TPTP Problem File: ITP288^2.p
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- Solve Problem
%------------------------------------------------------------------------------
% File : ITP288^2 : TPTP v8.2.0. Released v8.0.0.
% Domain : Interactive Theorem Proving
% Problem : Sledgehammer problem VEBT_DelImperative 00285_019298
% 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_00285_019298 [Des22]
% Status : Theorem
% Rating : 1.00 v8.1.0
% Syntax : Number of formulae : 10233 (3033 unt; 646 typ; 0 def)
% Number of atoms : 30704 (10184 equ; 0 cnn)
% Maximal formula atoms : 71 ( 3 avg)
% Number of connectives : 206373 (2551 ~; 339 |;2341 &;186866 @)
% ( 0 <=>;14276 =>; 0 <=; 0 <~>)
% Maximal formula depth : 40 ( 7 avg)
% Number of types : 20 ( 19 usr)
% Number of type conns : 3408 (3408 >; 0 *; 0 +; 0 <<)
% Number of symbols : 630 ( 627 usr; 23 con; 0-13 aty)
% Number of variables : 29593 (2085 ^;26132 !; 915 ?;29593 :)
% ( 461 !>; 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 21:37:39.061
%------------------------------------------------------------------------------
% Could-be-implicit typings (34)
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_Code__Evaluation_Oterm,type,
code_term: $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_String_Ochar,type,
char: $tType ).
thf(ty_t_Heap_Oarray,type,
array: $tType > $tType ).
thf(ty_t_Word_Oword,type,
word: $tType > $tType ).
thf(ty_t_Real_Oreal,type,
real: $tType ).
thf(ty_t_List_Olist,type,
list: $tType > $tType ).
thf(ty_t_Heap_Oref,type,
ref: $tType > $tType ).
thf(ty_t_Set_Oset,type,
set: $tType > $tType ).
thf(ty_t_Rat_Orat,type,
rat: $tType ).
thf(ty_t_Num_Onum,type,
num: $tType ).
thf(ty_t_Nat_Onat,type,
nat: $tType ).
thf(ty_t_Int_Oint,type,
int: $tType ).
thf(ty_t_itself,type,
itself: $tType > $tType ).
% Explicit typings (612)
thf(sy_cl_Enum_Oenum,type,
enum:
!>[A: $tType] : $o ).
thf(sy_cl_Typerep_Otyperep,type,
typerep:
!>[A: $tType] : $o ).
thf(sy_cl_HOL_Otype,type,
type:
!>[A: $tType] : $o ).
thf(sy_cl_Nat_Osize,type,
size:
!>[A: $tType] : $o ).
thf(sy_cl_HOL_Oequal,type,
cl_HOL_Oequal:
!>[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_Groups_Otimes,type,
times:
!>[A: $tType] : $o ).
thf(sy_cl_Rings_Oabs__if,type,
abs_if:
!>[A: $tType] : $o ).
thf(sy_cl_Rings_Oring__1,type,
ring_1:
!>[A: $tType] : $o ).
thf(sy_cl_Groups_Ouminus,type,
uminus:
!>[A: $tType] : $o ).
thf(sy_cl_Orderings_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_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_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_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_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_Semiring__Normalization_Ocomm__semiring__1__cancel__crossproduct,type,
semiri1453513574482234551roduct:
!>[A: $tType] : $o ).
thf(sy_cl_Bit__Operations_Ounique__euclidean__semiring__with__bit__operations,type,
bit_un5681908812861735899ations:
!>[A: $tType] : $o ).
thf(sy_cl_Conditionally__Complete__Lattices_Oconditionally__complete__lattice,type,
condit1219197933456340205attice:
!>[A: $tType] : $o ).
thf(sy_cl_Conditionally__Complete__Lattices_Oconditionally__complete__linorder,type,
condit6923001295902523014norder:
!>[A: $tType] : $o ).
thf(sy_c_Archimedean__Field_Oceiling,type,
archimedean_ceiling:
!>[A: $tType] : ( A > int ) ).
thf(sy_c_Archimedean__Field_Ofloor__ceiling__class_Ofloor,type,
archim6421214686448440834_floor:
!>[A: $tType] : ( A > int ) ).
thf(sy_c_Archimedean__Field_Ofrac,type,
archimedean_frac:
!>[A: $tType] : ( A > A ) ).
thf(sy_c_Archimedean__Field_Oround,type,
archimedean_round:
!>[A: $tType] : ( A > int ) ).
thf(sy_c_Array__Time_Olen,type,
array_len:
!>[A: $tType] : ( ( array @ A ) > ( heap_Time_Heap @ nat ) ) ).
thf(sy_c_Array__Time_Omake,type,
array_make:
!>[A: $tType] : ( nat > ( nat > A ) > ( heap_Time_Heap @ ( array @ A ) ) ) ).
thf(sy_c_Array__Time_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_Binomial_Obinomial,type,
binomial: nat > nat > nat ).
thf(sy_c_Binomial_Ogbinomial,type,
gbinomial:
!>[A: $tType] : ( A > nat > A ) ).
thf(sy_c_Bit__Comprehension_Obit__comprehension__class_Oset__bits,type,
bit_bi4170147762399595738t_bits:
!>[A: $tType] : ( ( nat > $o ) > A ) ).
thf(sy_c_Bit__Comprehension_Owf__set__bits__int,type,
bit_wf_set_bits_int: ( nat > $o ) > $o ).
thf(sy_c_Bit__Operations_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__Numeral_Ointeger__of__nat,type,
code_integer_of_nat: nat > code_integer ).
thf(sy_c_Code__Target__Word__Base_Oquickcheck__narrowing__samples_Onarrowing__samples,type,
code_T4080844693773952564amples:
!>[A: $tType] : ( ( code_integer > ( product_prod @ A @ A ) ) > A > code_integer > ( list @ A ) ) ).
thf(sy_c_Code__Target__Word__Base_Oquickcheck__narrowing__samples_Onarrowing__samples__rel,type,
code_T1710151556404007877es_rel:
!>[A: $tType] : ( ( code_integer > ( product_prod @ A @ A ) ) > code_integer > code_integer > $o ) ).
thf(sy_c_Code__Target__Word__Base_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_Filter_Omap__filter__on,type,
map_filter_on:
!>[A: $tType,B: $tType] : ( ( set @ A ) > ( A > B ) > ( filter @ A ) > ( filter @ B ) ) ).
thf(sy_c_Finite__Set_Ocard,type,
finite_card:
!>[B: $tType] : ( ( set @ B ) > nat ) ).
thf(sy_c_Finite__Set_Ofinite,type,
finite_finite2:
!>[A: $tType] : ( ( set @ A ) > $o ) ).
thf(sy_c_Finite__Set_Ofold,type,
finite_fold:
!>[A: $tType,B: $tType] : ( ( A > B > B ) > B > ( set @ A ) > B ) ).
thf(sy_c_Fun_Obij__betw,type,
bij_betw:
!>[A: $tType,B: $tType] : ( ( A > B ) > ( set @ A ) > ( set @ B ) > $o ) ).
thf(sy_c_Fun_Ocomp,type,
comp:
!>[B: $tType,C: $tType,A: $tType] : ( ( B > C ) > ( A > B ) > A > C ) ).
thf(sy_c_Fun_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_Oappend,type,
append:
!>[A: $tType] : ( ( list @ A ) > ( list @ A ) > ( list @ A ) ) ).
thf(sy_c_List_Oconcat,type,
concat:
!>[A: $tType] : ( ( list @ ( list @ A ) ) > ( list @ A ) ) ).
thf(sy_c_List_Ocount__list,type,
count_list:
!>[A: $tType] : ( ( list @ A ) > A > nat ) ).
thf(sy_c_List_Odistinct,type,
distinct:
!>[A: $tType] : ( ( list @ A ) > $o ) ).
thf(sy_c_List_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_ONil,type,
nil:
!>[A: $tType] : ( list @ A ) ).
thf(sy_c_List_Olist_Omap,type,
map:
!>[A: $tType,Aa: $tType] : ( ( A > Aa ) > ( list @ A ) > ( list @ Aa ) ) ).
thf(sy_c_List_Olist_Oset,type,
set2:
!>[A: $tType] : ( ( list @ A ) > ( set @ A ) ) ).
thf(sy_c_List_Olist_Osize__list,type,
size_list:
!>[A: $tType] : ( ( A > nat ) > ( list @ A ) > nat ) ).
thf(sy_c_List_Olist__update,type,
list_update:
!>[A: $tType] : ( ( list @ A ) > nat > A > ( list @ A ) ) ).
thf(sy_c_List_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_Oremdups,type,
remdups:
!>[A: $tType] : ( ( list @ A ) > ( 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_Otranspose,type,
transpose:
!>[A: $tType] : ( ( list @ ( list @ A ) ) > ( list @ ( list @ A ) ) ) ).
thf(sy_c_List_Oupt,type,
upt: nat > nat > ( list @ nat ) ).
thf(sy_c_List_Oupto__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_OAbs__bit0_H,type,
numeral_Abs_bit0:
!>[A: $tType] : ( int > ( numeral_bit0 @ A ) ) ).
thf(sy_c_Numeral__Type_OAbs__bit1_H,type,
numeral_Abs_bit1:
!>[A: $tType] : ( int > ( numeral_bit1 @ A ) ) ).
thf(sy_c_Numeral__Type_Obit0_OAbs__bit0,type,
numeral_Abs_bit02:
!>[A: $tType] : ( int > ( numeral_bit0 @ A ) ) ).
thf(sy_c_Numeral__Type_Obit0_ORep__bit0,type,
numeral_Rep_bit0:
!>[A: $tType] : ( ( numeral_bit0 @ A ) > int ) ).
thf(sy_c_Numeral__Type_Obit1_OAbs__bit1,type,
numeral_Abs_bit12:
!>[A: $tType] : ( int > ( numeral_bit1 @ A ) ) ).
thf(sy_c_Numeral__Type_Obit1_ORep__bit1,type,
numeral_Rep_bit1:
!>[A: $tType] : ( ( numeral_bit1 @ A ) > int ) ).
thf(sy_c_Numeral__Type_Omod__type,type,
numeral_mod_type:
!>[A: $tType] : ( int > ( A > int ) > ( int > A ) > $o ) ).
thf(sy_c_Option_Ooption_ONone,type,
none:
!>[A: $tType] : ( option @ A ) ).
thf(sy_c_Option_Ooption_OSome,type,
some:
!>[A: $tType] : ( A > ( option @ A ) ) ).
thf(sy_c_Option_Ooption_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_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_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__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_String_Oascii__of,type,
ascii_of: char > char ).
thf(sy_c_String_Ochar_OChar,type,
char2: $o > $o > $o > $o > $o > $o > $o > $o > char ).
thf(sy_c_String_Ocomm__semiring__1__class_Oof__char,type,
comm_s6883823935334413003f_char:
!>[A: $tType] : ( char > A ) ).
thf(sy_c_String_Ointeger__of__char,type,
integer_of_char: char > code_integer ).
thf(sy_c_String_Ounique__euclidean__semiring__with__bit__operations__class_Ochar__of,type,
unique5772411509450598832har_of:
!>[A: $tType] : ( A > char ) ).
thf(sy_c_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_Typedef_Otype__definition,type,
type_definition:
!>[B: $tType,A: $tType] : ( ( B > A ) > ( A > B ) > ( set @ A ) > $o ) ).
thf(sy_c_Uint32_Odiv0__uint32,type,
div0_uint32: uint32 > uint32 ).
thf(sy_c_Uint32_Ointeger__of__uint32,type,
integer_of_uint32: uint32 > code_integer ).
thf(sy_c_Uint32_Ointeger__of__uint32__signed,type,
intege5370686899274169573signed: uint32 > code_integer ).
thf(sy_c_Uint32_Omod0__uint32,type,
mod0_uint32: uint32 > uint32 ).
thf(sy_c_Uint32_Oset__bits__aux__uint32,type,
set_bits_aux_uint32: ( nat > $o ) > nat > uint32 > uint32 ).
thf(sy_c_Uint32_Osigned__drop__bit__uint32,type,
signed489701013188660438uint32: nat > uint32 > uint32 ).
thf(sy_c_Uint32_Ouint32__divmod,type,
uint32_divmod: uint32 > uint32 > ( product_prod @ uint32 @ uint32 ) ).
thf(sy_c_Uint32_Ouint32__sdiv,type,
uint32_sdiv: uint32 > uint32 > uint32 ).
thf(sy_c_Uint32_Ouint32__set__bit,type,
uint32_set_bit: uint32 > code_integer > $o > uint32 ).
thf(sy_c_Uint32_Ouint32__shiftl,type,
uint32_shiftl: uint32 > code_integer > uint32 ).
thf(sy_c_Uint32_Ouint32__shiftr,type,
uint32_shiftr: uint32 > code_integer > uint32 ).
thf(sy_c_Uint32_Ouint32__sshiftr,type,
uint32_sshiftr: uint32 > code_integer > uint32 ).
thf(sy_c_Uint32_Ouint32__test__bit,type,
uint32_test_bit: uint32 > code_integer > $o ).
thf(sy_c_VEBT__Bounds_OT_092_060_094sub_062i_092_060_094sub_062n_092_060_094sub_062s_092_060_094sub_062e_092_060_094sub_062r_092_060_094sub_062t,type,
vEBT_T_i_n_s_e_r_t: vEBT_VEBT > nat > nat ).
thf(sy_c_VEBT__Bounds_OT_092_060_094sub_062i_092_060_094sub_062n_092_060_094sub_062s_092_060_094sub_062e_092_060_094sub_062r_092_060_094sub_062t_H,type,
vEBT_T_i_n_s_e_r_t2: vEBT_VEBT > nat > nat ).
thf(sy_c_VEBT__Bounds_OT_092_060_094sub_062i_092_060_094sub_062n_092_060_094sub_062s_092_060_094sub_062e_092_060_094sub_062r_092_060_094sub_062t_H__rel,type,
vEBT_T5076183648494686801_t_rel: ( product_prod @ vEBT_VEBT @ nat ) > ( product_prod @ vEBT_VEBT @ nat ) > $o ).
thf(sy_c_VEBT__Bounds_OT_092_060_094sub_062i_092_060_094sub_062n_092_060_094sub_062s_092_060_094sub_062e_092_060_094sub_062r_092_060_094sub_062t__rel,type,
vEBT_T9217963907923527482_t_rel: ( product_prod @ vEBT_VEBT @ nat ) > ( product_prod @ vEBT_VEBT @ nat ) > $o ).
thf(sy_c_VEBT__Bounds_OT_092_060_094sub_062m_092_060_094sub_062a_092_060_094sub_062x_092_060_094sub_062t,type,
vEBT_T_m_a_x_t: vEBT_VEBT > nat ).
thf(sy_c_VEBT__Bounds_OT_092_060_094sub_062m_092_060_094sub_062a_092_060_094sub_062x_092_060_094sub_062t__rel,type,
vEBT_T_m_a_x_t_rel: vEBT_VEBT > vEBT_VEBT > $o ).
thf(sy_c_VEBT__Bounds_OT_092_060_094sub_062m_092_060_094sub_062e_092_060_094sub_062m_092_060_094sub_062b_092_060_094sub_062e_092_060_094sub_062r,type,
vEBT_T_m_e_m_b_e_r: vEBT_VEBT > nat > nat ).
thf(sy_c_VEBT__Bounds_OT_092_060_094sub_062m_092_060_094sub_062e_092_060_094sub_062m_092_060_094sub_062b_092_060_094sub_062e_092_060_094sub_062r_H,type,
vEBT_T_m_e_m_b_e_r2: vEBT_VEBT > nat > nat ).
thf(sy_c_VEBT__Bounds_OT_092_060_094sub_062m_092_060_094sub_062e_092_060_094sub_062m_092_060_094sub_062b_092_060_094sub_062e_092_060_094sub_062r_H__rel,type,
vEBT_T8099345112685741742_r_rel: ( product_prod @ vEBT_VEBT @ nat ) > ( product_prod @ vEBT_VEBT @ nat ) > $o ).
thf(sy_c_VEBT__Bounds_OT_092_060_094sub_062m_092_060_094sub_062e_092_060_094sub_062m_092_060_094sub_062b_092_060_094sub_062e_092_060_094sub_062r__rel,type,
vEBT_T5837161174952499735_r_rel: ( product_prod @ vEBT_VEBT @ nat ) > ( product_prod @ vEBT_VEBT @ nat ) > $o ).
thf(sy_c_VEBT__Bounds_OT_092_060_094sub_062m_092_060_094sub_062i_092_060_094sub_062n_092_060_094sub_062N_092_060_094sub_062u_092_060_094sub_062l_092_060_094sub_062l,type,
vEBT_T_m_i_n_N_u_l_l: vEBT_VEBT > nat ).
thf(sy_c_VEBT__Bounds_OT_092_060_094sub_062m_092_060_094sub_062i_092_060_094sub_062n_092_060_094sub_062N_092_060_094sub_062u_092_060_094sub_062l_092_060_094sub_062l__rel,type,
vEBT_T5462971552011256508_l_rel: vEBT_VEBT > vEBT_VEBT > $o ).
thf(sy_c_VEBT__Bounds_OT_092_060_094sub_062m_092_060_094sub_062i_092_060_094sub_062n_092_060_094sub_062t,type,
vEBT_T_m_i_n_t: vEBT_VEBT > nat ).
thf(sy_c_VEBT__Bounds_OT_092_060_094sub_062m_092_060_094sub_062i_092_060_094sub_062n_092_060_094sub_062t__rel,type,
vEBT_T_m_i_n_t_rel: vEBT_VEBT > vEBT_VEBT > $o ).
thf(sy_c_VEBT__Bounds_OT_092_060_094sub_062p_092_060_094sub_062r_092_060_094sub_062e_092_060_094sub_062d,type,
vEBT_T_p_r_e_d: vEBT_VEBT > nat > nat ).
thf(sy_c_VEBT__Bounds_OT_092_060_094sub_062p_092_060_094sub_062r_092_060_094sub_062e_092_060_094sub_062d_H,type,
vEBT_T_p_r_e_d2: vEBT_VEBT > nat > nat ).
thf(sy_c_VEBT__Bounds_OT_092_060_094sub_062p_092_060_094sub_062r_092_060_094sub_062e_092_060_094sub_062d_H__rel,type,
vEBT_T_p_r_e_d_rel: ( product_prod @ vEBT_VEBT @ nat ) > ( product_prod @ vEBT_VEBT @ nat ) > $o ).
thf(sy_c_VEBT__Bounds_OT_092_060_094sub_062p_092_060_094sub_062r_092_060_094sub_062e_092_060_094sub_062d__rel,type,
vEBT_T_p_r_e_d_rel2: ( product_prod @ vEBT_VEBT @ nat ) > ( product_prod @ vEBT_VEBT @ nat ) > $o ).
thf(sy_c_VEBT__Bounds_OT_092_060_094sub_062s_092_060_094sub_062u_092_060_094sub_062c_092_060_094sub_062c,type,
vEBT_T_s_u_c_c: vEBT_VEBT > nat > nat ).
thf(sy_c_VEBT__Bounds_OT_092_060_094sub_062s_092_060_094sub_062u_092_060_094sub_062c_092_060_094sub_062c_H,type,
vEBT_T_s_u_c_c2: vEBT_VEBT > nat > nat ).
thf(sy_c_VEBT__Bounds_OT_092_060_094sub_062s_092_060_094sub_062u_092_060_094sub_062c_092_060_094sub_062c_H__rel,type,
vEBT_T_s_u_c_c_rel: ( product_prod @ vEBT_VEBT @ nat ) > ( product_prod @ vEBT_VEBT @ nat ) > $o ).
thf(sy_c_VEBT__Bounds_OT_092_060_094sub_062s_092_060_094sub_062u_092_060_094sub_062c_092_060_094sub_062c__rel,type,
vEBT_T_s_u_c_c_rel2: ( product_prod @ vEBT_VEBT @ nat ) > ( product_prod @ vEBT_VEBT @ nat ) > $o ).
thf(sy_c_VEBT__BuildupMemImp_OVEBT__internal_OT__vebt__buildupi,type,
vEBT_V441764108873111860ildupi: nat > nat ).
thf(sy_c_VEBT__BuildupMemImp_OVEBT__internal_OT__vebt__buildupi_H,type,
vEBT_V9176841429113362141ildupi: nat > int ).
thf(sy_c_VEBT__BuildupMemImp_OVEBT__internal_OT__vebt__buildupi_H__rel,type,
vEBT_V3352910403632780892pi_rel: nat > nat > $o ).
thf(sy_c_VEBT__BuildupMemImp_OVEBT__internal_OT__vebt__buildupi__rel,type,
vEBT_V2957053500504383685pi_rel: nat > nat > $o ).
thf(sy_c_VEBT__BuildupMemImp_OVEBT__internal_OTb,type,
vEBT_VEBT_Tb: nat > int ).
thf(sy_c_VEBT__BuildupMemImp_OVEBT__internal_OTb_H,type,
vEBT_VEBT_Tb2: nat > nat ).
thf(sy_c_VEBT__BuildupMemImp_OVEBT__internal_OTb_H__rel,type,
vEBT_VEBT_Tb_rel: nat > nat > $o ).
thf(sy_c_VEBT__BuildupMemImp_OVEBT__internal_OTb__rel,type,
vEBT_VEBT_Tb_rel2: nat > nat > $o ).
thf(sy_c_VEBT__BuildupMemImp_OVEBT__internal_Ohighi,type,
vEBT_VEBT_highi: nat > nat > ( heap_Time_Heap @ nat ) ).
thf(sy_c_VEBT__BuildupMemImp_OVEBT__internal_Olowi,type,
vEBT_VEBT_lowi: nat > nat > ( heap_Time_Heap @ nat ) ).
thf(sy_c_VEBT__BuildupMemImp_OVEBT__internal_OminNulli,type,
vEBT_VEBT_minNulli: vEBT_VEBTi > ( heap_Time_Heap @ $o ) ).
thf(sy_c_VEBT__BuildupMemImp_OVEBT__internal_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_Osize__VEBT,type,
vEBT_size_VEBT: vEBT_VEBT > nat ).
thf(sy_c_VEBT__Definitions_OVEBT__internal_Oboth__member__options,type,
vEBT_V8194947554948674370ptions: vEBT_VEBT > nat > $o ).
thf(sy_c_VEBT__Definitions_OVEBT__internal_Ohigh,type,
vEBT_VEBT_high: nat > nat > nat ).
thf(sy_c_VEBT__Definitions_OVEBT__internal_Oin__children,type,
vEBT_V5917875025757280293ildren: nat > ( list @ vEBT_VEBT ) > nat > $o ).
thf(sy_c_VEBT__Definitions_OVEBT__internal_Olow,type,
vEBT_VEBT_low: nat > nat > nat ).
thf(sy_c_VEBT__Definitions_OVEBT__internal_Omembermima,type,
vEBT_VEBT_membermima: vEBT_VEBT > nat > $o ).
thf(sy_c_VEBT__Definitions_OVEBT__internal_Omembermima__rel,type,
vEBT_V4351362008482014158ma_rel: ( product_prod @ vEBT_VEBT @ nat ) > ( product_prod @ vEBT_VEBT @ nat ) > $o ).
thf(sy_c_VEBT__Definitions_OVEBT__internal_Onaive__member,type,
vEBT_V5719532721284313246member: vEBT_VEBT > nat > $o ).
thf(sy_c_VEBT__Definitions_OVEBT__internal_Onaive__member__rel,type,
vEBT_V5765760719290551771er_rel: ( product_prod @ vEBT_VEBT @ nat ) > ( product_prod @ vEBT_VEBT @ nat ) > $o ).
thf(sy_c_VEBT__Definitions_OVEBT__internal_Ovalid_H,type,
vEBT_VEBT_valid: vEBT_VEBT > nat > $o ).
thf(sy_c_VEBT__Definitions_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__List__Assn_Olist__assn__rel,type,
vEBT_L4249061453398456502sn_rel:
!>[A: $tType,C: $tType] : ( ( product_prod @ ( A > C > assn ) @ ( product_prod @ ( list @ A ) @ ( list @ C ) ) ) > ( product_prod @ ( A > C > assn ) @ ( product_prod @ ( list @ A ) @ ( list @ C ) ) ) > $o ) ).
thf(sy_c_VEBT__Member_OVEBT__internal_Obit__concat,type,
vEBT_VEBT_bit_concat: nat > nat > nat > nat ).
thf(sy_c_VEBT__Member_OVEBT__internal_OminNull,type,
vEBT_VEBT_minNull: vEBT_VEBT > $o ).
thf(sy_c_VEBT__Member_OVEBT__internal_OminNull__rel,type,
vEBT_V6963167321098673237ll_rel: vEBT_VEBT > vEBT_VEBT > $o ).
thf(sy_c_VEBT__Member_OVEBT__internal_Oset__vebt_H,type,
vEBT_VEBT_set_vebt: vEBT_VEBT > ( set @ nat ) ).
thf(sy_c_VEBT__Member_Ovebt__member,type,
vEBT_vebt_member: vEBT_VEBT > nat > $o ).
thf(sy_c_VEBT__Member_Ovebt__member__rel,type,
vEBT_vebt_member_rel: ( product_prod @ vEBT_VEBT @ nat ) > ( product_prod @ vEBT_VEBT @ nat ) > $o ).
thf(sy_c_VEBT__MinMax_OVEBT__internal_Oadd,type,
vEBT_VEBT_add: ( option @ nat ) > ( option @ nat ) > ( option @ nat ) ).
thf(sy_c_VEBT__MinMax_OVEBT__internal_Ogreater,type,
vEBT_VEBT_greater: ( option @ nat ) > ( option @ nat ) > $o ).
thf(sy_c_VEBT__MinMax_OVEBT__internal_Oless,type,
vEBT_VEBT_less: ( option @ nat ) > ( option @ nat ) > $o ).
thf(sy_c_VEBT__MinMax_OVEBT__internal_Olesseq,type,
vEBT_VEBT_lesseq: ( option @ nat ) > ( option @ nat ) > $o ).
thf(sy_c_VEBT__MinMax_OVEBT__internal_Omax__in__set,type,
vEBT_VEBT_max_in_set: ( set @ nat ) > nat > $o ).
thf(sy_c_VEBT__MinMax_OVEBT__internal_Omin__in__set,type,
vEBT_VEBT_min_in_set: ( set @ nat ) > nat > $o ).
thf(sy_c_VEBT__MinMax_OVEBT__internal_Omul,type,
vEBT_VEBT_mul: ( option @ nat ) > ( option @ nat ) > ( option @ nat ) ).
thf(sy_c_VEBT__MinMax_OVEBT__internal_Ooption__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_Ofinite__psubset,type,
finite_psubset:
!>[A: $tType] : ( set @ ( product_prod @ ( set @ A ) @ ( set @ A ) ) ) ).
thf(sy_c_Wellfounded_Omeasure,type,
measure:
!>[A: $tType] : ( ( A > nat ) > ( set @ ( product_prod @ A @ A ) ) ) ).
thf(sy_c_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_Word__Type__Copies_Oword__type__copy__misc,type,
word_T1870391057261780392y_misc:
!>[B: $tType,A: $tType] : ( ( ( word @ B ) > A ) > ( A > ( word @ B ) ) > ( nat > A > A ) > ( nat > A ) > ( A > nat ) > ( int > A ) > ( A > int ) > ( code_integer > A ) > ( A > code_integer ) > nat > ( ( nat > $o ) > nat > A > A ) > $o ) ).
thf(sy_c_Word__Type__Copies_Oword__type__copy__misc__axioms,type,
word_T8964210463127689547axioms:
!>[A: $tType,B: $tType] : ( ( A > ( word @ B ) ) > nat > ( ( nat > $o ) > nat > A > A ) > $o ) ).
thf(sy_c_fChoice,type,
fChoice:
!>[A: $tType] : ( ( A > $o ) > A ) ).
thf(sy_c_member,type,
member:
!>[A: $tType] : ( A > ( set @ A ) > $o ) ).
thf(sy_v_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_x11______,type,
x11: option @ ( product_prod @ nat @ 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: option @ nat ).
thf(sy_v_xe__7_058ATP,type,
xe_7_ATP: vEBT_VEBT ).
thf(sy_v_xnew____,type,
xnew: nat ).
% Relevant facts (8179)
thf(fact_0_even__odd__cases,axiom,
! [X: nat] :
( ! [N: nat] :
( X
!= ( plus_plus @ nat @ N @ N ) )
=> ~ ! [N: nat] :
( X
!= ( plus_plus @ nat @ N @ ( suc @ N ) ) ) ) ).
% even_odd_cases
thf(fact_1_bit__split__inv,axiom,
! [X: nat,D: nat] :
( ( vEBT_VEBT_bit_concat @ ( vEBT_VEBT_high @ X @ D ) @ ( vEBT_VEBT_low @ X @ D ) @ D )
= X ) ).
% bit_split_inv
thf(fact_2_pow__sum,axiom,
! [A2: nat,B2: nat] :
( ( divide_divide @ nat @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( plus_plus @ nat @ A2 @ B2 ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ A2 ) )
= ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ B2 ) ) ).
% pow_sum
thf(fact_3_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_4_high__def,axiom,
( vEBT_VEBT_high
= ( ^ [X2: nat,N2: nat] : ( divide_divide @ nat @ X2 @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N2 ) ) ) ) ).
% high_def
thf(fact_5_high__bound__aux,axiom,
! [Ma: nat,N3: nat,M: nat] :
( ( ord_less @ nat @ Ma @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( plus_plus @ nat @ N3 @ M ) ) )
=> ( ord_less @ nat @ ( vEBT_VEBT_high @ Ma @ N3 ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ M ) ) ) ).
% high_bound_aux
thf(fact_6_high__inv,axiom,
! [X: nat,N3: nat,Y: nat] :
( ( ord_less @ nat @ X @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N3 ) )
=> ( ( vEBT_VEBT_high @ ( plus_plus @ nat @ ( times_times @ nat @ Y @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N3 ) ) @ X ) @ N3 )
= Y ) ) ).
% high_inv
thf(fact_7_low__inv,axiom,
! [X: nat,N3: nat,Y: nat] :
( ( ord_less @ nat @ X @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N3 ) )
=> ( ( vEBT_VEBT_low @ ( plus_plus @ nat @ ( times_times @ nat @ Y @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N3 ) ) @ X ) @ N3 )
= X ) ) ).
% low_inv
thf(fact_8_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_9_bit__concat__def,axiom,
( vEBT_VEBT_bit_concat
= ( ^ [H: nat,L: nat,D2: nat] : ( plus_plus @ nat @ ( times_times @ nat @ H @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ D2 ) ) @ L ) ) ) ).
% bit_concat_def
thf(fact_10_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_11_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_12_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_13__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_14_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_15_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_16_add__2__eq__Suc,axiom,
! [N3: nat] :
( ( plus_plus @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N3 )
= ( suc @ ( suc @ N3 ) ) ) ).
% add_2_eq_Suc
thf(fact_17_add__2__eq__Suc_H,axiom,
! [N3: nat] :
( ( plus_plus @ nat @ N3 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) )
= ( suc @ ( suc @ N3 ) ) ) ).
% add_2_eq_Suc'
thf(fact_18_divide__less__eq__numeral1_I1_J,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [B2: A,W: num,A2: A] :
( ( ord_less @ A @ ( divide_divide @ A @ B2 @ ( numeral_numeral @ A @ W ) ) @ A2 )
= ( ord_less @ A @ B2 @ ( times_times @ A @ A2 @ ( numeral_numeral @ A @ W ) ) ) ) ) ).
% divide_less_eq_numeral1(1)
thf(fact_19_less__divide__eq__numeral1_I1_J,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [A2: A,B2: A,W: num] :
( ( ord_less @ A @ A2 @ ( divide_divide @ A @ B2 @ ( numeral_numeral @ A @ W ) ) )
= ( ord_less @ A @ ( times_times @ A @ A2 @ ( numeral_numeral @ A @ W ) ) @ B2 ) ) ) ).
% less_divide_eq_numeral1(1)
thf(fact_20_option_Ocollapse,axiom,
! [A: $tType,Option: option @ A] :
( ( Option
!= ( none @ A ) )
=> ( ( some @ A @ ( the2 @ A @ Option ) )
= Option ) ) ).
% option.collapse
thf(fact_21_mult__Suc__right,axiom,
! [M: nat,N3: nat] :
( ( times_times @ nat @ M @ ( suc @ N3 ) )
= ( plus_plus @ nat @ M @ ( times_times @ nat @ M @ N3 ) ) ) ).
% mult_Suc_right
thf(fact_22_distrib__left__numeral,axiom,
! [A: $tType] :
( ( ( numeral @ A )
& ( semiring @ A ) )
=> ! [V: num,B2: A,C2: A] :
( ( times_times @ A @ ( numeral_numeral @ A @ V ) @ ( plus_plus @ A @ B2 @ C2 ) )
= ( plus_plus @ A @ ( times_times @ A @ ( numeral_numeral @ A @ V ) @ B2 ) @ ( times_times @ A @ ( numeral_numeral @ A @ V ) @ C2 ) ) ) ) ).
% distrib_left_numeral
thf(fact_23_distrib__right__numeral,axiom,
! [A: $tType] :
( ( ( numeral @ A )
& ( semiring @ A ) )
=> ! [A2: A,B2: A,V: num] :
( ( times_times @ A @ ( plus_plus @ A @ A2 @ B2 ) @ ( numeral_numeral @ A @ V ) )
= ( plus_plus @ A @ ( times_times @ A @ A2 @ ( numeral_numeral @ A @ V ) ) @ ( times_times @ A @ B2 @ ( numeral_numeral @ A @ V ) ) ) ) ) ).
% distrib_right_numeral
thf(fact_24_power__shift,axiom,
! [X: nat,Y: nat,Z: nat] :
( ( ( power_power @ nat @ X @ Y )
= Z )
= ( ( vEBT_VEBT_power @ ( some @ nat @ X ) @ ( some @ nat @ Y ) )
= ( some @ nat @ Z ) ) ) ).
% power_shift
thf(fact_25_max__in__set__def,axiom,
( vEBT_VEBT_max_in_set
= ( ^ [Xs: set @ nat,X2: nat] :
( ( member @ nat @ X2 @ Xs )
& ! [Y2: nat] :
( ( member @ nat @ Y2 @ Xs )
=> ( ord_less_eq @ nat @ Y2 @ X2 ) ) ) ) ) ).
% max_in_set_def
thf(fact_26_min__in__set__def,axiom,
( vEBT_VEBT_min_in_set
= ( ^ [Xs: set @ nat,X2: nat] :
( ( member @ nat @ X2 @ Xs )
& ! [Y2: nat] :
( ( member @ nat @ Y2 @ Xs )
=> ( ord_less_eq @ nat @ X2 @ Y2 ) ) ) ) ) ).
% min_in_set_def
thf(fact_27_numeral__eq__iff,axiom,
! [A: $tType] :
( ( semiring_char_0 @ A )
=> ! [M: num,N3: num] :
( ( ( numeral_numeral @ A @ M )
= ( numeral_numeral @ A @ N3 ) )
= ( M = N3 ) ) ) ).
% numeral_eq_iff
thf(fact_28_old_Onat_Oinject,axiom,
! [Nat: nat,Nat2: nat] :
( ( ( suc @ Nat )
= ( suc @ Nat2 ) )
= ( Nat = Nat2 ) ) ).
% old.nat.inject
thf(fact_29_nat_Oinject,axiom,
! [X22: nat,Y22: nat] :
( ( ( suc @ X22 )
= ( suc @ Y22 ) )
= ( X22 = Y22 ) ) ).
% nat.inject
thf(fact_30_option_Oinject,axiom,
! [A: $tType,X22: A,Y22: A] :
( ( ( some @ A @ X22 )
= ( some @ A @ Y22 ) )
= ( X22 = Y22 ) ) ).
% option.inject
thf(fact_31_numeral__le__iff,axiom,
! [A: $tType] :
( ( linord181362715937106298miring @ A )
=> ! [M: num,N3: num] :
( ( ord_less_eq @ A @ ( numeral_numeral @ A @ M ) @ ( numeral_numeral @ A @ N3 ) )
= ( ord_less_eq @ num @ M @ N3 ) ) ) ).
% numeral_le_iff
thf(fact_32_numeral__less__iff,axiom,
! [A: $tType] :
( ( linord181362715937106298miring @ A )
=> ! [M: num,N3: num] :
( ( ord_less @ A @ ( numeral_numeral @ A @ M ) @ ( numeral_numeral @ A @ N3 ) )
= ( ord_less @ num @ M @ N3 ) ) ) ).
% numeral_less_iff
thf(fact_33_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_34_numeral__plus__numeral,axiom,
! [A: $tType] :
( ( numeral @ A )
=> ! [M: num,N3: num] :
( ( plus_plus @ A @ ( numeral_numeral @ A @ M ) @ ( numeral_numeral @ A @ N3 ) )
= ( numeral_numeral @ A @ ( plus_plus @ num @ M @ N3 ) ) ) ) ).
% numeral_plus_numeral
thf(fact_35_num__double,axiom,
! [N3: num] :
( ( times_times @ num @ ( bit0 @ one2 ) @ N3 )
= ( bit0 @ N3 ) ) ).
% num_double
thf(fact_36_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_37_numeral__times__numeral,axiom,
! [A: $tType] :
( ( semiring_numeral @ A )
=> ! [M: num,N3: num] :
( ( times_times @ A @ ( numeral_numeral @ A @ M ) @ ( numeral_numeral @ A @ N3 ) )
= ( numeral_numeral @ A @ ( times_times @ num @ M @ N3 ) ) ) ) ).
% numeral_times_numeral
thf(fact_38_Suc__less__eq,axiom,
! [M: nat,N3: nat] :
( ( ord_less @ nat @ ( suc @ M ) @ ( suc @ N3 ) )
= ( ord_less @ nat @ M @ N3 ) ) ).
% Suc_less_eq
thf(fact_39_Suc__mono,axiom,
! [M: nat,N3: nat] :
( ( ord_less @ nat @ M @ N3 )
=> ( ord_less @ nat @ ( suc @ M ) @ ( suc @ N3 ) ) ) ).
% Suc_mono
thf(fact_40_lessI,axiom,
! [N3: nat] : ( ord_less @ nat @ N3 @ ( suc @ N3 ) ) ).
% lessI
thf(fact_41_mem__Collect__eq,axiom,
! [A: $tType,A2: A,P: A > $o] :
( ( member @ A @ A2 @ ( collect @ A @ P ) )
= ( P @ A2 ) ) ).
% mem_Collect_eq
thf(fact_42_Collect__mem__eq,axiom,
! [A: $tType,A3: set @ A] :
( ( collect @ A
@ ^ [X2: A] : ( member @ A @ X2 @ A3 ) )
= A3 ) ).
% Collect_mem_eq
thf(fact_43_Collect__cong,axiom,
! [A: $tType,P: A > $o,Q: A > $o] :
( ! [X3: A] :
( ( P @ X3 )
= ( Q @ X3 ) )
=> ( ( collect @ A @ P )
= ( collect @ A @ Q ) ) ) ).
% Collect_cong
thf(fact_44_HOL_Oext,axiom,
! [B: $tType,A: $tType,F2: A > B,G: A > B] :
( ! [X3: A] :
( ( F2 @ X3 )
= ( G @ X3 ) )
=> ( F2 = G ) ) ).
% HOL.ext
thf(fact_45_add__Suc__right,axiom,
! [M: nat,N3: nat] :
( ( plus_plus @ nat @ M @ ( suc @ N3 ) )
= ( suc @ ( plus_plus @ nat @ M @ N3 ) ) ) ).
% add_Suc_right
thf(fact_46_Suc__le__mono,axiom,
! [N3: nat,M: nat] :
( ( ord_less_eq @ nat @ ( suc @ N3 ) @ ( suc @ M ) )
= ( ord_less_eq @ nat @ N3 @ M ) ) ).
% Suc_le_mono
thf(fact_47_nat__add__left__cancel__less,axiom,
! [K: nat,M: nat,N3: nat] :
( ( ord_less @ nat @ ( plus_plus @ nat @ K @ M ) @ ( plus_plus @ nat @ K @ N3 ) )
= ( ord_less @ nat @ M @ N3 ) ) ).
% nat_add_left_cancel_less
thf(fact_48_nat__add__left__cancel__le,axiom,
! [K: nat,M: nat,N3: nat] :
( ( ord_less_eq @ nat @ ( plus_plus @ nat @ K @ M ) @ ( plus_plus @ nat @ K @ N3 ) )
= ( ord_less_eq @ nat @ M @ N3 ) ) ).
% nat_add_left_cancel_le
thf(fact_49_not__Some__eq,axiom,
! [A: $tType,X: option @ A] :
( ( ! [Y2: A] :
( X
!= ( some @ A @ Y2 ) ) )
= ( X
= ( none @ A ) ) ) ).
% not_Some_eq
thf(fact_50_not__None__eq,axiom,
! [A: $tType,X: option @ A] :
( ( X
!= ( none @ A ) )
= ( ? [Y2: A] :
( X
= ( some @ A @ Y2 ) ) ) ) ).
% not_None_eq
thf(fact_51_lesseq__shift,axiom,
( ( ord_less_eq @ nat )
= ( ^ [X2: nat,Y2: nat] : ( vEBT_VEBT_lesseq @ ( some @ nat @ X2 ) @ ( some @ nat @ Y2 ) ) ) ) ).
% lesseq_shift
thf(fact_52_Suc__numeral,axiom,
! [N3: num] :
( ( suc @ ( numeral_numeral @ nat @ N3 ) )
= ( numeral_numeral @ nat @ ( plus_plus @ num @ N3 @ one2 ) ) ) ).
% Suc_numeral
thf(fact_53_le__divide__eq__numeral1_I1_J,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [A2: A,B2: A,W: num] :
( ( ord_less_eq @ A @ A2 @ ( divide_divide @ A @ B2 @ ( numeral_numeral @ A @ W ) ) )
= ( ord_less_eq @ A @ ( times_times @ A @ A2 @ ( numeral_numeral @ A @ W ) ) @ B2 ) ) ) ).
% le_divide_eq_numeral1(1)
thf(fact_54_divide__le__eq__numeral1_I1_J,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [B2: A,W: num,A2: A] :
( ( ord_less_eq @ A @ ( divide_divide @ A @ B2 @ ( numeral_numeral @ A @ W ) ) @ A2 )
= ( ord_less_eq @ A @ B2 @ ( times_times @ A @ A2 @ ( numeral_numeral @ A @ W ) ) ) ) ) ).
% divide_le_eq_numeral1(1)
thf(fact_55_lift__Suc__mono__le,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [F2: nat > A,N3: nat,N4: nat] :
( ! [N: nat] : ( ord_less_eq @ A @ ( F2 @ N ) @ ( F2 @ ( suc @ N ) ) )
=> ( ( ord_less_eq @ nat @ N3 @ N4 )
=> ( ord_less_eq @ A @ ( F2 @ N3 ) @ ( F2 @ N4 ) ) ) ) ) ).
% lift_Suc_mono_le
thf(fact_56_lift__Suc__antimono__le,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [F2: nat > A,N3: nat,N4: nat] :
( ! [N: nat] : ( ord_less_eq @ A @ ( F2 @ ( suc @ N ) ) @ ( F2 @ N ) )
=> ( ( ord_less_eq @ nat @ N3 @ N4 )
=> ( ord_less_eq @ A @ ( F2 @ N4 ) @ ( F2 @ N3 ) ) ) ) ) ).
% lift_Suc_antimono_le
thf(fact_57_le__refl,axiom,
! [N3: nat] : ( ord_less_eq @ nat @ N3 @ N3 ) ).
% le_refl
thf(fact_58_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_59_eq__imp__le,axiom,
! [M: nat,N3: nat] :
( ( M = N3 )
=> ( ord_less_eq @ nat @ M @ N3 ) ) ).
% eq_imp_le
thf(fact_60_le__antisym,axiom,
! [M: nat,N3: nat] :
( ( ord_less_eq @ nat @ M @ N3 )
=> ( ( ord_less_eq @ nat @ N3 @ M )
=> ( M = N3 ) ) ) ).
% le_antisym
thf(fact_61_nat__le__linear,axiom,
! [M: nat,N3: nat] :
( ( ord_less_eq @ nat @ M @ N3 )
| ( ord_less_eq @ nat @ N3 @ M ) ) ).
% nat_le_linear
thf(fact_62_Nat_Oex__has__greatest__nat,axiom,
! [P: nat > $o,K: nat,B2: nat] :
( ( P @ K )
=> ( ! [Y3: nat] :
( ( P @ Y3 )
=> ( ord_less_eq @ nat @ Y3 @ B2 ) )
=> ? [X3: nat] :
( ( P @ X3 )
& ! [Y4: nat] :
( ( P @ Y4 )
=> ( ord_less_eq @ nat @ Y4 @ X3 ) ) ) ) ) ).
% Nat.ex_has_greatest_nat
thf(fact_63_add__One__commute,axiom,
! [N3: num] :
( ( plus_plus @ num @ one2 @ N3 )
= ( plus_plus @ num @ N3 @ one2 ) ) ).
% add_One_commute
thf(fact_64_transitive__stepwise__le,axiom,
! [M: nat,N3: nat,R: nat > nat > $o] :
( ( ord_less_eq @ nat @ M @ N3 )
=> ( ! [X3: nat] : ( R @ X3 @ X3 )
=> ( ! [X3: nat,Y3: nat,Z2: nat] :
( ( R @ X3 @ Y3 )
=> ( ( R @ Y3 @ Z2 )
=> ( R @ X3 @ Z2 ) ) )
=> ( ! [N: nat] : ( R @ N @ ( suc @ N ) )
=> ( R @ M @ N3 ) ) ) ) ) ).
% transitive_stepwise_le
thf(fact_65_nat__induct__at__least,axiom,
! [M: nat,N3: nat,P: nat > $o] :
( ( ord_less_eq @ nat @ M @ N3 )
=> ( ( P @ M )
=> ( ! [N: nat] :
( ( ord_less_eq @ nat @ M @ N )
=> ( ( P @ N )
=> ( P @ ( suc @ N ) ) ) )
=> ( P @ N3 ) ) ) ) ).
% nat_induct_at_least
thf(fact_66_full__nat__induct,axiom,
! [P: nat > $o,N3: nat] :
( ! [N: nat] :
( ! [M2: nat] :
( ( ord_less_eq @ nat @ ( suc @ M2 ) @ N )
=> ( P @ M2 ) )
=> ( P @ N ) )
=> ( P @ N3 ) ) ).
% full_nat_induct
thf(fact_67_not__less__eq__eq,axiom,
! [M: nat,N3: nat] :
( ( ~ ( ord_less_eq @ nat @ M @ N3 ) )
= ( ord_less_eq @ nat @ ( suc @ N3 ) @ M ) ) ).
% not_less_eq_eq
thf(fact_68_Suc__n__not__le__n,axiom,
! [N3: nat] :
~ ( ord_less_eq @ nat @ ( suc @ N3 ) @ N3 ) ).
% Suc_n_not_le_n
thf(fact_69_le__Suc__eq,axiom,
! [M: nat,N3: nat] :
( ( ord_less_eq @ nat @ M @ ( suc @ N3 ) )
= ( ( ord_less_eq @ nat @ M @ N3 )
| ( M
= ( suc @ N3 ) ) ) ) ).
% le_Suc_eq
thf(fact_70_Suc__le__D,axiom,
! [N3: nat,M3: nat] :
( ( ord_less_eq @ nat @ ( suc @ N3 ) @ M3 )
=> ? [M4: nat] :
( M3
= ( suc @ M4 ) ) ) ).
% Suc_le_D
thf(fact_71_le__SucI,axiom,
! [M: nat,N3: nat] :
( ( ord_less_eq @ nat @ M @ N3 )
=> ( ord_less_eq @ nat @ M @ ( suc @ N3 ) ) ) ).
% le_SucI
thf(fact_72_le__SucE,axiom,
! [M: nat,N3: nat] :
( ( ord_less_eq @ nat @ M @ ( suc @ N3 ) )
=> ( ~ ( ord_less_eq @ nat @ M @ N3 )
=> ( M
= ( suc @ N3 ) ) ) ) ).
% le_SucE
thf(fact_73_Suc__leD,axiom,
! [M: nat,N3: nat] :
( ( ord_less_eq @ nat @ ( suc @ M ) @ N3 )
=> ( ord_less_eq @ nat @ M @ N3 ) ) ).
% Suc_leD
thf(fact_74_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_75_le__neq__implies__less,axiom,
! [M: nat,N3: nat] :
( ( ord_less_eq @ nat @ M @ N3 )
=> ( ( M != N3 )
=> ( ord_less @ nat @ M @ N3 ) ) ) ).
% le_neq_implies_less
thf(fact_76_less__or__eq__imp__le,axiom,
! [M: nat,N3: nat] :
( ( ( ord_less @ nat @ M @ N3 )
| ( M = N3 ) )
=> ( ord_less_eq @ nat @ M @ N3 ) ) ).
% less_or_eq_imp_le
thf(fact_77_le__eq__less__or__eq,axiom,
( ( ord_less_eq @ nat )
= ( ^ [M5: nat,N2: nat] :
( ( ord_less @ nat @ M5 @ N2 )
| ( M5 = N2 ) ) ) ) ).
% le_eq_less_or_eq
thf(fact_78_less__imp__le__nat,axiom,
! [M: nat,N3: nat] :
( ( ord_less @ nat @ M @ N3 )
=> ( ord_less_eq @ nat @ M @ N3 ) ) ).
% less_imp_le_nat
thf(fact_79_nat__less__le,axiom,
( ( ord_less @ nat )
= ( ^ [M5: nat,N2: nat] :
( ( ord_less_eq @ nat @ M5 @ N2 )
& ( M5 != N2 ) ) ) ) ).
% nat_less_le
thf(fact_80_nat__le__iff__add,axiom,
( ( ord_less_eq @ nat )
= ( ^ [M5: nat,N2: nat] :
? [K2: nat] :
( N2
= ( plus_plus @ nat @ M5 @ K2 ) ) ) ) ).
% nat_le_iff_add
thf(fact_81_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_82_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_83_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_84_add__le__mono,axiom,
! [I: nat,J: nat,K: nat,L2: nat] :
( ( ord_less_eq @ nat @ I @ J )
=> ( ( ord_less_eq @ nat @ K @ L2 )
=> ( ord_less_eq @ nat @ ( plus_plus @ nat @ I @ K ) @ ( plus_plus @ nat @ J @ L2 ) ) ) ) ).
% add_le_mono
thf(fact_85_le__Suc__ex,axiom,
! [K: nat,L2: nat] :
( ( ord_less_eq @ nat @ K @ L2 )
=> ? [N: nat] :
( L2
= ( plus_plus @ nat @ K @ N ) ) ) ).
% le_Suc_ex
thf(fact_86_add__leD2,axiom,
! [M: nat,K: nat,N3: nat] :
( ( ord_less_eq @ nat @ ( plus_plus @ nat @ M @ K ) @ N3 )
=> ( ord_less_eq @ nat @ K @ N3 ) ) ).
% add_leD2
thf(fact_87_add__leD1,axiom,
! [M: nat,K: nat,N3: nat] :
( ( ord_less_eq @ nat @ ( plus_plus @ nat @ M @ K ) @ N3 )
=> ( ord_less_eq @ nat @ M @ N3 ) ) ).
% add_leD1
thf(fact_88_le__add2,axiom,
! [N3: nat,M: nat] : ( ord_less_eq @ nat @ N3 @ ( plus_plus @ nat @ M @ N3 ) ) ).
% le_add2
thf(fact_89_le__add1,axiom,
! [N3: nat,M: nat] : ( ord_less_eq @ nat @ N3 @ ( plus_plus @ nat @ N3 @ M ) ) ).
% le_add1
thf(fact_90_add__leE,axiom,
! [M: nat,K: nat,N3: nat] :
( ( ord_less_eq @ nat @ ( plus_plus @ nat @ M @ K ) @ N3 )
=> ~ ( ( ord_less_eq @ nat @ M @ N3 )
=> ~ ( ord_less_eq @ nat @ K @ N3 ) ) ) ).
% add_leE
thf(fact_91_div__le__dividend,axiom,
! [M: nat,N3: nat] : ( ord_less_eq @ nat @ ( divide_divide @ nat @ M @ N3 ) @ M ) ).
% div_le_dividend
thf(fact_92_div__le__mono,axiom,
! [M: nat,N3: nat,K: nat] :
( ( ord_less_eq @ nat @ M @ N3 )
=> ( ord_less_eq @ nat @ ( divide_divide @ nat @ M @ K ) @ ( divide_divide @ nat @ N3 @ K ) ) ) ).
% div_le_mono
thf(fact_93_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_94_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_95_mult__le__mono,axiom,
! [I: nat,J: nat,K: nat,L2: nat] :
( ( ord_less_eq @ nat @ I @ J )
=> ( ( ord_less_eq @ nat @ K @ L2 )
=> ( ord_less_eq @ nat @ ( times_times @ nat @ I @ K ) @ ( times_times @ nat @ J @ L2 ) ) ) ) ).
% mult_le_mono
thf(fact_96_le__square,axiom,
! [M: nat] : ( ord_less_eq @ nat @ M @ ( times_times @ nat @ M @ M ) ) ).
% le_square
thf(fact_97_le__cube,axiom,
! [M: nat] : ( ord_less_eq @ nat @ M @ ( times_times @ nat @ M @ ( times_times @ nat @ M @ M ) ) ) ).
% le_cube
thf(fact_98_div__mult2__numeral__eq,axiom,
! [A: $tType] :
( ( euclid5411537665997757685th_nat @ A )
=> ! [A2: A,K: num,L2: num] :
( ( divide_divide @ A @ ( divide_divide @ A @ A2 @ ( numeral_numeral @ A @ K ) ) @ ( numeral_numeral @ A @ L2 ) )
= ( divide_divide @ A @ A2 @ ( numeral_numeral @ A @ ( times_times @ num @ K @ L2 ) ) ) ) ) ).
% div_mult2_numeral_eq
thf(fact_99_le__imp__less__Suc,axiom,
! [M: nat,N3: nat] :
( ( ord_less_eq @ nat @ M @ N3 )
=> ( ord_less @ nat @ M @ ( suc @ N3 ) ) ) ).
% le_imp_less_Suc
thf(fact_100_less__eq__Suc__le,axiom,
( ( ord_less @ nat )
= ( ^ [N2: nat] : ( ord_less_eq @ nat @ ( suc @ N2 ) ) ) ) ).
% less_eq_Suc_le
thf(fact_101_less__Suc__eq__le,axiom,
! [M: nat,N3: nat] :
( ( ord_less @ nat @ M @ ( suc @ N3 ) )
= ( ord_less_eq @ nat @ M @ N3 ) ) ).
% less_Suc_eq_le
thf(fact_102_le__less__Suc__eq,axiom,
! [M: nat,N3: nat] :
( ( ord_less_eq @ nat @ M @ N3 )
=> ( ( ord_less @ nat @ N3 @ ( suc @ M ) )
= ( N3 = M ) ) ) ).
% le_less_Suc_eq
thf(fact_103_Suc__le__lessD,axiom,
! [M: nat,N3: nat] :
( ( ord_less_eq @ nat @ ( suc @ M ) @ N3 )
=> ( ord_less @ nat @ M @ N3 ) ) ).
% Suc_le_lessD
thf(fact_104_inc__induct,axiom,
! [I: nat,J: nat,P: nat > $o] :
( ( ord_less_eq @ nat @ I @ J )
=> ( ( P @ J )
=> ( ! [N: nat] :
( ( ord_less_eq @ nat @ I @ N )
=> ( ( ord_less @ nat @ N @ J )
=> ( ( P @ ( suc @ N ) )
=> ( P @ N ) ) ) )
=> ( P @ I ) ) ) ) ).
% inc_induct
thf(fact_105_dec__induct,axiom,
! [I: nat,J: nat,P: nat > $o] :
( ( ord_less_eq @ nat @ I @ J )
=> ( ( P @ I )
=> ( ! [N: nat] :
( ( ord_less_eq @ nat @ I @ N )
=> ( ( ord_less @ nat @ N @ J )
=> ( ( P @ N )
=> ( P @ ( suc @ N ) ) ) ) )
=> ( P @ J ) ) ) ) ).
% dec_induct
thf(fact_106_Suc__le__eq,axiom,
! [M: nat,N3: nat] :
( ( ord_less_eq @ nat @ ( suc @ M ) @ N3 )
= ( ord_less @ nat @ M @ N3 ) ) ).
% Suc_le_eq
thf(fact_107_Suc__leI,axiom,
! [M: nat,N3: nat] :
( ( ord_less @ nat @ M @ N3 )
=> ( ord_less_eq @ nat @ ( suc @ M ) @ N3 ) ) ).
% Suc_leI
thf(fact_108_Suc__div__le__mono,axiom,
! [M: nat,N3: nat] : ( ord_less_eq @ nat @ ( divide_divide @ nat @ M @ N3 ) @ ( divide_divide @ nat @ ( suc @ M ) @ N3 ) ) ).
% Suc_div_le_mono
thf(fact_109_mono__nat__linear__lb,axiom,
! [F2: nat > nat,M: nat,K: nat] :
( ! [M4: nat,N: nat] :
( ( ord_less @ nat @ M4 @ N )
=> ( ord_less @ nat @ ( F2 @ M4 ) @ ( F2 @ N ) ) )
=> ( ord_less_eq @ nat @ ( plus_plus @ nat @ ( F2 @ M ) @ K ) @ ( F2 @ ( plus_plus @ nat @ M @ K ) ) ) ) ).
% mono_nat_linear_lb
thf(fact_110_Suc__mult__le__cancel1,axiom,
! [K: nat,M: nat,N3: nat] :
( ( ord_less_eq @ nat @ ( times_times @ nat @ ( suc @ K ) @ M ) @ ( times_times @ nat @ ( suc @ K ) @ N3 ) )
= ( ord_less_eq @ nat @ M @ N3 ) ) ).
% Suc_mult_le_cancel1
thf(fact_111_times__div__less__eq__dividend,axiom,
! [N3: nat,M: nat] : ( ord_less_eq @ nat @ ( times_times @ nat @ N3 @ ( divide_divide @ nat @ M @ N3 ) ) @ M ) ).
% times_div_less_eq_dividend
thf(fact_112_div__times__less__eq__dividend,axiom,
! [M: nat,N3: nat] : ( ord_less_eq @ nat @ ( times_times @ nat @ ( divide_divide @ nat @ M @ N3 ) @ N3 ) @ M ) ).
% div_times_less_eq_dividend
thf(fact_113_Suc__nat__number__of__add,axiom,
! [V: num,N3: nat] :
( ( suc @ ( plus_plus @ nat @ ( numeral_numeral @ nat @ V ) @ N3 ) )
= ( plus_plus @ nat @ ( numeral_numeral @ nat @ ( plus_plus @ num @ V @ one2 ) ) @ N3 ) ) ).
% Suc_nat_number_of_add
thf(fact_114_div__nat__eqI,axiom,
! [N3: nat,Q2: nat,M: nat] :
( ( ord_less_eq @ nat @ ( times_times @ nat @ N3 @ Q2 ) @ M )
=> ( ( ord_less @ nat @ M @ ( times_times @ nat @ N3 @ ( suc @ Q2 ) ) )
=> ( ( divide_divide @ nat @ M @ N3 )
= Q2 ) ) ) ).
% div_nat_eqI
thf(fact_115_measure__induct__rule,axiom,
! [B: $tType,A: $tType] :
( ( wellorder @ B )
=> ! [F2: A > B,P: A > $o,A2: A] :
( ! [X3: A] :
( ! [Y4: A] :
( ( ord_less @ B @ ( F2 @ Y4 ) @ ( F2 @ X3 ) )
=> ( P @ Y4 ) )
=> ( P @ X3 ) )
=> ( P @ A2 ) ) ) ).
% measure_induct_rule
thf(fact_116_measure__induct,axiom,
! [B: $tType,A: $tType] :
( ( wellorder @ B )
=> ! [F2: A > B,P: A > $o,A2: A] :
( ! [X3: A] :
( ! [Y4: A] :
( ( ord_less @ B @ ( F2 @ Y4 ) @ ( F2 @ X3 ) )
=> ( P @ Y4 ) )
=> ( P @ X3 ) )
=> ( P @ A2 ) ) ) ).
% measure_induct
thf(fact_117_is__num__normalize_I1_J,axiom,
! [A: $tType] :
( ( neg_numeral @ A )
=> ! [A2: A,B2: A,C2: A] :
( ( plus_plus @ A @ ( plus_plus @ A @ A2 @ B2 ) @ C2 )
= ( plus_plus @ A @ A2 @ ( plus_plus @ A @ B2 @ C2 ) ) ) ) ).
% is_num_normalize(1)
thf(fact_118_n__not__Suc__n,axiom,
! [N3: nat] :
( N3
!= ( suc @ N3 ) ) ).
% n_not_Suc_n
thf(fact_119_Suc__inject,axiom,
! [X: nat,Y: nat] :
( ( ( suc @ X )
= ( suc @ Y ) )
=> ( X = Y ) ) ).
% Suc_inject
thf(fact_120_infinite__descent__measure,axiom,
! [A: $tType,P: A > $o,V2: A > nat,X: A] :
( ! [X3: A] :
( ~ ( P @ X3 )
=> ? [Y4: A] :
( ( ord_less @ nat @ ( V2 @ Y4 ) @ ( V2 @ X3 ) )
& ~ ( P @ Y4 ) ) )
=> ( P @ X ) ) ).
% infinite_descent_measure
thf(fact_121_linorder__neqE__nat,axiom,
! [X: nat,Y: nat] :
( ( X != Y )
=> ( ~ ( ord_less @ nat @ X @ Y )
=> ( ord_less @ nat @ Y @ X ) ) ) ).
% linorder_neqE_nat
thf(fact_122_infinite__descent,axiom,
! [P: nat > $o,N3: nat] :
( ! [N: nat] :
( ~ ( P @ N )
=> ? [M2: nat] :
( ( ord_less @ nat @ M2 @ N )
& ~ ( P @ M2 ) ) )
=> ( P @ N3 ) ) ).
% infinite_descent
thf(fact_123_nat__less__induct,axiom,
! [P: nat > $o,N3: nat] :
( ! [N: nat] :
( ! [M2: nat] :
( ( ord_less @ nat @ M2 @ N )
=> ( P @ M2 ) )
=> ( P @ N ) )
=> ( P @ N3 ) ) ).
% nat_less_induct
thf(fact_124_less__irrefl__nat,axiom,
! [N3: nat] :
~ ( ord_less @ nat @ N3 @ N3 ) ).
% less_irrefl_nat
thf(fact_125_less__not__refl3,axiom,
! [S: nat,T2: nat] :
( ( ord_less @ nat @ S @ T2 )
=> ( S != T2 ) ) ).
% less_not_refl3
thf(fact_126_less__not__refl2,axiom,
! [N3: nat,M: nat] :
( ( ord_less @ nat @ N3 @ M )
=> ( M != N3 ) ) ).
% less_not_refl2
thf(fact_127_less__not__refl,axiom,
! [N3: nat] :
~ ( ord_less @ nat @ N3 @ N3 ) ).
% less_not_refl
thf(fact_128_nat__neq__iff,axiom,
! [M: nat,N3: nat] :
( ( M != N3 )
= ( ( ord_less @ nat @ M @ N3 )
| ( ord_less @ nat @ N3 @ M ) ) ) ).
% nat_neq_iff
thf(fact_129_size__neq__size__imp__neq,axiom,
! [A: $tType] :
( ( size @ A )
=> ! [X: A,Y: A] :
( ( ( size_size @ A @ X )
!= ( size_size @ A @ Y ) )
=> ( X != Y ) ) ) ).
% size_neq_size_imp_neq
thf(fact_130_not__less__less__Suc__eq,axiom,
! [N3: nat,M: nat] :
( ~ ( ord_less @ nat @ N3 @ M )
=> ( ( ord_less @ nat @ N3 @ ( suc @ M ) )
= ( N3 = M ) ) ) ).
% not_less_less_Suc_eq
thf(fact_131_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_132_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,K3: nat] :
( ( ord_less @ nat @ I2 @ J2 )
=> ( ( ord_less @ nat @ J2 @ K3 )
=> ( ( P @ I2 @ J2 )
=> ( ( P @ J2 @ K3 )
=> ( P @ I2 @ K3 ) ) ) ) )
=> ( P @ I @ J ) ) ) ) ).
% less_Suc_induct
thf(fact_133_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_134_Suc__less__SucD,axiom,
! [M: nat,N3: nat] :
( ( ord_less @ nat @ ( suc @ M ) @ ( suc @ N3 ) )
=> ( ord_less @ nat @ M @ N3 ) ) ).
% Suc_less_SucD
thf(fact_135_less__antisym,axiom,
! [N3: nat,M: nat] :
( ~ ( ord_less @ nat @ N3 @ M )
=> ( ( ord_less @ nat @ N3 @ ( suc @ M ) )
=> ( M = N3 ) ) ) ).
% less_antisym
thf(fact_136_Suc__less__eq2,axiom,
! [N3: nat,M: nat] :
( ( ord_less @ nat @ ( suc @ N3 ) @ M )
= ( ? [M6: nat] :
( ( M
= ( suc @ M6 ) )
& ( ord_less @ nat @ N3 @ M6 ) ) ) ) ).
% Suc_less_eq2
thf(fact_137_Nat_OAll__less__Suc,axiom,
! [N3: nat,P: nat > $o] :
( ( ! [I3: nat] :
( ( ord_less @ nat @ I3 @ ( suc @ N3 ) )
=> ( P @ I3 ) ) )
= ( ( P @ N3 )
& ! [I3: nat] :
( ( ord_less @ nat @ I3 @ N3 )
=> ( P @ I3 ) ) ) ) ).
% Nat.All_less_Suc
thf(fact_138_not__less__eq,axiom,
! [M: nat,N3: nat] :
( ( ~ ( ord_less @ nat @ M @ N3 ) )
= ( ord_less @ nat @ N3 @ ( suc @ M ) ) ) ).
% not_less_eq
thf(fact_139_less__Suc__eq,axiom,
! [M: nat,N3: nat] :
( ( ord_less @ nat @ M @ ( suc @ N3 ) )
= ( ( ord_less @ nat @ M @ N3 )
| ( M = N3 ) ) ) ).
% less_Suc_eq
thf(fact_140_Ex__less__Suc,axiom,
! [N3: nat,P: nat > $o] :
( ( ? [I3: nat] :
( ( ord_less @ nat @ I3 @ ( suc @ N3 ) )
& ( P @ I3 ) ) )
= ( ( P @ N3 )
| ? [I3: nat] :
( ( ord_less @ nat @ I3 @ N3 )
& ( P @ I3 ) ) ) ) ).
% Ex_less_Suc
thf(fact_141_less__SucI,axiom,
! [M: nat,N3: nat] :
( ( ord_less @ nat @ M @ N3 )
=> ( ord_less @ nat @ M @ ( suc @ N3 ) ) ) ).
% less_SucI
thf(fact_142_less__SucE,axiom,
! [M: nat,N3: nat] :
( ( ord_less @ nat @ M @ ( suc @ N3 ) )
=> ( ~ ( ord_less @ nat @ M @ N3 )
=> ( M = N3 ) ) ) ).
% less_SucE
thf(fact_143_Suc__lessI,axiom,
! [M: nat,N3: nat] :
( ( ord_less @ nat @ M @ N3 )
=> ( ( ( suc @ M )
!= N3 )
=> ( ord_less @ nat @ ( suc @ M ) @ N3 ) ) ) ).
% Suc_lessI
thf(fact_144_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_145_Suc__lessD,axiom,
! [M: nat,N3: nat] :
( ( ord_less @ nat @ ( suc @ M ) @ N3 )
=> ( ord_less @ nat @ M @ N3 ) ) ).
% Suc_lessD
thf(fact_146_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_147_add__Suc__shift,axiom,
! [M: nat,N3: nat] :
( ( plus_plus @ nat @ ( suc @ M ) @ N3 )
= ( plus_plus @ nat @ M @ ( suc @ N3 ) ) ) ).
% add_Suc_shift
thf(fact_148_add__Suc,axiom,
! [M: nat,N3: nat] :
( ( plus_plus @ nat @ ( suc @ M ) @ N3 )
= ( suc @ ( plus_plus @ nat @ M @ N3 ) ) ) ).
% add_Suc
thf(fact_149_nat__arith_Osuc1,axiom,
! [A3: nat,K: nat,A2: nat] :
( ( A3
= ( plus_plus @ nat @ K @ A2 ) )
=> ( ( suc @ A3 )
= ( plus_plus @ nat @ K @ ( suc @ A2 ) ) ) ) ).
% nat_arith.suc1
thf(fact_150_less__add__eq__less,axiom,
! [K: nat,L2: nat,M: nat,N3: nat] :
( ( ord_less @ nat @ K @ L2 )
=> ( ( ( plus_plus @ nat @ M @ L2 )
= ( plus_plus @ nat @ K @ N3 ) )
=> ( ord_less @ nat @ M @ N3 ) ) ) ).
% less_add_eq_less
thf(fact_151_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_152_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_153_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_154_not__add__less2,axiom,
! [J: nat,I: nat] :
~ ( ord_less @ nat @ ( plus_plus @ nat @ J @ I ) @ I ) ).
% not_add_less2
thf(fact_155_not__add__less1,axiom,
! [I: nat,J: nat] :
~ ( ord_less @ nat @ ( plus_plus @ nat @ I @ J ) @ I ) ).
% not_add_less1
thf(fact_156_add__less__mono,axiom,
! [I: nat,J: nat,K: nat,L2: nat] :
( ( ord_less @ nat @ I @ J )
=> ( ( ord_less @ nat @ K @ L2 )
=> ( ord_less @ nat @ ( plus_plus @ nat @ I @ K ) @ ( plus_plus @ nat @ J @ L2 ) ) ) ) ).
% add_less_mono
thf(fact_157_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_158_Suc__mult__cancel1,axiom,
! [K: nat,M: nat,N3: nat] :
( ( ( times_times @ nat @ ( suc @ K ) @ M )
= ( times_times @ nat @ ( suc @ K ) @ N3 ) )
= ( M = N3 ) ) ).
% Suc_mult_cancel1
thf(fact_159_combine__options__cases,axiom,
! [A: $tType,B: $tType,X: option @ A,P: ( option @ A ) > ( option @ B ) > $o,Y: option @ B] :
( ( ( X
= ( none @ A ) )
=> ( P @ X @ Y ) )
=> ( ( ( Y
= ( none @ B ) )
=> ( P @ X @ Y ) )
=> ( ! [A4: A,B3: B] :
( ( X
= ( some @ A @ A4 ) )
=> ( ( Y
= ( some @ B @ B3 ) )
=> ( P @ X @ Y ) ) )
=> ( P @ X @ Y ) ) ) ) ).
% combine_options_cases
thf(fact_160_split__option__all,axiom,
! [A: $tType] :
( ( ^ [P2: ( option @ A ) > $o] :
! [X4: option @ A] : ( P2 @ X4 ) )
= ( ^ [P3: ( option @ A ) > $o] :
( ( P3 @ ( none @ A ) )
& ! [X2: A] : ( P3 @ ( some @ A @ X2 ) ) ) ) ) ).
% split_option_all
thf(fact_161_split__option__ex,axiom,
! [A: $tType] :
( ( ^ [P2: ( option @ A ) > $o] :
? [X4: option @ A] : ( P2 @ X4 ) )
= ( ^ [P3: ( option @ A ) > $o] :
( ( P3 @ ( none @ A ) )
| ? [X2: A] : ( P3 @ ( some @ A @ X2 ) ) ) ) ) ).
% split_option_ex
thf(fact_162_option_Oexhaust,axiom,
! [A: $tType,Y: option @ A] :
( ( Y
!= ( none @ A ) )
=> ~ ! [X23: A] :
( Y
!= ( some @ A @ X23 ) ) ) ).
% option.exhaust
thf(fact_163_option_OdiscI,axiom,
! [A: $tType,Option: option @ A,X22: A] :
( ( Option
= ( some @ A @ X22 ) )
=> ( Option
!= ( none @ A ) ) ) ).
% option.discI
thf(fact_164_option_Odistinct_I1_J,axiom,
! [A: $tType,X22: A] :
( ( none @ A )
!= ( some @ A @ X22 ) ) ).
% option.distinct(1)
thf(fact_165_add__mult__distrib2,axiom,
! [K: nat,M: nat,N3: nat] :
( ( times_times @ nat @ K @ ( plus_plus @ nat @ M @ N3 ) )
= ( plus_plus @ nat @ ( times_times @ nat @ K @ M ) @ ( times_times @ nat @ K @ N3 ) ) ) ).
% add_mult_distrib2
thf(fact_166_add__mult__distrib,axiom,
! [M: nat,N3: nat,K: nat] :
( ( times_times @ nat @ ( plus_plus @ nat @ M @ N3 ) @ K )
= ( plus_plus @ nat @ ( times_times @ nat @ M @ K ) @ ( times_times @ nat @ N3 @ K ) ) ) ).
% add_mult_distrib
thf(fact_167_div__mult2__eq,axiom,
! [M: nat,N3: nat,Q2: nat] :
( ( divide_divide @ nat @ M @ ( times_times @ nat @ N3 @ Q2 ) )
= ( divide_divide @ nat @ ( divide_divide @ nat @ M @ N3 ) @ Q2 ) ) ).
% div_mult2_eq
thf(fact_168_option_Osel,axiom,
! [A: $tType,X22: A] :
( ( the2 @ A @ ( some @ A @ X22 ) )
= X22 ) ).
% option.sel
thf(fact_169_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_170_numeral__Bit0,axiom,
! [A: $tType] :
( ( numeral @ A )
=> ! [N3: num] :
( ( numeral_numeral @ A @ ( bit0 @ N3 ) )
= ( plus_plus @ A @ ( numeral_numeral @ A @ N3 ) @ ( numeral_numeral @ A @ N3 ) ) ) ) ).
% numeral_Bit0
thf(fact_171_mult__numeral__1__right,axiom,
! [A: $tType] :
( ( semiring_numeral @ A )
=> ! [A2: A] :
( ( times_times @ A @ A2 @ ( numeral_numeral @ A @ one2 ) )
= A2 ) ) ).
% mult_numeral_1_right
thf(fact_172_mult__numeral__1,axiom,
! [A: $tType] :
( ( semiring_numeral @ A )
=> ! [A2: A] :
( ( times_times @ A @ ( numeral_numeral @ A @ one2 ) @ A2 )
= A2 ) ) ).
% mult_numeral_1
thf(fact_173_divide__numeral__1,axiom,
! [A: $tType] :
( ( field @ A )
=> ! [A2: A] :
( ( divide_divide @ A @ A2 @ ( numeral_numeral @ A @ one2 ) )
= A2 ) ) ).
% divide_numeral_1
thf(fact_174_lift__Suc__mono__less__iff,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [F2: nat > A,N3: nat,M: nat] :
( ! [N: nat] : ( ord_less @ A @ ( F2 @ N ) @ ( F2 @ ( suc @ N ) ) )
=> ( ( ord_less @ A @ ( F2 @ N3 ) @ ( F2 @ M ) )
= ( ord_less @ nat @ N3 @ M ) ) ) ) ).
% lift_Suc_mono_less_iff
thf(fact_175_lift__Suc__mono__less,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [F2: nat > A,N3: nat,N4: nat] :
( ! [N: nat] : ( ord_less @ A @ ( F2 @ N ) @ ( F2 @ ( suc @ N ) ) )
=> ( ( ord_less @ nat @ N3 @ N4 )
=> ( ord_less @ A @ ( F2 @ N3 ) @ ( F2 @ N4 ) ) ) ) ) ).
% lift_Suc_mono_less
thf(fact_176_less__imp__Suc__add,axiom,
! [M: nat,N3: nat] :
( ( ord_less @ nat @ M @ N3 )
=> ? [K3: nat] :
( N3
= ( suc @ ( plus_plus @ nat @ M @ K3 ) ) ) ) ).
% less_imp_Suc_add
thf(fact_177_less__iff__Suc__add,axiom,
( ( ord_less @ nat )
= ( ^ [M5: nat,N2: nat] :
? [K2: nat] :
( N2
= ( suc @ ( plus_plus @ nat @ M5 @ K2 ) ) ) ) ) ).
% less_iff_Suc_add
thf(fact_178_less__add__Suc2,axiom,
! [I: nat,M: nat] : ( ord_less @ nat @ I @ ( suc @ ( plus_plus @ nat @ M @ I ) ) ) ).
% less_add_Suc2
thf(fact_179_less__add__Suc1,axiom,
! [I: nat,M: nat] : ( ord_less @ nat @ I @ ( suc @ ( plus_plus @ nat @ I @ M ) ) ) ).
% less_add_Suc1
thf(fact_180_less__natE,axiom,
! [M: nat,N3: nat] :
( ( ord_less @ nat @ M @ N3 )
=> ~ ! [Q3: nat] :
( N3
!= ( suc @ ( plus_plus @ nat @ M @ Q3 ) ) ) ) ).
% less_natE
thf(fact_181_Suc__mult__less__cancel1,axiom,
! [K: nat,M: nat,N3: nat] :
( ( ord_less @ nat @ ( times_times @ nat @ ( suc @ K ) @ M ) @ ( times_times @ nat @ ( suc @ K ) @ N3 ) )
= ( ord_less @ nat @ M @ N3 ) ) ).
% Suc_mult_less_cancel1
thf(fact_182_mult__Suc,axiom,
! [M: nat,N3: nat] :
( ( times_times @ nat @ ( suc @ M ) @ N3 )
= ( plus_plus @ nat @ N3 @ ( times_times @ nat @ M @ N3 ) ) ) ).
% mult_Suc
thf(fact_183_less__mult__imp__div__less,axiom,
! [M: nat,I: nat,N3: nat] :
( ( ord_less @ nat @ M @ ( times_times @ nat @ I @ N3 ) )
=> ( ord_less @ nat @ ( divide_divide @ nat @ M @ N3 ) @ I ) ) ).
% less_mult_imp_div_less
thf(fact_184_option_Oexhaust__sel,axiom,
! [A: $tType,Option: option @ A] :
( ( Option
!= ( none @ A ) )
=> ( Option
= ( some @ A @ ( the2 @ A @ Option ) ) ) ) ).
% option.exhaust_sel
thf(fact_185_numeral__Bit0__div__2,axiom,
! [A: $tType] :
( ( euclid5411537665997757685th_nat @ A )
=> ! [N3: num] :
( ( divide_divide @ A @ ( numeral_numeral @ A @ ( bit0 @ N3 ) ) @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) )
= ( numeral_numeral @ A @ N3 ) ) ) ).
% numeral_Bit0_div_2
thf(fact_186_left__add__twice,axiom,
! [A: $tType] :
( ( semiring_numeral @ A )
=> ! [A2: A,B2: A] :
( ( plus_plus @ A @ A2 @ ( plus_plus @ A @ A2 @ B2 ) )
= ( plus_plus @ A @ ( times_times @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ A2 ) @ B2 ) ) ) ).
% left_add_twice
thf(fact_187_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_188_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_189_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_190_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_191_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_192_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_193_sum__squares__bound,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [X: A,Y: A] : ( ord_less_eq @ A @ ( times_times @ A @ ( times_times @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ X ) @ Y ) @ ( plus_plus @ A @ ( power_power @ A @ X @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ ( power_power @ A @ Y @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) ).
% sum_squares_bound
thf(fact_194_power__add__numeral2,axiom,
! [A: $tType] :
( ( monoid_mult @ A )
=> ! [A2: A,M: num,N3: num,B2: A] :
( ( times_times @ A @ ( power_power @ A @ A2 @ ( numeral_numeral @ nat @ M ) ) @ ( times_times @ A @ ( power_power @ A @ A2 @ ( numeral_numeral @ nat @ N3 ) ) @ B2 ) )
= ( times_times @ A @ ( power_power @ A @ A2 @ ( numeral_numeral @ nat @ ( plus_plus @ num @ M @ N3 ) ) ) @ B2 ) ) ) ).
% power_add_numeral2
thf(fact_195_power__add__numeral,axiom,
! [A: $tType] :
( ( monoid_mult @ A )
=> ! [A2: A,M: num,N3: num] :
( ( times_times @ A @ ( power_power @ A @ A2 @ ( numeral_numeral @ nat @ M ) ) @ ( power_power @ A @ A2 @ ( numeral_numeral @ nat @ N3 ) ) )
= ( power_power @ A @ A2 @ ( numeral_numeral @ nat @ ( plus_plus @ num @ M @ N3 ) ) ) ) ) ).
% power_add_numeral
thf(fact_196_nat__add__offset__less,axiom,
! [Y: nat,N3: nat,X: nat,M: nat,Sz: nat] :
( ( ord_less @ nat @ Y @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N3 ) )
=> ( ( ord_less @ nat @ X @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ M ) )
=> ( ( Sz
= ( plus_plus @ nat @ M @ N3 ) )
=> ( ord_less @ nat @ ( plus_plus @ nat @ ( times_times @ nat @ X @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N3 ) ) @ Y ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ Sz ) ) ) ) ) ).
% nat_add_offset_less
thf(fact_197_two__pow__div__gt__le,axiom,
! [V: nat,N3: nat,M: nat] :
( ( ord_less @ nat @ V @ ( divide_divide @ nat @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N3 ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ M ) ) )
=> ( ord_less_eq @ nat @ M @ N3 ) ) ).
% two_pow_div_gt_le
thf(fact_198_power__odd__eq,axiom,
! [A: $tType] :
( ( monoid_mult @ A )
=> ! [A2: A,N3: nat] :
( ( power_power @ A @ A2 @ ( suc @ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N3 ) ) )
= ( times_times @ A @ A2 @ ( power_power @ A @ ( power_power @ A @ A2 @ N3 ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) ).
% power_odd_eq
thf(fact_199_power2__sum,axiom,
! [A: $tType] :
( ( comm_semiring_1 @ A )
=> ! [X: A,Y: A] :
( ( power_power @ A @ ( plus_plus @ A @ X @ Y ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) )
= ( plus_plus @ A @ ( plus_plus @ A @ ( power_power @ A @ X @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ ( power_power @ A @ Y @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( times_times @ A @ ( times_times @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ X ) @ Y ) ) ) ) ).
% power2_sum
thf(fact_200_delt__out__of__range,axiom,
! [X: nat,Mi: nat,Ma: nat,Deg: nat,TreeList: list @ vEBT_VEBT,Summary: vEBT_VEBT] :
( ( ( ord_less @ nat @ X @ Mi )
| ( ord_less @ nat @ Ma @ X ) )
=> ( ( ord_less_eq @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ Deg )
=> ( ( vEBT_vebt_delete @ ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi @ Ma ) ) @ Deg @ TreeList @ Summary ) @ X )
= ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi @ Ma ) ) @ Deg @ TreeList @ Summary ) ) ) ) ).
% delt_out_of_range
thf(fact_201_del__single__cont,axiom,
! [X: nat,Mi: nat,Ma: nat,Deg: nat,TreeList: list @ vEBT_VEBT,Summary: vEBT_VEBT] :
( ( ( X = Mi )
& ( X = Ma ) )
=> ( ( ord_less_eq @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ Deg )
=> ( ( vEBT_vebt_delete @ ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi @ Ma ) ) @ Deg @ TreeList @ Summary ) @ X )
= ( vEBT_Node @ ( none @ ( product_prod @ nat @ nat ) ) @ Deg @ TreeList @ Summary ) ) ) ) ).
% del_single_cont
thf(fact_202_power__mult__numeral,axiom,
! [A: $tType] :
( ( monoid_mult @ A )
=> ! [A2: A,M: num,N3: num] :
( ( power_power @ A @ ( power_power @ A @ A2 @ ( numeral_numeral @ nat @ M ) ) @ ( numeral_numeral @ nat @ N3 ) )
= ( power_power @ A @ A2 @ ( numeral_numeral @ nat @ ( times_times @ num @ M @ N3 ) ) ) ) ) ).
% power_mult_numeral
thf(fact_203_local_Opower__def,axiom,
( vEBT_VEBT_power
= ( vEBT_V2048590022279873568_shift @ nat @ ( power_power @ nat ) ) ) ).
% local.power_def
thf(fact_204__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_205_L2__set__mult__ineq__lemma,axiom,
! [A2: real,C2: real,B2: real,D: real] : ( ord_less_eq @ real @ ( times_times @ real @ ( times_times @ real @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) @ ( times_times @ real @ A2 @ C2 ) ) @ ( times_times @ real @ B2 @ D ) ) @ ( plus_plus @ real @ ( times_times @ real @ ( power_power @ real @ A2 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ ( power_power @ real @ D @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( times_times @ real @ ( power_power @ real @ B2 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ ( power_power @ real @ C2 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) ).
% L2_set_mult_ineq_lemma
thf(fact_206_le__num__One__iff,axiom,
! [X: num] :
( ( ord_less_eq @ num @ X @ one2 )
= ( X = one2 ) ) ).
% le_num_One_iff
thf(fact_207_four__x__squared,axiom,
! [X: real] :
( ( times_times @ real @ ( numeral_numeral @ real @ ( bit0 @ ( bit0 @ one2 ) ) ) @ ( power_power @ real @ X @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) )
= ( power_power @ real @ ( times_times @ real @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) @ X ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ).
% four_x_squared
thf(fact_208_power__commutes,axiom,
! [A: $tType] :
( ( monoid_mult @ A )
=> ! [A2: A,N3: nat] :
( ( times_times @ A @ ( power_power @ A @ A2 @ N3 ) @ A2 )
= ( times_times @ A @ A2 @ ( power_power @ A @ A2 @ N3 ) ) ) ) ).
% power_commutes
thf(fact_209_power__mult__distrib,axiom,
! [A: $tType] :
( ( comm_monoid_mult @ A )
=> ! [A2: A,B2: A,N3: nat] :
( ( power_power @ A @ ( times_times @ A @ A2 @ B2 ) @ N3 )
= ( times_times @ A @ ( power_power @ A @ A2 @ N3 ) @ ( power_power @ A @ B2 @ N3 ) ) ) ) ).
% power_mult_distrib
thf(fact_210_power__commuting__commutes,axiom,
! [A: $tType] :
( ( monoid_mult @ A )
=> ! [X: A,Y: A,N3: nat] :
( ( ( times_times @ A @ X @ Y )
= ( times_times @ A @ Y @ X ) )
=> ( ( times_times @ A @ ( power_power @ A @ X @ N3 ) @ Y )
= ( times_times @ A @ Y @ ( power_power @ A @ X @ N3 ) ) ) ) ) ).
% power_commuting_commutes
thf(fact_211_power__divide,axiom,
! [A: $tType] :
( ( field @ A )
=> ! [A2: A,B2: A,N3: nat] :
( ( power_power @ A @ ( divide_divide @ A @ A2 @ B2 ) @ N3 )
= ( divide_divide @ A @ ( power_power @ A @ A2 @ N3 ) @ ( power_power @ A @ B2 @ N3 ) ) ) ) ).
% power_divide
thf(fact_212_power__mult,axiom,
! [A: $tType] :
( ( monoid_mult @ A )
=> ! [A2: A,M: nat,N3: nat] :
( ( power_power @ A @ A2 @ ( times_times @ nat @ M @ N3 ) )
= ( power_power @ A @ ( power_power @ A @ A2 @ M ) @ N3 ) ) ) ).
% power_mult
thf(fact_213_power__Suc,axiom,
! [A: $tType] :
( ( power @ A )
=> ! [A2: A,N3: nat] :
( ( power_power @ A @ A2 @ ( suc @ N3 ) )
= ( times_times @ A @ A2 @ ( power_power @ A @ A2 @ N3 ) ) ) ) ).
% power_Suc
thf(fact_214_power__Suc2,axiom,
! [A: $tType] :
( ( monoid_mult @ A )
=> ! [A2: A,N3: nat] :
( ( power_power @ A @ A2 @ ( suc @ N3 ) )
= ( times_times @ A @ ( power_power @ A @ A2 @ N3 ) @ A2 ) ) ) ).
% power_Suc2
thf(fact_215_power__add,axiom,
! [A: $tType] :
( ( monoid_mult @ A )
=> ! [A2: A,M: nat,N3: nat] :
( ( power_power @ A @ A2 @ ( plus_plus @ nat @ M @ N3 ) )
= ( times_times @ A @ ( power_power @ A @ A2 @ M ) @ ( power_power @ A @ A2 @ N3 ) ) ) ) ).
% power_add
thf(fact_216_div__mult__le,axiom,
! [A2: nat,B2: nat] : ( ord_less_eq @ nat @ ( times_times @ nat @ ( divide_divide @ nat @ A2 @ B2 ) @ B2 ) @ A2 ) ).
% div_mult_le
thf(fact_217_power4__eq__xxxx,axiom,
! [A: $tType] :
( ( monoid_mult @ A )
=> ! [X: A] :
( ( power_power @ A @ X @ ( numeral_numeral @ nat @ ( bit0 @ ( bit0 @ one2 ) ) ) )
= ( times_times @ A @ ( times_times @ A @ ( times_times @ A @ X @ X ) @ X ) @ X ) ) ) ).
% power4_eq_xxxx
thf(fact_218_power2__eq__square,axiom,
! [A: $tType] :
( ( monoid_mult @ A )
=> ! [A2: A] :
( ( power_power @ A @ A2 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) )
= ( times_times @ A @ A2 @ A2 ) ) ) ).
% power2_eq_square
thf(fact_219_power__even__eq,axiom,
! [A: $tType] :
( ( monoid_mult @ A )
=> ! [A2: A,N3: nat] :
( ( power_power @ A @ A2 @ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N3 ) )
= ( power_power @ A @ ( power_power @ A @ A2 @ N3 ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ).
% power_even_eq
thf(fact_220_n__less__equal__power__2,axiom,
! [N3: nat] : ( ord_less @ nat @ N3 @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N3 ) ) ).
% n_less_equal_power_2
thf(fact_221_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_222_power2__nat__le__eq__le,axiom,
! [M: nat,N3: nat] :
( ( ord_less_eq @ nat @ ( power_power @ nat @ M @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ ( power_power @ nat @ N3 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) )
= ( ord_less_eq @ nat @ M @ N3 ) ) ).
% power2_nat_le_eq_le
thf(fact_223_power2__nat__le__imp__le,axiom,
! [M: nat,N3: nat] :
( ( ord_less_eq @ nat @ ( power_power @ nat @ M @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ N3 )
=> ( ord_less_eq @ nat @ M @ N3 ) ) ).
% power2_nat_le_imp_le
thf(fact_224_member__inv,axiom,
! [Mi: nat,Ma: nat,Deg: nat,TreeList: list @ vEBT_VEBT,Summary: vEBT_VEBT,X: nat] :
( ( vEBT_vebt_member @ ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi @ Ma ) ) @ Deg @ TreeList @ Summary ) @ X )
=> ( ( ord_less_eq @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ Deg )
& ( ( X = Mi )
| ( X = Ma )
| ( ( ord_less @ nat @ X @ Ma )
& ( ord_less @ nat @ Mi @ X )
& ( ord_less @ nat @ ( vEBT_VEBT_high @ X @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( size_size @ ( list @ vEBT_VEBT ) @ TreeList ) )
& ( vEBT_vebt_member @ ( nth @ vEBT_VEBT @ TreeList @ ( vEBT_VEBT_high @ X @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( vEBT_VEBT_low @ X @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) ) ) ) ).
% member_inv
thf(fact_225_pred__list__to__short,axiom,
! [Deg: nat,X: nat,Ma: nat,TreeList: list @ vEBT_VEBT,Mi: nat,Summary: vEBT_VEBT] :
( ( ord_less_eq @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ Deg )
=> ( ( ord_less_eq @ nat @ X @ Ma )
=> ( ( ord_less_eq @ nat @ ( size_size @ ( list @ vEBT_VEBT ) @ TreeList ) @ ( vEBT_VEBT_high @ X @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) )
=> ( ( vEBT_vebt_pred @ ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi @ Ma ) ) @ Deg @ TreeList @ Summary ) @ X )
= ( none @ nat ) ) ) ) ) ).
% pred_list_to_short
thf(fact_226_succ__list__to__short,axiom,
! [Deg: nat,Mi: nat,X: nat,TreeList: list @ vEBT_VEBT,Ma: nat,Summary: vEBT_VEBT] :
( ( ord_less_eq @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ Deg )
=> ( ( ord_less_eq @ nat @ Mi @ X )
=> ( ( ord_less_eq @ nat @ ( size_size @ ( list @ vEBT_VEBT ) @ TreeList ) @ ( vEBT_VEBT_high @ X @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) )
=> ( ( vEBT_vebt_succ @ ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi @ Ma ) ) @ Deg @ TreeList @ Summary ) @ X )
= ( none @ nat ) ) ) ) ) ).
% succ_list_to_short
thf(fact_227_nested__mint,axiom,
! [Mi: nat,Ma: nat,Deg: nat,TreeList: list @ vEBT_VEBT,Summary: vEBT_VEBT,N3: nat,Va: nat] :
( ( vEBT_invar_vebt @ ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi @ Ma ) ) @ Deg @ TreeList @ Summary ) @ N3 )
=> ( ( N3
= ( 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_228_insert__simp__mima,axiom,
! [X: nat,Mi: nat,Ma: nat,Deg: nat,TreeList: list @ vEBT_VEBT,Summary: vEBT_VEBT] :
( ( ( X = Mi )
| ( X = Ma ) )
=> ( ( ord_less_eq @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ Deg )
=> ( ( vEBT_vebt_insert @ ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi @ Ma ) ) @ Deg @ TreeList @ Summary ) @ X )
= ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi @ Ma ) ) @ Deg @ TreeList @ Summary ) ) ) ) ).
% insert_simp_mima
thf(fact_229_pred__max,axiom,
! [Deg: nat,Ma: nat,X: nat,Mi: nat,TreeList: list @ vEBT_VEBT,Summary: vEBT_VEBT] :
( ( ord_less_eq @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ Deg )
=> ( ( ord_less @ nat @ Ma @ X )
=> ( ( vEBT_vebt_pred @ ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi @ Ma ) ) @ Deg @ TreeList @ Summary ) @ X )
= ( some @ nat @ Ma ) ) ) ) ).
% pred_max
thf(fact_230_succ__min,axiom,
! [Deg: nat,X: nat,Mi: nat,Ma: nat,TreeList: list @ vEBT_VEBT,Summary: vEBT_VEBT] :
( ( ord_less_eq @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ Deg )
=> ( ( ord_less @ nat @ X @ Mi )
=> ( ( vEBT_vebt_succ @ ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi @ Ma ) ) @ Deg @ TreeList @ Summary ) @ X )
= ( some @ nat @ Mi ) ) ) ) ).
% succ_min
thf(fact_231_semiring__norm_I69_J,axiom,
! [M: num] :
~ ( ord_less_eq @ num @ ( bit0 @ M ) @ one2 ) ).
% semiring_norm(69)
thf(fact_232_semiring__norm_I76_J,axiom,
! [N3: num] : ( ord_less @ num @ one2 @ ( bit0 @ N3 ) ) ).
% semiring_norm(76)
thf(fact_233_semiring__norm_I2_J,axiom,
( ( plus_plus @ num @ one2 @ one2 )
= ( bit0 @ one2 ) ) ).
% semiring_norm(2)
thf(fact_234_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_235_deg__deg__n,axiom,
! [Info: option @ ( product_prod @ nat @ nat ),Deg: nat,TreeList: list @ vEBT_VEBT,Summary: vEBT_VEBT,N3: nat] :
( ( vEBT_invar_vebt @ ( vEBT_Node @ Info @ Deg @ TreeList @ Summary ) @ N3 )
=> ( Deg = N3 ) ) ).
% deg_deg_n
thf(fact_236_delete__pres__valid,axiom,
! [T2: vEBT_VEBT,N3: nat,X: nat] :
( ( vEBT_invar_vebt @ T2 @ N3 )
=> ( vEBT_invar_vebt @ ( vEBT_vebt_delete @ T2 @ X ) @ N3 ) ) ).
% delete_pres_valid
thf(fact_237_deg__SUcn__Node,axiom,
! [Tree: vEBT_VEBT,N3: nat] :
( ( vEBT_invar_vebt @ Tree @ ( suc @ ( suc @ N3 ) ) )
=> ? [Info2: option @ ( product_prod @ nat @ nat ),TreeList2: list @ vEBT_VEBT,S2: vEBT_VEBT] :
( Tree
= ( vEBT_Node @ Info2 @ ( suc @ ( suc @ N3 ) ) @ TreeList2 @ S2 ) ) ) ).
% deg_SUcn_Node
thf(fact_238_dele__member__cont__corr,axiom,
! [T2: vEBT_VEBT,N3: nat,X: nat,Y: nat] :
( ( vEBT_invar_vebt @ T2 @ N3 )
=> ( ( vEBT_vebt_member @ ( vEBT_vebt_delete @ T2 @ X ) @ Y )
= ( ( X != Y )
& ( vEBT_vebt_member @ T2 @ Y ) ) ) ) ).
% dele_member_cont_corr
thf(fact_239_mint__member,axiom,
! [T2: vEBT_VEBT,N3: nat,Maxi: nat] :
( ( vEBT_invar_vebt @ T2 @ N3 )
=> ( ( ( vEBT_vebt_mint @ T2 )
= ( some @ nat @ Maxi ) )
=> ( vEBT_vebt_member @ T2 @ Maxi ) ) ) ).
% mint_member
thf(fact_240_semiring__norm_I87_J,axiom,
! [M: num,N3: num] :
( ( ( bit0 @ M )
= ( bit0 @ N3 ) )
= ( M = N3 ) ) ).
% semiring_norm(87)
thf(fact_241_real__divide__square__eq,axiom,
! [R2: real,A2: real] :
( ( divide_divide @ real @ ( times_times @ real @ R2 @ A2 ) @ ( times_times @ real @ R2 @ R2 ) )
= ( divide_divide @ real @ A2 @ R2 ) ) ).
% real_divide_square_eq
thf(fact_242_mint__corr__help,axiom,
! [T2: vEBT_VEBT,N3: nat,Mini: nat,X: nat] :
( ( vEBT_invar_vebt @ T2 @ N3 )
=> ( ( ( vEBT_vebt_mint @ T2 )
= ( some @ nat @ Mini ) )
=> ( ( vEBT_vebt_member @ T2 @ X )
=> ( ord_less_eq @ nat @ Mini @ X ) ) ) ) ).
% mint_corr_help
thf(fact_243_member__bound,axiom,
! [Tree: vEBT_VEBT,X: nat,N3: nat] :
( ( vEBT_vebt_member @ Tree @ X )
=> ( ( vEBT_invar_vebt @ Tree @ N3 )
=> ( ord_less @ nat @ X @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N3 ) ) ) ) ).
% member_bound
thf(fact_244_geqmaxNone,axiom,
! [Mi: nat,Ma: nat,Deg: nat,TreeList: list @ vEBT_VEBT,Summary: vEBT_VEBT,N3: nat,X: nat] :
( ( vEBT_invar_vebt @ ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi @ Ma ) ) @ Deg @ TreeList @ Summary ) @ N3 )
=> ( ( ord_less_eq @ nat @ Ma @ X )
=> ( ( vEBT_vebt_succ @ ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi @ Ma ) ) @ Deg @ TreeList @ Summary ) @ X )
= ( none @ nat ) ) ) ) ).
% geqmaxNone
thf(fact_245_semiring__norm_I83_J,axiom,
! [N3: num] :
( one2
!= ( bit0 @ N3 ) ) ).
% semiring_norm(83)
thf(fact_246_semiring__norm_I85_J,axiom,
! [M: num] :
( ( bit0 @ M )
!= one2 ) ).
% semiring_norm(85)
thf(fact_247_misiz,axiom,
! [T2: vEBT_VEBT,N3: nat,M: nat] :
( ( vEBT_invar_vebt @ T2 @ N3 )
=> ( ( ( some @ nat @ M )
= ( vEBT_vebt_mint @ T2 ) )
=> ( ord_less @ nat @ M @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N3 ) ) ) ) ).
% misiz
thf(fact_248_helpypredd,axiom,
! [T2: vEBT_VEBT,N3: nat,X: nat,Y: nat] :
( ( vEBT_invar_vebt @ T2 @ N3 )
=> ( ( ( vEBT_vebt_pred @ T2 @ X )
= ( some @ nat @ Y ) )
=> ( ord_less @ nat @ Y @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N3 ) ) ) ) ).
% helpypredd
thf(fact_249_helpyd,axiom,
! [T2: vEBT_VEBT,N3: nat,X: nat,Y: nat] :
( ( vEBT_invar_vebt @ T2 @ N3 )
=> ( ( ( vEBT_vebt_succ @ T2 @ X )
= ( some @ nat @ Y ) )
=> ( ord_less @ nat @ Y @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N3 ) ) ) ) ).
% helpyd
thf(fact_250_Suc__diff__diff,axiom,
! [M: nat,N3: nat,K: nat] :
( ( minus_minus @ nat @ ( minus_minus @ nat @ ( suc @ M ) @ N3 ) @ ( suc @ K ) )
= ( minus_minus @ nat @ ( minus_minus @ nat @ M @ N3 ) @ K ) ) ).
% Suc_diff_diff
thf(fact_251_diff__Suc__Suc,axiom,
! [M: nat,N3: nat] :
( ( minus_minus @ nat @ ( suc @ M ) @ ( suc @ N3 ) )
= ( minus_minus @ nat @ M @ N3 ) ) ).
% diff_Suc_Suc
thf(fact_252_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_253_diff__diff__cancel,axiom,
! [I: nat,N3: nat] :
( ( ord_less_eq @ nat @ I @ N3 )
=> ( ( minus_minus @ nat @ N3 @ ( minus_minus @ nat @ N3 @ I ) )
= I ) ) ).
% diff_diff_cancel
thf(fact_254_post__member__pre__member,axiom,
! [T2: vEBT_VEBT,N3: nat,X: nat,Y: nat] :
( ( vEBT_invar_vebt @ T2 @ N3 )
=> ( ( ord_less @ nat @ X @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N3 ) )
=> ( ( ord_less @ nat @ Y @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N3 ) )
=> ( ( vEBT_vebt_member @ ( vEBT_vebt_insert @ T2 @ X ) @ Y )
=> ( ( vEBT_vebt_member @ T2 @ Y )
| ( X = Y ) ) ) ) ) ) ).
% post_member_pre_member
thf(fact_255_semiring__norm_I6_J,axiom,
! [M: num,N3: num] :
( ( plus_plus @ num @ ( bit0 @ M ) @ ( bit0 @ N3 ) )
= ( bit0 @ ( plus_plus @ num @ M @ N3 ) ) ) ).
% semiring_norm(6)
thf(fact_256_semiring__norm_I13_J,axiom,
! [M: num,N3: num] :
( ( times_times @ num @ ( bit0 @ M ) @ ( bit0 @ N3 ) )
= ( bit0 @ ( bit0 @ ( times_times @ num @ M @ N3 ) ) ) ) ).
% semiring_norm(13)
thf(fact_257_semiring__norm_I11_J,axiom,
! [M: num] :
( ( times_times @ num @ M @ one2 )
= M ) ).
% semiring_norm(11)
thf(fact_258_semiring__norm_I12_J,axiom,
! [N3: num] :
( ( times_times @ num @ one2 @ N3 )
= N3 ) ).
% semiring_norm(12)
thf(fact_259_semiring__norm_I78_J,axiom,
! [M: num,N3: num] :
( ( ord_less @ num @ ( bit0 @ M ) @ ( bit0 @ N3 ) )
= ( ord_less @ num @ M @ N3 ) ) ).
% semiring_norm(78)
thf(fact_260_semiring__norm_I71_J,axiom,
! [M: num,N3: num] :
( ( ord_less_eq @ num @ ( bit0 @ M ) @ ( bit0 @ N3 ) )
= ( ord_less_eq @ num @ M @ N3 ) ) ).
% semiring_norm(71)
thf(fact_261_semiring__norm_I75_J,axiom,
! [M: num] :
~ ( ord_less @ num @ M @ one2 ) ).
% semiring_norm(75)
thf(fact_262_semiring__norm_I68_J,axiom,
! [N3: num] : ( ord_less_eq @ num @ one2 @ N3 ) ).
% semiring_norm(68)
thf(fact_263_sumprop,axiom,
vEBT_invar_vebt @ summary @ ( minus_minus @ nat @ na @ ( divide_divide @ nat @ na @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ).
% sumprop
thf(fact_264_mi__ma__2__deg,axiom,
! [Mi: nat,Ma: nat,Deg: nat,TreeList: list @ vEBT_VEBT,Summary: vEBT_VEBT,N3: nat] :
( ( vEBT_invar_vebt @ ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi @ Ma ) ) @ Deg @ TreeList @ Summary ) @ N3 )
=> ( ( 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_265_member__correct,axiom,
! [T2: vEBT_VEBT,N3: nat,X: nat] :
( ( vEBT_invar_vebt @ T2 @ N3 )
=> ( ( vEBT_vebt_member @ T2 @ X )
= ( member @ nat @ X @ ( vEBT_set_vebt @ T2 ) ) ) ) ).
% member_correct
thf(fact_266_right__diff__distrib__numeral,axiom,
! [A: $tType] :
( ( ( numeral @ A )
& ( ring @ A ) )
=> ! [V: num,B2: A,C2: A] :
( ( times_times @ A @ ( numeral_numeral @ A @ V ) @ ( minus_minus @ A @ B2 @ C2 ) )
= ( minus_minus @ A @ ( times_times @ A @ ( numeral_numeral @ A @ V ) @ B2 ) @ ( times_times @ A @ ( numeral_numeral @ A @ V ) @ C2 ) ) ) ) ).
% right_diff_distrib_numeral
thf(fact_267_left__diff__distrib__numeral,axiom,
! [A: $tType] :
( ( ( numeral @ A )
& ( ring @ A ) )
=> ! [A2: A,B2: A,V: num] :
( ( times_times @ A @ ( minus_minus @ A @ A2 @ B2 ) @ ( numeral_numeral @ A @ V ) )
= ( minus_minus @ A @ ( times_times @ A @ A2 @ ( numeral_numeral @ A @ V ) ) @ ( times_times @ A @ B2 @ ( numeral_numeral @ A @ V ) ) ) ) ) ).
% left_diff_distrib_numeral
thf(fact_268_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_269_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_270_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_271_mintlistlength,axiom,
! [Mi: nat,Ma: nat,Deg: nat,TreeList: list @ vEBT_VEBT,Summary: vEBT_VEBT,N3: nat] :
( ( vEBT_invar_vebt @ ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi @ Ma ) ) @ Deg @ TreeList @ Summary ) @ N3 )
=> ( ( Mi != Ma )
=> ( ( ord_less @ nat @ Mi @ Ma )
& ? [M4: nat] :
( ( ( some @ nat @ M4 )
= ( vEBT_vebt_mint @ Summary ) )
& ( ord_less @ nat @ M4 @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( minus_minus @ nat @ N3 @ ( divide_divide @ nat @ N3 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) ) ) ) ) ).
% mintlistlength
thf(fact_272_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_273_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_274_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_275_zero__induct__lemma,axiom,
! [P: nat > $o,K: nat,I: nat] :
( ( P @ K )
=> ( ! [N: nat] :
( ( P @ ( suc @ N ) )
=> ( P @ N ) )
=> ( P @ ( minus_minus @ nat @ K @ I ) ) ) ) ).
% zero_induct_lemma
thf(fact_276_less__imp__diff__less,axiom,
! [J: nat,K: nat,N3: nat] :
( ( ord_less @ nat @ J @ K )
=> ( ord_less @ nat @ ( minus_minus @ nat @ J @ N3 ) @ K ) ) ).
% less_imp_diff_less
thf(fact_277_diff__less__mono2,axiom,
! [M: nat,N3: nat,L2: nat] :
( ( ord_less @ nat @ M @ N3 )
=> ( ( ord_less @ nat @ M @ L2 )
=> ( ord_less @ nat @ ( minus_minus @ nat @ L2 @ N3 ) @ ( minus_minus @ nat @ L2 @ M ) ) ) ) ).
% diff_less_mono2
thf(fact_278_diff__diff__less,axiom,
! [I: nat,M: nat,N3: nat] :
( ( ord_less @ nat @ I @ ( minus_minus @ nat @ M @ ( minus_minus @ nat @ M @ N3 ) ) )
= ( ( ord_less @ nat @ I @ M )
& ( ord_less @ nat @ I @ N3 ) ) ) ).
% diff_diff_less
thf(fact_279_diff__add__inverse2,axiom,
! [M: nat,N3: nat] :
( ( minus_minus @ nat @ ( plus_plus @ nat @ M @ N3 ) @ N3 )
= M ) ).
% diff_add_inverse2
thf(fact_280_diff__add__inverse,axiom,
! [N3: nat,M: nat] :
( ( minus_minus @ nat @ ( plus_plus @ nat @ N3 @ M ) @ N3 )
= M ) ).
% diff_add_inverse
thf(fact_281_diff__cancel2,axiom,
! [M: nat,K: nat,N3: nat] :
( ( minus_minus @ nat @ ( plus_plus @ nat @ M @ K ) @ ( plus_plus @ nat @ N3 @ K ) )
= ( minus_minus @ nat @ M @ N3 ) ) ).
% diff_cancel2
thf(fact_282_Nat_Odiff__cancel,axiom,
! [K: nat,M: nat,N3: nat] :
( ( minus_minus @ nat @ ( plus_plus @ nat @ K @ M ) @ ( plus_plus @ nat @ K @ N3 ) )
= ( minus_minus @ nat @ M @ N3 ) ) ).
% Nat.diff_cancel
thf(fact_283_eq__diff__iff,axiom,
! [K: nat,M: nat,N3: nat] :
( ( ord_less_eq @ nat @ K @ M )
=> ( ( ord_less_eq @ nat @ K @ N3 )
=> ( ( ( minus_minus @ nat @ M @ K )
= ( minus_minus @ nat @ N3 @ K ) )
= ( M = N3 ) ) ) ) ).
% eq_diff_iff
thf(fact_284_le__diff__iff,axiom,
! [K: nat,M: nat,N3: nat] :
( ( ord_less_eq @ nat @ K @ M )
=> ( ( ord_less_eq @ nat @ K @ N3 )
=> ( ( ord_less_eq @ nat @ ( minus_minus @ nat @ M @ K ) @ ( minus_minus @ nat @ N3 @ K ) )
= ( ord_less_eq @ nat @ M @ N3 ) ) ) ) ).
% le_diff_iff
thf(fact_285_Nat_Odiff__diff__eq,axiom,
! [K: nat,M: nat,N3: nat] :
( ( ord_less_eq @ nat @ K @ M )
=> ( ( ord_less_eq @ nat @ K @ N3 )
=> ( ( minus_minus @ nat @ ( minus_minus @ nat @ M @ K ) @ ( minus_minus @ nat @ N3 @ K ) )
= ( minus_minus @ nat @ M @ N3 ) ) ) ) ).
% Nat.diff_diff_eq
thf(fact_286_diff__le__mono,axiom,
! [M: nat,N3: nat,L2: nat] :
( ( ord_less_eq @ nat @ M @ N3 )
=> ( ord_less_eq @ nat @ ( minus_minus @ nat @ M @ L2 ) @ ( minus_minus @ nat @ N3 @ L2 ) ) ) ).
% diff_le_mono
thf(fact_287_diff__le__self,axiom,
! [M: nat,N3: nat] : ( ord_less_eq @ nat @ ( minus_minus @ nat @ M @ N3 ) @ M ) ).
% diff_le_self
thf(fact_288_le__diff__iff_H,axiom,
! [A2: nat,C2: nat,B2: nat] :
( ( ord_less_eq @ nat @ A2 @ C2 )
=> ( ( ord_less_eq @ nat @ B2 @ C2 )
=> ( ( ord_less_eq @ nat @ ( minus_minus @ nat @ C2 @ A2 ) @ ( minus_minus @ nat @ C2 @ B2 ) )
= ( ord_less_eq @ nat @ B2 @ A2 ) ) ) ) ).
% le_diff_iff'
thf(fact_289_diff__le__mono2,axiom,
! [M: nat,N3: nat,L2: nat] :
( ( ord_less_eq @ nat @ M @ N3 )
=> ( ord_less_eq @ nat @ ( minus_minus @ nat @ L2 @ N3 ) @ ( minus_minus @ nat @ L2 @ M ) ) ) ).
% diff_le_mono2
thf(fact_290_diff__mult__distrib2,axiom,
! [K: nat,M: nat,N3: nat] :
( ( times_times @ nat @ K @ ( minus_minus @ nat @ M @ N3 ) )
= ( minus_minus @ nat @ ( times_times @ nat @ K @ M ) @ ( times_times @ nat @ K @ N3 ) ) ) ).
% diff_mult_distrib2
thf(fact_291_diff__mult__distrib,axiom,
! [M: nat,N3: nat,K: nat] :
( ( times_times @ nat @ ( minus_minus @ nat @ M @ N3 ) @ K )
= ( minus_minus @ nat @ ( times_times @ nat @ M @ K ) @ ( times_times @ nat @ N3 @ K ) ) ) ).
% diff_mult_distrib
thf(fact_292_diff__less__Suc,axiom,
! [M: nat,N3: nat] : ( ord_less @ nat @ ( minus_minus @ nat @ M @ N3 ) @ ( suc @ M ) ) ).
% diff_less_Suc
thf(fact_293_Suc__diff__Suc,axiom,
! [N3: nat,M: nat] :
( ( ord_less @ nat @ N3 @ M )
=> ( ( suc @ ( minus_minus @ nat @ M @ ( suc @ N3 ) ) )
= ( minus_minus @ nat @ M @ N3 ) ) ) ).
% Suc_diff_Suc
thf(fact_294_nat__diff__add__eq2,axiom,
! [I: nat,J: nat,U: nat,M: nat,N3: 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 ) @ N3 ) )
= ( minus_minus @ nat @ M @ ( plus_plus @ nat @ ( times_times @ nat @ ( minus_minus @ nat @ J @ I ) @ U ) @ N3 ) ) ) ) ).
% nat_diff_add_eq2
thf(fact_295_nat__diff__add__eq1,axiom,
! [J: nat,I: nat,U: nat,M: nat,N3: 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 ) @ N3 ) )
= ( minus_minus @ nat @ ( plus_plus @ nat @ ( times_times @ nat @ ( minus_minus @ nat @ I @ J ) @ U ) @ M ) @ N3 ) ) ) ).
% nat_diff_add_eq1
thf(fact_296_nat__le__add__iff2,axiom,
! [I: nat,J: nat,U: nat,M: nat,N3: 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 ) @ N3 ) )
= ( ord_less_eq @ nat @ M @ ( plus_plus @ nat @ ( times_times @ nat @ ( minus_minus @ nat @ J @ I ) @ U ) @ N3 ) ) ) ) ).
% nat_le_add_iff2
thf(fact_297_nat__le__add__iff1,axiom,
! [J: nat,I: nat,U: nat,M: nat,N3: 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 ) @ N3 ) )
= ( ord_less_eq @ nat @ ( plus_plus @ nat @ ( times_times @ nat @ ( minus_minus @ nat @ I @ J ) @ U ) @ M ) @ N3 ) ) ) ).
% nat_le_add_iff1
thf(fact_298_nat__eq__add__iff2,axiom,
! [I: nat,J: nat,U: nat,M: nat,N3: nat] :
( ( ord_less_eq @ nat @ I @ J )
=> ( ( ( plus_plus @ nat @ ( times_times @ nat @ I @ U ) @ M )
= ( plus_plus @ nat @ ( times_times @ nat @ J @ U ) @ N3 ) )
= ( M
= ( plus_plus @ nat @ ( times_times @ nat @ ( minus_minus @ nat @ J @ I ) @ U ) @ N3 ) ) ) ) ).
% nat_eq_add_iff2
thf(fact_299_nat__eq__add__iff1,axiom,
! [J: nat,I: nat,U: nat,M: nat,N3: nat] :
( ( ord_less_eq @ nat @ J @ I )
=> ( ( ( plus_plus @ nat @ ( times_times @ nat @ I @ U ) @ M )
= ( plus_plus @ nat @ ( times_times @ nat @ J @ U ) @ N3 ) )
= ( ( plus_plus @ nat @ ( times_times @ nat @ ( minus_minus @ nat @ I @ J ) @ U ) @ M )
= N3 ) ) ) ).
% nat_eq_add_iff1
thf(fact_300_Suc__diff__le,axiom,
! [N3: nat,M: nat] :
( ( ord_less_eq @ nat @ N3 @ M )
=> ( ( minus_minus @ nat @ ( suc @ M ) @ N3 )
= ( suc @ ( minus_minus @ nat @ M @ N3 ) ) ) ) ).
% Suc_diff_le
thf(fact_301_add__diff__inverse__nat,axiom,
! [M: nat,N3: nat] :
( ~ ( ord_less @ nat @ M @ N3 )
=> ( ( plus_plus @ nat @ N3 @ ( minus_minus @ nat @ M @ N3 ) )
= M ) ) ).
% add_diff_inverse_nat
thf(fact_302_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_303_less__diff__iff,axiom,
! [K: nat,M: nat,N3: nat] :
( ( ord_less_eq @ nat @ K @ M )
=> ( ( ord_less_eq @ nat @ K @ N3 )
=> ( ( ord_less @ nat @ ( minus_minus @ nat @ M @ K ) @ ( minus_minus @ nat @ N3 @ K ) )
= ( ord_less @ nat @ M @ N3 ) ) ) ) ).
% less_diff_iff
thf(fact_304_diff__less__mono,axiom,
! [A2: nat,B2: nat,C2: nat] :
( ( ord_less @ nat @ A2 @ B2 )
=> ( ( ord_less_eq @ nat @ C2 @ A2 )
=> ( ord_less @ nat @ ( minus_minus @ nat @ A2 @ C2 ) @ ( minus_minus @ nat @ B2 @ C2 ) ) ) ) ).
% diff_less_mono
thf(fact_305_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_306_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_307_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_308_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_309_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_310_nat__less__add__iff2,axiom,
! [I: nat,J: nat,U: nat,M: nat,N3: 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 ) @ N3 ) )
= ( ord_less @ nat @ M @ ( plus_plus @ nat @ ( times_times @ nat @ ( minus_minus @ nat @ J @ I ) @ U ) @ N3 ) ) ) ) ).
% nat_less_add_iff2
thf(fact_311_nat__less__add__iff1,axiom,
! [J: nat,I: nat,U: nat,M: nat,N3: 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 ) @ N3 ) )
= ( ord_less @ nat @ ( plus_plus @ nat @ ( times_times @ nat @ ( minus_minus @ nat @ I @ J ) @ U ) @ M ) @ N3 ) ) ) ).
% nat_less_add_iff1
thf(fact_312_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_313_power2__commute,axiom,
! [A: $tType] :
( ( comm_ring_1 @ A )
=> ! [X: A,Y: A] :
( ( power_power @ A @ ( minus_minus @ A @ X @ Y ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) )
= ( power_power @ A @ ( minus_minus @ A @ Y @ X ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ).
% power2_commute
thf(fact_314_diff__le__diff__pow,axiom,
! [K: nat,M: nat,N3: nat] :
( ( ord_less_eq @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ K )
=> ( ord_less_eq @ nat @ ( minus_minus @ nat @ M @ N3 ) @ ( minus_minus @ nat @ ( power_power @ nat @ K @ M ) @ ( power_power @ nat @ K @ N3 ) ) ) ) ).
% diff_le_diff_pow
thf(fact_315_nat__power__less__diff,axiom,
! [N3: nat,Q2: nat,M: nat] :
( ( ord_less @ nat @ ( times_times @ nat @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N3 ) @ Q2 ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ M ) )
=> ( ord_less @ nat @ Q2 @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( minus_minus @ nat @ M @ N3 ) ) ) ) ).
% nat_power_less_diff
thf(fact_316_power__minus__is__div,axiom,
! [B2: nat,A2: nat] :
( ( ord_less_eq @ nat @ B2 @ A2 )
=> ( ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( minus_minus @ nat @ A2 @ B2 ) )
= ( divide_divide @ nat @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ A2 ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ B2 ) ) ) ) ).
% power_minus_is_div
thf(fact_317_nat__le__power__trans,axiom,
! [N3: nat,M: nat,K: nat] :
( ( ord_less_eq @ nat @ N3 @ ( 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 ) @ N3 ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ M ) ) ) ) ).
% nat_le_power_trans
thf(fact_318_power2__diff,axiom,
! [A: $tType] :
( ( comm_ring_1 @ A )
=> ! [X: A,Y: A] :
( ( power_power @ A @ ( minus_minus @ A @ X @ Y ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) )
= ( minus_minus @ A @ ( plus_plus @ A @ ( power_power @ A @ X @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ ( power_power @ A @ Y @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( times_times @ A @ ( times_times @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ X ) @ Y ) ) ) ) ).
% power2_diff
thf(fact_319_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_320_less__two__pow__divI,axiom,
! [X: nat,N3: nat,M: nat] :
( ( ord_less @ nat @ X @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( minus_minus @ nat @ N3 @ M ) ) )
=> ( ( ord_less_eq @ nat @ M @ N3 )
=> ( ord_less @ nat @ X @ ( divide_divide @ nat @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N3 ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ M ) ) ) ) ) ).
% less_two_pow_divI
thf(fact_321_less__two__pow__divD,axiom,
! [X: nat,N3: nat,M: nat] :
( ( ord_less @ nat @ X @ ( divide_divide @ nat @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N3 ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ M ) ) )
=> ( ( ord_less_eq @ nat @ M @ N3 )
& ( ord_less @ nat @ X @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( minus_minus @ nat @ N3 @ M ) ) ) ) ) ).
% less_two_pow_divD
thf(fact_322_nat__less__power__trans,axiom,
! [N3: nat,M: nat,K: nat] :
( ( ord_less @ nat @ N3 @ ( 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 ) @ N3 ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ M ) ) ) ) ).
% nat_less_power_trans
thf(fact_323_mint__corr,axiom,
! [T2: vEBT_VEBT,N3: nat,X: nat] :
( ( vEBT_invar_vebt @ T2 @ N3 )
=> ( ( ( vEBT_vebt_mint @ T2 )
= ( some @ nat @ X ) )
=> ( vEBT_VEBT_min_in_set @ ( vEBT_VEBT_set_vebt @ T2 ) @ X ) ) ) ).
% mint_corr
thf(fact_324_mint__sound,axiom,
! [T2: vEBT_VEBT,N3: nat,X: nat] :
( ( vEBT_invar_vebt @ T2 @ N3 )
=> ( ( vEBT_VEBT_min_in_set @ ( vEBT_VEBT_set_vebt @ T2 ) @ X )
=> ( ( vEBT_vebt_mint @ T2 )
= ( some @ nat @ X ) ) ) ) ).
% mint_sound
thf(fact_325_not__Some__eq2,axiom,
! [B: $tType,A: $tType,V: option @ ( product_prod @ A @ B )] :
( ( ! [X2: A,Y2: B] :
( V
!= ( some @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X2 @ Y2 ) ) ) )
= ( V
= ( none @ ( product_prod @ A @ B ) ) ) ) ).
% not_Some_eq2
thf(fact_326_le__add__diff__inverse,axiom,
! [A: $tType] :
( ( linordered_semidom @ A )
=> ! [B2: A,A2: A] :
( ( ord_less_eq @ A @ B2 @ A2 )
=> ( ( plus_plus @ A @ B2 @ ( minus_minus @ A @ A2 @ B2 ) )
= A2 ) ) ) ).
% le_add_diff_inverse
thf(fact_327_le__add__diff__inverse2,axiom,
! [A: $tType] :
( ( linordered_semidom @ A )
=> ! [B2: A,A2: A] :
( ( ord_less_eq @ A @ B2 @ A2 )
=> ( ( plus_plus @ A @ ( minus_minus @ A @ A2 @ B2 ) @ B2 )
= A2 ) ) ) ).
% le_add_diff_inverse2
thf(fact_328_two__powr__height__bound__deg,axiom,
! [T2: vEBT_VEBT,N3: nat] :
( ( vEBT_invar_vebt @ T2 @ N3 )
=> ( ord_less_eq @ nat @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( vEBT_VEBT_height @ T2 ) ) @ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N3 ) ) ) ).
% two_powr_height_bound_deg
thf(fact_329_both__member__options__ding,axiom,
! [Info: option @ ( product_prod @ nat @ nat ),Deg: nat,TreeList: list @ vEBT_VEBT,Summary: vEBT_VEBT,N3: nat,X: nat] :
( ( vEBT_invar_vebt @ ( vEBT_Node @ Info @ Deg @ TreeList @ Summary ) @ N3 )
=> ( ( ord_less @ nat @ X @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ Deg ) )
=> ( ( vEBT_V8194947554948674370ptions @ ( nth @ vEBT_VEBT @ TreeList @ ( vEBT_VEBT_high @ X @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( vEBT_VEBT_low @ X @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) )
=> ( vEBT_V8194947554948674370ptions @ ( vEBT_Node @ Info @ Deg @ TreeList @ Summary ) @ X ) ) ) ) ).
% both_member_options_ding
thf(fact_330_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_331_vebt__insert_Osimps_I4_J,axiom,
! [V: nat,TreeList: list @ vEBT_VEBT,Summary: vEBT_VEBT,X: nat] :
( ( vEBT_vebt_insert @ ( vEBT_Node @ ( none @ ( product_prod @ nat @ nat ) ) @ ( suc @ ( suc @ V ) ) @ TreeList @ Summary ) @ X )
= ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ X @ X ) ) @ ( suc @ ( suc @ V ) ) @ TreeList @ Summary ) ) ).
% vebt_insert.simps(4)
thf(fact_332_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_333_div__exp__eq,axiom,
! [A: $tType] :
( ( bit_semiring_bits @ A )
=> ! [A2: A,M: nat,N3: nat] :
( ( divide_divide @ A @ ( divide_divide @ A @ A2 @ ( power_power @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ M ) ) @ ( power_power @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ N3 ) )
= ( divide_divide @ A @ A2 @ ( power_power @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ ( plus_plus @ nat @ M @ N3 ) ) ) ) ) ).
% div_exp_eq
thf(fact_334_maxbmo,axiom,
! [T2: vEBT_VEBT,X: nat] :
( ( ( vEBT_vebt_maxt @ T2 )
= ( some @ nat @ X ) )
=> ( vEBT_V8194947554948674370ptions @ T2 @ X ) ) ).
% maxbmo
thf(fact_335_dele__bmo__cont__corr,axiom,
! [T2: vEBT_VEBT,N3: nat,X: nat,Y: nat] :
( ( vEBT_invar_vebt @ T2 @ N3 )
=> ( ( vEBT_V8194947554948674370ptions @ ( vEBT_vebt_delete @ T2 @ X ) @ Y )
= ( ( X != Y )
& ( vEBT_V8194947554948674370ptions @ T2 @ Y ) ) ) ) ).
% dele_bmo_cont_corr
thf(fact_336_valid__member__both__member__options,axiom,
! [T2: vEBT_VEBT,N3: nat,X: nat] :
( ( vEBT_invar_vebt @ T2 @ N3 )
=> ( ( vEBT_V8194947554948674370ptions @ T2 @ X )
=> ( vEBT_vebt_member @ T2 @ X ) ) ) ).
% valid_member_both_member_options
thf(fact_337_both__member__options__equiv__member,axiom,
! [T2: vEBT_VEBT,N3: nat,X: nat] :
( ( vEBT_invar_vebt @ T2 @ N3 )
=> ( ( vEBT_V8194947554948674370ptions @ T2 @ X )
= ( vEBT_vebt_member @ T2 @ X ) ) ) ).
% both_member_options_equiv_member
thf(fact_338_set__vebt__set__vebt_H__valid,axiom,
! [T2: vEBT_VEBT,N3: nat] :
( ( vEBT_invar_vebt @ T2 @ N3 )
=> ( ( vEBT_set_vebt @ T2 )
= ( vEBT_VEBT_set_vebt @ T2 ) ) ) ).
% set_vebt_set_vebt'_valid
thf(fact_339_inthall,axiom,
! [A: $tType,Xs2: list @ A,P: A > $o,N3: nat] :
( ! [X3: A] :
( ( member @ A @ X3 @ ( set2 @ A @ Xs2 ) )
=> ( P @ X3 ) )
=> ( ( ord_less @ nat @ N3 @ ( size_size @ ( list @ A ) @ Xs2 ) )
=> ( P @ ( nth @ A @ Xs2 @ N3 ) ) ) ) ).
% inthall
thf(fact_340_maxt__member,axiom,
! [T2: vEBT_VEBT,N3: nat,Maxi: nat] :
( ( vEBT_invar_vebt @ T2 @ N3 )
=> ( ( ( vEBT_vebt_maxt @ T2 )
= ( some @ nat @ Maxi ) )
=> ( vEBT_vebt_member @ T2 @ Maxi ) ) ) ).
% maxt_member
thf(fact_341_maxt__corr__help,axiom,
! [T2: vEBT_VEBT,N3: nat,Maxi: nat,X: nat] :
( ( vEBT_invar_vebt @ T2 @ N3 )
=> ( ( ( vEBT_vebt_maxt @ T2 )
= ( some @ nat @ Maxi ) )
=> ( ( vEBT_vebt_member @ T2 @ X )
=> ( ord_less_eq @ nat @ X @ Maxi ) ) ) ) ).
% maxt_corr_help
thf(fact_342_maxt__corr,axiom,
! [T2: vEBT_VEBT,N3: nat,X: nat] :
( ( vEBT_invar_vebt @ T2 @ N3 )
=> ( ( ( vEBT_vebt_maxt @ T2 )
= ( some @ nat @ X ) )
=> ( vEBT_VEBT_max_in_set @ ( vEBT_VEBT_set_vebt @ T2 ) @ X ) ) ) ).
% maxt_corr
thf(fact_343_maxt__sound,axiom,
! [T2: vEBT_VEBT,N3: nat,X: nat] :
( ( vEBT_invar_vebt @ T2 @ N3 )
=> ( ( vEBT_VEBT_max_in_set @ ( vEBT_VEBT_set_vebt @ T2 ) @ X )
=> ( ( vEBT_vebt_maxt @ T2 )
= ( some @ nat @ X ) ) ) ) ).
% maxt_sound
thf(fact_344_mi__eq__ma__no__ch,axiom,
! [Mi: nat,Ma: nat,Deg: nat,TreeList: list @ vEBT_VEBT,Summary: vEBT_VEBT] :
( ( vEBT_invar_vebt @ ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi @ Ma ) ) @ Deg @ TreeList @ Summary ) @ Deg )
=> ( ( Mi = Ma )
=> ( ! [X5: vEBT_VEBT] :
( ( member @ vEBT_VEBT @ X5 @ ( set2 @ vEBT_VEBT @ TreeList ) )
=> ~ ? [X_1: nat] : ( vEBT_V8194947554948674370ptions @ X5 @ X_1 ) )
& ~ ? [X_1: nat] : ( vEBT_V8194947554948674370ptions @ Summary @ X_1 ) ) ) ) ).
% mi_eq_ma_no_ch
thf(fact_345_valid__insert__both__member__options__add,axiom,
! [T2: vEBT_VEBT,N3: nat,X: nat] :
( ( vEBT_invar_vebt @ T2 @ N3 )
=> ( ( ord_less @ nat @ X @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N3 ) )
=> ( vEBT_V8194947554948674370ptions @ ( vEBT_vebt_insert @ T2 @ X ) @ X ) ) ) ).
% valid_insert_both_member_options_add
thf(fact_346_valid__insert__both__member__options__pres,axiom,
! [T2: vEBT_VEBT,N3: nat,X: nat,Y: nat] :
( ( vEBT_invar_vebt @ T2 @ N3 )
=> ( ( ord_less @ nat @ X @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N3 ) )
=> ( ( ord_less @ nat @ Y @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N3 ) )
=> ( ( vEBT_V8194947554948674370ptions @ T2 @ X )
=> ( vEBT_V8194947554948674370ptions @ ( vEBT_vebt_insert @ T2 @ Y ) @ X ) ) ) ) ) ).
% valid_insert_both_member_options_pres
thf(fact_347_succ__member,axiom,
! [T2: vEBT_VEBT,X: nat,Y: nat] :
( ( vEBT_is_succ_in_set @ ( vEBT_VEBT_set_vebt @ T2 ) @ X @ Y )
= ( ( vEBT_vebt_member @ T2 @ Y )
& ( ord_less @ nat @ X @ Y )
& ! [Z3: nat] :
( ( ( vEBT_vebt_member @ T2 @ Z3 )
& ( ord_less @ nat @ X @ Z3 ) )
=> ( ord_less_eq @ nat @ Y @ Z3 ) ) ) ) ).
% succ_member
thf(fact_348_pred__member,axiom,
! [T2: vEBT_VEBT,X: nat,Y: nat] :
( ( vEBT_is_pred_in_set @ ( vEBT_VEBT_set_vebt @ T2 ) @ X @ Y )
= ( ( vEBT_vebt_member @ T2 @ Y )
& ( ord_less @ nat @ Y @ X )
& ! [Z3: nat] :
( ( ( vEBT_vebt_member @ T2 @ Z3 )
& ( ord_less @ nat @ Z3 @ X ) )
=> ( ord_less_eq @ nat @ Z3 @ Y ) ) ) ) ).
% pred_member
thf(fact_349_succ__corr,axiom,
! [T2: vEBT_VEBT,N3: nat,X: nat,Sx: nat] :
( ( vEBT_invar_vebt @ T2 @ N3 )
=> ( ( ( vEBT_vebt_succ @ T2 @ X )
= ( some @ nat @ Sx ) )
= ( vEBT_is_succ_in_set @ ( vEBT_VEBT_set_vebt @ T2 ) @ X @ Sx ) ) ) ).
% succ_corr
thf(fact_350_pred__corr,axiom,
! [T2: vEBT_VEBT,N3: nat,X: nat,Px: nat] :
( ( vEBT_invar_vebt @ T2 @ N3 )
=> ( ( ( vEBT_vebt_pred @ T2 @ X )
= ( some @ nat @ Px ) )
= ( vEBT_is_pred_in_set @ ( vEBT_VEBT_set_vebt @ T2 ) @ X @ Px ) ) ) ).
% pred_corr
thf(fact_351_pred__correct,axiom,
! [T2: vEBT_VEBT,N3: nat,X: nat,Sx: nat] :
( ( vEBT_invar_vebt @ T2 @ N3 )
=> ( ( ( vEBT_vebt_pred @ T2 @ X )
= ( some @ nat @ Sx ) )
= ( vEBT_is_pred_in_set @ ( vEBT_set_vebt @ T2 ) @ X @ Sx ) ) ) ).
% pred_correct
thf(fact_352_succ__correct,axiom,
! [T2: vEBT_VEBT,N3: nat,X: nat,Sx: nat] :
( ( vEBT_invar_vebt @ T2 @ N3 )
=> ( ( ( vEBT_vebt_succ @ T2 @ X )
= ( some @ nat @ Sx ) )
= ( vEBT_is_succ_in_set @ ( vEBT_set_vebt @ T2 ) @ X @ Sx ) ) ) ).
% succ_correct
thf(fact_353_all__set__conv__nth,axiom,
! [A: $tType,L2: list @ A,P: A > $o] :
( ( ! [X2: A] :
( ( member @ A @ X2 @ ( set2 @ A @ L2 ) )
=> ( P @ X2 ) ) )
= ( ! [I3: nat] :
( ( ord_less @ nat @ I3 @ ( size_size @ ( list @ A ) @ L2 ) )
=> ( P @ ( nth @ A @ L2 @ I3 ) ) ) ) ) ).
% all_set_conv_nth
thf(fact_354_ord__eq__le__eq__trans,axiom,
! [A: $tType] :
( ( ord @ A )
=> ! [A2: A,B2: A,C2: A,D: A] :
( ( A2 = B2 )
=> ( ( ord_less_eq @ A @ B2 @ C2 )
=> ( ( C2 = D )
=> ( ord_less_eq @ A @ A2 @ D ) ) ) ) ) ).
% ord_eq_le_eq_trans
thf(fact_355_linorder__neqE__linordered__idom,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [X: A,Y: A] :
( ( X != Y )
=> ( ~ ( ord_less @ A @ X @ Y )
=> ( ord_less @ A @ Y @ X ) ) ) ) ).
% linorder_neqE_linordered_idom
thf(fact_356_pairself_Ocases,axiom,
! [B: $tType,A: $tType,X: product_prod @ ( A > B ) @ ( product_prod @ A @ A )] :
~ ! [F3: A > B,A4: A,B3: A] :
( X
!= ( product_Pair @ ( A > B ) @ ( product_prod @ A @ A ) @ F3 @ ( product_Pair @ A @ A @ A4 @ B3 ) ) ) ).
% pairself.cases
thf(fact_357_bex2I,axiom,
! [A: $tType,B: $tType,A2: A,B2: B,S3: set @ ( product_prod @ A @ B ),P: A > B > $o] :
( ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ A2 @ B2 ) @ S3 )
=> ( ( ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ A2 @ B2 ) @ S3 )
=> ( P @ A2 @ B2 ) )
=> ? [A4: A,B3: B] :
( ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ A4 @ B3 ) @ S3 )
& ( P @ A4 @ B3 ) ) ) ) ).
% bex2I
thf(fact_358_ring__class_Oring__distribs_I2_J,axiom,
! [A: $tType] :
( ( ring @ A )
=> ! [A2: A,B2: A,C2: A] :
( ( times_times @ A @ ( plus_plus @ A @ A2 @ B2 ) @ C2 )
= ( plus_plus @ A @ ( times_times @ A @ A2 @ C2 ) @ ( times_times @ A @ B2 @ C2 ) ) ) ) ).
% ring_class.ring_distribs(2)
thf(fact_359_ring__class_Oring__distribs_I1_J,axiom,
! [A: $tType] :
( ( ring @ A )
=> ! [A2: A,B2: A,C2: A] :
( ( times_times @ A @ A2 @ ( plus_plus @ A @ B2 @ C2 ) )
= ( plus_plus @ A @ ( times_times @ A @ A2 @ B2 ) @ ( times_times @ A @ A2 @ C2 ) ) ) ) ).
% ring_class.ring_distribs(1)
thf(fact_360_comm__semiring__class_Odistrib,axiom,
! [A: $tType] :
( ( comm_semiring @ A )
=> ! [A2: A,B2: A,C2: A] :
( ( times_times @ A @ ( plus_plus @ A @ A2 @ B2 ) @ C2 )
= ( plus_plus @ A @ ( times_times @ A @ A2 @ C2 ) @ ( times_times @ A @ B2 @ C2 ) ) ) ) ).
% comm_semiring_class.distrib
thf(fact_361_distrib__left,axiom,
! [A: $tType] :
( ( semiring @ A )
=> ! [A2: A,B2: A,C2: A] :
( ( times_times @ A @ A2 @ ( plus_plus @ A @ B2 @ C2 ) )
= ( plus_plus @ A @ ( times_times @ A @ A2 @ B2 ) @ ( times_times @ A @ A2 @ C2 ) ) ) ) ).
% distrib_left
thf(fact_362_distrib__right,axiom,
! [A: $tType] :
( ( semiring @ A )
=> ! [A2: A,B2: A,C2: A] :
( ( times_times @ A @ ( plus_plus @ A @ A2 @ B2 ) @ C2 )
= ( plus_plus @ A @ ( times_times @ A @ A2 @ C2 ) @ ( times_times @ A @ B2 @ C2 ) ) ) ) ).
% distrib_right
thf(fact_363_combine__common__factor,axiom,
! [A: $tType] :
( ( semiring @ A )
=> ! [A2: A,E: A,B2: A,C2: A] :
( ( plus_plus @ A @ ( times_times @ A @ A2 @ E ) @ ( plus_plus @ A @ ( times_times @ A @ B2 @ E ) @ C2 ) )
= ( plus_plus @ A @ ( times_times @ A @ ( plus_plus @ A @ A2 @ B2 ) @ E ) @ C2 ) ) ) ).
% combine_common_factor
thf(fact_364_left__diff__distrib,axiom,
! [A: $tType] :
( ( ring @ A )
=> ! [A2: A,B2: A,C2: A] :
( ( times_times @ A @ ( minus_minus @ A @ A2 @ B2 ) @ C2 )
= ( minus_minus @ A @ ( times_times @ A @ A2 @ C2 ) @ ( times_times @ A @ B2 @ C2 ) ) ) ) ).
% left_diff_distrib
thf(fact_365_right__diff__distrib,axiom,
! [A: $tType] :
( ( ring @ A )
=> ! [A2: A,B2: A,C2: A] :
( ( times_times @ A @ A2 @ ( minus_minus @ A @ B2 @ C2 ) )
= ( minus_minus @ A @ ( times_times @ A @ A2 @ B2 ) @ ( times_times @ A @ A2 @ C2 ) ) ) ) ).
% right_diff_distrib
thf(fact_366_left__diff__distrib_H,axiom,
! [A: $tType] :
( ( comm_s4317794764714335236cancel @ A )
=> ! [B2: A,C2: A,A2: A] :
( ( times_times @ A @ ( minus_minus @ A @ B2 @ C2 ) @ A2 )
= ( minus_minus @ A @ ( times_times @ A @ B2 @ A2 ) @ ( times_times @ A @ C2 @ A2 ) ) ) ) ).
% left_diff_distrib'
thf(fact_367_right__diff__distrib_H,axiom,
! [A: $tType] :
( ( comm_s4317794764714335236cancel @ A )
=> ! [A2: A,B2: A,C2: A] :
( ( times_times @ A @ A2 @ ( minus_minus @ A @ B2 @ C2 ) )
= ( minus_minus @ A @ ( times_times @ A @ A2 @ B2 ) @ ( times_times @ A @ A2 @ C2 ) ) ) ) ).
% right_diff_distrib'
thf(fact_368_le__some__optE,axiom,
! [A: $tType] :
( ( preorder @ A )
=> ! [M: A,X: option @ A] :
( ( ord_less_eq @ ( option @ A ) @ ( some @ A @ M ) @ X )
=> ~ ! [M7: A] :
( ( X
= ( some @ A @ M7 ) )
=> ~ ( ord_less_eq @ A @ M @ M7 ) ) ) ) ).
% le_some_optE
thf(fact_369_exists__leI,axiom,
! [N3: nat,P: nat > $o] :
( ( ! [N5: nat] :
( ( ord_less @ nat @ N5 @ N3 )
=> ~ ( P @ N5 ) )
=> ( P @ N3 ) )
=> ? [N6: nat] :
( ( ord_less_eq @ nat @ N6 @ N3 )
& ( P @ N6 ) ) ) ).
% exists_leI
thf(fact_370_add__le__add__imp__diff__le,axiom,
! [A: $tType] :
( ( linordered_semidom @ A )
=> ! [I: A,K: A,N3: A,J: A] :
( ( ord_less_eq @ A @ ( plus_plus @ A @ I @ K ) @ N3 )
=> ( ( ord_less_eq @ A @ N3 @ ( plus_plus @ A @ J @ K ) )
=> ( ( ord_less_eq @ A @ ( plus_plus @ A @ I @ K ) @ N3 )
=> ( ( ord_less_eq @ A @ N3 @ ( plus_plus @ A @ J @ K ) )
=> ( ord_less_eq @ A @ ( minus_minus @ A @ N3 @ K ) @ J ) ) ) ) ) ) ).
% add_le_add_imp_diff_le
thf(fact_371_add__le__imp__le__diff,axiom,
! [A: $tType] :
( ( linordered_semidom @ A )
=> ! [I: A,K: A,N3: A] :
( ( ord_less_eq @ A @ ( plus_plus @ A @ I @ K ) @ N3 )
=> ( ord_less_eq @ A @ I @ ( minus_minus @ A @ N3 @ K ) ) ) ) ).
% add_le_imp_le_diff
thf(fact_372_linordered__semidom__class_Oadd__diff__inverse,axiom,
! [A: $tType] :
( ( linordered_semidom @ A )
=> ! [A2: A,B2: A] :
( ~ ( ord_less @ A @ A2 @ B2 )
=> ( ( plus_plus @ A @ B2 @ ( minus_minus @ A @ A2 @ B2 ) )
= A2 ) ) ) ).
% linordered_semidom_class.add_diff_inverse
thf(fact_373_eq__add__iff1,axiom,
! [A: $tType] :
( ( ring @ A )
=> ! [A2: A,E: A,C2: A,B2: A,D: A] :
( ( ( plus_plus @ A @ ( times_times @ A @ A2 @ E ) @ C2 )
= ( plus_plus @ A @ ( times_times @ A @ B2 @ E ) @ D ) )
= ( ( plus_plus @ A @ ( times_times @ A @ ( minus_minus @ A @ A2 @ B2 ) @ E ) @ C2 )
= D ) ) ) ).
% eq_add_iff1
thf(fact_374_eq__add__iff2,axiom,
! [A: $tType] :
( ( ring @ A )
=> ! [A2: A,E: A,C2: A,B2: A,D: A] :
( ( ( plus_plus @ A @ ( times_times @ A @ A2 @ E ) @ C2 )
= ( plus_plus @ A @ ( times_times @ A @ B2 @ E ) @ D ) )
= ( C2
= ( plus_plus @ A @ ( times_times @ A @ ( minus_minus @ A @ B2 @ A2 ) @ E ) @ D ) ) ) ) ).
% eq_add_iff2
thf(fact_375_square__diff__square__factored,axiom,
! [A: $tType] :
( ( comm_ring @ A )
=> ! [X: A,Y: A] :
( ( minus_minus @ A @ ( times_times @ A @ X @ X ) @ ( times_times @ A @ Y @ Y ) )
= ( times_times @ A @ ( plus_plus @ A @ X @ Y ) @ ( minus_minus @ A @ X @ Y ) ) ) ) ).
% square_diff_square_factored
thf(fact_376_nat__in__between__eq_I2_J,axiom,
! [A2: nat,B2: nat] :
( ( ( ord_less_eq @ nat @ A2 @ B2 )
& ( ord_less @ nat @ B2 @ ( suc @ A2 ) ) )
= ( B2 = A2 ) ) ).
% nat_in_between_eq(2)
thf(fact_377_nat__in__between__eq_I1_J,axiom,
! [A2: nat,B2: nat] :
( ( ( ord_less @ nat @ A2 @ B2 )
& ( ord_less_eq @ nat @ B2 @ ( suc @ A2 ) ) )
= ( B2
= ( suc @ A2 ) ) ) ).
% nat_in_between_eq(1)
thf(fact_378_mlex__bound,axiom,
! [A2: nat,A3: nat,B2: nat,N7: nat] :
( ( ord_less @ nat @ A2 @ A3 )
=> ( ( ord_less @ nat @ B2 @ N7 )
=> ( ord_less @ nat @ ( plus_plus @ nat @ ( times_times @ nat @ A2 @ N7 ) @ B2 ) @ ( times_times @ nat @ A3 @ N7 ) ) ) ) ).
% mlex_bound
thf(fact_379_mlex__fst__decrI,axiom,
! [A2: nat,A5: nat,B2: nat,N7: nat,B4: nat] :
( ( ord_less @ nat @ A2 @ A5 )
=> ( ( ord_less @ nat @ B2 @ N7 )
=> ( ( ord_less @ nat @ B4 @ N7 )
=> ( ord_less @ nat @ ( plus_plus @ nat @ ( times_times @ nat @ A2 @ N7 ) @ B2 ) @ ( plus_plus @ nat @ ( times_times @ nat @ A5 @ N7 ) @ B4 ) ) ) ) ) ).
% mlex_fst_decrI
thf(fact_380_mlex__snd__decrI,axiom,
! [A2: nat,A5: nat,B2: nat,B4: nat,N7: nat] :
( ( A2 = A5 )
=> ( ( ord_less @ nat @ B2 @ B4 )
=> ( ord_less @ nat @ ( plus_plus @ nat @ ( times_times @ nat @ A2 @ N7 ) @ B2 ) @ ( plus_plus @ nat @ ( times_times @ nat @ A5 @ N7 ) @ B4 ) ) ) ) ).
% mlex_snd_decrI
thf(fact_381_obtain__list__from__elements,axiom,
! [A: $tType,N3: nat,P: A > nat > $o] :
( ! [I2: nat] :
( ( ord_less @ nat @ I2 @ N3 )
=> ? [Li: A] : ( P @ Li @ I2 ) )
=> ~ ! [L3: list @ A] :
( ( ( size_size @ ( list @ A ) @ L3 )
= N3 )
=> ~ ! [I4: nat] :
( ( ord_less @ nat @ I4 @ N3 )
=> ( P @ ( nth @ A @ L3 @ I4 ) @ I4 ) ) ) ) ).
% obtain_list_from_elements
thf(fact_382_mlex__leI,axiom,
! [A2: nat,A5: nat,B2: nat,B4: nat,N7: nat] :
( ( ord_less_eq @ nat @ A2 @ A5 )
=> ( ( ord_less_eq @ nat @ B2 @ B4 )
=> ( ord_less_eq @ nat @ ( plus_plus @ nat @ ( times_times @ nat @ A2 @ N7 ) @ B2 ) @ ( plus_plus @ nat @ ( times_times @ nat @ A5 @ N7 ) @ B4 ) ) ) ) ).
% mlex_leI
thf(fact_383_ordered__ring__class_Ole__add__iff1,axiom,
! [A: $tType] :
( ( ordered_ring @ A )
=> ! [A2: A,E: A,C2: A,B2: A,D: A] :
( ( ord_less_eq @ A @ ( plus_plus @ A @ ( times_times @ A @ A2 @ E ) @ C2 ) @ ( plus_plus @ A @ ( times_times @ A @ B2 @ E ) @ D ) )
= ( ord_less_eq @ A @ ( plus_plus @ A @ ( times_times @ A @ ( minus_minus @ A @ A2 @ B2 ) @ E ) @ C2 ) @ D ) ) ) ).
% ordered_ring_class.le_add_iff1
thf(fact_384_ordered__ring__class_Ole__add__iff2,axiom,
! [A: $tType] :
( ( ordered_ring @ A )
=> ! [A2: A,E: A,C2: A,B2: A,D: A] :
( ( ord_less_eq @ A @ ( plus_plus @ A @ ( times_times @ A @ A2 @ E ) @ C2 ) @ ( plus_plus @ A @ ( times_times @ A @ B2 @ E ) @ D ) )
= ( ord_less_eq @ A @ C2 @ ( plus_plus @ A @ ( times_times @ A @ ( minus_minus @ A @ B2 @ A2 ) @ E ) @ D ) ) ) ) ).
% ordered_ring_class.le_add_iff2
thf(fact_385_less__add__iff1,axiom,
! [A: $tType] :
( ( ordered_ring @ A )
=> ! [A2: A,E: A,C2: A,B2: A,D: A] :
( ( ord_less @ A @ ( plus_plus @ A @ ( times_times @ A @ A2 @ E ) @ C2 ) @ ( plus_plus @ A @ ( times_times @ A @ B2 @ E ) @ D ) )
= ( ord_less @ A @ ( plus_plus @ A @ ( times_times @ A @ ( minus_minus @ A @ A2 @ B2 ) @ E ) @ C2 ) @ D ) ) ) ).
% less_add_iff1
thf(fact_386_less__add__iff2,axiom,
! [A: $tType] :
( ( ordered_ring @ A )
=> ! [A2: A,E: A,C2: A,B2: A,D: A] :
( ( ord_less @ A @ ( plus_plus @ A @ ( times_times @ A @ A2 @ E ) @ C2 ) @ ( plus_plus @ A @ ( times_times @ A @ B2 @ E ) @ D ) )
= ( ord_less @ A @ C2 @ ( plus_plus @ A @ ( times_times @ A @ ( minus_minus @ A @ B2 @ A2 ) @ E ) @ D ) ) ) ) ).
% less_add_iff2
thf(fact_387_in__children__def,axiom,
( vEBT_V5917875025757280293ildren
= ( ^ [N2: nat,TreeList3: list @ vEBT_VEBT,X2: nat] : ( vEBT_V8194947554948674370ptions @ ( nth @ vEBT_VEBT @ TreeList3 @ ( vEBT_VEBT_high @ X2 @ N2 ) ) @ ( vEBT_VEBT_low @ X2 @ N2 ) ) ) ) ).
% in_children_def
thf(fact_388_invar__vebt_Ointros_I5_J,axiom,
! [TreeList: list @ vEBT_VEBT,N3: nat,Summary: vEBT_VEBT,M: nat,Deg: nat,Mi: nat,Ma: nat] :
( ! [X3: vEBT_VEBT] :
( ( member @ vEBT_VEBT @ X3 @ ( set2 @ vEBT_VEBT @ TreeList ) )
=> ( vEBT_invar_vebt @ X3 @ N3 ) )
=> ( ( vEBT_invar_vebt @ Summary @ M )
=> ( ( ( size_size @ ( list @ vEBT_VEBT ) @ TreeList )
= ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ M ) )
=> ( ( M
= ( suc @ N3 ) )
=> ( ( Deg
= ( plus_plus @ nat @ N3 @ M ) )
=> ( ! [I2: nat] :
( ( ord_less @ nat @ I2 @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ M ) )
=> ( ( ? [X6: nat] : ( vEBT_V8194947554948674370ptions @ ( nth @ vEBT_VEBT @ TreeList @ I2 ) @ X6 ) )
= ( vEBT_V8194947554948674370ptions @ Summary @ I2 ) ) )
=> ( ( ( Mi = Ma )
=> ! [X3: vEBT_VEBT] :
( ( member @ vEBT_VEBT @ X3 @ ( set2 @ vEBT_VEBT @ TreeList ) )
=> ~ ? [X_12: nat] : ( vEBT_V8194947554948674370ptions @ X3 @ X_12 ) ) )
=> ( ( ord_less_eq @ nat @ Mi @ Ma )
=> ( ( ord_less @ nat @ Ma @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ Deg ) )
=> ( ( ( Mi != Ma )
=> ! [I2: nat] :
( ( ord_less @ nat @ I2 @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ M ) )
=> ( ( ( ( vEBT_VEBT_high @ Ma @ N3 )
= I2 )
=> ( vEBT_V8194947554948674370ptions @ ( nth @ vEBT_VEBT @ TreeList @ I2 ) @ ( vEBT_VEBT_low @ Ma @ N3 ) ) )
& ! [X3: nat] :
( ( ( ( vEBT_VEBT_high @ X3 @ N3 )
= I2 )
& ( vEBT_V8194947554948674370ptions @ ( nth @ vEBT_VEBT @ TreeList @ I2 ) @ ( vEBT_VEBT_low @ X3 @ N3 ) ) )
=> ( ( ord_less @ nat @ Mi @ X3 )
& ( ord_less_eq @ nat @ X3 @ Ma ) ) ) ) ) )
=> ( vEBT_invar_vebt @ ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi @ Ma ) ) @ Deg @ TreeList @ Summary ) @ Deg ) ) ) ) ) ) ) ) ) ) ) ).
% invar_vebt.intros(5)
thf(fact_389_invar__vebt_Ointros_I4_J,axiom,
! [TreeList: list @ vEBT_VEBT,N3: nat,Summary: vEBT_VEBT,M: nat,Deg: nat,Mi: nat,Ma: nat] :
( ! [X3: vEBT_VEBT] :
( ( member @ vEBT_VEBT @ X3 @ ( set2 @ vEBT_VEBT @ TreeList ) )
=> ( vEBT_invar_vebt @ X3 @ N3 ) )
=> ( ( vEBT_invar_vebt @ Summary @ M )
=> ( ( ( size_size @ ( list @ vEBT_VEBT ) @ TreeList )
= ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ M ) )
=> ( ( M = N3 )
=> ( ( Deg
= ( plus_plus @ nat @ N3 @ M ) )
=> ( ! [I2: nat] :
( ( ord_less @ nat @ I2 @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ M ) )
=> ( ( ? [X6: nat] : ( vEBT_V8194947554948674370ptions @ ( nth @ vEBT_VEBT @ TreeList @ I2 ) @ X6 ) )
= ( vEBT_V8194947554948674370ptions @ Summary @ I2 ) ) )
=> ( ( ( Mi = Ma )
=> ! [X3: vEBT_VEBT] :
( ( member @ vEBT_VEBT @ X3 @ ( set2 @ vEBT_VEBT @ TreeList ) )
=> ~ ? [X_12: nat] : ( vEBT_V8194947554948674370ptions @ X3 @ X_12 ) ) )
=> ( ( ord_less_eq @ nat @ Mi @ Ma )
=> ( ( ord_less @ nat @ Ma @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ Deg ) )
=> ( ( ( Mi != Ma )
=> ! [I2: nat] :
( ( ord_less @ nat @ I2 @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ M ) )
=> ( ( ( ( vEBT_VEBT_high @ Ma @ N3 )
= I2 )
=> ( vEBT_V8194947554948674370ptions @ ( nth @ vEBT_VEBT @ TreeList @ I2 ) @ ( vEBT_VEBT_low @ Ma @ N3 ) ) )
& ! [X3: nat] :
( ( ( ( vEBT_VEBT_high @ X3 @ N3 )
= I2 )
& ( vEBT_V8194947554948674370ptions @ ( nth @ vEBT_VEBT @ TreeList @ I2 ) @ ( vEBT_VEBT_low @ X3 @ N3 ) ) )
=> ( ( ord_less @ nat @ Mi @ X3 )
& ( ord_less_eq @ nat @ X3 @ Ma ) ) ) ) ) )
=> ( vEBT_invar_vebt @ ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi @ Ma ) ) @ Deg @ TreeList @ Summary ) @ Deg ) ) ) ) ) ) ) ) ) ) ) ).
% invar_vebt.intros(4)
thf(fact_390_both__member__options__from__chilf__to__complete__tree,axiom,
! [X: nat,Deg: nat,TreeList: list @ vEBT_VEBT,Mi: nat,Ma: nat,Summary: vEBT_VEBT] :
( ( ord_less @ nat @ ( vEBT_VEBT_high @ X @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( size_size @ ( list @ vEBT_VEBT ) @ TreeList ) )
=> ( ( ord_less_eq @ nat @ ( one_one @ nat ) @ Deg )
=> ( ( vEBT_V8194947554948674370ptions @ ( nth @ vEBT_VEBT @ TreeList @ ( vEBT_VEBT_high @ X @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( vEBT_VEBT_low @ X @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) )
=> ( vEBT_V8194947554948674370ptions @ ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi @ Ma ) ) @ Deg @ TreeList @ Summary ) @ X ) ) ) ) ).
% both_member_options_from_chilf_to_complete_tree
thf(fact_391_real__average__minus__second,axiom,
! [B2: real,A2: real] :
( ( minus_minus @ real @ ( divide_divide @ real @ ( plus_plus @ real @ B2 @ A2 ) @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) @ A2 )
= ( divide_divide @ real @ ( minus_minus @ real @ B2 @ A2 ) @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) ) ).
% real_average_minus_second
thf(fact_392_real__average__minus__first,axiom,
! [A2: real,B2: real] :
( ( minus_minus @ real @ ( divide_divide @ real @ ( plus_plus @ real @ A2 @ B2 ) @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) @ A2 )
= ( divide_divide @ real @ ( minus_minus @ real @ B2 @ A2 ) @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) ) ).
