list @ vEBT_VEBT,Uz2: vEBT_VEBT] :
( ( X2
= ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi2 @ Ma2 ) ) @ Ux2 @ Uy2 @ Uz2 ) )
=> ( ( Y3
= ( some @ nat @ Mi2 ) )
=> ~ ( accp @ vEBT_VEBT @ vEBT_vebt_mint_rel @ ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi2 @ Ma2 ) ) @ Ux2 @ Uy2 @ Uz2 ) ) ) ) ) ) ) ) ).
% vebt_mint.pelims
thf(fact_5487_T_092_060_094sub_062m_092_060_094sub_062i_092_060_094sub_062n_092_060_094sub_062t_Opelims,axiom,
! [X2: vEBT_VEBT,Y3: nat] :
( ( ( vEBT_T_m_i_n_t @ X2 )
= Y3 )
=> ( ( accp @ vEBT_VEBT @ vEBT_T_m_i_n_t_rel @ X2 )
=> ( ! [A4: $o,B4: $o] :
( ( X2
= ( vEBT_Leaf @ A4 @ B4 ) )
=> ( ( Y3
= ( 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 ) ) ) )
=> ( ! [Uu2: nat,Uv2: list @ vEBT_VEBT,Uw2: vEBT_VEBT] :
( ( X2
= ( vEBT_Node @ ( none @ ( product_prod @ nat @ nat ) ) @ Uu2 @ Uv2 @ Uw2 ) )
=> ( ( Y3
= ( one_one @ nat ) )
=> ~ ( accp @ vEBT_VEBT @ vEBT_T_m_i_n_t_rel @ ( vEBT_Node @ ( none @ ( product_prod @ nat @ nat ) ) @ Uu2 @ Uv2 @ Uw2 ) ) ) )
=> ~ ! [Mi2: nat,Ma2: nat,Ux2: nat,Uy2: list @ vEBT_VEBT,Uz2: vEBT_VEBT] :
( ( X2
= ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi2 @ Ma2 ) ) @ Ux2 @ Uy2 @ Uz2 ) )
=> ( ( Y3
= ( one_one @ nat ) )
=> ~ ( accp @ vEBT_VEBT @ vEBT_T_m_i_n_t_rel @ ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi2 @ Ma2 ) ) @ Ux2 @ Uy2 @ Uz2 ) ) ) ) ) ) ) ) ).
% T\<^sub>m\<^sub>i\<^sub>n\<^sub>t.pelims
thf(fact_5488_T_092_060_094sub_062m_092_060_094sub_062a_092_060_094sub_062x_092_060_094sub_062t_Opelims,axiom,
! [X2: vEBT_VEBT,Y3: nat] :
( ( ( vEBT_T_m_a_x_t @ X2 )
= Y3 )
=> ( ( accp @ vEBT_VEBT @ vEBT_T_m_a_x_t_rel @ X2 )
=> ( ! [A4: $o,B4: $o] :
( ( X2
= ( vEBT_Leaf @ A4 @ B4 ) )
=> ( ( Y3
= ( 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 ) ) ) )
=> ( ! [Uu2: nat,Uv2: list @ vEBT_VEBT,Uw2: vEBT_VEBT] :
( ( X2
= ( vEBT_Node @ ( none @ ( product_prod @ nat @ nat ) ) @ Uu2 @ Uv2 @ Uw2 ) )
=> ( ( Y3
= ( one_one @ nat ) )
=> ~ ( accp @ vEBT_VEBT @ vEBT_T_m_a_x_t_rel @ ( vEBT_Node @ ( none @ ( product_prod @ nat @ nat ) ) @ Uu2 @ Uv2 @ Uw2 ) ) ) )
=> ~ ! [Mi2: nat,Ma2: nat,Ux2: nat,Uy2: list @ vEBT_VEBT,Uz2: vEBT_VEBT] :
( ( X2
= ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi2 @ Ma2 ) ) @ Ux2 @ Uy2 @ Uz2 ) )
=> ( ( Y3
= ( one_one @ nat ) )
=> ~ ( accp @ vEBT_VEBT @ vEBT_T_m_a_x_t_rel @ ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi2 @ Ma2 ) ) @ Ux2 @ Uy2 @ Uz2 ) ) ) ) ) ) ) ) ).
% T\<^sub>m\<^sub>a\<^sub>x\<^sub>t.pelims
thf(fact_5489_fold__atLeastAtMost__nat_Opsimps,axiom,
! [A: $tType,F2: nat > A > A,A3: nat,B3: nat,Acc3: A] :
( ( accp @ ( product_prod @ ( nat > A > A ) @ ( product_prod @ nat @ ( product_prod @ nat @ A ) ) ) @ ( set_fo1817059534552279752at_rel @ A ) @ ( product_Pair @ ( nat > A > A ) @ ( product_prod @ nat @ ( product_prod @ nat @ A ) ) @ F2 @ ( product_Pair @ nat @ ( product_prod @ nat @ A ) @ A3 @ ( product_Pair @ nat @ A @ B3 @ Acc3 ) ) ) )
=> ( ( ( ord_less @ nat @ B3 @ A3 )
=> ( ( set_fo6178422350223883121st_nat @ A @ F2 @ A3 @ B3 @ Acc3 )
= Acc3 ) )
& ( ~ ( ord_less @ nat @ B3 @ A3 )
=> ( ( set_fo6178422350223883121st_nat @ A @ F2 @ A3 @ B3 @ Acc3 )
= ( set_fo6178422350223883121st_nat @ A @ F2 @ ( plus_plus @ nat @ A3 @ ( one_one @ nat ) ) @ B3 @ ( F2 @ A3 @ Acc3 ) ) ) ) ) ) ).
% fold_atLeastAtMost_nat.psimps
thf(fact_5490_fold__atLeastAtMost__nat_Opinduct,axiom,
! [A: $tType,A0: nat > A > A,A12: nat,A23: nat,A32: A,P: ( nat > A > A ) > nat > nat > A > $o] :
( ( accp @ ( product_prod @ ( nat > A > A ) @ ( product_prod @ nat @ ( product_prod @ nat @ A ) ) ) @ ( set_fo1817059534552279752at_rel @ A ) @ ( product_Pair @ ( nat > A > A ) @ ( product_prod @ nat @ ( product_prod @ nat @ A ) ) @ A0 @ ( product_Pair @ nat @ ( product_prod @ nat @ A ) @ A12 @ ( product_Pair @ nat @ A @ A23 @ A32 ) ) ) )
=> ( ! [F3: 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 ) ) @ F3 @ ( product_Pair @ nat @ ( product_prod @ nat @ A ) @ A4 @ ( product_Pair @ nat @ A @ B4 @ Acc ) ) ) )
=> ( ( ~ ( ord_less @ nat @ B4 @ A4 )
=> ( P @ F3 @ ( plus_plus @ nat @ A4 @ ( one_one @ nat ) ) @ B4 @ ( F3 @ A4 @ Acc ) ) )
=> ( P @ F3 @ A4 @ B4 @ Acc ) ) )
=> ( P @ A0 @ A12 @ A23 @ A32 ) ) ) ).
% fold_atLeastAtMost_nat.pinduct
thf(fact_5491_fold__atLeastAtMost__nat_Opelims,axiom,
! [A: $tType,X2: nat > A > A,Xa: nat,Xb: nat,Xc: A,Y3: A] :
( ( ( set_fo6178422350223883121st_nat @ A @ X2 @ Xa @ Xb @ Xc )
= Y3 )
=> ( ( 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 ) ) @ X2 @ ( product_Pair @ nat @ ( product_prod @ nat @ A ) @ Xa @ ( product_Pair @ nat @ A @ Xb @ Xc ) ) ) )
=> ~ ( ( ( ( ord_less @ nat @ Xb @ Xa )
=> ( Y3 = Xc ) )
& ( ~ ( ord_less @ nat @ Xb @ Xa )
=> ( Y3
= ( set_fo6178422350223883121st_nat @ A @ X2 @ ( plus_plus @ nat @ Xa @ ( one_one @ nat ) ) @ Xb @ ( X2 @ Xa @ Xc ) ) ) ) )
=> ~ ( accp @ ( product_prod @ ( nat > A > A ) @ ( product_prod @ nat @ ( product_prod @ nat @ A ) ) ) @ ( set_fo1817059534552279752at_rel @ A ) @ ( product_Pair @ ( nat > A > A ) @ ( product_prod @ nat @ ( product_prod @ nat @ A ) ) @ X2 @ ( product_Pair @ nat @ ( product_prod @ nat @ A ) @ Xa @ ( product_Pair @ nat @ A @ Xb @ Xc ) ) ) ) ) ) ) ).
% fold_atLeastAtMost_nat.pelims
thf(fact_5492_VEBT__internal_OminNulli_Opelims,axiom,
! [X2: vEBT_VEBTi,Y3: heap_Time_Heap @ $o] :
( ( ( vEBT_VEBT_minNulli @ X2 )
= Y3 )
=> ( ( accp @ vEBT_VEBTi @ vEBT_V5740978063120863272li_rel @ X2 )
=> ( ( ( X2
= ( vEBT_Leafi @ $false @ $false ) )
=> ( ( Y3
= ( heap_Time_return @ $o @ $true ) )
=> ~ ( accp @ vEBT_VEBTi @ vEBT_V5740978063120863272li_rel @ ( vEBT_Leafi @ $false @ $false ) ) ) )
=> ( ! [Uv2: $o] :
( ( X2
= ( vEBT_Leafi @ $true @ Uv2 ) )
=> ( ( Y3
= ( heap_Time_return @ $o @ $false ) )
=> ~ ( accp @ vEBT_VEBTi @ vEBT_V5740978063120863272li_rel @ ( vEBT_Leafi @ $true @ Uv2 ) ) ) )
=> ( ! [Uu2: $o] :
( ( X2
= ( vEBT_Leafi @ Uu2 @ $true ) )
=> ( ( Y3
= ( heap_Time_return @ $o @ $false ) )
=> ~ ( accp @ vEBT_VEBTi @ vEBT_V5740978063120863272li_rel @ ( vEBT_Leafi @ Uu2 @ $true ) ) ) )
=> ( ! [Uw2: nat,Ux2: array @ vEBT_VEBTi,Uy2: vEBT_VEBTi] :
( ( X2
= ( vEBT_Nodei @ ( none @ ( product_prod @ nat @ nat ) ) @ Uw2 @ Ux2 @ Uy2 ) )
=> ( ( Y3
= ( heap_Time_return @ $o @ $true ) )
=> ~ ( accp @ vEBT_VEBTi @ vEBT_V5740978063120863272li_rel @ ( vEBT_Nodei @ ( none @ ( product_prod @ nat @ nat ) ) @ Uw2 @ Ux2 @ Uy2 ) ) ) )
=> ~ ! [Uz2: product_prod @ nat @ nat,Va2: nat,Vb2: array @ vEBT_VEBTi,Vc2: vEBT_VEBTi] :
( ( X2
= ( vEBT_Nodei @ ( some @ ( product_prod @ nat @ nat ) @ Uz2 ) @ Va2 @ Vb2 @ Vc2 ) )
=> ( ( Y3
= ( heap_Time_return @ $o @ $false ) )
=> ~ ( accp @ vEBT_VEBTi @ vEBT_V5740978063120863272li_rel @ ( vEBT_Nodei @ ( some @ ( product_prod @ nat @ nat ) @ Uz2 ) @ Va2 @ Vb2 @ Vc2 ) ) ) ) ) ) ) ) ) ) ).
% VEBT_internal.minNulli.pelims
thf(fact_5493_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,
! [X2: vEBT_VEBT,Y3: nat] :
( ( ( vEBT_T_m_i_n_N_u_l_l @ X2 )
= Y3 )
=> ( ( accp @ vEBT_VEBT @ vEBT_T5462971552011256508_l_rel @ X2 )
=> ( ( ( X2
= ( vEBT_Leaf @ $false @ $false ) )
=> ( ( Y3
= ( one_one @ nat ) )
=> ~ ( accp @ vEBT_VEBT @ vEBT_T5462971552011256508_l_rel @ ( vEBT_Leaf @ $false @ $false ) ) ) )
=> ( ! [Uv2: $o] :
( ( X2
= ( vEBT_Leaf @ $true @ Uv2 ) )
=> ( ( Y3
= ( one_one @ nat ) )
=> ~ ( accp @ vEBT_VEBT @ vEBT_T5462971552011256508_l_rel @ ( vEBT_Leaf @ $true @ Uv2 ) ) ) )
=> ( ! [Uu2: $o] :
( ( X2
= ( vEBT_Leaf @ Uu2 @ $true ) )
=> ( ( Y3
= ( one_one @ nat ) )
=> ~ ( accp @ vEBT_VEBT @ vEBT_T5462971552011256508_l_rel @ ( vEBT_Leaf @ Uu2 @ $true ) ) ) )
=> ( ! [Uw2: nat,Ux2: list @ vEBT_VEBT,Uy2: vEBT_VEBT] :
( ( X2
= ( vEBT_Node @ ( none @ ( product_prod @ nat @ nat ) ) @ Uw2 @ Ux2 @ Uy2 ) )
=> ( ( Y3
= ( one_one @ nat ) )
=> ~ ( accp @ vEBT_VEBT @ vEBT_T5462971552011256508_l_rel @ ( vEBT_Node @ ( none @ ( product_prod @ nat @ nat ) ) @ Uw2 @ Ux2 @ Uy2 ) ) ) )
=> ~ ! [Uz2: product_prod @ nat @ nat,Va2: nat,Vb2: list @ vEBT_VEBT,Vc2: vEBT_VEBT] :
( ( X2
= ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ Uz2 ) @ Va2 @ Vb2 @ Vc2 ) )
=> ( ( Y3
= ( one_one @ nat ) )
=> ~ ( accp @ vEBT_VEBT @ vEBT_T5462971552011256508_l_rel @ ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ Uz2 ) @ Va2 @ Vb2 @ Vc2 ) ) ) ) ) ) ) ) ) ) ).
% T\<^sub>m\<^sub>i\<^sub>n\<^sub>N\<^sub>u\<^sub>l\<^sub>l.pelims
thf(fact_5494_VEBT__internal_Ooption__shift_Opelims,axiom,
! [A: $tType,X2: A > A > A,Xa: option @ A,Xb: option @ A,Y3: option @ A] :
( ( ( vEBT_V2048590022279873568_shift @ A @ X2 @ Xa @ Xb )
= Y3 )
=> ( ( accp @ ( product_prod @ ( A > A > A ) @ ( product_prod @ ( option @ A ) @ ( option @ A ) ) ) @ ( vEBT_V459564278314245337ft_rel @ A ) @ ( product_Pair @ ( A > A > A ) @ ( product_prod @ ( option @ A ) @ ( option @ A ) ) @ X2 @ ( product_Pair @ ( option @ A ) @ ( option @ A ) @ Xa @ Xb ) ) )
=> ( ( ( Xa
= ( none @ A ) )
=> ( ( Y3
= ( none @ A ) )
=> ~ ( accp @ ( product_prod @ ( A > A > A ) @ ( product_prod @ ( option @ A ) @ ( option @ A ) ) ) @ ( vEBT_V459564278314245337ft_rel @ A ) @ ( product_Pair @ ( A > A > A ) @ ( product_prod @ ( option @ A ) @ ( option @ A ) ) @ X2 @ ( product_Pair @ ( option @ A ) @ ( option @ A ) @ ( none @ A ) @ Xb ) ) ) ) )
=> ( ! [V3: A] :
( ( Xa
= ( some @ A @ V3 ) )
=> ( ( Xb
= ( none @ A ) )
=> ( ( Y3
= ( none @ A ) )
=> ~ ( accp @ ( product_prod @ ( A > A > A ) @ ( product_prod @ ( option @ A ) @ ( option @ A ) ) ) @ ( vEBT_V459564278314245337ft_rel @ A ) @ ( product_Pair @ ( A > A > A ) @ ( product_prod @ ( option @ A ) @ ( option @ A ) ) @ X2 @ ( product_Pair @ ( option @ A ) @ ( option @ A ) @ ( some @ A @ V3 ) @ ( none @ A ) ) ) ) ) ) )
=> ~ ! [A4: A] :
( ( Xa
= ( some @ A @ A4 ) )
=> ! [B4: A] :
( ( Xb
= ( some @ A @ B4 ) )
=> ( ( Y3
= ( some @ A @ ( X2 @ A4 @ B4 ) ) )
=> ~ ( accp @ ( product_prod @ ( A > A > A ) @ ( product_prod @ ( option @ A ) @ ( option @ A ) ) ) @ ( vEBT_V459564278314245337ft_rel @ A ) @ ( product_Pair @ ( A > A > A ) @ ( product_prod @ ( option @ A ) @ ( option @ A ) ) @ X2 @ ( product_Pair @ ( option @ A ) @ ( option @ A ) @ ( some @ A @ A4 ) @ ( some @ A @ B4 ) ) ) ) ) ) ) ) ) ) ) ).
% VEBT_internal.option_shift.pelims
thf(fact_5495_quickcheck__narrowing__samples_Onarrowing__samples_Opinduct,axiom,
! [A: $tType] :
( ( ( code_term_of @ A )
& ( quickc6926020345158392990erm_of @ A ) )
=> ! [A_of_integer: code_integer > ( product_prod @ A @ A ),A0: code_integer,P: code_integer > $o] :
( ( accp @ code_integer @ ( code_T1710151556404007877es_rel @ A @ A_of_integer ) @ A0 )
=> ( ! [I2: code_integer] :
( ( accp @ code_integer @ ( code_T1710151556404007877es_rel @ A @ A_of_integer ) @ I2 )
=> ( ! [X8: product_prod @ A @ A,Xa2: A,Y5: A] :
( ( ord_less @ code_integer @ ( zero_zero @ code_integer ) @ I2 )
=> ( ( X8
= ( A_of_integer @ I2 ) )
=> ( ( ( product_Pair @ A @ A @ Xa2 @ Y5 )
= X8 )
=> ( P @ ( minus_minus @ code_integer @ I2 @ ( one_one @ code_integer ) ) ) ) ) )
=> ( P @ I2 ) ) )
=> ( P @ A0 ) ) ) ) ).
% quickcheck_narrowing_samples.narrowing_samples.pinduct
thf(fact_5496_in__measure,axiom,
! [A: $tType,X2: A,Y3: A,F2: A > nat] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X2 @ Y3 ) @ ( measure @ A @ F2 ) )
= ( ord_less @ nat @ ( F2 @ X2 ) @ ( F2 @ Y3 ) ) ) ).
% in_measure
thf(fact_5497_or__not__num__neg_Opelims,axiom,
! [X2: num,Xa: num,Y3: num] :
( ( ( bit_or_not_num_neg @ X2 @ Xa )
= Y3 )
=> ( ( accp @ ( product_prod @ num @ num ) @ bit_or3848514188828904588eg_rel @ ( product_Pair @ num @ num @ X2 @ Xa ) )
=> ( ( ( X2 = one2 )
=> ( ( Xa = one2 )
=> ( ( Y3 = one2 )
=> ~ ( accp @ ( product_prod @ num @ num ) @ bit_or3848514188828904588eg_rel @ ( product_Pair @ num @ num @ one2 @ one2 ) ) ) ) )
=> ( ( ( X2 = one2 )
=> ! [M5: num] :
( ( Xa
= ( bit0 @ M5 ) )
=> ( ( Y3
= ( bit1 @ M5 ) )
=> ~ ( accp @ ( product_prod @ num @ num ) @ bit_or3848514188828904588eg_rel @ ( product_Pair @ num @ num @ one2 @ ( bit0 @ M5 ) ) ) ) ) )
=> ( ( ( X2 = one2 )
=> ! [M5: num] :
( ( Xa
= ( bit1 @ M5 ) )
=> ( ( Y3
= ( bit1 @ M5 ) )
=> ~ ( accp @ ( product_prod @ num @ num ) @ bit_or3848514188828904588eg_rel @ ( product_Pair @ num @ num @ one2 @ ( bit1 @ M5 ) ) ) ) ) )
=> ( ! [N3: num] :
( ( X2
= ( bit0 @ N3 ) )
=> ( ( Xa = one2 )
=> ( ( Y3
= ( bit0 @ one2 ) )
=> ~ ( accp @ ( product_prod @ num @ num ) @ bit_or3848514188828904588eg_rel @ ( product_Pair @ num @ num @ ( bit0 @ N3 ) @ one2 ) ) ) ) )
=> ( ! [N3: num] :
( ( X2
= ( bit0 @ N3 ) )
=> ! [M5: num] :
( ( Xa
= ( bit0 @ M5 ) )
=> ( ( Y3
= ( bitM @ ( bit_or_not_num_neg @ N3 @ M5 ) ) )
=> ~ ( accp @ ( product_prod @ num @ num ) @ bit_or3848514188828904588eg_rel @ ( product_Pair @ num @ num @ ( bit0 @ N3 ) @ ( bit0 @ M5 ) ) ) ) ) )
=> ( ! [N3: num] :
( ( X2
= ( bit0 @ N3 ) )
=> ! [M5: num] :
( ( Xa
= ( bit1 @ M5 ) )
=> ( ( Y3
= ( bit0 @ ( bit_or_not_num_neg @ N3 @ M5 ) ) )
=> ~ ( accp @ ( product_prod @ num @ num ) @ bit_or3848514188828904588eg_rel @ ( product_Pair @ num @ num @ ( bit0 @ N3 ) @ ( bit1 @ M5 ) ) ) ) ) )
=> ( ! [N3: num] :
( ( X2
= ( bit1 @ N3 ) )
=> ( ( Xa = one2 )
=> ( ( Y3 = one2 )
=> ~ ( accp @ ( product_prod @ num @ num ) @ bit_or3848514188828904588eg_rel @ ( product_Pair @ num @ num @ ( bit1 @ N3 ) @ one2 ) ) ) ) )
=> ( ! [N3: num] :
( ( X2
= ( bit1 @ N3 ) )
=> ! [M5: num] :
( ( Xa
= ( bit0 @ M5 ) )
=> ( ( Y3
= ( bitM @ ( bit_or_not_num_neg @ N3 @ M5 ) ) )
=> ~ ( accp @ ( product_prod @ num @ num ) @ bit_or3848514188828904588eg_rel @ ( product_Pair @ num @ num @ ( bit1 @ N3 ) @ ( bit0 @ M5 ) ) ) ) ) )
=> ~ ! [N3: num] :
( ( X2
= ( bit1 @ N3 ) )
=> ! [M5: num] :
( ( Xa
= ( bit1 @ M5 ) )
=> ( ( Y3
= ( bitM @ ( bit_or_not_num_neg @ N3 @ M5 ) ) )
=> ~ ( accp @ ( product_prod @ num @ num ) @ bit_or3848514188828904588eg_rel @ ( product_Pair @ num @ num @ ( bit1 @ N3 ) @ ( bit1 @ M5 ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).
% or_not_num_neg.pelims
thf(fact_5498_VEBT__internal_OminNull_Opelims_I1_J,axiom,
! [X2: vEBT_VEBT,Y3: $o] :
( ( ( vEBT_VEBT_minNull @ X2 )
= Y3 )
=> ( ( accp @ vEBT_VEBT @ vEBT_V6963167321098673237ll_rel @ X2 )
=> ( ( ( X2
= ( vEBT_Leaf @ $false @ $false ) )
=> ( Y3
=> ~ ( accp @ vEBT_VEBT @ vEBT_V6963167321098673237ll_rel @ ( vEBT_Leaf @ $false @ $false ) ) ) )
=> ( ! [Uv2: $o] :
( ( X2
= ( vEBT_Leaf @ $true @ Uv2 ) )
=> ( ~ Y3
=> ~ ( accp @ vEBT_VEBT @ vEBT_V6963167321098673237ll_rel @ ( vEBT_Leaf @ $true @ Uv2 ) ) ) )
=> ( ! [Uu2: $o] :
( ( X2
= ( vEBT_Leaf @ Uu2 @ $true ) )
=> ( ~ Y3
=> ~ ( accp @ vEBT_VEBT @ vEBT_V6963167321098673237ll_rel @ ( vEBT_Leaf @ Uu2 @ $true ) ) ) )
=> ( ! [Uw2: nat,Ux2: list @ vEBT_VEBT,Uy2: vEBT_VEBT] :
( ( X2
= ( vEBT_Node @ ( none @ ( product_prod @ nat @ nat ) ) @ Uw2 @ Ux2 @ Uy2 ) )
=> ( Y3
=> ~ ( accp @ vEBT_VEBT @ vEBT_V6963167321098673237ll_rel @ ( vEBT_Node @ ( none @ ( product_prod @ nat @ nat ) ) @ Uw2 @ Ux2 @ Uy2 ) ) ) )
=> ~ ! [Uz2: product_prod @ nat @ nat,Va2: nat,Vb2: list @ vEBT_VEBT,Vc2: vEBT_VEBT] :
( ( X2
= ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ Uz2 ) @ Va2 @ Vb2 @ Vc2 ) )
=> ( ~ Y3
=> ~ ( accp @ vEBT_VEBT @ vEBT_V6963167321098673237ll_rel @ ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ Uz2 ) @ Va2 @ Vb2 @ Vc2 ) ) ) ) ) ) ) ) ) ) ).
% VEBT_internal.minNull.pelims(1)
thf(fact_5499_arctan__def,axiom,
( arctan
= ( ^ [Y: real] :
( the @ 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 ) ) ) ) ) ).
% arctan_def
thf(fact_5500_modulo__int__def,axiom,
( ( modulo_modulo @ int )
= ( ^ [K3: int,L3: int] :
( if @ int
@ ( L3
= ( zero_zero @ int ) )
@ K3
@ ( if @ int
@ ( ( sgn_sgn @ int @ K3 )
= ( sgn_sgn @ int @ L3 ) )
@ ( times_times @ int @ ( sgn_sgn @ int @ L3 ) @ ( semiring_1_of_nat @ int @ ( modulo_modulo @ nat @ ( nat2 @ ( abs_abs @ int @ K3 ) ) @ ( nat2 @ ( abs_abs @ int @ L3 ) ) ) ) )
@ ( times_times @ int @ ( sgn_sgn @ int @ L3 )
@ ( minus_minus @ int
@ ( times_times @ int @ ( abs_abs @ int @ L3 )
@ ( zero_neq_one_of_bool @ int
@ ~ ( dvd_dvd @ int @ L3 @ K3 ) ) )
@ ( semiring_1_of_nat @ int @ ( modulo_modulo @ nat @ ( nat2 @ ( abs_abs @ int @ K3 ) ) @ ( nat2 @ ( abs_abs @ int @ L3 ) ) ) ) ) ) ) ) ) ) ).
% modulo_int_def
thf(fact_5501_sgn__one,axiom,
! [A: $tType] :
( ( real_V2822296259951069270ebra_1 @ A )
=> ( ( sgn_sgn @ A @ ( one_one @ A ) )
= ( one_one @ A ) ) ) ).
% sgn_one
thf(fact_5502_sgn__1,axiom,
! [A: $tType] :
( ( idom_abs_sgn @ A )
=> ( ( sgn_sgn @ A @ ( one_one @ A ) )
= ( one_one @ A ) ) ) ).
% sgn_1
thf(fact_5503_sgn__divide,axiom,
! [A: $tType] :
( ( field_abs_sgn @ A )
=> ! [A3: A,B3: A] :
( ( sgn_sgn @ A @ ( divide_divide @ A @ A3 @ B3 ) )
= ( divide_divide @ A @ ( sgn_sgn @ A @ A3 ) @ ( sgn_sgn @ A @ B3 ) ) ) ) ).
% sgn_divide
thf(fact_5504_power__sgn,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [A3: A,N: nat] :
( ( sgn_sgn @ A @ ( power_power @ A @ A3 @ N ) )
= ( power_power @ A @ ( sgn_sgn @ A @ A3 ) @ N ) ) ) ).
% power_sgn
thf(fact_5505_sgn__greater,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [A3: A] :
( ( ord_less @ A @ ( zero_zero @ A ) @ ( sgn_sgn @ A @ A3 ) )
= ( ord_less @ A @ ( zero_zero @ A ) @ A3 ) ) ) ).
% sgn_greater
thf(fact_5506_sgn__less,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [A3: A] :
( ( ord_less @ A @ ( sgn_sgn @ A @ A3 ) @ ( zero_zero @ A ) )
= ( ord_less @ A @ A3 @ ( zero_zero @ A ) ) ) ) ).
% sgn_less
thf(fact_5507_divide__sgn,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [A3: A,B3: A] :
( ( divide_divide @ A @ A3 @ ( sgn_sgn @ A @ B3 ) )
= ( times_times @ A @ A3 @ ( sgn_sgn @ A @ B3 ) ) ) ) ).
% divide_sgn
thf(fact_5508_sgn__pos,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [A3: A] :
( ( ord_less @ A @ ( zero_zero @ A ) @ A3 )
=> ( ( sgn_sgn @ A @ A3 )
= ( one_one @ A ) ) ) ) ).
% sgn_pos
thf(fact_5509_abs__sgn__eq__1,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [A3: A] :
( ( A3
!= ( zero_zero @ A ) )
=> ( ( abs_abs @ A @ ( sgn_sgn @ A @ A3 ) )
= ( one_one @ A ) ) ) ) ).
% abs_sgn_eq_1
thf(fact_5510_sgn__mult__self__eq,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [A3: A] :
( ( times_times @ A @ ( sgn_sgn @ A @ A3 ) @ ( sgn_sgn @ A @ A3 ) )
= ( zero_neq_one_of_bool @ A
@ ( A3
!= ( zero_zero @ A ) ) ) ) ) ).
% sgn_mult_self_eq
thf(fact_5511_sgn__neg,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [A3: A] :
( ( ord_less @ A @ A3 @ ( zero_zero @ A ) )
=> ( ( sgn_sgn @ A @ A3 )
= ( uminus_uminus @ A @ ( one_one @ A ) ) ) ) ) ).
% sgn_neg
thf(fact_5512_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_5513_sgn__minus__1,axiom,
! [A: $tType] :
( ( idom_abs_sgn @ A )
=> ( ( sgn_sgn @ A @ ( uminus_uminus @ A @ ( one_one @ A ) ) )
= ( uminus_uminus @ A @ ( one_one @ A ) ) ) ) ).
% sgn_minus_1
thf(fact_5514_same__sgn__sgn__add,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [B3: A,A3: A] :
( ( ( sgn_sgn @ A @ B3 )
= ( sgn_sgn @ A @ A3 ) )
=> ( ( sgn_sgn @ A @ ( plus_plus @ A @ A3 @ B3 ) )
= ( sgn_sgn @ A @ A3 ) ) ) ) ).
% same_sgn_sgn_add
thf(fact_5515_Real__Vector__Spaces_Osgn__mult,axiom,
! [A: $tType] :
( ( real_V8999393235501362500lgebra @ A )
=> ! [X2: A,Y3: A] :
( ( sgn_sgn @ A @ ( times_times @ A @ X2 @ Y3 ) )
= ( times_times @ A @ ( sgn_sgn @ A @ X2 ) @ ( sgn_sgn @ A @ Y3 ) ) ) ) ).
% Real_Vector_Spaces.sgn_mult
thf(fact_5516_sgn__mult,axiom,
! [A: $tType] :
( ( idom_abs_sgn @ A )
=> ! [A3: A,B3: A] :
( ( sgn_sgn @ A @ ( times_times @ A @ A3 @ B3 ) )
= ( times_times @ A @ ( sgn_sgn @ A @ A3 ) @ ( sgn_sgn @ A @ B3 ) ) ) ) ).
% sgn_mult
thf(fact_5517_mult__sgn__abs,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [X2: A] :
( ( times_times @ A @ ( sgn_sgn @ A @ X2 ) @ ( abs_abs @ A @ X2 ) )
= X2 ) ) ).
% mult_sgn_abs
thf(fact_5518_sgn__mult__abs,axiom,
! [A: $tType] :
( ( idom_abs_sgn @ A )
=> ! [A3: A] :
( ( times_times @ A @ ( sgn_sgn @ A @ A3 ) @ ( abs_abs @ A @ A3 ) )
= A3 ) ) ).
% sgn_mult_abs
thf(fact_5519_abs__mult__sgn,axiom,
! [A: $tType] :
( ( idom_abs_sgn @ A )
=> ! [A3: A] :
( ( times_times @ A @ ( abs_abs @ A @ A3 ) @ ( sgn_sgn @ A @ A3 ) )
= A3 ) ) ).
% abs_mult_sgn
thf(fact_5520_linordered__idom__class_Oabs__sgn,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ( ( abs_abs @ A )
= ( ^ [K3: A] : ( times_times @ A @ K3 @ ( sgn_sgn @ A @ K3 ) ) ) ) ) ).
% linordered_idom_class.abs_sgn
thf(fact_5521_same__sgn__abs__add,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [B3: A,A3: A] :
( ( ( sgn_sgn @ A @ B3 )
= ( sgn_sgn @ A @ A3 ) )
=> ( ( abs_abs @ A @ ( plus_plus @ A @ A3 @ B3 ) )
= ( plus_plus @ A @ ( abs_abs @ A @ A3 ) @ ( abs_abs @ A @ B3 ) ) ) ) ) ).
% same_sgn_abs_add
thf(fact_5522_ln__real__def,axiom,
( ( ln_ln @ real )
= ( ^ [X3: real] :
( the @ real
@ ^ [U2: real] :
( ( exp @ real @ U2 )
= X3 ) ) ) ) ).
% ln_real_def
thf(fact_5523_suminf__def,axiom,
! [A: $tType] :
( ( ( comm_monoid_add @ A )
& ( topolo4958980785337419405_space @ A ) )
=> ( ( suminf @ A )
= ( ^ [F5: nat > A] : ( the @ A @ ( sums @ A @ F5 ) ) ) ) ) ).
% suminf_def
thf(fact_5524_sgn__1__pos,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [A3: A] :
( ( ( sgn_sgn @ A @ A3 )
= ( one_one @ A ) )
= ( ord_less @ A @ ( zero_zero @ A ) @ A3 ) ) ) ).
% sgn_1_pos
thf(fact_5525_abs__sgn__eq,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [A3: A] :
( ( ( A3
= ( zero_zero @ A ) )
=> ( ( abs_abs @ A @ ( sgn_sgn @ A @ A3 ) )
= ( zero_zero @ A ) ) )
& ( ( A3
!= ( zero_zero @ A ) )
=> ( ( abs_abs @ A @ ( sgn_sgn @ A @ A3 ) )
= ( one_one @ A ) ) ) ) ) ).
% abs_sgn_eq
thf(fact_5526_sgn__mod,axiom,
! [L: int,K: int] :
( ( L
!= ( zero_zero @ int ) )
=> ( ~ ( dvd_dvd @ int @ L @ K )
=> ( ( sgn_sgn @ int @ ( modulo_modulo @ int @ K @ L ) )
= ( sgn_sgn @ int @ L ) ) ) ) ).
% sgn_mod
thf(fact_5527_ln__neg__is__const,axiom,
! [X2: real] :
( ( ord_less_eq @ real @ X2 @ ( zero_zero @ real ) )
=> ( ( ln_ln @ real @ X2 )
= ( the @ real
@ ^ [X3: real] : $false ) ) ) ).
% ln_neg_is_const
thf(fact_5528_sgn__1__neg,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [A3: A] :
( ( ( sgn_sgn @ A @ A3 )
= ( uminus_uminus @ A @ ( one_one @ A ) ) )
= ( ord_less @ A @ A3 @ ( zero_zero @ A ) ) ) ) ).
% sgn_1_neg
thf(fact_5529_sgn__if,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ( ( sgn_sgn @ A )
= ( ^ [X3: A] :
( if @ A
@ ( X3
= ( zero_zero @ A ) )
@ ( zero_zero @ A )
@ ( if @ A @ ( ord_less @ A @ ( zero_zero @ A ) @ X3 ) @ ( one_one @ A ) @ ( uminus_uminus @ A @ ( one_one @ A ) ) ) ) ) ) ) ).
% sgn_if
thf(fact_5530_zsgn__def,axiom,
( ( sgn_sgn @ int )
= ( ^ [I4: int] :
( if @ int
@ ( I4
= ( zero_zero @ int ) )
@ ( zero_zero @ int )
@ ( if @ int @ ( ord_less @ int @ ( zero_zero @ int ) @ I4 ) @ ( one_one @ int ) @ ( uminus_uminus @ int @ ( one_one @ int ) ) ) ) ) ) ).
% zsgn_def
thf(fact_5531_norm__sgn,axiom,
! [A: $tType] :
( ( real_V822414075346904944vector @ A )
=> ! [X2: A] :
( ( ( X2
= ( zero_zero @ A ) )
=> ( ( real_V7770717601297561774m_norm @ A @ ( sgn_sgn @ A @ X2 ) )
= ( zero_zero @ real ) ) )
& ( ( X2
!= ( zero_zero @ A ) )
=> ( ( real_V7770717601297561774m_norm @ A @ ( sgn_sgn @ A @ X2 ) )
= ( one_one @ real ) ) ) ) ) ).
% norm_sgn
thf(fact_5532_arccos__def,axiom,
( arccos
= ( ^ [Y: real] :
( the @ real
@ ^ [X3: real] :
( ( ord_less_eq @ real @ ( zero_zero @ real ) @ X3 )
& ( ord_less_eq @ real @ X3 @ pi )
& ( ( cos @ real @ X3 )
= Y ) ) ) ) ) ).
% arccos_def
thf(fact_5533_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_5534_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_5535_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 ) ) )
=> ( ! [Q4: int] :
( ( A32
= ( product_Pair @ int @ int @ Q4 @ ( zero_zero @ int ) ) )
=> ( ( A23
!= ( zero_zero @ int ) )
=> ( A12
!= ( times_times @ int @ Q4 @ A23 ) ) ) )
=> ~ ! [R3: int,Q4: int] :
( ( A32
= ( product_Pair @ int @ int @ Q4 @ R3 ) )
=> ( ( ( sgn_sgn @ int @ R3 )
= ( sgn_sgn @ int @ A23 ) )
=> ( ( ord_less @ int @ ( abs_abs @ int @ R3 ) @ ( abs_abs @ int @ A23 ) )
=> ( A12
!= ( plus_plus @ int @ ( times_times @ int @ Q4 @ A23 ) @ R3 ) ) ) ) ) ) ) ) ).
% eucl_rel_int.cases
thf(fact_5536_eucl__rel__int_Osimps,axiom,
( eucl_rel_int
= ( ^ [A1: int,A22: int,A33: product_prod @ int @ int] :
( ? [K3: int] :
( ( A1 = K3 )
& ( A22
= ( zero_zero @ int ) )
& ( A33
= ( product_Pair @ int @ int @ ( zero_zero @ int ) @ K3 ) ) )
| ? [L3: int,K3: int,Q5: int] :
( ( A1 = K3 )
& ( A22 = L3 )
& ( A33
= ( product_Pair @ int @ int @ Q5 @ ( zero_zero @ int ) ) )
& ( L3
!= ( zero_zero @ int ) )
& ( K3
= ( times_times @ int @ Q5 @ L3 ) ) )
| ? [R5: int,L3: int,K3: int,Q5: int] :
( ( A1 = K3 )
& ( A22 = L3 )
& ( A33
= ( product_Pair @ int @ int @ Q5 @ R5 ) )
& ( ( sgn_sgn @ int @ R5 )
= ( sgn_sgn @ int @ L3 ) )
& ( ord_less @ int @ ( abs_abs @ int @ R5 ) @ ( abs_abs @ int @ L3 ) )
& ( K3
= ( plus_plus @ int @ ( times_times @ int @ Q5 @ L3 ) @ R5 ) ) ) ) ) ) ).
% eucl_rel_int.simps
thf(fact_5537_pi__half,axiom,
( ( divide_divide @ real @ pi @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) )
= ( the @ real
@ ^ [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 ) ) ) ) ) ).
% pi_half
thf(fact_5538_pi__def,axiom,
( pi
= ( times_times @ real @ ( numeral_numeral @ real @ ( bit0 @ one2 ) )
@ ( the @ real
@ ^ [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 ) ) ) ) ) ) ).
% pi_def
thf(fact_5539_VEBT__internal_OminNull_Opelims_I3_J,axiom,
! [X2: vEBT_VEBT] :
( ~ ( vEBT_VEBT_minNull @ X2 )
=> ( ( accp @ vEBT_VEBT @ vEBT_V6963167321098673237ll_rel @ X2 )
=> ( ! [Uv2: $o] :
( ( X2
= ( vEBT_Leaf @ $true @ Uv2 ) )
=> ~ ( accp @ vEBT_VEBT @ vEBT_V6963167321098673237ll_rel @ ( vEBT_Leaf @ $true @ Uv2 ) ) )
=> ( ! [Uu2: $o] :
( ( X2
= ( vEBT_Leaf @ Uu2 @ $true ) )
=> ~ ( accp @ vEBT_VEBT @ vEBT_V6963167321098673237ll_rel @ ( vEBT_Leaf @ Uu2 @ $true ) ) )
=> ~ ! [Uz2: product_prod @ nat @ nat,Va2: nat,Vb2: list @ vEBT_VEBT,Vc2: vEBT_VEBT] :
( ( X2
= ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ Uz2 ) @ Va2 @ Vb2 @ Vc2 ) )
=> ~ ( accp @ vEBT_VEBT @ vEBT_V6963167321098673237ll_rel @ ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ Uz2 ) @ Va2 @ Vb2 @ Vc2 ) ) ) ) ) ) ) ).
% VEBT_internal.minNull.pelims(3)
thf(fact_5540_VEBT__internal_OminNull_Opelims_I2_J,axiom,
! [X2: vEBT_VEBT] :
( ( vEBT_VEBT_minNull @ X2 )
=> ( ( accp @ vEBT_VEBT @ vEBT_V6963167321098673237ll_rel @ X2 )
=> ( ( ( X2
= ( vEBT_Leaf @ $false @ $false ) )
=> ~ ( accp @ vEBT_VEBT @ vEBT_V6963167321098673237ll_rel @ ( vEBT_Leaf @ $false @ $false ) ) )
=> ~ ! [Uw2: nat,Ux2: list @ vEBT_VEBT,Uy2: vEBT_VEBT] :
( ( X2
= ( vEBT_Node @ ( none @ ( product_prod @ nat @ nat ) ) @ Uw2 @ Ux2 @ Uy2 ) )
=> ~ ( accp @ vEBT_VEBT @ vEBT_V6963167321098673237ll_rel @ ( vEBT_Node @ ( none @ ( product_prod @ nat @ nat ) ) @ Uw2 @ Ux2 @ Uy2 ) ) ) ) ) ) ).
% VEBT_internal.minNull.pelims(2)
thf(fact_5541_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_5542_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_5543_divide__int__def,axiom,
( ( divide_divide @ int )
= ( ^ [K3: int,L3: int] :
( if @ int
@ ( L3
= ( zero_zero @ int ) )
@ ( zero_zero @ int )
@ ( if @ int
@ ( ( sgn_sgn @ int @ K3 )
= ( sgn_sgn @ int @ L3 ) )
@ ( semiring_1_of_nat @ int @ ( divide_divide @ nat @ ( nat2 @ ( abs_abs @ int @ K3 ) ) @ ( nat2 @ ( abs_abs @ int @ L3 ) ) ) )
@ ( uminus_uminus @ int
@ ( semiring_1_of_nat @ int
@ ( plus_plus @ nat @ ( divide_divide @ nat @ ( nat2 @ ( abs_abs @ int @ K3 ) ) @ ( nat2 @ ( abs_abs @ int @ L3 ) ) )
@ ( zero_neq_one_of_bool @ nat
@ ~ ( dvd_dvd @ int @ L3 @ K3 ) ) ) ) ) ) ) ) ) ).
% divide_int_def
thf(fact_5544_arcsin__def,axiom,
( arcsin
= ( ^ [Y: real] :
( the @ 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 ) ) ) ) ) ).
% arcsin_def
thf(fact_5545_old_Orec__prod__def,axiom,
! [T: $tType,B: $tType,A: $tType] :
( ( product_rec_prod @ A @ B @ T )
= ( ^ [F13: A > B > T,X3: product_prod @ A @ B] : ( the @ T @ ( product_rec_set_prod @ A @ B @ T @ F13 @ X3 ) ) ) ) ).
% old.rec_prod_def
thf(fact_5546_the__sym__eq__trivial,axiom,
! [A: $tType,X2: A] :
( ( the @ A
@ ( ^ [Y6: A,Z3: A] : Y6 = Z3
@ X2 ) )
= X2 ) ).
% the_sym_eq_trivial
thf(fact_5547_the__eq__trivial,axiom,
! [A: $tType,A3: A] :
( ( the @ A
@ ^ [X3: A] : X3 = A3 )
= A3 ) ).
% the_eq_trivial
thf(fact_5548_the__equality,axiom,
! [A: $tType,P: A > $o,A3: A] :
( ( P @ A3 )
=> ( ! [X4: A] :
( ( P @ X4 )
=> ( X4 = A3 ) )
=> ( ( the @ A @ P )
= A3 ) ) ) ).
% the_equality
thf(fact_5549_zero__le__sgn__iff,axiom,
! [X2: real] :
( ( ord_less_eq @ real @ ( zero_zero @ real ) @ ( sgn_sgn @ real @ X2 ) )
= ( ord_less_eq @ real @ ( zero_zero @ real ) @ X2 ) ) ).
% zero_le_sgn_iff
thf(fact_5550_sgn__le__0__iff,axiom,
! [X2: real] :
( ( ord_less_eq @ real @ ( sgn_sgn @ real @ X2 ) @ ( zero_zero @ real ) )
= ( ord_less_eq @ real @ X2 @ ( zero_zero @ real ) ) ) ).
% sgn_le_0_iff
thf(fact_5551_The__split__eq,axiom,
! [A: $tType,B: $tType,X2: A,Y3: B] :
( ( the @ ( product_prod @ A @ B )
@ ( product_case_prod @ A @ B @ $o
@ ^ [X10: A,Y9: B] :
( ( X2 = X10 )
& ( Y3 = Y9 ) ) ) )
= ( product_Pair @ A @ B @ X2 @ Y3 ) ) ).
% The_split_eq
thf(fact_5552_sgn__root,axiom,
! [N: nat,X2: real] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ N )
=> ( ( sgn_sgn @ real @ ( root @ N @ X2 ) )
= ( sgn_sgn @ real @ X2 ) ) ) ).
% sgn_root
thf(fact_5553_sgn__real__def,axiom,
( ( sgn_sgn @ real )
= ( ^ [A6: real] :
( if @ real
@ ( A6
= ( zero_zero @ real ) )
@ ( zero_zero @ real )
@ ( if @ real @ ( ord_less @ real @ ( zero_zero @ real ) @ A6 ) @ ( one_one @ real ) @ ( uminus_uminus @ real @ ( one_one @ real ) ) ) ) ) ) ).
% sgn_real_def
thf(fact_5554_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_5555_sgn__power__injE,axiom,
! [A3: real,N: nat,X2: real,B3: real] :
( ( ( times_times @ real @ ( sgn_sgn @ real @ A3 ) @ ( power_power @ real @ ( abs_abs @ real @ A3 ) @ N ) )
= X2 )
=> ( ( X2
= ( times_times @ real @ ( sgn_sgn @ real @ B3 ) @ ( power_power @ real @ ( abs_abs @ real @ B3 ) @ N ) ) )
=> ( ( ord_less @ nat @ ( zero_zero @ nat ) @ N )
=> ( A3 = B3 ) ) ) ) ).
% sgn_power_injE
thf(fact_5556_theI,axiom,
! [A: $tType,P: A > $o,A3: A] :
( ( P @ A3 )
=> ( ! [X4: A] :
( ( P @ X4 )
=> ( X4 = A3 ) )
=> ( P @ ( the @ A @ P ) ) ) ) ).
% theI
thf(fact_5557_theI_H,axiom,
! [A: $tType,P: A > $o] :
( ? [X8: A] :
( ( P @ X8 )
& ! [Y4: A] :
( ( P @ Y4 )
=> ( Y4 = X8 ) ) )
=> ( P @ ( the @ A @ P ) ) ) ).
% theI'
thf(fact_5558_theI2,axiom,
! [A: $tType,P: A > $o,A3: A,Q: A > $o] :
( ( P @ A3 )
=> ( ! [X4: A] :
( ( P @ X4 )
=> ( X4 = A3 ) )
=> ( ! [X4: A] :
( ( P @ X4 )
=> ( Q @ X4 ) )
=> ( Q @ ( the @ A @ P ) ) ) ) ) ).
% theI2
thf(fact_5559_If__def,axiom,
! [A: $tType] :
( ( if @ A )
= ( ^ [P3: $o,X3: A,Y: A] :
( the @ A
@ ^ [Z4: A] :
( ( P3
=> ( Z4 = X3 ) )
& ( ~ P3
=> ( Z4 = Y ) ) ) ) ) ) ).
% If_def
thf(fact_5560_the1I2,axiom,
! [A: $tType,P: A > $o,Q: A > $o] :
( ? [X8: A] :
( ( P @ X8 )
& ! [Y4: A] :
( ( P @ Y4 )
=> ( Y4 = X8 ) ) )
=> ( ! [X4: A] :
( ( P @ X4 )
=> ( Q @ X4 ) )
=> ( Q @ ( the @ A @ P ) ) ) ) ).
% the1I2
thf(fact_5561_the1__equality,axiom,
! [A: $tType,P: A > $o,A3: A] :
( ? [X8: A] :
( ( P @ X8 )
& ! [Y4: A] :
( ( P @ Y4 )
=> ( Y4 = X8 ) ) )
=> ( ( P @ A3 )
=> ( ( the @ A @ P )
= A3 ) ) ) ).
% the1_equality
thf(fact_5562_sgn__power__root,axiom,
! [N: nat,X2: real] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ N )
=> ( ( times_times @ real @ ( sgn_sgn @ real @ ( root @ N @ X2 ) ) @ ( power_power @ real @ ( abs_abs @ real @ ( root @ N @ X2 ) ) @ N ) )
= X2 ) ) ).
% sgn_power_root
thf(fact_5563_root__sgn__power,axiom,
! [N: nat,Y3: real] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ N )
=> ( ( root @ N @ ( times_times @ real @ ( sgn_sgn @ real @ Y3 ) @ ( power_power @ real @ ( abs_abs @ real @ Y3 ) @ N ) ) )
= Y3 ) ) ).
% root_sgn_power
thf(fact_5564_cis__Arg__unique,axiom,
! [Z: complex,X2: real] :
( ( ( sgn_sgn @ complex @ Z )
= ( cis @ X2 ) )
=> ( ( ord_less @ real @ ( uminus_uminus @ real @ pi ) @ X2 )
=> ( ( ord_less_eq @ real @ X2 @ pi )
=> ( ( arg @ Z )
= X2 ) ) ) ) ).
% cis_Arg_unique
thf(fact_5565_split__root,axiom,
! [P: real > $o,N: nat,X2: real] :
( ( P @ ( root @ N @ X2 ) )
= ( ( ( N
= ( zero_zero @ nat ) )
=> ( P @ ( zero_zero @ real ) ) )
& ( ( ord_less @ nat @ ( zero_zero @ nat ) @ N )
=> ! [Y: real] :
( ( ( times_times @ real @ ( sgn_sgn @ real @ Y ) @ ( power_power @ real @ ( abs_abs @ real @ Y ) @ N ) )
= X2 )
=> ( P @ Y ) ) ) ) ) ).
% split_root
thf(fact_5566_floor__real__def,axiom,
( ( archim6421214686448440834_floor @ real )
= ( ^ [X3: real] :
( the @ int
@ ^ [Z4: int] :
( ( ord_less_eq @ real @ ( ring_1_of_int @ real @ Z4 ) @ X3 )
& ( ord_less @ real @ X3 @ ( ring_1_of_int @ real @ ( plus_plus @ int @ Z4 @ ( one_one @ int ) ) ) ) ) ) ) ) ).
% floor_real_def
thf(fact_5567_Arg__correct,axiom,
! [Z: complex] :
( ( Z
!= ( zero_zero @ complex ) )
=> ( ( ( sgn_sgn @ complex @ Z )
= ( cis @ ( arg @ Z ) ) )
& ( ord_less @ real @ ( uminus_uminus @ real @ pi ) @ ( arg @ Z ) )
& ( ord_less_eq @ real @ ( arg @ Z ) @ pi ) ) ) ).
% Arg_correct
thf(fact_5568_arctan__inverse,axiom,
! [X2: real] :
( ( X2
!= ( zero_zero @ real ) )
=> ( ( arctan @ ( divide_divide @ real @ ( one_one @ real ) @ X2 ) )
= ( minus_minus @ real @ ( divide_divide @ real @ ( times_times @ real @ ( sgn_sgn @ real @ X2 ) @ pi ) @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) @ ( arctan @ X2 ) ) ) ) ).
% arctan_inverse
thf(fact_5569_neg__mask__is__div_H,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [N: nat,W: word @ A] :
( ( ord_less @ nat @ N @ ( size_size @ ( word @ A ) @ W ) )
=> ( ( bit_se5824344872417868541ns_and @ ( word @ A ) @ W @ ( bit_ri4277139882892585799ns_not @ ( word @ A ) @ ( bit_se2239418461657761734s_mask @ ( word @ A ) @ N ) ) )
= ( times_times @ ( word @ A ) @ ( divide_divide @ ( word @ A ) @ W @ ( 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_5570_signed__take__bit__eq__take__bit__minus,axiom,
( ( bit_ri4674362597316999326ke_bit @ int )
= ( ^ [N4: nat,K3: int] : ( minus_minus @ int @ ( bit_se2584673776208193580ke_bit @ int @ ( suc @ N4 ) @ K3 ) @ ( times_times @ int @ ( power_power @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ ( suc @ N4 ) ) @ ( zero_neq_one_of_bool @ int @ ( bit_se5641148757651400278ts_bit @ int @ K3 @ N4 ) ) ) ) ) ) ).
% signed_take_bit_eq_take_bit_minus
thf(fact_5571_neg__mask__add__mask,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [X2: word @ A,N: nat] :
( ( plus_plus @ ( word @ A ) @ ( bit_se5824344872417868541ns_and @ ( word @ A ) @ X2 @ ( bit_ri4277139882892585799ns_not @ ( word @ A ) @ ( 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 ) ) ) )
= ( bit_se1065995026697491101ons_or @ ( word @ A ) @ X2 @ ( bit_se2239418461657761734s_mask @ ( word @ A ) @ N ) ) ) ) ).
% neg_mask_add_mask
thf(fact_5572_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_5573_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_5574_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_5575_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_5576_bit_Odisj__cancel__left,axiom,
! [A: $tType] :
( ( bit_ri3973907225187159222ations @ A )
=> ! [X2: A] :
( ( bit_se1065995026697491101ons_or @ A @ ( bit_ri4277139882892585799ns_not @ A @ X2 ) @ X2 )
= ( uminus_uminus @ A @ ( one_one @ A ) ) ) ) ).
% bit.disj_cancel_left
thf(fact_5577_bit_Odisj__cancel__right,axiom,
! [A: $tType] :
( ( bit_ri3973907225187159222ations @ A )
=> ! [X2: A] :
( ( bit_se1065995026697491101ons_or @ A @ X2 @ ( bit_ri4277139882892585799ns_not @ A @ X2 ) )
= ( uminus_uminus @ A @ ( one_one @ A ) ) ) ) ).
% bit.disj_cancel_right
thf(fact_5578_minus__not__numeral__eq,axiom,
! [A: $tType] :
( ( bit_ri3973907225187159222ations @ A )
=> ! [N: num] :
( ( uminus_uminus @ A @ ( bit_ri4277139882892585799ns_not @ A @ ( numeral_numeral @ A @ N ) ) )
= ( numeral_numeral @ A @ ( inc @ N ) ) ) ) ).
% minus_not_numeral_eq
thf(fact_5579_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_5580_word__bitwise__m1__simps_I1_J,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ( ( bit_ri4277139882892585799ns_not @ ( word @ A ) @ ( uminus_uminus @ ( word @ A ) @ ( one_one @ ( word @ A ) ) ) )
= ( zero_zero @ ( word @ A ) ) ) ) ).
% word_bitwise_m1_simps(1)
thf(fact_5581_word__add__not,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [X2: word @ A] :
( ( plus_plus @ ( word @ A ) @ X2 @ ( bit_ri4277139882892585799ns_not @ ( word @ A ) @ X2 ) )
= ( uminus_uminus @ ( word @ A ) @ ( one_one @ ( word @ A ) ) ) ) ) ).
% word_add_not
thf(fact_5582_word__or__not,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [X2: word @ A] :
( ( bit_se1065995026697491101ons_or @ ( word @ A ) @ X2 @ ( bit_ri4277139882892585799ns_not @ ( word @ A ) @ X2 ) )
= ( uminus_uminus @ ( word @ A ) @ ( one_one @ ( word @ A ) ) ) ) ) ).
% word_or_not
thf(fact_5583_even__not__iff,axiom,
! [A: $tType] :
( ( bit_ri3973907225187159222ations @ A )
=> ! [A3: A] :
( ( dvd_dvd @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ ( bit_ri4277139882892585799ns_not @ A @ A3 ) )
= ( ~ ( dvd_dvd @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ A3 ) ) ) ) ).
% even_not_iff
thf(fact_5584_bit__minus__numeral__Bit0__Suc__iff,axiom,
! [W: num,N: nat] :
( ( bit_se5641148757651400278ts_bit @ int @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ ( bit0 @ W ) ) ) @ ( suc @ N ) )
= ( bit_se5641148757651400278ts_bit @ int @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ W ) ) @ N ) ) ).
% bit_minus_numeral_Bit0_Suc_iff
thf(fact_5585_bit__numeral__simps_I2_J,axiom,
! [A: $tType] :
( ( bit_un5681908812861735899ations @ A )
=> ! [W: num,N: num] :
( ( bit_se5641148757651400278ts_bit @ A @ ( numeral_numeral @ A @ ( bit0 @ W ) ) @ ( numeral_numeral @ nat @ N ) )
= ( bit_se5641148757651400278ts_bit @ A @ ( numeral_numeral @ A @ W ) @ ( pred_numeral @ N ) ) ) ) ).
% bit_numeral_simps(2)
thf(fact_5586_bit__minus__numeral__Bit1__Suc__iff,axiom,
! [W: num,N: nat] :
( ( bit_se5641148757651400278ts_bit @ int @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ ( bit1 @ W ) ) ) @ ( suc @ N ) )
= ( ~ ( bit_se5641148757651400278ts_bit @ int @ ( numeral_numeral @ int @ W ) @ N ) ) ) ).
% bit_minus_numeral_Bit1_Suc_iff
thf(fact_5587_bit__numeral__simps_I3_J,axiom,
! [A: $tType] :
( ( bit_un5681908812861735899ations @ A )
=> ! [W: num,N: num] :
( ( bit_se5641148757651400278ts_bit @ A @ ( numeral_numeral @ A @ ( bit1 @ W ) ) @ ( numeral_numeral @ nat @ N ) )
= ( bit_se5641148757651400278ts_bit @ A @ ( numeral_numeral @ A @ W ) @ ( pred_numeral @ N ) ) ) ) ).
% bit_numeral_simps(3)
thf(fact_5588_not__one__eq,axiom,
! [A: $tType] :
( ( bit_ri3973907225187159222ations @ A )
=> ( ( bit_ri4277139882892585799ns_not @ A @ ( one_one @ A ) )
= ( uminus_uminus @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) ) ) ).
% not_one_eq
thf(fact_5589_bit__0,axiom,
! [A: $tType] :
( ( bit_semiring_bits @ A )
=> ! [A3: A] :
( ( bit_se5641148757651400278ts_bit @ A @ A3 @ ( zero_zero @ nat ) )
= ( ~ ( dvd_dvd @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ A3 ) ) ) ) ).
% bit_0
thf(fact_5590_compl__of__1,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ( ( bit_ri4277139882892585799ns_not @ ( word @ A ) @ ( one_one @ ( word @ A ) ) )
= ( uminus_uminus @ ( word @ A ) @ ( numeral_numeral @ ( word @ A ) @ ( bit0 @ one2 ) ) ) ) ) ).
% compl_of_1
thf(fact_5591_bit__minus__numeral__int_I1_J,axiom,
! [W: num,N: num] :
( ( bit_se5641148757651400278ts_bit @ int @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ ( bit0 @ W ) ) ) @ ( numeral_numeral @ nat @ N ) )
= ( bit_se5641148757651400278ts_bit @ int @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ W ) ) @ ( pred_numeral @ N ) ) ) ).
% bit_minus_numeral_int(1)
thf(fact_5592_bit__minus__numeral__int_I2_J,axiom,
! [W: num,N: num] :
( ( bit_se5641148757651400278ts_bit @ int @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ ( bit1 @ W ) ) ) @ ( numeral_numeral @ nat @ N ) )
= ( ~ ( bit_se5641148757651400278ts_bit @ int @ ( numeral_numeral @ int @ W ) @ ( pred_numeral @ N ) ) ) ) ).
% bit_minus_numeral_int(2)
thf(fact_5593_bit__mod__2__iff,axiom,
! [A: $tType] :
( ( bit_semiring_bits @ A )
=> ! [A3: A,N: nat] :
( ( bit_se5641148757651400278ts_bit @ A @ ( modulo_modulo @ A @ A3 @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) @ N )
= ( ( N
= ( zero_zero @ nat ) )
& ~ ( dvd_dvd @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ A3 ) ) ) ) ).
% bit_mod_2_iff
thf(fact_5594_bin__nth__minus__Bit0,axiom,
! [N: nat,W: num] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ N )
=> ( ( bit_se5641148757651400278ts_bit @ int @ ( numeral_numeral @ int @ ( bit0 @ W ) ) @ N )
= ( bit_se5641148757651400278ts_bit @ int @ ( numeral_numeral @ int @ W ) @ ( minus_minus @ nat @ N @ ( one_one @ nat ) ) ) ) ) ).
% bin_nth_minus_Bit0
thf(fact_5595_bin__nth__minus__Bit1,axiom,
! [N: nat,W: num] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ N )
=> ( ( bit_se5641148757651400278ts_bit @ int @ ( numeral_numeral @ int @ ( bit1 @ W ) ) @ N )
= ( bit_se5641148757651400278ts_bit @ int @ ( numeral_numeral @ int @ W ) @ ( minus_minus @ nat @ N @ ( one_one @ nat ) ) ) ) ) ).
% bin_nth_minus_Bit1
thf(fact_5596_word__plus__and__or__coroll2,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [X2: word @ A,W: word @ A] :
( ( plus_plus @ ( word @ A ) @ ( bit_se5824344872417868541ns_and @ ( word @ A ) @ X2 @ W ) @ ( bit_se5824344872417868541ns_and @ ( word @ A ) @ X2 @ ( bit_ri4277139882892585799ns_not @ ( word @ A ) @ W ) ) )
= X2 ) ) ).
% word_plus_and_or_coroll2
thf(fact_5597_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_5598_bit__disjunctive__add__iff,axiom,
! [A: $tType] :
( ( bit_semiring_bits @ A )
=> ! [A3: A,B3: A,N: nat] :
( ! [N3: nat] :
( ~ ( bit_se5641148757651400278ts_bit @ A @ A3 @ N3 )
| ~ ( bit_se5641148757651400278ts_bit @ A @ B3 @ N3 ) )
=> ( ( bit_se5641148757651400278ts_bit @ A @ ( plus_plus @ A @ A3 @ B3 ) @ N )
= ( ( bit_se5641148757651400278ts_bit @ A @ A3 @ N )
| ( bit_se5641148757651400278ts_bit @ A @ B3 @ N ) ) ) ) ) ).
% bit_disjunctive_add_iff
thf(fact_5599_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_5600_disjunctive__diff,axiom,
! [A: $tType] :
( ( bit_ri3973907225187159222ations @ A )
=> ! [B3: A,A3: A] :
( ! [N3: nat] :
( ( bit_se5641148757651400278ts_bit @ A @ B3 @ N3 )
=> ( bit_se5641148757651400278ts_bit @ A @ A3 @ N3 ) )
=> ( ( minus_minus @ A @ A3 @ B3 )
= ( bit_se5824344872417868541ns_and @ A @ A3 @ ( bit_ri4277139882892585799ns_not @ A @ B3 ) ) ) ) ) ).
% disjunctive_diff
thf(fact_5601_disjunctive__add,axiom,
! [A: $tType] :
( ( bit_se359711467146920520ations @ A )
=> ! [A3: A,B3: A] :
( ! [N3: nat] :
( ~ ( bit_se5641148757651400278ts_bit @ A @ A3 @ N3 )
| ~ ( bit_se5641148757651400278ts_bit @ A @ B3 @ N3 ) )
=> ( ( plus_plus @ A @ A3 @ B3 )
= ( bit_se1065995026697491101ons_or @ A @ A3 @ B3 ) ) ) ) ).
% disjunctive_add
thf(fact_5602_not__diff__distrib,axiom,
! [A: $tType] :
( ( bit_ri3973907225187159222ations @ A )
=> ! [A3: A,B3: A] :
( ( bit_ri4277139882892585799ns_not @ A @ ( minus_minus @ A @ A3 @ B3 ) )
= ( plus_plus @ A @ ( bit_ri4277139882892585799ns_not @ A @ A3 ) @ B3 ) ) ) ).
% not_diff_distrib
thf(fact_5603_not__add__distrib,axiom,
! [A: $tType] :
( ( bit_ri3973907225187159222ations @ A )
=> ! [A3: A,B3: A] :
( ( bit_ri4277139882892585799ns_not @ A @ ( plus_plus @ A @ A3 @ B3 ) )
= ( minus_minus @ A @ ( bit_ri4277139882892585799ns_not @ A @ A3 ) @ B3 ) ) ) ).
% not_add_distrib
thf(fact_5604_bit__numeral__simps_I1_J,axiom,
! [A: $tType] :
( ( bit_un5681908812861735899ations @ A )
=> ! [N: num] :
~ ( bit_se5641148757651400278ts_bit @ A @ ( one_one @ A ) @ ( numeral_numeral @ nat @ N ) ) ) ).
% bit_numeral_simps(1)
thf(fact_5605_bit__numeral__iff,axiom,
! [A: $tType] :
( ( bit_un5681908812861735899ations @ A )
=> ! [M: num,N: nat] :
( ( bit_se5641148757651400278ts_bit @ A @ ( numeral_numeral @ A @ M ) @ N )
= ( bit_se5641148757651400278ts_bit @ nat @ ( numeral_numeral @ nat @ M ) @ N ) ) ) ).
% bit_numeral_iff
thf(fact_5606_of__int__not__numeral,axiom,
! [A: $tType] :
( ( bit_ri3973907225187159222ations @ A )
=> ! [K: num] :
( ( ring_1_of_int @ A @ ( bit_ri4277139882892585799ns_not @ int @ ( numeral_numeral @ int @ K ) ) )
= ( bit_ri4277139882892585799ns_not @ A @ ( numeral_numeral @ A @ K ) ) ) ) ).
% of_int_not_numeral
thf(fact_5607_bit__take__bit__iff,axiom,
! [A: $tType] :
( ( bit_se359711467146920520ations @ A )
=> ! [M: nat,A3: A,N: nat] :
( ( bit_se5641148757651400278ts_bit @ A @ ( bit_se2584673776208193580ke_bit @ A @ M @ A3 ) @ N )
= ( ( ord_less @ nat @ N @ M )
& ( bit_se5641148757651400278ts_bit @ A @ A3 @ N ) ) ) ) ).
% bit_take_bit_iff
thf(fact_5608_bit__not__iff__eq,axiom,
! [A: $tType] :
( ( bit_ri3973907225187159222ations @ A )
=> ! [A3: A,N: nat] :
( ( bit_se5641148757651400278ts_bit @ A @ ( bit_ri4277139882892585799ns_not @ A @ A3 ) @ N )
= ( ( ( power_power @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ N )
!= ( zero_zero @ A ) )
& ~ ( bit_se5641148757651400278ts_bit @ A @ A3 @ N ) ) ) ) ).
% bit_not_iff_eq
thf(fact_5609_signed__take__bit__def,axiom,
! [A: $tType] :
( ( bit_ri3973907225187159222ations @ A )
=> ( ( bit_ri4674362597316999326ke_bit @ A )
= ( ^ [N4: nat,A6: A] : ( bit_se1065995026697491101ons_or @ A @ ( bit_se2584673776208193580ke_bit @ A @ N4 @ A6 ) @ ( times_times @ A @ ( zero_neq_one_of_bool @ A @ ( bit_se5641148757651400278ts_bit @ A @ A6 @ N4 ) ) @ ( bit_ri4277139882892585799ns_not @ A @ ( bit_se2239418461657761734s_mask @ A @ N4 ) ) ) ) ) ) ) ).
% signed_take_bit_def
thf(fact_5610_minus__eq__not__plus__1,axiom,
! [A: $tType] :
( ( bit_ri3973907225187159222ations @ A )
=> ( ( uminus_uminus @ A )
= ( ^ [A6: A] : ( plus_plus @ A @ ( bit_ri4277139882892585799ns_not @ A @ A6 ) @ ( one_one @ A ) ) ) ) ) ).
% minus_eq_not_plus_1
thf(fact_5611_not__eq__complement,axiom,
! [A: $tType] :
( ( bit_ri3973907225187159222ations @ A )
=> ( ( bit_ri4277139882892585799ns_not @ A )
= ( ^ [A6: A] : ( minus_minus @ A @ ( uminus_uminus @ A @ A6 ) @ ( one_one @ A ) ) ) ) ) ).
% not_eq_complement
thf(fact_5612_minus__eq__not__minus__1,axiom,
! [A: $tType] :
( ( bit_ri3973907225187159222ations @ A )
=> ( ( uminus_uminus @ A )
= ( ^ [A6: A] : ( bit_ri4277139882892585799ns_not @ A @ ( minus_minus @ A @ A6 @ ( one_one @ A ) ) ) ) ) ) ).
% minus_eq_not_minus_1
thf(fact_5613_take__bit__not__eq__mask__diff,axiom,
! [A: $tType] :
( ( bit_ri3973907225187159222ations @ A )
=> ! [N: nat,A3: A] :
( ( bit_se2584673776208193580ke_bit @ A @ N @ ( bit_ri4277139882892585799ns_not @ A @ A3 ) )
= ( minus_minus @ A @ ( bit_se2239418461657761734s_mask @ A @ N ) @ ( bit_se2584673776208193580ke_bit @ A @ N @ A3 ) ) ) ) ).
% take_bit_not_eq_mask_diff
thf(fact_5614_mask__lower__twice,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [N: nat,M: nat,X2: word @ A] :
( ( ord_less_eq @ nat @ N @ M )
=> ( ( bit_se5824344872417868541ns_and @ ( word @ A ) @ ( bit_se5824344872417868541ns_and @ ( word @ A ) @ X2 @ ( 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 ) @ X2 @ ( bit_ri4277139882892585799ns_not @ ( word @ A ) @ ( bit_se2239418461657761734s_mask @ ( word @ A ) @ M ) ) ) ) ) ) ).
% mask_lower_twice
thf(fact_5615_mask__out__first__mask__some,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [X2: word @ A,N: nat,Y3: word @ A,M: nat] :
( ( ( bit_se5824344872417868541ns_and @ ( word @ A ) @ X2 @ ( bit_ri4277139882892585799ns_not @ ( word @ A ) @ ( bit_se2239418461657761734s_mask @ ( word @ A ) @ N ) ) )
= Y3 )
=> ( ( ord_less_eq @ nat @ N @ M )
=> ( ( bit_se5824344872417868541ns_and @ ( word @ A ) @ X2 @ ( bit_ri4277139882892585799ns_not @ ( word @ A ) @ ( bit_se2239418461657761734s_mask @ ( word @ A ) @ M ) ) )
= ( bit_se5824344872417868541ns_and @ ( word @ A ) @ Y3 @ ( bit_ri4277139882892585799ns_not @ ( word @ A ) @ ( bit_se2239418461657761734s_mask @ ( word @ A ) @ M ) ) ) ) ) ) ) ).
% mask_out_first_mask_some
thf(fact_5616_bit__not__int__iff_H,axiom,
! [K: int,N: nat] :
( ( bit_se5641148757651400278ts_bit @ int @ ( minus_minus @ int @ ( uminus_uminus @ int @ K ) @ ( one_one @ int ) ) @ N )
= ( ~ ( bit_se5641148757651400278ts_bit @ int @ K @ N ) ) ) ).
% bit_not_int_iff'
thf(fact_5617_NOT__eq,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ( ( bit_ri4277139882892585799ns_not @ ( word @ A ) )
= ( ^ [X3: word @ A] : ( minus_minus @ ( word @ A ) @ ( uminus_uminus @ ( word @ A ) @ X3 ) @ ( one_one @ ( word @ A ) ) ) ) ) ) ).
% NOT_eq
thf(fact_5618_AND__NOT__mask__plus__AND__mask__eq,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [W: word @ A,N: nat] :
( ( plus_plus @ ( word @ A ) @ ( bit_se5824344872417868541ns_and @ ( word @ A ) @ W @ ( bit_ri4277139882892585799ns_not @ ( word @ A ) @ ( bit_se2239418461657761734s_mask @ ( word @ A ) @ N ) ) ) @ ( bit_se5824344872417868541ns_and @ ( word @ A ) @ W @ ( bit_se2239418461657761734s_mask @ ( word @ A ) @ N ) ) )
= W ) ) ).
% AND_NOT_mask_plus_AND_mask_eq
thf(fact_5619_minus__numeral__inc__eq,axiom,
! [A: $tType] :
( ( bit_ri3973907225187159222ations @ A )
=> ! [N: num] :
( ( uminus_uminus @ A @ ( numeral_numeral @ A @ ( inc @ N ) ) )
= ( bit_ri4277139882892585799ns_not @ A @ ( numeral_numeral @ A @ N ) ) ) ) ).
% minus_numeral_inc_eq
thf(fact_5620_subtract__mask_I2_J,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [P6: word @ A,N: nat] :
( ( minus_minus @ ( word @ A ) @ P6 @ ( bit_se5824344872417868541ns_and @ ( word @ A ) @ P6 @ ( bit_ri4277139882892585799ns_not @ ( word @ A ) @ ( bit_se2239418461657761734s_mask @ ( word @ A ) @ N ) ) ) )
= ( bit_se5824344872417868541ns_and @ ( word @ A ) @ P6 @ ( bit_se2239418461657761734s_mask @ ( word @ A ) @ N ) ) ) ) ).
% subtract_mask(2)
thf(fact_5621_subtract__mask_I1_J,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [P6: word @ A,N: nat] :
( ( minus_minus @ ( word @ A ) @ P6 @ ( bit_se5824344872417868541ns_and @ ( word @ A ) @ P6 @ ( bit_se2239418461657761734s_mask @ ( word @ A ) @ N ) ) )
= ( bit_se5824344872417868541ns_and @ ( word @ A ) @ P6 @ ( bit_ri4277139882892585799ns_not @ ( word @ A ) @ ( bit_se2239418461657761734s_mask @ ( word @ A ) @ N ) ) ) ) ) ).
% subtract_mask(1)
thf(fact_5622_mask__out__sub__mask,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [X2: word @ A,N: nat] :
( ( bit_se5824344872417868541ns_and @ ( word @ A ) @ X2 @ ( bit_ri4277139882892585799ns_not @ ( word @ A ) @ ( bit_se2239418461657761734s_mask @ ( word @ A ) @ N ) ) )
= ( minus_minus @ ( word @ A ) @ X2 @ ( bit_se5824344872417868541ns_and @ ( word @ A ) @ X2 @ ( bit_se2239418461657761734s_mask @ ( word @ A ) @ N ) ) ) ) ) ).
% mask_out_sub_mask
thf(fact_5623_and__not__mask__twice,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [W: word @ A,N: nat,M: nat] :
( ( bit_se5824344872417868541ns_and @ ( word @ A ) @ ( bit_se5824344872417868541ns_and @ ( word @ A ) @ W @ ( 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 ) @ W @ ( bit_ri4277139882892585799ns_not @ ( word @ A ) @ ( bit_se2239418461657761734s_mask @ ( word @ A ) @ ( ord_max @ nat @ M @ N ) ) ) ) ) ) ).
% and_not_mask_twice
thf(fact_5624_mask__lower__twice2,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [A3: word @ A,N: nat,M: nat] :
( ( bit_se5824344872417868541ns_and @ ( word @ A ) @ ( bit_se5824344872417868541ns_and @ ( word @ A ) @ A3 @ ( 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 ) @ A3 @ ( bit_ri4277139882892585799ns_not @ ( word @ A ) @ ( bit_se2239418461657761734s_mask @ ( word @ A ) @ ( ord_max @ nat @ N @ M ) ) ) ) ) ) ).
% mask_lower_twice2
thf(fact_5625_neg__mask__combine,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [A3: nat,B3: nat] :
( ( bit_se5824344872417868541ns_and @ ( word @ A ) @ ( bit_ri4277139882892585799ns_not @ ( word @ A ) @ ( bit_se2239418461657761734s_mask @ ( word @ A ) @ A3 ) ) @ ( bit_ri4277139882892585799ns_not @ ( word @ A ) @ ( bit_se2239418461657761734s_mask @ ( word @ A ) @ B3 ) ) )
= ( bit_ri4277139882892585799ns_not @ ( word @ A ) @ ( bit_se2239418461657761734s_mask @ ( word @ A ) @ ( ord_max @ nat @ A3 @ B3 ) ) ) ) ) ).
% neg_mask_combine
thf(fact_5626_neg__mask__twice,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [X2: word @ A,N: nat,M: nat] :
( ( bit_se5824344872417868541ns_and @ ( word @ A ) @ X2 @ ( bit_se5824344872417868541ns_and @ ( word @ A ) @ ( 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 ) @ X2 @ ( bit_ri4277139882892585799ns_not @ ( word @ A ) @ ( bit_se2239418461657761734s_mask @ ( word @ A ) @ ( ord_max @ nat @ N @ M ) ) ) ) ) ) ).
% neg_mask_twice
thf(fact_5627_not__numeral__Bit0__eq,axiom,
! [A: $tType] :
( ( bit_ri3973907225187159222ations @ A )
=> ! [N: num] :
( ( bit_ri4277139882892585799ns_not @ A @ ( numeral_numeral @ A @ ( bit0 @ N ) ) )
= ( uminus_uminus @ A @ ( numeral_numeral @ A @ ( bit1 @ N ) ) ) ) ) ).
% not_numeral_Bit0_eq
thf(fact_5628_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_5629_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_5630_not__numeral__BitM__eq,axiom,
! [A: $tType] :
( ( bit_ri3973907225187159222ations @ A )
=> ! [N: num] :
( ( bit_ri4277139882892585799ns_not @ A @ ( numeral_numeral @ A @ ( bitM @ N ) ) )
= ( uminus_uminus @ A @ ( numeral_numeral @ A @ ( bit0 @ N ) ) ) ) ) ).
% not_numeral_BitM_eq
thf(fact_5631_multiple__mask__trivia,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [M: nat,N: nat,X2: word @ A] :
( ( ord_less_eq @ nat @ M @ N )
=> ( ( plus_plus @ ( word @ A ) @ ( bit_se5824344872417868541ns_and @ ( word @ A ) @ X2 @ ( bit_ri4277139882892585799ns_not @ ( word @ A ) @ ( bit_se2239418461657761734s_mask @ ( word @ A ) @ N ) ) ) @ ( bit_se5824344872417868541ns_and @ ( word @ A ) @ X2 @ ( 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 ) @ X2 @ ( bit_ri4277139882892585799ns_not @ ( word @ A ) @ ( bit_se2239418461657761734s_mask @ ( word @ A ) @ M ) ) ) ) ) ) ).
% multiple_mask_trivia
thf(fact_5632_and__mask__0__iff__le__mask,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [W: word @ A,N: nat] :
( ( ( bit_se5824344872417868541ns_and @ ( word @ A ) @ W @ ( bit_ri4277139882892585799ns_not @ ( word @ A ) @ ( bit_se2239418461657761734s_mask @ ( word @ A ) @ N ) ) )
= ( zero_zero @ ( word @ A ) ) )
= ( ord_less_eq @ ( word @ A ) @ W @ ( bit_se2239418461657761734s_mask @ ( word @ A ) @ N ) ) ) ) ).
% and_mask_0_iff_le_mask
thf(fact_5633_mask__subsume,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [N: nat,M: nat,X2: word @ A,Y3: word @ A] :
( ( ord_less_eq @ nat @ N @ M )
=> ( ( bit_se5824344872417868541ns_and @ ( word @ A ) @ ( bit_se1065995026697491101ons_or @ ( word @ A ) @ X2 @ ( bit_se5824344872417868541ns_and @ ( word @ A ) @ Y3 @ ( bit_se2239418461657761734s_mask @ ( word @ A ) @ N ) ) ) @ ( bit_ri4277139882892585799ns_not @ ( word @ A ) @ ( bit_se2239418461657761734s_mask @ ( word @ A ) @ M ) ) )
= ( bit_se5824344872417868541ns_and @ ( word @ A ) @ X2 @ ( bit_ri4277139882892585799ns_not @ ( word @ A ) @ ( bit_se2239418461657761734s_mask @ ( word @ A ) @ M ) ) ) ) ) ) ).
% mask_subsume
thf(fact_5634_exp__eq__0__imp__not__bit,axiom,
! [A: $tType] :
( ( bit_semiring_bits @ A )
=> ! [N: nat,A3: A] :
( ( ( power_power @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ N )
= ( zero_zero @ A ) )
=> ~ ( bit_se5641148757651400278ts_bit @ A @ A3 @ N ) ) ) ).
% exp_eq_0_imp_not_bit
thf(fact_5635_bit__Suc,axiom,
! [A: $tType] :
( ( bit_semiring_bits @ A )
=> ! [A3: A,N: nat] :
( ( bit_se5641148757651400278ts_bit @ A @ A3 @ ( suc @ N ) )
= ( bit_se5641148757651400278ts_bit @ A @ ( divide_divide @ A @ A3 @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) @ N ) ) ) ).
% bit_Suc
thf(fact_5636_bit__iff__idd__imp__stable,axiom,
! [A: $tType] :
( ( bit_semiring_bits @ A )
=> ! [A3: A] :
( ! [N3: nat] :
( ( bit_se5641148757651400278ts_bit @ A @ A3 @ N3 )
= ( ~ ( dvd_dvd @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ A3 ) ) )
=> ( ( divide_divide @ A @ A3 @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) )
= A3 ) ) ) ).
% bit_iff_idd_imp_stable
thf(fact_5637_stable__imp__bit__iff__odd,axiom,
! [A: $tType] :
( ( bit_semiring_bits @ A )
=> ! [A3: A,N: nat] :
( ( ( divide_divide @ A @ A3 @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) )
= A3 )
=> ( ( bit_se5641148757651400278ts_bit @ A @ A3 @ N )
= ( ~ ( dvd_dvd @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ A3 ) ) ) ) ) ).
% stable_imp_bit_iff_odd
thf(fact_5638_int__bit__bound,axiom,
! [K: int] :
~ ! [N3: nat] :
( ! [M3: nat] :
( ( ord_less_eq @ nat @ N3 @ M3 )
=> ( ( bit_se5641148757651400278ts_bit @ int @ K @ M3 )
= ( bit_se5641148757651400278ts_bit @ int @ K @ N3 ) ) )
=> ~ ( ( ord_less @ nat @ ( zero_zero @ nat ) @ N3 )
=> ( ( bit_se5641148757651400278ts_bit @ int @ K @ ( minus_minus @ nat @ N3 @ ( one_one @ nat ) ) )
= ( ~ ( bit_se5641148757651400278ts_bit @ int @ K @ N3 ) ) ) ) ) ).
% int_bit_bound
thf(fact_5639_bit_Ocompl__unique,axiom,
! [A: $tType] :
( ( bit_ri3973907225187159222ations @ A )
=> ! [X2: A,Y3: A] :
( ( ( bit_se5824344872417868541ns_and @ A @ X2 @ Y3 )
= ( zero_zero @ A ) )
=> ( ( ( bit_se1065995026697491101ons_or @ A @ X2 @ Y3 )
= ( uminus_uminus @ A @ ( one_one @ A ) ) )
=> ( ( bit_ri4277139882892585799ns_not @ A @ X2 )
= Y3 ) ) ) ) ).
% bit.compl_unique
thf(fact_5640_bit__iff__odd,axiom,
! [A: $tType] :
( ( bit_semiring_bits @ A )
=> ( ( bit_se5641148757651400278ts_bit @ A )
= ( ^ [A6: A,N4: nat] :
~ ( dvd_dvd @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ ( divide_divide @ A @ A6 @ ( power_power @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ N4 ) ) ) ) ) ) ).
% bit_iff_odd
thf(fact_5641_and__exp__eq__0__iff__not__bit,axiom,
! [A: $tType] :
( ( bit_se359711467146920520ations @ A )
=> ! [A3: A,N: nat] :
( ( ( bit_se5824344872417868541ns_and @ A @ A3 @ ( power_power @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ N ) )
= ( zero_zero @ A ) )
= ( ~ ( bit_se5641148757651400278ts_bit @ A @ A3 @ N ) ) ) ) ).
% and_exp_eq_0_iff_not_bit
thf(fact_5642_minus__exp__eq__not__mask,axiom,
! [A: $tType] :
( ( bit_ri3973907225187159222ations @ A )
=> ! [N: nat] :
( ( uminus_uminus @ A @ ( power_power @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ N ) )
= ( bit_ri4277139882892585799ns_not @ A @ ( bit_se2239418461657761734s_mask @ A @ N ) ) ) ) ).
% minus_exp_eq_not_mask
thf(fact_5643_NOT__mask,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [N: nat] :
( ( bit_ri4277139882892585799ns_not @ ( word @ A ) @ ( bit_se2239418461657761734s_mask @ ( word @ A ) @ N ) )
= ( uminus_uminus @ ( word @ A ) @ ( power_power @ ( word @ A ) @ ( numeral_numeral @ ( word @ A ) @ ( bit0 @ one2 ) ) @ N ) ) ) ) ).
% NOT_mask
thf(fact_5644_bit__int__def,axiom,
( ( bit_se5641148757651400278ts_bit @ int )
= ( ^ [K3: int,N4: nat] :
~ ( dvd_dvd @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ ( divide_divide @ int @ K3 @ ( power_power @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ N4 ) ) ) ) ) ).
% bit_int_def
thf(fact_5645_even__bit__succ__iff,axiom,
! [A: $tType] :
( ( bit_semiring_bits @ A )
=> ! [A3: A,N: nat] :
( ( dvd_dvd @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ A3 )
=> ( ( bit_se5641148757651400278ts_bit @ A @ ( plus_plus @ A @ ( one_one @ A ) @ A3 ) @ N )
= ( ( bit_se5641148757651400278ts_bit @ A @ A3 @ N )
| ( N
= ( zero_zero @ nat ) ) ) ) ) ) ).
% even_bit_succ_iff
thf(fact_5646_odd__bit__iff__bit__pred,axiom,
! [A: $tType] :
( ( bit_semiring_bits @ A )
=> ! [A3: A,N: nat] :
( ~ ( dvd_dvd @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ A3 )
=> ( ( bit_se5641148757651400278ts_bit @ A @ A3 @ N )
= ( ( bit_se5641148757651400278ts_bit @ A @ ( minus_minus @ A @ A3 @ ( one_one @ A ) ) @ N )
| ( N
= ( zero_zero @ nat ) ) ) ) ) ) ).
% odd_bit_iff_bit_pred
thf(fact_5647_bit__sum__mult__2__cases,axiom,
! [A: $tType] :
( ( bit_se359711467146920520ations @ A )
=> ! [A3: A,B3: A,N: nat] :
( ! [J2: nat] :
~ ( bit_se5641148757651400278ts_bit @ A @ A3 @ ( suc @ J2 ) )
=> ( ( bit_se5641148757651400278ts_bit @ A @ ( plus_plus @ A @ A3 @ ( times_times @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ B3 ) ) @ N )
= ( ( ( N
= ( zero_zero @ nat ) )
=> ~ ( dvd_dvd @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ A3 ) )
& ( ( N
!= ( zero_zero @ nat ) )
=> ( bit_se5641148757651400278ts_bit @ A @ ( times_times @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ B3 ) @ N ) ) ) ) ) ) ).
% bit_sum_mult_2_cases
thf(fact_5648_bit__rec,axiom,
! [A: $tType] :
( ( bit_semiring_bits @ A )
=> ( ( bit_se5641148757651400278ts_bit @ A )
= ( ^ [A6: A,N4: nat] :
( ( ( N4
= ( zero_zero @ nat ) )
=> ~ ( dvd_dvd @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ A6 ) )
& ( ( N4
!= ( zero_zero @ nat ) )
=> ( bit_se5641148757651400278ts_bit @ A @ ( divide_divide @ A @ A6 @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) @ ( minus_minus @ nat @ N4 @ ( one_one @ nat ) ) ) ) ) ) ) ) ).
% bit_rec
thf(fact_5649_neg__mask__is__div,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [W: word @ A,N: nat] :
( ( bit_se5824344872417868541ns_and @ ( word @ A ) @ W @ ( bit_ri4277139882892585799ns_not @ ( word @ A ) @ ( bit_se2239418461657761734s_mask @ ( word @ A ) @ N ) ) )
= ( times_times @ ( word @ A ) @ ( divide_divide @ ( word @ A ) @ W @ ( 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_5650_Bit__Operations_Oset__bit__eq,axiom,
( ( bit_se5668285175392031749et_bit @ int )
= ( ^ [N4: nat,K3: int] :
( plus_plus @ int @ K3
@ ( times_times @ int
@ ( zero_neq_one_of_bool @ int
@ ~ ( bit_se5641148757651400278ts_bit @ int @ K3 @ N4 ) )
@ ( power_power @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ N4 ) ) ) ) ) ).
% Bit_Operations.set_bit_eq
thf(fact_5651_unset__bit__eq,axiom,
( ( bit_se2638667681897837118et_bit @ int )
= ( ^ [N4: nat,K3: int] : ( minus_minus @ int @ K3 @ ( times_times @ int @ ( zero_neq_one_of_bool @ int @ ( bit_se5641148757651400278ts_bit @ int @ K3 @ N4 ) ) @ ( power_power @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ N4 ) ) ) ) ) ).
% unset_bit_eq
thf(fact_5652_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_5653_floor__rat__def,axiom,
( ( archim6421214686448440834_floor @ rat )
= ( ^ [X3: rat] :
( the @ int
@ ^ [Z4: int] :
( ( ord_less_eq @ rat @ ( ring_1_of_int @ rat @ Z4 ) @ X3 )
& ( ord_less @ rat @ X3 @ ( ring_1_of_int @ rat @ ( plus_plus @ int @ Z4 @ ( one_one @ int ) ) ) ) ) ) ) ) ).
% floor_rat_def
thf(fact_5654_root__def,axiom,
( root
= ( ^ [N4: nat,X3: real] :
( if @ real
@ ( N4
= ( zero_zero @ nat ) )
@ ( zero_zero @ real )
@ ( the_inv_into @ real @ real @ ( top_top @ ( set @ real ) )
@ ^ [Y: real] : ( times_times @ real @ ( sgn_sgn @ real @ Y ) @ ( power_power @ real @ ( abs_abs @ real @ Y ) @ N4 ) )
@ X3 ) ) ) ) ).
% root_def
thf(fact_5655_Arg__def,axiom,
( arg
= ( ^ [Z4: complex] :
( if @ real
@ ( Z4
= ( zero_zero @ complex ) )
@ ( zero_zero @ real )
@ ( fChoice @ real
@ ^ [A6: real] :
( ( ( sgn_sgn @ complex @ Z4 )
= ( cis @ A6 ) )
& ( ord_less @ real @ ( uminus_uminus @ real @ pi ) @ A6 )
& ( ord_less_eq @ real @ A6 @ pi ) ) ) ) ) ) ).
% Arg_def
thf(fact_5656_some__sym__eq__trivial,axiom,
! [A: $tType,X2: A] :
( ( fChoice @ A
@ ( ^ [Y6: A,Z3: A] : Y6 = Z3
@ X2 ) )
= X2 ) ).
% some_sym_eq_trivial
thf(fact_5657_some__eq__trivial,axiom,
! [A: $tType,X2: A] :
( ( fChoice @ A
@ ^ [Y: A] : Y = X2 )
= X2 ) ).
% some_eq_trivial
thf(fact_5658_some__equality,axiom,
! [A: $tType,P: A > $o,A3: A] :
( ( P @ A3 )
=> ( ! [X4: A] :
( ( P @ X4 )
=> ( X4 = A3 ) )
=> ( ( fChoice @ A @ P )
= A3 ) ) ) ).
% some_equality
thf(fact_5659_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_5660_some__insert__self,axiom,
! [A: $tType,S: set @ A] :
( ( S
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( insert @ A
@ ( fChoice @ A
@ ^ [X3: A] : ( member @ A @ X3 @ S ) )
@ S )
= S ) ) ).
% some_insert_self
thf(fact_5661_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_5662_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_5663_word__of__int__not__numeral__eq,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [Bin: num] :
( ( ring_1_of_int @ ( word @ A ) @ ( bit_ri4277139882892585799ns_not @ int @ ( numeral_numeral @ int @ Bin ) ) )
= ( minus_minus @ ( word @ A ) @ ( uminus_uminus @ ( word @ A ) @ ( numeral_numeral @ ( word @ A ) @ Bin ) ) @ ( one_one @ ( word @ A ) ) ) ) ) ).
% word_of_int_not_numeral_eq
thf(fact_5664_and__minus__minus__numerals,axiom,
! [M: num,N: num] :
( ( bit_se5824344872417868541ns_and @ int @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ M ) ) @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ N ) ) )
= ( bit_ri4277139882892585799ns_not @ int @ ( bit_se1065995026697491101ons_or @ int @ ( minus_minus @ int @ ( numeral_numeral @ int @ M ) @ ( one_one @ int ) ) @ ( minus_minus @ int @ ( numeral_numeral @ int @ N ) @ ( one_one @ int ) ) ) ) ) ).
% and_minus_minus_numerals
thf(fact_5665_or__minus__minus__numerals,axiom,
! [M: num,N: num] :
( ( bit_se1065995026697491101ons_or @ int @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ M ) ) @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ N ) ) )
= ( bit_ri4277139882892585799ns_not @ int @ ( bit_se5824344872417868541ns_and @ int @ ( minus_minus @ int @ ( numeral_numeral @ int @ M ) @ ( one_one @ int ) ) @ ( minus_minus @ int @ ( numeral_numeral @ int @ N ) @ ( one_one @ int ) ) ) ) ) ).
% or_minus_minus_numerals
thf(fact_5666_sgn__rat__def,axiom,
( ( sgn_sgn @ rat )
= ( ^ [A6: rat] :
( if @ rat
@ ( A6
= ( zero_zero @ rat ) )
@ ( zero_zero @ rat )
@ ( if @ rat @ ( ord_less @ rat @ ( zero_zero @ rat ) @ A6 ) @ ( one_one @ rat ) @ ( uminus_uminus @ rat @ ( one_one @ rat ) ) ) ) ) ) ).
% sgn_rat_def
thf(fact_5667_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_5668_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_5669_finite__bit__word,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [W: word @ A] : ( finite_finite2 @ nat @ ( collect @ nat @ ( bit_se5641148757651400278ts_bit @ ( word @ A ) @ W ) ) ) ) ).
% finite_bit_word
thf(fact_5670_some__in__eq,axiom,
! [A: $tType,A2: set @ A] :
( ( member @ A
@ ( fChoice @ A
@ ^ [X3: A] : ( member @ A @ X3 @ A2 ) )
@ A2 )
= ( A2
!= ( bot_bot @ ( set @ A ) ) ) ) ).
% some_in_eq
thf(fact_5671_some__elem,axiom,
! [A: $tType,S: set @ A] :
( ( S
!= ( bot_bot @ ( set @ A ) ) )
=> ( member @ A
@ ( fChoice @ A
@ ^ [X3: A] : ( member @ A @ X3 @ S ) )
@ S ) ) ).
% some_elem
thf(fact_5672_less__eq__rat__def,axiom,
( ( ord_less_eq @ rat )
= ( ^ [X3: rat,Y: rat] :
( ( ord_less @ rat @ X3 @ Y )
| ( X3 = Y ) ) ) ) ).
% less_eq_rat_def
thf(fact_5673_obtain__pos__sum,axiom,
! [R2: rat] :
( ( ord_less @ rat @ ( zero_zero @ rat ) @ R2 )
=> ~ ! [S3: rat] :
( ( ord_less @ rat @ ( zero_zero @ rat ) @ S3 )
=> ! [T8: rat] :
( ( ord_less @ rat @ ( zero_zero @ rat ) @ T8 )
=> ( R2
!= ( plus_plus @ rat @ S3 @ T8 ) ) ) ) ) ).
% obtain_pos_sum
thf(fact_5674_verit__sko__forall__indirect2,axiom,
! [A: $tType,X2: A,P: A > $o,P5: A > $o] :
( ( X2
= ( fChoice @ A
@ ^ [X3: A] :
~ ( P @ X3 ) ) )
=> ( ! [X4: A] :
( ( P @ X4 )
= ( P5 @ X4 ) )
=> ( ( ! [X7: A] : ( P5 @ X7 ) )
= ( P @ X2 ) ) ) ) ).
% verit_sko_forall_indirect2
thf(fact_5675_verit__sko__forall__indirect,axiom,
! [A: $tType,X2: A,P: A > $o] :
( ( X2
= ( fChoice @ A
@ ^ [X3: A] :
~ ( P @ X3 ) ) )
=> ( ( ! [X7: A] : ( P @ X7 ) )
= ( P @ X2 ) ) ) ).
% verit_sko_forall_indirect
thf(fact_5676_verit__sko__ex__indirect2,axiom,
! [A: $tType,X2: A,P: A > $o,P5: A > $o] :
( ( X2
= ( fChoice @ A @ P ) )
=> ( ! [X4: A] :
( ( P @ X4 )
= ( P5 @ X4 ) )
=> ( ( ? [X7: A] : ( P5 @ X7 ) )
= ( P @ X2 ) ) ) ) ).
% verit_sko_ex_indirect2
thf(fact_5677_verit__sko__ex__indirect,axiom,
! [A: $tType,X2: A,P: A > $o] :
( ( X2
= ( fChoice @ A @ P ) )
=> ( ( ? [X7: A] : ( P @ X7 ) )
= ( P @ X2 ) ) ) ).
% verit_sko_ex_indirect
thf(fact_5678_verit__sko__forall_H_H,axiom,
! [A: $tType,B2: A,A2: A,P: A > $o] :
( ( B2 = A2 )
=> ( ( ( fChoice @ A @ P )
= A2 )
= ( ( fChoice @ A @ P )
= B2 ) ) ) ).
% verit_sko_forall''
thf(fact_5679_verit__sko__forall_H,axiom,
! [A: $tType,P: A > $o,A2: $o] :
( ( ( P
@ ( fChoice @ A
@ ^ [X3: A] :
~ ( P @ X3 ) ) )
= A2 )
=> ( ( ! [X7: A] : ( P @ X7 ) )
= A2 ) ) ).
% verit_sko_forall'
thf(fact_5680_verit__sko__forall,axiom,
! [A: $tType] :
( ( ^ [P2: A > $o] :
! [X6: A] : ( P2 @ X6 ) )
= ( ^ [P3: A > $o] :
( P3
@ ( fChoice @ A
@ ^ [X3: A] :
~ ( P3 @ X3 ) ) ) ) ) ).
% verit_sko_forall
thf(fact_5681_verit__sko__ex_H,axiom,
! [A: $tType,P: A > $o,A2: $o] :
( ( ( P @ ( fChoice @ A @ P ) )
= A2 )
=> ( ( ? [X7: A] : ( P @ X7 ) )
= A2 ) ) ).
% verit_sko_ex'
thf(fact_5682_some1__equality,axiom,
! [A: $tType,P: A > $o,A3: A] :
( ? [X8: A] :
( ( P @ X8 )
& ! [Y4: A] :
( ( P @ Y4 )
=> ( Y4 = X8 ) ) )
=> ( ( P @ A3 )
=> ( ( fChoice @ A @ P )
= A3 ) ) ) ).
% some1_equality
thf(fact_5683_some__eq__ex,axiom,
! [A: $tType,P: A > $o] :
( ( P @ ( fChoice @ A @ P ) )
= ( ? [X7: A] : ( P @ X7 ) ) ) ).
% some_eq_ex
thf(fact_5684_someI2__bex,axiom,
! [A: $tType,A2: set @ A,P: A > $o,Q: A > $o] :
( ? [X8: A] :
( ( member @ A @ X8 @ A2 )
& ( P @ X8 ) )
=> ( ! [X4: A] :
( ( ( member @ A @ X4 @ A2 )
& ( P @ X4 ) )
=> ( Q @ X4 ) )
=> ( Q
@ ( fChoice @ A
@ ^ [X3: A] :
( ( member @ A @ X3 @ A2 )
& ( P @ X3 ) ) ) ) ) ) ).
% someI2_bex
thf(fact_5685_someI2__ex,axiom,
! [A: $tType,P: A > $o,Q: A > $o] :
( ? [X_12: A] : ( P @ X_12 )
=> ( ! [X4: A] :
( ( P @ X4 )
=> ( Q @ X4 ) )
=> ( Q @ ( fChoice @ A @ P ) ) ) ) ).
% someI2_ex
thf(fact_5686_someI__ex,axiom,
! [A: $tType,P: A > $o] :
( ? [X_12: A] : ( P @ X_12 )
=> ( P @ ( fChoice @ A @ P ) ) ) ).
% someI_ex
thf(fact_5687_someI2,axiom,
! [A: $tType,P: A > $o,A3: A,Q: A > $o] :
( ( P @ A3 )
=> ( ! [X4: A] :
( ( P @ X4 )
=> ( Q @ X4 ) )
=> ( Q @ ( fChoice @ A @ P ) ) ) ) ).
% someI2
thf(fact_5688_word__eqI,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [U: word @ A,V: word @ A] :
( ! [N3: nat] :
( ( ord_less @ nat @ N3 @ ( size_size @ ( word @ A ) @ U ) )
=> ( ( bit_se5641148757651400278ts_bit @ ( word @ A ) @ U @ N3 )
= ( bit_se5641148757651400278ts_bit @ ( word @ A ) @ V @ N3 ) ) )
=> ( U = V ) ) ) ).
% word_eqI
thf(fact_5689_test__bit__size,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [W: word @ A,N: nat] :
( ( bit_se5641148757651400278ts_bit @ ( word @ A ) @ W @ N )
=> ( ord_less @ nat @ N @ ( size_size @ ( word @ A ) @ W ) ) ) ) ).
% test_bit_size
thf(fact_5690_test__bit__over,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [X2: word @ A,N: nat] :
( ( ord_less_eq @ nat @ ( size_size @ ( word @ A ) @ X2 ) @ N )
=> ~ ( bit_se5641148757651400278ts_bit @ ( word @ A ) @ X2 @ N ) ) ) ).
% test_bit_over
thf(fact_5691_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_5692_not__int__def,axiom,
( ( bit_ri4277139882892585799ns_not @ int )
= ( ^ [K3: int] : ( minus_minus @ int @ ( uminus_uminus @ int @ K3 ) @ ( one_one @ int ) ) ) ) ).
% not_int_def
thf(fact_5693_and__not__numerals_I1_J,axiom,
( ( bit_se5824344872417868541ns_and @ int @ ( one_one @ int ) @ ( bit_ri4277139882892585799ns_not @ int @ ( one_one @ int ) ) )
= ( zero_zero @ int ) ) ).
% and_not_numerals(1)
thf(fact_5694_or__not__numerals_I1_J,axiom,
( ( bit_se1065995026697491101ons_or @ int @ ( one_one @ int ) @ ( bit_ri4277139882892585799ns_not @ int @ ( one_one @ int ) ) )
= ( bit_ri4277139882892585799ns_not @ int @ ( zero_zero @ int ) ) ) ).
% or_not_numerals(1)
thf(fact_5695_lsb__this__or__next,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [X2: word @ A] :
( ~ ( bit_se5641148757651400278ts_bit @ ( word @ A ) @ ( plus_plus @ ( word @ A ) @ X2 @ ( one_one @ ( word @ A ) ) ) @ ( zero_zero @ nat ) )
=> ( bit_se5641148757651400278ts_bit @ ( word @ A ) @ X2 @ ( zero_zero @ nat ) ) ) ) ).
% lsb_this_or_next
thf(fact_5696_word__leI,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [U: word @ A,V: word @ A] :
( ! [N3: nat] :
( ( ord_less @ nat @ N3 @ ( size_size @ ( word @ A ) @ U ) )
=> ( ( bit_se5641148757651400278ts_bit @ ( word @ A ) @ U @ N3 )
=> ( bit_se5641148757651400278ts_bit @ ( word @ A ) @ V @ N3 ) ) )
=> ( ord_less_eq @ ( word @ A ) @ U @ V ) ) ) ).
% word_leI
thf(fact_5697_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_5698_not__int__div__2,axiom,
! [K: int] :
( ( divide_divide @ int @ ( bit_ri4277139882892585799ns_not @ int @ K ) @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) )
= ( bit_ri4277139882892585799ns_not @ int @ ( divide_divide @ int @ K @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) ) ) ) ).
% not_int_div_2
thf(fact_5699_even__not__iff__int,axiom,
! [K: int] :
( ( dvd_dvd @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ ( bit_ri4277139882892585799ns_not @ int @ K ) )
= ( ~ ( dvd_dvd @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ K ) ) ) ).
% even_not_iff_int
thf(fact_5700_and__not__numerals_I4_J,axiom,
! [M: num] :
( ( bit_se5824344872417868541ns_and @ int @ ( numeral_numeral @ int @ ( bit0 @ M ) ) @ ( bit_ri4277139882892585799ns_not @ int @ ( one_one @ int ) ) )
= ( numeral_numeral @ int @ ( bit0 @ M ) ) ) ).
% and_not_numerals(4)
thf(fact_5701_and__not__numerals_I2_J,axiom,
! [N: num] :
( ( bit_se5824344872417868541ns_and @ int @ ( one_one @ int ) @ ( bit_ri4277139882892585799ns_not @ int @ ( numeral_numeral @ int @ ( bit0 @ N ) ) ) )
= ( one_one @ int ) ) ).
% and_not_numerals(2)
thf(fact_5702_or__not__numerals_I2_J,axiom,
! [N: num] :
( ( bit_se1065995026697491101ons_or @ int @ ( one_one @ int ) @ ( bit_ri4277139882892585799ns_not @ int @ ( numeral_numeral @ int @ ( bit0 @ N ) ) ) )
= ( bit_ri4277139882892585799ns_not @ int @ ( numeral_numeral @ int @ ( bit0 @ N ) ) ) ) ).
% or_not_numerals(2)
thf(fact_5703_or__not__numerals_I4_J,axiom,
! [M: num] :
( ( bit_se1065995026697491101ons_or @ int @ ( numeral_numeral @ int @ ( bit0 @ M ) ) @ ( bit_ri4277139882892585799ns_not @ int @ ( one_one @ int ) ) )
= ( bit_ri4277139882892585799ns_not @ int @ ( one_one @ int ) ) ) ).
% or_not_numerals(4)
thf(fact_5704_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_5705_bit__minus__int__iff,axiom,
! [K: int,N: nat] :
( ( bit_se5641148757651400278ts_bit @ int @ ( uminus_uminus @ int @ K ) @ N )
= ( bit_se5641148757651400278ts_bit @ int @ ( bit_ri4277139882892585799ns_not @ int @ ( minus_minus @ int @ K @ ( one_one @ int ) ) ) @ N ) ) ).
% bit_minus_int_iff
thf(fact_5706_overflow__imp__lsb,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [X2: word @ A] :
( ( ( plus_plus @ ( word @ A ) @ X2 @ ( one_one @ ( word @ A ) ) )
= ( zero_zero @ ( word @ A ) ) )
=> ( bit_se5641148757651400278ts_bit @ ( word @ A ) @ X2 @ ( zero_zero @ nat ) ) ) ) ).
% overflow_imp_lsb
thf(fact_5707_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_5708_test__bit__bin_H,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ( ( bit_se5641148757651400278ts_bit @ ( word @ A ) )
= ( ^ [W2: word @ A,N4: nat] :
( ( ord_less @ nat @ N4 @ ( size_size @ ( word @ A ) @ W2 ) )
& ( bit_se5641148757651400278ts_bit @ int @ ( semiring_1_unsigned @ A @ int @ W2 ) @ N4 ) ) ) ) ) ).
% test_bit_bin'
thf(fact_5709_le__mask__high__bits,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [W: word @ A,N: nat] :
( ( ord_less_eq @ ( word @ A ) @ W @ ( bit_se2239418461657761734s_mask @ ( word @ A ) @ N ) )
= ( ! [X3: nat] :
( ( member @ nat @ X3 @ ( set_or7035219750837199246ssThan @ nat @ N @ ( size_size @ ( word @ A ) @ W ) ) )
=> ~ ( bit_se5641148757651400278ts_bit @ ( word @ A ) @ W @ X3 ) ) ) ) ) ).
% le_mask_high_bits
thf(fact_5710_and__not__numerals_I5_J,axiom,
! [M: num,N: num] :
( ( bit_se5824344872417868541ns_and @ int @ ( numeral_numeral @ int @ ( bit0 @ M ) ) @ ( bit_ri4277139882892585799ns_not @ int @ ( numeral_numeral @ int @ ( bit0 @ N ) ) ) )
= ( times_times @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ ( bit_se5824344872417868541ns_and @ int @ ( numeral_numeral @ int @ M ) @ ( bit_ri4277139882892585799ns_not @ int @ ( numeral_numeral @ int @ N ) ) ) ) ) ).
% and_not_numerals(5)
thf(fact_5711_and__not__numerals_I7_J,axiom,
! [M: num] :
( ( bit_se5824344872417868541ns_and @ int @ ( numeral_numeral @ int @ ( bit1 @ M ) ) @ ( bit_ri4277139882892585799ns_not @ int @ ( one_one @ int ) ) )
= ( numeral_numeral @ int @ ( bit0 @ M ) ) ) ).
% and_not_numerals(7)
thf(fact_5712_or__not__numerals_I3_J,axiom,
! [N: num] :
( ( bit_se1065995026697491101ons_or @ int @ ( one_one @ int ) @ ( bit_ri4277139882892585799ns_not @ int @ ( numeral_numeral @ int @ ( bit1 @ N ) ) ) )
= ( bit_ri4277139882892585799ns_not @ int @ ( numeral_numeral @ int @ ( bit0 @ N ) ) ) ) ).
% or_not_numerals(3)
thf(fact_5713_and__not__numerals_I3_J,axiom,
! [N: num] :
( ( bit_se5824344872417868541ns_and @ int @ ( one_one @ int ) @ ( bit_ri4277139882892585799ns_not @ int @ ( numeral_numeral @ int @ ( bit1 @ N ) ) ) )
= ( zero_zero @ int ) ) ).
% and_not_numerals(3)
thf(fact_5714_or__not__numerals_I7_J,axiom,
! [M: num] :
( ( bit_se1065995026697491101ons_or @ int @ ( numeral_numeral @ int @ ( bit1 @ M ) ) @ ( bit_ri4277139882892585799ns_not @ int @ ( one_one @ int ) ) )
= ( bit_ri4277139882892585799ns_not @ int @ ( zero_zero @ int ) ) ) ).
% or_not_numerals(7)
thf(fact_5715_bang__is__le,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [X2: word @ A,M: nat] :
( ( bit_se5641148757651400278ts_bit @ ( word @ A ) @ X2 @ M )
=> ( ord_less_eq @ ( word @ A ) @ ( power_power @ ( word @ A ) @ ( numeral_numeral @ ( word @ A ) @ ( bit0 @ one2 ) ) @ M ) @ X2 ) ) ) ).
% bang_is_le
thf(fact_5716_bit__nat__def,axiom,
( ( bit_se5641148757651400278ts_bit @ nat )
= ( ^ [M6: nat,N4: nat] :
~ ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ M6 @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N4 ) ) ) ) ) ).
% bit_nat_def
thf(fact_5717_and__not__numerals_I9_J,axiom,
! [M: num,N: num] :
( ( bit_se5824344872417868541ns_and @ int @ ( numeral_numeral @ int @ ( bit1 @ M ) ) @ ( bit_ri4277139882892585799ns_not @ int @ ( numeral_numeral @ int @ ( bit1 @ N ) ) ) )
= ( times_times @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ ( bit_se5824344872417868541ns_and @ int @ ( numeral_numeral @ int @ M ) @ ( bit_ri4277139882892585799ns_not @ int @ ( numeral_numeral @ int @ N ) ) ) ) ) ).
% and_not_numerals(9)
thf(fact_5718_and__not__numerals_I6_J,axiom,
! [M: num,N: num] :
( ( bit_se5824344872417868541ns_and @ int @ ( numeral_numeral @ int @ ( bit0 @ M ) ) @ ( bit_ri4277139882892585799ns_not @ int @ ( numeral_numeral @ int @ ( bit1 @ N ) ) ) )
= ( times_times @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ ( bit_se5824344872417868541ns_and @ int @ ( numeral_numeral @ int @ M ) @ ( bit_ri4277139882892585799ns_not @ int @ ( numeral_numeral @ int @ N ) ) ) ) ) ).
% and_not_numerals(6)
thf(fact_5719_or__not__numerals_I6_J,axiom,
! [M: num,N: num] :
( ( bit_se1065995026697491101ons_or @ int @ ( numeral_numeral @ int @ ( bit0 @ M ) ) @ ( bit_ri4277139882892585799ns_not @ int @ ( numeral_numeral @ int @ ( bit1 @ N ) ) ) )
= ( times_times @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ ( bit_se1065995026697491101ons_or @ int @ ( numeral_numeral @ int @ M ) @ ( bit_ri4277139882892585799ns_not @ int @ ( numeral_numeral @ int @ N ) ) ) ) ) ).
% or_not_numerals(6)
thf(fact_5720_odd__iff__lsb,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [X2: word @ A] :
( ( ~ ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( semiring_1_unsigned @ A @ nat @ X2 ) ) )
= ( bit_se5641148757651400278ts_bit @ ( word @ A ) @ X2 @ ( zero_zero @ nat ) ) ) ) ).
% odd_iff_lsb
thf(fact_5721_and__neq__0__is__nth,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [Y3: word @ A,N: nat,X2: word @ A] :
( ( Y3
= ( power_power @ ( word @ A ) @ ( numeral_numeral @ ( word @ A ) @ ( bit0 @ one2 ) ) @ N ) )
=> ( ( ( bit_se5824344872417868541ns_and @ ( word @ A ) @ X2 @ Y3 )
!= ( zero_zero @ ( word @ A ) ) )
= ( bit_se5641148757651400278ts_bit @ ( word @ A ) @ X2 @ N ) ) ) ) ).
% and_neq_0_is_nth
thf(fact_5722_nth__is__and__neq__0,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ( ( bit_se5641148757651400278ts_bit @ ( word @ A ) )
= ( ^ [X3: word @ A,N4: nat] :
( ( bit_se5824344872417868541ns_and @ ( word @ A ) @ X3 @ ( power_power @ ( word @ A ) @ ( numeral_numeral @ ( word @ A ) @ ( bit0 @ one2 ) ) @ N4 ) )
!= ( zero_zero @ ( word @ A ) ) ) ) ) ) ).
% nth_is_and_neq_0
thf(fact_5723_or__not__numerals_I5_J,axiom,
! [M: num,N: num] :
( ( bit_se1065995026697491101ons_or @ int @ ( numeral_numeral @ int @ ( bit0 @ M ) ) @ ( bit_ri4277139882892585799ns_not @ int @ ( numeral_numeral @ int @ ( bit0 @ N ) ) ) )
= ( plus_plus @ int @ ( one_one @ int ) @ ( times_times @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ ( bit_se1065995026697491101ons_or @ int @ ( numeral_numeral @ int @ M ) @ ( bit_ri4277139882892585799ns_not @ int @ ( numeral_numeral @ int @ N ) ) ) ) ) ) ).
% or_not_numerals(5)
thf(fact_5724_and__not__numerals_I8_J,axiom,
! [M: num,N: num] :
( ( bit_se5824344872417868541ns_and @ int @ ( numeral_numeral @ int @ ( bit1 @ M ) ) @ ( bit_ri4277139882892585799ns_not @ int @ ( numeral_numeral @ int @ ( bit0 @ N ) ) ) )
= ( plus_plus @ int @ ( one_one @ int ) @ ( times_times @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ ( bit_se5824344872417868541ns_and @ int @ ( numeral_numeral @ int @ M ) @ ( bit_ri4277139882892585799ns_not @ int @ ( numeral_numeral @ int @ N ) ) ) ) ) ) ).
% and_not_numerals(8)
thf(fact_5725_or__not__numerals_I9_J,axiom,
! [M: num,N: num] :
( ( bit_se1065995026697491101ons_or @ int @ ( numeral_numeral @ int @ ( bit1 @ M ) ) @ ( bit_ri4277139882892585799ns_not @ int @ ( numeral_numeral @ int @ ( bit1 @ N ) ) ) )
= ( plus_plus @ int @ ( one_one @ int ) @ ( times_times @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ ( bit_se1065995026697491101ons_or @ int @ ( numeral_numeral @ int @ M ) @ ( bit_ri4277139882892585799ns_not @ int @ ( numeral_numeral @ int @ N ) ) ) ) ) ) ).
% or_not_numerals(9)
thf(fact_5726_or__not__numerals_I8_J,axiom,
! [M: num,N: num] :
( ( bit_se1065995026697491101ons_or @ int @ ( numeral_numeral @ int @ ( bit1 @ M ) ) @ ( bit_ri4277139882892585799ns_not @ int @ ( numeral_numeral @ int @ ( bit0 @ N ) ) ) )
= ( plus_plus @ int @ ( one_one @ int ) @ ( times_times @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ ( bit_se1065995026697491101ons_or @ int @ ( numeral_numeral @ int @ M ) @ ( bit_ri4277139882892585799ns_not @ int @ ( numeral_numeral @ int @ N ) ) ) ) ) ) ).
% or_not_numerals(8)
thf(fact_5727_not__int__rec,axiom,
( ( bit_ri4277139882892585799ns_not @ int )
= ( ^ [K3: int] : ( plus_plus @ int @ ( zero_neq_one_of_bool @ int @ ( dvd_dvd @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ K3 ) ) @ ( times_times @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ ( bit_ri4277139882892585799ns_not @ int @ ( divide_divide @ int @ K3 @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) ) ) ) ) ) ) ).
% not_int_rec
thf(fact_5728_int__not__code_I1_J,axiom,
( ( bit_ri4277139882892585799ns_not @ int @ ( zero_zero @ int ) )
= ( uminus_uminus @ int @ ( one_one @ int ) ) ) ).
% int_not_code(1)
thf(fact_5729_bitNOT__integer__code,axiom,
( ( bit_ri4277139882892585799ns_not @ code_integer )
= ( ^ [I4: code_integer] : ( minus_minus @ code_integer @ ( uminus_uminus @ code_integer @ I4 ) @ ( one_one @ code_integer ) ) ) ) ).
% bitNOT_integer_code
thf(fact_5730_xor__int__unfold,axiom,
( ( bit_se5824344971392196577ns_xor @ int )
= ( ^ [K3: int,L3: int] :
( if @ int
@ ( K3
= ( uminus_uminus @ int @ ( one_one @ int ) ) )
@ ( bit_ri4277139882892585799ns_not @ int @ L3 )
@ ( if @ int
@ ( L3
= ( uminus_uminus @ int @ ( one_one @ int ) ) )
@ ( bit_ri4277139882892585799ns_not @ int @ K3 )
@ ( if @ int
@ ( K3
= ( zero_zero @ int ) )
@ L3
@ ( if @ int
@ ( L3
= ( zero_zero @ int ) )
@ K3
@ ( plus_plus @ int @ ( abs_abs @ int @ ( minus_minus @ int @ ( modulo_modulo @ int @ K3 @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) ) @ ( modulo_modulo @ int @ L3 @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) ) ) ) @ ( times_times @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ ( bit_se5824344971392196577ns_xor @ int @ ( divide_divide @ int @ K3 @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) ) @ ( divide_divide @ int @ L3 @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) ) ) ) ) ) ) ) ) ) ) ).
% xor_int_unfold
thf(fact_5731_Eps__case__prod__eq,axiom,
! [A: $tType,B: $tType,X2: A,Y3: B] :
( ( fChoice @ ( product_prod @ A @ B )
@ ( product_case_prod @ A @ B @ $o
@ ^ [X10: A,Y9: B] :
( ( X2 = X10 )
& ( Y3 = Y9 ) ) ) )
= ( product_Pair @ A @ B @ X2 @ Y3 ) ) ).
% Eps_case_prod_eq
thf(fact_5732_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_5733_bit_Oxor__one__left,axiom,
! [A: $tType] :
( ( bit_ri3973907225187159222ations @ A )
=> ! [X2: A] :
( ( bit_se5824344971392196577ns_xor @ A @ ( uminus_uminus @ A @ ( one_one @ A ) ) @ X2 )
= ( bit_ri4277139882892585799ns_not @ A @ X2 ) ) ) ).
% bit.xor_one_left
thf(fact_5734_bit_Oxor__one__right,axiom,
! [A: $tType] :
( ( bit_ri3973907225187159222ations @ A )
=> ! [X2: A] :
( ( bit_se5824344971392196577ns_xor @ A @ X2 @ ( uminus_uminus @ A @ ( one_one @ A ) ) )
= ( bit_ri4277139882892585799ns_not @ A @ X2 ) ) ) ).
% bit.xor_one_right
thf(fact_5735_bit_Oxor__cancel__left,axiom,
! [A: $tType] :
( ( bit_ri3973907225187159222ations @ A )
=> ! [X2: A] :
( ( bit_se5824344971392196577ns_xor @ A @ ( bit_ri4277139882892585799ns_not @ A @ X2 ) @ X2 )
= ( uminus_uminus @ A @ ( one_one @ A ) ) ) ) ).
% bit.xor_cancel_left
thf(fact_5736_bit_Oxor__cancel__right,axiom,
! [A: $tType] :
( ( bit_ri3973907225187159222ations @ A )
=> ! [X2: A] :
( ( bit_se5824344971392196577ns_xor @ A @ X2 @ ( bit_ri4277139882892585799ns_not @ A @ X2 ) )
= ( uminus_uminus @ A @ ( one_one @ A ) ) ) ) ).
% bit.xor_cancel_right
thf(fact_5737_xor__numerals_I3_J,axiom,
! [A: $tType] :
( ( bit_un5681908812861735899ations @ A )
=> ! [X2: num,Y3: num] :
( ( bit_se5824344971392196577ns_xor @ A @ ( numeral_numeral @ A @ ( bit0 @ X2 ) ) @ ( numeral_numeral @ A @ ( bit0 @ Y3 ) ) )
= ( times_times @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ ( bit_se5824344971392196577ns_xor @ A @ ( numeral_numeral @ A @ X2 ) @ ( numeral_numeral @ A @ Y3 ) ) ) ) ) ).
% xor_numerals(3)
thf(fact_5738_xor__numerals_I8_J,axiom,
! [A: $tType] :
( ( bit_un5681908812861735899ations @ A )
=> ! [X2: num] :
( ( bit_se5824344971392196577ns_xor @ A @ ( numeral_numeral @ A @ ( bit1 @ X2 ) ) @ ( one_one @ A ) )
= ( numeral_numeral @ A @ ( bit0 @ X2 ) ) ) ) ).
% xor_numerals(8)
thf(fact_5739_xor__numerals_I5_J,axiom,
! [A: $tType] :
( ( bit_un5681908812861735899ations @ A )
=> ! [X2: num] :
( ( bit_se5824344971392196577ns_xor @ A @ ( numeral_numeral @ A @ ( bit0 @ X2 ) ) @ ( one_one @ A ) )
= ( numeral_numeral @ A @ ( bit1 @ X2 ) ) ) ) ).
% xor_numerals(5)
thf(fact_5740_xor__numerals_I2_J,axiom,
! [A: $tType] :
( ( bit_un5681908812861735899ations @ A )
=> ! [Y3: num] :
( ( bit_se5824344971392196577ns_xor @ A @ ( one_one @ A ) @ ( numeral_numeral @ A @ ( bit1 @ Y3 ) ) )
= ( numeral_numeral @ A @ ( bit0 @ Y3 ) ) ) ) ).
% xor_numerals(2)
thf(fact_5741_xor__numerals_I1_J,axiom,
! [A: $tType] :
( ( bit_un5681908812861735899ations @ A )
=> ! [Y3: num] :
( ( bit_se5824344971392196577ns_xor @ A @ ( one_one @ A ) @ ( numeral_numeral @ A @ ( bit0 @ Y3 ) ) )
= ( numeral_numeral @ A @ ( bit1 @ Y3 ) ) ) ) ).
% xor_numerals(1)
thf(fact_5742_xor__numerals_I7_J,axiom,
! [A: $tType] :
( ( bit_un5681908812861735899ations @ A )
=> ! [X2: num,Y3: num] :
( ( bit_se5824344971392196577ns_xor @ A @ ( numeral_numeral @ A @ ( bit1 @ X2 ) ) @ ( numeral_numeral @ A @ ( bit1 @ Y3 ) ) )
= ( times_times @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ ( bit_se5824344971392196577ns_xor @ A @ ( numeral_numeral @ A @ X2 ) @ ( numeral_numeral @ A @ Y3 ) ) ) ) ) ).
% xor_numerals(7)
thf(fact_5743_xor__numerals_I6_J,axiom,
! [A: $tType] :
( ( bit_un5681908812861735899ations @ A )
=> ! [X2: num,Y3: num] :
( ( bit_se5824344971392196577ns_xor @ A @ ( numeral_numeral @ A @ ( bit1 @ X2 ) ) @ ( numeral_numeral @ A @ ( bit0 @ Y3 ) ) )
= ( plus_plus @ A @ ( one_one @ A ) @ ( times_times @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ ( bit_se5824344971392196577ns_xor @ A @ ( numeral_numeral @ A @ X2 ) @ ( numeral_numeral @ A @ Y3 ) ) ) ) ) ) ).
% xor_numerals(6)
thf(fact_5744_xor__numerals_I4_J,axiom,
! [A: $tType] :
( ( bit_un5681908812861735899ations @ A )
=> ! [X2: num,Y3: num] :
( ( bit_se5824344971392196577ns_xor @ A @ ( numeral_numeral @ A @ ( bit0 @ X2 ) ) @ ( numeral_numeral @ A @ ( bit1 @ Y3 ) ) )
= ( plus_plus @ A @ ( one_one @ A ) @ ( times_times @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ ( bit_se5824344971392196577ns_xor @ A @ ( numeral_numeral @ A @ X2 ) @ ( numeral_numeral @ A @ Y3 ) ) ) ) ) ) ).
% xor_numerals(4)
thf(fact_5745_XOR__lower,axiom,
! [X2: int,Y3: int] :
( ( ord_less_eq @ int @ ( zero_zero @ int ) @ X2 )
=> ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ Y3 )
=> ( ord_less_eq @ int @ ( zero_zero @ int ) @ ( bit_se5824344971392196577ns_xor @ int @ X2 @ Y3 ) ) ) ) ).
% XOR_lower
thf(fact_5746_diff__rat__def,axiom,
( ( minus_minus @ rat )
= ( ^ [Q5: rat,R5: rat] : ( plus_plus @ rat @ Q5 @ ( uminus_uminus @ rat @ R5 ) ) ) ) ).
% diff_rat_def
thf(fact_5747_split__paired__Eps,axiom,
! [B: $tType,A: $tType] :
( ( fChoice @ ( product_prod @ A @ B ) )
= ( ^ [P3: ( product_prod @ A @ B ) > $o] :
( fChoice @ ( product_prod @ A @ B )
@ ( product_case_prod @ A @ B @ $o
@ ^ [A6: A,B7: B] : ( P3 @ ( product_Pair @ A @ B @ A6 @ B7 ) ) ) ) ) ) ).
% split_paired_Eps
thf(fact_5748_even__xor__iff,axiom,
! [A: $tType] :
( ( bit_se359711467146920520ations @ A )
=> ! [A3: A,B3: A] :
( ( dvd_dvd @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ ( bit_se5824344971392196577ns_xor @ A @ A3 @ B3 ) )
= ( ( dvd_dvd @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ A3 )
= ( dvd_dvd @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ B3 ) ) ) ) ).
% even_xor_iff
thf(fact_5749_XOR__upper,axiom,
! [X2: int,N: nat,Y3: int] :
( ( ord_less_eq @ int @ ( zero_zero @ int ) @ X2 )
=> ( ( ord_less @ int @ X2 @ ( power_power @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ N ) )
=> ( ( ord_less @ int @ Y3 @ ( power_power @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ N ) )
=> ( ord_less @ int @ ( bit_se5824344971392196577ns_xor @ int @ X2 @ Y3 ) @ ( power_power @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ N ) ) ) ) ) ).
% XOR_upper
thf(fact_5750_xor__int__rec,axiom,
( ( bit_se5824344971392196577ns_xor @ int )
= ( ^ [K3: int,L3: int] :
( plus_plus @ int
@ ( zero_neq_one_of_bool @ int
@ ( ( ~ ( dvd_dvd @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ K3 ) )
!= ( ~ ( dvd_dvd @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ L3 ) ) ) )
@ ( times_times @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ ( bit_se5824344971392196577ns_xor @ int @ ( divide_divide @ int @ K3 @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) ) @ ( divide_divide @ int @ L3 @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) ) ) ) ) ) ) ).
% xor_int_rec
thf(fact_5751_xor__one__eq,axiom,
! [A: $tType] :
( ( bit_se359711467146920520ations @ A )
=> ! [A3: A] :
( ( bit_se5824344971392196577ns_xor @ A @ A3 @ ( one_one @ A ) )
= ( minus_minus @ A @ ( plus_plus @ A @ A3 @ ( zero_neq_one_of_bool @ A @ ( dvd_dvd @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ A3 ) ) )
@ ( zero_neq_one_of_bool @ A
@ ~ ( dvd_dvd @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ A3 ) ) ) ) ) ).
% xor_one_eq
thf(fact_5752_one__xor__eq,axiom,
! [A: $tType] :
( ( bit_se359711467146920520ations @ A )
=> ! [A3: A] :
( ( bit_se5824344971392196577ns_xor @ A @ ( one_one @ A ) @ A3 )
= ( minus_minus @ A @ ( plus_plus @ A @ A3 @ ( zero_neq_one_of_bool @ A @ ( dvd_dvd @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ A3 ) ) )
@ ( zero_neq_one_of_bool @ A
@ ~ ( dvd_dvd @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ A3 ) ) ) ) ) ).
% one_xor_eq
thf(fact_5753_Bit__integer_Oabs__eq,axiom,
! [Xa: int,X2: $o] :
( ( bits_Bit_integer @ ( code_integer_of_int @ Xa ) @ X2 )
= ( code_integer_of_int @ ( plus_plus @ int @ ( zero_neq_one_of_bool @ int @ X2 ) @ ( times_times @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ Xa ) ) ) ) ).
% Bit_integer.abs_eq
thf(fact_5754_fun__of__rel__def,axiom,
! [A: $tType,B: $tType] :
( ( fun_of_rel @ B @ A )
= ( ^ [R6: set @ ( product_prod @ B @ A ),X3: B] :
( fChoice @ A
@ ^ [Y: A] : ( member @ ( product_prod @ B @ A ) @ ( product_Pair @ B @ A @ X3 @ Y ) @ R6 ) ) ) ) ).
% fun_of_rel_def
thf(fact_5755_the__inv__into__def,axiom,
! [B: $tType,A: $tType] :
( ( the_inv_into @ A @ B )
= ( ^ [A7: set @ A,F5: A > B,X3: B] :
( the @ A
@ ^ [Y: A] :
( ( member @ A @ Y @ A7 )
& ( ( F5 @ Y )
= X3 ) ) ) ) ) ).
% the_inv_into_def
thf(fact_5756_word__bitwise__m1__simps_I6_J,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [X2: word @ A] :
( ( bit_se5824344971392196577ns_xor @ ( word @ A ) @ ( uminus_uminus @ ( word @ A ) @ ( one_one @ ( word @ A ) ) ) @ X2 )
= ( bit_ri4277139882892585799ns_not @ ( word @ A ) @ X2 ) ) ) ).
% word_bitwise_m1_simps(6)
thf(fact_5757_word__bitwise__m1__simps_I7_J,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [X2: word @ A] :
( ( bit_se5824344971392196577ns_xor @ ( word @ A ) @ X2 @ ( uminus_uminus @ ( word @ A ) @ ( one_one @ ( word @ A ) ) ) )
= ( bit_ri4277139882892585799ns_not @ ( word @ A ) @ X2 ) ) ) ).
% word_bitwise_m1_simps(7)
thf(fact_5758_xor__nat__numerals_I1_J,axiom,
! [Y3: num] :
( ( bit_se5824344971392196577ns_xor @ nat @ ( suc @ ( zero_zero @ nat ) ) @ ( numeral_numeral @ nat @ ( bit0 @ Y3 ) ) )
= ( numeral_numeral @ nat @ ( bit1 @ Y3 ) ) ) ).
% xor_nat_numerals(1)
thf(fact_5759_xor__nat__numerals_I2_J,axiom,
! [Y3: num] :
( ( bit_se5824344971392196577ns_xor @ nat @ ( suc @ ( zero_zero @ nat ) ) @ ( numeral_numeral @ nat @ ( bit1 @ Y3 ) ) )
= ( numeral_numeral @ nat @ ( bit0 @ Y3 ) ) ) ).
% xor_nat_numerals(2)
thf(fact_5760_xor__nat__numerals_I3_J,axiom,
! [X2: num] :
( ( bit_se5824344971392196577ns_xor @ nat @ ( numeral_numeral @ nat @ ( bit0 @ X2 ) ) @ ( suc @ ( zero_zero @ nat ) ) )
= ( numeral_numeral @ nat @ ( bit1 @ X2 ) ) ) ).
% xor_nat_numerals(3)
thf(fact_5761_xor__nat__numerals_I4_J,axiom,
! [X2: num] :
( ( bit_se5824344971392196577ns_xor @ nat @ ( numeral_numeral @ nat @ ( bit1 @ X2 ) ) @ ( suc @ ( zero_zero @ nat ) ) )
= ( numeral_numeral @ nat @ ( bit0 @ X2 ) ) ) ).
% xor_nat_numerals(4)
thf(fact_5762_word__bitwise__1__simps_I10_J,axiom,
! [J4: $tType] :
( ( type_len @ J4 )
=> ! [B3: num] :
( ( bit_se5824344971392196577ns_xor @ ( word @ J4 ) @ ( one_one @ ( word @ J4 ) ) @ ( numeral_numeral @ ( word @ J4 ) @ B3 ) )
= ( ring_1_of_int @ ( word @ J4 ) @ ( bit_se5824344971392196577ns_xor @ int @ ( one_one @ int ) @ ( numeral_numeral @ int @ B3 ) ) ) ) ) ).
% word_bitwise_1_simps(10)
thf(fact_5763_word__bitwise__1__simps_I12_J,axiom,
! [L5: $tType] :
( ( type_len @ L5 )
=> ! [A3: num] :
( ( bit_se5824344971392196577ns_xor @ ( word @ L5 ) @ ( numeral_numeral @ ( word @ L5 ) @ A3 ) @ ( one_one @ ( word @ L5 ) ) )
= ( ring_1_of_int @ ( word @ L5 ) @ ( bit_se5824344971392196577ns_xor @ int @ ( numeral_numeral @ int @ A3 ) @ ( one_one @ int ) ) ) ) ) ).
% word_bitwise_1_simps(12)
thf(fact_5764_word__bitwise__1__simps_I13_J,axiom,
! [M12: $tType] :
( ( type_len @ M12 )
=> ! [A3: num] :
( ( bit_se5824344971392196577ns_xor @ ( word @ M12 ) @ ( uminus_uminus @ ( word @ M12 ) @ ( numeral_numeral @ ( word @ M12 ) @ A3 ) ) @ ( one_one @ ( word @ M12 ) ) )
= ( ring_1_of_int @ ( word @ M12 ) @ ( bit_se5824344971392196577ns_xor @ int @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ A3 ) ) @ ( one_one @ int ) ) ) ) ) ).
% word_bitwise_1_simps(13)
thf(fact_5765_word__bitwise__1__simps_I11_J,axiom,
! [K8: $tType] :
( ( type_len @ K8 )
=> ! [B3: num] :
( ( bit_se5824344971392196577ns_xor @ ( word @ K8 ) @ ( one_one @ ( word @ K8 ) ) @ ( uminus_uminus @ ( word @ K8 ) @ ( numeral_numeral @ ( word @ K8 ) @ B3 ) ) )
= ( ring_1_of_int @ ( word @ K8 ) @ ( bit_se5824344971392196577ns_xor @ int @ ( one_one @ int ) @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ B3 ) ) ) ) ) ) ).
% word_bitwise_1_simps(11)
thf(fact_5766_word__ops__nth__size,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [N: nat,X2: word @ A,Y3: word @ A] :
( ( ord_less @ nat @ N @ ( size_size @ ( word @ A ) @ X2 ) )
=> ( ( ( bit_se5641148757651400278ts_bit @ ( word @ A ) @ ( bit_se1065995026697491101ons_or @ ( word @ A ) @ X2 @ Y3 ) @ N )
= ( ( bit_se5641148757651400278ts_bit @ ( word @ A ) @ X2 @ N )
| ( bit_se5641148757651400278ts_bit @ ( word @ A ) @ Y3 @ N ) ) )
& ( ( bit_se5641148757651400278ts_bit @ ( word @ A ) @ ( bit_se5824344872417868541ns_and @ ( word @ A ) @ X2 @ Y3 ) @ N )
= ( ( bit_se5641148757651400278ts_bit @ ( word @ A ) @ X2 @ N )
& ( bit_se5641148757651400278ts_bit @ ( word @ A ) @ Y3 @ N ) ) )
& ( ( bit_se5641148757651400278ts_bit @ ( word @ A ) @ ( bit_se5824344971392196577ns_xor @ ( word @ A ) @ X2 @ Y3 ) @ N )
= ( ( bit_se5641148757651400278ts_bit @ ( word @ A ) @ X2 @ N )
!= ( bit_se5641148757651400278ts_bit @ ( word @ A ) @ Y3 @ N ) ) )
& ( ( bit_se5641148757651400278ts_bit @ ( word @ A ) @ ( bit_ri4277139882892585799ns_not @ ( word @ A ) @ X2 ) @ N )
= ( ~ ( bit_se5641148757651400278ts_bit @ ( word @ A ) @ X2 @ N ) ) ) ) ) ) ).
% word_ops_nth_size
thf(fact_5767_bit__twiddle__max,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [X2: word @ A,Y3: word @ A] :
( ( bit_se5824344971392196577ns_xor @ ( word @ A ) @ X2 @ ( bit_se5824344872417868541ns_and @ ( word @ A ) @ ( bit_se5824344971392196577ns_xor @ ( word @ A ) @ X2 @ Y3 ) @ ( if @ ( word @ A ) @ ( ord_less @ ( word @ A ) @ X2 @ Y3 ) @ ( uminus_uminus @ ( word @ A ) @ ( one_one @ ( word @ A ) ) ) @ ( zero_zero @ ( word @ A ) ) ) ) )
= ( ord_max @ ( word @ A ) @ X2 @ Y3 ) ) ) ).
% bit_twiddle_max
thf(fact_5768_xor__nat__unfold,axiom,
( ( bit_se5824344971392196577ns_xor @ nat )
= ( ^ [M6: nat,N4: nat] :
( if @ nat
@ ( M6
= ( zero_zero @ nat ) )
@ N4
@ ( if @ nat
@ ( N4
= ( zero_zero @ nat ) )
@ M6
@ ( plus_plus @ nat @ ( modulo_modulo @ nat @ ( plus_plus @ nat @ ( modulo_modulo @ nat @ M6 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ ( modulo_modulo @ nat @ N4 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( bit_se5824344971392196577ns_xor @ nat @ ( divide_divide @ nat @ M6 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ ( divide_divide @ nat @ N4 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) ) ) ) ) ).
% xor_nat_unfold
thf(fact_5769_xor__nat__rec,axiom,
( ( bit_se5824344971392196577ns_xor @ nat )
= ( ^ [M6: nat,N4: nat] :
( plus_plus @ nat
@ ( zero_neq_one_of_bool @ nat
@ ( ( ~ ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ M6 ) )
!= ( ~ ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N4 ) ) ) )
@ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( bit_se5824344971392196577ns_xor @ nat @ ( divide_divide @ nat @ M6 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ ( divide_divide @ nat @ N4 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) ) ) ).
% xor_nat_rec
thf(fact_5770_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_5771_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_5772_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_5773_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_5774_new__rule,axiom,
! [A: $tType] :
( ( heap @ A )
=> ! [N: nat,X2: A] :
( hoare_hoare_triple @ ( array @ A ) @ ( one_one @ assn ) @ ( array_new @ A @ N @ X2 )
@ ^ [R5: array @ A] : ( snga_assn @ A @ R5 @ ( replicate @ A @ N @ X2 ) ) ) ) ).
% new_rule
thf(fact_5775_time__array__new,axiom,
! [A: $tType] :
( ( heap @ A )
=> ! [N: nat,X2: A,H2: heap_ext @ product_unit] :
( ( time_time @ ( array @ A ) @ ( array_new @ A @ N @ X2 ) @ H2 )
= ( plus_plus @ nat @ N @ ( one_one @ nat ) ) ) ) ).
% time_array_new
thf(fact_5776_num_Osize__gen_I1_J,axiom,
( ( size_num @ one2 )
= ( zero_zero @ nat ) ) ).
% num.size_gen(1)
thf(fact_5777_TBOUND__new,axiom,
! [A: $tType] :
( ( heap @ A )
=> ! [N: nat,X2: A] : ( time_TBOUND @ ( array @ A ) @ ( array_new @ A @ N @ X2 ) @ ( plus_plus @ nat @ N @ ( one_one @ nat ) ) ) ) ).
% TBOUND_new
thf(fact_5778_num_Osize__gen_I2_J,axiom,
! [X22: num] :
( ( size_num @ ( bit0 @ X22 ) )
= ( plus_plus @ nat @ ( size_num @ X22 ) @ ( suc @ ( zero_zero @ nat ) ) ) ) ).
% num.size_gen(2)
thf(fact_5779_aux,axiom,
! [B: $tType,A: $tType,P: A > B > assn,A3: A,As3: list @ A,C3: B,Cs: list @ B] :
( ( finite_fold @ nat @ assn
@ ^ [I4: nat,Aa2: assn] : ( times_times @ assn @ Aa2 @ ( P @ ( nth @ A @ ( cons @ A @ A3 @ As3 ) @ I4 ) @ ( nth @ B @ ( cons @ B @ C3 @ Cs ) @ I4 ) ) )
@ ( one_one @ assn )
@ ( set_or7035219750837199246ssThan @ nat @ ( zero_zero @ nat ) @ ( suc @ ( size_size @ ( list @ A ) @ As3 ) ) ) )
= ( times_times @ assn @ ( P @ A3 @ C3 )
@ ( finite_fold @ nat @ assn
@ ^ [I4: nat,Aa2: assn] : ( times_times @ assn @ Aa2 @ ( P @ ( nth @ A @ As3 @ I4 ) @ ( nth @ B @ Cs @ I4 ) ) )
@ ( one_one @ assn )
@ ( set_or7035219750837199246ssThan @ nat @ ( zero_zero @ nat ) @ ( size_size @ ( list @ A ) @ As3 ) ) ) ) ) ).
% aux
thf(fact_5780_hash__code__prod__simps,axiom,
! [A: $tType,B: $tType,H_a: A > uint32,H_b: B > uint32,X2: A,Xa: B] :
( ( hash_hash_code_prod @ A @ B @ H_a @ H_b @ ( product_Pair @ A @ B @ X2 @ Xa ) )
= ( plus_plus @ uint32 @ ( times_times @ uint32 @ ( H_a @ X2 ) @ ( 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 @ Xa ) @ ( 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_5781_sum__count__set,axiom,
! [A: $tType,Xs: list @ A,X: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ ( set2 @ A @ Xs ) @ X )
=> ( ( finite_finite2 @ A @ X )
=> ( ( groups7311177749621191930dd_sum @ A @ nat @ ( count_list @ A @ Xs ) @ X )
= ( size_size @ ( list @ A ) @ Xs ) ) ) ) ).
% sum_count_set
thf(fact_5782_list_Oinject,axiom,
! [A: $tType,X213: A,X224: list @ A,Y21: A,Y222: list @ A] :
( ( ( cons @ A @ X213 @ X224 )
= ( cons @ A @ Y21 @ Y222 ) )
= ( ( X213 = Y21 )
& ( X224 = Y222 ) ) ) ).
% list.inject
thf(fact_5783_length__nth__simps_I4_J,axiom,
! [B: $tType,X2: B,Xs: list @ B,N: nat] :
( ( nth @ B @ ( cons @ B @ X2 @ Xs ) @ ( suc @ N ) )
= ( nth @ B @ Xs @ N ) ) ).
% length_nth_simps(4)
thf(fact_5784_nth__Cons__Suc,axiom,
! [A: $tType,X2: A,Xs: list @ A,N: nat] :
( ( nth @ A @ ( cons @ A @ X2 @ Xs ) @ ( suc @ N ) )
= ( nth @ A @ Xs @ N ) ) ).
% nth_Cons_Suc
thf(fact_5785_length__nth__simps_I3_J,axiom,
! [B: $tType,X2: B,Xs: list @ B] :
( ( nth @ B @ ( cons @ B @ X2 @ Xs ) @ ( zero_zero @ nat ) )
= X2 ) ).
% length_nth_simps(3)
thf(fact_5786_nth__Cons__0,axiom,
! [A: $tType,X2: A,Xs: list @ A] :
( ( nth @ A @ ( cons @ A @ X2 @ Xs ) @ ( zero_zero @ nat ) )
= X2 ) ).
% nth_Cons_0
thf(fact_5787_list_Osimps_I15_J,axiom,
! [A: $tType,X213: A,X224: list @ A] :
( ( set2 @ A @ ( cons @ A @ X213 @ X224 ) )
= ( insert @ A @ X213 @ ( set2 @ A @ X224 ) ) ) ).
% list.simps(15)
thf(fact_5788_count__notin,axiom,
! [A: $tType,X2: A,Xs: list @ A] :
( ~ ( member @ A @ X2 @ ( set2 @ A @ Xs ) )
=> ( ( count_list @ A @ Xs @ X2 )
= ( zero_zero @ nat ) ) ) ).
% count_notin
thf(fact_5789_nth__Cons__numeral,axiom,
! [A: $tType,X2: A,Xs: list @ A,V: num] :
( ( nth @ A @ ( cons @ A @ X2 @ Xs ) @ ( numeral_numeral @ nat @ V ) )
= ( nth @ A @ Xs @ ( minus_minus @ nat @ ( numeral_numeral @ nat @ V ) @ ( one_one @ nat ) ) ) ) ).
% nth_Cons_numeral
thf(fact_5790_nth__Cons__pos,axiom,
! [A: $tType,N: nat,X2: A,Xs: list @ A] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ N )
=> ( ( nth @ A @ ( cons @ A @ X2 @ Xs ) @ N )
= ( nth @ A @ Xs @ ( minus_minus @ nat @ N @ ( one_one @ nat ) ) ) ) ) ).
% nth_Cons_pos
thf(fact_5791_count__list_Osimps_I2_J,axiom,
! [A: $tType,X2: A,Y3: A,Xs: list @ A] :
( ( ( X2 = Y3 )
=> ( ( count_list @ A @ ( cons @ A @ X2 @ Xs ) @ Y3 )
= ( plus_plus @ nat @ ( count_list @ A @ Xs @ Y3 ) @ ( one_one @ nat ) ) ) )
& ( ( X2 != Y3 )
=> ( ( count_list @ A @ ( cons @ A @ X2 @ Xs ) @ Y3 )
= ( count_list @ A @ Xs @ Y3 ) ) ) ) ).
% count_list.simps(2)
thf(fact_5792_impossible__Cons,axiom,
! [A: $tType,Xs: list @ A,Ys: list @ A,X2: A] :
( ( ord_less_eq @ nat @ ( size_size @ ( list @ A ) @ Xs ) @ ( size_size @ ( list @ A ) @ Ys ) )
=> ( Xs
!= ( cons @ A @ X2 @ Ys ) ) ) ).
% impossible_Cons
thf(fact_5793_set__subset__Cons,axiom,
! [A: $tType,Xs: list @ A,X2: A] : ( ord_less_eq @ ( set @ A ) @ ( set2 @ A @ Xs ) @ ( set2 @ A @ ( cons @ A @ X2 @ Xs ) ) ) ).
% set_subset_Cons
thf(fact_5794_list__tail__coinc,axiom,
! [A: $tType,N1: A,R12: list @ A,N22: A,R23: list @ A] :
( ( ( cons @ A @ N1 @ R12 )
= ( cons @ A @ N22 @ R23 ) )
=> ( ( N1 = N22 )
& ( R12 = R23 ) ) ) ).
% list_tail_coinc
thf(fact_5795_not__Cons__self2,axiom,
! [A: $tType,X2: A,Xs: list @ A] :
( ( cons @ A @ X2 @ Xs )
!= Xs ) ).
% not_Cons_self2
thf(fact_5796_map__consI_I1_J,axiom,
! [A: $tType,B: $tType,W: list @ A,F2: B > A,Ww: list @ B,A3: B] :
( ( W
= ( map @ B @ A @ F2 @ Ww ) )
=> ( ( cons @ A @ ( F2 @ A3 ) @ W )
= ( map @ B @ A @ F2 @ ( cons @ B @ A3 @ Ww ) ) ) ) ).
% map_consI(1)
thf(fact_5797_list_Osimps_I9_J,axiom,
! [B: $tType,A: $tType,F2: A > B,X213: A,X224: list @ A] :
( ( map @ A @ B @ F2 @ ( cons @ A @ X213 @ X224 ) )
= ( cons @ B @ ( F2 @ X213 ) @ ( map @ A @ B @ F2 @ X224 ) ) ) ).
% list.simps(9)
thf(fact_5798_map__eq__consE,axiom,
! [B: $tType,A: $tType,F2: B > A,Ls: list @ B,Fa: A,Fl: list @ A] :
( ( ( map @ B @ A @ F2 @ Ls )
= ( cons @ A @ Fa @ Fl ) )
=> ~ ! [A4: B,L2: list @ B] :
( ( Ls
= ( cons @ B @ A4 @ L2 ) )
=> ( ( ( F2 @ A4 )
= Fa )
=> ( ( map @ B @ A @ F2 @ L2 )
!= Fl ) ) ) ) ).
% map_eq_consE
thf(fact_5799_Cons__eq__map__D,axiom,
! [A: $tType,B: $tType,X2: A,Xs: list @ A,F2: B > A,Ys: list @ B] :
( ( ( cons @ A @ X2 @ Xs )
= ( map @ B @ A @ F2 @ Ys ) )
=> ? [Z2: B,Zs: list @ B] :
( ( Ys
= ( cons @ B @ Z2 @ Zs ) )
& ( X2
= ( F2 @ Z2 ) )
& ( Xs
= ( map @ B @ A @ F2 @ Zs ) ) ) ) ).
% Cons_eq_map_D
thf(fact_5800_map__eq__Cons__D,axiom,
! [B: $tType,A: $tType,F2: B > A,Xs: list @ B,Y3: A,Ys: list @ A] :
( ( ( map @ B @ A @ F2 @ Xs )
= ( cons @ A @ Y3 @ Ys ) )
=> ? [Z2: B,Zs: list @ B] :
( ( Xs
= ( cons @ B @ Z2 @ Zs ) )
& ( ( F2 @ Z2 )
= Y3 )
& ( ( map @ B @ A @ F2 @ Zs )
= Ys ) ) ) ).
% map_eq_Cons_D
thf(fact_5801_Cons__eq__map__conv,axiom,
! [A: $tType,B: $tType,X2: A,Xs: list @ A,F2: B > A,Ys: list @ B] :
( ( ( cons @ A @ X2 @ Xs )
= ( map @ B @ A @ F2 @ Ys ) )
= ( ? [Z4: B,Zs2: list @ B] :
( ( Ys
= ( cons @ B @ Z4 @ Zs2 ) )
& ( X2
= ( F2 @ Z4 ) )
& ( Xs
= ( map @ B @ A @ F2 @ Zs2 ) ) ) ) ) ).
% Cons_eq_map_conv
thf(fact_5802_map__eq__Cons__conv,axiom,
! [A: $tType,B: $tType,F2: B > A,Xs: list @ B,Y3: A,Ys: list @ A] :
( ( ( map @ B @ A @ F2 @ Xs )
= ( cons @ A @ Y3 @ Ys ) )
= ( ? [Z4: B,Zs2: list @ B] :
( ( Xs
= ( cons @ B @ Z4 @ Zs2 ) )
& ( ( F2 @ Z4 )
= Y3 )
& ( ( map @ B @ A @ F2 @ Zs2 )
= Ys ) ) ) ) ).
% map_eq_Cons_conv
thf(fact_5803_list_Oset__intros_I2_J,axiom,
! [A: $tType,Y3: A,X224: list @ A,X213: A] :
( ( member @ A @ Y3 @ ( set2 @ A @ X224 ) )
=> ( member @ A @ Y3 @ ( set2 @ A @ ( cons @ A @ X213 @ X224 ) ) ) ) ).
% list.set_intros(2)
thf(fact_5804_list_Oset__intros_I1_J,axiom,
! [A: $tType,X213: A,X224: list @ A] : ( member @ A @ X213 @ ( set2 @ A @ ( cons @ A @ X213 @ X224 ) ) ) ).
% list.set_intros(1)
thf(fact_5805_list_Oset__cases,axiom,
! [A: $tType,E: A,A3: list @ A] :
( ( member @ A @ E @ ( set2 @ A @ A3 ) )
=> ( ! [Z23: list @ A] :
( A3
!= ( cons @ A @ E @ Z23 ) )
=> ~ ! [Z12: A,Z23: list @ A] :
( ( A3
= ( cons @ A @ Z12 @ Z23 ) )
=> ~ ( member @ A @ E @ ( set2 @ A @ Z23 ) ) ) ) ) ).
% list.set_cases
thf(fact_5806_set__ConsD,axiom,
! [A: $tType,Y3: A,X2: A,Xs: list @ A] :
( ( member @ A @ Y3 @ ( set2 @ A @ ( cons @ A @ X2 @ Xs ) ) )
=> ( ( Y3 = X2 )
| ( member @ A @ Y3 @ ( set2 @ A @ Xs ) ) ) ) ).
% set_ConsD
thf(fact_5807_Suc__length__conv,axiom,
! [A: $tType,N: nat,Xs: list @ A] :
( ( ( suc @ N )
= ( size_size @ ( list @ A ) @ Xs ) )
= ( ? [Y: A,Ys3: list @ A] :
( ( Xs
= ( cons @ A @ Y @ Ys3 ) )
& ( ( size_size @ ( list @ A ) @ Ys3 )
= N ) ) ) ) ).
% Suc_length_conv
thf(fact_5808_length__Suc__conv,axiom,
! [A: $tType,Xs: list @ A,N: nat] :
( ( ( size_size @ ( list @ A ) @ Xs )
= ( suc @ N ) )
= ( ? [Y: A,Ys3: list @ A] :
( ( Xs
= ( cons @ A @ Y @ Ys3 ) )
& ( ( size_size @ ( list @ A ) @ Ys3 )
= N ) ) ) ) ).
% length_Suc_conv
thf(fact_5809_length__nth__simps_I2_J,axiom,
! [B: $tType,X2: B,Xs: list @ B] :
( ( size_size @ ( list @ B ) @ ( cons @ B @ X2 @ Xs ) )
= ( suc @ ( size_size @ ( list @ B ) @ Xs ) ) ) ).
% length_nth_simps(2)
thf(fact_5810_list__update__code_I3_J,axiom,
! [A: $tType,X2: A,Xs: list @ A,I: nat,Y3: A] :
( ( list_update @ A @ ( cons @ A @ X2 @ Xs ) @ ( suc @ I ) @ Y3 )
= ( cons @ A @ X2 @ ( list_update @ A @ Xs @ I @ Y3 ) ) ) ).
% list_update_code(3)
thf(fact_5811_list__update__code_I2_J,axiom,
! [A: $tType,X2: A,Xs: list @ A,Y3: A] :
( ( list_update @ A @ ( cons @ A @ X2 @ Xs ) @ ( zero_zero @ nat ) @ Y3 )
= ( cons @ A @ Y3 @ Xs ) ) ).
% list_update_code(2)
thf(fact_5812_replicate__Suc,axiom,
! [A: $tType,N: nat,X2: A] :
( ( replicate @ A @ ( suc @ N ) @ X2 )
= ( cons @ A @ X2 @ ( replicate @ A @ N @ X2 ) ) ) ).
% replicate_Suc
thf(fact_5813_list__assn__simps_I2_J,axiom,
! [A: $tType,B: $tType,P: A > B > assn,A3: A,As3: list @ A,C3: B,Cs: list @ B] :
( ( vEBT_List_list_assn @ A @ B @ P @ ( cons @ A @ A3 @ As3 ) @ ( cons @ B @ C3 @ Cs ) )
= ( times_times @ assn @ ( P @ A3 @ C3 ) @ ( vEBT_List_list_assn @ A @ B @ P @ As3 @ Cs ) ) ) ).
% list_assn_simps(2)
thf(fact_5814_list__assn_Osimps_I2_J,axiom,
! [A: $tType,C: $tType,P: A > C > assn,A3: A,As3: list @ A,C3: C,Cs: list @ C] :
( ( vEBT_List_list_assn @ A @ C @ P @ ( cons @ A @ A3 @ As3 ) @ ( cons @ C @ C3 @ Cs ) )
= ( times_times @ assn @ ( P @ A3 @ C3 ) @ ( vEBT_List_list_assn @ A @ C @ P @ As3 @ Cs ) ) ) ).
% list_assn.simps(2)
thf(fact_5815_Suc__le__length__iff,axiom,
! [A: $tType,N: nat,Xs: list @ A] :
( ( ord_less_eq @ nat @ ( suc @ N ) @ ( size_size @ ( list @ A ) @ Xs ) )
= ( ? [X3: A,Ys3: list @ A] :
( ( Xs
= ( cons @ A @ X3 @ Ys3 ) )
& ( ord_less_eq @ nat @ N @ ( size_size @ ( list @ A ) @ Ys3 ) ) ) ) ) ).
% Suc_le_length_iff
thf(fact_5816_count__le__length,axiom,
! [A: $tType,Xs: list @ A,X2: A] : ( ord_less_eq @ nat @ ( count_list @ A @ Xs @ X2 ) @ ( size_size @ ( list @ A ) @ Xs ) ) ).
% count_le_length
thf(fact_5817_list_Osize_I4_J,axiom,
! [A: $tType,X213: A,X224: list @ A] :
( ( size_size @ ( list @ A ) @ ( cons @ A @ X213 @ X224 ) )
= ( plus_plus @ nat @ ( size_size @ ( list @ A ) @ X224 ) @ ( suc @ ( zero_zero @ nat ) ) ) ) ).
% list.size(4)
thf(fact_5818_nth__Cons_H,axiom,
! [A: $tType,N: nat,X2: A,Xs: list @ A] :
( ( ( N
= ( zero_zero @ nat ) )
=> ( ( nth @ A @ ( cons @ A @ X2 @ Xs ) @ N )
= X2 ) )
& ( ( N
!= ( zero_zero @ nat ) )
=> ( ( nth @ A @ ( cons @ A @ X2 @ Xs ) @ N )
= ( nth @ A @ Xs @ ( minus_minus @ nat @ N @ ( one_one @ nat ) ) ) ) ) ) ).
% nth_Cons'
thf(fact_5819_nth__non__equal__first__eq,axiom,
! [A: $tType,X2: A,Y3: A,Xs: list @ A,N: nat] :
( ( X2 != Y3 )
=> ( ( ( nth @ A @ ( cons @ A @ X2 @ Xs ) @ N )
= Y3 )
= ( ( ( nth @ A @ Xs @ ( minus_minus @ nat @ N @ ( one_one @ nat ) ) )
= Y3 )
& ( ord_less @ nat @ ( zero_zero @ nat ) @ N ) ) ) ) ).
% nth_non_equal_first_eq
thf(fact_5820_nth__equal__first__eq,axiom,
! [A: $tType,X2: A,Xs: list @ A,N: nat] :
( ~ ( member @ A @ X2 @ ( set2 @ A @ Xs ) )
=> ( ( ord_less_eq @ nat @ N @ ( size_size @ ( list @ A ) @ Xs ) )
=> ( ( ( nth @ A @ ( cons @ A @ X2 @ Xs ) @ N )
= X2 )
= ( N
= ( zero_zero @ nat ) ) ) ) ) ).
% nth_equal_first_eq
thf(fact_5821_Cons__replicate__eq,axiom,
! [A: $tType,X2: A,Xs: list @ A,N: nat,Y3: A] :
( ( ( cons @ A @ X2 @ Xs )
= ( replicate @ A @ N @ Y3 ) )
= ( ( X2 = Y3 )
& ( ord_less @ nat @ ( zero_zero @ nat ) @ N )
& ( Xs
= ( replicate @ A @ ( minus_minus @ nat @ N @ ( one_one @ nat ) ) @ X2 ) ) ) ) ).
% Cons_replicate_eq
thf(fact_5822_slice__Cons,axiom,
! [A: $tType,Begin: nat,End: nat,X2: A,Xs: list @ A] :
( ( ( ( Begin
= ( zero_zero @ nat ) )
& ( ord_less @ nat @ ( zero_zero @ nat ) @ End ) )
=> ( ( slice @ A @ Begin @ End @ ( cons @ A @ X2 @ Xs ) )
= ( cons @ A @ X2 @ ( 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 @ X2 @ Xs ) )
= ( slice @ A @ ( minus_minus @ nat @ Begin @ ( one_one @ nat ) ) @ ( minus_minus @ nat @ End @ ( one_one @ nat ) ) @ Xs ) ) ) ) ).
% slice_Cons
thf(fact_5823_hash__code__option__simps_I2_J,axiom,
! [A: $tType,H_a: A > uint32,X2: A] :
( ( hash_h1887023736457453652option @ A @ H_a @ ( some @ A @ X2 ) )
= ( plus_plus @ uint32 @ ( times_times @ uint32 @ ( H_a @ X2 ) @ ( numeral_numeral @ uint32 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit0 @ ( bit1 @ ( bit0 @ ( bit0 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit0 @ ( bit0 @ ( bit1 @ ( bit1 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit1 @ ( bit0 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit1 @ ( bit0 @ one2 ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) @ ( numeral_numeral @ uint32 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit1 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit1 @ ( bit1 @ ( bit0 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit0 @ ( bit1 @ ( bit1 @ ( bit0 @ ( bit1 @ ( bit0 @ ( bit1 @ ( bit1 @ ( bit0 @ ( bit1 @ ( bit0 @ ( bit0 @ ( bit0 @ one2 ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).
% hash_code_option_simps(2)
thf(fact_5824_hash__code__list__simps_I2_J,axiom,
! [A: $tType,H_a: A > uint32,X2: A,Xa: list @ A] :
( ( hash_hash_code_list @ A @ H_a @ ( cons @ A @ X2 @ Xa ) )
= ( plus_plus @ uint32 @ ( times_times @ uint32 @ ( H_a @ X2 ) @ ( numeral_numeral @ uint32 @ ( bit0 @ ( bit0 @ ( bit1 @ ( bit1 @ ( bit0 @ ( bit1 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit1 @ ( bit0 @ ( bit1 @ ( bit0 @ ( bit1 @ ( bit0 @ ( bit0 @ ( bit1 @ ( bit0 @ ( bit0 @ ( bit1 @ ( bit0 @ ( bit1 @ ( bit0 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit0 @ ( bit0 @ one2 ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) @ ( plus_plus @ uint32 @ ( times_times @ uint32 @ ( hash_hash_code_list @ A @ H_a @ Xa ) @ ( numeral_numeral @ uint32 @ ( bit1 @ ( bit0 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit0 @ ( bit0 @ ( bit1 @ ( bit0 @ ( bit1 @ ( bit1 @ ( bit0 @ ( bit1 @ ( bit1 @ ( bit0 @ ( bit0 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit1 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit1 @ ( bit0 @ ( bit0 @ one2 ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) @ ( numeral_numeral @ uint32 @ ( bit0 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit1 @ ( bit0 @ ( bit1 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit0 @ ( bit0 @ ( bit1 @ ( bit0 @ ( bit1 @ ( bit1 @ ( bit0 @ ( bit0 @ ( bit1 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ one2 ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).
% hash_code_list_simps(2)
thf(fact_5825_hash__code__option__simps_I1_J,axiom,
! [A: $tType,H_a: A > uint32] :
( ( hash_h1887023736457453652option @ A @ H_a @ ( none @ A ) )
= ( numeral_numeral @ uint32 @ ( bit1 @ ( bit1 @ ( bit0 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit0 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit1 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit0 @ ( bit1 @ ( bit0 @ ( bit1 @ one2 ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).
% hash_code_option_simps(1)
thf(fact_5826_length__Cons,axiom,
! [A: $tType,X2: A,Xs: list @ A] :
( ( size_size @ ( list @ A ) @ ( cons @ A @ X2 @ Xs ) )
= ( suc @ ( size_size @ ( list @ A ) @ Xs ) ) ) ).
% length_Cons
thf(fact_5827_dup__1,axiom,
( ( code_dup @ ( one_one @ code_integer ) )
= ( numeral_numeral @ code_integer @ ( bit0 @ one2 ) ) ) ).
% dup_1
thf(fact_5828_power__numeral,axiom,
! [A: $tType] :
( ( semiring_numeral @ A )
=> ! [K: num,L: num] :
( ( power_power @ A @ ( numeral_numeral @ A @ K ) @ ( numeral_numeral @ nat @ L ) )
= ( numeral_numeral @ A @ ( pow @ K @ L ) ) ) ) ).
% power_numeral
thf(fact_5829_pow_Osimps_I1_J,axiom,
! [X2: num] :
( ( pow @ X2 @ one2 )
= X2 ) ).
% pow.simps(1)
thf(fact_5830_dup_Oabs__eq,axiom,
! [X2: int] :
( ( code_dup @ ( code_integer_of_int @ X2 ) )
= ( code_integer_of_int @ ( plus_plus @ int @ X2 @ X2 ) ) ) ).
% dup.abs_eq
thf(fact_5831_subset__Collect__iff,axiom,
! [A: $tType,B2: set @ A,A2: set @ A,P: A > $o] :
( ( ord_less_eq @ ( set @ A ) @ B2 @ A2 )
=> ( ( ord_less_eq @ ( set @ A ) @ B2
@ ( collect @ A
@ ^ [X3: A] :
( ( member @ A @ X3 @ A2 )
& ( P @ X3 ) ) ) )
= ( ! [X3: A] :
( ( member @ A @ X3 @ B2 )
=> ( P @ X3 ) ) ) ) ) ).
% subset_Collect_iff
thf(fact_5832_subset__CollectI,axiom,
! [A: $tType,B2: set @ A,A2: set @ A,Q: A > $o,P: A > $o] :
( ( ord_less_eq @ ( set @ A ) @ B2 @ A2 )
=> ( ! [X4: A] :
( ( member @ A @ X4 @ B2 )
=> ( ( Q @ X4 )
=> ( P @ X4 ) ) )
=> ( ord_less_eq @ ( set @ A )
@ ( collect @ A
@ ^ [X3: A] :
( ( member @ A @ X3 @ B2 )
& ( Q @ X3 ) ) )
@ ( collect @ A
@ ^ [X3: A] :
( ( member @ A @ X3 @ A2 )
& ( P @ X3 ) ) ) ) ) ) ).
% subset_CollectI
thf(fact_5833_prod__case__refines,axiom,
! [C: $tType,B: $tType,A: $tType,P6: product_prod @ A @ B,P8: product_prod @ A @ B,F2: A > B > ( heap_Time_Heap @ C ),F7: A > B > ( heap_Time_Heap @ C )] :
( ( P6 = P8 )
=> ( ! [A4: A,B4: B] : ( refine_Imp_refines @ C @ ( F2 @ A4 @ B4 ) @ ( F7 @ A4 @ B4 ) )
=> ( refine_Imp_refines @ C @ ( product_case_prod @ A @ B @ ( heap_Time_Heap @ C ) @ F2 @ P6 ) @ ( product_case_prod @ A @ B @ ( heap_Time_Heap @ C ) @ F7 @ P8 ) ) ) ) ).
% prod_case_refines
thf(fact_5834_refines__case__prod__right,axiom,
! [C: $tType,B: $tType,A: $tType,M: heap_Time_Heap @ C,M4: A > B > ( heap_Time_Heap @ C ),T2: product_prod @ A @ B] :
( ! [A4: A,B4: B] : ( refine_Imp_refines @ C @ M @ ( M4 @ A4 @ B4 ) )
=> ( refine_Imp_refines @ C @ M @ ( product_case_prod @ A @ B @ ( heap_Time_Heap @ C ) @ M4 @ T2 ) ) ) ).
% refines_case_prod_right
thf(fact_5835_cis__multiple__2pi,axiom,
! [N: real] :
( ( member @ real @ N @ ( ring_1_Ints @ real ) )
=> ( ( cis @ ( times_times @ real @ ( times_times @ real @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) @ pi ) @ N ) )
= ( one_one @ complex ) ) ) ).
% cis_multiple_2pi
thf(fact_5836_floor__add2,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [X2: A,Y3: A] :
( ( ( member @ A @ X2 @ ( ring_1_Ints @ A ) )
| ( member @ A @ Y3 @ ( ring_1_Ints @ A ) ) )
=> ( ( archim6421214686448440834_floor @ A @ ( plus_plus @ A @ X2 @ Y3 ) )
= ( plus_plus @ int @ ( archim6421214686448440834_floor @ A @ X2 ) @ ( archim6421214686448440834_floor @ A @ Y3 ) ) ) ) ) ).
% floor_add2
thf(fact_5837_frac__gt__0__iff,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [X2: A] :
( ( ord_less @ A @ ( zero_zero @ A ) @ ( archimedean_frac @ A @ X2 ) )
= ( ~ ( member @ A @ X2 @ ( ring_1_Ints @ A ) ) ) ) ) ).
% frac_gt_0_iff
thf(fact_5838_Ints__double__eq__0__iff,axiom,
! [A: $tType] :
( ( ring_char_0 @ A )
=> ! [A3: A] :
( ( member @ A @ A3 @ ( ring_1_Ints @ A ) )
=> ( ( ( plus_plus @ A @ A3 @ A3 )
= ( zero_zero @ A ) )
= ( A3
= ( zero_zero @ A ) ) ) ) ) ).
% Ints_double_eq_0_iff
thf(fact_5839_Ints__add,axiom,
! [A: $tType] :
( ( ring_1 @ A )
=> ! [A3: A,B3: A] :
( ( member @ A @ A3 @ ( ring_1_Ints @ A ) )
=> ( ( member @ A @ B3 @ ( ring_1_Ints @ A ) )
=> ( member @ A @ ( plus_plus @ A @ A3 @ B3 ) @ ( ring_1_Ints @ A ) ) ) ) ) ).
% Ints_add
thf(fact_5840_Ints__mult,axiom,
! [A: $tType] :
( ( ring_1 @ A )
=> ! [A3: A,B3: A] :
( ( member @ A @ A3 @ ( ring_1_Ints @ A ) )
=> ( ( member @ A @ B3 @ ( ring_1_Ints @ A ) )
=> ( member @ A @ ( times_times @ A @ A3 @ B3 ) @ ( ring_1_Ints @ A ) ) ) ) ) ).
% Ints_mult
thf(fact_5841_Ints__numeral,axiom,
! [A: $tType] :
( ( ring_1 @ A )
=> ! [N: num] : ( member @ A @ ( numeral_numeral @ A @ N ) @ ( ring_1_Ints @ A ) ) ) ).
% Ints_numeral
thf(fact_5842_Ints__diff,axiom,
! [A: $tType] :
( ( ring_1 @ A )
=> ! [A3: A,B3: A] :
( ( member @ A @ A3 @ ( ring_1_Ints @ A ) )
=> ( ( member @ A @ B3 @ ( ring_1_Ints @ A ) )
=> ( member @ A @ ( minus_minus @ A @ A3 @ B3 ) @ ( ring_1_Ints @ A ) ) ) ) ) ).
% Ints_diff
thf(fact_5843_Ints__power,axiom,
! [A: $tType] :
( ( ring_1 @ A )
=> ! [A3: A,N: nat] :
( ( member @ A @ A3 @ ( ring_1_Ints @ A ) )
=> ( member @ A @ ( power_power @ A @ A3 @ N ) @ ( ring_1_Ints @ A ) ) ) ) ).
% Ints_power
thf(fact_5844_Ints__1,axiom,
! [A: $tType] :
( ( ring_1 @ A )
=> ( member @ A @ ( one_one @ A ) @ ( ring_1_Ints @ A ) ) ) ).
% Ints_1
thf(fact_5845_finite__int__segment,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [A3: A,B3: A] :
( finite_finite2 @ A
@ ( collect @ A
@ ^ [X3: A] :
( ( member @ A @ X3 @ ( ring_1_Ints @ A ) )
& ( ord_less_eq @ A @ A3 @ X3 )
& ( ord_less_eq @ A @ X3 @ B3 ) ) ) ) ) ).
% finite_int_segment
thf(fact_5846_Ints__odd__nonzero,axiom,
! [A: $tType] :
( ( ring_char_0 @ A )
=> ! [A3: A] :
( ( member @ A @ A3 @ ( ring_1_Ints @ A ) )
=> ( ( plus_plus @ A @ ( plus_plus @ A @ ( one_one @ A ) @ A3 ) @ A3 )
!= ( zero_zero @ A ) ) ) ) ).
% Ints_odd_nonzero
thf(fact_5847_of__int__divide__in__Ints,axiom,
! [A: $tType] :
( ( idom_divide @ A )
=> ! [B3: int,A3: int] :
( ( dvd_dvd @ int @ B3 @ A3 )
=> ( member @ A @ ( divide_divide @ A @ ( ring_1_of_int @ A @ A3 ) @ ( ring_1_of_int @ A @ B3 ) ) @ ( ring_1_Ints @ A ) ) ) ) ).
% of_int_divide_in_Ints
thf(fact_5848_finite__abs__int__segment,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [A3: A] :
( finite_finite2 @ A
@ ( collect @ A
@ ^ [K3: A] :
( ( member @ A @ K3 @ ( ring_1_Ints @ A ) )
& ( ord_less_eq @ A @ ( abs_abs @ A @ K3 ) @ A3 ) ) ) ) ) ).
% finite_abs_int_segment
thf(fact_5849_Ints__odd__less__0,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [A3: A] :
( ( member @ A @ A3 @ ( ring_1_Ints @ A ) )
=> ( ( ord_less @ A @ ( plus_plus @ A @ ( plus_plus @ A @ ( one_one @ A ) @ A3 ) @ A3 ) @ ( zero_zero @ A ) )
= ( ord_less @ A @ A3 @ ( zero_zero @ A ) ) ) ) ) ).
% Ints_odd_less_0
thf(fact_5850_Ints__nonzero__abs__ge1,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [X2: A] :
( ( member @ A @ X2 @ ( ring_1_Ints @ A ) )
=> ( ( X2
!= ( zero_zero @ A ) )
=> ( ord_less_eq @ A @ ( one_one @ A ) @ ( abs_abs @ A @ X2 ) ) ) ) ) ).
% Ints_nonzero_abs_ge1
thf(fact_5851_Ints__nonzero__abs__less1,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [X2: A] :
( ( member @ A @ X2 @ ( ring_1_Ints @ A ) )
=> ( ( ord_less @ A @ ( abs_abs @ A @ X2 ) @ ( one_one @ A ) )
=> ( X2
= ( zero_zero @ A ) ) ) ) ) ).
% Ints_nonzero_abs_less1
thf(fact_5852_Ints__eq__abs__less1,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [X2: A,Y3: A] :
( ( member @ A @ X2 @ ( ring_1_Ints @ A ) )
=> ( ( member @ A @ Y3 @ ( ring_1_Ints @ A ) )
=> ( ( X2 = Y3 )
= ( ord_less @ A @ ( abs_abs @ A @ ( minus_minus @ A @ X2 @ Y3 ) ) @ ( one_one @ A ) ) ) ) ) ) ).
% Ints_eq_abs_less1
thf(fact_5853_frac__neg,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [X2: A] :
( ( ( member @ A @ X2 @ ( ring_1_Ints @ A ) )
=> ( ( archimedean_frac @ A @ ( uminus_uminus @ A @ X2 ) )
= ( zero_zero @ A ) ) )
& ( ~ ( member @ A @ X2 @ ( ring_1_Ints @ A ) )
=> ( ( archimedean_frac @ A @ ( uminus_uminus @ A @ X2 ) )
= ( minus_minus @ A @ ( one_one @ A ) @ ( archimedean_frac @ A @ X2 ) ) ) ) ) ) ).
% frac_neg
thf(fact_5854_le__mult__floor__Ints,axiom,
! [A: $tType,B: $tType] :
( ( ( archim2362893244070406136eiling @ B )
& ( linordered_idom @ A ) )
=> ! [A3: B,B3: B] :
( ( ord_less_eq @ B @ ( zero_zero @ B ) @ A3 )
=> ( ( member @ B @ A3 @ ( ring_1_Ints @ B ) )
=> ( ord_less_eq @ A @ ( ring_1_of_int @ A @ ( times_times @ int @ ( archim6421214686448440834_floor @ B @ A3 ) @ ( archim6421214686448440834_floor @ B @ B3 ) ) ) @ ( ring_1_of_int @ A @ ( archim6421214686448440834_floor @ B @ ( times_times @ B @ A3 @ B3 ) ) ) ) ) ) ) ).
% le_mult_floor_Ints
thf(fact_5855_mult__ceiling__le__Ints,axiom,
! [A: $tType,B: $tType] :
( ( ( archim2362893244070406136eiling @ B )
& ( linordered_idom @ A ) )
=> ! [A3: B,B3: B] :
( ( ord_less_eq @ B @ ( zero_zero @ B ) @ A3 )
=> ( ( member @ B @ A3 @ ( ring_1_Ints @ B ) )
=> ( ord_less_eq @ A @ ( ring_1_of_int @ A @ ( archimedean_ceiling @ B @ ( times_times @ B @ A3 @ B3 ) ) ) @ ( ring_1_of_int @ A @ ( times_times @ int @ ( archimedean_ceiling @ B @ A3 ) @ ( archimedean_ceiling @ B @ B3 ) ) ) ) ) ) ) ).
% mult_ceiling_le_Ints
thf(fact_5856_frac__unique__iff,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [X2: A,A3: A] :
( ( ( archimedean_frac @ A @ X2 )
= A3 )
= ( ( member @ A @ ( minus_minus @ A @ X2 @ A3 ) @ ( ring_1_Ints @ A ) )
& ( ord_less_eq @ A @ ( zero_zero @ A ) @ A3 )
& ( ord_less @ A @ A3 @ ( one_one @ A ) ) ) ) ) ).
% frac_unique_iff
thf(fact_5857_sin__integer__2pi,axiom,
! [N: real] :
( ( member @ real @ N @ ( ring_1_Ints @ real ) )
=> ( ( sin @ real @ ( times_times @ real @ ( times_times @ real @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) @ pi ) @ N ) )
= ( zero_zero @ real ) ) ) ).
% sin_integer_2pi
thf(fact_5858_cos__integer__2pi,axiom,
! [N: real] :
( ( member @ real @ N @ ( ring_1_Ints @ real ) )
=> ( ( cos @ real @ ( times_times @ real @ ( times_times @ real @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) @ pi ) @ N ) )
= ( one_one @ real ) ) ) ).
% cos_integer_2pi
thf(fact_5859_ran__nth__set__encoding__conv,axiom,
! [A: $tType,L: list @ A] :
( ( ran @ nat @ A
@ ^ [I4: nat] : ( if @ ( option @ A ) @ ( ord_less @ nat @ I4 @ ( size_size @ ( list @ A ) @ L ) ) @ ( some @ A @ ( nth @ A @ L @ I4 ) ) @ ( none @ A ) ) )
= ( set2 @ A @ L ) ) ).
% ran_nth_set_encoding_conv
thf(fact_5860_rat__inverse__code,axiom,
! [P6: rat] :
( ( quotient_of @ ( inverse_inverse @ rat @ P6 ) )
= ( product_case_prod @ int @ int @ ( product_prod @ int @ int )
@ ^ [A6: int,B7: int] :
( if @ ( product_prod @ int @ int )
@ ( A6
= ( zero_zero @ int ) )
@ ( product_Pair @ int @ int @ ( zero_zero @ int ) @ ( one_one @ int ) )
@ ( product_Pair @ int @ int @ ( times_times @ int @ ( sgn_sgn @ int @ A6 ) @ B7 ) @ ( abs_abs @ int @ A6 ) ) )
@ ( quotient_of @ P6 ) ) ) ).
% rat_inverse_code
thf(fact_5861_setceilmax,axiom,
! [S2: vEBT_VEBT,M: nat,Listy: list @ vEBT_VEBT,N: nat] :
( ( vEBT_invar_vebt @ S2 @ M )
=> ( ! [X4: vEBT_VEBT] :
( ( member @ vEBT_VEBT @ X4 @ ( set2 @ vEBT_VEBT @ Listy ) )
=> ( vEBT_invar_vebt @ X4 @ N ) )
=> ( ( M
= ( suc @ N ) )
=> ( ! [X4: vEBT_VEBT] :
( ( member @ vEBT_VEBT @ X4 @ ( set2 @ vEBT_VEBT @ Listy ) )
=> ( ( semiring_1_of_nat @ int @ ( vEBT_VEBT_height @ X4 ) )
= ( archimedean_ceiling @ real @ ( log @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) @ ( semiring_1_of_nat @ real @ N ) ) ) ) )
=> ( ( ( semiring_1_of_nat @ int @ ( vEBT_VEBT_height @ S2 ) )
= ( 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 @ S2 @ ( set2 @ vEBT_VEBT @ Listy ) ) ) ) )
= ( archimedean_ceiling @ real @ ( log @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) @ ( semiring_1_of_nat @ real @ M ) ) ) ) ) ) ) ) ) ).
% setceilmax
thf(fact_5862_image__ident,axiom,
! [A: $tType,Y8: set @ A] :
( ( image @ A @ A
@ ^ [X3: A] : X3
@ Y8 )
= Y8 ) ).
% image_ident
thf(fact_5863_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_5864_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_5865_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_5866_image__add__0,axiom,
! [A: $tType] :
( ( comm_monoid_add @ A )
=> ! [S: set @ A] :
( ( image @ A @ A @ ( plus_plus @ A @ ( zero_zero @ A ) ) @ S )
= S ) ) ).
% image_add_0
thf(fact_5867_surj__plus,axiom,
! [A: $tType] :
( ( ab_group_add @ A )
=> ! [A3: A] :
( ( image @ A @ A @ ( plus_plus @ A @ A3 ) @ ( top_top @ ( set @ A ) ) )
= ( top_top @ ( set @ A ) ) ) ) ).
% surj_plus
thf(fact_5868_range__add,axiom,
! [A: $tType] :
( ( group_add @ A )
=> ! [A3: A] :
( ( image @ A @ A @ ( plus_plus @ A @ A3 ) @ ( top_top @ ( set @ A ) ) )
= ( top_top @ ( set @ A ) ) ) ) ).
% range_add
thf(fact_5869_range__diff,axiom,
! [A: $tType] :
( ( group_add @ A )
=> ! [A3: A] :
( ( image @ A @ A @ ( minus_minus @ A @ A3 ) @ ( top_top @ ( set @ A ) ) )
= ( top_top @ ( set @ A ) ) ) ) ).
% range_diff
thf(fact_5870_image__add__atLeastAtMost,axiom,
! [A: $tType] :
( ( linordered_semidom @ A )
=> ! [K: A,I: A,J: A] :
( ( image @ A @ A @ ( plus_plus @ A @ K ) @ ( set_or1337092689740270186AtMost @ A @ I @ J ) )
= ( set_or1337092689740270186AtMost @ A @ ( plus_plus @ A @ I @ K ) @ ( plus_plus @ A @ J @ K ) ) ) ) ).
% image_add_atLeastAtMost
thf(fact_5871_image__add__atLeastLessThan,axiom,
! [A: $tType] :
( ( linordered_semidom @ A )
=> ! [K: A,I: A,J: A] :
( ( image @ A @ A @ ( plus_plus @ A @ K ) @ ( set_or7035219750837199246ssThan @ A @ I @ J ) )
= ( set_or7035219750837199246ssThan @ A @ ( plus_plus @ A @ I @ K ) @ ( plus_plus @ A @ J @ K ) ) ) ) ).
% image_add_atLeastLessThan
thf(fact_5872_image__diff__atLeastAtMost,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [D2: A,A3: A,B3: A] :
( ( image @ A @ A @ ( minus_minus @ A @ D2 ) @ ( set_or1337092689740270186AtMost @ A @ A3 @ B3 ) )
= ( set_or1337092689740270186AtMost @ A @ ( minus_minus @ A @ D2 @ B3 ) @ ( minus_minus @ A @ D2 @ A3 ) ) ) ) ).
% image_diff_atLeastAtMost
thf(fact_5873_list_Oset__map,axiom,
! [B: $tType,A: $tType,F2: A > B,V: list @ A] :
( ( set2 @ B @ ( map @ A @ B @ F2 @ V ) )
= ( image @ A @ B @ F2 @ ( set2 @ A @ V ) ) ) ).
% list.set_map
thf(fact_5874_image__add__atMost,axiom,
! [A: $tType] :
( ( ordered_ab_group_add @ A )
=> ! [C3: A,A3: A] :
( ( image @ A @ A @ ( plus_plus @ A @ C3 ) @ ( set_ord_atMost @ A @ A3 ) )
= ( set_ord_atMost @ A @ ( plus_plus @ A @ C3 @ A3 ) ) ) ) ).
% image_add_atMost
thf(fact_5875_bij__betw__add,axiom,
! [A: $tType] :
( ( cancel_semigroup_add @ A )
=> ! [A3: A,A2: set @ A,B2: set @ A] :
( ( bij_betw @ A @ A @ ( plus_plus @ A @ A3 ) @ A2 @ B2 )
= ( ( image @ A @ A @ ( plus_plus @ A @ A3 ) @ A2 )
= B2 ) ) ) ).
% bij_betw_add
thf(fact_5876_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_5877_surj__diff__right,axiom,
! [A: $tType] :
( ( ab_group_add @ A )
=> ! [A3: A] :
( ( image @ A @ A
@ ^ [X3: A] : ( minus_minus @ A @ X3 @ A3 )
@ ( top_top @ ( set @ A ) ) )
= ( top_top @ ( set @ A ) ) ) ) ).
% surj_diff_right
thf(fact_5878_image__add__atLeastAtMost_H,axiom,
! [A: $tType] :
( ( linordered_semidom @ A )
=> ! [K: A,I: A,J: A] :
( ( image @ A @ A
@ ^ [N4: A] : ( plus_plus @ A @ N4 @ K )
@ ( set_or1337092689740270186AtMost @ A @ I @ J ) )
= ( set_or1337092689740270186AtMost @ A @ ( plus_plus @ A @ I @ K ) @ ( plus_plus @ A @ J @ K ) ) ) ) ).
% image_add_atLeastAtMost'
thf(fact_5879_image__add__atLeastLessThan_H,axiom,
! [A: $tType] :
( ( linordered_semidom @ A )
=> ! [K: A,I: A,J: A] :
( ( image @ A @ A
@ ^ [N4: A] : ( plus_plus @ A @ N4 @ K )
@ ( set_or7035219750837199246ssThan @ A @ I @ J ) )
= ( set_or7035219750837199246ssThan @ A @ ( plus_plus @ A @ I @ K ) @ ( plus_plus @ A @ J @ K ) ) ) ) ).
% image_add_atLeastLessThan'
thf(fact_5880_image__minus__const__atLeastAtMost_H,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [D2: A,A3: A,B3: A] :
( ( image @ A @ A
@ ^ [T3: A] : ( minus_minus @ A @ T3 @ D2 )
@ ( set_or1337092689740270186AtMost @ A @ A3 @ B3 ) )
= ( set_or1337092689740270186AtMost @ A @ ( minus_minus @ A @ A3 @ D2 ) @ ( minus_minus @ A @ B3 @ D2 ) ) ) ) ).
% image_minus_const_atLeastAtMost'
thf(fact_5881_Max__divisors__self__nat,axiom,
! [N: nat] :
( ( N
!= ( zero_zero @ nat ) )
=> ( ( lattic643756798349783984er_Max @ nat
@ ( collect @ nat
@ ^ [D3: nat] : ( dvd_dvd @ nat @ D3 @ N ) ) )
= N ) ) ).
% Max_divisors_self_nat
thf(fact_5882_Max_Obounded__iff,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A2: set @ A,X2: A] :
( ( finite_finite2 @ A @ A2 )
=> ( ( A2
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( ord_less_eq @ A @ ( lattic643756798349783984er_Max @ A @ A2 ) @ X2 )
= ( ! [X3: A] :
( ( member @ A @ X3 @ A2 )
=> ( ord_less_eq @ A @ X3 @ X2 ) ) ) ) ) ) ) ).
% Max.bounded_iff
thf(fact_5883_Max__less__iff,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A2: set @ A,X2: A] :
( ( finite_finite2 @ A @ A2 )
=> ( ( A2
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( ord_less @ A @ ( lattic643756798349783984er_Max @ A @ A2 ) @ X2 )
= ( ! [X3: A] :
( ( member @ A @ X3 @ A2 )
=> ( ord_less @ A @ X3 @ X2 ) ) ) ) ) ) ) ).
% Max_less_iff
thf(fact_5884_quotient__of__number_I3_J,axiom,
! [K: num] :
( ( quotient_of @ ( numeral_numeral @ rat @ K ) )
= ( product_Pair @ int @ int @ ( numeral_numeral @ int @ K ) @ ( one_one @ int ) ) ) ).
% quotient_of_number(3)
thf(fact_5885_range__constant,axiom,
! [B: $tType,A: $tType,X2: A] :
( ( image @ B @ A
@ ^ [Uu3: B] : X2
@ ( top_top @ ( set @ B ) ) )
= ( insert @ A @ X2 @ ( bot_bot @ ( set @ A ) ) ) ) ).
% range_constant
thf(fact_5886_Max__const,axiom,
! [B: $tType,A: $tType] :
( ( linorder @ A )
=> ! [A2: set @ B,C3: A] :
( ( finite_finite2 @ B @ A2 )
=> ( ( A2
!= ( bot_bot @ ( set @ B ) ) )
=> ( ( lattic643756798349783984er_Max @ A
@ ( image @ B @ A
@ ^ [Uu3: B] : C3
@ A2 ) )
= C3 ) ) ) ) ).
% Max_const
thf(fact_5887_rat__one__code,axiom,
( ( quotient_of @ ( one_one @ rat ) )
= ( product_Pair @ int @ int @ ( one_one @ int ) @ ( one_one @ int ) ) ) ).
% rat_one_code
thf(fact_5888_map__update__eta__repair_I2_J,axiom,
! [B: $tType,A: $tType,M: A > ( option @ B ),K: A,V: B] :
( ( ( M @ K )
= ( none @ B ) )
=> ( ( ran @ A @ B
@ ^ [X3: A] : ( if @ ( option @ B ) @ ( X3 = K ) @ ( some @ B @ V ) @ ( M @ X3 ) ) )
= ( insert @ B @ V @ ( ran @ A @ B @ M ) ) ) ) ).
% map_update_eta_repair(2)
thf(fact_5889_image__mult__atLeastAtMost,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [D2: A,A3: A,B3: A] :
( ( ord_less @ A @ ( zero_zero @ A ) @ D2 )
=> ( ( image @ A @ A @ ( times_times @ A @ D2 ) @ ( set_or1337092689740270186AtMost @ A @ A3 @ B3 ) )
= ( set_or1337092689740270186AtMost @ A @ ( times_times @ A @ D2 @ A3 ) @ ( times_times @ A @ D2 @ B3 ) ) ) ) ) ).
% image_mult_atLeastAtMost
thf(fact_5890_Max__insert,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A2: set @ A,X2: A] :
( ( finite_finite2 @ A @ A2 )
=> ( ( A2
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( lattic643756798349783984er_Max @ A @ ( insert @ A @ X2 @ A2 ) )
= ( ord_max @ A @ X2 @ ( lattic643756798349783984er_Max @ A @ A2 ) ) ) ) ) ) ).
% Max_insert
thf(fact_5891_image__divide__atLeastAtMost,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [D2: A,A3: A,B3: A] :
( ( ord_less @ A @ ( zero_zero @ A ) @ D2 )
=> ( ( image @ A @ A
@ ^ [C5: A] : ( divide_divide @ A @ C5 @ D2 )
@ ( set_or1337092689740270186AtMost @ A @ A3 @ B3 ) )
= ( set_or1337092689740270186AtMost @ A @ ( divide_divide @ A @ A3 @ D2 ) @ ( divide_divide @ A @ B3 @ D2 ) ) ) ) ) ).
% image_divide_atLeastAtMost
thf(fact_5892_rat__zero__code,axiom,
( ( quotient_of @ ( zero_zero @ rat ) )
= ( product_Pair @ int @ int @ ( zero_zero @ int ) @ ( one_one @ int ) ) ) ).
% rat_zero_code
thf(fact_5893_quotient__of__number_I5_J,axiom,
! [K: num] :
( ( quotient_of @ ( uminus_uminus @ rat @ ( numeral_numeral @ rat @ K ) ) )
= ( product_Pair @ int @ int @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ K ) ) @ ( one_one @ int ) ) ) ).
% quotient_of_number(5)
thf(fact_5894_quotient__of__number_I4_J,axiom,
( ( quotient_of @ ( uminus_uminus @ rat @ ( one_one @ rat ) ) )
= ( product_Pair @ int @ int @ ( uminus_uminus @ int @ ( one_one @ int ) ) @ ( one_one @ int ) ) ) ).
% quotient_of_number(4)
thf(fact_5895_Compr__image__eq,axiom,
! [A: $tType,B: $tType,F2: B > A,A2: set @ B,P: A > $o] :
( ( collect @ A
@ ^ [X3: A] :
( ( member @ A @ X3 @ ( image @ B @ A @ F2 @ A2 ) )
& ( P @ X3 ) ) )
= ( image @ B @ A @ F2
@ ( collect @ B
@ ^ [X3: B] :
( ( member @ B @ X3 @ A2 )
& ( P @ ( F2 @ X3 ) ) ) ) ) ) ).
% Compr_image_eq
thf(fact_5896_image__image,axiom,
! [A: $tType,B: $tType,C: $tType,F2: B > A,G: C > B,A2: set @ C] :
( ( image @ B @ A @ F2 @ ( image @ C @ B @ G @ A2 ) )
= ( image @ C @ A
@ ^ [X3: C] : ( F2 @ ( G @ X3 ) )
@ A2 ) ) ).
% image_image
thf(fact_5897_imageE,axiom,
! [A: $tType,B: $tType,B3: A,F2: B > A,A2: set @ B] :
( ( member @ A @ B3 @ ( image @ B @ A @ F2 @ A2 ) )
=> ~ ! [X4: B] :
( ( B3
= ( F2 @ X4 ) )
=> ~ ( member @ B @ X4 @ A2 ) ) ) ).
% imageE
thf(fact_5898_pigeonhole__infinite,axiom,
! [B: $tType,A: $tType,A2: set @ A,F2: A > B] :
( ~ ( finite_finite2 @ A @ A2 )
=> ( ( finite_finite2 @ B @ ( image @ A @ B @ F2 @ A2 ) )
=> ? [X4: A] :
( ( member @ A @ X4 @ A2 )
& ~ ( finite_finite2 @ A
@ ( collect @ A
@ ^ [A6: A] :
( ( member @ A @ A6 @ A2 )
& ( ( F2 @ A6 )
= ( F2 @ X4 ) ) ) ) ) ) ) ) ).
% pigeonhole_infinite
thf(fact_5899_image__set,axiom,
! [A: $tType,B: $tType,F2: B > A,Xs: list @ B] :
( ( image @ B @ A @ F2 @ ( set2 @ B @ Xs ) )
= ( set2 @ A @ ( map @ B @ A @ F2 @ Xs ) ) ) ).
% image_set
thf(fact_5900_hom__Max__commute,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [H2: A > A,N7: set @ A] :
( ! [X4: A,Y4: A] :
( ( H2 @ ( ord_max @ A @ X4 @ Y4 ) )
= ( ord_max @ A @ ( H2 @ X4 ) @ ( H2 @ Y4 ) ) )
=> ( ( finite_finite2 @ A @ N7 )
=> ( ( N7
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( H2 @ ( lattic643756798349783984er_Max @ A @ N7 ) )
= ( lattic643756798349783984er_Max @ A @ ( image @ A @ A @ H2 @ N7 ) ) ) ) ) ) ) ).
% hom_Max_commute
thf(fact_5901_Max_OcoboundedI,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A2: set @ A,A3: A] :
( ( finite_finite2 @ A @ A2 )
=> ( ( member @ A @ A3 @ A2 )
=> ( ord_less_eq @ A @ A3 @ ( lattic643756798349783984er_Max @ A @ A2 ) ) ) ) ) ).
% Max.coboundedI
thf(fact_5902_Max__eq__if,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A2: set @ A,B2: set @ A] :
( ( finite_finite2 @ A @ A2 )
=> ( ( finite_finite2 @ A @ B2 )
=> ( ! [X4: A] :
( ( member @ A @ X4 @ A2 )
=> ? [Xa2: A] :
( ( member @ A @ Xa2 @ B2 )
& ( ord_less_eq @ A @ X4 @ Xa2 ) ) )
=> ( ! [X4: A] :
( ( member @ A @ X4 @ B2 )
=> ? [Xa2: A] :
( ( member @ A @ Xa2 @ A2 )
& ( ord_less_eq @ A @ X4 @ Xa2 ) ) )
=> ( ( lattic643756798349783984er_Max @ A @ A2 )
= ( lattic643756798349783984er_Max @ A @ B2 ) ) ) ) ) ) ) ).
% Max_eq_if
thf(fact_5903_Max__eqI,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A2: set @ A,X2: A] :
( ( finite_finite2 @ A @ A2 )
=> ( ! [Y4: A] :
( ( member @ A @ Y4 @ A2 )
=> ( ord_less_eq @ A @ Y4 @ X2 ) )
=> ( ( member @ A @ X2 @ A2 )
=> ( ( lattic643756798349783984er_Max @ A @ A2 )
= X2 ) ) ) ) ) ).
% Max_eqI
thf(fact_5904_Max__ge,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A2: set @ A,X2: A] :
( ( finite_finite2 @ A @ A2 )
=> ( ( member @ A @ X2 @ A2 )
=> ( ord_less_eq @ A @ X2 @ ( lattic643756798349783984er_Max @ A @ A2 ) ) ) ) ) ).
% Max_ge
thf(fact_5905_finite__surj,axiom,
! [A: $tType,B: $tType,A2: set @ A,B2: set @ B,F2: A > B] :
( ( finite_finite2 @ A @ A2 )
=> ( ( ord_less_eq @ ( set @ B ) @ B2 @ ( image @ A @ B @ F2 @ A2 ) )
=> ( finite_finite2 @ B @ B2 ) ) ) ).
% finite_surj
thf(fact_5906_finite__subset__image,axiom,
! [A: $tType,B: $tType,B2: set @ A,F2: B > A,A2: set @ B] :
( ( finite_finite2 @ A @ B2 )
=> ( ( ord_less_eq @ ( set @ A ) @ B2 @ ( image @ B @ A @ F2 @ A2 ) )
=> ? [C7: set @ B] :
( ( ord_less_eq @ ( set @ B ) @ C7 @ A2 )
& ( finite_finite2 @ B @ C7 )
& ( B2
= ( image @ B @ A @ F2 @ C7 ) ) ) ) ) ).
% finite_subset_image
thf(fact_5907_ex__finite__subset__image,axiom,
! [A: $tType,B: $tType,F2: B > A,A2: set @ B,P: ( set @ A ) > $o] :
( ( ? [B6: set @ A] :
( ( finite_finite2 @ A @ B6 )
& ( ord_less_eq @ ( set @ A ) @ B6 @ ( image @ B @ A @ F2 @ A2 ) )
& ( P @ B6 ) ) )
= ( ? [B6: set @ B] :
( ( finite_finite2 @ B @ B6 )
& ( ord_less_eq @ ( set @ B ) @ B6 @ A2 )
& ( P @ ( image @ B @ A @ F2 @ B6 ) ) ) ) ) ).
% ex_finite_subset_image
thf(fact_5908_all__finite__subset__image,axiom,
! [A: $tType,B: $tType,F2: B > A,A2: set @ B,P: ( set @ A ) > $o] :
( ( ! [B6: set @ A] :
( ( ( finite_finite2 @ A @ B6 )
& ( ord_less_eq @ ( set @ A ) @ B6 @ ( image @ B @ A @ F2 @ A2 ) ) )
=> ( P @ B6 ) ) )
= ( ! [B6: set @ B] :
( ( ( finite_finite2 @ B @ B6 )
& ( ord_less_eq @ ( set @ B ) @ B6 @ A2 ) )
=> ( P @ ( image @ B @ A @ F2 @ B6 ) ) ) ) ) ).
% all_finite_subset_image
thf(fact_5909_all__subset__image,axiom,
! [A: $tType,B: $tType,F2: B > A,A2: set @ B,P: ( set @ A ) > $o] :
( ( ! [B6: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ B6 @ ( image @ B @ A @ F2 @ A2 ) )
=> ( P @ B6 ) ) )
= ( ! [B6: set @ B] :
( ( ord_less_eq @ ( set @ B ) @ B6 @ A2 )
=> ( P @ ( image @ B @ A @ F2 @ B6 ) ) ) ) ) ).
% all_subset_image
thf(fact_5910_image__mono,axiom,
! [B: $tType,A: $tType,A2: set @ A,B2: set @ A,F2: A > B] :
( ( ord_less_eq @ ( set @ A ) @ A2 @ B2 )
=> ( ord_less_eq @ ( set @ B ) @ ( image @ A @ B @ F2 @ A2 ) @ ( image @ A @ B @ F2 @ B2 ) ) ) ).
% image_mono
thf(fact_5911_image__subsetI,axiom,
! [A: $tType,B: $tType,A2: set @ A,F2: A > B,B2: set @ B] :
( ! [X4: A] :
( ( member @ A @ X4 @ A2 )
=> ( member @ B @ ( F2 @ X4 ) @ B2 ) )
=> ( ord_less_eq @ ( set @ B ) @ ( image @ A @ B @ F2 @ A2 ) @ B2 ) ) ).
% image_subsetI
thf(fact_5912_subset__imageE,axiom,
! [A: $tType,B: $tType,B2: set @ A,F2: B > A,A2: set @ B] :
( ( ord_less_eq @ ( set @ A ) @ B2 @ ( image @ B @ A @ F2 @ A2 ) )
=> ~ ! [C7: set @ B] :
( ( ord_less_eq @ ( set @ B ) @ C7 @ A2 )
=> ( B2
!= ( image @ B @ A @ F2 @ C7 ) ) ) ) ).
% subset_imageE
thf(fact_5913_image__subset__iff,axiom,
! [A: $tType,B: $tType,F2: B > A,A2: set @ B,B2: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ ( image @ B @ A @ F2 @ A2 ) @ B2 )
= ( ! [X3: B] :
( ( member @ B @ X3 @ A2 )
=> ( member @ A @ ( F2 @ X3 ) @ B2 ) ) ) ) ).
% image_subset_iff
thf(fact_5914_subset__image__iff,axiom,
! [A: $tType,B: $tType,B2: set @ A,F2: B > A,A2: set @ B] :
( ( ord_less_eq @ ( set @ A ) @ B2 @ ( image @ B @ A @ F2 @ A2 ) )
= ( ? [AA: set @ B] :
( ( ord_less_eq @ ( set @ B ) @ AA @ A2 )
& ( B2
= ( image @ B @ A @ F2 @ AA ) ) ) ) ) ).
% subset_image_iff
thf(fact_5915_range__subsetD,axiom,
! [B: $tType,A: $tType,F2: B > A,B2: set @ A,I: B] :
( ( ord_less_eq @ ( set @ A ) @ ( image @ B @ A @ F2 @ ( top_top @ ( set @ B ) ) ) @ B2 )
=> ( member @ A @ ( F2 @ I ) @ B2 ) ) ).
% range_subsetD
thf(fact_5916_range__composition,axiom,
! [A: $tType,C: $tType,B: $tType,F2: C > A,G: B > C] :
( ( image @ B @ A
@ ^ [X3: B] : ( F2 @ ( G @ X3 ) )
@ ( top_top @ ( set @ B ) ) )
= ( image @ C @ A @ F2 @ ( image @ B @ C @ G @ ( top_top @ ( set @ B ) ) ) ) ) ).
% range_composition
thf(fact_5917_rangeE,axiom,
! [A: $tType,B: $tType,B3: A,F2: B > A] :
( ( member @ A @ B3 @ ( image @ B @ A @ F2 @ ( top_top @ ( set @ B ) ) ) )
=> ~ ! [X4: B] :
( B3
!= ( F2 @ X4 ) ) ) ).
% rangeE
thf(fact_5918_bij__betw__subset,axiom,
! [A: $tType,B: $tType,F2: A > B,A2: set @ A,A8: set @ B,B2: set @ A,B9: set @ B] :
( ( bij_betw @ A @ B @ F2 @ A2 @ A8 )
=> ( ( ord_less_eq @ ( set @ A ) @ B2 @ A2 )
=> ( ( ( image @ A @ B @ F2 @ B2 )
= B9 )
=> ( bij_betw @ A @ B @ F2 @ B2 @ B9 ) ) ) ) ).
% bij_betw_subset
thf(fact_5919_bij__betw__byWitness,axiom,
! [A: $tType,B: $tType,A2: set @ A,F7: B > A,F2: A > B,A8: set @ B] :
( ! [X4: A] :
( ( member @ A @ X4 @ A2 )
=> ( ( F7 @ ( F2 @ X4 ) )
= X4 ) )
=> ( ! [X4: B] :
( ( member @ B @ X4 @ A8 )
=> ( ( F2 @ ( F7 @ X4 ) )
= X4 ) )
=> ( ( ord_less_eq @ ( set @ B ) @ ( image @ A @ B @ F2 @ A2 ) @ A8 )
=> ( ( ord_less_eq @ ( set @ A ) @ ( image @ B @ A @ F7 @ A8 ) @ A2 )
=> ( bij_betw @ A @ B @ F2 @ A2 @ A8 ) ) ) ) ) ).
% bij_betw_byWitness
thf(fact_5920_image__diff__subset,axiom,
! [A: $tType,B: $tType,F2: B > A,A2: set @ B,B2: set @ B] : ( ord_less_eq @ ( set @ A ) @ ( minus_minus @ ( set @ A ) @ ( image @ B @ A @ F2 @ A2 ) @ ( image @ B @ A @ F2 @ B2 ) ) @ ( image @ B @ A @ F2 @ ( minus_minus @ ( set @ B ) @ A2 @ B2 ) ) ) ).
% image_diff_subset
thf(fact_5921_translation__Compl,axiom,
! [A: $tType] :
( ( ab_group_add @ A )
=> ! [A3: A,T2: set @ A] :
( ( image @ A @ A @ ( plus_plus @ A @ A3 ) @ ( uminus_uminus @ ( set @ A ) @ T2 ) )
= ( uminus_uminus @ ( set @ A ) @ ( image @ A @ A @ ( plus_plus @ A @ A3 ) @ T2 ) ) ) ) ).
% translation_Compl
thf(fact_5922_translation__diff,axiom,
! [A: $tType] :
( ( ab_group_add @ A )
=> ! [A3: A,S2: set @ A,T2: set @ A] :
( ( image @ A @ A @ ( plus_plus @ A @ A3 ) @ ( minus_minus @ ( set @ A ) @ S2 @ T2 ) )
= ( minus_minus @ ( set @ A ) @ ( image @ A @ A @ ( plus_plus @ A @ A3 ) @ S2 ) @ ( image @ A @ A @ ( plus_plus @ A @ A3 ) @ T2 ) ) ) ) ).
% translation_diff
thf(fact_5923_Max__add__commute,axiom,
! [B: $tType,A: $tType] :
( ( linord4140545234300271783up_add @ A )
=> ! [S: set @ B,F2: B > A,K: A] :
( ( finite_finite2 @ B @ S )
=> ( ( S
!= ( bot_bot @ ( set @ B ) ) )
=> ( ( lattic643756798349783984er_Max @ A
@ ( image @ B @ A
@ ^ [X3: B] : ( plus_plus @ A @ ( F2 @ X3 ) @ K )
@ S ) )
= ( plus_plus @ A @ ( lattic643756798349783984er_Max @ A @ ( image @ B @ A @ F2 @ S ) ) @ K ) ) ) ) ) ).
% Max_add_commute
thf(fact_5924_Max_Oin__idem,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A2: set @ A,X2: A] :
( ( finite_finite2 @ A @ A2 )
=> ( ( member @ A @ X2 @ A2 )
=> ( ( ord_max @ A @ X2 @ ( lattic643756798349783984er_Max @ A @ A2 ) )
= ( lattic643756798349783984er_Max @ A @ A2 ) ) ) ) ) ).
% Max.in_idem
thf(fact_5925_finite__range__imageI,axiom,
! [C: $tType,A: $tType,B: $tType,G: B > A,F2: A > C] :
( ( finite_finite2 @ A @ ( image @ B @ A @ G @ ( top_top @ ( set @ B ) ) ) )
=> ( finite_finite2 @ C
@ ( image @ B @ C
@ ^ [X3: B] : ( F2 @ ( G @ X3 ) )
@ ( top_top @ ( set @ B ) ) ) ) ) ).