% real_average_minus_first
thf(fact_393_invar__vebt_Ointros_I3_J,axiom,
! [TreeList: list @ vEBT_VEBT,N3: nat,Summary: vEBT_VEBT,M: nat,Deg: nat] :
( ! [X3: vEBT_VEBT] :
( ( member @ vEBT_VEBT @ X3 @ ( set2 @ vEBT_VEBT @ TreeList ) )
=> ( vEBT_invar_vebt @ X3 @ N3 ) )
=> ( ( vEBT_invar_vebt @ Summary @ M )
=> ( ( ( size_size @ ( list @ vEBT_VEBT ) @ TreeList )
= ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ M ) )
=> ( ( M
= ( suc @ N3 ) )
=> ( ( Deg
= ( plus_plus @ nat @ N3 @ M ) )
=> ( ~ ? [X_12: nat] : ( vEBT_V8194947554948674370ptions @ Summary @ X_12 )
=> ( ! [X3: vEBT_VEBT] :
( ( member @ vEBT_VEBT @ X3 @ ( set2 @ vEBT_VEBT @ TreeList ) )
=> ~ ? [X_12: nat] : ( vEBT_V8194947554948674370ptions @ X3 @ X_12 ) )
=> ( vEBT_invar_vebt @ ( vEBT_Node @ ( none @ ( product_prod @ nat @ nat ) ) @ Deg @ TreeList @ Summary ) @ Deg ) ) ) ) ) ) ) ) ).
% invar_vebt.intros(3)
thf(fact_394_invar__vebt_Ointros_I2_J,axiom,
! [TreeList: list @ vEBT_VEBT,N3: nat,Summary: vEBT_VEBT,M: nat,Deg: nat] :
( ! [X3: vEBT_VEBT] :
( ( member @ vEBT_VEBT @ X3 @ ( set2 @ vEBT_VEBT @ TreeList ) )
=> ( vEBT_invar_vebt @ X3 @ N3 ) )
=> ( ( vEBT_invar_vebt @ Summary @ M )
=> ( ( ( size_size @ ( list @ vEBT_VEBT ) @ TreeList )
= ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ M ) )
=> ( ( M = N3 )
=> ( ( Deg
= ( plus_plus @ nat @ N3 @ M ) )
=> ( ~ ? [X_12: nat] : ( vEBT_V8194947554948674370ptions @ Summary @ X_12 )
=> ( ! [X3: vEBT_VEBT] :
( ( member @ vEBT_VEBT @ X3 @ ( set2 @ vEBT_VEBT @ TreeList ) )
=> ~ ? [X_12: nat] : ( vEBT_V8194947554948674370ptions @ X3 @ X_12 ) )
=> ( vEBT_invar_vebt @ ( vEBT_Node @ ( none @ ( product_prod @ nat @ nat ) ) @ Deg @ TreeList @ Summary ) @ Deg ) ) ) ) ) ) ) ) ).
% invar_vebt.intros(2)
thf(fact_395_both__member__options__from__complete__tree__to__child,axiom,
! [Deg: nat,Mi: nat,Ma: nat,TreeList: list @ vEBT_VEBT,Summary: vEBT_VEBT,X: nat] :
( ( ord_less_eq @ nat @ ( one_one @ nat ) @ Deg )
=> ( ( vEBT_V8194947554948674370ptions @ ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi @ Ma ) ) @ Deg @ TreeList @ Summary ) @ X )
=> ( ( vEBT_V8194947554948674370ptions @ ( nth @ vEBT_VEBT @ TreeList @ ( vEBT_VEBT_high @ X @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( vEBT_VEBT_low @ X @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) )
| ( X = Mi )
| ( X = Ma ) ) ) ) ).
% both_member_options_from_complete_tree_to_child
thf(fact_396_set__n__deg__not__0,axiom,
! [TreeList: list @ vEBT_VEBT,N3: nat,M: nat] :
( ! [X3: vEBT_VEBT] :
( ( member @ vEBT_VEBT @ X3 @ ( set2 @ vEBT_VEBT @ TreeList ) )
=> ( vEBT_invar_vebt @ X3 @ N3 ) )
=> ( ( ( size_size @ ( list @ vEBT_VEBT ) @ TreeList )
= ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ M ) )
=> ( ord_less_eq @ nat @ ( one_one @ nat ) @ N3 ) ) ) ).
% set_n_deg_not_0
thf(fact_397_field__less__half__sum,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [X: A,Y: A] :
( ( ord_less @ A @ X @ Y )
=> ( ord_less @ A @ X @ ( divide_divide @ A @ ( plus_plus @ A @ X @ Y ) @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) ) ) ) ).
% field_less_half_sum
thf(fact_398_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_399_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_400_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_401_bits__div__by__1,axiom,
! [A: $tType] :
( ( bit_semiring_bits @ A )
=> ! [A2: A] :
( ( divide_divide @ A @ A2 @ ( one_one @ A ) )
= A2 ) ) ).
% bits_div_by_1
thf(fact_402_div__by__1,axiom,
! [A: $tType] :
( ( semidom_divide @ A )
=> ! [A2: A] :
( ( divide_divide @ A @ A2 @ ( one_one @ A ) )
= A2 ) ) ).
% div_by_1
thf(fact_403_power__one,axiom,
! [A: $tType] :
( ( monoid_mult @ A )
=> ! [N3: nat] :
( ( power_power @ A @ ( one_one @ A ) @ N3 )
= ( one_one @ A ) ) ) ).
% power_one
thf(fact_404_power__one__right,axiom,
! [A: $tType] :
( ( monoid_mult @ A )
=> ! [A2: A] :
( ( power_power @ A @ A2 @ ( one_one @ nat ) )
= A2 ) ) ).
% power_one_right
thf(fact_405_nat__mult__eq__1__iff,axiom,
! [M: nat,N3: nat] :
( ( ( times_times @ nat @ M @ N3 )
= ( one_one @ nat ) )
= ( ( M
= ( one_one @ nat ) )
& ( N3
= ( one_one @ nat ) ) ) ) ).
% nat_mult_eq_1_iff
thf(fact_406_nat__1__eq__mult__iff,axiom,
! [M: nat,N3: nat] :
( ( ( one_one @ nat )
= ( times_times @ nat @ M @ N3 ) )
= ( ( M
= ( one_one @ nat ) )
& ( N3
= ( one_one @ nat ) ) ) ) ).
% nat_1_eq_mult_iff
thf(fact_407_one__eq__numeral__iff,axiom,
! [A: $tType] :
( ( semiring_char_0 @ A )
=> ! [N3: num] :
( ( ( one_one @ A )
= ( numeral_numeral @ A @ N3 ) )
= ( one2 = N3 ) ) ) ).
% one_eq_numeral_iff
thf(fact_408_numeral__eq__one__iff,axiom,
! [A: $tType] :
( ( semiring_char_0 @ A )
=> ! [N3: num] :
( ( ( numeral_numeral @ A @ N3 )
= ( one_one @ A ) )
= ( N3 = one2 ) ) ) ).
% numeral_eq_one_iff
thf(fact_409_power__inject__exp,axiom,
! [A: $tType] :
( ( linordered_semidom @ A )
=> ! [A2: A,M: nat,N3: nat] :
( ( ord_less @ A @ ( one_one @ A ) @ A2 )
=> ( ( ( power_power @ A @ A2 @ M )
= ( power_power @ A @ A2 @ N3 ) )
= ( M = N3 ) ) ) ) ).
% power_inject_exp
thf(fact_410_diff__Suc__1,axiom,
! [N3: nat] :
( ( minus_minus @ nat @ ( suc @ N3 ) @ ( one_one @ nat ) )
= N3 ) ).
% diff_Suc_1
thf(fact_411_power__strict__increasing__iff,axiom,
! [A: $tType] :
( ( linordered_semidom @ A )
=> ! [B2: A,X: nat,Y: nat] :
( ( ord_less @ A @ ( one_one @ A ) @ B2 )
=> ( ( ord_less @ A @ ( power_power @ A @ B2 @ X ) @ ( power_power @ A @ B2 @ Y ) )
= ( ord_less @ nat @ X @ Y ) ) ) ) ).
% power_strict_increasing_iff
thf(fact_412_Suc__diff,axiom,
! [M: nat,N3: nat] :
( ( ord_less_eq @ nat @ M @ N3 )
=> ( ( ord_less_eq @ nat @ ( one_one @ nat ) @ M )
=> ( ( suc @ ( minus_minus @ nat @ N3 @ M ) )
= ( minus_minus @ nat @ N3 @ ( minus_minus @ nat @ M @ ( one_one @ nat ) ) ) ) ) ) ).
% Suc_diff
thf(fact_413_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_414_power__increasing__iff,axiom,
! [A: $tType] :
( ( linordered_semidom @ A )
=> ! [B2: A,X: nat,Y: nat] :
( ( ord_less @ A @ ( one_one @ A ) @ B2 )
=> ( ( ord_less_eq @ A @ ( power_power @ A @ B2 @ X ) @ ( power_power @ A @ B2 @ Y ) )
= ( ord_less_eq @ nat @ X @ Y ) ) ) ) ).
% power_increasing_iff
thf(fact_415_Suc__1,axiom,
( ( suc @ ( one_one @ nat ) )
= ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ).
% Suc_1
thf(fact_416_one__plus__numeral,axiom,
! [A: $tType] :
( ( numeral @ A )
=> ! [N3: num] :
( ( plus_plus @ A @ ( one_one @ A ) @ ( numeral_numeral @ A @ N3 ) )
= ( numeral_numeral @ A @ ( plus_plus @ num @ one2 @ N3 ) ) ) ) ).
% one_plus_numeral
thf(fact_417_numeral__plus__one,axiom,
! [A: $tType] :
( ( numeral @ A )
=> ! [N3: num] :
( ( plus_plus @ A @ ( numeral_numeral @ A @ N3 ) @ ( one_one @ A ) )
= ( numeral_numeral @ A @ ( plus_plus @ num @ N3 @ one2 ) ) ) ) ).
% numeral_plus_one
thf(fact_418_numeral__le__one__iff,axiom,
! [A: $tType] :
( ( linord181362715937106298miring @ A )
=> ! [N3: num] :
( ( ord_less_eq @ A @ ( numeral_numeral @ A @ N3 ) @ ( one_one @ A ) )
= ( ord_less_eq @ num @ N3 @ one2 ) ) ) ).
% numeral_le_one_iff
thf(fact_419_one__less__numeral__iff,axiom,
! [A: $tType] :
( ( linord181362715937106298miring @ A )
=> ! [N3: num] :
( ( ord_less @ A @ ( one_one @ A ) @ ( numeral_numeral @ A @ N3 ) )
= ( ord_less @ num @ one2 @ N3 ) ) ) ).
% one_less_numeral_iff
thf(fact_420_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_421_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_422_nat__mult__1__right,axiom,
! [N3: nat] :
( ( times_times @ nat @ N3 @ ( one_one @ nat ) )
= N3 ) ).
% nat_mult_1_right
thf(fact_423_nat__mult__1,axiom,
! [N3: nat] :
( ( times_times @ nat @ ( one_one @ nat ) @ N3 )
= N3 ) ).
% nat_mult_1
thf(fact_424_one__le__numeral,axiom,
! [A: $tType] :
( ( linord181362715937106298miring @ A )
=> ! [N3: num] : ( ord_less_eq @ A @ ( one_one @ A ) @ ( numeral_numeral @ A @ N3 ) ) ) ).
% one_le_numeral
thf(fact_425_not__numeral__less__one,axiom,
! [A: $tType] :
( ( linord181362715937106298miring @ A )
=> ! [N3: num] :
~ ( ord_less @ A @ ( numeral_numeral @ A @ N3 ) @ ( one_one @ A ) ) ) ).
% not_numeral_less_one
thf(fact_426_numeral__One,axiom,
! [A: $tType] :
( ( numeral @ A )
=> ( ( numeral_numeral @ A @ one2 )
= ( one_one @ A ) ) ) ).
% numeral_One
thf(fact_427_add__mono1,axiom,
! [A: $tType] :
( ( linord181362715937106298miring @ A )
=> ! [A2: A,B2: A] :
( ( ord_less @ A @ A2 @ B2 )
=> ( ord_less @ A @ ( plus_plus @ A @ A2 @ ( one_one @ A ) ) @ ( plus_plus @ A @ B2 @ ( one_one @ A ) ) ) ) ) ).
% add_mono1
thf(fact_428_less__add__one,axiom,
! [A: $tType] :
( ( linordered_semidom @ A )
=> ! [A2: A] : ( ord_less @ A @ A2 @ ( plus_plus @ A @ A2 @ ( one_one @ A ) ) ) ) ).
% less_add_one
thf(fact_429_one__plus__numeral__commute,axiom,
! [A: $tType] :
( ( numeral @ A )
=> ! [X: num] :
( ( plus_plus @ A @ ( one_one @ A ) @ ( numeral_numeral @ A @ X ) )
= ( plus_plus @ A @ ( numeral_numeral @ A @ X ) @ ( one_one @ A ) ) ) ) ).
% one_plus_numeral_commute
thf(fact_430_less__1__mult,axiom,
! [A: $tType] :
( ( linordered_semidom @ A )
=> ! [M: A,N3: A] :
( ( ord_less @ A @ ( one_one @ A ) @ M )
=> ( ( ord_less @ A @ ( one_one @ A ) @ N3 )
=> ( ord_less @ A @ ( one_one @ A ) @ ( times_times @ A @ M @ N3 ) ) ) ) ) ).
% less_1_mult
thf(fact_431_one__le__power,axiom,
! [A: $tType] :
( ( linordered_semidom @ A )
=> ! [A2: A,N3: nat] :
( ( ord_less_eq @ A @ ( one_one @ A ) @ A2 )
=> ( ord_less_eq @ A @ ( one_one @ A ) @ ( power_power @ A @ A2 @ N3 ) ) ) ) ).
% one_le_power
thf(fact_432_left__right__inverse__power,axiom,
! [A: $tType] :
( ( monoid_mult @ A )
=> ! [X: A,Y: A,N3: nat] :
( ( ( times_times @ A @ X @ Y )
= ( one_one @ A ) )
=> ( ( times_times @ A @ ( power_power @ A @ X @ N3 ) @ ( power_power @ A @ Y @ N3 ) )
= ( one_one @ A ) ) ) ) ).
% left_right_inverse_power
thf(fact_433_power__one__over,axiom,
! [A: $tType] :
( ( division_ring @ A )
=> ! [A2: A,N3: nat] :
( ( power_power @ A @ ( divide_divide @ A @ ( one_one @ A ) @ A2 ) @ N3 )
= ( divide_divide @ A @ ( one_one @ A ) @ ( power_power @ A @ A2 @ N3 ) ) ) ) ).
% power_one_over
thf(fact_434_numerals_I1_J,axiom,
( ( numeral_numeral @ nat @ one2 )
= ( one_one @ nat ) ) ).
% numerals(1)
thf(fact_435_Suc__eq__plus1__left,axiom,
( suc
= ( plus_plus @ nat @ ( one_one @ nat ) ) ) ).
% Suc_eq_plus1_left
thf(fact_436_plus__1__eq__Suc,axiom,
( ( plus_plus @ nat @ ( one_one @ nat ) )
= suc ) ).
% plus_1_eq_Suc
thf(fact_437_Suc__eq__plus1,axiom,
( suc
= ( ^ [N2: nat] : ( plus_plus @ nat @ N2 @ ( one_one @ nat ) ) ) ) ).
% Suc_eq_plus1
thf(fact_438_diff__Suc__eq__diff__pred,axiom,
! [M: nat,N3: nat] :
( ( minus_minus @ nat @ M @ ( suc @ N3 ) )
= ( minus_minus @ nat @ ( minus_minus @ nat @ M @ ( one_one @ nat ) ) @ N3 ) ) ).
% diff_Suc_eq_diff_pred
thf(fact_439_square__diff__one__factored,axiom,
! [A: $tType] :
( ( ring_1 @ A )
=> ! [X: A] :
( ( minus_minus @ A @ ( times_times @ A @ X @ X ) @ ( one_one @ A ) )
= ( times_times @ A @ ( plus_plus @ A @ X @ ( one_one @ A ) ) @ ( minus_minus @ A @ X @ ( one_one @ A ) ) ) ) ) ).
% square_diff_one_factored
thf(fact_440_power__less__power__Suc,axiom,
! [A: $tType] :
( ( linordered_semidom @ A )
=> ! [A2: A,N3: nat] :
( ( ord_less @ A @ ( one_one @ A ) @ A2 )
=> ( ord_less @ A @ ( power_power @ A @ A2 @ N3 ) @ ( times_times @ A @ A2 @ ( power_power @ A @ A2 @ N3 ) ) ) ) ) ).
% power_less_power_Suc
thf(fact_441_power__gt1__lemma,axiom,
! [A: $tType] :
( ( linordered_semidom @ A )
=> ! [A2: A,N3: nat] :
( ( ord_less @ A @ ( one_one @ A ) @ A2 )
=> ( ord_less @ A @ ( one_one @ A ) @ ( times_times @ A @ A2 @ ( power_power @ A @ A2 @ N3 ) ) ) ) ) ).
% power_gt1_lemma
thf(fact_442_power__gt1,axiom,
! [A: $tType] :
( ( linordered_semidom @ A )
=> ! [A2: A,N3: nat] :
( ( ord_less @ A @ ( one_one @ A ) @ A2 )
=> ( ord_less @ A @ ( one_one @ A ) @ ( power_power @ A @ A2 @ ( suc @ N3 ) ) ) ) ) ).
% power_gt1
thf(fact_443_power__strict__increasing,axiom,
! [A: $tType] :
( ( linordered_semidom @ A )
=> ! [N3: nat,N7: nat,A2: A] :
( ( ord_less @ nat @ N3 @ N7 )
=> ( ( ord_less @ A @ ( one_one @ A ) @ A2 )
=> ( ord_less @ A @ ( power_power @ A @ A2 @ N3 ) @ ( power_power @ A @ A2 @ N7 ) ) ) ) ) ).
% power_strict_increasing
thf(fact_444_power__less__imp__less__exp,axiom,
! [A: $tType] :
( ( linordered_semidom @ A )
=> ! [A2: A,M: nat,N3: nat] :
( ( ord_less @ A @ ( one_one @ A ) @ A2 )
=> ( ( ord_less @ A @ ( power_power @ A @ A2 @ M ) @ ( power_power @ A @ A2 @ N3 ) )
=> ( ord_less @ nat @ M @ N3 ) ) ) ) ).
% power_less_imp_less_exp
thf(fact_445_power__increasing,axiom,
! [A: $tType] :
( ( linordered_semidom @ A )
=> ! [N3: nat,N7: nat,A2: A] :
( ( ord_less_eq @ nat @ N3 @ N7 )
=> ( ( ord_less_eq @ A @ ( one_one @ A ) @ A2 )
=> ( ord_less_eq @ A @ ( power_power @ A @ A2 @ N3 ) @ ( power_power @ A @ A2 @ N7 ) ) ) ) ) ).
% power_increasing
thf(fact_446_complete__real,axiom,
! [S3: set @ real] :
( ? [X5: real] : ( member @ real @ X5 @ S3 )
=> ( ? [Z4: real] :
! [X3: real] :
( ( member @ real @ X3 @ S3 )
=> ( ord_less_eq @ real @ X3 @ Z4 ) )
=> ? [Y3: real] :
( ! [X5: real] :
( ( member @ real @ X5 @ S3 )
=> ( ord_less_eq @ real @ X5 @ Y3 ) )
& ! [Z4: real] :
( ! [X3: real] :
( ( member @ real @ X3 @ S3 )
=> ( ord_less_eq @ real @ X3 @ Z4 ) )
=> ( ord_less_eq @ real @ Y3 @ Z4 ) ) ) ) ) ).
% complete_real
thf(fact_447_less__eq__real__def,axiom,
( ( ord_less_eq @ real )
= ( ^ [X2: real,Y2: real] :
( ( ord_less @ real @ X2 @ Y2 )
| ( X2 = Y2 ) ) ) ) ).
% less_eq_real_def
thf(fact_448_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_449_power__le__imp__le__exp,axiom,
! [A: $tType] :
( ( linordered_semidom @ A )
=> ! [A2: A,M: nat,N3: nat] :
( ( ord_less @ A @ ( one_one @ A ) @ A2 )
=> ( ( ord_less_eq @ A @ ( power_power @ A @ A2 @ M ) @ ( power_power @ A @ A2 @ N3 ) )
=> ( ord_less_eq @ nat @ M @ N3 ) ) ) ) ).
% power_le_imp_le_exp
thf(fact_450_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_451_Suc__n__minus__m__eq,axiom,
! [M: nat,N3: nat] :
( ( ord_less_eq @ nat @ M @ N3 )
=> ( ( ord_less @ nat @ ( one_one @ nat ) @ M )
=> ( ( suc @ ( minus_minus @ nat @ N3 @ M ) )
= ( minus_minus @ nat @ N3 @ ( minus_minus @ nat @ M @ ( one_one @ nat ) ) ) ) ) ) ).
% Suc_n_minus_m_eq
thf(fact_452_add__diff__add,axiom,
! [A: $tType] :
( ( ab_group_add @ A )
=> ! [A2: A,C2: A,B2: A,D: A] :
( ( minus_minus @ A @ ( plus_plus @ A @ A2 @ C2 ) @ ( plus_plus @ A @ B2 @ D ) )
= ( plus_plus @ A @ ( minus_minus @ A @ A2 @ B2 ) @ ( minus_minus @ A @ C2 @ D ) ) ) ) ).
% add_diff_add
thf(fact_453_ex__power__ivl2,axiom,
! [B2: nat,K: nat] :
( ( ord_less_eq @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ B2 )
=> ( ( ord_less_eq @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ K )
=> ? [N: nat] :
( ( ord_less @ nat @ ( power_power @ nat @ B2 @ N ) @ K )
& ( ord_less_eq @ nat @ K @ ( power_power @ nat @ B2 @ ( plus_plus @ nat @ N @ ( one_one @ nat ) ) ) ) ) ) ) ).
% ex_power_ivl2
thf(fact_454_ex__power__ivl1,axiom,
! [B2: nat,K: nat] :
( ( ord_less_eq @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ B2 )
=> ( ( ord_less_eq @ nat @ ( one_one @ nat ) @ K )
=> ? [N: nat] :
( ( ord_less_eq @ nat @ ( power_power @ nat @ B2 @ N ) @ K )
& ( ord_less @ nat @ K @ ( power_power @ nat @ B2 @ ( plus_plus @ nat @ N @ ( one_one @ nat ) ) ) ) ) ) ) ).
% ex_power_ivl1
thf(fact_455_power__2__mult__step__le,axiom,
! [N4: nat,N3: nat,K4: nat,K: nat] :
( ( ord_less_eq @ nat @ N4 @ N3 )
=> ( ( ord_less @ nat @ ( times_times @ nat @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N4 ) @ K4 ) @ ( times_times @ nat @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N3 ) @ K ) )
=> ( ord_less_eq @ nat @ ( times_times @ nat @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N4 ) @ ( plus_plus @ nat @ K4 @ ( one_one @ nat ) ) ) @ ( times_times @ nat @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N3 ) @ K ) ) ) ) ).
% power_2_mult_step_le
thf(fact_456_mult__diff__mult,axiom,
! [A: $tType] :
( ( ring @ A )
=> ! [X: A,Y: A,A2: A,B2: A] :
( ( minus_minus @ A @ ( times_times @ A @ X @ Y ) @ ( times_times @ A @ A2 @ B2 ) )
= ( plus_plus @ A @ ( times_times @ A @ X @ ( minus_minus @ A @ Y @ B2 ) ) @ ( times_times @ A @ ( minus_minus @ A @ X @ A2 ) @ B2 ) ) ) ) ).
% mult_diff_mult
thf(fact_457_field__sum__of__halves,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [X: A] :
( ( plus_plus @ A @ ( divide_divide @ A @ X @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) @ ( divide_divide @ A @ X @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) )
= X ) ) ).
% field_sum_of_halves
thf(fact_458_tdeletemimi_H,axiom,
! [Deg: nat,Mi: nat,TreeList: list @ vEBT_VEBT,Summary: vEBT_VEBT,X: nat] :
( ( ord_less_eq @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ Deg )
=> ( ord_less_eq @ nat @ ( vEBT_V1232361888498592333_e_t_e @ ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi @ Mi ) ) @ Deg @ TreeList @ Summary ) @ X ) @ ( one_one @ nat ) ) ) ).
% tdeletemimi'
thf(fact_459_delete__bound__height_H,axiom,
! [T2: vEBT_VEBT,N3: nat,X: nat] :
( ( vEBT_invar_vebt @ T2 @ N3 )
=> ( ord_less_eq @ nat @ ( vEBT_V1232361888498592333_e_t_e @ T2 @ X ) @ ( plus_plus @ nat @ ( one_one @ nat ) @ ( vEBT_VEBT_height @ T2 ) ) ) ) ).
% delete_bound_height'
thf(fact_460_less__eq__option__Some__None,axiom,
! [A: $tType] :
( ( preorder @ A )
=> ! [X: A] :
~ ( ord_less_eq @ ( option @ A ) @ ( some @ A @ X ) @ ( none @ A ) ) ) ).
% less_eq_option_Some_None
thf(fact_461_less__eq__option__Some,axiom,
! [A: $tType] :
( ( preorder @ A )
=> ! [X: A,Y: A] :
( ( ord_less_eq @ ( option @ A ) @ ( some @ A @ X ) @ ( some @ A @ Y ) )
= ( ord_less_eq @ A @ X @ Y ) ) ) ).
% less_eq_option_Some
thf(fact_462_height__node,axiom,
! [Mi: nat,Ma: nat,Deg: nat,TreeList: list @ vEBT_VEBT,Summary: vEBT_VEBT,N3: nat] :
( ( vEBT_invar_vebt @ ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi @ Ma ) ) @ Deg @ TreeList @ Summary ) @ N3 )
=> ( 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_463_less__eq__option__None__code,axiom,
! [A: $tType] :
( ( preorder @ A )
=> ! [X: option @ A] : ( ord_less_eq @ ( option @ A ) @ ( none @ A ) @ X ) ) ).
% less_eq_option_None_code
thf(fact_464_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_465_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_466_less__option__None__Some__code,axiom,
! [A: $tType] :
( ( preorder @ A )
=> ! [X: A] : ( ord_less @ ( option @ A ) @ ( none @ A ) @ ( some @ A @ X ) ) ) ).
% less_option_None_Some_code
thf(fact_467_enat__ord__number_I1_J,axiom,
! [M: num,N3: num] :
( ( ord_less_eq @ extended_enat @ ( numeral_numeral @ extended_enat @ M ) @ ( numeral_numeral @ extended_enat @ N3 ) )
= ( ord_less_eq @ nat @ ( numeral_numeral @ nat @ M ) @ ( numeral_numeral @ nat @ N3 ) ) ) ).
% enat_ord_number(1)
thf(fact_468_enat__ord__number_I2_J,axiom,
! [M: num,N3: num] :
( ( ord_less @ extended_enat @ ( numeral_numeral @ extended_enat @ M ) @ ( numeral_numeral @ extended_enat @ N3 ) )
= ( ord_less @ nat @ ( numeral_numeral @ nat @ M ) @ ( numeral_numeral @ nat @ N3 ) ) ) ).
% enat_ord_number(2)
thf(fact_469_less__option__None,axiom,
! [A: $tType] :
( ( preorder @ A )
=> ! [X: option @ A] :
~ ( ord_less @ ( option @ A ) @ X @ ( none @ A ) ) ) ).
% less_option_None
thf(fact_470_less__option__Some,axiom,
! [A: $tType] :
( ( preorder @ A )
=> ! [X: A,Y: A] :
( ( ord_less @ ( option @ A ) @ ( some @ A @ X ) @ ( some @ A @ Y ) )
= ( ord_less @ A @ X @ Y ) ) ) ).
% less_option_Some
thf(fact_471_add__diff__assoc__enat,axiom,
! [Z: extended_enat,Y: extended_enat,X: extended_enat] :
( ( ord_less_eq @ extended_enat @ Z @ Y )
=> ( ( plus_plus @ extended_enat @ X @ ( minus_minus @ extended_enat @ Y @ Z ) )
= ( minus_minus @ extended_enat @ ( plus_plus @ extended_enat @ X @ Y ) @ Z ) ) ) ).
% add_diff_assoc_enat
thf(fact_472_real__arch__pow,axiom,
! [X: real,Y: real] :
( ( ord_less @ real @ ( one_one @ real ) @ X )
=> ? [N: nat] : ( ord_less @ real @ Y @ ( power_power @ real @ X @ N ) ) ) ).
% real_arch_pow
thf(fact_473_z1pdiv2,axiom,
! [B2: int] :
( ( divide_divide @ int @ ( plus_plus @ int @ ( times_times @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ B2 ) @ ( one_one @ int ) ) @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) )
= B2 ) ).
% z1pdiv2
thf(fact_474_two__realpow__ge__one,axiom,
! [N3: nat] : ( ord_less_eq @ real @ ( one_one @ real ) @ ( power_power @ real @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) @ N3 ) ) ).
% two_realpow_ge_one
thf(fact_475_VEBT__internal_Ooption__shift_Ocases,axiom,
! [A: $tType,X: product_prod @ ( A > A > A ) @ ( product_prod @ ( option @ A ) @ ( option @ A ) )] :
( ! [Uu: A > A > A,Uv: option @ A] :
( X
!= ( product_Pair @ ( A > A > A ) @ ( product_prod @ ( option @ A ) @ ( option @ A ) ) @ Uu @ ( product_Pair @ ( option @ A ) @ ( option @ A ) @ ( none @ A ) @ Uv ) ) )
=> ( ! [Uw: A > A > A,V3: A] :
( X
!= ( product_Pair @ ( A > A > A ) @ ( product_prod @ ( option @ A ) @ ( option @ A ) ) @ Uw @ ( product_Pair @ ( option @ A ) @ ( option @ A ) @ ( some @ A @ V3 ) @ ( none @ A ) ) ) )
=> ~ ! [F3: A > A > A,A4: A,B3: A] :
( X
!= ( product_Pair @ ( A > A > A ) @ ( product_prod @ ( option @ A ) @ ( option @ A ) ) @ F3 @ ( product_Pair @ ( option @ A ) @ ( option @ A ) @ ( some @ A @ A4 ) @ ( some @ A @ B3 ) ) ) ) ) ) ).
% VEBT_internal.option_shift.cases
thf(fact_476_VEBT__internal_Ooption__comp__shift_Ocases,axiom,
! [A: $tType,X: product_prod @ ( A > A > $o ) @ ( product_prod @ ( option @ A ) @ ( option @ A ) )] :
( ! [Uu: A > A > $o,Uv: option @ A] :
( X
!= ( product_Pair @ ( A > A > $o ) @ ( product_prod @ ( option @ A ) @ ( option @ A ) ) @ Uu @ ( product_Pair @ ( option @ A ) @ ( option @ A ) @ ( none @ A ) @ Uv ) ) )
=> ( ! [Uw: A > A > $o,V3: A] :
( X
!= ( product_Pair @ ( A > A > $o ) @ ( product_prod @ ( option @ A ) @ ( option @ A ) ) @ Uw @ ( product_Pair @ ( option @ A ) @ ( option @ A ) @ ( some @ A @ V3 ) @ ( none @ A ) ) ) )
=> ~ ! [F3: A > A > $o,X3: A,Y3: A] :
( X
!= ( product_Pair @ ( A > A > $o ) @ ( product_prod @ ( option @ A ) @ ( option @ A ) ) @ F3 @ ( product_Pair @ ( option @ A ) @ ( option @ A ) @ ( some @ A @ X3 ) @ ( some @ A @ Y3 ) ) ) ) ) ) ).
% VEBT_internal.option_comp_shift.cases
thf(fact_477_less__option__None__Some,axiom,
! [A: $tType] :
( ( preorder @ A )
=> ! [X: A] : ( ord_less @ ( option @ A ) @ ( none @ A ) @ ( some @ A @ X ) ) ) ).
% less_option_None_Some
thf(fact_478_less__option__None__is__Some,axiom,
! [A: $tType] :
( ( preorder @ A )
=> ! [X: option @ A] :
( ( ord_less @ ( option @ A ) @ ( none @ A ) @ X )
=> ? [Z2: A] :
( X
= ( some @ A @ Z2 ) ) ) ) ).
% less_option_None_is_Some
thf(fact_479_VEBT__internal_Ooption__shift_Osimps_I3_J,axiom,
! [A: $tType,F2: A > A > A,A2: A,B2: A] :
( ( vEBT_V2048590022279873568_shift @ A @ F2 @ ( some @ A @ A2 ) @ ( some @ A @ B2 ) )
= ( some @ A @ ( F2 @ A2 @ B2 ) ) ) ).
% VEBT_internal.option_shift.simps(3)
thf(fact_480_VEBT__internal_Ooption__shift_Osimps_I1_J,axiom,
! [A: $tType,Uu2: A > A > A,Uv2: option @ A] :
( ( vEBT_V2048590022279873568_shift @ A @ Uu2 @ ( none @ A ) @ Uv2 )
= ( none @ A ) ) ).
% VEBT_internal.option_shift.simps(1)
thf(fact_481_less__eq__option__None,axiom,
! [A: $tType] :
( ( preorder @ A )
=> ! [X: option @ A] : ( ord_less_eq @ ( option @ A ) @ ( none @ A ) @ X ) ) ).
% less_eq_option_None
thf(fact_482_less__eq__option__None__is__None,axiom,
! [A: $tType] :
( ( preorder @ A )
=> ! [X: option @ A] :
( ( ord_less_eq @ ( option @ A ) @ X @ ( none @ A ) )
=> ( X
= ( none @ A ) ) ) ) ).
% less_eq_option_None_is_None
thf(fact_483_VEBT__internal_Ooption__shift_Osimps_I2_J,axiom,
! [A: $tType,Uw2: A > A > A,V: A] :
( ( vEBT_V2048590022279873568_shift @ A @ Uw2 @ ( some @ A @ V ) @ ( none @ A ) )
= ( none @ A ) ) ).
% VEBT_internal.option_shift.simps(2)
thf(fact_484_VEBT__internal_Ooption__shift_Oelims,axiom,
! [A: $tType,X: A > A > A,Xa: option @ A,Xb: option @ A,Y: option @ A] :
( ( ( vEBT_V2048590022279873568_shift @ A @ X @ Xa @ Xb )
= Y )
=> ( ( ( Xa
= ( none @ A ) )
=> ( Y
!= ( none @ A ) ) )
=> ( ( ? [V3: A] :
( Xa
= ( some @ A @ V3 ) )
=> ( ( Xb
= ( none @ A ) )
=> ( Y
!= ( none @ A ) ) ) )
=> ~ ! [A4: A] :
( ( Xa
= ( some @ A @ A4 ) )
=> ! [B3: A] :
( ( Xb
= ( some @ A @ B3 ) )
=> ( Y
!= ( some @ A @ ( X @ A4 @ B3 ) ) ) ) ) ) ) ) ).
% VEBT_internal.option_shift.elims
thf(fact_485_vebt__mint_Osimps_I2_J,axiom,
! [Uu2: nat,Uv2: list @ vEBT_VEBT,Uw2: vEBT_VEBT] :
( ( vEBT_vebt_mint @ ( vEBT_Node @ ( none @ ( product_prod @ nat @ nat ) ) @ Uu2 @ Uv2 @ Uw2 ) )
= ( none @ nat ) ) ).
% vebt_mint.simps(2)
thf(fact_486_vebt__maxt_Osimps_I2_J,axiom,
! [Uu2: nat,Uv2: list @ vEBT_VEBT,Uw2: vEBT_VEBT] :
( ( vEBT_vebt_maxt @ ( vEBT_Node @ ( none @ ( product_prod @ nat @ nat ) ) @ Uu2 @ Uv2 @ Uw2 ) )
= ( none @ nat ) ) ).
% vebt_maxt.simps(2)
thf(fact_487_cnt__bound_H,axiom,
! [T2: vEBT_VEBT,N3: nat] :
( ( vEBT_invar_vebt @ T2 @ N3 )
=> ( 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 ) ) @ N3 ) @ ( one_one @ real ) ) ) ) ) ).
% cnt_bound'
thf(fact_488_Suc__double__not__eq__double,axiom,
! [M: nat,N3: nat] :
( ( suc @ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ M ) )
!= ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N3 ) ) ).
% Suc_double_not_eq_double
thf(fact_489_double__not__eq__Suc__double,axiom,
! [M: nat,N3: nat] :
( ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ M )
!= ( suc @ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N3 ) ) ) ).
% double_not_eq_Suc_double
thf(fact_490_times__divide__eq__left,axiom,
! [A: $tType] :
( ( field @ A )
=> ! [B2: A,C2: A,A2: A] :
( ( times_times @ A @ ( divide_divide @ A @ B2 @ C2 ) @ A2 )
= ( divide_divide @ A @ ( times_times @ A @ B2 @ A2 ) @ C2 ) ) ) ).
% times_divide_eq_left
thf(fact_491_divide__divide__eq__left,axiom,
! [A: $tType] :
( ( field @ A )
=> ! [A2: A,B2: A,C2: A] :
( ( divide_divide @ A @ ( divide_divide @ A @ A2 @ B2 ) @ C2 )
= ( divide_divide @ A @ A2 @ ( times_times @ A @ B2 @ C2 ) ) ) ) ).
% divide_divide_eq_left
thf(fact_492_divide__divide__eq__right,axiom,
! [A: $tType] :
( ( field @ A )
=> ! [A2: A,B2: A,C2: A] :
( ( divide_divide @ A @ A2 @ ( divide_divide @ A @ B2 @ C2 ) )
= ( divide_divide @ A @ ( times_times @ A @ A2 @ C2 ) @ B2 ) ) ) ).
% divide_divide_eq_right
thf(fact_493_times__divide__eq__right,axiom,
! [A: $tType] :
( ( division_ring @ A )
=> ! [A2: A,B2: A,C2: A] :
( ( times_times @ A @ A2 @ ( divide_divide @ A @ B2 @ C2 ) )
= ( divide_divide @ A @ ( times_times @ A @ A2 @ B2 ) @ C2 ) ) ) ).
% times_divide_eq_right
thf(fact_494_add__diff__cancel,axiom,
! [A: $tType] :
( ( group_add @ A )
=> ! [A2: A,B2: A] :
( ( minus_minus @ A @ ( plus_plus @ A @ A2 @ B2 ) @ B2 )
= A2 ) ) ).
% add_diff_cancel
thf(fact_495_diff__add__cancel,axiom,
! [A: $tType] :
( ( group_add @ A )
=> ! [A2: A,B2: A] :
( ( plus_plus @ A @ ( minus_minus @ A @ A2 @ B2 ) @ B2 )
= A2 ) ) ).
% diff_add_cancel
thf(fact_496_add__diff__cancel__left,axiom,
! [A: $tType] :
( ( cancel2418104881723323429up_add @ A )
=> ! [C2: A,A2: A,B2: A] :
( ( minus_minus @ A @ ( plus_plus @ A @ C2 @ A2 ) @ ( plus_plus @ A @ C2 @ B2 ) )
= ( minus_minus @ A @ A2 @ B2 ) ) ) ).
% add_diff_cancel_left
thf(fact_497_add__diff__cancel__left_H,axiom,
! [A: $tType] :
( ( cancel2418104881723323429up_add @ A )
=> ! [A2: A,B2: A] :
( ( minus_minus @ A @ ( plus_plus @ A @ A2 @ B2 ) @ A2 )
= B2 ) ) ).
% add_diff_cancel_left'
thf(fact_498_add__right__cancel,axiom,
! [A: $tType] :
( ( cancel_semigroup_add @ A )
=> ! [B2: A,A2: A,C2: A] :
( ( ( plus_plus @ A @ B2 @ A2 )
= ( plus_plus @ A @ C2 @ A2 ) )
= ( B2 = C2 ) ) ) ).
% add_right_cancel
thf(fact_499_add__left__cancel,axiom,
! [A: $tType] :
( ( cancel_semigroup_add @ A )
=> ! [A2: A,B2: A,C2: A] :
( ( ( plus_plus @ A @ A2 @ B2 )
= ( plus_plus @ A @ A2 @ C2 ) )
= ( B2 = C2 ) ) ) ).
% add_left_cancel
thf(fact_500_add__le__cancel__right,axiom,
! [A: $tType] :
( ( ordere2412721322843649153imp_le @ A )
=> ! [A2: A,C2: A,B2: A] :
( ( ord_less_eq @ A @ ( plus_plus @ A @ A2 @ C2 ) @ ( plus_plus @ A @ B2 @ C2 ) )
= ( ord_less_eq @ A @ A2 @ B2 ) ) ) ).
% add_le_cancel_right
thf(fact_501_add__le__cancel__left,axiom,
! [A: $tType] :
( ( ordere2412721322843649153imp_le @ A )
=> ! [C2: A,A2: A,B2: A] :
( ( ord_less_eq @ A @ ( plus_plus @ A @ C2 @ A2 ) @ ( plus_plus @ A @ C2 @ B2 ) )
= ( ord_less_eq @ A @ A2 @ B2 ) ) ) ).
% add_le_cancel_left
thf(fact_502_add__less__cancel__right,axiom,
! [A: $tType] :
( ( ordere2412721322843649153imp_le @ A )
=> ! [A2: A,C2: A,B2: A] :
( ( ord_less @ A @ ( plus_plus @ A @ A2 @ C2 ) @ ( plus_plus @ A @ B2 @ C2 ) )
= ( ord_less @ A @ A2 @ B2 ) ) ) ).
% add_less_cancel_right
thf(fact_503_add__less__cancel__left,axiom,
! [A: $tType] :
( ( ordere2412721322843649153imp_le @ A )
=> ! [C2: A,A2: A,B2: A] :
( ( ord_less @ A @ ( plus_plus @ A @ C2 @ A2 ) @ ( plus_plus @ A @ C2 @ B2 ) )
= ( ord_less @ A @ A2 @ B2 ) ) ) ).
% add_less_cancel_left
thf(fact_504_mult__1,axiom,
! [A: $tType] :
( ( monoid_mult @ A )
=> ! [A2: A] :
( ( times_times @ A @ ( one_one @ A ) @ A2 )
= A2 ) ) ).
% mult_1
thf(fact_505_mult_Oright__neutral,axiom,
! [A: $tType] :
( ( monoid_mult @ A )
=> ! [A2: A] :
( ( times_times @ A @ A2 @ ( one_one @ A ) )
= A2 ) ) ).
% mult.right_neutral
thf(fact_506_add__diff__cancel__right_H,axiom,
! [A: $tType] :
( ( cancel2418104881723323429up_add @ A )
=> ! [A2: A,B2: A] :
( ( minus_minus @ A @ ( plus_plus @ A @ A2 @ B2 ) @ B2 )
= A2 ) ) ).
% add_diff_cancel_right'
thf(fact_507_add__diff__cancel__right,axiom,
! [A: $tType] :
( ( cancel2418104881723323429up_add @ A )
=> ! [A2: A,C2: A,B2: A] :
( ( minus_minus @ A @ ( plus_plus @ A @ A2 @ C2 ) @ ( plus_plus @ A @ B2 @ C2 ) )
= ( minus_minus @ A @ A2 @ B2 ) ) ) ).
% add_diff_cancel_right
thf(fact_508_linordered__field__no__ub,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [X5: A] :
? [X_12: A] : ( ord_less @ A @ X5 @ X_12 ) ) ).
% linordered_field_no_ub
thf(fact_509_linordered__field__no__lb,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [X5: A] :
? [Y3: A] : ( ord_less @ A @ Y3 @ X5 ) ) ).
% linordered_field_no_lb
thf(fact_510_add__right__imp__eq,axiom,
! [A: $tType] :
( ( cancel_semigroup_add @ A )
=> ! [B2: A,A2: A,C2: A] :
( ( ( plus_plus @ A @ B2 @ A2 )
= ( plus_plus @ A @ C2 @ A2 ) )
=> ( B2 = C2 ) ) ) ).
% add_right_imp_eq
thf(fact_511_add__left__imp__eq,axiom,
! [A: $tType] :
( ( cancel_semigroup_add @ A )
=> ! [A2: A,B2: A,C2: A] :
( ( ( plus_plus @ A @ A2 @ B2 )
= ( plus_plus @ A @ A2 @ C2 ) )
=> ( B2 = C2 ) ) ) ).
% add_left_imp_eq
thf(fact_512_ab__semigroup__add__class_Oadd_Oleft__commute,axiom,
! [A: $tType] :
( ( ab_semigroup_add @ A )
=> ! [B2: A,A2: A,C2: A] :
( ( plus_plus @ A @ B2 @ ( plus_plus @ A @ A2 @ C2 ) )
= ( plus_plus @ A @ A2 @ ( plus_plus @ A @ B2 @ C2 ) ) ) ) ).
% ab_semigroup_add_class.add.left_commute
thf(fact_513_ab__semigroup__add__class_Oadd_Ocommute,axiom,
! [A: $tType] :
( ( ab_semigroup_add @ A )
=> ( ( plus_plus @ A )
= ( ^ [A6: A,B5: A] : ( plus_plus @ A @ B5 @ A6 ) ) ) ) ).
% ab_semigroup_add_class.add.commute
thf(fact_514_add_Oright__cancel,axiom,
! [A: $tType] :
( ( group_add @ A )
=> ! [B2: A,A2: A,C2: A] :
( ( ( plus_plus @ A @ B2 @ A2 )
= ( plus_plus @ A @ C2 @ A2 ) )
= ( B2 = C2 ) ) ) ).
% add.right_cancel
thf(fact_515_add_Oleft__cancel,axiom,
! [A: $tType] :
( ( group_add @ A )
=> ! [A2: A,B2: A,C2: A] :
( ( ( plus_plus @ A @ A2 @ B2 )
= ( plus_plus @ A @ A2 @ C2 ) )
= ( B2 = C2 ) ) ) ).
% add.left_cancel
thf(fact_516_add_Oassoc,axiom,
! [A: $tType] :
( ( semigroup_add @ A )
=> ! [A2: A,B2: A,C2: A] :
( ( plus_plus @ A @ ( plus_plus @ A @ A2 @ B2 ) @ C2 )
= ( plus_plus @ A @ A2 @ ( plus_plus @ A @ B2 @ C2 ) ) ) ) ).
% add.assoc
thf(fact_517_group__cancel_Oadd2,axiom,
! [A: $tType] :
( ( comm_monoid_add @ A )
=> ! [B6: A,K: A,B2: A,A2: A] :
( ( B6
= ( plus_plus @ A @ K @ B2 ) )
=> ( ( plus_plus @ A @ A2 @ B6 )
= ( plus_plus @ A @ K @ ( plus_plus @ A @ A2 @ B2 ) ) ) ) ) ).
% group_cancel.add2
thf(fact_518_group__cancel_Oadd1,axiom,
! [A: $tType] :
( ( comm_monoid_add @ A )
=> ! [A3: A,K: A,A2: A,B2: A] :
( ( A3
= ( plus_plus @ A @ K @ A2 ) )
=> ( ( plus_plus @ A @ A3 @ B2 )
= ( plus_plus @ A @ K @ ( plus_plus @ A @ A2 @ B2 ) ) ) ) ) ).
% group_cancel.add1
thf(fact_519_add__mono__thms__linordered__semiring_I4_J,axiom,
! [A: $tType] :
( ( ordere6658533253407199908up_add @ A )
=> ! [I: A,J: A,K: A,L2: A] :
( ( ( I = J )
& ( K = L2 ) )
=> ( ( plus_plus @ A @ I @ K )
= ( plus_plus @ A @ J @ L2 ) ) ) ) ).
% add_mono_thms_linordered_semiring(4)
thf(fact_520_one__reorient,axiom,
! [A: $tType] :
( ( one @ A )
=> ! [X: A] :
( ( ( one_one @ A )
= X )
= ( X
= ( one_one @ A ) ) ) ) ).
% one_reorient
thf(fact_521_ab__semigroup__mult__class_Omult_Oleft__commute,axiom,
! [A: $tType] :
( ( ab_semigroup_mult @ A )
=> ! [B2: A,A2: A,C2: A] :
( ( times_times @ A @ B2 @ ( times_times @ A @ A2 @ C2 ) )
= ( times_times @ A @ A2 @ ( times_times @ A @ B2 @ C2 ) ) ) ) ).
% ab_semigroup_mult_class.mult.left_commute
thf(fact_522_ab__semigroup__mult__class_Omult_Ocommute,axiom,
! [A: $tType] :
( ( ab_semigroup_mult @ A )
=> ( ( times_times @ A )
= ( ^ [A6: A,B5: A] : ( times_times @ A @ B5 @ A6 ) ) ) ) ).
% ab_semigroup_mult_class.mult.commute
thf(fact_523_mult_Oassoc,axiom,
! [A: $tType] :
( ( semigroup_mult @ A )
=> ! [A2: A,B2: A,C2: A] :
( ( times_times @ A @ ( times_times @ A @ A2 @ B2 ) @ C2 )
= ( times_times @ A @ A2 @ ( times_times @ A @ B2 @ C2 ) ) ) ) ).
% mult.assoc
thf(fact_524_diff__eq__diff__eq,axiom,
! [A: $tType] :
( ( group_add @ A )
=> ! [A2: A,B2: A,C2: A,D: A] :
( ( ( minus_minus @ A @ A2 @ B2 )
= ( minus_minus @ A @ C2 @ D ) )
=> ( ( A2 = B2 )
= ( C2 = D ) ) ) ) ).
% diff_eq_diff_eq
thf(fact_525_cancel__ab__semigroup__add__class_Odiff__right__commute,axiom,
! [A: $tType] :
( ( cancel2418104881723323429up_add @ A )
=> ! [A2: A,C2: A,B2: A] :
( ( minus_minus @ A @ ( minus_minus @ A @ A2 @ C2 ) @ B2 )
= ( minus_minus @ A @ ( minus_minus @ A @ A2 @ B2 ) @ C2 ) ) ) ).
% cancel_ab_semigroup_add_class.diff_right_commute
thf(fact_526_add__le__imp__le__right,axiom,
! [A: $tType] :
( ( ordere2412721322843649153imp_le @ A )
=> ! [A2: A,C2: A,B2: A] :
( ( ord_less_eq @ A @ ( plus_plus @ A @ A2 @ C2 ) @ ( plus_plus @ A @ B2 @ C2 ) )
=> ( ord_less_eq @ A @ A2 @ B2 ) ) ) ).
% add_le_imp_le_right
thf(fact_527_add__le__imp__le__left,axiom,
! [A: $tType] :
( ( ordere2412721322843649153imp_le @ A )
=> ! [C2: A,A2: A,B2: A] :
( ( ord_less_eq @ A @ ( plus_plus @ A @ C2 @ A2 ) @ ( plus_plus @ A @ C2 @ B2 ) )
=> ( ord_less_eq @ A @ A2 @ B2 ) ) ) ).
% add_le_imp_le_left
thf(fact_528_le__iff__add,axiom,
! [A: $tType] :
( ( canoni5634975068530333245id_add @ A )
=> ( ( ord_less_eq @ A )
= ( ^ [A6: A,B5: A] :
? [C3: A] :
( B5
= ( plus_plus @ A @ A6 @ C3 ) ) ) ) ) ).
% le_iff_add
thf(fact_529_add__right__mono,axiom,
! [A: $tType] :
( ( ordere6658533253407199908up_add @ A )
=> ! [A2: A,B2: A,C2: A] :
( ( ord_less_eq @ A @ A2 @ B2 )
=> ( ord_less_eq @ A @ ( plus_plus @ A @ A2 @ C2 ) @ ( plus_plus @ A @ B2 @ C2 ) ) ) ) ).
% add_right_mono
thf(fact_530_less__eqE,axiom,
! [A: $tType] :
( ( canoni5634975068530333245id_add @ A )
=> ! [A2: A,B2: A] :
( ( ord_less_eq @ A @ A2 @ B2 )
=> ~ ! [C4: A] :
( B2
!= ( plus_plus @ A @ A2 @ C4 ) ) ) ) ).
% less_eqE
thf(fact_531_add__left__mono,axiom,
! [A: $tType] :
( ( ordere6658533253407199908up_add @ A )
=> ! [A2: A,B2: A,C2: A] :
( ( ord_less_eq @ A @ A2 @ B2 )
=> ( ord_less_eq @ A @ ( plus_plus @ A @ C2 @ A2 ) @ ( plus_plus @ A @ C2 @ B2 ) ) ) ) ).
% add_left_mono
thf(fact_532_add__mono,axiom,
! [A: $tType] :
( ( ordere6658533253407199908up_add @ A )
=> ! [A2: A,B2: A,C2: A,D: A] :
( ( ord_less_eq @ A @ A2 @ B2 )
=> ( ( ord_less_eq @ A @ C2 @ D )
=> ( ord_less_eq @ A @ ( plus_plus @ A @ A2 @ C2 ) @ ( plus_plus @ A @ B2 @ D ) ) ) ) ) ).
% add_mono
thf(fact_533_add__mono__thms__linordered__semiring_I1_J,axiom,
! [A: $tType] :
( ( ordere6658533253407199908up_add @ A )
=> ! [I: A,J: A,K: A,L2: A] :
( ( ( ord_less_eq @ A @ I @ J )
& ( ord_less_eq @ A @ K @ L2 ) )
=> ( ord_less_eq @ A @ ( plus_plus @ A @ I @ K ) @ ( plus_plus @ A @ J @ L2 ) ) ) ) ).
% add_mono_thms_linordered_semiring(1)
thf(fact_534_add__mono__thms__linordered__semiring_I2_J,axiom,
! [A: $tType] :
( ( ordere6658533253407199908up_add @ A )
=> ! [I: A,J: A,K: A,L2: A] :
( ( ( I = J )
& ( ord_less_eq @ A @ K @ L2 ) )
=> ( ord_less_eq @ A @ ( plus_plus @ A @ I @ K ) @ ( plus_plus @ A @ J @ L2 ) ) ) ) ).
% add_mono_thms_linordered_semiring(2)
thf(fact_535_add__mono__thms__linordered__semiring_I3_J,axiom,
! [A: $tType] :
( ( ordere6658533253407199908up_add @ A )
=> ! [I: A,J: A,K: A,L2: A] :
( ( ( ord_less_eq @ A @ I @ J )
& ( K = L2 ) )
=> ( ord_less_eq @ A @ ( plus_plus @ A @ I @ K ) @ ( plus_plus @ A @ J @ L2 ) ) ) ) ).
% add_mono_thms_linordered_semiring(3)
thf(fact_536_add__less__imp__less__right,axiom,
! [A: $tType] :
( ( ordere2412721322843649153imp_le @ A )
=> ! [A2: A,C2: A,B2: A] :
( ( ord_less @ A @ ( plus_plus @ A @ A2 @ C2 ) @ ( plus_plus @ A @ B2 @ C2 ) )
=> ( ord_less @ A @ A2 @ B2 ) ) ) ).
% add_less_imp_less_right
thf(fact_537_add__less__imp__less__left,axiom,
! [A: $tType] :
( ( ordere2412721322843649153imp_le @ A )
=> ! [C2: A,A2: A,B2: A] :
( ( ord_less @ A @ ( plus_plus @ A @ C2 @ A2 ) @ ( plus_plus @ A @ C2 @ B2 ) )
=> ( ord_less @ A @ A2 @ B2 ) ) ) ).
% add_less_imp_less_left
thf(fact_538_add__strict__right__mono,axiom,
! [A: $tType] :
( ( ordere580206878836729694up_add @ A )
=> ! [A2: A,B2: A,C2: A] :
( ( ord_less @ A @ A2 @ B2 )
=> ( ord_less @ A @ ( plus_plus @ A @ A2 @ C2 ) @ ( plus_plus @ A @ B2 @ C2 ) ) ) ) ).
% add_strict_right_mono
thf(fact_539_add__strict__left__mono,axiom,
! [A: $tType] :
( ( ordere580206878836729694up_add @ A )
=> ! [A2: A,B2: A,C2: A] :
( ( ord_less @ A @ A2 @ B2 )
=> ( ord_less @ A @ ( plus_plus @ A @ C2 @ A2 ) @ ( plus_plus @ A @ C2 @ B2 ) ) ) ) ).
% add_strict_left_mono
thf(fact_540_add__strict__mono,axiom,
! [A: $tType] :
( ( strict9044650504122735259up_add @ A )
=> ! [A2: A,B2: A,C2: A,D: A] :
( ( ord_less @ A @ A2 @ B2 )
=> ( ( ord_less @ A @ C2 @ D )
=> ( ord_less @ A @ ( plus_plus @ A @ A2 @ C2 ) @ ( plus_plus @ A @ B2 @ D ) ) ) ) ) ).
% add_strict_mono
thf(fact_541_add__mono__thms__linordered__field_I1_J,axiom,
! [A: $tType] :
( ( ordere580206878836729694up_add @ A )
=> ! [I: A,J: A,K: A,L2: A] :
( ( ( ord_less @ A @ I @ J )
& ( K = L2 ) )
=> ( ord_less @ A @ ( plus_plus @ A @ I @ K ) @ ( plus_plus @ A @ J @ L2 ) ) ) ) ).
% add_mono_thms_linordered_field(1)
thf(fact_542_add__mono__thms__linordered__field_I2_J,axiom,
! [A: $tType] :
( ( ordere580206878836729694up_add @ A )
=> ! [I: A,J: A,K: A,L2: A] :
( ( ( I = J )
& ( ord_less @ A @ K @ L2 ) )
=> ( ord_less @ A @ ( plus_plus @ A @ I @ K ) @ ( plus_plus @ A @ J @ L2 ) ) ) ) ).
% add_mono_thms_linordered_field(2)
thf(fact_543_add__mono__thms__linordered__field_I5_J,axiom,
! [A: $tType] :
( ( ordere580206878836729694up_add @ A )
=> ! [I: A,J: A,K: A,L2: A] :
( ( ( ord_less @ A @ I @ J )
& ( ord_less @ A @ K @ L2 ) )
=> ( ord_less @ A @ ( plus_plus @ A @ I @ K ) @ ( plus_plus @ A @ J @ L2 ) ) ) ) ).
% add_mono_thms_linordered_field(5)
thf(fact_544_diff__mono,axiom,
! [A: $tType] :
( ( ordered_ab_group_add @ A )
=> ! [A2: A,B2: A,D: A,C2: A] :
( ( ord_less_eq @ A @ A2 @ B2 )
=> ( ( ord_less_eq @ A @ D @ C2 )
=> ( ord_less_eq @ A @ ( minus_minus @ A @ A2 @ C2 ) @ ( minus_minus @ A @ B2 @ D ) ) ) ) ) ).
% diff_mono
thf(fact_545_diff__left__mono,axiom,
! [A: $tType] :
( ( ordered_ab_group_add @ A )
=> ! [B2: A,A2: A,C2: A] :
( ( ord_less_eq @ A @ B2 @ A2 )
=> ( ord_less_eq @ A @ ( minus_minus @ A @ C2 @ A2 ) @ ( minus_minus @ A @ C2 @ B2 ) ) ) ) ).
% diff_left_mono
thf(fact_546_diff__right__mono,axiom,
! [A: $tType] :
( ( ordered_ab_group_add @ A )
=> ! [A2: A,B2: A,C2: A] :
( ( ord_less_eq @ A @ A2 @ B2 )
=> ( ord_less_eq @ A @ ( minus_minus @ A @ A2 @ C2 ) @ ( minus_minus @ A @ B2 @ C2 ) ) ) ) ).
% diff_right_mono
thf(fact_547_diff__eq__diff__less__eq,axiom,
! [A: $tType] :
( ( ordered_ab_group_add @ A )
=> ! [A2: A,B2: A,C2: A,D: A] :
( ( ( minus_minus @ A @ A2 @ B2 )
= ( minus_minus @ A @ C2 @ D ) )
=> ( ( ord_less_eq @ A @ A2 @ B2 )
= ( ord_less_eq @ A @ C2 @ D ) ) ) ) ).
% diff_eq_diff_less_eq
thf(fact_548_diff__strict__right__mono,axiom,
! [A: $tType] :
( ( ordered_ab_group_add @ A )
=> ! [A2: A,B2: A,C2: A] :
( ( ord_less @ A @ A2 @ B2 )
=> ( ord_less @ A @ ( minus_minus @ A @ A2 @ C2 ) @ ( minus_minus @ A @ B2 @ C2 ) ) ) ) ).
% diff_strict_right_mono
thf(fact_549_diff__strict__left__mono,axiom,
! [A: $tType] :
( ( ordered_ab_group_add @ A )
=> ! [B2: A,A2: A,C2: A] :
( ( ord_less @ A @ B2 @ A2 )
=> ( ord_less @ A @ ( minus_minus @ A @ C2 @ A2 ) @ ( minus_minus @ A @ C2 @ B2 ) ) ) ) ).
% diff_strict_left_mono
thf(fact_550_diff__eq__diff__less,axiom,
! [A: $tType] :
( ( ordered_ab_group_add @ A )
=> ! [A2: A,B2: A,C2: A,D: A] :
( ( ( minus_minus @ A @ A2 @ B2 )
= ( minus_minus @ A @ C2 @ D ) )
=> ( ( ord_less @ A @ A2 @ B2 )
= ( ord_less @ A @ C2 @ D ) ) ) ) ).
% diff_eq_diff_less
thf(fact_551_diff__strict__mono,axiom,
! [A: $tType] :
( ( ordered_ab_group_add @ A )
=> ! [A2: A,B2: A,D: A,C2: A] :
( ( ord_less @ A @ A2 @ B2 )
=> ( ( ord_less @ A @ D @ C2 )
=> ( ord_less @ A @ ( minus_minus @ A @ A2 @ C2 ) @ ( minus_minus @ A @ B2 @ D ) ) ) ) ) ).
% diff_strict_mono
thf(fact_552_mult_Ocomm__neutral,axiom,
! [A: $tType] :
( ( comm_monoid_mult @ A )
=> ! [A2: A] :
( ( times_times @ A @ A2 @ ( one_one @ A ) )
= A2 ) ) ).
% mult.comm_neutral
thf(fact_553_comm__monoid__mult__class_Omult__1,axiom,
! [A: $tType] :
( ( comm_monoid_mult @ A )
=> ! [A2: A] :
( ( times_times @ A @ ( one_one @ A ) @ A2 )
= A2 ) ) ).
% comm_monoid_mult_class.mult_1
thf(fact_554_diff__diff__eq,axiom,
! [A: $tType] :
( ( cancel2418104881723323429up_add @ A )
=> ! [A2: A,B2: A,C2: A] :
( ( minus_minus @ A @ ( minus_minus @ A @ A2 @ B2 ) @ C2 )
= ( minus_minus @ A @ A2 @ ( plus_plus @ A @ B2 @ C2 ) ) ) ) ).
% diff_diff_eq
thf(fact_555_add__implies__diff,axiom,
! [A: $tType] :
( ( cancel1802427076303600483id_add @ A )
=> ! [C2: A,B2: A,A2: A] :
( ( ( plus_plus @ A @ C2 @ B2 )
= A2 )
=> ( C2
= ( minus_minus @ A @ A2 @ B2 ) ) ) ) ).
% add_implies_diff
thf(fact_556_diff__add__eq__diff__diff__swap,axiom,
! [A: $tType] :
( ( group_add @ A )
=> ! [A2: A,B2: A,C2: A] :
( ( minus_minus @ A @ A2 @ ( plus_plus @ A @ B2 @ C2 ) )
= ( minus_minus @ A @ ( minus_minus @ A @ A2 @ C2 ) @ B2 ) ) ) ).
% diff_add_eq_diff_diff_swap
thf(fact_557_diff__add__eq,axiom,
! [A: $tType] :
( ( ab_group_add @ A )
=> ! [A2: A,B2: A,C2: A] :
( ( plus_plus @ A @ ( minus_minus @ A @ A2 @ B2 ) @ C2 )
= ( minus_minus @ A @ ( plus_plus @ A @ A2 @ C2 ) @ B2 ) ) ) ).
% diff_add_eq
thf(fact_558_diff__diff__eq2,axiom,
! [A: $tType] :
( ( group_add @ A )
=> ! [A2: A,B2: A,C2: A] :
( ( minus_minus @ A @ A2 @ ( minus_minus @ A @ B2 @ C2 ) )
= ( minus_minus @ A @ ( plus_plus @ A @ A2 @ C2 ) @ B2 ) ) ) ).
% diff_diff_eq2
thf(fact_559_add__diff__eq,axiom,
! [A: $tType] :
( ( group_add @ A )
=> ! [A2: A,B2: A,C2: A] :
( ( plus_plus @ A @ A2 @ ( minus_minus @ A @ B2 @ C2 ) )
= ( minus_minus @ A @ ( plus_plus @ A @ A2 @ B2 ) @ C2 ) ) ) ).
% add_diff_eq
thf(fact_560_eq__diff__eq,axiom,
! [A: $tType] :
( ( group_add @ A )
=> ! [A2: A,C2: A,B2: A] :
( ( A2
= ( minus_minus @ A @ C2 @ B2 ) )
= ( ( plus_plus @ A @ A2 @ B2 )
= C2 ) ) ) ).
% eq_diff_eq
thf(fact_561_diff__eq__eq,axiom,
! [A: $tType] :
( ( group_add @ A )
=> ! [A2: A,B2: A,C2: A] :
( ( ( minus_minus @ A @ A2 @ B2 )
= C2 )
= ( A2
= ( plus_plus @ A @ C2 @ B2 ) ) ) ) ).
% diff_eq_eq
thf(fact_562_group__cancel_Osub1,axiom,
! [A: $tType] :
( ( ab_group_add @ A )
=> ! [A3: A,K: A,A2: A,B2: A] :
( ( A3
= ( plus_plus @ A @ K @ A2 ) )
=> ( ( minus_minus @ A @ A3 @ B2 )
= ( plus_plus @ A @ K @ ( minus_minus @ A @ A2 @ B2 ) ) ) ) ) ).
% group_cancel.sub1
thf(fact_563_add__divide__distrib,axiom,
! [A: $tType] :
( ( division_ring @ A )
=> ! [A2: A,B2: A,C2: A] :
( ( divide_divide @ A @ ( plus_plus @ A @ A2 @ B2 ) @ C2 )
= ( plus_plus @ A @ ( divide_divide @ A @ A2 @ C2 ) @ ( divide_divide @ A @ B2 @ C2 ) ) ) ) ).
% add_divide_distrib
thf(fact_564_divide__divide__eq__left_H,axiom,
! [A: $tType] :
( ( division_ring @ A )
=> ! [A2: A,B2: A,C2: A] :
( ( divide_divide @ A @ ( divide_divide @ A @ A2 @ B2 ) @ C2 )
= ( divide_divide @ A @ A2 @ ( times_times @ A @ C2 @ B2 ) ) ) ) ).
% divide_divide_eq_left'
thf(fact_565_divide__divide__times__eq,axiom,
! [A: $tType] :
( ( field @ A )
=> ! [X: A,Y: A,Z: A,W: A] :
( ( divide_divide @ A @ ( divide_divide @ A @ X @ Y ) @ ( divide_divide @ A @ Z @ W ) )
= ( divide_divide @ A @ ( times_times @ A @ X @ W ) @ ( times_times @ A @ Y @ Z ) ) ) ) ).
% divide_divide_times_eq
thf(fact_566_times__divide__times__eq,axiom,
! [A: $tType] :
( ( field @ A )
=> ! [X: A,Y: A,Z: A,W: A] :
( ( times_times @ A @ ( divide_divide @ A @ X @ Y ) @ ( divide_divide @ A @ Z @ W ) )
= ( divide_divide @ A @ ( times_times @ A @ X @ Z ) @ ( times_times @ A @ Y @ W ) ) ) ) ).
% times_divide_times_eq
thf(fact_567_diff__divide__distrib,axiom,
! [A: $tType] :
( ( division_ring @ A )
=> ! [A2: A,B2: A,C2: A] :
( ( divide_divide @ A @ ( minus_minus @ A @ A2 @ B2 ) @ C2 )
= ( minus_minus @ A @ ( divide_divide @ A @ A2 @ C2 ) @ ( divide_divide @ A @ B2 @ C2 ) ) ) ) ).
% diff_divide_distrib
thf(fact_568_add__less__le__mono,axiom,
! [A: $tType] :
( ( ordere580206878836729694up_add @ A )
=> ! [A2: A,B2: A,C2: A,D: A] :
( ( ord_less @ A @ A2 @ B2 )
=> ( ( ord_less_eq @ A @ C2 @ D )
=> ( ord_less @ A @ ( plus_plus @ A @ A2 @ C2 ) @ ( plus_plus @ A @ B2 @ D ) ) ) ) ) ).
% add_less_le_mono
thf(fact_569_add__le__less__mono,axiom,
! [A: $tType] :
( ( ordere580206878836729694up_add @ A )
=> ! [A2: A,B2: A,C2: A,D: A] :
( ( ord_less_eq @ A @ A2 @ B2 )
=> ( ( ord_less @ A @ C2 @ D )
=> ( ord_less @ A @ ( plus_plus @ A @ A2 @ C2 ) @ ( plus_plus @ A @ B2 @ D ) ) ) ) ) ).
% add_le_less_mono
thf(fact_570_add__mono__thms__linordered__field_I3_J,axiom,
! [A: $tType] :
( ( ordere580206878836729694up_add @ A )
=> ! [I: A,J: A,K: A,L2: A] :
( ( ( ord_less @ A @ I @ J )
& ( ord_less_eq @ A @ K @ L2 ) )
=> ( ord_less @ A @ ( plus_plus @ A @ I @ K ) @ ( plus_plus @ A @ J @ L2 ) ) ) ) ).
% add_mono_thms_linordered_field(3)
thf(fact_571_add__mono__thms__linordered__field_I4_J,axiom,
! [A: $tType] :
( ( ordere580206878836729694up_add @ A )
=> ! [I: A,J: A,K: A,L2: A] :
( ( ( ord_less_eq @ A @ I @ J )
& ( ord_less @ A @ K @ L2 ) )
=> ( ord_less @ A @ ( plus_plus @ A @ I @ K ) @ ( plus_plus @ A @ J @ L2 ) ) ) ) ).
% add_mono_thms_linordered_field(4)
thf(fact_572_diff__le__eq,axiom,
! [A: $tType] :
( ( ordered_ab_group_add @ A )
=> ! [A2: A,B2: A,C2: A] :
( ( ord_less_eq @ A @ ( minus_minus @ A @ A2 @ B2 ) @ C2 )
= ( ord_less_eq @ A @ A2 @ ( plus_plus @ A @ C2 @ B2 ) ) ) ) ).
% diff_le_eq
thf(fact_573_le__diff__eq,axiom,
! [A: $tType] :
( ( ordered_ab_group_add @ A )
=> ! [A2: A,C2: A,B2: A] :
( ( ord_less_eq @ A @ A2 @ ( minus_minus @ A @ C2 @ B2 ) )
= ( ord_less_eq @ A @ ( plus_plus @ A @ A2 @ B2 ) @ C2 ) ) ) ).
% le_diff_eq
thf(fact_574_ordered__cancel__comm__monoid__diff__class_Odiff__add,axiom,
! [A: $tType] :
( ( ordere1170586879665033532d_diff @ A )
=> ! [A2: A,B2: A] :
( ( ord_less_eq @ A @ A2 @ B2 )
=> ( ( plus_plus @ A @ ( minus_minus @ A @ B2 @ A2 ) @ A2 )
= B2 ) ) ) ).
% ordered_cancel_comm_monoid_diff_class.diff_add
thf(fact_575_le__add__diff,axiom,
! [A: $tType] :
( ( ordere1170586879665033532d_diff @ A )
=> ! [A2: A,B2: A,C2: A] :
( ( ord_less_eq @ A @ A2 @ B2 )
=> ( ord_less_eq @ A @ C2 @ ( minus_minus @ A @ ( plus_plus @ A @ B2 @ C2 ) @ A2 ) ) ) ) ).
% le_add_diff
thf(fact_576_ordered__cancel__comm__monoid__diff__class_Ole__diff__conv2,axiom,
! [A: $tType] :
( ( ordere1170586879665033532d_diff @ A )
=> ! [A2: A,B2: A,C2: A] :
( ( ord_less_eq @ A @ A2 @ B2 )
=> ( ( ord_less_eq @ A @ C2 @ ( minus_minus @ A @ B2 @ A2 ) )
= ( ord_less_eq @ A @ ( plus_plus @ A @ C2 @ A2 ) @ B2 ) ) ) ) ).
% ordered_cancel_comm_monoid_diff_class.le_diff_conv2
thf(fact_577_ordered__cancel__comm__monoid__diff__class_Oadd__diff__assoc,axiom,
! [A: $tType] :
( ( ordere1170586879665033532d_diff @ A )
=> ! [A2: A,B2: A,C2: A] :
( ( ord_less_eq @ A @ A2 @ B2 )
=> ( ( plus_plus @ A @ C2 @ ( minus_minus @ A @ B2 @ A2 ) )
= ( minus_minus @ A @ ( plus_plus @ A @ C2 @ B2 ) @ A2 ) ) ) ) ).
% ordered_cancel_comm_monoid_diff_class.add_diff_assoc
thf(fact_578_ordered__cancel__comm__monoid__diff__class_Odiff__add__assoc,axiom,
! [A: $tType] :
( ( ordere1170586879665033532d_diff @ A )
=> ! [A2: A,B2: A,C2: A] :
( ( ord_less_eq @ A @ A2 @ B2 )
=> ( ( minus_minus @ A @ ( plus_plus @ A @ C2 @ B2 ) @ A2 )
= ( plus_plus @ A @ C2 @ ( minus_minus @ A @ B2 @ A2 ) ) ) ) ) ).
% ordered_cancel_comm_monoid_diff_class.diff_add_assoc
thf(fact_579_ordered__cancel__comm__monoid__diff__class_Oadd__diff__assoc2,axiom,
! [A: $tType] :
( ( ordere1170586879665033532d_diff @ A )
=> ! [A2: A,B2: A,C2: A] :
( ( ord_less_eq @ A @ A2 @ B2 )
=> ( ( plus_plus @ A @ ( minus_minus @ A @ B2 @ A2 ) @ C2 )
= ( minus_minus @ A @ ( plus_plus @ A @ B2 @ C2 ) @ A2 ) ) ) ) ).
% ordered_cancel_comm_monoid_diff_class.add_diff_assoc2
thf(fact_580_ordered__cancel__comm__monoid__diff__class_Odiff__add__assoc2,axiom,
! [A: $tType] :
( ( ordere1170586879665033532d_diff @ A )
=> ! [A2: A,B2: A,C2: A] :
( ( ord_less_eq @ A @ A2 @ B2 )
=> ( ( minus_minus @ A @ ( plus_plus @ A @ B2 @ C2 ) @ A2 )
= ( plus_plus @ A @ ( minus_minus @ A @ B2 @ A2 ) @ C2 ) ) ) ) ).
% ordered_cancel_comm_monoid_diff_class.diff_add_assoc2
thf(fact_581_ordered__cancel__comm__monoid__diff__class_Odiff__diff__right,axiom,
! [A: $tType] :
( ( ordere1170586879665033532d_diff @ A )
=> ! [A2: A,B2: A,C2: A] :
( ( ord_less_eq @ A @ A2 @ B2 )
=> ( ( minus_minus @ A @ C2 @ ( minus_minus @ A @ B2 @ A2 ) )
= ( minus_minus @ A @ ( plus_plus @ A @ C2 @ A2 ) @ B2 ) ) ) ) ).
% ordered_cancel_comm_monoid_diff_class.diff_diff_right
thf(fact_582_ordered__cancel__comm__monoid__diff__class_Oadd__diff__inverse,axiom,
! [A: $tType] :
( ( ordere1170586879665033532d_diff @ A )
=> ! [A2: A,B2: A] :
( ( ord_less_eq @ A @ A2 @ B2 )
=> ( ( plus_plus @ A @ A2 @ ( minus_minus @ A @ B2 @ A2 ) )
= B2 ) ) ) ).
% ordered_cancel_comm_monoid_diff_class.add_diff_inverse
thf(fact_583_ordered__cancel__comm__monoid__diff__class_Ole__imp__diff__is__add,axiom,
! [A: $tType] :
( ( ordere1170586879665033532d_diff @ A )
=> ! [A2: A,B2: A,C2: A] :
( ( ord_less_eq @ A @ A2 @ B2 )
=> ( ( ord_less_eq @ A @ A2 @ B2 )
=> ( ( ( minus_minus @ A @ B2 @ A2 )
= C2 )
= ( B2
= ( plus_plus @ A @ C2 @ A2 ) ) ) ) ) ) ).
% ordered_cancel_comm_monoid_diff_class.le_imp_diff_is_add
thf(fact_584_less__diff__eq,axiom,
! [A: $tType] :
( ( ordered_ab_group_add @ A )
=> ! [A2: A,C2: A,B2: A] :
( ( ord_less @ A @ A2 @ ( minus_minus @ A @ C2 @ B2 ) )
= ( ord_less @ A @ ( plus_plus @ A @ A2 @ B2 ) @ C2 ) ) ) ).
% less_diff_eq
thf(fact_585_diff__less__eq,axiom,
! [A: $tType] :
( ( ordered_ab_group_add @ A )
=> ! [A2: A,B2: A,C2: A] :
( ( ord_less @ A @ ( minus_minus @ A @ A2 @ B2 ) @ C2 )
= ( ord_less @ A @ A2 @ ( plus_plus @ A @ C2 @ B2 ) ) ) ) ).
% diff_less_eq
thf(fact_586_less__half__sum,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [A2: A,B2: A] :
( ( ord_less @ A @ A2 @ B2 )
=> ( ord_less @ A @ A2 @ ( divide_divide @ A @ ( plus_plus @ A @ A2 @ B2 ) @ ( plus_plus @ A @ ( one_one @ A ) @ ( one_one @ A ) ) ) ) ) ) ).
% less_half_sum
thf(fact_587_gt__half__sum,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [A2: A,B2: A] :
( ( ord_less @ A @ A2 @ B2 )
=> ( ord_less @ A @ ( divide_divide @ A @ ( plus_plus @ A @ A2 @ B2 ) @ ( plus_plus @ A @ ( one_one @ A ) @ ( one_one @ A ) ) ) @ B2 ) ) ) ).
% gt_half_sum
thf(fact_588_count__buildup,axiom,
! [N3: nat] : ( ord_less_eq @ real @ ( vEBT_VEBT_cnt @ ( vEBT_vebt_buildup @ N3 ) ) @ ( times_times @ real @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) @ ( power_power @ real @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) @ N3 ) ) ) ).
% count_buildup
thf(fact_589_cnt__bound,axiom,
! [T2: vEBT_VEBT,N3: nat] :
( ( vEBT_invar_vebt @ T2 @ N3 )
=> ( 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 ) ) @ N3 ) @ ( divide_divide @ real @ ( numeral_numeral @ real @ ( bit1 @ ( bit1 @ ( bit1 @ one2 ) ) ) ) @ ( numeral_numeral @ real @ ( bit0 @ ( bit1 @ ( bit0 @ one2 ) ) ) ) ) ) ) ) ) ).
% cnt_bound
thf(fact_590_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_591_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_592_divmod__step__eq,axiom,
! [A: $tType] :
( ( unique1627219031080169319umeral @ A )
=> ! [L2: num,R2: A,Q2: A] :
( ( ( ord_less_eq @ A @ ( numeral_numeral @ A @ L2 ) @ R2 )
=> ( ( unique1321980374590559556d_step @ A @ L2 @ ( product_Pair @ A @ A @ Q2 @ R2 ) )
= ( product_Pair @ A @ A @ ( plus_plus @ A @ ( times_times @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ Q2 ) @ ( one_one @ A ) ) @ ( minus_minus @ A @ R2 @ ( numeral_numeral @ A @ L2 ) ) ) ) )
& ( ~ ( ord_less_eq @ A @ ( numeral_numeral @ A @ L2 ) @ R2 )
=> ( ( unique1321980374590559556d_step @ A @ L2 @ ( product_Pair @ A @ A @ Q2 @ R2 ) )
= ( product_Pair @ A @ A @ ( times_times @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ Q2 ) @ R2 ) ) ) ) ) ).
% divmod_step_eq
thf(fact_593_VEBT__internal_OT_092_060_094sub_062d_092_060_094sub_062e_092_060_094sub_062l_092_060_094sub_062e_092_060_094sub_062t_092_060_094sub_062e_H_Osimps_I4_J,axiom,
! [Deg: nat,TreeList: list @ vEBT_VEBT,Summary: vEBT_VEBT,Uu2: nat] :
( ( vEBT_V1232361888498592333_e_t_e @ ( vEBT_Node @ ( none @ ( product_prod @ nat @ nat ) ) @ Deg @ TreeList @ Summary ) @ Uu2 )
= ( one_one @ nat ) ) ).
% VEBT_internal.T\<^sub>d\<^sub>e\<^sub>l\<^sub>e\<^sub>t\<^sub>e'.simps(4)
thf(fact_594_two__realpow__ge__two,axiom,
! [N3: nat] :
( ( ord_less_eq @ real @ ( one_one @ real ) @ ( semiring_1_of_nat @ real @ N3 ) )
=> ( ord_less_eq @ real @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) @ ( power_power @ real @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) @ N3 ) ) ) ).
% two_realpow_ge_two
thf(fact_595_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_596_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_597_minNull__delete__time__bound_H,axiom,
! [T2: vEBT_VEBT,N3: nat,X: nat] :
( ( vEBT_invar_vebt @ T2 @ N3 )
=> ( ( vEBT_VEBT_minNull @ ( vEBT_vebt_delete @ T2 @ X ) )
=> ( ord_less_eq @ nat @ ( vEBT_V1232361888498592333_e_t_e @ T2 @ X ) @ ( one_one @ nat ) ) ) ) ).
% minNull_delete_time_bound'
thf(fact_598_not__min__Null__member,axiom,
! [T2: vEBT_VEBT] :
( ~ ( vEBT_VEBT_minNull @ T2 )
=> ? [X_12: nat] : ( vEBT_V8194947554948674370ptions @ T2 @ X_12 ) ) ).
% not_min_Null_member
thf(fact_599_min__Null__member,axiom,
! [T2: vEBT_VEBT,X: nat] :
( ( vEBT_VEBT_minNull @ T2 )
=> ~ ( vEBT_vebt_member @ T2 @ X ) ) ).
% min_Null_member
thf(fact_600_minNullmin,axiom,
! [T2: vEBT_VEBT] :
( ( vEBT_VEBT_minNull @ T2 )
=> ( ( vEBT_vebt_mint @ T2 )
= ( none @ nat ) ) ) ).
% minNullmin
thf(fact_601_minminNull,axiom,
! [T2: vEBT_VEBT] :
( ( ( vEBT_vebt_mint @ T2 )
= ( none @ nat ) )
=> ( vEBT_VEBT_minNull @ T2 ) ) ).
% minminNull
thf(fact_602_semiring__norm_I90_J,axiom,
! [M: num,N3: num] :
( ( ( bit1 @ M )
= ( bit1 @ N3 ) )
= ( M = N3 ) ) ).
% semiring_norm(90)
thf(fact_603_of__nat__eq__iff,axiom,
! [A: $tType] :
( ( semiring_char_0 @ A )
=> ! [M: nat,N3: nat] :
( ( ( semiring_1_of_nat @ A @ M )
= ( semiring_1_of_nat @ A @ N3 ) )
= ( M = N3 ) ) ) ).
% of_nat_eq_iff
thf(fact_604_semiring__norm_I89_J,axiom,
! [M: num,N3: num] :
( ( bit1 @ M )
!= ( bit0 @ N3 ) ) ).
% semiring_norm(89)
thf(fact_605_semiring__norm_I88_J,axiom,
! [M: num,N3: num] :
( ( bit0 @ M )
!= ( bit1 @ N3 ) ) ).
% semiring_norm(88)
thf(fact_606_semiring__norm_I86_J,axiom,
! [M: num] :
( ( bit1 @ M )
!= one2 ) ).
% semiring_norm(86)
thf(fact_607_semiring__norm_I84_J,axiom,
! [N3: num] :
( one2
!= ( bit1 @ N3 ) ) ).
% semiring_norm(84)
thf(fact_608_semiring__norm_I80_J,axiom,
! [M: num,N3: num] :
( ( ord_less @ num @ ( bit1 @ M ) @ ( bit1 @ N3 ) )
= ( ord_less @ num @ M @ N3 ) ) ).
% semiring_norm(80)
thf(fact_609_semiring__norm_I73_J,axiom,
! [M: num,N3: num] :
( ( ord_less_eq @ num @ ( bit1 @ M ) @ ( bit1 @ N3 ) )
= ( ord_less_eq @ num @ M @ N3 ) ) ).
% semiring_norm(73)
thf(fact_610_count__buildup_H,axiom,
! [N3: nat] : ( ord_less_eq @ real @ ( vEBT_VEBT_cnt @ ( vEBT_vebt_buildup @ N3 ) ) @ ( semiring_1_of_nat @ real @ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N3 ) ) ) ) ).
% count_buildup'
thf(fact_611_of__nat__numeral,axiom,
! [A: $tType] :
( ( semiring_1 @ A )
=> ! [N3: num] :
( ( semiring_1_of_nat @ A @ ( numeral_numeral @ nat @ N3 ) )
= ( numeral_numeral @ A @ N3 ) ) ) ).
% of_nat_numeral
thf(fact_612_of__nat__less__iff,axiom,
! [A: $tType] :
( ( linord181362715937106298miring @ A )
=> ! [M: nat,N3: nat] :
( ( ord_less @ A @ ( semiring_1_of_nat @ A @ M ) @ ( semiring_1_of_nat @ A @ N3 ) )
= ( ord_less @ nat @ M @ N3 ) ) ) ).
% of_nat_less_iff
thf(fact_613_of__nat__le__iff,axiom,
! [A: $tType] :
( ( linord181362715937106298miring @ A )
=> ! [M: nat,N3: nat] :
( ( ord_less_eq @ A @ ( semiring_1_of_nat @ A @ M ) @ ( semiring_1_of_nat @ A @ N3 ) )
= ( ord_less_eq @ nat @ M @ N3 ) ) ) ).
% of_nat_le_iff
thf(fact_614_of__nat__add,axiom,
! [A: $tType] :
( ( semiring_1 @ A )
=> ! [M: nat,N3: nat] :
( ( semiring_1_of_nat @ A @ ( plus_plus @ nat @ M @ N3 ) )
= ( plus_plus @ A @ ( semiring_1_of_nat @ A @ M ) @ ( semiring_1_of_nat @ A @ N3 ) ) ) ) ).
% of_nat_add
thf(fact_615_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_616_of__nat__1__eq__iff,axiom,
! [A: $tType] :
( ( semiring_char_0 @ A )
=> ! [N3: nat] :
( ( ( one_one @ A )
= ( semiring_1_of_nat @ A @ N3 ) )
= ( N3
= ( one_one @ nat ) ) ) ) ).
% of_nat_1_eq_iff
thf(fact_617_of__nat__eq__1__iff,axiom,
! [A: $tType] :
( ( semiring_char_0 @ A )
=> ! [N3: nat] :
( ( ( semiring_1_of_nat @ A @ N3 )
= ( one_one @ A ) )
= ( N3
= ( one_one @ nat ) ) ) ) ).
% of_nat_eq_1_iff
thf(fact_618_of__nat__mult,axiom,
! [A: $tType] :
( ( semiring_1 @ A )
=> ! [M: nat,N3: nat] :
( ( semiring_1_of_nat @ A @ ( times_times @ nat @ M @ N3 ) )
= ( times_times @ A @ ( semiring_1_of_nat @ A @ M ) @ ( semiring_1_of_nat @ A @ N3 ) ) ) ) ).
% of_nat_mult
thf(fact_619_semiring__1__class_Oof__nat__power,axiom,
! [A: $tType] :
( ( semiring_1 @ A )
=> ! [M: nat,N3: nat] :
( ( semiring_1_of_nat @ A @ ( power_power @ nat @ M @ N3 ) )
= ( power_power @ A @ ( semiring_1_of_nat @ A @ M ) @ N3 ) ) ) ).
% semiring_1_class.of_nat_power
thf(fact_620_of__nat__eq__of__nat__power__cancel__iff,axiom,
! [A: $tType] :
( ( semiring_char_0 @ A )
=> ! [B2: nat,W: nat,X: nat] :
( ( ( power_power @ A @ ( semiring_1_of_nat @ A @ B2 ) @ W )
= ( semiring_1_of_nat @ A @ X ) )
= ( ( power_power @ nat @ B2 @ W )
= X ) ) ) ).
% of_nat_eq_of_nat_power_cancel_iff
thf(fact_621_of__nat__power__eq__of__nat__cancel__iff,axiom,
! [A: $tType] :
( ( semiring_char_0 @ A )
=> ! [X: nat,B2: nat,W: nat] :
( ( ( semiring_1_of_nat @ A @ X )
= ( power_power @ A @ ( semiring_1_of_nat @ A @ B2 ) @ W ) )
= ( X
= ( power_power @ nat @ B2 @ W ) ) ) ) ).
% of_nat_power_eq_of_nat_cancel_iff
thf(fact_622_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_623_semiring__norm_I7_J,axiom,
! [M: num,N3: num] :
( ( plus_plus @ num @ ( bit0 @ M ) @ ( bit1 @ N3 ) )
= ( bit1 @ ( plus_plus @ num @ M @ N3 ) ) ) ).
% semiring_norm(7)
thf(fact_624_semiring__norm_I9_J,axiom,
! [M: num,N3: num] :
( ( plus_plus @ num @ ( bit1 @ M ) @ ( bit0 @ N3 ) )
= ( bit1 @ ( plus_plus @ num @ M @ N3 ) ) ) ).
% semiring_norm(9)
thf(fact_625_semiring__norm_I14_J,axiom,
! [M: num,N3: num] :
( ( times_times @ num @ ( bit0 @ M ) @ ( bit1 @ N3 ) )
= ( bit0 @ ( times_times @ num @ M @ ( bit1 @ N3 ) ) ) ) ).
% semiring_norm(14)
thf(fact_626_semiring__norm_I15_J,axiom,
! [M: num,N3: num] :
( ( times_times @ num @ ( bit1 @ M ) @ ( bit0 @ N3 ) )
= ( bit0 @ ( times_times @ num @ ( bit1 @ M ) @ N3 ) ) ) ).
% semiring_norm(15)
thf(fact_627_semiring__norm_I72_J,axiom,
! [M: num,N3: num] :
( ( ord_less_eq @ num @ ( bit0 @ M ) @ ( bit1 @ N3 ) )
= ( ord_less_eq @ num @ M @ N3 ) ) ).
% semiring_norm(72)
thf(fact_628_semiring__norm_I81_J,axiom,
! [M: num,N3: num] :
( ( ord_less @ num @ ( bit1 @ M ) @ ( bit0 @ N3 ) )
= ( ord_less @ num @ M @ N3 ) ) ).
% semiring_norm(81)
thf(fact_629_semiring__norm_I70_J,axiom,
! [M: num] :
~ ( ord_less_eq @ num @ ( bit1 @ M ) @ one2 ) ).
% semiring_norm(70)
thf(fact_630_semiring__norm_I77_J,axiom,
! [N3: num] : ( ord_less @ num @ one2 @ ( bit1 @ N3 ) ) ).
% semiring_norm(77)
thf(fact_631_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_632_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_633_semiring__norm_I10_J,axiom,
! [M: num,N3: num] :
( ( plus_plus @ num @ ( bit1 @ M ) @ ( bit1 @ N3 ) )
= ( bit0 @ ( plus_plus @ num @ ( plus_plus @ num @ M @ N3 ) @ one2 ) ) ) ).
% semiring_norm(10)
thf(fact_634_semiring__norm_I8_J,axiom,
! [M: num] :
( ( plus_plus @ num @ ( bit1 @ M ) @ one2 )
= ( bit0 @ ( plus_plus @ num @ M @ one2 ) ) ) ).
% semiring_norm(8)
thf(fact_635_semiring__norm_I5_J,axiom,
! [M: num] :
( ( plus_plus @ num @ ( bit0 @ M ) @ one2 )
= ( bit1 @ M ) ) ).
% semiring_norm(5)
thf(fact_636_semiring__norm_I4_J,axiom,
! [N3: num] :
( ( plus_plus @ num @ one2 @ ( bit1 @ N3 ) )
= ( bit0 @ ( plus_plus @ num @ N3 @ one2 ) ) ) ).
% semiring_norm(4)
thf(fact_637_semiring__norm_I3_J,axiom,
! [N3: num] :
( ( plus_plus @ num @ one2 @ ( bit0 @ N3 ) )
= ( bit1 @ N3 ) ) ).
% semiring_norm(3)
thf(fact_638_semiring__norm_I16_J,axiom,
! [M: num,N3: num] :
( ( times_times @ num @ ( bit1 @ M ) @ ( bit1 @ N3 ) )
= ( bit1 @ ( plus_plus @ num @ ( plus_plus @ num @ M @ N3 ) @ ( bit0 @ ( times_times @ num @ M @ N3 ) ) ) ) ) ).
% semiring_norm(16)
thf(fact_639_semiring__norm_I79_J,axiom,
! [M: num,N3: num] :
( ( ord_less @ num @ ( bit0 @ M ) @ ( bit1 @ N3 ) )
= ( ord_less_eq @ num @ M @ N3 ) ) ).
% semiring_norm(79)
thf(fact_640_semiring__norm_I74_J,axiom,
! [M: num,N3: num] :
( ( ord_less_eq @ num @ ( bit1 @ M ) @ ( bit0 @ N3 ) )
= ( ord_less @ num @ M @ N3 ) ) ).
% semiring_norm(74)
thf(fact_641_numeral__power__eq__of__nat__cancel__iff,axiom,
! [A: $tType] :
( ( semiring_char_0 @ A )
=> ! [X: num,N3: nat,Y: nat] :
( ( ( power_power @ A @ ( numeral_numeral @ A @ X ) @ N3 )
= ( semiring_1_of_nat @ A @ Y ) )
= ( ( power_power @ nat @ ( numeral_numeral @ nat @ X ) @ N3 )
= Y ) ) ) ).
% numeral_power_eq_of_nat_cancel_iff
thf(fact_642_real__of__nat__eq__numeral__power__cancel__iff,axiom,
! [A: $tType] :
( ( semiring_char_0 @ A )
=> ! [Y: nat,X: num,N3: nat] :
( ( ( semiring_1_of_nat @ A @ Y )
= ( power_power @ A @ ( numeral_numeral @ A @ X ) @ N3 ) )
= ( Y
= ( power_power @ nat @ ( numeral_numeral @ nat @ X ) @ N3 ) ) ) ) ).
% real_of_nat_eq_numeral_power_cancel_iff
thf(fact_643_of__nat__less__of__nat__power__cancel__iff,axiom,
! [A: $tType] :
( ( linordered_semidom @ A )
=> ! [B2: nat,W: nat,X: nat] :
( ( ord_less @ A @ ( power_power @ A @ ( semiring_1_of_nat @ A @ B2 ) @ W ) @ ( semiring_1_of_nat @ A @ X ) )
= ( ord_less @ nat @ ( power_power @ nat @ B2 @ W ) @ X ) ) ) ).
% of_nat_less_of_nat_power_cancel_iff
thf(fact_644_of__nat__power__less__of__nat__cancel__iff,axiom,
! [A: $tType] :
( ( linordered_semidom @ A )
=> ! [X: nat,B2: nat,W: nat] :
( ( ord_less @ A @ ( semiring_1_of_nat @ A @ X ) @ ( power_power @ A @ ( semiring_1_of_nat @ A @ B2 ) @ W ) )
= ( ord_less @ nat @ X @ ( power_power @ nat @ B2 @ W ) ) ) ) ).
% of_nat_power_less_of_nat_cancel_iff
thf(fact_645_of__nat__le__of__nat__power__cancel__iff,axiom,
! [A: $tType] :
( ( linordered_semidom @ A )
=> ! [B2: nat,W: nat,X: nat] :
( ( ord_less_eq @ A @ ( power_power @ A @ ( semiring_1_of_nat @ A @ B2 ) @ W ) @ ( semiring_1_of_nat @ A @ X ) )
= ( ord_less_eq @ nat @ ( power_power @ nat @ B2 @ W ) @ X ) ) ) ).
% of_nat_le_of_nat_power_cancel_iff
thf(fact_646_of__nat__power__le__of__nat__cancel__iff,axiom,
! [A: $tType] :
( ( linordered_semidom @ A )
=> ! [X: nat,B2: nat,W: nat] :
( ( ord_less_eq @ A @ ( semiring_1_of_nat @ A @ X ) @ ( power_power @ A @ ( semiring_1_of_nat @ A @ B2 ) @ W ) )
= ( ord_less_eq @ nat @ X @ ( power_power @ nat @ B2 @ W ) ) ) ) ).
% of_nat_power_le_of_nat_cancel_iff
thf(fact_647_numeral__less__real__of__nat__iff,axiom,
! [W: num,N3: nat] :
( ( ord_less @ real @ ( numeral_numeral @ real @ W ) @ ( semiring_1_of_nat @ real @ N3 ) )
= ( ord_less @ nat @ ( numeral_numeral @ nat @ W ) @ N3 ) ) ).
% numeral_less_real_of_nat_iff
thf(fact_648_real__of__nat__less__numeral__iff,axiom,
! [N3: nat,W: num] :
( ( ord_less @ real @ ( semiring_1_of_nat @ real @ N3 ) @ ( numeral_numeral @ real @ W ) )
= ( ord_less @ nat @ N3 @ ( numeral_numeral @ nat @ W ) ) ) ).
% real_of_nat_less_numeral_iff
thf(fact_649_numeral__le__real__of__nat__iff,axiom,
! [N3: num,M: nat] :
( ( ord_less_eq @ real @ ( numeral_numeral @ real @ N3 ) @ ( semiring_1_of_nat @ real @ M ) )
= ( ord_less_eq @ nat @ ( numeral_numeral @ nat @ N3 ) @ M ) ) ).
% numeral_le_real_of_nat_iff
thf(fact_650_div__Suc__eq__div__add3,axiom,
! [M: nat,N3: nat] :
( ( divide_divide @ nat @ M @ ( suc @ ( suc @ ( suc @ N3 ) ) ) )
= ( divide_divide @ nat @ M @ ( plus_plus @ nat @ ( numeral_numeral @ nat @ ( bit1 @ one2 ) ) @ N3 ) ) ) ).
% div_Suc_eq_div_add3
thf(fact_651_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_652_numeral__power__less__of__nat__cancel__iff,axiom,
! [A: $tType] :
( ( linordered_semidom @ A )
=> ! [I: num,N3: nat,X: nat] :
( ( ord_less @ A @ ( power_power @ A @ ( numeral_numeral @ A @ I ) @ N3 ) @ ( semiring_1_of_nat @ A @ X ) )
= ( ord_less @ nat @ ( power_power @ nat @ ( numeral_numeral @ nat @ I ) @ N3 ) @ X ) ) ) ).
% numeral_power_less_of_nat_cancel_iff
thf(fact_653_of__nat__less__numeral__power__cancel__iff,axiom,
! [A: $tType] :
( ( linordered_semidom @ A )
=> ! [X: nat,I: num,N3: nat] :
( ( ord_less @ A @ ( semiring_1_of_nat @ A @ X ) @ ( power_power @ A @ ( numeral_numeral @ A @ I ) @ N3 ) )
= ( ord_less @ nat @ X @ ( power_power @ nat @ ( numeral_numeral @ nat @ I ) @ N3 ) ) ) ) ).
% of_nat_less_numeral_power_cancel_iff
thf(fact_654_numeral__power__le__of__nat__cancel__iff,axiom,
! [A: $tType] :
( ( linordered_semidom @ A )
=> ! [I: num,N3: nat,X: nat] :
( ( ord_less_eq @ A @ ( power_power @ A @ ( numeral_numeral @ A @ I ) @ N3 ) @ ( semiring_1_of_nat @ A @ X ) )
= ( ord_less_eq @ nat @ ( power_power @ nat @ ( numeral_numeral @ nat @ I ) @ N3 ) @ X ) ) ) ).
% numeral_power_le_of_nat_cancel_iff
thf(fact_655_of__nat__le__numeral__power__cancel__iff,axiom,
! [A: $tType] :
( ( linordered_semidom @ A )
=> ! [X: nat,I: num,N3: nat] :
( ( ord_less_eq @ A @ ( semiring_1_of_nat @ A @ X ) @ ( power_power @ A @ ( numeral_numeral @ A @ I ) @ N3 ) )
= ( ord_less_eq @ nat @ X @ ( power_power @ nat @ ( numeral_numeral @ nat @ I ) @ N3 ) ) ) ) ).
% of_nat_le_numeral_power_cancel_iff
thf(fact_656_mult__of__nat__commute,axiom,
! [A: $tType] :
( ( semiring_1 @ A )
=> ! [X: nat,Y: A] :
( ( times_times @ A @ ( semiring_1_of_nat @ A @ X ) @ Y )
= ( times_times @ A @ Y @ ( semiring_1_of_nat @ A @ X ) ) ) ) ).
% mult_of_nat_commute
thf(fact_657_less__imp__of__nat__less,axiom,
! [A: $tType] :
( ( linord181362715937106298miring @ A )
=> ! [M: nat,N3: nat] :
( ( ord_less @ nat @ M @ N3 )
=> ( ord_less @ A @ ( semiring_1_of_nat @ A @ M ) @ ( semiring_1_of_nat @ A @ N3 ) ) ) ) ).
% less_imp_of_nat_less
thf(fact_658_of__nat__less__imp__less,axiom,
! [A: $tType] :
( ( linord181362715937106298miring @ A )
=> ! [M: nat,N3: nat] :
( ( ord_less @ A @ ( semiring_1_of_nat @ A @ M ) @ ( semiring_1_of_nat @ A @ N3 ) )
=> ( ord_less @ nat @ M @ N3 ) ) ) ).
% of_nat_less_imp_less
thf(fact_659_div__mult2__eq_H,axiom,
! [A: $tType] :
( ( euclid5411537665997757685th_nat @ A )
=> ! [A2: A,M: nat,N3: nat] :
( ( divide_divide @ A @ A2 @ ( times_times @ A @ ( semiring_1_of_nat @ A @ M ) @ ( semiring_1_of_nat @ A @ N3 ) ) )
= ( divide_divide @ A @ ( divide_divide @ A @ A2 @ ( semiring_1_of_nat @ A @ M ) ) @ ( semiring_1_of_nat @ A @ N3 ) ) ) ) ).
% div_mult2_eq'
thf(fact_660_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_661_unique__euclidean__semiring__with__nat__class_Oof__nat__div,axiom,
! [A: $tType] :
( ( euclid5411537665997757685th_nat @ A )
=> ! [M: nat,N3: nat] :
( ( semiring_1_of_nat @ A @ ( divide_divide @ nat @ M @ N3 ) )
= ( divide_divide @ A @ ( semiring_1_of_nat @ A @ M ) @ ( semiring_1_of_nat @ A @ N3 ) ) ) ) ).
% unique_euclidean_semiring_with_nat_class.of_nat_div
thf(fact_662_xor__num_Ocases,axiom,
! [X: product_prod @ num @ num] :
( ( X
!= ( product_Pair @ num @ num @ one2 @ one2 ) )
=> ( ! [N: num] :
( X
!= ( product_Pair @ num @ num @ one2 @ ( bit0 @ N ) ) )
=> ( ! [N: num] :
( X
!= ( product_Pair @ num @ num @ one2 @ ( bit1 @ N ) ) )
=> ( ! [M4: num] :
( X
!= ( product_Pair @ num @ num @ ( bit0 @ M4 ) @ one2 ) )
=> ( ! [M4: num,N: num] :
( X
!= ( product_Pair @ num @ num @ ( bit0 @ M4 ) @ ( bit0 @ N ) ) )
=> ( ! [M4: num,N: num] :
( X
!= ( product_Pair @ num @ num @ ( bit0 @ M4 ) @ ( bit1 @ N ) ) )
=> ( ! [M4: num] :
( X
!= ( product_Pair @ num @ num @ ( bit1 @ M4 ) @ one2 ) )
=> ( ! [M4: num,N: num] :
( X
!= ( product_Pair @ num @ num @ ( bit1 @ M4 ) @ ( bit0 @ N ) ) )
=> ~ ! [M4: num,N: num] :
( X
!= ( product_Pair @ num @ num @ ( bit1 @ M4 ) @ ( bit1 @ N ) ) ) ) ) ) ) ) ) ) ) ).
% xor_num.cases
thf(fact_663_num_Oexhaust,axiom,
! [Y: num] :
( ( Y != one2 )
=> ( ! [X23: num] :
( Y
!= ( bit0 @ X23 ) )
=> ~ ! [X32: num] :
( Y
!= ( bit1 @ X32 ) ) ) ) ).
% num.exhaust
thf(fact_664_of__nat__diff,axiom,
! [A: $tType] :
( ( semiring_1_cancel @ A )
=> ! [N3: nat,M: nat] :
( ( ord_less_eq @ nat @ N3 @ M )
=> ( ( semiring_1_of_nat @ A @ ( minus_minus @ nat @ M @ N3 ) )
= ( minus_minus @ A @ ( semiring_1_of_nat @ A @ M ) @ ( semiring_1_of_nat @ A @ N3 ) ) ) ) ) ).
% of_nat_diff
thf(fact_665_numeral__Bit1,axiom,
! [A: $tType] :
( ( numeral @ A )
=> ! [N3: num] :
( ( numeral_numeral @ A @ ( bit1 @ N3 ) )
= ( plus_plus @ A @ ( plus_plus @ A @ ( numeral_numeral @ A @ N3 ) @ ( numeral_numeral @ A @ N3 ) ) @ ( one_one @ A ) ) ) ) ).
% numeral_Bit1
thf(fact_666_eval__nat__numeral_I3_J,axiom,
! [N3: num] :
( ( numeral_numeral @ nat @ ( bit1 @ N3 ) )
= ( suc @ ( numeral_numeral @ nat @ ( bit0 @ N3 ) ) ) ) ).
% eval_nat_numeral(3)
thf(fact_667_real__of__nat__div4,axiom,
! [N3: nat,X: nat] : ( ord_less_eq @ real @ ( semiring_1_of_nat @ real @ ( divide_divide @ nat @ N3 @ X ) ) @ ( divide_divide @ real @ ( semiring_1_of_nat @ real @ N3 ) @ ( semiring_1_of_nat @ real @ X ) ) ) ).
% real_of_nat_div4
thf(fact_668_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_669_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_670_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_671_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_672_numeral__Bit1__div__2,axiom,
! [A: $tType] :
( ( euclid5411537665997757685th_nat @ A )
=> ! [N3: num] :
( ( divide_divide @ A @ ( numeral_numeral @ A @ ( bit1 @ N3 ) ) @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) )
= ( numeral_numeral @ A @ N3 ) ) ) ).
% numeral_Bit1_div_2
thf(fact_673_power3__eq__cube,axiom,
! [A: $tType] :
( ( monoid_mult @ A )
=> ! [A2: A] :
( ( power_power @ A @ A2 @ ( numeral_numeral @ nat @ ( bit1 @ one2 ) ) )
= ( times_times @ A @ ( times_times @ A @ A2 @ A2 ) @ A2 ) ) ) ).
% power3_eq_cube
thf(fact_674_Suc3__eq__add__3,axiom,
! [N3: nat] :
( ( suc @ ( suc @ ( suc @ N3 ) ) )
= ( plus_plus @ nat @ ( numeral_numeral @ nat @ ( bit1 @ one2 ) ) @ N3 ) ) ).
% Suc3_eq_add_3
thf(fact_675_nat__less__real__le,axiom,
( ( ord_less @ nat )
= ( ^ [N2: nat,M5: nat] : ( ord_less_eq @ real @ ( plus_plus @ real @ ( semiring_1_of_nat @ real @ N2 ) @ ( one_one @ real ) ) @ ( semiring_1_of_nat @ real @ M5 ) ) ) ) ).
% nat_less_real_le
thf(fact_676_nat__le__real__less,axiom,
( ( ord_less_eq @ nat )
= ( ^ [N2: nat,M5: nat] : ( ord_less @ real @ ( semiring_1_of_nat @ real @ N2 ) @ ( plus_plus @ real @ ( semiring_1_of_nat @ real @ M5 ) @ ( one_one @ real ) ) ) ) ) ).
% nat_le_real_less
thf(fact_677_of__nat__less__two__power,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [N3: nat] : ( ord_less @ A @ ( semiring_1_of_nat @ A @ N3 ) @ ( power_power @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ N3 ) ) ) ).
% of_nat_less_two_power
thf(fact_678_Suc__div__eq__add3__div,axiom,
! [M: nat,N3: nat] :
( ( divide_divide @ nat @ ( suc @ ( suc @ ( suc @ M ) ) ) @ N3 )
= ( divide_divide @ nat @ ( plus_plus @ nat @ ( numeral_numeral @ nat @ ( bit1 @ one2 ) ) @ M ) @ N3 ) ) ).
% Suc_div_eq_add3_div
thf(fact_679_real__of__nat__div3,axiom,
! [N3: nat,X: nat] : ( ord_less_eq @ real @ ( minus_minus @ real @ ( divide_divide @ real @ ( semiring_1_of_nat @ real @ N3 ) @ ( semiring_1_of_nat @ real @ X ) ) @ ( semiring_1_of_nat @ real @ ( divide_divide @ nat @ N3 @ X ) ) ) @ ( one_one @ real ) ) ).
% real_of_nat_div3
thf(fact_680_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_681_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_682_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_683_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_684_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_685_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_686_small__powers__of__2,axiom,
! [X: nat] :
( ( ord_less_eq @ nat @ ( numeral_numeral @ nat @ ( bit1 @ one2 ) ) @ X )
=> ( ord_less @ nat @ X @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( minus_minus @ nat @ X @ ( one_one @ nat ) ) ) ) ) ).
% small_powers_of_2
thf(fact_687_is__pred__in__set__def,axiom,
( vEBT_is_pred_in_set
= ( ^ [Xs: set @ nat,X2: nat,Y2: nat] :
( ( member @ nat @ Y2 @ Xs )
& ( ord_less @ nat @ Y2 @ X2 )
& ! [Z3: nat] :
( ( member @ nat @ Z3 @ Xs )
=> ( ( ord_less @ nat @ Z3 @ X2 )
=> ( ord_less_eq @ nat @ Z3 @ Y2 ) ) ) ) ) ) ).
% is_pred_in_set_def
thf(fact_688_is__succ__in__set__def,axiom,
( vEBT_is_succ_in_set
= ( ^ [Xs: set @ nat,X2: nat,Y2: nat] :
( ( member @ nat @ Y2 @ Xs )
& ( ord_less @ nat @ X2 @ Y2 )
& ! [Z3: nat] :
( ( member @ nat @ Z3 @ Xs )
=> ( ( ord_less @ nat @ X2 @ Z3 )
=> ( ord_less_eq @ nat @ Y2 @ Z3 ) ) ) ) ) ) ).
% is_succ_in_set_def
thf(fact_689_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_690_space__bound,axiom,
! [T2: vEBT_VEBT,N3: nat,U: nat] :
( ( vEBT_invar_vebt @ T2 @ N3 )
=> ( ( U
= ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N3 ) )
=> ( ord_less_eq @ nat @ ( vEBT_VEBT_space @ T2 ) @ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ ( bit0 @ ( bit1 @ one2 ) ) ) ) @ U ) ) ) ) ).
% space_bound
thf(fact_691_cnt__cnt__eq,axiom,
( vEBT_VEBT_cnt
= ( ^ [T3: vEBT_VEBT] : ( semiring_1_of_nat @ real @ ( vEBT_VEBT_cnt2 @ T3 ) ) ) ) ).
% cnt_cnt_eq
thf(fact_692_space__2__pow__bound,axiom,
! [T2: vEBT_VEBT,N3: nat] :
( ( vEBT_invar_vebt @ T2 @ N3 )
=> ( 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 ) ) @ N3 ) @ ( one_one @ real ) ) ) ) ) ).
% space_2_pow_bound
thf(fact_693_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_694_space_H__bound,axiom,
! [T2: vEBT_VEBT,N3: nat,U: nat] :
( ( vEBT_invar_vebt @ T2 @ N3 )
=> ( ( U
= ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N3 ) )
=> ( ord_less_eq @ nat @ ( vEBT_VEBT_space2 @ T2 ) @ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ ( bit0 @ ( bit1 @ one2 ) ) ) ) @ U ) ) ) ) ).
% space'_bound
thf(fact_695_delete__bound__height,axiom,
! [T2: vEBT_VEBT,N3: nat,X: nat] :
( ( vEBT_invar_vebt @ T2 @ N3 )
=> ( ord_less_eq @ nat @ ( vEBT_T_d_e_l_e_t_e @ T2 @ X ) @ ( times_times @ nat @ ( plus_plus @ nat @ ( one_one @ nat ) @ ( vEBT_VEBT_height @ T2 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ ( bit1 @ ( bit1 @ ( bit0 @ ( bit0 @ ( bit0 @ one2 ) ) ) ) ) ) ) ) ) ) ).
% delete_bound_height
thf(fact_696_t__build__cnt,axiom,
! [N3: nat] : ( ord_less_eq @ real @ ( semiring_1_of_nat @ real @ ( vEBT_V8646137997579335489_i_l_d @ N3 ) ) @ ( times_times @ real @ ( vEBT_VEBT_cnt @ ( vEBT_vebt_buildup @ N3 ) ) @ ( numeral_numeral @ real @ ( bit1 @ ( bit0 @ ( bit1 @ one2 ) ) ) ) ) ) ).
% t_build_cnt
thf(fact_697_t__buildup__cnt,axiom,
! [N3: nat] : ( ord_less_eq @ real @ ( semiring_1_of_nat @ real @ ( vEBT_V8346862874174094_d_u_p @ N3 ) ) @ ( times_times @ real @ ( vEBT_VEBT_cnt @ ( vEBT_vebt_buildup @ N3 ) ) @ ( numeral_numeral @ real @ ( bit1 @ ( bit0 @ ( bit1 @ one2 ) ) ) ) ) ) ).
% t_buildup_cnt
thf(fact_698_minNull__delete__time__bound,axiom,
! [T2: vEBT_VEBT,N3: nat,X: nat] :
( ( vEBT_invar_vebt @ T2 @ N3 )
=> ( ( vEBT_VEBT_minNull @ ( vEBT_vebt_delete @ T2 @ X ) )
=> ( ord_less_eq @ nat @ ( vEBT_T_d_e_l_e_t_e @ T2 @ X ) @ ( numeral_numeral @ nat @ ( bit1 @ ( bit0 @ ( bit0 @ one2 ) ) ) ) ) ) ) ).
% minNull_delete_time_bound
thf(fact_699_tdeletemimi,axiom,
! [Deg: nat,Mi: nat,TreeList: list @ vEBT_VEBT,Summary: vEBT_VEBT,X: nat] :
( ( ord_less_eq @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ Deg )
=> ( ord_less_eq @ nat @ ( vEBT_T_d_e_l_e_t_e @ ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi @ Mi ) ) @ Deg @ TreeList @ Summary ) @ X ) @ ( numeral_numeral @ nat @ ( bit1 @ ( bit0 @ ( bit0 @ one2 ) ) ) ) ) ) ).
% tdeletemimi
thf(fact_700_buildup__build__time,axiom,
! [N3: nat] : ( ord_less @ nat @ ( vEBT_V8346862874174094_d_u_p @ N3 ) @ ( vEBT_V8646137997579335489_i_l_d @ N3 ) ) ).
% buildup_build_time
thf(fact_701_space__space_H,axiom,
! [T2: vEBT_VEBT] : ( ord_less @ nat @ ( vEBT_VEBT_space @ T2 ) @ ( vEBT_VEBT_space2 @ T2 ) ) ).
% space_space'
thf(fact_702_vebt__buildup__bound,axiom,
! [U: nat,N3: nat] :
( ( U
= ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N3 ) )
=> ( ord_less_eq @ nat @ ( vEBT_V8346862874174094_d_u_p @ N3 ) @ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ ( bit1 @ ( bit0 @ ( bit1 @ one2 ) ) ) ) ) @ U ) ) ) ).
% vebt_buildup_bound
thf(fact_703_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_704_int__diff__cases,axiom,
! [Z: int] :
~ ! [M4: nat,N: nat] :
( Z
!= ( minus_minus @ int @ ( semiring_1_of_nat @ int @ M4 ) @ ( semiring_1_of_nat @ int @ N ) ) ) ).
% int_diff_cases
thf(fact_705_zle__int,axiom,
! [M: nat,N3: nat] :
( ( ord_less_eq @ int @ ( semiring_1_of_nat @ int @ M ) @ ( semiring_1_of_nat @ int @ N3 ) )
= ( ord_less_eq @ nat @ M @ N3 ) ) ).
% zle_int
thf(fact_706_zadd__int__left,axiom,
! [M: nat,N3: nat,Z: int] :
( ( plus_plus @ int @ ( semiring_1_of_nat @ int @ M ) @ ( plus_plus @ int @ ( semiring_1_of_nat @ int @ N3 ) @ Z ) )
= ( plus_plus @ int @ ( semiring_1_of_nat @ int @ ( plus_plus @ nat @ M @ N3 ) ) @ Z ) ) ).
% zadd_int_left
thf(fact_707_zle__iff__zadd,axiom,
( ( ord_less_eq @ int )
= ( ^ [W2: int,Z3: int] :
? [N2: nat] :
( Z3
= ( plus_plus @ int @ W2 @ ( semiring_1_of_nat @ int @ N2 ) ) ) ) ) ).
% zle_iff_zadd
thf(fact_708_zdiv__int,axiom,
! [A2: nat,B2: nat] :
( ( semiring_1_of_nat @ int @ ( divide_divide @ nat @ A2 @ B2 ) )
= ( divide_divide @ int @ ( semiring_1_of_nat @ int @ A2 ) @ ( semiring_1_of_nat @ int @ B2 ) ) ) ).
% zdiv_int
thf(fact_709_zless__iff__Suc__zadd,axiom,
( ( ord_less @ int )
= ( ^ [W2: int,Z3: int] :
? [N2: nat] :
( Z3
= ( plus_plus @ int @ W2 @ ( semiring_1_of_nat @ int @ ( suc @ N2 ) ) ) ) ) ) ).
% zless_iff_Suc_zadd
thf(fact_710_T_092_060_094sub_062d_092_060_094sub_062e_092_060_094sub_062l_092_060_094sub_062e_092_060_094sub_062t_092_060_094sub_062e_Osimps_I4_J,axiom,
! [Deg: nat,TreeList: list @ vEBT_VEBT,Summary: vEBT_VEBT,Uu2: nat] :
( ( vEBT_T_d_e_l_e_t_e @ ( vEBT_Node @ ( none @ ( product_prod @ nat @ nat ) ) @ Deg @ TreeList @ Summary ) @ Uu2 )
= ( one_one @ nat ) ) ).
% T\<^sub>d\<^sub>e\<^sub>l\<^sub>e\<^sub>t\<^sub>e.simps(4)
thf(fact_711_Tb_H__cnt,axiom,
! [N3: nat] : ( ord_less_eq @ nat @ ( vEBT_VEBT_Tb2 @ N3 ) @ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit1 @ ( bit0 @ one2 ) ) ) @ ( vEBT_VEBT_cnt2 @ ( vEBT_vebt_buildup @ N3 ) ) ) ) ).
% Tb'_cnt
thf(fact_712_T__vebt__buildupi__cnt_H,axiom,
! [N3: nat] : ( ord_less_eq @ real @ ( semiring_1_of_nat @ real @ ( vEBT_V441764108873111860ildupi @ N3 ) ) @ ( times_times @ real @ ( numeral_numeral @ real @ ( bit1 @ ( bit0 @ one2 ) ) ) @ ( vEBT_VEBT_cnt @ ( vEBT_vebt_buildup @ N3 ) ) ) ) ).
% T_vebt_buildupi_cnt'
thf(fact_713_del__x__mi__lets__in__not__minNull,axiom,
! [X: nat,Mi: nat,Ma: nat,Deg: nat,Xn: nat,H2: nat,Summary: vEBT_VEBT,TreeList: list @ vEBT_VEBT,L2: nat,Newnode: vEBT_VEBT,Newlist: list @ vEBT_VEBT] :
( ( ( X = Mi )
& ( ord_less @ nat @ X @ Ma ) )
=> ( ( Mi != Ma )
=> ( ( ord_less_eq @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ Deg )
=> ( ( ( vEBT_VEBT_high @ Xn @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) )
= H2 )
=> ( ( Xn
= ( plus_plus @ nat @ ( times_times @ nat @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) )
=> ( ( ( vEBT_VEBT_low @ Xn @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) )
= L2 )
=> ( ( 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 ) @ L2 ) )
=> ( ( Newlist
= ( list_update @ vEBT_VEBT @ TreeList @ H2 @ Newnode ) )
=> ( ~ ( vEBT_VEBT_minNull @ Newnode )
=> ( ( vEBT_vebt_delete @ ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi @ Ma ) ) @ Deg @ TreeList @ Summary ) @ X )
= ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Xn @ ( if @ nat @ ( Xn = Ma ) @ ( plus_plus @ nat @ ( times_times @ nat @ H2 @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_maxt @ ( nth @ vEBT_VEBT @ Newlist @ H2 ) ) ) ) @ Ma ) ) ) @ Deg @ Newlist @ Summary ) ) ) ) ) ) ) ) ) ) ) ) ).
% del_x_mi_lets_in_not_minNull
thf(fact_714_del__x__not__mi__newnode__not__nil,axiom,
! [Mi: nat,X: nat,Ma: nat,Deg: nat,H2: nat,L2: nat,Newnode: vEBT_VEBT,TreeList: list @ vEBT_VEBT,Newlist: list @ vEBT_VEBT,Summary: vEBT_VEBT] :
( ( ( ord_less @ nat @ Mi @ X )
& ( ord_less_eq @ nat @ X @ Ma ) )
=> ( ( Mi != Ma )
=> ( ( ord_less_eq @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ Deg )
=> ( ( ( vEBT_VEBT_high @ X @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) )
= H2 )
=> ( ( ( vEBT_VEBT_low @ X @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) )
= L2 )
=> ( ( Newnode
= ( vEBT_vebt_delete @ ( nth @ vEBT_VEBT @ TreeList @ H2 ) @ L2 ) )
=> ( ~ ( vEBT_VEBT_minNull @ Newnode )
=> ( ( Newlist
= ( list_update @ vEBT_VEBT @ TreeList @ H2 @ Newnode ) )
=> ( ( ord_less @ nat @ ( vEBT_VEBT_high @ X @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( size_size @ ( list @ vEBT_VEBT ) @ TreeList ) )
=> ( ( vEBT_vebt_delete @ ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi @ Ma ) ) @ Deg @ TreeList @ Summary ) @ X )
= ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi @ ( if @ nat @ ( X = Ma ) @ ( plus_plus @ nat @ ( times_times @ nat @ H2 @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_maxt @ ( nth @ vEBT_VEBT @ Newlist @ H2 ) ) ) ) @ Ma ) ) ) @ Deg @ Newlist @ Summary ) ) ) ) ) ) ) ) ) ) ) ).
% del_x_not_mi_newnode_not_nil
thf(fact_715_T__vebt__buildupi__univ,axiom,
! [U: nat,N3: nat] :
( ( U
= ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N3 ) )
=> ( ord_less_eq @ nat @ ( vEBT_V441764108873111860ildupi @ N3 ) @ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ ( bit1 @ ( bit0 @ one2 ) ) ) ) @ U ) ) ) ).
% T_vebt_buildupi_univ
thf(fact_716_delete__bound__size__univ,axiom,
! [T2: vEBT_VEBT,N3: nat,U: real,X: nat] :
( ( vEBT_invar_vebt @ T2 @ N3 )
=> ( ( U
= ( power_power @ real @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) @ N3 ) )
=> ( ord_less_eq @ real @ ( semiring_1_of_nat @ real @ ( vEBT_T_d_e_l_e_t_e @ T2 @ X ) ) @ ( plus_plus @ real @ ( numeral_numeral @ real @ ( bit0 @ ( bit0 @ ( bit1 @ ( bit1 @ ( bit0 @ ( bit0 @ ( bit0 @ one2 ) ) ) ) ) ) ) ) @ ( times_times @ real @ ( numeral_numeral @ real @ ( bit0 @ ( bit1 @ ( bit1 @ ( bit0 @ ( bit0 @ ( bit0 @ one2 ) ) ) ) ) ) ) @ ( log @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) @ ( log @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) @ U ) ) ) ) ) ) ) ).
% delete_bound_size_univ
thf(fact_717_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_718_delete__bound__size__univ_H,axiom,
! [T2: vEBT_VEBT,N3: nat,U: real,X: nat] :
( ( vEBT_invar_vebt @ T2 @ N3 )
=> ( ( U
= ( power_power @ real @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) @ N3 ) )
=> ( ord_less_eq @ real @ ( semiring_1_of_nat @ real @ ( vEBT_V1232361888498592333_e_t_e @ T2 @ X ) ) @ ( plus_plus @ real @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) @ ( log @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) @ ( log @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) @ U ) ) ) ) ) ) ).
% delete_bound_size_univ'
thf(fact_719_member__bound__height,axiom,
! [T2: vEBT_VEBT,N3: nat,X: nat] :
( ( vEBT_invar_vebt @ T2 @ N3 )
=> ( ord_less_eq @ nat @ ( vEBT_T_m_e_m_b_e_r @ T2 @ X ) @ ( times_times @ nat @ ( plus_plus @ nat @ ( one_one @ nat ) @ ( vEBT_VEBT_height @ T2 ) ) @ ( numeral_numeral @ nat @ ( bit1 @ ( bit1 @ ( bit1 @ one2 ) ) ) ) ) ) ) ).
% member_bound_height
thf(fact_720_succ__bound__height,axiom,
! [T2: vEBT_VEBT,N3: nat,X: nat] :
( ( vEBT_invar_vebt @ T2 @ N3 )
=> ( ord_less_eq @ nat @ ( vEBT_T_s_u_c_c @ T2 @ X ) @ ( times_times @ nat @ ( plus_plus @ nat @ ( one_one @ nat ) @ ( vEBT_VEBT_height @ T2 ) ) @ ( numeral_numeral @ nat @ ( bit1 @ ( bit1 @ ( bit0 @ ( bit1 @ one2 ) ) ) ) ) ) ) ) ).
% succ_bound_height
thf(fact_721_Tb__T__vebt__buildupi_H_H,axiom,
! [N3: nat] : ( ord_less_eq @ nat @ ( vEBT_V441764108873111860ildupi @ N3 ) @ ( minus_minus @ nat @ ( vEBT_VEBT_Tb2 @ N3 ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ).
% Tb_T_vebt_buildupi''
thf(fact_722_nth__update__invalid,axiom,
! [A: $tType,I: nat,L2: list @ A,J: nat,X: A] :
( ~ ( ord_less @ nat @ I @ ( size_size @ ( list @ A ) @ L2 ) )
=> ( ( nth @ A @ ( list_update @ A @ L2 @ J @ X ) @ I )
= ( nth @ A @ L2 @ I ) ) ) ).
% nth_update_invalid
thf(fact_723_in__set__upd__eq,axiom,
! [A: $tType,I: nat,L2: list @ A,X: A,Y: A] :
( ( ord_less @ nat @ I @ ( size_size @ ( list @ A ) @ L2 ) )
=> ( ( member @ A @ X @ ( set2 @ A @ ( list_update @ A @ L2 @ I @ Y ) ) )
= ( ( X = Y )
| ( ( member @ A @ X @ ( set2 @ A @ L2 ) )
& ! [Y2: A] : ( member @ A @ X @ ( set2 @ A @ ( list_update @ A @ L2 @ I @ Y2 ) ) ) ) ) ) ) ).
% in_set_upd_eq
thf(fact_724_in__set__upd__cases,axiom,
! [A: $tType,X: A,L2: list @ A,I: nat,Y: A] :
( ( member @ A @ X @ ( set2 @ A @ ( list_update @ A @ L2 @ I @ Y ) ) )
=> ( ( ( ord_less @ nat @ I @ ( size_size @ ( list @ A ) @ L2 ) )
=> ( X != Y ) )
=> ( member @ A @ X @ ( set2 @ A @ L2 ) ) ) ) ).
% in_set_upd_cases
thf(fact_725_in__set__upd__eq__aux,axiom,
! [A: $tType,I: nat,L2: list @ A,X: A,Y: A] :
( ( ord_less @ nat @ I @ ( size_size @ ( list @ A ) @ L2 ) )
=> ( ( member @ A @ X @ ( set2 @ A @ ( list_update @ A @ L2 @ I @ Y ) ) )
= ( ( X = Y )
| ! [Y2: A] : ( member @ A @ X @ ( set2 @ A @ ( list_update @ A @ L2 @ I @ Y2 ) ) ) ) ) ) ).
% in_set_upd_eq_aux
thf(fact_726_nth__list__update_H,axiom,
! [A: $tType,I: nat,J: nat,L2: list @ A,X: A] :
( ( ( ( I = J )
& ( ord_less @ nat @ I @ ( size_size @ ( list @ A ) @ L2 ) ) )
=> ( ( nth @ A @ ( list_update @ A @ L2 @ I @ X ) @ J )
= X ) )
& ( ~ ( ( I = J )
& ( ord_less @ nat @ I @ ( size_size @ ( list @ A ) @ L2 ) ) )
=> ( ( nth @ A @ ( list_update @ A @ L2 @ I @ X ) @ J )
= ( nth @ A @ L2 @ J ) ) ) ) ).
% nth_list_update'
thf(fact_727_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_728_succ__bound__size__univ,axiom,
! [T2: vEBT_VEBT,N3: nat,U: real,X: nat] :
( ( vEBT_invar_vebt @ T2 @ N3 )
=> ( ( U
= ( power_power @ real @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) @ N3 ) )
=> ( ord_less_eq @ real @ ( semiring_1_of_nat @ real @ ( vEBT_T_s_u_c_c @ T2 @ X ) ) @ ( plus_plus @ real @ ( numeral_numeral @ real @ ( bit0 @ ( bit1 @ ( bit1 @ ( bit0 @ ( bit1 @ one2 ) ) ) ) ) ) @ ( times_times @ real @ ( numeral_numeral @ real @ ( bit1 @ ( bit1 @ ( bit0 @ ( bit1 @ one2 ) ) ) ) ) @ ( log @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) @ ( log @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) @ U ) ) ) ) ) ) ) ).
% succ_bound_size_univ
thf(fact_729_member__bound__size__univ,axiom,
! [T2: vEBT_VEBT,N3: nat,U: real,X: nat] :
( ( vEBT_invar_vebt @ T2 @ N3 )
=> ( ( U
= ( power_power @ real @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) @ N3 ) )
=> ( ord_less_eq @ real @ ( semiring_1_of_nat @ real @ ( vEBT_T_m_e_m_b_e_r @ T2 @ X ) ) @ ( plus_plus @ real @ ( numeral_numeral @ real @ ( bit0 @ ( bit1 @ ( bit1 @ ( bit1 @ one2 ) ) ) ) ) @ ( times_times @ real @ ( numeral_numeral @ real @ ( bit1 @ ( bit1 @ ( bit1 @ one2 ) ) ) ) @ ( log @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) @ ( log @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) @ U ) ) ) ) ) ) ) ).
% member_bound_size_univ
thf(fact_730_T_092_060_094sub_062m_092_060_094sub_062e_092_060_094sub_062m_092_060_094sub_062b_092_060_094sub_062e_092_060_094sub_062r_Osimps_I2_J,axiom,
! [Uu2: nat,Uv2: list @ vEBT_VEBT,Uw2: vEBT_VEBT,X: nat] :
( ( vEBT_T_m_e_m_b_e_r @ ( vEBT_Node @ ( none @ ( product_prod @ nat @ nat ) ) @ Uu2 @ Uv2 @ Uw2 ) @ X )
= ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ).
% T\<^sub>m\<^sub>e\<^sub>m\<^sub>b\<^sub>e\<^sub>r.simps(2)
thf(fact_731_Tb__T__vebt__buildupi,axiom,
! [N3: nat] : ( ord_less_eq @ int @ ( semiring_1_of_nat @ int @ ( vEBT_V441764108873111860ildupi @ N3 ) ) @ ( minus_minus @ int @ ( vEBT_VEBT_Tb @ N3 ) @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) ) ) ).
% Tb_T_vebt_buildupi
thf(fact_732_set__swap,axiom,
! [A: $tType,I: nat,Xs2: list @ A,J: nat] :
( ( ord_less @ nat @ I @ ( size_size @ ( list @ A ) @ Xs2 ) )
=> ( ( ord_less @ nat @ J @ ( size_size @ ( list @ A ) @ Xs2 ) )
=> ( ( set2 @ A @ ( list_update @ A @ ( list_update @ A @ Xs2 @ I @ ( nth @ A @ Xs2 @ J ) ) @ J @ ( nth @ A @ Xs2 @ I ) ) )
= ( set2 @ A @ Xs2 ) ) ) ) ).
% set_swap
thf(fact_733_insert__simp__excp,axiom,
! [Mi: nat,Deg: nat,TreeList: list @ vEBT_VEBT,X: nat,Ma: nat,Summary: vEBT_VEBT] :
( ( ord_less @ nat @ ( vEBT_VEBT_high @ Mi @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( size_size @ ( list @ vEBT_VEBT ) @ TreeList ) )
=> ( ( ord_less @ nat @ X @ Mi )
=> ( ( ord_less_eq @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ Deg )
=> ( ( X != Ma )
=> ( ( vEBT_vebt_insert @ ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi @ Ma ) ) @ Deg @ TreeList @ Summary ) @ X )
= ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ X @ ( ord_max @ nat @ Mi @ Ma ) ) ) @ Deg @ ( list_update @ vEBT_VEBT @ TreeList @ ( vEBT_VEBT_high @ Mi @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( vEBT_vebt_insert @ ( nth @ vEBT_VEBT @ TreeList @ ( vEBT_VEBT_high @ Mi @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( vEBT_VEBT_low @ Mi @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) @ ( if @ vEBT_VEBT @ ( vEBT_VEBT_minNull @ ( nth @ vEBT_VEBT @ TreeList @ ( vEBT_VEBT_high @ Mi @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) @ ( vEBT_vebt_insert @ Summary @ ( vEBT_VEBT_high @ Mi @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ Summary ) ) ) ) ) ) ) ).
% insert_simp_excp
thf(fact_734_insert__simp__norm,axiom,
! [X: nat,Deg: nat,TreeList: list @ vEBT_VEBT,Mi: nat,Ma: nat,Summary: vEBT_VEBT] :
( ( ord_less @ nat @ ( vEBT_VEBT_high @ X @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( size_size @ ( list @ vEBT_VEBT ) @ TreeList ) )
=> ( ( ord_less @ nat @ Mi @ X )
=> ( ( ord_less_eq @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ Deg )
=> ( ( X != Ma )
=> ( ( vEBT_vebt_insert @ ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi @ Ma ) ) @ Deg @ TreeList @ Summary ) @ X )
= ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi @ ( ord_max @ nat @ X @ Ma ) ) ) @ Deg @ ( list_update @ vEBT_VEBT @ TreeList @ ( vEBT_VEBT_high @ X @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( vEBT_vebt_insert @ ( nth @ vEBT_VEBT @ TreeList @ ( vEBT_VEBT_high @ X @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( vEBT_VEBT_low @ X @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) @ ( if @ vEBT_VEBT @ ( vEBT_VEBT_minNull @ ( nth @ vEBT_VEBT @ TreeList @ ( vEBT_VEBT_high @ X @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) @ ( vEBT_vebt_insert @ Summary @ ( vEBT_VEBT_high @ X @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ Summary ) ) ) ) ) ) ) ).
% insert_simp_norm
thf(fact_735_nth__list__update__eq,axiom,
! [A: $tType,I: nat,Xs2: list @ A,X: A] :
( ( ord_less @ nat @ I @ ( size_size @ ( list @ A ) @ Xs2 ) )
=> ( ( nth @ A @ ( list_update @ A @ Xs2 @ I @ X ) @ I )
= X ) ) ).
% nth_list_update_eq
thf(fact_736_le__log2__of__power,axiom,
! [N3: nat,M: nat] :
( ( ord_less_eq @ nat @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N3 ) @ M )
=> ( ord_less_eq @ real @ ( semiring_1_of_nat @ real @ N3 ) @ ( log @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) @ ( semiring_1_of_nat @ real @ M ) ) ) ) ).
% le_log2_of_power
thf(fact_737_less__log2__of__power,axiom,
! [N3: nat,M: nat] :
( ( ord_less @ nat @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N3 ) @ M )
=> ( ord_less @ real @ ( semiring_1_of_nat @ real @ N3 ) @ ( log @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) @ ( semiring_1_of_nat @ real @ M ) ) ) ) ).
% less_log2_of_power
thf(fact_738_list__update__beyond,axiom,
! [A: $tType,Xs2: list @ A,I: nat,X: A] :
( ( ord_less_eq @ nat @ ( size_size @ ( list @ A ) @ Xs2 ) @ I )
=> ( ( list_update @ A @ Xs2 @ I @ X )
= Xs2 ) ) ).
% list_update_beyond
thf(fact_739_log2__of__power__eq,axiom,
! [M: nat,N3: nat] :
( ( M
= ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N3 ) )
=> ( ( semiring_1_of_nat @ real @ N3 )
= ( log @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) @ ( semiring_1_of_nat @ real @ M ) ) ) ) ).
% log2_of_power_eq
thf(fact_740_Tbuildupi__buildupi_H,axiom,
! [N3: nat] :
( ( semiring_1_of_nat @ int @ ( vEBT_V441764108873111860ildupi @ N3 ) )
= ( vEBT_V9176841429113362141ildupi @ N3 ) ) ).
% Tbuildupi_buildupi'
thf(fact_741_Tb__Tb_H,axiom,
( vEBT_VEBT_Tb
= ( ^ [T3: nat] : ( semiring_1_of_nat @ int @ ( vEBT_VEBT_Tb2 @ T3 ) ) ) ) ).
% Tb_Tb'
thf(fact_742_Tb__T__vebt__buildupi_H,axiom,
! [N3: nat] : ( ord_less_eq @ int @ ( vEBT_V9176841429113362141ildupi @ N3 ) @ ( minus_minus @ int @ ( vEBT_VEBT_Tb @ N3 ) @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) ) ) ).
% Tb_T_vebt_buildupi'
thf(fact_743_length__list__update,axiom,
! [A: $tType,Xs2: list @ A,I: nat,X: A] :
( ( size_size @ ( list @ A ) @ ( list_update @ A @ Xs2 @ I @ X ) )
= ( size_size @ ( list @ A ) @ Xs2 ) ) ).
% length_list_update
thf(fact_744_nth__list__update__neq,axiom,
! [A: $tType,I: nat,J: nat,Xs2: list @ A,X: A] :
( ( I != J )
=> ( ( nth @ A @ ( list_update @ A @ Xs2 @ I @ X ) @ J )
= ( nth @ A @ Xs2 @ J ) ) ) ).
% nth_list_update_neq
thf(fact_745_list__update__id,axiom,
! [A: $tType,Xs2: list @ A,I: nat] :
( ( list_update @ A @ Xs2 @ I @ ( nth @ A @ Xs2 @ I ) )
= Xs2 ) ).
% list_update_id
thf(fact_746_max__Suc__Suc,axiom,
! [M: nat,N3: nat] :
( ( ord_max @ nat @ ( suc @ M ) @ ( suc @ N3 ) )
= ( suc @ ( ord_max @ nat @ M @ N3 ) ) ) ).
% max_Suc_Suc
thf(fact_747_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_748_max__0__1_I6_J,axiom,
! [A: $tType] :
( ( linord181362715937106298miring @ A )
=> ! [X: num] :
( ( ord_max @ A @ ( numeral_numeral @ A @ X ) @ ( one_one @ A ) )
= ( numeral_numeral @ A @ X ) ) ) ).
% max_0_1(6)
thf(fact_749_max__0__1_I5_J,axiom,
! [A: $tType] :
( ( linord181362715937106298miring @ A )
=> ! [X: num] :
( ( ord_max @ A @ ( one_one @ A ) @ ( numeral_numeral @ A @ X ) )
= ( numeral_numeral @ A @ X ) ) ) ).
% max_0_1(5)
thf(fact_750_of__nat__max,axiom,
! [A: $tType] :
( ( linord181362715937106298miring @ A )
=> ! [X: nat,Y: nat] :
( ( semiring_1_of_nat @ A @ ( ord_max @ nat @ X @ Y ) )
= ( ord_max @ A @ ( semiring_1_of_nat @ A @ X ) @ ( semiring_1_of_nat @ A @ Y ) ) ) ) ).
% of_nat_max
thf(fact_751_max__add__distrib__right,axiom,
! [A: $tType] :
( ( ordere2412721322843649153imp_le @ A )
=> ! [X: A,Y: A,Z: A] :
( ( plus_plus @ A @ X @ ( ord_max @ A @ Y @ Z ) )
= ( ord_max @ A @ ( plus_plus @ A @ X @ Y ) @ ( plus_plus @ A @ X @ Z ) ) ) ) ).
% max_add_distrib_right
thf(fact_752_max__add__distrib__left,axiom,
! [A: $tType] :
( ( ordere2412721322843649153imp_le @ A )
=> ! [X: A,Y: A,Z: A] :
( ( plus_plus @ A @ ( ord_max @ A @ X @ Y ) @ Z )
= ( ord_max @ A @ ( plus_plus @ A @ X @ Z ) @ ( plus_plus @ A @ Y @ Z ) ) ) ) ).
% max_add_distrib_left
thf(fact_753_max__diff__distrib__left,axiom,
! [A: $tType] :
( ( ordered_ab_group_add @ A )
=> ! [X: A,Y: A,Z: A] :
( ( minus_minus @ A @ ( ord_max @ A @ X @ Y ) @ Z )
= ( ord_max @ A @ ( minus_minus @ A @ X @ Z ) @ ( minus_minus @ A @ Y @ Z ) ) ) ) ).
% max_diff_distrib_left
thf(fact_754_nat__add__max__right,axiom,
! [M: nat,N3: nat,Q2: nat] :
( ( plus_plus @ nat @ M @ ( ord_max @ nat @ N3 @ Q2 ) )
= ( ord_max @ nat @ ( plus_plus @ nat @ M @ N3 ) @ ( plus_plus @ nat @ M @ Q2 ) ) ) ).
% nat_add_max_right
thf(fact_755_nat__add__max__left,axiom,
! [M: nat,N3: nat,Q2: nat] :
( ( plus_plus @ nat @ ( ord_max @ nat @ M @ N3 ) @ Q2 )
= ( ord_max @ nat @ ( plus_plus @ nat @ M @ Q2 ) @ ( plus_plus @ nat @ N3 @ Q2 ) ) ) ).
% nat_add_max_left
thf(fact_756_nat__mult__max__right,axiom,
! [M: nat,N3: nat,Q2: nat] :
( ( times_times @ nat @ M @ ( ord_max @ nat @ N3 @ Q2 ) )
= ( ord_max @ nat @ ( times_times @ nat @ M @ N3 ) @ ( times_times @ nat @ M @ Q2 ) ) ) ).
% nat_mult_max_right
thf(fact_757_nat__mult__max__left,axiom,
! [M: nat,N3: nat,Q2: nat] :
( ( times_times @ nat @ ( ord_max @ nat @ M @ N3 ) @ Q2 )
= ( ord_max @ nat @ ( times_times @ nat @ M @ Q2 ) @ ( times_times @ nat @ N3 @ Q2 ) ) ) ).
% nat_mult_max_left
thf(fact_758_nat__minus__add__max,axiom,
! [N3: nat,M: nat] :
( ( plus_plus @ nat @ ( minus_minus @ nat @ N3 @ M ) @ M )
= ( ord_max @ nat @ N3 @ M ) ) ).
% nat_minus_add_max
thf(fact_759_neq__if__length__neq,axiom,
! [A: $tType,Xs2: list @ A,Ys: list @ A] :
( ( ( size_size @ ( list @ A ) @ Xs2 )
!= ( size_size @ ( list @ A ) @ Ys ) )
=> ( Xs2 != Ys ) ) ).
% neq_if_length_neq
thf(fact_760_Ex__list__of__length,axiom,
! [A: $tType,N3: nat] :
? [Xs3: list @ A] :
( ( size_size @ ( list @ A ) @ Xs3 )
= N3 ) ).
% Ex_list_of_length
thf(fact_761_subset__code_I1_J,axiom,
! [A: $tType,Xs2: list @ A,B6: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ ( set2 @ A @ Xs2 ) @ B6 )
= ( ! [X2: A] :
( ( member @ A @ X2 @ ( set2 @ A @ Xs2 ) )
=> ( member @ A @ X2 @ B6 ) ) ) ) ).
% subset_code(1)
thf(fact_762_length__induct,axiom,
! [A: $tType,P: ( list @ A ) > $o,Xs2: list @ A] :
( ! [Xs3: list @ A] :
( ! [Ys2: list @ A] :
( ( ord_less @ nat @ ( size_size @ ( list @ A ) @ Ys2 ) @ ( size_size @ ( list @ A ) @ Xs3 ) )
=> ( P @ Ys2 ) )
=> ( P @ Xs3 ) )
=> ( P @ Xs2 ) ) ).
% length_induct
thf(fact_763_set__update__subsetI,axiom,
! [A: $tType,Xs2: list @ A,A3: set @ A,X: A,I: nat] :
( ( ord_less_eq @ ( set @ A ) @ ( set2 @ A @ Xs2 ) @ A3 )
=> ( ( member @ A @ X @ A3 )
=> ( ord_less_eq @ ( set @ A ) @ ( set2 @ A @ ( list_update @ A @ Xs2 @ I @ X ) ) @ A3 ) ) ) ).
% set_update_subsetI
thf(fact_764_list__eq__iff__nth__eq,axiom,
! [A: $tType] :
( ( ^ [Y5: list @ A,Z5: list @ A] : Y5 = Z5 )
= ( ^ [Xs: list @ A,Ys3: list @ A] :
( ( ( size_size @ ( list @ A ) @ Xs )
= ( size_size @ ( list @ A ) @ Ys3 ) )
& ! [I3: nat] :
( ( ord_less @ nat @ I3 @ ( size_size @ ( list @ A ) @ Xs ) )
=> ( ( nth @ A @ Xs @ I3 )
= ( nth @ A @ Ys3 @ I3 ) ) ) ) ) ) ).
% list_eq_iff_nth_eq
thf(fact_765_Skolem__list__nth,axiom,
! [A: $tType,K: nat,P: nat > A > $o] :
( ( ! [I3: nat] :
( ( ord_less @ nat @ I3 @ K )
=> ? [X6: A] : ( P @ I3 @ X6 ) ) )
= ( ? [Xs: list @ A] :
( ( ( size_size @ ( list @ A ) @ Xs )
= K )
& ! [I3: nat] :
( ( ord_less @ nat @ I3 @ K )
=> ( P @ I3 @ ( nth @ A @ Xs @ I3 ) ) ) ) ) ) ).
% Skolem_list_nth
thf(fact_766_nth__equalityI,axiom,
! [A: $tType,Xs2: list @ A,Ys: list @ A] :
( ( ( size_size @ ( list @ A ) @ Xs2 )
= ( size_size @ ( list @ A ) @ Ys ) )
=> ( ! [I2: nat] :
( ( ord_less @ nat @ I2 @ ( size_size @ ( list @ A ) @ Xs2 ) )
=> ( ( nth @ A @ Xs2 @ I2 )
= ( nth @ A @ Ys @ I2 ) ) )
=> ( Xs2 = Ys ) ) ) ).
% nth_equalityI
thf(fact_767_all__set__conv__all__nth,axiom,
! [A: $tType,Xs2: list @ A,P: A > $o] :
( ( ! [X2: A] :
( ( member @ A @ X2 @ ( set2 @ A @ Xs2 ) )
=> ( P @ X2 ) ) )
= ( ! [I3: nat] :
( ( ord_less @ nat @ I3 @ ( size_size @ ( list @ A ) @ Xs2 ) )
=> ( P @ ( nth @ A @ Xs2 @ I3 ) ) ) ) ) ).
% all_set_conv_all_nth
thf(fact_768_all__nth__imp__all__set,axiom,
! [A: $tType,Xs2: list @ A,P: A > $o,X: A] :
( ! [I2: nat] :
( ( ord_less @ nat @ I2 @ ( size_size @ ( list @ A ) @ Xs2 ) )
=> ( P @ ( nth @ A @ Xs2 @ I2 ) ) )
=> ( ( member @ A @ X @ ( set2 @ A @ Xs2 ) )
=> ( P @ X ) ) ) ).
% all_nth_imp_all_set
thf(fact_769_in__set__conv__nth,axiom,
! [A: $tType,X: A,Xs2: list @ A] :
( ( member @ A @ X @ ( set2 @ A @ Xs2 ) )
= ( ? [I3: nat] :
( ( ord_less @ nat @ I3 @ ( size_size @ ( list @ A ) @ Xs2 ) )
& ( ( nth @ A @ Xs2 @ I3 )
= X ) ) ) ) ).
% in_set_conv_nth
thf(fact_770_list__ball__nth,axiom,
! [A: $tType,N3: nat,Xs2: list @ A,P: A > $o] :
( ( ord_less @ nat @ N3 @ ( size_size @ ( list @ A ) @ Xs2 ) )
=> ( ! [X3: A] :
( ( member @ A @ X3 @ ( set2 @ A @ Xs2 ) )
=> ( P @ X3 ) )
=> ( P @ ( nth @ A @ Xs2 @ N3 ) ) ) ) ).
% list_ball_nth
thf(fact_771_nth__mem,axiom,
! [A: $tType,N3: nat,Xs2: list @ A] :
( ( ord_less @ nat @ N3 @ ( size_size @ ( list @ A ) @ Xs2 ) )
=> ( member @ A @ ( nth @ A @ Xs2 @ N3 ) @ ( set2 @ A @ Xs2 ) ) ) ).
% nth_mem
thf(fact_772_set__update__memI,axiom,
! [A: $tType,N3: nat,Xs2: list @ A,X: A] :
( ( ord_less @ nat @ N3 @ ( size_size @ ( list @ A ) @ Xs2 ) )
=> ( member @ A @ X @ ( set2 @ A @ ( list_update @ A @ Xs2 @ N3 @ X ) ) ) ) ).
% set_update_memI
thf(fact_773_list__update__same__conv,axiom,
! [A: $tType,I: nat,Xs2: list @ A,X: A] :
( ( ord_less @ nat @ I @ ( size_size @ ( list @ A ) @ Xs2 ) )
=> ( ( ( list_update @ A @ Xs2 @ I @ X )
= Xs2 )
= ( ( nth @ A @ Xs2 @ I )
= X ) ) ) ).
% list_update_same_conv
thf(fact_774_nth__list__update,axiom,
! [A: $tType,I: nat,Xs2: list @ A,J: nat,X: A] :
( ( ord_less @ nat @ I @ ( size_size @ ( list @ A ) @ Xs2 ) )
=> ( ( ( I = J )
=> ( ( nth @ A @ ( list_update @ A @ Xs2 @ I @ X ) @ J )
= X ) )
& ( ( I != J )
=> ( ( nth @ A @ ( list_update @ A @ Xs2 @ I @ X ) @ J )
= ( nth @ A @ Xs2 @ J ) ) ) ) ) ).
% nth_list_update
thf(fact_775_log__of__power__eq,axiom,
! [M: nat,B2: real,N3: nat] :
( ( ( semiring_1_of_nat @ real @ M )
= ( power_power @ real @ B2 @ N3 ) )
=> ( ( ord_less @ real @ ( one_one @ real ) @ B2 )
=> ( ( semiring_1_of_nat @ real @ N3 )
= ( log @ B2 @ ( semiring_1_of_nat @ real @ M ) ) ) ) ) ).
% log_of_power_eq
thf(fact_776_less__log__of__power,axiom,
! [B2: real,N3: nat,M: real] :
( ( ord_less @ real @ ( power_power @ real @ B2 @ N3 ) @ M )
=> ( ( ord_less @ real @ ( one_one @ real ) @ B2 )
=> ( ord_less @ real @ ( semiring_1_of_nat @ real @ N3 ) @ ( log @ B2 @ M ) ) ) ) ).
% less_log_of_power
thf(fact_777_le__log__of__power,axiom,
! [B2: real,N3: nat,M: real] :
( ( ord_less_eq @ real @ ( power_power @ real @ B2 @ N3 ) @ M )
=> ( ( ord_less @ real @ ( one_one @ real ) @ B2 )
=> ( ord_less_eq @ real @ ( semiring_1_of_nat @ real @ N3 ) @ ( log @ B2 @ M ) ) ) ) ).
% le_log_of_power
thf(fact_778_max__less__iff__conj,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [X: A,Y: A,Z: A] :
( ( ord_less @ A @ ( ord_max @ A @ X @ Y ) @ Z )
= ( ( ord_less @ A @ X @ Z )
& ( ord_less @ A @ Y @ Z ) ) ) ) ).
% max_less_iff_conj
thf(fact_779_max_Oabsorb4,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A2: A,B2: A] :
( ( ord_less @ A @ A2 @ B2 )
=> ( ( ord_max @ A @ A2 @ B2 )
= B2 ) ) ) ).
% max.absorb4
thf(fact_780_max_Oabsorb3,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [B2: A,A2: A] :
( ( ord_less @ A @ B2 @ A2 )
=> ( ( ord_max @ A @ A2 @ B2 )
= A2 ) ) ) ).
% max.absorb3
thf(fact_781_max_Oabsorb1,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [B2: A,A2: A] :
( ( ord_less_eq @ A @ B2 @ A2 )
=> ( ( ord_max @ A @ A2 @ B2 )
= A2 ) ) ) ).
% max.absorb1
thf(fact_782_max_Oabsorb2,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A2: A,B2: A] :
( ( ord_less_eq @ A @ A2 @ B2 )
=> ( ( ord_max @ A @ A2 @ B2 )
= B2 ) ) ) ).
% max.absorb2
thf(fact_783_max_Obounded__iff,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [B2: A,C2: A,A2: A] :
( ( ord_less_eq @ A @ ( ord_max @ A @ B2 @ C2 ) @ A2 )
= ( ( ord_less_eq @ A @ B2 @ A2 )
& ( ord_less_eq @ A @ C2 @ A2 ) ) ) ) ).
% max.bounded_iff
thf(fact_784_heigt__uplog__rel,axiom,
! [T2: vEBT_VEBT,N3: nat] :
( ( vEBT_invar_vebt @ T2 @ N3 )
=> ( ( semiring_1_of_nat @ int @ ( vEBT_VEBT_height @ T2 ) )
= ( archimedean_ceiling @ real @ ( log @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) @ ( semiring_1_of_nat @ real @ N3 ) ) ) ) ) ).
% heigt_uplog_rel
thf(fact_785_pred__bound__size__univ,axiom,
! [T2: vEBT_VEBT,N3: nat,U: real,X: nat] :
( ( vEBT_invar_vebt @ T2 @ N3 )
=> ( ( U
= ( power_power @ real @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) @ N3 ) )
=> ( ord_less_eq @ real @ ( semiring_1_of_nat @ real @ ( vEBT_T_p_r_e_d @ T2 @ X ) ) @ ( plus_plus @ real @ ( numeral_numeral @ real @ ( bit0 @ ( bit1 @ ( bit0 @ ( bit1 @ ( bit1 @ one2 ) ) ) ) ) ) @ ( times_times @ real @ ( numeral_numeral @ real @ ( bit1 @ ( bit0 @ ( bit1 @ ( bit1 @ one2 ) ) ) ) ) @ ( log @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) @ ( log @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) @ U ) ) ) ) ) ) ) ).
% pred_bound_size_univ
thf(fact_786_insert__bound__size__univ,axiom,
! [T2: vEBT_VEBT,N3: nat,U: real,X: nat] :
( ( vEBT_invar_vebt @ T2 @ N3 )
=> ( ( U
= ( power_power @ real @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) @ N3 ) )
=> ( ord_less_eq @ real @ ( semiring_1_of_nat @ real @ ( vEBT_T_i_n_s_e_r_t @ T2 @ X ) ) @ ( plus_plus @ real @ ( numeral_numeral @ real @ ( bit0 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit0 @ one2 ) ) ) ) ) ) @ ( times_times @ real @ ( numeral_numeral @ real @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit0 @ one2 ) ) ) ) ) @ ( log @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) @ ( log @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) @ U ) ) ) ) ) ) ) ).
% insert_bound_size_univ
thf(fact_787_del__x__mia,axiom,
! [X: nat,Mi: nat,Ma: nat,Deg: nat,TreeList: list @ vEBT_VEBT,Summary: vEBT_VEBT] :
( ( ( X = Mi )
& ( ord_less @ nat @ X @ Ma ) )
=> ( ( Mi != Ma )
=> ( ( ord_less_eq @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ Deg )
=> ( ( vEBT_vebt_delete @ ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi @ Ma ) ) @ Deg @ TreeList @ Summary ) @ X )
= ( if @ vEBT_VEBT @ ( ord_less @ nat @ ( vEBT_VEBT_high @ ( plus_plus @ nat @ ( times_times @ nat @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( size_size @ ( list @ vEBT_VEBT ) @ TreeList ) )
@ ( if @ vEBT_VEBT @ ( vEBT_VEBT_minNull @ ( vEBT_vebt_delete @ ( nth @ vEBT_VEBT @ TreeList @ ( vEBT_VEBT_high @ ( plus_plus @ nat @ ( times_times @ nat @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( vEBT_VEBT_low @ ( plus_plus @ nat @ ( times_times @ nat @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) )
@ ( vEBT_Node
@ ( some @ ( product_prod @ nat @ nat )
@ ( product_Pair @ nat @ nat @ ( plus_plus @ nat @ ( times_times @ nat @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) )
@ ( if @ nat
@ ( ( plus_plus @ nat @ ( times_times @ nat @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) )
= Ma )
@ ( if @ nat
@ ( ( vEBT_vebt_maxt @ ( vEBT_vebt_delete @ Summary @ ( vEBT_VEBT_high @ ( plus_plus @ nat @ ( times_times @ nat @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) )
= ( none @ nat ) )
@ ( plus_plus @ nat @ ( times_times @ nat @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) )
@ ( plus_plus @ nat @ ( times_times @ nat @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_maxt @ ( vEBT_vebt_delete @ Summary @ ( vEBT_VEBT_high @ ( plus_plus @ nat @ ( times_times @ nat @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_maxt @ ( nth @ vEBT_VEBT @ ( list_update @ vEBT_VEBT @ TreeList @ ( vEBT_VEBT_high @ ( plus_plus @ nat @ ( times_times @ nat @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( vEBT_vebt_delete @ ( nth @ vEBT_VEBT @ TreeList @ ( vEBT_VEBT_high @ ( plus_plus @ nat @ ( times_times @ nat @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( vEBT_VEBT_low @ ( plus_plus @ nat @ ( times_times @ nat @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_maxt @ ( vEBT_vebt_delete @ Summary @ ( vEBT_VEBT_high @ ( plus_plus @ nat @ ( times_times @ nat @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) ) ) ) ) ) )
@ Ma ) ) )
@ Deg
@ ( list_update @ vEBT_VEBT @ TreeList @ ( vEBT_VEBT_high @ ( plus_plus @ nat @ ( times_times @ nat @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( vEBT_vebt_delete @ ( nth @ vEBT_VEBT @ TreeList @ ( vEBT_VEBT_high @ ( plus_plus @ nat @ ( times_times @ nat @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( vEBT_VEBT_low @ ( plus_plus @ nat @ ( times_times @ nat @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) )
@ ( vEBT_vebt_delete @ Summary @ ( vEBT_VEBT_high @ ( plus_plus @ nat @ ( times_times @ nat @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) )
@ ( vEBT_Node
@ ( some @ ( product_prod @ nat @ nat )
@ ( product_Pair @ nat @ nat @ ( plus_plus @ nat @ ( times_times @ nat @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) )
@ ( if @ nat
@ ( ( plus_plus @ nat @ ( times_times @ nat @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) )
= Ma )
@ ( plus_plus @ nat @ ( times_times @ nat @ ( vEBT_VEBT_high @ ( plus_plus @ nat @ ( times_times @ nat @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_maxt @ ( nth @ vEBT_VEBT @ ( list_update @ vEBT_VEBT @ TreeList @ ( vEBT_VEBT_high @ ( plus_plus @ nat @ ( times_times @ nat @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( vEBT_vebt_delete @ ( nth @ vEBT_VEBT @ TreeList @ ( vEBT_VEBT_high @ ( plus_plus @ nat @ ( times_times @ nat @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( vEBT_VEBT_low @ ( plus_plus @ nat @ ( times_times @ nat @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) @ ( vEBT_VEBT_high @ ( plus_plus @ nat @ ( times_times @ nat @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) ) )
@ Ma ) ) )
@ Deg
@ ( list_update @ vEBT_VEBT @ TreeList @ ( vEBT_VEBT_high @ ( plus_plus @ nat @ ( times_times @ nat @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( vEBT_vebt_delete @ ( nth @ vEBT_VEBT @ TreeList @ ( vEBT_VEBT_high @ ( plus_plus @ nat @ ( times_times @ nat @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( vEBT_VEBT_low @ ( plus_plus @ nat @ ( times_times @ nat @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) )
@ Summary ) )
@ ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi @ Ma ) ) @ Deg @ TreeList @ Summary ) ) ) ) ) ) ).