% finite_range_imageI
thf(fact_5926_image__constant__conv,axiom,
! [B: $tType,A: $tType,A2: set @ B,C3: A] :
( ( ( A2
= ( bot_bot @ ( set @ B ) ) )
=> ( ( image @ B @ A
@ ^ [X3: B] : C3
@ A2 )
= ( bot_bot @ ( set @ A ) ) ) )
& ( ( A2
!= ( bot_bot @ ( set @ B ) ) )
=> ( ( image @ B @ A
@ ^ [X3: B] : C3
@ A2 )
= ( insert @ A @ C3 @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ).
% image_constant_conv
thf(fact_5927_image__constant,axiom,
! [A: $tType,B: $tType,X2: A,A2: set @ A,C3: B] :
( ( member @ A @ X2 @ A2 )
=> ( ( image @ A @ B
@ ^ [X3: A] : C3
@ A2 )
= ( insert @ B @ C3 @ ( bot_bot @ ( set @ B ) ) ) ) ) ).
% image_constant
thf(fact_5928_sum_Oimage__gen,axiom,
! [A: $tType,C: $tType,B: $tType] :
( ( comm_monoid_add @ A )
=> ! [S: set @ B,H2: B > A,G: B > C] :
( ( finite_finite2 @ B @ S )
=> ( ( groups7311177749621191930dd_sum @ B @ A @ H2 @ S )
= ( groups7311177749621191930dd_sum @ C @ A
@ ^ [Y: C] :
( groups7311177749621191930dd_sum @ B @ A @ H2
@ ( collect @ B
@ ^ [X3: B] :
( ( member @ B @ X3 @ S )
& ( ( G @ X3 )
= Y ) ) ) )
@ ( image @ B @ C @ G @ S ) ) ) ) ) ).
% sum.image_gen
thf(fact_5929_translation__subtract__diff,axiom,
! [A: $tType] :
( ( ab_group_add @ A )
=> ! [A3: A,S2: set @ A,T2: set @ A] :
( ( image @ A @ A
@ ^ [X3: A] : ( minus_minus @ A @ X3 @ A3 )
@ ( minus_minus @ ( set @ A ) @ S2 @ T2 ) )
= ( minus_minus @ ( set @ A )
@ ( image @ A @ A
@ ^ [X3: A] : ( minus_minus @ A @ X3 @ A3 )
@ S2 )
@ ( image @ A @ A
@ ^ [X3: A] : ( minus_minus @ A @ X3 @ A3 )
@ T2 ) ) ) ) ).
% translation_subtract_diff
thf(fact_5930_prod_Oimage__gen,axiom,
! [A: $tType,C: $tType,B: $tType] :
( ( comm_monoid_mult @ A )
=> ! [S: set @ B,H2: B > A,G: B > C] :
( ( finite_finite2 @ B @ S )
=> ( ( groups7121269368397514597t_prod @ B @ A @ H2 @ S )
= ( groups7121269368397514597t_prod @ C @ A
@ ^ [Y: C] :
( groups7121269368397514597t_prod @ B @ A @ H2
@ ( collect @ B
@ ^ [X3: B] :
( ( member @ B @ X3 @ S )
& ( ( G @ X3 )
= Y ) ) ) )
@ ( image @ B @ C @ G @ S ) ) ) ) ) ).
% prod.image_gen
thf(fact_5931_translation__subtract__Compl,axiom,
! [A: $tType] :
( ( ab_group_add @ A )
=> ! [A3: A,T2: set @ A] :
( ( image @ A @ A
@ ^ [X3: A] : ( minus_minus @ A @ X3 @ A3 )
@ ( uminus_uminus @ ( set @ A ) @ T2 ) )
= ( uminus_uminus @ ( set @ A )
@ ( image @ A @ A
@ ^ [X3: A] : ( minus_minus @ A @ X3 @ A3 )
@ T2 ) ) ) ) ).
% translation_subtract_Compl
thf(fact_5932_surj__Compl__image__subset,axiom,
! [A: $tType,B: $tType,F2: B > A,A2: set @ B] :
( ( ( image @ B @ A @ F2 @ ( top_top @ ( set @ B ) ) )
= ( top_top @ ( set @ A ) ) )
=> ( ord_less_eq @ ( set @ A ) @ ( uminus_uminus @ ( set @ A ) @ ( image @ B @ A @ F2 @ A2 ) ) @ ( image @ B @ A @ F2 @ ( uminus_uminus @ ( set @ B ) @ A2 ) ) ) ) ).
% surj_Compl_image_subset
thf(fact_5933_Max__eq__iff,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A2: set @ A,M: A] :
( ( finite_finite2 @ A @ A2 )
=> ( ( A2
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( ( lattic643756798349783984er_Max @ A @ A2 )
= M )
= ( ( member @ A @ M @ A2 )
& ! [X3: A] :
( ( member @ A @ X3 @ A2 )
=> ( ord_less_eq @ A @ X3 @ M ) ) ) ) ) ) ) ).
% Max_eq_iff
thf(fact_5934_Max__ge__iff,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A2: set @ A,X2: A] :
( ( finite_finite2 @ A @ A2 )
=> ( ( A2
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( ord_less_eq @ A @ X2 @ ( lattic643756798349783984er_Max @ A @ A2 ) )
= ( ? [X3: A] :
( ( member @ A @ X3 @ A2 )
& ( ord_less_eq @ A @ X2 @ X3 ) ) ) ) ) ) ) ).
% Max_ge_iff
thf(fact_5935_eq__Max__iff,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A2: set @ A,M: A] :
( ( finite_finite2 @ A @ A2 )
=> ( ( A2
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( M
= ( lattic643756798349783984er_Max @ A @ A2 ) )
= ( ( member @ A @ M @ A2 )
& ! [X3: A] :
( ( member @ A @ X3 @ A2 )
=> ( ord_less_eq @ A @ X3 @ M ) ) ) ) ) ) ) ).
% eq_Max_iff
thf(fact_5936_Max_OboundedE,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A2: set @ A,X2: A] :
( ( finite_finite2 @ A @ A2 )
=> ( ( A2
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( ord_less_eq @ A @ ( lattic643756798349783984er_Max @ A @ A2 ) @ X2 )
=> ! [A10: A] :
( ( member @ A @ A10 @ A2 )
=> ( ord_less_eq @ A @ A10 @ X2 ) ) ) ) ) ) ).
% Max.boundedE
thf(fact_5937_Max_OboundedI,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A2: set @ A,X2: A] :
( ( finite_finite2 @ A @ A2 )
=> ( ( A2
!= ( bot_bot @ ( set @ A ) ) )
=> ( ! [A4: A] :
( ( member @ A @ A4 @ A2 )
=> ( ord_less_eq @ A @ A4 @ X2 ) )
=> ( ord_less_eq @ A @ ( lattic643756798349783984er_Max @ A @ A2 ) @ X2 ) ) ) ) ) ).
% Max.boundedI
thf(fact_5938_Max__gr__iff,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A2: set @ A,X2: A] :
( ( finite_finite2 @ A @ A2 )
=> ( ( A2
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( ord_less @ A @ X2 @ ( lattic643756798349783984er_Max @ A @ A2 ) )
= ( ? [X3: A] :
( ( member @ A @ X3 @ A2 )
& ( ord_less @ A @ X2 @ X3 ) ) ) ) ) ) ) ).
% Max_gr_iff
thf(fact_5939_Max__insert2,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A2: set @ A,A3: A] :
( ( finite_finite2 @ A @ A2 )
=> ( ! [B4: A] :
( ( member @ A @ B4 @ A2 )
=> ( ord_less_eq @ A @ B4 @ A3 ) )
=> ( ( lattic643756798349783984er_Max @ A @ ( insert @ A @ A3 @ A2 ) )
= A3 ) ) ) ) ).
% Max_insert2
thf(fact_5940_Max_Oinfinite,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A2: set @ A] :
( ~ ( finite_finite2 @ A @ A2 )
=> ( ( lattic643756798349783984er_Max @ A @ A2 )
= ( the2 @ A @ ( none @ A ) ) ) ) ) ).
% Max.infinite
thf(fact_5941_VEBT__internal_Oheight_Osimps_I2_J,axiom,
! [Uu: option @ ( product_prod @ nat @ nat ),Deg: nat,TreeList: list @ vEBT_VEBT,Summary: vEBT_VEBT] :
( ( vEBT_VEBT_height @ ( vEBT_Node @ Uu @ 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_5942_sum_Ogroup,axiom,
! [C: $tType,A: $tType,B: $tType] :
( ( comm_monoid_add @ A )
=> ! [S: set @ B,T6: set @ C,G: B > C,H2: B > A] :
( ( finite_finite2 @ B @ S )
=> ( ( finite_finite2 @ C @ T6 )
=> ( ( ord_less_eq @ ( set @ C ) @ ( image @ B @ C @ G @ S ) @ T6 )
=> ( ( groups7311177749621191930dd_sum @ C @ A
@ ^ [Y: C] :
( groups7311177749621191930dd_sum @ B @ A @ H2
@ ( collect @ B
@ ^ [X3: B] :
( ( member @ B @ X3 @ S )
& ( ( G @ X3 )
= Y ) ) ) )
@ T6 )
= ( groups7311177749621191930dd_sum @ B @ A @ H2 @ S ) ) ) ) ) ) ).
% sum.group
thf(fact_5943_prod_Ogroup,axiom,
! [C: $tType,A: $tType,B: $tType] :
( ( comm_monoid_mult @ A )
=> ! [S: set @ B,T6: set @ C,G: B > C,H2: B > A] :
( ( finite_finite2 @ B @ S )
=> ( ( finite_finite2 @ C @ T6 )
=> ( ( ord_less_eq @ ( set @ C ) @ ( image @ B @ C @ G @ S ) @ T6 )
=> ( ( groups7121269368397514597t_prod @ C @ A
@ ^ [Y: C] :
( groups7121269368397514597t_prod @ B @ A @ H2
@ ( collect @ B
@ ^ [X3: B] :
( ( member @ B @ X3 @ S )
& ( ( G @ X3 )
= Y ) ) ) )
@ T6 )
= ( groups7121269368397514597t_prod @ B @ A @ H2 @ S ) ) ) ) ) ) ).
% prod.group
thf(fact_5944_set__image__eq__pointwiseI,axiom,
! [B: $tType,A: $tType,L: list @ A,L4: list @ A,F2: A > B] :
( ( ( size_size @ ( list @ A ) @ L )
= ( size_size @ ( list @ A ) @ L4 ) )
=> ( ! [I2: nat] :
( ( ord_less @ nat @ I2 @ ( size_size @ ( list @ A ) @ L ) )
=> ( ( F2 @ ( nth @ A @ L @ I2 ) )
= ( F2 @ ( nth @ A @ L4 @ I2 ) ) ) )
=> ( ( image @ A @ B @ F2 @ ( set2 @ A @ L ) )
= ( image @ A @ B @ F2 @ ( set2 @ A @ L4 ) ) ) ) ) ).
% set_image_eq_pointwiseI
thf(fact_5945_in__set__image__conv__nth,axiom,
! [B: $tType,A: $tType,F2: B > A,X2: B,L: list @ B] :
( ( member @ A @ ( F2 @ X2 ) @ ( image @ B @ A @ F2 @ ( set2 @ B @ L ) ) )
= ( ? [I4: nat] :
( ( ord_less @ nat @ I4 @ ( size_size @ ( list @ B ) @ L ) )
& ( ( F2 @ ( nth @ B @ L @ I4 ) )
= ( F2 @ X2 ) ) ) ) ) ).
% in_set_image_conv_nth
thf(fact_5946_Max__mono,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [M8: set @ A,N7: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ M8 @ N7 )
=> ( ( M8
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( finite_finite2 @ A @ N7 )
=> ( ord_less_eq @ A @ ( lattic643756798349783984er_Max @ A @ M8 ) @ ( lattic643756798349783984er_Max @ A @ N7 ) ) ) ) ) ) ).
% Max_mono
thf(fact_5947_Max_Osubset__imp,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A2: set @ A,B2: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ A2 @ B2 )
=> ( ( A2
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( finite_finite2 @ A @ B2 )
=> ( ord_less_eq @ A @ ( lattic643756798349783984er_Max @ A @ A2 ) @ ( lattic643756798349783984er_Max @ A @ B2 ) ) ) ) ) ) ).
% Max.subset_imp
thf(fact_5948_VEBT__internal_Oheight_Oelims,axiom,
! [X2: vEBT_VEBT,Y3: nat] :
( ( ( vEBT_VEBT_height @ X2 )
= Y3 )
=> ( ( ? [A4: $o,B4: $o] :
( X2
= ( vEBT_Leaf @ A4 @ B4 ) )
=> ( Y3
!= ( zero_zero @ nat ) ) )
=> ~ ! [Uu2: option @ ( product_prod @ nat @ nat ),Deg2: nat,TreeList2: list @ vEBT_VEBT,Summary2: vEBT_VEBT] :
( ( X2
= ( vEBT_Node @ Uu2 @ Deg2 @ TreeList2 @ Summary2 ) )
=> ( Y3
!= ( plus_plus @ nat @ ( one_one @ nat ) @ ( lattic643756798349783984er_Max @ nat @ ( image @ vEBT_VEBT @ nat @ vEBT_VEBT_height @ ( insert @ vEBT_VEBT @ Summary2 @ ( set2 @ vEBT_VEBT @ TreeList2 ) ) ) ) ) ) ) ) ) ).
% VEBT_internal.height.elims
thf(fact_5949_Max_Osubset,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A2: set @ A,B2: set @ A] :
( ( finite_finite2 @ A @ A2 )
=> ( ( B2
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( ord_less_eq @ ( set @ A ) @ B2 @ A2 )
=> ( ( ord_max @ A @ ( lattic643756798349783984er_Max @ A @ B2 ) @ ( lattic643756798349783984er_Max @ A @ A2 ) )
= ( lattic643756798349783984er_Max @ A @ A2 ) ) ) ) ) ) ).
% Max.subset
thf(fact_5950_Max_Oclosed,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A2: set @ A] :
( ( finite_finite2 @ A @ A2 )
=> ( ( A2
!= ( bot_bot @ ( set @ A ) ) )
=> ( ! [X4: A,Y4: A] : ( member @ A @ ( ord_max @ A @ X4 @ Y4 ) @ ( insert @ A @ X4 @ ( insert @ A @ Y4 @ ( bot_bot @ ( set @ A ) ) ) ) )
=> ( member @ A @ ( lattic643756798349783984er_Max @ A @ A2 ) @ A2 ) ) ) ) ) ).
% Max.closed
thf(fact_5951_Max_Oinsert__not__elem,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A2: set @ A,X2: A] :
( ( finite_finite2 @ A @ A2 )
=> ( ~ ( member @ A @ X2 @ A2 )
=> ( ( A2
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( lattic643756798349783984er_Max @ A @ ( insert @ A @ X2 @ A2 ) )
= ( ord_max @ A @ X2 @ ( lattic643756798349783984er_Max @ A @ A2 ) ) ) ) ) ) ) ).
% Max.insert_not_elem
thf(fact_5952_divide__nat__def,axiom,
( ( divide_divide @ nat )
= ( ^ [M6: nat,N4: nat] :
( if @ nat
@ ( N4
= ( zero_zero @ nat ) )
@ ( zero_zero @ nat )
@ ( lattic643756798349783984er_Max @ nat
@ ( collect @ nat
@ ^ [K3: nat] : ( ord_less_eq @ nat @ ( times_times @ nat @ K3 @ N4 ) @ M6 ) ) ) ) ) ) ).
% divide_nat_def
thf(fact_5953_Max_Oeq__fold,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A2: set @ A,X2: A] :
( ( finite_finite2 @ A @ A2 )
=> ( ( lattic643756798349783984er_Max @ A @ ( insert @ A @ X2 @ A2 ) )
= ( finite_fold @ A @ A @ ( ord_max @ A ) @ X2 @ A2 ) ) ) ) ).
% Max.eq_fold
thf(fact_5954_rat__abs__code,axiom,
! [P6: rat] :
( ( quotient_of @ ( abs_abs @ rat @ P6 ) )
= ( product_case_prod @ int @ int @ ( product_prod @ int @ int )
@ ^ [A6: int] : ( product_Pair @ int @ int @ ( abs_abs @ int @ A6 ) )
@ ( quotient_of @ P6 ) ) ) ).
% rat_abs_code
thf(fact_5955_Max_Oinsert__remove,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A2: set @ A,X2: A] :
( ( finite_finite2 @ A @ A2 )
=> ( ( ( ( minus_minus @ ( set @ A ) @ A2 @ ( insert @ A @ X2 @ ( bot_bot @ ( set @ A ) ) ) )
= ( bot_bot @ ( set @ A ) ) )
=> ( ( lattic643756798349783984er_Max @ A @ ( insert @ A @ X2 @ A2 ) )
= X2 ) )
& ( ( ( minus_minus @ ( set @ A ) @ A2 @ ( insert @ A @ X2 @ ( bot_bot @ ( set @ A ) ) ) )
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( lattic643756798349783984er_Max @ A @ ( insert @ A @ X2 @ A2 ) )
= ( ord_max @ A @ X2 @ ( lattic643756798349783984er_Max @ A @ ( minus_minus @ ( set @ A ) @ A2 @ ( insert @ A @ X2 @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ) ) ) ) ) ).
% Max.insert_remove
thf(fact_5956_Max_Oremove,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A2: set @ A,X2: A] :
( ( finite_finite2 @ A @ A2 )
=> ( ( member @ A @ X2 @ A2 )
=> ( ( ( ( minus_minus @ ( set @ A ) @ A2 @ ( insert @ A @ X2 @ ( bot_bot @ ( set @ A ) ) ) )
= ( bot_bot @ ( set @ A ) ) )
=> ( ( lattic643756798349783984er_Max @ A @ A2 )
= X2 ) )
& ( ( ( minus_minus @ ( set @ A ) @ A2 @ ( insert @ A @ X2 @ ( bot_bot @ ( set @ A ) ) ) )
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( lattic643756798349783984er_Max @ A @ A2 )
= ( ord_max @ A @ X2 @ ( lattic643756798349783984er_Max @ A @ ( minus_minus @ ( set @ A ) @ A2 @ ( insert @ A @ X2 @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ) ) ) ) ) ) ).
% Max.remove
thf(fact_5957_image__mult__atLeastAtMost__if,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [C3: A,X2: A,Y3: A] :
( ( ( ord_less @ A @ ( zero_zero @ A ) @ C3 )
=> ( ( image @ A @ A @ ( times_times @ A @ C3 ) @ ( set_or1337092689740270186AtMost @ A @ X2 @ Y3 ) )
= ( set_or1337092689740270186AtMost @ A @ ( times_times @ A @ C3 @ X2 ) @ ( times_times @ A @ C3 @ Y3 ) ) ) )
& ( ~ ( ord_less @ A @ ( zero_zero @ A ) @ C3 )
=> ( ( ( ord_less_eq @ A @ X2 @ Y3 )
=> ( ( image @ A @ A @ ( times_times @ A @ C3 ) @ ( set_or1337092689740270186AtMost @ A @ X2 @ Y3 ) )
= ( set_or1337092689740270186AtMost @ A @ ( times_times @ A @ C3 @ Y3 ) @ ( times_times @ A @ C3 @ X2 ) ) ) )
& ( ~ ( ord_less_eq @ A @ X2 @ Y3 )
=> ( ( image @ A @ A @ ( times_times @ A @ C3 ) @ ( set_or1337092689740270186AtMost @ A @ X2 @ Y3 ) )
= ( bot_bot @ ( set @ A ) ) ) ) ) ) ) ) ).
% image_mult_atLeastAtMost_if
thf(fact_5958_rat__uminus__code,axiom,
! [P6: rat] :
( ( quotient_of @ ( uminus_uminus @ rat @ P6 ) )
= ( product_case_prod @ int @ int @ ( product_prod @ int @ int )
@ ^ [A6: int] : ( product_Pair @ int @ int @ ( uminus_uminus @ int @ A6 ) )
@ ( quotient_of @ P6 ) ) ) ).
% rat_uminus_code
thf(fact_5959_rat__less__code,axiom,
( ( ord_less @ rat )
= ( ^ [P4: rat,Q5: rat] :
( product_case_prod @ int @ int @ $o
@ ^ [A6: int,C5: int] :
( product_case_prod @ int @ int @ $o
@ ^ [B7: int,D3: int] : ( ord_less @ int @ ( times_times @ int @ A6 @ D3 ) @ ( times_times @ int @ C5 @ B7 ) )
@ ( quotient_of @ Q5 ) )
@ ( quotient_of @ P4 ) ) ) ) ).
% rat_less_code
thf(fact_5960_rat__floor__code,axiom,
( ( archim6421214686448440834_floor @ rat )
= ( ^ [P4: rat] : ( product_case_prod @ int @ int @ int @ ( divide_divide @ int ) @ ( quotient_of @ P4 ) ) ) ) ).
% rat_floor_code
thf(fact_5961_image__mult__atLeastAtMost__if_H,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [X2: A,Y3: A,C3: A] :
( ( ( ord_less_eq @ A @ X2 @ Y3 )
=> ( ( ( ord_less @ A @ ( zero_zero @ A ) @ C3 )
=> ( ( image @ A @ A
@ ^ [X3: A] : ( times_times @ A @ X3 @ C3 )
@ ( set_or1337092689740270186AtMost @ A @ X2 @ Y3 ) )
= ( set_or1337092689740270186AtMost @ A @ ( times_times @ A @ X2 @ C3 ) @ ( times_times @ A @ Y3 @ C3 ) ) ) )
& ( ~ ( ord_less @ A @ ( zero_zero @ A ) @ C3 )
=> ( ( image @ A @ A
@ ^ [X3: A] : ( times_times @ A @ X3 @ C3 )
@ ( set_or1337092689740270186AtMost @ A @ X2 @ Y3 ) )
= ( set_or1337092689740270186AtMost @ A @ ( times_times @ A @ Y3 @ C3 ) @ ( times_times @ A @ X2 @ C3 ) ) ) ) ) )
& ( ~ ( ord_less_eq @ A @ X2 @ Y3 )
=> ( ( image @ A @ A
@ ^ [X3: A] : ( times_times @ A @ X3 @ C3 )
@ ( set_or1337092689740270186AtMost @ A @ X2 @ Y3 ) )
= ( bot_bot @ ( set @ A ) ) ) ) ) ) ).
% image_mult_atLeastAtMost_if'
thf(fact_5962_rat__less__eq__code,axiom,
( ( ord_less_eq @ rat )
= ( ^ [P4: rat,Q5: rat] :
( product_case_prod @ int @ int @ $o
@ ^ [A6: int,C5: int] :
( product_case_prod @ int @ int @ $o
@ ^ [B7: int,D3: int] : ( ord_less_eq @ int @ ( times_times @ int @ A6 @ D3 ) @ ( times_times @ int @ C5 @ B7 ) )
@ ( quotient_of @ Q5 ) )
@ ( quotient_of @ P4 ) ) ) ) ).
% rat_less_eq_code
thf(fact_5963_image__affinity__atLeastAtMost,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [A3: A,B3: A,M: A,C3: A] :
( ( ( ( set_or1337092689740270186AtMost @ A @ A3 @ B3 )
= ( bot_bot @ ( set @ A ) ) )
=> ( ( image @ A @ A
@ ^ [X3: A] : ( plus_plus @ A @ ( times_times @ A @ M @ X3 ) @ C3 )
@ ( set_or1337092689740270186AtMost @ A @ A3 @ B3 ) )
= ( bot_bot @ ( set @ A ) ) ) )
& ( ( ( set_or1337092689740270186AtMost @ A @ A3 @ B3 )
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ M )
=> ( ( image @ A @ A
@ ^ [X3: A] : ( plus_plus @ A @ ( times_times @ A @ M @ X3 ) @ C3 )
@ ( set_or1337092689740270186AtMost @ A @ A3 @ B3 ) )
= ( set_or1337092689740270186AtMost @ A @ ( plus_plus @ A @ ( times_times @ A @ M @ A3 ) @ C3 ) @ ( plus_plus @ A @ ( times_times @ A @ M @ B3 ) @ C3 ) ) ) )
& ( ~ ( ord_less_eq @ A @ ( zero_zero @ A ) @ M )
=> ( ( image @ A @ A
@ ^ [X3: A] : ( plus_plus @ A @ ( times_times @ A @ M @ X3 ) @ C3 )
@ ( set_or1337092689740270186AtMost @ A @ A3 @ B3 ) )
= ( set_or1337092689740270186AtMost @ A @ ( plus_plus @ A @ ( times_times @ A @ M @ B3 ) @ C3 ) @ ( plus_plus @ A @ ( times_times @ A @ M @ A3 ) @ C3 ) ) ) ) ) ) ) ) ).
% image_affinity_atLeastAtMost
thf(fact_5964_image__affinity__atLeastAtMost__diff,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [A3: A,B3: A,M: A,C3: A] :
( ( ( ( set_or1337092689740270186AtMost @ A @ A3 @ B3 )
= ( bot_bot @ ( set @ A ) ) )
=> ( ( image @ A @ A
@ ^ [X3: A] : ( minus_minus @ A @ ( times_times @ A @ M @ X3 ) @ C3 )
@ ( set_or1337092689740270186AtMost @ A @ A3 @ B3 ) )
= ( bot_bot @ ( set @ A ) ) ) )
& ( ( ( set_or1337092689740270186AtMost @ A @ A3 @ B3 )
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ M )
=> ( ( image @ A @ A
@ ^ [X3: A] : ( minus_minus @ A @ ( times_times @ A @ M @ X3 ) @ C3 )
@ ( set_or1337092689740270186AtMost @ A @ A3 @ B3 ) )
= ( set_or1337092689740270186AtMost @ A @ ( minus_minus @ A @ ( times_times @ A @ M @ A3 ) @ C3 ) @ ( minus_minus @ A @ ( times_times @ A @ M @ B3 ) @ C3 ) ) ) )
& ( ~ ( ord_less_eq @ A @ ( zero_zero @ A ) @ M )
=> ( ( image @ A @ A
@ ^ [X3: A] : ( minus_minus @ A @ ( times_times @ A @ M @ X3 ) @ C3 )
@ ( set_or1337092689740270186AtMost @ A @ A3 @ B3 ) )
= ( set_or1337092689740270186AtMost @ A @ ( minus_minus @ A @ ( times_times @ A @ M @ B3 ) @ C3 ) @ ( minus_minus @ A @ ( times_times @ A @ M @ A3 ) @ C3 ) ) ) ) ) ) ) ) ).
% image_affinity_atLeastAtMost_diff
thf(fact_5965_image__affinity__atLeastAtMost__div,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [A3: A,B3: A,M: A,C3: A] :
( ( ( ( set_or1337092689740270186AtMost @ A @ A3 @ B3 )
= ( bot_bot @ ( set @ A ) ) )
=> ( ( image @ A @ A
@ ^ [X3: A] : ( plus_plus @ A @ ( divide_divide @ A @ X3 @ M ) @ C3 )
@ ( set_or1337092689740270186AtMost @ A @ A3 @ B3 ) )
= ( bot_bot @ ( set @ A ) ) ) )
& ( ( ( set_or1337092689740270186AtMost @ A @ A3 @ B3 )
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ M )
=> ( ( image @ A @ A
@ ^ [X3: A] : ( plus_plus @ A @ ( divide_divide @ A @ X3 @ M ) @ C3 )
@ ( set_or1337092689740270186AtMost @ A @ A3 @ B3 ) )
= ( set_or1337092689740270186AtMost @ A @ ( plus_plus @ A @ ( divide_divide @ A @ A3 @ M ) @ C3 ) @ ( plus_plus @ A @ ( divide_divide @ A @ B3 @ M ) @ C3 ) ) ) )
& ( ~ ( ord_less_eq @ A @ ( zero_zero @ A ) @ M )
=> ( ( image @ A @ A
@ ^ [X3: A] : ( plus_plus @ A @ ( divide_divide @ A @ X3 @ M ) @ C3 )
@ ( set_or1337092689740270186AtMost @ A @ A3 @ B3 ) )
= ( set_or1337092689740270186AtMost @ A @ ( plus_plus @ A @ ( divide_divide @ A @ B3 @ M ) @ C3 ) @ ( plus_plus @ A @ ( divide_divide @ A @ A3 @ M ) @ C3 ) ) ) ) ) ) ) ) ).
% image_affinity_atLeastAtMost_div
thf(fact_5966_image__affinity__atLeastAtMost__div__diff,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [A3: A,B3: A,M: A,C3: A] :
( ( ( ( set_or1337092689740270186AtMost @ A @ A3 @ B3 )
= ( bot_bot @ ( set @ A ) ) )
=> ( ( image @ A @ A
@ ^ [X3: A] : ( minus_minus @ A @ ( divide_divide @ A @ X3 @ M ) @ C3 )
@ ( set_or1337092689740270186AtMost @ A @ A3 @ B3 ) )
= ( bot_bot @ ( set @ A ) ) ) )
& ( ( ( set_or1337092689740270186AtMost @ A @ A3 @ B3 )
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ M )
=> ( ( image @ A @ A
@ ^ [X3: A] : ( minus_minus @ A @ ( divide_divide @ A @ X3 @ M ) @ C3 )
@ ( set_or1337092689740270186AtMost @ A @ A3 @ B3 ) )
= ( set_or1337092689740270186AtMost @ A @ ( minus_minus @ A @ ( divide_divide @ A @ A3 @ M ) @ C3 ) @ ( minus_minus @ A @ ( divide_divide @ A @ B3 @ M ) @ C3 ) ) ) )
& ( ~ ( ord_less_eq @ A @ ( zero_zero @ A ) @ M )
=> ( ( image @ A @ A
@ ^ [X3: A] : ( minus_minus @ A @ ( divide_divide @ A @ X3 @ M ) @ C3 )
@ ( set_or1337092689740270186AtMost @ A @ A3 @ B3 ) )
= ( set_or1337092689740270186AtMost @ A @ ( minus_minus @ A @ ( divide_divide @ A @ B3 @ M ) @ C3 ) @ ( minus_minus @ A @ ( divide_divide @ A @ A3 @ M ) @ C3 ) ) ) ) ) ) ) ) ).
% image_affinity_atLeastAtMost_div_diff
thf(fact_5967_VEBT__internal_Oheight_Opelims,axiom,
! [X2: vEBT_VEBT,Y3: nat] :
( ( ( vEBT_VEBT_height @ X2 )
= Y3 )
=> ( ( accp @ vEBT_VEBT @ vEBT_VEBT_height_rel @ X2 )
=> ( ! [A4: $o,B4: $o] :
( ( X2
= ( vEBT_Leaf @ A4 @ B4 ) )
=> ( ( Y3
= ( zero_zero @ nat ) )
=> ~ ( accp @ vEBT_VEBT @ vEBT_VEBT_height_rel @ ( vEBT_Leaf @ A4 @ B4 ) ) ) )
=> ~ ! [Uu2: option @ ( product_prod @ nat @ nat ),Deg2: nat,TreeList2: list @ vEBT_VEBT,Summary2: vEBT_VEBT] :
( ( X2
= ( vEBT_Node @ Uu2 @ Deg2 @ TreeList2 @ Summary2 ) )
=> ( ( Y3
= ( plus_plus @ nat @ ( one_one @ nat ) @ ( lattic643756798349783984er_Max @ nat @ ( image @ vEBT_VEBT @ nat @ vEBT_VEBT_height @ ( insert @ vEBT_VEBT @ Summary2 @ ( set2 @ vEBT_VEBT @ TreeList2 ) ) ) ) ) )
=> ~ ( accp @ vEBT_VEBT @ vEBT_VEBT_height_rel @ ( vEBT_Node @ Uu2 @ Deg2 @ TreeList2 @ Summary2 ) ) ) ) ) ) ) ).
% VEBT_internal.height.pelims
thf(fact_5968_ran__empty,axiom,
! [B: $tType,A: $tType] :
( ( ran @ B @ A
@ ^ [X3: B] : ( none @ A ) )
= ( bot_bot @ ( set @ A ) ) ) ).
% ran_empty
thf(fact_5969_quotient__of__int,axiom,
! [A3: int] :
( ( quotient_of @ ( of_int @ A3 ) )
= ( product_Pair @ int @ int @ A3 @ ( one_one @ int ) ) ) ).
% quotient_of_int
thf(fact_5970_bij__betw__Suc,axiom,
! [M8: set @ nat,N7: set @ nat] :
( ( bij_betw @ nat @ nat @ suc @ M8 @ N7 )
= ( ( image @ nat @ nat @ suc @ M8 )
= N7 ) ) ).
% bij_betw_Suc
thf(fact_5971_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_5972_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_5973_pair__imageI,axiom,
! [C: $tType,B: $tType,A: $tType,A3: A,B3: B,A2: set @ ( product_prod @ A @ B ),F2: A > B > C] :
( ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ A3 @ B3 ) @ A2 )
=> ( member @ C @ ( F2 @ A3 @ B3 ) @ ( image @ ( product_prod @ A @ B ) @ C @ ( product_case_prod @ A @ B @ C @ F2 ) @ A2 ) ) ) ).
% pair_imageI
thf(fact_5974_Max__divisors__self__int,axiom,
! [N: int] :
( ( N
!= ( zero_zero @ int ) )
=> ( ( lattic643756798349783984er_Max @ int
@ ( collect @ int
@ ^ [D3: int] : ( dvd_dvd @ int @ D3 @ N ) ) )
= ( abs_abs @ int @ N ) ) ) ).
% Max_divisors_self_int
thf(fact_5975_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_5976_zero__notin__Suc__image,axiom,
! [A2: set @ nat] :
~ ( member @ nat @ ( zero_zero @ nat ) @ ( image @ nat @ nat @ suc @ A2 ) ) ).
% zero_notin_Suc_image
thf(fact_5977_None__notin__image__Some,axiom,
! [A: $tType,A2: set @ A] :
~ ( member @ ( option @ A ) @ ( none @ A ) @ ( image @ A @ ( option @ A ) @ ( some @ A ) @ A2 ) ) ).
% None_notin_image_Some
thf(fact_5978_finite__conv__nat__seg__image,axiom,
! [A: $tType] :
( ( finite_finite2 @ A )
= ( ^ [A7: set @ A] :
? [N4: nat,F5: nat > A] :
( A7
= ( image @ nat @ A @ F5
@ ( collect @ nat
@ ^ [I4: nat] : ( ord_less @ nat @ I4 @ N4 ) ) ) ) ) ) ).
% finite_conv_nat_seg_image
thf(fact_5979_nat__seg__image__imp__finite,axiom,
! [A: $tType,A2: set @ A,F2: nat > A,N: nat] :
( ( A2
= ( image @ nat @ A @ F2
@ ( collect @ nat
@ ^ [I4: nat] : ( ord_less @ nat @ I4 @ N ) ) ) )
=> ( finite_finite2 @ A @ A2 ) ) ).
% nat_seg_image_imp_finite
thf(fact_5980_finite__int__iff__bounded,axiom,
( ( finite_finite2 @ int )
= ( ^ [S8: set @ int] :
? [K3: int] : ( ord_less_eq @ ( set @ int ) @ ( image @ int @ int @ ( abs_abs @ int ) @ S8 ) @ ( set_ord_lessThan @ int @ K3 ) ) ) ) ).
% finite_int_iff_bounded
thf(fact_5981_finite__int__iff__bounded__le,axiom,
( ( finite_finite2 @ int )
= ( ^ [S8: set @ int] :
? [K3: int] : ( ord_less_eq @ ( set @ int ) @ ( image @ int @ int @ ( abs_abs @ int ) @ S8 ) @ ( set_ord_atMost @ int @ K3 ) ) ) ) ).
% finite_int_iff_bounded_le
thf(fact_5982_notin__range__Some,axiom,
! [A: $tType,X2: option @ A] :
( ( ~ ( member @ ( option @ A ) @ X2 @ ( image @ A @ ( option @ A ) @ ( some @ A ) @ ( top_top @ ( set @ A ) ) ) ) )
= ( X2
= ( none @ A ) ) ) ).
% notin_range_Some
thf(fact_5983_in__image__insert__iff,axiom,
! [A: $tType,B2: set @ ( set @ A ),X2: A,A2: set @ A] :
( ! [C7: set @ A] :
( ( member @ ( set @ A ) @ C7 @ B2 )
=> ~ ( member @ A @ X2 @ C7 ) )
=> ( ( member @ ( set @ A ) @ A2 @ ( image @ ( set @ A ) @ ( set @ A ) @ ( insert @ A @ X2 ) @ B2 ) )
= ( ( member @ A @ X2 @ A2 )
& ( member @ ( set @ A ) @ ( minus_minus @ ( set @ A ) @ A2 @ ( insert @ A @ X2 @ ( bot_bot @ ( set @ A ) ) ) ) @ B2 ) ) ) ) ).
% in_image_insert_iff
thf(fact_5984_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_5985_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_5986_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_5987_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_5988_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_5989_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_5990_UNIV__option__conv,axiom,
! [A: $tType] :
( ( top_top @ ( set @ ( option @ A ) ) )
= ( insert @ ( option @ A ) @ ( none @ A ) @ ( image @ A @ ( option @ A ) @ ( some @ A ) @ ( top_top @ ( set @ A ) ) ) ) ) ).
% UNIV_option_conv
thf(fact_5991_image__fold__insert,axiom,
! [B: $tType,A: $tType,A2: set @ A,F2: A > B] :
( ( finite_finite2 @ A @ A2 )
=> ( ( image @ A @ B @ F2 @ A2 )
= ( finite_fold @ A @ ( set @ B )
@ ^ [K3: A] : ( insert @ B @ ( F2 @ K3 ) )
@ ( bot_bot @ ( set @ B ) )
@ A2 ) ) ) ).
% image_fold_insert
thf(fact_5992_range__mod,axiom,
! [N: nat] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ N )
=> ( ( image @ nat @ nat
@ ^ [M6: nat] : ( modulo_modulo @ nat @ M6 @ N )
@ ( top_top @ ( set @ nat ) ) )
= ( set_or7035219750837199246ssThan @ nat @ ( zero_zero @ nat ) @ N ) ) ) ).
% range_mod
thf(fact_5993_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_5994_image__add__int__atLeastLessThan,axiom,
! [L: int,U: int] :
( ( image @ int @ int
@ ^ [X3: int] : ( plus_plus @ int @ X3 @ L )
@ ( set_or7035219750837199246ssThan @ int @ ( zero_zero @ int ) @ ( minus_minus @ int @ U @ L ) ) )
= ( set_or7035219750837199246ssThan @ int @ L @ U ) ) ).
% image_add_int_atLeastLessThan
thf(fact_5995_image__add__integer__atLeastLessThan,axiom,
! [L: code_integer,U: code_integer] :
( ( image @ code_integer @ code_integer
@ ^ [X3: code_integer] : ( plus_plus @ code_integer @ X3 @ 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_5996_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_5997_image__minus__const__atLeastLessThan__nat,axiom,
! [C3: nat,Y3: nat,X2: nat] :
( ( ( ord_less @ nat @ C3 @ Y3 )
=> ( ( image @ nat @ nat
@ ^ [I4: nat] : ( minus_minus @ nat @ I4 @ C3 )
@ ( set_or7035219750837199246ssThan @ nat @ X2 @ Y3 ) )
= ( set_or7035219750837199246ssThan @ nat @ ( minus_minus @ nat @ X2 @ C3 ) @ ( minus_minus @ nat @ Y3 @ C3 ) ) ) )
& ( ~ ( ord_less @ nat @ C3 @ Y3 )
=> ( ( ( ord_less @ nat @ X2 @ Y3 )
=> ( ( image @ nat @ nat
@ ^ [I4: nat] : ( minus_minus @ nat @ I4 @ C3 )
@ ( set_or7035219750837199246ssThan @ nat @ X2 @ Y3 ) )
= ( insert @ nat @ ( zero_zero @ nat ) @ ( bot_bot @ ( set @ nat ) ) ) ) )
& ( ~ ( ord_less @ nat @ X2 @ Y3 )
=> ( ( image @ nat @ nat
@ ^ [I4: nat] : ( minus_minus @ nat @ I4 @ C3 )
@ ( set_or7035219750837199246ssThan @ nat @ X2 @ Y3 ) )
= ( bot_bot @ ( set @ nat ) ) ) ) ) ) ) ).
% image_minus_const_atLeastLessThan_nat
thf(fact_5998_ranI,axiom,
! [A: $tType,B: $tType,M: B > ( option @ A ),A3: B,B3: A] :
( ( ( M @ A3 )
= ( some @ A @ B3 ) )
=> ( member @ A @ B3 @ ( ran @ B @ A @ M ) ) ) ).
% ranI
thf(fact_5999_rat__minus__code,axiom,
! [P6: rat,Q3: rat] :
( ( quotient_of @ ( minus_minus @ rat @ P6 @ Q3 ) )
= ( product_case_prod @ int @ int @ ( product_prod @ int @ int )
@ ^ [A6: int,C5: int] :
( product_case_prod @ int @ int @ ( product_prod @ int @ int )
@ ^ [B7: int,D3: int] : ( normalize @ ( product_Pair @ int @ int @ ( minus_minus @ int @ ( times_times @ int @ A6 @ D3 ) @ ( times_times @ int @ B7 @ C5 ) ) @ ( times_times @ int @ C5 @ D3 ) ) )
@ ( quotient_of @ Q3 ) )
@ ( quotient_of @ P6 ) ) ) ).
% rat_minus_code
thf(fact_6000_rat__plus__code,axiom,
! [P6: rat,Q3: rat] :
( ( quotient_of @ ( plus_plus @ rat @ P6 @ Q3 ) )
= ( product_case_prod @ int @ int @ ( product_prod @ int @ int )
@ ^ [A6: int,C5: int] :
( product_case_prod @ int @ int @ ( product_prod @ int @ int )
@ ^ [B7: int,D3: int] : ( normalize @ ( product_Pair @ int @ int @ ( plus_plus @ int @ ( times_times @ int @ A6 @ D3 ) @ ( times_times @ int @ B7 @ C5 ) ) @ ( times_times @ int @ C5 @ D3 ) ) )
@ ( quotient_of @ Q3 ) )
@ ( quotient_of @ P6 ) ) ) ).
% rat_plus_code
thf(fact_6001_normalize__denom__zero,axiom,
! [P6: int] :
( ( normalize @ ( product_Pair @ int @ int @ P6 @ ( zero_zero @ int ) ) )
= ( product_Pair @ int @ int @ ( zero_zero @ int ) @ ( one_one @ int ) ) ) ).
% normalize_denom_zero
thf(fact_6002_rat__times__code,axiom,
! [P6: rat,Q3: rat] :
( ( quotient_of @ ( times_times @ rat @ P6 @ Q3 ) )
= ( product_case_prod @ int @ int @ ( product_prod @ int @ int )
@ ^ [A6: int,C5: int] :
( product_case_prod @ int @ int @ ( product_prod @ int @ int )
@ ^ [B7: int,D3: int] : ( normalize @ ( product_Pair @ int @ int @ ( times_times @ int @ A6 @ B7 ) @ ( times_times @ int @ C5 @ D3 ) ) )
@ ( quotient_of @ Q3 ) )
@ ( quotient_of @ P6 ) ) ) ).
% rat_times_code
thf(fact_6003_rat__divide__code,axiom,
! [P6: rat,Q3: rat] :
( ( quotient_of @ ( divide_divide @ rat @ P6 @ Q3 ) )
= ( product_case_prod @ int @ int @ ( product_prod @ int @ int )
@ ^ [A6: int,C5: int] :
( product_case_prod @ int @ int @ ( product_prod @ int @ int )
@ ^ [B7: int,D3: int] : ( normalize @ ( product_Pair @ int @ int @ ( times_times @ int @ A6 @ D3 ) @ ( times_times @ int @ C5 @ B7 ) ) )
@ ( quotient_of @ Q3 ) )
@ ( quotient_of @ P6 ) ) ) ).
% rat_divide_code
thf(fact_6004_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_6005_Frct__code__post_I5_J,axiom,
! [K: num] :
( ( frct @ ( product_Pair @ int @ int @ ( one_one @ int ) @ ( numeral_numeral @ int @ K ) ) )
= ( divide_divide @ rat @ ( one_one @ rat ) @ ( numeral_numeral @ rat @ K ) ) ) ).
% Frct_code_post(5)
thf(fact_6006_Frct__code__post_I3_J,axiom,
( ( frct @ ( product_Pair @ int @ int @ ( one_one @ int ) @ ( one_one @ int ) ) )
= ( one_one @ rat ) ) ).
% Frct_code_post(3)
thf(fact_6007_Frct__code__post_I4_J,axiom,
! [K: num] :
( ( frct @ ( product_Pair @ int @ int @ ( numeral_numeral @ int @ K ) @ ( one_one @ int ) ) )
= ( numeral_numeral @ rat @ K ) ) ).
% Frct_code_post(4)
thf(fact_6008_sofl__test,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [X2: word @ A,Y3: word @ A] :
( ( ( plus_plus @ int @ ( ring_1_signed @ A @ int @ X2 ) @ ( ring_1_signed @ A @ int @ Y3 ) )
= ( ring_1_signed @ A @ int @ ( plus_plus @ ( word @ A ) @ X2 @ Y3 ) ) )
= ( ( bit_se4197421643247451524op_bit @ ( word @ A ) @ ( minus_minus @ nat @ ( size_size @ ( word @ A ) @ X2 ) @ ( one_one @ nat ) ) @ ( bit_se5824344872417868541ns_and @ ( word @ A ) @ ( bit_se5824344971392196577ns_xor @ ( word @ A ) @ ( plus_plus @ ( word @ A ) @ X2 @ Y3 ) @ X2 ) @ ( bit_se5824344971392196577ns_xor @ ( word @ A ) @ ( plus_plus @ ( word @ A ) @ X2 @ Y3 ) @ Y3 ) ) )
= ( zero_zero @ ( word @ A ) ) ) ) ) ).
% sofl_test
thf(fact_6009_bin__rest__integer_Oabs__eq,axiom,
! [X2: int] :
( ( bits_b2549910563261871055nteger @ ( code_integer_of_int @ X2 ) )
= ( code_integer_of_int @ ( divide_divide @ int @ X2 @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) ) ) ) ).
% bin_rest_integer.abs_eq
thf(fact_6010_drop__bit__drop__bit,axiom,
! [A: $tType] :
( ( bit_se359711467146920520ations @ A )
=> ! [M: nat,N: nat,A3: A] :
( ( bit_se4197421643247451524op_bit @ A @ M @ ( bit_se4197421643247451524op_bit @ A @ N @ A3 ) )
= ( bit_se4197421643247451524op_bit @ A @ ( plus_plus @ nat @ M @ N ) @ A3 ) ) ) ).
% drop_bit_drop_bit
thf(fact_6011_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_6012_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_6013_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_6014_drop__bit__numeral__bit0,axiom,
! [A: $tType] :
( ( bit_un5681908812861735899ations @ A )
=> ! [L: num,K: num] :
( ( bit_se4197421643247451524op_bit @ A @ ( numeral_numeral @ nat @ L ) @ ( numeral_numeral @ A @ ( bit0 @ K ) ) )
= ( bit_se4197421643247451524op_bit @ A @ ( pred_numeral @ L ) @ ( numeral_numeral @ A @ K ) ) ) ) ).
% drop_bit_numeral_bit0
thf(fact_6015_drop__bit__numeral__bit1,axiom,
! [A: $tType] :
( ( bit_un5681908812861735899ations @ A )
=> ! [L: num,K: num] :
( ( bit_se4197421643247451524op_bit @ A @ ( numeral_numeral @ nat @ L ) @ ( numeral_numeral @ A @ ( bit1 @ K ) ) )
= ( bit_se4197421643247451524op_bit @ A @ ( pred_numeral @ L ) @ ( numeral_numeral @ A @ K ) ) ) ) ).
% drop_bit_numeral_bit1
thf(fact_6016_drop__bit__mask__eq,axiom,
! [A: $tType] :
( ( bit_un5681908812861735899ations @ A )
=> ! [M: nat,N: nat] :
( ( bit_se4197421643247451524op_bit @ A @ M @ ( bit_se2239418461657761734s_mask @ A @ N ) )
= ( bit_se2239418461657761734s_mask @ A @ ( minus_minus @ nat @ N @ M ) ) ) ) ).
% drop_bit_mask_eq
thf(fact_6017_drop__bit__take__bit,axiom,
! [A: $tType] :
( ( bit_se359711467146920520ations @ A )
=> ! [M: nat,N: nat,A3: A] :
( ( bit_se4197421643247451524op_bit @ A @ M @ ( bit_se2584673776208193580ke_bit @ A @ N @ A3 ) )
= ( bit_se2584673776208193580ke_bit @ A @ ( minus_minus @ nat @ N @ M ) @ ( bit_se4197421643247451524op_bit @ A @ M @ A3 ) ) ) ) ).
% drop_bit_take_bit
thf(fact_6018_take__bit__drop__bit,axiom,
! [A: $tType] :
( ( bit_se359711467146920520ations @ A )
=> ! [M: nat,N: nat,A3: A] :
( ( bit_se2584673776208193580ke_bit @ A @ M @ ( bit_se4197421643247451524op_bit @ A @ N @ A3 ) )
= ( bit_se4197421643247451524op_bit @ A @ N @ ( bit_se2584673776208193580ke_bit @ A @ ( plus_plus @ nat @ M @ N ) @ A3 ) ) ) ) ).
% take_bit_drop_bit
thf(fact_6019_bit__word__iff__drop__bit__and,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ( ( bit_se5641148757651400278ts_bit @ ( word @ A ) )
= ( ^ [A6: word @ A,N4: nat] :
( ( bit_se5824344872417868541ns_and @ ( word @ A ) @ ( bit_se4197421643247451524op_bit @ ( word @ A ) @ N4 @ A6 ) @ ( one_one @ ( word @ A ) ) )
= ( one_one @ ( word @ A ) ) ) ) ) ) ).
% bit_word_iff_drop_bit_and
thf(fact_6020_bit__iff__and__drop__bit__eq__1,axiom,
! [A: $tType] :
( ( bit_se359711467146920520ations @ A )
=> ( ( bit_se5641148757651400278ts_bit @ A )
= ( ^ [A6: A,N4: nat] :
( ( bit_se5824344872417868541ns_and @ A @ ( bit_se4197421643247451524op_bit @ A @ N4 @ A6 ) @ ( one_one @ A ) )
= ( one_one @ A ) ) ) ) ) ).
% bit_iff_and_drop_bit_eq_1
thf(fact_6021_stable__imp__drop__bit__eq,axiom,
! [A: $tType] :
( ( bit_se359711467146920520ations @ A )
=> ! [A3: A,N: nat] :
( ( ( divide_divide @ A @ A3 @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) )
= A3 )
=> ( ( bit_se4197421643247451524op_bit @ A @ N @ A3 )
= A3 ) ) ) ).
% stable_imp_drop_bit_eq
thf(fact_6022_drop__bit__half,axiom,
! [A: $tType] :
( ( bit_se359711467146920520ations @ A )
=> ! [N: nat,A3: A] :
( ( bit_se4197421643247451524op_bit @ A @ N @ ( divide_divide @ A @ A3 @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) )
= ( divide_divide @ A @ ( bit_se4197421643247451524op_bit @ A @ N @ A3 ) @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) ) ) ).
% drop_bit_half
thf(fact_6023_drop__bit__Suc,axiom,
! [A: $tType] :
( ( bit_se359711467146920520ations @ A )
=> ! [N: nat,A3: A] :
( ( bit_se4197421643247451524op_bit @ A @ ( suc @ N ) @ A3 )
= ( bit_se4197421643247451524op_bit @ A @ N @ ( divide_divide @ A @ A3 @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) ) ) ) ).
% drop_bit_Suc
thf(fact_6024_drop__bit__eq__div,axiom,
! [A: $tType] :
( ( bit_se359711467146920520ations @ A )
=> ( ( bit_se4197421643247451524op_bit @ A )
= ( ^ [N4: nat,A6: A] : ( divide_divide @ A @ A6 @ ( power_power @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ N4 ) ) ) ) ) ).
% drop_bit_eq_div
thf(fact_6025_even__drop__bit__iff__not__bit,axiom,
! [A: $tType] :
( ( bit_se359711467146920520ations @ A )
=> ! [N: nat,A3: A] :
( ( dvd_dvd @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ ( bit_se4197421643247451524op_bit @ A @ N @ A3 ) )
= ( ~ ( bit_se5641148757651400278ts_bit @ A @ A3 @ N ) ) ) ) ).
% even_drop_bit_iff_not_bit
thf(fact_6026_bit__iff__odd__drop__bit,axiom,
! [A: $tType] :
( ( bit_se359711467146920520ations @ A )
=> ( ( bit_se5641148757651400278ts_bit @ A )
= ( ^ [A6: A,N4: nat] :
~ ( dvd_dvd @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ ( bit_se4197421643247451524op_bit @ A @ N4 @ A6 ) ) ) ) ) ).
% bit_iff_odd_drop_bit
thf(fact_6027_bin__rest__integer__code,axiom,
( bits_b2549910563261871055nteger
= ( ^ [I4: code_integer] : ( divide_divide @ code_integer @ I4 @ ( numeral_numeral @ code_integer @ ( bit0 @ one2 ) ) ) ) ) ).
% bin_rest_integer_code
thf(fact_6028_drop__bit__rec,axiom,
! [A: $tType] :
( ( bit_se359711467146920520ations @ A )
=> ( ( bit_se4197421643247451524op_bit @ A )
= ( ^ [N4: nat,A6: A] :
( if @ A
@ ( N4
= ( zero_zero @ nat ) )
@ A6
@ ( bit_se4197421643247451524op_bit @ A @ ( minus_minus @ nat @ N4 @ ( one_one @ nat ) ) @ ( divide_divide @ A @ A6 @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) ) ) ) ) ) ).
% drop_bit_rec
thf(fact_6029_bitXOR__integer__unfold,axiom,
( ( bit_se5824344971392196577ns_xor @ code_integer )
= ( ^ [X3: code_integer,Y: code_integer] :
( if @ code_integer
@ ( X3
= ( zero_zero @ code_integer ) )
@ Y
@ ( if @ code_integer
@ ( X3
= ( uminus_uminus @ code_integer @ ( one_one @ code_integer ) ) )
@ ( bit_ri4277139882892585799ns_not @ code_integer @ Y )
@ ( bits_Bit_integer @ ( bit_se5824344971392196577ns_xor @ code_integer @ ( bits_b2549910563261871055nteger @ X3 ) @ ( bits_b2549910563261871055nteger @ Y ) )
@ ( ( ~ ( bits_b8758750999018896077nteger @ X3 ) )
= ( bits_b8758750999018896077nteger @ Y ) ) ) ) ) ) ) ).
% bitXOR_integer_unfold
thf(fact_6030_div__half__word,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [Y3: word @ A,X2: word @ A] :
( ( Y3
!= ( zero_zero @ ( word @ A ) ) )
=> ( ( product_Pair @ ( word @ A ) @ ( word @ A ) @ ( divide_divide @ ( word @ A ) @ X2 @ Y3 ) @ ( modulo_modulo @ ( word @ A ) @ X2 @ Y3 ) )
= ( if @ ( product_prod @ ( word @ A ) @ ( word @ A ) ) @ ( ord_less_eq @ ( word @ A ) @ Y3 @ ( minus_minus @ ( word @ A ) @ X2 @ ( times_times @ ( word @ A ) @ ( bit_se4730199178511100633sh_bit @ ( word @ A ) @ ( one_one @ nat ) @ ( divide_divide @ ( word @ A ) @ ( bit_se4197421643247451524op_bit @ ( word @ A ) @ ( one_one @ nat ) @ X2 ) @ Y3 ) ) @ Y3 ) ) ) @ ( 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 ) @ X2 ) @ Y3 ) ) @ ( one_one @ ( word @ A ) ) ) @ ( minus_minus @ ( word @ A ) @ ( minus_minus @ ( word @ A ) @ X2 @ ( times_times @ ( word @ A ) @ ( bit_se4730199178511100633sh_bit @ ( word @ A ) @ ( one_one @ nat ) @ ( divide_divide @ ( word @ A ) @ ( bit_se4197421643247451524op_bit @ ( word @ A ) @ ( one_one @ nat ) @ X2 ) @ Y3 ) ) @ Y3 ) ) @ Y3 ) ) @ ( 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 ) @ X2 ) @ Y3 ) ) @ ( minus_minus @ ( word @ A ) @ X2 @ ( times_times @ ( word @ A ) @ ( bit_se4730199178511100633sh_bit @ ( word @ A ) @ ( one_one @ nat ) @ ( divide_divide @ ( word @ A ) @ ( bit_se4197421643247451524op_bit @ ( word @ A ) @ ( one_one @ nat ) @ X2 ) @ Y3 ) ) @ Y3 ) ) ) ) ) ) ) ).
% div_half_word
thf(fact_6031_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_6032_drop__bit__minus__one,axiom,
! [N: nat] :
( ( bit_se4197421643247451524op_bit @ int @ N @ ( uminus_uminus @ int @ ( one_one @ int ) ) )
= ( uminus_uminus @ int @ ( one_one @ int ) ) ) ).
% drop_bit_minus_one
thf(fact_6033_push__bit__push__bit,axiom,
! [A: $tType] :
( ( bit_se359711467146920520ations @ A )
=> ! [M: nat,N: nat,A3: A] :
( ( bit_se4730199178511100633sh_bit @ A @ M @ ( bit_se4730199178511100633sh_bit @ A @ N @ A3 ) )
= ( bit_se4730199178511100633sh_bit @ A @ ( plus_plus @ nat @ M @ N ) @ A3 ) ) ) ).
% push_bit_push_bit
thf(fact_6034_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_6035_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_6036_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_6037_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_6038_push__bit__minus__one__eq__not__mask,axiom,
! [A: $tType] :
( ( bit_ri3973907225187159222ations @ A )
=> ! [N: nat] :
( ( bit_se4730199178511100633sh_bit @ A @ N @ ( uminus_uminus @ A @ ( one_one @ A ) ) )
= ( bit_ri4277139882892585799ns_not @ A @ ( bit_se2239418461657761734s_mask @ A @ N ) ) ) ) ).
% push_bit_minus_one_eq_not_mask
thf(fact_6039_push__bit__numeral,axiom,
! [A: $tType] :
( ( bit_se359711467146920520ations @ A )
=> ! [L: num,K: num] :
( ( bit_se4730199178511100633sh_bit @ A @ ( numeral_numeral @ nat @ L ) @ ( numeral_numeral @ A @ K ) )
= ( bit_se4730199178511100633sh_bit @ A @ ( pred_numeral @ L ) @ ( numeral_numeral @ A @ ( bit0 @ K ) ) ) ) ) ).
% push_bit_numeral
thf(fact_6040_drop__bit__numeral__minus__bit0,axiom,
! [L: num,K: num] :
( ( bit_se4197421643247451524op_bit @ int @ ( numeral_numeral @ nat @ L ) @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ ( bit0 @ K ) ) ) )
= ( bit_se4197421643247451524op_bit @ int @ ( pred_numeral @ L ) @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ K ) ) ) ) ).
% drop_bit_numeral_minus_bit0
thf(fact_6041_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_6042_push__bit__Suc,axiom,
! [A: $tType] :
( ( bit_se359711467146920520ations @ A )
=> ! [N: nat,A3: A] :
( ( bit_se4730199178511100633sh_bit @ A @ ( suc @ N ) @ A3 )
= ( bit_se4730199178511100633sh_bit @ A @ N @ ( times_times @ A @ A3 @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) ) ) ) ).
% push_bit_Suc
thf(fact_6043_push__bit__of__1,axiom,
! [A: $tType] :
( ( bit_se359711467146920520ations @ A )
=> ! [N: nat] :
( ( bit_se4730199178511100633sh_bit @ A @ N @ ( one_one @ A ) )
= ( power_power @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ N ) ) ) ).
% push_bit_of_1
thf(fact_6044_even__push__bit__iff,axiom,
! [A: $tType] :
( ( bit_se359711467146920520ations @ A )
=> ! [N: nat,A3: A] :
( ( dvd_dvd @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ ( bit_se4730199178511100633sh_bit @ A @ N @ A3 ) )
= ( ( N
!= ( zero_zero @ nat ) )
| ( dvd_dvd @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ A3 ) ) ) ) ).
% even_push_bit_iff
thf(fact_6045_push__bit__minus__numeral,axiom,
! [A: $tType] :
( ( bit_ri3973907225187159222ations @ A )
=> ! [L: num,K: num] :
( ( bit_se4730199178511100633sh_bit @ A @ ( numeral_numeral @ nat @ L ) @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ K ) ) )
= ( bit_se4730199178511100633sh_bit @ A @ ( pred_numeral @ L ) @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ ( bit0 @ K ) ) ) ) ) ) ).
% push_bit_minus_numeral
thf(fact_6046_drop__bit__numeral__minus__bit1,axiom,
! [L: num,K: num] :
( ( bit_se4197421643247451524op_bit @ int @ ( numeral_numeral @ nat @ L ) @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ ( bit1 @ K ) ) ) )
= ( bit_se4197421643247451524op_bit @ int @ ( pred_numeral @ L ) @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ ( inc @ K ) ) ) ) ) ).
% drop_bit_numeral_minus_bit1
thf(fact_6047_push__bit__numeral__minus__1,axiom,
! [A: $tType] :
( ( bit_ri3973907225187159222ations @ A )
=> ! [N: num] :
( ( bit_se4730199178511100633sh_bit @ A @ ( numeral_numeral @ nat @ N ) @ ( uminus_uminus @ A @ ( one_one @ A ) ) )
= ( uminus_uminus @ A @ ( power_power @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ ( numeral_numeral @ nat @ N ) ) ) ) ) ).
% push_bit_numeral_minus_1
thf(fact_6048_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_6049_push__bit__take__bit,axiom,
! [A: $tType] :
( ( bit_se359711467146920520ations @ A )
=> ! [M: nat,N: nat,A3: A] :
( ( bit_se4730199178511100633sh_bit @ A @ M @ ( bit_se2584673776208193580ke_bit @ A @ N @ A3 ) )
= ( bit_se2584673776208193580ke_bit @ A @ ( plus_plus @ nat @ M @ N ) @ ( bit_se4730199178511100633sh_bit @ A @ M @ A3 ) ) ) ) ).
% push_bit_take_bit
thf(fact_6050_push__bit__add,axiom,
! [A: $tType] :
( ( bit_se359711467146920520ations @ A )
=> ! [N: nat,A3: A,B3: A] :
( ( bit_se4730199178511100633sh_bit @ A @ N @ ( plus_plus @ A @ A3 @ B3 ) )
= ( plus_plus @ A @ ( bit_se4730199178511100633sh_bit @ A @ N @ A3 ) @ ( bit_se4730199178511100633sh_bit @ A @ N @ B3 ) ) ) ) ).
% push_bit_add
thf(fact_6051_take__bit__push__bit,axiom,
! [A: $tType] :
( ( bit_se359711467146920520ations @ A )
=> ! [M: nat,N: nat,A3: A] :
( ( bit_se2584673776208193580ke_bit @ A @ M @ ( bit_se4730199178511100633sh_bit @ A @ N @ A3 ) )
= ( bit_se4730199178511100633sh_bit @ A @ N @ ( bit_se2584673776208193580ke_bit @ A @ ( minus_minus @ nat @ M @ N ) @ A3 ) ) ) ) ).
% take_bit_push_bit
thf(fact_6052_bitval__bin__last__integer,axiom,
! [I: code_integer] :
( ( zero_neq_one_of_bool @ code_integer @ ( bits_b8758750999018896077nteger @ I ) )
= ( bit_se5824344872417868541ns_and @ code_integer @ I @ ( one_one @ code_integer ) ) ) ).
% bitval_bin_last_integer
thf(fact_6053_div__push__bit__of__1__eq__drop__bit,axiom,
! [A: $tType] :
( ( bit_se359711467146920520ations @ A )
=> ! [A3: A,N: nat] :
( ( divide_divide @ A @ A3 @ ( bit_se4730199178511100633sh_bit @ A @ N @ ( one_one @ A ) ) )
= ( bit_se4197421643247451524op_bit @ A @ N @ A3 ) ) ) ).
% div_push_bit_of_1_eq_drop_bit
thf(fact_6054_bits__ident,axiom,
! [A: $tType] :
( ( bit_se359711467146920520ations @ A )
=> ! [N: nat,A3: A] :
( ( plus_plus @ A @ ( bit_se4730199178511100633sh_bit @ A @ N @ ( bit_se4197421643247451524op_bit @ A @ N @ A3 ) ) @ ( bit_se2584673776208193580ke_bit @ A @ N @ A3 ) )
= A3 ) ) ).
% bits_ident
thf(fact_6055_shiftr__integer__conv__div__pow2,axiom,
( ( bit_se4197421643247451524op_bit @ code_integer )
= ( ^ [N4: nat,X3: code_integer] : ( divide_divide @ code_integer @ X3 @ ( power_power @ code_integer @ ( numeral_numeral @ code_integer @ ( bit0 @ one2 ) ) @ N4 ) ) ) ) ).
% shiftr_integer_conv_div_pow2
thf(fact_6056_set__bit__eq__or,axiom,
! [A: $tType] :
( ( bit_se359711467146920520ations @ A )
=> ( ( bit_se5668285175392031749et_bit @ A )
= ( ^ [N4: nat,A6: A] : ( bit_se1065995026697491101ons_or @ A @ A6 @ ( bit_se4730199178511100633sh_bit @ A @ N4 @ ( one_one @ A ) ) ) ) ) ) ).
% set_bit_eq_or
thf(fact_6057_flip__bit__eq__xor,axiom,
! [A: $tType] :
( ( bit_se359711467146920520ations @ A )
=> ( ( bit_se8732182000553998342ip_bit @ A )
= ( ^ [N4: nat,A6: A] : ( bit_se5824344971392196577ns_xor @ A @ A6 @ ( bit_se4730199178511100633sh_bit @ A @ N4 @ ( one_one @ A ) ) ) ) ) ) ).
% flip_bit_eq_xor
thf(fact_6058_push__bit__double,axiom,
! [A: $tType] :
( ( bit_se359711467146920520ations @ A )
=> ! [N: nat,A3: A] :
( ( bit_se4730199178511100633sh_bit @ A @ N @ ( times_times @ A @ A3 @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) )
= ( times_times @ A @ ( bit_se4730199178511100633sh_bit @ A @ N @ A3 ) @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) ) ) ).
% push_bit_double
thf(fact_6059_bin__last__integer__code,axiom,
( bits_b8758750999018896077nteger
= ( ^ [I4: code_integer] :
( ( bit_se5824344872417868541ns_and @ code_integer @ I4 @ ( one_one @ code_integer ) )
!= ( zero_zero @ code_integer ) ) ) ) ).
% bin_last_integer_code
thf(fact_6060_bit__iff__and__push__bit__not__eq__0,axiom,
! [A: $tType] :
( ( bit_se359711467146920520ations @ A )
=> ( ( bit_se5641148757651400278ts_bit @ A )
= ( ^ [A6: A,N4: nat] :
( ( bit_se5824344872417868541ns_and @ A @ A6 @ ( bit_se4730199178511100633sh_bit @ A @ N4 @ ( one_one @ A ) ) )
!= ( zero_zero @ A ) ) ) ) ) ).
% bit_iff_and_push_bit_not_eq_0
thf(fact_6061_push__bit__mask__eq,axiom,
! [A: $tType] :
( ( bit_ri3973907225187159222ations @ A )
=> ! [M: nat,N: nat] :
( ( bit_se4730199178511100633sh_bit @ A @ M @ ( bit_se2239418461657761734s_mask @ A @ N ) )
= ( bit_se5824344872417868541ns_and @ A @ ( bit_se2239418461657761734s_mask @ A @ ( plus_plus @ nat @ N @ M ) ) @ ( bit_ri4277139882892585799ns_not @ A @ ( bit_se2239418461657761734s_mask @ A @ M ) ) ) ) ) ).
% push_bit_mask_eq
thf(fact_6062_unset__bit__eq__and__not,axiom,
! [A: $tType] :
( ( bit_ri3973907225187159222ations @ A )
=> ( ( bit_se2638667681897837118et_bit @ A )
= ( ^ [N4: nat,A6: A] : ( bit_se5824344872417868541ns_and @ A @ A6 @ ( bit_ri4277139882892585799ns_not @ A @ ( bit_se4730199178511100633sh_bit @ A @ N4 @ ( one_one @ A ) ) ) ) ) ) ) ).
% unset_bit_eq_and_not
thf(fact_6063_bin__rest__code,axiom,
! [I: int] :
( ( divide_divide @ int @ I @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) )
= ( bit_se4197421643247451524op_bit @ int @ ( one_one @ nat ) @ I ) ) ).
% bin_rest_code
thf(fact_6064_drop__bit__int__def,axiom,
( ( bit_se4197421643247451524op_bit @ int )
= ( ^ [N4: nat,K3: int] : ( divide_divide @ int @ K3 @ ( power_power @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ N4 ) ) ) ) ).
% drop_bit_int_def
thf(fact_6065_push__bit__eq__mult,axiom,
! [A: $tType] :
( ( bit_se359711467146920520ations @ A )
=> ( ( bit_se4730199178511100633sh_bit @ A )
= ( ^ [N4: nat,A6: A] : ( times_times @ A @ A6 @ ( power_power @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ N4 ) ) ) ) ) ).
% push_bit_eq_mult
thf(fact_6066_exp__dvdE,axiom,
! [A: $tType] :
( ( bit_se359711467146920520ations @ A )
=> ! [N: nat,A3: A] :
( ( dvd_dvd @ A @ ( power_power @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ N ) @ A3 )
=> ~ ! [B4: A] :
( A3
!= ( bit_se4730199178511100633sh_bit @ A @ N @ B4 ) ) ) ) ).
% exp_dvdE
thf(fact_6067_drop__bit__nat__def,axiom,
( ( bit_se4197421643247451524op_bit @ nat )
= ( ^ [N4: nat,M6: nat] : ( divide_divide @ nat @ M6 @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N4 ) ) ) ) ).
% drop_bit_nat_def
thf(fact_6068_slice__eq__mask,axiom,
! [A: $tType] :
( ( bit_ri3973907225187159222ations @ A )
=> ! [N: nat,M: nat,A3: A] :
( ( bit_se4730199178511100633sh_bit @ A @ N @ ( bit_se2584673776208193580ke_bit @ A @ M @ ( bit_se4197421643247451524op_bit @ A @ N @ A3 ) ) )
= ( bit_se5824344872417868541ns_and @ A @ A3 @ ( bit_se5824344872417868541ns_and @ A @ ( bit_se2239418461657761734s_mask @ A @ ( plus_plus @ nat @ M @ N ) ) @ ( bit_ri4277139882892585799ns_not @ A @ ( bit_se2239418461657761734s_mask @ A @ N ) ) ) ) ) ) ).
% slice_eq_mask
thf(fact_6069_take__bit__sum,axiom,
! [A: $tType] :
( ( bit_un5681908812861735899ations @ A )
=> ( ( bit_se2584673776208193580ke_bit @ A )
= ( ^ [N4: nat,A6: A] :
( groups7311177749621191930dd_sum @ nat @ A
@ ^ [K3: nat] : ( bit_se4730199178511100633sh_bit @ A @ K3 @ ( zero_neq_one_of_bool @ A @ ( bit_se5641148757651400278ts_bit @ A @ A6 @ K3 ) ) )
@ ( set_or7035219750837199246ssThan @ nat @ ( zero_zero @ nat ) @ N4 ) ) ) ) ) ).
% take_bit_sum
thf(fact_6070_word__and__mask__or__conv__and__mask,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [N: word @ A,Index: nat] :
( ( bit_se5641148757651400278ts_bit @ ( word @ A ) @ N @ Index )
=> ( ( bit_se1065995026697491101ons_or @ ( word @ A ) @ ( bit_se5824344872417868541ns_and @ ( word @ A ) @ N @ ( bit_se2239418461657761734s_mask @ ( word @ A ) @ Index ) ) @ ( bit_se4730199178511100633sh_bit @ ( word @ A ) @ Index @ ( one_one @ ( word @ A ) ) ) )
= ( bit_se5824344872417868541ns_and @ ( word @ A ) @ N @ ( bit_se2239418461657761734s_mask @ ( word @ A ) @ ( plus_plus @ nat @ Index @ ( one_one @ nat ) ) ) ) ) ) ) ).
% word_and_mask_or_conv_and_mask
thf(fact_6071_bin__last__integer__nbe,axiom,
( bits_b8758750999018896077nteger
= ( ^ [I4: code_integer] :
( ( modulo_modulo @ code_integer @ I4 @ ( numeral_numeral @ code_integer @ ( bit0 @ one2 ) ) )
!= ( zero_zero @ code_integer ) ) ) ) ).
% bin_last_integer_nbe
thf(fact_6072_signed__take__bit__code,axiom,
! [A: $tType] :
( ( bit_ri3973907225187159222ations @ A )
=> ( ( bit_ri4674362597316999326ke_bit @ A )
= ( ^ [N4: nat,A6: A] : ( if @ A @ ( bit_se5641148757651400278ts_bit @ A @ ( bit_se2584673776208193580ke_bit @ A @ ( suc @ N4 ) @ A6 ) @ N4 ) @ ( plus_plus @ A @ ( bit_se2584673776208193580ke_bit @ A @ ( suc @ N4 ) @ A6 ) @ ( bit_se4730199178511100633sh_bit @ A @ ( suc @ N4 ) @ ( uminus_uminus @ A @ ( one_one @ A ) ) ) ) @ ( bit_se2584673776208193580ke_bit @ A @ ( suc @ N4 ) @ A6 ) ) ) ) ) ).
% signed_take_bit_code
thf(fact_6073_bin__last__integer_Oabs__eq,axiom,
! [X2: int] :
( ( bits_b8758750999018896077nteger @ ( code_integer_of_int @ X2 ) )
= ( ~ ( dvd_dvd @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ X2 ) ) ) ).
% bin_last_integer.abs_eq
thf(fact_6074_bitOR__integer__unfold,axiom,
( ( bit_se1065995026697491101ons_or @ code_integer )
= ( ^ [X3: code_integer,Y: code_integer] :
( if @ code_integer
@ ( X3
= ( zero_zero @ code_integer ) )
@ Y
@ ( if @ code_integer
@ ( X3
= ( uminus_uminus @ code_integer @ ( one_one @ code_integer ) ) )
@ ( uminus_uminus @ code_integer @ ( one_one @ code_integer ) )
@ ( bits_Bit_integer @ ( bit_se1065995026697491101ons_or @ code_integer @ ( bits_b2549910563261871055nteger @ X3 ) @ ( bits_b2549910563261871055nteger @ Y ) )
@ ( ( bits_b8758750999018896077nteger @ X3 )
| ( bits_b8758750999018896077nteger @ Y ) ) ) ) ) ) ) ).
% bitOR_integer_unfold
thf(fact_6075_bitAND__integer__unfold,axiom,
( ( bit_se5824344872417868541ns_and @ code_integer )
= ( ^ [X3: code_integer,Y: code_integer] :
( if @ code_integer
@ ( X3
= ( zero_zero @ code_integer ) )
@ ( zero_zero @ code_integer )
@ ( if @ code_integer
@ ( X3
= ( uminus_uminus @ code_integer @ ( one_one @ code_integer ) ) )
@ Y
@ ( bits_Bit_integer @ ( bit_se5824344872417868541ns_and @ code_integer @ ( bits_b2549910563261871055nteger @ X3 ) @ ( bits_b2549910563261871055nteger @ Y ) )
@ ( ( bits_b8758750999018896077nteger @ X3 )
& ( bits_b8758750999018896077nteger @ Y ) ) ) ) ) ) ) ).
% bitAND_integer_unfold
thf(fact_6076_set__bits__aux__Suc,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [F2: nat > $o,N: nat,W: word @ A] :
( ( code_T2661198915054445665ts_aux @ A @ F2 @ ( suc @ N ) @ W )
= ( code_T2661198915054445665ts_aux @ A @ F2 @ N @ ( bit_se1065995026697491101ons_or @ ( word @ A ) @ ( bit_se4730199178511100633sh_bit @ ( word @ A ) @ ( one_one @ nat ) @ W ) @ ( if @ ( word @ A ) @ ( F2 @ N ) @ ( one_one @ ( word @ A ) ) @ ( zero_zero @ ( word @ A ) ) ) ) ) ) ) ).
% set_bits_aux_Suc
thf(fact_6077_set__bits__aux__rec,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ( ( code_T2661198915054445665ts_aux @ A )
= ( ^ [F5: nat > $o,N4: nat,W2: word @ A] :
( if @ ( word @ A )
@ ( N4
= ( zero_zero @ nat ) )
@ W2
@ ( code_T2661198915054445665ts_aux @ A @ F5 @ ( minus_minus @ nat @ N4 @ ( one_one @ nat ) ) @ ( bit_se1065995026697491101ons_or @ ( word @ A ) @ ( bit_se4730199178511100633sh_bit @ ( word @ A ) @ ( one_one @ nat ) @ W2 ) @ ( if @ ( word @ A ) @ ( F5 @ ( minus_minus @ nat @ N4 @ ( one_one @ nat ) ) ) @ ( one_one @ ( word @ A ) ) @ ( zero_zero @ ( word @ A ) ) ) ) ) ) ) ) ) ).
% set_bits_aux_rec
thf(fact_6078_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_6079_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_6080_drop__bit__push__bit__int,axiom,
! [M: nat,N: nat,K: int] :
( ( bit_se4197421643247451524op_bit @ int @ M @ ( bit_se4730199178511100633sh_bit @ int @ N @ K ) )
= ( bit_se4197421643247451524op_bit @ int @ ( minus_minus @ nat @ M @ N ) @ ( bit_se4730199178511100633sh_bit @ int @ ( minus_minus @ nat @ N @ M ) @ K ) ) ) ).
% drop_bit_push_bit_int
thf(fact_6081_set__bit__nat__def,axiom,
( ( bit_se5668285175392031749et_bit @ nat )
= ( ^ [M6: nat,N4: nat] : ( bit_se1065995026697491101ons_or @ nat @ N4 @ ( bit_se4730199178511100633sh_bit @ nat @ M6 @ ( one_one @ nat ) ) ) ) ) ).
% set_bit_nat_def
thf(fact_6082_flip__bit__nat__def,axiom,
( ( bit_se8732182000553998342ip_bit @ nat )
= ( ^ [M6: nat,N4: nat] : ( bit_se5824344971392196577ns_xor @ nat @ N4 @ ( bit_se4730199178511100633sh_bit @ nat @ M6 @ ( one_one @ nat ) ) ) ) ) ).
% flip_bit_nat_def
thf(fact_6083_Bit__integer__code_I1_J,axiom,
! [I: code_integer] :
( ( bits_Bit_integer @ I @ $false )
= ( bit_se4730199178511100633sh_bit @ code_integer @ ( one_one @ nat ) @ I ) ) ).
% Bit_integer_code(1)
thf(fact_6084_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_6085_Bit__Operations_Oset__bit__int__def,axiom,
( ( bit_se5668285175392031749et_bit @ int )
= ( ^ [N4: nat,K3: int] : ( bit_se1065995026697491101ons_or @ int @ K3 @ ( bit_se4730199178511100633sh_bit @ int @ N4 @ ( one_one @ int ) ) ) ) ) ).
% Bit_Operations.set_bit_int_def
thf(fact_6086_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_6087_flip__bit__int__def,axiom,
( ( bit_se8732182000553998342ip_bit @ int )
= ( ^ [N4: nat,K3: int] : ( bit_se5824344971392196577ns_xor @ int @ K3 @ ( bit_se4730199178511100633sh_bit @ int @ N4 @ ( one_one @ int ) ) ) ) ) ).
% flip_bit_int_def
thf(fact_6088_shiftl__integer__conv__mult__pow2,axiom,
( ( bit_se4730199178511100633sh_bit @ code_integer )
= ( ^ [N4: nat,X3: code_integer] : ( times_times @ code_integer @ X3 @ ( power_power @ code_integer @ ( numeral_numeral @ code_integer @ ( bit0 @ one2 ) ) @ N4 ) ) ) ) ).
% shiftl_integer_conv_mult_pow2
thf(fact_6089_unset__bit__int__def,axiom,
( ( bit_se2638667681897837118et_bit @ int )
= ( ^ [N4: nat,K3: int] : ( bit_se5824344872417868541ns_and @ int @ K3 @ ( bit_ri4277139882892585799ns_not @ int @ ( bit_se4730199178511100633sh_bit @ int @ N4 @ ( one_one @ int ) ) ) ) ) ) ).
% unset_bit_int_def
thf(fact_6090_push__bit__int__def,axiom,
( ( bit_se4730199178511100633sh_bit @ int )
= ( ^ [N4: nat,K3: int] : ( times_times @ int @ K3 @ ( power_power @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ N4 ) ) ) ) ).
% push_bit_int_def
thf(fact_6091_Bit__integer__code_I2_J,axiom,
! [I: code_integer] :
( ( bits_Bit_integer @ I @ $true )
= ( plus_plus @ code_integer @ ( bit_se4730199178511100633sh_bit @ code_integer @ ( one_one @ nat ) @ I ) @ ( one_one @ code_integer ) ) ) ).
% Bit_integer_code(2)
thf(fact_6092_push__bit__nat__def,axiom,
( ( bit_se4730199178511100633sh_bit @ nat )
= ( ^ [N4: nat,M6: nat] : ( times_times @ nat @ M6 @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N4 ) ) ) ) ).
% push_bit_nat_def
thf(fact_6093_push__bit__minus__one,axiom,
! [N: nat] :
( ( bit_se4730199178511100633sh_bit @ int @ N @ ( uminus_uminus @ int @ ( one_one @ int ) ) )
= ( uminus_uminus @ int @ ( power_power @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ N ) ) ) ).
% push_bit_minus_one
thf(fact_6094_test__bit__split,axiom,
! [A: $tType,C: $tType,B: $tType] :
( ( ( type_len @ B )
& ( type_len @ C )
& ( type_len @ A ) )
=> ! [C3: word @ C,A3: word @ A,B3: word @ B] :
( ( ( word_split @ C @ A @ B @ C3 )
= ( product_Pair @ ( word @ A ) @ ( word @ B ) @ A3 @ B3 ) )
=> ( ! [N9: nat] :
( ( bit_se5641148757651400278ts_bit @ ( word @ B ) @ B3 @ N9 )
= ( ( ord_less @ nat @ N9 @ ( size_size @ ( word @ B ) @ B3 ) )
& ( bit_se5641148757651400278ts_bit @ ( word @ C ) @ C3 @ N9 ) ) )
& ! [M3: nat] :
( ( bit_se5641148757651400278ts_bit @ ( word @ A ) @ A3 @ M3 )
= ( ( ord_less @ nat @ M3 @ ( size_size @ ( word @ A ) @ A3 ) )
& ( bit_se5641148757651400278ts_bit @ ( word @ C ) @ C3 @ ( plus_plus @ nat @ M3 @ ( size_size @ ( word @ B ) @ B3 ) ) ) ) ) ) ) ) ).
% test_bit_split
thf(fact_6095_test__bit__split_H,axiom,
! [A: $tType,C: $tType,B: $tType] :
( ( ( type_len @ B )
& ( type_len @ C )
& ( type_len @ A ) )
=> ! [C3: word @ C,A3: word @ A,B3: word @ B] :
( ( ( word_split @ C @ A @ B @ C3 )
= ( product_Pair @ ( word @ A ) @ ( word @ B ) @ A3 @ B3 ) )
=> ! [N9: nat,M3: nat] :
( ( ( bit_se5641148757651400278ts_bit @ ( word @ B ) @ B3 @ N9 )
= ( ( ord_less @ nat @ N9 @ ( size_size @ ( word @ B ) @ B3 ) )
& ( bit_se5641148757651400278ts_bit @ ( word @ C ) @ C3 @ N9 ) ) )
& ( ( bit_se5641148757651400278ts_bit @ ( word @ A ) @ A3 @ M3 )
= ( ( ord_less @ nat @ M3 @ ( size_size @ ( word @ A ) @ A3 ) )
& ( bit_se5641148757651400278ts_bit @ ( word @ C ) @ C3 @ ( plus_plus @ nat @ M3 @ ( size_size @ ( word @ B ) @ B3 ) ) ) ) ) ) ) ) ).
% test_bit_split'
thf(fact_6096_test__bit__split__eq,axiom,
! [B: $tType,C: $tType,A: $tType] :
( ( ( type_len @ A )
& ( type_len @ C )
& ( type_len @ B ) )
=> ! [C3: word @ C,A3: word @ A,B3: word @ B] :
( ( ( word_split @ C @ A @ B @ C3 )
= ( product_Pair @ ( word @ A ) @ ( word @ B ) @ A3 @ B3 ) )
= ( ! [N4: nat] :
( ( bit_se5641148757651400278ts_bit @ ( word @ B ) @ B3 @ N4 )
= ( ( ord_less @ nat @ N4 @ ( size_size @ ( word @ B ) @ B3 ) )
& ( bit_se5641148757651400278ts_bit @ ( word @ C ) @ C3 @ N4 ) ) )
& ! [M6: nat] :
( ( bit_se5641148757651400278ts_bit @ ( word @ A ) @ A3 @ M6 )
= ( ( ord_less @ nat @ M6 @ ( size_size @ ( word @ A ) @ A3 ) )
& ( bit_se5641148757651400278ts_bit @ ( word @ C ) @ C3 @ ( plus_plus @ nat @ M6 @ ( size_size @ ( word @ B ) @ B3 ) ) ) ) ) ) ) ) ).
% test_bit_split_eq
thf(fact_6097_set__bit__integer__conv__masks,axiom,
( ( generi7602027413899671122et_bit @ code_integer )
= ( ^ [X3: code_integer,I4: nat,B7: $o] : ( if @ code_integer @ B7 @ ( bit_se1065995026697491101ons_or @ code_integer @ X3 @ ( bit_se4730199178511100633sh_bit @ code_integer @ I4 @ ( one_one @ code_integer ) ) ) @ ( bit_se5824344872417868541ns_and @ code_integer @ X3 @ ( bit_ri4277139882892585799ns_not @ code_integer @ ( bit_se4730199178511100633sh_bit @ code_integer @ I4 @ ( one_one @ code_integer ) ) ) ) ) ) ) ).
% set_bit_integer_conv_masks
thf(fact_6098_case__prod__app,axiom,
! [A: $tType,D6: $tType,C: $tType,B: $tType] :
( ( product_case_prod @ B @ C @ ( D6 > A ) )
= ( ^ [F5: B > C > D6 > A,X3: product_prod @ B @ C,Y: D6] :
( product_case_prod @ B @ C @ A
@ ^ [L3: B,R5: C] : ( F5 @ L3 @ R5 @ Y )
@ X3 ) ) ) ).
% case_prod_app
thf(fact_6099_predicate2D__conj,axiom,
! [A: $tType,B: $tType,P: A > B > $o,Q: A > B > $o,R: $o,X2: A,Y3: B] :
( ( ( ord_less_eq @ ( A > B > $o ) @ P @ Q )
& R )
=> ( R
& ( ( P @ X2 @ Y3 )
=> ( Q @ X2 @ Y3 ) ) ) ) ).
% predicate2D_conj
thf(fact_6100_fun__cong__unused__0,axiom,
! [A: $tType,B: $tType,C: $tType] :
( ( zero @ B )
=> ! [F2: ( A > B ) > C,G: C] :
( ( F2
= ( ^ [X3: A > B] : G ) )
=> ( ( F2
@ ^ [X3: A] : ( zero_zero @ B ) )
= G ) ) ) ).
% fun_cong_unused_0
thf(fact_6101_eq__subset,axiom,
! [A: $tType,P: A > A > $o] :
( ord_less_eq @ ( A > A > $o )
@ ^ [Y6: A,Z3: A] : Y6 = Z3
@ ^ [A6: A,B7: A] :
( ( P @ A6 @ B7 )
| ( A6 = B7 ) ) ) ).
% eq_subset
thf(fact_6102_case__prod__Pair__iden,axiom,
! [B: $tType,A: $tType,P6: product_prod @ A @ B] :
( ( product_case_prod @ A @ B @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B ) @ P6 )
= P6 ) ).
% case_prod_Pair_iden
thf(fact_6103_bit__set__bit__iff__2n,axiom,
! [A: $tType] :
( ( generic_set_set_bit @ A )
=> ! [A3: A,M: nat,B3: $o,N: nat] :
( ( bit_se5641148757651400278ts_bit @ A @ ( generi7602027413899671122et_bit @ A @ A3 @ M @ B3 ) @ N )
= ( ( ( M = N )
=> B3 )
& ( ( M != N )
=> ( bit_se5641148757651400278ts_bit @ A @ A3 @ N ) )
& ( ( power_power @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ N )
!= ( zero_zero @ A ) ) ) ) ) ).
% bit_set_bit_iff_2n
thf(fact_6104_Set__filter__fold,axiom,
! [A: $tType,A2: set @ A,P: A > $o] :
( ( finite_finite2 @ A @ A2 )
=> ( ( filter2 @ A @ P @ A2 )
= ( finite_fold @ A @ ( set @ A )
@ ^ [X3: A,A11: set @ A] : ( if @ ( set @ A ) @ ( P @ X3 ) @ ( insert @ A @ X3 @ A11 ) @ A11 )
@ ( bot_bot @ ( set @ A ) )
@ A2 ) ) ) ).
% Set_filter_fold
thf(fact_6105_set__bit__beyond,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [X2: word @ A,N: nat,B3: $o] :
( ( ord_less_eq @ nat @ ( size_size @ ( word @ A ) @ X2 ) @ N )
=> ( ( generi7602027413899671122et_bit @ ( word @ A ) @ X2 @ N @ B3 )
= X2 ) ) ) ).
% set_bit_beyond
thf(fact_6106_word__set__ge,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [W: word @ A,N: nat] : ( ord_less_eq @ ( word @ A ) @ W @ ( generi7602027413899671122et_bit @ ( word @ A ) @ W @ N @ $true ) ) ) ).
% word_set_ge
thf(fact_6107_word__clr__le,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [W: word @ A,N: nat] : ( ord_less_eq @ ( word @ A ) @ ( generi7602027413899671122et_bit @ ( word @ A ) @ W @ N @ $false ) @ W ) ) ).
% word_clr_le
thf(fact_6108_Set_Ofilter__def,axiom,
! [A: $tType] :
( ( filter2 @ A )
= ( ^ [P3: A > $o,A7: set @ A] :
( collect @ A
@ ^ [A6: A] :
( ( member @ A @ A6 @ A7 )
& ( P3 @ A6 ) ) ) ) ) ).
% Set.filter_def
thf(fact_6109_test__bit__set,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [W: word @ A,N: nat,X2: $o] :
( ( bit_se5641148757651400278ts_bit @ ( word @ A ) @ ( generi7602027413899671122et_bit @ ( word @ A ) @ W @ N @ X2 ) @ N )
= ( ( ord_less @ nat @ N @ ( size_size @ ( word @ A ) @ W ) )
& X2 ) ) ) ).
% test_bit_set
thf(fact_6110_test__bit__set__gen,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [W: word @ A,N: nat,X2: $o,M: nat] :
( ( bit_se5641148757651400278ts_bit @ ( word @ A ) @ ( generi7602027413899671122et_bit @ ( word @ A ) @ W @ N @ X2 ) @ M )
= ( ( ( M = N )
=> ( ( ord_less @ nat @ N @ ( size_size @ ( word @ A ) @ W ) )
& X2 ) )
& ( ( M != N )
=> ( bit_se5641148757651400278ts_bit @ ( word @ A ) @ W @ M ) ) ) ) ) ).
% test_bit_set_gen
thf(fact_6111_word__set__nth__iff,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [W: word @ A,N: nat,B3: $o] :
( ( ( generi7602027413899671122et_bit @ ( word @ A ) @ W @ N @ B3 )
= W )
= ( ( ( bit_se5641148757651400278ts_bit @ ( word @ A ) @ W @ N )
= B3 )
| ( ord_less_eq @ nat @ ( size_size @ ( word @ A ) @ W ) @ N ) ) ) ) ).
% word_set_nth_iff
thf(fact_6112_one__bit__pow,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [N: nat] :
( ( generi7602027413899671122et_bit @ ( word @ A ) @ ( zero_zero @ ( word @ A ) ) @ N @ $true )
= ( power_power @ ( word @ A ) @ ( numeral_numeral @ ( word @ A ) @ ( bit0 @ one2 ) ) @ N ) ) ) ).
% one_bit_pow
thf(fact_6113_int__set__bit__True__conv__OR,axiom,
! [I: int,N: nat] :
( ( generi7602027413899671122et_bit @ int @ I @ N @ $true )
= ( bit_se1065995026697491101ons_or @ int @ I @ ( bit_se4730199178511100633sh_bit @ int @ N @ ( one_one @ int ) ) ) ) ).
% int_set_bit_True_conv_OR
thf(fact_6114_int__set__bit__False__conv__NAND,axiom,
! [I: int,N: nat] :
( ( generi7602027413899671122et_bit @ int @ I @ N @ $false )
= ( bit_se5824344872417868541ns_and @ int @ I @ ( bit_ri4277139882892585799ns_not @ int @ ( bit_se4730199178511100633sh_bit @ int @ N @ ( one_one @ int ) ) ) ) ) ).
% int_set_bit_False_conv_NAND
thf(fact_6115_int__set__bit__conv__ops,axiom,
( ( generi7602027413899671122et_bit @ int )
= ( ^ [I4: int,N4: nat,B7: $o] : ( if @ int @ B7 @ ( bit_se1065995026697491101ons_or @ int @ I4 @ ( bit_se4730199178511100633sh_bit @ int @ N4 @ ( one_one @ int ) ) ) @ ( bit_se5824344872417868541ns_and @ int @ I4 @ ( bit_ri4277139882892585799ns_not @ int @ ( bit_se4730199178511100633sh_bit @ int @ N4 @ ( one_one @ int ) ) ) ) ) ) ) ).
% int_set_bit_conv_ops
thf(fact_6116_the__elem__def,axiom,
! [A: $tType] :
( ( the_elem @ A )
= ( ^ [X7: set @ A] :
( the @ A
@ ^ [X3: A] :
( X7
= ( insert @ A @ X3 @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ) ).
% the_elem_def
thf(fact_6117_upto__aux__rec,axiom,
( upto_aux
= ( ^ [I4: int,J3: int,Js: list @ int] : ( if @ ( list @ int ) @ ( ord_less @ int @ J3 @ I4 ) @ Js @ ( upto_aux @ I4 @ ( minus_minus @ int @ J3 @ ( one_one @ int ) ) @ ( cons @ int @ J3 @ Js ) ) ) ) ) ).
% upto_aux_rec
thf(fact_6118_word__lsb__neg__numeral,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [Bin: num] :
( ( least_8051144512741203767sb_lsb @ ( word @ A ) @ ( uminus_uminus @ ( word @ A ) @ ( numeral_numeral @ ( word @ A ) @ Bin ) ) )
= ( ~ ( dvd_dvd @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ Bin ) ) ) ) ) ) ).
% word_lsb_neg_numeral
thf(fact_6119_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_6120_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_6121_word__lsb__numeral,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [Bin: num] :
( ( least_8051144512741203767sb_lsb @ ( word @ A ) @ ( numeral_numeral @ ( word @ A ) @ Bin ) )
= ( ~ ( dvd_dvd @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ ( numeral_numeral @ int @ Bin ) ) ) ) ) ).
% word_lsb_numeral
thf(fact_6122_concat__bit__assoc,axiom,
! [N: nat,K: int,M: nat,L: int,R2: int] :
( ( bit_concat_bit @ N @ K @ ( bit_concat_bit @ M @ L @ R2 ) )
= ( bit_concat_bit @ ( plus_plus @ nat @ M @ N ) @ ( bit_concat_bit @ N @ K @ L ) @ R2 ) ) ).
% concat_bit_assoc
thf(fact_6123_word__lsb__1__0,axiom,
! [A: $tType,B: $tType] :
( ( ( type_len @ B )
& ( type_len @ A ) )
=> ( ( least_8051144512741203767sb_lsb @ ( word @ A ) @ ( one_one @ ( word @ A ) ) )
& ~ ( least_8051144512741203767sb_lsb @ ( word @ B ) @ ( zero_zero @ ( word @ B ) ) ) ) ) ).
% word_lsb_1_0
thf(fact_6124_concat__bit__eq,axiom,
( bit_concat_bit
= ( ^ [N4: nat,K3: int,L3: int] : ( plus_plus @ int @ ( bit_se2584673776208193580ke_bit @ int @ N4 @ K3 ) @ ( bit_se4730199178511100633sh_bit @ int @ N4 @ L3 ) ) ) ) ).
% concat_bit_eq
thf(fact_6125_lsb__word__eq,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ( ( least_8051144512741203767sb_lsb @ ( word @ A ) )
= ( ^ [A6: word @ A] :
~ ( dvd_dvd @ ( word @ A ) @ ( numeral_numeral @ ( word @ A ) @ ( bit0 @ one2 ) ) @ A6 ) ) ) ) ).
% lsb_word_eq
thf(fact_6126_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_6127_word__lsb__def,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ( ( least_8051144512741203767sb_lsb @ ( word @ A ) )
= ( ^ [A6: word @ A] :
~ ( dvd_dvd @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ ( semiring_1_unsigned @ A @ int @ A6 ) ) ) ) ) ).
% word_lsb_def
thf(fact_6128_lsb__odd,axiom,
! [A: $tType] :
( ( least_6119777620449941438nt_lsb @ A )
=> ( ( least_8051144512741203767sb_lsb @ A )
= ( ^ [A6: A] :
~ ( dvd_dvd @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ A6 ) ) ) ) ).
% lsb_odd
thf(fact_6129_word__lsb__nat,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ( ( least_8051144512741203767sb_lsb @ ( word @ A ) )
= ( ^ [W2: word @ A] :
( ( modulo_modulo @ nat @ ( semiring_1_unsigned @ A @ nat @ W2 ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) )
= ( one_one @ nat ) ) ) ) ) ).
% word_lsb_nat
thf(fact_6130_word__lsb__int,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ( ( least_8051144512741203767sb_lsb @ ( word @ A ) )
= ( ^ [W2: word @ A] :
( ( modulo_modulo @ int @ ( semiring_1_unsigned @ A @ int @ W2 ) @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) )
= ( one_one @ int ) ) ) ) ) ).
% word_lsb_int
thf(fact_6131_word__cat__split__alt,axiom,
! [C: $tType,B: $tType,A: $tType] :
( ( ( type_len @ A )
& ( type_len @ B )
& ( type_len @ C ) )
=> ! [W: word @ A,U: word @ B,V: word @ C] :
( ( ord_less_eq @ nat @ ( size_size @ ( word @ A ) @ W ) @ ( plus_plus @ nat @ ( size_size @ ( word @ B ) @ U ) @ ( size_size @ ( word @ C ) @ V ) ) )
=> ( ( ( word_split @ A @ B @ C @ W )
= ( product_Pair @ ( word @ B ) @ ( word @ C ) @ U @ V ) )
=> ( ( word_cat @ B @ C @ A @ U @ V )
= W ) ) ) ) ).
% word_cat_split_alt
thf(fact_6132_word__split__cat__alt,axiom,
! [A: $tType,B: $tType,C: $tType] :
( ( ( type_len @ C )
& ( type_len @ B )
& ( type_len @ A ) )
=> ! [W: word @ A,U: word @ B,V: word @ C] :
( ( W
= ( word_cat @ B @ C @ A @ U @ V ) )
=> ( ( ord_less_eq @ nat @ ( plus_plus @ nat @ ( size_size @ ( word @ B ) @ U ) @ ( size_size @ ( word @ C ) @ V ) ) @ ( size_size @ ( word @ A ) @ W ) )
=> ( ( word_split @ A @ B @ C @ W )
= ( product_Pair @ ( word @ B ) @ ( word @ C ) @ U @ V ) ) ) ) ) ).
% word_split_cat_alt
thf(fact_6133_int__lsb__numeral_I2_J,axiom,
least_8051144512741203767sb_lsb @ int @ ( one_one @ int ) ).
% int_lsb_numeral(2)
thf(fact_6134_int__lsb__numeral_I6_J,axiom,
! [W: num] :
~ ( least_8051144512741203767sb_lsb @ int @ ( numeral_numeral @ int @ ( bit0 @ W ) ) ) ).
% int_lsb_numeral(6)
thf(fact_6135_int__lsb__numeral_I3_J,axiom,
least_8051144512741203767sb_lsb @ int @ ( numeral_numeral @ int @ one2 ) ).
% int_lsb_numeral(3)
thf(fact_6136_int__lsb__numeral_I4_J,axiom,
least_8051144512741203767sb_lsb @ int @ ( uminus_uminus @ int @ ( one_one @ int ) ) ).
% int_lsb_numeral(4)
thf(fact_6137_int__lsb__numeral_I8_J,axiom,
! [W: num] :
~ ( least_8051144512741203767sb_lsb @ int @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ ( bit0 @ W ) ) ) ) ).
% int_lsb_numeral(8)
thf(fact_6138_int__lsb__numeral_I5_J,axiom,
least_8051144512741203767sb_lsb @ int @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ one2 ) ) ).
% int_lsb_numeral(5)
thf(fact_6139_bin__last__conv__lsb,axiom,
( ( ^ [A6: int] :
~ ( dvd_dvd @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ A6 ) )
= ( least_8051144512741203767sb_lsb @ int ) ) ).
% bin_last_conv_lsb
thf(fact_6140_test__bit__cat,axiom,
! [A: $tType,B: $tType,C: $tType] :
( ( ( type_len @ C )
& ( type_len @ B )
& ( type_len @ A ) )
=> ! [A3: word @ B,B3: word @ C,N: nat] :
( ( bit_se5641148757651400278ts_bit @ ( word @ A ) @ ( word_cat @ B @ C @ A @ A3 @ B3 ) @ N )
= ( ( ord_less @ nat @ N @ ( size_size @ ( word @ A ) @ ( word_cat @ B @ C @ A @ A3 @ B3 ) ) )
& ( ( ord_less @ nat @ N @ ( size_size @ ( word @ C ) @ B3 ) )
=> ( bit_se5641148757651400278ts_bit @ ( word @ C ) @ B3 @ N ) )
& ( ~ ( ord_less @ nat @ N @ ( size_size @ ( word @ C ) @ B3 ) )
=> ( bit_se5641148757651400278ts_bit @ ( word @ B ) @ A3 @ ( minus_minus @ nat @ N @ ( size_size @ ( word @ C ) @ B3 ) ) ) ) ) ) ) ).
% test_bit_cat
thf(fact_6141_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,V: word @ C] :
( ( ord_less_eq @ nat @ ( size_size @ ( word @ A ) @ T2 ) @ ( plus_plus @ nat @ ( size_size @ ( word @ B ) @ U ) @ ( size_size @ ( word @ C ) @ V ) ) )
=> ( ( ( product_Pair @ ( word @ B ) @ ( word @ C ) @ U @ V )
= ( word_split @ A @ B @ C @ T2 ) )
=> ( T2
= ( word_cat @ B @ C @ A @ U @ V ) ) ) ) ) ).
% word_cat_split_size
thf(fact_6142_cat__slices,axiom,
! [C: $tType,A: $tType,B: $tType] :
( ( ( type_len @ B )
& ( type_len @ A )
& ( type_len @ C ) )
=> ! [A3: word @ A,N: nat,C3: word @ B,B3: word @ C] :
( ( A3
= ( slice2 @ B @ A @ N @ C3 ) )
=> ( ( B3
= ( slice2 @ B @ C @ ( zero_zero @ nat ) @ C3 ) )
=> ( ( N
= ( size_size @ ( word @ C ) @ B3 ) )
=> ( ( ord_less_eq @ nat @ ( size_size @ ( word @ B ) @ C3 ) @ ( plus_plus @ nat @ ( size_size @ ( word @ A ) @ A3 ) @ ( size_size @ ( word @ C ) @ B3 ) ) )
=> ( ( word_cat @ A @ C @ B @ A3 @ B3 )
= C3 ) ) ) ) ) ) ).
% cat_slices
thf(fact_6143_sum__diff1_H__aux,axiom,
! [B: $tType,A: $tType] :
( ( ab_group_add @ B )
=> ! [F4: set @ A,I6: set @ A,F2: A > B,I: A] :
( ( finite_finite2 @ A @ F4 )
=> ( ( ord_less_eq @ ( set @ A )
@ ( collect @ A
@ ^ [I4: A] :
( ( member @ A @ I4 @ I6 )
& ( ( F2 @ I4 )
!= ( zero_zero @ B ) ) ) )
@ F4 )
=> ( ( ( member @ A @ I @ I6 )
=> ( ( groups1027152243600224163dd_sum @ A @ B @ F2 @ ( minus_minus @ ( set @ A ) @ I6 @ ( insert @ A @ I @ ( bot_bot @ ( set @ A ) ) ) ) )
= ( minus_minus @ B @ ( groups1027152243600224163dd_sum @ A @ B @ F2 @ I6 ) @ ( F2 @ I ) ) ) )
& ( ~ ( member @ A @ I @ I6 )
=> ( ( groups1027152243600224163dd_sum @ A @ B @ F2 @ ( minus_minus @ ( set @ A ) @ I6 @ ( insert @ A @ I @ ( bot_bot @ ( set @ A ) ) ) ) )
= ( groups1027152243600224163dd_sum @ A @ B @ F2 @ I6 ) ) ) ) ) ) ) ).
% sum_diff1'_aux
thf(fact_6144_sum_Oinsert_H,axiom,
! [A: $tType,B: $tType] :
( ( comm_monoid_add @ A )
=> ! [I6: set @ B,P6: B > A,I: B] :
( ( finite_finite2 @ B
@ ( collect @ B
@ ^ [X3: B] :
( ( member @ B @ X3 @ I6 )
& ( ( P6 @ X3 )
!= ( zero_zero @ A ) ) ) ) )
=> ( ( ( member @ B @ I @ I6 )
=> ( ( groups1027152243600224163dd_sum @ B @ A @ P6 @ ( insert @ B @ I @ I6 ) )
= ( groups1027152243600224163dd_sum @ B @ A @ P6 @ I6 ) ) )
& ( ~ ( member @ B @ I @ I6 )
=> ( ( groups1027152243600224163dd_sum @ B @ A @ P6 @ ( insert @ B @ I @ I6 ) )
= ( plus_plus @ A @ ( P6 @ I ) @ ( groups1027152243600224163dd_sum @ B @ A @ P6 @ I6 ) ) ) ) ) ) ) ).
% sum.insert'
thf(fact_6145_sum_Onon__neutral_H,axiom,
! [A: $tType,B: $tType] :
( ( comm_monoid_add @ A )
=> ! [G: B > A,I6: set @ B] :
( ( groups1027152243600224163dd_sum @ B @ A @ G
@ ( collect @ B
@ ^ [X3: B] :
( ( member @ B @ X3 @ I6 )
& ( ( G @ X3 )
!= ( zero_zero @ A ) ) ) ) )
= ( groups1027152243600224163dd_sum @ B @ A @ G @ I6 ) ) ) ).
% sum.non_neutral'
thf(fact_6146_sum_Odistrib__triv_H,axiom,
! [A: $tType,B: $tType] :
( ( comm_monoid_add @ A )
=> ! [I6: set @ B,G: B > A,H2: B > A] :
( ( finite_finite2 @ B @ I6 )
=> ( ( groups1027152243600224163dd_sum @ B @ A
@ ^ [I4: B] : ( plus_plus @ A @ ( G @ I4 ) @ ( H2 @ I4 ) )
@ I6 )
= ( plus_plus @ A @ ( groups1027152243600224163dd_sum @ B @ A @ G @ I6 ) @ ( groups1027152243600224163dd_sum @ B @ A @ H2 @ I6 ) ) ) ) ) ).
% sum.distrib_triv'
thf(fact_6147_sum_Omono__neutral__left_H,axiom,
! [A: $tType,B: $tType] :
( ( comm_monoid_add @ A )
=> ! [S: set @ B,T6: set @ B,G: B > A] :
( ( ord_less_eq @ ( set @ B ) @ S @ T6 )
=> ( ! [X4: B] :
( ( member @ B @ X4 @ ( minus_minus @ ( set @ B ) @ T6 @ S ) )
=> ( ( G @ X4 )
= ( zero_zero @ A ) ) )
=> ( ( groups1027152243600224163dd_sum @ B @ A @ G @ S )
= ( groups1027152243600224163dd_sum @ B @ A @ G @ T6 ) ) ) ) ) ).
% sum.mono_neutral_left'
thf(fact_6148_sum_Omono__neutral__right_H,axiom,
! [A: $tType,B: $tType] :
( ( comm_monoid_add @ A )
=> ! [S: set @ B,T6: set @ B,G: B > A] :
( ( ord_less_eq @ ( set @ B ) @ S @ T6 )
=> ( ! [X4: B] :
( ( member @ B @ X4 @ ( minus_minus @ ( set @ B ) @ T6 @ S ) )
=> ( ( G @ X4 )
= ( zero_zero @ A ) ) )
=> ( ( groups1027152243600224163dd_sum @ B @ A @ G @ T6 )
= ( groups1027152243600224163dd_sum @ B @ A @ G @ S ) ) ) ) ) ).
% sum.mono_neutral_right'
thf(fact_6149_sum_Omono__neutral__cong__left_H,axiom,
! [A: $tType,B: $tType] :
( ( comm_monoid_add @ A )
=> ! [S: set @ B,T6: set @ B,H2: B > A,G: B > A] :
( ( ord_less_eq @ ( set @ B ) @ S @ T6 )
=> ( ! [I2: B] :
( ( member @ B @ I2 @ ( minus_minus @ ( set @ B ) @ T6 @ S ) )
=> ( ( H2 @ I2 )
= ( zero_zero @ A ) ) )
=> ( ! [X4: B] :
( ( member @ B @ X4 @ S )
=> ( ( G @ X4 )
= ( H2 @ X4 ) ) )
=> ( ( groups1027152243600224163dd_sum @ B @ A @ G @ S )
= ( groups1027152243600224163dd_sum @ B @ A @ H2 @ T6 ) ) ) ) ) ) ).
% sum.mono_neutral_cong_left'
thf(fact_6150_sum_Omono__neutral__cong__right_H,axiom,
! [A: $tType,B: $tType] :
( ( comm_monoid_add @ A )
=> ! [S: set @ B,T6: set @ B,G: B > A,H2: B > A] :
( ( ord_less_eq @ ( set @ B ) @ S @ T6 )
=> ( ! [X4: B] :
( ( member @ B @ X4 @ ( minus_minus @ ( set @ B ) @ T6 @ S ) )
=> ( ( G @ X4 )
= ( zero_zero @ A ) ) )
=> ( ! [X4: B] :
( ( member @ B @ X4 @ S )
=> ( ( G @ X4 )
= ( H2 @ X4 ) ) )
=> ( ( groups1027152243600224163dd_sum @ B @ A @ G @ T6 )
= ( groups1027152243600224163dd_sum @ B @ A @ H2 @ S ) ) ) ) ) ) ).
% sum.mono_neutral_cong_right'
thf(fact_6151_sum_Odistrib_H,axiom,
! [A: $tType,B: $tType] :
( ( comm_monoid_add @ A )
=> ! [I6: set @ B,G: B > A,H2: B > A] :
( ( finite_finite2 @ B
@ ( collect @ B
@ ^ [X3: B] :
( ( member @ B @ X3 @ I6 )
& ( ( G @ X3 )
!= ( zero_zero @ A ) ) ) ) )
=> ( ( finite_finite2 @ B
@ ( collect @ B
@ ^ [X3: B] :
( ( member @ B @ X3 @ I6 )
& ( ( H2 @ X3 )
!= ( zero_zero @ A ) ) ) ) )
=> ( ( groups1027152243600224163dd_sum @ B @ A
@ ^ [I4: B] : ( plus_plus @ A @ ( G @ I4 ) @ ( H2 @ I4 ) )
@ I6 )
= ( plus_plus @ A @ ( groups1027152243600224163dd_sum @ B @ A @ G @ I6 ) @ ( groups1027152243600224163dd_sum @ B @ A @ H2 @ I6 ) ) ) ) ) ) ).
% sum.distrib'
thf(fact_6152_sum_OG__def,axiom,
! [A: $tType,B: $tType] :
( ( comm_monoid_add @ A )
=> ( ( groups1027152243600224163dd_sum @ B @ A )
= ( ^ [P4: B > A,I9: set @ B] :
( if @ A
@ ( finite_finite2 @ B
@ ( collect @ B
@ ^ [X3: B] :
( ( member @ B @ X3 @ I9 )
& ( ( P4 @ X3 )
!= ( zero_zero @ A ) ) ) ) )
@ ( groups7311177749621191930dd_sum @ B @ A @ P4
@ ( collect @ B
@ ^ [X3: B] :
( ( member @ B @ X3 @ I9 )
& ( ( P4 @ X3 )
!= ( zero_zero @ A ) ) ) ) )
@ ( zero_zero @ A ) ) ) ) ) ).
% sum.G_def
thf(fact_6153_slice__cat1,axiom,
! [C: $tType,A: $tType,B: $tType] :
( ( ( type_len @ B )
& ( type_len @ A )
& ( type_len @ C ) )
=> ! [A3: word @ B,B3: word @ C] :
( ( ord_less_eq @ nat @ ( plus_plus @ nat @ ( size_size @ ( word @ B ) @ A3 ) @ ( size_size @ ( word @ C ) @ B3 ) ) @ ( size_size @ ( word @ A ) @ ( word_cat @ B @ C @ A @ A3 @ B3 ) ) )
=> ( ( slice2 @ A @ B @ ( size_size @ ( word @ C ) @ B3 ) @ ( word_cat @ B @ C @ A @ A3 @ B3 ) )
= A3 ) ) ) ).
% slice_cat1
thf(fact_6154_sum__diff1_H,axiom,
! [B: $tType,A: $tType] :
( ( ab_group_add @ B )
=> ! [I6: set @ A,F2: A > B,I: A] :
( ( finite_finite2 @ A
@ ( collect @ A
@ ^ [I4: A] :
( ( member @ A @ I4 @ I6 )
& ( ( F2 @ I4 )
!= ( zero_zero @ B ) ) ) ) )
=> ( ( ( member @ A @ I @ I6 )
=> ( ( groups1027152243600224163dd_sum @ A @ B @ F2 @ ( minus_minus @ ( set @ A ) @ I6 @ ( insert @ A @ I @ ( bot_bot @ ( set @ A ) ) ) ) )
= ( minus_minus @ B @ ( groups1027152243600224163dd_sum @ A @ B @ F2 @ I6 ) @ ( F2 @ I ) ) ) )
& ( ~ ( member @ A @ I @ I6 )
=> ( ( groups1027152243600224163dd_sum @ A @ B @ F2 @ ( minus_minus @ ( set @ A ) @ I6 @ ( insert @ A @ I @ ( bot_bot @ ( set @ A ) ) ) ) )
= ( groups1027152243600224163dd_sum @ A @ B @ F2 @ I6 ) ) ) ) ) ) ).
% sum_diff1'
thf(fact_6155_bit__horner__sum__bit__iff,axiom,
! [A: $tType] :
( ( bit_un5681908812861735899ations @ A )
=> ! [Bs: list @ $o,N: nat] :
( ( bit_se5641148757651400278ts_bit @ A @ ( groups4207007520872428315er_sum @ $o @ A @ ( zero_neq_one_of_bool @ A ) @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ Bs ) @ N )
= ( ( ord_less @ nat @ N @ ( size_size @ ( list @ $o ) @ Bs ) )
& ( nth @ $o @ Bs @ N ) ) ) ) ).
% bit_horner_sum_bit_iff
thf(fact_6156_time__array__of__list,axiom,
! [A: $tType] :
( ( heap @ A )
=> ! [Xs: list @ A,H2: heap_ext @ product_unit] :
( ( time_time @ ( array @ A ) @ ( array_of_list @ A @ Xs ) @ H2 )
= ( plus_plus @ nat @ ( size_size @ ( list @ A ) @ Xs ) @ ( one_one @ nat ) ) ) ) ).
% time_array_of_list
thf(fact_6157_TBOUND__of__list,axiom,
! [A: $tType] :
( ( heap @ A )
=> ! [Xs: list @ A] : ( time_TBOUND @ ( array @ A ) @ ( array_of_list @ A @ Xs ) @ ( suc @ ( size_size @ ( list @ A ) @ Xs ) ) ) ) ).
% TBOUND_of_list
thf(fact_6158_of__list__rule,axiom,
! [A: $tType] :
( ( heap @ A )
=> ! [Xs: list @ A] :
( hoare_hoare_triple @ ( array @ A ) @ ( one_one @ assn ) @ ( array_of_list @ A @ Xs )
@ ^ [R5: array @ A] : ( snga_assn @ A @ R5 @ Xs ) ) ) ).
% of_list_rule
thf(fact_6159_horner__sum__of__bool__2__less,axiom,
! [Bs: list @ $o] : ( ord_less @ int @ ( groups4207007520872428315er_sum @ $o @ int @ ( zero_neq_one_of_bool @ int ) @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ Bs ) @ ( power_power @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ ( size_size @ ( list @ $o ) @ Bs ) ) ) ).
% horner_sum_of_bool_2_less
thf(fact_6160_horner__sum__simps_I2_J,axiom,
! [A: $tType,B: $tType] :
( ( comm_semiring_0 @ A )
=> ! [F2: B > A,A3: A,X2: B,Xs: list @ B] :
( ( groups4207007520872428315er_sum @ B @ A @ F2 @ A3 @ ( cons @ B @ X2 @ Xs ) )
= ( plus_plus @ A @ ( F2 @ X2 ) @ ( times_times @ A @ A3 @ ( groups4207007520872428315er_sum @ B @ A @ F2 @ A3 @ Xs ) ) ) ) ) ).
% horner_sum_simps(2)
thf(fact_6161_horner__sum__eq__sum,axiom,
! [A: $tType,B: $tType] :
( ( comm_semiring_1 @ A )
=> ( ( groups4207007520872428315er_sum @ B @ A )
= ( ^ [F5: B > A,A6: A,Xs2: list @ B] :
( groups7311177749621191930dd_sum @ nat @ A
@ ^ [N4: nat] : ( times_times @ A @ ( F5 @ ( nth @ B @ Xs2 @ N4 ) ) @ ( power_power @ A @ A6 @ N4 ) )
@ ( set_or7035219750837199246ssThan @ nat @ ( zero_zero @ nat ) @ ( size_size @ ( list @ B ) @ Xs2 ) ) ) ) ) ) ).
% horner_sum_eq_sum
thf(fact_6162_horner__sum__foldr,axiom,
! [A: $tType,B: $tType] :
( ( comm_semiring_0 @ A )
=> ( ( groups4207007520872428315er_sum @ B @ A )
= ( ^ [F5: B > A,A6: A,Xs2: list @ B] :
( foldr @ B @ A
@ ^ [X3: B,B7: A] : ( plus_plus @ A @ ( F5 @ X3 ) @ ( times_times @ A @ A6 @ B7 ) )
@ Xs2
@ ( zero_zero @ A ) ) ) ) ) ).
% horner_sum_foldr
thf(fact_6163_sdiv__word__min,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [A3: word @ A,B3: word @ A] : ( ord_less_eq @ int @ ( uminus_uminus @ int @ ( power_power @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ ( minus_minus @ nat @ ( size_size @ ( word @ A ) @ A3 ) @ ( one_one @ nat ) ) ) ) @ ( signed7115095781618012415divide @ int @ ( ring_1_signed @ A @ int @ A3 ) @ ( ring_1_signed @ A @ int @ B3 ) ) ) ) ).
% sdiv_word_min
thf(fact_6164_Cauchy__iff2,axiom,
( ( topolo3814608138187158403Cauchy @ real )
= ( ^ [X7: nat > real] :
! [J3: nat] :
? [M10: nat] :
! [M6: nat] :
( ( ord_less_eq @ nat @ M10 @ M6 )
=> ! [N4: nat] :
( ( ord_less_eq @ nat @ M10 @ N4 )
=> ( ord_less @ real @ ( abs_abs @ real @ ( minus_minus @ real @ ( X7 @ M6 ) @ ( X7 @ N4 ) ) ) @ ( inverse_inverse @ real @ ( semiring_1_of_nat @ real @ ( suc @ J3 ) ) ) ) ) ) ) ) ).
% Cauchy_iff2
thf(fact_6165_CauchyD,axiom,
! [A: $tType] :
( ( real_V822414075346904944vector @ A )
=> ! [X: nat > A,E: real] :
( ( topolo3814608138187158403Cauchy @ A @ X )
=> ( ( ord_less @ real @ ( zero_zero @ real ) @ E )
=> ? [M11: nat] :
! [M3: nat] :
( ( ord_less_eq @ nat @ M11 @ M3 )
=> ! [N9: nat] :
( ( ord_less_eq @ nat @ M11 @ N9 )
=> ( ord_less @ real @ ( real_V7770717601297561774m_norm @ A @ ( minus_minus @ A @ ( X @ M3 ) @ ( X @ N9 ) ) ) @ E ) ) ) ) ) ) ).
% CauchyD
thf(fact_6166_CauchyI,axiom,
! [A: $tType] :
( ( real_V822414075346904944vector @ A )
=> ! [X: nat > A] :
( ! [E2: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ E2 )
=> ? [M13: nat] :
! [M5: nat] :
( ( ord_less_eq @ nat @ M13 @ M5 )
=> ! [N3: nat] :
( ( ord_less_eq @ nat @ M13 @ N3 )
=> ( ord_less @ real @ ( real_V7770717601297561774m_norm @ A @ ( minus_minus @ A @ ( X @ M5 ) @ ( X @ N3 ) ) ) @ E2 ) ) ) )
=> ( topolo3814608138187158403Cauchy @ A @ X ) ) ) ).
% CauchyI
thf(fact_6167_Cauchy__iff,axiom,
! [A: $tType] :
( ( real_V822414075346904944vector @ A )
=> ( ( topolo3814608138187158403Cauchy @ A )
= ( ^ [X7: nat > A] :
! [E4: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ E4 )
=> ? [M10: nat] :
! [M6: nat] :
( ( ord_less_eq @ nat @ M10 @ M6 )
=> ! [N4: nat] :
( ( ord_less_eq @ nat @ M10 @ N4 )
=> ( ord_less @ real @ ( real_V7770717601297561774m_norm @ A @ ( minus_minus @ A @ ( X7 @ M6 ) @ ( X7 @ N4 ) ) ) @ E4 ) ) ) ) ) ) ) ).
% Cauchy_iff
thf(fact_6168_sdiv__word__max,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [A3: word @ A,B3: word @ A] : ( ord_less_eq @ int @ ( signed7115095781618012415divide @ int @ ( ring_1_signed @ A @ int @ A3 ) @ ( ring_1_signed @ A @ int @ B3 ) ) @ ( power_power @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ ( minus_minus @ nat @ ( size_size @ ( word @ A ) @ A3 ) @ ( suc @ ( zero_zero @ nat ) ) ) ) ) ) ).
% sdiv_word_max
thf(fact_6169_int__sdiv__negated__is__minus1,axiom,
! [A3: int,B3: int] :
( ( A3
!= ( zero_zero @ int ) )
=> ( ( ( signed7115095781618012415divide @ int @ A3 @ B3 )
= ( uminus_uminus @ int @ A3 ) )
= ( B3
= ( uminus_uminus @ int @ ( one_one @ int ) ) ) ) ) ).
% int_sdiv_negated_is_minus1
thf(fact_6170_int__sdiv__simps_I3_J,axiom,
! [A3: int] :
( ( signed7115095781618012415divide @ int @ A3 @ ( uminus_uminus @ int @ ( one_one @ int ) ) )
= ( uminus_uminus @ int @ A3 ) ) ).
% int_sdiv_simps(3)
thf(fact_6171_int__sdiv__simps_I1_J,axiom,
! [A3: int] :
( ( signed7115095781618012415divide @ int @ A3 @ ( one_one @ int ) )
= A3 ) ).
% int_sdiv_simps(1)
thf(fact_6172_int__sdiv__same__is__1,axiom,
! [A3: int,B3: int] :
( ( A3
!= ( zero_zero @ int ) )
=> ( ( ( signed7115095781618012415divide @ int @ A3 @ B3 )
= A3 )
= ( B3
= ( one_one @ int ) ) ) ) ).
% int_sdiv_same_is_1
thf(fact_6173_entails__solve__init_I1_J,axiom,
! [P: assn,Q: assn] :
( ( fI_QUERY @ P @ Q @ ( top_top @ assn ) )
=> ( entails @ P @ ( times_times @ assn @ Q @ ( top_top @ assn ) ) ) ) ).
% entails_solve_init(1)
thf(fact_6174_image__Collect__subsetI,axiom,
! [A: $tType,B: $tType,P: A > $o,F2: A > B,B2: set @ B] :
( ! [X4: A] :
( ( P @ X4 )
=> ( member @ B @ ( F2 @ X4 ) @ B2 ) )
=> ( ord_less_eq @ ( set @ B ) @ ( image @ A @ B @ F2 @ ( collect @ A @ P ) ) @ B2 ) ) ).
% image_Collect_subsetI
thf(fact_6175_ssubst__Pair__rhs,axiom,
! [B: $tType,A: $tType,R2: A,S2: B,R: set @ ( product_prod @ A @ B ),S4: B] :
( ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ R2 @ S2 ) @ R )
=> ( ( S4 = S2 )
=> ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ R2 @ S4 ) @ R ) ) ) ).
% ssubst_Pair_rhs
thf(fact_6176_Collect__restrict,axiom,
! [A: $tType,X: set @ A,P: A > $o] :
( ord_less_eq @ ( set @ A )
@ ( collect @ A
@ ^ [X3: A] :
( ( member @ A @ X3 @ X )
& ( P @ X3 ) ) )
@ X ) ).
% Collect_restrict
thf(fact_6177_prop__restrict,axiom,
! [A: $tType,X2: A,Z8: set @ A,X: set @ A,P: A > $o] :
( ( member @ A @ X2 @ Z8 )
=> ( ( ord_less_eq @ ( set @ A ) @ Z8
@ ( collect @ A
@ ^ [X3: A] :
( ( member @ A @ X3 @ X )
& ( P @ X3 ) ) ) )
=> ( P @ X2 ) ) ) ).
% prop_restrict
thf(fact_6178_frame__inference__init,axiom,
! [P: assn,Q: assn,F4: assn] :
( ( fI_QUERY @ P @ Q @ F4 )
=> ( entails @ P @ ( times_times @ assn @ Q @ F4 ) ) ) ).
% frame_inference_init
thf(fact_6179_FI__QUERY__def,axiom,
( fI_QUERY
= ( ^ [P3: assn,Q7: assn,F9: assn] : ( entails @ P3 @ ( times_times @ assn @ Q7 @ F9 ) ) ) ) ).
% FI_QUERY_def
thf(fact_6180_entails__solve__init_I2_J,axiom,
! [P: assn,Q: assn] :
( ( fI_QUERY @ P @ Q @ ( one_one @ assn ) )
=> ( entails @ P @ Q ) ) ).
% entails_solve_init(2)
thf(fact_6181_subset__emptyI,axiom,
! [A: $tType,A2: set @ A] :
( ! [X4: A] :
~ ( member @ A @ X4 @ A2 )
=> ( ord_less_eq @ ( set @ A ) @ A2 @ ( bot_bot @ ( set @ A ) ) ) ) ).
% subset_emptyI
thf(fact_6182_insert__subsetI,axiom,
! [A: $tType,X2: A,A2: set @ A,X: set @ A] :
( ( member @ A @ X2 @ A2 )
=> ( ( ord_less_eq @ ( set @ A ) @ X @ A2 )
=> ( ord_less_eq @ ( set @ A ) @ ( insert @ A @ X2 @ X ) @ A2 ) ) ) ).
% insert_subsetI
thf(fact_6183_length__product__lists,axiom,
! [B: $tType,Xss: list @ ( list @ B )] :
( ( size_size @ ( list @ ( list @ B ) ) @ ( product_lists @ B @ Xss ) )
= ( foldr @ nat @ nat @ ( times_times @ nat ) @ ( map @ ( list @ B ) @ nat @ ( size_size @ ( list @ B ) ) @ Xss ) @ ( one_one @ nat ) ) ) ).
% length_product_lists
thf(fact_6184_smod__int__range,axiom,
! [B3: int,A3: int] :
( ( B3
!= ( zero_zero @ int ) )
=> ( member @ int @ ( signed6721504322012087516modulo @ int @ A3 @ B3 ) @ ( set_or1337092689740270186AtMost @ int @ ( plus_plus @ int @ ( uminus_uminus @ int @ ( abs_abs @ int @ B3 ) ) @ ( one_one @ int ) ) @ ( minus_minus @ int @ ( abs_abs @ int @ B3 ) @ ( one_one @ int ) ) ) ) ) ).
% smod_int_range
thf(fact_6185_in__set__product__lists__length,axiom,
! [A: $tType,Xs: list @ A,Xss: list @ ( list @ A )] :
( ( member @ ( list @ A ) @ Xs @ ( set2 @ ( list @ A ) @ ( product_lists @ A @ Xss ) ) )
=> ( ( size_size @ ( list @ A ) @ Xs )
= ( size_size @ ( list @ ( list @ A ) ) @ Xss ) ) ) ).
% in_set_product_lists_length
thf(fact_6186_smod__int__compares_I1_J,axiom,
! [A3: int,B3: int] :
( ( ord_less_eq @ int @ ( zero_zero @ int ) @ A3 )
=> ( ( ord_less @ int @ ( zero_zero @ int ) @ B3 )
=> ( ord_less @ int @ ( signed6721504322012087516modulo @ int @ A3 @ B3 ) @ B3 ) ) ) ).
% smod_int_compares(1)
thf(fact_6187_smod__int__compares_I2_J,axiom,
! [A3: int,B3: int] :
( ( ord_less_eq @ int @ ( zero_zero @ int ) @ A3 )
=> ( ( ord_less @ int @ ( zero_zero @ int ) @ B3 )
=> ( ord_less_eq @ int @ ( zero_zero @ int ) @ ( signed6721504322012087516modulo @ int @ A3 @ B3 ) ) ) ) ).
% smod_int_compares(2)
thf(fact_6188_smod__int__compares_I4_J,axiom,
! [A3: int,B3: int] :
( ( ord_less_eq @ int @ A3 @ ( zero_zero @ int ) )
=> ( ( ord_less @ int @ ( zero_zero @ int ) @ B3 )
=> ( ord_less_eq @ int @ ( signed6721504322012087516modulo @ int @ A3 @ B3 ) @ ( zero_zero @ int ) ) ) ) ).
% smod_int_compares(4)
thf(fact_6189_smod__int__compares_I6_J,axiom,
! [A3: int,B3: int] :
( ( ord_less_eq @ int @ ( zero_zero @ int ) @ A3 )
=> ( ( ord_less @ int @ B3 @ ( zero_zero @ int ) )
=> ( ord_less_eq @ int @ ( zero_zero @ int ) @ ( signed6721504322012087516modulo @ int @ A3 @ B3 ) ) ) ) ).
% smod_int_compares(6)
thf(fact_6190_smod__int__compares_I7_J,axiom,
! [A3: int,B3: int] :
( ( ord_less_eq @ int @ A3 @ ( zero_zero @ int ) )
=> ( ( ord_less @ int @ B3 @ ( zero_zero @ int ) )
=> ( ord_less_eq @ int @ ( signed6721504322012087516modulo @ int @ A3 @ B3 ) @ ( zero_zero @ int ) ) ) ) ).
% smod_int_compares(7)
thf(fact_6191_smod__int__compares_I8_J,axiom,
! [A3: int,B3: int] :
( ( ord_less_eq @ int @ A3 @ ( zero_zero @ int ) )
=> ( ( ord_less @ int @ B3 @ ( zero_zero @ int ) )
=> ( ord_less_eq @ int @ B3 @ ( signed6721504322012087516modulo @ int @ A3 @ B3 ) ) ) ) ).
% smod_int_compares(8)
thf(fact_6192_smod__mod__positive,axiom,
! [A3: int,B3: int] :
( ( ord_less_eq @ int @ ( zero_zero @ int ) @ A3 )
=> ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ B3 )
=> ( ( signed6721504322012087516modulo @ int @ A3 @ B3 )
= ( modulo_modulo @ int @ A3 @ B3 ) ) ) ) ).
% smod_mod_positive
thf(fact_6193_signed__modulo__int__def,axiom,
( ( signed6721504322012087516modulo @ int )
= ( ^ [K3: int,L3: int] : ( minus_minus @ int @ K3 @ ( times_times @ int @ ( signed7115095781618012415divide @ int @ K3 @ L3 ) @ L3 ) ) ) ) ).
% signed_modulo_int_def
thf(fact_6194_smod__int__compares_I5_J,axiom,
! [A3: int,B3: int] :
( ( ord_less_eq @ int @ ( zero_zero @ int ) @ A3 )
=> ( ( ord_less @ int @ B3 @ ( zero_zero @ int ) )
=> ( ord_less @ int @ ( signed6721504322012087516modulo @ int @ A3 @ B3 ) @ ( uminus_uminus @ int @ B3 ) ) ) ) ).
% smod_int_compares(5)
thf(fact_6195_smod__int__compares_I3_J,axiom,
! [A3: int,B3: int] :
( ( ord_less_eq @ int @ A3 @ ( zero_zero @ int ) )
=> ( ( ord_less @ int @ ( zero_zero @ int ) @ B3 )
=> ( ord_less @ int @ ( uminus_uminus @ int @ B3 ) @ ( signed6721504322012087516modulo @ int @ A3 @ B3 ) ) ) ) ).
% smod_int_compares(3)
thf(fact_6196_length__subseqs,axiom,
! [A: $tType,Xs: list @ A] :
( ( size_size @ ( list @ ( list @ A ) ) @ ( subseqs @ A @ Xs ) )
= ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( size_size @ ( list @ A ) @ Xs ) ) ) ).
% length_subseqs
thf(fact_6197_length__mul__elem,axiom,
! [A: $tType,Xs: list @ ( list @ A ),N: nat] :
( ! [X4: list @ A] :
( ( member @ ( list @ A ) @ X4 @ ( set2 @ ( list @ A ) @ Xs ) )
=> ( ( size_size @ ( list @ A ) @ X4 )
= N ) )
=> ( ( size_size @ ( list @ A ) @ ( concat @ A @ Xs ) )
= ( times_times @ nat @ ( size_size @ ( list @ ( list @ A ) ) @ Xs ) @ N ) ) ) ).
% length_mul_elem
thf(fact_6198_map__concat,axiom,
! [A: $tType,B: $tType,F2: B > A,Xs: list @ ( list @ B )] :
( ( map @ B @ A @ F2 @ ( concat @ B @ Xs ) )
= ( concat @ A @ ( map @ ( list @ B ) @ ( list @ A ) @ ( map @ B @ A @ F2 ) @ Xs ) ) ) ).
% map_concat
thf(fact_6199_subseqs__refl,axiom,
! [A: $tType,Xs: list @ A] : ( member @ ( list @ A ) @ Xs @ ( set2 @ ( list @ A ) @ ( subseqs @ A @ Xs ) ) ) ).
% subseqs_refl
thf(fact_6200_Cons__in__subseqsD,axiom,
! [A: $tType,Y3: A,Ys: list @ A,Xs: list @ A] :
( ( member @ ( list @ A ) @ ( cons @ A @ Y3 @ Ys ) @ ( set2 @ ( list @ A ) @ ( subseqs @ A @ Xs ) ) )
=> ( member @ ( list @ A ) @ Ys @ ( set2 @ ( list @ A ) @ ( subseqs @ A @ Xs ) ) ) ) ).
% Cons_in_subseqsD
thf(fact_6201_product__lists_Osimps_I2_J,axiom,
! [A: $tType,Xs: list @ A,Xss: list @ ( list @ A )] :
( ( product_lists @ A @ ( cons @ ( list @ A ) @ Xs @ Xss ) )
= ( concat @ ( list @ A )
@ ( map @ A @ ( list @ ( list @ A ) )
@ ^ [X3: A] : ( map @ ( list @ A ) @ ( list @ A ) @ ( cons @ A @ X3 ) @ ( product_lists @ A @ Xss ) )
@ Xs ) ) ) ).
% product_lists.simps(2)
thf(fact_6202_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 ) )
@ ^ [X3: A] : ( map @ B @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X3 ) @ Ys3 )
@ Xs2 ) ) ) ) ).
% product_concat_map
thf(fact_6203_subset__subseqs,axiom,
! [A: $tType,X: set @ A,Xs: list @ A] :
( ( ord_less_eq @ ( set @ A ) @ X @ ( set2 @ A @ Xs ) )
=> ( member @ ( set @ A ) @ X @ ( image @ ( list @ A ) @ ( set @ A ) @ ( set2 @ A ) @ ( set2 @ ( list @ A ) @ ( subseqs @ A @ Xs ) ) ) ) ) ).
% subset_subseqs
thf(fact_6204_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_6205_mod__word__minus__1__minus__numeral,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [B3: num] :
( ( modulo_modulo @ ( word @ A ) @ ( uminus_uminus @ ( word @ A ) @ ( one_one @ ( word @ A ) ) ) @ ( uminus_uminus @ ( word @ A ) @ ( numeral_numeral @ ( word @ A ) @ B3 ) ) )
= ( ring_1_of_int @ ( word @ A ) @ ( modulo_modulo @ int @ ( minus_minus @ int @ ( power_power @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ ( type_len0_len_of @ A @ ( type2 @ A ) ) ) @ ( one_one @ int ) ) @ ( bit_se2584673776208193580ke_bit @ int @ ( type_len0_len_of @ A @ ( type2 @ A ) ) @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ B3 ) ) ) ) ) ) ) ).
% mod_word_minus_1_minus_numeral
thf(fact_6206_drop__bit__word__beyond,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [N: nat,W: word @ A] :
( ( ord_less_eq @ nat @ ( type_len0_len_of @ A @ ( type2 @ A ) ) @ N )
=> ( ( bit_se4197421643247451524op_bit @ ( word @ A ) @ N @ W )
= ( zero_zero @ ( word @ A ) ) ) ) ) ).
% drop_bit_word_beyond
thf(fact_6207_push__bit__word__beyond,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [N: nat,W: word @ A] :
( ( ord_less_eq @ nat @ ( type_len0_len_of @ A @ ( type2 @ A ) ) @ N )
=> ( ( bit_se4730199178511100633sh_bit @ ( word @ A ) @ N @ W )
= ( zero_zero @ ( word @ A ) ) ) ) ) ).
% push_bit_word_beyond
thf(fact_6208_signed__1,axiom,
! [B: $tType,A: $tType] :
( ( ( ring_1 @ A )
& ( type_len @ B ) )
=> ( ( ( ( type_len0_len_of @ B @ ( type2 @ B ) )
= ( one_one @ nat ) )
=> ( ( ring_1_signed @ B @ A @ ( one_one @ ( word @ B ) ) )
= ( uminus_uminus @ A @ ( one_one @ A ) ) ) )
& ( ( ( type_len0_len_of @ B @ ( type2 @ B ) )
!= ( one_one @ nat ) )
=> ( ( ring_1_signed @ B @ A @ ( one_one @ ( word @ B ) ) )
= ( one_one @ A ) ) ) ) ) ).
% signed_1
thf(fact_6209_word__exp__length__eq__0,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ( ( power_power @ ( word @ A ) @ ( numeral_numeral @ ( word @ A ) @ ( bit0 @ one2 ) ) @ ( type_len0_len_of @ A @ ( type2 @ A ) ) )
= ( zero_zero @ ( word @ A ) ) ) ) ).
% word_exp_length_eq_0
thf(fact_6210_less__eq__word__numeral__numeral,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [A3: num,B3: num] :
( ( ord_less_eq @ ( word @ A ) @ ( numeral_numeral @ ( word @ A ) @ A3 ) @ ( numeral_numeral @ ( word @ A ) @ B3 ) )
= ( ord_less_eq @ int @ ( bit_se2584673776208193580ke_bit @ int @ ( type_len0_len_of @ A @ ( type2 @ A ) ) @ ( numeral_numeral @ int @ A3 ) ) @ ( bit_se2584673776208193580ke_bit @ int @ ( type_len0_len_of @ A @ ( type2 @ A ) ) @ ( numeral_numeral @ int @ B3 ) ) ) ) ) ).
% less_eq_word_numeral_numeral
thf(fact_6211_bit__numeral__word__iff,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [W: num,N: nat] :
( ( bit_se5641148757651400278ts_bit @ ( word @ A ) @ ( numeral_numeral @ ( word @ A ) @ W ) @ N )
= ( ( ord_less @ nat @ N @ ( type_len0_len_of @ A @ ( type2 @ A ) ) )
& ( bit_se5641148757651400278ts_bit @ int @ ( numeral_numeral @ int @ W ) @ N ) ) ) ) ).
% bit_numeral_word_iff
thf(fact_6212_unsigned__numeral,axiom,
! [A: $tType,B: $tType] :
( ( ( type_len @ B )
& ( semiring_1 @ A ) )
=> ! [N: num] :
( ( semiring_1_unsigned @ B @ A @ ( numeral_numeral @ ( word @ B ) @ N ) )
= ( semiring_1_of_nat @ A @ ( bit_se2584673776208193580ke_bit @ nat @ ( type_len0_len_of @ B @ ( type2 @ B ) ) @ ( numeral_numeral @ nat @ N ) ) ) ) ) ).
% unsigned_numeral
thf(fact_6213_unat__lt2p,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [X2: word @ A] : ( ord_less @ nat @ ( semiring_1_unsigned @ A @ nat @ X2 ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( type_len0_len_of @ A @ ( type2 @ A ) ) ) ) ) ).
% unat_lt2p
thf(fact_6214_uint__lt2p,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [X2: word @ A] : ( ord_less @ int @ ( semiring_1_unsigned @ A @ int @ X2 ) @ ( power_power @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ ( type_len0_len_of @ A @ ( type2 @ A ) ) ) ) ) ).
% uint_lt2p
thf(fact_6215_uint__bounded,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [W: word @ A] : ( ord_less @ int @ ( semiring_1_unsigned @ A @ int @ W ) @ ( power_power @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ ( type_len0_len_of @ A @ ( type2 @ A ) ) ) ) ) ).
% uint_bounded
thf(fact_6216_of__nat__2p,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ( ( semiring_1_of_nat @ ( word @ A ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( type_len0_len_of @ A @ ( type2 @ A ) ) ) )
= ( zero_zero @ ( word @ A ) ) ) ) ).
% of_nat_2p
thf(fact_6217_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_6218_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_6219_signed__take__bit__word__numeral,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [N: num,K: num] :
( ( bit_ri4674362597316999326ke_bit @ ( word @ A ) @ ( numeral_numeral @ nat @ N ) @ ( numeral_numeral @ ( word @ A ) @ K ) )
= ( ring_1_of_int @ ( word @ A ) @ ( bit_ri4674362597316999326ke_bit @ int @ ( numeral_numeral @ nat @ N ) @ ( bit_se2584673776208193580ke_bit @ int @ ( type_len0_len_of @ A @ ( type2 @ A ) ) @ ( numeral_numeral @ int @ K ) ) ) ) ) ) ).
% signed_take_bit_word_numeral
thf(fact_6220_sint__sbintrunc,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [Bin: num] :
( ( ring_1_signed @ A @ int @ ( numeral_numeral @ ( word @ A ) @ Bin ) )
= ( bit_ri4674362597316999326ke_bit @ int @ ( minus_minus @ nat @ ( type_len0_len_of @ A @ ( type2 @ A ) ) @ ( one_one @ nat ) ) @ ( numeral_numeral @ int @ Bin ) ) ) ) ).
% sint_sbintrunc
thf(fact_6221_signed__numeral,axiom,
! [A: $tType,B: $tType] :
( ( ( type_len @ B )
& ( ring_1 @ A ) )
=> ! [N: num] :
( ( ring_1_signed @ B @ A @ ( numeral_numeral @ ( word @ B ) @ N ) )
= ( ring_1_of_int @ A @ ( bit_ri4674362597316999326ke_bit @ int @ ( minus_minus @ nat @ ( type_len0_len_of @ B @ ( type2 @ B ) ) @ ( one_one @ nat ) ) @ ( numeral_numeral @ int @ N ) ) ) ) ) ).
% signed_numeral
thf(fact_6222_less__eq__word__minus__numeral__minus__numeral,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [A3: num,B3: num] :
( ( ord_less_eq @ ( word @ A ) @ ( uminus_uminus @ ( word @ A ) @ ( numeral_numeral @ ( word @ A ) @ A3 ) ) @ ( uminus_uminus @ ( word @ A ) @ ( numeral_numeral @ ( word @ A ) @ B3 ) ) )
= ( ord_less_eq @ int @ ( bit_se2584673776208193580ke_bit @ int @ ( type_len0_len_of @ A @ ( type2 @ A ) ) @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ A3 ) ) ) @ ( bit_se2584673776208193580ke_bit @ int @ ( type_len0_len_of @ A @ ( type2 @ A ) ) @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ B3 ) ) ) ) ) ) ).
% less_eq_word_minus_numeral_minus_numeral
thf(fact_6223_less__eq__word__numeral__minus__numeral,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [A3: num,B3: num] :
( ( ord_less_eq @ ( word @ A ) @ ( numeral_numeral @ ( word @ A ) @ A3 ) @ ( uminus_uminus @ ( word @ A ) @ ( numeral_numeral @ ( word @ A ) @ B3 ) ) )
= ( ord_less_eq @ int @ ( bit_se2584673776208193580ke_bit @ int @ ( type_len0_len_of @ A @ ( type2 @ A ) ) @ ( numeral_numeral @ int @ A3 ) ) @ ( bit_se2584673776208193580ke_bit @ int @ ( type_len0_len_of @ A @ ( type2 @ A ) ) @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ B3 ) ) ) ) ) ) ).
% less_eq_word_numeral_minus_numeral
thf(fact_6224_less__eq__word__minus__numeral__numeral,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [A3: num,B3: num] :
( ( ord_less_eq @ ( word @ A ) @ ( uminus_uminus @ ( word @ A ) @ ( numeral_numeral @ ( word @ A ) @ A3 ) ) @ ( numeral_numeral @ ( word @ A ) @ B3 ) )
= ( ord_less_eq @ int @ ( bit_se2584673776208193580ke_bit @ int @ ( type_len0_len_of @ A @ ( type2 @ A ) ) @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ A3 ) ) ) @ ( bit_se2584673776208193580ke_bit @ int @ ( type_len0_len_of @ A @ ( type2 @ A ) ) @ ( numeral_numeral @ int @ B3 ) ) ) ) ) ).
% less_eq_word_minus_numeral_numeral
thf(fact_6225_bit__neg__numeral__word__iff,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [W: num,N: nat] :
( ( bit_se5641148757651400278ts_bit @ ( word @ A ) @ ( uminus_uminus @ ( word @ A ) @ ( numeral_numeral @ ( word @ A ) @ W ) ) @ N )
= ( ( ord_less @ nat @ N @ ( type_len0_len_of @ A @ ( type2 @ A ) ) )
& ( bit_se5641148757651400278ts_bit @ int @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ W ) ) @ N ) ) ) ) ).
% bit_neg_numeral_word_iff
thf(fact_6226_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_6227_drop__bit__word__numeral,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [N: num,K: num] :
( ( bit_se4197421643247451524op_bit @ ( word @ A ) @ ( numeral_numeral @ nat @ N ) @ ( numeral_numeral @ ( word @ A ) @ K ) )
= ( ring_1_of_int @ ( word @ A ) @ ( bit_se4197421643247451524op_bit @ int @ ( numeral_numeral @ nat @ N ) @ ( bit_se2584673776208193580ke_bit @ int @ ( type_len0_len_of @ A @ ( type2 @ A ) ) @ ( numeral_numeral @ int @ K ) ) ) ) ) ) ).
% drop_bit_word_numeral
thf(fact_6228_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_6229_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_6230_signed__take__bit__word__minus__numeral,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [N: num,K: num] :
( ( bit_ri4674362597316999326ke_bit @ ( word @ A ) @ ( numeral_numeral @ nat @ N ) @ ( uminus_uminus @ ( word @ A ) @ ( numeral_numeral @ ( word @ A ) @ K ) ) )
= ( ring_1_of_int @ ( word @ A ) @ ( bit_ri4674362597316999326ke_bit @ int @ ( numeral_numeral @ nat @ N ) @ ( bit_se2584673776208193580ke_bit @ int @ ( type_len0_len_of @ A @ ( type2 @ A ) ) @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ K ) ) ) ) ) ) ) ).
% signed_take_bit_word_minus_numeral
thf(fact_6231_sint__sbintrunc__neg,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [Bin: num] :
( ( ring_1_signed @ A @ int @ ( uminus_uminus @ ( word @ A ) @ ( numeral_numeral @ ( word @ A ) @ Bin ) ) )
= ( bit_ri4674362597316999326ke_bit @ int @ ( minus_minus @ nat @ ( type_len0_len_of @ A @ ( type2 @ A ) ) @ ( one_one @ nat ) ) @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ Bin ) ) ) ) ) ).
% sint_sbintrunc_neg
thf(fact_6232_scast__sbintr,axiom,
! [B: $tType,A: $tType] :
( ( ( type_len @ A )
& ( type_len @ B ) )
=> ! [W: num] :
( ( ring_1_signed @ A @ ( word @ B ) @ ( numeral_numeral @ ( word @ A ) @ W ) )
= ( 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 @ W ) ) ) ) ) ).
% scast_sbintr
thf(fact_6233_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_6234_word__less__sub__le,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [N: nat,X2: word @ A] :
( ( ord_less @ nat @ N @ ( type_len0_len_of @ A @ ( type2 @ A ) ) )
=> ( ( ord_less_eq @ ( word @ A ) @ X2 @ ( minus_minus @ ( word @ A ) @ ( power_power @ ( word @ A ) @ ( numeral_numeral @ ( word @ A ) @ ( bit0 @ one2 ) ) @ N ) @ ( one_one @ ( word @ A ) ) ) )
= ( ord_less @ ( word @ A ) @ X2 @ ( power_power @ ( word @ A ) @ ( numeral_numeral @ ( word @ A ) @ ( bit0 @ one2 ) ) @ N ) ) ) ) ) ).
% word_less_sub_le
thf(fact_6235_signed__neg__numeral,axiom,
! [A: $tType,B: $tType] :
( ( ( type_len @ B )
& ( ring_1 @ A ) )
=> ! [N: num] :
( ( ring_1_signed @ B @ A @ ( uminus_uminus @ ( word @ B ) @ ( numeral_numeral @ ( word @ B ) @ N ) ) )
= ( ring_1_of_int @ A @ ( bit_ri4674362597316999326ke_bit @ int @ ( minus_minus @ nat @ ( type_len0_len_of @ B @ ( type2 @ B ) ) @ ( one_one @ nat ) ) @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ N ) ) ) ) ) ) ).
% signed_neg_numeral
thf(fact_6236_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_6237_drop__bit__word__minus__numeral,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [N: num,K: num] :
( ( bit_se4197421643247451524op_bit @ ( word @ A ) @ ( numeral_numeral @ nat @ N ) @ ( uminus_uminus @ ( word @ A ) @ ( numeral_numeral @ ( word @ A ) @ K ) ) )
= ( ring_1_of_int @ ( word @ A ) @ ( bit_se4197421643247451524op_bit @ int @ ( numeral_numeral @ nat @ N ) @ ( bit_se2584673776208193580ke_bit @ int @ ( type_len0_len_of @ A @ ( type2 @ A ) ) @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ K ) ) ) ) ) ) ) ).
% drop_bit_word_minus_numeral
thf(fact_6238_less__word__numeral__minus__1,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [A3: num] :
( ( ord_less @ ( word @ A ) @ ( numeral_numeral @ ( word @ A ) @ A3 ) @ ( 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 @ A3 ) ) @ ( 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_6239_less__word__minus__numeral__minus__1,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [A3: num] :
( ( ord_less @ ( word @ A ) @ ( uminus_uminus @ ( word @ A ) @ ( numeral_numeral @ ( word @ A ) @ A3 ) ) @ ( 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 @ A3 ) ) ) @ ( 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_6240_div__word__minus__1__numeral,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [B3: num] :
( ( divide_divide @ ( word @ A ) @ ( uminus_uminus @ ( word @ A ) @ ( one_one @ ( word @ A ) ) ) @ ( numeral_numeral @ ( word @ A ) @ B3 ) )
= ( ring_1_of_int @ ( word @ A ) @ ( divide_divide @ int @ ( minus_minus @ int @ ( power_power @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ ( type_len0_len_of @ A @ ( type2 @ A ) ) ) @ ( one_one @ int ) ) @ ( bit_se2584673776208193580ke_bit @ int @ ( type_len0_len_of @ A @ ( type2 @ A ) ) @ ( numeral_numeral @ int @ B3 ) ) ) ) ) ) ).
% div_word_minus_1_numeral
thf(fact_6241_mod__word__minus__1__numeral,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [B3: num] :
( ( modulo_modulo @ ( word @ A ) @ ( uminus_uminus @ ( word @ A ) @ ( one_one @ ( word @ A ) ) ) @ ( numeral_numeral @ ( word @ A ) @ B3 ) )
= ( ring_1_of_int @ ( word @ A ) @ ( modulo_modulo @ int @ ( minus_minus @ int @ ( power_power @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ ( type_len0_len_of @ A @ ( type2 @ A ) ) ) @ ( one_one @ int ) ) @ ( bit_se2584673776208193580ke_bit @ int @ ( type_len0_len_of @ A @ ( type2 @ A ) ) @ ( numeral_numeral @ int @ B3 ) ) ) ) ) ) ).
% mod_word_minus_1_numeral
thf(fact_6242_div__word__minus__1__minus__numeral,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [B3: num] :
( ( divide_divide @ ( word @ A ) @ ( uminus_uminus @ ( word @ A ) @ ( one_one @ ( word @ A ) ) ) @ ( uminus_uminus @ ( word @ A ) @ ( numeral_numeral @ ( word @ A ) @ B3 ) ) )
= ( ring_1_of_int @ ( word @ A ) @ ( divide_divide @ int @ ( minus_minus @ int @ ( power_power @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ ( type_len0_len_of @ A @ ( type2 @ A ) ) ) @ ( one_one @ int ) ) @ ( bit_se2584673776208193580ke_bit @ int @ ( type_len0_len_of @ A @ ( type2 @ A ) ) @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ B3 ) ) ) ) ) ) ) ).
% div_word_minus_1_minus_numeral
thf(fact_6243_word__cat__bin_H,axiom,
! [B: $tType,C: $tType,A: $tType] :
( ( ( type_len @ A )
& ( type_len @ C )
& ( type_len @ B ) )
=> ( ( word_cat @ A @ B @ C )
= ( ^ [V5: word @ A,W2: word @ B] : ( plus_plus @ ( word @ C ) @ ( bit_se4730199178511100633sh_bit @ ( word @ C ) @ ( type_len0_len_of @ B @ ( type2 @ B ) ) @ ( semiring_1_unsigned @ A @ ( word @ C ) @ V5 ) ) @ ( semiring_1_unsigned @ B @ ( word @ C ) @ W2 ) ) ) ) ) ).
% word_cat_bin'
thf(fact_6244_bit__word__cat__iff,axiom,
! [C: $tType,A: $tType,B: $tType] :
( ( ( type_len @ B )
& ( type_len @ A )
& ( type_len @ C ) )
=> ! [V: word @ A,W: word @ B,N: nat] :
( ( bit_se5641148757651400278ts_bit @ ( word @ C ) @ ( word_cat @ A @ B @ C @ V @ W ) @ 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 ) @ W @ N ) )
& ( ~ ( ord_less @ nat @ N @ ( type_len0_len_of @ B @ ( type2 @ B ) ) )
=> ( bit_se5641148757651400278ts_bit @ ( word @ A ) @ V @ ( minus_minus @ nat @ N @ ( type_len0_len_of @ B @ ( type2 @ B ) ) ) ) ) ) ) ) ).
% bit_word_cat_iff
thf(fact_6245_word__cat__inj,axiom,
! [C: $tType,A: $tType,B: $tType] :
( ( ( type_len @ B )
& ( type_len @ A )
& ( type_len @ C ) )
=> ! [A3: word @ A,B3: 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 @ A3 @ B3 )
= ( word_cat @ A @ B @ C @ C3 @ D2 ) )
= ( ( A3 = C3 )
& ( B3 = D2 ) ) ) ) ) ).
% word_cat_inj
thf(fact_6246_bit__set__bit__word__iff,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [W: word @ A,M: nat,B3: $o,N: nat] :
( ( bit_se5641148757651400278ts_bit @ ( word @ A ) @ ( generi7602027413899671122et_bit @ ( word @ A ) @ W @ M @ B3 ) @ N )
= ( ( ( M = N )
=> ( ( ord_less @ nat @ N @ ( type_len0_len_of @ A @ ( type2 @ A ) ) )
& B3 ) )
& ( ( M != N )
=> ( bit_se5641148757651400278ts_bit @ ( word @ A ) @ W @ N ) ) ) ) ) ).
% bit_set_bit_word_iff
thf(fact_6247_ucast__drop__bit__eq,axiom,
! [B: $tType,A: $tType] :
( ( ( type_len @ A )
& ( type_len @ B ) )
=> ! [N: nat,W: 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 @ W ) )
= ( bit_se4197421643247451524op_bit @ ( word @ B ) @ N @ ( semiring_1_unsigned @ A @ ( word @ B ) @ W ) ) ) ) ) ).
% ucast_drop_bit_eq
thf(fact_6248_bit__set__bit__aux,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [F2: nat > $o,N: nat,W: word @ A,M: nat] :
( ( bit_se5641148757651400278ts_bit @ ( word @ A ) @ ( code_T2661198915054445665ts_aux @ A @ F2 @ N @ W ) @ M )
= ( ( ord_less @ nat @ M @ ( type_len0_len_of @ A @ ( type2 @ A ) ) )
& ( ( ord_less @ nat @ M @ N )
=> ( F2 @ M ) )
& ( ~ ( ord_less @ nat @ M @ N )
=> ( bit_se5641148757651400278ts_bit @ ( word @ A ) @ W @ ( minus_minus @ nat @ M @ N ) ) ) ) ) ) ).
% bit_set_bit_aux
thf(fact_6249_ucast__mask__drop,axiom,
! [A: $tType,B: $tType] :
( ( ( type_len @ B )
& ( type_len @ A ) )
=> ! [N: nat,X2: 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 ) @ X2 @ ( bit_se2239418461657761734s_mask @ ( word @ B ) @ N ) ) )
= ( semiring_1_unsigned @ B @ ( word @ A ) @ X2 ) ) ) ) ).
% ucast_mask_drop
thf(fact_6250_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_6251_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_6252_ucast__up__mono,axiom,
! [B: $tType,A: $tType] :
( ( ( type_len @ A )
& ( type_len @ B ) )
=> ! [X2: word @ A,Y3: word @ A] :
( ( ord_less_eq @ nat @ ( type_len0_len_of @ A @ ( type2 @ A ) ) @ ( type_len0_len_of @ B @ ( type2 @ B ) ) )
=> ( ( ord_less @ ( word @ A ) @ X2 @ Y3 )
=> ( ord_less @ ( word @ B ) @ ( semiring_1_unsigned @ A @ ( word @ B ) @ X2 ) @ ( semiring_1_unsigned @ A @ ( word @ B ) @ Y3 ) ) ) ) ) ).
% ucast_up_mono
thf(fact_6253_ucast__less__ucast,axiom,
! [B: $tType,A: $tType] :
( ( ( type_len @ A )
& ( type_len @ B ) )
=> ! [X2: word @ A,Y3: 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 ) @ X2 ) @ ( semiring_1_unsigned @ A @ ( word @ B ) @ Y3 ) )
= ( ord_less @ ( word @ A ) @ X2 @ Y3 ) ) ) ) ).
% ucast_less_ucast
thf(fact_6254_less__ucast__ucast__less,axiom,
! [A: $tType,B: $tType] :
( ( ( type_len @ B )
& ( type_len @ A ) )
=> ! [X2: word @ A,Y3: word @ B] :
( ( ord_less_eq @ nat @ ( type_len0_len_of @ B @ ( type2 @ B ) ) @ ( type_len0_len_of @ A @ ( type2 @ A ) ) )
=> ( ( ord_less @ ( word @ A ) @ X2 @ ( semiring_1_unsigned @ B @ ( word @ A ) @ Y3 ) )
=> ( ord_less @ ( word @ B ) @ ( semiring_1_unsigned @ A @ ( word @ B ) @ X2 ) @ Y3 ) ) ) ) ).
% less_ucast_ucast_less
thf(fact_6255_up__ucast__inj,axiom,
! [B: $tType,A: $tType] :
( ( ( type_len @ A )
& ( type_len @ B ) )
=> ! [X2: word @ A,Y3: word @ A] :
( ( ( semiring_1_unsigned @ A @ ( word @ B ) @ X2 )
= ( semiring_1_unsigned @ A @ ( word @ B ) @ Y3 ) )
=> ( ( ord_less_eq @ nat @ ( type_len0_len_of @ A @ ( type2 @ A ) ) @ ( type_len0_len_of @ B @ ( type2 @ B ) ) )
=> ( X2 = Y3 ) ) ) ) ).
% up_ucast_inj
thf(fact_6256_ucast__le__ucast,axiom,
! [B: $tType,A: $tType] :
( ( ( type_len @ A )
& ( type_len @ B ) )
=> ! [X2: word @ A,Y3: 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 ) @ X2 ) @ ( semiring_1_unsigned @ A @ ( word @ B ) @ Y3 ) )
= ( ord_less_eq @ ( word @ A ) @ X2 @ Y3 ) ) ) ) ).
% ucast_le_ucast
thf(fact_6257_ucast__ucast__eq,axiom,
! [C: $tType,A: $tType,B: $tType] :
( ( ( type_len @ B )
& ( type_len @ A )
& ( type_len @ C ) )
=> ! [X2: word @ A,Y3: word @ B] :
( ( ( semiring_1_unsigned @ A @ ( word @ C ) @ X2 )
= ( semiring_1_unsigned @ A @ ( word @ C ) @ ( semiring_1_unsigned @ B @ ( word @ A ) @ Y3 ) ) )
=> ( ( 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 ) ) )
=> ( X2
= ( semiring_1_unsigned @ B @ ( word @ A ) @ Y3 ) ) ) ) ) ) ).
% ucast_ucast_eq
thf(fact_6258_up__ucast__inj__eq,axiom,
! [B: $tType,A: $tType] :
( ( ( type_len @ A )
& ( type_len @ B ) )
=> ! [X2: word @ A,Y3: 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 ) @ X2 )
= ( semiring_1_unsigned @ A @ ( word @ B ) @ Y3 ) )
= ( X2 = Y3 ) ) ) ) ).
% up_ucast_inj_eq
thf(fact_6259_ucast__up__mono__le,axiom,
! [B: $tType,A: $tType] :
( ( ( type_len @ A )
& ( type_len @ B ) )
=> ! [X2: word @ A,Y3: 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 ) @ X2 @ Y3 )
=> ( ord_less_eq @ ( word @ B ) @ ( semiring_1_unsigned @ A @ ( word @ B ) @ X2 ) @ ( semiring_1_unsigned @ A @ ( word @ B ) @ Y3 ) ) ) ) ) ).
% ucast_up_mono_le
thf(fact_6260_eq__ucast__ucast__eq,axiom,
! [A: $tType,B: $tType] :
( ( ( type_len @ B )
& ( type_len @ A ) )
=> ! [X2: word @ A,Y3: word @ B] :
( ( ord_less_eq @ nat @ ( type_len0_len_of @ B @ ( type2 @ B ) ) @ ( type_len0_len_of @ A @ ( type2 @ A ) ) )
=> ( ( X2
= ( semiring_1_unsigned @ B @ ( word @ A ) @ Y3 ) )
=> ( ( semiring_1_unsigned @ A @ ( word @ B ) @ X2 )
= Y3 ) ) ) ) ).
% eq_ucast_ucast_eq
thf(fact_6261_unat__ucast__up__simp,axiom,
! [B: $tType,A: $tType] :
( ( ( type_len @ A )
& ( type_len @ B ) )
=> ! [X2: 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 ) @ X2 ) )
= ( semiring_1_unsigned @ A @ nat @ X2 ) ) ) ) ).
% unat_ucast_up_simp
thf(fact_6262_bintr__uint,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [N: nat,W: word @ A] :
( ( ord_less_eq @ nat @ ( type_len0_len_of @ A @ ( type2 @ A ) ) @ N )
=> ( ( bit_se2584673776208193580ke_bit @ int @ N @ ( semiring_1_unsigned @ A @ int @ W ) )
= ( semiring_1_unsigned @ A @ int @ W ) ) ) ) ).
% bintr_uint
thf(fact_6263_wi__bintr,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [N: nat,W: int] :
( ( ord_less_eq @ nat @ ( type_len0_len_of @ A @ ( type2 @ A ) ) @ N )
=> ( ( ring_1_of_int @ ( word @ A ) @ ( bit_se2584673776208193580ke_bit @ int @ N @ W ) )
= ( ring_1_of_int @ ( word @ A ) @ W ) ) ) ) ).
% wi_bintr
thf(fact_6264_take__bit__word__beyond__length__eq,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [N: nat,W: word @ A] :
( ( ord_less_eq @ nat @ ( type_len0_len_of @ A @ ( type2 @ A ) ) @ N )
=> ( ( bit_se2584673776208193580ke_bit @ ( word @ A ) @ N @ W )
= W ) ) ) ).
% take_bit_word_beyond_length_eq
thf(fact_6265_up__scast__inj__eq,axiom,
! [B: $tType,A: $tType] :
( ( ( type_len @ A )
& ( type_len @ B ) )
=> ! [X2: word @ A,Y3: 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 ) @ X2 )
= ( ring_1_signed @ A @ ( word @ B ) @ Y3 ) )
= ( X2 = Y3 ) ) ) ) ).
% up_scast_inj_eq
thf(fact_6266_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_6267_ucast__sub__ucast,axiom,
! [B: $tType,A: $tType] :
( ( ( type_len @ A )
& ( type_len @ B ) )
=> ! [Y3: word @ A,X2: word @ A] :
( ( ord_less_eq @ ( word @ A ) @ Y3 @ X2 )
=> ( ( 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 ) @ X2 @ Y3 ) )
= ( minus_minus @ ( word @ B ) @ ( semiring_1_unsigned @ A @ ( word @ B ) @ X2 ) @ ( semiring_1_unsigned @ A @ ( word @ B ) @ Y3 ) ) ) ) ) ) ).
% ucast_sub_ucast
thf(fact_6268_ucast__ucast__add,axiom,
! [A: $tType,B: $tType] :
( ( ( type_len @ B )
& ( type_len @ A ) )
=> ! [X2: word @ A,Y3: 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 ) @ X2 ) @ Y3 ) )
= ( plus_plus @ ( word @ A ) @ X2 @ ( semiring_1_unsigned @ B @ ( word @ A ) @ Y3 ) ) ) ) ) ).
% ucast_ucast_add
thf(fact_6269_uint__word__arith__bintrs_I1_J,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [A3: word @ A,B3: word @ A] :
( ( semiring_1_unsigned @ A @ int @ ( plus_plus @ ( word @ A ) @ A3 @ B3 ) )
= ( bit_se2584673776208193580ke_bit @ int @ ( type_len0_len_of @ A @ ( type2 @ A ) ) @ ( plus_plus @ int @ ( semiring_1_unsigned @ A @ int @ A3 ) @ ( semiring_1_unsigned @ A @ int @ B3 ) ) ) ) ) ).
% uint_word_arith_bintrs(1)
thf(fact_6270_uint__word__arith__bintrs_I2_J,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [A3: word @ A,B3: word @ A] :
( ( semiring_1_unsigned @ A @ int @ ( minus_minus @ ( word @ A ) @ A3 @ B3 ) )
= ( bit_se2584673776208193580ke_bit @ int @ ( type_len0_len_of @ A @ ( type2 @ A ) ) @ ( minus_minus @ int @ ( semiring_1_unsigned @ A @ int @ A3 ) @ ( semiring_1_unsigned @ A @ int @ B3 ) ) ) ) ) ).
% uint_word_arith_bintrs(2)
thf(fact_6271_scast__1,axiom,
! [A: $tType,B: $tType] :
( ( ( type_len @ B )
& ( type_len @ A ) )
=> ( ( ( ( type_len0_len_of @ A @ ( type2 @ A ) )
= ( one_one @ nat ) )
=> ( ( ring_1_signed @ A @ ( word @ B ) @ ( one_one @ ( word @ A ) ) )
= ( uminus_uminus @ ( word @ B ) @ ( one_one @ ( word @ B ) ) ) ) )
& ( ( ( type_len0_len_of @ A @ ( type2 @ A ) )
!= ( one_one @ nat ) )
=> ( ( ring_1_signed @ A @ ( word @ B ) @ ( one_one @ ( word @ A ) ) )
= ( one_one @ ( word @ B ) ) ) ) ) ) ).
% scast_1
thf(fact_6272_ucast__ucast__id,axiom,
! [B: $tType,A: $tType] :
( ( ( type_len @ A )
& ( type_len @ B ) )
=> ! [X2: 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 ) @ X2 ) )
= X2 ) ) ) ).
% ucast_ucast_id
thf(fact_6273_ucast__less__ucast__weak,axiom,
! [B: $tType,A: $tType] :
( ( ( type_len @ A )
& ( type_len @ B ) )
=> ! [X2: word @ A,Y3: 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 ) @ X2 ) @ ( semiring_1_unsigned @ A @ ( word @ B ) @ Y3 ) )
= ( ord_less @ ( word @ A ) @ X2 @ Y3 ) ) ) ) ).
% ucast_less_ucast_weak
thf(fact_6274_signed__take__bit__eq,axiom,
! [A: $tType,B: $tType] :
( ( ( type_len @ B )
& ( bit_ri3973907225187159222ations @ A ) )
=> ! [N: nat,W: word @ B] :
( ( ( ord_less @ nat @ N @ ( type_len0_len_of @ B @ ( type2 @ B ) ) )
=> ( ( ring_1_signed @ B @ A @ ( bit_se2584673776208193580ke_bit @ ( word @ B ) @ N @ W ) )
= ( bit_se2584673776208193580ke_bit @ A @ N @ ( ring_1_signed @ B @ A @ W ) ) ) )
& ( ~ ( ord_less @ nat @ N @ ( type_len0_len_of @ B @ ( type2 @ B ) ) )
=> ( ( ring_1_signed @ B @ A @ ( bit_se2584673776208193580ke_bit @ ( word @ B ) @ N @ W ) )
= ( ring_1_signed @ B @ A @ W ) ) ) ) ) ).
% signed_take_bit_eq
thf(fact_6275_one__word_Orsp,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ( ( bit_se2584673776208193580ke_bit @ int @ ( type_len0_len_of @ A @ ( type2 @ A ) ) @ ( one_one @ int ) )
= ( bit_se2584673776208193580ke_bit @ int @ ( type_len0_len_of @ A @ ( type2 @ A ) ) @ ( one_one @ int ) ) ) ) ).
% one_word.rsp
thf(fact_6276_degenerate__word,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [X2: word @ A] :
( ( ( type_len0_len_of @ A @ ( type2 @ A ) )
= ( one_one @ nat ) )
=> ( ( X2
= ( zero_zero @ ( word @ A ) ) )
| ( X2
= ( one_one @ ( word @ A ) ) ) ) ) ) ).
% degenerate_word
thf(fact_6277_max__word__mask,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ( ( uminus_uminus @ ( word @ A ) @ ( one_one @ ( word @ A ) ) )
= ( bit_se2239418461657761734s_mask @ ( word @ A ) @ ( type_len0_len_of @ A @ ( type2 @ A ) ) ) ) ) ).
% max_word_mask
thf(fact_6278_uint__word__arith__bintrs_I8_J,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ( ( semiring_1_unsigned @ A @ int @ ( one_one @ ( word @ A ) ) )
= ( bit_se2584673776208193580ke_bit @ int @ ( type_len0_len_of @ A @ ( type2 @ A ) ) @ ( one_one @ int ) ) ) ) ).
% uint_word_arith_bintrs(8)
thf(fact_6279_unsigned__minus__1__eq__mask,axiom,
! [A: $tType,B: $tType] :
( ( ( type_len @ B )
& ( bit_se359711467146920520ations @ A ) )
=> ( ( semiring_1_unsigned @ B @ A @ ( uminus_uminus @ ( word @ B ) @ ( one_one @ ( word @ B ) ) ) )
= ( bit_se2239418461657761734s_mask @ A @ ( type_len0_len_of @ B @ ( type2 @ B ) ) ) ) ) ).
% unsigned_minus_1_eq_mask
thf(fact_6280_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_6281_test__bit__wi,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [X2: int,N: nat] :
( ( bit_se5641148757651400278ts_bit @ ( word @ A ) @ ( ring_1_of_int @ ( word @ A ) @ X2 ) @ N )
= ( ( ord_less @ nat @ N @ ( type_len0_len_of @ A @ ( type2 @ A ) ) )
& ( bit_se5641148757651400278ts_bit @ int @ X2 @ N ) ) ) ) ).
% test_bit_wi
thf(fact_6282_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_6283_test__bit__conj__lt,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [X2: word @ A,M: nat] :
( ( ( bit_se5641148757651400278ts_bit @ ( word @ A ) @ X2 @ M )
& ( ord_less @ nat @ M @ ( type_len0_len_of @ A @ ( type2 @ A ) ) ) )
= ( bit_se5641148757651400278ts_bit @ ( word @ A ) @ X2 @ M ) ) ) ).
% test_bit_conj_lt
thf(fact_6284_neg__test__bit,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [X2: word @ A,N: nat] :
( ( bit_se5641148757651400278ts_bit @ ( word @ A ) @ ( bit_ri4277139882892585799ns_not @ ( word @ A ) @ X2 ) @ N )
= ( ~ ( bit_se5641148757651400278ts_bit @ ( word @ A ) @ X2 @ N )
& ( ord_less @ nat @ N @ ( type_len0_len_of @ A @ ( type2 @ A ) ) ) ) ) ) ).
% neg_test_bit
thf(fact_6285_word__eq__iff,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ( ( ^ [Y6: word @ A,Z3: word @ A] : Y6 = Z3 )
= ( ^ [X3: word @ A,Y: word @ A] :
! [N4: nat] :
( ( ord_less @ nat @ N4 @ ( type_len0_len_of @ A @ ( type2 @ A ) ) )
=> ( ( bit_se5641148757651400278ts_bit @ ( word @ A ) @ X3 @ N4 )
= ( bit_se5641148757651400278ts_bit @ ( word @ A ) @ Y @ N4 ) ) ) ) ) ) ).
% word_eq_iff
thf(fact_6286_bit__word__eqI,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [A3: word @ A,B3: word @ A] :
( ! [N3: nat] :
( ( ord_less @ nat @ N3 @ ( type_len0_len_of @ A @ ( type2 @ A ) ) )
=> ( ( bit_se5641148757651400278ts_bit @ ( word @ A ) @ A3 @ N3 )
= ( bit_se5641148757651400278ts_bit @ ( word @ A ) @ B3 @ N3 ) ) )
=> ( A3 = B3 ) ) ) ).
% bit_word_eqI
thf(fact_6287_bit__imp__le__length,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [W: word @ A,N: nat] :
( ( bit_se5641148757651400278ts_bit @ ( word @ A ) @ W @ N )
=> ( ord_less @ nat @ N @ ( type_len0_len_of @ A @ ( type2 @ A ) ) ) ) ) ).
% bit_imp_le_length
thf(fact_6288_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_6289_bit__word__ucast__iff,axiom,
! [B: $tType,A: $tType] :
( ( ( type_len @ A )
& ( type_len @ B ) )
=> ! [W: word @ A,N: nat] :
( ( bit_se5641148757651400278ts_bit @ ( word @ B ) @ ( semiring_1_unsigned @ A @ ( word @ B ) @ W ) @ 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 ) @ W @ N ) ) ) ) ).
% bit_word_ucast_iff
thf(fact_6290_bin__nth__uint__imp,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [W: word @ A,N: nat] :
( ( bit_se5641148757651400278ts_bit @ int @ ( semiring_1_unsigned @ A @ int @ W ) @ N )
=> ( ord_less @ nat @ N @ ( type_len0_len_of @ A @ ( type2 @ A ) ) ) ) ) ).
% bin_nth_uint_imp
thf(fact_6291_bit__ucast__iff,axiom,
! [A: $tType,B: $tType] :
( ( ( type_len @ B )
& ( type_len @ A ) )
=> ! [A3: word @ B,N: nat] :
( ( bit_se5641148757651400278ts_bit @ ( word @ A ) @ ( semiring_1_unsigned @ B @ ( word @ A ) @ A3 ) @ N )
= ( ( ord_less @ nat @ N @ ( type_len0_len_of @ A @ ( type2 @ A ) ) )
& ( bit_se5641148757651400278ts_bit @ ( word @ B ) @ A3 @ N ) ) ) ) ).
% bit_ucast_iff
thf(fact_6292_bit__uint__iff,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [W: word @ A,N: nat] :
( ( bit_se5641148757651400278ts_bit @ int @ ( semiring_1_unsigned @ A @ int @ W ) @ N )
= ( ( ord_less @ nat @ N @ ( type_len0_len_of @ A @ ( type2 @ A ) ) )
& ( bit_se5641148757651400278ts_bit @ ( word @ A ) @ W @ N ) ) ) ) ).
% bit_uint_iff
thf(fact_6293_nth__ucast,axiom,
! [B: $tType,A: $tType] :
( ( ( type_len @ A )
& ( type_len @ B ) )
=> ! [W: word @ B,N: nat] :
( ( bit_se5641148757651400278ts_bit @ ( word @ A ) @ ( semiring_1_unsigned @ B @ ( word @ A ) @ W ) @ N )
= ( ( bit_se5641148757651400278ts_bit @ ( word @ B ) @ W @ N )
& ( ord_less @ nat @ N @ ( type_len0_len_of @ A @ ( type2 @ A ) ) ) ) ) ) ).
% nth_ucast
thf(fact_6294_test__bit__bin,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ( ( bit_se5641148757651400278ts_bit @ ( word @ A ) )
= ( ^ [W2: word @ A,N4: nat] :
( ( ord_less @ nat @ N4 @ ( type_len0_len_of @ A @ ( type2 @ A ) ) )
& ( bit_se5641148757651400278ts_bit @ int @ ( semiring_1_unsigned @ A @ int @ W2 ) @ N4 ) ) ) ) ) ).
% test_bit_bin
thf(fact_6295_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_6296_uint32_Osize__eq__length,axiom,
( ( numeral_numeral @ nat @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ one2 ) ) ) ) ) )
= ( type_len0_len_of @ ( numeral_bit0 @ ( numeral_bit0 @ ( numeral_bit0 @ ( numeral_bit0 @ ( numeral_bit0 @ numeral_num1 ) ) ) ) ) @ ( type2 @ ( numeral_bit0 @ ( numeral_bit0 @ ( numeral_bit0 @ ( numeral_bit0 @ ( numeral_bit0 @ numeral_num1 ) ) ) ) ) ) ) ) ).
% uint32.size_eq_length
thf(fact_6297_nth__slice,axiom,
! [B: $tType,A: $tType] :
( ( ( type_len @ A )
& ( type_len @ B ) )
=> ! [N: nat,W: word @ B,M: nat] :
( ( bit_se5641148757651400278ts_bit @ ( word @ A ) @ ( slice2 @ B @ A @ N @ W ) @ M )
= ( ( bit_se5641148757651400278ts_bit @ ( word @ B ) @ W @ ( plus_plus @ nat @ M @ N ) )
& ( ord_less @ nat @ M @ ( type_len0_len_of @ A @ ( type2 @ A ) ) ) ) ) ) ).
% nth_slice
thf(fact_6298_up__scast__inj,axiom,
! [B: $tType,A: $tType] :
( ( ( type_len @ A )
& ( type_len @ B ) )
=> ! [X2: word @ A,Y3: word @ A] :
( ( ( ring_1_signed @ A @ ( word @ B ) @ X2 )
= ( ring_1_signed @ A @ ( word @ B ) @ Y3 ) )
=> ( ( ord_less_eq @ nat @ ( size_size @ ( word @ A ) @ X2 ) @ ( type_len0_len_of @ B @ ( type2 @ B ) ) )
=> ( X2 = Y3 ) ) ) ) ).
% up_scast_inj
thf(fact_6299_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_6300_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_6301_num__of__sbintr_H,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [A3: num,B3: num] :
( ( ( bit_ri4674362597316999326ke_bit @ int @ ( minus_minus @ nat @ ( type_len0_len_of @ A @ ( type2 @ A ) ) @ ( one_one @ nat ) ) @ ( numeral_numeral @ int @ A3 ) )
= ( numeral_numeral @ int @ B3 ) )
=> ( ( numeral_numeral @ ( word @ A ) @ A3 )
= ( numeral_numeral @ ( word @ A ) @ B3 ) ) ) ) ).
% num_of_sbintr'
thf(fact_6302_word__sint__1,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ( ( ( ( type_len0_len_of @ A @ ( type2 @ A ) )
= ( one_one @ nat ) )
=> ( ( ring_1_signed @ A @ int @ ( one_one @ ( word @ A ) ) )
= ( uminus_uminus @ int @ ( one_one @ int ) ) ) )
& ( ( ( type_len0_len_of @ A @ ( type2 @ A ) )
!= ( one_one @ nat ) )
=> ( ( ring_1_signed @ A @ int @ ( one_one @ ( word @ A ) ) )
= ( one_one @ int ) ) ) ) ) ).
% word_sint_1
thf(fact_6303_bin__nth__sint,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [N: nat,W: word @ A] :
( ( ord_less_eq @ nat @ ( type_len0_len_of @ A @ ( type2 @ A ) ) @ N )
=> ( ( bit_se5641148757651400278ts_bit @ int @ ( ring_1_signed @ A @ int @ W ) @ N )
= ( bit_se5641148757651400278ts_bit @ int @ ( ring_1_signed @ A @ int @ W ) @ ( minus_minus @ nat @ ( type_len0_len_of @ A @ ( type2 @ A ) ) @ ( one_one @ nat ) ) ) ) ) ) ).
% bin_nth_sint
thf(fact_6304_sint__sbintrunc_H,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [Bin: int] :
( ( ring_1_signed @ A @ int @ ( ring_1_of_int @ ( word @ A ) @ Bin ) )
= ( bit_ri4674362597316999326ke_bit @ int @ ( minus_minus @ nat @ ( type_len0_len_of @ A @ ( type2 @ A ) ) @ ( one_one @ nat ) ) @ Bin ) ) ) ).
% sint_sbintrunc'
thf(fact_6305_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_6306_word__of__int__bin__cat__eq__iff,axiom,
! [C: $tType,B: $tType,A: $tType] :
( ( ( type_len @ A )
& ( type_len @ B )
& ( type_len @ C ) )
=> ! [B3: word @ B,A3: 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 @ B3 ) @ ( semiring_1_unsigned @ A @ int @ A3 ) ) )
= ( 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 ) ) ) )
= ( ( B3 = D2 )
& ( A3 = C3 ) ) ) ) ) ).
% word_of_int_bin_cat_eq_iff
thf(fact_6307_mask__exceed,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [N: nat,X2: word @ A] :
( ( ord_less_eq @ nat @ ( type_len0_len_of @ A @ ( type2 @ A ) ) @ N )
=> ( ( bit_se5824344872417868541ns_and @ ( word @ A ) @ X2 @ ( bit_ri4277139882892585799ns_not @ ( word @ A ) @ ( bit_se2239418461657761734s_mask @ ( word @ A ) @ N ) ) )
= ( zero_zero @ ( word @ A ) ) ) ) ) ).
% mask_exceed
thf(fact_6308_unsigned__less,axiom,
! [A: $tType,B: $tType] :
( ( ( type_len @ B )
& ( unique1627219031080169319umeral @ A ) )
=> ! [W: word @ B] : ( ord_less @ A @ ( semiring_1_unsigned @ B @ A @ W ) @ ( power_power @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ ( type_len0_len_of @ B @ ( type2 @ B ) ) ) ) ) ).
% unsigned_less
thf(fact_6309_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_6310_word__nchotomy,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [W4: word @ A] :
? [N3: nat] :
( ( W4
= ( semiring_1_of_nat @ ( word @ A ) @ N3 ) )
& ( ord_less @ nat @ N3 @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( type_len0_len_of @ A @ ( type2 @ A ) ) ) ) ) ) ).
% word_nchotomy
thf(fact_6311_word__nat__cases,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [X2: word @ A] :
~ ! [N3: nat] :
( ( X2
= ( semiring_1_of_nat @ ( word @ A ) @ N3 ) )
=> ~ ( ord_less @ nat @ N3 @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( type_len0_len_of @ A @ ( type2 @ A ) ) ) ) ) ) ).
% word_nat_cases
thf(fact_6312_of__nat__inj,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [X2: nat,Y3: nat] :
( ( ord_less @ nat @ X2 @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( type_len0_len_of @ A @ ( type2 @ A ) ) ) )
=> ( ( ord_less @ nat @ Y3 @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( type_len0_len_of @ A @ ( type2 @ A ) ) ) )
=> ( ( ( semiring_1_of_nat @ ( word @ A ) @ X2 )
= ( semiring_1_of_nat @ ( word @ A ) @ Y3 ) )
= ( X2 = Y3 ) ) ) ) ) ).
% of_nat_inj
thf(fact_6313_word__of__nat__inj,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [X2: nat,Y3: nat] :
( ( ord_less @ nat @ X2 @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( type_len0_len_of @ A @ ( type2 @ A ) ) ) )
=> ( ( ord_less @ nat @ Y3 @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( type_len0_len_of @ A @ ( type2 @ A ) ) ) )
=> ( ( ( semiring_1_of_nat @ ( word @ A ) @ X2 )
= ( semiring_1_of_nat @ ( word @ A ) @ Y3 ) )
=> ( X2 = Y3 ) ) ) ) ) ).
% word_of_nat_inj
thf(fact_6314_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_6315_word__power__increasing,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [X2: nat,Y3: nat] :
( ( ord_less @ ( word @ A ) @ ( power_power @ ( word @ A ) @ ( numeral_numeral @ ( word @ A ) @ ( bit0 @ one2 ) ) @ X2 ) @ ( power_power @ ( word @ A ) @ ( numeral_numeral @ ( word @ A ) @ ( bit0 @ one2 ) ) @ Y3 ) )
=> ( ( ord_less @ nat @ X2 @ ( type_len0_len_of @ A @ ( type2 @ A ) ) )
=> ( ( ord_less @ nat @ Y3 @ ( type_len0_len_of @ A @ ( type2 @ A ) ) )
=> ( ord_less @ nat @ X2 @ Y3 ) ) ) ) ) ).
% word_power_increasing
thf(fact_6316_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_6317_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_6318_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_6319_uint__idem,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [W: word @ A] :
( ( modulo_modulo @ int @ ( semiring_1_unsigned @ A @ int @ W ) @ ( power_power @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ ( type_len0_len_of @ A @ ( type2 @ A ) ) ) )
= ( semiring_1_unsigned @ A @ int @ W ) ) ) ).
% uint_idem
thf(fact_6320_word__of__int__2p__len,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ( ( ring_1_of_int @ ( word @ A ) @ ( power_power @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ ( type_len0_len_of @ A @ ( type2 @ A ) ) ) )
= ( zero_zero @ ( word @ A ) ) ) ) ).
% word_of_int_2p_len
thf(fact_6321_of__nat__neq__iff__word,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [X2: nat,Y3: nat] :
( ( ( modulo_modulo @ nat @ X2 @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( type_len0_len_of @ A @ ( type2 @ A ) ) ) )
!= ( modulo_modulo @ nat @ Y3 @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( type_len0_len_of @ A @ ( type2 @ A ) ) ) ) )
=> ( ( ( semiring_1_of_nat @ ( word @ A ) @ X2 )
!= ( semiring_1_of_nat @ ( word @ A ) @ Y3 ) )
= ( X2 != Y3 ) ) ) ) ).
% of_nat_neq_iff_word
thf(fact_6322_sint__uint,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ( ( ring_1_signed @ A @ int )
= ( ^ [W2: 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 @ W2 ) ) ) ) ) ).
% sint_uint
thf(fact_6323_num__abs__sbintr,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ( ( numeral_numeral @ ( word @ A ) )
= ( ^ [X3: num] : ( ring_1_of_int @ ( word @ A ) @ ( bit_ri4674362597316999326ke_bit @ int @ ( minus_minus @ nat @ ( type_len0_len_of @ A @ ( type2 @ A ) ) @ ( one_one @ nat ) ) @ ( numeral_numeral @ int @ X3 ) ) ) ) ) ) ).
% num_abs_sbintr
thf(fact_6324_ucast__ucast__len,axiom,
! [B: $tType,A: $tType] :
( ( ( type_len @ A )
& ( type_len @ B ) )
=> ! [X2: word @ A] :
( ( ord_less @ ( word @ A ) @ X2 @ ( 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 ) @ X2 ) )
= X2 ) ) ) ).
% ucast_ucast_len
thf(fact_6325_ucast__mono,axiom,
! [A: $tType,B: $tType] :
( ( ( type_len @ B )
& ( type_len @ A ) )
=> ! [X2: word @ B,Y3: word @ B] :
( ( ord_less @ ( word @ B ) @ X2 @ Y3 )
=> ( ( ord_less @ ( word @ B ) @ Y3 @ ( 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 ) @ X2 ) @ ( semiring_1_unsigned @ B @ ( word @ A ) @ Y3 ) ) ) ) ) ).
% ucast_mono
thf(fact_6326_horner__sum__uint__exp__Cons__eq,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [W: word @ A,Ws: list @ ( word @ A )] :
( ( 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 ) ) ) @ ( cons @ ( word @ A ) @ W @ Ws ) )
= ( bit_concat_bit @ ( type_len0_len_of @ A @ ( type2 @ A ) ) @ ( semiring_1_unsigned @ A @ int @ W ) @ ( 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 ) ) ) ) ).
% horner_sum_uint_exp_Cons_eq
thf(fact_6327_sint__word__ariths_I7_J,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ( ( ring_1_signed @ A @ int @ ( zero_zero @ ( word @ A ) ) )
= ( bit_ri4674362597316999326ke_bit @ int @ ( minus_minus @ nat @ ( type_len0_len_of @ A @ ( type2 @ A ) ) @ ( one_one @ nat ) ) @ ( zero_zero @ int ) ) ) ) ).
% sint_word_ariths(7)
thf(fact_6328_sint__word__ariths_I8_J,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ( ( ring_1_signed @ A @ int @ ( one_one @ ( word @ A ) ) )
= ( bit_ri4674362597316999326ke_bit @ int @ ( minus_minus @ nat @ ( type_len0_len_of @ A @ ( type2 @ A ) ) @ ( one_one @ nat ) ) @ ( one_one @ int ) ) ) ) ).
% sint_word_ariths(8)
thf(fact_6329_sint__word__ariths_I1_J,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [A3: word @ A,B3: word @ A] :
( ( ring_1_signed @ A @ int @ ( plus_plus @ ( word @ A ) @ A3 @ B3 ) )
= ( bit_ri4674362597316999326ke_bit @ int @ ( minus_minus @ nat @ ( type_len0_len_of @ A @ ( type2 @ A ) ) @ ( one_one @ nat ) ) @ ( plus_plus @ int @ ( ring_1_signed @ A @ int @ A3 ) @ ( ring_1_signed @ A @ int @ B3 ) ) ) ) ) ).
% sint_word_ariths(1)
thf(fact_6330_nth__sint,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [W: word @ A,N: nat] :
( ( bit_se5641148757651400278ts_bit @ int @ ( ring_1_signed @ A @ int @ W ) @ N )
= ( ( ( ord_less @ nat @ N @ ( minus_minus @ nat @ ( type_len0_len_of @ A @ ( type2 @ A ) ) @ ( one_one @ nat ) ) )
=> ( bit_se5641148757651400278ts_bit @ ( word @ A ) @ W @ N ) )
& ( ~ ( ord_less @ nat @ N @ ( minus_minus @ nat @ ( type_len0_len_of @ A @ ( type2 @ A ) ) @ ( one_one @ nat ) ) )
=> ( bit_se5641148757651400278ts_bit @ ( word @ A ) @ W @ ( minus_minus @ nat @ ( type_len0_len_of @ A @ ( type2 @ A ) ) @ ( one_one @ nat ) ) ) ) ) ) ) ).
% nth_sint
thf(fact_6331_bit__sint__iff,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [W: word @ A,N: nat] :
( ( bit_se5641148757651400278ts_bit @ int @ ( ring_1_signed @ A @ int @ W ) @ N )
= ( ( ( ord_less_eq @ nat @ ( type_len0_len_of @ A @ ( type2 @ A ) ) @ N )
& ( bit_se5641148757651400278ts_bit @ ( word @ A ) @ W @ ( minus_minus @ nat @ ( type_len0_len_of @ A @ ( type2 @ A ) ) @ ( one_one @ nat ) ) ) )
| ( bit_se5641148757651400278ts_bit @ ( word @ A ) @ W @ N ) ) ) ) ).
% bit_sint_iff
thf(fact_6332_sint__word__ariths_I4_J,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [A3: word @ A] :
( ( ring_1_signed @ A @ int @ ( uminus_uminus @ ( word @ A ) @ A3 ) )
= ( bit_ri4674362597316999326ke_bit @ int @ ( minus_minus @ nat @ ( type_len0_len_of @ A @ ( type2 @ A ) ) @ ( one_one @ nat ) ) @ ( uminus_uminus @ int @ ( ring_1_signed @ A @ int @ A3 ) ) ) ) ) ).
% sint_word_ariths(4)
thf(fact_6333_drop__bit__eq__zero__iff__not__bit__last,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [W: word @ A] :
( ( ( bit_se4197421643247451524op_bit @ ( word @ A ) @ ( minus_minus @ nat @ ( type_len0_len_of @ A @ ( type2 @ A ) ) @ ( suc @ ( zero_zero @ nat ) ) ) @ W )
= ( zero_zero @ ( word @ A ) ) )
= ( ~ ( bit_se5641148757651400278ts_bit @ ( word @ A ) @ W @ ( minus_minus @ nat @ ( type_len0_len_of @ A @ ( type2 @ A ) ) @ ( suc @ ( zero_zero @ nat ) ) ) ) ) ) ) ).
% drop_bit_eq_zero_iff_not_bit_last
thf(fact_6334_sint__word__ariths_I2_J,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [A3: word @ A,B3: word @ A] :
( ( ring_1_signed @ A @ int @ ( minus_minus @ ( word @ A ) @ A3 @ B3 ) )
= ( bit_ri4674362597316999326ke_bit @ int @ ( minus_minus @ nat @ ( type_len0_len_of @ A @ ( type2 @ A ) ) @ ( one_one @ nat ) ) @ ( minus_minus @ int @ ( ring_1_signed @ A @ int @ A3 ) @ ( ring_1_signed @ A @ int @ B3 ) ) ) ) ) ).
% sint_word_ariths(2)
thf(fact_6335_signed__scast__eq,axiom,
! [C: $tType,A: $tType,B: $tType] :
( ( ( type_len @ B )
& ( bit_ri3973907225187159222ations @ A )
& ( type_len @ C ) )
=> ! [W: word @ B] :
( ( ring_1_signed @ C @ A @ ( ring_1_signed @ B @ ( word @ C ) @ W ) )
= ( bit_ri4674362597316999326ke_bit @ A @ ( minus_minus @ nat @ ( type_len0_len_of @ C @ ( type2 @ C ) ) @ ( suc @ ( zero_zero @ nat ) ) ) @ ( ring_1_signed @ B @ A @ W ) ) ) ) ).
% signed_scast_eq
thf(fact_6336_sint__word__ariths_I3_J,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [A3: word @ A,B3: word @ A] :
( ( ring_1_signed @ A @ int @ ( times_times @ ( word @ A ) @ A3 @ B3 ) )
= ( bit_ri4674362597316999326ke_bit @ int @ ( minus_minus @ nat @ ( type_len0_len_of @ A @ ( type2 @ A ) ) @ ( one_one @ nat ) ) @ ( times_times @ int @ ( ring_1_signed @ A @ int @ A3 ) @ ( ring_1_signed @ A @ int @ B3 ) ) ) ) ) ).
% sint_word_ariths(3)
thf(fact_6337_less__Suc__unat__less__bound,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [N: nat,X2: word @ A] :
( ( ord_less @ nat @ N @ ( suc @ ( semiring_1_unsigned @ A @ nat @ X2 ) ) )
=> ( 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_6338_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_6339_lt2p__lem,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [N: nat,W: word @ A] :
( ( ord_less_eq @ nat @ ( type_len0_len_of @ A @ ( type2 @ A ) ) @ N )
=> ( ord_less @ int @ ( semiring_1_unsigned @ A @ int @ W ) @ ( power_power @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ N ) ) ) ) ).
% lt2p_lem
thf(fact_6340_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_6341_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_6342_unat__numeral,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [B3: num] :
( ( semiring_1_unsigned @ A @ nat @ ( numeral_numeral @ ( word @ A ) @ B3 ) )
= ( modulo_modulo @ nat @ ( numeral_numeral @ nat @ B3 ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( type_len0_len_of @ A @ ( type2 @ A ) ) ) ) ) ) ).
% unat_numeral
thf(fact_6343_of__nat__mono__maybe,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [X2: nat,Y3: nat] :
( ( ord_less @ nat @ X2 @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( type_len0_len_of @ A @ ( type2 @ A ) ) ) )
=> ( ( ord_less @ nat @ Y3 @ X2 )
=> ( ord_less @ ( word @ A ) @ ( semiring_1_of_nat @ ( word @ A ) @ Y3 ) @ ( semiring_1_of_nat @ ( word @ A ) @ X2 ) ) ) ) ) ).
% of_nat_mono_maybe
thf(fact_6344_of__nat__mono__maybe_H,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [X2: nat,Y3: nat] :
( ( ord_less @ nat @ X2 @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( type_len0_len_of @ A @ ( type2 @ A ) ) ) )
=> ( ( ord_less @ nat @ Y3 @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( type_len0_len_of @ A @ ( type2 @ A ) ) ) )
=> ( ( ord_less @ nat @ Y3 @ X2 )
= ( ord_less @ ( word @ A ) @ ( semiring_1_of_nat @ ( word @ A ) @ Y3 ) @ ( semiring_1_of_nat @ ( word @ A ) @ X2 ) ) ) ) ) ) ).
% of_nat_mono_maybe'
thf(fact_6345_unat__of__nat__len,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [X2: nat] :
( ( ord_less @ nat @ X2 @ ( 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 ) @ X2 ) )
= X2 ) ) ) ).
% unat_of_nat_len
thf(fact_6346_unat__eq__of__nat,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [N: nat,X2: 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 @ X2 )
= N )
= ( X2
= ( semiring_1_of_nat @ ( word @ A ) @ N ) ) ) ) ) ).
% unat_eq_of_nat
thf(fact_6347_unat__split__asm,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [P: nat > $o,X2: word @ A] :
( ( P @ ( semiring_1_unsigned @ A @ nat @ X2 ) )
= ( ~ ? [N4: nat] :
( ( ( semiring_1_of_nat @ ( word @ A ) @ N4 )
= X2 )
& ( ord_less @ nat @ N4 @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( type_len0_len_of @ A @ ( type2 @ A ) ) ) )
& ~ ( P @ N4 ) ) ) ) ) ).
% unat_split_asm
thf(fact_6348_of__nat__inverse,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [R2: nat,A3: word @ A] :
( ( ( semiring_1_of_nat @ ( word @ A ) @ R2 )
= A3 )
=> ( ( ord_less @ nat @ R2 @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( type_len0_len_of @ A @ ( type2 @ A ) ) ) )
=> ( ( semiring_1_unsigned @ A @ nat @ A3 )
= R2 ) ) ) ) ).
% of_nat_inverse
thf(fact_6349_unat__split,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [P: nat > $o,X2: word @ A] :
( ( P @ ( semiring_1_unsigned @ A @ nat @ X2 ) )
= ( ! [N4: nat] :
( ( ( ( semiring_1_of_nat @ ( word @ A ) @ N4 )
= X2 )
& ( ord_less @ nat @ N4 @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( type_len0_len_of @ A @ ( type2 @ A ) ) ) ) )
=> ( P @ N4 ) ) ) ) ) ).
% unat_split
thf(fact_6350_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_6351_x__less__2__0__1_H,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [X2: word @ A] :
( ( ( type_len0_len_of @ A @ ( type2 @ A ) )
!= ( one_one @ nat ) )
=> ( ( ord_less @ ( word @ A ) @ X2 @ ( numeral_numeral @ ( word @ A ) @ ( bit0 @ one2 ) ) )
=> ( ( X2
= ( zero_zero @ ( word @ A ) ) )
| ( X2
= ( one_one @ ( word @ A ) ) ) ) ) ) ) ).
% x_less_2_0_1'
thf(fact_6352_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_6353_Word_Oof__nat__0,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [M: nat] :
( ( ( semiring_1_of_nat @ ( word @ A ) @ M )
= ( zero_zero @ ( word @ A ) ) )
= ( ? [Q5: nat] :
( M
= ( times_times @ nat @ Q5 @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( type_len0_len_of @ A @ ( type2 @ A ) ) ) ) ) ) ) ) ).
% Word.of_nat_0
thf(fact_6354_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_6355_ucast__of__nat__small,axiom,
! [A: $tType,B: $tType] :
( ( ( type_len @ B )
& ( type_len @ A ) )
=> ! [X2: nat] :
( ( ord_less @ nat @ X2 @ ( 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 ) @ X2 ) )
= ( semiring_1_of_nat @ ( word @ B ) @ X2 ) ) ) ) ).
% ucast_of_nat_small
thf(fact_6356_uint__sub__lt2p,axiom,
! [B: $tType,A: $tType] :
( ( ( type_len @ A )
& ( type_len @ B ) )
=> ! [X2: word @ A,Y3: word @ B] : ( ord_less @ int @ ( minus_minus @ int @ ( semiring_1_unsigned @ A @ int @ X2 ) @ ( semiring_1_unsigned @ B @ int @ Y3 ) ) @ ( power_power @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ ( type_len0_len_of @ A @ ( type2 @ A ) ) ) ) ) ).
% uint_sub_lt2p
thf(fact_6357_uint__numeral,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [B3: num] :
( ( semiring_1_unsigned @ A @ int @ ( numeral_numeral @ ( word @ A ) @ B3 ) )
= ( modulo_modulo @ int @ ( numeral_numeral @ int @ B3 ) @ ( power_power @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ ( type_len0_len_of @ A @ ( type2 @ A ) ) ) ) ) ) ).
% uint_numeral
thf(fact_6358_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_6359_word__of__nat__eq__0__iff,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [N: nat] :
( ( ( semiring_1_of_nat @ ( word @ A ) @ N )
= ( zero_zero @ ( word @ A ) ) )
= ( dvd_dvd @ nat @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( type_len0_len_of @ A @ ( type2 @ A ) ) ) @ N ) ) ) ).
% word_of_nat_eq_0_iff
thf(fact_6360_unat__ucast,axiom,
! [B: $tType,A: $tType] :
( ( ( type_len @ A )
& ( type_len @ B ) )
=> ! [X2: word @ B] :
( ( semiring_1_unsigned @ A @ nat @ ( semiring_1_unsigned @ B @ ( word @ A ) @ X2 ) )
= ( modulo_modulo @ nat @ ( semiring_1_unsigned @ B @ nat @ X2 ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( type_len0_len_of @ A @ ( type2 @ A ) ) ) ) ) ) ).
% unat_ucast
thf(fact_6361_word__of__int__minus,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [I: int] :
( ( ring_1_of_int @ ( word @ A ) @ ( minus_minus @ int @ ( power_power @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ ( type_len0_len_of @ A @ ( type2 @ A ) ) ) @ I ) )
= ( ring_1_of_int @ ( word @ A ) @ ( uminus_uminus @ int @ I ) ) ) ) ).
% word_of_int_minus
thf(fact_6362_bit__last__iff,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [W: word @ A] :
( ( bit_se5641148757651400278ts_bit @ ( word @ A ) @ W @ ( minus_minus @ nat @ ( type_len0_len_of @ A @ ( type2 @ A ) ) @ ( suc @ ( zero_zero @ nat ) ) ) )
= ( ord_less @ int @ ( ring_1_signed @ A @ int @ W ) @ ( zero_zero @ int ) ) ) ) ).
% bit_last_iff
thf(fact_6363_unat__of__nat,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [X2: nat] :
( ( semiring_1_unsigned @ A @ nat @ ( semiring_1_of_nat @ ( word @ A ) @ X2 ) )
= ( modulo_modulo @ nat @ X2 @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( type_len0_len_of @ A @ ( type2 @ A ) ) ) ) ) ) ).
% unat_of_nat
thf(fact_6364_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_6365_uint__word__of__int,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [K: int] :
( ( semiring_1_unsigned @ A @ int @ ( ring_1_of_int @ ( word @ A ) @ K ) )
= ( modulo_modulo @ int @ K @ ( power_power @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ ( type_len0_len_of @ A @ ( type2 @ A ) ) ) ) ) ) ).
% uint_word_of_int
thf(fact_6366_ucast__less,axiom,
! [A: $tType,B: $tType] :
( ( ( type_len @ B )
& ( type_len @ A ) )
=> ! [X2: 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 ) @ X2 ) @ ( power_power @ ( word @ A ) @ ( numeral_numeral @ ( word @ A ) @ ( bit0 @ one2 ) ) @ ( type_len0_len_of @ B @ ( type2 @ B ) ) ) ) ) ) ).
% ucast_less
thf(fact_6367_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_6368_word__of__int__eq__0__iff,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [K: int] :
( ( ( ring_1_of_int @ ( word @ A ) @ K )
= ( zero_zero @ ( word @ A ) ) )
= ( dvd_dvd @ int @ ( power_power @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ ( type_len0_len_of @ A @ ( type2 @ A ) ) ) @ K ) ) ) ).
% word_of_int_eq_0_iff
thf(fact_6369_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_6370_signed__ucast__eq,axiom,
! [C: $tType,A: $tType,B: $tType] :
( ( ( type_len @ B )
& ( bit_ri3973907225187159222ations @ A )
& ( type_len @ C ) )
=> ! [W: word @ B] :
( ( ring_1_signed @ C @ A @ ( semiring_1_unsigned @ B @ ( word @ C ) @ W ) )
= ( bit_ri4674362597316999326ke_bit @ A @ ( minus_minus @ nat @ ( type_len0_len_of @ C @ ( type2 @ C ) ) @ ( suc @ ( zero_zero @ nat ) ) ) @ ( semiring_1_unsigned @ B @ A @ W ) ) ) ) ).
% signed_ucast_eq
thf(fact_6371_complement__nth__w2p,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [N2: nat,N: nat] :
( ( ord_less @ nat @ N2 @ ( 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 ) ) @ N2 )
= ( N2 != N ) ) ) ) ).
% complement_nth_w2p
thf(fact_6372_upper__trivial,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [X2: word @ A] :
( ( X2
!= ( 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 ) @ X2 @ ( 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_6373_range__uint,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ( ( image @ ( word @ A ) @ int @ ( semiring_1_unsigned @ A @ int ) @ ( top_top @ ( set @ ( word @ A ) ) ) )
= ( set_or7035219750837199246ssThan @ int @ ( zero_zero @ int ) @ ( power_power @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ ( type_len0_len_of @ A @ ( type2 @ A ) ) ) ) ) ) ).
% range_uint
thf(fact_6374_bit__word__scast__iff,axiom,
! [B: $tType,A: $tType] :
( ( ( type_len @ A )
& ( type_len @ B ) )
=> ! [W: word @ A,N: nat] :
( ( bit_se5641148757651400278ts_bit @ ( word @ B ) @ ( ring_1_signed @ A @ ( word @ B ) @ W ) @ N )
= ( ( ord_less @ nat @ N @ ( type_len0_len_of @ B @ ( type2 @ B ) ) )
& ( ( bit_se5641148757651400278ts_bit @ ( word @ A ) @ W @ N )
| ( ( ord_less_eq @ nat @ ( type_len0_len_of @ A @ ( type2 @ A ) ) @ N )
& ( bit_se5641148757651400278ts_bit @ ( word @ A ) @ W @ ( minus_minus @ nat @ ( type_len0_len_of @ A @ ( type2 @ A ) ) @ ( suc @ ( zero_zero @ nat ) ) ) ) ) ) ) ) ) ).
% bit_word_scast_iff
thf(fact_6375_minus__one__word,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ( ( uminus_uminus @ ( word @ A ) @ ( one_one @ ( word @ A ) ) )
= ( 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 ) ) ) ) ) ).
% minus_one_word
thf(fact_6376_UNIV__eq,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ( ( top_top @ ( set @ ( word @ A ) ) )
= ( image @ int @ ( word @ A ) @ ( ring_1_of_int @ ( word @ A ) ) @ ( set_or7035219750837199246ssThan @ int @ ( zero_zero @ int ) @ ( power_power @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ ( type_len0_len_of @ A @ ( type2 @ A ) ) ) ) ) ) ) ).
% UNIV_eq
thf(fact_6377_unat__word__ariths_I7_J,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [A3: word @ A,B3: word @ A] :
( ( semiring_1_unsigned @ A @ nat @ ( modulo_modulo @ ( word @ A ) @ A3 @ B3 ) )
= ( modulo_modulo @ nat @ ( modulo_modulo @ nat @ ( semiring_1_unsigned @ A @ nat @ A3 ) @ ( semiring_1_unsigned @ A @ nat @ B3 ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( type_len0_len_of @ A @ ( type2 @ A ) ) ) ) ) ) ).
% unat_word_ariths(7)
thf(fact_6378_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_6379_ucast__mono__le,axiom,
! [B: $tType,A: $tType] :
( ( ( type_len @ A )
& ( type_len @ B ) )
=> ! [X2: word @ A,Y3: word @ A] :
( ( ord_less_eq @ ( word @ A ) @ X2 @ Y3 )
=> ( ( ord_less @ ( word @ A ) @ Y3 @ ( 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 ) @ X2 ) @ ( semiring_1_unsigned @ A @ ( word @ B ) @ Y3 ) ) ) ) ) ).
% ucast_mono_le
thf(fact_6380_take__bit__word__eq__self__iff,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [N: nat,W: word @ A] :
( ( ( bit_se2584673776208193580ke_bit @ ( word @ A ) @ N @ W )
= W )
= ( ( ord_less_eq @ nat @ ( type_len0_len_of @ A @ ( type2 @ A ) ) @ N )
| ( ord_less @ ( word @ A ) @ W @ ( power_power @ ( word @ A ) @ ( numeral_numeral @ ( word @ A ) @ ( bit0 @ one2 ) ) @ N ) ) ) ) ) ).
% take_bit_word_eq_self_iff
thf(fact_6381_signed__push__bit__eq,axiom,
! [A: $tType,B: $tType] :
( ( ( type_len @ B )
& ( bit_ri3973907225187159222ations @ A ) )
=> ! [N: nat,W: word @ B] :
( ( ring_1_signed @ B @ A @ ( bit_se4730199178511100633sh_bit @ ( word @ B ) @ N @ W ) )
= ( 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 @ W ) ) ) ) ) ).
% signed_push_bit_eq
thf(fact_6382_msb0,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [X2: word @ A,Y3: word @ A,I: nat] :
( ( ( bit_se5641148757651400278ts_bit @ ( word @ A ) @ ( bit_se1065995026697491101ons_or @ ( word @ A ) @ X2 @ Y3 ) @ ( minus_minus @ nat @ ( type_len0_len_of @ A @ ( type2 @ A ) ) @ ( suc @ I ) ) )
= ( ( bit_se5641148757651400278ts_bit @ ( word @ A ) @ X2 @ ( minus_minus @ nat @ ( type_len0_len_of @ A @ ( type2 @ A ) ) @ ( suc @ I ) ) )
| ( bit_se5641148757651400278ts_bit @ ( word @ A ) @ Y3 @ ( minus_minus @ nat @ ( type_len0_len_of @ A @ ( type2 @ A ) ) @ ( suc @ I ) ) ) ) )
& ( ( bit_se5641148757651400278ts_bit @ ( word @ A ) @ ( bit_se5824344872417868541ns_and @ ( word @ A ) @ X2 @ Y3 ) @ ( minus_minus @ nat @ ( type_len0_len_of @ A @ ( type2 @ A ) ) @ ( suc @ I ) ) )
= ( ( bit_se5641148757651400278ts_bit @ ( word @ A ) @ X2 @ ( minus_minus @ nat @ ( type_len0_len_of @ A @ ( type2 @ A ) ) @ ( suc @ I ) ) )
& ( bit_se5641148757651400278ts_bit @ ( word @ A ) @ Y3 @ ( minus_minus @ nat @ ( type_len0_len_of @ A @ ( type2 @ A ) ) @ ( suc @ I ) ) ) ) )
& ( ( bit_se5641148757651400278ts_bit @ ( word @ A ) @ ( bit_se5824344971392196577ns_xor @ ( word @ A ) @ X2 @ Y3 ) @ ( minus_minus @ nat @ ( type_len0_len_of @ A @ ( type2 @ A ) ) @ ( suc @ I ) ) )
= ( ( bit_se5641148757651400278ts_bit @ ( word @ A ) @ X2 @ ( minus_minus @ nat @ ( type_len0_len_of @ A @ ( type2 @ A ) ) @ ( suc @ I ) ) )
!= ( bit_se5641148757651400278ts_bit @ ( word @ A ) @ Y3 @ ( minus_minus @ nat @ ( type_len0_len_of @ A @ ( type2 @ A ) ) @ ( suc @ I ) ) ) ) )
& ( ( bit_se5641148757651400278ts_bit @ ( word @ A ) @ ( bit_ri4277139882892585799ns_not @ ( word @ A ) @ X2 ) @ ( minus_minus @ nat @ ( type_len0_len_of @ A @ ( type2 @ A ) ) @ ( suc @ I ) ) )
= ( ~ ( bit_se5641148757651400278ts_bit @ ( word @ A ) @ X2 @ ( minus_minus @ nat @ ( type_len0_len_of @ A @ ( type2 @ A ) ) @ ( suc @ I ) ) ) ) ) ) ) ).
% msb0
thf(fact_6383_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 )
@ ^ [X3: word @ B] : ( ord_less @ ( word @ B ) @ X3 @ ( power_power @ ( word @ B ) @ ( numeral_numeral @ ( word @ B ) @ ( bit0 @ one2 ) ) @ ( type_len0_len_of @ A @ ( type2 @ A ) ) ) ) ) ) ) ) ).
% ucast_range_less
thf(fact_6384_unat__add__lem,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [X2: word @ A,Y3: word @ A] :
( ( ord_less @ nat @ ( plus_plus @ nat @ ( semiring_1_unsigned @ A @ nat @ X2 ) @ ( semiring_1_unsigned @ A @ nat @ Y3 ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( type_len0_len_of @ A @ ( type2 @ A ) ) ) )
= ( ( semiring_1_unsigned @ A @ nat @ ( plus_plus @ ( word @ A ) @ X2 @ Y3 ) )
= ( plus_plus @ nat @ ( semiring_1_unsigned @ A @ nat @ X2 ) @ ( semiring_1_unsigned @ A @ nat @ Y3 ) ) ) ) ) ).
% unat_add_lem
thf(fact_6385_unat__add__lem_H,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [X2: word @ A,Y3: word @ A] :
( ( ord_less @ nat @ ( plus_plus @ nat @ ( semiring_1_unsigned @ A @ nat @ X2 ) @ ( semiring_1_unsigned @ A @ nat @ Y3 ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( type_len0_len_of @ A @ ( type2 @ A ) ) ) )
=> ( ( semiring_1_unsigned @ A @ nat @ ( plus_plus @ ( word @ A ) @ X2 @ Y3 ) )
= ( plus_plus @ nat @ ( semiring_1_unsigned @ A @ nat @ X2 ) @ ( semiring_1_unsigned @ A @ nat @ Y3 ) ) ) ) ) ).
% unat_add_lem'
thf(fact_6386_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_6387_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_6388_of__nat__mono__maybe__le,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [X2: nat,Y3: nat] :
( ( ord_less @ nat @ X2 @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( type_len0_len_of @ A @ ( type2 @ A ) ) ) )
=> ( ( ord_less @ nat @ Y3 @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( type_len0_len_of @ A @ ( type2 @ A ) ) ) )
=> ( ( ord_less_eq @ nat @ Y3 @ X2 )
= ( ord_less_eq @ ( word @ A ) @ ( semiring_1_of_nat @ ( word @ A ) @ Y3 ) @ ( semiring_1_of_nat @ ( word @ A ) @ X2 ) ) ) ) ) ) ).
% of_nat_mono_maybe_le
thf(fact_6389_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_6390_unat__word__ariths_I1_J,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [A3: word @ A,B3: word @ A] :
( ( semiring_1_unsigned @ A @ nat @ ( plus_plus @ ( word @ A ) @ A3 @ B3 ) )
= ( modulo_modulo @ nat @ ( plus_plus @ nat @ ( semiring_1_unsigned @ A @ nat @ A3 ) @ ( semiring_1_unsigned @ A @ nat @ B3 ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( type_len0_len_of @ A @ ( type2 @ A ) ) ) ) ) ) ).
% unat_word_ariths(1)
thf(fact_6391_bool__mask_H,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [X2: 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 ) @ X2 @ ( one_one @ ( word @ A ) ) ) )
= ( ( bit_se5824344872417868541ns_and @ ( word @ A ) @ X2 @ ( one_one @ ( word @ A ) ) )
= ( one_one @ ( word @ A ) ) ) ) ) ) ).
% bool_mask'
thf(fact_6392_uint__range_H,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [X2: word @ A] :
( ( ord_less_eq @ int @ ( zero_zero @ int ) @ ( semiring_1_unsigned @ A @ int @ X2 ) )
& ( ord_less @ int @ ( semiring_1_unsigned @ A @ int @ X2 ) @ ( power_power @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ ( type_len0_len_of @ A @ ( type2 @ A ) ) ) ) ) ) ).
% uint_range'
thf(fact_6393_ucast__mono__le_H,axiom,
! [B: $tType,A: $tType] :
( ( ( type_len @ A )
& ( type_len @ B ) )
=> ! [Y3: word @ A,X2: word @ A] :
( ( ord_less @ nat @ ( semiring_1_unsigned @ A @ nat @ Y3 ) @ ( 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 ) @ X2 @ Y3 )
=> ( ord_less_eq @ ( word @ B ) @ ( semiring_1_unsigned @ A @ ( word @ B ) @ X2 ) @ ( semiring_1_unsigned @ A @ ( word @ B ) @ Y3 ) ) ) ) ) ) ).
% ucast_mono_le'
thf(fact_6394_sint__lt,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [X2: word @ A] : ( ord_less @ int @ ( ring_1_signed @ A @ int @ X2 ) @ ( 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_6395_of__nat__eq,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [N: nat,W: word @ A] :
( ( ( semiring_1_of_nat @ ( word @ A ) @ N )
= W )
= ( ? [Q5: nat] :
( N
= ( plus_plus @ nat @ ( semiring_1_unsigned @ A @ nat @ W ) @ ( times_times @ nat @ Q5 @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( type_len0_len_of @ A @ ( type2 @ A ) ) ) ) ) ) ) ) ) ).
% of_nat_eq
thf(fact_6396_unat__mult__lem,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [X2: word @ A,Y3: word @ A] :
( ( ord_less @ nat @ ( times_times @ nat @ ( semiring_1_unsigned @ A @ nat @ X2 ) @ ( semiring_1_unsigned @ A @ nat @ Y3 ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( type_len0_len_of @ A @ ( type2 @ A ) ) ) )
= ( ( semiring_1_unsigned @ A @ nat @ ( times_times @ ( word @ A ) @ X2 @ Y3 ) )
= ( times_times @ nat @ ( semiring_1_unsigned @ A @ nat @ X2 ) @ ( semiring_1_unsigned @ A @ nat @ Y3 ) ) ) ) ) ).
% unat_mult_lem
thf(fact_6397_word__int__cases,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [X2: word @ A] :
~ ! [N3: int] :
( ( X2
= ( ring_1_of_int @ ( word @ A ) @ N3 ) )
=> ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ N3 )
=> ~ ( ord_less @ int @ N3 @ ( power_power @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ ( type_len0_len_of @ A @ ( type2 @ A ) ) ) ) ) ) ) ).
% word_int_cases
thf(fact_6398_word__of__int__inj,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [X2: int,Y3: int] :
( ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ X2 )
& ( ord_less @ int @ X2 @ ( power_power @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ ( type_len0_len_of @ A @ ( type2 @ A ) ) ) ) )
=> ( ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ Y3 )
& ( ord_less @ int @ Y3 @ ( power_power @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ ( type_len0_len_of @ A @ ( type2 @ A ) ) ) ) )
=> ( ( ( ring_1_of_int @ ( word @ A ) @ X2 )
= ( ring_1_of_int @ ( word @ A ) @ Y3 ) )
= ( X2 = Y3 ) ) ) ) ) ).
% word_of_int_inj
thf(fact_6399_unat__ucast__no__overflow__le,axiom,
! [A: $tType,B: $tType] :
( ( ( type_len @ B )
& ( type_len @ A ) )
=> ! [B3: word @ B,F2: word @ A] :
( ( ord_less @ nat @ ( semiring_1_unsigned @ B @ nat @ B3 ) @ ( 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 ) @ F2 ) @ B3 )
= ( ord_less @ nat @ ( semiring_1_unsigned @ A @ nat @ F2 ) @ ( semiring_1_unsigned @ B @ nat @ B3 ) ) ) ) ) ) ).
% unat_ucast_no_overflow_le
thf(fact_6400_uint__m2p__not__non__neg,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [X2: word @ A] :
~ ( ord_less_eq @ int @ ( zero_zero @ int ) @ ( minus_minus @ int @ ( semiring_1_unsigned @ A @ int @ X2 ) @ ( power_power @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ ( type_len0_len_of @ A @ ( type2 @ A ) ) ) ) ) ) ).
% uint_m2p_not_non_neg
thf(fact_6401_unat__ucast__less__no__overflow,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [N: nat,F2: 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 @ F2 ) @ N )
=> ( ord_less @ ( word @ A ) @ F2 @ ( semiring_1_of_nat @ ( word @ A ) @ N ) ) ) ) ) ).
% unat_ucast_less_no_overflow
thf(fact_6402_unat__ucast__less__no__overflow__simp,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [N: nat,F2: 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 @ F2 ) @ N )
= ( ord_less @ ( word @ A ) @ F2 @ ( semiring_1_of_nat @ ( word @ A ) @ N ) ) ) ) ) ).
% unat_ucast_less_no_overflow_simp
thf(fact_6403_uint__m2p__neg,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [X2: word @ A] : ( ord_less @ int @ ( minus_minus @ int @ ( semiring_1_unsigned @ A @ int @ X2 ) @ ( power_power @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ ( type_len0_len_of @ A @ ( type2 @ A ) ) ) ) @ ( zero_zero @ int ) ) ) ).
% uint_m2p_neg
thf(fact_6404_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_6405_less__2p__is__upper__bits__unset,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [P6: word @ A,N: nat] :
( ( ord_less @ ( word @ A ) @ P6 @ ( 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 ) @ P6 @ N12 ) ) ) ) ) ) ).
% less_2p_is_upper_bits_unset
thf(fact_6406_upper__bits__unset__is__l2p,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [N: nat,P6: 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 ) @ P6 @ N12 ) ) ) )
= ( ord_less @ ( word @ A ) @ P6 @ ( power_power @ ( word @ A ) @ ( numeral_numeral @ ( word @ A ) @ ( bit0 @ one2 ) ) @ N ) ) ) ) ) ).
% upper_bits_unset_is_l2p
thf(fact_6407_nth__bounded,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [X2: word @ A,N: nat,M: nat] :
( ( bit_se5641148757651400278ts_bit @ ( word @ A ) @ X2 @ N )
=> ( ( ord_less @ ( word @ A ) @ X2 @ ( 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_6408_uint__add__lem,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [X2: word @ A,Y3: word @ A] :
( ( ord_less @ int @ ( plus_plus @ int @ ( semiring_1_unsigned @ A @ int @ X2 ) @ ( semiring_1_unsigned @ A @ int @ Y3 ) ) @ ( power_power @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ ( type_len0_len_of @ A @ ( type2 @ A ) ) ) )
= ( ( semiring_1_unsigned @ A @ int @ ( plus_plus @ ( word @ A ) @ X2 @ Y3 ) )
= ( plus_plus @ int @ ( semiring_1_unsigned @ A @ int @ X2 ) @ ( semiring_1_unsigned @ A @ int @ Y3 ) ) ) ) ) ).
% uint_add_lem
thf(fact_6409_uint__word__ariths_I7_J,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ( ( semiring_1_unsigned @ A @ int @ ( zero_zero @ ( word @ A ) ) )
= ( modulo_modulo @ int @ ( zero_zero @ int ) @ ( power_power @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ ( type_len0_len_of @ A @ ( type2 @ A ) ) ) ) ) ) ).
% uint_word_ariths(7)
thf(fact_6410_uint__word__ariths_I8_J,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ( ( semiring_1_unsigned @ A @ int @ ( one_one @ ( word @ A ) ) )
= ( modulo_modulo @ int @ ( one_one @ int ) @ ( power_power @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ ( type_len0_len_of @ A @ ( type2 @ A ) ) ) ) ) ) ).
% uint_word_ariths(8)
thf(fact_6411_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_6412_uint__word__ariths_I1_J,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [A3: word @ A,B3: word @ A] :
( ( semiring_1_unsigned @ A @ int @ ( plus_plus @ ( word @ A ) @ A3 @ B3 ) )
= ( modulo_modulo @ int @ ( plus_plus @ int @ ( semiring_1_unsigned @ A @ int @ A3 ) @ ( semiring_1_unsigned @ A @ int @ B3 ) ) @ ( power_power @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ ( type_len0_len_of @ A @ ( type2 @ A ) ) ) ) ) ) ).
% uint_word_ariths(1)
thf(fact_6413_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_6414_unat__word__ariths_I2_J,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [A3: word @ A,B3: word @ A] :
( ( semiring_1_unsigned @ A @ nat @ ( times_times @ ( word @ A ) @ A3 @ B3 ) )
= ( modulo_modulo @ nat @ ( times_times @ nat @ ( semiring_1_unsigned @ A @ nat @ A3 ) @ ( semiring_1_unsigned @ A @ nat @ B3 ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( type_len0_len_of @ A @ ( type2 @ A ) ) ) ) ) ) ).
% unat_word_ariths(2)
thf(fact_6415_unat__word__ariths_I6_J,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [A3: word @ A,B3: word @ A] :
( ( semiring_1_unsigned @ A @ nat @ ( divide_divide @ ( word @ A ) @ A3 @ B3 ) )
= ( modulo_modulo @ nat @ ( divide_divide @ nat @ ( semiring_1_unsigned @ A @ nat @ A3 ) @ ( semiring_1_unsigned @ A @ nat @ B3 ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( type_len0_len_of @ A @ ( type2 @ A ) ) ) ) ) ) ).
% unat_word_ariths(6)
thf(fact_6416_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_6417_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_6418_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_6419_uint__word__ariths_I4_J,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [A3: word @ A] :
( ( semiring_1_unsigned @ A @ int @ ( uminus_uminus @ ( word @ A ) @ A3 ) )
= ( modulo_modulo @ int @ ( uminus_uminus @ int @ ( semiring_1_unsigned @ A @ int @ A3 ) ) @ ( power_power @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ ( type_len0_len_of @ A @ ( type2 @ A ) ) ) ) ) ) ).
% uint_word_ariths(4)
thf(fact_6420_le__mask__iff__lt__2n,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [N: nat,W: word @ A] :
( ( ord_less @ nat @ N @ ( type_len0_len_of @ A @ ( type2 @ A ) ) )
= ( ( ord_less_eq @ ( word @ A ) @ W @ ( bit_se2239418461657761734s_mask @ ( word @ A ) @ N ) )
= ( ord_less @ ( word @ A ) @ W @ ( power_power @ ( word @ A ) @ ( numeral_numeral @ ( word @ A ) @ ( bit0 @ one2 ) ) @ N ) ) ) ) ) ).
% le_mask_iff_lt_2n
thf(fact_6421_eq__mask__less,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [W: word @ A,N: nat] :
( ( W
= ( bit_se5824344872417868541ns_and @ ( word @ A ) @ W @ ( bit_se2239418461657761734s_mask @ ( word @ A ) @ N ) ) )
=> ( ( ord_less @ nat @ N @ ( type_len0_len_of @ A @ ( type2 @ A ) ) )
=> ( ord_less @ ( word @ A ) @ W @ ( power_power @ ( word @ A ) @ ( numeral_numeral @ ( word @ A ) @ ( bit0 @ one2 ) ) @ N ) ) ) ) ) ).
% eq_mask_less
thf(fact_6422_and__mask__less_H,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [N: nat,W: word @ A] :
( ( ord_less @ nat @ N @ ( type_len0_len_of @ A @ ( type2 @ A ) ) )
=> ( ord_less @ ( word @ A ) @ ( bit_se5824344872417868541ns_and @ ( word @ A ) @ W @ ( bit_se2239418461657761734s_mask @ ( word @ A ) @ N ) ) @ ( power_power @ ( word @ A ) @ ( numeral_numeral @ ( word @ A ) @ ( bit0 @ one2 ) ) @ N ) ) ) ) ).
% and_mask_less'
thf(fact_6423_sint__1__cases,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [A3: word @ A] :
( ( ( ( type_len0_len_of @ A @ ( type2 @ A ) )
= ( one_one @ nat ) )
=> ( ( A3
= ( zero_zero @ ( word @ A ) ) )
=> ( ( ring_1_signed @ A @ int @ A3 )
!= ( zero_zero @ int ) ) ) )
=> ( ( ( ( type_len0_len_of @ A @ ( type2 @ A ) )
= ( one_one @ nat ) )
=> ( ( A3
= ( 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_6424_uint__word__ariths_I2_J,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [A3: word @ A,B3: word @ A] :
( ( semiring_1_unsigned @ A @ int @ ( minus_minus @ ( word @ A ) @ A3 @ B3 ) )
= ( modulo_modulo @ int @ ( minus_minus @ int @ ( semiring_1_unsigned @ A @ int @ A3 ) @ ( semiring_1_unsigned @ A @ int @ B3 ) ) @ ( power_power @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ ( type_len0_len_of @ A @ ( type2 @ A ) ) ) ) ) ) ).
% uint_word_ariths(2)
thf(fact_6425_uint__mult__lem,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [X2: word @ A,Y3: word @ A] :
( ( ord_less @ int @ ( times_times @ int @ ( semiring_1_unsigned @ A @ int @ X2 ) @ ( semiring_1_unsigned @ A @ int @ Y3 ) ) @ ( power_power @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ ( type_len0_len_of @ A @ ( type2 @ A ) ) ) )
= ( ( semiring_1_unsigned @ A @ int @ ( times_times @ ( word @ A ) @ X2 @ Y3 ) )
= ( times_times @ int @ ( semiring_1_unsigned @ A @ int @ X2 ) @ ( semiring_1_unsigned @ A @ int @ Y3 ) ) ) ) ) ).
% uint_mult_lem
thf(fact_6426_uint__word__ariths_I3_J,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [A3: word @ A,B3: word @ A] :
( ( semiring_1_unsigned @ A @ int @ ( times_times @ ( word @ A ) @ A3 @ B3 ) )
= ( modulo_modulo @ int @ ( times_times @ int @ ( semiring_1_unsigned @ A @ int @ A3 ) @ ( semiring_1_unsigned @ A @ int @ B3 ) ) @ ( power_power @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ ( type_len0_len_of @ A @ ( type2 @ A ) ) ) ) ) ) ).
% uint_word_ariths(3)
thf(fact_6427_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_6428_word__power__mod__div,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [N: nat,M: nat,X2: 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 ) @ X2 @ ( 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 ) @ X2 @ ( 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_6429_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_6430_msb1,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [X2: word @ A,Y3: word @ A] :
( ( ( bit_se5641148757651400278ts_bit @ ( word @ A ) @ ( bit_se1065995026697491101ons_or @ ( word @ A ) @ X2 @ Y3 ) @ ( minus_minus @ nat @ ( type_len0_len_of @ A @ ( type2 @ A ) ) @ ( suc @ ( zero_zero @ nat ) ) ) )
= ( ( bit_se5641148757651400278ts_bit @ ( word @ A ) @ X2 @ ( minus_minus @ nat @ ( type_len0_len_of @ A @ ( type2 @ A ) ) @ ( suc @ ( zero_zero @ nat ) ) ) )
| ( bit_se5641148757651400278ts_bit @ ( word @ A ) @ Y3 @ ( minus_minus @ nat @ ( type_len0_len_of @ A @ ( type2 @ A ) ) @ ( suc @ ( zero_zero @ nat ) ) ) ) ) )
& ( ( bit_se5641148757651400278ts_bit @ ( word @ A ) @ ( bit_se5824344872417868541ns_and @ ( word @ A ) @ X2 @ Y3 ) @ ( minus_minus @ nat @ ( type_len0_len_of @ A @ ( type2 @ A ) ) @ ( suc @ ( zero_zero @ nat ) ) ) )
= ( ( bit_se5641148757651400278ts_bit @ ( word @ A ) @ X2 @ ( minus_minus @ nat @ ( type_len0_len_of @ A @ ( type2 @ A ) ) @ ( suc @ ( zero_zero @ nat ) ) ) )
& ( bit_se5641148757651400278ts_bit @ ( word @ A ) @ Y3 @ ( minus_minus @ nat @ ( type_len0_len_of @ A @ ( type2 @ A ) ) @ ( suc @ ( zero_zero @ nat ) ) ) ) ) )
& ( ( bit_se5641148757651400278ts_bit @ ( word @ A ) @ ( bit_se5824344971392196577ns_xor @ ( word @ A ) @ X2 @ Y3 ) @ ( minus_minus @ nat @ ( type_len0_len_of @ A @ ( type2 @ A ) ) @ ( suc @ ( zero_zero @ nat ) ) ) )
= ( ( bit_se5641148757651400278ts_bit @ ( word @ A ) @ X2 @ ( minus_minus @ nat @ ( type_len0_len_of @ A @ ( type2 @ A ) ) @ ( suc @ ( zero_zero @ nat ) ) ) )
!= ( bit_se5641148757651400278ts_bit @ ( word @ A ) @ Y3 @ ( minus_minus @ nat @ ( type_len0_len_of @ A @ ( type2 @ A ) ) @ ( suc @ ( zero_zero @ nat ) ) ) ) ) )
& ( ( bit_se5641148757651400278ts_bit @ ( word @ A ) @ ( bit_ri4277139882892585799ns_not @ ( word @ A ) @ X2 ) @ ( minus_minus @ nat @ ( type_len0_len_of @ A @ ( type2 @ A ) ) @ ( suc @ ( zero_zero @ nat ) ) ) )
= ( ~ ( bit_se5641148757651400278ts_bit @ ( word @ A ) @ X2 @ ( minus_minus @ nat @ ( type_len0_len_of @ A @ ( type2 @ A ) ) @ ( suc @ ( zero_zero @ nat ) ) ) ) ) ) ) ) ).
% msb1
thf(fact_6431_unat__plus__if_H,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [A3: word @ A,B3: word @ A] :
( ( ( ord_less @ nat @ ( plus_plus @ nat @ ( semiring_1_unsigned @ A @ nat @ A3 ) @ ( semiring_1_unsigned @ A @ nat @ B3 ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( type_len0_len_of @ A @ ( type2 @ A ) ) ) )
=> ( ( semiring_1_unsigned @ A @ nat @ ( plus_plus @ ( word @ A ) @ A3 @ B3 ) )
= ( plus_plus @ nat @ ( semiring_1_unsigned @ A @ nat @ A3 ) @ ( semiring_1_unsigned @ A @ nat @ B3 ) ) ) )
& ( ~ ( ord_less @ nat @ ( plus_plus @ nat @ ( semiring_1_unsigned @ A @ nat @ A3 ) @ ( semiring_1_unsigned @ A @ nat @ B3 ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( type_len0_len_of @ A @ ( type2 @ A ) ) ) )
=> ( ( semiring_1_unsigned @ A @ nat @ ( plus_plus @ ( word @ A ) @ A3 @ B3 ) )
= ( minus_minus @ nat @ ( plus_plus @ nat @ ( semiring_1_unsigned @ A @ nat @ A3 ) @ ( semiring_1_unsigned @ A @ nat @ B3 ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( type_len0_len_of @ A @ ( type2 @ A ) ) ) ) ) ) ) ) ).
% unat_plus_if'
thf(fact_6432_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_6433_unat__sub__if_H,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [Y3: word @ A,X2: word @ A] :
( ( ( ord_less_eq @ nat @ ( semiring_1_unsigned @ A @ nat @ Y3 ) @ ( semiring_1_unsigned @ A @ nat @ X2 ) )
=> ( ( semiring_1_unsigned @ A @ nat @ ( minus_minus @ ( word @ A ) @ X2 @ Y3 ) )
= ( minus_minus @ nat @ ( semiring_1_unsigned @ A @ nat @ X2 ) @ ( semiring_1_unsigned @ A @ nat @ Y3 ) ) ) )
& ( ~ ( ord_less_eq @ nat @ ( semiring_1_unsigned @ A @ nat @ Y3 ) @ ( semiring_1_unsigned @ A @ nat @ X2 ) )
=> ( ( semiring_1_unsigned @ A @ nat @ ( minus_minus @ ( word @ A ) @ X2 @ Y3 ) )
= ( minus_minus @ nat @ ( plus_plus @ nat @ ( semiring_1_unsigned @ A @ nat @ X2 ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( type_len0_len_of @ A @ ( type2 @ A ) ) ) ) @ ( semiring_1_unsigned @ A @ nat @ Y3 ) ) ) ) ) ) ).
% unat_sub_if'
thf(fact_6434_no__olen__add__nat,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [X2: word @ A,Y3: word @ A] :
( ( ord_less_eq @ ( word @ A ) @ X2 @ ( plus_plus @ ( word @ A ) @ X2 @ Y3 ) )
= ( ord_less @ nat @ ( plus_plus @ nat @ ( semiring_1_unsigned @ A @ nat @ X2 ) @ ( semiring_1_unsigned @ A @ nat @ Y3 ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( type_len0_len_of @ A @ ( type2 @ A ) ) ) ) ) ) ).
% no_olen_add_nat
thf(fact_6435_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_6436_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_6437_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_6438_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_6439_sint__less,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [W: word @ A] : ( ord_less @ int @ ( ring_1_signed @ A @ int @ W ) @ ( 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_6440_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_6441_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_6442_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_6443_unat__minus__one__word,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ( ( semiring_1_unsigned @ A @ nat @ ( uminus_uminus @ ( word @ A ) @ ( one_one @ ( word @ A ) ) ) )
= ( minus_minus @ nat @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( type_len0_len_of @ A @ ( type2 @ A ) ) ) @ ( one_one @ nat ) ) ) ) ).
% unat_minus_one_word
thf(fact_6444_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_6445_sint__ge,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [X2: 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 @ X2 ) ) ) ).
% sint_ge
thf(fact_6446_word__of__int__inverse,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [R2: int,A3: word @ A] :
( ( ( ring_1_of_int @ ( word @ A ) @ R2 )
= A3 )
=> ( ( 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 @ A3 )
= R2 ) ) ) ) ) ).
% word_of_int_inverse
thf(fact_6447_uint__split__asm,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [P: int > $o,X2: word @ A] :
( ( P @ ( semiring_1_unsigned @ A @ int @ X2 ) )
= ( ~ ? [I4: int] :
( ( ( ring_1_of_int @ ( word @ A ) @ I4 )
= X2 )
& ( ord_less_eq @ int @ ( zero_zero @ int ) @ I4 )
& ( ord_less @ int @ I4 @ ( power_power @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ ( type_len0_len_of @ A @ ( type2 @ A ) ) ) )
& ~ ( P @ I4 ) ) ) ) ) ).
% uint_split_asm
thf(fact_6448_uint__split,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [P: int > $o,X2: word @ A] :
( ( P @ ( semiring_1_unsigned @ A @ int @ X2 ) )
= ( ! [I4: int] :
( ( ( ( ring_1_of_int @ ( word @ A ) @ I4 )
= X2 )
& ( ord_less_eq @ int @ ( zero_zero @ int ) @ I4 )
& ( ord_less @ int @ I4 @ ( power_power @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ ( type_len0_len_of @ A @ ( type2 @ A ) ) ) ) )
=> ( P @ I4 ) ) ) ) ) ).
% uint_split
thf(fact_6449_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_6450_div__lt_H,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [I: word @ A,K: word @ A,X2: word @ A] :
( ( ord_less_eq @ ( word @ A ) @ I @ ( divide_divide @ ( word @ A ) @ K @ X2 ) )
=> ( ord_less @ nat @ ( times_times @ nat @ ( semiring_1_unsigned @ A @ nat @ I ) @ ( semiring_1_unsigned @ A @ nat @ X2 ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( type_len0_len_of @ A @ ( type2 @ A ) ) ) ) ) ) ).
% div_lt'
thf(fact_6451_div__lt_H_H,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [I: word @ A,K: word @ A,X2: word @ A] :
( ( ord_less @ ( word @ A ) @ I @ ( divide_divide @ ( word @ A ) @ K @ X2 ) )
=> ( ord_less @ nat @ ( times_times @ nat @ ( semiring_1_unsigned @ A @ nat @ I ) @ ( semiring_1_unsigned @ A @ nat @ X2 ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( type_len0_len_of @ A @ ( type2 @ A ) ) ) ) ) ) ).
% div_lt''
thf(fact_6452_double__eq__zero__iff,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [A3: word @ A] :
( ( ( times_times @ ( word @ A ) @ ( numeral_numeral @ ( word @ A ) @ ( bit0 @ one2 ) ) @ A3 )
= ( zero_zero @ ( word @ A ) ) )
= ( ( A3
= ( zero_zero @ ( word @ A ) ) )
| ( A3
= ( 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_6453_More__Word_Oof__nat__power,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [P6: nat,X2: nat] :
( ( ord_less @ nat @ P6 @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ X2 ) )
=> ( ( ord_less @ nat @ X2 @ ( type_len0_len_of @ A @ ( type2 @ A ) ) )
=> ( ord_less @ ( word @ A ) @ ( semiring_1_of_nat @ ( word @ A ) @ P6 ) @ ( power_power @ ( word @ A ) @ ( numeral_numeral @ ( word @ A ) @ ( bit0 @ one2 ) ) @ X2 ) ) ) ) ) ).
% More_Word.of_nat_power
thf(fact_6454_word__le__exists_H,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [X2: word @ A,Y3: word @ A] :
( ( ord_less_eq @ ( word @ A ) @ X2 @ Y3 )
=> ? [Z2: word @ A] :
( ( Y3
= ( plus_plus @ ( word @ A ) @ X2 @ Z2 ) )
& ( ord_less @ int @ ( plus_plus @ int @ ( semiring_1_unsigned @ A @ int @ X2 ) @ ( semiring_1_unsigned @ A @ int @ Z2 ) ) @ ( power_power @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ ( type_len0_len_of @ A @ ( type2 @ A ) ) ) ) ) ) ) ).
% word_le_exists'
thf(fact_6455_no__olen__add_H,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [X2: word @ A,Y3: word @ A] :
( ( ord_less_eq @ ( word @ A ) @ X2 @ ( plus_plus @ ( word @ A ) @ Y3 @ X2 ) )
= ( ord_less @ int @ ( plus_plus @ int @ ( semiring_1_unsigned @ A @ int @ Y3 ) @ ( semiring_1_unsigned @ A @ int @ X2 ) ) @ ( power_power @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ ( type_len0_len_of @ A @ ( type2 @ A ) ) ) ) ) ) ).
% no_olen_add'
thf(fact_6456_no__olen__add,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [X2: word @ A,Y3: word @ A] :
( ( ord_less_eq @ ( word @ A ) @ X2 @ ( plus_plus @ ( word @ A ) @ X2 @ Y3 ) )
= ( ord_less @ int @ ( plus_plus @ int @ ( semiring_1_unsigned @ A @ int @ X2 ) @ ( semiring_1_unsigned @ A @ int @ Y3 ) ) @ ( power_power @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ ( type_len0_len_of @ A @ ( type2 @ A ) ) ) ) ) ) ).
% no_olen_add
thf(fact_6457_uint__plus__if_H,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [A3: word @ A,B3: word @ A] :
( ( ( ord_less @ int @ ( plus_plus @ int @ ( semiring_1_unsigned @ A @ int @ A3 ) @ ( semiring_1_unsigned @ A @ int @ B3 ) ) @ ( power_power @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ ( type_len0_len_of @ A @ ( type2 @ A ) ) ) )
=> ( ( semiring_1_unsigned @ A @ int @ ( plus_plus @ ( word @ A ) @ A3 @ B3 ) )
= ( plus_plus @ int @ ( semiring_1_unsigned @ A @ int @ A3 ) @ ( semiring_1_unsigned @ A @ int @ B3 ) ) ) )
& ( ~ ( ord_less @ int @ ( plus_plus @ int @ ( semiring_1_unsigned @ A @ int @ A3 ) @ ( semiring_1_unsigned @ A @ int @ B3 ) ) @ ( power_power @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ ( type_len0_len_of @ A @ ( type2 @ A ) ) ) )
=> ( ( semiring_1_unsigned @ A @ int @ ( plus_plus @ ( word @ A ) @ A3 @ B3 ) )
= ( minus_minus @ int @ ( plus_plus @ int @ ( semiring_1_unsigned @ A @ int @ A3 ) @ ( semiring_1_unsigned @ A @ int @ B3 ) ) @ ( power_power @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ ( type_len0_len_of @ A @ ( type2 @ A ) ) ) ) ) ) ) ) ).
% uint_plus_if'
thf(fact_6458_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_6459_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_6460_length__n__lists__elem,axiom,
! [A: $tType,Ys: list @ A,N: nat,Xs: list @ A] :
( ( member @ ( list @ A ) @ Ys @ ( set2 @ ( list @ A ) @ ( n_lists @ A @ N @ Xs ) ) )
=> ( ( size_size @ ( list @ A ) @ Ys )
= N ) ) ).
% length_n_lists_elem
thf(fact_6461_uint__sub__if_H,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [B3: word @ A,A3: word @ A] :
( ( ( ord_less_eq @ int @ ( semiring_1_unsigned @ A @ int @ B3 ) @ ( semiring_1_unsigned @ A @ int @ A3 ) )
=> ( ( semiring_1_unsigned @ A @ int @ ( minus_minus @ ( word @ A ) @ A3 @ B3 ) )
= ( minus_minus @ int @ ( semiring_1_unsigned @ A @ int @ A3 ) @ ( semiring_1_unsigned @ A @ int @ B3 ) ) ) )
& ( ~ ( ord_less_eq @ int @ ( semiring_1_unsigned @ A @ int @ B3 ) @ ( semiring_1_unsigned @ A @ int @ A3 ) )
=> ( ( semiring_1_unsigned @ A @ int @ ( minus_minus @ ( word @ A ) @ A3 @ B3 ) )
= ( plus_plus @ int @ ( minus_minus @ int @ ( semiring_1_unsigned @ A @ int @ A3 ) @ ( semiring_1_unsigned @ A @ int @ B3 ) ) @ ( power_power @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ ( type_len0_len_of @ A @ ( type2 @ A ) ) ) ) ) ) ) ) ).
% uint_sub_if'
thf(fact_6462_word__less__two__pow__divI,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [X2: word @ A,N: nat,M: nat] :
( ( ord_less @ ( word @ A ) @ X2 @ ( 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 ) @ X2 @ ( 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_6463_uint__neg__numeral,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [B3: num] :
( ( semiring_1_unsigned @ A @ int @ ( uminus_uminus @ ( word @ A ) @ ( numeral_numeral @ ( word @ A ) @ B3 ) ) )
= ( modulo_modulo @ int @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ B3 ) ) @ ( power_power @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ ( type_len0_len_of @ A @ ( type2 @ A ) ) ) ) ) ) ).
% uint_neg_numeral
thf(fact_6464_word__power__nonzero,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [X2: word @ A,N: nat] :
( ( ord_less @ ( word @ A ) @ X2 @ ( 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 ) ) )
=> ( ( X2
!= ( zero_zero @ ( word @ A ) ) )
=> ( ( times_times @ ( word @ A ) @ X2 @ ( power_power @ ( word @ A ) @ ( numeral_numeral @ ( word @ A ) @ ( bit0 @ one2 ) ) @ N ) )
!= ( zero_zero @ ( word @ A ) ) ) ) ) ) ) ).
% word_power_nonzero
thf(fact_6465_mult__pow2__inj,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [M: nat,N: nat,X2: word @ A,Y3: word @ A] :
( ( ord_less_eq @ nat @ ( plus_plus @ nat @ M @ N ) @ ( type_len0_len_of @ A @ ( type2 @ A ) ) )
=> ( ( ord_less_eq @ ( word @ A ) @ X2 @ ( bit_se2239418461657761734s_mask @ ( word @ A ) @ M ) )
=> ( ( ord_less_eq @ ( word @ A ) @ Y3 @ ( bit_se2239418461657761734s_mask @ ( word @ A ) @ M ) )
=> ( ( ( times_times @ ( word @ A ) @ X2 @ ( power_power @ ( word @ A ) @ ( numeral_numeral @ ( word @ A ) @ ( bit0 @ one2 ) ) @ N ) )
= ( times_times @ ( word @ A ) @ Y3 @ ( power_power @ ( word @ A ) @ ( numeral_numeral @ ( word @ A ) @ ( bit0 @ one2 ) ) @ N ) ) )
=> ( X2 = Y3 ) ) ) ) ) ) ).
% mult_pow2_inj
thf(fact_6466_div__lt__uint_H,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [I: word @ A,K: word @ A,X2: word @ A] :
( ( ord_less_eq @ ( word @ A ) @ I @ ( divide_divide @ ( word @ A ) @ K @ X2 ) )
=> ( ord_less @ int @ ( times_times @ int @ ( semiring_1_unsigned @ A @ int @ I ) @ ( semiring_1_unsigned @ A @ int @ X2 ) ) @ ( power_power @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ ( type_len0_len_of @ A @ ( type2 @ A ) ) ) ) ) ) ).
% div_lt_uint'
thf(fact_6467_div__lt__uint_H_H,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [I: word @ A,K: word @ A,X2: word @ A] :
( ( ord_less @ ( word @ A ) @ I @ ( divide_divide @ ( word @ A ) @ K @ X2 ) )
=> ( ord_less @ int @ ( times_times @ int @ ( semiring_1_unsigned @ A @ int @ I ) @ ( semiring_1_unsigned @ A @ int @ X2 ) ) @ ( power_power @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ ( type_len0_len_of @ A @ ( type2 @ A ) ) ) ) ) ) ).
% div_lt_uint''
thf(fact_6468_push__bit__word__eq__nonzero,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [W: word @ A,M: nat,N: nat] :
( ( ord_less @ ( word @ A ) @ W @ ( 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 ) ) )
=> ( ( W
!= ( zero_zero @ ( word @ A ) ) )
=> ( ( bit_se4730199178511100633sh_bit @ ( word @ A ) @ N @ W )
!= ( zero_zero @ ( word @ A ) ) ) ) ) ) ) ).
% push_bit_word_eq_nonzero
thf(fact_6469_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 )
@ ^ [Y: A] : ( cons @ A @ Y @ Ys3 )
@ Xs )
@ ( n_lists @ A @ N @ Xs ) ) ) ) ).
% n_lists.simps(2)
thf(fact_6470_uint__and__mask__or__full,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [N: word @ A,Mask1: word @ A,Mask2: int] :
( ( bit_se5641148757651400278ts_bit @ ( word @ A ) @ N @ ( minus_minus @ nat @ ( type_len0_len_of @ A @ ( type2 @ A ) ) @ ( one_one @ nat ) ) )
=> ( ( Mask1
= ( bit_se2239418461657761734s_mask @ ( word @ A ) @ ( minus_minus @ nat @ ( type_len0_len_of @ A @ ( type2 @ A ) ) @ ( one_one @ nat ) ) ) )
=> ( ( Mask2
= ( bit_se4730199178511100633sh_bit @ int @ ( minus_minus @ nat @ ( type_len0_len_of @ A @ ( type2 @ A ) ) @ ( one_one @ nat ) ) @ ( one_one @ int ) ) )
=> ( ( bit_se1065995026697491101ons_or @ int @ ( semiring_1_unsigned @ A @ int @ ( bit_se5824344872417868541ns_and @ ( word @ A ) @ N @ Mask1 ) ) @ Mask2 )
= ( semiring_1_unsigned @ A @ int @ N ) ) ) ) ) ) ).
% uint_and_mask_or_full
thf(fact_6471_sint__greater__eq,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [W: 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 @ W ) ) ) ).
% sint_greater_eq
thf(fact_6472_int__eq__sint,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [X2: nat] :
( ( ord_less @ nat @ X2 @ ( 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 ) @ X2 ) )
= ( semiring_1_of_nat @ int @ X2 ) ) ) ) ).
% int_eq_sint
thf(fact_6473_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_6474_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_6475_smod__word__max,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [A3: word @ A,B3: word @ A] : ( ord_less @ int @ ( signed6721504322012087516modulo @ int @ ( ring_1_signed @ A @ int @ A3 ) @ ( ring_1_signed @ A @ int @ B3 ) ) @ ( 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_6476_le2p__bits__unset,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [P6: word @ A,N: nat] :
( ( ord_less_eq @ ( word @ A ) @ P6 @ ( minus_minus @ ( word @ A ) @ ( power_power @ ( word @ A ) @ ( numeral_numeral @ ( word @ A ) @ ( bit0 @ one2 ) ) @ N ) @ ( one_one @ ( word @ A ) ) ) )
=> ! [N5: nat] :
( ( ord_less_eq @ nat @ N @ N5 )
=> ( ( ord_less @ nat @ N5 @ ( type_len0_len_of @ A @ ( type2 @ A ) ) )
=> ~ ( bit_se5641148757651400278ts_bit @ ( word @ A ) @ P6 @ N5 ) ) ) ) ) ).
% le2p_bits_unset
thf(fact_6477_le__2p__upper__bits,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [P6: word @ A,N: nat] :
( ( ord_less_eq @ ( word @ A ) @ P6 @ ( 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 ) ) )
=> ! [N5: nat] :
( ( ord_less_eq @ nat @ N @ N5 )
=> ( ( ord_less @ nat @ N5 @ ( type_len0_len_of @ A @ ( type2 @ A ) ) )
=> ~ ( bit_se5641148757651400278ts_bit @ ( word @ A ) @ P6 @ N5 ) ) ) ) ) ) ).
% le_2p_upper_bits
thf(fact_6478_word__add__offset__less,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [Y3: word @ A,N: nat,X2: word @ A,M: nat,Sz: nat] :
( ( ord_less @ ( word @ A ) @ Y3 @ ( power_power @ ( word @ A ) @ ( numeral_numeral @ ( word @ A ) @ ( bit0 @ one2 ) ) @ N ) )
=> ( ( ord_less @ ( word @ A ) @ X2 @ ( 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 ) @ X2 @ ( 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 ) @ X2 @ ( power_power @ ( word @ A ) @ ( numeral_numeral @ ( word @ A ) @ ( bit0 @ one2 ) ) @ N ) ) @ Y3 ) @ ( power_power @ ( word @ A ) @ ( numeral_numeral @ ( word @ A ) @ ( bit0 @ one2 ) ) @ Sz ) ) ) ) ) ) ) ) ).
% word_add_offset_less
thf(fact_6479_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_6480_div__power__helper,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [X2: nat,Y3: nat] :
( ( ord_less_eq @ nat @ X2 @ Y3 )
=> ( ( ord_less @ nat @ Y3 @ ( 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 ) ) @ Y3 ) @ ( one_one @ ( word @ A ) ) ) @ ( power_power @ ( word @ A ) @ ( numeral_numeral @ ( word @ A ) @ ( bit0 @ one2 ) ) @ X2 ) )
= ( minus_minus @ ( word @ A ) @ ( power_power @ ( word @ A ) @ ( numeral_numeral @ ( word @ A ) @ ( bit0 @ one2 ) ) @ ( minus_minus @ nat @ Y3 @ X2 ) ) @ ( one_one @ ( word @ A ) ) ) ) ) ) ) ).
% div_power_helper
thf(fact_6481_even__mult__exp__div__word__iff,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [A3: word @ A,M: nat,N: nat] :
( ( dvd_dvd @ ( word @ A ) @ ( numeral_numeral @ ( word @ A ) @ ( bit0 @ one2 ) ) @ ( divide_divide @ ( word @ A ) @ ( times_times @ ( word @ A ) @ A3 @ ( 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 ) @ A3 @ ( power_power @ ( word @ A ) @ ( numeral_numeral @ ( word @ A ) @ ( bit0 @ one2 ) ) @ ( minus_minus @ nat @ N @ M ) ) ) ) ) ) ) ) ).
% even_mult_exp_div_word_iff
thf(fact_6482_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_6483_sint__of__nat__le,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [B3: nat,A3: nat] :
( ( ord_less @ nat @ B3 @ ( 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 @ A3 @ B3 )
=> ( ord_less_eq @ int @ ( ring_1_signed @ A @ int @ ( semiring_1_of_nat @ ( word @ A ) @ A3 ) ) @ ( ring_1_signed @ A @ int @ ( semiring_1_of_nat @ ( word @ A ) @ B3 ) ) ) ) ) ) ).
% sint_of_nat_le
thf(fact_6484_sint__of__nat__ge__zero,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [X2: nat] :
( ( ord_less @ nat @ X2 @ ( 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 ) @ X2 ) ) ) ) ) ).
% sint_of_nat_ge_zero
thf(fact_6485_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_6486_sint__of__int__eq,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [X2: 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 ) ) ) ) @ X2 )
=> ( ( ord_less @ int @ X2 @ ( 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 ) @ X2 ) )
= X2 ) ) ) ) ).
% sint_of_int_eq
thf(fact_6487_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_6488_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_6489_sint__numeral,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [B3: num] :
( ( ring_1_signed @ A @ int @ ( numeral_numeral @ ( word @ A ) @ B3 ) )
= ( minus_minus @ int @ ( modulo_modulo @ int @ ( plus_plus @ int @ ( numeral_numeral @ int @ B3 ) @ ( power_power @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ ( minus_minus @ nat @ ( type_len0_len_of @ A @ ( type2 @ A ) ) @ ( one_one @ nat ) ) ) ) @ ( power_power @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ ( type_len0_len_of @ A @ ( type2 @ A ) ) ) ) @ ( power_power @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ ( minus_minus @ nat @ ( type_len0_len_of @ A @ ( type2 @ A ) ) @ ( one_one @ nat ) ) ) ) ) ) ).
% sint_numeral
thf(fact_6490_smod__word__min,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [A3: word @ A,B3: 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 @ A3 ) @ ( ring_1_signed @ A @ int @ B3 ) ) ) ) ).
% smod_word_min
thf(fact_6491_int__word__sint,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [X2: int] :
( ( ring_1_signed @ A @ int @ ( ring_1_of_int @ ( word @ A ) @ X2 ) )
= ( minus_minus @ int @ ( modulo_modulo @ int @ ( plus_plus @ int @ X2 @ ( power_power @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ ( minus_minus @ nat @ ( type_len0_len_of @ A @ ( type2 @ A ) ) @ ( one_one @ nat ) ) ) ) @ ( power_power @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ ( type_len0_len_of @ A @ ( type2 @ A ) ) ) ) @ ( power_power @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ ( minus_minus @ nat @ ( type_len0_len_of @ A @ ( type2 @ A ) ) @ ( one_one @ nat ) ) ) ) ) ) ).
% int_word_sint
thf(fact_6492_Word_Oword__div__mult,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [Y3: word @ A,X2: word @ A] :
( ( ord_less @ ( word @ A ) @ ( zero_zero @ ( word @ A ) ) @ Y3 )
=> ( ( ord_less @ nat @ ( times_times @ nat @ ( semiring_1_unsigned @ A @ nat @ X2 ) @ ( semiring_1_unsigned @ A @ nat @ Y3 ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( type_len0_len_of @ A @ ( type2 @ A ) ) ) )
=> ( ( divide_divide @ ( word @ A ) @ ( times_times @ ( word @ A ) @ X2 @ Y3 ) @ Y3 )
= X2 ) ) ) ) ).
% Word.word_div_mult
thf(fact_6493_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 )
@ ^ [X3: word @ A] : ( ord_less @ ( word @ A ) @ X3 @ ( 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_6494_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_6495_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_6496_word__bit__induct,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [P: ( word @ A ) > $o,A3: word @ A] :
( ( P @ ( zero_zero @ ( word @ A ) ) )
=> ( ! [A4: word @ A] :
( ( P @ 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 ) ) ) ) )
=> ( P @ ( times_times @ ( word @ A ) @ ( numeral_numeral @ ( word @ A ) @ ( bit0 @ one2 ) ) @ A4 ) ) ) ) )
=> ( ! [A4: word @ A] :
( ( P @ 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 ) ) ) ) )
=> ( P @ ( plus_plus @ ( word @ A ) @ ( one_one @ ( word @ A ) ) @ ( times_times @ ( word @ A ) @ ( numeral_numeral @ ( word @ A ) @ ( bit0 @ one2 ) ) @ A4 ) ) ) ) )
=> ( P @ A3 ) ) ) ) ) ).
% word_bit_induct
thf(fact_6497_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_6498_bit__word__half__eq,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [A3: word @ A,B3: $o] :
( ( ord_less @ ( word @ A ) @ A3 @ ( 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 ) @ B3 ) @ ( times_times @ ( word @ A ) @ A3 @ ( numeral_numeral @ ( word @ A ) @ ( bit0 @ one2 ) ) ) ) @ ( numeral_numeral @ ( word @ A ) @ ( bit0 @ one2 ) ) )
= A3 ) ) ) ).
% bit_word_half_eq
thf(fact_6499_length__n__lists,axiom,
! [A: $tType,N: nat,Xs: list @ A] :
( ( size_size @ ( list @ ( list @ A ) ) @ ( n_lists @ A @ N @ Xs ) )
= ( power_power @ nat @ ( size_size @ ( list @ A ) @ Xs ) @ N ) ) ).
% length_n_lists
thf(fact_6500_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_6501_Suc__div__unat__helper,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [Sz: nat,Us: nat] :
( ( ord_less @ nat @ Sz @ ( type_len0_len_of @ A @ ( type2 @ A ) ) )
=> ( ( ord_less_eq @ nat @ Us @ Sz )
=> ( ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( minus_minus @ nat @ Sz @ Us ) )
= ( 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 ) ) @ Us ) ) ) ) ) ) ) ) ).
% Suc_div_unat_helper
thf(fact_6502_alignUp__div__helper,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [K: nat,N: nat,X2: word @ A,A3: 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 ) ) )
=> ( ( X2
= ( 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 ) @ A3 @ X2 )
=> ( ( ord_less @ nat @ N @ ( type_len0_len_of @ A @ ( type2 @ A ) ) )
=> ( ( ( modulo_modulo @ ( word @ A ) @ A3 @ ( power_power @ ( word @ A ) @ ( numeral_numeral @ ( word @ A ) @ ( bit0 @ one2 ) ) @ N ) )
!= ( zero_zero @ ( word @ A ) ) )
=> ( ord_less @ ( word @ A ) @ ( divide_divide @ ( word @ A ) @ A3 @ ( power_power @ ( word @ A ) @ ( numeral_numeral @ ( word @ A ) @ ( bit0 @ one2 ) ) @ N ) ) @ ( semiring_1_of_nat @ ( word @ A ) @ K ) ) ) ) ) ) ) ) ).
% alignUp_div_helper
thf(fact_6503_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_6504_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_6505_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_6506_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_6507_len__num1,axiom,
( ( type_len0_len_of @ numeral_num1 )
= ( ^ [Uu4: itself @ numeral_num1] : ( one_one @ nat ) ) ) ).
% len_num1
thf(fact_6508_len__of__finite__1__def,axiom,
( ( type_len0_len_of @ finite_1 )
= ( ^ [X3: itself @ finite_1] : ( one_one @ nat ) ) ) ).
% len_of_finite_1_def
thf(fact_6509_len__of__finite__2__def,axiom,
( ( type_len0_len_of @ finite_2 )
= ( ^ [X3: itself @ finite_2] : ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ).
% len_of_finite_2_def
thf(fact_6510_len__of__finite__3__def,axiom,
( ( type_len0_len_of @ finite_3 )
= ( ^ [X3: itself @ finite_3] : ( numeral_numeral @ nat @ ( bit0 @ ( bit0 @ one2 ) ) ) ) ) ).
% len_of_finite_3_def
thf(fact_6511_len__bit0,axiom,
! [A: $tType] :
( ( type_len0 @ A )
=> ( ( type_len0_len_of @ ( numeral_bit0 @ A ) )
= ( ^ [Uu4: itself @ ( numeral_bit0 @ A )] : ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( type_len0_len_of @ A @ ( type2 @ A ) ) ) ) ) ) ).
% len_bit0
thf(fact_6512_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_6513_len__bit1,axiom,
! [A: $tType] :
( ( type_len0 @ A )
=> ( ( type_len0_len_of @ ( numeral_bit1 @ A ) )
= ( ^ [Uu4: itself @ ( numeral_bit1 @ A )] : ( plus_plus @ nat @ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( type_len0_len_of @ A @ ( type2 @ A ) ) ) @ ( one_one @ nat ) ) ) ) ) ).
% len_bit1
thf(fact_6514_divmod__via__sdivmod,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [Y3: word @ A,X2: word @ A] :
( ( Y3
!= ( 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 ) ) ) @ Y3 )
=> ( ( ( ord_less @ ( word @ A ) @ X2 @ Y3 )
=> ( ( product_Pair @ ( word @ A ) @ ( word @ A ) @ ( divide_divide @ ( word @ A ) @ X2 @ Y3 ) @ ( modulo_modulo @ ( word @ A ) @ X2 @ Y3 ) )
= ( product_Pair @ ( word @ A ) @ ( word @ A ) @ ( zero_zero @ ( word @ A ) ) @ X2 ) ) )
& ( ~ ( ord_less @ ( word @ A ) @ X2 @ Y3 )
=> ( ( product_Pair @ ( word @ A ) @ ( word @ A ) @ ( divide_divide @ ( word @ A ) @ X2 @ Y3 ) @ ( modulo_modulo @ ( word @ A ) @ X2 @ Y3 ) )
= ( product_Pair @ ( word @ A ) @ ( word @ A ) @ ( one_one @ ( word @ A ) ) @ ( minus_minus @ ( word @ A ) @ X2 @ Y3 ) ) ) ) ) )
& ( ~ ( 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 ) ) ) @ Y3 )
=> ( ( product_Pair @ ( word @ A ) @ ( word @ A ) @ ( divide_divide @ ( word @ A ) @ X2 @ Y3 ) @ ( modulo_modulo @ ( word @ A ) @ X2 @ Y3 ) )
= ( if @ ( product_prod @ ( word @ A ) @ ( word @ A ) ) @ ( ord_less_eq @ ( word @ A ) @ Y3 @ ( minus_minus @ ( word @ A ) @ X2 @ ( times_times @ ( word @ A ) @ ( bit_se4730199178511100633sh_bit @ ( word @ A ) @ ( one_one @ nat ) @ ( signed7115095781618012415divide @ ( word @ A ) @ ( bit_se4197421643247451524op_bit @ ( word @ A ) @ ( one_one @ nat ) @ X2 ) @ Y3 ) ) @ Y3 ) ) ) @ ( 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 ) @ X2 ) @ Y3 ) ) @ ( one_one @ ( word @ A ) ) ) @ ( minus_minus @ ( word @ A ) @ ( minus_minus @ ( word @ A ) @ X2 @ ( times_times @ ( word @ A ) @ ( bit_se4730199178511100633sh_bit @ ( word @ A ) @ ( one_one @ nat ) @ ( signed7115095781618012415divide @ ( word @ A ) @ ( bit_se4197421643247451524op_bit @ ( word @ A ) @ ( one_one @ nat ) @ X2 ) @ Y3 ) ) @ Y3 ) ) @ Y3 ) ) @ ( 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 ) @ X2 ) @ Y3 ) ) @ ( minus_minus @ ( word @ A ) @ X2 @ ( times_times @ ( word @ A ) @ ( bit_se4730199178511100633sh_bit @ ( word @ A ) @ ( one_one @ nat ) @ ( signed7115095781618012415divide @ ( word @ A ) @ ( bit_se4197421643247451524op_bit @ ( word @ A ) @ ( one_one @ nat ) @ X2 ) @ Y3 ) ) @ Y3 ) ) ) ) ) ) ) ) ) ).
% divmod_via_sdivmod
thf(fact_6515_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 ) )
@ ^ [X3: A] : ( map @ B @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X3 ) @ Ys )
@ Xs ) ) ) ) ).
% product_code
thf(fact_6516_word__sdiv__div1,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [A3: word @ A] :
( ( signed7115095781618012415divide @ ( word @ A ) @ A3 @ ( one_one @ ( word @ A ) ) )
= A3 ) ) ).
% word_sdiv_div1
thf(fact_6517_minus__one__sdiv__word__eq,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [W: word @ A] :
( ( signed7115095781618012415divide @ ( word @ A ) @ ( uminus_uminus @ ( word @ A ) @ ( one_one @ ( word @ A ) ) ) @ W )
= ( uminus_uminus @ ( word @ A ) @ ( signed7115095781618012415divide @ ( word @ A ) @ ( one_one @ ( word @ A ) ) @ W ) ) ) ) ).
% minus_one_sdiv_word_eq
thf(fact_6518_word__sdiv__div__minus1,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [A3: word @ A] :
( ( signed7115095781618012415divide @ ( word @ A ) @ A3 @ ( uminus_uminus @ ( word @ A ) @ ( one_one @ ( word @ A ) ) ) )
= ( uminus_uminus @ ( word @ A ) @ A3 ) ) ) ).
% word_sdiv_div_minus1
thf(fact_6519_one__sdiv__word__eq,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [W: word @ A] :
( ( signed7115095781618012415divide @ ( word @ A ) @ ( one_one @ ( word @ A ) ) @ W )
= ( times_times @ ( word @ A )
@ ( zero_neq_one_of_bool @ ( word @ A )
@ ( ( W
= ( one_one @ ( word @ A ) ) )
| ( W
= ( uminus_uminus @ ( word @ A ) @ ( one_one @ ( word @ A ) ) ) ) ) )
@ W ) ) ) ).
% one_sdiv_word_eq
thf(fact_6520_sdiv__word__numeral__numeral,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [A3: num,B3: num] :
( ( signed7115095781618012415divide @ ( word @ A ) @ ( numeral_numeral @ ( word @ A ) @ A3 ) @ ( numeral_numeral @ ( word @ A ) @ B3 ) )
= ( ring_1_of_int @ ( word @ A ) @ ( signed7115095781618012415divide @ int @ ( bit_ri4674362597316999326ke_bit @ int @ ( minus_minus @ nat @ ( type_len0_len_of @ A @ ( type2 @ A ) ) @ ( one_one @ nat ) ) @ ( numeral_numeral @ int @ A3 ) ) @ ( bit_ri4674362597316999326ke_bit @ int @ ( minus_minus @ nat @ ( type_len0_len_of @ A @ ( type2 @ A ) ) @ ( one_one @ nat ) ) @ ( numeral_numeral @ int @ B3 ) ) ) ) ) ) ).
% sdiv_word_numeral_numeral
thf(fact_6521_sdiv__word__minus__numeral__numeral,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [A3: num,B3: num] :
( ( signed7115095781618012415divide @ ( word @ A ) @ ( uminus_uminus @ ( word @ A ) @ ( numeral_numeral @ ( word @ A ) @ A3 ) ) @ ( numeral_numeral @ ( word @ A ) @ B3 ) )
= ( ring_1_of_int @ ( word @ A ) @ ( signed7115095781618012415divide @ int @ ( bit_ri4674362597316999326ke_bit @ int @ ( minus_minus @ nat @ ( type_len0_len_of @ A @ ( type2 @ A ) ) @ ( one_one @ nat ) ) @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ A3 ) ) ) @ ( bit_ri4674362597316999326ke_bit @ int @ ( minus_minus @ nat @ ( type_len0_len_of @ A @ ( type2 @ A ) ) @ ( one_one @ nat ) ) @ ( numeral_numeral @ int @ B3 ) ) ) ) ) ) ).
% sdiv_word_minus_numeral_numeral
thf(fact_6522_sdiv__word__numeral__minus__numeral,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [A3: num,B3: num] :
( ( signed7115095781618012415divide @ ( word @ A ) @ ( numeral_numeral @ ( word @ A ) @ A3 ) @ ( uminus_uminus @ ( word @ A ) @ ( numeral_numeral @ ( word @ A ) @ B3 ) ) )
= ( ring_1_of_int @ ( word @ A ) @ ( signed7115095781618012415divide @ int @ ( bit_ri4674362597316999326ke_bit @ int @ ( minus_minus @ nat @ ( type_len0_len_of @ A @ ( type2 @ A ) ) @ ( one_one @ nat ) ) @ ( numeral_numeral @ int @ A3 ) ) @ ( bit_ri4674362597316999326ke_bit @ int @ ( minus_minus @ nat @ ( type_len0_len_of @ A @ ( type2 @ A ) ) @ ( one_one @ nat ) ) @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ B3 ) ) ) ) ) ) ) ).
% sdiv_word_numeral_minus_numeral
thf(fact_6523_sdiv__word__minus__numeral__minus__numeral,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [A3: num,B3: num] :
( ( signed7115095781618012415divide @ ( word @ A ) @ ( uminus_uminus @ ( word @ A ) @ ( numeral_numeral @ ( word @ A ) @ A3 ) ) @ ( uminus_uminus @ ( word @ A ) @ ( numeral_numeral @ ( word @ A ) @ B3 ) ) )
= ( ring_1_of_int @ ( word @ A ) @ ( signed7115095781618012415divide @ int @ ( bit_ri4674362597316999326ke_bit @ int @ ( minus_minus @ nat @ ( type_len0_len_of @ A @ ( type2 @ A ) ) @ ( one_one @ nat ) ) @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ A3 ) ) ) @ ( bit_ri4674362597316999326ke_bit @ int @ ( minus_minus @ nat @ ( type_len0_len_of @ A @ ( type2 @ A ) ) @ ( one_one @ nat ) ) @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ B3 ) ) ) ) ) ) ) ).
% sdiv_word_minus_numeral_minus_numeral
thf(fact_6524_word__sdiv__numerals_I9_J,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ( ( signed7115095781618012415divide @ ( word @ A ) @ ( one_one @ ( word @ A ) ) @ ( one_one @ ( word @ A ) ) )
= ( ring_1_of_int @ ( word @ A ) @ ( signed7115095781618012415divide @ int @ ( ring_1_signed @ A @ int @ ( one_one @ ( word @ A ) ) ) @ ( ring_1_signed @ A @ int @ ( one_one @ ( word @ A ) ) ) ) ) ) ) ).
% word_sdiv_numerals(9)
thf(fact_6525_word__sdiv__numerals__lhs_I3_J,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [W: word @ A] :
( ( signed7115095781618012415divide @ ( word @ A ) @ ( one_one @ ( word @ A ) ) @ W )
= ( ring_1_of_int @ ( word @ A ) @ ( signed7115095781618012415divide @ int @ ( ring_1_signed @ A @ int @ ( one_one @ ( word @ A ) ) ) @ ( ring_1_signed @ A @ int @ W ) ) ) ) ) ).
% word_sdiv_numerals_lhs(3)
thf(fact_6526_word__sdiv__numerals_I3_J,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [Y3: num] :
( ( signed7115095781618012415divide @ ( word @ A ) @ ( one_one @ ( word @ A ) ) @ ( numeral_numeral @ ( word @ A ) @ Y3 ) )
= ( ring_1_of_int @ ( word @ A ) @ ( signed7115095781618012415divide @ int @ ( ring_1_signed @ A @ int @ ( one_one @ ( word @ A ) ) ) @ ( ring_1_signed @ A @ int @ ( numeral_numeral @ ( word @ A ) @ Y3 ) ) ) ) ) ) ).
% word_sdiv_numerals(3)
thf(fact_6527_word__sdiv__numerals_I7_J,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [X2: num] :
( ( signed7115095781618012415divide @ ( word @ A ) @ ( numeral_numeral @ ( word @ A ) @ X2 ) @ ( one_one @ ( word @ A ) ) )
= ( ring_1_of_int @ ( word @ A ) @ ( signed7115095781618012415divide @ int @ ( ring_1_signed @ A @ int @ ( numeral_numeral @ ( word @ A ) @ X2 ) ) @ ( ring_1_signed @ A @ int @ ( one_one @ ( word @ A ) ) ) ) ) ) ) ).
% word_sdiv_numerals(7)
thf(fact_6528_word__sdiv__numerals_I8_J,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ( ( signed7115095781618012415divide @ ( word @ A ) @ ( zero_zero @ ( word @ A ) ) @ ( one_one @ ( word @ A ) ) )
= ( ring_1_of_int @ ( word @ A ) @ ( signed7115095781618012415divide @ int @ ( ring_1_signed @ A @ int @ ( zero_zero @ ( word @ A ) ) ) @ ( ring_1_signed @ A @ int @ ( one_one @ ( word @ A ) ) ) ) ) ) ) ).
% word_sdiv_numerals(8)
thf(fact_6529_word__sdiv__numerals_I6_J,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ( ( signed7115095781618012415divide @ ( word @ A ) @ ( one_one @ ( word @ A ) ) @ ( zero_zero @ ( word @ A ) ) )
= ( ring_1_of_int @ ( word @ A ) @ ( signed7115095781618012415divide @ int @ ( ring_1_signed @ A @ int @ ( one_one @ ( word @ A ) ) ) @ ( ring_1_signed @ A @ int @ ( zero_zero @ ( word @ A ) ) ) ) ) ) ) ).
% word_sdiv_numerals(6)
thf(fact_6530_signed__div__arith,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [A3: word @ A,B3: word @ A] :
( ( ring_1_signed @ A @ int @ ( signed7115095781618012415divide @ ( word @ A ) @ A3 @ B3 ) )
= ( bit_ri4674362597316999326ke_bit @ int @ ( minus_minus @ nat @ ( type_len0_len_of @ A @ ( type2 @ A ) ) @ ( one_one @ nat ) ) @ ( signed7115095781618012415divide @ int @ ( ring_1_signed @ A @ int @ A3 ) @ ( ring_1_signed @ A @ int @ B3 ) ) ) ) ) ).
% signed_div_arith
thf(fact_6531_signed__drop__bit__word__minus__numeral,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [N: num,K: num] :
( ( signed_drop_bit @ A @ ( numeral_numeral @ nat @ N ) @ ( uminus_uminus @ ( word @ A ) @ ( numeral_numeral @ ( word @ A ) @ K ) ) )
= ( ring_1_of_int @ ( word @ A ) @ ( bit_se4197421643247451524op_bit @ int @ ( numeral_numeral @ nat @ 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_minus_numeral
thf(fact_6532_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_6533_signed__drop__bit__signed__drop__bit,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [M: nat,N: nat,W: word @ A] :
( ( signed_drop_bit @ A @ M @ ( signed_drop_bit @ A @ N @ W ) )
= ( signed_drop_bit @ A @ ( plus_plus @ nat @ M @ N ) @ W ) ) ) ).
% signed_drop_bit_signed_drop_bit
thf(fact_6534_signed__drop__bit__of__minus__1,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [N: nat] :
( ( signed_drop_bit @ A @ N @ ( uminus_uminus @ ( word @ A ) @ ( one_one @ ( word @ A ) ) ) )
= ( uminus_uminus @ ( word @ A ) @ ( one_one @ ( word @ A ) ) ) ) ) ).
% signed_drop_bit_of_minus_1
thf(fact_6535_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_6536_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_6537_signed__drop__bit__word__numeral,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [N: num,K: num] :
( ( signed_drop_bit @ A @ ( numeral_numeral @ nat @ N ) @ ( numeral_numeral @ ( word @ A ) @ K ) )
= ( ring_1_of_int @ ( word @ A ) @ ( bit_se4197421643247451524op_bit @ int @ ( numeral_numeral @ nat @ 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_numeral
thf(fact_6538_bit__signed__drop__bit__iff,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [M: nat,W: word @ A,N: nat] :
( ( bit_se5641148757651400278ts_bit @ ( word @ A ) @ ( signed_drop_bit @ A @ M @ W ) @ N )
= ( bit_se5641148757651400278ts_bit @ ( word @ A ) @ W
@ ( 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_6539_signed__drop__bit__beyond,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [N: nat,W: word @ A] :
( ( ord_less_eq @ nat @ ( type_len0_len_of @ A @ ( type2 @ A ) ) @ N )
=> ( ( ( bit_se5641148757651400278ts_bit @ ( word @ A ) @ W @ ( minus_minus @ nat @ ( type_len0_len_of @ A @ ( type2 @ A ) ) @ ( suc @ ( zero_zero @ nat ) ) ) )
=> ( ( signed_drop_bit @ A @ N @ W )
= ( uminus_uminus @ ( word @ A ) @ ( one_one @ ( word @ A ) ) ) ) )
& ( ~ ( bit_se5641148757651400278ts_bit @ ( word @ A ) @ W @ ( minus_minus @ nat @ ( type_len0_len_of @ A @ ( type2 @ A ) ) @ ( suc @ ( zero_zero @ nat ) ) ) )
=> ( ( signed_drop_bit @ A @ N @ W )
= ( zero_zero @ ( word @ A ) ) ) ) ) ) ) ).
% signed_drop_bit_beyond
thf(fact_6540_word__int__split,axiom,
! [A: $tType,B: $tType] :
( ( type_len @ B )
=> ! [P: A > $o,F2: int > A,X2: word @ B] :
( ( P @ ( word_int_case @ A @ B @ F2 @ X2 ) )
= ( ! [I4: int] :
( ( ( X2
= ( ring_1_of_int @ ( word @ B ) @ I4 ) )
& ( ord_less_eq @ int @ ( zero_zero @ int ) @ I4 )
& ( ord_less @ int @ I4 @ ( power_power @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ ( type_len0_len_of @ B @ ( type2 @ B ) ) ) ) )
=> ( P @ ( F2 @ I4 ) ) ) ) ) ) ).
% word_int_split
thf(fact_6541_word__int__split__asm,axiom,
! [A: $tType,B: $tType] :
( ( type_len @ B )
=> ! [P: A > $o,F2: int > A,X2: word @ B] :
( ( P @ ( word_int_case @ A @ B @ F2 @ X2 ) )
= ( ~ ? [N4: int] :
( ( X2
= ( ring_1_of_int @ ( word @ B ) @ N4 ) )
& ( ord_less_eq @ int @ ( zero_zero @ int ) @ N4 )
& ( ord_less @ int @ N4 @ ( power_power @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ ( type_len0_len_of @ B @ ( type2 @ B ) ) ) )
& ~ ( P @ ( F2 @ N4 ) ) ) ) ) ) ).
% word_int_split_asm
thf(fact_6542_word__int__case__wi,axiom,
! [A: $tType,B: $tType] :
( ( type_len @ B )
=> ! [F2: int > A,I: int] :
( ( word_int_case @ A @ B @ F2 @ ( ring_1_of_int @ ( word @ B ) @ I ) )
= ( F2 @ ( modulo_modulo @ int @ I @ ( power_power @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ ( type_len0_len_of @ B @ ( type2 @ B ) ) ) ) ) ) ) ).
% word_int_case_wi
thf(fact_6543_smod__word__minus__numeral__minus__numeral,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [A3: num,B3: num] :
( ( signed6721504322012087516modulo @ ( word @ A ) @ ( uminus_uminus @ ( word @ A ) @ ( numeral_numeral @ ( word @ A ) @ A3 ) ) @ ( uminus_uminus @ ( word @ A ) @ ( numeral_numeral @ ( word @ A ) @ B3 ) ) )
= ( ring_1_of_int @ ( word @ A ) @ ( signed6721504322012087516modulo @ int @ ( bit_ri4674362597316999326ke_bit @ int @ ( minus_minus @ nat @ ( type_len0_len_of @ A @ ( type2 @ A ) ) @ ( one_one @ nat ) ) @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ A3 ) ) ) @ ( bit_ri4674362597316999326ke_bit @ int @ ( minus_minus @ nat @ ( type_len0_len_of @ A @ ( type2 @ A ) ) @ ( one_one @ nat ) ) @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ B3 ) ) ) ) ) ) ) ).
% smod_word_minus_numeral_minus_numeral
thf(fact_6544_smod__word__numeral__minus__numeral,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [A3: num,B3: num] :
( ( signed6721504322012087516modulo @ ( word @ A ) @ ( numeral_numeral @ ( word @ A ) @ A3 ) @ ( uminus_uminus @ ( word @ A ) @ ( numeral_numeral @ ( word @ A ) @ B3 ) ) )
= ( ring_1_of_int @ ( word @ A ) @ ( signed6721504322012087516modulo @ int @ ( bit_ri4674362597316999326ke_bit @ int @ ( minus_minus @ nat @ ( type_len0_len_of @ A @ ( type2 @ A ) ) @ ( one_one @ nat ) ) @ ( numeral_numeral @ int @ A3 ) ) @ ( bit_ri4674362597316999326ke_bit @ int @ ( minus_minus @ nat @ ( type_len0_len_of @ A @ ( type2 @ A ) ) @ ( one_one @ nat ) ) @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ B3 ) ) ) ) ) ) ) ).
% smod_word_numeral_minus_numeral
thf(fact_6545_smod__word__one,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [W: word @ A] :
( ( signed6721504322012087516modulo @ ( word @ A ) @ W @ ( one_one @ ( word @ A ) ) )
= ( zero_zero @ ( word @ A ) ) ) ) ).
% smod_word_one
thf(fact_6546_minus__one__smod__word__eq,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [W: word @ A] :
( ( signed6721504322012087516modulo @ ( word @ A ) @ ( uminus_uminus @ ( word @ A ) @ ( one_one @ ( word @ A ) ) ) @ W )
= ( uminus_uminus @ ( word @ A ) @ ( signed6721504322012087516modulo @ ( word @ A ) @ ( one_one @ ( word @ A ) ) @ W ) ) ) ) ).
% minus_one_smod_word_eq
thf(fact_6547_smod__word__minus__one,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [W: word @ A] :
( ( signed6721504322012087516modulo @ ( word @ A ) @ W @ ( uminus_uminus @ ( word @ A ) @ ( one_one @ ( word @ A ) ) ) )
= ( zero_zero @ ( word @ A ) ) ) ) ).
% smod_word_minus_one
thf(fact_6548_one__smod__word__eq,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [W: word @ A] :
( ( signed6721504322012087516modulo @ ( word @ A ) @ ( one_one @ ( word @ A ) ) @ W )
= ( minus_minus @ ( word @ A ) @ ( one_one @ ( word @ A ) )
@ ( zero_neq_one_of_bool @ ( word @ A )
@ ( ( W
= ( one_one @ ( word @ A ) ) )
| ( W
= ( uminus_uminus @ ( word @ A ) @ ( one_one @ ( word @ A ) ) ) ) ) ) ) ) ) ).
% one_smod_word_eq
thf(fact_6549_smod__word__numeral__numeral,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [A3: num,B3: num] :
( ( signed6721504322012087516modulo @ ( word @ A ) @ ( numeral_numeral @ ( word @ A ) @ A3 ) @ ( numeral_numeral @ ( word @ A ) @ B3 ) )
= ( ring_1_of_int @ ( word @ A ) @ ( signed6721504322012087516modulo @ int @ ( bit_ri4674362597316999326ke_bit @ int @ ( minus_minus @ nat @ ( type_len0_len_of @ A @ ( type2 @ A ) ) @ ( one_one @ nat ) ) @ ( numeral_numeral @ int @ A3 ) ) @ ( bit_ri4674362597316999326ke_bit @ int @ ( minus_minus @ nat @ ( type_len0_len_of @ A @ ( type2 @ A ) ) @ ( one_one @ nat ) ) @ ( numeral_numeral @ int @ B3 ) ) ) ) ) ) ).
% smod_word_numeral_numeral
thf(fact_6550_smod__word__minus__numeral__numeral,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [A3: num,B3: num] :
( ( signed6721504322012087516modulo @ ( word @ A ) @ ( uminus_uminus @ ( word @ A ) @ ( numeral_numeral @ ( word @ A ) @ A3 ) ) @ ( numeral_numeral @ ( word @ A ) @ B3 ) )
= ( ring_1_of_int @ ( word @ A ) @ ( signed6721504322012087516modulo @ int @ ( bit_ri4674362597316999326ke_bit @ int @ ( minus_minus @ nat @ ( type_len0_len_of @ A @ ( type2 @ A ) ) @ ( one_one @ nat ) ) @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ A3 ) ) ) @ ( bit_ri4674362597316999326ke_bit @ int @ ( minus_minus @ nat @ ( type_len0_len_of @ A @ ( type2 @ A ) ) @ ( one_one @ nat ) ) @ ( numeral_numeral @ int @ B3 ) ) ) ) ) ) ).
% smod_word_minus_numeral_numeral
thf(fact_6551_sdiv__smod__id,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [A3: word @ A,B3: word @ A] :
( ( plus_plus @ ( word @ A ) @ ( times_times @ ( word @ A ) @ ( signed7115095781618012415divide @ ( word @ A ) @ A3 @ B3 ) @ B3 ) @ ( signed6721504322012087516modulo @ ( word @ A ) @ A3 @ B3 ) )
= A3 ) ) ).
% sdiv_smod_id
thf(fact_6552_smod__word__alt__def,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ( ( signed6721504322012087516modulo @ ( word @ A ) )
= ( ^ [A6: word @ A,B7: word @ A] : ( minus_minus @ ( word @ A ) @ A6 @ ( times_times @ ( word @ A ) @ ( signed7115095781618012415divide @ ( word @ A ) @ A6 @ B7 ) @ B7 ) ) ) ) ) ).
% smod_word_alt_def
thf(fact_6553_word__smod__numerals_I9_J,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ( ( signed6721504322012087516modulo @ ( word @ A ) @ ( one_one @ ( word @ A ) ) @ ( one_one @ ( word @ A ) ) )
= ( ring_1_of_int @ ( word @ A ) @ ( signed6721504322012087516modulo @ int @ ( ring_1_signed @ A @ int @ ( one_one @ ( word @ A ) ) ) @ ( ring_1_signed @ A @ int @ ( one_one @ ( word @ A ) ) ) ) ) ) ) ).
% word_smod_numerals(9)
thf(fact_6554_word__smod__numerals__lhs_I3_J,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [W: word @ A] :
( ( signed6721504322012087516modulo @ ( word @ A ) @ ( one_one @ ( word @ A ) ) @ W )
= ( ring_1_of_int @ ( word @ A ) @ ( signed6721504322012087516modulo @ int @ ( ring_1_signed @ A @ int @ ( one_one @ ( word @ A ) ) ) @ ( ring_1_signed @ A @ int @ W ) ) ) ) ) ).
% word_smod_numerals_lhs(3)
thf(fact_6555_word__smod__numerals_I7_J,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [X2: num] :
( ( signed6721504322012087516modulo @ ( word @ A ) @ ( numeral_numeral @ ( word @ A ) @ X2 ) @ ( one_one @ ( word @ A ) ) )
= ( ring_1_of_int @ ( word @ A ) @ ( signed6721504322012087516modulo @ int @ ( ring_1_signed @ A @ int @ ( numeral_numeral @ ( word @ A ) @ X2 ) ) @ ( ring_1_signed @ A @ int @ ( one_one @ ( word @ A ) ) ) ) ) ) ) ).
% word_smod_numerals(7)
thf(fact_6556_word__smod__numerals_I3_J,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [Y3: num] :
( ( signed6721504322012087516modulo @ ( word @ A ) @ ( one_one @ ( word @ A ) ) @ ( numeral_numeral @ ( word @ A ) @ Y3 ) )
= ( ring_1_of_int @ ( word @ A ) @ ( signed6721504322012087516modulo @ int @ ( ring_1_signed @ A @ int @ ( one_one @ ( word @ A ) ) ) @ ( ring_1_signed @ A @ int @ ( numeral_numeral @ ( word @ A ) @ Y3 ) ) ) ) ) ) ).
% word_smod_numerals(3)
thf(fact_6557_word__smod__numerals_I8_J,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ( ( signed6721504322012087516modulo @ ( word @ A ) @ ( zero_zero @ ( word @ A ) ) @ ( one_one @ ( word @ A ) ) )
= ( ring_1_of_int @ ( word @ A ) @ ( signed6721504322012087516modulo @ int @ ( ring_1_signed @ A @ int @ ( zero_zero @ ( word @ A ) ) ) @ ( ring_1_signed @ A @ int @ ( one_one @ ( word @ A ) ) ) ) ) ) ) ).
% word_smod_numerals(8)
thf(fact_6558_word__smod__numerals_I6_J,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ( ( signed6721504322012087516modulo @ ( word @ A ) @ ( one_one @ ( word @ A ) ) @ ( zero_zero @ ( word @ A ) ) )
= ( ring_1_of_int @ ( word @ A ) @ ( signed6721504322012087516modulo @ int @ ( ring_1_signed @ A @ int @ ( one_one @ ( word @ A ) ) ) @ ( ring_1_signed @ A @ int @ ( zero_zero @ ( word @ A ) ) ) ) ) ) ) ).
% word_smod_numerals(6)
thf(fact_6559_signed__mod__arith,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [A3: word @ A,B3: word @ A] :
( ( ring_1_signed @ A @ int @ ( signed6721504322012087516modulo @ ( word @ A ) @ A3 @ B3 ) )
= ( bit_ri4674362597316999326ke_bit @ int @ ( minus_minus @ nat @ ( type_len0_len_of @ A @ ( type2 @ A ) ) @ ( one_one @ nat ) ) @ ( signed6721504322012087516modulo @ int @ ( ring_1_signed @ A @ int @ A3 ) @ ( ring_1_signed @ A @ int @ B3 ) ) ) ) ) ).
% signed_mod_arith
thf(fact_6560_uint__word__ariths_I6_J,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [A3: word @ A] :
( ( semiring_1_unsigned @ A @ int @ ( word_pred @ A @ A3 ) )
= ( modulo_modulo @ int @ ( minus_minus @ int @ ( semiring_1_unsigned @ A @ int @ A3 ) @ ( one_one @ int ) ) @ ( power_power @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ ( type_len0_len_of @ A @ ( type2 @ A ) ) ) ) ) ) ).
% uint_word_ariths(6)
thf(fact_6561_slice1__def,axiom,
! [A: $tType,B: $tType] :
( ( ( type_len @ B )
& ( type_len @ A ) )
=> ( ( slice1 @ A @ B )
= ( ^ [N4: nat,W2: word @ A] : ( if @ ( word @ B ) @ ( ord_less @ nat @ N4 @ ( 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 ) ) @ N4 ) @ W2 ) ) @ ( bit_se4730199178511100633sh_bit @ ( word @ B ) @ ( minus_minus @ nat @ N4 @ ( type_len0_len_of @ A @ ( type2 @ A ) ) ) @ ( semiring_1_unsigned @ A @ ( word @ B ) @ W2 ) ) ) ) ) ) ).
% slice1_def
thf(fact_6562_succ__pred__no_I2_J,axiom,
! [B: $tType] :
( ( type_len @ B )
=> ! [W: num] :
( ( word_pred @ B @ ( numeral_numeral @ ( word @ B ) @ W ) )
= ( minus_minus @ ( word @ B ) @ ( numeral_numeral @ ( word @ B ) @ W ) @ ( one_one @ ( word @ B ) ) ) ) ) ).
% succ_pred_no(2)
thf(fact_6563_succ__pred__no_I4_J,axiom,
! [D6: $tType] :
( ( type_len @ D6 )
=> ! [W: num] :
( ( word_pred @ D6 @ ( uminus_uminus @ ( word @ D6 ) @ ( numeral_numeral @ ( word @ D6 ) @ W ) ) )
= ( minus_minus @ ( word @ D6 ) @ ( uminus_uminus @ ( word @ D6 ) @ ( numeral_numeral @ ( word @ D6 ) @ W ) ) @ ( one_one @ ( word @ D6 ) ) ) ) ) ).
% succ_pred_no(4)
thf(fact_6564_word__m1__ge,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [Y3: word @ A] : ( ord_less_eq @ ( word @ A ) @ Y3 @ ( word_pred @ A @ ( zero_zero @ ( word @ A ) ) ) ) ) ).
% word_m1_ge
thf(fact_6565_word__pred__m1,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ( ( word_pred @ A )
= ( ^ [A6: word @ A] : ( minus_minus @ ( word @ A ) @ A6 @ ( one_one @ ( word @ A ) ) ) ) ) ) ).
% word_pred_m1
thf(fact_6566_wi__hom__pred,axiom,
! [F: $tType] :
( ( type_len @ F )
=> ! [A3: int] :
( ( word_pred @ F @ ( ring_1_of_int @ ( word @ F ) @ A3 ) )
= ( ring_1_of_int @ ( word @ F ) @ ( minus_minus @ int @ A3 @ ( one_one @ int ) ) ) ) ) ).
% wi_hom_pred
thf(fact_6567_word__pred__0__n1,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ( ( word_pred @ A @ ( zero_zero @ ( word @ A ) ) )
= ( ring_1_of_int @ ( word @ A ) @ ( uminus_uminus @ int @ ( one_one @ int ) ) ) ) ) ).
% word_pred_0_n1
thf(fact_6568_word__pred__alt,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ( ( word_pred @ A )
= ( ^ [A6: word @ A] : ( ring_1_of_int @ ( word @ A ) @ ( minus_minus @ int @ ( semiring_1_unsigned @ A @ int @ A6 ) @ ( one_one @ int ) ) ) ) ) ) ).
% word_pred_alt
thf(fact_6569_Word_Oslice__def,axiom,
! [A: $tType,B: $tType] :
( ( ( type_len @ B )
& ( type_len @ A ) )
=> ( ( slice2 @ A @ B )
= ( ^ [N4: nat] : ( slice1 @ A @ B @ ( minus_minus @ nat @ ( type_len0_len_of @ A @ ( type2 @ A ) ) @ N4 ) ) ) ) ) ).
% Word.slice_def
thf(fact_6570_uint__word__arith__bintrs_I6_J,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [A3: word @ A] :
( ( semiring_1_unsigned @ A @ int @ ( word_pred @ A @ A3 ) )
= ( bit_se2584673776208193580ke_bit @ int @ ( type_len0_len_of @ A @ ( type2 @ A ) ) @ ( minus_minus @ int @ ( semiring_1_unsigned @ A @ int @ A3 ) @ ( one_one @ int ) ) ) ) ) ).
% uint_word_arith_bintrs(6)
thf(fact_6571_sint__word__ariths_I6_J,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [A3: word @ A] :
( ( ring_1_signed @ A @ int @ ( word_pred @ A @ A3 ) )
= ( bit_ri4674362597316999326ke_bit @ int @ ( minus_minus @ nat @ ( type_len0_len_of @ A @ ( type2 @ A ) ) @ ( one_one @ nat ) ) @ ( minus_minus @ int @ ( ring_1_signed @ A @ int @ A3 ) @ ( one_one @ int ) ) ) ) ) ).
% sint_word_ariths(6)
thf(fact_6572_uint__word__ariths_I5_J,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [A3: word @ A] :
( ( semiring_1_unsigned @ A @ int @ ( word_succ @ A @ A3 ) )
= ( modulo_modulo @ int @ ( plus_plus @ int @ ( semiring_1_unsigned @ A @ int @ A3 ) @ ( one_one @ int ) ) @ ( power_power @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ ( type_len0_len_of @ A @ ( type2 @ A ) ) ) ) ) ) ).
% uint_word_ariths(5)
thf(fact_6573_bit__word__roti__iff,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [K: int,W: word @ A,N: nat] :
( ( bit_se5641148757651400278ts_bit @ ( word @ A ) @ ( word_roti @ A @ K @ W ) @ N )
= ( ( ord_less @ nat @ N @ ( type_len0_len_of @ A @ ( type2 @ A ) ) )
& ( bit_se5641148757651400278ts_bit @ ( word @ A ) @ W @ ( 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_6574_succ__pred__no_I1_J,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [W: num] :
( ( word_succ @ A @ ( numeral_numeral @ ( word @ A ) @ W ) )
= ( plus_plus @ ( word @ A ) @ ( numeral_numeral @ ( word @ A ) @ W ) @ ( one_one @ ( word @ A ) ) ) ) ) ).
% succ_pred_no(1)
thf(fact_6575_succ__pred__no_I3_J,axiom,
! [C: $tType] :
( ( type_len @ C )
=> ! [W: num] :
( ( word_succ @ C @ ( uminus_uminus @ ( word @ C ) @ ( numeral_numeral @ ( word @ C ) @ W ) ) )
= ( plus_plus @ ( word @ C ) @ ( uminus_uminus @ ( word @ C ) @ ( numeral_numeral @ ( word @ C ) @ W ) ) @ ( one_one @ ( word @ C ) ) ) ) ) ).
% succ_pred_no(3)
thf(fact_6576_word__roti__add,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [M: int,N: int,W: word @ A] :
( ( word_roti @ A @ ( plus_plus @ int @ M @ N ) @ W )
= ( word_roti @ A @ M @ ( word_roti @ A @ N @ W ) ) ) ) ).
% word_roti_add
thf(fact_6577_Abs__fnat__hom__Suc,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [A3: nat] :
( ( word_succ @ A @ ( semiring_1_of_nat @ ( word @ A ) @ A3 ) )
= ( semiring_1_of_nat @ ( word @ A ) @ ( suc @ A3 ) ) ) ) ).
% Abs_fnat_hom_Suc
thf(fact_6578_word__succ__p1,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ( ( word_succ @ A )
= ( ^ [A6: word @ A] : ( plus_plus @ ( word @ A ) @ A6 @ ( one_one @ ( word @ A ) ) ) ) ) ) ).
% word_succ_p1
thf(fact_6579_word__mult__succ,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [A3: word @ A,B3: word @ A] :
( ( times_times @ ( word @ A ) @ ( word_succ @ A @ A3 ) @ B3 )
= ( plus_plus @ ( word @ A ) @ B3 @ ( times_times @ ( word @ A ) @ A3 @ B3 ) ) ) ) ).
% word_mult_succ
thf(fact_6580_wi__hom__succ,axiom,
! [E3: $tType] :
( ( type_len @ E3 )
=> ! [A3: int] :
( ( word_succ @ E3 @ ( ring_1_of_int @ ( word @ E3 ) @ A3 ) )
= ( ring_1_of_int @ ( word @ E3 ) @ ( plus_plus @ int @ A3 @ ( one_one @ int ) ) ) ) ) ).
% wi_hom_succ
thf(fact_6581_word__arith__nat__Suc,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ( ( word_succ @ A )
= ( ^ [A6: word @ A] : ( semiring_1_of_nat @ ( word @ A ) @ ( suc @ ( semiring_1_unsigned @ A @ nat @ A6 ) ) ) ) ) ) ).
% word_arith_nat_Suc
thf(fact_6582_word__succ__alt,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ( ( word_succ @ A )
= ( ^ [A6: word @ A] : ( ring_1_of_int @ ( word @ A ) @ ( plus_plus @ int @ ( semiring_1_unsigned @ A @ int @ A6 ) @ ( one_one @ int ) ) ) ) ) ) ).
% word_succ_alt
thf(fact_6583_word__sp__01,axiom,
! [C: $tType,A: $tType,B: $tType,D6: $tType] :
( ( ( type_len @ D6 )
& ( type_len @ B )
& ( type_len @ A )
& ( type_len @ C ) )
=> ( ( ( word_succ @ A @ ( uminus_uminus @ ( word @ A ) @ ( one_one @ ( word @ A ) ) ) )
= ( zero_zero @ ( word @ A ) ) )
& ( ( word_succ @ B @ ( zero_zero @ ( word @ B ) ) )
= ( one_one @ ( word @ B ) ) )
& ( ( word_pred @ C @ ( zero_zero @ ( word @ C ) ) )
= ( uminus_uminus @ ( word @ C ) @ ( one_one @ ( word @ C ) ) ) )
& ( ( word_pred @ D6 @ ( one_one @ ( word @ D6 ) ) )
= ( zero_zero @ ( word @ D6 ) ) ) ) ) ).
% word_sp_01
thf(fact_6584_uint__word__arith__bintrs_I5_J,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [A3: word @ A] :
( ( semiring_1_unsigned @ A @ int @ ( word_succ @ A @ A3 ) )
= ( bit_se2584673776208193580ke_bit @ int @ ( type_len0_len_of @ A @ ( type2 @ A ) ) @ ( plus_plus @ int @ ( semiring_1_unsigned @ A @ int @ A3 ) @ ( one_one @ int ) ) ) ) ) ).
% uint_word_arith_bintrs(5)
thf(fact_6585_sint__word__ariths_I5_J,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [A3: word @ A] :
( ( ring_1_signed @ A @ int @ ( word_succ @ A @ A3 ) )
= ( bit_ri4674362597316999326ke_bit @ int @ ( minus_minus @ nat @ ( type_len0_len_of @ A @ ( type2 @ A ) ) @ ( one_one @ nat ) ) @ ( plus_plus @ int @ ( ring_1_signed @ A @ int @ A3 ) @ ( one_one @ int ) ) ) ) ) ).
% sint_word_ariths(5)
thf(fact_6586_unat__word__ariths_I3_J,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [A3: word @ A] :
( ( semiring_1_unsigned @ A @ nat @ ( word_succ @ A @ A3 ) )
= ( modulo_modulo @ nat @ ( suc @ ( semiring_1_unsigned @ A @ nat @ A3 ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( type_len0_len_of @ A @ ( type2 @ A ) ) ) ) ) ) ).
% unat_word_ariths(3)
thf(fact_6587_sless__eq__word__minus__numeral__minus__numeral,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [A3: num,B3: num] :
( ( word_sle @ A @ ( uminus_uminus @ ( word @ A ) @ ( numeral_numeral @ ( word @ A ) @ A3 ) ) @ ( uminus_uminus @ ( word @ A ) @ ( numeral_numeral @ ( word @ A ) @ B3 ) ) )
= ( 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 @ A3 ) ) ) @ ( bit_ri4674362597316999326ke_bit @ int @ ( minus_minus @ nat @ ( type_len0_len_of @ A @ ( type2 @ A ) ) @ ( one_one @ nat ) ) @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ B3 ) ) ) ) ) ) ).
% sless_eq_word_minus_numeral_minus_numeral
thf(fact_6588_sless__eq__word__numeral__minus__numeral,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [A3: num,B3: num] :
( ( word_sle @ A @ ( numeral_numeral @ ( word @ A ) @ A3 ) @ ( uminus_uminus @ ( word @ A ) @ ( numeral_numeral @ ( word @ A ) @ B3 ) ) )
= ( ord_less_eq @ int @ ( bit_ri4674362597316999326ke_bit @ int @ ( minus_minus @ nat @ ( type_len0_len_of @ A @ ( type2 @ A ) ) @ ( one_one @ nat ) ) @ ( numeral_numeral @ int @ A3 ) ) @ ( bit_ri4674362597316999326ke_bit @ int @ ( minus_minus @ nat @ ( type_len0_len_of @ A @ ( type2 @ A ) ) @ ( one_one @ nat ) ) @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ B3 ) ) ) ) ) ) ).
% sless_eq_word_numeral_minus_numeral
thf(fact_6589_word__sle__no,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [A3: num,B3: num] :
( ( word_sle @ A @ ( numeral_numeral @ ( word @ A ) @ A3 ) @ ( numeral_numeral @ ( word @ A ) @ B3 ) )
= ( ord_less_eq @ int @ ( ring_1_signed @ A @ int @ ( numeral_numeral @ ( word @ A ) @ A3 ) ) @ ( ring_1_signed @ A @ int @ ( numeral_numeral @ ( word @ A ) @ B3 ) ) ) ) ) ).
% word_sle_no
thf(fact_6590_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_6591_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_6592_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_6593_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_6594_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_6595_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_6596_sless__eq__word__numeral__numeral,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [A3: num,B3: num] :
( ( word_sle @ A @ ( numeral_numeral @ ( word @ A ) @ A3 ) @ ( numeral_numeral @ ( word @ A ) @ B3 ) )
= ( ord_less_eq @ int @ ( bit_ri4674362597316999326ke_bit @ int @ ( minus_minus @ nat @ ( type_len0_len_of @ A @ ( type2 @ A ) ) @ ( one_one @ nat ) ) @ ( numeral_numeral @ int @ A3 ) ) @ ( bit_ri4674362597316999326ke_bit @ int @ ( minus_minus @ nat @ ( type_len0_len_of @ A @ ( type2 @ A ) ) @ ( one_one @ nat ) ) @ ( numeral_numeral @ int @ B3 ) ) ) ) ) ).
% sless_eq_word_numeral_numeral
thf(fact_6597_sless__eq__word__minus__numeral__numeral,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [A3: num,B3: num] :
( ( word_sle @ A @ ( uminus_uminus @ ( word @ A ) @ ( numeral_numeral @ ( word @ A ) @ A3 ) ) @ ( numeral_numeral @ ( word @ A ) @ B3 ) )
= ( 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 @ A3 ) ) ) @ ( bit_ri4674362597316999326ke_bit @ int @ ( minus_minus @ nat @ ( type_len0_len_of @ A @ ( type2 @ A ) ) @ ( one_one @ nat ) ) @ ( numeral_numeral @ int @ B3 ) ) ) ) ) ).
% sless_eq_word_minus_numeral_numeral
thf(fact_6598_signed_Olift__Suc__mono__le,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [F2: nat > ( word @ A ),N: nat,N2: nat] :
( ! [N3: nat] : ( word_sle @ A @ ( F2 @ N3 ) @ ( F2 @ ( suc @ N3 ) ) )
=> ( ( ord_less_eq @ nat @ N @ N2 )
=> ( word_sle @ A @ ( F2 @ N ) @ ( F2 @ N2 ) ) ) ) ) ).
% signed.lift_Suc_mono_le
thf(fact_6599_signed_Olift__Suc__antimono__le,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [F2: nat > ( word @ A ),N: nat,N2: nat] :
( ! [N3: nat] : ( word_sle @ A @ ( F2 @ ( suc @ N3 ) ) @ ( F2 @ N3 ) )
=> ( ( ord_less_eq @ nat @ N @ N2 )
=> ( word_sle @ A @ ( F2 @ N2 ) @ ( F2 @ N ) ) ) ) ) ).
% signed.lift_Suc_antimono_le
thf(fact_6600_word__sle__eq,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ( ( word_sle @ A )
= ( ^ [A6: word @ A,B7: word @ A] : ( ord_less_eq @ int @ ( ring_1_signed @ A @ int @ A6 ) @ ( ring_1_signed @ A @ int @ B7 ) ) ) ) ) ).
% word_sle_eq
thf(fact_6601_word__0__sle__from__less,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [X2: word @ A] :
( ( ord_less @ ( word @ A ) @ X2 @ ( 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 ) ) @ X2 ) ) ) ).
% word_0_sle_from_less
thf(fact_6602_sless__word__minus__numeral__minus__numeral,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [A3: num,B3: num] :
( ( word_sless @ A @ ( uminus_uminus @ ( word @ A ) @ ( numeral_numeral @ ( word @ A ) @ A3 ) ) @ ( uminus_uminus @ ( word @ A ) @ ( numeral_numeral @ ( word @ A ) @ B3 ) ) )
= ( 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 @ A3 ) ) ) @ ( bit_ri4674362597316999326ke_bit @ int @ ( minus_minus @ nat @ ( type_len0_len_of @ A @ ( type2 @ A ) ) @ ( one_one @ nat ) ) @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ B3 ) ) ) ) ) ) ).
% sless_word_minus_numeral_minus_numeral
thf(fact_6603_sless__word__numeral__minus__numeral,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [A3: num,B3: num] :
( ( word_sless @ A @ ( numeral_numeral @ ( word @ A ) @ A3 ) @ ( uminus_uminus @ ( word @ A ) @ ( numeral_numeral @ ( word @ A ) @ B3 ) ) )
= ( ord_less @ int @ ( bit_ri4674362597316999326ke_bit @ int @ ( minus_minus @ nat @ ( type_len0_len_of @ A @ ( type2 @ A ) ) @ ( one_one @ nat ) ) @ ( numeral_numeral @ int @ A3 ) ) @ ( bit_ri4674362597316999326ke_bit @ int @ ( minus_minus @ nat @ ( type_len0_len_of @ A @ ( type2 @ A ) ) @ ( one_one @ nat ) ) @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ B3 ) ) ) ) ) ) ).
% sless_word_numeral_minus_numeral
thf(fact_6604_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_6605_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_6606_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_6607_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_6608_sless__word__numeral__numeral,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [A3: num,B3: num] :
( ( word_sless @ A @ ( numeral_numeral @ ( word @ A ) @ A3 ) @ ( numeral_numeral @ ( word @ A ) @ B3 ) )
= ( ord_less @ int @ ( bit_ri4674362597316999326ke_bit @ int @ ( minus_minus @ nat @ ( type_len0_len_of @ A @ ( type2 @ A ) ) @ ( one_one @ nat ) ) @ ( numeral_numeral @ int @ A3 ) ) @ ( bit_ri4674362597316999326ke_bit @ int @ ( minus_minus @ nat @ ( type_len0_len_of @ A @ ( type2 @ A ) ) @ ( one_one @ nat ) ) @ ( numeral_numeral @ int @ B3 ) ) ) ) ) ).
% sless_word_numeral_numeral
thf(fact_6609_sless__word__minus__numeral__numeral,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [A3: num,B3: num] :
( ( word_sless @ A @ ( uminus_uminus @ ( word @ A ) @ ( numeral_numeral @ ( word @ A ) @ A3 ) ) @ ( numeral_numeral @ ( word @ A ) @ B3 ) )
= ( 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 @ A3 ) ) ) @ ( bit_ri4674362597316999326ke_bit @ int @ ( minus_minus @ nat @ ( type_len0_len_of @ A @ ( type2 @ A ) ) @ ( one_one @ nat ) ) @ ( numeral_numeral @ int @ B3 ) ) ) ) ) ).
% sless_word_minus_numeral_numeral
thf(fact_6610_signed_Olift__Suc__mono__less__iff,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [F2: nat > ( word @ A ),N: nat,M: nat] :
( ! [N3: nat] : ( word_sless @ A @ ( F2 @ N3 ) @ ( F2 @ ( suc @ N3 ) ) )
=> ( ( word_sless @ A @ ( F2 @ N ) @ ( F2 @ M ) )
= ( ord_less @ nat @ N @ M ) ) ) ) ).
% signed.lift_Suc_mono_less_iff
thf(fact_6611_signed_Olift__Suc__mono__less,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [F2: nat > ( word @ A ),N: nat,N2: nat] :
( ! [N3: nat] : ( word_sless @ A @ ( F2 @ N3 ) @ ( F2 @ ( suc @ N3 ) ) )
=> ( ( ord_less @ nat @ N @ N2 )
=> ( word_sless @ A @ ( F2 @ N ) @ ( F2 @ N2 ) ) ) ) ) ).
% signed.lift_Suc_mono_less
thf(fact_6612_word__sless__sint__le,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [X2: word @ A,Y3: word @ A] :
( ( word_sless @ A @ X2 @ Y3 )
=> ( ord_less_eq @ int @ ( ring_1_signed @ A @ int @ X2 ) @ ( minus_minus @ int @ ( ring_1_signed @ A @ int @ Y3 ) @ ( one_one @ int ) ) ) ) ) ).
% word_sless_sint_le
thf(fact_6613_take__bit__word__minus__numeral,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [N: num,K: num] :
( ( bit_se2584673776208193580ke_bit @ ( word @ A ) @ ( numeral_numeral @ nat @ 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 ) ) @ ( numeral_numeral @ nat @ N ) ) @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ K ) ) ) ) ) ) ).
% take_bit_word_minus_numeral
thf(fact_6614_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_6615_min_Obounded__iff,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A3: A,B3: A,C3: A] :
( ( ord_less_eq @ A @ A3 @ ( ord_min @ A @ B3 @ C3 ) )
= ( ( ord_less_eq @ A @ A3 @ B3 )
& ( ord_less_eq @ A @ A3 @ C3 ) ) ) ) ).
% min.bounded_iff
thf(fact_6616_min_Oabsorb2,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [B3: A,A3: A] :
( ( ord_less_eq @ A @ B3 @ A3 )
=> ( ( ord_min @ A @ A3 @ B3 )
= B3 ) ) ) ).
% min.absorb2
thf(fact_6617_min_Oabsorb1,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A3: A,B3: A] :
( ( ord_less_eq @ A @ A3 @ B3 )
=> ( ( ord_min @ A @ A3 @ B3 )
= A3 ) ) ) ).
% min.absorb1
thf(fact_6618_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_6619_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_6620_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_6621_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_6622_min__simps_I2_J,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [B3: A,A3: A] :
( ( ord_less @ A @ B3 @ A3 )
=> ( ( ord_min @ A @ A3 @ B3 )
= B3 ) ) ) ).
% min_simps(2)
thf(fact_6623_min__simps_I1_J,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [A3: A,B3: A] :
( ( ord_less @ A @ A3 @ B3 )
=> ( ( ord_min @ A @ A3 @ B3 )
= A3 ) ) ) ).
% min_simps(1)
thf(fact_6624_min__less__self__conv_I2_J,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A3: A,B3: A] :
( ( ord_less @ A @ ( ord_min @ A @ A3 @ B3 ) @ B3 )
= ( ord_less @ A @ A3 @ B3 ) ) ) ).
% min_less_self_conv(2)
thf(fact_6625_min__less__self__conv_I1_J,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A3: A,B3: A] :
( ( ord_less @ A @ ( ord_min @ A @ A3 @ B3 ) @ A3 )
= ( ord_less @ A @ B3 @ A3 ) ) ) ).
% min_less_self_conv(1)
thf(fact_6626_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_6627_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_6628_min__less__iff__conj,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [Z: A,X2: A,Y3: A] :
( ( ord_less @ A @ Z @ ( ord_min @ A @ X2 @ Y3 ) )
= ( ( ord_less @ A @ Z @ X2 )
& ( ord_less @ A @ Z @ Y3 ) ) ) ) ).
% min_less_iff_conj
thf(fact_6629_min_Oabsorb4,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [B3: A,A3: A] :
( ( ord_less @ A @ B3 @ A3 )
=> ( ( ord_min @ A @ A3 @ B3 )
= B3 ) ) ) ).
% min.absorb4
thf(fact_6630_min_Oabsorb3,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A3: A,B3: A] :
( ( ord_less @ A @ A3 @ B3 )
=> ( ( ord_min @ A @ A3 @ B3 )
= A3 ) ) ) ).
% min.absorb3
thf(fact_6631_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_6632_min__0L,axiom,
! [N: nat] :
( ( ord_min @ nat @ ( zero_zero @ nat ) @ N )
= ( zero_zero @ nat ) ) ).
% min_0L
thf(fact_6633_min__0R,axiom,
! [N: nat] :
( ( ord_min @ nat @ N @ ( zero_zero @ nat ) )
= ( zero_zero @ nat ) ) ).
% min_0R
thf(fact_6634_max__min__same_I4_J,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [Y3: A,X2: A] :
( ( ord_max @ A @ Y3 @ ( ord_min @ A @ X2 @ Y3 ) )
= Y3 ) ) ).
% max_min_same(4)
thf(fact_6635_max__min__same_I3_J,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [X2: A,Y3: A] :
( ( ord_max @ A @ ( ord_min @ A @ X2 @ Y3 ) @ Y3 )
= Y3 ) ) ).
% max_min_same(3)
thf(fact_6636_max__min__same_I2_J,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [X2: A,Y3: A] :
( ( ord_max @ A @ ( ord_min @ A @ X2 @ Y3 ) @ X2 )
= X2 ) ) ).
% max_min_same(2)
thf(fact_6637_max__min__same_I1_J,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [X2: A,Y3: A] :
( ( ord_max @ A @ X2 @ ( ord_min @ A @ X2 @ Y3 ) )
= X2 ) ) ).
% max_min_same(1)
thf(fact_6638_min__minus,axiom,
! [M: nat,K: nat] :
( ( ord_min @ nat @ M @ ( minus_minus @ nat @ M @ K ) )
= ( minus_minus @ nat @ M @ K ) ) ).
% min_minus
thf(fact_6639_min__minus_H,axiom,
! [M: nat,K: nat] :
( ( ord_min @ nat @ ( minus_minus @ nat @ M @ K ) @ M )
= ( minus_minus @ nat @ M @ K ) ) ).
% min_minus'
thf(fact_6640_min__0__1_I4_J,axiom,
! [A: $tType] :
( ( linord181362715937106298miring @ A )
=> ! [X2: num] :
( ( ord_min @ A @ ( numeral_numeral @ A @ X2 ) @ ( zero_zero @ A ) )
= ( zero_zero @ A ) ) ) ).
% min_0_1(4)
thf(fact_6641_min__0__1_I3_J,axiom,
! [A: $tType] :
( ( linord181362715937106298miring @ A )
=> ! [X2: num] :
( ( ord_min @ A @ ( zero_zero @ A ) @ ( numeral_numeral @ A @ X2 ) )
= ( zero_zero @ A ) ) ) ).
% min_0_1(3)
thf(fact_6642_min__number__of_I1_J,axiom,
! [A: $tType] :
( ( ( numeral @ A )
& ( ord @ A ) )
=> ! [U: num,V: num] :
( ( ( ord_less_eq @ A @ ( numeral_numeral @ A @ U ) @ ( numeral_numeral @ A @ V ) )
=> ( ( ord_min @ A @ ( numeral_numeral @ A @ U ) @ ( numeral_numeral @ A @ V ) )
= ( numeral_numeral @ A @ U ) ) )
& ( ~ ( ord_less_eq @ A @ ( numeral_numeral @ A @ U ) @ ( numeral_numeral @ A @ V ) )
=> ( ( ord_min @ A @ ( numeral_numeral @ A @ U ) @ ( numeral_numeral @ A @ V ) )
= ( numeral_numeral @ A @ V ) ) ) ) ) ).
% min_number_of(1)
thf(fact_6643_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_6644_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_6645_min__0__1_I6_J,axiom,
! [A: $tType] :
( ( linord181362715937106298miring @ A )
=> ! [X2: num] :
( ( ord_min @ A @ ( numeral_numeral @ A @ X2 ) @ ( one_one @ A ) )
= ( one_one @ A ) ) ) ).
% min_0_1(6)
thf(fact_6646_min__0__1_I5_J,axiom,
! [A: $tType] :
( ( linord181362715937106298miring @ A )
=> ! [X2: num] :
( ( ord_min @ A @ ( one_one @ A ) @ ( numeral_numeral @ A @ X2 ) )
= ( one_one @ A ) ) ) ).
% min_0_1(5)
thf(fact_6647_min__Suc__gt_I1_J,axiom,
! [A3: nat,B3: nat] :
( ( ord_less @ nat @ A3 @ B3 )
=> ( ( ord_min @ nat @ ( suc @ A3 ) @ B3 )
= ( suc @ A3 ) ) ) ).
% min_Suc_gt(1)
thf(fact_6648_min__Suc__gt_I2_J,axiom,
! [A3: nat,B3: nat] :
( ( ord_less @ nat @ A3 @ B3 )
=> ( ( ord_min @ nat @ B3 @ ( suc @ A3 ) )
= ( suc @ A3 ) ) ) ).
% min_Suc_gt(2)
thf(fact_6649_rev__min__pm1,axiom,
! [A3: nat,B3: nat] :
( ( plus_plus @ nat @ ( minus_minus @ nat @ A3 @ B3 ) @ ( ord_min @ nat @ B3 @ A3 ) )
= A3 ) ).
% rev_min_pm1
thf(fact_6650_rev__min__pm,axiom,
! [B3: nat,A3: nat] :
( ( plus_plus @ nat @ ( ord_min @ nat @ B3 @ A3 ) @ ( minus_minus @ nat @ A3 @ B3 ) )
= A3 ) ).
% rev_min_pm
thf(fact_6651_min__pm1,axiom,
! [A3: nat,B3: nat] :
( ( plus_plus @ nat @ ( minus_minus @ nat @ A3 @ B3 ) @ ( ord_min @ nat @ A3 @ B3 ) )
= A3 ) ).
% min_pm1
thf(fact_6652_min__pm,axiom,
! [A3: nat,B3: nat] :
( ( plus_plus @ nat @ ( ord_min @ nat @ A3 @ B3 ) @ ( minus_minus @ nat @ A3 @ B3 ) )
= A3 ) ).
% min_pm
thf(fact_6653_min__number__of_I2_J,axiom,
! [A: $tType] :
( ( ( uminus @ A )
& ( numeral @ A )
& ( ord @ A ) )
=> ! [U: num,V: num] :
( ( ( ord_less_eq @ A @ ( numeral_numeral @ A @ U ) @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ V ) ) )
=> ( ( ord_min @ A @ ( numeral_numeral @ A @ U ) @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ V ) ) )
= ( numeral_numeral @ A @ U ) ) )
& ( ~ ( ord_less_eq @ A @ ( numeral_numeral @ A @ U ) @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ V ) ) )
=> ( ( ord_min @ A @ ( numeral_numeral @ A @ U ) @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ V ) ) )
= ( uminus_uminus @ A @ ( numeral_numeral @ A @ V ) ) ) ) ) ) ).
% min_number_of(2)
thf(fact_6654_min__number__of_I3_J,axiom,
! [A: $tType] :
( ( ( uminus @ A )
& ( numeral @ A )
& ( ord @ A ) )
=> ! [U: num,V: num] :
( ( ( ord_less_eq @ A @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ U ) ) @ ( numeral_numeral @ A @ V ) )
=> ( ( ord_min @ A @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ U ) ) @ ( numeral_numeral @ A @ V ) )
= ( uminus_uminus @ A @ ( numeral_numeral @ A @ U ) ) ) )
& ( ~ ( ord_less_eq @ A @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ U ) ) @ ( numeral_numeral @ A @ V ) )
=> ( ( ord_min @ A @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ U ) ) @ ( numeral_numeral @ A @ V ) )
= ( numeral_numeral @ A @ V ) ) ) ) ) ).
% min_number_of(3)
thf(fact_6655_min__number__of_I4_J,axiom,
! [A: $tType] :
( ( ( uminus @ A )
& ( numeral @ A )
& ( ord @ A ) )
=> ! [U: num,V: num] :
( ( ( ord_less_eq @ A @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ U ) ) @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ V ) ) )
=> ( ( ord_min @ A @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ U ) ) @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ V ) ) )
= ( uminus_uminus @ A @ ( numeral_numeral @ A @ U ) ) ) )
& ( ~ ( ord_less_eq @ A @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ U ) ) @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ V ) ) )
=> ( ( ord_min @ A @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ U ) ) @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ V ) ) )
= ( uminus_uminus @ A @ ( numeral_numeral @ A @ V ) ) ) ) ) ) ).
% min_number_of(4)
thf(fact_6656_min__numeral__Suc,axiom,
! [K: num,N: nat] :
( ( ord_min @ nat @ ( numeral_numeral @ nat @ K ) @ ( suc @ N ) )
= ( suc @ ( ord_min @ nat @ ( pred_numeral @ K ) @ N ) ) ) ).
% min_numeral_Suc
thf(fact_6657_min__Suc__numeral,axiom,
! [N: nat,K: num] :
( ( ord_min @ nat @ ( suc @ N ) @ ( numeral_numeral @ nat @ K ) )
= ( suc @ ( ord_min @ nat @ N @ ( pred_numeral @ K ) ) ) ) ).
% min_Suc_numeral
thf(fact_6658_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_6659_take__bit__word__numeral,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [N: num,K: num] :
( ( bit_se2584673776208193580ke_bit @ ( word @ A ) @ ( numeral_numeral @ nat @ 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 ) ) @ ( numeral_numeral @ nat @ N ) ) @ ( numeral_numeral @ int @ K ) ) ) ) ) ).
% take_bit_word_numeral
thf(fact_6660_min__def__raw,axiom,
! [A: $tType] :
( ( ord @ A )
=> ( ( ord_min @ A )
= ( ^ [A6: A,B7: A] : ( if @ A @ ( ord_less_eq @ A @ A6 @ B7 ) @ A6 @ B7 ) ) ) ) ).
% min_def_raw
thf(fact_6661_min__le__iff__disj,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [X2: A,Y3: A,Z: A] :
( ( ord_less_eq @ A @ ( ord_min @ A @ X2 @ Y3 ) @ Z )
= ( ( ord_less_eq @ A @ X2 @ Z )
| ( ord_less_eq @ A @ Y3 @ Z ) ) ) ) ).
% min_le_iff_disj
thf(fact_6662_min_OcoboundedI2,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [B3: A,C3: A,A3: A] :
( ( ord_less_eq @ A @ B3 @ C3 )
=> ( ord_less_eq @ A @ ( ord_min @ A @ A3 @ B3 ) @ C3 ) ) ) ).
% min.coboundedI2
thf(fact_6663_min_OcoboundedI1,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A3: A,C3: A,B3: A] :
( ( ord_less_eq @ A @ A3 @ C3 )
=> ( ord_less_eq @ A @ ( ord_min @ A @ A3 @ B3 ) @ C3 ) ) ) ).
% min.coboundedI1
thf(fact_6664_min_Oabsorb__iff2,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ( ( ord_less_eq @ A )
= ( ^ [B7: A,A6: A] :
( ( ord_min @ A @ A6 @ B7 )
= B7 ) ) ) ) ).
% min.absorb_iff2
thf(fact_6665_min_Oabsorb__iff1,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ( ( ord_less_eq @ A )
= ( ^ [A6: A,B7: A] :
( ( ord_min @ A @ A6 @ B7 )
= A6 ) ) ) ) ).
% min.absorb_iff1
thf(fact_6666_min_Ocobounded2,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A3: A,B3: A] : ( ord_less_eq @ A @ ( ord_min @ A @ A3 @ B3 ) @ B3 ) ) ).
% min.cobounded2
thf(fact_6667_min_Ocobounded1,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A3: A,B3: A] : ( ord_less_eq @ A @ ( ord_min @ A @ A3 @ B3 ) @ A3 ) ) ).
% min.cobounded1
thf(fact_6668_min_Oorder__iff,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ( ( ord_less_eq @ A )
= ( ^ [A6: A,B7: A] :
( A6
= ( ord_min @ A @ A6 @ B7 ) ) ) ) ) ).
% min.order_iff
thf(fact_6669_min_OboundedI,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A3: A,B3: A,C3: A] :
( ( ord_less_eq @ A @ A3 @ B3 )
=> ( ( ord_less_eq @ A @ A3 @ C3 )
=> ( ord_less_eq @ A @ A3 @ ( ord_min @ A @ B3 @ C3 ) ) ) ) ) ).
% min.boundedI
thf(fact_6670_min_OboundedE,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A3: A,B3: A,C3: A] :
( ( ord_less_eq @ A @ A3 @ ( ord_min @ A @ B3 @ C3 ) )
=> ~ ( ( ord_less_eq @ A @ A3 @ B3 )
=> ~ ( ord_less_eq @ A @ A3 @ C3 ) ) ) ) ).
% min.boundedE
thf(fact_6671_min_OorderI,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A3: A,B3: A] :
( ( A3
= ( ord_min @ A @ A3 @ B3 ) )
=> ( ord_less_eq @ A @ A3 @ B3 ) ) ) ).
% min.orderI
thf(fact_6672_min_OorderE,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A3: A,B3: A] :
( ( ord_less_eq @ A @ A3 @ B3 )
=> ( A3
= ( ord_min @ A @ A3 @ B3 ) ) ) ) ).
% min.orderE
thf(fact_6673_min_Omono,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A3: A,C3: A,B3: A,D2: A] :
( ( ord_less_eq @ A @ A3 @ C3 )
=> ( ( ord_less_eq @ A @ B3 @ D2 )
=> ( ord_less_eq @ A @ ( ord_min @ A @ A3 @ B3 ) @ ( ord_min @ A @ C3 @ D2 ) ) ) ) ) ).
% min.mono
thf(fact_6674_min__def,axiom,
! [A: $tType] :
( ( ord @ A )
=> ( ( ord_min @ A )
= ( ^ [A6: A,B7: A] : ( if @ A @ ( ord_less_eq @ A @ A6 @ B7 ) @ A6 @ B7 ) ) ) ) ).
% min_def
thf(fact_6675_min__absorb1,axiom,
! [A: $tType] :
( ( ord @ A )
=> ! [X2: A,Y3: A] :
( ( ord_less_eq @ A @ X2 @ Y3 )
=> ( ( ord_min @ A @ X2 @ Y3 )
= X2 ) ) ) ).
% min_absorb1
thf(fact_6676_min__absorb2,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [Y3: A,X2: A] :
( ( ord_less_eq @ A @ Y3 @ X2 )
=> ( ( ord_min @ A @ X2 @ Y3 )
= Y3 ) ) ) ).
% min_absorb2
thf(fact_6677_min__add__distrib__right,axiom,
! [A: $tType] :
( ( ordere2412721322843649153imp_le @ A )
=> ! [X2: A,Y3: A,Z: A] :
( ( plus_plus @ A @ X2 @ ( ord_min @ A @ Y3 @ Z ) )
= ( ord_min @ A @ ( plus_plus @ A @ X2 @ Y3 ) @ ( plus_plus @ A @ X2 @ Z ) ) ) ) ).
% min_add_distrib_right
thf(fact_6678_min__add__distrib__left,axiom,
! [A: $tType] :
( ( ordere2412721322843649153imp_le @ A )
=> ! [X2: A,Y3: A,Z: A] :
( ( plus_plus @ A @ ( ord_min @ A @ X2 @ Y3 ) @ Z )
= ( ord_min @ A @ ( plus_plus @ A @ X2 @ Z ) @ ( plus_plus @ A @ Y3 @ Z ) ) ) ) ).
% min_add_distrib_left
thf(fact_6679_min__diff__distrib__left,axiom,
! [A: $tType] :
( ( ordered_ab_group_add @ A )
=> ! [X2: A,Y3: A,Z: A] :
( ( minus_minus @ A @ ( ord_min @ A @ X2 @ Y3 ) @ Z )
= ( ord_min @ A @ ( minus_minus @ A @ X2 @ Z ) @ ( minus_minus @ A @ Y3 @ Z ) ) ) ) ).
% min_diff_distrib_left
thf(fact_6680_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_6681_min__less__iff__disj,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [X2: A,Y3: A,Z: A] :
( ( ord_less @ A @ ( ord_min @ A @ X2 @ Y3 ) @ Z )
= ( ( ord_less @ A @ X2 @ Z )
| ( ord_less @ A @ Y3 @ Z ) ) ) ) ).
% min_less_iff_disj
thf(fact_6682_min_Ostrict__boundedE,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A3: A,B3: A,C3: A] :
( ( ord_less @ A @ A3 @ ( ord_min @ A @ B3 @ C3 ) )
=> ~ ( ( ord_less @ A @ A3 @ B3 )
=> ~ ( ord_less @ A @ A3 @ C3 ) ) ) ) ).
% min.strict_boundedE
thf(fact_6683_min_Ostrict__order__iff,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ( ( ord_less @ A )
= ( ^ [A6: A,B7: A] :
( ( A6
= ( ord_min @ A @ A6 @ B7 ) )
& ( A6 != B7 ) ) ) ) ) ).
% min.strict_order_iff
thf(fact_6684_min_Ostrict__coboundedI1,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A3: A,C3: A,B3: A] :
( ( ord_less @ A @ A3 @ C3 )
=> ( ord_less @ A @ ( ord_min @ A @ A3 @ B3 ) @ C3 ) ) ) ).
% min.strict_coboundedI1
thf(fact_6685_min_Ostrict__coboundedI2,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [B3: A,C3: A,A3: A] :
( ( ord_less @ A @ B3 @ C3 )
=> ( ord_less @ A @ ( ord_min @ A @ A3 @ B3 ) @ C3 ) ) ) ).
% min.strict_coboundedI2
thf(fact_6686_min__max__distrib2,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A3: A,B3: A,C3: A] :
( ( ord_min @ A @ A3 @ ( ord_max @ A @ B3 @ C3 ) )
= ( ord_max @ A @ ( ord_min @ A @ A3 @ B3 ) @ ( ord_min @ A @ A3 @ C3 ) ) ) ) ).
% min_max_distrib2
thf(fact_6687_min__max__distrib1,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [B3: A,C3: A,A3: A] :
( ( ord_min @ A @ ( ord_max @ A @ B3 @ C3 ) @ A3 )
= ( ord_max @ A @ ( ord_min @ A @ B3 @ A3 ) @ ( ord_min @ A @ C3 @ A3 ) ) ) ) ).
% min_max_distrib1
thf(fact_6688_max__min__distrib2,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A3: A,B3: A,C3: A] :
( ( ord_max @ A @ A3 @ ( ord_min @ A @ B3 @ C3 ) )
= ( ord_min @ A @ ( ord_max @ A @ A3 @ B3 ) @ ( ord_max @ A @ A3 @ C3 ) ) ) ) ).
% max_min_distrib2
thf(fact_6689_max__min__distrib1,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [B3: A,C3: A,A3: A] :
( ( ord_max @ A @ ( ord_min @ A @ B3 @ C3 ) @ A3 )
= ( ord_min @ A @ ( ord_max @ A @ B3 @ A3 ) @ ( ord_max @ A @ C3 @ A3 ) ) ) ) ).
% max_min_distrib1
thf(fact_6690_of__nat__min,axiom,
! [A: $tType] :
( ( linord181362715937106298miring @ A )
=> ! [X2: nat,Y3: nat] :
( ( semiring_1_of_nat @ A @ ( ord_min @ nat @ X2 @ Y3 ) )
= ( ord_min @ A @ ( semiring_1_of_nat @ A @ X2 ) @ ( semiring_1_of_nat @ A @ Y3 ) ) ) ) ).
% of_nat_min
thf(fact_6691_minus__min__eq__max,axiom,
! [A: $tType] :
( ( linord5086331880401160121up_add @ A )
=> ! [X2: A,Y3: A] :
( ( uminus_uminus @ A @ ( ord_min @ A @ X2 @ Y3 ) )
= ( ord_max @ A @ ( uminus_uminus @ A @ X2 ) @ ( uminus_uminus @ A @ Y3 ) ) ) ) ).
% minus_min_eq_max
thf(fact_6692_minus__max__eq__min,axiom,
! [A: $tType] :
( ( linord5086331880401160121up_add @ A )
=> ! [X2: A,Y3: A] :
( ( uminus_uminus @ A @ ( ord_max @ A @ X2 @ Y3 ) )
= ( ord_min @ A @ ( uminus_uminus @ A @ X2 ) @ ( uminus_uminus @ A @ Y3 ) ) ) ) ).
% minus_max_eq_min
thf(fact_6693_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_6694_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_6695_concat__bit__assoc__sym,axiom,
! [M: nat,N: nat,K: int,L: int,R2: int] :
( ( bit_concat_bit @ M @ ( bit_concat_bit @ N @ K @ L ) @ R2 )
= ( bit_concat_bit @ ( ord_min @ nat @ M @ N ) @ K @ ( bit_concat_bit @ ( minus_minus @ nat @ M @ N ) @ L @ R2 ) ) ) ).
% concat_bit_assoc_sym
thf(fact_6696_take__bit__concat__bit__eq,axiom,
! [M: nat,N: nat,K: int,L: int] :
( ( bit_se2584673776208193580ke_bit @ int @ M @ ( bit_concat_bit @ N @ K @ L ) )
= ( bit_concat_bit @ ( ord_min @ nat @ M @ N ) @ K @ ( bit_se2584673776208193580ke_bit @ int @ ( minus_minus @ nat @ M @ N ) @ L ) ) ) ).
% take_bit_concat_bit_eq
thf(fact_6697_min__mult__distrib__right,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [P6: A,X2: A,Y3: A] :
( ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ P6 )
=> ( ( times_times @ A @ ( ord_min @ A @ X2 @ Y3 ) @ P6 )
= ( ord_min @ A @ ( times_times @ A @ X2 @ P6 ) @ ( times_times @ A @ Y3 @ P6 ) ) ) )
& ( ~ ( ord_less_eq @ A @ ( zero_zero @ A ) @ P6 )
=> ( ( times_times @ A @ ( ord_min @ A @ X2 @ Y3 ) @ P6 )
= ( ord_max @ A @ ( times_times @ A @ X2 @ P6 ) @ ( times_times @ A @ Y3 @ P6 ) ) ) ) ) ) ).
% min_mult_distrib_right
thf(fact_6698_max__mult__distrib__right,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [P6: A,X2: A,Y3: A] :
( ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ P6 )
=> ( ( times_times @ A @ ( ord_max @ A @ X2 @ Y3 ) @ P6 )
= ( ord_max @ A @ ( times_times @ A @ X2 @ P6 ) @ ( times_times @ A @ Y3 @ P6 ) ) ) )
& ( ~ ( ord_less_eq @ A @ ( zero_zero @ A ) @ P6 )
=> ( ( times_times @ A @ ( ord_max @ A @ X2 @ Y3 ) @ P6 )
= ( ord_min @ A @ ( times_times @ A @ X2 @ P6 ) @ ( times_times @ A @ Y3 @ P6 ) ) ) ) ) ) ).
% max_mult_distrib_right
thf(fact_6699_min__mult__distrib__left,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [P6: A,X2: A,Y3: A] :
( ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ P6 )
=> ( ( times_times @ A @ P6 @ ( ord_min @ A @ X2 @ Y3 ) )
= ( ord_min @ A @ ( times_times @ A @ P6 @ X2 ) @ ( times_times @ A @ P6 @ Y3 ) ) ) )
& ( ~ ( ord_less_eq @ A @ ( zero_zero @ A ) @ P6 )
=> ( ( times_times @ A @ P6 @ ( ord_min @ A @ X2 @ Y3 ) )
= ( ord_max @ A @ ( times_times @ A @ P6 @ X2 ) @ ( times_times @ A @ P6 @ Y3 ) ) ) ) ) ) ).
% min_mult_distrib_left
thf(fact_6700_max__mult__distrib__left,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [P6: A,X2: A,Y3: A] :
( ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ P6 )
=> ( ( times_times @ A @ P6 @ ( ord_max @ A @ X2 @ Y3 ) )
= ( ord_max @ A @ ( times_times @ A @ P6 @ X2 ) @ ( times_times @ A @ P6 @ Y3 ) ) ) )
& ( ~ ( ord_less_eq @ A @ ( zero_zero @ A ) @ P6 )
=> ( ( times_times @ A @ P6 @ ( ord_max @ A @ X2 @ Y3 ) )
= ( ord_min @ A @ ( times_times @ A @ P6 @ X2 ) @ ( times_times @ A @ P6 @ Y3 ) ) ) ) ) ) ).
% max_mult_distrib_left
thf(fact_6701_min__divide__distrib__right,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [P6: A,X2: A,Y3: A] :
( ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ P6 )
=> ( ( divide_divide @ A @ ( ord_min @ A @ X2 @ Y3 ) @ P6 )
= ( ord_min @ A @ ( divide_divide @ A @ X2 @ P6 ) @ ( divide_divide @ A @ Y3 @ P6 ) ) ) )
& ( ~ ( ord_less_eq @ A @ ( zero_zero @ A ) @ P6 )
=> ( ( divide_divide @ A @ ( ord_min @ A @ X2 @ Y3 ) @ P6 )
= ( ord_max @ A @ ( divide_divide @ A @ X2 @ P6 ) @ ( divide_divide @ A @ Y3 @ P6 ) ) ) ) ) ) ).
% min_divide_distrib_right
thf(fact_6702_max__divide__distrib__right,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [P6: A,X2: A,Y3: A] :
( ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ P6 )
=> ( ( divide_divide @ A @ ( ord_max @ A @ X2 @ Y3 ) @ P6 )
= ( ord_max @ A @ ( divide_divide @ A @ X2 @ P6 ) @ ( divide_divide @ A @ Y3 @ P6 ) ) ) )
& ( ~ ( ord_less_eq @ A @ ( zero_zero @ A ) @ P6 )
=> ( ( divide_divide @ A @ ( ord_max @ A @ X2 @ Y3 ) @ P6 )
= ( ord_min @ A @ ( divide_divide @ A @ X2 @ P6 ) @ ( divide_divide @ A @ Y3 @ P6 ) ) ) ) ) ) ).
% max_divide_distrib_right
thf(fact_6703_mod__exp__eq,axiom,
! [A: $tType] :
( ( bit_semiring_bits @ A )
=> ! [A3: A,M: nat,N: nat] :
( ( modulo_modulo @ A @ ( modulo_modulo @ A @ A3 @ ( power_power @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ M ) ) @ ( power_power @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ N ) )
= ( modulo_modulo @ A @ A3 @ ( power_power @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ ( ord_min @ nat @ M @ N ) ) ) ) ) ).
% mod_exp_eq
thf(fact_6704_mod__mod__power,axiom,
! [K: nat,M: nat,N: nat] :
( ( modulo_modulo @ nat @ ( modulo_modulo @ nat @ K @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ M ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N ) )
= ( modulo_modulo @ nat @ K @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( ord_min @ nat @ M @ N ) ) ) ) ).
% mod_mod_power
thf(fact_6705_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_6706_mask__mod__exp,axiom,
! [A: $tType] :
( ( euclid5411537665997757685th_nat @ A )
=> ! [N: nat,M: nat] :
( ( modulo_modulo @ A @ ( minus_minus @ A @ ( power_power @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ N ) @ ( one_one @ A ) ) @ ( power_power @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ M ) )
= ( minus_minus @ A @ ( power_power @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ ( ord_min @ nat @ M @ N ) ) @ ( one_one @ A ) ) ) ) ).
% mask_mod_exp
thf(fact_6707_bit__slice__iff,axiom,
! [B: $tType,A: $tType] :
( ( ( type_len @ A )
& ( type_len @ B ) )
=> ! [M: nat,W: word @ A,N: nat] :
( ( bit_se5641148757651400278ts_bit @ ( word @ B ) @ ( slice2 @ A @ B @ M @ W ) @ 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 ) @ W @ ( 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_6708_bit__slice1__iff,axiom,
! [B: $tType,A: $tType] :
( ( ( type_len @ A )
& ( type_len @ B ) )
=> ! [M: nat,W: word @ A,N: nat] :
( ( bit_se5641148757651400278ts_bit @ ( word @ B ) @ ( slice1 @ A @ B @ M @ W ) @ 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 ) @ W @ ( 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_6709_unat__mask,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [N: nat] :
( ( semiring_1_unsigned @ A @ nat @ ( bit_se2239418461657761734s_mask @ ( word @ A ) @ N ) )
= ( minus_minus @ nat @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( ord_min @ nat @ N @ ( type_len0_len_of @ A @ ( type2 @ A ) ) ) ) @ ( one_one @ nat ) ) ) ) ).
% unat_mask
thf(fact_6710_bit__horner__sum__bit__word__iff,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [Bs: 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 ) ) @ Bs ) @ N )
= ( ( ord_less @ nat @ N @ ( ord_min @ nat @ ( type_len0_len_of @ A @ ( type2 @ A ) ) @ ( size_size @ ( list @ $o ) @ Bs ) ) )
& ( nth @ $o @ Bs @ N ) ) ) ) ).
% bit_horner_sum_bit_word_iff
thf(fact_6711_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_6712_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_6713_min__enat__simps_I3_J,axiom,
! [Q3: extended_enat] :
( ( ord_min @ extended_enat @ ( zero_zero @ extended_enat ) @ Q3 )
= ( zero_zero @ extended_enat ) ) ).
% min_enat_simps(3)
thf(fact_6714_min__enat__simps_I2_J,axiom,
! [Q3: extended_enat] :
( ( ord_min @ extended_enat @ Q3 @ ( zero_zero @ extended_enat ) )
= ( zero_zero @ extended_enat ) ) ).
% min_enat_simps(2)
thf(fact_6715_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_6716_bit__minus__1__iff,axiom,
! [A: $tType] :
( ( bit_ri3973907225187159222ations @ A )
=> ! [N: nat] :
( ( bit_se5641148757651400278ts_bit @ A @ ( uminus_uminus @ A @ ( one_one @ A ) ) @ N )
= ( bit_se6407376104438227557le_bit @ A @ ( type2 @ A ) @ N ) ) ) ).
% bit_minus_1_iff
thf(fact_6717_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_6718_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_6719_possible__bit__def,axiom,
! [A: $tType] :
( ( bit_semiring_bits @ A )
=> ( ( bit_se6407376104438227557le_bit @ A )
= ( ^ [Tyrep2: itself @ A,N4: nat] :
( ( power_power @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ N4 )
!= ( zero_zero @ A ) ) ) ) ) ).
% possible_bit_def
thf(fact_6720_bit__minus__iff,axiom,
! [A: $tType] :
( ( bit_ri3973907225187159222ations @ A )
=> ! [A3: A,N: nat] :
( ( bit_se5641148757651400278ts_bit @ A @ ( uminus_uminus @ A @ A3 ) @ N )
= ( ( bit_se6407376104438227557le_bit @ A @ ( type2 @ A ) @ N )
& ~ ( bit_se5641148757651400278ts_bit @ A @ ( minus_minus @ A @ A3 @ ( one_one @ A ) ) @ N ) ) ) ) ).
% bit_minus_iff
thf(fact_6721_bit__twiddle__min,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [Y3: word @ A,X2: word @ A] :
( ( bit_se5824344971392196577ns_xor @ ( word @ A ) @ Y3 @ ( bit_se5824344872417868541ns_and @ ( word @ A ) @ ( bit_se5824344971392196577ns_xor @ ( word @ A ) @ X2 @ Y3 ) @ ( if @ ( word @ A ) @ ( ord_less @ ( word @ A ) @ X2 @ Y3 ) @ ( uminus_uminus @ ( word @ A ) @ ( one_one @ ( word @ A ) ) ) @ ( zero_zero @ ( word @ A ) ) ) ) )
= ( ord_min @ ( word @ A ) @ X2 @ Y3 ) ) ) ).
% bit_twiddle_min
thf(fact_6722_bit__push__bit__iff,axiom,
! [A: $tType] :
( ( bit_se359711467146920520ations @ A )
=> ! [M: nat,A3: A,N: nat] :
( ( bit_se5641148757651400278ts_bit @ A @ ( bit_se4730199178511100633sh_bit @ A @ M @ A3 ) @ N )
= ( ( ord_less_eq @ nat @ M @ N )
& ( bit_se6407376104438227557le_bit @ A @ ( type2 @ A ) @ N )
& ( bit_se5641148757651400278ts_bit @ A @ A3 @ ( minus_minus @ nat @ N @ M ) ) ) ) ) ).
% bit_push_bit_iff
thf(fact_6723_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_6724_bit__exp__iff,axiom,
! [A: $tType] :
( ( bit_semiring_bits @ A )
=> ! [M: nat,N: nat] :
( ( bit_se5641148757651400278ts_bit @ A @ ( power_power @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ M ) @ N )
= ( ( bit_se6407376104438227557le_bit @ A @ ( type2 @ A ) @ N )
& ( M = N ) ) ) ) ).
% bit_exp_iff
thf(fact_6725_bit__2__iff,axiom,
! [A: $tType] :
( ( bit_semiring_bits @ A )
=> ! [N: nat] :
( ( bit_se5641148757651400278ts_bit @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ N )
= ( ( bit_se6407376104438227557le_bit @ A @ ( type2 @ A ) @ ( one_one @ nat ) )
& ( N
= ( one_one @ nat ) ) ) ) ) ).
% bit_2_iff
thf(fact_6726_bit__not__exp__iff,axiom,
! [A: $tType] :
( ( bit_ri3973907225187159222ations @ A )
=> ! [M: nat,N: nat] :
( ( bit_se5641148757651400278ts_bit @ A @ ( bit_ri4277139882892585799ns_not @ A @ ( power_power @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ M ) ) @ N )
= ( ( bit_se6407376104438227557le_bit @ A @ ( type2 @ A ) @ N )
& ( N != M ) ) ) ) ).
% bit_not_exp_iff
thf(fact_6727_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_6728_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_6729_bit__double__iff,axiom,
! [A: $tType] :
( ( bit_semiring_bits @ A )
=> ! [A3: A,N: nat] :
( ( bit_se5641148757651400278ts_bit @ A @ ( times_times @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ A3 ) @ N )
= ( ( bit_se5641148757651400278ts_bit @ A @ A3 @ ( minus_minus @ nat @ N @ ( one_one @ nat ) ) )
& ( N
!= ( zero_zero @ nat ) )
& ( bit_se6407376104438227557le_bit @ A @ ( type2 @ A ) @ N ) ) ) ) ).
% bit_double_iff
thf(fact_6730_bit__signed__iff,axiom,
! [A: $tType,B: $tType] :
( ( ( type_len @ B )
& ( bit_ri3973907225187159222ations @ A ) )
=> ! [W: word @ B,N: nat] :
( ( bit_se5641148757651400278ts_bit @ A @ ( ring_1_signed @ B @ A @ W ) @ N )
= ( ( bit_se6407376104438227557le_bit @ A @ ( type2 @ A ) @ N )
& ( bit_se5641148757651400278ts_bit @ ( word @ B ) @ W @ ( ord_min @ nat @ ( minus_minus @ nat @ ( type_len0_len_of @ B @ ( type2 @ B ) ) @ ( suc @ ( zero_zero @ nat ) ) ) @ N ) ) ) ) ) ).
% bit_signed_iff
thf(fact_6731_uint__word__rotr__eq,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [N: nat,W: word @ A] :
( ( semiring_1_unsigned @ A @ int @ ( word_rotr @ A @ N @ W ) )
= ( bit_concat_bit @ ( minus_minus @ nat @ ( type_len0_len_of @ A @ ( type2 @ A ) ) @ ( modulo_modulo @ nat @ N @ ( type_len0_len_of @ A @ ( type2 @ A ) ) ) ) @ ( bit_se4197421643247451524op_bit @ int @ ( modulo_modulo @ nat @ N @ ( type_len0_len_of @ A @ ( type2 @ A ) ) ) @ ( semiring_1_unsigned @ A @ int @ W ) ) @ ( semiring_1_unsigned @ A @ int @ ( bit_se2584673776208193580ke_bit @ ( word @ A ) @ ( modulo_modulo @ nat @ N @ ( type_len0_len_of @ A @ ( type2 @ A ) ) ) @ W ) ) ) ) ) ).
% uint_word_rotr_eq
thf(fact_6732_word__roti_Oabs__eq,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [Xa: int,X2: int] :
( ( word_roti @ A @ Xa @ ( word2 @ A @ X2 ) )
= ( word2 @ A @ ( bit_concat_bit @ ( minus_minus @ nat @ ( type_len0_len_of @ A @ ( type2 @ A ) ) @ ( nat2 @ ( modulo_modulo @ int @ Xa @ ( semiring_1_of_nat @ int @ ( type_len0_len_of @ A @ ( type2 @ A ) ) ) ) ) ) @ ( bit_se4197421643247451524op_bit @ int @ ( nat2 @ ( modulo_modulo @ int @ Xa @ ( semiring_1_of_nat @ int @ ( type_len0_len_of @ A @ ( type2 @ A ) ) ) ) ) @ ( bit_se2584673776208193580ke_bit @ int @ ( type_len0_len_of @ A @ ( type2 @ A ) ) @ X2 ) ) @ ( bit_se2584673776208193580ke_bit @ int @ ( nat2 @ ( modulo_modulo @ int @ Xa @ ( semiring_1_of_nat @ int @ ( type_len0_len_of @ A @ ( type2 @ A ) ) ) ) ) @ X2 ) ) ) ) ) ).
% word_roti.abs_eq
thf(fact_6733_word__rotr__word__rotr__eq,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [M: nat,N: nat,W: word @ A] :
( ( word_rotr @ A @ M @ ( word_rotr @ A @ N @ W ) )
= ( word_rotr @ A @ ( plus_plus @ nat @ M @ N ) @ W ) ) ) ).
% word_rotr_word_rotr_eq
thf(fact_6734_plus__word_Oabs__eq,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [Xa: int,X2: int] :
( ( plus_plus @ ( word @ A ) @ ( word2 @ A @ Xa ) @ ( word2 @ A @ X2 ) )
= ( word2 @ A @ ( plus_plus @ int @ Xa @ X2 ) ) ) ) ).
% plus_word.abs_eq
thf(fact_6735_minus__word_Oabs__eq,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [Xa: int,X2: int] :
( ( minus_minus @ ( word @ A ) @ ( word2 @ A @ Xa ) @ ( word2 @ A @ X2 ) )
= ( word2 @ A @ ( minus_minus @ int @ Xa @ X2 ) ) ) ) ).
% minus_word.abs_eq
thf(fact_6736_one__word__def,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ( ( one_one @ ( word @ A ) )
= ( word2 @ A @ ( one_one @ int ) ) ) ) ).
% one_word_def
thf(fact_6737_word__succ_Oabs__eq,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [X2: int] :
( ( word_succ @ A @ ( word2 @ A @ X2 ) )
= ( word2 @ A @ ( plus_plus @ int @ X2 @ ( one_one @ int ) ) ) ) ) ).
% word_succ.abs_eq
thf(fact_6738_word__pred_Oabs__eq,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [X2: int] :
( ( word_pred @ A @ ( word2 @ A @ X2 ) )
= ( word2 @ A @ ( minus_minus @ int @ X2 @ ( one_one @ int ) ) ) ) ) ).
% word_pred.abs_eq
thf(fact_6739_word__rotr_Oabs__eq,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [Xa: nat,X2: int] :
( ( word_rotr @ A @ Xa @ ( word2 @ A @ X2 ) )
= ( word2 @ A @ ( bit_concat_bit @ ( minus_minus @ nat @ ( type_len0_len_of @ A @ ( type2 @ A ) ) @ ( modulo_modulo @ nat @ Xa @ ( type_len0_len_of @ A @ ( type2 @ A ) ) ) ) @ ( bit_se4197421643247451524op_bit @ int @ ( modulo_modulo @ nat @ Xa @ ( type_len0_len_of @ A @ ( type2 @ A ) ) ) @ ( bit_se2584673776208193580ke_bit @ int @ ( type_len0_len_of @ A @ ( type2 @ A ) ) @ X2 ) ) @ ( bit_se2584673776208193580ke_bit @ int @ ( modulo_modulo @ nat @ Xa @ ( type_len0_len_of @ A @ ( type2 @ A ) ) ) @ X2 ) ) ) ) ) ).
% word_rotr.abs_eq
thf(fact_6740_less__eq__word_Oabs__eq,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [Xa: int,X2: int] :
( ( ord_less_eq @ ( word @ A ) @ ( word2 @ A @ Xa ) @ ( word2 @ A @ X2 ) )
= ( ord_less_eq @ int @ ( bit_se2584673776208193580ke_bit @ int @ ( type_len0_len_of @ A @ ( type2 @ A ) ) @ Xa ) @ ( bit_se2584673776208193580ke_bit @ int @ ( type_len0_len_of @ A @ ( type2 @ A ) ) @ X2 ) ) ) ) ).
% less_eq_word.abs_eq
thf(fact_6741_bit__word_Oabs__eq,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [X2: int] :
( ( bit_se5641148757651400278ts_bit @ ( word @ A ) @ ( word2 @ A @ X2 ) )
= ( ^ [N4: nat] :
( ( ord_less @ nat @ N4 @ ( type_len0_len_of @ A @ ( type2 @ A ) ) )
& ( bit_se5641148757651400278ts_bit @ int @ X2 @ N4 ) ) ) ) ) ).
% bit_word.abs_eq
thf(fact_6742_word__sle_Oabs__eq,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [Xa: int,X2: int] :
( ( word_sle @ A @ ( word2 @ A @ Xa ) @ ( word2 @ A @ X2 ) )
= ( ord_less_eq @ int @ ( bit_ri4674362597316999326ke_bit @ int @ ( minus_minus @ nat @ ( type_len0_len_of @ A @ ( type2 @ A ) ) @ ( suc @ ( zero_zero @ nat ) ) ) @ Xa ) @ ( bit_ri4674362597316999326ke_bit @ int @ ( minus_minus @ nat @ ( type_len0_len_of @ A @ ( type2 @ A ) ) @ ( suc @ ( zero_zero @ nat ) ) ) @ X2 ) ) ) ) ).
% word_sle.abs_eq
thf(fact_6743_word__sless_Oabs__eq,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [Xa: int,X2: int] :
( ( word_sless @ A @ ( word2 @ A @ Xa ) @ ( word2 @ A @ X2 ) )
= ( ord_less @ int @ ( bit_ri4674362597316999326ke_bit @ int @ ( minus_minus @ nat @ ( type_len0_len_of @ A @ ( type2 @ A ) ) @ ( suc @ ( zero_zero @ nat ) ) ) @ Xa ) @ ( bit_ri4674362597316999326ke_bit @ int @ ( minus_minus @ nat @ ( type_len0_len_of @ A @ ( type2 @ A ) ) @ ( suc @ ( zero_zero @ nat ) ) ) @ X2 ) ) ) ) ).
% word_sless.abs_eq
thf(fact_6744_signed_Oabs__eq,axiom,
! [A: $tType,B: $tType] :
( ( ( type_len @ B )
& ( ring_1 @ A ) )
=> ! [X2: int] :
( ( ring_1_signed @ B @ A @ ( word2 @ B @ X2 ) )
= ( ring_1_of_int @ A @ ( bit_ri4674362597316999326ke_bit @ int @ ( minus_minus @ nat @ ( type_len0_len_of @ B @ ( type2 @ B ) ) @ ( suc @ ( zero_zero @ nat ) ) ) @ X2 ) ) ) ) ).
% signed.abs_eq
thf(fact_6745_signed__drop__bit_Oabs__eq,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [Xa: nat,X2: int] :
( ( signed_drop_bit @ A @ Xa @ ( word2 @ A @ X2 ) )
= ( word2 @ A @ ( bit_se4197421643247451524op_bit @ int @ Xa @ ( bit_ri4674362597316999326ke_bit @ int @ ( minus_minus @ nat @ ( type_len0_len_of @ A @ ( type2 @ A ) ) @ ( suc @ ( zero_zero @ nat ) ) ) @ X2 ) ) ) ) ) ).
% signed_drop_bit.abs_eq
thf(fact_6746_bit__word__rotr__iff,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [M: nat,W: word @ A,N: nat] :
( ( bit_se5641148757651400278ts_bit @ ( word @ A ) @ ( word_rotr @ A @ M @ W ) @ N )
= ( ( ord_less @ nat @ N @ ( type_len0_len_of @ A @ ( type2 @ A ) ) )
& ( bit_se5641148757651400278ts_bit @ ( word @ A ) @ W @ ( modulo_modulo @ nat @ ( plus_plus @ nat @ N @ M ) @ ( type_len0_len_of @ A @ ( type2 @ A ) ) ) ) ) ) ) ).
% bit_word_rotr_iff
thf(fact_6747_signed__modulo__word_Oabs__eq,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [Xa: int,X2: int] :
( ( signed6721504322012087516modulo @ ( word @ A ) @ ( word2 @ A @ Xa ) @ ( word2 @ A @ X2 ) )
= ( word2 @ A @ ( signed6721504322012087516modulo @ int @ ( bit_ri4674362597316999326ke_bit @ int @ ( minus_minus @ nat @ ( type_len0_len_of @ A @ ( type2 @ A ) ) @ ( suc @ ( zero_zero @ nat ) ) ) @ Xa ) @ ( bit_ri4674362597316999326ke_bit @ int @ ( minus_minus @ nat @ ( type_len0_len_of @ A @ ( type2 @ A ) ) @ ( suc @ ( zero_zero @ nat ) ) ) @ X2 ) ) ) ) ) ).
% signed_modulo_word.abs_eq
thf(fact_6748_signed__divide__word_Oabs__eq,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [Xa: int,X2: int] :
( ( signed7115095781618012415divide @ ( word @ A ) @ ( word2 @ A @ Xa ) @ ( word2 @ A @ X2 ) )
= ( word2 @ A @ ( signed7115095781618012415divide @ int @ ( bit_ri4674362597316999326ke_bit @ int @ ( minus_minus @ nat @ ( type_len0_len_of @ A @ ( type2 @ A ) ) @ ( suc @ ( zero_zero @ nat ) ) ) @ Xa ) @ ( bit_ri4674362597316999326ke_bit @ int @ ( minus_minus @ nat @ ( type_len0_len_of @ A @ ( type2 @ A ) ) @ ( suc @ ( zero_zero @ nat ) ) ) @ X2 ) ) ) ) ) ).
% signed_divide_word.abs_eq
thf(fact_6749_signed__cast_Oabs__eq,axiom,
! [B: $tType,A: $tType] :
( ( ( type_len @ A )
& ( type_len @ B ) )
=> ! [X2: int] :
( ( signed_cast @ A @ B @ ( word2 @ A @ X2 ) )
= ( word2 @ B @ ( bit_ri4674362597316999326ke_bit @ int @ ( minus_minus @ nat @ ( type_len0_len_of @ A @ ( type2 @ A ) ) @ ( suc @ ( zero_zero @ nat ) ) ) @ X2 ) ) ) ) ).
% signed_cast.abs_eq
thf(fact_6750_the__signed__int_Oabs__eq,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [X2: int] :
( ( the_signed_int @ A @ ( word2 @ A @ X2 ) )
= ( bit_ri4674362597316999326ke_bit @ int @ ( minus_minus @ nat @ ( type_len0_len_of @ A @ ( type2 @ A ) ) @ ( suc @ ( zero_zero @ nat ) ) ) @ X2 ) ) ) ).
% the_signed_int.abs_eq
thf(fact_6751_word__roti__eq__word__rotr__word__rotl,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ( ( word_roti @ A )
= ( ^ [I4: int,W2: word @ A] : ( if @ ( word @ A ) @ ( ord_less_eq @ int @ ( zero_zero @ int ) @ I4 ) @ ( word_rotr @ A @ ( nat2 @ I4 ) @ W2 ) @ ( word_rotl @ A @ ( nat2 @ ( uminus_uminus @ int @ I4 ) ) @ W2 ) ) ) ) ) ).
% word_roti_eq_word_rotr_word_rotl
thf(fact_6752_word__rotl_Oabs__eq,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [Xa: nat,X2: int] :
( ( word_rotl @ A @ Xa @ ( word2 @ A @ X2 ) )
= ( word2 @ A @ ( bit_concat_bit @ ( modulo_modulo @ nat @ Xa @ ( type_len0_len_of @ A @ ( type2 @ A ) ) ) @ ( bit_se4197421643247451524op_bit @ int @ ( minus_minus @ nat @ ( type_len0_len_of @ A @ ( type2 @ A ) ) @ ( modulo_modulo @ nat @ Xa @ ( type_len0_len_of @ A @ ( type2 @ A ) ) ) ) @ ( bit_se2584673776208193580ke_bit @ int @ ( type_len0_len_of @ A @ ( type2 @ A ) ) @ X2 ) ) @ ( bit_se2584673776208193580ke_bit @ int @ ( minus_minus @ nat @ ( type_len0_len_of @ A @ ( type2 @ A ) ) @ ( modulo_modulo @ nat @ Xa @ ( type_len0_len_of @ A @ ( type2 @ A ) ) ) ) @ X2 ) ) ) ) ) ).
% word_rotl.abs_eq
thf(fact_6753_word__rotl__eq__word__rotr,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ( ( word_rotl @ A )
= ( ^ [N4: nat] : ( word_rotr @ A @ ( minus_minus @ nat @ ( type_len0_len_of @ A @ ( type2 @ A ) ) @ ( modulo_modulo @ nat @ N4 @ ( type_len0_len_of @ A @ ( type2 @ A ) ) ) ) ) ) ) ) ).
% word_rotl_eq_word_rotr
thf(fact_6754_bit__word__rotl__iff,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [M: nat,W: word @ A,N: nat] :
( ( bit_se5641148757651400278ts_bit @ ( word @ A ) @ ( word_rotl @ A @ M @ W ) @ N )
= ( ( ord_less @ nat @ N @ ( type_len0_len_of @ A @ ( type2 @ A ) ) )
& ( bit_se5641148757651400278ts_bit @ ( word @ A ) @ W @ ( 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_6755_CHAR__pos__iff,axiom,
! [A: $tType] :
( ( semiring_1 @ A )
=> ( ( ord_less @ nat @ ( zero_zero @ nat ) @ ( semiri4206861660011772517g_char @ A @ ( type2 @ A ) ) )
= ( ? [N4: nat] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ N4 )
& ( ( semiring_1_of_nat @ A @ N4 )
= ( zero_zero @ A ) ) ) ) ) ) ).
% CHAR_pos_iff
thf(fact_6756_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 ) )
=> ( ! [X4: nat] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ X4 )
=> ( ( ord_less @ nat @ X4 @ C3 )
=> ( ( semiring_1_of_nat @ A @ X4 )
!= ( zero_zero @ A ) ) ) )
=> ( ( semiri4206861660011772517g_char @ A @ ( type2 @ A ) )
= C3 ) ) ) ) ) ).
% CHAR_eq_posI
thf(fact_6757_CHAR__eq0__iff,axiom,
! [A: $tType] :
( ( semiring_1 @ A )
=> ( ( ( semiri4206861660011772517g_char @ A @ ( type2 @ A ) )
= ( zero_zero @ nat ) )
= ( ! [N4: nat] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ N4 )
=> ( ( semiring_1_of_nat @ A @ N4 )
!= ( zero_zero @ A ) ) ) ) ) ) ).
% CHAR_eq0_iff
thf(fact_6758_bit__sshiftr__iff,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [W: word @ A,M: nat,N: nat] :
( ( bit_se5641148757651400278ts_bit @ ( word @ A ) @ ( bit_Sh8784991116023147202shiftr @ A @ W @ M ) @ N )
= ( bit_se5641148757651400278ts_bit @ ( word @ A ) @ W
@ ( 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_6759_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_6760_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_6761_sshiftr__numeral__numeral,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [M: num,N: num] :
( ( bit_Sh8784991116023147202shiftr @ A @ ( numeral_numeral @ ( word @ A ) @ M ) @ ( numeral_numeral @ nat @ N ) )
= ( signed_drop_bit @ A @ ( numeral_numeral @ nat @ N ) @ ( numeral_numeral @ ( word @ A ) @ M ) ) ) ) ).
% sshiftr_numeral_numeral
thf(fact_6762_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_6763_sshiftr__minus__numeral__numeral,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [M: num,N: num] :
( ( bit_Sh8784991116023147202shiftr @ A @ ( uminus_uminus @ ( word @ A ) @ ( numeral_numeral @ ( word @ A ) @ M ) ) @ ( numeral_numeral @ nat @ N ) )
= ( signed_drop_bit @ A @ ( numeral_numeral @ nat @ N ) @ ( uminus_uminus @ ( word @ A ) @ ( numeral_numeral @ ( word @ A ) @ M ) ) ) ) ) ).
% sshiftr_minus_numeral_numeral
thf(fact_6764_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_6765_shiftl__minus__1__numeral,axiom,
! [A: $tType] :
( ( bit_ri3973907225187159222ations @ A )
=> ! [N: num] :
( ( bit_Sh4282982442137083160shiftl @ A @ ( uminus_uminus @ A @ ( one_one @ A ) ) @ ( numeral_numeral @ nat @ N ) )
= ( bit_ri4277139882892585799ns_not @ A @ ( bit_se2239418461657761734s_mask @ A @ ( numeral_numeral @ nat @ N ) ) ) ) ) ).
% shiftl_minus_1_numeral
thf(fact_6766_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_6767_shiftl__numeral__numeral,axiom,
! [A: $tType] :
( ( bit_se359711467146920520ations @ A )
=> ! [M: num,N: num] :
( ( bit_Sh4282982442137083160shiftl @ A @ ( numeral_numeral @ A @ M ) @ ( numeral_numeral @ nat @ N ) )
= ( bit_se4730199178511100633sh_bit @ A @ ( numeral_numeral @ nat @ N ) @ ( numeral_numeral @ A @ M ) ) ) ) ).
% shiftl_numeral_numeral
thf(fact_6768_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_6769_shiftl__minus__numeral__numeral,axiom,
! [A: $tType] :
( ( bit_ri3973907225187159222ations @ A )
=> ! [M: num,N: num] :
( ( bit_Sh4282982442137083160shiftl @ A @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ M ) ) @ ( numeral_numeral @ nat @ N ) )
= ( bit_se4730199178511100633sh_bit @ A @ ( numeral_numeral @ nat @ N ) @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ M ) ) ) ) ) ).
% shiftl_minus_numeral_numeral
thf(fact_6770_shiftl__of__Suc,axiom,
! [A: $tType] :
( ( bit_se359711467146920520ations @ A )
=> ! [A3: A,N: nat] :
( ( bit_Sh4282982442137083160shiftl @ A @ A3 @ ( suc @ N ) )
= ( bit_Sh4282982442137083160shiftl @ A @ ( times_times @ A @ A3 @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) @ N ) ) ) ).
% shiftl_of_Suc
thf(fact_6771_shiftl__1,axiom,
! [A: $tType] :
( ( bit_se359711467146920520ations @ A )
=> ! [N: nat] :
( ( bit_Sh4282982442137083160shiftl @ A @ ( one_one @ A ) @ N )
= ( power_power @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ N ) ) ) ).
% shiftl_1
thf(fact_6772_one__bit__shiftl,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [N: nat] :
( ( generi7602027413899671122et_bit @ ( word @ A ) @ ( zero_zero @ ( word @ A ) ) @ N @ $true )
= ( bit_Sh4282982442137083160shiftl @ ( word @ A ) @ ( one_one @ ( word @ A ) ) @ N ) ) ) ).
% one_bit_shiftl
thf(fact_6773_shiftl__eq__mult,axiom,
! [A: $tType] :
( ( bit_se359711467146920520ations @ A )
=> ( ( bit_Sh4282982442137083160shiftl @ A )
= ( ^ [X3: A,N4: nat] : ( times_times @ A @ X3 @ ( power_power @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ N4 ) ) ) ) ) ).
% shiftl_eq_mult
thf(fact_6774_bit__shiftl__iff,axiom,
! [A: $tType] :
( ( bit_se359711467146920520ations @ A )
=> ! [A3: A,M: nat,N: nat] :
( ( bit_se5641148757651400278ts_bit @ A @ ( bit_Sh4282982442137083160shiftl @ A @ A3 @ M ) @ N )
= ( ( ord_less_eq @ nat @ M @ N )
& ( bit_se6407376104438227557le_bit @ A @ ( type2 @ A ) @ N )
& ( bit_se5641148757651400278ts_bit @ A @ A3 @ ( minus_minus @ nat @ N @ M ) ) ) ) ) ).
% bit_shiftl_iff
thf(fact_6775_shiftr__minus__numeral__numeral,axiom,
! [A: $tType] :
( ( bit_ri3973907225187159222ations @ A )
=> ! [M: num,N: num] :
( ( bit_Sh4282982442137083166shiftr @ A @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ M ) ) @ ( numeral_numeral @ nat @ N ) )
= ( bit_se4197421643247451524op_bit @ A @ ( numeral_numeral @ nat @ N ) @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ M ) ) ) ) ) ).
% shiftr_minus_numeral_numeral
thf(fact_6776_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_6777_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_6778_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_6779_shiftr__numeral__numeral,axiom,
! [A: $tType] :
( ( bit_se359711467146920520ations @ A )
=> ! [M: num,N: num] :
( ( bit_Sh4282982442137083166shiftr @ A @ ( numeral_numeral @ A @ M ) @ ( numeral_numeral @ nat @ N ) )
= ( bit_se4197421643247451524op_bit @ A @ ( numeral_numeral @ nat @ N ) @ ( numeral_numeral @ A @ M ) ) ) ) ).
% shiftr_numeral_numeral
thf(fact_6780_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_6781_shiftr__eq__div,axiom,
! [A: $tType] :
( ( bit_se359711467146920520ations @ A )
=> ( ( bit_Sh4282982442137083166shiftr @ A )
= ( ^ [X3: A,N4: nat] : ( divide_divide @ A @ X3 @ ( power_power @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ N4 ) ) ) ) ) ).
% shiftr_eq_div
thf(fact_6782_bit__revcast__iff,axiom,
! [B: $tType,A: $tType] :
( ( ( type_len @ A )
& ( type_len @ B ) )
=> ! [W: word @ A,N: nat] :
( ( bit_se5641148757651400278ts_bit @ ( word @ B ) @ ( revcast @ A @ B @ W ) @ 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 ) @ W @ ( 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_6783_word__msb__neg__numeral,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [W: num] :
( ( most_s684356279273892711sb_msb @ ( word @ A ) @ ( uminus_uminus @ ( word @ A ) @ ( numeral_numeral @ ( word @ A ) @ W ) ) )
= ( bit_se5641148757651400278ts_bit @ int @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ W ) ) @ ( minus_minus @ nat @ ( type_len0_len_of @ A @ ( type2 @ A ) ) @ ( one_one @ nat ) ) ) ) ) ).
% word_msb_neg_numeral
thf(fact_6784_word__msb__1,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ( ( most_s684356279273892711sb_msb @ ( word @ A ) @ ( one_one @ ( word @ A ) ) )
= ( ( type_len0_len_of @ A @ ( type2 @ A ) )
= ( one_one @ nat ) ) ) ) ).
% word_msb_1
thf(fact_6785_word__msb__numeral,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [W: num] :
( ( most_s684356279273892711sb_msb @ ( word @ A ) @ ( numeral_numeral @ ( word @ A ) @ W ) )
= ( bit_se5641148757651400278ts_bit @ int @ ( numeral_numeral @ int @ W ) @ ( minus_minus @ nat @ ( type_len0_len_of @ A @ ( type2 @ A ) ) @ ( one_one @ nat ) ) ) ) ) ).
% word_msb_numeral
thf(fact_6786_word__sle__msb__le,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ( ( word_sle @ A )
= ( ^ [X3: word @ A,Y: word @ A] :
( ( ( most_s684356279273892711sb_msb @ ( word @ A ) @ Y )
=> ( most_s684356279273892711sb_msb @ ( word @ A ) @ X3 ) )
& ( ( ( most_s684356279273892711sb_msb @ ( word @ A ) @ X3 )
& ~ ( most_s684356279273892711sb_msb @ ( word @ A ) @ Y ) )
| ( ord_less_eq @ ( word @ A ) @ X3 @ Y ) ) ) ) ) ) ).
% word_sle_msb_le
thf(fact_6787_word__msb__n1,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ( most_s684356279273892711sb_msb @ ( word @ A ) @ ( uminus_uminus @ ( word @ A ) @ ( one_one @ ( word @ A ) ) ) ) ) ).
% word_msb_n1
thf(fact_6788_msb__shift,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ( ( most_s684356279273892711sb_msb @ ( word @ A ) )
= ( ^ [W2: word @ A] :
( ( bit_Sh4282982442137083166shiftr @ ( word @ A ) @ W2 @ ( minus_minus @ nat @ ( type_len0_len_of @ A @ ( type2 @ A ) ) @ ( one_one @ nat ) ) )
!= ( zero_zero @ ( word @ A ) ) ) ) ) ) ).
% msb_shift
thf(fact_6789_msb__word__eq,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ( ( most_s684356279273892711sb_msb @ ( word @ A ) )
= ( ^ [W2: word @ A] : ( bit_se5641148757651400278ts_bit @ ( word @ A ) @ W2 @ ( minus_minus @ nat @ ( type_len0_len_of @ A @ ( type2 @ A ) ) @ ( one_one @ nat ) ) ) ) ) ) ).
% msb_word_eq
thf(fact_6790_msb__word__iff__bit,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ( ( most_s684356279273892711sb_msb @ ( word @ A ) )
= ( ^ [W2: word @ A] : ( bit_se5641148757651400278ts_bit @ ( word @ A ) @ W2 @ ( minus_minus @ nat @ ( type_len0_len_of @ A @ ( type2 @ A ) ) @ ( suc @ ( zero_zero @ nat ) ) ) ) ) ) ) ).
% msb_word_iff_bit
thf(fact_6791_word__msb__nth,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ( ( most_s684356279273892711sb_msb @ ( word @ A ) )
= ( ^ [W2: word @ A] : ( bit_se5641148757651400278ts_bit @ int @ ( semiring_1_unsigned @ A @ int @ W2 ) @ ( minus_minus @ nat @ ( type_len0_len_of @ A @ ( type2 @ A ) ) @ ( one_one @ nat ) ) ) ) ) ) ).
% word_msb_nth
thf(fact_6792_msb__word__of__int,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [X2: int] :
( ( most_s684356279273892711sb_msb @ ( word @ A ) @ ( ring_1_of_int @ ( word @ A ) @ X2 ) )
= ( bit_se5641148757651400278ts_bit @ int @ X2 @ ( minus_minus @ nat @ ( type_len0_len_of @ A @ ( type2 @ A ) ) @ ( one_one @ nat ) ) ) ) ) ).
% msb_word_of_int
thf(fact_6793_not__msb__from__less,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [V: word @ A] :
( ( ord_less @ ( word @ A ) @ V @ ( 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 ) @ V ) ) ) ).
% not_msb_from_less
thf(fact_6794_word__sint__msb__eq,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ( ( ring_1_signed @ A @ int )
= ( ^ [X3: word @ A] : ( minus_minus @ int @ ( semiring_1_unsigned @ A @ int @ X3 ) @ ( if @ int @ ( most_s684356279273892711sb_msb @ ( word @ A ) @ X3 ) @ ( power_power @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ ( size_size @ ( word @ A ) @ X3 ) ) @ ( zero_zero @ int ) ) ) ) ) ) ).
% word_sint_msb_eq
thf(fact_6795_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_6796_inj__on__word__of__int,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ( inj_on @ int @ ( word @ A ) @ ( ring_1_of_int @ ( word @ A ) ) @ ( set_or7035219750837199246ssThan @ int @ ( zero_zero @ int ) @ ( power_power @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ ( type_len0_len_of @ A @ ( type2 @ A ) ) ) ) ) ) ).
% inj_on_word_of_int
thf(fact_6797_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_6798_inj__map__eq__map,axiom,
! [B: $tType,A: $tType,F2: A > B,Xs: list @ A,Ys: list @ A] :
( ( inj_on @ A @ B @ F2 @ ( top_top @ ( set @ A ) ) )
=> ( ( ( map @ A @ B @ F2 @ Xs )
= ( map @ A @ B @ F2 @ Ys ) )
= ( Xs = Ys ) ) ) ).
% inj_map_eq_map
thf(fact_6799_inj__mult__left,axiom,
! [A: $tType] :
( ( idom @ A )
=> ! [A3: A] :
( ( inj_on @ A @ A @ ( times_times @ A @ A3 ) @ ( top_top @ ( set @ A ) ) )
= ( A3
!= ( zero_zero @ A ) ) ) ) ).
% inj_mult_left
thf(fact_6800_inj__divide__right,axiom,
! [A: $tType] :
( ( field @ A )
=> ! [A3: A] :
( ( inj_on @ A @ A
@ ^ [B7: A] : ( divide_divide @ A @ B7 @ A3 )
@ ( top_top @ ( set @ A ) ) )
= ( A3
!= ( zero_zero @ A ) ) ) ) ).
% inj_divide_right
thf(fact_6801_inj__map,axiom,
! [B: $tType,A: $tType,F2: A > B] :
( ( inj_on @ ( list @ A ) @ ( list @ B ) @ ( map @ A @ B @ F2 ) @ ( top_top @ ( set @ ( list @ A ) ) ) )
= ( inj_on @ A @ B @ F2 @ ( top_top @ ( set @ A ) ) ) ) ).
% inj_map
thf(fact_6802_inj__mapI,axiom,
! [B: $tType,A: $tType,F2: A > B] :
( ( inj_on @ A @ B @ F2 @ ( top_top @ ( set @ A ) ) )
=> ( inj_on @ ( list @ A ) @ ( list @ B ) @ ( map @ A @ B @ F2 ) @ ( top_top @ ( set @ ( list @ A ) ) ) ) ) ).
% inj_mapI
thf(fact_6803_inj__on__insert,axiom,
! [B: $tType,A: $tType,F2: A > B,A3: A,A2: set @ A] :
( ( inj_on @ A @ B @ F2 @ ( insert @ A @ A3 @ A2 ) )
= ( ( inj_on @ A @ B @ F2 @ A2 )
& ~ ( member @ B @ ( F2 @ A3 ) @ ( image @ A @ B @ F2 @ ( minus_minus @ ( set @ A ) @ A2 @ ( insert @ A @ A3 @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ) ) ).
% inj_on_insert
thf(fact_6804_subset__image__inj,axiom,
! [A: $tType,B: $tType,S: set @ A,F2: B > A,T6: set @ B] :
( ( ord_less_eq @ ( set @ A ) @ S @ ( image @ B @ A @ F2 @ T6 ) )
= ( ? [U3: set @ B] :
( ( ord_less_eq @ ( set @ B ) @ U3 @ T6 )
& ( inj_on @ B @ A @ F2 @ U3 )
& ( S
= ( image @ B @ A @ F2 @ U3 ) ) ) ) ) ).
% subset_image_inj
thf(fact_6805_inj__on__image__mem__iff,axiom,
! [B: $tType,A: $tType,F2: A > B,B2: set @ A,A3: A,A2: set @ A] :
( ( inj_on @ A @ B @ F2 @ B2 )
=> ( ( member @ A @ A3 @ B2 )
=> ( ( ord_less_eq @ ( set @ A ) @ A2 @ B2 )
=> ( ( member @ B @ ( F2 @ A3 ) @ ( image @ A @ B @ F2 @ A2 ) )
= ( member @ A @ A3 @ A2 ) ) ) ) ) ).
% inj_on_image_mem_iff
thf(fact_6806_inj__on__image__eq__iff,axiom,
! [B: $tType,A: $tType,F2: A > B,C2: set @ A,A2: set @ A,B2: set @ A] :
( ( inj_on @ A @ B @ F2 @ C2 )
=> ( ( ord_less_eq @ ( set @ A ) @ A2 @ C2 )
=> ( ( ord_less_eq @ ( set @ A ) @ B2 @ C2 )
=> ( ( ( image @ A @ B @ F2 @ A2 )
= ( image @ A @ B @ F2 @ B2 ) )
= ( A2 = B2 ) ) ) ) ) ).
% inj_on_image_eq_iff
thf(fact_6807_sum_Oimage__eq,axiom,
! [A: $tType,B: $tType] :
( ( comm_monoid_add @ A )
=> ! [G: B > A,A2: set @ B] :
( ( inj_on @ B @ A @ G @ A2 )
=> ( ( groups7311177749621191930dd_sum @ A @ A
@ ^ [X3: A] : X3
@ ( image @ B @ A @ G @ A2 ) )
= ( groups7311177749621191930dd_sum @ B @ A @ G @ A2 ) ) ) ) ).
% sum.image_eq
thf(fact_6808_prod_Oimage__eq,axiom,
! [A: $tType,B: $tType] :
( ( comm_monoid_mult @ A )
=> ! [G: B > A,A2: set @ B] :
( ( inj_on @ B @ A @ G @ A2 )
=> ( ( groups7121269368397514597t_prod @ A @ A
@ ^ [X3: A] : X3
@ ( image @ B @ A @ G @ A2 ) )
= ( groups7121269368397514597t_prod @ B @ A @ G @ A2 ) ) ) ) ).
% prod.image_eq
thf(fact_6809_inj__mapD,axiom,
! [B: $tType,A: $tType,F2: A > B] :
( ( inj_on @ ( list @ A ) @ ( list @ B ) @ ( map @ A @ B @ F2 ) @ ( top_top @ ( set @ ( list @ A ) ) ) )
=> ( inj_on @ A @ B @ F2 @ ( top_top @ ( set @ A ) ) ) ) ).
% inj_mapD
thf(fact_6810_map__injective,axiom,
! [A: $tType,B: $tType,F2: B > A,Xs: list @ B,Ys: list @ B] :
( ( ( map @ B @ A @ F2 @ Xs )
= ( map @ B @ A @ F2 @ Ys ) )
=> ( ( inj_on @ B @ A @ F2 @ ( top_top @ ( set @ B ) ) )
=> ( Xs = Ys ) ) ) ).
% map_injective
thf(fact_6811_inj__add__left,axiom,
! [A: $tType] :
( ( cancel_semigroup_add @ A )
=> ! [A3: A] : ( inj_on @ A @ A @ ( plus_plus @ A @ A3 ) @ ( top_top @ ( set @ A ) ) ) ) ).
% inj_add_left
thf(fact_6812_inj__fun,axiom,
! [B: $tType,C: $tType,A: $tType,F2: A > B] :
( ( inj_on @ A @ B @ F2 @ ( top_top @ ( set @ A ) ) )
=> ( inj_on @ A @ ( C > B )
@ ^ [X3: A,Y: C] : ( F2 @ X3 )
@ ( top_top @ ( set @ A ) ) ) ) ).
% inj_fun
thf(fact_6813_linorder__injI,axiom,
! [B: $tType,A: $tType] :
( ( linorder @ A )
=> ! [F2: A > B] :
( ! [X4: A,Y4: A] :
( ( ord_less @ A @ X4 @ Y4 )
=> ( ( F2 @ X4 )
!= ( F2 @ Y4 ) ) )
=> ( inj_on @ A @ B @ F2 @ ( top_top @ ( set @ A ) ) ) ) ) ).
% linorder_injI
thf(fact_6814_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_6815_sorted__list__of__set_Oinj__on,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ( inj_on @ A @ A
@ ^ [X3: A] : X3
@ ( top_top @ ( set @ A ) ) ) ) ).
% sorted_list_of_set.inj_on
thf(fact_6816_finite__inverse__image,axiom,
! [A: $tType,B: $tType,A2: set @ A,F2: B > A] :
( ( finite_finite2 @ A @ A2 )
=> ( ( inj_on @ B @ A @ F2 @ ( top_top @ ( set @ B ) ) )
=> ( finite_finite2 @ B
@ ( collect @ B
@ ^ [J3: B] : ( member @ A @ ( F2 @ J3 ) @ A2 ) ) ) ) ) ).
% finite_inverse_image
thf(fact_6817_finite__Collect,axiom,
! [A: $tType,B: $tType,S: set @ A,F2: B > A] :
( ( finite_finite2 @ A @ S )
=> ( ( inj_on @ B @ A @ F2 @ ( top_top @ ( set @ B ) ) )
=> ( finite_finite2 @ B
@ ( collect @ B
@ ^ [A6: B] : ( member @ A @ ( F2 @ A6 ) @ S ) ) ) ) ) ).
% finite_Collect
thf(fact_6818_inj__diff__right,axiom,
! [A: $tType] :
( ( ab_group_add @ A )
=> ! [A3: A] :
( inj_on @ A @ A
@ ^ [B7: A] : ( minus_minus @ A @ B7 @ A3 )
@ ( top_top @ ( set @ A ) ) ) ) ).
% inj_diff_right
thf(fact_6819_linorder__inj__onI,axiom,
! [B: $tType,A: $tType] :
( ( order @ A )
=> ! [A2: set @ A,F2: A > B] :
( ! [X4: A,Y4: A] :
( ( ord_less @ A @ X4 @ Y4 )
=> ( ( member @ A @ X4 @ A2 )
=> ( ( member @ A @ Y4 @ A2 )
=> ( ( F2 @ X4 )
!= ( F2 @ Y4 ) ) ) ) )
=> ( ! [X4: A,Y4: A] :
( ( member @ A @ X4 @ A2 )
=> ( ( member @ A @ Y4 @ A2 )
=> ( ( ord_less_eq @ A @ X4 @ Y4 )
| ( ord_less_eq @ A @ Y4 @ X4 ) ) ) )
=> ( inj_on @ A @ B @ F2 @ A2 ) ) ) ) ).
% linorder_inj_onI
thf(fact_6820_inj__on__subset,axiom,
! [B: $tType,A: $tType,F2: A > B,A2: set @ A,B2: set @ A] :
( ( inj_on @ A @ B @ F2 @ A2 )
=> ( ( ord_less_eq @ ( set @ A ) @ B2 @ A2 )
=> ( inj_on @ A @ B @ F2 @ B2 ) ) ) ).
% inj_on_subset
thf(fact_6821_subset__inj__on,axiom,
! [B: $tType,A: $tType,F2: A > B,B2: set @ A,A2: set @ A] :
( ( inj_on @ A @ B @ F2 @ B2 )
=> ( ( ord_less_eq @ ( set @ A ) @ A2 @ B2 )
=> ( inj_on @ A @ B @ F2 @ A2 ) ) ) ).
% subset_inj_on
thf(fact_6822_inj__on__add_H,axiom,
! [A: $tType] :
( ( cancel_semigroup_add @ A )
=> ! [A3: A,A2: set @ A] :
( inj_on @ A @ A
@ ^ [B7: A] : ( plus_plus @ A @ B7 @ A3 )
@ A2 ) ) ).
% inj_on_add'
thf(fact_6823_inj__on__add,axiom,
! [A: $tType] :
( ( cancel_semigroup_add @ A )
=> ! [A3: A,A2: set @ A] : ( inj_on @ A @ A @ ( plus_plus @ A @ A3 ) @ A2 ) ) ).
% inj_on_add
thf(fact_6824_inj__on__id2,axiom,
! [A: $tType,A2: set @ A] :
( inj_on @ A @ A
@ ^ [X3: A] : X3
@ A2 ) ).
% inj_on_id2
thf(fact_6825_inj__on__diff,axiom,
! [B: $tType,A: $tType,F2: A > B,A2: set @ A,B2: set @ A] :
( ( inj_on @ A @ B @ F2 @ A2 )
=> ( inj_on @ A @ B @ F2 @ ( minus_minus @ ( set @ A ) @ A2 @ B2 ) ) ) ).
% inj_on_diff
thf(fact_6826_finite__inverse__image__gen,axiom,
! [A: $tType,B: $tType,A2: set @ A,F2: B > A,D: set @ B] :
( ( finite_finite2 @ A @ A2 )
=> ( ( inj_on @ B @ A @ F2 @ D )
=> ( finite_finite2 @ B
@ ( collect @ B
@ ^ [J3: B] :
( ( member @ B @ J3 @ D )
& ( member @ A @ ( F2 @ J3 ) @ A2 ) ) ) ) ) ) ).
% finite_inverse_image_gen
thf(fact_6827_inj__on__mult,axiom,
! [A: $tType] :
( ( semidom_divide @ A )
=> ! [A3: A,A2: set @ A] :
( ( A3
!= ( zero_zero @ A ) )
=> ( inj_on @ A @ A @ ( times_times @ A @ A3 ) @ A2 ) ) ) ).
% inj_on_mult
thf(fact_6828_inj__on__iff__surj,axiom,
! [A: $tType,B: $tType,A2: set @ A,A8: set @ B] :
( ( A2
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( ? [F5: A > B] :
( ( inj_on @ A @ B @ F5 @ A2 )
& ( ord_less_eq @ ( set @ B ) @ ( image @ A @ B @ F5 @ A2 ) @ A8 ) ) )
= ( ? [G2: B > A] :
( ( image @ B @ A @ G2 @ A8 )
= A2 ) ) ) ) ).
% inj_on_iff_surj
thf(fact_6829_inj__image__subset__iff,axiom,
! [B: $tType,A: $tType,F2: A > B,A2: set @ A,B2: set @ A] :
( ( inj_on @ A @ B @ F2 @ ( top_top @ ( set @ A ) ) )
=> ( ( ord_less_eq @ ( set @ B ) @ ( image @ A @ B @ F2 @ A2 ) @ ( image @ A @ B @ F2 @ B2 ) )
= ( ord_less_eq @ ( set @ A ) @ A2 @ B2 ) ) ) ).
% inj_image_subset_iff
thf(fact_6830_finite__surj__inj,axiom,
! [A: $tType,A2: set @ A,F2: A > A] :
( ( finite_finite2 @ A @ A2 )
=> ( ( ord_less_eq @ ( set @ A ) @ A2 @ ( image @ A @ A @ F2 @ A2 ) )
=> ( inj_on @ A @ A @ F2 @ A2 ) ) ) ).
% finite_surj_inj
thf(fact_6831_inj__on__finite,axiom,
! [B: $tType,A: $tType,F2: A > B,A2: set @ A,B2: set @ B] :
( ( inj_on @ A @ B @ F2 @ A2 )
=> ( ( ord_less_eq @ ( set @ B ) @ ( image @ A @ B @ F2 @ A2 ) @ B2 )
=> ( ( finite_finite2 @ B @ B2 )
=> ( finite_finite2 @ A @ A2 ) ) ) ) ).
% inj_on_finite
thf(fact_6832_endo__inj__surj,axiom,
! [A: $tType,A2: set @ A,F2: A > A] :
( ( finite_finite2 @ A @ A2 )
=> ( ( ord_less_eq @ ( set @ A ) @ ( image @ A @ A @ F2 @ A2 ) @ A2 )
=> ( ( inj_on @ A @ A @ F2 @ A2 )
=> ( ( image @ A @ A @ F2 @ A2 )
= A2 ) ) ) ) ).
% endo_inj_surj
thf(fact_6833_image__set__diff,axiom,
! [B: $tType,A: $tType,F2: A > B,A2: set @ A,B2: set @ A] :
( ( inj_on @ A @ B @ F2 @ ( top_top @ ( set @ A ) ) )
=> ( ( image @ A @ B @ F2 @ ( minus_minus @ ( set @ A ) @ A2 @ B2 ) )
= ( minus_minus @ ( set @ B ) @ ( image @ A @ B @ F2 @ A2 ) @ ( image @ A @ B @ F2 @ B2 ) ) ) ) ).
% image_set_diff
thf(fact_6834_inj__on__image__set__diff,axiom,
! [B: $tType,A: $tType,F2: A > B,C2: set @ A,A2: set @ A,B2: set @ A] :
( ( inj_on @ A @ B @ F2 @ C2 )
=> ( ( ord_less_eq @ ( set @ A ) @ ( minus_minus @ ( set @ A ) @ A2 @ B2 ) @ C2 )
=> ( ( ord_less_eq @ ( set @ A ) @ B2 @ C2 )
=> ( ( image @ A @ B @ F2 @ ( minus_minus @ ( set @ A ) @ A2 @ B2 ) )
= ( minus_minus @ ( set @ B ) @ ( image @ A @ B @ F2 @ A2 ) @ ( image @ A @ B @ F2 @ B2 ) ) ) ) ) ) ).
% inj_on_image_set_diff
thf(fact_6835_the__inv__into__into,axiom,
! [B: $tType,A: $tType,F2: A > B,A2: set @ A,X2: B,B2: set @ A] :
( ( inj_on @ A @ B @ F2 @ A2 )
=> ( ( member @ B @ X2 @ ( image @ A @ B @ F2 @ A2 ) )
=> ( ( ord_less_eq @ ( set @ A ) @ A2 @ B2 )
=> ( member @ A @ ( the_inv_into @ A @ B @ A2 @ F2 @ X2 ) @ B2 ) ) ) ) ).
% the_inv_into_into
thf(fact_6836_injective__scaleR,axiom,
! [A: $tType] :
( ( real_V4867850818363320053vector @ A )
=> ! [C3: real] :
( ( C3
!= ( zero_zero @ real ) )
=> ( inj_on @ A @ A @ ( real_V8093663219630862766scaleR @ A @ C3 ) @ ( top_top @ ( set @ A ) ) ) ) ) ).
% injective_scaleR
thf(fact_6837_finite__imp__nat__seg__image__inj__on,axiom,
! [A: $tType,A2: set @ A] :
( ( finite_finite2 @ A @ A2 )
=> ? [N3: nat,F3: nat > A] :
( ( A2
= ( image @ nat @ A @ F3
@ ( collect @ nat
@ ^ [I4: nat] : ( ord_less @ nat @ I4 @ N3 ) ) ) )
& ( inj_on @ nat @ A @ F3
@ ( collect @ nat
@ ^ [I4: nat] : ( ord_less @ nat @ I4 @ N3 ) ) ) ) ) ).
% finite_imp_nat_seg_image_inj_on
thf(fact_6838_inj__image__Compl__subset,axiom,
! [B: $tType,A: $tType,F2: A > B,A2: set @ A] :
( ( inj_on @ A @ B @ F2 @ ( top_top @ ( set @ A ) ) )
=> ( ord_less_eq @ ( set @ B ) @ ( image @ A @ B @ F2 @ ( uminus_uminus @ ( set @ A ) @ A2 ) ) @ ( uminus_uminus @ ( set @ B ) @ ( image @ A @ B @ F2 @ A2 ) ) ) ) ).
% inj_image_Compl_subset
thf(fact_6839_infinite__iff__countable__subset,axiom,
! [A: $tType,S: set @ A] :
( ( ~ ( finite_finite2 @ A @ S ) )
= ( ? [F5: nat > A] :
( ( inj_on @ nat @ A @ F5 @ ( top_top @ ( set @ nat ) ) )
& ( ord_less_eq @ ( set @ A ) @ ( image @ nat @ A @ F5 @ ( top_top @ ( set @ nat ) ) ) @ S ) ) ) ) ).
% infinite_iff_countable_subset
thf(fact_6840_infinite__countable__subset,axiom,
! [A: $tType,S: set @ A] :
( ~ ( finite_finite2 @ A @ S )
=> ? [F3: nat > A] :
( ( inj_on @ nat @ A @ F3 @ ( top_top @ ( set @ nat ) ) )
& ( ord_less_eq @ ( set @ A ) @ ( image @ nat @ A @ F3 @ ( top_top @ ( set @ nat ) ) ) @ S ) ) ) ).
% infinite_countable_subset
thf(fact_6841_inj__on__map__inv__f,axiom,
! [B: $tType,A: $tType,L: list @ A,A2: set @ A,F2: A > B] :
( ( ord_less_eq @ ( set @ A ) @ ( set2 @ A @ L ) @ A2 )
=> ( ( inj_on @ A @ B @ F2 @ A2 )
=> ( ( map @ B @ A @ ( inv_on @ A @ B @ F2 @ A2 ) @ ( map @ A @ B @ F2 @ L ) )
= L ) ) ) ).
% inj_on_map_inv_f
thf(fact_6842_Schroeder__Bernstein,axiom,
! [A: $tType,B: $tType,F2: A > B,A2: set @ A,B2: set @ B,G: B > A] :
( ( inj_on @ A @ B @ F2 @ A2 )
=> ( ( ord_less_eq @ ( set @ B ) @ ( image @ A @ B @ F2 @ A2 ) @ B2 )
=> ( ( inj_on @ B @ A @ G @ B2 )
=> ( ( ord_less_eq @ ( set @ A ) @ ( image @ B @ A @ G @ B2 ) @ A2 )
=> ? [H3: A > B] : ( bij_betw @ A @ B @ H3 @ A2 @ B2 ) ) ) ) ) ).
% Schroeder_Bernstein
thf(fact_6843_inv__on__f__f,axiom,
! [B: $tType,A: $tType,F2: A > B,A2: set @ A,X2: A] :
( ( inj_on @ A @ B @ F2 @ A2 )
=> ( ( member @ A @ X2 @ A2 )
=> ( ( inv_on @ A @ B @ F2 @ A2 @ ( F2 @ X2 ) )
= X2 ) ) ) ).
% inv_on_f_f
thf(fact_6844_inj__split__Cons,axiom,
! [A: $tType,X: set @ ( product_prod @ ( list @ A ) @ A )] :
( inj_on @ ( product_prod @ ( list @ A ) @ A ) @ ( list @ A )
@ ( product_case_prod @ ( list @ A ) @ A @ ( list @ A )
@ ^ [Xs2: list @ A,N4: A] : ( cons @ A @ N4 @ Xs2 ) )
@ X ) ).
% inj_split_Cons
thf(fact_6845_inj__on__Cons1,axiom,
! [A: $tType,X2: A,A2: set @ ( list @ A )] : ( inj_on @ ( list @ A ) @ ( list @ A ) @ ( cons @ A @ X2 ) @ A2 ) ).
% inj_on_Cons1
thf(fact_6846_inj__graph,axiom,
! [B: $tType,A: $tType] :
( inj_on @ ( A > B ) @ ( set @ ( product_prod @ A @ B ) )
@ ^ [F5: A > B] :
( collect @ ( product_prod @ A @ B )
@ ( product_case_prod @ A @ B @ $o
@ ^ [X3: A,Y: B] :
( Y
= ( F5 @ X3 ) ) ) )
@ ( top_top @ ( set @ ( A > B ) ) ) ) ).
% inj_graph
thf(fact_6847_signed_Osorted__list__of__set_Oinj__on,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ( inj_on @ ( word @ A ) @ ( word @ A )
@ ^ [X3: word @ A] : X3
@ ( top_top @ ( set @ ( word @ A ) ) ) ) ) ).
% signed.sorted_list_of_set.inj_on
thf(fact_6848_swap__inj__on,axiom,
! [B: $tType,A: $tType,A2: set @ ( product_prod @ A @ B )] :
( inj_on @ ( product_prod @ A @ B ) @ ( product_prod @ B @ A )
@ ( product_case_prod @ A @ B @ ( product_prod @ B @ A )
@ ^ [I4: A,J3: B] : ( product_Pair @ B @ A @ J3 @ I4 ) )
@ A2 ) ).
% swap_inj_on
thf(fact_6849_inj__on__convol__ident,axiom,
! [B: $tType,A: $tType,F2: A > B,X: set @ A] :
( inj_on @ A @ ( product_prod @ A @ B )
@ ^ [X3: A] : ( product_Pair @ A @ B @ X3 @ ( F2 @ X3 ) )
@ X ) ).
% inj_on_convol_ident
thf(fact_6850_inj__singleton,axiom,
! [A: $tType,A2: set @ A] :
( inj_on @ A @ ( set @ A )
@ ^ [X3: A] : ( insert @ A @ X3 @ ( bot_bot @ ( set @ A ) ) )
@ A2 ) ).
% inj_singleton
thf(fact_6851_inj__on__diff__nat,axiom,
! [N7: set @ nat,K: nat] :
( ! [N3: nat] :
( ( member @ nat @ N3 @ N7 )
=> ( ord_less_eq @ nat @ K @ N3 ) )
=> ( inj_on @ nat @ nat
@ ^ [N4: nat] : ( minus_minus @ nat @ N4 @ K )
@ N7 ) ) ).
% inj_on_diff_nat
thf(fact_6852_inj__Suc,axiom,
! [N7: set @ nat] : ( inj_on @ nat @ nat @ suc @ N7 ) ).
% inj_Suc
thf(fact_6853_inj__Some,axiom,
! [A: $tType,A2: set @ A] : ( inj_on @ A @ ( option @ A ) @ ( some @ A ) @ A2 ) ).
% inj_Some
thf(fact_6854_inj__Pair_I1_J,axiom,
! [B: $tType,A: $tType,C3: A > B,S: set @ A] :
( inj_on @ A @ ( product_prod @ A @ B )
@ ^ [X3: A] : ( product_Pair @ A @ B @ X3 @ ( C3 @ X3 ) )
@ S ) ).
% inj_Pair(1)
thf(fact_6855_inj__Pair_I2_J,axiom,
! [B: $tType,A: $tType,C3: A > B,S: set @ A] :
( inj_on @ A @ ( product_prod @ B @ A )
@ ^ [X3: A] : ( product_Pair @ B @ A @ ( C3 @ X3 ) @ X3 )
@ S ) ).
% inj_Pair(2)
thf(fact_6856_inv__on__f__range,axiom,
! [A: $tType,B: $tType,Y3: A,F2: B > A,A2: set @ B] :
( ( member @ A @ Y3 @ ( image @ B @ A @ F2 @ A2 ) )
=> ( member @ B @ ( inv_on @ B @ A @ F2 @ A2 @ Y3 ) @ A2 ) ) ).
% inv_on_f_range
thf(fact_6857_f__inv__on__f,axiom,
! [B: $tType,A: $tType,Y3: A,F2: B > A,A2: set @ B] :
( ( member @ A @ Y3 @ ( image @ B @ A @ F2 @ A2 ) )
=> ( ( F2 @ ( inv_on @ B @ A @ F2 @ A2 @ Y3 ) )
= Y3 ) ) ).
% f_inv_on_f
thf(fact_6858_le__rel__bool__arg__iff,axiom,
! [A: $tType] :
( ( ord @ A )
=> ( ( ord_less_eq @ ( $o > A ) )
= ( ^ [X7: $o > A,Y10: $o > A] :
( ( ord_less_eq @ A @ ( X7 @ $false ) @ ( Y10 @ $false ) )
& ( ord_less_eq @ A @ ( X7 @ $true ) @ ( Y10 @ $true ) ) ) ) ) ) ).
% le_rel_bool_arg_iff
thf(fact_6859_finite__imp__inj__to__nat__seg_H,axiom,
! [A: $tType,A2: set @ A] :
( ( finite_finite2 @ A @ A2 )
=> ~ ! [F3: A > nat] :
( ? [N3: nat] :
( ( image @ A @ nat @ F3 @ A2 )
= ( collect @ nat
@ ^ [I4: nat] : ( ord_less @ nat @ I4 @ N3 ) ) )
=> ~ ( inj_on @ A @ nat @ F3 @ A2 ) ) ) ).
% finite_imp_inj_to_nat_seg'
thf(fact_6860_finite__imp__inj__to__nat__seg,axiom,
! [A: $tType,A2: set @ A] :
( ( finite_finite2 @ A @ A2 )
=> ? [F3: A > nat,N3: nat] :
( ( ( image @ A @ nat @ F3 @ A2 )
= ( collect @ nat
@ ^ [I4: nat] : ( ord_less @ nat @ I4 @ N3 ) ) )
& ( inj_on @ A @ nat @ F3 @ A2 ) ) ) ).
% finite_imp_inj_to_nat_seg
thf(fact_6861_inv__on__def,axiom,
! [B: $tType,A: $tType] :
( ( inv_on @ A @ B )
= ( ^ [F5: A > B,A7: set @ A,X3: B] :
( fChoice @ A
@ ^ [Y: A] :
( ( member @ A @ Y @ A7 )
& ( ( F5 @ Y )
= X3 ) ) ) ) ) ).
% inv_on_def
thf(fact_6862_inj__sgn__power,axiom,
! [N: nat] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ N )
=> ( inj_on @ real @ real
@ ^ [Y: real] : ( times_times @ real @ ( sgn_sgn @ real @ Y ) @ ( power_power @ real @ ( abs_abs @ real @ Y ) @ N ) )
@ ( top_top @ ( set @ real ) ) ) ) ).
% inj_sgn_power
thf(fact_6863_all__subset__image__inj,axiom,
! [A: $tType,B: $tType,F2: B > A,S: set @ B,P: ( set @ A ) > $o] :
( ( ! [T9: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ T9 @ ( image @ B @ A @ F2 @ S ) )
=> ( P @ T9 ) ) )
= ( ! [T9: set @ B] :
( ( ( ord_less_eq @ ( set @ B ) @ T9 @ S )
& ( inj_on @ B @ A @ F2 @ T9 ) )
=> ( P @ ( image @ B @ A @ F2 @ T9 ) ) ) ) ) ).
% all_subset_image_inj
thf(fact_6864_ex__subset__image__inj,axiom,
! [A: $tType,B: $tType,F2: B > A,S: set @ B,P: ( set @ A ) > $o] :
( ( ? [T9: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ T9 @ ( image @ B @ A @ F2 @ S ) )
& ( P @ T9 ) ) )
= ( ? [T9: set @ B] :
( ( ord_less_eq @ ( set @ B ) @ T9 @ S )
& ( inj_on @ B @ A @ F2 @ T9 )
& ( P @ ( image @ B @ A @ F2 @ T9 ) ) ) ) ) ).
% ex_subset_image_inj
thf(fact_6865_valid__eq2,axiom,
! [T2: vEBT_VEBT,D2: nat] :
( ( vEBT_VEBT_valid @ T2 @ D2 )
=> ( vEBT_invar_vebt @ T2 @ D2 ) ) ).
% valid_eq2
thf(fact_6866_valid__eq1,axiom,
! [T2: vEBT_VEBT,D2: nat] :
( ( vEBT_invar_vebt @ T2 @ D2 )
=> ( vEBT_VEBT_valid @ T2 @ D2 ) ) ).
% valid_eq1
thf(fact_6867_valid__eq,axiom,
vEBT_VEBT_valid = vEBT_invar_vebt ).
% valid_eq
thf(fact_6868_VEBT__internal_Ovalid_H_Osimps_I1_J,axiom,
! [Uu: $o,Uv: $o,D2: nat] :
( ( vEBT_VEBT_valid @ ( vEBT_Leaf @ Uu @ Uv ) @ D2 )
= ( D2
= ( one_one @ nat ) ) ) ).
% VEBT_internal.valid'.simps(1)
thf(fact_6869_DERIV__real__root__generic,axiom,
! [N: nat,X2: real,D: real] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ N )
=> ( ( X2
!= ( zero_zero @ real ) )
=> ( ( ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N )
=> ( ( ord_less @ real @ ( zero_zero @ real ) @ X2 )
=> ( D
= ( inverse_inverse @ real @ ( times_times @ real @ ( semiring_1_of_nat @ real @ N ) @ ( power_power @ real @ ( root @ N @ X2 ) @ ( minus_minus @ nat @ N @ ( suc @ ( zero_zero @ nat ) ) ) ) ) ) ) ) )
=> ( ( ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N )
=> ( ( ord_less @ real @ X2 @ ( zero_zero @ real ) )
=> ( D
= ( uminus_uminus @ real @ ( inverse_inverse @ real @ ( times_times @ real @ ( semiring_1_of_nat @ real @ N ) @ ( power_power @ real @ ( root @ N @ X2 ) @ ( minus_minus @ nat @ N @ ( suc @ ( zero_zero @ nat ) ) ) ) ) ) ) ) ) )
=> ( ( ~ ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N )
=> ( D
= ( inverse_inverse @ real @ ( times_times @ real @ ( semiring_1_of_nat @ real @ N ) @ ( power_power @ real @ ( root @ N @ X2 ) @ ( minus_minus @ nat @ N @ ( suc @ ( zero_zero @ nat ) ) ) ) ) ) ) )
=> ( has_field_derivative @ real @ ( root @ N ) @ D @ ( topolo174197925503356063within @ real @ X2 @ ( top_top @ ( set @ real ) ) ) ) ) ) ) ) ) ).
% DERIV_real_root_generic
thf(fact_6870_DERIV__even__real__root,axiom,
! [N: nat,X2: real] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ N )
=> ( ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N )
=> ( ( ord_less @ real @ X2 @ ( 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 @ X2 ) @ ( minus_minus @ nat @ N @ ( suc @ ( zero_zero @ nat ) ) ) ) ) ) @ ( topolo174197925503356063within @ real @ X2 @ ( top_top @ ( set @ real ) ) ) ) ) ) ) ).
% DERIV_even_real_root
thf(fact_6871_has__real__derivative__pos__inc__right,axiom,
! [F2: real > real,L: real,X2: real,S: set @ real] :
( ( has_field_derivative @ real @ F2 @ L @ ( topolo174197925503356063within @ real @ X2 @ S ) )
=> ( ( ord_less @ real @ ( zero_zero @ real ) @ L )
=> ? [D4: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ D4 )
& ! [H7: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ H7 )
=> ( ( member @ real @ ( plus_plus @ real @ X2 @ H7 ) @ S )
=> ( ( ord_less @ real @ H7 @ D4 )
=> ( ord_less @ real @ ( F2 @ X2 ) @ ( F2 @ ( plus_plus @ real @ X2 @ H7 ) ) ) ) ) ) ) ) ) ).
% has_real_derivative_pos_inc_right
thf(fact_6872_has__real__derivative__neg__dec__right,axiom,
! [F2: real > real,L: real,X2: real,S: set @ real] :
( ( has_field_derivative @ real @ F2 @ L @ ( topolo174197925503356063within @ real @ X2 @ S ) )
=> ( ( ord_less @ real @ L @ ( zero_zero @ real ) )
=> ? [D4: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ D4 )
& ! [H7: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ H7 )
=> ( ( member @ real @ ( plus_plus @ real @ X2 @ H7 ) @ S )
=> ( ( ord_less @ real @ H7 @ D4 )
=> ( ord_less @ real @ ( F2 @ ( plus_plus @ real @ X2 @ H7 ) ) @ ( F2 @ X2 ) ) ) ) ) ) ) ) ).
% has_real_derivative_neg_dec_right
thf(fact_6873_has__real__derivative__neg__dec__left,axiom,
! [F2: real > real,L: real,X2: real,S: set @ real] :
( ( has_field_derivative @ real @ F2 @ L @ ( topolo174197925503356063within @ real @ X2 @ S ) )
=> ( ( ord_less @ real @ L @ ( zero_zero @ real ) )
=> ? [D4: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ D4 )
& ! [H7: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ H7 )
=> ( ( member @ real @ ( minus_minus @ real @ X2 @ H7 ) @ S )
=> ( ( ord_less @ real @ H7 @ D4 )
=> ( ord_less @ real @ ( F2 @ X2 ) @ ( F2 @ ( minus_minus @ real @ X2 @ H7 ) ) ) ) ) ) ) ) ) ).
% has_real_derivative_neg_dec_left
thf(fact_6874_has__real__derivative__pos__inc__left,axiom,
! [F2: real > real,L: real,X2: real,S: set @ real] :
( ( has_field_derivative @ real @ F2 @ L @ ( topolo174197925503356063within @ real @ X2 @ S ) )
=> ( ( ord_less @ real @ ( zero_zero @ real ) @ L )
=> ? [D4: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ D4 )
& ! [H7: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ H7 )
=> ( ( member @ real @ ( minus_minus @ real @ X2 @ H7 ) @ S )
=> ( ( ord_less @ real @ H7 @ D4 )
=> ( ord_less @ real @ ( F2 @ ( minus_minus @ real @ X2 @ H7 ) ) @ ( F2 @ X2 ) ) ) ) ) ) ) ) ).
% has_real_derivative_pos_inc_left
thf(fact_6875_DERIV__cmult__Id,axiom,
! [A: $tType] :
( ( real_V3459762299906320749_field @ A )
=> ! [C3: A,X2: A,S2: set @ A] : ( has_field_derivative @ A @ ( times_times @ A @ C3 ) @ C3 @ ( topolo174197925503356063within @ A @ X2 @ S2 ) ) ) ).
% DERIV_cmult_Id
thf(fact_6876_DERIV__cdivide,axiom,
! [A: $tType] :
( ( real_V3459762299906320749_field @ A )
=> ! [F2: A > A,D: A,X2: A,S2: set @ A,C3: A] :
( ( has_field_derivative @ A @ F2 @ D @ ( topolo174197925503356063within @ A @ X2 @ S2 ) )
=> ( has_field_derivative @ A
@ ^ [X3: A] : ( divide_divide @ A @ ( F2 @ X3 ) @ C3 )
@ ( divide_divide @ A @ D @ C3 )
@ ( topolo174197925503356063within @ A @ X2 @ S2 ) ) ) ) ).