% del_x_mia
thf(fact_788_set__vebt_H__def,axiom,
( vEBT_VEBT_set_vebt
= ( ^ [T3: vEBT_VEBT] : ( collect @ nat @ ( vEBT_vebt_member @ T3 ) ) ) ) ).
% set_vebt'_def
thf(fact_789_minus__apply,axiom,
! [B: $tType,A: $tType] :
( ( minus @ B )
=> ( ( minus_minus @ ( A > B ) )
= ( ^ [A7: A > B,B7: A > B,X2: A] : ( minus_minus @ B @ ( A7 @ X2 ) @ ( B7 @ X2 ) ) ) ) ) ).
% minus_apply
thf(fact_790_ceiling__numeral__power,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [X: num,N3: nat] :
( ( archimedean_ceiling @ A @ ( power_power @ A @ ( numeral_numeral @ A @ X ) @ N3 ) )
= ( power_power @ int @ ( numeral_numeral @ int @ X ) @ N3 ) ) ) ).
% ceiling_numeral_power
thf(fact_791_del__x__not__mi,axiom,
! [Mi: nat,X: nat,Ma: nat,Deg: nat,H2: nat,L2: nat,Newnode: vEBT_VEBT,TreeList: list @ vEBT_VEBT,Newlist: list @ vEBT_VEBT,Summary: vEBT_VEBT] :
( ( ( ord_less @ nat @ Mi @ X )
& ( ord_less_eq @ nat @ X @ Ma ) )
=> ( ( Mi != Ma )
=> ( ( ord_less_eq @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ Deg )
=> ( ( ( vEBT_VEBT_high @ X @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) )
= H2 )
=> ( ( ( vEBT_VEBT_low @ X @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) )
= L2 )
=> ( ( Newnode
= ( vEBT_vebt_delete @ ( nth @ vEBT_VEBT @ TreeList @ H2 ) @ L2 ) )
=> ( ( Newlist
= ( list_update @ vEBT_VEBT @ TreeList @ H2 @ Newnode ) )
=> ( ( ord_less @ nat @ ( vEBT_VEBT_high @ X @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( size_size @ ( list @ vEBT_VEBT ) @ TreeList ) )
=> ( ( ( vEBT_VEBT_minNull @ Newnode )
=> ( ( vEBT_vebt_delete @ ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi @ Ma ) ) @ Deg @ TreeList @ Summary ) @ X )
= ( vEBT_Node
@ ( some @ ( product_prod @ nat @ nat )
@ ( product_Pair @ nat @ nat @ Mi
@ ( if @ nat @ ( X = Ma )
@ ( if @ nat
@ ( ( vEBT_vebt_maxt @ ( vEBT_vebt_delete @ Summary @ H2 ) )
= ( none @ nat ) )
@ Mi
@ ( plus_plus @ nat @ ( times_times @ nat @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_maxt @ ( vEBT_vebt_delete @ Summary @ H2 ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_maxt @ ( nth @ vEBT_VEBT @ Newlist @ ( the2 @ nat @ ( vEBT_vebt_maxt @ ( vEBT_vebt_delete @ Summary @ H2 ) ) ) ) ) ) ) )
@ Ma ) ) )
@ Deg
@ Newlist
@ ( vEBT_vebt_delete @ Summary @ H2 ) ) ) )
& ( ~ ( vEBT_VEBT_minNull @ Newnode )
=> ( ( vEBT_vebt_delete @ ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi @ Ma ) ) @ Deg @ TreeList @ Summary ) @ X )
= ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi @ ( if @ nat @ ( X = Ma ) @ ( plus_plus @ nat @ ( times_times @ nat @ H2 @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_maxt @ ( nth @ vEBT_VEBT @ Newlist @ H2 ) ) ) ) @ Ma ) ) ) @ Deg @ Newlist @ Summary ) ) ) ) ) ) ) ) ) ) ) ) ).
% del_x_not_mi
thf(fact_792_del__x__not__mi__new__node__nil,axiom,
! [Mi: nat,X: nat,Ma: nat,Deg: nat,H2: nat,L2: nat,Newnode: vEBT_VEBT,TreeList: list @ vEBT_VEBT,Sn: vEBT_VEBT,Summary: vEBT_VEBT,Newlist: list @ vEBT_VEBT] :
( ( ( ord_less @ nat @ Mi @ X )
& ( ord_less_eq @ nat @ X @ Ma ) )
=> ( ( Mi != Ma )
=> ( ( ord_less_eq @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ Deg )
=> ( ( ( vEBT_VEBT_high @ X @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) )
= H2 )
=> ( ( ( vEBT_VEBT_low @ X @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) )
= L2 )
=> ( ( Newnode
= ( vEBT_vebt_delete @ ( nth @ vEBT_VEBT @ TreeList @ H2 ) @ L2 ) )
=> ( ( vEBT_VEBT_minNull @ Newnode )
=> ( ( Sn
= ( vEBT_vebt_delete @ Summary @ H2 ) )
=> ( ( Newlist
= ( list_update @ vEBT_VEBT @ TreeList @ H2 @ Newnode ) )
=> ( ( ord_less @ nat @ ( vEBT_VEBT_high @ X @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( size_size @ ( list @ vEBT_VEBT ) @ TreeList ) )
=> ( ( vEBT_vebt_delete @ ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi @ Ma ) ) @ Deg @ TreeList @ Summary ) @ X )
= ( vEBT_Node
@ ( some @ ( product_prod @ nat @ nat )
@ ( product_Pair @ nat @ nat @ Mi
@ ( if @ nat @ ( X = Ma )
@ ( if @ nat
@ ( ( vEBT_vebt_maxt @ Sn )
= ( none @ nat ) )
@ Mi
@ ( plus_plus @ nat @ ( times_times @ nat @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_maxt @ Sn ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_maxt @ ( nth @ vEBT_VEBT @ Newlist @ ( the2 @ nat @ ( vEBT_vebt_maxt @ Sn ) ) ) ) ) ) )
@ Ma ) ) )
@ Deg
@ Newlist
@ Sn ) ) ) ) ) ) ) ) ) ) ) ) ).
% del_x_not_mi_new_node_nil
thf(fact_793_del__x__not__mia,axiom,
! [Mi: nat,X: nat,Ma: nat,Deg: nat,H2: nat,L2: nat,TreeList: list @ vEBT_VEBT,Summary: vEBT_VEBT] :
( ( ( ord_less @ nat @ Mi @ X )
& ( ord_less_eq @ nat @ X @ Ma ) )
=> ( ( Mi != Ma )
=> ( ( ord_less_eq @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ Deg )
=> ( ( ( vEBT_VEBT_high @ X @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) )
= H2 )
=> ( ( ( vEBT_VEBT_low @ X @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) )
= L2 )
=> ( ( ord_less @ nat @ ( vEBT_VEBT_high @ X @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( size_size @ ( list @ vEBT_VEBT ) @ TreeList ) )
=> ( ( vEBT_vebt_delete @ ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi @ Ma ) ) @ Deg @ TreeList @ Summary ) @ X )
= ( if @ vEBT_VEBT @ ( vEBT_VEBT_minNull @ ( vEBT_vebt_delete @ ( nth @ vEBT_VEBT @ TreeList @ H2 ) @ L2 ) )
@ ( vEBT_Node
@ ( some @ ( product_prod @ nat @ nat )
@ ( product_Pair @ nat @ nat @ Mi
@ ( if @ nat @ ( X = Ma )
@ ( if @ nat
@ ( ( vEBT_vebt_maxt @ ( vEBT_vebt_delete @ Summary @ H2 ) )
= ( none @ nat ) )
@ Mi
@ ( plus_plus @ nat @ ( times_times @ nat @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_maxt @ ( vEBT_vebt_delete @ Summary @ H2 ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_maxt @ ( nth @ vEBT_VEBT @ ( list_update @ vEBT_VEBT @ TreeList @ H2 @ ( vEBT_vebt_delete @ ( nth @ vEBT_VEBT @ TreeList @ H2 ) @ L2 ) ) @ ( 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 ) @ L2 ) )
@ ( vEBT_vebt_delete @ Summary @ H2 ) )
@ ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi @ ( if @ nat @ ( X = Ma ) @ ( plus_plus @ nat @ ( times_times @ nat @ H2 @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_maxt @ ( nth @ vEBT_VEBT @ ( list_update @ vEBT_VEBT @ TreeList @ H2 @ ( vEBT_vebt_delete @ ( nth @ vEBT_VEBT @ TreeList @ H2 ) @ L2 ) ) @ H2 ) ) ) ) @ Ma ) ) ) @ Deg @ ( list_update @ vEBT_VEBT @ TreeList @ H2 @ ( vEBT_vebt_delete @ ( nth @ vEBT_VEBT @ TreeList @ H2 ) @ L2 ) ) @ Summary ) ) ) ) ) ) ) ) ) ).
% del_x_not_mia
thf(fact_794_del__in__range,axiom,
! [Mi: nat,X: nat,Ma: nat,Deg: nat,TreeList: list @ vEBT_VEBT,Summary: vEBT_VEBT] :
( ( ( ord_less_eq @ nat @ Mi @ X )
& ( ord_less_eq @ nat @ X @ Ma ) )
=> ( ( Mi != Ma )
=> ( ( ord_less_eq @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ Deg )
=> ( ( vEBT_vebt_delete @ ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi @ Ma ) ) @ Deg @ TreeList @ Summary ) @ X )
= ( if @ vEBT_VEBT @ ( ord_less @ nat @ ( vEBT_VEBT_high @ ( if @ nat @ ( X = Mi ) @ ( plus_plus @ nat @ ( times_times @ nat @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ X ) @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( size_size @ ( list @ vEBT_VEBT ) @ TreeList ) )
@ ( if @ vEBT_VEBT @ ( vEBT_VEBT_minNull @ ( vEBT_vebt_delete @ ( nth @ vEBT_VEBT @ TreeList @ ( vEBT_VEBT_high @ ( if @ nat @ ( X = Mi ) @ ( plus_plus @ nat @ ( times_times @ nat @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ X ) @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( vEBT_VEBT_low @ ( if @ nat @ ( X = Mi ) @ ( plus_plus @ nat @ ( times_times @ nat @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ X ) @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) )
@ ( vEBT_Node
@ ( some @ ( product_prod @ nat @ nat )
@ ( product_Pair @ nat @ nat @ ( if @ nat @ ( X = Mi ) @ ( if @ nat @ ( X = Mi ) @ ( plus_plus @ nat @ ( times_times @ nat @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ X ) @ Mi )
@ ( if @ nat
@ ( ( ( X = Mi )
=> ( ( plus_plus @ nat @ ( times_times @ nat @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) )
= Ma ) )
& ( ( X != Mi )
=> ( X = Ma ) ) )
@ ( if @ nat
@ ( ( vEBT_vebt_maxt @ ( vEBT_vebt_delete @ Summary @ ( vEBT_VEBT_high @ ( if @ nat @ ( X = Mi ) @ ( plus_plus @ nat @ ( times_times @ nat @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ X ) @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) )
= ( none @ nat ) )
@ ( if @ nat @ ( X = Mi ) @ ( if @ nat @ ( X = Mi ) @ ( plus_plus @ nat @ ( times_times @ nat @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ X ) @ Mi )
@ ( plus_plus @ nat @ ( times_times @ nat @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_maxt @ ( vEBT_vebt_delete @ Summary @ ( vEBT_VEBT_high @ ( if @ nat @ ( X = Mi ) @ ( plus_plus @ nat @ ( times_times @ nat @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ X ) @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_maxt @ ( nth @ vEBT_VEBT @ ( list_update @ vEBT_VEBT @ TreeList @ ( vEBT_VEBT_high @ ( if @ nat @ ( X = Mi ) @ ( plus_plus @ nat @ ( times_times @ nat @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ X ) @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( vEBT_vebt_delete @ ( nth @ vEBT_VEBT @ TreeList @ ( vEBT_VEBT_high @ ( if @ nat @ ( X = Mi ) @ ( plus_plus @ nat @ ( times_times @ nat @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ X ) @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( vEBT_VEBT_low @ ( if @ nat @ ( X = Mi ) @ ( plus_plus @ nat @ ( times_times @ nat @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ X ) @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_maxt @ ( vEBT_vebt_delete @ Summary @ ( vEBT_VEBT_high @ ( if @ nat @ ( X = Mi ) @ ( plus_plus @ nat @ ( times_times @ nat @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ X ) @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) ) ) ) ) ) )
@ Ma ) ) )
@ Deg
@ ( list_update @ vEBT_VEBT @ TreeList @ ( vEBT_VEBT_high @ ( if @ nat @ ( X = Mi ) @ ( plus_plus @ nat @ ( times_times @ nat @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ X ) @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( vEBT_vebt_delete @ ( nth @ vEBT_VEBT @ TreeList @ ( vEBT_VEBT_high @ ( if @ nat @ ( X = Mi ) @ ( plus_plus @ nat @ ( times_times @ nat @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ X ) @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( vEBT_VEBT_low @ ( if @ nat @ ( X = Mi ) @ ( plus_plus @ nat @ ( times_times @ nat @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ X ) @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) )
@ ( vEBT_vebt_delete @ Summary @ ( vEBT_VEBT_high @ ( if @ nat @ ( X = Mi ) @ ( plus_plus @ nat @ ( times_times @ nat @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ X ) @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) )
@ ( vEBT_Node
@ ( some @ ( product_prod @ nat @ nat )
@ ( product_Pair @ nat @ nat @ ( if @ nat @ ( X = Mi ) @ ( if @ nat @ ( X = Mi ) @ ( plus_plus @ nat @ ( times_times @ nat @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ X ) @ Mi )
@ ( if @ nat
@ ( ( ( X = Mi )
=> ( ( plus_plus @ nat @ ( times_times @ nat @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) )
= Ma ) )
& ( ( X != Mi )
=> ( X = Ma ) ) )
@ ( plus_plus @ nat @ ( times_times @ nat @ ( vEBT_VEBT_high @ ( if @ nat @ ( X = Mi ) @ ( plus_plus @ nat @ ( times_times @ nat @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ X ) @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_maxt @ ( nth @ vEBT_VEBT @ ( list_update @ vEBT_VEBT @ TreeList @ ( vEBT_VEBT_high @ ( if @ nat @ ( X = Mi ) @ ( plus_plus @ nat @ ( times_times @ nat @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ X ) @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( vEBT_vebt_delete @ ( nth @ vEBT_VEBT @ TreeList @ ( vEBT_VEBT_high @ ( if @ nat @ ( X = Mi ) @ ( plus_plus @ nat @ ( times_times @ nat @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ X ) @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( vEBT_VEBT_low @ ( if @ nat @ ( X = Mi ) @ ( plus_plus @ nat @ ( times_times @ nat @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ X ) @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) @ ( vEBT_VEBT_high @ ( if @ nat @ ( X = Mi ) @ ( plus_plus @ nat @ ( times_times @ nat @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ X ) @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) ) )
@ Ma ) ) )
@ Deg
@ ( list_update @ vEBT_VEBT @ TreeList @ ( vEBT_VEBT_high @ ( if @ nat @ ( X = Mi ) @ ( plus_plus @ nat @ ( times_times @ nat @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ X ) @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( vEBT_vebt_delete @ ( nth @ vEBT_VEBT @ TreeList @ ( vEBT_VEBT_high @ ( if @ nat @ ( X = Mi ) @ ( plus_plus @ nat @ ( times_times @ nat @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ X ) @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( vEBT_VEBT_low @ ( if @ nat @ ( X = Mi ) @ ( plus_plus @ nat @ ( times_times @ nat @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ X ) @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) )
@ Summary ) )
@ ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi @ Ma ) ) @ Deg @ TreeList @ Summary ) ) ) ) ) ) ).
% del_in_range
thf(fact_795_del__x__mi,axiom,
! [X: nat,Mi: nat,Ma: nat,Deg: nat,Xn: nat,H2: nat,Summary: vEBT_VEBT,TreeList: list @ vEBT_VEBT,L2: nat] :
( ( ( X = Mi )
& ( ord_less @ nat @ X @ Ma ) )
=> ( ( Mi != Ma )
=> ( ( ord_less_eq @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ Deg )
=> ( ( ( vEBT_VEBT_high @ Xn @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) )
= H2 )
=> ( ( Xn
= ( plus_plus @ nat @ ( times_times @ nat @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) )
=> ( ( ( vEBT_VEBT_low @ Xn @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) )
= L2 )
=> ( ( ord_less @ nat @ ( vEBT_VEBT_high @ Xn @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( size_size @ ( list @ vEBT_VEBT ) @ TreeList ) )
=> ( ( vEBT_vebt_delete @ ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi @ Ma ) ) @ Deg @ TreeList @ Summary ) @ X )
= ( if @ vEBT_VEBT @ ( vEBT_VEBT_minNull @ ( vEBT_vebt_delete @ ( nth @ vEBT_VEBT @ TreeList @ H2 ) @ L2 ) )
@ ( 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 ) @ L2 ) ) @ ( 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 ) @ L2 ) )
@ ( 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 ) @ L2 ) ) @ H2 ) ) ) ) @ Ma ) ) ) @ Deg @ ( list_update @ vEBT_VEBT @ TreeList @ H2 @ ( vEBT_vebt_delete @ ( nth @ vEBT_VEBT @ TreeList @ H2 ) @ L2 ) ) @ Summary ) ) ) ) ) ) ) ) ) ) ).
% del_x_mi
thf(fact_796_del__x__mi__lets__in,axiom,
! [X: nat,Mi: nat,Ma: nat,Deg: nat,Xn: nat,H2: nat,Summary: vEBT_VEBT,TreeList: list @ vEBT_VEBT,L2: nat,Newnode: vEBT_VEBT,Newlist: list @ vEBT_VEBT] :
( ( ( X = Mi )
& ( ord_less @ nat @ X @ Ma ) )
=> ( ( Mi != Ma )
=> ( ( ord_less_eq @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ Deg )
=> ( ( ( vEBT_VEBT_high @ Xn @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) )
= H2 )
=> ( ( Xn
= ( plus_plus @ nat @ ( times_times @ nat @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) )
=> ( ( ( vEBT_VEBT_low @ Xn @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) )
= L2 )
=> ( ( 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 ) @ L2 ) )
=> ( ( Newlist
= ( list_update @ vEBT_VEBT @ TreeList @ H2 @ Newnode ) )
=> ( ( ( vEBT_VEBT_minNull @ Newnode )
=> ( ( vEBT_vebt_delete @ ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi @ Ma ) ) @ Deg @ TreeList @ Summary ) @ X )
= ( vEBT_Node
@ ( some @ ( product_prod @ nat @ nat )
@ ( product_Pair @ nat @ nat @ Xn
@ ( if @ nat @ ( Xn = Ma )
@ ( if @ nat
@ ( ( vEBT_vebt_maxt @ ( vEBT_vebt_delete @ Summary @ H2 ) )
= ( none @ nat ) )
@ Xn
@ ( plus_plus @ nat @ ( times_times @ nat @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_maxt @ ( vEBT_vebt_delete @ Summary @ H2 ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_maxt @ ( nth @ vEBT_VEBT @ Newlist @ ( the2 @ nat @ ( vEBT_vebt_maxt @ ( vEBT_vebt_delete @ Summary @ H2 ) ) ) ) ) ) ) )
@ Ma ) ) )
@ Deg
@ Newlist
@ ( vEBT_vebt_delete @ Summary @ H2 ) ) ) )
& ( ~ ( vEBT_VEBT_minNull @ Newnode )
=> ( ( vEBT_vebt_delete @ ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi @ Ma ) ) @ Deg @ TreeList @ Summary ) @ X )
= ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Xn @ ( if @ nat @ ( Xn = Ma ) @ ( plus_plus @ nat @ ( times_times @ nat @ H2 @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_maxt @ ( nth @ vEBT_VEBT @ Newlist @ H2 ) ) ) ) @ Ma ) ) ) @ Deg @ Newlist @ Summary ) ) ) ) ) ) ) ) ) ) ) ) ) ).
% del_x_mi_lets_in
thf(fact_797_del__x__mi__lets__in__minNull,axiom,
! [X: nat,Mi: nat,Ma: nat,Deg: nat,Xn: nat,H2: nat,Summary: vEBT_VEBT,TreeList: list @ vEBT_VEBT,L2: nat,Newnode: vEBT_VEBT,Newlist: list @ vEBT_VEBT,Sn: vEBT_VEBT] :
( ( ( X = Mi )
& ( ord_less @ nat @ X @ Ma ) )
=> ( ( Mi != Ma )
=> ( ( ord_less_eq @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ Deg )
=> ( ( ( vEBT_VEBT_high @ Xn @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) )
= H2 )
=> ( ( Xn
= ( plus_plus @ nat @ ( times_times @ nat @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) )
=> ( ( ( vEBT_VEBT_low @ Xn @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) )
= L2 )
=> ( ( 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 ) @ L2 ) )
=> ( ( Newlist
= ( list_update @ vEBT_VEBT @ TreeList @ H2 @ Newnode ) )
=> ( ( vEBT_VEBT_minNull @ Newnode )
=> ( ( Sn
= ( vEBT_vebt_delete @ Summary @ H2 ) )
=> ( ( vEBT_vebt_delete @ ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi @ Ma ) ) @ Deg @ TreeList @ Summary ) @ X )
= ( vEBT_Node
@ ( some @ ( product_prod @ nat @ nat )
@ ( product_Pair @ nat @ nat @ Xn
@ ( if @ nat @ ( Xn = Ma )
@ ( if @ nat
@ ( ( vEBT_vebt_maxt @ Sn )
= ( none @ nat ) )
@ Xn
@ ( plus_plus @ nat @ ( times_times @ nat @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_maxt @ Sn ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_maxt @ ( nth @ vEBT_VEBT @ Newlist @ ( the2 @ nat @ ( vEBT_vebt_maxt @ Sn ) ) ) ) ) ) )
@ Ma ) ) )
@ Deg
@ Newlist
@ Sn ) ) ) ) ) ) ) ) ) ) ) ) ) ).
% del_x_mi_lets_in_minNull
thf(fact_798_mult__commute__abs,axiom,
! [A: $tType] :
( ( ab_semigroup_mult @ A )
=> ! [C2: A] :
( ( ^ [X2: A] : ( times_times @ A @ X2 @ C2 ) )
= ( times_times @ A @ C2 ) ) ) ).
% mult_commute_abs
thf(fact_799_subset__Collect__conv,axiom,
! [A: $tType,S3: set @ A,P: A > $o] :
( ( ord_less_eq @ ( set @ A ) @ S3 @ ( collect @ A @ P ) )
= ( ! [X2: A] :
( ( member @ A @ X2 @ S3 )
=> ( P @ X2 ) ) ) ) ).
% subset_Collect_conv
thf(fact_800_lambda__one,axiom,
! [A: $tType] :
( ( monoid_mult @ A )
=> ( ( ^ [X2: A] : X2 )
= ( times_times @ A @ ( one_one @ A ) ) ) ) ).
% lambda_one
thf(fact_801_set__vebt__def,axiom,
( vEBT_set_vebt
= ( ^ [T3: vEBT_VEBT] : ( collect @ nat @ ( vEBT_V8194947554948674370ptions @ T3 ) ) ) ) ).
% set_vebt_def
thf(fact_802_numeral__code_I2_J,axiom,
! [A: $tType] :
( ( numeral @ A )
=> ! [N3: num] :
( ( numeral_numeral @ A @ ( bit0 @ N3 ) )
= ( plus_plus @ A @ ( numeral_numeral @ A @ N3 ) @ ( numeral_numeral @ A @ N3 ) ) ) ) ).
% numeral_code(2)
thf(fact_803_numeral__code_I3_J,axiom,
! [A: $tType] :
( ( numeral @ A )
=> ! [N3: num] :
( ( numeral_numeral @ A @ ( bit1 @ N3 ) )
= ( plus_plus @ A @ ( plus_plus @ A @ ( numeral_numeral @ A @ N3 ) @ ( numeral_numeral @ A @ N3 ) ) @ ( one_one @ A ) ) ) ) ).
% numeral_code(3)
thf(fact_804_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_805_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_806_fun__diff__def,axiom,
! [B: $tType,A: $tType] :
( ( minus @ B )
=> ( ( minus_minus @ ( A > B ) )
= ( ^ [A7: A > B,B7: A > B,X2: A] : ( minus_minus @ B @ ( A7 @ X2 ) @ ( B7 @ X2 ) ) ) ) ) ).
% fun_diff_def
thf(fact_807_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_808_max_Omono,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [C2: A,A2: A,D: A,B2: A] :
( ( ord_less_eq @ A @ C2 @ A2 )
=> ( ( ord_less_eq @ A @ D @ B2 )
=> ( ord_less_eq @ A @ ( ord_max @ A @ C2 @ D ) @ ( ord_max @ A @ A2 @ B2 ) ) ) ) ) ).
% max.mono
thf(fact_809_max_OorderE,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [B2: A,A2: A] :
( ( ord_less_eq @ A @ B2 @ A2 )
=> ( A2
= ( ord_max @ A @ A2 @ B2 ) ) ) ) ).
% max.orderE
thf(fact_810_max_OorderI,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A2: A,B2: A] :
( ( A2
= ( ord_max @ A @ A2 @ B2 ) )
=> ( ord_less_eq @ A @ B2 @ A2 ) ) ) ).
% max.orderI
thf(fact_811_max_OboundedE,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [B2: A,C2: A,A2: A] :
( ( ord_less_eq @ A @ ( ord_max @ A @ B2 @ C2 ) @ A2 )
=> ~ ( ( ord_less_eq @ A @ B2 @ A2 )
=> ~ ( ord_less_eq @ A @ C2 @ A2 ) ) ) ) ).
% max.boundedE
thf(fact_812_max_OboundedI,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [B2: A,A2: A,C2: A] :
( ( ord_less_eq @ A @ B2 @ A2 )
=> ( ( ord_less_eq @ A @ C2 @ A2 )
=> ( ord_less_eq @ A @ ( ord_max @ A @ B2 @ C2 ) @ A2 ) ) ) ) ).
% max.boundedI
thf(fact_813_max_Oorder__iff,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ( ( ord_less_eq @ A )
= ( ^ [B5: A,A6: A] :
( A6
= ( ord_max @ A @ A6 @ B5 ) ) ) ) ) ).
% max.order_iff
thf(fact_814_max_Ocobounded1,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A2: A,B2: A] : ( ord_less_eq @ A @ A2 @ ( ord_max @ A @ A2 @ B2 ) ) ) ).
% max.cobounded1
thf(fact_815_max_Ocobounded2,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [B2: A,A2: A] : ( ord_less_eq @ A @ B2 @ ( ord_max @ A @ A2 @ B2 ) ) ) ).
% max.cobounded2
thf(fact_816_le__max__iff__disj,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [Z: A,X: A,Y: A] :
( ( ord_less_eq @ A @ Z @ ( ord_max @ A @ X @ Y ) )
= ( ( ord_less_eq @ A @ Z @ X )
| ( ord_less_eq @ A @ Z @ Y ) ) ) ) ).
% le_max_iff_disj
thf(fact_817_max_Oabsorb__iff1,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ( ( ord_less_eq @ A )
= ( ^ [B5: A,A6: A] :
( ( ord_max @ A @ A6 @ B5 )
= A6 ) ) ) ) ).
% max.absorb_iff1
thf(fact_818_max_Oabsorb__iff2,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ( ( ord_less_eq @ A )
= ( ^ [A6: A,B5: A] :
( ( ord_max @ A @ A6 @ B5 )
= B5 ) ) ) ) ).
% max.absorb_iff2
thf(fact_819_max_OcoboundedI1,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [C2: A,A2: A,B2: A] :
( ( ord_less_eq @ A @ C2 @ A2 )
=> ( ord_less_eq @ A @ C2 @ ( ord_max @ A @ A2 @ B2 ) ) ) ) ).
% max.coboundedI1
thf(fact_820_max_OcoboundedI2,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [C2: A,B2: A,A2: A] :
( ( ord_less_eq @ A @ C2 @ B2 )
=> ( ord_less_eq @ A @ C2 @ ( ord_max @ A @ A2 @ B2 ) ) ) ) ).
% max.coboundedI2
thf(fact_821_max_Ostrict__coboundedI2,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [C2: A,B2: A,A2: A] :
( ( ord_less @ A @ C2 @ B2 )
=> ( ord_less @ A @ C2 @ ( ord_max @ A @ A2 @ B2 ) ) ) ) ).
% max.strict_coboundedI2
thf(fact_822_max_Ostrict__coboundedI1,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [C2: A,A2: A,B2: A] :
( ( ord_less @ A @ C2 @ A2 )
=> ( ord_less @ A @ C2 @ ( ord_max @ A @ A2 @ B2 ) ) ) ) ).
% max.strict_coboundedI1
thf(fact_823_max_Ostrict__order__iff,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ( ( ord_less @ A )
= ( ^ [B5: A,A6: A] :
( ( A6
= ( ord_max @ A @ A6 @ B5 ) )
& ( A6 != B5 ) ) ) ) ) ).
% max.strict_order_iff
thf(fact_824_max_Ostrict__boundedE,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [B2: A,C2: A,A2: A] :
( ( ord_less @ A @ ( ord_max @ A @ B2 @ C2 ) @ A2 )
=> ~ ( ( ord_less @ A @ B2 @ A2 )
=> ~ ( ord_less @ A @ C2 @ A2 ) ) ) ) ).
% max.strict_boundedE
thf(fact_825_less__max__iff__disj,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [Z: A,X: A,Y: A] :
( ( ord_less @ A @ Z @ ( ord_max @ A @ X @ Y ) )
= ( ( ord_less @ A @ Z @ X )
| ( ord_less @ A @ Z @ Y ) ) ) ) ).
% less_max_iff_disj
thf(fact_826_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,X: nat] :
( ( vEBT_T_m_e_m_b_e_r @ ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi @ Ma ) ) @ ( suc @ ( suc @ Va ) ) @ TreeList @ Summary ) @ X )
= ( plus_plus @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( if @ nat @ ( X = Mi ) @ ( one_one @ nat ) @ ( plus_plus @ nat @ ( one_one @ nat ) @ ( if @ nat @ ( X = Ma ) @ ( one_one @ nat ) @ ( plus_plus @ nat @ ( one_one @ nat ) @ ( if @ nat @ ( ord_less @ nat @ X @ Mi ) @ ( one_one @ nat ) @ ( plus_plus @ nat @ ( one_one @ nat ) @ ( if @ nat @ ( ord_less @ nat @ Ma @ X ) @ ( one_one @ nat ) @ ( plus_plus @ nat @ ( numeral_numeral @ nat @ ( bit1 @ ( bit0 @ ( bit0 @ one2 ) ) ) ) @ ( if @ nat @ ( ord_less @ nat @ ( vEBT_VEBT_high @ X @ ( divide_divide @ nat @ ( suc @ ( suc @ 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 @ X @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( vEBT_VEBT_low @ X @ ( 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_827_vebt__insert_Osimps_I5_J,axiom,
! [Mi: nat,Ma: nat,Va: nat,TreeList: list @ vEBT_VEBT,Summary: vEBT_VEBT,X: nat] :
( ( vEBT_vebt_insert @ ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi @ Ma ) ) @ ( suc @ ( suc @ Va ) ) @ TreeList @ Summary ) @ X )
= ( if @ vEBT_VEBT
@ ( ( ord_less @ nat @ ( vEBT_VEBT_high @ ( if @ nat @ ( ord_less @ nat @ X @ Mi ) @ Mi @ X ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( size_size @ ( list @ vEBT_VEBT ) @ TreeList ) )
& ~ ( ( X = Mi )
| ( X = Ma ) ) )
@ ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ ( if @ nat @ ( ord_less @ nat @ X @ Mi ) @ X @ Mi ) @ ( ord_max @ nat @ ( if @ nat @ ( ord_less @ nat @ X @ Mi ) @ Mi @ X ) @ Ma ) ) ) @ ( suc @ ( suc @ Va ) ) @ ( list_update @ vEBT_VEBT @ TreeList @ ( vEBT_VEBT_high @ ( if @ nat @ ( ord_less @ nat @ X @ Mi ) @ Mi @ X ) @ ( 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 @ X @ Mi ) @ Mi @ X ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( vEBT_VEBT_low @ ( if @ nat @ ( ord_less @ nat @ X @ Mi ) @ Mi @ X ) @ ( 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 @ X @ Mi ) @ Mi @ X ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) @ ( vEBT_vebt_insert @ Summary @ ( vEBT_VEBT_high @ ( if @ nat @ ( ord_less @ nat @ X @ Mi ) @ Mi @ X ) @ ( divide_divide @ nat @ ( suc @ ( suc @ 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_828_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,X: nat] :
( ( vEBT_T_i_n_s_e_r_t @ ( vEBT_Node @ ( none @ ( product_prod @ nat @ nat ) ) @ ( suc @ ( suc @ V ) ) @ TreeList @ Summary ) @ X )
= ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ).
% T\<^sub>i\<^sub>n\<^sub>s\<^sub>e\<^sub>r\<^sub>t.simps(4)
thf(fact_829_insersimp,axiom,
! [T2: vEBT_VEBT,N3: nat,Y: nat] :
( ( vEBT_invar_vebt @ T2 @ N3 )
=> ( ~ ? [X_12: nat] : ( vEBT_V8194947554948674370ptions @ T2 @ X_12 )
=> ( ord_less_eq @ nat @ ( vEBT_T_i_n_s_e_r_t @ T2 @ Y ) @ ( numeral_numeral @ nat @ ( bit1 @ one2 ) ) ) ) ) ).
% insersimp
thf(fact_830_insertsimp,axiom,
! [T2: vEBT_VEBT,N3: nat,L2: nat] :
( ( vEBT_invar_vebt @ T2 @ N3 )
=> ( ( vEBT_VEBT_minNull @ T2 )
=> ( ord_less_eq @ nat @ ( vEBT_T_i_n_s_e_r_t @ T2 @ L2 ) @ ( numeral_numeral @ nat @ ( bit1 @ one2 ) ) ) ) ) ).
% insertsimp
thf(fact_831_VEBT__internal_OT_092_060_094sub_062d_092_060_094sub_062e_092_060_094sub_062l_092_060_094sub_062e_092_060_094sub_062t_092_060_094sub_062e_H_Osimps_I7_J,axiom,
! [X: nat,Mi: nat,Ma: nat,Va: nat,TreeList: list @ vEBT_VEBT,Summary: vEBT_VEBT] :
( ( ( ( ord_less @ nat @ X @ Mi )
| ( ord_less @ nat @ Ma @ X ) )
=> ( ( vEBT_V1232361888498592333_e_t_e @ ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi @ Ma ) ) @ ( suc @ ( suc @ Va ) ) @ TreeList @ Summary ) @ X )
= ( one_one @ nat ) ) )
& ( ~ ( ( ord_less @ nat @ X @ Mi )
| ( ord_less @ nat @ Ma @ X ) )
=> ( ( ( ( X = Mi )
& ( X = Ma ) )
=> ( ( vEBT_V1232361888498592333_e_t_e @ ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi @ Ma ) ) @ ( suc @ ( suc @ Va ) ) @ TreeList @ Summary ) @ X )
= ( one_one @ nat ) ) )
& ( ~ ( ( X = Mi )
& ( X = Ma ) )
=> ( ( vEBT_V1232361888498592333_e_t_e @ ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi @ Ma ) ) @ ( suc @ ( suc @ Va ) ) @ TreeList @ Summary ) @ X )
= ( if @ nat @ ( ord_less @ nat @ ( vEBT_VEBT_high @ ( if @ nat @ ( X = Mi ) @ ( plus_plus @ nat @ ( times_times @ nat @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ X ) @ ( divide_divide @ nat @ ( suc @ ( suc @ 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 @ ( X = Mi ) @ ( plus_plus @ nat @ ( times_times @ nat @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ X ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( vEBT_VEBT_low @ ( if @ nat @ ( X = Mi ) @ ( plus_plus @ nat @ ( times_times @ nat @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ X ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( if @ nat @ ( vEBT_VEBT_minNull @ ( vEBT_vebt_delete @ ( nth @ vEBT_VEBT @ TreeList @ ( vEBT_VEBT_high @ ( if @ nat @ ( X = Mi ) @ ( plus_plus @ nat @ ( times_times @ nat @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ X ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( vEBT_VEBT_low @ ( if @ nat @ ( X = Mi ) @ ( plus_plus @ nat @ ( times_times @ nat @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ X ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) @ ( vEBT_V1232361888498592333_e_t_e @ Summary @ ( vEBT_VEBT_high @ ( if @ nat @ ( X = Mi ) @ ( plus_plus @ nat @ ( times_times @ nat @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ X ) @ ( divide_divide @ nat @ ( suc @ ( suc @ 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_832_ceiling__log__nat__eq__if,axiom,
! [B2: nat,N3: nat,K: nat] :
( ( ord_less @ nat @ ( power_power @ nat @ B2 @ N3 ) @ K )
=> ( ( ord_less_eq @ nat @ K @ ( power_power @ nat @ B2 @ ( plus_plus @ nat @ N3 @ ( one_one @ nat ) ) ) )
=> ( ( ord_less_eq @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ B2 )
=> ( ( archimedean_ceiling @ real @ ( log @ ( semiring_1_of_nat @ real @ B2 ) @ ( semiring_1_of_nat @ real @ K ) ) )
= ( plus_plus @ int @ ( semiring_1_of_nat @ int @ N3 ) @ ( one_one @ int ) ) ) ) ) ) ).
% ceiling_log_nat_eq_if
thf(fact_833_ceiling__log2__div2,axiom,
! [N3: nat] :
( ( ord_less_eq @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N3 )
=> ( ( archimedean_ceiling @ real @ ( log @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) @ ( semiring_1_of_nat @ real @ N3 ) ) )
= ( 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 @ N3 @ ( one_one @ nat ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ ( one_one @ nat ) ) ) ) ) @ ( one_one @ int ) ) ) ) ).
% ceiling_log2_div2
thf(fact_834_pred__bound__height,axiom,
! [T2: vEBT_VEBT,N3: nat,X: nat] :
( ( vEBT_invar_vebt @ T2 @ N3 )
=> ( ord_less_eq @ nat @ ( vEBT_T_p_r_e_d @ T2 @ X ) @ ( times_times @ nat @ ( plus_plus @ nat @ ( one_one @ nat ) @ ( vEBT_VEBT_height @ T2 ) ) @ ( numeral_numeral @ nat @ ( bit1 @ ( bit0 @ ( bit1 @ ( bit1 @ one2 ) ) ) ) ) ) ) ) ).
% pred_bound_height
thf(fact_835_insert__bound__height,axiom,
! [T2: vEBT_VEBT,N3: nat,X: nat] :
( ( vEBT_invar_vebt @ T2 @ N3 )
=> ( ord_less_eq @ nat @ ( vEBT_T_i_n_s_e_r_t @ T2 @ X ) @ ( times_times @ nat @ ( plus_plus @ nat @ ( one_one @ nat ) @ ( vEBT_VEBT_height @ T2 ) ) @ ( numeral_numeral @ nat @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit0 @ one2 ) ) ) ) ) ) ) ) ).
% insert_bound_height
thf(fact_836_vebt__delete_Osimps_I7_J,axiom,
! [X: nat,Mi: nat,Ma: nat,Va: nat,TreeList: list @ vEBT_VEBT,Summary: vEBT_VEBT] :
( ( ( ( ord_less @ nat @ X @ Mi )
| ( ord_less @ nat @ Ma @ X ) )
=> ( ( vEBT_vebt_delete @ ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi @ Ma ) ) @ ( suc @ ( suc @ Va ) ) @ TreeList @ Summary ) @ X )
= ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi @ Ma ) ) @ ( suc @ ( suc @ Va ) ) @ TreeList @ Summary ) ) )
& ( ~ ( ( ord_less @ nat @ X @ Mi )
| ( ord_less @ nat @ Ma @ X ) )
=> ( ( ( ( X = Mi )
& ( X = Ma ) )
=> ( ( vEBT_vebt_delete @ ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi @ Ma ) ) @ ( suc @ ( suc @ Va ) ) @ TreeList @ Summary ) @ X )
= ( vEBT_Node @ ( none @ ( product_prod @ nat @ nat ) ) @ ( suc @ ( suc @ Va ) ) @ TreeList @ Summary ) ) )
& ( ~ ( ( X = Mi )
& ( X = Ma ) )
=> ( ( vEBT_vebt_delete @ ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi @ Ma ) ) @ ( suc @ ( suc @ Va ) ) @ TreeList @ Summary ) @ X )
= ( if @ vEBT_VEBT @ ( ord_less @ nat @ ( vEBT_VEBT_high @ ( if @ nat @ ( X = Mi ) @ ( plus_plus @ nat @ ( times_times @ nat @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ X ) @ ( divide_divide @ nat @ ( suc @ ( suc @ 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 @ ( X = Mi ) @ ( plus_plus @ nat @ ( times_times @ nat @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ X ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( vEBT_VEBT_low @ ( if @ nat @ ( X = Mi ) @ ( plus_plus @ nat @ ( times_times @ nat @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ X ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) )
@ ( vEBT_Node
@ ( some @ ( product_prod @ nat @ nat )
@ ( product_Pair @ nat @ nat @ ( if @ nat @ ( X = Mi ) @ ( if @ nat @ ( X = Mi ) @ ( plus_plus @ nat @ ( times_times @ nat @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ X ) @ Mi )
@ ( if @ nat
@ ( ( ( X = Mi )
=> ( ( plus_plus @ nat @ ( times_times @ nat @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) )
= Ma ) )
& ( ( X != Mi )
=> ( X = Ma ) ) )
@ ( if @ nat
@ ( ( vEBT_vebt_maxt @ ( vEBT_vebt_delete @ Summary @ ( vEBT_VEBT_high @ ( if @ nat @ ( X = Mi ) @ ( plus_plus @ nat @ ( times_times @ nat @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ X ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) )
= ( none @ nat ) )
@ ( if @ nat @ ( X = Mi ) @ ( if @ nat @ ( X = Mi ) @ ( plus_plus @ nat @ ( times_times @ nat @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ X ) @ Mi )
@ ( plus_plus @ nat @ ( times_times @ nat @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_maxt @ ( vEBT_vebt_delete @ Summary @ ( vEBT_VEBT_high @ ( if @ nat @ ( X = Mi ) @ ( plus_plus @ nat @ ( times_times @ nat @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ X ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_maxt @ ( nth @ vEBT_VEBT @ ( list_update @ vEBT_VEBT @ TreeList @ ( vEBT_VEBT_high @ ( if @ nat @ ( X = Mi ) @ ( plus_plus @ nat @ ( times_times @ nat @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ X ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( vEBT_vebt_delete @ ( nth @ vEBT_VEBT @ TreeList @ ( vEBT_VEBT_high @ ( if @ nat @ ( X = Mi ) @ ( plus_plus @ nat @ ( times_times @ nat @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ X ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( vEBT_VEBT_low @ ( if @ nat @ ( X = Mi ) @ ( plus_plus @ nat @ ( times_times @ nat @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ X ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_maxt @ ( vEBT_vebt_delete @ Summary @ ( vEBT_VEBT_high @ ( if @ nat @ ( X = Mi ) @ ( plus_plus @ nat @ ( times_times @ nat @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ X ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) ) ) ) ) ) )
@ Ma ) ) )
@ ( suc @ ( suc @ Va ) )
@ ( list_update @ vEBT_VEBT @ TreeList @ ( vEBT_VEBT_high @ ( if @ nat @ ( X = Mi ) @ ( plus_plus @ nat @ ( times_times @ nat @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ X ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( vEBT_vebt_delete @ ( nth @ vEBT_VEBT @ TreeList @ ( vEBT_VEBT_high @ ( if @ nat @ ( X = Mi ) @ ( plus_plus @ nat @ ( times_times @ nat @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ X ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( vEBT_VEBT_low @ ( if @ nat @ ( X = Mi ) @ ( plus_plus @ nat @ ( times_times @ nat @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ X ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) )
@ ( vEBT_vebt_delete @ Summary @ ( vEBT_VEBT_high @ ( if @ nat @ ( X = Mi ) @ ( plus_plus @ nat @ ( times_times @ nat @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ X ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) )
@ ( vEBT_Node
@ ( some @ ( product_prod @ nat @ nat )
@ ( product_Pair @ nat @ nat @ ( if @ nat @ ( X = Mi ) @ ( if @ nat @ ( X = Mi ) @ ( plus_plus @ nat @ ( times_times @ nat @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ X ) @ Mi )
@ ( if @ nat
@ ( ( ( X = Mi )
=> ( ( plus_plus @ nat @ ( times_times @ nat @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) )
= Ma ) )
& ( ( X != Mi )
=> ( X = Ma ) ) )
@ ( plus_plus @ nat @ ( times_times @ nat @ ( vEBT_VEBT_high @ ( if @ nat @ ( X = Mi ) @ ( plus_plus @ nat @ ( times_times @ nat @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ X ) @ ( divide_divide @ nat @ ( suc @ ( suc @ 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 @ ( X = Mi ) @ ( plus_plus @ nat @ ( times_times @ nat @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ X ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( vEBT_vebt_delete @ ( nth @ vEBT_VEBT @ TreeList @ ( vEBT_VEBT_high @ ( if @ nat @ ( X = Mi ) @ ( plus_plus @ nat @ ( times_times @ nat @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ X ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( vEBT_VEBT_low @ ( if @ nat @ ( X = Mi ) @ ( plus_plus @ nat @ ( times_times @ nat @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ X ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) @ ( vEBT_VEBT_high @ ( if @ nat @ ( X = Mi ) @ ( plus_plus @ nat @ ( times_times @ nat @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ X ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) ) )
@ Ma ) ) )
@ ( suc @ ( suc @ Va ) )
@ ( list_update @ vEBT_VEBT @ TreeList @ ( vEBT_VEBT_high @ ( if @ nat @ ( X = Mi ) @ ( plus_plus @ nat @ ( times_times @ nat @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ X ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( vEBT_vebt_delete @ ( nth @ vEBT_VEBT @ TreeList @ ( vEBT_VEBT_high @ ( if @ nat @ ( X = Mi ) @ ( plus_plus @ nat @ ( times_times @ nat @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ X ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( vEBT_VEBT_low @ ( if @ nat @ ( X = Mi ) @ ( plus_plus @ nat @ ( times_times @ nat @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ X ) @ ( divide_divide @ nat @ ( suc @ ( suc @ 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_837_numeral__le__ceiling,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [V: num,X: A] :
( ( ord_less_eq @ int @ ( numeral_numeral @ int @ V ) @ ( archimedean_ceiling @ A @ X ) )
= ( ord_less @ A @ ( minus_minus @ A @ ( numeral_numeral @ A @ V ) @ ( one_one @ A ) ) @ X ) ) ) ).
% numeral_le_ceiling
thf(fact_838_ceiling__less__numeral,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [X: A,V: num] :
( ( ord_less @ int @ ( archimedean_ceiling @ A @ X ) @ ( numeral_numeral @ int @ V ) )
= ( ord_less_eq @ A @ X @ ( minus_minus @ A @ ( numeral_numeral @ A @ V ) @ ( one_one @ A ) ) ) ) ) ).
% ceiling_less_numeral
thf(fact_839_vebt__member_Osimps_I5_J,axiom,
! [Mi: nat,Ma: nat,Va: nat,TreeList: list @ vEBT_VEBT,Summary: vEBT_VEBT,X: nat] :
( ( vEBT_vebt_member @ ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi @ Ma ) ) @ ( suc @ ( suc @ Va ) ) @ TreeList @ Summary ) @ X )
= ( ( X != Mi )
=> ( ( X != Ma )
=> ( ~ ( ord_less @ nat @ X @ Mi )
& ( ~ ( ord_less @ nat @ X @ Mi )
=> ( ~ ( ord_less @ nat @ Ma @ X )
& ( ~ ( ord_less @ nat @ Ma @ X )
=> ( ( ( ord_less @ nat @ ( vEBT_VEBT_high @ X @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( size_size @ ( list @ vEBT_VEBT ) @ TreeList ) )
=> ( vEBT_vebt_member @ ( nth @ vEBT_VEBT @ TreeList @ ( vEBT_VEBT_high @ X @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( vEBT_VEBT_low @ X @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) )
& ( ord_less @ nat @ ( vEBT_VEBT_high @ X @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( size_size @ ( list @ vEBT_VEBT ) @ TreeList ) ) ) ) ) ) ) ) ) ) ).
% vebt_member.simps(5)
thf(fact_840_ceiling__diff__one,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [X: A] :
( ( archimedean_ceiling @ A @ ( minus_minus @ A @ X @ ( one_one @ A ) ) )
= ( minus_minus @ int @ ( archimedean_ceiling @ A @ X ) @ ( one_one @ int ) ) ) ) ).
% ceiling_diff_one
thf(fact_841_ceiling__diff__numeral,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [X: A,V: num] :
( ( archimedean_ceiling @ A @ ( minus_minus @ A @ X @ ( numeral_numeral @ A @ V ) ) )
= ( minus_minus @ int @ ( archimedean_ceiling @ A @ X ) @ ( numeral_numeral @ int @ V ) ) ) ) ).
% ceiling_diff_numeral
thf(fact_842_ceiling__add__one,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [X: A] :
( ( archimedean_ceiling @ A @ ( plus_plus @ A @ X @ ( one_one @ A ) ) )
= ( plus_plus @ int @ ( archimedean_ceiling @ A @ X ) @ ( one_one @ int ) ) ) ) ).
% ceiling_add_one
thf(fact_843_one__less__ceiling,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [X: A] :
( ( ord_less @ int @ ( one_one @ int ) @ ( archimedean_ceiling @ A @ X ) )
= ( ord_less @ A @ ( one_one @ A ) @ X ) ) ) ).
% one_less_ceiling
thf(fact_844_ceiling__le__one,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [X: A] :
( ( ord_less_eq @ int @ ( archimedean_ceiling @ A @ X ) @ ( one_one @ int ) )
= ( ord_less_eq @ A @ X @ ( one_one @ A ) ) ) ) ).
% ceiling_le_one
thf(fact_845_ceiling__add__numeral,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [X: A,V: num] :
( ( archimedean_ceiling @ A @ ( plus_plus @ A @ X @ ( numeral_numeral @ A @ V ) ) )
= ( plus_plus @ int @ ( archimedean_ceiling @ A @ X ) @ ( numeral_numeral @ int @ V ) ) ) ) ).
% ceiling_add_numeral
thf(fact_846_ceiling__numeral,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [V: num] :
( ( archimedean_ceiling @ A @ ( numeral_numeral @ A @ V ) )
= ( numeral_numeral @ int @ V ) ) ) ).
% ceiling_numeral
thf(fact_847_ceiling__one,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ( ( archimedean_ceiling @ A @ ( one_one @ A ) )
= ( one_one @ int ) ) ) ).
% ceiling_one
thf(fact_848_ceiling__le__numeral,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [X: A,V: num] :
( ( ord_less_eq @ int @ ( archimedean_ceiling @ A @ X ) @ ( numeral_numeral @ int @ V ) )
= ( ord_less_eq @ A @ X @ ( numeral_numeral @ A @ V ) ) ) ) ).
% ceiling_le_numeral
thf(fact_849_numeral__less__ceiling,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [V: num,X: A] :
( ( ord_less @ int @ ( numeral_numeral @ int @ V ) @ ( archimedean_ceiling @ A @ X ) )
= ( ord_less @ A @ ( numeral_numeral @ A @ V ) @ X ) ) ) ).
% numeral_less_ceiling
thf(fact_850_real__arch__simple,axiom,
! [A: $tType] :
( ( archim462609752435547400_field @ A )
=> ! [X: A] :
? [N: nat] : ( ord_less_eq @ A @ X @ ( semiring_1_of_nat @ A @ N ) ) ) ).
% real_arch_simple
thf(fact_851_reals__Archimedean2,axiom,
! [A: $tType] :
( ( archim462609752435547400_field @ A )
=> ! [X: A] :
? [N: nat] : ( ord_less @ A @ X @ ( semiring_1_of_nat @ A @ N ) ) ) ).
% reals_Archimedean2
thf(fact_852_ceiling__mono,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [Y: A,X: A] :
( ( ord_less_eq @ A @ Y @ X )
=> ( ord_less_eq @ int @ ( archimedean_ceiling @ A @ Y ) @ ( archimedean_ceiling @ A @ X ) ) ) ) ).
% ceiling_mono
thf(fact_853_ceiling__less__cancel,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [X: A,Y: A] :
( ( ord_less @ int @ ( archimedean_ceiling @ A @ X ) @ ( archimedean_ceiling @ A @ Y ) )
=> ( ord_less @ A @ X @ Y ) ) ) ).
% ceiling_less_cancel
thf(fact_854_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_855_vebt__delete_Osimps_I4_J,axiom,
! [Deg: nat,TreeList: list @ vEBT_VEBT,Summary: vEBT_VEBT,Uu2: nat] :
( ( vEBT_vebt_delete @ ( vEBT_Node @ ( none @ ( product_prod @ nat @ nat ) ) @ Deg @ TreeList @ Summary ) @ Uu2 )
= ( vEBT_Node @ ( none @ ( product_prod @ nat @ nat ) ) @ Deg @ TreeList @ Summary ) ) ).
% vebt_delete.simps(4)
thf(fact_856_VEBT__internal_OminNull_Osimps_I4_J,axiom,
! [Uw2: nat,Ux: list @ vEBT_VEBT,Uy: vEBT_VEBT] : ( vEBT_VEBT_minNull @ ( vEBT_Node @ ( none @ ( product_prod @ nat @ nat ) ) @ Uw2 @ Ux @ Uy ) ) ).
% VEBT_internal.minNull.simps(4)
thf(fact_857_vebt__member_Osimps_I2_J,axiom,
! [Uu2: nat,Uv2: list @ vEBT_VEBT,Uw2: vEBT_VEBT,X: nat] :
~ ( vEBT_vebt_member @ ( vEBT_Node @ ( none @ ( product_prod @ nat @ nat ) ) @ Uu2 @ Uv2 @ Uw2 ) @ X ) ).
% vebt_member.simps(2)
thf(fact_858_ceiling__add__le,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [X: A,Y: A] : ( ord_less_eq @ int @ ( archimedean_ceiling @ A @ ( plus_plus @ A @ X @ Y ) ) @ ( plus_plus @ int @ ( archimedean_ceiling @ A @ X ) @ ( archimedean_ceiling @ A @ Y ) ) ) ) ).
% ceiling_add_le
thf(fact_859_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,X: nat] :
( ( vEBT_T_i_n_s_e_r_t @ ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi @ Ma ) ) @ ( suc @ ( suc @ Va ) ) @ TreeList @ Summary ) @ X )
= ( plus_plus @ nat @ ( numeral_numeral @ nat @ ( bit1 @ ( bit1 @ ( bit0 @ ( bit0 @ one2 ) ) ) ) )
@ ( if @ nat
@ ( ( ord_less @ nat @ ( vEBT_VEBT_high @ ( if @ nat @ ( ord_less @ nat @ X @ Mi ) @ Mi @ X ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( size_size @ ( list @ vEBT_VEBT ) @ TreeList ) )
& ~ ( ( X = Mi )
| ( X = Ma ) ) )
@ ( plus_plus @ nat @ ( plus_plus @ nat @ ( vEBT_T_i_n_s_e_r_t @ ( nth @ vEBT_VEBT @ TreeList @ ( vEBT_VEBT_high @ ( if @ nat @ ( ord_less @ nat @ X @ Mi ) @ Mi @ X ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( vEBT_VEBT_low @ ( if @ nat @ ( ord_less @ nat @ X @ Mi ) @ Mi @ X ) @ ( 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 @ X @ Mi ) @ Mi @ X ) @ ( 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 @ X @ Mi ) @ Mi @ X ) @ ( 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 @ X @ Mi ) @ Mi @ X ) @ ( 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_860_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,X: nat] :
( ( vEBT_T_i_n_s_e_r_t2 @ ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi @ Ma ) ) @ ( suc @ ( suc @ Va ) ) @ TreeList @ Summary ) @ X )
= ( if @ nat
@ ( ( ord_less @ nat @ ( vEBT_VEBT_high @ ( if @ nat @ ( ord_less @ nat @ X @ Mi ) @ Mi @ X ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( size_size @ ( list @ vEBT_VEBT ) @ TreeList ) )
& ~ ( ( X = Mi )
| ( X = Ma ) ) )
@ ( plus_plus @ nat @ ( vEBT_T_i_n_s_e_r_t2 @ ( nth @ vEBT_VEBT @ TreeList @ ( vEBT_VEBT_high @ ( if @ nat @ ( ord_less @ nat @ X @ Mi ) @ Mi @ X ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( vEBT_VEBT_low @ ( if @ nat @ ( ord_less @ nat @ X @ Mi ) @ Mi @ X ) @ ( 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 @ X @ Mi ) @ Mi @ X ) @ ( 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 @ X @ Mi ) @ Mi @ X ) @ ( 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_861_pred__empty,axiom,
! [T2: vEBT_VEBT,N3: nat,X: nat] :
( ( vEBT_invar_vebt @ T2 @ N3 )
=> ( ( ( vEBT_vebt_pred @ T2 @ X )
= ( none @ nat ) )
= ( ( collect @ nat
@ ^ [Y2: nat] :
( ( vEBT_vebt_member @ T2 @ Y2 )
& ( ord_less @ nat @ Y2 @ X ) ) )
= ( bot_bot @ ( set @ nat ) ) ) ) ) ).
% pred_empty
thf(fact_862_succ__empty,axiom,
! [T2: vEBT_VEBT,N3: nat,X: nat] :
( ( vEBT_invar_vebt @ T2 @ N3 )
=> ( ( ( vEBT_vebt_succ @ T2 @ X )
= ( none @ nat ) )
= ( ( collect @ nat
@ ^ [Y2: nat] :
( ( vEBT_vebt_member @ T2 @ Y2 )
& ( ord_less @ nat @ X @ Y2 ) ) )
= ( bot_bot @ ( set @ nat ) ) ) ) ) ).
% succ_empty
thf(fact_863_log__ceil__idem,axiom,
! [X: real] :
( ( ord_less_eq @ real @ ( one_one @ real ) @ X )
=> ( ( archimedean_ceiling @ real @ ( log @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) @ X ) )
= ( archimedean_ceiling @ real @ ( log @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) @ ( ring_1_of_int @ real @ ( archimedean_ceiling @ real @ X ) ) ) ) ) ) ).
% log_ceil_idem
thf(fact_864_VEBT__internal_Omembermima_Osimps_I4_J,axiom,
! [Mi: nat,Ma: nat,V: nat,TreeList: list @ vEBT_VEBT,Vc: vEBT_VEBT,X: nat] :
( ( vEBT_VEBT_membermima @ ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi @ Ma ) ) @ ( suc @ V ) @ TreeList @ Vc ) @ X )
= ( ( X = Mi )
| ( X = Ma )
| ( ( ( ord_less @ nat @ ( vEBT_VEBT_high @ X @ ( 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 @ X @ ( divide_divide @ nat @ ( suc @ V ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( vEBT_VEBT_low @ X @ ( divide_divide @ nat @ ( suc @ V ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) )
& ( ord_less @ nat @ ( vEBT_VEBT_high @ X @ ( divide_divide @ nat @ ( suc @ V ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( size_size @ ( list @ vEBT_VEBT ) @ TreeList ) ) ) ) ) ).
% VEBT_internal.membermima.simps(4)
thf(fact_865_VEBT__internal_Onaive__member_Osimps_I3_J,axiom,
! [Uy: option @ ( product_prod @ nat @ nat ),V: nat,TreeList: list @ vEBT_VEBT,S: vEBT_VEBT,X: nat] :
( ( vEBT_V5719532721284313246member @ ( vEBT_Node @ Uy @ ( suc @ V ) @ TreeList @ S ) @ X )
= ( ( ( ord_less @ nat @ ( vEBT_VEBT_high @ X @ ( 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 @ X @ ( divide_divide @ nat @ ( suc @ V ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( vEBT_VEBT_low @ X @ ( divide_divide @ nat @ ( suc @ V ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) )
& ( ord_less @ nat @ ( vEBT_VEBT_high @ X @ ( divide_divide @ nat @ ( suc @ V ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( size_size @ ( list @ vEBT_VEBT ) @ TreeList ) ) ) ) ).
% VEBT_internal.naive_member.simps(3)
thf(fact_866_ceiling__log__nat__eq__powr__iff,axiom,
! [B2: nat,K: nat,N3: nat] :
( ( ord_less_eq @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ B2 )
=> ( ( ord_less @ nat @ ( zero_zero @ nat ) @ K )
=> ( ( ( archimedean_ceiling @ real @ ( log @ ( semiring_1_of_nat @ real @ B2 ) @ ( semiring_1_of_nat @ real @ K ) ) )
= ( plus_plus @ int @ ( semiring_1_of_nat @ int @ N3 ) @ ( one_one @ int ) ) )
= ( ( ord_less @ nat @ ( power_power @ nat @ B2 @ N3 ) @ K )
& ( ord_less_eq @ nat @ K @ ( power_power @ nat @ B2 @ ( plus_plus @ nat @ N3 @ ( one_one @ nat ) ) ) ) ) ) ) ) ).
% ceiling_log_nat_eq_powr_iff
thf(fact_867_VEBT__internal_Omembermima_Osimps_I5_J,axiom,
! [V: nat,TreeList: list @ vEBT_VEBT,Vd: vEBT_VEBT,X: nat] :
( ( vEBT_VEBT_membermima @ ( vEBT_Node @ ( none @ ( product_prod @ nat @ nat ) ) @ ( suc @ V ) @ TreeList @ Vd ) @ X )
= ( ( ( ord_less @ nat @ ( vEBT_VEBT_high @ X @ ( 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 @ X @ ( divide_divide @ nat @ ( suc @ V ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( vEBT_VEBT_low @ X @ ( divide_divide @ nat @ ( suc @ V ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) )
& ( ord_less @ nat @ ( vEBT_VEBT_high @ X @ ( divide_divide @ nat @ ( suc @ V ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( size_size @ ( list @ vEBT_VEBT ) @ TreeList ) ) ) ) ).
% VEBT_internal.membermima.simps(5)
thf(fact_868_valid__tree__deg__neq__0,axiom,
! [T2: vEBT_VEBT] :
~ ( vEBT_invar_vebt @ T2 @ ( zero_zero @ nat ) ) ).
% valid_tree_deg_neq_0
thf(fact_869_valid__0__not,axiom,
! [T2: vEBT_VEBT] :
~ ( vEBT_invar_vebt @ T2 @ ( zero_zero @ nat ) ) ).
% valid_0_not
thf(fact_870_buildup__nothing__in__min__max,axiom,
! [N3: nat,X: nat] :
~ ( vEBT_VEBT_membermima @ ( vEBT_vebt_buildup @ N3 ) @ X ) ).
% buildup_nothing_in_min_max
thf(fact_871_buildup__nothing__in__leaf,axiom,
! [N3: nat,X: nat] :
~ ( vEBT_V5719532721284313246member @ ( vEBT_vebt_buildup @ N3 ) @ X ) ).
% buildup_nothing_in_leaf
thf(fact_872_deg__not__0,axiom,
! [T2: vEBT_VEBT,N3: nat] :
( ( vEBT_invar_vebt @ T2 @ N3 )
=> ( ord_less @ nat @ ( zero_zero @ nat ) @ N3 ) ) ).
% deg_not_0
thf(fact_873_T__vebt__buildupi__gq__0,axiom,
! [N3: nat] : ( ord_less @ nat @ ( zero_zero @ nat ) @ ( vEBT_V441764108873111860ildupi @ N3 ) ) ).
% T_vebt_buildupi_gq_0
thf(fact_874_buildup__gives__empty,axiom,
! [N3: nat] :
( ( vEBT_VEBT_set_vebt @ ( vEBT_vebt_buildup @ N3 ) )
= ( bot_bot @ ( set @ nat ) ) ) ).
% buildup_gives_empty
thf(fact_875_both__member__options__def,axiom,
( vEBT_V8194947554948674370ptions
= ( ^ [T3: vEBT_VEBT,X2: nat] :
( ( vEBT_V5719532721284313246member @ T3 @ X2 )
| ( vEBT_VEBT_membermima @ T3 @ X2 ) ) ) ) ).
% both_member_options_def
thf(fact_876_buildup__gives__valid,axiom,
! [N3: nat] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ N3 )
=> ( vEBT_invar_vebt @ ( vEBT_vebt_buildup @ N3 ) @ N3 ) ) ).
% buildup_gives_valid
thf(fact_877_member__valid__both__member__options,axiom,
! [Tree: vEBT_VEBT,N3: nat,X: nat] :
( ( vEBT_invar_vebt @ Tree @ N3 )
=> ( ( vEBT_vebt_member @ Tree @ X )
=> ( ( vEBT_V5719532721284313246member @ Tree @ X )
| ( vEBT_VEBT_membermima @ Tree @ X ) ) ) ) ).
% member_valid_both_member_options
thf(fact_878_mint__corr__help__empty,axiom,
! [T2: vEBT_VEBT,N3: nat] :
( ( vEBT_invar_vebt @ T2 @ N3 )
=> ( ( ( vEBT_vebt_mint @ T2 )
= ( none @ nat ) )
=> ( ( vEBT_VEBT_set_vebt @ T2 )
= ( bot_bot @ ( set @ nat ) ) ) ) ) ).
% mint_corr_help_empty
thf(fact_879_maxt__corr__help__empty,axiom,
! [T2: vEBT_VEBT,N3: nat] :
( ( vEBT_invar_vebt @ T2 @ N3 )
=> ( ( ( vEBT_vebt_maxt @ T2 )
= ( none @ nat ) )
=> ( ( vEBT_VEBT_set_vebt @ T2 )
= ( bot_bot @ ( set @ nat ) ) ) ) ) ).
% maxt_corr_help_empty
thf(fact_880_le__zero__eq,axiom,
! [A: $tType] :
( ( canoni5634975068530333245id_add @ A )
=> ! [N3: A] :
( ( ord_less_eq @ A @ N3 @ ( zero_zero @ A ) )
= ( N3
= ( zero_zero @ A ) ) ) ) ).
% le_zero_eq
thf(fact_881_not__gr__zero,axiom,
! [A: $tType] :
( ( canoni5634975068530333245id_add @ A )
=> ! [N3: A] :
( ( ~ ( ord_less @ A @ ( zero_zero @ A ) @ N3 ) )
= ( N3
= ( zero_zero @ A ) ) ) ) ).
% not_gr_zero
thf(fact_882_double__eq__0__iff,axiom,
! [A: $tType] :
( ( linord5086331880401160121up_add @ A )
=> ! [A2: A] :
( ( ( plus_plus @ A @ A2 @ A2 )
= ( zero_zero @ A ) )
= ( A2
= ( zero_zero @ A ) ) ) ) ).
% double_eq_0_iff
thf(fact_883_add_Oright__neutral,axiom,
! [A: $tType] :
( ( monoid_add @ A )
=> ! [A2: A] :
( ( plus_plus @ A @ A2 @ ( zero_zero @ A ) )
= A2 ) ) ).
% add.right_neutral
thf(fact_884_double__zero__sym,axiom,
! [A: $tType] :
( ( linord5086331880401160121up_add @ A )
=> ! [A2: A] :
( ( ( zero_zero @ A )
= ( plus_plus @ A @ A2 @ A2 ) )
= ( A2
= ( zero_zero @ A ) ) ) ) ).
% double_zero_sym
thf(fact_885_add__cancel__left__left,axiom,
! [A: $tType] :
( ( cancel1802427076303600483id_add @ A )
=> ! [B2: A,A2: A] :
( ( ( plus_plus @ A @ B2 @ A2 )
= A2 )
= ( B2
= ( zero_zero @ A ) ) ) ) ).
% add_cancel_left_left
thf(fact_886_add__cancel__left__right,axiom,
! [A: $tType] :
( ( cancel1802427076303600483id_add @ A )
=> ! [A2: A,B2: A] :
( ( ( plus_plus @ A @ A2 @ B2 )
= A2 )
= ( B2
= ( zero_zero @ A ) ) ) ) ).
% add_cancel_left_right
thf(fact_887_add__cancel__right__left,axiom,
! [A: $tType] :
( ( cancel1802427076303600483id_add @ A )
=> ! [A2: A,B2: A] :
( ( A2
= ( plus_plus @ A @ B2 @ A2 ) )
= ( B2
= ( zero_zero @ A ) ) ) ) ).
% add_cancel_right_left
thf(fact_888_add__cancel__right__right,axiom,
! [A: $tType] :
( ( cancel1802427076303600483id_add @ A )
=> ! [A2: A,B2: A] :
( ( A2
= ( plus_plus @ A @ A2 @ B2 ) )
= ( B2
= ( zero_zero @ A ) ) ) ) ).
% add_cancel_right_right
thf(fact_889_add__eq__0__iff__both__eq__0,axiom,
! [A: $tType] :
( ( canoni5634975068530333245id_add @ A )
=> ! [X: A,Y: A] :
( ( ( plus_plus @ A @ X @ Y )
= ( zero_zero @ A ) )
= ( ( X
= ( zero_zero @ A ) )
& ( Y
= ( zero_zero @ A ) ) ) ) ) ).
% add_eq_0_iff_both_eq_0
thf(fact_890_zero__eq__add__iff__both__eq__0,axiom,
! [A: $tType] :
( ( canoni5634975068530333245id_add @ A )
=> ! [X: A,Y: A] :
( ( ( zero_zero @ A )
= ( plus_plus @ A @ X @ Y ) )
= ( ( X
= ( zero_zero @ A ) )
& ( Y
= ( zero_zero @ A ) ) ) ) ) ).
% zero_eq_add_iff_both_eq_0
thf(fact_891_add__0,axiom,
! [A: $tType] :
( ( monoid_add @ A )
=> ! [A2: A] :
( ( plus_plus @ A @ ( zero_zero @ A ) @ A2 )
= A2 ) ) ).
% add_0
thf(fact_892_mult__zero__left,axiom,
! [A: $tType] :
( ( mult_zero @ A )
=> ! [A2: A] :
( ( times_times @ A @ ( zero_zero @ A ) @ A2 )
= ( zero_zero @ A ) ) ) ).
% mult_zero_left
thf(fact_893_mult__zero__right,axiom,
! [A: $tType] :
( ( mult_zero @ A )
=> ! [A2: A] :
( ( times_times @ A @ A2 @ ( zero_zero @ A ) )
= ( zero_zero @ A ) ) ) ).
% mult_zero_right
thf(fact_894_mult__eq__0__iff,axiom,
! [A: $tType] :
( ( semiri3467727345109120633visors @ A )
=> ! [A2: A,B2: A] :
( ( ( times_times @ A @ A2 @ B2 )
= ( zero_zero @ A ) )
= ( ( A2
= ( zero_zero @ A ) )
| ( B2
= ( zero_zero @ A ) ) ) ) ) ).
% mult_eq_0_iff
thf(fact_895_mult__cancel__left,axiom,
! [A: $tType] :
( ( semiri6575147826004484403cancel @ A )
=> ! [C2: A,A2: A,B2: A] :
( ( ( times_times @ A @ C2 @ A2 )
= ( times_times @ A @ C2 @ B2 ) )
= ( ( C2
= ( zero_zero @ A ) )
| ( A2 = B2 ) ) ) ) ).
% mult_cancel_left
thf(fact_896_mult__cancel__right,axiom,
! [A: $tType] :
( ( semiri6575147826004484403cancel @ A )
=> ! [A2: A,C2: A,B2: A] :
( ( ( times_times @ A @ A2 @ C2 )
= ( times_times @ A @ B2 @ C2 ) )
= ( ( C2
= ( zero_zero @ A ) )
| ( A2 = B2 ) ) ) ) ).
% mult_cancel_right
thf(fact_897_cancel__comm__monoid__add__class_Odiff__cancel,axiom,
! [A: $tType] :
( ( cancel1802427076303600483id_add @ A )
=> ! [A2: A] :
( ( minus_minus @ A @ A2 @ A2 )
= ( zero_zero @ A ) ) ) ).
% cancel_comm_monoid_add_class.diff_cancel
thf(fact_898_diff__zero,axiom,
! [A: $tType] :
( ( cancel1802427076303600483id_add @ A )
=> ! [A2: A] :
( ( minus_minus @ A @ A2 @ ( zero_zero @ A ) )
= A2 ) ) ).
% diff_zero
thf(fact_899_zero__diff,axiom,
! [A: $tType] :
( ( comm_monoid_diff @ A )
=> ! [A2: A] :
( ( minus_minus @ A @ ( zero_zero @ A ) @ A2 )
= ( zero_zero @ A ) ) ) ).
% zero_diff
thf(fact_900_diff__0__right,axiom,
! [A: $tType] :
( ( group_add @ A )
=> ! [A2: A] :
( ( minus_minus @ A @ A2 @ ( zero_zero @ A ) )
= A2 ) ) ).
% diff_0_right
thf(fact_901_diff__self,axiom,
! [A: $tType] :
( ( group_add @ A )
=> ! [A2: A] :
( ( minus_minus @ A @ A2 @ A2 )
= ( zero_zero @ A ) ) ) ).
% diff_self
thf(fact_902_div__0,axiom,
! [A: $tType] :
( ( semidom_divide @ A )
=> ! [A2: A] :
( ( divide_divide @ A @ ( zero_zero @ A ) @ A2 )
= ( zero_zero @ A ) ) ) ).
% div_0
thf(fact_903_div__by__0,axiom,
! [A: $tType] :
( ( semidom_divide @ A )
=> ! [A2: A] :
( ( divide_divide @ A @ A2 @ ( zero_zero @ A ) )
= ( zero_zero @ A ) ) ) ).
% div_by_0
thf(fact_904_bits__div__0,axiom,
! [A: $tType] :
( ( bit_semiring_bits @ A )
=> ! [A2: A] :
( ( divide_divide @ A @ ( zero_zero @ A ) @ A2 )
= ( zero_zero @ A ) ) ) ).
% bits_div_0
thf(fact_905_bits__div__by__0,axiom,
! [A: $tType] :
( ( bit_semiring_bits @ A )
=> ! [A2: A] :
( ( divide_divide @ A @ A2 @ ( zero_zero @ A ) )
= ( zero_zero @ A ) ) ) ).
% bits_div_by_0
thf(fact_906_divide__eq__0__iff,axiom,
! [A: $tType] :
( ( field @ A )
=> ! [A2: A,B2: A] :
( ( ( divide_divide @ A @ A2 @ B2 )
= ( zero_zero @ A ) )
= ( ( A2
= ( zero_zero @ A ) )
| ( B2
= ( zero_zero @ A ) ) ) ) ) ).
% divide_eq_0_iff
thf(fact_907_divide__cancel__left,axiom,
! [A: $tType] :
( ( field @ A )
=> ! [C2: A,A2: A,B2: A] :
( ( ( divide_divide @ A @ C2 @ A2 )
= ( divide_divide @ A @ C2 @ B2 ) )
= ( ( C2
= ( zero_zero @ A ) )
| ( A2 = B2 ) ) ) ) ).
% divide_cancel_left
thf(fact_908_divide__cancel__right,axiom,
! [A: $tType] :
( ( field @ A )
=> ! [A2: A,C2: A,B2: A] :
( ( ( divide_divide @ A @ A2 @ C2 )
= ( divide_divide @ A @ B2 @ C2 ) )
= ( ( C2
= ( zero_zero @ A ) )
| ( A2 = B2 ) ) ) ) ).
% divide_cancel_right
thf(fact_909_division__ring__divide__zero,axiom,
! [A: $tType] :
( ( division_ring @ A )
=> ! [A2: A] :
( ( divide_divide @ A @ A2 @ ( zero_zero @ A ) )
= ( zero_zero @ A ) ) ) ).
% division_ring_divide_zero
thf(fact_910_bot__nat__0_Onot__eq__extremum,axiom,
! [A2: nat] :
( ( A2
!= ( zero_zero @ nat ) )
= ( ord_less @ nat @ ( zero_zero @ nat ) @ A2 ) ) ).
% bot_nat_0.not_eq_extremum
thf(fact_911_neq0__conv,axiom,
! [N3: nat] :
( ( N3
!= ( zero_zero @ nat ) )
= ( ord_less @ nat @ ( zero_zero @ nat ) @ N3 ) ) ).
% neq0_conv
thf(fact_912_less__nat__zero__code,axiom,
! [N3: nat] :
~ ( ord_less @ nat @ N3 @ ( zero_zero @ nat ) ) ).
% less_nat_zero_code
thf(fact_913_add__is__0,axiom,
! [M: nat,N3: nat] :
( ( ( plus_plus @ nat @ M @ N3 )
= ( zero_zero @ nat ) )
= ( ( M
= ( zero_zero @ nat ) )
& ( N3
= ( zero_zero @ nat ) ) ) ) ).
% add_is_0
thf(fact_914_Nat_Oadd__0__right,axiom,
! [M: nat] :
( ( plus_plus @ nat @ M @ ( zero_zero @ nat ) )
= M ) ).
% Nat.add_0_right
thf(fact_915_le0,axiom,
! [N3: nat] : ( ord_less_eq @ nat @ ( zero_zero @ nat ) @ N3 ) ).
% le0
thf(fact_916_bot__nat__0_Oextremum,axiom,
! [A2: nat] : ( ord_less_eq @ nat @ ( zero_zero @ nat ) @ A2 ) ).
% bot_nat_0.extremum
thf(fact_917_diff__0__eq__0,axiom,
! [N3: nat] :
( ( minus_minus @ nat @ ( zero_zero @ nat ) @ N3 )
= ( zero_zero @ nat ) ) ).
% diff_0_eq_0
thf(fact_918_diff__self__eq__0,axiom,
! [M: nat] :
( ( minus_minus @ nat @ M @ M )
= ( zero_zero @ nat ) ) ).
% diff_self_eq_0
thf(fact_919_mult__is__0,axiom,
! [M: nat,N3: nat] :
( ( ( times_times @ nat @ M @ N3 )
= ( zero_zero @ nat ) )
= ( ( M
= ( zero_zero @ nat ) )
| ( N3
= ( zero_zero @ nat ) ) ) ) ).
% mult_is_0
thf(fact_920_mult__0__right,axiom,
! [M: nat] :
( ( times_times @ nat @ M @ ( zero_zero @ nat ) )
= ( zero_zero @ nat ) ) ).
% mult_0_right
thf(fact_921_mult__cancel1,axiom,
! [K: nat,M: nat,N3: nat] :
( ( ( times_times @ nat @ K @ M )
= ( times_times @ nat @ K @ N3 ) )
= ( ( M = N3 )
| ( K
= ( zero_zero @ nat ) ) ) ) ).
% mult_cancel1
thf(fact_922_mult__cancel2,axiom,
! [M: nat,K: nat,N3: nat] :
( ( ( times_times @ nat @ M @ K )
= ( times_times @ nat @ N3 @ K ) )
= ( ( M = N3 )
| ( K
= ( zero_zero @ nat ) ) ) ) ).
% mult_cancel2
thf(fact_923_of__int__ceiling__cancel,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [X: A] :
( ( ( ring_1_of_int @ A @ ( archimedean_ceiling @ A @ X ) )
= X )
= ( ? [N2: int] :
( X
= ( ring_1_of_int @ A @ N2 ) ) ) ) ) ).
% of_int_ceiling_cancel
thf(fact_924_max__nat_Oeq__neutr__iff,axiom,
! [A2: nat,B2: nat] :
( ( ( ord_max @ nat @ A2 @ B2 )
= ( zero_zero @ nat ) )
= ( ( A2
= ( zero_zero @ nat ) )
& ( B2
= ( zero_zero @ nat ) ) ) ) ).
% max_nat.eq_neutr_iff
thf(fact_925_max__nat_Oleft__neutral,axiom,
! [A2: nat] :
( ( ord_max @ nat @ ( zero_zero @ nat ) @ A2 )
= A2 ) ).
% max_nat.left_neutral
thf(fact_926_max__nat_Oneutr__eq__iff,axiom,
! [A2: nat,B2: nat] :
( ( ( zero_zero @ nat )
= ( ord_max @ nat @ A2 @ B2 ) )
= ( ( A2
= ( zero_zero @ nat ) )
& ( B2
= ( zero_zero @ nat ) ) ) ) ).
% max_nat.neutr_eq_iff
thf(fact_927_max__nat_Oright__neutral,axiom,
! [A2: nat] :
( ( ord_max @ nat @ A2 @ ( zero_zero @ nat ) )
= A2 ) ).
% max_nat.right_neutral
thf(fact_928_max__0L,axiom,
! [N3: nat] :
( ( ord_max @ nat @ ( zero_zero @ nat ) @ N3 )
= N3 ) ).
% max_0L
thf(fact_929_max__0R,axiom,
! [N3: nat] :
( ( ord_max @ nat @ N3 @ ( zero_zero @ nat ) )
= N3 ) ).
% max_0R
thf(fact_930_zero__le__double__add__iff__zero__le__single__add,axiom,
! [A: $tType] :
( ( linord5086331880401160121up_add @ A )
=> ! [A2: A] :
( ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( plus_plus @ A @ A2 @ A2 ) )
= ( ord_less_eq @ A @ ( zero_zero @ A ) @ A2 ) ) ) ).
% zero_le_double_add_iff_zero_le_single_add
thf(fact_931_double__add__le__zero__iff__single__add__le__zero,axiom,
! [A: $tType] :
( ( linord5086331880401160121up_add @ A )
=> ! [A2: A] :
( ( ord_less_eq @ A @ ( plus_plus @ A @ A2 @ A2 ) @ ( zero_zero @ A ) )
= ( ord_less_eq @ A @ A2 @ ( zero_zero @ A ) ) ) ) ).
% double_add_le_zero_iff_single_add_le_zero
thf(fact_932_le__add__same__cancel2,axiom,
! [A: $tType] :
( ( ordere1937475149494474687imp_le @ A )
=> ! [A2: A,B2: A] :
( ( ord_less_eq @ A @ A2 @ ( plus_plus @ A @ B2 @ A2 ) )
= ( ord_less_eq @ A @ ( zero_zero @ A ) @ B2 ) ) ) ).
% le_add_same_cancel2
thf(fact_933_le__add__same__cancel1,axiom,
! [A: $tType] :
( ( ordere1937475149494474687imp_le @ A )
=> ! [A2: A,B2: A] :
( ( ord_less_eq @ A @ A2 @ ( plus_plus @ A @ A2 @ B2 ) )
= ( ord_less_eq @ A @ ( zero_zero @ A ) @ B2 ) ) ) ).
% le_add_same_cancel1
thf(fact_934_add__le__same__cancel2,axiom,
! [A: $tType] :
( ( ordere1937475149494474687imp_le @ A )
=> ! [A2: A,B2: A] :
( ( ord_less_eq @ A @ ( plus_plus @ A @ A2 @ B2 ) @ B2 )
= ( ord_less_eq @ A @ A2 @ ( zero_zero @ A ) ) ) ) ).
% add_le_same_cancel2
thf(fact_935_add__le__same__cancel1,axiom,
! [A: $tType] :
( ( ordere1937475149494474687imp_le @ A )
=> ! [B2: A,A2: A] :
( ( ord_less_eq @ A @ ( plus_plus @ A @ B2 @ A2 ) @ B2 )
= ( ord_less_eq @ A @ A2 @ ( zero_zero @ A ) ) ) ) ).
% add_le_same_cancel1
thf(fact_936_add__less__same__cancel1,axiom,
! [A: $tType] :
( ( ordere1937475149494474687imp_le @ A )
=> ! [B2: A,A2: A] :
( ( ord_less @ A @ ( plus_plus @ A @ B2 @ A2 ) @ B2 )
= ( ord_less @ A @ A2 @ ( zero_zero @ A ) ) ) ) ).
% add_less_same_cancel1
thf(fact_937_add__less__same__cancel2,axiom,
! [A: $tType] :
( ( ordere1937475149494474687imp_le @ A )
=> ! [A2: A,B2: A] :
( ( ord_less @ A @ ( plus_plus @ A @ A2 @ B2 ) @ B2 )
= ( ord_less @ A @ A2 @ ( zero_zero @ A ) ) ) ) ).
% add_less_same_cancel2
thf(fact_938_less__add__same__cancel1,axiom,
! [A: $tType] :
( ( ordere1937475149494474687imp_le @ A )
=> ! [A2: A,B2: A] :
( ( ord_less @ A @ A2 @ ( plus_plus @ A @ A2 @ B2 ) )
= ( ord_less @ A @ ( zero_zero @ A ) @ B2 ) ) ) ).
% less_add_same_cancel1
thf(fact_939_less__add__same__cancel2,axiom,
! [A: $tType] :
( ( ordere1937475149494474687imp_le @ A )
=> ! [A2: A,B2: A] :
( ( ord_less @ A @ A2 @ ( plus_plus @ A @ B2 @ A2 ) )
= ( ord_less @ A @ ( zero_zero @ A ) @ B2 ) ) ) ).
% less_add_same_cancel2
thf(fact_940_double__add__less__zero__iff__single__add__less__zero,axiom,
! [A: $tType] :
( ( linord5086331880401160121up_add @ A )
=> ! [A2: A] :
( ( ord_less @ A @ ( plus_plus @ A @ A2 @ A2 ) @ ( zero_zero @ A ) )
= ( ord_less @ A @ A2 @ ( zero_zero @ A ) ) ) ) ).
% double_add_less_zero_iff_single_add_less_zero
thf(fact_941_zero__less__double__add__iff__zero__less__single__add,axiom,
! [A: $tType] :
( ( linord5086331880401160121up_add @ A )
=> ! [A2: A] :
( ( ord_less @ A @ ( zero_zero @ A ) @ ( plus_plus @ A @ A2 @ A2 ) )
= ( ord_less @ A @ ( zero_zero @ A ) @ A2 ) ) ) ).
% zero_less_double_add_iff_zero_less_single_add
thf(fact_942_zero__comp__diff__simps_I1_J,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [A2: A,B2: A] :
( ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( minus_minus @ A @ A2 @ B2 ) )
= ( ord_less_eq @ A @ B2 @ A2 ) ) ) ).
% zero_comp_diff_simps(1)
thf(fact_943_diff__ge__0__iff__ge,axiom,
! [A: $tType] :
( ( ordered_ab_group_add @ A )
=> ! [A2: A,B2: A] :
( ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( minus_minus @ A @ A2 @ B2 ) )
= ( ord_less_eq @ A @ B2 @ A2 ) ) ) ).
% diff_ge_0_iff_ge
thf(fact_944_zero__comp__diff__simps_I2_J,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [A2: A,B2: A] :
( ( ord_less @ A @ ( zero_zero @ A ) @ ( minus_minus @ A @ A2 @ B2 ) )
= ( ord_less @ A @ B2 @ A2 ) ) ) ).
% zero_comp_diff_simps(2)
thf(fact_945_diff__gt__0__iff__gt,axiom,
! [A: $tType] :
( ( ordered_ab_group_add @ A )
=> ! [A2: A,B2: A] :
( ( ord_less @ A @ ( zero_zero @ A ) @ ( minus_minus @ A @ A2 @ B2 ) )
= ( ord_less @ A @ B2 @ A2 ) ) ) ).
% diff_gt_0_iff_gt
thf(fact_946_sum__squares__eq__zero__iff,axiom,
! [A: $tType] :
( ( linord4710134922213307826strict @ A )
=> ! [X: A,Y: A] :
( ( ( plus_plus @ A @ ( times_times @ A @ X @ X ) @ ( times_times @ A @ Y @ Y ) )
= ( zero_zero @ A ) )
= ( ( X
= ( zero_zero @ A ) )
& ( Y
= ( zero_zero @ A ) ) ) ) ) ).
% sum_squares_eq_zero_iff
thf(fact_947_mult__cancel__left1,axiom,
! [A: $tType] :
( ( ring_15535105094025558882visors @ A )
=> ! [C2: A,B2: A] :
( ( C2
= ( times_times @ A @ C2 @ B2 ) )
= ( ( C2
= ( zero_zero @ A ) )
| ( B2
= ( one_one @ A ) ) ) ) ) ).
% mult_cancel_left1
thf(fact_948_mult__cancel__left2,axiom,
! [A: $tType] :
( ( ring_15535105094025558882visors @ A )
=> ! [C2: A,A2: A] :
( ( ( times_times @ A @ C2 @ A2 )
= C2 )
= ( ( C2
= ( zero_zero @ A ) )
| ( A2
= ( one_one @ A ) ) ) ) ) ).
% mult_cancel_left2
thf(fact_949_mult__cancel__right1,axiom,
! [A: $tType] :
( ( ring_15535105094025558882visors @ A )
=> ! [C2: A,B2: A] :
( ( C2
= ( times_times @ A @ B2 @ C2 ) )
= ( ( C2
= ( zero_zero @ A ) )
| ( B2
= ( one_one @ A ) ) ) ) ) ).
% mult_cancel_right1
thf(fact_950_mult__cancel__right2,axiom,
! [A: $tType] :
( ( ring_15535105094025558882visors @ A )
=> ! [A2: A,C2: A] :
( ( ( times_times @ A @ A2 @ C2 )
= C2 )
= ( ( C2
= ( zero_zero @ A ) )
| ( A2
= ( one_one @ A ) ) ) ) ) ).
% mult_cancel_right2
thf(fact_951_diff__add__zero,axiom,
! [A: $tType] :
( ( comm_monoid_diff @ A )
=> ! [A2: A,B2: A] :
( ( minus_minus @ A @ A2 @ ( plus_plus @ A @ A2 @ B2 ) )
= ( zero_zero @ A ) ) ) ).
% diff_add_zero
thf(fact_952_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_953_div__self,axiom,
! [A: $tType] :
( ( semidom_divide @ A )
=> ! [A2: A] :
( ( A2
!= ( zero_zero @ A ) )
=> ( ( divide_divide @ A @ A2 @ A2 )
= ( one_one @ A ) ) ) ) ).
% div_self
thf(fact_954_zero__eq__1__divide__iff,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [A2: A] :
( ( ( zero_zero @ A )
= ( divide_divide @ A @ ( one_one @ A ) @ A2 ) )
= ( A2
= ( zero_zero @ A ) ) ) ) ).
% zero_eq_1_divide_iff
thf(fact_955_one__divide__eq__0__iff,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [A2: A] :
( ( ( divide_divide @ A @ ( one_one @ A ) @ A2 )
= ( zero_zero @ A ) )
= ( A2
= ( zero_zero @ A ) ) ) ) ).
% one_divide_eq_0_iff
thf(fact_956_eq__divide__eq__1,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [B2: A,A2: A] :
( ( ( one_one @ A )
= ( divide_divide @ A @ B2 @ A2 ) )
= ( ( A2
!= ( zero_zero @ A ) )
& ( A2 = B2 ) ) ) ) ).
% eq_divide_eq_1
thf(fact_957_divide__eq__eq__1,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [B2: A,A2: A] :
( ( ( divide_divide @ A @ B2 @ A2 )
= ( one_one @ A ) )
= ( ( A2
!= ( zero_zero @ A ) )
& ( A2 = B2 ) ) ) ) ).
% divide_eq_eq_1
thf(fact_958_divide__self__if,axiom,
! [A: $tType] :
( ( division_ring @ A )
=> ! [A2: A] :
( ( ( A2
= ( zero_zero @ A ) )
=> ( ( divide_divide @ A @ A2 @ A2 )
= ( zero_zero @ A ) ) )
& ( ( A2
!= ( zero_zero @ A ) )
=> ( ( divide_divide @ A @ A2 @ A2 )
= ( one_one @ A ) ) ) ) ) ).
% divide_self_if
thf(fact_959_divide__self,axiom,
! [A: $tType] :
( ( division_ring @ A )
=> ! [A2: A] :
( ( A2
!= ( zero_zero @ A ) )
=> ( ( divide_divide @ A @ A2 @ A2 )
= ( one_one @ A ) ) ) ) ).
% divide_self
thf(fact_960_one__eq__divide__iff,axiom,
! [A: $tType] :
( ( field @ A )
=> ! [A2: A,B2: A] :
( ( ( one_one @ A )
= ( divide_divide @ A @ A2 @ B2 ) )
= ( ( B2
!= ( zero_zero @ A ) )
& ( A2 = B2 ) ) ) ) ).
% one_eq_divide_iff
thf(fact_961_divide__eq__1__iff,axiom,
! [A: $tType] :
( ( field @ A )
=> ! [A2: A,B2: A] :
( ( ( divide_divide @ A @ A2 @ B2 )
= ( one_one @ A ) )
= ( ( B2
!= ( zero_zero @ A ) )
& ( A2 = B2 ) ) ) ) ).
% divide_eq_1_iff
thf(fact_962_nonzero__mult__div__cancel__left,axiom,
! [A: $tType] :
( ( semidom_divide @ A )
=> ! [A2: A,B2: A] :
( ( A2
!= ( zero_zero @ A ) )
=> ( ( divide_divide @ A @ ( times_times @ A @ A2 @ B2 ) @ A2 )
= B2 ) ) ) ).
% nonzero_mult_div_cancel_left
thf(fact_963_nonzero__mult__div__cancel__right,axiom,
! [A: $tType] :
( ( semidom_divide @ A )
=> ! [B2: A,A2: A] :
( ( B2
!= ( zero_zero @ A ) )
=> ( ( divide_divide @ A @ ( times_times @ A @ A2 @ B2 ) @ B2 )
= A2 ) ) ) ).
% nonzero_mult_div_cancel_right
thf(fact_964_div__mult__mult1,axiom,
! [A: $tType] :
( ( euclid4440199948858584721cancel @ A )
=> ! [C2: A,A2: A,B2: A] :
( ( C2
!= ( zero_zero @ A ) )
=> ( ( divide_divide @ A @ ( times_times @ A @ C2 @ A2 ) @ ( times_times @ A @ C2 @ B2 ) )
= ( divide_divide @ A @ A2 @ B2 ) ) ) ) ).
% div_mult_mult1
thf(fact_965_div__mult__mult2,axiom,
! [A: $tType] :
( ( euclid4440199948858584721cancel @ A )
=> ! [C2: A,A2: A,B2: A] :
( ( C2
!= ( zero_zero @ A ) )
=> ( ( divide_divide @ A @ ( times_times @ A @ A2 @ C2 ) @ ( times_times @ A @ B2 @ C2 ) )
= ( divide_divide @ A @ A2 @ B2 ) ) ) ) ).
% div_mult_mult2
thf(fact_966_div__mult__mult1__if,axiom,
! [A: $tType] :
( ( euclid4440199948858584721cancel @ A )
=> ! [C2: A,A2: A,B2: A] :
( ( ( C2
= ( zero_zero @ A ) )
=> ( ( divide_divide @ A @ ( times_times @ A @ C2 @ A2 ) @ ( times_times @ A @ C2 @ B2 ) )
= ( zero_zero @ A ) ) )
& ( ( C2
!= ( zero_zero @ A ) )
=> ( ( divide_divide @ A @ ( times_times @ A @ C2 @ A2 ) @ ( times_times @ A @ C2 @ B2 ) )
= ( divide_divide @ A @ A2 @ B2 ) ) ) ) ) ).
% div_mult_mult1_if
thf(fact_967_mult__divide__mult__cancel__left__if,axiom,
! [A: $tType] :
( ( field @ A )
=> ! [C2: A,A2: A,B2: A] :
( ( ( C2
= ( zero_zero @ A ) )
=> ( ( divide_divide @ A @ ( times_times @ A @ C2 @ A2 ) @ ( times_times @ A @ C2 @ B2 ) )
= ( zero_zero @ A ) ) )
& ( ( C2
!= ( zero_zero @ A ) )
=> ( ( divide_divide @ A @ ( times_times @ A @ C2 @ A2 ) @ ( times_times @ A @ C2 @ B2 ) )
= ( divide_divide @ A @ A2 @ B2 ) ) ) ) ) ).
% mult_divide_mult_cancel_left_if
thf(fact_968_nonzero__mult__divide__mult__cancel__left,axiom,
! [A: $tType] :
( ( field @ A )
=> ! [C2: A,A2: A,B2: A] :
( ( C2
!= ( zero_zero @ A ) )
=> ( ( divide_divide @ A @ ( times_times @ A @ C2 @ A2 ) @ ( times_times @ A @ C2 @ B2 ) )
= ( divide_divide @ A @ A2 @ B2 ) ) ) ) ).
% nonzero_mult_divide_mult_cancel_left
thf(fact_969_nonzero__mult__divide__mult__cancel__left2,axiom,
! [A: $tType] :
( ( field @ A )
=> ! [C2: A,A2: A,B2: A] :
( ( C2
!= ( zero_zero @ A ) )
=> ( ( divide_divide @ A @ ( times_times @ A @ C2 @ A2 ) @ ( times_times @ A @ B2 @ C2 ) )
= ( divide_divide @ A @ A2 @ B2 ) ) ) ) ).
% nonzero_mult_divide_mult_cancel_left2
thf(fact_970_nonzero__mult__divide__mult__cancel__right,axiom,
! [A: $tType] :
( ( field @ A )
=> ! [C2: A,A2: A,B2: A] :
( ( C2
!= ( zero_zero @ A ) )
=> ( ( divide_divide @ A @ ( times_times @ A @ A2 @ C2 ) @ ( times_times @ A @ B2 @ C2 ) )
= ( divide_divide @ A @ A2 @ B2 ) ) ) ) ).
% nonzero_mult_divide_mult_cancel_right
thf(fact_971_nonzero__mult__divide__mult__cancel__right2,axiom,
! [A: $tType] :
( ( field @ A )
=> ! [C2: A,A2: A,B2: A] :
( ( C2
!= ( zero_zero @ A ) )
=> ( ( divide_divide @ A @ ( times_times @ A @ A2 @ C2 ) @ ( times_times @ A @ C2 @ B2 ) )
= ( divide_divide @ A @ A2 @ B2 ) ) ) ) ).
% nonzero_mult_divide_mult_cancel_right2
thf(fact_972_power__0__Suc,axiom,
! [A: $tType] :
( ( semiring_1 @ A )
=> ! [N3: nat] :
( ( power_power @ A @ ( zero_zero @ A ) @ ( suc @ N3 ) )
= ( zero_zero @ A ) ) ) ).
% power_0_Suc
thf(fact_973_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_974_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_975_of__nat__0__eq__iff,axiom,
! [A: $tType] :
( ( semiring_char_0 @ A )
=> ! [N3: nat] :
( ( ( zero_zero @ A )
= ( semiring_1_of_nat @ A @ N3 ) )
= ( ( zero_zero @ nat )
= N3 ) ) ) ).
% of_nat_0_eq_iff
thf(fact_976_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_977_power__Suc0__right,axiom,
! [A: $tType] :
( ( monoid_mult @ A )
=> ! [A2: A] :
( ( power_power @ A @ A2 @ ( suc @ ( zero_zero @ nat ) ) )
= A2 ) ) ).
% power_Suc0_right
thf(fact_978_less__Suc0,axiom,
! [N3: nat] :
( ( ord_less @ nat @ N3 @ ( suc @ ( zero_zero @ nat ) ) )
= ( N3
= ( zero_zero @ nat ) ) ) ).
% less_Suc0
thf(fact_979_zero__less__Suc,axiom,
! [N3: nat] : ( ord_less @ nat @ ( zero_zero @ nat ) @ ( suc @ N3 ) ) ).
% zero_less_Suc
thf(fact_980_div__by__Suc__0,axiom,
! [M: nat] :
( ( divide_divide @ nat @ M @ ( suc @ ( zero_zero @ nat ) ) )
= M ) ).
% div_by_Suc_0
thf(fact_981_add__gr__0,axiom,
! [M: nat,N3: nat] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ ( plus_plus @ nat @ M @ N3 ) )
= ( ( ord_less @ nat @ ( zero_zero @ nat ) @ M )
| ( ord_less @ nat @ ( zero_zero @ nat ) @ N3 ) ) ) ).
% add_gr_0
thf(fact_982_less__one,axiom,
! [N3: nat] :
( ( ord_less @ nat @ N3 @ ( one_one @ nat ) )
= ( N3
= ( zero_zero @ nat ) ) ) ).
% less_one
thf(fact_983_mult__eq__1__iff,axiom,
! [M: nat,N3: nat] :
( ( ( times_times @ nat @ M @ N3 )
= ( suc @ ( zero_zero @ nat ) ) )
= ( ( M
= ( suc @ ( zero_zero @ nat ) ) )
& ( N3
= ( suc @ ( zero_zero @ nat ) ) ) ) ) ).
% mult_eq_1_iff
thf(fact_984_one__eq__mult__iff,axiom,
! [M: nat,N3: nat] :
( ( ( suc @ ( zero_zero @ nat ) )
= ( times_times @ nat @ M @ N3 ) )
= ( ( M
= ( suc @ ( zero_zero @ nat ) ) )
& ( N3
= ( suc @ ( zero_zero @ nat ) ) ) ) ) ).
% one_eq_mult_iff
thf(fact_985_div__less,axiom,
! [M: nat,N3: nat] :
( ( ord_less @ nat @ M @ N3 )
=> ( ( divide_divide @ nat @ M @ N3 )
= ( zero_zero @ nat ) ) ) ).
% div_less
thf(fact_986_zero__less__diff,axiom,
! [N3: nat,M: nat] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ ( minus_minus @ nat @ N3 @ M ) )
= ( ord_less @ nat @ M @ N3 ) ) ).
% zero_less_diff
thf(fact_987_power__Suc__0,axiom,
! [N3: nat] :
( ( power_power @ nat @ ( suc @ ( zero_zero @ nat ) ) @ N3 )
= ( suc @ ( zero_zero @ nat ) ) ) ).
% power_Suc_0
thf(fact_988_nat__power__eq__Suc__0__iff,axiom,
! [X: nat,M: nat] :
( ( ( power_power @ nat @ X @ M )
= ( suc @ ( zero_zero @ nat ) ) )
= ( ( M
= ( zero_zero @ nat ) )
| ( X
= ( suc @ ( zero_zero @ nat ) ) ) ) ) ).
% nat_power_eq_Suc_0_iff
thf(fact_989_max__0__1_I3_J,axiom,
! [A: $tType] :
( ( linord181362715937106298miring @ A )
=> ! [X: num] :
( ( ord_max @ A @ ( zero_zero @ A ) @ ( numeral_numeral @ A @ X ) )
= ( numeral_numeral @ A @ X ) ) ) ).
% max_0_1(3)
thf(fact_990_max__0__1_I4_J,axiom,
! [A: $tType] :
( ( linord181362715937106298miring @ A )
=> ! [X: num] :
( ( ord_max @ A @ ( numeral_numeral @ A @ X ) @ ( zero_zero @ A ) )
= ( numeral_numeral @ A @ X ) ) ) ).
% max_0_1(4)
thf(fact_991_mult__less__cancel2,axiom,
! [M: nat,K: nat,N3: nat] :
( ( ord_less @ nat @ ( times_times @ nat @ M @ K ) @ ( times_times @ nat @ N3 @ K ) )
= ( ( ord_less @ nat @ ( zero_zero @ nat ) @ K )
& ( ord_less @ nat @ M @ N3 ) ) ) ).
% mult_less_cancel2
thf(fact_992_nat__0__less__mult__iff,axiom,
! [M: nat,N3: nat] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ ( times_times @ nat @ M @ N3 ) )
= ( ( ord_less @ nat @ ( zero_zero @ nat ) @ M )
& ( ord_less @ nat @ ( zero_zero @ nat ) @ N3 ) ) ) ).
% nat_0_less_mult_iff
thf(fact_993_nat__mult__less__cancel__disj,axiom,
! [K: nat,M: nat,N3: nat] :
( ( ord_less @ nat @ ( times_times @ nat @ K @ M ) @ ( times_times @ nat @ K @ N3 ) )
= ( ( ord_less @ nat @ ( zero_zero @ nat ) @ K )
& ( ord_less @ nat @ M @ N3 ) ) ) ).
% nat_mult_less_cancel_disj
thf(fact_994_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_995_of__int__eq__numeral__iff,axiom,
! [A: $tType] :
( ( ring_char_0 @ A )
=> ! [Z: int,N3: num] :
( ( ( ring_1_of_int @ A @ Z )
= ( numeral_numeral @ A @ N3 ) )
= ( Z
= ( numeral_numeral @ int @ N3 ) ) ) ) ).
% of_int_eq_numeral_iff
thf(fact_996_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_997_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_998_diff__is__0__eq_H,axiom,
! [M: nat,N3: nat] :
( ( ord_less_eq @ nat @ M @ N3 )
=> ( ( minus_minus @ nat @ M @ N3 )
= ( zero_zero @ nat ) ) ) ).
% diff_is_0_eq'
thf(fact_999_diff__is__0__eq,axiom,
! [M: nat,N3: nat] :
( ( ( minus_minus @ nat @ M @ N3 )
= ( zero_zero @ nat ) )
= ( ord_less_eq @ nat @ M @ N3 ) ) ).
% diff_is_0_eq
thf(fact_1000_nat__zero__less__power__iff,axiom,
! [X: nat,N3: nat] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ ( power_power @ nat @ X @ N3 ) )
= ( ( ord_less @ nat @ ( zero_zero @ nat ) @ X )
| ( N3
= ( zero_zero @ nat ) ) ) ) ).
% nat_zero_less_power_iff
thf(fact_1001_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_1002_nat__mult__div__cancel__disj,axiom,
! [K: nat,M: nat,N3: nat] :
( ( ( K
= ( zero_zero @ nat ) )
=> ( ( divide_divide @ nat @ ( times_times @ nat @ K @ M ) @ ( times_times @ nat @ K @ N3 ) )
= ( zero_zero @ nat ) ) )
& ( ( K
!= ( zero_zero @ nat ) )
=> ( ( divide_divide @ nat @ ( times_times @ nat @ K @ M ) @ ( times_times @ nat @ K @ N3 ) )
= ( divide_divide @ nat @ M @ N3 ) ) ) ) ).
% nat_mult_div_cancel_disj
thf(fact_1003_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_1004_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_1005_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_1006_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_1007_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_1008_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_1009_of__int__power,axiom,
! [A: $tType] :
( ( ring_1 @ A )
=> ! [Z: int,N3: nat] :
( ( ring_1_of_int @ A @ ( power_power @ int @ Z @ N3 ) )
= ( power_power @ A @ ( ring_1_of_int @ A @ Z ) @ N3 ) ) ) ).
% of_int_power
thf(fact_1010_of__int__eq__of__int__power__cancel__iff,axiom,
! [A: $tType] :
( ( ring_char_0 @ A )
=> ! [B2: int,W: nat,X: int] :
( ( ( power_power @ A @ ( ring_1_of_int @ A @ B2 ) @ W )
= ( ring_1_of_int @ A @ X ) )
= ( ( power_power @ int @ B2 @ W )
= X ) ) ) ).
% of_int_eq_of_int_power_cancel_iff
thf(fact_1011_of__int__power__eq__of__int__cancel__iff,axiom,
! [A: $tType] :
( ( ring_char_0 @ A )
=> ! [X: int,B2: int,W: nat] :
( ( ( ring_1_of_int @ A @ X )
= ( power_power @ A @ ( ring_1_of_int @ A @ B2 ) @ W ) )
= ( X
= ( power_power @ int @ B2 @ W ) ) ) ) ).
% of_int_power_eq_of_int_cancel_iff
thf(fact_1012_zero__le__divide__1__iff,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [A2: A] :
( ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( divide_divide @ A @ ( one_one @ A ) @ A2 ) )
= ( ord_less_eq @ A @ ( zero_zero @ A ) @ A2 ) ) ) ).
% zero_le_divide_1_iff
thf(fact_1013_divide__le__0__1__iff,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [A2: A] :
( ( ord_less_eq @ A @ ( divide_divide @ A @ ( one_one @ A ) @ A2 ) @ ( zero_zero @ A ) )
= ( ord_less_eq @ A @ A2 @ ( zero_zero @ A ) ) ) ) ).
% divide_le_0_1_iff
thf(fact_1014_divide__less__0__1__iff,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [A2: A] :
( ( ord_less @ A @ ( divide_divide @ A @ ( one_one @ A ) @ A2 ) @ ( zero_zero @ A ) )
= ( ord_less @ A @ A2 @ ( zero_zero @ A ) ) ) ) ).
% divide_less_0_1_iff
thf(fact_1015_divide__less__eq__1__neg,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [A2: A,B2: A] :
( ( ord_less @ A @ A2 @ ( zero_zero @ A ) )
=> ( ( ord_less @ A @ ( divide_divide @ A @ B2 @ A2 ) @ ( one_one @ A ) )
= ( ord_less @ A @ A2 @ B2 ) ) ) ) ).
% divide_less_eq_1_neg
thf(fact_1016_divide__less__eq__1__pos,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [A2: A,B2: A] :
( ( ord_less @ A @ ( zero_zero @ A ) @ A2 )
=> ( ( ord_less @ A @ ( divide_divide @ A @ B2 @ A2 ) @ ( one_one @ A ) )
= ( ord_less @ A @ B2 @ A2 ) ) ) ) ).
% divide_less_eq_1_pos
thf(fact_1017_less__divide__eq__1__neg,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [A2: A,B2: A] :
( ( ord_less @ A @ A2 @ ( zero_zero @ A ) )
=> ( ( ord_less @ A @ ( one_one @ A ) @ ( divide_divide @ A @ B2 @ A2 ) )
= ( ord_less @ A @ B2 @ A2 ) ) ) ) ).
% less_divide_eq_1_neg
thf(fact_1018_less__divide__eq__1__pos,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [A2: A,B2: A] :
( ( ord_less @ A @ ( zero_zero @ A ) @ A2 )
=> ( ( ord_less @ A @ ( one_one @ A ) @ ( divide_divide @ A @ B2 @ A2 ) )
= ( ord_less @ A @ A2 @ B2 ) ) ) ) ).
% less_divide_eq_1_pos
thf(fact_1019_zero__less__divide__1__iff,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [A2: A] :
( ( ord_less @ A @ ( zero_zero @ A ) @ ( divide_divide @ A @ ( one_one @ A ) @ A2 ) )
= ( ord_less @ A @ ( zero_zero @ A ) @ A2 ) ) ) ).
% zero_less_divide_1_iff
thf(fact_1020_divide__eq__eq__numeral1_I1_J,axiom,
! [A: $tType] :
( ( division_ring @ A )
=> ! [B2: A,W: num,A2: A] :
( ( ( divide_divide @ A @ B2 @ ( numeral_numeral @ A @ W ) )
= A2 )
= ( ( ( ( numeral_numeral @ A @ W )
!= ( zero_zero @ A ) )
=> ( B2
= ( times_times @ A @ A2 @ ( numeral_numeral @ A @ W ) ) ) )
& ( ( ( numeral_numeral @ A @ W )
= ( zero_zero @ A ) )
=> ( A2
= ( zero_zero @ A ) ) ) ) ) ) ).
% divide_eq_eq_numeral1(1)
thf(fact_1021_eq__divide__eq__numeral1_I1_J,axiom,
! [A: $tType] :
( ( division_ring @ A )
=> ! [A2: A,B2: A,W: num] :
( ( A2
= ( divide_divide @ A @ B2 @ ( numeral_numeral @ A @ W ) ) )
= ( ( ( ( numeral_numeral @ A @ W )
!= ( zero_zero @ A ) )
=> ( ( times_times @ A @ A2 @ ( numeral_numeral @ A @ W ) )
= B2 ) )
& ( ( ( numeral_numeral @ A @ W )
= ( zero_zero @ A ) )
=> ( A2
= ( zero_zero @ A ) ) ) ) ) ) ).
% eq_divide_eq_numeral1(1)
thf(fact_1022_div__mult__self1,axiom,
! [A: $tType] :
( ( euclid4440199948858584721cancel @ A )
=> ! [B2: A,A2: A,C2: A] :
( ( B2
!= ( zero_zero @ A ) )
=> ( ( divide_divide @ A @ ( plus_plus @ A @ A2 @ ( times_times @ A @ C2 @ B2 ) ) @ B2 )
= ( plus_plus @ A @ C2 @ ( divide_divide @ A @ A2 @ B2 ) ) ) ) ) ).
% div_mult_self1
thf(fact_1023_div__mult__self2,axiom,
! [A: $tType] :
( ( euclid4440199948858584721cancel @ A )
=> ! [B2: A,A2: A,C2: A] :
( ( B2
!= ( zero_zero @ A ) )
=> ( ( divide_divide @ A @ ( plus_plus @ A @ A2 @ ( times_times @ A @ B2 @ C2 ) ) @ B2 )
= ( plus_plus @ A @ C2 @ ( divide_divide @ A @ A2 @ B2 ) ) ) ) ) ).
% div_mult_self2
thf(fact_1024_div__mult__self3,axiom,
! [A: $tType] :
( ( euclid4440199948858584721cancel @ A )
=> ! [B2: A,C2: A,A2: A] :
( ( B2
!= ( zero_zero @ A ) )
=> ( ( divide_divide @ A @ ( plus_plus @ A @ ( times_times @ A @ C2 @ B2 ) @ A2 ) @ B2 )
= ( plus_plus @ A @ C2 @ ( divide_divide @ A @ A2 @ B2 ) ) ) ) ) ).
% div_mult_self3
thf(fact_1025_div__mult__self4,axiom,
! [A: $tType] :
( ( euclid4440199948858584721cancel @ A )
=> ! [B2: A,C2: A,A2: A] :
( ( B2
!= ( zero_zero @ A ) )
=> ( ( divide_divide @ A @ ( plus_plus @ A @ ( times_times @ A @ B2 @ C2 ) @ A2 ) @ B2 )
= ( plus_plus @ A @ C2 @ ( divide_divide @ A @ A2 @ B2 ) ) ) ) ) ).
% div_mult_self4
thf(fact_1026_nonzero__divide__mult__cancel__left,axiom,
! [A: $tType] :
( ( field @ A )
=> ! [A2: A,B2: A] :
( ( A2
!= ( zero_zero @ A ) )
=> ( ( divide_divide @ A @ A2 @ ( times_times @ A @ A2 @ B2 ) )
= ( divide_divide @ A @ ( one_one @ A ) @ B2 ) ) ) ) ).
% nonzero_divide_mult_cancel_left
thf(fact_1027_nonzero__divide__mult__cancel__right,axiom,
! [A: $tType] :
( ( field @ A )
=> ! [B2: A,A2: A] :
( ( B2
!= ( zero_zero @ A ) )
=> ( ( divide_divide @ A @ B2 @ ( times_times @ A @ A2 @ B2 ) )
= ( divide_divide @ A @ ( one_one @ A ) @ A2 ) ) ) ) ).
% nonzero_divide_mult_cancel_right
thf(fact_1028_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_1029_power__eq__0__iff,axiom,
! [A: $tType] :
( ( semiri2026040879449505780visors @ A )
=> ! [A2: A,N3: nat] :
( ( ( power_power @ A @ A2 @ N3 )
= ( zero_zero @ A ) )
= ( ( A2
= ( zero_zero @ A ) )
& ( ord_less @ nat @ ( zero_zero @ nat ) @ N3 ) ) ) ) ).
% power_eq_0_iff
thf(fact_1030_Suc__pred,axiom,
! [N3: nat] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ N3 )
=> ( ( suc @ ( minus_minus @ nat @ N3 @ ( suc @ ( zero_zero @ nat ) ) ) )
= N3 ) ) ).
% Suc_pred
thf(fact_1031_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_1032_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_1033_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_1034_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_1035_one__le__mult__iff,axiom,
! [M: nat,N3: nat] :
( ( ord_less_eq @ nat @ ( suc @ ( zero_zero @ nat ) ) @ ( times_times @ nat @ M @ N3 ) )
= ( ( ord_less_eq @ nat @ ( suc @ ( zero_zero @ nat ) ) @ M )
& ( ord_less_eq @ nat @ ( suc @ ( zero_zero @ nat ) ) @ N3 ) ) ) ).
% one_le_mult_iff
thf(fact_1036_div__eq__dividend__iff,axiom,
! [M: nat,N3: nat] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ M )
=> ( ( ( divide_divide @ nat @ M @ N3 )
= M )
= ( N3
= ( one_one @ nat ) ) ) ) ).
% div_eq_dividend_iff
thf(fact_1037_mult__le__cancel2,axiom,
! [M: nat,K: nat,N3: nat] :
( ( ord_less_eq @ nat @ ( times_times @ nat @ M @ K ) @ ( times_times @ nat @ N3 @ K ) )
= ( ( ord_less @ nat @ ( zero_zero @ nat ) @ K )
=> ( ord_less_eq @ nat @ M @ N3 ) ) ) ).
% mult_le_cancel2
thf(fact_1038_nat__mult__le__cancel__disj,axiom,
! [K: nat,M: nat,N3: nat] :
( ( ord_less_eq @ nat @ ( times_times @ nat @ K @ M ) @ ( times_times @ nat @ K @ N3 ) )
= ( ( ord_less @ nat @ ( zero_zero @ nat ) @ K )
=> ( ord_less_eq @ nat @ M @ N3 ) ) ) ).
% nat_mult_le_cancel_disj
thf(fact_1039_div__mult__self__is__m,axiom,
! [N3: nat,M: nat] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ N3 )
=> ( ( divide_divide @ nat @ ( times_times @ nat @ M @ N3 ) @ N3 )
= M ) ) ).
% div_mult_self_is_m
thf(fact_1040_div__mult__self1__is__m,axiom,
! [N3: nat,M: nat] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ N3 )
=> ( ( divide_divide @ nat @ ( times_times @ nat @ N3 @ M ) @ N3 )
= M ) ) ).
% div_mult_self1_is_m
thf(fact_1041_ceiling__le__zero,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [X: A] :
( ( ord_less_eq @ int @ ( archimedean_ceiling @ A @ X ) @ ( zero_zero @ int ) )
= ( ord_less_eq @ A @ X @ ( zero_zero @ A ) ) ) ) ).
% ceiling_le_zero
thf(fact_1042_zero__less__ceiling,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [X: A] :
( ( ord_less @ int @ ( zero_zero @ int ) @ ( archimedean_ceiling @ A @ X ) )
= ( ord_less @ A @ ( zero_zero @ A ) @ X ) ) ) ).
% zero_less_ceiling
thf(fact_1043_ceiling__add__of__int,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [X: A,Z: int] :
( ( archimedean_ceiling @ A @ ( plus_plus @ A @ X @ ( ring_1_of_int @ A @ Z ) ) )
= ( plus_plus @ int @ ( archimedean_ceiling @ A @ X ) @ Z ) ) ) ).
% ceiling_add_of_int
thf(fact_1044_ceiling__diff__of__int,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [X: A,Z: int] :
( ( archimedean_ceiling @ A @ ( minus_minus @ A @ X @ ( ring_1_of_int @ A @ Z ) ) )
= ( minus_minus @ int @ ( archimedean_ceiling @ A @ X ) @ Z ) ) ) ).
% ceiling_diff_of_int
thf(fact_1045_divide__le__eq__1__neg,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [A2: A,B2: A] :
( ( ord_less @ A @ A2 @ ( zero_zero @ A ) )
=> ( ( ord_less_eq @ A @ ( divide_divide @ A @ B2 @ A2 ) @ ( one_one @ A ) )
= ( ord_less_eq @ A @ A2 @ B2 ) ) ) ) ).
% divide_le_eq_1_neg
thf(fact_1046_divide__le__eq__1__pos,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [A2: A,B2: A] :
( ( ord_less @ A @ ( zero_zero @ A ) @ A2 )
=> ( ( ord_less_eq @ A @ ( divide_divide @ A @ B2 @ A2 ) @ ( one_one @ A ) )
= ( ord_less_eq @ A @ B2 @ A2 ) ) ) ) ).
% divide_le_eq_1_pos
thf(fact_1047_le__divide__eq__1__neg,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [A2: A,B2: A] :
( ( ord_less @ A @ A2 @ ( zero_zero @ A ) )
=> ( ( ord_less_eq @ A @ ( one_one @ A ) @ ( divide_divide @ A @ B2 @ A2 ) )
= ( ord_less_eq @ A @ B2 @ A2 ) ) ) ) ).
% le_divide_eq_1_neg
thf(fact_1048_le__divide__eq__1__pos,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [A2: A,B2: A] :
( ( ord_less @ A @ ( zero_zero @ A ) @ A2 )
=> ( ( ord_less_eq @ A @ ( one_one @ A ) @ ( divide_divide @ A @ B2 @ A2 ) )
= ( ord_less_eq @ A @ A2 @ B2 ) ) ) ) ).
% le_divide_eq_1_pos
thf(fact_1049_zero__eq__power2,axiom,
! [A: $tType] :
( ( semiri2026040879449505780visors @ A )
=> ! [A2: A] :
( ( ( power_power @ A @ A2 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) )
= ( zero_zero @ A ) )
= ( A2
= ( zero_zero @ A ) ) ) ) ).
% zero_eq_power2
thf(fact_1050_power__strict__decreasing__iff,axiom,
! [A: $tType] :
( ( linordered_semidom @ A )
=> ! [B2: A,M: nat,N3: nat] :
( ( ord_less @ A @ ( zero_zero @ A ) @ B2 )
=> ( ( ord_less @ A @ B2 @ ( one_one @ A ) )
=> ( ( ord_less @ A @ ( power_power @ A @ B2 @ M ) @ ( power_power @ A @ B2 @ N3 ) )
= ( ord_less @ nat @ N3 @ M ) ) ) ) ) ).
% power_strict_decreasing_iff
thf(fact_1051_power__mono__iff,axiom,
! [A: $tType] :
( ( linordered_semidom @ A )
=> ! [A2: A,B2: A,N3: nat] :
( ( ord_less_eq @ A @ ( zero_zero @ A ) @ A2 )
=> ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ B2 )
=> ( ( ord_less @ nat @ ( zero_zero @ nat ) @ N3 )
=> ( ( ord_less_eq @ A @ ( power_power @ A @ A2 @ N3 ) @ ( power_power @ A @ B2 @ N3 ) )
= ( ord_less_eq @ A @ A2 @ B2 ) ) ) ) ) ) ).
% power_mono_iff
thf(fact_1052_of__nat__0__less__iff,axiom,
! [A: $tType] :
( ( linord181362715937106298miring @ A )
=> ! [N3: nat] :
( ( ord_less @ A @ ( zero_zero @ A ) @ ( semiring_1_of_nat @ A @ N3 ) )
= ( ord_less @ nat @ ( zero_zero @ nat ) @ N3 ) ) ) ).
% of_nat_0_less_iff
thf(fact_1053_Suc__diff__1,axiom,
! [N3: nat] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ N3 )
=> ( ( suc @ ( minus_minus @ nat @ N3 @ ( one_one @ nat ) ) )
= N3 ) ) ).
% Suc_diff_1
thf(fact_1054_of__int__numeral__le__iff,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [N3: num,Z: int] :
( ( ord_less_eq @ A @ ( numeral_numeral @ A @ N3 ) @ ( ring_1_of_int @ A @ Z ) )
= ( ord_less_eq @ int @ ( numeral_numeral @ int @ N3 ) @ Z ) ) ) ).
% of_int_numeral_le_iff
thf(fact_1055_of__int__le__numeral__iff,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [Z: int,N3: num] :
( ( ord_less_eq @ A @ ( ring_1_of_int @ A @ Z ) @ ( numeral_numeral @ A @ N3 ) )
= ( ord_less_eq @ int @ Z @ ( numeral_numeral @ int @ N3 ) ) ) ) ).
% of_int_le_numeral_iff
thf(fact_1056_of__int__less__numeral__iff,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [Z: int,N3: num] :
( ( ord_less @ A @ ( ring_1_of_int @ A @ Z ) @ ( numeral_numeral @ A @ N3 ) )
= ( ord_less @ int @ Z @ ( numeral_numeral @ int @ N3 ) ) ) ) ).
% of_int_less_numeral_iff
thf(fact_1057_of__int__numeral__less__iff,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [N3: num,Z: int] :
( ( ord_less @ A @ ( numeral_numeral @ A @ N3 ) @ ( ring_1_of_int @ A @ Z ) )
= ( ord_less @ int @ ( numeral_numeral @ int @ N3 ) @ Z ) ) ) ).
% of_int_numeral_less_iff
thf(fact_1058_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_1059_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_1060_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_1061_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_1062_numeral__power__eq__of__int__cancel__iff,axiom,
! [A: $tType] :
( ( ring_char_0 @ A )
=> ! [X: num,N3: nat,Y: int] :
( ( ( power_power @ A @ ( numeral_numeral @ A @ X ) @ N3 )
= ( ring_1_of_int @ A @ Y ) )
= ( ( power_power @ int @ ( numeral_numeral @ int @ X ) @ N3 )
= Y ) ) ) ).
% numeral_power_eq_of_int_cancel_iff
thf(fact_1063_of__int__eq__numeral__power__cancel__iff,axiom,
! [A: $tType] :
( ( ring_char_0 @ A )
=> ! [Y: int,X: num,N3: nat] :
( ( ( ring_1_of_int @ A @ Y )
= ( power_power @ A @ ( numeral_numeral @ A @ X ) @ N3 ) )
= ( Y
= ( power_power @ int @ ( numeral_numeral @ int @ X ) @ N3 ) ) ) ) ).
% of_int_eq_numeral_power_cancel_iff
thf(fact_1064_of__int__power__le__of__int__cancel__iff,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [X: int,B2: int,W: nat] :
( ( ord_less_eq @ A @ ( ring_1_of_int @ A @ X ) @ ( power_power @ A @ ( ring_1_of_int @ A @ B2 ) @ W ) )
= ( ord_less_eq @ int @ X @ ( power_power @ int @ B2 @ W ) ) ) ) ).
% of_int_power_le_of_int_cancel_iff
thf(fact_1065_of__int__le__of__int__power__cancel__iff,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [B2: int,W: nat,X: int] :
( ( ord_less_eq @ A @ ( power_power @ A @ ( ring_1_of_int @ A @ B2 ) @ W ) @ ( ring_1_of_int @ A @ X ) )
= ( ord_less_eq @ int @ ( power_power @ int @ B2 @ W ) @ X ) ) ) ).
% of_int_le_of_int_power_cancel_iff
thf(fact_1066_of__int__less__of__int__power__cancel__iff,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [B2: int,W: nat,X: int] :
( ( ord_less @ A @ ( power_power @ A @ ( ring_1_of_int @ A @ B2 ) @ W ) @ ( ring_1_of_int @ A @ X ) )
= ( ord_less @ int @ ( power_power @ int @ B2 @ W ) @ X ) ) ) ).
% of_int_less_of_int_power_cancel_iff
thf(fact_1067_of__int__power__less__of__int__cancel__iff,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [X: int,B2: int,W: nat] :
( ( ord_less @ A @ ( ring_1_of_int @ A @ X ) @ ( power_power @ A @ ( ring_1_of_int @ A @ B2 ) @ W ) )
= ( ord_less @ int @ X @ ( power_power @ int @ B2 @ W ) ) ) ) ).
% of_int_power_less_of_int_cancel_iff
thf(fact_1068_ceiling__less__one,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [X: A] :
( ( ord_less @ int @ ( archimedean_ceiling @ A @ X ) @ ( one_one @ int ) )
= ( ord_less_eq @ A @ X @ ( zero_zero @ A ) ) ) ) ).
% ceiling_less_one
thf(fact_1069_one__le__ceiling,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [X: A] :
( ( ord_less_eq @ int @ ( one_one @ int ) @ ( archimedean_ceiling @ A @ X ) )
= ( ord_less @ A @ ( zero_zero @ A ) @ X ) ) ) ).
% one_le_ceiling
thf(fact_1070_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_1071_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_1072_power2__less__eq__zero__iff,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [A2: A] :
( ( ord_less_eq @ A @ ( power_power @ A @ A2 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ ( zero_zero @ A ) )
= ( A2
= ( zero_zero @ A ) ) ) ) ).
% power2_less_eq_zero_iff
thf(fact_1073_power2__eq__iff__nonneg,axiom,
! [A: $tType] :
( ( linordered_semidom @ A )
=> ! [X: A,Y: A] :
( ( ord_less_eq @ A @ ( zero_zero @ A ) @ X )
=> ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ Y )
=> ( ( ( power_power @ A @ X @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) )
= ( power_power @ A @ Y @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) )
= ( X = Y ) ) ) ) ) ).
% power2_eq_iff_nonneg
thf(fact_1074_zero__less__power2,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [A2: A] :
( ( ord_less @ A @ ( zero_zero @ A ) @ ( power_power @ A @ A2 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) )
= ( A2
!= ( zero_zero @ A ) ) ) ) ).
% zero_less_power2
thf(fact_1075_sum__power2__eq__zero__iff,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [X: A,Y: A] :
( ( ( plus_plus @ A @ ( power_power @ A @ X @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ ( power_power @ A @ Y @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) )
= ( zero_zero @ A ) )
= ( ( X
= ( zero_zero @ A ) )
& ( Y
= ( zero_zero @ A ) ) ) ) ) ).
% sum_power2_eq_zero_iff
thf(fact_1076_power__decreasing__iff,axiom,
! [A: $tType] :
( ( linordered_semidom @ A )
=> ! [B2: A,M: nat,N3: nat] :
( ( ord_less @ A @ ( zero_zero @ A ) @ B2 )
=> ( ( ord_less @ A @ B2 @ ( one_one @ A ) )
=> ( ( ord_less_eq @ A @ ( power_power @ A @ B2 @ M ) @ ( power_power @ A @ B2 @ N3 ) )
= ( ord_less_eq @ nat @ N3 @ M ) ) ) ) ) ).
% power_decreasing_iff
thf(fact_1077_of__nat__zero__less__power__iff,axiom,
! [A: $tType] :
( ( linordered_semidom @ A )
=> ! [X: nat,N3: nat] :
( ( ord_less @ A @ ( zero_zero @ A ) @ ( power_power @ A @ ( semiring_1_of_nat @ A @ X ) @ N3 ) )
= ( ( ord_less @ nat @ ( zero_zero @ nat ) @ X )
| ( N3
= ( zero_zero @ nat ) ) ) ) ) ).
% of_nat_zero_less_power_iff
thf(fact_1078_numeral__power__le__of__int__cancel__iff,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [X: num,N3: nat,A2: int] :
( ( ord_less_eq @ A @ ( power_power @ A @ ( numeral_numeral @ A @ X ) @ N3 ) @ ( ring_1_of_int @ A @ A2 ) )
= ( ord_less_eq @ int @ ( power_power @ int @ ( numeral_numeral @ int @ X ) @ N3 ) @ A2 ) ) ) ).
% numeral_power_le_of_int_cancel_iff
thf(fact_1079_of__int__le__numeral__power__cancel__iff,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [A2: int,X: num,N3: nat] :
( ( ord_less_eq @ A @ ( ring_1_of_int @ A @ A2 ) @ ( power_power @ A @ ( numeral_numeral @ A @ X ) @ N3 ) )
= ( ord_less_eq @ int @ A2 @ ( power_power @ int @ ( numeral_numeral @ int @ X ) @ N3 ) ) ) ) ).
% of_int_le_numeral_power_cancel_iff
thf(fact_1080_numeral__power__less__of__int__cancel__iff,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [X: num,N3: nat,A2: int] :
( ( ord_less @ A @ ( power_power @ A @ ( numeral_numeral @ A @ X ) @ N3 ) @ ( ring_1_of_int @ A @ A2 ) )
= ( ord_less @ int @ ( power_power @ int @ ( numeral_numeral @ int @ X ) @ N3 ) @ A2 ) ) ) ).
% numeral_power_less_of_int_cancel_iff
thf(fact_1081_of__int__less__numeral__power__cancel__iff,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [A2: int,X: num,N3: nat] :
( ( ord_less @ A @ ( ring_1_of_int @ A @ A2 ) @ ( power_power @ A @ ( numeral_numeral @ A @ X ) @ N3 ) )
= ( ord_less @ int @ A2 @ ( power_power @ int @ ( numeral_numeral @ int @ X ) @ N3 ) ) ) ) ).
% of_int_less_numeral_power_cancel_iff
thf(fact_1082_option_Osize__neq,axiom,
! [A: $tType,X: option @ A] :
( ( size_size @ ( option @ A ) @ X )
!= ( zero_zero @ nat ) ) ).
% option.size_neq
thf(fact_1083_set__notEmptyE,axiom,
! [A: $tType,S3: set @ A] :
( ( S3
!= ( bot_bot @ ( set @ A ) ) )
=> ~ ! [X3: A] :
~ ( member @ A @ X3 @ S3 ) ) ).
% set_notEmptyE
thf(fact_1084_memb__imp__not__empty,axiom,
! [A: $tType,X: A,S3: set @ A] :
( ( member @ A @ X @ S3 )
=> ( S3
!= ( bot_bot @ ( set @ A ) ) ) ) ).
% memb_imp_not_empty
thf(fact_1085_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_1086_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_1087_VEBT__internal_Onaive__member_Osimps_I2_J,axiom,
! [Uu2: option @ ( product_prod @ nat @ nat ),Uv2: list @ vEBT_VEBT,Uw2: vEBT_VEBT,Ux: nat] :
~ ( vEBT_V5719532721284313246member @ ( vEBT_Node @ Uu2 @ ( zero_zero @ nat ) @ Uv2 @ Uw2 ) @ Ux ) ).
% VEBT_internal.naive_member.simps(2)
thf(fact_1088_power__0__left,axiom,
! [A: $tType] :
( ( semiring_1 @ A )
=> ! [N3: nat] :
( ( ( N3
= ( zero_zero @ nat ) )
=> ( ( power_power @ A @ ( zero_zero @ A ) @ N3 )
= ( one_one @ A ) ) )
& ( ( N3
!= ( zero_zero @ nat ) )
=> ( ( power_power @ A @ ( zero_zero @ A ) @ N3 )
= ( zero_zero @ A ) ) ) ) ) ).
% power_0_left
thf(fact_1089_ex__le__of__int,axiom,
! [A: $tType] :
( ( archim462609752435547400_field @ A )
=> ! [X: A] :
? [Z2: int] : ( ord_less_eq @ A @ X @ ( ring_1_of_int @ A @ Z2 ) ) ) ).
% ex_le_of_int
thf(fact_1090_zero__power,axiom,
! [A: $tType] :
( ( semiring_1 @ A )
=> ! [N3: nat] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ N3 )
=> ( ( power_power @ A @ ( zero_zero @ A ) @ N3 )
= ( zero_zero @ A ) ) ) ) ).
% zero_power
thf(fact_1091_ex__of__int__less,axiom,
! [A: $tType] :
( ( archim462609752435547400_field @ A )
=> ! [X: A] :
? [Z2: int] : ( ord_less @ A @ ( ring_1_of_int @ A @ Z2 ) @ X ) ) ).
% ex_of_int_less
thf(fact_1092_ex__less__of__int,axiom,
! [A: $tType] :
( ( archim462609752435547400_field @ A )
=> ! [X: A] :
? [Z2: int] : ( ord_less @ A @ X @ ( ring_1_of_int @ A @ Z2 ) ) ) ).
% ex_less_of_int
thf(fact_1093_mult__of__int__commute,axiom,
! [A: $tType] :
( ( ring_1 @ A )
=> ! [X: int,Y: A] :
( ( times_times @ A @ ( ring_1_of_int @ A @ X ) @ Y )
= ( times_times @ A @ Y @ ( ring_1_of_int @ A @ X ) ) ) ) ).
% mult_of_int_commute
thf(fact_1094_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_1095_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_1096_zero__le,axiom,
! [A: $tType] :
( ( canoni5634975068530333245id_add @ A )
=> ! [X: A] : ( ord_less_eq @ A @ ( zero_zero @ A ) @ X ) ) ).
% zero_le
thf(fact_1097_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_1098_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_1099_gr__zeroI,axiom,
! [A: $tType] :
( ( canoni5634975068530333245id_add @ A )
=> ! [N3: A] :
( ( N3
!= ( zero_zero @ A ) )
=> ( ord_less @ A @ ( zero_zero @ A ) @ N3 ) ) ) ).
% gr_zeroI
thf(fact_1100_not__less__zero,axiom,
! [A: $tType] :
( ( canoni5634975068530333245id_add @ A )
=> ! [N3: A] :
~ ( ord_less @ A @ N3 @ ( zero_zero @ A ) ) ) ).
% not_less_zero
thf(fact_1101_gr__implies__not__zero,axiom,
! [A: $tType] :
( ( canoni5634975068530333245id_add @ A )
=> ! [M: A,N3: A] :
( ( ord_less @ A @ M @ N3 )
=> ( N3
!= ( zero_zero @ A ) ) ) ) ).
% gr_implies_not_zero
thf(fact_1102_zero__less__iff__neq__zero,axiom,
! [A: $tType] :
( ( canoni5634975068530333245id_add @ A )
=> ! [N3: A] :
( ( ord_less @ A @ ( zero_zero @ A ) @ N3 )
= ( N3
!= ( zero_zero @ A ) ) ) ) ).
% zero_less_iff_neq_zero
thf(fact_1103_zero__neq__numeral,axiom,
! [A: $tType] :
( ( semiring_char_0 @ A )
=> ! [N3: num] :
( ( zero_zero @ A )
!= ( numeral_numeral @ A @ N3 ) ) ) ).
% zero_neq_numeral
thf(fact_1104_comm__monoid__add__class_Oadd__0,axiom,
! [A: $tType] :
( ( comm_monoid_add @ A )
=> ! [A2: A] :
( ( plus_plus @ A @ ( zero_zero @ A ) @ A2 )
= A2 ) ) ).
% comm_monoid_add_class.add_0
thf(fact_1105_add_Ocomm__neutral,axiom,
! [A: $tType] :
( ( comm_monoid_add @ A )
=> ! [A2: A] :
( ( plus_plus @ A @ A2 @ ( zero_zero @ A ) )
= A2 ) ) ).
% add.comm_neutral
thf(fact_1106_add_Ogroup__left__neutral,axiom,
! [A: $tType] :
( ( group_add @ A )
=> ! [A2: A] :
( ( plus_plus @ A @ ( zero_zero @ A ) @ A2 )
= A2 ) ) ).
% add.group_left_neutral
thf(fact_1107_zero__neq__one,axiom,
! [A: $tType] :
( ( zero_neq_one @ A )
=> ( ( zero_zero @ A )
!= ( one_one @ A ) ) ) ).
% zero_neq_one
thf(fact_1108_mult__not__zero,axiom,
! [A: $tType] :
( ( mult_zero @ A )
=> ! [A2: A,B2: A] :
( ( ( times_times @ A @ A2 @ B2 )
!= ( zero_zero @ A ) )
=> ( ( A2
!= ( zero_zero @ A ) )
& ( B2
!= ( zero_zero @ A ) ) ) ) ) ).
% mult_not_zero
thf(fact_1109_divisors__zero,axiom,
! [A: $tType] :
( ( semiri3467727345109120633visors @ A )
=> ! [A2: A,B2: A] :
( ( ( times_times @ A @ A2 @ B2 )
= ( zero_zero @ A ) )
=> ( ( A2
= ( zero_zero @ A ) )
| ( B2
= ( zero_zero @ A ) ) ) ) ) ).
% divisors_zero
thf(fact_1110_no__zero__divisors,axiom,
! [A: $tType] :
( ( semiri3467727345109120633visors @ A )
=> ! [A2: A,B2: A] :
( ( A2
!= ( zero_zero @ A ) )
=> ( ( B2
!= ( zero_zero @ A ) )
=> ( ( times_times @ A @ A2 @ B2 )
!= ( zero_zero @ A ) ) ) ) ) ).
% no_zero_divisors
thf(fact_1111_mult__left__cancel,axiom,
! [A: $tType] :
( ( semiri6575147826004484403cancel @ A )
=> ! [C2: A,A2: A,B2: A] :
( ( C2
!= ( zero_zero @ A ) )
=> ( ( ( times_times @ A @ C2 @ A2 )
= ( times_times @ A @ C2 @ B2 ) )
= ( A2 = B2 ) ) ) ) ).
% mult_left_cancel
thf(fact_1112_mult__right__cancel,axiom,
! [A: $tType] :
( ( semiri6575147826004484403cancel @ A )
=> ! [C2: A,A2: A,B2: A] :
( ( C2
!= ( zero_zero @ A ) )
=> ( ( ( times_times @ A @ A2 @ C2 )
= ( times_times @ A @ B2 @ C2 ) )
= ( A2 = B2 ) ) ) ) ).
% mult_right_cancel
thf(fact_1113_eq__iff__diff__eq__0,axiom,
! [A: $tType] :
( ( group_add @ A )
=> ( ( ^ [Y5: A,Z5: A] : Y5 = Z5 )
= ( ^ [A6: A,B5: A] :
( ( minus_minus @ A @ A6 @ B5 )
= ( zero_zero @ A ) ) ) ) ) ).
% eq_iff_diff_eq_0
thf(fact_1114_subset__minus__empty,axiom,
! [A: $tType,A3: set @ A,B6: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ A3 @ B6 )
=> ( ( minus_minus @ ( set @ A ) @ A3 @ B6 )
= ( bot_bot @ ( set @ A ) ) ) ) ).
% subset_minus_empty
thf(fact_1115_semiring__1__no__zero__divisors__class_Opower__not__zero,axiom,
! [A: $tType] :
( ( semiri2026040879449505780visors @ A )
=> ! [A2: A,N3: nat] :
( ( A2
!= ( zero_zero @ A ) )
=> ( ( power_power @ A @ A2 @ N3 )
!= ( zero_zero @ A ) ) ) ) ).
% semiring_1_no_zero_divisors_class.power_not_zero
thf(fact_1116_num_Osize_I4_J,axiom,
( ( size_size @ num @ one2 )
= ( zero_zero @ nat ) ) ).
% num.size(4)
thf(fact_1117_nat_Odistinct_I1_J,axiom,
! [X22: nat] :
( ( zero_zero @ nat )
!= ( suc @ X22 ) ) ).
% nat.distinct(1)
thf(fact_1118_old_Onat_Odistinct_I2_J,axiom,
! [Nat3: nat] :
( ( suc @ Nat3 )
!= ( zero_zero @ nat ) ) ).
% old.nat.distinct(2)
thf(fact_1119_old_Onat_Odistinct_I1_J,axiom,
! [Nat2: nat] :
( ( zero_zero @ nat )
!= ( suc @ Nat2 ) ) ).
% old.nat.distinct(1)
thf(fact_1120_nat_OdiscI,axiom,
! [Nat: nat,X22: nat] :
( ( Nat
= ( suc @ X22 ) )
=> ( Nat
!= ( zero_zero @ nat ) ) ) ).
% nat.discI
thf(fact_1121_old_Onat_Oexhaust,axiom,
! [Y: nat] :
( ( Y
!= ( zero_zero @ nat ) )
=> ~ ! [Nat4: nat] :
( Y
!= ( suc @ Nat4 ) ) ) ).
% old.nat.exhaust
thf(fact_1122_nat__induct,axiom,
! [P: nat > $o,N3: nat] :
( ( P @ ( zero_zero @ nat ) )
=> ( ! [N: nat] :
( ( P @ N )
=> ( P @ ( suc @ N ) ) )
=> ( P @ N3 ) ) ) ).
% nat_induct
thf(fact_1123_diff__induct,axiom,
! [P: nat > nat > $o,M: nat,N3: nat] :
( ! [X3: nat] : ( P @ X3 @ ( zero_zero @ nat ) )
=> ( ! [Y3: nat] : ( P @ ( zero_zero @ nat ) @ ( suc @ Y3 ) )
=> ( ! [X3: nat,Y3: nat] :
( ( P @ X3 @ Y3 )
=> ( P @ ( suc @ X3 ) @ ( suc @ Y3 ) ) )
=> ( P @ M @ N3 ) ) ) ) ).
% diff_induct
thf(fact_1124_zero__induct,axiom,
! [P: nat > $o,K: nat] :
( ( P @ K )
=> ( ! [N: nat] :
( ( P @ ( suc @ N ) )
=> ( P @ N ) )
=> ( P @ ( zero_zero @ nat ) ) ) ) ).
% zero_induct
thf(fact_1125_Suc__neq__Zero,axiom,
! [M: nat] :
( ( suc @ M )
!= ( zero_zero @ nat ) ) ).
% Suc_neq_Zero
thf(fact_1126_Zero__neq__Suc,axiom,
! [M: nat] :
( ( zero_zero @ nat )
!= ( suc @ M ) ) ).
% Zero_neq_Suc
thf(fact_1127_Zero__not__Suc,axiom,
! [M: nat] :
( ( zero_zero @ nat )
!= ( suc @ M ) ) ).
% Zero_not_Suc
thf(fact_1128_not0__implies__Suc,axiom,
! [N3: nat] :
( ( N3
!= ( zero_zero @ nat ) )
=> ? [M4: nat] :
( N3
= ( suc @ M4 ) ) ) ).
% not0_implies_Suc
thf(fact_1129_exists__least__lemma,axiom,
! [P: nat > $o] :
( ~ ( P @ ( zero_zero @ nat ) )
=> ( ? [X_1: nat] : ( P @ X_1 )
=> ? [N: nat] :
( ~ ( P @ N )
& ( P @ ( suc @ N ) ) ) ) ) ).
% exists_least_lemma
thf(fact_1130_bot__nat__0_Oextremum__strict,axiom,
! [A2: nat] :
~ ( ord_less @ nat @ A2 @ ( zero_zero @ nat ) ) ).
% bot_nat_0.extremum_strict
thf(fact_1131_gr0I,axiom,
! [N3: nat] :
( ( N3
!= ( zero_zero @ nat ) )
=> ( ord_less @ nat @ ( zero_zero @ nat ) @ N3 ) ) ).
% gr0I
thf(fact_1132_not__gr0,axiom,
! [N3: nat] :
( ( ~ ( ord_less @ nat @ ( zero_zero @ nat ) @ N3 ) )
= ( N3
= ( zero_zero @ nat ) ) ) ).
% not_gr0
thf(fact_1133_not__less0,axiom,
! [N3: nat] :
~ ( ord_less @ nat @ N3 @ ( zero_zero @ nat ) ) ).
% not_less0
thf(fact_1134_less__zeroE,axiom,
! [N3: nat] :
~ ( ord_less @ nat @ N3 @ ( zero_zero @ nat ) ) ).
% less_zeroE
thf(fact_1135_gr__implies__not0,axiom,
! [M: nat,N3: nat] :
( ( ord_less @ nat @ M @ N3 )
=> ( N3
!= ( zero_zero @ nat ) ) ) ).
% gr_implies_not0
thf(fact_1136_infinite__descent0,axiom,
! [P: nat > $o,N3: nat] :
( ( P @ ( zero_zero @ nat ) )
=> ( ! [N: nat] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ N )
=> ( ~ ( P @ N )
=> ? [M2: nat] :
( ( ord_less @ nat @ M2 @ N )
& ~ ( P @ M2 ) ) ) )
=> ( P @ N3 ) ) ) ).
% infinite_descent0
thf(fact_1137_infinite__descent0__measure,axiom,
! [A: $tType,V2: A > nat,P: A > $o,X: A] :
( ! [X3: A] :
( ( ( V2 @ X3 )
= ( zero_zero @ nat ) )
=> ( P @ X3 ) )
=> ( ! [X3: A] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ ( V2 @ X3 ) )
=> ( ~ ( P @ X3 )
=> ? [Y4: A] :
( ( ord_less @ nat @ ( V2 @ Y4 ) @ ( V2 @ X3 ) )
& ~ ( P @ Y4 ) ) ) )
=> ( P @ X ) ) ) ).
% infinite_descent0_measure
thf(fact_1138_T_092_060_094sub_062i_092_060_094sub_062n_092_060_094sub_062s_092_060_094sub_062e_092_060_094sub_062r_092_060_094sub_062t_H_Osimps_I2_J,axiom,
! [Info: option @ ( product_prod @ nat @ nat ),Ts: list @ vEBT_VEBT,S: vEBT_VEBT,X: nat] :
( ( vEBT_T_i_n_s_e_r_t2 @ ( vEBT_Node @ Info @ ( zero_zero @ nat ) @ Ts @ S ) @ X )
= ( one_one @ nat ) ) ).
% T\<^sub>i\<^sub>n\<^sub>s\<^sub>e\<^sub>r\<^sub>t'.simps(2)
thf(fact_1139_plus__nat_Oadd__0,axiom,
! [N3: nat] :
( ( plus_plus @ nat @ ( zero_zero @ nat ) @ N3 )
= N3 ) ).
% plus_nat.add_0
thf(fact_1140_add__eq__self__zero,axiom,
! [M: nat,N3: nat] :
( ( ( plus_plus @ nat @ M @ N3 )
= M )
=> ( N3
= ( zero_zero @ nat ) ) ) ).
% add_eq_self_zero
thf(fact_1141_le__0__eq,axiom,
! [N3: nat] :
( ( ord_less_eq @ nat @ N3 @ ( zero_zero @ nat ) )
= ( N3
= ( zero_zero @ nat ) ) ) ).
% le_0_eq
thf(fact_1142_bot__nat__0_Oextremum__uniqueI,axiom,
! [A2: nat] :
( ( ord_less_eq @ nat @ A2 @ ( zero_zero @ nat ) )
=> ( A2
= ( zero_zero @ nat ) ) ) ).
% bot_nat_0.extremum_uniqueI
thf(fact_1143_bot__nat__0_Oextremum__unique,axiom,
! [A2: nat] :
( ( ord_less_eq @ nat @ A2 @ ( zero_zero @ nat ) )
= ( A2
= ( zero_zero @ nat ) ) ) ).
% bot_nat_0.extremum_unique
thf(fact_1144_less__eq__nat_Osimps_I1_J,axiom,
! [N3: nat] : ( ord_less_eq @ nat @ ( zero_zero @ nat ) @ N3 ) ).
% less_eq_nat.simps(1)
thf(fact_1145_minus__nat_Odiff__0,axiom,
! [M: nat] :
( ( minus_minus @ nat @ M @ ( zero_zero @ nat ) )
= M ) ).
% minus_nat.diff_0
thf(fact_1146_diffs0__imp__equal,axiom,
! [M: nat,N3: nat] :
( ( ( minus_minus @ nat @ M @ N3 )
= ( zero_zero @ nat ) )
=> ( ( ( minus_minus @ nat @ N3 @ M )
= ( zero_zero @ nat ) )
=> ( M = N3 ) ) ) ).
% diffs0_imp_equal
thf(fact_1147_mult__0,axiom,
! [N3: nat] :
( ( times_times @ nat @ ( zero_zero @ nat ) @ N3 )
= ( zero_zero @ nat ) ) ).
% mult_0
thf(fact_1148_nat__mult__eq__cancel__disj,axiom,
! [K: nat,M: nat,N3: nat] :
( ( ( times_times @ nat @ K @ M )
= ( times_times @ nat @ K @ N3 ) )
= ( ( K
= ( zero_zero @ nat ) )
| ( M = N3 ) ) ) ).
% nat_mult_eq_cancel_disj
thf(fact_1149_power__eq__iff__eq__base,axiom,
! [A: $tType] :
( ( linordered_semidom @ A )
=> ! [N3: nat,A2: A,B2: A] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ N3 )
=> ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ A2 )
=> ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ B2 )
=> ( ( ( power_power @ A @ A2 @ N3 )
= ( power_power @ A @ B2 @ N3 ) )
= ( A2 = B2 ) ) ) ) ) ) ).
% power_eq_iff_eq_base
thf(fact_1150_power__eq__imp__eq__base,axiom,
! [A: $tType] :
( ( linordered_semidom @ A )
=> ! [A2: A,N3: nat,B2: A] :
( ( ( power_power @ A @ A2 @ N3 )
= ( power_power @ A @ B2 @ N3 ) )
=> ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ A2 )
=> ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ B2 )
=> ( ( ord_less @ nat @ ( zero_zero @ nat ) @ N3 )
=> ( A2 = B2 ) ) ) ) ) ) ).
% power_eq_imp_eq_base
thf(fact_1151_lambda__zero,axiom,
! [A: $tType] :
( ( mult_zero @ A )
=> ( ( ^ [H: A] : ( zero_zero @ A ) )
= ( times_times @ A @ ( zero_zero @ A ) ) ) ) ).
% lambda_zero
thf(fact_1152_VEBT__internal_Omembermima_Osimps_I3_J,axiom,
! [Mi: nat,Ma: nat,Va: list @ vEBT_VEBT,Vb: vEBT_VEBT,X: nat] :
( ( vEBT_VEBT_membermima @ ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi @ Ma ) ) @ ( zero_zero @ nat ) @ Va @ Vb ) @ X )
= ( ( X = Mi )
| ( X = Ma ) ) ) ).
% VEBT_internal.membermima.simps(3)
thf(fact_1153_T_092_060_094sub_062i_092_060_094sub_062n_092_060_094sub_062s_092_060_094sub_062e_092_060_094sub_062r_092_060_094sub_062t_H_Osimps_I3_J,axiom,
! [Info: option @ ( product_prod @ nat @ nat ),Ts: list @ vEBT_VEBT,S: vEBT_VEBT,X: nat] :
( ( vEBT_T_i_n_s_e_r_t2 @ ( vEBT_Node @ Info @ ( suc @ ( zero_zero @ nat ) ) @ Ts @ S ) @ X )
= ( one_one @ nat ) ) ).
% T\<^sub>i\<^sub>n\<^sub>s\<^sub>e\<^sub>r\<^sub>t'.simps(3)
thf(fact_1154_power__strict__mono,axiom,
! [A: $tType] :
( ( linordered_semidom @ A )
=> ! [A2: A,B2: A,N3: nat] :
( ( ord_less @ A @ A2 @ B2 )
=> ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ A2 )
=> ( ( ord_less @ nat @ ( zero_zero @ nat ) @ N3 )
=> ( ord_less @ A @ ( power_power @ A @ A2 @ N3 ) @ ( power_power @ A @ B2 @ N3 ) ) ) ) ) ) ).
% power_strict_mono
thf(fact_1155_le__of__int__ceiling,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [X: A] : ( ord_less_eq @ A @ X @ ( ring_1_of_int @ A @ ( archimedean_ceiling @ A @ X ) ) ) ) ).
% le_of_int_ceiling
thf(fact_1156_not__numeral__le__zero,axiom,
! [A: $tType] :
( ( linord181362715937106298miring @ A )
=> ! [N3: num] :
~ ( ord_less_eq @ A @ ( numeral_numeral @ A @ N3 ) @ ( zero_zero @ A ) ) ) ).
% not_numeral_le_zero
thf(fact_1157_zero__le__numeral,axiom,
! [A: $tType] :
( ( linord181362715937106298miring @ A )
=> ! [N3: num] : ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( numeral_numeral @ A @ N3 ) ) ) ).
% zero_le_numeral
thf(fact_1158_zero__less__numeral,axiom,
! [A: $tType] :
( ( linord181362715937106298miring @ A )
=> ! [N3: num] : ( ord_less @ A @ ( zero_zero @ A ) @ ( numeral_numeral @ A @ N3 ) ) ) ).
% zero_less_numeral
thf(fact_1159_not__numeral__less__zero,axiom,
! [A: $tType] :
( ( linord181362715937106298miring @ A )
=> ! [N3: num] :
~ ( ord_less @ A @ ( numeral_numeral @ A @ N3 ) @ ( zero_zero @ A ) ) ) ).
% not_numeral_less_zero
thf(fact_1160_add__nonpos__eq__0__iff,axiom,
! [A: $tType] :
( ( ordere6911136660526730532id_add @ A )
=> ! [X: A,Y: A] :
( ( ord_less_eq @ A @ X @ ( zero_zero @ A ) )
=> ( ( ord_less_eq @ A @ Y @ ( zero_zero @ A ) )
=> ( ( ( plus_plus @ A @ X @ Y )
= ( zero_zero @ A ) )
= ( ( X
= ( zero_zero @ A ) )
& ( Y
= ( zero_zero @ A ) ) ) ) ) ) ) ).
% add_nonpos_eq_0_iff
thf(fact_1161_add__nonneg__eq__0__iff,axiom,
! [A: $tType] :
( ( ordere6911136660526730532id_add @ A )
=> ! [X: A,Y: A] :
( ( ord_less_eq @ A @ ( zero_zero @ A ) @ X )
=> ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ Y )
=> ( ( ( plus_plus @ A @ X @ Y )
= ( zero_zero @ A ) )
= ( ( X
= ( zero_zero @ A ) )
& ( Y
= ( zero_zero @ A ) ) ) ) ) ) ) ).
% add_nonneg_eq_0_iff
thf(fact_1162_add__nonpos__nonpos,axiom,
! [A: $tType] :
( ( ordere6911136660526730532id_add @ A )
=> ! [A2: A,B2: A] :
( ( ord_less_eq @ A @ A2 @ ( zero_zero @ A ) )
=> ( ( ord_less_eq @ A @ B2 @ ( zero_zero @ A ) )
=> ( ord_less_eq @ A @ ( plus_plus @ A @ A2 @ B2 ) @ ( zero_zero @ A ) ) ) ) ) ).
% add_nonpos_nonpos
thf(fact_1163_add__nonneg__nonneg,axiom,
! [A: $tType] :
( ( ordere6911136660526730532id_add @ A )
=> ! [A2: A,B2: A] :
( ( ord_less_eq @ A @ ( zero_zero @ A ) @ A2 )
=> ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ B2 )
=> ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( plus_plus @ A @ A2 @ B2 ) ) ) ) ) ).