% DERIV_cdivide
thf(fact_6877_DERIV__divide,axiom,
! [A: $tType] :
( ( real_V3459762299906320749_field @ A )
=> ! [F2: A > A,D: A,X2: A,S2: set @ A,G: A > A,E5: A] :
( ( has_field_derivative @ A @ F2 @ D @ ( topolo174197925503356063within @ A @ X2 @ S2 ) )
=> ( ( has_field_derivative @ A @ G @ E5 @ ( topolo174197925503356063within @ A @ X2 @ S2 ) )
=> ( ( ( G @ X2 )
!= ( zero_zero @ A ) )
=> ( has_field_derivative @ A
@ ^ [X3: A] : ( divide_divide @ A @ ( F2 @ X3 ) @ ( G @ X3 ) )
@ ( divide_divide @ A @ ( minus_minus @ A @ ( times_times @ A @ D @ ( G @ X2 ) ) @ ( times_times @ A @ ( F2 @ X2 ) @ E5 ) ) @ ( times_times @ A @ ( G @ X2 ) @ ( G @ X2 ) ) )
@ ( topolo174197925503356063within @ A @ X2 @ S2 ) ) ) ) ) ) ).
% DERIV_divide
thf(fact_6878_DERIV__inverse_H,axiom,
! [A: $tType] :
( ( real_V3459762299906320749_field @ A )
=> ! [F2: A > A,D: A,X2: A,S2: set @ A] :
( ( has_field_derivative @ A @ F2 @ D @ ( topolo174197925503356063within @ A @ X2 @ S2 ) )
=> ( ( ( F2 @ X2 )
!= ( zero_zero @ A ) )
=> ( has_field_derivative @ A
@ ^ [X3: A] : ( inverse_inverse @ A @ ( F2 @ X3 ) )
@ ( uminus_uminus @ A @ ( times_times @ A @ ( times_times @ A @ ( inverse_inverse @ A @ ( F2 @ X2 ) ) @ D ) @ ( inverse_inverse @ A @ ( F2 @ X2 ) ) ) )
@ ( topolo174197925503356063within @ A @ X2 @ S2 ) ) ) ) ) ).
% DERIV_inverse'
thf(fact_6879_DERIV__mult,axiom,
! [A: $tType] :
( ( real_V3459762299906320749_field @ A )
=> ! [F2: A > A,Da: A,X2: A,S2: set @ A,G: A > A,Db: A] :
( ( has_field_derivative @ A @ F2 @ Da @ ( topolo174197925503356063within @ A @ X2 @ S2 ) )
=> ( ( has_field_derivative @ A @ G @ Db @ ( topolo174197925503356063within @ A @ X2 @ S2 ) )
=> ( has_field_derivative @ A
@ ^ [X3: A] : ( times_times @ A @ ( F2 @ X3 ) @ ( G @ X3 ) )
@ ( plus_plus @ A @ ( times_times @ A @ Da @ ( G @ X2 ) ) @ ( times_times @ A @ Db @ ( F2 @ X2 ) ) )
@ ( topolo174197925503356063within @ A @ X2 @ S2 ) ) ) ) ) ).
% DERIV_mult
thf(fact_6880_DERIV__mult_H,axiom,
! [A: $tType] :
( ( real_V3459762299906320749_field @ A )
=> ! [F2: A > A,D: A,X2: A,S2: set @ A,G: A > A,E5: A] :
( ( has_field_derivative @ A @ F2 @ D @ ( topolo174197925503356063within @ A @ X2 @ S2 ) )
=> ( ( has_field_derivative @ A @ G @ E5 @ ( topolo174197925503356063within @ A @ X2 @ S2 ) )
=> ( has_field_derivative @ A
@ ^ [X3: A] : ( times_times @ A @ ( F2 @ X3 ) @ ( G @ X3 ) )
@ ( plus_plus @ A @ ( times_times @ A @ ( F2 @ X2 ) @ E5 ) @ ( times_times @ A @ D @ ( G @ X2 ) ) )
@ ( topolo174197925503356063within @ A @ X2 @ S2 ) ) ) ) ) ).
% DERIV_mult'
thf(fact_6881_has__field__derivative__cosh,axiom,
! [A13: $tType] :
( ( ( real_Vector_banach @ A13 )
& ( real_V3459762299906320749_field @ A13 ) )
=> ! [G: A13 > A13,Db: A13,X2: A13,S2: set @ A13] :
( ( has_field_derivative @ A13 @ G @ Db @ ( topolo174197925503356063within @ A13 @ X2 @ S2 ) )
=> ( has_field_derivative @ A13
@ ^ [X3: A13] : ( cosh @ A13 @ ( G @ X3 ) )
@ ( times_times @ A13 @ ( sinh @ A13 @ ( G @ X2 ) ) @ Db )
@ ( topolo174197925503356063within @ A13 @ X2 @ S2 ) ) ) ) ).
% has_field_derivative_cosh
thf(fact_6882_has__field__derivative__sinh,axiom,
! [A13: $tType] :
( ( ( real_Vector_banach @ A13 )
& ( real_V3459762299906320749_field @ A13 ) )
=> ! [G: A13 > A13,Db: A13,X2: A13,S2: set @ A13] :
( ( has_field_derivative @ A13 @ G @ Db @ ( topolo174197925503356063within @ A13 @ X2 @ S2 ) )
=> ( has_field_derivative @ A13
@ ^ [X3: A13] : ( sinh @ A13 @ ( G @ X3 ) )
@ ( times_times @ A13 @ ( cosh @ A13 @ ( G @ X2 ) ) @ Db )
@ ( topolo174197925503356063within @ A13 @ X2 @ S2 ) ) ) ) ).
% has_field_derivative_sinh
thf(fact_6883_DERIV__cmult,axiom,
! [A: $tType] :
( ( real_V3459762299906320749_field @ A )
=> ! [F2: A > A,D: A,X2: A,S2: set @ A,C3: A] :
( ( has_field_derivative @ A @ F2 @ D @ ( topolo174197925503356063within @ A @ X2 @ S2 ) )
=> ( has_field_derivative @ A
@ ^ [X3: A] : ( times_times @ A @ C3 @ ( F2 @ X3 ) )
@ ( times_times @ A @ C3 @ D )
@ ( topolo174197925503356063within @ A @ X2 @ S2 ) ) ) ) ).
% DERIV_cmult
thf(fact_6884_DERIV__cmult__right,axiom,
! [A: $tType] :
( ( real_V3459762299906320749_field @ A )
=> ! [F2: A > A,D: A,X2: A,S2: set @ A,C3: A] :
( ( has_field_derivative @ A @ F2 @ D @ ( topolo174197925503356063within @ A @ X2 @ S2 ) )
=> ( has_field_derivative @ A
@ ^ [X3: A] : ( times_times @ A @ ( F2 @ X3 ) @ C3 )
@ ( times_times @ A @ D @ C3 )
@ ( topolo174197925503356063within @ A @ X2 @ S2 ) ) ) ) ).
% DERIV_cmult_right
thf(fact_6885_field__differentiable__minus,axiom,
! [A: $tType] :
( ( real_V3459762299906320749_field @ A )
=> ! [F2: A > A,F7: A,F4: filter @ A] :
( ( has_field_derivative @ A @ F2 @ F7 @ F4 )
=> ( has_field_derivative @ A
@ ^ [Z4: A] : ( uminus_uminus @ A @ ( F2 @ Z4 ) )
@ ( uminus_uminus @ A @ F7 )
@ F4 ) ) ) ).
% field_differentiable_minus
thf(fact_6886_DERIV__minus,axiom,
! [A: $tType] :
( ( real_V3459762299906320749_field @ A )
=> ! [F2: A > A,D: A,X2: A,S2: set @ A] :
( ( has_field_derivative @ A @ F2 @ D @ ( topolo174197925503356063within @ A @ X2 @ S2 ) )
=> ( has_field_derivative @ A
@ ^ [X3: A] : ( uminus_uminus @ A @ ( F2 @ X3 ) )
@ ( uminus_uminus @ A @ D )
@ ( topolo174197925503356063within @ A @ X2 @ S2 ) ) ) ) ).
% DERIV_minus
thf(fact_6887_DERIV__ident,axiom,
! [A: $tType] :
( ( real_V3459762299906320749_field @ A )
=> ! [F4: filter @ A] :
( has_field_derivative @ A
@ ^ [X3: A] : X3
@ ( one_one @ A )
@ F4 ) ) ).
% DERIV_ident
thf(fact_6888_DERIV__const,axiom,
! [A: $tType] :
( ( real_V3459762299906320749_field @ A )
=> ! [K: A,F4: filter @ A] :
( has_field_derivative @ A
@ ^ [X3: A] : K
@ ( zero_zero @ A )
@ F4 ) ) ).
% DERIV_const
thf(fact_6889_has__field__derivative__scaleR__right,axiom,
! [A: $tType] :
( ( real_V3459762299906320749_field @ A )
=> ! [F2: A > A,D: A,F4: filter @ A,C3: real] :
( ( has_field_derivative @ A @ F2 @ D @ F4 )
=> ( has_field_derivative @ A
@ ^ [X3: A] : ( real_V8093663219630862766scaleR @ A @ C3 @ ( F2 @ X3 ) )
@ ( real_V8093663219630862766scaleR @ A @ C3 @ D )
@ F4 ) ) ) ).
% has_field_derivative_scaleR_right
thf(fact_6890_DERIV__sum,axiom,
! [A: $tType,C: $tType,B: $tType] :
( ( real_V3459762299906320749_field @ B )
=> ! [S: set @ A,F2: B > A > B,F7: C > A > B,X2: C,F4: filter @ B] :
( ! [N3: A] :
( ( member @ A @ N3 @ S )
=> ( has_field_derivative @ B
@ ^ [X3: B] : ( F2 @ X3 @ N3 )
@ ( F7 @ X2 @ N3 )
@ F4 ) )
=> ( has_field_derivative @ B
@ ^ [X3: B] : ( groups7311177749621191930dd_sum @ A @ B @ ( F2 @ X3 ) @ S )
@ ( groups7311177749621191930dd_sum @ A @ B @ ( F7 @ X2 ) @ S )
@ F4 ) ) ) ).
% DERIV_sum
thf(fact_6891_field__differentiable__diff,axiom,
! [A: $tType] :
( ( real_V3459762299906320749_field @ A )
=> ! [F2: A > A,F7: A,F4: filter @ A,G: A > A,G5: A] :
( ( has_field_derivative @ A @ F2 @ F7 @ F4 )
=> ( ( has_field_derivative @ A @ G @ G5 @ F4 )
=> ( has_field_derivative @ A
@ ^ [Z4: A] : ( minus_minus @ A @ ( F2 @ Z4 ) @ ( G @ Z4 ) )
@ ( minus_minus @ A @ F7 @ G5 )
@ F4 ) ) ) ) ).
% field_differentiable_diff
thf(fact_6892_DERIV__diff,axiom,
! [A: $tType] :
( ( real_V3459762299906320749_field @ A )
=> ! [F2: A > A,D: A,X2: A,S2: set @ A,G: A > A,E5: A] :
( ( has_field_derivative @ A @ F2 @ D @ ( topolo174197925503356063within @ A @ X2 @ S2 ) )
=> ( ( has_field_derivative @ A @ G @ E5 @ ( topolo174197925503356063within @ A @ X2 @ S2 ) )
=> ( has_field_derivative @ A
@ ^ [X3: A] : ( minus_minus @ A @ ( F2 @ X3 ) @ ( G @ X3 ) )
@ ( minus_minus @ A @ D @ E5 )
@ ( topolo174197925503356063within @ A @ X2 @ S2 ) ) ) ) ) ).
% DERIV_diff
thf(fact_6893_field__differentiable__add,axiom,
! [A: $tType] :
( ( real_V3459762299906320749_field @ A )
=> ! [F2: A > A,F7: A,F4: filter @ A,G: A > A,G5: A] :
( ( has_field_derivative @ A @ F2 @ F7 @ F4 )
=> ( ( has_field_derivative @ A @ G @ G5 @ F4 )
=> ( has_field_derivative @ A
@ ^ [Z4: A] : ( plus_plus @ A @ ( F2 @ Z4 ) @ ( G @ Z4 ) )
@ ( plus_plus @ A @ F7 @ G5 )
@ F4 ) ) ) ) ).
% field_differentiable_add
thf(fact_6894_DERIV__add,axiom,
! [A: $tType] :
( ( real_V3459762299906320749_field @ A )
=> ! [F2: A > A,D: A,X2: A,S2: set @ A,G: A > A,E5: A] :
( ( has_field_derivative @ A @ F2 @ D @ ( topolo174197925503356063within @ A @ X2 @ S2 ) )
=> ( ( has_field_derivative @ A @ G @ E5 @ ( topolo174197925503356063within @ A @ X2 @ S2 ) )
=> ( has_field_derivative @ A
@ ^ [X3: A] : ( plus_plus @ A @ ( F2 @ X3 ) @ ( G @ X3 ) )
@ ( plus_plus @ A @ D @ E5 )
@ ( topolo174197925503356063within @ A @ X2 @ S2 ) ) ) ) ) ).
% DERIV_add
thf(fact_6895_at__le,axiom,
! [A: $tType] :
( ( topolo4958980785337419405_space @ A )
=> ! [S2: set @ A,T2: set @ A,X2: A] :
( ( ord_less_eq @ ( set @ A ) @ S2 @ T2 )
=> ( ord_less_eq @ ( filter @ A ) @ ( topolo174197925503356063within @ A @ X2 @ S2 ) @ ( topolo174197925503356063within @ A @ X2 @ T2 ) ) ) ) ).
% at_le
thf(fact_6896_has__field__derivative__subset,axiom,
! [A: $tType] :
( ( real_V3459762299906320749_field @ A )
=> ! [F2: A > A,Y3: A,X2: A,S2: set @ A,T2: set @ A] :
( ( has_field_derivative @ A @ F2 @ Y3 @ ( topolo174197925503356063within @ A @ X2 @ S2 ) )
=> ( ( ord_less_eq @ ( set @ A ) @ T2 @ S2 )
=> ( has_field_derivative @ A @ F2 @ Y3 @ ( topolo174197925503356063within @ A @ X2 @ T2 ) ) ) ) ) ).
% has_field_derivative_subset
thf(fact_6897_DERIV__subset,axiom,
! [A: $tType] :
( ( real_V3459762299906320749_field @ A )
=> ! [F2: A > A,F7: A,X2: A,S2: set @ A,T2: set @ A] :
( ( has_field_derivative @ A @ F2 @ F7 @ ( topolo174197925503356063within @ A @ X2 @ S2 ) )
=> ( ( ord_less_eq @ ( set @ A ) @ T2 @ S2 )
=> ( has_field_derivative @ A @ F2 @ F7 @ ( topolo174197925503356063within @ A @ X2 @ T2 ) ) ) ) ) ).
% DERIV_subset
thf(fact_6898_DERIV__fun__cos,axiom,
! [A: $tType] :
( ( ( real_Vector_banach @ A )
& ( real_V3459762299906320749_field @ A ) )
=> ! [G: A > A,M: A,X2: A] :
( ( has_field_derivative @ A @ G @ M @ ( topolo174197925503356063within @ A @ X2 @ ( top_top @ ( set @ A ) ) ) )
=> ( has_field_derivative @ A
@ ^ [X3: A] : ( cos @ A @ ( G @ X3 ) )
@ ( times_times @ A @ ( uminus_uminus @ A @ ( sin @ A @ ( G @ X2 ) ) ) @ M )
@ ( topolo174197925503356063within @ A @ X2 @ ( top_top @ ( set @ A ) ) ) ) ) ) ).
% DERIV_fun_cos
thf(fact_6899_DERIV__cos__add,axiom,
! [A: $tType] :
( ( ( real_Vector_banach @ A )
& ( real_V3459762299906320749_field @ A ) )
=> ! [K: A,Xa: A] :
( has_field_derivative @ A
@ ^ [X3: A] : ( cos @ A @ ( plus_plus @ A @ X3 @ K ) )
@ ( uminus_uminus @ A @ ( sin @ A @ ( plus_plus @ A @ Xa @ K ) ) )
@ ( topolo174197925503356063within @ A @ Xa @ ( top_top @ ( set @ A ) ) ) ) ) ).
% DERIV_cos_add
thf(fact_6900_DERIV__local__const,axiom,
! [F2: real > real,L: real,X2: real,D2: real] :
( ( has_field_derivative @ real @ F2 @ L @ ( topolo174197925503356063within @ real @ X2 @ ( top_top @ ( set @ real ) ) ) )
=> ( ( ord_less @ real @ ( zero_zero @ real ) @ D2 )
=> ( ! [Y4: real] :
( ( ord_less @ real @ ( abs_abs @ real @ ( minus_minus @ real @ X2 @ Y4 ) ) @ D2 )
=> ( ( F2 @ X2 )
= ( F2 @ Y4 ) ) )
=> ( L
= ( zero_zero @ real ) ) ) ) ) ).
% DERIV_local_const
thf(fact_6901_MVT2,axiom,
! [A3: real,B3: real,F2: real > real,F7: real > real] :
( ( ord_less @ real @ A3 @ B3 )
=> ( ! [X4: real] :
( ( ord_less_eq @ real @ A3 @ X4 )
=> ( ( ord_less_eq @ real @ X4 @ B3 )
=> ( has_field_derivative @ real @ F2 @ ( F7 @ X4 ) @ ( topolo174197925503356063within @ real @ X4 @ ( top_top @ ( set @ real ) ) ) ) ) )
=> ? [Z2: real] :
( ( ord_less @ real @ A3 @ Z2 )
& ( ord_less @ real @ Z2 @ B3 )
& ( ( minus_minus @ real @ ( F2 @ B3 ) @ ( F2 @ A3 ) )
= ( times_times @ real @ ( minus_minus @ real @ B3 @ A3 ) @ ( F7 @ Z2 ) ) ) ) ) ) ).
% MVT2
thf(fact_6902_DERIV__mirror,axiom,
! [F2: real > real,Y3: real,X2: real] :
( ( has_field_derivative @ real @ F2 @ Y3 @ ( topolo174197925503356063within @ real @ ( uminus_uminus @ real @ X2 ) @ ( top_top @ ( set @ real ) ) ) )
= ( has_field_derivative @ real
@ ^ [X3: real] : ( F2 @ ( uminus_uminus @ real @ X3 ) )
@ ( uminus_uminus @ real @ Y3 )
@ ( topolo174197925503356063within @ real @ X2 @ ( top_top @ ( set @ real ) ) ) ) ) ).
% DERIV_mirror
thf(fact_6903_deriv__nonneg__imp__mono,axiom,
! [A3: real,B3: real,G: real > real,G5: real > real] :
( ! [X4: real] :
( ( member @ real @ X4 @ ( set_or1337092689740270186AtMost @ real @ A3 @ B3 ) )
=> ( has_field_derivative @ real @ G @ ( G5 @ X4 ) @ ( topolo174197925503356063within @ real @ X4 @ ( top_top @ ( set @ real ) ) ) ) )
=> ( ! [X4: real] :
( ( member @ real @ X4 @ ( set_or1337092689740270186AtMost @ real @ A3 @ B3 ) )
=> ( ord_less_eq @ real @ ( zero_zero @ real ) @ ( G5 @ X4 ) ) )
=> ( ( ord_less_eq @ real @ A3 @ B3 )
=> ( ord_less_eq @ real @ ( G @ A3 ) @ ( G @ B3 ) ) ) ) ) ).
% deriv_nonneg_imp_mono
thf(fact_6904_DERIV__nonpos__imp__nonincreasing,axiom,
! [A3: real,B3: real,F2: real > real] :
( ( ord_less_eq @ real @ A3 @ B3 )
=> ( ! [X4: real] :
( ( ord_less_eq @ real @ A3 @ X4 )
=> ( ( ord_less_eq @ real @ X4 @ B3 )
=> ? [Y5: real] :
( ( has_field_derivative @ real @ F2 @ Y5 @ ( topolo174197925503356063within @ real @ X4 @ ( top_top @ ( set @ real ) ) ) )
& ( ord_less_eq @ real @ Y5 @ ( zero_zero @ real ) ) ) ) )
=> ( ord_less_eq @ real @ ( F2 @ B3 ) @ ( F2 @ A3 ) ) ) ) ).
% DERIV_nonpos_imp_nonincreasing
thf(fact_6905_DERIV__nonneg__imp__nondecreasing,axiom,
! [A3: real,B3: real,F2: real > real] :
( ( ord_less_eq @ real @ A3 @ B3 )
=> ( ! [X4: real] :
( ( ord_less_eq @ real @ A3 @ X4 )
=> ( ( ord_less_eq @ real @ X4 @ B3 )
=> ? [Y5: real] :
( ( has_field_derivative @ real @ F2 @ Y5 @ ( topolo174197925503356063within @ real @ X4 @ ( top_top @ ( set @ real ) ) ) )
& ( ord_less_eq @ real @ ( zero_zero @ real ) @ Y5 ) ) ) )
=> ( ord_less_eq @ real @ ( F2 @ A3 ) @ ( F2 @ B3 ) ) ) ) ).
% DERIV_nonneg_imp_nondecreasing
thf(fact_6906_DERIV__neg__imp__decreasing,axiom,
! [A3: real,B3: real,F2: real > real] :
( ( ord_less @ real @ A3 @ B3 )
=> ( ! [X4: real] :
( ( ord_less_eq @ real @ A3 @ X4 )
=> ( ( ord_less_eq @ real @ X4 @ B3 )
=> ? [Y5: real] :
( ( has_field_derivative @ real @ F2 @ Y5 @ ( topolo174197925503356063within @ real @ X4 @ ( top_top @ ( set @ real ) ) ) )
& ( ord_less @ real @ Y5 @ ( zero_zero @ real ) ) ) ) )
=> ( ord_less @ real @ ( F2 @ B3 ) @ ( F2 @ A3 ) ) ) ) ).
% DERIV_neg_imp_decreasing
thf(fact_6907_DERIV__pos__imp__increasing,axiom,
! [A3: real,B3: real,F2: real > real] :
( ( ord_less @ real @ A3 @ B3 )
=> ( ! [X4: real] :
( ( ord_less_eq @ real @ A3 @ X4 )
=> ( ( ord_less_eq @ real @ X4 @ B3 )
=> ? [Y5: real] :
( ( has_field_derivative @ real @ F2 @ Y5 @ ( topolo174197925503356063within @ real @ X4 @ ( top_top @ ( set @ real ) ) ) )
& ( ord_less @ real @ ( zero_zero @ real ) @ Y5 ) ) ) )
=> ( ord_less @ real @ ( F2 @ A3 ) @ ( F2 @ B3 ) ) ) ) ).
% DERIV_pos_imp_increasing
thf(fact_6908_DERIV__neg__dec__left,axiom,
! [F2: real > real,L: real,X2: real] :
( ( has_field_derivative @ real @ F2 @ L @ ( topolo174197925503356063within @ real @ X2 @ ( top_top @ ( set @ real ) ) ) )
=> ( ( ord_less @ real @ L @ ( zero_zero @ real ) )
=> ? [D4: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ D4 )
& ! [H7: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ H7 )
=> ( ( ord_less @ real @ H7 @ D4 )
=> ( ord_less @ real @ ( F2 @ X2 ) @ ( F2 @ ( minus_minus @ real @ X2 @ H7 ) ) ) ) ) ) ) ) ).
% DERIV_neg_dec_left
thf(fact_6909_DERIV__pos__inc__left,axiom,
! [F2: real > real,L: real,X2: real] :
( ( has_field_derivative @ real @ F2 @ L @ ( topolo174197925503356063within @ real @ X2 @ ( top_top @ ( set @ real ) ) ) )
=> ( ( ord_less @ real @ ( zero_zero @ real ) @ L )
=> ? [D4: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ D4 )
& ! [H7: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ H7 )
=> ( ( ord_less @ real @ H7 @ D4 )
=> ( ord_less @ real @ ( F2 @ ( minus_minus @ real @ X2 @ H7 ) ) @ ( F2 @ X2 ) ) ) ) ) ) ) ).
% DERIV_pos_inc_left
thf(fact_6910_DERIV__neg__dec__right,axiom,
! [F2: real > real,L: real,X2: real] :
( ( has_field_derivative @ real @ F2 @ L @ ( topolo174197925503356063within @ real @ X2 @ ( top_top @ ( set @ real ) ) ) )
=> ( ( ord_less @ real @ L @ ( zero_zero @ real ) )
=> ? [D4: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ D4 )
& ! [H7: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ H7 )
=> ( ( ord_less @ real @ H7 @ D4 )
=> ( ord_less @ real @ ( F2 @ ( plus_plus @ real @ X2 @ H7 ) ) @ ( F2 @ X2 ) ) ) ) ) ) ) ).
% DERIV_neg_dec_right
thf(fact_6911_DERIV__pos__inc__right,axiom,
! [F2: real > real,L: real,X2: real] :
( ( has_field_derivative @ real @ F2 @ L @ ( topolo174197925503356063within @ real @ X2 @ ( top_top @ ( set @ real ) ) ) )
=> ( ( ord_less @ real @ ( zero_zero @ real ) @ L )
=> ? [D4: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ D4 )
& ! [H7: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ H7 )
=> ( ( ord_less @ real @ H7 @ D4 )
=> ( ord_less @ real @ ( F2 @ X2 ) @ ( F2 @ ( plus_plus @ real @ X2 @ H7 ) ) ) ) ) ) ) ) ).
% DERIV_pos_inc_right
thf(fact_6912_DERIV__shift,axiom,
! [A: $tType] :
( ( real_V3459762299906320749_field @ A )
=> ! [F2: A > A,Y3: A,X2: A,Z: A] :
( ( has_field_derivative @ A @ F2 @ Y3 @ ( topolo174197925503356063within @ A @ ( plus_plus @ A @ X2 @ Z ) @ ( top_top @ ( set @ A ) ) ) )
= ( has_field_derivative @ A
@ ^ [X3: A] : ( F2 @ ( plus_plus @ A @ X3 @ Z ) )
@ Y3
@ ( topolo174197925503356063within @ A @ X2 @ ( top_top @ ( set @ A ) ) ) ) ) ) ).
% DERIV_shift
thf(fact_6913_DERIV__fun__sin,axiom,
! [A: $tType] :
( ( ( real_Vector_banach @ A )
& ( real_V3459762299906320749_field @ A ) )
=> ! [G: A > A,M: A,X2: A] :
( ( has_field_derivative @ A @ G @ M @ ( topolo174197925503356063within @ A @ X2 @ ( top_top @ ( set @ A ) ) ) )
=> ( has_field_derivative @ A
@ ^ [X3: A] : ( sin @ A @ ( G @ X3 ) )
@ ( times_times @ A @ ( cos @ A @ ( G @ X2 ) ) @ M )
@ ( topolo174197925503356063within @ A @ X2 @ ( top_top @ ( set @ A ) ) ) ) ) ) ).
% DERIV_fun_sin
thf(fact_6914_DERIV__fun__exp,axiom,
! [A: $tType] :
( ( ( real_Vector_banach @ A )
& ( real_V3459762299906320749_field @ A ) )
=> ! [G: A > A,M: A,X2: A] :
( ( has_field_derivative @ A @ G @ M @ ( topolo174197925503356063within @ A @ X2 @ ( top_top @ ( set @ A ) ) ) )
=> ( has_field_derivative @ A
@ ^ [X3: A] : ( exp @ A @ ( G @ X3 ) )
@ ( times_times @ A @ ( exp @ A @ ( G @ X2 ) ) @ M )
@ ( topolo174197925503356063within @ A @ X2 @ ( top_top @ ( set @ A ) ) ) ) ) ) ).
% DERIV_fun_exp
thf(fact_6915_DERIV__chain_H,axiom,
! [A: $tType] :
( ( real_V3459762299906320749_field @ A )
=> ! [F2: A > A,D: A,X2: A,S2: set @ A,G: A > A,E5: A] :
( ( has_field_derivative @ A @ F2 @ D @ ( topolo174197925503356063within @ A @ X2 @ S2 ) )
=> ( ( has_field_derivative @ A @ G @ E5 @ ( topolo174197925503356063within @ A @ ( F2 @ X2 ) @ ( top_top @ ( set @ A ) ) ) )
=> ( has_field_derivative @ A
@ ^ [X3: A] : ( G @ ( F2 @ X3 ) )
@ ( times_times @ A @ E5 @ D )
@ ( topolo174197925503356063within @ A @ X2 @ S2 ) ) ) ) ) ).
% DERIV_chain'
thf(fact_6916_DERIV__chain2,axiom,
! [A: $tType] :
( ( real_V3459762299906320749_field @ A )
=> ! [F2: A > A,Da: A,G: A > A,X2: A,Db: A,S2: set @ A] :
( ( has_field_derivative @ A @ F2 @ Da @ ( topolo174197925503356063within @ A @ ( G @ X2 ) @ ( top_top @ ( set @ A ) ) ) )
=> ( ( has_field_derivative @ A @ G @ Db @ ( topolo174197925503356063within @ A @ X2 @ S2 ) )
=> ( has_field_derivative @ A
@ ^ [X3: A] : ( F2 @ ( G @ X3 ) )
@ ( times_times @ A @ Da @ Db )
@ ( topolo174197925503356063within @ A @ X2 @ S2 ) ) ) ) ) ).
% DERIV_chain2
thf(fact_6917_DERIV__chain3,axiom,
! [A: $tType] :
( ( real_V3459762299906320749_field @ A )
=> ! [G: A > A,G5: A > A,F2: A > A,F7: A,X2: A] :
( ! [X4: A] : ( has_field_derivative @ A @ G @ ( G5 @ X4 ) @ ( topolo174197925503356063within @ A @ X4 @ ( top_top @ ( set @ A ) ) ) )
=> ( ( has_field_derivative @ A @ F2 @ F7 @ ( topolo174197925503356063within @ A @ X2 @ ( top_top @ ( set @ A ) ) ) )
=> ( has_field_derivative @ A
@ ^ [X3: A] : ( G @ ( F2 @ X3 ) )
@ ( times_times @ A @ F7 @ ( G5 @ ( F2 @ X2 ) ) )
@ ( topolo174197925503356063within @ A @ X2 @ ( top_top @ ( set @ A ) ) ) ) ) ) ) ).
% DERIV_chain3
thf(fact_6918_DERIV__chain__s,axiom,
! [A: $tType] :
( ( real_V3459762299906320749_field @ A )
=> ! [S2: set @ A,G: A > A,G5: A > A,F2: A > A,F7: A,X2: A] :
( ! [X4: A] :
( ( member @ A @ X4 @ S2 )
=> ( has_field_derivative @ A @ G @ ( G5 @ X4 ) @ ( topolo174197925503356063within @ A @ X4 @ ( top_top @ ( set @ A ) ) ) ) )
=> ( ( has_field_derivative @ A @ F2 @ F7 @ ( topolo174197925503356063within @ A @ X2 @ ( top_top @ ( set @ A ) ) ) )
=> ( ( member @ A @ ( F2 @ X2 ) @ S2 )
=> ( has_field_derivative @ A
@ ^ [X3: A] : ( G @ ( F2 @ X3 ) )
@ ( times_times @ A @ F7 @ ( G5 @ ( F2 @ X2 ) ) )
@ ( topolo174197925503356063within @ A @ X2 @ ( top_top @ ( set @ A ) ) ) ) ) ) ) ) ).
% DERIV_chain_s
thf(fact_6919_DERIV__const__ratio__const,axiom,
! [A3: real,B3: real,F2: real > real,K: real] :
( ( A3 != B3 )
=> ( ! [X4: real] : ( has_field_derivative @ real @ F2 @ K @ ( topolo174197925503356063within @ real @ X4 @ ( top_top @ ( set @ real ) ) ) )
=> ( ( minus_minus @ real @ ( F2 @ B3 ) @ ( F2 @ A3 ) )
= ( times_times @ real @ ( minus_minus @ real @ B3 @ A3 ) @ K ) ) ) ) ).
% DERIV_const_ratio_const
thf(fact_6920_DERIV__const__ratio__const2,axiom,
! [A3: real,B3: real,F2: real > real,K: real] :
( ( A3 != B3 )
=> ( ! [X4: real] : ( has_field_derivative @ real @ F2 @ K @ ( topolo174197925503356063within @ real @ X4 @ ( top_top @ ( set @ real ) ) ) )
=> ( ( divide_divide @ real @ ( minus_minus @ real @ ( F2 @ B3 ) @ ( F2 @ A3 ) ) @ ( minus_minus @ real @ B3 @ A3 ) )
= K ) ) ) ).
% DERIV_const_ratio_const2
thf(fact_6921_DERIV__at__within__shift,axiom,
! [A: $tType] :
( ( real_V3459762299906320749_field @ A )
=> ! [F2: A > A,Y3: A,Z: A,X2: A,S: set @ A] :
( ( has_field_derivative @ A @ F2 @ Y3 @ ( topolo174197925503356063within @ A @ ( plus_plus @ A @ Z @ X2 ) @ ( image @ A @ A @ ( plus_plus @ A @ Z ) @ S ) ) )
= ( has_field_derivative @ A
@ ^ [X3: A] : ( F2 @ ( plus_plus @ A @ Z @ X3 ) )
@ Y3
@ ( topolo174197925503356063within @ A @ X2 @ S ) ) ) ) ).
% DERIV_at_within_shift
thf(fact_6922_DERIV__power__Suc,axiom,
! [A: $tType] :
( ( real_V3459762299906320749_field @ A )
=> ! [F2: A > A,D: A,X2: A,S2: set @ A,N: nat] :
( ( has_field_derivative @ A @ F2 @ D @ ( topolo174197925503356063within @ A @ X2 @ S2 ) )
=> ( has_field_derivative @ A
@ ^ [X3: A] : ( power_power @ A @ ( F2 @ X3 ) @ ( suc @ N ) )
@ ( times_times @ A @ ( plus_plus @ A @ ( one_one @ A ) @ ( semiring_1_of_nat @ A @ N ) ) @ ( times_times @ A @ D @ ( power_power @ A @ ( F2 @ X2 ) @ N ) ) )
@ ( topolo174197925503356063within @ A @ X2 @ S2 ) ) ) ) ).
% DERIV_power_Suc
thf(fact_6923_DERIV__const__average,axiom,
! [A3: real,B3: real,V: real > real,K: real] :
( ( A3 != B3 )
=> ( ! [X4: real] : ( has_field_derivative @ real @ V @ K @ ( topolo174197925503356063within @ real @ X4 @ ( top_top @ ( set @ real ) ) ) )
=> ( ( V @ ( divide_divide @ real @ ( plus_plus @ real @ A3 @ B3 ) @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) )
= ( divide_divide @ real @ ( plus_plus @ real @ ( V @ A3 ) @ ( V @ B3 ) ) @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) ) ) ) ).
% DERIV_const_average
thf(fact_6924_DERIV__power,axiom,
! [A: $tType] :
( ( real_V3459762299906320749_field @ A )
=> ! [F2: A > A,D: A,X2: A,S2: set @ A,N: nat] :
( ( has_field_derivative @ A @ F2 @ D @ ( topolo174197925503356063within @ A @ X2 @ S2 ) )
=> ( has_field_derivative @ A
@ ^ [X3: A] : ( power_power @ A @ ( F2 @ X3 ) @ N )
@ ( times_times @ A @ ( semiring_1_of_nat @ A @ N ) @ ( times_times @ A @ D @ ( power_power @ A @ ( F2 @ X2 ) @ ( minus_minus @ nat @ N @ ( suc @ ( zero_zero @ nat ) ) ) ) ) )
@ ( topolo174197925503356063within @ A @ X2 @ S2 ) ) ) ) ).
% DERIV_power
thf(fact_6925_DERIV__inverse,axiom,
! [A: $tType] :
( ( real_V3459762299906320749_field @ A )
=> ! [X2: A,S2: set @ A] :
( ( X2
!= ( zero_zero @ A ) )
=> ( has_field_derivative @ A @ ( inverse_inverse @ A ) @ ( uminus_uminus @ A @ ( power_power @ A @ ( inverse_inverse @ A @ X2 ) @ ( suc @ ( suc @ ( zero_zero @ nat ) ) ) ) ) @ ( topolo174197925503356063within @ A @ X2 @ S2 ) ) ) ) ).
% DERIV_inverse
thf(fact_6926_DERIV__local__min,axiom,
! [F2: real > real,L: real,X2: real,D2: real] :
( ( has_field_derivative @ real @ F2 @ L @ ( topolo174197925503356063within @ real @ X2 @ ( top_top @ ( set @ real ) ) ) )
=> ( ( ord_less @ real @ ( zero_zero @ real ) @ D2 )
=> ( ! [Y4: real] :
( ( ord_less @ real @ ( abs_abs @ real @ ( minus_minus @ real @ X2 @ Y4 ) ) @ D2 )
=> ( ord_less_eq @ real @ ( F2 @ X2 ) @ ( F2 @ Y4 ) ) )
=> ( L
= ( zero_zero @ real ) ) ) ) ) ).
% DERIV_local_min
thf(fact_6927_DERIV__local__max,axiom,
! [F2: real > real,L: real,X2: real,D2: real] :
( ( has_field_derivative @ real @ F2 @ L @ ( topolo174197925503356063within @ real @ X2 @ ( top_top @ ( set @ real ) ) ) )
=> ( ( ord_less @ real @ ( zero_zero @ real ) @ D2 )
=> ( ! [Y4: real] :
( ( ord_less @ real @ ( abs_abs @ real @ ( minus_minus @ real @ X2 @ Y4 ) ) @ D2 )
=> ( ord_less_eq @ real @ ( F2 @ Y4 ) @ ( F2 @ X2 ) ) )
=> ( L
= ( zero_zero @ real ) ) ) ) ) ).
% DERIV_local_max
thf(fact_6928_DERIV__ln__divide,axiom,
! [X2: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ X2 )
=> ( has_field_derivative @ real @ ( ln_ln @ real ) @ ( divide_divide @ real @ ( one_one @ real ) @ X2 ) @ ( topolo174197925503356063within @ real @ X2 @ ( top_top @ ( set @ real ) ) ) ) ) ).
% DERIV_ln_divide
thf(fact_6929_DERIV__pow,axiom,
! [N: nat,X2: real,S2: set @ real] :
( has_field_derivative @ real
@ ^ [X3: real] : ( power_power @ real @ X3 @ N )
@ ( times_times @ real @ ( semiring_1_of_nat @ real @ N ) @ ( power_power @ real @ X2 @ ( minus_minus @ nat @ N @ ( suc @ ( zero_zero @ nat ) ) ) ) )
@ ( topolo174197925503356063within @ real @ X2 @ S2 ) ) ).
% DERIV_pow
thf(fact_6930_termdiffs__strong__converges__everywhere,axiom,
! [A: $tType] :
( ( ( real_Vector_banach @ A )
& ( real_V3459762299906320749_field @ A ) )
=> ! [C3: nat > A,X2: A] :
( ! [Y4: A] :
( summable @ A
@ ^ [N4: nat] : ( times_times @ A @ ( C3 @ N4 ) @ ( power_power @ A @ Y4 @ N4 ) ) )
=> ( has_field_derivative @ A
@ ^ [X3: A] :
( suminf @ A
@ ^ [N4: nat] : ( times_times @ A @ ( C3 @ N4 ) @ ( power_power @ A @ X3 @ N4 ) ) )
@ ( suminf @ A
@ ^ [N4: nat] : ( times_times @ A @ ( diffs @ A @ C3 @ N4 ) @ ( power_power @ A @ X2 @ N4 ) ) )
@ ( topolo174197925503356063within @ A @ X2 @ ( top_top @ ( set @ A ) ) ) ) ) ) ).
% termdiffs_strong_converges_everywhere
thf(fact_6931_at__within__Icc__at,axiom,
! [A: $tType] :
( ( topolo2564578578187576103pology @ A )
=> ! [A3: A,X2: A,B3: A] :
( ( ord_less @ A @ A3 @ X2 )
=> ( ( ord_less @ A @ X2 @ B3 )
=> ( ( topolo174197925503356063within @ A @ X2 @ ( set_or1337092689740270186AtMost @ A @ A3 @ B3 ) )
= ( topolo174197925503356063within @ A @ X2 @ ( top_top @ ( set @ A ) ) ) ) ) ) ) ).
% at_within_Icc_at
thf(fact_6932_DERIV__fun__pow,axiom,
! [G: real > real,M: real,X2: real,N: nat] :
( ( has_field_derivative @ real @ G @ M @ ( topolo174197925503356063within @ real @ X2 @ ( top_top @ ( set @ real ) ) ) )
=> ( has_field_derivative @ real
@ ^ [X3: real] : ( power_power @ real @ ( G @ X3 ) @ N )
@ ( times_times @ real @ ( times_times @ real @ ( semiring_1_of_nat @ real @ N ) @ ( power_power @ real @ ( G @ X2 ) @ ( minus_minus @ nat @ N @ ( one_one @ nat ) ) ) ) @ M )
@ ( topolo174197925503356063within @ real @ X2 @ ( top_top @ ( set @ real ) ) ) ) ) ).
% DERIV_fun_pow
thf(fact_6933_at__within__Icc__at__left,axiom,
! [A: $tType] :
( ( topolo2564578578187576103pology @ A )
=> ! [A3: A,B3: A] :
( ( ord_less @ A @ A3 @ B3 )
=> ( ( topolo174197925503356063within @ A @ B3 @ ( set_or1337092689740270186AtMost @ A @ A3 @ B3 ) )
= ( topolo174197925503356063within @ A @ B3 @ ( set_ord_lessThan @ A @ B3 ) ) ) ) ) ).
% at_within_Icc_at_left
thf(fact_6934_DERIV__quotient,axiom,
! [A: $tType] :
( ( real_V3459762299906320749_field @ A )
=> ! [F2: A > A,D2: A,X2: A,S2: set @ A,G: A > A,E: A] :
( ( has_field_derivative @ A @ F2 @ D2 @ ( topolo174197925503356063within @ A @ X2 @ S2 ) )
=> ( ( has_field_derivative @ A @ G @ E @ ( topolo174197925503356063within @ A @ X2 @ S2 ) )
=> ( ( ( G @ X2 )
!= ( zero_zero @ A ) )
=> ( has_field_derivative @ A
@ ^ [Y: A] : ( divide_divide @ A @ ( F2 @ Y ) @ ( G @ Y ) )
@ ( divide_divide @ A @ ( minus_minus @ A @ ( times_times @ A @ D2 @ ( G @ X2 ) ) @ ( times_times @ A @ E @ ( F2 @ X2 ) ) ) @ ( power_power @ A @ ( G @ X2 ) @ ( suc @ ( suc @ ( zero_zero @ nat ) ) ) ) )
@ ( topolo174197925503356063within @ A @ X2 @ S2 ) ) ) ) ) ) ).
% DERIV_quotient
thf(fact_6935_DERIV__inverse__fun,axiom,
! [A: $tType] :
( ( real_V3459762299906320749_field @ A )
=> ! [F2: A > A,D2: A,X2: A,S2: set @ A] :
( ( has_field_derivative @ A @ F2 @ D2 @ ( topolo174197925503356063within @ A @ X2 @ S2 ) )
=> ( ( ( F2 @ X2 )
!= ( zero_zero @ A ) )
=> ( has_field_derivative @ A
@ ^ [X3: A] : ( inverse_inverse @ A @ ( F2 @ X3 ) )
@ ( uminus_uminus @ A @ ( times_times @ A @ D2 @ ( inverse_inverse @ A @ ( power_power @ A @ ( F2 @ X2 ) @ ( suc @ ( suc @ ( zero_zero @ nat ) ) ) ) ) ) )
@ ( topolo174197925503356063within @ A @ X2 @ S2 ) ) ) ) ) ).
% DERIV_inverse_fun
thf(fact_6936_termdiffs__sums__strong,axiom,
! [A: $tType] :
( ( ( real_Vector_banach @ A )
& ( real_V3459762299906320749_field @ A ) )
=> ! [K6: real,C3: nat > A,F2: A > A,F7: A,Z: A] :
( ! [Z2: A] :
( ( ord_less @ real @ ( real_V7770717601297561774m_norm @ A @ Z2 ) @ K6 )
=> ( sums @ A
@ ^ [N4: nat] : ( times_times @ A @ ( C3 @ N4 ) @ ( power_power @ A @ Z2 @ N4 ) )
@ ( F2 @ Z2 ) ) )
=> ( ( has_field_derivative @ A @ F2 @ F7 @ ( topolo174197925503356063within @ A @ Z @ ( top_top @ ( set @ A ) ) ) )
=> ( ( ord_less @ real @ ( real_V7770717601297561774m_norm @ A @ Z ) @ K6 )
=> ( sums @ A
@ ^ [N4: nat] : ( times_times @ A @ ( diffs @ A @ C3 @ N4 ) @ ( power_power @ A @ Z @ N4 ) )
@ F7 ) ) ) ) ) ).
% termdiffs_sums_strong
thf(fact_6937_has__real__derivative__powr,axiom,
! [Z: real,R2: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ Z )
=> ( has_field_derivative @ real
@ ^ [Z4: real] : ( powr @ real @ Z4 @ 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_6938_termdiffs,axiom,
! [A: $tType] :
( ( ( real_Vector_banach @ A )
& ( real_V3459762299906320749_field @ A ) )
=> ! [C3: nat > A,K6: A,X2: A] :
( ( summable @ A
@ ^ [N4: nat] : ( times_times @ A @ ( C3 @ N4 ) @ ( power_power @ A @ K6 @ N4 ) ) )
=> ( ( summable @ A
@ ^ [N4: nat] : ( times_times @ A @ ( diffs @ A @ C3 @ N4 ) @ ( power_power @ A @ K6 @ N4 ) ) )
=> ( ( summable @ A
@ ^ [N4: nat] : ( times_times @ A @ ( diffs @ A @ ( diffs @ A @ C3 ) @ N4 ) @ ( power_power @ A @ K6 @ N4 ) ) )
=> ( ( ord_less @ real @ ( real_V7770717601297561774m_norm @ A @ X2 ) @ ( real_V7770717601297561774m_norm @ A @ K6 ) )
=> ( has_field_derivative @ A
@ ^ [X3: A] :
( suminf @ A
@ ^ [N4: nat] : ( times_times @ A @ ( C3 @ N4 ) @ ( power_power @ A @ X3 @ N4 ) ) )
@ ( suminf @ A
@ ^ [N4: nat] : ( times_times @ A @ ( diffs @ A @ C3 @ N4 ) @ ( power_power @ A @ X2 @ N4 ) ) )
@ ( topolo174197925503356063within @ A @ X2 @ ( top_top @ ( set @ A ) ) ) ) ) ) ) ) ) ).
% termdiffs
thf(fact_6939_termdiffs__strong,axiom,
! [A: $tType] :
( ( ( real_Vector_banach @ A )
& ( real_V3459762299906320749_field @ A ) )
=> ! [C3: nat > A,K6: A,X2: A] :
( ( summable @ A
@ ^ [N4: nat] : ( times_times @ A @ ( C3 @ N4 ) @ ( power_power @ A @ K6 @ N4 ) ) )
=> ( ( ord_less @ real @ ( real_V7770717601297561774m_norm @ A @ X2 ) @ ( real_V7770717601297561774m_norm @ A @ K6 ) )
=> ( has_field_derivative @ A
@ ^ [X3: A] :
( suminf @ A
@ ^ [N4: nat] : ( times_times @ A @ ( C3 @ N4 ) @ ( power_power @ A @ X3 @ N4 ) ) )
@ ( suminf @ A
@ ^ [N4: nat] : ( times_times @ A @ ( diffs @ A @ C3 @ N4 ) @ ( power_power @ A @ X2 @ N4 ) ) )
@ ( topolo174197925503356063within @ A @ X2 @ ( top_top @ ( set @ A ) ) ) ) ) ) ) ).
% termdiffs_strong
thf(fact_6940_termdiffs__strong_H,axiom,
! [A: $tType] :
( ( ( real_Vector_banach @ A )
& ( real_V3459762299906320749_field @ A ) )
=> ! [K6: real,C3: nat > A,Z: A] :
( ! [Z2: A] :
( ( ord_less @ real @ ( real_V7770717601297561774m_norm @ A @ Z2 ) @ K6 )
=> ( summable @ A
@ ^ [N4: nat] : ( times_times @ A @ ( C3 @ N4 ) @ ( power_power @ A @ Z2 @ N4 ) ) ) )
=> ( ( ord_less @ real @ ( real_V7770717601297561774m_norm @ A @ Z ) @ K6 )
=> ( has_field_derivative @ A
@ ^ [Z4: A] :
( suminf @ A
@ ^ [N4: nat] : ( times_times @ A @ ( C3 @ N4 ) @ ( power_power @ A @ Z4 @ N4 ) ) )
@ ( suminf @ A
@ ^ [N4: nat] : ( times_times @ A @ ( diffs @ A @ C3 @ N4 ) @ ( power_power @ A @ Z @ N4 ) ) )
@ ( topolo174197925503356063within @ A @ Z @ ( top_top @ ( set @ A ) ) ) ) ) ) ) ).
% termdiffs_strong'
thf(fact_6941_DERIV__fun__powr,axiom,
! [G: real > real,M: real,X2: real,R2: real] :
( ( has_field_derivative @ real @ G @ M @ ( topolo174197925503356063within @ real @ X2 @ ( top_top @ ( set @ real ) ) ) )
=> ( ( ord_less @ real @ ( zero_zero @ real ) @ ( G @ X2 ) )
=> ( has_field_derivative @ real
@ ^ [X3: real] : ( powr @ real @ ( G @ X3 ) @ R2 )
@ ( times_times @ real @ ( times_times @ real @ R2 @ ( powr @ real @ ( G @ X2 ) @ ( minus_minus @ real @ R2 @ ( semiring_1_of_nat @ real @ ( one_one @ nat ) ) ) ) ) @ M )
@ ( topolo174197925503356063within @ real @ X2 @ ( top_top @ ( set @ real ) ) ) ) ) ) ).
% DERIV_fun_powr
thf(fact_6942_DERIV__log,axiom,
! [X2: real,B3: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ X2 )
=> ( has_field_derivative @ real @ ( log @ B3 ) @ ( divide_divide @ real @ ( one_one @ real ) @ ( times_times @ real @ ( ln_ln @ real @ B3 ) @ X2 ) ) @ ( topolo174197925503356063within @ real @ X2 @ ( top_top @ ( set @ real ) ) ) ) ) ).
% DERIV_log
thf(fact_6943_DERIV__powr,axiom,
! [G: real > real,M: real,X2: real,F2: real > real,R2: real] :
( ( has_field_derivative @ real @ G @ M @ ( topolo174197925503356063within @ real @ X2 @ ( top_top @ ( set @ real ) ) ) )
=> ( ( ord_less @ real @ ( zero_zero @ real ) @ ( G @ X2 ) )
=> ( ( has_field_derivative @ real @ F2 @ R2 @ ( topolo174197925503356063within @ real @ X2 @ ( top_top @ ( set @ real ) ) ) )
=> ( has_field_derivative @ real
@ ^ [X3: real] : ( powr @ real @ ( G @ X3 ) @ ( F2 @ X3 ) )
@ ( times_times @ real @ ( powr @ real @ ( G @ X2 ) @ ( F2 @ X2 ) ) @ ( plus_plus @ real @ ( times_times @ real @ R2 @ ( ln_ln @ real @ ( G @ X2 ) ) ) @ ( divide_divide @ real @ ( times_times @ real @ M @ ( F2 @ X2 ) ) @ ( G @ X2 ) ) ) )
@ ( topolo174197925503356063within @ real @ X2 @ ( top_top @ ( set @ real ) ) ) ) ) ) ) ).
% DERIV_powr
thf(fact_6944_DERIV__tan,axiom,
! [A: $tType] :
( ( ( real_Vector_banach @ A )
& ( real_V3459762299906320749_field @ A ) )
=> ! [X2: A] :
( ( ( cos @ A @ X2 )
!= ( zero_zero @ A ) )
=> ( has_field_derivative @ A @ ( tan @ A ) @ ( inverse_inverse @ A @ ( power_power @ A @ ( cos @ A @ X2 ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( topolo174197925503356063within @ A @ X2 @ ( top_top @ ( set @ A ) ) ) ) ) ) ).
% DERIV_tan
thf(fact_6945_DERIV__real__sqrt,axiom,
! [X2: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ X2 )
=> ( has_field_derivative @ real @ sqrt @ ( divide_divide @ real @ ( inverse_inverse @ real @ ( sqrt @ X2 ) ) @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) @ ( topolo174197925503356063within @ real @ X2 @ ( top_top @ ( set @ real ) ) ) ) ) ).
% DERIV_real_sqrt
thf(fact_6946_DERIV__arctan,axiom,
! [X2: real] : ( has_field_derivative @ real @ arctan @ ( inverse_inverse @ real @ ( plus_plus @ real @ ( one_one @ real ) @ ( power_power @ real @ X2 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( topolo174197925503356063within @ real @ X2 @ ( top_top @ ( set @ real ) ) ) ) ).
% DERIV_arctan
thf(fact_6947_arsinh__real__has__field__derivative,axiom,
! [X2: real,A2: set @ real] : ( has_field_derivative @ real @ ( arsinh @ real ) @ ( divide_divide @ real @ ( one_one @ real ) @ ( sqrt @ ( plus_plus @ real @ ( power_power @ real @ X2 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ ( one_one @ real ) ) ) ) @ ( topolo174197925503356063within @ real @ X2 @ A2 ) ) ).
% arsinh_real_has_field_derivative
thf(fact_6948_DERIV__cot,axiom,
! [A: $tType] :
( ( ( real_Vector_banach @ A )
& ( real_V3459762299906320749_field @ A ) )
=> ! [X2: A] :
( ( ( sin @ A @ X2 )
!= ( zero_zero @ A ) )
=> ( has_field_derivative @ A @ ( cot @ A ) @ ( uminus_uminus @ A @ ( inverse_inverse @ A @ ( power_power @ A @ ( sin @ A @ X2 ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( topolo174197925503356063within @ A @ X2 @ ( top_top @ ( set @ A ) ) ) ) ) ) ).
% DERIV_cot
thf(fact_6949_has__field__derivative__tanh,axiom,
! [A13: $tType] :
( ( ( real_Vector_banach @ A13 )
& ( real_V3459762299906320749_field @ A13 ) )
=> ! [G: A13 > A13,X2: A13,Db: A13,S2: set @ A13] :
( ( ( cosh @ A13 @ ( G @ X2 ) )
!= ( zero_zero @ A13 ) )
=> ( ( has_field_derivative @ A13 @ G @ Db @ ( topolo174197925503356063within @ A13 @ X2 @ S2 ) )
=> ( has_field_derivative @ A13
@ ^ [X3: A13] : ( tanh @ A13 @ ( G @ X3 ) )
@ ( times_times @ A13 @ ( minus_minus @ A13 @ ( one_one @ A13 ) @ ( power_power @ A13 @ ( tanh @ A13 @ ( G @ X2 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ Db )
@ ( topolo174197925503356063within @ A13 @ X2 @ S2 ) ) ) ) ) ).
% has_field_derivative_tanh
thf(fact_6950_DERIV__real__sqrt__generic,axiom,
! [X2: real,D: real] :
( ( X2
!= ( zero_zero @ real ) )
=> ( ( ( ord_less @ real @ ( zero_zero @ real ) @ X2 )
=> ( D
= ( divide_divide @ real @ ( inverse_inverse @ real @ ( sqrt @ X2 ) ) @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) ) )
=> ( ( ( ord_less @ real @ X2 @ ( zero_zero @ real ) )
=> ( D
= ( divide_divide @ real @ ( uminus_uminus @ real @ ( inverse_inverse @ real @ ( sqrt @ X2 ) ) ) @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) ) )
=> ( has_field_derivative @ real @ sqrt @ D @ ( topolo174197925503356063within @ real @ X2 @ ( top_top @ ( set @ real ) ) ) ) ) ) ) ).
% DERIV_real_sqrt_generic
thf(fact_6951_arcosh__real__has__field__derivative,axiom,
! [X2: real,A2: set @ real] :
( ( ord_less @ real @ ( one_one @ real ) @ X2 )
=> ( has_field_derivative @ real @ ( arcosh @ real ) @ ( divide_divide @ real @ ( one_one @ real ) @ ( sqrt @ ( minus_minus @ real @ ( power_power @ real @ X2 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ ( one_one @ real ) ) ) ) @ ( topolo174197925503356063within @ real @ X2 @ A2 ) ) ) ).
% arcosh_real_has_field_derivative
thf(fact_6952_artanh__real__has__field__derivative,axiom,
! [X2: real,A2: set @ real] :
( ( ord_less @ real @ ( abs_abs @ real @ X2 ) @ ( one_one @ real ) )
=> ( has_field_derivative @ real @ ( artanh @ real ) @ ( divide_divide @ real @ ( one_one @ real ) @ ( minus_minus @ real @ ( one_one @ real ) @ ( power_power @ real @ X2 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( topolo174197925503356063within @ real @ X2 @ A2 ) ) ) ).
% artanh_real_has_field_derivative
thf(fact_6953_DERIV__real__root,axiom,
! [N: nat,X2: real] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ N )
=> ( ( ord_less @ real @ ( zero_zero @ real ) @ X2 )
=> ( has_field_derivative @ real @ ( root @ N ) @ ( inverse_inverse @ real @ ( times_times @ real @ ( semiring_1_of_nat @ real @ N ) @ ( power_power @ real @ ( root @ N @ X2 ) @ ( minus_minus @ nat @ N @ ( suc @ ( zero_zero @ nat ) ) ) ) ) ) @ ( topolo174197925503356063within @ real @ X2 @ ( top_top @ ( set @ real ) ) ) ) ) ) ).
% DERIV_real_root
thf(fact_6954_DERIV__arccos,axiom,
! [X2: real] :
( ( ord_less @ real @ ( uminus_uminus @ real @ ( one_one @ real ) ) @ X2 )
=> ( ( ord_less @ real @ X2 @ ( one_one @ real ) )
=> ( has_field_derivative @ real @ arccos @ ( inverse_inverse @ real @ ( uminus_uminus @ real @ ( sqrt @ ( minus_minus @ real @ ( one_one @ real ) @ ( power_power @ real @ X2 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) ) @ ( topolo174197925503356063within @ real @ X2 @ ( top_top @ ( set @ real ) ) ) ) ) ) ).
% DERIV_arccos
thf(fact_6955_DERIV__arcsin,axiom,
! [X2: real] :
( ( ord_less @ real @ ( uminus_uminus @ real @ ( one_one @ real ) ) @ X2 )
=> ( ( ord_less @ real @ X2 @ ( one_one @ real ) )
=> ( has_field_derivative @ real @ arcsin @ ( inverse_inverse @ real @ ( sqrt @ ( minus_minus @ real @ ( one_one @ real ) @ ( power_power @ real @ X2 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) @ ( topolo174197925503356063within @ real @ X2 @ ( top_top @ ( set @ real ) ) ) ) ) ) ).
% DERIV_arcsin
thf(fact_6956_Maclaurin__all__le,axiom,
! [Diff: nat > real > real,F2: real > real,X2: real,N: nat] :
( ( ( Diff @ ( zero_zero @ nat ) )
= F2 )
=> ( ! [M5: nat,X4: real] : ( has_field_derivative @ real @ ( Diff @ M5 ) @ ( Diff @ ( suc @ M5 ) @ X4 ) @ ( topolo174197925503356063within @ real @ X4 @ ( top_top @ ( set @ real ) ) ) )
=> ? [T8: real] :
( ( ord_less_eq @ real @ ( abs_abs @ real @ T8 ) @ ( abs_abs @ real @ X2 ) )
& ( ( F2 @ X2 )
= ( plus_plus @ real
@ ( groups7311177749621191930dd_sum @ nat @ real
@ ^ [M6: nat] : ( times_times @ real @ ( divide_divide @ real @ ( Diff @ M6 @ ( zero_zero @ real ) ) @ ( semiring_char_0_fact @ real @ M6 ) ) @ ( power_power @ real @ X2 @ M6 ) )
@ ( set_ord_lessThan @ nat @ N ) )
@ ( times_times @ real @ ( divide_divide @ real @ ( Diff @ N @ T8 ) @ ( semiring_char_0_fact @ real @ N ) ) @ ( power_power @ real @ X2 @ N ) ) ) ) ) ) ) ).
% Maclaurin_all_le
thf(fact_6957_Maclaurin__all__le__objl,axiom,
! [Diff: nat > real > real,F2: real > real,X2: real,N: nat] :
( ( ( ( Diff @ ( zero_zero @ nat ) )
= F2 )
& ! [M5: nat,X4: real] : ( has_field_derivative @ real @ ( Diff @ M5 ) @ ( Diff @ ( suc @ M5 ) @ X4 ) @ ( topolo174197925503356063within @ real @ X4 @ ( top_top @ ( set @ real ) ) ) ) )
=> ? [T8: real] :
( ( ord_less_eq @ real @ ( abs_abs @ real @ T8 ) @ ( abs_abs @ real @ X2 ) )
& ( ( F2 @ X2 )
= ( plus_plus @ real
@ ( groups7311177749621191930dd_sum @ nat @ real
@ ^ [M6: nat] : ( times_times @ real @ ( divide_divide @ real @ ( Diff @ M6 @ ( zero_zero @ real ) ) @ ( semiring_char_0_fact @ real @ M6 ) ) @ ( power_power @ real @ X2 @ M6 ) )
@ ( set_ord_lessThan @ nat @ N ) )
@ ( times_times @ real @ ( divide_divide @ real @ ( Diff @ N @ T8 ) @ ( semiring_char_0_fact @ real @ N ) ) @ ( power_power @ real @ X2 @ N ) ) ) ) ) ) ).
% Maclaurin_all_le_objl
thf(fact_6958_DERIV__odd__real__root,axiom,
! [N: nat,X2: real] :
( ~ ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N )
=> ( ( X2
!= ( 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 @ X2 ) @ ( minus_minus @ nat @ N @ ( suc @ ( zero_zero @ nat ) ) ) ) ) ) @ ( topolo174197925503356063within @ real @ X2 @ ( top_top @ ( set @ real ) ) ) ) ) ) ).
% DERIV_odd_real_root
thf(fact_6959_Maclaurin,axiom,
! [H2: real,N: nat,Diff: nat > real > real,F2: real > real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ H2 )
=> ( ( ord_less @ nat @ ( zero_zero @ nat ) @ N )
=> ( ( ( Diff @ ( zero_zero @ nat ) )
= F2 )
=> ( ! [M5: nat,T8: real] :
( ( ( ord_less @ nat @ M5 @ N )
& ( ord_less_eq @ real @ ( zero_zero @ real ) @ T8 )
& ( ord_less_eq @ real @ T8 @ H2 ) )
=> ( has_field_derivative @ real @ ( Diff @ M5 ) @ ( Diff @ ( suc @ M5 ) @ T8 ) @ ( topolo174197925503356063within @ real @ T8 @ ( top_top @ ( set @ real ) ) ) ) )
=> ? [T8: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ T8 )
& ( ord_less @ real @ T8 @ H2 )
& ( ( F2 @ H2 )
= ( plus_plus @ real
@ ( groups7311177749621191930dd_sum @ nat @ real
@ ^ [M6: nat] : ( times_times @ real @ ( divide_divide @ real @ ( Diff @ M6 @ ( zero_zero @ real ) ) @ ( semiring_char_0_fact @ real @ M6 ) ) @ ( power_power @ real @ H2 @ M6 ) )
@ ( set_ord_lessThan @ nat @ N ) )
@ ( times_times @ real @ ( divide_divide @ real @ ( Diff @ N @ T8 ) @ ( semiring_char_0_fact @ real @ N ) ) @ ( power_power @ real @ H2 @ N ) ) ) ) ) ) ) ) ) ).
% Maclaurin
thf(fact_6960_Maclaurin2,axiom,
! [H2: real,Diff: nat > real > real,F2: real > real,N: nat] :
( ( ord_less @ real @ ( zero_zero @ real ) @ H2 )
=> ( ( ( Diff @ ( zero_zero @ nat ) )
= F2 )
=> ( ! [M5: nat,T8: real] :
( ( ( ord_less @ nat @ M5 @ N )
& ( ord_less_eq @ real @ ( zero_zero @ real ) @ T8 )
& ( ord_less_eq @ real @ T8 @ H2 ) )
=> ( has_field_derivative @ real @ ( Diff @ M5 ) @ ( Diff @ ( suc @ M5 ) @ T8 ) @ ( topolo174197925503356063within @ real @ T8 @ ( top_top @ ( set @ real ) ) ) ) )
=> ? [T8: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ T8 )
& ( ord_less_eq @ real @ T8 @ H2 )
& ( ( F2 @ H2 )
= ( plus_plus @ real
@ ( groups7311177749621191930dd_sum @ nat @ real
@ ^ [M6: nat] : ( times_times @ real @ ( divide_divide @ real @ ( Diff @ M6 @ ( zero_zero @ real ) ) @ ( semiring_char_0_fact @ real @ M6 ) ) @ ( power_power @ real @ H2 @ M6 ) )
@ ( set_ord_lessThan @ nat @ N ) )
@ ( times_times @ real @ ( divide_divide @ real @ ( Diff @ N @ T8 ) @ ( semiring_char_0_fact @ real @ N ) ) @ ( power_power @ real @ H2 @ N ) ) ) ) ) ) ) ) ).
% Maclaurin2
thf(fact_6961_Maclaurin__minus,axiom,
! [H2: real,N: nat,Diff: nat > real > real,F2: real > real] :
( ( ord_less @ real @ H2 @ ( zero_zero @ real ) )
=> ( ( ord_less @ nat @ ( zero_zero @ nat ) @ N )
=> ( ( ( Diff @ ( zero_zero @ nat ) )
= F2 )
=> ( ! [M5: nat,T8: real] :
( ( ( ord_less @ nat @ M5 @ N )
& ( ord_less_eq @ real @ H2 @ T8 )
& ( ord_less_eq @ real @ T8 @ ( zero_zero @ real ) ) )
=> ( has_field_derivative @ real @ ( Diff @ M5 ) @ ( Diff @ ( suc @ M5 ) @ T8 ) @ ( topolo174197925503356063within @ real @ T8 @ ( top_top @ ( set @ real ) ) ) ) )
=> ? [T8: real] :
( ( ord_less @ real @ H2 @ T8 )
& ( ord_less @ real @ T8 @ ( zero_zero @ real ) )
& ( ( F2 @ H2 )
= ( plus_plus @ real
@ ( groups7311177749621191930dd_sum @ nat @ real
@ ^ [M6: nat] : ( times_times @ real @ ( divide_divide @ real @ ( Diff @ M6 @ ( zero_zero @ real ) ) @ ( semiring_char_0_fact @ real @ M6 ) ) @ ( power_power @ real @ H2 @ M6 ) )
@ ( set_ord_lessThan @ nat @ N ) )
@ ( times_times @ real @ ( divide_divide @ real @ ( Diff @ N @ T8 ) @ ( semiring_char_0_fact @ real @ N ) ) @ ( power_power @ real @ H2 @ N ) ) ) ) ) ) ) ) ) ).
% Maclaurin_minus
thf(fact_6962_Maclaurin__all__lt,axiom,
! [Diff: nat > real > real,F2: real > real,N: nat,X2: real] :
( ( ( Diff @ ( zero_zero @ nat ) )
= F2 )
=> ( ( ord_less @ nat @ ( zero_zero @ nat ) @ N )
=> ( ( X2
!= ( zero_zero @ real ) )
=> ( ! [M5: nat,X4: real] : ( has_field_derivative @ real @ ( Diff @ M5 ) @ ( Diff @ ( suc @ M5 ) @ X4 ) @ ( topolo174197925503356063within @ real @ X4 @ ( top_top @ ( set @ real ) ) ) )
=> ? [T8: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ ( abs_abs @ real @ T8 ) )
& ( ord_less @ real @ ( abs_abs @ real @ T8 ) @ ( abs_abs @ real @ X2 ) )
& ( ( F2 @ X2 )
= ( plus_plus @ real
@ ( groups7311177749621191930dd_sum @ nat @ real
@ ^ [M6: nat] : ( times_times @ real @ ( divide_divide @ real @ ( Diff @ M6 @ ( zero_zero @ real ) ) @ ( semiring_char_0_fact @ real @ M6 ) ) @ ( power_power @ real @ X2 @ M6 ) )
@ ( set_ord_lessThan @ nat @ N ) )
@ ( times_times @ real @ ( divide_divide @ real @ ( Diff @ N @ T8 ) @ ( semiring_char_0_fact @ real @ N ) ) @ ( power_power @ real @ X2 @ N ) ) ) ) ) ) ) ) ) ).
% Maclaurin_all_lt
thf(fact_6963_Maclaurin__bi__le,axiom,
! [Diff: nat > real > real,F2: real > real,N: nat,X2: real] :
( ( ( Diff @ ( zero_zero @ nat ) )
= F2 )
=> ( ! [M5: nat,T8: real] :
( ( ( ord_less @ nat @ M5 @ N )
& ( ord_less_eq @ real @ ( abs_abs @ real @ T8 ) @ ( abs_abs @ real @ X2 ) ) )
=> ( has_field_derivative @ real @ ( Diff @ M5 ) @ ( Diff @ ( suc @ M5 ) @ T8 ) @ ( topolo174197925503356063within @ real @ T8 @ ( top_top @ ( set @ real ) ) ) ) )
=> ? [T8: real] :
( ( ord_less_eq @ real @ ( abs_abs @ real @ T8 ) @ ( abs_abs @ real @ X2 ) )
& ( ( F2 @ X2 )
= ( plus_plus @ real
@ ( groups7311177749621191930dd_sum @ nat @ real
@ ^ [M6: nat] : ( times_times @ real @ ( divide_divide @ real @ ( Diff @ M6 @ ( zero_zero @ real ) ) @ ( semiring_char_0_fact @ real @ M6 ) ) @ ( power_power @ real @ X2 @ M6 ) )
@ ( set_ord_lessThan @ nat @ N ) )
@ ( times_times @ real @ ( divide_divide @ real @ ( Diff @ N @ T8 ) @ ( semiring_char_0_fact @ real @ N ) ) @ ( power_power @ real @ X2 @ N ) ) ) ) ) ) ) ).
% Maclaurin_bi_le
thf(fact_6964_Taylor__down,axiom,
! [N: nat,Diff: nat > real > real,F2: real > real,A3: real,B3: real,C3: real] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ N )
=> ( ( ( Diff @ ( zero_zero @ nat ) )
= F2 )
=> ( ! [M5: nat,T8: real] :
( ( ( ord_less @ nat @ M5 @ N )
& ( ord_less_eq @ real @ A3 @ T8 )
& ( ord_less_eq @ real @ T8 @ B3 ) )
=> ( has_field_derivative @ real @ ( Diff @ M5 ) @ ( Diff @ ( suc @ M5 ) @ T8 ) @ ( topolo174197925503356063within @ real @ T8 @ ( top_top @ ( set @ real ) ) ) ) )
=> ( ( ord_less @ real @ A3 @ C3 )
=> ( ( ord_less_eq @ real @ C3 @ B3 )
=> ? [T8: real] :
( ( ord_less @ real @ A3 @ T8 )
& ( ord_less @ real @ T8 @ C3 )
& ( ( F2 @ A3 )
= ( plus_plus @ real
@ ( groups7311177749621191930dd_sum @ nat @ real
@ ^ [M6: nat] : ( times_times @ real @ ( divide_divide @ real @ ( Diff @ M6 @ C3 ) @ ( semiring_char_0_fact @ real @ M6 ) ) @ ( power_power @ real @ ( minus_minus @ real @ A3 @ C3 ) @ M6 ) )
@ ( set_ord_lessThan @ nat @ N ) )
@ ( times_times @ real @ ( divide_divide @ real @ ( Diff @ N @ T8 ) @ ( semiring_char_0_fact @ real @ N ) ) @ ( power_power @ real @ ( minus_minus @ real @ A3 @ C3 ) @ N ) ) ) ) ) ) ) ) ) ) ).
% Taylor_down
thf(fact_6965_Taylor__up,axiom,
! [N: nat,Diff: nat > real > real,F2: real > real,A3: real,B3: real,C3: real] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ N )
=> ( ( ( Diff @ ( zero_zero @ nat ) )
= F2 )
=> ( ! [M5: nat,T8: real] :
( ( ( ord_less @ nat @ M5 @ N )
& ( ord_less_eq @ real @ A3 @ T8 )
& ( ord_less_eq @ real @ T8 @ B3 ) )
=> ( has_field_derivative @ real @ ( Diff @ M5 ) @ ( Diff @ ( suc @ M5 ) @ T8 ) @ ( topolo174197925503356063within @ real @ T8 @ ( top_top @ ( set @ real ) ) ) ) )
=> ( ( ord_less_eq @ real @ A3 @ C3 )
=> ( ( ord_less @ real @ C3 @ B3 )
=> ? [T8: real] :
( ( ord_less @ real @ C3 @ T8 )
& ( ord_less @ real @ T8 @ B3 )
& ( ( F2 @ B3 )
= ( plus_plus @ real
@ ( groups7311177749621191930dd_sum @ nat @ real
@ ^ [M6: nat] : ( times_times @ real @ ( divide_divide @ real @ ( Diff @ M6 @ C3 ) @ ( semiring_char_0_fact @ real @ M6 ) ) @ ( power_power @ real @ ( minus_minus @ real @ B3 @ C3 ) @ M6 ) )
@ ( set_ord_lessThan @ nat @ N ) )
@ ( times_times @ real @ ( divide_divide @ real @ ( Diff @ N @ T8 ) @ ( semiring_char_0_fact @ real @ N ) ) @ ( power_power @ real @ ( minus_minus @ real @ B3 @ C3 ) @ N ) ) ) ) ) ) ) ) ) ) ).
% Taylor_up
thf(fact_6966_Taylor,axiom,
! [N: nat,Diff: nat > real > real,F2: real > real,A3: real,B3: real,C3: real,X2: real] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ N )
=> ( ( ( Diff @ ( zero_zero @ nat ) )
= F2 )
=> ( ! [M5: nat,T8: real] :
( ( ( ord_less @ nat @ M5 @ N )
& ( ord_less_eq @ real @ A3 @ T8 )
& ( ord_less_eq @ real @ T8 @ B3 ) )
=> ( has_field_derivative @ real @ ( Diff @ M5 ) @ ( Diff @ ( suc @ M5 ) @ T8 ) @ ( topolo174197925503356063within @ real @ T8 @ ( top_top @ ( set @ real ) ) ) ) )
=> ( ( ord_less_eq @ real @ A3 @ C3 )
=> ( ( ord_less_eq @ real @ C3 @ B3 )
=> ( ( ord_less_eq @ real @ A3 @ X2 )
=> ( ( ord_less_eq @ real @ X2 @ B3 )
=> ( ( X2 != C3 )
=> ? [T8: real] :
( ( ( ord_less @ real @ X2 @ C3 )
=> ( ( ord_less @ real @ X2 @ T8 )
& ( ord_less @ real @ T8 @ C3 ) ) )
& ( ~ ( ord_less @ real @ X2 @ C3 )
=> ( ( ord_less @ real @ C3 @ T8 )
& ( ord_less @ real @ T8 @ X2 ) ) )
& ( ( F2 @ X2 )
= ( plus_plus @ real
@ ( groups7311177749621191930dd_sum @ nat @ real
@ ^ [M6: nat] : ( times_times @ real @ ( divide_divide @ real @ ( Diff @ M6 @ C3 ) @ ( semiring_char_0_fact @ real @ M6 ) ) @ ( power_power @ real @ ( minus_minus @ real @ X2 @ C3 ) @ M6 ) )
@ ( set_ord_lessThan @ nat @ N ) )
@ ( times_times @ real @ ( divide_divide @ real @ ( Diff @ N @ T8 ) @ ( semiring_char_0_fact @ real @ N ) ) @ ( power_power @ real @ ( minus_minus @ real @ X2 @ C3 ) @ N ) ) ) ) ) ) ) ) ) ) ) ) ) ).
% Taylor
thf(fact_6967_Maclaurin__lemma2,axiom,
! [N: nat,H2: real,Diff: nat > real > real,K: nat,B2: real] :
( ! [M5: nat,T8: real] :
( ( ( ord_less @ nat @ M5 @ N )
& ( ord_less_eq @ real @ ( zero_zero @ real ) @ T8 )
& ( ord_less_eq @ real @ T8 @ H2 ) )
=> ( has_field_derivative @ real @ ( Diff @ M5 ) @ ( Diff @ ( suc @ M5 ) @ T8 ) @ ( topolo174197925503356063within @ real @ T8 @ ( top_top @ ( set @ real ) ) ) ) )
=> ( ( N
= ( suc @ K ) )
=> ! [M3: nat,T10: real] :
( ( ( ord_less @ nat @ M3 @ N )
& ( ord_less_eq @ real @ ( zero_zero @ real ) @ T10 )
& ( ord_less_eq @ real @ T10 @ H2 ) )
=> ( has_field_derivative @ real
@ ^ [U2: real] :
( minus_minus @ real @ ( Diff @ M3 @ U2 )
@ ( plus_plus @ real
@ ( groups7311177749621191930dd_sum @ nat @ real
@ ^ [P4: nat] : ( times_times @ real @ ( divide_divide @ real @ ( Diff @ ( plus_plus @ nat @ M3 @ P4 ) @ ( zero_zero @ real ) ) @ ( semiring_char_0_fact @ real @ P4 ) ) @ ( power_power @ real @ U2 @ P4 ) )
@ ( set_ord_lessThan @ nat @ ( minus_minus @ nat @ N @ M3 ) ) )
@ ( times_times @ real @ B2 @ ( divide_divide @ real @ ( power_power @ real @ U2 @ ( minus_minus @ nat @ N @ M3 ) ) @ ( semiring_char_0_fact @ real @ ( minus_minus @ nat @ N @ M3 ) ) ) ) ) )
@ ( minus_minus @ real @ ( Diff @ ( suc @ M3 ) @ T10 )
@ ( plus_plus @ real
@ ( groups7311177749621191930dd_sum @ nat @ real
@ ^ [P4: nat] : ( times_times @ real @ ( divide_divide @ real @ ( Diff @ ( plus_plus @ nat @ ( suc @ M3 ) @ P4 ) @ ( zero_zero @ real ) ) @ ( semiring_char_0_fact @ real @ P4 ) ) @ ( power_power @ real @ T10 @ P4 ) )
@ ( set_ord_lessThan @ nat @ ( minus_minus @ nat @ N @ ( suc @ M3 ) ) ) )
@ ( times_times @ real @ B2 @ ( divide_divide @ real @ ( power_power @ real @ T10 @ ( minus_minus @ nat @ N @ ( suc @ M3 ) ) ) @ ( semiring_char_0_fact @ real @ ( minus_minus @ nat @ N @ ( suc @ M3 ) ) ) ) ) ) )
@ ( topolo174197925503356063within @ real @ T10 @ ( top_top @ ( set @ real ) ) ) ) ) ) ) ).
% Maclaurin_lemma2
thf(fact_6968_DERIV__arctan__series,axiom,
! [X2: real] :
( ( ord_less @ real @ ( abs_abs @ real @ X2 ) @ ( one_one @ real ) )
=> ( has_field_derivative @ real
@ ^ [X10: real] :
( suminf @ real
@ ^ [K3: nat] : ( times_times @ real @ ( power_power @ real @ ( uminus_uminus @ real @ ( one_one @ real ) ) @ K3 ) @ ( times_times @ real @ ( divide_divide @ real @ ( one_one @ real ) @ ( semiring_1_of_nat @ real @ ( plus_plus @ nat @ ( times_times @ nat @ K3 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ ( one_one @ nat ) ) ) ) @ ( power_power @ real @ X10 @ ( plus_plus @ nat @ ( times_times @ nat @ K3 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ ( one_one @ nat ) ) ) ) ) )
@ ( suminf @ real
@ ^ [K3: nat] : ( times_times @ real @ ( power_power @ real @ ( uminus_uminus @ real @ ( one_one @ real ) ) @ K3 ) @ ( power_power @ real @ X2 @ ( times_times @ nat @ K3 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) )
@ ( topolo174197925503356063within @ real @ X2 @ ( top_top @ ( set @ real ) ) ) ) ) ).
% DERIV_arctan_series
thf(fact_6969_DERIV__power__series_H,axiom,
! [R: real,F2: nat > real,X0: real] :
( ! [X4: real] :
( ( member @ real @ X4 @ ( set_or5935395276787703475ssThan @ real @ ( uminus_uminus @ real @ R ) @ R ) )
=> ( summable @ real
@ ^ [N4: nat] : ( times_times @ real @ ( times_times @ real @ ( F2 @ N4 ) @ ( semiring_1_of_nat @ real @ ( suc @ N4 ) ) ) @ ( power_power @ real @ X4 @ N4 ) ) ) )
=> ( ( member @ real @ X0 @ ( set_or5935395276787703475ssThan @ real @ ( uminus_uminus @ real @ R ) @ R ) )
=> ( ( ord_less @ real @ ( zero_zero @ real ) @ R )
=> ( has_field_derivative @ real
@ ^ [X3: real] :
( suminf @ real
@ ^ [N4: nat] : ( times_times @ real @ ( F2 @ N4 ) @ ( power_power @ real @ X3 @ ( suc @ N4 ) ) ) )
@ ( suminf @ real
@ ^ [N4: nat] : ( times_times @ real @ ( times_times @ real @ ( F2 @ N4 ) @ ( semiring_1_of_nat @ real @ ( suc @ N4 ) ) ) @ ( power_power @ real @ X0 @ N4 ) ) )
@ ( topolo174197925503356063within @ real @ X0 @ ( top_top @ ( set @ real ) ) ) ) ) ) ) ).
% DERIV_power_series'
thf(fact_6970_has__derivative__arcsin,axiom,
! [A: $tType] :
( ( real_V822414075346904944vector @ A )
=> ! [G: A > real,X2: A,G5: A > real,S2: set @ A] :
( ( ord_less @ real @ ( uminus_uminus @ real @ ( one_one @ real ) ) @ ( G @ X2 ) )
=> ( ( ord_less @ real @ ( G @ X2 ) @ ( one_one @ real ) )
=> ( ( has_derivative @ A @ real @ G @ G5 @ ( topolo174197925503356063within @ A @ X2 @ S2 ) )
=> ( has_derivative @ A @ real
@ ^ [X3: A] : ( arcsin @ ( G @ X3 ) )
@ ^ [X3: A] : ( times_times @ real @ ( G5 @ X3 ) @ ( inverse_inverse @ real @ ( sqrt @ ( minus_minus @ real @ ( one_one @ real ) @ ( power_power @ real @ ( G @ X2 ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) )
@ ( topolo174197925503356063within @ A @ X2 @ S2 ) ) ) ) ) ) ).
% has_derivative_arcsin
thf(fact_6971_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_6972_greaterThanLessThan__empty__iff2,axiom,
! [A: $tType] :
( ( dense_linorder @ A )
=> ! [A3: A,B3: A] :
( ( ( bot_bot @ ( set @ A ) )
= ( set_or5935395276787703475ssThan @ A @ A3 @ B3 ) )
= ( ord_less_eq @ A @ B3 @ A3 ) ) ) ).
% greaterThanLessThan_empty_iff2
thf(fact_6973_greaterThanLessThan__empty__iff,axiom,
! [A: $tType] :
( ( dense_linorder @ A )
=> ! [A3: A,B3: A] :
( ( ( set_or5935395276787703475ssThan @ A @ A3 @ B3 )
= ( bot_bot @ ( set @ A ) ) )
= ( ord_less_eq @ A @ B3 @ A3 ) ) ) ).
% greaterThanLessThan_empty_iff
thf(fact_6974_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_6975_infinite__Ioo__iff,axiom,
! [A: $tType] :
( ( dense_linorder @ A )
=> ! [A3: A,B3: A] :
( ( ~ ( finite_finite2 @ A @ ( set_or5935395276787703475ssThan @ A @ A3 @ B3 ) ) )
= ( ord_less @ A @ A3 @ B3 ) ) ) ).
% infinite_Ioo_iff
thf(fact_6976_has__field__derivative__imp__has__derivative,axiom,
! [A: $tType] :
( ( real_V3459762299906320749_field @ A )
=> ! [F2: A > A,D: A,F4: filter @ A] :
( ( has_field_derivative @ A @ F2 @ D @ F4 )
=> ( has_derivative @ A @ A @ F2 @ ( times_times @ A @ D ) @ F4 ) ) ) ).
% has_field_derivative_imp_has_derivative
thf(fact_6977_has__derivative__imp__has__field__derivative,axiom,
! [A: $tType] :
( ( real_V3459762299906320749_field @ A )
=> ! [F2: A > A,D: A > A,F4: filter @ A,D7: A] :
( ( has_derivative @ A @ A @ F2 @ D @ F4 )
=> ( ! [X4: A] :
( ( times_times @ A @ X4 @ D7 )
= ( D @ X4 ) )
=> ( has_field_derivative @ A @ F2 @ D7 @ F4 ) ) ) ) ).
% has_derivative_imp_has_field_derivative
thf(fact_6978_has__field__derivative__def,axiom,
! [A: $tType] :
( ( real_V3459762299906320749_field @ A )
=> ( ( has_field_derivative @ A )
= ( ^ [F5: A > A,D8: A] : ( has_derivative @ A @ A @ F5 @ ( times_times @ A @ D8 ) ) ) ) ) ).
% has_field_derivative_def
thf(fact_6979_has__derivative__scaleR,axiom,
! [C: $tType,D6: $tType] :
( ( ( real_V822414075346904944vector @ D6 )
& ( real_V822414075346904944vector @ C ) )
=> ! [F2: D6 > real,F7: D6 > real,X2: D6,S2: set @ D6,G: D6 > C,G5: D6 > C] :
( ( has_derivative @ D6 @ real @ F2 @ F7 @ ( topolo174197925503356063within @ D6 @ X2 @ S2 ) )
=> ( ( has_derivative @ D6 @ C @ G @ G5 @ ( topolo174197925503356063within @ D6 @ X2 @ S2 ) )
=> ( has_derivative @ D6 @ C
@ ^ [X3: D6] : ( real_V8093663219630862766scaleR @ C @ ( F2 @ X3 ) @ ( G @ X3 ) )
@ ^ [H: D6] : ( plus_plus @ C @ ( real_V8093663219630862766scaleR @ C @ ( F2 @ X2 ) @ ( G5 @ H ) ) @ ( real_V8093663219630862766scaleR @ C @ ( F7 @ H ) @ ( G @ X2 ) ) )
@ ( topolo174197925503356063within @ D6 @ X2 @ S2 ) ) ) ) ) ).
% has_derivative_scaleR
thf(fact_6980_has__derivative__subset,axiom,
! [B: $tType,A: $tType] :
( ( ( real_V822414075346904944vector @ A )
& ( real_V822414075346904944vector @ B ) )
=> ! [F2: A > B,F7: A > B,X2: A,S2: set @ A,T2: set @ A] :
( ( has_derivative @ A @ B @ F2 @ F7 @ ( topolo174197925503356063within @ A @ X2 @ S2 ) )
=> ( ( ord_less_eq @ ( set @ A ) @ T2 @ S2 )
=> ( has_derivative @ A @ B @ F2 @ F7 @ ( topolo174197925503356063within @ A @ X2 @ T2 ) ) ) ) ) ).
% has_derivative_subset
thf(fact_6981_has__derivative__compose,axiom,
! [C: $tType,B: $tType,A: $tType] :
( ( ( real_V822414075346904944vector @ A )
& ( real_V822414075346904944vector @ B )
& ( real_V822414075346904944vector @ C ) )
=> ! [F2: A > B,F7: A > B,X2: A,S2: set @ A,G: B > C,G5: B > C] :
( ( has_derivative @ A @ B @ F2 @ F7 @ ( topolo174197925503356063within @ A @ X2 @ S2 ) )
=> ( ( has_derivative @ B @ C @ G @ G5 @ ( topolo174197925503356063within @ B @ ( F2 @ X2 ) @ ( top_top @ ( set @ B ) ) ) )
=> ( has_derivative @ A @ C
@ ^ [X3: A] : ( G @ ( F2 @ X3 ) )
@ ^ [X3: A] : ( G5 @ ( F7 @ X3 ) )
@ ( topolo174197925503356063within @ A @ X2 @ S2 ) ) ) ) ) ).
% has_derivative_compose
thf(fact_6982_has__derivative__in__compose,axiom,
! [C: $tType,B: $tType,A: $tType] :
( ( ( real_V822414075346904944vector @ A )
& ( real_V822414075346904944vector @ B )
& ( real_V822414075346904944vector @ C ) )
=> ! [F2: A > B,F7: A > B,X2: A,S2: set @ A,G: B > C,G5: B > C] :
( ( has_derivative @ A @ B @ F2 @ F7 @ ( topolo174197925503356063within @ A @ X2 @ S2 ) )
=> ( ( has_derivative @ B @ C @ G @ G5 @ ( topolo174197925503356063within @ B @ ( F2 @ X2 ) @ ( image @ A @ B @ F2 @ S2 ) ) )
=> ( has_derivative @ A @ C
@ ^ [X3: A] : ( G @ ( F2 @ X3 ) )
@ ^ [X3: A] : ( G5 @ ( F7 @ X3 ) )
@ ( topolo174197925503356063within @ A @ X2 @ S2 ) ) ) ) ) ).
% has_derivative_in_compose
thf(fact_6983_has__derivative__add,axiom,
! [B: $tType,A: $tType] :
( ( ( real_V822414075346904944vector @ A )
& ( real_V822414075346904944vector @ B ) )
=> ! [F2: A > B,F7: A > B,F4: filter @ A,G: A > B,G5: A > B] :
( ( has_derivative @ A @ B @ F2 @ F7 @ F4 )
=> ( ( has_derivative @ A @ B @ G @ G5 @ F4 )
=> ( has_derivative @ A @ B
@ ^ [X3: A] : ( plus_plus @ B @ ( F2 @ X3 ) @ ( G @ X3 ) )
@ ^ [X3: A] : ( plus_plus @ B @ ( F7 @ X3 ) @ ( G5 @ X3 ) )
@ F4 ) ) ) ) ).
% has_derivative_add
thf(fact_6984_has__derivative__diff,axiom,
! [B: $tType,A: $tType] :
( ( ( real_V822414075346904944vector @ A )
& ( real_V822414075346904944vector @ B ) )
=> ! [F2: A > B,F7: A > B,F4: filter @ A,G: A > B,G5: A > B] :
( ( has_derivative @ A @ B @ F2 @ F7 @ F4 )
=> ( ( has_derivative @ A @ B @ G @ G5 @ F4 )
=> ( has_derivative @ A @ B
@ ^ [X3: A] : ( minus_minus @ B @ ( F2 @ X3 ) @ ( G @ X3 ) )
@ ^ [X3: A] : ( minus_minus @ B @ ( F7 @ X3 ) @ ( G5 @ X3 ) )
@ F4 ) ) ) ) ).
% has_derivative_diff
thf(fact_6985_has__derivative__sum,axiom,
! [C: $tType,A: $tType,B: $tType] :
( ( ( real_V822414075346904944vector @ B )
& ( real_V822414075346904944vector @ C ) )
=> ! [I6: set @ A,F2: A > B > C,F7: A > B > C,F4: filter @ B] :
( ! [I2: A] :
( ( member @ A @ I2 @ I6 )
=> ( has_derivative @ B @ C @ ( F2 @ I2 ) @ ( F7 @ I2 ) @ F4 ) )
=> ( has_derivative @ B @ C
@ ^ [X3: B] :
( groups7311177749621191930dd_sum @ A @ C
@ ^ [I4: A] : ( F2 @ I4 @ X3 )
@ I6 )
@ ^ [X3: B] :
( groups7311177749621191930dd_sum @ A @ C
@ ^ [I4: A] : ( F7 @ I4 @ X3 )
@ I6 )
@ F4 ) ) ) ).
% has_derivative_sum
thf(fact_6986_has__derivative__of__real,axiom,
! [A: $tType,C: $tType] :
( ( ( real_V822414075346904944vector @ C )
& ( real_V2191834092415804123ebra_1 @ A )
& ( real_V822414075346904944vector @ A ) )
=> ! [G: C > real,G5: C > real,F4: filter @ C] :
( ( has_derivative @ C @ real @ G @ G5 @ F4 )
=> ( has_derivative @ C @ A
@ ^ [X3: C] : ( real_Vector_of_real @ A @ ( G @ X3 ) )
@ ^ [X3: C] : ( real_Vector_of_real @ A @ ( G5 @ X3 ) )
@ F4 ) ) ) ).
% has_derivative_of_real
thf(fact_6987_has__derivative__ident,axiom,
! [A: $tType] :
( ( real_V822414075346904944vector @ A )
=> ! [F4: filter @ A] :
( has_derivative @ A @ A
@ ^ [X3: A] : X3
@ ^ [X3: A] : X3
@ F4 ) ) ).
% has_derivative_ident
thf(fact_6988_has__derivative__scaleR__right,axiom,
! [B: $tType,C: $tType] :
( ( ( real_V822414075346904944vector @ C )
& ( real_V822414075346904944vector @ B ) )
=> ! [G: C > B,G5: C > B,F4: filter @ C,R2: real] :
( ( has_derivative @ C @ B @ G @ G5 @ F4 )
=> ( has_derivative @ C @ B
@ ^ [X3: C] : ( real_V8093663219630862766scaleR @ B @ R2 @ ( G @ X3 ) )
@ ^ [X3: C] : ( real_V8093663219630862766scaleR @ B @ R2 @ ( G5 @ X3 ) )
@ F4 ) ) ) ).
% has_derivative_scaleR_right
thf(fact_6989_has__derivative__scaleR__left,axiom,
! [B: $tType,C: $tType] :
( ( ( real_V822414075346904944vector @ C )
& ( real_V822414075346904944vector @ B ) )
=> ! [G: C > real,G5: C > real,F4: filter @ C,X2: B] :
( ( has_derivative @ C @ real @ G @ G5 @ F4 )
=> ( has_derivative @ C @ B
@ ^ [X3: C] : ( real_V8093663219630862766scaleR @ B @ ( G @ X3 ) @ X2 )
@ ^ [X3: C] : ( real_V8093663219630862766scaleR @ B @ ( G5 @ X3 ) @ X2 )
@ F4 ) ) ) ).
% has_derivative_scaleR_left
thf(fact_6990_has__derivative__const,axiom,
! [B: $tType,A: $tType] :
( ( ( real_V822414075346904944vector @ A )
& ( real_V822414075346904944vector @ B ) )
=> ! [C3: B,F4: filter @ A] :
( has_derivative @ A @ B
@ ^ [X3: A] : C3
@ ^ [X3: A] : ( zero_zero @ B )
@ F4 ) ) ).
% has_derivative_const
thf(fact_6991_has__derivative__minus,axiom,
! [B: $tType,A: $tType] :
( ( ( real_V822414075346904944vector @ A )
& ( real_V822414075346904944vector @ B ) )
=> ! [F2: A > B,F7: A > B,F4: filter @ A] :
( ( has_derivative @ A @ B @ F2 @ F7 @ F4 )
=> ( has_derivative @ A @ B
@ ^ [X3: A] : ( uminus_uminus @ B @ ( F2 @ X3 ) )
@ ^ [X3: A] : ( uminus_uminus @ B @ ( F7 @ X3 ) )
@ F4 ) ) ) ).
% has_derivative_minus
thf(fact_6992_has__derivative__mult__right,axiom,
! [A: $tType,C: $tType] :
( ( ( real_V822414075346904944vector @ C )
& ( real_V4412858255891104859lgebra @ A ) )
=> ! [G: C > A,G5: C > A,F4: filter @ C,X2: A] :
( ( has_derivative @ C @ A @ G @ G5 @ F4 )
=> ( has_derivative @ C @ A
@ ^ [X3: C] : ( times_times @ A @ X2 @ ( G @ X3 ) )
@ ^ [X3: C] : ( times_times @ A @ X2 @ ( G5 @ X3 ) )
@ F4 ) ) ) ).
% has_derivative_mult_right
thf(fact_6993_has__derivative__mult__left,axiom,
! [A: $tType,C: $tType] :
( ( ( real_V822414075346904944vector @ C )
& ( real_V4412858255891104859lgebra @ A ) )
=> ! [G: C > A,G5: C > A,F4: filter @ C,Y3: A] :
( ( has_derivative @ C @ A @ G @ G5 @ F4 )
=> ( has_derivative @ C @ A
@ ^ [X3: C] : ( times_times @ A @ ( G @ X3 ) @ Y3 )
@ ^ [X3: C] : ( times_times @ A @ ( G5 @ X3 ) @ Y3 )
@ F4 ) ) ) ).
% has_derivative_mult_left
thf(fact_6994_infinite__Ioo,axiom,
! [A: $tType] :
( ( dense_linorder @ A )
=> ! [A3: A,B3: A] :
( ( ord_less @ A @ A3 @ B3 )
=> ~ ( finite_finite2 @ A @ ( set_or5935395276787703475ssThan @ A @ A3 @ B3 ) ) ) ) ).
% infinite_Ioo
thf(fact_6995_has__derivative__mult,axiom,
! [A: $tType,D6: $tType] :
( ( ( real_V822414075346904944vector @ D6 )
& ( real_V4412858255891104859lgebra @ A ) )
=> ! [F2: D6 > A,F7: D6 > A,X2: D6,S2: set @ D6,G: D6 > A,G5: D6 > A] :
( ( has_derivative @ D6 @ A @ F2 @ F7 @ ( topolo174197925503356063within @ D6 @ X2 @ S2 ) )
=> ( ( has_derivative @ D6 @ A @ G @ G5 @ ( topolo174197925503356063within @ D6 @ X2 @ S2 ) )
=> ( has_derivative @ D6 @ A
@ ^ [X3: D6] : ( times_times @ A @ ( F2 @ X3 ) @ ( G @ X3 ) )
@ ^ [H: D6] : ( plus_plus @ A @ ( times_times @ A @ ( F2 @ X2 ) @ ( G5 @ H ) ) @ ( times_times @ A @ ( F7 @ H ) @ ( G @ X2 ) ) )
@ ( topolo174197925503356063within @ D6 @ X2 @ S2 ) ) ) ) ) ).
% has_derivative_mult
thf(fact_6996_has__derivative__zero__unique,axiom,
! [B: $tType,A: $tType] :
( ( ( real_V822414075346904944vector @ A )
& ( real_V822414075346904944vector @ B ) )
=> ! [F4: A > B,X2: A] :
( ( has_derivative @ A @ B
@ ^ [X3: A] : ( zero_zero @ B )
@ F4
@ ( topolo174197925503356063within @ A @ X2 @ ( top_top @ ( set @ A ) ) ) )
=> ( F4
= ( ^ [H: A] : ( zero_zero @ B ) ) ) ) ) ).
% has_derivative_zero_unique
thf(fact_6997_has__derivative__in__compose2,axiom,
! [B: $tType,A: $tType,C: $tType] :
( ( ( real_V822414075346904944vector @ C )
& ( real_V822414075346904944vector @ A )
& ( real_V822414075346904944vector @ B ) )
=> ! [T2: set @ A,G: A > B,G5: A > A > B,F2: C > A,S2: set @ C,X2: C,F7: C > A] :
( ! [X4: A] :
( ( member @ A @ X4 @ T2 )
=> ( has_derivative @ A @ B @ G @ ( G5 @ X4 ) @ ( topolo174197925503356063within @ A @ X4 @ T2 ) ) )
=> ( ( ord_less_eq @ ( set @ A ) @ ( image @ C @ A @ F2 @ S2 ) @ T2 )
=> ( ( member @ C @ X2 @ S2 )
=> ( ( has_derivative @ C @ A @ F2 @ F7 @ ( topolo174197925503356063within @ C @ X2 @ S2 ) )
=> ( has_derivative @ C @ B
@ ^ [X3: C] : ( G @ ( F2 @ X3 ) )
@ ^ [Y: C] : ( G5 @ ( F2 @ X2 ) @ ( F7 @ Y ) )
@ ( topolo174197925503356063within @ C @ X2 @ S2 ) ) ) ) ) ) ) ).
% has_derivative_in_compose2
thf(fact_6998_has__derivative__exp,axiom,
! [A: $tType] :
( ( real_V822414075346904944vector @ A )
=> ! [G: A > real,G5: A > real,X2: A,S2: set @ A] :
( ( has_derivative @ A @ real @ G @ G5 @ ( topolo174197925503356063within @ A @ X2 @ S2 ) )
=> ( has_derivative @ A @ real
@ ^ [X3: A] : ( exp @ real @ ( G @ X3 ) )
@ ^ [X3: A] : ( times_times @ real @ ( G5 @ X3 ) @ ( exp @ real @ ( G @ X2 ) ) )
@ ( topolo174197925503356063within @ A @ X2 @ S2 ) ) ) ) ).
% has_derivative_exp
thf(fact_6999_greaterThanLessThan__subseteq__greaterThanLessThan,axiom,
! [A: $tType] :
( ( dense_linorder @ A )
=> ! [A3: A,B3: A,C3: A,D2: A] :
( ( ord_less_eq @ ( set @ A ) @ ( set_or5935395276787703475ssThan @ A @ A3 @ B3 ) @ ( set_or5935395276787703475ssThan @ A @ C3 @ D2 ) )
= ( ( ord_less @ A @ A3 @ B3 )
=> ( ( ord_less_eq @ A @ C3 @ A3 )
& ( ord_less_eq @ A @ B3 @ D2 ) ) ) ) ) ).
% greaterThanLessThan_subseteq_greaterThanLessThan
thf(fact_7000_tanh__real__bounds,axiom,
! [X2: real] : ( member @ real @ ( tanh @ real @ X2 ) @ ( set_or5935395276787703475ssThan @ real @ ( uminus_uminus @ real @ ( one_one @ real ) ) @ ( one_one @ real ) ) ) ).
% tanh_real_bounds
thf(fact_7001_has__derivative__sinh,axiom,
! [A: $tType] :
( ( ( real_Vector_banach @ A )
& ( real_V3459762299906320749_field @ A ) )
=> ! [G: A > A,Db: A,X2: A,S2: set @ A] :
( ( has_derivative @ A @ A @ G @ ( times_times @ A @ Db ) @ ( topolo174197925503356063within @ A @ X2 @ S2 ) )
=> ( has_derivative @ A @ A
@ ^ [X3: A] : ( sinh @ A @ ( G @ X3 ) )
@ ( times_times @ A @ ( times_times @ A @ ( cosh @ A @ ( G @ X2 ) ) @ Db ) )
@ ( topolo174197925503356063within @ A @ X2 @ S2 ) ) ) ) ).
% has_derivative_sinh
thf(fact_7002_has__derivative__cosh,axiom,
! [A: $tType] :
( ( ( real_Vector_banach @ A )
& ( real_V3459762299906320749_field @ A ) )
=> ! [G: A > A,Db: A,X2: A,S2: set @ A] :
( ( has_derivative @ A @ A @ G @ ( times_times @ A @ Db ) @ ( topolo174197925503356063within @ A @ X2 @ S2 ) )
=> ( has_derivative @ A @ A
@ ^ [X3: A] : ( cosh @ A @ ( G @ X3 ) )
@ ( times_times @ A @ ( times_times @ A @ ( sinh @ A @ ( G @ X2 ) ) @ Db ) )
@ ( topolo174197925503356063within @ A @ X2 @ S2 ) ) ) ) ).
% has_derivative_cosh
thf(fact_7003_has__derivative__sin,axiom,
! [A: $tType] :
( ( real_V822414075346904944vector @ A )
=> ! [G: A > real,G5: A > real,X2: A,S2: set @ A] :
( ( has_derivative @ A @ real @ G @ G5 @ ( topolo174197925503356063within @ A @ X2 @ S2 ) )
=> ( has_derivative @ A @ real
@ ^ [X3: A] : ( sin @ real @ ( G @ X3 ) )
@ ^ [X3: A] : ( times_times @ real @ ( G5 @ X3 ) @ ( cos @ real @ ( G @ X2 ) ) )
@ ( topolo174197925503356063within @ A @ X2 @ S2 ) ) ) ) ).
% has_derivative_sin
thf(fact_7004_greaterThanLessThan__subseteq__atLeastAtMost__iff,axiom,
! [A: $tType] :
( ( dense_linorder @ A )
=> ! [A3: A,B3: A,C3: A,D2: A] :
( ( ord_less_eq @ ( set @ A ) @ ( set_or5935395276787703475ssThan @ A @ A3 @ B3 ) @ ( set_or1337092689740270186AtMost @ A @ C3 @ D2 ) )
= ( ( ord_less @ A @ A3 @ B3 )
=> ( ( ord_less_eq @ A @ C3 @ A3 )
& ( ord_less_eq @ A @ B3 @ D2 ) ) ) ) ) ).
% greaterThanLessThan_subseteq_atLeastAtMost_iff
thf(fact_7005_greaterThanLessThan__subseteq__atLeastLessThan__iff,axiom,
! [A: $tType] :
( ( dense_linorder @ A )
=> ! [A3: A,B3: A,C3: A,D2: A] :
( ( ord_less_eq @ ( set @ A ) @ ( set_or5935395276787703475ssThan @ A @ A3 @ B3 ) @ ( set_or7035219750837199246ssThan @ A @ C3 @ D2 ) )
= ( ( ord_less @ A @ A3 @ B3 )
=> ( ( ord_less_eq @ A @ C3 @ A3 )
& ( ord_less_eq @ A @ B3 @ D2 ) ) ) ) ) ).
% greaterThanLessThan_subseteq_atLeastLessThan_iff
thf(fact_7006_atLeastAtMost__diff__ends,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A3: A,B3: A] :
( ( minus_minus @ ( set @ A ) @ ( set_or1337092689740270186AtMost @ A @ A3 @ B3 ) @ ( insert @ A @ A3 @ ( insert @ A @ B3 @ ( bot_bot @ ( set @ A ) ) ) ) )
= ( set_or5935395276787703475ssThan @ A @ A3 @ B3 ) ) ) ).
% atLeastAtMost_diff_ends
thf(fact_7007_has__derivative__divide_H,axiom,
! [A: $tType,C: $tType] :
( ( ( real_V822414075346904944vector @ C )
& ( real_V3459762299906320749_field @ A ) )
=> ! [F2: C > A,F7: C > A,X2: C,S: set @ C,G: C > A,G5: C > A] :
( ( has_derivative @ C @ A @ F2 @ F7 @ ( topolo174197925503356063within @ C @ X2 @ S ) )
=> ( ( has_derivative @ C @ A @ G @ G5 @ ( topolo174197925503356063within @ C @ X2 @ S ) )
=> ( ( ( G @ X2 )
!= ( zero_zero @ A ) )
=> ( has_derivative @ C @ A
@ ^ [X3: C] : ( divide_divide @ A @ ( F2 @ X3 ) @ ( G @ X3 ) )
@ ^ [H: C] : ( divide_divide @ A @ ( minus_minus @ A @ ( times_times @ A @ ( F7 @ H ) @ ( G @ X2 ) ) @ ( times_times @ A @ ( F2 @ X2 ) @ ( G5 @ H ) ) ) @ ( times_times @ A @ ( G @ X2 ) @ ( G @ X2 ) ) )
@ ( topolo174197925503356063within @ C @ X2 @ S ) ) ) ) ) ) ).
% has_derivative_divide'
thf(fact_7008_has__derivative__inverse_H,axiom,
! [A: $tType] :
( ( real_V8999393235501362500lgebra @ A )
=> ! [X2: A,S: set @ A] :
( ( X2
!= ( zero_zero @ A ) )
=> ( has_derivative @ A @ A @ ( inverse_inverse @ A )
@ ^ [H: A] : ( uminus_uminus @ A @ ( times_times @ A @ ( times_times @ A @ ( inverse_inverse @ A @ X2 ) @ H ) @ ( inverse_inverse @ A @ X2 ) ) )
@ ( topolo174197925503356063within @ A @ X2 @ S ) ) ) ) ).
% has_derivative_inverse'
thf(fact_7009_has__derivative__inverse,axiom,
! [A: $tType,C: $tType] :
( ( ( real_V822414075346904944vector @ C )
& ( real_V8999393235501362500lgebra @ A ) )
=> ! [F2: C > A,X2: C,F7: C > A,S: set @ C] :
( ( ( F2 @ X2 )
!= ( zero_zero @ A ) )
=> ( ( has_derivative @ C @ A @ F2 @ F7 @ ( topolo174197925503356063within @ C @ X2 @ S ) )
=> ( has_derivative @ C @ A
@ ^ [X3: C] : ( inverse_inverse @ A @ ( F2 @ X3 ) )
@ ^ [H: C] : ( uminus_uminus @ A @ ( times_times @ A @ ( times_times @ A @ ( inverse_inverse @ A @ ( F2 @ X2 ) ) @ ( F7 @ H ) ) @ ( inverse_inverse @ A @ ( F2 @ X2 ) ) ) )
@ ( topolo174197925503356063within @ C @ X2 @ S ) ) ) ) ) ).
% has_derivative_inverse
thf(fact_7010_DERIV__compose__FDERIV,axiom,
! [A: $tType] :
( ( real_V822414075346904944vector @ A )
=> ! [F2: real > real,F7: real,G: A > real,X2: A,G5: A > real,S2: set @ A] :
( ( has_field_derivative @ real @ F2 @ F7 @ ( topolo174197925503356063within @ real @ ( G @ X2 ) @ ( top_top @ ( set @ real ) ) ) )
=> ( ( has_derivative @ A @ real @ G @ G5 @ ( topolo174197925503356063within @ A @ X2 @ S2 ) )
=> ( has_derivative @ A @ real
@ ^ [X3: A] : ( F2 @ ( G @ X3 ) )
@ ^ [X3: A] : ( times_times @ real @ ( G5 @ X3 ) @ F7 )
@ ( topolo174197925503356063within @ A @ X2 @ S2 ) ) ) ) ) ).
% DERIV_compose_FDERIV
thf(fact_7011_has__derivative__cos,axiom,
! [A: $tType] :
( ( real_V822414075346904944vector @ A )
=> ! [G: A > real,G5: A > real,X2: A,S2: set @ A] :
( ( has_derivative @ A @ real @ G @ G5 @ ( topolo174197925503356063within @ A @ X2 @ S2 ) )
=> ( has_derivative @ A @ real
@ ^ [X3: A] : ( cos @ real @ ( G @ X3 ) )
@ ^ [X3: A] : ( times_times @ real @ ( G5 @ X3 ) @ ( uminus_uminus @ real @ ( sin @ real @ ( G @ X2 ) ) ) )
@ ( topolo174197925503356063within @ A @ X2 @ S2 ) ) ) ) ).
% has_derivative_cos
thf(fact_7012_has__derivative__power,axiom,
! [B: $tType,A: $tType] :
( ( ( real_V822414075346904944vector @ A )
& ( real_V3459762299906320749_field @ B ) )
=> ! [F2: A > B,F7: A > B,X2: A,S: set @ A,N: nat] :
( ( has_derivative @ A @ B @ F2 @ F7 @ ( topolo174197925503356063within @ A @ X2 @ S ) )
=> ( has_derivative @ A @ B
@ ^ [X3: A] : ( power_power @ B @ ( F2 @ X3 ) @ N )
@ ^ [Y: A] : ( times_times @ B @ ( times_times @ B @ ( semiring_1_of_nat @ B @ N ) @ ( F7 @ Y ) ) @ ( power_power @ B @ ( F2 @ X2 ) @ ( minus_minus @ nat @ N @ ( one_one @ nat ) ) ) )
@ ( topolo174197925503356063within @ A @ X2 @ S ) ) ) ) ).
% has_derivative_power
thf(fact_7013_has__derivative__ln,axiom,
! [A: $tType] :
( ( real_V822414075346904944vector @ A )
=> ! [G: A > real,X2: A,G5: A > real,S2: set @ A] :
( ( ord_less @ real @ ( zero_zero @ real ) @ ( G @ X2 ) )
=> ( ( has_derivative @ A @ real @ G @ G5 @ ( topolo174197925503356063within @ A @ X2 @ S2 ) )
=> ( has_derivative @ A @ real
@ ^ [X3: A] : ( ln_ln @ real @ ( G @ X3 ) )
@ ^ [X3: A] : ( times_times @ real @ ( G5 @ X3 ) @ ( inverse_inverse @ real @ ( G @ X2 ) ) )
@ ( topolo174197925503356063within @ A @ X2 @ S2 ) ) ) ) ) ).
% has_derivative_ln
thf(fact_7014_has__derivative__divide,axiom,
! [A: $tType,C: $tType] :
( ( ( real_V822414075346904944vector @ C )
& ( real_V8999393235501362500lgebra @ A ) )
=> ! [F2: C > A,F7: C > A,X2: C,S: set @ C,G: C > A,G5: C > A] :
( ( has_derivative @ C @ A @ F2 @ F7 @ ( topolo174197925503356063within @ C @ X2 @ S ) )
=> ( ( has_derivative @ C @ A @ G @ G5 @ ( topolo174197925503356063within @ C @ X2 @ S ) )
=> ( ( ( G @ X2 )
!= ( zero_zero @ A ) )
=> ( has_derivative @ C @ A
@ ^ [X3: C] : ( divide_divide @ A @ ( F2 @ X3 ) @ ( G @ X3 ) )
@ ^ [H: C] : ( plus_plus @ A @ ( times_times @ A @ ( uminus_uminus @ A @ ( F2 @ X2 ) ) @ ( times_times @ A @ ( times_times @ A @ ( inverse_inverse @ A @ ( G @ X2 ) ) @ ( G5 @ H ) ) @ ( inverse_inverse @ A @ ( G @ X2 ) ) ) ) @ ( divide_divide @ A @ ( F7 @ H ) @ ( G @ X2 ) ) )
@ ( topolo174197925503356063within @ C @ X2 @ S ) ) ) ) ) ) ).
% has_derivative_divide
thf(fact_7015_has__derivative__prod,axiom,
! [B: $tType,I8: $tType,A: $tType] :
( ( ( real_V822414075346904944vector @ A )
& ( real_V3459762299906320749_field @ B ) )
=> ! [I6: set @ I8,F2: I8 > A > B,F7: I8 > A > B,X2: A,S: set @ A] :
( ! [I2: I8] :
( ( member @ I8 @ I2 @ I6 )
=> ( has_derivative @ A @ B @ ( F2 @ I2 ) @ ( F7 @ I2 ) @ ( topolo174197925503356063within @ A @ X2 @ S ) ) )
=> ( has_derivative @ A @ B
@ ^ [X3: A] :
( groups7121269368397514597t_prod @ I8 @ B
@ ^ [I4: I8] : ( F2 @ I4 @ X3 )
@ I6 )
@ ^ [Y: A] :
( groups7311177749621191930dd_sum @ I8 @ B
@ ^ [I4: I8] :
( times_times @ B @ ( F7 @ I4 @ Y )
@ ( groups7121269368397514597t_prod @ I8 @ B
@ ^ [J3: I8] : ( F2 @ J3 @ X2 )
@ ( minus_minus @ ( set @ I8 ) @ I6 @ ( insert @ I8 @ I4 @ ( bot_bot @ ( set @ I8 ) ) ) ) ) )
@ I6 )
@ ( topolo174197925503356063within @ A @ X2 @ S ) ) ) ) ).
% has_derivative_prod
thf(fact_7016_has__derivative__powr,axiom,
! [A: $tType] :
( ( real_V822414075346904944vector @ A )
=> ! [G: A > real,G5: A > real,X2: A,X: set @ A,F2: A > real,F7: A > real] :
( ( has_derivative @ A @ real @ G @ G5 @ ( topolo174197925503356063within @ A @ X2 @ X ) )
=> ( ( has_derivative @ A @ real @ F2 @ F7 @ ( topolo174197925503356063within @ A @ X2 @ X ) )
=> ( ( ord_less @ real @ ( zero_zero @ real ) @ ( G @ X2 ) )
=> ( ( member @ A @ X2 @ X )
=> ( has_derivative @ A @ real
@ ^ [X3: A] : ( powr @ real @ ( G @ X3 ) @ ( F2 @ X3 ) )
@ ^ [H: A] : ( times_times @ real @ ( powr @ real @ ( G @ X2 ) @ ( F2 @ X2 ) ) @ ( plus_plus @ real @ ( times_times @ real @ ( F7 @ H ) @ ( ln_ln @ real @ ( G @ X2 ) ) ) @ ( divide_divide @ real @ ( times_times @ real @ ( G5 @ H ) @ ( F2 @ X2 ) ) @ ( G @ X2 ) ) ) )
@ ( topolo174197925503356063within @ A @ X2 @ X ) ) ) ) ) ) ) ).
% has_derivative_powr
thf(fact_7017_has__derivative__real__sqrt,axiom,
! [A: $tType] :
( ( real_V822414075346904944vector @ A )
=> ! [G: A > real,X2: A,G5: A > real,S2: set @ A] :
( ( ord_less @ real @ ( zero_zero @ real ) @ ( G @ X2 ) )
=> ( ( has_derivative @ A @ real @ G @ G5 @ ( topolo174197925503356063within @ A @ X2 @ S2 ) )
=> ( has_derivative @ A @ real
@ ^ [X3: A] : ( sqrt @ ( G @ X3 ) )
@ ^ [X3: A] : ( times_times @ real @ ( G5 @ X3 ) @ ( divide_divide @ real @ ( inverse_inverse @ real @ ( sqrt @ ( G @ X2 ) ) ) @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) )
@ ( topolo174197925503356063within @ A @ X2 @ S2 ) ) ) ) ) ).
% has_derivative_real_sqrt
thf(fact_7018_has__derivative__arctan,axiom,
! [A: $tType] :
( ( real_V822414075346904944vector @ A )
=> ! [G: A > real,G5: A > real,X2: A,S2: set @ A] :
( ( has_derivative @ A @ real @ G @ G5 @ ( topolo174197925503356063within @ A @ X2 @ S2 ) )
=> ( has_derivative @ A @ real
@ ^ [X3: A] : ( arctan @ ( G @ X3 ) )
@ ^ [X3: A] : ( times_times @ real @ ( G5 @ X3 ) @ ( inverse_inverse @ real @ ( plus_plus @ real @ ( one_one @ real ) @ ( power_power @ real @ ( G @ X2 ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) )
@ ( topolo174197925503356063within @ A @ X2 @ S2 ) ) ) ) ).
% has_derivative_arctan
thf(fact_7019_has__derivative__tan,axiom,
! [A: $tType] :
( ( real_V822414075346904944vector @ A )
=> ! [G: A > real,X2: A,G5: A > real,S2: set @ A] :
( ( ( cos @ real @ ( G @ X2 ) )
!= ( zero_zero @ real ) )
=> ( ( has_derivative @ A @ real @ G @ G5 @ ( topolo174197925503356063within @ A @ X2 @ S2 ) )
=> ( has_derivative @ A @ real
@ ^ [X3: A] : ( tan @ real @ ( G @ X3 ) )
@ ^ [X3: A] : ( times_times @ real @ ( G5 @ X3 ) @ ( inverse_inverse @ real @ ( power_power @ real @ ( cos @ real @ ( G @ X2 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) )
@ ( topolo174197925503356063within @ A @ X2 @ S2 ) ) ) ) ) ).
% has_derivative_tan
thf(fact_7020_DERIV__series_H,axiom,
! [F2: real > nat > real,F7: real > nat > real,X0: real,A3: real,B3: real,L6: nat > real] :
( ! [N3: nat] :
( has_field_derivative @ real
@ ^ [X3: real] : ( F2 @ X3 @ N3 )
@ ( F7 @ X0 @ N3 )
@ ( topolo174197925503356063within @ real @ X0 @ ( top_top @ ( set @ real ) ) ) )
=> ( ! [X4: real] :
( ( member @ real @ X4 @ ( set_or5935395276787703475ssThan @ real @ A3 @ B3 ) )
=> ( summable @ real @ ( F2 @ X4 ) ) )
=> ( ( member @ real @ X0 @ ( set_or5935395276787703475ssThan @ real @ A3 @ B3 ) )
=> ( ( summable @ real @ ( F7 @ X0 ) )
=> ( ( summable @ real @ L6 )
=> ( ! [N3: nat,X4: real,Y4: real] :
( ( member @ real @ X4 @ ( set_or5935395276787703475ssThan @ real @ A3 @ B3 ) )
=> ( ( member @ real @ Y4 @ ( set_or5935395276787703475ssThan @ real @ A3 @ B3 ) )
=> ( ord_less_eq @ real @ ( abs_abs @ real @ ( minus_minus @ real @ ( F2 @ X4 @ N3 ) @ ( F2 @ Y4 @ N3 ) ) ) @ ( times_times @ real @ ( L6 @ N3 ) @ ( abs_abs @ real @ ( minus_minus @ real @ X4 @ Y4 ) ) ) ) ) )
=> ( has_field_derivative @ real
@ ^ [X3: real] : ( suminf @ real @ ( F2 @ X3 ) )
@ ( suminf @ real @ ( F7 @ X0 ) )
@ ( topolo174197925503356063within @ real @ X0 @ ( top_top @ ( set @ real ) ) ) ) ) ) ) ) ) ) ).
% DERIV_series'
thf(fact_7021_has__derivative__arccos,axiom,
! [A: $tType] :
( ( real_V822414075346904944vector @ A )
=> ! [G: A > real,X2: A,G5: A > real,S2: set @ A] :
( ( ord_less @ real @ ( uminus_uminus @ real @ ( one_one @ real ) ) @ ( G @ X2 ) )
=> ( ( ord_less @ real @ ( G @ X2 ) @ ( one_one @ real ) )
=> ( ( has_derivative @ A @ real @ G @ G5 @ ( topolo174197925503356063within @ A @ X2 @ S2 ) )
=> ( has_derivative @ A @ real
@ ^ [X3: A] : ( arccos @ ( G @ X3 ) )
@ ^ [X3: A] : ( times_times @ real @ ( G5 @ X3 ) @ ( inverse_inverse @ real @ ( uminus_uminus @ real @ ( sqrt @ ( minus_minus @ real @ ( one_one @ real ) @ ( power_power @ real @ ( G @ X2 ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) ) )
@ ( topolo174197925503356063within @ A @ X2 @ S2 ) ) ) ) ) ) ).
% has_derivative_arccos
thf(fact_7022_has__derivative__floor,axiom,
! [Aa: $tType,A: $tType] :
( ( ( real_V822414075346904944vector @ A )
& ( archim2362893244070406136eiling @ Aa )
& ( topolo2564578578187576103pology @ Aa ) )
=> ! [G: A > real,X2: A,F2: real > Aa,G5: A > real,S2: set @ A] :
( ( topolo3448309680560233919inuous @ real @ Aa @ ( topolo174197925503356063within @ real @ ( G @ X2 ) @ ( top_top @ ( set @ real ) ) ) @ F2 )
=> ( ~ ( member @ Aa @ ( F2 @ ( G @ X2 ) ) @ ( ring_1_Ints @ Aa ) )
=> ( ( has_derivative @ A @ real @ G @ G5 @ ( topolo174197925503356063within @ A @ X2 @ S2 ) )
=> ( has_derivative @ A @ real
@ ^ [X3: A] : ( ring_1_of_int @ real @ ( archim6421214686448440834_floor @ Aa @ ( F2 @ ( G @ X3 ) ) ) )
@ ^ [X3: A] : ( times_times @ real @ ( G5 @ X3 ) @ ( zero_zero @ real ) )
@ ( topolo174197925503356063within @ A @ X2 @ S2 ) ) ) ) ) ) ).
% has_derivative_floor
thf(fact_7023_termdiffs__aux,axiom,
! [A: $tType] :
( ( ( real_Vector_banach @ A )
& ( real_V3459762299906320749_field @ A ) )
=> ! [C3: nat > A,K6: A,X2: A] :
( ( summable @ A
@ ^ [N4: nat] : ( times_times @ A @ ( diffs @ A @ ( diffs @ A @ C3 ) @ N4 ) @ ( power_power @ A @ K6 @ N4 ) ) )
=> ( ( ord_less @ real @ ( real_V7770717601297561774m_norm @ A @ X2 ) @ ( real_V7770717601297561774m_norm @ A @ K6 ) )
=> ( filterlim @ A @ A
@ ^ [H: A] :
( suminf @ A
@ ^ [N4: nat] : ( times_times @ A @ ( C3 @ N4 ) @ ( minus_minus @ A @ ( divide_divide @ A @ ( minus_minus @ A @ ( power_power @ A @ ( plus_plus @ A @ X2 @ H ) @ N4 ) @ ( power_power @ A @ X2 @ N4 ) ) @ H ) @ ( times_times @ A @ ( semiring_1_of_nat @ A @ N4 ) @ ( power_power @ A @ X2 @ ( minus_minus @ nat @ N4 @ ( suc @ ( zero_zero @ nat ) ) ) ) ) ) ) )
@ ( topolo7230453075368039082e_nhds @ A @ ( zero_zero @ A ) )
@ ( topolo174197925503356063within @ A @ ( zero_zero @ A ) @ ( top_top @ ( set @ A ) ) ) ) ) ) ) ).
% termdiffs_aux
thf(fact_7024_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_7025_tendsto__const,axiom,
! [A: $tType,B: $tType] :
( ( topolo4958980785337419405_space @ A )
=> ! [K: A,F4: filter @ B] :
( filterlim @ B @ A
@ ^ [X3: B] : K
@ ( topolo7230453075368039082e_nhds @ A @ K )
@ F4 ) ) ).
% tendsto_const
thf(fact_7026_tendsto__ident__at,axiom,
! [A: $tType] :
( ( topolo4958980785337419405_space @ A )
=> ! [A3: A,S2: set @ A] :
( filterlim @ A @ A
@ ^ [X3: A] : X3
@ ( topolo7230453075368039082e_nhds @ A @ A3 )
@ ( topolo174197925503356063within @ A @ A3 @ S2 ) ) ) ).
% tendsto_ident_at
thf(fact_7027_tendsto__mult__left__iff,axiom,
! [A: $tType,B: $tType] :
( ( ( field @ A )
& ( topolo4211221413907600880p_mult @ A ) )
=> ! [C3: A,F2: B > A,L: A,F4: filter @ B] :
( ( C3
!= ( zero_zero @ A ) )
=> ( ( filterlim @ B @ A
@ ^ [X3: B] : ( times_times @ A @ C3 @ ( F2 @ X3 ) )
@ ( topolo7230453075368039082e_nhds @ A @ ( times_times @ A @ C3 @ L ) )
@ F4 )
= ( filterlim @ B @ A @ F2 @ ( topolo7230453075368039082e_nhds @ A @ L ) @ F4 ) ) ) ) ).
% tendsto_mult_left_iff
thf(fact_7028_tendsto__mult__right__iff,axiom,
! [A: $tType,B: $tType] :
( ( ( field @ A )
& ( topolo4211221413907600880p_mult @ A ) )
=> ! [C3: A,F2: B > A,L: A,F4: filter @ B] :
( ( C3
!= ( zero_zero @ A ) )
=> ( ( filterlim @ B @ A
@ ^ [X3: B] : ( times_times @ A @ ( F2 @ X3 ) @ C3 )
@ ( topolo7230453075368039082e_nhds @ A @ ( times_times @ A @ L @ C3 ) )
@ F4 )
= ( filterlim @ B @ A @ F2 @ ( topolo7230453075368039082e_nhds @ A @ L ) @ F4 ) ) ) ) ).
% tendsto_mult_right_iff
thf(fact_7029_power__tendsto__0__iff,axiom,
! [A: $tType,N: nat,F2: A > real,F4: filter @ A] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ N )
=> ( ( filterlim @ A @ real
@ ^ [X3: A] : ( power_power @ real @ ( F2 @ X3 ) @ N )
@ ( topolo7230453075368039082e_nhds @ real @ ( zero_zero @ real ) )
@ F4 )
= ( filterlim @ A @ real @ F2 @ ( topolo7230453075368039082e_nhds @ real @ ( zero_zero @ real ) ) @ F4 ) ) ) ).
% power_tendsto_0_iff
thf(fact_7030_has__field__derivativeD,axiom,
! [A: $tType] :
( ( real_V3459762299906320749_field @ A )
=> ! [F2: A > A,D: A,X2: A,S: set @ A] :
( ( has_field_derivative @ A @ F2 @ D @ ( topolo174197925503356063within @ A @ X2 @ S ) )
=> ( filterlim @ A @ A
@ ^ [Y: A] : ( divide_divide @ A @ ( minus_minus @ A @ ( F2 @ Y ) @ ( F2 @ X2 ) ) @ ( minus_minus @ A @ Y @ X2 ) )
@ ( topolo7230453075368039082e_nhds @ A @ D )
@ ( topolo174197925503356063within @ A @ X2 @ S ) ) ) ) ).
% has_field_derivativeD
thf(fact_7031_has__field__derivative__iff,axiom,
! [A: $tType] :
( ( real_V3459762299906320749_field @ A )
=> ! [F2: A > A,D: A,X2: A,S: set @ A] :
( ( has_field_derivative @ A @ F2 @ D @ ( topolo174197925503356063within @ A @ X2 @ S ) )
= ( filterlim @ A @ A
@ ^ [Y: A] : ( divide_divide @ A @ ( minus_minus @ A @ ( F2 @ Y ) @ ( F2 @ X2 ) ) @ ( minus_minus @ A @ Y @ X2 ) )
@ ( topolo7230453075368039082e_nhds @ A @ D )
@ ( topolo174197925503356063within @ A @ X2 @ S ) ) ) ) ).
% has_field_derivative_iff
thf(fact_7032_continuous__ident,axiom,
! [A: $tType] :
( ( topological_t2_space @ A )
=> ! [X2: A,S: set @ A] :
( topolo3448309680560233919inuous @ A @ A @ ( topolo174197925503356063within @ A @ X2 @ S )
@ ^ [X3: A] : X3 ) ) ).
% continuous_ident
thf(fact_7033_tendsto__within__subset,axiom,
! [B: $tType,A: $tType] :
( ( topolo4958980785337419405_space @ A )
=> ! [F2: A > B,L: filter @ B,X2: A,S: set @ A,T6: set @ A] :
( ( filterlim @ A @ B @ F2 @ L @ ( topolo174197925503356063within @ A @ X2 @ S ) )
=> ( ( ord_less_eq @ ( set @ A ) @ T6 @ S )
=> ( filterlim @ A @ B @ F2 @ L @ ( topolo174197925503356063within @ A @ X2 @ T6 ) ) ) ) ) ).
% tendsto_within_subset
thf(fact_7034_LIM__not__zero,axiom,
! [Aa: $tType,A: $tType] :
( ( ( topolo8386298272705272623_space @ A )
& ( zero @ Aa )
& ( topological_t2_space @ Aa ) )
=> ! [K: Aa,A3: A] :
( ( K
!= ( zero_zero @ Aa ) )
=> ~ ( filterlim @ A @ Aa
@ ^ [X3: A] : K
@ ( topolo7230453075368039082e_nhds @ Aa @ ( zero_zero @ Aa ) )
@ ( topolo174197925503356063within @ A @ A3 @ ( top_top @ ( set @ A ) ) ) ) ) ) ).
% LIM_not_zero
thf(fact_7035_real__LIM__sandwich__zero,axiom,
! [A: $tType] :
( ( topolo4958980785337419405_space @ A )
=> ! [F2: A > real,A3: A,G: A > real] :
( ( filterlim @ A @ real @ F2 @ ( topolo7230453075368039082e_nhds @ real @ ( zero_zero @ real ) ) @ ( topolo174197925503356063within @ A @ A3 @ ( top_top @ ( set @ A ) ) ) )
=> ( ! [X4: A] :
( ( X4 != A3 )
=> ( ord_less_eq @ real @ ( zero_zero @ real ) @ ( G @ X4 ) ) )
=> ( ! [X4: A] :
( ( X4 != A3 )
=> ( ord_less_eq @ real @ ( G @ X4 ) @ ( F2 @ X4 ) ) )
=> ( filterlim @ A @ real @ G @ ( topolo7230453075368039082e_nhds @ real @ ( zero_zero @ real ) ) @ ( topolo174197925503356063within @ A @ A3 @ ( top_top @ ( set @ A ) ) ) ) ) ) ) ) ).
% real_LIM_sandwich_zero
thf(fact_7036_IVT2,axiom,
! [A: $tType,B: $tType] :
( ( ( topolo1944317154257567458pology @ B )
& ( topolo8458572112393995274pology @ A ) )
=> ! [F2: A > B,B3: A,Y3: B,A3: A] :
( ( ord_less_eq @ B @ ( F2 @ B3 ) @ Y3 )
=> ( ( ord_less_eq @ B @ Y3 @ ( F2 @ A3 ) )
=> ( ( ord_less_eq @ A @ A3 @ B3 )
=> ( ! [X4: A] :
( ( ( ord_less_eq @ A @ A3 @ X4 )
& ( ord_less_eq @ A @ X4 @ B3 ) )
=> ( topolo3448309680560233919inuous @ A @ B @ ( topolo174197925503356063within @ A @ X4 @ ( top_top @ ( set @ A ) ) ) @ F2 ) )
=> ? [X4: A] :
( ( ord_less_eq @ A @ A3 @ X4 )
& ( ord_less_eq @ A @ X4 @ B3 )
& ( ( F2 @ X4 )
= Y3 ) ) ) ) ) ) ) ).
% IVT2
thf(fact_7037_IVT,axiom,
! [A: $tType,B: $tType] :
( ( ( topolo1944317154257567458pology @ B )
& ( topolo8458572112393995274pology @ A ) )
=> ! [F2: A > B,A3: A,Y3: B,B3: A] :
( ( ord_less_eq @ B @ ( F2 @ A3 ) @ Y3 )
=> ( ( ord_less_eq @ B @ Y3 @ ( F2 @ B3 ) )
=> ( ( ord_less_eq @ A @ A3 @ B3 )
=> ( ! [X4: A] :
( ( ( ord_less_eq @ A @ A3 @ X4 )
& ( ord_less_eq @ A @ X4 @ B3 ) )
=> ( topolo3448309680560233919inuous @ A @ B @ ( topolo174197925503356063within @ A @ X4 @ ( top_top @ ( set @ A ) ) ) @ F2 ) )
=> ? [X4: A] :
( ( ord_less_eq @ A @ A3 @ X4 )
& ( ord_less_eq @ A @ X4 @ B3 )
& ( ( F2 @ X4 )
= Y3 ) ) ) ) ) ) ) ).
% IVT
thf(fact_7038_isCont__of__real,axiom,
! [A: $tType,C: $tType] :
( ( ( topological_t2_space @ C )
& ( real_V2191834092415804123ebra_1 @ A )
& ( real_V822414075346904944vector @ A ) )
=> ! [A3: C,G: C > real] :
( ( topolo3448309680560233919inuous @ C @ real @ ( topolo174197925503356063within @ C @ A3 @ ( top_top @ ( set @ C ) ) ) @ G )
=> ( topolo3448309680560233919inuous @ C @ A @ ( topolo174197925503356063within @ C @ A3 @ ( top_top @ ( set @ C ) ) )
@ ^ [X3: C] : ( real_Vector_of_real @ A @ ( G @ X3 ) ) ) ) ) ).
% isCont_of_real
thf(fact_7039_isCont__norm,axiom,
! [B: $tType,A: $tType] :
( ( ( topological_t2_space @ A )
& ( real_V822414075346904944vector @ B ) )
=> ! [A3: A,F2: A > B] :
( ( topolo3448309680560233919inuous @ A @ B @ ( topolo174197925503356063within @ A @ A3 @ ( top_top @ ( set @ A ) ) ) @ F2 )
=> ( topolo3448309680560233919inuous @ A @ real @ ( topolo174197925503356063within @ A @ A3 @ ( top_top @ ( set @ A ) ) )
@ ^ [X3: A] : ( real_V7770717601297561774m_norm @ B @ ( F2 @ X3 ) ) ) ) ) ).
% isCont_norm
thf(fact_7040_isCont__scaleR,axiom,
! [C: $tType,D6: $tType] :
( ( ( topological_t2_space @ D6 )
& ( real_V822414075346904944vector @ C ) )
=> ! [A3: D6,F2: D6 > real,G: D6 > C] :
( ( topolo3448309680560233919inuous @ D6 @ real @ ( topolo174197925503356063within @ D6 @ A3 @ ( top_top @ ( set @ D6 ) ) ) @ F2 )
=> ( ( topolo3448309680560233919inuous @ D6 @ C @ ( topolo174197925503356063within @ D6 @ A3 @ ( top_top @ ( set @ D6 ) ) ) @ G )
=> ( topolo3448309680560233919inuous @ D6 @ C @ ( topolo174197925503356063within @ D6 @ A3 @ ( top_top @ ( set @ D6 ) ) )
@ ^ [X3: D6] : ( real_V8093663219630862766scaleR @ C @ ( F2 @ X3 ) @ ( G @ X3 ) ) ) ) ) ) ).
% isCont_scaleR
thf(fact_7041_continuous__within__compose3,axiom,
! [A: $tType,B: $tType,C: $tType] :
( ( ( topological_t2_space @ C )
& ( topolo4958980785337419405_space @ B )
& ( topological_t2_space @ A ) )
=> ! [F2: C > A,X2: C,G: A > B,S2: set @ C] :
( ( topolo3448309680560233919inuous @ A @ B @ ( topolo174197925503356063within @ A @ ( F2 @ X2 ) @ ( top_top @ ( set @ A ) ) ) @ G )
=> ( ( topolo3448309680560233919inuous @ C @ A @ ( topolo174197925503356063within @ C @ X2 @ S2 ) @ F2 )
=> ( topolo3448309680560233919inuous @ C @ B @ ( topolo174197925503356063within @ C @ X2 @ S2 )
@ ^ [X3: C] : ( G @ ( F2 @ X3 ) ) ) ) ) ) ).
% continuous_within_compose3
thf(fact_7042_isCont__tendsto__compose,axiom,
! [A: $tType,B: $tType,C: $tType] :
( ( ( topolo4958980785337419405_space @ B )
& ( topological_t2_space @ A ) )
=> ! [L: A,G: A > B,F2: C > A,F4: filter @ C] :
( ( topolo3448309680560233919inuous @ A @ B @ ( topolo174197925503356063within @ A @ L @ ( top_top @ ( set @ A ) ) ) @ G )
=> ( ( filterlim @ C @ A @ F2 @ ( topolo7230453075368039082e_nhds @ A @ L ) @ F4 )
=> ( filterlim @ C @ B
@ ^ [X3: C] : ( G @ ( F2 @ X3 ) )
@ ( topolo7230453075368039082e_nhds @ B @ ( G @ L ) )
@ F4 ) ) ) ) ).
% isCont_tendsto_compose
thf(fact_7043_LIM__const__not__eq,axiom,
! [B: $tType,A: $tType] :
( ( ( topolo8386298272705272623_space @ A )
& ( topological_t2_space @ B ) )
=> ! [K: B,L6: B,A3: A] :
( ( K != L6 )
=> ~ ( filterlim @ A @ B
@ ^ [X3: A] : K
@ ( topolo7230453075368039082e_nhds @ B @ L6 )
@ ( topolo174197925503356063within @ A @ A3 @ ( top_top @ ( set @ A ) ) ) ) ) ) ).
% LIM_const_not_eq
thf(fact_7044_tendsto__compose,axiom,
! [A: $tType,B: $tType,C: $tType] :
( ( ( topolo4958980785337419405_space @ B )
& ( topolo4958980785337419405_space @ A ) )
=> ! [G: A > B,L: A,F2: C > A,F4: filter @ C] :
( ( filterlim @ A @ B @ G @ ( topolo7230453075368039082e_nhds @ B @ ( G @ L ) ) @ ( topolo174197925503356063within @ A @ L @ ( top_top @ ( set @ A ) ) ) )
=> ( ( filterlim @ C @ A @ F2 @ ( topolo7230453075368039082e_nhds @ A @ L ) @ F4 )
=> ( filterlim @ C @ B
@ ^ [X3: C] : ( G @ ( F2 @ X3 ) )
@ ( topolo7230453075368039082e_nhds @ B @ ( G @ L ) )
@ F4 ) ) ) ) ).
% tendsto_compose
thf(fact_7045_LIM__const__eq,axiom,
! [A: $tType,B: $tType] :
( ( ( topological_t2_space @ B )
& ( topolo8386298272705272623_space @ A ) )
=> ! [K: B,L6: B,A3: A] :
( ( filterlim @ A @ B
@ ^ [X3: A] : K
@ ( topolo7230453075368039082e_nhds @ B @ L6 )
@ ( topolo174197925503356063within @ A @ A3 @ ( top_top @ ( set @ A ) ) ) )
=> ( K = L6 ) ) ) ).
% LIM_const_eq
thf(fact_7046_isCont__o2,axiom,
! [B: $tType,C: $tType,A: $tType] :
( ( ( topological_t2_space @ A )
& ( topolo4958980785337419405_space @ C )
& ( topological_t2_space @ B ) )
=> ! [A3: A,F2: A > B,G: B > C] :
( ( topolo3448309680560233919inuous @ A @ B @ ( topolo174197925503356063within @ A @ A3 @ ( top_top @ ( set @ A ) ) ) @ F2 )
=> ( ( topolo3448309680560233919inuous @ B @ C @ ( topolo174197925503356063within @ B @ ( F2 @ A3 ) @ ( top_top @ ( set @ B ) ) ) @ G )
=> ( topolo3448309680560233919inuous @ A @ C @ ( topolo174197925503356063within @ A @ A3 @ ( top_top @ ( set @ A ) ) )
@ ^ [X3: A] : ( G @ ( F2 @ X3 ) ) ) ) ) ) ).
% isCont_o2
thf(fact_7047_filterlim__at__If,axiom,
! [B: $tType,A: $tType] :
( ( topolo4958980785337419405_space @ A )
=> ! [F2: A > B,G6: filter @ B,X2: A,P: A > $o,G: A > B] :
( ( filterlim @ A @ B @ F2 @ G6 @ ( topolo174197925503356063within @ A @ X2 @ ( collect @ A @ P ) ) )
=> ( ( filterlim @ A @ B @ G @ G6
@ ( topolo174197925503356063within @ A @ X2
@ ( collect @ A
@ ^ [X3: A] :
~ ( P @ X3 ) ) ) )
=> ( filterlim @ A @ B
@ ^ [X3: A] : ( if @ B @ ( P @ X3 ) @ ( F2 @ X3 ) @ ( G @ X3 ) )
@ G6
@ ( topolo174197925503356063within @ A @ X2 @ ( top_top @ ( set @ A ) ) ) ) ) ) ) ).
% filterlim_at_If
thf(fact_7048_LIM__imp__LIM,axiom,
! [B: $tType,C: $tType,A: $tType] :
( ( ( topolo4958980785337419405_space @ A )
& ( real_V822414075346904944vector @ C )
& ( real_V822414075346904944vector @ B ) )
=> ! [F2: A > B,L: B,A3: A,G: A > C,M: C] :
( ( filterlim @ A @ B @ F2 @ ( topolo7230453075368039082e_nhds @ B @ L ) @ ( topolo174197925503356063within @ A @ A3 @ ( top_top @ ( set @ A ) ) ) )
=> ( ! [X4: A] :
( ( X4 != A3 )
=> ( ord_less_eq @ real @ ( real_V7770717601297561774m_norm @ C @ ( minus_minus @ C @ ( G @ X4 ) @ M ) ) @ ( real_V7770717601297561774m_norm @ B @ ( minus_minus @ B @ ( F2 @ X4 ) @ L ) ) ) )
=> ( filterlim @ A @ C @ G @ ( topolo7230453075368039082e_nhds @ C @ M ) @ ( topolo174197925503356063within @ A @ A3 @ ( top_top @ ( set @ A ) ) ) ) ) ) ) ).
% LIM_imp_LIM
thf(fact_7049_isCont__LIM__compose2,axiom,
! [C: $tType,B: $tType,A: $tType] :
( ( ( real_V822414075346904944vector @ A )
& ( topolo4958980785337419405_space @ B )
& ( topolo4958980785337419405_space @ C ) )
=> ! [A3: A,F2: A > B,G: B > C,L: C] :
( ( topolo3448309680560233919inuous @ A @ B @ ( topolo174197925503356063within @ A @ A3 @ ( top_top @ ( set @ A ) ) ) @ F2 )
=> ( ( filterlim @ B @ C @ G @ ( topolo7230453075368039082e_nhds @ C @ L ) @ ( topolo174197925503356063within @ B @ ( F2 @ A3 ) @ ( top_top @ ( set @ B ) ) ) )
=> ( ? [D5: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ D5 )
& ! [X4: A] :
( ( ( X4 != A3 )
& ( ord_less @ real @ ( real_V7770717601297561774m_norm @ A @ ( minus_minus @ A @ X4 @ A3 ) ) @ D5 ) )
=> ( ( F2 @ X4 )
!= ( F2 @ A3 ) ) ) )
=> ( filterlim @ A @ C
@ ^ [X3: A] : ( G @ ( F2 @ X3 ) )
@ ( topolo7230453075368039082e_nhds @ C @ L )
@ ( topolo174197925503356063within @ A @ A3 @ ( top_top @ ( set @ A ) ) ) ) ) ) ) ) ).
% isCont_LIM_compose2
thf(fact_7050_LIM__equal2,axiom,
! [B: $tType,A: $tType] :
( ( ( real_V822414075346904944vector @ A )
& ( topolo4958980785337419405_space @ B ) )
=> ! [R: real,A3: A,F2: A > B,G: A > B,L: B] :
( ( ord_less @ real @ ( zero_zero @ real ) @ R )
=> ( ! [X4: A] :
( ( X4 != A3 )
=> ( ( ord_less @ real @ ( real_V7770717601297561774m_norm @ A @ ( minus_minus @ A @ X4 @ A3 ) ) @ R )
=> ( ( F2 @ X4 )
= ( G @ X4 ) ) ) )
=> ( ( filterlim @ A @ B @ G @ ( topolo7230453075368039082e_nhds @ B @ L ) @ ( topolo174197925503356063within @ A @ A3 @ ( top_top @ ( set @ A ) ) ) )
=> ( filterlim @ A @ B @ F2 @ ( topolo7230453075368039082e_nhds @ B @ L ) @ ( topolo174197925503356063within @ A @ A3 @ ( top_top @ ( set @ A ) ) ) ) ) ) ) ) ).
% LIM_equal2
thf(fact_7051_LIM__eq,axiom,
! [B: $tType,A: $tType] :
( ( ( real_V822414075346904944vector @ A )
& ( real_V822414075346904944vector @ B ) )
=> ! [F2: A > B,L6: B,A3: A] :
( ( filterlim @ A @ B @ F2 @ ( topolo7230453075368039082e_nhds @ B @ L6 ) @ ( topolo174197925503356063within @ A @ A3 @ ( top_top @ ( set @ A ) ) ) )
= ( ! [R5: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ R5 )
=> ? [S5: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ S5 )
& ! [X3: A] :
( ( ( X3 != A3 )
& ( ord_less @ real @ ( real_V7770717601297561774m_norm @ A @ ( minus_minus @ A @ X3 @ A3 ) ) @ S5 ) )
=> ( ord_less @ real @ ( real_V7770717601297561774m_norm @ B @ ( minus_minus @ B @ ( F2 @ X3 ) @ L6 ) ) @ R5 ) ) ) ) ) ) ) ).
% LIM_eq
thf(fact_7052_LIM__I,axiom,
! [B: $tType,A: $tType] :
( ( ( real_V822414075346904944vector @ A )
& ( real_V822414075346904944vector @ B ) )
=> ! [A3: A,F2: A > B,L6: B] :
( ! [R3: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ R3 )
=> ? [S9: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ S9 )
& ! [X4: A] :
( ( ( X4 != A3 )
& ( ord_less @ real @ ( real_V7770717601297561774m_norm @ A @ ( minus_minus @ A @ X4 @ A3 ) ) @ S9 ) )
=> ( ord_less @ real @ ( real_V7770717601297561774m_norm @ B @ ( minus_minus @ B @ ( F2 @ X4 ) @ L6 ) ) @ R3 ) ) ) )
=> ( filterlim @ A @ B @ F2 @ ( topolo7230453075368039082e_nhds @ B @ L6 ) @ ( topolo174197925503356063within @ A @ A3 @ ( top_top @ ( set @ A ) ) ) ) ) ) ).
% LIM_I
thf(fact_7053_LIM__D,axiom,
! [B: $tType,A: $tType] :
( ( ( real_V822414075346904944vector @ A )
& ( real_V822414075346904944vector @ B ) )
=> ! [F2: A > B,L6: B,A3: A,R2: real] :
( ( filterlim @ A @ B @ F2 @ ( topolo7230453075368039082e_nhds @ B @ L6 ) @ ( topolo174197925503356063within @ A @ A3 @ ( top_top @ ( set @ A ) ) ) )
=> ( ( ord_less @ real @ ( zero_zero @ real ) @ R2 )
=> ? [S3: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ S3 )
& ! [X8: A] :
( ( ( X8 != A3 )
& ( ord_less @ real @ ( real_V7770717601297561774m_norm @ A @ ( minus_minus @ A @ X8 @ A3 ) ) @ S3 ) )
=> ( ord_less @ real @ ( real_V7770717601297561774m_norm @ B @ ( minus_minus @ B @ ( F2 @ X8 ) @ L6 ) ) @ R2 ) ) ) ) ) ) ).
% LIM_D
thf(fact_7054_LIM__offset,axiom,
! [B: $tType,A: $tType] :
( ( ( real_V822414075346904944vector @ A )
& ( topolo4958980785337419405_space @ B ) )
=> ! [F2: A > B,L6: B,A3: A,K: A] :
( ( filterlim @ A @ B @ F2 @ ( topolo7230453075368039082e_nhds @ B @ L6 ) @ ( topolo174197925503356063within @ A @ A3 @ ( top_top @ ( set @ A ) ) ) )
=> ( filterlim @ A @ B
@ ^ [X3: A] : ( F2 @ ( plus_plus @ A @ X3 @ K ) )
@ ( topolo7230453075368039082e_nhds @ B @ L6 )
@ ( topolo174197925503356063within @ A @ ( minus_minus @ A @ A3 @ K ) @ ( top_top @ ( set @ A ) ) ) ) ) ) ).
% LIM_offset
thf(fact_7055_isCont__iff,axiom,
! [B: $tType,A: $tType] :
( ( ( real_V822414075346904944vector @ A )
& ( topolo4958980785337419405_space @ B ) )
=> ! [X2: A,F2: A > B] :
( ( topolo3448309680560233919inuous @ A @ B @ ( topolo174197925503356063within @ A @ X2 @ ( top_top @ ( set @ A ) ) ) @ F2 )
= ( filterlim @ A @ B
@ ^ [H: A] : ( F2 @ ( plus_plus @ A @ X2 @ H ) )
@ ( topolo7230453075368039082e_nhds @ B @ ( F2 @ X2 ) )
@ ( topolo174197925503356063within @ A @ ( zero_zero @ A ) @ ( top_top @ ( set @ A ) ) ) ) ) ) ).
% isCont_iff
thf(fact_7056_LIM__isCont__iff,axiom,
! [B: $tType,A: $tType] :
( ( ( real_V822414075346904944vector @ A )
& ( topolo4958980785337419405_space @ B ) )
=> ! [F2: A > B,A3: A] :
( ( filterlim @ A @ B @ F2 @ ( topolo7230453075368039082e_nhds @ B @ ( F2 @ A3 ) ) @ ( topolo174197925503356063within @ A @ A3 @ ( top_top @ ( set @ A ) ) ) )
= ( filterlim @ A @ B
@ ^ [H: A] : ( F2 @ ( plus_plus @ A @ A3 @ H ) )
@ ( topolo7230453075368039082e_nhds @ B @ ( F2 @ A3 ) )
@ ( topolo174197925503356063within @ A @ ( zero_zero @ A ) @ ( top_top @ ( set @ A ) ) ) ) ) ) ).
% LIM_isCont_iff
thf(fact_7057_LIM__offset__zero,axiom,
! [B: $tType,A: $tType] :
( ( ( real_V822414075346904944vector @ A )
& ( topolo4958980785337419405_space @ B ) )
=> ! [F2: A > B,L6: B,A3: A] :
( ( filterlim @ A @ B @ F2 @ ( topolo7230453075368039082e_nhds @ B @ L6 ) @ ( topolo174197925503356063within @ A @ A3 @ ( top_top @ ( set @ A ) ) ) )
=> ( filterlim @ A @ B
@ ^ [H: A] : ( F2 @ ( plus_plus @ A @ A3 @ H ) )
@ ( topolo7230453075368039082e_nhds @ B @ L6 )
@ ( topolo174197925503356063within @ A @ ( zero_zero @ A ) @ ( top_top @ ( set @ A ) ) ) ) ) ) ).
% LIM_offset_zero
thf(fact_7058_LIM__offset__zero__cancel,axiom,
! [B: $tType,A: $tType] :
( ( ( real_V822414075346904944vector @ A )
& ( topolo4958980785337419405_space @ B ) )
=> ! [F2: A > B,A3: A,L6: B] :
( ( filterlim @ A @ B
@ ^ [H: A] : ( F2 @ ( plus_plus @ A @ A3 @ H ) )
@ ( topolo7230453075368039082e_nhds @ B @ L6 )
@ ( topolo174197925503356063within @ A @ ( zero_zero @ A ) @ ( top_top @ ( set @ A ) ) ) )
=> ( filterlim @ A @ B @ F2 @ ( topolo7230453075368039082e_nhds @ B @ L6 ) @ ( topolo174197925503356063within @ A @ A3 @ ( top_top @ ( set @ A ) ) ) ) ) ) ).
% LIM_offset_zero_cancel
thf(fact_7059_isCont__Pair,axiom,
! [C: $tType,B: $tType,A: $tType] :
( ( ( topological_t2_space @ A )
& ( topolo4958980785337419405_space @ B )
& ( topolo4958980785337419405_space @ C ) )
=> ! [A3: A,F2: A > B,G: A > C] :
( ( topolo3448309680560233919inuous @ A @ B @ ( topolo174197925503356063within @ A @ A3 @ ( top_top @ ( set @ A ) ) ) @ F2 )
=> ( ( topolo3448309680560233919inuous @ A @ C @ ( topolo174197925503356063within @ A @ A3 @ ( top_top @ ( set @ A ) ) ) @ G )
=> ( topolo3448309680560233919inuous @ A @ ( product_prod @ B @ C ) @ ( topolo174197925503356063within @ A @ A3 @ ( top_top @ ( set @ A ) ) )
@ ^ [X3: A] : ( product_Pair @ B @ C @ ( F2 @ X3 ) @ ( G @ X3 ) ) ) ) ) ) ).
% isCont_Pair
thf(fact_7060_continuous__within__compose2,axiom,
! [B: $tType,C: $tType,A: $tType] :
( ( ( topological_t2_space @ A )
& ( topolo4958980785337419405_space @ C )
& ( topological_t2_space @ B ) )
=> ! [X2: A,S2: set @ A,F2: A > B,G: B > C] :
( ( topolo3448309680560233919inuous @ A @ B @ ( topolo174197925503356063within @ A @ X2 @ S2 ) @ F2 )
=> ( ( topolo3448309680560233919inuous @ B @ C @ ( topolo174197925503356063within @ B @ ( F2 @ X2 ) @ ( image @ A @ B @ F2 @ S2 ) ) @ G )
=> ( topolo3448309680560233919inuous @ A @ C @ ( topolo174197925503356063within @ A @ X2 @ S2 )
@ ^ [X3: A] : ( G @ ( F2 @ X3 ) ) ) ) ) ) ).
% continuous_within_compose2
thf(fact_7061_tendsto__mono,axiom,
! [A: $tType,B: $tType] :
( ( topolo4958980785337419405_space @ A )
=> ! [F4: filter @ B,F6: filter @ B,F2: B > A,L: A] :
( ( ord_less_eq @ ( filter @ B ) @ F4 @ F6 )
=> ( ( filterlim @ B @ A @ F2 @ ( topolo7230453075368039082e_nhds @ A @ L ) @ F6 )
=> ( filterlim @ B @ A @ F2 @ ( topolo7230453075368039082e_nhds @ A @ L ) @ F4 ) ) ) ) ).
% tendsto_mono
thf(fact_7062_atLeastSucLessThan__greaterThanLessThan,axiom,
! [L: nat,U: nat] :
( ( set_or7035219750837199246ssThan @ nat @ ( suc @ L ) @ U )
= ( set_or5935395276787703475ssThan @ nat @ L @ U ) ) ).
% atLeastSucLessThan_greaterThanLessThan
thf(fact_7063_tendsto__min,axiom,
! [A: $tType,B: $tType] :
( ( topolo1944317154257567458pology @ A )
=> ! [X: B > A,X2: A,Net: filter @ B,Y8: B > A,Y3: A] :
( ( filterlim @ B @ A @ X @ ( topolo7230453075368039082e_nhds @ A @ X2 ) @ Net )
=> ( ( filterlim @ B @ A @ Y8 @ ( topolo7230453075368039082e_nhds @ A @ Y3 ) @ Net )
=> ( filterlim @ B @ A
@ ^ [X3: B] : ( ord_min @ A @ ( X @ X3 ) @ ( Y8 @ X3 ) )
@ ( topolo7230453075368039082e_nhds @ A @ ( ord_min @ A @ X2 @ Y3 ) )
@ Net ) ) ) ) ).
% tendsto_min
thf(fact_7064_continuous__min,axiom,
! [B: $tType,A: $tType] :
( ( ( topological_t2_space @ A )
& ( topolo1944317154257567458pology @ B ) )
=> ! [F4: filter @ A,F2: A > B,G: A > B] :
( ( topolo3448309680560233919inuous @ A @ B @ F4 @ F2 )
=> ( ( topolo3448309680560233919inuous @ A @ B @ F4 @ G )
=> ( topolo3448309680560233919inuous @ A @ B @ F4
@ ^ [X3: A] : ( ord_min @ B @ ( F2 @ X3 ) @ ( G @ X3 ) ) ) ) ) ) ).
% continuous_min
thf(fact_7065_tendsto__Pair,axiom,
! [C: $tType,B: $tType,A: $tType] :
( ( ( topolo4958980785337419405_space @ B )
& ( topolo4958980785337419405_space @ C ) )
=> ! [F2: A > B,A3: B,F4: filter @ A,G: A > C,B3: C] :
( ( filterlim @ A @ B @ F2 @ ( topolo7230453075368039082e_nhds @ B @ A3 ) @ F4 )
=> ( ( filterlim @ A @ C @ G @ ( topolo7230453075368039082e_nhds @ C @ B3 ) @ F4 )
=> ( filterlim @ A @ ( product_prod @ B @ C )
@ ^ [X3: A] : ( product_Pair @ B @ C @ ( F2 @ X3 ) @ ( G @ X3 ) )
@ ( topolo7230453075368039082e_nhds @ ( product_prod @ B @ C ) @ ( product_Pair @ B @ C @ A3 @ B3 ) )
@ F4 ) ) ) ) ).
% tendsto_Pair
thf(fact_7066_continuous__Pair,axiom,
! [C: $tType,B: $tType,A: $tType] :
( ( ( topological_t2_space @ A )
& ( topolo4958980785337419405_space @ B )
& ( topolo4958980785337419405_space @ C ) )
=> ! [F4: filter @ A,F2: A > B,G: A > C] :
( ( topolo3448309680560233919inuous @ A @ B @ F4 @ F2 )
=> ( ( topolo3448309680560233919inuous @ A @ C @ F4 @ G )
=> ( topolo3448309680560233919inuous @ A @ ( product_prod @ B @ C ) @ F4
@ ^ [X3: A] : ( product_Pair @ B @ C @ ( F2 @ X3 ) @ ( G @ X3 ) ) ) ) ) ) ).
% continuous_Pair
thf(fact_7067_tendsto__divide__zero,axiom,
! [A: $tType,B: $tType] :
( ( real_V3459762299906320749_field @ A )
=> ! [F2: B > A,F4: filter @ B,C3: A] :
( ( filterlim @ B @ A @ F2 @ ( topolo7230453075368039082e_nhds @ A @ ( zero_zero @ A ) ) @ F4 )
=> ( filterlim @ B @ A
@ ^ [X3: B] : ( divide_divide @ A @ ( F2 @ X3 ) @ C3 )
@ ( topolo7230453075368039082e_nhds @ A @ ( zero_zero @ A ) )
@ F4 ) ) ) ).
% tendsto_divide_zero
thf(fact_7068_tendsto__divide,axiom,
! [A: $tType,B: $tType] :
( ( real_V3459762299906320749_field @ A )
=> ! [F2: B > A,A3: A,F4: filter @ B,G: B > A,B3: A] :
( ( filterlim @ B @ A @ F2 @ ( topolo7230453075368039082e_nhds @ A @ A3 ) @ F4 )
=> ( ( filterlim @ B @ A @ G @ ( topolo7230453075368039082e_nhds @ A @ B3 ) @ F4 )
=> ( ( B3
!= ( zero_zero @ A ) )
=> ( filterlim @ B @ A
@ ^ [X3: B] : ( divide_divide @ A @ ( F2 @ X3 ) @ ( G @ X3 ) )
@ ( topolo7230453075368039082e_nhds @ A @ ( divide_divide @ A @ A3 @ B3 ) )
@ F4 ) ) ) ) ) ).
% tendsto_divide
thf(fact_7069_tendsto__mult__zero,axiom,
! [A: $tType,D6: $tType] :
( ( real_V4412858255891104859lgebra @ A )
=> ! [F2: D6 > A,F4: filter @ D6,G: D6 > A] :
( ( filterlim @ D6 @ A @ F2 @ ( topolo7230453075368039082e_nhds @ A @ ( zero_zero @ A ) ) @ F4 )
=> ( ( filterlim @ D6 @ A @ G @ ( topolo7230453075368039082e_nhds @ A @ ( zero_zero @ A ) ) @ F4 )
=> ( filterlim @ D6 @ A
@ ^ [X3: D6] : ( times_times @ A @ ( F2 @ X3 ) @ ( G @ X3 ) )
@ ( topolo7230453075368039082e_nhds @ A @ ( zero_zero @ A ) )
@ F4 ) ) ) ) ).
% tendsto_mult_zero
thf(fact_7070_tendsto__mult__left__zero,axiom,
! [A: $tType,D6: $tType] :
( ( real_V4412858255891104859lgebra @ A )
=> ! [F2: D6 > A,F4: filter @ D6,C3: A] :
( ( filterlim @ D6 @ A @ F2 @ ( topolo7230453075368039082e_nhds @ A @ ( zero_zero @ A ) ) @ F4 )
=> ( filterlim @ D6 @ A
@ ^ [X3: D6] : ( times_times @ A @ ( F2 @ X3 ) @ C3 )
@ ( topolo7230453075368039082e_nhds @ A @ ( zero_zero @ A ) )
@ F4 ) ) ) ).
% tendsto_mult_left_zero
thf(fact_7071_tendsto__mult__right__zero,axiom,
! [A: $tType,D6: $tType] :
( ( real_V4412858255891104859lgebra @ A )
=> ! [F2: D6 > A,F4: filter @ D6,C3: A] :
( ( filterlim @ D6 @ A @ F2 @ ( topolo7230453075368039082e_nhds @ A @ ( zero_zero @ A ) ) @ F4 )
=> ( filterlim @ D6 @ A
@ ^ [X3: D6] : ( times_times @ A @ C3 @ ( F2 @ X3 ) )
@ ( topolo7230453075368039082e_nhds @ A @ ( zero_zero @ A ) )
@ F4 ) ) ) ).
% tendsto_mult_right_zero
thf(fact_7072_tendsto__mult__one,axiom,
! [B: $tType,D6: $tType] :
( ( topolo1898628316856586783d_mult @ B )
=> ! [F2: D6 > B,F4: filter @ D6,G: D6 > B] :
( ( filterlim @ D6 @ B @ F2 @ ( topolo7230453075368039082e_nhds @ B @ ( one_one @ B ) ) @ F4 )
=> ( ( filterlim @ D6 @ B @ G @ ( topolo7230453075368039082e_nhds @ B @ ( one_one @ B ) ) @ F4 )
=> ( filterlim @ D6 @ B
@ ^ [X3: D6] : ( times_times @ B @ ( F2 @ X3 ) @ ( G @ X3 ) )
@ ( topolo7230453075368039082e_nhds @ B @ ( one_one @ B ) )
@ F4 ) ) ) ) ).
% tendsto_mult_one
thf(fact_7073_tendsto__mult,axiom,
! [A: $tType,B: $tType] :
( ( topolo4211221413907600880p_mult @ A )
=> ! [F2: B > A,A3: A,F4: filter @ B,G: B > A,B3: A] :
( ( filterlim @ B @ A @ F2 @ ( topolo7230453075368039082e_nhds @ A @ A3 ) @ F4 )
=> ( ( filterlim @ B @ A @ G @ ( topolo7230453075368039082e_nhds @ A @ B3 ) @ F4 )
=> ( filterlim @ B @ A
@ ^ [X3: B] : ( times_times @ A @ ( F2 @ X3 ) @ ( G @ X3 ) )
@ ( topolo7230453075368039082e_nhds @ A @ ( times_times @ A @ A3 @ B3 ) )
@ F4 ) ) ) ) ).
% tendsto_mult
thf(fact_7074_tendsto__mult__left,axiom,
! [A: $tType,B: $tType] :
( ( topolo4211221413907600880p_mult @ A )
=> ! [F2: B > A,L: A,F4: filter @ B,C3: A] :
( ( filterlim @ B @ A @ F2 @ ( topolo7230453075368039082e_nhds @ A @ L ) @ F4 )
=> ( filterlim @ B @ A
@ ^ [X3: B] : ( times_times @ A @ C3 @ ( F2 @ X3 ) )
@ ( topolo7230453075368039082e_nhds @ A @ ( times_times @ A @ C3 @ L ) )
@ F4 ) ) ) ).
% tendsto_mult_left
thf(fact_7075_tendsto__mult__right,axiom,
! [A: $tType,B: $tType] :
( ( topolo4211221413907600880p_mult @ A )
=> ! [F2: B > A,L: A,F4: filter @ B,C3: A] :
( ( filterlim @ B @ A @ F2 @ ( topolo7230453075368039082e_nhds @ A @ L ) @ F4 )
=> ( filterlim @ B @ A
@ ^ [X3: B] : ( times_times @ A @ ( F2 @ X3 ) @ C3 )
@ ( topolo7230453075368039082e_nhds @ A @ ( times_times @ A @ L @ C3 ) )
@ F4 ) ) ) ).
% tendsto_mult_right
thf(fact_7076_continuous__mult,axiom,
! [A: $tType,D6: $tType] :
( ( ( topological_t2_space @ D6 )
& ( real_V4412858255891104859lgebra @ A ) )
=> ! [F4: filter @ D6,F2: D6 > A,G: D6 > A] :
( ( topolo3448309680560233919inuous @ D6 @ A @ F4 @ F2 )
=> ( ( topolo3448309680560233919inuous @ D6 @ A @ F4 @ G )
=> ( topolo3448309680560233919inuous @ D6 @ A @ F4
@ ^ [X3: D6] : ( times_times @ A @ ( F2 @ X3 ) @ ( G @ X3 ) ) ) ) ) ) ).
% continuous_mult
thf(fact_7077_continuous__mult_H,axiom,
! [B: $tType,D6: $tType] :
( ( ( topological_t2_space @ D6 )
& ( topolo4211221413907600880p_mult @ B ) )
=> ! [F4: filter @ D6,F2: D6 > B,G: D6 > B] :
( ( topolo3448309680560233919inuous @ D6 @ B @ F4 @ F2 )
=> ( ( topolo3448309680560233919inuous @ D6 @ B @ F4 @ G )
=> ( topolo3448309680560233919inuous @ D6 @ B @ F4
@ ^ [X3: D6] : ( times_times @ B @ ( F2 @ X3 ) @ ( G @ X3 ) ) ) ) ) ) ).
% continuous_mult'
thf(fact_7078_continuous__mult__left,axiom,
! [A: $tType,B: $tType] :
( ( ( topological_t2_space @ B )
& ( real_V4412858255891104859lgebra @ A ) )
=> ! [F4: filter @ B,F2: B > A,C3: A] :
( ( topolo3448309680560233919inuous @ B @ A @ F4 @ F2 )
=> ( topolo3448309680560233919inuous @ B @ A @ F4
@ ^ [X3: B] : ( times_times @ A @ C3 @ ( F2 @ X3 ) ) ) ) ) ).
% continuous_mult_left
thf(fact_7079_continuous__mult__right,axiom,
! [A: $tType,B: $tType] :
( ( ( topological_t2_space @ B )
& ( real_V4412858255891104859lgebra @ A ) )
=> ! [F4: filter @ B,F2: B > A,C3: A] :
( ( topolo3448309680560233919inuous @ B @ A @ F4 @ F2 )
=> ( topolo3448309680560233919inuous @ B @ A @ F4
@ ^ [X3: B] : ( times_times @ A @ ( F2 @ X3 ) @ C3 ) ) ) ) ).
% continuous_mult_right
thf(fact_7080_tendsto__rabs,axiom,
! [A: $tType,F2: A > real,L: real,F4: filter @ A] :
( ( filterlim @ A @ real @ F2 @ ( topolo7230453075368039082e_nhds @ real @ L ) @ F4 )
=> ( filterlim @ A @ real
@ ^ [X3: A] : ( abs_abs @ real @ ( F2 @ X3 ) )
@ ( topolo7230453075368039082e_nhds @ real @ ( abs_abs @ real @ L ) )
@ F4 ) ) ).
% tendsto_rabs
thf(fact_7081_tendsto__rabs__zero__cancel,axiom,
! [A: $tType,F2: A > real,F4: filter @ A] :
( ( filterlim @ A @ real
@ ^ [X3: A] : ( abs_abs @ real @ ( F2 @ X3 ) )
@ ( topolo7230453075368039082e_nhds @ real @ ( zero_zero @ real ) )
@ F4 )
=> ( filterlim @ A @ real @ F2 @ ( topolo7230453075368039082e_nhds @ real @ ( zero_zero @ real ) ) @ F4 ) ) ).
% tendsto_rabs_zero_cancel
thf(fact_7082_tendsto__rabs__zero__iff,axiom,
! [A: $tType,F2: A > real,F4: filter @ A] :
( ( filterlim @ A @ real
@ ^ [X3: A] : ( abs_abs @ real @ ( F2 @ X3 ) )
@ ( topolo7230453075368039082e_nhds @ real @ ( zero_zero @ real ) )
@ F4 )
= ( filterlim @ A @ real @ F2 @ ( topolo7230453075368039082e_nhds @ real @ ( zero_zero @ real ) ) @ F4 ) ) ).
% tendsto_rabs_zero_iff
thf(fact_7083_tendsto__rabs__zero,axiom,
! [A: $tType,F2: A > real,F4: filter @ A] :
( ( filterlim @ A @ real @ F2 @ ( topolo7230453075368039082e_nhds @ real @ ( zero_zero @ real ) ) @ F4 )
=> ( filterlim @ A @ real
@ ^ [X3: A] : ( abs_abs @ real @ ( F2 @ X3 ) )
@ ( topolo7230453075368039082e_nhds @ real @ ( zero_zero @ real ) )
@ F4 ) ) ).
% tendsto_rabs_zero
thf(fact_7084_tendsto__artanh,axiom,
! [A: $tType,F2: A > real,A3: real,F4: filter @ A] :
( ( filterlim @ A @ real @ F2 @ ( topolo7230453075368039082e_nhds @ real @ A3 ) @ F4 )
=> ( ( ord_less @ real @ ( uminus_uminus @ real @ ( one_one @ real ) ) @ A3 )
=> ( ( ord_less @ real @ A3 @ ( one_one @ real ) )
=> ( filterlim @ A @ real
@ ^ [X3: A] : ( artanh @ real @ ( F2 @ X3 ) )
@ ( topolo7230453075368039082e_nhds @ real @ ( artanh @ real @ A3 ) )
@ F4 ) ) ) ) ).
% tendsto_artanh
thf(fact_7085_tendsto__arcosh,axiom,
! [B: $tType,F2: B > real,A3: real,F4: filter @ B] :
( ( filterlim @ B @ real @ F2 @ ( topolo7230453075368039082e_nhds @ real @ A3 ) @ F4 )
=> ( ( ord_less @ real @ ( one_one @ real ) @ A3 )
=> ( filterlim @ B @ real
@ ^ [X3: B] : ( arcosh @ real @ ( F2 @ X3 ) )
@ ( topolo7230453075368039082e_nhds @ real @ ( arcosh @ real @ A3 ) )
@ F4 ) ) ) ).
% tendsto_arcosh
thf(fact_7086_tendsto__log,axiom,
! [A: $tType,F2: A > real,A3: real,F4: filter @ A,G: A > real,B3: real] :
( ( filterlim @ A @ real @ F2 @ ( topolo7230453075368039082e_nhds @ real @ A3 ) @ F4 )
=> ( ( filterlim @ A @ real @ G @ ( topolo7230453075368039082e_nhds @ real @ B3 ) @ F4 )
=> ( ( ord_less @ real @ ( zero_zero @ real ) @ A3 )
=> ( ( A3
!= ( one_one @ real ) )
=> ( ( ord_less @ real @ ( zero_zero @ real ) @ B3 )
=> ( filterlim @ A @ real
@ ^ [X3: A] : ( log @ ( F2 @ X3 ) @ ( G @ X3 ) )
@ ( topolo7230453075368039082e_nhds @ real @ ( log @ A3 @ B3 ) )
@ F4 ) ) ) ) ) ) ).
% tendsto_log
thf(fact_7087_tendsto__one__prod_H,axiom,
! [C: $tType,B: $tType,A: $tType] :
( ( topolo4987421752381908075d_mult @ C )
=> ! [I6: set @ B,F2: A > B > C,F4: filter @ A] :
( ! [I2: B] :
( ( member @ B @ I2 @ I6 )
=> ( filterlim @ A @ C
@ ^ [X3: A] : ( F2 @ X3 @ I2 )
@ ( topolo7230453075368039082e_nhds @ C @ ( one_one @ C ) )
@ F4 ) )
=> ( filterlim @ A @ C
@ ^ [I4: A] : ( groups7121269368397514597t_prod @ B @ C @ ( F2 @ I4 ) @ I6 )
@ ( topolo7230453075368039082e_nhds @ C @ ( one_one @ C ) )
@ F4 ) ) ) ).
% tendsto_one_prod'
thf(fact_7088_tendsto__minus__cancel__left,axiom,
! [B: $tType,A: $tType] :
( ( topolo1633459387980952147up_add @ B )
=> ! [F2: A > B,Y3: B,F4: filter @ A] :
( ( filterlim @ A @ B @ F2 @ ( topolo7230453075368039082e_nhds @ B @ ( uminus_uminus @ B @ Y3 ) ) @ F4 )
= ( filterlim @ A @ B
@ ^ [X3: A] : ( uminus_uminus @ B @ ( F2 @ X3 ) )
@ ( topolo7230453075368039082e_nhds @ B @ Y3 )
@ F4 ) ) ) ).
% tendsto_minus_cancel_left
thf(fact_7089_tendsto__minus__cancel,axiom,
! [A: $tType,B: $tType] :
( ( topolo1633459387980952147up_add @ A )
=> ! [F2: B > A,A3: A,F4: filter @ B] :
( ( filterlim @ B @ A
@ ^ [X3: B] : ( uminus_uminus @ A @ ( F2 @ X3 ) )
@ ( topolo7230453075368039082e_nhds @ A @ ( uminus_uminus @ A @ A3 ) )
@ F4 )
=> ( filterlim @ B @ A @ F2 @ ( topolo7230453075368039082e_nhds @ A @ A3 ) @ F4 ) ) ) ).
% tendsto_minus_cancel
thf(fact_7090_continuous__minus,axiom,
! [B: $tType,A: $tType] :
( ( ( topological_t2_space @ A )
& ( topolo1633459387980952147up_add @ B ) )
=> ! [F4: filter @ A,F2: A > B] :
( ( topolo3448309680560233919inuous @ A @ B @ F4 @ F2 )
=> ( topolo3448309680560233919inuous @ A @ B @ F4
@ ^ [X3: A] : ( uminus_uminus @ B @ ( F2 @ X3 ) ) ) ) ) ).
% continuous_minus
thf(fact_7091_tendsto__minus,axiom,
! [A: $tType,B: $tType] :
( ( topolo1633459387980952147up_add @ A )
=> ! [F2: B > A,A3: A,F4: filter @ B] :
( ( filterlim @ B @ A @ F2 @ ( topolo7230453075368039082e_nhds @ A @ A3 ) @ F4 )
=> ( filterlim @ B @ A
@ ^ [X3: B] : ( uminus_uminus @ A @ ( F2 @ X3 ) )
@ ( topolo7230453075368039082e_nhds @ A @ ( uminus_uminus @ A @ A3 ) )
@ F4 ) ) ) ).
% tendsto_minus
thf(fact_7092_tendsto__null__sum,axiom,
! [C: $tType,B: $tType,A: $tType] :
( ( topolo5987344860129210374id_add @ C )
=> ! [I6: set @ B,F2: A > B > C,F4: filter @ A] :
( ! [I2: B] :
( ( member @ B @ I2 @ I6 )
=> ( filterlim @ A @ C
@ ^ [X3: A] : ( F2 @ X3 @ I2 )
@ ( topolo7230453075368039082e_nhds @ C @ ( zero_zero @ C ) )
@ F4 ) )
=> ( filterlim @ A @ C
@ ^ [I4: A] : ( groups7311177749621191930dd_sum @ B @ C @ ( F2 @ I4 ) @ I6 )
@ ( topolo7230453075368039082e_nhds @ C @ ( zero_zero @ C ) )
@ F4 ) ) ) ).
% tendsto_null_sum
thf(fact_7093_tendsto__tanh,axiom,
! [A: $tType,C: $tType] :
( ( ( real_Vector_banach @ A )
& ( real_V3459762299906320749_field @ A ) )
=> ! [F2: C > A,A3: A,F4: filter @ C] :
( ( filterlim @ C @ A @ F2 @ ( topolo7230453075368039082e_nhds @ A @ A3 ) @ F4 )
=> ( ( ( cosh @ A @ A3 )
!= ( zero_zero @ A ) )
=> ( filterlim @ C @ A
@ ^ [X3: C] : ( tanh @ A @ ( F2 @ X3 ) )
@ ( topolo7230453075368039082e_nhds @ A @ ( tanh @ A @ A3 ) )
@ F4 ) ) ) ) ).
% tendsto_tanh
thf(fact_7094_tendsto__ln,axiom,
! [A: $tType,F2: A > real,A3: real,F4: filter @ A] :
( ( filterlim @ A @ real @ F2 @ ( topolo7230453075368039082e_nhds @ real @ A3 ) @ F4 )
=> ( ( A3
!= ( zero_zero @ real ) )
=> ( filterlim @ A @ real
@ ^ [X3: A] : ( ln_ln @ real @ ( F2 @ X3 ) )
@ ( topolo7230453075368039082e_nhds @ real @ ( ln_ln @ real @ A3 ) )
@ F4 ) ) ) ).
% tendsto_ln
thf(fact_7095_tendsto__powr,axiom,
! [A: $tType,F2: A > real,A3: real,F4: filter @ A,G: A > real,B3: real] :
( ( filterlim @ A @ real @ F2 @ ( topolo7230453075368039082e_nhds @ real @ A3 ) @ F4 )
=> ( ( filterlim @ A @ real @ G @ ( topolo7230453075368039082e_nhds @ real @ B3 ) @ F4 )
=> ( ( A3
!= ( zero_zero @ real ) )
=> ( filterlim @ A @ real
@ ^ [X3: A] : ( powr @ real @ ( F2 @ X3 ) @ ( G @ X3 ) )
@ ( topolo7230453075368039082e_nhds @ real @ ( powr @ real @ A3 @ B3 ) )
@ F4 ) ) ) ) ).
% tendsto_powr
thf(fact_7096_tendsto__norm__zero,axiom,
! [B: $tType,A: $tType] :
( ( real_V822414075346904944vector @ B )
=> ! [F2: A > B,F4: filter @ A] :
( ( filterlim @ A @ B @ F2 @ ( topolo7230453075368039082e_nhds @ B @ ( zero_zero @ B ) ) @ F4 )
=> ( filterlim @ A @ real
@ ^ [X3: A] : ( real_V7770717601297561774m_norm @ B @ ( F2 @ X3 ) )
@ ( topolo7230453075368039082e_nhds @ real @ ( zero_zero @ real ) )
@ F4 ) ) ) ).
% tendsto_norm_zero
thf(fact_7097_tendsto__norm__zero__iff,axiom,
! [B: $tType,A: $tType] :
( ( real_V822414075346904944vector @ B )
=> ! [F2: A > B,F4: filter @ A] :
( ( filterlim @ A @ real
@ ^ [X3: A] : ( real_V7770717601297561774m_norm @ B @ ( F2 @ X3 ) )
@ ( topolo7230453075368039082e_nhds @ real @ ( zero_zero @ real ) )
@ F4 )
= ( filterlim @ A @ B @ F2 @ ( topolo7230453075368039082e_nhds @ B @ ( zero_zero @ B ) ) @ F4 ) ) ) ).
% tendsto_norm_zero_iff
thf(fact_7098_tendsto__norm__zero__cancel,axiom,
! [B: $tType,A: $tType] :
( ( real_V822414075346904944vector @ B )
=> ! [F2: A > B,F4: filter @ A] :
( ( filterlim @ A @ real
@ ^ [X3: A] : ( real_V7770717601297561774m_norm @ B @ ( F2 @ X3 ) )
@ ( topolo7230453075368039082e_nhds @ real @ ( zero_zero @ real ) )
@ F4 )
=> ( filterlim @ A @ B @ F2 @ ( topolo7230453075368039082e_nhds @ B @ ( zero_zero @ B ) ) @ F4 ) ) ) ).
% tendsto_norm_zero_cancel
thf(fact_7099_tendsto__tan,axiom,
! [A: $tType] :
( ( ( real_Vector_banach @ A )
& ( real_V3459762299906320749_field @ A ) )
=> ! [F2: A > A,A3: A,F4: filter @ A] :
( ( filterlim @ A @ A @ F2 @ ( topolo7230453075368039082e_nhds @ A @ A3 ) @ F4 )
=> ( ( ( cos @ A @ A3 )
!= ( zero_zero @ A ) )
=> ( filterlim @ A @ A
@ ^ [X3: A] : ( tan @ A @ ( F2 @ X3 ) )
@ ( topolo7230453075368039082e_nhds @ A @ ( tan @ A @ A3 ) )
@ F4 ) ) ) ) ).
% tendsto_tan
thf(fact_7100_tendsto__cot,axiom,
! [A: $tType] :
( ( ( real_Vector_banach @ A )
& ( real_V3459762299906320749_field @ A ) )
=> ! [F2: A > A,A3: A,F4: filter @ A] :
( ( filterlim @ A @ A @ F2 @ ( topolo7230453075368039082e_nhds @ A @ A3 ) @ F4 )
=> ( ( ( sin @ A @ A3 )
!= ( zero_zero @ A ) )
=> ( filterlim @ A @ A
@ ^ [X3: A] : ( cot @ A @ ( F2 @ X3 ) )
@ ( topolo7230453075368039082e_nhds @ A @ ( cot @ A @ A3 ) )
@ F4 ) ) ) ) ).
% tendsto_cot
thf(fact_7101_tendsto__sgn,axiom,
! [A: $tType,B: $tType] :
( ( real_V822414075346904944vector @ A )
=> ! [F2: B > A,L: A,F4: filter @ B] :
( ( filterlim @ B @ A @ F2 @ ( topolo7230453075368039082e_nhds @ A @ L ) @ F4 )
=> ( ( L
!= ( zero_zero @ A ) )
=> ( filterlim @ B @ A
@ ^ [X3: B] : ( sgn_sgn @ A @ ( F2 @ X3 ) )
@ ( topolo7230453075368039082e_nhds @ A @ ( sgn_sgn @ A @ L ) )
@ F4 ) ) ) ) ).
% tendsto_sgn
thf(fact_7102_tendsto__const__iff,axiom,
! [B: $tType,A: $tType] :
( ( topological_t2_space @ A )
=> ! [F4: filter @ B,A3: A,B3: A] :
( ( F4
!= ( bot_bot @ ( filter @ B ) ) )
=> ( ( filterlim @ B @ A
@ ^ [X3: B] : A3
@ ( topolo7230453075368039082e_nhds @ A @ B3 )
@ F4 )
= ( A3 = B3 ) ) ) ) ).
% tendsto_const_iff
thf(fact_7103_tendsto__real__root,axiom,
! [A: $tType,F2: A > real,X2: real,F4: filter @ A,N: nat] :
( ( filterlim @ A @ real @ F2 @ ( topolo7230453075368039082e_nhds @ real @ X2 ) @ F4 )
=> ( filterlim @ A @ real
@ ^ [X3: A] : ( root @ N @ ( F2 @ X3 ) )
@ ( topolo7230453075368039082e_nhds @ real @ ( root @ N @ X2 ) )
@ F4 ) ) ).
% tendsto_real_root
thf(fact_7104_tendsto__exp,axiom,
! [A: $tType,C: $tType] :
( ( ( real_Vector_banach @ A )
& ( real_V3459762299906320749_field @ A ) )
=> ! [F2: C > A,A3: A,F4: filter @ C] :
( ( filterlim @ C @ A @ F2 @ ( topolo7230453075368039082e_nhds @ A @ A3 ) @ F4 )
=> ( filterlim @ C @ A
@ ^ [X3: C] : ( exp @ A @ ( F2 @ X3 ) )
@ ( topolo7230453075368039082e_nhds @ A @ ( exp @ A @ A3 ) )
@ F4 ) ) ) ).
% tendsto_exp
thf(fact_7105_continuous__exp,axiom,
! [A: $tType,C: $tType] :
( ( ( topological_t2_space @ C )
& ( real_Vector_banach @ A )
& ( real_V3459762299906320749_field @ A ) )
=> ! [F4: filter @ C,F2: C > A] :
( ( topolo3448309680560233919inuous @ C @ A @ F4 @ F2 )
=> ( topolo3448309680560233919inuous @ C @ A @ F4
@ ^ [X3: C] : ( exp @ A @ ( F2 @ X3 ) ) ) ) ) ).
% continuous_exp
thf(fact_7106_continuous__const,axiom,
! [B: $tType,A: $tType] :
( ( ( topological_t2_space @ A )
& ( topolo4958980785337419405_space @ B ) )
=> ! [F4: filter @ A,C3: B] :
( topolo3448309680560233919inuous @ A @ B @ F4
@ ^ [X3: A] : C3 ) ) ).
% continuous_const
thf(fact_7107_tendsto__Complex,axiom,
! [A: $tType,F2: A > real,A3: real,F4: filter @ A,G: A > real,B3: real] :
( ( filterlim @ A @ real @ F2 @ ( topolo7230453075368039082e_nhds @ real @ A3 ) @ F4 )
=> ( ( filterlim @ A @ real @ G @ ( topolo7230453075368039082e_nhds @ real @ B3 ) @ F4 )
=> ( filterlim @ A @ complex
@ ^ [X3: A] : ( complex2 @ ( F2 @ X3 ) @ ( G @ X3 ) )
@ ( topolo7230453075368039082e_nhds @ complex @ ( complex2 @ A3 @ B3 ) )
@ F4 ) ) ) ).
% tendsto_Complex
thf(fact_7108_tendsto__arctan,axiom,
! [A: $tType,F2: A > real,X2: real,F4: filter @ A] :
( ( filterlim @ A @ real @ F2 @ ( topolo7230453075368039082e_nhds @ real @ X2 ) @ F4 )
=> ( filterlim @ A @ real
@ ^ [X3: A] : ( arctan @ ( F2 @ X3 ) )
@ ( topolo7230453075368039082e_nhds @ real @ ( arctan @ X2 ) )
@ F4 ) ) ).
% tendsto_arctan
thf(fact_7109_continuous__sin,axiom,
! [B: $tType,A: $tType] :
( ( ( topological_t2_space @ A )
& ( real_Vector_banach @ B )
& ( real_V3459762299906320749_field @ B ) )
=> ! [F4: filter @ A,F2: A > B] :
( ( topolo3448309680560233919inuous @ A @ B @ F4 @ F2 )
=> ( topolo3448309680560233919inuous @ A @ B @ F4
@ ^ [X3: A] : ( sin @ B @ ( F2 @ X3 ) ) ) ) ) ).
% continuous_sin
thf(fact_7110_continuous__cos,axiom,
! [B: $tType,A: $tType] :
( ( ( topological_t2_space @ A )
& ( real_Vector_banach @ B )
& ( real_V3459762299906320749_field @ B ) )
=> ! [F4: filter @ A,F2: A > B] :
( ( topolo3448309680560233919inuous @ A @ B @ F4 @ F2 )
=> ( topolo3448309680560233919inuous @ A @ B @ F4
@ ^ [X3: A] : ( cos @ B @ ( F2 @ X3 ) ) ) ) ) ).
% continuous_cos
thf(fact_7111_tendsto__sin,axiom,
! [B: $tType,A: $tType] :
( ( ( topological_t2_space @ A )
& ( real_Vector_banach @ B )
& ( real_V3459762299906320749_field @ B ) )
=> ! [F2: A > B,A3: B,F4: filter @ A] :
( ( filterlim @ A @ B @ F2 @ ( topolo7230453075368039082e_nhds @ B @ A3 ) @ F4 )
=> ( filterlim @ A @ B
@ ^ [X3: A] : ( sin @ B @ ( F2 @ X3 ) )
@ ( topolo7230453075368039082e_nhds @ B @ ( sin @ B @ A3 ) )
@ F4 ) ) ) ).
% tendsto_sin
thf(fact_7112_tendsto__cos,axiom,
! [B: $tType,A: $tType] :
( ( ( topological_t2_space @ A )
& ( real_Vector_banach @ B )
& ( real_V3459762299906320749_field @ B ) )
=> ! [F2: A > B,A3: B,F4: filter @ A] :
( ( filterlim @ A @ B @ F2 @ ( topolo7230453075368039082e_nhds @ B @ A3 ) @ F4 )
=> ( filterlim @ A @ B
@ ^ [X3: A] : ( cos @ B @ ( F2 @ X3 ) )
@ ( topolo7230453075368039082e_nhds @ B @ ( cos @ B @ A3 ) )
@ F4 ) ) ) ).
% tendsto_cos
thf(fact_7113_tendsto__scaleR,axiom,
! [C: $tType,D6: $tType] :
( ( real_V822414075346904944vector @ C )
=> ! [F2: D6 > real,A3: real,F4: filter @ D6,G: D6 > C,B3: C] :
( ( filterlim @ D6 @ real @ F2 @ ( topolo7230453075368039082e_nhds @ real @ A3 ) @ F4 )
=> ( ( filterlim @ D6 @ C @ G @ ( topolo7230453075368039082e_nhds @ C @ B3 ) @ F4 )
=> ( filterlim @ D6 @ C
@ ^ [X3: D6] : ( real_V8093663219630862766scaleR @ C @ ( F2 @ X3 ) @ ( G @ X3 ) )
@ ( topolo7230453075368039082e_nhds @ C @ ( real_V8093663219630862766scaleR @ C @ A3 @ B3 ) )
@ F4 ) ) ) ) ).
% tendsto_scaleR
thf(fact_7114_continuous__scaleR,axiom,
! [C: $tType,D6: $tType] :
( ( ( topological_t2_space @ D6 )
& ( real_V822414075346904944vector @ C ) )
=> ! [F4: filter @ D6,F2: D6 > real,G: D6 > C] :
( ( topolo3448309680560233919inuous @ D6 @ real @ F4 @ F2 )
=> ( ( topolo3448309680560233919inuous @ D6 @ C @ F4 @ G )
=> ( topolo3448309680560233919inuous @ D6 @ C @ F4
@ ^ [X3: D6] : ( real_V8093663219630862766scaleR @ C @ ( F2 @ X3 ) @ ( G @ X3 ) ) ) ) ) ) ).
% continuous_scaleR
thf(fact_7115_continuous__norm,axiom,
! [B: $tType,A: $tType] :
( ( ( topological_t2_space @ A )
& ( real_V822414075346904944vector @ B ) )
=> ! [F4: filter @ A,F2: A > B] :
( ( topolo3448309680560233919inuous @ A @ B @ F4 @ F2 )
=> ( topolo3448309680560233919inuous @ A @ real @ F4
@ ^ [X3: A] : ( real_V7770717601297561774m_norm @ B @ ( F2 @ X3 ) ) ) ) ) ).
% continuous_norm
thf(fact_7116_tendsto__norm,axiom,
! [B: $tType,A: $tType] :
( ( real_V822414075346904944vector @ B )
=> ! [F2: A > B,A3: B,F4: filter @ A] :
( ( filterlim @ A @ B @ F2 @ ( topolo7230453075368039082e_nhds @ B @ A3 ) @ F4 )
=> ( filterlim @ A @ real
@ ^ [X3: A] : ( real_V7770717601297561774m_norm @ B @ ( F2 @ X3 ) )
@ ( topolo7230453075368039082e_nhds @ real @ ( real_V7770717601297561774m_norm @ B @ A3 ) )
@ F4 ) ) ) ).
% tendsto_norm
thf(fact_7117_tendsto__real__sqrt,axiom,
! [A: $tType,F2: A > real,X2: real,F4: filter @ A] :
( ( filterlim @ A @ real @ F2 @ ( topolo7230453075368039082e_nhds @ real @ X2 ) @ F4 )
=> ( filterlim @ A @ real
@ ^ [X3: A] : ( sqrt @ ( F2 @ X3 ) )
@ ( topolo7230453075368039082e_nhds @ real @ ( sqrt @ X2 ) )
@ F4 ) ) ).
% tendsto_real_sqrt
thf(fact_7118_tendsto__arsinh,axiom,
! [B: $tType,F2: B > real,A3: real,F4: filter @ B] :
( ( filterlim @ B @ real @ F2 @ ( topolo7230453075368039082e_nhds @ real @ A3 ) @ F4 )
=> ( filterlim @ B @ real
@ ^ [X3: B] : ( arsinh @ real @ ( F2 @ X3 ) )
@ ( topolo7230453075368039082e_nhds @ real @ ( arsinh @ real @ A3 ) )
@ F4 ) ) ).
% tendsto_arsinh
thf(fact_7119_tendsto__of__real__iff,axiom,
! [A: $tType,B: $tType] :
( ( real_V8999393235501362500lgebra @ A )
=> ! [F2: B > real,C3: real,F4: filter @ B] :
( ( filterlim @ B @ A
@ ^ [X3: B] : ( real_Vector_of_real @ A @ ( F2 @ X3 ) )
@ ( topolo7230453075368039082e_nhds @ A @ ( real_Vector_of_real @ A @ C3 ) )
@ F4 )
= ( filterlim @ B @ real @ F2 @ ( topolo7230453075368039082e_nhds @ real @ C3 ) @ F4 ) ) ) ).
% tendsto_of_real_iff
thf(fact_7120_continuous__of__real,axiom,
! [A: $tType,C: $tType] :
( ( ( topological_t2_space @ C )
& ( real_V2191834092415804123ebra_1 @ A )
& ( real_V822414075346904944vector @ A ) )
=> ! [F4: filter @ C,G: C > real] :
( ( topolo3448309680560233919inuous @ C @ real @ F4 @ G )
=> ( topolo3448309680560233919inuous @ C @ A @ F4
@ ^ [X3: C] : ( real_Vector_of_real @ A @ ( G @ X3 ) ) ) ) ) ).
% continuous_of_real
thf(fact_7121_tendsto__of__real,axiom,
! [A: $tType,C: $tType] :
( ( ( real_V2191834092415804123ebra_1 @ A )
& ( real_V822414075346904944vector @ A ) )
=> ! [G: C > real,A3: real,F4: filter @ C] :
( ( filterlim @ C @ real @ G @ ( topolo7230453075368039082e_nhds @ real @ A3 ) @ F4 )
=> ( filterlim @ C @ A
@ ^ [X3: C] : ( real_Vector_of_real @ A @ ( G @ X3 ) )
@ ( topolo7230453075368039082e_nhds @ A @ ( real_Vector_of_real @ A @ A3 ) )
@ F4 ) ) ) ).
% tendsto_of_real
thf(fact_7122_continuous__sinh,axiom,
! [A: $tType,C: $tType] :
( ( ( topological_t2_space @ C )
& ( real_Vector_banach @ A )
& ( real_V3459762299906320749_field @ A ) )
=> ! [F4: filter @ C,F2: C > A] :
( ( topolo3448309680560233919inuous @ C @ A @ F4 @ F2 )
=> ( topolo3448309680560233919inuous @ C @ A @ F4
@ ^ [X3: C] : ( sinh @ A @ ( F2 @ X3 ) ) ) ) ) ).
% continuous_sinh
thf(fact_7123_continuous__cosh,axiom,
! [A: $tType,C: $tType] :
( ( ( topological_t2_space @ C )
& ( real_Vector_banach @ A )
& ( real_V3459762299906320749_field @ A ) )
=> ! [F4: filter @ C,F2: C > A] :
( ( topolo3448309680560233919inuous @ C @ A @ F4 @ F2 )
=> ( topolo3448309680560233919inuous @ C @ A @ F4
@ ^ [X3: C] : ( cosh @ A @ ( F2 @ X3 ) ) ) ) ) ).
% continuous_cosh
thf(fact_7124_tendsto__sinh,axiom,
! [A: $tType,C: $tType] :
( ( ( real_Vector_banach @ A )
& ( real_V3459762299906320749_field @ A ) )
=> ! [F2: C > A,A3: A,F4: filter @ C] :
( ( filterlim @ C @ A @ F2 @ ( topolo7230453075368039082e_nhds @ A @ A3 ) @ F4 )
=> ( filterlim @ C @ A
@ ^ [X3: C] : ( sinh @ A @ ( F2 @ X3 ) )
@ ( topolo7230453075368039082e_nhds @ A @ ( sinh @ A @ A3 ) )
@ F4 ) ) ) ).
% tendsto_sinh
thf(fact_7125_tendsto__cosh,axiom,
! [A: $tType,C: $tType] :
( ( ( real_Vector_banach @ A )
& ( real_V3459762299906320749_field @ A ) )
=> ! [F2: C > A,A3: A,F4: filter @ C] :
( ( filterlim @ C @ A @ F2 @ ( topolo7230453075368039082e_nhds @ A @ A3 ) @ F4 )
=> ( filterlim @ C @ A
@ ^ [X3: C] : ( cosh @ A @ ( F2 @ X3 ) )
@ ( topolo7230453075368039082e_nhds @ A @ ( cosh @ A @ A3 ) )
@ F4 ) ) ) ).
% tendsto_cosh
thf(fact_7126_tendsto__sum,axiom,
! [A: $tType,C: $tType,B: $tType] :
( ( topolo5987344860129210374id_add @ C )
=> ! [I6: set @ A,F2: A > B > C,A3: A > C,F4: filter @ B] :
( ! [I2: A] :
( ( member @ A @ I2 @ I6 )
=> ( filterlim @ B @ C @ ( F2 @ I2 ) @ ( topolo7230453075368039082e_nhds @ C @ ( A3 @ I2 ) ) @ F4 ) )
=> ( filterlim @ B @ C
@ ^ [X3: B] :
( groups7311177749621191930dd_sum @ A @ C
@ ^ [I4: A] : ( F2 @ I4 @ X3 )
@ I6 )
@ ( topolo7230453075368039082e_nhds @ C @ ( groups7311177749621191930dd_sum @ A @ C @ A3 @ I6 ) )
@ F4 ) ) ) ).
% tendsto_sum
thf(fact_7127_continuous__sum,axiom,
! [A: $tType,C: $tType,B: $tType] :
( ( ( topological_t2_space @ B )
& ( topolo5987344860129210374id_add @ C ) )
=> ! [I6: set @ A,F4: filter @ B,F2: A > B > C] :
( ! [I2: A] :
( ( member @ A @ I2 @ I6 )
=> ( topolo3448309680560233919inuous @ B @ C @ F4 @ ( F2 @ I2 ) ) )
=> ( topolo3448309680560233919inuous @ B @ C @ F4
@ ^ [X3: B] :
( groups7311177749621191930dd_sum @ A @ C
@ ^ [I4: A] : ( F2 @ I4 @ X3 )
@ I6 ) ) ) ) ).
% continuous_sum
thf(fact_7128_tendsto__prod,axiom,
! [A: $tType,C: $tType,B: $tType] :
( ( ( real_V4412858255891104859lgebra @ C )
& ( comm_ring_1 @ C ) )
=> ! [S: set @ A,F2: A > B > C,L6: A > C,F4: filter @ B] :
( ! [I2: A] :
( ( member @ A @ I2 @ S )
=> ( filterlim @ B @ C @ ( F2 @ I2 ) @ ( topolo7230453075368039082e_nhds @ C @ ( L6 @ I2 ) ) @ F4 ) )
=> ( filterlim @ B @ C
@ ^ [X3: B] :
( groups7121269368397514597t_prod @ A @ C
@ ^ [I4: A] : ( F2 @ I4 @ X3 )
@ S )
@ ( topolo7230453075368039082e_nhds @ C @ ( groups7121269368397514597t_prod @ A @ C @ L6 @ S ) )
@ F4 ) ) ) ).
% tendsto_prod
thf(fact_7129_tendsto__prod_H,axiom,
! [A: $tType,C: $tType,B: $tType] :
( ( topolo4987421752381908075d_mult @ C )
=> ! [I6: set @ A,F2: A > B > C,A3: A > C,F4: filter @ B] :
( ! [I2: A] :
( ( member @ A @ I2 @ I6 )
=> ( filterlim @ B @ C @ ( F2 @ I2 ) @ ( topolo7230453075368039082e_nhds @ C @ ( A3 @ I2 ) ) @ F4 ) )
=> ( filterlim @ B @ C
@ ^ [X3: B] :
( groups7121269368397514597t_prod @ A @ C
@ ^ [I4: A] : ( F2 @ I4 @ X3 )
@ I6 )
@ ( topolo7230453075368039082e_nhds @ C @ ( groups7121269368397514597t_prod @ A @ C @ A3 @ I6 ) )
@ F4 ) ) ) ).
% tendsto_prod'
thf(fact_7130_continuous__prod_H,axiom,
! [A: $tType,C: $tType,B: $tType] :
( ( ( topological_t2_space @ B )
& ( topolo4987421752381908075d_mult @ C ) )
=> ! [I6: set @ A,F4: filter @ B,F2: A > B > C] :
( ! [I2: A] :
( ( member @ A @ I2 @ I6 )
=> ( topolo3448309680560233919inuous @ B @ C @ F4 @ ( F2 @ I2 ) ) )
=> ( topolo3448309680560233919inuous @ B @ C @ F4
@ ^ [X3: B] :
( groups7121269368397514597t_prod @ A @ C
@ ^ [I4: A] : ( F2 @ I4 @ X3 )
@ I6 ) ) ) ) ).
% continuous_prod'
thf(fact_7131_continuous__prod,axiom,
! [A: $tType,C: $tType,B: $tType] :
( ( ( topological_t2_space @ B )
& ( real_V4412858255891104859lgebra @ C )
& ( comm_ring_1 @ C ) )
=> ! [S: set @ A,F4: filter @ B,F2: A > B > C] :
( ! [I2: A] :
( ( member @ A @ I2 @ S )
=> ( topolo3448309680560233919inuous @ B @ C @ F4 @ ( F2 @ I2 ) ) )
=> ( topolo3448309680560233919inuous @ B @ C @ F4
@ ^ [X3: B] :
( groups7121269368397514597t_prod @ A @ C
@ ^ [I4: A] : ( F2 @ I4 @ X3 )
@ S ) ) ) ) ).
% continuous_prod
thf(fact_7132_tendsto__max,axiom,
! [A: $tType,B: $tType] :
( ( topolo1944317154257567458pology @ A )
=> ! [X: B > A,X2: A,Net: filter @ B,Y8: B > A,Y3: A] :
( ( filterlim @ B @ A @ X @ ( topolo7230453075368039082e_nhds @ A @ X2 ) @ Net )
=> ( ( filterlim @ B @ A @ Y8 @ ( topolo7230453075368039082e_nhds @ A @ Y3 ) @ Net )
=> ( filterlim @ B @ A
@ ^ [X3: B] : ( ord_max @ A @ ( X @ X3 ) @ ( Y8 @ X3 ) )
@ ( topolo7230453075368039082e_nhds @ A @ ( ord_max @ A @ X2 @ Y3 ) )
@ Net ) ) ) ) ).
% tendsto_max
thf(fact_7133_continuous__max,axiom,
! [B: $tType,A: $tType] :
( ( ( topological_t2_space @ A )
& ( topolo1944317154257567458pology @ B ) )
=> ! [F4: filter @ A,F2: A > B,G: A > B] :
( ( topolo3448309680560233919inuous @ A @ B @ F4 @ F2 )
=> ( ( topolo3448309680560233919inuous @ A @ B @ F4 @ G )
=> ( topolo3448309680560233919inuous @ A @ B @ F4
@ ^ [X3: A] : ( ord_max @ B @ ( F2 @ X3 ) @ ( G @ X3 ) ) ) ) ) ) ).
% continuous_max
thf(fact_7134_continuous__diff,axiom,
! [B: $tType,A: $tType] :
( ( ( topological_t2_space @ A )
& ( topolo1633459387980952147up_add @ B ) )
=> ! [F4: filter @ A,F2: A > B,G: A > B] :
( ( topolo3448309680560233919inuous @ A @ B @ F4 @ F2 )
=> ( ( topolo3448309680560233919inuous @ A @ B @ F4 @ G )
=> ( topolo3448309680560233919inuous @ A @ B @ F4
@ ^ [X3: A] : ( minus_minus @ B @ ( F2 @ X3 ) @ ( G @ X3 ) ) ) ) ) ) ).
% continuous_diff
thf(fact_7135_tendsto__diff,axiom,
! [A: $tType,B: $tType] :
( ( topolo1633459387980952147up_add @ A )
=> ! [F2: B > A,A3: A,F4: filter @ B,G: B > A,B3: A] :
( ( filterlim @ B @ A @ F2 @ ( topolo7230453075368039082e_nhds @ A @ A3 ) @ F4 )
=> ( ( filterlim @ B @ A @ G @ ( topolo7230453075368039082e_nhds @ A @ B3 ) @ F4 )
=> ( filterlim @ B @ A
@ ^ [X3: B] : ( minus_minus @ A @ ( F2 @ X3 ) @ ( G @ X3 ) )
@ ( topolo7230453075368039082e_nhds @ A @ ( minus_minus @ A @ A3 @ B3 ) )
@ F4 ) ) ) ) ).
% tendsto_diff
thf(fact_7136_Lim__transform__eq,axiom,
! [A: $tType,B: $tType] :
( ( real_V822414075346904944vector @ A )
=> ! [F2: B > A,G: B > A,F4: filter @ B,A3: A] :
( ( filterlim @ B @ A
@ ^ [X3: B] : ( minus_minus @ A @ ( F2 @ X3 ) @ ( G @ X3 ) )
@ ( topolo7230453075368039082e_nhds @ A @ ( zero_zero @ A ) )
@ F4 )
=> ( ( filterlim @ B @ A @ F2 @ ( topolo7230453075368039082e_nhds @ A @ A3 ) @ F4 )
= ( filterlim @ B @ A @ G @ ( topolo7230453075368039082e_nhds @ A @ A3 ) @ F4 ) ) ) ) ).
% Lim_transform_eq
thf(fact_7137_LIM__zero__cancel,axiom,
! [B: $tType,A: $tType] :
( ( real_V822414075346904944vector @ B )
=> ! [F2: A > B,L: B,F4: filter @ A] :
( ( filterlim @ A @ B
@ ^ [X3: A] : ( minus_minus @ B @ ( F2 @ X3 ) @ L )
@ ( topolo7230453075368039082e_nhds @ B @ ( zero_zero @ B ) )
@ F4 )
=> ( filterlim @ A @ B @ F2 @ ( topolo7230453075368039082e_nhds @ B @ L ) @ F4 ) ) ) ).
% LIM_zero_cancel
thf(fact_7138_Lim__transform2,axiom,
! [A: $tType,B: $tType] :
( ( real_V822414075346904944vector @ A )
=> ! [F2: B > A,A3: A,F4: filter @ B,G: B > A] :
( ( filterlim @ B @ A @ F2 @ ( topolo7230453075368039082e_nhds @ A @ A3 ) @ F4 )
=> ( ( filterlim @ B @ A
@ ^ [X3: B] : ( minus_minus @ A @ ( F2 @ X3 ) @ ( G @ X3 ) )
@ ( topolo7230453075368039082e_nhds @ A @ ( zero_zero @ A ) )
@ F4 )
=> ( filterlim @ B @ A @ G @ ( topolo7230453075368039082e_nhds @ A @ A3 ) @ F4 ) ) ) ) ).
% Lim_transform2
thf(fact_7139_Lim__transform,axiom,
! [A: $tType,B: $tType] :
( ( real_V822414075346904944vector @ A )
=> ! [G: B > A,A3: A,F4: filter @ B,F2: B > A] :
( ( filterlim @ B @ A @ G @ ( topolo7230453075368039082e_nhds @ A @ A3 ) @ F4 )
=> ( ( filterlim @ B @ A
@ ^ [X3: B] : ( minus_minus @ A @ ( F2 @ X3 ) @ ( G @ X3 ) )
@ ( topolo7230453075368039082e_nhds @ A @ ( zero_zero @ A ) )
@ F4 )
=> ( filterlim @ B @ A @ F2 @ ( topolo7230453075368039082e_nhds @ A @ A3 ) @ F4 ) ) ) ) ).
% Lim_transform
thf(fact_7140_LIM__zero__iff,axiom,
! [B: $tType,A: $tType] :
( ( real_V822414075346904944vector @ B )
=> ! [F2: A > B,L: B,F4: filter @ A] :
( ( filterlim @ A @ B
@ ^ [X3: A] : ( minus_minus @ B @ ( F2 @ X3 ) @ L )
@ ( topolo7230453075368039082e_nhds @ B @ ( zero_zero @ B ) )
@ F4 )
= ( filterlim @ A @ B @ F2 @ ( topolo7230453075368039082e_nhds @ B @ L ) @ F4 ) ) ) ).
% LIM_zero_iff
thf(fact_7141_LIM__zero,axiom,
! [B: $tType,A: $tType] :
( ( real_V822414075346904944vector @ B )
=> ! [F2: A > B,L: B,F4: filter @ A] :
( ( filterlim @ A @ B @ F2 @ ( topolo7230453075368039082e_nhds @ B @ L ) @ F4 )
=> ( filterlim @ A @ B
@ ^ [X3: A] : ( minus_minus @ B @ ( F2 @ X3 ) @ L )
@ ( topolo7230453075368039082e_nhds @ B @ ( zero_zero @ B ) )
@ F4 ) ) ) ).