% add_nonneg_nonneg
thf(fact_1164_add__increasing2,axiom,
! [A: $tType] :
( ( ordere6911136660526730532id_add @ A )
=> ! [C2: A,B2: A,A2: A] :
( ( ord_less_eq @ A @ ( zero_zero @ A ) @ C2 )
=> ( ( ord_less_eq @ A @ B2 @ A2 )
=> ( ord_less_eq @ A @ B2 @ ( plus_plus @ A @ A2 @ C2 ) ) ) ) ) ).
% add_increasing2
thf(fact_1165_add__decreasing2,axiom,
! [A: $tType] :
( ( ordere6911136660526730532id_add @ A )
=> ! [C2: A,A2: A,B2: A] :
( ( ord_less_eq @ A @ C2 @ ( zero_zero @ A ) )
=> ( ( ord_less_eq @ A @ A2 @ B2 )
=> ( ord_less_eq @ A @ ( plus_plus @ A @ A2 @ C2 ) @ B2 ) ) ) ) ).
% add_decreasing2
thf(fact_1166_add__increasing,axiom,
! [A: $tType] :
( ( ordere6911136660526730532id_add @ A )
=> ! [A2: A,B2: A,C2: A] :
( ( ord_less_eq @ A @ ( zero_zero @ A ) @ A2 )
=> ( ( ord_less_eq @ A @ B2 @ C2 )
=> ( ord_less_eq @ A @ B2 @ ( plus_plus @ A @ A2 @ C2 ) ) ) ) ) ).
% add_increasing
thf(fact_1167_add__decreasing,axiom,
! [A: $tType] :
( ( ordere6911136660526730532id_add @ A )
=> ! [A2: A,C2: A,B2: A] :
( ( ord_less_eq @ A @ A2 @ ( zero_zero @ A ) )
=> ( ( ord_less_eq @ A @ C2 @ B2 )
=> ( ord_less_eq @ A @ ( plus_plus @ A @ A2 @ C2 ) @ B2 ) ) ) ) ).
% add_decreasing
thf(fact_1168_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_1169_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_1170_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_1171_ordered__comm__semiring__class_Ocomm__mult__left__mono,axiom,
! [A: $tType] :
( ( ordere2520102378445227354miring @ A )
=> ! [A2: A,B2: A,C2: A] :
( ( ord_less_eq @ A @ A2 @ B2 )
=> ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ C2 )
=> ( ord_less_eq @ A @ ( times_times @ A @ C2 @ A2 ) @ ( times_times @ A @ C2 @ B2 ) ) ) ) ) ).
% ordered_comm_semiring_class.comm_mult_left_mono
thf(fact_1172_zero__le__mult__iff,axiom,
! [A: $tType] :
( ( linord4710134922213307826strict @ A )
=> ! [A2: A,B2: A] :
( ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( times_times @ A @ A2 @ B2 ) )
= ( ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ A2 )
& ( ord_less_eq @ A @ ( zero_zero @ A ) @ B2 ) )
| ( ( ord_less_eq @ A @ A2 @ ( zero_zero @ A ) )
& ( ord_less_eq @ A @ B2 @ ( zero_zero @ A ) ) ) ) ) ) ).
% zero_le_mult_iff
thf(fact_1173_mult__nonneg__nonpos2,axiom,
! [A: $tType] :
( ( ordered_semiring_0 @ A )
=> ! [A2: A,B2: A] :
( ( ord_less_eq @ A @ ( zero_zero @ A ) @ A2 )
=> ( ( ord_less_eq @ A @ B2 @ ( zero_zero @ A ) )
=> ( ord_less_eq @ A @ ( times_times @ A @ B2 @ A2 ) @ ( zero_zero @ A ) ) ) ) ) ).
% mult_nonneg_nonpos2
thf(fact_1174_mult__nonpos__nonneg,axiom,
! [A: $tType] :
( ( ordered_semiring_0 @ A )
=> ! [A2: A,B2: A] :
( ( ord_less_eq @ A @ A2 @ ( zero_zero @ A ) )
=> ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ B2 )
=> ( ord_less_eq @ A @ ( times_times @ A @ A2 @ B2 ) @ ( zero_zero @ A ) ) ) ) ) ).
% mult_nonpos_nonneg
thf(fact_1175_mult__nonneg__nonpos,axiom,
! [A: $tType] :
( ( ordered_semiring_0 @ A )
=> ! [A2: A,B2: A] :
( ( ord_less_eq @ A @ ( zero_zero @ A ) @ A2 )
=> ( ( ord_less_eq @ A @ B2 @ ( zero_zero @ A ) )
=> ( ord_less_eq @ A @ ( times_times @ A @ A2 @ B2 ) @ ( zero_zero @ A ) ) ) ) ) ).
% mult_nonneg_nonpos
thf(fact_1176_mult__nonneg__nonneg,axiom,
! [A: $tType] :
( ( ordered_semiring_0 @ A )
=> ! [A2: A,B2: A] :
( ( ord_less_eq @ A @ ( zero_zero @ A ) @ A2 )
=> ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ B2 )
=> ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( times_times @ A @ A2 @ B2 ) ) ) ) ) ).
% mult_nonneg_nonneg
thf(fact_1177_split__mult__neg__le,axiom,
! [A: $tType] :
( ( ordered_semiring_0 @ A )
=> ! [A2: A,B2: A] :
( ( ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ A2 )
& ( ord_less_eq @ A @ B2 @ ( zero_zero @ A ) ) )
| ( ( ord_less_eq @ A @ A2 @ ( zero_zero @ A ) )
& ( ord_less_eq @ A @ ( zero_zero @ A ) @ B2 ) ) )
=> ( ord_less_eq @ A @ ( times_times @ A @ A2 @ B2 ) @ ( zero_zero @ A ) ) ) ) ).
% split_mult_neg_le
thf(fact_1178_mult__le__0__iff,axiom,
! [A: $tType] :
( ( linord4710134922213307826strict @ A )
=> ! [A2: A,B2: A] :
( ( ord_less_eq @ A @ ( times_times @ A @ A2 @ B2 ) @ ( zero_zero @ A ) )
= ( ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ A2 )
& ( ord_less_eq @ A @ B2 @ ( zero_zero @ A ) ) )
| ( ( ord_less_eq @ A @ A2 @ ( zero_zero @ A ) )
& ( ord_less_eq @ A @ ( zero_zero @ A ) @ B2 ) ) ) ) ) ).
% mult_le_0_iff
thf(fact_1179_mult__right__mono,axiom,
! [A: $tType] :
( ( ordered_semiring @ A )
=> ! [A2: A,B2: A,C2: A] :
( ( ord_less_eq @ A @ A2 @ B2 )
=> ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ C2 )
=> ( ord_less_eq @ A @ ( times_times @ A @ A2 @ C2 ) @ ( times_times @ A @ B2 @ C2 ) ) ) ) ) ).
% mult_right_mono
thf(fact_1180_mult__right__mono__neg,axiom,
! [A: $tType] :
( ( ordered_ring @ A )
=> ! [B2: A,A2: A,C2: A] :
( ( ord_less_eq @ A @ B2 @ A2 )
=> ( ( ord_less_eq @ A @ C2 @ ( zero_zero @ A ) )
=> ( ord_less_eq @ A @ ( times_times @ A @ A2 @ C2 ) @ ( times_times @ A @ B2 @ C2 ) ) ) ) ) ).
% mult_right_mono_neg
thf(fact_1181_mult__left__mono,axiom,
! [A: $tType] :
( ( ordered_semiring @ A )
=> ! [A2: A,B2: A,C2: A] :
( ( ord_less_eq @ A @ A2 @ B2 )
=> ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ C2 )
=> ( ord_less_eq @ A @ ( times_times @ A @ C2 @ A2 ) @ ( times_times @ A @ C2 @ B2 ) ) ) ) ) ).
% mult_left_mono
thf(fact_1182_mult__nonpos__nonpos,axiom,
! [A: $tType] :
( ( ordered_ring @ A )
=> ! [A2: A,B2: A] :
( ( ord_less_eq @ A @ A2 @ ( zero_zero @ A ) )
=> ( ( ord_less_eq @ A @ B2 @ ( zero_zero @ A ) )
=> ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( times_times @ A @ A2 @ B2 ) ) ) ) ) ).
% mult_nonpos_nonpos
thf(fact_1183_mult__left__mono__neg,axiom,
! [A: $tType] :
( ( ordered_ring @ A )
=> ! [B2: A,A2: A,C2: A] :
( ( ord_less_eq @ A @ B2 @ A2 )
=> ( ( ord_less_eq @ A @ C2 @ ( zero_zero @ A ) )
=> ( ord_less_eq @ A @ ( times_times @ A @ C2 @ A2 ) @ ( times_times @ A @ C2 @ B2 ) ) ) ) ) ).
% mult_left_mono_neg
thf(fact_1184_split__mult__pos__le,axiom,
! [A: $tType] :
( ( ordered_ring @ A )
=> ! [A2: A,B2: A] :
( ( ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ A2 )
& ( ord_less_eq @ A @ ( zero_zero @ A ) @ B2 ) )
| ( ( ord_less_eq @ A @ A2 @ ( zero_zero @ A ) )
& ( ord_less_eq @ A @ B2 @ ( zero_zero @ A ) ) ) )
=> ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( times_times @ A @ A2 @ B2 ) ) ) ) ).
% split_mult_pos_le
thf(fact_1185_zero__le__square,axiom,
! [A: $tType] :
( ( linordered_ring @ A )
=> ! [A2: A] : ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( times_times @ A @ A2 @ A2 ) ) ) ).
% zero_le_square
thf(fact_1186_mult__mono_H,axiom,
! [A: $tType] :
( ( ordered_semiring @ A )
=> ! [A2: A,B2: A,C2: A,D: A] :
( ( ord_less_eq @ A @ A2 @ B2 )
=> ( ( ord_less_eq @ A @ C2 @ D )
=> ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ A2 )
=> ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ C2 )
=> ( ord_less_eq @ A @ ( times_times @ A @ A2 @ C2 ) @ ( times_times @ A @ B2 @ D ) ) ) ) ) ) ) ).
% mult_mono'
thf(fact_1187_mult__mono,axiom,
! [A: $tType] :
( ( ordered_semiring @ A )
=> ! [A2: A,B2: A,C2: A,D: A] :
( ( ord_less_eq @ A @ A2 @ B2 )
=> ( ( ord_less_eq @ A @ C2 @ D )
=> ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ B2 )
=> ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ C2 )
=> ( ord_less_eq @ A @ ( times_times @ A @ A2 @ C2 ) @ ( times_times @ A @ B2 @ D ) ) ) ) ) ) ) ).
% mult_mono
thf(fact_1188_add__less__zeroD,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [X: A,Y: A] :
( ( ord_less @ A @ ( plus_plus @ A @ X @ Y ) @ ( zero_zero @ A ) )
=> ( ( ord_less @ A @ X @ ( zero_zero @ A ) )
| ( ord_less @ A @ Y @ ( zero_zero @ A ) ) ) ) ) ).
% add_less_zeroD
thf(fact_1189_add__neg__neg,axiom,
! [A: $tType] :
( ( ordere6911136660526730532id_add @ A )
=> ! [A2: A,B2: A] :
( ( ord_less @ A @ A2 @ ( zero_zero @ A ) )
=> ( ( ord_less @ A @ B2 @ ( zero_zero @ A ) )
=> ( ord_less @ A @ ( plus_plus @ A @ A2 @ B2 ) @ ( zero_zero @ A ) ) ) ) ) ).
% add_neg_neg
thf(fact_1190_add__pos__pos,axiom,
! [A: $tType] :
( ( ordere6911136660526730532id_add @ A )
=> ! [A2: A,B2: A] :
( ( ord_less @ A @ ( zero_zero @ A ) @ A2 )
=> ( ( ord_less @ A @ ( zero_zero @ A ) @ B2 )
=> ( ord_less @ A @ ( zero_zero @ A ) @ ( plus_plus @ A @ A2 @ B2 ) ) ) ) ) ).
% add_pos_pos
thf(fact_1191_canonically__ordered__monoid__add__class_OlessE,axiom,
! [A: $tType] :
( ( canoni5634975068530333245id_add @ A )
=> ! [A2: A,B2: A] :
( ( ord_less @ A @ A2 @ B2 )
=> ~ ! [C4: A] :
( ( B2
= ( plus_plus @ A @ A2 @ C4 ) )
=> ( C4
= ( zero_zero @ A ) ) ) ) ) ).
% canonically_ordered_monoid_add_class.lessE
thf(fact_1192_pos__add__strict,axiom,
! [A: $tType] :
( ( strict7427464778891057005id_add @ A )
=> ! [A2: A,B2: A,C2: A] :
( ( ord_less @ A @ ( zero_zero @ A ) @ A2 )
=> ( ( ord_less @ A @ B2 @ C2 )
=> ( ord_less @ A @ B2 @ ( plus_plus @ A @ A2 @ C2 ) ) ) ) ) ).
% pos_add_strict
thf(fact_1193_not__one__less__zero,axiom,
! [A: $tType] :
( ( linord181362715937106298miring @ A )
=> ~ ( ord_less @ A @ ( one_one @ A ) @ ( zero_zero @ A ) ) ) ).
% not_one_less_zero
thf(fact_1194_zero__less__one,axiom,
! [A: $tType] :
( ( zero_less_one @ A )
=> ( ord_less @ A @ ( zero_zero @ A ) @ ( one_one @ A ) ) ) ).
% zero_less_one
thf(fact_1195_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_1196_mult__neg__neg,axiom,
! [A: $tType] :
( ( linord4710134922213307826strict @ A )
=> ! [A2: A,B2: A] :
( ( ord_less @ A @ A2 @ ( zero_zero @ A ) )
=> ( ( ord_less @ A @ B2 @ ( zero_zero @ A ) )
=> ( ord_less @ A @ ( zero_zero @ A ) @ ( times_times @ A @ A2 @ B2 ) ) ) ) ) ).
% mult_neg_neg
thf(fact_1197_not__square__less__zero,axiom,
! [A: $tType] :
( ( linordered_ring @ A )
=> ! [A2: A] :
~ ( ord_less @ A @ ( times_times @ A @ A2 @ A2 ) @ ( zero_zero @ A ) ) ) ).
% not_square_less_zero
thf(fact_1198_mult__less__0__iff,axiom,
! [A: $tType] :
( ( linord4710134922213307826strict @ A )
=> ! [A2: A,B2: A] :
( ( ord_less @ A @ ( times_times @ A @ A2 @ B2 ) @ ( zero_zero @ A ) )
= ( ( ( ord_less @ A @ ( zero_zero @ A ) @ A2 )
& ( ord_less @ A @ B2 @ ( zero_zero @ A ) ) )
| ( ( ord_less @ A @ A2 @ ( zero_zero @ A ) )
& ( ord_less @ A @ ( zero_zero @ A ) @ B2 ) ) ) ) ) ).
% mult_less_0_iff
thf(fact_1199_mult__neg__pos,axiom,
! [A: $tType] :
( ( linord8928482502909563296strict @ A )
=> ! [A2: A,B2: A] :
( ( ord_less @ A @ A2 @ ( zero_zero @ A ) )
=> ( ( ord_less @ A @ ( zero_zero @ A ) @ B2 )
=> ( ord_less @ A @ ( times_times @ A @ A2 @ B2 ) @ ( zero_zero @ A ) ) ) ) ) ).
% mult_neg_pos
thf(fact_1200_mult__pos__neg,axiom,
! [A: $tType] :
( ( linord8928482502909563296strict @ A )
=> ! [A2: A,B2: A] :
( ( ord_less @ A @ ( zero_zero @ A ) @ A2 )
=> ( ( ord_less @ A @ B2 @ ( zero_zero @ A ) )
=> ( ord_less @ A @ ( times_times @ A @ A2 @ B2 ) @ ( zero_zero @ A ) ) ) ) ) ).
% mult_pos_neg
thf(fact_1201_mult__pos__pos,axiom,
! [A: $tType] :
( ( linord8928482502909563296strict @ A )
=> ! [A2: A,B2: A] :
( ( ord_less @ A @ ( zero_zero @ A ) @ A2 )
=> ( ( ord_less @ A @ ( zero_zero @ A ) @ B2 )
=> ( ord_less @ A @ ( zero_zero @ A ) @ ( times_times @ A @ A2 @ B2 ) ) ) ) ) ).
% mult_pos_pos
thf(fact_1202_mult__pos__neg2,axiom,
! [A: $tType] :
( ( linord8928482502909563296strict @ A )
=> ! [A2: A,B2: A] :
( ( ord_less @ A @ ( zero_zero @ A ) @ A2 )
=> ( ( ord_less @ A @ B2 @ ( zero_zero @ A ) )
=> ( ord_less @ A @ ( times_times @ A @ B2 @ A2 ) @ ( zero_zero @ A ) ) ) ) ) ).
% mult_pos_neg2
thf(fact_1203_zero__less__mult__iff,axiom,
! [A: $tType] :
( ( linord4710134922213307826strict @ A )
=> ! [A2: A,B2: A] :
( ( ord_less @ A @ ( zero_zero @ A ) @ ( times_times @ A @ A2 @ B2 ) )
= ( ( ( ord_less @ A @ ( zero_zero @ A ) @ A2 )
& ( ord_less @ A @ ( zero_zero @ A ) @ B2 ) )
| ( ( ord_less @ A @ A2 @ ( zero_zero @ A ) )
& ( ord_less @ A @ B2 @ ( zero_zero @ A ) ) ) ) ) ) ).
% zero_less_mult_iff
thf(fact_1204_zero__less__mult__pos,axiom,
! [A: $tType] :
( ( linord8928482502909563296strict @ A )
=> ! [A2: A,B2: A] :
( ( ord_less @ A @ ( zero_zero @ A ) @ ( times_times @ A @ A2 @ B2 ) )
=> ( ( ord_less @ A @ ( zero_zero @ A ) @ A2 )
=> ( ord_less @ A @ ( zero_zero @ A ) @ B2 ) ) ) ) ).
% zero_less_mult_pos
thf(fact_1205_zero__less__mult__pos2,axiom,
! [A: $tType] :
( ( linord8928482502909563296strict @ A )
=> ! [B2: A,A2: A] :
( ( ord_less @ A @ ( zero_zero @ A ) @ ( times_times @ A @ B2 @ A2 ) )
=> ( ( ord_less @ A @ ( zero_zero @ A ) @ A2 )
=> ( ord_less @ A @ ( zero_zero @ A ) @ B2 ) ) ) ) ).
% zero_less_mult_pos2
thf(fact_1206_mult__less__cancel__left__neg,axiom,
! [A: $tType] :
( ( linord4710134922213307826strict @ A )
=> ! [C2: A,A2: A,B2: A] :
( ( ord_less @ A @ C2 @ ( zero_zero @ A ) )
=> ( ( ord_less @ A @ ( times_times @ A @ C2 @ A2 ) @ ( times_times @ A @ C2 @ B2 ) )
= ( ord_less @ A @ B2 @ A2 ) ) ) ) ).
% mult_less_cancel_left_neg
thf(fact_1207_mult__less__cancel__left__pos,axiom,
! [A: $tType] :
( ( linord4710134922213307826strict @ A )
=> ! [C2: A,A2: A,B2: A] :
( ( ord_less @ A @ ( zero_zero @ A ) @ C2 )
=> ( ( ord_less @ A @ ( times_times @ A @ C2 @ A2 ) @ ( times_times @ A @ C2 @ B2 ) )
= ( ord_less @ A @ A2 @ B2 ) ) ) ) ).
% mult_less_cancel_left_pos
thf(fact_1208_mult__strict__left__mono__neg,axiom,
! [A: $tType] :
( ( linord4710134922213307826strict @ A )
=> ! [B2: A,A2: A,C2: A] :
( ( ord_less @ A @ B2 @ A2 )
=> ( ( ord_less @ A @ C2 @ ( zero_zero @ A ) )
=> ( ord_less @ A @ ( times_times @ A @ C2 @ A2 ) @ ( times_times @ A @ C2 @ B2 ) ) ) ) ) ).
% mult_strict_left_mono_neg
thf(fact_1209_mult__strict__left__mono,axiom,
! [A: $tType] :
( ( linord8928482502909563296strict @ A )
=> ! [A2: A,B2: A,C2: A] :
( ( ord_less @ A @ A2 @ B2 )
=> ( ( ord_less @ A @ ( zero_zero @ A ) @ C2 )
=> ( ord_less @ A @ ( times_times @ A @ C2 @ A2 ) @ ( times_times @ A @ C2 @ B2 ) ) ) ) ) ).
% mult_strict_left_mono
thf(fact_1210_mult__less__cancel__left__disj,axiom,
! [A: $tType] :
( ( linord4710134922213307826strict @ A )
=> ! [C2: A,A2: A,B2: A] :
( ( ord_less @ A @ ( times_times @ A @ C2 @ A2 ) @ ( times_times @ A @ C2 @ B2 ) )
= ( ( ( ord_less @ A @ ( zero_zero @ A ) @ C2 )
& ( ord_less @ A @ A2 @ B2 ) )
| ( ( ord_less @ A @ C2 @ ( zero_zero @ A ) )
& ( ord_less @ A @ B2 @ A2 ) ) ) ) ) ).
% mult_less_cancel_left_disj
thf(fact_1211_mult__strict__right__mono__neg,axiom,
! [A: $tType] :
( ( linord4710134922213307826strict @ A )
=> ! [B2: A,A2: A,C2: A] :
( ( ord_less @ A @ B2 @ A2 )
=> ( ( ord_less @ A @ C2 @ ( zero_zero @ A ) )
=> ( ord_less @ A @ ( times_times @ A @ A2 @ C2 ) @ ( times_times @ A @ B2 @ C2 ) ) ) ) ) ).
% mult_strict_right_mono_neg
thf(fact_1212_mult__strict__right__mono,axiom,
! [A: $tType] :
( ( linord8928482502909563296strict @ A )
=> ! [A2: A,B2: A,C2: A] :
( ( ord_less @ A @ A2 @ B2 )
=> ( ( ord_less @ A @ ( zero_zero @ A ) @ C2 )
=> ( ord_less @ A @ ( times_times @ A @ A2 @ C2 ) @ ( times_times @ A @ B2 @ C2 ) ) ) ) ) ).
% mult_strict_right_mono
thf(fact_1213_mult__less__cancel__right__disj,axiom,
! [A: $tType] :
( ( linord4710134922213307826strict @ A )
=> ! [A2: A,C2: A,B2: A] :
( ( ord_less @ A @ ( times_times @ A @ A2 @ C2 ) @ ( times_times @ A @ B2 @ C2 ) )
= ( ( ( ord_less @ A @ ( zero_zero @ A ) @ C2 )
& ( ord_less @ A @ A2 @ B2 ) )
| ( ( ord_less @ A @ C2 @ ( zero_zero @ A ) )
& ( ord_less @ A @ B2 @ A2 ) ) ) ) ) ).
% mult_less_cancel_right_disj
thf(fact_1214_linordered__comm__semiring__strict__class_Ocomm__mult__strict__left__mono,axiom,
! [A: $tType] :
( ( linord2810124833399127020strict @ A )
=> ! [A2: A,B2: A,C2: A] :
( ( ord_less @ A @ A2 @ B2 )
=> ( ( ord_less @ A @ ( zero_zero @ A ) @ C2 )
=> ( ord_less @ A @ ( times_times @ A @ C2 @ A2 ) @ ( times_times @ A @ C2 @ B2 ) ) ) ) ) ).
% linordered_comm_semiring_strict_class.comm_mult_strict_left_mono
thf(fact_1215_le__iff__diff__le__0,axiom,
! [A: $tType] :
( ( ordered_ab_group_add @ A )
=> ( ( ord_less_eq @ A )
= ( ^ [A6: A,B5: A] : ( ord_less_eq @ A @ ( minus_minus @ A @ A6 @ B5 ) @ ( zero_zero @ A ) ) ) ) ) ).
% le_iff_diff_le_0
thf(fact_1216_less__iff__diff__less__0,axiom,
! [A: $tType] :
( ( ordered_ab_group_add @ A )
=> ( ( ord_less @ A )
= ( ^ [A6: A,B5: A] : ( ord_less @ A @ ( minus_minus @ A @ A6 @ B5 ) @ ( zero_zero @ A ) ) ) ) ) ).
% less_iff_diff_less_0
thf(fact_1217_divide__right__mono__neg,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [A2: A,B2: A,C2: A] :
( ( ord_less_eq @ A @ A2 @ B2 )
=> ( ( ord_less_eq @ A @ C2 @ ( zero_zero @ A ) )
=> ( ord_less_eq @ A @ ( divide_divide @ A @ B2 @ C2 ) @ ( divide_divide @ A @ A2 @ C2 ) ) ) ) ) ).
% divide_right_mono_neg
thf(fact_1218_divide__nonpos__nonpos,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [X: A,Y: A] :
( ( ord_less_eq @ A @ X @ ( zero_zero @ A ) )
=> ( ( ord_less_eq @ A @ Y @ ( zero_zero @ A ) )
=> ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( divide_divide @ A @ X @ Y ) ) ) ) ) ).
% divide_nonpos_nonpos
thf(fact_1219_divide__nonpos__nonneg,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [X: A,Y: A] :
( ( ord_less_eq @ A @ X @ ( zero_zero @ A ) )
=> ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ Y )
=> ( ord_less_eq @ A @ ( divide_divide @ A @ X @ Y ) @ ( zero_zero @ A ) ) ) ) ) ).
% divide_nonpos_nonneg
thf(fact_1220_divide__nonneg__nonpos,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [X: A,Y: A] :
( ( ord_less_eq @ A @ ( zero_zero @ A ) @ X )
=> ( ( ord_less_eq @ A @ Y @ ( zero_zero @ A ) )
=> ( ord_less_eq @ A @ ( divide_divide @ A @ X @ Y ) @ ( zero_zero @ A ) ) ) ) ) ).
% divide_nonneg_nonpos
thf(fact_1221_divide__nonneg__nonneg,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [X: A,Y: A] :
( ( ord_less_eq @ A @ ( zero_zero @ A ) @ X )
=> ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ Y )
=> ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( divide_divide @ A @ X @ Y ) ) ) ) ) ).
% divide_nonneg_nonneg
thf(fact_1222_zero__le__divide__iff,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [A2: A,B2: A] :
( ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( divide_divide @ A @ A2 @ B2 ) )
= ( ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ A2 )
& ( ord_less_eq @ A @ ( zero_zero @ A ) @ B2 ) )
| ( ( ord_less_eq @ A @ A2 @ ( zero_zero @ A ) )
& ( ord_less_eq @ A @ B2 @ ( zero_zero @ A ) ) ) ) ) ) ).
% zero_le_divide_iff
thf(fact_1223_divide__right__mono,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [A2: A,B2: A,C2: A] :
( ( ord_less_eq @ A @ A2 @ B2 )
=> ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ C2 )
=> ( ord_less_eq @ A @ ( divide_divide @ A @ A2 @ C2 ) @ ( divide_divide @ A @ B2 @ C2 ) ) ) ) ) ).
% divide_right_mono
thf(fact_1224_divide__le__0__iff,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [A2: A,B2: A] :
( ( ord_less_eq @ A @ ( divide_divide @ A @ A2 @ B2 ) @ ( zero_zero @ A ) )
= ( ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ A2 )
& ( ord_less_eq @ A @ B2 @ ( zero_zero @ A ) ) )
| ( ( ord_less_eq @ A @ A2 @ ( zero_zero @ A ) )
& ( ord_less_eq @ A @ ( zero_zero @ A ) @ B2 ) ) ) ) ) ).
% divide_le_0_iff
thf(fact_1225_divide__neg__neg,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [X: A,Y: A] :
( ( ord_less @ A @ X @ ( zero_zero @ A ) )
=> ( ( ord_less @ A @ Y @ ( zero_zero @ A ) )
=> ( ord_less @ A @ ( zero_zero @ A ) @ ( divide_divide @ A @ X @ Y ) ) ) ) ) ).
% divide_neg_neg
thf(fact_1226_divide__neg__pos,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [X: A,Y: A] :
( ( ord_less @ A @ X @ ( zero_zero @ A ) )
=> ( ( ord_less @ A @ ( zero_zero @ A ) @ Y )
=> ( ord_less @ A @ ( divide_divide @ A @ X @ Y ) @ ( zero_zero @ A ) ) ) ) ) ).
% divide_neg_pos
thf(fact_1227_divide__pos__neg,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [X: A,Y: A] :
( ( ord_less @ A @ ( zero_zero @ A ) @ X )
=> ( ( ord_less @ A @ Y @ ( zero_zero @ A ) )
=> ( ord_less @ A @ ( divide_divide @ A @ X @ Y ) @ ( zero_zero @ A ) ) ) ) ) ).
% divide_pos_neg
thf(fact_1228_divide__pos__pos,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [X: A,Y: A] :
( ( ord_less @ A @ ( zero_zero @ A ) @ X )
=> ( ( ord_less @ A @ ( zero_zero @ A ) @ Y )
=> ( ord_less @ A @ ( zero_zero @ A ) @ ( divide_divide @ A @ X @ Y ) ) ) ) ) ).
% divide_pos_pos
thf(fact_1229_divide__less__0__iff,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [A2: A,B2: A] :
( ( ord_less @ A @ ( divide_divide @ A @ A2 @ B2 ) @ ( zero_zero @ A ) )
= ( ( ( ord_less @ A @ ( zero_zero @ A ) @ A2 )
& ( ord_less @ A @ B2 @ ( zero_zero @ A ) ) )
| ( ( ord_less @ A @ A2 @ ( zero_zero @ A ) )
& ( ord_less @ A @ ( zero_zero @ A ) @ B2 ) ) ) ) ) ).
% divide_less_0_iff
thf(fact_1230_divide__less__cancel,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [A2: A,C2: A,B2: A] :
( ( ord_less @ A @ ( divide_divide @ A @ A2 @ C2 ) @ ( divide_divide @ A @ B2 @ C2 ) )
= ( ( ( ord_less @ A @ ( zero_zero @ A ) @ C2 )
=> ( ord_less @ A @ A2 @ B2 ) )
& ( ( ord_less @ A @ C2 @ ( zero_zero @ A ) )
=> ( ord_less @ A @ B2 @ A2 ) )
& ( C2
!= ( zero_zero @ A ) ) ) ) ) ).
% divide_less_cancel
thf(fact_1231_zero__less__divide__iff,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [A2: A,B2: A] :
( ( ord_less @ A @ ( zero_zero @ A ) @ ( divide_divide @ A @ A2 @ B2 ) )
= ( ( ( ord_less @ A @ ( zero_zero @ A ) @ A2 )
& ( ord_less @ A @ ( zero_zero @ A ) @ B2 ) )
| ( ( ord_less @ A @ A2 @ ( zero_zero @ A ) )
& ( ord_less @ A @ B2 @ ( zero_zero @ A ) ) ) ) ) ) ).
% zero_less_divide_iff
thf(fact_1232_divide__strict__right__mono,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [A2: A,B2: A,C2: A] :
( ( ord_less @ A @ A2 @ B2 )
=> ( ( ord_less @ A @ ( zero_zero @ A ) @ C2 )
=> ( ord_less @ A @ ( divide_divide @ A @ A2 @ C2 ) @ ( divide_divide @ A @ B2 @ C2 ) ) ) ) ) ).
% divide_strict_right_mono
thf(fact_1233_divide__strict__right__mono__neg,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [B2: A,A2: A,C2: A] :
( ( ord_less @ A @ B2 @ A2 )
=> ( ( ord_less @ A @ C2 @ ( zero_zero @ A ) )
=> ( ord_less @ A @ ( divide_divide @ A @ A2 @ C2 ) @ ( divide_divide @ A @ B2 @ C2 ) ) ) ) ) ).
% divide_strict_right_mono_neg
thf(fact_1234_zero__le__power,axiom,
! [A: $tType] :
( ( linordered_semidom @ A )
=> ! [A2: A,N3: nat] :
( ( ord_less_eq @ A @ ( zero_zero @ A ) @ A2 )
=> ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( power_power @ A @ A2 @ N3 ) ) ) ) ).
% zero_le_power
thf(fact_1235_power__mono,axiom,
! [A: $tType] :
( ( linordered_semidom @ A )
=> ! [A2: A,B2: A,N3: nat] :
( ( ord_less_eq @ A @ A2 @ B2 )
=> ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ A2 )
=> ( ord_less_eq @ A @ ( power_power @ A @ A2 @ N3 ) @ ( power_power @ A @ B2 @ N3 ) ) ) ) ) ).
% power_mono
thf(fact_1236_zero__less__power,axiom,
! [A: $tType] :
( ( linordered_semidom @ A )
=> ! [A2: A,N3: nat] :
( ( ord_less @ A @ ( zero_zero @ A ) @ A2 )
=> ( ord_less @ A @ ( zero_zero @ A ) @ ( power_power @ A @ A2 @ N3 ) ) ) ) ).
% zero_less_power
thf(fact_1237_right__inverse__eq,axiom,
! [A: $tType] :
( ( division_ring @ A )
=> ! [B2: A,A2: A] :
( ( B2
!= ( zero_zero @ A ) )
=> ( ( ( divide_divide @ A @ A2 @ B2 )
= ( one_one @ A ) )
= ( A2 = B2 ) ) ) ) ).
% right_inverse_eq
thf(fact_1238_frac__eq__eq,axiom,
! [A: $tType] :
( ( field @ A )
=> ! [Y: A,Z: A,X: A,W: A] :
( ( Y
!= ( zero_zero @ A ) )
=> ( ( Z
!= ( zero_zero @ A ) )
=> ( ( ( divide_divide @ A @ X @ Y )
= ( divide_divide @ A @ W @ Z ) )
= ( ( times_times @ A @ X @ Z )
= ( times_times @ A @ W @ Y ) ) ) ) ) ) ).
% frac_eq_eq
thf(fact_1239_divide__eq__eq,axiom,
! [A: $tType] :
( ( division_ring @ A )
=> ! [B2: A,C2: A,A2: A] :
( ( ( divide_divide @ A @ B2 @ C2 )
= A2 )
= ( ( ( C2
!= ( zero_zero @ A ) )
=> ( B2
= ( times_times @ A @ A2 @ C2 ) ) )
& ( ( C2
= ( zero_zero @ A ) )
=> ( A2
= ( zero_zero @ A ) ) ) ) ) ) ).
% divide_eq_eq
thf(fact_1240_eq__divide__eq,axiom,
! [A: $tType] :
( ( division_ring @ A )
=> ! [A2: A,B2: A,C2: A] :
( ( A2
= ( divide_divide @ A @ B2 @ C2 ) )
= ( ( ( C2
!= ( zero_zero @ A ) )
=> ( ( times_times @ A @ A2 @ C2 )
= B2 ) )
& ( ( C2
= ( zero_zero @ A ) )
=> ( A2
= ( zero_zero @ A ) ) ) ) ) ) ).
% eq_divide_eq
thf(fact_1241_divide__eq__imp,axiom,
! [A: $tType] :
( ( division_ring @ A )
=> ! [C2: A,B2: A,A2: A] :
( ( C2
!= ( zero_zero @ A ) )
=> ( ( B2
= ( times_times @ A @ A2 @ C2 ) )
=> ( ( divide_divide @ A @ B2 @ C2 )
= A2 ) ) ) ) ).
% divide_eq_imp
thf(fact_1242_eq__divide__imp,axiom,
! [A: $tType] :
( ( division_ring @ A )
=> ! [C2: A,A2: A,B2: A] :
( ( C2
!= ( zero_zero @ A ) )
=> ( ( ( times_times @ A @ A2 @ C2 )
= B2 )
=> ( A2
= ( divide_divide @ A @ B2 @ C2 ) ) ) ) ) ).
% eq_divide_imp
thf(fact_1243_nonzero__divide__eq__eq,axiom,
! [A: $tType] :
( ( division_ring @ A )
=> ! [C2: A,B2: A,A2: A] :
( ( C2
!= ( zero_zero @ A ) )
=> ( ( ( divide_divide @ A @ B2 @ C2 )
= A2 )
= ( B2
= ( times_times @ A @ A2 @ C2 ) ) ) ) ) ).
% nonzero_divide_eq_eq
thf(fact_1244_nonzero__eq__divide__eq,axiom,
! [A: $tType] :
( ( division_ring @ A )
=> ! [C2: A,A2: A,B2: A] :
( ( C2
!= ( zero_zero @ A ) )
=> ( ( A2
= ( divide_divide @ A @ B2 @ C2 ) )
= ( ( times_times @ A @ A2 @ C2 )
= B2 ) ) ) ) ).
% nonzero_eq_divide_eq
thf(fact_1245_of__nat__0__le__iff,axiom,
! [A: $tType] :
( ( linord181362715937106298miring @ A )
=> ! [N3: nat] : ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( semiring_1_of_nat @ A @ N3 ) ) ) ).
% of_nat_0_le_iff
thf(fact_1246_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_1247_semiring__char__0__class_Oof__nat__neq__0,axiom,
! [A: $tType] :
( ( semiring_char_0 @ A )
=> ! [N3: nat] :
( ( semiring_1_of_nat @ A @ ( suc @ N3 ) )
!= ( zero_zero @ A ) ) ) ).
% semiring_char_0_class.of_nat_neq_0
thf(fact_1248_power__0,axiom,
! [A: $tType] :
( ( power @ A )
=> ! [A2: A] :
( ( power_power @ A @ A2 @ ( zero_zero @ nat ) )
= ( one_one @ A ) ) ) ).
% power_0
thf(fact_1249_Ex__less__Suc2,axiom,
! [N3: nat,P: nat > $o] :
( ( ? [I3: nat] :
( ( ord_less @ nat @ I3 @ ( suc @ N3 ) )
& ( P @ I3 ) ) )
= ( ( P @ ( zero_zero @ nat ) )
| ? [I3: nat] :
( ( ord_less @ nat @ I3 @ N3 )
& ( P @ ( suc @ I3 ) ) ) ) ) ).
% Ex_less_Suc2
thf(fact_1250_gr0__conv__Suc,axiom,
! [N3: nat] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ N3 )
= ( ? [M5: nat] :
( N3
= ( suc @ M5 ) ) ) ) ).
% gr0_conv_Suc
thf(fact_1251_All__less__Suc2,axiom,
! [N3: nat,P: nat > $o] :
( ( ! [I3: nat] :
( ( ord_less @ nat @ I3 @ ( suc @ N3 ) )
=> ( P @ I3 ) ) )
= ( ( P @ ( zero_zero @ nat ) )
& ! [I3: nat] :
( ( ord_less @ nat @ I3 @ N3 )
=> ( P @ ( suc @ I3 ) ) ) ) ) ).
% All_less_Suc2
thf(fact_1252_gr0__implies__Suc,axiom,
! [N3: nat] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ N3 )
=> ? [M4: nat] :
( N3
= ( suc @ M4 ) ) ) ).
% gr0_implies_Suc
thf(fact_1253_less__Suc__eq__0__disj,axiom,
! [M: nat,N3: nat] :
( ( ord_less @ nat @ M @ ( suc @ N3 ) )
= ( ( M
= ( zero_zero @ nat ) )
| ? [J3: nat] :
( ( M
= ( suc @ J3 ) )
& ( ord_less @ nat @ J3 @ N3 ) ) ) ) ).
% less_Suc_eq_0_disj
thf(fact_1254_add__is__1,axiom,
! [M: nat,N3: nat] :
( ( ( plus_plus @ nat @ M @ N3 )
= ( suc @ ( zero_zero @ nat ) ) )
= ( ( ( M
= ( suc @ ( zero_zero @ nat ) ) )
& ( N3
= ( zero_zero @ nat ) ) )
| ( ( M
= ( zero_zero @ nat ) )
& ( N3
= ( suc @ ( zero_zero @ nat ) ) ) ) ) ) ).
% add_is_1
thf(fact_1255_one__is__add,axiom,
! [M: nat,N3: nat] :
( ( ( suc @ ( zero_zero @ nat ) )
= ( plus_plus @ nat @ M @ N3 ) )
= ( ( ( M
= ( suc @ ( zero_zero @ nat ) ) )
& ( N3
= ( zero_zero @ nat ) ) )
| ( ( M
= ( zero_zero @ nat ) )
& ( N3
= ( suc @ ( zero_zero @ nat ) ) ) ) ) ) ).
% one_is_add
thf(fact_1256_nat__compl__induct_H,axiom,
! [P: nat > $o,N3: nat] :
( ( P @ ( zero_zero @ nat ) )
=> ( ! [N: nat] :
( ! [Nn: nat] :
( ( ord_less_eq @ nat @ Nn @ N )
=> ( P @ Nn ) )
=> ( P @ ( suc @ N ) ) )
=> ( P @ N3 ) ) ) ).
% nat_compl_induct'
thf(fact_1257_nat__compl__induct,axiom,
! [P: nat > $o,N3: nat] :
( ( P @ ( zero_zero @ nat ) )
=> ( ! [N: nat] :
( ! [Nn: nat] :
( ( ord_less_eq @ nat @ Nn @ N )
=> ( P @ Nn ) )
=> ( P @ ( suc @ N ) ) )
=> ( P @ N3 ) ) ) ).
% nat_compl_induct
thf(fact_1258_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_1259_One__nat__def,axiom,
( ( one_one @ nat )
= ( suc @ ( zero_zero @ nat ) ) ) ).
% One_nat_def
thf(fact_1260_option_Osize_I3_J,axiom,
! [A: $tType] :
( ( size_size @ ( option @ A ) @ ( none @ A ) )
= ( suc @ ( zero_zero @ nat ) ) ) ).
% option.size(3)
thf(fact_1261_less__imp__add__positive,axiom,
! [I: nat,J: nat] :
( ( ord_less @ nat @ I @ J )
=> ? [K3: nat] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ K3 )
& ( ( plus_plus @ nat @ I @ K3 )
= J ) ) ) ).
% less_imp_add_positive
thf(fact_1262_ex__least__nat__le,axiom,
! [P: nat > $o,N3: nat] :
( ( P @ N3 )
=> ( ~ ( P @ ( zero_zero @ nat ) )
=> ? [K3: nat] :
( ( ord_less_eq @ nat @ K3 @ N3 )
& ! [I4: nat] :
( ( ord_less @ nat @ I4 @ K3 )
=> ~ ( P @ I4 ) )
& ( P @ K3 ) ) ) ) ).
% ex_least_nat_le
thf(fact_1263_Suc__to__right,axiom,
! [N3: nat,M: nat] :
( ( ( suc @ N3 )
= M )
=> ( N3
= ( minus_minus @ nat @ M @ ( suc @ ( zero_zero @ nat ) ) ) ) ) ).
% Suc_to_right
thf(fact_1264_Euclidean__Division_Odiv__eq__0__iff,axiom,
! [M: nat,N3: nat] :
( ( ( divide_divide @ nat @ M @ N3 )
= ( zero_zero @ nat ) )
= ( ( ord_less @ nat @ M @ N3 )
| ( N3
= ( zero_zero @ nat ) ) ) ) ).
% Euclidean_Division.div_eq_0_iff
thf(fact_1265_diff__less,axiom,
! [N3: nat,M: nat] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ N3 )
=> ( ( ord_less @ nat @ ( zero_zero @ nat ) @ M )
=> ( ord_less @ nat @ ( minus_minus @ nat @ M @ N3 ) @ M ) ) ) ).
% diff_less
thf(fact_1266_nat__geq__1__eq__neqz,axiom,
! [X: nat] :
( ( ord_less_eq @ nat @ ( one_one @ nat ) @ X )
= ( X
!= ( zero_zero @ nat ) ) ) ).
% nat_geq_1_eq_neqz
thf(fact_1267_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_1268_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_1269_nat__mult__eq__cancel1,axiom,
! [K: nat,M: nat,N3: nat] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ K )
=> ( ( ( times_times @ nat @ K @ M )
= ( times_times @ nat @ K @ N3 ) )
= ( M = N3 ) ) ) ).
% nat_mult_eq_cancel1
thf(fact_1270_nat__mult__less__cancel1,axiom,
! [K: nat,M: nat,N3: nat] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ K )
=> ( ( ord_less @ nat @ ( times_times @ nat @ K @ M ) @ ( times_times @ nat @ K @ N3 ) )
= ( ord_less @ nat @ M @ N3 ) ) ) ).
% nat_mult_less_cancel1
thf(fact_1271_diff__add__0,axiom,
! [N3: nat,M: nat] :
( ( minus_minus @ nat @ N3 @ ( plus_plus @ nat @ N3 @ M ) )
= ( zero_zero @ nat ) ) ).
% diff_add_0
thf(fact_1272_nat__power__less__imp__less,axiom,
! [I: nat,M: nat,N3: nat] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ I )
=> ( ( ord_less @ nat @ ( power_power @ nat @ I @ M ) @ ( power_power @ nat @ I @ N3 ) )
=> ( ord_less @ nat @ M @ N3 ) ) ) ).
% nat_power_less_imp_less
thf(fact_1273_mult__eq__self__implies__10,axiom,
! [M: nat,N3: nat] :
( ( M
= ( times_times @ nat @ M @ N3 ) )
=> ( ( N3
= ( one_one @ nat ) )
| ( M
= ( zero_zero @ nat ) ) ) ) ).
% mult_eq_self_implies_10
thf(fact_1274_ceiling__divide__upper,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [Q2: A,P4: A] :
( ( ord_less @ A @ ( zero_zero @ A ) @ Q2 )
=> ( ord_less_eq @ A @ P4 @ ( times_times @ A @ ( ring_1_of_int @ A @ ( archimedean_ceiling @ A @ ( divide_divide @ A @ P4 @ Q2 ) ) ) @ Q2 ) ) ) ) ).
% ceiling_divide_upper
thf(fact_1275_vebt__insert_Osimps_I2_J,axiom,
! [Info: option @ ( product_prod @ nat @ nat ),Ts: list @ vEBT_VEBT,S: vEBT_VEBT,X: nat] :
( ( vEBT_vebt_insert @ ( vEBT_Node @ Info @ ( zero_zero @ nat ) @ Ts @ S ) @ X )
= ( vEBT_Node @ Info @ ( zero_zero @ nat ) @ Ts @ S ) ) ).
% vebt_insert.simps(2)
thf(fact_1276_ceiling__divide__lower,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [Q2: A,P4: A] :
( ( ord_less @ A @ ( zero_zero @ A ) @ Q2 )
=> ( ord_less @ A @ ( times_times @ A @ ( minus_minus @ A @ ( ring_1_of_int @ A @ ( archimedean_ceiling @ A @ ( divide_divide @ A @ P4 @ Q2 ) ) ) @ ( one_one @ A ) ) @ Q2 ) @ P4 ) ) ) ).
% ceiling_divide_lower
thf(fact_1277_ceiling__le,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [X: A,A2: int] :
( ( ord_less_eq @ A @ X @ ( ring_1_of_int @ A @ A2 ) )
=> ( ord_less_eq @ int @ ( archimedean_ceiling @ A @ X ) @ A2 ) ) ) ).
% ceiling_le
thf(fact_1278_ceiling__le__iff,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [X: A,Z: int] :
( ( ord_less_eq @ int @ ( archimedean_ceiling @ A @ X ) @ Z )
= ( ord_less_eq @ A @ X @ ( ring_1_of_int @ A @ Z ) ) ) ) ).
% ceiling_le_iff
thf(fact_1279_less__ceiling__iff,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [Z: int,X: A] :
( ( ord_less @ int @ Z @ ( archimedean_ceiling @ A @ X ) )
= ( ord_less @ A @ ( ring_1_of_int @ A @ Z ) @ X ) ) ) ).
% less_ceiling_iff
thf(fact_1280_real__of__int__div4,axiom,
! [N3: int,X: int] : ( ord_less_eq @ real @ ( ring_1_of_int @ real @ ( divide_divide @ int @ N3 @ X ) ) @ ( divide_divide @ real @ ( ring_1_of_int @ real @ N3 ) @ ( ring_1_of_int @ real @ X ) ) ) ).
% real_of_int_div4
thf(fact_1281_field__le__epsilon,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [X: A,Y: A] :
( ! [E2: A] :
( ( ord_less @ A @ ( zero_zero @ A ) @ E2 )
=> ( ord_less_eq @ A @ X @ ( plus_plus @ A @ Y @ E2 ) ) )
=> ( ord_less_eq @ A @ X @ Y ) ) ) ).
% field_le_epsilon
thf(fact_1282_add__neg__nonpos,axiom,
! [A: $tType] :
( ( ordere6911136660526730532id_add @ A )
=> ! [A2: A,B2: A] :
( ( ord_less @ A @ A2 @ ( zero_zero @ A ) )
=> ( ( ord_less_eq @ A @ B2 @ ( zero_zero @ A ) )
=> ( ord_less @ A @ ( plus_plus @ A @ A2 @ B2 ) @ ( zero_zero @ A ) ) ) ) ) ).
% add_neg_nonpos
thf(fact_1283_add__nonneg__pos,axiom,
! [A: $tType] :
( ( ordere6911136660526730532id_add @ A )
=> ! [A2: A,B2: A] :
( ( ord_less_eq @ A @ ( zero_zero @ A ) @ A2 )
=> ( ( ord_less @ A @ ( zero_zero @ A ) @ B2 )
=> ( ord_less @ A @ ( zero_zero @ A ) @ ( plus_plus @ A @ A2 @ B2 ) ) ) ) ) ).
% add_nonneg_pos
thf(fact_1284_add__nonpos__neg,axiom,
! [A: $tType] :
( ( ordere6911136660526730532id_add @ A )
=> ! [A2: A,B2: A] :
( ( ord_less_eq @ A @ A2 @ ( zero_zero @ A ) )
=> ( ( ord_less @ A @ B2 @ ( zero_zero @ A ) )
=> ( ord_less @ A @ ( plus_plus @ A @ A2 @ B2 ) @ ( zero_zero @ A ) ) ) ) ) ).
% add_nonpos_neg
thf(fact_1285_add__pos__nonneg,axiom,
! [A: $tType] :
( ( ordere6911136660526730532id_add @ A )
=> ! [A2: A,B2: A] :
( ( ord_less @ A @ ( zero_zero @ A ) @ A2 )
=> ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ B2 )
=> ( ord_less @ A @ ( zero_zero @ A ) @ ( plus_plus @ A @ A2 @ B2 ) ) ) ) ) ).
% add_pos_nonneg
thf(fact_1286_add__strict__increasing,axiom,
! [A: $tType] :
( ( ordere8940638589300402666id_add @ A )
=> ! [A2: A,B2: A,C2: A] :
( ( ord_less @ A @ ( zero_zero @ A ) @ A2 )
=> ( ( ord_less_eq @ A @ B2 @ C2 )
=> ( ord_less @ A @ B2 @ ( plus_plus @ A @ A2 @ C2 ) ) ) ) ) ).
% add_strict_increasing
thf(fact_1287_add__strict__increasing2,axiom,
! [A: $tType] :
( ( ordere8940638589300402666id_add @ A )
=> ! [A2: A,B2: A,C2: A] :
( ( ord_less_eq @ A @ ( zero_zero @ A ) @ A2 )
=> ( ( ord_less @ A @ B2 @ C2 )
=> ( ord_less @ A @ B2 @ ( plus_plus @ A @ A2 @ C2 ) ) ) ) ) ).
% add_strict_increasing2
thf(fact_1288_mult__le__cancel__left,axiom,
! [A: $tType] :
( ( linord4710134922213307826strict @ A )
=> ! [C2: A,A2: A,B2: A] :
( ( ord_less_eq @ A @ ( times_times @ A @ C2 @ A2 ) @ ( times_times @ A @ C2 @ B2 ) )
= ( ( ( ord_less @ A @ ( zero_zero @ A ) @ C2 )
=> ( ord_less_eq @ A @ A2 @ B2 ) )
& ( ( ord_less @ A @ C2 @ ( zero_zero @ A ) )
=> ( ord_less_eq @ A @ B2 @ A2 ) ) ) ) ) ).
% mult_le_cancel_left
thf(fact_1289_mult__le__cancel__right,axiom,
! [A: $tType] :
( ( linord4710134922213307826strict @ A )
=> ! [A2: A,C2: A,B2: A] :
( ( ord_less_eq @ A @ ( times_times @ A @ A2 @ C2 ) @ ( times_times @ A @ B2 @ C2 ) )
= ( ( ( ord_less @ A @ ( zero_zero @ A ) @ C2 )
=> ( ord_less_eq @ A @ A2 @ B2 ) )
& ( ( ord_less @ A @ C2 @ ( zero_zero @ A ) )
=> ( ord_less_eq @ A @ B2 @ A2 ) ) ) ) ) ).
% mult_le_cancel_right
thf(fact_1290_mult__left__less__imp__less,axiom,
! [A: $tType] :
( ( linordered_semiring @ A )
=> ! [C2: A,A2: A,B2: A] :
( ( ord_less @ A @ ( times_times @ A @ C2 @ A2 ) @ ( times_times @ A @ C2 @ B2 ) )
=> ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ C2 )
=> ( ord_less @ A @ A2 @ B2 ) ) ) ) ).
% mult_left_less_imp_less
thf(fact_1291_mult__strict__mono,axiom,
! [A: $tType] :
( ( linord8928482502909563296strict @ A )
=> ! [A2: A,B2: A,C2: A,D: A] :
( ( ord_less @ A @ A2 @ B2 )
=> ( ( ord_less @ A @ C2 @ D )
=> ( ( ord_less @ A @ ( zero_zero @ A ) @ B2 )
=> ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ C2 )
=> ( ord_less @ A @ ( times_times @ A @ A2 @ C2 ) @ ( times_times @ A @ B2 @ D ) ) ) ) ) ) ) ).
% mult_strict_mono
thf(fact_1292_mult__less__cancel__left,axiom,
! [A: $tType] :
( ( linord4710134922213307826strict @ A )
=> ! [C2: A,A2: A,B2: A] :
( ( ord_less @ A @ ( times_times @ A @ C2 @ A2 ) @ ( times_times @ A @ C2 @ B2 ) )
= ( ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ C2 )
=> ( ord_less @ A @ A2 @ B2 ) )
& ( ( ord_less_eq @ A @ C2 @ ( zero_zero @ A ) )
=> ( ord_less @ A @ B2 @ A2 ) ) ) ) ) ).
% mult_less_cancel_left
thf(fact_1293_mult__right__less__imp__less,axiom,
! [A: $tType] :
( ( linordered_semiring @ A )
=> ! [A2: A,C2: A,B2: A] :
( ( ord_less @ A @ ( times_times @ A @ A2 @ C2 ) @ ( times_times @ A @ B2 @ C2 ) )
=> ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ C2 )
=> ( ord_less @ A @ A2 @ B2 ) ) ) ) ).
% mult_right_less_imp_less
thf(fact_1294_mult__strict__mono_H,axiom,
! [A: $tType] :
( ( linord8928482502909563296strict @ A )
=> ! [A2: A,B2: A,C2: A,D: A] :
( ( ord_less @ A @ A2 @ B2 )
=> ( ( ord_less @ A @ C2 @ D )
=> ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ A2 )
=> ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ C2 )
=> ( ord_less @ A @ ( times_times @ A @ A2 @ C2 ) @ ( times_times @ A @ B2 @ D ) ) ) ) ) ) ) ).
% mult_strict_mono'
thf(fact_1295_mult__less__cancel__right,axiom,
! [A: $tType] :
( ( linord4710134922213307826strict @ A )
=> ! [A2: A,C2: A,B2: A] :
( ( ord_less @ A @ ( times_times @ A @ A2 @ C2 ) @ ( times_times @ A @ B2 @ C2 ) )
= ( ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ C2 )
=> ( ord_less @ A @ A2 @ B2 ) )
& ( ( ord_less_eq @ A @ C2 @ ( zero_zero @ A ) )
=> ( ord_less @ A @ B2 @ A2 ) ) ) ) ) ).
% mult_less_cancel_right
thf(fact_1296_mult__le__cancel__left__neg,axiom,
! [A: $tType] :
( ( linord4710134922213307826strict @ A )
=> ! [C2: A,A2: A,B2: A] :
( ( ord_less @ A @ C2 @ ( zero_zero @ A ) )
=> ( ( ord_less_eq @ A @ ( times_times @ A @ C2 @ A2 ) @ ( times_times @ A @ C2 @ B2 ) )
= ( ord_less_eq @ A @ B2 @ A2 ) ) ) ) ).
% mult_le_cancel_left_neg
thf(fact_1297_mult__le__cancel__left__pos,axiom,
! [A: $tType] :
( ( linord4710134922213307826strict @ A )
=> ! [C2: A,A2: A,B2: A] :
( ( ord_less @ A @ ( zero_zero @ A ) @ C2 )
=> ( ( ord_less_eq @ A @ ( times_times @ A @ C2 @ A2 ) @ ( times_times @ A @ C2 @ B2 ) )
= ( ord_less_eq @ A @ A2 @ B2 ) ) ) ) ).
% mult_le_cancel_left_pos
thf(fact_1298_mult__left__le__imp__le,axiom,
! [A: $tType] :
( ( linord8928482502909563296strict @ A )
=> ! [C2: A,A2: A,B2: A] :
( ( ord_less_eq @ A @ ( times_times @ A @ C2 @ A2 ) @ ( times_times @ A @ C2 @ B2 ) )
=> ( ( ord_less @ A @ ( zero_zero @ A ) @ C2 )
=> ( ord_less_eq @ A @ A2 @ B2 ) ) ) ) ).
% mult_left_le_imp_le
thf(fact_1299_mult__right__le__imp__le,axiom,
! [A: $tType] :
( ( linord8928482502909563296strict @ A )
=> ! [A2: A,C2: A,B2: A] :
( ( ord_less_eq @ A @ ( times_times @ A @ A2 @ C2 ) @ ( times_times @ A @ B2 @ C2 ) )
=> ( ( ord_less @ A @ ( zero_zero @ A ) @ C2 )
=> ( ord_less_eq @ A @ A2 @ B2 ) ) ) ) ).
% mult_right_le_imp_le
thf(fact_1300_mult__le__less__imp__less,axiom,
! [A: $tType] :
( ( linord8928482502909563296strict @ A )
=> ! [A2: A,B2: A,C2: A,D: A] :
( ( ord_less_eq @ A @ A2 @ B2 )
=> ( ( ord_less @ A @ C2 @ D )
=> ( ( ord_less @ A @ ( zero_zero @ A ) @ A2 )
=> ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ C2 )
=> ( ord_less @ A @ ( times_times @ A @ A2 @ C2 ) @ ( times_times @ A @ B2 @ D ) ) ) ) ) ) ) ).
% mult_le_less_imp_less
thf(fact_1301_mult__less__le__imp__less,axiom,
! [A: $tType] :
( ( linord8928482502909563296strict @ A )
=> ! [A2: A,B2: A,C2: A,D: A] :
( ( ord_less @ A @ A2 @ B2 )
=> ( ( ord_less_eq @ A @ C2 @ D )
=> ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ A2 )
=> ( ( ord_less @ A @ ( zero_zero @ A ) @ C2 )
=> ( ord_less @ A @ ( times_times @ A @ A2 @ C2 ) @ ( times_times @ A @ B2 @ D ) ) ) ) ) ) ) ).
% mult_less_le_imp_less
thf(fact_1302_sum__squares__ge__zero,axiom,
! [A: $tType] :
( ( linordered_ring @ A )
=> ! [X: A,Y: A] : ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( plus_plus @ A @ ( times_times @ A @ X @ X ) @ ( times_times @ A @ Y @ Y ) ) ) ) ).
% sum_squares_ge_zero
thf(fact_1303_sum__squares__le__zero__iff,axiom,
! [A: $tType] :
( ( linord4710134922213307826strict @ A )
=> ! [X: A,Y: A] :
( ( ord_less_eq @ A @ ( plus_plus @ A @ ( times_times @ A @ X @ X ) @ ( times_times @ A @ Y @ Y ) ) @ ( zero_zero @ A ) )
= ( ( X
= ( zero_zero @ A ) )
& ( Y
= ( zero_zero @ A ) ) ) ) ) ).
% sum_squares_le_zero_iff
thf(fact_1304_mult__left__le,axiom,
! [A: $tType] :
( ( linord181362715937106298miring @ A )
=> ! [C2: A,A2: A] :
( ( ord_less_eq @ A @ C2 @ ( one_one @ A ) )
=> ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ A2 )
=> ( ord_less_eq @ A @ ( times_times @ A @ A2 @ C2 ) @ A2 ) ) ) ) ).
% mult_left_le
thf(fact_1305_mult__le__one,axiom,
! [A: $tType] :
( ( linord181362715937106298miring @ A )
=> ! [A2: A,B2: A] :
( ( ord_less_eq @ A @ A2 @ ( one_one @ A ) )
=> ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ B2 )
=> ( ( ord_less_eq @ A @ B2 @ ( one_one @ A ) )
=> ( ord_less_eq @ A @ ( times_times @ A @ A2 @ B2 ) @ ( one_one @ A ) ) ) ) ) ) ).
% mult_le_one
thf(fact_1306_mult__right__le__one__le,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [X: A,Y: A] :
( ( ord_less_eq @ A @ ( zero_zero @ A ) @ X )
=> ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ Y )
=> ( ( ord_less_eq @ A @ Y @ ( one_one @ A ) )
=> ( ord_less_eq @ A @ ( times_times @ A @ X @ Y ) @ X ) ) ) ) ) ).
% mult_right_le_one_le
thf(fact_1307_mult__left__le__one__le,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [X: A,Y: A] :
( ( ord_less_eq @ A @ ( zero_zero @ A ) @ X )
=> ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ Y )
=> ( ( ord_less_eq @ A @ Y @ ( one_one @ A ) )
=> ( ord_less_eq @ A @ ( times_times @ A @ Y @ X ) @ X ) ) ) ) ) ).
% mult_left_le_one_le
thf(fact_1308_unique__euclidean__semiring__numeral__class_Odiv__less,axiom,
! [A: $tType] :
( ( unique1627219031080169319umeral @ A )
=> ! [A2: A,B2: A] :
( ( ord_less_eq @ A @ ( zero_zero @ A ) @ A2 )
=> ( ( ord_less @ A @ A2 @ B2 )
=> ( ( divide_divide @ A @ A2 @ B2 )
= ( zero_zero @ A ) ) ) ) ) ).
% unique_euclidean_semiring_numeral_class.div_less
thf(fact_1309_div__positive,axiom,
! [A: $tType] :
( ( unique1627219031080169319umeral @ A )
=> ! [B2: A,A2: A] :
( ( ord_less @ A @ ( zero_zero @ A ) @ B2 )
=> ( ( ord_less_eq @ A @ B2 @ A2 )
=> ( ord_less @ A @ ( zero_zero @ A ) @ ( divide_divide @ A @ A2 @ B2 ) ) ) ) ) ).
% div_positive
thf(fact_1310_frac__le,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [Y: A,X: A,W: A,Z: A] :
( ( ord_less_eq @ A @ ( zero_zero @ A ) @ Y )
=> ( ( ord_less_eq @ A @ X @ Y )
=> ( ( ord_less @ A @ ( zero_zero @ A ) @ W )
=> ( ( ord_less_eq @ A @ W @ Z )
=> ( ord_less_eq @ A @ ( divide_divide @ A @ X @ Z ) @ ( divide_divide @ A @ Y @ W ) ) ) ) ) ) ) ).
% frac_le
thf(fact_1311_frac__less,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [X: A,Y: A,W: A,Z: A] :
( ( ord_less_eq @ A @ ( zero_zero @ A ) @ X )
=> ( ( ord_less @ A @ X @ Y )
=> ( ( ord_less @ A @ ( zero_zero @ A ) @ W )
=> ( ( ord_less_eq @ A @ W @ Z )
=> ( ord_less @ A @ ( divide_divide @ A @ X @ Z ) @ ( divide_divide @ A @ Y @ W ) ) ) ) ) ) ) ).
% frac_less
thf(fact_1312_frac__less2,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [X: A,Y: A,W: A,Z: A] :
( ( ord_less @ A @ ( zero_zero @ A ) @ X )
=> ( ( ord_less_eq @ A @ X @ Y )
=> ( ( ord_less @ A @ ( zero_zero @ A ) @ W )
=> ( ( ord_less @ A @ W @ Z )
=> ( ord_less @ A @ ( divide_divide @ A @ X @ Z ) @ ( divide_divide @ A @ Y @ W ) ) ) ) ) ) ) ).
% frac_less2
thf(fact_1313_divide__le__cancel,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [A2: A,C2: A,B2: A] :
( ( ord_less_eq @ A @ ( divide_divide @ A @ A2 @ C2 ) @ ( divide_divide @ A @ B2 @ C2 ) )
= ( ( ( ord_less @ A @ ( zero_zero @ A ) @ C2 )
=> ( ord_less_eq @ A @ A2 @ B2 ) )
& ( ( ord_less @ A @ C2 @ ( zero_zero @ A ) )
=> ( ord_less_eq @ A @ B2 @ A2 ) ) ) ) ) ).
% divide_le_cancel
thf(fact_1314_divide__nonneg__neg,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [X: A,Y: A] :
( ( ord_less_eq @ A @ ( zero_zero @ A ) @ X )
=> ( ( ord_less @ A @ Y @ ( zero_zero @ A ) )
=> ( ord_less_eq @ A @ ( divide_divide @ A @ X @ Y ) @ ( zero_zero @ A ) ) ) ) ) ).
% divide_nonneg_neg
thf(fact_1315_divide__nonneg__pos,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [X: A,Y: A] :
( ( ord_less_eq @ A @ ( zero_zero @ A ) @ X )
=> ( ( ord_less @ A @ ( zero_zero @ A ) @ Y )
=> ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( divide_divide @ A @ X @ Y ) ) ) ) ) ).
% divide_nonneg_pos
thf(fact_1316_divide__nonpos__neg,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [X: A,Y: A] :
( ( ord_less_eq @ A @ X @ ( zero_zero @ A ) )
=> ( ( ord_less @ A @ Y @ ( zero_zero @ A ) )
=> ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( divide_divide @ A @ X @ Y ) ) ) ) ) ).
% divide_nonpos_neg
thf(fact_1317_divide__nonpos__pos,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [X: A,Y: A] :
( ( ord_less_eq @ A @ X @ ( zero_zero @ A ) )
=> ( ( ord_less @ A @ ( zero_zero @ A ) @ Y )
=> ( ord_less_eq @ A @ ( divide_divide @ A @ X @ Y ) @ ( zero_zero @ A ) ) ) ) ) ).
% divide_nonpos_pos
thf(fact_1318_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_1319_not__sum__squares__lt__zero,axiom,
! [A: $tType] :
( ( linordered_ring @ A )
=> ! [X: A,Y: A] :
~ ( ord_less @ A @ ( plus_plus @ A @ ( times_times @ A @ X @ X ) @ ( times_times @ A @ Y @ Y ) ) @ ( zero_zero @ A ) ) ) ).
% not_sum_squares_lt_zero
thf(fact_1320_sum__squares__gt__zero__iff,axiom,
! [A: $tType] :
( ( linord4710134922213307826strict @ A )
=> ! [X: A,Y: A] :
( ( ord_less @ A @ ( zero_zero @ A ) @ ( plus_plus @ A @ ( times_times @ A @ X @ X ) @ ( times_times @ A @ Y @ Y ) ) )
= ( ( X
!= ( zero_zero @ A ) )
| ( Y
!= ( zero_zero @ A ) ) ) ) ) ).
% sum_squares_gt_zero_iff
thf(fact_1321_power__less__imp__less__base,axiom,
! [A: $tType] :
( ( linordered_semidom @ A )
=> ! [A2: A,N3: nat,B2: A] :
( ( ord_less @ A @ ( power_power @ A @ A2 @ N3 ) @ ( power_power @ A @ B2 @ N3 ) )
=> ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ B2 )
=> ( ord_less @ A @ A2 @ B2 ) ) ) ) ).
% power_less_imp_less_base
thf(fact_1322_unique__euclidean__semiring__numeral__class_Odiv__mult2__eq,axiom,
! [A: $tType] :
( ( unique1627219031080169319umeral @ A )
=> ! [C2: A,A2: A,B2: A] :
( ( ord_less_eq @ A @ ( zero_zero @ A ) @ C2 )
=> ( ( divide_divide @ A @ A2 @ ( times_times @ A @ B2 @ C2 ) )
= ( divide_divide @ A @ ( divide_divide @ A @ A2 @ B2 ) @ C2 ) ) ) ) ).
% unique_euclidean_semiring_numeral_class.div_mult2_eq
thf(fact_1323_divide__less__eq__1,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [B2: A,A2: A] :
( ( ord_less @ A @ ( divide_divide @ A @ B2 @ A2 ) @ ( one_one @ A ) )
= ( ( ( ord_less @ A @ ( zero_zero @ A ) @ A2 )
& ( ord_less @ A @ B2 @ A2 ) )
| ( ( ord_less @ A @ A2 @ ( zero_zero @ A ) )
& ( ord_less @ A @ A2 @ B2 ) )
| ( A2
= ( zero_zero @ A ) ) ) ) ) ).
% divide_less_eq_1
thf(fact_1324_less__divide__eq__1,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [B2: A,A2: A] :
( ( ord_less @ A @ ( one_one @ A ) @ ( divide_divide @ A @ B2 @ A2 ) )
= ( ( ( ord_less @ A @ ( zero_zero @ A ) @ A2 )
& ( ord_less @ A @ A2 @ B2 ) )
| ( ( ord_less @ A @ A2 @ ( zero_zero @ A ) )
& ( ord_less @ A @ B2 @ A2 ) ) ) ) ) ).
% less_divide_eq_1
thf(fact_1325_power__le__one,axiom,
! [A: $tType] :
( ( linordered_semidom @ A )
=> ! [A2: A,N3: nat] :
( ( ord_less_eq @ A @ ( zero_zero @ A ) @ A2 )
=> ( ( ord_less_eq @ A @ A2 @ ( one_one @ A ) )
=> ( ord_less_eq @ A @ ( power_power @ A @ A2 @ N3 ) @ ( one_one @ A ) ) ) ) ) ).
% power_le_one
thf(fact_1326_divide__less__eq,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [B2: A,C2: A,A2: A] :
( ( ord_less @ A @ ( divide_divide @ A @ B2 @ C2 ) @ A2 )
= ( ( ( ord_less @ A @ ( zero_zero @ A ) @ C2 )
=> ( ord_less @ A @ B2 @ ( times_times @ A @ A2 @ C2 ) ) )
& ( ~ ( ord_less @ A @ ( zero_zero @ A ) @ C2 )
=> ( ( ( ord_less @ A @ C2 @ ( zero_zero @ A ) )
=> ( ord_less @ A @ ( times_times @ A @ A2 @ C2 ) @ B2 ) )
& ( ~ ( ord_less @ A @ C2 @ ( zero_zero @ A ) )
=> ( ord_less @ A @ ( zero_zero @ A ) @ A2 ) ) ) ) ) ) ) ).
% divide_less_eq
thf(fact_1327_less__divide__eq,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [A2: A,B2: A,C2: A] :
( ( ord_less @ A @ A2 @ ( divide_divide @ A @ B2 @ C2 ) )
= ( ( ( ord_less @ A @ ( zero_zero @ A ) @ C2 )
=> ( ord_less @ A @ ( times_times @ A @ A2 @ C2 ) @ B2 ) )
& ( ~ ( ord_less @ A @ ( zero_zero @ A ) @ C2 )
=> ( ( ( ord_less @ A @ C2 @ ( zero_zero @ A ) )
=> ( ord_less @ A @ B2 @ ( times_times @ A @ A2 @ C2 ) ) )
& ( ~ ( ord_less @ A @ C2 @ ( zero_zero @ A ) )
=> ( ord_less @ A @ A2 @ ( zero_zero @ A ) ) ) ) ) ) ) ) ).
% less_divide_eq
thf(fact_1328_neg__divide__less__eq,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [C2: A,B2: A,A2: A] :
( ( ord_less @ A @ C2 @ ( zero_zero @ A ) )
=> ( ( ord_less @ A @ ( divide_divide @ A @ B2 @ C2 ) @ A2 )
= ( ord_less @ A @ ( times_times @ A @ A2 @ C2 ) @ B2 ) ) ) ) ).
% neg_divide_less_eq
thf(fact_1329_neg__less__divide__eq,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [C2: A,A2: A,B2: A] :
( ( ord_less @ A @ C2 @ ( zero_zero @ A ) )
=> ( ( ord_less @ A @ A2 @ ( divide_divide @ A @ B2 @ C2 ) )
= ( ord_less @ A @ B2 @ ( times_times @ A @ A2 @ C2 ) ) ) ) ) ).
% neg_less_divide_eq
thf(fact_1330_pos__divide__less__eq,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [C2: A,B2: A,A2: A] :
( ( ord_less @ A @ ( zero_zero @ A ) @ C2 )
=> ( ( ord_less @ A @ ( divide_divide @ A @ B2 @ C2 ) @ A2 )
= ( ord_less @ A @ B2 @ ( times_times @ A @ A2 @ C2 ) ) ) ) ) ).
% pos_divide_less_eq
thf(fact_1331_pos__less__divide__eq,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [C2: A,A2: A,B2: A] :
( ( ord_less @ A @ ( zero_zero @ A ) @ C2 )
=> ( ( ord_less @ A @ A2 @ ( divide_divide @ A @ B2 @ C2 ) )
= ( ord_less @ A @ ( times_times @ A @ A2 @ C2 ) @ B2 ) ) ) ) ).
% pos_less_divide_eq
thf(fact_1332_mult__imp__div__pos__less,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [Y: A,X: A,Z: A] :
( ( ord_less @ A @ ( zero_zero @ A ) @ Y )
=> ( ( ord_less @ A @ X @ ( times_times @ A @ Z @ Y ) )
=> ( ord_less @ A @ ( divide_divide @ A @ X @ Y ) @ Z ) ) ) ) ).
% mult_imp_div_pos_less
thf(fact_1333_mult__imp__less__div__pos,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [Y: A,Z: A,X: A] :
( ( ord_less @ A @ ( zero_zero @ A ) @ Y )
=> ( ( ord_less @ A @ ( times_times @ A @ Z @ Y ) @ X )
=> ( ord_less @ A @ Z @ ( divide_divide @ A @ X @ Y ) ) ) ) ) ).
% mult_imp_less_div_pos
thf(fact_1334_divide__strict__left__mono,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [B2: A,A2: A,C2: A] :
( ( ord_less @ A @ B2 @ A2 )
=> ( ( ord_less @ A @ ( zero_zero @ A ) @ C2 )
=> ( ( ord_less @ A @ ( zero_zero @ A ) @ ( times_times @ A @ A2 @ B2 ) )
=> ( ord_less @ A @ ( divide_divide @ A @ C2 @ A2 ) @ ( divide_divide @ A @ C2 @ B2 ) ) ) ) ) ) ).
% divide_strict_left_mono
thf(fact_1335_divide__strict__left__mono__neg,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [A2: A,B2: A,C2: A] :
( ( ord_less @ A @ A2 @ B2 )
=> ( ( ord_less @ A @ C2 @ ( zero_zero @ A ) )
=> ( ( ord_less @ A @ ( zero_zero @ A ) @ ( times_times @ A @ A2 @ B2 ) )
=> ( ord_less @ A @ ( divide_divide @ A @ C2 @ A2 ) @ ( divide_divide @ A @ C2 @ B2 ) ) ) ) ) ) ).
% divide_strict_left_mono_neg
thf(fact_1336_divide__eq__eq__numeral_I1_J,axiom,
! [A: $tType] :
( ( division_ring @ A )
=> ! [B2: A,C2: A,W: num] :
( ( ( divide_divide @ A @ B2 @ C2 )
= ( numeral_numeral @ A @ W ) )
= ( ( ( C2
!= ( zero_zero @ A ) )
=> ( B2
= ( times_times @ A @ ( numeral_numeral @ A @ W ) @ C2 ) ) )
& ( ( C2
= ( zero_zero @ A ) )
=> ( ( numeral_numeral @ A @ W )
= ( zero_zero @ A ) ) ) ) ) ) ).
% divide_eq_eq_numeral(1)
thf(fact_1337_eq__divide__eq__numeral_I1_J,axiom,
! [A: $tType] :
( ( division_ring @ A )
=> ! [W: num,B2: A,C2: A] :
( ( ( numeral_numeral @ A @ W )
= ( divide_divide @ A @ B2 @ C2 ) )
= ( ( ( C2
!= ( zero_zero @ A ) )
=> ( ( times_times @ A @ ( numeral_numeral @ A @ W ) @ C2 )
= B2 ) )
& ( ( C2
= ( zero_zero @ A ) )
=> ( ( numeral_numeral @ A @ W )
= ( zero_zero @ A ) ) ) ) ) ) ).
% eq_divide_eq_numeral(1)
thf(fact_1338_power__inject__base,axiom,
! [A: $tType] :
( ( linordered_semidom @ A )
=> ! [A2: A,N3: nat,B2: A] :
( ( ( power_power @ A @ A2 @ ( suc @ N3 ) )
= ( power_power @ A @ B2 @ ( suc @ N3 ) ) )
=> ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ A2 )
=> ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ B2 )
=> ( A2 = B2 ) ) ) ) ) ).
% power_inject_base
thf(fact_1339_power__le__imp__le__base,axiom,
! [A: $tType] :
( ( linordered_semidom @ A )
=> ! [A2: A,N3: nat,B2: A] :
( ( ord_less_eq @ A @ ( power_power @ A @ A2 @ ( suc @ N3 ) ) @ ( power_power @ A @ B2 @ ( suc @ N3 ) ) )
=> ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ B2 )
=> ( ord_less_eq @ A @ A2 @ B2 ) ) ) ) ).
% power_le_imp_le_base
thf(fact_1340_div__add__self1,axiom,
! [A: $tType] :
( ( euclid4440199948858584721cancel @ A )
=> ! [B2: A,A2: A] :
( ( B2
!= ( zero_zero @ A ) )
=> ( ( divide_divide @ A @ ( plus_plus @ A @ B2 @ A2 ) @ B2 )
= ( plus_plus @ A @ ( divide_divide @ A @ A2 @ B2 ) @ ( one_one @ A ) ) ) ) ) ).
% div_add_self1
thf(fact_1341_div__add__self2,axiom,
! [A: $tType] :
( ( euclid4440199948858584721cancel @ A )
=> ! [B2: A,A2: A] :
( ( B2
!= ( zero_zero @ A ) )
=> ( ( divide_divide @ A @ ( plus_plus @ A @ A2 @ B2 ) @ B2 )
= ( plus_plus @ A @ ( divide_divide @ A @ A2 @ B2 ) @ ( one_one @ A ) ) ) ) ) ).
% div_add_self2
thf(fact_1342_add__divide__eq__if__simps_I2_J,axiom,
! [A: $tType] :
( ( division_ring @ A )
=> ! [Z: A,A2: A,B2: A] :
( ( ( Z
= ( zero_zero @ A ) )
=> ( ( plus_plus @ A @ ( divide_divide @ A @ A2 @ Z ) @ B2 )
= B2 ) )
& ( ( Z
!= ( zero_zero @ A ) )
=> ( ( plus_plus @ A @ ( divide_divide @ A @ A2 @ Z ) @ B2 )
= ( divide_divide @ A @ ( plus_plus @ A @ A2 @ ( times_times @ A @ B2 @ Z ) ) @ Z ) ) ) ) ) ).
% add_divide_eq_if_simps(2)
thf(fact_1343_add__divide__eq__if__simps_I1_J,axiom,
! [A: $tType] :
( ( division_ring @ A )
=> ! [Z: A,A2: A,B2: A] :
( ( ( Z
= ( zero_zero @ A ) )
=> ( ( plus_plus @ A @ A2 @ ( divide_divide @ A @ B2 @ Z ) )
= A2 ) )
& ( ( Z
!= ( zero_zero @ A ) )
=> ( ( plus_plus @ A @ A2 @ ( divide_divide @ A @ B2 @ Z ) )
= ( divide_divide @ A @ ( plus_plus @ A @ ( times_times @ A @ A2 @ Z ) @ B2 ) @ Z ) ) ) ) ) ).
% add_divide_eq_if_simps(1)
thf(fact_1344_add__frac__eq,axiom,
! [A: $tType] :
( ( field @ A )
=> ! [Y: A,Z: A,X: A,W: A] :
( ( Y
!= ( zero_zero @ A ) )
=> ( ( Z
!= ( zero_zero @ A ) )
=> ( ( plus_plus @ A @ ( divide_divide @ A @ X @ Y ) @ ( divide_divide @ A @ W @ Z ) )
= ( divide_divide @ A @ ( plus_plus @ A @ ( times_times @ A @ X @ Z ) @ ( times_times @ A @ W @ Y ) ) @ ( times_times @ A @ Y @ Z ) ) ) ) ) ) ).
% add_frac_eq
thf(fact_1345_add__frac__num,axiom,
! [A: $tType] :
( ( field @ A )
=> ! [Y: A,X: A,Z: A] :
( ( Y
!= ( zero_zero @ A ) )
=> ( ( plus_plus @ A @ ( divide_divide @ A @ X @ Y ) @ Z )
= ( divide_divide @ A @ ( plus_plus @ A @ X @ ( times_times @ A @ Z @ Y ) ) @ Y ) ) ) ) ).
% add_frac_num
thf(fact_1346_add__num__frac,axiom,
! [A: $tType] :
( ( field @ A )
=> ! [Y: A,Z: A,X: A] :
( ( Y
!= ( zero_zero @ A ) )
=> ( ( plus_plus @ A @ Z @ ( divide_divide @ A @ X @ Y ) )
= ( divide_divide @ A @ ( plus_plus @ A @ X @ ( times_times @ A @ Z @ Y ) ) @ Y ) ) ) ) ).
% add_num_frac
thf(fact_1347_add__divide__eq__iff,axiom,
! [A: $tType] :
( ( division_ring @ A )
=> ! [Z: A,X: A,Y: A] :
( ( Z
!= ( zero_zero @ A ) )
=> ( ( plus_plus @ A @ X @ ( divide_divide @ A @ Y @ Z ) )
= ( divide_divide @ A @ ( plus_plus @ A @ ( times_times @ A @ X @ Z ) @ Y ) @ Z ) ) ) ) ).
% add_divide_eq_iff
thf(fact_1348_divide__add__eq__iff,axiom,
! [A: $tType] :
( ( division_ring @ A )
=> ! [Z: A,X: A,Y: A] :
( ( Z
!= ( zero_zero @ A ) )
=> ( ( plus_plus @ A @ ( divide_divide @ A @ X @ Z ) @ Y )
= ( divide_divide @ A @ ( plus_plus @ A @ X @ ( times_times @ A @ Y @ Z ) ) @ Z ) ) ) ) ).
% divide_add_eq_iff
thf(fact_1349_add__divide__eq__if__simps_I4_J,axiom,
! [A: $tType] :
( ( division_ring @ A )
=> ! [Z: A,A2: A,B2: A] :
( ( ( Z
= ( zero_zero @ A ) )
=> ( ( minus_minus @ A @ A2 @ ( divide_divide @ A @ B2 @ Z ) )
= A2 ) )
& ( ( Z
!= ( zero_zero @ A ) )
=> ( ( minus_minus @ A @ A2 @ ( divide_divide @ A @ B2 @ Z ) )
= ( divide_divide @ A @ ( minus_minus @ A @ ( times_times @ A @ A2 @ Z ) @ B2 ) @ Z ) ) ) ) ) ).
% add_divide_eq_if_simps(4)
thf(fact_1350_diff__frac__eq,axiom,
! [A: $tType] :
( ( field @ A )
=> ! [Y: A,Z: A,X: A,W: A] :
( ( Y
!= ( zero_zero @ A ) )
=> ( ( Z
!= ( zero_zero @ A ) )
=> ( ( minus_minus @ A @ ( divide_divide @ A @ X @ Y ) @ ( divide_divide @ A @ W @ Z ) )
= ( divide_divide @ A @ ( minus_minus @ A @ ( times_times @ A @ X @ Z ) @ ( times_times @ A @ W @ Y ) ) @ ( times_times @ A @ Y @ Z ) ) ) ) ) ) ).
% diff_frac_eq
thf(fact_1351_diff__divide__eq__iff,axiom,
! [A: $tType] :
( ( division_ring @ A )
=> ! [Z: A,X: A,Y: A] :
( ( Z
!= ( zero_zero @ A ) )
=> ( ( minus_minus @ A @ X @ ( divide_divide @ A @ Y @ Z ) )
= ( divide_divide @ A @ ( minus_minus @ A @ ( times_times @ A @ X @ Z ) @ Y ) @ Z ) ) ) ) ).
% diff_divide_eq_iff
thf(fact_1352_divide__diff__eq__iff,axiom,
! [A: $tType] :
( ( division_ring @ A )
=> ! [Z: A,X: A,Y: A] :
( ( Z
!= ( zero_zero @ A ) )
=> ( ( minus_minus @ A @ ( divide_divide @ A @ X @ Z ) @ Y )
= ( divide_divide @ A @ ( minus_minus @ A @ X @ ( times_times @ A @ Y @ Z ) ) @ Z ) ) ) ) ).
% divide_diff_eq_iff
thf(fact_1353_ex__less__of__nat__mult,axiom,
! [A: $tType] :
( ( archim462609752435547400_field @ A )
=> ! [X: A,Y: A] :
( ( ord_less @ A @ ( zero_zero @ A ) @ X )
=> ? [N: nat] : ( ord_less @ A @ Y @ ( times_times @ A @ ( semiring_1_of_nat @ A @ N ) @ X ) ) ) ) ).
% ex_less_of_nat_mult
thf(fact_1354_numeral__1__eq__Suc__0,axiom,
( ( numeral_numeral @ nat @ one2 )
= ( suc @ ( zero_zero @ nat ) ) ) ).
% numeral_1_eq_Suc_0
thf(fact_1355_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_1356_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_1357_ex__least__nat__less,axiom,
! [P: nat > $o,N3: nat] :
( ( P @ N3 )
=> ( ~ ( P @ ( zero_zero @ nat ) )
=> ? [K3: nat] :
( ( ord_less @ nat @ K3 @ N3 )
& ! [I4: nat] :
( ( ord_less_eq @ nat @ I4 @ K3 )
=> ~ ( P @ I4 ) )
& ( P @ ( suc @ K3 ) ) ) ) ) ).
% ex_least_nat_less
thf(fact_1358_nat__induct__non__zero,axiom,
! [N3: nat,P: nat > $o] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ N3 )
=> ( ( P @ ( one_one @ nat ) )
=> ( ! [N: nat] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ N )
=> ( ( P @ N )
=> ( P @ ( suc @ N ) ) ) )
=> ( P @ N3 ) ) ) ) ).
% nat_induct_non_zero
thf(fact_1359_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_1360_diff__Suc__less,axiom,
! [N3: nat,I: nat] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ N3 )
=> ( ord_less @ nat @ ( minus_minus @ nat @ N3 @ ( suc @ I ) ) @ N3 ) ) ).
% diff_Suc_less
thf(fact_1361_one__less__mult,axiom,
! [N3: nat,M: nat] :
( ( ord_less @ nat @ ( suc @ ( zero_zero @ nat ) ) @ N3 )
=> ( ( ord_less @ nat @ ( suc @ ( zero_zero @ nat ) ) @ M )
=> ( ord_less @ nat @ ( suc @ ( zero_zero @ nat ) ) @ ( times_times @ nat @ M @ N3 ) ) ) ) ).
% one_less_mult
thf(fact_1362_n__less__m__mult__n,axiom,
! [N3: nat,M: nat] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ N3 )
=> ( ( ord_less @ nat @ ( suc @ ( zero_zero @ nat ) ) @ M )
=> ( ord_less @ nat @ N3 @ ( times_times @ nat @ M @ N3 ) ) ) ) ).
% n_less_m_mult_n
thf(fact_1363_n__less__n__mult__m,axiom,
! [N3: nat,M: nat] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ N3 )
=> ( ( ord_less @ nat @ ( suc @ ( zero_zero @ nat ) ) @ M )
=> ( ord_less @ nat @ N3 @ ( times_times @ nat @ N3 @ M ) ) ) ) ).
% n_less_n_mult_m
thf(fact_1364_power__gt__expt,axiom,
! [N3: nat,K: nat] :
( ( ord_less @ nat @ ( suc @ ( zero_zero @ nat ) ) @ N3 )
=> ( ord_less @ nat @ K @ ( power_power @ nat @ N3 @ K ) ) ) ).
% power_gt_expt
thf(fact_1365_div__le__mono2,axiom,
! [M: nat,N3: nat,K: nat] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ M )
=> ( ( ord_less_eq @ nat @ M @ N3 )
=> ( ord_less_eq @ nat @ ( divide_divide @ nat @ K @ N3 ) @ ( divide_divide @ nat @ K @ M ) ) ) ) ).
% div_le_mono2
thf(fact_1366_div__greater__zero__iff,axiom,
! [M: nat,N3: nat] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ ( divide_divide @ nat @ M @ N3 ) )
= ( ( ord_less_eq @ nat @ N3 @ M )
& ( ord_less @ nat @ ( zero_zero @ nat ) @ N3 ) ) ) ).
% div_greater_zero_iff
thf(fact_1367_nat__diff__split,axiom,
! [P: nat > $o,A2: nat,B2: nat] :
( ( P @ ( minus_minus @ nat @ A2 @ B2 ) )
= ( ( ( ord_less @ nat @ A2 @ B2 )
=> ( P @ ( zero_zero @ nat ) ) )
& ! [D2: nat] :
( ( A2
= ( plus_plus @ nat @ B2 @ D2 ) )
=> ( P @ D2 ) ) ) ) ).
% nat_diff_split
thf(fact_1368_nat__diff__split__asm,axiom,
! [P: nat > $o,A2: nat,B2: nat] :
( ( P @ ( minus_minus @ nat @ A2 @ B2 ) )
= ( ~ ( ( ( ord_less @ nat @ A2 @ B2 )
& ~ ( P @ ( zero_zero @ nat ) ) )
| ? [D2: nat] :
( ( A2
= ( plus_plus @ nat @ B2 @ D2 ) )
& ~ ( P @ D2 ) ) ) ) ) ).
% nat_diff_split_asm
thf(fact_1369_div__less__dividend,axiom,
! [N3: nat,M: nat] :
( ( ord_less @ nat @ ( one_one @ nat ) @ N3 )
=> ( ( ord_less @ nat @ ( zero_zero @ nat ) @ M )
=> ( ord_less @ nat @ ( divide_divide @ nat @ M @ N3 ) @ M ) ) ) ).
% div_less_dividend
thf(fact_1370_length__pos__if__in__set,axiom,
! [A: $tType,X: A,Xs2: list @ A] :
( ( member @ A @ X @ ( set2 @ A @ Xs2 ) )
=> ( ord_less @ nat @ ( zero_zero @ nat ) @ ( size_size @ ( list @ A ) @ Xs2 ) ) ) ).
% length_pos_if_in_set
thf(fact_1371_nat__one__le__power,axiom,
! [I: nat,N3: nat] :
( ( ord_less_eq @ nat @ ( suc @ ( zero_zero @ nat ) ) @ I )
=> ( ord_less_eq @ nat @ ( suc @ ( zero_zero @ nat ) ) @ ( power_power @ nat @ I @ N3 ) ) ) ).
% nat_one_le_power
thf(fact_1372_nat__mult__le__cancel1,axiom,
! [K: nat,M: nat,N3: nat] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ K )
=> ( ( ord_less_eq @ nat @ ( times_times @ nat @ K @ M ) @ ( times_times @ nat @ K @ N3 ) )
= ( ord_less_eq @ nat @ M @ N3 ) ) ) ).
% nat_mult_le_cancel1
thf(fact_1373_td__gal__lt,axiom,
! [C2: nat,A2: nat,B2: nat] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ C2 )
=> ( ( ord_less @ nat @ A2 @ ( times_times @ nat @ B2 @ C2 ) )
= ( ord_less @ nat @ ( divide_divide @ nat @ A2 @ C2 ) @ B2 ) ) ) ).
% td_gal_lt
thf(fact_1374_div__less__iff__less__mult,axiom,
! [Q2: nat,M: nat,N3: nat] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ Q2 )
=> ( ( ord_less @ nat @ ( divide_divide @ nat @ M @ Q2 ) @ N3 )
= ( ord_less @ nat @ M @ ( times_times @ nat @ N3 @ Q2 ) ) ) ) ).
% div_less_iff_less_mult
thf(fact_1375_nat__mult__div__cancel1,axiom,
! [K: nat,M: nat,N3: nat] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ K )
=> ( ( divide_divide @ nat @ ( times_times @ nat @ K @ M ) @ ( times_times @ nat @ K @ N3 ) )
= ( divide_divide @ nat @ M @ N3 ) ) ) ).
% nat_mult_div_cancel1
thf(fact_1376_vebt__insert_Osimps_I3_J,axiom,
! [Info: option @ ( product_prod @ nat @ nat ),Ts: list @ vEBT_VEBT,S: vEBT_VEBT,X: nat] :
( ( vEBT_vebt_insert @ ( vEBT_Node @ Info @ ( suc @ ( zero_zero @ nat ) ) @ Ts @ S ) @ X )
= ( vEBT_Node @ Info @ ( suc @ ( zero_zero @ nat ) ) @ Ts @ S ) ) ).
% vebt_insert.simps(3)
thf(fact_1377_vebt__member_Osimps_I3_J,axiom,
! [V: product_prod @ nat @ nat,Uy: list @ vEBT_VEBT,Uz: vEBT_VEBT,X: nat] :
~ ( vEBT_vebt_member @ ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ V ) @ ( zero_zero @ nat ) @ Uy @ Uz ) @ X ) ).
% vebt_member.simps(3)
thf(fact_1378_T_092_060_094sub_062i_092_060_094sub_062n_092_060_094sub_062s_092_060_094sub_062e_092_060_094sub_062r_092_060_094sub_062t_Osimps_I2_J,axiom,
! [Info: option @ ( product_prod @ nat @ nat ),Ts: list @ vEBT_VEBT,S: vEBT_VEBT,X: nat] :
( ( vEBT_T_i_n_s_e_r_t @ ( vEBT_Node @ Info @ ( zero_zero @ nat ) @ Ts @ S ) @ X )
= ( one_one @ nat ) ) ).
% T\<^sub>i\<^sub>n\<^sub>s\<^sub>e\<^sub>r\<^sub>t.simps(2)
thf(fact_1379_floor__exists1,axiom,
! [A: $tType] :
( ( archim462609752435547400_field @ A )
=> ! [X: A] :
? [X3: int] :
( ( ord_less_eq @ A @ ( ring_1_of_int @ A @ X3 ) @ X )
& ( ord_less @ A @ X @ ( ring_1_of_int @ A @ ( plus_plus @ int @ X3 @ ( one_one @ int ) ) ) )
& ! [Y4: int] :
( ( ( ord_less_eq @ A @ ( ring_1_of_int @ A @ Y4 ) @ X )
& ( ord_less @ A @ X @ ( ring_1_of_int @ A @ ( plus_plus @ int @ Y4 @ ( one_one @ int ) ) ) ) )
=> ( Y4 = X3 ) ) ) ) ).
% floor_exists1
thf(fact_1380_floor__exists,axiom,
! [A: $tType] :
( ( archim462609752435547400_field @ A )
=> ! [X: A] :
? [Z2: int] :
( ( ord_less_eq @ A @ ( ring_1_of_int @ A @ Z2 ) @ X )
& ( ord_less @ A @ X @ ( ring_1_of_int @ A @ ( plus_plus @ int @ Z2 @ ( one_one @ int ) ) ) ) ) ) ).
% floor_exists
thf(fact_1381_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_1382_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_1383_of__nat__less__of__int__iff,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [N3: nat,X: int] :
( ( ord_less @ A @ ( semiring_1_of_nat @ A @ N3 ) @ ( ring_1_of_int @ A @ X ) )
= ( ord_less @ int @ ( semiring_1_of_nat @ int @ N3 ) @ X ) ) ) ).
% of_nat_less_of_int_iff
thf(fact_1384_int__le__real__less,axiom,
( ( ord_less_eq @ int )
= ( ^ [N2: int,M5: int] : ( ord_less @ real @ ( ring_1_of_int @ real @ N2 ) @ ( plus_plus @ real @ ( ring_1_of_int @ real @ M5 ) @ ( one_one @ real ) ) ) ) ) ).
% int_le_real_less
thf(fact_1385_int__less__real__le,axiom,
( ( ord_less @ int )
= ( ^ [N2: int,M5: int] : ( ord_less_eq @ real @ ( plus_plus @ real @ ( ring_1_of_int @ real @ N2 ) @ ( one_one @ real ) ) @ ( ring_1_of_int @ real @ M5 ) ) ) ) ).
% int_less_real_le
thf(fact_1386_mult__le__cancel__left1,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [C2: A,B2: A] :
( ( ord_less_eq @ A @ C2 @ ( times_times @ A @ C2 @ B2 ) )
= ( ( ( ord_less @ A @ ( zero_zero @ A ) @ C2 )
=> ( ord_less_eq @ A @ ( one_one @ A ) @ B2 ) )
& ( ( ord_less @ A @ C2 @ ( zero_zero @ A ) )
=> ( ord_less_eq @ A @ B2 @ ( one_one @ A ) ) ) ) ) ) ).
% mult_le_cancel_left1
thf(fact_1387_mult__le__cancel__left2,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [C2: A,A2: A] :
( ( ord_less_eq @ A @ ( times_times @ A @ C2 @ A2 ) @ C2 )
= ( ( ( ord_less @ A @ ( zero_zero @ A ) @ C2 )
=> ( ord_less_eq @ A @ A2 @ ( one_one @ A ) ) )
& ( ( ord_less @ A @ C2 @ ( zero_zero @ A ) )
=> ( ord_less_eq @ A @ ( one_one @ A ) @ A2 ) ) ) ) ) ).
% mult_le_cancel_left2
thf(fact_1388_mult__le__cancel__right1,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [C2: A,B2: A] :
( ( ord_less_eq @ A @ C2 @ ( times_times @ A @ B2 @ C2 ) )
= ( ( ( ord_less @ A @ ( zero_zero @ A ) @ C2 )
=> ( ord_less_eq @ A @ ( one_one @ A ) @ B2 ) )
& ( ( ord_less @ A @ C2 @ ( zero_zero @ A ) )
=> ( ord_less_eq @ A @ B2 @ ( one_one @ A ) ) ) ) ) ) ).
% mult_le_cancel_right1
thf(fact_1389_mult__le__cancel__right2,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [A2: A,C2: A] :
( ( ord_less_eq @ A @ ( times_times @ A @ A2 @ C2 ) @ C2 )
= ( ( ( ord_less @ A @ ( zero_zero @ A ) @ C2 )
=> ( ord_less_eq @ A @ A2 @ ( one_one @ A ) ) )
& ( ( ord_less @ A @ C2 @ ( zero_zero @ A ) )
=> ( ord_less_eq @ A @ ( one_one @ A ) @ A2 ) ) ) ) ) ).
% mult_le_cancel_right2
thf(fact_1390_mult__less__cancel__left1,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [C2: A,B2: A] :
( ( ord_less @ A @ C2 @ ( times_times @ A @ C2 @ B2 ) )
= ( ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ C2 )
=> ( ord_less @ A @ ( one_one @ A ) @ B2 ) )
& ( ( ord_less_eq @ A @ C2 @ ( zero_zero @ A ) )
=> ( ord_less @ A @ B2 @ ( one_one @ A ) ) ) ) ) ) ).
% mult_less_cancel_left1
thf(fact_1391_mult__less__cancel__left2,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [C2: A,A2: A] :
( ( ord_less @ A @ ( times_times @ A @ C2 @ A2 ) @ C2 )
= ( ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ C2 )
=> ( ord_less @ A @ A2 @ ( one_one @ A ) ) )
& ( ( ord_less_eq @ A @ C2 @ ( zero_zero @ A ) )
=> ( ord_less @ A @ ( one_one @ A ) @ A2 ) ) ) ) ) ).
% mult_less_cancel_left2
thf(fact_1392_mult__less__cancel__right1,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [C2: A,B2: A] :
( ( ord_less @ A @ C2 @ ( times_times @ A @ B2 @ C2 ) )
= ( ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ C2 )
=> ( ord_less @ A @ ( one_one @ A ) @ B2 ) )
& ( ( ord_less_eq @ A @ C2 @ ( zero_zero @ A ) )
=> ( ord_less @ A @ B2 @ ( one_one @ A ) ) ) ) ) ) ).
% mult_less_cancel_right1
thf(fact_1393_mult__less__cancel__right2,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [A2: A,C2: A] :
( ( ord_less @ A @ ( times_times @ A @ A2 @ C2 ) @ C2 )
= ( ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ C2 )
=> ( ord_less @ A @ A2 @ ( one_one @ A ) ) )
& ( ( ord_less_eq @ A @ C2 @ ( zero_zero @ A ) )
=> ( ord_less @ A @ ( one_one @ A ) @ A2 ) ) ) ) ) ).
% mult_less_cancel_right2
thf(fact_1394_field__le__mult__one__interval,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [X: A,Y: 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 @ X ) @ Y ) ) )
=> ( ord_less_eq @ A @ X @ Y ) ) ) ).
% field_le_mult_one_interval
thf(fact_1395_convex__bound__le,axiom,
! [A: $tType] :
( ( linord6961819062388156250ring_1 @ A )
=> ! [X: A,A2: A,Y: A,U: A,V: A] :
( ( ord_less_eq @ A @ X @ A2 )
=> ( ( ord_less_eq @ A @ Y @ A2 )
=> ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ U )
=> ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ V )
=> ( ( ( plus_plus @ A @ U @ V )
= ( one_one @ A ) )
=> ( ord_less_eq @ A @ ( plus_plus @ A @ ( times_times @ A @ U @ X ) @ ( times_times @ A @ V @ Y ) ) @ A2 ) ) ) ) ) ) ) ).
% convex_bound_le
thf(fact_1396_divide__le__eq__1,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [B2: A,A2: A] :
( ( ord_less_eq @ A @ ( divide_divide @ A @ B2 @ A2 ) @ ( one_one @ A ) )
= ( ( ( ord_less @ A @ ( zero_zero @ A ) @ A2 )
& ( ord_less_eq @ A @ B2 @ A2 ) )
| ( ( ord_less @ A @ A2 @ ( zero_zero @ A ) )
& ( ord_less_eq @ A @ A2 @ B2 ) )
| ( A2
= ( zero_zero @ A ) ) ) ) ) ).
% divide_le_eq_1
thf(fact_1397_le__divide__eq__1,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [B2: A,A2: A] :
( ( ord_less_eq @ A @ ( one_one @ A ) @ ( divide_divide @ A @ B2 @ A2 ) )
= ( ( ( ord_less @ A @ ( zero_zero @ A ) @ A2 )
& ( ord_less_eq @ A @ A2 @ B2 ) )
| ( ( ord_less @ A @ A2 @ ( zero_zero @ A ) )
& ( ord_less_eq @ A @ B2 @ A2 ) ) ) ) ) ).
% le_divide_eq_1
thf(fact_1398_divide__le__eq,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [B2: A,C2: A,A2: A] :
( ( ord_less_eq @ A @ ( divide_divide @ A @ B2 @ C2 ) @ A2 )
= ( ( ( ord_less @ A @ ( zero_zero @ A ) @ C2 )
=> ( ord_less_eq @ A @ B2 @ ( times_times @ A @ A2 @ C2 ) ) )
& ( ~ ( ord_less @ A @ ( zero_zero @ A ) @ C2 )
=> ( ( ( ord_less @ A @ C2 @ ( zero_zero @ A ) )
=> ( ord_less_eq @ A @ ( times_times @ A @ A2 @ C2 ) @ B2 ) )
& ( ~ ( ord_less @ A @ C2 @ ( zero_zero @ A ) )
=> ( ord_less_eq @ A @ ( zero_zero @ A ) @ A2 ) ) ) ) ) ) ) ).
% divide_le_eq
thf(fact_1399_le__divide__eq,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [A2: A,B2: A,C2: A] :
( ( ord_less_eq @ A @ A2 @ ( divide_divide @ A @ B2 @ C2 ) )
= ( ( ( ord_less @ A @ ( zero_zero @ A ) @ C2 )
=> ( ord_less_eq @ A @ ( times_times @ A @ A2 @ C2 ) @ B2 ) )
& ( ~ ( ord_less @ A @ ( zero_zero @ A ) @ C2 )
=> ( ( ( ord_less @ A @ C2 @ ( zero_zero @ A ) )
=> ( ord_less_eq @ A @ B2 @ ( times_times @ A @ A2 @ C2 ) ) )
& ( ~ ( ord_less @ A @ C2 @ ( zero_zero @ A ) )
=> ( ord_less_eq @ A @ A2 @ ( zero_zero @ A ) ) ) ) ) ) ) ) ).
% le_divide_eq
thf(fact_1400_divide__left__mono,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [B2: A,A2: A,C2: A] :
( ( ord_less_eq @ A @ B2 @ A2 )
=> ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ C2 )
=> ( ( ord_less @ A @ ( zero_zero @ A ) @ ( times_times @ A @ A2 @ B2 ) )
=> ( ord_less_eq @ A @ ( divide_divide @ A @ C2 @ A2 ) @ ( divide_divide @ A @ C2 @ B2 ) ) ) ) ) ) ).
% divide_left_mono
thf(fact_1401_neg__divide__le__eq,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [C2: A,B2: A,A2: A] :
( ( ord_less @ A @ C2 @ ( zero_zero @ A ) )
=> ( ( ord_less_eq @ A @ ( divide_divide @ A @ B2 @ C2 ) @ A2 )
= ( ord_less_eq @ A @ ( times_times @ A @ A2 @ C2 ) @ B2 ) ) ) ) ).
% neg_divide_le_eq
thf(fact_1402_neg__le__divide__eq,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [C2: A,A2: A,B2: A] :
( ( ord_less @ A @ C2 @ ( zero_zero @ A ) )
=> ( ( ord_less_eq @ A @ A2 @ ( divide_divide @ A @ B2 @ C2 ) )
= ( ord_less_eq @ A @ B2 @ ( times_times @ A @ A2 @ C2 ) ) ) ) ) ).
% neg_le_divide_eq
thf(fact_1403_pos__divide__le__eq,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [C2: A,B2: A,A2: A] :
( ( ord_less @ A @ ( zero_zero @ A ) @ C2 )
=> ( ( ord_less_eq @ A @ ( divide_divide @ A @ B2 @ C2 ) @ A2 )
= ( ord_less_eq @ A @ B2 @ ( times_times @ A @ A2 @ C2 ) ) ) ) ) ).
% pos_divide_le_eq
thf(fact_1404_pos__le__divide__eq,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [C2: A,A2: A,B2: A] :
( ( ord_less @ A @ ( zero_zero @ A ) @ C2 )
=> ( ( ord_less_eq @ A @ A2 @ ( divide_divide @ A @ B2 @ C2 ) )
= ( ord_less_eq @ A @ ( times_times @ A @ A2 @ C2 ) @ B2 ) ) ) ) ).
% pos_le_divide_eq
thf(fact_1405_mult__imp__div__pos__le,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [Y: A,X: A,Z: A] :
( ( ord_less @ A @ ( zero_zero @ A ) @ Y )
=> ( ( ord_less_eq @ A @ X @ ( times_times @ A @ Z @ Y ) )
=> ( ord_less_eq @ A @ ( divide_divide @ A @ X @ Y ) @ Z ) ) ) ) ).
% mult_imp_div_pos_le
thf(fact_1406_mult__imp__le__div__pos,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [Y: A,Z: A,X: A] :
( ( ord_less @ A @ ( zero_zero @ A ) @ Y )
=> ( ( ord_less_eq @ A @ ( times_times @ A @ Z @ Y ) @ X )
=> ( ord_less_eq @ A @ Z @ ( divide_divide @ A @ X @ Y ) ) ) ) ) ).
% mult_imp_le_div_pos
thf(fact_1407_divide__left__mono__neg,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [A2: A,B2: A,C2: A] :
( ( ord_less_eq @ A @ A2 @ B2 )
=> ( ( ord_less_eq @ A @ C2 @ ( zero_zero @ A ) )
=> ( ( ord_less @ A @ ( zero_zero @ A ) @ ( times_times @ A @ A2 @ B2 ) )
=> ( ord_less_eq @ A @ ( divide_divide @ A @ C2 @ A2 ) @ ( divide_divide @ A @ C2 @ B2 ) ) ) ) ) ) ).
% divide_left_mono_neg
thf(fact_1408_divide__less__eq__numeral_I1_J,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [B2: A,C2: A,W: num] :
( ( ord_less @ A @ ( divide_divide @ A @ B2 @ C2 ) @ ( numeral_numeral @ A @ W ) )
= ( ( ( ord_less @ A @ ( zero_zero @ A ) @ C2 )
=> ( ord_less @ A @ B2 @ ( times_times @ A @ ( numeral_numeral @ A @ W ) @ C2 ) ) )
& ( ~ ( ord_less @ A @ ( zero_zero @ A ) @ C2 )
=> ( ( ( ord_less @ A @ C2 @ ( zero_zero @ A ) )
=> ( ord_less @ A @ ( times_times @ A @ ( numeral_numeral @ A @ W ) @ C2 ) @ B2 ) )
& ( ~ ( ord_less @ A @ C2 @ ( zero_zero @ A ) )
=> ( ord_less @ A @ ( zero_zero @ A ) @ ( numeral_numeral @ A @ W ) ) ) ) ) ) ) ) ).
% divide_less_eq_numeral(1)
thf(fact_1409_less__divide__eq__numeral_I1_J,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [W: num,B2: A,C2: A] :
( ( ord_less @ A @ ( numeral_numeral @ A @ W ) @ ( divide_divide @ A @ B2 @ C2 ) )
= ( ( ( ord_less @ A @ ( zero_zero @ A ) @ C2 )
=> ( ord_less @ A @ ( times_times @ A @ ( numeral_numeral @ A @ W ) @ C2 ) @ B2 ) )
& ( ~ ( ord_less @ A @ ( zero_zero @ A ) @ C2 )
=> ( ( ( ord_less @ A @ C2 @ ( zero_zero @ A ) )
=> ( ord_less @ A @ B2 @ ( times_times @ A @ ( numeral_numeral @ A @ W ) @ C2 ) ) )
& ( ~ ( ord_less @ A @ C2 @ ( zero_zero @ A ) )
=> ( ord_less @ A @ ( numeral_numeral @ A @ W ) @ ( zero_zero @ A ) ) ) ) ) ) ) ) ).
% less_divide_eq_numeral(1)
thf(fact_1410_frac__le__eq,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [Y: A,Z: A,X: A,W: A] :
( ( Y
!= ( zero_zero @ A ) )
=> ( ( Z
!= ( zero_zero @ A ) )
=> ( ( ord_less_eq @ A @ ( divide_divide @ A @ X @ Y ) @ ( divide_divide @ A @ W @ Z ) )
= ( ord_less_eq @ A @ ( divide_divide @ A @ ( minus_minus @ A @ ( times_times @ A @ X @ Z ) @ ( times_times @ A @ W @ Y ) ) @ ( times_times @ A @ Y @ Z ) ) @ ( zero_zero @ A ) ) ) ) ) ) ).
% frac_le_eq
thf(fact_1411_power__Suc__less,axiom,
! [A: $tType] :
( ( linordered_semidom @ A )
=> ! [A2: A,N3: nat] :
( ( ord_less @ A @ ( zero_zero @ A ) @ A2 )
=> ( ( ord_less @ A @ A2 @ ( one_one @ A ) )
=> ( ord_less @ A @ ( times_times @ A @ A2 @ ( power_power @ A @ A2 @ N3 ) ) @ ( power_power @ A @ A2 @ N3 ) ) ) ) ) ).
% power_Suc_less
thf(fact_1412_power__Suc__le__self,axiom,
! [A: $tType] :
( ( linordered_semidom @ A )
=> ! [A2: A,N3: nat] :
( ( ord_less_eq @ A @ ( zero_zero @ A ) @ A2 )
=> ( ( ord_less_eq @ A @ A2 @ ( one_one @ A ) )
=> ( ord_less_eq @ A @ ( power_power @ A @ A2 @ ( suc @ N3 ) ) @ A2 ) ) ) ) ).
% power_Suc_le_self
thf(fact_1413_frac__less__eq,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [Y: A,Z: A,X: A,W: A] :
( ( Y
!= ( zero_zero @ A ) )
=> ( ( Z
!= ( zero_zero @ A ) )
=> ( ( ord_less @ A @ ( divide_divide @ A @ X @ Y ) @ ( divide_divide @ A @ W @ Z ) )
= ( ord_less @ A @ ( divide_divide @ A @ ( minus_minus @ A @ ( times_times @ A @ X @ Z ) @ ( times_times @ A @ W @ Y ) ) @ ( times_times @ A @ Y @ Z ) ) @ ( zero_zero @ A ) ) ) ) ) ) ).
% frac_less_eq
thf(fact_1414_power__Suc__less__one,axiom,
! [A: $tType] :
( ( linordered_semidom @ A )
=> ! [A2: A,N3: nat] :
( ( ord_less @ A @ ( zero_zero @ A ) @ A2 )
=> ( ( ord_less @ A @ A2 @ ( one_one @ A ) )
=> ( ord_less @ A @ ( power_power @ A @ A2 @ ( suc @ N3 ) ) @ ( one_one @ A ) ) ) ) ) ).
% power_Suc_less_one
thf(fact_1415_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_1416_power__strict__decreasing,axiom,
! [A: $tType] :
( ( linordered_semidom @ A )
=> ! [N3: nat,N7: nat,A2: A] :
( ( ord_less @ nat @ N3 @ N7 )
=> ( ( ord_less @ A @ ( zero_zero @ A ) @ A2 )
=> ( ( ord_less @ A @ A2 @ ( one_one @ A ) )
=> ( ord_less @ A @ ( power_power @ A @ A2 @ N7 ) @ ( power_power @ A @ A2 @ N3 ) ) ) ) ) ) ).
% power_strict_decreasing
thf(fact_1417_power__decreasing,axiom,
! [A: $tType] :
( ( linordered_semidom @ A )
=> ! [N3: nat,N7: nat,A2: A] :
( ( ord_less_eq @ nat @ N3 @ N7 )
=> ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ A2 )
=> ( ( ord_less_eq @ A @ A2 @ ( one_one @ A ) )
=> ( ord_less_eq @ A @ ( power_power @ A @ A2 @ N7 ) @ ( power_power @ A @ A2 @ N3 ) ) ) ) ) ) ).
% power_decreasing
thf(fact_1418_numeral__2__eq__2,axiom,
( ( numeral_numeral @ nat @ ( bit0 @ one2 ) )
= ( suc @ ( suc @ ( zero_zero @ nat ) ) ) ) ).
% numeral_2_eq_2
thf(fact_1419_self__le__power,axiom,
! [A: $tType] :
( ( linordered_semidom @ A )
=> ! [A2: A,N3: nat] :
( ( ord_less_eq @ A @ ( one_one @ A ) @ A2 )
=> ( ( ord_less @ nat @ ( zero_zero @ nat ) @ N3 )
=> ( ord_less_eq @ A @ A2 @ ( power_power @ A @ A2 @ N3 ) ) ) ) ) ).
% self_le_power
thf(fact_1420_pos2,axiom,
ord_less @ nat @ ( zero_zero @ nat ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ).
% pos2
thf(fact_1421_power__diff,axiom,
! [A: $tType] :
( ( semidom_divide @ A )
=> ! [A2: A,N3: nat,M: nat] :
( ( A2
!= ( zero_zero @ A ) )
=> ( ( ord_less_eq @ nat @ N3 @ M )
=> ( ( power_power @ A @ A2 @ ( minus_minus @ nat @ M @ N3 ) )
= ( divide_divide @ A @ ( power_power @ A @ A2 @ M ) @ ( power_power @ A @ A2 @ N3 ) ) ) ) ) ) ).
% power_diff
thf(fact_1422_one__less__power,axiom,
! [A: $tType] :
( ( linordered_semidom @ A )
=> ! [A2: A,N3: nat] :
( ( ord_less @ A @ ( one_one @ A ) @ A2 )
=> ( ( ord_less @ nat @ ( zero_zero @ nat ) @ N3 )
=> ( ord_less @ A @ ( one_one @ A ) @ ( power_power @ A @ A2 @ N3 ) ) ) ) ) ).
% one_less_power
thf(fact_1423_numeral__3__eq__3,axiom,
( ( numeral_numeral @ nat @ ( bit1 @ one2 ) )
= ( suc @ ( suc @ ( suc @ ( zero_zero @ nat ) ) ) ) ) ).
% numeral_3_eq_3
thf(fact_1424_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_1425_Suc__pred_H,axiom,
! [N3: nat] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ N3 )
=> ( N3
= ( suc @ ( minus_minus @ nat @ N3 @ ( one_one @ nat ) ) ) ) ) ).
% Suc_pred'
thf(fact_1426_Suc__diff__eq__diff__pred,axiom,
! [N3: nat,M: nat] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ N3 )
=> ( ( minus_minus @ nat @ ( suc @ M ) @ N3 )
= ( minus_minus @ nat @ M @ ( minus_minus @ nat @ N3 @ ( one_one @ nat ) ) ) ) ) ).
% Suc_diff_eq_diff_pred
thf(fact_1427_div__geq,axiom,
! [N3: nat,M: nat] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ N3 )
=> ( ~ ( ord_less @ nat @ M @ N3 )
=> ( ( divide_divide @ nat @ M @ N3 )
= ( suc @ ( divide_divide @ nat @ ( minus_minus @ nat @ M @ N3 ) @ N3 ) ) ) ) ) ).
% div_geq
thf(fact_1428_div__if,axiom,
( ( divide_divide @ nat )
= ( ^ [M5: nat,N2: nat] :
( if @ nat
@ ( ( ord_less @ nat @ M5 @ N2 )
| ( N2
= ( zero_zero @ nat ) ) )
@ ( zero_zero @ nat )
@ ( suc @ ( divide_divide @ nat @ ( minus_minus @ nat @ M5 @ N2 ) @ N2 ) ) ) ) ) ).
% div_if
thf(fact_1429_add__eq__if,axiom,
( ( plus_plus @ nat )
= ( ^ [M5: nat,N2: nat] :
( if @ nat
@ ( M5
= ( zero_zero @ nat ) )
@ N2
@ ( suc @ ( plus_plus @ nat @ ( minus_minus @ nat @ M5 @ ( one_one @ nat ) ) @ N2 ) ) ) ) ) ).
% add_eq_if
thf(fact_1430_msrevs_I1_J,axiom,
! [N3: nat,K: nat,M: nat] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ N3 )
=> ( ( divide_divide @ nat @ ( plus_plus @ nat @ ( times_times @ nat @ K @ N3 ) @ M ) @ N3 )
= ( plus_plus @ nat @ ( divide_divide @ nat @ M @ N3 ) @ K ) ) ) ).
% msrevs(1)
thf(fact_1431_split__div,axiom,
! [P: nat > $o,M: nat,N3: nat] :
( ( P @ ( divide_divide @ nat @ M @ N3 ) )
= ( ( ( N3
= ( zero_zero @ nat ) )
=> ( P @ ( zero_zero @ nat ) ) )
& ( ( N3
!= ( zero_zero @ nat ) )
=> ! [I3: nat,J3: nat] :
( ( ord_less @ nat @ J3 @ N3 )
=> ( ( M
= ( plus_plus @ nat @ ( times_times @ nat @ N3 @ I3 ) @ J3 ) )
=> ( P @ I3 ) ) ) ) ) ) ).
% split_div
thf(fact_1432_dividend__less__div__times,axiom,
! [N3: nat,M: nat] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ N3 )
=> ( ord_less @ nat @ M @ ( plus_plus @ nat @ N3 @ ( times_times @ nat @ ( divide_divide @ nat @ M @ N3 ) @ N3 ) ) ) ) ).
% dividend_less_div_times
thf(fact_1433_dividend__less__times__div,axiom,
! [N3: nat,M: nat] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ N3 )
=> ( ord_less @ nat @ M @ ( plus_plus @ nat @ N3 @ ( times_times @ nat @ N3 @ ( divide_divide @ nat @ M @ N3 ) ) ) ) ) ).
% dividend_less_times_div
thf(fact_1434_td__gal,axiom,
! [C2: nat,B2: nat,A2: nat] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ C2 )
=> ( ( ord_less_eq @ nat @ ( times_times @ nat @ B2 @ C2 ) @ A2 )
= ( ord_less_eq @ nat @ B2 @ ( divide_divide @ nat @ A2 @ C2 ) ) ) ) ).
% td_gal
thf(fact_1435_less__eq__div__iff__mult__less__eq,axiom,
! [Q2: nat,M: nat,N3: nat] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ Q2 )
=> ( ( ord_less_eq @ nat @ M @ ( divide_divide @ nat @ N3 @ Q2 ) )
= ( ord_less_eq @ nat @ ( times_times @ nat @ M @ Q2 ) @ N3 ) ) ) ).
% less_eq_div_iff_mult_less_eq
thf(fact_1436_mult__eq__if,axiom,
( ( times_times @ nat )
= ( ^ [M5: nat,N2: nat] :
( if @ nat
@ ( M5
= ( zero_zero @ nat ) )
@ ( zero_zero @ nat )
@ ( plus_plus @ nat @ N2 @ ( times_times @ nat @ ( minus_minus @ nat @ M5 @ ( one_one @ nat ) ) @ N2 ) ) ) ) ) ).
% mult_eq_if
thf(fact_1437_nat__mult__power__less__eq,axiom,
! [B2: nat,A2: nat,N3: nat,M: nat] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ B2 )
=> ( ( ord_less @ nat @ ( times_times @ nat @ A2 @ ( power_power @ nat @ B2 @ N3 ) ) @ ( power_power @ nat @ B2 @ M ) )
= ( ord_less @ nat @ A2 @ ( power_power @ nat @ B2 @ ( minus_minus @ nat @ M @ N3 ) ) ) ) ) ).
% nat_mult_power_less_eq
thf(fact_1438_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_1439_vebt__delete_Osimps_I5_J,axiom,
! [Mi: nat,Ma: nat,TrLst: list @ vEBT_VEBT,Smry: vEBT_VEBT,X: nat] :
( ( vEBT_vebt_delete @ ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi @ Ma ) ) @ ( zero_zero @ nat ) @ TrLst @ Smry ) @ X )
= ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi @ Ma ) ) @ ( zero_zero @ nat ) @ TrLst @ Smry ) ) ).
% vebt_delete.simps(5)
thf(fact_1440_vebt__member_Osimps_I4_J,axiom,
! [V: product_prod @ nat @ nat,Vb: list @ vEBT_VEBT,Vc: vEBT_VEBT,X: nat] :
~ ( vEBT_vebt_member @ ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ V ) @ ( suc @ ( zero_zero @ nat ) ) @ Vb @ Vc ) @ X ) ).
% vebt_member.simps(4)
thf(fact_1441_ceiling__correct,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [X: A] :
( ( ord_less @ A @ ( minus_minus @ A @ ( ring_1_of_int @ A @ ( archimedean_ceiling @ A @ X ) ) @ ( one_one @ A ) ) @ X )
& ( ord_less_eq @ A @ X @ ( ring_1_of_int @ A @ ( archimedean_ceiling @ A @ X ) ) ) ) ) ).
% ceiling_correct
thf(fact_1442_ceiling__unique,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [Z: int,X: A] :
( ( ord_less @ A @ ( minus_minus @ A @ ( ring_1_of_int @ A @ Z ) @ ( one_one @ A ) ) @ X )
=> ( ( ord_less_eq @ A @ X @ ( ring_1_of_int @ A @ Z ) )
=> ( ( archimedean_ceiling @ A @ X )
= Z ) ) ) ) ).
% ceiling_unique
thf(fact_1443_ceiling__eq__iff,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [X: A,A2: int] :
( ( ( archimedean_ceiling @ A @ X )
= A2 )
= ( ( ord_less @ A @ ( minus_minus @ A @ ( ring_1_of_int @ A @ A2 ) @ ( one_one @ A ) ) @ X )
& ( ord_less_eq @ A @ X @ ( ring_1_of_int @ A @ A2 ) ) ) ) ) ).
% ceiling_eq_iff
thf(fact_1444_ceiling__split,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [P: int > $o,T2: A] :
( ( P @ ( archimedean_ceiling @ A @ T2 ) )
= ( ! [I3: int] :
( ( ( ord_less @ A @ ( minus_minus @ A @ ( ring_1_of_int @ A @ I3 ) @ ( one_one @ A ) ) @ T2 )
& ( ord_less_eq @ A @ T2 @ ( ring_1_of_int @ A @ I3 ) ) )
=> ( P @ I3 ) ) ) ) ) ).
% ceiling_split
thf(fact_1445_vebt__pred_Osimps_I5_J,axiom,
! [V: product_prod @ nat @ nat,Vd: list @ vEBT_VEBT,Ve: vEBT_VEBT,Vf: nat] :
( ( vEBT_vebt_pred @ ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ V ) @ ( zero_zero @ nat ) @ Vd @ Ve ) @ Vf )
= ( none @ nat ) ) ).
% vebt_pred.simps(5)
thf(fact_1446_vebt__succ_Osimps_I4_J,axiom,
! [V: product_prod @ nat @ nat,Vc: list @ vEBT_VEBT,Vd: vEBT_VEBT,Ve: nat] :
( ( vEBT_vebt_succ @ ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ V ) @ ( zero_zero @ nat ) @ Vc @ Vd ) @ Ve )
= ( none @ nat ) ) ).
% vebt_succ.simps(4)
thf(fact_1447_T_092_060_094sub_062i_092_060_094sub_062n_092_060_094sub_062s_092_060_094sub_062e_092_060_094sub_062r_092_060_094sub_062t_Osimps_I3_J,axiom,
! [Info: option @ ( product_prod @ nat @ nat ),Ts: list @ vEBT_VEBT,S: vEBT_VEBT,X: nat] :
( ( vEBT_T_i_n_s_e_r_t @ ( vEBT_Node @ Info @ ( suc @ ( zero_zero @ nat ) ) @ Ts @ S ) @ X )
= ( one_one @ nat ) ) ).
% T\<^sub>i\<^sub>n\<^sub>s\<^sub>e\<^sub>r\<^sub>t.simps(3)
thf(fact_1448_ceiling__less__iff,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [X: A,Z: int] :
( ( ord_less @ int @ ( archimedean_ceiling @ A @ X ) @ Z )
= ( ord_less_eq @ A @ X @ ( minus_minus @ A @ ( ring_1_of_int @ A @ Z ) @ ( one_one @ A ) ) ) ) ) ).
% ceiling_less_iff
thf(fact_1449_le__ceiling__iff,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [Z: int,X: A] :
( ( ord_less_eq @ int @ Z @ ( archimedean_ceiling @ A @ X ) )
= ( ord_less @ A @ ( minus_minus @ A @ ( ring_1_of_int @ A @ Z ) @ ( one_one @ A ) ) @ X ) ) ) ).
% le_ceiling_iff
thf(fact_1450_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,Vd: list @ vEBT_VEBT,Ve: vEBT_VEBT,Vf: nat] :
( ( vEBT_T_p_r_e_d @ ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ V ) @ ( zero_zero @ nat ) @ Vd @ Ve ) @ Vf )
= ( one_one @ nat ) ) ).
% T\<^sub>p\<^sub>r\<^sub>e\<^sub>d.simps(5)
thf(fact_1451_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,Vd: vEBT_VEBT,Ve: nat] :
( ( vEBT_T_s_u_c_c @ ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ V ) @ ( zero_zero @ nat ) @ Vc @ Vd ) @ Ve )
= ( one_one @ nat ) ) ).
% T\<^sub>s\<^sub>u\<^sub>c\<^sub>c.simps(4)
thf(fact_1452_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_1453_T_092_060_094sub_062m_092_060_094sub_062i_092_060_094sub_062n_092_060_094sub_062N_092_060_094sub_062u_092_060_094sub_062l_092_060_094sub_062l_Osimps_I4_J,axiom,
! [Uw2: nat,Ux: list @ vEBT_VEBT,Uy: vEBT_VEBT] :
( ( vEBT_T_m_i_n_N_u_l_l @ ( vEBT_Node @ ( none @ ( product_prod @ nat @ nat ) ) @ Uw2 @ Ux @ Uy ) )
= ( one_one @ nat ) ) ).
% T\<^sub>m\<^sub>i\<^sub>n\<^sub>N\<^sub>u\<^sub>l\<^sub>l.simps(4)
thf(fact_1454_real__of__int__div3,axiom,
! [N3: int,X: int] : ( ord_less_eq @ real @ ( minus_minus @ real @ ( divide_divide @ real @ ( ring_1_of_int @ real @ N3 ) @ ( ring_1_of_int @ real @ X ) ) @ ( ring_1_of_int @ real @ ( divide_divide @ int @ N3 @ X ) ) ) @ ( one_one @ real ) ) ).
% real_of_int_div3
thf(fact_1455_convex__bound__lt,axiom,
! [A: $tType] :
( ( linord715952674999750819strict @ A )
=> ! [X: A,A2: A,Y: A,U: A,V: A] :
( ( ord_less @ A @ X @ A2 )
=> ( ( ord_less @ A @ Y @ A2 )
=> ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ U )
=> ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ V )
=> ( ( ( plus_plus @ A @ U @ V )
= ( one_one @ A ) )
=> ( ord_less @ A @ ( plus_plus @ A @ ( times_times @ A @ U @ X ) @ ( times_times @ A @ V @ Y ) ) @ A2 ) ) ) ) ) ) ) ).
% convex_bound_lt
thf(fact_1456_divide__le__eq__numeral_I1_J,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [B2: A,C2: A,W: num] :
( ( ord_less_eq @ A @ ( divide_divide @ A @ B2 @ C2 ) @ ( numeral_numeral @ A @ W ) )
= ( ( ( ord_less @ A @ ( zero_zero @ A ) @ C2 )
=> ( ord_less_eq @ A @ B2 @ ( times_times @ A @ ( numeral_numeral @ A @ W ) @ C2 ) ) )
& ( ~ ( ord_less @ A @ ( zero_zero @ A ) @ C2 )
=> ( ( ( ord_less @ A @ C2 @ ( zero_zero @ A ) )
=> ( ord_less_eq @ A @ ( times_times @ A @ ( numeral_numeral @ A @ W ) @ C2 ) @ B2 ) )
& ( ~ ( ord_less @ A @ C2 @ ( zero_zero @ A ) )
=> ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( numeral_numeral @ A @ W ) ) ) ) ) ) ) ) ).
% divide_le_eq_numeral(1)
thf(fact_1457_le__divide__eq__numeral_I1_J,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [W: num,B2: A,C2: A] :
( ( ord_less_eq @ A @ ( numeral_numeral @ A @ W ) @ ( divide_divide @ A @ B2 @ C2 ) )
= ( ( ( ord_less @ A @ ( zero_zero @ A ) @ C2 )
=> ( ord_less_eq @ A @ ( times_times @ A @ ( numeral_numeral @ A @ W ) @ C2 ) @ B2 ) )
& ( ~ ( ord_less @ A @ ( zero_zero @ A ) @ C2 )
=> ( ( ( ord_less @ A @ C2 @ ( zero_zero @ A ) )
=> ( ord_less_eq @ A @ B2 @ ( times_times @ A @ ( numeral_numeral @ A @ W ) @ C2 ) ) )
& ( ~ ( ord_less @ A @ C2 @ ( zero_zero @ A ) )
=> ( ord_less_eq @ A @ ( numeral_numeral @ A @ W ) @ ( zero_zero @ A ) ) ) ) ) ) ) ) ).
% le_divide_eq_numeral(1)
thf(fact_1458_half__gt__zero,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [A2: A] :
( ( ord_less @ A @ ( zero_zero @ A ) @ A2 )
=> ( ord_less @ A @ ( zero_zero @ A ) @ ( divide_divide @ A @ A2 @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) ) ) ) ).
% half_gt_zero
thf(fact_1459_half__gt__zero__iff,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [A2: A] :
( ( ord_less @ A @ ( zero_zero @ A ) @ ( divide_divide @ A @ A2 @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) )
= ( ord_less @ A @ ( zero_zero @ A ) @ A2 ) ) ) ).
% half_gt_zero_iff
thf(fact_1460_scaling__mono,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [U: A,V: A,R2: A,S: A] :
( ( ord_less_eq @ A @ U @ V )
=> ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ R2 )
=> ( ( ord_less_eq @ A @ R2 @ S )
=> ( ord_less_eq @ A @ ( plus_plus @ A @ U @ ( divide_divide @ A @ ( times_times @ A @ R2 @ ( minus_minus @ A @ V @ U ) ) @ S ) ) @ V ) ) ) ) ) ).
% scaling_mono
thf(fact_1461_zero__le__power2,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [A2: A] : ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( power_power @ A @ A2 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ).
% zero_le_power2
thf(fact_1462_power2__eq__imp__eq,axiom,
! [A: $tType] :
( ( linordered_semidom @ A )
=> ! [X: A,Y: A] :
( ( ( power_power @ A @ X @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) )
= ( power_power @ A @ Y @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) )
=> ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ X )
=> ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ Y )
=> ( X = Y ) ) ) ) ) ).
% power2_eq_imp_eq
thf(fact_1463_power2__le__imp__le,axiom,
! [A: $tType] :
( ( linordered_semidom @ A )
=> ! [X: A,Y: A] :
( ( ord_less_eq @ A @ ( power_power @ A @ X @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ ( power_power @ A @ Y @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) )
=> ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ Y )
=> ( ord_less_eq @ A @ X @ Y ) ) ) ) ).
% power2_le_imp_le
thf(fact_1464_power2__less__0,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [A2: A] :
~ ( ord_less @ A @ ( power_power @ A @ A2 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ ( zero_zero @ A ) ) ) ).
% power2_less_0
thf(fact_1465_exp__add__not__zero__imp__left,axiom,
! [A: $tType] :
( ( bit_semiring_bits @ A )
=> ! [M: nat,N3: nat] :
( ( ( power_power @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ ( plus_plus @ nat @ M @ N3 ) )
!= ( zero_zero @ A ) )
=> ( ( power_power @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ M )
!= ( zero_zero @ A ) ) ) ) ).
% exp_add_not_zero_imp_left
thf(fact_1466_exp__add__not__zero__imp__right,axiom,
! [A: $tType] :
( ( bit_semiring_bits @ A )
=> ! [M: nat,N3: nat] :
( ( ( power_power @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ ( plus_plus @ nat @ M @ N3 ) )
!= ( zero_zero @ A ) )
=> ( ( power_power @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ N3 )
!= ( zero_zero @ A ) ) ) ) ).
% exp_add_not_zero_imp_right
thf(fact_1467_exp__not__zero__imp__exp__diff__not__zero,axiom,
! [A: $tType] :
( ( bit_semiring_bits @ A )
=> ! [N3: nat,M: nat] :
( ( ( power_power @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ N3 )
!= ( zero_zero @ A ) )
=> ( ( power_power @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ ( minus_minus @ nat @ N3 @ M ) )
!= ( zero_zero @ A ) ) ) ) ).
% exp_not_zero_imp_exp_diff_not_zero
thf(fact_1468_nat__approx__posE,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [E: A] :
( ( ord_less @ A @ ( zero_zero @ A ) @ E )
=> ~ ! [N: nat] :
~ ( ord_less @ A @ ( divide_divide @ A @ ( one_one @ A ) @ ( semiring_1_of_nat @ A @ ( suc @ N ) ) ) @ E ) ) ) ).
% nat_approx_posE
thf(fact_1469_power__diff__power__eq,axiom,
! [A: $tType] :
( ( euclid4440199948858584721cancel @ A )
=> ! [A2: A,N3: nat,M: nat] :
( ( A2
!= ( zero_zero @ A ) )
=> ( ( ( ord_less_eq @ nat @ N3 @ M )
=> ( ( divide_divide @ A @ ( power_power @ A @ A2 @ M ) @ ( power_power @ A @ A2 @ N3 ) )
= ( power_power @ A @ A2 @ ( minus_minus @ nat @ M @ N3 ) ) ) )
& ( ~ ( ord_less_eq @ nat @ N3 @ M )
=> ( ( divide_divide @ A @ ( power_power @ A @ A2 @ M ) @ ( power_power @ A @ A2 @ N3 ) )
= ( divide_divide @ A @ ( one_one @ A ) @ ( power_power @ A @ A2 @ ( minus_minus @ nat @ N3 @ M ) ) ) ) ) ) ) ) ).
% power_diff_power_eq
thf(fact_1470_less__2__cases,axiom,
! [N3: nat] :
( ( ord_less @ nat @ N3 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) )
=> ( ( N3
= ( zero_zero @ nat ) )
| ( N3
= ( suc @ ( zero_zero @ nat ) ) ) ) ) ).
% less_2_cases
thf(fact_1471_less__2__cases__iff,axiom,
! [N3: nat] :
( ( ord_less @ nat @ N3 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) )
= ( ( N3
= ( zero_zero @ nat ) )
| ( N3
= ( suc @ ( zero_zero @ nat ) ) ) ) ) ).
% less_2_cases_iff
thf(fact_1472_inverse__of__nat__le,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [N3: nat,M: nat] :
( ( ord_less_eq @ nat @ N3 @ M )
=> ( ( N3
!= ( 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 @ N3 ) ) ) ) ) ) ).
% inverse_of_nat_le
thf(fact_1473_nat__induct2,axiom,
! [P: nat > $o,N3: nat] :
( ( P @ ( zero_zero @ nat ) )
=> ( ( P @ ( one_one @ nat ) )
=> ( ! [N: nat] :
( ( P @ N )
=> ( P @ ( plus_plus @ nat @ N @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) )
=> ( P @ N3 ) ) ) ) ).
% nat_induct2
thf(fact_1474_power__eq__if,axiom,
! [A: $tType] :
( ( power @ A )
=> ( ( power_power @ A )
= ( ^ [P5: A,M5: nat] :
( if @ A
@ ( M5
= ( zero_zero @ nat ) )
@ ( one_one @ A )
@ ( times_times @ A @ P5 @ ( power_power @ A @ P5 @ ( minus_minus @ nat @ M5 @ ( one_one @ nat ) ) ) ) ) ) ) ) ).
% power_eq_if
thf(fact_1475_power__minus__mult,axiom,
! [A: $tType] :
( ( monoid_mult @ A )
=> ! [N3: nat,A2: A] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ N3 )
=> ( ( times_times @ A @ ( power_power @ A @ A2 @ ( minus_minus @ nat @ N3 @ ( one_one @ nat ) ) ) @ A2 )
= ( power_power @ A @ A2 @ N3 ) ) ) ) ).
% power_minus_mult
thf(fact_1476_le__div__geq,axiom,
! [N3: nat,M: nat] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ N3 )
=> ( ( ord_less_eq @ nat @ N3 @ M )
=> ( ( divide_divide @ nat @ M @ N3 )
= ( suc @ ( divide_divide @ nat @ ( minus_minus @ nat @ M @ N3 ) @ N3 ) ) ) ) ) ).
% le_div_geq
thf(fact_1477_split__div_H,axiom,
! [P: nat > $o,M: nat,N3: nat] :
( ( P @ ( divide_divide @ nat @ M @ N3 ) )
= ( ( ( N3
= ( zero_zero @ nat ) )
& ( P @ ( zero_zero @ nat ) ) )
| ? [Q4: nat] :
( ( ord_less_eq @ nat @ ( times_times @ nat @ N3 @ Q4 ) @ M )
& ( ord_less @ nat @ M @ ( times_times @ nat @ N3 @ ( suc @ Q4 ) ) )
& ( P @ Q4 ) ) ) ) ).
% split_div'
thf(fact_1478_mult__ceiling__le,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [A2: A,B2: A] :
( ( ord_less_eq @ A @ ( zero_zero @ A ) @ A2 )
=> ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ B2 )
=> ( ord_less_eq @ int @ ( archimedean_ceiling @ A @ ( times_times @ A @ A2 @ B2 ) ) @ ( times_times @ int @ ( archimedean_ceiling @ A @ A2 ) @ ( archimedean_ceiling @ A @ B2 ) ) ) ) ) ) ).
% mult_ceiling_le
thf(fact_1479_power__sub,axiom,
! [N3: nat,M: nat,A2: nat] :
( ( ord_less_eq @ nat @ N3 @ M )
=> ( ( ord_less @ nat @ ( zero_zero @ nat ) @ A2 )
=> ( ( power_power @ nat @ A2 @ ( minus_minus @ nat @ M @ N3 ) )
= ( divide_divide @ nat @ ( power_power @ nat @ A2 @ M ) @ ( power_power @ nat @ A2 @ N3 ) ) ) ) ) ).
% power_sub
thf(fact_1480_vebt__delete_Osimps_I6_J,axiom,
! [Mi: nat,Ma: nat,Tr: list @ vEBT_VEBT,Sm: vEBT_VEBT,X: nat] :
( ( vEBT_vebt_delete @ ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi @ Ma ) ) @ ( suc @ ( zero_zero @ nat ) ) @ Tr @ Sm ) @ X )
= ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi @ Ma ) ) @ ( suc @ ( zero_zero @ nat ) ) @ Tr @ Sm ) ) ).
% vebt_delete.simps(6)
thf(fact_1481_vebt__pred_Osimps_I6_J,axiom,
! [V: product_prod @ nat @ nat,Vh: list @ vEBT_VEBT,Vi: vEBT_VEBT,Vj: nat] :
( ( vEBT_vebt_pred @ ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ V ) @ ( suc @ ( zero_zero @ nat ) ) @ Vh @ Vi ) @ Vj )
= ( none @ nat ) ) ).
% vebt_pred.simps(6)
thf(fact_1482_vebt__succ_Osimps_I5_J,axiom,
! [V: product_prod @ nat @ nat,Vg: list @ vEBT_VEBT,Vh: vEBT_VEBT,Vi: nat] :
( ( vEBT_vebt_succ @ ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ V ) @ ( suc @ ( zero_zero @ nat ) ) @ Vg @ Vh ) @ Vi )
= ( none @ nat ) ) ).
% vebt_succ.simps(5)
thf(fact_1483_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,Vh: list @ vEBT_VEBT,Vi: vEBT_VEBT,Vj: nat] :
( ( vEBT_T_p_r_e_d @ ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ V ) @ ( suc @ ( zero_zero @ nat ) ) @ Vh @ Vi ) @ Vj )
= ( one_one @ nat ) ) ).
% T\<^sub>p\<^sub>r\<^sub>e\<^sub>d.simps(6)
thf(fact_1484_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,Vg: list @ vEBT_VEBT,Vh: vEBT_VEBT,Vi: nat] :
( ( vEBT_T_s_u_c_c @ ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ V ) @ ( suc @ ( zero_zero @ nat ) ) @ Vg @ Vh ) @ Vi )
= ( one_one @ nat ) ) ).
% T\<^sub>s\<^sub>u\<^sub>c\<^sub>c.simps(5)
thf(fact_1485_VEBT__internal_OT_092_060_094sub_062d_092_060_094sub_062e_092_060_094sub_062l_092_060_094sub_062e_092_060_094sub_062t_092_060_094sub_062e_H_Osimps_I5_J,axiom,
! [Mi: nat,Ma: nat,TreeList: list @ vEBT_VEBT,Summary: vEBT_VEBT,X: nat] :
( ( vEBT_V1232361888498592333_e_t_e @ ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi @ Ma ) ) @ ( zero_zero @ nat ) @ TreeList @ Summary ) @ X )
= ( one_one @ nat ) ) ).
% VEBT_internal.T\<^sub>d\<^sub>e\<^sub>l\<^sub>e\<^sub>t\<^sub>e'.simps(5)
thf(fact_1486_T_092_060_094sub_062d_092_060_094sub_062e_092_060_094sub_062l_092_060_094sub_062e_092_060_094sub_062t_092_060_094sub_062e_Osimps_I5_J,axiom,
! [Mi: nat,Ma: nat,TreeList: list @ vEBT_VEBT,Summary: vEBT_VEBT,X: nat] :
( ( vEBT_T_d_e_l_e_t_e @ ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi @ Ma ) ) @ ( zero_zero @ nat ) @ TreeList @ Summary ) @ X )
= ( one_one @ nat ) ) ).
% T\<^sub>d\<^sub>e\<^sub>l\<^sub>e\<^sub>t\<^sub>e.simps(5)
thf(fact_1487_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,X: nat] :
( ( vEBT_T_i_n_s_e_r_t2 @ ( vEBT_Node @ ( none @ ( product_prod @ nat @ nat ) ) @ ( suc @ ( suc @ V ) ) @ TreeList @ Summary ) @ X )
= ( one_one @ nat ) ) ).
% T\<^sub>i\<^sub>n\<^sub>s\<^sub>e\<^sub>r\<^sub>t'.simps(4)
thf(fact_1488_power2__less__imp__less,axiom,
! [A: $tType] :
( ( linordered_semidom @ A )
=> ! [X: A,Y: A] :
( ( ord_less @ A @ ( power_power @ A @ X @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ ( power_power @ A @ Y @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) )
=> ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ Y )
=> ( ord_less @ A @ X @ Y ) ) ) ) ).
% power2_less_imp_less
thf(fact_1489_sum__power2__ge__zero,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [X: A,Y: A] : ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( plus_plus @ A @ ( power_power @ A @ X @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ ( power_power @ A @ Y @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) ).
% sum_power2_ge_zero
thf(fact_1490_sum__power2__le__zero__iff,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [X: A,Y: A] :
( ( ord_less_eq @ A @ ( plus_plus @ A @ ( power_power @ A @ X @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ ( power_power @ A @ Y @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( zero_zero @ A ) )
= ( ( X
= ( zero_zero @ A ) )
& ( Y
= ( zero_zero @ A ) ) ) ) ) ).
% sum_power2_le_zero_iff
thf(fact_1491_not__sum__power2__lt__zero,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [X: A,Y: A] :
~ ( ord_less @ A @ ( plus_plus @ A @ ( power_power @ A @ X @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ ( power_power @ A @ Y @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( zero_zero @ A ) ) ) ).
% not_sum_power2_lt_zero
thf(fact_1492_sum__power2__gt__zero__iff,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [X: A,Y: A] :
( ( ord_less @ A @ ( zero_zero @ A ) @ ( plus_plus @ A @ ( power_power @ A @ X @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ ( power_power @ A @ Y @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) )
= ( ( X
!= ( zero_zero @ A ) )
| ( Y
!= ( zero_zero @ A ) ) ) ) ) ).
% sum_power2_gt_zero_iff
thf(fact_1493_zero__le__even__power_H,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [A2: A,N3: nat] : ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( power_power @ A @ A2 @ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N3 ) ) ) ) ).
% zero_le_even_power'
thf(fact_1494_insersimp_H,axiom,
! [T2: vEBT_VEBT,N3: nat,Y: nat] :
( ( vEBT_invar_vebt @ T2 @ N3 )
=> ( ~ ? [X_12: nat] : ( vEBT_V8194947554948674370ptions @ T2 @ X_12 )
=> ( ord_less_eq @ nat @ ( vEBT_T_i_n_s_e_r_t2 @ T2 @ Y ) @ ( one_one @ nat ) ) ) ) ).
% insersimp'
thf(fact_1495_insertsimp_H,axiom,
! [T2: vEBT_VEBT,N3: nat,L2: nat] :
( ( vEBT_invar_vebt @ T2 @ N3 )
=> ( ( vEBT_VEBT_minNull @ T2 )
=> ( ord_less_eq @ nat @ ( vEBT_T_i_n_s_e_r_t2 @ T2 @ L2 ) @ ( one_one @ nat ) ) ) ) ).
% insertsimp'
thf(fact_1496_div__2__gt__zero,axiom,
! [N3: nat] :
( ( ord_less @ nat @ ( suc @ ( zero_zero @ nat ) ) @ N3 )
=> ( ord_less @ nat @ ( zero_zero @ nat ) @ ( divide_divide @ nat @ N3 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ).
% div_2_gt_zero
thf(fact_1497_Suc__n__div__2__gt__zero,axiom,
! [N3: nat] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ N3 )
=> ( ord_less @ nat @ ( zero_zero @ nat ) @ ( divide_divide @ nat @ ( suc @ N3 ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ).
% Suc_n_div_2_gt_zero
thf(fact_1498_nat__bit__induct,axiom,
! [P: nat > $o,N3: nat] :
( ( P @ ( zero_zero @ nat ) )
=> ( ! [N: nat] :
( ( P @ N )
=> ( ( ord_less @ nat @ ( zero_zero @ nat ) @ N )
=> ( P @ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N ) ) ) )
=> ( ! [N: nat] :
( ( P @ N )
=> ( P @ ( suc @ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N ) ) ) )
=> ( P @ N3 ) ) ) ) ).
% nat_bit_induct
thf(fact_1499_ceiling__eq,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [N3: int,X: A] :
( ( ord_less @ A @ ( ring_1_of_int @ A @ N3 ) @ X )
=> ( ( ord_less_eq @ A @ X @ ( plus_plus @ A @ ( ring_1_of_int @ A @ N3 ) @ ( one_one @ A ) ) )
=> ( ( archimedean_ceiling @ A @ X )
= ( plus_plus @ int @ N3 @ ( one_one @ int ) ) ) ) ) ) ).
% ceiling_eq
thf(fact_1500_log__of__power__less,axiom,
! [M: nat,B2: real,N3: nat] :
( ( ord_less @ real @ ( semiring_1_of_nat @ real @ M ) @ ( power_power @ real @ B2 @ N3 ) )
=> ( ( ord_less @ real @ ( one_one @ real ) @ B2 )
=> ( ( ord_less @ nat @ ( zero_zero @ nat ) @ M )
=> ( ord_less @ real @ ( log @ B2 @ ( semiring_1_of_nat @ real @ M ) ) @ ( semiring_1_of_nat @ real @ N3 ) ) ) ) ) ).
% log_of_power_less
thf(fact_1501_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,X: nat] :
( ( vEBT_T_m_e_m_b_e_r @ ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ V ) @ ( zero_zero @ nat ) @ Uy @ Uz ) @ X )
= ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ).
% T\<^sub>m\<^sub>e\<^sub>m\<^sub>b\<^sub>e\<^sub>r.simps(3)
thf(fact_1502_VEBT__internal_OT_092_060_094sub_062d_092_060_094sub_062e_092_060_094sub_062l_092_060_094sub_062e_092_060_094sub_062t_092_060_094sub_062e_H_Osimps_I6_J,axiom,
! [Mi: nat,Ma: nat,TreeList: list @ vEBT_VEBT,Summary: vEBT_VEBT,X: nat] :
( ( vEBT_V1232361888498592333_e_t_e @ ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi @ Ma ) ) @ ( suc @ ( zero_zero @ nat ) ) @ TreeList @ Summary ) @ X )
= ( one_one @ nat ) ) ).
% VEBT_internal.T\<^sub>d\<^sub>e\<^sub>l\<^sub>e\<^sub>t\<^sub>e'.simps(6)
thf(fact_1503_T_092_060_094sub_062d_092_060_094sub_062e_092_060_094sub_062l_092_060_094sub_062e_092_060_094sub_062t_092_060_094sub_062e_Osimps_I6_J,axiom,
! [Mi: nat,Ma: nat,TreeList: list @ vEBT_VEBT,Summary: vEBT_VEBT,X: nat] :
( ( vEBT_T_d_e_l_e_t_e @ ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi @ Ma ) ) @ ( suc @ ( zero_zero @ nat ) ) @ TreeList @ Summary ) @ X )
= ( one_one @ nat ) ) ).
% T\<^sub>d\<^sub>e\<^sub>l\<^sub>e\<^sub>t\<^sub>e.simps(6)
thf(fact_1504_odd__0__le__power__imp__0__le,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [A2: A,N3: nat] :
( ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( power_power @ A @ A2 @ ( suc @ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N3 ) ) ) )
=> ( ord_less_eq @ A @ ( zero_zero @ A ) @ A2 ) ) ) ).
% odd_0_le_power_imp_0_le
thf(fact_1505_odd__power__less__zero,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [A2: A,N3: nat] :
( ( ord_less @ A @ A2 @ ( zero_zero @ A ) )
=> ( ord_less @ A @ ( power_power @ A @ A2 @ ( suc @ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N3 ) ) ) @ ( zero_zero @ A ) ) ) ) ).
% odd_power_less_zero
thf(fact_1506_VEBT__internal_Oexp__split__high__low_I1_J,axiom,
! [X: nat,N3: nat,M: nat] :
( ( ord_less @ nat @ X @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( plus_plus @ nat @ N3 @ M ) ) )
=> ( ( ord_less @ nat @ ( zero_zero @ nat ) @ N3 )
=> ( ( ord_less @ nat @ ( zero_zero @ nat ) @ M )
=> ( ord_less @ nat @ ( vEBT_VEBT_high @ X @ N3 ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ M ) ) ) ) ) ).
% VEBT_internal.exp_split_high_low(1)
thf(fact_1507_VEBT__internal_Oexp__split__high__low_I2_J,axiom,
! [X: nat,N3: nat,M: nat] :
( ( ord_less @ nat @ X @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( plus_plus @ nat @ N3 @ M ) ) )
=> ( ( ord_less @ nat @ ( zero_zero @ nat ) @ N3 )
=> ( ( ord_less @ nat @ ( zero_zero @ nat ) @ M )
=> ( ord_less @ nat @ ( vEBT_VEBT_low @ X @ N3 ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N3 ) ) ) ) ) ).
% VEBT_internal.exp_split_high_low(2)
thf(fact_1508_insert_H__bound__height,axiom,
! [T2: vEBT_VEBT,N3: nat,X: nat] :
( ( vEBT_invar_vebt @ T2 @ N3 )
=> ( ord_less_eq @ nat @ ( vEBT_T_i_n_s_e_r_t2 @ T2 @ X ) @ ( plus_plus @ nat @ ( one_one @ nat ) @ ( vEBT_VEBT_height @ T2 ) ) ) ) ).
% insert'_bound_height
thf(fact_1509_log__of__power__le,axiom,
! [M: nat,B2: real,N3: nat] :
( ( ord_less_eq @ real @ ( semiring_1_of_nat @ real @ M ) @ ( power_power @ real @ B2 @ N3 ) )
=> ( ( ord_less @ real @ ( one_one @ real ) @ B2 )
=> ( ( ord_less @ nat @ ( zero_zero @ nat ) @ M )
=> ( ord_less_eq @ real @ ( log @ B2 @ ( semiring_1_of_nat @ real @ M ) ) @ ( semiring_1_of_nat @ real @ N3 ) ) ) ) ) ).
% log_of_power_le
thf(fact_1510_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,X: nat] :
( ( vEBT_T_m_e_m_b_e_r @ ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ V ) @ ( suc @ ( zero_zero @ nat ) ) @ Vb @ Vc ) @ X )
= ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ).
% T\<^sub>m\<^sub>e\<^sub>m\<^sub>b\<^sub>e\<^sub>r.simps(4)
thf(fact_1511_VEBT__internal_Omembermima_Oelims_I2_J,axiom,
! [X: vEBT_VEBT,Xa: nat] :
( ( vEBT_VEBT_membermima @ X @ Xa )
=> ( ! [Mi2: nat,Ma2: nat] :
( ? [Va2: list @ vEBT_VEBT,Vb2: vEBT_VEBT] :
( X
= ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi2 @ Ma2 ) ) @ ( zero_zero @ nat ) @ Va2 @ Vb2 ) )
=> ~ ( ( Xa = Mi2 )
| ( Xa = Ma2 ) ) )
=> ( ! [Mi2: nat,Ma2: nat,V3: nat,TreeList2: list @ vEBT_VEBT] :
( ? [Vc2: vEBT_VEBT] :
( X
= ( 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] :
( X
= ( 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_1512_nat__div__eq__Suc__0__iff,axiom,
! [N3: nat,M: nat] :
( ( ( divide_divide @ nat @ N3 @ M )
= ( suc @ ( zero_zero @ nat ) ) )
= ( ( ord_less_eq @ nat @ M @ N3 )
& ( ord_less @ nat @ N3 @ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ M ) ) ) ) ).
% nat_div_eq_Suc_0_iff
thf(fact_1513_arith__geo__mean,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [U: A,X: A,Y: A] :
( ( ( power_power @ A @ U @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) )
= ( times_times @ A @ X @ Y ) )
=> ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ X )
=> ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ Y )
=> ( ord_less_eq @ A @ U @ ( divide_divide @ A @ ( plus_plus @ A @ X @ Y ) @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) ) ) ) ) ) ).
% arith_geo_mean
thf(fact_1514_log2__of__power__less,axiom,
! [M: nat,N3: nat] :
( ( ord_less @ nat @ M @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N3 ) )
=> ( ( 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 @ N3 ) ) ) ) ).
% log2_of_power_less
thf(fact_1515_log2__of__power__le,axiom,
! [M: nat,N3: nat] :
( ( ord_less_eq @ nat @ M @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N3 ) )
=> ( ( 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 @ N3 ) ) ) ) ).
% log2_of_power_le
thf(fact_1516_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,X: nat] :
( ( vEBT_T_m_e_m_b_e_r2 @ ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi @ Ma ) ) @ ( suc @ ( suc @ Va ) ) @ TreeList @ Summary ) @ X )
= ( plus_plus @ nat @ ( one_one @ nat )
@ ( if @ nat @ ( X = Mi ) @ ( zero_zero @ nat )
@ ( if @ nat @ ( X = Ma ) @ ( zero_zero @ nat )
@ ( if @ nat @ ( ord_less @ nat @ X @ Mi ) @ ( zero_zero @ nat )
@ ( if @ nat @ ( ord_less @ nat @ Ma @ X ) @ ( zero_zero @ nat )
@ ( if @ nat
@ ( ( ord_less @ nat @ Mi @ X )
& ( ord_less @ nat @ X @ Ma ) )
@ ( if @ nat @ ( ord_less @ nat @ ( vEBT_VEBT_high @ X @ ( divide_divide @ nat @ ( suc @ ( suc @ 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 @ X @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( vEBT_VEBT_low @ X @ ( 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_1517_VEBT__internal_OT_092_060_094sub_062d_092_060_094sub_062e_092_060_094sub_062l_092_060_094sub_062e_092_060_094sub_062t_092_060_094sub_062e_H_Oelims,axiom,
! [X: vEBT_VEBT,Xa: nat,Y: nat] :
( ( ( vEBT_V1232361888498592333_e_t_e @ X @ Xa )
= Y )
=> ( ( ? [A4: $o,B3: $o] :
( X
= ( vEBT_Leaf @ A4 @ B3 ) )
=> ( ( Xa
= ( zero_zero @ nat ) )
=> ( Y
!= ( one_one @ nat ) ) ) )
=> ( ( ? [A4: $o,B3: $o] :
( X
= ( vEBT_Leaf @ A4 @ B3 ) )
=> ( ( Xa
= ( suc @ ( zero_zero @ nat ) ) )
=> ( Y
!= ( one_one @ nat ) ) ) )
=> ( ( ? [A4: $o,B3: $o] :
( X
= ( vEBT_Leaf @ A4 @ B3 ) )
=> ( ? [N: nat] :
( Xa
= ( suc @ ( suc @ N ) ) )
=> ( Y
!= ( one_one @ nat ) ) ) )
=> ( ( ? [Deg2: nat,TreeList2: list @ vEBT_VEBT,Summary2: vEBT_VEBT] :
( X
= ( vEBT_Node @ ( none @ ( product_prod @ nat @ nat ) ) @ Deg2 @ TreeList2 @ Summary2 ) )
=> ( Y
!= ( one_one @ nat ) ) )
=> ( ( ? [Mi2: nat,Ma2: nat,TreeList2: list @ vEBT_VEBT,Summary2: vEBT_VEBT] :
( X
= ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi2 @ Ma2 ) ) @ ( zero_zero @ nat ) @ TreeList2 @ Summary2 ) )
=> ( Y
!= ( one_one @ nat ) ) )
=> ( ( ? [Mi2: nat,Ma2: nat,TreeList2: list @ vEBT_VEBT,Summary2: vEBT_VEBT] :
( X
= ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( zero_zero @ nat ) ) @ TreeList2 @ Summary2 ) )
=> ( Y
!= ( one_one @ nat ) ) )
=> ~ ! [Mi2: nat,Ma2: nat,Va3: nat,TreeList2: list @ vEBT_VEBT,Summary2: vEBT_VEBT] :
( ( X
= ( 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 ) )
=> ( Y
= ( one_one @ nat ) ) )
& ( ~ ( ( ord_less @ nat @ Xa @ Mi2 )
| ( ord_less @ nat @ Ma2 @ Xa ) )
=> ( ( ( ( Xa = Mi2 )
& ( Xa = Ma2 ) )
=> ( Y
= ( one_one @ nat ) ) )
& ( ~ ( ( Xa = Mi2 )
& ( Xa = Ma2 ) )
=> ( Y
= ( if @ nat @ ( ord_less @ nat @ ( vEBT_VEBT_high @ ( if @ nat @ ( Xa = Mi2 ) @ ( plus_plus @ nat @ ( times_times @ nat @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary2 ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ ( suc @ ( suc @ 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_1518_vebt__delete_Oelims,axiom,
! [X: vEBT_VEBT,Xa: nat,Y: vEBT_VEBT] :
( ( ( vEBT_vebt_delete @ X @ Xa )
= Y )
=> ( ! [A4: $o,B3: $o] :
( ( X
= ( vEBT_Leaf @ A4 @ B3 ) )
=> ( ( Xa
= ( zero_zero @ nat ) )
=> ( Y
!= ( vEBT_Leaf @ $false @ B3 ) ) ) )
=> ( ! [A4: $o] :
( ? [B3: $o] :
( X
= ( vEBT_Leaf @ A4 @ B3 ) )
=> ( ( Xa
= ( suc @ ( zero_zero @ nat ) ) )
=> ( Y
!= ( vEBT_Leaf @ A4 @ $false ) ) ) )
=> ( ! [A4: $o,B3: $o] :
( ( X
= ( vEBT_Leaf @ A4 @ B3 ) )
=> ( ? [N: nat] :
( Xa
= ( suc @ ( suc @ N ) ) )
=> ( Y
!= ( vEBT_Leaf @ A4 @ B3 ) ) ) )
=> ( ! [Deg2: nat,TreeList2: list @ vEBT_VEBT,Summary2: vEBT_VEBT] :
( ( X
= ( vEBT_Node @ ( none @ ( product_prod @ nat @ nat ) ) @ Deg2 @ TreeList2 @ Summary2 ) )
=> ( Y
!= ( vEBT_Node @ ( none @ ( product_prod @ nat @ nat ) ) @ Deg2 @ TreeList2 @ Summary2 ) ) )
=> ( ! [Mi2: nat,Ma2: nat,TrLst2: list @ vEBT_VEBT,Smry2: vEBT_VEBT] :
( ( X
= ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi2 @ Ma2 ) ) @ ( zero_zero @ nat ) @ TrLst2 @ Smry2 ) )
=> ( Y
!= ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi2 @ Ma2 ) ) @ ( zero_zero @ nat ) @ TrLst2 @ Smry2 ) ) )
=> ( ! [Mi2: nat,Ma2: nat,Tr2: list @ vEBT_VEBT,Sm2: vEBT_VEBT] :
( ( X
= ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( zero_zero @ nat ) ) @ Tr2 @ Sm2 ) )
=> ( Y
!= ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( zero_zero @ nat ) ) @ Tr2 @ Sm2 ) ) )
=> ~ ! [Mi2: nat,Ma2: nat,Va3: nat,TreeList2: list @ vEBT_VEBT,Summary2: vEBT_VEBT] :
( ( X
= ( 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 ) )
=> ( Y
= ( 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 ) )
=> ( Y
= ( vEBT_Node @ ( none @ ( product_prod @ nat @ nat ) ) @ ( suc @ ( suc @ Va3 ) ) @ TreeList2 @ Summary2 ) ) )
& ( ~ ( ( Xa = Mi2 )
& ( Xa = Ma2 ) )
=> ( Y
= ( if @ vEBT_VEBT @ ( ord_less @ nat @ ( vEBT_VEBT_high @ ( if @ nat @ ( Xa = Mi2 ) @ ( plus_plus @ nat @ ( times_times @ nat @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary2 ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ ( suc @ ( suc @ 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_1519_arcosh__1,axiom,
! [A: $tType] :
( ( ln @ A )
=> ( ( arcosh @ A @ ( one_one @ A ) )
= ( zero_zero @ A ) ) ) ).
% arcosh_1
thf(fact_1520_inrange,axiom,
! [T2: vEBT_VEBT,N3: nat] :
( ( vEBT_invar_vebt @ T2 @ N3 )
=> ( 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 ) ) @ N3 ) @ ( one_one @ nat ) ) ) ) ) ).
% inrange
thf(fact_1521_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_1522_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_1523_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_1524_cnt__non__neg,axiom,
! [T2: vEBT_VEBT] : ( ord_less_eq @ real @ ( zero_zero @ real ) @ ( vEBT_VEBT_cnt @ T2 ) ) ).
% cnt_non_neg
thf(fact_1525_Leaf__0__not,axiom,
! [A2: $o,B2: $o] :
~ ( vEBT_invar_vebt @ ( vEBT_Leaf @ A2 @ B2 ) @ ( zero_zero @ nat ) ) ).
% Leaf_0_not
thf(fact_1526_deg1Leaf,axiom,
! [T2: vEBT_VEBT] :
( ( vEBT_invar_vebt @ T2 @ ( one_one @ nat ) )
= ( ? [A6: $o,B5: $o] :
( T2
= ( vEBT_Leaf @ A6 @ B5 ) ) ) ) ).
% deg1Leaf
thf(fact_1527_deg__1__Leaf,axiom,
! [T2: vEBT_VEBT] :
( ( vEBT_invar_vebt @ T2 @ ( one_one @ nat ) )
=> ? [A4: $o,B3: $o] :
( T2
= ( vEBT_Leaf @ A4 @ B3 ) ) ) ).
% deg_1_Leaf
thf(fact_1528_deg__1__Leafy,axiom,
! [T2: vEBT_VEBT,N3: nat] :
( ( vEBT_invar_vebt @ T2 @ N3 )
=> ( ( N3
= ( one_one @ nat ) )
=> ? [A4: $o,B3: $o] :
( T2
= ( vEBT_Leaf @ A4 @ B3 ) ) ) ) ).
% deg_1_Leafy
thf(fact_1529_VEBT_Oinject_I2_J,axiom,
! [X21: $o,X222: $o,Y21: $o,Y222: $o] :
( ( ( vEBT_Leaf @ X21 @ X222 )
= ( vEBT_Leaf @ Y21 @ Y222 ) )
= ( ( X21 = Y21 )
& ( X222 = Y222 ) ) ) ).
% VEBT.inject(2)
thf(fact_1530_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_1531_idiff__0,axiom,
! [N3: extended_enat] :
( ( minus_minus @ extended_enat @ ( zero_zero @ extended_enat ) @ N3 )
= ( zero_zero @ extended_enat ) ) ).
% idiff_0
thf(fact_1532_idiff__0__right,axiom,
! [N3: extended_enat] :
( ( minus_minus @ extended_enat @ N3 @ ( zero_zero @ extended_enat ) )
= N3 ) ).
% idiff_0_right
thf(fact_1533_not__real__square__gt__zero,axiom,
! [X: real] :
( ( ~ ( ord_less @ real @ ( zero_zero @ real ) @ ( times_times @ real @ X @ X ) ) )
= ( X
= ( zero_zero @ real ) ) ) ).
% not_real_square_gt_zero
thf(fact_1534_log__one,axiom,
! [A2: real] :
( ( log @ A2 @ ( one_one @ real ) )
= ( zero_zero @ real ) ) ).
% log_one
thf(fact_1535_log__eq__one,axiom,
! [A2: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ A2 )
=> ( ( A2
!= ( one_one @ real ) )
=> ( ( log @ A2 @ A2 )
= ( one_one @ real ) ) ) ) ).
% log_eq_one
thf(fact_1536_log__less__cancel__iff,axiom,
! [A2: real,X: real,Y: real] :
( ( ord_less @ real @ ( one_one @ real ) @ A2 )
=> ( ( ord_less @ real @ ( zero_zero @ real ) @ X )
=> ( ( ord_less @ real @ ( zero_zero @ real ) @ Y )
=> ( ( ord_less @ real @ ( log @ A2 @ X ) @ ( log @ A2 @ Y ) )
= ( ord_less @ real @ X @ Y ) ) ) ) ) ).
% log_less_cancel_iff
thf(fact_1537_log__less__one__cancel__iff,axiom,
! [A2: real,X: real] :
( ( ord_less @ real @ ( one_one @ real ) @ A2 )
=> ( ( ord_less @ real @ ( zero_zero @ real ) @ X )
=> ( ( ord_less @ real @ ( log @ A2 @ X ) @ ( one_one @ real ) )
= ( ord_less @ real @ X @ A2 ) ) ) ) ).
% log_less_one_cancel_iff
thf(fact_1538_one__less__log__cancel__iff,axiom,
! [A2: real,X: real] :
( ( ord_less @ real @ ( one_one @ real ) @ A2 )
=> ( ( ord_less @ real @ ( zero_zero @ real ) @ X )
=> ( ( ord_less @ real @ ( one_one @ real ) @ ( log @ A2 @ X ) )
= ( ord_less @ real @ A2 @ X ) ) ) ) ).
% one_less_log_cancel_iff
thf(fact_1539_log__less__zero__cancel__iff,axiom,
! [A2: real,X: real] :
( ( ord_less @ real @ ( one_one @ real ) @ A2 )
=> ( ( ord_less @ real @ ( zero_zero @ real ) @ X )
=> ( ( ord_less @ real @ ( log @ A2 @ X ) @ ( zero_zero @ real ) )
= ( ord_less @ real @ X @ ( one_one @ real ) ) ) ) ) ).
% log_less_zero_cancel_iff
thf(fact_1540_zero__less__log__cancel__iff,axiom,
! [A2: real,X: real] :
( ( ord_less @ real @ ( one_one @ real ) @ A2 )
=> ( ( ord_less @ real @ ( zero_zero @ real ) @ X )
=> ( ( ord_less @ real @ ( zero_zero @ real ) @ ( log @ A2 @ X ) )
= ( ord_less @ real @ ( one_one @ real ) @ X ) ) ) ) ).
% zero_less_log_cancel_iff
thf(fact_1541_div__neg__neg__trivial,axiom,
! [K: int,L2: int] :
( ( ord_less_eq @ int @ K @ ( zero_zero @ int ) )
=> ( ( ord_less @ int @ L2 @ K )
=> ( ( divide_divide @ int @ K @ L2 )
= ( zero_zero @ int ) ) ) ) ).
% div_neg_neg_trivial
thf(fact_1542_div__pos__pos__trivial,axiom,
! [K: int,L2: int] :
( ( ord_less_eq @ int @ ( zero_zero @ int ) @ K )
=> ( ( ord_less @ int @ K @ L2 )
=> ( ( divide_divide @ int @ K @ L2 )
= ( zero_zero @ int ) ) ) ) ).
% div_pos_pos_trivial
thf(fact_1543_log__le__cancel__iff,axiom,
! [A2: real,X: real,Y: real] :
( ( ord_less @ real @ ( one_one @ real ) @ A2 )
=> ( ( ord_less @ real @ ( zero_zero @ real ) @ X )
=> ( ( ord_less @ real @ ( zero_zero @ real ) @ Y )
=> ( ( ord_less_eq @ real @ ( log @ A2 @ X ) @ ( log @ A2 @ Y ) )
= ( ord_less_eq @ real @ X @ Y ) ) ) ) ) ).
% log_le_cancel_iff
thf(fact_1544_log__le__one__cancel__iff,axiom,
! [A2: real,X: real] :
( ( ord_less @ real @ ( one_one @ real ) @ A2 )
=> ( ( ord_less @ real @ ( zero_zero @ real ) @ X )
=> ( ( ord_less_eq @ real @ ( log @ A2 @ X ) @ ( one_one @ real ) )
= ( ord_less_eq @ real @ X @ A2 ) ) ) ) ).
% log_le_one_cancel_iff
thf(fact_1545_one__le__log__cancel__iff,axiom,
! [A2: real,X: real] :
( ( ord_less @ real @ ( one_one @ real ) @ A2 )
=> ( ( ord_less @ real @ ( zero_zero @ real ) @ X )
=> ( ( ord_less_eq @ real @ ( one_one @ real ) @ ( log @ A2 @ X ) )
= ( ord_less_eq @ real @ A2 @ X ) ) ) ) ).
% one_le_log_cancel_iff
thf(fact_1546_log__le__zero__cancel__iff,axiom,
! [A2: real,X: real] :
( ( ord_less @ real @ ( one_one @ real ) @ A2 )
=> ( ( ord_less @ real @ ( zero_zero @ real ) @ X )
=> ( ( ord_less_eq @ real @ ( log @ A2 @ X ) @ ( zero_zero @ real ) )
= ( ord_less_eq @ real @ X @ ( one_one @ real ) ) ) ) ) ).
% log_le_zero_cancel_iff
thf(fact_1547_zero__le__log__cancel__iff,axiom,
! [A2: real,X: real] :
( ( ord_less @ real @ ( one_one @ real ) @ A2 )
=> ( ( ord_less @ real @ ( zero_zero @ real ) @ X )
=> ( ( ord_less_eq @ real @ ( zero_zero @ real ) @ ( log @ A2 @ X ) )
= ( ord_less_eq @ real @ ( one_one @ real ) @ X ) ) ) ) ).
% zero_le_log_cancel_iff
thf(fact_1548_int__div__same__is__1,axiom,
! [A2: int,B2: int] :
( ( ord_less @ int @ ( zero_zero @ int ) @ A2 )
=> ( ( ( divide_divide @ int @ A2 @ B2 )
= A2 )
= ( B2
= ( one_one @ int ) ) ) ) ).
% int_div_same_is_1
thf(fact_1549_log__pow__cancel,axiom,
! [A2: real,B2: nat] :
( ( ord_less @ real @ ( zero_zero @ real ) @ A2 )
=> ( ( A2
!= ( one_one @ real ) )
=> ( ( log @ A2 @ ( power_power @ real @ A2 @ B2 ) )
= ( semiring_1_of_nat @ real @ B2 ) ) ) ) ).
% log_pow_cancel
thf(fact_1550_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_1551_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_1552_T_092_060_094sub_062m_092_060_094sub_062e_092_060_094sub_062m_092_060_094sub_062b_092_060_094sub_062e_092_060_094sub_062r_H_Osimps_I1_J,axiom,
! [A2: $o,B2: $o,X: nat] :
( ( vEBT_T_m_e_m_b_e_r2 @ ( vEBT_Leaf @ A2 @ B2 ) @ X )
= ( one_one @ nat ) ) ).
% T\<^sub>m\<^sub>e\<^sub>m\<^sub>b\<^sub>e\<^sub>r'.simps(1)
thf(fact_1553_VEBT_Osize_I4_J,axiom,
! [X21: $o,X222: $o] :
( ( size_size @ vEBT_VEBT @ ( vEBT_Leaf @ X21 @ X222 ) )
= ( zero_zero @ nat ) ) ).
% VEBT.size(4)
thf(fact_1554_VEBT_Odistinct_I1_J,axiom,
! [X11: option @ ( product_prod @ nat @ nat ),X12: nat,X13: list @ vEBT_VEBT,X14: vEBT_VEBT,X21: $o,X222: $o] :
( ( vEBT_Node @ X11 @ X12 @ X13 @ X14 )
!= ( vEBT_Leaf @ X21 @ X222 ) ) ).
% VEBT.distinct(1)
thf(fact_1555_VEBT_Oexhaust,axiom,
! [Y: vEBT_VEBT] :
( ! [X112: option @ ( product_prod @ nat @ nat ),X122: nat,X132: list @ vEBT_VEBT,X142: vEBT_VEBT] :
( Y
!= ( vEBT_Node @ X112 @ X122 @ X132 @ X142 ) )
=> ~ ! [X212: $o,X223: $o] :
( Y
!= ( vEBT_Leaf @ X212 @ X223 ) ) ) ).
% VEBT.exhaust
thf(fact_1556_VEBT__internal_Ovalid_H_Ocases,axiom,
! [X: product_prod @ vEBT_VEBT @ nat] :
( ! [Uu: $o,Uv: $o,D3: nat] :
( X
!= ( product_Pair @ vEBT_VEBT @ nat @ ( vEBT_Leaf @ Uu @ Uv ) @ D3 ) )
=> ~ ! [Mima: option @ ( product_prod @ nat @ nat ),Deg2: nat,TreeList2: list @ vEBT_VEBT,Summary2: vEBT_VEBT,Deg3: nat] :
( X
!= ( product_Pair @ vEBT_VEBT @ nat @ ( vEBT_Node @ Mima @ Deg2 @ TreeList2 @ Summary2 ) @ Deg3 ) ) ) ).
% VEBT_internal.valid'.cases
thf(fact_1557_less__eq__int__code_I1_J,axiom,
ord_less_eq @ int @ ( zero_zero @ int ) @ ( zero_zero @ int ) ).
% less_eq_int_code(1)
thf(fact_1558_VEBT__internal_Omembermima_Osimps_I1_J,axiom,
! [Uu2: $o,Uv2: $o,Uw2: nat] :
~ ( vEBT_VEBT_membermima @ ( vEBT_Leaf @ Uu2 @ Uv2 ) @ Uw2 ) ).
% VEBT_internal.membermima.simps(1)
thf(fact_1559_minus__int__code_I1_J,axiom,
! [K: int] :
( ( minus_minus @ int @ K @ ( zero_zero @ int ) )
= K ) ).
% minus_int_code(1)
thf(fact_1560_VEBT__internal_Ocnt_Osimps_I1_J,axiom,
! [A2: $o,B2: $o] :
( ( vEBT_VEBT_cnt @ ( vEBT_Leaf @ A2 @ B2 ) )
= ( one_one @ real ) ) ).
% VEBT_internal.cnt.simps(1)
thf(fact_1561_i0__lb,axiom,
! [N3: extended_enat] : ( ord_less_eq @ extended_enat @ ( zero_zero @ extended_enat ) @ N3 ) ).
% i0_lb
thf(fact_1562_ile0__eq,axiom,
! [N3: extended_enat] :
( ( ord_less_eq @ extended_enat @ N3 @ ( zero_zero @ extended_enat ) )
= ( N3
= ( zero_zero @ extended_enat ) ) ) ).
% ile0_eq
thf(fact_1563_VEBT__internal_Ocnt_H_Osimps_I1_J,axiom,
! [A2: $o,B2: $o] :
( ( vEBT_VEBT_cnt2 @ ( vEBT_Leaf @ A2 @ B2 ) )
= ( one_one @ nat ) ) ).
% VEBT_internal.cnt'.simps(1)
thf(fact_1564_ex__nat__less,axiom,
! [N3: nat,P: nat > $o] :
( ( ? [M5: nat] :
( ( ord_less_eq @ nat @ M5 @ N3 )
& ( P @ M5 ) ) )
= ( ? [X2: nat] :
( ( member @ nat @ X2 @ ( set_or1337092689740270186AtMost @ nat @ ( zero_zero @ nat ) @ N3 ) )
& ( P @ X2 ) ) ) ) ).
% ex_nat_less
thf(fact_1565_all__nat__less,axiom,
! [N3: nat,P: nat > $o] :
( ( ! [M5: nat] :
( ( ord_less_eq @ nat @ M5 @ N3 )
=> ( P @ M5 ) ) )
= ( ! [X2: nat] :
( ( member @ nat @ X2 @ ( set_or1337092689740270186AtMost @ nat @ ( zero_zero @ nat ) @ N3 ) )
=> ( P @ X2 ) ) ) ) ).
% all_nat_less
thf(fact_1566_vebt__delete_Osimps_I3_J,axiom,
! [A2: $o,B2: $o,N3: nat] :
( ( vEBT_vebt_delete @ ( vEBT_Leaf @ A2 @ B2 ) @ ( suc @ ( suc @ N3 ) ) )
= ( vEBT_Leaf @ A2 @ B2 ) ) ).
% vebt_delete.simps(3)
thf(fact_1567_vebt__delete_Osimps_I1_J,axiom,
! [A2: $o,B2: $o] :
( ( vEBT_vebt_delete @ ( vEBT_Leaf @ A2 @ B2 ) @ ( zero_zero @ nat ) )
= ( vEBT_Leaf @ $false @ B2 ) ) ).
% vebt_delete.simps(1)
thf(fact_1568_vebt__buildup_Osimps_I1_J,axiom,
( ( vEBT_vebt_buildup @ ( zero_zero @ nat ) )
= ( vEBT_Leaf @ $false @ $false ) ) ).
% vebt_buildup.simps(1)
thf(fact_1569_T_092_060_094sub_062i_092_060_094sub_062n_092_060_094sub_062s_092_060_094sub_062e_092_060_094sub_062r_092_060_094sub_062t_H_Osimps_I1_J,axiom,
! [A2: $o,B2: $o,X: nat] :
( ( vEBT_T_i_n_s_e_r_t2 @ ( vEBT_Leaf @ A2 @ B2 ) @ X )
= ( one_one @ nat ) ) ).
% T\<^sub>i\<^sub>n\<^sub>s\<^sub>e\<^sub>r\<^sub>t'.simps(1)
thf(fact_1570_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_1571_T_092_060_094sub_062m_092_060_094sub_062i_092_060_094sub_062n_092_060_094sub_062N_092_060_094sub_062u_092_060_094sub_062l_092_060_094sub_062l_Osimps_I2_J,axiom,
! [Uv2: $o] :
( ( vEBT_T_m_i_n_N_u_l_l @ ( vEBT_Leaf @ $true @ Uv2 ) )
= ( one_one @ nat ) ) ).
% T\<^sub>m\<^sub>i\<^sub>n\<^sub>N\<^sub>u\<^sub>l\<^sub>l.simps(2)
thf(fact_1572_T_092_060_094sub_062m_092_060_094sub_062i_092_060_094sub_062n_092_060_094sub_062N_092_060_094sub_062u_092_060_094sub_062l_092_060_094sub_062l_Osimps_I3_J,axiom,
! [Uu2: $o] :
( ( vEBT_T_m_i_n_N_u_l_l @ ( vEBT_Leaf @ Uu2 @ $true ) )
= ( one_one @ nat ) ) ).
% T\<^sub>m\<^sub>i\<^sub>n\<^sub>N\<^sub>u\<^sub>l\<^sub>l.simps(3)
thf(fact_1573_odd__nonzero,axiom,
! [Z: int] :
( ( plus_plus @ int @ ( plus_plus @ int @ ( one_one @ int ) @ Z ) @ Z )
!= ( zero_zero @ int ) ) ).
% odd_nonzero
thf(fact_1574_zero__le__imp__eq__int,axiom,
! [K: int] :
( ( ord_less_eq @ int @ ( zero_zero @ int ) @ K )
=> ? [N: nat] :
( K
= ( semiring_1_of_nat @ int @ N ) ) ) ).
% zero_le_imp_eq_int
thf(fact_1575_nonneg__int__cases,axiom,
! [K: int] :
( ( ord_less_eq @ int @ ( zero_zero @ int ) @ K )
=> ~ ! [N: nat] :
( K
!= ( semiring_1_of_nat @ int @ N ) ) ) ).
% nonneg_int_cases
thf(fact_1576_VEBT__internal_Ospace_H_Osimps_I1_J,axiom,
! [A2: $o,B2: $o] :
( ( vEBT_VEBT_space2 @ ( vEBT_Leaf @ A2 @ B2 ) )
= ( numeral_numeral @ nat @ ( bit0 @ ( bit0 @ one2 ) ) ) ) ).
% VEBT_internal.space'.simps(1)
thf(fact_1577_VEBT__internal_Onaive__member_Ocases,axiom,
! [X: product_prod @ vEBT_VEBT @ nat] :
( ! [A4: $o,B3: $o,X3: nat] :
( X
!= ( product_Pair @ vEBT_VEBT @ nat @ ( vEBT_Leaf @ A4 @ B3 ) @ X3 ) )
=> ( ! [Uu: option @ ( product_prod @ nat @ nat ),Uv: list @ vEBT_VEBT,Uw: vEBT_VEBT,Ux2: nat] :
( X
!= ( product_Pair @ vEBT_VEBT @ nat @ ( vEBT_Node @ Uu @ ( zero_zero @ nat ) @ Uv @ Uw ) @ Ux2 ) )
=> ~ ! [Uy2: option @ ( product_prod @ nat @ nat ),V3: nat,TreeList2: list @ vEBT_VEBT,S2: vEBT_VEBT,X3: nat] :
( X
!= ( product_Pair @ vEBT_VEBT @ nat @ ( vEBT_Node @ Uy2 @ ( suc @ V3 ) @ TreeList2 @ S2 ) @ X3 ) ) ) ) ).
% VEBT_internal.naive_member.cases
thf(fact_1578_VEBT__internal_Ospace_Osimps_I1_J,axiom,
! [A2: $o,B2: $o] :
( ( vEBT_VEBT_space @ ( vEBT_Leaf @ A2 @ B2 ) )
= ( numeral_numeral @ nat @ ( bit1 @ one2 ) ) ) ).
% VEBT_internal.space.simps(1)
thf(fact_1579_invar__vebt_Ointros_I1_J,axiom,
! [A2: $o,B2: $o] : ( vEBT_invar_vebt @ ( vEBT_Leaf @ A2 @ B2 ) @ ( suc @ ( zero_zero @ nat ) ) ) ).
% invar_vebt.intros(1)
thf(fact_1580_vebt__delete_Osimps_I2_J,axiom,
! [A2: $o,B2: $o] :
( ( vEBT_vebt_delete @ ( vEBT_Leaf @ A2 @ B2 ) @ ( suc @ ( zero_zero @ nat ) ) )
= ( vEBT_Leaf @ A2 @ $false ) ) ).
% vebt_delete.simps(2)
thf(fact_1581_T_092_060_094sub_062m_092_060_094sub_062i_092_060_094sub_062n_092_060_094sub_062N_092_060_094sub_062u_092_060_094sub_062l_092_060_094sub_062l_Ocases,axiom,
! [X: vEBT_VEBT] :
( ( X
!= ( vEBT_Leaf @ $false @ $false ) )
=> ( ! [Uv: $o] :
( X
!= ( vEBT_Leaf @ $true @ Uv ) )
=> ( ! [Uu: $o] :
( X
!= ( vEBT_Leaf @ Uu @ $true ) )
=> ( ! [Uw: nat,Ux2: list @ vEBT_VEBT,Uy2: vEBT_VEBT] :
( X
!= ( vEBT_Node @ ( none @ ( product_prod @ nat @ nat ) ) @ Uw @ Ux2 @ Uy2 ) )
=> ~ ! [Uz2: product_prod @ nat @ nat,Va2: nat,Vb2: list @ vEBT_VEBT,Vc2: vEBT_VEBT] :
( X
!= ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ Uz2 ) @ Va2 @ Vb2 @ Vc2 ) ) ) ) ) ) ).
% T\<^sub>m\<^sub>i\<^sub>n\<^sub>N\<^sub>u\<^sub>l\<^sub>l.cases
thf(fact_1582_vebt__member_Osimps_I1_J,axiom,
! [A2: $o,B2: $o,X: nat] :
( ( vEBT_vebt_member @ ( vEBT_Leaf @ A2 @ B2 ) @ X )
= ( ( ( X
= ( zero_zero @ nat ) )
=> A2 )
& ( ( X
!= ( zero_zero @ nat ) )
=> ( ( ( X
= ( one_one @ nat ) )
=> B2 )
& ( X
= ( one_one @ nat ) ) ) ) ) ) ).
% vebt_member.simps(1)
thf(fact_1583_vebt__buildup_Osimps_I2_J,axiom,
( ( vEBT_vebt_buildup @ ( suc @ ( zero_zero @ nat ) ) )
= ( vEBT_Leaf @ $false @ $false ) ) ).
% vebt_buildup.simps(2)
thf(fact_1584_VEBT__internal_OminNull_Oelims_I3_J,axiom,
! [X: vEBT_VEBT] :
( ~ ( vEBT_VEBT_minNull @ X )
=> ( ! [Uv: $o] :
( X
!= ( vEBT_Leaf @ $true @ Uv ) )
=> ( ! [Uu: $o] :
( X
!= ( vEBT_Leaf @ Uu @ $true ) )
=> ~ ! [Uz2: product_prod @ nat @ nat,Va2: nat,Vb2: list @ vEBT_VEBT,Vc2: vEBT_VEBT] :
( X
!= ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ Uz2 ) @ Va2 @ Vb2 @ Vc2 ) ) ) ) ) ).
% VEBT_internal.minNull.elims(3)
thf(fact_1585_vebt__succ_Osimps_I2_J,axiom,
! [Uv2: $o,Uw2: $o,N3: nat] :
( ( vEBT_vebt_succ @ ( vEBT_Leaf @ Uv2 @ Uw2 ) @ ( suc @ N3 ) )
= ( none @ nat ) ) ).
% vebt_succ.simps(2)
thf(fact_1586_vebt__insert_Osimps_I1_J,axiom,
! [X: nat,A2: $o,B2: $o] :
( ( ( X
= ( zero_zero @ nat ) )
=> ( ( vEBT_vebt_insert @ ( vEBT_Leaf @ A2 @ B2 ) @ X )
= ( vEBT_Leaf @ $true @ B2 ) ) )
& ( ( X
!= ( zero_zero @ nat ) )
=> ( ( ( X
= ( one_one @ nat ) )
=> ( ( vEBT_vebt_insert @ ( vEBT_Leaf @ A2 @ B2 ) @ X )
= ( vEBT_Leaf @ A2 @ $true ) ) )
& ( ( X
!= ( one_one @ nat ) )
=> ( ( vEBT_vebt_insert @ ( vEBT_Leaf @ A2 @ B2 ) @ X )
= ( vEBT_Leaf @ A2 @ B2 ) ) ) ) ) ) ).
% vebt_insert.simps(1)
thf(fact_1587_vebt__pred_Osimps_I1_J,axiom,
! [Uu2: $o,Uv2: $o] :
( ( vEBT_vebt_pred @ ( vEBT_Leaf @ Uu2 @ Uv2 ) @ ( zero_zero @ nat ) )
= ( none @ nat ) ) ).
% vebt_pred.simps(1)
thf(fact_1588_VEBT__internal_OminNull_Oelims_I2_J,axiom,
! [X: vEBT_VEBT] :
( ( vEBT_VEBT_minNull @ X )
=> ( ( X
!= ( vEBT_Leaf @ $false @ $false ) )
=> ~ ! [Uw: nat,Ux2: list @ vEBT_VEBT,Uy2: vEBT_VEBT] :
( X
!= ( vEBT_Node @ ( none @ ( product_prod @ nat @ nat ) ) @ Uw @ Ux2 @ Uy2 ) ) ) ) ).
% VEBT_internal.minNull.elims(2)
thf(fact_1589_T_092_060_094sub_062d_092_060_094sub_062e_092_060_094sub_062l_092_060_094sub_062e_092_060_094sub_062t_092_060_094sub_062e_Osimps_I3_J,axiom,
! [A2: $o,B2: $o,N3: nat] :
( ( vEBT_T_d_e_l_e_t_e @ ( vEBT_Leaf @ A2 @ B2 ) @ ( suc @ ( suc @ N3 ) ) )
= ( one_one @ nat ) ) ).
% T\<^sub>d\<^sub>e\<^sub>l\<^sub>e\<^sub>t\<^sub>e.simps(3)
thf(fact_1590_VEBT__internal_OT_092_060_094sub_062d_092_060_094sub_062e_092_060_094sub_062l_092_060_094sub_062e_092_060_094sub_062t_092_060_094sub_062e_H_Osimps_I3_J,axiom,
! [A2: $o,B2: $o,N3: nat] :
( ( vEBT_V1232361888498592333_e_t_e @ ( vEBT_Leaf @ A2 @ B2 ) @ ( suc @ ( suc @ N3 ) ) )
= ( one_one @ nat ) ) ).
% VEBT_internal.T\<^sub>d\<^sub>e\<^sub>l\<^sub>e\<^sub>t\<^sub>e'.simps(3)
thf(fact_1591_T_092_060_094sub_062d_092_060_094sub_062e_092_060_094sub_062l_092_060_094sub_062e_092_060_094sub_062t_092_060_094sub_062e_Osimps_I1_J,axiom,
! [A2: $o,B2: $o] :
( ( vEBT_T_d_e_l_e_t_e @ ( vEBT_Leaf @ A2 @ B2 ) @ ( zero_zero @ nat ) )
= ( one_one @ nat ) ) ).
% T\<^sub>d\<^sub>e\<^sub>l\<^sub>e\<^sub>t\<^sub>e.simps(1)
thf(fact_1592_VEBT__internal_OT_092_060_094sub_062d_092_060_094sub_062e_092_060_094sub_062l_092_060_094sub_062e_092_060_094sub_062t_092_060_094sub_062e_H_Osimps_I1_J,axiom,
! [A2: $o,B2: $o] :
( ( vEBT_V1232361888498592333_e_t_e @ ( vEBT_Leaf @ A2 @ B2 ) @ ( zero_zero @ nat ) )
= ( one_one @ nat ) ) ).
% VEBT_internal.T\<^sub>d\<^sub>e\<^sub>l\<^sub>e\<^sub>t\<^sub>e'.simps(1)
thf(fact_1593_VEBT__internal_Onaive__member_Osimps_I1_J,axiom,
! [A2: $o,B2: $o,X: nat] :
( ( vEBT_V5719532721284313246member @ ( vEBT_Leaf @ A2 @ B2 ) @ X )
= ( ( ( X
= ( zero_zero @ nat ) )
=> A2 )
& ( ( X
!= ( zero_zero @ nat ) )
=> ( ( ( X
= ( one_one @ nat ) )
=> B2 )
& ( X
= ( one_one @ nat ) ) ) ) ) ) ).
% VEBT_internal.naive_member.simps(1)
thf(fact_1594_T_092_060_094sub_062p_092_060_094sub_062r_092_060_094sub_062e_092_060_094sub_062d_Osimps_I1_J,axiom,
! [Uu2: $o,Uv2: $o] :
( ( vEBT_T_p_r_e_d @ ( vEBT_Leaf @ Uu2 @ Uv2 ) @ ( zero_zero @ nat ) )
= ( one_one @ nat ) ) ).
% T\<^sub>p\<^sub>r\<^sub>e\<^sub>d.simps(1)
thf(fact_1595_T_092_060_094sub_062s_092_060_094sub_062u_092_060_094sub_062c_092_060_094sub_062c_Osimps_I2_J,axiom,
! [Uv2: $o,Uw2: $o,N3: nat] :
( ( vEBT_T_s_u_c_c @ ( vEBT_Leaf @ Uv2 @ Uw2 ) @ ( suc @ N3 ) )
= ( one_one @ nat ) ) ).
% T\<^sub>s\<^sub>u\<^sub>c\<^sub>c.simps(2)
thf(fact_1596_realpow__pos__nth2,axiom,
! [A2: real,N3: nat] :
( ( ord_less @ real @ ( zero_zero @ real ) @ A2 )
=> ? [R3: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ R3 )
& ( ( power_power @ real @ R3 @ ( suc @ N3 ) )
= A2 ) ) ) ).
% realpow_pos_nth2
thf(fact_1597_real__arch__pow__inv,axiom,
! [Y: real,X: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ Y )
=> ( ( ord_less @ real @ X @ ( one_one @ real ) )
=> ? [N: nat] : ( ord_less @ real @ ( power_power @ real @ X @ N ) @ Y ) ) ) ).
% real_arch_pow_inv
thf(fact_1598_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_1599_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_1600_pos__zmult__eq__1__iff,axiom,
! [M: int,N3: int] :
( ( ord_less @ int @ ( zero_zero @ int ) @ M )
=> ( ( ( times_times @ int @ M @ N3 )
= ( one_one @ int ) )
= ( ( M
= ( one_one @ int ) )
& ( N3
= ( one_one @ int ) ) ) ) ) ).
% pos_zmult_eq_1_iff
thf(fact_1601_reals__Archimedean3,axiom,
! [X: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ X )
=> ! [Y4: real] :
? [N: nat] : ( ord_less @ real @ Y4 @ ( times_times @ real @ ( semiring_1_of_nat @ real @ N ) @ X ) ) ) ).
% reals_Archimedean3
thf(fact_1602_zdiv__le__dividend,axiom,
! [A2: int,B2: int] :
( ( ord_less_eq @ int @ ( zero_zero @ int ) @ A2 )
=> ( ( ord_less @ int @ ( zero_zero @ int ) @ B2 )
=> ( ord_less_eq @ int @ ( divide_divide @ int @ A2 @ B2 ) @ A2 ) ) ) ).
% zdiv_le_dividend
thf(fact_1603_zdiv__mono1,axiom,
! [A2: int,A5: int,B2: int] :
( ( ord_less_eq @ int @ A2 @ A5 )
=> ( ( ord_less @ int @ ( zero_zero @ int ) @ B2 )
=> ( ord_less_eq @ int @ ( divide_divide @ int @ A2 @ B2 ) @ ( divide_divide @ int @ A5 @ B2 ) ) ) ) ).
% zdiv_mono1
thf(fact_1604_zdiv__mono2,axiom,
! [A2: int,B4: int,B2: int] :
( ( ord_less_eq @ int @ ( zero_zero @ int ) @ A2 )
=> ( ( ord_less @ int @ ( zero_zero @ int ) @ B4 )
=> ( ( ord_less_eq @ int @ B4 @ B2 )
=> ( ord_less_eq @ int @ ( divide_divide @ int @ A2 @ B2 ) @ ( divide_divide @ int @ A2 @ B4 ) ) ) ) ) ).
% zdiv_mono2
thf(fact_1605_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_1606_zdiv__mono1__neg,axiom,
! [A2: int,A5: int,B2: int] :
( ( ord_less_eq @ int @ A2 @ A5 )
=> ( ( ord_less @ int @ B2 @ ( zero_zero @ int ) )
=> ( ord_less_eq @ int @ ( divide_divide @ int @ A5 @ B2 ) @ ( divide_divide @ int @ A2 @ B2 ) ) ) ) ).
% zdiv_mono1_neg
thf(fact_1607_zdiv__mono2__neg,axiom,
! [A2: int,B4: int,B2: int] :
( ( ord_less @ int @ A2 @ ( zero_zero @ int ) )
=> ( ( ord_less @ int @ ( zero_zero @ int ) @ B4 )
=> ( ( ord_less_eq @ int @ B4 @ B2 )
=> ( ord_less_eq @ int @ ( divide_divide @ int @ A2 @ B4 ) @ ( divide_divide @ int @ A2 @ B2 ) ) ) ) ) ).
% zdiv_mono2_neg
thf(fact_1608_div__int__pos__iff,axiom,
! [K: int,L2: int] :
( ( ord_less_eq @ int @ ( zero_zero @ int ) @ ( divide_divide @ int @ K @ L2 ) )
= ( ( K
= ( zero_zero @ int ) )
| ( L2
= ( zero_zero @ int ) )
| ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ K )
& ( ord_less_eq @ int @ ( zero_zero @ int ) @ L2 ) )
| ( ( ord_less @ int @ K @ ( zero_zero @ int ) )
& ( ord_less @ int @ L2 @ ( zero_zero @ int ) ) ) ) ) ).
% div_int_pos_iff
thf(fact_1609_div__positive__int,axiom,
! [L2: int,K: int] :
( ( ord_less_eq @ int @ L2 @ K )
=> ( ( ord_less @ int @ ( zero_zero @ int ) @ L2 )
=> ( ord_less @ int @ ( zero_zero @ int ) @ ( divide_divide @ int @ K @ L2 ) ) ) ) ).
% div_positive_int
thf(fact_1610_div__nonneg__neg__le0,axiom,
! [A2: int,B2: int] :
( ( ord_less_eq @ int @ ( zero_zero @ int ) @ A2 )
=> ( ( ord_less @ int @ B2 @ ( zero_zero @ int ) )
=> ( ord_less_eq @ int @ ( divide_divide @ int @ A2 @ B2 ) @ ( zero_zero @ int ) ) ) ) ).
% div_nonneg_neg_le0
thf(fact_1611_div__nonpos__pos__le0,axiom,
! [A2: int,B2: int] :
( ( ord_less_eq @ int @ A2 @ ( zero_zero @ int ) )
=> ( ( ord_less @ int @ ( zero_zero @ int ) @ B2 )
=> ( ord_less_eq @ int @ ( divide_divide @ int @ A2 @ B2 ) @ ( zero_zero @ int ) ) ) ) ).
% div_nonpos_pos_le0
thf(fact_1612_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_1613_neg__imp__zdiv__nonneg__iff,axiom,
! [B2: int,A2: int] :
( ( ord_less @ int @ B2 @ ( zero_zero @ int ) )
=> ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ ( divide_divide @ int @ A2 @ B2 ) )
= ( ord_less_eq @ int @ A2 @ ( zero_zero @ int ) ) ) ) ).
% neg_imp_zdiv_nonneg_iff
thf(fact_1614_pos__imp__zdiv__nonneg__iff,axiom,
! [B2: int,A2: int] :
( ( ord_less @ int @ ( zero_zero @ int ) @ B2 )
=> ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ ( divide_divide @ int @ A2 @ B2 ) )
= ( ord_less_eq @ int @ ( zero_zero @ int ) @ A2 ) ) ) ).
% pos_imp_zdiv_nonneg_iff
thf(fact_1615_nonneg1__imp__zdiv__pos__iff,axiom,
! [A2: int,B2: int] :
( ( ord_less_eq @ int @ ( zero_zero @ int ) @ A2 )
=> ( ( ord_less @ int @ ( zero_zero @ int ) @ ( divide_divide @ int @ A2 @ B2 ) )
= ( ( ord_less_eq @ int @ B2 @ A2 )
& ( ord_less @ int @ ( zero_zero @ int ) @ B2 ) ) ) ) ).
% nonneg1_imp_zdiv_pos_iff
thf(fact_1616_int__div__less__self,axiom,
! [X: int,K: int] :
( ( ord_less @ int @ ( zero_zero @ int ) @ X )
=> ( ( ord_less @ int @ ( one_one @ int ) @ K )
=> ( ord_less @ int @ ( divide_divide @ int @ X @ K ) @ X ) ) ) ).
% int_div_less_self
thf(fact_1617_zdiv__zmult2__eq,axiom,
! [C2: int,A2: int,B2: int] :
( ( ord_less_eq @ int @ ( zero_zero @ int ) @ C2 )
=> ( ( divide_divide @ int @ A2 @ ( times_times @ int @ B2 @ C2 ) )
= ( divide_divide @ int @ ( divide_divide @ int @ A2 @ B2 ) @ C2 ) ) ) ).
% zdiv_zmult2_eq
thf(fact_1618_zdiv__mult__self,axiom,
! [M: int,A2: int,N3: int] :
( ( M
!= ( zero_zero @ int ) )
=> ( ( divide_divide @ int @ ( plus_plus @ int @ A2 @ ( times_times @ int @ M @ N3 ) ) @ M )
= ( plus_plus @ int @ ( divide_divide @ int @ A2 @ M ) @ N3 ) ) ) ).
% zdiv_mult_self
thf(fact_1619_T_092_060_094sub_062m_092_060_094sub_062i_092_060_094sub_062n_092_060_094sub_062t_Ocases,axiom,
! [X: vEBT_VEBT] :
( ! [A4: $o,B3: $o] :
( X
!= ( vEBT_Leaf @ A4 @ B3 ) )
=> ( ! [Uu: nat,Uv: list @ vEBT_VEBT,Uw: vEBT_VEBT] :
( X
!= ( vEBT_Node @ ( none @ ( product_prod @ nat @ nat ) ) @ Uu @ Uv @ Uw ) )
=> ~ ! [Mi2: nat,Ma2: nat,Ux2: nat,Uy2: list @ vEBT_VEBT,Uz2: vEBT_VEBT] :
( X
!= ( 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_1620_T_092_060_094sub_062d_092_060_094sub_062e_092_060_094sub_062l_092_060_094sub_062e_092_060_094sub_062t_092_060_094sub_062e_Osimps_I2_J,axiom,
! [A2: $o,B2: $o] :
( ( vEBT_T_d_e_l_e_t_e @ ( vEBT_Leaf @ A2 @ B2 ) @ ( suc @ ( zero_zero @ nat ) ) )
= ( one_one @ nat ) ) ).
% T\<^sub>d\<^sub>e\<^sub>l\<^sub>e\<^sub>t\<^sub>e.simps(2)
thf(fact_1621_VEBT__internal_OT_092_060_094sub_062d_092_060_094sub_062e_092_060_094sub_062l_092_060_094sub_062e_092_060_094sub_062t_092_060_094sub_062e_H_Osimps_I2_J,axiom,
! [A2: $o,B2: $o] :
( ( vEBT_V1232361888498592333_e_t_e @ ( vEBT_Leaf @ A2 @ B2 ) @ ( suc @ ( zero_zero @ nat ) ) )
= ( one_one @ nat ) ) ).
% VEBT_internal.T\<^sub>d\<^sub>e\<^sub>l\<^sub>e\<^sub>t\<^sub>e'.simps(2)
thf(fact_1622_VEBT__internal_OminNull_Oelims_I1_J,axiom,
! [X: vEBT_VEBT,Y: $o] :
( ( ( vEBT_VEBT_minNull @ X )
= Y )
=> ( ( ( X
= ( vEBT_Leaf @ $false @ $false ) )
=> ~ Y )
=> ( ( ? [Uv: $o] :
( X
= ( vEBT_Leaf @ $true @ Uv ) )
=> Y )
=> ( ( ? [Uu: $o] :
( X
= ( vEBT_Leaf @ Uu @ $true ) )
=> Y )
=> ( ( ? [Uw: nat,Ux2: list @ vEBT_VEBT,Uy2: vEBT_VEBT] :
( X
= ( vEBT_Node @ ( none @ ( product_prod @ nat @ nat ) ) @ Uw @ Ux2 @ Uy2 ) )
=> ~ Y )
=> ~ ( ? [Uz2: product_prod @ nat @ nat,Va2: nat,Vb2: list @ vEBT_VEBT,Vc2: vEBT_VEBT] :
( X
= ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ Uz2 ) @ Va2 @ Vb2 @ Vc2 ) )
=> Y ) ) ) ) ) ) ).
% VEBT_internal.minNull.elims(1)
thf(fact_1623_T_092_060_094sub_062p_092_060_094sub_062r_092_060_094sub_062e_092_060_094sub_062d_Osimps_I3_J,axiom,
! [A2: $o,B2: $o,Va: nat] :
( ( vEBT_T_p_r_e_d @ ( vEBT_Leaf @ A2 @ B2 ) @ ( suc @ ( suc @ Va ) ) )
= ( plus_plus @ nat @ ( one_one @ nat ) @ ( if @ nat @ B2 @ ( one_one @ nat ) @ ( plus_plus @ nat @ ( one_one @ nat ) @ ( one_one @ nat ) ) ) ) ) ).
% T\<^sub>p\<^sub>r\<^sub>e\<^sub>d.simps(3)
thf(fact_1624_T_092_060_094sub_062i_092_060_094sub_062n_092_060_094sub_062s_092_060_094sub_062e_092_060_094sub_062r_092_060_094sub_062t_Osimps_I1_J,axiom,
! [A2: $o,B2: $o,X: nat] :
( ( vEBT_T_i_n_s_e_r_t @ ( vEBT_Leaf @ A2 @ B2 ) @ X )
= ( plus_plus @ nat @ ( one_one @ nat )
@ ( if @ nat
@ ( X
= ( zero_zero @ nat ) )
@ ( one_one @ nat )
@ ( plus_plus @ nat @ ( one_one @ nat ) @ ( one_one @ nat ) ) ) ) ) ).
% T\<^sub>i\<^sub>n\<^sub>s\<^sub>e\<^sub>r\<^sub>t.simps(1)
thf(fact_1625_T_092_060_094sub_062s_092_060_094sub_062u_092_060_094sub_062c_092_060_094sub_062c_Osimps_I1_J,axiom,
! [Uu2: $o,B2: $o] :
( ( vEBT_T_s_u_c_c @ ( vEBT_Leaf @ Uu2 @ B2 ) @ ( zero_zero @ nat ) )
= ( plus_plus @ nat @ ( one_one @ nat ) @ ( one_one @ nat ) ) ) ).
% T\<^sub>s\<^sub>u\<^sub>c\<^sub>c.simps(1)
thf(fact_1626_realpow__pos__nth__unique,axiom,
! [N3: nat,A2: real] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ N3 )
=> ( ( ord_less @ real @ ( zero_zero @ real ) @ A2 )
=> ? [X3: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ X3 )
& ( ( power_power @ real @ X3 @ N3 )
= A2 )
& ! [Y4: real] :
( ( ( ord_less @ real @ ( zero_zero @ real ) @ Y4 )
& ( ( power_power @ real @ Y4 @ N3 )
= A2 ) )
=> ( Y4 = X3 ) ) ) ) ) ).
% realpow_pos_nth_unique
thf(fact_1627_realpow__pos__nth,axiom,
! [N3: nat,A2: real] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ N3 )
=> ( ( ord_less @ real @ ( zero_zero @ real ) @ A2 )
=> ? [R3: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ R3 )
& ( ( power_power @ real @ R3 @ N3 )
= A2 ) ) ) ) ).
% realpow_pos_nth
thf(fact_1628_zero__less__imp__eq__int,axiom,
! [K: int] :
( ( ord_less @ int @ ( zero_zero @ int ) @ K )
=> ? [N: nat] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ N )
& ( K
= ( semiring_1_of_nat @ int @ N ) ) ) ) ).
% zero_less_imp_eq_int
thf(fact_1629_pos__int__cases,axiom,
! [K: int] :
( ( ord_less @ int @ ( zero_zero @ int ) @ K )
=> ~ ! [N: nat] :
( ( K
= ( semiring_1_of_nat @ int @ N ) )
=> ~ ( ord_less @ nat @ ( zero_zero @ nat ) @ N ) ) ) ).
% pos_int_cases
thf(fact_1630_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_1631_unique__quotient__lemma__neg,axiom,
! [B2: int,Q5: int,R4: int,Q2: int,R2: int] :
( ( ord_less_eq @ int @ ( plus_plus @ int @ ( times_times @ int @ B2 @ Q5 ) @ R4 ) @ ( plus_plus @ int @ ( times_times @ int @ B2 @ Q2 ) @ R2 ) )
=> ( ( ord_less_eq @ int @ R2 @ ( zero_zero @ int ) )
=> ( ( ord_less @ int @ B2 @ R2 )
=> ( ( ord_less @ int @ B2 @ R4 )
=> ( ord_less_eq @ int @ Q2 @ Q5 ) ) ) ) ) ).
% unique_quotient_lemma_neg
thf(fact_1632_unique__quotient__lemma,axiom,
! [B2: int,Q5: int,R4: int,Q2: int,R2: int] :
( ( ord_less_eq @ int @ ( plus_plus @ int @ ( times_times @ int @ B2 @ Q5 ) @ R4 ) @ ( plus_plus @ int @ ( times_times @ int @ B2 @ Q2 ) @ R2 ) )
=> ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ R4 )
=> ( ( ord_less @ int @ R4 @ B2 )
=> ( ( ord_less @ int @ R2 @ B2 )
=> ( ord_less_eq @ int @ Q5 @ Q2 ) ) ) ) ) ).
% unique_quotient_lemma
thf(fact_1633_zdiv__mono2__neg__lemma,axiom,
! [B2: int,Q2: int,R2: int,B4: int,Q5: int,R4: int] :
( ( ( plus_plus @ int @ ( times_times @ int @ B2 @ Q2 ) @ R2 )
= ( plus_plus @ int @ ( times_times @ int @ B4 @ Q5 ) @ R4 ) )
=> ( ( ord_less @ int @ ( plus_plus @ int @ ( times_times @ int @ B4 @ Q5 ) @ R4 ) @ ( zero_zero @ int ) )
=> ( ( ord_less @ int @ R2 @ B2 )
=> ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ R4 )
=> ( ( ord_less @ int @ ( zero_zero @ int ) @ B4 )
=> ( ( ord_less_eq @ int @ B4 @ B2 )
=> ( ord_less_eq @ int @ Q5 @ Q2 ) ) ) ) ) ) ) ).
% zdiv_mono2_neg_lemma
thf(fact_1634_zdiv__mono2__lemma,axiom,
! [B2: int,Q2: int,R2: int,B4: int,Q5: int,R4: int] :
( ( ( plus_plus @ int @ ( times_times @ int @ B2 @ Q2 ) @ R2 )
= ( plus_plus @ int @ ( times_times @ int @ B4 @ Q5 ) @ R4 ) )
=> ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ ( plus_plus @ int @ ( times_times @ int @ B4 @ Q5 ) @ R4 ) )
=> ( ( ord_less @ int @ R4 @ B4 )
=> ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ R2 )
=> ( ( ord_less @ int @ ( zero_zero @ int ) @ B4 )
=> ( ( ord_less_eq @ int @ B4 @ B2 )
=> ( ord_less_eq @ int @ Q2 @ Q5 ) ) ) ) ) ) ) ).
% zdiv_mono2_lemma
thf(fact_1635_q__pos__lemma,axiom,
! [B4: int,Q5: int,R4: int] :
( ( ord_less_eq @ int @ ( zero_zero @ int ) @ ( plus_plus @ int @ ( times_times @ int @ B4 @ Q5 ) @ R4 ) )
=> ( ( ord_less @ int @ R4 @ B4 )
=> ( ( ord_less @ int @ ( zero_zero @ int ) @ B4 )
=> ( ord_less_eq @ int @ ( zero_zero @ int ) @ Q5 ) ) ) ) ).
% q_pos_lemma
thf(fact_1636_log__base__change,axiom,
! [A2: real,B2: real,X: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ A2 )
=> ( ( A2
!= ( one_one @ real ) )
=> ( ( log @ B2 @ X )
= ( divide_divide @ real @ ( log @ A2 @ X ) @ ( log @ A2 @ B2 ) ) ) ) ) ).
% log_base_change
thf(fact_1637_vebt__mint_Osimps_I1_J,axiom,
! [A2: $o,B2: $o] :
( ( A2
=> ( ( vEBT_vebt_mint @ ( vEBT_Leaf @ A2 @ B2 ) )
= ( some @ nat @ ( zero_zero @ nat ) ) ) )
& ( ~ A2
=> ( ( B2
=> ( ( vEBT_vebt_mint @ ( vEBT_Leaf @ A2 @ B2 ) )
= ( some @ nat @ ( one_one @ nat ) ) ) )
& ( ~ B2
=> ( ( vEBT_vebt_mint @ ( vEBT_Leaf @ A2 @ B2 ) )
= ( none @ nat ) ) ) ) ) ) ).
% vebt_mint.simps(1)
thf(fact_1638_vebt__maxt_Osimps_I1_J,axiom,
! [B2: $o,A2: $o] :
( ( B2
=> ( ( vEBT_vebt_maxt @ ( vEBT_Leaf @ A2 @ B2 ) )
= ( some @ nat @ ( one_one @ nat ) ) ) )
& ( ~ B2
=> ( ( A2
=> ( ( vEBT_vebt_maxt @ ( vEBT_Leaf @ A2 @ B2 ) )
= ( some @ nat @ ( zero_zero @ nat ) ) ) )
& ( ~ A2
=> ( ( vEBT_vebt_maxt @ ( vEBT_Leaf @ A2 @ B2 ) )
= ( none @ nat ) ) ) ) ) ) ).
% vebt_maxt.simps(1)
thf(fact_1639_vebt__pred_Osimps_I2_J,axiom,
! [A2: $o,Uw2: $o] :
( ( A2
=> ( ( vEBT_vebt_pred @ ( vEBT_Leaf @ A2 @ Uw2 ) @ ( suc @ ( zero_zero @ nat ) ) )
= ( some @ nat @ ( zero_zero @ nat ) ) ) )
& ( ~ A2
=> ( ( vEBT_vebt_pred @ ( vEBT_Leaf @ A2 @ Uw2 ) @ ( suc @ ( zero_zero @ nat ) ) )
= ( none @ nat ) ) ) ) ).
% vebt_pred.simps(2)
thf(fact_1640_T_092_060_094sub_062p_092_060_094sub_062r_092_060_094sub_062e_092_060_094sub_062d_Osimps_I2_J,axiom,
! [A2: $o,Uw2: $o] :
( ( vEBT_T_p_r_e_d @ ( vEBT_Leaf @ A2 @ Uw2 ) @ ( suc @ ( zero_zero @ nat ) ) )
= ( plus_plus @ nat @ ( one_one @ nat ) @ ( one_one @ nat ) ) ) ).
% T\<^sub>p\<^sub>r\<^sub>e\<^sub>d.simps(2)
thf(fact_1641_vebt__succ_Osimps_I1_J,axiom,
! [B2: $o,Uu2: $o] :
( ( B2
=> ( ( vEBT_vebt_succ @ ( vEBT_Leaf @ Uu2 @ B2 ) @ ( zero_zero @ nat ) )
= ( some @ nat @ ( one_one @ nat ) ) ) )
& ( ~ B2
=> ( ( vEBT_vebt_succ @ ( vEBT_Leaf @ Uu2 @ B2 ) @ ( zero_zero @ nat ) )
= ( none @ nat ) ) ) ) ).
% vebt_succ.simps(1)
thf(fact_1642_T_092_060_094sub_062m_092_060_094sub_062i_092_060_094sub_062n_092_060_094sub_062N_092_060_094sub_062u_092_060_094sub_062l_092_060_094sub_062l_Oelims,axiom,
! [X: vEBT_VEBT,Y: nat] :
( ( ( vEBT_T_m_i_n_N_u_l_l @ X )
= Y )
=> ( ( ( X
= ( vEBT_Leaf @ $false @ $false ) )
=> ( Y
!= ( one_one @ nat ) ) )
=> ( ( ? [Uv: $o] :
( X
= ( vEBT_Leaf @ $true @ Uv ) )
=> ( Y
!= ( one_one @ nat ) ) )
=> ( ( ? [Uu: $o] :
( X
= ( vEBT_Leaf @ Uu @ $true ) )
=> ( Y
!= ( one_one @ nat ) ) )
=> ( ( ? [Uw: nat,Ux2: list @ vEBT_VEBT,Uy2: vEBT_VEBT] :
( X
= ( vEBT_Node @ ( none @ ( product_prod @ nat @ nat ) ) @ Uw @ Ux2 @ Uy2 ) )
=> ( Y
!= ( one_one @ nat ) ) )
=> ~ ( ? [Uz2: product_prod @ nat @ nat,Va2: nat,Vb2: list @ vEBT_VEBT,Vc2: vEBT_VEBT] :
( X
= ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ Uz2 ) @ Va2 @ Vb2 @ Vc2 ) )
=> ( Y
!= ( one_one @ nat ) ) ) ) ) ) ) ) ).
% T\<^sub>m\<^sub>i\<^sub>n\<^sub>N\<^sub>u\<^sub>l\<^sub>l.elims
thf(fact_1643_T_092_060_094sub_062m_092_060_094sub_062e_092_060_094sub_062m_092_060_094sub_062b_092_060_094sub_062e_092_060_094sub_062r_H_Osimps_I2_J,axiom,
! [Uu2: nat,Uv2: list @ vEBT_VEBT,Uw2: vEBT_VEBT,X: nat] :
( ( vEBT_T_m_e_m_b_e_r2 @ ( vEBT_Node @ ( none @ ( product_prod @ nat @ nat ) ) @ Uu2 @ Uv2 @ Uw2 ) @ X )
= ( one_one @ nat ) ) ).
% T\<^sub>m\<^sub>e\<^sub>m\<^sub>b\<^sub>e\<^sub>r'.simps(2)
thf(fact_1644_not__exp__less__eq__0__int,axiom,
! [N3: nat] :
~ ( ord_less_eq @ int @ ( power_power @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ N3 ) @ ( zero_zero @ int ) ) ).
% not_exp_less_eq_0_int
thf(fact_1645_real__archimedian__rdiv__eq__0,axiom,
! [X: real,C2: real] :
( ( ord_less_eq @ real @ ( zero_zero @ real ) @ X )
=> ( ( ord_less_eq @ real @ ( zero_zero @ real ) @ C2 )
=> ( ! [M4: nat] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ M4 )
=> ( ord_less_eq @ real @ ( times_times @ real @ ( semiring_1_of_nat @ real @ M4 ) @ X ) @ C2 ) )
=> ( X
= ( zero_zero @ real ) ) ) ) ) ).
% real_archimedian_rdiv_eq_0
thf(fact_1646_split__zdiv,axiom,
! [P: int > $o,N3: int,K: int] :
( ( P @ ( divide_divide @ int @ N3 @ K ) )
= ( ( ( K
= ( zero_zero @ int ) )
=> ( P @ ( zero_zero @ int ) ) )
& ( ( ord_less @ int @ ( zero_zero @ int ) @ K )
=> ! [I3: int,J3: int] :
( ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ J3 )
& ( ord_less @ int @ J3 @ K )
& ( N3
= ( plus_plus @ int @ ( times_times @ int @ K @ I3 ) @ J3 ) ) )
=> ( P @ I3 ) ) )
& ( ( ord_less @ int @ K @ ( zero_zero @ int ) )
=> ! [I3: int,J3: int] :
( ( ( ord_less @ int @ K @ J3 )
& ( ord_less_eq @ int @ J3 @ ( zero_zero @ int ) )
& ( N3
= ( plus_plus @ int @ ( times_times @ int @ K @ I3 ) @ J3 ) ) )
=> ( P @ I3 ) ) ) ) ) ).
% split_zdiv
thf(fact_1647_int__div__neg__eq,axiom,
! [A2: int,B2: int,Q2: int,R2: int] :
( ( A2
= ( plus_plus @ int @ ( times_times @ int @ B2 @ Q2 ) @ R2 ) )
=> ( ( ord_less_eq @ int @ R2 @ ( zero_zero @ int ) )
=> ( ( ord_less @ int @ B2 @ R2 )
=> ( ( divide_divide @ int @ A2 @ B2 )
= Q2 ) ) ) ) ).
% int_div_neg_eq
thf(fact_1648_int__div__pos__eq,axiom,
! [A2: int,B2: int,Q2: int,R2: int] :
( ( A2
= ( plus_plus @ int @ ( times_times @ int @ B2 @ Q2 ) @ R2 ) )
=> ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ R2 )
=> ( ( ord_less @ int @ R2 @ B2 )
=> ( ( divide_divide @ int @ A2 @ B2 )
= Q2 ) ) ) ) ).
% int_div_pos_eq
thf(fact_1649_real__of__nat__div2,axiom,
! [N3: nat,X: nat] : ( ord_less_eq @ real @ ( zero_zero @ real ) @ ( minus_minus @ real @ ( divide_divide @ real @ ( semiring_1_of_nat @ real @ N3 ) @ ( semiring_1_of_nat @ real @ X ) ) @ ( semiring_1_of_nat @ real @ ( divide_divide @ nat @ N3 @ X ) ) ) ) ).
% real_of_nat_div2
thf(fact_1650_log__mult,axiom,
! [A2: real,X: real,Y: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ A2 )
=> ( ( A2
!= ( one_one @ real ) )
=> ( ( ord_less @ real @ ( zero_zero @ real ) @ X )
=> ( ( ord_less @ real @ ( zero_zero @ real ) @ Y )
=> ( ( log @ A2 @ ( times_times @ real @ X @ Y ) )
= ( plus_plus @ real @ ( log @ A2 @ X ) @ ( log @ A2 @ Y ) ) ) ) ) ) ) ).
% log_mult
thf(fact_1651_log__divide,axiom,
! [A2: real,X: real,Y: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ A2 )
=> ( ( A2
!= ( one_one @ real ) )
=> ( ( ord_less @ real @ ( zero_zero @ real ) @ X )
=> ( ( ord_less @ real @ ( zero_zero @ real ) @ Y )
=> ( ( log @ A2 @ ( divide_divide @ real @ X @ Y ) )
= ( minus_minus @ real @ ( log @ A2 @ X ) @ ( log @ A2 @ Y ) ) ) ) ) ) ) ).
% log_divide
thf(fact_1652_log__base__pow,axiom,
! [A2: real,N3: nat,X: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ A2 )
=> ( ( log @ ( power_power @ real @ A2 @ N3 ) @ X )
= ( divide_divide @ real @ ( log @ A2 @ X ) @ ( semiring_1_of_nat @ real @ N3 ) ) ) ) ).
% log_base_pow
thf(fact_1653_log__nat__power,axiom,
! [X: real,B2: real,N3: nat] :
( ( ord_less @ real @ ( zero_zero @ real ) @ X )
=> ( ( log @ B2 @ ( power_power @ real @ X @ N3 ) )
= ( times_times @ real @ ( semiring_1_of_nat @ real @ N3 ) @ ( log @ B2 @ X ) ) ) ) ).
% log_nat_power
thf(fact_1654_real__of__int__div2,axiom,
! [N3: int,X: int] : ( ord_less_eq @ real @ ( zero_zero @ real ) @ ( minus_minus @ real @ ( divide_divide @ real @ ( ring_1_of_int @ real @ N3 ) @ ( ring_1_of_int @ real @ X ) ) @ ( ring_1_of_int @ real @ ( divide_divide @ int @ N3 @ X ) ) ) ) ).
% real_of_int_div2
thf(fact_1655_VEBT__internal_Omembermima_Ocases,axiom,
! [X: product_prod @ vEBT_VEBT @ nat] :
( ! [Uu: $o,Uv: $o,Uw: nat] :
( X
!= ( product_Pair @ vEBT_VEBT @ nat @ ( vEBT_Leaf @ Uu @ Uv ) @ Uw ) )
=> ( ! [Ux2: list @ vEBT_VEBT,Uy2: vEBT_VEBT,Uz2: nat] :
( X
!= ( 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,X3: nat] :
( X
!= ( product_Pair @ vEBT_VEBT @ nat @ ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi2 @ Ma2 ) ) @ ( zero_zero @ nat ) @ Va2 @ Vb2 ) @ X3 ) )
=> ( ! [Mi2: nat,Ma2: nat,V3: nat,TreeList2: list @ vEBT_VEBT,Vc2: vEBT_VEBT,X3: nat] :
( X
!= ( product_Pair @ vEBT_VEBT @ nat @ ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi2 @ Ma2 ) ) @ ( suc @ V3 ) @ TreeList2 @ Vc2 ) @ X3 ) )
=> ~ ! [V3: nat,TreeList2: list @ vEBT_VEBT,Vd2: vEBT_VEBT,X3: nat] :
( X
!= ( product_Pair @ vEBT_VEBT @ nat @ ( vEBT_Node @ ( none @ ( product_prod @ nat @ nat ) ) @ ( suc @ V3 ) @ TreeList2 @ Vd2 ) @ X3 ) ) ) ) ) ) ).
% VEBT_internal.membermima.cases
thf(fact_1656_T_092_060_094sub_062m_092_060_094sub_062e_092_060_094sub_062m_092_060_094sub_062b_092_060_094sub_062e_092_060_094sub_062r_H_Ocases,axiom,
! [X: product_prod @ vEBT_VEBT @ nat] :
( ! [A4: $o,B3: $o,X3: nat] :
( X
!= ( product_Pair @ vEBT_VEBT @ nat @ ( vEBT_Leaf @ A4 @ B3 ) @ X3 ) )
=> ( ! [Uu: nat,Uv: list @ vEBT_VEBT,Uw: vEBT_VEBT,X3: nat] :
( X
!= ( product_Pair @ vEBT_VEBT @ nat @ ( vEBT_Node @ ( none @ ( product_prod @ nat @ nat ) ) @ Uu @ Uv @ Uw ) @ X3 ) )
=> ( ! [V3: product_prod @ nat @ nat,Uy2: list @ vEBT_VEBT,Uz2: vEBT_VEBT,X3: nat] :
( X
!= ( product_Pair @ vEBT_VEBT @ nat @ ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ V3 ) @ ( zero_zero @ nat ) @ Uy2 @ Uz2 ) @ X3 ) )
=> ( ! [V3: product_prod @ nat @ nat,Vb2: list @ vEBT_VEBT,Vc2: vEBT_VEBT,X3: nat] :
( X
!= ( product_Pair @ vEBT_VEBT @ nat @ ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ V3 ) @ ( suc @ ( zero_zero @ nat ) ) @ Vb2 @ Vc2 ) @ X3 ) )
=> ~ ! [Mi2: nat,Ma2: nat,Va3: nat,TreeList2: list @ vEBT_VEBT,Summary2: vEBT_VEBT,X3: nat] :
( X
!= ( product_Pair @ vEBT_VEBT @ nat @ ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( suc @ Va3 ) ) @ TreeList2 @ Summary2 ) @ X3 ) ) ) ) ) ) ).
% T\<^sub>m\<^sub>e\<^sub>m\<^sub>b\<^sub>e\<^sub>r'.cases
thf(fact_1657_T_092_060_094sub_062i_092_060_094sub_062n_092_060_094sub_062s_092_060_094sub_062e_092_060_094sub_062r_092_060_094sub_062t_H_Ocases,axiom,
! [X: product_prod @ vEBT_VEBT @ nat] :
( ! [A4: $o,B3: $o,X3: nat] :
( X
!= ( product_Pair @ vEBT_VEBT @ nat @ ( vEBT_Leaf @ A4 @ B3 ) @ X3 ) )
=> ( ! [Info2: option @ ( product_prod @ nat @ nat ),Ts2: list @ vEBT_VEBT,S2: vEBT_VEBT,X3: nat] :
( X
!= ( product_Pair @ vEBT_VEBT @ nat @ ( vEBT_Node @ Info2 @ ( zero_zero @ nat ) @ Ts2 @ S2 ) @ X3 ) )
=> ( ! [Info2: option @ ( product_prod @ nat @ nat ),Ts2: list @ vEBT_VEBT,S2: vEBT_VEBT,X3: nat] :
( X
!= ( product_Pair @ vEBT_VEBT @ nat @ ( vEBT_Node @ Info2 @ ( suc @ ( zero_zero @ nat ) ) @ Ts2 @ S2 ) @ X3 ) )
=> ( ! [V3: nat,TreeList2: list @ vEBT_VEBT,Summary2: vEBT_VEBT,X3: nat] :
( X
!= ( product_Pair @ vEBT_VEBT @ nat @ ( vEBT_Node @ ( none @ ( product_prod @ nat @ nat ) ) @ ( suc @ ( suc @ V3 ) ) @ TreeList2 @ Summary2 ) @ X3 ) )
=> ~ ! [Mi2: nat,Ma2: nat,Va3: nat,TreeList2: list @ vEBT_VEBT,Summary2: vEBT_VEBT,X3: nat] :
( X
!= ( product_Pair @ vEBT_VEBT @ nat @ ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( suc @ Va3 ) ) @ TreeList2 @ Summary2 ) @ X3 ) ) ) ) ) ) ).
% T\<^sub>i\<^sub>n\<^sub>s\<^sub>e\<^sub>r\<^sub>t'.cases
thf(fact_1658_T_092_060_094sub_062s_092_060_094sub_062u_092_060_094sub_062c_092_060_094sub_062c_H_Ocases,axiom,
! [X: product_prod @ vEBT_VEBT @ nat] :
( ! [Uu: $o,B3: $o] :
( X
!= ( product_Pair @ vEBT_VEBT @ nat @ ( vEBT_Leaf @ Uu @ B3 ) @ ( zero_zero @ nat ) ) )
=> ( ! [Uv: $o,Uw: $o,N: nat] :
( X
!= ( product_Pair @ vEBT_VEBT @ nat @ ( vEBT_Leaf @ Uv @ Uw ) @ ( suc @ N ) ) )
=> ( ! [Ux2: nat,Uy2: list @ vEBT_VEBT,Uz2: vEBT_VEBT,Va2: nat] :
( X
!= ( 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] :
( X
!= ( product_Pair @ vEBT_VEBT @ nat @ ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ V3 ) @ ( zero_zero @ nat ) @ Vc2 @ Vd2 ) @ Ve2 ) )
=> ( ! [V3: product_prod @ nat @ nat,Vg2: list @ vEBT_VEBT,Vh2: vEBT_VEBT,Vi2: nat] :
( X
!= ( product_Pair @ vEBT_VEBT @ nat @ ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ V3 ) @ ( suc @ ( zero_zero @ nat ) ) @ Vg2 @ Vh2 ) @ Vi2 ) )
=> ~ ! [Mi2: nat,Ma2: nat,Va3: nat,TreeList2: list @ vEBT_VEBT,Summary2: vEBT_VEBT,X3: nat] :
( X
!= ( product_Pair @ vEBT_VEBT @ nat @ ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( suc @ Va3 ) ) @ TreeList2 @ Summary2 ) @ X3 ) ) ) ) ) ) ) ).
% T\<^sub>s\<^sub>u\<^sub>c\<^sub>c'.cases
thf(fact_1659_T_092_060_094sub_062p_092_060_094sub_062r_092_060_094sub_062e_092_060_094sub_062d_H_Ocases,axiom,
! [X: product_prod @ vEBT_VEBT @ nat] :
( ! [Uu: $o,Uv: $o] :
( X
!= ( product_Pair @ vEBT_VEBT @ nat @ ( vEBT_Leaf @ Uu @ Uv ) @ ( zero_zero @ nat ) ) )
=> ( ! [A4: $o,Uw: $o] :
( X
!= ( product_Pair @ vEBT_VEBT @ nat @ ( vEBT_Leaf @ A4 @ Uw ) @ ( suc @ ( zero_zero @ nat ) ) ) )
=> ( ! [A4: $o,B3: $o,Va3: nat] :
( X
!= ( product_Pair @ vEBT_VEBT @ nat @ ( vEBT_Leaf @ A4 @ B3 ) @ ( suc @ ( suc @ Va3 ) ) ) )
=> ( ! [Uy2: nat,Uz2: list @ vEBT_VEBT,Va2: vEBT_VEBT,Vb2: nat] :
( X
!= ( 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,Vf2: nat] :
( X
!= ( product_Pair @ vEBT_VEBT @ nat @ ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ V3 ) @ ( zero_zero @ nat ) @ Vd2 @ Ve2 ) @ Vf2 ) )
=> ( ! [V3: product_prod @ nat @ nat,Vh2: list @ vEBT_VEBT,Vi2: vEBT_VEBT,Vj2: nat] :
( X
!= ( product_Pair @ vEBT_VEBT @ nat @ ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ V3 ) @ ( suc @ ( zero_zero @ nat ) ) @ Vh2 @ Vi2 ) @ Vj2 ) )
=> ~ ! [Mi2: nat,Ma2: nat,Va3: nat,TreeList2: list @ vEBT_VEBT,Summary2: vEBT_VEBT,X3: nat] :
( X
!= ( product_Pair @ vEBT_VEBT @ nat @ ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( suc @ Va3 ) ) @ TreeList2 @ Summary2 ) @ X3 ) ) ) ) ) ) ) ) ).
% T\<^sub>p\<^sub>r\<^sub>e\<^sub>d'.cases
thf(fact_1660_VEBT__internal_OT_092_060_094sub_062d_092_060_094sub_062e_092_060_094sub_062l_092_060_094sub_062e_092_060_094sub_062t_092_060_094sub_062e_H_Ocases,axiom,
! [X: product_prod @ vEBT_VEBT @ nat] :
( ! [A4: $o,B3: $o] :
( X
!= ( product_Pair @ vEBT_VEBT @ nat @ ( vEBT_Leaf @ A4 @ B3 ) @ ( zero_zero @ nat ) ) )
=> ( ! [A4: $o,B3: $o] :
( X
!= ( product_Pair @ vEBT_VEBT @ nat @ ( vEBT_Leaf @ A4 @ B3 ) @ ( suc @ ( zero_zero @ nat ) ) ) )
=> ( ! [A4: $o,B3: $o,N: nat] :
( X
!= ( product_Pair @ vEBT_VEBT @ nat @ ( vEBT_Leaf @ A4 @ B3 ) @ ( suc @ ( suc @ N ) ) ) )
=> ( ! [Deg2: nat,TreeList2: list @ vEBT_VEBT,Summary2: vEBT_VEBT,Uu: nat] :
( X
!= ( product_Pair @ vEBT_VEBT @ nat @ ( vEBT_Node @ ( none @ ( product_prod @ nat @ nat ) ) @ Deg2 @ TreeList2 @ Summary2 ) @ Uu ) )
=> ( ! [Mi2: nat,Ma2: nat,TreeList2: list @ vEBT_VEBT,Summary2: vEBT_VEBT,X3: nat] :
( X
!= ( product_Pair @ vEBT_VEBT @ nat @ ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi2 @ Ma2 ) ) @ ( zero_zero @ nat ) @ TreeList2 @ Summary2 ) @ X3 ) )
=> ( ! [Mi2: nat,Ma2: nat,TreeList2: list @ vEBT_VEBT,Summary2: vEBT_VEBT,X3: nat] :
( X
!= ( product_Pair @ vEBT_VEBT @ nat @ ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( zero_zero @ nat ) ) @ TreeList2 @ Summary2 ) @ X3 ) )
=> ~ ! [Mi2: nat,Ma2: nat,Va3: nat,TreeList2: list @ vEBT_VEBT,Summary2: vEBT_VEBT,X3: nat] :
( X
!= ( product_Pair @ vEBT_VEBT @ nat @ ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( suc @ Va3 ) ) @ TreeList2 @ Summary2 ) @ X3 ) ) ) ) ) ) ) ) ).
% VEBT_internal.T\<^sub>d\<^sub>e\<^sub>l\<^sub>e\<^sub>t\<^sub>e'.cases
thf(fact_1661_vebt__pred_Osimps_I3_J,axiom,
! [B2: $o,A2: $o,Va: nat] :
( ( B2
=> ( ( vEBT_vebt_pred @ ( vEBT_Leaf @ A2 @ B2 ) @ ( suc @ ( suc @ Va ) ) )
= ( some @ nat @ ( one_one @ nat ) ) ) )
& ( ~ B2
=> ( ( A2
=> ( ( vEBT_vebt_pred @ ( vEBT_Leaf @ A2 @ B2 ) @ ( suc @ ( suc @ Va ) ) )
= ( some @ nat @ ( zero_zero @ nat ) ) ) )
& ( ~ A2
=> ( ( vEBT_vebt_pred @ ( vEBT_Leaf @ A2 @ B2 ) @ ( suc @ ( suc @ Va ) ) )
= ( none @ nat ) ) ) ) ) ) ).
% vebt_pred.simps(3)
thf(fact_1662_axxdiv2,axiom,
! [X: int] :
( ( ( divide_divide @ int @ ( plus_plus @ int @ ( plus_plus @ int @ ( one_one @ int ) @ X ) @ X ) @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) )
= X )
& ( ( divide_divide @ int @ ( plus_plus @ int @ ( plus_plus @ int @ ( zero_zero @ int ) @ X ) @ X ) @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) )
= X ) ) ).
% axxdiv2
thf(fact_1663_VEBT__internal_OT_092_060_094sub_062b_092_060_094sub_062u_092_060_094sub_062i_092_060_094sub_062l_092_060_094sub_062d_092_060_094sub_062u_092_060_094sub_062p_Ocases,axiom,
! [X: nat] :
( ( X
!= ( zero_zero @ nat ) )
=> ( ( X
!= ( suc @ ( zero_zero @ nat ) ) )
=> ~ ! [Va3: nat] :
( X
!= ( suc @ ( suc @ Va3 ) ) ) ) ) ).
% VEBT_internal.T\<^sub>b\<^sub>u\<^sub>i\<^sub>l\<^sub>d\<^sub>u\<^sub>p.cases
thf(fact_1664_VEBT__internal_Oreplicatei_Ocases,axiom,
! [A: $tType,X: product_prod @ nat @ ( heap_Time_Heap @ A )] :
( ! [X3: heap_Time_Heap @ A] :
( X
!= ( product_Pair @ nat @ ( heap_Time_Heap @ A ) @ ( zero_zero @ nat ) @ X3 ) )
=> ~ ! [N: nat,X3: heap_Time_Heap @ A] :
( X
!= ( product_Pair @ nat @ ( heap_Time_Heap @ A ) @ ( suc @ N ) @ X3 ) ) ) ).
% VEBT_internal.replicatei.cases
thf(fact_1665_div__pos__geq,axiom,
! [L2: int,K: int] :
( ( ord_less @ int @ ( zero_zero @ int ) @ L2 )
=> ( ( ord_less_eq @ int @ L2 @ K )
=> ( ( divide_divide @ int @ K @ L2 )
= ( plus_plus @ int @ ( divide_divide @ int @ ( minus_minus @ int @ K @ L2 ) @ L2 ) @ ( one_one @ int ) ) ) ) ) ).
% div_pos_geq
thf(fact_1666_T_092_060_094sub_062m_092_060_094sub_062e_092_060_094sub_062m_092_060_094sub_062b_092_060_094sub_062e_092_060_094sub_062r_Osimps_I1_J,axiom,
! [A2: $o,B2: $o,X: nat] :
( ( vEBT_T_m_e_m_b_e_r @ ( vEBT_Leaf @ A2 @ B2 ) @ X )
= ( plus_plus @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) )
@ ( if @ nat
@ ( X
= ( zero_zero @ nat ) )
@ ( one_one @ nat )
@ ( plus_plus @ nat @ ( one_one @ nat ) @ ( one_one @ nat ) ) ) ) ) ).
% T\<^sub>m\<^sub>e\<^sub>m\<^sub>b\<^sub>e\<^sub>r.simps(1)
thf(fact_1667_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,X: nat] :
( ( vEBT_T_m_e_m_b_e_r2 @ ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ V ) @ ( zero_zero @ nat ) @ Uy @ Uz ) @ X )
= ( one_one @ nat ) ) ).
% T\<^sub>m\<^sub>e\<^sub>m\<^sub>b\<^sub>e\<^sub>r'.simps(3)
thf(fact_1668_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_1669_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,X: nat] :
( ( vEBT_T_m_e_m_b_e_r2 @ ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ V ) @ ( suc @ ( zero_zero @ nat ) ) @ Vb @ Vc ) @ X )
= ( one_one @ nat ) ) ).
% T\<^sub>m\<^sub>e\<^sub>m\<^sub>b\<^sub>e\<^sub>r'.simps(4)
thf(fact_1670_neg__zdiv__mult__2,axiom,
! [A2: int,B2: int] :
( ( ord_less_eq @ int @ A2 @ ( zero_zero @ int ) )
=> ( ( divide_divide @ int @ ( plus_plus @ int @ ( one_one @ int ) @ ( times_times @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ B2 ) ) @ ( times_times @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ A2 ) )
= ( divide_divide @ int @ ( plus_plus @ int @ B2 @ ( one_one @ int ) ) @ A2 ) ) ) ).
% neg_zdiv_mult_2
thf(fact_1671_pos__zdiv__mult__2,axiom,
! [A2: int,B2: int] :
( ( ord_less_eq @ int @ ( zero_zero @ int ) @ A2 )
=> ( ( divide_divide @ int @ ( plus_plus @ int @ ( one_one @ int ) @ ( times_times @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ B2 ) ) @ ( times_times @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ A2 ) )
= ( divide_divide @ int @ B2 @ A2 ) ) ) ).
% pos_zdiv_mult_2
thf(fact_1672_vebt__mint_Oelims,axiom,
! [X: vEBT_VEBT,Y: option @ nat] :
( ( ( vEBT_vebt_mint @ X )
= Y )
=> ( ! [A4: $o,B3: $o] :
( ( X
= ( vEBT_Leaf @ A4 @ B3 ) )
=> ~ ( ( A4
=> ( Y
= ( some @ nat @ ( zero_zero @ nat ) ) ) )
& ( ~ A4
=> ( ( B3
=> ( Y
= ( some @ nat @ ( one_one @ nat ) ) ) )
& ( ~ B3
=> ( Y
= ( none @ nat ) ) ) ) ) ) )
=> ( ( ? [Uu: nat,Uv: list @ vEBT_VEBT,Uw: vEBT_VEBT] :
( X
= ( vEBT_Node @ ( none @ ( product_prod @ nat @ nat ) ) @ Uu @ Uv @ Uw ) )
=> ( Y
!= ( none @ nat ) ) )
=> ~ ! [Mi2: nat] :
( ? [Ma2: nat,Ux2: nat,Uy2: list @ vEBT_VEBT,Uz2: vEBT_VEBT] :
( X
= ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi2 @ Ma2 ) ) @ Ux2 @ Uy2 @ Uz2 ) )
=> ( Y
!= ( some @ nat @ Mi2 ) ) ) ) ) ) ).
% vebt_mint.elims
thf(fact_1673_vebt__maxt_Oelims,axiom,
! [X: vEBT_VEBT,Y: option @ nat] :
( ( ( vEBT_vebt_maxt @ X )
= Y )
=> ( ! [A4: $o,B3: $o] :
( ( X
= ( vEBT_Leaf @ A4 @ B3 ) )
=> ~ ( ( B3
=> ( Y
= ( some @ nat @ ( one_one @ nat ) ) ) )
& ( ~ B3
=> ( ( A4
=> ( Y
= ( some @ nat @ ( zero_zero @ nat ) ) ) )
& ( ~ A4
=> ( Y
= ( none @ nat ) ) ) ) ) ) )
=> ( ( ? [Uu: nat,Uv: list @ vEBT_VEBT,Uw: vEBT_VEBT] :
( X
= ( vEBT_Node @ ( none @ ( product_prod @ nat @ nat ) ) @ Uu @ Uv @ Uw ) )
=> ( Y
!= ( none @ nat ) ) )
=> ~ ! [Mi2: nat,Ma2: nat] :
( ? [Ux2: nat,Uy2: list @ vEBT_VEBT,Uz2: vEBT_VEBT] :
( X
= ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi2 @ Ma2 ) ) @ Ux2 @ Uy2 @ Uz2 ) )
=> ( Y
!= ( some @ nat @ Ma2 ) ) ) ) ) ) ).
% vebt_maxt.elims
thf(fact_1674_member__bound__height_H,axiom,
! [T2: vEBT_VEBT,N3: nat,X: nat] :
( ( vEBT_invar_vebt @ T2 @ N3 )
=> ( ord_less_eq @ nat @ ( vEBT_T_m_e_m_b_e_r2 @ T2 @ X ) @ ( plus_plus @ nat @ ( one_one @ nat ) @ ( vEBT_VEBT_height @ T2 ) ) ) ) ).
% member_bound_height'
thf(fact_1675_T_092_060_094sub_062m_092_060_094sub_062e_092_060_094sub_062m_092_060_094sub_062b_092_060_094sub_062e_092_060_094sub_062r_H_Oelims,axiom,
! [X: vEBT_VEBT,Xa: nat,Y: nat] :
( ( ( vEBT_T_m_e_m_b_e_r2 @ X @ Xa )
= Y )
=> ( ( ? [A4: $o,B3: $o] :
( X
= ( vEBT_Leaf @ A4 @ B3 ) )
=> ( Y
!= ( one_one @ nat ) ) )
=> ( ( ? [Uu: nat,Uv: list @ vEBT_VEBT,Uw: vEBT_VEBT] :
( X
= ( vEBT_Node @ ( none @ ( product_prod @ nat @ nat ) ) @ Uu @ Uv @ Uw ) )
=> ( Y
!= ( one_one @ nat ) ) )
=> ( ( ? [V3: product_prod @ nat @ nat,Uy2: list @ vEBT_VEBT,Uz2: vEBT_VEBT] :
( X
= ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ V3 ) @ ( zero_zero @ nat ) @ Uy2 @ Uz2 ) )
=> ( Y
!= ( one_one @ nat ) ) )
=> ( ( ? [V3: product_prod @ nat @ nat,Vb2: list @ vEBT_VEBT,Vc2: vEBT_VEBT] :
( X
= ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ V3 ) @ ( suc @ ( zero_zero @ nat ) ) @ Vb2 @ Vc2 ) )
=> ( Y
!= ( one_one @ nat ) ) )
=> ~ ! [Mi2: nat,Ma2: nat,Va3: nat,TreeList2: list @ vEBT_VEBT] :
( ? [Summary2: vEBT_VEBT] :
( X
= ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( suc @ Va3 ) ) @ TreeList2 @ Summary2 ) )
=> ( Y
!= ( plus_plus @ nat @ ( one_one @ nat )
@ ( if @ nat @ ( Xa = Mi2 ) @ ( zero_zero @ nat )
@ ( if @ nat @ ( Xa = Ma2 ) @ ( zero_zero @ nat )
@ ( if @ nat @ ( ord_less @ nat @ Xa @ Mi2 ) @ ( zero_zero @ nat )
@ ( if @ nat @ ( ord_less @ nat @ Ma2 @ Xa ) @ ( zero_zero @ nat )
@ ( if @ nat
@ ( ( ord_less @ nat @ Mi2 @ Xa )
& ( ord_less @ nat @ Xa @ Ma2 ) )
@ ( if @ nat @ ( ord_less @ nat @ ( vEBT_VEBT_high @ Xa @ ( divide_divide @ nat @ ( suc @ ( suc @ 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_1676_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_1677_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_1678_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_1679_VEBT__internal_Onaive__member_Oelims_I1_J,axiom,
! [X: vEBT_VEBT,Xa: nat,Y: $o] :
( ( ( vEBT_V5719532721284313246member @ X @ Xa )
= Y )
=> ( ! [A4: $o,B3: $o] :
( ( X
= ( vEBT_Leaf @ A4 @ B3 ) )
=> ( Y
= ( ~ ( ( ( Xa
= ( zero_zero @ nat ) )
=> A4 )
& ( ( Xa
!= ( zero_zero @ nat ) )
=> ( ( ( Xa
= ( one_one @ nat ) )
=> B3 )
& ( Xa
= ( one_one @ nat ) ) ) ) ) ) ) )
=> ( ( ? [Uu: option @ ( product_prod @ nat @ nat ),Uv: list @ vEBT_VEBT,Uw: vEBT_VEBT] :
( X
= ( vEBT_Node @ Uu @ ( zero_zero @ nat ) @ Uv @ Uw ) )
=> Y )
=> ~ ! [Uy2: option @ ( product_prod @ nat @ nat ),V3: nat,TreeList2: list @ vEBT_VEBT] :
( ? [S2: vEBT_VEBT] :
( X
= ( vEBT_Node @ Uy2 @ ( suc @ V3 ) @ TreeList2 @ S2 ) )
=> ( Y
= ( ~ ( ( ( 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_1680_VEBT__internal_Onaive__member_Oelims_I2_J,axiom,
! [X: vEBT_VEBT,Xa: nat] :
( ( vEBT_V5719532721284313246member @ X @ Xa )
=> ( ! [A4: $o,B3: $o] :
( ( X
= ( vEBT_Leaf @ A4 @ B3 ) )
=> ~ ( ( ( Xa
= ( zero_zero @ nat ) )
=> A4 )
& ( ( Xa
!= ( zero_zero @ nat ) )
=> ( ( ( Xa
= ( one_one @ nat ) )
=> B3 )
& ( Xa
= ( one_one @ nat ) ) ) ) ) )
=> ~ ! [Uy2: option @ ( product_prod @ nat @ nat ),V3: nat,TreeList2: list @ vEBT_VEBT] :
( ? [S2: vEBT_VEBT] :
( X
= ( vEBT_Node @ Uy2 @ ( suc @ V3 ) @ TreeList2 @ S2 ) )
=> ~ ( ( ( 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_1681_VEBT__internal_Onaive__member_Oelims_I3_J,axiom,
! [X: vEBT_VEBT,Xa: nat] :
( ~ ( vEBT_V5719532721284313246member @ X @ Xa )
=> ( ! [A4: $o,B3: $o] :
( ( X
= ( vEBT_Leaf @ A4 @ B3 ) )
=> ( ( ( Xa
= ( zero_zero @ nat ) )
=> A4 )
& ( ( Xa
!= ( zero_zero @ nat ) )
=> ( ( ( Xa
= ( one_one @ nat ) )
=> B3 )
& ( Xa
= ( one_one @ nat ) ) ) ) ) )
=> ( ! [Uu: option @ ( product_prod @ nat @ nat ),Uv: list @ vEBT_VEBT,Uw: vEBT_VEBT] :
( X
!= ( vEBT_Node @ Uu @ ( zero_zero @ nat ) @ Uv @ Uw ) )
=> ~ ! [Uy2: option @ ( product_prod @ nat @ nat ),V3: nat,TreeList2: list @ vEBT_VEBT] :
( ? [S2: vEBT_VEBT] :
( X
= ( vEBT_Node @ Uy2 @ ( suc @ V3 ) @ TreeList2 @ S2 ) )
=> ( ( ( 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_1682_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_1683_vebt__member_Oelims_I2_J,axiom,
! [X: vEBT_VEBT,Xa: nat] :
( ( vEBT_vebt_member @ X @ Xa )
=> ( ! [A4: $o,B3: $o] :
( ( X
= ( vEBT_Leaf @ A4 @ B3 ) )
=> ~ ( ( ( Xa
= ( zero_zero @ nat ) )
=> A4 )
& ( ( Xa
!= ( zero_zero @ nat ) )
=> ( ( ( Xa
= ( one_one @ nat ) )
=> B3 )
& ( Xa
= ( one_one @ nat ) ) ) ) ) )
=> ~ ! [Mi2: nat,Ma2: nat,Va3: nat,TreeList2: list @ vEBT_VEBT] :
( ? [Summary2: vEBT_VEBT] :
( X
= ( 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_1684_VEBT__internal_Omembermima_Oelims_I1_J,axiom,
! [X: vEBT_VEBT,Xa: nat,Y: $o] :
( ( ( vEBT_VEBT_membermima @ X @ Xa )
= Y )
=> ( ( ? [Uu: $o,Uv: $o] :
( X
= ( vEBT_Leaf @ Uu @ Uv ) )
=> Y )
=> ( ( ? [Ux2: list @ vEBT_VEBT,Uy2: vEBT_VEBT] :
( X
= ( vEBT_Node @ ( none @ ( product_prod @ nat @ nat ) ) @ ( zero_zero @ nat ) @ Ux2 @ Uy2 ) )
=> Y )
=> ( ! [Mi2: nat,Ma2: nat] :
( ? [Va2: list @ vEBT_VEBT,Vb2: vEBT_VEBT] :
( X
= ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi2 @ Ma2 ) ) @ ( zero_zero @ nat ) @ Va2 @ Vb2 ) )
=> ( Y
= ( ~ ( ( Xa = Mi2 )
| ( Xa = Ma2 ) ) ) ) )
=> ( ! [Mi2: nat,Ma2: nat,V3: nat,TreeList2: list @ vEBT_VEBT] :
( ? [Vc2: vEBT_VEBT] :
( X
= ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi2 @ Ma2 ) ) @ ( suc @ V3 ) @ TreeList2 @ Vc2 ) )
=> ( Y
= ( ~ ( ( 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] :
( X
= ( vEBT_Node @ ( none @ ( product_prod @ nat @ nat ) ) @ ( suc @ V3 ) @ TreeList2 @ Vd2 ) )
=> ( Y
= ( ~ ( ( ( 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_1685_VEBT__internal_Omembermima_Oelims_I3_J,axiom,
! [X: vEBT_VEBT,Xa: nat] :
( ~ ( vEBT_VEBT_membermima @ X @ Xa )
=> ( ! [Uu: $o,Uv: $o] :
( X
!= ( vEBT_Leaf @ Uu @ Uv ) )
=> ( ! [Ux2: list @ vEBT_VEBT,Uy2: vEBT_VEBT] :
( X
!= ( vEBT_Node @ ( none @ ( product_prod @ nat @ nat ) ) @ ( zero_zero @ nat ) @ Ux2 @ Uy2 ) )
=> ( ! [Mi2: nat,Ma2: nat] :
( ? [Va2: list @ vEBT_VEBT,Vb2: vEBT_VEBT] :
( X
= ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi2 @ Ma2 ) ) @ ( zero_zero @ nat ) @ Va2 @ Vb2 ) )
=> ( ( Xa = Mi2 )
| ( Xa = Ma2 ) ) )
=> ( ! [Mi2: nat,Ma2: nat,V3: nat,TreeList2: list @ vEBT_VEBT] :
( ? [Vc2: vEBT_VEBT] :
( X
= ( 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] :
( X
= ( 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_1686_vebt__member_Oelims_I1_J,axiom,
! [X: vEBT_VEBT,Xa: nat,Y: $o] :
( ( ( vEBT_vebt_member @ X @ Xa )
= Y )
=> ( ! [A4: $o,B3: $o] :
( ( X
= ( vEBT_Leaf @ A4 @ B3 ) )
=> ( Y
= ( ~ ( ( ( Xa
= ( zero_zero @ nat ) )
=> A4 )
& ( ( Xa
!= ( zero_zero @ nat ) )
=> ( ( ( Xa
= ( one_one @ nat ) )
=> B3 )
& ( Xa
= ( one_one @ nat ) ) ) ) ) ) ) )
=> ( ( ? [Uu: nat,Uv: list @ vEBT_VEBT,Uw: vEBT_VEBT] :
( X
= ( vEBT_Node @ ( none @ ( product_prod @ nat @ nat ) ) @ Uu @ Uv @ Uw ) )
=> Y )
=> ( ( ? [V3: product_prod @ nat @ nat,Uy2: list @ vEBT_VEBT,Uz2: vEBT_VEBT] :
( X
= ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ V3 ) @ ( zero_zero @ nat ) @ Uy2 @ Uz2 ) )
=> Y )
=> ( ( ? [V3: product_prod @ nat @ nat,Vb2: list @ vEBT_VEBT,Vc2: vEBT_VEBT] :
( X
= ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ V3 ) @ ( suc @ ( zero_zero @ nat ) ) @ Vb2 @ Vc2 ) )
=> Y )
=> ~ ! [Mi2: nat,Ma2: nat,Va3: nat,TreeList2: list @ vEBT_VEBT] :
( ? [Summary2: vEBT_VEBT] :
( X
= ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( suc @ Va3 ) ) @ TreeList2 @ Summary2 ) )
=> ( Y
= ( ~ ( ( Xa != Mi2 )
=> ( ( Xa != Ma2 )
=> ( ~ ( ord_less @ nat @ Xa @ Mi2 )
& ( ~ ( ord_less @ nat @ Xa @ Mi2 )
=> ( ~ ( ord_less @ nat @ Ma2 @ Xa )
& ( ~ ( ord_less @ nat @ Ma2 @ Xa )
=> ( ( ( ord_less @ nat @ ( vEBT_VEBT_high @ Xa @ ( divide_divide @ nat @ ( suc @ ( suc @ 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_1687_vebt__member_Oelims_I3_J,axiom,
! [X: vEBT_VEBT,Xa: nat] :
( ~ ( vEBT_vebt_member @ X @ Xa )
=> ( ! [A4: $o,B3: $o] :
( ( X
= ( vEBT_Leaf @ A4 @ B3 ) )
=> ( ( ( Xa
= ( zero_zero @ nat ) )
=> A4 )
& ( ( Xa
!= ( zero_zero @ nat ) )
=> ( ( ( Xa
= ( one_one @ nat ) )
=> B3 )
& ( Xa
= ( one_one @ nat ) ) ) ) ) )
=> ( ! [Uu: nat,Uv: list @ vEBT_VEBT,Uw: vEBT_VEBT] :
( X
!= ( vEBT_Node @ ( none @ ( product_prod @ nat @ nat ) ) @ Uu @ Uv @ Uw ) )
=> ( ! [V3: product_prod @ nat @ nat,Uy2: list @ vEBT_VEBT,Uz2: vEBT_VEBT] :
( X
!= ( 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] :
( X
!= ( 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] :
( X
= ( 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_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_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_1689_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_1690_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_1691_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_1692_T_092_060_094sub_062m_092_060_094sub_062e_092_060_094sub_062m_092_060_094sub_062b_092_060_094sub_062e_092_060_094sub_062r_Oelims,axiom,
! [X: vEBT_VEBT,Xa: nat,Y: nat] :
( ( ( vEBT_T_m_e_m_b_e_r @ X @ Xa )
= Y )
=> ( ( ? [A4: $o,B3: $o] :
( X
= ( vEBT_Leaf @ A4 @ B3 ) )
=> ( Y
!= ( plus_plus @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) )
@ ( if @ nat
@ ( Xa
= ( zero_zero @ nat ) )
@ ( one_one @ nat )
@ ( plus_plus @ nat @ ( one_one @ nat ) @ ( one_one @ nat ) ) ) ) ) )
=> ( ( ? [Uu: nat,Uv: list @ vEBT_VEBT,Uw: vEBT_VEBT] :
( X
= ( vEBT_Node @ ( none @ ( product_prod @ nat @ nat ) ) @ Uu @ Uv @ Uw ) )
=> ( Y
!= ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) )
=> ( ( ? [V3: product_prod @ nat @ nat,Uy2: list @ vEBT_VEBT,Uz2: vEBT_VEBT] :
( X
= ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ V3 ) @ ( zero_zero @ nat ) @ Uy2 @ Uz2 ) )
=> ( Y
!= ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) )
=> ( ( ? [V3: product_prod @ nat @ nat,Vb2: list @ vEBT_VEBT,Vc2: vEBT_VEBT] :
( X
= ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ V3 ) @ ( suc @ ( zero_zero @ nat ) ) @ Vb2 @ Vc2 ) )
=> ( Y
!= ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) )
=> ~ ! [Mi2: nat,Ma2: nat,Va3: nat,TreeList2: list @ vEBT_VEBT] :
( ? [Summary2: vEBT_VEBT] :
( X
= ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( suc @ Va3 ) ) @ TreeList2 @ Summary2 ) )
=> ( Y
!= ( plus_plus @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( if @ nat @ ( Xa = Mi2 ) @ ( one_one @ nat ) @ ( plus_plus @ nat @ ( one_one @ nat ) @ ( if @ nat @ ( Xa = Ma2 ) @ ( one_one @ nat ) @ ( plus_plus @ nat @ ( one_one @ nat ) @ ( if @ nat @ ( ord_less @ nat @ Xa @ Mi2 ) @ ( one_one @ nat ) @ ( plus_plus @ nat @ ( one_one @ nat ) @ ( if @ nat @ ( ord_less @ nat @ Ma2 @ Xa ) @ ( one_one @ nat ) @ ( plus_plus @ nat @ ( numeral_numeral @ nat @ ( bit1 @ ( bit0 @ ( bit0 @ one2 ) ) ) ) @ ( if @ nat @ ( ord_less @ nat @ ( vEBT_VEBT_high @ Xa @ ( divide_divide @ nat @ ( suc @ ( suc @ 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_1693_T_092_060_094sub_062i_092_060_094sub_062n_092_060_094sub_062s_092_060_094sub_062e_092_060_094sub_062r_092_060_094sub_062t_H_Oelims,axiom,
! [X: vEBT_VEBT,Xa: nat,Y: nat] :
( ( ( vEBT_T_i_n_s_e_r_t2 @ X @ Xa )
= Y )
=> ( ( ? [A4: $o,B3: $o] :
( X
= ( vEBT_Leaf @ A4 @ B3 ) )
=> ( Y
!= ( one_one @ nat ) ) )
=> ( ( ? [Info2: option @ ( product_prod @ nat @ nat ),Ts2: list @ vEBT_VEBT,S2: vEBT_VEBT] :
( X
= ( vEBT_Node @ Info2 @ ( zero_zero @ nat ) @ Ts2 @ S2 ) )
=> ( Y
!= ( one_one @ nat ) ) )
=> ( ( ? [Info2: option @ ( product_prod @ nat @ nat ),Ts2: list @ vEBT_VEBT,S2: vEBT_VEBT] :
( X
= ( vEBT_Node @ Info2 @ ( suc @ ( zero_zero @ nat ) ) @ Ts2 @ S2 ) )
=> ( Y
!= ( one_one @ nat ) ) )
=> ( ( ? [V3: nat,TreeList2: list @ vEBT_VEBT,Summary2: vEBT_VEBT] :
( X
= ( vEBT_Node @ ( none @ ( product_prod @ nat @ nat ) ) @ ( suc @ ( suc @ V3 ) ) @ TreeList2 @ Summary2 ) )
=> ( Y
!= ( one_one @ nat ) ) )
=> ~ ! [Mi2: nat,Ma2: nat,Va3: nat,TreeList2: list @ vEBT_VEBT,Summary2: vEBT_VEBT] :
( ( X
= ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( suc @ Va3 ) ) @ TreeList2 @ Summary2 ) )
=> ( Y
!= ( if @ nat
@ ( ( ord_less @ nat @ ( vEBT_VEBT_high @ ( if @ nat @ ( ord_less @ nat @ Xa @ Mi2 ) @ Mi2 @ Xa ) @ ( divide_divide @ nat @ ( suc @ ( suc @ 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_1694_invar__vebt_Osimps,axiom,
( vEBT_invar_vebt
= ( ^ [A1: vEBT_VEBT,A22: nat] :
( ( ? [A6: $o,B5: $o] :
( A1
= ( vEBT_Leaf @ A6 @ B5 ) )
& ( A22
= ( suc @ ( zero_zero @ nat ) ) ) )
| ? [TreeList3: list @ vEBT_VEBT,N2: nat,Summary3: vEBT_VEBT] :
( ( A1
= ( vEBT_Node @ ( none @ ( product_prod @ nat @ nat ) ) @ A22 @ TreeList3 @ Summary3 ) )
& ! [X2: vEBT_VEBT] :
( ( member @ vEBT_VEBT @ X2 @ ( set2 @ vEBT_VEBT @ TreeList3 ) )
=> ( vEBT_invar_vebt @ X2 @ N2 ) )
& ( vEBT_invar_vebt @ Summary3 @ N2 )
& ( ( size_size @ ( list @ vEBT_VEBT ) @ TreeList3 )
= ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N2 ) )
& ( A22
= ( plus_plus @ nat @ N2 @ N2 ) )
& ~ ? [X6: nat] : ( vEBT_V8194947554948674370ptions @ Summary3 @ X6 )
& ! [X2: vEBT_VEBT] :
( ( member @ vEBT_VEBT @ X2 @ ( set2 @ vEBT_VEBT @ TreeList3 ) )
=> ~ ? [X6: nat] : ( vEBT_V8194947554948674370ptions @ X2 @ X6 ) ) )
| ? [TreeList3: list @ vEBT_VEBT,N2: nat,Summary3: vEBT_VEBT] :
( ( A1
= ( vEBT_Node @ ( none @ ( product_prod @ nat @ nat ) ) @ A22 @ TreeList3 @ Summary3 ) )
& ! [X2: vEBT_VEBT] :
( ( member @ vEBT_VEBT @ X2 @ ( set2 @ vEBT_VEBT @ TreeList3 ) )
=> ( vEBT_invar_vebt @ X2 @ N2 ) )
& ( vEBT_invar_vebt @ Summary3 @ ( suc @ N2 ) )
& ( ( size_size @ ( list @ vEBT_VEBT ) @ TreeList3 )
= ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( suc @ N2 ) ) )
& ( A22
= ( plus_plus @ nat @ N2 @ ( suc @ N2 ) ) )
& ~ ? [X6: nat] : ( vEBT_V8194947554948674370ptions @ Summary3 @ X6 )
& ! [X2: vEBT_VEBT] :
( ( member @ vEBT_VEBT @ X2 @ ( set2 @ vEBT_VEBT @ TreeList3 ) )
=> ~ ? [X6: nat] : ( vEBT_V8194947554948674370ptions @ X2 @ X6 ) ) )
| ? [TreeList3: list @ vEBT_VEBT,N2: 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 ) )
& ! [X2: vEBT_VEBT] :
( ( member @ vEBT_VEBT @ X2 @ ( set2 @ vEBT_VEBT @ TreeList3 ) )
=> ( vEBT_invar_vebt @ X2 @ N2 ) )
& ( vEBT_invar_vebt @ Summary3 @ N2 )
& ( ( size_size @ ( list @ vEBT_VEBT ) @ TreeList3 )
= ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N2 ) )
& ( A22
= ( plus_plus @ nat @ N2 @ N2 ) )
& ! [I3: nat] :
( ( ord_less @ nat @ I3 @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N2 ) )
=> ( ( ? [X6: nat] : ( vEBT_V8194947554948674370ptions @ ( nth @ vEBT_VEBT @ TreeList3 @ I3 ) @ X6 ) )
= ( vEBT_V8194947554948674370ptions @ Summary3 @ I3 ) ) )
& ( ( Mi3 = Ma3 )
=> ! [X2: vEBT_VEBT] :
( ( member @ vEBT_VEBT @ X2 @ ( set2 @ vEBT_VEBT @ TreeList3 ) )
=> ~ ? [X6: nat] : ( vEBT_V8194947554948674370ptions @ X2 @ X6 ) ) )
& ( ord_less_eq @ nat @ Mi3 @ Ma3 )
& ( ord_less @ nat @ Ma3 @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ A22 ) )
& ( ( Mi3 != Ma3 )
=> ! [I3: nat] :
( ( ord_less @ nat @ I3 @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N2 ) )
=> ( ( ( ( vEBT_VEBT_high @ Ma3 @ N2 )
= I3 )
=> ( vEBT_V8194947554948674370ptions @ ( nth @ vEBT_VEBT @ TreeList3 @ I3 ) @ ( vEBT_VEBT_low @ Ma3 @ N2 ) ) )
& ! [X2: nat] :
( ( ( ( vEBT_VEBT_high @ X2 @ N2 )
= I3 )
& ( vEBT_V8194947554948674370ptions @ ( nth @ vEBT_VEBT @ TreeList3 @ I3 ) @ ( vEBT_VEBT_low @ X2 @ N2 ) ) )
=> ( ( ord_less @ nat @ Mi3 @ X2 )
& ( ord_less_eq @ nat @ X2 @ Ma3 ) ) ) ) ) ) )
| ? [TreeList3: list @ vEBT_VEBT,N2: 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 ) )
& ! [X2: vEBT_VEBT] :
( ( member @ vEBT_VEBT @ X2 @ ( set2 @ vEBT_VEBT @ TreeList3 ) )
=> ( vEBT_invar_vebt @ X2 @ N2 ) )
& ( vEBT_invar_vebt @ Summary3 @ ( suc @ N2 ) )
& ( ( size_size @ ( list @ vEBT_VEBT ) @ TreeList3 )
= ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( suc @ N2 ) ) )
& ( A22
= ( plus_plus @ nat @ N2 @ ( suc @ N2 ) ) )
& ! [I3: nat] :
( ( ord_less @ nat @ I3 @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( suc @ N2 ) ) )
=> ( ( ? [X6: nat] : ( vEBT_V8194947554948674370ptions @ ( nth @ vEBT_VEBT @ TreeList3 @ I3 ) @ X6 ) )
= ( vEBT_V8194947554948674370ptions @ Summary3 @ I3 ) ) )
& ( ( Mi3 = Ma3 )
=> ! [X2: vEBT_VEBT] :
( ( member @ vEBT_VEBT @ X2 @ ( set2 @ vEBT_VEBT @ TreeList3 ) )
=> ~ ? [X6: nat] : ( vEBT_V8194947554948674370ptions @ X2 @ X6 ) ) )
& ( ord_less_eq @ nat @ Mi3 @ Ma3 )
& ( ord_less @ nat @ Ma3 @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ A22 ) )
& ( ( Mi3 != Ma3 )
=> ! [I3: nat] :
( ( ord_less @ nat @ I3 @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( suc @ N2 ) ) )
=> ( ( ( ( vEBT_VEBT_high @ Ma3 @ N2 )
= I3 )
=> ( vEBT_V8194947554948674370ptions @ ( nth @ vEBT_VEBT @ TreeList3 @ I3 ) @ ( vEBT_VEBT_low @ Ma3 @ N2 ) ) )
& ! [X2: nat] :
( ( ( ( vEBT_VEBT_high @ X2 @ N2 )
= I3 )
& ( vEBT_V8194947554948674370ptions @ ( nth @ vEBT_VEBT @ TreeList3 @ I3 ) @ ( vEBT_VEBT_low @ X2 @ N2 ) ) )
=> ( ( ord_less @ nat @ Mi3 @ X2 )
& ( ord_less_eq @ nat @ X2 @ Ma3 ) ) ) ) ) ) ) ) ) ) ).
% invar_vebt.simps
thf(fact_1695_invar__vebt_Ocases,axiom,
! [A12: vEBT_VEBT,A23: nat] :
( ( vEBT_invar_vebt @ A12 @ A23 )
=> ( ( ? [A4: $o,B3: $o] :
( A12
= ( vEBT_Leaf @ A4 @ B3 ) )
=> ( A23
!= ( suc @ ( zero_zero @ nat ) ) ) )
=> ( ! [TreeList2: list @ vEBT_VEBT,N: nat,Summary2: vEBT_VEBT,M4: nat,Deg2: nat] :
( ( A12
= ( vEBT_Node @ ( none @ ( product_prod @ nat @ nat ) ) @ Deg2 @ TreeList2 @ Summary2 ) )
=> ( ( A23 = Deg2 )
=> ( ! [X5: vEBT_VEBT] :
( ( member @ vEBT_VEBT @ X5 @ ( set2 @ vEBT_VEBT @ TreeList2 ) )
=> ( vEBT_invar_vebt @ X5 @ N ) )
=> ( ( vEBT_invar_vebt @ Summary2 @ M4 )
=> ( ( ( size_size @ ( list @ vEBT_VEBT ) @ TreeList2 )
= ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ M4 ) )
=> ( ( M4 = N )
=> ( ( Deg2
= ( plus_plus @ nat @ N @ M4 ) )
=> ( ~ ? [X_1: nat] : ( vEBT_V8194947554948674370ptions @ Summary2 @ X_1 )
=> ~ ! [X5: vEBT_VEBT] :
( ( member @ vEBT_VEBT @ X5 @ ( set2 @ vEBT_VEBT @ TreeList2 ) )
=> ~ ? [X_1: nat] : ( vEBT_V8194947554948674370ptions @ X5 @ X_1 ) ) ) ) ) ) ) ) ) )
=> ( ! [TreeList2: list @ vEBT_VEBT,N: nat,Summary2: vEBT_VEBT,M4: nat,Deg2: nat] :
( ( A12
= ( vEBT_Node @ ( none @ ( product_prod @ nat @ nat ) ) @ Deg2 @ TreeList2 @ Summary2 ) )
=> ( ( A23 = Deg2 )
=> ( ! [X5: vEBT_VEBT] :
( ( member @ vEBT_VEBT @ X5 @ ( set2 @ vEBT_VEBT @ TreeList2 ) )
=> ( vEBT_invar_vebt @ X5 @ N ) )
=> ( ( vEBT_invar_vebt @ Summary2 @ M4 )
=> ( ( ( size_size @ ( list @ vEBT_VEBT ) @ TreeList2 )
= ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ M4 ) )
=> ( ( M4
= ( suc @ N ) )
=> ( ( Deg2
= ( plus_plus @ nat @ N @ M4 ) )
=> ( ~ ? [X_1: nat] : ( vEBT_V8194947554948674370ptions @ Summary2 @ X_1 )
=> ~ ! [X5: vEBT_VEBT] :
( ( member @ vEBT_VEBT @ X5 @ ( set2 @ vEBT_VEBT @ TreeList2 ) )
=> ~ ? [X_1: nat] : ( vEBT_V8194947554948674370ptions @ X5 @ X_1 ) ) ) ) ) ) ) ) ) )
=> ( ! [TreeList2: list @ vEBT_VEBT,N: nat,Summary2: vEBT_VEBT,M4: nat,Deg2: nat,Mi2: nat,Ma2: nat] :
( ( A12
= ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi2 @ Ma2 ) ) @ Deg2 @ TreeList2 @ Summary2 ) )
=> ( ( A23 = Deg2 )
=> ( ! [X5: vEBT_VEBT] :
( ( member @ vEBT_VEBT @ X5 @ ( set2 @ vEBT_VEBT @ TreeList2 ) )
=> ( vEBT_invar_vebt @ X5 @ N ) )
=> ( ( vEBT_invar_vebt @ Summary2 @ M4 )
=> ( ( ( size_size @ ( list @ vEBT_VEBT ) @ TreeList2 )
= ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ M4 ) )
=> ( ( M4 = N )
=> ( ( Deg2
= ( plus_plus @ nat @ N @ M4 ) )
=> ( ! [I4: nat] :
( ( ord_less @ nat @ I4 @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ M4 ) )
=> ( ( ? [X6: nat] : ( vEBT_V8194947554948674370ptions @ ( nth @ vEBT_VEBT @ TreeList2 @ I4 ) @ X6 ) )
= ( vEBT_V8194947554948674370ptions @ Summary2 @ I4 ) ) )
=> ( ( ( Mi2 = Ma2 )
=> ! [X5: vEBT_VEBT] :
( ( member @ vEBT_VEBT @ X5 @ ( set2 @ vEBT_VEBT @ TreeList2 ) )
=> ~ ? [X_1: nat] : ( vEBT_V8194947554948674370ptions @ X5 @ X_1 ) ) )
=> ( ( ord_less_eq @ nat @ Mi2 @ Ma2 )
=> ( ( ord_less @ nat @ Ma2 @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ Deg2 ) )
=> ~ ( ( Mi2 != Ma2 )
=> ! [I4: nat] :
( ( ord_less @ nat @ I4 @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ M4 ) )
=> ( ( ( ( vEBT_VEBT_high @ Ma2 @ N )
= I4 )
=> ( vEBT_V8194947554948674370ptions @ ( nth @ vEBT_VEBT @ TreeList2 @ I4 ) @ ( vEBT_VEBT_low @ Ma2 @ N ) ) )
& ! [X5: nat] :
( ( ( ( vEBT_VEBT_high @ X5 @ N )
= I4 )
& ( vEBT_V8194947554948674370ptions @ ( nth @ vEBT_VEBT @ TreeList2 @ I4 ) @ ( vEBT_VEBT_low @ X5 @ N ) ) )
=> ( ( ord_less @ nat @ Mi2 @ X5 )
& ( ord_less_eq @ nat @ X5 @ Ma2 ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) )
=> ~ ! [TreeList2: list @ vEBT_VEBT,N: nat,Summary2: vEBT_VEBT,M4: nat,Deg2: nat,Mi2: nat,Ma2: nat] :
( ( A12
= ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi2 @ Ma2 ) ) @ Deg2 @ TreeList2 @ Summary2 ) )
=> ( ( A23 = Deg2 )
=> ( ! [X5: vEBT_VEBT] :
( ( member @ vEBT_VEBT @ X5 @ ( set2 @ vEBT_VEBT @ TreeList2 ) )
=> ( vEBT_invar_vebt @ X5 @ N ) )
=> ( ( vEBT_invar_vebt @ Summary2 @ M4 )
=> ( ( ( size_size @ ( list @ vEBT_VEBT ) @ TreeList2 )
= ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ M4 ) )
=> ( ( M4
= ( suc @ N ) )
=> ( ( Deg2
= ( plus_plus @ nat @ N @ M4 ) )
=> ( ! [I4: nat] :
( ( ord_less @ nat @ I4 @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ M4 ) )
=> ( ( ? [X6: nat] : ( vEBT_V8194947554948674370ptions @ ( nth @ vEBT_VEBT @ TreeList2 @ I4 ) @ X6 ) )
= ( vEBT_V8194947554948674370ptions @ Summary2 @ I4 ) ) )
=> ( ( ( Mi2 = Ma2 )
=> ! [X5: vEBT_VEBT] :
( ( member @ vEBT_VEBT @ X5 @ ( set2 @ vEBT_VEBT @ TreeList2 ) )
=> ~ ? [X_1: nat] : ( vEBT_V8194947554948674370ptions @ X5 @ X_1 ) ) )
=> ( ( ord_less_eq @ nat @ Mi2 @ Ma2 )
=> ( ( ord_less @ nat @ Ma2 @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ Deg2 ) )
=> ~ ( ( Mi2 != Ma2 )
=> ! [I4: nat] :
( ( ord_less @ nat @ I4 @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ M4 ) )
=> ( ( ( ( vEBT_VEBT_high @ Ma2 @ N )
= I4 )
=> ( vEBT_V8194947554948674370ptions @ ( nth @ vEBT_VEBT @ TreeList2 @ I4 ) @ ( vEBT_VEBT_low @ Ma2 @ N ) ) )
& ! [X5: nat] :
( ( ( ( vEBT_VEBT_high @ X5 @ N )
= I4 )
& ( vEBT_V8194947554948674370ptions @ ( nth @ vEBT_VEBT @ TreeList2 @ I4 ) @ ( vEBT_VEBT_low @ X5 @ N ) ) )
=> ( ( ord_less @ nat @ Mi2 @ X5 )
& ( ord_less_eq @ nat @ X5 @ Ma2 ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).
% invar_vebt.cases
thf(fact_1696_vebt__insert_Oelims,axiom,
! [X: vEBT_VEBT,Xa: nat,Y: vEBT_VEBT] :
( ( ( vEBT_vebt_insert @ X @ Xa )
= Y )
=> ( ! [A4: $o,B3: $o] :
( ( X
= ( vEBT_Leaf @ A4 @ B3 ) )
=> ~ ( ( ( Xa
= ( zero_zero @ nat ) )
=> ( Y
= ( vEBT_Leaf @ $true @ B3 ) ) )
& ( ( Xa
!= ( zero_zero @ nat ) )
=> ( ( ( Xa
= ( one_one @ nat ) )
=> ( Y
= ( vEBT_Leaf @ A4 @ $true ) ) )
& ( ( Xa
!= ( one_one @ nat ) )
=> ( Y
= ( vEBT_Leaf @ A4 @ B3 ) ) ) ) ) ) )
=> ( ! [Info2: option @ ( product_prod @ nat @ nat ),Ts2: list @ vEBT_VEBT,S2: vEBT_VEBT] :
( ( X
= ( vEBT_Node @ Info2 @ ( zero_zero @ nat ) @ Ts2 @ S2 ) )
=> ( Y
!= ( vEBT_Node @ Info2 @ ( zero_zero @ nat ) @ Ts2 @ S2 ) ) )
=> ( ! [Info2: option @ ( product_prod @ nat @ nat ),Ts2: list @ vEBT_VEBT,S2: vEBT_VEBT] :
( ( X
= ( vEBT_Node @ Info2 @ ( suc @ ( zero_zero @ nat ) ) @ Ts2 @ S2 ) )
=> ( Y
!= ( vEBT_Node @ Info2 @ ( suc @ ( zero_zero @ nat ) ) @ Ts2 @ S2 ) ) )
=> ( ! [V3: nat,TreeList2: list @ vEBT_VEBT,Summary2: vEBT_VEBT] :
( ( X
= ( vEBT_Node @ ( none @ ( product_prod @ nat @ nat ) ) @ ( suc @ ( suc @ V3 ) ) @ TreeList2 @ Summary2 ) )
=> ( Y
!= ( 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] :
( ( X
= ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( suc @ Va3 ) ) @ TreeList2 @ Summary2 ) )
=> ( Y
!= ( if @ vEBT_VEBT
@ ( ( ord_less @ nat @ ( vEBT_VEBT_high @ ( if @ nat @ ( ord_less @ nat @ Xa @ Mi2 ) @ Mi2 @ Xa ) @ ( divide_divide @ nat @ ( suc @ ( suc @ 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_1697_T_092_060_094sub_062i_092_060_094sub_062n_092_060_094sub_062s_092_060_094sub_062e_092_060_094sub_062r_092_060_094sub_062t_Oelims,axiom,
! [X: vEBT_VEBT,Xa: nat,Y: nat] :
( ( ( vEBT_T_i_n_s_e_r_t @ X @ Xa )
= Y )
=> ( ( ? [A4: $o,B3: $o] :
( X
= ( vEBT_Leaf @ A4 @ B3 ) )
=> ( Y
!= ( plus_plus @ nat @ ( one_one @ nat )
@ ( if @ nat
@ ( Xa
= ( zero_zero @ nat ) )
@ ( one_one @ nat )
@ ( plus_plus @ nat @ ( one_one @ nat ) @ ( one_one @ nat ) ) ) ) ) )
=> ( ( ? [Info2: option @ ( product_prod @ nat @ nat ),Ts2: list @ vEBT_VEBT,S2: vEBT_VEBT] :
( X
= ( vEBT_Node @ Info2 @ ( zero_zero @ nat ) @ Ts2 @ S2 ) )
=> ( Y
!= ( one_one @ nat ) ) )
=> ( ( ? [Info2: option @ ( product_prod @ nat @ nat ),Ts2: list @ vEBT_VEBT,S2: vEBT_VEBT] :
( X
= ( vEBT_Node @ Info2 @ ( suc @ ( zero_zero @ nat ) ) @ Ts2 @ S2 ) )
=> ( Y
!= ( one_one @ nat ) ) )
=> ( ( ? [V3: nat,TreeList2: list @ vEBT_VEBT,Summary2: vEBT_VEBT] :
( X
= ( vEBT_Node @ ( none @ ( product_prod @ nat @ nat ) ) @ ( suc @ ( suc @ V3 ) ) @ TreeList2 @ Summary2 ) )
=> ( Y
!= ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) )
=> ~ ! [Mi2: nat,Ma2: nat,Va3: nat,TreeList2: list @ vEBT_VEBT,Summary2: vEBT_VEBT] :
( ( X
= ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( suc @ Va3 ) ) @ TreeList2 @ Summary2 ) )
=> ( Y
!= ( plus_plus @ nat @ ( numeral_numeral @ nat @ ( bit1 @ ( bit1 @ ( bit0 @ ( bit0 @ one2 ) ) ) ) )
@ ( if @ nat
@ ( ( ord_less @ nat @ ( vEBT_VEBT_high @ ( if @ nat @ ( ord_less @ nat @ Xa @ Mi2 ) @ Mi2 @ Xa ) @ ( divide_divide @ nat @ ( suc @ ( suc @ 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_1698_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_1699_Diff__eq__empty__iff,axiom,
! [A: $tType,A3: set @ A,B6: set @ A] :
( ( ( minus_minus @ ( set @ A ) @ A3 @ B6 )
= ( bot_bot @ ( set @ A ) ) )
= ( ord_less_eq @ ( set @ A ) @ A3 @ B6 ) ) ).
% Diff_eq_empty_iff
thf(fact_1700_atLeastatMost__subset__iff,axiom,
! [A: $tType] :
( ( preorder @ A )
=> ! [A2: A,B2: A,C2: A,D: A] :
( ( ord_less_eq @ ( set @ A ) @ ( set_or1337092689740270186AtMost @ A @ A2 @ B2 ) @ ( set_or1337092689740270186AtMost @ A @ C2 @ D ) )
= ( ~ ( ord_less_eq @ A @ A2 @ B2 )
| ( ( ord_less_eq @ A @ C2 @ A2 )
& ( ord_less_eq @ A @ B2 @ D ) ) ) ) ) ).
% atLeastatMost_subset_iff
thf(fact_1701_atLeastatMost__empty,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [B2: A,A2: A] :
( ( ord_less @ A @ B2 @ A2 )
=> ( ( set_or1337092689740270186AtMost @ A @ A2 @ B2 )
= ( bot_bot @ ( set @ A ) ) ) ) ) ).
% atLeastatMost_empty
thf(fact_1702_atLeastatMost__empty__iff2,axiom,
! [A: $tType] :
( ( preorder @ A )
=> ! [A2: A,B2: A] :
( ( ( bot_bot @ ( set @ A ) )
= ( set_or1337092689740270186AtMost @ A @ A2 @ B2 ) )
= ( ~ ( ord_less_eq @ A @ A2 @ B2 ) ) ) ) ).
% atLeastatMost_empty_iff2
thf(fact_1703_atLeastatMost__empty__iff,axiom,
! [A: $tType] :
( ( preorder @ A )
=> ! [A2: A,B2: A] :
( ( ( set_or1337092689740270186AtMost @ A @ A2 @ B2 )
= ( bot_bot @ ( set @ A ) ) )
= ( ~ ( ord_less_eq @ A @ A2 @ B2 ) ) ) ) ).
% atLeastatMost_empty_iff
thf(fact_1704_greater__shift,axiom,
( ( ord_less @ nat )
= ( ^ [Y2: nat,X2: nat] : ( vEBT_VEBT_greater @ ( some @ nat @ X2 ) @ ( some @ nat @ Y2 ) ) ) ) ).
% greater_shift
thf(fact_1705_less__shift,axiom,
( ( ord_less @ nat )
= ( ^ [X2: nat,Y2: nat] : ( vEBT_VEBT_less @ ( some @ nat @ X2 ) @ ( some @ nat @ Y2 ) ) ) ) ).
% less_shift
thf(fact_1706_subset__antisym,axiom,
! [A: $tType,A3: set @ A,B6: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ A3 @ B6 )
=> ( ( ord_less_eq @ ( set @ A ) @ B6 @ A3 )
=> ( A3 = B6 ) ) ) ).
% subset_antisym
thf(fact_1707_psubsetI,axiom,
! [A: $tType,A3: set @ A,B6: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ A3 @ B6 )
=> ( ( A3 != B6 )
=> ( ord_less @ ( set @ A ) @ A3 @ B6 ) ) ) ).
% psubsetI
thf(fact_1708_subsetI,axiom,
! [A: $tType,A3: set @ A,B6: set @ A] :
( ! [X3: A] :
( ( member @ A @ X3 @ A3 )
=> ( member @ A @ X3 @ B6 ) )
=> ( ord_less_eq @ ( set @ A ) @ A3 @ B6 ) ) ).
% subsetI
thf(fact_1709_Diff__idemp,axiom,
! [A: $tType,A3: set @ A,B6: set @ A] :
( ( minus_minus @ ( set @ A ) @ ( minus_minus @ ( set @ A ) @ A3 @ B6 ) @ B6 )
= ( minus_minus @ ( set @ A ) @ A3 @ B6 ) ) ).
% Diff_idemp
thf(fact_1710_Diff__iff,axiom,
! [A: $tType,C2: A,A3: set @ A,B6: set @ A] :
( ( member @ A @ C2 @ ( minus_minus @ ( set @ A ) @ A3 @ B6 ) )
= ( ( member @ A @ C2 @ A3 )
& ~ ( member @ A @ C2 @ B6 ) ) ) ).
% Diff_iff
thf(fact_1711_DiffI,axiom,
! [A: $tType,C2: A,A3: set @ A,B6: set @ A] :
( ( member @ A @ C2 @ A3 )
=> ( ~ ( member @ A @ C2 @ B6 )
=> ( member @ A @ C2 @ ( minus_minus @ ( set @ A ) @ A3 @ B6 ) ) ) ) ).
% DiffI
thf(fact_1712_empty__subsetI,axiom,
! [A: $tType,A3: set @ A] : ( ord_less_eq @ ( set @ A ) @ ( bot_bot @ ( set @ A ) ) @ A3 ) ).
% empty_subsetI
thf(fact_1713_subset__empty,axiom,
! [A: $tType,A3: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ A3 @ ( bot_bot @ ( set @ A ) ) )
= ( A3
= ( bot_bot @ ( set @ A ) ) ) ) ).
% subset_empty
thf(fact_1714_atLeastAtMost__iff,axiom,
! [A: $tType] :
( ( ord @ A )
=> ! [I: A,L2: A,U: A] :
( ( member @ A @ I @ ( set_or1337092689740270186AtMost @ A @ L2 @ U ) )
= ( ( ord_less_eq @ A @ L2 @ I )
& ( ord_less_eq @ A @ I @ U ) ) ) ) ).
% atLeastAtMost_iff
thf(fact_1715_Icc__eq__Icc,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [L2: A,H2: A,L4: A,H3: A] :
( ( ( set_or1337092689740270186AtMost @ A @ L2 @ H2 )
= ( set_or1337092689740270186AtMost @ A @ L4 @ H3 ) )
= ( ( ( L2 = L4 )
& ( H2 = H3 ) )
| ( ~ ( ord_less_eq @ A @ L2 @ H2 )
& ~ ( ord_less_eq @ A @ L4 @ H3 ) ) ) ) ) ).
% Icc_eq_Icc
thf(fact_1716_Diff__cancel,axiom,
! [A: $tType,A3: set @ A] :
( ( minus_minus @ ( set @ A ) @ A3 @ A3 )
= ( bot_bot @ ( set @ A ) ) ) ).
% Diff_cancel
thf(fact_1717_empty__Diff,axiom,
! [A: $tType,A3: set @ A] :
( ( minus_minus @ ( set @ A ) @ ( bot_bot @ ( set @ A ) ) @ A3 )
= ( bot_bot @ ( set @ A ) ) ) ).
% empty_Diff
thf(fact_1718_Diff__empty,axiom,
! [A: $tType,A3: set @ A] :
( ( minus_minus @ ( set @ A ) @ A3 @ ( bot_bot @ ( set @ A ) ) )
= A3 ) ).
% Diff_empty
thf(fact_1719_zero__one__enat__neq_I1_J,axiom,
( ( zero_zero @ extended_enat )
!= ( one_one @ extended_enat ) ) ).
% zero_one_enat_neq(1)
thf(fact_1720_subset__iff__psubset__eq,axiom,
! [A: $tType] :
( ( ord_less_eq @ ( set @ A ) )
= ( ^ [A7: set @ A,B7: set @ A] :
( ( ord_less @ ( set @ A ) @ A7 @ B7 )
| ( A7 = B7 ) ) ) ) ).
% subset_iff_psubset_eq
thf(fact_1721_subset__psubset__trans,axiom,
! [A: $tType,A3: set @ A,B6: set @ A,C5: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ A3 @ B6 )
=> ( ( ord_less @ ( set @ A ) @ B6 @ C5 )
=> ( ord_less @ ( set @ A ) @ A3 @ C5 ) ) ) ).
% subset_psubset_trans
thf(fact_1722_subset__not__subset__eq,axiom,
! [A: $tType] :
( ( ord_less @ ( set @ A ) )
= ( ^ [A7: set @ A,B7: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ A7 @ B7 )
& ~ ( ord_less_eq @ ( set @ A ) @ B7 @ A7 ) ) ) ) ).
% subset_not_subset_eq
thf(fact_1723_psubset__subset__trans,axiom,
! [A: $tType,A3: set @ A,B6: set @ A,C5: set @ A] :
( ( ord_less @ ( set @ A ) @ A3 @ B6 )
=> ( ( ord_less_eq @ ( set @ A ) @ B6 @ C5 )
=> ( ord_less @ ( set @ A ) @ A3 @ C5 ) ) ) ).
% psubset_subset_trans
thf(fact_1724_psubset__imp__subset,axiom,
! [A: $tType,A3: set @ A,B6: set @ A] :
( ( ord_less @ ( set @ A ) @ A3 @ B6 )
=> ( ord_less_eq @ ( set @ A ) @ A3 @ B6 ) ) ).
% psubset_imp_subset
thf(fact_1725_Collect__mono__iff,axiom,
! [A: $tType,P: A > $o,Q: A > $o] :
( ( ord_less_eq @ ( set @ A ) @ ( collect @ A @ P ) @ ( collect @ A @ Q ) )
= ( ! [X2: A] :
( ( P @ X2 )
=> ( Q @ X2 ) ) ) ) ).
% Collect_mono_iff
thf(fact_1726_set__eq__subset,axiom,
! [A: $tType] :
( ( ^ [Y5: set @ A,Z5: set @ A] : Y5 = Z5 )
= ( ^ [A7: set @ A,B7: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ A7 @ B7 )
& ( ord_less_eq @ ( set @ A ) @ B7 @ A7 ) ) ) ) ).
% set_eq_subset
thf(fact_1727_subset__trans,axiom,
! [A: $tType,A3: set @ A,B6: set @ A,C5: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ A3 @ B6 )
=> ( ( ord_less_eq @ ( set @ A ) @ B6 @ C5 )
=> ( ord_less_eq @ ( set @ A ) @ A3 @ C5 ) ) ) ).
% subset_trans
thf(fact_1728_Collect__mono,axiom,
! [A: $tType,P: A > $o,Q: A > $o] :
( ! [X3: A] :
( ( P @ X3 )
=> ( Q @ X3 ) )
=> ( ord_less_eq @ ( set @ A ) @ ( collect @ A @ P ) @ ( collect @ A @ Q ) ) ) ).
% Collect_mono
thf(fact_1729_subset__refl,axiom,
! [A: $tType,A3: set @ A] : ( ord_less_eq @ ( set @ A ) @ A3 @ A3 ) ).
% subset_refl
thf(fact_1730_subset__iff,axiom,
! [A: $tType] :
( ( ord_less_eq @ ( set @ A ) )
= ( ^ [A7: set @ A,B7: set @ A] :
! [T3: A] :
( ( member @ A @ T3 @ A7 )
=> ( member @ A @ T3 @ B7 ) ) ) ) ).
% subset_iff
thf(fact_1731_psubset__eq,axiom,
! [A: $tType] :
( ( ord_less @ ( set @ A ) )
= ( ^ [A7: set @ A,B7: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ A7 @ B7 )
& ( A7 != B7 ) ) ) ) ).
% psubset_eq
thf(fact_1732_Set_OequalityD2,axiom,
! [A: $tType,A3: set @ A,B6: set @ A] :
( ( A3 = B6 )
=> ( ord_less_eq @ ( set @ A ) @ B6 @ A3 ) ) ).
% Set.equalityD2
thf(fact_1733_equalityD1,axiom,
! [A: $tType,A3: set @ A,B6: set @ A] :
( ( A3 = B6 )
=> ( ord_less_eq @ ( set @ A ) @ A3 @ B6 ) ) ).
% equalityD1
thf(fact_1734_subset__eq,axiom,
! [A: $tType] :
( ( ord_less_eq @ ( set @ A ) )
= ( ^ [A7: set @ A,B7: set @ A] :
! [X2: A] :
( ( member @ A @ X2 @ A7 )
=> ( member @ A @ X2 @ B7 ) ) ) ) ).
% subset_eq
thf(fact_1735_equalityE,axiom,
! [A: $tType,A3: set @ A,B6: set @ A] :
( ( A3 = B6 )
=> ~ ( ( ord_less_eq @ ( set @ A ) @ A3 @ B6 )
=> ~ ( ord_less_eq @ ( set @ A ) @ B6 @ A3 ) ) ) ).
% equalityE
thf(fact_1736_psubsetE,axiom,
! [A: $tType,A3: set @ A,B6: set @ A] :
( ( ord_less @ ( set @ A ) @ A3 @ B6 )
=> ~ ( ( ord_less_eq @ ( set @ A ) @ A3 @ B6 )
=> ( ord_less_eq @ ( set @ A ) @ B6 @ A3 ) ) ) ).
% psubsetE
thf(fact_1737_subsetD,axiom,
! [A: $tType,A3: set @ A,B6: set @ A,C2: A] :
( ( ord_less_eq @ ( set @ A ) @ A3 @ B6 )
=> ( ( member @ A @ C2 @ A3 )
=> ( member @ A @ C2 @ B6 ) ) ) ).
% subsetD
thf(fact_1738_in__mono,axiom,
! [A: $tType,A3: set @ A,B6: set @ A,X: A] :
( ( ord_less_eq @ ( set @ A ) @ A3 @ B6 )
=> ( ( member @ A @ X @ A3 )
=> ( member @ A @ X @ B6 ) ) ) ).
% in_mono
thf(fact_1739_bounded__Max__nat,axiom,
! [P: nat > $o,X: nat,M8: nat] :
( ( P @ X )
=> ( ! [X3: nat] :
( ( P @ X3 )
=> ( ord_less_eq @ nat @ X3 @ M8 ) )
=> ~ ! [M4: nat] :
( ( P @ M4 )
=> ~ ! [X5: nat] :
( ( P @ X5 )
=> ( ord_less_eq @ nat @ X5 @ M4 ) ) ) ) ) ).
% bounded_Max_nat
thf(fact_1740_fold__atLeastAtMost__nat_Ocases,axiom,
! [A: $tType,X: product_prod @ ( nat > A > A ) @ ( product_prod @ nat @ ( product_prod @ nat @ A ) )] :
~ ! [F3: nat > A > A,A4: nat,B3: nat,Acc: A] :
( X
!= ( 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 @ B3 @ Acc ) ) ) ) ).
% fold_atLeastAtMost_nat.cases
thf(fact_1741_DiffD2,axiom,
! [A: $tType,C2: A,A3: set @ A,B6: set @ A] :
( ( member @ A @ C2 @ ( minus_minus @ ( set @ A ) @ A3 @ B6 ) )
=> ~ ( member @ A @ C2 @ B6 ) ) ).
% DiffD2
thf(fact_1742_DiffD1,axiom,
! [A: $tType,C2: A,A3: set @ A,B6: set @ A] :
( ( member @ A @ C2 @ ( minus_minus @ ( set @ A ) @ A3 @ B6 ) )
=> ( member @ A @ C2 @ A3 ) ) ).
% DiffD1
thf(fact_1743_DiffE,axiom,
! [A: $tType,C2: A,A3: set @ A,B6: set @ A] :
( ( member @ A @ C2 @ ( minus_minus @ ( set @ A ) @ A3 @ B6 ) )
=> ~ ( ( member @ A @ C2 @ A3 )
=> ( member @ A @ C2 @ B6 ) ) ) ).
% DiffE
thf(fact_1744_psubset__imp__ex__mem,axiom,
! [A: $tType,A3: set @ A,B6: set @ A] :
( ( ord_less @ ( set @ A ) @ A3 @ B6 )
=> ? [B3: A] : ( member @ A @ B3 @ ( minus_minus @ ( set @ A ) @ B6 @ A3 ) ) ) ).
% psubset_imp_ex_mem
thf(fact_1745_less__eq__set__def,axiom,
! [A: $tType] :
( ( ord_less_eq @ ( set @ A ) )
= ( ^ [A7: set @ A,B7: set @ A] :
( ord_less_eq @ ( A > $o )
@ ^ [X2: A] : ( member @ A @ X2 @ A7 )
@ ^ [X2: A] : ( member @ A @ X2 @ B7 ) ) ) ) ).
% less_eq_set_def
thf(fact_1746_Collect__subset,axiom,
! [A: $tType,A3: set @ A,P: A > $o] :
( ord_less_eq @ ( set @ A )
@ ( collect @ A
@ ^ [X2: A] :
( ( member @ A @ X2 @ A3 )
& ( P @ X2 ) ) )
@ A3 ) ).
% Collect_subset
thf(fact_1747_set__diff__eq,axiom,
! [A: $tType] :
( ( minus_minus @ ( set @ A ) )
= ( ^ [A7: set @ A,B7: set @ A] :
( collect @ A
@ ^ [X2: A] :
( ( member @ A @ X2 @ A7 )
& ~ ( member @ A @ X2 @ B7 ) ) ) ) ) ).
% set_diff_eq
thf(fact_1748_minus__set__def,axiom,
! [A: $tType] :
( ( minus_minus @ ( set @ A ) )
= ( ^ [A7: set @ A,B7: set @ A] :
( collect @ A
@ ( minus_minus @ ( A > $o )
@ ^ [X2: A] : ( member @ A @ X2 @ A7 )
@ ^ [X2: A] : ( member @ A @ X2 @ B7 ) ) ) ) ) ).
% minus_set_def
thf(fact_1749_Diff__mono,axiom,
! [A: $tType,A3: set @ A,C5: set @ A,D4: set @ A,B6: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ A3 @ C5 )
=> ( ( ord_less_eq @ ( set @ A ) @ D4 @ B6 )
=> ( ord_less_eq @ ( set @ A ) @ ( minus_minus @ ( set @ A ) @ A3 @ B6 ) @ ( minus_minus @ ( set @ A ) @ C5 @ D4 ) ) ) ) ).
% Diff_mono
thf(fact_1750_Diff__subset,axiom,
! [A: $tType,A3: set @ A,B6: set @ A] : ( ord_less_eq @ ( set @ A ) @ ( minus_minus @ ( set @ A ) @ A3 @ B6 ) @ A3 ) ).
% Diff_subset
thf(fact_1751_double__diff,axiom,
! [A: $tType,A3: set @ A,B6: set @ A,C5: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ A3 @ B6 )
=> ( ( ord_less_eq @ ( set @ A ) @ B6 @ C5 )
=> ( ( minus_minus @ ( set @ A ) @ B6 @ ( minus_minus @ ( set @ A ) @ C5 @ A3 ) )
= A3 ) ) ) ).
% double_diff
thf(fact_1752_atLeastatMost__psubset__iff,axiom,
! [A: $tType] :
( ( preorder @ A )
=> ! [A2: A,B2: A,C2: A,D: A] :
( ( ord_less @ ( set @ A ) @ ( set_or1337092689740270186AtMost @ A @ A2 @ B2 ) @ ( set_or1337092689740270186AtMost @ A @ C2 @ D ) )
= ( ( ~ ( ord_less_eq @ A @ A2 @ B2 )
| ( ( ord_less_eq @ A @ C2 @ A2 )
& ( ord_less_eq @ A @ B2 @ D )
& ( ( ord_less @ A @ C2 @ A2 )
| ( ord_less @ A @ B2 @ D ) ) ) )
& ( ord_less_eq @ A @ C2 @ D ) ) ) ) ).
% atLeastatMost_psubset_iff
thf(fact_1753_T_092_060_094sub_062s_092_060_094sub_062u_092_060_094sub_062c_092_060_094sub_062c_H_Oelims,axiom,
! [X: vEBT_VEBT,Xa: nat,Y: nat] :
( ( ( vEBT_T_s_u_c_c2 @ X @ Xa )
= Y )
=> ( ( ? [Uu: $o,B3: $o] :
( X
= ( vEBT_Leaf @ Uu @ B3 ) )
=> ( ( Xa
= ( zero_zero @ nat ) )
=> ( Y
!= ( one_one @ nat ) ) ) )
=> ( ( ? [Uv: $o,Uw: $o] :
( X
= ( vEBT_Leaf @ Uv @ Uw ) )
=> ( ? [N: nat] :
( Xa
= ( suc @ N ) )
=> ( Y
!= ( one_one @ nat ) ) ) )
=> ( ( ? [Ux2: nat,Uy2: list @ vEBT_VEBT,Uz2: vEBT_VEBT] :
( X
= ( vEBT_Node @ ( none @ ( product_prod @ nat @ nat ) ) @ Ux2 @ Uy2 @ Uz2 ) )
=> ( Y
!= ( one_one @ nat ) ) )
=> ( ( ? [V3: product_prod @ nat @ nat,Vc2: list @ vEBT_VEBT,Vd2: vEBT_VEBT] :
( X
= ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ V3 ) @ ( zero_zero @ nat ) @ Vc2 @ Vd2 ) )
=> ( Y
!= ( one_one @ nat ) ) )
=> ( ( ? [V3: product_prod @ nat @ nat,Vg2: list @ vEBT_VEBT,Vh2: vEBT_VEBT] :
( X
= ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ V3 ) @ ( suc @ ( zero_zero @ nat ) ) @ Vg2 @ Vh2 ) )
=> ( Y
!= ( one_one @ nat ) ) )
=> ~ ! [Mi2: nat,Ma2: nat,Va3: nat,TreeList2: list @ vEBT_VEBT,Summary2: vEBT_VEBT] :
( ( X
= ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( suc @ Va3 ) ) @ TreeList2 @ Summary2 ) )
=> ~ ( ( ( ord_less @ nat @ Xa @ Mi2 )
=> ( Y
= ( one_one @ nat ) ) )
& ( ~ ( ord_less @ nat @ Xa @ Mi2 )
=> ( Y
= ( if @ nat @ ( ord_less @ nat @ ( vEBT_VEBT_high @ Xa @ ( divide_divide @ nat @ ( suc @ ( suc @ 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_1754_T_092_060_094sub_062p_092_060_094sub_062r_092_060_094sub_062e_092_060_094sub_062d_H_Oelims,axiom,
! [X: vEBT_VEBT,Xa: nat,Y: nat] :
( ( ( vEBT_T_p_r_e_d2 @ X @ Xa )
= Y )
=> ( ( ? [Uu: $o,Uv: $o] :
( X
= ( vEBT_Leaf @ Uu @ Uv ) )
=> ( ( Xa
= ( zero_zero @ nat ) )
=> ( Y
!= ( one_one @ nat ) ) ) )
=> ( ( ? [A4: $o,Uw: $o] :
( X
= ( vEBT_Leaf @ A4 @ Uw ) )
=> ( ( Xa
= ( suc @ ( zero_zero @ nat ) ) )
=> ( Y
!= ( one_one @ nat ) ) ) )
=> ( ( ? [A4: $o,B3: $o] :
( X
= ( vEBT_Leaf @ A4 @ B3 ) )
=> ( ? [Va3: nat] :
( Xa
= ( suc @ ( suc @ Va3 ) ) )
=> ( Y
!= ( one_one @ nat ) ) ) )
=> ( ( ? [Uy2: nat,Uz2: list @ vEBT_VEBT,Va2: vEBT_VEBT] :
( X
= ( vEBT_Node @ ( none @ ( product_prod @ nat @ nat ) ) @ Uy2 @ Uz2 @ Va2 ) )
=> ( Y
!= ( one_one @ nat ) ) )
=> ( ( ? [V3: product_prod @ nat @ nat,Vd2: list @ vEBT_VEBT,Ve2: vEBT_VEBT] :
( X
= ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ V3 ) @ ( zero_zero @ nat ) @ Vd2 @ Ve2 ) )
=> ( Y
!= ( one_one @ nat ) ) )
=> ( ( ? [V3: product_prod @ nat @ nat,Vh2: list @ vEBT_VEBT,Vi2: vEBT_VEBT] :
( X
= ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ V3 ) @ ( suc @ ( zero_zero @ nat ) ) @ Vh2 @ Vi2 ) )
=> ( Y
!= ( one_one @ nat ) ) )
=> ~ ! [Mi2: nat,Ma2: nat,Va3: nat,TreeList2: list @ vEBT_VEBT,Summary2: vEBT_VEBT] :
( ( X
= ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( suc @ Va3 ) ) @ TreeList2 @ Summary2 ) )
=> ~ ( ( ( ord_less @ nat @ Ma2 @ Xa )
=> ( Y
= ( one_one @ nat ) ) )
& ( ~ ( ord_less @ nat @ Ma2 @ Xa )
=> ( Y
= ( if @ nat @ ( ord_less @ nat @ ( vEBT_VEBT_high @ Xa @ ( divide_divide @ nat @ ( suc @ ( suc @ 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_1755_T_092_060_094sub_062s_092_060_094sub_062u_092_060_094sub_062c_092_060_094sub_062c_H_Osimps_I6_J,axiom,
! [X: nat,Mi: nat,Ma: nat,Va: nat,TreeList: list @ vEBT_VEBT,Summary: vEBT_VEBT] :
( ( ( ord_less @ nat @ X @ Mi )
=> ( ( vEBT_T_s_u_c_c2 @ ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi @ Ma ) ) @ ( suc @ ( suc @ Va ) ) @ TreeList @ Summary ) @ X )
= ( one_one @ nat ) ) )
& ( ~ ( ord_less @ nat @ X @ Mi )
=> ( ( vEBT_T_s_u_c_c2 @ ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi @ Ma ) ) @ ( suc @ ( suc @ Va ) ) @ TreeList @ Summary ) @ X )
= ( if @ nat @ ( ord_less @ nat @ ( vEBT_VEBT_high @ X @ ( 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 @ X @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) )
!= ( none @ nat ) )
& ( vEBT_VEBT_less @ ( some @ nat @ ( vEBT_VEBT_low @ X @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( vEBT_vebt_maxt @ ( nth @ vEBT_VEBT @ TreeList @ ( vEBT_VEBT_high @ X @ ( 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 @ X @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( vEBT_VEBT_low @ X @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) )
@ ( plus_plus @ nat @ ( vEBT_T_s_u_c_c2 @ Summary @ ( vEBT_VEBT_high @ X @ ( 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_1756_T_092_060_094sub_062p_092_060_094sub_062r_092_060_094sub_062e_092_060_094sub_062d_H_Osimps_I7_J,axiom,
! [Ma: nat,X: nat,Mi: nat,Va: nat,TreeList: list @ vEBT_VEBT,Summary: vEBT_VEBT] :
( ( ( ord_less @ nat @ Ma @ X )
=> ( ( vEBT_T_p_r_e_d2 @ ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi @ Ma ) ) @ ( suc @ ( suc @ Va ) ) @ TreeList @ Summary ) @ X )
= ( one_one @ nat ) ) )
& ( ~ ( ord_less @ nat @ Ma @ X )
=> ( ( vEBT_T_p_r_e_d2 @ ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi @ Ma ) ) @ ( suc @ ( suc @ Va ) ) @ TreeList @ Summary ) @ X )
= ( if @ nat @ ( ord_less @ nat @ ( vEBT_VEBT_high @ X @ ( 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 @ X @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) )
!= ( none @ nat ) )
& ( vEBT_VEBT_greater @ ( some @ nat @ ( vEBT_VEBT_low @ X @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList @ ( vEBT_VEBT_high @ X @ ( 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 @ X @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( vEBT_VEBT_low @ X @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) )
@ ( plus_plus @ nat @ ( vEBT_T_p_r_e_d2 @ Summary @ ( vEBT_VEBT_high @ X @ ( 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_1757_linear__plus__1__le__power,axiom,
! [X: real,N3: nat] :
( ( ord_less_eq @ real @ ( zero_zero @ real ) @ X )
=> ( ord_less_eq @ real @ ( plus_plus @ real @ ( times_times @ real @ ( semiring_1_of_nat @ real @ N3 ) @ X ) @ ( one_one @ real ) ) @ ( power_power @ real @ ( plus_plus @ real @ X @ ( one_one @ real ) ) @ N3 ) ) ) ).
% linear_plus_1_le_power
thf(fact_1758_T_092_060_094sub_062d_092_060_094sub_062e_092_060_094sub_062l_092_060_094sub_062e_092_060_094sub_062t_092_060_094sub_062e_Oelims,axiom,
! [X: vEBT_VEBT,Xa: nat,Y: nat] :
( ( ( vEBT_T_d_e_l_e_t_e @ X @ Xa )
= Y )
=> ( ( ? [A4: $o,B3: $o] :
( X
= ( vEBT_Leaf @ A4 @ B3 ) )
=> ( ( Xa
= ( zero_zero @ nat ) )
=> ( Y
!= ( one_one @ nat ) ) ) )
=> ( ( ? [A4: $o,B3: $o] :
( X
= ( vEBT_Leaf @ A4 @ B3 ) )
=> ( ( Xa
= ( suc @ ( zero_zero @ nat ) ) )
=> ( Y
!= ( one_one @ nat ) ) ) )
=> ( ( ? [A4: $o,B3: $o] :
( X
= ( vEBT_Leaf @ A4 @ B3 ) )
=> ( ? [N: nat] :
( Xa
= ( suc @ ( suc @ N ) ) )
=> ( Y
!= ( one_one @ nat ) ) ) )
=> ( ( ? [Deg2: nat,TreeList2: list @ vEBT_VEBT,Summary2: vEBT_VEBT] :
( X
= ( vEBT_Node @ ( none @ ( product_prod @ nat @ nat ) ) @ Deg2 @ TreeList2 @ Summary2 ) )
=> ( Y
!= ( one_one @ nat ) ) )
=> ( ( ? [Mi2: nat,Ma2: nat,TreeList2: list @ vEBT_VEBT,Summary2: vEBT_VEBT] :
( X
= ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi2 @ Ma2 ) ) @ ( zero_zero @ nat ) @ TreeList2 @ Summary2 ) )
=> ( Y
!= ( one_one @ nat ) ) )
=> ( ( ? [Mi2: nat,Ma2: nat,TreeList2: list @ vEBT_VEBT,Summary2: vEBT_VEBT] :
( X
= ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( zero_zero @ nat ) ) @ TreeList2 @ Summary2 ) )
=> ( Y
!= ( one_one @ nat ) ) )
=> ~ ! [Mi2: nat,Ma2: nat,Va3: nat,TreeList2: list @ vEBT_VEBT,Summary2: vEBT_VEBT] :
( ( X
= ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( suc @ Va3 ) ) @ TreeList2 @ Summary2 ) )
=> ( Y
!= ( plus_plus @ nat @ ( numeral_numeral @ nat @ ( bit1 @ one2 ) )
@ ( if @ nat
@ ( ( ord_less @ nat @ Xa @ Mi2 )
| ( ord_less @ nat @ Ma2 @ Xa ) )
@ ( one_one @ nat )
@ ( plus_plus @ nat @ ( numeral_numeral @ nat @ ( bit1 @ one2 ) )
@ ( if @ nat
@ ( ( Xa = Mi2 )
& ( Xa = Ma2 ) )
@ ( numeral_numeral @ nat @ ( bit1 @ one2 ) )
@ ( plus_plus @ nat @ ( plus_plus @ nat @ ( plus_plus @ nat @ ( numeral_numeral @ nat @ ( bit1 @ ( bit0 @ ( bit1 @ one2 ) ) ) ) @ ( if @ nat @ ( Xa = Mi2 ) @ ( plus_plus @ nat @ ( plus_plus @ nat @ ( vEBT_T_m_i_n_t @ Summary2 ) @ ( vEBT_T_m_i_n_t @ ( nth @ vEBT_VEBT @ TreeList2 @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary2 ) ) ) ) ) @ ( numeral_numeral @ nat @ ( bit1 @ ( bit1 @ one2 ) ) ) ) @ ( one_one @ nat ) ) ) @ ( one_one @ nat ) )
@ ( if @ nat @ ( ord_less @ nat @ ( vEBT_VEBT_high @ ( if @ nat @ ( Xa = Mi2 ) @ ( plus_plus @ nat @ ( times_times @ nat @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary2 ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ ( suc @ ( suc @ 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_1759_cppi,axiom,
! [D4: int,P: int > $o,P6: int > $o,A3: set @ int] :
( ( ord_less @ int @ ( zero_zero @ int ) @ D4 )
=> ( ? [Z4: int] :
! [X3: int] :
( ( ord_less @ int @ Z4 @ X3 )
=> ( ( P @ X3 )
= ( P6 @ X3 ) ) )
=> ( ! [X3: int] :
( ! [Xa2: int] :
( ( member @ int @ Xa2 @ ( set_or1337092689740270186AtMost @ int @ ( one_one @ int ) @ D4 ) )
=> ! [Xb2: int] :
( ( member @ int @ Xb2 @ A3 )
=> ( X3
!= ( minus_minus @ int @ Xb2 @ Xa2 ) ) ) )
=> ( ( P @ X3 )
=> ( P @ ( plus_plus @ int @ X3 @ D4 ) ) ) )
=> ( ! [X3: int,K3: int] :
( ( P6 @ X3 )
= ( P6 @ ( minus_minus @ int @ X3 @ ( times_times @ int @ K3 @ D4 ) ) ) )
=> ( ( ? [X6: int] : ( P @ X6 ) )
= ( ? [X2: int] :
( ( member @ int @ X2 @ ( set_or1337092689740270186AtMost @ int @ ( one_one @ int ) @ D4 ) )
& ( P6 @ X2 ) )
| ? [X2: int] :
( ( member @ int @ X2 @ ( set_or1337092689740270186AtMost @ int @ ( one_one @ int ) @ D4 ) )
& ? [Y2: int] :
( ( member @ int @ Y2 @ A3 )
& ( P @ ( minus_minus @ int @ Y2 @ X2 ) ) ) ) ) ) ) ) ) ) ).