% LIM_zero
thf(fact_7142_tendsto__add__zero,axiom,
! [B: $tType,D6: $tType] :
( ( topolo6943815403480290642id_add @ B )
=> ! [F2: D6 > B,F4: filter @ D6,G: D6 > B] :
( ( filterlim @ D6 @ B @ F2 @ ( topolo7230453075368039082e_nhds @ B @ ( zero_zero @ B ) ) @ F4 )
=> ( ( filterlim @ D6 @ B @ G @ ( topolo7230453075368039082e_nhds @ B @ ( zero_zero @ B ) ) @ F4 )
=> ( filterlim @ D6 @ B
@ ^ [X3: D6] : ( plus_plus @ B @ ( F2 @ X3 ) @ ( G @ X3 ) )
@ ( topolo7230453075368039082e_nhds @ B @ ( zero_zero @ B ) )
@ F4 ) ) ) ) ).
% tendsto_add_zero
thf(fact_7143_tendsto__add,axiom,
! [A: $tType,B: $tType] :
( ( topolo6943815403480290642id_add @ A )
=> ! [F2: B > A,A3: A,F4: filter @ B,G: B > A,B3: A] :
( ( filterlim @ B @ A @ F2 @ ( topolo7230453075368039082e_nhds @ A @ A3 ) @ F4 )
=> ( ( filterlim @ B @ A @ G @ ( topolo7230453075368039082e_nhds @ A @ B3 ) @ F4 )
=> ( filterlim @ B @ A
@ ^ [X3: B] : ( plus_plus @ A @ ( F2 @ X3 ) @ ( G @ X3 ) )
@ ( topolo7230453075368039082e_nhds @ A @ ( plus_plus @ A @ A3 @ B3 ) )
@ F4 ) ) ) ) ).
% tendsto_add
thf(fact_7144_tendsto__add__const__iff,axiom,
! [A: $tType,B: $tType] :
( ( topolo1633459387980952147up_add @ A )
=> ! [C3: A,F2: B > A,D2: A,F4: filter @ B] :
( ( filterlim @ B @ A
@ ^ [X3: B] : ( plus_plus @ A @ C3 @ ( F2 @ X3 ) )
@ ( topolo7230453075368039082e_nhds @ A @ ( plus_plus @ A @ C3 @ D2 ) )
@ F4 )
= ( filterlim @ B @ A @ F2 @ ( topolo7230453075368039082e_nhds @ A @ D2 ) @ F4 ) ) ) ).
% tendsto_add_const_iff
thf(fact_7145_continuous__add,axiom,
! [B: $tType,D6: $tType] :
( ( ( topological_t2_space @ D6 )
& ( topolo6943815403480290642id_add @ B ) )
=> ! [F4: filter @ D6,F2: D6 > B,G: D6 > B] :
( ( topolo3448309680560233919inuous @ D6 @ B @ F4 @ F2 )
=> ( ( topolo3448309680560233919inuous @ D6 @ B @ F4 @ G )
=> ( topolo3448309680560233919inuous @ D6 @ B @ F4
@ ^ [X3: D6] : ( plus_plus @ B @ ( F2 @ X3 ) @ ( G @ X3 ) ) ) ) ) ) ).
% continuous_add
thf(fact_7146_tendsto__inverse,axiom,
! [A: $tType,B: $tType] :
( ( real_V8999393235501362500lgebra @ A )
=> ! [F2: B > A,A3: A,F4: filter @ B] :
( ( filterlim @ B @ A @ F2 @ ( topolo7230453075368039082e_nhds @ A @ A3 ) @ F4 )
=> ( ( A3
!= ( zero_zero @ A ) )
=> ( filterlim @ B @ A
@ ^ [X3: B] : ( inverse_inverse @ A @ ( F2 @ X3 ) )
@ ( topolo7230453075368039082e_nhds @ A @ ( inverse_inverse @ A @ A3 ) )
@ F4 ) ) ) ) ).
% tendsto_inverse
thf(fact_7147_tendsto__of__int__ceiling,axiom,
! [B: $tType,C: $tType,A: $tType] :
( ( ( ring_1 @ C )
& ( topolo4958980785337419405_space @ C )
& ( archim2362893244070406136eiling @ B )
& ( topolo2564578578187576103pology @ B ) )
=> ! [F2: A > B,L: B,F4: filter @ A] :
( ( filterlim @ A @ B @ F2 @ ( topolo7230453075368039082e_nhds @ B @ L ) @ F4 )
=> ( ~ ( member @ B @ L @ ( ring_1_Ints @ B ) )
=> ( filterlim @ A @ C
@ ^ [X3: A] : ( ring_1_of_int @ C @ ( archimedean_ceiling @ B @ ( F2 @ X3 ) ) )
@ ( topolo7230453075368039082e_nhds @ C @ ( ring_1_of_int @ C @ ( archimedean_ceiling @ B @ L ) ) )
@ F4 ) ) ) ) ).
% tendsto_of_int_ceiling
thf(fact_7148_tendsto__of__int__floor,axiom,
! [B: $tType,C: $tType,A: $tType] :
( ( ( ring_1 @ C )
& ( topolo4958980785337419405_space @ C )
& ( archim2362893244070406136eiling @ B )
& ( topolo2564578578187576103pology @ B ) )
=> ! [F2: A > B,L: B,F4: filter @ A] :
( ( filterlim @ A @ B @ F2 @ ( topolo7230453075368039082e_nhds @ B @ L ) @ F4 )
=> ( ~ ( member @ B @ L @ ( ring_1_Ints @ B ) )
=> ( filterlim @ A @ C
@ ^ [X3: A] : ( ring_1_of_int @ C @ ( archim6421214686448440834_floor @ B @ ( F2 @ X3 ) ) )
@ ( topolo7230453075368039082e_nhds @ C @ ( ring_1_of_int @ C @ ( archim6421214686448440834_floor @ B @ L ) ) )
@ F4 ) ) ) ) ).
% tendsto_of_int_floor
thf(fact_7149_tendsto__power,axiom,
! [B: $tType,A: $tType] :
( ( ( power @ B )
& ( real_V4412858255891104859lgebra @ B ) )
=> ! [F2: A > B,A3: B,F4: filter @ A,N: nat] :
( ( filterlim @ A @ B @ F2 @ ( topolo7230453075368039082e_nhds @ B @ A3 ) @ F4 )
=> ( filterlim @ A @ B
@ ^ [X3: A] : ( power_power @ B @ ( F2 @ X3 ) @ N )
@ ( topolo7230453075368039082e_nhds @ B @ ( power_power @ B @ A3 @ N ) )
@ F4 ) ) ) ).
% tendsto_power
thf(fact_7150_continuous__power_H,axiom,
! [B: $tType,C: $tType] :
( ( ( topological_t2_space @ C )
& ( topolo1898628316856586783d_mult @ B ) )
=> ! [F4: filter @ C,F2: C > B,G: C > nat] :
( ( topolo3448309680560233919inuous @ C @ B @ F4 @ F2 )
=> ( ( topolo3448309680560233919inuous @ C @ nat @ F4 @ G )
=> ( topolo3448309680560233919inuous @ C @ B @ F4
@ ^ [X3: C] : ( power_power @ B @ ( F2 @ X3 ) @ ( G @ X3 ) ) ) ) ) ) ).
% continuous_power'
thf(fact_7151_tendsto__power__strong,axiom,
! [B: $tType,C: $tType] :
( ( topolo1898628316856586783d_mult @ B )
=> ! [F2: C > B,A3: B,F4: filter @ C,G: C > nat,B3: nat] :
( ( filterlim @ C @ B @ F2 @ ( topolo7230453075368039082e_nhds @ B @ A3 ) @ F4 )
=> ( ( filterlim @ C @ nat @ G @ ( topolo7230453075368039082e_nhds @ nat @ B3 ) @ F4 )
=> ( filterlim @ C @ B
@ ^ [X3: C] : ( power_power @ B @ ( F2 @ X3 ) @ ( G @ X3 ) )
@ ( topolo7230453075368039082e_nhds @ B @ ( power_power @ B @ A3 @ B3 ) )
@ F4 ) ) ) ) ).
% tendsto_power_strong
thf(fact_7152_continuous__power,axiom,
! [B: $tType,A: $tType] :
( ( ( topological_t2_space @ A )
& ( power @ B )
& ( real_V4412858255891104859lgebra @ B ) )
=> ! [F4: filter @ A,F2: A > B,N: nat] :
( ( topolo3448309680560233919inuous @ A @ B @ F4 @ F2 )
=> ( topolo3448309680560233919inuous @ A @ B @ F4
@ ^ [X3: A] : ( power_power @ B @ ( F2 @ X3 ) @ N ) ) ) ) ).
% continuous_power
thf(fact_7153_tendsto__null__power,axiom,
! [B: $tType,A: $tType] :
( ( real_V2822296259951069270ebra_1 @ B )
=> ! [F2: A > B,F4: filter @ A,N: nat] :
( ( filterlim @ A @ B @ F2 @ ( topolo7230453075368039082e_nhds @ B @ ( zero_zero @ B ) ) @ F4 )
=> ( ( ord_less @ nat @ ( zero_zero @ nat ) @ N )
=> ( filterlim @ A @ B
@ ^ [X3: A] : ( power_power @ B @ ( F2 @ X3 ) @ N )
@ ( topolo7230453075368039082e_nhds @ B @ ( zero_zero @ B ) )
@ F4 ) ) ) ) ).
% tendsto_null_power
thf(fact_7154_DERIV__LIM__iff,axiom,
! [A: $tType] :
( ( ( inverse @ A )
& ( real_V822414075346904944vector @ A ) )
=> ! [F2: A > A,A3: A,D: A] :
( ( filterlim @ A @ A
@ ^ [H: A] : ( divide_divide @ A @ ( minus_minus @ A @ ( F2 @ ( plus_plus @ A @ A3 @ H ) ) @ ( F2 @ A3 ) ) @ H )
@ ( topolo7230453075368039082e_nhds @ A @ D )
@ ( topolo174197925503356063within @ A @ ( zero_zero @ A ) @ ( top_top @ ( set @ A ) ) ) )
= ( filterlim @ A @ A
@ ^ [X3: A] : ( divide_divide @ A @ ( minus_minus @ A @ ( F2 @ X3 ) @ ( F2 @ A3 ) ) @ ( minus_minus @ A @ X3 @ A3 ) )
@ ( topolo7230453075368039082e_nhds @ A @ D )
@ ( topolo174197925503356063within @ A @ A3 @ ( top_top @ ( set @ A ) ) ) ) ) ) ).
% DERIV_LIM_iff
thf(fact_7155_isCont__Lb__Ub,axiom,
! [A3: real,B3: real,F2: real > real] :
( ( ord_less_eq @ real @ A3 @ B3 )
=> ( ! [X4: real] :
( ( ( ord_less_eq @ real @ A3 @ X4 )
& ( ord_less_eq @ real @ X4 @ B3 ) )
=> ( topolo3448309680560233919inuous @ real @ real @ ( topolo174197925503356063within @ real @ X4 @ ( top_top @ ( set @ real ) ) ) @ F2 ) )
=> ? [L7: real,M11: real] :
( ! [X8: real] :
( ( ( ord_less_eq @ real @ A3 @ X8 )
& ( ord_less_eq @ real @ X8 @ B3 ) )
=> ( ( ord_less_eq @ real @ L7 @ ( F2 @ X8 ) )
& ( ord_less_eq @ real @ ( F2 @ X8 ) @ M11 ) ) )
& ! [Y5: real] :
( ( ( ord_less_eq @ real @ L7 @ Y5 )
& ( ord_less_eq @ real @ Y5 @ M11 ) )
=> ? [X4: real] :
( ( ord_less_eq @ real @ A3 @ X4 )
& ( ord_less_eq @ real @ X4 @ B3 )
& ( ( F2 @ X4 )
= Y5 ) ) ) ) ) ) ).
% isCont_Lb_Ub
thf(fact_7156_LIM__fun__gt__zero,axiom,
! [F2: real > real,L: real,C3: real] :
( ( filterlim @ real @ real @ F2 @ ( topolo7230453075368039082e_nhds @ real @ L ) @ ( topolo174197925503356063within @ real @ C3 @ ( top_top @ ( set @ real ) ) ) )
=> ( ( ord_less @ real @ ( zero_zero @ real ) @ L )
=> ? [R3: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ R3 )
& ! [X8: real] :
( ( ( X8 != C3 )
& ( ord_less @ real @ ( abs_abs @ real @ ( minus_minus @ real @ C3 @ X8 ) ) @ R3 ) )
=> ( ord_less @ real @ ( zero_zero @ real ) @ ( F2 @ X8 ) ) ) ) ) ) ).
% LIM_fun_gt_zero
thf(fact_7157_LIM__fun__not__zero,axiom,
! [F2: real > real,L: real,C3: real] :
( ( filterlim @ real @ real @ F2 @ ( topolo7230453075368039082e_nhds @ real @ L ) @ ( topolo174197925503356063within @ real @ C3 @ ( top_top @ ( set @ real ) ) ) )
=> ( ( L
!= ( zero_zero @ real ) )
=> ? [R3: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ R3 )
& ! [X8: real] :
( ( ( X8 != C3 )
& ( ord_less @ real @ ( abs_abs @ real @ ( minus_minus @ real @ C3 @ X8 ) ) @ R3 ) )
=> ( ( F2 @ X8 )
!= ( zero_zero @ real ) ) ) ) ) ) ).
% LIM_fun_not_zero
thf(fact_7158_LIM__fun__less__zero,axiom,
! [F2: real > real,L: real,C3: real] :
( ( filterlim @ real @ real @ F2 @ ( topolo7230453075368039082e_nhds @ real @ L ) @ ( topolo174197925503356063within @ real @ C3 @ ( top_top @ ( set @ real ) ) ) )
=> ( ( ord_less @ real @ L @ ( zero_zero @ real ) )
=> ? [R3: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ R3 )
& ! [X8: real] :
( ( ( X8 != C3 )
& ( ord_less @ real @ ( abs_abs @ real @ ( minus_minus @ real @ C3 @ X8 ) ) @ R3 ) )
=> ( ord_less @ real @ ( F2 @ X8 ) @ ( zero_zero @ real ) ) ) ) ) ) ).
% LIM_fun_less_zero
thf(fact_7159_LIM__compose2,axiom,
! [C: $tType,B: $tType,A: $tType] :
( ( ( real_V822414075346904944vector @ A )
& ( topolo4958980785337419405_space @ B )
& ( topolo4958980785337419405_space @ C ) )
=> ! [F2: A > B,B3: B,A3: A,G: B > C,C3: C] :
( ( filterlim @ A @ B @ F2 @ ( topolo7230453075368039082e_nhds @ B @ B3 ) @ ( topolo174197925503356063within @ A @ A3 @ ( top_top @ ( set @ A ) ) ) )
=> ( ( filterlim @ B @ C @ G @ ( topolo7230453075368039082e_nhds @ C @ C3 ) @ ( topolo174197925503356063within @ B @ B3 @ ( top_top @ ( set @ B ) ) ) )
=> ( ? [D5: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ D5 )
& ! [X4: A] :
( ( ( X4 != A3 )
& ( ord_less @ real @ ( real_V7770717601297561774m_norm @ A @ ( minus_minus @ A @ X4 @ A3 ) ) @ D5 ) )
=> ( ( F2 @ X4 )
!= B3 ) ) )
=> ( filterlim @ A @ C
@ ^ [X3: A] : ( G @ ( F2 @ X3 ) )
@ ( topolo7230453075368039082e_nhds @ C @ C3 )
@ ( topolo174197925503356063within @ A @ A3 @ ( top_top @ ( set @ A ) ) ) ) ) ) ) ) ).
% LIM_compose2
thf(fact_7160_continuous__at__within__divide,axiom,
! [B: $tType,A: $tType] :
( ( ( topological_t2_space @ A )
& ( real_V3459762299906320749_field @ B ) )
=> ! [A3: A,S2: set @ A,F2: A > B,G: A > B] :
( ( topolo3448309680560233919inuous @ A @ B @ ( topolo174197925503356063within @ A @ A3 @ S2 ) @ F2 )
=> ( ( topolo3448309680560233919inuous @ A @ B @ ( topolo174197925503356063within @ A @ A3 @ S2 ) @ G )
=> ( ( ( G @ A3 )
!= ( zero_zero @ B ) )
=> ( topolo3448309680560233919inuous @ A @ B @ ( topolo174197925503356063within @ A @ A3 @ S2 )
@ ^ [X3: A] : ( divide_divide @ B @ ( F2 @ X3 ) @ ( G @ X3 ) ) ) ) ) ) ) ).
% continuous_at_within_divide
thf(fact_7161_isCont__add,axiom,
! [B: $tType,A: $tType] :
( ( ( topological_t2_space @ A )
& ( topolo6943815403480290642id_add @ B ) )
=> ! [A3: A,F2: A > B,G: A > B] :
( ( topolo3448309680560233919inuous @ A @ B @ ( topolo174197925503356063within @ A @ A3 @ ( top_top @ ( set @ A ) ) ) @ F2 )
=> ( ( topolo3448309680560233919inuous @ A @ B @ ( topolo174197925503356063within @ A @ A3 @ ( top_top @ ( set @ A ) ) ) @ G )
=> ( topolo3448309680560233919inuous @ A @ B @ ( topolo174197925503356063within @ A @ A3 @ ( top_top @ ( set @ A ) ) )
@ ^ [X3: A] : ( plus_plus @ B @ ( F2 @ X3 ) @ ( G @ X3 ) ) ) ) ) ) ).
% isCont_add
thf(fact_7162_isCont__mult,axiom,
! [B: $tType,A: $tType] :
( ( ( topological_t2_space @ A )
& ( real_V4412858255891104859lgebra @ B ) )
=> ! [A3: A,F2: A > B,G: A > B] :
( ( topolo3448309680560233919inuous @ A @ B @ ( topolo174197925503356063within @ A @ A3 @ ( top_top @ ( set @ A ) ) ) @ F2 )
=> ( ( topolo3448309680560233919inuous @ A @ B @ ( topolo174197925503356063within @ A @ A3 @ ( top_top @ ( set @ A ) ) ) @ G )
=> ( topolo3448309680560233919inuous @ A @ B @ ( topolo174197925503356063within @ A @ A3 @ ( top_top @ ( set @ A ) ) )
@ ^ [X3: A] : ( times_times @ B @ ( F2 @ X3 ) @ ( G @ X3 ) ) ) ) ) ) ).
% isCont_mult
thf(fact_7163_isCont__diff,axiom,
! [B: $tType,A: $tType] :
( ( ( topological_t2_space @ A )
& ( real_V822414075346904944vector @ B ) )
=> ! [A3: A,F2: A > B,G: A > B] :
( ( topolo3448309680560233919inuous @ A @ B @ ( topolo174197925503356063within @ A @ A3 @ ( top_top @ ( set @ A ) ) ) @ F2 )
=> ( ( topolo3448309680560233919inuous @ A @ B @ ( topolo174197925503356063within @ A @ A3 @ ( top_top @ ( set @ A ) ) ) @ G )
=> ( topolo3448309680560233919inuous @ A @ B @ ( topolo174197925503356063within @ A @ A3 @ ( top_top @ ( set @ A ) ) )
@ ^ [X3: A] : ( minus_minus @ B @ ( F2 @ X3 ) @ ( G @ X3 ) ) ) ) ) ) ).
% isCont_diff
thf(fact_7164_isCont__power,axiom,
! [B: $tType,A: $tType] :
( ( ( topological_t2_space @ A )
& ( power @ B )
& ( real_V4412858255891104859lgebra @ B ) )
=> ! [A3: A,F2: A > B,N: nat] :
( ( topolo3448309680560233919inuous @ A @ B @ ( topolo174197925503356063within @ A @ A3 @ ( top_top @ ( set @ A ) ) ) @ F2 )
=> ( topolo3448309680560233919inuous @ A @ B @ ( topolo174197925503356063within @ A @ A3 @ ( top_top @ ( set @ A ) ) )
@ ^ [X3: A] : ( power_power @ B @ ( F2 @ X3 ) @ N ) ) ) ) ).
% isCont_power
thf(fact_7165_isCont__minus,axiom,
! [B: $tType,A: $tType] :
( ( ( topological_t2_space @ A )
& ( real_V822414075346904944vector @ B ) )
=> ! [A3: A,F2: A > B] :
( ( topolo3448309680560233919inuous @ A @ B @ ( topolo174197925503356063within @ A @ A3 @ ( top_top @ ( set @ A ) ) ) @ F2 )
=> ( topolo3448309680560233919inuous @ A @ B @ ( topolo174197925503356063within @ A @ A3 @ ( top_top @ ( set @ A ) ) )
@ ^ [X3: A] : ( uminus_uminus @ B @ ( F2 @ X3 ) ) ) ) ) ).
% isCont_minus
thf(fact_7166_continuous__at__within__inverse,axiom,
! [B: $tType,A: $tType] :
( ( ( topological_t2_space @ A )
& ( real_V8999393235501362500lgebra @ B ) )
=> ! [A3: A,S2: set @ A,F2: A > B] :
( ( topolo3448309680560233919inuous @ A @ B @ ( topolo174197925503356063within @ A @ A3 @ S2 ) @ F2 )
=> ( ( ( F2 @ A3 )
!= ( zero_zero @ B ) )
=> ( topolo3448309680560233919inuous @ A @ B @ ( topolo174197925503356063within @ A @ A3 @ S2 )
@ ^ [X3: A] : ( inverse_inverse @ B @ ( F2 @ X3 ) ) ) ) ) ) ).
% continuous_at_within_inverse
thf(fact_7167_isCont__sum,axiom,
! [A: $tType,C: $tType,B: $tType] :
( ( ( topological_t2_space @ B )
& ( topolo5987344860129210374id_add @ C ) )
=> ! [A2: set @ A,A3: B,F2: A > B > C] :
( ! [X4: A] :
( ( member @ A @ X4 @ A2 )
=> ( topolo3448309680560233919inuous @ B @ C @ ( topolo174197925503356063within @ B @ A3 @ ( top_top @ ( set @ B ) ) ) @ ( F2 @ X4 ) ) )
=> ( topolo3448309680560233919inuous @ B @ C @ ( topolo174197925503356063within @ B @ A3 @ ( top_top @ ( set @ B ) ) )
@ ^ [X3: B] :
( groups7311177749621191930dd_sum @ A @ C
@ ^ [I4: A] : ( F2 @ I4 @ X3 )
@ A2 ) ) ) ) ).
% isCont_sum
thf(fact_7168_continuous__at__within__sgn,axiom,
! [B: $tType,A: $tType] :
( ( ( topological_t2_space @ A )
& ( real_V822414075346904944vector @ B ) )
=> ! [A3: A,S2: set @ A,F2: A > B] :
( ( topolo3448309680560233919inuous @ A @ B @ ( topolo174197925503356063within @ A @ A3 @ S2 ) @ F2 )
=> ( ( ( F2 @ A3 )
!= ( zero_zero @ B ) )
=> ( topolo3448309680560233919inuous @ A @ B @ ( topolo174197925503356063within @ A @ A3 @ S2 )
@ ^ [X3: A] : ( sgn_sgn @ B @ ( F2 @ X3 ) ) ) ) ) ) ).
% continuous_at_within_sgn
thf(fact_7169_isCont__cos_H,axiom,
! [B: $tType,A: $tType] :
( ( ( topological_t2_space @ A )
& ( real_Vector_banach @ B )
& ( real_V3459762299906320749_field @ B ) )
=> ! [A3: A,F2: A > B] :
( ( topolo3448309680560233919inuous @ A @ B @ ( topolo174197925503356063within @ A @ A3 @ ( top_top @ ( set @ A ) ) ) @ F2 )
=> ( topolo3448309680560233919inuous @ A @ B @ ( topolo174197925503356063within @ A @ A3 @ ( top_top @ ( set @ A ) ) )
@ ^ [X3: A] : ( cos @ B @ ( F2 @ X3 ) ) ) ) ) ).
% isCont_cos'
thf(fact_7170_isCont__sin_H,axiom,
! [B: $tType,A: $tType] :
( ( ( topological_t2_space @ A )
& ( real_Vector_banach @ B )
& ( real_V3459762299906320749_field @ B ) )
=> ! [A3: A,F2: A > B] :
( ( topolo3448309680560233919inuous @ A @ B @ ( topolo174197925503356063within @ A @ A3 @ ( top_top @ ( set @ A ) ) ) @ F2 )
=> ( topolo3448309680560233919inuous @ A @ B @ ( topolo174197925503356063within @ A @ A3 @ ( top_top @ ( set @ A ) ) )
@ ^ [X3: A] : ( sin @ B @ ( F2 @ X3 ) ) ) ) ) ).
% isCont_sin'
thf(fact_7171_isCont__exp_H,axiom,
! [A: $tType,C: $tType] :
( ( ( topological_t2_space @ C )
& ( real_Vector_banach @ A )
& ( real_V3459762299906320749_field @ A ) )
=> ! [A3: C,F2: C > A] :
( ( topolo3448309680560233919inuous @ C @ A @ ( topolo174197925503356063within @ C @ A3 @ ( top_top @ ( set @ C ) ) ) @ F2 )
=> ( topolo3448309680560233919inuous @ C @ A @ ( topolo174197925503356063within @ C @ A3 @ ( top_top @ ( set @ C ) ) )
@ ^ [X3: C] : ( exp @ A @ ( F2 @ X3 ) ) ) ) ) ).
% isCont_exp'
thf(fact_7172_isCont__pochhammer,axiom,
! [A: $tType] :
( ( real_V3459762299906320749_field @ A )
=> ! [Z: A,N: nat] :
( topolo3448309680560233919inuous @ A @ A @ ( topolo174197925503356063within @ A @ Z @ ( top_top @ ( set @ A ) ) )
@ ^ [Z4: A] : ( comm_s3205402744901411588hammer @ A @ Z4 @ N ) ) ) ).
% isCont_pochhammer
thf(fact_7173_DERIV__D,axiom,
! [A: $tType] :
( ( real_V3459762299906320749_field @ A )
=> ! [F2: A > A,D: A,X2: A] :
( ( has_field_derivative @ A @ F2 @ D @ ( topolo174197925503356063within @ A @ X2 @ ( top_top @ ( set @ A ) ) ) )
=> ( filterlim @ A @ A
@ ^ [H: A] : ( divide_divide @ A @ ( minus_minus @ A @ ( F2 @ ( plus_plus @ A @ X2 @ H ) ) @ ( F2 @ X2 ) ) @ H )
@ ( topolo7230453075368039082e_nhds @ A @ D )
@ ( topolo174197925503356063within @ A @ ( zero_zero @ A ) @ ( top_top @ ( set @ A ) ) ) ) ) ) ).
% DERIV_D
thf(fact_7174_DERIV__def,axiom,
! [A: $tType] :
( ( real_V3459762299906320749_field @ A )
=> ! [F2: A > A,D: A,X2: A] :
( ( has_field_derivative @ A @ F2 @ D @ ( topolo174197925503356063within @ A @ X2 @ ( top_top @ ( set @ A ) ) ) )
= ( filterlim @ A @ A
@ ^ [H: A] : ( divide_divide @ A @ ( minus_minus @ A @ ( F2 @ ( plus_plus @ A @ X2 @ H ) ) @ ( F2 @ X2 ) ) @ H )
@ ( topolo7230453075368039082e_nhds @ A @ D )
@ ( topolo174197925503356063within @ A @ ( zero_zero @ A ) @ ( top_top @ ( set @ A ) ) ) ) ) ) ).
% DERIV_def
thf(fact_7175_lim__exp__minus__1,axiom,
! [A: $tType] :
( ( ( real_Vector_banach @ A )
& ( real_V3459762299906320749_field @ A ) )
=> ( filterlim @ A @ A
@ ^ [Z4: A] : ( divide_divide @ A @ ( minus_minus @ A @ ( exp @ A @ Z4 ) @ ( one_one @ A ) ) @ Z4 )
@ ( topolo7230453075368039082e_nhds @ A @ ( one_one @ A ) )
@ ( topolo174197925503356063within @ A @ ( zero_zero @ A ) @ ( top_top @ ( set @ A ) ) ) ) ) ).
% lim_exp_minus_1
thf(fact_7176_atLeastPlusOneLessThan__greaterThanLessThan__int,axiom,
! [L: int,U: int] :
( ( set_or7035219750837199246ssThan @ int @ ( plus_plus @ int @ L @ ( one_one @ int ) ) @ U )
= ( set_or5935395276787703475ssThan @ int @ L @ U ) ) ).
% atLeastPlusOneLessThan_greaterThanLessThan_int
thf(fact_7177_lemma__termdiff4,axiom,
! [B: $tType,A: $tType] :
( ( ( real_V822414075346904944vector @ A )
& ( real_V822414075346904944vector @ B ) )
=> ! [K: real,F2: A > B,K6: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ K )
=> ( ! [H3: A] :
( ( H3
!= ( zero_zero @ A ) )
=> ( ( ord_less @ real @ ( real_V7770717601297561774m_norm @ A @ H3 ) @ K )
=> ( ord_less_eq @ real @ ( real_V7770717601297561774m_norm @ B @ ( F2 @ H3 ) ) @ ( times_times @ real @ K6 @ ( real_V7770717601297561774m_norm @ A @ H3 ) ) ) ) )
=> ( filterlim @ A @ B @ F2 @ ( topolo7230453075368039082e_nhds @ B @ ( zero_zero @ B ) ) @ ( topolo174197925503356063within @ A @ ( zero_zero @ A ) @ ( top_top @ ( set @ A ) ) ) ) ) ) ) ).
% lemma_termdiff4
thf(fact_7178_isCont__bounded,axiom,
! [A: $tType] :
( ( topolo1944317154257567458pology @ A )
=> ! [A3: real,B3: real,F2: real > A] :
( ( ord_less_eq @ real @ A3 @ B3 )
=> ( ! [X4: real] :
( ( ( ord_less_eq @ real @ A3 @ X4 )
& ( ord_less_eq @ real @ X4 @ B3 ) )
=> ( topolo3448309680560233919inuous @ real @ A @ ( topolo174197925503356063within @ real @ X4 @ ( top_top @ ( set @ real ) ) ) @ F2 ) )
=> ? [M11: A] :
! [X8: real] :
( ( ( ord_less_eq @ real @ A3 @ X8 )
& ( ord_less_eq @ real @ X8 @ B3 ) )
=> ( ord_less_eq @ A @ ( F2 @ X8 ) @ M11 ) ) ) ) ) ).
% isCont_bounded
thf(fact_7179_isCont__eq__Ub,axiom,
! [A: $tType] :
( ( topolo1944317154257567458pology @ A )
=> ! [A3: real,B3: real,F2: real > A] :
( ( ord_less_eq @ real @ A3 @ B3 )
=> ( ! [X4: real] :
( ( ( ord_less_eq @ real @ A3 @ X4 )
& ( ord_less_eq @ real @ X4 @ B3 ) )
=> ( topolo3448309680560233919inuous @ real @ A @ ( topolo174197925503356063within @ real @ X4 @ ( top_top @ ( set @ real ) ) ) @ F2 ) )
=> ? [M11: A] :
( ! [X8: real] :
( ( ( ord_less_eq @ real @ A3 @ X8 )
& ( ord_less_eq @ real @ X8 @ B3 ) )
=> ( ord_less_eq @ A @ ( F2 @ X8 ) @ M11 ) )
& ? [X4: real] :
( ( ord_less_eq @ real @ A3 @ X4 )
& ( ord_less_eq @ real @ X4 @ B3 )
& ( ( F2 @ X4 )
= M11 ) ) ) ) ) ) ).
% isCont_eq_Ub
thf(fact_7180_isCont__eq__Lb,axiom,
! [A: $tType] :
( ( topolo1944317154257567458pology @ A )
=> ! [A3: real,B3: real,F2: real > A] :
( ( ord_less_eq @ real @ A3 @ B3 )
=> ( ! [X4: real] :
( ( ( ord_less_eq @ real @ A3 @ X4 )
& ( ord_less_eq @ real @ X4 @ B3 ) )
=> ( topolo3448309680560233919inuous @ real @ A @ ( topolo174197925503356063within @ real @ X4 @ ( top_top @ ( set @ real ) ) ) @ F2 ) )
=> ? [M11: A] :
( ! [X8: real] :
( ( ( ord_less_eq @ real @ A3 @ X8 )
& ( ord_less_eq @ real @ X8 @ B3 ) )
=> ( ord_less_eq @ A @ M11 @ ( F2 @ X8 ) ) )
& ? [X4: real] :
( ( ord_less_eq @ real @ A3 @ X4 )
& ( ord_less_eq @ real @ X4 @ B3 )
& ( ( F2 @ X4 )
= M11 ) ) ) ) ) ) ).
% isCont_eq_Lb
thf(fact_7181_isCont__inverse__function2,axiom,
! [A3: real,X2: real,B3: real,G: real > real,F2: real > real] :
( ( ord_less @ real @ A3 @ X2 )
=> ( ( ord_less @ real @ X2 @ B3 )
=> ( ! [Z2: real] :
( ( ord_less_eq @ real @ A3 @ Z2 )
=> ( ( ord_less_eq @ real @ Z2 @ B3 )
=> ( ( G @ ( F2 @ Z2 ) )
= Z2 ) ) )
=> ( ! [Z2: real] :
( ( ord_less_eq @ real @ A3 @ Z2 )
=> ( ( ord_less_eq @ real @ Z2 @ B3 )
=> ( topolo3448309680560233919inuous @ real @ real @ ( topolo174197925503356063within @ real @ Z2 @ ( top_top @ ( set @ real ) ) ) @ F2 ) ) )
=> ( topolo3448309680560233919inuous @ real @ real @ ( topolo174197925503356063within @ real @ ( F2 @ X2 ) @ ( top_top @ ( set @ real ) ) ) @ G ) ) ) ) ) ).
% isCont_inverse_function2
thf(fact_7182_field__has__derivative__at,axiom,
! [A: $tType] :
( ( real_V3459762299906320749_field @ A )
=> ! [F2: A > A,D: A,X2: A] :
( ( has_derivative @ A @ A @ F2 @ ( times_times @ A @ D ) @ ( topolo174197925503356063within @ A @ X2 @ ( top_top @ ( set @ A ) ) ) )
= ( filterlim @ A @ A
@ ^ [H: A] : ( divide_divide @ A @ ( minus_minus @ A @ ( F2 @ ( plus_plus @ A @ X2 @ H ) ) @ ( F2 @ X2 ) ) @ H )
@ ( topolo7230453075368039082e_nhds @ A @ D )
@ ( topolo174197925503356063within @ A @ ( zero_zero @ A ) @ ( top_top @ ( set @ A ) ) ) ) ) ) ).
% field_has_derivative_at
thf(fact_7183_isCont__divide,axiom,
! [B: $tType,A: $tType] :
( ( ( topological_t2_space @ A )
& ( real_V3459762299906320749_field @ B ) )
=> ! [A3: A,F2: A > B,G: A > B] :
( ( topolo3448309680560233919inuous @ A @ B @ ( topolo174197925503356063within @ A @ A3 @ ( top_top @ ( set @ A ) ) ) @ F2 )
=> ( ( topolo3448309680560233919inuous @ A @ B @ ( topolo174197925503356063within @ A @ A3 @ ( top_top @ ( set @ A ) ) ) @ G )
=> ( ( ( G @ A3 )
!= ( zero_zero @ B ) )
=> ( topolo3448309680560233919inuous @ A @ B @ ( topolo174197925503356063within @ A @ A3 @ ( top_top @ ( set @ A ) ) )
@ ^ [X3: A] : ( divide_divide @ B @ ( F2 @ X3 ) @ ( G @ X3 ) ) ) ) ) ) ) ).
% isCont_divide
thf(fact_7184_isCont__sgn,axiom,
! [B: $tType,A: $tType] :
( ( ( topological_t2_space @ A )
& ( real_V822414075346904944vector @ B ) )
=> ! [A3: A,F2: A > B] :
( ( topolo3448309680560233919inuous @ A @ B @ ( topolo174197925503356063within @ A @ A3 @ ( top_top @ ( set @ A ) ) ) @ F2 )
=> ( ( ( F2 @ A3 )
!= ( zero_zero @ B ) )
=> ( topolo3448309680560233919inuous @ A @ B @ ( topolo174197925503356063within @ A @ A3 @ ( top_top @ ( set @ A ) ) )
@ ^ [X3: A] : ( sgn_sgn @ B @ ( F2 @ X3 ) ) ) ) ) ) ).
% isCont_sgn
thf(fact_7185_filterlim__at__to__0,axiom,
! [B: $tType,A: $tType] :
( ( real_V822414075346904944vector @ A )
=> ! [F2: A > B,F4: filter @ B,A3: A] :
( ( filterlim @ A @ B @ F2 @ F4 @ ( topolo174197925503356063within @ A @ A3 @ ( top_top @ ( set @ A ) ) ) )
= ( filterlim @ A @ B
@ ^ [X3: A] : ( F2 @ ( plus_plus @ A @ X3 @ A3 ) )
@ F4
@ ( topolo174197925503356063within @ A @ ( zero_zero @ A ) @ ( top_top @ ( set @ A ) ) ) ) ) ) ).
% filterlim_at_to_0
thf(fact_7186_continuous__within__tan,axiom,
! [A: $tType] :
( ( ( real_Vector_banach @ A )
& ( real_V3459762299906320749_field @ A ) )
=> ! [X2: A,S2: set @ A,F2: A > A] :
( ( topolo3448309680560233919inuous @ A @ A @ ( topolo174197925503356063within @ A @ X2 @ S2 ) @ F2 )
=> ( ( ( cos @ A @ ( F2 @ X2 ) )
!= ( zero_zero @ A ) )
=> ( topolo3448309680560233919inuous @ A @ A @ ( topolo174197925503356063within @ A @ X2 @ S2 )
@ ^ [X3: A] : ( tan @ A @ ( F2 @ X3 ) ) ) ) ) ) ).
% continuous_within_tan
thf(fact_7187_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_7188_continuous__within__cot,axiom,
! [A: $tType] :
( ( ( real_Vector_banach @ A )
& ( real_V3459762299906320749_field @ A ) )
=> ! [X2: A,S2: set @ A,F2: A > A] :
( ( topolo3448309680560233919inuous @ A @ A @ ( topolo174197925503356063within @ A @ X2 @ S2 ) @ F2 )
=> ( ( ( sin @ A @ ( F2 @ X2 ) )
!= ( zero_zero @ A ) )
=> ( topolo3448309680560233919inuous @ A @ A @ ( topolo174197925503356063within @ A @ X2 @ S2 )
@ ^ [X3: A] : ( cot @ A @ ( F2 @ X3 ) ) ) ) ) ) ).
% continuous_within_cot
thf(fact_7189_continuous__at__within__tanh,axiom,
! [A: $tType,C: $tType] :
( ( ( topological_t2_space @ C )
& ( real_Vector_banach @ A )
& ( real_V3459762299906320749_field @ A ) )
=> ! [X2: C,A2: set @ C,F2: C > A] :
( ( topolo3448309680560233919inuous @ C @ A @ ( topolo174197925503356063within @ C @ X2 @ A2 ) @ F2 )
=> ( ( ( cosh @ A @ ( F2 @ X2 ) )
!= ( zero_zero @ A ) )
=> ( topolo3448309680560233919inuous @ C @ A @ ( topolo174197925503356063within @ C @ X2 @ A2 )
@ ^ [X3: C] : ( tanh @ A @ ( F2 @ X3 ) ) ) ) ) ) ).
% continuous_at_within_tanh
thf(fact_7190_CARAT__DERIV,axiom,
! [A: $tType] :
( ( real_V3459762299906320749_field @ A )
=> ! [F2: A > A,L: A,X2: A] :
( ( has_field_derivative @ A @ F2 @ L @ ( topolo174197925503356063within @ A @ X2 @ ( top_top @ ( set @ A ) ) ) )
= ( ? [G2: A > A] :
( ! [Z4: A] :
( ( minus_minus @ A @ ( F2 @ Z4 ) @ ( F2 @ X2 ) )
= ( times_times @ A @ ( G2 @ Z4 ) @ ( minus_minus @ A @ Z4 @ X2 ) ) )
& ( topolo3448309680560233919inuous @ A @ A @ ( topolo174197925503356063within @ A @ X2 @ ( top_top @ ( set @ A ) ) ) @ G2 )
& ( ( G2 @ X2 )
= L ) ) ) ) ) ).
% CARAT_DERIV
thf(fact_7191_isCont__has__Ub,axiom,
! [A: $tType] :
( ( topolo1944317154257567458pology @ A )
=> ! [A3: real,B3: real,F2: real > A] :
( ( ord_less_eq @ real @ A3 @ B3 )
=> ( ! [X4: real] :
( ( ( ord_less_eq @ real @ A3 @ X4 )
& ( ord_less_eq @ real @ X4 @ B3 ) )
=> ( topolo3448309680560233919inuous @ real @ A @ ( topolo174197925503356063within @ real @ X4 @ ( top_top @ ( set @ real ) ) ) @ F2 ) )
=> ? [M11: A] :
( ! [X8: real] :
( ( ( ord_less_eq @ real @ A3 @ X8 )
& ( ord_less_eq @ real @ X8 @ B3 ) )
=> ( ord_less_eq @ A @ ( F2 @ X8 ) @ M11 ) )
& ! [N10: A] :
( ( ord_less @ A @ N10 @ M11 )
=> ? [X4: real] :
( ( ord_less_eq @ real @ A3 @ X4 )
& ( ord_less_eq @ real @ X4 @ B3 )
& ( ord_less @ A @ N10 @ ( F2 @ X4 ) ) ) ) ) ) ) ) ).
% isCont_has_Ub
thf(fact_7192_powser__limit__0__strong,axiom,
! [A: $tType] :
( ( ( real_Vector_banach @ A )
& ( real_V3459762299906320749_field @ A ) )
=> ! [S2: real,A3: nat > A,F2: A > A] :
( ( ord_less @ real @ ( zero_zero @ real ) @ S2 )
=> ( ! [X4: A] :
( ( X4
!= ( zero_zero @ A ) )
=> ( ( ord_less @ real @ ( real_V7770717601297561774m_norm @ A @ X4 ) @ S2 )
=> ( sums @ A
@ ^ [N4: nat] : ( times_times @ A @ ( A3 @ N4 ) @ ( power_power @ A @ X4 @ N4 ) )
@ ( F2 @ X4 ) ) ) )
=> ( filterlim @ A @ A @ F2 @ ( topolo7230453075368039082e_nhds @ A @ ( A3 @ ( zero_zero @ nat ) ) ) @ ( topolo174197925503356063within @ A @ ( zero_zero @ A ) @ ( top_top @ ( set @ A ) ) ) ) ) ) ) ).
% powser_limit_0_strong
thf(fact_7193_powser__limit__0,axiom,
! [A: $tType] :
( ( ( real_Vector_banach @ A )
& ( real_V3459762299906320749_field @ A ) )
=> ! [S2: real,A3: nat > A,F2: A > A] :
( ( ord_less @ real @ ( zero_zero @ real ) @ S2 )
=> ( ! [X4: A] :
( ( ord_less @ real @ ( real_V7770717601297561774m_norm @ A @ X4 ) @ S2 )
=> ( sums @ A
@ ^ [N4: nat] : ( times_times @ A @ ( A3 @ N4 ) @ ( power_power @ A @ X4 @ N4 ) )
@ ( F2 @ X4 ) ) )
=> ( filterlim @ A @ A @ F2 @ ( topolo7230453075368039082e_nhds @ A @ ( A3 @ ( zero_zero @ nat ) ) ) @ ( topolo174197925503356063within @ A @ ( zero_zero @ A ) @ ( top_top @ ( set @ A ) ) ) ) ) ) ) ).
% powser_limit_0
thf(fact_7194_lemma__termdiff5,axiom,
! [B: $tType,A: $tType] :
( ( ( real_V822414075346904944vector @ A )
& ( real_Vector_banach @ B ) )
=> ! [K: real,F2: nat > real,G: A > nat > B] :
( ( ord_less @ real @ ( zero_zero @ real ) @ K )
=> ( ( summable @ real @ F2 )
=> ( ! [H3: A,N3: nat] :
( ( H3
!= ( zero_zero @ A ) )
=> ( ( ord_less @ real @ ( real_V7770717601297561774m_norm @ A @ H3 ) @ K )
=> ( ord_less_eq @ real @ ( real_V7770717601297561774m_norm @ B @ ( G @ H3 @ N3 ) ) @ ( times_times @ real @ ( F2 @ N3 ) @ ( real_V7770717601297561774m_norm @ A @ H3 ) ) ) ) )
=> ( filterlim @ A @ B
@ ^ [H: A] : ( suminf @ B @ ( G @ H ) )
@ ( topolo7230453075368039082e_nhds @ B @ ( zero_zero @ B ) )
@ ( topolo174197925503356063within @ A @ ( zero_zero @ A ) @ ( top_top @ ( set @ A ) ) ) ) ) ) ) ) ).
% lemma_termdiff5
thf(fact_7195_isCont__tan_H,axiom,
! [A: $tType] :
( ( ( real_Vector_banach @ A )
& ( real_V3459762299906320749_field @ A ) )
=> ! [A3: A,F2: A > A] :
( ( topolo3448309680560233919inuous @ A @ A @ ( topolo174197925503356063within @ A @ A3 @ ( top_top @ ( set @ A ) ) ) @ F2 )
=> ( ( ( cos @ A @ ( F2 @ A3 ) )
!= ( zero_zero @ A ) )
=> ( topolo3448309680560233919inuous @ A @ A @ ( topolo174197925503356063within @ A @ A3 @ ( top_top @ ( set @ A ) ) )
@ ^ [X3: A] : ( tan @ A @ ( F2 @ X3 ) ) ) ) ) ) ).
% isCont_tan'
thf(fact_7196_isCont__arcosh,axiom,
! [X2: real] :
( ( ord_less @ real @ ( one_one @ real ) @ X2 )
=> ( topolo3448309680560233919inuous @ real @ real @ ( topolo174197925503356063within @ real @ X2 @ ( top_top @ ( set @ real ) ) ) @ ( arcosh @ real ) ) ) ).
% isCont_arcosh
thf(fact_7197_LIM__cos__div__sin,axiom,
( filterlim @ real @ real
@ ^ [X3: real] : ( divide_divide @ real @ ( cos @ real @ X3 ) @ ( sin @ real @ X3 ) )
@ ( topolo7230453075368039082e_nhds @ real @ ( zero_zero @ real ) )
@ ( topolo174197925503356063within @ real @ ( divide_divide @ real @ pi @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) @ ( top_top @ ( set @ real ) ) ) ) ).
% LIM_cos_div_sin
thf(fact_7198_isCont__cot_H,axiom,
! [A: $tType] :
( ( ( real_Vector_banach @ A )
& ( real_V3459762299906320749_field @ A ) )
=> ! [A3: A,F2: A > A] :
( ( topolo3448309680560233919inuous @ A @ A @ ( topolo174197925503356063within @ A @ A3 @ ( top_top @ ( set @ A ) ) ) @ F2 )
=> ( ( ( sin @ A @ ( F2 @ A3 ) )
!= ( zero_zero @ A ) )
=> ( topolo3448309680560233919inuous @ A @ A @ ( topolo174197925503356063within @ A @ A3 @ ( top_top @ ( set @ A ) ) )
@ ^ [X3: A] : ( cot @ A @ ( F2 @ X3 ) ) ) ) ) ) ).
% isCont_cot'
thf(fact_7199_isCont__polynom,axiom,
! [A: $tType] :
( ( real_V8999393235501362500lgebra @ A )
=> ! [A3: A,C3: nat > A,N: nat] :
( topolo3448309680560233919inuous @ A @ A @ ( topolo174197925503356063within @ A @ A3 @ ( top_top @ ( set @ A ) ) )
@ ^ [W2: A] :
( groups7311177749621191930dd_sum @ nat @ A
@ ^ [I4: nat] : ( times_times @ A @ ( C3 @ I4 ) @ ( power_power @ A @ W2 @ I4 ) )
@ ( set_ord_atMost @ nat @ N ) ) ) ) ).
% isCont_polynom
thf(fact_7200_isCont__powser__converges__everywhere,axiom,
! [A: $tType] :
( ( ( real_Vector_banach @ A )
& ( real_V3459762299906320749_field @ A ) )
=> ! [C3: nat > A,X2: A] :
( ! [Y4: A] :
( summable @ A
@ ^ [N4: nat] : ( times_times @ A @ ( C3 @ N4 ) @ ( power_power @ A @ Y4 @ N4 ) ) )
=> ( topolo3448309680560233919inuous @ A @ A @ ( topolo174197925503356063within @ A @ X2 @ ( top_top @ ( set @ A ) ) )
@ ^ [X3: A] :
( suminf @ A
@ ^ [N4: nat] : ( times_times @ A @ ( C3 @ N4 ) @ ( power_power @ A @ X3 @ N4 ) ) ) ) ) ) ).
% isCont_powser_converges_everywhere
thf(fact_7201_isCont__arccos,axiom,
! [X2: real] :
( ( ord_less @ real @ ( uminus_uminus @ real @ ( one_one @ real ) ) @ X2 )
=> ( ( ord_less @ real @ X2 @ ( one_one @ real ) )
=> ( topolo3448309680560233919inuous @ real @ real @ ( topolo174197925503356063within @ real @ X2 @ ( top_top @ ( set @ real ) ) ) @ arccos ) ) ) ).
% isCont_arccos
thf(fact_7202_isCont__arcsin,axiom,
! [X2: real] :
( ( ord_less @ real @ ( uminus_uminus @ real @ ( one_one @ real ) ) @ X2 )
=> ( ( ord_less @ real @ X2 @ ( one_one @ real ) )
=> ( topolo3448309680560233919inuous @ real @ real @ ( topolo174197925503356063within @ real @ X2 @ ( top_top @ ( set @ real ) ) ) @ arcsin ) ) ) ).
% isCont_arcsin
thf(fact_7203_LIM__less__bound,axiom,
! [B3: real,X2: real,F2: real > real] :
( ( ord_less @ real @ B3 @ X2 )
=> ( ! [X4: real] :
( ( member @ real @ X4 @ ( set_or5935395276787703475ssThan @ real @ B3 @ X2 ) )
=> ( ord_less_eq @ real @ ( zero_zero @ real ) @ ( F2 @ X4 ) ) )
=> ( ( topolo3448309680560233919inuous @ real @ real @ ( topolo174197925503356063within @ real @ X2 @ ( top_top @ ( set @ real ) ) ) @ F2 )
=> ( ord_less_eq @ real @ ( zero_zero @ real ) @ ( F2 @ X2 ) ) ) ) ) ).
% LIM_less_bound
thf(fact_7204_isCont__artanh,axiom,
! [X2: real] :
( ( ord_less @ real @ ( uminus_uminus @ real @ ( one_one @ real ) ) @ X2 )
=> ( ( ord_less @ real @ X2 @ ( one_one @ real ) )
=> ( topolo3448309680560233919inuous @ real @ real @ ( topolo174197925503356063within @ real @ X2 @ ( top_top @ ( set @ real ) ) ) @ ( artanh @ real ) ) ) ) ).
% isCont_artanh
thf(fact_7205_isCont__inverse__function,axiom,
! [D2: real,X2: real,G: real > real,F2: real > real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ D2 )
=> ( ! [Z2: real] :
( ( ord_less_eq @ real @ ( abs_abs @ real @ ( minus_minus @ real @ Z2 @ X2 ) ) @ D2 )
=> ( ( G @ ( F2 @ Z2 ) )
= Z2 ) )
=> ( ! [Z2: real] :
( ( ord_less_eq @ real @ ( abs_abs @ real @ ( minus_minus @ real @ Z2 @ X2 ) ) @ D2 )
=> ( topolo3448309680560233919inuous @ real @ real @ ( topolo174197925503356063within @ real @ Z2 @ ( top_top @ ( set @ real ) ) ) @ F2 ) )
=> ( topolo3448309680560233919inuous @ real @ real @ ( topolo174197925503356063within @ real @ ( F2 @ X2 ) @ ( top_top @ ( set @ real ) ) ) @ G ) ) ) ) ).
% isCont_inverse_function
thf(fact_7206_GMVT_H,axiom,
! [A3: real,B3: real,F2: real > real,G: real > real,G5: real > real,F7: real > real] :
( ( ord_less @ real @ A3 @ B3 )
=> ( ! [Z2: real] :
( ( ord_less_eq @ real @ A3 @ Z2 )
=> ( ( ord_less_eq @ real @ Z2 @ B3 )
=> ( topolo3448309680560233919inuous @ real @ real @ ( topolo174197925503356063within @ real @ Z2 @ ( top_top @ ( set @ real ) ) ) @ F2 ) ) )
=> ( ! [Z2: real] :
( ( ord_less_eq @ real @ A3 @ Z2 )
=> ( ( ord_less_eq @ real @ Z2 @ B3 )
=> ( topolo3448309680560233919inuous @ real @ real @ ( topolo174197925503356063within @ real @ Z2 @ ( top_top @ ( set @ real ) ) ) @ G ) ) )
=> ( ! [Z2: real] :
( ( ord_less @ real @ A3 @ Z2 )
=> ( ( ord_less @ real @ Z2 @ B3 )
=> ( has_field_derivative @ real @ G @ ( G5 @ Z2 ) @ ( topolo174197925503356063within @ real @ Z2 @ ( top_top @ ( set @ real ) ) ) ) ) )
=> ( ! [Z2: real] :
( ( ord_less @ real @ A3 @ Z2 )
=> ( ( ord_less @ real @ Z2 @ B3 )
=> ( has_field_derivative @ real @ F2 @ ( F7 @ Z2 ) @ ( topolo174197925503356063within @ real @ Z2 @ ( top_top @ ( set @ real ) ) ) ) ) )
=> ? [C4: real] :
( ( ord_less @ real @ A3 @ C4 )
& ( ord_less @ real @ C4 @ B3 )
& ( ( times_times @ real @ ( minus_minus @ real @ ( F2 @ B3 ) @ ( F2 @ A3 ) ) @ ( G5 @ C4 ) )
= ( times_times @ real @ ( minus_minus @ real @ ( G @ B3 ) @ ( G @ A3 ) ) @ ( F7 @ C4 ) ) ) ) ) ) ) ) ) ).
% GMVT'
thf(fact_7207_floor__has__real__derivative,axiom,
! [A: $tType] :
( ( ( archim2362893244070406136eiling @ A )
& ( topolo2564578578187576103pology @ A ) )
=> ! [X2: real,F2: real > A] :
( ( topolo3448309680560233919inuous @ real @ A @ ( topolo174197925503356063within @ real @ X2 @ ( top_top @ ( set @ real ) ) ) @ F2 )
=> ( ~ ( member @ A @ ( F2 @ X2 ) @ ( ring_1_Ints @ A ) )
=> ( has_field_derivative @ real
@ ^ [X3: real] : ( ring_1_of_int @ real @ ( archim6421214686448440834_floor @ A @ ( F2 @ X3 ) ) )
@ ( zero_zero @ real )
@ ( topolo174197925503356063within @ real @ X2 @ ( top_top @ ( set @ real ) ) ) ) ) ) ) ).
% floor_has_real_derivative
thf(fact_7208_isCont__powser_H,axiom,
! [Aa: $tType,A: $tType] :
( ( ( topological_t2_space @ A )
& ( real_Vector_banach @ Aa )
& ( real_V3459762299906320749_field @ Aa ) )
=> ! [A3: A,F2: A > Aa,C3: nat > Aa,K6: Aa] :
( ( topolo3448309680560233919inuous @ A @ Aa @ ( topolo174197925503356063within @ A @ A3 @ ( top_top @ ( set @ A ) ) ) @ F2 )
=> ( ( summable @ Aa
@ ^ [N4: nat] : ( times_times @ Aa @ ( C3 @ N4 ) @ ( power_power @ Aa @ K6 @ N4 ) ) )
=> ( ( ord_less @ real @ ( real_V7770717601297561774m_norm @ Aa @ ( F2 @ A3 ) ) @ ( real_V7770717601297561774m_norm @ Aa @ K6 ) )
=> ( topolo3448309680560233919inuous @ A @ Aa @ ( topolo174197925503356063within @ A @ A3 @ ( top_top @ ( set @ A ) ) )
@ ^ [X3: A] :
( suminf @ Aa
@ ^ [N4: nat] : ( times_times @ Aa @ ( C3 @ N4 ) @ ( power_power @ Aa @ ( F2 @ X3 ) @ N4 ) ) ) ) ) ) ) ) ).
% isCont_powser'
thf(fact_7209_isCont__powser,axiom,
! [A: $tType] :
( ( ( real_Vector_banach @ A )
& ( real_V3459762299906320749_field @ A ) )
=> ! [C3: nat > A,K6: A,X2: A] :
( ( summable @ A
@ ^ [N4: nat] : ( times_times @ A @ ( C3 @ N4 ) @ ( power_power @ A @ K6 @ N4 ) ) )
=> ( ( ord_less @ real @ ( real_V7770717601297561774m_norm @ A @ X2 ) @ ( real_V7770717601297561774m_norm @ A @ K6 ) )
=> ( topolo3448309680560233919inuous @ A @ A @ ( topolo174197925503356063within @ A @ X2 @ ( top_top @ ( set @ A ) ) )
@ ^ [X3: A] :
( suminf @ A
@ ^ [N4: nat] : ( times_times @ A @ ( C3 @ N4 ) @ ( power_power @ A @ X3 @ N4 ) ) ) ) ) ) ) ).
% isCont_powser
thf(fact_7210_summable__Leibniz_I2_J,axiom,
! [A3: nat > real] :
( ( filterlim @ nat @ real @ A3 @ ( topolo7230453075368039082e_nhds @ real @ ( zero_zero @ real ) ) @ ( at_top @ nat ) )
=> ( ( topological_monoseq @ real @ A3 )
=> ( ( ord_less @ real @ ( zero_zero @ real ) @ ( A3 @ ( zero_zero @ nat ) ) )
=> ! [N9: nat] :
( member @ real
@ ( suminf @ real
@ ^ [I4: nat] : ( times_times @ real @ ( power_power @ real @ ( uminus_uminus @ real @ ( one_one @ real ) ) @ I4 ) @ ( A3 @ I4 ) ) )
@ ( set_or1337092689740270186AtMost @ real
@ ( groups7311177749621191930dd_sum @ nat @ real
@ ^ [I4: nat] : ( times_times @ real @ ( power_power @ real @ ( uminus_uminus @ real @ ( one_one @ real ) ) @ I4 ) @ ( A3 @ I4 ) )
@ ( set_ord_lessThan @ nat @ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N9 ) ) )
@ ( groups7311177749621191930dd_sum @ nat @ real
@ ^ [I4: nat] : ( times_times @ real @ ( power_power @ real @ ( uminus_uminus @ real @ ( one_one @ real ) ) @ I4 ) @ ( A3 @ I4 ) )
@ ( set_ord_lessThan @ nat @ ( plus_plus @ nat @ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N9 ) @ ( one_one @ nat ) ) ) ) ) ) ) ) ) ).
% summable_Leibniz(2)
thf(fact_7211_summable__Leibniz_I3_J,axiom,
! [A3: nat > real] :
( ( filterlim @ nat @ real @ A3 @ ( topolo7230453075368039082e_nhds @ real @ ( zero_zero @ real ) ) @ ( at_top @ nat ) )
=> ( ( topological_monoseq @ real @ A3 )
=> ( ( ord_less @ real @ ( A3 @ ( zero_zero @ nat ) ) @ ( zero_zero @ real ) )
=> ! [N9: nat] :
( member @ real
@ ( suminf @ real
@ ^ [I4: nat] : ( times_times @ real @ ( power_power @ real @ ( uminus_uminus @ real @ ( one_one @ real ) ) @ I4 ) @ ( A3 @ I4 ) ) )
@ ( set_or1337092689740270186AtMost @ real
@ ( groups7311177749621191930dd_sum @ nat @ real
@ ^ [I4: nat] : ( times_times @ real @ ( power_power @ real @ ( uminus_uminus @ real @ ( one_one @ real ) ) @ I4 ) @ ( A3 @ I4 ) )
@ ( set_ord_lessThan @ nat @ ( plus_plus @ nat @ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N9 ) @ ( one_one @ nat ) ) ) )
@ ( groups7311177749621191930dd_sum @ nat @ real
@ ^ [I4: nat] : ( times_times @ real @ ( power_power @ real @ ( uminus_uminus @ real @ ( one_one @ real ) ) @ I4 ) @ ( A3 @ I4 ) )
@ ( set_ord_lessThan @ nat @ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N9 ) ) ) ) ) ) ) ) ).
% summable_Leibniz(3)
thf(fact_7212_tendsto__zero__mult__right__iff,axiom,
! [A: $tType] :
( ( ( field @ A )
& ( topolo4211221413907600880p_mult @ A ) )
=> ! [C3: A,A3: nat > A] :
( ( C3
!= ( zero_zero @ A ) )
=> ( ( filterlim @ nat @ A
@ ^ [N4: nat] : ( times_times @ A @ ( A3 @ N4 ) @ C3 )
@ ( topolo7230453075368039082e_nhds @ A @ ( zero_zero @ A ) )
@ ( at_top @ nat ) )
= ( filterlim @ nat @ A @ A3 @ ( topolo7230453075368039082e_nhds @ A @ ( zero_zero @ A ) ) @ ( at_top @ nat ) ) ) ) ) ).
% tendsto_zero_mult_right_iff
thf(fact_7213_tendsto__zero__mult__left__iff,axiom,
! [A: $tType] :
( ( ( field @ A )
& ( topolo4211221413907600880p_mult @ A ) )
=> ! [C3: A,A3: nat > A] :
( ( C3
!= ( zero_zero @ A ) )
=> ( ( filterlim @ nat @ A
@ ^ [N4: nat] : ( times_times @ A @ C3 @ ( A3 @ N4 ) )
@ ( topolo7230453075368039082e_nhds @ A @ ( zero_zero @ A ) )
@ ( at_top @ nat ) )
= ( filterlim @ nat @ A @ A3 @ ( topolo7230453075368039082e_nhds @ A @ ( zero_zero @ A ) ) @ ( at_top @ nat ) ) ) ) ) ).
% tendsto_zero_mult_left_iff
thf(fact_7214_tendsto__zero__divide__iff,axiom,
! [A: $tType] :
( ( ( field @ A )
& ( topolo4211221413907600880p_mult @ A ) )
=> ! [C3: A,A3: nat > A] :
( ( C3
!= ( zero_zero @ A ) )
=> ( ( filterlim @ nat @ A
@ ^ [N4: nat] : ( divide_divide @ A @ ( A3 @ N4 ) @ C3 )
@ ( topolo7230453075368039082e_nhds @ A @ ( zero_zero @ A ) )
@ ( at_top @ nat ) )
= ( filterlim @ nat @ A @ A3 @ ( topolo7230453075368039082e_nhds @ A @ ( zero_zero @ A ) ) @ ( at_top @ nat ) ) ) ) ) ).
% tendsto_zero_divide_iff
thf(fact_7215_approx__from__below__dense__linorder,axiom,
! [A: $tType] :
( ( ( dense_linorder @ A )
& ( topolo3112930676232923870pology @ A )
& ( topolo1944317154257567458pology @ A ) )
=> ! [Y3: A,X2: A] :
( ( ord_less @ A @ Y3 @ X2 )
=> ? [U4: nat > A] :
( ! [N9: nat] : ( ord_less @ A @ ( U4 @ N9 ) @ X2 )
& ( filterlim @ nat @ A @ U4 @ ( topolo7230453075368039082e_nhds @ A @ X2 ) @ ( at_top @ nat ) ) ) ) ) ).
% approx_from_below_dense_linorder
thf(fact_7216_approx__from__above__dense__linorder,axiom,
! [A: $tType] :
( ( ( dense_linorder @ A )
& ( topolo3112930676232923870pology @ A )
& ( topolo1944317154257567458pology @ A ) )
=> ! [X2: A,Y3: A] :
( ( ord_less @ A @ X2 @ Y3 )
=> ? [U4: nat > A] :
( ! [N9: nat] : ( ord_less @ A @ X2 @ ( U4 @ N9 ) )
& ( filterlim @ nat @ A @ U4 @ ( topolo7230453075368039082e_nhds @ A @ X2 ) @ ( at_top @ nat ) ) ) ) ) ).
% approx_from_above_dense_linorder
thf(fact_7217_continuous__rabs,axiom,
! [A: $tType] :
( ( topological_t2_space @ A )
=> ! [F4: filter @ A,F2: A > real] :
( ( topolo3448309680560233919inuous @ A @ real @ F4 @ F2 )
=> ( topolo3448309680560233919inuous @ A @ real @ F4
@ ^ [X3: A] : ( abs_abs @ real @ ( F2 @ X3 ) ) ) ) ) ).
% continuous_rabs
thf(fact_7218_continuous__real__root,axiom,
! [A: $tType] :
( ( topological_t2_space @ A )
=> ! [F4: filter @ A,F2: A > real,N: nat] :
( ( topolo3448309680560233919inuous @ A @ real @ F4 @ F2 )
=> ( topolo3448309680560233919inuous @ A @ real @ F4
@ ^ [X3: A] : ( root @ N @ ( F2 @ X3 ) ) ) ) ) ).
% continuous_real_root
thf(fact_7219_LIMSEQ__const__iff,axiom,
! [A: $tType] :
( ( topological_t2_space @ A )
=> ! [K: A,L: A] :
( ( filterlim @ nat @ A
@ ^ [N4: nat] : K
@ ( topolo7230453075368039082e_nhds @ A @ L )
@ ( at_top @ nat ) )
= ( K = L ) ) ) ).
% LIMSEQ_const_iff
thf(fact_7220_LIMSEQ__imp__Suc,axiom,
! [A: $tType] :
( ( topolo4958980785337419405_space @ A )
=> ! [F2: nat > A,L: A] :
( ( filterlim @ nat @ A
@ ^ [N4: nat] : ( F2 @ ( suc @ N4 ) )
@ ( topolo7230453075368039082e_nhds @ A @ L )
@ ( at_top @ nat ) )
=> ( filterlim @ nat @ A @ F2 @ ( topolo7230453075368039082e_nhds @ A @ L ) @ ( at_top @ nat ) ) ) ) ).
% LIMSEQ_imp_Suc
thf(fact_7221_LIMSEQ__Suc,axiom,
! [A: $tType] :
( ( topolo4958980785337419405_space @ A )
=> ! [F2: nat > A,L: A] :
( ( filterlim @ nat @ A @ F2 @ ( topolo7230453075368039082e_nhds @ A @ L ) @ ( at_top @ nat ) )
=> ( filterlim @ nat @ A
@ ^ [N4: nat] : ( F2 @ ( suc @ N4 ) )
@ ( topolo7230453075368039082e_nhds @ A @ L )
@ ( at_top @ nat ) ) ) ) ).
% LIMSEQ_Suc
thf(fact_7222_continuous__arctan,axiom,
! [A: $tType] :
( ( topological_t2_space @ A )
=> ! [F4: filter @ A,F2: A > real] :
( ( topolo3448309680560233919inuous @ A @ real @ F4 @ F2 )
=> ( topolo3448309680560233919inuous @ A @ real @ F4
@ ^ [X3: A] : ( arctan @ ( F2 @ X3 ) ) ) ) ) ).
% continuous_arctan
thf(fact_7223_continuous__real__sqrt,axiom,
! [A: $tType] :
( ( topological_t2_space @ A )
=> ! [F4: filter @ A,F2: A > real] :
( ( topolo3448309680560233919inuous @ A @ real @ F4 @ F2 )
=> ( topolo3448309680560233919inuous @ A @ real @ F4
@ ^ [X3: A] : ( sqrt @ ( F2 @ X3 ) ) ) ) ) ).
% continuous_real_sqrt
thf(fact_7224_continuous__arsinh,axiom,
! [A: $tType] :
( ( topological_t2_space @ A )
=> ! [F4: filter @ A,F2: A > real] :
( ( topolo3448309680560233919inuous @ A @ real @ F4 @ F2 )
=> ( topolo3448309680560233919inuous @ A @ real @ F4
@ ^ [X3: A] : ( arsinh @ real @ ( F2 @ X3 ) ) ) ) ) ).
% continuous_arsinh
thf(fact_7225_seq__offset__neg,axiom,
! [A: $tType] :
( ( topolo4958980785337419405_space @ A )
=> ! [F2: nat > A,L: A,K: nat] :
( ( filterlim @ nat @ A @ F2 @ ( topolo7230453075368039082e_nhds @ A @ L ) @ ( at_top @ nat ) )
=> ( filterlim @ nat @ A
@ ^ [I4: nat] : ( F2 @ ( minus_minus @ nat @ I4 @ K ) )
@ ( topolo7230453075368039082e_nhds @ A @ L )
@ ( at_top @ nat ) ) ) ) ).
% seq_offset_neg
thf(fact_7226_LIMSEQ__offset,axiom,
! [A: $tType] :
( ( topolo4958980785337419405_space @ A )
=> ! [F2: nat > A,K: nat,A3: A] :
( ( filterlim @ nat @ A
@ ^ [N4: nat] : ( F2 @ ( plus_plus @ nat @ N4 @ K ) )
@ ( topolo7230453075368039082e_nhds @ A @ A3 )
@ ( at_top @ nat ) )
=> ( filterlim @ nat @ A @ F2 @ ( topolo7230453075368039082e_nhds @ A @ A3 ) @ ( at_top @ nat ) ) ) ) ).
% LIMSEQ_offset
thf(fact_7227_LIMSEQ__ignore__initial__segment,axiom,
! [A: $tType] :
( ( topolo4958980785337419405_space @ A )
=> ! [F2: nat > A,A3: A,K: nat] :
( ( filterlim @ nat @ A @ F2 @ ( topolo7230453075368039082e_nhds @ A @ A3 ) @ ( at_top @ nat ) )
=> ( filterlim @ nat @ A
@ ^ [N4: nat] : ( F2 @ ( plus_plus @ nat @ N4 @ K ) )
@ ( topolo7230453075368039082e_nhds @ A @ A3 )
@ ( at_top @ nat ) ) ) ) ).
% LIMSEQ_ignore_initial_segment
thf(fact_7228_LIMSEQ__le__const2,axiom,
! [A: $tType] :
( ( topolo1944317154257567458pology @ A )
=> ! [X: nat > A,X2: A,A3: A] :
( ( filterlim @ nat @ A @ X @ ( topolo7230453075368039082e_nhds @ A @ X2 ) @ ( at_top @ nat ) )
=> ( ? [N10: nat] :
! [N3: nat] :
( ( ord_less_eq @ nat @ N10 @ N3 )
=> ( ord_less_eq @ A @ ( X @ N3 ) @ A3 ) )
=> ( ord_less_eq @ A @ X2 @ A3 ) ) ) ) ).
% LIMSEQ_le_const2
thf(fact_7229_LIMSEQ__le__const,axiom,
! [A: $tType] :
( ( topolo1944317154257567458pology @ A )
=> ! [X: nat > A,X2: A,A3: A] :
( ( filterlim @ nat @ A @ X @ ( topolo7230453075368039082e_nhds @ A @ X2 ) @ ( at_top @ nat ) )
=> ( ? [N10: nat] :
! [N3: nat] :
( ( ord_less_eq @ nat @ N10 @ N3 )
=> ( ord_less_eq @ A @ A3 @ ( X @ N3 ) ) )
=> ( ord_less_eq @ A @ A3 @ X2 ) ) ) ) ).
% LIMSEQ_le_const
thf(fact_7230_Lim__bounded2,axiom,
! [A: $tType] :
( ( topolo1944317154257567458pology @ A )
=> ! [F2: nat > A,L: A,N7: nat,C2: A] :
( ( filterlim @ nat @ A @ F2 @ ( topolo7230453075368039082e_nhds @ A @ L ) @ ( at_top @ nat ) )
=> ( ! [N3: nat] :
( ( ord_less_eq @ nat @ N7 @ N3 )
=> ( ord_less_eq @ A @ C2 @ ( F2 @ N3 ) ) )
=> ( ord_less_eq @ A @ C2 @ L ) ) ) ) ).
% Lim_bounded2
thf(fact_7231_Lim__bounded,axiom,
! [A: $tType] :
( ( topolo1944317154257567458pology @ A )
=> ! [F2: nat > A,L: A,M8: nat,C2: A] :
( ( filterlim @ nat @ A @ F2 @ ( topolo7230453075368039082e_nhds @ A @ L ) @ ( at_top @ nat ) )
=> ( ! [N3: nat] :
( ( ord_less_eq @ nat @ M8 @ N3 )
=> ( ord_less_eq @ A @ ( F2 @ N3 ) @ C2 ) )
=> ( ord_less_eq @ A @ L @ C2 ) ) ) ) ).
% Lim_bounded
thf(fact_7232_LIMSEQ__le,axiom,
! [A: $tType] :
( ( topolo1944317154257567458pology @ A )
=> ! [X: nat > A,X2: A,Y8: nat > A,Y3: A] :
( ( filterlim @ nat @ A @ X @ ( topolo7230453075368039082e_nhds @ A @ X2 ) @ ( at_top @ nat ) )
=> ( ( filterlim @ nat @ A @ Y8 @ ( topolo7230453075368039082e_nhds @ A @ Y3 ) @ ( at_top @ nat ) )
=> ( ? [N10: nat] :
! [N3: nat] :
( ( ord_less_eq @ nat @ N10 @ N3 )
=> ( ord_less_eq @ A @ ( X @ N3 ) @ ( Y8 @ N3 ) ) )
=> ( ord_less_eq @ A @ X2 @ Y3 ) ) ) ) ) ).
% LIMSEQ_le
thf(fact_7233_lim__mono,axiom,
! [A: $tType] :
( ( topolo1944317154257567458pology @ A )
=> ! [N7: nat,X: nat > A,Y8: nat > A,X2: A,Y3: A] :
( ! [N3: nat] :
( ( ord_less_eq @ nat @ N7 @ N3 )
=> ( ord_less_eq @ A @ ( X @ N3 ) @ ( Y8 @ N3 ) ) )
=> ( ( filterlim @ nat @ A @ X @ ( topolo7230453075368039082e_nhds @ A @ X2 ) @ ( at_top @ nat ) )
=> ( ( filterlim @ nat @ A @ Y8 @ ( topolo7230453075368039082e_nhds @ A @ Y3 ) @ ( at_top @ nat ) )
=> ( ord_less_eq @ A @ X2 @ Y3 ) ) ) ) ) ).
% lim_mono
thf(fact_7234_isCont__rabs,axiom,
! [A: $tType] :
( ( topological_t2_space @ A )
=> ! [A3: A,F2: A > real] :
( ( topolo3448309680560233919inuous @ A @ real @ ( topolo174197925503356063within @ A @ A3 @ ( top_top @ ( set @ A ) ) ) @ F2 )
=> ( topolo3448309680560233919inuous @ A @ real @ ( topolo174197925503356063within @ A @ A3 @ ( top_top @ ( set @ A ) ) )
@ ^ [X3: A] : ( abs_abs @ real @ ( F2 @ X3 ) ) ) ) ) ).
% isCont_rabs
thf(fact_7235_continuous__at__within__powr,axiom,
! [C: $tType] :
( ( topological_t2_space @ C )
=> ! [A3: C,S2: set @ C,F2: C > real,G: C > real] :
( ( topolo3448309680560233919inuous @ C @ real @ ( topolo174197925503356063within @ C @ A3 @ S2 ) @ F2 )
=> ( ( topolo3448309680560233919inuous @ C @ real @ ( topolo174197925503356063within @ C @ A3 @ S2 ) @ G )
=> ( ( ( F2 @ A3 )
!= ( zero_zero @ real ) )
=> ( topolo3448309680560233919inuous @ C @ real @ ( topolo174197925503356063within @ C @ A3 @ S2 )
@ ^ [X3: C] : ( powr @ real @ ( F2 @ X3 ) @ ( G @ X3 ) ) ) ) ) ) ) ).
% continuous_at_within_powr
thf(fact_7236_continuous__within__ln,axiom,
! [A: $tType] :
( ( topological_t2_space @ A )
=> ! [X2: A,S2: set @ A,F2: A > real] :
( ( topolo3448309680560233919inuous @ A @ real @ ( topolo174197925503356063within @ A @ X2 @ S2 ) @ F2 )
=> ( ( ( F2 @ X2 )
!= ( zero_zero @ real ) )
=> ( topolo3448309680560233919inuous @ A @ real @ ( topolo174197925503356063within @ A @ X2 @ S2 )
@ ^ [X3: A] : ( ln_ln @ real @ ( F2 @ X3 ) ) ) ) ) ) ).
% continuous_within_ln
thf(fact_7237_mult__nat__right__at__top,axiom,
! [C3: nat] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ C3 )
=> ( filterlim @ nat @ nat
@ ^ [X3: nat] : ( times_times @ nat @ X3 @ C3 )
@ ( at_top @ nat )
@ ( at_top @ nat ) ) ) ).
% mult_nat_right_at_top
thf(fact_7238_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_7239_monoseq__convergent,axiom,
! [X: nat > real,B2: real] :
( ( topological_monoseq @ real @ X )
=> ( ! [I2: nat] : ( ord_less_eq @ real @ ( abs_abs @ real @ ( X @ I2 ) ) @ B2 )
=> ~ ! [L7: real] :
~ ( filterlim @ nat @ real @ X @ ( topolo7230453075368039082e_nhds @ real @ L7 ) @ ( at_top @ nat ) ) ) ) ).
% monoseq_convergent
thf(fact_7240_LIMSEQ__lessThan__iff__atMost,axiom,
! [A: $tType] :
( ( topolo4958980785337419405_space @ A )
=> ! [F2: ( set @ nat ) > A,X2: A] :
( ( filterlim @ nat @ A
@ ^ [N4: nat] : ( F2 @ ( set_ord_lessThan @ nat @ N4 ) )
@ ( topolo7230453075368039082e_nhds @ A @ X2 )
@ ( at_top @ nat ) )
= ( filterlim @ nat @ A
@ ^ [N4: nat] : ( F2 @ ( set_ord_atMost @ nat @ N4 ) )
@ ( topolo7230453075368039082e_nhds @ A @ X2 )
@ ( at_top @ nat ) ) ) ) ).
% LIMSEQ_lessThan_iff_atMost
thf(fact_7241_LIMSEQ__root,axiom,
( filterlim @ nat @ real
@ ^ [N4: nat] : ( root @ N4 @ ( semiring_1_of_nat @ real @ N4 ) )
@ ( topolo7230453075368039082e_nhds @ real @ ( one_one @ real ) )
@ ( at_top @ nat ) ) ).
% LIMSEQ_root
thf(fact_7242_isCont__powr,axiom,
! [C: $tType] :
( ( topological_t2_space @ C )
=> ! [A3: C,F2: C > real,G: C > real] :
( ( topolo3448309680560233919inuous @ C @ real @ ( topolo174197925503356063within @ C @ A3 @ ( top_top @ ( set @ C ) ) ) @ F2 )
=> ( ( topolo3448309680560233919inuous @ C @ real @ ( topolo174197925503356063within @ C @ A3 @ ( top_top @ ( set @ C ) ) ) @ G )
=> ( ( ( F2 @ A3 )
!= ( zero_zero @ real ) )
=> ( topolo3448309680560233919inuous @ C @ real @ ( topolo174197925503356063within @ C @ A3 @ ( top_top @ ( set @ C ) ) )
@ ^ [X3: C] : ( powr @ real @ ( F2 @ X3 ) @ ( G @ X3 ) ) ) ) ) ) ) ).
% isCont_powr
thf(fact_7243_isCont__ln_H,axiom,
! [A: $tType] :
( ( topological_t2_space @ A )
=> ! [X2: A,F2: A > real] :
( ( topolo3448309680560233919inuous @ A @ real @ ( topolo174197925503356063within @ A @ X2 @ ( top_top @ ( set @ A ) ) ) @ F2 )
=> ( ( ( F2 @ X2 )
!= ( zero_zero @ real ) )
=> ( topolo3448309680560233919inuous @ A @ real @ ( topolo174197925503356063within @ A @ X2 @ ( top_top @ ( set @ A ) ) )
@ ^ [X3: A] : ( ln_ln @ real @ ( F2 @ X3 ) ) ) ) ) ) ).
% isCont_ln'
thf(fact_7244_monoseq__le,axiom,
! [A: $tType] :
( ( topolo1944317154257567458pology @ A )
=> ! [A3: nat > A,X2: A] :
( ( topological_monoseq @ A @ A3 )
=> ( ( filterlim @ nat @ A @ A3 @ ( topolo7230453075368039082e_nhds @ A @ X2 ) @ ( at_top @ nat ) )
=> ( ( ! [N9: nat] : ( ord_less_eq @ A @ ( A3 @ N9 ) @ X2 )
& ! [M3: nat,N9: nat] :
( ( ord_less_eq @ nat @ M3 @ N9 )
=> ( ord_less_eq @ A @ ( A3 @ M3 ) @ ( A3 @ N9 ) ) ) )
| ( ! [N9: nat] : ( ord_less_eq @ A @ X2 @ ( A3 @ N9 ) )
& ! [M3: nat,N9: nat] :
( ( ord_less_eq @ nat @ M3 @ N9 )
=> ( ord_less_eq @ A @ ( A3 @ N9 ) @ ( A3 @ M3 ) ) ) ) ) ) ) ) ).
% monoseq_le
thf(fact_7245_lim__const__over__n,axiom,
! [A: $tType] :
( ( real_V3459762299906320749_field @ A )
=> ! [A3: A] :
( filterlim @ nat @ A
@ ^ [N4: nat] : ( divide_divide @ A @ A3 @ ( semiring_1_of_nat @ A @ N4 ) )
@ ( topolo7230453075368039082e_nhds @ A @ ( zero_zero @ A ) )
@ ( at_top @ nat ) ) ) ).
% lim_const_over_n
thf(fact_7246_LIMSEQ__linear,axiom,
! [A: $tType] :
( ( topolo4958980785337419405_space @ A )
=> ! [X: nat > A,X2: A,L: nat] :
( ( filterlim @ nat @ A @ X @ ( topolo7230453075368039082e_nhds @ A @ X2 ) @ ( at_top @ nat ) )
=> ( ( ord_less @ nat @ ( zero_zero @ nat ) @ L )
=> ( filterlim @ nat @ A
@ ^ [N4: nat] : ( X @ ( times_times @ nat @ N4 @ L ) )
@ ( topolo7230453075368039082e_nhds @ A @ X2 )
@ ( at_top @ nat ) ) ) ) ) ).
% LIMSEQ_linear
thf(fact_7247_lim__inverse__n,axiom,
! [A: $tType] :
( ( real_V3459762299906320749_field @ A )
=> ( filterlim @ nat @ A
@ ^ [N4: nat] : ( inverse_inverse @ A @ ( semiring_1_of_nat @ A @ N4 ) )
@ ( topolo7230453075368039082e_nhds @ A @ ( zero_zero @ A ) )
@ ( at_top @ nat ) ) ) ).
% lim_inverse_n
thf(fact_7248_LIMSEQ__SEQ__conv,axiom,
! [B: $tType,A: $tType] :
( ( ( topolo3112930676232923870pology @ A )
& ( topolo4958980785337419405_space @ B ) )
=> ! [A3: A,X: A > B,L6: B] :
( ( ! [S8: nat > A] :
( ( ! [N4: nat] :
( ( S8 @ N4 )
!= A3 )
& ( filterlim @ nat @ A @ S8 @ ( topolo7230453075368039082e_nhds @ A @ A3 ) @ ( at_top @ nat ) ) )
=> ( filterlim @ nat @ B
@ ^ [N4: nat] : ( X @ ( S8 @ N4 ) )
@ ( topolo7230453075368039082e_nhds @ B @ L6 )
@ ( at_top @ nat ) ) ) )
= ( filterlim @ A @ B @ X @ ( topolo7230453075368039082e_nhds @ B @ L6 ) @ ( topolo174197925503356063within @ A @ A3 @ ( top_top @ ( set @ A ) ) ) ) ) ) ).
% LIMSEQ_SEQ_conv
thf(fact_7249_LIMSEQ__SEQ__conv1,axiom,
! [B: $tType,A: $tType] :
( ( ( topolo4958980785337419405_space @ A )
& ( topolo4958980785337419405_space @ B ) )
=> ! [F2: A > B,L: B,A3: A] :
( ( filterlim @ A @ B @ F2 @ ( topolo7230453075368039082e_nhds @ B @ L ) @ ( topolo174197925503356063within @ A @ A3 @ ( top_top @ ( set @ A ) ) ) )
=> ! [S10: nat > A] :
( ( ! [N3: nat] :
( ( S10 @ N3 )
!= A3 )
& ( filterlim @ nat @ A @ S10 @ ( topolo7230453075368039082e_nhds @ A @ A3 ) @ ( at_top @ nat ) ) )
=> ( filterlim @ nat @ B
@ ^ [N4: nat] : ( F2 @ ( S10 @ N4 ) )
@ ( topolo7230453075368039082e_nhds @ B @ L )
@ ( at_top @ nat ) ) ) ) ) ).
% LIMSEQ_SEQ_conv1
thf(fact_7250_LIMSEQ__SEQ__conv2,axiom,
! [B: $tType,A: $tType] :
( ( ( topolo3112930676232923870pology @ A )
& ( topolo4958980785337419405_space @ B ) )
=> ! [A3: A,F2: A > B,L: B] :
( ! [S6: nat > A] :
( ( ! [N9: nat] :
( ( S6 @ N9 )
!= A3 )
& ( filterlim @ nat @ A @ S6 @ ( topolo7230453075368039082e_nhds @ A @ A3 ) @ ( at_top @ nat ) ) )
=> ( filterlim @ nat @ B
@ ^ [N4: nat] : ( F2 @ ( S6 @ N4 ) )
@ ( topolo7230453075368039082e_nhds @ B @ L )
@ ( at_top @ nat ) ) )
=> ( filterlim @ A @ B @ F2 @ ( topolo7230453075368039082e_nhds @ B @ L ) @ ( topolo174197925503356063within @ A @ A3 @ ( top_top @ ( set @ A ) ) ) ) ) ) ).
% LIMSEQ_SEQ_conv2
thf(fact_7251_telescope__summable,axiom,
! [A: $tType] :
( ( real_V822414075346904944vector @ A )
=> ! [F2: nat > A,C3: A] :
( ( filterlim @ nat @ A @ F2 @ ( topolo7230453075368039082e_nhds @ A @ C3 ) @ ( at_top @ nat ) )
=> ( summable @ A
@ ^ [N4: nat] : ( minus_minus @ A @ ( F2 @ ( suc @ N4 ) ) @ ( F2 @ N4 ) ) ) ) ) ).
% telescope_summable
thf(fact_7252_telescope__summable_H,axiom,
! [A: $tType] :
( ( real_V822414075346904944vector @ A )
=> ! [F2: nat > A,C3: A] :
( ( filterlim @ nat @ A @ F2 @ ( topolo7230453075368039082e_nhds @ A @ C3 ) @ ( at_top @ nat ) )
=> ( summable @ A
@ ^ [N4: nat] : ( minus_minus @ A @ ( F2 @ N4 ) @ ( F2 @ ( suc @ N4 ) ) ) ) ) ) ).
% telescope_summable'
thf(fact_7253_nested__sequence__unique,axiom,
! [F2: nat > real,G: nat > real] :
( ! [N3: nat] : ( ord_less_eq @ real @ ( F2 @ N3 ) @ ( F2 @ ( suc @ N3 ) ) )
=> ( ! [N3: nat] : ( ord_less_eq @ real @ ( G @ ( suc @ N3 ) ) @ ( G @ N3 ) )
=> ( ! [N3: nat] : ( ord_less_eq @ real @ ( F2 @ N3 ) @ ( G @ N3 ) )
=> ( ( filterlim @ nat @ real
@ ^ [N4: nat] : ( minus_minus @ real @ ( F2 @ N4 ) @ ( G @ N4 ) )
@ ( topolo7230453075368039082e_nhds @ real @ ( zero_zero @ real ) )
@ ( at_top @ nat ) )
=> ? [L2: real] :
( ! [N9: nat] : ( ord_less_eq @ real @ ( F2 @ N9 ) @ L2 )
& ( filterlim @ nat @ real @ F2 @ ( topolo7230453075368039082e_nhds @ real @ L2 ) @ ( at_top @ nat ) )
& ! [N9: nat] : ( ord_less_eq @ real @ L2 @ ( G @ N9 ) )
& ( filterlim @ nat @ real @ G @ ( topolo7230453075368039082e_nhds @ real @ L2 ) @ ( at_top @ nat ) ) ) ) ) ) ) ).
% nested_sequence_unique
thf(fact_7254_LIMSEQ__inverse__zero,axiom,
! [X: nat > real] :
( ! [R3: real] :
? [N10: nat] :
! [N3: nat] :
( ( ord_less_eq @ nat @ N10 @ N3 )
=> ( ord_less @ real @ R3 @ ( X @ N3 ) ) )
=> ( filterlim @ nat @ real
@ ^ [N4: nat] : ( inverse_inverse @ real @ ( X @ N4 ) )
@ ( topolo7230453075368039082e_nhds @ real @ ( zero_zero @ real ) )
@ ( at_top @ nat ) ) ) ).
% LIMSEQ_inverse_zero
thf(fact_7255_lim__inverse__n_H,axiom,
( filterlim @ nat @ real
@ ^ [N4: nat] : ( divide_divide @ real @ ( one_one @ real ) @ ( semiring_1_of_nat @ real @ N4 ) )
@ ( topolo7230453075368039082e_nhds @ real @ ( zero_zero @ real ) )
@ ( at_top @ nat ) ) ).
% lim_inverse_n'
thf(fact_7256_LIMSEQ__inverse__real__of__nat,axiom,
( filterlim @ nat @ real
@ ^ [N4: nat] : ( inverse_inverse @ real @ ( semiring_1_of_nat @ real @ ( suc @ N4 ) ) )
@ ( topolo7230453075368039082e_nhds @ real @ ( zero_zero @ real ) )
@ ( at_top @ nat ) ) ).
% LIMSEQ_inverse_real_of_nat
thf(fact_7257_LIMSEQ__root__const,axiom,
! [C3: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ C3 )
=> ( filterlim @ nat @ real
@ ^ [N4: nat] : ( root @ N4 @ C3 )
@ ( topolo7230453075368039082e_nhds @ real @ ( one_one @ real ) )
@ ( at_top @ nat ) ) ) ).
% LIMSEQ_root_const
thf(fact_7258_LIMSEQ__inverse__real__of__nat__add,axiom,
! [R2: real] :
( filterlim @ nat @ real
@ ^ [N4: nat] : ( plus_plus @ real @ R2 @ ( inverse_inverse @ real @ ( semiring_1_of_nat @ real @ ( suc @ N4 ) ) ) )
@ ( topolo7230453075368039082e_nhds @ real @ R2 )
@ ( at_top @ nat ) ) ).
% LIMSEQ_inverse_real_of_nat_add
thf(fact_7259_sums__def,axiom,
! [A: $tType] :
( ( ( comm_monoid_add @ A )
& ( topolo4958980785337419405_space @ A ) )
=> ( ( sums @ A )
= ( ^ [F5: nat > A,S5: A] :
( filterlim @ nat @ A
@ ^ [N4: nat] : ( groups7311177749621191930dd_sum @ nat @ A @ F5 @ ( set_ord_lessThan @ nat @ N4 ) )
@ ( topolo7230453075368039082e_nhds @ A @ S5 )
@ ( at_top @ nat ) ) ) ) ) ).
% sums_def
thf(fact_7260_sums__def__le,axiom,
! [A: $tType] :
( ( ( comm_monoid_add @ A )
& ( topolo4958980785337419405_space @ A ) )
=> ( ( sums @ A )
= ( ^ [F5: nat > A,S5: A] :
( filterlim @ nat @ A
@ ^ [N4: nat] : ( groups7311177749621191930dd_sum @ nat @ A @ F5 @ ( set_ord_atMost @ nat @ N4 ) )
@ ( topolo7230453075368039082e_nhds @ A @ S5 )
@ ( at_top @ nat ) ) ) ) ) ).
% sums_def_le
thf(fact_7261_continuous__at__within__log,axiom,
! [A: $tType] :
( ( topological_t2_space @ A )
=> ! [A3: A,S2: set @ A,F2: A > real,G: A > real] :
( ( topolo3448309680560233919inuous @ A @ real @ ( topolo174197925503356063within @ A @ A3 @ S2 ) @ F2 )
=> ( ( topolo3448309680560233919inuous @ A @ real @ ( topolo174197925503356063within @ A @ A3 @ S2 ) @ G )
=> ( ( ord_less @ real @ ( zero_zero @ real ) @ ( F2 @ A3 ) )
=> ( ( ( F2 @ A3 )
!= ( one_one @ real ) )
=> ( ( ord_less @ real @ ( zero_zero @ real ) @ ( G @ A3 ) )
=> ( topolo3448309680560233919inuous @ A @ real @ ( topolo174197925503356063within @ A @ A3 @ S2 )
@ ^ [X3: A] : ( log @ ( F2 @ X3 ) @ ( G @ X3 ) ) ) ) ) ) ) ) ) ).
% continuous_at_within_log
thf(fact_7262_increasing__LIMSEQ,axiom,
! [F2: nat > real,L: real] :
( ! [N3: nat] : ( ord_less_eq @ real @ ( F2 @ N3 ) @ ( F2 @ ( suc @ N3 ) ) )
=> ( ! [N3: nat] : ( ord_less_eq @ real @ ( F2 @ N3 ) @ L )
=> ( ! [E2: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ E2 )
=> ? [N9: nat] : ( ord_less_eq @ real @ L @ ( plus_plus @ real @ ( F2 @ N9 ) @ E2 ) ) )
=> ( filterlim @ nat @ real @ F2 @ ( topolo7230453075368039082e_nhds @ real @ L ) @ ( at_top @ nat ) ) ) ) ) ).
% increasing_LIMSEQ
thf(fact_7263_lim__1__over__n,axiom,
! [A: $tType] :
( ( real_V3459762299906320749_field @ A )
=> ( filterlim @ nat @ A
@ ^ [N4: nat] : ( divide_divide @ A @ ( one_one @ A ) @ ( semiring_1_of_nat @ A @ N4 ) )
@ ( topolo7230453075368039082e_nhds @ A @ ( zero_zero @ A ) )
@ ( at_top @ nat ) ) ) ).
% lim_1_over_n
thf(fact_7264_LIMSEQ__Suc__n__over__n,axiom,
! [A: $tType] :
( ( real_V3459762299906320749_field @ A )
=> ( filterlim @ nat @ A
@ ^ [N4: nat] : ( divide_divide @ A @ ( semiring_1_of_nat @ A @ ( suc @ N4 ) ) @ ( semiring_1_of_nat @ A @ N4 ) )
@ ( topolo7230453075368039082e_nhds @ A @ ( one_one @ A ) )
@ ( at_top @ nat ) ) ) ).
% LIMSEQ_Suc_n_over_n
thf(fact_7265_LIMSEQ__n__over__Suc__n,axiom,
! [A: $tType] :
( ( real_V3459762299906320749_field @ A )
=> ( filterlim @ nat @ A
@ ^ [N4: nat] : ( divide_divide @ A @ ( semiring_1_of_nat @ A @ N4 ) @ ( semiring_1_of_nat @ A @ ( suc @ N4 ) ) )
@ ( topolo7230453075368039082e_nhds @ A @ ( one_one @ A ) )
@ ( at_top @ nat ) ) ) ).
% LIMSEQ_n_over_Suc_n
thf(fact_7266_telescope__sums_H,axiom,
! [A: $tType] :
( ( real_V822414075346904944vector @ A )
=> ! [F2: nat > A,C3: A] :
( ( filterlim @ nat @ A @ F2 @ ( topolo7230453075368039082e_nhds @ A @ C3 ) @ ( at_top @ nat ) )
=> ( sums @ A
@ ^ [N4: nat] : ( minus_minus @ A @ ( F2 @ N4 ) @ ( F2 @ ( suc @ N4 ) ) )
@ ( minus_minus @ A @ ( F2 @ ( zero_zero @ nat ) ) @ C3 ) ) ) ) ).
% telescope_sums'
thf(fact_7267_telescope__sums,axiom,
! [A: $tType] :
( ( real_V822414075346904944vector @ A )
=> ! [F2: nat > A,C3: A] :
( ( filterlim @ nat @ A @ F2 @ ( topolo7230453075368039082e_nhds @ A @ C3 ) @ ( at_top @ nat ) )
=> ( sums @ A
@ ^ [N4: nat] : ( minus_minus @ A @ ( F2 @ ( suc @ N4 ) ) @ ( F2 @ N4 ) )
@ ( minus_minus @ A @ C3 @ ( F2 @ ( zero_zero @ nat ) ) ) ) ) ) ).
% telescope_sums
thf(fact_7268_LIMSEQ__realpow__zero,axiom,
! [X2: real] :
( ( ord_less_eq @ real @ ( zero_zero @ real ) @ X2 )
=> ( ( ord_less @ real @ X2 @ ( one_one @ real ) )
=> ( filterlim @ nat @ real @ ( power_power @ real @ X2 ) @ ( topolo7230453075368039082e_nhds @ real @ ( zero_zero @ real ) ) @ ( at_top @ nat ) ) ) ) ).
% LIMSEQ_realpow_zero
thf(fact_7269_LIMSEQ__divide__realpow__zero,axiom,
! [X2: real,A3: real] :
( ( ord_less @ real @ ( one_one @ real ) @ X2 )
=> ( filterlim @ nat @ real
@ ^ [N4: nat] : ( divide_divide @ real @ A3 @ ( power_power @ real @ X2 @ N4 ) )
@ ( topolo7230453075368039082e_nhds @ real @ ( zero_zero @ real ) )
@ ( at_top @ nat ) ) ) ).
% LIMSEQ_divide_realpow_zero
thf(fact_7270_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_7271_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_7272_LIMSEQ__inverse__realpow__zero,axiom,
! [X2: real] :
( ( ord_less @ real @ ( one_one @ real ) @ X2 )
=> ( filterlim @ nat @ real
@ ^ [N4: nat] : ( inverse_inverse @ real @ ( power_power @ real @ X2 @ N4 ) )
@ ( topolo7230453075368039082e_nhds @ real @ ( zero_zero @ real ) )
@ ( at_top @ nat ) ) ) ).
% LIMSEQ_inverse_realpow_zero
thf(fact_7273_sums__def_H,axiom,
! [A: $tType] :
( ( ( comm_monoid_add @ A )
& ( topolo4958980785337419405_space @ A ) )
=> ( ( sums @ A )
= ( ^ [F5: nat > A,S5: A] :
( filterlim @ nat @ A
@ ^ [N4: nat] : ( groups7311177749621191930dd_sum @ nat @ A @ F5 @ ( set_or1337092689740270186AtMost @ nat @ ( zero_zero @ nat ) @ N4 ) )
@ ( topolo7230453075368039082e_nhds @ A @ S5 )
@ ( at_top @ nat ) ) ) ) ) ).
% sums_def'
thf(fact_7274_LIMSEQ__inverse__real__of__nat__add__minus,axiom,
! [R2: real] :
( filterlim @ nat @ real
@ ^ [N4: nat] : ( plus_plus @ real @ R2 @ ( uminus_uminus @ real @ ( inverse_inverse @ real @ ( semiring_1_of_nat @ real @ ( suc @ N4 ) ) ) ) )
@ ( topolo7230453075368039082e_nhds @ real @ R2 )
@ ( at_top @ nat ) ) ).
% LIMSEQ_inverse_real_of_nat_add_minus
thf(fact_7275_root__test__convergence,axiom,
! [A: $tType] :
( ( real_Vector_banach @ A )
=> ! [F2: nat > A,X2: real] :
( ( filterlim @ nat @ real
@ ^ [N4: nat] : ( root @ N4 @ ( real_V7770717601297561774m_norm @ A @ ( F2 @ N4 ) ) )
@ ( topolo7230453075368039082e_nhds @ real @ X2 )
@ ( at_top @ nat ) )
=> ( ( ord_less @ real @ X2 @ ( one_one @ real ) )
=> ( summable @ A @ F2 ) ) ) ) ).
% root_test_convergence
thf(fact_7276_summable__LIMSEQ,axiom,
! [A: $tType] :
( ( ( comm_monoid_add @ A )
& ( topological_t2_space @ A ) )
=> ! [F2: nat > A] :
( ( summable @ A @ F2 )
=> ( filterlim @ nat @ A
@ ^ [N4: nat] : ( groups7311177749621191930dd_sum @ nat @ A @ F2 @ ( set_ord_lessThan @ nat @ N4 ) )
@ ( topolo7230453075368039082e_nhds @ A @ ( suminf @ A @ F2 ) )
@ ( at_top @ nat ) ) ) ) ).
% summable_LIMSEQ
thf(fact_7277_summable__LIMSEQ_H,axiom,
! [A: $tType] :
( ( ( comm_monoid_add @ A )
& ( topological_t2_space @ A ) )
=> ! [F2: nat > A] :
( ( summable @ A @ F2 )
=> ( filterlim @ nat @ A
@ ^ [N4: nat] : ( groups7311177749621191930dd_sum @ nat @ A @ F2 @ ( set_ord_atMost @ nat @ N4 ) )
@ ( topolo7230453075368039082e_nhds @ A @ ( suminf @ A @ F2 ) )
@ ( at_top @ nat ) ) ) ) ).
% summable_LIMSEQ'
thf(fact_7278_isCont__log,axiom,
! [A: $tType] :
( ( topological_t2_space @ A )
=> ! [A3: A,F2: A > real,G: A > real] :
( ( topolo3448309680560233919inuous @ A @ real @ ( topolo174197925503356063within @ A @ A3 @ ( top_top @ ( set @ A ) ) ) @ F2 )
=> ( ( topolo3448309680560233919inuous @ A @ real @ ( topolo174197925503356063within @ A @ A3 @ ( top_top @ ( set @ A ) ) ) @ G )
=> ( ( ord_less @ real @ ( zero_zero @ real ) @ ( F2 @ A3 ) )
=> ( ( ( F2 @ A3 )
!= ( one_one @ real ) )
=> ( ( ord_less @ real @ ( zero_zero @ real ) @ ( G @ A3 ) )
=> ( topolo3448309680560233919inuous @ A @ real @ ( topolo174197925503356063within @ A @ A3 @ ( top_top @ ( set @ A ) ) )
@ ^ [X3: A] : ( log @ ( F2 @ X3 ) @ ( G @ X3 ) ) ) ) ) ) ) ) ) ).
% isCont_log
thf(fact_7279_LIMSEQ__D,axiom,
! [A: $tType] :
( ( real_V822414075346904944vector @ A )
=> ! [X: nat > A,L6: A,R2: real] :
( ( filterlim @ nat @ A @ X @ ( topolo7230453075368039082e_nhds @ A @ L6 ) @ ( 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 @ ( X @ N9 ) @ L6 ) ) @ R2 ) ) ) ) ) ).
% LIMSEQ_D
thf(fact_7280_LIMSEQ__I,axiom,
! [A: $tType] :
( ( real_V822414075346904944vector @ A )
=> ! [X: nat > A,L6: A] :
( ! [R3: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ R3 )
=> ? [No2: nat] :
! [N3: nat] :
( ( ord_less_eq @ nat @ No2 @ N3 )
=> ( ord_less @ real @ ( real_V7770717601297561774m_norm @ A @ ( minus_minus @ A @ ( X @ N3 ) @ L6 ) ) @ R3 ) ) )
=> ( filterlim @ nat @ A @ X @ ( topolo7230453075368039082e_nhds @ A @ L6 ) @ ( at_top @ nat ) ) ) ) ).
% LIMSEQ_I
thf(fact_7281_LIMSEQ__iff,axiom,
! [A: $tType] :
( ( real_V822414075346904944vector @ A )
=> ! [X: nat > A,L6: A] :
( ( filterlim @ nat @ A @ X @ ( topolo7230453075368039082e_nhds @ A @ L6 ) @ ( at_top @ nat ) )
= ( ! [R5: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ R5 )
=> ? [No3: nat] :
! [N4: nat] :
( ( ord_less_eq @ nat @ No3 @ N4 )
=> ( ord_less @ real @ ( real_V7770717601297561774m_norm @ A @ ( minus_minus @ A @ ( X @ N4 ) @ L6 ) ) @ R5 ) ) ) ) ) ) ).
% LIMSEQ_iff
thf(fact_7282_LIMSEQ__power__zero,axiom,
! [A: $tType] :
( ( real_V2822296259951069270ebra_1 @ A )
=> ! [X2: A] :
( ( ord_less @ real @ ( real_V7770717601297561774m_norm @ A @ X2 ) @ ( one_one @ real ) )
=> ( filterlim @ nat @ A @ ( power_power @ A @ X2 ) @ ( topolo7230453075368039082e_nhds @ A @ ( zero_zero @ A ) ) @ ( at_top @ nat ) ) ) ) ).
% LIMSEQ_power_zero
thf(fact_7283_tendsto__exp__limit__sequentially,axiom,
! [X2: real] :
( filterlim @ nat @ real
@ ^ [N4: nat] : ( power_power @ real @ ( plus_plus @ real @ ( one_one @ real ) @ ( divide_divide @ real @ X2 @ ( semiring_1_of_nat @ real @ N4 ) ) ) @ N4 )
@ ( topolo7230453075368039082e_nhds @ real @ ( exp @ real @ X2 ) )
@ ( at_top @ nat ) ) ).
% tendsto_exp_limit_sequentially
thf(fact_7284_tendsto__power__zero,axiom,
! [A: $tType,B: $tType] :
( ( real_V2822296259951069270ebra_1 @ A )
=> ! [F2: B > nat,F4: filter @ B,X2: A] :
( ( filterlim @ B @ nat @ F2 @ ( at_top @ nat ) @ F4 )
=> ( ( ord_less @ real @ ( real_V7770717601297561774m_norm @ A @ X2 ) @ ( one_one @ real ) )
=> ( filterlim @ B @ A
@ ^ [Y: B] : ( power_power @ A @ X2 @ ( F2 @ Y ) )
@ ( topolo7230453075368039082e_nhds @ A @ ( zero_zero @ A ) )
@ F4 ) ) ) ) ).
% tendsto_power_zero
thf(fact_7285_LIMSEQ__inverse__real__of__nat__add__minus__mult,axiom,
! [R2: real] :
( filterlim @ nat @ real
@ ^ [N4: nat] : ( times_times @ real @ R2 @ ( plus_plus @ real @ ( one_one @ real ) @ ( uminus_uminus @ real @ ( inverse_inverse @ real @ ( semiring_1_of_nat @ real @ ( suc @ N4 ) ) ) ) ) )
@ ( topolo7230453075368039082e_nhds @ real @ R2 )
@ ( at_top @ nat ) ) ).
% LIMSEQ_inverse_real_of_nat_add_minus_mult
thf(fact_7286_LIMSEQ__norm__0,axiom,
! [A: $tType] :
( ( real_V822414075346904944vector @ A )
=> ! [F2: nat > A] :
( ! [N3: nat] : ( ord_less @ real @ ( real_V7770717601297561774m_norm @ A @ ( F2 @ N3 ) ) @ ( divide_divide @ real @ ( one_one @ real ) @ ( semiring_1_of_nat @ real @ ( suc @ N3 ) ) ) )
=> ( filterlim @ nat @ A @ F2 @ ( topolo7230453075368039082e_nhds @ A @ ( zero_zero @ A ) ) @ ( at_top @ nat ) ) ) ) ).
% LIMSEQ_norm_0
thf(fact_7287_summable__Leibniz_I1_J,axiom,
! [A3: nat > real] :
( ( filterlim @ nat @ real @ A3 @ ( topolo7230453075368039082e_nhds @ real @ ( zero_zero @ real ) ) @ ( at_top @ nat ) )
=> ( ( topological_monoseq @ real @ A3 )
=> ( summable @ real
@ ^ [N4: nat] : ( times_times @ real @ ( power_power @ real @ ( uminus_uminus @ real @ ( one_one @ real ) ) @ N4 ) @ ( A3 @ N4 ) ) ) ) ) ).
% summable_Leibniz(1)
thf(fact_7288_field__derivative__lim__unique,axiom,
! [A: $tType] :
( ( real_V3459762299906320749_field @ A )
=> ! [F2: A > A,Df: A,Z: A,S2: nat > A,A3: A] :
( ( has_field_derivative @ A @ F2 @ Df @ ( topolo174197925503356063within @ A @ Z @ ( top_top @ ( set @ A ) ) ) )
=> ( ( filterlim @ nat @ A @ S2 @ ( topolo7230453075368039082e_nhds @ A @ ( zero_zero @ A ) ) @ ( at_top @ nat ) )
=> ( ! [N3: nat] :
( ( S2 @ N3 )
!= ( zero_zero @ A ) )
=> ( ( filterlim @ nat @ A
@ ^ [N4: nat] : ( divide_divide @ A @ ( minus_minus @ A @ ( F2 @ ( plus_plus @ A @ Z @ ( S2 @ N4 ) ) ) @ ( F2 @ Z ) ) @ ( S2 @ N4 ) )
@ ( topolo7230453075368039082e_nhds @ A @ A3 )
@ ( at_top @ nat ) )
=> ( Df = A3 ) ) ) ) ) ) ).
% field_derivative_lim_unique
thf(fact_7289_powser__times__n__limit__0,axiom,
! [A: $tType] :
( ( ( real_Vector_banach @ A )
& ( real_V8999393235501362500lgebra @ A ) )
=> ! [X2: A] :
( ( ord_less @ real @ ( real_V7770717601297561774m_norm @ A @ X2 ) @ ( one_one @ real ) )
=> ( filterlim @ nat @ A
@ ^ [N4: nat] : ( times_times @ A @ ( semiring_1_of_nat @ A @ N4 ) @ ( power_power @ A @ X2 @ N4 ) )
@ ( topolo7230453075368039082e_nhds @ A @ ( zero_zero @ A ) )
@ ( at_top @ nat ) ) ) ) ).
% powser_times_n_limit_0
thf(fact_7290_lim__n__over__pown,axiom,
! [A: $tType] :
( ( ( real_Vector_banach @ A )
& ( real_V3459762299906320749_field @ A ) )
=> ! [X2: A] :
( ( ord_less @ real @ ( one_one @ real ) @ ( real_V7770717601297561774m_norm @ A @ X2 ) )
=> ( filterlim @ nat @ A
@ ^ [N4: nat] : ( divide_divide @ A @ ( semiring_1_of_nat @ A @ N4 ) @ ( power_power @ A @ X2 @ N4 ) )
@ ( topolo7230453075368039082e_nhds @ A @ ( zero_zero @ A ) )
@ ( at_top @ nat ) ) ) ) ).
% lim_n_over_pown
thf(fact_7291_summable,axiom,
! [A3: nat > real] :
( ( filterlim @ nat @ real @ A3 @ ( topolo7230453075368039082e_nhds @ real @ ( zero_zero @ real ) ) @ ( at_top @ nat ) )
=> ( ! [N3: nat] : ( ord_less_eq @ real @ ( zero_zero @ real ) @ ( A3 @ N3 ) )
=> ( ! [N3: nat] : ( ord_less_eq @ real @ ( A3 @ ( suc @ N3 ) ) @ ( A3 @ N3 ) )
=> ( summable @ real
@ ^ [N4: nat] : ( times_times @ real @ ( power_power @ real @ ( uminus_uminus @ real @ ( one_one @ real ) ) @ N4 ) @ ( A3 @ N4 ) ) ) ) ) ) ).
% summable
thf(fact_7292_cos__diff__limit__1,axiom,
! [Theta: nat > real,Theta2: real] :
( ( filterlim @ nat @ real
@ ^ [J3: nat] : ( cos @ real @ ( minus_minus @ real @ ( Theta @ J3 ) @ Theta2 ) )
@ ( topolo7230453075368039082e_nhds @ real @ ( one_one @ real ) )
@ ( at_top @ nat ) )
=> ~ ! [K2: nat > int] :
~ ( filterlim @ nat @ real
@ ^ [J3: nat] : ( minus_minus @ real @ ( Theta @ J3 ) @ ( times_times @ real @ ( ring_1_of_int @ real @ ( K2 @ J3 ) ) @ ( times_times @ real @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) @ pi ) ) )
@ ( topolo7230453075368039082e_nhds @ real @ Theta2 )
@ ( at_top @ nat ) ) ) ).
% cos_diff_limit_1
thf(fact_7293_cos__limit__1,axiom,
! [Theta: nat > real] :
( ( filterlim @ nat @ real
@ ^ [J3: nat] : ( cos @ real @ ( Theta @ J3 ) )
@ ( topolo7230453075368039082e_nhds @ real @ ( one_one @ real ) )
@ ( at_top @ nat ) )
=> ? [K2: nat > int] :
( filterlim @ nat @ real
@ ^ [J3: nat] : ( minus_minus @ real @ ( Theta @ J3 ) @ ( times_times @ real @ ( ring_1_of_int @ real @ ( K2 @ J3 ) ) @ ( times_times @ real @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) @ pi ) ) )
@ ( topolo7230453075368039082e_nhds @ real @ ( zero_zero @ real ) )
@ ( at_top @ nat ) ) ) ).
% cos_limit_1
thf(fact_7294_summable__Leibniz_I4_J,axiom,
! [A3: nat > real] :
( ( filterlim @ nat @ real @ A3 @ ( topolo7230453075368039082e_nhds @ real @ ( zero_zero @ real ) ) @ ( at_top @ nat ) )
=> ( ( topological_monoseq @ real @ A3 )
=> ( filterlim @ nat @ real
@ ^ [N4: nat] :
( groups7311177749621191930dd_sum @ nat @ real
@ ^ [I4: nat] : ( times_times @ real @ ( power_power @ real @ ( uminus_uminus @ real @ ( one_one @ real ) ) @ I4 ) @ ( A3 @ I4 ) )
@ ( set_ord_lessThan @ nat @ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N4 ) ) )
@ ( topolo7230453075368039082e_nhds @ real
@ ( suminf @ real
@ ^ [I4: nat] : ( times_times @ real @ ( power_power @ real @ ( uminus_uminus @ real @ ( one_one @ real ) ) @ I4 ) @ ( A3 @ I4 ) ) ) )
@ ( at_top @ nat ) ) ) ) ).
% summable_Leibniz(4)
thf(fact_7295_zeroseq__arctan__series,axiom,
! [X2: real] :
( ( ord_less_eq @ real @ ( abs_abs @ real @ X2 ) @ ( one_one @ real ) )
=> ( filterlim @ nat @ real
@ ^ [N4: nat] : ( times_times @ real @ ( divide_divide @ real @ ( one_one @ real ) @ ( semiring_1_of_nat @ real @ ( plus_plus @ nat @ ( times_times @ nat @ N4 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ ( one_one @ nat ) ) ) ) @ ( power_power @ real @ X2 @ ( plus_plus @ nat @ ( times_times @ nat @ N4 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ ( one_one @ nat ) ) ) )
@ ( topolo7230453075368039082e_nhds @ real @ ( zero_zero @ real ) )
@ ( at_top @ nat ) ) ) ).
% zeroseq_arctan_series
thf(fact_7296_summable__Leibniz_H_I3_J,axiom,
! [A3: nat > real] :
( ( filterlim @ nat @ real @ A3 @ ( topolo7230453075368039082e_nhds @ real @ ( zero_zero @ real ) ) @ ( at_top @ nat ) )
=> ( ! [N3: nat] : ( ord_less_eq @ real @ ( zero_zero @ real ) @ ( A3 @ N3 ) )
=> ( ! [N3: nat] : ( ord_less_eq @ real @ ( A3 @ ( suc @ N3 ) ) @ ( A3 @ N3 ) )
=> ( filterlim @ nat @ real
@ ^ [N4: nat] :
( groups7311177749621191930dd_sum @ nat @ real
@ ^ [I4: nat] : ( times_times @ real @ ( power_power @ real @ ( uminus_uminus @ real @ ( one_one @ real ) ) @ I4 ) @ ( A3 @ I4 ) )
@ ( set_ord_lessThan @ nat @ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N4 ) ) )
@ ( topolo7230453075368039082e_nhds @ real
@ ( suminf @ real
@ ^ [I4: nat] : ( times_times @ real @ ( power_power @ real @ ( uminus_uminus @ real @ ( one_one @ real ) ) @ I4 ) @ ( A3 @ I4 ) ) ) )
@ ( at_top @ nat ) ) ) ) ) ).
% summable_Leibniz'(3)
thf(fact_7297_summable__Leibniz_H_I2_J,axiom,
! [A3: nat > real,N: nat] :
( ( filterlim @ nat @ real @ A3 @ ( topolo7230453075368039082e_nhds @ real @ ( zero_zero @ real ) ) @ ( at_top @ nat ) )
=> ( ! [N3: nat] : ( ord_less_eq @ real @ ( zero_zero @ real ) @ ( A3 @ N3 ) )
=> ( ! [N3: nat] : ( ord_less_eq @ real @ ( A3 @ ( suc @ N3 ) ) @ ( A3 @ N3 ) )
=> ( ord_less_eq @ real
@ ( groups7311177749621191930dd_sum @ nat @ real
@ ^ [I4: nat] : ( times_times @ real @ ( power_power @ real @ ( uminus_uminus @ real @ ( one_one @ real ) ) @ I4 ) @ ( A3 @ I4 ) )
@ ( set_ord_lessThan @ nat @ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N ) ) )
@ ( suminf @ real
@ ^ [I4: nat] : ( times_times @ real @ ( power_power @ real @ ( uminus_uminus @ real @ ( one_one @ real ) ) @ I4 ) @ ( A3 @ I4 ) ) ) ) ) ) ) ).
% summable_Leibniz'(2)
thf(fact_7298_sums__alternating__upper__lower,axiom,
! [A3: nat > real] :
( ! [N3: nat] : ( ord_less_eq @ real @ ( A3 @ ( suc @ N3 ) ) @ ( A3 @ N3 ) )
=> ( ! [N3: nat] : ( ord_less_eq @ real @ ( zero_zero @ real ) @ ( A3 @ N3 ) )
=> ( ( filterlim @ nat @ real @ A3 @ ( topolo7230453075368039082e_nhds @ real @ ( zero_zero @ real ) ) @ ( at_top @ nat ) )
=> ? [L2: real] :
( ! [N9: nat] :
( ord_less_eq @ real
@ ( groups7311177749621191930dd_sum @ nat @ real
@ ^ [I4: nat] : ( times_times @ real @ ( power_power @ real @ ( uminus_uminus @ real @ ( one_one @ real ) ) @ I4 ) @ ( A3 @ I4 ) )
@ ( set_ord_lessThan @ nat @ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N9 ) ) )
@ L2 )
& ( filterlim @ nat @ real
@ ^ [N4: nat] :
( groups7311177749621191930dd_sum @ nat @ real
@ ^ [I4: nat] : ( times_times @ real @ ( power_power @ real @ ( uminus_uminus @ real @ ( one_one @ real ) ) @ I4 ) @ ( A3 @ I4 ) )
@ ( set_ord_lessThan @ nat @ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N4 ) ) )
@ ( topolo7230453075368039082e_nhds @ real @ L2 )
@ ( at_top @ nat ) )
& ! [N9: nat] :
( ord_less_eq @ real @ L2
@ ( groups7311177749621191930dd_sum @ nat @ real
@ ^ [I4: nat] : ( times_times @ real @ ( power_power @ real @ ( uminus_uminus @ real @ ( one_one @ real ) ) @ I4 ) @ ( A3 @ I4 ) )
@ ( set_ord_lessThan @ nat @ ( plus_plus @ nat @ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N9 ) @ ( one_one @ nat ) ) ) ) )
& ( filterlim @ nat @ real
@ ^ [N4: nat] :
( groups7311177749621191930dd_sum @ nat @ real
@ ^ [I4: nat] : ( times_times @ real @ ( power_power @ real @ ( uminus_uminus @ real @ ( one_one @ real ) ) @ I4 ) @ ( A3 @ I4 ) )
@ ( set_ord_lessThan @ nat @ ( plus_plus @ nat @ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N4 ) @ ( one_one @ nat ) ) ) )
@ ( topolo7230453075368039082e_nhds @ real @ L2 )
@ ( at_top @ nat ) ) ) ) ) ) ).
% sums_alternating_upper_lower
thf(fact_7299_summable__Leibniz_I5_J,axiom,
! [A3: nat > real] :
( ( filterlim @ nat @ real @ A3 @ ( topolo7230453075368039082e_nhds @ real @ ( zero_zero @ real ) ) @ ( at_top @ nat ) )
=> ( ( topological_monoseq @ real @ A3 )
=> ( filterlim @ nat @ real
@ ^ [N4: nat] :
( groups7311177749621191930dd_sum @ nat @ real
@ ^ [I4: nat] : ( times_times @ real @ ( power_power @ real @ ( uminus_uminus @ real @ ( one_one @ real ) ) @ I4 ) @ ( A3 @ I4 ) )
@ ( set_ord_lessThan @ nat @ ( plus_plus @ nat @ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N4 ) @ ( one_one @ nat ) ) ) )
@ ( topolo7230453075368039082e_nhds @ real
@ ( suminf @ real
@ ^ [I4: nat] : ( times_times @ real @ ( power_power @ real @ ( uminus_uminus @ real @ ( one_one @ real ) ) @ I4 ) @ ( A3 @ I4 ) ) ) )
@ ( at_top @ nat ) ) ) ) ).
% summable_Leibniz(5)
thf(fact_7300_summable__Leibniz_H_I5_J,axiom,
! [A3: nat > real] :
( ( filterlim @ nat @ real @ A3 @ ( topolo7230453075368039082e_nhds @ real @ ( zero_zero @ real ) ) @ ( at_top @ nat ) )
=> ( ! [N3: nat] : ( ord_less_eq @ real @ ( zero_zero @ real ) @ ( A3 @ N3 ) )
=> ( ! [N3: nat] : ( ord_less_eq @ real @ ( A3 @ ( suc @ N3 ) ) @ ( A3 @ N3 ) )
=> ( filterlim @ nat @ real
@ ^ [N4: nat] :
( groups7311177749621191930dd_sum @ nat @ real
@ ^ [I4: nat] : ( times_times @ real @ ( power_power @ real @ ( uminus_uminus @ real @ ( one_one @ real ) ) @ I4 ) @ ( A3 @ I4 ) )
@ ( set_ord_lessThan @ nat @ ( plus_plus @ nat @ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N4 ) @ ( one_one @ nat ) ) ) )
@ ( topolo7230453075368039082e_nhds @ real
@ ( suminf @ real
@ ^ [I4: nat] : ( times_times @ real @ ( power_power @ real @ ( uminus_uminus @ real @ ( one_one @ real ) ) @ I4 ) @ ( A3 @ I4 ) ) ) )
@ ( at_top @ nat ) ) ) ) ) ).
% summable_Leibniz'(5)
thf(fact_7301_summable__Leibniz_H_I4_J,axiom,
! [A3: nat > real,N: nat] :
( ( filterlim @ nat @ real @ A3 @ ( topolo7230453075368039082e_nhds @ real @ ( zero_zero @ real ) ) @ ( at_top @ nat ) )
=> ( ! [N3: nat] : ( ord_less_eq @ real @ ( zero_zero @ real ) @ ( A3 @ N3 ) )
=> ( ! [N3: nat] : ( ord_less_eq @ real @ ( A3 @ ( suc @ N3 ) ) @ ( A3 @ N3 ) )
=> ( ord_less_eq @ real
@ ( suminf @ real
@ ^ [I4: nat] : ( times_times @ real @ ( power_power @ real @ ( uminus_uminus @ real @ ( one_one @ real ) ) @ I4 ) @ ( A3 @ I4 ) ) )
@ ( groups7311177749621191930dd_sum @ nat @ real
@ ^ [I4: nat] : ( times_times @ real @ ( power_power @ real @ ( uminus_uminus @ real @ ( one_one @ real ) ) @ I4 ) @ ( A3 @ I4 ) )
@ ( set_ord_lessThan @ nat @ ( plus_plus @ nat @ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N ) @ ( one_one @ nat ) ) ) ) ) ) ) ) ).
% summable_Leibniz'(4)
thf(fact_7302_filterlim__sequentially__Suc,axiom,
! [A: $tType,F2: nat > A,F4: filter @ A] :
( ( filterlim @ nat @ A
@ ^ [X3: nat] : ( F2 @ ( suc @ X3 ) )
@ F4
@ ( at_top @ nat ) )
= ( filterlim @ nat @ A @ F2 @ F4 @ ( at_top @ nat ) ) ) ).
% filterlim_sequentially_Suc
thf(fact_7303_has__derivative__at2,axiom,
! [B: $tType,A: $tType] :
( ( ( real_V822414075346904944vector @ A )
& ( real_V822414075346904944vector @ B ) )
=> ! [F2: A > B,F7: A > B,X2: A] :
( ( has_derivative @ A @ B @ F2 @ F7 @ ( topolo174197925503356063within @ A @ X2 @ ( top_top @ ( set @ A ) ) ) )
= ( ( real_V3181309239436604168linear @ A @ B @ F7 )
& ( filterlim @ A @ B
@ ^ [Y: A] : ( real_V8093663219630862766scaleR @ B @ ( divide_divide @ real @ ( one_one @ real ) @ ( real_V7770717601297561774m_norm @ A @ ( minus_minus @ A @ Y @ X2 ) ) ) @ ( minus_minus @ B @ ( F2 @ Y ) @ ( plus_plus @ B @ ( F2 @ X2 ) @ ( F7 @ ( minus_minus @ A @ Y @ X2 ) ) ) ) )
@ ( topolo7230453075368039082e_nhds @ B @ ( zero_zero @ B ) )
@ ( topolo174197925503356063within @ A @ X2 @ ( top_top @ ( set @ A ) ) ) ) ) ) ) ).
% has_derivative_at2
thf(fact_7304_bounded__linear_Ocontinuous,axiom,
! [A: $tType,B: $tType,C: $tType] :
( ( ( topological_t2_space @ C )
& ( real_V822414075346904944vector @ B )
& ( real_V822414075346904944vector @ A ) )
=> ! [F2: A > B,F4: filter @ C,G: C > A] :
( ( real_V3181309239436604168linear @ A @ B @ F2 )
=> ( ( topolo3448309680560233919inuous @ C @ A @ F4 @ G )
=> ( topolo3448309680560233919inuous @ C @ B @ F4
@ ^ [X3: C] : ( F2 @ ( G @ X3 ) ) ) ) ) ) ).
% bounded_linear.continuous
thf(fact_7305_bounded__linear_Otendsto,axiom,
! [A: $tType,B: $tType,C: $tType] :
( ( ( real_V822414075346904944vector @ B )
& ( real_V822414075346904944vector @ A ) )
=> ! [F2: A > B,G: C > A,A3: A,F4: filter @ C] :
( ( real_V3181309239436604168linear @ A @ B @ F2 )
=> ( ( filterlim @ C @ A @ G @ ( topolo7230453075368039082e_nhds @ A @ A3 ) @ F4 )
=> ( filterlim @ C @ B
@ ^ [X3: C] : ( F2 @ ( G @ X3 ) )
@ ( topolo7230453075368039082e_nhds @ B @ ( F2 @ A3 ) )
@ F4 ) ) ) ) ).
% bounded_linear.tendsto
thf(fact_7306_bounded__linear_OCauchy,axiom,
! [A: $tType,B: $tType] :
( ( ( real_V822414075346904944vector @ B )
& ( real_V822414075346904944vector @ A ) )
=> ! [F2: A > B,X: nat > A] :
( ( real_V3181309239436604168linear @ A @ B @ F2 )
=> ( ( topolo3814608138187158403Cauchy @ A @ X )
=> ( topolo3814608138187158403Cauchy @ B
@ ^ [N4: nat] : ( F2 @ ( X @ N4 ) ) ) ) ) ) ).
% bounded_linear.Cauchy
thf(fact_7307_bounded__linear__divide,axiom,
! [A: $tType] :
( ( real_V3459762299906320749_field @ A )
=> ! [Y3: A] :
( real_V3181309239436604168linear @ A @ A
@ ^ [X3: A] : ( divide_divide @ A @ X3 @ Y3 ) ) ) ).
% bounded_linear_divide
thf(fact_7308_bounded__linear__mult__left,axiom,
! [A: $tType] :
( ( real_V4412858255891104859lgebra @ A )
=> ! [Y3: A] :
( real_V3181309239436604168linear @ A @ A
@ ^ [X3: A] : ( times_times @ A @ X3 @ Y3 ) ) ) ).
% bounded_linear_mult_left
thf(fact_7309_bounded__linear__const__mult,axiom,
! [A: $tType,C: $tType] :
( ( ( real_V822414075346904944vector @ C )
& ( real_V4412858255891104859lgebra @ A ) )
=> ! [G: C > A,X2: A] :
( ( real_V3181309239436604168linear @ C @ A @ G )
=> ( real_V3181309239436604168linear @ C @ A
@ ^ [X3: C] : ( times_times @ A @ X2 @ ( G @ X3 ) ) ) ) ) ).
% bounded_linear_const_mult
thf(fact_7310_bounded__linear__mult__const,axiom,
! [A: $tType,C: $tType] :
( ( ( real_V822414075346904944vector @ C )
& ( real_V4412858255891104859lgebra @ A ) )
=> ! [G: C > A,Y3: A] :
( ( real_V3181309239436604168linear @ C @ A @ G )
=> ( real_V3181309239436604168linear @ C @ A
@ ^ [X3: C] : ( times_times @ A @ ( G @ X3 ) @ Y3 ) ) ) ) ).
% bounded_linear_mult_const
thf(fact_7311_bounded__linear__mult__right,axiom,
! [A: $tType] :
( ( real_V4412858255891104859lgebra @ A )
=> ! [X2: A] : ( real_V3181309239436604168linear @ A @ A @ ( times_times @ A @ X2 ) ) ) ).
% bounded_linear_mult_right
thf(fact_7312_bounded__linear__minus,axiom,
! [B: $tType,A: $tType] :
( ( ( real_V822414075346904944vector @ A )
& ( real_V822414075346904944vector @ B ) )
=> ! [F2: A > B] :
( ( real_V3181309239436604168linear @ A @ B @ F2 )
=> ( real_V3181309239436604168linear @ A @ B
@ ^ [X3: A] : ( uminus_uminus @ B @ ( F2 @ X3 ) ) ) ) ) ).
% bounded_linear_minus
thf(fact_7313_bounded__linear__zero,axiom,
! [B: $tType,A: $tType] :
( ( ( real_V822414075346904944vector @ A )
& ( real_V822414075346904944vector @ B ) )
=> ( real_V3181309239436604168linear @ A @ B
@ ^ [X3: A] : ( zero_zero @ B ) ) ) ).
% bounded_linear_zero
thf(fact_7314_bounded__linear__scaleR__left,axiom,
! [A: $tType] :
( ( real_V822414075346904944vector @ A )
=> ! [X2: A] :
( real_V3181309239436604168linear @ real @ A
@ ^ [R5: real] : ( real_V8093663219630862766scaleR @ A @ R5 @ X2 ) ) ) ).
% bounded_linear_scaleR_left
thf(fact_7315_bounded__linear__const__scaleR,axiom,
! [B: $tType,C: $tType] :
( ( ( real_V822414075346904944vector @ C )
& ( real_V822414075346904944vector @ B ) )
=> ! [G: C > B,R2: real] :
( ( real_V3181309239436604168linear @ C @ B @ G )
=> ( real_V3181309239436604168linear @ C @ B
@ ^ [X3: C] : ( real_V8093663219630862766scaleR @ B @ R2 @ ( G @ X3 ) ) ) ) ) ).
% bounded_linear_const_scaleR
thf(fact_7316_bounded__linear__scaleR__const,axiom,
! [B: $tType,C: $tType] :
( ( ( real_V822414075346904944vector @ C )
& ( real_V822414075346904944vector @ B ) )
=> ! [G: C > real,X2: B] :
( ( real_V3181309239436604168linear @ C @ real @ G )
=> ( real_V3181309239436604168linear @ C @ B
@ ^ [X3: C] : ( real_V8093663219630862766scaleR @ B @ ( G @ X3 ) @ X2 ) ) ) ) ).
% bounded_linear_scaleR_const
thf(fact_7317_bounded__linear__scaleR__right,axiom,
! [A: $tType] :
( ( real_V822414075346904944vector @ A )
=> ! [R2: real] : ( real_V3181309239436604168linear @ A @ A @ ( real_V8093663219630862766scaleR @ A @ R2 ) ) ) ).
% bounded_linear_scaleR_right
thf(fact_7318_bounded__linear_Osums,axiom,
! [B: $tType,A: $tType] :
( ( ( real_V822414075346904944vector @ A )
& ( real_V822414075346904944vector @ B ) )
=> ! [F2: A > B,X: nat > A,A3: A] :
( ( real_V3181309239436604168linear @ A @ B @ F2 )
=> ( ( sums @ A @ X @ A3 )
=> ( sums @ B
@ ^ [N4: nat] : ( F2 @ ( X @ N4 ) )
@ ( F2 @ A3 ) ) ) ) ) ).
% bounded_linear.sums
thf(fact_7319_bounded__linear__compose,axiom,
! [A: $tType,B: $tType,C: $tType] :
( ( ( real_V822414075346904944vector @ C )
& ( real_V822414075346904944vector @ B )
& ( real_V822414075346904944vector @ A ) )
=> ! [F2: A > B,G: C > A] :
( ( real_V3181309239436604168linear @ A @ B @ F2 )
=> ( ( real_V3181309239436604168linear @ C @ A @ G )
=> ( real_V3181309239436604168linear @ C @ B
@ ^ [X3: C] : ( F2 @ ( G @ X3 ) ) ) ) ) ) ).
% bounded_linear_compose
thf(fact_7320_bounded__linear__ident,axiom,
! [A: $tType] :
( ( real_V822414075346904944vector @ A )
=> ( real_V3181309239436604168linear @ A @ A
@ ^ [X3: A] : X3 ) ) ).
% bounded_linear_ident
thf(fact_7321_bounded__linear_Osummable,axiom,
! [A: $tType,B: $tType] :
( ( ( real_V822414075346904944vector @ B )
& ( real_V822414075346904944vector @ A ) )
=> ! [F2: A > B,X: nat > A] :
( ( real_V3181309239436604168linear @ A @ B @ F2 )
=> ( ( summable @ A @ X )
=> ( summable @ B
@ ^ [N4: nat] : ( F2 @ ( X @ N4 ) ) ) ) ) ) ).
% bounded_linear.summable
thf(fact_7322_bounded__linear__of__real,axiom,
! [A: $tType] :
( ( ( real_V2191834092415804123ebra_1 @ A )
& ( real_V822414075346904944vector @ A ) )
=> ( real_V3181309239436604168linear @ real @ A @ ( real_Vector_of_real @ A ) ) ) ).
% bounded_linear_of_real
thf(fact_7323_bounded__linear__sum,axiom,
! [I8: $tType,B: $tType,A: $tType] :
( ( ( real_V822414075346904944vector @ A )
& ( real_V822414075346904944vector @ B ) )
=> ! [I6: set @ I8,F2: I8 > A > B] :
( ! [I2: I8] :
( ( member @ I8 @ I2 @ I6 )
=> ( real_V3181309239436604168linear @ A @ B @ ( F2 @ I2 ) ) )
=> ( real_V3181309239436604168linear @ A @ B
@ ^ [X3: A] :
( groups7311177749621191930dd_sum @ I8 @ B
@ ^ [I4: I8] : ( F2 @ I4 @ X3 )
@ I6 ) ) ) ) ).
% bounded_linear_sum
thf(fact_7324_bounded__linear__sub,axiom,
! [B: $tType,A: $tType] :
( ( ( real_V822414075346904944vector @ A )
& ( real_V822414075346904944vector @ B ) )
=> ! [F2: A > B,G: A > B] :
( ( real_V3181309239436604168linear @ A @ B @ F2 )
=> ( ( real_V3181309239436604168linear @ A @ B @ G )
=> ( real_V3181309239436604168linear @ A @ B
@ ^ [X3: A] : ( minus_minus @ B @ ( F2 @ X3 ) @ ( G @ X3 ) ) ) ) ) ) ).
% bounded_linear_sub
thf(fact_7325_bounded__linear__add,axiom,
! [B: $tType,A: $tType] :
( ( ( real_V822414075346904944vector @ A )
& ( real_V822414075346904944vector @ B ) )
=> ! [F2: A > B,G: A > B] :
( ( real_V3181309239436604168linear @ A @ B @ F2 )
=> ( ( real_V3181309239436604168linear @ A @ B @ G )
=> ( real_V3181309239436604168linear @ A @ B
@ ^ [X3: A] : ( plus_plus @ B @ ( F2 @ X3 ) @ ( G @ X3 ) ) ) ) ) ) ).
% bounded_linear_add
thf(fact_7326_bounded__linear_Osuminf,axiom,
! [A: $tType,B: $tType] :
( ( ( real_V822414075346904944vector @ B )
& ( real_V822414075346904944vector @ A ) )
=> ! [F2: A > B,X: nat > A] :
( ( real_V3181309239436604168linear @ A @ B @ F2 )
=> ( ( summable @ A @ X )
=> ( ( F2 @ ( suminf @ A @ X ) )
= ( suminf @ B
@ ^ [N4: nat] : ( F2 @ ( X @ N4 ) ) ) ) ) ) ) ).
% bounded_linear.suminf
thf(fact_7327_bounded__linear_Ohas__derivative,axiom,
! [B: $tType,A: $tType,C: $tType] :
( ( ( real_V822414075346904944vector @ C )
& ( real_V822414075346904944vector @ A )
& ( real_V822414075346904944vector @ B ) )
=> ! [F2: A > B,G: C > A,G5: C > A,F4: filter @ C] :
( ( real_V3181309239436604168linear @ A @ B @ F2 )
=> ( ( has_derivative @ C @ A @ G @ G5 @ F4 )
=> ( has_derivative @ C @ B
@ ^ [X3: C] : ( F2 @ ( G @ X3 ) )
@ ^ [X3: C] : ( F2 @ ( G5 @ X3 ) )
@ F4 ) ) ) ) ).
% bounded_linear.has_derivative
thf(fact_7328_bounded__linear_Obounded,axiom,
! [B: $tType,A: $tType] :
( ( ( real_V822414075346904944vector @ A )
& ( real_V822414075346904944vector @ B ) )
=> ! [F2: A > B] :
( ( real_V3181309239436604168linear @ A @ B @ F2 )
=> ? [K9: real] :
! [X8: A] : ( ord_less_eq @ real @ ( real_V7770717601297561774m_norm @ B @ ( F2 @ X8 ) ) @ ( times_times @ real @ ( real_V7770717601297561774m_norm @ A @ X8 ) @ K9 ) ) ) ) ).
% bounded_linear.bounded
thf(fact_7329_bounded__linear_Otendsto__zero,axiom,
! [B: $tType,A: $tType,C: $tType] :
( ( ( real_V822414075346904944vector @ A )
& ( real_V822414075346904944vector @ B ) )
=> ! [F2: A > B,G: C > A,F4: filter @ C] :
( ( real_V3181309239436604168linear @ A @ B @ F2 )
=> ( ( filterlim @ C @ A @ G @ ( topolo7230453075368039082e_nhds @ A @ ( zero_zero @ A ) ) @ F4 )
=> ( filterlim @ C @ B
@ ^ [X3: C] : ( F2 @ ( G @ X3 ) )
@ ( topolo7230453075368039082e_nhds @ B @ ( zero_zero @ B ) )
@ F4 ) ) ) ) ).
% bounded_linear.tendsto_zero
thf(fact_7330_bounded__linear_OisCont,axiom,
! [A: $tType,B: $tType,C: $tType] :
( ( ( topological_t2_space @ C )
& ( real_V822414075346904944vector @ B )
& ( real_V822414075346904944vector @ A ) )
=> ! [F2: A > B,A3: C,G: C > A] :
( ( real_V3181309239436604168linear @ A @ B @ F2 )
=> ( ( topolo3448309680560233919inuous @ C @ A @ ( topolo174197925503356063within @ C @ A3 @ ( top_top @ ( set @ C ) ) ) @ G )
=> ( topolo3448309680560233919inuous @ C @ B @ ( topolo174197925503356063within @ C @ A3 @ ( top_top @ ( set @ C ) ) )
@ ^ [X3: C] : ( F2 @ ( G @ X3 ) ) ) ) ) ) ).
% bounded_linear.isCont
thf(fact_7331_bounded__linear_Ononneg__bounded,axiom,
! [B: $tType,A: $tType] :
( ( ( real_V822414075346904944vector @ A )
& ( real_V822414075346904944vector @ B ) )
=> ! [F2: A > B] :
( ( real_V3181309239436604168linear @ A @ B @ F2 )
=> ? [K9: real] :
( ( ord_less_eq @ real @ ( zero_zero @ real ) @ K9 )
& ! [X8: A] : ( ord_less_eq @ real @ ( real_V7770717601297561774m_norm @ B @ ( F2 @ X8 ) ) @ ( times_times @ real @ ( real_V7770717601297561774m_norm @ A @ X8 ) @ K9 ) ) ) ) ) ).
% bounded_linear.nonneg_bounded
thf(fact_7332_less__filter__def,axiom,
! [A: $tType] :
( ( ord_less @ ( filter @ A ) )
= ( ^ [F9: filter @ A,F10: filter @ A] :
( ( ord_less_eq @ ( filter @ A ) @ F9 @ F10 )
& ~ ( ord_less_eq @ ( filter @ A ) @ F10 @ F9 ) ) ) ) ).
% less_filter_def
thf(fact_7333_filterlim__ident,axiom,
! [A: $tType,F4: filter @ A] :
( filterlim @ A @ A
@ ^ [X3: A] : X3
@ F4
@ F4 ) ).
% filterlim_ident
thf(fact_7334_filterlim__compose,axiom,
! [B: $tType,A: $tType,C: $tType,G: A > B,F32: filter @ B,F24: filter @ A,F2: C > A,F14: filter @ C] :
( ( filterlim @ A @ B @ G @ F32 @ F24 )
=> ( ( filterlim @ C @ A @ F2 @ F24 @ F14 )
=> ( filterlim @ C @ B
@ ^ [X3: C] : ( G @ ( F2 @ X3 ) )
@ F32
@ F14 ) ) ) ).
% filterlim_compose
thf(fact_7335_bounded__linear_Opos__bounded,axiom,
! [B: $tType,A: $tType] :
( ( ( real_V822414075346904944vector @ A )
& ( real_V822414075346904944vector @ B ) )
=> ! [F2: A > B] :
( ( real_V3181309239436604168linear @ A @ B @ F2 )
=> ? [K9: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ K9 )
& ! [X8: A] : ( ord_less_eq @ real @ ( real_V7770717601297561774m_norm @ B @ ( F2 @ X8 ) ) @ ( times_times @ real @ ( real_V7770717601297561774m_norm @ A @ X8 ) @ K9 ) ) ) ) ) ).
% bounded_linear.pos_bounded
thf(fact_7336_bounded__linear__intro,axiom,
! [B: $tType,A: $tType] :
( ( ( real_V822414075346904944vector @ A )
& ( real_V822414075346904944vector @ B ) )
=> ! [F2: A > B,K6: real] :
( ! [X4: A,Y4: A] :
( ( F2 @ ( plus_plus @ A @ X4 @ Y4 ) )
= ( plus_plus @ B @ ( F2 @ X4 ) @ ( F2 @ Y4 ) ) )
=> ( ! [R3: real,X4: A] :
( ( F2 @ ( real_V8093663219630862766scaleR @ A @ R3 @ X4 ) )
= ( real_V8093663219630862766scaleR @ B @ R3 @ ( F2 @ X4 ) ) )
=> ( ! [X4: A] : ( ord_less_eq @ real @ ( real_V7770717601297561774m_norm @ B @ ( F2 @ X4 ) ) @ ( times_times @ real @ ( real_V7770717601297561774m_norm @ A @ X4 ) @ K6 ) )
=> ( real_V3181309239436604168linear @ A @ B @ F2 ) ) ) ) ) ).
% bounded_linear_intro
thf(fact_7337_filterlim__Suc,axiom,
filterlim @ nat @ nat @ suc @ ( at_top @ nat ) @ ( at_top @ nat ) ).
% filterlim_Suc
thf(fact_7338_filterlim__mono,axiom,
! [B: $tType,A: $tType,F2: A > B,F24: filter @ B,F14: filter @ A,F25: filter @ B,F15: filter @ A] :
( ( filterlim @ A @ B @ F2 @ F24 @ F14 )
=> ( ( ord_less_eq @ ( filter @ B ) @ F24 @ F25 )
=> ( ( ord_less_eq @ ( filter @ A ) @ F15 @ F14 )
=> ( filterlim @ A @ B @ F2 @ F25 @ F15 ) ) ) ) ).
% filterlim_mono
thf(fact_7339_has__derivative__iff__norm,axiom,
! [B: $tType,A: $tType] :
( ( ( real_V822414075346904944vector @ A )
& ( real_V822414075346904944vector @ B ) )
=> ! [F2: A > B,F7: A > B,X2: A,S2: set @ A] :
( ( has_derivative @ A @ B @ F2 @ F7 @ ( topolo174197925503356063within @ A @ X2 @ S2 ) )
= ( ( real_V3181309239436604168linear @ A @ B @ F7 )
& ( filterlim @ A @ real
@ ^ [Y: A] : ( divide_divide @ real @ ( real_V7770717601297561774m_norm @ B @ ( minus_minus @ B @ ( minus_minus @ B @ ( F2 @ Y ) @ ( F2 @ X2 ) ) @ ( F7 @ ( minus_minus @ A @ Y @ X2 ) ) ) ) @ ( real_V7770717601297561774m_norm @ A @ ( minus_minus @ A @ Y @ X2 ) ) )
@ ( topolo7230453075368039082e_nhds @ real @ ( zero_zero @ real ) )
@ ( topolo174197925503356063within @ A @ X2 @ S2 ) ) ) ) ) ).
% has_derivative_iff_norm
thf(fact_7340_has__derivative__at__within,axiom,
! [B: $tType,A: $tType] :
( ( ( real_V822414075346904944vector @ A )
& ( real_V822414075346904944vector @ B ) )
=> ! [F2: A > B,F7: A > B,X2: A,S2: set @ A] :
( ( has_derivative @ A @ B @ F2 @ F7 @ ( topolo174197925503356063within @ A @ X2 @ S2 ) )
= ( ( real_V3181309239436604168linear @ A @ B @ F7 )
& ( filterlim @ A @ B
@ ^ [Y: A] : ( real_V8093663219630862766scaleR @ B @ ( inverse_inverse @ real @ ( real_V7770717601297561774m_norm @ A @ ( minus_minus @ A @ Y @ X2 ) ) ) @ ( minus_minus @ B @ ( minus_minus @ B @ ( F2 @ Y ) @ ( F2 @ X2 ) ) @ ( F7 @ ( minus_minus @ A @ Y @ X2 ) ) ) )
@ ( topolo7230453075368039082e_nhds @ B @ ( zero_zero @ B ) )
@ ( topolo174197925503356063within @ A @ X2 @ S2 ) ) ) ) ) ).
% has_derivative_at_within
thf(fact_7341_has__derivativeI,axiom,
! [B: $tType,A: $tType] :
( ( ( real_V822414075346904944vector @ A )
& ( real_V822414075346904944vector @ B ) )
=> ! [F7: A > B,X2: A,F2: A > B,S2: set @ A] :
( ( real_V3181309239436604168linear @ A @ B @ F7 )
=> ( ( filterlim @ A @ B
@ ^ [Y: A] : ( real_V8093663219630862766scaleR @ B @ ( inverse_inverse @ real @ ( real_V7770717601297561774m_norm @ A @ ( minus_minus @ A @ Y @ X2 ) ) ) @ ( minus_minus @ B @ ( minus_minus @ B @ ( F2 @ Y ) @ ( F2 @ X2 ) ) @ ( F7 @ ( minus_minus @ A @ Y @ X2 ) ) ) )
@ ( topolo7230453075368039082e_nhds @ B @ ( zero_zero @ B ) )
@ ( topolo174197925503356063within @ A @ X2 @ S2 ) )
=> ( has_derivative @ A @ B @ F2 @ F7 @ ( topolo174197925503356063within @ A @ X2 @ S2 ) ) ) ) ) ).
% has_derivativeI
thf(fact_7342_has__derivative__iff__Ex,axiom,
! [B: $tType,A: $tType] :
( ( ( real_V822414075346904944vector @ A )
& ( real_V822414075346904944vector @ B ) )
=> ! [F2: A > B,F7: A > B,X2: A] :
( ( has_derivative @ A @ B @ F2 @ F7 @ ( topolo174197925503356063within @ A @ X2 @ ( top_top @ ( set @ A ) ) ) )
= ( ( real_V3181309239436604168linear @ A @ B @ F7 )
& ? [E4: A > B] :
( ! [H: A] :
( ( F2 @ ( plus_plus @ A @ X2 @ H ) )
= ( plus_plus @ B @ ( plus_plus @ B @ ( F2 @ X2 ) @ ( F7 @ H ) ) @ ( E4 @ H ) ) )
& ( filterlim @ A @ real
@ ^ [H: A] : ( divide_divide @ real @ ( real_V7770717601297561774m_norm @ B @ ( E4 @ H ) ) @ ( real_V7770717601297561774m_norm @ A @ H ) )
@ ( topolo7230453075368039082e_nhds @ real @ ( zero_zero @ real ) )
@ ( topolo174197925503356063within @ A @ ( zero_zero @ A ) @ ( top_top @ ( set @ A ) ) ) ) ) ) ) ) ).
% has_derivative_iff_Ex
thf(fact_7343_has__derivative__within,axiom,
! [B: $tType,A: $tType] :
( ( ( real_V822414075346904944vector @ A )
& ( real_V822414075346904944vector @ B ) )
=> ! [F2: A > B,F7: A > B,X2: A,S2: set @ A] :
( ( has_derivative @ A @ B @ F2 @ F7 @ ( topolo174197925503356063within @ A @ X2 @ S2 ) )
= ( ( real_V3181309239436604168linear @ A @ B @ F7 )
& ( filterlim @ A @ B
@ ^ [Y: A] : ( real_V8093663219630862766scaleR @ B @ ( divide_divide @ real @ ( one_one @ real ) @ ( real_V7770717601297561774m_norm @ A @ ( minus_minus @ A @ Y @ X2 ) ) ) @ ( minus_minus @ B @ ( F2 @ Y ) @ ( plus_plus @ B @ ( F2 @ X2 ) @ ( F7 @ ( minus_minus @ A @ Y @ X2 ) ) ) ) )
@ ( topolo7230453075368039082e_nhds @ B @ ( zero_zero @ B ) )
@ ( topolo174197925503356063within @ A @ X2 @ S2 ) ) ) ) ) ).
% has_derivative_within
thf(fact_7344_has__derivative__at,axiom,
! [B: $tType,A: $tType] :
( ( ( real_V822414075346904944vector @ A )
& ( real_V822414075346904944vector @ B ) )
=> ! [F2: A > B,D: A > B,X2: A] :
( ( has_derivative @ A @ B @ F2 @ D @ ( topolo174197925503356063within @ A @ X2 @ ( top_top @ ( set @ A ) ) ) )
= ( ( real_V3181309239436604168linear @ A @ B @ D )
& ( filterlim @ A @ real
@ ^ [H: A] : ( divide_divide @ real @ ( real_V7770717601297561774m_norm @ B @ ( minus_minus @ B @ ( minus_minus @ B @ ( F2 @ ( plus_plus @ A @ X2 @ H ) ) @ ( F2 @ X2 ) ) @ ( D @ H ) ) ) @ ( real_V7770717601297561774m_norm @ A @ H ) )
@ ( topolo7230453075368039082e_nhds @ real @ ( zero_zero @ real ) )
@ ( topolo174197925503356063within @ A @ ( zero_zero @ A ) @ ( top_top @ ( set @ A ) ) ) ) ) ) ) ).
% has_derivative_at
thf(fact_7345_has__derivative__def,axiom,
! [B: $tType,A: $tType] :
( ( ( real_V822414075346904944vector @ A )
& ( real_V822414075346904944vector @ B ) )
=> ( ( has_derivative @ A @ B )
= ( ^ [F5: A > B,F11: A > B,F9: filter @ A] :
( ( real_V3181309239436604168linear @ A @ B @ F11 )
& ( filterlim @ A @ B
@ ^ [Y: A] :
( real_V8093663219630862766scaleR @ B
@ ( inverse_inverse @ real
@ ( real_V7770717601297561774m_norm @ A
@ ( minus_minus @ A @ Y
@ ( topolo3827282254853284352ce_Lim @ A @ A @ F9
@ ^ [X3: A] : X3 ) ) ) )
@ ( minus_minus @ B
@ ( minus_minus @ B @ ( F5 @ Y )
@ ( F5
@ ( topolo3827282254853284352ce_Lim @ A @ A @ F9
@ ^ [X3: A] : X3 ) ) )
@ ( F11
@ ( minus_minus @ A @ Y
@ ( topolo3827282254853284352ce_Lim @ A @ A @ F9
@ ^ [X3: A] : X3 ) ) ) ) )
@ ( topolo7230453075368039082e_nhds @ B @ ( zero_zero @ B ) )
@ F9 ) ) ) ) ) ).
% has_derivative_def
thf(fact_7346_has__derivative__at__within__iff__Ex,axiom,
! [B: $tType,A: $tType] :
( ( ( real_V822414075346904944vector @ A )
& ( real_V822414075346904944vector @ B ) )
=> ! [X2: A,S: set @ A,F2: A > B,F7: A > B] :
( ( member @ A @ X2 @ S )
=> ( ( topolo1002775350975398744n_open @ A @ S )
=> ( ( has_derivative @ A @ B @ F2 @ F7 @ ( topolo174197925503356063within @ A @ X2 @ S ) )
= ( ( real_V3181309239436604168linear @ A @ B @ F7 )
& ? [E4: A > B] :
( ! [H: A] :
( ( member @ A @ ( plus_plus @ A @ X2 @ H ) @ S )
=> ( ( F2 @ ( plus_plus @ A @ X2 @ H ) )
= ( plus_plus @ B @ ( plus_plus @ B @ ( F2 @ X2 ) @ ( F7 @ H ) ) @ ( E4 @ H ) ) ) )
& ( filterlim @ A @ real
@ ^ [H: A] : ( divide_divide @ real @ ( real_V7770717601297561774m_norm @ B @ ( E4 @ H ) ) @ ( real_V7770717601297561774m_norm @ A @ H ) )
@ ( topolo7230453075368039082e_nhds @ real @ ( zero_zero @ real ) )
@ ( topolo174197925503356063within @ A @ ( zero_zero @ A ) @ ( top_top @ ( set @ A ) ) ) ) ) ) ) ) ) ) ).
% has_derivative_at_within_iff_Ex
thf(fact_7347_lim__const,axiom,
! [A: $tType] :
( ( topological_t2_space @ A )
=> ! [A3: A] :
( ( topolo3827282254853284352ce_Lim @ nat @ A @ ( at_top @ nat )
@ ^ [M6: nat] : A3 )
= A3 ) ) ).
% lim_const
thf(fact_7348_at__within__open__subset,axiom,
! [A: $tType] :
( ( topolo4958980785337419405_space @ A )
=> ! [A3: A,S: set @ A,T6: set @ A] :
( ( member @ A @ A3 @ S )
=> ( ( topolo1002775350975398744n_open @ A @ S )
=> ( ( ord_less_eq @ ( set @ A ) @ S @ T6 )
=> ( ( topolo174197925503356063within @ A @ A3 @ T6 )
= ( topolo174197925503356063within @ A @ A3 @ ( top_top @ ( set @ A ) ) ) ) ) ) ) ) ).
% at_within_open_subset
thf(fact_7349_open__right,axiom,
! [A: $tType] :
( ( topolo1944317154257567458pology @ A )
=> ! [S: set @ A,X2: A,Y3: A] :
( ( topolo1002775350975398744n_open @ A @ S )
=> ( ( member @ A @ X2 @ S )
=> ( ( ord_less @ A @ X2 @ Y3 )
=> ? [B4: A] :
( ( ord_less @ A @ X2 @ B4 )
& ( ord_less_eq @ ( set @ A ) @ ( set_or7035219750837199246ssThan @ A @ X2 @ B4 ) @ S ) ) ) ) ) ) ).
% open_right
thf(fact_7350_open__Collect__conj,axiom,
! [A: $tType] :
( ( topolo4958980785337419405_space @ A )
=> ! [P: A > $o,Q: A > $o] :
( ( topolo1002775350975398744n_open @ A @ ( collect @ A @ P ) )
=> ( ( topolo1002775350975398744n_open @ A @ ( collect @ A @ Q ) )
=> ( topolo1002775350975398744n_open @ A
@ ( collect @ A
@ ^ [X3: A] :
( ( P @ X3 )
& ( Q @ X3 ) ) ) ) ) ) ) ).
% open_Collect_conj
thf(fact_7351_open__Collect__disj,axiom,
! [A: $tType] :
( ( topolo4958980785337419405_space @ A )
=> ! [P: A > $o,Q: A > $o] :
( ( topolo1002775350975398744n_open @ A @ ( collect @ A @ P ) )
=> ( ( topolo1002775350975398744n_open @ A @ ( collect @ A @ Q ) )
=> ( topolo1002775350975398744n_open @ A
@ ( collect @ A
@ ^ [X3: A] :
( ( P @ X3 )
| ( Q @ X3 ) ) ) ) ) ) ) ).
% open_Collect_disj
thf(fact_7352_open__Collect__const,axiom,
! [A: $tType] :
( ( topolo4958980785337419405_space @ A )
=> ! [P: $o] :
( topolo1002775350975398744n_open @ A
@ ( collect @ A
@ ^ [X3: A] : P ) ) ) ).
% open_Collect_const
thf(fact_7353_first__countable__basis,axiom,
! [A: $tType] :
( ( topolo3112930676232923870pology @ A )
=> ! [X2: A] :
? [A9: nat > ( set @ A )] :
( ! [I5: nat] :
( ( member @ A @ X2 @ ( A9 @ I5 ) )
& ( topolo1002775350975398744n_open @ A @ ( A9 @ I5 ) ) )
& ! [S10: set @ A] :
( ( ( topolo1002775350975398744n_open @ A @ S10 )
& ( member @ A @ X2 @ S10 ) )
=> ? [I2: nat] : ( ord_less_eq @ ( set @ A ) @ ( A9 @ I2 ) @ S10 ) ) ) ) ).
% first_countable_basis
thf(fact_7354_open__subopen,axiom,
! [A: $tType] :
( ( topolo4958980785337419405_space @ A )
=> ( ( topolo1002775350975398744n_open @ A )
= ( ^ [S8: set @ A] :
! [X3: A] :
( ( member @ A @ X3 @ S8 )
=> ? [T9: set @ A] :
( ( topolo1002775350975398744n_open @ A @ T9 )
& ( member @ A @ X3 @ T9 )
& ( ord_less_eq @ ( set @ A ) @ T9 @ S8 ) ) ) ) ) ) ).
% open_subopen
thf(fact_7355_openI,axiom,
! [A: $tType] :
( ( topolo4958980785337419405_space @ A )
=> ! [S: set @ A] :
( ! [X4: A] :
( ( member @ A @ X4 @ S )
=> ? [T11: set @ A] :
( ( topolo1002775350975398744n_open @ A @ T11 )
& ( member @ A @ X4 @ T11 )
& ( ord_less_eq @ ( set @ A ) @ T11 @ S ) ) )
=> ( topolo1002775350975398744n_open @ A @ S ) ) ) ).
% openI
thf(fact_7356_Lim__ident__at,axiom,
! [A: $tType] :
( ( topological_t2_space @ A )
=> ! [X2: A,S2: set @ A] :
( ( ( topolo174197925503356063within @ A @ X2 @ S2 )
!= ( bot_bot @ ( filter @ A ) ) )
=> ( ( topolo3827282254853284352ce_Lim @ A @ A @ ( topolo174197925503356063within @ A @ X2 @ S2 )
@ ^ [X3: A] : X3 )
= X2 ) ) ) ).
% Lim_ident_at
thf(fact_7357_lim__explicit,axiom,
! [A: $tType] :
( ( topolo4958980785337419405_space @ A )
=> ! [F2: nat > A,F0: A] :
( ( filterlim @ nat @ A @ F2 @ ( topolo7230453075368039082e_nhds @ A @ F0 ) @ ( at_top @ nat ) )
= ( ! [S8: set @ A] :
( ( topolo1002775350975398744n_open @ A @ S8 )
=> ( ( member @ A @ F0 @ S8 )
=> ? [N8: nat] :
! [N4: nat] :
( ( ord_less_eq @ nat @ N8 @ N4 )
=> ( member @ A @ ( F2 @ N4 ) @ S8 ) ) ) ) ) ) ) ).
% lim_explicit
thf(fact_7358_continuous__divide,axiom,
! [B: $tType,A: $tType] :
( ( ( topological_t2_space @ A )
& ( real_V3459762299906320749_field @ B ) )
=> ! [F4: filter @ A,F2: A > B,G: A > B] :
( ( topolo3448309680560233919inuous @ A @ B @ F4 @ F2 )
=> ( ( topolo3448309680560233919inuous @ A @ B @ F4 @ G )
=> ( ( ( G
@ ( topolo3827282254853284352ce_Lim @ A @ A @ F4
@ ^ [X3: A] : X3 ) )
!= ( zero_zero @ B ) )
=> ( topolo3448309680560233919inuous @ A @ B @ F4
@ ^ [X3: A] : ( divide_divide @ B @ ( F2 @ X3 ) @ ( G @ X3 ) ) ) ) ) ) ) ).
% continuous_divide
thf(fact_7359_continuous__inverse,axiom,
! [B: $tType,A: $tType] :
( ( ( topological_t2_space @ A )
& ( real_V8999393235501362500lgebra @ B ) )
=> ! [F4: filter @ A,F2: A > B] :
( ( topolo3448309680560233919inuous @ A @ B @ F4 @ F2 )
=> ( ( ( F2
@ ( topolo3827282254853284352ce_Lim @ A @ A @ F4
@ ^ [X3: A] : X3 ) )
!= ( zero_zero @ B ) )
=> ( topolo3448309680560233919inuous @ A @ B @ F4
@ ^ [X3: A] : ( inverse_inverse @ B @ ( F2 @ X3 ) ) ) ) ) ) ).
% continuous_inverse
thf(fact_7360_continuous__def,axiom,
! [B: $tType,A: $tType] :
( ( ( topological_t2_space @ A )
& ( topolo4958980785337419405_space @ B ) )
=> ( ( topolo3448309680560233919inuous @ A @ B )
= ( ^ [F9: filter @ A,F5: A > B] :
( filterlim @ A @ B @ F5
@ ( topolo7230453075368039082e_nhds @ B
@ ( F5
@ ( topolo3827282254853284352ce_Lim @ A @ A @ F9
@ ^ [X3: A] : X3 ) ) )
@ F9 ) ) ) ) ).
% continuous_def
thf(fact_7361_continuous__sgn,axiom,
! [B: $tType,A: $tType] :
( ( ( topological_t2_space @ A )
& ( real_V822414075346904944vector @ B ) )
=> ! [F4: filter @ A,F2: A > B] :
( ( topolo3448309680560233919inuous @ A @ B @ F4 @ F2 )
=> ( ( ( F2
@ ( topolo3827282254853284352ce_Lim @ A @ A @ F4
@ ^ [X3: A] : X3 ) )
!= ( zero_zero @ B ) )
=> ( topolo3448309680560233919inuous @ A @ B @ F4
@ ^ [X3: A] : ( sgn_sgn @ B @ ( F2 @ X3 ) ) ) ) ) ) ).
% continuous_sgn
thf(fact_7362_t2__space__class_OLim__def,axiom,
! [A: $tType,F: $tType] :
( ( topological_t2_space @ A )
=> ( ( topolo3827282254853284352ce_Lim @ F @ A )
= ( ^ [A7: filter @ F,F5: F > A] :
( the @ A
@ ^ [L3: A] : ( filterlim @ F @ A @ F5 @ ( topolo7230453075368039082e_nhds @ A @ L3 ) @ A7 ) ) ) ) ) ).
% t2_space_class.Lim_def
thf(fact_7363_continuous__powr,axiom,
! [A: $tType] :
( ( topological_t2_space @ A )
=> ! [F4: filter @ A,F2: A > real,G: A > real] :
( ( topolo3448309680560233919inuous @ A @ real @ F4 @ F2 )
=> ( ( topolo3448309680560233919inuous @ A @ real @ F4 @ G )
=> ( ( ( F2
@ ( topolo3827282254853284352ce_Lim @ A @ A @ F4
@ ^ [X3: A] : X3 ) )
!= ( zero_zero @ real ) )
=> ( topolo3448309680560233919inuous @ A @ real @ F4
@ ^ [X3: A] : ( powr @ real @ ( F2 @ X3 ) @ ( G @ X3 ) ) ) ) ) ) ) ).
% continuous_powr
thf(fact_7364_continuous__ln,axiom,
! [A: $tType] :
( ( topological_t2_space @ A )
=> ! [F4: filter @ A,F2: A > real] :
( ( topolo3448309680560233919inuous @ A @ real @ F4 @ F2 )
=> ( ( ( F2
@ ( topolo3827282254853284352ce_Lim @ A @ A @ F4
@ ^ [X3: A] : X3 ) )
!= ( zero_zero @ real ) )
=> ( topolo3448309680560233919inuous @ A @ real @ F4
@ ^ [X3: A] : ( ln_ln @ real @ ( F2 @ X3 ) ) ) ) ) ) ).
% continuous_ln
thf(fact_7365_suminf__eq__lim,axiom,
! [A: $tType] :
( ( ( comm_monoid_add @ A )
& ( topological_t2_space @ A ) )
=> ( ( suminf @ A )
= ( ^ [F5: nat > A] :
( topolo3827282254853284352ce_Lim @ nat @ A @ ( at_top @ nat )
@ ^ [N4: nat] : ( groups7311177749621191930dd_sum @ nat @ A @ F5 @ ( set_ord_lessThan @ nat @ N4 ) ) ) ) ) ) ).
% suminf_eq_lim
thf(fact_7366_Topological__Spaces_Olim__def,axiom,
! [A: $tType] :
( ( topological_t2_space @ A )
=> ! [X: nat > A] :
( ( topolo3827282254853284352ce_Lim @ nat @ A @ ( at_top @ nat ) @ X )
= ( the @ A
@ ^ [L8: A] : ( filterlim @ nat @ A @ X @ ( topolo7230453075368039082e_nhds @ A @ L8 ) @ ( at_top @ nat ) ) ) ) ) ).
% Topological_Spaces.lim_def
thf(fact_7367_continuous__tan,axiom,
! [A: $tType] :
( ( ( real_Vector_banach @ A )
& ( real_V3459762299906320749_field @ A ) )
=> ! [F4: filter @ A,F2: A > A] :
( ( topolo3448309680560233919inuous @ A @ A @ F4 @ F2 )
=> ( ( ( cos @ A
@ ( F2
@ ( topolo3827282254853284352ce_Lim @ A @ A @ F4
@ ^ [X3: A] : X3 ) ) )
!= ( zero_zero @ A ) )
=> ( topolo3448309680560233919inuous @ A @ A @ F4
@ ^ [X3: A] : ( tan @ A @ ( F2 @ X3 ) ) ) ) ) ) ).
% continuous_tan
thf(fact_7368_continuous__cot,axiom,
! [A: $tType] :
( ( ( real_Vector_banach @ A )
& ( real_V3459762299906320749_field @ A ) )
=> ! [F4: filter @ A,F2: A > A] :
( ( topolo3448309680560233919inuous @ A @ A @ F4 @ F2 )
=> ( ( ( sin @ A
@ ( F2
@ ( topolo3827282254853284352ce_Lim @ A @ A @ F4
@ ^ [X3: A] : X3 ) ) )
!= ( zero_zero @ A ) )
=> ( topolo3448309680560233919inuous @ A @ A @ F4
@ ^ [X3: A] : ( cot @ A @ ( F2 @ X3 ) ) ) ) ) ) ).
% continuous_cot
thf(fact_7369_continuous__tanh,axiom,
! [A: $tType,C: $tType] :
( ( ( topological_t2_space @ C )
& ( real_Vector_banach @ A )
& ( real_V3459762299906320749_field @ A ) )
=> ! [F4: filter @ C,F2: C > A] :
( ( topolo3448309680560233919inuous @ C @ A @ F4 @ F2 )
=> ( ( ( cosh @ A
@ ( F2
@ ( topolo3827282254853284352ce_Lim @ C @ C @ F4
@ ^ [X3: C] : X3 ) ) )
!= ( zero_zero @ A ) )
=> ( topolo3448309680560233919inuous @ C @ A @ F4
@ ^ [X3: C] : ( tanh @ A @ ( F2 @ X3 ) ) ) ) ) ) ).
% continuous_tanh
thf(fact_7370_continuous__arcosh,axiom,
! [A: $tType] :
( ( topological_t2_space @ A )
=> ! [F4: filter @ A,F2: A > real] :
( ( topolo3448309680560233919inuous @ A @ real @ F4 @ F2 )
=> ( ( ord_less @ real @ ( one_one @ real )
@ ( F2
@ ( topolo3827282254853284352ce_Lim @ A @ A @ F4
@ ^ [X3: A] : X3 ) ) )
=> ( topolo3448309680560233919inuous @ A @ real @ F4
@ ^ [X3: A] : ( arcosh @ real @ ( F2 @ X3 ) ) ) ) ) ) ).
% continuous_arcosh
thf(fact_7371_continuous__log,axiom,
! [A: $tType] :
( ( topological_t2_space @ A )
=> ! [F4: filter @ A,F2: A > real,G: A > real] :
( ( topolo3448309680560233919inuous @ A @ real @ F4 @ F2 )
=> ( ( topolo3448309680560233919inuous @ A @ real @ F4 @ G )
=> ( ( ord_less @ real @ ( zero_zero @ real )
@ ( F2
@ ( topolo3827282254853284352ce_Lim @ A @ A @ F4
@ ^ [X3: A] : X3 ) ) )
=> ( ( ( F2
@ ( topolo3827282254853284352ce_Lim @ A @ A @ F4
@ ^ [X3: A] : X3 ) )
!= ( one_one @ real ) )
=> ( ( ord_less @ real @ ( zero_zero @ real )
@ ( G
@ ( topolo3827282254853284352ce_Lim @ A @ A @ F4
@ ^ [X3: A] : X3 ) ) )
=> ( topolo3448309680560233919inuous @ A @ real @ F4
@ ^ [X3: A] : ( log @ ( F2 @ X3 ) @ ( G @ X3 ) ) ) ) ) ) ) ) ) ).
% continuous_log
thf(fact_7372_continuous__artanh,axiom,
! [A: $tType] :
( ( topological_t2_space @ A )
=> ! [F4: filter @ A,F2: A > real] :
( ( topolo3448309680560233919inuous @ A @ real @ F4 @ F2 )
=> ( ( member @ real
@ ( F2
@ ( topolo3827282254853284352ce_Lim @ A @ A @ F4
@ ^ [X3: A] : X3 ) )
@ ( set_or5935395276787703475ssThan @ real @ ( uminus_uminus @ real @ ( one_one @ real ) ) @ ( one_one @ real ) ) )
=> ( topolo3448309680560233919inuous @ A @ real @ F4
@ ^ [X3: A] : ( artanh @ real @ ( F2 @ X3 ) ) ) ) ) ) ).
% continuous_artanh
thf(fact_7373_tendsto__offset__zero__iff,axiom,
! [C: $tType,D6: $tType,A: $tType] :
( ( ( real_V822414075346904944vector @ A )
& ( topolo4958980785337419405_space @ D6 )
& ( zero @ C ) )
=> ! [A3: A,S: set @ A,F2: A > D6,L6: D6] :
( ( nO_MATCH @ C @ A @ ( zero_zero @ C ) @ A3 )
=> ( ( member @ A @ A3 @ S )
=> ( ( topolo1002775350975398744n_open @ A @ S )
=> ( ( filterlim @ A @ D6 @ F2 @ ( topolo7230453075368039082e_nhds @ D6 @ L6 ) @ ( topolo174197925503356063within @ A @ A3 @ S ) )
= ( filterlim @ A @ D6
@ ^ [H: A] : ( F2 @ ( plus_plus @ A @ A3 @ H ) )
@ ( topolo7230453075368039082e_nhds @ D6 @ L6 )
@ ( topolo174197925503356063within @ A @ ( zero_zero @ A ) @ ( top_top @ ( set @ A ) ) ) ) ) ) ) ) ) ).
% tendsto_offset_zero_iff
thf(fact_7374_has__derivativeI__sandwich,axiom,
! [B: $tType,A: $tType] :
( ( ( real_V822414075346904944vector @ A )
& ( real_V822414075346904944vector @ B ) )
=> ! [E: real,F7: A > B,S2: set @ A,X2: A,F2: A > B,H8: A > real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ E )
=> ( ( real_V3181309239436604168linear @ A @ B @ F7 )
=> ( ! [Y4: A] :
( ( member @ A @ Y4 @ S2 )
=> ( ( Y4 != X2 )
=> ( ( ord_less @ real @ ( real_V557655796197034286t_dist @ A @ Y4 @ X2 ) @ E )
=> ( ord_less_eq @ real @ ( divide_divide @ real @ ( real_V7770717601297561774m_norm @ B @ ( minus_minus @ B @ ( minus_minus @ B @ ( F2 @ Y4 ) @ ( F2 @ X2 ) ) @ ( F7 @ ( minus_minus @ A @ Y4 @ X2 ) ) ) ) @ ( real_V7770717601297561774m_norm @ A @ ( minus_minus @ A @ Y4 @ X2 ) ) ) @ ( H8 @ Y4 ) ) ) ) )
=> ( ( filterlim @ A @ real @ H8 @ ( topolo7230453075368039082e_nhds @ real @ ( zero_zero @ real ) ) @ ( topolo174197925503356063within @ A @ X2 @ S2 ) )
=> ( has_derivative @ A @ B @ F2 @ F7 @ ( topolo174197925503356063within @ A @ X2 @ S2 ) ) ) ) ) ) ) ).
% has_derivativeI_sandwich
thf(fact_7375_dist__add__cancel,axiom,
! [A: $tType] :
( ( real_V822414075346904944vector @ A )
=> ! [A3: A,B3: A,C3: A] :
( ( real_V557655796197034286t_dist @ A @ ( plus_plus @ A @ A3 @ B3 ) @ ( plus_plus @ A @ A3 @ C3 ) )
= ( real_V557655796197034286t_dist @ A @ B3 @ C3 ) ) ) ).
% dist_add_cancel
thf(fact_7376_dist__add__cancel2,axiom,
! [A: $tType] :
( ( real_V822414075346904944vector @ A )
=> ! [B3: A,A3: A,C3: A] :
( ( real_V557655796197034286t_dist @ A @ ( plus_plus @ A @ B3 @ A3 ) @ ( plus_plus @ A @ C3 @ A3 ) )
= ( real_V557655796197034286t_dist @ A @ B3 @ C3 ) ) ) ).
% dist_add_cancel2
thf(fact_7377_dist__le__zero__iff,axiom,
! [A: $tType] :
( ( real_V7819770556892013058_space @ A )
=> ! [X2: A,Y3: A] :
( ( ord_less_eq @ real @ ( real_V557655796197034286t_dist @ A @ X2 @ Y3 ) @ ( zero_zero @ real ) )
= ( X2 = Y3 ) ) ) ).
% dist_le_zero_iff
thf(fact_7378_dist__diff_I2_J,axiom,
! [A: $tType] :
( ( real_V822414075346904944vector @ A )
=> ! [A3: A,B3: A] :
( ( real_V557655796197034286t_dist @ A @ ( minus_minus @ A @ A3 @ B3 ) @ A3 )
= ( real_V7770717601297561774m_norm @ A @ B3 ) ) ) ).
% dist_diff(2)
thf(fact_7379_dist__diff_I1_J,axiom,
! [A: $tType] :
( ( real_V822414075346904944vector @ A )
=> ! [A3: A,B3: A] :
( ( real_V557655796197034286t_dist @ A @ A3 @ ( minus_minus @ A @ A3 @ B3 ) )
= ( real_V7770717601297561774m_norm @ A @ B3 ) ) ) ).
% dist_diff(1)
thf(fact_7380_div__add__self1__no__field,axiom,
! [B: $tType,A: $tType] :
( ( ( euclid4440199948858584721cancel @ A )
& ( field @ B ) )
=> ! [X2: B,B3: A,A3: A] :
( ( nO_MATCH @ B @ A @ X2 @ B3 )
=> ( ( B3
!= ( zero_zero @ A ) )
=> ( ( divide_divide @ A @ ( plus_plus @ A @ B3 @ A3 ) @ B3 )
= ( plus_plus @ A @ ( divide_divide @ A @ A3 @ B3 ) @ ( one_one @ A ) ) ) ) ) ) ).
% div_add_self1_no_field
thf(fact_7381_div__add__self2__no__field,axiom,
! [B: $tType,A: $tType] :
( ( ( euclid4440199948858584721cancel @ A )
& ( field @ B ) )
=> ! [X2: B,B3: A,A3: A] :
( ( nO_MATCH @ B @ A @ X2 @ B3 )
=> ( ( B3
!= ( zero_zero @ A ) )
=> ( ( divide_divide @ A @ ( plus_plus @ A @ A3 @ B3 ) @ B3 )
= ( plus_plus @ A @ ( divide_divide @ A @ A3 @ B3 ) @ ( one_one @ A ) ) ) ) ) ) ).
% div_add_self2_no_field
thf(fact_7382_dist__scaleR,axiom,
! [A: $tType] :
( ( real_V822414075346904944vector @ A )
=> ! [X2: real,A3: A,Y3: real] :
( ( real_V557655796197034286t_dist @ A @ ( real_V8093663219630862766scaleR @ A @ X2 @ A3 ) @ ( real_V8093663219630862766scaleR @ A @ Y3 @ A3 ) )
= ( times_times @ real @ ( abs_abs @ real @ ( minus_minus @ real @ X2 @ Y3 ) ) @ ( real_V7770717601297561774m_norm @ A @ A3 ) ) ) ) ).
% dist_scaleR
thf(fact_7383_open__ball,axiom,
! [A: $tType] :
( ( real_V7819770556892013058_space @ A )
=> ! [X2: A,D2: real] :
( topolo1002775350975398744n_open @ A
@ ( collect @ A
@ ^ [Y: A] : ( ord_less @ real @ ( real_V557655796197034286t_dist @ A @ X2 @ Y ) @ D2 ) ) ) ) ).
% open_ball
thf(fact_7384_dist__triangle__less__add,axiom,
! [A: $tType] :
( ( real_V7819770556892013058_space @ A )
=> ! [X1: A,Y3: A,E1: real,X22: A,E22: real] :
( ( ord_less @ real @ ( real_V557655796197034286t_dist @ A @ X1 @ Y3 ) @ E1 )
=> ( ( ord_less @ real @ ( real_V557655796197034286t_dist @ A @ X22 @ Y3 ) @ E22 )
=> ( ord_less @ real @ ( real_V557655796197034286t_dist @ A @ X1 @ X22 ) @ ( plus_plus @ real @ E1 @ E22 ) ) ) ) ) ).
% dist_triangle_less_add
thf(fact_7385_dist__triangle__lt,axiom,
! [A: $tType] :
( ( real_V7819770556892013058_space @ A )
=> ! [X2: A,Z: A,Y3: A,E: real] :
( ( ord_less @ real @ ( plus_plus @ real @ ( real_V557655796197034286t_dist @ A @ X2 @ Z ) @ ( real_V557655796197034286t_dist @ A @ Y3 @ Z ) ) @ E )
=> ( ord_less @ real @ ( real_V557655796197034286t_dist @ A @ X2 @ Y3 ) @ E ) ) ) ).
% dist_triangle_lt
thf(fact_7386_continuous__dist,axiom,
! [A: $tType,D6: $tType] :
( ( ( topological_t2_space @ D6 )
& ( real_V7819770556892013058_space @ A ) )
=> ! [F4: filter @ D6,F2: D6 > A,G: D6 > A] :
( ( topolo3448309680560233919inuous @ D6 @ A @ F4 @ F2 )
=> ( ( topolo3448309680560233919inuous @ D6 @ A @ F4 @ G )
=> ( topolo3448309680560233919inuous @ D6 @ real @ F4
@ ^ [X3: D6] : ( real_V557655796197034286t_dist @ A @ ( F2 @ X3 ) @ ( G @ X3 ) ) ) ) ) ) ).
% continuous_dist
thf(fact_7387_abs__dist__diff__le,axiom,
! [A: $tType] :
( ( real_V7819770556892013058_space @ A )
=> ! [A3: A,B3: A,C3: A] : ( ord_less_eq @ real @ ( abs_abs @ real @ ( minus_minus @ real @ ( real_V557655796197034286t_dist @ A @ A3 @ B3 ) @ ( real_V557655796197034286t_dist @ A @ B3 @ C3 ) ) ) @ ( real_V557655796197034286t_dist @ A @ A3 @ C3 ) ) ) ).
% abs_dist_diff_le
thf(fact_7388_zero__le__dist,axiom,
! [A: $tType] :
( ( real_V7819770556892013058_space @ A )
=> ! [X2: A,Y3: A] : ( ord_less_eq @ real @ ( zero_zero @ real ) @ ( real_V557655796197034286t_dist @ A @ X2 @ Y3 ) ) ) ).
% zero_le_dist
thf(fact_7389_dist__triangle,axiom,
! [A: $tType] :
( ( real_V7819770556892013058_space @ A )
=> ! [X2: A,Z: A,Y3: A] : ( ord_less_eq @ real @ ( real_V557655796197034286t_dist @ A @ X2 @ Z ) @ ( plus_plus @ real @ ( real_V557655796197034286t_dist @ A @ X2 @ Y3 ) @ ( real_V557655796197034286t_dist @ A @ Y3 @ Z ) ) ) ) ).
% dist_triangle
thf(fact_7390_dist__triangle2,axiom,
! [A: $tType] :
( ( real_V7819770556892013058_space @ A )
=> ! [X2: A,Y3: A,Z: A] : ( ord_less_eq @ real @ ( real_V557655796197034286t_dist @ A @ X2 @ Y3 ) @ ( plus_plus @ real @ ( real_V557655796197034286t_dist @ A @ X2 @ Z ) @ ( real_V557655796197034286t_dist @ A @ Y3 @ Z ) ) ) ) ).
% dist_triangle2
thf(fact_7391_dist__triangle3,axiom,
! [A: $tType] :
( ( real_V7819770556892013058_space @ A )
=> ! [X2: A,Y3: A,A3: A] : ( ord_less_eq @ real @ ( real_V557655796197034286t_dist @ A @ X2 @ Y3 ) @ ( plus_plus @ real @ ( real_V557655796197034286t_dist @ A @ A3 @ X2 ) @ ( real_V557655796197034286t_dist @ A @ A3 @ Y3 ) ) ) ) ).
% dist_triangle3
thf(fact_7392_dist__triangle__le,axiom,
! [A: $tType] :
( ( real_V7819770556892013058_space @ A )
=> ! [X2: A,Z: A,Y3: A,E: real] :
( ( ord_less_eq @ real @ ( plus_plus @ real @ ( real_V557655796197034286t_dist @ A @ X2 @ Z ) @ ( real_V557655796197034286t_dist @ A @ Y3 @ Z ) ) @ E )
=> ( ord_less_eq @ real @ ( real_V557655796197034286t_dist @ A @ X2 @ Y3 ) @ E ) ) ) ).
% dist_triangle_le
thf(fact_7393_dist__complex__def,axiom,
( ( real_V557655796197034286t_dist @ complex )
= ( ^ [X3: complex,Y: complex] : ( real_V7770717601297561774m_norm @ complex @ ( minus_minus @ complex @ X3 @ Y ) ) ) ) ).
% dist_complex_def
thf(fact_7394_dist__norm,axiom,
! [A: $tType] :
( ( real_V6936659425649961206t_norm @ A )
=> ( ( real_V557655796197034286t_dist @ A )
= ( ^ [X3: A,Y: A] : ( real_V7770717601297561774m_norm @ A @ ( minus_minus @ A @ X3 @ Y ) ) ) ) ) ).
% dist_norm
thf(fact_7395_dist__real__def,axiom,
( ( real_V557655796197034286t_dist @ real )
= ( ^ [X3: real,Y: real] : ( abs_abs @ real @ ( minus_minus @ real @ X3 @ Y ) ) ) ) ).
% dist_real_def
thf(fact_7396_Cauchy__def,axiom,
! [A: $tType] :
( ( real_V7819770556892013058_space @ A )
=> ( ( topolo3814608138187158403Cauchy @ A )
= ( ^ [X7: nat > A] :
! [E4: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ E4 )
=> ? [M10: nat] :
! [M6: nat] :
( ( ord_less_eq @ nat @ M10 @ M6 )
=> ! [N4: nat] :
( ( ord_less_eq @ nat @ M10 @ N4 )
=> ( ord_less @ real @ ( real_V557655796197034286t_dist @ A @ ( X7 @ M6 ) @ ( X7 @ N4 ) ) @ E4 ) ) ) ) ) ) ) ).
% Cauchy_def
thf(fact_7397_Cauchy__altdef2,axiom,
! [A: $tType] :
( ( real_V7819770556892013058_space @ A )
=> ( ( topolo3814608138187158403Cauchy @ A )
= ( ^ [S5: nat > A] :
! [E4: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ E4 )
=> ? [N8: nat] :
! [N4: nat] :
( ( ord_less_eq @ nat @ N8 @ N4 )
=> ( ord_less @ real @ ( real_V557655796197034286t_dist @ A @ ( S5 @ N4 ) @ ( S5 @ N8 ) ) @ E4 ) ) ) ) ) ) ).
% Cauchy_altdef2
thf(fact_7398_metric__CauchyD,axiom,
! [A: $tType] :
( ( real_V7819770556892013058_space @ A )
=> ! [X: nat > A,E: real] :
( ( topolo3814608138187158403Cauchy @ A @ X )
=> ( ( ord_less @ real @ ( zero_zero @ real ) @ E )
=> ? [M11: nat] :
! [M3: nat] :
( ( ord_less_eq @ nat @ M11 @ M3 )
=> ! [N9: nat] :
( ( ord_less_eq @ nat @ M11 @ N9 )
=> ( ord_less @ real @ ( real_V557655796197034286t_dist @ A @ ( X @ M3 ) @ ( X @ N9 ) ) @ E ) ) ) ) ) ) ).
% metric_CauchyD
thf(fact_7399_metric__CauchyI,axiom,
! [A: $tType] :
( ( real_V7819770556892013058_space @ A )
=> ! [X: nat > A] :
( ! [E2: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ E2 )
=> ? [M13: nat] :
! [M5: nat] :
( ( ord_less_eq @ nat @ M13 @ M5 )
=> ! [N3: nat] :
( ( ord_less_eq @ nat @ M13 @ N3 )
=> ( ord_less @ real @ ( real_V557655796197034286t_dist @ A @ ( X @ M5 ) @ ( X @ N3 ) ) @ E2 ) ) ) )
=> ( topolo3814608138187158403Cauchy @ A @ X ) ) ) ).
% metric_CauchyI
thf(fact_7400_dist__of__int,axiom,
! [A: $tType] :
( ( real_V2822296259951069270ebra_1 @ A )
=> ! [M: int,N: int] :
( ( real_V557655796197034286t_dist @ A @ ( ring_1_of_int @ A @ M ) @ ( ring_1_of_int @ A @ N ) )
= ( ring_1_of_int @ real @ ( abs_abs @ int @ ( minus_minus @ int @ M @ N ) ) ) ) ) ).
% dist_of_int
thf(fact_7401_tendsto__dist,axiom,
! [A: $tType,B: $tType] :
( ( real_V7819770556892013058_space @ A )
=> ! [F2: B > A,L: A,F4: filter @ B,G: B > A,M: A] :
( ( filterlim @ B @ A @ F2 @ ( topolo7230453075368039082e_nhds @ A @ L ) @ F4 )
=> ( ( filterlim @ B @ A @ G @ ( topolo7230453075368039082e_nhds @ A @ M ) @ F4 )
=> ( filterlim @ B @ real
@ ^ [X3: B] : ( real_V557655796197034286t_dist @ A @ ( F2 @ X3 ) @ ( G @ X3 ) )
@ ( topolo7230453075368039082e_nhds @ real @ ( real_V557655796197034286t_dist @ A @ L @ M ) )
@ F4 ) ) ) ) ).
% tendsto_dist
thf(fact_7402_scale__right__distrib__NO__MATCH,axiom,
! [A: $tType] :
( ( real_V4867850818363320053vector @ A )
=> ! [X2: A,Y3: A,A3: real] :
( ( nO_MATCH @ A @ real @ ( divide_divide @ A @ X2 @ Y3 ) @ A3 )
=> ( ( real_V8093663219630862766scaleR @ A @ A3 @ ( plus_plus @ A @ X2 @ Y3 ) )
= ( plus_plus @ A @ ( real_V8093663219630862766scaleR @ A @ A3 @ X2 ) @ ( real_V8093663219630862766scaleR @ A @ A3 @ Y3 ) ) ) ) ) ).
% scale_right_distrib_NO_MATCH
thf(fact_7403_scale__right__diff__distrib__NO__MATCH,axiom,
! [A: $tType] :
( ( real_V4867850818363320053vector @ A )
=> ! [X2: A,Y3: A,A3: real] :
( ( nO_MATCH @ A @ real @ ( divide_divide @ A @ X2 @ Y3 ) @ A3 )
=> ( ( real_V8093663219630862766scaleR @ A @ A3 @ ( minus_minus @ A @ X2 @ Y3 ) )
= ( minus_minus @ A @ ( real_V8093663219630862766scaleR @ A @ A3 @ X2 ) @ ( real_V8093663219630862766scaleR @ A @ A3 @ Y3 ) ) ) ) ) ).
% scale_right_diff_distrib_NO_MATCH
thf(fact_7404_distrib__left__NO__MATCH,axiom,
! [B: $tType,A: $tType] :
( ( semiring @ A )
=> ! [X2: B,Y3: B,A3: A,B3: A,C3: A] :
( ( nO_MATCH @ B @ A @ ( divide_divide @ B @ X2 @ Y3 ) @ A3 )
=> ( ( times_times @ A @ A3 @ ( plus_plus @ A @ B3 @ C3 ) )
= ( plus_plus @ A @ ( times_times @ A @ A3 @ B3 ) @ ( times_times @ A @ A3 @ C3 ) ) ) ) ) ).
% distrib_left_NO_MATCH
thf(fact_7405_distrib__right__NO__MATCH,axiom,
! [B: $tType,A: $tType] :
( ( semiring @ A )
=> ! [X2: B,Y3: B,C3: A,A3: A,B3: A] :
( ( nO_MATCH @ B @ A @ ( divide_divide @ B @ X2 @ Y3 ) @ C3 )
=> ( ( times_times @ A @ ( plus_plus @ A @ A3 @ B3 ) @ C3 )
= ( plus_plus @ A @ ( times_times @ A @ A3 @ C3 ) @ ( times_times @ A @ B3 @ C3 ) ) ) ) ) ).
% distrib_right_NO_MATCH
thf(fact_7406_left__diff__distrib__NO__MATCH,axiom,
! [B: $tType,A: $tType] :
( ( ring @ A )
=> ! [X2: B,Y3: B,C3: A,A3: A,B3: A] :
( ( nO_MATCH @ B @ A @ ( divide_divide @ B @ X2 @ Y3 ) @ C3 )
=> ( ( times_times @ A @ ( minus_minus @ A @ A3 @ B3 ) @ C3 )
= ( minus_minus @ A @ ( times_times @ A @ A3 @ C3 ) @ ( times_times @ A @ B3 @ C3 ) ) ) ) ) ).
% left_diff_distrib_NO_MATCH
thf(fact_7407_right__diff__distrib__NO__MATCH,axiom,
! [B: $tType,A: $tType] :
( ( ring @ A )
=> ! [X2: B,Y3: B,A3: A,B3: A,C3: A] :
( ( nO_MATCH @ B @ A @ ( divide_divide @ B @ X2 @ Y3 ) @ A3 )
=> ( ( times_times @ A @ A3 @ ( minus_minus @ A @ B3 @ C3 ) )
= ( minus_minus @ A @ ( times_times @ A @ A3 @ B3 ) @ ( times_times @ A @ A3 @ C3 ) ) ) ) ) ).
% right_diff_distrib_NO_MATCH
thf(fact_7408_dist__triangle__half__r,axiom,
! [A: $tType] :
( ( real_V7819770556892013058_space @ A )
=> ! [Y3: A,X1: A,E: real,X22: A] :
( ( ord_less @ real @ ( real_V557655796197034286t_dist @ A @ Y3 @ X1 ) @ ( divide_divide @ real @ E @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) )
=> ( ( ord_less @ real @ ( real_V557655796197034286t_dist @ A @ Y3 @ X22 ) @ ( divide_divide @ real @ E @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) )
=> ( ord_less @ real @ ( real_V557655796197034286t_dist @ A @ X1 @ X22 ) @ E ) ) ) ) ).
% dist_triangle_half_r
thf(fact_7409_dist__triangle__half__l,axiom,
! [A: $tType] :
( ( real_V7819770556892013058_space @ A )
=> ! [X1: A,Y3: A,E: real,X22: A] :
( ( ord_less @ real @ ( real_V557655796197034286t_dist @ A @ X1 @ Y3 ) @ ( divide_divide @ real @ E @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) )
=> ( ( ord_less @ real @ ( real_V557655796197034286t_dist @ A @ X22 @ Y3 ) @ ( divide_divide @ real @ E @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) )
=> ( ord_less @ real @ ( real_V557655796197034286t_dist @ A @ X1 @ X22 ) @ E ) ) ) ) ).
% dist_triangle_half_l
thf(fact_7410_metric__LIM__imp__LIM,axiom,
! [A: $tType,B: $tType,C: $tType] :
( ( ( topolo4958980785337419405_space @ C )
& ( real_V7819770556892013058_space @ B )
& ( real_V7819770556892013058_space @ A ) )
=> ! [F2: C > A,L: A,A3: C,G: C > B,M: B] :
( ( filterlim @ C @ A @ F2 @ ( topolo7230453075368039082e_nhds @ A @ L ) @ ( topolo174197925503356063within @ C @ A3 @ ( top_top @ ( set @ C ) ) ) )
=> ( ! [X4: C] :
( ( X4 != A3 )
=> ( ord_less_eq @ real @ ( real_V557655796197034286t_dist @ B @ ( G @ X4 ) @ M ) @ ( real_V557655796197034286t_dist @ A @ ( F2 @ X4 ) @ L ) ) )
=> ( filterlim @ C @ B @ G @ ( topolo7230453075368039082e_nhds @ B @ M ) @ ( topolo174197925503356063within @ C @ A3 @ ( top_top @ ( set @ C ) ) ) ) ) ) ) ).
% metric_LIM_imp_LIM
thf(fact_7411_dist__triangle__third,axiom,
! [A: $tType] :
( ( real_V7819770556892013058_space @ A )
=> ! [X1: A,X22: A,E: real,X33: A,X43: A] :
( ( ord_less @ real @ ( real_V557655796197034286t_dist @ A @ X1 @ X22 ) @ ( divide_divide @ real @ E @ ( numeral_numeral @ real @ ( bit1 @ one2 ) ) ) )
=> ( ( ord_less @ real @ ( real_V557655796197034286t_dist @ A @ X22 @ X33 ) @ ( divide_divide @ real @ E @ ( numeral_numeral @ real @ ( bit1 @ one2 ) ) ) )
=> ( ( ord_less @ real @ ( real_V557655796197034286t_dist @ A @ X33 @ X43 ) @ ( divide_divide @ real @ E @ ( numeral_numeral @ real @ ( bit1 @ one2 ) ) ) )
=> ( ord_less @ real @ ( real_V557655796197034286t_dist @ A @ X1 @ X43 ) @ E ) ) ) ) ) ).
% dist_triangle_third
thf(fact_7412_filterlim__transform__within,axiom,
! [B: $tType,A: $tType] :
( ( real_V7819770556892013058_space @ A )
=> ! [G: A > B,G6: filter @ B,X2: A,S: set @ A,F4: filter @ B,D2: real,F2: A > B] :
( ( filterlim @ A @ B @ G @ G6 @ ( topolo174197925503356063within @ A @ X2 @ S ) )
=> ( ( ord_less_eq @ ( filter @ B ) @ G6 @ F4 )
=> ( ( ord_less @ real @ ( zero_zero @ real ) @ D2 )
=> ( ! [X9: A] :
( ( member @ A @ X9 @ S )
=> ( ( ord_less @ real @ ( zero_zero @ real ) @ ( real_V557655796197034286t_dist @ A @ X9 @ X2 ) )
=> ( ( ord_less @ real @ ( real_V557655796197034286t_dist @ A @ X9 @ X2 ) @ D2 )
=> ( ( F2 @ X9 )
= ( G @ X9 ) ) ) ) )
=> ( filterlim @ A @ B @ F2 @ F4 @ ( topolo174197925503356063within @ A @ X2 @ S ) ) ) ) ) ) ) ).
% filterlim_transform_within
thf(fact_7413_Cauchy__altdef,axiom,
! [A: $tType] :
( ( real_V7819770556892013058_space @ A )
=> ( ( topolo3814608138187158403Cauchy @ A )
= ( ^ [F5: nat > A] :
! [E4: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ E4 )
=> ? [M10: nat] :
! [M6: nat] :
( ( ord_less_eq @ nat @ M10 @ M6 )
=> ! [N4: nat] :
( ( ord_less @ nat @ M6 @ N4 )
=> ( ord_less @ real @ ( real_V557655796197034286t_dist @ A @ ( F5 @ M6 ) @ ( F5 @ N4 ) ) @ E4 ) ) ) ) ) ) ) ).
% Cauchy_altdef
thf(fact_7414_CauchyI_H,axiom,
! [A: $tType] :
( ( real_V7819770556892013058_space @ A )
=> ! [X: nat > A] :
( ! [E2: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ E2 )
=> ? [M13: nat] :
! [M5: nat] :
( ( ord_less_eq @ nat @ M13 @ M5 )
=> ! [N3: nat] :
( ( ord_less @ nat @ M5 @ N3 )
=> ( ord_less @ real @ ( real_V557655796197034286t_dist @ A @ ( X @ M5 ) @ ( X @ N3 ) ) @ E2 ) ) ) )
=> ( topolo3814608138187158403Cauchy @ A @ X ) ) ) ).
% CauchyI'
thf(fact_7415_dist__of__nat,axiom,
! [A: $tType] :
( ( real_V2822296259951069270ebra_1 @ A )
=> ! [M: nat,N: nat] :
( ( real_V557655796197034286t_dist @ A @ ( semiring_1_of_nat @ A @ M ) @ ( semiring_1_of_nat @ A @ N ) )
= ( ring_1_of_int @ real @ ( abs_abs @ int @ ( minus_minus @ int @ ( semiring_1_of_nat @ int @ M ) @ ( semiring_1_of_nat @ int @ N ) ) ) ) ) ) ).
% dist_of_nat
thf(fact_7416_tendsto__dist__iff,axiom,
! [B: $tType,A: $tType] :
( ( real_V7819770556892013058_space @ B )
=> ! [F2: A > B,L: B,F4: filter @ A] :
( ( filterlim @ A @ B @ F2 @ ( topolo7230453075368039082e_nhds @ B @ L ) @ F4 )
= ( filterlim @ A @ real
@ ^ [X3: A] : ( real_V557655796197034286t_dist @ B @ ( F2 @ X3 ) @ L )
@ ( topolo7230453075368039082e_nhds @ real @ ( zero_zero @ real ) )
@ F4 ) ) ) ).
% tendsto_dist_iff
thf(fact_7417_metric__LIMSEQ__D,axiom,
! [A: $tType] :
( ( real_V7819770556892013058_space @ A )
=> ! [X: nat > A,L6: A,R2: real] :
( ( filterlim @ nat @ A @ X @ ( topolo7230453075368039082e_nhds @ A @ L6 ) @ ( 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 @ ( X @ N9 ) @ L6 ) @ R2 ) ) ) ) ) ).
% metric_LIMSEQ_D
thf(fact_7418_metric__LIMSEQ__I,axiom,
! [A: $tType] :
( ( real_V7819770556892013058_space @ A )
=> ! [X: nat > A,L6: A] :
( ! [R3: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ R3 )
=> ? [No2: nat] :
! [N3: nat] :
( ( ord_less_eq @ nat @ No2 @ N3 )
=> ( ord_less @ real @ ( real_V557655796197034286t_dist @ A @ ( X @ N3 ) @ L6 ) @ R3 ) ) )
=> ( filterlim @ nat @ A @ X @ ( topolo7230453075368039082e_nhds @ A @ L6 ) @ ( at_top @ nat ) ) ) ) ).
% metric_LIMSEQ_I
thf(fact_7419_lim__sequentially,axiom,
! [A: $tType] :
( ( real_V7819770556892013058_space @ A )
=> ! [X: nat > A,L6: A] :
( ( filterlim @ nat @ A @ X @ ( topolo7230453075368039082e_nhds @ A @ L6 ) @ ( at_top @ nat ) )
= ( ! [R5: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ R5 )
=> ? [No3: nat] :
! [N4: nat] :
( ( ord_less_eq @ nat @ No3 @ N4 )
=> ( ord_less @ real @ ( real_V557655796197034286t_dist @ A @ ( X @ N4 ) @ L6 ) @ R5 ) ) ) ) ) ) ).
% lim_sequentially
thf(fact_7420_power__minus_H,axiom,
! [A: $tType] :
( ( ring_1 @ A )
=> ! [X2: A,N: nat] :
( ( nO_MATCH @ A @ A @ ( one_one @ A ) @ X2 )
=> ( ( power_power @ A @ ( uminus_uminus @ A @ X2 ) @ N )
= ( times_times @ A @ ( power_power @ A @ ( uminus_uminus @ A @ ( one_one @ A ) ) @ N ) @ ( power_power @ A @ X2 @ N ) ) ) ) ) ).
% power_minus'
thf(fact_7421_metric__Cauchy__iff2,axiom,
! [A: $tType] :
( ( real_V7819770556892013058_space @ A )
=> ( ( topolo3814608138187158403Cauchy @ A )
= ( ^ [X7: nat > A] :
! [J3: nat] :
? [M10: nat] :
! [M6: nat] :
( ( ord_less_eq @ nat @ M10 @ M6 )
=> ! [N4: nat] :
( ( ord_less_eq @ nat @ M10 @ N4 )
=> ( ord_less @ real @ ( real_V557655796197034286t_dist @ A @ ( X7 @ M6 ) @ ( X7 @ N4 ) ) @ ( inverse_inverse @ real @ ( semiring_1_of_nat @ real @ ( suc @ J3 ) ) ) ) ) ) ) ) ) ).
% metric_Cauchy_iff2
thf(fact_7422_scale__left__distrib__NO__MATCH,axiom,
! [C: $tType,A: $tType] :
( ( real_V4867850818363320053vector @ A )
=> ! [X2: A,Y3: A,C3: C,A3: real,B3: real] :
( ( nO_MATCH @ A @ C @ ( divide_divide @ A @ X2 @ Y3 ) @ C3 )
=> ( ( real_V8093663219630862766scaleR @ A @ ( plus_plus @ real @ A3 @ B3 ) @ X2 )
= ( plus_plus @ A @ ( real_V8093663219630862766scaleR @ A @ A3 @ X2 ) @ ( real_V8093663219630862766scaleR @ A @ B3 @ X2 ) ) ) ) ) ).
% scale_left_distrib_NO_MATCH
thf(fact_7423_metric__LIM__compose2,axiom,
! [C: $tType,B: $tType,A: $tType] :
( ( ( real_V7819770556892013058_space @ A )
& ( topolo4958980785337419405_space @ B )
& ( topolo4958980785337419405_space @ C ) )
=> ! [F2: A > B,B3: B,A3: A,G: B > C,C3: C] :
( ( filterlim @ A @ B @ F2 @ ( topolo7230453075368039082e_nhds @ B @ B3 ) @ ( topolo174197925503356063within @ A @ A3 @ ( top_top @ ( set @ A ) ) ) )
=> ( ( filterlim @ B @ C @ G @ ( topolo7230453075368039082e_nhds @ C @ C3 ) @ ( topolo174197925503356063within @ B @ B3 @ ( top_top @ ( set @ B ) ) ) )
=> ( ? [D5: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ D5 )
& ! [X4: A] :
( ( ( X4 != A3 )
& ( ord_less @ real @ ( real_V557655796197034286t_dist @ A @ X4 @ A3 ) @ D5 ) )
=> ( ( F2 @ X4 )
!= B3 ) ) )
=> ( filterlim @ A @ C
@ ^ [X3: A] : ( G @ ( F2 @ X3 ) )
@ ( topolo7230453075368039082e_nhds @ C @ C3 )
@ ( topolo174197925503356063within @ A @ A3 @ ( top_top @ ( set @ A ) ) ) ) ) ) ) ) ).
% metric_LIM_compose2
thf(fact_7424_scale__left__diff__distrib__NO__MATCH,axiom,
! [C: $tType,A: $tType] :
( ( real_V4867850818363320053vector @ A )
=> ! [X2: A,Y3: A,C3: C,A3: real,B3: real] :
( ( nO_MATCH @ A @ C @ ( divide_divide @ A @ X2 @ Y3 ) @ C3 )
=> ( ( real_V8093663219630862766scaleR @ A @ ( minus_minus @ real @ A3 @ B3 ) @ X2 )
= ( minus_minus @ A @ ( real_V8093663219630862766scaleR @ A @ A3 @ X2 ) @ ( real_V8093663219630862766scaleR @ A @ B3 @ X2 ) ) ) ) ) ).
% scale_left_diff_distrib_NO_MATCH
thf(fact_7425_metric__isCont__LIM__compose2,axiom,
! [D6: $tType,C: $tType,A: $tType] :
( ( ( real_V7819770556892013058_space @ A )
& ( topolo4958980785337419405_space @ C )
& ( topolo4958980785337419405_space @ D6 ) )
=> ! [A3: A,F2: A > C,G: C > D6,L: D6] :
( ( topolo3448309680560233919inuous @ A @ C @ ( topolo174197925503356063within @ A @ A3 @ ( top_top @ ( set @ A ) ) ) @ F2 )
=> ( ( filterlim @ C @ D6 @ G @ ( topolo7230453075368039082e_nhds @ D6 @ L ) @ ( topolo174197925503356063within @ C @ ( F2 @ A3 ) @ ( top_top @ ( set @ C ) ) ) )
=> ( ? [D5: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ D5 )
& ! [X4: A] :
( ( ( X4 != A3 )
& ( ord_less @ real @ ( real_V557655796197034286t_dist @ A @ X4 @ A3 ) @ D5 ) )
=> ( ( F2 @ X4 )
!= ( F2 @ A3 ) ) ) )
=> ( filterlim @ A @ D6
@ ^ [X3: A] : ( G @ ( F2 @ X3 ) )
@ ( topolo7230453075368039082e_nhds @ D6 @ L )
@ ( topolo174197925503356063within @ A @ A3 @ ( top_top @ ( set @ A ) ) ) ) ) ) ) ) ).
% metric_isCont_LIM_compose2
thf(fact_7426_LIMSEQ__iff__nz,axiom,
! [A: $tType] :
( ( real_V7819770556892013058_space @ A )
=> ! [X: nat > A,L6: A] :
( ( filterlim @ nat @ A @ X @ ( topolo7230453075368039082e_nhds @ A @ L6 ) @ ( at_top @ nat ) )
= ( ! [R5: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ R5 )
=> ? [No3: nat] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ No3 )
& ! [N4: nat] :
( ( ord_less_eq @ nat @ No3 @ N4 )
=> ( ord_less @ real @ ( real_V557655796197034286t_dist @ A @ ( X @ N4 ) @ L6 ) @ R5 ) ) ) ) ) ) ) ).
% LIMSEQ_iff_nz
thf(fact_7427_LIM__offset__zero__iff,axiom,
! [C: $tType,D6: $tType,A: $tType] :
( ( ( real_V822414075346904944vector @ A )
& ( topolo4958980785337419405_space @ D6 )
& ( zero @ C ) )
=> ! [A3: A,F2: A > D6,L6: D6] :
( ( nO_MATCH @ C @ A @ ( zero_zero @ C ) @ A3 )
=> ( ( filterlim @ A @ D6 @ F2 @ ( topolo7230453075368039082e_nhds @ D6 @ L6 ) @ ( topolo174197925503356063within @ A @ A3 @ ( top_top @ ( set @ A ) ) ) )
= ( filterlim @ A @ D6
@ ^ [H: A] : ( F2 @ ( plus_plus @ A @ A3 @ H ) )
@ ( topolo7230453075368039082e_nhds @ D6 @ L6 )
@ ( topolo174197925503356063within @ A @ ( zero_zero @ A ) @ ( top_top @ ( set @ A ) ) ) ) ) ) ) ).
% LIM_offset_zero_iff
thf(fact_7428_tendsto__exp__limit__at__right,axiom,
! [X2: real] :
( filterlim @ real @ real
@ ^ [Y: real] : ( powr @ real @ ( plus_plus @ real @ ( one_one @ real ) @ ( times_times @ real @ X2 @ Y ) ) @ ( divide_divide @ real @ ( one_one @ real ) @ Y ) )
@ ( topolo7230453075368039082e_nhds @ real @ ( exp @ real @ X2 ) )
@ ( topolo174197925503356063within @ real @ ( zero_zero @ real ) @ ( set_ord_greaterThan @ real @ ( zero_zero @ real ) ) ) ) ).
% tendsto_exp_limit_at_right
thf(fact_7429_tendsto__arctan__at__bot,axiom,
filterlim @ real @ real @ arctan @ ( topolo7230453075368039082e_nhds @ real @ ( uminus_uminus @ real @ ( divide_divide @ real @ pi @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) ) ) @ ( at_bot @ real ) ).
% tendsto_arctan_at_bot
thf(fact_7430_greaterThan__iff,axiom,
! [A: $tType] :
( ( ord @ A )
=> ! [I: A,K: A] :
( ( member @ A @ I @ ( set_ord_greaterThan @ A @ K ) )
= ( ord_less @ A @ K @ I ) ) ) ).
% greaterThan_iff
thf(fact_7431_greaterThan__subset__iff,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [X2: A,Y3: A] :
( ( ord_less_eq @ ( set @ A ) @ ( set_ord_greaterThan @ A @ X2 ) @ ( set_ord_greaterThan @ A @ Y3 ) )
= ( ord_less_eq @ A @ Y3 @ X2 ) ) ) ).
% greaterThan_subset_iff
thf(fact_7432_artanh__real__at__right__1,axiom,
filterlim @ real @ real @ ( artanh @ real ) @ ( at_bot @ real ) @ ( topolo174197925503356063within @ real @ ( uminus_uminus @ real @ ( one_one @ real ) ) @ ( set_ord_greaterThan @ real @ ( uminus_uminus @ real @ ( one_one @ real ) ) ) ) ).
% artanh_real_at_right_1
thf(fact_7433_ln__at__0,axiom,
filterlim @ real @ real @ ( ln_ln @ real ) @ ( at_bot @ real ) @ ( topolo174197925503356063within @ real @ ( zero_zero @ real ) @ ( set_ord_greaterThan @ real @ ( zero_zero @ real ) ) ) ).
% ln_at_0
thf(fact_7434_at__within__Icc__at__right,axiom,
! [A: $tType] :
( ( topolo2564578578187576103pology @ A )
=> ! [A3: A,B3: A] :
( ( ord_less @ A @ A3 @ B3 )
=> ( ( topolo174197925503356063within @ A @ A3 @ ( set_or1337092689740270186AtMost @ A @ A3 @ B3 ) )
= ( topolo174197925503356063within @ A @ A3 @ ( set_ord_greaterThan @ A @ A3 ) ) ) ) ) ).
% at_within_Icc_at_right
thf(fact_7435_greaterThan__def,axiom,
! [A: $tType] :
( ( ord @ A )
=> ( ( set_ord_greaterThan @ A )
= ( ^ [L3: A] : ( collect @ A @ ( ord_less @ A @ L3 ) ) ) ) ) ).
% greaterThan_def
thf(fact_7436_filterlim__at__left__to__right,axiom,
! [A: $tType,F2: real > A,F4: filter @ A,A3: real] :
( ( filterlim @ real @ A @ F2 @ F4 @ ( topolo174197925503356063within @ real @ A3 @ ( set_ord_lessThan @ real @ A3 ) ) )
= ( filterlim @ real @ A
@ ^ [X3: real] : ( F2 @ ( uminus_uminus @ real @ X3 ) )
@ F4
@ ( topolo174197925503356063within @ real @ ( uminus_uminus @ real @ A3 ) @ ( set_ord_greaterThan @ real @ ( uminus_uminus @ real @ A3 ) ) ) ) ) ).
% filterlim_at_left_to_right
thf(fact_7437_filterlim__at__right__to__0,axiom,
! [A: $tType,F2: real > A,F4: filter @ A,A3: real] :
( ( filterlim @ real @ A @ F2 @ F4 @ ( topolo174197925503356063within @ real @ A3 @ ( set_ord_greaterThan @ real @ A3 ) ) )
= ( filterlim @ real @ A
@ ^ [X3: real] : ( F2 @ ( plus_plus @ real @ X3 @ A3 ) )
@ F4
@ ( topolo174197925503356063within @ real @ ( zero_zero @ real ) @ ( set_ord_greaterThan @ real @ ( zero_zero @ real ) ) ) ) ) ).
% filterlim_at_right_to_0
thf(fact_7438_filterlim__tan__at__right,axiom,
filterlim @ real @ real @ ( tan @ real ) @ ( at_bot @ real ) @ ( topolo174197925503356063within @ real @ ( uminus_uminus @ real @ ( divide_divide @ real @ pi @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) ) @ ( set_ord_greaterThan @ real @ ( uminus_uminus @ real @ ( divide_divide @ real @ pi @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) ) ) ) ).
% filterlim_tan_at_right
thf(fact_7439_filterlim__inverse__at__bot__neg,axiom,
filterlim @ real @ real @ ( inverse_inverse @ real ) @ ( at_bot @ real ) @ ( topolo174197925503356063within @ real @ ( zero_zero @ real ) @ ( set_ord_lessThan @ real @ ( zero_zero @ real ) ) ) ).
% filterlim_inverse_at_bot_neg
thf(fact_7440_filterlim__times__pos,axiom,
! [A: $tType,B: $tType] :
( ( ( linordered_field @ A )
& ( topolo1944317154257567458pology @ A ) )
=> ! [F2: B > A,P6: A,F14: filter @ B,C3: A,L: A] :
( ( filterlim @ B @ A @ F2 @ ( topolo174197925503356063within @ A @ P6 @ ( set_ord_greaterThan @ A @ P6 ) ) @ F14 )
=> ( ( ord_less @ A @ ( zero_zero @ A ) @ C3 )
=> ( ( L
= ( times_times @ A @ C3 @ P6 ) )
=> ( filterlim @ B @ A
@ ^ [X3: B] : ( times_times @ A @ C3 @ ( F2 @ X3 ) )
@ ( topolo174197925503356063within @ A @ L @ ( set_ord_greaterThan @ A @ L ) )
@ F14 ) ) ) ) ) ).
% filterlim_times_pos
thf(fact_7441_log__inj,axiom,
! [B3: real] :
( ( ord_less @ real @ ( one_one @ real ) @ B3 )
=> ( inj_on @ real @ real @ ( log @ B3 ) @ ( set_ord_greaterThan @ real @ ( zero_zero @ real ) ) ) ) ).
% log_inj
thf(fact_7442_tanh__real__at__bot,axiom,
filterlim @ real @ real @ ( tanh @ real ) @ ( topolo7230453075368039082e_nhds @ real @ ( uminus_uminus @ real @ ( one_one @ real ) ) ) @ ( at_bot @ real ) ).
% tanh_real_at_bot
thf(fact_7443_filterlim__tendsto__pos__mult__at__bot,axiom,
! [A: $tType,F2: A > real,C3: real,F4: filter @ A,G: A > real] :
( ( filterlim @ A @ real @ F2 @ ( topolo7230453075368039082e_nhds @ real @ C3 ) @ F4 )
=> ( ( ord_less @ real @ ( zero_zero @ real ) @ C3 )
=> ( ( filterlim @ A @ real @ G @ ( at_bot @ real ) @ F4 )
=> ( filterlim @ A @ real
@ ^ [X3: A] : ( times_times @ real @ ( F2 @ X3 ) @ ( G @ X3 ) )
@ ( at_bot @ real )
@ F4 ) ) ) ) ).
% filterlim_tendsto_pos_mult_at_bot
thf(fact_7444_tendsto__arcosh__at__left__1,axiom,
filterlim @ real @ real @ ( arcosh @ real ) @ ( topolo7230453075368039082e_nhds @ real @ ( zero_zero @ real ) ) @ ( topolo174197925503356063within @ real @ ( one_one @ real ) @ ( set_ord_greaterThan @ real @ ( one_one @ real ) ) ) ).
% tendsto_arcosh_at_left_1
thf(fact_7445_tendsto__at__botI__sequentially,axiom,
! [B: $tType] :
( ( topolo3112930676232923870pology @ B )
=> ! [F2: real > B,Y3: B] :
( ! [X17: nat > real] :
( ( filterlim @ nat @ real @ X17 @ ( at_bot @ real ) @ ( at_top @ nat ) )
=> ( filterlim @ nat @ B
@ ^ [N4: nat] : ( F2 @ ( X17 @ N4 ) )
@ ( topolo7230453075368039082e_nhds @ B @ Y3 )
@ ( at_top @ nat ) ) )
=> ( filterlim @ real @ B @ F2 @ ( topolo7230453075368039082e_nhds @ B @ Y3 ) @ ( at_bot @ real ) ) ) ) ).
% tendsto_at_botI_sequentially
thf(fact_7446_isCont__If__ge,axiom,
! [B: $tType,A: $tType] :
( ( ( topolo1944317154257567458pology @ A )
& ( topolo4958980785337419405_space @ B ) )
=> ! [A3: A,G: A > B,F2: A > B] :
( ( topolo3448309680560233919inuous @ A @ B @ ( topolo174197925503356063within @ A @ A3 @ ( set_ord_lessThan @ A @ A3 ) ) @ G )
=> ( ( filterlim @ A @ B @ F2 @ ( topolo7230453075368039082e_nhds @ B @ ( G @ A3 ) ) @ ( topolo174197925503356063within @ A @ A3 @ ( set_ord_greaterThan @ A @ A3 ) ) )
=> ( topolo3448309680560233919inuous @ A @ B @ ( topolo174197925503356063within @ A @ A3 @ ( top_top @ ( set @ A ) ) )
@ ^ [X3: A] : ( if @ B @ ( ord_less_eq @ A @ X3 @ A3 ) @ ( G @ X3 ) @ ( F2 @ X3 ) ) ) ) ) ) ).
% isCont_If_ge
thf(fact_7447_DERIV__pos__imp__increasing__at__bot,axiom,
! [B3: real,F2: real > real,Flim: real] :
( ! [X4: real] :
( ( ord_less_eq @ real @ X4 @ B3 )
=> ? [Y5: real] :
( ( has_field_derivative @ real @ F2 @ Y5 @ ( topolo174197925503356063within @ real @ X4 @ ( top_top @ ( set @ real ) ) ) )
& ( ord_less @ real @ ( zero_zero @ real ) @ Y5 ) ) )
=> ( ( filterlim @ real @ real @ F2 @ ( topolo7230453075368039082e_nhds @ real @ Flim ) @ ( at_bot @ real ) )
=> ( ord_less @ real @ Flim @ ( F2 @ B3 ) ) ) ) ).
% DERIV_pos_imp_increasing_at_bot
thf(fact_7448_filterlim__pow__at__bot__odd,axiom,
! [N: nat,F2: real > real,F4: filter @ real] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ N )
=> ( ( filterlim @ real @ real @ F2 @ ( at_bot @ real ) @ F4 )
=> ( ~ ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N )
=> ( filterlim @ real @ real
@ ^ [X3: real] : ( power_power @ real @ ( F2 @ X3 ) @ N )
@ ( at_bot @ real )
@ F4 ) ) ) ) ).
% filterlim_pow_at_bot_odd
thf(fact_7449_filterlim__pow__at__bot__even,axiom,
! [N: nat,F2: real > real,F4: filter @ real] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ N )
=> ( ( filterlim @ real @ real @ F2 @ ( at_bot @ real ) @ F4 )
=> ( ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N )
=> ( filterlim @ real @ real
@ ^ [X3: real] : ( power_power @ real @ ( F2 @ X3 ) @ N )
@ ( at_top @ real )
@ F4 ) ) ) ) ).
% filterlim_pow_at_bot_even
thf(fact_7450_lim__zero__infinity,axiom,
! [A: $tType] :
( ( real_V3459762299906320749_field @ A )
=> ! [F2: A > A,L: A] :
( ( filterlim @ A @ A
@ ^ [X3: A] : ( F2 @ ( divide_divide @ A @ ( one_one @ A ) @ X3 ) )
@ ( topolo7230453075368039082e_nhds @ A @ L )
@ ( topolo174197925503356063within @ A @ ( zero_zero @ A ) @ ( top_top @ ( set @ A ) ) ) )
=> ( filterlim @ A @ A @ F2 @ ( topolo7230453075368039082e_nhds @ A @ L ) @ ( at_infinity @ A ) ) ) ) ).
% lim_zero_infinity
thf(fact_7451_at__bot__le__at__infinity,axiom,
ord_less_eq @ ( filter @ real ) @ ( at_bot @ real ) @ ( at_infinity @ real ) ).
% at_bot_le_at_infinity
thf(fact_7452_at__top__le__at__infinity,axiom,
ord_less_eq @ ( filter @ real ) @ ( at_top @ real ) @ ( at_infinity @ real ) ).
% at_top_le_at_infinity
thf(fact_7453_filterlim__at__top__add__at__top,axiom,
! [A: $tType,F2: A > real,F4: filter @ A,G: A > real] :
( ( filterlim @ A @ real @ F2 @ ( at_top @ real ) @ F4 )
=> ( ( filterlim @ A @ real @ G @ ( at_top @ real ) @ F4 )
=> ( filterlim @ A @ real
@ ^ [X3: A] : ( plus_plus @ real @ ( F2 @ X3 ) @ ( G @ X3 ) )
@ ( at_top @ real )
@ F4 ) ) ) ).
% filterlim_at_top_add_at_top
thf(fact_7454_ln__at__top,axiom,
filterlim @ real @ real @ ( ln_ln @ real ) @ ( at_top @ real ) @ ( at_top @ real ) ).
% ln_at_top
thf(fact_7455_sqrt__at__top,axiom,
filterlim @ real @ real @ sqrt @ ( at_top @ real ) @ ( at_top @ real ) ).
% sqrt_at_top
thf(fact_7456_filterlim__at__infinity__imp__norm__at__top,axiom,
! [B: $tType,A: $tType] :
( ( real_V822414075346904944vector @ B )
=> ! [F2: A > B,F4: filter @ A] :
( ( filterlim @ A @ B @ F2 @ ( at_infinity @ B ) @ F4 )
=> ( filterlim @ A @ real
@ ^ [X3: A] : ( real_V7770717601297561774m_norm @ B @ ( F2 @ X3 ) )
@ ( at_top @ real )
@ F4 ) ) ) ).
% filterlim_at_infinity_imp_norm_at_top
thf(fact_7457_filterlim__norm__at__top__imp__at__infinity,axiom,
! [B: $tType,A: $tType] :
( ( real_V822414075346904944vector @ B )
=> ! [F2: A > B,F4: filter @ A] :
( ( filterlim @ A @ real
@ ^ [X3: A] : ( real_V7770717601297561774m_norm @ B @ ( F2 @ X3 ) )
@ ( at_top @ real )
@ F4 )
=> ( filterlim @ A @ B @ F2 @ ( at_infinity @ B ) @ F4 ) ) ) ).
% filterlim_norm_at_top_imp_at_infinity
thf(fact_7458_filterlim__at__infinity__conv__norm__at__top,axiom,
! [B: $tType,A: $tType] :
( ( real_V822414075346904944vector @ B )
=> ! [F2: A > B,G6: filter @ A] :
( ( filterlim @ A @ B @ F2 @ ( at_infinity @ B ) @ G6 )
= ( filterlim @ A @ real
@ ^ [X3: A] : ( real_V7770717601297561774m_norm @ B @ ( F2 @ X3 ) )
@ ( at_top @ real )
@ G6 ) ) ) ).
% filterlim_at_infinity_conv_norm_at_top
thf(fact_7459_exp__at__top,axiom,
filterlim @ real @ real @ ( exp @ real ) @ ( at_top @ real ) @ ( at_top @ real ) ).
% exp_at_top
thf(fact_7460_filterlim__at__top__mult__at__top,axiom,
! [A: $tType,F2: A > real,F4: filter @ A,G: A > real] :
( ( filterlim @ A @ real @ F2 @ ( at_top @ real ) @ F4 )
=> ( ( filterlim @ A @ real @ G @ ( at_top @ real ) @ F4 )
=> ( filterlim @ A @ real
@ ^ [X3: A] : ( times_times @ real @ ( F2 @ X3 ) @ ( G @ X3 ) )
@ ( at_top @ real )
@ F4 ) ) ) ).
% filterlim_at_top_mult_at_top
thf(fact_7461_filterlim__tendsto__add__at__top,axiom,
! [A: $tType,F2: A > real,C3: real,F4: filter @ A,G: A > real] :
( ( filterlim @ A @ real @ F2 @ ( topolo7230453075368039082e_nhds @ real @ C3 ) @ F4 )
=> ( ( filterlim @ A @ real @ G @ ( at_top @ real ) @ F4 )
=> ( filterlim @ A @ real
@ ^ [X3: A] : ( plus_plus @ real @ ( F2 @ X3 ) @ ( G @ X3 ) )
@ ( at_top @ real )
@ F4 ) ) ) ).
% filterlim_tendsto_add_at_top
thf(fact_7462_filterlim__real__sequentially,axiom,
filterlim @ nat @ real @ ( semiring_1_of_nat @ real ) @ ( at_top @ real ) @ ( at_top @ nat ) ).
% filterlim_real_sequentially
thf(fact_7463_filterlim__uminus__at__bot,axiom,
! [A: $tType,F2: A > real,F4: filter @ A] :
( ( filterlim @ A @ real @ F2 @ ( at_bot @ real ) @ F4 )
= ( filterlim @ A @ real
@ ^ [X3: A] : ( uminus_uminus @ real @ ( F2 @ X3 ) )
@ ( at_top @ real )
@ F4 ) ) ).
% filterlim_uminus_at_bot
thf(fact_7464_filterlim__uminus__at__top,axiom,
! [A: $tType,F2: A > real,F4: filter @ A] :
( ( filterlim @ A @ real @ F2 @ ( at_top @ real ) @ F4 )
= ( filterlim @ A @ real
@ ^ [X3: A] : ( uminus_uminus @ real @ ( F2 @ X3 ) )
@ ( at_bot @ real )
@ F4 ) ) ).
% filterlim_uminus_at_top
thf(fact_7465_filterlim__at__bot__mirror,axiom,
! [A: $tType,F2: real > A,F4: filter @ A] :
( ( filterlim @ real @ A @ F2 @ F4 @ ( at_bot @ real ) )
= ( filterlim @ real @ A
@ ^ [X3: real] : ( F2 @ ( uminus_uminus @ real @ X3 ) )
@ F4
@ ( at_top @ real ) ) ) ).
% filterlim_at_bot_mirror
thf(fact_7466_filterlim__at__top__mirror,axiom,
! [A: $tType,F2: real > A,F4: filter @ A] :
( ( filterlim @ real @ A @ F2 @ F4 @ ( at_top @ real ) )
= ( filterlim @ real @ A
@ ^ [X3: real] : ( F2 @ ( uminus_uminus @ real @ X3 ) )
@ F4
@ ( at_bot @ real ) ) ) ).
% filterlim_at_top_mirror
thf(fact_7467_filterlim__uminus__at__bot__at__top,axiom,
filterlim @ real @ real @ ( uminus_uminus @ real ) @ ( at_bot @ real ) @ ( at_top @ real ) ).
% filterlim_uminus_at_bot_at_top
thf(fact_7468_filterlim__uminus__at__top__at__bot,axiom,
filterlim @ real @ real @ ( uminus_uminus @ real ) @ ( at_top @ real ) @ ( at_bot @ real ) ).
% filterlim_uminus_at_top_at_bot
thf(fact_7469_greaterThan__0,axiom,
( ( set_ord_greaterThan @ nat @ ( zero_zero @ nat ) )
= ( image @ nat @ nat @ suc @ ( top_top @ ( set @ nat ) ) ) ) ).
% greaterThan_0
thf(fact_7470_filterlim__pow__at__top,axiom,
! [A: $tType,N: nat,F2: A > real,F4: filter @ A] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ N )
=> ( ( filterlim @ A @ real @ F2 @ ( at_top @ real ) @ F4 )
=> ( filterlim @ A @ real
@ ^ [X3: A] : ( power_power @ real @ ( F2 @ X3 ) @ N )
@ ( at_top @ real )
@ F4 ) ) ) ).
% filterlim_pow_at_top
thf(fact_7471_greaterThan__Suc,axiom,
! [K: nat] :
( ( set_ord_greaterThan @ nat @ ( suc @ K ) )
= ( minus_minus @ ( set @ nat ) @ ( set_ord_greaterThan @ nat @ K ) @ ( insert @ nat @ ( suc @ K ) @ ( bot_bot @ ( set @ nat ) ) ) ) ) ).
% greaterThan_Suc
thf(fact_7472_tendsto__add__filterlim__at__infinity,axiom,
! [B: $tType,A: $tType] :
( ( real_V822414075346904944vector @ B )
=> ! [F2: A > B,C3: B,F4: filter @ A,G: A > B] :
( ( filterlim @ A @ B @ F2 @ ( topolo7230453075368039082e_nhds @ B @ C3 ) @ F4 )
=> ( ( filterlim @ A @ B @ G @ ( at_infinity @ B ) @ F4 )
=> ( filterlim @ A @ B
@ ^ [X3: A] : ( plus_plus @ B @ ( F2 @ X3 ) @ ( G @ X3 ) )
@ ( at_infinity @ B )
@ F4 ) ) ) ) ).
% tendsto_add_filterlim_at_infinity
thf(fact_7473_tendsto__add__filterlim__at__infinity_H,axiom,
! [B: $tType,A: $tType] :
( ( real_V822414075346904944vector @ B )
=> ! [F2: A > B,F4: filter @ A,G: A > B,C3: B] :
( ( filterlim @ A @ B @ F2 @ ( at_infinity @ B ) @ F4 )
=> ( ( filterlim @ A @ B @ G @ ( topolo7230453075368039082e_nhds @ B @ C3 ) @ F4 )
=> ( filterlim @ A @ B
@ ^ [X3: A] : ( plus_plus @ B @ ( F2 @ X3 ) @ ( G @ X3 ) )
@ ( at_infinity @ B )
@ F4 ) ) ) ) ).
% tendsto_add_filterlim_at_infinity'
thf(fact_7474_tanh__real__at__top,axiom,
filterlim @ real @ real @ ( tanh @ real ) @ ( topolo7230453075368039082e_nhds @ real @ ( one_one @ real ) ) @ ( at_top @ real ) ).
% tanh_real_at_top
thf(fact_7475_real__tendsto__divide__at__top,axiom,
! [A: $tType,F2: A > real,C3: real,F4: filter @ A,G: A > real] :
( ( filterlim @ A @ real @ F2 @ ( topolo7230453075368039082e_nhds @ real @ C3 ) @ F4 )
=> ( ( filterlim @ A @ real @ G @ ( at_top @ real ) @ F4 )
=> ( filterlim @ A @ real
@ ^ [X3: A] : ( divide_divide @ real @ ( F2 @ X3 ) @ ( G @ X3 ) )
@ ( topolo7230453075368039082e_nhds @ real @ ( zero_zero @ real ) )
@ F4 ) ) ) ).
% real_tendsto_divide_at_top
thf(fact_7476_artanh__real__at__left__1,axiom,
filterlim @ real @ real @ ( artanh @ real ) @ ( at_top @ real ) @ ( topolo174197925503356063within @ real @ ( one_one @ real ) @ ( set_ord_lessThan @ real @ ( one_one @ real ) ) ) ).
% artanh_real_at_left_1
thf(fact_7477_tendsto__inverse__0__at__top,axiom,
! [A: $tType,F2: A > real,F4: filter @ A] :
( ( filterlim @ A @ real @ F2 @ ( at_top @ real ) @ F4 )
=> ( filterlim @ A @ real
@ ^ [X3: A] : ( inverse_inverse @ real @ ( F2 @ X3 ) )
@ ( topolo7230453075368039082e_nhds @ real @ ( zero_zero @ real ) )
@ F4 ) ) ).
% tendsto_inverse_0_at_top
thf(fact_7478_filterlim__sequentially__iff__filterlim__real,axiom,
! [A: $tType,F2: A > nat,F4: filter @ A] :
( ( filterlim @ A @ nat @ F2 @ ( at_top @ nat ) @ F4 )
= ( filterlim @ A @ real
@ ^ [X3: A] : ( semiring_1_of_nat @ real @ ( F2 @ X3 ) )
@ ( at_top @ real )
@ F4 ) ) ).
% filterlim_sequentially_iff_filterlim_real
thf(fact_7479_filterlim__at__top__mult__tendsto__pos,axiom,
! [A: $tType,F2: A > real,C3: real,F4: filter @ A,G: A > real] :
( ( filterlim @ A @ real @ F2 @ ( topolo7230453075368039082e_nhds @ real @ C3 ) @ F4 )
=> ( ( ord_less @ real @ ( zero_zero @ real ) @ C3 )
=> ( ( filterlim @ A @ real @ G @ ( at_top @ real ) @ F4 )
=> ( filterlim @ A @ real
@ ^ [X3: A] : ( times_times @ real @ ( G @ X3 ) @ ( F2 @ X3 ) )
@ ( at_top @ real )
@ F4 ) ) ) ) ).
% filterlim_at_top_mult_tendsto_pos
thf(fact_7480_filterlim__tendsto__pos__mult__at__top,axiom,
! [A: $tType,F2: A > real,C3: real,F4: filter @ A,G: A > real] :
( ( filterlim @ A @ real @ F2 @ ( topolo7230453075368039082e_nhds @ real @ C3 ) @ F4 )
=> ( ( ord_less @ real @ ( zero_zero @ real ) @ C3 )
=> ( ( filterlim @ A @ real @ G @ ( at_top @ real ) @ F4 )
=> ( filterlim @ A @ real
@ ^ [X3: A] : ( times_times @ real @ ( F2 @ X3 ) @ ( G @ X3 ) )
@ ( at_top @ real )
@ F4 ) ) ) ) ).
% filterlim_tendsto_pos_mult_at_top
thf(fact_7481_tendsto__neg__powr,axiom,
! [A: $tType,S2: real,F2: A > real,F4: filter @ A] :
( ( ord_less @ real @ S2 @ ( zero_zero @ real ) )
=> ( ( filterlim @ A @ real @ F2 @ ( at_top @ real ) @ F4 )
=> ( filterlim @ A @ real
@ ^ [X3: A] : ( powr @ real @ ( F2 @ X3 ) @ S2 )
@ ( topolo7230453075368039082e_nhds @ real @ ( zero_zero @ real ) )
@ F4 ) ) ) ).
% tendsto_neg_powr
thf(fact_7482_tendsto__mult__filterlim__at__infinity,axiom,
! [A: $tType,B: $tType] :
( ( real_V3459762299906320749_field @ A )
=> ! [F2: B > A,C3: A,F4: filter @ B,G: B > A] :
( ( filterlim @ B @ A @ F2 @ ( topolo7230453075368039082e_nhds @ A @ C3 ) @ F4 )
=> ( ( C3
!= ( zero_zero @ A ) )
=> ( ( filterlim @ B @ A @ G @ ( at_infinity @ A ) @ F4 )
=> ( filterlim @ B @ A
@ ^ [X3: B] : ( times_times @ A @ ( F2 @ X3 ) @ ( G @ X3 ) )
@ ( at_infinity @ A )
@ F4 ) ) ) ) ) ).
% tendsto_mult_filterlim_at_infinity
thf(fact_7483_filterlim__power__at__infinity,axiom,
! [B: $tType,A: $tType] :
( ( real_V8999393235501362500lgebra @ B )
=> ! [F2: A > B,F4: filter @ A,N: nat] :
( ( filterlim @ A @ B @ F2 @ ( at_infinity @ B ) @ F4 )
=> ( ( ord_less @ nat @ ( zero_zero @ nat ) @ N )
=> ( filterlim @ A @ B
@ ^ [X3: A] : ( power_power @ B @ ( F2 @ X3 ) @ N )
@ ( at_infinity @ B )
@ F4 ) ) ) ) ).
% filterlim_power_at_infinity
thf(fact_7484_tendsto__divide__0,axiom,
! [A: $tType,C: $tType] :
( ( real_V8999393235501362500lgebra @ A )
=> ! [F2: C > A,C3: A,F4: filter @ C,G: C > A] :
( ( filterlim @ C @ A @ F2 @ ( topolo7230453075368039082e_nhds @ A @ C3 ) @ F4 )
=> ( ( filterlim @ C @ A @ G @ ( at_infinity @ A ) @ F4 )
=> ( filterlim @ C @ A
@ ^ [X3: C] : ( divide_divide @ A @ ( F2 @ X3 ) @ ( G @ X3 ) )
@ ( topolo7230453075368039082e_nhds @ A @ ( zero_zero @ A ) )
@ F4 ) ) ) ) ).
% tendsto_divide_0
thf(fact_7485_ln__x__over__x__tendsto__0,axiom,
( filterlim @ real @ real
@ ^ [X3: real] : ( divide_divide @ real @ ( ln_ln @ real @ X3 ) @ X3 )
@ ( topolo7230453075368039082e_nhds @ real @ ( zero_zero @ real ) )
@ ( at_top @ real ) ) ).
% ln_x_over_x_tendsto_0
thf(fact_7486_filterlim__at__right__to__top,axiom,
! [A: $tType,F2: real > A,F4: filter @ A] :
( ( filterlim @ real @ A @ F2 @ F4 @ ( topolo174197925503356063within @ real @ ( zero_zero @ real ) @ ( set_ord_greaterThan @ real @ ( zero_zero @ real ) ) ) )
= ( filterlim @ real @ A
@ ^ [X3: real] : ( F2 @ ( inverse_inverse @ real @ X3 ) )
@ F4
@ ( at_top @ real ) ) ) ).
% filterlim_at_right_to_top
thf(fact_7487_filterlim__at__top__to__right,axiom,
! [A: $tType,F2: real > A,F4: filter @ A] :
( ( filterlim @ real @ A @ F2 @ F4 @ ( at_top @ real ) )
= ( filterlim @ real @ A
@ ^ [X3: real] : ( F2 @ ( inverse_inverse @ real @ X3 ) )
@ F4
@ ( topolo174197925503356063within @ real @ ( zero_zero @ real ) @ ( set_ord_greaterThan @ real @ ( zero_zero @ real ) ) ) ) ) ).
% filterlim_at_top_to_right
thf(fact_7488_filterlim__inverse__at__right__top,axiom,
filterlim @ real @ real @ ( inverse_inverse @ real ) @ ( topolo174197925503356063within @ real @ ( zero_zero @ real ) @ ( set_ord_greaterThan @ real @ ( zero_zero @ real ) ) ) @ ( at_top @ real ) ).
% filterlim_inverse_at_right_top
thf(fact_7489_filterlim__inverse__at__top__right,axiom,
filterlim @ real @ real @ ( inverse_inverse @ real ) @ ( at_top @ real ) @ ( topolo174197925503356063within @ real @ ( zero_zero @ real ) @ ( set_ord_greaterThan @ real @ ( zero_zero @ real ) ) ) ).
% filterlim_inverse_at_top_right
thf(fact_7490_tendsto__at__topI__sequentially,axiom,
! [B: $tType] :
( ( topolo3112930676232923870pology @ B )
=> ! [F2: real > B,Y3: B] :
( ! [X17: nat > real] :
( ( filterlim @ nat @ real @ X17 @ ( at_top @ real ) @ ( at_top @ nat ) )
=> ( filterlim @ nat @ B
@ ^ [N4: nat] : ( F2 @ ( X17 @ N4 ) )
@ ( topolo7230453075368039082e_nhds @ B @ Y3 )
@ ( at_top @ nat ) ) )
=> ( filterlim @ real @ B @ F2 @ ( topolo7230453075368039082e_nhds @ B @ Y3 ) @ ( at_top @ real ) ) ) ) ).
% tendsto_at_topI_sequentially
thf(fact_7491_filterlim__tendsto__neg__mult__at__bot,axiom,
! [A: $tType,F2: A > real,C3: real,F4: filter @ A,G: A > real] :
( ( filterlim @ A @ real @ F2 @ ( topolo7230453075368039082e_nhds @ real @ C3 ) @ F4 )
=> ( ( ord_less @ real @ C3 @ ( zero_zero @ real ) )
=> ( ( filterlim @ A @ real @ G @ ( at_top @ real ) @ F4 )
=> ( filterlim @ A @ real
@ ^ [X3: A] : ( times_times @ real @ ( F2 @ X3 ) @ ( G @ X3 ) )
@ ( at_bot @ real )
@ F4 ) ) ) ) ).
% filterlim_tendsto_neg_mult_at_bot
thf(fact_7492_tendsto__power__div__exp__0,axiom,
! [K: nat] :
( filterlim @ real @ real
@ ^ [X3: real] : ( divide_divide @ real @ ( power_power @ real @ X3 @ K ) @ ( exp @ real @ X3 ) )
@ ( topolo7230453075368039082e_nhds @ real @ ( zero_zero @ real ) )
@ ( at_top @ real ) ) ).
% tendsto_power_div_exp_0
thf(fact_7493_lim__infinity__imp__sequentially,axiom,
! [A: $tType] :
( ( topolo4958980785337419405_space @ A )
=> ! [F2: real > A,L: A] :
( ( filterlim @ real @ A @ F2 @ ( topolo7230453075368039082e_nhds @ A @ L ) @ ( at_infinity @ real ) )
=> ( filterlim @ nat @ A
@ ^ [N4: nat] : ( F2 @ ( semiring_1_of_nat @ real @ N4 ) )
@ ( topolo7230453075368039082e_nhds @ A @ L )
@ ( at_top @ nat ) ) ) ) ).
% lim_infinity_imp_sequentially
thf(fact_7494_filterlim__inverse__at__iff,axiom,
! [B: $tType,A: $tType] :
( ( real_V8999393235501362500lgebra @ B )
=> ! [G: A > B,F4: filter @ A] :
( ( filterlim @ A @ B
@ ^ [X3: A] : ( inverse_inverse @ B @ ( G @ X3 ) )
@ ( topolo174197925503356063within @ B @ ( zero_zero @ B ) @ ( top_top @ ( set @ B ) ) )
@ F4 )
= ( filterlim @ A @ B @ G @ ( at_infinity @ B ) @ F4 ) ) ) ).
% filterlim_inverse_at_iff
thf(fact_7495_tendsto__exp__limit__at__top,axiom,
! [X2: real] :
( filterlim @ real @ real
@ ^ [Y: real] : ( powr @ real @ ( plus_plus @ real @ ( one_one @ real ) @ ( divide_divide @ real @ X2 @ Y ) ) @ Y )
@ ( topolo7230453075368039082e_nhds @ real @ ( exp @ real @ X2 ) )
@ ( at_top @ real ) ) ).
% tendsto_exp_limit_at_top
thf(fact_7496_filterlim__divide__at__infinity,axiom,
! [A: $tType] :
( ( real_V3459762299906320749_field @ A )
=> ! [F2: A > A,C3: A,F4: filter @ A,G: A > A] :
( ( filterlim @ A @ A @ F2 @ ( topolo7230453075368039082e_nhds @ A @ C3 ) @ F4 )
=> ( ( filterlim @ A @ A @ G @ ( topolo174197925503356063within @ A @ ( zero_zero @ A ) @ ( top_top @ ( set @ A ) ) ) @ F4 )
=> ( ( C3
!= ( zero_zero @ A ) )
=> ( filterlim @ A @ A
@ ^ [X3: A] : ( divide_divide @ A @ ( F2 @ X3 ) @ ( G @ X3 ) )
@ ( at_infinity @ A )
@ F4 ) ) ) ) ) ).
% filterlim_divide_at_infinity
thf(fact_7497_filterlim__tan__at__left,axiom,
filterlim @ real @ real @ ( tan @ real ) @ ( at_top @ real ) @ ( topolo174197925503356063within @ real @ ( divide_divide @ real @ pi @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) @ ( set_ord_lessThan @ real @ ( divide_divide @ real @ pi @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) ) ) ).
% filterlim_tan_at_left
thf(fact_7498_tendsto__arctan__at__top,axiom,
filterlim @ real @ real @ arctan @ ( topolo7230453075368039082e_nhds @ real @ ( divide_divide @ real @ pi @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) ) @ ( at_top @ real ) ).
% tendsto_arctan_at_top
thf(fact_7499_DERIV__neg__imp__decreasing__at__top,axiom,
! [B3: real,F2: real > real,Flim: real] :
( ! [X4: real] :
( ( ord_less_eq @ real @ B3 @ X4 )
=> ? [Y5: real] :
( ( has_field_derivative @ real @ F2 @ Y5 @ ( topolo174197925503356063within @ real @ X4 @ ( top_top @ ( set @ real ) ) ) )
& ( ord_less @ real @ Y5 @ ( zero_zero @ real ) ) ) )
=> ( ( filterlim @ real @ real @ F2 @ ( topolo7230453075368039082e_nhds @ real @ Flim ) @ ( at_top @ real ) )
=> ( ord_less @ real @ Flim @ ( F2 @ B3 ) ) ) ) ).
% DERIV_neg_imp_decreasing_at_top
thf(fact_7500_filterlim__realpow__sequentially__gt1,axiom,
! [A: $tType] :
( ( real_V8999393235501362500lgebra @ A )
=> ! [X2: A] :
( ( ord_less @ real @ ( one_one @ real ) @ ( real_V7770717601297561774m_norm @ A @ X2 ) )
=> ( filterlim @ nat @ A @ ( power_power @ A @ X2 ) @ ( at_infinity @ A ) @ ( at_top @ nat ) ) ) ) ).
% filterlim_realpow_sequentially_gt1
thf(fact_7501_polyfun__extremal,axiom,
! [A: $tType] :
( ( real_V8999393235501362500lgebra @ A )
=> ! [C3: nat > A,K: nat,N: nat,B2: real] :
( ( ( C3 @ K )
!= ( zero_zero @ A ) )
=> ( ( ord_less_eq @ nat @ ( one_one @ nat ) @ K )
=> ( ( ord_less_eq @ nat @ K @ N )
=> ( eventually @ A
@ ^ [Z4: A] :
( ord_less_eq @ real @ B2
@ ( real_V7770717601297561774m_norm @ A
@ ( groups7311177749621191930dd_sum @ nat @ A
@ ^ [I4: nat] : ( times_times @ A @ ( C3 @ I4 ) @ ( power_power @ A @ Z4 @ I4 ) )
@ ( set_ord_atMost @ nat @ N ) ) ) )
@ ( at_infinity @ A ) ) ) ) ) ) ).
% polyfun_extremal
thf(fact_7502_lhopital__left__at__top,axiom,
! [G: real > real,X2: real,G5: real > real,F2: real > real,F7: real > real,Y3: real] :
( ( filterlim @ real @ real @ G @ ( at_top @ real ) @ ( topolo174197925503356063within @ real @ X2 @ ( set_ord_lessThan @ real @ X2 ) ) )
=> ( ( eventually @ real
@ ^ [X3: real] :
( ( G5 @ X3 )
!= ( zero_zero @ real ) )
@ ( topolo174197925503356063within @ real @ X2 @ ( set_ord_lessThan @ real @ X2 ) ) )
=> ( ( eventually @ real
@ ^ [X3: real] : ( has_field_derivative @ real @ F2 @ ( F7 @ X3 ) @ ( topolo174197925503356063within @ real @ X3 @ ( top_top @ ( set @ real ) ) ) )
@ ( topolo174197925503356063within @ real @ X2 @ ( set_ord_lessThan @ real @ X2 ) ) )
=> ( ( eventually @ real
@ ^ [X3: real] : ( has_field_derivative @ real @ G @ ( G5 @ X3 ) @ ( topolo174197925503356063within @ real @ X3 @ ( top_top @ ( set @ real ) ) ) )
@ ( topolo174197925503356063within @ real @ X2 @ ( set_ord_lessThan @ real @ X2 ) ) )
=> ( ( filterlim @ real @ real
@ ^ [X3: real] : ( divide_divide @ real @ ( F7 @ X3 ) @ ( G5 @ X3 ) )
@ ( topolo7230453075368039082e_nhds @ real @ Y3 )
@ ( topolo174197925503356063within @ real @ X2 @ ( set_ord_lessThan @ real @ X2 ) ) )
=> ( filterlim @ real @ real
@ ^ [X3: real] : ( divide_divide @ real @ ( F2 @ X3 ) @ ( G @ X3 ) )
@ ( topolo7230453075368039082e_nhds @ real @ Y3 )
@ ( topolo174197925503356063within @ real @ X2 @ ( set_ord_lessThan @ real @ X2 ) ) ) ) ) ) ) ) ).
% lhopital_left_at_top
thf(fact_7503_eventually__sequentially__Suc,axiom,
! [P: nat > $o] :
( ( eventually @ nat
@ ^ [I4: nat] : ( P @ ( suc @ I4 ) )
@ ( at_top @ nat ) )
= ( eventually @ nat @ P @ ( at_top @ nat ) ) ) ).
% eventually_sequentially_Suc
thf(fact_7504_eventually__sequentially__seg,axiom,
! [P: nat > $o,K: nat] :
( ( eventually @ nat
@ ^ [N4: nat] : ( P @ ( plus_plus @ nat @ N4 @ K ) )
@ ( at_top @ nat ) )
= ( eventually @ nat @ P @ ( at_top @ nat ) ) ) ).
% eventually_sequentially_seg
thf(fact_7505_eventually__const,axiom,
! [A: $tType,F4: filter @ A,P: $o] :
( ( F4
!= ( bot_bot @ ( filter @ A ) ) )
=> ( ( eventually @ A
@ ^ [X3: A] : P
@ F4 )
= P ) ) ).
% eventually_const
thf(fact_7506_eventually__at__right__field,axiom,
! [A: $tType] :
( ( ( linordered_field @ A )
& ( topolo1944317154257567458pology @ A ) )
=> ! [P: A > $o,X2: A] :
( ( eventually @ A @ P @ ( topolo174197925503356063within @ A @ X2 @ ( set_ord_greaterThan @ A @ X2 ) ) )
= ( ? [B7: A] :
( ( ord_less @ A @ X2 @ B7 )
& ! [Y: A] :
( ( ord_less @ A @ X2 @ Y )
=> ( ( ord_less @ A @ Y @ B7 )
=> ( P @ Y ) ) ) ) ) ) ) ).
% eventually_at_right_field
thf(fact_7507_eventually__at__right,axiom,
! [A: $tType] :
( ( topolo1944317154257567458pology @ A )
=> ! [X2: A,Y3: A,P: A > $o] :
( ( ord_less @ A @ X2 @ Y3 )
=> ( ( eventually @ A @ P @ ( topolo174197925503356063within @ A @ X2 @ ( set_ord_greaterThan @ A @ X2 ) ) )
= ( ? [B7: A] :
( ( ord_less @ A @ X2 @ B7 )
& ! [Y: A] :
( ( ord_less @ A @ X2 @ Y )
=> ( ( ord_less @ A @ Y @ B7 )
=> ( P @ Y ) ) ) ) ) ) ) ) ).
% eventually_at_right
thf(fact_7508_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_7509_eventually__at__top__dense,axiom,
! [A: $tType] :
( ( ( linorder @ A )
& ( no_top @ A ) )
=> ! [P: A > $o] :
( ( eventually @ A @ P @ ( at_top @ A ) )
= ( ? [N8: A] :
! [N4: A] :
( ( ord_less @ A @ N8 @ N4 )
=> ( P @ N4 ) ) ) ) ) ).
% eventually_at_top_dense
thf(fact_7510_eventually__at__left,axiom,
! [A: $tType] :
( ( topolo1944317154257567458pology @ A )
=> ! [Y3: A,X2: A,P: A > $o] :
( ( ord_less @ A @ Y3 @ X2 )
=> ( ( eventually @ A @ P @ ( topolo174197925503356063within @ A @ X2 @ ( set_ord_lessThan @ A @ X2 ) ) )
= ( ? [B7: A] :
( ( ord_less @ A @ B7 @ X2 )
& ! [Y: A] :
( ( ord_less @ A @ B7 @ Y )
=> ( ( ord_less @ A @ Y @ X2 )
=> ( P @ Y ) ) ) ) ) ) ) ) ).
% eventually_at_left
thf(fact_7511_eventually__at__left__field,axiom,
! [A: $tType] :
( ( ( linordered_field @ A )
& ( topolo1944317154257567458pology @ A ) )
=> ! [P: A > $o,X2: A] :
( ( eventually @ A @ P @ ( topolo174197925503356063within @ A @ X2 @ ( set_ord_lessThan @ A @ X2 ) ) )
= ( ? [B7: A] :
( ( ord_less @ A @ B7 @ X2 )
& ! [Y: A] :
( ( ord_less @ A @ B7 @ Y )
=> ( ( ord_less @ A @ Y @ X2 )
=> ( P @ Y ) ) ) ) ) ) ) ).
% eventually_at_left_field
thf(fact_7512_sequentially__offset,axiom,
! [P: nat > $o,K: nat] :
( ( eventually @ nat @ P @ ( at_top @ nat ) )
=> ( eventually @ nat
@ ^ [I4: nat] : ( P @ ( plus_plus @ nat @ I4 @ K ) )
@ ( at_top @ nat ) ) ) ).
% sequentially_offset
thf(fact_7513_summable__cong,axiom,
! [A: $tType] :
( ( real_V822414075346904944vector @ A )
=> ! [F2: nat > A,G: nat > A] :
( ( eventually @ nat
@ ^ [X3: nat] :
( ( F2 @ X3 )
= ( G @ X3 ) )
@ ( at_top @ nat ) )
=> ( ( summable @ A @ F2 )
= ( summable @ A @ G ) ) ) ) ).
% summable_cong
thf(fact_7514_eventually__at__infinity,axiom,
! [A: $tType] :
( ( real_V822414075346904944vector @ A )
=> ! [P: A > $o] :
( ( eventually @ A @ P @ ( at_infinity @ A ) )
= ( ? [B7: real] :
! [X3: A] :
( ( ord_less_eq @ real @ B7 @ ( real_V7770717601297561774m_norm @ A @ X3 ) )
=> ( P @ X3 ) ) ) ) ) ).
% eventually_at_infinity
thf(fact_7515_eventually__not__equal__at__infinity,axiom,
! [A: $tType] :
( ( real_V822414075346904944vector @ A )
=> ! [A3: A] :
( eventually @ A
@ ^ [X3: A] : X3 != A3
@ ( at_infinity @ A ) ) ) ).
% eventually_not_equal_at_infinity
thf(fact_7516_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_7517_le__sequentially,axiom,
! [F4: filter @ nat] :
( ( ord_less_eq @ ( filter @ nat ) @ F4 @ ( at_top @ nat ) )
= ( ! [N8: nat] : ( eventually @ nat @ ( ord_less_eq @ nat @ N8 ) @ F4 ) ) ) ).
% le_sequentially
thf(fact_7518_eventually__sequentially,axiom,
! [P: nat > $o] :
( ( eventually @ nat @ P @ ( at_top @ nat ) )
= ( ? [N8: nat] :
! [N4: nat] :
( ( ord_less_eq @ nat @ N8 @ N4 )
=> ( P @ N4 ) ) ) ) ).
% eventually_sequentially
thf(fact_7519_eventually__sequentiallyI,axiom,
! [C3: nat,P: nat > $o] :
( ! [X4: nat] :
( ( ord_less_eq @ nat @ C3 @ X4 )
=> ( P @ X4 ) )
=> ( eventually @ nat @ P @ ( at_top @ nat ) ) ) ).
% eventually_sequentiallyI
thf(fact_7520_eventually__at__top__linorder,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [P: A > $o] :
( ( eventually @ A @ P @ ( at_top @ A ) )
= ( ? [N8: A] :
! [N4: A] :
( ( ord_less_eq @ A @ N8 @ N4 )
=> ( P @ N4 ) ) ) ) ) ).
% eventually_at_top_linorder
thf(fact_7521_eventually__at__top__linorderI,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [C3: A,P: A > $o] :
( ! [X4: A] :
( ( ord_less_eq @ A @ C3 @ X4 )
=> ( P @ X4 ) )
=> ( eventually @ A @ P @ ( at_top @ A ) ) ) ) ).
% eventually_at_top_linorderI
thf(fact_7522_eventually__at__top__not__equal,axiom,
! [A: $tType] :
( ( ( linorder @ A )
& ( no_top @ A ) )
=> ! [C3: A] :
( eventually @ A
@ ^ [X3: A] : X3 != C3
@ ( at_top @ A ) ) ) ).
% eventually_at_top_not_equal
thf(fact_7523_eventually__False__sequentially,axiom,
~ ( eventually @ nat
@ ^ [N4: nat] : $false
@ ( at_top @ nat ) ) ).
% eventually_False_sequentially
thf(fact_7524_Lim__transform__eventually,axiom,
! [B: $tType,A: $tType] :
( ( topolo4958980785337419405_space @ B )
=> ! [F2: A > B,L: B,F4: filter @ A,G: A > B] :
( ( filterlim @ A @ B @ F2 @ ( topolo7230453075368039082e_nhds @ B @ L ) @ F4 )
=> ( ( eventually @ A
@ ^ [X3: A] :
( ( F2 @ X3 )
= ( G @ X3 ) )
@ F4 )
=> ( filterlim @ A @ B @ G @ ( topolo7230453075368039082e_nhds @ B @ L ) @ F4 ) ) ) ) ).
% Lim_transform_eventually
thf(fact_7525_filterlim__at__infinity__imp__eventually__ne,axiom,
! [B: $tType,A: $tType] :
( ( real_V822414075346904944vector @ B )
=> ! [F2: A > B,F4: filter @ A,C3: B] :
( ( filterlim @ A @ B @ F2 @ ( at_infinity @ B ) @ F4 )
=> ( eventually @ A
@ ^ [Z4: A] :
( ( F2 @ Z4 )
!= C3 )
@ F4 ) ) ) ).
% filterlim_at_infinity_imp_eventually_ne
thf(fact_7526_eventually__compose__filterlim,axiom,
! [A: $tType,B: $tType,P: A > $o,F4: filter @ A,F2: B > A,G6: filter @ B] :
( ( eventually @ A @ P @ F4 )
=> ( ( filterlim @ B @ A @ F2 @ F4 @ G6 )
=> ( eventually @ B
@ ^ [X3: B] : ( P @ ( F2 @ X3 ) )
@ G6 ) ) ) ).
% eventually_compose_filterlim
thf(fact_7527_filterlim__cong,axiom,
! [A: $tType,B: $tType,F14: filter @ A,F15: filter @ A,F24: filter @ B,F25: filter @ B,F2: B > A,G: B > A] :
( ( F14 = F15 )
=> ( ( F24 = F25 )
=> ( ( eventually @ B
@ ^ [X3: B] :
( ( F2 @ X3 )
= ( G @ X3 ) )
@ F24 )
=> ( ( filterlim @ B @ A @ F2 @ F14 @ F24 )
= ( filterlim @ B @ A @ G @ F15 @ F25 ) ) ) ) ) ).
% filterlim_cong
thf(fact_7528_filterlim__iff,axiom,
! [B: $tType,A: $tType] :
( ( filterlim @ A @ B )
= ( ^ [F5: A > B,F26: filter @ B,F16: filter @ A] :
! [P3: B > $o] :
( ( eventually @ B @ P3 @ F26 )
=> ( eventually @ A
@ ^ [X3: A] : ( P3 @ ( F5 @ X3 ) )
@ F16 ) ) ) ) ).
% filterlim_iff
thf(fact_7529_tendsto__cong,axiom,
! [A: $tType,B: $tType] :
( ( topolo4958980785337419405_space @ A )
=> ! [F2: B > A,G: B > A,F4: filter @ B,C3: A] :
( ( eventually @ B
@ ^ [X3: B] :
( ( F2 @ X3 )
= ( G @ X3 ) )
@ F4 )
=> ( ( filterlim @ B @ A @ F2 @ ( topolo7230453075368039082e_nhds @ A @ C3 ) @ F4 )
= ( filterlim @ B @ A @ G @ ( topolo7230453075368039082e_nhds @ A @ C3 ) @ F4 ) ) ) ) ).
% tendsto_cong
thf(fact_7530_tendsto__discrete,axiom,
! [A: $tType,B: $tType] :
( ( topolo8865339358273720382pology @ A )
=> ! [F2: B > A,Y3: A,F4: filter @ B] :
( ( filterlim @ B @ A @ F2 @ ( topolo7230453075368039082e_nhds @ A @ Y3 ) @ F4 )
= ( eventually @ B
@ ^ [X3: B] :
( ( F2 @ X3 )
= Y3 )
@ F4 ) ) ) ).
% tendsto_discrete
thf(fact_7531_tendsto__imp__eventually__ne,axiom,
! [A: $tType,B: $tType] :
( ( topological_t1_space @ A )
=> ! [F2: B > A,C3: A,F4: filter @ B,C8: A] :
( ( filterlim @ B @ A @ F2 @ ( topolo7230453075368039082e_nhds @ A @ C3 ) @ F4 )
=> ( ( C3 != C8 )
=> ( eventually @ B
@ ^ [Z4: B] :
( ( F2 @ Z4 )
!= C8 )
@ F4 ) ) ) ) ).
% tendsto_imp_eventually_ne
thf(fact_7532_tendsto__eventually,axiom,
! [A: $tType,B: $tType] :
( ( topolo4958980785337419405_space @ A )
=> ! [F2: B > A,L: A,Net: filter @ B] :
( ( eventually @ B
@ ^ [X3: B] :
( ( F2 @ X3 )
= L )
@ Net )
=> ( filterlim @ B @ A @ F2 @ ( topolo7230453075368039082e_nhds @ A @ L ) @ Net ) ) ) ).
% tendsto_eventually
thf(fact_7533_eventually__nhds__iff__sequentially,axiom,
! [A: $tType] :
( ( topolo3112930676232923870pology @ A )
=> ! [P: A > $o,A3: A] :
( ( eventually @ A @ P @ ( topolo7230453075368039082e_nhds @ A @ A3 ) )
= ( ! [F5: nat > A] :
( ( filterlim @ nat @ A @ F5 @ ( topolo7230453075368039082e_nhds @ A @ A3 ) @ ( at_top @ nat ) )
=> ( eventually @ nat
@ ^ [N4: nat] : ( P @ ( F5 @ N4 ) )
@ ( at_top @ nat ) ) ) ) ) ) ).
% eventually_nhds_iff_sequentially
thf(fact_7534_filterlim__mono__eventually,axiom,
! [B: $tType,A: $tType,F2: A > B,F4: filter @ B,G6: filter @ A,F6: filter @ B,G7: filter @ A,F7: A > B] :
( ( filterlim @ A @ B @ F2 @ F4 @ G6 )
=> ( ( ord_less_eq @ ( filter @ B ) @ F4 @ F6 )
=> ( ( ord_less_eq @ ( filter @ A ) @ G7 @ G6 )
=> ( ( eventually @ A
@ ^ [X3: A] :
( ( F2 @ X3 )
= ( F7 @ X3 ) )
@ G7 )
=> ( filterlim @ A @ B @ F7 @ F6 @ G7 ) ) ) ) ) ).
% filterlim_mono_eventually
thf(fact_7535_filterlim__at__top__at__top,axiom,
! [B: $tType,A: $tType] :
( ( ( linorder @ A )
& ( linorder @ B ) )
=> ! [Q: A > $o,F2: A > B,P: B > $o,G: B > A] :
( ! [X4: A,Y4: A] :
( ( Q @ X4 )
=> ( ( Q @ Y4 )
=> ( ( ord_less_eq @ A @ X4 @ Y4 )
=> ( ord_less_eq @ B @ ( F2 @ X4 ) @ ( F2 @ Y4 ) ) ) ) )
=> ( ! [X4: B] :
( ( P @ X4 )
=> ( ( F2 @ ( G @ X4 ) )
= X4 ) )
=> ( ! [X4: B] :
( ( P @ X4 )
=> ( Q @ ( G @ X4 ) ) )
=> ( ( eventually @ A @ Q @ ( at_top @ A ) )
=> ( ( eventually @ B @ P @ ( at_top @ B ) )
=> ( filterlim @ A @ B @ F2 @ ( at_top @ B ) @ ( at_top @ A ) ) ) ) ) ) ) ) ).
% filterlim_at_top_at_top
thf(fact_7536_sequentially__imp__eventually__at,axiom,
! [A: $tType] :
( ( topolo3112930676232923870pology @ A )
=> ! [A3: A,P: A > $o] :
( ! [F3: nat > A] :
( ( ! [N9: nat] :
( ( F3 @ N9 )
!= A3 )
& ( filterlim @ nat @ A @ F3 @ ( topolo7230453075368039082e_nhds @ A @ A3 ) @ ( at_top @ nat ) ) )
=> ( eventually @ nat
@ ^ [N4: nat] : ( P @ ( F3 @ N4 ) )
@ ( at_top @ nat ) ) )
=> ( eventually @ A @ P @ ( topolo174197925503356063within @ A @ A3 @ ( top_top @ ( set @ A ) ) ) ) ) ) ).
% sequentially_imp_eventually_at
thf(fact_7537_sequentially__imp__eventually__within,axiom,
! [A: $tType] :
( ( topolo3112930676232923870pology @ A )
=> ! [S2: set @ A,A3: A,P: A > $o] :
( ! [F3: nat > A] :
( ( ! [N9: nat] :
( ( member @ A @ ( F3 @ N9 ) @ S2 )
& ( ( F3 @ N9 )
!= A3 ) )
& ( filterlim @ nat @ A @ F3 @ ( topolo7230453075368039082e_nhds @ A @ A3 ) @ ( at_top @ nat ) ) )
=> ( eventually @ nat
@ ^ [N4: nat] : ( P @ ( F3 @ N4 ) )
@ ( at_top @ nat ) ) )
=> ( eventually @ A @ P @ ( topolo174197925503356063within @ A @ A3 @ S2 ) ) ) ) ).
% sequentially_imp_eventually_within
thf(fact_7538_filterlim__at__within__not__equal,axiom,
! [B: $tType,A: $tType] :
( ( topological_t2_space @ B )
=> ! [F2: A > B,A3: B,S2: set @ B,F4: filter @ A,B3: B] :
( ( filterlim @ A @ B @ F2 @ ( topolo174197925503356063within @ B @ A3 @ S2 ) @ F4 )
=> ( eventually @ A
@ ^ [W2: A] :
( ( member @ B @ ( F2 @ W2 ) @ S2 )
& ( ( F2 @ W2 )
!= B3 ) )
@ F4 ) ) ) ).
% filterlim_at_within_not_equal
thf(fact_7539_eventually__Lim__ident__at,axiom,
! [A: $tType] :
( ( topological_t2_space @ A )
=> ! [P: A > A > $o,X2: A,X: set @ A] :
( ( eventually @ A
@ ( P
@ ( topolo3827282254853284352ce_Lim @ A @ A @ ( topolo174197925503356063within @ A @ X2 @ X )
@ ^ [X3: A] : X3 ) )
@ ( topolo174197925503356063within @ A @ X2 @ X ) )
= ( eventually @ A @ ( P @ X2 ) @ ( topolo174197925503356063within @ A @ X2 @ X ) ) ) ) ).
% eventually_Lim_ident_at
thf(fact_7540_trivial__limit__def,axiom,
! [A: $tType,F4: filter @ A] :
( ( F4
= ( bot_bot @ ( filter @ A ) ) )
= ( eventually @ A
@ ^ [X3: A] : $false
@ F4 ) ) ).
% trivial_limit_def
thf(fact_7541_eventually__const__iff,axiom,
! [A: $tType,P: $o,F4: filter @ A] :
( ( eventually @ A
@ ^ [X3: A] : P
@ F4 )
= ( P
| ( F4
= ( bot_bot @ ( filter @ A ) ) ) ) ) ).
% eventually_const_iff
thf(fact_7542_False__imp__not__eventually,axiom,
! [A: $tType,P: A > $o,Net: filter @ A] :
( ! [X4: A] :
~ ( P @ X4 )
=> ( ( Net
!= ( bot_bot @ ( filter @ A ) ) )
=> ~ ( eventually @ A @ P @ Net ) ) ) ).
% False_imp_not_eventually
thf(fact_7543_filter__leD,axiom,
! [A: $tType,F4: filter @ A,F6: filter @ A,P: A > $o] :
( ( ord_less_eq @ ( filter @ A ) @ F4 @ F6 )
=> ( ( eventually @ A @ P @ F6 )
=> ( eventually @ A @ P @ F4 ) ) ) ).
% filter_leD
thf(fact_7544_filter__leI,axiom,
! [A: $tType,F6: filter @ A,F4: filter @ A] :
( ! [P9: A > $o] :
( ( eventually @ A @ P9 @ F6 )
=> ( eventually @ A @ P9 @ F4 ) )
=> ( ord_less_eq @ ( filter @ A ) @ F4 @ F6 ) ) ).
% filter_leI
thf(fact_7545_le__filter__def,axiom,
! [A: $tType] :
( ( ord_less_eq @ ( filter @ A ) )
= ( ^ [F9: filter @ A,F10: filter @ A] :
! [P3: A > $o] :
( ( eventually @ A @ P3 @ F10 )
=> ( eventually @ A @ P3 @ F9 ) ) ) ) ).
% le_filter_def
thf(fact_7546_eventually__frequently__const__simps_I6_J,axiom,
! [A: $tType,C2: $o,P: A > $o,F4: filter @ A] :
( ( eventually @ A
@ ^ [X3: A] :
( C2
=> ( P @ X3 ) )
@ F4 )
= ( C2
=> ( eventually @ A @ P @ F4 ) ) ) ).
% eventually_frequently_const_simps(6)
thf(fact_7547_eventually__frequently__const__simps_I4_J,axiom,
! [A: $tType,C2: $o,P: A > $o,F4: filter @ A] :
( ( eventually @ A
@ ^ [X3: A] :
( C2
| ( P @ X3 ) )
@ F4 )
= ( C2
| ( eventually @ A @ P @ F4 ) ) ) ).
% eventually_frequently_const_simps(4)
thf(fact_7548_eventually__frequently__const__simps_I3_J,axiom,
! [A: $tType,P: A > $o,C2: $o,F4: filter @ A] :
( ( eventually @ A
@ ^ [X3: A] :
( ( P @ X3 )
| C2 )
@ F4 )
= ( ( eventually @ A @ P @ F4 )
| C2 ) ) ).
% eventually_frequently_const_simps(3)
thf(fact_7549_eventually__mp,axiom,
! [A: $tType,P: A > $o,Q: A > $o,F4: filter @ A] :
( ( eventually @ A
@ ^ [X3: A] :
( ( P @ X3 )
=> ( Q @ X3 ) )
@ F4 )
=> ( ( eventually @ A @ P @ F4 )
=> ( eventually @ A @ Q @ F4 ) ) ) ).
% eventually_mp
thf(fact_7550_eventually__True,axiom,
! [A: $tType,F4: filter @ A] :
( eventually @ A
@ ^ [X3: A] : $true
@ F4 ) ).
% eventually_True
thf(fact_7551_eventually__conj,axiom,
! [A: $tType,P: A > $o,F4: filter @ A,Q: A > $o] :
( ( eventually @ A @ P @ F4 )
=> ( ( eventually @ A @ Q @ F4 )
=> ( eventually @ A
@ ^ [X3: A] :
( ( P @ X3 )
& ( Q @ X3 ) )
@ F4 ) ) ) ).
% eventually_conj
thf(fact_7552_eventually__elim2,axiom,
! [A: $tType,P: A > $o,F4: filter @ A,Q: A > $o,R: A > $o] :
( ( eventually @ A @ P @ F4 )
=> ( ( eventually @ A @ Q @ F4 )
=> ( ! [I2: A] :
( ( P @ I2 )
=> ( ( Q @ I2 )
=> ( R @ I2 ) ) )
=> ( eventually @ A @ R @ F4 ) ) ) ) ).
% eventually_elim2
thf(fact_7553_eventually__subst,axiom,
! [A: $tType,P: A > $o,Q: A > $o,F4: filter @ A] :
( ( eventually @ A
@ ^ [N4: A] :
( ( P @ N4 )
= ( Q @ N4 ) )
@ F4 )
=> ( ( eventually @ A @ P @ F4 )
= ( eventually @ A @ Q @ F4 ) ) ) ).
% eventually_subst
thf(fact_7554_eventually__rev__mp,axiom,
! [A: $tType,P: A > $o,F4: filter @ A,Q: A > $o] :
( ( eventually @ A @ P @ F4 )
=> ( ( eventually @ A
@ ^ [X3: A] :
( ( P @ X3 )
=> ( Q @ X3 ) )
@ F4 )
=> ( eventually @ A @ Q @ F4 ) ) ) ).
% eventually_rev_mp
thf(fact_7555_eventually__conj__iff,axiom,
! [A: $tType,P: A > $o,Q: A > $o,F4: filter @ A] :
( ( eventually @ A
@ ^ [X3: A] :
( ( P @ X3 )
& ( Q @ X3 ) )
@ F4 )
= ( ( eventually @ A @ P @ F4 )
& ( eventually @ A @ Q @ F4 ) ) ) ).
% eventually_conj_iff
thf(fact_7556_not__eventually__impI,axiom,
! [A: $tType,P: A > $o,F4: filter @ A,Q: A > $o] :
( ( eventually @ A @ P @ F4 )
=> ( ~ ( eventually @ A @ Q @ F4 )
=> ~ ( eventually @ A
@ ^ [X3: A] :
( ( P @ X3 )
=> ( Q @ X3 ) )
@ F4 ) ) ) ).
% not_eventually_impI
thf(fact_7557_has__derivative__transform__eventually,axiom,
! [B: $tType,A: $tType] :
( ( ( real_V822414075346904944vector @ A )
& ( real_V822414075346904944vector @ B ) )
=> ! [F2: A > B,F7: A > B,X2: A,S2: set @ A,G: A > B] :
( ( has_derivative @ A @ B @ F2 @ F7 @ ( topolo174197925503356063within @ A @ X2 @ S2 ) )
=> ( ( eventually @ A
@ ^ [X10: A] :
( ( F2 @ X10 )
= ( G @ X10 ) )
@ ( topolo174197925503356063within @ A @ X2 @ S2 ) )
=> ( ( ( F2 @ X2 )
= ( G @ X2 ) )
=> ( ( member @ A @ X2 @ S2 )
=> ( has_derivative @ A @ B @ G @ F7 @ ( topolo174197925503356063within @ A @ X2 @ S2 ) ) ) ) ) ) ) ).
% has_derivative_transform_eventually
thf(fact_7558_has__field__derivative__cong__eventually,axiom,
! [A: $tType] :
( ( real_V3459762299906320749_field @ A )
=> ! [F2: A > A,G: A > A,X2: A,S: set @ A,U: A] :
( ( eventually @ A
@ ^ [X3: A] :
( ( F2 @ X3 )
= ( G @ X3 ) )
@ ( topolo174197925503356063within @ A @ X2 @ S ) )
=> ( ( ( F2 @ X2 )
= ( G @ X2 ) )
=> ( ( has_field_derivative @ A @ F2 @ U @ ( topolo174197925503356063within @ A @ X2 @ S ) )
= ( has_field_derivative @ A @ G @ U @ ( topolo174197925503356063within @ A @ X2 @ S ) ) ) ) ) ) ).
% has_field_derivative_cong_eventually
thf(fact_7559_t1__space__nhds,axiom,
! [A: $tType] :
( ( topological_t1_space @ A )
=> ! [X2: A,Y3: A] :
( ( X2 != Y3 )
=> ( eventually @ A
@ ^ [X3: A] : X3 != Y3
@ ( topolo7230453075368039082e_nhds @ A @ X2 ) ) ) ) ).
% t1_space_nhds
thf(fact_7560_eventually__eventually,axiom,
! [A: $tType] :
( ( topolo4958980785337419405_space @ A )
=> ! [P: A > $o,X2: A] :
( ( eventually @ A
@ ^ [Y: A] : ( eventually @ A @ P @ ( topolo7230453075368039082e_nhds @ A @ Y ) )
@ ( topolo7230453075368039082e_nhds @ A @ X2 ) )
= ( eventually @ A @ P @ ( topolo7230453075368039082e_nhds @ A @ X2 ) ) ) ) ).
% eventually_eventually
thf(fact_7561_eventually__at__filter,axiom,
! [A: $tType] :
( ( topolo4958980785337419405_space @ A )
=> ! [P: A > $o,A3: A,S2: set @ A] :
( ( eventually @ A @ P @ ( topolo174197925503356063within @ A @ A3 @ S2 ) )
= ( eventually @ A
@ ^ [X3: A] :
( ( X3 != A3 )
=> ( ( member @ A @ X3 @ S2 )
=> ( P @ X3 ) ) )
@ ( topolo7230453075368039082e_nhds @ A @ A3 ) ) ) ) ).
% eventually_at_filter
thf(fact_7562_eventually__nhds__in__open,axiom,
! [A: $tType] :
( ( topolo4958980785337419405_space @ A )
=> ! [S2: set @ A,X2: A] :
( ( topolo1002775350975398744n_open @ A @ S2 )
=> ( ( member @ A @ X2 @ S2 )
=> ( eventually @ A
@ ^ [Y: A] : ( member @ A @ Y @ S2 )
@ ( topolo7230453075368039082e_nhds @ A @ X2 ) ) ) ) ) ).
% eventually_nhds_in_open
thf(fact_7563_eventually__nhds__top,axiom,
! [A: $tType] :
( ( ( order_top @ A )
& ( topolo1944317154257567458pology @ A ) )
=> ! [B3: A,P: A > $o] :
( ( ord_less @ A @ B3 @ ( top_top @ A ) )
=> ( ( eventually @ A @ P @ ( topolo7230453075368039082e_nhds @ A @ ( top_top @ A ) ) )
= ( ? [B7: A] :
( ( ord_less @ A @ B7 @ ( top_top @ A ) )
& ! [Z4: A] :
( ( ord_less @ A @ B7 @ Z4 )
=> ( P @ Z4 ) ) ) ) ) ) ) ).
% eventually_nhds_top
thf(fact_7564_eventually__at__bot__dense,axiom,
! [A: $tType] :
( ( ( linorder @ A )
& ( no_bot @ A ) )
=> ! [P: A > $o] :
( ( eventually @ A @ P @ ( at_bot @ A ) )
= ( ? [N8: A] :
! [N4: A] :
( ( ord_less @ A @ N4 @ N8 )
=> ( P @ N4 ) ) ) ) ) ).
% eventually_at_bot_dense
thf(fact_7565_eventually__gt__at__bot,axiom,
! [A: $tType] :
( ( unboun7993243217541854897norder @ A )
=> ! [C3: A] :
( eventually @ A
@ ^ [X3: A] : ( ord_less @ A @ X3 @ C3 )
@ ( at_bot @ A ) ) ) ).
% eventually_gt_at_bot
thf(fact_7566_eventually__at__bot__not__equal,axiom,
! [A: $tType] :
( ( ( linorder @ A )
& ( no_bot @ A ) )
=> ! [C3: A] :
( eventually @ A
@ ^ [X3: A] : X3 != C3
@ ( at_bot @ A ) ) ) ).
% eventually_at_bot_not_equal
thf(fact_7567_eventually__at__bot__linorder,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [P: A > $o] :
( ( eventually @ A @ P @ ( at_bot @ A ) )
= ( ? [N8: A] :
! [N4: A] :
( ( ord_less_eq @ A @ N4 @ N8 )
=> ( P @ N4 ) ) ) ) ) ).
% eventually_at_bot_linorder
thf(fact_7568_eventually__le__at__bot,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [C3: A] :
( eventually @ A
@ ^ [X3: A] : ( ord_less_eq @ A @ X3 @ C3 )
@ ( at_bot @ A ) ) ) ).
% eventually_le_at_bot
thf(fact_7569_tendsto__sandwich,axiom,
! [A: $tType,B: $tType] :
( ( topolo2564578578187576103pology @ A )
=> ! [F2: B > A,G: B > A,Net: filter @ B,H2: B > A,C3: A] :
( ( eventually @ B
@ ^ [N4: B] : ( ord_less_eq @ A @ ( F2 @ N4 ) @ ( G @ N4 ) )
@ Net )
=> ( ( eventually @ B
@ ^ [N4: B] : ( ord_less_eq @ A @ ( G @ N4 ) @ ( H2 @ N4 ) )
@ Net )
=> ( ( filterlim @ B @ A @ F2 @ ( topolo7230453075368039082e_nhds @ A @ C3 ) @ Net )
=> ( ( filterlim @ B @ A @ H2 @ ( topolo7230453075368039082e_nhds @ A @ C3 ) @ Net )
=> ( filterlim @ B @ A @ G @ ( topolo7230453075368039082e_nhds @ A @ C3 ) @ Net ) ) ) ) ) ) ).
% tendsto_sandwich
thf(fact_7570_order__tendsto__iff,axiom,
! [B: $tType,A: $tType] :
( ( topolo2564578578187576103pology @ A )
=> ! [F2: B > A,X2: A,F4: filter @ B] :
( ( filterlim @ B @ A @ F2 @ ( topolo7230453075368039082e_nhds @ A @ X2 ) @ F4 )
= ( ! [L3: A] :
( ( ord_less @ A @ L3 @ X2 )
=> ( eventually @ B
@ ^ [X3: B] : ( ord_less @ A @ L3 @ ( F2 @ X3 ) )
@ F4 ) )
& ! [U2: A] :
( ( ord_less @ A @ X2 @ U2 )
=> ( eventually @ B
@ ^ [X3: B] : ( ord_less @ A @ ( F2 @ X3 ) @ U2 )
@ F4 ) ) ) ) ) ).
% order_tendsto_iff
thf(fact_7571_order__tendstoI,axiom,
! [A: $tType,B: $tType] :
( ( topolo2564578578187576103pology @ A )
=> ! [Y3: A,F2: B > A,F4: filter @ B] :
( ! [A4: A] :
( ( ord_less @ A @ A4 @ Y3 )
=> ( eventually @ B
@ ^ [X3: B] : ( ord_less @ A @ A4 @ ( F2 @ X3 ) )
@ F4 ) )
=> ( ! [A4: A] :
( ( ord_less @ A @ Y3 @ A4 )
=> ( eventually @ B
@ ^ [X3: B] : ( ord_less @ A @ ( F2 @ X3 ) @ A4 )
@ F4 ) )
=> ( filterlim @ B @ A @ F2 @ ( topolo7230453075368039082e_nhds @ A @ Y3 ) @ F4 ) ) ) ) ).
% order_tendstoI
thf(fact_7572_order__tendstoD_I1_J,axiom,
! [A: $tType,B: $tType] :
( ( topolo2564578578187576103pology @ A )
=> ! [F2: B > A,Y3: A,F4: filter @ B,A3: A] :
( ( filterlim @ B @ A @ F2 @ ( topolo7230453075368039082e_nhds @ A @ Y3 ) @ F4 )
=> ( ( ord_less @ A @ A3 @ Y3 )
=> ( eventually @ B
@ ^ [X3: B] : ( ord_less @ A @ A3 @ ( F2 @ X3 ) )
@ F4 ) ) ) ) ).
% order_tendstoD(1)
thf(fact_7573_order__tendstoD_I2_J,axiom,
! [A: $tType,B: $tType] :
( ( topolo2564578578187576103pology @ A )
=> ! [F2: B > A,Y3: A,F4: filter @ B,A3: A] :
( ( filterlim @ B @ A @ F2 @ ( topolo7230453075368039082e_nhds @ A @ Y3 ) @ F4 )
=> ( ( ord_less @ A @ Y3 @ A3 )
=> ( eventually @ B
@ ^ [X3: B] : ( ord_less @ A @ ( F2 @ X3 ) @ A3 )
@ F4 ) ) ) ) ).
% order_tendstoD(2)
thf(fact_7574_filterlim__at__top__mono,axiom,
! [A: $tType,B: $tType] :
( ( linorder @ A )
=> ! [F2: B > A,F4: filter @ B,G: B > A] :
( ( filterlim @ B @ A @ F2 @ ( at_top @ A ) @ F4 )
=> ( ( eventually @ B
@ ^ [X3: B] : ( ord_less_eq @ A @ ( F2 @ X3 ) @ ( G @ X3 ) )
@ F4 )
=> ( filterlim @ B @ A @ G @ ( at_top @ A ) @ F4 ) ) ) ) ).
% filterlim_at_top_mono
thf(fact_7575_filterlim__at__top__ge,axiom,
! [A: $tType,B: $tType] :
( ( linorder @ B )
=> ! [F2: A > B,F4: filter @ A,C3: B] :
( ( filterlim @ A @ B @ F2 @ ( at_top @ B ) @ F4 )
= ( ! [Z10: B] :
( ( ord_less_eq @ B @ C3 @ Z10 )
=> ( eventually @ A
@ ^ [X3: A] : ( ord_less_eq @ B @ Z10 @ ( F2 @ X3 ) )
@ F4 ) ) ) ) ) ).
% filterlim_at_top_ge
thf(fact_7576_filterlim__at__top,axiom,
! [A: $tType,B: $tType] :
( ( linorder @ B )
=> ! [F2: A > B,F4: filter @ A] :
( ( filterlim @ A @ B @ F2 @ ( at_top @ B ) @ F4 )
= ( ! [Z10: B] :
( eventually @ A
@ ^ [X3: A] : ( ord_less_eq @ B @ Z10 @ ( F2 @ X3 ) )
@ F4 ) ) ) ) ).
% filterlim_at_top
thf(fact_7577_filterlim__at__top__dense,axiom,
! [A: $tType,B: $tType] :
( ( unboun7993243217541854897norder @ B )
=> ! [F2: A > B,F4: filter @ A] :
( ( filterlim @ A @ B @ F2 @ ( at_top @ B ) @ F4 )
= ( ! [Z10: B] :
( eventually @ A
@ ^ [X3: A] : ( ord_less @ B @ Z10 @ ( F2 @ X3 ) )
@ F4 ) ) ) ) ).
% filterlim_at_top_dense
thf(fact_7578_filterlim__at,axiom,
! [A: $tType,B: $tType] :
( ( topolo4958980785337419405_space @ A )
=> ! [F2: B > A,B3: A,S2: set @ A,F4: filter @ B] :
( ( filterlim @ B @ A @ F2 @ ( topolo174197925503356063within @ A @ B3 @ S2 ) @ F4 )
= ( ( eventually @ B
@ ^ [X3: B] :
( ( member @ A @ ( F2 @ X3 ) @ S2 )
& ( ( F2 @ X3 )
!= B3 ) )
@ F4 )
& ( filterlim @ B @ A @ F2 @ ( topolo7230453075368039082e_nhds @ A @ B3 ) @ F4 ) ) ) ) ).
% filterlim_at
thf(fact_7579_eventually__at__right__less,axiom,
! [A: $tType] :
( ( ( no_top @ A )
& ( topolo1944317154257567458pology @ A ) )
=> ! [X2: A] : ( eventually @ A @ ( ord_less @ A @ X2 ) @ ( topolo174197925503356063within @ A @ X2 @ ( set_ord_greaterThan @ A @ X2 ) ) ) ) ).
% eventually_at_right_less
thf(fact_7580_has__field__derivative__cong__ev,axiom,
! [A: $tType] :
( ( real_V3459762299906320749_field @ A )
=> ! [X2: A,Y3: A,S: set @ A,F2: A > A,G: A > A,U: A,V: A,T2: set @ A] :
( ( X2 = Y3 )
=> ( ( eventually @ A
@ ^ [X3: A] :
( ( member @ A @ X3 @ S )
=> ( ( F2 @ X3 )
= ( G @ X3 ) ) )
@ ( topolo7230453075368039082e_nhds @ A @ X2 ) )
=> ( ( U = V )
=> ( ( S = T2 )
=> ( ( member @ A @ X2 @ S )
=> ( ( has_field_derivative @ A @ F2 @ U @ ( topolo174197925503356063within @ A @ X2 @ S ) )
= ( has_field_derivative @ A @ G @ V @ ( topolo174197925503356063within @ A @ Y3 @ T2 ) ) ) ) ) ) ) ) ) ).
% has_field_derivative_cong_ev
thf(fact_7581_topological__tendstoI,axiom,
! [A: $tType,B: $tType] :
( ( topolo4958980785337419405_space @ A )
=> ! [L: A,F2: B > A,F4: filter @ B] :
( ! [S6: set @ A] :
( ( topolo1002775350975398744n_open @ A @ S6 )
=> ( ( member @ A @ L @ S6 )
=> ( eventually @ B
@ ^ [X3: B] : ( member @ A @ ( F2 @ X3 ) @ S6 )
@ F4 ) ) )
=> ( filterlim @ B @ A @ F2 @ ( topolo7230453075368039082e_nhds @ A @ L ) @ F4 ) ) ) ).
% topological_tendstoI
thf(fact_7582_topological__tendstoD,axiom,
! [A: $tType,B: $tType] :
( ( topolo4958980785337419405_space @ A )
=> ! [F2: B > A,L: A,F4: filter @ B,S: set @ A] :
( ( filterlim @ B @ A @ F2 @ ( topolo7230453075368039082e_nhds @ A @ L ) @ F4 )
=> ( ( topolo1002775350975398744n_open @ A @ S )
=> ( ( member @ A @ L @ S )
=> ( eventually @ B
@ ^ [X3: B] : ( member @ A @ ( F2 @ X3 ) @ S )
@ F4 ) ) ) ) ) ).
% topological_tendstoD
thf(fact_7583_tendsto__def,axiom,
! [B: $tType,A: $tType] :
( ( topolo4958980785337419405_space @ A )
=> ! [F2: B > A,L: A,F4: filter @ B] :
( ( filterlim @ B @ A @ F2 @ ( topolo7230453075368039082e_nhds @ A @ L ) @ F4 )
= ( ! [S8: set @ A] :
( ( topolo1002775350975398744n_open @ A @ S8 )
=> ( ( member @ A @ L @ S8 )
=> ( eventually @ B
@ ^ [X3: B] : ( member @ A @ ( F2 @ X3 ) @ S8 )
@ F4 ) ) ) ) ) ) ).
% tendsto_def
thf(fact_7584_filterlim__at__bot,axiom,
! [A: $tType,B: $tType] :
( ( linorder @ B )
=> ! [F2: A > B,F4: filter @ A] :
( ( filterlim @ A @ B @ F2 @ ( at_bot @ B ) @ F4 )
= ( ! [Z10: B] :
( eventually @ A
@ ^ [X3: A] : ( ord_less_eq @ B @ ( F2 @ X3 ) @ Z10 )
@ F4 ) ) ) ) ).
% filterlim_at_bot
thf(fact_7585_filterlim__at__bot__le,axiom,
! [A: $tType,B: $tType] :
( ( linorder @ B )
=> ! [F2: A > B,F4: filter @ A,C3: B] :
( ( filterlim @ A @ B @ F2 @ ( at_bot @ B ) @ F4 )
= ( ! [Z10: B] :
( ( ord_less_eq @ B @ Z10 @ C3 )
=> ( eventually @ A
@ ^ [X3: A] : ( ord_less_eq @ B @ ( F2 @ X3 ) @ Z10 )
@ F4 ) ) ) ) ) ).
% filterlim_at_bot_le
thf(fact_7586_filterlim__at__bot__dense,axiom,
! [A: $tType,B: $tType] :
( ( ( dense_linorder @ B )
& ( no_bot @ B ) )
=> ! [F2: A > B,F4: filter @ A] :
( ( filterlim @ A @ B @ F2 @ ( at_bot @ B ) @ F4 )
= ( ! [Z10: B] :
( eventually @ A
@ ^ [X3: A] : ( ord_less @ B @ ( F2 @ X3 ) @ Z10 )
@ F4 ) ) ) ) ).
% filterlim_at_bot_dense
thf(fact_7587_real__tendsto__sandwich,axiom,
! [B: $tType,F2: B > real,G: B > real,Net: filter @ B,H2: B > real,C3: real] :
( ( eventually @ B
@ ^ [N4: B] : ( ord_less_eq @ real @ ( F2 @ N4 ) @ ( G @ N4 ) )
@ Net )
=> ( ( eventually @ B
@ ^ [N4: B] : ( ord_less_eq @ real @ ( G @ N4 ) @ ( H2 @ N4 ) )
@ Net )
=> ( ( filterlim @ B @ real @ F2 @ ( topolo7230453075368039082e_nhds @ real @ C3 ) @ Net )
=> ( ( filterlim @ B @ real @ H2 @ ( topolo7230453075368039082e_nhds @ real @ C3 ) @ Net )
=> ( filterlim @ B @ real @ G @ ( topolo7230453075368039082e_nhds @ real @ C3 ) @ Net ) ) ) ) ) ).
% real_tendsto_sandwich
thf(fact_7588_countable__basis__at__decseq,axiom,
! [A: $tType] :
( ( topolo3112930676232923870pology @ A )
=> ! [X2: A] :
~ ! [A9: nat > ( set @ A )] :
( ! [I5: nat] : ( topolo1002775350975398744n_open @ A @ ( A9 @ I5 ) )
=> ( ! [I5: nat] : ( member @ A @ X2 @ ( A9 @ I5 ) )
=> ~ ! [S10: set @ A] :
( ( topolo1002775350975398744n_open @ A @ S10 )
=> ( ( member @ A @ X2 @ S10 )
=> ( eventually @ nat
@ ^ [I4: nat] : ( ord_less_eq @ ( set @ A ) @ ( A9 @ I4 ) @ S10 )
@ ( at_top @ nat ) ) ) ) ) ) ) ).
% countable_basis_at_decseq
thf(fact_7589_eventually__at__leftI,axiom,
! [A: $tType] :
( ( topolo2564578578187576103pology @ A )
=> ! [A3: A,B3: A,P: A > $o] :
( ! [X4: A] :
( ( member @ A @ X4 @ ( set_or5935395276787703475ssThan @ A @ A3 @ B3 ) )
=> ( P @ X4 ) )
=> ( ( ord_less @ A @ A3 @ B3 )
=> ( eventually @ A @ P @ ( topolo174197925503356063within @ A @ B3 @ ( set_ord_lessThan @ A @ B3 ) ) ) ) ) ) ).
% eventually_at_leftI
thf(fact_7590_eventually__at__rightI,axiom,
! [A: $tType] :
( ( topolo2564578578187576103pology @ A )
=> ! [A3: A,B3: A,P: A > $o] :
( ! [X4: A] :
( ( member @ A @ X4 @ ( set_or5935395276787703475ssThan @ A @ A3 @ B3 ) )
=> ( P @ X4 ) )
=> ( ( ord_less @ A @ A3 @ B3 )
=> ( eventually @ A @ P @ ( topolo174197925503356063within @ A @ A3 @ ( set_ord_greaterThan @ A @ A3 ) ) ) ) ) ) ).
% eventually_at_rightI
thf(fact_7591_eventually__at__to__0,axiom,
! [A: $tType] :
( ( real_V822414075346904944vector @ A )
=> ! [P: A > $o,A3: A] :
( ( eventually @ A @ P @ ( topolo174197925503356063within @ A @ A3 @ ( top_top @ ( set @ A ) ) ) )
= ( eventually @ A
@ ^ [X3: A] : ( P @ ( plus_plus @ A @ X3 @ A3 ) )
@ ( topolo174197925503356063within @ A @ ( zero_zero @ A ) @ ( top_top @ ( set @ A ) ) ) ) ) ) ).
% eventually_at_to_0
thf(fact_7592_increasing__tendsto,axiom,
! [A: $tType,B: $tType] :
( ( topolo2564578578187576103pology @ A )
=> ! [F2: B > A,L: A,F4: filter @ B] :
( ( eventually @ B
@ ^ [N4: B] : ( ord_less_eq @ A @ ( F2 @ N4 ) @ L )
@ F4 )
=> ( ! [X4: A] :
( ( ord_less @ A @ X4 @ L )
=> ( eventually @ B
@ ^ [N4: B] : ( ord_less @ A @ X4 @ ( F2 @ N4 ) )
@ F4 ) )
=> ( filterlim @ B @ A @ F2 @ ( topolo7230453075368039082e_nhds @ A @ L ) @ F4 ) ) ) ) ).
% increasing_tendsto
thf(fact_7593_decreasing__tendsto,axiom,
! [A: $tType,B: $tType] :
( ( topolo2564578578187576103pology @ A )
=> ! [L: A,F2: B > A,F4: filter @ B] :
( ( eventually @ B
@ ^ [N4: B] : ( ord_less_eq @ A @ L @ ( F2 @ N4 ) )
@ F4 )
=> ( ! [X4: A] :
( ( ord_less @ A @ L @ X4 )
=> ( eventually @ B
@ ^ [N4: B] : ( ord_less @ A @ ( F2 @ N4 ) @ X4 )
@ F4 ) )
=> ( filterlim @ B @ A @ F2 @ ( topolo7230453075368039082e_nhds @ A @ L ) @ F4 ) ) ) ) ).
% decreasing_tendsto
thf(fact_7594_filterlim__at__top__gt,axiom,
! [A: $tType,B: $tType] :
( ( unboun7993243217541854897norder @ B )
=> ! [F2: A > B,F4: filter @ A,C3: B] :
( ( filterlim @ A @ B @ F2 @ ( at_top @ B ) @ F4 )
= ( ! [Z10: B] :
( ( ord_less @ B @ C3 @ Z10 )
=> ( eventually @ A
@ ^ [X3: A] : ( ord_less_eq @ B @ Z10 @ ( F2 @ X3 ) )
@ F4 ) ) ) ) ) ).
% filterlim_at_top_gt
thf(fact_7595_filterlim__atI,axiom,
! [A: $tType,B: $tType] :
( ( topolo4958980785337419405_space @ A )
=> ! [F2: B > A,C3: A,F4: filter @ B] :
( ( filterlim @ B @ A @ F2 @ ( topolo7230453075368039082e_nhds @ A @ C3 ) @ F4 )
=> ( ( eventually @ B
@ ^ [X3: B] :
( ( F2 @ X3 )
!= C3 )
@ F4 )
=> ( filterlim @ B @ A @ F2 @ ( topolo174197925503356063within @ A @ C3 @ ( top_top @ ( set @ A ) ) ) @ F4 ) ) ) ) ).
% filterlim_atI
thf(fact_7596_LIM__compose__eventually,axiom,
! [C: $tType,B: $tType,A: $tType] :
( ( ( topolo4958980785337419405_space @ A )
& ( topolo4958980785337419405_space @ B )
& ( topolo4958980785337419405_space @ C ) )
=> ! [F2: A > B,B3: B,A3: A,G: B > C,C3: C] :
( ( filterlim @ A @ B @ F2 @ ( topolo7230453075368039082e_nhds @ B @ B3 ) @ ( topolo174197925503356063within @ A @ A3 @ ( top_top @ ( set @ A ) ) ) )
=> ( ( filterlim @ B @ C @ G @ ( topolo7230453075368039082e_nhds @ C @ C3 ) @ ( topolo174197925503356063within @ B @ B3 @ ( top_top @ ( set @ B ) ) ) )
=> ( ( eventually @ A
@ ^ [X3: A] :
( ( F2 @ X3 )
!= B3 )
@ ( topolo174197925503356063within @ A @ A3 @ ( top_top @ ( set @ A ) ) ) )
=> ( filterlim @ A @ C
@ ^ [X3: A] : ( G @ ( F2 @ X3 ) )
@ ( topolo7230453075368039082e_nhds @ C @ C3 )
@ ( topolo174197925503356063within @ A @ A3 @ ( top_top @ ( set @ A ) ) ) ) ) ) ) ) ).
% LIM_compose_eventually
thf(fact_7597_tendsto__compose__eventually,axiom,
! [B: $tType,A: $tType,C: $tType] :
( ( ( topolo4958980785337419405_space @ A )
& ( topolo4958980785337419405_space @ B ) )
=> ! [G: A > B,M: B,L: A,F2: C > A,F4: filter @ C] :
( ( filterlim @ A @ B @ G @ ( topolo7230453075368039082e_nhds @ B @ M ) @ ( topolo174197925503356063within @ A @ L @ ( top_top @ ( set @ A ) ) ) )
=> ( ( filterlim @ C @ A @ F2 @ ( topolo7230453075368039082e_nhds @ A @ L ) @ F4 )
=> ( ( eventually @ C
@ ^ [X3: C] :
( ( F2 @ X3 )
!= L )
@ F4 )
=> ( filterlim @ C @ B
@ ^ [X3: C] : ( G @ ( F2 @ X3 ) )
@ ( topolo7230453075368039082e_nhds @ B @ M )
@ F4 ) ) ) ) ) ).
% tendsto_compose_eventually
thf(fact_7598_isCont__cong,axiom,
! [B: $tType,A: $tType] :
( ( ( topological_t2_space @ A )
& ( topolo4958980785337419405_space @ B ) )
=> ! [F2: A > B,G: A > B,X2: A] :
( ( eventually @ A
@ ^ [X3: A] :
( ( F2 @ X3 )
= ( G @ X3 ) )
@ ( topolo7230453075368039082e_nhds @ A @ X2 ) )
=> ( ( topolo3448309680560233919inuous @ A @ B @ ( topolo174197925503356063within @ A @ X2 @ ( top_top @ ( set @ A ) ) ) @ F2 )
= ( topolo3448309680560233919inuous @ A @ B @ ( topolo174197925503356063within @ A @ X2 @ ( top_top @ ( set @ A ) ) ) @ G ) ) ) ) ).
% isCont_cong
thf(fact_7599_DERIV__cong__ev,axiom,
! [A: $tType] :
( ( real_V3459762299906320749_field @ A )
=> ! [X2: A,Y3: A,F2: A > A,G: A > A,U: A,V: A] :
( ( X2 = Y3 )
=> ( ( eventually @ A
@ ^ [X3: A] :
( ( F2 @ X3 )
= ( G @ X3 ) )
@ ( topolo7230453075368039082e_nhds @ A @ X2 ) )
=> ( ( U = V )
=> ( ( has_field_derivative @ A @ F2 @ U @ ( topolo174197925503356063within @ A @ X2 @ ( top_top @ ( set @ A ) ) ) )
= ( has_field_derivative @ A @ G @ V @ ( topolo174197925503356063within @ A @ Y3 @ ( top_top @ ( set @ A ) ) ) ) ) ) ) ) ) ).
% DERIV_cong_ev
thf(fact_7600_filterlim__at__bot__lt,axiom,
! [A: $tType,B: $tType] :
( ( unboun7993243217541854897norder @ B )
=> ! [F2: A > B,F4: filter @ A,C3: B] :
( ( filterlim @ A @ B @ F2 @ ( at_bot @ B ) @ F4 )
= ( ! [Z10: B] :
( ( ord_less @ B @ Z10 @ C3 )
=> ( eventually @ A
@ ^ [X3: A] : ( ord_less_eq @ B @ ( F2 @ X3 ) @ Z10 )
@ F4 ) ) ) ) ) ).
% filterlim_at_bot_lt
thf(fact_7601_tendsto__upperbound,axiom,
! [B: $tType,A: $tType] :
( ( topolo1944317154257567458pology @ A )
=> ! [F2: B > A,X2: A,F4: filter @ B,A3: A] :
( ( filterlim @ B @ A @ F2 @ ( topolo7230453075368039082e_nhds @ A @ X2 ) @ F4 )
=> ( ( eventually @ B
@ ^ [I4: B] : ( ord_less_eq @ A @ ( F2 @ I4 ) @ A3 )
@ F4 )
=> ( ( F4
!= ( bot_bot @ ( filter @ B ) ) )
=> ( ord_less_eq @ A @ X2 @ A3 ) ) ) ) ) ).
% tendsto_upperbound
thf(fact_7602_tendsto__lowerbound,axiom,
! [B: $tType,A: $tType] :
( ( topolo1944317154257567458pology @ A )
=> ! [F2: B > A,X2: A,F4: filter @ B,A3: A] :
( ( filterlim @ B @ A @ F2 @ ( topolo7230453075368039082e_nhds @ A @ X2 ) @ F4 )
=> ( ( eventually @ B
@ ^ [I4: B] : ( ord_less_eq @ A @ A3 @ ( F2 @ I4 ) )
@ F4 )
=> ( ( F4
!= ( bot_bot @ ( filter @ B ) ) )
=> ( ord_less_eq @ A @ A3 @ X2 ) ) ) ) ) ).
% tendsto_lowerbound
thf(fact_7603_tendsto__le,axiom,
! [B: $tType,A: $tType] :
( ( topolo1944317154257567458pology @ A )
=> ! [F4: filter @ B,F2: B > A,X2: A,G: B > A,Y3: A] :
( ( F4
!= ( bot_bot @ ( filter @ B ) ) )
=> ( ( filterlim @ B @ A @ F2 @ ( topolo7230453075368039082e_nhds @ A @ X2 ) @ F4 )
=> ( ( filterlim @ B @ A @ G @ ( topolo7230453075368039082e_nhds @ A @ Y3 ) @ F4 )
=> ( ( eventually @ B
@ ^ [X3: B] : ( ord_less_eq @ A @ ( G @ X3 ) @ ( F2 @ X3 ) )
@ F4 )
=> ( ord_less_eq @ A @ Y3 @ X2 ) ) ) ) ) ) ).
% tendsto_le
thf(fact_7604_metric__tendsto__imp__tendsto,axiom,
! [A: $tType,B: $tType,C: $tType] :
( ( ( real_V7819770556892013058_space @ B )
& ( real_V7819770556892013058_space @ A ) )
=> ! [F2: C > A,A3: A,F4: filter @ C,G: C > B,B3: B] :
( ( filterlim @ C @ A @ F2 @ ( topolo7230453075368039082e_nhds @ A @ A3 ) @ F4 )
=> ( ( eventually @ C
@ ^ [X3: C] : ( ord_less_eq @ real @ ( real_V557655796197034286t_dist @ B @ ( G @ X3 ) @ B3 ) @ ( real_V557655796197034286t_dist @ A @ ( F2 @ X3 ) @ A3 ) )
@ F4 )
=> ( filterlim @ C @ B @ G @ ( topolo7230453075368039082e_nhds @ B @ B3 ) @ F4 ) ) ) ) ).
% metric_tendsto_imp_tendsto
thf(fact_7605_filterlim__at__infinity__imp__filterlim__at__top,axiom,
! [A: $tType,F2: A > real,F4: filter @ A] :
( ( filterlim @ A @ real @ F2 @ ( at_infinity @ real ) @ F4 )
=> ( ( eventually @ A
@ ^ [X3: A] : ( ord_less @ real @ ( zero_zero @ real ) @ ( F2 @ X3 ) )
@ F4 )
=> ( filterlim @ A @ real @ F2 @ ( at_top @ real ) @ F4 ) ) ) ).
% filterlim_at_infinity_imp_filterlim_at_top
thf(fact_7606_filterlim__at__infinity__imp__filterlim__at__bot,axiom,
! [A: $tType,F2: A > real,F4: filter @ A] :
( ( filterlim @ A @ real @ F2 @ ( at_infinity @ real ) @ F4 )
=> ( ( eventually @ A
@ ^ [X3: A] : ( ord_less @ real @ ( F2 @ X3 ) @ ( zero_zero @ real ) )
@ F4 )
=> ( filterlim @ A @ real @ F2 @ ( at_bot @ real ) @ F4 ) ) ) ).
% filterlim_at_infinity_imp_filterlim_at_bot
thf(fact_7607_eventually__floor__eq,axiom,
! [B: $tType,A: $tType] :
( ( ( archim2362893244070406136eiling @ B )
& ( topolo2564578578187576103pology @ B ) )
=> ! [F2: A > B,L: B,F4: filter @ A] :
( ( filterlim @ A @ B @ F2 @ ( topolo7230453075368039082e_nhds @ B @ L ) @ F4 )
=> ( ~ ( member @ B @ L @ ( ring_1_Ints @ B ) )
=> ( eventually @ A
@ ^ [X3: A] :
( ( archim6421214686448440834_floor @ B @ ( F2 @ X3 ) )
= ( archim6421214686448440834_floor @ B @ L ) )
@ F4 ) ) ) ) ).
% eventually_floor_eq
thf(fact_7608_eventually__ceiling__eq,axiom,
! [B: $tType,A: $tType] :
( ( ( archim2362893244070406136eiling @ B )
& ( topolo2564578578187576103pology @ B ) )
=> ! [F2: A > B,L: B,F4: filter @ A] :
( ( filterlim @ A @ B @ F2 @ ( topolo7230453075368039082e_nhds @ B @ L ) @ F4 )
=> ( ~ ( member @ B @ L @ ( ring_1_Ints @ B ) )
=> ( eventually @ A
@ ^ [X3: A] :
( ( archimedean_ceiling @ B @ ( F2 @ X3 ) )
= ( archimedean_ceiling @ B @ L ) )
@ F4 ) ) ) ) ).
% eventually_ceiling_eq
thf(fact_7609_eventually__at__right__to__0,axiom,
! [P: real > $o,A3: real] :
( ( eventually @ real @ P @ ( topolo174197925503356063within @ real @ A3 @ ( set_ord_greaterThan @ real @ A3 ) ) )
= ( eventually @ real
@ ^ [X3: real] : ( P @ ( plus_plus @ real @ X3 @ A3 ) )
@ ( topolo174197925503356063within @ real @ ( zero_zero @ real ) @ ( set_ord_greaterThan @ real @ ( zero_zero @ real ) ) ) ) ) ).
% eventually_at_right_to_0
thf(fact_7610_eventually__at__left__to__right,axiom,
! [P: real > $o,A3: real] :
( ( eventually @ real @ P @ ( topolo174197925503356063within @ real @ A3 @ ( set_ord_lessThan @ real @ A3 ) ) )
= ( eventually @ real
@ ^ [X3: real] : ( P @ ( uminus_uminus @ real @ X3 ) )
@ ( topolo174197925503356063within @ real @ ( uminus_uminus @ real @ A3 ) @ ( set_ord_greaterThan @ real @ ( uminus_uminus @ real @ A3 ) ) ) ) ) ).
% eventually_at_left_to_right
thf(fact_7611_continuous__arcosh__strong,axiom,
! [A: $tType] :
( ( topological_t2_space @ A )
=> ! [F4: filter @ A,F2: A > real] :
( ( topolo3448309680560233919inuous @ A @ real @ F4 @ F2 )
=> ( ( eventually @ A
@ ^ [X3: A] : ( ord_less_eq @ real @ ( one_one @ real ) @ ( F2 @ X3 ) )
@ F4 )
=> ( topolo3448309680560233919inuous @ A @ real @ F4
@ ^ [X3: A] : ( arcosh @ real @ ( F2 @ X3 ) ) ) ) ) ) ).
% continuous_arcosh_strong
thf(fact_7612_eventually__at__right__real,axiom,
! [A3: real,B3: real] :
( ( ord_less @ real @ A3 @ B3 )
=> ( eventually @ real
@ ^ [X3: real] : ( member @ real @ X3 @ ( set_or5935395276787703475ssThan @ real @ A3 @ B3 ) )
@ ( topolo174197925503356063within @ real @ A3 @ ( set_ord_greaterThan @ real @ A3 ) ) ) ) ).
% eventually_at_right_real
thf(fact_7613_eventually__at__left__real,axiom,
! [B3: real,A3: real] :
( ( ord_less @ real @ B3 @ A3 )
=> ( eventually @ real
@ ^ [X3: real] : ( member @ real @ X3 @ ( set_or5935395276787703475ssThan @ real @ B3 @ A3 ) )
@ ( topolo174197925503356063within @ real @ A3 @ ( set_ord_lessThan @ real @ A3 ) ) ) ) ).
% eventually_at_left_real
thf(fact_7614_eventually__at__le,axiom,
! [A: $tType] :
( ( real_V7819770556892013058_space @ A )
=> ! [P: A > $o,A3: A,S: set @ A] :
( ( eventually @ A @ P @ ( topolo174197925503356063within @ A @ A3 @ S ) )
= ( ? [D3: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ D3 )
& ! [X3: A] :
( ( member @ A @ X3 @ S )
=> ( ( ( X3 != A3 )
& ( ord_less_eq @ real @ ( real_V557655796197034286t_dist @ A @ X3 @ A3 ) @ D3 ) )
=> ( P @ X3 ) ) ) ) ) ) ) ).
% eventually_at_le
thf(fact_7615_eventually__at__infinity__pos,axiom,
! [A: $tType] :
( ( real_V822414075346904944vector @ A )
=> ! [P6: A > $o] :
( ( eventually @ A @ P6 @ ( at_infinity @ A ) )
= ( ? [B7: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ B7 )
& ! [X3: A] :
( ( ord_less_eq @ real @ B7 @ ( real_V7770717601297561774m_norm @ A @ X3 ) )
=> ( P6 @ X3 ) ) ) ) ) ) ).
% eventually_at_infinity_pos
thf(fact_7616_tendsto__imp__filterlim__at__left,axiom,
! [B: $tType,A: $tType] :
( ( topolo2564578578187576103pology @ B )
=> ! [F2: A > B,L6: B,F4: filter @ A] :
( ( filterlim @ A @ B @ F2 @ ( topolo7230453075368039082e_nhds @ B @ L6 ) @ F4 )
=> ( ( eventually @ A
@ ^ [X3: A] : ( ord_less @ B @ ( F2 @ X3 ) @ L6 )
@ F4 )
=> ( filterlim @ A @ B @ F2 @ ( topolo174197925503356063within @ B @ L6 @ ( set_ord_lessThan @ B @ L6 ) ) @ F4 ) ) ) ) ).
% tendsto_imp_filterlim_at_left
thf(fact_7617_tendsto__imp__filterlim__at__right,axiom,
! [B: $tType,A: $tType] :
( ( topolo2564578578187576103pology @ B )
=> ! [F2: A > B,L6: B,F4: filter @ A] :
( ( filterlim @ A @ B @ F2 @ ( topolo7230453075368039082e_nhds @ B @ L6 ) @ F4 )
=> ( ( eventually @ A
@ ^ [X3: A] : ( ord_less @ B @ L6 @ ( F2 @ X3 ) )
@ F4 )
=> ( filterlim @ A @ B @ F2 @ ( topolo174197925503356063within @ B @ L6 @ ( set_ord_greaterThan @ B @ L6 ) ) @ F4 ) ) ) ) ).
% tendsto_imp_filterlim_at_right
thf(fact_7618_tendsto__iff,axiom,
! [B: $tType,A: $tType] :
( ( real_V7819770556892013058_space @ A )
=> ! [F2: B > A,L: A,F4: filter @ B] :
( ( filterlim @ B @ A @ F2 @ ( topolo7230453075368039082e_nhds @ A @ L ) @ F4 )
= ( ! [E4: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ E4 )
=> ( eventually @ B
@ ^ [X3: B] : ( ord_less @ real @ ( real_V557655796197034286t_dist @ A @ ( F2 @ X3 ) @ L ) @ E4 )
@ F4 ) ) ) ) ) ).
% tendsto_iff
thf(fact_7619_tendstoI,axiom,
! [A: $tType,B: $tType] :
( ( real_V7819770556892013058_space @ A )
=> ! [F2: B > A,L: A,F4: filter @ B] :
( ! [E2: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ E2 )
=> ( eventually @ B
@ ^ [X3: B] : ( ord_less @ real @ ( real_V557655796197034286t_dist @ A @ ( F2 @ X3 ) @ L ) @ E2 )
@ F4 ) )
=> ( filterlim @ B @ A @ F2 @ ( topolo7230453075368039082e_nhds @ A @ L ) @ F4 ) ) ) ).
% tendstoI
thf(fact_7620_tendstoD,axiom,
! [A: $tType,B: $tType] :
( ( real_V7819770556892013058_space @ A )
=> ! [F2: B > A,L: A,F4: filter @ B,E: real] :
( ( filterlim @ B @ A @ F2 @ ( topolo7230453075368039082e_nhds @ A @ L ) @ F4 )
=> ( ( ord_less @ real @ ( zero_zero @ real ) @ E )
=> ( eventually @ B
@ ^ [X3: B] : ( ord_less @ real @ ( real_V557655796197034286t_dist @ A @ ( F2 @ X3 ) @ L ) @ E )
@ F4 ) ) ) ) ).
% tendstoD
thf(fact_7621_summable__comparison__test__ev,axiom,
! [A: $tType] :
( ( real_Vector_banach @ A )
=> ! [F2: nat > A,G: nat > real] :
( ( eventually @ nat
@ ^ [N4: nat] : ( ord_less_eq @ real @ ( real_V7770717601297561774m_norm @ A @ ( F2 @ N4 ) ) @ ( G @ N4 ) )
@ ( at_top @ nat ) )
=> ( ( summable @ real @ G )
=> ( summable @ A @ F2 ) ) ) ) ).
% summable_comparison_test_ev
thf(fact_7622_eventually__at__right__to__top,axiom,
! [P: real > $o] :
( ( eventually @ real @ P @ ( topolo174197925503356063within @ real @ ( zero_zero @ real ) @ ( set_ord_greaterThan @ real @ ( zero_zero @ real ) ) ) )
= ( eventually @ real
@ ^ [X3: real] : ( P @ ( inverse_inverse @ real @ X3 ) )
@ ( at_top @ real ) ) ) ).
% eventually_at_right_to_top
thf(fact_7623_eventually__at__top__to__right,axiom,
! [P: real > $o] :
( ( eventually @ real @ P @ ( at_top @ real ) )
= ( eventually @ real
@ ^ [X3: real] : ( P @ ( inverse_inverse @ real @ X3 ) )
@ ( topolo174197925503356063within @ real @ ( zero_zero @ real ) @ ( set_ord_greaterThan @ real @ ( zero_zero @ real ) ) ) ) ) ).
% eventually_at_top_to_right
thf(fact_7624_tendsto__arcosh__strong,axiom,
! [B: $tType,F2: B > real,A3: real,F4: filter @ B] :
( ( filterlim @ B @ real @ F2 @ ( topolo7230453075368039082e_nhds @ real @ A3 ) @ F4 )
=> ( ( ord_less_eq @ real @ ( one_one @ real ) @ A3 )
=> ( ( eventually @ B
@ ^ [X3: B] : ( ord_less_eq @ real @ ( one_one @ real ) @ ( F2 @ X3 ) )
@ F4 )
=> ( filterlim @ B @ real
@ ^ [X3: B] : ( arcosh @ real @ ( F2 @ X3 ) )
@ ( topolo7230453075368039082e_nhds @ real @ ( arcosh @ real @ A3 ) )
@ F4 ) ) ) ) ).
% tendsto_arcosh_strong
thf(fact_7625_filterlim__at__top__at__left,axiom,
! [B: $tType,A: $tType] :
( ( ( topolo1944317154257567458pology @ A )
& ( linorder @ B ) )
=> ! [Q: A > $o,F2: A > B,P: B > $o,G: B > A,A3: A] :
( ! [X4: A,Y4: A] :
( ( Q @ X4 )
=> ( ( Q @ Y4 )
=> ( ( ord_less_eq @ A @ X4 @ Y4 )
=> ( ord_less_eq @ B @ ( F2 @ X4 ) @ ( F2 @ Y4 ) ) ) ) )
=> ( ! [X4: B] :
( ( P @ X4 )
=> ( ( F2 @ ( G @ X4 ) )
= X4 ) )
=> ( ! [X4: B] :
( ( P @ X4 )
=> ( Q @ ( G @ X4 ) ) )
=> ( ( eventually @ A @ Q @ ( topolo174197925503356063within @ A @ A3 @ ( set_ord_lessThan @ A @ A3 ) ) )
=> ( ! [B4: A] :
( ( Q @ B4 )
=> ( ord_less @ A @ B4 @ A3 ) )
=> ( ( eventually @ B @ P @ ( at_top @ B ) )
=> ( filterlim @ A @ B @ F2 @ ( at_top @ B ) @ ( topolo174197925503356063within @ A @ A3 @ ( set_ord_lessThan @ A @ A3 ) ) ) ) ) ) ) ) ) ) ).
% filterlim_at_top_at_left
thf(fact_7626_filterlim__at__bot__at__right,axiom,
! [B: $tType,A: $tType] :
( ( ( topolo1944317154257567458pology @ A )
& ( linorder @ B ) )
=> ! [Q: A > $o,F2: A > B,P: B > $o,G: B > A,A3: A] :
( ! [X4: A,Y4: A] :
( ( Q @ X4 )
=> ( ( Q @ Y4 )
=> ( ( ord_less_eq @ A @ X4 @ Y4 )
=> ( ord_less_eq @ B @ ( F2 @ X4 ) @ ( F2 @ Y4 ) ) ) ) )
=> ( ! [X4: B] :
( ( P @ X4 )
=> ( ( F2 @ ( G @ X4 ) )
= X4 ) )
=> ( ! [X4: B] :
( ( P @ X4 )
=> ( Q @ ( G @ X4 ) ) )
=> ( ( eventually @ A @ Q @ ( topolo174197925503356063within @ A @ A3 @ ( set_ord_greaterThan @ A @ A3 ) ) )
=> ( ! [B4: A] :
( ( Q @ B4 )
=> ( ord_less @ A @ A3 @ B4 ) )
=> ( ( eventually @ B @ P @ ( at_bot @ B ) )
=> ( filterlim @ A @ B @ F2 @ ( at_bot @ B ) @ ( topolo174197925503356063within @ A @ A3 @ ( set_ord_greaterThan @ A @ A3 ) ) ) ) ) ) ) ) ) ) ).
% filterlim_at_bot_at_right
thf(fact_7627_filterlim__at__withinI,axiom,
! [A: $tType,B: $tType] :
( ( topolo4958980785337419405_space @ A )
=> ! [F2: B > A,C3: A,F4: filter @ B,A2: set @ A] :
( ( filterlim @ B @ A @ F2 @ ( topolo7230453075368039082e_nhds @ A @ C3 ) @ F4 )
=> ( ( eventually @ B
@ ^ [X3: B] : ( member @ A @ ( F2 @ X3 ) @ ( minus_minus @ ( set @ A ) @ A2 @ ( insert @ A @ C3 @ ( bot_bot @ ( set @ A ) ) ) ) )
@ F4 )
=> ( filterlim @ B @ A @ F2 @ ( topolo174197925503356063within @ A @ C3 @ A2 ) @ F4 ) ) ) ) ).
% filterlim_at_withinI
thf(fact_7628_tendsto__0__le,axiom,
! [B: $tType,C: $tType,A: $tType] :
( ( ( real_V822414075346904944vector @ C )
& ( real_V822414075346904944vector @ B ) )
=> ! [F2: A > B,F4: filter @ A,G: A > C,K6: real] :
( ( filterlim @ A @ B @ F2 @ ( topolo7230453075368039082e_nhds @ B @ ( zero_zero @ B ) ) @ F4 )
=> ( ( eventually @ A
@ ^ [X3: A] : ( ord_less_eq @ real @ ( real_V7770717601297561774m_norm @ C @ ( G @ X3 ) ) @ ( times_times @ real @ ( real_V7770717601297561774m_norm @ B @ ( F2 @ X3 ) ) @ K6 ) )
@ F4 )
=> ( filterlim @ A @ C @ G @ ( topolo7230453075368039082e_nhds @ C @ ( zero_zero @ C ) ) @ F4 ) ) ) ) ).
% tendsto_0_le
thf(fact_7629_eventually__floor__less,axiom,
! [B: $tType,A: $tType] :
( ( ( archim2362893244070406136eiling @ B )
& ( topolo2564578578187576103pology @ B ) )
=> ! [F2: A > B,L: B,F4: filter @ A] :
( ( filterlim @ A @ B @ F2 @ ( topolo7230453075368039082e_nhds @ B @ L ) @ F4 )
=> ( ~ ( member @ B @ L @ ( ring_1_Ints @ B ) )
=> ( eventually @ A
@ ^ [X3: A] : ( ord_less @ B @ ( ring_1_of_int @ B @ ( archim6421214686448440834_floor @ B @ L ) ) @ ( F2 @ X3 ) )
@ F4 ) ) ) ) ).
% eventually_floor_less
thf(fact_7630_eventually__less__ceiling,axiom,
! [B: $tType,A: $tType] :
( ( ( archim2362893244070406136eiling @ B )
& ( topolo2564578578187576103pology @ B ) )
=> ! [F2: A > B,L: B,F4: filter @ A] :
( ( filterlim @ A @ B @ F2 @ ( topolo7230453075368039082e_nhds @ B @ L ) @ F4 )
=> ( ~ ( member @ B @ L @ ( ring_1_Ints @ B ) )
=> ( eventually @ A
@ ^ [X3: A] : ( ord_less @ B @ ( F2 @ X3 ) @ ( ring_1_of_int @ B @ ( archimedean_ceiling @ B @ L ) ) )
@ F4 ) ) ) ) ).
% eventually_less_ceiling
thf(fact_7631_filterlim__at__infinity,axiom,
! [C: $tType,A: $tType] :
( ( real_V822414075346904944vector @ A )
=> ! [C3: real,F2: C > A,F4: filter @ C] :
( ( ord_less_eq @ real @ ( zero_zero @ real ) @ C3 )
=> ( ( filterlim @ C @ A @ F2 @ ( at_infinity @ A ) @ F4 )
= ( ! [R5: real] :
( ( ord_less @ real @ C3 @ R5 )
=> ( eventually @ C
@ ^ [X3: C] : ( ord_less_eq @ real @ R5 @ ( real_V7770717601297561774m_norm @ A @ ( F2 @ X3 ) ) )
@ F4 ) ) ) ) ) ) ).
% filterlim_at_infinity
thf(fact_7632_tendsto__powr_H,axiom,
! [A: $tType,F2: A > real,A3: real,F4: filter @ A,G: A > real,B3: real] :
( ( filterlim @ A @ real @ F2 @ ( topolo7230453075368039082e_nhds @ real @ A3 ) @ F4 )
=> ( ( filterlim @ A @ real @ G @ ( topolo7230453075368039082e_nhds @ real @ B3 ) @ F4 )
=> ( ( ( A3
!= ( zero_zero @ real ) )
| ( ( ord_less @ real @ ( zero_zero @ real ) @ B3 )
& ( eventually @ A
@ ^ [X3: A] : ( ord_less_eq @ real @ ( zero_zero @ real ) @ ( F2 @ X3 ) )
@ F4 ) ) )
=> ( filterlim @ A @ real
@ ^ [X3: A] : ( powr @ real @ ( F2 @ X3 ) @ ( G @ X3 ) )
@ ( topolo7230453075368039082e_nhds @ real @ ( powr @ real @ A3 @ B3 ) )
@ F4 ) ) ) ) ).
% tendsto_powr'
thf(fact_7633_tendsto__powr2,axiom,
! [A: $tType,F2: A > real,A3: real,F4: filter @ A,G: A > real,B3: real] :
( ( filterlim @ A @ real @ F2 @ ( topolo7230453075368039082e_nhds @ real @ A3 ) @ F4 )
=> ( ( filterlim @ A @ real @ G @ ( topolo7230453075368039082e_nhds @ real @ B3 ) @ F4 )
=> ( ( eventually @ A
@ ^ [X3: A] : ( ord_less_eq @ real @ ( zero_zero @ real ) @ ( F2 @ X3 ) )
@ F4 )
=> ( ( ord_less @ real @ ( zero_zero @ real ) @ B3 )
=> ( filterlim @ A @ real
@ ^ [X3: A] : ( powr @ real @ ( F2 @ X3 ) @ ( G @ X3 ) )
@ ( topolo7230453075368039082e_nhds @ real @ ( powr @ real @ A3 @ B3 ) )
@ F4 ) ) ) ) ) ).
% tendsto_powr2
thf(fact_7634_tendsto__zero__powrI,axiom,
! [A: $tType,F2: A > real,F4: filter @ A,G: A > real,B3: real] :
( ( filterlim @ A @ real @ F2 @ ( topolo7230453075368039082e_nhds @ real @ ( zero_zero @ real ) ) @ F4 )
=> ( ( filterlim @ A @ real @ G @ ( topolo7230453075368039082e_nhds @ real @ B3 ) @ F4 )
=> ( ( eventually @ A
@ ^ [X3: A] : ( ord_less_eq @ real @ ( zero_zero @ real ) @ ( F2 @ X3 ) )
@ F4 )
=> ( ( ord_less @ real @ ( zero_zero @ real ) @ B3 )
=> ( filterlim @ A @ real
@ ^ [X3: A] : ( powr @ real @ ( F2 @ X3 ) @ ( G @ X3 ) )
@ ( topolo7230453075368039082e_nhds @ real @ ( zero_zero @ real ) )
@ F4 ) ) ) ) ) ).
% tendsto_zero_powrI
thf(fact_7635_LIM__at__top__divide,axiom,
! [A: $tType,F2: A > real,A3: real,F4: filter @ A,G: A > real] :
( ( filterlim @ A @ real @ F2 @ ( topolo7230453075368039082e_nhds @ real @ A3 ) @ F4 )
=> ( ( ord_less @ real @ ( zero_zero @ real ) @ A3 )
=> ( ( filterlim @ A @ real @ G @ ( topolo7230453075368039082e_nhds @ real @ ( zero_zero @ real ) ) @ F4 )
=> ( ( eventually @ A
@ ^ [X3: A] : ( ord_less @ real @ ( zero_zero @ real ) @ ( G @ X3 ) )
@ F4 )
=> ( filterlim @ A @ real
@ ^ [X3: A] : ( divide_divide @ real @ ( F2 @ X3 ) @ ( G @ X3 ) )
@ ( at_top @ real )
@ F4 ) ) ) ) ) ).
% LIM_at_top_divide
thf(fact_7636_filterlim__inverse__at__top__iff,axiom,
! [A: $tType,F2: A > real,F4: filter @ A] :
( ( eventually @ A
@ ^ [X3: A] : ( ord_less @ real @ ( zero_zero @ real ) @ ( F2 @ X3 ) )
@ F4 )
=> ( ( filterlim @ A @ real
@ ^ [X3: A] : ( inverse_inverse @ real @ ( F2 @ X3 ) )
@ ( at_top @ real )
@ F4 )
= ( filterlim @ A @ real @ F2 @ ( topolo7230453075368039082e_nhds @ real @ ( zero_zero @ real ) ) @ F4 ) ) ) ).
% filterlim_inverse_at_top_iff
thf(fact_7637_filterlim__inverse__at__top,axiom,
! [A: $tType,F2: A > real,F4: filter @ A] :
( ( filterlim @ A @ real @ F2 @ ( topolo7230453075368039082e_nhds @ real @ ( zero_zero @ real ) ) @ F4 )
=> ( ( eventually @ A
@ ^ [X3: A] : ( ord_less @ real @ ( zero_zero @ real ) @ ( F2 @ X3 ) )
@ F4 )
=> ( filterlim @ A @ real
@ ^ [X3: A] : ( inverse_inverse @ real @ ( F2 @ X3 ) )
@ ( at_top @ real )
@ F4 ) ) ) ).
% filterlim_inverse_at_top
thf(fact_7638_filterlim__inverse__at__bot,axiom,
! [A: $tType,F2: A > real,F4: filter @ A] :
( ( filterlim @ A @ real @ F2 @ ( topolo7230453075368039082e_nhds @ real @ ( zero_zero @ real ) ) @ F4 )
=> ( ( eventually @ A
@ ^ [X3: A] : ( ord_less @ real @ ( F2 @ X3 ) @ ( zero_zero @ real ) )
@ F4 )
=> ( filterlim @ A @ real
@ ^ [X3: A] : ( inverse_inverse @ real @ ( F2 @ X3 ) )
@ ( at_bot @ real )
@ F4 ) ) ) ).
% filterlim_inverse_at_bot
thf(fact_7639_lhopital__at__top__at__top,axiom,
! [F2: real > real,A3: real,G: real > real,F7: real > real,G5: real > real] :
( ( filterlim @ real @ real @ F2 @ ( at_top @ real ) @ ( topolo174197925503356063within @ real @ A3 @ ( top_top @ ( set @ real ) ) ) )
=> ( ( filterlim @ real @ real @ G @ ( at_top @ real ) @ ( topolo174197925503356063within @ real @ A3 @ ( top_top @ ( set @ real ) ) ) )
=> ( ( eventually @ real
@ ^ [X3: real] : ( has_field_derivative @ real @ F2 @ ( F7 @ X3 ) @ ( topolo174197925503356063within @ real @ X3 @ ( top_top @ ( set @ real ) ) ) )
@ ( topolo174197925503356063within @ real @ A3 @ ( top_top @ ( set @ real ) ) ) )
=> ( ( eventually @ real
@ ^ [X3: real] : ( has_field_derivative @ real @ G @ ( G5 @ X3 ) @ ( topolo174197925503356063within @ real @ X3 @ ( top_top @ ( set @ real ) ) ) )
@ ( topolo174197925503356063within @ real @ A3 @ ( top_top @ ( set @ real ) ) ) )
=> ( ( filterlim @ real @ real
@ ^ [X3: real] : ( divide_divide @ real @ ( F7 @ X3 ) @ ( G5 @ X3 ) )
@ ( at_top @ real )
@ ( topolo174197925503356063within @ real @ A3 @ ( top_top @ ( set @ real ) ) ) )
=> ( filterlim @ real @ real
@ ^ [X3: real] : ( divide_divide @ real @ ( F2 @ X3 ) @ ( G @ X3 ) )
@ ( at_top @ real )
@ ( topolo174197925503356063within @ real @ A3 @ ( top_top @ ( set @ real ) ) ) ) ) ) ) ) ) ).
% lhopital_at_top_at_top
thf(fact_7640_lhopital,axiom,
! [F2: real > real,X2: real,G: real > real,G5: real > real,F7: real > real,F4: filter @ real] :
( ( filterlim @ real @ real @ F2 @ ( topolo7230453075368039082e_nhds @ real @ ( zero_zero @ real ) ) @ ( topolo174197925503356063within @ real @ X2 @ ( top_top @ ( set @ real ) ) ) )
=> ( ( filterlim @ real @ real @ G @ ( topolo7230453075368039082e_nhds @ real @ ( zero_zero @ real ) ) @ ( topolo174197925503356063within @ real @ X2 @ ( top_top @ ( set @ real ) ) ) )
=> ( ( eventually @ real
@ ^ [X3: real] :
( ( G @ X3 )
!= ( zero_zero @ real ) )
@ ( topolo174197925503356063within @ real @ X2 @ ( top_top @ ( set @ real ) ) ) )
=> ( ( eventually @ real
@ ^ [X3: real] :
( ( G5 @ X3 )
!= ( zero_zero @ real ) )
@ ( topolo174197925503356063within @ real @ X2 @ ( top_top @ ( set @ real ) ) ) )
=> ( ( eventually @ real
@ ^ [X3: real] : ( has_field_derivative @ real @ F2 @ ( F7 @ X3 ) @ ( topolo174197925503356063within @ real @ X3 @ ( top_top @ ( set @ real ) ) ) )
@ ( topolo174197925503356063within @ real @ X2 @ ( top_top @ ( set @ real ) ) ) )
=> ( ( eventually @ real
@ ^ [X3: real] : ( has_field_derivative @ real @ G @ ( G5 @ X3 ) @ ( topolo174197925503356063within @ real @ X3 @ ( top_top @ ( set @ real ) ) ) )
@ ( topolo174197925503356063within @ real @ X2 @ ( top_top @ ( set @ real ) ) ) )
=> ( ( filterlim @ real @ real
@ ^ [X3: real] : ( divide_divide @ real @ ( F7 @ X3 ) @ ( G5 @ X3 ) )
@ F4
@ ( topolo174197925503356063within @ real @ X2 @ ( top_top @ ( set @ real ) ) ) )
=> ( filterlim @ real @ real
@ ^ [X3: real] : ( divide_divide @ real @ ( F2 @ X3 ) @ ( G @ X3 ) )
@ F4
@ ( topolo174197925503356063within @ real @ X2 @ ( top_top @ ( set @ real ) ) ) ) ) ) ) ) ) ) ) ).
% lhopital
thf(fact_7641_lhopital__right__at__top__at__top,axiom,
! [F2: real > real,A3: real,G: real > real,F7: real > real,G5: real > real] :
( ( filterlim @ real @ real @ F2 @ ( at_top @ real ) @ ( topolo174197925503356063within @ real @ A3 @ ( set_ord_greaterThan @ real @ A3 ) ) )
=> ( ( filterlim @ real @ real @ G @ ( at_top @ real ) @ ( topolo174197925503356063within @ real @ A3 @ ( set_ord_greaterThan @ real @ A3 ) ) )
=> ( ( eventually @ real
@ ^ [X3: real] : ( has_field_derivative @ real @ F2 @ ( F7 @ X3 ) @ ( topolo174197925503356063within @ real @ X3 @ ( top_top @ ( set @ real ) ) ) )
@ ( topolo174197925503356063within @ real @ A3 @ ( set_ord_greaterThan @ real @ A3 ) ) )
=> ( ( eventually @ real
@ ^ [X3: real] : ( has_field_derivative @ real @ G @ ( G5 @ X3 ) @ ( topolo174197925503356063within @ real @ X3 @ ( top_top @ ( set @ real ) ) ) )
@ ( topolo174197925503356063within @ real @ A3 @ ( set_ord_greaterThan @ real @ A3 ) ) )
=> ( ( filterlim @ real @ real
@ ^ [X3: real] : ( divide_divide @ real @ ( F7 @ X3 ) @ ( G5 @ X3 ) )
@ ( at_top @ real )
@ ( topolo174197925503356063within @ real @ A3 @ ( set_ord_greaterThan @ real @ A3 ) ) )
=> ( filterlim @ real @ real
@ ^ [X3: real] : ( divide_divide @ real @ ( F2 @ X3 ) @ ( G @ X3 ) )
@ ( at_top @ real )
@ ( topolo174197925503356063within @ real @ A3 @ ( set_ord_greaterThan @ real @ A3 ) ) ) ) ) ) ) ) ).
% lhopital_right_at_top_at_top
thf(fact_7642_lhopital__at__top__at__bot,axiom,
! [F2: real > real,A3: real,G: real > real,F7: real > real,G5: real > real] :
( ( filterlim @ real @ real @ F2 @ ( at_top @ real ) @ ( topolo174197925503356063within @ real @ A3 @ ( top_top @ ( set @ real ) ) ) )
=> ( ( filterlim @ real @ real @ G @ ( at_bot @ real ) @ ( topolo174197925503356063within @ real @ A3 @ ( top_top @ ( set @ real ) ) ) )
=> ( ( eventually @ real
@ ^ [X3: real] : ( has_field_derivative @ real @ F2 @ ( F7 @ X3 ) @ ( topolo174197925503356063within @ real @ X3 @ ( top_top @ ( set @ real ) ) ) )
@ ( topolo174197925503356063within @ real @ A3 @ ( top_top @ ( set @ real ) ) ) )
=> ( ( eventually @ real
@ ^ [X3: real] : ( has_field_derivative @ real @ G @ ( G5 @ X3 ) @ ( topolo174197925503356063within @ real @ X3 @ ( top_top @ ( set @ real ) ) ) )
@ ( topolo174197925503356063within @ real @ A3 @ ( top_top @ ( set @ real ) ) ) )
=> ( ( filterlim @ real @ real
@ ^ [X3: real] : ( divide_divide @ real @ ( F7 @ X3 ) @ ( G5 @ X3 ) )
@ ( at_bot @ real )
@ ( topolo174197925503356063within @ real @ A3 @ ( top_top @ ( set @ real ) ) ) )
=> ( filterlim @ real @ real
@ ^ [X3: real] : ( divide_divide @ real @ ( F2 @ X3 ) @ ( G @ X3 ) )
@ ( at_bot @ real )
@ ( topolo174197925503356063within @ real @ A3 @ ( top_top @ ( set @ real ) ) ) ) ) ) ) ) ) ).
% lhopital_at_top_at_bot
thf(fact_7643_lhopital__left__at__top__at__top,axiom,
! [F2: real > real,A3: real,G: real > real,F7: real > real,G5: real > real] :
( ( filterlim @ real @ real @ F2 @ ( at_top @ real ) @ ( topolo174197925503356063within @ real @ A3 @ ( set_ord_lessThan @ real @ A3 ) ) )
=> ( ( filterlim @ real @ real @ G @ ( at_top @ real ) @ ( topolo174197925503356063within @ real @ A3 @ ( set_ord_lessThan @ real @ A3 ) ) )
=> ( ( eventually @ real
@ ^ [X3: real] : ( has_field_derivative @ real @ F2 @ ( F7 @ X3 ) @ ( topolo174197925503356063within @ real @ X3 @ ( top_top @ ( set @ real ) ) ) )
@ ( topolo174197925503356063within @ real @ A3 @ ( set_ord_lessThan @ real @ A3 ) ) )
=> ( ( eventually @ real
@ ^ [X3: real] : ( has_field_derivative @ real @ G @ ( G5 @ X3 ) @ ( topolo174197925503356063within @ real @ X3 @ ( top_top @ ( set @ real ) ) ) )
@ ( topolo174197925503356063within @ real @ A3 @ ( set_ord_lessThan @ real @ A3 ) ) )
=> ( ( filterlim @ real @ real
@ ^ [X3: real] : ( divide_divide @ real @ ( F7 @ X3 ) @ ( G5 @ X3 ) )
@ ( at_top @ real )
@ ( topolo174197925503356063within @ real @ A3 @ ( set_ord_lessThan @ real @ A3 ) ) )
=> ( filterlim @ real @ real
@ ^ [X3: real] : ( divide_divide @ real @ ( F2 @ X3 ) @ ( G @ X3 ) )
@ ( at_top @ real )
@ ( topolo174197925503356063within @ real @ A3 @ ( set_ord_lessThan @ real @ A3 ) ) ) ) ) ) ) ) ).
% lhopital_left_at_top_at_top
thf(fact_7644_lhopital__at__top,axiom,
! [G: real > real,X2: real,G5: real > real,F2: real > real,F7: real > real,Y3: real] :
( ( filterlim @ real @ real @ G @ ( at_top @ real ) @ ( topolo174197925503356063within @ real @ X2 @ ( top_top @ ( set @ real ) ) ) )
=> ( ( eventually @ real
@ ^ [X3: real] :
( ( G5 @ X3 )
!= ( zero_zero @ real ) )
@ ( topolo174197925503356063within @ real @ X2 @ ( top_top @ ( set @ real ) ) ) )
=> ( ( eventually @ real
@ ^ [X3: real] : ( has_field_derivative @ real @ F2 @ ( F7 @ X3 ) @ ( topolo174197925503356063within @ real @ X3 @ ( top_top @ ( set @ real ) ) ) )
@ ( topolo174197925503356063within @ real @ X2 @ ( top_top @ ( set @ real ) ) ) )
=> ( ( eventually @ real
@ ^ [X3: real] : ( has_field_derivative @ real @ G @ ( G5 @ X3 ) @ ( topolo174197925503356063within @ real @ X3 @ ( top_top @ ( set @ real ) ) ) )
@ ( topolo174197925503356063within @ real @ X2 @ ( top_top @ ( set @ real ) ) ) )
=> ( ( filterlim @ real @ real
@ ^ [X3: real] : ( divide_divide @ real @ ( F7 @ X3 ) @ ( G5 @ X3 ) )
@ ( topolo7230453075368039082e_nhds @ real @ Y3 )
@ ( topolo174197925503356063within @ real @ X2 @ ( top_top @ ( set @ real ) ) ) )
=> ( filterlim @ real @ real
@ ^ [X3: real] : ( divide_divide @ real @ ( F2 @ X3 ) @ ( G @ X3 ) )
@ ( topolo7230453075368039082e_nhds @ real @ Y3 )
@ ( topolo174197925503356063within @ real @ X2 @ ( top_top @ ( set @ real ) ) ) ) ) ) ) ) ) ).
% lhopital_at_top
thf(fact_7645_lhospital__at__top__at__top,axiom,
! [G: real > real,G5: real > real,F2: real > real,F7: real > real,X2: real] :
( ( filterlim @ real @ real @ G @ ( at_top @ real ) @ ( at_top @ real ) )
=> ( ( eventually @ real
@ ^ [X3: real] :
( ( G5 @ X3 )
!= ( zero_zero @ real ) )
@ ( at_top @ real ) )
=> ( ( eventually @ real
@ ^ [X3: real] : ( has_field_derivative @ real @ F2 @ ( F7 @ X3 ) @ ( topolo174197925503356063within @ real @ X3 @ ( top_top @ ( set @ real ) ) ) )
@ ( at_top @ real ) )
=> ( ( eventually @ real
@ ^ [X3: real] : ( has_field_derivative @ real @ G @ ( G5 @ X3 ) @ ( topolo174197925503356063within @ real @ X3 @ ( top_top @ ( set @ real ) ) ) )
@ ( at_top @ real ) )
=> ( ( filterlim @ real @ real
@ ^ [X3: real] : ( divide_divide @ real @ ( F7 @ X3 ) @ ( G5 @ X3 ) )
@ ( topolo7230453075368039082e_nhds @ real @ X2 )
@ ( at_top @ real ) )
=> ( filterlim @ real @ real
@ ^ [X3: real] : ( divide_divide @ real @ ( F2 @ X3 ) @ ( G @ X3 ) )
@ ( topolo7230453075368039082e_nhds @ real @ X2 )
@ ( at_top @ real ) ) ) ) ) ) ) ).
% lhospital_at_top_at_top
thf(fact_7646_lhopital__right,axiom,
! [F2: real > real,X2: real,G: real > real,G5: real > real,F7: real > real,F4: filter @ real] :
( ( filterlim @ real @ real @ F2 @ ( topolo7230453075368039082e_nhds @ real @ ( zero_zero @ real ) ) @ ( topolo174197925503356063within @ real @ X2 @ ( set_ord_greaterThan @ real @ X2 ) ) )
=> ( ( filterlim @ real @ real @ G @ ( topolo7230453075368039082e_nhds @ real @ ( zero_zero @ real ) ) @ ( topolo174197925503356063within @ real @ X2 @ ( set_ord_greaterThan @ real @ X2 ) ) )
=> ( ( eventually @ real
@ ^ [X3: real] :
( ( G @ X3 )
!= ( zero_zero @ real ) )
@ ( topolo174197925503356063within @ real @ X2 @ ( set_ord_greaterThan @ real @ X2 ) ) )
=> ( ( eventually @ real
@ ^ [X3: real] :
( ( G5 @ X3 )
!= ( zero_zero @ real ) )
@ ( topolo174197925503356063within @ real @ X2 @ ( set_ord_greaterThan @ real @ X2 ) ) )
=> ( ( eventually @ real
@ ^ [X3: real] : ( has_field_derivative @ real @ F2 @ ( F7 @ X3 ) @ ( topolo174197925503356063within @ real @ X3 @ ( top_top @ ( set @ real ) ) ) )
@ ( topolo174197925503356063within @ real @ X2 @ ( set_ord_greaterThan @ real @ X2 ) ) )
=> ( ( eventually @ real
@ ^ [X3: real] : ( has_field_derivative @ real @ G @ ( G5 @ X3 ) @ ( topolo174197925503356063within @ real @ X3 @ ( top_top @ ( set @ real ) ) ) )
@ ( topolo174197925503356063within @ real @ X2 @ ( set_ord_greaterThan @ real @ X2 ) ) )
=> ( ( filterlim @ real @ real
@ ^ [X3: real] : ( divide_divide @ real @ ( F7 @ X3 ) @ ( G5 @ X3 ) )
@ F4
@ ( topolo174197925503356063within @ real @ X2 @ ( set_ord_greaterThan @ real @ X2 ) ) )
=> ( filterlim @ real @ real
@ ^ [X3: real] : ( divide_divide @ real @ ( F2 @ X3 ) @ ( G @ X3 ) )
@ F4
@ ( topolo174197925503356063within @ real @ X2 @ ( set_ord_greaterThan @ real @ X2 ) ) ) ) ) ) ) ) ) ) ).
% lhopital_right
thf(fact_7647_lhopital__right__0,axiom,
! [F0: real > real,G0: real > real,G5: real > real,F7: real > real,F4: filter @ real] :
( ( filterlim @ real @ real @ F0 @ ( topolo7230453075368039082e_nhds @ real @ ( zero_zero @ real ) ) @ ( topolo174197925503356063within @ real @ ( zero_zero @ real ) @ ( set_ord_greaterThan @ real @ ( zero_zero @ real ) ) ) )
=> ( ( filterlim @ real @ real @ G0 @ ( topolo7230453075368039082e_nhds @ real @ ( zero_zero @ real ) ) @ ( topolo174197925503356063within @ real @ ( zero_zero @ real ) @ ( set_ord_greaterThan @ real @ ( zero_zero @ real ) ) ) )
=> ( ( eventually @ real
@ ^ [X3: real] :
( ( G0 @ X3 )
!= ( zero_zero @ real ) )
@ ( topolo174197925503356063within @ real @ ( zero_zero @ real ) @ ( set_ord_greaterThan @ real @ ( zero_zero @ real ) ) ) )
=> ( ( eventually @ real
@ ^ [X3: real] :
( ( G5 @ X3 )
!= ( zero_zero @ real ) )
@ ( topolo174197925503356063within @ real @ ( zero_zero @ real ) @ ( set_ord_greaterThan @ real @ ( zero_zero @ real ) ) ) )
=> ( ( eventually @ real
@ ^ [X3: real] : ( has_field_derivative @ real @ F0 @ ( F7 @ X3 ) @ ( topolo174197925503356063within @ real @ X3 @ ( top_top @ ( set @ real ) ) ) )
@ ( topolo174197925503356063within @ real @ ( zero_zero @ real ) @ ( set_ord_greaterThan @ real @ ( zero_zero @ real ) ) ) )
=> ( ( eventually @ real
@ ^ [X3: real] : ( has_field_derivative @ real @ G0 @ ( G5 @ X3 ) @ ( topolo174197925503356063within @ real @ X3 @ ( top_top @ ( set @ real ) ) ) )
@ ( topolo174197925503356063within @ real @ ( zero_zero @ real ) @ ( set_ord_greaterThan @ real @ ( zero_zero @ real ) ) ) )
=> ( ( filterlim @ real @ real
@ ^ [X3: real] : ( divide_divide @ real @ ( F7 @ X3 ) @ ( G5 @ X3 ) )
@ F4
@ ( topolo174197925503356063within @ real @ ( zero_zero @ real ) @ ( set_ord_greaterThan @ real @ ( zero_zero @ real ) ) ) )
=> ( filterlim @ real @ real
@ ^ [X3: real] : ( divide_divide @ real @ ( F0 @ X3 ) @ ( G0 @ X3 ) )
@ F4
@ ( topolo174197925503356063within @ real @ ( zero_zero @ real ) @ ( set_ord_greaterThan @ real @ ( zero_zero @ real ) ) ) ) ) ) ) ) ) ) ) ).
% lhopital_right_0
thf(fact_7648_lhopital__left,axiom,
! [F2: real > real,X2: real,G: real > real,G5: real > real,F7: real > real,F4: filter @ real] :
( ( filterlim @ real @ real @ F2 @ ( topolo7230453075368039082e_nhds @ real @ ( zero_zero @ real ) ) @ ( topolo174197925503356063within @ real @ X2 @ ( set_ord_lessThan @ real @ X2 ) ) )
=> ( ( filterlim @ real @ real @ G @ ( topolo7230453075368039082e_nhds @ real @ ( zero_zero @ real ) ) @ ( topolo174197925503356063within @ real @ X2 @ ( set_ord_lessThan @ real @ X2 ) ) )
=> ( ( eventually @ real
@ ^ [X3: real] :
( ( G @ X3 )
!= ( zero_zero @ real ) )
@ ( topolo174197925503356063within @ real @ X2 @ ( set_ord_lessThan @ real @ X2 ) ) )
=> ( ( eventually @ real
@ ^ [X3: real] :
( ( G5 @ X3 )
!= ( zero_zero @ real ) )
@ ( topolo174197925503356063within @ real @ X2 @ ( set_ord_lessThan @ real @ X2 ) ) )
=> ( ( eventually @ real
@ ^ [X3: real] : ( has_field_derivative @ real @ F2 @ ( F7 @ X3 ) @ ( topolo174197925503356063within @ real @ X3 @ ( top_top @ ( set @ real ) ) ) )
@ ( topolo174197925503356063within @ real @ X2 @ ( set_ord_lessThan @ real @ X2 ) ) )
=> ( ( eventually @ real
@ ^ [X3: real] : ( has_field_derivative @ real @ G @ ( G5 @ X3 ) @ ( topolo174197925503356063within @ real @ X3 @ ( top_top @ ( set @ real ) ) ) )
@ ( topolo174197925503356063within @ real @ X2 @ ( set_ord_lessThan @ real @ X2 ) ) )
=> ( ( filterlim @ real @ real
@ ^ [X3: real] : ( divide_divide @ real @ ( F7 @ X3 ) @ ( G5 @ X3 ) )
@ F4
@ ( topolo174197925503356063within @ real @ X2 @ ( set_ord_lessThan @ real @ X2 ) ) )
=> ( filterlim @ real @ real
@ ^ [X3: real] : ( divide_divide @ real @ ( F2 @ X3 ) @ ( G @ X3 ) )
@ F4
@ ( topolo174197925503356063within @ real @ X2 @ ( set_ord_lessThan @ real @ X2 ) ) ) ) ) ) ) ) ) ) ).
% lhopital_left
thf(fact_7649_lhopital__right__at__top__at__bot,axiom,
! [F2: real > real,A3: real,G: real > real,F7: real > real,G5: real > real] :
( ( filterlim @ real @ real @ F2 @ ( at_top @ real ) @ ( topolo174197925503356063within @ real @ A3 @ ( set_ord_greaterThan @ real @ A3 ) ) )
=> ( ( filterlim @ real @ real @ G @ ( at_bot @ real ) @ ( topolo174197925503356063within @ real @ A3 @ ( set_ord_greaterThan @ real @ A3 ) ) )
=> ( ( eventually @ real
@ ^ [X3: real] : ( has_field_derivative @ real @ F2 @ ( F7 @ X3 ) @ ( topolo174197925503356063within @ real @ X3 @ ( top_top @ ( set @ real ) ) ) )
@ ( topolo174197925503356063within @ real @ A3 @ ( set_ord_greaterThan @ real @ A3 ) ) )
=> ( ( eventually @ real
@ ^ [X3: real] : ( has_field_derivative @ real @ G @ ( G5 @ X3 ) @ ( topolo174197925503356063within @ real @ X3 @ ( top_top @ ( set @ real ) ) ) )
@ ( topolo174197925503356063within @ real @ A3 @ ( set_ord_greaterThan @ real @ A3 ) ) )
=> ( ( filterlim @ real @ real
@ ^ [X3: real] : ( divide_divide @ real @ ( F7 @ X3 ) @ ( G5 @ X3 ) )
@ ( at_bot @ real )
@ ( topolo174197925503356063within @ real @ A3 @ ( set_ord_greaterThan @ real @ A3 ) ) )
=> ( filterlim @ real @ real
@ ^ [X3: real] : ( divide_divide @ real @ ( F2 @ X3 ) @ ( G @ X3 ) )
@ ( at_bot @ real )
@ ( topolo174197925503356063within @ real @ A3 @ ( set_ord_greaterThan @ real @ A3 ) ) ) ) ) ) ) ) ).
% lhopital_right_at_top_at_bot
thf(fact_7650_lhopital__left__at__top__at__bot,axiom,
! [F2: real > real,A3: real,G: real > real,F7: real > real,G5: real > real] :
( ( filterlim @ real @ real @ F2 @ ( at_top @ real ) @ ( topolo174197925503356063within @ real @ A3 @ ( set_ord_lessThan @ real @ A3 ) ) )
=> ( ( filterlim @ real @ real @ G @ ( at_bot @ real ) @ ( topolo174197925503356063within @ real @ A3 @ ( set_ord_lessThan @ real @ A3 ) ) )
=> ( ( eventually @ real
@ ^ [X3: real] : ( has_field_derivative @ real @ F2 @ ( F7 @ X3 ) @ ( topolo174197925503356063within @ real @ X3 @ ( top_top @ ( set @ real ) ) ) )
@ ( topolo174197925503356063within @ real @ A3 @ ( set_ord_lessThan @ real @ A3 ) ) )
=> ( ( eventually @ real
@ ^ [X3: real] : ( has_field_derivative @ real @ G @ ( G5 @ X3 ) @ ( topolo174197925503356063within @ real @ X3 @ ( top_top @ ( set @ real ) ) ) )
@ ( topolo174197925503356063within @ real @ A3 @ ( set_ord_lessThan @ real @ A3 ) ) )
=> ( ( filterlim @ real @ real
@ ^ [X3: real] : ( divide_divide @ real @ ( F7 @ X3 ) @ ( G5 @ X3 ) )
@ ( at_bot @ real )
@ ( topolo174197925503356063within @ real @ A3 @ ( set_ord_lessThan @ real @ A3 ) ) )
=> ( filterlim @ real @ real
@ ^ [X3: real] : ( divide_divide @ real @ ( F2 @ X3 ) @ ( G @ X3 ) )
@ ( at_bot @ real )
@ ( topolo174197925503356063within @ real @ A3 @ ( set_ord_lessThan @ real @ A3 ) ) ) ) ) ) ) ) ).
% lhopital_left_at_top_at_bot
thf(fact_7651_lhopital__right__at__top,axiom,
! [G: real > real,X2: real,G5: real > real,F2: real > real,F7: real > real,Y3: real] :
( ( filterlim @ real @ real @ G @ ( at_top @ real ) @ ( topolo174197925503356063within @ real @ X2 @ ( set_ord_greaterThan @ real @ X2 ) ) )
=> ( ( eventually @ real
@ ^ [X3: real] :
( ( G5 @ X3 )
!= ( zero_zero @ real ) )
@ ( topolo174197925503356063within @ real @ X2 @ ( set_ord_greaterThan @ real @ X2 ) ) )
=> ( ( eventually @ real
@ ^ [X3: real] : ( has_field_derivative @ real @ F2 @ ( F7 @ X3 ) @ ( topolo174197925503356063within @ real @ X3 @ ( top_top @ ( set @ real ) ) ) )
@ ( topolo174197925503356063within @ real @ X2 @ ( set_ord_greaterThan @ real @ X2 ) ) )
=> ( ( eventually @ real
@ ^ [X3: real] : ( has_field_derivative @ real @ G @ ( G5 @ X3 ) @ ( topolo174197925503356063within @ real @ X3 @ ( top_top @ ( set @ real ) ) ) )
@ ( topolo174197925503356063within @ real @ X2 @ ( set_ord_greaterThan @ real @ X2 ) ) )
=> ( ( filterlim @ real @ real
@ ^ [X3: real] : ( divide_divide @ real @ ( F7 @ X3 ) @ ( G5 @ X3 ) )
@ ( topolo7230453075368039082e_nhds @ real @ Y3 )
@ ( topolo174197925503356063within @ real @ X2 @ ( set_ord_greaterThan @ real @ X2 ) ) )
=> ( filterlim @ real @ real
@ ^ [X3: real] : ( divide_divide @ real @ ( F2 @ X3 ) @ ( G @ X3 ) )
@ ( topolo7230453075368039082e_nhds @ real @ Y3 )
@ ( topolo174197925503356063within @ real @ X2 @ ( set_ord_greaterThan @ real @ X2 ) ) ) ) ) ) ) ) ).
% lhopital_right_at_top
thf(fact_7652_lhopital__right__0__at__top,axiom,
! [G: real > real,G5: real > real,F2: real > real,F7: real > real,X2: real] :
( ( filterlim @ real @ real @ G @ ( at_top @ real ) @ ( topolo174197925503356063within @ real @ ( zero_zero @ real ) @ ( set_ord_greaterThan @ real @ ( zero_zero @ real ) ) ) )
=> ( ( eventually @ real
@ ^ [X3: real] :
( ( G5 @ X3 )
!= ( zero_zero @ real ) )
@ ( topolo174197925503356063within @ real @ ( zero_zero @ real ) @ ( set_ord_greaterThan @ real @ ( zero_zero @ real ) ) ) )
=> ( ( eventually @ real
@ ^ [X3: real] : ( has_field_derivative @ real @ F2 @ ( F7 @ X3 ) @ ( topolo174197925503356063within @ real @ X3 @ ( top_top @ ( set @ real ) ) ) )
@ ( topolo174197925503356063within @ real @ ( zero_zero @ real ) @ ( set_ord_greaterThan @ real @ ( zero_zero @ real ) ) ) )
=> ( ( eventually @ real
@ ^ [X3: real] : ( has_field_derivative @ real @ G @ ( G5 @ X3 ) @ ( topolo174197925503356063within @ real @ X3 @ ( top_top @ ( set @ real ) ) ) )
@ ( topolo174197925503356063within @ real @ ( zero_zero @ real ) @ ( set_ord_greaterThan @ real @ ( zero_zero @ real ) ) ) )
=> ( ( filterlim @ real @ real
@ ^ [X3: real] : ( divide_divide @ real @ ( F7 @ X3 ) @ ( G5 @ X3 ) )
@ ( topolo7230453075368039082e_nhds @ real @ X2 )
@ ( topolo174197925503356063within @ real @ ( zero_zero @ real ) @ ( set_ord_greaterThan @ real @ ( zero_zero @ real ) ) ) )
=> ( filterlim @ real @ real
@ ^ [X3: real] : ( divide_divide @ real @ ( F2 @ X3 ) @ ( G @ X3 ) )
@ ( topolo7230453075368039082e_nhds @ real @ X2 )
@ ( topolo174197925503356063within @ real @ ( zero_zero @ real ) @ ( set_ord_greaterThan @ real @ ( zero_zero @ real ) ) ) ) ) ) ) ) ) ).
% lhopital_right_0_at_top
thf(fact_7653_summable__Cauchy_H,axiom,
! [A: $tType] :
( ( real_Vector_banach @ A )
=> ! [F2: nat > A,G: nat > real] :
( ( eventually @ nat
@ ^ [M6: nat] :
! [N4: nat] :
( ( ord_less_eq @ nat @ M6 @ N4 )
=> ( ord_less_eq @ real @ ( real_V7770717601297561774m_norm @ A @ ( groups7311177749621191930dd_sum @ nat @ A @ F2 @ ( set_or7035219750837199246ssThan @ nat @ M6 @ N4 ) ) ) @ ( G @ M6 ) ) )
@ ( at_top @ nat ) )
=> ( ( filterlim @ nat @ real @ G @ ( topolo7230453075368039082e_nhds @ real @ ( zero_zero @ real ) ) @ ( at_top @ nat ) )
=> ( summable @ A @ F2 ) ) ) ) ).
% summable_Cauchy'
thf(fact_7654_Bfun__metric__def,axiom,
! [B: $tType,A: $tType] :
( ( real_V7819770556892013058_space @ B )
=> ( ( bfun @ A @ B )
= ( ^ [F5: A > B,F9: filter @ A] :
? [Y: B,K7: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ K7 )
& ( eventually @ A
@ ^ [X3: A] : ( ord_less_eq @ real @ ( real_V557655796197034286t_dist @ B @ ( F5 @ X3 ) @ Y ) @ K7 )
@ F9 ) ) ) ) ) ).
% Bfun_metric_def
thf(fact_7655_eventually__all__finite,axiom,
! [B: $tType,A: $tType] :
( ( finite_finite @ B )
=> ! [P: A > B > $o,Net: filter @ A] :
( ! [Y4: B] :
( eventually @ A
@ ^ [X3: A] : ( P @ X3 @ Y4 )
@ Net )
=> ( eventually @ A
@ ^ [X3: A] :
! [X7: B] : ( P @ X3 @ X7 )
@ Net ) ) ) ).
% eventually_all_finite
thf(fact_7656_finite__set__of__finite__funs,axiom,
! [B: $tType,A: $tType,A2: set @ A,B2: set @ B,D2: B] :
( ( finite_finite2 @ A @ A2 )
=> ( ( finite_finite2 @ B @ B2 )
=> ( finite_finite2 @ ( A > B )
@ ( collect @ ( A > B )
@ ^ [F5: A > B] :
! [X3: A] :
( ( ( member @ A @ X3 @ A2 )
=> ( member @ B @ ( F5 @ X3 ) @ B2 ) )
& ( ~ ( member @ A @ X3 @ A2 )
=> ( ( F5 @ X3 )
= D2 ) ) ) ) ) ) ) ).
% finite_set_of_finite_funs
thf(fact_7657_Bfun__const,axiom,
! [B: $tType,A: $tType] :
( ( real_V7819770556892013058_space @ B )
=> ! [C3: B,F4: filter @ A] :
( bfun @ A @ B
@ ^ [Uu3: A] : C3
@ F4 ) ) ).
% Bfun_const
thf(fact_7658_Bseq__mult,axiom,
! [A: $tType] :
( ( real_V3459762299906320749_field @ A )
=> ! [F2: nat > A,G: nat > A] :
( ( bfun @ nat @ A @ F2 @ ( at_top @ nat ) )
=> ( ( bfun @ nat @ A @ G @ ( at_top @ nat ) )
=> ( bfun @ nat @ A
@ ^ [X3: nat] : ( times_times @ A @ ( F2 @ X3 ) @ ( G @ X3 ) )
@ ( at_top @ nat ) ) ) ) ) ).
% Bseq_mult
thf(fact_7659_Bseq__minus__iff,axiom,
! [A: $tType] :
( ( real_V822414075346904944vector @ A )
=> ! [X: nat > A] :
( ( bfun @ nat @ A
@ ^ [N4: nat] : ( uminus_uminus @ A @ ( X @ N4 ) )
@ ( at_top @ nat ) )
= ( bfun @ nat @ A @ X @ ( at_top @ nat ) ) ) ) ).
% Bseq_minus_iff
thf(fact_7660_Bseq__subseq,axiom,
! [A: $tType] :
( ( real_V822414075346904944vector @ A )
=> ! [F2: nat > A,G: nat > nat] :
( ( bfun @ nat @ A @ F2 @ ( at_top @ nat ) )
=> ( bfun @ nat @ A
@ ^ [X3: nat] : ( F2 @ ( G @ X3 ) )
@ ( at_top @ nat ) ) ) ) ).
% Bseq_subseq
thf(fact_7661_Bseq__Suc__iff,axiom,
! [A: $tType] :
( ( real_V822414075346904944vector @ A )
=> ! [F2: nat > A] :
( ( bfun @ nat @ A
@ ^ [N4: nat] : ( F2 @ ( suc @ N4 ) )
@ ( at_top @ nat ) )
= ( bfun @ nat @ A @ F2 @ ( at_top @ nat ) ) ) ) ).
% Bseq_Suc_iff
thf(fact_7662_Bseq__add,axiom,
! [A: $tType] :
( ( real_V822414075346904944vector @ A )
=> ! [F2: nat > A,C3: A] :
( ( bfun @ nat @ A @ F2 @ ( at_top @ nat ) )
=> ( bfun @ nat @ A
@ ^ [X3: nat] : ( plus_plus @ A @ ( F2 @ X3 ) @ C3 )
@ ( at_top @ nat ) ) ) ) ).
% Bseq_add
thf(fact_7663_Bseq__offset,axiom,
! [A: $tType] :
( ( real_V7819770556892013058_space @ A )
=> ! [X: nat > A,K: nat] :
( ( bfun @ nat @ A
@ ^ [N4: nat] : ( X @ ( plus_plus @ nat @ N4 @ K ) )
@ ( at_top @ nat ) )
=> ( bfun @ nat @ A @ X @ ( at_top @ nat ) ) ) ) ).
% Bseq_offset
thf(fact_7664_Bseq__add__iff,axiom,
! [A: $tType] :
( ( real_V822414075346904944vector @ A )
=> ! [F2: nat > A,C3: A] :
( ( bfun @ nat @ A
@ ^ [X3: nat] : ( plus_plus @ A @ ( F2 @ X3 ) @ C3 )
@ ( at_top @ nat ) )
= ( bfun @ nat @ A @ F2 @ ( at_top @ nat ) ) ) ) ).
% Bseq_add_iff
thf(fact_7665_Bseq__ignore__initial__segment,axiom,
! [A: $tType] :
( ( real_V7819770556892013058_space @ A )
=> ! [X: nat > A,K: nat] :
( ( bfun @ nat @ A @ X @ ( at_top @ nat ) )
=> ( bfun @ nat @ A
@ ^ [N4: nat] : ( X @ ( plus_plus @ nat @ N4 @ K ) )
@ ( at_top @ nat ) ) ) ) ).
% Bseq_ignore_initial_segment
thf(fact_7666_BseqI_H,axiom,
! [A: $tType] :
( ( real_V822414075346904944vector @ A )
=> ! [X: nat > A,K6: real] :
( ! [N3: nat] : ( ord_less_eq @ real @ ( real_V7770717601297561774m_norm @ A @ ( X @ N3 ) ) @ K6 )
=> ( bfun @ nat @ A @ X @ ( at_top @ nat ) ) ) ) ).
% BseqI'
thf(fact_7667_eventually__all__ge__at__top,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [P: A > $o] :
( ( eventually @ A @ P @ ( at_top @ A ) )
=> ( eventually @ A
@ ^ [X3: A] :
! [Y: A] :
( ( ord_less_eq @ A @ X3 @ Y )
=> ( P @ Y ) )
@ ( at_top @ A ) ) ) ) ).
% eventually_all_ge_at_top
thf(fact_7668_Bseq__eq__bounded,axiom,
! [F2: nat > real,A3: real,B3: real] :
( ( ord_less_eq @ ( set @ real ) @ ( image @ nat @ real @ F2 @ ( top_top @ ( set @ nat ) ) ) @ ( set_or1337092689740270186AtMost @ real @ A3 @ B3 ) )
=> ( bfun @ nat @ real @ F2 @ ( at_top @ nat ) ) ) ).
% Bseq_eq_bounded
thf(fact_7669_Bseq__cmult__iff,axiom,
! [A: $tType] :
( ( real_V3459762299906320749_field @ A )
=> ! [C3: A,F2: nat > A] :
( ( C3
!= ( zero_zero @ A ) )
=> ( ( bfun @ nat @ A
@ ^ [X3: nat] : ( times_times @ A @ C3 @ ( F2 @ X3 ) )
@ ( at_top @ nat ) )
= ( bfun @ nat @ A @ F2 @ ( at_top @ nat ) ) ) ) ) ).
% Bseq_cmult_iff
thf(fact_7670_Bseq__eventually__mono,axiom,
! [B: $tType,A: $tType] :
( ( ( real_V822414075346904944vector @ A )
& ( real_V822414075346904944vector @ B ) )
=> ! [F2: nat > A,G: nat > B] :
( ( eventually @ nat
@ ^ [N4: nat] : ( ord_less_eq @ real @ ( real_V7770717601297561774m_norm @ A @ ( F2 @ N4 ) ) @ ( real_V7770717601297561774m_norm @ B @ ( G @ N4 ) ) )
@ ( at_top @ nat ) )
=> ( ( bfun @ nat @ B @ G @ ( at_top @ nat ) )
=> ( bfun @ nat @ A @ F2 @ ( at_top @ nat ) ) ) ) ) ).
% Bseq_eventually_mono
thf(fact_7671_filterlim__int__of__nat__at__topD,axiom,
! [A: $tType,F2: int > A,F4: filter @ A] :
( ( filterlim @ nat @ A
@ ^ [X3: nat] : ( F2 @ ( semiring_1_of_nat @ int @ X3 ) )
@ F4
@ ( at_top @ nat ) )
=> ( filterlim @ int @ A @ F2 @ F4 @ ( at_top @ int ) ) ) ).
% filterlim_int_of_nat_at_topD
thf(fact_7672_BseqD,axiom,
! [A: $tType] :
( ( real_V822414075346904944vector @ A )
=> ! [X: nat > A] :
( ( bfun @ nat @ A @ X @ ( at_top @ nat ) )
=> ? [K9: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ K9 )
& ! [N9: nat] : ( ord_less_eq @ real @ ( real_V7770717601297561774m_norm @ A @ ( X @ N9 ) ) @ K9 ) ) ) ) ).
% BseqD
thf(fact_7673_BseqE,axiom,
! [A: $tType] :
( ( real_V822414075346904944vector @ A )
=> ! [X: nat > A] :
( ( bfun @ nat @ A @ X @ ( at_top @ nat ) )
=> ~ ! [K9: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ K9 )
=> ~ ! [N9: nat] : ( ord_less_eq @ real @ ( real_V7770717601297561774m_norm @ A @ ( X @ N9 ) ) @ K9 ) ) ) ) ).
% BseqE
thf(fact_7674_BseqI,axiom,
! [A: $tType] :
( ( real_V822414075346904944vector @ A )
=> ! [K6: real,X: nat > A] :
( ( ord_less @ real @ ( zero_zero @ real ) @ K6 )
=> ( ! [N3: nat] : ( ord_less_eq @ real @ ( real_V7770717601297561774m_norm @ A @ ( X @ N3 ) ) @ K6 )
=> ( bfun @ nat @ A @ X @ ( at_top @ nat ) ) ) ) ) ).
% BseqI
thf(fact_7675_Bseq__def,axiom,
! [A: $tType] :
( ( real_V822414075346904944vector @ A )
=> ! [X: nat > A] :
( ( bfun @ nat @ A @ X @ ( at_top @ nat ) )
= ( ? [K7: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ K7 )
& ! [N4: nat] : ( ord_less_eq @ real @ ( real_V7770717601297561774m_norm @ A @ ( X @ N4 ) ) @ K7 ) ) ) ) ) ).
% Bseq_def
thf(fact_7676_Bseq__iff1a,axiom,
! [A: $tType] :
( ( real_V822414075346904944vector @ A )
=> ! [X: nat > A] :
( ( bfun @ nat @ A @ X @ ( at_top @ nat ) )
= ( ? [N8: nat] :
! [N4: nat] : ( ord_less @ real @ ( real_V7770717601297561774m_norm @ A @ ( X @ N4 ) ) @ ( semiring_1_of_nat @ real @ ( suc @ N8 ) ) ) ) ) ) ).
% Bseq_iff1a
thf(fact_7677_Bseq__iff,axiom,
! [A: $tType] :
( ( real_V822414075346904944vector @ A )
=> ! [X: nat > A] :
( ( bfun @ nat @ A @ X @ ( at_top @ nat ) )
= ( ? [N8: nat] :
! [N4: nat] : ( ord_less_eq @ real @ ( real_V7770717601297561774m_norm @ A @ ( X @ N4 ) ) @ ( semiring_1_of_nat @ real @ ( suc @ N8 ) ) ) ) ) ) ).
% Bseq_iff
thf(fact_7678_Bseq__realpow,axiom,
! [X2: real] :
( ( ord_less_eq @ real @ ( zero_zero @ real ) @ X2 )
=> ( ( ord_less_eq @ real @ X2 @ ( one_one @ real ) )
=> ( bfun @ nat @ real @ ( power_power @ real @ X2 ) @ ( at_top @ nat ) ) ) ) ).
% Bseq_realpow
thf(fact_7679_BfunI,axiom,
! [B: $tType,A: $tType] :
( ( real_V822414075346904944vector @ B )
=> ! [F2: A > B,K6: real,F4: filter @ A] :
( ( eventually @ A
@ ^ [X3: A] : ( ord_less_eq @ real @ ( real_V7770717601297561774m_norm @ B @ ( F2 @ X3 ) ) @ K6 )
@ F4 )
=> ( bfun @ A @ B @ F2 @ F4 ) ) ) ).
% BfunI
thf(fact_7680_Bseq__iff2,axiom,
! [A: $tType] :
( ( real_V822414075346904944vector @ A )
=> ! [X: nat > A] :
( ( bfun @ nat @ A @ X @ ( at_top @ nat ) )
= ( ? [K3: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ K3 )
& ? [X3: A] :
! [N4: nat] : ( ord_less_eq @ real @ ( real_V7770717601297561774m_norm @ A @ ( plus_plus @ A @ ( X @ N4 ) @ ( uminus_uminus @ A @ X3 ) ) ) @ K3 ) ) ) ) ) ).
% Bseq_iff2
thf(fact_7681_Bseq__iff3,axiom,
! [A: $tType] :
( ( real_V822414075346904944vector @ A )
=> ! [X: nat > A] :
( ( bfun @ nat @ A @ X @ ( at_top @ nat ) )
= ( ? [K3: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ K3 )
& ? [N8: nat] :
! [N4: nat] : ( ord_less_eq @ real @ ( real_V7770717601297561774m_norm @ A @ ( plus_plus @ A @ ( X @ N4 ) @ ( uminus_uminus @ A @ ( X @ N8 ) ) ) ) @ K3 ) ) ) ) ) ).
% Bseq_iff3
thf(fact_7682_Bfun__inverse,axiom,
! [A: $tType,B: $tType] :
( ( real_V8999393235501362500lgebra @ A )
=> ! [F2: B > A,A3: A,F4: filter @ B] :
( ( filterlim @ B @ A @ F2 @ ( topolo7230453075368039082e_nhds @ A @ A3 ) @ F4 )
=> ( ( A3
!= ( zero_zero @ A ) )
=> ( bfun @ B @ A
@ ^ [X3: B] : ( inverse_inverse @ A @ ( F2 @ X3 ) )
@ F4 ) ) ) ) ).
% Bfun_inverse
thf(fact_7683_Bfun__def,axiom,
! [B: $tType,A: $tType] :
( ( real_V822414075346904944vector @ B )
=> ( ( bfun @ A @ B )
= ( ^ [F5: A > B,F9: filter @ A] :
? [K7: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ K7 )
& ( eventually @ A
@ ^ [X3: A] : ( ord_less_eq @ real @ ( real_V7770717601297561774m_norm @ B @ ( F5 @ X3 ) ) @ K7 )
@ F9 ) ) ) ) ) ).
% Bfun_def
thf(fact_7684_BfunE,axiom,
! [B: $tType,A: $tType] :
( ( real_V822414075346904944vector @ B )
=> ! [F2: A > B,F4: filter @ A] :
( ( bfun @ A @ B @ F2 @ F4 )
=> ~ ! [B10: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ B10 )
=> ~ ( eventually @ A
@ ^ [X3: A] : ( ord_less_eq @ real @ ( real_V7770717601297561774m_norm @ B @ ( F2 @ X3 ) ) @ B10 )
@ F4 ) ) ) ) ).
% BfunE
thf(fact_7685_summable__bounded__partials,axiom,
! [A: $tType] :
( ( ( real_V8037385150606011577_space @ A )
& ( real_V822414075346904944vector @ A ) )
=> ! [F2: nat > A,G: nat > real] :
( ( eventually @ nat
@ ^ [X02: nat] :
! [A6: nat] :
( ( ord_less_eq @ nat @ X02 @ A6 )
=> ! [B7: nat] :
( ( ord_less @ nat @ A6 @ B7 )
=> ( ord_less_eq @ real @ ( real_V7770717601297561774m_norm @ A @ ( groups7311177749621191930dd_sum @ nat @ A @ F2 @ ( set_or3652927894154168847AtMost @ nat @ A6 @ B7 ) ) ) @ ( G @ A6 ) ) ) )
@ ( at_top @ nat ) )
=> ( ( filterlim @ nat @ real @ G @ ( topolo7230453075368039082e_nhds @ real @ ( zero_zero @ real ) ) @ ( at_top @ nat ) )
=> ( summable @ A @ F2 ) ) ) ) ).
% summable_bounded_partials
thf(fact_7686_Greatest__def,axiom,
! [A: $tType] :
( ( order @ A )
=> ( ( order_Greatest @ A )
= ( ^ [P3: A > $o] :
( the @ A
@ ^ [X3: A] :
( ( P3 @ X3 )
& ! [Y: A] :
( ( P3 @ Y )
=> ( ord_less_eq @ A @ Y @ X3 ) ) ) ) ) ) ) ).
% Greatest_def
thf(fact_7687_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_7688_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_7689_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_7690_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_7691_infinite__Ioc__iff,axiom,
! [A: $tType] :
( ( dense_linorder @ A )
=> ! [A3: A,B3: A] :
( ( ~ ( finite_finite2 @ A @ ( set_or3652927894154168847AtMost @ A @ A3 @ B3 ) ) )
= ( ord_less @ A @ A3 @ B3 ) ) ) ).
% infinite_Ioc_iff
thf(fact_7692_image__add__greaterThanAtMost,axiom,
! [A: $tType] :
( ( linordered_semidom @ A )
=> ! [C3: A,A3: A,B3: A] :
( ( image @ A @ A @ ( plus_plus @ A @ C3 ) @ ( set_or3652927894154168847AtMost @ A @ A3 @ B3 ) )
= ( set_or3652927894154168847AtMost @ A @ ( plus_plus @ A @ C3 @ A3 ) @ ( plus_plus @ A @ C3 @ B3 ) ) ) ) ).
% image_add_greaterThanAtMost
thf(fact_7693_image__minus__const__greaterThanAtMost,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [C3: A,A3: A,B3: A] :
( ( image @ A @ A @ ( minus_minus @ A @ C3 ) @ ( set_or3652927894154168847AtMost @ A @ A3 @ B3 ) )
= ( set_or7035219750837199246ssThan @ A @ ( minus_minus @ A @ C3 @ B3 ) @ ( minus_minus @ A @ C3 @ A3 ) ) ) ) ).
% image_minus_const_greaterThanAtMost
thf(fact_7694_image__diff__atLeastLessThan,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [C3: A,A3: A,B3: A] :
( ( image @ A @ A @ ( minus_minus @ A @ C3 ) @ ( set_or7035219750837199246ssThan @ A @ A3 @ B3 ) )
= ( set_or3652927894154168847AtMost @ A @ ( minus_minus @ A @ C3 @ B3 ) @ ( minus_minus @ A @ C3 @ A3 ) ) ) ) ).
% image_diff_atLeastLessThan
thf(fact_7695_Ioc__inj,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A3: A,B3: A,C3: A,D2: A] :
( ( ( set_or3652927894154168847AtMost @ A @ A3 @ B3 )
= ( set_or3652927894154168847AtMost @ A @ C3 @ D2 ) )
= ( ( ( ord_less_eq @ A @ B3 @ A3 )
& ( ord_less_eq @ A @ D2 @ C3 ) )
| ( ( A3 = C3 )
& ( B3 = D2 ) ) ) ) ) ).
% Ioc_inj
thf(fact_7696_Ioc__subset__iff,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A3: A,B3: A,C3: A,D2: A] :
( ( ord_less_eq @ ( set @ A ) @ ( set_or3652927894154168847AtMost @ A @ A3 @ B3 ) @ ( set_or3652927894154168847AtMost @ A @ C3 @ D2 ) )
= ( ( ord_less_eq @ A @ B3 @ A3 )
| ( ( ord_less_eq @ A @ C3 @ A3 )
& ( ord_less_eq @ A @ B3 @ D2 ) ) ) ) ) ).
% Ioc_subset_iff
thf(fact_7697_GreatestI__ex__nat,axiom,
! [P: nat > $o,B3: nat] :
( ? [X_12: nat] : ( P @ X_12 )
=> ( ! [Y4: nat] :
( ( P @ Y4 )
=> ( ord_less_eq @ nat @ Y4 @ B3 ) )
=> ( P @ ( order_Greatest @ nat @ P ) ) ) ) ).
% GreatestI_ex_nat
thf(fact_7698_Greatest__le__nat,axiom,
! [P: nat > $o,K: nat,B3: nat] :
( ( P @ K )
=> ( ! [Y4: nat] :
( ( P @ Y4 )
=> ( ord_less_eq @ nat @ Y4 @ B3 ) )
=> ( ord_less_eq @ nat @ K @ ( order_Greatest @ nat @ P ) ) ) ) ).
% Greatest_le_nat
thf(fact_7699_GreatestI__nat,axiom,
! [P: nat > $o,K: nat,B3: nat] :
( ( P @ K )
=> ( ! [Y4: nat] :
( ( P @ Y4 )
=> ( ord_less_eq @ nat @ Y4 @ B3 ) )
=> ( P @ ( order_Greatest @ nat @ P ) ) ) ) ).
% GreatestI_nat
thf(fact_7700_atLeastSucAtMost__greaterThanAtMost,axiom,
! [L: nat,U: nat] :
( ( set_or1337092689740270186AtMost @ nat @ ( suc @ L ) @ U )
= ( set_or3652927894154168847AtMost @ nat @ L @ U ) ) ).
% atLeastSucAtMost_greaterThanAtMost
thf(fact_7701_infinite__Ioc,axiom,
! [A: $tType] :
( ( dense_linorder @ A )
=> ! [A3: A,B3: A] :
( ( ord_less @ A @ A3 @ B3 )
=> ~ ( finite_finite2 @ A @ ( set_or3652927894154168847AtMost @ A @ A3 @ B3 ) ) ) ) ).
% infinite_Ioc
thf(fact_7702_open__left,axiom,
! [A: $tType] :
( ( topolo1944317154257567458pology @ A )
=> ! [S: set @ A,X2: A,Y3: A] :
( ( topolo1002775350975398744n_open @ A @ S )
=> ( ( member @ A @ X2 @ S )
=> ( ( ord_less @ A @ Y3 @ X2 )
=> ? [B4: A] :
( ( ord_less @ A @ B4 @ X2 )
& ( ord_less_eq @ ( set @ A ) @ ( set_or3652927894154168847AtMost @ A @ B4 @ X2 ) @ S ) ) ) ) ) ) ).
% open_left
thf(fact_7703_GreatestI2__order,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [P: A > $o,X2: A,Q: A > $o] :
( ( P @ X2 )
=> ( ! [Y4: A] :
( ( P @ Y4 )
=> ( ord_less_eq @ A @ Y4 @ X2 ) )
=> ( ! [X4: A] :
( ( P @ X4 )
=> ( ! [Y5: A] :
( ( P @ Y5 )
=> ( ord_less_eq @ A @ Y5 @ X4 ) )
=> ( Q @ X4 ) ) )
=> ( Q @ ( order_Greatest @ A @ P ) ) ) ) ) ) ).
% GreatestI2_order
thf(fact_7704_Greatest__equality,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [P: A > $o,X2: A] :
( ( P @ X2 )
=> ( ! [Y4: A] :
( ( P @ Y4 )
=> ( ord_less_eq @ A @ Y4 @ X2 ) )
=> ( ( order_Greatest @ A @ P )
= X2 ) ) ) ) ).
% Greatest_equality
thf(fact_7705_sum_Ohead,axiom,
! [A: $tType] :
( ( comm_monoid_add @ A )
=> ! [M: nat,N: nat,G: nat > A] :
( ( ord_less_eq @ nat @ M @ N )
=> ( ( groups7311177749621191930dd_sum @ nat @ A @ G @ ( set_or1337092689740270186AtMost @ nat @ M @ N ) )
= ( plus_plus @ A @ ( G @ M ) @ ( groups7311177749621191930dd_sum @ nat @ A @ G @ ( set_or3652927894154168847AtMost @ nat @ M @ N ) ) ) ) ) ) ).
% sum.head
thf(fact_7706_prod_Ohead,axiom,
! [A: $tType] :
( ( comm_monoid_mult @ A )
=> ! [M: nat,N: nat,G: nat > A] :
( ( ord_less_eq @ nat @ M @ N )
=> ( ( groups7121269368397514597t_prod @ nat @ A @ G @ ( set_or1337092689740270186AtMost @ nat @ M @ N ) )
= ( times_times @ A @ ( G @ M ) @ ( groups7121269368397514597t_prod @ nat @ A @ G @ ( set_or3652927894154168847AtMost @ nat @ M @ N ) ) ) ) ) ) ).
% prod.head
thf(fact_7707_greaterThanAtMost__subseteq__atLeastAtMost__iff,axiom,
! [A: $tType] :
( ( dense_linorder @ A )
=> ! [A3: A,B3: A,C3: A,D2: A] :
( ( ord_less_eq @ ( set @ A ) @ ( set_or3652927894154168847AtMost @ A @ A3 @ B3 ) @ ( set_or1337092689740270186AtMost @ A @ C3 @ D2 ) )
= ( ( ord_less @ A @ A3 @ B3 )
=> ( ( ord_less_eq @ A @ C3 @ A3 )
& ( ord_less_eq @ A @ B3 @ D2 ) ) ) ) ) ).
% greaterThanAtMost_subseteq_atLeastAtMost_iff
thf(fact_7708_greaterThanAtMost__subseteq__atLeastLessThan__iff,axiom,
! [A: $tType] :
( ( dense_linorder @ A )
=> ! [A3: A,B3: A,C3: A,D2: A] :
( ( ord_less_eq @ ( set @ A ) @ ( set_or3652927894154168847AtMost @ A @ A3 @ B3 ) @ ( set_or7035219750837199246ssThan @ A @ C3 @ D2 ) )
= ( ( ord_less @ A @ A3 @ B3 )
=> ( ( ord_less_eq @ A @ C3 @ A3 )
& ( ord_less @ A @ B3 @ D2 ) ) ) ) ) ).
% greaterThanAtMost_subseteq_atLeastLessThan_iff
thf(fact_7709_greaterThanLessThan__subseteq__greaterThanAtMost__iff,axiom,
! [A: $tType] :
( ( dense_linorder @ A )
=> ! [A3: A,B3: A,C3: A,D2: A] :
( ( ord_less_eq @ ( set @ A ) @ ( set_or5935395276787703475ssThan @ A @ A3 @ B3 ) @ ( set_or3652927894154168847AtMost @ A @ C3 @ D2 ) )
= ( ( ord_less @ A @ A3 @ B3 )
=> ( ( ord_less_eq @ A @ C3 @ A3 )
& ( ord_less_eq @ A @ B3 @ D2 ) ) ) ) ) ).
% greaterThanLessThan_subseteq_greaterThanAtMost_iff
thf(fact_7710_greaterThanAtMost__eq__atLeastAtMost__diff,axiom,
! [A: $tType] :
( ( order @ A )
=> ( ( set_or3652927894154168847AtMost @ A )
= ( ^ [A6: A,B7: A] : ( minus_minus @ ( set @ A ) @ ( set_or1337092689740270186AtMost @ A @ A6 @ B7 ) @ ( insert @ A @ A6 @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ) ).
% greaterThanAtMost_eq_atLeastAtMost_diff
thf(fact_7711_sequentially__imp__eventually__at__right,axiom,
! [A: $tType] :
( ( ( topolo3112930676232923870pology @ A )
& ( topolo1944317154257567458pology @ A ) )
=> ! [A3: A,B3: A,P: A > $o] :
( ( ord_less @ A @ A3 @ B3 )
=> ( ! [F3: nat > A] :
( ! [N9: nat] : ( ord_less @ A @ A3 @ ( F3 @ N9 ) )
=> ( ! [N9: nat] : ( ord_less @ A @ ( F3 @ N9 ) @ B3 )
=> ( ( order_antimono @ nat @ A @ F3 )
=> ( ( filterlim @ nat @ A @ F3 @ ( topolo7230453075368039082e_nhds @ A @ A3 ) @ ( at_top @ nat ) )
=> ( eventually @ nat
@ ^ [N4: nat] : ( P @ ( F3 @ N4 ) )
@ ( at_top @ nat ) ) ) ) ) )
=> ( eventually @ A @ P @ ( topolo174197925503356063within @ A @ A3 @ ( set_ord_greaterThan @ A @ A3 ) ) ) ) ) ) ).
% sequentially_imp_eventually_at_right
thf(fact_7712_suminf__eq__SUP__real,axiom,
! [X: nat > real] :
( ( summable @ real @ X )
=> ( ! [I2: nat] : ( ord_less_eq @ real @ ( zero_zero @ real ) @ ( X @ I2 ) )
=> ( ( suminf @ real @ X )
= ( complete_Sup_Sup @ real
@ ( image @ nat @ real
@ ^ [I4: nat] : ( groups7311177749621191930dd_sum @ nat @ real @ X @ ( set_ord_lessThan @ nat @ I4 ) )
@ ( top_top @ ( set @ nat ) ) ) ) ) ) ) ).
% suminf_eq_SUP_real
thf(fact_7713_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_7714_decseq__const,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [K: A] :
( order_antimono @ nat @ A
@ ^ [X3: nat] : K ) ) ).
% decseq_const
thf(fact_7715_Sup__atLeastAtMost,axiom,
! [A: $tType] :
( ( comple6319245703460814977attice @ A )
=> ! [X2: A,Y3: A] :
( ( ord_less_eq @ A @ X2 @ Y3 )
=> ( ( complete_Sup_Sup @ A @ ( set_or1337092689740270186AtMost @ A @ X2 @ Y3 ) )
= Y3 ) ) ) ).
% Sup_atLeastAtMost
thf(fact_7716_Sup__atLeastLessThan,axiom,
! [A: $tType] :
( ( ( comple6319245703460814977attice @ A )
& ( dense_linorder @ A ) )
=> ! [X2: A,Y3: A] :
( ( ord_less @ A @ X2 @ Y3 )
=> ( ( complete_Sup_Sup @ A @ ( set_or7035219750837199246ssThan @ A @ X2 @ Y3 ) )
= Y3 ) ) ) ).
% Sup_atLeastLessThan
thf(fact_7717_Sup__greaterThanLessThan,axiom,
! [A: $tType] :
( ( ( comple6319245703460814977attice @ A )
& ( dense_linorder @ A ) )
=> ! [X2: A,Y3: A] :
( ( ord_less @ A @ X2 @ Y3 )
=> ( ( complete_Sup_Sup @ A @ ( set_or5935395276787703475ssThan @ A @ X2 @ Y3 ) )
= Y3 ) ) ) ).
% Sup_greaterThanLessThan
thf(fact_7718_Sup__greaterThanAtMost,axiom,
! [A: $tType] :
( ( comple6319245703460814977attice @ A )
=> ! [X2: A,Y3: A] :
( ( ord_less @ A @ X2 @ Y3 )
=> ( ( complete_Sup_Sup @ A @ ( set_or3652927894154168847AtMost @ A @ X2 @ Y3 ) )
= Y3 ) ) ) ).
% Sup_greaterThanAtMost
thf(fact_7719_ccSUP__bot,axiom,
! [B: $tType,A: $tType] :
( ( counta3822494911875563373attice @ A )
=> ! [A2: set @ B] :
( ( complete_Sup_Sup @ A
@ ( image @ B @ A
@ ^ [X3: B] : ( bot_bot @ A )
@ A2 ) )
= ( bot_bot @ A ) ) ) ).
% ccSUP_bot
thf(fact_7720_ccSUP__const,axiom,
! [B: $tType,A: $tType] :
( ( counta3822494911875563373attice @ A )
=> ! [A2: set @ B,F2: A] :
( ( A2
!= ( bot_bot @ ( set @ B ) ) )
=> ( ( complete_Sup_Sup @ A
@ ( image @ B @ A
@ ^ [I4: B] : F2
@ A2 ) )
= F2 ) ) ) ).
% ccSUP_const
thf(fact_7721_Sup__greaterThanAtLeast,axiom,
! [A: $tType] :
( ( comple6319245703460814977attice @ A )
=> ! [X2: A] :
( ( ord_less @ A @ X2 @ ( top_top @ A ) )
=> ( ( complete_Sup_Sup @ A @ ( set_ord_greaterThan @ A @ X2 ) )
= ( top_top @ A ) ) ) ) ).
% Sup_greaterThanAtLeast
thf(fact_7722_ccSUP__empty,axiom,
! [B: $tType,A: $tType] :
( ( counta3822494911875563373attice @ A )
=> ! [F2: B > A] :
( ( complete_Sup_Sup @ A @ ( image @ B @ A @ F2 @ ( bot_bot @ ( set @ B ) ) ) )
= ( bot_bot @ A ) ) ) ).
% ccSUP_empty
thf(fact_7723_sums__SUP,axiom,
! [A: $tType] :
( ( ( comple5582772986160207858norder @ A )
& ( canoni5634975068530333245id_add @ A )
& ( topolo1944317154257567458pology @ A ) )
=> ! [F2: nat > A] :
( sums @ A @ F2
@ ( complete_Sup_Sup @ A
@ ( image @ nat @ A
@ ^ [N4: nat] : ( groups7311177749621191930dd_sum @ nat @ A @ F2 @ ( set_ord_lessThan @ nat @ N4 ) )
@ ( top_top @ ( set @ nat ) ) ) ) ) ) ).
% sums_SUP
thf(fact_7724_antimonoD,axiom,
! [B: $tType,A: $tType] :
( ( ( order @ A )
& ( order @ B ) )
=> ! [F2: A > B,X2: A,Y3: A] :
( ( order_antimono @ A @ B @ F2 )
=> ( ( ord_less_eq @ A @ X2 @ Y3 )
=> ( ord_less_eq @ B @ ( F2 @ Y3 ) @ ( F2 @ X2 ) ) ) ) ) ).
% antimonoD
thf(fact_7725_antimonoE,axiom,
! [B: $tType,A: $tType] :
( ( ( order @ A )
& ( order @ B ) )
=> ! [F2: A > B,X2: A,Y3: A] :
( ( order_antimono @ A @ B @ F2 )
=> ( ( ord_less_eq @ A @ X2 @ Y3 )
=> ( ord_less_eq @ B @ ( F2 @ Y3 ) @ ( F2 @ X2 ) ) ) ) ) ).
% antimonoE
thf(fact_7726_antimonoI,axiom,
! [B: $tType,A: $tType] :
( ( ( order @ A )
& ( order @ B ) )
=> ! [F2: A > B] :
( ! [X4: A,Y4: A] :
( ( ord_less_eq @ A @ X4 @ Y4 )
=> ( ord_less_eq @ B @ ( F2 @ Y4 ) @ ( F2 @ X4 ) ) )
=> ( order_antimono @ A @ B @ F2 ) ) ) ).
% antimonoI
thf(fact_7727_antimono__def,axiom,
! [B: $tType,A: $tType] :
( ( ( order @ A )
& ( order @ B ) )
=> ( ( order_antimono @ A @ B )
= ( ^ [F5: A > B] :
! [X3: A,Y: A] :
( ( ord_less_eq @ A @ X3 @ Y )
=> ( ord_less_eq @ B @ ( F5 @ Y ) @ ( F5 @ X3 ) ) ) ) ) ) ).
% antimono_def
thf(fact_7728_decseq__SucD,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [A2: nat > A,I: nat] :
( ( order_antimono @ nat @ A @ A2 )
=> ( ord_less_eq @ A @ ( A2 @ ( suc @ I ) ) @ ( A2 @ I ) ) ) ) ).
% decseq_SucD
thf(fact_7729_decseq__SucI,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [X: nat > A] :
( ! [N3: nat] : ( ord_less_eq @ A @ ( X @ ( suc @ N3 ) ) @ ( X @ N3 ) )
=> ( order_antimono @ nat @ A @ X ) ) ) ).
% decseq_SucI
thf(fact_7730_decseq__Suc__iff,axiom,
! [A: $tType] :
( ( order @ A )
=> ( ( order_antimono @ nat @ A )
= ( ^ [F5: nat > A] :
! [N4: nat] : ( ord_less_eq @ A @ ( F5 @ ( suc @ N4 ) ) @ ( F5 @ N4 ) ) ) ) ) ).
% decseq_Suc_iff
thf(fact_7731_decseqD,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [F2: nat > A,I: nat,J: nat] :
( ( order_antimono @ nat @ A @ F2 )
=> ( ( ord_less_eq @ nat @ I @ J )
=> ( ord_less_eq @ A @ ( F2 @ J ) @ ( F2 @ I ) ) ) ) ) ).
% decseqD
thf(fact_7732_decseq__def,axiom,
! [A: $tType] :
( ( order @ A )
=> ( ( order_antimono @ nat @ A )
= ( ^ [X7: nat > A] :
! [M6: nat,N4: nat] :
( ( ord_less_eq @ nat @ M6 @ N4 )
=> ( ord_less_eq @ A @ ( X7 @ N4 ) @ ( X7 @ M6 ) ) ) ) ) ) ).
% decseq_def
thf(fact_7733_Some__Sup,axiom,
! [A: $tType] :
( ( comple6319245703460814977attice @ A )
=> ! [A2: set @ A] :
( ( A2
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( some @ A @ ( complete_Sup_Sup @ A @ A2 ) )
= ( complete_Sup_Sup @ ( option @ A ) @ ( image @ A @ ( option @ A ) @ ( some @ A ) @ A2 ) ) ) ) ) ).
% Some_Sup
thf(fact_7734_Some__SUP,axiom,
! [B: $tType,A: $tType] :
( ( comple6319245703460814977attice @ B )
=> ! [A2: set @ A,F2: A > B] :
( ( A2
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( some @ B @ ( complete_Sup_Sup @ B @ ( image @ A @ B @ F2 @ A2 ) ) )
= ( complete_Sup_Sup @ ( option @ B )
@ ( image @ A @ ( option @ B )
@ ^ [X3: A] : ( some @ B @ ( F2 @ X3 ) )
@ A2 ) ) ) ) ) ).
% Some_SUP
thf(fact_7735_Sup__notin__open,axiom,
! [A: $tType] :
( ( topolo8458572112393995274pology @ A )
=> ! [A2: set @ A,X2: A] :
( ( topolo1002775350975398744n_open @ A @ A2 )
=> ( ! [X4: A] :
( ( member @ A @ X4 @ A2 )
=> ( ord_less @ A @ X4 @ X2 ) )
=> ~ ( member @ A @ ( complete_Sup_Sup @ A @ A2 ) @ A2 ) ) ) ) ).
% Sup_notin_open
thf(fact_7736_atLeastPlusOneAtMost__greaterThanAtMost__int,axiom,
! [L: int,U: int] :
( ( set_or1337092689740270186AtMost @ int @ ( plus_plus @ int @ L @ ( one_one @ int ) ) @ U )
= ( set_or3652927894154168847AtMost @ int @ L @ U ) ) ).
% atLeastPlusOneAtMost_greaterThanAtMost_int
thf(fact_7737_min__of__antimono,axiom,
! [B: $tType,A: $tType] :
( ( ( linorder @ A )
& ( linorder @ B ) )
=> ! [F2: A > B,X2: A,Y3: A] :
( ( order_antimono @ A @ B @ F2 )
=> ( ( ord_min @ B @ ( F2 @ X2 ) @ ( F2 @ Y3 ) )
= ( F2 @ ( ord_max @ A @ X2 @ Y3 ) ) ) ) ) ).
% min_of_antimono
thf(fact_7738_max__of__antimono,axiom,
! [B: $tType,A: $tType] :
( ( ( linorder @ A )
& ( linorder @ B ) )
=> ! [F2: A > B,X2: A,Y3: A] :
( ( order_antimono @ A @ B @ F2 )
=> ( ( ord_max @ B @ ( F2 @ X2 ) @ ( F2 @ Y3 ) )
= ( F2 @ ( ord_min @ A @ X2 @ Y3 ) ) ) ) ) ).
% max_of_antimono
thf(fact_7739_cSup__asclose,axiom,
! [A: $tType] :
( ( ( condit6923001295902523014norder @ A )
& ( linordered_idom @ A ) )
=> ! [S: set @ A,L: A,E: A] :
( ( S
!= ( bot_bot @ ( set @ A ) ) )
=> ( ! [X4: A] :
( ( member @ A @ X4 @ S )
=> ( ord_less_eq @ A @ ( abs_abs @ A @ ( minus_minus @ A @ X4 @ L ) ) @ E ) )
=> ( ord_less_eq @ A @ ( abs_abs @ A @ ( minus_minus @ A @ ( complete_Sup_Sup @ A @ S ) @ L ) ) @ E ) ) ) ) ).
% cSup_asclose
thf(fact_7740_decseq__bounded,axiom,
! [X: nat > real,B2: real] :
( ( order_antimono @ nat @ real @ X )
=> ( ! [I2: nat] : ( ord_less_eq @ real @ B2 @ ( X @ I2 ) )
=> ( bfun @ nat @ real @ X @ ( at_top @ nat ) ) ) ) ).
% decseq_bounded
thf(fact_7741_Sup__lim,axiom,
! [A: $tType] :
( ( ( comple5582772986160207858norder @ A )
& ( topolo1944317154257567458pology @ A ) )
=> ! [B3: nat > A,S2: set @ A,A3: A] :
( ! [N3: nat] : ( member @ A @ ( B3 @ N3 ) @ S2 )
=> ( ( filterlim @ nat @ A @ B3 @ ( topolo7230453075368039082e_nhds @ A @ A3 ) @ ( at_top @ nat ) )
=> ( ord_less_eq @ A @ A3 @ ( complete_Sup_Sup @ A @ S2 ) ) ) ) ) ).
% Sup_lim
thf(fact_7742_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_7743_word__atLeastAtMost__Suc__greaterThanAtMost,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [M: word @ A,U: word @ A] :
( ( M
!= ( uminus_uminus @ ( word @ A ) @ ( one_one @ ( word @ A ) ) ) )
=> ( ( set_or3652927894154168847AtMost @ ( word @ A ) @ M @ U )
= ( set_or1337092689740270186AtMost @ ( word @ A ) @ ( plus_plus @ ( word @ A ) @ M @ ( one_one @ ( word @ A ) ) ) @ U ) ) ) ) ).
% word_atLeastAtMost_Suc_greaterThanAtMost
thf(fact_7744_word__range__minus__1_H,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [A3: word @ A,B3: word @ A] :
( ( A3
!= ( zero_zero @ ( word @ A ) ) )
=> ( ( set_or3652927894154168847AtMost @ ( word @ A ) @ ( minus_minus @ ( word @ A ) @ A3 @ ( one_one @ ( word @ A ) ) ) @ B3 )
= ( set_or1337092689740270186AtMost @ ( word @ A ) @ A3 @ B3 ) ) ) ) ).
% word_range_minus_1'
thf(fact_7745_decseq__ge,axiom,
! [A: $tType] :
( ( topolo1944317154257567458pology @ A )
=> ! [X: nat > A,L6: A,N: nat] :
( ( order_antimono @ nat @ A @ X )
=> ( ( filterlim @ nat @ A @ X @ ( topolo7230453075368039082e_nhds @ A @ L6 ) @ ( at_top @ nat ) )
=> ( ord_less_eq @ A @ L6 @ ( X @ N ) ) ) ) ) ).
% decseq_ge
thf(fact_7746_decseq__convergent,axiom,
! [X: nat > real,B2: real] :
( ( order_antimono @ nat @ real @ X )
=> ( ! [I2: nat] : ( ord_less_eq @ real @ B2 @ ( X @ I2 ) )
=> ~ ! [L7: real] :
( ( filterlim @ nat @ real @ X @ ( topolo7230453075368039082e_nhds @ real @ L7 ) @ ( at_top @ nat ) )
=> ~ ! [I5: nat] : ( ord_less_eq @ real @ L7 @ ( X @ I5 ) ) ) ) ) ).
% decseq_convergent
thf(fact_7747_suminf__eq__SUP,axiom,
! [A: $tType] :
( ( ( comple5582772986160207858norder @ A )
& ( canoni5634975068530333245id_add @ A )
& ( topolo1944317154257567458pology @ A ) )
=> ( ( suminf @ A )
= ( ^ [F5: nat > A] :
( complete_Sup_Sup @ A
@ ( image @ nat @ A
@ ^ [N4: nat] : ( groups7311177749621191930dd_sum @ nat @ A @ F5 @ ( set_ord_lessThan @ nat @ N4 ) )
@ ( top_top @ ( set @ nat ) ) ) ) ) ) ) ).
% suminf_eq_SUP
thf(fact_7748_tendsto__at__right__sequentially,axiom,
! [C: $tType,B: $tType] :
( ( ( topolo3112930676232923870pology @ B )
& ( topolo1944317154257567458pology @ B )
& ( topolo4958980785337419405_space @ C ) )
=> ! [A3: B,B3: B,X: B > C,L6: C] :
( ( ord_less @ B @ A3 @ B3 )
=> ( ! [S6: nat > B] :
( ! [N9: nat] : ( ord_less @ B @ A3 @ ( S6 @ N9 ) )
=> ( ! [N9: nat] : ( ord_less @ B @ ( S6 @ N9 ) @ B3 )
=> ( ( order_antimono @ nat @ B @ S6 )
=> ( ( filterlim @ nat @ B @ S6 @ ( topolo7230453075368039082e_nhds @ B @ A3 ) @ ( at_top @ nat ) )
=> ( filterlim @ nat @ C
@ ^ [N4: nat] : ( X @ ( S6 @ N4 ) )
@ ( topolo7230453075368039082e_nhds @ C @ L6 )
@ ( at_top @ nat ) ) ) ) ) )
=> ( filterlim @ B @ C @ X @ ( topolo7230453075368039082e_nhds @ C @ L6 ) @ ( topolo174197925503356063within @ B @ A3 @ ( set_ord_greaterThan @ B @ A3 ) ) ) ) ) ) ).
% tendsto_at_right_sequentially
thf(fact_7749_SUP__eq__top__iff,axiom,
! [B: $tType,A: $tType] :
( ( comple5582772986160207858norder @ A )
=> ! [F2: B > A,A2: set @ B] :
( ( ( complete_Sup_Sup @ A @ ( image @ B @ A @ F2 @ A2 ) )
= ( top_top @ A ) )
= ( ! [X3: A] :
( ( ord_less @ A @ X3 @ ( top_top @ A ) )
=> ? [Y: B] :
( ( member @ B @ Y @ A2 )
& ( ord_less @ A @ X3 @ ( F2 @ Y ) ) ) ) ) ) ) ).
% SUP_eq_top_iff
thf(fact_7750_SUP__const,axiom,
! [B: $tType,A: $tType] :
( ( comple6319245703460814977attice @ A )
=> ! [A2: set @ B,F2: A] :
( ( A2
!= ( bot_bot @ ( set @ B ) ) )
=> ( ( complete_Sup_Sup @ A
@ ( image @ B @ A
@ ^ [I4: B] : F2
@ A2 ) )
= F2 ) ) ) ).
% SUP_const
thf(fact_7751_subset__mset_OcSUP__const,axiom,
! [B: $tType,A: $tType,A2: set @ B,C3: multiset @ A] :
( ( A2
!= ( bot_bot @ ( set @ B ) ) )
=> ( ( complete_Sup_Sup @ ( multiset @ A )
@ ( image @ B @ ( multiset @ A )
@ ^ [X3: B] : C3
@ A2 ) )
= C3 ) ) ).
% subset_mset.cSUP_const
thf(fact_7752_Sup__apply,axiom,
! [B: $tType,A: $tType] :
( ( complete_Sup @ B )
=> ( ( complete_Sup_Sup @ ( A > B ) )
= ( ^ [A7: set @ ( A > B ),X3: A] :
( complete_Sup_Sup @ B
@ ( image @ ( A > B ) @ B
@ ^ [F5: A > B] : ( F5 @ X3 )
@ A7 ) ) ) ) ) ).
% Sup_apply
thf(fact_7753_UN__I,axiom,
! [B: $tType,A: $tType,A3: A,A2: set @ A,B3: B,B2: A > ( set @ B )] :
( ( member @ A @ A3 @ A2 )
=> ( ( member @ B @ B3 @ ( B2 @ A3 ) )
=> ( member @ B @ B3 @ ( complete_Sup_Sup @ ( set @ B ) @ ( image @ A @ ( set @ B ) @ B2 @ A2 ) ) ) ) ) ).
% UN_I
thf(fact_7754_UN__iff,axiom,
! [A: $tType,B: $tType,B3: A,B2: B > ( set @ A ),A2: set @ B] :
( ( member @ A @ B3 @ ( complete_Sup_Sup @ ( set @ A ) @ ( image @ B @ ( set @ A ) @ B2 @ A2 ) ) )
= ( ? [X3: B] :
( ( member @ B @ X3 @ A2 )
& ( member @ A @ B3 @ ( B2 @ X3 ) ) ) ) ) ).
% UN_iff
thf(fact_7755_SUP__apply,axiom,
! [A: $tType,B: $tType,C: $tType] :
( ( complete_Sup @ A )
=> ! [F2: C > B > A,A2: set @ C,X2: B] :
( ( complete_Sup_Sup @ ( B > A ) @ ( image @ C @ ( B > A ) @ F2 @ A2 ) @ X2 )
= ( complete_Sup_Sup @ A
@ ( image @ C @ A
@ ^ [Y: C] : ( F2 @ Y @ X2 )
@ A2 ) ) ) ) ).
% SUP_apply
thf(fact_7756_SUP__identity__eq,axiom,
! [A: $tType] :
( ( complete_Sup @ A )
=> ! [A2: set @ A] :
( ( complete_Sup_Sup @ A
@ ( image @ A @ A
@ ^ [X3: A] : X3
@ A2 ) )
= ( complete_Sup_Sup @ A @ A2 ) ) ) ).
% SUP_identity_eq
thf(fact_7757_UN__constant,axiom,
! [B: $tType,A: $tType,A2: set @ B,C3: set @ A] :
( ( ( A2
= ( bot_bot @ ( set @ B ) ) )
=> ( ( complete_Sup_Sup @ ( set @ A )
@ ( image @ B @ ( set @ A )
@ ^ [Y: B] : C3
@ A2 ) )
= ( bot_bot @ ( set @ A ) ) ) )
& ( ( A2
!= ( bot_bot @ ( set @ B ) ) )
=> ( ( complete_Sup_Sup @ ( set @ A )
@ ( image @ B @ ( set @ A )
@ ^ [Y: B] : C3
@ A2 ) )
= C3 ) ) ) ).
% UN_constant
thf(fact_7758_finite__UN__I,axiom,
! [B: $tType,A: $tType,A2: set @ A,B2: A > ( set @ B )] :
( ( finite_finite2 @ A @ A2 )
=> ( ! [A4: A] :
( ( member @ A @ A4 @ A2 )
=> ( finite_finite2 @ B @ ( B2 @ A4 ) ) )
=> ( finite_finite2 @ B @ ( complete_Sup_Sup @ ( set @ B ) @ ( image @ A @ ( set @ B ) @ B2 @ A2 ) ) ) ) ) ).
% finite_UN_I
thf(fact_7759_open__UN,axiom,
! [A: $tType,B: $tType] :
( ( topolo4958980785337419405_space @ A )
=> ! [A2: set @ B,B2: B > ( set @ A )] :
( ! [X4: B] :
( ( member @ B @ X4 @ A2 )
=> ( topolo1002775350975398744n_open @ A @ ( B2 @ X4 ) ) )
=> ( topolo1002775350975398744n_open @ A @ ( complete_Sup_Sup @ ( set @ A ) @ ( image @ B @ ( set @ A ) @ B2 @ A2 ) ) ) ) ) ).
% open_UN
thf(fact_7760_empty__Sup,axiom,
! [A: $tType] :
( ( comple6319245703460814977attice @ A )
=> ( ( complete_Sup_Sup @ ( option @ A ) @ ( bot_bot @ ( set @ ( option @ A ) ) ) )
= ( none @ A ) ) ) ).
% empty_Sup
thf(fact_7761_Sup__eq__top__iff,axiom,
! [A: $tType] :
( ( comple5582772986160207858norder @ A )
=> ! [A2: set @ A] :
( ( ( complete_Sup_Sup @ A @ A2 )
= ( top_top @ A ) )
= ( ! [X3: A] :
( ( ord_less @ A @ X3 @ ( top_top @ A ) )
=> ? [Y: A] :
( ( member @ A @ Y @ A2 )
& ( ord_less @ A @ X3 @ Y ) ) ) ) ) ) ).
% Sup_eq_top_iff
thf(fact_7762_SUP__bot,axiom,
! [B: $tType,A: $tType] :
( ( comple6319245703460814977attice @ A )
=> ! [A2: set @ B] :
( ( complete_Sup_Sup @ A
@ ( image @ B @ A
@ ^ [X3: B] : ( bot_bot @ A )
@ A2 ) )
= ( bot_bot @ A ) ) ) ).
% SUP_bot
thf(fact_7763_SUP__bot__conv_I1_J,axiom,
! [A: $tType,B: $tType] :
( ( comple6319245703460814977attice @ A )
=> ! [B2: B > A,A2: set @ B] :
( ( ( complete_Sup_Sup @ A @ ( image @ B @ A @ B2 @ A2 ) )
= ( bot_bot @ A ) )
= ( ! [X3: B] :
( ( member @ B @ X3 @ A2 )
=> ( ( B2 @ X3 )
= ( bot_bot @ A ) ) ) ) ) ) ).
% SUP_bot_conv(1)
thf(fact_7764_SUP__bot__conv_I2_J,axiom,
! [A: $tType,B: $tType] :
( ( comple6319245703460814977attice @ A )
=> ! [B2: B > A,A2: set @ B] :
( ( ( bot_bot @ A )
= ( complete_Sup_Sup @ A @ ( image @ B @ A @ B2 @ A2 ) ) )
= ( ! [X3: B] :
( ( member @ B @ X3 @ A2 )
=> ( ( B2 @ X3 )
= ( bot_bot @ A ) ) ) ) ) ) ).
% SUP_bot_conv(2)
thf(fact_7765_UN__simps_I1_J,axiom,
! [A: $tType,B: $tType,C2: set @ B,A3: A,B2: B > ( set @ A )] :
( ( ( C2
= ( bot_bot @ ( set @ B ) ) )
=> ( ( complete_Sup_Sup @ ( set @ A )
@ ( image @ B @ ( set @ A )
@ ^ [X3: B] : ( insert @ A @ A3 @ ( B2 @ X3 ) )
@ C2 ) )
= ( bot_bot @ ( set @ A ) ) ) )
& ( ( C2
!= ( bot_bot @ ( set @ B ) ) )
=> ( ( complete_Sup_Sup @ ( set @ A )
@ ( image @ B @ ( set @ A )
@ ^ [X3: B] : ( insert @ A @ A3 @ ( B2 @ X3 ) )
@ C2 ) )
= ( insert @ A @ A3 @ ( complete_Sup_Sup @ ( set @ A ) @ ( image @ B @ ( set @ A ) @ B2 @ C2 ) ) ) ) ) ) ).
% UN_simps(1)
thf(fact_7766_UN__singleton,axiom,
! [A: $tType,A2: set @ A] :
( ( complete_Sup_Sup @ ( set @ A )
@ ( image @ A @ ( set @ A )
@ ^ [X3: A] : ( insert @ A @ X3 @ ( bot_bot @ ( set @ A ) ) )
@ A2 ) )
= A2 ) ).
% UN_singleton
thf(fact_7767_singleton__None__Sup,axiom,
! [A: $tType] :
( ( comple6319245703460814977attice @ A )
=> ( ( complete_Sup_Sup @ ( option @ A ) @ ( insert @ ( option @ A ) @ ( none @ A ) @ ( bot_bot @ ( set @ ( option @ A ) ) ) ) )
= ( none @ A ) ) ) ).
% singleton_None_Sup
thf(fact_7768_set__concat,axiom,
! [A: $tType,Xs: list @ ( list @ A )] :
( ( set2 @ A @ ( concat @ A @ Xs ) )
= ( complete_Sup_Sup @ ( set @ A ) @ ( image @ ( list @ A ) @ ( set @ A ) @ ( set2 @ A ) @ ( set2 @ ( list @ A ) @ Xs ) ) ) ) ).