% cppi
thf(fact_1760_pinf_I1_J,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [P: A > $o,P6: A > $o,Q: A > $o,Q6: A > $o] :
( ? [Z4: A] :
! [X3: A] :
( ( ord_less @ A @ Z4 @ X3 )
=> ( ( P @ X3 )
= ( P6 @ X3 ) ) )
=> ( ? [Z4: A] :
! [X3: A] :
( ( ord_less @ A @ Z4 @ X3 )
=> ( ( Q @ X3 )
= ( Q6 @ X3 ) ) )
=> ? [Z2: A] :
! [X5: A] :
( ( ord_less @ A @ Z2 @ X5 )
=> ( ( ( P @ X5 )
& ( Q @ X5 ) )
= ( ( P6 @ X5 )
& ( Q6 @ X5 ) ) ) ) ) ) ) ).
% pinf(1)
thf(fact_1761_pinf_I2_J,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [P: A > $o,P6: A > $o,Q: A > $o,Q6: A > $o] :
( ? [Z4: A] :
! [X3: A] :
( ( ord_less @ A @ Z4 @ X3 )
=> ( ( P @ X3 )
= ( P6 @ X3 ) ) )
=> ( ? [Z4: A] :
! [X3: A] :
( ( ord_less @ A @ Z4 @ X3 )
=> ( ( Q @ X3 )
= ( Q6 @ X3 ) ) )
=> ? [Z2: A] :
! [X5: A] :
( ( ord_less @ A @ Z2 @ X5 )
=> ( ( ( P @ X5 )
| ( Q @ X5 ) )
= ( ( P6 @ X5 )
| ( Q6 @ X5 ) ) ) ) ) ) ) ).
% pinf(2)
thf(fact_1762_pinf_I3_J,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [T2: A] :
? [Z2: A] :
! [X5: A] :
( ( ord_less @ A @ Z2 @ X5 )
=> ( X5 != T2 ) ) ) ).
% pinf(3)
thf(fact_1763_pinf_I4_J,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [T2: A] :
? [Z2: A] :
! [X5: A] :
( ( ord_less @ A @ Z2 @ X5 )
=> ( X5 != T2 ) ) ) ).
% pinf(4)
thf(fact_1764_pinf_I5_J,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [T2: A] :
? [Z2: A] :
! [X5: A] :
( ( ord_less @ A @ Z2 @ X5 )
=> ~ ( ord_less @ A @ X5 @ T2 ) ) ) ).
% pinf(5)
thf(fact_1765_pinf_I7_J,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [T2: A] :
? [Z2: A] :
! [X5: A] :
( ( ord_less @ A @ Z2 @ X5 )
=> ( ord_less @ A @ T2 @ X5 ) ) ) ).
% pinf(7)
thf(fact_1766_pinf_I11_J,axiom,
! [C: $tType,D5: $tType] :
( ( ord @ C )
=> ! [F4: D5] :
? [Z2: C] :
! [X5: C] :
( ( ord_less @ C @ Z2 @ X5 )
=> ( F4 = F4 ) ) ) ).
% pinf(11)
thf(fact_1767_minf_I1_J,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [P: A > $o,P6: A > $o,Q: A > $o,Q6: A > $o] :
( ? [Z4: A] :
! [X3: A] :
( ( ord_less @ A @ X3 @ Z4 )
=> ( ( P @ X3 )
= ( P6 @ X3 ) ) )
=> ( ? [Z4: A] :
! [X3: A] :
( ( ord_less @ A @ X3 @ Z4 )
=> ( ( Q @ X3 )
= ( Q6 @ X3 ) ) )
=> ? [Z2: A] :
! [X5: A] :
( ( ord_less @ A @ X5 @ Z2 )
=> ( ( ( P @ X5 )
& ( Q @ X5 ) )
= ( ( P6 @ X5 )
& ( Q6 @ X5 ) ) ) ) ) ) ) ).
% minf(1)
thf(fact_1768_minf_I2_J,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [P: A > $o,P6: A > $o,Q: A > $o,Q6: A > $o] :
( ? [Z4: A] :
! [X3: A] :
( ( ord_less @ A @ X3 @ Z4 )
=> ( ( P @ X3 )
= ( P6 @ X3 ) ) )
=> ( ? [Z4: A] :
! [X3: A] :
( ( ord_less @ A @ X3 @ Z4 )
=> ( ( Q @ X3 )
= ( Q6 @ X3 ) ) )
=> ? [Z2: A] :
! [X5: A] :
( ( ord_less @ A @ X5 @ Z2 )
=> ( ( ( P @ X5 )
| ( Q @ X5 ) )
= ( ( P6 @ X5 )
| ( Q6 @ X5 ) ) ) ) ) ) ) ).
% minf(2)
thf(fact_1769_minf_I3_J,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [T2: A] :
? [Z2: A] :
! [X5: A] :
( ( ord_less @ A @ X5 @ Z2 )
=> ( X5 != T2 ) ) ) ).
% minf(3)
thf(fact_1770_minf_I4_J,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [T2: A] :
? [Z2: A] :
! [X5: A] :
( ( ord_less @ A @ X5 @ Z2 )
=> ( X5 != T2 ) ) ) ).
% minf(4)
thf(fact_1771_minf_I5_J,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [T2: A] :
? [Z2: A] :
! [X5: A] :
( ( ord_less @ A @ X5 @ Z2 )
=> ( ord_less @ A @ X5 @ T2 ) ) ) ).
% minf(5)
thf(fact_1772_minf_I7_J,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [T2: A] :
? [Z2: A] :
! [X5: A] :
( ( ord_less @ A @ X5 @ Z2 )
=> ~ ( ord_less @ A @ T2 @ X5 ) ) ) ).
% minf(7)
thf(fact_1773_minf_I11_J,axiom,
! [C: $tType,D5: $tType] :
( ( ord @ C )
=> ! [F4: D5] :
? [Z2: C] :
! [X5: C] :
( ( ord_less @ C @ X5 @ Z2 )
=> ( F4 = F4 ) ) ) ).
% minf(11)
thf(fact_1774_T_092_060_094sub_062m_092_060_094sub_062a_092_060_094sub_062x_092_060_094sub_062t_Osimps_I1_J,axiom,
! [A2: $o,B2: $o] :
( ( vEBT_T_m_a_x_t @ ( vEBT_Leaf @ A2 @ B2 ) )
= ( plus_plus @ nat @ ( one_one @ nat ) @ ( if @ nat @ B2 @ ( one_one @ nat ) @ ( plus_plus @ nat @ ( one_one @ nat ) @ ( one_one @ nat ) ) ) ) ) ).
% T\<^sub>m\<^sub>a\<^sub>x\<^sub>t.simps(1)
thf(fact_1775_T_092_060_094sub_062m_092_060_094sub_062a_092_060_094sub_062x_092_060_094sub_062t_Osimps_I2_J,axiom,
! [Uu2: nat,Uv2: list @ vEBT_VEBT,Uw2: vEBT_VEBT] :
( ( vEBT_T_m_a_x_t @ ( vEBT_Node @ ( none @ ( product_prod @ nat @ nat ) ) @ Uu2 @ Uv2 @ Uw2 ) )
= ( one_one @ nat ) ) ).
% T\<^sub>m\<^sub>a\<^sub>x\<^sub>t.simps(2)
thf(fact_1776_T_092_060_094sub_062p_092_060_094sub_062r_092_060_094sub_062e_092_060_094sub_062d_H_Osimps_I3_J,axiom,
! [A2: $o,B2: $o,Va: nat] :
( ( vEBT_T_p_r_e_d2 @ ( vEBT_Leaf @ A2 @ B2 ) @ ( suc @ ( suc @ Va ) ) )
= ( one_one @ nat ) ) ).
% T\<^sub>p\<^sub>r\<^sub>e\<^sub>d'.simps(3)
thf(fact_1777_T_092_060_094sub_062p_092_060_094sub_062r_092_060_094sub_062e_092_060_094sub_062d_H_Osimps_I1_J,axiom,
! [Uu2: $o,Uv2: $o] :
( ( vEBT_T_p_r_e_d2 @ ( vEBT_Leaf @ Uu2 @ Uv2 ) @ ( zero_zero @ nat ) )
= ( one_one @ nat ) ) ).
% T\<^sub>p\<^sub>r\<^sub>e\<^sub>d'.simps(1)
thf(fact_1778_T_092_060_094sub_062m_092_060_094sub_062i_092_060_094sub_062n_092_060_094sub_062t_Osimps_I2_J,axiom,
! [Uu2: nat,Uv2: list @ vEBT_VEBT,Uw2: vEBT_VEBT] :
( ( vEBT_T_m_i_n_t @ ( vEBT_Node @ ( none @ ( product_prod @ nat @ nat ) ) @ Uu2 @ Uv2 @ Uw2 ) )
= ( one_one @ nat ) ) ).
% T\<^sub>m\<^sub>i\<^sub>n\<^sub>t.simps(2)
thf(fact_1779_T_092_060_094sub_062s_092_060_094sub_062u_092_060_094sub_062c_092_060_094sub_062c_H_Osimps_I2_J,axiom,
! [Uv2: $o,Uw2: $o,N3: nat] :
( ( vEBT_T_s_u_c_c2 @ ( vEBT_Leaf @ Uv2 @ Uw2 ) @ ( suc @ N3 ) )
= ( one_one @ nat ) ) ).
% T\<^sub>s\<^sub>u\<^sub>c\<^sub>c'.simps(2)
thf(fact_1780_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_1781_T_092_060_094sub_062s_092_060_094sub_062u_092_060_094sub_062c_092_060_094sub_062c_H_Osimps_I1_J,axiom,
! [Uu2: $o,B2: $o] :
( ( vEBT_T_s_u_c_c2 @ ( vEBT_Leaf @ Uu2 @ B2 ) @ ( zero_zero @ nat ) )
= ( one_one @ nat ) ) ).
% T\<^sub>s\<^sub>u\<^sub>c\<^sub>c'.simps(1)
thf(fact_1782_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_1783_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_1784_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_1785_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_1786_T_092_060_094sub_062m_092_060_094sub_062i_092_060_094sub_062n_092_060_094sub_062t_Osimps_I1_J,axiom,
! [A2: $o,B2: $o] :
( ( vEBT_T_m_i_n_t @ ( vEBT_Leaf @ A2 @ B2 ) )
= ( plus_plus @ nat @ ( one_one @ nat ) @ ( if @ nat @ A2 @ ( zero_zero @ nat ) @ ( plus_plus @ nat @ ( one_one @ nat ) @ ( one_one @ nat ) ) ) ) ) ).
% T\<^sub>m\<^sub>i\<^sub>n\<^sub>t.simps(1)
thf(fact_1787_T_092_060_094sub_062p_092_060_094sub_062r_092_060_094sub_062e_092_060_094sub_062d_H_Osimps_I2_J,axiom,
! [A2: $o,Uw2: $o] :
( ( vEBT_T_p_r_e_d2 @ ( vEBT_Leaf @ A2 @ Uw2 ) @ ( suc @ ( zero_zero @ nat ) ) )
= ( one_one @ nat ) ) ).
% T\<^sub>p\<^sub>r\<^sub>e\<^sub>d'.simps(2)
thf(fact_1788_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_1789_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,Vd: list @ vEBT_VEBT,Ve: vEBT_VEBT,Vf: nat] :
( ( vEBT_T_p_r_e_d2 @ ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ V ) @ ( zero_zero @ nat ) @ Vd @ Ve ) @ Vf )
= ( one_one @ nat ) ) ).
% T\<^sub>p\<^sub>r\<^sub>e\<^sub>d'.simps(5)
thf(fact_1790_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,Vd: vEBT_VEBT,Ve: nat] :
( ( vEBT_T_s_u_c_c2 @ ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ V ) @ ( zero_zero @ nat ) @ Vc @ Vd ) @ Ve )
= ( one_one @ nat ) ) ).
% T\<^sub>s\<^sub>u\<^sub>c\<^sub>c'.simps(4)
thf(fact_1791_pinf_I6_J,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [T2: A] :
? [Z2: A] :
! [X5: A] :
( ( ord_less @ A @ Z2 @ X5 )
=> ~ ( ord_less_eq @ A @ X5 @ T2 ) ) ) ).
% pinf(6)
thf(fact_1792_pinf_I8_J,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [T2: A] :
? [Z2: A] :
! [X5: A] :
( ( ord_less @ A @ Z2 @ X5 )
=> ( ord_less_eq @ A @ T2 @ X5 ) ) ) ).
% pinf(8)
thf(fact_1793_minf_I6_J,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [T2: A] :
? [Z2: A] :
! [X5: A] :
( ( ord_less @ A @ X5 @ Z2 )
=> ( ord_less_eq @ A @ X5 @ T2 ) ) ) ).
% minf(6)
thf(fact_1794_minf_I8_J,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [T2: A] :
? [Z2: A] :
! [X5: A] :
( ( ord_less @ A @ X5 @ Z2 )
=> ~ ( ord_less_eq @ A @ T2 @ X5 ) ) ) ).
% minf(8)
thf(fact_1795_inf__period_I1_J,axiom,
! [A: $tType] :
( ( ( comm_ring @ A )
& ( dvd @ A ) )
=> ! [P: A > $o,D4: A,Q: A > $o] :
( ! [X3: A,K3: A] :
( ( P @ X3 )
= ( P @ ( minus_minus @ A @ X3 @ ( times_times @ A @ K3 @ D4 ) ) ) )
=> ( ! [X3: A,K3: A] :
( ( Q @ X3 )
= ( Q @ ( minus_minus @ A @ X3 @ ( times_times @ A @ K3 @ D4 ) ) ) )
=> ! [X5: A,K5: A] :
( ( ( P @ X5 )
& ( Q @ X5 ) )
= ( ( P @ ( minus_minus @ A @ X5 @ ( times_times @ A @ K5 @ D4 ) ) )
& ( Q @ ( minus_minus @ A @ X5 @ ( times_times @ A @ K5 @ D4 ) ) ) ) ) ) ) ) ).
% inf_period(1)
thf(fact_1796_inf__period_I2_J,axiom,
! [A: $tType] :
( ( ( comm_ring @ A )
& ( dvd @ A ) )
=> ! [P: A > $o,D4: A,Q: A > $o] :
( ! [X3: A,K3: A] :
( ( P @ X3 )
= ( P @ ( minus_minus @ A @ X3 @ ( times_times @ A @ K3 @ D4 ) ) ) )
=> ( ! [X3: A,K3: A] :
( ( Q @ X3 )
= ( Q @ ( minus_minus @ A @ X3 @ ( times_times @ A @ K3 @ D4 ) ) ) )
=> ! [X5: A,K5: A] :
( ( ( P @ X5 )
| ( Q @ X5 ) )
= ( ( P @ ( minus_minus @ A @ X5 @ ( times_times @ A @ K5 @ D4 ) ) )
| ( Q @ ( minus_minus @ A @ X5 @ ( times_times @ A @ K5 @ D4 ) ) ) ) ) ) ) ) ).
% inf_period(2)
thf(fact_1797_conj__le__cong,axiom,
! [X: int,X7: int,P: $o,P6: $o] :
( ( X = X7 )
=> ( ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ X7 )
=> ( P = P6 ) )
=> ( ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ X )
& P )
= ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ X7 )
& P6 ) ) ) ) ).
% conj_le_cong
thf(fact_1798_imp__le__cong,axiom,
! [X: int,X7: int,P: $o,P6: $o] :
( ( X = X7 )
=> ( ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ X7 )
=> ( P = P6 ) )
=> ( ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ X )
=> P )
= ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ X7 )
=> P6 ) ) ) ) ).
% imp_le_cong
thf(fact_1799_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,Vh: list @ vEBT_VEBT,Vi: vEBT_VEBT,Vj: nat] :
( ( vEBT_T_p_r_e_d2 @ ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ V ) @ ( suc @ ( zero_zero @ nat ) ) @ Vh @ Vi ) @ Vj )
= ( one_one @ nat ) ) ).
% T\<^sub>p\<^sub>r\<^sub>e\<^sub>d'.simps(6)
thf(fact_1800_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,Vg: list @ vEBT_VEBT,Vh: vEBT_VEBT,Vi: nat] :
( ( vEBT_T_s_u_c_c2 @ ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ V ) @ ( suc @ ( zero_zero @ nat ) ) @ Vg @ Vh ) @ Vi )
= ( one_one @ nat ) ) ).
% T\<^sub>s\<^sub>u\<^sub>c\<^sub>c'.simps(5)
thf(fact_1801_pred__bound__height_H,axiom,
! [T2: vEBT_VEBT,N3: nat,X: nat] :
( ( vEBT_invar_vebt @ T2 @ N3 )
=> ( ord_less_eq @ nat @ ( vEBT_T_p_r_e_d2 @ T2 @ X ) @ ( plus_plus @ nat @ ( one_one @ nat ) @ ( vEBT_VEBT_height @ T2 ) ) ) ) ).
% pred_bound_height'
thf(fact_1802_succ_H__bound__height,axiom,
! [T2: vEBT_VEBT,N3: nat,X: nat] :
( ( vEBT_invar_vebt @ T2 @ N3 )
=> ( ord_less_eq @ nat @ ( vEBT_T_s_u_c_c2 @ T2 @ X ) @ ( plus_plus @ nat @ ( one_one @ nat ) @ ( vEBT_VEBT_height @ T2 ) ) ) ) ).
% succ'_bound_height
thf(fact_1803_T_092_060_094sub_062m_092_060_094sub_062a_092_060_094sub_062x_092_060_094sub_062t_Oelims,axiom,
! [X: vEBT_VEBT,Y: nat] :
( ( ( vEBT_T_m_a_x_t @ X )
= Y )
=> ( ! [A4: $o,B3: $o] :
( ( X
= ( vEBT_Leaf @ A4 @ B3 ) )
=> ( Y
!= ( plus_plus @ nat @ ( one_one @ nat ) @ ( if @ nat @ B3 @ ( one_one @ nat ) @ ( plus_plus @ nat @ ( one_one @ nat ) @ ( one_one @ nat ) ) ) ) ) )
=> ( ( ? [Uu: nat,Uv: list @ vEBT_VEBT,Uw: vEBT_VEBT] :
( X
= ( vEBT_Node @ ( none @ ( product_prod @ nat @ nat ) ) @ Uu @ Uv @ Uw ) )
=> ( Y
!= ( one_one @ nat ) ) )
=> ~ ( ? [Mi2: nat,Ma2: nat,Ux2: nat,Uy2: list @ vEBT_VEBT,Uz2: vEBT_VEBT] :
( X
= ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi2 @ Ma2 ) ) @ Ux2 @ Uy2 @ Uz2 ) )
=> ( Y
!= ( one_one @ nat ) ) ) ) ) ) ).
% T\<^sub>m\<^sub>a\<^sub>x\<^sub>t.elims
thf(fact_1804_aset_I2_J,axiom,
! [D4: int,A3: set @ int,P: int > $o,Q: int > $o] :
( ! [X3: int] :
( ! [Xa2: int] :
( ( member @ int @ Xa2 @ ( set_or1337092689740270186AtMost @ int @ ( one_one @ int ) @ D4 ) )
=> ! [Xb2: int] :
( ( member @ int @ Xb2 @ A3 )
=> ( X3
!= ( minus_minus @ int @ Xb2 @ Xa2 ) ) ) )
=> ( ( P @ X3 )
=> ( P @ ( plus_plus @ int @ X3 @ D4 ) ) ) )
=> ( ! [X3: int] :
( ! [Xa2: int] :
( ( member @ int @ Xa2 @ ( set_or1337092689740270186AtMost @ int @ ( one_one @ int ) @ D4 ) )
=> ! [Xb2: int] :
( ( member @ int @ Xb2 @ A3 )
=> ( X3
!= ( minus_minus @ int @ Xb2 @ Xa2 ) ) ) )
=> ( ( Q @ X3 )
=> ( Q @ ( plus_plus @ int @ X3 @ D4 ) ) ) )
=> ! [X5: int] :
( ! [Xa3: int] :
( ( member @ int @ Xa3 @ ( set_or1337092689740270186AtMost @ int @ ( one_one @ int ) @ D4 ) )
=> ! [Xb3: int] :
( ( member @ int @ Xb3 @ A3 )
=> ( X5
!= ( minus_minus @ int @ Xb3 @ Xa3 ) ) ) )
=> ( ( ( P @ X5 )
| ( Q @ X5 ) )
=> ( ( P @ ( plus_plus @ int @ X5 @ D4 ) )
| ( Q @ ( plus_plus @ int @ X5 @ D4 ) ) ) ) ) ) ) ).
% aset(2)
thf(fact_1805_aset_I1_J,axiom,
! [D4: int,A3: set @ int,P: int > $o,Q: int > $o] :
( ! [X3: int] :
( ! [Xa2: int] :
( ( member @ int @ Xa2 @ ( set_or1337092689740270186AtMost @ int @ ( one_one @ int ) @ D4 ) )
=> ! [Xb2: int] :
( ( member @ int @ Xb2 @ A3 )
=> ( X3
!= ( minus_minus @ int @ Xb2 @ Xa2 ) ) ) )
=> ( ( P @ X3 )
=> ( P @ ( plus_plus @ int @ X3 @ D4 ) ) ) )
=> ( ! [X3: int] :
( ! [Xa2: int] :
( ( member @ int @ Xa2 @ ( set_or1337092689740270186AtMost @ int @ ( one_one @ int ) @ D4 ) )
=> ! [Xb2: int] :
( ( member @ int @ Xb2 @ A3 )
=> ( X3
!= ( minus_minus @ int @ Xb2 @ Xa2 ) ) ) )
=> ( ( Q @ X3 )
=> ( Q @ ( plus_plus @ int @ X3 @ D4 ) ) ) )
=> ! [X5: int] :
( ! [Xa3: int] :
( ( member @ int @ Xa3 @ ( set_or1337092689740270186AtMost @ int @ ( one_one @ int ) @ D4 ) )
=> ! [Xb3: int] :
( ( member @ int @ Xb3 @ A3 )
=> ( X5
!= ( minus_minus @ int @ Xb3 @ Xa3 ) ) ) )
=> ( ( ( P @ X5 )
& ( Q @ X5 ) )
=> ( ( P @ ( plus_plus @ int @ X5 @ D4 ) )
& ( Q @ ( plus_plus @ int @ X5 @ D4 ) ) ) ) ) ) ) ).
% aset(1)
thf(fact_1806_bset_I2_J,axiom,
! [D4: int,B6: set @ int,P: int > $o,Q: int > $o] :
( ! [X3: int] :
( ! [Xa2: int] :
( ( member @ int @ Xa2 @ ( set_or1337092689740270186AtMost @ int @ ( one_one @ int ) @ D4 ) )
=> ! [Xb2: int] :
( ( member @ int @ Xb2 @ B6 )
=> ( X3
!= ( plus_plus @ int @ Xb2 @ Xa2 ) ) ) )
=> ( ( P @ X3 )
=> ( P @ ( minus_minus @ int @ X3 @ D4 ) ) ) )
=> ( ! [X3: int] :
( ! [Xa2: int] :
( ( member @ int @ Xa2 @ ( set_or1337092689740270186AtMost @ int @ ( one_one @ int ) @ D4 ) )
=> ! [Xb2: int] :
( ( member @ int @ Xb2 @ B6 )
=> ( X3
!= ( plus_plus @ int @ Xb2 @ Xa2 ) ) ) )
=> ( ( Q @ X3 )
=> ( Q @ ( minus_minus @ int @ X3 @ D4 ) ) ) )
=> ! [X5: int] :
( ! [Xa3: int] :
( ( member @ int @ Xa3 @ ( set_or1337092689740270186AtMost @ int @ ( one_one @ int ) @ D4 ) )
=> ! [Xb3: int] :
( ( member @ int @ Xb3 @ B6 )
=> ( X5
!= ( plus_plus @ int @ Xb3 @ Xa3 ) ) ) )
=> ( ( ( P @ X5 )
| ( Q @ X5 ) )
=> ( ( P @ ( minus_minus @ int @ X5 @ D4 ) )
| ( Q @ ( minus_minus @ int @ X5 @ D4 ) ) ) ) ) ) ) ).
% bset(2)
thf(fact_1807_bset_I1_J,axiom,
! [D4: int,B6: set @ int,P: int > $o,Q: int > $o] :
( ! [X3: int] :
( ! [Xa2: int] :
( ( member @ int @ Xa2 @ ( set_or1337092689740270186AtMost @ int @ ( one_one @ int ) @ D4 ) )
=> ! [Xb2: int] :
( ( member @ int @ Xb2 @ B6 )
=> ( X3
!= ( plus_plus @ int @ Xb2 @ Xa2 ) ) ) )
=> ( ( P @ X3 )
=> ( P @ ( minus_minus @ int @ X3 @ D4 ) ) ) )
=> ( ! [X3: int] :
( ! [Xa2: int] :
( ( member @ int @ Xa2 @ ( set_or1337092689740270186AtMost @ int @ ( one_one @ int ) @ D4 ) )
=> ! [Xb2: int] :
( ( member @ int @ Xb2 @ B6 )
=> ( X3
!= ( plus_plus @ int @ Xb2 @ Xa2 ) ) ) )
=> ( ( Q @ X3 )
=> ( Q @ ( minus_minus @ int @ X3 @ D4 ) ) ) )
=> ! [X5: int] :
( ! [Xa3: int] :
( ( member @ int @ Xa3 @ ( set_or1337092689740270186AtMost @ int @ ( one_one @ int ) @ D4 ) )
=> ! [Xb3: int] :
( ( member @ int @ Xb3 @ B6 )
=> ( X5
!= ( plus_plus @ int @ Xb3 @ Xa3 ) ) ) )
=> ( ( ( P @ X5 )
& ( Q @ X5 ) )
=> ( ( P @ ( minus_minus @ int @ X5 @ D4 ) )
& ( Q @ ( minus_minus @ int @ X5 @ D4 ) ) ) ) ) ) ) ).
% bset(1)
thf(fact_1808_T_092_060_094sub_062m_092_060_094sub_062i_092_060_094sub_062n_092_060_094sub_062t_Oelims,axiom,
! [X: vEBT_VEBT,Y: nat] :
( ( ( vEBT_T_m_i_n_t @ X )
= Y )
=> ( ! [A4: $o] :
( ? [B3: $o] :
( X
= ( vEBT_Leaf @ A4 @ B3 ) )
=> ( Y
!= ( plus_plus @ nat @ ( one_one @ nat ) @ ( if @ nat @ A4 @ ( zero_zero @ nat ) @ ( plus_plus @ nat @ ( one_one @ nat ) @ ( one_one @ nat ) ) ) ) ) )
=> ( ( ? [Uu: nat,Uv: list @ vEBT_VEBT,Uw: vEBT_VEBT] :
( X
= ( vEBT_Node @ ( none @ ( product_prod @ nat @ nat ) ) @ Uu @ Uv @ Uw ) )
=> ( Y
!= ( one_one @ nat ) ) )
=> ~ ( ? [Mi2: nat,Ma2: nat,Ux2: nat,Uy2: list @ vEBT_VEBT,Uz2: vEBT_VEBT] :
( X
= ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi2 @ Ma2 ) ) @ Ux2 @ Uy2 @ Uz2 ) )
=> ( Y
!= ( one_one @ nat ) ) ) ) ) ) ).
% T\<^sub>m\<^sub>i\<^sub>n\<^sub>t.elims
thf(fact_1809_pred__bound__size__univ_H,axiom,
! [T2: vEBT_VEBT,N3: nat,U: real,X: nat] :
( ( vEBT_invar_vebt @ T2 @ N3 )
=> ( ( U
= ( power_power @ real @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) @ N3 ) )
=> ( ord_less_eq @ real @ ( semiring_1_of_nat @ real @ ( vEBT_T_p_r_e_d2 @ T2 @ X ) ) @ ( plus_plus @ real @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) @ ( log @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) @ ( log @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) @ U ) ) ) ) ) ) ).
% pred_bound_size_univ'
thf(fact_1810_succ__bound__size__univ_H,axiom,
! [T2: vEBT_VEBT,N3: nat,U: real,X: nat] :
( ( vEBT_invar_vebt @ T2 @ N3 )
=> ( ( U
= ( power_power @ real @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) @ N3 ) )
=> ( ord_less_eq @ real @ ( semiring_1_of_nat @ real @ ( vEBT_T_s_u_c_c2 @ T2 @ X ) ) @ ( plus_plus @ real @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) @ ( log @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) @ ( log @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) @ U ) ) ) ) ) ) ).
% succ_bound_size_univ'
thf(fact_1811_Bolzano,axiom,
! [A2: real,B2: real,P: real > real > $o] :
( ( ord_less_eq @ real @ A2 @ B2 )
=> ( ! [A4: real,B3: real,C4: real] :
( ( P @ A4 @ B3 )
=> ( ( P @ B3 @ C4 )
=> ( ( ord_less_eq @ real @ A4 @ B3 )
=> ( ( ord_less_eq @ real @ B3 @ C4 )
=> ( P @ A4 @ C4 ) ) ) ) )
=> ( ! [X3: real] :
( ( ord_less_eq @ real @ A2 @ X3 )
=> ( ( ord_less_eq @ real @ X3 @ B2 )
=> ? [D6: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ D6 )
& ! [A4: real,B3: real] :
( ( ( ord_less_eq @ real @ A4 @ X3 )
& ( ord_less_eq @ real @ X3 @ B3 )
& ( ord_less @ real @ ( minus_minus @ real @ B3 @ A4 ) @ D6 ) )
=> ( P @ A4 @ B3 ) ) ) ) )
=> ( P @ A2 @ B2 ) ) ) ) ).
% Bolzano
thf(fact_1812_plusinfinity,axiom,
! [D: int,P6: int > $o,P: int > $o] :
( ( ord_less @ int @ ( zero_zero @ int ) @ D )
=> ( ! [X3: int,K3: int] :
( ( P6 @ X3 )
= ( P6 @ ( minus_minus @ int @ X3 @ ( times_times @ int @ K3 @ D ) ) ) )
=> ( ? [Z4: int] :
! [X3: int] :
( ( ord_less @ int @ Z4 @ X3 )
=> ( ( P @ X3 )
= ( P6 @ X3 ) ) )
=> ( ? [X_1: int] : ( P6 @ X_1 )
=> ? [X_12: int] : ( P @ X_12 ) ) ) ) ) ).
% plusinfinity
thf(fact_1813_minusinfinity,axiom,
! [D: int,P1: int > $o,P: int > $o] :
( ( ord_less @ int @ ( zero_zero @ int ) @ D )
=> ( ! [X3: int,K3: int] :
( ( P1 @ X3 )
= ( P1 @ ( minus_minus @ int @ X3 @ ( times_times @ int @ K3 @ D ) ) ) )
=> ( ? [Z4: int] :
! [X3: int] :
( ( ord_less @ int @ X3 @ Z4 )
=> ( ( P @ X3 )
= ( P1 @ X3 ) ) )
=> ( ? [X_1: int] : ( P1 @ X_1 )
=> ? [X_12: int] : ( P @ X_12 ) ) ) ) ) ).
% minusinfinity
thf(fact_1814_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,X: 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 ) @ X )
= ( plus_plus @ nat @ ( one_one @ nat )
@ ( if @ nat @ ( ord_less @ nat @ Ma @ X ) @ ( one_one @ nat )
@ ( plus_plus @ nat @ ( plus_plus @ nat @ ( numeral_numeral @ nat @ ( bit0 @ ( bit1 @ ( bit0 @ one2 ) ) ) ) @ ( one_one @ nat ) )
@ ( if @ nat @ ( ord_less @ nat @ ( vEBT_VEBT_high @ X @ ( divide_divide @ nat @ ( suc @ ( suc @ 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 @ X @ ( 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 @ X @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) )
!= ( none @ nat ) )
& ( vEBT_VEBT_greater @ ( some @ nat @ ( vEBT_VEBT_low @ X @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList @ ( vEBT_VEBT_high @ X @ ( 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 @ X @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( vEBT_VEBT_low @ X @ ( 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 @ X @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) @ ( one_one @ nat ) )
@ ( if @ nat
@ ( ( vEBT_vebt_pred @ Summary @ ( vEBT_VEBT_high @ X @ ( 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 @ X @ ( 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_1815_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,X: 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 ) @ X )
= ( plus_plus @ nat @ ( one_one @ nat )
@ ( if @ nat @ ( ord_less @ nat @ X @ Mi ) @ ( one_one @ nat )
@ ( plus_plus @ nat @ ( numeral_numeral @ nat @ ( bit0 @ ( bit1 @ ( bit0 @ one2 ) ) ) )
@ ( if @ nat @ ( ord_less @ nat @ ( vEBT_VEBT_high @ X @ ( divide_divide @ nat @ ( suc @ ( suc @ 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 @ X @ ( 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 @ X @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) )
!= ( none @ nat ) )
& ( vEBT_VEBT_less @ ( some @ nat @ ( vEBT_VEBT_low @ X @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( vEBT_vebt_maxt @ ( nth @ vEBT_VEBT @ TreeList @ ( vEBT_VEBT_high @ X @ ( 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 @ X @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( vEBT_VEBT_low @ X @ ( 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 @ X @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) @ ( one_one @ nat ) )
@ ( if @ nat
@ ( ( vEBT_vebt_succ @ Summary @ ( vEBT_VEBT_high @ X @ ( 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 @ X @ ( 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_1816_incr__mult__lemma,axiom,
! [D: int,P: int > $o,K: int] :
( ( ord_less @ int @ ( zero_zero @ int ) @ D )
=> ( ! [X3: int] :
( ( P @ X3 )
=> ( P @ ( plus_plus @ int @ X3 @ D ) ) )
=> ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ K )
=> ! [X5: int] :
( ( P @ X5 )
=> ( P @ ( plus_plus @ int @ X5 @ ( times_times @ int @ K @ D ) ) ) ) ) ) ) ).
% incr_mult_lemma
thf(fact_1817_aset_I7_J,axiom,
! [D4: int,A3: set @ int,T2: int] :
( ( ord_less @ int @ ( zero_zero @ int ) @ D4 )
=> ! [X5: int] :
( ! [Xa3: int] :
( ( member @ int @ Xa3 @ ( set_or1337092689740270186AtMost @ int @ ( one_one @ int ) @ D4 ) )
=> ! [Xb3: int] :
( ( member @ int @ Xb3 @ A3 )
=> ( X5
!= ( minus_minus @ int @ Xb3 @ Xa3 ) ) ) )
=> ( ( ord_less @ int @ T2 @ X5 )
=> ( ord_less @ int @ T2 @ ( plus_plus @ int @ X5 @ D4 ) ) ) ) ) ).
% aset(7)
thf(fact_1818_aset_I5_J,axiom,
! [D4: int,T2: int,A3: set @ int] :
( ( ord_less @ int @ ( zero_zero @ int ) @ D4 )
=> ( ( member @ int @ T2 @ A3 )
=> ! [X5: int] :
( ! [Xa3: int] :
( ( member @ int @ Xa3 @ ( set_or1337092689740270186AtMost @ int @ ( one_one @ int ) @ D4 ) )
=> ! [Xb3: int] :
( ( member @ int @ Xb3 @ A3 )
=> ( X5
!= ( minus_minus @ int @ Xb3 @ Xa3 ) ) ) )
=> ( ( ord_less @ int @ X5 @ T2 )
=> ( ord_less @ int @ ( plus_plus @ int @ X5 @ D4 ) @ T2 ) ) ) ) ) ).
% aset(5)
thf(fact_1819_aset_I4_J,axiom,
! [D4: int,T2: int,A3: set @ int] :
( ( ord_less @ int @ ( zero_zero @ int ) @ D4 )
=> ( ( member @ int @ T2 @ A3 )
=> ! [X5: int] :
( ! [Xa3: int] :
( ( member @ int @ Xa3 @ ( set_or1337092689740270186AtMost @ int @ ( one_one @ int ) @ D4 ) )
=> ! [Xb3: int] :
( ( member @ int @ Xb3 @ A3 )
=> ( X5
!= ( minus_minus @ int @ Xb3 @ Xa3 ) ) ) )
=> ( ( X5 != T2 )
=> ( ( plus_plus @ int @ X5 @ D4 )
!= T2 ) ) ) ) ) ).
% aset(4)
thf(fact_1820_aset_I3_J,axiom,
! [D4: int,T2: int,A3: set @ int] :
( ( ord_less @ int @ ( zero_zero @ int ) @ D4 )
=> ( ( member @ int @ ( plus_plus @ int @ T2 @ ( one_one @ int ) ) @ A3 )
=> ! [X5: int] :
( ! [Xa3: int] :
( ( member @ int @ Xa3 @ ( set_or1337092689740270186AtMost @ int @ ( one_one @ int ) @ D4 ) )
=> ! [Xb3: int] :
( ( member @ int @ Xb3 @ A3 )
=> ( X5
!= ( minus_minus @ int @ Xb3 @ Xa3 ) ) ) )
=> ( ( X5 = T2 )
=> ( ( plus_plus @ int @ X5 @ D4 )
= T2 ) ) ) ) ) ).
% aset(3)
thf(fact_1821_bset_I7_J,axiom,
! [D4: int,T2: int,B6: set @ int] :
( ( ord_less @ int @ ( zero_zero @ int ) @ D4 )
=> ( ( member @ int @ T2 @ B6 )
=> ! [X5: int] :
( ! [Xa3: int] :
( ( member @ int @ Xa3 @ ( set_or1337092689740270186AtMost @ int @ ( one_one @ int ) @ D4 ) )
=> ! [Xb3: int] :
( ( member @ int @ Xb3 @ B6 )
=> ( X5
!= ( plus_plus @ int @ Xb3 @ Xa3 ) ) ) )
=> ( ( ord_less @ int @ T2 @ X5 )
=> ( ord_less @ int @ T2 @ ( minus_minus @ int @ X5 @ D4 ) ) ) ) ) ) ).
% bset(7)
thf(fact_1822_bset_I5_J,axiom,
! [D4: int,B6: set @ int,T2: int] :
( ( ord_less @ int @ ( zero_zero @ int ) @ D4 )
=> ! [X5: int] :
( ! [Xa3: int] :
( ( member @ int @ Xa3 @ ( set_or1337092689740270186AtMost @ int @ ( one_one @ int ) @ D4 ) )
=> ! [Xb3: int] :
( ( member @ int @ Xb3 @ B6 )
=> ( X5
!= ( plus_plus @ int @ Xb3 @ Xa3 ) ) ) )
=> ( ( ord_less @ int @ X5 @ T2 )
=> ( ord_less @ int @ ( minus_minus @ int @ X5 @ D4 ) @ T2 ) ) ) ) ).
% bset(5)
thf(fact_1823_bset_I4_J,axiom,
! [D4: int,T2: int,B6: set @ int] :
( ( ord_less @ int @ ( zero_zero @ int ) @ D4 )
=> ( ( member @ int @ T2 @ B6 )
=> ! [X5: int] :
( ! [Xa3: int] :
( ( member @ int @ Xa3 @ ( set_or1337092689740270186AtMost @ int @ ( one_one @ int ) @ D4 ) )
=> ! [Xb3: int] :
( ( member @ int @ Xb3 @ B6 )
=> ( X5
!= ( plus_plus @ int @ Xb3 @ Xa3 ) ) ) )
=> ( ( X5 != T2 )
=> ( ( minus_minus @ int @ X5 @ D4 )
!= T2 ) ) ) ) ) ).
% bset(4)
thf(fact_1824_bset_I3_J,axiom,
! [D4: int,T2: int,B6: set @ int] :
( ( ord_less @ int @ ( zero_zero @ int ) @ D4 )
=> ( ( member @ int @ ( minus_minus @ int @ T2 @ ( one_one @ int ) ) @ B6 )
=> ! [X5: int] :
( ! [Xa3: int] :
( ( member @ int @ Xa3 @ ( set_or1337092689740270186AtMost @ int @ ( one_one @ int ) @ D4 ) )
=> ! [Xb3: int] :
( ( member @ int @ Xb3 @ B6 )
=> ( X5
!= ( plus_plus @ int @ Xb3 @ Xa3 ) ) ) )
=> ( ( X5 = T2 )
=> ( ( minus_minus @ int @ X5 @ D4 )
= T2 ) ) ) ) ) ).
% bset(3)
thf(fact_1825_decr__mult__lemma,axiom,
! [D: int,P: int > $o,K: int] :
( ( ord_less @ int @ ( zero_zero @ int ) @ D )
=> ( ! [X3: int] :
( ( P @ X3 )
=> ( P @ ( minus_minus @ int @ X3 @ D ) ) )
=> ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ K )
=> ! [X5: int] :
( ( P @ X5 )
=> ( P @ ( minus_minus @ int @ X5 @ ( times_times @ int @ K @ D ) ) ) ) ) ) ) ).
% decr_mult_lemma
thf(fact_1826_periodic__finite__ex,axiom,
! [D: int,P: int > $o] :
( ( ord_less @ int @ ( zero_zero @ int ) @ D )
=> ( ! [X3: int,K3: int] :
( ( P @ X3 )
= ( P @ ( minus_minus @ int @ X3 @ ( times_times @ int @ K3 @ D ) ) ) )
=> ( ( ? [X6: int] : ( P @ X6 ) )
= ( ? [X2: int] :
( ( member @ int @ X2 @ ( set_or1337092689740270186AtMost @ int @ ( one_one @ int ) @ D ) )
& ( P @ X2 ) ) ) ) ) ) ).
% periodic_finite_ex
thf(fact_1827_T_092_060_094sub_062p_092_060_094sub_062r_092_060_094sub_062e_092_060_094sub_062d_Oelims,axiom,
! [X: vEBT_VEBT,Xa: nat,Y: nat] :
( ( ( vEBT_T_p_r_e_d @ X @ Xa )
= Y )
=> ( ( ? [Uu: $o,Uv: $o] :
( X
= ( vEBT_Leaf @ Uu @ Uv ) )
=> ( ( Xa
= ( zero_zero @ nat ) )
=> ( Y
!= ( one_one @ nat ) ) ) )
=> ( ( ? [A4: $o,Uw: $o] :
( X
= ( vEBT_Leaf @ A4 @ Uw ) )
=> ( ( Xa
= ( suc @ ( zero_zero @ nat ) ) )
=> ( Y
!= ( plus_plus @ nat @ ( one_one @ nat ) @ ( one_one @ nat ) ) ) ) )
=> ( ! [A4: $o,B3: $o] :
( ( X
= ( vEBT_Leaf @ A4 @ B3 ) )
=> ( ? [Va3: nat] :
( Xa
= ( suc @ ( suc @ Va3 ) ) )
=> ( Y
!= ( plus_plus @ nat @ ( one_one @ nat ) @ ( if @ nat @ B3 @ ( one_one @ nat ) @ ( plus_plus @ nat @ ( one_one @ nat ) @ ( one_one @ nat ) ) ) ) ) ) )
=> ( ( ? [Uy2: nat,Uz2: list @ vEBT_VEBT,Va2: vEBT_VEBT] :
( X
= ( vEBT_Node @ ( none @ ( product_prod @ nat @ nat ) ) @ Uy2 @ Uz2 @ Va2 ) )
=> ( Y
!= ( one_one @ nat ) ) )
=> ( ( ? [V3: product_prod @ nat @ nat,Vd2: list @ vEBT_VEBT,Ve2: vEBT_VEBT] :
( X
= ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ V3 ) @ ( zero_zero @ nat ) @ Vd2 @ Ve2 ) )
=> ( Y
!= ( one_one @ nat ) ) )
=> ( ( ? [V3: product_prod @ nat @ nat,Vh2: list @ vEBT_VEBT,Vi2: vEBT_VEBT] :
( X
= ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ V3 ) @ ( suc @ ( zero_zero @ nat ) ) @ Vh2 @ Vi2 ) )
=> ( Y
!= ( one_one @ nat ) ) )
=> ~ ! [Mi2: nat,Ma2: nat,Va3: nat,TreeList2: list @ vEBT_VEBT,Summary2: vEBT_VEBT] :
( ( X
= ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( suc @ Va3 ) ) @ TreeList2 @ Summary2 ) )
=> ( Y
!= ( plus_plus @ nat @ ( one_one @ nat )
@ ( if @ nat @ ( ord_less @ nat @ Ma2 @ Xa ) @ ( one_one @ nat )
@ ( plus_plus @ nat @ ( plus_plus @ nat @ ( numeral_numeral @ nat @ ( bit0 @ ( bit1 @ ( bit0 @ one2 ) ) ) ) @ ( one_one @ nat ) )
@ ( if @ nat @ ( ord_less @ nat @ ( vEBT_VEBT_high @ Xa @ ( divide_divide @ nat @ ( suc @ ( suc @ 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_1828_T_092_060_094sub_062s_092_060_094sub_062u_092_060_094sub_062c_092_060_094sub_062c_Oelims,axiom,
! [X: vEBT_VEBT,Xa: nat,Y: nat] :
( ( ( vEBT_T_s_u_c_c @ X @ Xa )
= Y )
=> ( ( ? [Uu: $o,B3: $o] :
( X
= ( vEBT_Leaf @ Uu @ B3 ) )
=> ( ( Xa
= ( zero_zero @ nat ) )
=> ( Y
!= ( plus_plus @ nat @ ( one_one @ nat ) @ ( one_one @ nat ) ) ) ) )
=> ( ( ? [Uv: $o,Uw: $o] :
( X
= ( vEBT_Leaf @ Uv @ Uw ) )
=> ( ? [N: nat] :
( Xa
= ( suc @ N ) )
=> ( Y
!= ( one_one @ nat ) ) ) )
=> ( ( ? [Ux2: nat,Uy2: list @ vEBT_VEBT,Uz2: vEBT_VEBT] :
( X
= ( vEBT_Node @ ( none @ ( product_prod @ nat @ nat ) ) @ Ux2 @ Uy2 @ Uz2 ) )
=> ( Y
!= ( one_one @ nat ) ) )
=> ( ( ? [V3: product_prod @ nat @ nat,Vc2: list @ vEBT_VEBT,Vd2: vEBT_VEBT] :
( X
= ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ V3 ) @ ( zero_zero @ nat ) @ Vc2 @ Vd2 ) )
=> ( Y
!= ( one_one @ nat ) ) )
=> ( ( ? [V3: product_prod @ nat @ nat,Vg2: list @ vEBT_VEBT,Vh2: vEBT_VEBT] :
( X
= ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ V3 ) @ ( suc @ ( zero_zero @ nat ) ) @ Vg2 @ Vh2 ) )
=> ( Y
!= ( one_one @ nat ) ) )
=> ~ ! [Mi2: nat,Ma2: nat,Va3: nat,TreeList2: list @ vEBT_VEBT,Summary2: vEBT_VEBT] :
( ( X
= ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( suc @ Va3 ) ) @ TreeList2 @ Summary2 ) )
=> ( Y
!= ( plus_plus @ nat @ ( one_one @ nat )
@ ( if @ nat @ ( ord_less @ nat @ Xa @ Mi2 ) @ ( one_one @ nat )
@ ( plus_plus @ nat @ ( numeral_numeral @ nat @ ( bit0 @ ( bit1 @ ( bit0 @ one2 ) ) ) )
@ ( if @ nat @ ( ord_less @ nat @ ( vEBT_VEBT_high @ Xa @ ( divide_divide @ nat @ ( suc @ ( suc @ 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_1829_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,X: nat] :
( ( vEBT_T_d_e_l_e_t_e @ ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi @ Ma ) ) @ ( suc @ ( suc @ Va ) ) @ TreeList @ Summary ) @ X )
= ( plus_plus @ nat @ ( numeral_numeral @ nat @ ( bit1 @ one2 ) )
@ ( if @ nat
@ ( ( ord_less @ nat @ X @ Mi )
| ( ord_less @ nat @ Ma @ X ) )
@ ( one_one @ nat )
@ ( plus_plus @ nat @ ( numeral_numeral @ nat @ ( bit1 @ one2 ) )
@ ( if @ nat
@ ( ( X = Mi )
& ( X = Ma ) )
@ ( numeral_numeral @ nat @ ( bit1 @ one2 ) )
@ ( plus_plus @ nat @ ( plus_plus @ nat @ ( plus_plus @ nat @ ( numeral_numeral @ nat @ ( bit1 @ ( bit0 @ ( bit1 @ one2 ) ) ) ) @ ( if @ nat @ ( X = Mi ) @ ( plus_plus @ nat @ ( plus_plus @ nat @ ( vEBT_T_m_i_n_t @ Summary ) @ ( vEBT_T_m_i_n_t @ ( nth @ vEBT_VEBT @ TreeList @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) @ ( numeral_numeral @ nat @ ( bit1 @ ( bit1 @ one2 ) ) ) ) @ ( one_one @ nat ) ) ) @ ( one_one @ nat ) )
@ ( if @ nat @ ( ord_less @ nat @ ( vEBT_VEBT_high @ ( if @ nat @ ( X = Mi ) @ ( plus_plus @ nat @ ( times_times @ nat @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ X ) @ ( divide_divide @ nat @ ( suc @ ( suc @ 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 @ ( X = Mi ) @ ( plus_plus @ nat @ ( times_times @ nat @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ X ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( vEBT_VEBT_low @ ( if @ nat @ ( X = Mi ) @ ( plus_plus @ nat @ ( times_times @ nat @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ X ) @ ( divide_divide @ nat @ ( suc @ ( suc @ 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 @ ( X = Mi ) @ ( plus_plus @ nat @ ( times_times @ nat @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ X ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( vEBT_VEBT_low @ ( if @ nat @ ( X = Mi ) @ ( plus_plus @ nat @ ( times_times @ nat @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ X ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) )
@ ( if @ nat @ ( vEBT_VEBT_minNull @ ( vEBT_vebt_delete @ ( nth @ vEBT_VEBT @ TreeList @ ( vEBT_VEBT_high @ ( if @ nat @ ( X = Mi ) @ ( plus_plus @ nat @ ( times_times @ nat @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ X ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( vEBT_VEBT_low @ ( if @ nat @ ( X = Mi ) @ ( plus_plus @ nat @ ( times_times @ nat @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ X ) @ ( divide_divide @ nat @ ( suc @ ( suc @ 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 @ ( X = Mi ) @ ( plus_plus @ nat @ ( times_times @ nat @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ X ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) )
@ ( plus_plus @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) )
@ ( if @ nat
@ ( ( ( X = Mi )
=> ( ( plus_plus @ nat @ ( times_times @ nat @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) )
= Ma ) )
& ( ( X != Mi )
=> ( X = Ma ) ) )
@ ( plus_plus @ nat @ ( plus_plus @ nat @ ( one_one @ nat ) @ ( vEBT_T_m_a_x_t @ ( vEBT_vebt_delete @ Summary @ ( vEBT_VEBT_high @ ( if @ nat @ ( X = Mi ) @ ( plus_plus @ nat @ ( times_times @ nat @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ X ) @ ( divide_divide @ nat @ ( suc @ ( suc @ 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 @ ( X = Mi ) @ ( plus_plus @ nat @ ( times_times @ nat @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ X ) @ ( divide_divide @ nat @ ( suc @ ( suc @ 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 @ ( X = Mi ) @ ( plus_plus @ nat @ ( times_times @ nat @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ X ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( vEBT_vebt_delete @ ( nth @ vEBT_VEBT @ TreeList @ ( vEBT_VEBT_high @ ( if @ nat @ ( X = Mi ) @ ( plus_plus @ nat @ ( times_times @ nat @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ X ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( vEBT_VEBT_low @ ( if @ nat @ ( X = Mi ) @ ( plus_plus @ nat @ ( times_times @ nat @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ X ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_maxt @ ( vEBT_vebt_delete @ Summary @ ( vEBT_VEBT_high @ ( if @ nat @ ( X = Mi ) @ ( plus_plus @ nat @ ( times_times @ nat @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ X ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) ) ) ) ) ) ) )
@ ( one_one @ nat ) ) ) )
@ ( plus_plus @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) )
@ ( if @ nat
@ ( ( ( X = Mi )
=> ( ( plus_plus @ nat @ ( times_times @ nat @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) )
= Ma ) )
& ( ( X != Mi )
=> ( X = Ma ) ) )
@ ( plus_plus @ nat @ ( numeral_numeral @ nat @ ( bit0 @ ( bit1 @ one2 ) ) ) @ ( vEBT_T_m_a_x_t @ ( nth @ vEBT_VEBT @ ( list_update @ vEBT_VEBT @ TreeList @ ( vEBT_VEBT_high @ ( if @ nat @ ( X = Mi ) @ ( plus_plus @ nat @ ( times_times @ nat @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ X ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( vEBT_vebt_delete @ ( nth @ vEBT_VEBT @ TreeList @ ( vEBT_VEBT_high @ ( if @ nat @ ( X = Mi ) @ ( plus_plus @ nat @ ( times_times @ nat @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ X ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( vEBT_VEBT_low @ ( if @ nat @ ( X = Mi ) @ ( plus_plus @ nat @ ( times_times @ nat @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ X ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) @ ( vEBT_VEBT_high @ ( if @ nat @ ( X = Mi ) @ ( plus_plus @ nat @ ( times_times @ nat @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ X ) @ ( divide_divide @ nat @ ( suc @ ( suc @ 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_1830_zdiff__int__split,axiom,
! [P: int > $o,X: nat,Y: nat] :
( ( P @ ( semiring_1_of_nat @ int @ ( minus_minus @ nat @ X @ Y ) ) )
= ( ( ( ord_less_eq @ nat @ Y @ X )
=> ( P @ ( minus_minus @ int @ ( semiring_1_of_nat @ int @ X ) @ ( semiring_1_of_nat @ int @ Y ) ) ) )
& ( ( ord_less @ nat @ X @ Y )
=> ( P @ ( zero_zero @ int ) ) ) ) ) ).
% zdiff_int_split
thf(fact_1831_aset_I8_J,axiom,
! [D4: int,A3: set @ int,T2: int] :
( ( ord_less @ int @ ( zero_zero @ int ) @ D4 )
=> ! [X5: int] :
( ! [Xa3: int] :
( ( member @ int @ Xa3 @ ( set_or1337092689740270186AtMost @ int @ ( one_one @ int ) @ D4 ) )
=> ! [Xb3: int] :
( ( member @ int @ Xb3 @ A3 )
=> ( X5
!= ( minus_minus @ int @ Xb3 @ Xa3 ) ) ) )
=> ( ( ord_less_eq @ int @ T2 @ X5 )
=> ( ord_less_eq @ int @ T2 @ ( plus_plus @ int @ X5 @ D4 ) ) ) ) ) ).
% aset(8)
thf(fact_1832_aset_I6_J,axiom,
! [D4: int,T2: int,A3: set @ int] :
( ( ord_less @ int @ ( zero_zero @ int ) @ D4 )
=> ( ( member @ int @ ( plus_plus @ int @ T2 @ ( one_one @ int ) ) @ A3 )
=> ! [X5: int] :
( ! [Xa3: int] :
( ( member @ int @ Xa3 @ ( set_or1337092689740270186AtMost @ int @ ( one_one @ int ) @ D4 ) )
=> ! [Xb3: int] :
( ( member @ int @ Xb3 @ A3 )
=> ( X5
!= ( minus_minus @ int @ Xb3 @ Xa3 ) ) ) )
=> ( ( ord_less_eq @ int @ X5 @ T2 )
=> ( ord_less_eq @ int @ ( plus_plus @ int @ X5 @ D4 ) @ T2 ) ) ) ) ) ).
% aset(6)
thf(fact_1833_bset_I8_J,axiom,
! [D4: int,T2: int,B6: set @ int] :
( ( ord_less @ int @ ( zero_zero @ int ) @ D4 )
=> ( ( member @ int @ ( minus_minus @ int @ T2 @ ( one_one @ int ) ) @ B6 )
=> ! [X5: int] :
( ! [Xa3: int] :
( ( member @ int @ Xa3 @ ( set_or1337092689740270186AtMost @ int @ ( one_one @ int ) @ D4 ) )
=> ! [Xb3: int] :
( ( member @ int @ Xb3 @ B6 )
=> ( X5
!= ( plus_plus @ int @ Xb3 @ Xa3 ) ) ) )
=> ( ( ord_less_eq @ int @ T2 @ X5 )
=> ( ord_less_eq @ int @ T2 @ ( minus_minus @ int @ X5 @ D4 ) ) ) ) ) ) ).
% bset(8)
thf(fact_1834_bset_I6_J,axiom,
! [D4: int,B6: set @ int,T2: int] :
( ( ord_less @ int @ ( zero_zero @ int ) @ D4 )
=> ! [X5: int] :
( ! [Xa3: int] :
( ( member @ int @ Xa3 @ ( set_or1337092689740270186AtMost @ int @ ( one_one @ int ) @ D4 ) )
=> ! [Xb3: int] :
( ( member @ int @ Xb3 @ B6 )
=> ( X5
!= ( plus_plus @ int @ Xb3 @ Xa3 ) ) ) )
=> ( ( ord_less_eq @ int @ X5 @ T2 )
=> ( ord_less_eq @ int @ ( minus_minus @ int @ X5 @ D4 ) @ T2 ) ) ) ) ).
% bset(6)
thf(fact_1835_cpmi,axiom,
! [D4: int,P: int > $o,P6: int > $o,B6: set @ int] :
( ( ord_less @ int @ ( zero_zero @ int ) @ D4 )
=> ( ? [Z4: int] :
! [X3: int] :
( ( ord_less @ int @ X3 @ Z4 )
=> ( ( P @ X3 )
= ( P6 @ X3 ) ) )
=> ( ! [X3: int] :
( ! [Xa2: int] :
( ( member @ int @ Xa2 @ ( set_or1337092689740270186AtMost @ int @ ( one_one @ int ) @ D4 ) )
=> ! [Xb2: int] :
( ( member @ int @ Xb2 @ B6 )
=> ( X3
!= ( plus_plus @ int @ Xb2 @ Xa2 ) ) ) )
=> ( ( P @ X3 )
=> ( P @ ( minus_minus @ int @ X3 @ D4 ) ) ) )
=> ( ! [X3: int,K3: int] :
( ( P6 @ X3 )
= ( P6 @ ( minus_minus @ int @ X3 @ ( times_times @ int @ K3 @ D4 ) ) ) )
=> ( ( ? [X6: int] : ( P @ X6 ) )
= ( ? [X2: int] :
( ( member @ int @ X2 @ ( set_or1337092689740270186AtMost @ int @ ( one_one @ int ) @ D4 ) )
& ( P6 @ X2 ) )
| ? [X2: int] :
( ( member @ int @ X2 @ ( set_or1337092689740270186AtMost @ int @ ( one_one @ int ) @ D4 ) )
& ? [Y2: int] :
( ( member @ int @ Y2 @ B6 )
& ( P @ ( plus_plus @ int @ Y2 @ X2 ) ) ) ) ) ) ) ) ) ) ).
% cpmi
thf(fact_1836_T_092_060_094sub_062d_092_060_094sub_062e_092_060_094sub_062l_092_060_094sub_062e_092_060_094sub_062t_092_060_094sub_062e_Opelims,axiom,
! [X: vEBT_VEBT,Xa: nat,Y: nat] :
( ( ( vEBT_T_d_e_l_e_t_e @ X @ Xa )
= Y )
=> ( ( accp @ ( product_prod @ vEBT_VEBT @ nat ) @ vEBT_T8441311223069195367_e_rel @ ( product_Pair @ vEBT_VEBT @ nat @ X @ Xa ) )
=> ( ! [A4: $o,B3: $o] :
( ( X
= ( vEBT_Leaf @ A4 @ B3 ) )
=> ( ( Xa
= ( zero_zero @ nat ) )
=> ( ( Y
= ( one_one @ nat ) )
=> ~ ( accp @ ( product_prod @ vEBT_VEBT @ nat ) @ vEBT_T8441311223069195367_e_rel @ ( product_Pair @ vEBT_VEBT @ nat @ ( vEBT_Leaf @ A4 @ B3 ) @ ( zero_zero @ nat ) ) ) ) ) )
=> ( ! [A4: $o,B3: $o] :
( ( X
= ( vEBT_Leaf @ A4 @ B3 ) )
=> ( ( Xa
= ( suc @ ( zero_zero @ nat ) ) )
=> ( ( Y
= ( one_one @ nat ) )
=> ~ ( accp @ ( product_prod @ vEBT_VEBT @ nat ) @ vEBT_T8441311223069195367_e_rel @ ( product_Pair @ vEBT_VEBT @ nat @ ( vEBT_Leaf @ A4 @ B3 ) @ ( suc @ ( zero_zero @ nat ) ) ) ) ) ) )
=> ( ! [A4: $o,B3: $o] :
( ( X
= ( vEBT_Leaf @ A4 @ B3 ) )
=> ! [N: nat] :
( ( Xa
= ( suc @ ( suc @ N ) ) )
=> ( ( Y
= ( one_one @ nat ) )
=> ~ ( accp @ ( product_prod @ vEBT_VEBT @ nat ) @ vEBT_T8441311223069195367_e_rel @ ( product_Pair @ vEBT_VEBT @ nat @ ( vEBT_Leaf @ A4 @ B3 ) @ ( suc @ ( suc @ N ) ) ) ) ) ) )
=> ( ! [Deg2: nat,TreeList2: list @ vEBT_VEBT,Summary2: vEBT_VEBT] :
( ( X
= ( vEBT_Node @ ( none @ ( product_prod @ nat @ nat ) ) @ Deg2 @ TreeList2 @ Summary2 ) )
=> ( ( Y
= ( one_one @ nat ) )
=> ~ ( accp @ ( product_prod @ vEBT_VEBT @ nat ) @ vEBT_T8441311223069195367_e_rel @ ( product_Pair @ vEBT_VEBT @ nat @ ( vEBT_Node @ ( none @ ( product_prod @ nat @ nat ) ) @ Deg2 @ TreeList2 @ Summary2 ) @ Xa ) ) ) )
=> ( ! [Mi2: nat,Ma2: nat,TreeList2: list @ vEBT_VEBT,Summary2: vEBT_VEBT] :
( ( X
= ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi2 @ Ma2 ) ) @ ( zero_zero @ nat ) @ TreeList2 @ Summary2 ) )
=> ( ( Y
= ( one_one @ nat ) )
=> ~ ( accp @ ( product_prod @ vEBT_VEBT @ nat ) @ vEBT_T8441311223069195367_e_rel @ ( product_Pair @ vEBT_VEBT @ nat @ ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi2 @ Ma2 ) ) @ ( zero_zero @ nat ) @ TreeList2 @ Summary2 ) @ Xa ) ) ) )
=> ( ! [Mi2: nat,Ma2: nat,TreeList2: list @ vEBT_VEBT,Summary2: vEBT_VEBT] :
( ( X
= ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( zero_zero @ nat ) ) @ TreeList2 @ Summary2 ) )
=> ( ( Y
= ( one_one @ nat ) )
=> ~ ( accp @ ( product_prod @ vEBT_VEBT @ nat ) @ vEBT_T8441311223069195367_e_rel @ ( product_Pair @ vEBT_VEBT @ nat @ ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( zero_zero @ nat ) ) @ TreeList2 @ Summary2 ) @ Xa ) ) ) )
=> ~ ! [Mi2: nat,Ma2: nat,Va3: nat,TreeList2: list @ vEBT_VEBT,Summary2: vEBT_VEBT] :
( ( X
= ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( suc @ Va3 ) ) @ TreeList2 @ Summary2 ) )
=> ( ( Y
= ( plus_plus @ nat @ ( numeral_numeral @ nat @ ( bit1 @ one2 ) )
@ ( if @ nat
@ ( ( ord_less @ nat @ Xa @ Mi2 )
| ( ord_less @ nat @ Ma2 @ Xa ) )
@ ( one_one @ nat )
@ ( plus_plus @ nat @ ( numeral_numeral @ nat @ ( bit1 @ one2 ) )
@ ( if @ nat
@ ( ( Xa = Mi2 )
& ( Xa = Ma2 ) )
@ ( numeral_numeral @ nat @ ( bit1 @ one2 ) )
@ ( plus_plus @ nat @ ( plus_plus @ nat @ ( plus_plus @ nat @ ( numeral_numeral @ nat @ ( bit1 @ ( bit0 @ ( bit1 @ one2 ) ) ) ) @ ( if @ nat @ ( Xa = Mi2 ) @ ( plus_plus @ nat @ ( plus_plus @ nat @ ( vEBT_T_m_i_n_t @ Summary2 ) @ ( vEBT_T_m_i_n_t @ ( nth @ vEBT_VEBT @ TreeList2 @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary2 ) ) ) ) ) @ ( numeral_numeral @ nat @ ( bit1 @ ( bit1 @ one2 ) ) ) ) @ ( one_one @ nat ) ) ) @ ( one_one @ nat ) )
@ ( if @ nat @ ( ord_less @ nat @ ( vEBT_VEBT_high @ ( if @ nat @ ( Xa = Mi2 ) @ ( plus_plus @ nat @ ( times_times @ nat @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary2 ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ ( suc @ ( suc @ 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_1837_T_092_060_094sub_062s_092_060_094sub_062u_092_060_094sub_062c_092_060_094sub_062c_Opelims,axiom,
! [X: vEBT_VEBT,Xa: nat,Y: nat] :
( ( ( vEBT_T_s_u_c_c @ X @ Xa )
= Y )
=> ( ( accp @ ( product_prod @ vEBT_VEBT @ nat ) @ vEBT_T_s_u_c_c_rel2 @ ( product_Pair @ vEBT_VEBT @ nat @ X @ Xa ) )
=> ( ! [Uu: $o,B3: $o] :
( ( X
= ( vEBT_Leaf @ Uu @ B3 ) )
=> ( ( Xa
= ( zero_zero @ nat ) )
=> ( ( Y
= ( plus_plus @ nat @ ( one_one @ nat ) @ ( one_one @ nat ) ) )
=> ~ ( accp @ ( product_prod @ vEBT_VEBT @ nat ) @ vEBT_T_s_u_c_c_rel2 @ ( product_Pair @ vEBT_VEBT @ nat @ ( vEBT_Leaf @ Uu @ B3 ) @ ( zero_zero @ nat ) ) ) ) ) )
=> ( ! [Uv: $o,Uw: $o] :
( ( X
= ( vEBT_Leaf @ Uv @ Uw ) )
=> ! [N: nat] :
( ( Xa
= ( suc @ N ) )
=> ( ( Y
= ( one_one @ nat ) )
=> ~ ( accp @ ( product_prod @ vEBT_VEBT @ nat ) @ vEBT_T_s_u_c_c_rel2 @ ( product_Pair @ vEBT_VEBT @ nat @ ( vEBT_Leaf @ Uv @ Uw ) @ ( suc @ N ) ) ) ) ) )
=> ( ! [Ux2: nat,Uy2: list @ vEBT_VEBT,Uz2: vEBT_VEBT] :
( ( X
= ( vEBT_Node @ ( none @ ( product_prod @ nat @ nat ) ) @ Ux2 @ Uy2 @ Uz2 ) )
=> ( ( Y
= ( one_one @ nat ) )
=> ~ ( accp @ ( product_prod @ vEBT_VEBT @ nat ) @ vEBT_T_s_u_c_c_rel2 @ ( product_Pair @ vEBT_VEBT @ nat @ ( vEBT_Node @ ( none @ ( product_prod @ nat @ nat ) ) @ Ux2 @ Uy2 @ Uz2 ) @ Xa ) ) ) )
=> ( ! [V3: product_prod @ nat @ nat,Vc2: list @ vEBT_VEBT,Vd2: vEBT_VEBT] :
( ( X
= ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ V3 ) @ ( zero_zero @ nat ) @ Vc2 @ Vd2 ) )
=> ( ( Y
= ( one_one @ nat ) )
=> ~ ( accp @ ( product_prod @ vEBT_VEBT @ nat ) @ vEBT_T_s_u_c_c_rel2 @ ( product_Pair @ vEBT_VEBT @ nat @ ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ V3 ) @ ( zero_zero @ nat ) @ Vc2 @ Vd2 ) @ Xa ) ) ) )
=> ( ! [V3: product_prod @ nat @ nat,Vg2: list @ vEBT_VEBT,Vh2: vEBT_VEBT] :
( ( X
= ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ V3 ) @ ( suc @ ( zero_zero @ nat ) ) @ Vg2 @ Vh2 ) )
=> ( ( Y
= ( one_one @ nat ) )
=> ~ ( accp @ ( product_prod @ vEBT_VEBT @ nat ) @ vEBT_T_s_u_c_c_rel2 @ ( product_Pair @ vEBT_VEBT @ nat @ ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ V3 ) @ ( suc @ ( zero_zero @ nat ) ) @ Vg2 @ Vh2 ) @ Xa ) ) ) )
=> ~ ! [Mi2: nat,Ma2: nat,Va3: nat,TreeList2: list @ vEBT_VEBT,Summary2: vEBT_VEBT] :
( ( X
= ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( suc @ Va3 ) ) @ TreeList2 @ Summary2 ) )
=> ( ( Y
= ( plus_plus @ nat @ ( one_one @ nat )
@ ( if @ nat @ ( ord_less @ nat @ Xa @ Mi2 ) @ ( one_one @ nat )
@ ( plus_plus @ nat @ ( numeral_numeral @ nat @ ( bit0 @ ( bit1 @ ( bit0 @ one2 ) ) ) )
@ ( if @ nat @ ( ord_less @ nat @ ( vEBT_VEBT_high @ Xa @ ( divide_divide @ nat @ ( suc @ ( suc @ 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_1838_T_092_060_094sub_062p_092_060_094sub_062r_092_060_094sub_062e_092_060_094sub_062d_Opelims,axiom,
! [X: vEBT_VEBT,Xa: nat,Y: nat] :
( ( ( vEBT_T_p_r_e_d @ X @ Xa )
= Y )
=> ( ( accp @ ( product_prod @ vEBT_VEBT @ nat ) @ vEBT_T_p_r_e_d_rel2 @ ( product_Pair @ vEBT_VEBT @ nat @ X @ Xa ) )
=> ( ! [Uu: $o,Uv: $o] :
( ( X
= ( vEBT_Leaf @ Uu @ Uv ) )
=> ( ( Xa
= ( zero_zero @ nat ) )
=> ( ( Y
= ( one_one @ nat ) )
=> ~ ( accp @ ( product_prod @ vEBT_VEBT @ nat ) @ vEBT_T_p_r_e_d_rel2 @ ( product_Pair @ vEBT_VEBT @ nat @ ( vEBT_Leaf @ Uu @ Uv ) @ ( zero_zero @ nat ) ) ) ) ) )
=> ( ! [A4: $o,Uw: $o] :
( ( X
= ( vEBT_Leaf @ A4 @ Uw ) )
=> ( ( Xa
= ( suc @ ( zero_zero @ nat ) ) )
=> ( ( Y
= ( plus_plus @ nat @ ( one_one @ nat ) @ ( one_one @ nat ) ) )
=> ~ ( accp @ ( product_prod @ vEBT_VEBT @ nat ) @ vEBT_T_p_r_e_d_rel2 @ ( product_Pair @ vEBT_VEBT @ nat @ ( vEBT_Leaf @ A4 @ Uw ) @ ( suc @ ( zero_zero @ nat ) ) ) ) ) ) )
=> ( ! [A4: $o,B3: $o] :
( ( X
= ( vEBT_Leaf @ A4 @ B3 ) )
=> ! [Va3: nat] :
( ( Xa
= ( suc @ ( suc @ Va3 ) ) )
=> ( ( Y
= ( plus_plus @ nat @ ( one_one @ nat ) @ ( if @ nat @ B3 @ ( 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 @ B3 ) @ ( suc @ ( suc @ Va3 ) ) ) ) ) ) )
=> ( ! [Uy2: nat,Uz2: list @ vEBT_VEBT,Va2: vEBT_VEBT] :
( ( X
= ( vEBT_Node @ ( none @ ( product_prod @ nat @ nat ) ) @ Uy2 @ Uz2 @ Va2 ) )
=> ( ( Y
= ( one_one @ nat ) )
=> ~ ( accp @ ( product_prod @ vEBT_VEBT @ nat ) @ vEBT_T_p_r_e_d_rel2 @ ( product_Pair @ vEBT_VEBT @ nat @ ( vEBT_Node @ ( none @ ( product_prod @ nat @ nat ) ) @ Uy2 @ Uz2 @ Va2 ) @ Xa ) ) ) )
=> ( ! [V3: product_prod @ nat @ nat,Vd2: list @ vEBT_VEBT,Ve2: vEBT_VEBT] :
( ( X
= ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ V3 ) @ ( zero_zero @ nat ) @ Vd2 @ Ve2 ) )
=> ( ( Y
= ( one_one @ nat ) )
=> ~ ( accp @ ( product_prod @ vEBT_VEBT @ nat ) @ vEBT_T_p_r_e_d_rel2 @ ( product_Pair @ vEBT_VEBT @ nat @ ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ V3 ) @ ( zero_zero @ nat ) @ Vd2 @ Ve2 ) @ Xa ) ) ) )
=> ( ! [V3: product_prod @ nat @ nat,Vh2: list @ vEBT_VEBT,Vi2: vEBT_VEBT] :
( ( X
= ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ V3 ) @ ( suc @ ( zero_zero @ nat ) ) @ Vh2 @ Vi2 ) )
=> ( ( Y
= ( one_one @ nat ) )
=> ~ ( accp @ ( product_prod @ vEBT_VEBT @ nat ) @ vEBT_T_p_r_e_d_rel2 @ ( product_Pair @ vEBT_VEBT @ nat @ ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ V3 ) @ ( suc @ ( zero_zero @ nat ) ) @ Vh2 @ Vi2 ) @ Xa ) ) ) )
=> ~ ! [Mi2: nat,Ma2: nat,Va3: nat,TreeList2: list @ vEBT_VEBT,Summary2: vEBT_VEBT] :
( ( X
= ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( suc @ Va3 ) ) @ TreeList2 @ Summary2 ) )
=> ( ( Y
= ( plus_plus @ nat @ ( one_one @ nat )
@ ( if @ nat @ ( ord_less @ nat @ Ma2 @ Xa ) @ ( one_one @ nat )
@ ( plus_plus @ nat @ ( plus_plus @ nat @ ( numeral_numeral @ nat @ ( bit0 @ ( bit1 @ ( bit0 @ one2 ) ) ) ) @ ( one_one @ nat ) )
@ ( if @ nat @ ( ord_less @ nat @ ( vEBT_VEBT_high @ Xa @ ( divide_divide @ nat @ ( suc @ ( suc @ 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_1839_int__ops_I6_J,axiom,
! [A2: nat,B2: nat] :
( ( ( ord_less @ int @ ( semiring_1_of_nat @ int @ A2 ) @ ( semiring_1_of_nat @ int @ B2 ) )
=> ( ( semiring_1_of_nat @ int @ ( minus_minus @ nat @ A2 @ B2 ) )
= ( zero_zero @ int ) ) )
& ( ~ ( ord_less @ int @ ( semiring_1_of_nat @ int @ A2 ) @ ( semiring_1_of_nat @ int @ B2 ) )
=> ( ( semiring_1_of_nat @ int @ ( minus_minus @ nat @ A2 @ B2 ) )
= ( minus_minus @ int @ ( semiring_1_of_nat @ int @ A2 ) @ ( semiring_1_of_nat @ int @ B2 ) ) ) ) ) ).
% int_ops(6)
thf(fact_1840_pos__mult__pos__ge,axiom,
! [X: int,N3: int] :
( ( ord_less @ int @ ( zero_zero @ int ) @ X )
=> ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ N3 )
=> ( ord_less_eq @ int @ ( times_times @ int @ N3 @ ( one_one @ int ) ) @ ( times_times @ int @ N3 @ X ) ) ) ) ).
% pos_mult_pos_ge
thf(fact_1841_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_1842_VEBT__internal_OT_092_060_094sub_062d_092_060_094sub_062e_092_060_094sub_062l_092_060_094sub_062e_092_060_094sub_062t_092_060_094sub_062e_H_Opelims,axiom,
! [X: vEBT_VEBT,Xa: nat,Y: nat] :
( ( ( vEBT_V1232361888498592333_e_t_e @ X @ Xa )
= Y )
=> ( ( accp @ ( product_prod @ vEBT_VEBT @ nat ) @ vEBT_V6368547301243506412_e_rel @ ( product_Pair @ vEBT_VEBT @ nat @ X @ Xa ) )
=> ( ! [A4: $o,B3: $o] :
( ( X
= ( vEBT_Leaf @ A4 @ B3 ) )
=> ( ( Xa
= ( zero_zero @ nat ) )
=> ( ( Y
= ( one_one @ nat ) )
=> ~ ( accp @ ( product_prod @ vEBT_VEBT @ nat ) @ vEBT_V6368547301243506412_e_rel @ ( product_Pair @ vEBT_VEBT @ nat @ ( vEBT_Leaf @ A4 @ B3 ) @ ( zero_zero @ nat ) ) ) ) ) )
=> ( ! [A4: $o,B3: $o] :
( ( X
= ( vEBT_Leaf @ A4 @ B3 ) )
=> ( ( Xa
= ( suc @ ( zero_zero @ nat ) ) )
=> ( ( Y
= ( one_one @ nat ) )
=> ~ ( accp @ ( product_prod @ vEBT_VEBT @ nat ) @ vEBT_V6368547301243506412_e_rel @ ( product_Pair @ vEBT_VEBT @ nat @ ( vEBT_Leaf @ A4 @ B3 ) @ ( suc @ ( zero_zero @ nat ) ) ) ) ) ) )
=> ( ! [A4: $o,B3: $o] :
( ( X
= ( vEBT_Leaf @ A4 @ B3 ) )
=> ! [N: nat] :
( ( Xa
= ( suc @ ( suc @ N ) ) )
=> ( ( Y
= ( one_one @ nat ) )
=> ~ ( accp @ ( product_prod @ vEBT_VEBT @ nat ) @ vEBT_V6368547301243506412_e_rel @ ( product_Pair @ vEBT_VEBT @ nat @ ( vEBT_Leaf @ A4 @ B3 ) @ ( suc @ ( suc @ N ) ) ) ) ) ) )
=> ( ! [Deg2: nat,TreeList2: list @ vEBT_VEBT,Summary2: vEBT_VEBT] :
( ( X
= ( vEBT_Node @ ( none @ ( product_prod @ nat @ nat ) ) @ Deg2 @ TreeList2 @ Summary2 ) )
=> ( ( Y
= ( one_one @ nat ) )
=> ~ ( accp @ ( product_prod @ vEBT_VEBT @ nat ) @ vEBT_V6368547301243506412_e_rel @ ( product_Pair @ vEBT_VEBT @ nat @ ( vEBT_Node @ ( none @ ( product_prod @ nat @ nat ) ) @ Deg2 @ TreeList2 @ Summary2 ) @ Xa ) ) ) )
=> ( ! [Mi2: nat,Ma2: nat,TreeList2: list @ vEBT_VEBT,Summary2: vEBT_VEBT] :
( ( X
= ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi2 @ Ma2 ) ) @ ( zero_zero @ nat ) @ TreeList2 @ Summary2 ) )
=> ( ( Y
= ( one_one @ nat ) )
=> ~ ( accp @ ( product_prod @ vEBT_VEBT @ nat ) @ vEBT_V6368547301243506412_e_rel @ ( product_Pair @ vEBT_VEBT @ nat @ ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi2 @ Ma2 ) ) @ ( zero_zero @ nat ) @ TreeList2 @ Summary2 ) @ Xa ) ) ) )
=> ( ! [Mi2: nat,Ma2: nat,TreeList2: list @ vEBT_VEBT,Summary2: vEBT_VEBT] :
( ( X
= ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( zero_zero @ nat ) ) @ TreeList2 @ Summary2 ) )
=> ( ( Y
= ( one_one @ nat ) )
=> ~ ( accp @ ( product_prod @ vEBT_VEBT @ nat ) @ vEBT_V6368547301243506412_e_rel @ ( product_Pair @ vEBT_VEBT @ nat @ ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( zero_zero @ nat ) ) @ TreeList2 @ Summary2 ) @ Xa ) ) ) )
=> ~ ! [Mi2: nat,Ma2: nat,Va3: nat,TreeList2: list @ vEBT_VEBT,Summary2: vEBT_VEBT] :
( ( X
= ( 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 ) )
=> ( Y
= ( one_one @ nat ) ) )
& ( ~ ( ( ord_less @ nat @ Xa @ Mi2 )
| ( ord_less @ nat @ Ma2 @ Xa ) )
=> ( ( ( ( Xa = Mi2 )
& ( Xa = Ma2 ) )
=> ( Y
= ( one_one @ nat ) ) )
& ( ~ ( ( Xa = Mi2 )
& ( Xa = Ma2 ) )
=> ( Y
= ( if @ nat @ ( ord_less @ nat @ ( vEBT_VEBT_high @ ( if @ nat @ ( Xa = Mi2 ) @ ( plus_plus @ nat @ ( times_times @ nat @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary2 ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ ( suc @ ( suc @ 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_1843_verit__eq__simplify_I8_J,axiom,
! [X22: num,Y22: num] :
( ( ( bit0 @ X22 )
= ( bit0 @ Y22 ) )
= ( X22 = Y22 ) ) ).
% verit_eq_simplify(8)
thf(fact_1844_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_1845_le__minus,axiom,
! [Aa: $tType,A: $tType] :
( ( ( type_len @ A )
& ( order @ Aa ) )
=> ! [Y: Aa,X: Aa,A2: word @ A,C2: word @ A,B2: word @ A] :
( ( ord_less_eq @ Aa @ Y @ X )
=> ( ( ord_less_eq @ ( word @ A ) @ ( plus_plus @ ( word @ A ) @ A2 @ C2 ) @ B2 )
=> ( ( ord_less_eq @ ( word @ A ) @ A2 @ ( plus_plus @ ( word @ A ) @ A2 @ C2 ) )
=> ( ord_less_eq @ ( word @ A ) @ C2 @ ( minus_minus @ ( word @ A ) @ B2 @ A2 ) ) ) ) ) ) ).
% le_minus
thf(fact_1846_verit__la__disequality,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A2: A,B2: A] :
( ( A2 = B2 )
| ~ ( ord_less_eq @ A @ A2 @ B2 )
| ~ ( ord_less_eq @ A @ B2 @ A2 ) ) ) ).
% verit_la_disequality
thf(fact_1847_verit__comp__simplify1_I2_J,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [A2: A] : ( ord_less_eq @ A @ A2 @ A2 ) ) ).
% verit_comp_simplify1(2)
thf(fact_1848_verit__comp__simplify1_I1_J,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [A2: A] :
~ ( ord_less @ A @ A2 @ A2 ) ) ).
% verit_comp_simplify1(1)
thf(fact_1849_size__0__same_H,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [W: word @ A,V: word @ A] :
( ( ( size_size @ ( word @ A ) @ W )
= ( zero_zero @ nat ) )
=> ( W = V ) ) ) ).
% size_0_same'
thf(fact_1850_lens__not__0,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [W: word @ A] :
( ( size_size @ ( word @ A ) @ W )
!= ( zero_zero @ nat ) ) ) ).
% lens_not_0
thf(fact_1851_size__0__eq,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [W: word @ A,V: word @ A] :
( ( ( size_size @ ( word @ A ) @ W )
= ( zero_zero @ nat ) )
=> ( V = W ) ) ) ).
% size_0_eq
thf(fact_1852_Abs__fnat__hom__mult,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [A2: nat,B2: nat] :
( ( times_times @ ( word @ A ) @ ( semiring_1_of_nat @ ( word @ A ) @ A2 ) @ ( semiring_1_of_nat @ ( word @ A ) @ B2 ) )
= ( semiring_1_of_nat @ ( word @ A ) @ ( times_times @ nat @ A2 @ B2 ) ) ) ) ).
% Abs_fnat_hom_mult
thf(fact_1853_verit__la__generic,axiom,
! [A2: int,X: int] :
( ( ord_less_eq @ int @ A2 @ X )
| ( A2 = X )
| ( ord_less_eq @ int @ X @ A2 ) ) ).
% verit_la_generic
thf(fact_1854_wi__hom__sub,axiom,
! [B: $tType] :
( ( type_len @ B )
=> ! [A2: int,B2: int] :
( ( minus_minus @ ( word @ B ) @ ( ring_1_of_int @ ( word @ B ) @ A2 ) @ ( ring_1_of_int @ ( word @ B ) @ B2 ) )
= ( ring_1_of_int @ ( word @ B ) @ ( minus_minus @ int @ A2 @ B2 ) ) ) ) ).
% wi_hom_sub
thf(fact_1855_verit__comp__simplify1_I3_J,axiom,
! [B: $tType] :
( ( linorder @ B )
=> ! [B4: B,A5: B] :
( ( ~ ( ord_less_eq @ B @ B4 @ A5 ) )
= ( ord_less @ B @ A5 @ B4 ) ) ) ).
% verit_comp_simplify1(3)
thf(fact_1856_verit__sum__simplify,axiom,
! [A: $tType] :
( ( cancel1802427076303600483id_add @ A )
=> ! [A2: A] :
( ( plus_plus @ A @ A2 @ ( zero_zero @ A ) )
= A2 ) ) ).
% verit_sum_simplify
thf(fact_1857_verit__eq__simplify_I10_J,axiom,
! [X22: num] :
( one2
!= ( bit0 @ X22 ) ) ).
% verit_eq_simplify(10)
thf(fact_1858_verit__eq__simplify_I14_J,axiom,
! [X22: num,X33: num] :
( ( bit0 @ X22 )
!= ( bit1 @ X33 ) ) ).
% verit_eq_simplify(14)
thf(fact_1859_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_1860_verit__eq__simplify_I12_J,axiom,
! [X33: num] :
( one2
!= ( bit1 @ X33 ) ) ).
% verit_eq_simplify(12)
thf(fact_1861_max__def__raw,axiom,
! [A: $tType] :
( ( ord @ A )
=> ( ( ord_max @ A )
= ( ^ [A6: A,B5: A] : ( if @ A @ ( ord_less_eq @ A @ A6 @ B5 ) @ B5 @ A6 ) ) ) ) ).
% max_def_raw
thf(fact_1862_p2__eq__1,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [N3: nat] :
( ( ( power_power @ ( word @ A ) @ ( numeral_numeral @ ( word @ A ) @ ( bit0 @ one2 ) ) @ N3 )
= ( one_one @ ( word @ A ) ) )
= ( N3
= ( zero_zero @ nat ) ) ) ) ).
% p2_eq_1
thf(fact_1863_Abs__fnat__hom__add,axiom,
! [A: $tType] :
( ( semiring_1 @ A )
=> ! [A2: nat,B2: nat] :
( ( plus_plus @ A @ ( semiring_1_of_nat @ A @ A2 ) @ ( semiring_1_of_nat @ A @ B2 ) )
= ( semiring_1_of_nat @ A @ ( plus_plus @ nat @ A2 @ B2 ) ) ) ) ).
% Abs_fnat_hom_add
thf(fact_1864_int__ops_I3_J,axiom,
! [N3: num] :
( ( semiring_1_of_nat @ int @ ( numeral_numeral @ nat @ N3 ) )
= ( numeral_numeral @ int @ N3 ) ) ).
% int_ops(3)
thf(fact_1865_nat__int__comparison_I2_J,axiom,
( ( ord_less @ nat )
= ( ^ [A6: nat,B5: nat] : ( ord_less @ int @ ( semiring_1_of_nat @ int @ A6 ) @ ( semiring_1_of_nat @ int @ B5 ) ) ) ) ).
% nat_int_comparison(2)
thf(fact_1866_nat__int__comparison_I3_J,axiom,
( ( ord_less_eq @ nat )
= ( ^ [A6: nat,B5: nat] : ( ord_less_eq @ int @ ( semiring_1_of_nat @ int @ A6 ) @ ( semiring_1_of_nat @ int @ B5 ) ) ) ) ).
% nat_int_comparison(3)
thf(fact_1867_int__ops_I2_J,axiom,
( ( semiring_1_of_nat @ int @ ( one_one @ nat ) )
= ( one_one @ int ) ) ).
% int_ops(2)
thf(fact_1868_int__plus,axiom,
! [N3: nat,M: nat] :
( ( semiring_1_of_nat @ int @ ( plus_plus @ nat @ N3 @ M ) )
= ( plus_plus @ int @ ( semiring_1_of_nat @ int @ N3 ) @ ( semiring_1_of_nat @ int @ M ) ) ) ).
% int_plus
thf(fact_1869_int__ops_I5_J,axiom,
! [A2: nat,B2: nat] :
( ( semiring_1_of_nat @ int @ ( plus_plus @ nat @ A2 @ B2 ) )
= ( plus_plus @ int @ ( semiring_1_of_nat @ int @ A2 ) @ ( semiring_1_of_nat @ int @ B2 ) ) ) ).
% int_ops(5)
thf(fact_1870_int__ops_I7_J,axiom,
! [A2: nat,B2: nat] :
( ( semiring_1_of_nat @ int @ ( times_times @ nat @ A2 @ B2 ) )
= ( times_times @ int @ ( semiring_1_of_nat @ int @ A2 ) @ ( semiring_1_of_nat @ int @ B2 ) ) ) ).
% int_ops(7)
thf(fact_1871_nat__less__as__int,axiom,
( ( ord_less @ nat )
= ( ^ [A6: nat,B5: nat] : ( ord_less @ int @ ( semiring_1_of_nat @ int @ A6 ) @ ( semiring_1_of_nat @ int @ B5 ) ) ) ) ).
% nat_less_as_int
thf(fact_1872_nat__leq__as__int,axiom,
( ( ord_less_eq @ nat )
= ( ^ [A6: nat,B5: nat] : ( ord_less_eq @ int @ ( semiring_1_of_nat @ int @ A6 ) @ ( semiring_1_of_nat @ int @ B5 ) ) ) ) ).
% nat_leq_as_int
thf(fact_1873_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_1874_word__unat__power,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [N3: nat] :
( ( power_power @ ( word @ A ) @ ( numeral_numeral @ ( word @ A ) @ ( bit0 @ one2 ) ) @ N3 )
= ( semiring_1_of_nat @ ( word @ A ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N3 ) ) ) ) ).
% word_unat_power
thf(fact_1875_word__less__two__pow__divD,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [X: word @ A,N3: nat,M: nat] :
( ( ord_less @ ( word @ A ) @ X @ ( divide_divide @ ( word @ A ) @ ( power_power @ ( word @ A ) @ ( numeral_numeral @ ( word @ A ) @ ( bit0 @ one2 ) ) @ N3 ) @ ( power_power @ ( word @ A ) @ ( numeral_numeral @ ( word @ A ) @ ( bit0 @ one2 ) ) @ M ) ) )
=> ( ( ord_less_eq @ nat @ M @ N3 )
& ( ord_less @ ( word @ A ) @ X @ ( power_power @ ( word @ A ) @ ( numeral_numeral @ ( word @ A ) @ ( bit0 @ one2 ) ) @ ( minus_minus @ nat @ N3 @ M ) ) ) ) ) ) ).
% word_less_two_pow_divD
thf(fact_1876_word__of__int__2p,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [N3: nat] :
( ( ring_1_of_int @ ( word @ A ) @ ( power_power @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ N3 ) )
= ( power_power @ ( word @ A ) @ ( numeral_numeral @ ( word @ A ) @ ( bit0 @ one2 ) ) @ N3 ) ) ) ).
% word_of_int_2p
thf(fact_1877_int__Suc,axiom,
! [N3: nat] :
( ( semiring_1_of_nat @ int @ ( suc @ N3 ) )
= ( plus_plus @ int @ ( semiring_1_of_nat @ int @ N3 ) @ ( one_one @ int ) ) ) ).
% int_Suc
thf(fact_1878_int__ops_I4_J,axiom,
! [A2: nat] :
( ( semiring_1_of_nat @ int @ ( suc @ A2 ) )
= ( plus_plus @ int @ ( semiring_1_of_nat @ int @ A2 ) @ ( one_one @ int ) ) ) ).
% int_ops(4)
thf(fact_1879_less__1__helper,axiom,
! [N3: int,M: int] :
( ( ord_less_eq @ int @ N3 @ M )
=> ( ord_less @ int @ ( minus_minus @ int @ N3 @ ( one_one @ int ) ) @ M ) ) ).
% less_1_helper
thf(fact_1880_vebt__delete_Opelims,axiom,
! [X: vEBT_VEBT,Xa: nat,Y: vEBT_VEBT] :
( ( ( vEBT_vebt_delete @ X @ Xa )
= Y )
=> ( ( accp @ ( product_prod @ vEBT_VEBT @ nat ) @ vEBT_vebt_delete_rel @ ( product_Pair @ vEBT_VEBT @ nat @ X @ Xa ) )
=> ( ! [A4: $o,B3: $o] :
( ( X
= ( vEBT_Leaf @ A4 @ B3 ) )
=> ( ( Xa
= ( zero_zero @ nat ) )
=> ( ( Y
= ( vEBT_Leaf @ $false @ B3 ) )
=> ~ ( accp @ ( product_prod @ vEBT_VEBT @ nat ) @ vEBT_vebt_delete_rel @ ( product_Pair @ vEBT_VEBT @ nat @ ( vEBT_Leaf @ A4 @ B3 ) @ ( zero_zero @ nat ) ) ) ) ) )
=> ( ! [A4: $o,B3: $o] :
( ( X
= ( vEBT_Leaf @ A4 @ B3 ) )
=> ( ( Xa
= ( suc @ ( zero_zero @ nat ) ) )
=> ( ( Y
= ( vEBT_Leaf @ A4 @ $false ) )
=> ~ ( accp @ ( product_prod @ vEBT_VEBT @ nat ) @ vEBT_vebt_delete_rel @ ( product_Pair @ vEBT_VEBT @ nat @ ( vEBT_Leaf @ A4 @ B3 ) @ ( suc @ ( zero_zero @ nat ) ) ) ) ) ) )
=> ( ! [A4: $o,B3: $o] :
( ( X
= ( vEBT_Leaf @ A4 @ B3 ) )
=> ! [N: nat] :
( ( Xa
= ( suc @ ( suc @ N ) ) )
=> ( ( Y
= ( vEBT_Leaf @ A4 @ B3 ) )
=> ~ ( accp @ ( product_prod @ vEBT_VEBT @ nat ) @ vEBT_vebt_delete_rel @ ( product_Pair @ vEBT_VEBT @ nat @ ( vEBT_Leaf @ A4 @ B3 ) @ ( suc @ ( suc @ N ) ) ) ) ) ) )
=> ( ! [Deg2: nat,TreeList2: list @ vEBT_VEBT,Summary2: vEBT_VEBT] :
( ( X
= ( vEBT_Node @ ( none @ ( product_prod @ nat @ nat ) ) @ Deg2 @ TreeList2 @ Summary2 ) )
=> ( ( Y
= ( vEBT_Node @ ( none @ ( product_prod @ nat @ nat ) ) @ Deg2 @ TreeList2 @ Summary2 ) )
=> ~ ( accp @ ( product_prod @ vEBT_VEBT @ nat ) @ vEBT_vebt_delete_rel @ ( product_Pair @ vEBT_VEBT @ nat @ ( vEBT_Node @ ( none @ ( product_prod @ nat @ nat ) ) @ Deg2 @ TreeList2 @ Summary2 ) @ Xa ) ) ) )
=> ( ! [Mi2: nat,Ma2: nat,TrLst2: list @ vEBT_VEBT,Smry2: vEBT_VEBT] :
( ( X
= ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi2 @ Ma2 ) ) @ ( zero_zero @ nat ) @ TrLst2 @ Smry2 ) )
=> ( ( Y
= ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi2 @ Ma2 ) ) @ ( zero_zero @ nat ) @ TrLst2 @ Smry2 ) )
=> ~ ( accp @ ( product_prod @ vEBT_VEBT @ nat ) @ vEBT_vebt_delete_rel @ ( product_Pair @ vEBT_VEBT @ nat @ ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi2 @ Ma2 ) ) @ ( zero_zero @ nat ) @ TrLst2 @ Smry2 ) @ Xa ) ) ) )
=> ( ! [Mi2: nat,Ma2: nat,Tr2: list @ vEBT_VEBT,Sm2: vEBT_VEBT] :
( ( X
= ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( zero_zero @ nat ) ) @ Tr2 @ Sm2 ) )
=> ( ( Y
= ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( zero_zero @ nat ) ) @ Tr2 @ Sm2 ) )
=> ~ ( accp @ ( product_prod @ vEBT_VEBT @ nat ) @ vEBT_vebt_delete_rel @ ( product_Pair @ vEBT_VEBT @ nat @ ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( zero_zero @ nat ) ) @ Tr2 @ Sm2 ) @ Xa ) ) ) )
=> ~ ! [Mi2: nat,Ma2: nat,Va3: nat,TreeList2: list @ vEBT_VEBT,Summary2: vEBT_VEBT] :
( ( X
= ( 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 ) )
=> ( Y
= ( 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 ) )
=> ( Y
= ( vEBT_Node @ ( none @ ( product_prod @ nat @ nat ) ) @ ( suc @ ( suc @ Va3 ) ) @ TreeList2 @ Summary2 ) ) )
& ( ~ ( ( Xa = Mi2 )
& ( Xa = Ma2 ) )
=> ( Y
= ( if @ vEBT_VEBT @ ( ord_less @ nat @ ( vEBT_VEBT_high @ ( if @ nat @ ( Xa = Mi2 ) @ ( plus_plus @ nat @ ( times_times @ nat @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary2 ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ ( suc @ ( suc @ 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_1881_T_092_060_094sub_062s_092_060_094sub_062u_092_060_094sub_062c_092_060_094sub_062c_H_Opelims,axiom,
! [X: vEBT_VEBT,Xa: nat,Y: nat] :
( ( ( vEBT_T_s_u_c_c2 @ X @ Xa )
= Y )
=> ( ( accp @ ( product_prod @ vEBT_VEBT @ nat ) @ vEBT_T_s_u_c_c_rel @ ( product_Pair @ vEBT_VEBT @ nat @ X @ Xa ) )
=> ( ! [Uu: $o,B3: $o] :
( ( X
= ( vEBT_Leaf @ Uu @ B3 ) )
=> ( ( Xa
= ( zero_zero @ nat ) )
=> ( ( Y
= ( one_one @ nat ) )
=> ~ ( accp @ ( product_prod @ vEBT_VEBT @ nat ) @ vEBT_T_s_u_c_c_rel @ ( product_Pair @ vEBT_VEBT @ nat @ ( vEBT_Leaf @ Uu @ B3 ) @ ( zero_zero @ nat ) ) ) ) ) )
=> ( ! [Uv: $o,Uw: $o] :
( ( X
= ( vEBT_Leaf @ Uv @ Uw ) )
=> ! [N: nat] :
( ( Xa
= ( suc @ N ) )
=> ( ( Y
= ( one_one @ nat ) )
=> ~ ( accp @ ( product_prod @ vEBT_VEBT @ nat ) @ vEBT_T_s_u_c_c_rel @ ( product_Pair @ vEBT_VEBT @ nat @ ( vEBT_Leaf @ Uv @ Uw ) @ ( suc @ N ) ) ) ) ) )
=> ( ! [Ux2: nat,Uy2: list @ vEBT_VEBT,Uz2: vEBT_VEBT] :
( ( X
= ( vEBT_Node @ ( none @ ( product_prod @ nat @ nat ) ) @ Ux2 @ Uy2 @ Uz2 ) )
=> ( ( Y
= ( one_one @ nat ) )
=> ~ ( accp @ ( product_prod @ vEBT_VEBT @ nat ) @ vEBT_T_s_u_c_c_rel @ ( product_Pair @ vEBT_VEBT @ nat @ ( vEBT_Node @ ( none @ ( product_prod @ nat @ nat ) ) @ Ux2 @ Uy2 @ Uz2 ) @ Xa ) ) ) )
=> ( ! [V3: product_prod @ nat @ nat,Vc2: list @ vEBT_VEBT,Vd2: vEBT_VEBT] :
( ( X
= ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ V3 ) @ ( zero_zero @ nat ) @ Vc2 @ Vd2 ) )
=> ( ( Y
= ( one_one @ nat ) )
=> ~ ( accp @ ( product_prod @ vEBT_VEBT @ nat ) @ vEBT_T_s_u_c_c_rel @ ( product_Pair @ vEBT_VEBT @ nat @ ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ V3 ) @ ( zero_zero @ nat ) @ Vc2 @ Vd2 ) @ Xa ) ) ) )
=> ( ! [V3: product_prod @ nat @ nat,Vg2: list @ vEBT_VEBT,Vh2: vEBT_VEBT] :
( ( X
= ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ V3 ) @ ( suc @ ( zero_zero @ nat ) ) @ Vg2 @ Vh2 ) )
=> ( ( Y
= ( one_one @ nat ) )
=> ~ ( accp @ ( product_prod @ vEBT_VEBT @ nat ) @ vEBT_T_s_u_c_c_rel @ ( product_Pair @ vEBT_VEBT @ nat @ ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ V3 ) @ ( suc @ ( zero_zero @ nat ) ) @ Vg2 @ Vh2 ) @ Xa ) ) ) )
=> ~ ! [Mi2: nat,Ma2: nat,Va3: nat,TreeList2: list @ vEBT_VEBT,Summary2: vEBT_VEBT] :
( ( X
= ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( suc @ Va3 ) ) @ TreeList2 @ Summary2 ) )
=> ( ( ( ( ord_less @ nat @ Xa @ Mi2 )
=> ( Y
= ( one_one @ nat ) ) )
& ( ~ ( ord_less @ nat @ Xa @ Mi2 )
=> ( Y
= ( if @ nat @ ( ord_less @ nat @ ( vEBT_VEBT_high @ Xa @ ( divide_divide @ nat @ ( suc @ ( suc @ 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_1882_T_092_060_094sub_062i_092_060_094sub_062n_092_060_094sub_062s_092_060_094sub_062e_092_060_094sub_062r_092_060_094sub_062t_Opelims,axiom,
! [X: vEBT_VEBT,Xa: nat,Y: nat] :
( ( ( vEBT_T_i_n_s_e_r_t @ X @ Xa )
= Y )
=> ( ( accp @ ( product_prod @ vEBT_VEBT @ nat ) @ vEBT_T9217963907923527482_t_rel @ ( product_Pair @ vEBT_VEBT @ nat @ X @ Xa ) )
=> ( ! [A4: $o,B3: $o] :
( ( X
= ( vEBT_Leaf @ A4 @ B3 ) )
=> ( ( Y
= ( plus_plus @ nat @ ( one_one @ nat )
@ ( if @ nat
@ ( Xa
= ( zero_zero @ nat ) )
@ ( one_one @ nat )
@ ( plus_plus @ nat @ ( one_one @ nat ) @ ( one_one @ nat ) ) ) ) )
=> ~ ( accp @ ( product_prod @ vEBT_VEBT @ nat ) @ vEBT_T9217963907923527482_t_rel @ ( product_Pair @ vEBT_VEBT @ nat @ ( vEBT_Leaf @ A4 @ B3 ) @ Xa ) ) ) )
=> ( ! [Info2: option @ ( product_prod @ nat @ nat ),Ts2: list @ vEBT_VEBT,S2: vEBT_VEBT] :
( ( X
= ( vEBT_Node @ Info2 @ ( zero_zero @ nat ) @ Ts2 @ S2 ) )
=> ( ( Y
= ( one_one @ nat ) )
=> ~ ( accp @ ( product_prod @ vEBT_VEBT @ nat ) @ vEBT_T9217963907923527482_t_rel @ ( product_Pair @ vEBT_VEBT @ nat @ ( vEBT_Node @ Info2 @ ( zero_zero @ nat ) @ Ts2 @ S2 ) @ Xa ) ) ) )
=> ( ! [Info2: option @ ( product_prod @ nat @ nat ),Ts2: list @ vEBT_VEBT,S2: vEBT_VEBT] :
( ( X
= ( vEBT_Node @ Info2 @ ( suc @ ( zero_zero @ nat ) ) @ Ts2 @ S2 ) )
=> ( ( Y
= ( one_one @ nat ) )
=> ~ ( accp @ ( product_prod @ vEBT_VEBT @ nat ) @ vEBT_T9217963907923527482_t_rel @ ( product_Pair @ vEBT_VEBT @ nat @ ( vEBT_Node @ Info2 @ ( suc @ ( zero_zero @ nat ) ) @ Ts2 @ S2 ) @ Xa ) ) ) )
=> ( ! [V3: nat,TreeList2: list @ vEBT_VEBT,Summary2: vEBT_VEBT] :
( ( X
= ( vEBT_Node @ ( none @ ( product_prod @ nat @ nat ) ) @ ( suc @ ( suc @ V3 ) ) @ TreeList2 @ Summary2 ) )
=> ( ( Y
= ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) )
=> ~ ( accp @ ( product_prod @ vEBT_VEBT @ nat ) @ vEBT_T9217963907923527482_t_rel @ ( product_Pair @ vEBT_VEBT @ nat @ ( vEBT_Node @ ( none @ ( product_prod @ nat @ nat ) ) @ ( suc @ ( suc @ V3 ) ) @ TreeList2 @ Summary2 ) @ Xa ) ) ) )
=> ~ ! [Mi2: nat,Ma2: nat,Va3: nat,TreeList2: list @ vEBT_VEBT,Summary2: vEBT_VEBT] :
( ( X
= ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( suc @ Va3 ) ) @ TreeList2 @ Summary2 ) )
=> ( ( Y
= ( plus_plus @ nat @ ( numeral_numeral @ nat @ ( bit1 @ ( bit1 @ ( bit0 @ ( bit0 @ one2 ) ) ) ) )
@ ( if @ nat
@ ( ( ord_less @ nat @ ( vEBT_VEBT_high @ ( if @ nat @ ( ord_less @ nat @ Xa @ Mi2 ) @ Mi2 @ Xa ) @ ( divide_divide @ nat @ ( suc @ ( suc @ 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_1883_vebt__insert_Opelims,axiom,
! [X: vEBT_VEBT,Xa: nat,Y: vEBT_VEBT] :
( ( ( vEBT_vebt_insert @ X @ Xa )
= Y )
=> ( ( accp @ ( product_prod @ vEBT_VEBT @ nat ) @ vEBT_vebt_insert_rel @ ( product_Pair @ vEBT_VEBT @ nat @ X @ Xa ) )
=> ( ! [A4: $o,B3: $o] :
( ( X
= ( vEBT_Leaf @ A4 @ B3 ) )
=> ( ( ( ( Xa
= ( zero_zero @ nat ) )
=> ( Y
= ( vEBT_Leaf @ $true @ B3 ) ) )
& ( ( Xa
!= ( zero_zero @ nat ) )
=> ( ( ( Xa
= ( one_one @ nat ) )
=> ( Y
= ( vEBT_Leaf @ A4 @ $true ) ) )
& ( ( Xa
!= ( one_one @ nat ) )
=> ( Y
= ( vEBT_Leaf @ A4 @ B3 ) ) ) ) ) )
=> ~ ( accp @ ( product_prod @ vEBT_VEBT @ nat ) @ vEBT_vebt_insert_rel @ ( product_Pair @ vEBT_VEBT @ nat @ ( vEBT_Leaf @ A4 @ B3 ) @ Xa ) ) ) )
=> ( ! [Info2: option @ ( product_prod @ nat @ nat ),Ts2: list @ vEBT_VEBT,S2: vEBT_VEBT] :
( ( X
= ( vEBT_Node @ Info2 @ ( zero_zero @ nat ) @ Ts2 @ S2 ) )
=> ( ( Y
= ( vEBT_Node @ Info2 @ ( zero_zero @ nat ) @ Ts2 @ S2 ) )
=> ~ ( accp @ ( product_prod @ vEBT_VEBT @ nat ) @ vEBT_vebt_insert_rel @ ( product_Pair @ vEBT_VEBT @ nat @ ( vEBT_Node @ Info2 @ ( zero_zero @ nat ) @ Ts2 @ S2 ) @ Xa ) ) ) )
=> ( ! [Info2: option @ ( product_prod @ nat @ nat ),Ts2: list @ vEBT_VEBT,S2: vEBT_VEBT] :
( ( X
= ( vEBT_Node @ Info2 @ ( suc @ ( zero_zero @ nat ) ) @ Ts2 @ S2 ) )
=> ( ( Y
= ( vEBT_Node @ Info2 @ ( suc @ ( zero_zero @ nat ) ) @ Ts2 @ S2 ) )
=> ~ ( accp @ ( product_prod @ vEBT_VEBT @ nat ) @ vEBT_vebt_insert_rel @ ( product_Pair @ vEBT_VEBT @ nat @ ( vEBT_Node @ Info2 @ ( suc @ ( zero_zero @ nat ) ) @ Ts2 @ S2 ) @ Xa ) ) ) )
=> ( ! [V3: nat,TreeList2: list @ vEBT_VEBT,Summary2: vEBT_VEBT] :
( ( X
= ( vEBT_Node @ ( none @ ( product_prod @ nat @ nat ) ) @ ( suc @ ( suc @ V3 ) ) @ TreeList2 @ Summary2 ) )
=> ( ( Y
= ( 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] :
( ( X
= ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( suc @ Va3 ) ) @ TreeList2 @ Summary2 ) )
=> ( ( Y
= ( if @ vEBT_VEBT
@ ( ( ord_less @ nat @ ( vEBT_VEBT_high @ ( if @ nat @ ( ord_less @ nat @ Xa @ Mi2 ) @ Mi2 @ Xa ) @ ( divide_divide @ nat @ ( suc @ ( suc @ 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_1884_T_092_060_094sub_062p_092_060_094sub_062r_092_060_094sub_062e_092_060_094sub_062d_H_Opelims,axiom,
! [X: vEBT_VEBT,Xa: nat,Y: nat] :
( ( ( vEBT_T_p_r_e_d2 @ X @ Xa )
= Y )
=> ( ( accp @ ( product_prod @ vEBT_VEBT @ nat ) @ vEBT_T_p_r_e_d_rel @ ( product_Pair @ vEBT_VEBT @ nat @ X @ Xa ) )
=> ( ! [Uu: $o,Uv: $o] :
( ( X
= ( vEBT_Leaf @ Uu @ Uv ) )
=> ( ( Xa
= ( zero_zero @ nat ) )
=> ( ( Y
= ( one_one @ nat ) )
=> ~ ( accp @ ( product_prod @ vEBT_VEBT @ nat ) @ vEBT_T_p_r_e_d_rel @ ( product_Pair @ vEBT_VEBT @ nat @ ( vEBT_Leaf @ Uu @ Uv ) @ ( zero_zero @ nat ) ) ) ) ) )
=> ( ! [A4: $o,Uw: $o] :
( ( X
= ( vEBT_Leaf @ A4 @ Uw ) )
=> ( ( Xa
= ( suc @ ( zero_zero @ nat ) ) )
=> ( ( Y
= ( one_one @ nat ) )
=> ~ ( accp @ ( product_prod @ vEBT_VEBT @ nat ) @ vEBT_T_p_r_e_d_rel @ ( product_Pair @ vEBT_VEBT @ nat @ ( vEBT_Leaf @ A4 @ Uw ) @ ( suc @ ( zero_zero @ nat ) ) ) ) ) ) )
=> ( ! [A4: $o,B3: $o] :
( ( X
= ( vEBT_Leaf @ A4 @ B3 ) )
=> ! [Va3: nat] :
( ( Xa
= ( suc @ ( suc @ Va3 ) ) )
=> ( ( Y
= ( one_one @ nat ) )
=> ~ ( accp @ ( product_prod @ vEBT_VEBT @ nat ) @ vEBT_T_p_r_e_d_rel @ ( product_Pair @ vEBT_VEBT @ nat @ ( vEBT_Leaf @ A4 @ B3 ) @ ( suc @ ( suc @ Va3 ) ) ) ) ) ) )
=> ( ! [Uy2: nat,Uz2: list @ vEBT_VEBT,Va2: vEBT_VEBT] :
( ( X
= ( vEBT_Node @ ( none @ ( product_prod @ nat @ nat ) ) @ Uy2 @ Uz2 @ Va2 ) )
=> ( ( Y
= ( one_one @ nat ) )
=> ~ ( accp @ ( product_prod @ vEBT_VEBT @ nat ) @ vEBT_T_p_r_e_d_rel @ ( product_Pair @ vEBT_VEBT @ nat @ ( vEBT_Node @ ( none @ ( product_prod @ nat @ nat ) ) @ Uy2 @ Uz2 @ Va2 ) @ Xa ) ) ) )
=> ( ! [V3: product_prod @ nat @ nat,Vd2: list @ vEBT_VEBT,Ve2: vEBT_VEBT] :
( ( X
= ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ V3 ) @ ( zero_zero @ nat ) @ Vd2 @ Ve2 ) )
=> ( ( Y
= ( one_one @ nat ) )
=> ~ ( accp @ ( product_prod @ vEBT_VEBT @ nat ) @ vEBT_T_p_r_e_d_rel @ ( product_Pair @ vEBT_VEBT @ nat @ ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ V3 ) @ ( zero_zero @ nat ) @ Vd2 @ Ve2 ) @ Xa ) ) ) )
=> ( ! [V3: product_prod @ nat @ nat,Vh2: list @ vEBT_VEBT,Vi2: vEBT_VEBT] :
( ( X
= ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ V3 ) @ ( suc @ ( zero_zero @ nat ) ) @ Vh2 @ Vi2 ) )
=> ( ( Y
= ( one_one @ nat ) )
=> ~ ( accp @ ( product_prod @ vEBT_VEBT @ nat ) @ vEBT_T_p_r_e_d_rel @ ( product_Pair @ vEBT_VEBT @ nat @ ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ V3 ) @ ( suc @ ( zero_zero @ nat ) ) @ Vh2 @ Vi2 ) @ Xa ) ) ) )
=> ~ ! [Mi2: nat,Ma2: nat,Va3: nat,TreeList2: list @ vEBT_VEBT,Summary2: vEBT_VEBT] :
( ( X
= ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( suc @ Va3 ) ) @ TreeList2 @ Summary2 ) )
=> ( ( ( ( ord_less @ nat @ Ma2 @ Xa )
=> ( Y
= ( one_one @ nat ) ) )
& ( ~ ( ord_less @ nat @ Ma2 @ Xa )
=> ( Y
= ( if @ nat @ ( ord_less @ nat @ ( vEBT_VEBT_high @ Xa @ ( divide_divide @ nat @ ( suc @ ( suc @ 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_1885_T_092_060_094sub_062i_092_060_094sub_062n_092_060_094sub_062s_092_060_094sub_062e_092_060_094sub_062r_092_060_094sub_062t_H_Opelims,axiom,
! [X: vEBT_VEBT,Xa: nat,Y: nat] :
( ( ( vEBT_T_i_n_s_e_r_t2 @ X @ Xa )
= Y )
=> ( ( accp @ ( product_prod @ vEBT_VEBT @ nat ) @ vEBT_T5076183648494686801_t_rel @ ( product_Pair @ vEBT_VEBT @ nat @ X @ Xa ) )
=> ( ! [A4: $o,B3: $o] :
( ( X
= ( vEBT_Leaf @ A4 @ B3 ) )
=> ( ( Y
= ( one_one @ nat ) )
=> ~ ( accp @ ( product_prod @ vEBT_VEBT @ nat ) @ vEBT_T5076183648494686801_t_rel @ ( product_Pair @ vEBT_VEBT @ nat @ ( vEBT_Leaf @ A4 @ B3 ) @ Xa ) ) ) )
=> ( ! [Info2: option @ ( product_prod @ nat @ nat ),Ts2: list @ vEBT_VEBT,S2: vEBT_VEBT] :
( ( X
= ( vEBT_Node @ Info2 @ ( zero_zero @ nat ) @ Ts2 @ S2 ) )
=> ( ( Y
= ( one_one @ nat ) )
=> ~ ( accp @ ( product_prod @ vEBT_VEBT @ nat ) @ vEBT_T5076183648494686801_t_rel @ ( product_Pair @ vEBT_VEBT @ nat @ ( vEBT_Node @ Info2 @ ( zero_zero @ nat ) @ Ts2 @ S2 ) @ Xa ) ) ) )
=> ( ! [Info2: option @ ( product_prod @ nat @ nat ),Ts2: list @ vEBT_VEBT,S2: vEBT_VEBT] :
( ( X
= ( vEBT_Node @ Info2 @ ( suc @ ( zero_zero @ nat ) ) @ Ts2 @ S2 ) )
=> ( ( Y
= ( one_one @ nat ) )
=> ~ ( accp @ ( product_prod @ vEBT_VEBT @ nat ) @ vEBT_T5076183648494686801_t_rel @ ( product_Pair @ vEBT_VEBT @ nat @ ( vEBT_Node @ Info2 @ ( suc @ ( zero_zero @ nat ) ) @ Ts2 @ S2 ) @ Xa ) ) ) )
=> ( ! [V3: nat,TreeList2: list @ vEBT_VEBT,Summary2: vEBT_VEBT] :
( ( X
= ( vEBT_Node @ ( none @ ( product_prod @ nat @ nat ) ) @ ( suc @ ( suc @ V3 ) ) @ TreeList2 @ Summary2 ) )
=> ( ( Y
= ( one_one @ nat ) )
=> ~ ( accp @ ( product_prod @ vEBT_VEBT @ nat ) @ vEBT_T5076183648494686801_t_rel @ ( product_Pair @ vEBT_VEBT @ nat @ ( vEBT_Node @ ( none @ ( product_prod @ nat @ nat ) ) @ ( suc @ ( suc @ V3 ) ) @ TreeList2 @ Summary2 ) @ Xa ) ) ) )
=> ~ ! [Mi2: nat,Ma2: nat,Va3: nat,TreeList2: list @ vEBT_VEBT,Summary2: vEBT_VEBT] :
( ( X
= ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( suc @ Va3 ) ) @ TreeList2 @ Summary2 ) )
=> ( ( Y
= ( if @ nat
@ ( ( ord_less @ nat @ ( vEBT_VEBT_high @ ( if @ nat @ ( ord_less @ nat @ Xa @ Mi2 ) @ Mi2 @ Xa ) @ ( divide_divide @ nat @ ( suc @ ( suc @ 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_1886_T_092_060_094sub_062m_092_060_094sub_062e_092_060_094sub_062m_092_060_094sub_062b_092_060_094sub_062e_092_060_094sub_062r_Opelims,axiom,
! [X: vEBT_VEBT,Xa: nat,Y: nat] :
( ( ( vEBT_T_m_e_m_b_e_r @ X @ Xa )
= Y )
=> ( ( accp @ ( product_prod @ vEBT_VEBT @ nat ) @ vEBT_T5837161174952499735_r_rel @ ( product_Pair @ vEBT_VEBT @ nat @ X @ Xa ) )
=> ( ! [A4: $o,B3: $o] :
( ( X
= ( vEBT_Leaf @ A4 @ B3 ) )
=> ( ( Y
= ( plus_plus @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) )
@ ( if @ nat
@ ( Xa
= ( zero_zero @ nat ) )
@ ( one_one @ nat )
@ ( plus_plus @ nat @ ( one_one @ nat ) @ ( one_one @ nat ) ) ) ) )
=> ~ ( accp @ ( product_prod @ vEBT_VEBT @ nat ) @ vEBT_T5837161174952499735_r_rel @ ( product_Pair @ vEBT_VEBT @ nat @ ( vEBT_Leaf @ A4 @ B3 ) @ Xa ) ) ) )
=> ( ! [Uu: nat,Uv: list @ vEBT_VEBT,Uw: vEBT_VEBT] :
( ( X
= ( vEBT_Node @ ( none @ ( product_prod @ nat @ nat ) ) @ Uu @ Uv @ Uw ) )
=> ( ( Y
= ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) )
=> ~ ( accp @ ( product_prod @ vEBT_VEBT @ nat ) @ vEBT_T5837161174952499735_r_rel @ ( product_Pair @ vEBT_VEBT @ nat @ ( vEBT_Node @ ( none @ ( product_prod @ nat @ nat ) ) @ Uu @ Uv @ Uw ) @ Xa ) ) ) )
=> ( ! [V3: product_prod @ nat @ nat,Uy2: list @ vEBT_VEBT,Uz2: vEBT_VEBT] :
( ( X
= ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ V3 ) @ ( zero_zero @ nat ) @ Uy2 @ Uz2 ) )
=> ( ( Y
= ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) )
=> ~ ( accp @ ( product_prod @ vEBT_VEBT @ nat ) @ vEBT_T5837161174952499735_r_rel @ ( product_Pair @ vEBT_VEBT @ nat @ ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ V3 ) @ ( zero_zero @ nat ) @ Uy2 @ Uz2 ) @ Xa ) ) ) )
=> ( ! [V3: product_prod @ nat @ nat,Vb2: list @ vEBT_VEBT,Vc2: vEBT_VEBT] :
( ( X
= ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ V3 ) @ ( suc @ ( zero_zero @ nat ) ) @ Vb2 @ Vc2 ) )
=> ( ( Y
= ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) )
=> ~ ( accp @ ( product_prod @ vEBT_VEBT @ nat ) @ vEBT_T5837161174952499735_r_rel @ ( product_Pair @ vEBT_VEBT @ nat @ ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ V3 ) @ ( suc @ ( zero_zero @ nat ) ) @ Vb2 @ Vc2 ) @ Xa ) ) ) )
=> ~ ! [Mi2: nat,Ma2: nat,Va3: nat,TreeList2: list @ vEBT_VEBT,Summary2: vEBT_VEBT] :
( ( X
= ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( suc @ Va3 ) ) @ TreeList2 @ Summary2 ) )
=> ( ( Y
= ( plus_plus @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( if @ nat @ ( Xa = Mi2 ) @ ( one_one @ nat ) @ ( plus_plus @ nat @ ( one_one @ nat ) @ ( if @ nat @ ( Xa = Ma2 ) @ ( one_one @ nat ) @ ( plus_plus @ nat @ ( one_one @ nat ) @ ( if @ nat @ ( ord_less @ nat @ Xa @ Mi2 ) @ ( one_one @ nat ) @ ( plus_plus @ nat @ ( one_one @ nat ) @ ( if @ nat @ ( ord_less @ nat @ Ma2 @ Xa ) @ ( one_one @ nat ) @ ( plus_plus @ nat @ ( numeral_numeral @ nat @ ( bit1 @ ( bit0 @ ( bit0 @ one2 ) ) ) ) @ ( if @ nat @ ( ord_less @ nat @ ( vEBT_VEBT_high @ Xa @ ( divide_divide @ nat @ ( suc @ ( suc @ 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_1887_word__coorder_Oextremum__unique,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [A2: word @ A] :
( ( ord_less_eq @ ( word @ A ) @ A2 @ ( zero_zero @ ( word @ A ) ) )
= ( A2
= ( zero_zero @ ( word @ A ) ) ) ) ) ).
% word_coorder.extremum_unique
thf(fact_1888_word__le__0__iff,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [X: word @ A] :
( ( ord_less_eq @ ( word @ A ) @ X @ ( zero_zero @ ( word @ A ) ) )
= ( X
= ( zero_zero @ ( word @ A ) ) ) ) ) ).
% word_le_0_iff
thf(fact_1889_word__less__1,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [X: word @ A] :
( ( ord_less @ ( word @ A ) @ X @ ( one_one @ ( word @ A ) ) )
= ( X
= ( zero_zero @ ( word @ A ) ) ) ) ) ).
% word_less_1
thf(fact_1890_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_1891_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_1892_sub__wrap,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [X: word @ A,Z: word @ A] :
( ( ord_less_eq @ ( word @ A ) @ X @ ( minus_minus @ ( word @ A ) @ X @ Z ) )
= ( ( Z
= ( zero_zero @ ( word @ A ) ) )
| ( ord_less @ ( word @ A ) @ X @ Z ) ) ) ) ).
% sub_wrap
thf(fact_1893_word__le__sub1,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [X: word @ A] :
( ( X
!= ( zero_zero @ ( word @ A ) ) )
=> ( ( ord_less_eq @ ( word @ A ) @ ( one_one @ ( word @ A ) ) @ X )
= ( ord_less_eq @ ( word @ A ) @ ( zero_zero @ ( word @ A ) ) @ ( minus_minus @ ( word @ A ) @ X @ ( one_one @ ( word @ A ) ) ) ) ) ) ) ).
% word_le_sub1
thf(fact_1894_less__1__simp,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [N3: word @ A,M: word @ A] :
( ( ord_less @ ( word @ A ) @ ( minus_minus @ ( word @ A ) @ N3 @ ( one_one @ ( word @ A ) ) ) @ M )
= ( ( ord_less_eq @ ( word @ A ) @ N3 @ M )
& ( N3
!= ( zero_zero @ ( word @ A ) ) ) ) ) ) ).
% less_1_simp
thf(fact_1895_le__m1__iff__lt,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [X: word @ A,Y: word @ A] :
( ( ord_less @ ( word @ A ) @ ( zero_zero @ ( word @ A ) ) @ X )
= ( ( ord_less_eq @ ( word @ A ) @ Y @ ( minus_minus @ ( word @ A ) @ X @ ( one_one @ ( word @ A ) ) ) )
= ( ord_less @ ( word @ A ) @ Y @ X ) ) ) ) ).
% le_m1_iff_lt
thf(fact_1896_word__div__sub,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [Y: word @ A,X: word @ A] :
( ( ord_less_eq @ ( word @ A ) @ Y @ X )
=> ( ( ord_less @ ( word @ A ) @ ( zero_zero @ ( word @ A ) ) @ Y )
=> ( ( divide_divide @ ( word @ A ) @ ( minus_minus @ ( word @ A ) @ X @ Y ) @ Y )
= ( minus_minus @ ( word @ A ) @ ( divide_divide @ ( word @ A ) @ X @ Y ) @ ( one_one @ ( word @ A ) ) ) ) ) ) ) ).
% word_div_sub
thf(fact_1897_word__less__nowrapI,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [X: word @ A,Z: word @ A,K: word @ A] :
( ( ord_less @ ( word @ A ) @ X @ ( minus_minus @ ( word @ A ) @ Z @ K ) )
=> ( ( ord_less_eq @ ( word @ A ) @ K @ Z )
=> ( ( ord_less @ ( word @ A ) @ ( zero_zero @ ( word @ A ) ) @ K )
=> ( ord_less @ ( word @ A ) @ X @ ( plus_plus @ ( word @ A ) @ X @ K ) ) ) ) ) ) ).
% word_less_nowrapI
thf(fact_1898_word__sub__1__le,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [X: word @ A] :
( ( X
!= ( zero_zero @ ( word @ A ) ) )
=> ( ord_less_eq @ ( word @ A ) @ ( minus_minus @ ( word @ A ) @ X @ ( one_one @ ( word @ A ) ) ) @ X ) ) ) ).
% word_sub_1_le
thf(fact_1899_word__diff__less,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [N3: word @ A,M: word @ A] :
( ( ord_less @ ( word @ A ) @ ( zero_zero @ ( word @ A ) ) @ N3 )
=> ( ( ord_less @ ( word @ A ) @ ( zero_zero @ ( word @ A ) ) @ M )
=> ( ( ord_less_eq @ ( word @ A ) @ N3 @ M )
=> ( ord_less @ ( word @ A ) @ ( minus_minus @ ( word @ A ) @ M @ N3 ) @ M ) ) ) ) ) ).
% word_diff_less
thf(fact_1900_word__must__wrap,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [X: word @ A,N3: word @ A] :
( ( ord_less_eq @ ( word @ A ) @ X @ ( minus_minus @ ( word @ A ) @ N3 @ ( one_one @ ( word @ A ) ) ) )
=> ( ( ord_less_eq @ ( word @ A ) @ N3 @ X )
=> ( N3
= ( zero_zero @ ( word @ A ) ) ) ) ) ) ).
% word_must_wrap
thf(fact_1901_plus__minus__not__NULL,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [X: word @ A,Ab: word @ A,C2: word @ A] :
( ( ord_less @ ( word @ A ) @ X @ ( minus_minus @ ( word @ A ) @ Ab @ C2 ) )
=> ( ( ord_less_eq @ ( word @ A ) @ C2 @ Ab )
=> ( ( C2
!= ( zero_zero @ ( word @ A ) ) )
=> ( ( plus_plus @ ( word @ A ) @ X @ C2 )
!= ( zero_zero @ ( word @ A ) ) ) ) ) ) ) ).
% plus_minus_not_NULL
thf(fact_1902_word__subset__less,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [X: word @ A,R2: word @ A,Y: word @ A,S: word @ A] :
( ( ord_less_eq @ ( set @ ( word @ A ) ) @ ( set_or1337092689740270186AtMost @ ( word @ A ) @ X @ ( minus_minus @ ( word @ A ) @ ( plus_plus @ ( word @ A ) @ X @ R2 ) @ ( one_one @ ( word @ A ) ) ) ) @ ( set_or1337092689740270186AtMost @ ( word @ A ) @ Y @ ( minus_minus @ ( word @ A ) @ ( plus_plus @ ( word @ A ) @ Y @ S ) @ ( one_one @ ( word @ A ) ) ) ) )
=> ( ( ord_less_eq @ ( word @ A ) @ X @ ( minus_minus @ ( word @ A ) @ ( plus_plus @ ( word @ A ) @ X @ R2 ) @ ( one_one @ ( word @ A ) ) ) )
=> ( ( ord_less_eq @ ( word @ A ) @ Y @ ( minus_minus @ ( word @ A ) @ ( plus_plus @ ( word @ A ) @ Y @ S ) @ ( one_one @ ( word @ A ) ) ) )
=> ( ( S
!= ( zero_zero @ ( word @ A ) ) )
=> ( ord_less_eq @ ( word @ A ) @ R2 @ S ) ) ) ) ) ) ).
% word_subset_less
thf(fact_1903_plus__minus__not__NULL__ab,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [X: word @ A,Ab: word @ A,C2: word @ A] :
( ( ord_less_eq @ ( word @ A ) @ X @ ( minus_minus @ ( word @ A ) @ Ab @ C2 ) )
=> ( ( ord_less_eq @ ( word @ A ) @ C2 @ Ab )
=> ( ( C2
!= ( zero_zero @ ( word @ A ) ) )
=> ( ( plus_plus @ ( word @ A ) @ X @ C2 )
!= ( zero_zero @ ( word @ A ) ) ) ) ) ) ) ).
% plus_minus_not_NULL_ab
thf(fact_1904_word__less__nowrapI_H,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [X: word @ A,Z: word @ A,K: word @ A] :
( ( ord_less_eq @ ( word @ A ) @ X @ ( minus_minus @ ( word @ A ) @ Z @ K ) )
=> ( ( ord_less_eq @ ( word @ A ) @ K @ Z )
=> ( ( ord_less @ ( word @ A ) @ ( zero_zero @ ( word @ A ) ) @ K )
=> ( ord_less @ ( word @ A ) @ X @ ( plus_plus @ ( word @ A ) @ X @ K ) ) ) ) ) ) ).
% word_less_nowrapI'
thf(fact_1905_word__leq__le__minus__one,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [X: word @ A,Y: word @ A] :
( ( ord_less_eq @ ( word @ A ) @ X @ Y )
=> ( ( X
!= ( zero_zero @ ( word @ A ) ) )
=> ( ord_less @ ( word @ A ) @ ( minus_minus @ ( word @ A ) @ X @ ( one_one @ ( word @ A ) ) ) @ Y ) ) ) ) ).
% word_leq_le_minus_one
thf(fact_1906_word__leq__minus__one__le,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [Y: word @ A,X: word @ A] :
( ( Y
!= ( zero_zero @ ( word @ A ) ) )
=> ( ( ord_less_eq @ ( word @ A ) @ X @ ( minus_minus @ ( word @ A ) @ Y @ ( one_one @ ( word @ A ) ) ) )
=> ( ord_less @ ( word @ A ) @ X @ Y ) ) ) ) ).
% word_leq_minus_one_le
thf(fact_1907_word__sub__plus__one__nonzero,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [N4: word @ A,N3: word @ A] :
( ( ord_less_eq @ ( word @ A ) @ N4 @ N3 )
=> ( ( N4
!= ( zero_zero @ ( word @ A ) ) )
=> ( ( plus_plus @ ( word @ A ) @ ( minus_minus @ ( word @ A ) @ N3 @ N4 ) @ ( one_one @ ( word @ A ) ) )
!= ( zero_zero @ ( word @ A ) ) ) ) ) ) ).
% word_sub_plus_one_nonzero
thf(fact_1908_gt0__iff__gem1,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [X: word @ A] :
( ( ord_less @ ( word @ A ) @ ( zero_zero @ ( word @ A ) ) @ X )
= ( ord_less @ ( word @ A ) @ ( minus_minus @ ( word @ A ) @ X @ ( one_one @ ( word @ A ) ) ) @ X ) ) ) ).
% gt0_iff_gem1
thf(fact_1909_word__less__sub1,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [X: word @ A] :
( ( X
!= ( zero_zero @ ( word @ A ) ) )
=> ( ( ord_less @ ( word @ A ) @ ( one_one @ ( word @ A ) ) @ X )
= ( ord_less @ ( word @ A ) @ ( zero_zero @ ( word @ A ) ) @ ( minus_minus @ ( word @ A ) @ X @ ( one_one @ ( word @ A ) ) ) ) ) ) ) ).
% word_less_sub1
thf(fact_1910_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_1911_word__induct2,axiom,
! [B: $tType] :
( ( type_len @ B )
=> ! [P: ( word @ B ) > $o,N3: word @ B] :
( ( P @ ( zero_zero @ ( word @ B ) ) )
=> ( ! [N: word @ B] :
( ( ( plus_plus @ ( word @ B ) @ ( one_one @ ( word @ B ) ) @ N )
!= ( zero_zero @ ( word @ B ) ) )
=> ( ( P @ N )
=> ( P @ ( plus_plus @ ( word @ B ) @ ( one_one @ ( word @ B ) ) @ N ) ) ) )
=> ( P @ N3 ) ) ) ) ).
% word_induct2
thf(fact_1912_word__induct,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [P: ( word @ A ) > $o,M: word @ A] :
( ( P @ ( zero_zero @ ( word @ A ) ) )
=> ( ! [N: word @ A] :
( ( P @ N )
=> ( P @ ( plus_plus @ ( word @ A ) @ ( one_one @ ( word @ A ) ) @ N ) ) )
=> ( P @ M ) ) ) ) ).
% word_induct
thf(fact_1913_word__induct__less,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [P: ( word @ A ) > $o,M: word @ A] :
( ( P @ ( zero_zero @ ( word @ A ) ) )
=> ( ! [N: word @ A] :
( ( ord_less @ ( word @ A ) @ N @ M )
=> ( ( P @ N )
=> ( P @ ( plus_plus @ ( word @ A ) @ ( one_one @ ( word @ A ) ) @ N ) ) ) )
=> ( P @ M ) ) ) ) ).
% word_induct_less
thf(fact_1914_word__overflow,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [X: word @ A] :
( ( ord_less @ ( word @ A ) @ X @ ( plus_plus @ ( word @ A ) @ X @ ( one_one @ ( word @ A ) ) ) )
| ( ( plus_plus @ ( word @ A ) @ X @ ( one_one @ ( word @ A ) ) )
= ( zero_zero @ ( word @ A ) ) ) ) ) ).
% word_overflow
thf(fact_1915_word__gr0__conv__Suc,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [M: word @ A] :
( ( ord_less @ ( word @ A ) @ ( zero_zero @ ( word @ A ) ) @ M )
=> ? [N: word @ A] :
( M
= ( plus_plus @ ( word @ A ) @ N @ ( one_one @ ( word @ A ) ) ) ) ) ) ).
% word_gr0_conv_Suc
thf(fact_1916_less__is__non__zero__p1,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [A2: word @ A,K: word @ A] :
( ( ord_less @ ( word @ A ) @ A2 @ K )
=> ( ( plus_plus @ ( word @ A ) @ A2 @ ( one_one @ ( word @ A ) ) )
!= ( zero_zero @ ( word @ A ) ) ) ) ) ).
% less_is_non_zero_p1
thf(fact_1917_div__less__dividend__word,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [X: word @ A,N3: word @ A] :
( ( X
!= ( zero_zero @ ( word @ A ) ) )
=> ( ( N3
!= ( one_one @ ( word @ A ) ) )
=> ( ord_less @ ( word @ A ) @ ( divide_divide @ ( word @ A ) @ X @ N3 ) @ X ) ) ) ) ).
% div_less_dividend_word
thf(fact_1918_div__le__mult,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [I: word @ A,K: word @ A,X: word @ A] :
( ( ord_less_eq @ ( word @ A ) @ I @ ( divide_divide @ ( word @ A ) @ K @ X ) )
=> ( ( ord_less @ ( word @ A ) @ ( zero_zero @ ( word @ A ) ) @ X )
=> ( ord_less_eq @ ( word @ A ) @ ( times_times @ ( word @ A ) @ I @ X ) @ K ) ) ) ) ).
% div_le_mult
thf(fact_1919_word__plus__one__nonzero,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [X: word @ A,Y: word @ A] :
( ( ord_less_eq @ ( word @ A ) @ X @ ( plus_plus @ ( word @ A ) @ X @ Y ) )
=> ( ( Y
!= ( zero_zero @ ( word @ A ) ) )
=> ( ( plus_plus @ ( word @ A ) @ X @ ( one_one @ ( word @ A ) ) )
!= ( zero_zero @ ( word @ A ) ) ) ) ) ) ).
% word_plus_one_nonzero
thf(fact_1920_plus__one__helper2,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [X: word @ A,N3: word @ A] :
( ( ord_less_eq @ ( word @ A ) @ X @ N3 )
=> ( ( ( plus_plus @ ( word @ A ) @ N3 @ ( one_one @ ( word @ A ) ) )
!= ( zero_zero @ ( word @ A ) ) )
=> ( ord_less @ ( word @ A ) @ X @ ( plus_plus @ ( word @ A ) @ N3 @ ( one_one @ ( word @ A ) ) ) ) ) ) ) ).
% plus_one_helper2
thf(fact_1921_neq__0__no__wrap,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [X: word @ A,Y: word @ A] :
( ( ord_less_eq @ ( word @ A ) @ X @ ( plus_plus @ ( word @ A ) @ X @ Y ) )
=> ( ( X
!= ( zero_zero @ ( word @ A ) ) )
=> ( ( plus_plus @ ( word @ A ) @ X @ Y )
!= ( zero_zero @ ( word @ A ) ) ) ) ) ) ).
% neq_0_no_wrap
thf(fact_1922_lt1__neq0,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [X: word @ A] :
( ( ord_less_eq @ ( word @ A ) @ ( one_one @ ( word @ A ) ) @ X )
= ( X
!= ( zero_zero @ ( word @ A ) ) ) ) ) ).
% lt1_neq0
thf(fact_1923_word__coorder_Oextremum,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [A2: word @ A] : ( ord_less_eq @ ( word @ A ) @ ( zero_zero @ ( word @ A ) ) @ A2 ) ) ).
% word_coorder.extremum
thf(fact_1924_word__coorder_Oextremum__uniqueI,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [A2: word @ A] :
( ( ord_less_eq @ ( word @ A ) @ A2 @ ( zero_zero @ ( word @ A ) ) )
=> ( A2
= ( zero_zero @ ( word @ A ) ) ) ) ) ).
% word_coorder.extremum_uniqueI
thf(fact_1925_word__zero__le,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [Y: word @ A] : ( ord_less_eq @ ( word @ A ) @ ( zero_zero @ ( word @ A ) ) @ Y ) ) ).
% word_zero_le
thf(fact_1926_range__subset__card,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [A2: word @ A,B2: word @ A,C2: word @ A,D: word @ A] :
( ( ord_less_eq @ ( set @ ( word @ A ) ) @ ( set_or1337092689740270186AtMost @ ( word @ A ) @ A2 @ B2 ) @ ( set_or1337092689740270186AtMost @ ( word @ A ) @ C2 @ D ) )
=> ( ( ord_less_eq @ ( word @ A ) @ A2 @ B2 )
=> ( ( ord_less_eq @ ( word @ A ) @ C2 @ D )
& ( ord_less_eq @ ( word @ A ) @ ( minus_minus @ ( word @ A ) @ B2 @ A2 ) @ ( minus_minus @ ( word @ A ) @ D @ C2 ) ) ) ) ) ) ).
% range_subset_card
thf(fact_1927_word__div__1,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [N3: word @ A] :
( ( divide_divide @ ( word @ A ) @ N3 @ ( one_one @ ( word @ A ) ) )
= N3 ) ) ).
% word_div_1
thf(fact_1928_word__diff__ls_I4_J,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [Y: word @ A,W: word @ A,Xa: word @ A,X: word @ A] :
( ( ord_less_eq @ ( word @ A ) @ Y @ ( plus_plus @ ( word @ A ) @ W @ Xa ) )
=> ( ( ord_less_eq @ ( word @ A ) @ X @ Y )
=> ( ord_less_eq @ ( word @ A ) @ ( minus_minus @ ( word @ A ) @ Y @ X ) @ ( minus_minus @ ( word @ A ) @ ( plus_plus @ ( word @ A ) @ W @ Xa ) @ X ) ) ) ) ) ).
% word_diff_ls(4)
thf(fact_1929_word__diff__ls_I3_J,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [Y: word @ A,X: word @ A,W: word @ A,Xa: word @ A] :
( ( ord_less_eq @ ( word @ A ) @ ( minus_minus @ ( word @ A ) @ Y @ X ) @ ( minus_minus @ ( word @ A ) @ ( plus_plus @ ( word @ A ) @ W @ Xa ) @ X ) )
=> ( ( ord_less_eq @ ( word @ A ) @ X @ ( plus_plus @ ( word @ A ) @ W @ Xa ) )
=> ( ord_less_eq @ ( word @ A ) @ Y @ ( plus_plus @ ( word @ A ) @ W @ Xa ) ) ) ) ) ).
% word_diff_ls(3)
thf(fact_1930_word__diff__ls_I2_J,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [Y: word @ A,W: word @ A,Xa: word @ A,X: word @ A] :
( ( ord_less @ ( word @ A ) @ Y @ ( plus_plus @ ( word @ A ) @ W @ Xa ) )
=> ( ( ord_less_eq @ ( word @ A ) @ X @ Y )
=> ( ord_less @ ( word @ A ) @ ( minus_minus @ ( word @ A ) @ Y @ X ) @ ( minus_minus @ ( word @ A ) @ ( plus_plus @ ( word @ A ) @ W @ Xa ) @ X ) ) ) ) ) ).
% word_diff_ls(2)
thf(fact_1931_word__diff__ls_I1_J,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [Y: word @ A,X: word @ A,W: word @ A,Xa: word @ A] :
( ( ord_less @ ( word @ A ) @ ( minus_minus @ ( word @ A ) @ Y @ X ) @ ( minus_minus @ ( word @ A ) @ ( plus_plus @ ( word @ A ) @ W @ Xa ) @ X ) )
=> ( ( ord_less_eq @ ( word @ A ) @ X @ ( plus_plus @ ( word @ A ) @ W @ Xa ) )
=> ( ord_less @ ( word @ A ) @ Y @ ( plus_plus @ ( word @ A ) @ W @ Xa ) ) ) ) ) ).
% word_diff_ls(1)
thf(fact_1932_Word_Oword__l__diffs_I4_J,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [W: word @ A,Xa: word @ A,Z: word @ A,X: word @ A] :
( ( ord_less_eq @ ( word @ A ) @ ( plus_plus @ ( word @ A ) @ W @ Xa ) @ Z )
=> ( ( ord_less_eq @ ( word @ A ) @ X @ ( plus_plus @ ( word @ A ) @ W @ Xa ) )
=> ( ord_less_eq @ ( word @ A ) @ ( minus_minus @ ( word @ A ) @ ( plus_plus @ ( word @ A ) @ W @ Xa ) @ X ) @ ( minus_minus @ ( word @ A ) @ Z @ X ) ) ) ) ) ).
% Word.word_l_diffs(4)
thf(fact_1933_Word_Oword__l__diffs_I3_J,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [W: word @ A,Xa: word @ A,X: word @ A,Z: word @ A] :
( ( ord_less_eq @ ( word @ A ) @ ( minus_minus @ ( word @ A ) @ ( plus_plus @ ( word @ A ) @ W @ Xa ) @ X ) @ ( minus_minus @ ( word @ A ) @ Z @ X ) )
=> ( ( ord_less_eq @ ( word @ A ) @ X @ Z )
=> ( ord_less_eq @ ( word @ A ) @ ( plus_plus @ ( word @ A ) @ W @ Xa ) @ Z ) ) ) ) ).
% Word.word_l_diffs(3)
thf(fact_1934_Word_Oword__l__diffs_I2_J,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [W: word @ A,Xa: word @ A,Z: word @ A,X: word @ A] :
( ( ord_less @ ( word @ A ) @ ( plus_plus @ ( word @ A ) @ W @ Xa ) @ Z )
=> ( ( ord_less_eq @ ( word @ A ) @ X @ ( plus_plus @ ( word @ A ) @ W @ Xa ) )
=> ( ord_less @ ( word @ A ) @ ( minus_minus @ ( word @ A ) @ ( plus_plus @ ( word @ A ) @ W @ Xa ) @ X ) @ ( minus_minus @ ( word @ A ) @ Z @ X ) ) ) ) ) ).
% Word.word_l_diffs(2)
thf(fact_1935_Word_Oword__l__diffs_I1_J,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [W: word @ A,Xa: word @ A,X: word @ A,Z: word @ A] :
( ( ord_less @ ( word @ A ) @ ( minus_minus @ ( word @ A ) @ ( plus_plus @ ( word @ A ) @ W @ Xa ) @ X ) @ ( minus_minus @ ( word @ A ) @ Z @ X ) )
=> ( ( ord_less_eq @ ( word @ A ) @ X @ Z )
=> ( ord_less @ ( word @ A ) @ ( plus_plus @ ( word @ A ) @ W @ Xa ) @ Z ) ) ) ) ).
% Word.word_l_diffs(1)
thf(fact_1936_word__plus__mcs_I4_J,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [V: word @ A,Xb: word @ A,W: word @ A,Xa: word @ A,X: word @ A] :
( ( ord_less_eq @ ( word @ A ) @ ( plus_plus @ ( word @ A ) @ V @ Xb ) @ ( plus_plus @ ( word @ A ) @ W @ Xa ) )
=> ( ( ord_less_eq @ ( word @ A ) @ X @ ( plus_plus @ ( word @ A ) @ V @ Xb ) )
=> ( ord_less_eq @ ( word @ A ) @ ( minus_minus @ ( word @ A ) @ ( plus_plus @ ( word @ A ) @ V @ Xb ) @ X ) @ ( minus_minus @ ( word @ A ) @ ( plus_plus @ ( word @ A ) @ W @ Xa ) @ X ) ) ) ) ) ).
% word_plus_mcs(4)
thf(fact_1937_word__plus__mcs_I3_J,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [V: word @ A,Xb: word @ A,X: word @ A,W: word @ A,Xa: word @ A] :
( ( ord_less_eq @ ( word @ A ) @ ( minus_minus @ ( word @ A ) @ ( plus_plus @ ( word @ A ) @ V @ Xb ) @ X ) @ ( minus_minus @ ( word @ A ) @ ( plus_plus @ ( word @ A ) @ W @ Xa ) @ X ) )
=> ( ( ord_less_eq @ ( word @ A ) @ X @ ( 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_1938_word__plus__mcs_I2_J,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [V: word @ A,Xb: word @ A,W: word @ A,Xa: word @ A,X: word @ A] :
( ( ord_less @ ( word @ A ) @ ( plus_plus @ ( word @ A ) @ V @ Xb ) @ ( plus_plus @ ( word @ A ) @ W @ Xa ) )
=> ( ( ord_less_eq @ ( word @ A ) @ X @ ( plus_plus @ ( word @ A ) @ V @ Xb ) )
=> ( ord_less @ ( word @ A ) @ ( minus_minus @ ( word @ A ) @ ( plus_plus @ ( word @ A ) @ V @ Xb ) @ X ) @ ( minus_minus @ ( word @ A ) @ ( plus_plus @ ( word @ A ) @ W @ Xa ) @ X ) ) ) ) ) ).
% word_plus_mcs(2)
thf(fact_1939_word__plus__mcs_I1_J,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [V: word @ A,Xb: word @ A,X: word @ A,W: word @ A,Xa: word @ A] :
( ( ord_less @ ( word @ A ) @ ( minus_minus @ ( word @ A ) @ ( plus_plus @ ( word @ A ) @ V @ Xb ) @ X ) @ ( minus_minus @ ( word @ A ) @ ( plus_plus @ ( word @ A ) @ W @ Xa ) @ X ) )
=> ( ( ord_less_eq @ ( word @ A ) @ X @ ( 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_1940_More__Word_Oword__l__diffs_I4_J,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [W: word @ A,X: word @ A,Z: word @ A] :
( ( ord_less_eq @ ( word @ A ) @ ( plus_plus @ ( word @ A ) @ W @ X ) @ Z )
=> ( ( ord_less_eq @ ( word @ A ) @ X @ ( plus_plus @ ( word @ A ) @ W @ X ) )
=> ( ord_less_eq @ ( word @ A ) @ W @ ( minus_minus @ ( word @ A ) @ Z @ X ) ) ) ) ) ).
% More_Word.word_l_diffs(4)
thf(fact_1941_More__Word_Oword__l__diffs_I3_J,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [W: word @ A,Z: word @ A,X: word @ A] :
( ( ord_less_eq @ ( word @ A ) @ W @ ( minus_minus @ ( word @ A ) @ Z @ X ) )
=> ( ( ord_less_eq @ ( word @ A ) @ X @ Z )
=> ( ord_less_eq @ ( word @ A ) @ ( plus_plus @ ( word @ A ) @ W @ X ) @ Z ) ) ) ) ).
% More_Word.word_l_diffs(3)
thf(fact_1942_More__Word_Oword__l__diffs_I2_J,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [W: word @ A,X: word @ A,Z: word @ A] :
( ( ord_less @ ( word @ A ) @ ( plus_plus @ ( word @ A ) @ W @ X ) @ Z )
=> ( ( ord_less_eq @ ( word @ A ) @ X @ ( plus_plus @ ( word @ A ) @ W @ X ) )
=> ( ord_less @ ( word @ A ) @ W @ ( minus_minus @ ( word @ A ) @ Z @ X ) ) ) ) ) ).
% More_Word.word_l_diffs(2)
thf(fact_1943_More__Word_Oword__l__diffs_I1_J,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [W: word @ A,Z: word @ A,X: word @ A] :
( ( ord_less @ ( word @ A ) @ W @ ( minus_minus @ ( word @ A ) @ Z @ X ) )
=> ( ( ord_less_eq @ ( word @ A ) @ X @ Z )
=> ( ord_less @ ( word @ A ) @ ( plus_plus @ ( word @ A ) @ W @ X ) @ Z ) ) ) ) ).
% More_Word.word_l_diffs(1)
thf(fact_1944_word__diff__ls_H_I4_J,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [Y: word @ A,W: word @ A,X: word @ A] :
( ( ord_less_eq @ ( word @ A ) @ Y @ ( plus_plus @ ( word @ A ) @ W @ X ) )
=> ( ( ord_less_eq @ ( word @ A ) @ X @ Y )
=> ( ord_less_eq @ ( word @ A ) @ ( minus_minus @ ( word @ A ) @ Y @ X ) @ W ) ) ) ) ).
% word_diff_ls'(4)
thf(fact_1945_word__diff__ls_H_I3_J,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [Y: word @ A,X: word @ A,W: word @ A] :
( ( ord_less_eq @ ( word @ A ) @ ( minus_minus @ ( word @ A ) @ Y @ X ) @ W )
=> ( ( ord_less_eq @ ( word @ A ) @ X @ ( plus_plus @ ( word @ A ) @ W @ X ) )
=> ( ord_less_eq @ ( word @ A ) @ Y @ ( plus_plus @ ( word @ A ) @ W @ X ) ) ) ) ) ).
% word_diff_ls'(3)
thf(fact_1946_word__diff__ls_H_I2_J,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [Y: word @ A,W: word @ A,X: word @ A] :
( ( ord_less @ ( word @ A ) @ Y @ ( plus_plus @ ( word @ A ) @ W @ X ) )
=> ( ( ord_less_eq @ ( word @ A ) @ X @ Y )
=> ( ord_less @ ( word @ A ) @ ( minus_minus @ ( word @ A ) @ Y @ X ) @ W ) ) ) ) ).
% word_diff_ls'(2)
thf(fact_1947_word__diff__ls_H_I1_J,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [Y: word @ A,X: word @ A,W: word @ A] :
( ( ord_less @ ( word @ A ) @ ( minus_minus @ ( word @ A ) @ Y @ X ) @ W )
=> ( ( ord_less_eq @ ( word @ A ) @ X @ ( plus_plus @ ( word @ A ) @ W @ X ) )
=> ( ord_less @ ( word @ A ) @ Y @ ( plus_plus @ ( word @ A ) @ W @ X ) ) ) ) ) ).
% word_diff_ls'(1)
thf(fact_1948_word__l__diffs_H_I4_J,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [W: word @ A,X: word @ A,Z: word @ A] :
( ( ord_less_eq @ ( word @ A ) @ ( plus_plus @ ( word @ A ) @ W @ X ) @ Z )
=> ( ( ord_less_eq @ ( word @ A ) @ X @ ( plus_plus @ ( word @ A ) @ W @ X ) )
=> ( ord_less_eq @ ( word @ A ) @ ( minus_minus @ ( word @ A ) @ ( plus_plus @ ( word @ A ) @ W @ X ) @ X ) @ ( minus_minus @ ( word @ A ) @ Z @ X ) ) ) ) ) ).
% word_l_diffs'(4)
thf(fact_1949_word__l__diffs_H_I3_J,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [W: word @ A,X: word @ A,Z: word @ A] :
( ( ord_less_eq @ ( word @ A ) @ ( minus_minus @ ( word @ A ) @ ( plus_plus @ ( word @ A ) @ W @ X ) @ X ) @ ( minus_minus @ ( word @ A ) @ Z @ X ) )
=> ( ( ord_less_eq @ ( word @ A ) @ X @ Z )
=> ( ord_less_eq @ ( word @ A ) @ ( plus_plus @ ( word @ A ) @ W @ X ) @ Z ) ) ) ) ).
% word_l_diffs'(3)
thf(fact_1950_word__l__diffs_H_I2_J,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [W: word @ A,X: word @ A,Z: word @ A] :
( ( ord_less @ ( word @ A ) @ ( plus_plus @ ( word @ A ) @ W @ X ) @ Z )
=> ( ( ord_less_eq @ ( word @ A ) @ X @ ( plus_plus @ ( word @ A ) @ W @ X ) )
=> ( ord_less @ ( word @ A ) @ ( minus_minus @ ( word @ A ) @ ( plus_plus @ ( word @ A ) @ W @ X ) @ X ) @ ( minus_minus @ ( word @ A ) @ Z @ X ) ) ) ) ) ).
% word_l_diffs'(2)
thf(fact_1951_word__l__diffs_H_I1_J,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [W: word @ A,X: word @ A,Z: word @ A] :
( ( ord_less @ ( word @ A ) @ ( minus_minus @ ( word @ A ) @ ( plus_plus @ ( word @ A ) @ W @ X ) @ X ) @ ( minus_minus @ ( word @ A ) @ Z @ X ) )
=> ( ( ord_less_eq @ ( word @ A ) @ X @ Z )
=> ( ord_less @ ( word @ A ) @ ( plus_plus @ ( word @ A ) @ W @ X ) @ Z ) ) ) ) ).
% word_l_diffs'(1)
thf(fact_1952_word__diff__ls_H_H_I4_J,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [Y: word @ A,W: word @ A,X: word @ A] :
( ( ord_less_eq @ ( word @ A ) @ Y @ ( plus_plus @ ( word @ A ) @ W @ X ) )
=> ( ( ord_less_eq @ ( word @ A ) @ X @ Y )
=> ( ord_less_eq @ ( word @ A ) @ ( minus_minus @ ( word @ A ) @ Y @ X ) @ ( minus_minus @ ( word @ A ) @ ( plus_plus @ ( word @ A ) @ W @ X ) @ X ) ) ) ) ) ).
% word_diff_ls''(4)
thf(fact_1953_word__diff__ls_H_H_I3_J,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [Y: word @ A,X: word @ A,W: word @ A] :
( ( ord_less_eq @ ( word @ A ) @ ( minus_minus @ ( word @ A ) @ Y @ X ) @ ( minus_minus @ ( word @ A ) @ ( plus_plus @ ( word @ A ) @ W @ X ) @ X ) )
=> ( ( ord_less_eq @ ( word @ A ) @ X @ ( plus_plus @ ( word @ A ) @ W @ X ) )
=> ( ord_less_eq @ ( word @ A ) @ Y @ ( plus_plus @ ( word @ A ) @ W @ X ) ) ) ) ) ).
% word_diff_ls''(3)
thf(fact_1954_word__diff__ls_H_H_I2_J,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [Y: word @ A,W: word @ A,X: word @ A] :
( ( ord_less @ ( word @ A ) @ Y @ ( plus_plus @ ( word @ A ) @ W @ X ) )
=> ( ( ord_less_eq @ ( word @ A ) @ X @ Y )
=> ( ord_less @ ( word @ A ) @ ( minus_minus @ ( word @ A ) @ Y @ X ) @ ( minus_minus @ ( word @ A ) @ ( plus_plus @ ( word @ A ) @ W @ X ) @ X ) ) ) ) ) ).
% word_diff_ls''(2)
thf(fact_1955_word__diff__ls_H_H_I1_J,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [Y: word @ A,X: word @ A,W: word @ A] :
( ( ord_less @ ( word @ A ) @ ( minus_minus @ ( word @ A ) @ Y @ X ) @ ( minus_minus @ ( word @ A ) @ ( plus_plus @ ( word @ A ) @ W @ X ) @ X ) )
=> ( ( ord_less_eq @ ( word @ A ) @ X @ ( plus_plus @ ( word @ A ) @ W @ X ) )
=> ( ord_less @ ( word @ A ) @ Y @ ( plus_plus @ ( word @ A ) @ W @ X ) ) ) ) ) ).
% word_diff_ls''(1)
thf(fact_1956_le__plus,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [C2: word @ A,B2: word @ A,A2: word @ A] :
( ( ord_less_eq @ ( word @ A ) @ C2 @ ( minus_minus @ ( word @ A ) @ B2 @ A2 ) )
=> ( ( ord_less_eq @ ( word @ A ) @ A2 @ B2 )
=> ( ord_less_eq @ ( word @ A ) @ ( plus_plus @ ( word @ A ) @ A2 @ C2 ) @ B2 ) ) ) ) ).
% le_plus
thf(fact_1957_le__plus_H,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [A2: word @ A,B2: word @ A,C2: word @ A] :
( ( ord_less_eq @ ( word @ A ) @ A2 @ B2 )
=> ( ( ord_less_eq @ ( word @ A ) @ C2 @ ( minus_minus @ ( word @ A ) @ B2 @ A2 ) )
=> ( ord_less_eq @ ( word @ A ) @ ( plus_plus @ ( word @ A ) @ A2 @ C2 ) @ B2 ) ) ) ) ).