% set_concat
thf(fact_7769_Union__mono,axiom,
! [A: $tType,A2: set @ ( set @ A ),B2: set @ ( set @ A )] :
( ( ord_less_eq @ ( set @ ( set @ A ) ) @ A2 @ B2 )
=> ( ord_less_eq @ ( set @ A ) @ ( complete_Sup_Sup @ ( set @ A ) @ A2 ) @ ( complete_Sup_Sup @ ( set @ A ) @ B2 ) ) ) ).
% Union_mono
thf(fact_7770_Union__least,axiom,
! [A: $tType,A2: set @ ( set @ A ),C2: set @ A] :
( ! [X17: set @ A] :
( ( member @ ( set @ A ) @ X17 @ A2 )
=> ( ord_less_eq @ ( set @ A ) @ X17 @ C2 ) )
=> ( ord_less_eq @ ( set @ A ) @ ( complete_Sup_Sup @ ( set @ A ) @ A2 ) @ C2 ) ) ).
% Union_least
thf(fact_7771_Union__upper,axiom,
! [A: $tType,B2: set @ A,A2: set @ ( set @ A )] :
( ( member @ ( set @ A ) @ B2 @ A2 )
=> ( ord_less_eq @ ( set @ A ) @ B2 @ ( complete_Sup_Sup @ ( set @ A ) @ A2 ) ) ) ).
% Union_upper
thf(fact_7772_Union__subsetI,axiom,
! [A: $tType,A2: set @ ( set @ A ),B2: set @ ( set @ A )] :
( ! [X4: set @ A] :
( ( member @ ( set @ A ) @ X4 @ A2 )
=> ? [Y5: set @ A] :
( ( member @ ( set @ A ) @ Y5 @ B2 )
& ( ord_less_eq @ ( set @ A ) @ X4 @ Y5 ) ) )
=> ( ord_less_eq @ ( set @ A ) @ ( complete_Sup_Sup @ ( set @ A ) @ A2 ) @ ( complete_Sup_Sup @ ( set @ A ) @ B2 ) ) ) ).
% Union_subsetI
thf(fact_7773_bij__betw__UNION__chain,axiom,
! [B: $tType,C: $tType,A: $tType,I6: set @ A,A2: A > ( set @ B ),F2: B > C,A8: A > ( set @ C )] :
( ! [I2: A,J2: A] :
( ( member @ A @ I2 @ I6 )
=> ( ( member @ A @ J2 @ I6 )
=> ( ( ord_less_eq @ ( set @ B ) @ ( A2 @ I2 ) @ ( A2 @ J2 ) )
| ( ord_less_eq @ ( set @ B ) @ ( A2 @ J2 ) @ ( A2 @ I2 ) ) ) ) )
=> ( ! [I2: A] :
( ( member @ A @ I2 @ I6 )
=> ( bij_betw @ B @ C @ F2 @ ( A2 @ I2 ) @ ( A8 @ I2 ) ) )
=> ( bij_betw @ B @ C @ F2 @ ( complete_Sup_Sup @ ( set @ B ) @ ( image @ A @ ( set @ B ) @ A2 @ I6 ) ) @ ( complete_Sup_Sup @ ( set @ C ) @ ( image @ A @ ( set @ C ) @ A8 @ I6 ) ) ) ) ) ).
% bij_betw_UNION_chain
thf(fact_7774_UN__subset__iff,axiom,
! [A: $tType,B: $tType,A2: B > ( set @ A ),I6: set @ B,B2: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ ( complete_Sup_Sup @ ( set @ A ) @ ( image @ B @ ( set @ A ) @ A2 @ I6 ) ) @ B2 )
= ( ! [X3: B] :
( ( member @ B @ X3 @ I6 )
=> ( ord_less_eq @ ( set @ A ) @ ( A2 @ X3 ) @ B2 ) ) ) ) ).
% UN_subset_iff
thf(fact_7775_UN__upper,axiom,
! [B: $tType,A: $tType,A3: A,A2: set @ A,B2: A > ( set @ B )] :
( ( member @ A @ A3 @ A2 )
=> ( ord_less_eq @ ( set @ B ) @ ( B2 @ A3 ) @ ( complete_Sup_Sup @ ( set @ B ) @ ( image @ A @ ( set @ B ) @ B2 @ A2 ) ) ) ) ).
% UN_upper
thf(fact_7776_UN__least,axiom,
! [A: $tType,B: $tType,A2: set @ A,B2: A > ( set @ B ),C2: set @ B] :
( ! [X4: A] :
( ( member @ A @ X4 @ A2 )
=> ( ord_less_eq @ ( set @ B ) @ ( B2 @ X4 ) @ C2 ) )
=> ( ord_less_eq @ ( set @ B ) @ ( complete_Sup_Sup @ ( set @ B ) @ ( image @ A @ ( set @ B ) @ B2 @ A2 ) ) @ C2 ) ) ).
% UN_least
thf(fact_7777_UN__mono,axiom,
! [B: $tType,A: $tType,A2: set @ A,B2: set @ A,F2: A > ( set @ B ),G: A > ( set @ B )] :
( ( ord_less_eq @ ( set @ A ) @ A2 @ B2 )
=> ( ! [X4: A] :
( ( member @ A @ X4 @ A2 )
=> ( ord_less_eq @ ( set @ B ) @ ( F2 @ X4 ) @ ( G @ X4 ) ) )
=> ( ord_less_eq @ ( set @ B ) @ ( complete_Sup_Sup @ ( set @ B ) @ ( image @ A @ ( set @ B ) @ F2 @ A2 ) ) @ ( complete_Sup_Sup @ ( set @ B ) @ ( image @ A @ ( set @ B ) @ G @ B2 ) ) ) ) ) ).
% UN_mono
thf(fact_7778_inj__on__UNION__chain,axiom,
! [C: $tType,B: $tType,A: $tType,I6: set @ A,A2: A > ( set @ B ),F2: B > C] :
( ! [I2: A,J2: A] :
( ( member @ A @ I2 @ I6 )
=> ( ( member @ A @ J2 @ I6 )
=> ( ( ord_less_eq @ ( set @ B ) @ ( A2 @ I2 ) @ ( A2 @ J2 ) )
| ( ord_less_eq @ ( set @ B ) @ ( A2 @ J2 ) @ ( A2 @ I2 ) ) ) ) )
=> ( ! [I2: A] :
( ( member @ A @ I2 @ I6 )
=> ( inj_on @ B @ C @ F2 @ ( A2 @ I2 ) ) )
=> ( inj_on @ B @ C @ F2 @ ( complete_Sup_Sup @ ( set @ B ) @ ( image @ A @ ( set @ B ) @ A2 @ I6 ) ) ) ) ) ).
% inj_on_UNION_chain
thf(fact_7779_UN__lessThan__UNIV,axiom,
( ( complete_Sup_Sup @ ( set @ nat ) @ ( image @ nat @ ( set @ nat ) @ ( set_ord_lessThan @ nat ) @ ( top_top @ ( set @ nat ) ) ) )
= ( top_top @ ( set @ nat ) ) ) ).
% UN_lessThan_UNIV
thf(fact_7780_UN__atMost__UNIV,axiom,
( ( complete_Sup_Sup @ ( set @ nat ) @ ( image @ nat @ ( set @ nat ) @ ( set_ord_atMost @ nat ) @ ( top_top @ ( set @ nat ) ) ) )
= ( top_top @ ( set @ nat ) ) ) ).
% UN_atMost_UNIV
thf(fact_7781_finite__UNION__then__finite,axiom,
! [A: $tType,B: $tType,B2: B > ( set @ A ),A2: set @ B,A3: B] :
( ( finite_finite2 @ A @ ( complete_Sup_Sup @ ( set @ A ) @ ( image @ B @ ( set @ A ) @ B2 @ A2 ) ) )
=> ( ( member @ B @ A3 @ A2 )
=> ( finite_finite2 @ A @ ( B2 @ A3 ) ) ) ) ).
% finite_UNION_then_finite
thf(fact_7782_UN__extend__simps_I10_J,axiom,
! [V6: $tType,U5: $tType,T: $tType,B2: U5 > ( set @ V6 ),F2: T > U5,A2: set @ T] :
( ( complete_Sup_Sup @ ( set @ V6 )
@ ( image @ T @ ( set @ V6 )
@ ^ [A6: T] : ( B2 @ ( F2 @ A6 ) )
@ A2 ) )
= ( complete_Sup_Sup @ ( set @ V6 ) @ ( image @ U5 @ ( set @ V6 ) @ B2 @ ( image @ T @ U5 @ F2 @ A2 ) ) ) ) ).
% UN_extend_simps(10)
thf(fact_7783_UN__extend__simps_I9_J,axiom,
! [S11: $tType,R7: $tType,Q8: $tType,C2: R7 > ( set @ S11 ),B2: Q8 > ( set @ R7 ),A2: set @ Q8] :
( ( complete_Sup_Sup @ ( set @ S11 )
@ ( image @ Q8 @ ( set @ S11 )
@ ^ [X3: Q8] : ( complete_Sup_Sup @ ( set @ S11 ) @ ( image @ R7 @ ( set @ S11 ) @ C2 @ ( B2 @ X3 ) ) )
@ A2 ) )
= ( complete_Sup_Sup @ ( set @ S11 ) @ ( image @ R7 @ ( set @ S11 ) @ C2 @ ( complete_Sup_Sup @ ( set @ R7 ) @ ( image @ Q8 @ ( set @ R7 ) @ B2 @ A2 ) ) ) ) ) ).
% UN_extend_simps(9)
thf(fact_7784_UN__extend__simps_I8_J,axiom,
! [P10: $tType,O2: $tType,B2: O2 > ( set @ P10 ),A2: set @ ( set @ O2 )] :
( ( complete_Sup_Sup @ ( set @ P10 )
@ ( image @ ( set @ O2 ) @ ( set @ P10 )
@ ^ [Y: set @ O2] : ( complete_Sup_Sup @ ( set @ P10 ) @ ( image @ O2 @ ( set @ P10 ) @ B2 @ Y ) )
@ A2 ) )
= ( complete_Sup_Sup @ ( set @ P10 ) @ ( image @ O2 @ ( set @ P10 ) @ B2 @ ( complete_Sup_Sup @ ( set @ O2 ) @ A2 ) ) ) ) ).
% UN_extend_simps(8)
thf(fact_7785_UN__E,axiom,
! [A: $tType,B: $tType,B3: A,B2: B > ( set @ A ),A2: set @ B] :
( ( member @ A @ B3 @ ( complete_Sup_Sup @ ( set @ A ) @ ( image @ B @ ( set @ A ) @ B2 @ A2 ) ) )
=> ~ ! [X4: B] :
( ( member @ B @ X4 @ A2 )
=> ~ ( member @ A @ B3 @ ( B2 @ X4 ) ) ) ) ).
% UN_E
thf(fact_7786_image__UN,axiom,
! [A: $tType,B: $tType,C: $tType,F2: B > A,B2: C > ( set @ B ),A2: set @ C] :
( ( image @ B @ A @ F2 @ ( complete_Sup_Sup @ ( set @ B ) @ ( image @ C @ ( set @ B ) @ B2 @ A2 ) ) )
= ( complete_Sup_Sup @ ( set @ A )
@ ( image @ C @ ( set @ A )
@ ^ [X3: C] : ( image @ B @ A @ F2 @ ( B2 @ X3 ) )
@ A2 ) ) ) ).
% image_UN
thf(fact_7787_image__Union,axiom,
! [A: $tType,B: $tType,F2: B > A,S: set @ ( set @ B )] :
( ( image @ B @ A @ F2 @ ( complete_Sup_Sup @ ( set @ B ) @ S ) )
= ( complete_Sup_Sup @ ( set @ A ) @ ( image @ ( set @ B ) @ ( set @ A ) @ ( image @ B @ A @ F2 ) @ S ) ) ) ).
% image_Union
thf(fact_7788_UN__UN__flatten,axiom,
! [A: $tType,B: $tType,C: $tType,C2: B > ( set @ A ),B2: C > ( set @ B ),A2: set @ C] :
( ( complete_Sup_Sup @ ( set @ A ) @ ( image @ B @ ( set @ A ) @ C2 @ ( complete_Sup_Sup @ ( set @ B ) @ ( image @ C @ ( set @ B ) @ B2 @ A2 ) ) ) )
= ( complete_Sup_Sup @ ( set @ A )
@ ( image @ C @ ( set @ A )
@ ^ [Y: C] : ( complete_Sup_Sup @ ( set @ A ) @ ( image @ B @ ( set @ A ) @ C2 @ ( B2 @ Y ) ) )
@ A2 ) ) ) ).
% UN_UN_flatten
thf(fact_7789_UN__insert__distrib,axiom,
! [B: $tType,A: $tType,U: A,A2: set @ A,A3: B,B2: A > ( set @ B )] :
( ( member @ A @ U @ A2 )
=> ( ( complete_Sup_Sup @ ( set @ B )
@ ( image @ A @ ( set @ B )
@ ^ [X3: A] : ( insert @ B @ A3 @ ( B2 @ X3 ) )
@ A2 ) )
= ( insert @ B @ A3 @ ( complete_Sup_Sup @ ( set @ B ) @ ( image @ A @ ( set @ B ) @ B2 @ A2 ) ) ) ) ) ).
% UN_insert_distrib
thf(fact_7790_SUP__UN__eq,axiom,
! [B: $tType,A: $tType,R2: B > ( set @ A ),S: set @ B] :
( ( complete_Sup_Sup @ ( A > $o )
@ ( image @ B @ ( A > $o )
@ ^ [I4: B,X3: A] : ( member @ A @ X3 @ ( R2 @ I4 ) )
@ S ) )
= ( ^ [X3: A] : ( member @ A @ X3 @ ( complete_Sup_Sup @ ( set @ A ) @ ( image @ B @ ( set @ A ) @ R2 @ S ) ) ) ) ) ).
% SUP_UN_eq
thf(fact_7791_UN__extend__simps_I6_J,axiom,
! [L5: $tType,K8: $tType,A2: K8 > ( set @ L5 ),C2: set @ K8,B2: set @ L5] :
( ( minus_minus @ ( set @ L5 ) @ ( complete_Sup_Sup @ ( set @ L5 ) @ ( image @ K8 @ ( set @ L5 ) @ A2 @ C2 ) ) @ B2 )
= ( complete_Sup_Sup @ ( set @ L5 )
@ ( image @ K8 @ ( set @ L5 )
@ ^ [X3: K8] : ( minus_minus @ ( set @ L5 ) @ ( A2 @ X3 ) @ B2 )
@ C2 ) ) ) ).
% UN_extend_simps(6)
thf(fact_7792_SUP__UN__eq2,axiom,
! [B: $tType,C: $tType,A: $tType,R2: C > ( set @ ( product_prod @ A @ B ) ),S: set @ C] :
( ( complete_Sup_Sup @ ( A > B > $o )
@ ( image @ C @ ( A > B > $o )
@ ^ [I4: C,X3: A,Y: B] : ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X3 @ Y ) @ ( R2 @ I4 ) )
@ S ) )
= ( ^ [X3: A,Y: B] : ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X3 @ Y ) @ ( complete_Sup_Sup @ ( set @ ( product_prod @ A @ B ) ) @ ( image @ C @ ( set @ ( product_prod @ A @ B ) ) @ R2 @ S ) ) ) ) ) ).
% SUP_UN_eq2
thf(fact_7793_SUP__Sup__eq2,axiom,
! [B: $tType,A: $tType,S: set @ ( set @ ( product_prod @ A @ B ) )] :
( ( complete_Sup_Sup @ ( A > B > $o )
@ ( image @ ( set @ ( product_prod @ A @ B ) ) @ ( A > B > $o )
@ ^ [I4: set @ ( product_prod @ A @ B ),X3: A,Y: B] : ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X3 @ Y ) @ I4 )
@ S ) )
= ( ^ [X3: A,Y: B] : ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X3 @ Y ) @ ( complete_Sup_Sup @ ( set @ ( product_prod @ A @ B ) ) @ S ) ) ) ) ).
% SUP_Sup_eq2
thf(fact_7794_Sup__SUP__eq2,axiom,
! [B: $tType,A: $tType] :
( ( complete_Sup_Sup @ ( A > B > $o ) )
= ( ^ [S8: set @ ( A > B > $o ),X3: A,Y: B] : ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X3 @ Y ) @ ( 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 ) @ S8 ) ) ) ) ) ) ).
% Sup_SUP_eq2
thf(fact_7795_UNION__empty__conv_I2_J,axiom,
! [A: $tType,B: $tType,B2: B > ( set @ A ),A2: set @ B] :
( ( ( complete_Sup_Sup @ ( set @ A ) @ ( image @ B @ ( set @ A ) @ B2 @ A2 ) )
= ( bot_bot @ ( set @ A ) ) )
= ( ! [X3: B] :
( ( member @ B @ X3 @ A2 )
=> ( ( B2 @ X3 )
= ( bot_bot @ ( set @ A ) ) ) ) ) ) ).
% UNION_empty_conv(2)
thf(fact_7796_UNION__empty__conv_I1_J,axiom,
! [A: $tType,B: $tType,B2: B > ( set @ A ),A2: set @ B] :
( ( ( bot_bot @ ( set @ A ) )
= ( complete_Sup_Sup @ ( set @ A ) @ ( image @ B @ ( set @ A ) @ B2 @ A2 ) ) )
= ( ! [X3: B] :
( ( member @ B @ X3 @ A2 )
=> ( ( B2 @ X3 )
= ( bot_bot @ ( set @ A ) ) ) ) ) ) ).
% UNION_empty_conv(1)
thf(fact_7797_UN__empty,axiom,
! [B: $tType,A: $tType,B2: B > ( set @ A )] :
( ( complete_Sup_Sup @ ( set @ A ) @ ( image @ B @ ( set @ A ) @ B2 @ ( bot_bot @ ( set @ B ) ) ) )
= ( bot_bot @ ( set @ A ) ) ) ).
% UN_empty
thf(fact_7798_UN__empty2,axiom,
! [B: $tType,A: $tType,A2: set @ B] :
( ( complete_Sup_Sup @ ( set @ A )
@ ( image @ B @ ( set @ A )
@ ^ [X3: B] : ( bot_bot @ ( set @ A ) )
@ A2 ) )
= ( bot_bot @ ( set @ A ) ) ) ).
% UN_empty2
thf(fact_7799_UN__extend__simps_I1_J,axiom,
! [A: $tType,B: $tType,C2: set @ B,A3: A,B2: B > ( set @ A )] :
( ( ( C2
= ( bot_bot @ ( set @ B ) ) )
=> ( ( insert @ A @ A3 @ ( complete_Sup_Sup @ ( set @ A ) @ ( image @ B @ ( set @ A ) @ B2 @ C2 ) ) )
= ( insert @ A @ A3 @ ( bot_bot @ ( set @ A ) ) ) ) )
& ( ( C2
!= ( bot_bot @ ( set @ B ) ) )
=> ( ( insert @ A @ A3 @ ( complete_Sup_Sup @ ( set @ A ) @ ( image @ B @ ( set @ A ) @ B2 @ C2 ) ) )
= ( complete_Sup_Sup @ ( set @ A )
@ ( image @ B @ ( set @ A )
@ ^ [X3: B] : ( insert @ A @ A3 @ ( B2 @ X3 ) )
@ C2 ) ) ) ) ) ).
% UN_extend_simps(1)
thf(fact_7800_UNION__singleton__eq__range,axiom,
! [A: $tType,B: $tType,F2: B > A,A2: set @ B] :
( ( complete_Sup_Sup @ ( set @ A )
@ ( image @ B @ ( set @ A )
@ ^ [X3: B] : ( insert @ A @ ( F2 @ X3 ) @ ( bot_bot @ ( set @ A ) ) )
@ A2 ) )
= ( image @ B @ A @ F2 @ A2 ) ) ).
% UNION_singleton_eq_range
thf(fact_7801_finite__subset__Union,axiom,
! [A: $tType,A2: set @ A,B11: set @ ( set @ A )] :
( ( finite_finite2 @ A @ A2 )
=> ( ( ord_less_eq @ ( set @ A ) @ A2 @ ( complete_Sup_Sup @ ( set @ A ) @ B11 ) )
=> ~ ! [F17: set @ ( set @ A )] :
( ( finite_finite2 @ ( set @ A ) @ F17 )
=> ( ( ord_less_eq @ ( set @ ( set @ A ) ) @ F17 @ B11 )
=> ~ ( ord_less_eq @ ( set @ A ) @ A2 @ ( complete_Sup_Sup @ ( set @ A ) @ F17 ) ) ) ) ) ) ).
% finite_subset_Union
thf(fact_7802_UN__UN__finite__eq,axiom,
! [A: $tType,A2: nat > ( set @ A )] :
( ( complete_Sup_Sup @ ( set @ A )
@ ( image @ nat @ ( set @ A )
@ ^ [N4: nat] : ( complete_Sup_Sup @ ( set @ A ) @ ( image @ nat @ ( set @ A ) @ A2 @ ( set_or7035219750837199246ssThan @ nat @ ( zero_zero @ nat ) @ N4 ) ) )
@ ( top_top @ ( set @ nat ) ) ) )
= ( complete_Sup_Sup @ ( set @ A ) @ ( image @ nat @ ( set @ A ) @ A2 @ ( top_top @ ( set @ nat ) ) ) ) ) ).
% UN_UN_finite_eq
thf(fact_7803_UN__le__add__shift__strict,axiom,
! [A: $tType,M8: nat > ( set @ A ),K: nat,N: nat] :
( ( complete_Sup_Sup @ ( set @ A )
@ ( image @ nat @ ( set @ A )
@ ^ [I4: nat] : ( M8 @ ( plus_plus @ nat @ I4 @ K ) )
@ ( set_ord_lessThan @ nat @ N ) ) )
= ( complete_Sup_Sup @ ( set @ A ) @ ( image @ nat @ ( set @ A ) @ M8 @ ( set_or7035219750837199246ssThan @ nat @ K @ ( plus_plus @ nat @ N @ K ) ) ) ) ) ).
% UN_le_add_shift_strict
thf(fact_7804_UN__le__add__shift,axiom,
! [A: $tType,M8: nat > ( set @ A ),K: nat,N: nat] :
( ( complete_Sup_Sup @ ( set @ A )
@ ( image @ nat @ ( set @ A )
@ ^ [I4: nat] : ( M8 @ ( plus_plus @ nat @ I4 @ K ) )
@ ( set_ord_atMost @ nat @ N ) ) )
= ( complete_Sup_Sup @ ( set @ A ) @ ( image @ nat @ ( set @ A ) @ M8 @ ( set_or1337092689740270186AtMost @ nat @ K @ ( plus_plus @ nat @ N @ K ) ) ) ) ) ).
% UN_le_add_shift
thf(fact_7805_UN__finite__subset,axiom,
! [A: $tType,A2: nat > ( set @ A ),C2: set @ A] :
( ! [N3: nat] : ( ord_less_eq @ ( set @ A ) @ ( complete_Sup_Sup @ ( set @ A ) @ ( image @ nat @ ( set @ A ) @ A2 @ ( set_or7035219750837199246ssThan @ nat @ ( zero_zero @ nat ) @ N3 ) ) ) @ C2 )
=> ( ord_less_eq @ ( set @ A ) @ ( complete_Sup_Sup @ ( set @ A ) @ ( image @ nat @ ( set @ A ) @ A2 @ ( top_top @ ( set @ nat ) ) ) ) @ C2 ) ) ).
% UN_finite_subset
thf(fact_7806_UN__finite2__eq,axiom,
! [A: $tType,A2: nat > ( set @ A ),B2: nat > ( set @ A ),K: nat] :
( ! [N3: nat] :
( ( complete_Sup_Sup @ ( set @ A ) @ ( image @ nat @ ( set @ A ) @ A2 @ ( set_or7035219750837199246ssThan @ nat @ ( zero_zero @ nat ) @ N3 ) ) )
= ( complete_Sup_Sup @ ( set @ A ) @ ( image @ nat @ ( set @ A ) @ B2 @ ( set_or7035219750837199246ssThan @ nat @ ( zero_zero @ nat ) @ ( plus_plus @ nat @ N3 @ K ) ) ) ) )
=> ( ( complete_Sup_Sup @ ( set @ A ) @ ( image @ nat @ ( set @ A ) @ A2 @ ( top_top @ ( set @ nat ) ) ) )
= ( complete_Sup_Sup @ ( set @ A ) @ ( image @ nat @ ( set @ A ) @ B2 @ ( top_top @ ( set @ nat ) ) ) ) ) ) ).
% UN_finite2_eq
thf(fact_7807_Inf_OINF__identity__eq,axiom,
! [A: $tType,Inf: ( set @ A ) > A,A2: set @ A] :
( ( Inf
@ ( image @ A @ A
@ ^ [X3: A] : X3
@ A2 ) )
= ( Inf @ A2 ) ) ).
% Inf.INF_identity_eq
thf(fact_7808_Sup_OSUP__identity__eq,axiom,
! [A: $tType,Sup: ( set @ A ) > A,A2: set @ A] :
( ( Sup
@ ( image @ A @ A
@ ^ [X3: A] : X3
@ A2 ) )
= ( Sup @ A2 ) ) ).
% Sup.SUP_identity_eq
thf(fact_7809_Sup__fun__def,axiom,
! [B: $tType,A: $tType] :
( ( complete_Sup @ B )
=> ( ( complete_Sup_Sup @ ( A > B ) )
= ( ^ [A7: set @ ( A > B ),X3: A] :
( complete_Sup_Sup @ B
@ ( image @ ( A > B ) @ B
@ ^ [F5: A > B] : ( F5 @ X3 )
@ A7 ) ) ) ) ) ).
% Sup_fun_def
thf(fact_7810_UN__finite2__subset,axiom,
! [A: $tType,A2: nat > ( set @ A ),B2: nat > ( set @ A ),K: nat] :
( ! [N3: nat] : ( ord_less_eq @ ( set @ A ) @ ( complete_Sup_Sup @ ( set @ A ) @ ( image @ nat @ ( set @ A ) @ A2 @ ( set_or7035219750837199246ssThan @ nat @ ( zero_zero @ nat ) @ N3 ) ) ) @ ( complete_Sup_Sup @ ( set @ A ) @ ( image @ nat @ ( set @ A ) @ B2 @ ( set_or7035219750837199246ssThan @ nat @ ( zero_zero @ nat ) @ ( plus_plus @ nat @ N3 @ K ) ) ) ) )
=> ( ord_less_eq @ ( set @ A ) @ ( complete_Sup_Sup @ ( set @ A ) @ ( image @ nat @ ( set @ A ) @ A2 @ ( top_top @ ( set @ nat ) ) ) ) @ ( complete_Sup_Sup @ ( set @ A ) @ ( image @ nat @ ( set @ A ) @ B2 @ ( top_top @ ( set @ nat ) ) ) ) ) ) ).
% UN_finite2_subset
thf(fact_7811_inj__on__mapI,axiom,
! [B: $tType,A: $tType,F2: A > B,A2: set @ ( list @ A )] :
( ( inj_on @ A @ B @ F2 @ ( complete_Sup_Sup @ ( set @ A ) @ ( image @ ( list @ A ) @ ( set @ A ) @ ( set2 @ A ) @ A2 ) ) )
=> ( inj_on @ ( list @ A ) @ ( list @ B ) @ ( map @ A @ B @ F2 ) @ A2 ) ) ).
% inj_on_mapI
thf(fact_7812_Sup__eqI,axiom,
! [A: $tType] :
( ( comple6319245703460814977attice @ A )
=> ! [A2: set @ A,X2: A] :
( ! [Y4: A] :
( ( member @ A @ Y4 @ A2 )
=> ( ord_less_eq @ A @ Y4 @ X2 ) )
=> ( ! [Y4: A] :
( ! [Z5: A] :
( ( member @ A @ Z5 @ A2 )
=> ( ord_less_eq @ A @ Z5 @ Y4 ) )
=> ( ord_less_eq @ A @ X2 @ Y4 ) )
=> ( ( complete_Sup_Sup @ A @ A2 )
= X2 ) ) ) ) ).
% Sup_eqI
thf(fact_7813_Sup__mono,axiom,
! [A: $tType] :
( ( comple6319245703460814977attice @ A )
=> ! [A2: set @ A,B2: set @ A] :
( ! [A4: A] :
( ( member @ A @ A4 @ A2 )
=> ? [X8: A] :
( ( member @ A @ X8 @ B2 )
& ( ord_less_eq @ A @ A4 @ X8 ) ) )
=> ( ord_less_eq @ A @ ( complete_Sup_Sup @ A @ A2 ) @ ( complete_Sup_Sup @ A @ B2 ) ) ) ) ).
% Sup_mono
thf(fact_7814_Sup__least,axiom,
! [A: $tType] :
( ( comple6319245703460814977attice @ A )
=> ! [A2: set @ A,Z: A] :
( ! [X4: A] :
( ( member @ A @ X4 @ A2 )
=> ( ord_less_eq @ A @ X4 @ Z ) )
=> ( ord_less_eq @ A @ ( complete_Sup_Sup @ A @ A2 ) @ Z ) ) ) ).
% Sup_least
thf(fact_7815_Sup__upper,axiom,
! [A: $tType] :
( ( comple6319245703460814977attice @ A )
=> ! [X2: A,A2: set @ A] :
( ( member @ A @ X2 @ A2 )
=> ( ord_less_eq @ A @ X2 @ ( complete_Sup_Sup @ A @ A2 ) ) ) ) ).
% Sup_upper
thf(fact_7816_Sup__le__iff,axiom,
! [A: $tType] :
( ( comple6319245703460814977attice @ A )
=> ! [A2: set @ A,B3: A] :
( ( ord_less_eq @ A @ ( complete_Sup_Sup @ A @ A2 ) @ B3 )
= ( ! [X3: A] :
( ( member @ A @ X3 @ A2 )
=> ( ord_less_eq @ A @ X3 @ B3 ) ) ) ) ) ).
% Sup_le_iff
thf(fact_7817_Sup__upper2,axiom,
! [A: $tType] :
( ( comple6319245703460814977attice @ A )
=> ! [U: A,A2: set @ A,V: A] :
( ( member @ A @ U @ A2 )
=> ( ( ord_less_eq @ A @ V @ U )
=> ( ord_less_eq @ A @ V @ ( complete_Sup_Sup @ A @ A2 ) ) ) ) ) ).
% Sup_upper2
thf(fact_7818_less__Sup__iff,axiom,
! [A: $tType] :
( ( comple5582772986160207858norder @ A )
=> ! [A3: A,S: set @ A] :
( ( ord_less @ A @ A3 @ ( complete_Sup_Sup @ A @ S ) )
= ( ? [X3: A] :
( ( member @ A @ X3 @ S )
& ( ord_less @ A @ A3 @ X3 ) ) ) ) ) ).
% less_Sup_iff
thf(fact_7819_SUP__commute,axiom,
! [A: $tType,B: $tType,C: $tType] :
( ( comple6319245703460814977attice @ A )
=> ! [F2: B > C > A,B2: set @ C,A2: set @ B] :
( ( complete_Sup_Sup @ A
@ ( image @ B @ A
@ ^ [I4: B] : ( complete_Sup_Sup @ A @ ( image @ C @ A @ ( F2 @ I4 ) @ B2 ) )
@ A2 ) )
= ( complete_Sup_Sup @ A
@ ( image @ C @ A
@ ^ [J3: C] :
( complete_Sup_Sup @ A
@ ( image @ B @ A
@ ^ [I4: B] : ( F2 @ I4 @ J3 )
@ A2 ) )
@ B2 ) ) ) ) ).
% SUP_commute
thf(fact_7820_le__Sup__iff,axiom,
! [A: $tType] :
( ( comple5582772986160207858norder @ A )
=> ! [X2: A,A2: set @ A] :
( ( ord_less_eq @ A @ X2 @ ( complete_Sup_Sup @ A @ A2 ) )
= ( ! [Y: A] :
( ( ord_less @ A @ Y @ X2 )
=> ? [X3: A] :
( ( member @ A @ X3 @ A2 )
& ( ord_less @ A @ Y @ X3 ) ) ) ) ) ) ).
% le_Sup_iff
thf(fact_7821_SUP__eq,axiom,
! [B: $tType,A: $tType,C: $tType] :
( ( comple6319245703460814977attice @ A )
=> ! [A2: set @ B,B2: set @ C,F2: B > A,G: C > A] :
( ! [I2: B] :
( ( member @ B @ I2 @ A2 )
=> ? [X8: C] :
( ( member @ C @ X8 @ B2 )
& ( ord_less_eq @ A @ ( F2 @ I2 ) @ ( G @ X8 ) ) ) )
=> ( ! [J2: C] :
( ( member @ C @ J2 @ B2 )
=> ? [X8: B] :
( ( member @ B @ X8 @ A2 )
& ( ord_less_eq @ A @ ( G @ J2 ) @ ( F2 @ X8 ) ) ) )
=> ( ( complete_Sup_Sup @ A @ ( image @ B @ A @ F2 @ A2 ) )
= ( complete_Sup_Sup @ A @ ( image @ C @ A @ G @ B2 ) ) ) ) ) ) ).
% SUP_eq
thf(fact_7822_less__eq__Sup,axiom,
! [A: $tType] :
( ( comple6319245703460814977attice @ A )
=> ! [A2: set @ A,U: A] :
( ! [V3: A] :
( ( member @ A @ V3 @ A2 )
=> ( ord_less_eq @ A @ U @ V3 ) )
=> ( ( A2
!= ( bot_bot @ ( set @ A ) ) )
=> ( ord_less_eq @ A @ U @ ( complete_Sup_Sup @ A @ A2 ) ) ) ) ) ).
% less_eq_Sup
thf(fact_7823_Sup__subset__mono,axiom,
! [A: $tType] :
( ( comple6319245703460814977attice @ A )
=> ! [A2: set @ A,B2: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ A2 @ B2 )
=> ( ord_less_eq @ A @ ( complete_Sup_Sup @ A @ A2 ) @ ( complete_Sup_Sup @ A @ B2 ) ) ) ) ).
% Sup_subset_mono
thf(fact_7824_SUP__eqI,axiom,
! [B: $tType,A: $tType] :
( ( comple6319245703460814977attice @ A )
=> ! [A2: set @ B,F2: B > A,X2: A] :
( ! [I2: B] :
( ( member @ B @ I2 @ A2 )
=> ( ord_less_eq @ A @ ( F2 @ I2 ) @ X2 ) )
=> ( ! [Y4: A] :
( ! [I5: B] :
( ( member @ B @ I5 @ A2 )
=> ( ord_less_eq @ A @ ( F2 @ I5 ) @ Y4 ) )
=> ( ord_less_eq @ A @ X2 @ Y4 ) )
=> ( ( complete_Sup_Sup @ A @ ( image @ B @ A @ F2 @ A2 ) )
= X2 ) ) ) ) ).
% SUP_eqI
thf(fact_7825_SUP__mono,axiom,
! [B: $tType,A: $tType,C: $tType] :
( ( comple6319245703460814977attice @ A )
=> ! [A2: set @ B,B2: set @ C,F2: B > A,G: C > A] :
( ! [N3: B] :
( ( member @ B @ N3 @ A2 )
=> ? [X8: C] :
( ( member @ C @ X8 @ B2 )
& ( ord_less_eq @ A @ ( F2 @ N3 ) @ ( G @ X8 ) ) ) )
=> ( ord_less_eq @ A @ ( complete_Sup_Sup @ A @ ( image @ B @ A @ F2 @ A2 ) ) @ ( complete_Sup_Sup @ A @ ( image @ C @ A @ G @ B2 ) ) ) ) ) ).
% SUP_mono
thf(fact_7826_SUP__least,axiom,
! [B: $tType,A: $tType] :
( ( comple6319245703460814977attice @ A )
=> ! [A2: set @ B,F2: B > A,U: A] :
( ! [I2: B] :
( ( member @ B @ I2 @ A2 )
=> ( ord_less_eq @ A @ ( F2 @ I2 ) @ U ) )
=> ( ord_less_eq @ A @ ( complete_Sup_Sup @ A @ ( image @ B @ A @ F2 @ A2 ) ) @ U ) ) ) ).
% SUP_least
thf(fact_7827_SUP__mono_H,axiom,
! [A: $tType,B: $tType] :
( ( comple6319245703460814977attice @ A )
=> ! [F2: B > A,G: B > A,A2: set @ B] :
( ! [X4: B] : ( ord_less_eq @ A @ ( F2 @ X4 ) @ ( G @ X4 ) )
=> ( ord_less_eq @ A @ ( complete_Sup_Sup @ A @ ( image @ B @ A @ F2 @ A2 ) ) @ ( complete_Sup_Sup @ A @ ( image @ B @ A @ G @ A2 ) ) ) ) ) ).
% SUP_mono'
thf(fact_7828_SUP__upper,axiom,
! [A: $tType,B: $tType] :
( ( comple6319245703460814977attice @ A )
=> ! [I: B,A2: set @ B,F2: B > A] :
( ( member @ B @ I @ A2 )
=> ( ord_less_eq @ A @ ( F2 @ I ) @ ( complete_Sup_Sup @ A @ ( image @ B @ A @ F2 @ A2 ) ) ) ) ) ).
% SUP_upper
thf(fact_7829_SUP__le__iff,axiom,
! [A: $tType,B: $tType] :
( ( comple6319245703460814977attice @ A )
=> ! [F2: B > A,A2: set @ B,U: A] :
( ( ord_less_eq @ A @ ( complete_Sup_Sup @ A @ ( image @ B @ A @ F2 @ A2 ) ) @ U )
= ( ! [X3: B] :
( ( member @ B @ X3 @ A2 )
=> ( ord_less_eq @ A @ ( F2 @ X3 ) @ U ) ) ) ) ) ).
% SUP_le_iff
thf(fact_7830_SUP__upper2,axiom,
! [A: $tType,B: $tType] :
( ( comple6319245703460814977attice @ A )
=> ! [I: B,A2: set @ B,U: A,F2: B > A] :
( ( member @ B @ I @ A2 )
=> ( ( ord_less_eq @ A @ U @ ( F2 @ I ) )
=> ( ord_less_eq @ A @ U @ ( complete_Sup_Sup @ A @ ( image @ B @ A @ F2 @ A2 ) ) ) ) ) ) ).
% SUP_upper2
thf(fact_7831_less__SUP__iff,axiom,
! [A: $tType,B: $tType] :
( ( comple5582772986160207858norder @ A )
=> ! [A3: A,F2: B > A,A2: set @ B] :
( ( ord_less @ A @ A3 @ ( complete_Sup_Sup @ A @ ( image @ B @ A @ F2 @ A2 ) ) )
= ( ? [X3: B] :
( ( member @ B @ X3 @ A2 )
& ( ord_less @ A @ A3 @ ( F2 @ X3 ) ) ) ) ) ) ).
% less_SUP_iff
thf(fact_7832_SUP__lessD,axiom,
! [B: $tType,A: $tType] :
( ( comple6319245703460814977attice @ A )
=> ! [F2: B > A,A2: set @ B,Y3: A,I: B] :
( ( ord_less @ A @ ( complete_Sup_Sup @ A @ ( image @ B @ A @ F2 @ A2 ) ) @ Y3 )
=> ( ( member @ B @ I @ A2 )
=> ( ord_less @ A @ ( F2 @ I ) @ Y3 ) ) ) ) ).
% SUP_lessD
thf(fact_7833_SUP__UNION,axiom,
! [A: $tType,B: $tType,C: $tType] :
( ( comple6319245703460814977attice @ A )
=> ! [F2: B > A,G: C > ( set @ B ),A2: set @ C] :
( ( complete_Sup_Sup @ A @ ( image @ B @ A @ F2 @ ( complete_Sup_Sup @ ( set @ B ) @ ( image @ C @ ( set @ B ) @ G @ A2 ) ) ) )
= ( complete_Sup_Sup @ A
@ ( image @ C @ A
@ ^ [Y: C] : ( complete_Sup_Sup @ A @ ( image @ B @ A @ F2 @ ( G @ Y ) ) )
@ A2 ) ) ) ) ).
% SUP_UNION
thf(fact_7834_le__SUP__iff,axiom,
! [B: $tType,A: $tType] :
( ( comple5582772986160207858norder @ A )
=> ! [X2: A,F2: B > A,A2: set @ B] :
( ( ord_less_eq @ A @ X2 @ ( complete_Sup_Sup @ A @ ( image @ B @ A @ F2 @ A2 ) ) )
= ( ! [Y: A] :
( ( ord_less @ A @ Y @ X2 )
=> ? [X3: B] :
( ( member @ B @ X3 @ A2 )
& ( ord_less @ A @ Y @ ( F2 @ X3 ) ) ) ) ) ) ) ).
% le_SUP_iff
thf(fact_7835_SUP__eq__iff,axiom,
! [A: $tType,B: $tType] :
( ( comple6319245703460814977attice @ A )
=> ! [I6: set @ B,C3: A,F2: B > A] :
( ( I6
!= ( bot_bot @ ( set @ B ) ) )
=> ( ! [I2: B] :
( ( member @ B @ I2 @ I6 )
=> ( ord_less_eq @ A @ C3 @ ( F2 @ I2 ) ) )
=> ( ( ( complete_Sup_Sup @ A @ ( image @ B @ A @ F2 @ I6 ) )
= C3 )
= ( ! [X3: B] :
( ( member @ B @ X3 @ I6 )
=> ( ( F2 @ X3 )
= C3 ) ) ) ) ) ) ) ).
% SUP_eq_iff
thf(fact_7836_SUP__subset__mono,axiom,
! [A: $tType,B: $tType] :
( ( comple6319245703460814977attice @ A )
=> ! [A2: set @ B,B2: set @ B,F2: B > A,G: B > A] :
( ( ord_less_eq @ ( set @ B ) @ A2 @ B2 )
=> ( ! [X4: B] :
( ( member @ B @ X4 @ A2 )
=> ( ord_less_eq @ A @ ( F2 @ X4 ) @ ( G @ X4 ) ) )
=> ( ord_less_eq @ A @ ( complete_Sup_Sup @ A @ ( image @ B @ A @ F2 @ A2 ) ) @ ( complete_Sup_Sup @ A @ ( image @ B @ A @ G @ B2 ) ) ) ) ) ) ).
% SUP_subset_mono
thf(fact_7837_SUP__constant,axiom,
! [B: $tType,A: $tType] :
( ( comple6319245703460814977attice @ A )
=> ! [A2: set @ B,C3: A] :
( ( ( A2
= ( bot_bot @ ( set @ B ) ) )
=> ( ( complete_Sup_Sup @ A
@ ( image @ B @ A
@ ^ [Y: B] : C3
@ A2 ) )
= ( bot_bot @ A ) ) )
& ( ( A2
!= ( bot_bot @ ( set @ B ) ) )
=> ( ( complete_Sup_Sup @ A
@ ( image @ B @ A
@ ^ [Y: B] : C3
@ A2 ) )
= C3 ) ) ) ) ).
% SUP_constant
thf(fact_7838_SUP__empty,axiom,
! [B: $tType,A: $tType] :
( ( comple6319245703460814977attice @ A )
=> ! [F2: B > A] :
( ( complete_Sup_Sup @ A @ ( image @ B @ A @ F2 @ ( bot_bot @ ( set @ B ) ) ) )
= ( bot_bot @ A ) ) ) ).
% SUP_empty
thf(fact_7839_cSUP__const,axiom,
! [B: $tType,A: $tType] :
( ( condit1219197933456340205attice @ A )
=> ! [A2: set @ B,C3: A] :
( ( A2
!= ( bot_bot @ ( set @ B ) ) )
=> ( ( complete_Sup_Sup @ A
@ ( image @ B @ A
@ ^ [X3: B] : C3
@ A2 ) )
= C3 ) ) ) ).
% cSUP_const
thf(fact_7840_cSup__greaterThanAtMost,axiom,
! [A: $tType] :
( ( condit1219197933456340205attice @ A )
=> ! [Y3: A,X2: A] :
( ( ord_less @ A @ Y3 @ X2 )
=> ( ( complete_Sup_Sup @ A @ ( set_or3652927894154168847AtMost @ A @ Y3 @ X2 ) )
= X2 ) ) ) ).
% cSup_greaterThanAtMost
thf(fact_7841_cSup__atLeastAtMost,axiom,
! [A: $tType] :
( ( condit1219197933456340205attice @ A )
=> ! [Y3: A,X2: A] :
( ( ord_less_eq @ A @ Y3 @ X2 )
=> ( ( complete_Sup_Sup @ A @ ( set_or1337092689740270186AtMost @ A @ Y3 @ X2 ) )
= X2 ) ) ) ).
% cSup_atLeastAtMost
thf(fact_7842_cSup__atLeastLessThan,axiom,
! [A: $tType] :
( ( ( condit6923001295902523014norder @ A )
& ( dense_linorder @ A ) )
=> ! [Y3: A,X2: A] :
( ( ord_less @ A @ Y3 @ X2 )
=> ( ( complete_Sup_Sup @ A @ ( set_or7035219750837199246ssThan @ A @ Y3 @ X2 ) )
= X2 ) ) ) ).
% cSup_atLeastLessThan
thf(fact_7843_SUP2__I,axiom,
! [B: $tType,A: $tType,C: $tType,A3: A,A2: set @ A,B2: A > B > C > $o,B3: B,C3: C] :
( ( member @ A @ A3 @ A2 )
=> ( ( B2 @ A3 @ B3 @ C3 )
=> ( complete_Sup_Sup @ ( B > C > $o ) @ ( image @ A @ ( B > C > $o ) @ B2 @ A2 ) @ B3 @ C3 ) ) ) ).
% SUP2_I
thf(fact_7844_cSup__greaterThanLessThan,axiom,
! [A: $tType] :
( ( ( condit6923001295902523014norder @ A )
& ( dense_linorder @ A ) )
=> ! [Y3: A,X2: A] :
( ( ord_less @ A @ Y3 @ X2 )
=> ( ( complete_Sup_Sup @ A @ ( set_or5935395276787703475ssThan @ A @ Y3 @ X2 ) )
= X2 ) ) ) ).
% cSup_greaterThanLessThan
thf(fact_7845_SUP1__I,axiom,
! [A: $tType,B: $tType,A3: A,A2: set @ A,B2: A > B > $o,B3: B] :
( ( member @ A @ A3 @ A2 )
=> ( ( B2 @ A3 @ B3 )
=> ( complete_Sup_Sup @ ( B > $o ) @ ( image @ A @ ( B > $o ) @ B2 @ A2 ) @ B3 ) ) ) ).
% SUP1_I
thf(fact_7846_SUP1__E,axiom,
! [B: $tType,A: $tType,B2: B > A > $o,A2: set @ B,B3: A] :
( ( complete_Sup_Sup @ ( A > $o ) @ ( image @ B @ ( A > $o ) @ B2 @ A2 ) @ B3 )
=> ~ ! [X4: B] :
( ( member @ B @ X4 @ A2 )
=> ~ ( B2 @ X4 @ B3 ) ) ) ).
% SUP1_E
thf(fact_7847_SUP2__E,axiom,
! [A: $tType,C: $tType,B: $tType,B2: C > A > B > $o,A2: set @ C,B3: A,C3: B] :
( ( complete_Sup_Sup @ ( A > B > $o ) @ ( image @ C @ ( A > B > $o ) @ B2 @ A2 ) @ B3 @ C3 )
=> ~ ! [X4: C] :
( ( member @ C @ X4 @ A2 )
=> ~ ( B2 @ X4 @ B3 @ C3 ) ) ) ).
% SUP2_E
thf(fact_7848_Sup__set__def,axiom,
! [A: $tType] :
( ( complete_Sup_Sup @ ( set @ A ) )
= ( ^ [A7: set @ ( set @ A )] :
( collect @ A
@ ^ [X3: A] : ( complete_Sup_Sup @ $o @ ( image @ ( set @ A ) @ $o @ ( member @ A @ X3 ) @ A7 ) ) ) ) ) ).
% Sup_set_def
thf(fact_7849_SUP__Sup__eq,axiom,
! [A: $tType,S: set @ ( set @ A )] :
( ( complete_Sup_Sup @ ( A > $o )
@ ( image @ ( set @ A ) @ ( A > $o )
@ ^ [I4: set @ A,X3: A] : ( member @ A @ X3 @ I4 )
@ S ) )
= ( ^ [X3: A] : ( member @ A @ X3 @ ( complete_Sup_Sup @ ( set @ A ) @ S ) ) ) ) ).
% SUP_Sup_eq
thf(fact_7850_ex__gt__or__lt,axiom,
! [A: $tType] :
( ( condit5016429287641298734tinuum @ A )
=> ! [A3: A] :
? [B4: A] :
( ( ord_less @ A @ A3 @ B4 )
| ( ord_less @ A @ B4 @ A3 ) ) ) ).
% ex_gt_or_lt
thf(fact_7851_complete__interval,axiom,
! [A: $tType] :
( ( condit6923001295902523014norder @ A )
=> ! [A3: A,B3: A,P: A > $o] :
( ( ord_less @ A @ A3 @ B3 )
=> ( ( P @ A3 )
=> ( ~ ( P @ B3 )
=> ? [C4: A] :
( ( ord_less_eq @ A @ A3 @ C4 )
& ( ord_less_eq @ A @ C4 @ B3 )
& ! [X8: A] :
( ( ( ord_less_eq @ A @ A3 @ X8 )
& ( ord_less @ A @ X8 @ C4 ) )
=> ( P @ X8 ) )
& ! [D5: A] :
( ! [X4: A] :
( ( ( ord_less_eq @ A @ A3 @ X4 )
& ( ord_less @ A @ X4 @ D5 ) )
=> ( P @ X4 ) )
=> ( ord_less_eq @ A @ D5 @ C4 ) ) ) ) ) ) ) ).
% complete_interval
thf(fact_7852_cSup__eq,axiom,
! [A: $tType] :
( ( ( condit1219197933456340205attice @ A )
& ( no_bot @ A ) )
=> ! [X: set @ A,A3: A] :
( ! [X4: A] :
( ( member @ A @ X4 @ X )
=> ( ord_less_eq @ A @ X4 @ A3 ) )
=> ( ! [Y4: A] :
( ! [X8: A] :
( ( member @ A @ X8 @ X )
=> ( ord_less_eq @ A @ X8 @ Y4 ) )
=> ( ord_less_eq @ A @ A3 @ Y4 ) )
=> ( ( complete_Sup_Sup @ A @ X )
= A3 ) ) ) ) ).
% cSup_eq
thf(fact_7853_cSup__eq__maximum,axiom,
! [A: $tType] :
( ( condit1219197933456340205attice @ A )
=> ! [Z: A,X: set @ A] :
( ( member @ A @ Z @ X )
=> ( ! [X4: A] :
( ( member @ A @ X4 @ X )
=> ( ord_less_eq @ A @ X4 @ Z ) )
=> ( ( complete_Sup_Sup @ A @ X )
= Z ) ) ) ) ).
% cSup_eq_maximum
thf(fact_7854_cSup__least,axiom,
! [A: $tType] :
( ( condit1219197933456340205attice @ A )
=> ! [X: set @ A,Z: A] :
( ( X
!= ( bot_bot @ ( set @ A ) ) )
=> ( ! [X4: A] :
( ( member @ A @ X4 @ X )
=> ( ord_less_eq @ A @ X4 @ Z ) )
=> ( ord_less_eq @ A @ ( complete_Sup_Sup @ A @ X ) @ Z ) ) ) ) ).
% cSup_least
thf(fact_7855_cSup__eq__non__empty,axiom,
! [A: $tType] :
( ( condit1219197933456340205attice @ A )
=> ! [X: set @ A,A3: A] :
( ( X
!= ( bot_bot @ ( set @ A ) ) )
=> ( ! [X4: A] :
( ( member @ A @ X4 @ X )
=> ( ord_less_eq @ A @ X4 @ A3 ) )
=> ( ! [Y4: A] :
( ! [X8: A] :
( ( member @ A @ X8 @ X )
=> ( ord_less_eq @ A @ X8 @ Y4 ) )
=> ( ord_less_eq @ A @ A3 @ Y4 ) )
=> ( ( complete_Sup_Sup @ A @ X )
= A3 ) ) ) ) ) ).
% cSup_eq_non_empty
thf(fact_7856_le__cSup__finite,axiom,
! [A: $tType] :
( ( condit1219197933456340205attice @ A )
=> ! [X: set @ A,X2: A] :
( ( finite_finite2 @ A @ X )
=> ( ( member @ A @ X2 @ X )
=> ( ord_less_eq @ A @ X2 @ ( complete_Sup_Sup @ A @ X ) ) ) ) ) ).
% le_cSup_finite
thf(fact_7857_less__cSupD,axiom,
! [A: $tType] :
( ( condit6923001295902523014norder @ A )
=> ! [X: set @ A,Z: A] :
( ( X
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( ord_less @ A @ Z @ ( complete_Sup_Sup @ A @ X ) )
=> ? [X4: A] :
( ( member @ A @ X4 @ X )
& ( ord_less @ A @ Z @ X4 ) ) ) ) ) ).
% less_cSupD
thf(fact_7858_less__cSupE,axiom,
! [A: $tType] :
( ( condit6923001295902523014norder @ A )
=> ! [Y3: A,X: set @ A] :
( ( ord_less @ A @ Y3 @ ( complete_Sup_Sup @ A @ X ) )
=> ( ( X
!= ( bot_bot @ ( set @ A ) ) )
=> ~ ! [X4: A] :
( ( member @ A @ X4 @ X )
=> ~ ( ord_less @ A @ Y3 @ X4 ) ) ) ) ) ).
% less_cSupE
thf(fact_7859_finite__imp__Sup__less,axiom,
! [A: $tType] :
( ( condit6923001295902523014norder @ A )
=> ! [X: set @ A,X2: A,A3: A] :
( ( finite_finite2 @ A @ X )
=> ( ( member @ A @ X2 @ X )
=> ( ! [X4: A] :
( ( member @ A @ X4 @ X )
=> ( ord_less @ A @ X4 @ A3 ) )
=> ( ord_less @ A @ ( complete_Sup_Sup @ A @ X ) @ A3 ) ) ) ) ) ).
% finite_imp_Sup_less
thf(fact_7860_cSUP__least,axiom,
! [B: $tType,A: $tType] :
( ( condit1219197933456340205attice @ A )
=> ! [A2: set @ B,F2: B > A,M8: A] :
( ( A2
!= ( bot_bot @ ( set @ B ) ) )
=> ( ! [X4: B] :
( ( member @ B @ X4 @ A2 )
=> ( ord_less_eq @ A @ ( F2 @ X4 ) @ M8 ) )
=> ( ord_less_eq @ A @ ( complete_Sup_Sup @ A @ ( image @ B @ A @ F2 @ A2 ) ) @ M8 ) ) ) ) ).
% cSUP_least
thf(fact_7861_finite__Sup__less__iff,axiom,
! [A: $tType] :
( ( condit6923001295902523014norder @ A )
=> ! [X: set @ A,A3: A] :
( ( finite_finite2 @ A @ X )
=> ( ( X
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( ord_less @ A @ ( complete_Sup_Sup @ A @ X ) @ A3 )
= ( ! [X3: A] :
( ( member @ A @ X3 @ X )
=> ( ord_less @ A @ X3 @ A3 ) ) ) ) ) ) ) ).
% finite_Sup_less_iff
thf(fact_7862_cSup__abs__le,axiom,
! [A: $tType] :
( ( ( condit6923001295902523014norder @ A )
& ( linordered_idom @ A ) )
=> ! [S: set @ A,A3: A] :
( ( S
!= ( bot_bot @ ( set @ A ) ) )
=> ( ! [X4: A] :
( ( member @ A @ X4 @ S )
=> ( ord_less_eq @ A @ ( abs_abs @ A @ X4 ) @ A3 ) )
=> ( ord_less_eq @ A @ ( abs_abs @ A @ ( complete_Sup_Sup @ A @ S ) ) @ A3 ) ) ) ) ).
% cSup_abs_le
thf(fact_7863_Sup__insert__finite,axiom,
! [A: $tType] :
( ( condit6923001295902523014norder @ A )
=> ! [S: set @ A,X2: A] :
( ( finite_finite2 @ A @ S )
=> ( ( ( S
= ( bot_bot @ ( set @ A ) ) )
=> ( ( complete_Sup_Sup @ A @ ( insert @ A @ X2 @ S ) )
= X2 ) )
& ( ( S
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( complete_Sup_Sup @ A @ ( insert @ A @ X2 @ S ) )
= ( ord_max @ A @ X2 @ ( complete_Sup_Sup @ A @ S ) ) ) ) ) ) ) ).
% Sup_insert_finite
thf(fact_7864_totally__bounded__metric,axiom,
! [A: $tType] :
( ( real_V7819770556892013058_space @ A )
=> ( ( topolo6688025880775521714ounded @ A )
= ( ^ [S8: set @ A] :
! [E4: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ E4 )
=> ? [K3: set @ A] :
( ( finite_finite2 @ A @ K3 )
& ( ord_less_eq @ ( set @ A ) @ S8
@ ( complete_Sup_Sup @ ( set @ A )
@ ( image @ A @ ( set @ A )
@ ^ [X3: A] :
( collect @ A
@ ^ [Y: A] : ( ord_less @ real @ ( real_V557655796197034286t_dist @ A @ X3 @ Y ) @ E4 ) )
@ K3 ) ) ) ) ) ) ) ) ).
% totally_bounded_metric
thf(fact_7865_UN__image__subset,axiom,
! [C: $tType,A: $tType,B: $tType,F2: B > ( set @ A ),G: C > ( set @ B ),X2: C,X: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ ( complete_Sup_Sup @ ( set @ A ) @ ( image @ B @ ( set @ A ) @ F2 @ ( G @ X2 ) ) ) @ X )
= ( ord_less_eq @ ( set @ B ) @ ( G @ X2 )
@ ( collect @ B
@ ^ [X3: B] : ( ord_less_eq @ ( set @ A ) @ ( F2 @ X3 ) @ X ) ) ) ) ).
% UN_image_subset
thf(fact_7866_totally__bounded__subset,axiom,
! [A: $tType] :
( ( topolo7287701948861334536_space @ A )
=> ! [S: set @ A,T6: set @ A] :
( ( topolo6688025880775521714ounded @ A @ S )
=> ( ( ord_less_eq @ ( set @ A ) @ T6 @ S )
=> ( topolo6688025880775521714ounded @ A @ T6 ) ) ) ) ).
% totally_bounded_subset
thf(fact_7867_conj__subset__def,axiom,
! [A: $tType,A2: set @ A,P: A > $o,Q: A > $o] :
( ( ord_less_eq @ ( set @ A ) @ A2
@ ( collect @ A
@ ^ [X3: A] :
( ( P @ X3 )
& ( Q @ X3 ) ) ) )
= ( ( ord_less_eq @ ( set @ A ) @ A2 @ ( collect @ A @ P ) )
& ( ord_less_eq @ ( set @ A ) @ A2 @ ( collect @ A @ Q ) ) ) ) ).
% conj_subset_def
thf(fact_7868_Sup__option__def,axiom,
! [A: $tType] :
( ( comple6319245703460814977attice @ A )
=> ( ( complete_Sup_Sup @ ( option @ A ) )
= ( ^ [A7: set @ ( option @ A )] :
( if @ ( option @ A )
@ ( ( A7
= ( bot_bot @ ( set @ ( option @ A ) ) ) )
| ( A7
= ( insert @ ( option @ A ) @ ( none @ A ) @ ( bot_bot @ ( set @ ( option @ A ) ) ) ) ) )
@ ( none @ A )
@ ( some @ A @ ( complete_Sup_Sup @ A @ ( these @ A @ A7 ) ) ) ) ) ) ) ).
% Sup_option_def
thf(fact_7869_interval__cases,axiom,
! [A: $tType] :
( ( condit6923001295902523014norder @ A )
=> ! [S: set @ A] :
( ! [A4: A,B4: A,X4: A] :
( ( member @ A @ A4 @ S )
=> ( ( member @ A @ B4 @ S )
=> ( ( ord_less_eq @ A @ A4 @ X4 )
=> ( ( ord_less_eq @ A @ X4 @ B4 )
=> ( member @ A @ X4 @ S ) ) ) ) )
=> ? [A4: A,B4: A] :
( ( S
= ( bot_bot @ ( set @ A ) ) )
| ( S
= ( top_top @ ( set @ A ) ) )
| ( S
= ( set_ord_lessThan @ A @ B4 ) )
| ( S
= ( set_ord_atMost @ A @ B4 ) )
| ( S
= ( set_ord_greaterThan @ A @ A4 ) )
| ( S
= ( set_ord_atLeast @ A @ A4 ) )
| ( S
= ( set_or5935395276787703475ssThan @ A @ A4 @ B4 ) )
| ( S
= ( set_or3652927894154168847AtMost @ A @ A4 @ B4 ) )
| ( S
= ( set_or7035219750837199246ssThan @ A @ A4 @ B4 ) )
| ( S
= ( set_or1337092689740270186AtMost @ A @ A4 @ B4 ) ) ) ) ) ).
% interval_cases
thf(fact_7870_atLeast__iff,axiom,
! [A: $tType] :
( ( ord @ A )
=> ! [I: A,K: A] :
( ( member @ A @ I @ ( set_ord_atLeast @ A @ K ) )
= ( ord_less_eq @ A @ K @ I ) ) ) ).
% atLeast_iff
thf(fact_7871_image__add__atLeast,axiom,
! [A: $tType] :
( ( linordered_semidom @ A )
=> ! [K: A,I: A] :
( ( image @ A @ A @ ( plus_plus @ A @ K ) @ ( set_ord_atLeast @ A @ I ) )
= ( set_ord_atLeast @ A @ ( plus_plus @ A @ K @ I ) ) ) ) ).
% image_add_atLeast
thf(fact_7872_atLeast__subset__iff,axiom,
! [A: $tType] :
( ( preorder @ A )
=> ! [X2: A,Y3: A] :
( ( ord_less_eq @ ( set @ A ) @ ( set_ord_atLeast @ A @ X2 ) @ ( set_ord_atLeast @ A @ Y3 ) )
= ( ord_less_eq @ A @ Y3 @ X2 ) ) ) ).
% atLeast_subset_iff
thf(fact_7873_these__image__Some__eq,axiom,
! [A: $tType,A2: set @ A] :
( ( these @ A @ ( image @ A @ ( option @ A ) @ ( some @ A ) @ A2 ) )
= A2 ) ).
% these_image_Some_eq
thf(fact_7874_these__empty,axiom,
! [A: $tType] :
( ( these @ A @ ( bot_bot @ ( set @ ( option @ A ) ) ) )
= ( bot_bot @ ( set @ A ) ) ) ).
% these_empty
thf(fact_7875_these__insert__None,axiom,
! [A: $tType,A2: set @ ( option @ A )] :
( ( these @ A @ ( insert @ ( option @ A ) @ ( none @ A ) @ A2 ) )
= ( these @ A @ A2 ) ) ).
% these_insert_None
thf(fact_7876_Icc__subset__Ici__iff,axiom,
! [A: $tType] :
( ( preorder @ A )
=> ! [L: A,H2: A,L4: A] :
( ( ord_less_eq @ ( set @ A ) @ ( set_or1337092689740270186AtMost @ A @ L @ H2 ) @ ( set_ord_atLeast @ A @ L4 ) )
= ( ~ ( ord_less_eq @ A @ L @ H2 )
| ( ord_less_eq @ A @ L4 @ L ) ) ) ) ).
% Icc_subset_Ici_iff
thf(fact_7877_image__minus__const__atLeast,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [C3: A,A3: A] :
( ( image @ A @ A @ ( minus_minus @ A @ C3 ) @ ( set_ord_atLeast @ A @ A3 ) )
= ( set_ord_atMost @ A @ ( minus_minus @ A @ C3 @ A3 ) ) ) ) ).
% image_minus_const_atLeast
thf(fact_7878_image__minus__const__AtMost,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [C3: A,B3: A] :
( ( image @ A @ A @ ( minus_minus @ A @ C3 ) @ ( set_ord_atMost @ A @ B3 ) )
= ( set_ord_atLeast @ A @ ( minus_minus @ A @ C3 @ B3 ) ) ) ) ).
% image_minus_const_AtMost
thf(fact_7879_these__insert__Some,axiom,
! [A: $tType,X2: A,A2: set @ ( option @ A )] :
( ( these @ A @ ( insert @ ( option @ A ) @ ( some @ A @ X2 ) @ A2 ) )
= ( insert @ A @ X2 @ ( these @ A @ A2 ) ) ) ).
% these_insert_Some
thf(fact_7880_atLeast__Suc__greaterThan,axiom,
! [K: nat] :
( ( set_ord_atLeast @ nat @ ( suc @ K ) )
= ( set_ord_greaterThan @ nat @ K ) ) ).
% atLeast_Suc_greaterThan
thf(fact_7881_in__these__eq,axiom,
! [A: $tType,X2: A,A2: set @ ( option @ A )] :
( ( member @ A @ X2 @ ( these @ A @ A2 ) )
= ( member @ ( option @ A ) @ ( some @ A @ X2 ) @ A2 ) ) ).
% in_these_eq
thf(fact_7882_Ici__subset__Ioi__iff,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A3: A,B3: A] :
( ( ord_less_eq @ ( set @ A ) @ ( set_ord_atLeast @ A @ A3 ) @ ( set_ord_greaterThan @ A @ B3 ) )
= ( ord_less @ A @ B3 @ A3 ) ) ) ).
% Ici_subset_Ioi_iff
thf(fact_7883_not__UNIV__le__Ici,axiom,
! [A: $tType] :
( ( no_bot @ A )
=> ! [L: A] :
~ ( ord_less_eq @ ( set @ A ) @ ( top_top @ ( set @ A ) ) @ ( set_ord_atLeast @ A @ L ) ) ) ).
% not_UNIV_le_Ici
thf(fact_7884_not__Iic__le__Ici,axiom,
! [A: $tType] :
( ( no_bot @ A )
=> ! [H2: A,L4: A] :
~ ( ord_less_eq @ ( set @ A ) @ ( set_ord_atMost @ A @ H2 ) @ ( set_ord_atLeast @ A @ L4 ) ) ) ).
% not_Iic_le_Ici
thf(fact_7885_not__Ici__le__Iic,axiom,
! [A: $tType] :
( ( no_top @ A )
=> ! [L: A,H4: A] :
~ ( ord_less_eq @ ( set @ A ) @ ( set_ord_atLeast @ A @ L ) @ ( set_ord_atMost @ A @ H4 ) ) ) ).
% not_Ici_le_Iic
thf(fact_7886_atLeast__def,axiom,
! [A: $tType] :
( ( ord @ A )
=> ( ( set_ord_atLeast @ A )
= ( ^ [L3: A] : ( collect @ A @ ( ord_less_eq @ A @ L3 ) ) ) ) ) ).
% atLeast_def
thf(fact_7887_not__Ici__le__Icc,axiom,
! [A: $tType] :
( ( no_top @ A )
=> ! [L: A,L4: A,H4: A] :
~ ( ord_less_eq @ ( set @ A ) @ ( set_ord_atLeast @ A @ L ) @ ( set_or1337092689740270186AtMost @ A @ L4 @ H4 ) ) ) ).
% not_Ici_le_Icc
thf(fact_7888_Ioi__le__Ico,axiom,
! [A: $tType] :
( ( preorder @ A )
=> ! [A3: A] : ( ord_less_eq @ ( set @ A ) @ ( set_ord_greaterThan @ A @ A3 ) @ ( set_ord_atLeast @ A @ A3 ) ) ) ).
% Ioi_le_Ico
thf(fact_7889_Option_Othese__def,axiom,
! [A: $tType] :
( ( these @ A )
= ( ^ [A7: set @ ( option @ A )] :
( image @ ( option @ A ) @ A @ ( the2 @ A )
@ ( collect @ ( option @ A )
@ ^ [X3: option @ A] :
( ( member @ ( option @ A ) @ X3 @ A7 )
& ( X3
!= ( none @ A ) ) ) ) ) ) ) ).
% Option.these_def
thf(fact_7890_atLeast__Suc,axiom,
! [K: nat] :
( ( set_ord_atLeast @ nat @ ( suc @ K ) )
= ( minus_minus @ ( set @ nat ) @ ( set_ord_atLeast @ nat @ K ) @ ( insert @ nat @ K @ ( bot_bot @ ( set @ nat ) ) ) ) ) ).
% atLeast_Suc
thf(fact_7891_UN__atLeast__UNIV,axiom,
( ( complete_Sup_Sup @ ( set @ nat ) @ ( image @ nat @ ( set @ nat ) @ ( set_ord_atLeast @ nat ) @ ( top_top @ ( set @ nat ) ) ) )
= ( top_top @ ( set @ nat ) ) ) ).
% UN_atLeast_UNIV
thf(fact_7892_Some__image__these__eq,axiom,
! [A: $tType,A2: set @ ( option @ A )] :
( ( image @ A @ ( option @ A ) @ ( some @ A ) @ ( these @ A @ A2 ) )
= ( collect @ ( option @ A )
@ ^ [X3: option @ A] :
( ( member @ ( option @ A ) @ X3 @ A2 )
& ( X3
!= ( none @ A ) ) ) ) ) ).
% Some_image_these_eq
thf(fact_7893_these__not__empty__eq,axiom,
! [A: $tType,B2: set @ ( option @ A )] :
( ( ( these @ A @ B2 )
!= ( bot_bot @ ( set @ A ) ) )
= ( ( B2
!= ( bot_bot @ ( set @ ( option @ A ) ) ) )
& ( B2
!= ( insert @ ( option @ A ) @ ( none @ A ) @ ( bot_bot @ ( set @ ( option @ A ) ) ) ) ) ) ) ).
% these_not_empty_eq
thf(fact_7894_these__empty__eq,axiom,
! [A: $tType,B2: set @ ( option @ A )] :
( ( ( these @ A @ B2 )
= ( bot_bot @ ( set @ A ) ) )
= ( ( B2
= ( bot_bot @ ( set @ ( option @ A ) ) ) )
| ( B2
= ( insert @ ( option @ A ) @ ( none @ A ) @ ( bot_bot @ ( set @ ( option @ A ) ) ) ) ) ) ) ).
% these_empty_eq
thf(fact_7895_UN__UN__split__split__eq,axiom,
! [A: $tType,E3: $tType,D6: $tType,C: $tType,B: $tType,A2: B > C > D6 > E3 > ( set @ A ),Y8: set @ ( product_prod @ D6 @ E3 ),X: set @ ( product_prod @ B @ C )] :
( ( complete_Sup_Sup @ ( set @ A )
@ ( image @ ( product_prod @ B @ C ) @ ( set @ A )
@ ( product_case_prod @ B @ C @ ( set @ A )
@ ^ [X15: B,X24: C] : ( complete_Sup_Sup @ ( set @ A ) @ ( image @ ( product_prod @ D6 @ E3 ) @ ( set @ A ) @ ( product_case_prod @ D6 @ E3 @ ( set @ A ) @ ( A2 @ X15 @ X24 ) ) @ Y8 ) ) )
@ X ) )
= ( complete_Sup_Sup @ ( set @ A )
@ ( image @ ( product_prod @ B @ C ) @ ( set @ A )
@ ^ [X3: product_prod @ B @ C] :
( complete_Sup_Sup @ ( set @ A )
@ ( image @ ( product_prod @ D6 @ E3 ) @ ( set @ A )
@ ^ [Y: product_prod @ D6 @ E3] :
( product_case_prod @ B @ C @ ( set @ A )
@ ^ [X15: B,X24: C] : ( product_case_prod @ D6 @ E3 @ ( set @ A ) @ ( A2 @ X15 @ X24 ) @ Y )
@ X3 )
@ Y8 ) )
@ X ) ) ) ).
% UN_UN_split_split_eq
thf(fact_7896_GMVT,axiom,
! [A3: real,B3: real,F2: real > real,G: real > real] :
( ( ord_less @ real @ A3 @ B3 )
=> ( ! [X4: real] :
( ( ( ord_less_eq @ real @ A3 @ X4 )
& ( ord_less_eq @ real @ X4 @ B3 ) )
=> ( topolo3448309680560233919inuous @ real @ real @ ( topolo174197925503356063within @ real @ X4 @ ( top_top @ ( set @ real ) ) ) @ F2 ) )
=> ( ! [X4: real] :
( ( ( ord_less @ real @ A3 @ X4 )
& ( ord_less @ real @ X4 @ B3 ) )
=> ( differentiable @ real @ real @ F2 @ ( topolo174197925503356063within @ real @ X4 @ ( top_top @ ( set @ real ) ) ) ) )
=> ( ! [X4: real] :
( ( ( ord_less_eq @ real @ A3 @ X4 )
& ( ord_less_eq @ real @ X4 @ B3 ) )
=> ( topolo3448309680560233919inuous @ real @ real @ ( topolo174197925503356063within @ real @ X4 @ ( top_top @ ( set @ real ) ) ) @ G ) )
=> ( ! [X4: real] :
( ( ( ord_less @ real @ A3 @ X4 )
& ( ord_less @ real @ X4 @ B3 ) )
=> ( differentiable @ real @ real @ G @ ( topolo174197925503356063within @ real @ X4 @ ( top_top @ ( set @ real ) ) ) ) )
=> ? [G_c: real,F_c: real,C4: real] :
( ( has_field_derivative @ real @ G @ G_c @ ( topolo174197925503356063within @ real @ C4 @ ( top_top @ ( set @ real ) ) ) )
& ( has_field_derivative @ real @ F2 @ F_c @ ( topolo174197925503356063within @ real @ C4 @ ( top_top @ ( set @ real ) ) ) )
& ( ord_less @ real @ A3 @ C4 )
& ( ord_less @ real @ C4 @ B3 )
& ( ( times_times @ real @ ( minus_minus @ real @ ( F2 @ B3 ) @ ( F2 @ A3 ) ) @ G_c )
= ( times_times @ real @ ( minus_minus @ real @ ( G @ B3 ) @ ( G @ A3 ) ) @ F_c ) ) ) ) ) ) ) ) ).
% GMVT
thf(fact_7897_differentiable__cmult__left__iff,axiom,
! [A: $tType,B: $tType] :
( ( ( real_V822414075346904944vector @ B )
& ( real_V3459762299906320749_field @ A ) )
=> ! [C3: A,Q3: B > A,T2: B] :
( ( differentiable @ B @ A
@ ^ [T3: B] : ( times_times @ A @ C3 @ ( Q3 @ T3 ) )
@ ( topolo174197925503356063within @ B @ T2 @ ( top_top @ ( set @ B ) ) ) )
= ( ( C3
= ( zero_zero @ A ) )
| ( differentiable @ B @ A @ Q3 @ ( topolo174197925503356063within @ B @ T2 @ ( top_top @ ( set @ B ) ) ) ) ) ) ) ).
% differentiable_cmult_left_iff
thf(fact_7898_differentiable__cmult__right__iff,axiom,
! [A: $tType,B: $tType] :
( ( ( real_V822414075346904944vector @ B )
& ( real_V3459762299906320749_field @ A ) )
=> ! [Q3: B > A,C3: A,T2: B] :
( ( differentiable @ B @ A
@ ^ [T3: B] : ( times_times @ A @ ( Q3 @ T3 ) @ C3 )
@ ( topolo174197925503356063within @ B @ T2 @ ( top_top @ ( set @ B ) ) ) )
= ( ( C3
= ( zero_zero @ A ) )
| ( differentiable @ B @ A @ Q3 @ ( topolo174197925503356063within @ B @ T2 @ ( top_top @ ( set @ B ) ) ) ) ) ) ) ).
% differentiable_cmult_right_iff
thf(fact_7899_differentiable__within__subset,axiom,
! [B: $tType,A: $tType] :
( ( ( real_V822414075346904944vector @ A )
& ( real_V822414075346904944vector @ B ) )
=> ! [F2: A > B,X2: A,S2: set @ A,T2: set @ A] :
( ( differentiable @ A @ B @ F2 @ ( topolo174197925503356063within @ A @ X2 @ S2 ) )
=> ( ( ord_less_eq @ ( set @ A ) @ T2 @ S2 )
=> ( differentiable @ A @ B @ F2 @ ( topolo174197925503356063within @ A @ X2 @ T2 ) ) ) ) ) ).
% differentiable_within_subset
thf(fact_7900_differentiable__add,axiom,
! [B: $tType,A: $tType] :
( ( ( real_V822414075346904944vector @ A )
& ( real_V822414075346904944vector @ B ) )
=> ! [F2: A > B,F4: filter @ A,G: A > B] :
( ( differentiable @ A @ B @ F2 @ F4 )
=> ( ( differentiable @ A @ B @ G @ F4 )
=> ( differentiable @ A @ B
@ ^ [X3: A] : ( plus_plus @ B @ ( F2 @ X3 ) @ ( G @ X3 ) )
@ F4 ) ) ) ) ).
% differentiable_add
thf(fact_7901_differentiable__diff,axiom,
! [B: $tType,A: $tType] :
( ( ( real_V822414075346904944vector @ A )
& ( real_V822414075346904944vector @ B ) )
=> ! [F2: A > B,F4: filter @ A,G: A > B] :
( ( differentiable @ A @ B @ F2 @ F4 )
=> ( ( differentiable @ A @ B @ G @ F4 )
=> ( differentiable @ A @ B
@ ^ [X3: A] : ( minus_minus @ B @ ( F2 @ X3 ) @ ( G @ X3 ) )
@ F4 ) ) ) ) ).
% differentiable_diff
thf(fact_7902_differentiable__const,axiom,
! [B: $tType,A: $tType] :
( ( ( real_V822414075346904944vector @ A )
& ( real_V822414075346904944vector @ B ) )
=> ! [A3: B,F4: filter @ A] :
( differentiable @ A @ B
@ ^ [Z4: A] : A3
@ F4 ) ) ).
% differentiable_const
thf(fact_7903_differentiable__ident,axiom,
! [A: $tType] :
( ( real_V822414075346904944vector @ A )
=> ! [F4: filter @ A] :
( differentiable @ A @ A
@ ^ [X3: A] : X3
@ F4 ) ) ).
% differentiable_ident
thf(fact_7904_differentiable__minus,axiom,
! [B: $tType,A: $tType] :
( ( ( real_V822414075346904944vector @ A )
& ( real_V822414075346904944vector @ B ) )
=> ! [F2: A > B,F4: filter @ A] :
( ( differentiable @ A @ B @ F2 @ F4 )
=> ( differentiable @ A @ B
@ ^ [X3: A] : ( uminus_uminus @ B @ ( F2 @ X3 ) )
@ F4 ) ) ) ).
% differentiable_minus
thf(fact_7905_differentiable__compose,axiom,
! [B: $tType,A: $tType,C: $tType] :
( ( ( real_V822414075346904944vector @ C )
& ( real_V822414075346904944vector @ A )
& ( real_V822414075346904944vector @ B ) )
=> ! [F2: A > B,G: C > A,X2: C,S2: set @ C] :
( ( differentiable @ A @ B @ F2 @ ( topolo174197925503356063within @ A @ ( G @ X2 ) @ ( top_top @ ( set @ A ) ) ) )
=> ( ( differentiable @ C @ A @ G @ ( topolo174197925503356063within @ C @ X2 @ S2 ) )
=> ( differentiable @ C @ B
@ ^ [X3: C] : ( F2 @ ( G @ X3 ) )
@ ( topolo174197925503356063within @ C @ X2 @ S2 ) ) ) ) ) ).
% differentiable_compose
thf(fact_7906_differentiable__power,axiom,
! [B: $tType,A: $tType] :
( ( ( real_V822414075346904944vector @ A )
& ( real_V3459762299906320749_field @ B ) )
=> ! [F2: A > B,X2: A,S2: set @ A,N: nat] :
( ( differentiable @ A @ B @ F2 @ ( topolo174197925503356063within @ A @ X2 @ S2 ) )
=> ( differentiable @ A @ B
@ ^ [X3: A] : ( power_power @ B @ ( F2 @ X3 ) @ N )
@ ( topolo174197925503356063within @ A @ X2 @ S2 ) ) ) ) ).
% differentiable_power
thf(fact_7907_differentiable__scaleR,axiom,
! [B: $tType,A: $tType] :
( ( ( real_V822414075346904944vector @ A )
& ( real_V822414075346904944vector @ B ) )
=> ! [F2: A > real,X2: A,S2: set @ A,G: A > B] :
( ( differentiable @ A @ real @ F2 @ ( topolo174197925503356063within @ A @ X2 @ S2 ) )
=> ( ( differentiable @ A @ B @ G @ ( topolo174197925503356063within @ A @ X2 @ S2 ) )
=> ( differentiable @ A @ B
@ ^ [X3: A] : ( real_V8093663219630862766scaleR @ B @ ( F2 @ X3 ) @ ( G @ X3 ) )
@ ( topolo174197925503356063within @ A @ X2 @ S2 ) ) ) ) ) ).
% differentiable_scaleR
thf(fact_7908_differentiable__mult,axiom,
! [B: $tType,A: $tType] :
( ( ( real_V822414075346904944vector @ A )
& ( real_V4412858255891104859lgebra @ B ) )
=> ! [F2: A > B,X2: A,S2: set @ A,G: A > B] :
( ( differentiable @ A @ B @ F2 @ ( topolo174197925503356063within @ A @ X2 @ S2 ) )
=> ( ( differentiable @ A @ B @ G @ ( topolo174197925503356063within @ A @ X2 @ S2 ) )
=> ( differentiable @ A @ B
@ ^ [X3: A] : ( times_times @ B @ ( F2 @ X3 ) @ ( G @ X3 ) )
@ ( topolo174197925503356063within @ A @ X2 @ S2 ) ) ) ) ) ).
% differentiable_mult
thf(fact_7909_differentiable__sum,axiom,
! [C: $tType,A: $tType,B: $tType] :
( ( ( real_V822414075346904944vector @ B )
& ( real_V822414075346904944vector @ C ) )
=> ! [S2: set @ A,F2: A > B > C,Net: filter @ B] :
( ( finite_finite2 @ A @ S2 )
=> ( ! [X4: A] :
( ( member @ A @ X4 @ S2 )
=> ( differentiable @ B @ C @ ( F2 @ X4 ) @ Net ) )
=> ( differentiable @ B @ C
@ ^ [X3: B] :
( groups7311177749621191930dd_sum @ A @ C
@ ^ [A6: A] : ( F2 @ A6 @ X3 )
@ S2 )
@ Net ) ) ) ) ).
% differentiable_sum
thf(fact_7910_differentiable__in__compose,axiom,
! [B: $tType,A: $tType,C: $tType] :
( ( ( real_V822414075346904944vector @ C )
& ( real_V822414075346904944vector @ A )
& ( real_V822414075346904944vector @ B ) )
=> ! [F2: A > B,G: C > A,X2: C,S2: set @ C] :
( ( differentiable @ A @ B @ F2 @ ( topolo174197925503356063within @ A @ ( G @ X2 ) @ ( image @ C @ A @ G @ S2 ) ) )
=> ( ( differentiable @ C @ A @ G @ ( topolo174197925503356063within @ C @ X2 @ S2 ) )
=> ( differentiable @ C @ B
@ ^ [X3: C] : ( F2 @ ( G @ X3 ) )
@ ( topolo174197925503356063within @ C @ X2 @ S2 ) ) ) ) ) ).
% differentiable_in_compose
thf(fact_7911_differentiable__divide,axiom,
! [B: $tType,A: $tType] :
( ( ( real_V822414075346904944vector @ A )
& ( real_V3459762299906320749_field @ B ) )
=> ! [F2: A > B,X2: A,S2: set @ A,G: A > B] :
( ( differentiable @ A @ B @ F2 @ ( topolo174197925503356063within @ A @ X2 @ S2 ) )
=> ( ( differentiable @ A @ B @ G @ ( topolo174197925503356063within @ A @ X2 @ S2 ) )
=> ( ( ( G @ X2 )
!= ( zero_zero @ B ) )
=> ( differentiable @ A @ B
@ ^ [X3: A] : ( divide_divide @ B @ ( F2 @ X3 ) @ ( G @ X3 ) )
@ ( topolo174197925503356063within @ A @ X2 @ S2 ) ) ) ) ) ) ).
% differentiable_divide
thf(fact_7912_differentiable__inverse,axiom,
! [B: $tType,A: $tType] :
( ( ( real_V822414075346904944vector @ A )
& ( real_V3459762299906320749_field @ B ) )
=> ! [F2: A > B,X2: A,S2: set @ A] :
( ( differentiable @ A @ B @ F2 @ ( topolo174197925503356063within @ A @ X2 @ S2 ) )
=> ( ( ( F2 @ X2 )
!= ( zero_zero @ B ) )
=> ( differentiable @ A @ B
@ ^ [X3: A] : ( inverse_inverse @ B @ ( F2 @ X3 ) )
@ ( topolo174197925503356063within @ A @ X2 @ S2 ) ) ) ) ) ).
% differentiable_inverse
thf(fact_7913_UN__constant__eq,axiom,
! [A: $tType,B: $tType,A3: A,A2: set @ A,F2: A > ( set @ B ),C3: set @ B] :
( ( member @ A @ A3 @ A2 )
=> ( ! [X4: A] :
( ( member @ A @ X4 @ A2 )
=> ( ( F2 @ X4 )
= C3 ) )
=> ( ( complete_Sup_Sup @ ( set @ B ) @ ( image @ A @ ( set @ B ) @ F2 @ A2 ) )
= C3 ) ) ) ).
% UN_constant_eq
thf(fact_7914_finite__mono__strict__prefix__implies__finite__fixpoint,axiom,
! [A: $tType,F2: nat > ( set @ A ),S: set @ A] :
( ! [I2: nat] : ( ord_less_eq @ ( set @ A ) @ ( F2 @ I2 ) @ S )
=> ( ( finite_finite2 @ A @ S )
=> ( ? [N10: nat] :
( ! [N3: nat] :
( ( ord_less_eq @ nat @ N3 @ N10 )
=> ! [M5: nat] :
( ( ord_less_eq @ nat @ M5 @ N10 )
=> ( ( ord_less @ nat @ M5 @ N3 )
=> ( ord_less @ ( set @ A ) @ ( F2 @ M5 ) @ ( F2 @ N3 ) ) ) ) )
& ! [N3: nat] :
( ( ord_less_eq @ nat @ N10 @ N3 )
=> ( ( F2 @ N10 )
= ( F2 @ N3 ) ) ) )
=> ( ( F2 @ ( finite_card @ A @ S ) )
= ( complete_Sup_Sup @ ( set @ A ) @ ( image @ nat @ ( set @ A ) @ F2 @ ( top_top @ ( set @ nat ) ) ) ) ) ) ) ) ).
% finite_mono_strict_prefix_implies_finite_fixpoint
thf(fact_7915_MVT,axiom,
! [A3: real,B3: real,F2: real > real] :
( ( ord_less @ real @ A3 @ B3 )
=> ( ( topolo81223032696312382ous_on @ real @ real @ ( set_or1337092689740270186AtMost @ real @ A3 @ B3 ) @ F2 )
=> ( ! [X4: real] :
( ( ord_less @ real @ A3 @ X4 )
=> ( ( ord_less @ real @ X4 @ B3 )
=> ( differentiable @ real @ real @ F2 @ ( topolo174197925503356063within @ real @ X4 @ ( top_top @ ( set @ real ) ) ) ) ) )
=> ? [L2: real,Z2: real] :
( ( ord_less @ real @ A3 @ Z2 )
& ( ord_less @ real @ Z2 @ B3 )
& ( has_field_derivative @ real @ F2 @ L2 @ ( topolo174197925503356063within @ real @ Z2 @ ( top_top @ ( set @ real ) ) ) )
& ( ( minus_minus @ real @ ( F2 @ B3 ) @ ( F2 @ A3 ) )
= ( times_times @ real @ ( minus_minus @ real @ B3 @ A3 ) @ L2 ) ) ) ) ) ) ).
% MVT
thf(fact_7916_card__UNIV__unit,axiom,
( ( finite_card @ product_unit @ ( top_top @ ( set @ product_unit ) ) )
= ( one_one @ nat ) ) ).
% card_UNIV_unit
thf(fact_7917_card__Collect__less__nat,axiom,
! [N: nat] :
( ( finite_card @ nat
@ ( collect @ nat
@ ^ [I4: nat] : ( ord_less @ nat @ I4 @ N ) ) )
= N ) ).
% card_Collect_less_nat
thf(fact_7918_card__eq__UNIV2,axiom,
! [A: $tType] :
( ( finite_finite @ A )
=> ! [S: set @ A] :
( ( ( finite_card @ A @ ( top_top @ ( set @ A ) ) )
= ( finite_card @ A @ S ) )
= ( S
= ( top_top @ ( set @ A ) ) ) ) ) ).
% card_eq_UNIV2
thf(fact_7919_card__eq__UNIV,axiom,
! [A: $tType] :
( ( finite_finite @ A )
=> ! [S: set @ A] :
( ( ( finite_card @ A @ S )
= ( finite_card @ A @ ( top_top @ ( set @ A ) ) ) )
= ( S
= ( top_top @ ( set @ A ) ) ) ) ) ).
% card_eq_UNIV
thf(fact_7920_card__atMost,axiom,
! [U: nat] :
( ( finite_card @ nat @ ( set_ord_atMost @ nat @ U ) )
= ( suc @ U ) ) ).
% card_atMost
thf(fact_7921_card__atLeastLessThan,axiom,
! [L: nat,U: nat] :
( ( finite_card @ nat @ ( set_or7035219750837199246ssThan @ nat @ L @ U ) )
= ( minus_minus @ nat @ U @ L ) ) ).
% card_atLeastLessThan
thf(fact_7922_card__Collect__le__nat,axiom,
! [N: nat] :
( ( finite_card @ nat
@ ( collect @ nat
@ ^ [I4: nat] : ( ord_less_eq @ nat @ I4 @ N ) ) )
= ( suc @ N ) ) ).
% card_Collect_le_nat
thf(fact_7923_card__greaterThanAtMost,axiom,
! [L: nat,U: nat] :
( ( finite_card @ nat @ ( set_or3652927894154168847AtMost @ nat @ L @ U ) )
= ( minus_minus @ nat @ U @ L ) ) ).
% card_greaterThanAtMost
thf(fact_7924_card__UNIV__bool,axiom,
( ( finite_card @ $o @ ( top_top @ ( set @ $o ) ) )
= ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ).
% card_UNIV_bool
thf(fact_7925_card__ge__UNIV,axiom,
! [A: $tType] :
( ( finite_finite @ A )
=> ! [S: set @ A] :
( ( ord_less_eq @ nat @ ( finite_card @ A @ ( top_top @ ( set @ A ) ) ) @ ( finite_card @ A @ S ) )
= ( S
= ( top_top @ ( set @ A ) ) ) ) ) ).
% card_ge_UNIV
thf(fact_7926_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_7927_card__atLeastLessThan__int,axiom,
! [L: int,U: int] :
( ( finite_card @ int @ ( set_or7035219750837199246ssThan @ int @ L @ U ) )
= ( nat2 @ ( minus_minus @ int @ U @ L ) ) ) ).
% card_atLeastLessThan_int
thf(fact_7928_prod__constant,axiom,
! [A: $tType,B: $tType] :
( ( comm_monoid_mult @ A )
=> ! [Y3: A,A2: set @ B] :
( ( groups7121269368397514597t_prod @ B @ A
@ ^ [X3: B] : Y3
@ A2 )
= ( power_power @ A @ Y3 @ ( finite_card @ B @ A2 ) ) ) ) ).
% prod_constant
thf(fact_7929_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_7930_card__greaterThanAtMost__int,axiom,
! [L: int,U: int] :
( ( finite_card @ int @ ( set_or3652927894154168847AtMost @ int @ L @ U ) )
= ( nat2 @ ( minus_minus @ int @ U @ L ) ) ) ).
% card_greaterThanAtMost_int
thf(fact_7931_card__insert__disjoint,axiom,
! [A: $tType,A2: set @ A,X2: A] :
( ( finite_finite2 @ A @ A2 )
=> ( ~ ( member @ A @ X2 @ A2 )
=> ( ( finite_card @ A @ ( insert @ A @ X2 @ A2 ) )
= ( suc @ ( finite_card @ A @ A2 ) ) ) ) ) ).
% card_insert_disjoint
thf(fact_7932_sum__constant,axiom,
! [B: $tType,A: $tType] :
( ( semiring_1 @ A )
=> ! [Y3: A,A2: set @ B] :
( ( groups7311177749621191930dd_sum @ B @ A
@ ^ [X3: B] : Y3
@ A2 )
= ( times_times @ A @ ( semiring_1_of_nat @ A @ ( finite_card @ B @ A2 ) ) @ Y3 ) ) ) ).
% sum_constant
thf(fact_7933_card__Diff__insert,axiom,
! [A: $tType,A3: A,A2: set @ A,B2: set @ A] :
( ( member @ A @ A3 @ A2 )
=> ( ~ ( member @ A @ A3 @ B2 )
=> ( ( finite_card @ A @ ( minus_minus @ ( set @ A ) @ A2 @ ( insert @ A @ A3 @ B2 ) ) )
= ( minus_minus @ nat @ ( finite_card @ A @ ( minus_minus @ ( set @ A ) @ A2 @ B2 ) ) @ ( one_one @ nat ) ) ) ) ) ).
% card_Diff_insert
thf(fact_7934_card__atLeastAtMost__int,axiom,
! [L: int,U: int] :
( ( finite_card @ int @ ( set_or1337092689740270186AtMost @ int @ L @ U ) )
= ( nat2 @ ( plus_plus @ int @ ( minus_minus @ int @ U @ L ) @ ( one_one @ int ) ) ) ) ).
% card_atLeastAtMost_int
thf(fact_7935_card__doubleton__eq__2__iff,axiom,
! [A: $tType,A3: A,B3: A] :
( ( ( finite_card @ A @ ( insert @ A @ A3 @ ( insert @ A @ B3 @ ( bot_bot @ ( set @ A ) ) ) ) )
= ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) )
= ( A3 != B3 ) ) ).
% card_doubleton_eq_2_iff
thf(fact_7936_card__greaterThanLessThan__int,axiom,
! [L: int,U: int] :
( ( finite_card @ int @ ( set_or5935395276787703475ssThan @ int @ L @ U ) )
= ( nat2 @ ( minus_minus @ int @ U @ ( plus_plus @ int @ L @ ( one_one @ int ) ) ) ) ) ).
% card_greaterThanLessThan_int
thf(fact_7937_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_7938_pigeonhole,axiom,
! [A: $tType,B: $tType,F2: B > A,A2: set @ B] :
( ( ord_less @ nat @ ( finite_card @ A @ ( image @ B @ A @ F2 @ A2 ) ) @ ( finite_card @ B @ A2 ) )
=> ~ ( inj_on @ B @ A @ F2 @ A2 ) ) ).
% pigeonhole
thf(fact_7939_continuous__onI__mono,axiom,
! [B: $tType,A: $tType] :
( ( ( topolo1944317154257567458pology @ A )
& ( dense_order @ B )
& ( topolo1944317154257567458pology @ B ) )
=> ! [F2: A > B,A2: set @ A] :
( ( topolo1002775350975398744n_open @ B @ ( image @ A @ B @ F2 @ A2 ) )
=> ( ! [X4: A,Y4: A] :
( ( member @ A @ X4 @ A2 )
=> ( ( member @ A @ Y4 @ A2 )
=> ( ( ord_less_eq @ A @ X4 @ Y4 )
=> ( ord_less_eq @ B @ ( F2 @ X4 ) @ ( F2 @ Y4 ) ) ) ) )
=> ( topolo81223032696312382ous_on @ A @ B @ A2 @ F2 ) ) ) ) ).
% continuous_onI_mono
thf(fact_7940_card__image__le,axiom,
! [B: $tType,A: $tType,A2: set @ A,F2: A > B] :
( ( finite_finite2 @ A @ A2 )
=> ( ord_less_eq @ nat @ ( finite_card @ B @ ( image @ A @ B @ F2 @ A2 ) ) @ ( finite_card @ A @ A2 ) ) ) ).
% card_image_le
thf(fact_7941_continuous__image__closed__interval,axiom,
! [A3: real,B3: real,F2: real > real] :
( ( ord_less_eq @ real @ A3 @ B3 )
=> ( ( topolo81223032696312382ous_on @ real @ real @ ( set_or1337092689740270186AtMost @ real @ A3 @ B3 ) @ F2 )
=> ? [C4: real,D4: real] :
( ( ( image @ real @ real @ F2 @ ( set_or1337092689740270186AtMost @ real @ A3 @ B3 ) )
= ( set_or1337092689740270186AtMost @ real @ C4 @ D4 ) )
& ( ord_less_eq @ real @ C4 @ D4 ) ) ) ) ).
% continuous_image_closed_interval
thf(fact_7942_continuous__on__compose2,axiom,
! [A: $tType,B: $tType,C: $tType] :
( ( ( topolo4958980785337419405_space @ C )
& ( topolo4958980785337419405_space @ B )
& ( topolo4958980785337419405_space @ A ) )
=> ! [T2: set @ A,G: A > B,S2: set @ C,F2: C > A] :
( ( topolo81223032696312382ous_on @ A @ B @ T2 @ G )
=> ( ( topolo81223032696312382ous_on @ C @ A @ S2 @ F2 )
=> ( ( ord_less_eq @ ( set @ A ) @ ( image @ C @ A @ F2 @ S2 ) @ T2 )
=> ( topolo81223032696312382ous_on @ C @ B @ S2
@ ^ [X3: C] : ( G @ ( F2 @ X3 ) ) ) ) ) ) ) ).
% continuous_on_compose2
thf(fact_7943_continuous__on__open__UN,axiom,
! [A: $tType,C: $tType,B: $tType] :
( ( ( topolo4958980785337419405_space @ B )
& ( topolo4958980785337419405_space @ C ) )
=> ! [S: set @ A,A2: A > ( set @ B ),F2: B > C] :
( ! [S3: A] :
( ( member @ A @ S3 @ S )
=> ( topolo1002775350975398744n_open @ B @ ( A2 @ S3 ) ) )
=> ( ! [S3: A] :
( ( member @ A @ S3 @ S )
=> ( topolo81223032696312382ous_on @ B @ C @ ( A2 @ S3 ) @ F2 ) )
=> ( topolo81223032696312382ous_on @ B @ C @ ( complete_Sup_Sup @ ( set @ B ) @ ( image @ A @ ( set @ B ) @ A2 @ S ) ) @ F2 ) ) ) ) ).
% continuous_on_open_UN
thf(fact_7944_card__Union__le__sum__card__weak,axiom,
! [A: $tType,U6: set @ ( set @ A )] :
( ! [X4: set @ A] :
( ( member @ ( set @ A ) @ X4 @ U6 )
=> ( finite_finite2 @ A @ X4 ) )
=> ( ord_less_eq @ nat @ ( finite_card @ A @ ( complete_Sup_Sup @ ( set @ A ) @ U6 ) ) @ ( groups7311177749621191930dd_sum @ ( set @ A ) @ nat @ ( finite_card @ A ) @ U6 ) ) ) ).
% card_Union_le_sum_card_weak
thf(fact_7945_card__Union__le__sum__card,axiom,
! [A: $tType,U6: set @ ( set @ A )] : ( ord_less_eq @ nat @ ( finite_card @ A @ ( complete_Sup_Sup @ ( set @ A ) @ U6 ) ) @ ( groups7311177749621191930dd_sum @ ( set @ A ) @ nat @ ( finite_card @ A ) @ U6 ) ) ).
% card_Union_le_sum_card
thf(fact_7946_continuous__on__cot,axiom,
! [A: $tType] :
( ( ( real_Vector_banach @ A )
& ( real_V3459762299906320749_field @ A ) )
=> ! [S2: set @ A,F2: A > A] :
( ( topolo81223032696312382ous_on @ A @ A @ S2 @ F2 )
=> ( ! [X4: A] :
( ( member @ A @ X4 @ S2 )
=> ( ( sin @ A @ ( F2 @ X4 ) )
!= ( zero_zero @ A ) ) )
=> ( topolo81223032696312382ous_on @ A @ A @ S2
@ ^ [X3: A] : ( cot @ A @ ( F2 @ X3 ) ) ) ) ) ) ).
% continuous_on_cot
thf(fact_7947_continuous__on__tendsto__compose,axiom,
! [A: $tType,B: $tType,C: $tType] :
( ( ( topolo4958980785337419405_space @ B )
& ( topolo4958980785337419405_space @ A ) )
=> ! [S2: set @ A,F2: A > B,G: C > A,L: A,F4: filter @ C] :
( ( topolo81223032696312382ous_on @ A @ B @ S2 @ F2 )
=> ( ( filterlim @ C @ A @ G @ ( topolo7230453075368039082e_nhds @ A @ L ) @ F4 )
=> ( ( member @ A @ L @ S2 )
=> ( ( eventually @ C
@ ^ [X3: C] : ( member @ A @ ( G @ X3 ) @ S2 )
@ F4 )
=> ( filterlim @ C @ B
@ ^ [X3: C] : ( F2 @ ( G @ X3 ) )
@ ( topolo7230453075368039082e_nhds @ B @ ( F2 @ L ) )
@ F4 ) ) ) ) ) ) ).
% continuous_on_tendsto_compose
thf(fact_7948_sum__multicount__gen,axiom,
! [A: $tType,B: $tType,S2: set @ A,T2: set @ B,R: A > B > $o,K: B > nat] :
( ( finite_finite2 @ A @ S2 )
=> ( ( finite_finite2 @ B @ T2 )
=> ( ! [X4: B] :
( ( member @ B @ X4 @ T2 )
=> ( ( finite_card @ A
@ ( collect @ A
@ ^ [I4: A] :
( ( member @ A @ I4 @ S2 )
& ( R @ I4 @ X4 ) ) ) )
= ( K @ X4 ) ) )
=> ( ( groups7311177749621191930dd_sum @ A @ nat
@ ^ [I4: A] :
( finite_card @ B
@ ( collect @ B
@ ^ [J3: B] :
( ( member @ B @ J3 @ T2 )
& ( R @ I4 @ J3 ) ) ) )
@ S2 )
= ( groups7311177749621191930dd_sum @ B @ nat @ K @ T2 ) ) ) ) ) ).
% sum_multicount_gen
thf(fact_7949_card__Suc__eq__finite,axiom,
! [A: $tType,A2: set @ A,K: nat] :
( ( ( finite_card @ A @ A2 )
= ( suc @ K ) )
= ( ? [B7: A,B6: set @ A] :
( ( A2
= ( insert @ A @ B7 @ B6 ) )
& ~ ( member @ A @ B7 @ B6 )
& ( ( finite_card @ A @ B6 )
= K )
& ( finite_finite2 @ A @ B6 ) ) ) ) ).
% card_Suc_eq_finite
thf(fact_7950_card__insert__if,axiom,
! [A: $tType,A2: set @ A,X2: A] :
( ( finite_finite2 @ A @ A2 )
=> ( ( ( member @ A @ X2 @ A2 )
=> ( ( finite_card @ A @ ( insert @ A @ X2 @ A2 ) )
= ( finite_card @ A @ A2 ) ) )
& ( ~ ( member @ A @ X2 @ A2 )
=> ( ( finite_card @ A @ ( insert @ A @ X2 @ A2 ) )
= ( suc @ ( finite_card @ A @ A2 ) ) ) ) ) ) ).
% card_insert_if
thf(fact_7951_psubset__card__mono,axiom,
! [A: $tType,B2: set @ A,A2: set @ A] :
( ( finite_finite2 @ A @ B2 )
=> ( ( ord_less @ ( set @ A ) @ A2 @ B2 )
=> ( ord_less @ nat @ ( finite_card @ A @ A2 ) @ ( finite_card @ A @ B2 ) ) ) ) ).
% psubset_card_mono
thf(fact_7952_card__ge__0__finite,axiom,
! [A: $tType,A2: set @ A] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ ( finite_card @ A @ A2 ) )
=> ( finite_finite2 @ A @ A2 ) ) ).
% card_ge_0_finite
thf(fact_7953_card__less__sym__Diff,axiom,
! [A: $tType,A2: set @ A,B2: set @ A] :
( ( finite_finite2 @ A @ A2 )
=> ( ( finite_finite2 @ A @ B2 )
=> ( ( ord_less @ nat @ ( finite_card @ A @ A2 ) @ ( finite_card @ A @ B2 ) )
=> ( ord_less @ nat @ ( finite_card @ A @ ( minus_minus @ ( set @ A ) @ A2 @ B2 ) ) @ ( finite_card @ A @ ( minus_minus @ ( set @ A ) @ B2 @ A2 ) ) ) ) ) ) ).
% card_less_sym_Diff
thf(fact_7954_continuous__on__dist,axiom,
! [A: $tType,D6: $tType] :
( ( ( topolo4958980785337419405_space @ D6 )
& ( real_V7819770556892013058_space @ A ) )
=> ! [S2: set @ D6,F2: D6 > A,G: D6 > A] :
( ( topolo81223032696312382ous_on @ D6 @ A @ S2 @ F2 )
=> ( ( topolo81223032696312382ous_on @ D6 @ A @ S2 @ G )
=> ( topolo81223032696312382ous_on @ D6 @ real @ S2
@ ^ [X3: D6] : ( real_V557655796197034286t_dist @ A @ ( F2 @ X3 ) @ ( G @ X3 ) ) ) ) ) ) ).
% continuous_on_dist
thf(fact_7955_open__Collect__less,axiom,
! [B: $tType,A: $tType] :
( ( ( topolo4958980785337419405_space @ A )
& ( topolo1944317154257567458pology @ B ) )
=> ! [F2: A > B,G: A > B] :
( ( topolo81223032696312382ous_on @ A @ B @ ( top_top @ ( set @ A ) ) @ F2 )
=> ( ( topolo81223032696312382ous_on @ A @ B @ ( top_top @ ( set @ A ) ) @ G )
=> ( topolo1002775350975398744n_open @ A
@ ( collect @ A
@ ^ [X3: A] : ( ord_less @ B @ ( F2 @ X3 ) @ ( G @ X3 ) ) ) ) ) ) ) ).
% open_Collect_less
thf(fact_7956_bounded__linear_Ocontinuous__on,axiom,
! [A: $tType,B: $tType,C: $tType] :
( ( ( topolo4958980785337419405_space @ C )
& ( real_V822414075346904944vector @ B )
& ( real_V822414075346904944vector @ A ) )
=> ! [F2: A > B,S2: set @ C,G: C > A] :
( ( real_V3181309239436604168linear @ A @ B @ F2 )
=> ( ( topolo81223032696312382ous_on @ C @ A @ S2 @ G )
=> ( topolo81223032696312382ous_on @ C @ B @ S2
@ ^ [X3: C] : ( F2 @ ( G @ X3 ) ) ) ) ) ) ).
% bounded_linear.continuous_on
thf(fact_7957_open__Collect__neq,axiom,
! [B: $tType,A: $tType] :
( ( ( topolo4958980785337419405_space @ A )
& ( topological_t2_space @ B ) )
=> ! [F2: A > B,G: A > B] :
( ( topolo81223032696312382ous_on @ A @ B @ ( top_top @ ( set @ A ) ) @ F2 )
=> ( ( topolo81223032696312382ous_on @ A @ B @ ( top_top @ ( set @ A ) ) @ G )
=> ( topolo1002775350975398744n_open @ A
@ ( collect @ A
@ ^ [X3: A] :
( ( F2 @ X3 )
!= ( G @ X3 ) ) ) ) ) ) ) ).
% open_Collect_neq
thf(fact_7958_continuous__inj__imp__mono,axiom,
! [B: $tType,A: $tType] :
( ( ( topolo8458572112393995274pology @ A )
& ( topolo1944317154257567458pology @ B ) )
=> ! [A3: A,X2: A,B3: A,F2: A > B] :
( ( ord_less @ A @ A3 @ X2 )
=> ( ( ord_less @ A @ X2 @ B3 )
=> ( ( topolo81223032696312382ous_on @ A @ B @ ( set_or1337092689740270186AtMost @ A @ A3 @ B3 ) @ F2 )
=> ( ( inj_on @ A @ B @ F2 @ ( set_or1337092689740270186AtMost @ A @ A3 @ B3 ) )
=> ( ( ( ord_less @ B @ ( F2 @ A3 ) @ ( F2 @ X2 ) )
& ( ord_less @ B @ ( F2 @ X2 ) @ ( F2 @ B3 ) ) )
| ( ( ord_less @ B @ ( F2 @ B3 ) @ ( F2 @ X2 ) )
& ( ord_less @ B @ ( F2 @ X2 ) @ ( F2 @ A3 ) ) ) ) ) ) ) ) ) ).
% continuous_inj_imp_mono
thf(fact_7959_card__2__iff_H,axiom,
! [A: $tType,S: set @ A] :
( ( ( finite_card @ A @ S )
= ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) )
= ( ? [X3: A] :
( ( member @ A @ X3 @ S )
& ? [Y: A] :
( ( member @ A @ Y @ S )
& ( X3 != Y )
& ! [Z4: A] :
( ( member @ A @ Z4 @ S )
=> ( ( Z4 = X3 )
| ( Z4 = Y ) ) ) ) ) ) ) ).
% card_2_iff'
thf(fact_7960_continuous__on__op__minus,axiom,
! [A: $tType] :
( ( topolo1633459387980952147up_add @ A )
=> ! [S2: set @ A,X2: A] : ( topolo81223032696312382ous_on @ A @ A @ S2 @ ( minus_minus @ A @ X2 ) ) ) ).
% continuous_on_op_minus
thf(fact_7961_continuous__on__mult__const,axiom,
! [A: $tType] :
( ( real_V4412858255891104859lgebra @ A )
=> ! [S2: set @ A,C3: A] : ( topolo81223032696312382ous_on @ A @ A @ S2 @ ( times_times @ A @ C3 ) ) ) ).
% continuous_on_mult_const
thf(fact_7962_continuous__on__min,axiom,
! [B: $tType,A: $tType] :
( ( ( topolo4958980785337419405_space @ A )
& ( topolo1944317154257567458pology @ B ) )
=> ! [A2: set @ A,F2: A > B,G: A > B] :
( ( topolo81223032696312382ous_on @ A @ B @ A2 @ F2 )
=> ( ( topolo81223032696312382ous_on @ A @ B @ A2 @ G )
=> ( topolo81223032696312382ous_on @ A @ B @ A2
@ ^ [X3: A] : ( ord_min @ B @ ( F2 @ X3 ) @ ( G @ X3 ) ) ) ) ) ) ).
% continuous_on_min
thf(fact_7963_continuous__on__Pair,axiom,
! [C: $tType,B: $tType,A: $tType] :
( ( ( topolo4958980785337419405_space @ A )
& ( topolo4958980785337419405_space @ B )
& ( topolo4958980785337419405_space @ C ) )
=> ! [S2: set @ A,F2: A > B,G: A > C] :
( ( topolo81223032696312382ous_on @ A @ B @ S2 @ F2 )
=> ( ( topolo81223032696312382ous_on @ A @ C @ S2 @ G )
=> ( topolo81223032696312382ous_on @ A @ ( product_prod @ B @ C ) @ S2
@ ^ [X3: A] : ( product_Pair @ B @ C @ ( F2 @ X3 ) @ ( G @ X3 ) ) ) ) ) ) ).
% continuous_on_Pair
thf(fact_7964_continuous__on__divide,axiom,
! [B: $tType,A: $tType] :
( ( ( topolo4958980785337419405_space @ A )
& ( real_V3459762299906320749_field @ B ) )
=> ! [S2: set @ A,F2: A > B,G: A > B] :
( ( topolo81223032696312382ous_on @ A @ B @ S2 @ F2 )
=> ( ( topolo81223032696312382ous_on @ A @ B @ S2 @ G )
=> ( ! [X4: A] :
( ( member @ A @ X4 @ S2 )
=> ( ( G @ X4 )
!= ( zero_zero @ B ) ) )
=> ( topolo81223032696312382ous_on @ A @ B @ S2
@ ^ [X3: A] : ( divide_divide @ B @ ( F2 @ X3 ) @ ( G @ X3 ) ) ) ) ) ) ) ).
% continuous_on_divide
thf(fact_7965_continuous__on__powr,axiom,
! [C: $tType] :
( ( topolo4958980785337419405_space @ C )
=> ! [S2: set @ C,F2: C > real,G: C > real] :
( ( topolo81223032696312382ous_on @ C @ real @ S2 @ F2 )
=> ( ( topolo81223032696312382ous_on @ C @ real @ S2 @ G )
=> ( ! [X4: C] :
( ( member @ C @ X4 @ S2 )
=> ( ( F2 @ X4 )
!= ( zero_zero @ real ) ) )
=> ( topolo81223032696312382ous_on @ C @ real @ S2
@ ^ [X3: C] : ( powr @ real @ ( F2 @ X3 ) @ ( G @ X3 ) ) ) ) ) ) ) ).
% continuous_on_powr
thf(fact_7966_continuous__on__mult,axiom,
! [A: $tType,D6: $tType] :
( ( ( topolo4958980785337419405_space @ D6 )
& ( real_V4412858255891104859lgebra @ A ) )
=> ! [S2: set @ D6,F2: D6 > A,G: D6 > A] :
( ( topolo81223032696312382ous_on @ D6 @ A @ S2 @ F2 )
=> ( ( topolo81223032696312382ous_on @ D6 @ A @ S2 @ G )
=> ( topolo81223032696312382ous_on @ D6 @ A @ S2
@ ^ [X3: D6] : ( times_times @ A @ ( F2 @ X3 ) @ ( G @ X3 ) ) ) ) ) ) ).
% continuous_on_mult
thf(fact_7967_continuous__on__mult_H,axiom,
! [B: $tType,D6: $tType] :
( ( ( topolo4958980785337419405_space @ D6 )
& ( topolo4211221413907600880p_mult @ B ) )
=> ! [A2: set @ D6,F2: D6 > B,G: D6 > B] :
( ( topolo81223032696312382ous_on @ D6 @ B @ A2 @ F2 )
=> ( ( topolo81223032696312382ous_on @ D6 @ B @ A2 @ G )
=> ( topolo81223032696312382ous_on @ D6 @ B @ A2
@ ^ [X3: D6] : ( times_times @ B @ ( F2 @ X3 ) @ ( G @ X3 ) ) ) ) ) ) ).
% continuous_on_mult'
thf(fact_7968_continuous__on__mult__left,axiom,
! [A: $tType,B: $tType] :
( ( ( topolo4958980785337419405_space @ B )
& ( real_V4412858255891104859lgebra @ A ) )
=> ! [S2: set @ B,F2: B > A,C3: A] :
( ( topolo81223032696312382ous_on @ B @ A @ S2 @ F2 )
=> ( topolo81223032696312382ous_on @ B @ A @ S2
@ ^ [X3: B] : ( times_times @ A @ C3 @ ( F2 @ X3 ) ) ) ) ) ).
% continuous_on_mult_left
thf(fact_7969_continuous__on__mult__right,axiom,
! [A: $tType,B: $tType] :
( ( ( topolo4958980785337419405_space @ B )
& ( real_V4412858255891104859lgebra @ A ) )
=> ! [S2: set @ B,F2: B > A,C3: A] :
( ( topolo81223032696312382ous_on @ B @ A @ S2 @ F2 )
=> ( topolo81223032696312382ous_on @ B @ A @ S2
@ ^ [X3: B] : ( times_times @ A @ ( F2 @ X3 ) @ C3 ) ) ) ) ).
% continuous_on_mult_right
thf(fact_7970_continuous__on__rabs,axiom,
! [A: $tType] :
( ( topolo4958980785337419405_space @ A )
=> ! [S2: set @ A,F2: A > real] :
( ( topolo81223032696312382ous_on @ A @ real @ S2 @ F2 )
=> ( topolo81223032696312382ous_on @ A @ real @ S2
@ ^ [X3: A] : ( abs_abs @ real @ ( F2 @ X3 ) ) ) ) ) ).
% continuous_on_rabs
thf(fact_7971_continuous__on__sgn,axiom,
! [B: $tType,A: $tType] :
( ( ( topolo4958980785337419405_space @ A )
& ( real_V822414075346904944vector @ B ) )
=> ! [S2: set @ A,F2: A > B] :
( ( topolo81223032696312382ous_on @ A @ B @ S2 @ F2 )
=> ( ! [X4: A] :
( ( member @ A @ X4 @ S2 )
=> ( ( F2 @ X4 )
!= ( zero_zero @ B ) ) )
=> ( topolo81223032696312382ous_on @ A @ B @ S2
@ ^ [X3: A] : ( sgn_sgn @ B @ ( F2 @ X3 ) ) ) ) ) ) ).
% continuous_on_sgn
thf(fact_7972_continuous__on__minus,axiom,
! [B: $tType,C: $tType] :
( ( ( topolo4958980785337419405_space @ C )
& ( topolo1633459387980952147up_add @ B ) )
=> ! [S2: set @ C,F2: C > B] :
( ( topolo81223032696312382ous_on @ C @ B @ S2 @ F2 )
=> ( topolo81223032696312382ous_on @ C @ B @ S2
@ ^ [X3: C] : ( uminus_uminus @ B @ ( F2 @ X3 ) ) ) ) ) ).
% continuous_on_minus
thf(fact_7973_continuous__on__real__root,axiom,
! [A: $tType] :
( ( topolo4958980785337419405_space @ A )
=> ! [S2: set @ A,F2: A > real,N: nat] :
( ( topolo81223032696312382ous_on @ A @ real @ S2 @ F2 )
=> ( topolo81223032696312382ous_on @ A @ real @ S2
@ ^ [X3: A] : ( root @ N @ ( F2 @ X3 ) ) ) ) ) ).
% continuous_on_real_root
thf(fact_7974_continuous__on__exp,axiom,
! [A: $tType,C: $tType] :
( ( ( topolo4958980785337419405_space @ C )
& ( real_Vector_banach @ A )
& ( real_V3459762299906320749_field @ A ) )
=> ! [S2: set @ C,F2: C > A] :
( ( topolo81223032696312382ous_on @ C @ A @ S2 @ F2 )
=> ( topolo81223032696312382ous_on @ C @ A @ S2
@ ^ [X3: C] : ( exp @ A @ ( F2 @ X3 ) ) ) ) ) ).
% continuous_on_exp
thf(fact_7975_continuous__on__const,axiom,
! [B: $tType,A: $tType] :
( ( ( topolo4958980785337419405_space @ A )
& ( topolo4958980785337419405_space @ B ) )
=> ! [S2: set @ A,C3: B] :
( topolo81223032696312382ous_on @ A @ B @ S2
@ ^ [X3: A] : C3 ) ) ).
% continuous_on_const
thf(fact_7976_continuous__on__id,axiom,
! [A: $tType] :
( ( topolo4958980785337419405_space @ A )
=> ! [S2: set @ A] :
( topolo81223032696312382ous_on @ A @ A @ S2
@ ^ [X3: A] : X3 ) ) ).
% continuous_on_id
thf(fact_7977_continuous__on__arctan,axiom,
! [A: $tType] :
( ( topolo4958980785337419405_space @ A )
=> ! [S2: set @ A,F2: A > real] :
( ( topolo81223032696312382ous_on @ A @ real @ S2 @ F2 )
=> ( topolo81223032696312382ous_on @ A @ real @ S2
@ ^ [X3: A] : ( arctan @ ( F2 @ X3 ) ) ) ) ) ).
% continuous_on_arctan
thf(fact_7978_continuous__on__sin,axiom,
! [B: $tType,A: $tType] :
( ( ( topological_t2_space @ A )
& ( real_Vector_banach @ B )
& ( real_V3459762299906320749_field @ B ) )
=> ! [S2: set @ A,F2: A > B] :
( ( topolo81223032696312382ous_on @ A @ B @ S2 @ F2 )
=> ( topolo81223032696312382ous_on @ A @ B @ S2
@ ^ [X3: A] : ( sin @ B @ ( F2 @ X3 ) ) ) ) ) ).
% continuous_on_sin
thf(fact_7979_continuous__on__cos,axiom,
! [B: $tType,A: $tType] :
( ( ( topological_t2_space @ A )
& ( real_Vector_banach @ B )
& ( real_V3459762299906320749_field @ B ) )
=> ! [S2: set @ A,F2: A > B] :
( ( topolo81223032696312382ous_on @ A @ B @ S2 @ F2 )
=> ( topolo81223032696312382ous_on @ A @ B @ S2
@ ^ [X3: A] : ( cos @ B @ ( F2 @ X3 ) ) ) ) ) ).
% continuous_on_cos
thf(fact_7980_continuous__on__scaleR,axiom,
! [C: $tType,D6: $tType] :
( ( ( topolo4958980785337419405_space @ D6 )
& ( real_V822414075346904944vector @ C ) )
=> ! [S2: set @ D6,F2: D6 > real,G: D6 > C] :
( ( topolo81223032696312382ous_on @ D6 @ real @ S2 @ F2 )
=> ( ( topolo81223032696312382ous_on @ D6 @ C @ S2 @ G )
=> ( topolo81223032696312382ous_on @ D6 @ C @ S2
@ ^ [X3: D6] : ( real_V8093663219630862766scaleR @ C @ ( F2 @ X3 ) @ ( G @ X3 ) ) ) ) ) ) ).
% continuous_on_scaleR
thf(fact_7981_continuous__on__pochhammer,axiom,
! [A: $tType] :
( ( real_V3459762299906320749_field @ A )
=> ! [A2: set @ A,N: nat] :
( topolo81223032696312382ous_on @ A @ A @ A2
@ ^ [Z4: A] : ( comm_s3205402744901411588hammer @ A @ Z4 @ N ) ) ) ).
% continuous_on_pochhammer
thf(fact_7982_continuous__on__pochhammer_H,axiom,
! [A: $tType,C: $tType] :
( ( ( topolo4958980785337419405_space @ C )
& ( real_V3459762299906320749_field @ A ) )
=> ! [S2: set @ C,F2: C > A,N: nat] :
( ( topolo81223032696312382ous_on @ C @ A @ S2 @ F2 )
=> ( topolo81223032696312382ous_on @ C @ A @ S2
@ ^ [X3: C] : ( comm_s3205402744901411588hammer @ A @ ( F2 @ X3 ) @ N ) ) ) ) ).
% continuous_on_pochhammer'
thf(fact_7983_continuous__on__norm,axiom,
! [B: $tType,A: $tType] :
( ( ( topolo4958980785337419405_space @ A )
& ( real_V822414075346904944vector @ B ) )
=> ! [S2: set @ A,F2: A > B] :
( ( topolo81223032696312382ous_on @ A @ B @ S2 @ F2 )
=> ( topolo81223032696312382ous_on @ A @ real @ S2
@ ^ [X3: A] : ( real_V7770717601297561774m_norm @ B @ ( F2 @ X3 ) ) ) ) ) ).
% continuous_on_norm
thf(fact_7984_continuous__on__real__sqrt,axiom,
! [A: $tType] :
( ( topolo4958980785337419405_space @ A )
=> ! [S2: set @ A,F2: A > real] :
( ( topolo81223032696312382ous_on @ A @ real @ S2 @ F2 )
=> ( topolo81223032696312382ous_on @ A @ real @ S2
@ ^ [X3: A] : ( sqrt @ ( F2 @ X3 ) ) ) ) ) ).
% continuous_on_real_sqrt
thf(fact_7985_continuous__on__arsinh_H,axiom,
! [A2: set @ real,F2: real > real] :
( ( topolo81223032696312382ous_on @ real @ real @ A2 @ F2 )
=> ( topolo81223032696312382ous_on @ real @ real @ A2
@ ^ [X3: real] : ( arsinh @ real @ ( F2 @ X3 ) ) ) ) ).
% continuous_on_arsinh'
thf(fact_7986_continuous__on__of__real,axiom,
! [A: $tType,C: $tType] :
( ( ( topolo4958980785337419405_space @ C )
& ( real_V2191834092415804123ebra_1 @ A )
& ( real_V822414075346904944vector @ A ) )
=> ! [S2: set @ C,G: C > real] :
( ( topolo81223032696312382ous_on @ C @ real @ S2 @ G )
=> ( topolo81223032696312382ous_on @ C @ A @ S2
@ ^ [X3: C] : ( real_Vector_of_real @ A @ ( G @ X3 ) ) ) ) ) ).
% continuous_on_of_real
thf(fact_7987_continuous__on__sinh,axiom,
! [A: $tType,C: $tType] :
( ( ( topolo4958980785337419405_space @ C )
& ( real_Vector_banach @ A )
& ( real_V3459762299906320749_field @ A ) )
=> ! [A2: set @ C,F2: C > A] :
( ( topolo81223032696312382ous_on @ C @ A @ A2 @ F2 )
=> ( topolo81223032696312382ous_on @ C @ A @ A2
@ ^ [X3: C] : ( sinh @ A @ ( F2 @ X3 ) ) ) ) ) ).
% continuous_on_sinh
thf(fact_7988_continuous__on__cosh,axiom,
! [A: $tType,C: $tType] :
( ( ( topolo4958980785337419405_space @ C )
& ( real_Vector_banach @ A )
& ( real_V3459762299906320749_field @ A ) )
=> ! [A2: set @ C,F2: C > A] :
( ( topolo81223032696312382ous_on @ C @ A @ A2 @ F2 )
=> ( topolo81223032696312382ous_on @ C @ A @ A2
@ ^ [X3: C] : ( cosh @ A @ ( F2 @ X3 ) ) ) ) ) ).
% continuous_on_cosh
thf(fact_7989_continuous__on__sum,axiom,
! [A: $tType,C: $tType,B: $tType] :
( ( ( topolo4958980785337419405_space @ B )
& ( topolo5987344860129210374id_add @ C ) )
=> ! [I6: set @ A,S: set @ B,F2: A > B > C] :
( ! [I2: A] :
( ( member @ A @ I2 @ I6 )
=> ( topolo81223032696312382ous_on @ B @ C @ S @ ( F2 @ I2 ) ) )
=> ( topolo81223032696312382ous_on @ B @ C @ S
@ ^ [X3: B] :
( groups7311177749621191930dd_sum @ A @ C
@ ^ [I4: A] : ( F2 @ I4 @ X3 )
@ I6 ) ) ) ) ).
% continuous_on_sum
thf(fact_7990_continuous__on__prod_H,axiom,
! [A: $tType,C: $tType,B: $tType] :
( ( ( topolo4958980785337419405_space @ B )
& ( topolo4987421752381908075d_mult @ C ) )
=> ! [I6: set @ A,S: set @ B,F2: A > B > C] :
( ! [I2: A] :
( ( member @ A @ I2 @ I6 )
=> ( topolo81223032696312382ous_on @ B @ C @ S @ ( F2 @ I2 ) ) )
=> ( topolo81223032696312382ous_on @ B @ C @ S
@ ^ [X3: B] :
( groups7121269368397514597t_prod @ A @ C
@ ^ [I4: A] : ( F2 @ I4 @ X3 )
@ I6 ) ) ) ) ).
% continuous_on_prod'
thf(fact_7991_continuous__on__prod,axiom,
! [A: $tType,C: $tType,D6: $tType] :
( ( ( topolo4958980785337419405_space @ D6 )
& ( real_V4412858255891104859lgebra @ C )
& ( comm_ring_1 @ C ) )
=> ! [S: set @ A,S2: set @ D6,F2: A > D6 > C] :
( ! [I2: A] :
( ( member @ A @ I2 @ S )
=> ( topolo81223032696312382ous_on @ D6 @ C @ S2 @ ( F2 @ I2 ) ) )
=> ( topolo81223032696312382ous_on @ D6 @ C @ S2
@ ^ [X3: D6] :
( groups7121269368397514597t_prod @ A @ C
@ ^ [I4: A] : ( F2 @ I4 @ X3 )
@ S ) ) ) ) ).
% continuous_on_prod
thf(fact_7992_continuous__on__max,axiom,
! [B: $tType,A: $tType] :
( ( ( topolo4958980785337419405_space @ A )
& ( topolo1944317154257567458pology @ B ) )
=> ! [A2: set @ A,F2: A > B,G: A > B] :
( ( topolo81223032696312382ous_on @ A @ B @ A2 @ F2 )
=> ( ( topolo81223032696312382ous_on @ A @ B @ A2 @ G )
=> ( topolo81223032696312382ous_on @ A @ B @ A2
@ ^ [X3: A] : ( ord_max @ B @ ( F2 @ X3 ) @ ( G @ X3 ) ) ) ) ) ) ).
% continuous_on_max
thf(fact_7993_continuous__on__diff,axiom,
! [B: $tType,D6: $tType] :
( ( ( topolo4958980785337419405_space @ D6 )
& ( topolo1633459387980952147up_add @ B ) )
=> ! [S2: set @ D6,F2: D6 > B,G: D6 > B] :
( ( topolo81223032696312382ous_on @ D6 @ B @ S2 @ F2 )
=> ( ( topolo81223032696312382ous_on @ D6 @ B @ S2 @ G )
=> ( topolo81223032696312382ous_on @ D6 @ B @ S2
@ ^ [X3: D6] : ( minus_minus @ B @ ( F2 @ X3 ) @ ( G @ X3 ) ) ) ) ) ) ).
% continuous_on_diff
thf(fact_7994_continuous__on__add,axiom,
! [B: $tType,D6: $tType] :
( ( ( topolo4958980785337419405_space @ D6 )
& ( topolo6943815403480290642id_add @ B ) )
=> ! [S2: set @ D6,F2: D6 > B,G: D6 > B] :
( ( topolo81223032696312382ous_on @ D6 @ B @ S2 @ F2 )
=> ( ( topolo81223032696312382ous_on @ D6 @ B @ S2 @ G )
=> ( topolo81223032696312382ous_on @ D6 @ B @ S2
@ ^ [X3: D6] : ( plus_plus @ B @ ( F2 @ X3 ) @ ( G @ X3 ) ) ) ) ) ) ).
% continuous_on_add
thf(fact_7995_continuous__on__inverse,axiom,
! [B: $tType,A: $tType] :
( ( ( topolo4958980785337419405_space @ A )
& ( real_V8999393235501362500lgebra @ B ) )
=> ! [S2: set @ A,F2: A > B] :
( ( topolo81223032696312382ous_on @ A @ B @ S2 @ F2 )
=> ( ! [X4: A] :
( ( member @ A @ X4 @ S2 )
=> ( ( F2 @ X4 )
!= ( zero_zero @ B ) ) )
=> ( topolo81223032696312382ous_on @ A @ B @ S2
@ ^ [X3: A] : ( inverse_inverse @ B @ ( F2 @ X3 ) ) ) ) ) ) ).
% continuous_on_inverse
thf(fact_7996_continuous__on__ln,axiom,
! [A: $tType] :
( ( topolo4958980785337419405_space @ A )
=> ! [S2: set @ A,F2: A > real] :
( ( topolo81223032696312382ous_on @ A @ real @ S2 @ F2 )
=> ( ! [X4: A] :
( ( member @ A @ X4 @ S2 )
=> ( ( F2 @ X4 )
!= ( zero_zero @ real ) ) )
=> ( topolo81223032696312382ous_on @ A @ real @ S2
@ ^ [X3: A] : ( ln_ln @ real @ ( F2 @ X3 ) ) ) ) ) ) ).
% continuous_on_ln
thf(fact_7997_continuous__on__power,axiom,
! [B: $tType,C: $tType] :
( ( ( topolo4958980785337419405_space @ C )
& ( power @ B )
& ( real_V4412858255891104859lgebra @ B ) )
=> ! [S2: set @ C,F2: C > B,N: nat] :
( ( topolo81223032696312382ous_on @ C @ B @ S2 @ F2 )
=> ( topolo81223032696312382ous_on @ C @ B @ S2
@ ^ [X3: C] : ( power_power @ B @ ( F2 @ X3 ) @ N ) ) ) ) ).
% continuous_on_power
thf(fact_7998_continuous__on__power_H,axiom,
! [B: $tType,C: $tType] :
( ( ( topolo4958980785337419405_space @ C )
& ( topolo1898628316856586783d_mult @ B ) )
=> ! [A2: set @ C,F2: C > B,G: C > nat] :
( ( topolo81223032696312382ous_on @ C @ B @ A2 @ F2 )
=> ( ( topolo81223032696312382ous_on @ C @ nat @ A2 @ G )
=> ( topolo81223032696312382ous_on @ C @ B @ A2
@ ^ [X3: C] : ( power_power @ B @ ( F2 @ X3 ) @ ( G @ X3 ) ) ) ) ) ) ).
% continuous_on_power'
thf(fact_7999_card_Oeq__fold,axiom,
! [A: $tType] :
( ( finite_card @ A )
= ( finite_fold @ A @ nat
@ ^ [Uu3: A] : suc
@ ( zero_zero @ nat ) ) ) ).
% card.eq_fold
thf(fact_8000_card__eq__sum,axiom,
! [A: $tType] :
( ( finite_card @ A )
= ( groups7311177749621191930dd_sum @ A @ nat
@ ^ [X3: A] : ( one_one @ nat ) ) ) ).
% card_eq_sum
thf(fact_8001_continuous__on__tan,axiom,
! [A: $tType] :
( ( ( real_Vector_banach @ A )
& ( real_V3459762299906320749_field @ A ) )
=> ! [S2: set @ A,F2: A > A] :
( ( topolo81223032696312382ous_on @ A @ A @ S2 @ F2 )
=> ( ! [X4: A] :
( ( member @ A @ X4 @ S2 )
=> ( ( cos @ A @ ( F2 @ X4 ) )
!= ( zero_zero @ A ) ) )
=> ( topolo81223032696312382ous_on @ A @ A @ S2
@ ^ [X3: A] : ( tan @ A @ ( F2 @ X3 ) ) ) ) ) ) ).
% continuous_on_tan
thf(fact_8002_card__less__Suc2,axiom,
! [M8: set @ nat,I: nat] :
( ~ ( member @ nat @ ( zero_zero @ nat ) @ M8 )
=> ( ( finite_card @ nat
@ ( collect @ nat
@ ^ [K3: nat] :
( ( member @ nat @ ( suc @ K3 ) @ M8 )
& ( ord_less @ nat @ K3 @ I ) ) ) )
= ( finite_card @ nat
@ ( collect @ nat
@ ^ [K3: nat] :
( ( member @ nat @ K3 @ M8 )
& ( ord_less @ nat @ K3 @ ( suc @ I ) ) ) ) ) ) ) ).
% card_less_Suc2
thf(fact_8003_card__less__Suc,axiom,
! [M8: set @ nat,I: nat] :
( ( member @ nat @ ( zero_zero @ nat ) @ M8 )
=> ( ( suc
@ ( finite_card @ nat
@ ( collect @ nat
@ ^ [K3: nat] :
( ( member @ nat @ ( suc @ K3 ) @ M8 )
& ( ord_less @ nat @ K3 @ I ) ) ) ) )
= ( finite_card @ nat
@ ( collect @ nat
@ ^ [K3: nat] :
( ( member @ nat @ K3 @ M8 )
& ( ord_less @ nat @ K3 @ ( suc @ I ) ) ) ) ) ) ) ).
% card_less_Suc
thf(fact_8004_card__less,axiom,
! [M8: set @ nat,I: nat] :
( ( member @ nat @ ( zero_zero @ nat ) @ M8 )
=> ( ( finite_card @ nat
@ ( collect @ nat
@ ^ [K3: nat] :
( ( member @ nat @ K3 @ M8 )
& ( ord_less @ nat @ K3 @ ( suc @ I ) ) ) ) )
!= ( zero_zero @ nat ) ) ) ).
% card_less
thf(fact_8005_continuous__on__tanh,axiom,
! [A: $tType,C: $tType] :
( ( ( topolo4958980785337419405_space @ C )
& ( real_Vector_banach @ A )
& ( real_V3459762299906320749_field @ A ) )
=> ! [A2: set @ C,F2: C > A] :
( ( topolo81223032696312382ous_on @ C @ A @ A2 @ F2 )
=> ( ! [X4: C] :
( ( member @ C @ X4 @ A2 )
=> ( ( cosh @ A @ ( F2 @ X4 ) )
!= ( zero_zero @ A ) ) )
=> ( topolo81223032696312382ous_on @ C @ A @ A2
@ ^ [X3: C] : ( tanh @ A @ ( F2 @ X3 ) ) ) ) ) ) ).
% continuous_on_tanh
thf(fact_8006_card__1__singletonE,axiom,
! [A: $tType,A2: set @ A] :
( ( ( finite_card @ A @ A2 )
= ( one_one @ nat ) )
=> ~ ! [X4: A] :
( A2
!= ( insert @ A @ X4 @ ( bot_bot @ ( set @ A ) ) ) ) ) ).
% card_1_singletonE
thf(fact_8007_card__subset__eq,axiom,
! [A: $tType,B2: set @ A,A2: set @ A] :
( ( finite_finite2 @ A @ B2 )
=> ( ( ord_less_eq @ ( set @ A ) @ A2 @ B2 )
=> ( ( ( finite_card @ A @ A2 )
= ( finite_card @ A @ B2 ) )
=> ( A2 = B2 ) ) ) ) ).
% card_subset_eq
thf(fact_8008_infinite__arbitrarily__large,axiom,
! [A: $tType,A2: set @ A,N: nat] :
( ~ ( finite_finite2 @ A @ A2 )
=> ? [B10: set @ A] :
( ( finite_finite2 @ A @ B10 )
& ( ( finite_card @ A @ B10 )
= N )
& ( ord_less_eq @ ( set @ A ) @ B10 @ A2 ) ) ) ).
% infinite_arbitrarily_large
thf(fact_8009_card__le__if__inj__on__rel,axiom,
! [B: $tType,A: $tType,B2: set @ A,A2: set @ B,R2: B > A > $o] :
( ( finite_finite2 @ A @ B2 )
=> ( ! [A4: B] :
( ( member @ B @ A4 @ A2 )
=> ? [B12: A] :
( ( member @ A @ B12 @ B2 )
& ( R2 @ A4 @ B12 ) ) )
=> ( ! [A14: B,A24: B,B4: A] :
( ( member @ B @ A14 @ A2 )
=> ( ( member @ B @ A24 @ A2 )
=> ( ( member @ A @ B4 @ B2 )
=> ( ( R2 @ A14 @ B4 )
=> ( ( R2 @ A24 @ B4 )
=> ( A14 = A24 ) ) ) ) ) )
=> ( ord_less_eq @ nat @ ( finite_card @ B @ A2 ) @ ( finite_card @ A @ B2 ) ) ) ) ) ).
% card_le_if_inj_on_rel
thf(fact_8010_card__insert__le,axiom,
! [A: $tType,A2: set @ A,X2: A] : ( ord_less_eq @ nat @ ( finite_card @ A @ A2 ) @ ( finite_card @ A @ ( insert @ A @ X2 @ A2 ) ) ) ).
% card_insert_le
thf(fact_8011_n__subsets,axiom,
! [A: $tType,A2: set @ A,K: nat] :
( ( finite_finite2 @ A @ A2 )
=> ( ( finite_card @ ( set @ A )
@ ( collect @ ( set @ A )
@ ^ [B6: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ B6 @ A2 )
& ( ( finite_card @ A @ B6 )
= K ) ) ) )
= ( binomial @ ( finite_card @ A @ A2 ) @ K ) ) ) ).
% n_subsets
thf(fact_8012_continuous__on__subset,axiom,
! [B: $tType,A: $tType] :
( ( ( topolo4958980785337419405_space @ A )
& ( topolo4958980785337419405_space @ B ) )
=> ! [S2: set @ A,F2: A > B,T2: set @ A] :
( ( topolo81223032696312382ous_on @ A @ B @ S2 @ F2 )
=> ( ( ord_less_eq @ ( set @ A ) @ T2 @ S2 )
=> ( topolo81223032696312382ous_on @ A @ B @ T2 @ F2 ) ) ) ) ).
% continuous_on_subset
thf(fact_8013_IVT2_H,axiom,
! [A: $tType,B: $tType] :
( ( ( topolo1944317154257567458pology @ B )
& ( topolo8458572112393995274pology @ A ) )
=> ! [F2: A > B,B3: A,Y3: B,A3: A] :
( ( ord_less_eq @ B @ ( F2 @ B3 ) @ Y3 )
=> ( ( ord_less_eq @ B @ Y3 @ ( F2 @ A3 ) )
=> ( ( ord_less_eq @ A @ A3 @ B3 )
=> ( ( topolo81223032696312382ous_on @ A @ B @ ( set_or1337092689740270186AtMost @ A @ A3 @ B3 ) @ F2 )
=> ? [X4: A] :
( ( ord_less_eq @ A @ A3 @ X4 )
& ( ord_less_eq @ A @ X4 @ B3 )
& ( ( F2 @ X4 )
= Y3 ) ) ) ) ) ) ) ).
% IVT2'
thf(fact_8014_IVT_H,axiom,
! [A: $tType,B: $tType] :
( ( ( topolo1944317154257567458pology @ B )
& ( topolo8458572112393995274pology @ A ) )
=> ! [F2: A > B,A3: A,Y3: B,B3: A] :
( ( ord_less_eq @ B @ ( F2 @ A3 ) @ Y3 )
=> ( ( ord_less_eq @ B @ Y3 @ ( F2 @ B3 ) )
=> ( ( ord_less_eq @ A @ A3 @ B3 )
=> ( ( topolo81223032696312382ous_on @ A @ B @ ( set_or1337092689740270186AtMost @ A @ A3 @ B3 ) @ F2 )
=> ? [X4: A] :
( ( ord_less_eq @ A @ A3 @ X4 )
& ( ord_less_eq @ A @ X4 @ B3 )
& ( ( F2 @ X4 )
= Y3 ) ) ) ) ) ) ) ).
% IVT'
thf(fact_8015_card__lists__length__eq,axiom,
! [A: $tType,A2: set @ A,N: nat] :
( ( finite_finite2 @ A @ A2 )
=> ( ( finite_card @ ( list @ A )
@ ( collect @ ( list @ A )
@ ^ [Xs2: list @ A] :
( ( ord_less_eq @ ( set @ A ) @ ( set2 @ A @ Xs2 ) @ A2 )
& ( ( size_size @ ( list @ A ) @ Xs2 )
= N ) ) ) )
= ( power_power @ nat @ ( finite_card @ A @ A2 ) @ N ) ) ) ).
% card_lists_length_eq
thf(fact_8016_card__le__sym__Diff,axiom,
! [A: $tType,A2: set @ A,B2: set @ A] :
( ( finite_finite2 @ A @ A2 )
=> ( ( finite_finite2 @ A @ B2 )
=> ( ( ord_less_eq @ nat @ ( finite_card @ A @ A2 ) @ ( finite_card @ A @ B2 ) )
=> ( ord_less_eq @ nat @ ( finite_card @ A @ ( minus_minus @ ( set @ A ) @ A2 @ B2 ) ) @ ( finite_card @ A @ ( minus_minus @ ( set @ A ) @ B2 @ A2 ) ) ) ) ) ) ).
% card_le_sym_Diff
thf(fact_8017_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_8018_obtain__subset__with__card__n,axiom,
! [A: $tType,N: nat,S: set @ A] :
( ( ord_less_eq @ nat @ N @ ( finite_card @ A @ S ) )
=> ~ ! [T5: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ T5 @ S )
=> ( ( ( finite_card @ A @ T5 )
= N )
=> ~ ( finite_finite2 @ A @ T5 ) ) ) ) ).
% obtain_subset_with_card_n
thf(fact_8019_finite__if__finite__subsets__card__bdd,axiom,
! [A: $tType,F4: set @ A,C2: nat] :
( ! [G8: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ G8 @ F4 )
=> ( ( finite_finite2 @ A @ G8 )
=> ( ord_less_eq @ nat @ ( finite_card @ A @ G8 ) @ C2 ) ) )
=> ( ( finite_finite2 @ A @ F4 )
& ( ord_less_eq @ nat @ ( finite_card @ A @ F4 ) @ C2 ) ) ) ).
% finite_if_finite_subsets_card_bdd
thf(fact_8020_card__seteq,axiom,
! [A: $tType,B2: set @ A,A2: set @ A] :
( ( finite_finite2 @ A @ B2 )
=> ( ( ord_less_eq @ ( set @ A ) @ A2 @ B2 )
=> ( ( ord_less_eq @ nat @ ( finite_card @ A @ B2 ) @ ( finite_card @ A @ A2 ) )
=> ( A2 = B2 ) ) ) ) ).
% card_seteq
thf(fact_8021_card__mono,axiom,
! [A: $tType,B2: set @ A,A2: set @ A] :
( ( finite_finite2 @ A @ B2 )
=> ( ( ord_less_eq @ ( set @ A ) @ A2 @ B2 )
=> ( ord_less_eq @ nat @ ( finite_card @ A @ A2 ) @ ( finite_card @ A @ B2 ) ) ) ) ).
% card_mono
thf(fact_8022_subset__card__intvl__is__intvl,axiom,
! [A2: set @ nat,K: nat] :
( ( ord_less_eq @ ( set @ nat ) @ A2 @ ( set_or7035219750837199246ssThan @ nat @ K @ ( plus_plus @ nat @ K @ ( finite_card @ nat @ A2 ) ) ) )
=> ( A2
= ( set_or7035219750837199246ssThan @ nat @ K @ ( plus_plus @ nat @ K @ ( finite_card @ nat @ A2 ) ) ) ) ) ).
% subset_card_intvl_is_intvl
thf(fact_8023_continuous__on__arcosh_H,axiom,
! [A2: set @ real,F2: real > real] :
( ( topolo81223032696312382ous_on @ real @ real @ A2 @ F2 )
=> ( ! [X4: real] :
( ( member @ real @ X4 @ A2 )
=> ( ord_less_eq @ real @ ( one_one @ real ) @ ( F2 @ X4 ) ) )
=> ( topolo81223032696312382ous_on @ real @ real @ A2
@ ^ [X3: real] : ( arcosh @ real @ ( F2 @ X3 ) ) ) ) ) ).
% continuous_on_arcosh'
thf(fact_8024_sum__Suc,axiom,
! [A: $tType,F2: A > nat,A2: set @ A] :
( ( groups7311177749621191930dd_sum @ A @ nat
@ ^ [X3: A] : ( suc @ ( F2 @ X3 ) )
@ A2 )
= ( plus_plus @ nat @ ( groups7311177749621191930dd_sum @ A @ nat @ F2 @ A2 ) @ ( finite_card @ A @ A2 ) ) ) ).
% sum_Suc
thf(fact_8025_sum__constant__scaleR,axiom,
! [C: $tType,A: $tType] :
( ( real_V4867850818363320053vector @ A )
=> ! [Y3: A,A2: set @ C] :
( ( groups7311177749621191930dd_sum @ C @ A
@ ^ [X3: C] : Y3
@ A2 )
= ( real_V8093663219630862766scaleR @ A @ ( semiring_1_of_nat @ real @ ( finite_card @ C @ A2 ) ) @ Y3 ) ) ) ).
% sum_constant_scaleR
thf(fact_8026_sum__multicount,axiom,
! [A: $tType,B: $tType,S: set @ A,T6: set @ B,R: A > B > $o,K: nat] :
( ( finite_finite2 @ A @ S )
=> ( ( finite_finite2 @ B @ T6 )
=> ( ! [X4: B] :
( ( member @ B @ X4 @ T6 )
=> ( ( finite_card @ A
@ ( collect @ A
@ ^ [I4: A] :
( ( member @ A @ I4 @ S )
& ( R @ I4 @ X4 ) ) ) )
= K ) )
=> ( ( groups7311177749621191930dd_sum @ A @ nat
@ ^ [I4: A] :
( finite_card @ B
@ ( collect @ B
@ ^ [J3: B] :
( ( member @ B @ J3 @ T6 )
& ( R @ I4 @ J3 ) ) ) )
@ S )
= ( times_times @ nat @ K @ ( finite_card @ B @ T6 ) ) ) ) ) ) ).
% sum_multicount
thf(fact_8027_real__of__card,axiom,
! [A: $tType,A2: set @ A] :
( ( semiring_1_of_nat @ real @ ( finite_card @ A @ A2 ) )
= ( groups7311177749621191930dd_sum @ A @ real
@ ^ [X3: A] : ( one_one @ real )
@ A2 ) ) ).
% real_of_card
thf(fact_8028_sum__bounded__above,axiom,
! [B: $tType,A: $tType] :
( ( ( ordere6911136660526730532id_add @ A )
& ( semiring_1 @ A ) )
=> ! [A2: set @ B,F2: B > A,K6: A] :
( ! [I2: B] :
( ( member @ B @ I2 @ A2 )
=> ( ord_less_eq @ A @ ( F2 @ I2 ) @ K6 ) )
=> ( ord_less_eq @ A @ ( groups7311177749621191930dd_sum @ B @ A @ F2 @ A2 ) @ ( times_times @ A @ ( semiring_1_of_nat @ A @ ( finite_card @ B @ A2 ) ) @ K6 ) ) ) ) ).
% sum_bounded_above
thf(fact_8029_sum__bounded__below,axiom,
! [A: $tType,B: $tType] :
( ( ( ordere6911136660526730532id_add @ A )
& ( semiring_1 @ A ) )
=> ! [A2: set @ B,K6: A,F2: B > A] :
( ! [I2: B] :
( ( member @ B @ I2 @ A2 )
=> ( ord_less_eq @ A @ K6 @ ( F2 @ I2 ) ) )
=> ( ord_less_eq @ A @ ( times_times @ A @ ( semiring_1_of_nat @ A @ ( finite_card @ B @ A2 ) ) @ K6 ) @ ( groups7311177749621191930dd_sum @ B @ A @ F2 @ A2 ) ) ) ) ).
% sum_bounded_below
thf(fact_8030_card__le__Suc0__iff__eq,axiom,
! [A: $tType,A2: set @ A] :
( ( finite_finite2 @ A @ A2 )
=> ( ( ord_less_eq @ nat @ ( finite_card @ A @ A2 ) @ ( suc @ ( zero_zero @ nat ) ) )
= ( ! [X3: A] :
( ( member @ A @ X3 @ A2 )
=> ! [Y: A] :
( ( member @ A @ Y @ A2 )
=> ( X3 = Y ) ) ) ) ) ) ).
% card_le_Suc0_iff_eq
thf(fact_8031_card__gt__0__iff,axiom,
! [A: $tType,A2: set @ A] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ ( finite_card @ A @ A2 ) )
= ( ( A2
!= ( bot_bot @ ( set @ A ) ) )
& ( finite_finite2 @ A @ A2 ) ) ) ).
% card_gt_0_iff
thf(fact_8032_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_8033_card__1__singleton__iff,axiom,
! [A: $tType,A2: set @ A] :
( ( ( finite_card @ A @ A2 )
= ( suc @ ( zero_zero @ nat ) ) )
= ( ? [X3: A] :
( A2
= ( insert @ A @ X3 @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ).
% card_1_singleton_iff
thf(fact_8034_card__eq__SucD,axiom,
! [A: $tType,A2: set @ A,K: nat] :
( ( ( finite_card @ A @ A2 )
= ( suc @ K ) )
=> ? [B4: A,B10: set @ A] :
( ( A2
= ( insert @ A @ B4 @ B10 ) )
& ~ ( member @ A @ B4 @ B10 )
& ( ( finite_card @ A @ B10 )
= K )
& ( ( K
= ( zero_zero @ nat ) )
=> ( B10
= ( bot_bot @ ( set @ A ) ) ) ) ) ) ).
% card_eq_SucD
thf(fact_8035_card__Suc__eq,axiom,
! [A: $tType,A2: set @ A,K: nat] :
( ( ( finite_card @ A @ A2 )
= ( suc @ K ) )
= ( ? [B7: A,B6: set @ A] :
( ( A2
= ( insert @ A @ B7 @ B6 ) )
& ~ ( member @ A @ B7 @ B6 )
& ( ( finite_card @ A @ B6 )
= K )
& ( ( K
= ( zero_zero @ nat ) )
=> ( B6
= ( bot_bot @ ( set @ A ) ) ) ) ) ) ) ).
% card_Suc_eq
thf(fact_8036_continuous__on__arcosh,axiom,
! [A2: set @ real] :
( ( ord_less_eq @ ( set @ real ) @ A2 @ ( set_ord_atLeast @ real @ ( one_one @ real ) ) )
=> ( topolo81223032696312382ous_on @ real @ real @ A2 @ ( arcosh @ real ) ) ) ).
% continuous_on_arcosh
thf(fact_8037_card__le__Suc__iff,axiom,
! [A: $tType,N: nat,A2: set @ A] :
( ( ord_less_eq @ nat @ ( suc @ N ) @ ( finite_card @ A @ A2 ) )
= ( ? [A6: A,B6: set @ A] :
( ( A2
= ( insert @ A @ A6 @ B6 ) )
& ~ ( member @ A @ A6 @ B6 )
& ( ord_less_eq @ nat @ N @ ( finite_card @ A @ B6 ) )
& ( finite_finite2 @ A @ B6 ) ) ) ) ).
% card_le_Suc_iff
thf(fact_8038_surj__card__le,axiom,
! [B: $tType,A: $tType,A2: set @ A,B2: set @ B,F2: A > B] :
( ( finite_finite2 @ A @ A2 )
=> ( ( ord_less_eq @ ( set @ B ) @ B2 @ ( image @ A @ B @ F2 @ A2 ) )
=> ( ord_less_eq @ nat @ ( finite_card @ B @ B2 ) @ ( finite_card @ A @ A2 ) ) ) ) ).
% surj_card_le
thf(fact_8039_card__1__singletonI,axiom,
! [A: $tType,S: set @ A,X2: A] :
( ( finite_finite2 @ A @ S )
=> ( ( ( finite_card @ A @ S )
= ( one_one @ nat ) )
=> ( ( member @ A @ X2 @ S )
=> ( S
= ( insert @ A @ X2 @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ) ).
% card_1_singletonI
thf(fact_8040_card__bij__eq,axiom,
! [A: $tType,B: $tType,F2: A > B,A2: set @ A,B2: set @ B,G: B > A] :
( ( inj_on @ A @ B @ F2 @ A2 )
=> ( ( ord_less_eq @ ( set @ B ) @ ( image @ A @ B @ F2 @ A2 ) @ B2 )
=> ( ( inj_on @ B @ A @ G @ B2 )
=> ( ( ord_less_eq @ ( set @ A ) @ ( image @ B @ A @ G @ B2 ) @ A2 )
=> ( ( finite_finite2 @ A @ A2 )
=> ( ( finite_finite2 @ B @ B2 )
=> ( ( finite_card @ A @ A2 )
= ( finite_card @ B @ B2 ) ) ) ) ) ) ) ) ).
% card_bij_eq
thf(fact_8041_surjective__iff__injective__gen,axiom,
! [B: $tType,A: $tType,S: set @ A,T6: set @ B,F2: A > B] :
( ( finite_finite2 @ A @ S )
=> ( ( finite_finite2 @ B @ T6 )
=> ( ( ( finite_card @ A @ S )
= ( finite_card @ B @ T6 ) )
=> ( ( ord_less_eq @ ( set @ B ) @ ( image @ A @ B @ F2 @ S ) @ T6 )
=> ( ( ! [X3: B] :
( ( member @ B @ X3 @ T6 )
=> ? [Y: A] :
( ( member @ A @ Y @ S )
& ( ( F2 @ Y )
= X3 ) ) ) )
= ( inj_on @ A @ B @ F2 @ S ) ) ) ) ) ) ).
% surjective_iff_injective_gen
thf(fact_8042_card__Diff1__le,axiom,
! [A: $tType,A2: set @ A,X2: A] : ( ord_less_eq @ nat @ ( finite_card @ A @ ( minus_minus @ ( set @ A ) @ A2 @ ( insert @ A @ X2 @ ( bot_bot @ ( set @ A ) ) ) ) ) @ ( finite_card @ A @ A2 ) ) ).
% card_Diff1_le
thf(fact_8043_card__Diff__subset,axiom,
! [A: $tType,B2: set @ A,A2: set @ A] :
( ( finite_finite2 @ A @ B2 )
=> ( ( ord_less_eq @ ( set @ A ) @ B2 @ A2 )
=> ( ( finite_card @ A @ ( minus_minus @ ( set @ A ) @ A2 @ B2 ) )
= ( minus_minus @ nat @ ( finite_card @ A @ A2 ) @ ( finite_card @ A @ B2 ) ) ) ) ) ).
% card_Diff_subset
thf(fact_8044_diff__card__le__card__Diff,axiom,
! [A: $tType,B2: set @ A,A2: set @ A] :
( ( finite_finite2 @ A @ B2 )
=> ( ord_less_eq @ nat @ ( minus_minus @ nat @ ( finite_card @ A @ A2 ) @ ( finite_card @ A @ B2 ) ) @ ( finite_card @ A @ ( minus_minus @ ( set @ A ) @ A2 @ B2 ) ) ) ) ).
% diff_card_le_card_Diff
thf(fact_8045_card__psubset,axiom,
! [A: $tType,B2: set @ A,A2: set @ A] :
( ( finite_finite2 @ A @ B2 )
=> ( ( ord_less_eq @ ( set @ A ) @ A2 @ B2 )
=> ( ( ord_less @ nat @ ( finite_card @ A @ A2 ) @ ( finite_card @ A @ B2 ) )
=> ( ord_less @ ( set @ A ) @ A2 @ B2 ) ) ) ) ).
% card_psubset
thf(fact_8046_continuous__on__powr_H,axiom,
! [C: $tType] :
( ( topolo4958980785337419405_space @ C )
=> ! [S2: set @ C,F2: C > real,G: C > real] :
( ( topolo81223032696312382ous_on @ C @ real @ S2 @ F2 )
=> ( ( topolo81223032696312382ous_on @ C @ real @ S2 @ G )
=> ( ! [X4: C] :
( ( member @ C @ X4 @ S2 )
=> ( ( ord_less_eq @ real @ ( zero_zero @ real ) @ ( F2 @ X4 ) )
& ( ( ( F2 @ X4 )
= ( zero_zero @ real ) )
=> ( ord_less @ real @ ( zero_zero @ real ) @ ( G @ X4 ) ) ) ) )
=> ( topolo81223032696312382ous_on @ C @ real @ S2
@ ^ [X3: C] : ( powr @ real @ ( F2 @ X3 ) @ ( G @ X3 ) ) ) ) ) ) ) ).
% continuous_on_powr'
thf(fact_8047_continuous__on__log,axiom,
! [A: $tType] :
( ( topolo4958980785337419405_space @ A )
=> ! [S2: set @ A,F2: A > real,G: A > real] :
( ( topolo81223032696312382ous_on @ A @ real @ S2 @ F2 )
=> ( ( topolo81223032696312382ous_on @ A @ real @ S2 @ G )
=> ( ! [X4: A] :
( ( member @ A @ X4 @ S2 )
=> ( ord_less @ real @ ( zero_zero @ real ) @ ( F2 @ X4 ) ) )
=> ( ! [X4: A] :
( ( member @ A @ X4 @ S2 )
=> ( ( F2 @ X4 )
!= ( one_one @ real ) ) )
=> ( ! [X4: A] :
( ( member @ A @ X4 @ S2 )
=> ( ord_less @ real @ ( zero_zero @ real ) @ ( G @ X4 ) ) )
=> ( topolo81223032696312382ous_on @ A @ real @ S2
@ ^ [X3: A] : ( log @ ( F2 @ X3 ) @ ( G @ X3 ) ) ) ) ) ) ) ) ) ).
% continuous_on_log
thf(fact_8048_card__lists__length__le,axiom,
! [A: $tType,A2: set @ A,N: nat] :
( ( finite_finite2 @ A @ A2 )
=> ( ( finite_card @ ( list @ A )
@ ( collect @ ( list @ A )
@ ^ [Xs2: list @ A] :
( ( ord_less_eq @ ( set @ A ) @ ( set2 @ A @ Xs2 ) @ A2 )
& ( ord_less_eq @ nat @ ( size_size @ ( list @ A ) @ Xs2 ) @ N ) ) ) )
= ( groups7311177749621191930dd_sum @ nat @ nat @ ( power_power @ nat @ ( finite_card @ A @ A2 ) ) @ ( set_ord_atMost @ nat @ N ) ) ) ) ).
% card_lists_length_le
thf(fact_8049_continuous__on__arccos_H,axiom,
topolo81223032696312382ous_on @ real @ real @ ( set_or1337092689740270186AtMost @ real @ ( uminus_uminus @ real @ ( one_one @ real ) ) @ ( one_one @ real ) ) @ arccos ).
% continuous_on_arccos'
thf(fact_8050_continuous__on__arcsin_H,axiom,
topolo81223032696312382ous_on @ real @ real @ ( set_or1337092689740270186AtMost @ real @ ( uminus_uminus @ real @ ( one_one @ real ) ) @ ( one_one @ real ) ) @ arcsin ).
% continuous_on_arcsin'
thf(fact_8051_continuous__on__arccos,axiom,
! [A: $tType] :
( ( topolo4958980785337419405_space @ A )
=> ! [S2: set @ A,F2: A > real] :
( ( topolo81223032696312382ous_on @ A @ real @ S2 @ F2 )
=> ( ! [X4: A] :
( ( member @ A @ X4 @ S2 )
=> ( ( ord_less_eq @ real @ ( uminus_uminus @ real @ ( one_one @ real ) ) @ ( F2 @ X4 ) )
& ( ord_less_eq @ real @ ( F2 @ X4 ) @ ( one_one @ real ) ) ) )
=> ( topolo81223032696312382ous_on @ A @ real @ S2
@ ^ [X3: A] : ( arccos @ ( F2 @ X3 ) ) ) ) ) ) ).
% continuous_on_arccos
thf(fact_8052_continuous__on__arcsin,axiom,
! [A: $tType] :
( ( topolo4958980785337419405_space @ A )
=> ! [S2: set @ A,F2: A > real] :
( ( topolo81223032696312382ous_on @ A @ real @ S2 @ F2 )
=> ( ! [X4: A] :
( ( member @ A @ X4 @ S2 )
=> ( ( ord_less_eq @ real @ ( uminus_uminus @ real @ ( one_one @ real ) ) @ ( F2 @ X4 ) )
& ( ord_less_eq @ real @ ( F2 @ X4 ) @ ( one_one @ real ) ) ) )
=> ( topolo81223032696312382ous_on @ A @ real @ S2
@ ^ [X3: A] : ( arcsin @ ( F2 @ X3 ) ) ) ) ) ) ).
% continuous_on_arcsin
thf(fact_8053_ex__bij__betw__nat__finite__1,axiom,
! [A: $tType,M8: set @ A] :
( ( finite_finite2 @ A @ M8 )
=> ? [H3: nat > A] : ( bij_betw @ nat @ A @ H3 @ ( set_or1337092689740270186AtMost @ nat @ ( one_one @ nat ) @ ( finite_card @ A @ M8 ) ) @ M8 ) ) ).
% ex_bij_betw_nat_finite_1
thf(fact_8054_card__roots__unity,axiom,
! [A: $tType] :
( ( ( real_V8999393235501362500lgebra @ A )
& ( idom @ A ) )
=> ! [N: nat] :
( ( ord_less_eq @ nat @ ( one_one @ nat ) @ N )
=> ( ord_less_eq @ nat
@ ( finite_card @ A
@ ( collect @ A
@ ^ [Z4: A] :
( ( power_power @ A @ Z4 @ N )
= ( one_one @ A ) ) ) )
@ N ) ) ) ).
% card_roots_unity
thf(fact_8055_subset__eq__atLeast0__lessThan__card,axiom,
! [N7: set @ nat,N: nat] :
( ( ord_less_eq @ ( set @ nat ) @ N7 @ ( set_or7035219750837199246ssThan @ nat @ ( zero_zero @ nat ) @ N ) )
=> ( ord_less_eq @ nat @ ( finite_card @ nat @ N7 ) @ N ) ) ).
% subset_eq_atLeast0_lessThan_card
thf(fact_8056_card__le__Suc__Max,axiom,
! [S: set @ nat] :
( ( finite_finite2 @ nat @ S )
=> ( ord_less_eq @ nat @ ( finite_card @ nat @ S ) @ ( suc @ ( lattic643756798349783984er_Max @ nat @ S ) ) ) ) ).
% card_le_Suc_Max
thf(fact_8057_DERIV__atLeastAtMost__imp__continuous__on,axiom,
! [A: $tType] :
( ( ( ord @ A )
& ( real_V3459762299906320749_field @ A ) )
=> ! [A3: A,B3: A,F2: A > A] :
( ! [X4: A] :
( ( ord_less_eq @ A @ A3 @ X4 )
=> ( ( ord_less_eq @ A @ X4 @ B3 )
=> ? [Y5: A] : ( has_field_derivative @ A @ F2 @ Y5 @ ( topolo174197925503356063within @ A @ X4 @ ( top_top @ ( set @ A ) ) ) ) ) )
=> ( topolo81223032696312382ous_on @ A @ A @ ( set_or1337092689740270186AtMost @ A @ A3 @ B3 ) @ F2 ) ) ) ).
% DERIV_atLeastAtMost_imp_continuous_on
thf(fact_8058_card__sum__le__nat__sum,axiom,
! [S: set @ nat] :
( ord_less_eq @ nat
@ ( groups7311177749621191930dd_sum @ nat @ nat
@ ^ [X3: nat] : X3
@ ( set_or7035219750837199246ssThan @ nat @ ( zero_zero @ nat ) @ ( finite_card @ nat @ S ) ) )
@ ( groups7311177749621191930dd_sum @ nat @ nat
@ ^ [X3: nat] : X3
@ S ) ) ).
% card_sum_le_nat_sum
thf(fact_8059_continuous__on__artanh_H,axiom,
! [A2: set @ real,F2: real > real] :
( ( topolo81223032696312382ous_on @ real @ real @ A2 @ F2 )
=> ( ! [X4: real] :
( ( member @ real @ X4 @ A2 )
=> ( member @ real @ ( F2 @ X4 ) @ ( set_or5935395276787703475ssThan @ real @ ( uminus_uminus @ real @ ( one_one @ real ) ) @ ( one_one @ real ) ) ) )
=> ( topolo81223032696312382ous_on @ real @ real @ A2
@ ^ [X3: real] : ( artanh @ real @ ( F2 @ X3 ) ) ) ) ) ).
% continuous_on_artanh'
thf(fact_8060_card__nth__roots,axiom,
! [C3: complex,N: nat] :
( ( C3
!= ( zero_zero @ complex ) )
=> ( ( ord_less @ nat @ ( zero_zero @ nat ) @ N )
=> ( ( finite_card @ complex
@ ( collect @ complex
@ ^ [Z4: complex] :
( ( power_power @ complex @ Z4 @ N )
= C3 ) ) )
= N ) ) ) ).
% card_nth_roots
thf(fact_8061_card__roots__unity__eq,axiom,
! [N: nat] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ N )
=> ( ( finite_card @ complex
@ ( collect @ complex
@ ^ [Z4: complex] :
( ( power_power @ complex @ Z4 @ N )
= ( one_one @ complex ) ) ) )
= N ) ) ).
% card_roots_unity_eq
thf(fact_8062_card__2__iff,axiom,
! [A: $tType,S: set @ A] :
( ( ( finite_card @ A @ S )
= ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) )
= ( ? [X3: A,Y: A] :
( ( S
= ( insert @ A @ X3 @ ( insert @ A @ Y @ ( bot_bot @ ( set @ A ) ) ) ) )
& ( X3 != Y ) ) ) ) ).
% card_2_iff
thf(fact_8063_card__3__iff,axiom,
! [A: $tType,S: set @ A] :
( ( ( finite_card @ A @ S )
= ( numeral_numeral @ nat @ ( bit1 @ one2 ) ) )
= ( ? [X3: A,Y: A,Z4: A] :
( ( S
= ( insert @ A @ X3 @ ( insert @ A @ Y @ ( insert @ A @ Z4 @ ( bot_bot @ ( set @ A ) ) ) ) ) )
& ( X3 != Y )
& ( Y != Z4 )
& ( X3 != Z4 ) ) ) ) ).
% card_3_iff
thf(fact_8064_card__range__greater__zero,axiom,
! [A: $tType,B: $tType,F2: B > A] :
( ( finite_finite2 @ A @ ( image @ B @ A @ F2 @ ( top_top @ ( set @ B ) ) ) )
=> ( ord_less @ nat @ ( zero_zero @ nat ) @ ( finite_card @ A @ ( image @ B @ A @ F2 @ ( top_top @ ( set @ B ) ) ) ) ) ) ).
% card_range_greater_zero
thf(fact_8065_odd__card__imp__not__empty,axiom,
! [A: $tType,A2: set @ A] :
( ~ ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( finite_card @ A @ A2 ) )
=> ( A2
!= ( bot_bot @ ( set @ A ) ) ) ) ).
% odd_card_imp_not_empty
thf(fact_8066_card__insert__disjoint_H,axiom,
! [A: $tType,A2: set @ A,X2: A] :
( ( finite_finite2 @ A @ A2 )
=> ( ~ ( member @ A @ X2 @ A2 )
=> ( ( minus_minus @ nat @ ( finite_card @ A @ ( insert @ A @ X2 @ A2 ) ) @ ( suc @ ( zero_zero @ nat ) ) )
= ( finite_card @ A @ A2 ) ) ) ) ).
% card_insert_disjoint'
thf(fact_8067_card_Oremove,axiom,
! [A: $tType,A2: set @ A,X2: A] :
( ( finite_finite2 @ A @ A2 )
=> ( ( member @ A @ X2 @ A2 )
=> ( ( finite_card @ A @ A2 )
= ( suc @ ( finite_card @ A @ ( minus_minus @ ( set @ A ) @ A2 @ ( insert @ A @ X2 @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ) ) ) ).
% card.remove
thf(fact_8068_card_Oinsert__remove,axiom,
! [A: $tType,A2: set @ A,X2: A] :
( ( finite_finite2 @ A @ A2 )
=> ( ( finite_card @ A @ ( insert @ A @ X2 @ A2 ) )
= ( suc @ ( finite_card @ A @ ( minus_minus @ ( set @ A ) @ A2 @ ( insert @ A @ X2 @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ) ) ).
% card.insert_remove
thf(fact_8069_card__Suc__Diff1,axiom,
! [A: $tType,A2: set @ A,X2: A] :
( ( finite_finite2 @ A @ A2 )
=> ( ( member @ A @ X2 @ A2 )
=> ( ( suc @ ( finite_card @ A @ ( minus_minus @ ( set @ A ) @ A2 @ ( insert @ A @ X2 @ ( bot_bot @ ( set @ A ) ) ) ) ) )
= ( finite_card @ A @ A2 ) ) ) ) ).
% card_Suc_Diff1
thf(fact_8070_card__Diff1__less,axiom,
! [A: $tType,A2: set @ A,X2: A] :
( ( finite_finite2 @ A @ A2 )
=> ( ( member @ A @ X2 @ A2 )
=> ( ord_less @ nat @ ( finite_card @ A @ ( minus_minus @ ( set @ A ) @ A2 @ ( insert @ A @ X2 @ ( bot_bot @ ( set @ A ) ) ) ) ) @ ( finite_card @ A @ A2 ) ) ) ) ).
% card_Diff1_less
thf(fact_8071_card__Diff2__less,axiom,
! [A: $tType,A2: set @ A,X2: A,Y3: A] :
( ( finite_finite2 @ A @ A2 )
=> ( ( member @ A @ X2 @ A2 )
=> ( ( member @ A @ Y3 @ A2 )
=> ( ord_less @ nat @ ( finite_card @ A @ ( minus_minus @ ( set @ A ) @ ( minus_minus @ ( set @ A ) @ A2 @ ( insert @ A @ X2 @ ( bot_bot @ ( set @ A ) ) ) ) @ ( insert @ A @ Y3 @ ( bot_bot @ ( set @ A ) ) ) ) ) @ ( finite_card @ A @ A2 ) ) ) ) ) ).
% card_Diff2_less
thf(fact_8072_card__Diff1__less__iff,axiom,
! [A: $tType,A2: set @ A,X2: A] :
( ( ord_less @ nat @ ( finite_card @ A @ ( minus_minus @ ( set @ A ) @ A2 @ ( insert @ A @ X2 @ ( bot_bot @ ( set @ A ) ) ) ) ) @ ( finite_card @ A @ A2 ) )
= ( ( finite_finite2 @ A @ A2 )
& ( member @ A @ X2 @ A2 ) ) ) ).
% card_Diff1_less_iff
thf(fact_8073_card__le__inj,axiom,
! [B: $tType,A: $tType,A2: set @ A,B2: set @ B] :
( ( finite_finite2 @ A @ A2 )
=> ( ( finite_finite2 @ B @ B2 )
=> ( ( ord_less_eq @ nat @ ( finite_card @ A @ A2 ) @ ( finite_card @ B @ B2 ) )
=> ? [F3: A > B] :
( ( ord_less_eq @ ( set @ B ) @ ( image @ A @ B @ F3 @ A2 ) @ B2 )
& ( inj_on @ A @ B @ F3 @ A2 ) ) ) ) ) ).
% card_le_inj
thf(fact_8074_card__inj__on__le,axiom,
! [A: $tType,B: $tType,F2: A > B,A2: set @ A,B2: set @ B] :
( ( inj_on @ A @ B @ F2 @ A2 )
=> ( ( ord_less_eq @ ( set @ B ) @ ( image @ A @ B @ F2 @ A2 ) @ B2 )
=> ( ( finite_finite2 @ B @ B2 )
=> ( ord_less_eq @ nat @ ( finite_card @ A @ A2 ) @ ( finite_card @ B @ B2 ) ) ) ) ) ).
% card_inj_on_le
thf(fact_8075_inj__on__iff__card__le,axiom,
! [A: $tType,B: $tType,A2: set @ A,B2: set @ B] :
( ( finite_finite2 @ A @ A2 )
=> ( ( finite_finite2 @ B @ B2 )
=> ( ( ? [F5: A > B] :
( ( inj_on @ A @ B @ F5 @ A2 )
& ( ord_less_eq @ ( set @ B ) @ ( image @ A @ B @ F5 @ A2 ) @ B2 ) ) )
= ( ord_less_eq @ nat @ ( finite_card @ A @ A2 ) @ ( finite_card @ B @ B2 ) ) ) ) ) ).
% inj_on_iff_card_le
thf(fact_8076_card__Diff__singleton__if,axiom,
! [A: $tType,X2: A,A2: set @ A] :
( ( ( member @ A @ X2 @ A2 )
=> ( ( finite_card @ A @ ( minus_minus @ ( set @ A ) @ A2 @ ( insert @ A @ X2 @ ( bot_bot @ ( set @ A ) ) ) ) )
= ( minus_minus @ nat @ ( finite_card @ A @ A2 ) @ ( one_one @ nat ) ) ) )
& ( ~ ( member @ A @ X2 @ A2 )
=> ( ( finite_card @ A @ ( minus_minus @ ( set @ A ) @ A2 @ ( insert @ A @ X2 @ ( bot_bot @ ( set @ A ) ) ) ) )
= ( finite_card @ A @ A2 ) ) ) ) ).
% card_Diff_singleton_if
thf(fact_8077_card__Diff__singleton,axiom,
! [A: $tType,X2: A,A2: set @ A] :
( ( member @ A @ X2 @ A2 )
=> ( ( finite_card @ A @ ( minus_minus @ ( set @ A ) @ A2 @ ( insert @ A @ X2 @ ( bot_bot @ ( set @ A ) ) ) ) )
= ( minus_minus @ nat @ ( finite_card @ A @ A2 ) @ ( one_one @ nat ) ) ) ) ).
% card_Diff_singleton
thf(fact_8078_mvt,axiom,
! [A3: real,B3: real,F2: real > real,F7: real > real > real] :
( ( ord_less @ real @ A3 @ B3 )
=> ( ( topolo81223032696312382ous_on @ real @ real @ ( set_or1337092689740270186AtMost @ real @ A3 @ B3 ) @ F2 )
=> ( ! [X4: real] :
( ( ord_less @ real @ A3 @ X4 )
=> ( ( ord_less @ real @ X4 @ B3 )
=> ( has_derivative @ real @ real @ F2 @ ( F7 @ X4 ) @ ( topolo174197925503356063within @ real @ X4 @ ( top_top @ ( set @ real ) ) ) ) ) )
=> ~ ! [Xi3: real] :
( ( ord_less @ real @ A3 @ Xi3 )
=> ( ( ord_less @ real @ Xi3 @ B3 )
=> ( ( minus_minus @ real @ ( F2 @ B3 ) @ ( F2 @ A3 ) )
!= ( F7 @ Xi3 @ ( minus_minus @ real @ B3 @ A3 ) ) ) ) ) ) ) ) ).
% mvt
thf(fact_8079_continuous__on__of__int__floor,axiom,
! [B: $tType,A: $tType] :
( ( ( archim2362893244070406136eiling @ A )
& ( topolo2564578578187576103pology @ A )
& ( ring_1 @ B )
& ( topolo4958980785337419405_space @ B ) )
=> ( topolo81223032696312382ous_on @ A @ B @ ( minus_minus @ ( set @ A ) @ ( top_top @ ( set @ A ) ) @ ( ring_1_Ints @ A ) )
@ ^ [X3: A] : ( ring_1_of_int @ B @ ( archim6421214686448440834_floor @ A @ X3 ) ) ) ) ).
% continuous_on_of_int_floor
thf(fact_8080_continuous__on__of__int__ceiling,axiom,
! [B: $tType,A: $tType] :
( ( ( archim2362893244070406136eiling @ A )
& ( topolo2564578578187576103pology @ A )
& ( ring_1 @ B )
& ( topolo4958980785337419405_space @ B ) )
=> ( topolo81223032696312382ous_on @ A @ B @ ( minus_minus @ ( set @ A ) @ ( top_top @ ( set @ A ) ) @ ( ring_1_Ints @ A ) )
@ ^ [X3: A] : ( ring_1_of_int @ B @ ( archimedean_ceiling @ A @ X3 ) ) ) ) ).
% continuous_on_of_int_ceiling
thf(fact_8081_sum__norm__bound,axiom,
! [A: $tType,B: $tType] :
( ( real_V822414075346904944vector @ A )
=> ! [S: set @ B,F2: B > A,K6: real] :
( ! [X4: B] :
( ( member @ B @ X4 @ S )
=> ( ord_less_eq @ real @ ( real_V7770717601297561774m_norm @ A @ ( F2 @ X4 ) ) @ K6 ) )
=> ( ord_less_eq @ real @ ( real_V7770717601297561774m_norm @ A @ ( groups7311177749621191930dd_sum @ B @ A @ F2 @ S ) ) @ ( times_times @ real @ ( semiring_1_of_nat @ real @ ( finite_card @ B @ S ) ) @ K6 ) ) ) ) ).
% sum_norm_bound
thf(fact_8082_continuous__on__Icc__at__leftD,axiom,
! [B: $tType,A: $tType] :
( ( ( topolo1944317154257567458pology @ A )
& ( topolo4958980785337419405_space @ B ) )
=> ! [A3: A,B3: A,F2: A > B] :
( ( topolo81223032696312382ous_on @ A @ B @ ( set_or1337092689740270186AtMost @ A @ A3 @ B3 ) @ F2 )
=> ( ( ord_less @ A @ A3 @ B3 )
=> ( filterlim @ A @ B @ F2 @ ( topolo7230453075368039082e_nhds @ B @ ( F2 @ B3 ) ) @ ( topolo174197925503356063within @ A @ B3 @ ( set_ord_lessThan @ A @ B3 ) ) ) ) ) ) ).
% continuous_on_Icc_at_leftD
thf(fact_8083_continuous__on__Icc__at__rightD,axiom,
! [B: $tType,A: $tType] :
( ( ( topolo1944317154257567458pology @ A )
& ( topolo4958980785337419405_space @ B ) )
=> ! [A3: A,B3: A,F2: A > B] :
( ( topolo81223032696312382ous_on @ A @ B @ ( set_or1337092689740270186AtMost @ A @ A3 @ B3 ) @ F2 )
=> ( ( ord_less @ A @ A3 @ B3 )
=> ( filterlim @ A @ B @ F2 @ ( topolo7230453075368039082e_nhds @ B @ ( F2 @ A3 ) ) @ ( topolo174197925503356063within @ A @ A3 @ ( set_ord_greaterThan @ A @ A3 ) ) ) ) ) ) ).
% continuous_on_Icc_at_rightD
thf(fact_8084_card__UN__le,axiom,
! [B: $tType,A: $tType,I6: set @ A,A2: A > ( set @ B )] :
( ( finite_finite2 @ A @ I6 )
=> ( ord_less_eq @ nat @ ( finite_card @ B @ ( complete_Sup_Sup @ ( set @ B ) @ ( image @ A @ ( set @ B ) @ A2 @ I6 ) ) )
@ ( groups7311177749621191930dd_sum @ A @ nat
@ ^ [I4: A] : ( finite_card @ B @ ( A2 @ I4 ) )
@ I6 ) ) ) ).
% card_UN_le
thf(fact_8085_prod__le__power,axiom,
! [B: $tType,A: $tType] :
( ( linordered_semidom @ A )
=> ! [A2: set @ B,F2: B > A,N: A,K: nat] :
( ! [I2: B] :
( ( member @ B @ I2 @ A2 )
=> ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( F2 @ I2 ) )
& ( ord_less_eq @ A @ ( F2 @ I2 ) @ N ) ) )
=> ( ( ord_less_eq @ nat @ ( finite_card @ B @ A2 ) @ K )
=> ( ( ord_less_eq @ A @ ( one_one @ A ) @ N )
=> ( ord_less_eq @ A @ ( groups7121269368397514597t_prod @ B @ A @ F2 @ A2 ) @ ( power_power @ A @ N @ K ) ) ) ) ) ) ).
% prod_le_power
thf(fact_8086_sum__bounded__above__strict,axiom,
! [B: $tType,A: $tType] :
( ( ( ordere8940638589300402666id_add @ A )
& ( semiring_1 @ A ) )
=> ! [A2: set @ B,F2: B > A,K6: A] :
( ! [I2: B] :
( ( member @ B @ I2 @ A2 )
=> ( ord_less @ A @ ( F2 @ I2 ) @ K6 ) )
=> ( ( ord_less @ nat @ ( zero_zero @ nat ) @ ( finite_card @ B @ A2 ) )
=> ( ord_less @ A @ ( groups7311177749621191930dd_sum @ B @ A @ F2 @ A2 ) @ ( times_times @ A @ ( semiring_1_of_nat @ A @ ( finite_card @ B @ A2 ) ) @ K6 ) ) ) ) ) ).
% sum_bounded_above_strict
thf(fact_8087_sum__bounded__above__divide,axiom,
! [B: $tType,A: $tType] :
( ( linordered_field @ A )
=> ! [A2: set @ B,F2: B > A,K6: A] :
( ! [I2: B] :
( ( member @ B @ I2 @ A2 )
=> ( ord_less_eq @ A @ ( F2 @ I2 ) @ ( divide_divide @ A @ K6 @ ( semiring_1_of_nat @ A @ ( finite_card @ B @ A2 ) ) ) ) )
=> ( ( finite_finite2 @ B @ A2 )
=> ( ( A2
!= ( bot_bot @ ( set @ B ) ) )
=> ( ord_less_eq @ A @ ( groups7311177749621191930dd_sum @ B @ A @ F2 @ A2 ) @ K6 ) ) ) ) ) ).
% sum_bounded_above_divide
thf(fact_8088_card__insert__le__m1,axiom,
! [A: $tType,N: nat,Y3: set @ A,X2: A] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ N )
=> ( ( ord_less_eq @ nat @ ( finite_card @ A @ Y3 ) @ ( minus_minus @ nat @ N @ ( one_one @ nat ) ) )
=> ( ord_less_eq @ nat @ ( finite_card @ A @ ( insert @ A @ X2 @ Y3 ) ) @ N ) ) ) ).
% card_insert_le_m1
thf(fact_8089_continuous__on__artanh,axiom,
! [A2: set @ real] :
( ( ord_less_eq @ ( set @ real ) @ A2 @ ( set_or5935395276787703475ssThan @ real @ ( uminus_uminus @ real @ ( one_one @ real ) ) @ ( one_one @ real ) ) )
=> ( topolo81223032696312382ous_on @ real @ real @ A2 @ ( artanh @ real ) ) ) ).
% continuous_on_artanh
thf(fact_8090_card__word__size,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [X2: word @ A] :
( ( finite_card @ ( word @ A ) @ ( top_top @ ( set @ ( word @ A ) ) ) )
= ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( size_size @ ( word @ A ) @ X2 ) ) ) ) ).
% card_word_size
thf(fact_8091_sum__fun__comp,axiom,
! [A: $tType,C: $tType,B: $tType] :
( ( semiring_1 @ C )
=> ! [S: set @ A,R: set @ B,G: A > B,F2: B > C] :
( ( finite_finite2 @ A @ S )
=> ( ( finite_finite2 @ B @ R )
=> ( ( ord_less_eq @ ( set @ B ) @ ( image @ A @ B @ G @ S ) @ R )
=> ( ( groups7311177749621191930dd_sum @ A @ C
@ ^ [X3: A] : ( F2 @ ( G @ X3 ) )
@ S )
= ( groups7311177749621191930dd_sum @ B @ C
@ ^ [Y: B] :
( times_times @ C
@ ( semiring_1_of_nat @ C
@ ( finite_card @ A
@ ( collect @ A
@ ^ [X3: A] :
( ( member @ A @ X3 @ S )
& ( ( G @ X3 )
= Y ) ) ) ) )
@ ( F2 @ Y ) )
@ R ) ) ) ) ) ) ).
% sum_fun_comp
thf(fact_8092_prod__gen__delta,axiom,
! [A: $tType,B: $tType] :
( ( comm_monoid_mult @ A )
=> ! [S: set @ B,A3: B,B3: B > A,C3: A] :
( ( finite_finite2 @ B @ S )
=> ( ( ( member @ B @ A3 @ S )
=> ( ( groups7121269368397514597t_prod @ B @ A
@ ^ [K3: B] : ( if @ A @ ( K3 = A3 ) @ ( B3 @ K3 ) @ C3 )
@ S )
= ( times_times @ A @ ( B3 @ A3 ) @ ( power_power @ A @ C3 @ ( minus_minus @ nat @ ( finite_card @ B @ S ) @ ( one_one @ nat ) ) ) ) ) )
& ( ~ ( member @ B @ A3 @ S )
=> ( ( groups7121269368397514597t_prod @ B @ A
@ ^ [K3: B] : ( if @ A @ ( K3 = A3 ) @ ( B3 @ K3 ) @ C3 )
@ S )
= ( power_power @ A @ C3 @ ( finite_card @ B @ S ) ) ) ) ) ) ) ).
% prod_gen_delta
thf(fact_8093_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
@ ^ [Z4: A] :
( ( groups7311177749621191930dd_sum @ nat @ A
@ ^ [I4: nat] : ( times_times @ A @ ( C3 @ I4 ) @ ( power_power @ A @ Z4 @ I4 ) )
@ ( set_ord_atMost @ nat @ N ) )
= ( zero_zero @ A ) ) ) )
@ N ) ) ) ) ).
% polyfun_roots_card
thf(fact_8094_sum__le__card__Max,axiom,
! [A: $tType,A2: set @ A,F2: A > nat] :
( ( finite_finite2 @ A @ A2 )
=> ( ord_less_eq @ nat @ ( groups7311177749621191930dd_sum @ A @ nat @ F2 @ A2 ) @ ( times_times @ nat @ ( finite_card @ A @ A2 ) @ ( lattic643756798349783984er_Max @ nat @ ( image @ A @ nat @ F2 @ A2 ) ) ) ) ) ).
% sum_le_card_Max
thf(fact_8095_DERIV__isconst2,axiom,
! [A3: real,B3: real,F2: real > real,X2: real] :
( ( ord_less @ real @ A3 @ B3 )
=> ( ( topolo81223032696312382ous_on @ real @ real @ ( set_or1337092689740270186AtMost @ real @ A3 @ B3 ) @ F2 )
=> ( ! [X4: real] :
( ( ord_less @ real @ A3 @ X4 )
=> ( ( ord_less @ real @ X4 @ B3 )
=> ( has_field_derivative @ real @ F2 @ ( zero_zero @ real ) @ ( topolo174197925503356063within @ real @ X4 @ ( top_top @ ( set @ real ) ) ) ) ) )
=> ( ( ord_less_eq @ real @ A3 @ X2 )
=> ( ( ord_less_eq @ real @ X2 @ B3 )
=> ( ( F2 @ X2 )
= ( F2 @ A3 ) ) ) ) ) ) ) ).
% DERIV_isconst2
thf(fact_8096_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_8097_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_8098_card__word,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ( ( finite_card @ ( word @ A ) @ ( top_top @ ( set @ ( word @ A ) ) ) )
= ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( type_len0_len_of @ A @ ( type2 @ A ) ) ) ) ) ).
% card_word
thf(fact_8099_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
@ ^ [Z4: A] :
( ( groups7311177749621191930dd_sum @ nat @ A
@ ^ [I4: nat] : ( times_times @ A @ ( C3 @ I4 ) @ ( power_power @ A @ Z4 @ I4 ) )
@ ( set_ord_atMost @ nat @ N ) )
= ( zero_zero @ A ) ) ) )
& ( ord_less_eq @ nat
@ ( finite_card @ A
@ ( collect @ A
@ ^ [Z4: A] :
( ( groups7311177749621191930dd_sum @ nat @ A
@ ^ [I4: nat] : ( times_times @ A @ ( C3 @ I4 ) @ ( power_power @ A @ Z4 @ I4 ) )
@ ( set_ord_atMost @ nat @ N ) )
= ( zero_zero @ A ) ) ) )
@ N ) ) ) ) ) ).
% polyfun_rootbound
thf(fact_8100_continuous__on__IccI,axiom,
! [B: $tType,A: $tType] :
( ( ( topolo1944317154257567458pology @ A )
& ( topolo4958980785337419405_space @ B ) )
=> ! [F2: A > B,A3: A,B3: A] :
( ( filterlim @ A @ B @ F2 @ ( topolo7230453075368039082e_nhds @ B @ ( F2 @ A3 ) ) @ ( topolo174197925503356063within @ A @ A3 @ ( set_ord_greaterThan @ A @ A3 ) ) )
=> ( ( filterlim @ A @ B @ F2 @ ( topolo7230453075368039082e_nhds @ B @ ( F2 @ B3 ) ) @ ( topolo174197925503356063within @ A @ B3 @ ( set_ord_lessThan @ A @ B3 ) ) )
=> ( ! [X4: A] :
( ( ord_less @ A @ A3 @ X4 )
=> ( ( ord_less @ A @ X4 @ B3 )
=> ( filterlim @ A @ B @ F2 @ ( topolo7230453075368039082e_nhds @ B @ ( F2 @ X4 ) ) @ ( topolo174197925503356063within @ A @ X4 @ ( top_top @ ( set @ A ) ) ) ) ) )
=> ( ( ord_less @ A @ A3 @ B3 )
=> ( topolo81223032696312382ous_on @ A @ B @ ( set_or1337092689740270186AtMost @ A @ A3 @ B3 ) @ F2 ) ) ) ) ) ) ).
% continuous_on_IccI
thf(fact_8101_even__sum__iff,axiom,
! [A: $tType,B: $tType] :
( ( semiring_parity @ A )
=> ! [A2: set @ B,F2: B > A] :
( ( finite_finite2 @ B @ A2 )
=> ( ( dvd_dvd @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ ( groups7311177749621191930dd_sum @ B @ A @ F2 @ A2 ) )
= ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) )
@ ( finite_card @ B
@ ( collect @ B
@ ^ [A6: B] :
( ( member @ B @ A6 @ A2 )
& ~ ( dvd_dvd @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ ( F2 @ A6 ) ) ) ) ) ) ) ) ) ).
% even_sum_iff
thf(fact_8102_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_8103_Cardinality_Ocard__set,axiom,
! [A: $tType] :
( ( finite_finite @ A )
=> ( ( finite_card @ ( set @ A ) @ ( top_top @ ( set @ ( set @ A ) ) ) )
= ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( finite_card @ A @ ( top_top @ ( set @ A ) ) ) ) ) ) ).
% Cardinality.card_set
thf(fact_8104_card__num1,axiom,
( ( finite_card @ numeral_num1 @ ( top_top @ ( set @ numeral_num1 ) ) )
= ( one_one @ nat ) ) ).
% card_num1
thf(fact_8105_continuous__on__of__real__o__iff,axiom,
! [A: $tType] :
( ( topolo4958980785337419405_space @ A )
=> ! [S: set @ A,G: A > real] :
( ( topolo81223032696312382ous_on @ A @ complex @ S
@ ^ [X3: A] : ( real_Vector_of_real @ complex @ ( G @ X3 ) ) )
= ( topolo81223032696312382ous_on @ A @ real @ S @ G ) ) ) ).
% continuous_on_of_real_o_iff
thf(fact_8106_card__sum,axiom,
! [A: $tType,B: $tType] :
( ( ( finite_finite @ B )
& ( finite_finite @ A ) )
=> ( ( 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 ) ) ) ) ) ) ).
% card_sum
thf(fact_8107_card__prod,axiom,
! [A: $tType,B: $tType] :
( ( finite_card @ ( product_prod @ A @ B ) @ ( top_top @ ( set @ ( product_prod @ A @ B ) ) ) )
= ( times_times @ nat @ ( finite_card @ A @ ( top_top @ ( set @ A ) ) ) @ ( finite_card @ B @ ( top_top @ ( set @ B ) ) ) ) ) ).
% card_prod
thf(fact_8108_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_8109_card__bit0,axiom,
! [A: $tType] :
( ( finite_finite @ A )
=> ( ( finite_card @ ( numeral_bit0 @ A ) @ ( top_top @ ( set @ ( numeral_bit0 @ A ) ) ) )
= ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( finite_card @ A @ ( top_top @ ( set @ A ) ) ) ) ) ) ).
% card_bit0
thf(fact_8110_continuous__on__cis,axiom,
! [A: $tType] :
( ( topolo4958980785337419405_space @ A )
=> ! [A2: set @ A,F2: A > real] :
( ( topolo81223032696312382ous_on @ A @ real @ A2 @ F2 )
=> ( topolo81223032696312382ous_on @ A @ complex @ A2
@ ^ [X3: A] : ( cis @ ( F2 @ X3 ) ) ) ) ) ).
% continuous_on_cis
thf(fact_8111_CARD__1,axiom,
! [A: $tType] :
( ( cARD_1 @ A )
=> ( ( finite_card @ A @ ( top_top @ ( set @ A ) ) )
= ( one_one @ nat ) ) ) ).
% CARD_1
thf(fact_8112_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_8113_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_8114_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_8115_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_8116_bit1__induct,axiom,
! [A: $tType] :
( ( finite_finite @ A )
=> ! [P: ( numeral_bit1 @ A ) > $o,X2: numeral_bit1 @ A] :
( ! [Z2: int] :
( ( ord_less_eq @ int @ ( zero_zero @ int ) @ Z2 )
=> ( ( ord_less @ int @ Z2 @ ( semiring_1_of_nat @ int @ ( finite_card @ ( numeral_bit1 @ A ) @ ( top_top @ ( set @ ( numeral_bit1 @ A ) ) ) ) ) )
=> ( P @ ( ring_1_of_int @ ( numeral_bit1 @ A ) @ Z2 ) ) ) )
=> ( P @ X2 ) ) ) ).
% bit1_induct
thf(fact_8117_bit0__induct,axiom,
! [A: $tType] :
( ( finite_finite @ A )
=> ! [P: ( numeral_bit0 @ A ) > $o,X2: numeral_bit0 @ A] :
( ! [Z2: int] :
( ( ord_less_eq @ int @ ( zero_zero @ int ) @ Z2 )
=> ( ( ord_less @ int @ Z2 @ ( semiring_1_of_nat @ int @ ( finite_card @ ( numeral_bit0 @ A ) @ ( top_top @ ( set @ ( numeral_bit0 @ A ) ) ) ) ) )
=> ( P @ ( ring_1_of_int @ ( numeral_bit0 @ A ) @ Z2 ) ) ) )
=> ( P @ X2 ) ) ) ).
% bit0_induct
thf(fact_8118_bit1__cases,axiom,
! [A: $tType] :
( ( finite_finite @ A )
=> ! [X2: numeral_bit1 @ A] :
~ ! [Z2: int] :
( ( X2
= ( ring_1_of_int @ ( numeral_bit1 @ A ) @ Z2 ) )
=> ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ Z2 )
=> ~ ( ord_less @ int @ Z2 @ ( semiring_1_of_nat @ int @ ( finite_card @ ( numeral_bit1 @ A ) @ ( top_top @ ( set @ ( numeral_bit1 @ A ) ) ) ) ) ) ) ) ) ).
% bit1_cases
thf(fact_8119_bit0__cases,axiom,
! [A: $tType] :
( ( finite_finite @ A )
=> ! [X2: numeral_bit0 @ A] :
~ ! [Z2: int] :
( ( X2
= ( ring_1_of_int @ ( numeral_bit0 @ A ) @ Z2 ) )
=> ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ Z2 )
=> ~ ( ord_less @ int @ Z2 @ ( semiring_1_of_nat @ int @ ( finite_card @ ( numeral_bit0 @ A ) @ ( top_top @ ( set @ ( numeral_bit0 @ A ) ) ) ) ) ) ) ) ) ).
% bit0_cases
thf(fact_8120_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_8121_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_8122_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_8123_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_8124_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_8125_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_8126_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_8127_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_8128_card__UNION,axiom,
! [A: $tType,A2: set @ ( set @ A )] :
( ( finite_finite2 @ ( set @ A ) @ A2 )
=> ( ! [X4: set @ A] :
( ( member @ ( set @ A ) @ X4 @ A2 )
=> ( finite_finite2 @ A @ X4 ) )
=> ( ( finite_card @ A @ ( complete_Sup_Sup @ ( set @ A ) @ A2 ) )
= ( nat2
@ ( groups7311177749621191930dd_sum @ ( set @ ( set @ A ) ) @ int
@ ^ [I9: set @ ( set @ A )] : ( times_times @ int @ ( power_power @ int @ ( uminus_uminus @ int @ ( one_one @ int ) ) @ ( plus_plus @ nat @ ( finite_card @ ( set @ A ) @ I9 ) @ ( one_one @ nat ) ) ) @ ( semiring_1_of_nat @ int @ ( finite_card @ A @ ( complete_Inf_Inf @ ( set @ A ) @ I9 ) ) ) )
@ ( collect @ ( set @ ( set @ A ) )
@ ^ [I9: set @ ( set @ A )] :
( ( ord_less_eq @ ( set @ ( set @ A ) ) @ I9 @ A2 )
& ( I9
!= ( bot_bot @ ( set @ ( set @ A ) ) ) ) ) ) ) ) ) ) ) ).
% card_UNION
thf(fact_8129_inj__on__Abs__bit1,axiom,
! [A: $tType] :
( ( finite_finite @ A )
=> ( inj_on @ int @ ( numeral_bit1 @ A ) @ ( numeral_Abs_bit1 @ A ) @ ( set_or7035219750837199246ssThan @ int @ ( zero_zero @ int ) @ ( plus_plus @ int @ ( one_one @ int ) @ ( times_times @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ ( semiring_1_of_nat @ int @ ( finite_card @ A @ ( top_top @ ( set @ A ) ) ) ) ) ) ) ) ) ).
% inj_on_Abs_bit1
thf(fact_8130_INF__identity__eq,axiom,
! [A: $tType] :
( ( complete_Inf @ A )
=> ! [A2: set @ A] :
( ( complete_Inf_Inf @ A
@ ( image @ A @ A
@ ^ [X3: A] : X3
@ A2 ) )
= ( complete_Inf_Inf @ A @ A2 ) ) ) ).
% INF_identity_eq
thf(fact_8131_INT__iff,axiom,
! [A: $tType,B: $tType,B3: A,B2: B > ( set @ A ),A2: set @ B] :
( ( member @ A @ B3 @ ( complete_Inf_Inf @ ( set @ A ) @ ( image @ B @ ( set @ A ) @ B2 @ A2 ) ) )
= ( ! [X3: B] :
( ( member @ B @ X3 @ A2 )
=> ( member @ A @ B3 @ ( B2 @ X3 ) ) ) ) ) ).
% INT_iff
thf(fact_8132_INT__I,axiom,
! [B: $tType,A: $tType,A2: set @ A,B3: B,B2: A > ( set @ B )] :
( ! [X4: A] :
( ( member @ A @ X4 @ A2 )
=> ( member @ B @ B3 @ ( B2 @ X4 ) ) )
=> ( member @ B @ B3 @ ( complete_Inf_Inf @ ( set @ B ) @ ( image @ A @ ( set @ B ) @ B2 @ A2 ) ) ) ) ).
% INT_I
thf(fact_8133_Inf__apply,axiom,
! [B: $tType,A: $tType] :
( ( complete_Inf @ B )
=> ( ( complete_Inf_Inf @ ( A > B ) )
= ( ^ [A7: set @ ( A > B ),X3: A] :
( complete_Inf_Inf @ B
@ ( image @ ( A > B ) @ B
@ ^ [F5: A > B] : ( F5 @ X3 )
@ A7 ) ) ) ) ) ).
% Inf_apply
thf(fact_8134_Inf__eq__bot__iff,axiom,
! [A: $tType] :
( ( comple5582772986160207858norder @ A )
=> ! [A2: set @ A] :
( ( ( complete_Inf_Inf @ A @ A2 )
= ( bot_bot @ A ) )
= ( ! [X3: A] :
( ( ord_less @ A @ ( bot_bot @ A ) @ X3 )
=> ? [Y: A] :
( ( member @ A @ Y @ A2 )
& ( ord_less @ A @ Y @ X3 ) ) ) ) ) ) ).
% Inf_eq_bot_iff
thf(fact_8135_Inf__atLeastAtMost,axiom,
! [A: $tType] :
( ( comple6319245703460814977attice @ A )
=> ! [X2: A,Y3: A] :
( ( ord_less_eq @ A @ X2 @ Y3 )
=> ( ( complete_Inf_Inf @ A @ ( set_or1337092689740270186AtMost @ A @ X2 @ Y3 ) )
= X2 ) ) ) ).
% Inf_atLeastAtMost
thf(fact_8136_cInf__atLeastAtMost,axiom,
! [A: $tType] :
( ( condit1219197933456340205attice @ A )
=> ! [Y3: A,X2: A] :
( ( ord_less_eq @ A @ Y3 @ X2 )
=> ( ( complete_Inf_Inf @ A @ ( set_or1337092689740270186AtMost @ A @ Y3 @ X2 ) )
= Y3 ) ) ) ).
% cInf_atLeastAtMost
thf(fact_8137_Inf__atLeastLessThan,axiom,
! [A: $tType] :
( ( comple6319245703460814977attice @ A )
=> ! [X2: A,Y3: A] :
( ( ord_less @ A @ X2 @ Y3 )
=> ( ( complete_Inf_Inf @ A @ ( set_or7035219750837199246ssThan @ A @ X2 @ Y3 ) )
= X2 ) ) ) ).
% Inf_atLeastLessThan
thf(fact_8138_cInf__atLeastLessThan,axiom,
! [A: $tType] :
( ( condit1219197933456340205attice @ A )
=> ! [Y3: A,X2: A] :
( ( ord_less @ A @ Y3 @ X2 )
=> ( ( complete_Inf_Inf @ A @ ( set_or7035219750837199246ssThan @ A @ Y3 @ X2 ) )
= Y3 ) ) ) ).
% cInf_atLeastLessThan
thf(fact_8139_Inf__greaterThanLessThan,axiom,
! [A: $tType] :
( ( ( comple6319245703460814977attice @ A )
& ( dense_linorder @ A ) )
=> ! [X2: A,Y3: A] :
( ( ord_less @ A @ X2 @ Y3 )
=> ( ( complete_Inf_Inf @ A @ ( set_or5935395276787703475ssThan @ A @ X2 @ Y3 ) )
= X2 ) ) ) ).
% Inf_greaterThanLessThan
thf(fact_8140_cInf__greaterThanLessThan,axiom,
! [A: $tType] :
( ( ( condit6923001295902523014norder @ A )
& ( dense_linorder @ A ) )
=> ! [Y3: A,X2: A] :
( ( ord_less @ A @ Y3 @ X2 )
=> ( ( complete_Inf_Inf @ A @ ( set_or5935395276787703475ssThan @ A @ Y3 @ X2 ) )
= Y3 ) ) ) ).
% cInf_greaterThanLessThan
thf(fact_8141_Inf__greaterThanAtMost,axiom,
! [A: $tType] :
( ( ( comple6319245703460814977attice @ A )
& ( dense_linorder @ A ) )
=> ! [X2: A,Y3: A] :
( ( ord_less @ A @ X2 @ Y3 )
=> ( ( complete_Inf_Inf @ A @ ( set_or3652927894154168847AtMost @ A @ X2 @ Y3 ) )
= X2 ) ) ) ).
% Inf_greaterThanAtMost
thf(fact_8142_cInf__greaterThanAtMost,axiom,
! [A: $tType] :
( ( ( condit6923001295902523014norder @ A )
& ( dense_linorder @ A ) )
=> ! [Y3: A,X2: A] :
( ( ord_less @ A @ Y3 @ X2 )
=> ( ( complete_Inf_Inf @ A @ ( set_or3652927894154168847AtMost @ A @ Y3 @ X2 ) )
= Y3 ) ) ) ).
% cInf_greaterThanAtMost
thf(fact_8143_INF__top,axiom,
! [B: $tType,A: $tType] :
( ( comple6319245703460814977attice @ A )
=> ! [A2: set @ B] :
( ( complete_Inf_Inf @ A
@ ( image @ B @ A
@ ^ [X3: B] : ( top_top @ A )
@ A2 ) )
= ( top_top @ A ) ) ) ).
% INF_top
thf(fact_8144_INF__top__conv_I1_J,axiom,
! [A: $tType,B: $tType] :
( ( comple6319245703460814977attice @ A )
=> ! [B2: B > A,A2: set @ B] :
( ( ( complete_Inf_Inf @ A @ ( image @ B @ A @ B2 @ A2 ) )
= ( top_top @ A ) )
= ( ! [X3: B] :
( ( member @ B @ X3 @ A2 )
=> ( ( B2 @ X3 )
= ( top_top @ A ) ) ) ) ) ) ).
% INF_top_conv(1)
thf(fact_8145_INF__top__conv_I2_J,axiom,
! [A: $tType,B: $tType] :
( ( comple6319245703460814977attice @ A )
=> ! [B2: B > A,A2: set @ B] :
( ( ( top_top @ A )
= ( complete_Inf_Inf @ A @ ( image @ B @ A @ B2 @ A2 ) ) )
= ( ! [X3: B] :
( ( member @ B @ X3 @ A2 )
=> ( ( B2 @ X3 )
= ( top_top @ A ) ) ) ) ) ) ).
% INF_top_conv(2)
thf(fact_8146_ccINF__top,axiom,
! [B: $tType,A: $tType] :
( ( counta3822494911875563373attice @ A )
=> ! [A2: set @ B] :
( ( complete_Inf_Inf @ A
@ ( image @ B @ A
@ ^ [X3: B] : ( top_top @ A )
@ A2 ) )
= ( top_top @ A ) ) ) ).
% ccINF_top
thf(fact_8147_INF__const,axiom,
! [B: $tType,A: $tType] :
( ( comple6319245703460814977attice @ A )
=> ! [A2: set @ B,F2: A] :
( ( A2
!= ( bot_bot @ ( set @ B ) ) )
=> ( ( complete_Inf_Inf @ A
@ ( image @ B @ A
@ ^ [I4: B] : F2
@ A2 ) )
= F2 ) ) ) ).
% INF_const
thf(fact_8148_cINF__const,axiom,
! [B: $tType,A: $tType] :
( ( condit1219197933456340205attice @ A )
=> ! [A2: set @ B,C3: A] :
( ( A2
!= ( bot_bot @ ( set @ B ) ) )
=> ( ( complete_Inf_Inf @ A
@ ( image @ B @ A
@ ^ [X3: B] : C3
@ A2 ) )
= C3 ) ) ) ).
% cINF_const
thf(fact_8149_ccINF__const,axiom,
! [B: $tType,A: $tType] :
( ( counta3822494911875563373attice @ A )
=> ! [A2: set @ B,F2: A] :
( ( A2
!= ( bot_bot @ ( set @ B ) ) )
=> ( ( complete_Inf_Inf @ A
@ ( image @ B @ A
@ ^ [I4: B] : F2
@ A2 ) )
= F2 ) ) ) ).
% ccINF_const
thf(fact_8150_finite__INT,axiom,
! [B: $tType,A: $tType,I6: set @ A,A2: A > ( set @ B )] :
( ? [X8: A] :
( ( member @ A @ X8 @ I6 )
& ( finite_finite2 @ B @ ( A2 @ X8 ) ) )
=> ( finite_finite2 @ B @ ( complete_Inf_Inf @ ( set @ B ) @ ( image @ A @ ( set @ B ) @ A2 @ I6 ) ) ) ) ).
% finite_INT
thf(fact_8151_INF__apply,axiom,
! [A: $tType,B: $tType,C: $tType] :
( ( complete_Inf @ A )
=> ! [F2: C > B > A,A2: set @ C,X2: B] :
( ( complete_Inf_Inf @ ( B > A ) @ ( image @ C @ ( B > A ) @ F2 @ A2 ) @ X2 )
= ( complete_Inf_Inf @ A
@ ( image @ C @ A
@ ^ [Y: C] : ( F2 @ Y @ X2 )
@ A2 ) ) ) ) ).
% INF_apply
thf(fact_8152_INF__eq__bot__iff,axiom,
! [B: $tType,A: $tType] :
( ( comple5582772986160207858norder @ A )
=> ! [F2: B > A,A2: set @ B] :
( ( ( complete_Inf_Inf @ A @ ( image @ B @ A @ F2 @ A2 ) )
= ( bot_bot @ A ) )
= ( ! [X3: A] :
( ( ord_less @ A @ ( bot_bot @ A ) @ X3 )
=> ? [Y: B] :
( ( member @ B @ Y @ A2 )
& ( ord_less @ A @ ( F2 @ Y ) @ X3 ) ) ) ) ) ) ).
% INF_eq_bot_iff
thf(fact_8153_ccINF__empty,axiom,
! [B: $tType,A: $tType] :
( ( counta3822494911875563373attice @ A )
=> ! [F2: B > A] :
( ( complete_Inf_Inf @ A @ ( image @ B @ A @ F2 @ ( bot_bot @ ( set @ B ) ) ) )
= ( top_top @ A ) ) ) ).
% ccINF_empty
thf(fact_8154_INT__constant,axiom,
! [B: $tType,A: $tType,A2: set @ B,C3: set @ A] :
( ( ( A2
= ( bot_bot @ ( set @ B ) ) )
=> ( ( complete_Inf_Inf @ ( set @ A )
@ ( image @ B @ ( set @ A )
@ ^ [Y: B] : C3
@ A2 ) )
= ( top_top @ ( set @ A ) ) ) )
& ( ( A2
!= ( bot_bot @ ( set @ B ) ) )
=> ( ( complete_Inf_Inf @ ( set @ A )
@ ( image @ B @ ( set @ A )
@ ^ [Y: B] : C3
@ A2 ) )
= C3 ) ) ) ).
% INT_constant
thf(fact_8155_open__INT,axiom,
! [A: $tType,B: $tType] :
( ( topolo4958980785337419405_space @ A )
=> ! [A2: set @ B,B2: B > ( set @ A )] :
( ( finite_finite2 @ B @ A2 )
=> ( ! [X4: B] :
( ( member @ B @ X4 @ A2 )
=> ( topolo1002775350975398744n_open @ A @ ( B2 @ X4 ) ) )
=> ( topolo1002775350975398744n_open @ A @ ( complete_Inf_Inf @ ( set @ A ) @ ( image @ B @ ( set @ A ) @ B2 @ A2 ) ) ) ) ) ) ).
% open_INT
thf(fact_8156_Compl__UN,axiom,
! [A: $tType,B: $tType,B2: B > ( set @ A ),A2: set @ B] :
( ( uminus_uminus @ ( set @ A ) @ ( complete_Sup_Sup @ ( set @ A ) @ ( image @ B @ ( set @ A ) @ B2 @ A2 ) ) )
= ( complete_Inf_Inf @ ( set @ A )
@ ( image @ B @ ( set @ A )
@ ^ [X3: B] : ( uminus_uminus @ ( set @ A ) @ ( B2 @ X3 ) )
@ A2 ) ) ) ).
% Compl_UN
thf(fact_8157_Compl__INT,axiom,
! [A: $tType,B: $tType,B2: B > ( set @ A ),A2: set @ B] :
( ( uminus_uminus @ ( set @ A ) @ ( complete_Inf_Inf @ ( set @ A ) @ ( image @ B @ ( set @ A ) @ B2 @ A2 ) ) )
= ( complete_Sup_Sup @ ( set @ A )
@ ( image @ B @ ( set @ A )
@ ^ [X3: B] : ( uminus_uminus @ ( set @ A ) @ ( B2 @ X3 ) )
@ A2 ) ) ) ).
% Compl_INT
thf(fact_8158_Inf__atMostLessThan,axiom,
! [A: $tType] :
( ( comple6319245703460814977attice @ A )
=> ! [X2: A] :
( ( ord_less @ A @ ( top_top @ A ) @ X2 )
=> ( ( complete_Inf_Inf @ A @ ( set_ord_lessThan @ A @ X2 ) )
= ( bot_bot @ A ) ) ) ) ).
% Inf_atMostLessThan
thf(fact_8159_INT__simps_I3_J,axiom,
! [E3: $tType,F: $tType,C2: set @ E3,A2: E3 > ( set @ F ),B2: set @ F] :
( ( ( C2
= ( bot_bot @ ( set @ E3 ) ) )
=> ( ( complete_Inf_Inf @ ( set @ F )
@ ( image @ E3 @ ( set @ F )
@ ^ [X3: E3] : ( minus_minus @ ( set @ F ) @ ( A2 @ X3 ) @ B2 )
@ C2 ) )
= ( top_top @ ( set @ F ) ) ) )
& ( ( C2
!= ( bot_bot @ ( set @ E3 ) ) )
=> ( ( complete_Inf_Inf @ ( set @ F )
@ ( image @ E3 @ ( set @ F )
@ ^ [X3: E3] : ( minus_minus @ ( set @ F ) @ ( A2 @ X3 ) @ B2 )
@ C2 ) )
= ( minus_minus @ ( set @ F ) @ ( complete_Inf_Inf @ ( set @ F ) @ ( image @ E3 @ ( set @ F ) @ A2 @ C2 ) ) @ B2 ) ) ) ) ).
% INT_simps(3)
thf(fact_8160_INT__simps_I4_J,axiom,
! [G3: $tType,H6: $tType,C2: set @ H6,A2: set @ G3,B2: H6 > ( set @ G3 )] :
( ( ( C2
= ( bot_bot @ ( set @ H6 ) ) )
=> ( ( complete_Inf_Inf @ ( set @ G3 )
@ ( image @ H6 @ ( set @ G3 )
@ ^ [X3: H6] : ( minus_minus @ ( set @ G3 ) @ A2 @ ( B2 @ X3 ) )
@ C2 ) )
= ( top_top @ ( set @ G3 ) ) ) )
& ( ( C2
!= ( bot_bot @ ( set @ H6 ) ) )
=> ( ( complete_Inf_Inf @ ( set @ G3 )
@ ( image @ H6 @ ( set @ G3 )
@ ^ [X3: H6] : ( minus_minus @ ( set @ G3 ) @ A2 @ ( B2 @ X3 ) )
@ C2 ) )
= ( minus_minus @ ( set @ G3 ) @ A2 @ ( complete_Sup_Sup @ ( set @ G3 ) @ ( image @ H6 @ ( set @ G3 ) @ B2 @ C2 ) ) ) ) ) ) ).
% INT_simps(4)
thf(fact_8161_INT__empty,axiom,
! [B: $tType,A: $tType,B2: B > ( set @ A )] :
( ( complete_Inf_Inf @ ( set @ A ) @ ( image @ B @ ( set @ A ) @ B2 @ ( bot_bot @ ( set @ B ) ) ) )
= ( top_top @ ( set @ A ) ) ) ).
% INT_empty
thf(fact_8162_Collect__all__eq,axiom,
! [A: $tType,B: $tType,P: A > B > $o] :
( ( collect @ A
@ ^ [X3: A] :
! [X7: B] : ( P @ X3 @ X7 ) )
= ( complete_Inf_Inf @ ( set @ A )
@ ( image @ B @ ( set @ A )
@ ^ [Y: B] :
( collect @ A
@ ^ [X3: A] : ( P @ X3 @ Y ) )
@ ( top_top @ ( set @ B ) ) ) ) ) ).
% Collect_all_eq
thf(fact_8163_one__bit1__def,axiom,
! [A: $tType] :
( ( finite_finite @ A )
=> ( ( one_one @ ( numeral_bit1 @ A ) )
= ( numeral_Abs_bit1 @ A @ ( one_one @ int ) ) ) ) ).
% one_bit1_def
thf(fact_8164_INT__extend__simps_I9_J,axiom,
! [S11: $tType,R7: $tType,Q8: $tType,C2: R7 > ( set @ S11 ),B2: Q8 > ( set @ R7 ),A2: set @ Q8] :
( ( complete_Inf_Inf @ ( set @ S11 )
@ ( image @ Q8 @ ( set @ S11 )
@ ^ [X3: Q8] : ( complete_Inf_Inf @ ( set @ S11 ) @ ( image @ R7 @ ( set @ S11 ) @ C2 @ ( B2 @ X3 ) ) )
@ A2 ) )
= ( complete_Inf_Inf @ ( set @ S11 ) @ ( image @ R7 @ ( set @ S11 ) @ C2 @ ( complete_Sup_Sup @ ( set @ R7 ) @ ( image @ Q8 @ ( set @ R7 ) @ B2 @ A2 ) ) ) ) ) ).
% INT_extend_simps(9)
thf(fact_8165_INF__less__iff,axiom,
! [A: $tType,B: $tType] :
( ( comple5582772986160207858norder @ A )
=> ! [F2: B > A,A2: set @ B,A3: A] :
( ( ord_less @ A @ ( complete_Inf_Inf @ A @ ( image @ B @ A @ F2 @ A2 ) ) @ A3 )
= ( ? [X3: B] :
( ( member @ B @ X3 @ A2 )
& ( ord_less @ A @ ( F2 @ X3 ) @ A3 ) ) ) ) ) ).
% INF_less_iff
thf(fact_8166_less__INF__D,axiom,
! [A: $tType,B: $tType] :
( ( comple6319245703460814977attice @ A )
=> ! [Y3: A,F2: B > A,A2: set @ B,I: B] :
( ( ord_less @ A @ Y3 @ ( complete_Inf_Inf @ A @ ( image @ B @ A @ F2 @ A2 ) ) )
=> ( ( member @ B @ I @ A2 )
=> ( ord_less @ A @ Y3 @ ( F2 @ I ) ) ) ) ) ).
% less_INF_D
thf(fact_8167_INF__constant,axiom,
! [B: $tType,A: $tType] :
( ( comple6319245703460814977attice @ A )
=> ! [A2: set @ B,C3: A] :
( ( ( A2
= ( bot_bot @ ( set @ B ) ) )
=> ( ( complete_Inf_Inf @ A
@ ( image @ B @ A
@ ^ [Y: B] : C3
@ A2 ) )
= ( top_top @ A ) ) )
& ( ( A2
!= ( bot_bot @ ( set @ B ) ) )
=> ( ( complete_Inf_Inf @ A
@ ( image @ B @ A
@ ^ [Y: B] : C3
@ A2 ) )
= C3 ) ) ) ) ).
% INF_constant
thf(fact_8168_INF__empty,axiom,
! [B: $tType,A: $tType] :
( ( comple6319245703460814977attice @ A )
=> ! [F2: B > A] :
( ( complete_Inf_Inf @ A @ ( image @ B @ A @ F2 @ ( bot_bot @ ( set @ B ) ) ) )
= ( top_top @ A ) ) ) ).
% INF_empty
thf(fact_8169_INTER__UNIV__conv_I1_J,axiom,
! [A: $tType,B: $tType,B2: B > ( set @ A ),A2: set @ B] :
( ( ( top_top @ ( set @ A ) )
= ( complete_Inf_Inf @ ( set @ A ) @ ( image @ B @ ( set @ A ) @ B2 @ A2 ) ) )
= ( ! [X3: B] :
( ( member @ B @ X3 @ A2 )
=> ( ( B2 @ X3 )
= ( top_top @ ( set @ A ) ) ) ) ) ) ).
% INTER_UNIV_conv(1)
thf(fact_8170_INTER__UNIV__conv_I2_J,axiom,
! [A: $tType,B: $tType,B2: B > ( set @ A ),A2: set @ B] :
( ( ( complete_Inf_Inf @ ( set @ A ) @ ( image @ B @ ( set @ A ) @ B2 @ A2 ) )
= ( top_top @ ( set @ A ) ) )
= ( ! [X3: B] :
( ( member @ B @ X3 @ A2 )
=> ( ( B2 @ X3 )
= ( top_top @ ( set @ A ) ) ) ) ) ) ).
% INTER_UNIV_conv(2)
thf(fact_8171_INT__extend__simps_I5_J,axiom,
! [I8: $tType,J4: $tType,A3: I8,B2: J4 > ( set @ I8 ),C2: set @ J4] :
( ( insert @ I8 @ A3 @ ( complete_Inf_Inf @ ( set @ I8 ) @ ( image @ J4 @ ( set @ I8 ) @ B2 @ C2 ) ) )
= ( complete_Inf_Inf @ ( set @ I8 )
@ ( image @ J4 @ ( set @ I8 )
@ ^ [X3: J4] : ( insert @ I8 @ A3 @ ( B2 @ X3 ) )
@ C2 ) ) ) ).
% INT_extend_simps(5)
thf(fact_8172_INT__insert__distrib,axiom,
! [B: $tType,A: $tType,U: A,A2: set @ A,A3: B,B2: A > ( set @ B )] :
( ( member @ A @ U @ A2 )
=> ( ( complete_Inf_Inf @ ( set @ B )
@ ( image @ A @ ( set @ B )
@ ^ [X3: A] : ( insert @ B @ A3 @ ( B2 @ X3 ) )
@ A2 ) )
= ( insert @ B @ A3 @ ( complete_Inf_Inf @ ( set @ B ) @ ( image @ A @ ( set @ B ) @ B2 @ A2 ) ) ) ) ) ).
% INT_insert_distrib
thf(fact_8173_Inf__fun__def,axiom,
! [B: $tType,A: $tType] :
( ( complete_Inf @ B )
=> ( ( complete_Inf_Inf @ ( A > B ) )
= ( ^ [A7: set @ ( A > B ),X3: A] :
( complete_Inf_Inf @ B
@ ( image @ ( A > B ) @ B
@ ^ [F5: A > B] : ( F5 @ X3 )
@ A7 ) ) ) ) ) ).
% Inf_fun_def
thf(fact_8174_INT__E,axiom,
! [A: $tType,B: $tType,B3: A,B2: B > ( set @ A ),A2: set @ B,A3: B] :
( ( member @ A @ B3 @ ( complete_Inf_Inf @ ( set @ A ) @ ( image @ B @ ( set @ A ) @ B2 @ A2 ) ) )
=> ( ~ ( member @ A @ B3 @ ( B2 @ A3 ) )
=> ~ ( member @ B @ A3 @ A2 ) ) ) ).
% INT_E
% Type constructors (1361)
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,
! [A13: $tType] :
( ( typerep @ A13 )
=> ( typerep @ ( heap_Time_Heap @ A13 ) ) ) ).
thf(tcon_Code__Numeral_Ointeger___Typerep_Otyperep_3,axiom,
typerep @ code_integer ).
thf(tcon_Heap_Oheap_Oheap__ext___Typerep_Otyperep_4,axiom,
! [A13: $tType] :
( ( typerep @ A13 )
=> ( typerep @ ( heap_ext @ A13 ) ) ) ).
thf(tcon_Product__Type_Ounit___Typerep_Otyperep_5,axiom,
typerep @ product_unit ).
thf(tcon_Product__Type_Oprod___Typerep_Otyperep_6,axiom,
! [A13: $tType,A15: $tType] :
( ( ( typerep @ A13 )
& ( typerep @ A15 ) )
=> ( typerep @ ( product_prod @ A13 @ A15 ) ) ) ).
thf(tcon_Numeral__Type_Onum1___Typerep_Otyperep_7,axiom,
typerep @ numeral_num1 ).
thf(tcon_Numeral__Type_Obit1___Typerep_Otyperep_8,axiom,
! [A13: $tType] :
( ( typerep @ A13 )
=> ( typerep @ ( numeral_bit1 @ A13 ) ) ) ).
thf(tcon_Numeral__Type_Obit0___Typerep_Otyperep_9,axiom,
! [A13: $tType] :
( ( typerep @ A13 )
=> ( typerep @ ( numeral_bit0 @ A13 ) ) ) ).
thf(tcon_Multiset_Omultiset___Typerep_Otyperep_10,axiom,
! [A13: $tType] :
( ( typerep @ A13 )
=> ( typerep @ ( multiset @ A13 ) ) ) ).
thf(tcon_Extended__Nat_Oenat___Typerep_Otyperep_11,axiom,
typerep @ extended_enat ).
thf(tcon_Complex_Ocomplex___Typerep_Otyperep_12,axiom,
typerep @ complex ).
thf(tcon_Assertions_Oassn___Typerep_Otyperep_13,axiom,
typerep @ assn ).
thf(tcon_Uint32_Ouint32___Typerep_Otyperep_14,axiom,
typerep @ uint32 ).
thf(tcon_Option_Ooption___Typerep_Otyperep_15,axiom,
! [A13: $tType] :
( ( typerep @ A13 )
=> ( typerep @ ( option @ A13 ) ) ) ).
thf(tcon_Filter_Ofilter___Typerep_Otyperep_16,axiom,
! [A13: $tType] :
( ( typerep @ A13 )
=> ( typerep @ ( filter @ A13 ) ) ) ).
thf(tcon_Enum_Ofinite__3___Typerep_Otyperep_17,axiom,
typerep @ finite_3 ).
thf(tcon_Enum_Ofinite__2___Typerep_Otyperep_18,axiom,
typerep @ finite_2 ).
thf(tcon_Enum_Ofinite__1___Typerep_Otyperep_19,axiom,
typerep @ finite_1 ).
thf(tcon_Sum__Type_Osum___Typerep_Otyperep_20,axiom,
! [A13: $tType,A15: $tType] :
( ( ( typerep @ A13 )
& ( typerep @ A15 ) )
=> ( typerep @ ( sum_sum @ A13 @ A15 ) ) ) ).
thf(tcon_Heap_Oarray___Typerep_Otyperep_21,axiom,
! [A13: $tType] : ( typerep @ ( array @ A13 ) ) ).
thf(tcon_Word_Oword___Typerep_Otyperep_22,axiom,
! [A13: $tType] :
( ( typerep @ A13 )
=> ( typerep @ ( word @ A13 ) ) ) ).
thf(tcon_Real_Oreal___Typerep_Otyperep_23,axiom,
typerep @ real ).
thf(tcon_List_Olist___Typerep_Otyperep_24,axiom,
! [A13: $tType] :
( ( typerep @ A13 )
=> ( typerep @ ( list @ A13 ) ) ) ).
thf(tcon_HOL_Obool___Typerep_Otyperep_25,axiom,
typerep @ $o ).
thf(tcon_Set_Oset___Typerep_Otyperep_26,axiom,
! [A13: $tType] :
( ( typerep @ A13 )
=> ( typerep @ ( set @ A13 ) ) ) ).
thf(tcon_Rat_Orat___Typerep_Otyperep_27,axiom,
typerep @ rat ).
thf(tcon_Num_Onum___Typerep_Otyperep_28,axiom,
typerep @ num ).
thf(tcon_Nat_Onat___Typerep_Otyperep_29,axiom,
typerep @ nat ).
thf(tcon_Int_Oint___Typerep_Otyperep_30,axiom,
typerep @ int ).
thf(tcon_fun___Typerep_Otyperep_31,axiom,
! [A13: $tType,A15: $tType] :
( ( ( typerep @ A13 )
& ( typerep @ A15 ) )
=> ( typerep @ ( A13 > A15 ) ) ) ).
thf(tcon_fun___Conditionally__Complete__Lattices_Oconditionally__complete__lattice,axiom,
! [A13: $tType,A15: $tType] :
( ( comple6319245703460814977attice @ A15 )
=> ( condit1219197933456340205attice @ ( A13 > A15 ) ) ) ).
thf(tcon_fun___Countable__Complete__Lattices_Ocountable__complete__lattice,axiom,
! [A13: $tType,A15: $tType] :
( ( counta3822494911875563373attice @ A15 )
=> ( counta3822494911875563373attice @ ( A13 > A15 ) ) ) ).
thf(tcon_fun___Complete__Lattices_Ocomplete__lattice,axiom,
! [A13: $tType,A15: $tType] :
( ( comple6319245703460814977attice @ A15 )
=> ( comple6319245703460814977attice @ ( A13 > A15 ) ) ) ).
thf(tcon_fun___Boolean__Algebras_Oboolean__algebra,axiom,
! [A13: $tType,A15: $tType] :
( ( boolea8198339166811842893lgebra @ A15 )
=> ( boolea8198339166811842893lgebra @ ( A13 > A15 ) ) ) ).
thf(tcon_fun___Code__Evaluation_Oterm__of,axiom,
! [A13: $tType,A15: $tType] :
( ( ( typerep @ A13 )
& ( typerep @ A15 ) )
=> ( code_term_of @ ( A13 > A15 ) ) ) ).
thf(tcon_fun___Complete__Lattices_OSup,axiom,
! [A13: $tType,A15: $tType] :
( ( complete_Sup @ A15 )
=> ( complete_Sup @ ( A13 > A15 ) ) ) ).
thf(tcon_fun___Complete__Lattices_OInf,axiom,
! [A13: $tType,A15: $tType] :
( ( complete_Inf @ A15 )
=> ( complete_Inf @ ( A13 > A15 ) ) ) ).
thf(tcon_fun___Orderings_Oorder__top,axiom,
! [A13: $tType,A15: $tType] :
( ( order_top @ A15 )
=> ( order_top @ ( A13 > A15 ) ) ) ).
thf(tcon_fun___Orderings_Oorder__bot,axiom,
! [A13: $tType,A15: $tType] :
( ( order_bot @ A15 )
=> ( order_bot @ ( A13 > A15 ) ) ) ).
thf(tcon_fun___Orderings_Opreorder,axiom,
! [A13: $tType,A15: $tType] :
( ( preorder @ A15 )
=> ( preorder @ ( A13 > A15 ) ) ) ).
thf(tcon_fun___Finite__Set_Ofinite,axiom,
! [A13: $tType,A15: $tType] :
( ( ( finite_finite @ A13 )
& ( finite_finite @ A15 ) )
=> ( finite_finite @ ( A13 > A15 ) ) ) ).
thf(tcon_fun___Orderings_Oorder,axiom,
! [A13: $tType,A15: $tType] :
( ( order @ A15 )
=> ( order @ ( A13 > A15 ) ) ) ).
thf(tcon_fun___Orderings_Oord,axiom,
! [A13: $tType,A15: $tType] :
( ( ord @ A15 )
=> ( ord @ ( A13 > A15 ) ) ) ).
thf(tcon_fun___Groups_Ouminus,axiom,
! [A13: $tType,A15: $tType] :
( ( uminus @ A15 )
=> ( uminus @ ( A13 > A15 ) ) ) ).
thf(tcon_fun___Groups_Ominus,axiom,
! [A13: $tType,A15: $tType] :
( ( minus @ A15 )
=> ( minus @ ( A13 > A15 ) ) ) ).
thf(tcon_Int_Oint___Conditionally__Complete__Lattices_Oconditionally__complete__linorder,axiom,
condit6923001295902523014norder @ int ).
thf(tcon_Int_Oint___Conditionally__Complete__Lattices_Oconditionally__complete__lattice_32,axiom,
condit1219197933456340205attice @ int ).
thf(tcon_Int_Oint___Bit__Operations_Ounique__euclidean__semiring__with__bit__operations,axiom,
bit_un5681908812861735899ations @ int ).
thf(tcon_Int_Oint___Semiring__Normalization_Ocomm__semiring__1__cancel__crossproduct,axiom,
semiri1453513574482234551roduct @ int ).
thf(tcon_Int_Oint___Euclidean__Division_Ounique__euclidean__semiring__with__nat,axiom,
euclid5411537665997757685th_nat @ int ).
thf(tcon_Int_Oint___Euclidean__Division_Ounique__euclidean__ring__with__nat,axiom,
euclid8789492081693882211th_nat @ int ).
thf(tcon_Int_Oint___Groups_Oordered__ab__semigroup__monoid__add__imp__le,axiom,
ordere1937475149494474687imp_le @ int ).
thf(tcon_Int_Oint___Euclidean__Division_Ounique__euclidean__semiring,axiom,
euclid3128863361964157862miring @ int ).
thf(tcon_Int_Oint___Euclidean__Division_Oeuclidean__semiring__cancel,axiom,
euclid4440199948858584721cancel @ int ).
thf(tcon_Int_Oint___Divides_Ounique__euclidean__semiring__numeral,axiom,
unique1627219031080169319umeral @ int ).
thf(tcon_Int_Oint___Euclidean__Division_Oeuclidean__ring__cancel,axiom,
euclid8851590272496341667cancel @ int ).
thf(tcon_Int_Oint___Rings_Osemiring__no__zero__divisors__cancel,axiom,
semiri6575147826004484403cancel @ int ).
thf(tcon_Int_Oint___Groups_Ostrict__ordered__ab__semigroup__add,axiom,
strict9044650504122735259up_add @ int ).
thf(tcon_Int_Oint___Groups_Oordered__cancel__ab__semigroup__add,axiom,
ordere580206878836729694up_add @ int ).
thf(tcon_Int_Oint___Groups_Oordered__ab__semigroup__add__imp__le,axiom,
ordere2412721322843649153imp_le @ int ).
thf(tcon_Int_Oint___Bit__Operations_Osemiring__bit__operations,axiom,
bit_se359711467146920520ations @ int ).
thf(tcon_Int_Oint___Rings_Olinordered__comm__semiring__strict,axiom,
linord2810124833399127020strict @ int ).
thf(tcon_Int_Oint___Groups_Ostrict__ordered__comm__monoid__add,axiom,
strict7427464778891057005id_add @ int ).
thf(tcon_Int_Oint___Groups_Oordered__cancel__comm__monoid__add,axiom,
ordere8940638589300402666id_add @ int ).
thf(tcon_Int_Oint___Euclidean__Division_Oeuclidean__semiring,axiom,
euclid3725896446679973847miring @ int ).
thf(tcon_Int_Oint___Topological__Spaces_Otopological__space,axiom,
topolo4958980785337419405_space @ int ).
thf(tcon_Int_Oint___Topological__Spaces_Olinorder__topology,axiom,
topolo1944317154257567458pology @ int ).
thf(tcon_Int_Oint___Topological__Spaces_Odiscrete__topology,axiom,
topolo8865339358273720382pology @ int ).
thf(tcon_Int_Oint___Quickcheck__Narrowing_Opartial__term__of,axiom,
quickc6926020345158392990erm_of @ int ).
thf(tcon_Int_Oint___Limits_Otopological__comm__monoid__mult,axiom,
topolo4987421752381908075d_mult @ int ).
thf(tcon_Int_Oint___Rings_Olinordered__semiring__1__strict,axiom,
linord715952674999750819strict @ int ).
thf(tcon_Int_Oint___Limits_Otopological__comm__monoid__add,axiom,
topolo5987344860129210374id_add @ int ).
thf(tcon_Int_Oint___Groups_Olinordered__ab__semigroup__add,axiom,
linord4140545234300271783up_add @ int ).
thf(tcon_Int_Oint___Bit__Operations_Oring__bit__operations,axiom,
bit_ri3973907225187159222ations @ int ).
thf(tcon_Int_Oint___Topological__Spaces_Oorder__topology,axiom,
topolo2564578578187576103pology @ int ).
thf(tcon_Int_Oint___Rings_Osemiring__1__no__zero__divisors,axiom,
semiri2026040879449505780visors @ int ).
thf(tcon_Int_Oint___Rings_Olinordered__nonzero__semiring,axiom,
linord181362715937106298miring @ int ).
thf(tcon_Int_Oint___Limits_Otopological__semigroup__mult,axiom,
topolo4211221413907600880p_mult @ int ).
thf(tcon_Int_Oint___Euclidean__Division_Oeuclidean__ring,axiom,
euclid5891614535332579305n_ring @ int ).
thf(tcon_Int_Oint___Rings_Olinordered__semiring__strict,axiom,
linord8928482502909563296strict @ int ).
thf(tcon_Int_Oint___Rings_Osemiring__no__zero__divisors,axiom,
semiri3467727345109120633visors @ int ).
thf(tcon_Int_Oint___Groups_Oordered__ab__semigroup__add,axiom,
ordere6658533253407199908up_add @ int ).
thf(tcon_Int_Oint___Groups_Oordered__ab__group__add__abs,axiom,
ordere166539214618696060dd_abs @ int ).
thf(tcon_Int_Oint___Limits_Otopological__monoid__mult,axiom,
topolo1898628316856586783d_mult @ int ).
thf(tcon_Int_Oint___Groups_Oordered__comm__monoid__add,axiom,
ordere6911136660526730532id_add @ int ).
thf(tcon_Int_Oint___Groups_Olinordered__ab__group__add,axiom,
linord5086331880401160121up_add @ int ).
thf(tcon_Int_Oint___Groups_Ocancel__ab__semigroup__add,axiom,
cancel2418104881723323429up_add @ int ).
thf(tcon_Int_Oint___Rings_Oring__1__no__zero__divisors,axiom,
ring_15535105094025558882visors @ int ).
thf(tcon_Int_Oint___Limits_Otopological__monoid__add,axiom,
topolo6943815403480290642id_add @ int ).
thf(tcon_Int_Oint___Groups_Ocancel__comm__monoid__add,axiom,
cancel1802427076303600483id_add @ int ).
thf(tcon_Int_Oint___Rings_Olinordered__ring__strict,axiom,
linord4710134922213307826strict @ int ).
thf(tcon_Int_Oint___Rings_Ocomm__semiring__1__cancel,axiom,
comm_s4317794764714335236cancel @ int ).
thf(tcon_Int_Oint___Bit__Operations_Osemiring__bits,axiom,
bit_semiring_bits @ int ).
thf(tcon_Int_Oint___Topological__Spaces_Ot2__space,axiom,
topological_t2_space @ int ).
thf(tcon_Int_Oint___Topological__Spaces_Ot1__space,axiom,
topological_t1_space @ int ).
thf(tcon_Int_Oint___Rings_Oordered__comm__semiring,axiom,
ordere2520102378445227354miring @ int ).
thf(tcon_Int_Oint___Rings_Olinordered__semiring__1,axiom,
linord6961819062388156250ring_1 @ int ).
thf(tcon_Int_Oint___Groups_Oordered__ab__group__add,axiom,
ordered_ab_group_add @ int ).
thf(tcon_Int_Oint___Groups_Ocancel__semigroup__add,axiom,
cancel_semigroup_add @ int ).
thf(tcon_Int_Oint___Rings_Olinordered__semiring,axiom,
linordered_semiring @ int ).
thf(tcon_Int_Oint___Least__significant__bit_Olsb,axiom,
least_6119777620449941438nt_lsb @ 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___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_33,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___Complete__Lattices_OSup_34,axiom,
complete_Sup @ int ).
thf(tcon_Int_Oint___Complete__Lattices_OInf_35,axiom,
complete_Inf @ int ).
thf(tcon_Int_Oint___Rings_Osemidom__modulo,axiom,
semidom_modulo @ int ).
thf(tcon_Int_Oint___Rings_Osemidom__divide,axiom,
semidom_divide @ int ).
thf(tcon_Int_Oint___Num_Osemiring__numeral,axiom,
semiring_numeral @ int ).
thf(tcon_Int_Oint___Groups_Osemigroup__add,axiom,
semigroup_add @ int ).
thf(tcon_Int_Oint___Rings_Ozero__less__one,axiom,
zero_less_one @ int ).
thf(tcon_Int_Oint___Rings_Ocomm__semiring,axiom,
comm_semiring @ int ).
thf(tcon_Int_Oint___Nat_Osemiring__char__0,axiom,
semiring_char_0 @ int ).
thf(tcon_Int_Oint___Groups_Oab__group__add,axiom,
ab_group_add @ int ).
thf(tcon_Int_Oint___Rings_Ozero__neq__one,axiom,
zero_neq_one @ int ).
thf(tcon_Int_Oint___Rings_Oordered__ring,axiom,
ordered_ring @ int ).
thf(tcon_Int_Oint___Rings_Oidom__abs__sgn,axiom,
idom_abs_sgn @ int ).
thf(tcon_Int_Oint___Parity_Oring__parity,axiom,
ring_parity @ int ).
thf(tcon_Int_Oint___Orderings_Opreorder_36,axiom,
preorder @ int ).
thf(tcon_Int_Oint___Orderings_Olinorder,axiom,
linorder @ int ).
thf(tcon_Int_Oint___Groups_Omonoid__mult,axiom,
monoid_mult @ int ).
thf(tcon_Int_Oint___Rings_Oidom__modulo,axiom,
idom_modulo @ int ).
thf(tcon_Int_Oint___Rings_Oidom__divide,axiom,
idom_divide @ int ).
thf(tcon_Int_Oint___Rings_Ocomm__ring__1,axiom,
comm_ring_1 @ int ).
thf(tcon_Int_Oint___Groups_Omonoid__add,axiom,
monoid_add @ int ).
thf(tcon_Int_Oint___Rings_Osemiring__1,axiom,
semiring_1 @ int ).
thf(tcon_Int_Oint___Rings_Osemiring__0,axiom,
semiring_0 @ int ).
thf(tcon_Int_Oint___Orderings_Ono__top,axiom,
no_top @ int ).
thf(tcon_Int_Oint___Orderings_Ono__bot,axiom,
no_bot @ int ).
thf(tcon_Int_Oint___Groups_Ogroup__add,axiom,
group_add @ 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_37,axiom,
order @ int ).
thf(tcon_Int_Oint___Num_Oneg__numeral,axiom,
neg_numeral @ int ).
thf(tcon_Int_Oint___Nat_Oring__char__0,axiom,
ring_char_0 @ int ).
thf(tcon_Int_Oint___Rings_Osemiring,axiom,
semiring @ int ).
thf(tcon_Int_Oint___Orderings_Oord_38,axiom,
ord @ int ).
thf(tcon_Int_Oint___Groups_Ouminus_39,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_Ominus_40,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_Oring,axiom,
ring @ int ).
thf(tcon_Int_Oint___Rings_Oidom,axiom,
idom @ int ).
thf(tcon_Int_Oint___Groups_Oone,axiom,
one @ int ).
thf(tcon_Int_Oint___Rings_Odvd,axiom,
dvd @ int ).
thf(tcon_Int_Oint___Heap_Oheap,axiom,
heap @ int ).
thf(tcon_Nat_Onat___Conditionally__Complete__Lattices_Oconditionally__complete__linorder_41,axiom,
condit6923001295902523014norder @ nat ).
thf(tcon_Nat_Onat___Conditionally__Complete__Lattices_Oconditionally__complete__lattice_42,axiom,
condit1219197933456340205attice @ nat ).
thf(tcon_Nat_Onat___Bit__Operations_Ounique__euclidean__semiring__with__bit__operations_43,axiom,
bit_un5681908812861735899ations @ nat ).
thf(tcon_Nat_Onat___Semiring__Normalization_Ocomm__semiring__1__cancel__crossproduct_44,axiom,
semiri1453513574482234551roduct @ nat ).
thf(tcon_Nat_Onat___Euclidean__Division_Ounique__euclidean__semiring__with__nat_45,axiom,
euclid5411537665997757685th_nat @ nat ).
thf(tcon_Nat_Onat___Groups_Oordered__ab__semigroup__monoid__add__imp__le_46,axiom,
ordere1937475149494474687imp_le @ nat ).
thf(tcon_Nat_Onat___Euclidean__Division_Ounique__euclidean__semiring_47,axiom,
euclid3128863361964157862miring @ nat ).
thf(tcon_Nat_Onat___Euclidean__Division_Oeuclidean__semiring__cancel_48,axiom,
euclid4440199948858584721cancel @ nat ).
thf(tcon_Nat_Onat___Divides_Ounique__euclidean__semiring__numeral_49,axiom,
unique1627219031080169319umeral @ nat ).
thf(tcon_Nat_Onat___Rings_Osemiring__no__zero__divisors__cancel_50,axiom,
semiri6575147826004484403cancel @ nat ).
thf(tcon_Nat_Onat___Groups_Ostrict__ordered__ab__semigroup__add_51,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_52,axiom,
ordere580206878836729694up_add @ nat ).
thf(tcon_Nat_Onat___Groups_Oordered__ab__semigroup__add__imp__le_53,axiom,
ordere2412721322843649153imp_le @ nat ).
thf(tcon_Nat_Onat___Bit__Operations_Osemiring__bit__operations_54,axiom,
bit_se359711467146920520ations @ nat ).
thf(tcon_Nat_Onat___Rings_Olinordered__comm__semiring__strict_55,axiom,
linord2810124833399127020strict @ nat ).
thf(tcon_Nat_Onat___Groups_Ostrict__ordered__comm__monoid__add_56,axiom,
strict7427464778891057005id_add @ nat ).
thf(tcon_Nat_Onat___Groups_Oordered__cancel__comm__monoid__add_57,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_58,axiom,
euclid3725896446679973847miring @ nat ).
thf(tcon_Nat_Onat___Topological__Spaces_Otopological__space_59,axiom,
topolo4958980785337419405_space @ nat ).
thf(tcon_Nat_Onat___Topological__Spaces_Olinorder__topology_60,axiom,
topolo1944317154257567458pology @ nat ).
thf(tcon_Nat_Onat___Topological__Spaces_Odiscrete__topology_61,axiom,
topolo8865339358273720382pology @ nat ).
thf(tcon_Nat_Onat___Quickcheck__Narrowing_Opartial__term__of_62,axiom,
quickc6926020345158392990erm_of @ nat ).
thf(tcon_Nat_Onat___Limits_Otopological__comm__monoid__mult_63,axiom,
topolo4987421752381908075d_mult @ nat ).
thf(tcon_Nat_Onat___Limits_Otopological__comm__monoid__add_64,axiom,
topolo5987344860129210374id_add @ nat ).
thf(tcon_Nat_Onat___Groups_Olinordered__ab__semigroup__add_65,axiom,
linord4140545234300271783up_add @ nat ).
thf(tcon_Nat_Onat___Topological__Spaces_Oorder__topology_66,axiom,
topolo2564578578187576103pology @ nat ).
thf(tcon_Nat_Onat___Rings_Osemiring__1__no__zero__divisors_67,axiom,
semiri2026040879449505780visors @ nat ).
thf(tcon_Nat_Onat___Rings_Olinordered__nonzero__semiring_68,axiom,
linord181362715937106298miring @ nat ).
thf(tcon_Nat_Onat___Limits_Otopological__semigroup__mult_69,axiom,
topolo4211221413907600880p_mult @ nat ).
thf(tcon_Nat_Onat___Rings_Olinordered__semiring__strict_70,axiom,
linord8928482502909563296strict @ nat ).
thf(tcon_Nat_Onat___Rings_Osemiring__no__zero__divisors_71,axiom,
semiri3467727345109120633visors @ nat ).
thf(tcon_Nat_Onat___Groups_Oordered__ab__semigroup__add_72,axiom,
ordere6658533253407199908up_add @ nat ).
thf(tcon_Nat_Onat___Limits_Otopological__monoid__mult_73,axiom,
topolo1898628316856586783d_mult @ nat ).
thf(tcon_Nat_Onat___Groups_Oordered__comm__monoid__add_74,axiom,
ordere6911136660526730532id_add @ nat ).
thf(tcon_Nat_Onat___Groups_Ocancel__ab__semigroup__add_75,axiom,
cancel2418104881723323429up_add @ nat ).
thf(tcon_Nat_Onat___Limits_Otopological__monoid__add_76,axiom,
topolo6943815403480290642id_add @ nat ).
thf(tcon_Nat_Onat___Groups_Ocancel__comm__monoid__add_77,axiom,
cancel1802427076303600483id_add @ nat ).
thf(tcon_Nat_Onat___Rings_Ocomm__semiring__1__cancel_78,axiom,
comm_s4317794764714335236cancel @ nat ).
thf(tcon_Nat_Onat___Bit__Operations_Osemiring__bits_79,axiom,
bit_semiring_bits @ nat ).
thf(tcon_Nat_Onat___Topological__Spaces_Ot2__space_80,axiom,
topological_t2_space @ nat ).
thf(tcon_Nat_Onat___Topological__Spaces_Ot1__space_81,axiom,
topological_t1_space @ nat ).
thf(tcon_Nat_Onat___Rings_Oordered__comm__semiring_82,axiom,
ordere2520102378445227354miring @ nat ).
thf(tcon_Nat_Onat___Groups_Ocancel__semigroup__add_83,axiom,
cancel_semigroup_add @ nat ).
thf(tcon_Nat_Onat___Rings_Olinordered__semiring_84,axiom,
linordered_semiring @ nat ).
thf(tcon_Nat_Onat___Rings_Oordered__semiring__0_85,axiom,
ordered_semiring_0 @ nat ).
thf(tcon_Nat_Onat___Rings_Olinordered__semidom_86,axiom,
linordered_semidom @ nat ).
thf(tcon_Nat_Onat___Groups_Oab__semigroup__mult_87,axiom,
ab_semigroup_mult @ nat ).
thf(tcon_Nat_Onat___Rings_Osemiring__1__cancel_88,axiom,
semiring_1_cancel @ nat ).
thf(tcon_Nat_Onat___Rings_Oalgebraic__semidom_89,axiom,
algebraic_semidom @ nat ).
thf(tcon_Nat_Onat___Groups_Ocomm__monoid__mult_90,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_91,axiom,
ab_semigroup_add @ nat ).
thf(tcon_Nat_Onat___Code__Evaluation_Oterm__of_92,axiom,
code_term_of @ nat ).
thf(tcon_Nat_Onat___Rings_Oordered__semiring_93,axiom,
ordered_semiring @ nat ).
thf(tcon_Nat_Onat___Parity_Osemiring__parity_94,axiom,
semiring_parity @ nat ).
thf(tcon_Nat_Onat___Groups_Ocomm__monoid__add_95,axiom,
comm_monoid_add @ nat ).
thf(tcon_Nat_Onat___Rings_Osemiring__modulo_96,axiom,
semiring_modulo @ nat ).
thf(tcon_Nat_Onat___Rings_Ocomm__semiring__1_97,axiom,
comm_semiring_1 @ nat ).
thf(tcon_Nat_Onat___Rings_Ocomm__semiring__0_98,axiom,
comm_semiring_0 @ nat ).
thf(tcon_Nat_Onat___Groups_Osemigroup__mult_99,axiom,
semigroup_mult @ nat ).
thf(tcon_Nat_Onat___Complete__Lattices_OSup_100,axiom,
complete_Sup @ nat ).
thf(tcon_Nat_Onat___Complete__Lattices_OInf_101,axiom,
complete_Inf @ nat ).
thf(tcon_Nat_Onat___Rings_Osemidom__modulo_102,axiom,
semidom_modulo @ nat ).
thf(tcon_Nat_Onat___Rings_Osemidom__divide_103,axiom,
semidom_divide @ nat ).
thf(tcon_Nat_Onat___Num_Osemiring__numeral_104,axiom,
semiring_numeral @ nat ).
thf(tcon_Nat_Onat___Groups_Osemigroup__add_105,axiom,
semigroup_add @ nat ).
thf(tcon_Nat_Onat___Rings_Ozero__less__one_106,axiom,
zero_less_one @ nat ).
thf(tcon_Nat_Onat___Rings_Ocomm__semiring_107,axiom,
comm_semiring @ nat ).
thf(tcon_Nat_Onat___Orderings_Owellorder,axiom,
wellorder @ nat ).
thf(tcon_Nat_Onat___Orderings_Oorder__bot_108,axiom,
order_bot @ nat ).
thf(tcon_Nat_Onat___Nat_Osemiring__char__0_109,axiom,
semiring_char_0 @ nat ).
thf(tcon_Nat_Onat___Rings_Ozero__neq__one_110,axiom,
zero_neq_one @ nat ).
thf(tcon_Nat_Onat___Orderings_Opreorder_111,axiom,
preorder @ nat ).
thf(tcon_Nat_Onat___Orderings_Olinorder_112,axiom,
linorder @ nat ).
thf(tcon_Nat_Onat___Groups_Omonoid__mult_113,axiom,
monoid_mult @ nat ).
thf(tcon_Nat_Onat___Groups_Omonoid__add_114,axiom,
monoid_add @ nat ).
thf(tcon_Nat_Onat___Rings_Osemiring__1_115,axiom,
semiring_1 @ nat ).
thf(tcon_Nat_Onat___Rings_Osemiring__0_116,axiom,
semiring_0 @ nat ).
thf(tcon_Nat_Onat___Orderings_Ono__top_117,axiom,
no_top @ nat ).
thf(tcon_Nat_Onat___GCD_Osemiring__gcd_118,axiom,
semiring_gcd @ nat ).
thf(tcon_Nat_Onat___Rings_Omult__zero_119,axiom,
mult_zero @ nat ).
thf(tcon_Nat_Onat___Orderings_Oorder_120,axiom,
order @ nat ).
thf(tcon_Nat_Onat___Rings_Osemiring_121,axiom,
semiring @ nat ).
thf(tcon_Nat_Onat___Orderings_Oord_122,axiom,
ord @ nat ).
thf(tcon_Nat_Onat___Groups_Ominus_123,axiom,
minus @ nat ).
thf(tcon_Nat_Onat___Power_Opower_124,axiom,
power @ nat ).
thf(tcon_Nat_Onat___Num_Onumeral_125,axiom,
numeral @ nat ).
thf(tcon_Nat_Onat___Groups_Ozero_126,axiom,
zero @ nat ).
thf(tcon_Nat_Onat___Groups_Oplus_127,axiom,
plus @ nat ).
thf(tcon_Nat_Onat___Groups_Oone_128,axiom,
one @ nat ).
thf(tcon_Nat_Onat___Rings_Odvd_129,axiom,
dvd @ nat ).
thf(tcon_Nat_Onat___Heap_Oheap_130,axiom,
heap @ nat ).
thf(tcon_Nat_Onat___Nat_Osize,axiom,
size @ nat ).
thf(tcon_Num_Onum___Quickcheck__Narrowing_Opartial__term__of_131,axiom,
quickc6926020345158392990erm_of @ num ).
thf(tcon_Num_Onum___Code__Evaluation_Oterm__of_132,axiom,
code_term_of @ num ).
thf(tcon_Num_Onum___Orderings_Opreorder_133,axiom,
preorder @ num ).
thf(tcon_Num_Onum___Orderings_Olinorder_134,axiom,
linorder @ num ).
thf(tcon_Num_Onum___Orderings_Oorder_135,axiom,
order @ num ).
thf(tcon_Num_Onum___Orderings_Oord_136,axiom,
ord @ num ).
thf(tcon_Num_Onum___Groups_Oplus_137,axiom,
plus @ num ).
thf(tcon_Num_Onum___Nat_Osize_138,axiom,
size @ num ).
thf(tcon_Rat_Orat___Semiring__Normalization_Ocomm__semiring__1__cancel__crossproduct_139,axiom,
semiri1453513574482234551roduct @ rat ).
thf(tcon_Rat_Orat___Groups_Oordered__ab__semigroup__monoid__add__imp__le_140,axiom,
ordere1937475149494474687imp_le @ rat ).
thf(tcon_Rat_Orat___Rings_Osemiring__no__zero__divisors__cancel_141,axiom,
semiri6575147826004484403cancel @ rat ).
thf(tcon_Rat_Orat___Groups_Ostrict__ordered__ab__semigroup__add_142,axiom,
strict9044650504122735259up_add @ rat ).
thf(tcon_Rat_Orat___Groups_Oordered__cancel__ab__semigroup__add_143,axiom,
ordere580206878836729694up_add @ rat ).
thf(tcon_Rat_Orat___Groups_Oordered__ab__semigroup__add__imp__le_144,axiom,
ordere2412721322843649153imp_le @ rat ).
thf(tcon_Rat_Orat___Rings_Olinordered__comm__semiring__strict_145,axiom,
linord2810124833399127020strict @ rat ).
thf(tcon_Rat_Orat___Groups_Ostrict__ordered__comm__monoid__add_146,axiom,
strict7427464778891057005id_add @ rat ).
thf(tcon_Rat_Orat___Groups_Oordered__cancel__comm__monoid__add_147,axiom,
ordere8940638589300402666id_add @ rat ).
thf(tcon_Rat_Orat___Quickcheck__Narrowing_Opartial__term__of_148,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_149,axiom,
linord715952674999750819strict @ rat ).
thf(tcon_Rat_Orat___Orderings_Ounbounded__dense__linorder,axiom,
unboun7993243217541854897norder @ rat ).
thf(tcon_Rat_Orat___Groups_Olinordered__ab__semigroup__add_150,axiom,
linord4140545234300271783up_add @ rat ).
thf(tcon_Rat_Orat___Rings_Osemiring__1__no__zero__divisors_151,axiom,
semiri2026040879449505780visors @ rat ).
thf(tcon_Rat_Orat___Rings_Olinordered__nonzero__semiring_152,axiom,
linord181362715937106298miring @ rat ).
thf(tcon_Rat_Orat___Rings_Olinordered__semiring__strict_153,axiom,
linord8928482502909563296strict @ rat ).
thf(tcon_Rat_Orat___Rings_Osemiring__no__zero__divisors_154,axiom,
semiri3467727345109120633visors @ rat ).
thf(tcon_Rat_Orat___Groups_Oordered__ab__semigroup__add_155,axiom,
ordere6658533253407199908up_add @ rat ).
thf(tcon_Rat_Orat___Groups_Oordered__ab__group__add__abs_156,axiom,
ordere166539214618696060dd_abs @ rat ).
thf(tcon_Rat_Orat___Archimedean__Field_Ofloor__ceiling,axiom,
archim2362893244070406136eiling @ rat ).
thf(tcon_Rat_Orat___Groups_Oordered__comm__monoid__add_157,axiom,
ordere6911136660526730532id_add @ rat ).
thf(tcon_Rat_Orat___Groups_Olinordered__ab__group__add_158,axiom,
linord5086331880401160121up_add @ rat ).
thf(tcon_Rat_Orat___Groups_Ocancel__ab__semigroup__add_159,axiom,
cancel2418104881723323429up_add @ rat ).
thf(tcon_Rat_Orat___Rings_Oring__1__no__zero__divisors_160,axiom,
ring_15535105094025558882visors @ rat ).
thf(tcon_Rat_Orat___Groups_Ocancel__comm__monoid__add_161,axiom,
cancel1802427076303600483id_add @ rat ).
thf(tcon_Rat_Orat___Rings_Olinordered__ring__strict_162,axiom,
linord4710134922213307826strict @ rat ).
thf(tcon_Rat_Orat___Rings_Ocomm__semiring__1__cancel_163,axiom,
comm_s4317794764714335236cancel @ rat ).
thf(tcon_Rat_Orat___Rings_Oordered__comm__semiring_164,axiom,
ordere2520102378445227354miring @ rat ).
thf(tcon_Rat_Orat___Rings_Olinordered__semiring__1_165,axiom,
linord6961819062388156250ring_1 @ rat ).
thf(tcon_Rat_Orat___Groups_Oordered__ab__group__add_166,axiom,
ordered_ab_group_add @ rat ).
thf(tcon_Rat_Orat___Groups_Ocancel__semigroup__add_167,axiom,
cancel_semigroup_add @ rat ).
thf(tcon_Rat_Orat___Rings_Olinordered__semiring_168,axiom,
linordered_semiring @ rat ).
thf(tcon_Rat_Orat___Rings_Oordered__semiring__0_169,axiom,
ordered_semiring_0 @ rat ).
thf(tcon_Rat_Orat___Rings_Olinordered__semidom_170,axiom,
linordered_semidom @ rat ).
thf(tcon_Rat_Orat___Orderings_Odense__linorder,axiom,
dense_linorder @ rat ).
thf(tcon_Rat_Orat___Groups_Oab__semigroup__mult_171,axiom,
ab_semigroup_mult @ rat ).
thf(tcon_Rat_Orat___Rings_Osemiring__1__cancel_172,axiom,
semiring_1_cancel @ rat ).
thf(tcon_Rat_Orat___Groups_Ocomm__monoid__mult_173,axiom,
comm_monoid_mult @ rat ).
thf(tcon_Rat_Orat___Groups_Oab__semigroup__add_174,axiom,
ab_semigroup_add @ rat ).
thf(tcon_Rat_Orat___Fields_Olinordered__field,axiom,
linordered_field @ rat ).
thf(tcon_Rat_Orat___Code__Evaluation_Oterm__of_175,axiom,
code_term_of @ rat ).
thf(tcon_Rat_Orat___Rings_Oordered__semiring_176,axiom,
ordered_semiring @ rat ).
thf(tcon_Rat_Orat___Rings_Oordered__ring__abs_177,axiom,
ordered_ring_abs @ rat ).
thf(tcon_Rat_Orat___Groups_Ocomm__monoid__add_178,axiom,
comm_monoid_add @ rat ).
thf(tcon_Rat_Orat___Rings_Olinordered__ring_179,axiom,
linordered_ring @ rat ).
thf(tcon_Rat_Orat___Rings_Olinordered__idom_180,axiom,
linordered_idom @ rat ).
thf(tcon_Rat_Orat___Rings_Ocomm__semiring__1_181,axiom,
comm_semiring_1 @ rat ).
thf(tcon_Rat_Orat___Rings_Ocomm__semiring__0_182,axiom,
comm_semiring_0 @ rat ).
thf(tcon_Rat_Orat___Orderings_Odense__order,axiom,
dense_order @ rat ).
thf(tcon_Rat_Orat___Groups_Osemigroup__mult_183,axiom,
semigroup_mult @ rat ).
thf(tcon_Rat_Orat___Rings_Osemidom__divide_184,axiom,
semidom_divide @ rat ).
thf(tcon_Rat_Orat___Num_Osemiring__numeral_185,axiom,
semiring_numeral @ rat ).
thf(tcon_Rat_Orat___Groups_Osemigroup__add_186,axiom,
semigroup_add @ rat ).
thf(tcon_Rat_Orat___Fields_Ofield__abs__sgn,axiom,
field_abs_sgn @ rat ).
thf(tcon_Rat_Orat___Fields_Odivision__ring,axiom,
division_ring @ rat ).
thf(tcon_Rat_Orat___Rings_Ozero__less__one_187,axiom,
zero_less_one @ rat ).
thf(tcon_Rat_Orat___Rings_Ocomm__semiring_188,axiom,
comm_semiring @ rat ).
thf(tcon_Rat_Orat___Nat_Osemiring__char__0_189,axiom,
semiring_char_0 @ rat ).
thf(tcon_Rat_Orat___Groups_Oab__group__add_190,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_191,axiom,
zero_neq_one @ rat ).
thf(tcon_Rat_Orat___Rings_Oordered__ring_192,axiom,
ordered_ring @ rat ).
thf(tcon_Rat_Orat___Rings_Oidom__abs__sgn_193,axiom,
idom_abs_sgn @ rat ).
thf(tcon_Rat_Orat___Orderings_Opreorder_194,axiom,
preorder @ rat ).
thf(tcon_Rat_Orat___Orderings_Olinorder_195,axiom,
linorder @ rat ).
thf(tcon_Rat_Orat___Groups_Omonoid__mult_196,axiom,
monoid_mult @ rat ).
thf(tcon_Rat_Orat___Rings_Oidom__divide_197,axiom,
idom_divide @ rat ).
thf(tcon_Rat_Orat___Rings_Ocomm__ring__1_198,axiom,
comm_ring_1 @ rat ).
thf(tcon_Rat_Orat___Groups_Omonoid__add_199,axiom,
monoid_add @ rat ).
thf(tcon_Rat_Orat___Rings_Osemiring__1_200,axiom,
semiring_1 @ rat ).
thf(tcon_Rat_Orat___Rings_Osemiring__0_201,axiom,
semiring_0 @ rat ).
thf(tcon_Rat_Orat___Orderings_Ono__top_202,axiom,
no_top @ rat ).
thf(tcon_Rat_Orat___Orderings_Ono__bot_203,axiom,
no_bot @ rat ).
thf(tcon_Rat_Orat___Groups_Ogroup__add_204,axiom,
group_add @ rat ).
thf(tcon_Rat_Orat___Rings_Omult__zero_205,axiom,
mult_zero @ rat ).
thf(tcon_Rat_Orat___Rings_Ocomm__ring_206,axiom,
comm_ring @ rat ).
thf(tcon_Rat_Orat___Orderings_Oorder_207,axiom,
order @ rat ).
thf(tcon_Rat_Orat___Num_Oneg__numeral_208,axiom,
neg_numeral @ rat ).
thf(tcon_Rat_Orat___Nat_Oring__char__0_209,axiom,
ring_char_0 @ rat ).
thf(tcon_Rat_Orat___Rings_Osemiring_210,axiom,
semiring @ rat ).
thf(tcon_Rat_Orat___Fields_Oinverse,axiom,
inverse @ rat ).
thf(tcon_Rat_Orat___Orderings_Oord_211,axiom,
ord @ rat ).
thf(tcon_Rat_Orat___Groups_Ouminus_212,axiom,
uminus @ rat ).
thf(tcon_Rat_Orat___Rings_Oring__1_213,axiom,
ring_1 @ rat ).
thf(tcon_Rat_Orat___Rings_Oabs__if_214,axiom,
abs_if @ rat ).
thf(tcon_Rat_Orat___Groups_Ominus_215,axiom,
minus @ rat ).
thf(tcon_Rat_Orat___Fields_Ofield,axiom,
field @ rat ).
thf(tcon_Rat_Orat___Power_Opower_216,axiom,
power @ rat ).
thf(tcon_Rat_Orat___Num_Onumeral_217,axiom,
numeral @ rat ).
thf(tcon_Rat_Orat___Groups_Ozero_218,axiom,
zero @ rat ).
thf(tcon_Rat_Orat___Groups_Oplus_219,axiom,
plus @ rat ).
thf(tcon_Rat_Orat___Rings_Oring_220,axiom,
ring @ rat ).
thf(tcon_Rat_Orat___Rings_Oidom_221,axiom,
idom @ rat ).
thf(tcon_Rat_Orat___Groups_Oone_222,axiom,
one @ rat ).
thf(tcon_Rat_Orat___Rings_Odvd_223,axiom,
dvd @ rat ).
thf(tcon_Set_Oset___Conditionally__Complete__Lattices_Oconditionally__complete__lattice_224,axiom,
! [A13: $tType] : ( condit1219197933456340205attice @ ( set @ A13 ) ) ).
thf(tcon_Set_Oset___Countable__Complete__Lattices_Ocountable__complete__lattice_225,axiom,
! [A13: $tType] : ( counta3822494911875563373attice @ ( set @ A13 ) ) ).
thf(tcon_Set_Oset___Quickcheck__Narrowing_Opartial__term__of_226,axiom,
! [A13: $tType] :
( ( typerep @ A13 )
=> ( quickc6926020345158392990erm_of @ ( set @ A13 ) ) ) ).
thf(tcon_Set_Oset___Complete__Lattices_Ocomplete__lattice_227,axiom,
! [A13: $tType] : ( comple6319245703460814977attice @ ( set @ A13 ) ) ).
thf(tcon_Set_Oset___Boolean__Algebras_Oboolean__algebra_228,axiom,
! [A13: $tType] : ( boolea8198339166811842893lgebra @ ( set @ A13 ) ) ).
thf(tcon_Set_Oset___Code__Evaluation_Oterm__of_229,axiom,
! [A13: $tType] :
( ( typerep @ A13 )
=> ( code_term_of @ ( set @ A13 ) ) ) ).
thf(tcon_Set_Oset___Complete__Lattices_OSup_230,axiom,
! [A13: $tType] : ( complete_Sup @ ( set @ A13 ) ) ).
thf(tcon_Set_Oset___Complete__Lattices_OInf_231,axiom,
! [A13: $tType] : ( complete_Inf @ ( set @ A13 ) ) ).
thf(tcon_Set_Oset___Orderings_Oorder__top_232,axiom,
! [A13: $tType] : ( order_top @ ( set @ A13 ) ) ).
thf(tcon_Set_Oset___Orderings_Oorder__bot_233,axiom,
! [A13: $tType] : ( order_bot @ ( set @ A13 ) ) ).
thf(tcon_Set_Oset___Orderings_Opreorder_234,axiom,
! [A13: $tType] : ( preorder @ ( set @ A13 ) ) ).
thf(tcon_Set_Oset___Finite__Set_Ofinite_235,axiom,
! [A13: $tType] :
( ( finite_finite @ A13 )
=> ( finite_finite @ ( set @ A13 ) ) ) ).
thf(tcon_Set_Oset___Orderings_Oorder_236,axiom,
! [A13: $tType] : ( order @ ( set @ A13 ) ) ).
thf(tcon_Set_Oset___Orderings_Oord_237,axiom,
! [A13: $tType] : ( ord @ ( set @ A13 ) ) ).
thf(tcon_Set_Oset___Groups_Ouminus_238,axiom,
! [A13: $tType] : ( uminus @ ( set @ A13 ) ) ).
thf(tcon_Set_Oset___Groups_Ominus_239,axiom,
! [A13: $tType] : ( minus @ ( set @ A13 ) ) ).
thf(tcon_HOL_Obool___Conditionally__Complete__Lattices_Oconditionally__complete__lattice_240,axiom,
condit1219197933456340205attice @ $o ).
thf(tcon_HOL_Obool___Countable__Complete__Lattices_Ocountable__complete__lattice_241,axiom,
counta3822494911875563373attice @ $o ).
thf(tcon_HOL_Obool___Topological__Spaces_Otopological__space_242,axiom,
topolo4958980785337419405_space @ $o ).
thf(tcon_HOL_Obool___Topological__Spaces_Olinorder__topology_243,axiom,
topolo1944317154257567458pology @ $o ).
thf(tcon_HOL_Obool___Topological__Spaces_Odiscrete__topology_244,axiom,
topolo8865339358273720382pology @ $o ).
thf(tcon_HOL_Obool___Quickcheck__Narrowing_Opartial__term__of_245,axiom,
quickc6926020345158392990erm_of @ $o ).
thf(tcon_HOL_Obool___Complete__Lattices_Ocomplete__lattice_246,axiom,
comple6319245703460814977attice @ $o ).
thf(tcon_HOL_Obool___Topological__Spaces_Oorder__topology_247,axiom,
topolo2564578578187576103pology @ $o ).
thf(tcon_HOL_Obool___Boolean__Algebras_Oboolean__algebra_248,axiom,
boolea8198339166811842893lgebra @ $o ).
thf(tcon_HOL_Obool___Topological__Spaces_Ot2__space_249,axiom,
topological_t2_space @ $o ).
thf(tcon_HOL_Obool___Topological__Spaces_Ot1__space_250,axiom,
topological_t1_space @ $o ).
thf(tcon_HOL_Obool___Code__Evaluation_Oterm__of_251,axiom,
code_term_of @ $o ).
thf(tcon_HOL_Obool___Complete__Lattices_OSup_252,axiom,
complete_Sup @ $o ).
thf(tcon_HOL_Obool___Complete__Lattices_OInf_253,axiom,
complete_Inf @ $o ).
thf(tcon_HOL_Obool___Orderings_Oorder__top_254,axiom,
order_top @ $o ).
thf(tcon_HOL_Obool___Orderings_Oorder__bot_255,axiom,
order_bot @ $o ).
thf(tcon_HOL_Obool___Orderings_Opreorder_256,axiom,
preorder @ $o ).
thf(tcon_HOL_Obool___Orderings_Olinorder_257,axiom,
linorder @ $o ).
thf(tcon_HOL_Obool___Finite__Set_Ofinite_258,axiom,
finite_finite @ $o ).
thf(tcon_HOL_Obool___Orderings_Oorder_259,axiom,
order @ $o ).
thf(tcon_HOL_Obool___Orderings_Oord_260,axiom,
ord @ $o ).
thf(tcon_HOL_Obool___Groups_Ouminus_261,axiom,
uminus @ $o ).
thf(tcon_HOL_Obool___Groups_Ominus_262,axiom,
minus @ $o ).
thf(tcon_HOL_Obool___Heap_Oheap_263,axiom,
heap @ $o ).
thf(tcon_List_Olist___Quickcheck__Narrowing_Opartial__term__of_264,axiom,
! [A13: $tType] :
( ( typerep @ A13 )
=> ( quickc6926020345158392990erm_of @ ( list @ A13 ) ) ) ).
thf(tcon_List_Olist___Code__Evaluation_Oterm__of_265,axiom,
! [A13: $tType] :
( ( typerep @ A13 )
=> ( code_term_of @ ( list @ A13 ) ) ) ).
thf(tcon_List_Olist___Heap_Oheap_266,axiom,
! [A13: $tType] :
( ( heap @ A13 )
=> ( heap @ ( list @ A13 ) ) ) ).
thf(tcon_List_Olist___Nat_Osize_267,axiom,
! [A13: $tType] : ( size @ ( list @ A13 ) ) ).
thf(tcon_Real_Oreal___Conditionally__Complete__Lattices_Oconditionally__complete__linorder_268,axiom,
condit6923001295902523014norder @ real ).
thf(tcon_Real_Oreal___Conditionally__Complete__Lattices_Oconditionally__complete__lattice_269,axiom,
condit1219197933456340205attice @ real ).
thf(tcon_Real_Oreal___Semiring__Normalization_Ocomm__semiring__1__cancel__crossproduct_270,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_271,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_272,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_273,axiom,
strict9044650504122735259up_add @ real ).
thf(tcon_Real_Oreal___Groups_Oordered__cancel__ab__semigroup__add_274,axiom,
ordere580206878836729694up_add @ real ).
thf(tcon_Real_Oreal___Groups_Oordered__ab__semigroup__add__imp__le_275,axiom,
ordere2412721322843649153imp_le @ real ).
thf(tcon_Real_Oreal___Rings_Olinordered__comm__semiring__strict_276,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_277,axiom,
strict7427464778891057005id_add @ real ).
thf(tcon_Real_Oreal___Groups_Oordered__cancel__comm__monoid__add_278,axiom,
ordere8940638589300402666id_add @ real ).
thf(tcon_Real_Oreal___Topological__Spaces_Otopological__space_279,axiom,
topolo4958980785337419405_space @ real ).
thf(tcon_Real_Oreal___Topological__Spaces_Olinorder__topology_280,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_281,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_282,axiom,
archim462609752435547400_field @ real ).
thf(tcon_Real_Oreal___Rings_Olinordered__semiring__1__strict_283,axiom,
linord715952674999750819strict @ real ).
thf(tcon_Real_Oreal___Orderings_Ounbounded__dense__linorder_284,axiom,
unboun7993243217541854897norder @ real ).
thf(tcon_Real_Oreal___Limits_Otopological__comm__monoid__add_285,axiom,
topolo5987344860129210374id_add @ real ).
thf(tcon_Real_Oreal___Groups_Olinordered__ab__semigroup__add_286,axiom,
linord4140545234300271783up_add @ real ).
thf(tcon_Real_Oreal___Topological__Spaces_Oorder__topology_287,axiom,
topolo2564578578187576103pology @ real ).
thf(tcon_Real_Oreal___Rings_Osemiring__1__no__zero__divisors_288,axiom,
semiri2026040879449505780visors @ real ).
thf(tcon_Real_Oreal___Rings_Olinordered__nonzero__semiring_289,axiom,
linord181362715937106298miring @ real ).
thf(tcon_Real_Oreal___Real__Vector__Spaces_Oreal__algebra__1,axiom,
real_V2191834092415804123ebra_1 @ real ).
thf(tcon_Real_Oreal___Real__Vector__Spaces_Ocomplete__space,axiom,
real_V8037385150606011577_space @ real ).
thf(tcon_Real_Oreal___Limits_Otopological__semigroup__mult_290,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_291,axiom,
linord8928482502909563296strict @ real ).
thf(tcon_Real_Oreal___Rings_Osemiring__no__zero__divisors_292,axiom,
semiri3467727345109120633visors @ real ).
thf(tcon_Real_Oreal___Real__Vector__Spaces_Oreal__algebra,axiom,
real_V6157519004096292374lgebra @ real ).
thf(tcon_Real_Oreal___Real__Vector__Spaces_Ometric__space,axiom,
real_V7819770556892013058_space @ real ).
thf(tcon_Real_Oreal___Limits_Otopological__ab__group__add,axiom,
topolo1287966508704411220up_add @ real ).
thf(tcon_Real_Oreal___Groups_Oordered__ab__semigroup__add_293,axiom,
ordere6658533253407199908up_add @ real ).
thf(tcon_Real_Oreal___Groups_Oordered__ab__group__add__abs_294,axiom,
ordere166539214618696060dd_abs @ real ).
thf(tcon_Real_Oreal___Archimedean__Field_Ofloor__ceiling_295,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_296,axiom,
ordere6911136660526730532id_add @ real ).
thf(tcon_Real_Oreal___Groups_Olinordered__ab__group__add_297,axiom,
linord5086331880401160121up_add @ real ).
thf(tcon_Real_Oreal___Groups_Ocancel__ab__semigroup__add_298,axiom,
cancel2418104881723323429up_add @ real ).
thf(tcon_Real_Oreal___Rings_Oring__1__no__zero__divisors_299,axiom,
ring_15535105094025558882visors @ real ).
thf(tcon_Real_Oreal___Real__Vector__Spaces_Oreal__field,axiom,
real_V7773925162809079976_field @ real ).
thf(tcon_Real_Oreal___Limits_Otopological__monoid__add_300,axiom,
topolo6943815403480290642id_add @ real ).
thf(tcon_Real_Oreal___Groups_Ocancel__comm__monoid__add_301,axiom,
cancel1802427076303600483id_add @ real ).
thf(tcon_Real_Oreal___Rings_Olinordered__ring__strict_302,axiom,
linord4710134922213307826strict @ real ).
thf(tcon_Real_Oreal___Rings_Ocomm__semiring__1__cancel_303,axiom,
comm_s4317794764714335236cancel @ real ).
thf(tcon_Real_Oreal___Real__Vector__Spaces_Odist__norm,axiom,
real_V6936659425649961206t_norm @ real ).
thf(tcon_Real_Oreal___Limits_Otopological__group__add,axiom,
topolo1633459387980952147up_add @ real ).
thf(tcon_Real_Oreal___Topological__Spaces_Ot2__space_304,axiom,
topological_t2_space @ real ).
thf(tcon_Real_Oreal___Topological__Spaces_Ot1__space_305,axiom,
topological_t1_space @ real ).
thf(tcon_Real_Oreal___Rings_Oordered__comm__semiring_306,axiom,
ordere2520102378445227354miring @ real ).
thf(tcon_Real_Oreal___Rings_Olinordered__semiring__1_307,axiom,
linord6961819062388156250ring_1 @ real ).
thf(tcon_Real_Oreal___Groups_Oordered__ab__group__add_308,axiom,
ordered_ab_group_add @ real ).
thf(tcon_Real_Oreal___Groups_Ocancel__semigroup__add_309,axiom,
cancel_semigroup_add @ real ).
thf(tcon_Real_Oreal___Rings_Olinordered__semiring_310,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_311,axiom,
ordered_semiring_0 @ real ).
thf(tcon_Real_Oreal___Rings_Olinordered__semidom_312,axiom,
linordered_semidom @ real ).
thf(tcon_Real_Oreal___Orderings_Odense__linorder_313,axiom,
dense_linorder @ real ).
thf(tcon_Real_Oreal___Groups_Oab__semigroup__mult_314,axiom,
ab_semigroup_mult @ real ).
thf(tcon_Real_Oreal___Rings_Osemiring__1__cancel_315,axiom,
semiring_1_cancel @ real ).
thf(tcon_Real_Oreal___Groups_Ocomm__monoid__mult_316,axiom,
comm_monoid_mult @ real ).
thf(tcon_Real_Oreal___Groups_Oab__semigroup__add_317,axiom,
ab_semigroup_add @ real ).
thf(tcon_Real_Oreal___Fields_Olinordered__field_318,axiom,
linordered_field @ real ).
thf(tcon_Real_Oreal___Code__Evaluation_Oterm__of_319,axiom,
code_term_of @ real ).
thf(tcon_Real_Oreal___Rings_Oordered__semiring_320,axiom,
ordered_semiring @ real ).
thf(tcon_Real_Oreal___Rings_Oordered__ring__abs_321,axiom,
ordered_ring_abs @ real ).
thf(tcon_Real_Oreal___Groups_Ocomm__monoid__add_322,axiom,
comm_monoid_add @ real ).
thf(tcon_Real_Oreal___Rings_Olinordered__ring_323,axiom,
linordered_ring @ real ).
thf(tcon_Real_Oreal___Rings_Olinordered__idom_324,axiom,
linordered_idom @ real ).
thf(tcon_Real_Oreal___Rings_Ocomm__semiring__1_325,axiom,
comm_semiring_1 @ real ).
thf(tcon_Real_Oreal___Rings_Ocomm__semiring__0_326,axiom,
comm_semiring_0 @ real ).
thf(tcon_Real_Oreal___Orderings_Odense__order_327,axiom,
dense_order @ real ).
thf(tcon_Real_Oreal___Groups_Osemigroup__mult_328,axiom,
semigroup_mult @ real ).
thf(tcon_Real_Oreal___Complete__Lattices_OSup_329,axiom,
complete_Sup @ real ).
thf(tcon_Real_Oreal___Complete__Lattices_OInf_330,axiom,
complete_Inf @ real ).
thf(tcon_Real_Oreal___Rings_Osemidom__divide_331,axiom,
semidom_divide @ real ).
thf(tcon_Real_Oreal___Num_Osemiring__numeral_332,axiom,
semiring_numeral @ real ).
thf(tcon_Real_Oreal___Groups_Osemigroup__add_333,axiom,
semigroup_add @ real ).
thf(tcon_Real_Oreal___Fields_Ofield__abs__sgn_334,axiom,
field_abs_sgn @ real ).
thf(tcon_Real_Oreal___Fields_Odivision__ring_335,axiom,
division_ring @ real ).
thf(tcon_Real_Oreal___Rings_Ozero__less__one_336,axiom,
zero_less_one @ real ).
thf(tcon_Real_Oreal___Rings_Ocomm__semiring_337,axiom,
comm_semiring @ real ).
thf(tcon_Real_Oreal___Nat_Osemiring__char__0_338,axiom,
semiring_char_0 @ real ).
thf(tcon_Real_Oreal___Groups_Oab__group__add_339,axiom,
ab_group_add @ real ).
thf(tcon_Real_Oreal___Fields_Ofield__char__0_340,axiom,
field_char_0 @ real ).
thf(tcon_Real_Oreal___Rings_Ozero__neq__one_341,axiom,
zero_neq_one @ real ).
thf(tcon_Real_Oreal___Rings_Oordered__ring_342,axiom,
ordered_ring @ real ).
thf(tcon_Real_Oreal___Rings_Oidom__abs__sgn_343,axiom,
idom_abs_sgn @ real ).
thf(tcon_Real_Oreal___Orderings_Opreorder_344,axiom,
preorder @ real ).
thf(tcon_Real_Oreal___Orderings_Olinorder_345,axiom,
linorder @ real ).
thf(tcon_Real_Oreal___Groups_Omonoid__mult_346,axiom,
monoid_mult @ real ).
thf(tcon_Real_Oreal___Transcendental_Oln,axiom,
ln @ real ).
thf(tcon_Real_Oreal___Rings_Oidom__divide_347,axiom,
idom_divide @ real ).
thf(tcon_Real_Oreal___Rings_Ocomm__ring__1_348,axiom,
comm_ring_1 @ real ).
thf(tcon_Real_Oreal___Groups_Omonoid__add_349,axiom,
monoid_add @ real ).
thf(tcon_Real_Oreal___Rings_Osemiring__1_350,axiom,
semiring_1 @ real ).
thf(tcon_Real_Oreal___Rings_Osemiring__0_351,axiom,
semiring_0 @ real ).
thf(tcon_Real_Oreal___Orderings_Ono__top_352,axiom,
no_top @ real ).
thf(tcon_Real_Oreal___Orderings_Ono__bot_353,axiom,
no_bot @ real ).
thf(tcon_Real_Oreal___Groups_Ogroup__add_354,axiom,
group_add @ real ).
thf(tcon_Real_Oreal___Rings_Omult__zero_355,axiom,
mult_zero @ real ).
thf(tcon_Real_Oreal___Rings_Ocomm__ring_356,axiom,
comm_ring @ real ).
thf(tcon_Real_Oreal___Orderings_Oorder_357,axiom,
order @ real ).
thf(tcon_Real_Oreal___Num_Oneg__numeral_358,axiom,
neg_numeral @ real ).
thf(tcon_Real_Oreal___Nat_Oring__char__0_359,axiom,
ring_char_0 @ real ).
thf(tcon_Real_Oreal___Rings_Osemiring_360,axiom,
semiring @ real ).
thf(tcon_Real_Oreal___Fields_Oinverse_361,axiom,
inverse @ real ).
thf(tcon_Real_Oreal___Orderings_Oord_362,axiom,
ord @ real ).
thf(tcon_Real_Oreal___Groups_Ouminus_363,axiom,
uminus @ real ).
thf(tcon_Real_Oreal___Rings_Oring__1_364,axiom,
ring_1 @ real ).
thf(tcon_Real_Oreal___Rings_Oabs__if_365,axiom,
abs_if @ real ).
thf(tcon_Real_Oreal___Groups_Ominus_366,axiom,
minus @ real ).
thf(tcon_Real_Oreal___Fields_Ofield_367,axiom,
field @ real ).
thf(tcon_Real_Oreal___Power_Opower_368,axiom,
power @ real ).
thf(tcon_Real_Oreal___Num_Onumeral_369,axiom,
numeral @ real ).
thf(tcon_Real_Oreal___Groups_Ozero_370,axiom,
zero @ real ).
thf(tcon_Real_Oreal___Groups_Oplus_371,axiom,
plus @ real ).
thf(tcon_Real_Oreal___Rings_Oring_372,axiom,
ring @ real ).
thf(tcon_Real_Oreal___Rings_Oidom_373,axiom,
idom @ real ).
thf(tcon_Real_Oreal___Groups_Oone_374,axiom,
one @ real ).
thf(tcon_Real_Oreal___Rings_Odvd_375,axiom,
dvd @ real ).
thf(tcon_Word_Oword___Bit__Operations_Osemiring__bit__operations_376,axiom,
! [A13: $tType] :
( ( type_len @ A13 )
=> ( bit_se359711467146920520ations @ ( word @ A13 ) ) ) ).
thf(tcon_Word_Oword___Quickcheck__Narrowing_Opartial__term__of_377,axiom,
! [A13: $tType] :
( ( typerep @ A13 )
=> ( quickc6926020345158392990erm_of @ ( word @ A13 ) ) ) ).
thf(tcon_Word_Oword___Bit__Operations_Oring__bit__operations_378,axiom,
! [A13: $tType] :
( ( type_len @ A13 )
=> ( bit_ri3973907225187159222ations @ ( word @ A13 ) ) ) ).
thf(tcon_Word_Oword___Groups_Ocancel__ab__semigroup__add_379,axiom,
! [A13: $tType] :
( ( type_len @ A13 )
=> ( cancel2418104881723323429up_add @ ( word @ A13 ) ) ) ).
thf(tcon_Word_Oword___Groups_Ocancel__comm__monoid__add_380,axiom,
! [A13: $tType] :
( ( type_len @ A13 )
=> ( cancel1802427076303600483id_add @ ( word @ A13 ) ) ) ).
thf(tcon_Word_Oword___Rings_Ocomm__semiring__1__cancel_381,axiom,
! [A13: $tType] :
( ( type_len @ A13 )
=> ( comm_s4317794764714335236cancel @ ( word @ A13 ) ) ) ).
thf(tcon_Word_Oword___Bit__Operations_Osemiring__bits_382,axiom,
! [A13: $tType] :
( ( type_len @ A13 )
=> ( bit_semiring_bits @ ( word @ A13 ) ) ) ).
thf(tcon_Word_Oword___Groups_Ocancel__semigroup__add_383,axiom,
! [A13: $tType] :
( ( type_len @ A13 )
=> ( cancel_semigroup_add @ ( word @ A13 ) ) ) ).
thf(tcon_Word_Oword___Least__significant__bit_Olsb_384,axiom,
! [A13: $tType] :
( ( type_len @ A13 )
=> ( least_6119777620449941438nt_lsb @ ( word @ A13 ) ) ) ).
thf(tcon_Word_Oword___Groups_Oab__semigroup__mult_385,axiom,
! [A13: $tType] :
( ( type_len @ A13 )
=> ( ab_semigroup_mult @ ( word @ A13 ) ) ) ).
thf(tcon_Word_Oword___Rings_Osemiring__1__cancel_386,axiom,
! [A13: $tType] :
( ( type_len @ A13 )
=> ( semiring_1_cancel @ ( word @ A13 ) ) ) ).
thf(tcon_Word_Oword___Groups_Ocomm__monoid__mult_387,axiom,
! [A13: $tType] :
( ( type_len @ A13 )
=> ( comm_monoid_mult @ ( word @ A13 ) ) ) ).
thf(tcon_Word_Oword___Groups_Oab__semigroup__add_388,axiom,
! [A13: $tType] :
( ( type_len @ A13 )
=> ( ab_semigroup_add @ ( word @ A13 ) ) ) ).
thf(tcon_Word_Oword___Generic__set__bit_Oset__bit_389,axiom,
! [A13: $tType] :
( ( type_len @ A13 )
=> ( generic_set_set_bit @ ( word @ A13 ) ) ) ).
thf(tcon_Word_Oword___Code__Evaluation_Oterm__of_390,axiom,
! [A13: $tType] :
( ( typerep @ A13 )
=> ( code_term_of @ ( word @ A13 ) ) ) ).
thf(tcon_Word_Oword___Parity_Osemiring__parity_391,axiom,
! [A13: $tType] :
( ( type_len @ A13 )
=> ( semiring_parity @ ( word @ A13 ) ) ) ).
thf(tcon_Word_Oword___Groups_Ocomm__monoid__add_392,axiom,
! [A13: $tType] :
( ( type_len @ A13 )
=> ( comm_monoid_add @ ( word @ A13 ) ) ) ).
thf(tcon_Word_Oword___Rings_Osemiring__modulo_393,axiom,
! [A13: $tType] :
( ( type_len @ A13 )
=> ( semiring_modulo @ ( word @ A13 ) ) ) ).
thf(tcon_Word_Oword___Rings_Ocomm__semiring__1_394,axiom,
! [A13: $tType] :
( ( type_len @ A13 )
=> ( comm_semiring_1 @ ( word @ A13 ) ) ) ).
thf(tcon_Word_Oword___Rings_Ocomm__semiring__0_395,axiom,
! [A13: $tType] :
( ( type_len @ A13 )
=> ( comm_semiring_0 @ ( word @ A13 ) ) ) ).
thf(tcon_Word_Oword___Groups_Osemigroup__mult_396,axiom,
! [A13: $tType] :
( ( type_len @ A13 )
=> ( semigroup_mult @ ( word @ A13 ) ) ) ).
thf(tcon_Word_Oword___Num_Osemiring__numeral_397,axiom,
! [A13: $tType] :
( ( type_len @ A13 )
=> ( semiring_numeral @ ( word @ A13 ) ) ) ).
thf(tcon_Word_Oword___Groups_Osemigroup__add_398,axiom,
! [A13: $tType] :
( ( type_len @ A13 )
=> ( semigroup_add @ ( word @ A13 ) ) ) ).
thf(tcon_Word_Oword___Rings_Ocomm__semiring_399,axiom,
! [A13: $tType] :
( ( type_len @ A13 )
=> ( comm_semiring @ ( word @ A13 ) ) ) ).
thf(tcon_Word_Oword___Orderings_Owellorder_400,axiom,
! [A13: $tType] :
( ( type_len @ A13 )
=> ( wellorder @ ( word @ A13 ) ) ) ).
thf(tcon_Word_Oword___Groups_Oab__group__add_401,axiom,
! [A13: $tType] :
( ( type_len @ A13 )
=> ( ab_group_add @ ( word @ A13 ) ) ) ).
thf(tcon_Word_Oword___Rings_Ozero__neq__one_402,axiom,
! [A13: $tType] :
( ( type_len @ A13 )
=> ( zero_neq_one @ ( word @ A13 ) ) ) ).
thf(tcon_Word_Oword___Parity_Oring__parity_403,axiom,
! [A13: $tType] :
( ( type_len @ A13 )
=> ( ring_parity @ ( word @ A13 ) ) ) ).
thf(tcon_Word_Oword___Orderings_Opreorder_404,axiom,
! [A13: $tType] :
( ( type_len @ A13 )
=> ( preorder @ ( word @ A13 ) ) ) ).
thf(tcon_Word_Oword___Orderings_Olinorder_405,axiom,
! [A13: $tType] :
( ( type_len @ A13 )
=> ( linorder @ ( word @ A13 ) ) ) ).
thf(tcon_Word_Oword___Groups_Omonoid__mult_406,axiom,
! [A13: $tType] :
( ( type_len @ A13 )
=> ( monoid_mult @ ( word @ A13 ) ) ) ).
thf(tcon_Word_Oword___Rings_Ocomm__ring__1_407,axiom,
! [A13: $tType] :
( ( type_len @ A13 )
=> ( comm_ring_1 @ ( word @ A13 ) ) ) ).
thf(tcon_Word_Oword___Groups_Omonoid__add_408,axiom,
! [A13: $tType] :
( ( type_len @ A13 )
=> ( monoid_add @ ( word @ A13 ) ) ) ).
thf(tcon_Word_Oword___Finite__Set_Ofinite_409,axiom,
! [A13: $tType] :
( ( type_len @ A13 )
=> ( finite_finite @ ( word @ A13 ) ) ) ).
thf(tcon_Word_Oword___Rings_Osemiring__1_410,axiom,
! [A13: $tType] :
( ( type_len @ A13 )
=> ( semiring_1 @ ( word @ A13 ) ) ) ).
thf(tcon_Word_Oword___Rings_Osemiring__0_411,axiom,
! [A13: $tType] :
( ( type_len @ A13 )
=> ( semiring_0 @ ( word @ A13 ) ) ) ).
thf(tcon_Word_Oword___Groups_Ogroup__add_412,axiom,
! [A13: $tType] :
( ( type_len @ A13 )
=> ( group_add @ ( word @ A13 ) ) ) ).
thf(tcon_Word_Oword___Rings_Omult__zero_413,axiom,
! [A13: $tType] :
( ( type_len @ A13 )
=> ( mult_zero @ ( word @ A13 ) ) ) ).
thf(tcon_Word_Oword___Rings_Ocomm__ring_414,axiom,
! [A13: $tType] :
( ( type_len @ A13 )
=> ( comm_ring @ ( word @ A13 ) ) ) ).
thf(tcon_Word_Oword___Orderings_Oorder_415,axiom,
! [A13: $tType] :
( ( type_len @ A13 )
=> ( order @ ( word @ A13 ) ) ) ).
thf(tcon_Word_Oword___Num_Oneg__numeral_416,axiom,
! [A13: $tType] :
( ( type_len @ A13 )
=> ( neg_numeral @ ( word @ A13 ) ) ) ).
thf(tcon_Word_Oword___Rings_Osemiring_417,axiom,
! [A13: $tType] :
( ( type_len @ A13 )
=> ( semiring @ ( word @ A13 ) ) ) ).
thf(tcon_Word_Oword___Orderings_Oord_418,axiom,
! [A13: $tType] :
( ( type_len @ A13 )
=> ( ord @ ( word @ A13 ) ) ) ).
thf(tcon_Word_Oword___Groups_Ouminus_419,axiom,
! [A13: $tType] :
( ( type_len @ A13 )
=> ( uminus @ ( word @ A13 ) ) ) ).
thf(tcon_Word_Oword___Rings_Oring__1_420,axiom,
! [A13: $tType] :
( ( type_len @ A13 )
=> ( ring_1 @ ( word @ A13 ) ) ) ).
thf(tcon_Word_Oword___Groups_Ominus_421,axiom,
! [A13: $tType] :
( ( type_len @ A13 )
=> ( minus @ ( word @ A13 ) ) ) ).
thf(tcon_Word_Oword___Power_Opower_422,axiom,
! [A13: $tType] :
( ( type_len @ A13 )
=> ( power @ ( word @ A13 ) ) ) ).
thf(tcon_Word_Oword___Num_Onumeral_423,axiom,
! [A13: $tType] :
( ( type_len @ A13 )
=> ( numeral @ ( word @ A13 ) ) ) ).
thf(tcon_Word_Oword___Groups_Ozero_424,axiom,
! [A13: $tType] :
( ( type_len @ A13 )
=> ( zero @ ( word @ A13 ) ) ) ).
thf(tcon_Word_Oword___Groups_Oplus_425,axiom,
! [A13: $tType] :
( ( type_len @ A13 )
=> ( plus @ ( word @ A13 ) ) ) ).
thf(tcon_Word_Oword___Rings_Oring_426,axiom,
! [A13: $tType] :
( ( type_len @ A13 )
=> ( ring @ ( word @ A13 ) ) ) ).
thf(tcon_Word_Oword___Groups_Oone_427,axiom,
! [A13: $tType] :
( ( type_len @ A13 )
=> ( one @ ( word @ A13 ) ) ) ).
thf(tcon_Word_Oword___Rings_Odvd_428,axiom,
! [A13: $tType] :
( ( type_len @ A13 )
=> ( dvd @ ( word @ A13 ) ) ) ).
thf(tcon_Word_Oword___Nat_Osize_429,axiom,
! [A13: $tType] :
( ( type_len @ A13 )
=> ( size @ ( word @ A13 ) ) ) ).
thf(tcon_Heap_Oarray___Quickcheck__Narrowing_Opartial__term__of_430,axiom,
! [A13: $tType] :
( ( typerep @ A13 )
=> ( quickc6926020345158392990erm_of @ ( array @ A13 ) ) ) ).
thf(tcon_Heap_Oarray___Code__Evaluation_Oterm__of_431,axiom,
! [A13: $tType] :
( ( typerep @ A13 )
=> ( code_term_of @ ( array @ A13 ) ) ) ).
thf(tcon_Heap_Oarray___Heap_Oheap_432,axiom,
! [A13: $tType] : ( heap @ ( array @ A13 ) ) ).
thf(tcon_Heap_Oarray___Nat_Osize_433,axiom,
! [A13: $tType] : ( size @ ( array @ A13 ) ) ).
thf(tcon_Sum__Type_Osum___Quickcheck__Narrowing_Opartial__term__of_434,axiom,
! [A13: $tType,A15: $tType] :
( ( ( typerep @ A13 )
& ( typerep @ A15 ) )
=> ( quickc6926020345158392990erm_of @ ( sum_sum @ A13 @ A15 ) ) ) ).
thf(tcon_Sum__Type_Osum___Code__Evaluation_Oterm__of_435,axiom,
! [A13: $tType,A15: $tType] :
( ( ( typerep @ A13 )
& ( typerep @ A15 ) )
=> ( code_term_of @ ( sum_sum @ A13 @ A15 ) ) ) ).
thf(tcon_Sum__Type_Osum___Finite__Set_Ofinite_436,axiom,
! [A13: $tType,A15: $tType] :
( ( ( finite_finite @ A13 )
& ( finite_finite @ A15 ) )
=> ( finite_finite @ ( sum_sum @ A13 @ A15 ) ) ) ).
thf(tcon_Sum__Type_Osum___Heap_Oheap_437,axiom,
! [A13: $tType,A15: $tType] :
( ( ( heap @ A13 )
& ( heap @ A15 ) )
=> ( heap @ ( sum_sum @ A13 @ A15 ) ) ) ).
thf(tcon_Sum__Type_Osum___Nat_Osize_438,axiom,
! [A13: $tType,A15: $tType] : ( size @ ( sum_sum @ A13 @ A15 ) ) ).
thf(tcon_Enum_Ofinite__1___Conditionally__Complete__Lattices_Oconditionally__complete__linorder_439,axiom,
condit6923001295902523014norder @ finite_1 ).
thf(tcon_Enum_Ofinite__1___Conditionally__Complete__Lattices_Oconditionally__complete__lattice_440,axiom,
condit1219197933456340205attice @ finite_1 ).
thf(tcon_Enum_Ofinite__1___Countable__Complete__Lattices_Ocountable__complete__lattice_441,axiom,
counta3822494911875563373attice @ finite_1 ).
thf(tcon_Enum_Ofinite__1___Groups_Oordered__ab__semigroup__monoid__add__imp__le_442,axiom,
ordere1937475149494474687imp_le @ finite_1 ).
thf(tcon_Enum_Ofinite__1___Rings_Osemiring__no__zero__divisors__cancel_443,axiom,
semiri6575147826004484403cancel @ finite_1 ).
thf(tcon_Enum_Ofinite__1___Groups_Ostrict__ordered__ab__semigroup__add_444,axiom,
strict9044650504122735259up_add @ finite_1 ).
thf(tcon_Enum_Ofinite__1___Groups_Oordered__cancel__comm__monoid__diff_445,axiom,
ordere1170586879665033532d_diff @ finite_1 ).
thf(tcon_Enum_Ofinite__1___Groups_Oordered__cancel__ab__semigroup__add_446,axiom,
ordere580206878836729694up_add @ finite_1 ).
thf(tcon_Enum_Ofinite__1___Groups_Oordered__ab__semigroup__add__imp__le_447,axiom,
ordere2412721322843649153imp_le @ finite_1 ).
thf(tcon_Enum_Ofinite__1___Rings_Olinordered__comm__semiring__strict_448,axiom,
linord2810124833399127020strict @ finite_1 ).
thf(tcon_Enum_Ofinite__1___Groups_Ostrict__ordered__comm__monoid__add_449,axiom,
strict7427464778891057005id_add @ finite_1 ).
thf(tcon_Enum_Ofinite__1___Groups_Oordered__cancel__comm__monoid__add_450,axiom,
ordere8940638589300402666id_add @ finite_1 ).
thf(tcon_Enum_Ofinite__1___Groups_Ocanonically__ordered__monoid__add_451,axiom,
canoni5634975068530333245id_add @ finite_1 ).
thf(tcon_Enum_Ofinite__1___Quickcheck__Narrowing_Opartial__term__of_452,axiom,
quickc6926020345158392990erm_of @ finite_1 ).
thf(tcon_Enum_Ofinite__1___Complete__Lattices_Ocomplete__linorder,axiom,
comple5582772986160207858norder @ finite_1 ).
thf(tcon_Enum_Ofinite__1___Groups_Olinordered__ab__semigroup__add_453,axiom,
linord4140545234300271783up_add @ finite_1 ).
thf(tcon_Enum_Ofinite__1___Complete__Lattices_Ocomplete__lattice_454,axiom,
comple6319245703460814977attice @ finite_1 ).
thf(tcon_Enum_Ofinite__1___Rings_Olinordered__semiring__strict_455,axiom,
linord8928482502909563296strict @ finite_1 ).
thf(tcon_Enum_Ofinite__1___Boolean__Algebras_Oboolean__algebra_456,axiom,
boolea8198339166811842893lgebra @ finite_1 ).
thf(tcon_Enum_Ofinite__1___Rings_Osemiring__no__zero__divisors_457,axiom,
semiri3467727345109120633visors @ finite_1 ).
thf(tcon_Enum_Ofinite__1___Groups_Oordered__ab__semigroup__add_458,axiom,
ordere6658533253407199908up_add @ finite_1 ).
thf(tcon_Enum_Ofinite__1___Groups_Oordered__ab__group__add__abs_459,axiom,
ordere166539214618696060dd_abs @ finite_1 ).
thf(tcon_Enum_Ofinite__1___Groups_Oordered__comm__monoid__add_460,axiom,
ordere6911136660526730532id_add @ finite_1 ).
thf(tcon_Enum_Ofinite__1___Groups_Olinordered__ab__group__add_461,axiom,
linord5086331880401160121up_add @ finite_1 ).
thf(tcon_Enum_Ofinite__1___Groups_Ocancel__ab__semigroup__add_462,axiom,
cancel2418104881723323429up_add @ finite_1 ).
thf(tcon_Enum_Ofinite__1___Groups_Ocancel__comm__monoid__add_463,axiom,
cancel1802427076303600483id_add @ finite_1 ).
thf(tcon_Enum_Ofinite__1___Rings_Olinordered__ring__strict_464,axiom,
linord4710134922213307826strict @ finite_1 ).
thf(tcon_Enum_Ofinite__1___Rings_Oordered__comm__semiring_465,axiom,
ordere2520102378445227354miring @ finite_1 ).
thf(tcon_Enum_Ofinite__1___Groups_Oordered__ab__group__add_466,axiom,
ordered_ab_group_add @ finite_1 ).
thf(tcon_Enum_Ofinite__1___Groups_Ocancel__semigroup__add_467,axiom,
cancel_semigroup_add @ finite_1 ).
thf(tcon_Enum_Ofinite__1___Rings_Olinordered__semiring_468,axiom,
linordered_semiring @ finite_1 ).
thf(tcon_Enum_Ofinite__1___Rings_Oordered__semiring__0_469,axiom,
ordered_semiring_0 @ finite_1 ).
thf(tcon_Enum_Ofinite__1___Orderings_Odense__linorder_470,axiom,
dense_linorder @ finite_1 ).
thf(tcon_Enum_Ofinite__1___Groups_Oab__semigroup__mult_471,axiom,
ab_semigroup_mult @ finite_1 ).
thf(tcon_Enum_Ofinite__1___Groups_Ocomm__monoid__mult_472,axiom,
comm_monoid_mult @ finite_1 ).
thf(tcon_Enum_Ofinite__1___Groups_Ocomm__monoid__diff_473,axiom,
comm_monoid_diff @ finite_1 ).
thf(tcon_Enum_Ofinite__1___Groups_Oab__semigroup__add_474,axiom,
ab_semigroup_add @ finite_1 ).
thf(tcon_Enum_Ofinite__1___Code__Evaluation_Oterm__of_475,axiom,
code_term_of @ finite_1 ).
thf(tcon_Enum_Ofinite__1___Rings_Oordered__semiring_476,axiom,
ordered_semiring @ finite_1 ).
thf(tcon_Enum_Ofinite__1___Rings_Oordered__ring__abs_477,axiom,
ordered_ring_abs @ finite_1 ).
thf(tcon_Enum_Ofinite__1___Groups_Ocomm__monoid__add_478,axiom,
comm_monoid_add @ finite_1 ).
thf(tcon_Enum_Ofinite__1___Rings_Olinordered__ring_479,axiom,
linordered_ring @ finite_1 ).
thf(tcon_Enum_Ofinite__1___Rings_Ocomm__semiring__0_480,axiom,
comm_semiring_0 @ finite_1 ).
thf(tcon_Enum_Ofinite__1___Orderings_Odense__order_481,axiom,
dense_order @ finite_1 ).
thf(tcon_Enum_Ofinite__1___Groups_Osemigroup__mult_482,axiom,
semigroup_mult @ finite_1 ).
thf(tcon_Enum_Ofinite__1___Complete__Lattices_OSup_483,axiom,
complete_Sup @ finite_1 ).
thf(tcon_Enum_Ofinite__1___Complete__Lattices_OInf_484,axiom,
complete_Inf @ finite_1 ).
thf(tcon_Enum_Ofinite__1___Groups_Osemigroup__add_485,axiom,
semigroup_add @ finite_1 ).
thf(tcon_Enum_Ofinite__1___Rings_Ocomm__semiring_486,axiom,
comm_semiring @ finite_1 ).
thf(tcon_Enum_Ofinite__1___Orderings_Owellorder_487,axiom,
wellorder @ finite_1 ).
thf(tcon_Enum_Ofinite__1___Orderings_Oorder__top_488,axiom,
order_top @ finite_1 ).
thf(tcon_Enum_Ofinite__1___Orderings_Oorder__bot_489,axiom,
order_bot @ finite_1 ).
thf(tcon_Enum_Ofinite__1___Groups_Oab__group__add_490,axiom,
ab_group_add @ finite_1 ).
thf(tcon_Enum_Ofinite__1___Rings_Oordered__ring_491,axiom,
ordered_ring @ finite_1 ).
thf(tcon_Enum_Ofinite__1___Orderings_Opreorder_492,axiom,
preorder @ finite_1 ).
thf(tcon_Enum_Ofinite__1___Orderings_Olinorder_493,axiom,
linorder @ finite_1 ).
thf(tcon_Enum_Ofinite__1___Groups_Omonoid__mult_494,axiom,
monoid_mult @ finite_1 ).
thf(tcon_Enum_Ofinite__1___Groups_Omonoid__add_495,axiom,
monoid_add @ finite_1 ).
thf(tcon_Enum_Ofinite__1___Finite__Set_Ofinite_496,axiom,
finite_finite @ finite_1 ).
thf(tcon_Enum_Ofinite__1___Type__Length_Olen0,axiom,
type_len0 @ finite_1 ).
thf(tcon_Enum_Ofinite__1___Rings_Osemiring__0_497,axiom,
semiring_0 @ finite_1 ).
thf(tcon_Enum_Ofinite__1___Groups_Ogroup__add_498,axiom,
group_add @ finite_1 ).
thf(tcon_Enum_Ofinite__1___Type__Length_Olen,axiom,
type_len @ finite_1 ).
thf(tcon_Enum_Ofinite__1___Rings_Omult__zero_499,axiom,
mult_zero @ finite_1 ).
thf(tcon_Enum_Ofinite__1___Rings_Ocomm__ring_500,axiom,
comm_ring @ finite_1 ).
thf(tcon_Enum_Ofinite__1___Orderings_Oorder_501,axiom,
order @ finite_1 ).
thf(tcon_Enum_Ofinite__1___Rings_Osemiring_502,axiom,
semiring @ finite_1 ).
thf(tcon_Enum_Ofinite__1___Fields_Oinverse_503,axiom,
inverse @ finite_1 ).
thf(tcon_Enum_Ofinite__1___Orderings_Oord_504,axiom,
ord @ finite_1 ).
thf(tcon_Enum_Ofinite__1___Groups_Ouminus_505,axiom,
uminus @ finite_1 ).
thf(tcon_Enum_Ofinite__1___Rings_Oabs__if_506,axiom,
abs_if @ finite_1 ).
thf(tcon_Enum_Ofinite__1___Groups_Ominus_507,axiom,
minus @ finite_1 ).
thf(tcon_Enum_Ofinite__1___Power_Opower_508,axiom,
power @ finite_1 ).
thf(tcon_Enum_Ofinite__1___Groups_Ozero_509,axiom,
zero @ finite_1 ).
thf(tcon_Enum_Ofinite__1___Groups_Oplus_510,axiom,
plus @ finite_1 ).
thf(tcon_Enum_Ofinite__1___Rings_Oring_511,axiom,
ring @ finite_1 ).
thf(tcon_Enum_Ofinite__1___Groups_Oone_512,axiom,
one @ finite_1 ).
thf(tcon_Enum_Ofinite__1___Rings_Odvd_513,axiom,
dvd @ finite_1 ).
thf(tcon_Enum_Ofinite__2___Conditionally__Complete__Lattices_Oconditionally__complete__linorder_514,axiom,
condit6923001295902523014norder @ finite_2 ).
thf(tcon_Enum_Ofinite__2___Conditionally__Complete__Lattices_Oconditionally__complete__lattice_515,axiom,
condit1219197933456340205attice @ finite_2 ).
thf(tcon_Enum_Ofinite__2___Semiring__Normalization_Ocomm__semiring__1__cancel__crossproduct_516,axiom,
semiri1453513574482234551roduct @ finite_2 ).
thf(tcon_Enum_Ofinite__2___Countable__Complete__Lattices_Ocountable__complete__lattice_517,axiom,
counta3822494911875563373attice @ finite_2 ).
thf(tcon_Enum_Ofinite__2___Euclidean__Division_Ounique__euclidean__semiring_518,axiom,
euclid3128863361964157862miring @ finite_2 ).
thf(tcon_Enum_Ofinite__2___Euclidean__Division_Oeuclidean__semiring__cancel_519,axiom,
euclid4440199948858584721cancel @ finite_2 ).
thf(tcon_Enum_Ofinite__2___Rings_Osemiring__no__zero__divisors__cancel_520,axiom,
semiri6575147826004484403cancel @ finite_2 ).
thf(tcon_Enum_Ofinite__2___Euclidean__Division_Oeuclidean__semiring_521,axiom,
euclid3725896446679973847miring @ finite_2 ).
thf(tcon_Enum_Ofinite__2___Quickcheck__Narrowing_Opartial__term__of_522,axiom,
quickc6926020345158392990erm_of @ finite_2 ).
thf(tcon_Enum_Ofinite__2___Complete__Lattices_Ocomplete__linorder_523,axiom,
comple5582772986160207858norder @ finite_2 ).
thf(tcon_Enum_Ofinite__2___Complete__Lattices_Ocomplete__lattice_524,axiom,
comple6319245703460814977attice @ finite_2 ).
thf(tcon_Enum_Ofinite__2___Rings_Osemiring__1__no__zero__divisors_525,axiom,
semiri2026040879449505780visors @ finite_2 ).
thf(tcon_Enum_Ofinite__2___Rings_Osemiring__no__zero__divisors_526,axiom,
semiri3467727345109120633visors @ finite_2 ).
thf(tcon_Enum_Ofinite__2___Groups_Ocancel__ab__semigroup__add_527,axiom,
cancel2418104881723323429up_add @ finite_2 ).
thf(tcon_Enum_Ofinite__2___Rings_Oring__1__no__zero__divisors_528,axiom,
ring_15535105094025558882visors @ finite_2 ).
thf(tcon_Enum_Ofinite__2___Groups_Ocancel__comm__monoid__add_529,axiom,
cancel1802427076303600483id_add @ finite_2 ).
thf(tcon_Enum_Ofinite__2___Rings_Ocomm__semiring__1__cancel_530,axiom,
comm_s4317794764714335236cancel @ finite_2 ).
thf(tcon_Enum_Ofinite__2___Groups_Ocancel__semigroup__add_531,axiom,
cancel_semigroup_add @ finite_2 ).
thf(tcon_Enum_Ofinite__2___Groups_Oab__semigroup__mult_532,axiom,
ab_semigroup_mult @ finite_2 ).
thf(tcon_Enum_Ofinite__2___Rings_Osemiring__1__cancel_533,axiom,
semiring_1_cancel @ finite_2 ).
thf(tcon_Enum_Ofinite__2___Rings_Oalgebraic__semidom_534,axiom,
algebraic_semidom @ finite_2 ).
thf(tcon_Enum_Ofinite__2___Groups_Ocomm__monoid__mult_535,axiom,
comm_monoid_mult @ finite_2 ).
thf(tcon_Enum_Ofinite__2___Groups_Oab__semigroup__add_536,axiom,
ab_semigroup_add @ finite_2 ).
thf(tcon_Enum_Ofinite__2___Code__Evaluation_Oterm__of_537,axiom,
code_term_of @ finite_2 ).
thf(tcon_Enum_Ofinite__2___Groups_Ocomm__monoid__add_538,axiom,
comm_monoid_add @ finite_2 ).
thf(tcon_Enum_Ofinite__2___Rings_Osemiring__modulo_539,axiom,
semiring_modulo @ finite_2 ).
thf(tcon_Enum_Ofinite__2___Rings_Ocomm__semiring__1_540,axiom,
comm_semiring_1 @ finite_2 ).
thf(tcon_Enum_Ofinite__2___Rings_Ocomm__semiring__0_541,axiom,
comm_semiring_0 @ finite_2 ).
thf(tcon_Enum_Ofinite__2___Groups_Osemigroup__mult_542,axiom,
semigroup_mult @ finite_2 ).
thf(tcon_Enum_Ofinite__2___Complete__Lattices_OSup_543,axiom,
complete_Sup @ finite_2 ).
thf(tcon_Enum_Ofinite__2___Complete__Lattices_OInf_544,axiom,
complete_Inf @ finite_2 ).
thf(tcon_Enum_Ofinite__2___Rings_Osemidom__modulo_545,axiom,
semidom_modulo @ finite_2 ).
thf(tcon_Enum_Ofinite__2___Rings_Osemidom__divide_546,axiom,
semidom_divide @ finite_2 ).
thf(tcon_Enum_Ofinite__2___Num_Osemiring__numeral_547,axiom,
semiring_numeral @ finite_2 ).
thf(tcon_Enum_Ofinite__2___Groups_Osemigroup__add_548,axiom,
semigroup_add @ finite_2 ).
thf(tcon_Enum_Ofinite__2___Fields_Odivision__ring_549,axiom,
division_ring @ finite_2 ).
thf(tcon_Enum_Ofinite__2___Rings_Ocomm__semiring_550,axiom,
comm_semiring @ finite_2 ).
thf(tcon_Enum_Ofinite__2___Orderings_Owellorder_551,axiom,
wellorder @ finite_2 ).
thf(tcon_Enum_Ofinite__2___Orderings_Oorder__top_552,axiom,
order_top @ finite_2 ).
thf(tcon_Enum_Ofinite__2___Orderings_Oorder__bot_553,axiom,
order_bot @ finite_2 ).
thf(tcon_Enum_Ofinite__2___Groups_Oab__group__add_554,axiom,
ab_group_add @ finite_2 ).
thf(tcon_Enum_Ofinite__2___Rings_Ozero__neq__one_555,axiom,
zero_neq_one @ finite_2 ).
thf(tcon_Enum_Ofinite__2___Rings_Oidom__abs__sgn_556,axiom,
idom_abs_sgn @ finite_2 ).
thf(tcon_Enum_Ofinite__2___Orderings_Opreorder_557,axiom,
preorder @ finite_2 ).
thf(tcon_Enum_Ofinite__2___Orderings_Olinorder_558,axiom,
linorder @ finite_2 ).
thf(tcon_Enum_Ofinite__2___Groups_Omonoid__mult_559,axiom,
monoid_mult @ finite_2 ).
thf(tcon_Enum_Ofinite__2___Rings_Oidom__modulo_560,axiom,
idom_modulo @ finite_2 ).
thf(tcon_Enum_Ofinite__2___Rings_Oidom__divide_561,axiom,
idom_divide @ finite_2 ).
thf(tcon_Enum_Ofinite__2___Rings_Ocomm__ring__1_562,axiom,
comm_ring_1 @ finite_2 ).
thf(tcon_Enum_Ofinite__2___Groups_Omonoid__add_563,axiom,
monoid_add @ finite_2 ).
thf(tcon_Enum_Ofinite__2___Finite__Set_Ofinite_564,axiom,
finite_finite @ finite_2 ).
thf(tcon_Enum_Ofinite__2___Type__Length_Olen0_565,axiom,
type_len0 @ finite_2 ).
thf(tcon_Enum_Ofinite__2___Rings_Osemiring__1_566,axiom,
semiring_1 @ finite_2 ).
thf(tcon_Enum_Ofinite__2___Rings_Osemiring__0_567,axiom,
semiring_0 @ finite_2 ).
thf(tcon_Enum_Ofinite__2___Groups_Ogroup__add_568,axiom,
group_add @ finite_2 ).
thf(tcon_Enum_Ofinite__2___Type__Length_Olen_569,axiom,
type_len @ finite_2 ).
thf(tcon_Enum_Ofinite__2___Rings_Omult__zero_570,axiom,
mult_zero @ finite_2 ).
thf(tcon_Enum_Ofinite__2___Rings_Ocomm__ring_571,axiom,
comm_ring @ finite_2 ).
thf(tcon_Enum_Ofinite__2___Orderings_Oorder_572,axiom,
order @ finite_2 ).
thf(tcon_Enum_Ofinite__2___Num_Oneg__numeral_573,axiom,
neg_numeral @ finite_2 ).
thf(tcon_Enum_Ofinite__2___Rings_Osemiring_574,axiom,
semiring @ finite_2 ).
thf(tcon_Enum_Ofinite__2___Fields_Oinverse_575,axiom,
inverse @ finite_2 ).
thf(tcon_Enum_Ofinite__2___Orderings_Oord_576,axiom,
ord @ finite_2 ).
thf(tcon_Enum_Ofinite__2___Groups_Ouminus_577,axiom,
uminus @ finite_2 ).
thf(tcon_Enum_Ofinite__2___Rings_Oring__1_578,axiom,
ring_1 @ finite_2 ).
thf(tcon_Enum_Ofinite__2___Groups_Ominus_579,axiom,
minus @ finite_2 ).
thf(tcon_Enum_Ofinite__2___Fields_Ofield_580,axiom,
field @ finite_2 ).
thf(tcon_Enum_Ofinite__2___Power_Opower_581,axiom,
power @ finite_2 ).
thf(tcon_Enum_Ofinite__2___Num_Onumeral_582,axiom,
numeral @ finite_2 ).
thf(tcon_Enum_Ofinite__2___Groups_Ozero_583,axiom,
zero @ finite_2 ).
thf(tcon_Enum_Ofinite__2___Groups_Oplus_584,axiom,
plus @ finite_2 ).
thf(tcon_Enum_Ofinite__2___Rings_Oring_585,axiom,
ring @ finite_2 ).
thf(tcon_Enum_Ofinite__2___Rings_Oidom_586,axiom,
idom @ finite_2 ).
thf(tcon_Enum_Ofinite__2___Groups_Oone_587,axiom,
one @ finite_2 ).
thf(tcon_Enum_Ofinite__2___Rings_Odvd_588,axiom,
dvd @ finite_2 ).
thf(tcon_Enum_Ofinite__3___Conditionally__Complete__Lattices_Oconditionally__complete__linorder_589,axiom,
condit6923001295902523014norder @ finite_3 ).
thf(tcon_Enum_Ofinite__3___Conditionally__Complete__Lattices_Oconditionally__complete__lattice_590,axiom,
condit1219197933456340205attice @ finite_3 ).
thf(tcon_Enum_Ofinite__3___Semiring__Normalization_Ocomm__semiring__1__cancel__crossproduct_591,axiom,
semiri1453513574482234551roduct @ finite_3 ).
thf(tcon_Enum_Ofinite__3___Countable__Complete__Lattices_Ocountable__complete__lattice_592,axiom,
counta3822494911875563373attice @ finite_3 ).
thf(tcon_Enum_Ofinite__3___Euclidean__Division_Ounique__euclidean__semiring_593,axiom,
euclid3128863361964157862miring @ finite_3 ).
thf(tcon_Enum_Ofinite__3___Euclidean__Division_Oeuclidean__semiring__cancel_594,axiom,
euclid4440199948858584721cancel @ finite_3 ).
thf(tcon_Enum_Ofinite__3___Rings_Osemiring__no__zero__divisors__cancel_595,axiom,
semiri6575147826004484403cancel @ finite_3 ).
thf(tcon_Enum_Ofinite__3___Euclidean__Division_Oeuclidean__semiring_596,axiom,
euclid3725896446679973847miring @ finite_3 ).
thf(tcon_Enum_Ofinite__3___Quickcheck__Narrowing_Opartial__term__of_597,axiom,
quickc6926020345158392990erm_of @ finite_3 ).
thf(tcon_Enum_Ofinite__3___Complete__Lattices_Ocomplete__linorder_598,axiom,
comple5582772986160207858norder @ finite_3 ).
thf(tcon_Enum_Ofinite__3___Complete__Lattices_Ocomplete__lattice_599,axiom,
comple6319245703460814977attice @ finite_3 ).
thf(tcon_Enum_Ofinite__3___Rings_Osemiring__1__no__zero__divisors_600,axiom,
semiri2026040879449505780visors @ finite_3 ).
thf(tcon_Enum_Ofinite__3___Rings_Osemiring__no__zero__divisors_601,axiom,
semiri3467727345109120633visors @ finite_3 ).
thf(tcon_Enum_Ofinite__3___Groups_Ocancel__ab__semigroup__add_602,axiom,
cancel2418104881723323429up_add @ finite_3 ).
thf(tcon_Enum_Ofinite__3___Rings_Oring__1__no__zero__divisors_603,axiom,
ring_15535105094025558882visors @ finite_3 ).
thf(tcon_Enum_Ofinite__3___Groups_Ocancel__comm__monoid__add_604,axiom,
cancel1802427076303600483id_add @ finite_3 ).
thf(tcon_Enum_Ofinite__3___Rings_Ocomm__semiring__1__cancel_605,axiom,
comm_s4317794764714335236cancel @ finite_3 ).
thf(tcon_Enum_Ofinite__3___Groups_Ocancel__semigroup__add_606,axiom,
cancel_semigroup_add @ finite_3 ).
thf(tcon_Enum_Ofinite__3___Groups_Oab__semigroup__mult_607,axiom,
ab_semigroup_mult @ finite_3 ).
thf(tcon_Enum_Ofinite__3___Rings_Osemiring__1__cancel_608,axiom,
semiring_1_cancel @ finite_3 ).
thf(tcon_Enum_Ofinite__3___Rings_Oalgebraic__semidom_609,axiom,
algebraic_semidom @ finite_3 ).
thf(tcon_Enum_Ofinite__3___Groups_Ocomm__monoid__mult_610,axiom,
comm_monoid_mult @ finite_3 ).
thf(tcon_Enum_Ofinite__3___Groups_Oab__semigroup__add_611,axiom,
ab_semigroup_add @ finite_3 ).
thf(tcon_Enum_Ofinite__3___Code__Evaluation_Oterm__of_612,axiom,
code_term_of @ finite_3 ).
thf(tcon_Enum_Ofinite__3___Groups_Ocomm__monoid__add_613,axiom,
comm_monoid_add @ finite_3 ).
thf(tcon_Enum_Ofinite__3___Rings_Osemiring__modulo_614,axiom,
semiring_modulo @ finite_3 ).
thf(tcon_Enum_Ofinite__3___Rings_Ocomm__semiring__1_615,axiom,
comm_semiring_1 @ finite_3 ).
thf(tcon_Enum_Ofinite__3___Rings_Ocomm__semiring__0_616,axiom,
comm_semiring_0 @ finite_3 ).
thf(tcon_Enum_Ofinite__3___Groups_Osemigroup__mult_617,axiom,
semigroup_mult @ finite_3 ).
thf(tcon_Enum_Ofinite__3___Complete__Lattices_OSup_618,axiom,
complete_Sup @ finite_3 ).
thf(tcon_Enum_Ofinite__3___Complete__Lattices_OInf_619,axiom,
complete_Inf @ finite_3 ).
thf(tcon_Enum_Ofinite__3___Rings_Osemidom__modulo_620,axiom,
semidom_modulo @ finite_3 ).
thf(tcon_Enum_Ofinite__3___Rings_Osemidom__divide_621,axiom,
semidom_divide @ finite_3 ).
thf(tcon_Enum_Ofinite__3___Num_Osemiring__numeral_622,axiom,
semiring_numeral @ finite_3 ).
thf(tcon_Enum_Ofinite__3___Groups_Osemigroup__add_623,axiom,
semigroup_add @ finite_3 ).
thf(tcon_Enum_Ofinite__3___Fields_Odivision__ring_624,axiom,
division_ring @ finite_3 ).
thf(tcon_Enum_Ofinite__3___Rings_Ocomm__semiring_625,axiom,
comm_semiring @ finite_3 ).
thf(tcon_Enum_Ofinite__3___Orderings_Owellorder_626,axiom,
wellorder @ finite_3 ).
thf(tcon_Enum_Ofinite__3___Orderings_Oorder__top_627,axiom,
order_top @ finite_3 ).
thf(tcon_Enum_Ofinite__3___Orderings_Oorder__bot_628,axiom,
order_bot @ finite_3 ).
thf(tcon_Enum_Ofinite__3___Groups_Oab__group__add_629,axiom,
ab_group_add @ finite_3 ).
thf(tcon_Enum_Ofinite__3___Rings_Ozero__neq__one_630,axiom,
zero_neq_one @ finite_3 ).
thf(tcon_Enum_Ofinite__3___Rings_Oidom__abs__sgn_631,axiom,
idom_abs_sgn @ finite_3 ).
thf(tcon_Enum_Ofinite__3___Orderings_Opreorder_632,axiom,
preorder @ finite_3 ).
thf(tcon_Enum_Ofinite__3___Orderings_Olinorder_633,axiom,
linorder @ finite_3 ).
thf(tcon_Enum_Ofinite__3___Groups_Omonoid__mult_634,axiom,
monoid_mult @ finite_3 ).
thf(tcon_Enum_Ofinite__3___Rings_Oidom__modulo_635,axiom,
idom_modulo @ finite_3 ).
thf(tcon_Enum_Ofinite__3___Rings_Oidom__divide_636,axiom,
idom_divide @ finite_3 ).
thf(tcon_Enum_Ofinite__3___Rings_Ocomm__ring__1_637,axiom,
comm_ring_1 @ finite_3 ).
thf(tcon_Enum_Ofinite__3___Groups_Omonoid__add_638,axiom,
monoid_add @ finite_3 ).
thf(tcon_Enum_Ofinite__3___Finite__Set_Ofinite_639,axiom,
finite_finite @ finite_3 ).
thf(tcon_Enum_Ofinite__3___Type__Length_Olen0_640,axiom,
type_len0 @ finite_3 ).
thf(tcon_Enum_Ofinite__3___Rings_Osemiring__1_641,axiom,
semiring_1 @ finite_3 ).
thf(tcon_Enum_Ofinite__3___Rings_Osemiring__0_642,axiom,
semiring_0 @ finite_3 ).
thf(tcon_Enum_Ofinite__3___Groups_Ogroup__add_643,axiom,
group_add @ finite_3 ).
thf(tcon_Enum_Ofinite__3___Type__Length_Olen_644,axiom,
type_len @ finite_3 ).
thf(tcon_Enum_Ofinite__3___Rings_Omult__zero_645,axiom,
mult_zero @ finite_3 ).
thf(tcon_Enum_Ofinite__3___Rings_Ocomm__ring_646,axiom,
comm_ring @ finite_3 ).
thf(tcon_Enum_Ofinite__3___Orderings_Oorder_647,axiom,
order @ finite_3 ).
thf(tcon_Enum_Ofinite__3___Num_Oneg__numeral_648,axiom,
neg_numeral @ finite_3 ).
thf(tcon_Enum_Ofinite__3___Rings_Osemiring_649,axiom,
semiring @ finite_3 ).
thf(tcon_Enum_Ofinite__3___Fields_Oinverse_650,axiom,
inverse @ finite_3 ).
thf(tcon_Enum_Ofinite__3___Orderings_Oord_651,axiom,
ord @ finite_3 ).
thf(tcon_Enum_Ofinite__3___Groups_Ouminus_652,axiom,
uminus @ finite_3 ).
thf(tcon_Enum_Ofinite__3___Rings_Oring__1_653,axiom,
ring_1 @ finite_3 ).
thf(tcon_Enum_Ofinite__3___Groups_Ominus_654,axiom,
minus @ finite_3 ).
thf(tcon_Enum_Ofinite__3___Fields_Ofield_655,axiom,
field @ finite_3 ).
thf(tcon_Enum_Ofinite__3___Power_Opower_656,axiom,
power @ finite_3 ).
thf(tcon_Enum_Ofinite__3___Num_Onumeral_657,axiom,
numeral @ finite_3 ).
thf(tcon_Enum_Ofinite__3___Groups_Ozero_658,axiom,
zero @ finite_3 ).
thf(tcon_Enum_Ofinite__3___Groups_Oplus_659,axiom,
plus @ finite_3 ).
thf(tcon_Enum_Ofinite__3___Rings_Oring_660,axiom,
ring @ finite_3 ).
thf(tcon_Enum_Ofinite__3___Rings_Oidom_661,axiom,
idom @ finite_3 ).
thf(tcon_Enum_Ofinite__3___Groups_Oone_662,axiom,
one @ finite_3 ).
thf(tcon_Enum_Ofinite__3___Rings_Odvd_663,axiom,
dvd @ finite_3 ).
thf(tcon_Filter_Ofilter___Conditionally__Complete__Lattices_Oconditionally__complete__lattice_664,axiom,
! [A13: $tType] : ( condit1219197933456340205attice @ ( filter @ A13 ) ) ).
thf(tcon_Filter_Ofilter___Countable__Complete__Lattices_Ocountable__complete__lattice_665,axiom,
! [A13: $tType] : ( counta3822494911875563373attice @ ( filter @ A13 ) ) ).
thf(tcon_Filter_Ofilter___Quickcheck__Narrowing_Opartial__term__of_666,axiom,
! [A13: $tType] :
( ( typerep @ A13 )
=> ( quickc6926020345158392990erm_of @ ( filter @ A13 ) ) ) ).
thf(tcon_Filter_Ofilter___Complete__Lattices_Ocomplete__lattice_667,axiom,
! [A13: $tType] : ( comple6319245703460814977attice @ ( filter @ A13 ) ) ).
thf(tcon_Filter_Ofilter___Code__Evaluation_Oterm__of_668,axiom,
! [A13: $tType] :
( ( typerep @ A13 )
=> ( code_term_of @ ( filter @ A13 ) ) ) ).
thf(tcon_Filter_Ofilter___Complete__Lattices_OSup_669,axiom,
! [A13: $tType] : ( complete_Sup @ ( filter @ A13 ) ) ).
thf(tcon_Filter_Ofilter___Complete__Lattices_OInf_670,axiom,
! [A13: $tType] : ( complete_Inf @ ( filter @ A13 ) ) ).
thf(tcon_Filter_Ofilter___Orderings_Oorder__top_671,axiom,
! [A13: $tType] : ( order_top @ ( filter @ A13 ) ) ).
thf(tcon_Filter_Ofilter___Orderings_Oorder__bot_672,axiom,
! [A13: $tType] : ( order_bot @ ( filter @ A13 ) ) ).
thf(tcon_Filter_Ofilter___Orderings_Opreorder_673,axiom,
! [A13: $tType] : ( preorder @ ( filter @ A13 ) ) ).
thf(tcon_Filter_Ofilter___Orderings_Oorder_674,axiom,
! [A13: $tType] : ( order @ ( filter @ A13 ) ) ).
thf(tcon_Filter_Ofilter___Orderings_Oord_675,axiom,
! [A13: $tType] : ( ord @ ( filter @ A13 ) ) ).
thf(tcon_Option_Ooption___Conditionally__Complete__Lattices_Oconditionally__complete__linorder_676,axiom,
! [A13: $tType] :
( ( comple5582772986160207858norder @ A13 )
=> ( condit6923001295902523014norder @ ( option @ A13 ) ) ) ).
thf(tcon_Option_Ooption___Conditionally__Complete__Lattices_Oconditionally__complete__lattice_677,axiom,
! [A13: $tType] :
( ( comple6319245703460814977attice @ A13 )
=> ( condit1219197933456340205attice @ ( option @ A13 ) ) ) ).
thf(tcon_Option_Ooption___Countable__Complete__Lattices_Ocountable__complete__lattice_678,axiom,
! [A13: $tType] :
( ( comple6319245703460814977attice @ A13 )
=> ( counta3822494911875563373attice @ ( option @ A13 ) ) ) ).
thf(tcon_Option_Ooption___Quickcheck__Narrowing_Opartial__term__of_679,axiom,
! [A13: $tType] :
( ( typerep @ A13 )
=> ( quickc6926020345158392990erm_of @ ( option @ A13 ) ) ) ).
thf(tcon_Option_Ooption___Complete__Lattices_Ocomplete__linorder_680,axiom,
! [A13: $tType] :
( ( comple5582772986160207858norder @ A13 )
=> ( comple5582772986160207858norder @ ( option @ A13 ) ) ) ).
thf(tcon_Option_Ooption___Complete__Lattices_Ocomplete__lattice_681,axiom,
! [A13: $tType] :
( ( comple6319245703460814977attice @ A13 )
=> ( comple6319245703460814977attice @ ( option @ A13 ) ) ) ).
thf(tcon_Option_Ooption___Code__Evaluation_Oterm__of_682,axiom,
! [A13: $tType] :
( ( typerep @ A13 )
=> ( code_term_of @ ( option @ A13 ) ) ) ).
thf(tcon_Option_Ooption___Complete__Lattices_OSup_683,axiom,
! [A13: $tType] :
( ( comple6319245703460814977attice @ A13 )
=> ( complete_Sup @ ( option @ A13 ) ) ) ).
thf(tcon_Option_Ooption___Complete__Lattices_OInf_684,axiom,
! [A13: $tType] :
( ( comple6319245703460814977attice @ A13 )
=> ( complete_Inf @ ( option @ A13 ) ) ) ).
thf(tcon_Option_Ooption___Orderings_Owellorder_685,axiom,
! [A13: $tType] :
( ( wellorder @ A13 )
=> ( wellorder @ ( option @ A13 ) ) ) ).
thf(tcon_Option_Ooption___Orderings_Oorder__top_686,axiom,
! [A13: $tType] :
( ( order_top @ A13 )
=> ( order_top @ ( option @ A13 ) ) ) ).
thf(tcon_Option_Ooption___Orderings_Oorder__bot_687,axiom,
! [A13: $tType] :
( ( order @ A13 )
=> ( order_bot @ ( option @ A13 ) ) ) ).
thf(tcon_Option_Ooption___Orderings_Opreorder_688,axiom,
! [A13: $tType] :
( ( preorder @ A13 )
=> ( preorder @ ( option @ A13 ) ) ) ).
thf(tcon_Option_Ooption___Orderings_Olinorder_689,axiom,
! [A13: $tType] :
( ( linorder @ A13 )
=> ( linorder @ ( option @ A13 ) ) ) ).
thf(tcon_Option_Ooption___Finite__Set_Ofinite_690,axiom,
! [A13: $tType] :
( ( finite_finite @ A13 )
=> ( finite_finite @ ( option @ A13 ) ) ) ).
thf(tcon_Option_Ooption___Orderings_Oorder_691,axiom,
! [A13: $tType] :
( ( order @ A13 )
=> ( order @ ( option @ A13 ) ) ) ).
thf(tcon_Option_Ooption___Orderings_Oord_692,axiom,
! [A13: $tType] :
( ( preorder @ A13 )
=> ( ord @ ( option @ A13 ) ) ) ).
thf(tcon_Option_Ooption___Heap_Oheap_693,axiom,
! [A13: $tType] :
( ( heap @ A13 )
=> ( heap @ ( option @ A13 ) ) ) ).
thf(tcon_Option_Ooption___Nat_Osize_694,axiom,
! [A13: $tType] : ( size @ ( option @ A13 ) ) ).
thf(tcon_Uint32_Ouint32___Bit__Operations_Osemiring__bit__operations_695,axiom,
bit_se359711467146920520ations @ uint32 ).
thf(tcon_Uint32_Ouint32___Quickcheck__Narrowing_Opartial__term__of_696,axiom,
quickc6926020345158392990erm_of @ uint32 ).
thf(tcon_Uint32_Ouint32___Bit__Operations_Oring__bit__operations_697,axiom,
bit_ri3973907225187159222ations @ uint32 ).
thf(tcon_Uint32_Ouint32___Groups_Ocancel__ab__semigroup__add_698,axiom,
cancel2418104881723323429up_add @ uint32 ).
thf(tcon_Uint32_Ouint32___Groups_Ocancel__comm__monoid__add_699,axiom,
cancel1802427076303600483id_add @ uint32 ).
thf(tcon_Uint32_Ouint32___Rings_Ocomm__semiring__1__cancel_700,axiom,
comm_s4317794764714335236cancel @ uint32 ).
thf(tcon_Uint32_Ouint32___Bit__Operations_Osemiring__bits_701,axiom,
bit_semiring_bits @ uint32 ).
thf(tcon_Uint32_Ouint32___Groups_Ocancel__semigroup__add_702,axiom,
cancel_semigroup_add @ uint32 ).
thf(tcon_Uint32_Ouint32___Least__significant__bit_Olsb_703,axiom,
least_6119777620449941438nt_lsb @ uint32 ).
thf(tcon_Uint32_Ouint32___Groups_Oab__semigroup__mult_704,axiom,
ab_semigroup_mult @ uint32 ).
thf(tcon_Uint32_Ouint32___Rings_Osemiring__1__cancel_705,axiom,
semiring_1_cancel @ uint32 ).
thf(tcon_Uint32_Ouint32___Groups_Ocomm__monoid__mult_706,axiom,
comm_monoid_mult @ uint32 ).
thf(tcon_Uint32_Ouint32___Groups_Oab__semigroup__add_707,axiom,
ab_semigroup_add @ uint32 ).
thf(tcon_Uint32_Ouint32___Generic__set__bit_Oset__bit_708,axiom,
generic_set_set_bit @ uint32 ).
thf(tcon_Uint32_Ouint32___Code__Evaluation_Oterm__of_709,axiom,
code_term_of @ uint32 ).
thf(tcon_Uint32_Ouint32___Parity_Osemiring__parity_710,axiom,
semiring_parity @ uint32 ).
thf(tcon_Uint32_Ouint32___Groups_Ocomm__monoid__add_711,axiom,
comm_monoid_add @ uint32 ).
thf(tcon_Uint32_Ouint32___Rings_Osemiring__modulo_712,axiom,
semiring_modulo @ uint32 ).
thf(tcon_Uint32_Ouint32___Rings_Ocomm__semiring__1_713,axiom,
comm_semiring_1 @ uint32 ).
thf(tcon_Uint32_Ouint32___Rings_Ocomm__semiring__0_714,axiom,
comm_semiring_0 @ uint32 ).
thf(tcon_Uint32_Ouint32___Groups_Osemigroup__mult_715,axiom,
semigroup_mult @ uint32 ).
thf(tcon_Uint32_Ouint32___Num_Osemiring__numeral_716,axiom,
semiring_numeral @ uint32 ).
thf(tcon_Uint32_Ouint32___Groups_Osemigroup__add_717,axiom,
semigroup_add @ uint32 ).
thf(tcon_Uint32_Ouint32___Rings_Ocomm__semiring_718,axiom,
comm_semiring @ uint32 ).
thf(tcon_Uint32_Ouint32___Groups_Oab__group__add_719,axiom,
ab_group_add @ uint32 ).
thf(tcon_Uint32_Ouint32___Rings_Ozero__neq__one_720,axiom,
zero_neq_one @ uint32 ).
thf(tcon_Uint32_Ouint32___Parity_Oring__parity_721,axiom,
ring_parity @ uint32 ).
thf(tcon_Uint32_Ouint32___Orderings_Opreorder_722,axiom,
preorder @ uint32 ).
thf(tcon_Uint32_Ouint32___Orderings_Olinorder_723,axiom,
linorder @ uint32 ).
thf(tcon_Uint32_Ouint32___Groups_Omonoid__mult_724,axiom,
monoid_mult @ uint32 ).
thf(tcon_Uint32_Ouint32___Rings_Ocomm__ring__1_725,axiom,
comm_ring_1 @ uint32 ).
thf(tcon_Uint32_Ouint32___Groups_Omonoid__add_726,axiom,
monoid_add @ uint32 ).
thf(tcon_Uint32_Ouint32___Rings_Osemiring__1_727,axiom,
semiring_1 @ uint32 ).
thf(tcon_Uint32_Ouint32___Rings_Osemiring__0_728,axiom,
semiring_0 @ uint32 ).
thf(tcon_Uint32_Ouint32___Groups_Ogroup__add_729,axiom,
group_add @ uint32 ).
thf(tcon_Uint32_Ouint32___Rings_Omult__zero_730,axiom,
mult_zero @ uint32 ).
thf(tcon_Uint32_Ouint32___Rings_Ocomm__ring_731,axiom,
comm_ring @ uint32 ).
thf(tcon_Uint32_Ouint32___Orderings_Oorder_732,axiom,
order @ uint32 ).
thf(tcon_Uint32_Ouint32___Num_Oneg__numeral_733,axiom,
neg_numeral @ uint32 ).
thf(tcon_Uint32_Ouint32___Rings_Osemiring_734,axiom,
semiring @ uint32 ).
thf(tcon_Uint32_Ouint32___Orderings_Oord_735,axiom,
ord @ uint32 ).
thf(tcon_Uint32_Ouint32___Groups_Ouminus_736,axiom,
uminus @ uint32 ).
thf(tcon_Uint32_Ouint32___Rings_Oring__1_737,axiom,
ring_1 @ uint32 ).
thf(tcon_Uint32_Ouint32___Groups_Ominus_738,axiom,
minus @ uint32 ).
thf(tcon_Uint32_Ouint32___Power_Opower_739,axiom,
power @ uint32 ).
thf(tcon_Uint32_Ouint32___Num_Onumeral_740,axiom,
numeral @ uint32 ).
thf(tcon_Uint32_Ouint32___Groups_Ozero_741,axiom,
zero @ uint32 ).
thf(tcon_Uint32_Ouint32___Groups_Oplus_742,axiom,
plus @ uint32 ).
thf(tcon_Uint32_Ouint32___Rings_Oring_743,axiom,
ring @ uint32 ).
thf(tcon_Uint32_Ouint32___Groups_Oone_744,axiom,
one @ uint32 ).
thf(tcon_Uint32_Ouint32___Rings_Odvd_745,axiom,
dvd @ uint32 ).
thf(tcon_Uint32_Ouint32___Nat_Osize_746,axiom,
size @ uint32 ).
thf(tcon_Assertions_Oassn___Boolean__Algebras_Oboolean__algebra_747,axiom,
boolea8198339166811842893lgebra @ assn ).
thf(tcon_Assertions_Oassn___Groups_Oab__semigroup__mult_748,axiom,
ab_semigroup_mult @ assn ).
thf(tcon_Assertions_Oassn___Groups_Ocomm__monoid__mult_749,axiom,
comm_monoid_mult @ assn ).
thf(tcon_Assertions_Oassn___Groups_Osemigroup__mult_750,axiom,
semigroup_mult @ assn ).
thf(tcon_Assertions_Oassn___Orderings_Oorder__top_751,axiom,
order_top @ assn ).
thf(tcon_Assertions_Oassn___Orderings_Oorder__bot_752,axiom,
order_bot @ assn ).
thf(tcon_Assertions_Oassn___Orderings_Opreorder_753,axiom,
preorder @ assn ).
thf(tcon_Assertions_Oassn___Groups_Omonoid__mult_754,axiom,
monoid_mult @ assn ).
thf(tcon_Assertions_Oassn___Orderings_Oorder_755,axiom,
order @ assn ).
thf(tcon_Assertions_Oassn___Orderings_Oord_756,axiom,
ord @ assn ).
thf(tcon_Assertions_Oassn___Groups_Ouminus_757,axiom,
uminus @ assn ).
thf(tcon_Assertions_Oassn___Groups_Ominus_758,axiom,
minus @ assn ).
thf(tcon_Assertions_Oassn___Power_Opower_759,axiom,
power @ assn ).
thf(tcon_Assertions_Oassn___Groups_Oone_760,axiom,
one @ assn ).
thf(tcon_Assertions_Oassn___Rings_Odvd_761,axiom,
dvd @ assn ).
thf(tcon_Complex_Ocomplex___Semiring__Normalization_Ocomm__semiring__1__cancel__crossproduct_762,axiom,
semiri1453513574482234551roduct @ complex ).
thf(tcon_Complex_Ocomplex___Topological__Spaces_Ofirst__countable__topology_763,axiom,
topolo3112930676232923870pology @ complex ).
thf(tcon_Complex_Ocomplex___Real__Vector__Spaces_Oreal__normed__div__algebra_764,axiom,
real_V8999393235501362500lgebra @ complex ).
thf(tcon_Complex_Ocomplex___Real__Vector__Spaces_Oreal__normed__algebra__1_765,axiom,
real_V2822296259951069270ebra_1 @ complex ).
thf(tcon_Complex_Ocomplex___Rings_Osemiring__no__zero__divisors__cancel_766,axiom,
semiri6575147826004484403cancel @ complex ).
thf(tcon_Complex_Ocomplex___Real__Vector__Spaces_Oreal__normed__algebra_767,axiom,
real_V4412858255891104859lgebra @ complex ).
thf(tcon_Complex_Ocomplex___Real__Vector__Spaces_Oreal__normed__vector_768,axiom,
real_V822414075346904944vector @ complex ).
thf(tcon_Complex_Ocomplex___Topological__Spaces_Otopological__space_769,axiom,
topolo4958980785337419405_space @ complex ).
thf(tcon_Complex_Ocomplex___Real__Vector__Spaces_Oreal__normed__field_770,axiom,
real_V3459762299906320749_field @ complex ).
thf(tcon_Complex_Ocomplex___Quickcheck__Narrowing_Opartial__term__of_771,axiom,
quickc6926020345158392990erm_of @ complex ).
thf(tcon_Complex_Ocomplex___Real__Vector__Spaces_Oreal__div__algebra_772,axiom,
real_V5047593784448816457lgebra @ complex ).
thf(tcon_Complex_Ocomplex___Limits_Otopological__comm__monoid__add_773,axiom,
topolo5987344860129210374id_add @ complex ).
thf(tcon_Complex_Ocomplex___Rings_Osemiring__1__no__zero__divisors_774,axiom,
semiri2026040879449505780visors @ complex ).
thf(tcon_Complex_Ocomplex___Real__Vector__Spaces_Oreal__algebra__1_775,axiom,
real_V2191834092415804123ebra_1 @ complex ).
thf(tcon_Complex_Ocomplex___Real__Vector__Spaces_Ocomplete__space_776,axiom,
real_V8037385150606011577_space @ complex ).
thf(tcon_Complex_Ocomplex___Limits_Otopological__semigroup__mult_777,axiom,
topolo4211221413907600880p_mult @ complex ).
thf(tcon_Complex_Ocomplex___Topological__Spaces_Ouniform__space_778,axiom,
topolo7287701948861334536_space @ complex ).
thf(tcon_Complex_Ocomplex___Topological__Spaces_Operfect__space_779,axiom,
topolo8386298272705272623_space @ complex ).
thf(tcon_Complex_Ocomplex___Rings_Osemiring__no__zero__divisors_780,axiom,
semiri3467727345109120633visors @ complex ).
thf(tcon_Complex_Ocomplex___Real__Vector__Spaces_Oreal__algebra_781,axiom,
real_V6157519004096292374lgebra @ complex ).
thf(tcon_Complex_Ocomplex___Real__Vector__Spaces_Ometric__space_782,axiom,
real_V7819770556892013058_space @ complex ).
thf(tcon_Complex_Ocomplex___Limits_Otopological__ab__group__add_783,axiom,
topolo1287966508704411220up_add @ complex ).
thf(tcon_Complex_Ocomplex___Real__Vector__Spaces_Oreal__vector_784,axiom,
real_V4867850818363320053vector @ complex ).
thf(tcon_Complex_Ocomplex___Groups_Ocancel__ab__semigroup__add_785,axiom,
cancel2418104881723323429up_add @ complex ).
thf(tcon_Complex_Ocomplex___Rings_Oring__1__no__zero__divisors_786,axiom,
ring_15535105094025558882visors @ complex ).
thf(tcon_Complex_Ocomplex___Real__Vector__Spaces_Oreal__field_787,axiom,
real_V7773925162809079976_field @ complex ).
thf(tcon_Complex_Ocomplex___Limits_Otopological__monoid__add_788,axiom,
topolo6943815403480290642id_add @ complex ).
thf(tcon_Complex_Ocomplex___Groups_Ocancel__comm__monoid__add_789,axiom,
cancel1802427076303600483id_add @ complex ).
thf(tcon_Complex_Ocomplex___Rings_Ocomm__semiring__1__cancel_790,axiom,
comm_s4317794764714335236cancel @ complex ).
thf(tcon_Complex_Ocomplex___Real__Vector__Spaces_Odist__norm_791,axiom,
real_V6936659425649961206t_norm @ complex ).
thf(tcon_Complex_Ocomplex___Limits_Otopological__group__add_792,axiom,
topolo1633459387980952147up_add @ complex ).
thf(tcon_Complex_Ocomplex___Topological__Spaces_Ot2__space_793,axiom,
topological_t2_space @ complex ).
thf(tcon_Complex_Ocomplex___Topological__Spaces_Ot1__space_794,axiom,
topological_t1_space @ complex ).
thf(tcon_Complex_Ocomplex___Groups_Ocancel__semigroup__add_795,axiom,
cancel_semigroup_add @ complex ).
thf(tcon_Complex_Ocomplex___Real__Vector__Spaces_Obanach_796,axiom,
real_Vector_banach @ complex ).
thf(tcon_Complex_Ocomplex___Groups_Oab__semigroup__mult_797,axiom,
ab_semigroup_mult @ complex ).
thf(tcon_Complex_Ocomplex___Rings_Osemiring__1__cancel_798,axiom,
semiring_1_cancel @ complex ).
thf(tcon_Complex_Ocomplex___Groups_Ocomm__monoid__mult_799,axiom,
comm_monoid_mult @ complex ).
thf(tcon_Complex_Ocomplex___Groups_Oab__semigroup__add_800,axiom,
ab_semigroup_add @ complex ).
thf(tcon_Complex_Ocomplex___Code__Evaluation_Oterm__of_801,axiom,
code_term_of @ complex ).
thf(tcon_Complex_Ocomplex___Groups_Ocomm__monoid__add_802,axiom,
comm_monoid_add @ complex ).
thf(tcon_Complex_Ocomplex___Rings_Ocomm__semiring__1_803,axiom,
comm_semiring_1 @ complex ).
thf(tcon_Complex_Ocomplex___Rings_Ocomm__semiring__0_804,axiom,
comm_semiring_0 @ complex ).
thf(tcon_Complex_Ocomplex___Groups_Osemigroup__mult_805,axiom,
semigroup_mult @ complex ).
thf(tcon_Complex_Ocomplex___Rings_Osemidom__divide_806,axiom,
semidom_divide @ complex ).
thf(tcon_Complex_Ocomplex___Num_Osemiring__numeral_807,axiom,
semiring_numeral @ complex ).
thf(tcon_Complex_Ocomplex___Groups_Osemigroup__add_808,axiom,
semigroup_add @ complex ).
thf(tcon_Complex_Ocomplex___Fields_Ofield__abs__sgn_809,axiom,
field_abs_sgn @ complex ).
thf(tcon_Complex_Ocomplex___Fields_Odivision__ring_810,axiom,
division_ring @ complex ).
thf(tcon_Complex_Ocomplex___Rings_Ocomm__semiring_811,axiom,
comm_semiring @ complex ).
thf(tcon_Complex_Ocomplex___Nat_Osemiring__char__0_812,axiom,
semiring_char_0 @ complex ).
thf(tcon_Complex_Ocomplex___Groups_Oab__group__add_813,axiom,
ab_group_add @ complex ).
thf(tcon_Complex_Ocomplex___Fields_Ofield__char__0_814,axiom,
field_char_0 @ complex ).
thf(tcon_Complex_Ocomplex___Rings_Ozero__neq__one_815,axiom,
zero_neq_one @ complex ).
thf(tcon_Complex_Ocomplex___Rings_Oidom__abs__sgn_816,axiom,
idom_abs_sgn @ complex ).
thf(tcon_Complex_Ocomplex___Groups_Omonoid__mult_817,axiom,
monoid_mult @ complex ).
thf(tcon_Complex_Ocomplex___Rings_Oidom__divide_818,axiom,
idom_divide @ complex ).
thf(tcon_Complex_Ocomplex___Rings_Ocomm__ring__1_819,axiom,
comm_ring_1 @ complex ).
thf(tcon_Complex_Ocomplex___Groups_Omonoid__add_820,axiom,
monoid_add @ complex ).
thf(tcon_Complex_Ocomplex___Rings_Osemiring__1_821,axiom,
semiring_1 @ complex ).
thf(tcon_Complex_Ocomplex___Rings_Osemiring__0_822,axiom,
semiring_0 @ complex ).
thf(tcon_Complex_Ocomplex___Groups_Ogroup__add_823,axiom,
group_add @ complex ).
thf(tcon_Complex_Ocomplex___Rings_Omult__zero_824,axiom,
mult_zero @ complex ).
thf(tcon_Complex_Ocomplex___Rings_Ocomm__ring_825,axiom,
comm_ring @ complex ).
thf(tcon_Complex_Ocomplex___Num_Oneg__numeral_826,axiom,
neg_numeral @ complex ).
thf(tcon_Complex_Ocomplex___Nat_Oring__char__0_827,axiom,
ring_char_0 @ complex ).
thf(tcon_Complex_Ocomplex___Rings_Osemiring_828,axiom,
semiring @ complex ).
thf(tcon_Complex_Ocomplex___Fields_Oinverse_829,axiom,
inverse @ complex ).
thf(tcon_Complex_Ocomplex___Groups_Ouminus_830,axiom,
uminus @ complex ).
thf(tcon_Complex_Ocomplex___Rings_Oring__1_831,axiom,
ring_1 @ complex ).
thf(tcon_Complex_Ocomplex___Groups_Ominus_832,axiom,
minus @ complex ).
thf(tcon_Complex_Ocomplex___Fields_Ofield_833,axiom,
field @ complex ).
thf(tcon_Complex_Ocomplex___Power_Opower_834,axiom,
power @ complex ).
thf(tcon_Complex_Ocomplex___Num_Onumeral_835,axiom,
numeral @ complex ).
thf(tcon_Complex_Ocomplex___Groups_Ozero_836,axiom,
zero @ complex ).
thf(tcon_Complex_Ocomplex___Groups_Oplus_837,axiom,
plus @ complex ).
thf(tcon_Complex_Ocomplex___Rings_Oring_838,axiom,
ring @ complex ).
thf(tcon_Complex_Ocomplex___Rings_Oidom_839,axiom,
idom @ complex ).
thf(tcon_Complex_Ocomplex___Groups_Oone_840,axiom,
one @ complex ).
thf(tcon_Complex_Ocomplex___Rings_Odvd_841,axiom,
dvd @ complex ).
thf(tcon_Extended__Nat_Oenat___Conditionally__Complete__Lattices_Oconditionally__complete__linorder_842,axiom,
condit6923001295902523014norder @ extended_enat ).
thf(tcon_Extended__Nat_Oenat___Conditionally__Complete__Lattices_Oconditionally__complete__lattice_843,axiom,
condit1219197933456340205attice @ extended_enat ).
thf(tcon_Extended__Nat_Oenat___Countable__Complete__Lattices_Ocountable__complete__lattice_844,axiom,
counta3822494911875563373attice @ extended_enat ).
thf(tcon_Extended__Nat_Oenat___Groups_Ostrict__ordered__ab__semigroup__add_845,axiom,
strict9044650504122735259up_add @ extended_enat ).
thf(tcon_Extended__Nat_Oenat___Groups_Ostrict__ordered__comm__monoid__add_846,axiom,
strict7427464778891057005id_add @ extended_enat ).
thf(tcon_Extended__Nat_Oenat___Groups_Ocanonically__ordered__monoid__add_847,axiom,
canoni5634975068530333245id_add @ extended_enat ).
thf(tcon_Extended__Nat_Oenat___Quickcheck__Narrowing_Opartial__term__of_848,axiom,
quickc6926020345158392990erm_of @ extended_enat ).
thf(tcon_Extended__Nat_Oenat___Complete__Lattices_Ocomplete__linorder_849,axiom,
comple5582772986160207858norder @ extended_enat ).
thf(tcon_Extended__Nat_Oenat___Groups_Olinordered__ab__semigroup__add_850,axiom,
linord4140545234300271783up_add @ extended_enat ).
thf(tcon_Extended__Nat_Oenat___Complete__Lattices_Ocomplete__lattice_851,axiom,
comple6319245703460814977attice @ extended_enat ).
thf(tcon_Extended__Nat_Oenat___Rings_Olinordered__nonzero__semiring_852,axiom,
linord181362715937106298miring @ extended_enat ).
thf(tcon_Extended__Nat_Oenat___Rings_Osemiring__no__zero__divisors_853,axiom,
semiri3467727345109120633visors @ extended_enat ).
thf(tcon_Extended__Nat_Oenat___Groups_Oordered__ab__semigroup__add_854,axiom,
ordere6658533253407199908up_add @ extended_enat ).
thf(tcon_Extended__Nat_Oenat___Groups_Oordered__comm__monoid__add_855,axiom,
ordere6911136660526730532id_add @ extended_enat ).
thf(tcon_Extended__Nat_Oenat___Rings_Oordered__comm__semiring_856,axiom,
ordere2520102378445227354miring @ extended_enat ).
thf(tcon_Extended__Nat_Oenat___Groups_Oab__semigroup__mult_857,axiom,
ab_semigroup_mult @ extended_enat ).
thf(tcon_Extended__Nat_Oenat___Groups_Ocomm__monoid__mult_858,axiom,
comm_monoid_mult @ extended_enat ).
thf(tcon_Extended__Nat_Oenat___Groups_Oab__semigroup__add_859,axiom,
ab_semigroup_add @ extended_enat ).
thf(tcon_Extended__Nat_Oenat___Code__Evaluation_Oterm__of_860,axiom,
code_term_of @ extended_enat ).
thf(tcon_Extended__Nat_Oenat___Rings_Oordered__semiring_861,axiom,
ordered_semiring @ extended_enat ).
thf(tcon_Extended__Nat_Oenat___Groups_Ocomm__monoid__add_862,axiom,
comm_monoid_add @ extended_enat ).
thf(tcon_Extended__Nat_Oenat___Rings_Ocomm__semiring__1_863,axiom,
comm_semiring_1 @ extended_enat ).
thf(tcon_Extended__Nat_Oenat___Rings_Ocomm__semiring__0_864,axiom,
comm_semiring_0 @ extended_enat ).
thf(tcon_Extended__Nat_Oenat___Groups_Osemigroup__mult_865,axiom,
semigroup_mult @ extended_enat ).
thf(tcon_Extended__Nat_Oenat___Complete__Lattices_OSup_866,axiom,
complete_Sup @ extended_enat ).
thf(tcon_Extended__Nat_Oenat___Complete__Lattices_OInf_867,axiom,
complete_Inf @ extended_enat ).
thf(tcon_Extended__Nat_Oenat___Num_Osemiring__numeral_868,axiom,
semiring_numeral @ extended_enat ).
thf(tcon_Extended__Nat_Oenat___Groups_Osemigroup__add_869,axiom,
semigroup_add @ extended_enat ).
thf(tcon_Extended__Nat_Oenat___Rings_Ozero__less__one_870,axiom,
zero_less_one @ extended_enat ).
thf(tcon_Extended__Nat_Oenat___Rings_Ocomm__semiring_871,axiom,
comm_semiring @ extended_enat ).
thf(tcon_Extended__Nat_Oenat___Orderings_Owellorder_872,axiom,
wellorder @ extended_enat ).
thf(tcon_Extended__Nat_Oenat___Orderings_Oorder__top_873,axiom,
order_top @ extended_enat ).
thf(tcon_Extended__Nat_Oenat___Orderings_Oorder__bot_874,axiom,
order_bot @ extended_enat ).
thf(tcon_Extended__Nat_Oenat___Nat_Osemiring__char__0_875,axiom,
semiring_char_0 @ extended_enat ).
thf(tcon_Extended__Nat_Oenat___Rings_Ozero__neq__one_876,axiom,
zero_neq_one @ extended_enat ).
thf(tcon_Extended__Nat_Oenat___Orderings_Opreorder_877,axiom,
preorder @ extended_enat ).
thf(tcon_Extended__Nat_Oenat___Orderings_Olinorder_878,axiom,
linorder @ extended_enat ).
thf(tcon_Extended__Nat_Oenat___Groups_Omonoid__mult_879,axiom,
monoid_mult @ extended_enat ).
thf(tcon_Extended__Nat_Oenat___Groups_Omonoid__add_880,axiom,
monoid_add @ extended_enat ).
thf(tcon_Extended__Nat_Oenat___Rings_Osemiring__1_881,axiom,
semiring_1 @ extended_enat ).
thf(tcon_Extended__Nat_Oenat___Rings_Osemiring__0_882,axiom,
semiring_0 @ extended_enat ).
thf(tcon_Extended__Nat_Oenat___Rings_Omult__zero_883,axiom,
mult_zero @ extended_enat ).
thf(tcon_Extended__Nat_Oenat___Orderings_Oorder_884,axiom,
order @ extended_enat ).
thf(tcon_Extended__Nat_Oenat___Rings_Osemiring_885,axiom,
semiring @ extended_enat ).
thf(tcon_Extended__Nat_Oenat___Orderings_Oord_886,axiom,
ord @ extended_enat ).
thf(tcon_Extended__Nat_Oenat___Groups_Ominus_887,axiom,
minus @ extended_enat ).
thf(tcon_Extended__Nat_Oenat___Power_Opower_888,axiom,
power @ extended_enat ).
thf(tcon_Extended__Nat_Oenat___Num_Onumeral_889,axiom,
numeral @ extended_enat ).
thf(tcon_Extended__Nat_Oenat___Groups_Ozero_890,axiom,
zero @ extended_enat ).
thf(tcon_Extended__Nat_Oenat___Groups_Oplus_891,axiom,
plus @ extended_enat ).
thf(tcon_Extended__Nat_Oenat___Groups_Oone_892,axiom,
one @ extended_enat ).
thf(tcon_Extended__Nat_Oenat___Rings_Odvd_893,axiom,
dvd @ extended_enat ).
thf(tcon_Multiset_Omultiset___Quickcheck__Narrowing_Opartial__term__of_894,axiom,
! [A13: $tType] :
( ( typerep @ A13 )
=> ( quickc6926020345158392990erm_of @ ( multiset @ A13 ) ) ) ).
thf(tcon_Multiset_Omultiset___Groups_Oordered__ab__semigroup__add_895,axiom,
! [A13: $tType] :
( ( preorder @ A13 )
=> ( ordere6658533253407199908up_add @ ( multiset @ A13 ) ) ) ).
thf(tcon_Multiset_Omultiset___Groups_Ocancel__ab__semigroup__add_896,axiom,
! [A13: $tType] : ( cancel2418104881723323429up_add @ ( multiset @ A13 ) ) ).
thf(tcon_Multiset_Omultiset___Groups_Ocancel__comm__monoid__add_897,axiom,
! [A13: $tType] : ( cancel1802427076303600483id_add @ ( multiset @ A13 ) ) ).
thf(tcon_Multiset_Omultiset___Groups_Ocancel__semigroup__add_898,axiom,
! [A13: $tType] : ( cancel_semigroup_add @ ( multiset @ A13 ) ) ).
thf(tcon_Multiset_Omultiset___Groups_Ocomm__monoid__diff_899,axiom,
! [A13: $tType] : ( comm_monoid_diff @ ( multiset @ A13 ) ) ).
thf(tcon_Multiset_Omultiset___Groups_Oab__semigroup__add_900,axiom,
! [A13: $tType] : ( ab_semigroup_add @ ( multiset @ A13 ) ) ).
thf(tcon_Multiset_Omultiset___Code__Evaluation_Oterm__of_901,axiom,
! [A13: $tType] :
( ( typerep @ A13 )
=> ( code_term_of @ ( multiset @ A13 ) ) ) ).
thf(tcon_Multiset_Omultiset___Groups_Ocomm__monoid__add_902,axiom,
! [A13: $tType] : ( comm_monoid_add @ ( multiset @ A13 ) ) ).
thf(tcon_Multiset_Omultiset___Complete__Lattices_OSup_903,axiom,
! [A13: $tType] : ( complete_Sup @ ( multiset @ A13 ) ) ).
thf(tcon_Multiset_Omultiset___Complete__Lattices_OInf_904,axiom,
! [A13: $tType] : ( complete_Inf @ ( multiset @ A13 ) ) ).
thf(tcon_Multiset_Omultiset___Groups_Osemigroup__add_905,axiom,
! [A13: $tType] : ( semigroup_add @ ( multiset @ A13 ) ) ).
thf(tcon_Multiset_Omultiset___Orderings_Opreorder_906,axiom,
! [A13: $tType] :
( ( preorder @ A13 )
=> ( preorder @ ( multiset @ A13 ) ) ) ).
thf(tcon_Multiset_Omultiset___Groups_Omonoid__add_907,axiom,
! [A13: $tType] : ( monoid_add @ ( multiset @ A13 ) ) ).
thf(tcon_Multiset_Omultiset___Orderings_Oorder_908,axiom,
! [A13: $tType] :
( ( preorder @ A13 )
=> ( order @ ( multiset @ A13 ) ) ) ).
thf(tcon_Multiset_Omultiset___Orderings_Oord_909,axiom,
! [A13: $tType] :
( ( preorder @ A13 )
=> ( ord @ ( multiset @ A13 ) ) ) ).
thf(tcon_Multiset_Omultiset___Groups_Ominus_910,axiom,
! [A13: $tType] : ( minus @ ( multiset @ A13 ) ) ).
thf(tcon_Multiset_Omultiset___Groups_Ozero_911,axiom,
! [A13: $tType] : ( zero @ ( multiset @ A13 ) ) ).
thf(tcon_Multiset_Omultiset___Groups_Oplus_912,axiom,
! [A13: $tType] : ( plus @ ( multiset @ A13 ) ) ).
thf(tcon_Multiset_Omultiset___Nat_Osize_913,axiom,
! [A13: $tType] : ( size @ ( multiset @ A13 ) ) ).
thf(tcon_Numeral__Type_Obit0___Groups_Ocancel__ab__semigroup__add_914,axiom,
! [A13: $tType] :
( ( finite_finite @ A13 )
=> ( cancel2418104881723323429up_add @ ( numeral_bit0 @ A13 ) ) ) ).
thf(tcon_Numeral__Type_Obit0___Groups_Ocancel__comm__monoid__add_915,axiom,
! [A13: $tType] :
( ( finite_finite @ A13 )
=> ( cancel1802427076303600483id_add @ ( numeral_bit0 @ A13 ) ) ) ).
thf(tcon_Numeral__Type_Obit0___Rings_Ocomm__semiring__1__cancel_916,axiom,
! [A13: $tType] :
( ( finite_finite @ A13 )
=> ( comm_s4317794764714335236cancel @ ( numeral_bit0 @ A13 ) ) ) ).
thf(tcon_Numeral__Type_Obit0___Groups_Ocancel__semigroup__add_917,axiom,
! [A13: $tType] :
( ( finite_finite @ A13 )
=> ( cancel_semigroup_add @ ( numeral_bit0 @ A13 ) ) ) ).
thf(tcon_Numeral__Type_Obit0___Groups_Oab__semigroup__mult_918,axiom,
! [A13: $tType] :
( ( finite_finite @ A13 )
=> ( ab_semigroup_mult @ ( numeral_bit0 @ A13 ) ) ) ).
thf(tcon_Numeral__Type_Obit0___Rings_Osemiring__1__cancel_919,axiom,
! [A13: $tType] :
( ( finite_finite @ A13 )
=> ( semiring_1_cancel @ ( numeral_bit0 @ A13 ) ) ) ).
thf(tcon_Numeral__Type_Obit0___Groups_Ocomm__monoid__mult_920,axiom,
! [A13: $tType] :
( ( finite_finite @ A13 )
=> ( comm_monoid_mult @ ( numeral_bit0 @ A13 ) ) ) ).
thf(tcon_Numeral__Type_Obit0___Groups_Oab__semigroup__add_921,axiom,
! [A13: $tType] :
( ( finite_finite @ A13 )
=> ( ab_semigroup_add @ ( numeral_bit0 @ A13 ) ) ) ).
thf(tcon_Numeral__Type_Obit0___Code__Evaluation_Oterm__of_922,axiom,
! [A13: $tType] :
( ( typerep @ A13 )
=> ( code_term_of @ ( numeral_bit0 @ A13 ) ) ) ).
thf(tcon_Numeral__Type_Obit0___Groups_Ocomm__monoid__add_923,axiom,
! [A13: $tType] :
( ( finite_finite @ A13 )
=> ( comm_monoid_add @ ( numeral_bit0 @ A13 ) ) ) ).
thf(tcon_Numeral__Type_Obit0___Rings_Ocomm__semiring__1_924,axiom,
! [A13: $tType] :
( ( finite_finite @ A13 )
=> ( comm_semiring_1 @ ( numeral_bit0 @ A13 ) ) ) ).
thf(tcon_Numeral__Type_Obit0___Rings_Ocomm__semiring__0_925,axiom,
! [A13: $tType] :
( ( finite_finite @ A13 )
=> ( comm_semiring_0 @ ( numeral_bit0 @ A13 ) ) ) ).
thf(tcon_Numeral__Type_Obit0___Groups_Osemigroup__mult_926,axiom,
! [A13: $tType] :
( ( finite_finite @ A13 )
=> ( semigroup_mult @ ( numeral_bit0 @ A13 ) ) ) ).
thf(tcon_Numeral__Type_Obit0___Num_Osemiring__numeral_927,axiom,
! [A13: $tType] :
( ( finite_finite @ A13 )
=> ( semiring_numeral @ ( numeral_bit0 @ A13 ) ) ) ).
thf(tcon_Numeral__Type_Obit0___Groups_Osemigroup__add_928,axiom,
! [A13: $tType] :
( ( finite_finite @ A13 )
=> ( semigroup_add @ ( numeral_bit0 @ A13 ) ) ) ).
thf(tcon_Numeral__Type_Obit0___Rings_Ocomm__semiring_929,axiom,
! [A13: $tType] :
( ( finite_finite @ A13 )
=> ( comm_semiring @ ( numeral_bit0 @ A13 ) ) ) ).
thf(tcon_Numeral__Type_Obit0___Orderings_Owellorder_930,axiom,
! [A13: $tType] :
( ( finite_finite @ A13 )
=> ( wellorder @ ( numeral_bit0 @ A13 ) ) ) ).
thf(tcon_Numeral__Type_Obit0___Groups_Oab__group__add_931,axiom,
! [A13: $tType] :
( ( finite_finite @ A13 )
=> ( ab_group_add @ ( numeral_bit0 @ A13 ) ) ) ).
thf(tcon_Numeral__Type_Obit0___Rings_Ozero__neq__one_932,axiom,
! [A13: $tType] :
( ( finite_finite @ A13 )
=> ( zero_neq_one @ ( numeral_bit0 @ A13 ) ) ) ).
thf(tcon_Numeral__Type_Obit0___Orderings_Opreorder_933,axiom,
! [A13: $tType] :
( ( finite_finite @ A13 )
=> ( preorder @ ( numeral_bit0 @ A13 ) ) ) ).
thf(tcon_Numeral__Type_Obit0___Orderings_Olinorder_934,axiom,
! [A13: $tType] :
( ( finite_finite @ A13 )
=> ( linorder @ ( numeral_bit0 @ A13 ) ) ) ).
thf(tcon_Numeral__Type_Obit0___Groups_Omonoid__mult_935,axiom,
! [A13: $tType] :
( ( finite_finite @ A13 )
=> ( monoid_mult @ ( numeral_bit0 @ A13 ) ) ) ).
thf(tcon_Numeral__Type_Obit0___Rings_Ocomm__ring__1_936,axiom,
! [A13: $tType] :
( ( finite_finite @ A13 )
=> ( comm_ring_1 @ ( numeral_bit0 @ A13 ) ) ) ).
thf(tcon_Numeral__Type_Obit0___Groups_Omonoid__add_937,axiom,
! [A13: $tType] :
( ( finite_finite @ A13 )
=> ( monoid_add @ ( numeral_bit0 @ A13 ) ) ) ).
thf(tcon_Numeral__Type_Obit0___Finite__Set_Ofinite_938,axiom,
! [A13: $tType] :
( ( finite_finite @ A13 )
=> ( finite_finite @ ( numeral_bit0 @ A13 ) ) ) ).
thf(tcon_Numeral__Type_Obit0___Cardinality_Ocard2,axiom,
! [A13: $tType] :
( ( finite_finite @ A13 )
=> ( card2 @ ( numeral_bit0 @ A13 ) ) ) ).
thf(tcon_Numeral__Type_Obit0___Type__Length_Olen0_939,axiom,
! [A13: $tType] :
( ( type_len0 @ A13 )
=> ( type_len0 @ ( numeral_bit0 @ A13 ) ) ) ).
thf(tcon_Numeral__Type_Obit0___Rings_Osemiring__1_940,axiom,
! [A13: $tType] :
( ( finite_finite @ A13 )
=> ( semiring_1 @ ( numeral_bit0 @ A13 ) ) ) ).
thf(tcon_Numeral__Type_Obit0___Rings_Osemiring__0_941,axiom,
! [A13: $tType] :
( ( finite_finite @ A13 )
=> ( semiring_0 @ ( numeral_bit0 @ A13 ) ) ) ).
thf(tcon_Numeral__Type_Obit0___Groups_Ogroup__add_942,axiom,
! [A13: $tType] :
( ( finite_finite @ A13 )
=> ( group_add @ ( numeral_bit0 @ A13 ) ) ) ).
thf(tcon_Numeral__Type_Obit0___Type__Length_Olen_943,axiom,
! [A13: $tType] :
( ( type_len @ A13 )
=> ( type_len @ ( numeral_bit0 @ A13 ) ) ) ).
thf(tcon_Numeral__Type_Obit0___Rings_Omult__zero_944,axiom,
! [A13: $tType] :
( ( finite_finite @ A13 )
=> ( mult_zero @ ( numeral_bit0 @ A13 ) ) ) ).
thf(tcon_Numeral__Type_Obit0___Rings_Ocomm__ring_945,axiom,
! [A13: $tType] :
( ( finite_finite @ A13 )
=> ( comm_ring @ ( numeral_bit0 @ A13 ) ) ) ).
thf(tcon_Numeral__Type_Obit0___Orderings_Oorder_946,axiom,
! [A13: $tType] :
( ( finite_finite @ A13 )
=> ( order @ ( numeral_bit0 @ A13 ) ) ) ).
thf(tcon_Numeral__Type_Obit0___Num_Oneg__numeral_947,axiom,
! [A13: $tType] :
( ( finite_finite @ A13 )
=> ( neg_numeral @ ( numeral_bit0 @ A13 ) ) ) ).
thf(tcon_Numeral__Type_Obit0___Rings_Osemiring_948,axiom,
! [A13: $tType] :
( ( finite_finite @ A13 )
=> ( semiring @ ( numeral_bit0 @ A13 ) ) ) ).
thf(tcon_Numeral__Type_Obit0___Orderings_Oord_949,axiom,
! [A13: $tType] :
( ( finite_finite @ A13 )
=> ( ord @ ( numeral_bit0 @ A13 ) ) ) ).
thf(tcon_Numeral__Type_Obit0___Groups_Ouminus_950,axiom,
! [A13: $tType] :
( ( finite_finite @ A13 )
=> ( uminus @ ( numeral_bit0 @ A13 ) ) ) ).
thf(tcon_Numeral__Type_Obit0___Rings_Oring__1_951,axiom,
! [A13: $tType] :
( ( finite_finite @ A13 )
=> ( ring_1 @ ( numeral_bit0 @ A13 ) ) ) ).
thf(tcon_Numeral__Type_Obit0___Groups_Ominus_952,axiom,
! [A13: $tType] :
( ( finite_finite @ A13 )
=> ( minus @ ( numeral_bit0 @ A13 ) ) ) ).
thf(tcon_Numeral__Type_Obit0___Power_Opower_953,axiom,
! [A13: $tType] :
( ( finite_finite @ A13 )
=> ( power @ ( numeral_bit0 @ A13 ) ) ) ).
thf(tcon_Numeral__Type_Obit0___Num_Onumeral_954,axiom,
! [A13: $tType] :
( ( finite_finite @ A13 )
=> ( numeral @ ( numeral_bit0 @ A13 ) ) ) ).
thf(tcon_Numeral__Type_Obit0___Groups_Ozero_955,axiom,
! [A13: $tType] :
( ( finite_finite @ A13 )
=> ( zero @ ( numeral_bit0 @ A13 ) ) ) ).
thf(tcon_Numeral__Type_Obit0___Groups_Oplus_956,axiom,
! [A13: $tType] :
( ( finite_finite @ A13 )
=> ( plus @ ( numeral_bit0 @ A13 ) ) ) ).
thf(tcon_Numeral__Type_Obit0___Rings_Oring_957,axiom,
! [A13: $tType] :
( ( finite_finite @ A13 )
=> ( ring @ ( numeral_bit0 @ A13 ) ) ) ).
thf(tcon_Numeral__Type_Obit0___Groups_Oone_958,axiom,
! [A13: $tType] :
( ( finite_finite @ A13 )
=> ( one @ ( numeral_bit0 @ A13 ) ) ) ).
thf(tcon_Numeral__Type_Obit0___Rings_Odvd_959,axiom,
! [A13: $tType] :
( ( finite_finite @ A13 )
=> ( dvd @ ( numeral_bit0 @ A13 ) ) ) ).
thf(tcon_Numeral__Type_Obit1___Groups_Ocancel__ab__semigroup__add_960,axiom,
! [A13: $tType] :
( ( finite_finite @ A13 )
=> ( cancel2418104881723323429up_add @ ( numeral_bit1 @ A13 ) ) ) ).
thf(tcon_Numeral__Type_Obit1___Groups_Ocancel__comm__monoid__add_961,axiom,
! [A13: $tType] :
( ( finite_finite @ A13 )
=> ( cancel1802427076303600483id_add @ ( numeral_bit1 @ A13 ) ) ) ).
thf(tcon_Numeral__Type_Obit1___Rings_Ocomm__semiring__1__cancel_962,axiom,
! [A13: $tType] :
( ( finite_finite @ A13 )
=> ( comm_s4317794764714335236cancel @ ( numeral_bit1 @ A13 ) ) ) ).
thf(tcon_Numeral__Type_Obit1___Groups_Ocancel__semigroup__add_963,axiom,
! [A13: $tType] :
( ( finite_finite @ A13 )
=> ( cancel_semigroup_add @ ( numeral_bit1 @ A13 ) ) ) ).
thf(tcon_Numeral__Type_Obit1___Groups_Oab__semigroup__mult_964,axiom,
! [A13: $tType] :
( ( finite_finite @ A13 )
=> ( ab_semigroup_mult @ ( numeral_bit1 @ A13 ) ) ) ).
thf(tcon_Numeral__Type_Obit1___Rings_Osemiring__1__cancel_965,axiom,
! [A13: $tType] :
( ( finite_finite @ A13 )
=> ( semiring_1_cancel @ ( numeral_bit1 @ A13 ) ) ) ).
thf(tcon_Numeral__Type_Obit1___Groups_Ocomm__monoid__mult_966,axiom,
! [A13: $tType] :
( ( finite_finite @ A13 )
=> ( comm_monoid_mult @ ( numeral_bit1 @ A13 ) ) ) ).
thf(tcon_Numeral__Type_Obit1___Groups_Oab__semigroup__add_967,axiom,
! [A13: $tType] :
( ( finite_finite @ A13 )
=> ( ab_semigroup_add @ ( numeral_bit1 @ A13 ) ) ) ).
thf(tcon_Numeral__Type_Obit1___Code__Evaluation_Oterm__of_968,axiom,
! [A13: $tType] :
( ( typerep @ A13 )
=> ( code_term_of @ ( numeral_bit1 @ A13 ) ) ) ).
thf(tcon_Numeral__Type_Obit1___Groups_Ocomm__monoid__add_969,axiom,
! [A13: $tType] :
( ( finite_finite @ A13 )
=> ( comm_monoid_add @ ( numeral_bit1 @ A13 ) ) ) ).
thf(tcon_Numeral__Type_Obit1___Rings_Ocomm__semiring__1_970,axiom,
! [A13: $tType] :
( ( finite_finite @ A13 )
=> ( comm_semiring_1 @ ( numeral_bit1 @ A13 ) ) ) ).
thf(tcon_Numeral__Type_Obit1___Rings_Ocomm__semiring__0_971,axiom,
! [A13: $tType] :
( ( finite_finite @ A13 )
=> ( comm_semiring_0 @ ( numeral_bit1 @ A13 ) ) ) ).
thf(tcon_Numeral__Type_Obit1___Groups_Osemigroup__mult_972,axiom,
! [A13: $tType] :
( ( finite_finite @ A13 )
=> ( semigroup_mult @ ( numeral_bit1 @ A13 ) ) ) ).
thf(tcon_Numeral__Type_Obit1___Num_Osemiring__numeral_973,axiom,
! [A13: $tType] :
( ( finite_finite @ A13 )
=> ( semiring_numeral @ ( numeral_bit1 @ A13 ) ) ) ).
thf(tcon_Numeral__Type_Obit1___Groups_Osemigroup__add_974,axiom,
! [A13: $tType] :
( ( finite_finite @ A13 )
=> ( semigroup_add @ ( numeral_bit1 @ A13 ) ) ) ).
thf(tcon_Numeral__Type_Obit1___Rings_Ocomm__semiring_975,axiom,
! [A13: $tType] :
( ( finite_finite @ A13 )
=> ( comm_semiring @ ( numeral_bit1 @ A13 ) ) ) ).
thf(tcon_Numeral__Type_Obit1___Orderings_Owellorder_976,axiom,
! [A13: $tType] :
( ( finite_finite @ A13 )
=> ( wellorder @ ( numeral_bit1 @ A13 ) ) ) ).
thf(tcon_Numeral__Type_Obit1___Groups_Oab__group__add_977,axiom,
! [A13: $tType] :
( ( finite_finite @ A13 )
=> ( ab_group_add @ ( numeral_bit1 @ A13 ) ) ) ).
thf(tcon_Numeral__Type_Obit1___Rings_Ozero__neq__one_978,axiom,
! [A13: $tType] :
( ( finite_finite @ A13 )
=> ( zero_neq_one @ ( numeral_bit1 @ A13 ) ) ) ).
thf(tcon_Numeral__Type_Obit1___Orderings_Opreorder_979,axiom,
! [A13: $tType] :
( ( finite_finite @ A13 )
=> ( preorder @ ( numeral_bit1 @ A13 ) ) ) ).
thf(tcon_Numeral__Type_Obit1___Orderings_Olinorder_980,axiom,
! [A13: $tType] :
( ( finite_finite @ A13 )
=> ( linorder @ ( numeral_bit1 @ A13 ) ) ) ).
thf(tcon_Numeral__Type_Obit1___Groups_Omonoid__mult_981,axiom,
! [A13: $tType] :
( ( finite_finite @ A13 )
=> ( monoid_mult @ ( numeral_bit1 @ A13 ) ) ) ).
thf(tcon_Numeral__Type_Obit1___Rings_Ocomm__ring__1_982,axiom,
! [A13: $tType] :
( ( finite_finite @ A13 )
=> ( comm_ring_1 @ ( numeral_bit1 @ A13 ) ) ) ).
thf(tcon_Numeral__Type_Obit1___Groups_Omonoid__add_983,axiom,
! [A13: $tType] :
( ( finite_finite @ A13 )
=> ( monoid_add @ ( numeral_bit1 @ A13 ) ) ) ).
thf(tcon_Numeral__Type_Obit1___Finite__Set_Ofinite_984,axiom,
! [A13: $tType] :
( ( finite_finite @ A13 )
=> ( finite_finite @ ( numeral_bit1 @ A13 ) ) ) ).
thf(tcon_Numeral__Type_Obit1___Cardinality_Ocard2_985,axiom,
! [A13: $tType] :
( ( finite_finite @ A13 )
=> ( card2 @ ( numeral_bit1 @ A13 ) ) ) ).
thf(tcon_Numeral__Type_Obit1___Type__Length_Olen0_986,axiom,
! [A13: $tType] :
( ( type_len0 @ A13 )
=> ( type_len0 @ ( numeral_bit1 @ A13 ) ) ) ).
thf(tcon_Numeral__Type_Obit1___Rings_Osemiring__1_987,axiom,
! [A13: $tType] :
( ( finite_finite @ A13 )
=> ( semiring_1 @ ( numeral_bit1 @ A13 ) ) ) ).
thf(tcon_Numeral__Type_Obit1___Rings_Osemiring__0_988,axiom,
! [A13: $tType] :
( ( finite_finite @ A13 )
=> ( semiring_0 @ ( numeral_bit1 @ A13 ) ) ) ).
thf(tcon_Numeral__Type_Obit1___Groups_Ogroup__add_989,axiom,
! [A13: $tType] :
( ( finite_finite @ A13 )
=> ( group_add @ ( numeral_bit1 @ A13 ) ) ) ).
thf(tcon_Numeral__Type_Obit1___Type__Length_Olen_990,axiom,
! [A13: $tType] :
( ( type_len0 @ A13 )
=> ( type_len @ ( numeral_bit1 @ A13 ) ) ) ).
thf(tcon_Numeral__Type_Obit1___Rings_Omult__zero_991,axiom,
! [A13: $tType] :
( ( finite_finite @ A13 )
=> ( mult_zero @ ( numeral_bit1 @ A13 ) ) ) ).
thf(tcon_Numeral__Type_Obit1___Rings_Ocomm__ring_992,axiom,
! [A13: $tType] :
( ( finite_finite @ A13 )
=> ( comm_ring @ ( numeral_bit1 @ A13 ) ) ) ).
thf(tcon_Numeral__Type_Obit1___Orderings_Oorder_993,axiom,
! [A13: $tType] :
( ( finite_finite @ A13 )
=> ( order @ ( numeral_bit1 @ A13 ) ) ) ).
thf(tcon_Numeral__Type_Obit1___Num_Oneg__numeral_994,axiom,
! [A13: $tType] :
( ( finite_finite @ A13 )
=> ( neg_numeral @ ( numeral_bit1 @ A13 ) ) ) ).
thf(tcon_Numeral__Type_Obit1___Rings_Osemiring_995,axiom,
! [A13: $tType] :
( ( finite_finite @ A13 )
=> ( semiring @ ( numeral_bit1 @ A13 ) ) ) ).
thf(tcon_Numeral__Type_Obit1___Orderings_Oord_996,axiom,
! [A13: $tType] :
( ( finite_finite @ A13 )
=> ( ord @ ( numeral_bit1 @ A13 ) ) ) ).
thf(tcon_Numeral__Type_Obit1___Groups_Ouminus_997,axiom,
! [A13: $tType] :
( ( finite_finite @ A13 )
=> ( uminus @ ( numeral_bit1 @ A13 ) ) ) ).
thf(tcon_Numeral__Type_Obit1___Rings_Oring__1_998,axiom,
! [A13: $tType] :
( ( finite_finite @ A13 )
=> ( ring_1 @ ( numeral_bit1 @ A13 ) ) ) ).
thf(tcon_Numeral__Type_Obit1___Groups_Ominus_999,axiom,
! [A13: $tType] :
( ( finite_finite @ A13 )
=> ( minus @ ( numeral_bit1 @ A13 ) ) ) ).
thf(tcon_Numeral__Type_Obit1___Power_Opower_1000,axiom,
! [A13: $tType] :
( ( finite_finite @ A13 )
=> ( power @ ( numeral_bit1 @ A13 ) ) ) ).
thf(tcon_Numeral__Type_Obit1___Num_Onumeral_1001,axiom,
! [A13: $tType] :
( ( finite_finite @ A13 )
=> ( numeral @ ( numeral_bit1 @ A13 ) ) ) ).
thf(tcon_Numeral__Type_Obit1___Groups_Ozero_1002,axiom,
! [A13: $tType] :
( ( finite_finite @ A13 )
=> ( zero @ ( numeral_bit1 @ A13 ) ) ) ).
thf(tcon_Numeral__Type_Obit1___Groups_Oplus_1003,axiom,
! [A13: $tType] :
( ( finite_finite @ A13 )
=> ( plus @ ( numeral_bit1 @ A13 ) ) ) ).
thf(tcon_Numeral__Type_Obit1___Rings_Oring_1004,axiom,
! [A13: $tType] :
( ( finite_finite @ A13 )
=> ( ring @ ( numeral_bit1 @ A13 ) ) ) ).
thf(tcon_Numeral__Type_Obit1___Groups_Oone_1005,axiom,
! [A13: $tType] :
( ( finite_finite @ A13 )
=> ( one @ ( numeral_bit1 @ A13 ) ) ) ).
thf(tcon_Numeral__Type_Obit1___Rings_Odvd_1006,axiom,
! [A13: $tType] :
( ( finite_finite @ A13 )
=> ( dvd @ ( numeral_bit1 @ A13 ) ) ) ).
thf(tcon_Numeral__Type_Onum1___Quickcheck__Narrowing_Opartial__term__of_1007,axiom,
quickc6926020345158392990erm_of @ numeral_num1 ).
thf(tcon_Numeral__Type_Onum1___Groups_Ocancel__ab__semigroup__add_1008,axiom,
cancel2418104881723323429up_add @ numeral_num1 ).
thf(tcon_Numeral__Type_Onum1___Groups_Ocancel__comm__monoid__add_1009,axiom,
cancel1802427076303600483id_add @ numeral_num1 ).
thf(tcon_Numeral__Type_Onum1___Groups_Ocancel__semigroup__add_1010,axiom,
cancel_semigroup_add @ numeral_num1 ).
thf(tcon_Numeral__Type_Onum1___Groups_Oab__semigroup__mult_1011,axiom,
ab_semigroup_mult @ numeral_num1 ).
thf(tcon_Numeral__Type_Onum1___Groups_Ocomm__monoid__mult_1012,axiom,
comm_monoid_mult @ numeral_num1 ).
thf(tcon_Numeral__Type_Onum1___Groups_Oab__semigroup__add_1013,axiom,
ab_semigroup_add @ numeral_num1 ).
thf(tcon_Numeral__Type_Onum1___Code__Evaluation_Oterm__of_1014,axiom,
code_term_of @ numeral_num1 ).
thf(tcon_Numeral__Type_Onum1___Groups_Ocomm__monoid__add_1015,axiom,
comm_monoid_add @ numeral_num1 ).
thf(tcon_Numeral__Type_Onum1___Rings_Ocomm__semiring__0_1016,axiom,
comm_semiring_0 @ numeral_num1 ).
thf(tcon_Numeral__Type_Onum1___Groups_Osemigroup__mult_1017,axiom,
semigroup_mult @ numeral_num1 ).
thf(tcon_Numeral__Type_Onum1___Groups_Osemigroup__add_1018,axiom,
semigroup_add @ numeral_num1 ).
thf(tcon_Numeral__Type_Onum1___Rings_Ocomm__semiring_1019,axiom,
comm_semiring @ numeral_num1 ).
thf(tcon_Numeral__Type_Onum1___Orderings_Owellorder_1020,axiom,
wellorder @ numeral_num1 ).
thf(tcon_Numeral__Type_Onum1___Groups_Oab__group__add_1021,axiom,
ab_group_add @ numeral_num1 ).
thf(tcon_Numeral__Type_Onum1___Orderings_Opreorder_1022,axiom,
preorder @ numeral_num1 ).
thf(tcon_Numeral__Type_Onum1___Orderings_Olinorder_1023,axiom,
linorder @ numeral_num1 ).
thf(tcon_Numeral__Type_Onum1___Groups_Omonoid__mult_1024,axiom,
monoid_mult @ numeral_num1 ).
thf(tcon_Numeral__Type_Onum1___Cardinality_OCARD__1,axiom,
cARD_1 @ numeral_num1 ).
thf(tcon_Numeral__Type_Onum1___Groups_Omonoid__add_1025,axiom,
monoid_add @ numeral_num1 ).
thf(tcon_Numeral__Type_Onum1___Finite__Set_Ofinite_1026,axiom,
finite_finite @ numeral_num1 ).
thf(tcon_Numeral__Type_Onum1___Type__Length_Olen0_1027,axiom,
type_len0 @ numeral_num1 ).
thf(tcon_Numeral__Type_Onum1___Rings_Osemiring__0_1028,axiom,
semiring_0 @ numeral_num1 ).
thf(tcon_Numeral__Type_Onum1___Groups_Ogroup__add_1029,axiom,
group_add @ numeral_num1 ).
thf(tcon_Numeral__Type_Onum1___Type__Length_Olen_1030,axiom,
type_len @ numeral_num1 ).
thf(tcon_Numeral__Type_Onum1___Rings_Omult__zero_1031,axiom,
mult_zero @ numeral_num1 ).
thf(tcon_Numeral__Type_Onum1___Rings_Ocomm__ring_1032,axiom,
comm_ring @ numeral_num1 ).
thf(tcon_Numeral__Type_Onum1___Orderings_Oorder_1033,axiom,
order @ numeral_num1 ).
thf(tcon_Numeral__Type_Onum1___Rings_Osemiring_1034,axiom,
semiring @ numeral_num1 ).
thf(tcon_Numeral__Type_Onum1___Orderings_Oord_1035,axiom,
ord @ numeral_num1 ).
thf(tcon_Numeral__Type_Onum1___Groups_Ouminus_1036,axiom,
uminus @ numeral_num1 ).
thf(tcon_Numeral__Type_Onum1___Groups_Ominus_1037,axiom,
minus @ numeral_num1 ).
thf(tcon_Numeral__Type_Onum1___Power_Opower_1038,axiom,
power @ numeral_num1 ).
thf(tcon_Numeral__Type_Onum1___Num_Onumeral_1039,axiom,
numeral @ numeral_num1 ).
thf(tcon_Numeral__Type_Onum1___Groups_Ozero_1040,axiom,
zero @ numeral_num1 ).
thf(tcon_Numeral__Type_Onum1___Groups_Oplus_1041,axiom,
plus @ numeral_num1 ).
thf(tcon_Numeral__Type_Onum1___Rings_Oring_1042,axiom,
ring @ numeral_num1 ).
thf(tcon_Numeral__Type_Onum1___Groups_Oone_1043,axiom,
one @ numeral_num1 ).
thf(tcon_Numeral__Type_Onum1___Rings_Odvd_1044,axiom,
dvd @ numeral_num1 ).
thf(tcon_Product__Type_Oprod___Topological__Spaces_Otopological__space_1045,axiom,
! [A13: $tType,A15: $tType] :
( ( ( topolo4958980785337419405_space @ A13 )
& ( topolo4958980785337419405_space @ A15 ) )
=> ( topolo4958980785337419405_space @ ( product_prod @ A13 @ A15 ) ) ) ).
thf(tcon_Product__Type_Oprod___Quickcheck__Narrowing_Opartial__term__of_1046,axiom,
! [A13: $tType,A15: $tType] :
( ( ( typerep @ A13 )
& ( typerep @ A15 ) )
=> ( quickc6926020345158392990erm_of @ ( product_prod @ A13 @ A15 ) ) ) ).
thf(tcon_Product__Type_Oprod___Topological__Spaces_Ot2__space_1047,axiom,
! [A13: $tType,A15: $tType] :
( ( ( topological_t2_space @ A13 )
& ( topological_t2_space @ A15 ) )
=> ( topological_t2_space @ ( product_prod @ A13 @ A15 ) ) ) ).
thf(tcon_Product__Type_Oprod___Topological__Spaces_Ot1__space_1048,axiom,
! [A13: $tType,A15: $tType] :
( ( ( topological_t1_space @ A13 )
& ( topological_t1_space @ A15 ) )
=> ( topological_t1_space @ ( product_prod @ A13 @ A15 ) ) ) ).
thf(tcon_Product__Type_Oprod___Code__Evaluation_Oterm__of_1049,axiom,
! [A13: $tType,A15: $tType] :
( ( ( typerep @ A13 )
& ( typerep @ A15 ) )
=> ( code_term_of @ ( product_prod @ A13 @ A15 ) ) ) ).
thf(tcon_Product__Type_Oprod___Finite__Set_Ofinite_1050,axiom,
! [A13: $tType,A15: $tType] :
( ( ( finite_finite @ A13 )
& ( finite_finite @ A15 ) )
=> ( finite_finite @ ( product_prod @ A13 @ A15 ) ) ) ).
thf(tcon_Product__Type_Oprod___Heap_Oheap_1051,axiom,
! [A13: $tType,A15: $tType] :
( ( ( heap @ A13 )
& ( heap @ A15 ) )
=> ( heap @ ( product_prod @ A13 @ A15 ) ) ) ).
thf(tcon_Product__Type_Oprod___Nat_Osize_1052,axiom,
! [A13: $tType,A15: $tType] : ( size @ ( product_prod @ A13 @ A15 ) ) ).
thf(tcon_Product__Type_Ounit___Conditionally__Complete__Lattices_Oconditionally__complete__linorder_1053,axiom,
condit6923001295902523014norder @ product_unit ).
thf(tcon_Product__Type_Ounit___Conditionally__Complete__Lattices_Oconditionally__complete__lattice_1054,axiom,
condit1219197933456340205attice @ product_unit ).
thf(tcon_Product__Type_Ounit___Countable__Complete__Lattices_Ocountable__complete__lattice_1055,axiom,
counta3822494911875563373attice @ product_unit ).
thf(tcon_Product__Type_Ounit___Quickcheck__Narrowing_Opartial__term__of_1056,axiom,
quickc6926020345158392990erm_of @ product_unit ).
thf(tcon_Product__Type_Ounit___Complete__Lattices_Ocomplete__linorder_1057,axiom,
comple5582772986160207858norder @ product_unit ).
thf(tcon_Product__Type_Ounit___Complete__Lattices_Ocomplete__lattice_1058,axiom,
comple6319245703460814977attice @ product_unit ).
thf(tcon_Product__Type_Ounit___Boolean__Algebras_Oboolean__algebra_1059,axiom,
boolea8198339166811842893lgebra @ product_unit ).
thf(tcon_Product__Type_Ounit___Code__Evaluation_Oterm__of_1060,axiom,
code_term_of @ product_unit ).
thf(tcon_Product__Type_Ounit___Complete__Lattices_OSup_1061,axiom,
complete_Sup @ product_unit ).
thf(tcon_Product__Type_Ounit___Complete__Lattices_OInf_1062,axiom,
complete_Inf @ product_unit ).
thf(tcon_Product__Type_Ounit___Orderings_Owellorder_1063,axiom,
wellorder @ product_unit ).
thf(tcon_Product__Type_Ounit___Orderings_Oorder__top_1064,axiom,
order_top @ product_unit ).
thf(tcon_Product__Type_Ounit___Orderings_Oorder__bot_1065,axiom,
order_bot @ product_unit ).
thf(tcon_Product__Type_Ounit___Orderings_Opreorder_1066,axiom,
preorder @ product_unit ).
thf(tcon_Product__Type_Ounit___Orderings_Olinorder_1067,axiom,
linorder @ product_unit ).
thf(tcon_Product__Type_Ounit___Finite__Set_Ofinite_1068,axiom,
finite_finite @ product_unit ).
thf(tcon_Product__Type_Ounit___Orderings_Oorder_1069,axiom,
order @ product_unit ).
thf(tcon_Product__Type_Ounit___Orderings_Oord_1070,axiom,
ord @ product_unit ).
thf(tcon_Product__Type_Ounit___Groups_Ouminus_1071,axiom,
uminus @ product_unit ).
thf(tcon_Product__Type_Ounit___Groups_Ominus_1072,axiom,
minus @ product_unit ).
thf(tcon_Product__Type_Ounit___Heap_Oheap_1073,axiom,
heap @ product_unit ).
thf(tcon_Heap_Oheap_Oheap__ext___Quickcheck__Narrowing_Opartial__term__of_1074,axiom,
! [A13: $tType] :
( ( typerep @ A13 )
=> ( quickc6926020345158392990erm_of @ ( heap_ext @ A13 ) ) ) ).
thf(tcon_Heap_Oheap_Oheap__ext___Code__Evaluation_Oterm__of_1075,axiom,
! [A13: $tType] :
( ( typerep @ A13 )
=> ( code_term_of @ ( heap_ext @ A13 ) ) ) ).
thf(tcon_Code__Numeral_Ointeger___Bit__Operations_Ounique__euclidean__semiring__with__bit__operations_1076,axiom,
bit_un5681908812861735899ations @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Semiring__Normalization_Ocomm__semiring__1__cancel__crossproduct_1077,axiom,
semiri1453513574482234551roduct @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Euclidean__Division_Ounique__euclidean__semiring__with__nat_1078,axiom,
euclid5411537665997757685th_nat @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Euclidean__Division_Ounique__euclidean__ring__with__nat_1079,axiom,
euclid8789492081693882211th_nat @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Groups_Oordered__ab__semigroup__monoid__add__imp__le_1080,axiom,
ordere1937475149494474687imp_le @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Euclidean__Division_Ounique__euclidean__semiring_1081,axiom,
euclid3128863361964157862miring @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Euclidean__Division_Oeuclidean__semiring__cancel_1082,axiom,
euclid4440199948858584721cancel @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Divides_Ounique__euclidean__semiring__numeral_1083,axiom,
unique1627219031080169319umeral @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Euclidean__Division_Oeuclidean__ring__cancel_1084,axiom,
euclid8851590272496341667cancel @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Rings_Osemiring__no__zero__divisors__cancel_1085,axiom,
semiri6575147826004484403cancel @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Groups_Ostrict__ordered__ab__semigroup__add_1086,axiom,
strict9044650504122735259up_add @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Groups_Oordered__cancel__ab__semigroup__add_1087,axiom,
ordere580206878836729694up_add @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Groups_Oordered__ab__semigroup__add__imp__le_1088,axiom,
ordere2412721322843649153imp_le @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Bit__Operations_Osemiring__bit__operations_1089,axiom,
bit_se359711467146920520ations @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Rings_Olinordered__comm__semiring__strict_1090,axiom,
linord2810124833399127020strict @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Groups_Ostrict__ordered__comm__monoid__add_1091,axiom,
strict7427464778891057005id_add @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Groups_Oordered__cancel__comm__monoid__add_1092,axiom,
ordere8940638589300402666id_add @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Euclidean__Division_Oeuclidean__semiring_1093,axiom,
euclid3725896446679973847miring @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Quickcheck__Narrowing_Opartial__term__of_1094,axiom,
quickc6926020345158392990erm_of @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Rings_Olinordered__semiring__1__strict_1095,axiom,
linord715952674999750819strict @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Groups_Olinordered__ab__semigroup__add_1096,axiom,
linord4140545234300271783up_add @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Bit__Operations_Oring__bit__operations_1097,axiom,
bit_ri3973907225187159222ations @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Rings_Osemiring__1__no__zero__divisors_1098,axiom,
semiri2026040879449505780visors @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Rings_Olinordered__nonzero__semiring_1099,axiom,
linord181362715937106298miring @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Euclidean__Division_Oeuclidean__ring_1100,axiom,
euclid5891614535332579305n_ring @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Rings_Olinordered__semiring__strict_1101,axiom,
linord8928482502909563296strict @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Rings_Osemiring__no__zero__divisors_1102,axiom,
semiri3467727345109120633visors @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Groups_Oordered__ab__semigroup__add_1103,axiom,
ordere6658533253407199908up_add @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Groups_Oordered__ab__group__add__abs_1104,axiom,
ordere166539214618696060dd_abs @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Groups_Oordered__comm__monoid__add_1105,axiom,
ordere6911136660526730532id_add @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Groups_Olinordered__ab__group__add_1106,axiom,
linord5086331880401160121up_add @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Groups_Ocancel__ab__semigroup__add_1107,axiom,
cancel2418104881723323429up_add @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Rings_Oring__1__no__zero__divisors_1108,axiom,
ring_15535105094025558882visors @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Groups_Ocancel__comm__monoid__add_1109,axiom,
cancel1802427076303600483id_add @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Rings_Olinordered__ring__strict_1110,axiom,
linord4710134922213307826strict @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Rings_Ocomm__semiring__1__cancel_1111,axiom,
comm_s4317794764714335236cancel @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Bit__Operations_Osemiring__bits_1112,axiom,
bit_semiring_bits @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Rings_Oordered__comm__semiring_1113,axiom,
ordere2520102378445227354miring @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Rings_Olinordered__semiring__1_1114,axiom,
linord6961819062388156250ring_1 @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Groups_Oordered__ab__group__add_1115,axiom,
ordered_ab_group_add @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Groups_Ocancel__semigroup__add_1116,axiom,
cancel_semigroup_add @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Rings_Olinordered__semiring_1117,axiom,
linordered_semiring @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Least__significant__bit_Olsb_1118,axiom,
least_6119777620449941438nt_lsb @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Rings_Oordered__semiring__0_1119,axiom,
ordered_semiring_0 @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Rings_Olinordered__semidom_1120,axiom,
linordered_semidom @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Groups_Oab__semigroup__mult_1121,axiom,
ab_semigroup_mult @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Rings_Osemiring__1__cancel_1122,axiom,
semiring_1_cancel @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Rings_Oalgebraic__semidom_1123,axiom,
algebraic_semidom @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Groups_Ocomm__monoid__mult_1124,axiom,
comm_monoid_mult @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Groups_Oab__semigroup__add_1125,axiom,
ab_semigroup_add @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Generic__set__bit_Oset__bit_1126,axiom,
generic_set_set_bit @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Code__Evaluation_Oterm__of_1127,axiom,
code_term_of @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Rings_Oordered__semiring_1128,axiom,
ordered_semiring @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Rings_Oordered__ring__abs_1129,axiom,
ordered_ring_abs @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Parity_Osemiring__parity_1130,axiom,
semiring_parity @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Groups_Ocomm__monoid__add_1131,axiom,
comm_monoid_add @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Rings_Osemiring__modulo_1132,axiom,
semiring_modulo @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Rings_Olinordered__ring_1133,axiom,
linordered_ring @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Rings_Olinordered__idom_1134,axiom,
linordered_idom @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Rings_Ocomm__semiring__1_1135,axiom,
comm_semiring_1 @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Rings_Ocomm__semiring__0_1136,axiom,
comm_semiring_0 @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Groups_Osemigroup__mult_1137,axiom,
semigroup_mult @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Rings_Osemidom__modulo_1138,axiom,
semidom_modulo @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Rings_Osemidom__divide_1139,axiom,
semidom_divide @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Num_Osemiring__numeral_1140,axiom,
semiring_numeral @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Groups_Osemigroup__add_1141,axiom,
semigroup_add @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Rings_Ozero__less__one_1142,axiom,
zero_less_one @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Rings_Ocomm__semiring_1143,axiom,
comm_semiring @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Nat_Osemiring__char__0_1144,axiom,
semiring_char_0 @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Groups_Oab__group__add_1145,axiom,
ab_group_add @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Rings_Ozero__neq__one_1146,axiom,
zero_neq_one @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Rings_Oordered__ring_1147,axiom,
ordered_ring @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Rings_Oidom__abs__sgn_1148,axiom,
idom_abs_sgn @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Parity_Oring__parity_1149,axiom,
ring_parity @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Orderings_Opreorder_1150,axiom,
preorder @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Orderings_Olinorder_1151,axiom,
linorder @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Groups_Omonoid__mult_1152,axiom,
monoid_mult @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Rings_Oidom__modulo_1153,axiom,
idom_modulo @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Rings_Oidom__divide_1154,axiom,
idom_divide @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Rings_Ocomm__ring__1_1155,axiom,
comm_ring_1 @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Groups_Omonoid__add_1156,axiom,
monoid_add @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Rings_Osemiring__1_1157,axiom,
semiring_1 @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Rings_Osemiring__0_1158,axiom,
semiring_0 @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Groups_Ogroup__add_1159,axiom,
group_add @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Rings_Omult__zero_1160,axiom,
mult_zero @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Rings_Ocomm__ring_1161,axiom,
comm_ring @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Orderings_Oorder_1162,axiom,
order @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Num_Oneg__numeral_1163,axiom,
neg_numeral @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Nat_Oring__char__0_1164,axiom,
ring_char_0 @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Rings_Osemiring_1165,axiom,
semiring @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Orderings_Oord_1166,axiom,
ord @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Groups_Ouminus_1167,axiom,
uminus @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Rings_Oring__1_1168,axiom,
ring_1 @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Rings_Oabs__if_1169,axiom,
abs_if @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Groups_Ominus_1170,axiom,
minus @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Power_Opower_1171,axiom,
power @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Num_Onumeral_1172,axiom,
numeral @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Groups_Ozero_1173,axiom,
zero @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Groups_Oplus_1174,axiom,
plus @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Rings_Oring_1175,axiom,
ring @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Rings_Oidom_1176,axiom,
idom @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Groups_Oone_1177,axiom,
one @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Rings_Odvd_1178,axiom,
dvd @ code_integer ).
thf(tcon_Heap__Time__Monad_OHeap___Quickcheck__Narrowing_Opartial__term__of_1179,axiom,
! [A13: $tType] :
( ( typerep @ A13 )
=> ( quickc6926020345158392990erm_of @ ( heap_Time_Heap @ A13 ) ) ) ).
thf(tcon_Heap__Time__Monad_OHeap___Code__Evaluation_Oterm__of_1180,axiom,
! [A13: $tType] :
( ( typerep @ A13 )
=> ( code_term_of @ ( heap_Time_Heap @ A13 ) ) ) ).
thf(tcon_Heap__Time__Monad_OHeap___Nat_Osize_1181,axiom,
! [A13: $tType] : ( size @ ( heap_Time_Heap @ A13 ) ) ).
thf(tcon_VEBT__Definitions_OVEBT___Quickcheck__Narrowing_Opartial__term__of_1182,axiom,
quickc6926020345158392990erm_of @ vEBT_VEBT ).
thf(tcon_VEBT__Definitions_OVEBT___Code__Evaluation_Oterm__of_1183,axiom,
code_term_of @ vEBT_VEBT ).
thf(tcon_VEBT__Definitions_OVEBT___Nat_Osize_1184,axiom,
size @ vEBT_VEBT ).
thf(tcon_VEBT__BuildupMemImp_OVEBTi___Quickcheck__Narrowing_Opartial__term__of_1185,axiom,
quickc6926020345158392990erm_of @ vEBT_VEBTi ).
thf(tcon_VEBT__BuildupMemImp_OVEBTi___Code__Evaluation_Oterm__of_1186,axiom,
code_term_of @ vEBT_VEBTi ).
thf(tcon_VEBT__BuildupMemImp_OVEBTi___Heap_Oheap_1187,axiom,
heap @ vEBT_VEBTi ).
thf(tcon_VEBT__BuildupMemImp_OVEBTi___Nat_Osize_1188,axiom,
size @ vEBT_VEBTi ).
% Helper facts (4)
thf(help_If_3_1_T,axiom,
! [P: $o] :
( ( P = $true )
| ( P = $false ) ) ).
thf(help_If_2_1_T,axiom,
! [A: $tType,X2: A,Y3: A] :
( ( if @ A @ $false @ X2 @ Y3 )
= Y3 ) ).
thf(help_If_1_1_T,axiom,
! [A: $tType,X2: A,Y3: A] :
( ( if @ A @ $true @ X2 @ Y3 )
= X2 ) ).
thf(help_fChoice_1_1_T,axiom,
! [A: $tType,P: A > $o] :
( ( P @ ( fChoice @ A @ P ) )
= ( ? [X7: A] : ( P @ X7 ) ) ) ).
% Conjectures (10)
thf(conj_0,hypothesis,
( tia
= ( vEBT_Nodei @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ mi @ ma ) ) @ ( suc @ ( suc @ va ) ) @ x13 @ x14 ) ) ).
thf(conj_1,hypothesis,
( ( size_size @ ( list @ vEBT_VEBTi ) @ tree_is )
= ( size_size @ ( list @ vEBT_VEBT ) @ treeList ) ) ).
thf(conj_2,hypothesis,
xa != mi ).
thf(conj_3,hypothesis,
~ ( ord_less @ nat @ xa @ mi ) ).
thf(conj_4,hypothesis,
~ ( ord_less @ nat @ ma @ xa ) ).
thf(conj_5,hypothesis,
( xaa
= ( vEBT_vebt_delete @ ( nth @ vEBT_VEBT @ treeList @ ( vEBT_VEBT_high @ xa @ ( suc @ ( divide_divide @ nat @ va @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) @ ( vEBT_VEBT_low @ xa @ ( suc @ ( divide_divide @ nat @ va @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) ) ).
thf(conj_6,hypothesis,
( ( vEBT_vebt_mint @ ( vEBT_vebt_delete @ ( nth @ vEBT_VEBT @ treeList @ ( vEBT_VEBT_high @ xa @ ( suc @ ( divide_divide @ nat @ va @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) @ ( vEBT_VEBT_low @ xa @ ( suc @ ( divide_divide @ nat @ va @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) )
= ( none @ nat ) ) ).
thf(conj_7,hypothesis,
( xba
= ( none @ nat ) ) ).
thf(conj_8,hypothesis,
xa != ma ).
thf(conj_9,conjecture,
entails @ ( times_times @ assn @ ( times_times @ assn @ ( vEBT_vebt_assn_raw @ ( vEBT_vebt_delete @ summary @ ( vEBT_VEBT_high @ xa @ ( suc @ ( divide_divide @ nat @ va @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) @ xc ) @ ( snga_assn @ vEBT_VEBTi @ x13 @ ( list_update @ vEBT_VEBTi @ tree_is @ ( vEBT_VEBT_high @ xa @ ( suc @ ( divide_divide @ nat @ va @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ xb ) ) ) @ ( vEBT_List_list_assn @ vEBT_VEBT @ vEBT_VEBTi @ vEBT_vebt_assn_raw @ ( list_update @ vEBT_VEBT @ treeList @ ( vEBT_VEBT_high @ xa @ ( suc @ ( divide_divide @ nat @ va @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( vEBT_vebt_delete @ ( nth @ vEBT_VEBT @ treeList @ ( vEBT_VEBT_high @ xa @ ( suc @ ( divide_divide @ nat @ va @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) @ ( vEBT_VEBT_low @ xa @ ( suc @ ( divide_divide @ nat @ va @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) ) @ ( list_update @ vEBT_VEBTi @ tree_is @ ( vEBT_VEBT_high @ xa @ ( suc @ ( divide_divide @ nat @ va @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ xb ) ) ) @ ( vEBT_vebt_assn_raw @ ( if @ vEBT_VEBT @ ( ord_less @ nat @ ( vEBT_VEBT_high @ xa @ ( suc @ ( divide_divide @ nat @ va @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( size_size @ ( list @ vEBT_VEBT ) @ treeList ) ) @ ( if @ vEBT_VEBT @ ( vEBT_VEBT_minNull @ ( vEBT_vebt_delete @ ( nth @ vEBT_VEBT @ treeList @ ( vEBT_VEBT_high @ xa @ ( suc @ ( divide_divide @ nat @ va @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) @ ( vEBT_VEBT_low @ xa @ ( divide_divide @ nat @ ( suc @ ( suc @ va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) @ ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ mi @ ma ) ) @ ( suc @ ( suc @ va ) ) @ ( list_update @ vEBT_VEBT @ treeList @ ( vEBT_VEBT_high @ xa @ ( suc @ ( divide_divide @ nat @ va @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( vEBT_vebt_delete @ ( nth @ vEBT_VEBT @ treeList @ ( vEBT_VEBT_high @ xa @ ( suc @ ( divide_divide @ nat @ va @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) @ ( vEBT_VEBT_low @ xa @ ( divide_divide @ nat @ ( suc @ ( suc @ va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) @ ( vEBT_vebt_delete @ summary @ ( vEBT_VEBT_high @ xa @ ( suc @ ( divide_divide @ nat @ va @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) ) @ ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ mi @ ma ) ) @ ( suc @ ( suc @ va ) ) @ ( list_update @ vEBT_VEBT @ treeList @ ( vEBT_VEBT_high @ xa @ ( suc @ ( divide_divide @ nat @ va @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( vEBT_vebt_delete @ ( nth @ vEBT_VEBT @ treeList @ ( vEBT_VEBT_high @ xa @ ( suc @ ( divide_divide @ nat @ va @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) @ ( vEBT_VEBT_low @ xa @ ( divide_divide @ nat @ ( suc @ ( suc @ va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) @ summary ) ) @ ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ mi @ ma ) ) @ ( suc @ ( suc @ va ) ) @ treeList @ summary ) ) @ ( vEBT_Nodei @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ mi @ ma ) ) @ ( suc @ ( suc @ va ) ) @ x13 @ xc ) ) ).
%------------------------------------------------------------------------------