% le_plus'
thf(fact_1958_le__minus_H,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [A2: word @ A,C2: word @ A,B2: word @ A] :
( ( ord_less_eq @ ( word @ A ) @ ( plus_plus @ ( word @ A ) @ A2 @ C2 ) @ B2 )
=> ( ( ord_less_eq @ ( word @ A ) @ A2 @ ( plus_plus @ ( word @ A ) @ A2 @ C2 ) )
=> ( ord_less_eq @ ( word @ A ) @ C2 @ ( minus_minus @ ( word @ A ) @ B2 @ A2 ) ) ) ) ) ).
% le_minus'
thf(fact_1959_word__sub__le,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [Y: word @ A,X: word @ A] :
( ( ord_less_eq @ ( word @ A ) @ Y @ X )
=> ( ord_less_eq @ ( word @ A ) @ ( minus_minus @ ( word @ A ) @ X @ Y ) @ X ) ) ) ).
% word_sub_le
thf(fact_1960_word__sub__le__iff,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [X: word @ A,Y: word @ A] :
( ( ord_less_eq @ ( word @ A ) @ ( minus_minus @ ( word @ A ) @ X @ Y ) @ X )
= ( ord_less_eq @ ( word @ A ) @ Y @ X ) ) ) ).
% word_sub_le_iff
thf(fact_1961_word__le__minus__mono,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [A2: word @ A,C2: word @ A,D: word @ A,B2: word @ A] :
( ( ord_less_eq @ ( word @ A ) @ A2 @ C2 )
=> ( ( ord_less_eq @ ( word @ A ) @ D @ B2 )
=> ( ( ord_less_eq @ ( word @ A ) @ ( minus_minus @ ( word @ A ) @ A2 @ B2 ) @ A2 )
=> ( ( ord_less_eq @ ( word @ A ) @ ( minus_minus @ ( word @ A ) @ C2 @ D ) @ C2 )
=> ( ord_less_eq @ ( word @ A ) @ ( minus_minus @ ( word @ A ) @ A2 @ B2 ) @ ( minus_minus @ ( word @ A ) @ C2 @ D ) ) ) ) ) ) ) ).
% word_le_minus_mono
thf(fact_1962_word__less__add__right,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [X: word @ A,Y: word @ A,Z: word @ A] :
( ( ord_less @ ( word @ A ) @ X @ ( minus_minus @ ( word @ A ) @ Y @ Z ) )
=> ( ( ord_less_eq @ ( word @ A ) @ Z @ Y )
=> ( ord_less @ ( word @ A ) @ ( plus_plus @ ( word @ A ) @ X @ Z ) @ Y ) ) ) ) ).
% word_less_add_right
thf(fact_1963_word__less__sub__right,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [X: word @ A,Y: word @ A,Z: word @ A] :
( ( ord_less @ ( word @ A ) @ X @ ( plus_plus @ ( word @ A ) @ Y @ Z ) )
=> ( ( ord_less_eq @ ( word @ A ) @ Y @ X )
=> ( ord_less @ ( word @ A ) @ ( minus_minus @ ( word @ A ) @ X @ Y ) @ Z ) ) ) ) ).
% word_less_sub_right
thf(fact_1964_word__le__minus__cancel,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [Y: word @ A,X: word @ A,Z: word @ A] :
( ( ord_less_eq @ ( word @ A ) @ ( minus_minus @ ( word @ A ) @ Y @ X ) @ ( minus_minus @ ( word @ A ) @ Z @ X ) )
=> ( ( ord_less_eq @ ( word @ A ) @ X @ Z )
=> ( ord_less_eq @ ( word @ A ) @ Y @ Z ) ) ) ) ).
% word_le_minus_cancel
thf(fact_1965_le__step__down__word,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [I: word @ A,N3: word @ A] :
( ( ord_less_eq @ ( word @ A ) @ I @ N3 )
=> ( ( I != N3 )
=> ( ord_less_eq @ ( word @ A ) @ I @ ( minus_minus @ ( word @ A ) @ N3 @ ( one_one @ ( word @ A ) ) ) ) ) ) ) ).
% le_step_down_word
thf(fact_1966_plus__minus__no__overflow,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [X: word @ A,Ab: word @ A,C2: word @ A] :
( ( ord_less @ ( word @ A ) @ X @ ( minus_minus @ ( word @ A ) @ Ab @ C2 ) )
=> ( ( ord_less_eq @ ( word @ A ) @ C2 @ Ab )
=> ( ord_less_eq @ ( word @ A ) @ X @ ( plus_plus @ ( word @ A ) @ X @ C2 ) ) ) ) ) ).
% plus_minus_no_overflow
thf(fact_1967_word__less__minus__cancel,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [Y: word @ A,X: word @ A,Z: word @ A] :
( ( ord_less @ ( word @ A ) @ ( minus_minus @ ( word @ A ) @ Y @ X ) @ ( minus_minus @ ( word @ A ) @ Z @ X ) )
=> ( ( ord_less_eq @ ( word @ A ) @ X @ Z )
=> ( ord_less @ ( word @ A ) @ Y @ Z ) ) ) ) ).
% word_less_minus_cancel
thf(fact_1968_word__le__minus__mono__left,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [Y: word @ A,Z: word @ A,X: word @ A] :
( ( ord_less_eq @ ( word @ A ) @ Y @ Z )
=> ( ( ord_less_eq @ ( word @ A ) @ X @ Y )
=> ( ord_less_eq @ ( word @ A ) @ ( minus_minus @ ( word @ A ) @ Y @ X ) @ ( minus_minus @ ( word @ A ) @ Z @ X ) ) ) ) ) ).
% word_le_minus_mono_left
thf(fact_1969_le__step__down__word__2,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [X: word @ A,Y: word @ A] :
( ( ord_less_eq @ ( word @ A ) @ X @ Y )
=> ( ( X != Y )
=> ( ord_less_eq @ ( word @ A ) @ X @ ( minus_minus @ ( word @ A ) @ Y @ ( one_one @ ( word @ A ) ) ) ) ) ) ) ).
% le_step_down_word_2
thf(fact_1970_word__le__imp__diff__le,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [K: word @ A,N3: word @ A,M: word @ A] :
( ( ord_less_eq @ ( word @ A ) @ K @ N3 )
=> ( ( ord_less_eq @ ( word @ A ) @ N3 @ M )
=> ( ord_less_eq @ ( word @ A ) @ ( minus_minus @ ( word @ A ) @ N3 @ K ) @ M ) ) ) ) ).
% word_le_imp_diff_le
thf(fact_1971_plus__minus__no__overflow__ab,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [X: word @ A,Ab: word @ A,C2: word @ A] :
( ( ord_less_eq @ ( word @ A ) @ X @ ( minus_minus @ ( word @ A ) @ Ab @ C2 ) )
=> ( ( ord_less_eq @ ( word @ A ) @ C2 @ Ab )
=> ( ord_less_eq @ ( word @ A ) @ X @ ( plus_plus @ ( word @ A ) @ X @ C2 ) ) ) ) ) ).
% plus_minus_no_overflow_ab
thf(fact_1972_word__less__minus__mono__left,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [Y: word @ A,Z: word @ A,X: word @ A] :
( ( ord_less @ ( word @ A ) @ Y @ Z )
=> ( ( ord_less_eq @ ( word @ A ) @ X @ Y )
=> ( ord_less @ ( word @ A ) @ ( minus_minus @ ( word @ A ) @ Y @ X ) @ ( minus_minus @ ( word @ A ) @ Z @ X ) ) ) ) ) ).
% word_less_minus_mono_left
thf(fact_1973_word__le__minus__one__leq,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [X: word @ A,Y: word @ A] :
( ( ord_less @ ( word @ A ) @ X @ Y )
=> ( ord_less_eq @ ( word @ A ) @ X @ ( minus_minus @ ( word @ A ) @ Y @ ( one_one @ ( word @ A ) ) ) ) ) ) ).
% word_le_minus_one_leq
thf(fact_1974_word__minus__one__le__leq,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [X: word @ A,Y: word @ A] :
( ( ord_less @ ( word @ A ) @ ( minus_minus @ ( word @ A ) @ X @ ( one_one @ ( word @ A ) ) ) @ Y )
=> ( ord_less_eq @ ( word @ A ) @ X @ Y ) ) ) ).
% word_minus_one_le_leq
thf(fact_1975_word__less__imp__diff__less,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [K: word @ A,N3: word @ A,M: word @ A] :
( ( ord_less_eq @ ( word @ A ) @ K @ N3 )
=> ( ( ord_less @ ( word @ A ) @ N3 @ M )
=> ( ord_less @ ( word @ A ) @ ( minus_minus @ ( word @ A ) @ N3 @ K ) @ M ) ) ) ) ).
% word_less_imp_diff_less
thf(fact_1976_word__le__minus__mono__right,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [Z: word @ A,Y: word @ A,X: word @ A] :
( ( ord_less_eq @ ( word @ A ) @ Z @ Y )
=> ( ( ord_less_eq @ ( word @ A ) @ Y @ X )
=> ( ( ord_less_eq @ ( word @ A ) @ Z @ X )
=> ( ord_less_eq @ ( word @ A ) @ ( minus_minus @ ( word @ A ) @ X @ Y ) @ ( minus_minus @ ( word @ A ) @ X @ Z ) ) ) ) ) ) ).
% word_le_minus_mono_right
thf(fact_1977_word__less__minus__mono,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [A2: word @ A,C2: word @ A,D: word @ A,B2: word @ A] :
( ( ord_less @ ( word @ A ) @ A2 @ C2 )
=> ( ( ord_less @ ( word @ A ) @ D @ B2 )
=> ( ( ord_less @ ( word @ A ) @ ( minus_minus @ ( word @ A ) @ A2 @ B2 ) @ A2 )
=> ( ( ord_less @ ( word @ A ) @ ( minus_minus @ ( word @ A ) @ C2 @ D ) @ C2 )
=> ( ord_less @ ( word @ A ) @ ( minus_minus @ ( word @ A ) @ A2 @ B2 ) @ ( minus_minus @ ( word @ A ) @ C2 @ D ) ) ) ) ) ) ) ).
% word_less_minus_mono
thf(fact_1978_word__less__cases,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [X: word @ A,Y: word @ A] :
( ( ord_less @ ( word @ A ) @ X @ Y )
=> ( ( X
= ( minus_minus @ ( word @ A ) @ Y @ ( one_one @ ( word @ A ) ) ) )
| ( ord_less @ ( word @ A ) @ X @ ( minus_minus @ ( word @ A ) @ Y @ ( one_one @ ( word @ A ) ) ) ) ) ) ) ).
% word_less_cases
thf(fact_1979_word__sub__less__iff,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [X: word @ A,Y: word @ A] :
( ( ord_less @ ( word @ A ) @ X @ ( minus_minus @ ( word @ A ) @ X @ Y ) )
= ( ord_less @ ( word @ A ) @ X @ Y ) ) ) ).
% word_sub_less_iff
thf(fact_1980_sub__wrap__lt,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [X: word @ A,Z: word @ A] :
( ( ord_less @ ( word @ A ) @ X @ ( minus_minus @ ( word @ A ) @ X @ Z ) )
= ( ord_less @ ( word @ A ) @ X @ Z ) ) ) ).
% sub_wrap_lt
thf(fact_1981_word__le__add,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [X: word @ A,Y: word @ A] :
( ( ord_less_eq @ ( word @ A ) @ X @ Y )
=> ? [N: nat] :
( Y
= ( plus_plus @ ( word @ A ) @ X @ ( semiring_1_of_nat @ ( word @ A ) @ N ) ) ) ) ) ).
% word_le_add
thf(fact_1982_word__1__0,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [A2: word @ A,B2: word @ A,X: nat] :
( ( ord_less_eq @ ( word @ A ) @ ( plus_plus @ ( word @ A ) @ A2 @ ( one_one @ ( word @ A ) ) ) @ B2 )
=> ( ( ord_less @ ( word @ A ) @ A2 @ ( semiring_1_of_nat @ ( word @ A ) @ X ) )
=> ( ord_less @ ( word @ A ) @ A2 @ B2 ) ) ) ) ).
% word_1_0
thf(fact_1983_word__div__mult__le,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [A2: word @ A,B2: word @ A] : ( ord_less_eq @ ( word @ A ) @ ( times_times @ ( word @ A ) @ ( divide_divide @ ( word @ A ) @ A2 @ B2 ) @ B2 ) @ A2 ) ) ).
% word_div_mult_le
thf(fact_1984_div__to__mult__word__lt,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [X: word @ A,Y: word @ A,Z: word @ A] :
( ( ord_less_eq @ ( word @ A ) @ X @ ( divide_divide @ ( word @ A ) @ Y @ Z ) )
=> ( ord_less_eq @ ( word @ A ) @ ( times_times @ ( word @ A ) @ X @ Z ) @ Y ) ) ) ).
% div_to_mult_word_lt
thf(fact_1985_word__plus__strict__mono__right,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [Y: word @ A,Z: word @ A,X: word @ A] :
( ( ord_less @ ( word @ A ) @ Y @ Z )
=> ( ( ord_less_eq @ ( word @ A ) @ X @ ( plus_plus @ ( word @ A ) @ X @ Z ) )
=> ( ord_less @ ( word @ A ) @ ( plus_plus @ ( word @ A ) @ X @ Y ) @ ( plus_plus @ ( word @ A ) @ X @ Z ) ) ) ) ) ).
% word_plus_strict_mono_right
thf(fact_1986_plus__le__left__cancel__nowrap,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [X: word @ A,Y6: word @ A,Y: word @ A] :
( ( ord_less_eq @ ( word @ A ) @ X @ ( plus_plus @ ( word @ A ) @ X @ Y6 ) )
=> ( ( ord_less_eq @ ( word @ A ) @ X @ ( plus_plus @ ( word @ A ) @ X @ Y ) )
=> ( ( ord_less @ ( word @ A ) @ ( plus_plus @ ( word @ A ) @ X @ Y6 ) @ ( plus_plus @ ( word @ A ) @ X @ Y ) )
= ( ord_less @ ( word @ A ) @ Y6 @ Y ) ) ) ) ) ).
% plus_le_left_cancel_nowrap
thf(fact_1987_word__plus__mono__left,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [Y: word @ A,Z: word @ A,X: word @ A] :
( ( ord_less_eq @ ( word @ A ) @ Y @ Z )
=> ( ( ord_less_eq @ ( word @ A ) @ X @ ( plus_plus @ ( word @ A ) @ X @ Z ) )
=> ( ord_less_eq @ ( word @ A ) @ ( plus_plus @ ( word @ A ) @ Y @ X ) @ ( plus_plus @ ( word @ A ) @ Z @ X ) ) ) ) ) ).
% word_plus_mono_left
thf(fact_1988_word__add__increasing,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [P4: word @ A,W: word @ A,X: word @ A] :
( ( ord_less_eq @ ( word @ A ) @ ( plus_plus @ ( word @ A ) @ P4 @ W ) @ X )
=> ( ( ord_less_eq @ ( word @ A ) @ P4 @ ( plus_plus @ ( word @ A ) @ P4 @ W ) )
=> ( ord_less_eq @ ( word @ A ) @ P4 @ X ) ) ) ) ).
% word_add_increasing
thf(fact_1989_word__plus__mono__right2,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [A2: word @ A,B2: word @ A,C2: word @ A] :
( ( ord_less_eq @ ( word @ A ) @ A2 @ ( plus_plus @ ( word @ A ) @ A2 @ B2 ) )
=> ( ( ord_less_eq @ ( word @ A ) @ C2 @ B2 )
=> ( ord_less_eq @ ( word @ A ) @ A2 @ ( plus_plus @ ( word @ A ) @ A2 @ C2 ) ) ) ) ) ).
% word_plus_mono_right2
thf(fact_1990_word__plus__mcs__4_H,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [X: word @ A,V: word @ A,W: word @ A] :
( ( ord_less_eq @ ( word @ A ) @ ( plus_plus @ ( word @ A ) @ X @ V ) @ ( plus_plus @ ( word @ A ) @ X @ W ) )
=> ( ( ord_less_eq @ ( word @ A ) @ X @ ( plus_plus @ ( word @ A ) @ X @ V ) )
=> ( ord_less_eq @ ( word @ A ) @ V @ W ) ) ) ) ).
% word_plus_mcs_4'
thf(fact_1991_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_1992_word__plus__mono__right,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [Y: word @ A,Z: word @ A,X: word @ A] :
( ( ord_less_eq @ ( word @ A ) @ Y @ Z )
=> ( ( ord_less_eq @ ( word @ A ) @ X @ ( plus_plus @ ( word @ A ) @ X @ Z ) )
=> ( ord_less_eq @ ( word @ A ) @ ( plus_plus @ ( word @ A ) @ X @ Y ) @ ( plus_plus @ ( word @ A ) @ X @ Z ) ) ) ) ) ).
% word_plus_mono_right
thf(fact_1993_word__plus__mcs__4,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [V: word @ A,X: word @ A,W: word @ A] :
( ( ord_less_eq @ ( word @ A ) @ ( plus_plus @ ( word @ A ) @ V @ X ) @ ( plus_plus @ ( word @ A ) @ W @ X ) )
=> ( ( ord_less_eq @ ( word @ A ) @ X @ ( plus_plus @ ( word @ A ) @ V @ X ) )
=> ( ord_less_eq @ ( word @ A ) @ V @ W ) ) ) ) ).
% word_plus_mcs_4
thf(fact_1994_word__plus__mcs__3,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [V: word @ A,W: word @ A,X: word @ A] :
( ( ord_less_eq @ ( word @ A ) @ V @ W )
=> ( ( ord_less_eq @ ( word @ A ) @ X @ ( plus_plus @ ( word @ A ) @ W @ X ) )
=> ( ord_less_eq @ ( word @ A ) @ ( plus_plus @ ( word @ A ) @ V @ X ) @ ( plus_plus @ ( word @ A ) @ W @ X ) ) ) ) ) ).
% word_plus_mcs_3
thf(fact_1995_plus__one__helper,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [X: word @ A,N3: word @ A] :
( ( ord_less @ ( word @ A ) @ X @ ( plus_plus @ ( word @ A ) @ N3 @ ( one_one @ ( word @ A ) ) ) )
=> ( ord_less_eq @ ( word @ A ) @ X @ N3 ) ) ) ).
% plus_one_helper
thf(fact_1996_word__le__plus__either,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [X: word @ A,Y: word @ A,Z: word @ A] :
( ( ( ord_less_eq @ ( word @ A ) @ X @ Y )
| ( ord_less_eq @ ( word @ A ) @ X @ Z ) )
=> ( ( ord_less_eq @ ( word @ A ) @ Y @ ( plus_plus @ ( word @ A ) @ Y @ Z ) )
=> ( ord_less_eq @ ( word @ A ) @ X @ ( plus_plus @ ( word @ A ) @ Y @ Z ) ) ) ) ) ).
% word_le_plus_either
thf(fact_1997_word__sub__mono2,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [A2: word @ A,B2: word @ A,C2: word @ A,D: word @ A] :
( ( ord_less_eq @ ( word @ A ) @ ( plus_plus @ ( word @ A ) @ A2 @ B2 ) @ ( plus_plus @ ( word @ A ) @ C2 @ D ) )
=> ( ( ord_less_eq @ ( word @ A ) @ C2 @ A2 )
=> ( ( ord_less_eq @ ( word @ A ) @ B2 @ ( plus_plus @ ( word @ A ) @ A2 @ B2 ) )
=> ( ( ord_less_eq @ ( word @ A ) @ D @ ( plus_plus @ ( word @ A ) @ C2 @ D ) )
=> ( ord_less_eq @ ( word @ A ) @ B2 @ D ) ) ) ) ) ) ).
% word_sub_mono2
thf(fact_1998_word__le__plus__1,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [Y: word @ A,N3: word @ A,A2: word @ A] :
( ( ord_less @ ( word @ A ) @ Y @ ( plus_plus @ ( word @ A ) @ Y @ N3 ) )
=> ( ( ord_less @ ( word @ A ) @ A2 @ N3 )
=> ( ord_less_eq @ ( word @ A ) @ ( plus_plus @ ( word @ A ) @ Y @ A2 ) @ ( plus_plus @ ( word @ A ) @ ( plus_plus @ ( word @ A ) @ Y @ A2 ) @ ( one_one @ ( word @ A ) ) ) ) ) ) ) ).
% word_le_plus_1
thf(fact_1999_word__le__plus,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [A2: word @ A,B2: word @ A,C2: word @ A] :
( ( ord_less @ ( word @ A ) @ A2 @ ( plus_plus @ ( word @ A ) @ A2 @ B2 ) )
=> ( ( ord_less @ ( word @ A ) @ C2 @ B2 )
=> ( ord_less_eq @ ( word @ A ) @ A2 @ ( plus_plus @ ( word @ A ) @ A2 @ C2 ) ) ) ) ) ).
% word_le_plus
thf(fact_2000_word__random,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [P4: word @ A,X7: word @ A,X: word @ A] :
( ( ord_less_eq @ ( word @ A ) @ P4 @ ( plus_plus @ ( word @ A ) @ P4 @ X7 ) )
=> ( ( ord_less_eq @ ( word @ A ) @ X @ X7 )
=> ( ord_less_eq @ ( word @ A ) @ P4 @ ( plus_plus @ ( word @ A ) @ P4 @ X ) ) ) ) ) ).
% word_random
thf(fact_2001_le__no__overflow,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [X: word @ A,B2: word @ A,A2: word @ A] :
( ( ord_less_eq @ ( word @ A ) @ X @ B2 )
=> ( ( ord_less_eq @ ( word @ A ) @ A2 @ ( plus_plus @ ( word @ A ) @ A2 @ B2 ) )
=> ( ord_less_eq @ ( word @ A ) @ X @ ( plus_plus @ ( word @ A ) @ A2 @ B2 ) ) ) ) ) ).
% le_no_overflow
thf(fact_2002_olen__add__eqv,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [Y: word @ A,X: word @ A] :
( ( ord_less_eq @ ( word @ A ) @ Y @ ( plus_plus @ ( word @ A ) @ Y @ X ) )
= ( ord_less_eq @ ( word @ A ) @ X @ ( plus_plus @ ( word @ A ) @ Y @ X ) ) ) ) ).
% olen_add_eqv
thf(fact_2003_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_2004_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_2005_word__le__less__eq,axiom,
! [Z6: $tType] :
( ( type_len @ Z6 )
=> ( ( ord_less_eq @ ( word @ Z6 ) )
= ( ^ [X2: word @ Z6,Y2: word @ Z6] :
( ( X2 = Y2 )
| ( ord_less @ ( word @ Z6 ) @ X2 @ Y2 ) ) ) ) ) ).
% word_le_less_eq
thf(fact_2006_word__not__le,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [X: word @ A,Y: word @ A] :
( ( ~ ( ord_less_eq @ ( word @ A ) @ X @ Y ) )
= ( ord_less @ ( word @ A ) @ Y @ X ) ) ) ).
% word_not_le
thf(fact_2007_word__le__not__less,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ( ( ord_less_eq @ ( word @ A ) )
= ( ^ [B5: word @ A,A6: word @ A] :
~ ( ord_less @ ( word @ A ) @ A6 @ B5 ) ) ) ) ).
% word_le_not_less
thf(fact_2008_constraint__expand,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [X: word @ A,Lower: word @ A,Upper: word @ A] :
( ( member @ ( word @ A ) @ X
@ ( collect @ ( word @ A )
@ ^ [Y2: word @ A] :
( ( ord_less_eq @ ( word @ A ) @ Lower @ Y2 )
& ( ord_less_eq @ ( word @ A ) @ Y2 @ Upper ) ) ) )
= ( ( ord_less_eq @ ( word @ A ) @ Lower @ X )
& ( ord_less_eq @ ( word @ A ) @ X @ Upper ) ) ) ) ).
% constraint_expand
thf(fact_2009_T_092_060_094sub_062m_092_060_094sub_062e_092_060_094sub_062m_092_060_094sub_062b_092_060_094sub_062e_092_060_094sub_062r_H_Opelims,axiom,
! [X: vEBT_VEBT,Xa: nat,Y: nat] :
( ( ( vEBT_T_m_e_m_b_e_r2 @ X @ Xa )
= Y )
=> ( ( accp @ ( product_prod @ vEBT_VEBT @ nat ) @ vEBT_T8099345112685741742_r_rel @ ( product_Pair @ vEBT_VEBT @ nat @ X @ Xa ) )
=> ( ! [A4: $o,B3: $o] :
( ( X
= ( vEBT_Leaf @ A4 @ B3 ) )
=> ( ( Y
= ( one_one @ nat ) )
=> ~ ( accp @ ( product_prod @ vEBT_VEBT @ nat ) @ vEBT_T8099345112685741742_r_rel @ ( product_Pair @ vEBT_VEBT @ nat @ ( vEBT_Leaf @ A4 @ B3 ) @ Xa ) ) ) )
=> ( ! [Uu: nat,Uv: list @ vEBT_VEBT,Uw: vEBT_VEBT] :
( ( X
= ( vEBT_Node @ ( none @ ( product_prod @ nat @ nat ) ) @ Uu @ Uv @ Uw ) )
=> ( ( Y
= ( one_one @ nat ) )
=> ~ ( accp @ ( product_prod @ vEBT_VEBT @ nat ) @ vEBT_T8099345112685741742_r_rel @ ( product_Pair @ vEBT_VEBT @ nat @ ( vEBT_Node @ ( none @ ( product_prod @ nat @ nat ) ) @ Uu @ Uv @ Uw ) @ Xa ) ) ) )
=> ( ! [V3: product_prod @ nat @ nat,Uy2: list @ vEBT_VEBT,Uz2: vEBT_VEBT] :
( ( X
= ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ V3 ) @ ( zero_zero @ nat ) @ Uy2 @ Uz2 ) )
=> ( ( Y
= ( one_one @ nat ) )
=> ~ ( accp @ ( product_prod @ vEBT_VEBT @ nat ) @ vEBT_T8099345112685741742_r_rel @ ( product_Pair @ vEBT_VEBT @ nat @ ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ V3 ) @ ( zero_zero @ nat ) @ Uy2 @ Uz2 ) @ Xa ) ) ) )
=> ( ! [V3: product_prod @ nat @ nat,Vb2: list @ vEBT_VEBT,Vc2: vEBT_VEBT] :
( ( X
= ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ V3 ) @ ( suc @ ( zero_zero @ nat ) ) @ Vb2 @ Vc2 ) )
=> ( ( Y
= ( one_one @ nat ) )
=> ~ ( accp @ ( product_prod @ vEBT_VEBT @ nat ) @ vEBT_T8099345112685741742_r_rel @ ( product_Pair @ vEBT_VEBT @ nat @ ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ V3 ) @ ( suc @ ( zero_zero @ nat ) ) @ Vb2 @ Vc2 ) @ Xa ) ) ) )
=> ~ ! [Mi2: nat,Ma2: nat,Va3: nat,TreeList2: list @ vEBT_VEBT,Summary2: vEBT_VEBT] :
( ( X
= ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( suc @ Va3 ) ) @ TreeList2 @ Summary2 ) )
=> ( ( Y
= ( plus_plus @ nat @ ( one_one @ nat )
@ ( if @ nat @ ( Xa = Mi2 ) @ ( zero_zero @ nat )
@ ( if @ nat @ ( Xa = Ma2 ) @ ( zero_zero @ nat )
@ ( if @ nat @ ( ord_less @ nat @ Xa @ Mi2 ) @ ( zero_zero @ nat )
@ ( if @ nat @ ( ord_less @ nat @ Ma2 @ Xa ) @ ( zero_zero @ nat )
@ ( if @ nat
@ ( ( ord_less @ nat @ Mi2 @ Xa )
& ( ord_less @ nat @ Xa @ Ma2 ) )
@ ( if @ nat @ ( ord_less @ nat @ ( vEBT_VEBT_high @ Xa @ ( divide_divide @ nat @ ( suc @ ( suc @ 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_2010_vebt__member_Opelims_I1_J,axiom,
! [X: vEBT_VEBT,Xa: nat,Y: $o] :
( ( ( vEBT_vebt_member @ X @ Xa )
= Y )
=> ( ( accp @ ( product_prod @ vEBT_VEBT @ nat ) @ vEBT_vebt_member_rel @ ( product_Pair @ vEBT_VEBT @ nat @ X @ Xa ) )
=> ( ! [A4: $o,B3: $o] :
( ( X
= ( vEBT_Leaf @ A4 @ B3 ) )
=> ( ( Y
= ( ( ( Xa
= ( zero_zero @ nat ) )
=> A4 )
& ( ( Xa
!= ( zero_zero @ nat ) )
=> ( ( ( Xa
= ( one_one @ nat ) )
=> B3 )
& ( Xa
= ( one_one @ nat ) ) ) ) ) )
=> ~ ( accp @ ( product_prod @ vEBT_VEBT @ nat ) @ vEBT_vebt_member_rel @ ( product_Pair @ vEBT_VEBT @ nat @ ( vEBT_Leaf @ A4 @ B3 ) @ Xa ) ) ) )
=> ( ! [Uu: nat,Uv: list @ vEBT_VEBT,Uw: vEBT_VEBT] :
( ( X
= ( vEBT_Node @ ( none @ ( product_prod @ nat @ nat ) ) @ Uu @ Uv @ Uw ) )
=> ( ~ Y
=> ~ ( accp @ ( product_prod @ vEBT_VEBT @ nat ) @ vEBT_vebt_member_rel @ ( product_Pair @ vEBT_VEBT @ nat @ ( vEBT_Node @ ( none @ ( product_prod @ nat @ nat ) ) @ Uu @ Uv @ Uw ) @ Xa ) ) ) )
=> ( ! [V3: product_prod @ nat @ nat,Uy2: list @ vEBT_VEBT,Uz2: vEBT_VEBT] :
( ( X
= ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ V3 ) @ ( zero_zero @ nat ) @ Uy2 @ Uz2 ) )
=> ( ~ Y
=> ~ ( 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] :
( ( X
= ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ V3 ) @ ( suc @ ( zero_zero @ nat ) ) @ Vb2 @ Vc2 ) )
=> ( ~ Y
=> ~ ( accp @ ( product_prod @ vEBT_VEBT @ nat ) @ vEBT_vebt_member_rel @ ( product_Pair @ vEBT_VEBT @ nat @ ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ V3 ) @ ( suc @ ( zero_zero @ nat ) ) @ Vb2 @ Vc2 ) @ Xa ) ) ) )
=> ~ ! [Mi2: nat,Ma2: nat,Va3: nat,TreeList2: list @ vEBT_VEBT,Summary2: vEBT_VEBT] :
( ( X
= ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( suc @ Va3 ) ) @ TreeList2 @ Summary2 ) )
=> ( ( Y
= ( ( Xa != Mi2 )
=> ( ( Xa != Ma2 )
=> ( ~ ( ord_less @ nat @ Xa @ Mi2 )
& ( ~ ( ord_less @ nat @ Xa @ Mi2 )
=> ( ~ ( ord_less @ nat @ Ma2 @ Xa )
& ( ~ ( ord_less @ nat @ Ma2 @ Xa )
=> ( ( ( ord_less @ nat @ ( vEBT_VEBT_high @ Xa @ ( divide_divide @ nat @ ( suc @ ( suc @ 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_2011_vebt__member_Opelims_I3_J,axiom,
! [X: vEBT_VEBT,Xa: nat] :
( ~ ( vEBT_vebt_member @ X @ Xa )
=> ( ( accp @ ( product_prod @ vEBT_VEBT @ nat ) @ vEBT_vebt_member_rel @ ( product_Pair @ vEBT_VEBT @ nat @ X @ Xa ) )
=> ( ! [A4: $o,B3: $o] :
( ( X
= ( vEBT_Leaf @ A4 @ B3 ) )
=> ( ( accp @ ( product_prod @ vEBT_VEBT @ nat ) @ vEBT_vebt_member_rel @ ( product_Pair @ vEBT_VEBT @ nat @ ( vEBT_Leaf @ A4 @ B3 ) @ Xa ) )
=> ( ( ( Xa
= ( zero_zero @ nat ) )
=> A4 )
& ( ( Xa
!= ( zero_zero @ nat ) )
=> ( ( ( Xa
= ( one_one @ nat ) )
=> B3 )
& ( Xa
= ( one_one @ nat ) ) ) ) ) ) )
=> ( ! [Uu: nat,Uv: list @ vEBT_VEBT,Uw: vEBT_VEBT] :
( ( X
= ( vEBT_Node @ ( none @ ( product_prod @ nat @ nat ) ) @ Uu @ Uv @ Uw ) )
=> ~ ( accp @ ( product_prod @ vEBT_VEBT @ nat ) @ vEBT_vebt_member_rel @ ( product_Pair @ vEBT_VEBT @ nat @ ( vEBT_Node @ ( none @ ( product_prod @ nat @ nat ) ) @ Uu @ Uv @ Uw ) @ Xa ) ) )
=> ( ! [V3: product_prod @ nat @ nat,Uy2: list @ vEBT_VEBT,Uz2: vEBT_VEBT] :
( ( X
= ( 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] :
( ( X
= ( 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] :
( ( X
= ( 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_2012_VEBT__internal_Onaive__member_Opelims_I1_J,axiom,
! [X: vEBT_VEBT,Xa: nat,Y: $o] :
( ( ( vEBT_V5719532721284313246member @ X @ Xa )
= Y )
=> ( ( accp @ ( product_prod @ vEBT_VEBT @ nat ) @ vEBT_V5765760719290551771er_rel @ ( product_Pair @ vEBT_VEBT @ nat @ X @ Xa ) )
=> ( ! [A4: $o,B3: $o] :
( ( X
= ( vEBT_Leaf @ A4 @ B3 ) )
=> ( ( Y
= ( ( ( Xa
= ( zero_zero @ nat ) )
=> A4 )
& ( ( Xa
!= ( zero_zero @ nat ) )
=> ( ( ( Xa
= ( one_one @ nat ) )
=> B3 )
& ( Xa
= ( one_one @ nat ) ) ) ) ) )
=> ~ ( accp @ ( product_prod @ vEBT_VEBT @ nat ) @ vEBT_V5765760719290551771er_rel @ ( product_Pair @ vEBT_VEBT @ nat @ ( vEBT_Leaf @ A4 @ B3 ) @ Xa ) ) ) )
=> ( ! [Uu: option @ ( product_prod @ nat @ nat ),Uv: list @ vEBT_VEBT,Uw: vEBT_VEBT] :
( ( X
= ( vEBT_Node @ Uu @ ( zero_zero @ nat ) @ Uv @ Uw ) )
=> ( ~ Y
=> ~ ( accp @ ( product_prod @ vEBT_VEBT @ nat ) @ vEBT_V5765760719290551771er_rel @ ( product_Pair @ vEBT_VEBT @ nat @ ( vEBT_Node @ Uu @ ( zero_zero @ nat ) @ Uv @ Uw ) @ Xa ) ) ) )
=> ~ ! [Uy2: option @ ( product_prod @ nat @ nat ),V3: nat,TreeList2: list @ vEBT_VEBT,S2: vEBT_VEBT] :
( ( X
= ( vEBT_Node @ Uy2 @ ( suc @ V3 ) @ TreeList2 @ S2 ) )
=> ( ( Y
= ( ( ( 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 @ S2 ) @ Xa ) ) ) ) ) ) ) ) ).
% VEBT_internal.naive_member.pelims(1)
thf(fact_2013_VEBT__internal_Onaive__member_Opelims_I2_J,axiom,
! [X: vEBT_VEBT,Xa: nat] :
( ( vEBT_V5719532721284313246member @ X @ Xa )
=> ( ( accp @ ( product_prod @ vEBT_VEBT @ nat ) @ vEBT_V5765760719290551771er_rel @ ( product_Pair @ vEBT_VEBT @ nat @ X @ Xa ) )
=> ( ! [A4: $o,B3: $o] :
( ( X
= ( vEBT_Leaf @ A4 @ B3 ) )
=> ( ( accp @ ( product_prod @ vEBT_VEBT @ nat ) @ vEBT_V5765760719290551771er_rel @ ( product_Pair @ vEBT_VEBT @ nat @ ( vEBT_Leaf @ A4 @ B3 ) @ Xa ) )
=> ~ ( ( ( Xa
= ( zero_zero @ nat ) )
=> A4 )
& ( ( Xa
!= ( zero_zero @ nat ) )
=> ( ( ( Xa
= ( one_one @ nat ) )
=> B3 )
& ( Xa
= ( one_one @ nat ) ) ) ) ) ) )
=> ~ ! [Uy2: option @ ( product_prod @ nat @ nat ),V3: nat,TreeList2: list @ vEBT_VEBT,S2: vEBT_VEBT] :
( ( X
= ( vEBT_Node @ Uy2 @ ( suc @ V3 ) @ TreeList2 @ S2 ) )
=> ( ( accp @ ( product_prod @ vEBT_VEBT @ nat ) @ vEBT_V5765760719290551771er_rel @ ( product_Pair @ vEBT_VEBT @ nat @ ( vEBT_Node @ Uy2 @ ( suc @ V3 ) @ TreeList2 @ S2 ) @ 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_2014_VEBT__internal_Onaive__member_Opelims_I3_J,axiom,
! [X: vEBT_VEBT,Xa: nat] :
( ~ ( vEBT_V5719532721284313246member @ X @ Xa )
=> ( ( accp @ ( product_prod @ vEBT_VEBT @ nat ) @ vEBT_V5765760719290551771er_rel @ ( product_Pair @ vEBT_VEBT @ nat @ X @ Xa ) )
=> ( ! [A4: $o,B3: $o] :
( ( X
= ( vEBT_Leaf @ A4 @ B3 ) )
=> ( ( accp @ ( product_prod @ vEBT_VEBT @ nat ) @ vEBT_V5765760719290551771er_rel @ ( product_Pair @ vEBT_VEBT @ nat @ ( vEBT_Leaf @ A4 @ B3 ) @ Xa ) )
=> ( ( ( Xa
= ( zero_zero @ nat ) )
=> A4 )
& ( ( Xa
!= ( zero_zero @ nat ) )
=> ( ( ( Xa
= ( one_one @ nat ) )
=> B3 )
& ( Xa
= ( one_one @ nat ) ) ) ) ) ) )
=> ( ! [Uu: option @ ( product_prod @ nat @ nat ),Uv: list @ vEBT_VEBT,Uw: vEBT_VEBT] :
( ( X
= ( vEBT_Node @ Uu @ ( zero_zero @ nat ) @ Uv @ Uw ) )
=> ~ ( accp @ ( product_prod @ vEBT_VEBT @ nat ) @ vEBT_V5765760719290551771er_rel @ ( product_Pair @ vEBT_VEBT @ nat @ ( vEBT_Node @ Uu @ ( zero_zero @ nat ) @ Uv @ Uw ) @ Xa ) ) )
=> ~ ! [Uy2: option @ ( product_prod @ nat @ nat ),V3: nat,TreeList2: list @ vEBT_VEBT,S2: vEBT_VEBT] :
( ( X
= ( vEBT_Node @ Uy2 @ ( suc @ V3 ) @ TreeList2 @ S2 ) )
=> ( ( accp @ ( product_prod @ vEBT_VEBT @ nat ) @ vEBT_V5765760719290551771er_rel @ ( product_Pair @ vEBT_VEBT @ nat @ ( vEBT_Node @ Uy2 @ ( suc @ V3 ) @ TreeList2 @ S2 ) @ 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_2015_vebt__member_Opelims_I2_J,axiom,
! [X: vEBT_VEBT,Xa: nat] :
( ( vEBT_vebt_member @ X @ Xa )
=> ( ( accp @ ( product_prod @ vEBT_VEBT @ nat ) @ vEBT_vebt_member_rel @ ( product_Pair @ vEBT_VEBT @ nat @ X @ Xa ) )
=> ( ! [A4: $o,B3: $o] :
( ( X
= ( vEBT_Leaf @ A4 @ B3 ) )
=> ( ( accp @ ( product_prod @ vEBT_VEBT @ nat ) @ vEBT_vebt_member_rel @ ( product_Pair @ vEBT_VEBT @ nat @ ( vEBT_Leaf @ A4 @ B3 ) @ Xa ) )
=> ~ ( ( ( Xa
= ( zero_zero @ nat ) )
=> A4 )
& ( ( Xa
!= ( zero_zero @ nat ) )
=> ( ( ( Xa
= ( one_one @ nat ) )
=> B3 )
& ( Xa
= ( one_one @ nat ) ) ) ) ) ) )
=> ~ ! [Mi2: nat,Ma2: nat,Va3: nat,TreeList2: list @ vEBT_VEBT,Summary2: vEBT_VEBT] :
( ( X
= ( 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_2016_VEBT__internal_Omembermima_Opelims_I3_J,axiom,
! [X: vEBT_VEBT,Xa: nat] :
( ~ ( vEBT_VEBT_membermima @ X @ Xa )
=> ( ( accp @ ( product_prod @ vEBT_VEBT @ nat ) @ vEBT_V4351362008482014158ma_rel @ ( product_Pair @ vEBT_VEBT @ nat @ X @ Xa ) )
=> ( ! [Uu: $o,Uv: $o] :
( ( X
= ( vEBT_Leaf @ Uu @ Uv ) )
=> ~ ( accp @ ( product_prod @ vEBT_VEBT @ nat ) @ vEBT_V4351362008482014158ma_rel @ ( product_Pair @ vEBT_VEBT @ nat @ ( vEBT_Leaf @ Uu @ Uv ) @ Xa ) ) )
=> ( ! [Ux2: list @ vEBT_VEBT,Uy2: vEBT_VEBT] :
( ( X
= ( 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] :
( ( X
= ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi2 @ Ma2 ) ) @ ( zero_zero @ nat ) @ Va2 @ Vb2 ) )
=> ( ( accp @ ( product_prod @ vEBT_VEBT @ nat ) @ vEBT_V4351362008482014158ma_rel @ ( product_Pair @ vEBT_VEBT @ nat @ ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi2 @ Ma2 ) ) @ ( zero_zero @ nat ) @ Va2 @ Vb2 ) @ Xa ) )
=> ( ( Xa = Mi2 )
| ( Xa = Ma2 ) ) ) )
=> ( ! [Mi2: nat,Ma2: nat,V3: nat,TreeList2: list @ vEBT_VEBT,Vc2: vEBT_VEBT] :
( ( X
= ( 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] :
( ( X
= ( 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_2017_VEBT__internal_Omembermima_Opelims_I1_J,axiom,
! [X: vEBT_VEBT,Xa: nat,Y: $o] :
( ( ( vEBT_VEBT_membermima @ X @ Xa )
= Y )
=> ( ( accp @ ( product_prod @ vEBT_VEBT @ nat ) @ vEBT_V4351362008482014158ma_rel @ ( product_Pair @ vEBT_VEBT @ nat @ X @ Xa ) )
=> ( ! [Uu: $o,Uv: $o] :
( ( X
= ( vEBT_Leaf @ Uu @ Uv ) )
=> ( ~ Y
=> ~ ( accp @ ( product_prod @ vEBT_VEBT @ nat ) @ vEBT_V4351362008482014158ma_rel @ ( product_Pair @ vEBT_VEBT @ nat @ ( vEBT_Leaf @ Uu @ Uv ) @ Xa ) ) ) )
=> ( ! [Ux2: list @ vEBT_VEBT,Uy2: vEBT_VEBT] :
( ( X
= ( vEBT_Node @ ( none @ ( product_prod @ nat @ nat ) ) @ ( zero_zero @ nat ) @ Ux2 @ Uy2 ) )
=> ( ~ Y
=> ~ ( accp @ ( product_prod @ vEBT_VEBT @ nat ) @ vEBT_V4351362008482014158ma_rel @ ( product_Pair @ vEBT_VEBT @ nat @ ( vEBT_Node @ ( none @ ( product_prod @ nat @ nat ) ) @ ( zero_zero @ nat ) @ Ux2 @ Uy2 ) @ Xa ) ) ) )
=> ( ! [Mi2: nat,Ma2: nat,Va2: list @ vEBT_VEBT,Vb2: vEBT_VEBT] :
( ( X
= ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi2 @ Ma2 ) ) @ ( zero_zero @ nat ) @ Va2 @ Vb2 ) )
=> ( ( Y
= ( ( 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] :
( ( X
= ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi2 @ Ma2 ) ) @ ( suc @ V3 ) @ TreeList2 @ Vc2 ) )
=> ( ( Y
= ( ( 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] :
( ( X
= ( vEBT_Node @ ( none @ ( product_prod @ nat @ nat ) ) @ ( suc @ V3 ) @ TreeList2 @ Vd2 ) )
=> ( ( Y
= ( ( ( 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_2018_VEBT__internal_Omembermima_Opelims_I2_J,axiom,
! [X: vEBT_VEBT,Xa: nat] :
( ( vEBT_VEBT_membermima @ X @ Xa )
=> ( ( accp @ ( product_prod @ vEBT_VEBT @ nat ) @ vEBT_V4351362008482014158ma_rel @ ( product_Pair @ vEBT_VEBT @ nat @ X @ Xa ) )
=> ( ! [Mi2: nat,Ma2: nat,Va2: list @ vEBT_VEBT,Vb2: vEBT_VEBT] :
( ( X
= ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi2 @ Ma2 ) ) @ ( zero_zero @ nat ) @ Va2 @ Vb2 ) )
=> ( ( accp @ ( product_prod @ vEBT_VEBT @ nat ) @ vEBT_V4351362008482014158ma_rel @ ( product_Pair @ vEBT_VEBT @ nat @ ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi2 @ Ma2 ) ) @ ( zero_zero @ nat ) @ Va2 @ Vb2 ) @ Xa ) )
=> ~ ( ( Xa = Mi2 )
| ( Xa = Ma2 ) ) ) )
=> ( ! [Mi2: nat,Ma2: nat,V3: nat,TreeList2: list @ vEBT_VEBT,Vc2: vEBT_VEBT] :
( ( X
= ( 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] :
( ( X
= ( 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_2019_set__bit__0,axiom,
! [A: $tType] :
( ( bit_se359711467146920520ations @ A )
=> ! [A2: A] :
( ( bit_se5668285175392031749et_bit @ A @ ( zero_zero @ nat ) @ A2 )
= ( plus_plus @ A @ ( one_one @ A ) @ ( times_times @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ ( divide_divide @ A @ A2 @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) ) ) ) ) ).
% set_bit_0
thf(fact_2020_pred__less__length__list,axiom,
! [Deg: nat,X: nat,Ma: nat,TreeList: list @ vEBT_VEBT,Mi: nat,Summary: vEBT_VEBT] :
( ( ord_less_eq @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ Deg )
=> ( ( ord_less_eq @ nat @ X @ Ma )
=> ( ( ord_less @ nat @ ( vEBT_VEBT_high @ X @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( size_size @ ( list @ vEBT_VEBT ) @ TreeList ) )
=> ( ( vEBT_vebt_pred @ ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi @ Ma ) ) @ Deg @ TreeList @ Summary ) @ X )
= ( if @ ( option @ nat )
@ ( ( ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList @ ( vEBT_VEBT_high @ X @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) )
!= ( none @ nat ) )
& ( vEBT_VEBT_greater @ ( some @ nat @ ( vEBT_VEBT_low @ X @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList @ ( vEBT_VEBT_high @ X @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) ) )
@ ( vEBT_VEBT_add @ ( vEBT_VEBT_mul @ ( some @ nat @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( some @ nat @ ( vEBT_VEBT_high @ X @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) @ ( vEBT_vebt_pred @ ( nth @ vEBT_VEBT @ TreeList @ ( vEBT_VEBT_high @ X @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( vEBT_VEBT_low @ X @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) )
@ ( if @ ( option @ nat )
@ ( ( vEBT_vebt_pred @ Summary @ ( vEBT_VEBT_high @ X @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) )
= ( none @ nat ) )
@ ( if @ ( option @ nat ) @ ( ord_less @ nat @ Mi @ X ) @ ( some @ nat @ Mi ) @ ( none @ nat ) )
@ ( vEBT_VEBT_add @ ( vEBT_VEBT_mul @ ( some @ nat @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( vEBT_vebt_pred @ Summary @ ( vEBT_VEBT_high @ X @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) @ ( vEBT_vebt_maxt @ ( nth @ vEBT_VEBT @ TreeList @ ( the2 @ nat @ ( vEBT_vebt_pred @ Summary @ ( vEBT_VEBT_high @ X @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).
% pred_less_length_list
thf(fact_2021_pred__lesseq__max,axiom,
! [Deg: nat,X: nat,Ma: nat,Mi: nat,TreeList: list @ vEBT_VEBT,Summary: vEBT_VEBT] :
( ( ord_less_eq @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ Deg )
=> ( ( ord_less_eq @ nat @ X @ Ma )
=> ( ( vEBT_vebt_pred @ ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi @ Ma ) ) @ Deg @ TreeList @ Summary ) @ X )
= ( if @ ( option @ nat ) @ ( ord_less @ nat @ ( vEBT_VEBT_high @ X @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( size_size @ ( list @ vEBT_VEBT ) @ TreeList ) )
@ ( if @ ( option @ nat )
@ ( ( ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList @ ( vEBT_VEBT_high @ X @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) )
!= ( none @ nat ) )
& ( vEBT_VEBT_greater @ ( some @ nat @ ( vEBT_VEBT_low @ X @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList @ ( vEBT_VEBT_high @ X @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) ) )
@ ( vEBT_VEBT_add @ ( vEBT_VEBT_mul @ ( some @ nat @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( some @ nat @ ( vEBT_VEBT_high @ X @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) @ ( vEBT_vebt_pred @ ( nth @ vEBT_VEBT @ TreeList @ ( vEBT_VEBT_high @ X @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( vEBT_VEBT_low @ X @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) )
@ ( if @ ( option @ nat )
@ ( ( vEBT_vebt_pred @ Summary @ ( vEBT_VEBT_high @ X @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) )
= ( none @ nat ) )
@ ( if @ ( option @ nat ) @ ( ord_less @ nat @ Mi @ X ) @ ( some @ nat @ Mi ) @ ( none @ nat ) )
@ ( vEBT_VEBT_add @ ( vEBT_VEBT_mul @ ( some @ nat @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( vEBT_vebt_pred @ Summary @ ( vEBT_VEBT_high @ X @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) @ ( vEBT_vebt_maxt @ ( nth @ vEBT_VEBT @ TreeList @ ( the2 @ nat @ ( vEBT_vebt_pred @ Summary @ ( vEBT_VEBT_high @ X @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) ) ) ) ) )
@ ( none @ nat ) ) ) ) ) ).
% pred_lesseq_max
thf(fact_2022_succ__greatereq__min,axiom,
! [Deg: nat,Mi: nat,X: nat,Ma: nat,TreeList: list @ vEBT_VEBT,Summary: vEBT_VEBT] :
( ( ord_less_eq @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ Deg )
=> ( ( ord_less_eq @ nat @ Mi @ X )
=> ( ( vEBT_vebt_succ @ ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi @ Ma ) ) @ Deg @ TreeList @ Summary ) @ X )
= ( if @ ( option @ nat ) @ ( ord_less @ nat @ ( vEBT_VEBT_high @ X @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( size_size @ ( list @ vEBT_VEBT ) @ TreeList ) )
@ ( if @ ( option @ nat )
@ ( ( ( vEBT_vebt_maxt @ ( nth @ vEBT_VEBT @ TreeList @ ( vEBT_VEBT_high @ X @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) )
!= ( none @ nat ) )
& ( vEBT_VEBT_less @ ( some @ nat @ ( vEBT_VEBT_low @ X @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( vEBT_vebt_maxt @ ( nth @ vEBT_VEBT @ TreeList @ ( vEBT_VEBT_high @ X @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) ) )
@ ( vEBT_VEBT_add @ ( vEBT_VEBT_mul @ ( some @ nat @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( some @ nat @ ( vEBT_VEBT_high @ X @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) @ ( vEBT_vebt_succ @ ( nth @ vEBT_VEBT @ TreeList @ ( vEBT_VEBT_high @ X @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( vEBT_VEBT_low @ X @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) )
@ ( if @ ( option @ nat )
@ ( ( vEBT_vebt_succ @ Summary @ ( vEBT_VEBT_high @ X @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) )
= ( none @ nat ) )
@ ( none @ nat )
@ ( vEBT_VEBT_add @ ( vEBT_VEBT_mul @ ( some @ nat @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( vEBT_vebt_succ @ Summary @ ( vEBT_VEBT_high @ X @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) @ ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList @ ( the2 @ nat @ ( vEBT_vebt_succ @ Summary @ ( vEBT_VEBT_high @ X @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) ) ) ) ) )
@ ( none @ nat ) ) ) ) ) ).
% succ_greatereq_min
thf(fact_2023_add__shift,axiom,
! [X: nat,Y: nat,Z: nat] :
( ( ( plus_plus @ nat @ X @ Y )
= Z )
= ( ( vEBT_VEBT_add @ ( some @ nat @ X ) @ ( some @ nat @ Y ) )
= ( some @ nat @ Z ) ) ) ).
% add_shift
thf(fact_2024_mul__shift,axiom,
! [X: nat,Y: nat,Z: nat] :
( ( ( times_times @ nat @ X @ Y )
= Z )
= ( ( vEBT_VEBT_mul @ ( some @ nat @ X ) @ ( some @ nat @ Y ) )
= ( some @ nat @ Z ) ) ) ).
% mul_shift
thf(fact_2025_add__def,axiom,
( vEBT_VEBT_add
= ( vEBT_V2048590022279873568_shift @ nat @ ( plus_plus @ nat ) ) ) ).
% add_def
thf(fact_2026_mul__def,axiom,
( vEBT_VEBT_mul
= ( vEBT_V2048590022279873568_shift @ nat @ ( times_times @ nat ) ) ) ).
% mul_def
thf(fact_2027_set__bit__nonnegative__int__iff,axiom,
! [N3: nat,K: int] :
( ( ord_less_eq @ int @ ( zero_zero @ int ) @ ( bit_se5668285175392031749et_bit @ int @ N3 @ K ) )
= ( ord_less_eq @ int @ ( zero_zero @ int ) @ K ) ) ).
% set_bit_nonnegative_int_iff
thf(fact_2028_set__bit__negative__int__iff,axiom,
! [N3: nat,K: int] :
( ( ord_less @ int @ ( bit_se5668285175392031749et_bit @ int @ N3 @ K ) @ ( zero_zero @ int ) )
= ( ord_less @ int @ K @ ( zero_zero @ int ) ) ) ).
% set_bit_negative_int_iff
thf(fact_2029_succ__less__length__list,axiom,
! [Deg: nat,Mi: nat,X: nat,TreeList: list @ vEBT_VEBT,Ma: nat,Summary: vEBT_VEBT] :
( ( ord_less_eq @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ Deg )
=> ( ( ord_less_eq @ nat @ Mi @ X )
=> ( ( ord_less @ nat @ ( vEBT_VEBT_high @ X @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( size_size @ ( list @ vEBT_VEBT ) @ TreeList ) )
=> ( ( vEBT_vebt_succ @ ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi @ Ma ) ) @ Deg @ TreeList @ Summary ) @ X )
= ( if @ ( option @ nat )
@ ( ( ( vEBT_vebt_maxt @ ( nth @ vEBT_VEBT @ TreeList @ ( vEBT_VEBT_high @ X @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) )
!= ( none @ nat ) )
& ( vEBT_VEBT_less @ ( some @ nat @ ( vEBT_VEBT_low @ X @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( vEBT_vebt_maxt @ ( nth @ vEBT_VEBT @ TreeList @ ( vEBT_VEBT_high @ X @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) ) )
@ ( vEBT_VEBT_add @ ( vEBT_VEBT_mul @ ( some @ nat @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( some @ nat @ ( vEBT_VEBT_high @ X @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) @ ( vEBT_vebt_succ @ ( nth @ vEBT_VEBT @ TreeList @ ( vEBT_VEBT_high @ X @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( vEBT_VEBT_low @ X @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) )
@ ( if @ ( option @ nat )
@ ( ( vEBT_vebt_succ @ Summary @ ( vEBT_VEBT_high @ X @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) )
= ( none @ nat ) )
@ ( none @ nat )
@ ( vEBT_VEBT_add @ ( vEBT_VEBT_mul @ ( some @ nat @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( vEBT_vebt_succ @ Summary @ ( vEBT_VEBT_high @ X @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) @ ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList @ ( the2 @ nat @ ( vEBT_vebt_succ @ Summary @ ( vEBT_VEBT_high @ X @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).
% succ_less_length_list
thf(fact_2030_set__bit__greater__eq,axiom,
! [K: int,N3: nat] : ( ord_less_eq @ int @ K @ ( bit_se5668285175392031749et_bit @ int @ N3 @ K ) ) ).
% set_bit_greater_eq
thf(fact_2031_vebt__succ_Osimps_I6_J,axiom,
! [X: nat,Mi: nat,Ma: nat,Va: nat,TreeList: list @ vEBT_VEBT,Summary: vEBT_VEBT] :
( ( ( ord_less @ nat @ X @ Mi )
=> ( ( vEBT_vebt_succ @ ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi @ Ma ) ) @ ( suc @ ( suc @ Va ) ) @ TreeList @ Summary ) @ X )
= ( some @ nat @ Mi ) ) )
& ( ~ ( ord_less @ nat @ X @ Mi )
=> ( ( vEBT_vebt_succ @ ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi @ Ma ) ) @ ( suc @ ( suc @ Va ) ) @ TreeList @ Summary ) @ X )
= ( if @ ( option @ nat ) @ ( ord_less @ nat @ ( vEBT_VEBT_high @ X @ ( 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 @ X @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) )
!= ( none @ nat ) )
& ( vEBT_VEBT_less @ ( some @ nat @ ( vEBT_VEBT_low @ X @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( vEBT_vebt_maxt @ ( nth @ vEBT_VEBT @ TreeList @ ( vEBT_VEBT_high @ X @ ( 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 @ X @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) @ ( vEBT_vebt_succ @ ( nth @ vEBT_VEBT @ TreeList @ ( vEBT_VEBT_high @ X @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( vEBT_VEBT_low @ X @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) )
@ ( if @ ( option @ nat )
@ ( ( vEBT_vebt_succ @ Summary @ ( vEBT_VEBT_high @ X @ ( 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 @ X @ ( 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 @ X @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) ) ) ) ) )
@ ( none @ nat ) ) ) ) ) ).
% vebt_succ.simps(6)
thf(fact_2032_vebt__pred_Osimps_I7_J,axiom,
! [Ma: nat,X: nat,Mi: nat,Va: nat,TreeList: list @ vEBT_VEBT,Summary: vEBT_VEBT] :
( ( ( ord_less @ nat @ Ma @ X )
=> ( ( vEBT_vebt_pred @ ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi @ Ma ) ) @ ( suc @ ( suc @ Va ) ) @ TreeList @ Summary ) @ X )
= ( some @ nat @ Ma ) ) )
& ( ~ ( ord_less @ nat @ Ma @ X )
=> ( ( vEBT_vebt_pred @ ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi @ Ma ) ) @ ( suc @ ( suc @ Va ) ) @ TreeList @ Summary ) @ X )
= ( if @ ( option @ nat ) @ ( ord_less @ nat @ ( vEBT_VEBT_high @ X @ ( 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 @ X @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) )
!= ( none @ nat ) )
& ( vEBT_VEBT_greater @ ( some @ nat @ ( vEBT_VEBT_low @ X @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList @ ( vEBT_VEBT_high @ X @ ( 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 @ X @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) @ ( vEBT_vebt_pred @ ( nth @ vEBT_VEBT @ TreeList @ ( vEBT_VEBT_high @ X @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( vEBT_VEBT_low @ X @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) )
@ ( if @ ( option @ nat )
@ ( ( vEBT_vebt_pred @ Summary @ ( vEBT_VEBT_high @ X @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) )
= ( none @ nat ) )
@ ( if @ ( option @ nat ) @ ( ord_less @ nat @ Mi @ X ) @ ( some @ nat @ Mi ) @ ( none @ nat ) )
@ ( vEBT_VEBT_add @ ( vEBT_VEBT_mul @ ( some @ nat @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( vEBT_vebt_pred @ Summary @ ( vEBT_VEBT_high @ X @ ( 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 @ X @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) ) ) ) ) )
@ ( none @ nat ) ) ) ) ) ).
% vebt_pred.simps(7)
thf(fact_2033_vebt__succ_Oelims,axiom,
! [X: vEBT_VEBT,Xa: nat,Y: option @ nat] :
( ( ( vEBT_vebt_succ @ X @ Xa )
= Y )
=> ( ! [Uu: $o,B3: $o] :
( ( X
= ( vEBT_Leaf @ Uu @ B3 ) )
=> ( ( Xa
= ( zero_zero @ nat ) )
=> ~ ( ( B3
=> ( Y
= ( some @ nat @ ( one_one @ nat ) ) ) )
& ( ~ B3
=> ( Y
= ( none @ nat ) ) ) ) ) )
=> ( ( ? [Uv: $o,Uw: $o] :
( X
= ( vEBT_Leaf @ Uv @ Uw ) )
=> ( ? [N: nat] :
( Xa
= ( suc @ N ) )
=> ( Y
!= ( none @ nat ) ) ) )
=> ( ( ? [Ux2: nat,Uy2: list @ vEBT_VEBT,Uz2: vEBT_VEBT] :
( X
= ( vEBT_Node @ ( none @ ( product_prod @ nat @ nat ) ) @ Ux2 @ Uy2 @ Uz2 ) )
=> ( Y
!= ( none @ nat ) ) )
=> ( ( ? [V3: product_prod @ nat @ nat,Vc2: list @ vEBT_VEBT,Vd2: vEBT_VEBT] :
( X
= ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ V3 ) @ ( zero_zero @ nat ) @ Vc2 @ Vd2 ) )
=> ( Y
!= ( none @ nat ) ) )
=> ( ( ? [V3: product_prod @ nat @ nat,Vg2: list @ vEBT_VEBT,Vh2: vEBT_VEBT] :
( X
= ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ V3 ) @ ( suc @ ( zero_zero @ nat ) ) @ Vg2 @ Vh2 ) )
=> ( Y
!= ( none @ nat ) ) )
=> ~ ! [Mi2: nat,Ma2: nat,Va3: nat,TreeList2: list @ vEBT_VEBT,Summary2: vEBT_VEBT] :
( ( X
= ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( suc @ Va3 ) ) @ TreeList2 @ Summary2 ) )
=> ~ ( ( ( ord_less @ nat @ Xa @ Mi2 )
=> ( Y
= ( some @ nat @ Mi2 ) ) )
& ( ~ ( ord_less @ nat @ Xa @ Mi2 )
=> ( Y
= ( if @ ( option @ nat ) @ ( ord_less @ nat @ ( vEBT_VEBT_high @ Xa @ ( divide_divide @ nat @ ( suc @ ( suc @ 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_2034_vebt__pred_Oelims,axiom,
! [X: vEBT_VEBT,Xa: nat,Y: option @ nat] :
( ( ( vEBT_vebt_pred @ X @ Xa )
= Y )
=> ( ( ? [Uu: $o,Uv: $o] :
( X
= ( vEBT_Leaf @ Uu @ Uv ) )
=> ( ( Xa
= ( zero_zero @ nat ) )
=> ( Y
!= ( none @ nat ) ) ) )
=> ( ! [A4: $o] :
( ? [Uw: $o] :
( X
= ( vEBT_Leaf @ A4 @ Uw ) )
=> ( ( Xa
= ( suc @ ( zero_zero @ nat ) ) )
=> ~ ( ( A4
=> ( Y
= ( some @ nat @ ( zero_zero @ nat ) ) ) )
& ( ~ A4
=> ( Y
= ( none @ nat ) ) ) ) ) )
=> ( ! [A4: $o,B3: $o] :
( ( X
= ( vEBT_Leaf @ A4 @ B3 ) )
=> ( ? [Va3: nat] :
( Xa
= ( suc @ ( suc @ Va3 ) ) )
=> ~ ( ( B3
=> ( Y
= ( some @ nat @ ( one_one @ nat ) ) ) )
& ( ~ B3
=> ( ( A4
=> ( Y
= ( some @ nat @ ( zero_zero @ nat ) ) ) )
& ( ~ A4
=> ( Y
= ( none @ nat ) ) ) ) ) ) ) )
=> ( ( ? [Uy2: nat,Uz2: list @ vEBT_VEBT,Va2: vEBT_VEBT] :
( X
= ( vEBT_Node @ ( none @ ( product_prod @ nat @ nat ) ) @ Uy2 @ Uz2 @ Va2 ) )
=> ( Y
!= ( none @ nat ) ) )
=> ( ( ? [V3: product_prod @ nat @ nat,Vd2: list @ vEBT_VEBT,Ve2: vEBT_VEBT] :
( X
= ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ V3 ) @ ( zero_zero @ nat ) @ Vd2 @ Ve2 ) )
=> ( Y
!= ( none @ nat ) ) )
=> ( ( ? [V3: product_prod @ nat @ nat,Vh2: list @ vEBT_VEBT,Vi2: vEBT_VEBT] :
( X
= ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ V3 ) @ ( suc @ ( zero_zero @ nat ) ) @ Vh2 @ Vi2 ) )
=> ( Y
!= ( none @ nat ) ) )
=> ~ ! [Mi2: nat,Ma2: nat,Va3: nat,TreeList2: list @ vEBT_VEBT,Summary2: vEBT_VEBT] :
( ( X
= ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( suc @ Va3 ) ) @ TreeList2 @ Summary2 ) )
=> ~ ( ( ( ord_less @ nat @ Ma2 @ Xa )
=> ( Y
= ( some @ nat @ Ma2 ) ) )
& ( ~ ( ord_less @ nat @ Ma2 @ Xa )
=> ( Y
= ( if @ ( option @ nat ) @ ( ord_less @ nat @ ( vEBT_VEBT_high @ Xa @ ( divide_divide @ nat @ ( suc @ ( suc @ 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_2035_vebt__succ_Opelims,axiom,
! [X: vEBT_VEBT,Xa: nat,Y: option @ nat] :
( ( ( vEBT_vebt_succ @ X @ Xa )
= Y )
=> ( ( accp @ ( product_prod @ vEBT_VEBT @ nat ) @ vEBT_vebt_succ_rel @ ( product_Pair @ vEBT_VEBT @ nat @ X @ Xa ) )
=> ( ! [Uu: $o,B3: $o] :
( ( X
= ( vEBT_Leaf @ Uu @ B3 ) )
=> ( ( Xa
= ( zero_zero @ nat ) )
=> ( ( ( B3
=> ( Y
= ( some @ nat @ ( one_one @ nat ) ) ) )
& ( ~ B3
=> ( Y
= ( none @ nat ) ) ) )
=> ~ ( accp @ ( product_prod @ vEBT_VEBT @ nat ) @ vEBT_vebt_succ_rel @ ( product_Pair @ vEBT_VEBT @ nat @ ( vEBT_Leaf @ Uu @ B3 ) @ ( zero_zero @ nat ) ) ) ) ) )
=> ( ! [Uv: $o,Uw: $o] :
( ( X
= ( vEBT_Leaf @ Uv @ Uw ) )
=> ! [N: nat] :
( ( Xa
= ( suc @ N ) )
=> ( ( Y
= ( none @ nat ) )
=> ~ ( accp @ ( product_prod @ vEBT_VEBT @ nat ) @ vEBT_vebt_succ_rel @ ( product_Pair @ vEBT_VEBT @ nat @ ( vEBT_Leaf @ Uv @ Uw ) @ ( suc @ N ) ) ) ) ) )
=> ( ! [Ux2: nat,Uy2: list @ vEBT_VEBT,Uz2: vEBT_VEBT] :
( ( X
= ( vEBT_Node @ ( none @ ( product_prod @ nat @ nat ) ) @ Ux2 @ Uy2 @ Uz2 ) )
=> ( ( Y
= ( none @ nat ) )
=> ~ ( accp @ ( product_prod @ vEBT_VEBT @ nat ) @ vEBT_vebt_succ_rel @ ( product_Pair @ vEBT_VEBT @ nat @ ( vEBT_Node @ ( none @ ( product_prod @ nat @ nat ) ) @ Ux2 @ Uy2 @ Uz2 ) @ Xa ) ) ) )
=> ( ! [V3: product_prod @ nat @ nat,Vc2: list @ vEBT_VEBT,Vd2: vEBT_VEBT] :
( ( X
= ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ V3 ) @ ( zero_zero @ nat ) @ Vc2 @ Vd2 ) )
=> ( ( Y
= ( none @ nat ) )
=> ~ ( accp @ ( product_prod @ vEBT_VEBT @ nat ) @ vEBT_vebt_succ_rel @ ( product_Pair @ vEBT_VEBT @ nat @ ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ V3 ) @ ( zero_zero @ nat ) @ Vc2 @ Vd2 ) @ Xa ) ) ) )
=> ( ! [V3: product_prod @ nat @ nat,Vg2: list @ vEBT_VEBT,Vh2: vEBT_VEBT] :
( ( X
= ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ V3 ) @ ( suc @ ( zero_zero @ nat ) ) @ Vg2 @ Vh2 ) )
=> ( ( Y
= ( none @ nat ) )
=> ~ ( accp @ ( product_prod @ vEBT_VEBT @ nat ) @ vEBT_vebt_succ_rel @ ( product_Pair @ vEBT_VEBT @ nat @ ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ V3 ) @ ( suc @ ( zero_zero @ nat ) ) @ Vg2 @ Vh2 ) @ Xa ) ) ) )
=> ~ ! [Mi2: nat,Ma2: nat,Va3: nat,TreeList2: list @ vEBT_VEBT,Summary2: vEBT_VEBT] :
( ( X
= ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( suc @ Va3 ) ) @ TreeList2 @ Summary2 ) )
=> ( ( ( ( ord_less @ nat @ Xa @ Mi2 )
=> ( Y
= ( some @ nat @ Mi2 ) ) )
& ( ~ ( ord_less @ nat @ Xa @ Mi2 )
=> ( Y
= ( if @ ( option @ nat ) @ ( ord_less @ nat @ ( vEBT_VEBT_high @ Xa @ ( divide_divide @ nat @ ( suc @ ( suc @ 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_2036_vebt__pred_Opelims,axiom,
! [X: vEBT_VEBT,Xa: nat,Y: option @ nat] :
( ( ( vEBT_vebt_pred @ X @ Xa )
= Y )
=> ( ( accp @ ( product_prod @ vEBT_VEBT @ nat ) @ vEBT_vebt_pred_rel @ ( product_Pair @ vEBT_VEBT @ nat @ X @ Xa ) )
=> ( ! [Uu: $o,Uv: $o] :
( ( X
= ( vEBT_Leaf @ Uu @ Uv ) )
=> ( ( Xa
= ( zero_zero @ nat ) )
=> ( ( Y
= ( none @ nat ) )
=> ~ ( accp @ ( product_prod @ vEBT_VEBT @ nat ) @ vEBT_vebt_pred_rel @ ( product_Pair @ vEBT_VEBT @ nat @ ( vEBT_Leaf @ Uu @ Uv ) @ ( zero_zero @ nat ) ) ) ) ) )
=> ( ! [A4: $o,Uw: $o] :
( ( X
= ( vEBT_Leaf @ A4 @ Uw ) )
=> ( ( Xa
= ( suc @ ( zero_zero @ nat ) ) )
=> ( ( ( A4
=> ( Y
= ( some @ nat @ ( zero_zero @ nat ) ) ) )
& ( ~ A4
=> ( Y
= ( none @ nat ) ) ) )
=> ~ ( accp @ ( product_prod @ vEBT_VEBT @ nat ) @ vEBT_vebt_pred_rel @ ( product_Pair @ vEBT_VEBT @ nat @ ( vEBT_Leaf @ A4 @ Uw ) @ ( suc @ ( zero_zero @ nat ) ) ) ) ) ) )
=> ( ! [A4: $o,B3: $o] :
( ( X
= ( vEBT_Leaf @ A4 @ B3 ) )
=> ! [Va3: nat] :
( ( Xa
= ( suc @ ( suc @ Va3 ) ) )
=> ( ( ( B3
=> ( Y
= ( some @ nat @ ( one_one @ nat ) ) ) )
& ( ~ B3
=> ( ( A4
=> ( Y
= ( some @ nat @ ( zero_zero @ nat ) ) ) )
& ( ~ A4
=> ( Y
= ( none @ nat ) ) ) ) ) )
=> ~ ( accp @ ( product_prod @ vEBT_VEBT @ nat ) @ vEBT_vebt_pred_rel @ ( product_Pair @ vEBT_VEBT @ nat @ ( vEBT_Leaf @ A4 @ B3 ) @ ( suc @ ( suc @ Va3 ) ) ) ) ) ) )
=> ( ! [Uy2: nat,Uz2: list @ vEBT_VEBT,Va2: vEBT_VEBT] :
( ( X
= ( vEBT_Node @ ( none @ ( product_prod @ nat @ nat ) ) @ Uy2 @ Uz2 @ Va2 ) )
=> ( ( Y
= ( none @ nat ) )
=> ~ ( accp @ ( product_prod @ vEBT_VEBT @ nat ) @ vEBT_vebt_pred_rel @ ( product_Pair @ vEBT_VEBT @ nat @ ( vEBT_Node @ ( none @ ( product_prod @ nat @ nat ) ) @ Uy2 @ Uz2 @ Va2 ) @ Xa ) ) ) )
=> ( ! [V3: product_prod @ nat @ nat,Vd2: list @ vEBT_VEBT,Ve2: vEBT_VEBT] :
( ( X
= ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ V3 ) @ ( zero_zero @ nat ) @ Vd2 @ Ve2 ) )
=> ( ( Y
= ( none @ nat ) )
=> ~ ( accp @ ( product_prod @ vEBT_VEBT @ nat ) @ vEBT_vebt_pred_rel @ ( product_Pair @ vEBT_VEBT @ nat @ ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ V3 ) @ ( zero_zero @ nat ) @ Vd2 @ Ve2 ) @ Xa ) ) ) )
=> ( ! [V3: product_prod @ nat @ nat,Vh2: list @ vEBT_VEBT,Vi2: vEBT_VEBT] :
( ( X
= ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ V3 ) @ ( suc @ ( zero_zero @ nat ) ) @ Vh2 @ Vi2 ) )
=> ( ( Y
= ( none @ nat ) )
=> ~ ( accp @ ( product_prod @ vEBT_VEBT @ nat ) @ vEBT_vebt_pred_rel @ ( product_Pair @ vEBT_VEBT @ nat @ ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ V3 ) @ ( suc @ ( zero_zero @ nat ) ) @ Vh2 @ Vi2 ) @ Xa ) ) ) )
=> ~ ! [Mi2: nat,Ma2: nat,Va3: nat,TreeList2: list @ vEBT_VEBT,Summary2: vEBT_VEBT] :
( ( X
= ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( suc @ Va3 ) ) @ TreeList2 @ Summary2 ) )
=> ( ( ( ( ord_less @ nat @ Ma2 @ Xa )
=> ( Y
= ( some @ nat @ Ma2 ) ) )
& ( ~ ( ord_less @ nat @ Ma2 @ Xa )
=> ( Y
= ( if @ ( option @ nat ) @ ( ord_less @ nat @ ( vEBT_VEBT_high @ Xa @ ( divide_divide @ nat @ ( suc @ ( suc @ 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_2037_delete__correct,axiom,
! [T2: vEBT_VEBT,N3: nat,X: nat] :
( ( vEBT_invar_vebt @ T2 @ N3 )
=> ( ( vEBT_VEBT_set_vebt @ ( vEBT_vebt_delete @ T2 @ X ) )
= ( minus_minus @ ( set @ nat ) @ ( vEBT_set_vebt @ T2 ) @ ( insert @ nat @ X @ ( bot_bot @ ( set @ nat ) ) ) ) ) ) ).
% delete_correct
thf(fact_2038_lemma__termdiff3,axiom,
! [A: $tType] :
( ( real_V3459762299906320749_field @ A )
=> ! [H2: A,Z: A,K6: real,N3: 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 ) @ N3 ) @ ( power_power @ A @ Z @ N3 ) ) @ H2 ) @ ( times_times @ A @ ( semiring_1_of_nat @ A @ N3 ) @ ( power_power @ A @ Z @ ( minus_minus @ nat @ N3 @ ( suc @ ( zero_zero @ nat ) ) ) ) ) ) ) @ ( times_times @ real @ ( times_times @ real @ ( times_times @ real @ ( semiring_1_of_nat @ real @ N3 ) @ ( semiring_1_of_nat @ real @ ( minus_minus @ nat @ N3 @ ( suc @ ( zero_zero @ nat ) ) ) ) ) @ ( power_power @ real @ K6 @ ( minus_minus @ nat @ N3 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( real_V7770717601297561774m_norm @ A @ H2 ) ) ) ) ) ) ) ).
% lemma_termdiff3
thf(fact_2039_foldr__zero,axiom,
! [Xs2: list @ nat,D: nat] :
( ! [I2: nat] :
( ( ord_less @ nat @ I2 @ ( size_size @ ( list @ nat ) @ Xs2 ) )
=> ( ord_less @ nat @ ( zero_zero @ nat ) @ ( nth @ nat @ Xs2 @ I2 ) ) )
=> ( ord_less_eq @ nat @ ( size_size @ ( list @ nat ) @ Xs2 ) @ ( minus_minus @ nat @ ( foldr @ nat @ nat @ ( plus_plus @ nat ) @ Xs2 @ D ) @ D ) ) ) ).
% foldr_zero
thf(fact_2040_rel__of__empty,axiom,
! [B: $tType,A: $tType,P: ( product_prod @ A @ B ) > $o] :
( ( rel_of @ A @ B
@ ^ [X2: A] : ( none @ B )
@ P )
= ( bot_bot @ ( set @ ( product_prod @ A @ B ) ) ) ) ).
% rel_of_empty
thf(fact_2041_foldr__same,axiom,
! [Xs2: list @ real,Y: real] :
( ! [X3: real,Y3: real] :
( ( member @ real @ X3 @ ( set2 @ real @ Xs2 ) )
=> ( ( member @ real @ Y3 @ ( set2 @ real @ Xs2 ) )
=> ( X3 = Y3 ) ) )
=> ( ! [X3: real] :
( ( member @ real @ X3 @ ( set2 @ real @ Xs2 ) )
=> ( X3 = Y ) )
=> ( ( foldr @ real @ real @ ( plus_plus @ real ) @ Xs2 @ ( zero_zero @ real ) )
= ( times_times @ real @ ( semiring_1_of_nat @ real @ ( size_size @ ( list @ real ) @ Xs2 ) ) @ Y ) ) ) ) ).
% foldr_same
thf(fact_2042_foldr0,axiom,
! [Xs2: list @ real,C2: real,D: real] :
( ( foldr @ real @ real @ ( plus_plus @ real ) @ Xs2 @ ( plus_plus @ real @ C2 @ D ) )
= ( plus_plus @ real @ ( foldr @ real @ real @ ( plus_plus @ real ) @ Xs2 @ D ) @ C2 ) ) ).
% foldr0
thf(fact_2043_foldr__one,axiom,
! [D: nat,Ys: list @ nat] : ( ord_less_eq @ nat @ D @ ( foldr @ nat @ nat @ ( plus_plus @ nat ) @ Ys @ D ) ) ).
% foldr_one
thf(fact_2044_foldr__same__int,axiom,
! [Xs2: list @ nat,Y: nat] :
( ! [X3: nat,Y3: nat] :
( ( member @ nat @ X3 @ ( set2 @ nat @ Xs2 ) )
=> ( ( member @ nat @ Y3 @ ( set2 @ nat @ Xs2 ) )
=> ( X3 = Y3 ) ) )
=> ( ! [X3: nat] :
( ( member @ nat @ X3 @ ( set2 @ nat @ Xs2 ) )
=> ( X3 = Y ) )
=> ( ( foldr @ nat @ nat @ ( plus_plus @ nat ) @ Xs2 @ ( zero_zero @ nat ) )
= ( times_times @ nat @ ( size_size @ ( list @ nat ) @ Xs2 ) @ Y ) ) ) ) ).
% foldr_same_int
thf(fact_2045_delete__correct_H,axiom,
! [T2: vEBT_VEBT,N3: nat,X: nat] :
( ( vEBT_invar_vebt @ T2 @ N3 )
=> ( ( vEBT_VEBT_set_vebt @ ( vEBT_vebt_delete @ T2 @ X ) )
= ( minus_minus @ ( set @ nat ) @ ( vEBT_VEBT_set_vebt @ T2 ) @ ( insert @ nat @ X @ ( bot_bot @ ( set @ nat ) ) ) ) ) ) ).
% delete_correct'
thf(fact_2046_foldr__mono,axiom,
! [Xs2: list @ nat,Ys: list @ nat,C2: nat,D: nat] :
( ( ( size_size @ ( list @ nat ) @ Xs2 )
= ( size_size @ ( list @ nat ) @ Ys ) )
=> ( ! [I2: nat] :
( ( ord_less @ nat @ I2 @ ( size_size @ ( list @ nat ) @ Xs2 ) )
=> ( ord_less @ nat @ ( nth @ nat @ Xs2 @ I2 ) @ ( nth @ nat @ Ys @ I2 ) ) )
=> ( ( ord_less_eq @ nat @ C2 @ D )
=> ( ord_less_eq @ nat @ ( plus_plus @ nat @ ( foldr @ nat @ nat @ ( plus_plus @ nat ) @ Xs2 @ C2 ) @ ( size_size @ ( list @ nat ) @ Ys ) ) @ ( foldr @ nat @ nat @ ( plus_plus @ nat ) @ Ys @ D ) ) ) ) ) ).
% foldr_mono
thf(fact_2047_insert__subset,axiom,
! [A: $tType,X: A,A3: set @ A,B6: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ ( insert @ A @ X @ A3 ) @ B6 )
= ( ( member @ A @ X @ B6 )
& ( ord_less_eq @ ( set @ A ) @ A3 @ B6 ) ) ) ).
% insert_subset
thf(fact_2048_insert__Diff1,axiom,
! [A: $tType,X: A,B6: set @ A,A3: set @ A] :
( ( member @ A @ X @ B6 )
=> ( ( minus_minus @ ( set @ A ) @ ( insert @ A @ X @ A3 ) @ B6 )
= ( minus_minus @ ( set @ A ) @ A3 @ B6 ) ) ) ).
% insert_Diff1
thf(fact_2049_Diff__insert0,axiom,
! [A: $tType,X: A,A3: set @ A,B6: set @ A] :
( ~ ( member @ A @ X @ A3 )
=> ( ( minus_minus @ ( set @ A ) @ A3 @ ( insert @ A @ X @ B6 ) )
= ( minus_minus @ ( set @ A ) @ A3 @ B6 ) ) ) ).
% Diff_insert0
thf(fact_2050_singleton__insert__inj__eq,axiom,
! [A: $tType,B2: A,A2: A,A3: set @ A] :
( ( ( insert @ A @ B2 @ ( bot_bot @ ( set @ A ) ) )
= ( insert @ A @ A2 @ A3 ) )
= ( ( A2 = B2 )
& ( ord_less_eq @ ( set @ A ) @ A3 @ ( insert @ A @ B2 @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ).
% singleton_insert_inj_eq
thf(fact_2051_singleton__insert__inj__eq_H,axiom,
! [A: $tType,A2: A,A3: set @ A,B2: A] :
( ( ( insert @ A @ A2 @ A3 )
= ( insert @ A @ B2 @ ( bot_bot @ ( set @ A ) ) ) )
= ( ( A2 = B2 )
& ( ord_less_eq @ ( set @ A ) @ A3 @ ( insert @ A @ B2 @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ).
% singleton_insert_inj_eq'
thf(fact_2052_insert__Diff__single,axiom,
! [A: $tType,A2: A,A3: set @ A] :
( ( insert @ A @ A2 @ ( minus_minus @ ( set @ A ) @ A3 @ ( insert @ A @ A2 @ ( bot_bot @ ( set @ A ) ) ) ) )
= ( insert @ A @ A2 @ A3 ) ) ).
% insert_Diff_single
thf(fact_2053_foldr__length,axiom,
! [A: $tType,L2: list @ A] :
( ( foldr @ A @ nat
@ ^ [X2: A] : suc
@ L2
@ ( zero_zero @ nat ) )
= ( size_size @ ( list @ A ) @ L2 ) ) ).
% foldr_length
thf(fact_2054_insert__mono,axiom,
! [A: $tType,C5: set @ A,D4: set @ A,A2: A] :
( ( ord_less_eq @ ( set @ A ) @ C5 @ D4 )
=> ( ord_less_eq @ ( set @ A ) @ ( insert @ A @ A2 @ C5 ) @ ( insert @ A @ A2 @ D4 ) ) ) ).
% insert_mono
thf(fact_2055_subset__insert,axiom,
! [A: $tType,X: A,A3: set @ A,B6: set @ A] :
( ~ ( member @ A @ X @ A3 )
=> ( ( ord_less_eq @ ( set @ A ) @ A3 @ ( insert @ A @ X @ B6 ) )
= ( ord_less_eq @ ( set @ A ) @ A3 @ B6 ) ) ) ).
% subset_insert
thf(fact_2056_subset__insertI,axiom,
! [A: $tType,B6: set @ A,A2: A] : ( ord_less_eq @ ( set @ A ) @ B6 @ ( insert @ A @ A2 @ B6 ) ) ).
% subset_insertI
thf(fact_2057_subset__insertI2,axiom,
! [A: $tType,A3: set @ A,B6: set @ A,B2: A] :
( ( ord_less_eq @ ( set @ A ) @ A3 @ B6 )
=> ( ord_less_eq @ ( set @ A ) @ A3 @ ( insert @ A @ B2 @ B6 ) ) ) ).
% subset_insertI2
thf(fact_2058_insert__Diff__if,axiom,
! [A: $tType,X: A,B6: set @ A,A3: set @ A] :
( ( ( member @ A @ X @ B6 )
=> ( ( minus_minus @ ( set @ A ) @ ( insert @ A @ X @ A3 ) @ B6 )
= ( minus_minus @ ( set @ A ) @ A3 @ B6 ) ) )
& ( ~ ( member @ A @ X @ B6 )
=> ( ( minus_minus @ ( set @ A ) @ ( insert @ A @ X @ A3 ) @ B6 )
= ( insert @ A @ X @ ( minus_minus @ ( set @ A ) @ A3 @ B6 ) ) ) ) ) ).
% insert_Diff_if
thf(fact_2059_foldr__cong,axiom,
! [B: $tType,A: $tType,A2: A,B2: A,L2: list @ B,K: list @ B,F2: B > A > A,G: B > A > A] :
( ( A2 = B2 )
=> ( ( L2 = K )
=> ( ! [A4: A,X3: B] :
( ( member @ B @ X3 @ ( set2 @ B @ L2 ) )
=> ( ( F2 @ X3 @ A4 )
= ( G @ X3 @ A4 ) ) )
=> ( ( foldr @ B @ A @ F2 @ L2 @ A2 )
= ( foldr @ B @ A @ G @ K @ B2 ) ) ) ) ) ).
% foldr_cong
thf(fact_2060_subset__singletonD,axiom,
! [A: $tType,A3: set @ A,X: A] :
( ( ord_less_eq @ ( set @ A ) @ A3 @ ( insert @ A @ X @ ( bot_bot @ ( set @ A ) ) ) )
=> ( ( A3
= ( bot_bot @ ( set @ A ) ) )
| ( A3
= ( insert @ A @ X @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ).
% subset_singletonD
thf(fact_2061_subset__singleton__iff,axiom,
! [A: $tType,X8: set @ A,A2: A] :
( ( ord_less_eq @ ( set @ A ) @ X8 @ ( insert @ A @ A2 @ ( bot_bot @ ( set @ A ) ) ) )
= ( ( X8
= ( bot_bot @ ( set @ A ) ) )
| ( X8
= ( insert @ A @ A2 @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ).
% subset_singleton_iff
thf(fact_2062_Diff__insert__absorb,axiom,
! [A: $tType,X: A,A3: set @ A] :
( ~ ( member @ A @ X @ A3 )
=> ( ( minus_minus @ ( set @ A ) @ ( insert @ A @ X @ A3 ) @ ( insert @ A @ X @ ( bot_bot @ ( set @ A ) ) ) )
= A3 ) ) ).
% Diff_insert_absorb
thf(fact_2063_Diff__insert2,axiom,
! [A: $tType,A3: set @ A,A2: A,B6: set @ A] :
( ( minus_minus @ ( set @ A ) @ A3 @ ( insert @ A @ A2 @ B6 ) )
= ( minus_minus @ ( set @ A ) @ ( minus_minus @ ( set @ A ) @ A3 @ ( insert @ A @ A2 @ ( bot_bot @ ( set @ A ) ) ) ) @ B6 ) ) ).
% Diff_insert2
thf(fact_2064_insert__Diff,axiom,
! [A: $tType,A2: A,A3: set @ A] :
( ( member @ A @ A2 @ A3 )
=> ( ( insert @ A @ A2 @ ( minus_minus @ ( set @ A ) @ A3 @ ( insert @ A @ A2 @ ( bot_bot @ ( set @ A ) ) ) ) )
= A3 ) ) ).
% insert_Diff
thf(fact_2065_Diff__insert,axiom,
! [A: $tType,A3: set @ A,A2: A,B6: set @ A] :
( ( minus_minus @ ( set @ A ) @ A3 @ ( insert @ A @ A2 @ B6 ) )
= ( minus_minus @ ( set @ A ) @ ( minus_minus @ ( set @ A ) @ A3 @ B6 ) @ ( insert @ A @ A2 @ ( bot_bot @ ( set @ A ) ) ) ) ) ).
% Diff_insert
thf(fact_2066_set__minus__singleton__eq,axiom,
! [A: $tType,X: A,X8: set @ A] :
( ~ ( member @ A @ X @ X8 )
=> ( ( minus_minus @ ( set @ A ) @ X8 @ ( insert @ A @ X @ ( bot_bot @ ( set @ A ) ) ) )
= X8 ) ) ).
% set_minus_singleton_eq
thf(fact_2067_insert__minus__eq,axiom,
! [A: $tType,X: A,Y: A,A3: set @ A] :
( ( X != Y )
=> ( ( minus_minus @ ( set @ A ) @ ( insert @ A @ X @ A3 ) @ ( insert @ A @ Y @ ( bot_bot @ ( set @ A ) ) ) )
= ( insert @ A @ X @ ( minus_minus @ ( set @ A ) @ A3 @ ( insert @ A @ Y @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ) ).
% insert_minus_eq
thf(fact_2068_subset__Diff__insert,axiom,
! [A: $tType,A3: set @ A,B6: set @ A,X: A,C5: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ A3 @ ( minus_minus @ ( set @ A ) @ B6 @ ( insert @ A @ X @ C5 ) ) )
= ( ( ord_less_eq @ ( set @ A ) @ A3 @ ( minus_minus @ ( set @ A ) @ B6 @ C5 ) )
& ~ ( member @ A @ X @ A3 ) ) ) ).
% subset_Diff_insert
thf(fact_2069_subset__insert__iff,axiom,
! [A: $tType,A3: set @ A,X: A,B6: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ A3 @ ( insert @ A @ X @ B6 ) )
= ( ( ( member @ A @ X @ A3 )
=> ( ord_less_eq @ ( set @ A ) @ ( minus_minus @ ( set @ A ) @ A3 @ ( insert @ A @ X @ ( bot_bot @ ( set @ A ) ) ) ) @ B6 ) )
& ( ~ ( member @ A @ X @ A3 )
=> ( ord_less_eq @ ( set @ A ) @ A3 @ B6 ) ) ) ) ).
% subset_insert_iff
thf(fact_2070_Diff__single__insert,axiom,
! [A: $tType,A3: set @ A,X: A,B6: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ ( minus_minus @ ( set @ A ) @ A3 @ ( insert @ A @ X @ ( bot_bot @ ( set @ A ) ) ) ) @ B6 )
=> ( ord_less_eq @ ( set @ A ) @ A3 @ ( insert @ A @ X @ B6 ) ) ) ).
% Diff_single_insert
thf(fact_2071_atLeast0__atMost__Suc,axiom,
! [N3: nat] :
( ( set_or1337092689740270186AtMost @ nat @ ( zero_zero @ nat ) @ ( suc @ N3 ) )
= ( insert @ nat @ ( suc @ N3 ) @ ( set_or1337092689740270186AtMost @ nat @ ( zero_zero @ nat ) @ N3 ) ) ) ).
% atLeast0_atMost_Suc
thf(fact_2072_atLeastAtMost__insertL,axiom,
! [M: nat,N3: nat] :
( ( ord_less_eq @ nat @ M @ N3 )
=> ( ( insert @ nat @ M @ ( set_or1337092689740270186AtMost @ nat @ ( suc @ M ) @ N3 ) )
= ( set_or1337092689740270186AtMost @ nat @ M @ N3 ) ) ) ).
% atLeastAtMost_insertL
thf(fact_2073_atLeastAtMostSuc__conv,axiom,
! [M: nat,N3: nat] :
( ( ord_less_eq @ nat @ M @ ( suc @ N3 ) )
=> ( ( set_or1337092689740270186AtMost @ nat @ M @ ( suc @ N3 ) )
= ( insert @ nat @ ( suc @ N3 ) @ ( set_or1337092689740270186AtMost @ nat @ M @ N3 ) ) ) ) ).
% atLeastAtMostSuc_conv
thf(fact_2074_Icc__eq__insert__lb__nat,axiom,
! [M: nat,N3: nat] :
( ( ord_less_eq @ nat @ M @ N3 )
=> ( ( set_or1337092689740270186AtMost @ nat @ M @ N3 )
= ( insert @ nat @ M @ ( set_or1337092689740270186AtMost @ nat @ ( suc @ M ) @ N3 ) ) ) ) ).
% Icc_eq_insert_lb_nat
thf(fact_2075_remove__subset,axiom,
! [A: $tType,X: A,S3: set @ A] :
( ( member @ A @ X @ S3 )
=> ( ord_less @ ( set @ A ) @ ( minus_minus @ ( set @ A ) @ S3 @ ( insert @ A @ X @ ( bot_bot @ ( set @ A ) ) ) ) @ S3 ) ) ).
% remove_subset
thf(fact_2076_set__update__subset__insert,axiom,
! [A: $tType,Xs2: list @ A,I: nat,X: A] : ( ord_less_eq @ ( set @ A ) @ ( set2 @ A @ ( list_update @ A @ Xs2 @ I @ X ) ) @ ( insert @ A @ X @ ( set2 @ A @ Xs2 ) ) ) ).
% set_update_subset_insert
thf(fact_2077_foldr__length__aux,axiom,
! [A: $tType,L2: list @ A,A2: nat] :
( ( foldr @ A @ nat
@ ^ [X2: A] : suc
@ L2
@ A2 )
= ( plus_plus @ nat @ A2 @ ( size_size @ ( list @ A ) @ L2 ) ) ) ).
% foldr_length_aux
thf(fact_2078_psubset__insert__iff,axiom,
! [A: $tType,A3: set @ A,X: A,B6: set @ A] :
( ( ord_less @ ( set @ A ) @ A3 @ ( insert @ A @ X @ B6 ) )
= ( ( ( member @ A @ X @ B6 )
=> ( ord_less @ ( set @ A ) @ A3 @ B6 ) )
& ( ~ ( member @ A @ X @ B6 )
=> ( ( ( member @ A @ X @ A3 )
=> ( ord_less @ ( set @ A ) @ ( minus_minus @ ( set @ A ) @ A3 @ ( insert @ A @ X @ ( bot_bot @ ( set @ A ) ) ) ) @ B6 ) )
& ( ~ ( member @ A @ X @ A3 )
=> ( ord_less_eq @ ( set @ A ) @ A3 @ B6 ) ) ) ) ) ) ).
% psubset_insert_iff
thf(fact_2079_lemma__NBseq__def2,axiom,
! [A: $tType,B: $tType] :
( ( real_V822414075346904944vector @ B )
=> ! [X8: A > B] :
( ( ? [K7: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ K7 )
& ! [N2: A] : ( ord_less_eq @ real @ ( real_V7770717601297561774m_norm @ B @ ( X8 @ N2 ) ) @ K7 ) ) )
= ( ? [N8: nat] :
! [N2: A] : ( ord_less @ real @ ( real_V7770717601297561774m_norm @ B @ ( X8 @ N2 ) ) @ ( semiring_1_of_nat @ real @ ( suc @ N8 ) ) ) ) ) ) ).
% lemma_NBseq_def2
thf(fact_2080_lemma__NBseq__def,axiom,
! [A: $tType,B: $tType] :
( ( real_V822414075346904944vector @ B )
=> ! [X8: A > B] :
( ( ? [K7: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ K7 )
& ! [N2: A] : ( ord_less_eq @ real @ ( real_V7770717601297561774m_norm @ B @ ( X8 @ N2 ) ) @ K7 ) ) )
= ( ? [N8: nat] :
! [N2: A] : ( ord_less_eq @ real @ ( real_V7770717601297561774m_norm @ B @ ( X8 @ N2 ) ) @ ( semiring_1_of_nat @ real @ ( suc @ N8 ) ) ) ) ) ) ).
% lemma_NBseq_def
thf(fact_2081_insert__swap__set__eq,axiom,
! [A: $tType,I: nat,L2: list @ A,X: A] :
( ( ord_less @ nat @ I @ ( size_size @ ( list @ A ) @ L2 ) )
=> ( ( insert @ A @ ( nth @ A @ L2 @ I ) @ ( set2 @ A @ ( list_update @ A @ L2 @ I @ X ) ) )
= ( insert @ A @ X @ ( set2 @ A @ L2 ) ) ) ) ).
% insert_swap_set_eq
thf(fact_2082_norm__divide__numeral,axiom,
! [A: $tType] :
( ( real_V3459762299906320749_field @ A )
=> ! [A2: A,W: num] :
( ( real_V7770717601297561774m_norm @ A @ ( divide_divide @ A @ A2 @ ( numeral_numeral @ A @ W ) ) )
= ( divide_divide @ real @ ( real_V7770717601297561774m_norm @ A @ A2 ) @ ( numeral_numeral @ real @ W ) ) ) ) ).
% norm_divide_numeral
thf(fact_2083_norm__mult__numeral1,axiom,
! [A: $tType] :
( ( real_V3459762299906320749_field @ A )
=> ! [W: num,A2: A] :
( ( real_V7770717601297561774m_norm @ A @ ( times_times @ A @ ( numeral_numeral @ A @ W ) @ A2 ) )
= ( times_times @ real @ ( numeral_numeral @ real @ W ) @ ( real_V7770717601297561774m_norm @ A @ A2 ) ) ) ) ).
% norm_mult_numeral1
thf(fact_2084_norm__mult__numeral2,axiom,
! [A: $tType] :
( ( real_V3459762299906320749_field @ A )
=> ! [A2: A,W: num] :
( ( real_V7770717601297561774m_norm @ A @ ( times_times @ A @ A2 @ ( numeral_numeral @ A @ W ) ) )
= ( times_times @ real @ ( real_V7770717601297561774m_norm @ A @ A2 ) @ ( numeral_numeral @ real @ W ) ) ) ) ).
% norm_mult_numeral2
thf(fact_2085_norm__le__zero__iff,axiom,
! [A: $tType] :
( ( real_V822414075346904944vector @ A )
=> ! [X: A] :
( ( ord_less_eq @ real @ ( real_V7770717601297561774m_norm @ A @ X ) @ ( zero_zero @ real ) )
= ( X
= ( zero_zero @ A ) ) ) ) ).
% norm_le_zero_iff
thf(fact_2086_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_2087_norm__one,axiom,
! [A: $tType] :
( ( real_V2822296259951069270ebra_1 @ A )
=> ( ( real_V7770717601297561774m_norm @ A @ ( one_one @ A ) )
= ( one_one @ real ) ) ) ).
% norm_one
thf(fact_2088_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_2089_atLeastAtMostPlus1__int__conv,axiom,
! [M: int,N3: int] :
( ( ord_less_eq @ int @ M @ ( plus_plus @ int @ ( one_one @ int ) @ N3 ) )
=> ( ( set_or1337092689740270186AtMost @ int @ M @ ( plus_plus @ int @ ( one_one @ int ) @ N3 ) )
= ( insert @ int @ ( plus_plus @ int @ ( one_one @ int ) @ N3 ) @ ( set_or1337092689740270186AtMost @ int @ M @ N3 ) ) ) ) ).
% atLeastAtMostPlus1_int_conv
thf(fact_2090_simp__from__to,axiom,
( ( set_or1337092689740270186AtMost @ int )
= ( ^ [I3: int,J3: int] : ( if @ ( set @ int ) @ ( ord_less @ int @ J3 @ I3 ) @ ( bot_bot @ ( set @ int ) ) @ ( insert @ int @ I3 @ ( set_or1337092689740270186AtMost @ int @ ( plus_plus @ int @ I3 @ ( one_one @ int ) ) @ J3 ) ) ) ) ) ).
% simp_from_to
thf(fact_2091_norm__minus__commute,axiom,
! [A: $tType] :
( ( real_V822414075346904944vector @ A )
=> ! [A2: A,B2: A] :
( ( real_V7770717601297561774m_norm @ A @ ( minus_minus @ A @ A2 @ B2 ) )
= ( real_V7770717601297561774m_norm @ A @ ( minus_minus @ A @ B2 @ A2 ) ) ) ) ).
% norm_minus_commute
thf(fact_2092_norm__mult,axiom,
! [A: $tType] :
( ( real_V8999393235501362500lgebra @ A )
=> ! [X: A,Y: A] :
( ( real_V7770717601297561774m_norm @ A @ ( times_times @ A @ X @ Y ) )
= ( times_times @ real @ ( real_V7770717601297561774m_norm @ A @ X ) @ ( real_V7770717601297561774m_norm @ A @ Y ) ) ) ) ).
% norm_mult
thf(fact_2093_norm__ge__zero,axiom,
! [A: $tType] :
( ( real_V822414075346904944vector @ A )
=> ! [X: A] : ( ord_less_eq @ real @ ( zero_zero @ real ) @ ( real_V7770717601297561774m_norm @ A @ X ) ) ) ).
% norm_ge_zero
thf(fact_2094_norm__divide,axiom,
! [A: $tType] :
( ( real_V3459762299906320749_field @ A )
=> ! [A2: A,B2: A] :
( ( real_V7770717601297561774m_norm @ A @ ( divide_divide @ A @ A2 @ B2 ) )
= ( divide_divide @ real @ ( real_V7770717601297561774m_norm @ A @ A2 ) @ ( real_V7770717601297561774m_norm @ A @ B2 ) ) ) ) ).
% norm_divide
thf(fact_2095_norm__power,axiom,
! [A: $tType] :
( ( real_V8999393235501362500lgebra @ A )
=> ! [X: A,N3: nat] :
( ( real_V7770717601297561774m_norm @ A @ ( power_power @ A @ X @ N3 ) )
= ( power_power @ real @ ( real_V7770717601297561774m_norm @ A @ X ) @ N3 ) ) ) ).
% norm_power
thf(fact_2096_power__eq__imp__eq__norm,axiom,
! [A: $tType] :
( ( real_V8999393235501362500lgebra @ A )
=> ! [W: A,N3: nat,Z: A] :
( ( ( power_power @ A @ W @ N3 )
= ( power_power @ A @ Z @ N3 ) )
=> ( ( ord_less @ nat @ ( zero_zero @ nat ) @ N3 )
=> ( ( real_V7770717601297561774m_norm @ A @ W )
= ( real_V7770717601297561774m_norm @ A @ Z ) ) ) ) ) ).
% power_eq_imp_eq_norm
thf(fact_2097_nonzero__norm__divide,axiom,
! [A: $tType] :
( ( real_V3459762299906320749_field @ A )
=> ! [B2: A,A2: A] :
( ( B2
!= ( zero_zero @ A ) )
=> ( ( real_V7770717601297561774m_norm @ A @ ( divide_divide @ A @ A2 @ B2 ) )
= ( divide_divide @ real @ ( real_V7770717601297561774m_norm @ A @ A2 ) @ ( real_V7770717601297561774m_norm @ A @ B2 ) ) ) ) ) ).
% nonzero_norm_divide
thf(fact_2098_norm__mult__less,axiom,
! [A: $tType] :
( ( real_V4412858255891104859lgebra @ A )
=> ! [X: A,R2: real,Y: A,S: real] :
( ( ord_less @ real @ ( real_V7770717601297561774m_norm @ A @ X ) @ R2 )
=> ( ( ord_less @ real @ ( real_V7770717601297561774m_norm @ A @ Y ) @ S )
=> ( ord_less @ real @ ( real_V7770717601297561774m_norm @ A @ ( times_times @ A @ X @ Y ) ) @ ( times_times @ real @ R2 @ S ) ) ) ) ) ).
% norm_mult_less
thf(fact_2099_norm__mult__ineq,axiom,
! [A: $tType] :
( ( real_V4412858255891104859lgebra @ A )
=> ! [X: A,Y: A] : ( ord_less_eq @ real @ ( real_V7770717601297561774m_norm @ A @ ( times_times @ A @ X @ Y ) ) @ ( times_times @ real @ ( real_V7770717601297561774m_norm @ A @ X ) @ ( real_V7770717601297561774m_norm @ A @ Y ) ) ) ) ).
% norm_mult_ineq
thf(fact_2100_norm__triangle__lt,axiom,
! [A: $tType] :
( ( real_V822414075346904944vector @ A )
=> ! [X: A,Y: A,E: real] :
( ( ord_less @ real @ ( plus_plus @ real @ ( real_V7770717601297561774m_norm @ A @ X ) @ ( real_V7770717601297561774m_norm @ A @ Y ) ) @ E )
=> ( ord_less @ real @ ( real_V7770717601297561774m_norm @ A @ ( plus_plus @ A @ X @ Y ) ) @ E ) ) ) ).
% norm_triangle_lt
thf(fact_2101_norm__add__less,axiom,
! [A: $tType] :
( ( real_V822414075346904944vector @ A )
=> ! [X: A,R2: real,Y: A,S: real] :
( ( ord_less @ real @ ( real_V7770717601297561774m_norm @ A @ X ) @ R2 )
=> ( ( ord_less @ real @ ( real_V7770717601297561774m_norm @ A @ Y ) @ S )
=> ( ord_less @ real @ ( real_V7770717601297561774m_norm @ A @ ( plus_plus @ A @ X @ Y ) ) @ ( plus_plus @ real @ R2 @ S ) ) ) ) ) ).
% norm_add_less
thf(fact_2102_norm__triangle__mono,axiom,
! [A: $tType] :
( ( real_V822414075346904944vector @ A )
=> ! [A2: A,R2: real,B2: A,S: real] :
( ( ord_less_eq @ real @ ( real_V7770717601297561774m_norm @ A @ A2 ) @ R2 )
=> ( ( ord_less_eq @ real @ ( real_V7770717601297561774m_norm @ A @ B2 ) @ S )
=> ( ord_less_eq @ real @ ( real_V7770717601297561774m_norm @ A @ ( plus_plus @ A @ A2 @ B2 ) ) @ ( plus_plus @ real @ R2 @ S ) ) ) ) ) ).
% norm_triangle_mono
thf(fact_2103_norm__triangle__ineq,axiom,
! [A: $tType] :
( ( real_V822414075346904944vector @ A )
=> ! [X: A,Y: A] : ( ord_less_eq @ real @ ( real_V7770717601297561774m_norm @ A @ ( plus_plus @ A @ X @ Y ) ) @ ( plus_plus @ real @ ( real_V7770717601297561774m_norm @ A @ X ) @ ( real_V7770717601297561774m_norm @ A @ Y ) ) ) ) ).
% norm_triangle_ineq
thf(fact_2104_norm__triangle__le,axiom,
! [A: $tType] :
( ( real_V822414075346904944vector @ A )
=> ! [X: A,Y: A,E: real] :
( ( ord_less_eq @ real @ ( plus_plus @ real @ ( real_V7770717601297561774m_norm @ A @ X ) @ ( real_V7770717601297561774m_norm @ A @ Y ) ) @ E )
=> ( ord_less_eq @ real @ ( real_V7770717601297561774m_norm @ A @ ( plus_plus @ A @ X @ Y ) ) @ E ) ) ) ).
% norm_triangle_le
thf(fact_2105_norm__add__leD,axiom,
! [A: $tType] :
( ( real_V822414075346904944vector @ A )
=> ! [A2: A,B2: A,C2: real] :
( ( ord_less_eq @ real @ ( real_V7770717601297561774m_norm @ A @ ( plus_plus @ A @ A2 @ B2 ) ) @ C2 )
=> ( ord_less_eq @ real @ ( real_V7770717601297561774m_norm @ A @ B2 ) @ ( plus_plus @ real @ ( real_V7770717601297561774m_norm @ A @ A2 ) @ C2 ) ) ) ) ).
% norm_add_leD
thf(fact_2106_norm__diff__triangle__less,axiom,
! [A: $tType] :
( ( real_V822414075346904944vector @ A )
=> ! [X: A,Y: A,E1: real,Z: A,E22: real] :
( ( ord_less @ real @ ( real_V7770717601297561774m_norm @ A @ ( minus_minus @ A @ X @ Y ) ) @ E1 )
=> ( ( ord_less @ real @ ( real_V7770717601297561774m_norm @ A @ ( minus_minus @ A @ Y @ Z ) ) @ E22 )
=> ( ord_less @ real @ ( real_V7770717601297561774m_norm @ A @ ( minus_minus @ A @ X @ Z ) ) @ ( plus_plus @ real @ E1 @ E22 ) ) ) ) ) ).
% norm_diff_triangle_less
thf(fact_2107_norm__power__ineq,axiom,
! [A: $tType] :
( ( real_V2822296259951069270ebra_1 @ A )
=> ! [X: A,N3: nat] : ( ord_less_eq @ real @ ( real_V7770717601297561774m_norm @ A @ ( power_power @ A @ X @ N3 ) ) @ ( power_power @ real @ ( real_V7770717601297561774m_norm @ A @ X ) @ N3 ) ) ) ).
% norm_power_ineq
thf(fact_2108_norm__triangle__sub,axiom,
! [A: $tType] :
( ( real_V822414075346904944vector @ A )
=> ! [X: A,Y: A] : ( ord_less_eq @ real @ ( real_V7770717601297561774m_norm @ A @ X ) @ ( plus_plus @ real @ ( real_V7770717601297561774m_norm @ A @ Y ) @ ( real_V7770717601297561774m_norm @ A @ ( minus_minus @ A @ X @ Y ) ) ) ) ) ).
% norm_triangle_sub
thf(fact_2109_norm__triangle__ineq4,axiom,
! [A: $tType] :
( ( real_V822414075346904944vector @ A )
=> ! [A2: A,B2: A] : ( ord_less_eq @ real @ ( real_V7770717601297561774m_norm @ A @ ( minus_minus @ A @ A2 @ B2 ) ) @ ( plus_plus @ real @ ( real_V7770717601297561774m_norm @ A @ A2 ) @ ( real_V7770717601297561774m_norm @ A @ B2 ) ) ) ) ).
% norm_triangle_ineq4
thf(fact_2110_norm__diff__triangle__le,axiom,
! [A: $tType] :
( ( real_V822414075346904944vector @ A )
=> ! [X: A,Y: A,E1: real,Z: A,E22: real] :
( ( ord_less_eq @ real @ ( real_V7770717601297561774m_norm @ A @ ( minus_minus @ A @ X @ Y ) ) @ E1 )
=> ( ( ord_less_eq @ real @ ( real_V7770717601297561774m_norm @ A @ ( minus_minus @ A @ Y @ Z ) ) @ E22 )
=> ( ord_less_eq @ real @ ( real_V7770717601297561774m_norm @ A @ ( minus_minus @ A @ X @ Z ) ) @ ( plus_plus @ real @ E1 @ E22 ) ) ) ) ) ).
% norm_diff_triangle_le
thf(fact_2111_norm__triangle__le__diff,axiom,
! [A: $tType] :
( ( real_V822414075346904944vector @ A )
=> ! [X: A,Y: A,E: real] :
( ( ord_less_eq @ real @ ( plus_plus @ real @ ( real_V7770717601297561774m_norm @ A @ X ) @ ( real_V7770717601297561774m_norm @ A @ Y ) ) @ E )
=> ( ord_less_eq @ real @ ( real_V7770717601297561774m_norm @ A @ ( minus_minus @ A @ X @ Y ) ) @ E ) ) ) ).
% norm_triangle_le_diff
thf(fact_2112_norm__diff__ineq,axiom,
! [A: $tType] :
( ( real_V822414075346904944vector @ A )
=> ! [A2: A,B2: A] : ( ord_less_eq @ real @ ( minus_minus @ real @ ( real_V7770717601297561774m_norm @ A @ A2 ) @ ( real_V7770717601297561774m_norm @ A @ B2 ) ) @ ( real_V7770717601297561774m_norm @ A @ ( plus_plus @ A @ A2 @ B2 ) ) ) ) ).
% norm_diff_ineq
thf(fact_2113_norm__triangle__ineq2,axiom,
! [A: $tType] :
( ( real_V822414075346904944vector @ A )
=> ! [A2: A,B2: A] : ( ord_less_eq @ real @ ( minus_minus @ real @ ( real_V7770717601297561774m_norm @ A @ A2 ) @ ( real_V7770717601297561774m_norm @ A @ B2 ) ) @ ( real_V7770717601297561774m_norm @ A @ ( minus_minus @ A @ A2 @ B2 ) ) ) ) ).
% norm_triangle_ineq2
thf(fact_2114_power__eq__1__iff,axiom,
! [A: $tType] :
( ( real_V8999393235501362500lgebra @ A )
=> ! [W: A,N3: nat] :
( ( ( power_power @ A @ W @ N3 )
= ( one_one @ A ) )
=> ( ( ( real_V7770717601297561774m_norm @ A @ W )
= ( one_one @ real ) )
| ( N3
= ( zero_zero @ nat ) ) ) ) ) ).
% power_eq_1_iff
thf(fact_2115_norm__diff__triangle__ineq,axiom,
! [A: $tType] :
( ( real_V822414075346904944vector @ A )
=> ! [A2: A,B2: A,C2: A,D: A] : ( ord_less_eq @ real @ ( real_V7770717601297561774m_norm @ A @ ( minus_minus @ A @ ( plus_plus @ A @ A2 @ B2 ) @ ( plus_plus @ A @ C2 @ D ) ) ) @ ( plus_plus @ real @ ( real_V7770717601297561774m_norm @ A @ ( minus_minus @ A @ A2 @ C2 ) ) @ ( real_V7770717601297561774m_norm @ A @ ( minus_minus @ A @ B2 @ D ) ) ) ) ) ).
% norm_diff_triangle_ineq
thf(fact_2116_square__norm__one,axiom,
! [A: $tType] :
( ( real_V8999393235501362500lgebra @ A )
=> ! [X: A] :
( ( ( power_power @ A @ X @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) )
= ( one_one @ A ) )
=> ( ( real_V7770717601297561774m_norm @ A @ X )
= ( one_one @ real ) ) ) ) ).
% square_norm_one
thf(fact_2117_list__every__elemnt__bound__sum__bound__real,axiom,
! [A: $tType,Xs2: list @ A,F2: A > real,Bound: real,I: real] :
( ! [X3: A] :
( ( member @ A @ X3 @ ( set2 @ A @ Xs2 ) )
=> ( ord_less_eq @ real @ ( F2 @ X3 ) @ Bound ) )
=> ( ord_less_eq @ real @ ( foldr @ real @ real @ ( plus_plus @ real ) @ ( map @ A @ real @ F2 @ Xs2 ) @ I ) @ ( plus_plus @ real @ ( times_times @ real @ ( semiring_1_of_nat @ real @ ( size_size @ ( list @ A ) @ Xs2 ) ) @ Bound ) @ I ) ) ) ).
% list_every_elemnt_bound_sum_bound_real
thf(fact_2118_f__g__map__foldr__bound,axiom,
! [A: $tType,Xs2: list @ A,F2: A > real,C2: real,G: A > real,D: real] :
( ! [X3: A] :
( ( member @ A @ X3 @ ( set2 @ A @ Xs2 ) )
=> ( ord_less_eq @ real @ ( F2 @ X3 ) @ ( times_times @ real @ C2 @ ( G @ X3 ) ) ) )
=> ( ord_less_eq @ real @ ( foldr @ real @ real @ ( plus_plus @ real ) @ ( map @ A @ real @ F2 @ Xs2 ) @ D ) @ ( plus_plus @ real @ ( times_times @ real @ C2 @ ( foldr @ real @ real @ ( plus_plus @ real ) @ ( map @ A @ real @ G @ Xs2 ) @ ( zero_zero @ real ) ) ) @ D ) ) ) ).
% f_g_map_foldr_bound
thf(fact_2119_list__every__elemnt__bound__sum__bound,axiom,
! [A: $tType,Xs2: list @ A,F2: A > nat,Bound: nat,I: nat] :
( ! [X3: A] :
( ( member @ A @ X3 @ ( set2 @ A @ Xs2 ) )
=> ( ord_less_eq @ nat @ ( F2 @ X3 ) @ Bound ) )
=> ( ord_less_eq @ nat @ ( foldr @ nat @ nat @ ( plus_plus @ nat ) @ ( map @ A @ nat @ F2 @ Xs2 ) @ I ) @ ( plus_plus @ nat @ ( times_times @ nat @ ( size_size @ ( list @ A ) @ Xs2 ) @ Bound ) @ I ) ) ) ).
% list_every_elemnt_bound_sum_bound
thf(fact_2120_Suc__if__eq,axiom,
! [A: $tType,F2: nat > A,H2: nat > A,G: A,N3: nat] :
( ! [N: nat] :
( ( F2 @ ( suc @ N ) )
= ( H2 @ N ) )
=> ( ( ( F2 @ ( zero_zero @ nat ) )
= G )
=> ( ( ( N3
= ( zero_zero @ nat ) )
=> ( ( F2 @ N3 )
= G ) )
& ( ( N3
!= ( zero_zero @ nat ) )
=> ( ( F2 @ N3 )
= ( H2 @ ( minus_minus @ nat @ N3 @ ( one_one @ nat ) ) ) ) ) ) ) ) ).
% Suc_if_eq
thf(fact_2121_round__unique,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [X: A,Y: int] :
( ( ord_less @ A @ ( minus_minus @ A @ X @ ( divide_divide @ A @ ( one_one @ A ) @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) ) @ ( ring_1_of_int @ A @ Y ) )
=> ( ( ord_less_eq @ A @ ( ring_1_of_int @ A @ Y ) @ ( plus_plus @ A @ X @ ( divide_divide @ A @ ( one_one @ A ) @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) ) )
=> ( ( archimedean_round @ A @ X )
= Y ) ) ) ) ).
% round_unique
thf(fact_2122_mult__le__cancel__iff2,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [Z: A,X: A,Y: A] :
( ( ord_less @ A @ ( zero_zero @ A ) @ Z )
=> ( ( ord_less_eq @ A @ ( times_times @ A @ Z @ X ) @ ( times_times @ A @ Z @ Y ) )
= ( ord_less_eq @ A @ X @ Y ) ) ) ) ).
% mult_le_cancel_iff2
thf(fact_2123_listsum__bound,axiom,
! [A: $tType,Xs2: list @ A,F2: A > real,Y: real] :
( ! [X3: A] :
( ( member @ A @ X3 @ ( set2 @ A @ Xs2 ) )
=> ( ord_less_eq @ real @ ( zero_zero @ real ) @ ( F2 @ X3 ) ) )
=> ( ord_less_eq @ real @ Y @ ( foldr @ real @ real @ ( plus_plus @ real ) @ ( map @ A @ real @ F2 @ Xs2 ) @ Y ) ) ) ).
% listsum_bound
thf(fact_2124_length__map,axiom,
! [A: $tType,B: $tType,F2: B > A,Xs2: list @ B] :
( ( size_size @ ( list @ A ) @ ( map @ B @ A @ F2 @ Xs2 ) )
= ( size_size @ ( list @ B ) @ Xs2 ) ) ).
% length_map
thf(fact_2125_map__eq__conv,axiom,
! [A: $tType,B: $tType,F2: B > A,Xs2: list @ B,G: B > A] :
( ( ( map @ B @ A @ F2 @ Xs2 )
= ( map @ B @ A @ G @ Xs2 ) )
= ( ! [X2: B] :
( ( member @ B @ X2 @ ( set2 @ B @ Xs2 ) )
=> ( ( F2 @ X2 )
= ( G @ X2 ) ) ) ) ) ).
% map_eq_conv
thf(fact_2126_real__nat__list,axiom,
! [A: $tType,F2: A > nat,Xs2: list @ A,C2: nat] :
( ( semiring_1_of_nat @ real @ ( foldr @ nat @ nat @ ( plus_plus @ nat ) @ ( map @ A @ nat @ F2 @ Xs2 ) @ C2 ) )
= ( foldr @ real @ real @ ( plus_plus @ real )
@ ( map @ A @ real
@ ^ [X2: A] : ( semiring_1_of_nat @ real @ ( F2 @ X2 ) )
@ Xs2 )
@ ( semiring_1_of_nat @ real @ C2 ) ) ) ).
% real_nat_list
thf(fact_2127_round__numeral,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [N3: num] :
( ( archimedean_round @ A @ ( numeral_numeral @ A @ N3 ) )
= ( numeral_numeral @ int @ N3 ) ) ) ).
% round_numeral
thf(fact_2128_round__1,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ( ( archimedean_round @ A @ ( one_one @ A ) )
= ( one_one @ int ) ) ) ).
% round_1
thf(fact_2129_nth__map,axiom,
! [B: $tType,A: $tType,N3: nat,Xs2: list @ A,F2: A > B] :
( ( ord_less @ nat @ N3 @ ( size_size @ ( list @ A ) @ Xs2 ) )
=> ( ( nth @ B @ ( map @ A @ B @ F2 @ Xs2 ) @ N3 )
= ( F2 @ ( nth @ A @ Xs2 @ N3 ) ) ) ) ).
% nth_map
thf(fact_2130_map__eq__imp__length__eq,axiom,
! [A: $tType,B: $tType,C: $tType,F2: B > A,Xs2: list @ B,G: C > A,Ys: list @ C] :
( ( ( map @ B @ A @ F2 @ Xs2 )
= ( map @ C @ A @ G @ Ys ) )
=> ( ( size_size @ ( list @ B ) @ Xs2 )
= ( size_size @ ( list @ C ) @ Ys ) ) ) ).
% map_eq_imp_length_eq
thf(fact_2131_ex__map__conv,axiom,
! [A: $tType,B: $tType,Ys: list @ B,F2: A > B] :
( ( ? [Xs: list @ A] :
( Ys
= ( map @ A @ B @ F2 @ Xs ) ) )
= ( ! [X2: B] :
( ( member @ B @ X2 @ ( set2 @ B @ Ys ) )
=> ? [Y2: A] :
( X2
= ( F2 @ Y2 ) ) ) ) ) ).
% ex_map_conv
thf(fact_2132_map__cong,axiom,
! [B: $tType,A: $tType,Xs2: list @ A,Ys: list @ A,F2: A > B,G: A > B] :
( ( Xs2 = Ys )
=> ( ! [X3: A] :
( ( member @ A @ X3 @ ( set2 @ A @ Ys ) )
=> ( ( F2 @ X3 )
= ( G @ X3 ) ) )
=> ( ( map @ A @ B @ F2 @ Xs2 )
= ( map @ A @ B @ G @ Ys ) ) ) ) ).
% map_cong
thf(fact_2133_map__idI,axiom,
! [A: $tType,Xs2: list @ A,F2: A > A] :
( ! [X3: A] :
( ( member @ A @ X3 @ ( set2 @ A @ Xs2 ) )
=> ( ( F2 @ X3 )
= X3 ) )
=> ( ( map @ A @ A @ F2 @ Xs2 )
= Xs2 ) ) ).
% map_idI
thf(fact_2134_map__ext,axiom,
! [B: $tType,A: $tType,Xs2: list @ A,F2: A > B,G: A > B] :
( ! [X3: A] :
( ( member @ A @ X3 @ ( set2 @ A @ Xs2 ) )
=> ( ( F2 @ X3 )
= ( G @ X3 ) ) )
=> ( ( map @ A @ B @ F2 @ Xs2 )
= ( map @ A @ B @ G @ Xs2 ) ) ) ).
% map_ext
thf(fact_2135_list_Oinj__map__strong,axiom,
! [B: $tType,A: $tType,X: list @ A,Xa: list @ A,F2: A > B,Fa: A > B] :
( ! [Z2: A,Za: A] :
( ( member @ A @ Z2 @ ( set2 @ A @ X ) )
=> ( ( member @ A @ Za @ ( set2 @ A @ Xa ) )
=> ( ( ( F2 @ Z2 )
= ( Fa @ Za ) )
=> ( Z2 = Za ) ) ) )
=> ( ( ( map @ A @ B @ F2 @ X )
= ( map @ A @ B @ Fa @ Xa ) )
=> ( X = Xa ) ) ) ).
% list.inj_map_strong
thf(fact_2136_list_Omap__cong0,axiom,
! [B: $tType,A: $tType,X: list @ A,F2: A > B,G: A > B] :
( ! [Z2: A] :
( ( member @ A @ Z2 @ ( set2 @ A @ X ) )
=> ( ( F2 @ Z2 )
= ( G @ Z2 ) ) )
=> ( ( map @ A @ B @ F2 @ X )
= ( map @ A @ B @ G @ X ) ) ) ).
% list.map_cong0
thf(fact_2137_list_Omap__cong,axiom,
! [B: $tType,A: $tType,X: list @ A,Ya: list @ A,F2: A > B,G: A > B] :
( ( X = Ya )
=> ( ! [Z2: A] :
( ( member @ A @ Z2 @ ( set2 @ A @ Ya ) )
=> ( ( F2 @ Z2 )
= ( G @ Z2 ) ) )
=> ( ( map @ A @ B @ F2 @ X )
= ( map @ A @ B @ G @ Ya ) ) ) ) ).
% list.map_cong
thf(fact_2138_map__eq__nth__eq,axiom,
! [A: $tType,B: $tType,F2: B > A,L2: list @ B,L4: list @ B,I: nat] :
( ( ( map @ B @ A @ F2 @ L2 )
= ( map @ B @ A @ F2 @ L4 ) )
=> ( ( F2 @ ( nth @ B @ L2 @ I ) )
= ( F2 @ ( nth @ B @ L4 @ I ) ) ) ) ).
% map_eq_nth_eq
thf(fact_2139_round__mono,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [X: A,Y: A] :
( ( ord_less_eq @ A @ X @ Y )
=> ( ord_less_eq @ int @ ( archimedean_round @ A @ X ) @ ( archimedean_round @ A @ Y ) ) ) ) ).
% round_mono
thf(fact_2140_ceiling__ge__round,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [X: A] : ( ord_less_eq @ int @ ( archimedean_round @ A @ X ) @ ( archimedean_ceiling @ A @ X ) ) ) ).
% ceiling_ge_round
thf(fact_2141_map__upd__eq,axiom,
! [B: $tType,A: $tType,I: nat,L2: list @ A,F2: A > B,X: A] :
( ( ( ord_less @ nat @ I @ ( size_size @ ( list @ A ) @ L2 ) )
=> ( ( F2 @ ( nth @ A @ L2 @ I ) )
= ( F2 @ X ) ) )
=> ( ( map @ A @ B @ F2 @ ( list_update @ A @ L2 @ I @ X ) )
= ( map @ A @ B @ F2 @ L2 ) ) ) ).
% map_upd_eq
thf(fact_2142_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_2143_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_2144_VEBT__internal_Ocnt_H_Oelims,axiom,
! [X: vEBT_VEBT,Y: nat] :
( ( ( vEBT_VEBT_cnt2 @ X )
= Y )
=> ( ( ? [A4: $o,B3: $o] :
( X
= ( vEBT_Leaf @ A4 @ B3 ) )
=> ( Y
!= ( one_one @ nat ) ) )
=> ~ ! [Info2: option @ ( product_prod @ nat @ nat ),Deg2: nat,TreeList2: list @ vEBT_VEBT,Summary2: vEBT_VEBT] :
( ( X
= ( vEBT_Node @ Info2 @ Deg2 @ TreeList2 @ Summary2 ) )
=> ( Y
!= ( plus_plus @ nat @ ( plus_plus @ nat @ ( one_one @ nat ) @ ( vEBT_VEBT_cnt2 @ Summary2 ) ) @ ( foldr @ nat @ nat @ ( plus_plus @ nat ) @ ( map @ vEBT_VEBT @ nat @ vEBT_VEBT_cnt2 @ TreeList2 ) @ ( zero_zero @ nat ) ) ) ) ) ) ) ).
% VEBT_internal.cnt'.elims
thf(fact_2145_VEBT__internal_Ocnt_Oelims,axiom,
! [X: vEBT_VEBT,Y: real] :
( ( ( vEBT_VEBT_cnt @ X )
= Y )
=> ( ( ? [A4: $o,B3: $o] :
( X
= ( vEBT_Leaf @ A4 @ B3 ) )
=> ( Y
!= ( one_one @ real ) ) )
=> ~ ! [Info2: option @ ( product_prod @ nat @ nat ),Deg2: nat,TreeList2: list @ vEBT_VEBT,Summary2: vEBT_VEBT] :
( ( X
= ( vEBT_Node @ Info2 @ Deg2 @ TreeList2 @ Summary2 ) )
=> ( Y
!= ( plus_plus @ real @ ( plus_plus @ real @ ( one_one @ real ) @ ( vEBT_VEBT_cnt @ Summary2 ) ) @ ( foldr @ real @ real @ ( plus_plus @ real ) @ ( map @ vEBT_VEBT @ real @ vEBT_VEBT_cnt @ TreeList2 ) @ ( zero_zero @ real ) ) ) ) ) ) ) ).
% VEBT_internal.cnt.elims
thf(fact_2146_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_2147_VEBT__internal_Ospace_H_Oelims,axiom,
! [X: vEBT_VEBT,Y: nat] :
( ( ( vEBT_VEBT_space2 @ X )
= Y )
=> ( ( ? [A4: $o,B3: $o] :
( X
= ( vEBT_Leaf @ A4 @ B3 ) )
=> ( Y
!= ( numeral_numeral @ nat @ ( bit0 @ ( bit0 @ one2 ) ) ) ) )
=> ~ ! [Info2: option @ ( product_prod @ nat @ nat ),Deg2: nat,TreeList2: list @ vEBT_VEBT,Summary2: vEBT_VEBT] :
( ( X
= ( vEBT_Node @ Info2 @ Deg2 @ TreeList2 @ Summary2 ) )
=> ( Y
!= ( plus_plus @ nat @ ( plus_plus @ nat @ ( numeral_numeral @ nat @ ( bit0 @ ( bit1 @ one2 ) ) ) @ ( vEBT_VEBT_space2 @ Summary2 ) ) @ ( foldr @ nat @ nat @ ( plus_plus @ nat ) @ ( map @ vEBT_VEBT @ nat @ vEBT_VEBT_space2 @ TreeList2 ) @ ( zero_zero @ nat ) ) ) ) ) ) ) ).
% VEBT_internal.space'.elims
thf(fact_2148_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_2149_mult__less__iff1,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [Z: A,X: A,Y: A] :
( ( ord_less @ A @ ( zero_zero @ A ) @ Z )
=> ( ( ord_less @ A @ ( times_times @ A @ X @ Z ) @ ( times_times @ A @ Y @ Z ) )
= ( ord_less @ A @ X @ Y ) ) ) ) ).
% mult_less_iff1
thf(fact_2150_VEBT__internal_Ospace_Oelims,axiom,
! [X: vEBT_VEBT,Y: nat] :
( ( ( vEBT_VEBT_space @ X )
= Y )
=> ( ( ? [A4: $o,B3: $o] :
( X
= ( vEBT_Leaf @ A4 @ B3 ) )
=> ( Y
!= ( numeral_numeral @ nat @ ( bit1 @ one2 ) ) ) )
=> ~ ! [Info2: option @ ( product_prod @ nat @ nat ),Deg2: nat,TreeList2: list @ vEBT_VEBT,Summary2: vEBT_VEBT] :
( ( X
= ( vEBT_Node @ Info2 @ Deg2 @ TreeList2 @ Summary2 ) )
=> ( Y
!= ( plus_plus @ nat @ ( plus_plus @ nat @ ( plus_plus @ nat @ ( numeral_numeral @ nat @ ( bit1 @ ( bit0 @ one2 ) ) ) @ ( vEBT_VEBT_space @ Summary2 ) ) @ ( size_size @ ( list @ vEBT_VEBT ) @ TreeList2 ) ) @ ( foldr @ nat @ nat @ ( plus_plus @ nat ) @ ( map @ vEBT_VEBT @ nat @ vEBT_VEBT_space @ TreeList2 ) @ ( zero_zero @ nat ) ) ) ) ) ) ) ).
% VEBT_internal.space.elims
thf(fact_2151_of__int__round__le,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [X: A] : ( ord_less_eq @ A @ ( ring_1_of_int @ A @ ( archimedean_round @ A @ X ) ) @ ( plus_plus @ A @ X @ ( divide_divide @ A @ ( one_one @ A ) @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) ) ) ) ).
% of_int_round_le
thf(fact_2152_of__int__round__ge,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [X: A] : ( ord_less_eq @ A @ ( minus_minus @ A @ X @ ( divide_divide @ A @ ( one_one @ A ) @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) ) @ ( ring_1_of_int @ A @ ( archimedean_round @ A @ X ) ) ) ) ).
% of_int_round_ge
thf(fact_2153_of__int__round__gt,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [X: A] : ( ord_less @ A @ ( minus_minus @ A @ X @ ( divide_divide @ A @ ( one_one @ A ) @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) ) @ ( ring_1_of_int @ A @ ( archimedean_round @ A @ X ) ) ) ) ).
% of_int_round_gt
thf(fact_2154_mult__le__cancel__iff1,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [Z: A,X: A,Y: A] :
( ( ord_less @ A @ ( zero_zero @ A ) @ Z )
=> ( ( ord_less_eq @ A @ ( times_times @ A @ X @ Z ) @ ( times_times @ A @ Y @ Z ) )
= ( ord_less_eq @ A @ X @ Y ) ) ) ) ).
% mult_le_cancel_iff1
thf(fact_2155_divmod__algorithm__code_I7_J,axiom,
! [A: $tType] :
( ( unique1627219031080169319umeral @ A )
=> ! [M: num,N3: num] :
( ( ( ord_less_eq @ num @ M @ N3 )
=> ( ( unique8689654367752047608divmod @ A @ ( bit0 @ M ) @ ( bit1 @ N3 ) )
= ( product_Pair @ A @ A @ ( zero_zero @ A ) @ ( numeral_numeral @ A @ ( bit0 @ M ) ) ) ) )
& ( ~ ( ord_less_eq @ num @ M @ N3 )
=> ( ( unique8689654367752047608divmod @ A @ ( bit0 @ M ) @ ( bit1 @ N3 ) )
= ( unique1321980374590559556d_step @ A @ ( bit1 @ N3 ) @ ( unique8689654367752047608divmod @ A @ ( bit0 @ M ) @ ( bit0 @ ( bit1 @ N3 ) ) ) ) ) ) ) ) ).
% divmod_algorithm_code(7)
thf(fact_2156_divmod__algorithm__code_I8_J,axiom,
! [A: $tType] :
( ( unique1627219031080169319umeral @ A )
=> ! [M: num,N3: num] :
( ( ( ord_less @ num @ M @ N3 )
=> ( ( unique8689654367752047608divmod @ A @ ( bit1 @ M ) @ ( bit1 @ N3 ) )
= ( product_Pair @ A @ A @ ( zero_zero @ A ) @ ( numeral_numeral @ A @ ( bit1 @ M ) ) ) ) )
& ( ~ ( ord_less @ num @ M @ N3 )
=> ( ( unique8689654367752047608divmod @ A @ ( bit1 @ M ) @ ( bit1 @ N3 ) )
= ( unique1321980374590559556d_step @ A @ ( bit1 @ N3 ) @ ( unique8689654367752047608divmod @ A @ ( bit1 @ M ) @ ( bit0 @ ( bit1 @ N3 ) ) ) ) ) ) ) ) ).
% divmod_algorithm_code(8)
thf(fact_2157_divides__aux__eq,axiom,
! [A: $tType] :
( ( unique1627219031080169319umeral @ A )
=> ! [Q2: A,R2: A] :
( ( unique5940410009612947441es_aux @ A @ ( product_Pair @ A @ A @ Q2 @ R2 ) )
= ( R2
= ( zero_zero @ A ) ) ) ) ).
% divides_aux_eq
thf(fact_2158_VEBT__internal_OT__vebt__buildupi_H_Oelims,axiom,
! [X: nat,Y: int] :
( ( ( vEBT_V9176841429113362141ildupi @ X )
= Y )
=> ( ( ( X
= ( zero_zero @ nat ) )
=> ( Y
!= ( one_one @ int ) ) )
=> ( ( ( X
= ( suc @ ( zero_zero @ nat ) ) )
=> ( Y
!= ( one_one @ int ) ) )
=> ~ ! [N: nat] :
( ( X
= ( suc @ ( suc @ N ) ) )
=> ~ ( ( ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N )
=> ( Y
= ( 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 )
=> ( Y
= ( 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'.elims
thf(fact_2159__C7_OIH_C_I2_J,axiom,
! [Xa: nat,Xb: nat,Xc: nat,Xd: nat,Xe: vEBT_VEBT,Xf: list @ vEBT_VEBT,N3: 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 @ N3 )
=> ( 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_2160_unset__bit__0,axiom,
! [A: $tType] :
( ( bit_se359711467146920520ations @ A )
=> ! [A2: A] :
( ( bit_se2638667681897837118et_bit @ A @ ( zero_zero @ nat ) @ A2 )
= ( times_times @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ ( divide_divide @ A @ A2 @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) ) ) ) ).
% unset_bit_0
thf(fact_2161_low__def,axiom,
( vEBT_VEBT_low
= ( ^ [X2: nat,N2: nat] : ( modulo_modulo @ nat @ X2 @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N2 ) ) ) ) ).
% low_def
thf(fact_2162_mod__mod__trivial,axiom,
! [A: $tType] :
( ( euclid4440199948858584721cancel @ A )
=> ! [A2: A,B2: A] :
( ( modulo_modulo @ A @ ( modulo_modulo @ A @ A2 @ B2 ) @ B2 )
= ( modulo_modulo @ A @ A2 @ B2 ) ) ) ).
% mod_mod_trivial
thf(fact_2163_dvd__0__left__iff,axiom,
! [A: $tType] :
( ( comm_semiring_1 @ A )
=> ! [A2: A] :
( ( dvd_dvd @ A @ ( zero_zero @ A ) @ A2 )
= ( A2
= ( zero_zero @ A ) ) ) ) ).
% dvd_0_left_iff
thf(fact_2164_dvd__0__right,axiom,
! [A: $tType] :
( ( comm_semiring_1 @ A )
=> ! [A2: A] : ( dvd_dvd @ A @ A2 @ ( zero_zero @ A ) ) ) ).
% dvd_0_right
thf(fact_2165_bits__mod__0,axiom,
! [A: $tType] :
( ( bit_semiring_bits @ A )
=> ! [A2: A] :
( ( modulo_modulo @ A @ ( zero_zero @ A ) @ A2 )
= ( zero_zero @ A ) ) ) ).
% bits_mod_0
thf(fact_2166_mod__self,axiom,
! [A: $tType] :
( ( semidom_modulo @ A )
=> ! [A2: A] :
( ( modulo_modulo @ A @ A2 @ A2 )
= ( zero_zero @ A ) ) ) ).
% mod_self
thf(fact_2167_mod__by__0,axiom,
! [A: $tType] :
( ( semidom_modulo @ A )
=> ! [A2: A] :
( ( modulo_modulo @ A @ A2 @ ( zero_zero @ A ) )
= A2 ) ) ).
% mod_by_0
thf(fact_2168_mod__0,axiom,
! [A: $tType] :
( ( semidom_modulo @ A )
=> ! [A2: A] :
( ( modulo_modulo @ A @ ( zero_zero @ A ) @ A2 )
= ( zero_zero @ A ) ) ) ).
% mod_0
thf(fact_2169_dvd__add__triv__right__iff,axiom,
! [A: $tType] :
( ( comm_s4317794764714335236cancel @ A )
=> ! [A2: A,B2: A] :
( ( dvd_dvd @ A @ A2 @ ( plus_plus @ A @ B2 @ A2 ) )
= ( dvd_dvd @ A @ A2 @ B2 ) ) ) ).
% dvd_add_triv_right_iff
thf(fact_2170_dvd__add__triv__left__iff,axiom,
! [A: $tType] :
( ( comm_s4317794764714335236cancel @ A )
=> ! [A2: A,B2: A] :
( ( dvd_dvd @ A @ A2 @ ( plus_plus @ A @ A2 @ B2 ) )
= ( dvd_dvd @ A @ A2 @ B2 ) ) ) ).
% dvd_add_triv_left_iff
thf(fact_2171_mod__add__self2,axiom,
! [A: $tType] :
( ( euclid4440199948858584721cancel @ A )
=> ! [A2: A,B2: A] :
( ( modulo_modulo @ A @ ( plus_plus @ A @ A2 @ B2 ) @ B2 )
= ( modulo_modulo @ A @ A2 @ B2 ) ) ) ).
% mod_add_self2
thf(fact_2172_mod__add__self1,axiom,
! [A: $tType] :
( ( euclid4440199948858584721cancel @ A )
=> ! [B2: A,A2: A] :
( ( modulo_modulo @ A @ ( plus_plus @ A @ B2 @ A2 ) @ B2 )
= ( modulo_modulo @ A @ A2 @ B2 ) ) ) ).
% mod_add_self1
thf(fact_2173_div__dvd__div,axiom,
! [A: $tType] :
( ( algebraic_semidom @ A )
=> ! [A2: A,B2: A,C2: A] :
( ( dvd_dvd @ A @ A2 @ B2 )
=> ( ( dvd_dvd @ A @ A2 @ C2 )
=> ( ( dvd_dvd @ A @ ( divide_divide @ A @ B2 @ A2 ) @ ( divide_divide @ A @ C2 @ A2 ) )
= ( dvd_dvd @ A @ B2 @ C2 ) ) ) ) ) ).
% div_dvd_div
thf(fact_2174_minus__mod__self2,axiom,
! [A: $tType] :
( ( euclid8851590272496341667cancel @ A )
=> ! [A2: A,B2: A] :
( ( modulo_modulo @ A @ ( minus_minus @ A @ A2 @ B2 ) @ B2 )
= ( modulo_modulo @ A @ A2 @ B2 ) ) ) ).
% minus_mod_self2
thf(fact_2175_unset__bit__nonnegative__int__iff,axiom,
! [N3: nat,K: int] :
( ( ord_less_eq @ int @ ( zero_zero @ int ) @ ( bit_se2638667681897837118et_bit @ int @ N3 @ K ) )
= ( ord_less_eq @ int @ ( zero_zero @ int ) @ K ) ) ).
% unset_bit_nonnegative_int_iff
thf(fact_2176_unset__bit__negative__int__iff,axiom,
! [N3: nat,K: int] :
( ( ord_less @ int @ ( bit_se2638667681897837118et_bit @ int @ N3 @ K ) @ ( zero_zero @ int ) )
= ( ord_less @ int @ K @ ( zero_zero @ int ) ) ) ).
% unset_bit_negative_int_iff
thf(fact_2177_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_2178_mod__less,axiom,
! [M: nat,N3: nat] :
( ( ord_less @ nat @ M @ N3 )
=> ( ( modulo_modulo @ nat @ M @ N3 )
= M ) ) ).
% mod_less
thf(fact_2179_nat__mod__eq_H,axiom,
! [A2: nat,N3: nat] :
( ( ord_less @ nat @ A2 @ N3 )
=> ( ( modulo_modulo @ nat @ A2 @ N3 )
= A2 ) ) ).
% nat_mod_eq'
thf(fact_2180_dvd__times__right__cancel__iff,axiom,
! [A: $tType] :
( ( algebraic_semidom @ A )
=> ! [A2: A,B2: A,C2: A] :
( ( A2
!= ( zero_zero @ A ) )
=> ( ( dvd_dvd @ A @ ( times_times @ A @ B2 @ A2 ) @ ( times_times @ A @ C2 @ A2 ) )
= ( dvd_dvd @ A @ B2 @ C2 ) ) ) ) ).
% dvd_times_right_cancel_iff
thf(fact_2181_dvd__times__left__cancel__iff,axiom,
! [A: $tType] :
( ( algebraic_semidom @ A )
=> ! [A2: A,B2: A,C2: A] :
( ( A2
!= ( zero_zero @ A ) )
=> ( ( dvd_dvd @ A @ ( times_times @ A @ A2 @ B2 ) @ ( times_times @ A @ A2 @ C2 ) )
= ( dvd_dvd @ A @ B2 @ C2 ) ) ) ) ).
% dvd_times_left_cancel_iff
thf(fact_2182_dvd__mult__cancel__right,axiom,
! [A: $tType] :
( ( idom @ A )
=> ! [A2: A,C2: A,B2: A] :
( ( dvd_dvd @ A @ ( times_times @ A @ A2 @ C2 ) @ ( times_times @ A @ B2 @ C2 ) )
= ( ( C2
= ( zero_zero @ A ) )
| ( dvd_dvd @ A @ A2 @ B2 ) ) ) ) ).
% dvd_mult_cancel_right
thf(fact_2183_dvd__mult__cancel__left,axiom,
! [A: $tType] :
( ( idom @ A )
=> ! [C2: A,A2: A,B2: A] :
( ( dvd_dvd @ A @ ( times_times @ A @ C2 @ A2 ) @ ( times_times @ A @ C2 @ B2 ) )
= ( ( C2
= ( zero_zero @ A ) )
| ( dvd_dvd @ A @ A2 @ B2 ) ) ) ) ).
% dvd_mult_cancel_left
thf(fact_2184_mod__by__1,axiom,
! [A: $tType] :
( ( semidom_modulo @ A )
=> ! [A2: A] :
( ( modulo_modulo @ A @ A2 @ ( one_one @ A ) )
= ( zero_zero @ A ) ) ) ).
% mod_by_1
thf(fact_2185_bits__mod__by__1,axiom,
! [A: $tType] :
( ( bit_semiring_bits @ A )
=> ! [A2: A] :
( ( modulo_modulo @ A @ A2 @ ( one_one @ A ) )
= ( zero_zero @ A ) ) ) ).
% bits_mod_by_1
thf(fact_2186_mod__mult__self2__is__0,axiom,
! [A: $tType] :
( ( euclid4440199948858584721cancel @ A )
=> ! [A2: A,B2: A] :
( ( modulo_modulo @ A @ ( times_times @ A @ A2 @ B2 ) @ B2 )
= ( zero_zero @ A ) ) ) ).
% mod_mult_self2_is_0
thf(fact_2187_mod__mult__self1__is__0,axiom,
! [A: $tType] :
( ( euclid4440199948858584721cancel @ A )
=> ! [B2: A,A2: A] :
( ( modulo_modulo @ A @ ( times_times @ A @ B2 @ A2 ) @ B2 )
= ( zero_zero @ A ) ) ) ).
% mod_mult_self1_is_0
thf(fact_2188_dvd__add__times__triv__right__iff,axiom,
! [A: $tType] :
( ( comm_s4317794764714335236cancel @ A )
=> ! [A2: A,B2: A,C2: A] :
( ( dvd_dvd @ A @ A2 @ ( plus_plus @ A @ B2 @ ( times_times @ A @ C2 @ A2 ) ) )
= ( dvd_dvd @ A @ A2 @ B2 ) ) ) ).
% dvd_add_times_triv_right_iff
thf(fact_2189_dvd__add__times__triv__left__iff,axiom,
! [A: $tType] :
( ( comm_s4317794764714335236cancel @ A )
=> ! [A2: A,C2: A,B2: A] :
( ( dvd_dvd @ A @ A2 @ ( plus_plus @ A @ ( times_times @ A @ C2 @ A2 ) @ B2 ) )
= ( dvd_dvd @ A @ A2 @ B2 ) ) ) ).
% dvd_add_times_triv_left_iff
thf(fact_2190_unit__prod,axiom,
! [A: $tType] :
( ( algebraic_semidom @ A )
=> ! [A2: A,B2: A] :
( ( dvd_dvd @ A @ A2 @ ( one_one @ A ) )
=> ( ( dvd_dvd @ A @ B2 @ ( one_one @ A ) )
=> ( dvd_dvd @ A @ ( times_times @ A @ A2 @ B2 ) @ ( one_one @ A ) ) ) ) ) ).
% unit_prod
thf(fact_2191_bits__mod__div__trivial,axiom,
! [A: $tType] :
( ( bit_semiring_bits @ A )
=> ! [A2: A,B2: A] :
( ( divide_divide @ A @ ( modulo_modulo @ A @ A2 @ B2 ) @ B2 )
= ( zero_zero @ A ) ) ) ).
% bits_mod_div_trivial
thf(fact_2192_mod__div__trivial,axiom,
! [A: $tType] :
( ( euclid4440199948858584721cancel @ A )
=> ! [A2: A,B2: A] :
( ( divide_divide @ A @ ( modulo_modulo @ A @ A2 @ B2 ) @ B2 )
= ( zero_zero @ A ) ) ) ).
% mod_div_trivial
thf(fact_2193_mod__mult__self4,axiom,
! [A: $tType] :
( ( euclid4440199948858584721cancel @ A )
=> ! [B2: A,C2: A,A2: A] :
( ( modulo_modulo @ A @ ( plus_plus @ A @ ( times_times @ A @ B2 @ C2 ) @ A2 ) @ B2 )
= ( modulo_modulo @ A @ A2 @ B2 ) ) ) ).
% mod_mult_self4
thf(fact_2194_mod__mult__self3,axiom,
! [A: $tType] :
( ( euclid4440199948858584721cancel @ A )
=> ! [C2: A,B2: A,A2: A] :
( ( modulo_modulo @ A @ ( plus_plus @ A @ ( times_times @ A @ C2 @ B2 ) @ A2 ) @ B2 )
= ( modulo_modulo @ A @ A2 @ B2 ) ) ) ).
% mod_mult_self3
thf(fact_2195_mod__mult__self2,axiom,
! [A: $tType] :
( ( euclid4440199948858584721cancel @ A )
=> ! [A2: A,B2: A,C2: A] :
( ( modulo_modulo @ A @ ( plus_plus @ A @ A2 @ ( times_times @ A @ B2 @ C2 ) ) @ B2 )
= ( modulo_modulo @ A @ A2 @ B2 ) ) ) ).
% mod_mult_self2
thf(fact_2196_mod__mult__self1,axiom,
! [A: $tType] :
( ( euclid4440199948858584721cancel @ A )
=> ! [A2: A,C2: A,B2: A] :
( ( modulo_modulo @ A @ ( plus_plus @ A @ A2 @ ( times_times @ A @ C2 @ B2 ) ) @ B2 )
= ( modulo_modulo @ A @ A2 @ B2 ) ) ) ).
% mod_mult_self1
thf(fact_2197_div__add,axiom,
! [A: $tType] :
( ( algebraic_semidom @ A )
=> ! [C2: A,A2: A,B2: A] :
( ( dvd_dvd @ A @ C2 @ A2 )
=> ( ( dvd_dvd @ A @ C2 @ B2 )
=> ( ( divide_divide @ A @ ( plus_plus @ A @ A2 @ B2 ) @ C2 )
= ( plus_plus @ A @ ( divide_divide @ A @ A2 @ C2 ) @ ( divide_divide @ A @ B2 @ C2 ) ) ) ) ) ) ).
% div_add
thf(fact_2198_unit__div__1__div__1,axiom,
! [A: $tType] :
( ( algebraic_semidom @ A )
=> ! [A2: A] :
( ( dvd_dvd @ A @ A2 @ ( one_one @ A ) )
=> ( ( divide_divide @ A @ ( one_one @ A ) @ ( divide_divide @ A @ ( one_one @ A ) @ A2 ) )
= A2 ) ) ) ).
% unit_div_1_div_1
thf(fact_2199_unit__div__1__unit,axiom,
! [A: $tType] :
( ( algebraic_semidom @ A )
=> ! [A2: A] :
( ( dvd_dvd @ A @ A2 @ ( one_one @ A ) )
=> ( dvd_dvd @ A @ ( divide_divide @ A @ ( one_one @ A ) @ A2 ) @ ( one_one @ A ) ) ) ) ).
% unit_div_1_unit
thf(fact_2200_unit__div,axiom,
! [A: $tType] :
( ( algebraic_semidom @ A )
=> ! [A2: A,B2: A] :
( ( dvd_dvd @ A @ A2 @ ( one_one @ A ) )
=> ( ( dvd_dvd @ A @ B2 @ ( one_one @ A ) )
=> ( dvd_dvd @ A @ ( divide_divide @ A @ A2 @ B2 ) @ ( one_one @ A ) ) ) ) ) ).
% unit_div
thf(fact_2201_dvd__mult__div__cancel,axiom,
! [A: $tType] :
( ( algebraic_semidom @ A )
=> ! [A2: A,B2: A] :
( ( dvd_dvd @ A @ A2 @ B2 )
=> ( ( times_times @ A @ A2 @ ( divide_divide @ A @ B2 @ A2 ) )
= B2 ) ) ) ).
% dvd_mult_div_cancel
thf(fact_2202_dvd__div__mult__self,axiom,
! [A: $tType] :
( ( algebraic_semidom @ A )
=> ! [A2: A,B2: A] :
( ( dvd_dvd @ A @ A2 @ B2 )
=> ( ( times_times @ A @ ( divide_divide @ A @ B2 @ A2 ) @ A2 )
= B2 ) ) ) ).
% dvd_div_mult_self
thf(fact_2203_div__diff,axiom,
! [A: $tType] :
( ( idom_modulo @ A )
=> ! [C2: A,A2: A,B2: A] :
( ( dvd_dvd @ A @ C2 @ A2 )
=> ( ( dvd_dvd @ A @ C2 @ B2 )
=> ( ( divide_divide @ A @ ( minus_minus @ A @ A2 @ B2 ) @ C2 )
= ( minus_minus @ A @ ( divide_divide @ A @ A2 @ C2 ) @ ( divide_divide @ A @ B2 @ C2 ) ) ) ) ) ) ).
% div_diff
thf(fact_2204_dvd__imp__mod__0,axiom,
! [A: $tType] :
( ( semidom_modulo @ A )
=> ! [A2: A,B2: A] :
( ( dvd_dvd @ A @ A2 @ B2 )
=> ( ( modulo_modulo @ A @ B2 @ A2 )
= ( zero_zero @ A ) ) ) ) ).
% dvd_imp_mod_0
thf(fact_2205_dvd__1__left,axiom,
! [K: nat] : ( dvd_dvd @ nat @ ( suc @ ( zero_zero @ nat ) ) @ K ) ).
% dvd_1_left
thf(fact_2206_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_2207_mod__by__Suc__0,axiom,
! [M: nat] :
( ( modulo_modulo @ nat @ M @ ( suc @ ( zero_zero @ nat ) ) )
= ( zero_zero @ nat ) ) ).
% mod_by_Suc_0
thf(fact_2208_nat__mult__dvd__cancel__disj,axiom,
! [K: nat,M: nat,N3: nat] :
( ( dvd_dvd @ nat @ ( times_times @ nat @ K @ M ) @ ( times_times @ nat @ K @ N3 ) )
= ( ( K
= ( zero_zero @ nat ) )
| ( dvd_dvd @ nat @ M @ N3 ) ) ) ).
% nat_mult_dvd_cancel_disj
thf(fact_2209_unit__div__mult__self,axiom,
! [A: $tType] :
( ( algebraic_semidom @ A )
=> ! [A2: A,B2: A] :
( ( dvd_dvd @ A @ A2 @ ( one_one @ A ) )
=> ( ( times_times @ A @ ( divide_divide @ A @ B2 @ A2 ) @ A2 )
= B2 ) ) ) ).
% unit_div_mult_self
thf(fact_2210_unit__mult__div__div,axiom,
! [A: $tType] :
( ( algebraic_semidom @ A )
=> ! [A2: A,B2: A] :
( ( dvd_dvd @ A @ A2 @ ( one_one @ A ) )
=> ( ( times_times @ A @ B2 @ ( divide_divide @ A @ ( one_one @ A ) @ A2 ) )
= ( divide_divide @ A @ B2 @ A2 ) ) ) ) ).
% unit_mult_div_div
thf(fact_2211_Suc__mod__mult__self4,axiom,
! [N3: nat,K: nat,M: nat] :
( ( modulo_modulo @ nat @ ( suc @ ( plus_plus @ nat @ ( times_times @ nat @ N3 @ K ) @ M ) ) @ N3 )
= ( modulo_modulo @ nat @ ( suc @ M ) @ N3 ) ) ).
% Suc_mod_mult_self4
thf(fact_2212_Suc__mod__mult__self3,axiom,
! [K: nat,N3: nat,M: nat] :
( ( modulo_modulo @ nat @ ( suc @ ( plus_plus @ nat @ ( times_times @ nat @ K @ N3 ) @ M ) ) @ N3 )
= ( modulo_modulo @ nat @ ( suc @ M ) @ N3 ) ) ).
% Suc_mod_mult_self3
thf(fact_2213_Suc__mod__mult__self2,axiom,
! [M: nat,N3: nat,K: nat] :
( ( modulo_modulo @ nat @ ( suc @ ( plus_plus @ nat @ M @ ( times_times @ nat @ N3 @ K ) ) ) @ N3 )
= ( modulo_modulo @ nat @ ( suc @ M ) @ N3 ) ) ).
% Suc_mod_mult_self2
thf(fact_2214_Suc__mod__mult__self1,axiom,
! [M: nat,K: nat,N3: nat] :
( ( modulo_modulo @ nat @ ( suc @ ( plus_plus @ nat @ M @ ( times_times @ nat @ K @ N3 ) ) ) @ N3 )
= ( modulo_modulo @ nat @ ( suc @ M ) @ N3 ) ) ).
% Suc_mod_mult_self1
thf(fact_2215_even__add,axiom,
! [A: $tType] :
( ( semiring_parity @ A )
=> ! [A2: A,B2: A] :
( ( dvd_dvd @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ ( plus_plus @ A @ A2 @ B2 ) )
= ( ( dvd_dvd @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ A2 )
= ( dvd_dvd @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ B2 ) ) ) ) ).
% even_add
thf(fact_2216_odd__add,axiom,
! [A: $tType] :
( ( semiring_parity @ A )
=> ! [A2: A,B2: A] :
( ( ~ ( dvd_dvd @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ ( plus_plus @ A @ A2 @ B2 ) ) )
= ( ( ~ ( dvd_dvd @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ A2 ) )
!= ( ~ ( dvd_dvd @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ B2 ) ) ) ) ) ).
% odd_add
thf(fact_2217_even__mult__iff,axiom,
! [A: $tType] :
( ( semiring_parity @ A )
=> ! [A2: A,B2: A] :
( ( dvd_dvd @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ ( times_times @ A @ A2 @ B2 ) )
= ( ( dvd_dvd @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ A2 )
| ( dvd_dvd @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ B2 ) ) ) ) ).
% even_mult_iff
thf(fact_2218_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_2219_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_2220_even__mod__2__iff,axiom,
! [A: $tType] :
( ( semiring_parity @ A )
=> ! [A2: A] :
( ( dvd_dvd @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ ( modulo_modulo @ A @ A2 @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) )
= ( dvd_dvd @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ A2 ) ) ) ).
% even_mod_2_iff
thf(fact_2221_even__Suc__Suc__iff,axiom,
! [N3: nat] :
( ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( suc @ ( suc @ N3 ) ) )
= ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N3 ) ) ).
% even_Suc_Suc_iff
thf(fact_2222_even__Suc,axiom,
! [N3: nat] :
( ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( suc @ N3 ) )
= ( ~ ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N3 ) ) ) ).
% even_Suc
thf(fact_2223_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_2224_Suc__times__numeral__mod__eq,axiom,
! [K: num,N3: nat] :
( ( ( numeral_numeral @ nat @ K )
!= ( one_one @ nat ) )
=> ( ( modulo_modulo @ nat @ ( suc @ ( times_times @ nat @ ( numeral_numeral @ nat @ K ) @ N3 ) ) @ ( numeral_numeral @ nat @ K ) )
= ( one_one @ nat ) ) ) ).
% Suc_times_numeral_mod_eq
thf(fact_2225_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_2226_dvd__numeral__simp,axiom,
! [A: $tType] :
( ( unique1627219031080169319umeral @ A )
=> ! [M: num,N3: num] :
( ( dvd_dvd @ A @ ( numeral_numeral @ A @ M ) @ ( numeral_numeral @ A @ N3 ) )
= ( unique5940410009612947441es_aux @ A @ ( unique8689654367752047608divmod @ A @ N3 @ M ) ) ) ) ).
% dvd_numeral_simp
thf(fact_2227_not__mod__2__eq__1__eq__0,axiom,
! [A: $tType] :
( ( semiring_parity @ A )
=> ! [A2: A] :
( ( ( modulo_modulo @ A @ A2 @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) )
!= ( one_one @ A ) )
= ( ( modulo_modulo @ A @ A2 @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) )
= ( zero_zero @ A ) ) ) ) ).
% not_mod_2_eq_1_eq_0
thf(fact_2228_not__mod__2__eq__0__eq__1,axiom,
! [A: $tType] :
( ( semiring_parity @ A )
=> ! [A2: A] :
( ( ( modulo_modulo @ A @ A2 @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) )
!= ( zero_zero @ A ) )
= ( ( modulo_modulo @ A @ A2 @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) )
= ( one_one @ A ) ) ) ) ).
% not_mod_2_eq_0_eq_1
thf(fact_2229_even__plus__one__iff,axiom,
! [A: $tType] :
( ( semiring_parity @ A )
=> ! [A2: A] :
( ( dvd_dvd @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ ( plus_plus @ A @ A2 @ ( one_one @ A ) ) )
= ( ~ ( dvd_dvd @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ A2 ) ) ) ) ).
% even_plus_one_iff
thf(fact_2230_even__diff,axiom,
! [A: $tType] :
( ( ring_parity @ A )
=> ! [A2: A,B2: A] :
( ( dvd_dvd @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ ( minus_minus @ A @ A2 @ B2 ) )
= ( dvd_dvd @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ ( plus_plus @ A @ A2 @ B2 ) ) ) ) ).
% even_diff
thf(fact_2231_not__mod2__eq__Suc__0__eq__0,axiom,
! [N3: nat] :
( ( ( modulo_modulo @ nat @ N3 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) )
!= ( suc @ ( zero_zero @ nat ) ) )
= ( ( modulo_modulo @ nat @ N3 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) )
= ( zero_zero @ nat ) ) ) ).
% not_mod2_eq_Suc_0_eq_0
thf(fact_2232_even__Suc__div__two,axiom,
! [N3: nat] :
( ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N3 )
=> ( ( divide_divide @ nat @ ( suc @ N3 ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) )
= ( divide_divide @ nat @ N3 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ).
% even_Suc_div_two
thf(fact_2233_odd__Suc__div__two,axiom,
! [N3: nat] :
( ~ ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N3 )
=> ( ( divide_divide @ nat @ ( suc @ N3 ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) )
= ( suc @ ( divide_divide @ nat @ N3 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) ).
% odd_Suc_div_two
thf(fact_2234_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_2235_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_2236_mod__Suc__eq__mod__add3,axiom,
! [M: nat,N3: nat] :
( ( modulo_modulo @ nat @ M @ ( suc @ ( suc @ ( suc @ N3 ) ) ) )
= ( modulo_modulo @ nat @ M @ ( plus_plus @ nat @ ( numeral_numeral @ nat @ ( bit1 @ one2 ) ) @ N3 ) ) ) ).
% mod_Suc_eq_mod_add3
thf(fact_2237_divmod__algorithm__code_I3_J,axiom,
! [A: $tType] :
( ( unique1627219031080169319umeral @ A )
=> ! [N3: num] :
( ( unique8689654367752047608divmod @ A @ one2 @ ( bit0 @ N3 ) )
= ( product_Pair @ A @ A @ ( zero_zero @ A ) @ ( numeral_numeral @ A @ one2 ) ) ) ) ).
% divmod_algorithm_code(3)
thf(fact_2238_divmod__algorithm__code_I4_J,axiom,
! [A: $tType] :
( ( unique1627219031080169319umeral @ A )
=> ! [N3: num] :
( ( unique8689654367752047608divmod @ A @ one2 @ ( bit1 @ N3 ) )
= ( product_Pair @ A @ A @ ( zero_zero @ A ) @ ( numeral_numeral @ A @ one2 ) ) ) ) ).
% divmod_algorithm_code(4)
thf(fact_2239_even__succ__div__two,axiom,
! [A: $tType] :
( ( euclid5411537665997757685th_nat @ A )
=> ! [A2: A] :
( ( dvd_dvd @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ A2 )
=> ( ( divide_divide @ A @ ( plus_plus @ A @ A2 @ ( one_one @ A ) ) @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) )
= ( divide_divide @ A @ A2 @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) ) ) ) ).
% even_succ_div_two
thf(fact_2240_odd__succ__div__two,axiom,
! [A: $tType] :
( ( euclid5411537665997757685th_nat @ A )
=> ! [A2: A] :
( ~ ( dvd_dvd @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ A2 )
=> ( ( divide_divide @ A @ ( plus_plus @ A @ A2 @ ( one_one @ A ) ) @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) )
= ( plus_plus @ A @ ( divide_divide @ A @ A2 @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) @ ( one_one @ A ) ) ) ) ) ).
% odd_succ_div_two
thf(fact_2241_even__succ__div__2,axiom,
! [A: $tType] :
( ( bit_semiring_bits @ A )
=> ! [A2: A] :
( ( dvd_dvd @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ A2 )
=> ( ( divide_divide @ A @ ( plus_plus @ A @ ( one_one @ A ) @ A2 ) @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) )
= ( divide_divide @ A @ A2 @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) ) ) ) ).
% even_succ_div_2
thf(fact_2242_even__power,axiom,
! [A: $tType] :
( ( semiring_parity @ A )
=> ! [A2: A,N3: nat] :
( ( dvd_dvd @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ ( power_power @ A @ A2 @ N3 ) )
= ( ( dvd_dvd @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ A2 )
& ( ord_less @ nat @ ( zero_zero @ nat ) @ N3 ) ) ) ) ).
% even_power
thf(fact_2243_zero__le__power__eq__numeral,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [A2: A,W: num] :
( ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( power_power @ A @ A2 @ ( numeral_numeral @ nat @ W ) ) )
= ( ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( numeral_numeral @ nat @ W ) )
| ( ~ ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( numeral_numeral @ nat @ W ) )
& ( ord_less_eq @ A @ ( zero_zero @ A ) @ A2 ) ) ) ) ) ).
% zero_le_power_eq_numeral
thf(fact_2244_power__less__zero__eq__numeral,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [A2: A,W: num] :
( ( ord_less @ A @ ( power_power @ A @ A2 @ ( numeral_numeral @ nat @ W ) ) @ ( zero_zero @ A ) )
= ( ~ ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( numeral_numeral @ nat @ W ) )
& ( ord_less @ A @ A2 @ ( zero_zero @ A ) ) ) ) ) ).
% power_less_zero_eq_numeral
thf(fact_2245_power__less__zero__eq,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [A2: A,N3: nat] :
( ( ord_less @ A @ ( power_power @ A @ A2 @ N3 ) @ ( zero_zero @ A ) )
= ( ~ ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N3 )
& ( ord_less @ A @ A2 @ ( zero_zero @ A ) ) ) ) ) ).
% power_less_zero_eq
thf(fact_2246_even__of__nat,axiom,
! [A: $tType] :
( ( semiring_parity @ A )
=> ! [N3: nat] :
( ( dvd_dvd @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ ( semiring_1_of_nat @ A @ N3 ) )
= ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N3 ) ) ) ).
% even_of_nat
thf(fact_2247_odd__Suc__minus__one,axiom,
! [N3: nat] :
( ~ ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N3 )
=> ( ( suc @ ( minus_minus @ nat @ N3 @ ( suc @ ( zero_zero @ nat ) ) ) )
= N3 ) ) ).
% odd_Suc_minus_one
thf(fact_2248_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_2249_even__diff__nat,axiom,
! [M: nat,N3: nat] :
( ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( minus_minus @ nat @ M @ N3 ) )
= ( ( ord_less @ nat @ M @ N3 )
| ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( plus_plus @ nat @ M @ N3 ) ) ) ) ).
% even_diff_nat
thf(fact_2250_odd__two__times__div__two__succ,axiom,
! [A: $tType] :
( ( euclid5411537665997757685th_nat @ A )
=> ! [A2: A] :
( ~ ( dvd_dvd @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ A2 )
=> ( ( plus_plus @ A @ ( times_times @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ ( divide_divide @ A @ A2 @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) ) @ ( one_one @ A ) )
= A2 ) ) ) ).
% odd_two_times_div_two_succ
thf(fact_2251_semiring__parity__class_Oeven__mask__iff,axiom,
! [A: $tType] :
( ( semiring_parity @ A )
=> ! [N3: nat] :
( ( dvd_dvd @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ ( minus_minus @ A @ ( power_power @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ N3 ) @ ( one_one @ A ) ) )
= ( N3
= ( zero_zero @ nat ) ) ) ) ).
% semiring_parity_class.even_mask_iff
thf(fact_2252_zero__less__power__eq__numeral,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [A2: A,W: num] :
( ( ord_less @ A @ ( zero_zero @ A ) @ ( power_power @ A @ A2 @ ( numeral_numeral @ nat @ W ) ) )
= ( ( ( numeral_numeral @ nat @ W )
= ( zero_zero @ nat ) )
| ( ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( numeral_numeral @ nat @ W ) )
& ( A2
!= ( zero_zero @ A ) ) )
| ( ~ ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( numeral_numeral @ nat @ W ) )
& ( ord_less @ A @ ( zero_zero @ A ) @ A2 ) ) ) ) ) ).
% zero_less_power_eq_numeral
thf(fact_2253_odd__two__times__div__two__nat,axiom,
! [N3: nat] :
( ~ ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N3 )
=> ( ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ N3 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) )
= ( minus_minus @ nat @ N3 @ ( one_one @ nat ) ) ) ) ).
% odd_two_times_div_two_nat
thf(fact_2254__C7_OIH_C_I1_J,axiom,
! [Xa: nat,Xb: nat,Xc: nat,Xd: nat,N3: 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 ) @ N3 )
=> ( 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_2255_power__le__zero__eq__numeral,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [A2: A,W: num] :
( ( ord_less_eq @ A @ ( power_power @ A @ A2 @ ( numeral_numeral @ nat @ W ) ) @ ( zero_zero @ A ) )
= ( ( ord_less @ nat @ ( zero_zero @ nat ) @ ( numeral_numeral @ nat @ W ) )
& ( ( ~ ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( numeral_numeral @ nat @ W ) )
& ( ord_less_eq @ A @ A2 @ ( zero_zero @ A ) ) )
| ( ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( numeral_numeral @ nat @ W ) )
& ( A2
= ( zero_zero @ A ) ) ) ) ) ) ) ).
% power_le_zero_eq_numeral
thf(fact_2256_even__succ__div__exp,axiom,
! [A: $tType] :
( ( bit_semiring_bits @ A )
=> ! [A2: A,N3: nat] :
( ( dvd_dvd @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ A2 )
=> ( ( ord_less @ nat @ ( zero_zero @ nat ) @ N3 )
=> ( ( divide_divide @ A @ ( plus_plus @ A @ ( one_one @ A ) @ A2 ) @ ( power_power @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ N3 ) )
= ( divide_divide @ A @ A2 @ ( power_power @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ N3 ) ) ) ) ) ) ).
% even_succ_div_exp
thf(fact_2257_even__succ__mod__exp,axiom,
! [A: $tType] :
( ( bit_semiring_bits @ A )
=> ! [A2: A,N3: nat] :
( ( dvd_dvd @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ A2 )
=> ( ( ord_less @ nat @ ( zero_zero @ nat ) @ N3 )
=> ( ( modulo_modulo @ A @ ( plus_plus @ A @ ( one_one @ A ) @ A2 ) @ ( power_power @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ N3 ) )
= ( plus_plus @ A @ ( one_one @ A ) @ ( modulo_modulo @ A @ A2 @ ( power_power @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ N3 ) ) ) ) ) ) ) ).
% even_succ_mod_exp
thf(fact_2258_vebt__inserti_H__rf__abstr,axiom,
! [T2: vEBT_VEBT,Ti: vEBT_VEBTi,X: nat] : ( hoare_hoare_triple @ vEBT_VEBTi @ ( vEBT_vebt_assn_raw @ T2 @ Ti ) @ ( vEBT_V3964819847710782039nserti @ T2 @ Ti @ X ) @ ( vEBT_vebt_assn_raw @ ( vEBT_vebt_insert @ T2 @ X ) ) ) ).
% vebt_inserti'_rf_abstr
thf(fact_2259_mod__eq__dvd__iff,axiom,
! [A: $tType] :
( ( euclid8851590272496341667cancel @ A )
=> ! [A2: A,C2: A,B2: A] :
( ( ( modulo_modulo @ A @ A2 @ C2 )
= ( modulo_modulo @ A @ B2 @ C2 ) )
= ( dvd_dvd @ A @ C2 @ ( minus_minus @ A @ A2 @ B2 ) ) ) ) ).
% mod_eq_dvd_iff
thf(fact_2260_dvd__minus__mod,axiom,
! [A: $tType] :
( ( semidom_modulo @ A )
=> ! [B2: A,A2: A] : ( dvd_dvd @ A @ B2 @ ( minus_minus @ A @ A2 @ ( modulo_modulo @ A @ A2 @ B2 ) ) ) ) ).
% dvd_minus_mod
thf(fact_2261_mod__eq__0__iff__dvd,axiom,
! [A: $tType] :
( ( semidom_modulo @ A )
=> ! [A2: A,B2: A] :
( ( ( modulo_modulo @ A @ A2 @ B2 )
= ( zero_zero @ A ) )
= ( dvd_dvd @ A @ B2 @ A2 ) ) ) ).
% mod_eq_0_iff_dvd
thf(fact_2262_dvd__eq__mod__eq__0,axiom,
! [A: $tType] :
( ( semidom_modulo @ A )
=> ( ( dvd_dvd @ A )
= ( ^ [A6: A,B5: A] :
( ( modulo_modulo @ A @ B5 @ A6 )
= ( zero_zero @ A ) ) ) ) ) ).
% dvd_eq_mod_eq_0
thf(fact_2263_mod__0__imp__dvd,axiom,
! [A: $tType] :
( ( semiring_modulo @ A )
=> ! [A2: A,B2: A] :
( ( ( modulo_modulo @ A @ A2 @ B2 )
= ( zero_zero @ A ) )
=> ( dvd_dvd @ A @ B2 @ A2 ) ) ) ).
% mod_0_imp_dvd
thf(fact_2264_dvd__mod__imp__dvd,axiom,
! [A: $tType] :
( ( semidom_modulo @ A )
=> ! [C2: A,A2: A,B2: A] :
( ( dvd_dvd @ A @ C2 @ ( modulo_modulo @ A @ A2 @ B2 ) )
=> ( ( dvd_dvd @ A @ C2 @ B2 )
=> ( dvd_dvd @ A @ C2 @ A2 ) ) ) ) ).
% dvd_mod_imp_dvd
thf(fact_2265_dvd__mod__iff,axiom,
! [A: $tType] :
( ( semidom_modulo @ A )
=> ! [C2: A,B2: A,A2: A] :
( ( dvd_dvd @ A @ C2 @ B2 )
=> ( ( dvd_dvd @ A @ C2 @ ( modulo_modulo @ A @ A2 @ B2 ) )
= ( dvd_dvd @ A @ C2 @ A2 ) ) ) ) ).
% dvd_mod_iff
thf(fact_2266_dvd__trans,axiom,
! [A: $tType] :
( ( comm_monoid_mult @ A )
=> ! [A2: A,B2: A,C2: A] :
( ( dvd_dvd @ A @ A2 @ B2 )
=> ( ( dvd_dvd @ A @ B2 @ C2 )
=> ( dvd_dvd @ A @ A2 @ C2 ) ) ) ) ).
% dvd_trans
thf(fact_2267_dvd__refl,axiom,
! [A: $tType] :
( ( comm_monoid_mult @ A )
=> ! [A2: A] : ( dvd_dvd @ A @ A2 @ A2 ) ) ).
% dvd_refl
thf(fact_2268_dvd__antisym,axiom,
! [M: nat,N3: nat] :
( ( dvd_dvd @ nat @ M @ N3 )
=> ( ( dvd_dvd @ nat @ N3 @ M )
=> ( M = N3 ) ) ) ).
% dvd_antisym
thf(fact_2269_mod__mod__cancel,axiom,
! [A: $tType] :
( ( euclid4440199948858584721cancel @ A )
=> ! [C2: A,B2: A,A2: A] :
( ( dvd_dvd @ A @ C2 @ B2 )
=> ( ( modulo_modulo @ A @ ( modulo_modulo @ A @ A2 @ B2 ) @ C2 )
= ( modulo_modulo @ A @ A2 @ C2 ) ) ) ) ).
% mod_mod_cancel
thf(fact_2270_dvd__mod,axiom,
! [A: $tType] :
( ( euclid4440199948858584721cancel @ A )
=> ! [K: A,M: A,N3: A] :
( ( dvd_dvd @ A @ K @ M )
=> ( ( dvd_dvd @ A @ K @ N3 )
=> ( dvd_dvd @ A @ K @ ( modulo_modulo @ A @ M @ N3 ) ) ) ) ) ).
% dvd_mod
thf(fact_2271_of__nat__dvd__iff,axiom,
! [A: $tType] :
( ( euclid5411537665997757685th_nat @ A )
=> ! [M: nat,N3: nat] :
( ( dvd_dvd @ A @ ( semiring_1_of_nat @ A @ M ) @ ( semiring_1_of_nat @ A @ N3 ) )
= ( dvd_dvd @ nat @ M @ N3 ) ) ) ).
% of_nat_dvd_iff
thf(fact_2272_of__nat__mod,axiom,
! [A: $tType] :
( ( euclid5411537665997757685th_nat @ A )
=> ! [M: nat,N3: nat] :
( ( semiring_1_of_nat @ A @ ( modulo_modulo @ nat @ M @ N3 ) )
= ( modulo_modulo @ A @ ( semiring_1_of_nat @ A @ M ) @ ( semiring_1_of_nat @ A @ N3 ) ) ) ) ).
% of_nat_mod
thf(fact_2273_unit__imp__mod__eq__0,axiom,
! [A: $tType] :
( ( euclid3725896446679973847miring @ A )
=> ! [B2: A,A2: A] :
( ( dvd_dvd @ A @ B2 @ ( one_one @ A ) )
=> ( ( modulo_modulo @ A @ A2 @ B2 )
= ( zero_zero @ A ) ) ) ) ).
% unit_imp_mod_eq_0
thf(fact_2274_mod__greater__zero__iff__not__dvd,axiom,
! [M: nat,N3: nat] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ ( modulo_modulo @ nat @ M @ N3 ) )
= ( ~ ( dvd_dvd @ nat @ N3 @ M ) ) ) ).
% mod_greater_zero_iff_not_dvd
thf(fact_2275_mod__eq__dvd__iff__nat,axiom,
! [N3: nat,M: nat,Q2: nat] :
( ( ord_less_eq @ nat @ N3 @ M )
=> ( ( ( modulo_modulo @ nat @ M @ Q2 )
= ( modulo_modulo @ nat @ N3 @ Q2 ) )
= ( dvd_dvd @ nat @ Q2 @ ( minus_minus @ nat @ M @ N3 ) ) ) ) ).
% mod_eq_dvd_iff_nat
thf(fact_2276_mod__add__right__eq,axiom,
! [A: $tType] :
( ( euclid4440199948858584721cancel @ A )
=> ! [A2: A,B2: A,C2: A] :
( ( modulo_modulo @ A @ ( plus_plus @ A @ A2 @ ( modulo_modulo @ A @ B2 @ C2 ) ) @ C2 )
= ( modulo_modulo @ A @ ( plus_plus @ A @ A2 @ B2 ) @ C2 ) ) ) ).
% mod_add_right_eq
thf(fact_2277_mod__add__left__eq,axiom,
! [A: $tType] :
( ( euclid4440199948858584721cancel @ A )
=> ! [A2: A,C2: A,B2: A] :
( ( modulo_modulo @ A @ ( plus_plus @ A @ ( modulo_modulo @ A @ A2 @ C2 ) @ B2 ) @ C2 )
= ( modulo_modulo @ A @ ( plus_plus @ A @ A2 @ B2 ) @ C2 ) ) ) ).
% mod_add_left_eq
thf(fact_2278_mod__add__cong,axiom,
! [A: $tType] :
( ( euclid4440199948858584721cancel @ A )
=> ! [A2: A,C2: A,A5: A,B2: A,B4: A] :
( ( ( modulo_modulo @ A @ A2 @ C2 )
= ( modulo_modulo @ A @ A5 @ C2 ) )
=> ( ( ( modulo_modulo @ A @ B2 @ C2 )
= ( modulo_modulo @ A @ B4 @ C2 ) )
=> ( ( modulo_modulo @ A @ ( plus_plus @ A @ A2 @ B2 ) @ C2 )
= ( modulo_modulo @ A @ ( plus_plus @ A @ A5 @ B4 ) @ C2 ) ) ) ) ) ).
% mod_add_cong
thf(fact_2279_mod__add__eq,axiom,
! [A: $tType] :
( ( euclid4440199948858584721cancel @ A )
=> ! [A2: A,C2: A,B2: A] :
( ( modulo_modulo @ A @ ( plus_plus @ A @ ( modulo_modulo @ A @ A2 @ C2 ) @ ( modulo_modulo @ A @ B2 @ C2 ) ) @ C2 )
= ( modulo_modulo @ A @ ( plus_plus @ A @ A2 @ B2 ) @ C2 ) ) ) ).
% mod_add_eq
thf(fact_2280_mod__mult__right__eq,axiom,
! [A: $tType] :
( ( euclid4440199948858584721cancel @ A )
=> ! [A2: A,B2: A,C2: A] :
( ( modulo_modulo @ A @ ( times_times @ A @ A2 @ ( modulo_modulo @ A @ B2 @ C2 ) ) @ C2 )
= ( modulo_modulo @ A @ ( times_times @ A @ A2 @ B2 ) @ C2 ) ) ) ).
% mod_mult_right_eq
thf(fact_2281_mod__mult__left__eq,axiom,
! [A: $tType] :
( ( euclid4440199948858584721cancel @ A )
=> ! [A2: A,C2: A,B2: A] :
( ( modulo_modulo @ A @ ( times_times @ A @ ( modulo_modulo @ A @ A2 @ C2 ) @ B2 ) @ C2 )
= ( modulo_modulo @ A @ ( times_times @ A @ A2 @ B2 ) @ C2 ) ) ) ).
% mod_mult_left_eq
thf(fact_2282_mult__mod__right,axiom,
! [A: $tType] :
( ( euclid4440199948858584721cancel @ A )
=> ! [C2: A,A2: A,B2: A] :
( ( times_times @ A @ C2 @ ( modulo_modulo @ A @ A2 @ B2 ) )
= ( modulo_modulo @ A @ ( times_times @ A @ C2 @ A2 ) @ ( times_times @ A @ C2 @ B2 ) ) ) ) ).
% mult_mod_right
thf(fact_2283_mod__mult__mult2,axiom,
! [A: $tType] :
( ( euclid4440199948858584721cancel @ A )
=> ! [A2: A,C2: A,B2: A] :
( ( modulo_modulo @ A @ ( times_times @ A @ A2 @ C2 ) @ ( times_times @ A @ B2 @ C2 ) )
= ( times_times @ A @ ( modulo_modulo @ A @ A2 @ B2 ) @ C2 ) ) ) ).
% mod_mult_mult2
thf(fact_2284_mod__mult__cong,axiom,
! [A: $tType] :
( ( euclid4440199948858584721cancel @ A )
=> ! [A2: A,C2: A,A5: A,B2: A,B4: A] :
( ( ( modulo_modulo @ A @ A2 @ C2 )
= ( modulo_modulo @ A @ A5 @ C2 ) )
=> ( ( ( modulo_modulo @ A @ B2 @ C2 )
= ( modulo_modulo @ A @ B4 @ C2 ) )
=> ( ( modulo_modulo @ A @ ( times_times @ A @ A2 @ B2 ) @ C2 )
= ( modulo_modulo @ A @ ( times_times @ A @ A5 @ B4 ) @ C2 ) ) ) ) ) ).
% mod_mult_cong
thf(fact_2285_mod__mult__eq,axiom,
! [A: $tType] :
( ( euclid4440199948858584721cancel @ A )
=> ! [A2: A,C2: A,B2: A] :
( ( modulo_modulo @ A @ ( times_times @ A @ ( modulo_modulo @ A @ A2 @ C2 ) @ ( modulo_modulo @ A @ B2 @ C2 ) ) @ C2 )
= ( modulo_modulo @ A @ ( times_times @ A @ A2 @ B2 ) @ C2 ) ) ) ).
% mod_mult_eq
thf(fact_2286_mod__diff__right__eq,axiom,
! [A: $tType] :
( ( euclid8851590272496341667cancel @ A )
=> ! [A2: A,B2: A,C2: A] :
( ( modulo_modulo @ A @ ( minus_minus @ A @ A2 @ ( modulo_modulo @ A @ B2 @ C2 ) ) @ C2 )
= ( modulo_modulo @ A @ ( minus_minus @ A @ A2 @ B2 ) @ C2 ) ) ) ).
% mod_diff_right_eq
thf(fact_2287_mod__diff__left__eq,axiom,
! [A: $tType] :
( ( euclid8851590272496341667cancel @ A )
=> ! [A2: A,C2: A,B2: A] :
( ( modulo_modulo @ A @ ( minus_minus @ A @ ( modulo_modulo @ A @ A2 @ C2 ) @ B2 ) @ C2 )
= ( modulo_modulo @ A @ ( minus_minus @ A @ A2 @ B2 ) @ C2 ) ) ) ).
% mod_diff_left_eq
thf(fact_2288_mod__diff__cong,axiom,
! [A: $tType] :
( ( euclid8851590272496341667cancel @ A )
=> ! [A2: A,C2: A,A5: A,B2: A,B4: A] :
( ( ( modulo_modulo @ A @ A2 @ C2 )
= ( modulo_modulo @ A @ A5 @ C2 ) )
=> ( ( ( modulo_modulo @ A @ B2 @ C2 )
= ( modulo_modulo @ A @ B4 @ C2 ) )
=> ( ( modulo_modulo @ A @ ( minus_minus @ A @ A2 @ B2 ) @ C2 )
= ( modulo_modulo @ A @ ( minus_minus @ A @ A5 @ B4 ) @ C2 ) ) ) ) ) ).
% mod_diff_cong
thf(fact_2289_mod__diff__eq,axiom,
! [A: $tType] :
( ( euclid8851590272496341667cancel @ A )
=> ! [A2: A,C2: A,B2: A] :
( ( modulo_modulo @ A @ ( minus_minus @ A @ ( modulo_modulo @ A @ A2 @ C2 ) @ ( modulo_modulo @ A @ B2 @ C2 ) ) @ C2 )
= ( modulo_modulo @ A @ ( minus_minus @ A @ A2 @ B2 ) @ C2 ) ) ) ).
% mod_diff_eq
thf(fact_2290_power__mod,axiom,
! [A: $tType] :
( ( euclid4440199948858584721cancel @ A )
=> ! [A2: A,B2: A,N3: nat] :
( ( modulo_modulo @ A @ ( power_power @ A @ ( modulo_modulo @ A @ A2 @ B2 ) @ N3 ) @ B2 )
= ( modulo_modulo @ A @ ( power_power @ A @ A2 @ N3 ) @ B2 ) ) ) ).
% power_mod
thf(fact_2291_dvd__0__left,axiom,
! [A: $tType] :
( ( comm_semiring_1 @ A )
=> ! [A2: A] :
( ( dvd_dvd @ A @ ( zero_zero @ A ) @ A2 )
=> ( A2
= ( zero_zero @ A ) ) ) ) ).
% dvd_0_left
thf(fact_2292_mod__Suc__Suc__eq,axiom,
! [M: nat,N3: nat] :
( ( modulo_modulo @ nat @ ( suc @ ( suc @ ( modulo_modulo @ nat @ M @ N3 ) ) ) @ N3 )
= ( modulo_modulo @ nat @ ( suc @ ( suc @ M ) ) @ N3 ) ) ).
% mod_Suc_Suc_eq
thf(fact_2293_mod__Suc__eq,axiom,
! [M: nat,N3: nat] :
( ( modulo_modulo @ nat @ ( suc @ ( modulo_modulo @ nat @ M @ N3 ) ) @ N3 )
= ( modulo_modulo @ nat @ ( suc @ M ) @ N3 ) ) ).
% mod_Suc_eq
thf(fact_2294_dvd__add__right__iff,axiom,
! [A: $tType] :
( ( comm_s4317794764714335236cancel @ A )
=> ! [A2: A,B2: A,C2: A] :
( ( dvd_dvd @ A @ A2 @ B2 )
=> ( ( dvd_dvd @ A @ A2 @ ( plus_plus @ A @ B2 @ C2 ) )
= ( dvd_dvd @ A @ A2 @ C2 ) ) ) ) ).
% dvd_add_right_iff
thf(fact_2295_dvd__add__left__iff,axiom,
! [A: $tType] :
( ( comm_s4317794764714335236cancel @ A )
=> ! [A2: A,C2: A,B2: A] :
( ( dvd_dvd @ A @ A2 @ C2 )
=> ( ( dvd_dvd @ A @ A2 @ ( plus_plus @ A @ B2 @ C2 ) )
= ( dvd_dvd @ A @ A2 @ B2 ) ) ) ) ).
% dvd_add_left_iff
thf(fact_2296_dvd__add,axiom,
! [A: $tType] :
( ( comm_semiring_1 @ A )
=> ! [A2: A,B2: A,C2: A] :
( ( dvd_dvd @ A @ A2 @ B2 )
=> ( ( dvd_dvd @ A @ A2 @ C2 )
=> ( dvd_dvd @ A @ A2 @ ( plus_plus @ A @ B2 @ C2 ) ) ) ) ) ).
% dvd_add
thf(fact_2297_dvd__unit__imp__unit,axiom,
! [A: $tType] :
( ( algebraic_semidom @ A )
=> ! [A2: A,B2: A] :
( ( dvd_dvd @ A @ A2 @ B2 )
=> ( ( dvd_dvd @ A @ B2 @ ( one_one @ A ) )
=> ( dvd_dvd @ A @ A2 @ ( one_one @ A ) ) ) ) ) ).
% dvd_unit_imp_unit
thf(fact_2298_unit__imp__dvd,axiom,
! [A: $tType] :
( ( algebraic_semidom @ A )
=> ! [B2: A,A2: A] :
( ( dvd_dvd @ A @ B2 @ ( one_one @ A ) )
=> ( dvd_dvd @ A @ B2 @ A2 ) ) ) ).
% unit_imp_dvd
thf(fact_2299_one__dvd,axiom,
! [A: $tType] :
( ( comm_monoid_mult @ A )
=> ! [A2: A] : ( dvd_dvd @ A @ ( one_one @ A ) @ A2 ) ) ).
% one_dvd
thf(fact_2300_dvd__triv__right,axiom,
! [A: $tType] :
( ( comm_monoid_mult @ A )
=> ! [A2: A,B2: A] : ( dvd_dvd @ A @ A2 @ ( times_times @ A @ B2 @ A2 ) ) ) ).
% dvd_triv_right
thf(fact_2301_dvd__mult__right,axiom,
! [A: $tType] :
( ( comm_monoid_mult @ A )
=> ! [A2: A,B2: A,C2: A] :
( ( dvd_dvd @ A @ ( times_times @ A @ A2 @ B2 ) @ C2 )
=> ( dvd_dvd @ A @ B2 @ C2 ) ) ) ).
% dvd_mult_right
thf(fact_2302_mult__dvd__mono,axiom,
! [A: $tType] :
( ( comm_monoid_mult @ A )
=> ! [A2: A,B2: A,C2: A,D: A] :
( ( dvd_dvd @ A @ A2 @ B2 )
=> ( ( dvd_dvd @ A @ C2 @ D )
=> ( dvd_dvd @ A @ ( times_times @ A @ A2 @ C2 ) @ ( times_times @ A @ B2 @ D ) ) ) ) ) ).
% mult_dvd_mono
thf(fact_2303_dvd__triv__left,axiom,
! [A: $tType] :
( ( comm_monoid_mult @ A )
=> ! [A2: A,B2: A] : ( dvd_dvd @ A @ A2 @ ( times_times @ A @ A2 @ B2 ) ) ) ).
% dvd_triv_left
thf(fact_2304_dvd__mult__left,axiom,
! [A: $tType] :
( ( comm_monoid_mult @ A )
=> ! [A2: A,B2: A,C2: A] :
( ( dvd_dvd @ A @ ( times_times @ A @ A2 @ B2 ) @ C2 )
=> ( dvd_dvd @ A @ A2 @ C2 ) ) ) ).
% dvd_mult_left
thf(fact_2305_dvd__mult2,axiom,
! [A: $tType] :
( ( comm_monoid_mult @ A )
=> ! [A2: A,B2: A,C2: A] :
( ( dvd_dvd @ A @ A2 @ B2 )
=> ( dvd_dvd @ A @ A2 @ ( times_times @ A @ B2 @ C2 ) ) ) ) ).
% dvd_mult2
thf(fact_2306_dvd__mult,axiom,
! [A: $tType] :
( ( comm_monoid_mult @ A )
=> ! [A2: A,C2: A,B2: A] :
( ( dvd_dvd @ A @ A2 @ C2 )
=> ( dvd_dvd @ A @ A2 @ ( times_times @ A @ B2 @ C2 ) ) ) ) ).
% dvd_mult
thf(fact_2307_dvd__def,axiom,
! [A: $tType] :
( ( dvd @ A )
=> ( ( dvd_dvd @ A )
= ( ^ [B5: A,A6: A] :
? [K2: A] :
( A6
= ( times_times @ A @ B5 @ K2 ) ) ) ) ) ).
% dvd_def
thf(fact_2308_dvdI,axiom,
! [A: $tType] :
( ( dvd @ A )
=> ! [A2: A,B2: A,K: A] :
( ( A2
= ( times_times @ A @ B2 @ K ) )
=> ( dvd_dvd @ A @ B2 @ A2 ) ) ) ).
% dvdI
thf(fact_2309_dvdE,axiom,
! [A: $tType] :
( ( dvd @ A )
=> ! [B2: A,A2: A] :
( ( dvd_dvd @ A @ B2 @ A2 )
=> ~ ! [K3: A] :
( A2
!= ( times_times @ A @ B2 @ K3 ) ) ) ) ).
% dvdE
thf(fact_2310_dvd__diff__commute,axiom,
! [A: $tType] :
( ( euclid5891614535332579305n_ring @ A )
=> ! [A2: A,C2: A,B2: A] :
( ( dvd_dvd @ A @ A2 @ ( minus_minus @ A @ C2 @ B2 ) )
= ( dvd_dvd @ A @ A2 @ ( minus_minus @ A @ B2 @ C2 ) ) ) ) ).
% dvd_diff_commute
thf(fact_2311_dvd__diff,axiom,
! [A: $tType] :
( ( comm_ring_1 @ A )
=> ! [X: A,Y: A,Z: A] :
( ( dvd_dvd @ A @ X @ Y )
=> ( ( dvd_dvd @ A @ X @ Z )
=> ( dvd_dvd @ A @ X @ ( minus_minus @ A @ Y @ Z ) ) ) ) ) ).
% dvd_diff
thf(fact_2312_nat__mod__eq,axiom,
! [B2: nat,N3: nat,A2: nat] :
( ( ord_less @ nat @ B2 @ N3 )
=> ( ( ( modulo_modulo @ nat @ A2 @ N3 )
= ( modulo_modulo @ nat @ B2 @ N3 ) )
=> ( ( modulo_modulo @ nat @ A2 @ N3 )
= B2 ) ) ) ).
% nat_mod_eq
thf(fact_2313_div__div__div__same,axiom,
! [A: $tType] :
( ( algebraic_semidom @ A )
=> ! [D: A,B2: A,A2: A] :
( ( dvd_dvd @ A @ D @ B2 )
=> ( ( dvd_dvd @ A @ B2 @ A2 )
=> ( ( divide_divide @ A @ ( divide_divide @ A @ A2 @ D ) @ ( divide_divide @ A @ B2 @ D ) )
= ( divide_divide @ A @ A2 @ B2 ) ) ) ) ) ).
% div_div_div_same
thf(fact_2314_dvd__div__eq__cancel,axiom,
! [A: $tType] :
( ( semidom_divide @ A )
=> ! [A2: A,C2: A,B2: A] :
( ( ( divide_divide @ A @ A2 @ C2 )
= ( divide_divide @ A @ B2 @ C2 ) )
=> ( ( dvd_dvd @ A @ C2 @ A2 )
=> ( ( dvd_dvd @ A @ C2 @ B2 )
=> ( A2 = B2 ) ) ) ) ) ).
% dvd_div_eq_cancel
thf(fact_2315_dvd__div__eq__iff,axiom,
! [A: $tType] :
( ( semidom_divide @ A )
=> ! [C2: A,A2: A,B2: A] :
( ( dvd_dvd @ A @ C2 @ A2 )
=> ( ( dvd_dvd @ A @ C2 @ B2 )
=> ( ( ( divide_divide @ A @ A2 @ C2 )
= ( divide_divide @ A @ B2 @ C2 ) )
= ( A2 = B2 ) ) ) ) ) ).
% dvd_div_eq_iff
thf(fact_2316_dvd__power__same,axiom,
! [A: $tType] :
( ( comm_semiring_1 @ A )
=> ! [X: A,Y: A,N3: nat] :
( ( dvd_dvd @ A @ X @ Y )
=> ( dvd_dvd @ A @ ( power_power @ A @ X @ N3 ) @ ( power_power @ A @ Y @ N3 ) ) ) ) ).
% dvd_power_same
thf(fact_2317_mod__plus__right,axiom,
! [A2: nat,X: nat,M: nat,B2: nat] :
( ( ( modulo_modulo @ nat @ ( plus_plus @ nat @ A2 @ X ) @ M )
= ( modulo_modulo @ nat @ ( plus_plus @ nat @ B2 @ X ) @ M ) )
= ( ( modulo_modulo @ nat @ A2 @ M )
= ( modulo_modulo @ nat @ B2 @ M ) ) ) ).
% mod_plus_right
thf(fact_2318_mod__less__eq__dividend,axiom,
! [M: nat,N3: nat] : ( ord_less_eq @ nat @ ( modulo_modulo @ nat @ M @ N3 ) @ M ) ).
% mod_less_eq_dividend
thf(fact_2319_dvd__diff__nat,axiom,
! [K: nat,M: nat,N3: nat] :
( ( dvd_dvd @ nat @ K @ M )
=> ( ( dvd_dvd @ nat @ K @ N3 )
=> ( dvd_dvd @ nat @ K @ ( minus_minus @ nat @ M @ N3 ) ) ) ) ).
% dvd_diff_nat
thf(fact_2320_even__even__mod__4__iff,axiom,
! [N3: nat] :
( ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N3 )
= ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( modulo_modulo @ nat @ N3 @ ( numeral_numeral @ nat @ ( bit0 @ ( bit0 @ one2 ) ) ) ) ) ) ).
% even_even_mod_4_iff
thf(fact_2321_mod__nat__eqI,axiom,
! [R2: nat,N3: nat,M: nat] :
( ( ord_less @ nat @ R2 @ N3 )
=> ( ( ord_less_eq @ nat @ R2 @ M )
=> ( ( dvd_dvd @ nat @ N3 @ ( minus_minus @ nat @ M @ R2 ) )
=> ( ( modulo_modulo @ nat @ M @ N3 )
= R2 ) ) ) ) ).
% mod_nat_eqI
thf(fact_2322_diff__mod__le,axiom,
! [A2: nat,D: nat,B2: nat] :
( ( ord_less @ nat @ A2 @ D )
=> ( ( dvd_dvd @ nat @ B2 @ D )
=> ( ord_less_eq @ nat @ ( minus_minus @ nat @ A2 @ ( modulo_modulo @ nat @ A2 @ B2 ) ) @ ( minus_minus @ nat @ D @ B2 ) ) ) ) ).
% diff_mod_le
thf(fact_2323_unset__bit__less__eq,axiom,
! [N3: nat,K: int] : ( ord_less_eq @ int @ ( bit_se2638667681897837118et_bit @ int @ N3 @ K ) @ K ) ).
% unset_bit_less_eq
thf(fact_2324_subset__divisors__dvd,axiom,
! [A: $tType] :
( ( comm_monoid_mult @ A )
=> ! [A2: A,B2: A] :
( ( ord_less_eq @ ( set @ A )
@ ( collect @ A
@ ^ [C3: A] : ( dvd_dvd @ A @ C3 @ A2 ) )
@ ( collect @ A
@ ^ [C3: A] : ( dvd_dvd @ A @ C3 @ B2 ) ) )
= ( dvd_dvd @ A @ A2 @ B2 ) ) ) ).
% subset_divisors_dvd
thf(fact_2325_strict__subset__divisors__dvd,axiom,
! [A: $tType] :
( ( comm_monoid_mult @ A )
=> ! [A2: A,B2: A] :
( ( ord_less @ ( set @ A )
@ ( collect @ A
@ ^ [C3: A] : ( dvd_dvd @ A @ C3 @ A2 ) )
@ ( collect @ A
@ ^ [C3: A] : ( dvd_dvd @ A @ C3 @ B2 ) ) )
= ( ( dvd_dvd @ A @ A2 @ B2 )
& ~ ( dvd_dvd @ A @ B2 @ A2 ) ) ) ) ).
% strict_subset_divisors_dvd
thf(fact_2326_divmod__def,axiom,
! [A: $tType] :
( ( unique1627219031080169319umeral @ A )
=> ( ( unique8689654367752047608divmod @ A )
= ( ^ [M5: num,N2: num] : ( product_Pair @ A @ A @ ( divide_divide @ A @ ( numeral_numeral @ A @ M5 ) @ ( numeral_numeral @ A @ N2 ) ) @ ( modulo_modulo @ A @ ( numeral_numeral @ A @ M5 ) @ ( numeral_numeral @ A @ N2 ) ) ) ) ) ) ).
% divmod_def
thf(fact_2327_divmod_H__nat__def,axiom,
( ( unique8689654367752047608divmod @ nat )
= ( ^ [M5: num,N2: num] : ( product_Pair @ nat @ nat @ ( divide_divide @ nat @ ( numeral_numeral @ nat @ M5 ) @ ( numeral_numeral @ nat @ N2 ) ) @ ( modulo_modulo @ nat @ ( numeral_numeral @ nat @ M5 ) @ ( numeral_numeral @ nat @ N2 ) ) ) ) ) ).
% divmod'_nat_def
thf(fact_2328_even__iff__mod__2__eq__zero,axiom,
! [A: $tType] :
( ( semiring_parity @ A )
=> ! [A2: A] :
( ( dvd_dvd @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ A2 )
= ( ( modulo_modulo @ A @ A2 @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) )
= ( zero_zero @ A ) ) ) ) ).
% even_iff_mod_2_eq_zero
thf(fact_2329_odd__iff__mod__2__eq__one,axiom,
! [A: $tType] :
( ( semiring_parity @ A )
=> ! [A2: A] :
( ( ~ ( dvd_dvd @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ A2 ) )
= ( ( modulo_modulo @ A @ A2 @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) )
= ( one_one @ A ) ) ) ) ).
% odd_iff_mod_2_eq_one
thf(fact_2330_unique__euclidean__semiring__numeral__class_Omod__less__eq__dividend,axiom,
! [A: $tType] :
( ( unique1627219031080169319umeral @ A )
=> ! [A2: A,B2: A] :
( ( ord_less_eq @ A @ ( zero_zero @ A ) @ A2 )
=> ( ord_less_eq @ A @ ( modulo_modulo @ A @ A2 @ B2 ) @ A2 ) ) ) ).
% unique_euclidean_semiring_numeral_class.mod_less_eq_dividend
thf(fact_2331_unique__euclidean__semiring__numeral__class_Opos__mod__bound,axiom,
! [A: $tType] :
( ( unique1627219031080169319umeral @ A )
=> ! [B2: A,A2: A] :
( ( ord_less @ A @ ( zero_zero @ A ) @ B2 )
=> ( ord_less @ A @ ( modulo_modulo @ A @ A2 @ B2 ) @ B2 ) ) ) ).
% unique_euclidean_semiring_numeral_class.pos_mod_bound
thf(fact_2332_cong__exp__iff__simps_I9_J,axiom,
! [A: $tType] :
( ( unique1627219031080169319umeral @ A )
=> ! [M: num,Q2: num,N3: num] :
( ( ( modulo_modulo @ A @ ( numeral_numeral @ A @ ( bit0 @ M ) ) @ ( numeral_numeral @ A @ ( bit0 @ Q2 ) ) )
= ( modulo_modulo @ A @ ( numeral_numeral @ A @ ( bit0 @ N3 ) ) @ ( numeral_numeral @ A @ ( bit0 @ Q2 ) ) ) )
= ( ( modulo_modulo @ A @ ( numeral_numeral @ A @ M ) @ ( numeral_numeral @ A @ Q2 ) )
= ( modulo_modulo @ A @ ( numeral_numeral @ A @ N3 ) @ ( numeral_numeral @ A @ Q2 ) ) ) ) ) ).
% cong_exp_iff_simps(9)
thf(fact_2333_cong__exp__iff__simps_I4_J,axiom,
! [A: $tType] :
( ( unique1627219031080169319umeral @ A )
=> ! [M: num,N3: num] :
( ( modulo_modulo @ A @ ( numeral_numeral @ A @ M ) @ ( numeral_numeral @ A @ one2 ) )
= ( modulo_modulo @ A @ ( numeral_numeral @ A @ N3 ) @ ( numeral_numeral @ A @ one2 ) ) ) ) ).
% cong_exp_iff_simps(4)
thf(fact_2334_mod__eq__self__iff__div__eq__0,axiom,
! [A: $tType] :
( ( euclid3725896446679973847miring @ A )
=> ! [A2: A,B2: A] :
( ( ( modulo_modulo @ A @ A2 @ B2 )
= A2 )
= ( ( divide_divide @ A @ A2 @ B2 )
= ( zero_zero @ A ) ) ) ) ).
% mod_eq_self_iff_div_eq_0
thf(fact_2335_mod__eqE,axiom,
! [A: $tType] :
( ( euclid8851590272496341667cancel @ A )
=> ! [A2: A,C2: A,B2: A] :
( ( ( modulo_modulo @ A @ A2 @ C2 )
= ( modulo_modulo @ A @ B2 @ C2 ) )
=> ~ ! [D3: A] :
( B2
!= ( plus_plus @ A @ A2 @ ( times_times @ A @ C2 @ D3 ) ) ) ) ) ).
% mod_eqE
thf(fact_2336_div__add1__eq,axiom,
! [A: $tType] :
( ( euclid3128863361964157862miring @ A )
=> ! [A2: A,B2: A,C2: A] :
( ( divide_divide @ A @ ( plus_plus @ A @ A2 @ B2 ) @ C2 )
= ( plus_plus @ A @ ( plus_plus @ A @ ( divide_divide @ A @ A2 @ C2 ) @ ( divide_divide @ A @ B2 @ C2 ) ) @ ( divide_divide @ A @ ( plus_plus @ A @ ( modulo_modulo @ A @ A2 @ C2 ) @ ( modulo_modulo @ A @ B2 @ C2 ) ) @ C2 ) ) ) ) ).
% div_add1_eq
thf(fact_2337_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_2338_pinf_I9_J,axiom,
! [B: $tType] :
( ( ( plus @ B )
& ( linorder @ B )
& ( dvd @ B ) )
=> ! [D: B,S: B] :
? [Z2: B] :
! [X5: B] :
( ( ord_less @ B @ Z2 @ X5 )
=> ( ( dvd_dvd @ B @ D @ ( plus_plus @ B @ X5 @ S ) )
= ( dvd_dvd @ B @ D @ ( plus_plus @ B @ X5 @ S ) ) ) ) ) ).
% pinf(9)
thf(fact_2339_pinf_I10_J,axiom,
! [B: $tType] :
( ( ( plus @ B )
& ( linorder @ B )
& ( dvd @ B ) )
=> ! [D: B,S: B] :
? [Z2: B] :
! [X5: B] :
( ( ord_less @ B @ Z2 @ X5 )
=> ( ( ~ ( dvd_dvd @ B @ D @ ( plus_plus @ B @ X5 @ S ) ) )
= ( ~ ( dvd_dvd @ B @ D @ ( plus_plus @ B @ X5 @ S ) ) ) ) ) ) ).
% pinf(10)
thf(fact_2340_minf_I9_J,axiom,
! [B: $tType] :
( ( ( plus @ B )
& ( linorder @ B )
& ( dvd @ B ) )
=> ! [D: B,S: B] :
? [Z2: B] :
! [X5: B] :
( ( ord_less @ B @ X5 @ Z2 )
=> ( ( dvd_dvd @ B @ D @ ( plus_plus @ B @ X5 @ S ) )
= ( dvd_dvd @ B @ D @ ( plus_plus @ B @ X5 @ S ) ) ) ) ) ).
% minf(9)
thf(fact_2341_minf_I10_J,axiom,
! [B: $tType] :
( ( ( plus @ B )
& ( linorder @ B )
& ( dvd @ B ) )
=> ! [D: B,S: B] :
? [Z2: B] :
! [X5: B] :
( ( ord_less @ B @ X5 @ Z2 )
=> ( ( ~ ( dvd_dvd @ B @ D @ ( plus_plus @ B @ X5 @ S ) ) )
= ( ~ ( dvd_dvd @ B @ D @ ( plus_plus @ B @ X5 @ S ) ) ) ) ) ) ).
% minf(10)
thf(fact_2342_dvd__div__eq__0__iff,axiom,
! [A: $tType] :
( ( semidom_divide @ A )
=> ! [B2: A,A2: A] :
( ( dvd_dvd @ A @ B2 @ A2 )
=> ( ( ( divide_divide @ A @ A2 @ B2 )
= ( zero_zero @ A ) )
= ( A2
= ( zero_zero @ A ) ) ) ) ) ).
% dvd_div_eq_0_iff
thf(fact_2343_unit__mult__right__cancel,axiom,
! [A: $tType] :
( ( algebraic_semidom @ A )
=> ! [A2: A,B2: A,C2: A] :
( ( dvd_dvd @ A @ A2 @ ( one_one @ A ) )
=> ( ( ( times_times @ A @ B2 @ A2 )
= ( times_times @ A @ C2 @ A2 ) )
= ( B2 = C2 ) ) ) ) ).
% unit_mult_right_cancel
thf(fact_2344_unit__mult__left__cancel,axiom,
! [A: $tType] :
( ( algebraic_semidom @ A )
=> ! [A2: A,B2: A,C2: A] :
( ( dvd_dvd @ A @ A2 @ ( one_one @ A ) )
=> ( ( ( times_times @ A @ A2 @ B2 )
= ( times_times @ A @ A2 @ C2 ) )
= ( B2 = C2 ) ) ) ) ).
% unit_mult_left_cancel
thf(fact_2345_mult__unit__dvd__iff_H,axiom,
! [A: $tType] :
( ( algebraic_semidom @ A )
=> ! [A2: A,B2: A,C2: A] :
( ( dvd_dvd @ A @ A2 @ ( one_one @ A ) )
=> ( ( dvd_dvd @ A @ ( times_times @ A @ A2 @ B2 ) @ C2 )
= ( dvd_dvd @ A @ B2 @ C2 ) ) ) ) ).
% mult_unit_dvd_iff'
thf(fact_2346_dvd__mult__unit__iff_H,axiom,
! [A: $tType] :
( ( algebraic_semidom @ A )
=> ! [B2: A,A2: A,C2: A] :
( ( dvd_dvd @ A @ B2 @ ( one_one @ A ) )
=> ( ( dvd_dvd @ A @ A2 @ ( times_times @ A @ B2 @ C2 ) )
= ( dvd_dvd @ A @ A2 @ C2 ) ) ) ) ).
% dvd_mult_unit_iff'
thf(fact_2347_mult__unit__dvd__iff,axiom,
! [A: $tType] :
( ( algebraic_semidom @ A )
=> ! [B2: A,A2: A,C2: A] :
( ( dvd_dvd @ A @ B2 @ ( one_one @ A ) )
=> ( ( dvd_dvd @ A @ ( times_times @ A @ A2 @ B2 ) @ C2 )
= ( dvd_dvd @ A @ A2 @ C2 ) ) ) ) ).
% mult_unit_dvd_iff
thf(fact_2348_dvd__mult__unit__iff,axiom,
! [A: $tType] :
( ( algebraic_semidom @ A )
=> ! [B2: A,A2: A,C2: A] :
( ( dvd_dvd @ A @ B2 @ ( one_one @ A ) )
=> ( ( dvd_dvd @ A @ A2 @ ( times_times @ A @ C2 @ B2 ) )
= ( dvd_dvd @ A @ A2 @ C2 ) ) ) ) ).
% dvd_mult_unit_iff
thf(fact_2349_is__unit__mult__iff,axiom,
! [A: $tType] :
( ( algebraic_semidom @ A )
=> ! [A2: A,B2: A] :
( ( dvd_dvd @ A @ ( times_times @ A @ A2 @ B2 ) @ ( one_one @ A ) )
= ( ( dvd_dvd @ A @ A2 @ ( one_one @ A ) )
& ( dvd_dvd @ A @ B2 @ ( one_one @ A ) ) ) ) ) ).
% is_unit_mult_iff
thf(fact_2350_mod__Suc,axiom,
! [M: nat,N3: nat] :
( ( ( ( suc @ ( modulo_modulo @ nat @ M @ N3 ) )
= N3 )
=> ( ( modulo_modulo @ nat @ ( suc @ M ) @ N3 )
= ( zero_zero @ nat ) ) )
& ( ( ( suc @ ( modulo_modulo @ nat @ M @ N3 ) )
!= N3 )
=> ( ( modulo_modulo @ nat @ ( suc @ M ) @ N3 )
= ( suc @ ( modulo_modulo @ nat @ M @ N3 ) ) ) ) ) ).
% mod_Suc
thf(fact_2351_mod__induct,axiom,
! [P: nat > $o,N3: nat,P4: nat,M: nat] :
( ( P @ N3 )
=> ( ( ord_less @ nat @ N3 @ P4 )
=> ( ( ord_less @ nat @ M @ P4 )
=> ( ! [N: nat] :
( ( ord_less @ nat @ N @ P4 )
=> ( ( P @ N )
=> ( P @ ( modulo_modulo @ nat @ ( suc @ N ) @ P4 ) ) ) )
=> ( P @ M ) ) ) ) ) ).
% mod_induct
thf(fact_2352_div__plus__div__distrib__dvd__right,axiom,
! [A: $tType] :
( ( euclid4440199948858584721cancel @ A )
=> ! [C2: A,B2: A,A2: A] :
( ( dvd_dvd @ A @ C2 @ B2 )
=> ( ( divide_divide @ A @ ( plus_plus @ A @ A2 @ B2 ) @ C2 )
= ( plus_plus @ A @ ( divide_divide @ A @ A2 @ C2 ) @ ( divide_divide @ A @ B2 @ C2 ) ) ) ) ) ).
% div_plus_div_distrib_dvd_right
thf(fact_2353_div__plus__div__distrib__dvd__left,axiom,
! [A: $tType] :
( ( euclid4440199948858584721cancel @ A )
=> ! [C2: A,A2: A,B2: A] :
( ( dvd_dvd @ A @ C2 @ A2 )
=> ( ( divide_divide @ A @ ( plus_plus @ A @ A2 @ B2 ) @ C2 )
= ( plus_plus @ A @ ( divide_divide @ A @ A2 @ C2 ) @ ( divide_divide @ A @ B2 @ C2 ) ) ) ) ) ).
% div_plus_div_distrib_dvd_left
thf(fact_2354_dvd__div__unit__iff,axiom,
! [A: $tType] :
( ( algebraic_semidom @ A )
=> ! [B2: A,A2: A,C2: A] :
( ( dvd_dvd @ A @ B2 @ ( one_one @ A ) )
=> ( ( dvd_dvd @ A @ A2 @ ( divide_divide @ A @ C2 @ B2 ) )
= ( dvd_dvd @ A @ A2 @ C2 ) ) ) ) ).
% dvd_div_unit_iff
thf(fact_2355_div__unit__dvd__iff,axiom,
! [A: $tType] :
( ( algebraic_semidom @ A )
=> ! [B2: A,A2: A,C2: A] :
( ( dvd_dvd @ A @ B2 @ ( one_one @ A ) )
=> ( ( dvd_dvd @ A @ ( divide_divide @ A @ A2 @ B2 ) @ C2 )
= ( dvd_dvd @ A @ A2 @ C2 ) ) ) ) ).
% div_unit_dvd_iff
thf(fact_2356_unit__div__cancel,axiom,
! [A: $tType] :
( ( algebraic_semidom @ A )
=> ! [A2: A,B2: A,C2: A] :
( ( dvd_dvd @ A @ A2 @ ( one_one @ A ) )
=> ( ( ( divide_divide @ A @ B2 @ A2 )
= ( divide_divide @ A @ C2 @ A2 ) )
= ( B2 = C2 ) ) ) ) ).
% unit_div_cancel
thf(fact_2357_div__mult__div__if__dvd,axiom,
! [A: $tType] :
( ( algebraic_semidom @ A )
=> ! [B2: A,A2: A,D: A,C2: A] :
( ( dvd_dvd @ A @ B2 @ A2 )
=> ( ( dvd_dvd @ A @ D @ C2 )
=> ( ( times_times @ A @ ( divide_divide @ A @ A2 @ B2 ) @ ( divide_divide @ A @ C2 @ D ) )
= ( divide_divide @ A @ ( times_times @ A @ A2 @ C2 ) @ ( times_times @ A @ B2 @ D ) ) ) ) ) ) ).
% div_mult_div_if_dvd
thf(fact_2358_dvd__mult__imp__div,axiom,
! [A: $tType] :
( ( algebraic_semidom @ A )
=> ! [A2: A,C2: A,B2: A] :
( ( dvd_dvd @ A @ ( times_times @ A @ A2 @ C2 ) @ B2 )
=> ( dvd_dvd @ A @ A2 @ ( divide_divide @ A @ B2 @ C2 ) ) ) ) ).
% dvd_mult_imp_div
thf(fact_2359_dvd__div__mult2__eq,axiom,
! [A: $tType] :
( ( algebraic_semidom @ A )
=> ! [B2: A,C2: A,A2: A] :
( ( dvd_dvd @ A @ ( times_times @ A @ B2 @ C2 ) @ A2 )
=> ( ( divide_divide @ A @ A2 @ ( times_times @ A @ B2 @ C2 ) )
= ( divide_divide @ A @ ( divide_divide @ A @ A2 @ B2 ) @ C2 ) ) ) ) ).
% dvd_div_mult2_eq
thf(fact_2360_div__div__eq__right,axiom,
! [A: $tType] :
( ( algebraic_semidom @ A )
=> ! [C2: A,B2: A,A2: A] :
( ( dvd_dvd @ A @ C2 @ B2 )
=> ( ( dvd_dvd @ A @ B2 @ A2 )
=> ( ( divide_divide @ A @ A2 @ ( divide_divide @ A @ B2 @ C2 ) )
= ( times_times @ A @ ( divide_divide @ A @ A2 @ B2 ) @ C2 ) ) ) ) ) ).
% div_div_eq_right
thf(fact_2361_div__mult__swap,axiom,
! [A: $tType] :
( ( algebraic_semidom @ A )
=> ! [C2: A,B2: A,A2: A] :
( ( dvd_dvd @ A @ C2 @ B2 )
=> ( ( times_times @ A @ A2 @ ( divide_divide @ A @ B2 @ C2 ) )
= ( divide_divide @ A @ ( times_times @ A @ A2 @ B2 ) @ C2 ) ) ) ) ).
% div_mult_swap
thf(fact_2362_dvd__div__mult,axiom,
! [A: $tType] :
( ( algebraic_semidom @ A )
=> ! [C2: A,B2: A,A2: A] :
( ( dvd_dvd @ A @ C2 @ B2 )
=> ( ( times_times @ A @ ( divide_divide @ A @ B2 @ C2 ) @ A2 )
= ( divide_divide @ A @ ( times_times @ A @ B2 @ A2 ) @ C2 ) ) ) ) ).
% dvd_div_mult
thf(fact_2363_nat__mod__lem,axiom,
! [N3: nat,B2: nat] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ N3 )
=> ( ( ord_less @ nat @ B2 @ N3 )
= ( ( modulo_modulo @ nat @ B2 @ N3 )
= B2 ) ) ) ).
% nat_mod_lem
thf(fact_2364_mod__less__divisor,axiom,
! [N3: nat,M: nat] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ N3 )
=> ( ord_less @ nat @ ( modulo_modulo @ nat @ M @ N3 ) @ N3 ) ) ).
% mod_less_divisor
thf(fact_2365_mod__Suc__le__divisor,axiom,
! [M: nat,N3: nat] : ( ord_less_eq @ nat @ ( modulo_modulo @ nat @ M @ ( suc @ N3 ) ) @ N3 ) ).
% mod_Suc_le_divisor
thf(fact_2366_div__power,axiom,
! [A: $tType] :
( ( algebraic_semidom @ A )
=> ! [B2: A,A2: A,N3: nat] :
( ( dvd_dvd @ A @ B2 @ A2 )
=> ( ( power_power @ A @ ( divide_divide @ A @ A2 @ B2 ) @ N3 )
= ( divide_divide @ A @ ( power_power @ A @ A2 @ N3 ) @ ( power_power @ A @ B2 @ N3 ) ) ) ) ) ).
% div_power
thf(fact_2367_word__rot__lem,axiom,
! [L2: nat,K: nat,D: nat,N3: nat] :
( ( ( plus_plus @ nat @ L2 @ K )
= ( plus_plus @ nat @ D @ ( modulo_modulo @ nat @ K @ L2 ) ) )
=> ( ( ord_less @ nat @ N3 @ L2 )
=> ( ( modulo_modulo @ nat @ ( plus_plus @ nat @ D @ N3 ) @ L2 )
= N3 ) ) ) ).
% word_rot_lem
thf(fact_2368_nat__minus__mod,axiom,
! [N3: nat,M: nat] :
( ( modulo_modulo @ nat @ ( minus_minus @ nat @ N3 @ ( modulo_modulo @ nat @ N3 @ M ) ) @ M )
= ( zero_zero @ nat ) ) ).
% nat_minus_mod
thf(fact_2369_mod__if,axiom,
( ( modulo_modulo @ nat )
= ( ^ [M5: nat,N2: nat] : ( if @ nat @ ( ord_less @ nat @ M5 @ N2 ) @ M5 @ ( modulo_modulo @ nat @ ( minus_minus @ nat @ M5 @ N2 ) @ N2 ) ) ) ) ).
% mod_if
thf(fact_2370_mod__nat__sub,axiom,
! [X: nat,Z: nat,Y: nat] :
( ( ord_less @ nat @ X @ Z )
=> ( ( modulo_modulo @ nat @ ( minus_minus @ nat @ X @ Y ) @ Z )
= ( minus_minus @ nat @ X @ Y ) ) ) ).
% mod_nat_sub
thf(fact_2371_mod__geq,axiom,
! [M: nat,N3: nat] :
( ~ ( ord_less @ nat @ M @ N3 )
=> ( ( modulo_modulo @ nat @ M @ N3 )
= ( modulo_modulo @ nat @ ( minus_minus @ nat @ M @ N3 ) @ N3 ) ) ) ).
% mod_geq
thf(fact_2372_dvd__power__le,axiom,
! [A: $tType] :
( ( comm_semiring_1 @ A )
=> ! [X: A,Y: A,N3: nat,M: nat] :
( ( dvd_dvd @ A @ X @ Y )
=> ( ( ord_less_eq @ nat @ N3 @ M )
=> ( dvd_dvd @ A @ ( power_power @ A @ X @ N3 ) @ ( power_power @ A @ Y @ M ) ) ) ) ) ).
% dvd_power_le
thf(fact_2373_power__le__dvd,axiom,
! [A: $tType] :
( ( comm_semiring_1 @ A )
=> ! [A2: A,N3: nat,B2: A,M: nat] :
( ( dvd_dvd @ A @ ( power_power @ A @ A2 @ N3 ) @ B2 )
=> ( ( ord_less_eq @ nat @ M @ N3 )
=> ( dvd_dvd @ A @ ( power_power @ A @ A2 @ M ) @ B2 ) ) ) ) ).
% power_le_dvd
thf(fact_2374_le__imp__power__dvd,axiom,
! [A: $tType] :
( ( comm_semiring_1 @ A )
=> ! [M: nat,N3: nat,A2: A] :
( ( ord_less_eq @ nat @ M @ N3 )
=> ( dvd_dvd @ A @ ( power_power @ A @ A2 @ M ) @ ( power_power @ A @ A2 @ N3 ) ) ) ) ).
% le_imp_power_dvd
thf(fact_2375_mod__eq__0D,axiom,
! [M: nat,D: nat] :
( ( ( modulo_modulo @ nat @ M @ D )
= ( zero_zero @ nat ) )
=> ? [Q3: nat] :
( M
= ( times_times @ nat @ D @ Q3 ) ) ) ).
% mod_eq_0D
thf(fact_2376_nat__minus__mod__plus__right,axiom,
! [N3: nat,X: nat,M: nat] :
( ( modulo_modulo @ nat @ ( minus_minus @ nat @ ( plus_plus @ nat @ N3 @ X ) @ ( modulo_modulo @ nat @ N3 @ M ) ) @ M )
= ( modulo_modulo @ nat @ X @ M ) ) ).
% nat_minus_mod_plus_right
thf(fact_2377_le__mod__geq,axiom,
! [N3: nat,M: nat] :
( ( ord_less_eq @ nat @ N3 @ M )
=> ( ( modulo_modulo @ nat @ M @ N3 )
= ( modulo_modulo @ nat @ ( minus_minus @ nat @ M @ N3 ) @ N3 ) ) ) ).
% le_mod_geq
thf(fact_2378_msrevs_I2_J,axiom,
! [K: nat,N3: nat,M: nat] :
( ( modulo_modulo @ nat @ ( plus_plus @ nat @ ( times_times @ nat @ K @ N3 ) @ M ) @ N3 )
= ( modulo_modulo @ nat @ M @ N3 ) ) ).
% msrevs(2)
thf(fact_2379_nat__mod__eq__iff,axiom,
! [X: nat,N3: nat,Y: nat] :
( ( ( modulo_modulo @ nat @ X @ N3 )
= ( modulo_modulo @ nat @ Y @ N3 ) )
= ( ? [Q1: nat,Q22: nat] :
( ( plus_plus @ nat @ X @ ( times_times @ nat @ N3 @ Q1 ) )
= ( plus_plus @ nat @ Y @ ( times_times @ nat @ N3 @ Q22 ) ) ) ) ) ).
% nat_mod_eq_iff
thf(fact_2380_nat__dvd__not__less,axiom,
! [M: nat,N3: nat] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ M )
=> ( ( ord_less @ nat @ M @ N3 )
=> ~ ( dvd_dvd @ nat @ N3 @ M ) ) ) ).
% nat_dvd_not_less
thf(fact_2381_dvd__minus__self,axiom,
! [M: nat,N3: nat] :
( ( dvd_dvd @ nat @ M @ ( minus_minus @ nat @ N3 @ M ) )
= ( ( ord_less @ nat @ N3 @ M )
| ( dvd_dvd @ nat @ M @ N3 ) ) ) ).
% dvd_minus_self
thf(fact_2382_dvd__diffD,axiom,
! [K: nat,M: nat,N3: nat] :
( ( dvd_dvd @ nat @ K @ ( minus_minus @ nat @ M @ N3 ) )
=> ( ( dvd_dvd @ nat @ K @ N3 )
=> ( ( ord_less_eq @ nat @ N3 @ M )
=> ( dvd_dvd @ nat @ K @ M ) ) ) ) ).
% dvd_diffD
thf(fact_2383_dvd__diffD1,axiom,
! [K: nat,M: nat,N3: nat] :
( ( dvd_dvd @ nat @ K @ ( minus_minus @ nat @ M @ N3 ) )
=> ( ( dvd_dvd @ nat @ K @ M )
=> ( ( ord_less_eq @ nat @ N3 @ M )
=> ( dvd_dvd @ nat @ K @ N3 ) ) ) ) ).
% dvd_diffD1
thf(fact_2384_less__eq__dvd__minus,axiom,
! [M: nat,N3: nat] :
( ( ord_less_eq @ nat @ M @ N3 )
=> ( ( dvd_dvd @ nat @ M @ N3 )
= ( dvd_dvd @ nat @ M @ ( minus_minus @ nat @ N3 @ M ) ) ) ) ).
% less_eq_dvd_minus
thf(fact_2385_parity__cases,axiom,
! [A: $tType] :
( ( semiring_parity @ A )
=> ! [A2: A] :
( ( ( dvd_dvd @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ A2 )
=> ( ( modulo_modulo @ A @ A2 @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) )
!= ( zero_zero @ A ) ) )
=> ~ ( ~ ( dvd_dvd @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ A2 )
=> ( ( modulo_modulo @ A @ A2 @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) )
!= ( one_one @ A ) ) ) ) ) ).
% parity_cases
thf(fact_2386_mod2__eq__if,axiom,
! [A: $tType] :
( ( semiring_parity @ A )
=> ! [A2: A] :
( ( ( dvd_dvd @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ A2 )
=> ( ( modulo_modulo @ A @ A2 @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) )
= ( zero_zero @ A ) ) )
& ( ~ ( dvd_dvd @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ A2 )
=> ( ( modulo_modulo @ A @ A2 @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) )
= ( one_one @ A ) ) ) ) ) ).
% mod2_eq_if
thf(fact_2387_even__unset__bit__iff,axiom,
! [A: $tType] :
( ( bit_se359711467146920520ations @ A )
=> ! [M: nat,A2: A] :
( ( dvd_dvd @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ ( bit_se2638667681897837118et_bit @ A @ M @ A2 ) )
= ( ( dvd_dvd @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ A2 )
| ( M
= ( zero_zero @ nat ) ) ) ) ) ).
% even_unset_bit_iff
thf(fact_2388_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_2389_unique__euclidean__semiring__numeral__class_Omod__less,axiom,
! [A: $tType] :
( ( unique1627219031080169319umeral @ A )
=> ! [A2: A,B2: A] :
( ( ord_less_eq @ A @ ( zero_zero @ A ) @ A2 )
=> ( ( ord_less @ A @ A2 @ B2 )
=> ( ( modulo_modulo @ A @ A2 @ B2 )
= A2 ) ) ) ) ).
% unique_euclidean_semiring_numeral_class.mod_less
thf(fact_2390_unique__euclidean__semiring__numeral__class_Opos__mod__sign,axiom,
! [A: $tType] :
( ( unique1627219031080169319umeral @ A )
=> ! [B2: A,A2: A] :
( ( ord_less @ A @ ( zero_zero @ A ) @ B2 )
=> ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( modulo_modulo @ A @ A2 @ B2 ) ) ) ) ).
% unique_euclidean_semiring_numeral_class.pos_mod_sign
thf(fact_2391_cong__exp__iff__simps_I2_J,axiom,
! [A: $tType] :
( ( unique1627219031080169319umeral @ A )
=> ! [N3: num,Q2: num] :
( ( ( modulo_modulo @ A @ ( numeral_numeral @ A @ ( bit0 @ N3 ) ) @ ( numeral_numeral @ A @ ( bit0 @ Q2 ) ) )
= ( zero_zero @ A ) )
= ( ( modulo_modulo @ A @ ( numeral_numeral @ A @ N3 ) @ ( numeral_numeral @ A @ Q2 ) )
= ( zero_zero @ A ) ) ) ) ).
% cong_exp_iff_simps(2)
thf(fact_2392_cong__exp__iff__simps_I1_J,axiom,
! [A: $tType] :
( ( unique1627219031080169319umeral @ A )
=> ! [N3: num] :
( ( modulo_modulo @ A @ ( numeral_numeral @ A @ N3 ) @ ( numeral_numeral @ A @ one2 ) )
= ( zero_zero @ A ) ) ) ).
% cong_exp_iff_simps(1)
thf(fact_2393_cong__exp__iff__simps_I6_J,axiom,
! [A: $tType] :
( ( unique1627219031080169319umeral @ A )
=> ! [Q2: num,N3: num] :
( ( modulo_modulo @ A @ ( numeral_numeral @ A @ one2 ) @ ( numeral_numeral @ A @ ( bit0 @ Q2 ) ) )
!= ( modulo_modulo @ A @ ( numeral_numeral @ A @ ( bit0 @ N3 ) ) @ ( numeral_numeral @ A @ ( bit0 @ Q2 ) ) ) ) ) ).
% cong_exp_iff_simps(6)
thf(fact_2394_cong__exp__iff__simps_I8_J,axiom,
! [A: $tType] :
( ( unique1627219031080169319umeral @ A )
=> ! [M: num,Q2: num] :
( ( modulo_modulo @ A @ ( numeral_numeral @ A @ ( bit0 @ M ) ) @ ( numeral_numeral @ A @ ( bit0 @ Q2 ) ) )
!= ( modulo_modulo @ A @ ( numeral_numeral @ A @ one2 ) @ ( numeral_numeral @ A @ ( bit0 @ Q2 ) ) ) ) ) ).
% cong_exp_iff_simps(8)
thf(fact_2395_cong__exp__iff__simps_I10_J,axiom,
! [A: $tType] :
( ( unique1627219031080169319umeral @ A )
=> ! [M: num,Q2: num,N3: num] :
( ( modulo_modulo @ A @ ( numeral_numeral @ A @ ( bit0 @ M ) ) @ ( numeral_numeral @ A @ ( bit0 @ Q2 ) ) )
!= ( modulo_modulo @ A @ ( numeral_numeral @ A @ ( bit1 @ N3 ) ) @ ( numeral_numeral @ A @ ( bit0 @ Q2 ) ) ) ) ) ).
% cong_exp_iff_simps(10)
thf(fact_2396_cong__exp__iff__simps_I12_J,axiom,
! [A: $tType] :
( ( unique1627219031080169319umeral @ A )
=> ! [M: num,Q2: num,N3: num] :
( ( modulo_modulo @ A @ ( numeral_numeral @ A @ ( bit1 @ M ) ) @ ( numeral_numeral @ A @ ( bit0 @ Q2 ) ) )
!= ( modulo_modulo @ A @ ( numeral_numeral @ A @ ( bit0 @ N3 ) ) @ ( numeral_numeral @ A @ ( bit0 @ Q2 ) ) ) ) ) ).
% cong_exp_iff_simps(12)
thf(fact_2397_cong__exp__iff__simps_I13_J,axiom,
! [A: $tType] :
( ( unique1627219031080169319umeral @ A )
=> ! [M: num,Q2: num,N3: num] :
( ( ( modulo_modulo @ A @ ( numeral_numeral @ A @ ( bit1 @ M ) ) @ ( numeral_numeral @ A @ ( bit0 @ Q2 ) ) )
= ( modulo_modulo @ A @ ( numeral_numeral @ A @ ( bit1 @ N3 ) ) @ ( numeral_numeral @ A @ ( bit0 @ Q2 ) ) ) )
= ( ( modulo_modulo @ A @ ( numeral_numeral @ A @ M ) @ ( numeral_numeral @ A @ Q2 ) )
= ( modulo_modulo @ A @ ( numeral_numeral @ A @ N3 ) @ ( numeral_numeral @ A @ Q2 ) ) ) ) ) ).
% cong_exp_iff_simps(13)
thf(fact_2398_div__mult1__eq,axiom,
! [A: $tType] :
( ( euclid3128863361964157862miring @ A )
=> ! [A2: A,B2: A,C2: A] :
( ( divide_divide @ A @ ( times_times @ A @ A2 @ B2 ) @ C2 )
= ( plus_plus @ A @ ( times_times @ A @ A2 @ ( divide_divide @ A @ B2 @ C2 ) ) @ ( divide_divide @ A @ ( times_times @ A @ A2 @ ( modulo_modulo @ A @ B2 @ C2 ) ) @ C2 ) ) ) ) ).
% div_mult1_eq
thf(fact_2399_mult__div__mod__eq,axiom,
! [A: $tType] :
( ( semiring_modulo @ A )
=> ! [B2: A,A2: A] :
( ( plus_plus @ A @ ( times_times @ A @ B2 @ ( divide_divide @ A @ A2 @ B2 ) ) @ ( modulo_modulo @ A @ A2 @ B2 ) )
= A2 ) ) ).
% mult_div_mod_eq
thf(fact_2400_mod__mult__div__eq,axiom,
! [A: $tType] :
( ( semiring_modulo @ A )
=> ! [A2: A,B2: A] :
( ( plus_plus @ A @ ( modulo_modulo @ A @ A2 @ B2 ) @ ( times_times @ A @ B2 @ ( divide_divide @ A @ A2 @ B2 ) ) )
= A2 ) ) ).
% mod_mult_div_eq
thf(fact_2401_mod__div__mult__eq,axiom,
! [A: $tType] :
( ( semiring_modulo @ A )
=> ! [A2: A,B2: A] :
( ( plus_plus @ A @ ( modulo_modulo @ A @ A2 @ B2 ) @ ( times_times @ A @ ( divide_divide @ A @ A2 @ B2 ) @ B2 ) )
= A2 ) ) ).
% mod_div_mult_eq
thf(fact_2402_div__mult__mod__eq,axiom,
! [A: $tType] :
( ( semiring_modulo @ A )
=> ! [A2: A,B2: A] :
( ( plus_plus @ A @ ( times_times @ A @ ( divide_divide @ A @ A2 @ B2 ) @ B2 ) @ ( modulo_modulo @ A @ A2 @ B2 ) )
= A2 ) ) ).
% div_mult_mod_eq
thf(fact_2403_mod__div__decomp,axiom,
! [A: $tType] :
( ( semiring_modulo @ A )
=> ! [A2: A,B2: A] :
( A2
= ( plus_plus @ A @ ( times_times @ A @ ( divide_divide @ A @ A2 @ B2 ) @ B2 ) @ ( modulo_modulo @ A @ A2 @ B2 ) ) ) ) ).
% mod_div_decomp
thf(fact_2404_cancel__div__mod__rules_I1_J,axiom,
! [A: $tType] :
( ( semidom_modulo @ A )
=> ! [A2: A,B2: A,C2: A] :
( ( plus_plus @ A @ ( plus_plus @ A @ ( times_times @ A @ ( divide_divide @ A @ A2 @ B2 ) @ B2 ) @ ( modulo_modulo @ A @ A2 @ B2 ) ) @ C2 )
= ( plus_plus @ A @ A2 @ C2 ) ) ) ).
% cancel_div_mod_rules(1)
thf(fact_2405_cancel__div__mod__rules_I2_J,axiom,
! [A: $tType] :
( ( semidom_modulo @ A )
=> ! [B2: A,A2: A,C2: A] :
( ( plus_plus @ A @ ( plus_plus @ A @ ( times_times @ A @ B2 @ ( divide_divide @ A @ A2 @ B2 ) ) @ ( modulo_modulo @ A @ A2 @ B2 ) ) @ C2 )
= ( plus_plus @ A @ A2 @ C2 ) ) ) ).
% cancel_div_mod_rules(2)
thf(fact_2406_zmde,axiom,
! [A: $tType] :
( ( ( group_add @ A )
& ( semiring_modulo @ A ) )
=> ! [B2: A,A2: A] :
( ( times_times @ A @ B2 @ ( divide_divide @ A @ A2 @ B2 ) )
= ( minus_minus @ A @ A2 @ ( modulo_modulo @ A @ A2 @ B2 ) ) ) ) ).
% zmde
thf(fact_2407_minus__mult__div__eq__mod,axiom,
! [A: $tType] :
( ( semiring_modulo @ A )
=> ! [A2: A,B2: A] :
( ( minus_minus @ A @ A2 @ ( times_times @ A @ B2 @ ( divide_divide @ A @ A2 @ B2 ) ) )
= ( modulo_modulo @ A @ A2 @ B2 ) ) ) ).
% minus_mult_div_eq_mod
thf(fact_2408_minus__mod__eq__mult__div,axiom,
! [A: $tType] :
( ( semiring_modulo @ A )
=> ! [A2: A,B2: A] :
( ( minus_minus @ A @ A2 @ ( modulo_modulo @ A @ A2 @ B2 ) )
= ( times_times @ A @ B2 @ ( divide_divide @ A @ A2 @ B2 ) ) ) ) ).
% minus_mod_eq_mult_div
thf(fact_2409_minus__mod__eq__div__mult,axiom,
! [A: $tType] :
( ( semiring_modulo @ A )
=> ! [A2: A,B2: A] :
( ( minus_minus @ A @ A2 @ ( modulo_modulo @ A @ A2 @ B2 ) )
= ( times_times @ A @ ( divide_divide @ A @ A2 @ B2 ) @ B2 ) ) ) ).
% minus_mod_eq_div_mult
thf(fact_2410_minus__div__mult__eq__mod,axiom,
! [A: $tType] :
( ( semiring_modulo @ A )
=> ! [A2: A,B2: A] :
( ( minus_minus @ A @ A2 @ ( times_times @ A @ ( divide_divide @ A @ A2 @ B2 ) @ B2 ) )
= ( modulo_modulo @ A @ A2 @ B2 ) ) ) ).
% minus_div_mult_eq_mod
thf(fact_2411_even__numeral,axiom,
! [A: $tType] :
( ( semiring_parity @ A )
=> ! [N3: num] : ( dvd_dvd @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ ( numeral_numeral @ A @ ( bit0 @ N3 ) ) ) ) ).
% even_numeral
thf(fact_2412_unity__coeff__ex,axiom,
! [A: $tType] :
( ( ( dvd @ A )
& ( semiring_0 @ A ) )
=> ! [P: A > $o,L2: A] :
( ( ? [X2: A] : ( P @ ( times_times @ A @ L2 @ X2 ) ) )
= ( ? [X2: A] :
( ( dvd_dvd @ A @ L2 @ ( plus_plus @ A @ X2 @ ( zero_zero @ A ) ) )
& ( P @ X2 ) ) ) ) ) ).
% unity_coeff_ex
thf(fact_2413_unit__dvdE,axiom,
! [A: $tType] :
( ( algebraic_semidom @ A )
=> ! [A2: A,B2: A] :
( ( dvd_dvd @ A @ A2 @ ( one_one @ A ) )
=> ~ ( ( A2
!= ( zero_zero @ A ) )
=> ! [C4: A] :
( B2
!= ( times_times @ A @ A2 @ C4 ) ) ) ) ) ).
% unit_dvdE
thf(fact_2414_unit__div__eq__0__iff,axiom,
! [A: $tType] :
( ( algebraic_semidom @ A )
=> ! [B2: A,A2: A] :
( ( dvd_dvd @ A @ B2 @ ( one_one @ A ) )
=> ( ( ( divide_divide @ A @ A2 @ B2 )
= ( zero_zero @ A ) )
= ( A2
= ( zero_zero @ A ) ) ) ) ) ).
% unit_div_eq_0_iff
thf(fact_2415_dvd__div__div__eq__mult,axiom,
! [A: $tType] :
( ( algebraic_semidom @ A )
=> ! [A2: A,C2: A,B2: A,D: A] :
( ( A2
!= ( zero_zero @ A ) )
=> ( ( C2
!= ( zero_zero @ A ) )
=> ( ( dvd_dvd @ A @ A2 @ B2 )
=> ( ( dvd_dvd @ A @ C2 @ D )
=> ( ( ( divide_divide @ A @ B2 @ A2 )
= ( divide_divide @ A @ D @ C2 ) )
= ( ( times_times @ A @ B2 @ C2 )
= ( times_times @ A @ A2 @ D ) ) ) ) ) ) ) ) ).
% dvd_div_div_eq_mult
thf(fact_2416_dvd__div__iff__mult,axiom,
! [A: $tType] :
( ( algebraic_semidom @ A )
=> ! [C2: A,B2: A,A2: A] :
( ( C2
!= ( zero_zero @ A ) )
=> ( ( dvd_dvd @ A @ C2 @ B2 )
=> ( ( dvd_dvd @ A @ A2 @ ( divide_divide @ A @ B2 @ C2 ) )
= ( dvd_dvd @ A @ ( times_times @ A @ A2 @ C2 ) @ B2 ) ) ) ) ) ).
% dvd_div_iff_mult
thf(fact_2417_div__dvd__iff__mult,axiom,
! [A: $tType] :
( ( algebraic_semidom @ A )
=> ! [B2: A,A2: A,C2: A] :
( ( B2
!= ( zero_zero @ A ) )
=> ( ( dvd_dvd @ A @ B2 @ A2 )
=> ( ( dvd_dvd @ A @ ( divide_divide @ A @ A2 @ B2 ) @ C2 )
= ( dvd_dvd @ A @ A2 @ ( times_times @ A @ C2 @ B2 ) ) ) ) ) ) ).
% div_dvd_iff_mult
thf(fact_2418_dvd__div__eq__mult,axiom,
! [A: $tType] :
( ( algebraic_semidom @ A )
=> ! [A2: A,B2: A,C2: A] :
( ( A2
!= ( zero_zero @ A ) )
=> ( ( dvd_dvd @ A @ A2 @ B2 )
=> ( ( ( divide_divide @ A @ B2 @ A2 )
= C2 )
= ( B2
= ( times_times @ A @ C2 @ A2 ) ) ) ) ) ) ).
% dvd_div_eq_mult
thf(fact_2419_inf__period_I3_J,axiom,
! [A: $tType] :
( ( ( comm_ring @ A )
& ( dvd @ A ) )
=> ! [D: A,D4: A,T2: A] :
( ( dvd_dvd @ A @ D @ D4 )
=> ! [X5: A,K5: A] :
( ( dvd_dvd @ A @ D @ ( plus_plus @ A @ X5 @ T2 ) )
= ( dvd_dvd @ A @ D @ ( plus_plus @ A @ ( minus_minus @ A @ X5 @ ( times_times @ A @ K5 @ D4 ) ) @ T2 ) ) ) ) ) ).
% inf_period(3)
thf(fact_2420_inf__period_I4_J,axiom,
! [A: $tType] :
( ( ( comm_ring @ A )
& ( dvd @ A ) )
=> ! [D: A,D4: A,T2: A] :
( ( dvd_dvd @ A @ D @ D4 )
=> ! [X5: A,K5: A] :
( ( ~ ( dvd_dvd @ A @ D @ ( plus_plus @ A @ X5 @ T2 ) ) )
= ( ~ ( dvd_dvd @ A @ D @ ( plus_plus @ A @ ( minus_minus @ A @ X5 @ ( times_times @ A @ K5 @ D4 ) ) @ T2 ) ) ) ) ) ) ).
% inf_period(4)
thf(fact_2421_is__unit__div__mult2__eq,axiom,
! [A: $tType] :
( ( algebraic_semidom @ A )
=> ! [B2: A,C2: A,A2: A] :
( ( dvd_dvd @ A @ B2 @ ( one_one @ A ) )
=> ( ( dvd_dvd @ A @ C2 @ ( one_one @ A ) )
=> ( ( divide_divide @ A @ A2 @ ( times_times @ A @ B2 @ C2 ) )
= ( divide_divide @ A @ ( divide_divide @ A @ A2 @ B2 ) @ C2 ) ) ) ) ) ).
% is_unit_div_mult2_eq
thf(fact_2422_unit__div__mult__swap,axiom,
! [A: $tType] :
( ( algebraic_semidom @ A )
=> ! [C2: A,A2: A,B2: A] :
( ( dvd_dvd @ A @ C2 @ ( one_one @ A ) )
=> ( ( times_times @ A @ A2 @ ( divide_divide @ A @ B2 @ C2 ) )
= ( divide_divide @ A @ ( times_times @ A @ A2 @ B2 ) @ C2 ) ) ) ) ).
% unit_div_mult_swap
thf(fact_2423_unit__div__commute,axiom,
! [A: $tType] :
( ( algebraic_semidom @ A )
=> ! [B2: A,A2: A,C2: A] :
( ( dvd_dvd @ A @ B2 @ ( one_one @ A ) )
=> ( ( times_times @ A @ ( divide_divide @ A @ A2 @ B2 ) @ C2 )
= ( divide_divide @ A @ ( times_times @ A @ A2 @ C2 ) @ B2 ) ) ) ) ).
% unit_div_commute
thf(fact_2424_div__mult__unit2,axiom,
! [A: $tType] :
( ( algebraic_semidom @ A )
=> ! [C2: A,B2: A,A2: A] :
( ( dvd_dvd @ A @ C2 @ ( one_one @ A ) )
=> ( ( dvd_dvd @ A @ B2 @ A2 )
=> ( ( divide_divide @ A @ A2 @ ( times_times @ A @ B2 @ C2 ) )
= ( divide_divide @ A @ ( divide_divide @ A @ A2 @ B2 ) @ C2 ) ) ) ) ) ).
% div_mult_unit2
thf(fact_2425_unit__eq__div2,axiom,
! [A: $tType] :
( ( algebraic_semidom @ A )
=> ! [B2: A,A2: A,C2: A] :
( ( dvd_dvd @ A @ B2 @ ( one_one @ A ) )
=> ( ( A2
= ( divide_divide @ A @ C2 @ B2 ) )
= ( ( times_times @ A @ A2 @ B2 )
= C2 ) ) ) ) ).
% unit_eq_div2
thf(fact_2426_unit__eq__div1,axiom,
! [A: $tType] :
( ( algebraic_semidom @ A )
=> ! [B2: A,A2: A,C2: A] :
( ( dvd_dvd @ A @ B2 @ ( one_one @ A ) )
=> ( ( ( divide_divide @ A @ A2 @ B2 )
= C2 )
= ( A2
= ( times_times @ A @ C2 @ B2 ) ) ) ) ) ).
% unit_eq_div1
thf(fact_2427_is__unit__power__iff,axiom,
! [A: $tType] :
( ( algebraic_semidom @ A )
=> ! [A2: A,N3: nat] :
( ( dvd_dvd @ A @ ( power_power @ A @ A2 @ N3 ) @ ( one_one @ A ) )
= ( ( dvd_dvd @ A @ A2 @ ( one_one @ A ) )
| ( N3
= ( zero_zero @ nat ) ) ) ) ) ).
% is_unit_power_iff
thf(fact_2428_mod__le__divisor,axiom,
! [N3: nat,M: nat] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ N3 )
=> ( ord_less_eq @ nat @ ( modulo_modulo @ nat @ M @ N3 ) @ N3 ) ) ).
% mod_le_divisor
thf(fact_2429_div__less__mono,axiom,
! [A3: nat,B6: nat,N3: nat] :
( ( ord_less @ nat @ A3 @ B6 )
=> ( ( ord_less @ nat @ ( zero_zero @ nat ) @ N3 )
=> ( ( ( modulo_modulo @ nat @ A3 @ N3 )
= ( zero_zero @ nat ) )
=> ( ( ( modulo_modulo @ nat @ B6 @ N3 )
= ( zero_zero @ nat ) )
=> ( ord_less @ nat @ ( divide_divide @ nat @ A3 @ N3 ) @ ( divide_divide @ nat @ B6 @ N3 ) ) ) ) ) ) ).
% div_less_mono
thf(fact_2430_mod__nat__add,axiom,
! [X: nat,Z: nat,Y: nat] :
( ( ord_less @ nat @ X @ Z )
=> ( ( ord_less @ nat @ Y @ Z )
=> ( ( ( ord_less @ nat @ ( plus_plus @ nat @ X @ Y ) @ Z )
=> ( ( modulo_modulo @ nat @ ( plus_plus @ nat @ X @ Y ) @ Z )
= ( plus_plus @ nat @ X @ Y ) ) )
& ( ~ ( ord_less @ nat @ ( plus_plus @ nat @ X @ Y ) @ Z )
=> ( ( modulo_modulo @ nat @ ( plus_plus @ nat @ X @ Y ) @ Z )
= ( minus_minus @ nat @ ( plus_plus @ nat @ X @ Y ) @ Z ) ) ) ) ) ) ).
% mod_nat_add
thf(fact_2431_nat__mod__eq__lemma,axiom,
! [X: nat,N3: nat,Y: nat] :
( ( ( modulo_modulo @ nat @ X @ N3 )
= ( modulo_modulo @ nat @ Y @ N3 ) )
=> ( ( ord_less_eq @ nat @ Y @ X )
=> ? [Q3: nat] :
( X
= ( plus_plus @ nat @ Y @ ( times_times @ nat @ N3 @ Q3 ) ) ) ) ) ).
% nat_mod_eq_lemma
thf(fact_2432_mod__eq__nat2E,axiom,
! [M: nat,Q2: nat,N3: nat] :
( ( ( modulo_modulo @ nat @ M @ Q2 )
= ( modulo_modulo @ nat @ N3 @ Q2 ) )
=> ( ( ord_less_eq @ nat @ M @ N3 )
=> ~ ! [S2: nat] :
( N3
!= ( plus_plus @ nat @ M @ ( times_times @ nat @ Q2 @ S2 ) ) ) ) ) ).
% mod_eq_nat2E
thf(fact_2433_mod__eq__nat1E,axiom,
! [M: nat,Q2: nat,N3: nat] :
( ( ( modulo_modulo @ nat @ M @ Q2 )
= ( modulo_modulo @ nat @ N3 @ Q2 ) )
=> ( ( ord_less_eq @ nat @ N3 @ M )
=> ~ ! [S2: nat] :
( M
!= ( plus_plus @ nat @ N3 @ ( times_times @ nat @ Q2 @ S2 ) ) ) ) ) ).
% mod_eq_nat1E
thf(fact_2434_mod__mult2__eq,axiom,
! [M: nat,N3: nat,Q2: nat] :
( ( modulo_modulo @ nat @ M @ ( times_times @ nat @ N3 @ Q2 ) )
= ( plus_plus @ nat @ ( times_times @ nat @ N3 @ ( modulo_modulo @ nat @ ( divide_divide @ nat @ M @ N3 ) @ Q2 ) ) @ ( modulo_modulo @ nat @ M @ N3 ) ) ) ).
% mod_mult2_eq
thf(fact_2435_div__mod__decomp,axiom,
! [A3: nat,N3: nat] :
( A3
= ( plus_plus @ nat @ ( times_times @ nat @ ( divide_divide @ nat @ A3 @ N3 ) @ N3 ) @ ( modulo_modulo @ nat @ A3 @ N3 ) ) ) ).
% div_mod_decomp
thf(fact_2436_modulo__nat__def,axiom,
( ( modulo_modulo @ nat )
= ( ^ [M5: nat,N2: nat] : ( minus_minus @ nat @ M5 @ ( times_times @ nat @ ( divide_divide @ nat @ M5 @ N2 ) @ N2 ) ) ) ) ).
% modulo_nat_def
thf(fact_2437_dvd__imp__le,axiom,
! [K: nat,N3: nat] :
( ( dvd_dvd @ nat @ K @ N3 )
=> ( ( ord_less @ nat @ ( zero_zero @ nat ) @ N3 )
=> ( ord_less_eq @ nat @ K @ N3 ) ) ) ).
% dvd_imp_le
thf(fact_2438_nat__mult__dvd__cancel1,axiom,
! [K: nat,M: nat,N3: nat] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ K )
=> ( ( dvd_dvd @ nat @ ( times_times @ nat @ K @ M ) @ ( times_times @ nat @ K @ N3 ) )
= ( dvd_dvd @ nat @ M @ N3 ) ) ) ).
% nat_mult_dvd_cancel1
thf(fact_2439_dvd__mult__cancel,axiom,
! [K: nat,M: nat,N3: nat] :
( ( dvd_dvd @ nat @ ( times_times @ nat @ K @ M ) @ ( times_times @ nat @ K @ N3 ) )
=> ( ( ord_less @ nat @ ( zero_zero @ nat ) @ K )
=> ( dvd_dvd @ nat @ M @ N3 ) ) ) ).
% dvd_mult_cancel
thf(fact_2440_real__of__nat__div,axiom,
! [D: nat,N3: nat] :
( ( dvd_dvd @ nat @ D @ N3 )
=> ( ( semiring_1_of_nat @ real @ ( divide_divide @ nat @ N3 @ D ) )
= ( divide_divide @ real @ ( semiring_1_of_nat @ real @ N3 ) @ ( semiring_1_of_nat @ real @ D ) ) ) ) ).
% real_of_nat_div
thf(fact_2441_even__mod__4__div__2,axiom,
! [N3: nat] :
( ( ( modulo_modulo @ nat @ N3 @ ( numeral_numeral @ nat @ ( bit0 @ ( bit0 @ one2 ) ) ) )
= ( suc @ ( zero_zero @ nat ) ) )
=> ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ ( minus_minus @ nat @ N3 @ ( suc @ ( zero_zero @ nat ) ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ).
% even_mod_4_div_2
thf(fact_2442_cong__exp__iff__simps_I3_J,axiom,
! [A: $tType] :
( ( unique1627219031080169319umeral @ A )
=> ! [N3: num,Q2: num] :
( ( modulo_modulo @ A @ ( numeral_numeral @ A @ ( bit1 @ N3 ) ) @ ( numeral_numeral @ A @ ( bit0 @ Q2 ) ) )
!= ( zero_zero @ A ) ) ) ).
% cong_exp_iff_simps(3)
thf(fact_2443_odd__mod__4__div__2,axiom,
! [N3: nat] :
( ( ( modulo_modulo @ nat @ N3 @ ( 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 @ N3 @ ( suc @ ( zero_zero @ nat ) ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ).
% odd_mod_4_div_2
thf(fact_2444_mod__mult2__eq_H,axiom,
! [A: $tType] :
( ( euclid5411537665997757685th_nat @ A )
=> ! [A2: A,M: nat,N3: nat] :
( ( modulo_modulo @ A @ A2 @ ( times_times @ A @ ( semiring_1_of_nat @ A @ M ) @ ( semiring_1_of_nat @ A @ N3 ) ) )
= ( plus_plus @ A @ ( times_times @ A @ ( semiring_1_of_nat @ A @ M ) @ ( modulo_modulo @ A @ ( divide_divide @ A @ A2 @ ( semiring_1_of_nat @ A @ M ) ) @ ( semiring_1_of_nat @ A @ N3 ) ) ) @ ( modulo_modulo @ A @ A2 @ ( semiring_1_of_nat @ A @ M ) ) ) ) ) ).
% mod_mult2_eq'
thf(fact_2445_even__zero,axiom,
! [A: $tType] :
( ( semiring_parity @ A )
=> ( dvd_dvd @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ ( zero_zero @ A ) ) ) ).
% even_zero
thf(fact_2446_odd__even__add,axiom,
! [A: $tType] :
( ( semiring_parity @ A )
=> ! [A2: A,B2: A] :
( ~ ( dvd_dvd @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ A2 )
=> ( ~ ( dvd_dvd @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ B2 )
=> ( dvd_dvd @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ ( plus_plus @ A @ A2 @ B2 ) ) ) ) ) ).
% odd_even_add
thf(fact_2447_odd__one,axiom,
! [A: $tType] :
( ( semiring_parity @ A )
=> ~ ( dvd_dvd @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ ( one_one @ A ) ) ) ).
% odd_one
thf(fact_2448_evenE,axiom,
! [A: $tType] :
( ( semiring_parity @ A )
=> ! [A2: A] :
( ( dvd_dvd @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ A2 )
=> ~ ! [B3: A] :
( A2
!= ( times_times @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ B3 ) ) ) ) ).
% evenE
thf(fact_2449_bit__eq__rec,axiom,
! [A: $tType] :
( ( bit_semiring_bits @ A )
=> ( ( ^ [Y5: A,Z5: A] : Y5 = Z5 )
= ( ^ [A6: A,B5: A] :
( ( ( dvd_dvd @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ A6 )
= ( dvd_dvd @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ B5 ) )
& ( ( divide_divide @ A @ A6 @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) )
= ( divide_divide @ A @ B5 @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) ) ) ) ) ) ).
% bit_eq_rec
thf(fact_2450_is__unit__div__mult__cancel__right,axiom,
! [A: $tType] :
( ( algebraic_semidom @ A )
=> ! [A2: A,B2: A] :
( ( A2
!= ( zero_zero @ A ) )
=> ( ( dvd_dvd @ A @ B2 @ ( one_one @ A ) )
=> ( ( divide_divide @ A @ A2 @ ( times_times @ A @ B2 @ A2 ) )
= ( divide_divide @ A @ ( one_one @ A ) @ B2 ) ) ) ) ) ).
% is_unit_div_mult_cancel_right
thf(fact_2451_is__unit__div__mult__cancel__left,axiom,
! [A: $tType] :
( ( algebraic_semidom @ A )
=> ! [A2: A,B2: A] :
( ( A2
!= ( zero_zero @ A ) )
=> ( ( dvd_dvd @ A @ B2 @ ( one_one @ A ) )
=> ( ( divide_divide @ A @ A2 @ ( times_times @ A @ A2 @ B2 ) )
= ( divide_divide @ A @ ( one_one @ A ) @ B2 ) ) ) ) ) ).
% is_unit_div_mult_cancel_left
thf(fact_2452_is__unitE,axiom,
! [A: $tType] :
( ( algebraic_semidom @ A )
=> ! [A2: A,C2: A] :
( ( dvd_dvd @ A @ A2 @ ( one_one @ A ) )
=> ~ ( ( A2
!= ( zero_zero @ A ) )
=> ! [B3: A] :
( ( B3
!= ( zero_zero @ A ) )
=> ( ( dvd_dvd @ A @ B3 @ ( one_one @ A ) )
=> ( ( ( divide_divide @ A @ ( one_one @ A ) @ A2 )
= B3 )
=> ( ( ( divide_divide @ A @ ( one_one @ A ) @ B3 )
= A2 )
=> ( ( ( times_times @ A @ A2 @ B3 )
= ( one_one @ A ) )
=> ( ( divide_divide @ A @ C2 @ A2 )
!= ( times_times @ A @ C2 @ B3 ) ) ) ) ) ) ) ) ) ) ).
% is_unitE
thf(fact_2453_odd__numeral,axiom,
! [A: $tType] :
( ( semiring_parity @ A )
=> ! [N3: num] :
~ ( dvd_dvd @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ ( numeral_numeral @ A @ ( bit1 @ N3 ) ) ) ) ).
% odd_numeral
thf(fact_2454_dvd__power__iff,axiom,
! [A: $tType] :
( ( algebraic_semidom @ A )
=> ! [X: A,M: nat,N3: nat] :
( ( X
!= ( zero_zero @ A ) )
=> ( ( dvd_dvd @ A @ ( power_power @ A @ X @ M ) @ ( power_power @ A @ X @ N3 ) )
= ( ( dvd_dvd @ A @ X @ ( one_one @ A ) )
| ( ord_less_eq @ nat @ M @ N3 ) ) ) ) ) ).
% dvd_power_iff
thf(fact_2455_dvd__power,axiom,
! [A: $tType] :
( ( comm_semiring_1 @ A )
=> ! [N3: nat,X: A] :
( ( ( ord_less @ nat @ ( zero_zero @ nat ) @ N3 )
| ( X
= ( one_one @ A ) ) )
=> ( dvd_dvd @ A @ X @ ( power_power @ A @ X @ N3 ) ) ) ) ).
% dvd_power
thf(fact_2456_unset__bit__Suc,axiom,
! [A: $tType] :
( ( bit_se359711467146920520ations @ A )
=> ! [N3: nat,A2: A] :
( ( bit_se2638667681897837118et_bit @ A @ ( suc @ N3 ) @ A2 )
= ( plus_plus @ A @ ( modulo_modulo @ A @ A2 @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) @ ( times_times @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ ( bit_se2638667681897837118et_bit @ A @ N3 @ ( divide_divide @ A @ A2 @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) ) ) ) ) ) ).
% unset_bit_Suc
thf(fact_2457_field__char__0__class_Oof__nat__div,axiom,
! [A: $tType] :
( ( field_char_0 @ A )
=> ! [M: nat,N3: nat] :
( ( semiring_1_of_nat @ A @ ( divide_divide @ nat @ M @ N3 ) )
= ( divide_divide @ A @ ( minus_minus @ A @ ( semiring_1_of_nat @ A @ M ) @ ( semiring_1_of_nat @ A @ ( modulo_modulo @ nat @ M @ N3 ) ) ) @ ( semiring_1_of_nat @ A @ N3 ) ) ) ) ).
% field_char_0_class.of_nat_div
thf(fact_2458_split__mod,axiom,
! [P: nat > $o,M: nat,N3: nat] :
( ( P @ ( modulo_modulo @ nat @ M @ N3 ) )
= ( ( ( N3
= ( zero_zero @ nat ) )
=> ( P @ M ) )
& ( ( N3
!= ( zero_zero @ nat ) )
=> ! [I3: nat,J3: nat] :
( ( ord_less @ nat @ J3 @ N3 )
=> ( ( M
= ( plus_plus @ nat @ ( times_times @ nat @ N3 @ I3 ) @ J3 ) )
=> ( P @ J3 ) ) ) ) ) ) ).
% split_mod
thf(fact_2459_mod__lemma,axiom,
! [C2: nat,R2: nat,B2: nat,Q2: nat] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ C2 )
=> ( ( ord_less @ nat @ R2 @ B2 )
=> ( ord_less @ nat @ ( plus_plus @ nat @ ( times_times @ nat @ B2 @ ( modulo_modulo @ nat @ Q2 @ C2 ) ) @ R2 ) @ ( times_times @ nat @ B2 @ C2 ) ) ) ) ).
% mod_lemma
thf(fact_2460_dvd__mult__cancel2,axiom,
! [M: nat,N3: nat] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ M )
=> ( ( dvd_dvd @ nat @ ( times_times @ nat @ N3 @ M ) @ M )
= ( N3
= ( one_one @ nat ) ) ) ) ).
% dvd_mult_cancel2
thf(fact_2461_dvd__mult__cancel1,axiom,
! [M: nat,N3: nat] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ M )
=> ( ( dvd_dvd @ nat @ ( times_times @ nat @ M @ N3 ) @ M )
= ( N3
= ( one_one @ nat ) ) ) ) ).
% dvd_mult_cancel1
thf(fact_2462_power__dvd__imp__le,axiom,
! [I: nat,M: nat,N3: nat] :
( ( dvd_dvd @ nat @ ( power_power @ nat @ I @ M ) @ ( power_power @ nat @ I @ N3 ) )
=> ( ( ord_less @ nat @ ( one_one @ nat ) @ I )
=> ( ord_less_eq @ nat @ M @ N3 ) ) ) ).
% power_dvd_imp_le
thf(fact_2463_dvd__minus__add,axiom,
! [Q2: nat,N3: nat,R2: nat,M: nat] :
( ( ord_less_eq @ nat @ Q2 @ N3 )
=> ( ( ord_less_eq @ nat @ Q2 @ ( times_times @ nat @ R2 @ M ) )
=> ( ( dvd_dvd @ nat @ M @ ( minus_minus @ nat @ N3 @ Q2 ) )
= ( dvd_dvd @ nat @ M @ ( plus_plus @ nat @ N3 @ ( minus_minus @ nat @ ( times_times @ nat @ R2 @ M ) @ Q2 ) ) ) ) ) ) ).
% dvd_minus_add
thf(fact_2464_real__of__nat__div__aux,axiom,
! [X: nat,D: nat] :
( ( divide_divide @ real @ ( semiring_1_of_nat @ real @ X ) @ ( semiring_1_of_nat @ real @ D ) )
= ( plus_plus @ real @ ( semiring_1_of_nat @ real @ ( divide_divide @ nat @ X @ D ) ) @ ( divide_divide @ real @ ( semiring_1_of_nat @ real @ ( modulo_modulo @ nat @ X @ D ) ) @ ( semiring_1_of_nat @ real @ D ) ) ) ) ).
% real_of_nat_div_aux
thf(fact_2465_unique__euclidean__semiring__numeral__class_Omod__mult2__eq,axiom,
! [A: $tType] :
( ( unique1627219031080169319umeral @ A )
=> ! [C2: A,A2: A,B2: A] :
( ( ord_less_eq @ A @ ( zero_zero @ A ) @ C2 )
=> ( ( modulo_modulo @ A @ A2 @ ( times_times @ A @ B2 @ C2 ) )
= ( plus_plus @ A @ ( times_times @ A @ B2 @ ( modulo_modulo @ A @ ( divide_divide @ A @ A2 @ B2 ) @ C2 ) ) @ ( modulo_modulo @ A @ A2 @ B2 ) ) ) ) ) ).
% unique_euclidean_semiring_numeral_class.mod_mult2_eq
thf(fact_2466_cong__exp__iff__simps_I7_J,axiom,
! [A: $tType] :
( ( unique1627219031080169319umeral @ A )
=> ! [Q2: num,N3: num] :
( ( ( modulo_modulo @ A @ ( numeral_numeral @ A @ one2 ) @ ( numeral_numeral @ A @ ( bit0 @ Q2 ) ) )
= ( modulo_modulo @ A @ ( numeral_numeral @ A @ ( bit1 @ N3 ) ) @ ( numeral_numeral @ A @ ( bit0 @ Q2 ) ) ) )
= ( ( modulo_modulo @ A @ ( numeral_numeral @ A @ N3 ) @ ( numeral_numeral @ A @ Q2 ) )
= ( zero_zero @ A ) ) ) ) ).
% cong_exp_iff_simps(7)
thf(fact_2467_cong__exp__iff__simps_I11_J,axiom,
! [A: $tType] :
( ( unique1627219031080169319umeral @ A )
=> ! [M: num,Q2: num] :
( ( ( modulo_modulo @ A @ ( numeral_numeral @ A @ ( bit1 @ M ) ) @ ( numeral_numeral @ A @ ( bit0 @ Q2 ) ) )
= ( modulo_modulo @ A @ ( numeral_numeral @ A @ one2 ) @ ( numeral_numeral @ A @ ( bit0 @ Q2 ) ) ) )
= ( ( modulo_modulo @ A @ ( numeral_numeral @ A @ M ) @ ( numeral_numeral @ A @ Q2 ) )
= ( zero_zero @ A ) ) ) ) ).
% cong_exp_iff_simps(11)
thf(fact_2468_even__two__times__div__two,axiom,
! [A: $tType] :
( ( euclid5411537665997757685th_nat @ A )
=> ! [A2: A] :
( ( dvd_dvd @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ A2 )
=> ( ( times_times @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ ( divide_divide @ A @ A2 @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) )
= A2 ) ) ) ).
% even_two_times_div_two
thf(fact_2469_Suc__mod__eq__add3__mod,axiom,
! [M: nat,N3: nat] :
( ( modulo_modulo @ nat @ ( suc @ ( suc @ ( suc @ M ) ) ) @ N3 )
= ( modulo_modulo @ nat @ ( plus_plus @ nat @ ( numeral_numeral @ nat @ ( bit1 @ one2 ) ) @ M ) @ N3 ) ) ).
% Suc_mod_eq_add3_mod
thf(fact_2470_Suc__times__mod__eq,axiom,
! [M: nat,N3: nat] :
( ( ord_less @ nat @ ( suc @ ( zero_zero @ nat ) ) @ M )
=> ( ( modulo_modulo @ nat @ ( suc @ ( times_times @ nat @ M @ N3 ) ) @ M )
= ( one_one @ nat ) ) ) ).
% Suc_times_mod_eq
thf(fact_2471_power__mono__odd,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [N3: nat,A2: A,B2: A] :
( ~ ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N3 )
=> ( ( ord_less_eq @ A @ A2 @ B2 )
=> ( ord_less_eq @ A @ ( power_power @ A @ A2 @ N3 ) @ ( power_power @ A @ B2 @ N3 ) ) ) ) ) ).
% power_mono_odd
thf(fact_2472_odd__pos,axiom,
! [N3: nat] :
( ~ ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N3 )
=> ( ord_less @ nat @ ( zero_zero @ nat ) @ N3 ) ) ).
% odd_pos
thf(fact_2473_dvd__power__iff__le,axiom,
! [K: nat,M: nat,N3: nat] :
( ( ord_less_eq @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ K )
=> ( ( dvd_dvd @ nat @ ( power_power @ nat @ K @ M ) @ ( power_power @ nat @ K @ N3 ) )
= ( ord_less_eq @ nat @ M @ N3 ) ) ) ).
% dvd_power_iff_le
thf(fact_2474_even__set__bit__iff,axiom,
! [A: $tType] :
( ( bit_se359711467146920520ations @ A )
=> ! [M: nat,A2: A] :
( ( dvd_dvd @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ ( bit_se5668285175392031749et_bit @ A @ M @ A2 ) )
= ( ( dvd_dvd @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ A2 )
& ( M
!= ( zero_zero @ nat ) ) ) ) ) ).
% even_set_bit_iff
thf(fact_2475_divmod__digit__0_I2_J,axiom,
! [A: $tType] :
( ( unique1627219031080169319umeral @ A )
=> ! [B2: A,A2: A] :
( ( ord_less @ A @ ( zero_zero @ A ) @ B2 )
=> ( ( ord_less @ A @ ( modulo_modulo @ A @ A2 @ ( times_times @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ B2 ) ) @ B2 )
=> ( ( modulo_modulo @ A @ A2 @ ( times_times @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ B2 ) )
= ( modulo_modulo @ A @ A2 @ B2 ) ) ) ) ) ).
% divmod_digit_0(2)
thf(fact_2476_bits__stable__imp__add__self,axiom,
! [A: $tType] :
( ( bit_semiring_bits @ A )
=> ! [A2: A] :
( ( ( divide_divide @ A @ A2 @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) )
= A2 )
=> ( ( plus_plus @ A @ A2 @ ( modulo_modulo @ A @ A2 @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) )
= ( zero_zero @ A ) ) ) ) ).
% bits_stable_imp_add_self
thf(fact_2477_div__exp__mod__exp__eq,axiom,
! [A: $tType] :
( ( bit_semiring_bits @ A )
=> ! [A2: A,N3: nat,M: nat] :
( ( modulo_modulo @ A @ ( divide_divide @ A @ A2 @ ( power_power @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ N3 ) ) @ ( power_power @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ M ) )
= ( divide_divide @ A @ ( modulo_modulo @ A @ A2 @ ( power_power @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ ( plus_plus @ nat @ N3 @ M ) ) ) @ ( power_power @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ N3 ) ) ) ) ).
% div_exp_mod_exp_eq
thf(fact_2478_oddE,axiom,
! [A: $tType] :
( ( semiring_parity @ A )
=> ! [A2: A] :
( ~ ( dvd_dvd @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ A2 )
=> ~ ! [B3: A] :
( A2
!= ( plus_plus @ A @ ( times_times @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ B3 ) @ ( one_one @ A ) ) ) ) ) ).
% oddE
thf(fact_2479_zero__le__even__power,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [N3: nat,A2: A] :
( ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N3 )
=> ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( power_power @ A @ A2 @ N3 ) ) ) ) ).
% zero_le_even_power
thf(fact_2480_zero__le__odd__power,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [N3: nat,A2: A] :
( ~ ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N3 )
=> ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( power_power @ A @ A2 @ N3 ) )
= ( ord_less_eq @ A @ ( zero_zero @ A ) @ A2 ) ) ) ) ).
% zero_le_odd_power
thf(fact_2481_zero__le__power__eq,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [A2: A,N3: nat] :
( ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( power_power @ A @ A2 @ N3 ) )
= ( ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N3 )
| ( ~ ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N3 )
& ( ord_less_eq @ A @ ( zero_zero @ A ) @ A2 ) ) ) ) ) ).
% zero_le_power_eq
thf(fact_2482_power__mod__div,axiom,
! [X: nat,N3: nat,M: nat] :
( ( divide_divide @ nat @ ( modulo_modulo @ nat @ X @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N3 ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ M ) )
= ( modulo_modulo @ nat @ ( divide_divide @ nat @ X @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ M ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( minus_minus @ nat @ N3 @ M ) ) ) ) ).
% power_mod_div
thf(fact_2483_verit__le__mono__div,axiom,
! [A3: nat,B6: nat,N3: nat] :
( ( ord_less @ nat @ A3 @ B6 )
=> ( ( ord_less @ nat @ ( zero_zero @ nat ) @ N3 )
=> ( ord_less_eq @ nat
@ ( plus_plus @ nat @ ( divide_divide @ nat @ A3 @ N3 )
@ ( if @ nat
@ ( ( modulo_modulo @ nat @ B6 @ N3 )
= ( zero_zero @ nat ) )
@ ( one_one @ nat )
@ ( zero_zero @ nat ) ) )
@ ( divide_divide @ nat @ B6 @ N3 ) ) ) ) ).
% verit_le_mono_div
thf(fact_2484_divmod__digit__0_I1_J,axiom,
! [A: $tType] :
( ( unique1627219031080169319umeral @ A )
=> ! [B2: A,A2: A] :
( ( ord_less @ A @ ( zero_zero @ A ) @ B2 )
=> ( ( ord_less @ A @ ( modulo_modulo @ A @ A2 @ ( times_times @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ B2 ) ) @ B2 )
=> ( ( times_times @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ ( divide_divide @ A @ A2 @ ( times_times @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ B2 ) ) )
= ( divide_divide @ A @ A2 @ B2 ) ) ) ) ) ).
% divmod_digit_0(1)
thf(fact_2485_mult__exp__mod__exp__eq,axiom,
! [A: $tType] :
( ( bit_semiring_bits @ A )
=> ! [M: nat,N3: nat,A2: A] :
( ( ord_less_eq @ nat @ M @ N3 )
=> ( ( modulo_modulo @ A @ ( times_times @ A @ A2 @ ( power_power @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ M ) ) @ ( power_power @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ N3 ) )
= ( times_times @ A @ ( modulo_modulo @ A @ A2 @ ( power_power @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ ( minus_minus @ nat @ N3 @ M ) ) ) @ ( power_power @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ M ) ) ) ) ) ).
% mult_exp_mod_exp_eq
thf(fact_2486_zero__less__power__eq,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [A2: A,N3: nat] :
( ( ord_less @ A @ ( zero_zero @ A ) @ ( power_power @ A @ A2 @ N3 ) )
= ( ( N3
= ( zero_zero @ nat ) )
| ( ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N3 )
& ( A2
!= ( zero_zero @ A ) ) )
| ( ~ ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N3 )
& ( ord_less @ A @ ( zero_zero @ A ) @ A2 ) ) ) ) ) ).
% zero_less_power_eq
thf(fact_2487_mod__double__modulus,axiom,
! [A: $tType] :
( ( unique1627219031080169319umeral @ A )
=> ! [M: A,X: A] :
( ( ord_less @ A @ ( zero_zero @ A ) @ M )
=> ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ X )
=> ( ( ( modulo_modulo @ A @ X @ ( times_times @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ M ) )
= ( modulo_modulo @ A @ X @ M ) )
| ( ( modulo_modulo @ A @ X @ ( times_times @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ M ) )
= ( plus_plus @ A @ ( modulo_modulo @ A @ X @ M ) @ M ) ) ) ) ) ) ).
% mod_double_modulus
thf(fact_2488_divmod__digit__1_I2_J,axiom,
! [A: $tType] :
( ( unique1627219031080169319umeral @ A )
=> ! [A2: A,B2: A] :
( ( ord_less_eq @ A @ ( zero_zero @ A ) @ A2 )
=> ( ( ord_less @ A @ ( zero_zero @ A ) @ B2 )
=> ( ( ord_less_eq @ A @ B2 @ ( modulo_modulo @ A @ A2 @ ( times_times @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ B2 ) ) )
=> ( ( minus_minus @ A @ ( modulo_modulo @ A @ A2 @ ( times_times @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ B2 ) ) @ B2 )
= ( modulo_modulo @ A @ A2 @ B2 ) ) ) ) ) ) ).
% divmod_digit_1(2)
thf(fact_2489_set__bit__Suc,axiom,
! [A: $tType] :
( ( bit_se359711467146920520ations @ A )
=> ! [N3: nat,A2: A] :
( ( bit_se5668285175392031749et_bit @ A @ ( suc @ N3 ) @ A2 )
= ( plus_plus @ A @ ( modulo_modulo @ A @ A2 @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) @ ( times_times @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ ( bit_se5668285175392031749et_bit @ A @ N3 @ ( divide_divide @ A @ A2 @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) ) ) ) ) ) ).
% set_bit_Suc
thf(fact_2490_even__mask__div__iff_H,axiom,
! [A: $tType] :
( ( euclid5411537665997757685th_nat @ A )
=> ! [M: nat,N3: 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 ) ) @ N3 ) ) )
= ( ord_less_eq @ nat @ M @ N3 ) ) ) ).
% even_mask_div_iff'
thf(fact_2491_power__le__zero__eq,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [A2: A,N3: nat] :
( ( ord_less_eq @ A @ ( power_power @ A @ A2 @ N3 ) @ ( zero_zero @ A ) )
= ( ( ord_less @ nat @ ( zero_zero @ nat ) @ N3 )
& ( ( ~ ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N3 )
& ( ord_less_eq @ A @ A2 @ ( zero_zero @ A ) ) )
| ( ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N3 )
& ( A2
= ( zero_zero @ A ) ) ) ) ) ) ) ).
% power_le_zero_eq
thf(fact_2492_even__mask__div__iff,axiom,
! [A: $tType] :
( ( bit_semiring_bits @ A )
=> ! [M: nat,N3: 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 ) ) @ N3 ) ) )
= ( ( ( power_power @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ N3 )
= ( zero_zero @ A ) )
| ( ord_less_eq @ nat @ M @ N3 ) ) ) ) ).
% even_mask_div_iff
thf(fact_2493_divmod__divmod__step,axiom,
! [A: $tType] :
( ( unique1627219031080169319umeral @ A )
=> ( ( unique8689654367752047608divmod @ A )
= ( ^ [M5: num,N2: num] : ( if @ ( product_prod @ A @ A ) @ ( ord_less @ num @ M5 @ N2 ) @ ( product_Pair @ A @ A @ ( zero_zero @ A ) @ ( numeral_numeral @ A @ M5 ) ) @ ( unique1321980374590559556d_step @ A @ N2 @ ( unique8689654367752047608divmod @ A @ M5 @ ( bit0 @ N2 ) ) ) ) ) ) ) ).
% divmod_divmod_step
thf(fact_2494_Bernoulli__inequality__even,axiom,
! [N3: nat,X: real] :
( ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N3 )
=> ( ord_less_eq @ real @ ( plus_plus @ real @ ( one_one @ real ) @ ( times_times @ real @ ( semiring_1_of_nat @ real @ N3 ) @ X ) ) @ ( power_power @ real @ ( plus_plus @ real @ ( one_one @ real ) @ X ) @ N3 ) ) ) ).
% Bernoulli_inequality_even
thf(fact_2495_divmod__digit__1_I1_J,axiom,
! [A: $tType] :
( ( unique1627219031080169319umeral @ A )
=> ! [A2: A,B2: A] :
( ( ord_less_eq @ A @ ( zero_zero @ A ) @ A2 )
=> ( ( ord_less @ A @ ( zero_zero @ A ) @ B2 )
=> ( ( ord_less_eq @ A @ B2 @ ( modulo_modulo @ A @ A2 @ ( times_times @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ B2 ) ) )
=> ( ( plus_plus @ A @ ( times_times @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ ( divide_divide @ A @ A2 @ ( times_times @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ B2 ) ) ) @ ( one_one @ A ) )
= ( divide_divide @ A @ A2 @ B2 ) ) ) ) ) ) ).
% divmod_digit_1(1)
thf(fact_2496_VEBT__internal_OT__vebt__buildupi_Osimps_I3_J,axiom,
! [N3: nat] :
( ( ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N3 )
=> ( ( vEBT_V441764108873111860ildupi @ ( suc @ ( suc @ N3 ) ) )
= ( 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 )
=> ( ( vEBT_V441764108873111860ildupi @ ( suc @ ( suc @ N3 ) ) )
= ( 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.simps(3)
thf(fact_2497_even__mult__exp__div__exp__iff,axiom,
! [A: $tType] :
( ( bit_semiring_bits @ A )
=> ! [A2: A,M: nat,N3: nat] :
( ( dvd_dvd @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ ( divide_divide @ A @ ( times_times @ A @ A2 @ ( power_power @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ M ) ) @ ( power_power @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ N3 ) ) )
= ( ( ord_less @ nat @ N3 @ M )
| ( ( power_power @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ N3 )
= ( zero_zero @ A ) )
| ( ( ord_less_eq @ nat @ M @ N3 )
& ( dvd_dvd @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ ( divide_divide @ A @ A2 @ ( power_power @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ ( minus_minus @ nat @ N3 @ M ) ) ) ) ) ) ) ) ).
% even_mult_exp_div_exp_iff
thf(fact_2498_VEBT__internal_OT__vebt__buildupi_Oelims,axiom,
! [X: nat,Y: nat] :
( ( ( vEBT_V441764108873111860ildupi @ X )
= Y )
=> ( ( ( X
= ( zero_zero @ nat ) )
=> ( Y
!= ( suc @ ( zero_zero @ nat ) ) ) )
=> ( ( ( X
= ( suc @ ( zero_zero @ nat ) ) )
=> ( Y
!= ( suc @ ( zero_zero @ nat ) ) ) )
=> ~ ! [N: nat] :
( ( X
= ( suc @ ( suc @ N ) ) )
=> ~ ( ( ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N )
=> ( Y
= ( 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 )
=> ( Y
= ( 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.elims
thf(fact_2499_VEBT__internal_OTb_H_Osimps_I3_J,axiom,
! [N3: nat] :
( ( ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N3 )
=> ( ( vEBT_VEBT_Tb2 @ ( suc @ ( suc @ N3 ) ) )
= ( 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 )
=> ( ( vEBT_VEBT_Tb2 @ ( suc @ ( suc @ N3 ) ) )
= ( 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'.simps(3)
thf(fact_2500_VEBT__internal_OTb_H_Oelims,axiom,
! [X: nat,Y: nat] :
( ( ( vEBT_VEBT_Tb2 @ X )
= Y )
=> ( ( ( X
= ( zero_zero @ nat ) )
=> ( Y
!= ( numeral_numeral @ nat @ ( bit1 @ one2 ) ) ) )
=> ( ( ( X
= ( suc @ ( zero_zero @ nat ) ) )
=> ( Y
!= ( numeral_numeral @ nat @ ( bit1 @ one2 ) ) ) )
=> ~ ! [N: nat] :
( ( X
= ( suc @ ( suc @ N ) ) )
=> ~ ( ( ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N )
=> ( Y
= ( 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 )
=> ( Y
= ( 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'.elims
thf(fact_2501_VEBT__internal_OTb_Osimps_I3_J,axiom,
! [N3: nat] :
( ( ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N3 )
=> ( ( vEBT_VEBT_Tb @ ( suc @ ( suc @ N3 ) ) )
= ( 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 )
=> ( ( vEBT_VEBT_Tb @ ( suc @ ( suc @ N3 ) ) )
= ( 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.simps(3)
thf(fact_2502_VEBT__internal_OT__vebt__buildupi_H_Osimps_I3_J,axiom,
! [N3: nat] :
( ( ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N3 )
=> ( ( vEBT_V9176841429113362141ildupi @ ( suc @ ( suc @ N3 ) ) )
= ( 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 )
=> ( ( vEBT_V9176841429113362141ildupi @ ( suc @ ( suc @ N3 ) ) )
= ( 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'.simps(3)
thf(fact_2503_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_2504_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_2505_VEBT__internal_OTb_Oelims,axiom,
! [X: nat,Y: int] :
( ( ( vEBT_VEBT_Tb @ X )
= Y )
=> ( ( ( X
= ( zero_zero @ nat ) )
=> ( Y
!= ( numeral_numeral @ int @ ( bit1 @ one2 ) ) ) )
=> ( ( ( X
= ( suc @ ( zero_zero @ nat ) ) )
=> ( Y
!= ( numeral_numeral @ int @ ( bit1 @ one2 ) ) ) )
=> ~ ! [N: nat] :
( ( X
= ( suc @ ( suc @ N ) ) )
=> ~ ( ( ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N )
=> ( Y
= ( 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 )
=> ( Y
= ( 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.elims
thf(fact_2506_VEBT__internal_OT_092_060_094sub_062b_092_060_094sub_062u_092_060_094sub_062i_092_060_094sub_062l_092_060_094sub_062d_092_060_094sub_062u_092_060_094sub_062p_Oelims,axiom,
! [X: nat,Y: nat] :
( ( ( vEBT_V8346862874174094_d_u_p @ X )
= Y )
=> ( ( ( X
= ( zero_zero @ nat ) )
=> ( Y
!= ( numeral_numeral @ nat @ ( bit1 @ one2 ) ) ) )
=> ( ( ( X
= ( suc @ ( zero_zero @ nat ) ) )
=> ( Y
!= ( numeral_numeral @ nat @ ( bit1 @ one2 ) ) ) )
=> ~ ! [Va3: nat] :
( ( X
= ( suc @ ( suc @ Va3 ) ) )
=> ~ ( ( ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( suc @ ( suc @ Va3 ) ) )
=> ( Y
= ( plus_plus @ nat @ ( one_one @ nat ) @ ( plus_plus @ nat @ ( plus_plus @ nat @ ( numeral_numeral @ nat @ ( bit1 @ ( bit0 @ ( bit0 @ one2 ) ) ) ) @ ( vEBT_V8346862874174094_d_u_p @ ( divide_divide @ nat @ ( suc @ ( suc @ 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 ) ) )
=> ( Y
= ( plus_plus @ nat @ ( plus_plus @ nat @ ( numeral_numeral @ nat @ ( bit1 @ ( bit1 @ ( bit0 @ one2 ) ) ) ) @ ( vEBT_V8346862874174094_d_u_p @ ( suc @ ( divide_divide @ nat @ ( suc @ ( suc @ 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_2507_VEBT__internal_OT_092_060_094sub_062b_092_060_094sub_062u_092_060_094sub_062i_092_060_094sub_062l_092_060_094sub_062d_Oelims,axiom,
! [X: nat,Y: nat] :
( ( ( vEBT_V8646137997579335489_i_l_d @ X )
= Y )
=> ( ( ( X
= ( zero_zero @ nat ) )
=> ( Y
!= ( numeral_numeral @ nat @ ( bit0 @ ( bit0 @ one2 ) ) ) ) )
=> ( ( ( X
= ( suc @ ( zero_zero @ nat ) ) )
=> ( Y
!= ( numeral_numeral @ nat @ ( bit0 @ ( bit0 @ one2 ) ) ) ) )
=> ~ ! [Va3: nat] :
( ( X
= ( suc @ ( suc @ Va3 ) ) )
=> ~ ( ( ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( suc @ ( suc @ Va3 ) ) )
=> ( Y
= ( plus_plus @ nat @ ( one_one @ nat ) @ ( plus_plus @ nat @ ( plus_plus @ nat @ ( numeral_numeral @ nat @ ( bit0 @ ( bit1 @ ( bit0 @ one2 ) ) ) ) @ ( vEBT_V8646137997579335489_i_l_d @ ( divide_divide @ nat @ ( suc @ ( suc @ 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 ) ) )
=> ( Y
= ( plus_plus @ nat @ ( plus_plus @ nat @ ( numeral_numeral @ nat @ ( bit0 @ ( bit0 @ ( bit1 @ one2 ) ) ) ) @ ( vEBT_V8646137997579335489_i_l_d @ ( suc @ ( divide_divide @ nat @ ( suc @ ( suc @ 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_2508__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,
! [X213: $o,X224: $o] :
( ( tia
= ( vEBT_Leafi @ X213 @ X224 ) )
=> ( 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_2509_div__half__nat,axiom,
! [Y: nat,X: nat] :
( ( Y
!= ( zero_zero @ nat ) )
=> ( ( product_Pair @ nat @ nat @ ( divide_divide @ nat @ X @ Y ) @ ( modulo_modulo @ nat @ X @ Y ) )
= ( if @ ( product_prod @ nat @ nat ) @ ( ord_less_eq @ nat @ Y @ ( minus_minus @ nat @ X @ ( times_times @ nat @ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ ( divide_divide @ nat @ X @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ Y ) ) @ Y ) ) ) @ ( product_Pair @ nat @ nat @ ( plus_plus @ nat @ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ ( divide_divide @ nat @ X @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ Y ) ) @ ( one_one @ nat ) ) @ ( minus_minus @ nat @ ( minus_minus @ nat @ X @ ( times_times @ nat @ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ ( divide_divide @ nat @ X @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ Y ) ) @ Y ) ) @ Y ) ) @ ( product_Pair @ nat @ nat @ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ ( divide_divide @ nat @ X @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ Y ) ) @ ( minus_minus @ nat @ X @ ( times_times @ nat @ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ ( divide_divide @ nat @ X @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ Y ) ) @ Y ) ) ) ) ) ) ).
% div_half_nat
thf(fact_2510_pow__divides__pow__iff,axiom,
! [A: $tType] :
( ( semiring_gcd @ A )
=> ! [N3: nat,A2: A,B2: A] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ N3 )
=> ( ( dvd_dvd @ A @ ( power_power @ A @ A2 @ N3 ) @ ( power_power @ A @ B2 @ N3 ) )
= ( dvd_dvd @ A @ A2 @ B2 ) ) ) ) ).
% pow_divides_pow_iff
thf(fact_2511_builupicorr,axiom,
! [N3: nat] : ( hoare_hoare_triple @ vEBT_VEBTi @ ( one_one @ assn ) @ ( vEBT_vebt_buildupi @ N3 ) @ ( vEBT_vebt_assn_raw @ ( vEBT_vebt_buildup @ N3 ) ) ) ).
% builupicorr
thf(fact_2512_builupi_Hcorr,axiom,
! [N3: nat] : ( hoare_hoare_triple @ vEBT_VEBTi @ ( one_one @ assn ) @ ( vEBT_V739175172307565963ildupi @ N3 ) @ ( vEBT_vebt_assn_raw @ ( vEBT_vebt_buildup @ N3 ) ) ) ).
% builupi'corr
thf(fact_2513_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_2514_div2__even__ext__nat,axiom,
! [X: nat,Y: nat] :
( ( ( divide_divide @ nat @ X @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) )
= ( divide_divide @ nat @ Y @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) )
=> ( ( ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ X )
= ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ Y ) )
=> ( X = Y ) ) ) ).
% div2_even_ext_nat
thf(fact_2515_mod__neg__neg__trivial,axiom,
! [K: int,L2: int] :
( ( ord_less_eq @ int @ K @ ( zero_zero @ int ) )
=> ( ( ord_less @ int @ L2 @ K )
=> ( ( modulo_modulo @ int @ K @ L2 )
= K ) ) ) ).
% mod_neg_neg_trivial
thf(fact_2516_mod__pos__pos__trivial,axiom,
! [K: int,L2: int] :
( ( ord_less_eq @ int @ ( zero_zero @ int ) @ K )
=> ( ( ord_less @ int @ K @ L2 )
=> ( ( modulo_modulo @ int @ K @ L2 )
= K ) ) ) ).
% mod_pos_pos_trivial
thf(fact_2517_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_2518_one__mod__exp__eq__one,axiom,
! [N3: nat] :
( ( modulo_modulo @ int @ ( one_one @ int ) @ ( times_times @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ ( power_power @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ N3 ) ) )
= ( one_one @ int ) ) ).
% one_mod_exp_eq_one
thf(fact_2519_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_2520_VEBTi_Osize_I4_J,axiom,
! [X21: $o,X222: $o] :
( ( size_size @ vEBT_VEBTi @ ( vEBT_Leafi @ X21 @ X222 ) )
= ( zero_zero @ nat ) ) ).
% VEBTi.size(4)
thf(fact_2521_VEBTi_Odistinct_I1_J,axiom,
! [X11: option @ ( product_prod @ nat @ nat ),X12: nat,X13: array @ vEBT_VEBTi,X14: vEBT_VEBTi,X21: $o,X222: $o] :
( ( vEBT_Nodei @ X11 @ X12 @ X13 @ X14 )
!= ( vEBT_Leafi @ X21 @ X222 ) ) ).
% VEBTi.distinct(1)
thf(fact_2522_VEBTi_Oexhaust,axiom,
! [Y: vEBT_VEBTi] :
( ! [X112: option @ ( product_prod @ nat @ nat ),X122: nat,X132: array @ vEBT_VEBTi,X142: vEBT_VEBTi] :
( Y
!= ( vEBT_Nodei @ X112 @ X122 @ X132 @ X142 ) )
=> ~ ! [X212: $o,X223: $o] :
( Y
!= ( vEBT_Leafi @ X212 @ X223 ) ) ) ).
% VEBTi.exhaust
thf(fact_2523_zmod__helper,axiom,
! [N3: int,M: int,K: int,A2: int] :
( ( ( modulo_modulo @ int @ N3 @ M )
= K )
=> ( ( modulo_modulo @ int @ ( plus_plus @ int @ N3 @ A2 ) @ M )
= ( modulo_modulo @ int @ ( plus_plus @ int @ K @ A2 ) @ M ) ) ) ).
% zmod_helper
thf(fact_2524_Word_Omod__minus__cong,axiom,
! [B2: int,B4: int,X: int,X7: int,Y: int,Y6: int,Z7: int] :
( ( B2 = B4 )
=> ( ( ( modulo_modulo @ int @ X @ B4 )
= ( modulo_modulo @ int @ X7 @ B4 ) )
=> ( ( ( modulo_modulo @ int @ Y @ B4 )
= ( modulo_modulo @ int @ Y6 @ B4 ) )
=> ( ( ( minus_minus @ int @ X7 @ Y6 )
= Z7 )
=> ( ( modulo_modulo @ int @ ( minus_minus @ int @ X @ Y ) @ B2 )
= ( modulo_modulo @ int @ Z7 @ B4 ) ) ) ) ) ) ).
% Word.mod_minus_cong
thf(fact_2525_zdvd__zdiffD,axiom,
! [K: int,M: int,N3: int] :
( ( dvd_dvd @ int @ K @ ( minus_minus @ int @ M @ N3 ) )
=> ( ( dvd_dvd @ int @ K @ N3 )
=> ( dvd_dvd @ int @ K @ M ) ) ) ).
% zdvd_zdiffD
thf(fact_2526_mod__int__pos__iff,axiom,
! [K: int,L2: int] :
( ( ord_less_eq @ int @ ( zero_zero @ int ) @ ( modulo_modulo @ int @ K @ L2 ) )
= ( ( dvd_dvd @ int @ L2 @ K )
| ( ( L2
= ( zero_zero @ int ) )
& ( ord_less_eq @ int @ ( zero_zero @ int ) @ K ) )
| ( ord_less @ int @ ( zero_zero @ int ) @ L2 ) ) ) ).
% mod_int_pos_iff
thf(fact_2527_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_2528_neg__mod__bound,axiom,
! [L2: int,K: int] :
( ( ord_less @ int @ L2 @ ( zero_zero @ int ) )
=> ( ord_less @ int @ L2 @ ( modulo_modulo @ int @ K @ L2 ) ) ) ).
% neg_mod_bound
thf(fact_2529_Euclidean__Division_Opos__mod__bound,axiom,
! [L2: int,K: int] :
( ( ord_less @ int @ ( zero_zero @ int ) @ L2 )
=> ( ord_less @ int @ ( modulo_modulo @ int @ K @ L2 ) @ L2 ) ) ).
% Euclidean_Division.pos_mod_bound
thf(fact_2530_zdvd__antisym__nonneg,axiom,
! [M: int,N3: int] :
( ( ord_less_eq @ int @ ( zero_zero @ int ) @ M )
=> ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ N3 )
=> ( ( dvd_dvd @ int @ M @ N3 )
=> ( ( dvd_dvd @ int @ N3 @ M )
=> ( M = N3 ) ) ) ) ) ).
% zdvd_antisym_nonneg
thf(fact_2531_divmod__int__def,axiom,
( ( unique8689654367752047608divmod @ int )
= ( ^ [M5: num,N2: num] : ( product_Pair @ int @ int @ ( divide_divide @ int @ ( numeral_numeral @ int @ M5 ) @ ( numeral_numeral @ int @ N2 ) ) @ ( modulo_modulo @ int @ ( numeral_numeral @ int @ M5 ) @ ( numeral_numeral @ int @ N2 ) ) ) ) ) ).
% divmod_int_def
thf(fact_2532_vebt__assn__raw_Ocases,axiom,
! [X: product_prod @ vEBT_VEBT @ vEBT_VEBTi] :
( ! [A4: $o,B3: $o,Ai: $o,Bi: $o] :
( X
!= ( product_Pair @ vEBT_VEBT @ vEBT_VEBTi @ ( vEBT_Leaf @ A4 @ B3 ) @ ( vEBT_Leafi @ Ai @ Bi ) ) )
=> ( ! [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] :
( X
!= ( 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,Vd3: $o,Ve3: $o] :
( X
!= ( product_Pair @ vEBT_VEBT @ vEBT_VEBTi @ ( vEBT_Node @ V3 @ Va3 @ Vb3 @ Vc3 ) @ ( vEBT_Leafi @ Vd3 @ Ve3 ) ) )
=> ~ ! [Vd3: $o,Ve3: $o,V3: option @ ( product_prod @ nat @ nat ),Va3: nat,Vb3: array @ vEBT_VEBTi,Vc3: vEBT_VEBTi] :
( X
!= ( product_Pair @ vEBT_VEBT @ vEBT_VEBTi @ ( vEBT_Leaf @ Vd3 @ Ve3 ) @ ( vEBT_Nodei @ V3 @ Va3 @ Vb3 @ Vc3 ) ) ) ) ) ) ).
% vebt_assn_raw.cases
thf(fact_2533_VEBT__internal_OminNulli_Ocases,axiom,
! [X: vEBT_VEBTi] :
( ( X
!= ( vEBT_Leafi @ $false @ $false ) )
=> ( ! [Uv: $o] :
( X
!= ( vEBT_Leafi @ $true @ Uv ) )
=> ( ! [Uu: $o] :
( X
!= ( vEBT_Leafi @ Uu @ $true ) )
=> ( ! [Uw: nat,Ux2: array @ vEBT_VEBTi,Uy2: vEBT_VEBTi] :
( X
!= ( vEBT_Nodei @ ( none @ ( product_prod @ nat @ nat ) ) @ Uw @ Ux2 @ Uy2 ) )
=> ~ ! [Uz2: product_prod @ nat @ nat,Va2: nat,Vb2: array @ vEBT_VEBTi,Vc2: vEBT_VEBTi] :
( X
!= ( vEBT_Nodei @ ( some @ ( product_prod @ nat @ nat ) @ Uz2 ) @ Va2 @ Vb2 @ Vc2 ) ) ) ) ) ) ).
% VEBT_internal.minNulli.cases
thf(fact_2534_int__mod__ge,axiom,
! [A2: int,N3: int] :
( ( ord_less @ int @ A2 @ N3 )
=> ( ( ord_less @ int @ ( zero_zero @ int ) @ N3 )
=> ( ord_less_eq @ int @ A2 @ ( modulo_modulo @ int @ A2 @ N3 ) ) ) ) ).
% int_mod_ge
thf(fact_2535_neg__mod__conj,axiom,
! [B2: int,A2: int] :
( ( ord_less @ int @ B2 @ ( zero_zero @ int ) )
=> ( ( ord_less_eq @ int @ ( modulo_modulo @ int @ A2 @ B2 ) @ ( zero_zero @ int ) )
& ( ord_less @ int @ B2 @ ( modulo_modulo @ int @ A2 @ B2 ) ) ) ) ).
% neg_mod_conj
thf(fact_2536_pos__mod__conj,axiom,
! [B2: int,A2: int] :
( ( ord_less @ int @ ( zero_zero @ int ) @ B2 )
=> ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ ( modulo_modulo @ int @ A2 @ B2 ) )
& ( ord_less @ int @ ( modulo_modulo @ int @ A2 @ B2 ) @ B2 ) ) ) ).
% pos_mod_conj
thf(fact_2537_int__mod__eq,axiom,
! [B2: int,N3: int,A2: int] :
( ( ord_less_eq @ int @ ( zero_zero @ int ) @ B2 )
=> ( ( ord_less @ int @ B2 @ N3 )
=> ( ( ( modulo_modulo @ int @ A2 @ N3 )
= ( modulo_modulo @ int @ B2 @ N3 ) )
=> ( ( modulo_modulo @ int @ A2 @ N3 )
= B2 ) ) ) ) ).
% int_mod_eq
thf(fact_2538_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_2539_int__mod__lem,axiom,
! [N3: int,B2: int] :
( ( ord_less @ int @ ( zero_zero @ int ) @ N3 )
=> ( ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ B2 )
& ( ord_less @ int @ B2 @ N3 ) )
= ( ( modulo_modulo @ int @ B2 @ N3 )
= B2 ) ) ) ).
% int_mod_lem
thf(fact_2540_neg__mod__sign,axiom,
! [L2: int,K: int] :
( ( ord_less @ int @ L2 @ ( zero_zero @ int ) )
=> ( ord_less_eq @ int @ ( modulo_modulo @ int @ K @ L2 ) @ ( zero_zero @ int ) ) ) ).
% neg_mod_sign
thf(fact_2541_Euclidean__Division_Opos__mod__sign,axiom,
! [L2: int,K: int] :
( ( ord_less @ int @ ( zero_zero @ int ) @ L2 )
=> ( ord_less_eq @ int @ ( zero_zero @ int ) @ ( modulo_modulo @ int @ K @ L2 ) ) ) ).
% Euclidean_Division.pos_mod_sign
thf(fact_2542_int__mod__le_H,axiom,
! [B2: int,N3: int] :
( ( ord_less_eq @ int @ ( zero_zero @ int ) @ ( minus_minus @ int @ B2 @ N3 ) )
=> ( ord_less_eq @ int @ ( modulo_modulo @ int @ B2 @ N3 ) @ ( minus_minus @ int @ B2 @ N3 ) ) ) ).
% int_mod_le'
thf(fact_2543_nonneg__mod__div,axiom,
! [A2: int,B2: int] :
( ( ord_less_eq @ int @ ( zero_zero @ int ) @ A2 )
=> ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ B2 )
=> ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ ( modulo_modulo @ int @ A2 @ B2 ) )
& ( ord_less_eq @ int @ ( zero_zero @ int ) @ ( divide_divide @ int @ A2 @ B2 ) ) ) ) ) ).
% nonneg_mod_div
thf(fact_2544_zdvd__imp__le,axiom,
! [Z: int,N3: int] :
( ( dvd_dvd @ int @ Z @ N3 )
=> ( ( ord_less @ int @ ( zero_zero @ int ) @ N3 )
=> ( ord_less_eq @ int @ Z @ N3 ) ) ) ).
% zdvd_imp_le
thf(fact_2545_mod__div__equality__div__eq,axiom,
! [A2: int,B2: int] :
( ( times_times @ int @ ( divide_divide @ int @ A2 @ B2 ) @ B2 )
= ( minus_minus @ int @ A2 @ ( modulo_modulo @ int @ A2 @ B2 ) ) ) ).
% mod_div_equality_div_eq
thf(fact_2546_real__of__int__div,axiom,
! [D: int,N3: int] :
( ( dvd_dvd @ int @ D @ N3 )
=> ( ( ring_1_of_int @ real @ ( divide_divide @ int @ N3 @ D ) )
= ( divide_divide @ real @ ( ring_1_of_int @ real @ N3 ) @ ( ring_1_of_int @ real @ D ) ) ) ) ).
% real_of_int_div
thf(fact_2547_emep1,axiom,
! [N3: int,D: int] :
( ( dvd_dvd @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ N3 )
=> ( ( dvd_dvd @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ D )
=> ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ D )
=> ( ( modulo_modulo @ int @ ( plus_plus @ int @ N3 @ ( one_one @ int ) ) @ D )
= ( plus_plus @ int @ ( modulo_modulo @ int @ N3 @ D ) @ ( one_one @ int ) ) ) ) ) ) ).
% emep1
thf(fact_2548_eme1p,axiom,
! [N3: int,D: int] :
( ( dvd_dvd @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ N3 )
=> ( ( dvd_dvd @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ D )
=> ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ D )
=> ( ( modulo_modulo @ int @ ( plus_plus @ int @ ( one_one @ int ) @ N3 ) @ D )
= ( plus_plus @ int @ ( one_one @ int ) @ ( modulo_modulo @ int @ N3 @ D ) ) ) ) ) ) ).
% eme1p
thf(fact_2549_prod__decode__aux_Ocases,axiom,
! [X: product_prod @ nat @ nat] :
~ ! [K3: nat,M4: nat] :
( X
!= ( product_Pair @ nat @ nat @ K3 @ M4 ) ) ).
% prod_decode_aux.cases
thf(fact_2550_vebt__minti_Ocases,axiom,
! [X: vEBT_VEBTi] :
( ! [A4: $o,B3: $o] :
( X
!= ( vEBT_Leafi @ A4 @ B3 ) )
=> ( ! [Uu: nat,Uv: array @ vEBT_VEBTi,Uw: vEBT_VEBTi] :
( X
!= ( vEBT_Nodei @ ( none @ ( product_prod @ nat @ nat ) ) @ Uu @ Uv @ Uw ) )
=> ~ ! [Mi2: nat,Ma2: nat,Ux2: nat,Uy2: array @ vEBT_VEBTi,Uz2: vEBT_VEBTi] :
( X
!= ( vEBT_Nodei @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi2 @ Ma2 ) ) @ Ux2 @ Uy2 @ Uz2 ) ) ) ) ).
% vebt_minti.cases
thf(fact_2551_pos__mod__bound2,axiom,
! [A2: int] : ( ord_less @ int @ ( modulo_modulo @ int @ A2 @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) ) @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) ) ).
% pos_mod_bound2
thf(fact_2552_int__mod__ge_H,axiom,
! [B2: int,N3: int] :
( ( ord_less @ int @ B2 @ ( zero_zero @ int ) )
=> ( ( ord_less @ int @ ( zero_zero @ int ) @ N3 )
=> ( ord_less_eq @ int @ ( plus_plus @ int @ B2 @ N3 ) @ ( modulo_modulo @ int @ B2 @ N3 ) ) ) ) ).
% int_mod_ge'
thf(fact_2553_mod__pos__neg__trivial,axiom,
! [K: int,L2: int] :
( ( ord_less @ int @ ( zero_zero @ int ) @ K )
=> ( ( ord_less_eq @ int @ ( plus_plus @ int @ K @ L2 ) @ ( zero_zero @ int ) )
=> ( ( modulo_modulo @ int @ K @ L2 )
= ( plus_plus @ int @ K @ L2 ) ) ) ) ).
% mod_pos_neg_trivial
thf(fact_2554_mod__pos__geq,axiom,
! [L2: int,K: int] :
( ( ord_less @ int @ ( zero_zero @ int ) @ L2 )
=> ( ( ord_less_eq @ int @ L2 @ K )
=> ( ( modulo_modulo @ int @ K @ L2 )
= ( modulo_modulo @ int @ ( minus_minus @ int @ K @ L2 ) @ L2 ) ) ) ) ).
% mod_pos_geq
thf(fact_2555_int__div__sub__1,axiom,
! [M: int,N3: int] :
( ( ord_less_eq @ int @ ( one_one @ int ) @ M )
=> ( ( ( dvd_dvd @ int @ M @ N3 )
=> ( ( divide_divide @ int @ ( minus_minus @ int @ N3 @ ( one_one @ int ) ) @ M )
= ( minus_minus @ int @ ( divide_divide @ int @ N3 @ M ) @ ( one_one @ int ) ) ) )
& ( ~ ( dvd_dvd @ int @ M @ N3 )
=> ( ( divide_divide @ int @ ( minus_minus @ int @ N3 @ ( one_one @ int ) ) @ M )
= ( divide_divide @ int @ N3 @ M ) ) ) ) ) ).
% int_div_sub_1
thf(fact_2556_real__of__int__div__aux,axiom,
! [X: int,D: int] :
( ( divide_divide @ real @ ( ring_1_of_int @ real @ X ) @ ( ring_1_of_int @ real @ D ) )
= ( plus_plus @ real @ ( ring_1_of_int @ real @ ( divide_divide @ int @ X @ D ) ) @ ( divide_divide @ real @ ( ring_1_of_int @ real @ ( modulo_modulo @ int @ X @ D ) ) @ ( ring_1_of_int @ real @ D ) ) ) ) ).
% real_of_int_div_aux
thf(fact_2557_bset_I9_J,axiom,
! [D: int,D4: int,B6: set @ int,T2: int] :
( ( dvd_dvd @ int @ D @ D4 )
=> ! [X5: int] :
( ! [Xa3: int] :
( ( member @ int @ Xa3 @ ( set_or1337092689740270186AtMost @ int @ ( one_one @ int ) @ D4 ) )
=> ! [Xb3: int] :
( ( member @ int @ Xb3 @ B6 )
=> ( X5
!= ( plus_plus @ int @ Xb3 @ Xa3 ) ) ) )
=> ( ( dvd_dvd @ int @ D @ ( plus_plus @ int @ X5 @ T2 ) )
=> ( dvd_dvd @ int @ D @ ( plus_plus @ int @ ( minus_minus @ int @ X5 @ D4 ) @ T2 ) ) ) ) ) ).
% bset(9)
thf(fact_2558_bset_I10_J,axiom,
! [D: int,D4: int,B6: set @ int,T2: int] :
( ( dvd_dvd @ int @ D @ D4 )
=> ! [X5: int] :
( ! [Xa3: int] :
( ( member @ int @ Xa3 @ ( set_or1337092689740270186AtMost @ int @ ( one_one @ int ) @ D4 ) )
=> ! [Xb3: int] :
( ( member @ int @ Xb3 @ B6 )
=> ( X5
!= ( plus_plus @ int @ Xb3 @ Xa3 ) ) ) )
=> ( ~ ( dvd_dvd @ int @ D @ ( plus_plus @ int @ X5 @ T2 ) )
=> ~ ( dvd_dvd @ int @ D @ ( plus_plus @ int @ ( minus_minus @ int @ X5 @ D4 ) @ T2 ) ) ) ) ) ).
% bset(10)
thf(fact_2559_aset_I9_J,axiom,
! [D: int,D4: int,A3: set @ int,T2: int] :
( ( dvd_dvd @ int @ D @ D4 )
=> ! [X5: int] :
( ! [Xa3: int] :
( ( member @ int @ Xa3 @ ( set_or1337092689740270186AtMost @ int @ ( one_one @ int ) @ D4 ) )
=> ! [Xb3: int] :
( ( member @ int @ Xb3 @ A3 )
=> ( X5
!= ( minus_minus @ int @ Xb3 @ Xa3 ) ) ) )
=> ( ( dvd_dvd @ int @ D @ ( plus_plus @ int @ X5 @ T2 ) )
=> ( dvd_dvd @ int @ D @ ( plus_plus @ int @ ( plus_plus @ int @ X5 @ D4 ) @ T2 ) ) ) ) ) ).
% aset(9)
thf(fact_2560_aset_I10_J,axiom,
! [D: int,D4: int,A3: set @ int,T2: int] :
( ( dvd_dvd @ int @ D @ D4 )
=> ! [X5: int] :
( ! [Xa3: int] :
( ( member @ int @ Xa3 @ ( set_or1337092689740270186AtMost @ int @ ( one_one @ int ) @ D4 ) )
=> ! [Xb3: int] :
( ( member @ int @ Xb3 @ A3 )
=> ( X5
!= ( minus_minus @ int @ Xb3 @ Xa3 ) ) ) )
=> ( ~ ( dvd_dvd @ int @ D @ ( plus_plus @ int @ X5 @ T2 ) )
=> ~ ( dvd_dvd @ int @ D @ ( plus_plus @ int @ ( plus_plus @ int @ X5 @ D4 ) @ T2 ) ) ) ) ) ).
% aset(10)
thf(fact_2561_pos__mod__sign2,axiom,
! [A2: int] : ( ord_less_eq @ int @ ( zero_zero @ int ) @ ( modulo_modulo @ int @ A2 @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) ) ) ).
% pos_mod_sign2
thf(fact_2562_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_2563_nmod2,axiom,
! [N3: int] :
( ( ( modulo_modulo @ int @ N3 @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) )
= ( zero_zero @ int ) )
| ( ( modulo_modulo @ int @ N3 @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) )
= ( one_one @ int ) ) ) ).
% nmod2
thf(fact_2564_mod__exp__less__eq__exp,axiom,
! [A2: int,N3: nat] : ( ord_less @ int @ ( modulo_modulo @ int @ A2 @ ( power_power @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ N3 ) ) @ ( power_power @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ N3 ) ) ).
% mod_exp_less_eq_exp
thf(fact_2565_even__diff__iff,axiom,
! [K: int,L2: int] :
( ( dvd_dvd @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ ( minus_minus @ int @ K @ L2 ) )
= ( dvd_dvd @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ ( plus_plus @ int @ K @ L2 ) ) ) ).
% even_diff_iff
thf(fact_2566_mod__power__lem,axiom,
! [A2: int,M: nat,N3: nat] :
( ( ord_less @ int @ ( one_one @ int ) @ A2 )
=> ( ( ( ord_less_eq @ nat @ M @ N3 )
=> ( ( modulo_modulo @ int @ ( power_power @ int @ A2 @ N3 ) @ ( power_power @ int @ A2 @ M ) )
= ( zero_zero @ int ) ) )
& ( ~ ( ord_less_eq @ nat @ M @ N3 )
=> ( ( modulo_modulo @ int @ ( power_power @ int @ A2 @ N3 ) @ ( power_power @ int @ A2 @ M ) )
= ( power_power @ int @ A2 @ N3 ) ) ) ) ) ).
% mod_power_lem
thf(fact_2567_split__zmod,axiom,
! [P: int > $o,N3: int,K: int] :
( ( P @ ( modulo_modulo @ int @ N3 @ K ) )
= ( ( ( K
= ( zero_zero @ int ) )
=> ( P @ N3 ) )
& ( ( ord_less @ int @ ( zero_zero @ int ) @ K )
=> ! [I3: int,J3: int] :
( ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ J3 )
& ( ord_less @ int @ J3 @ K )
& ( N3
= ( plus_plus @ int @ ( times_times @ int @ K @ I3 ) @ J3 ) ) )
=> ( P @ J3 ) ) )
& ( ( ord_less @ int @ K @ ( zero_zero @ int ) )
=> ! [I3: int,J3: int] :
( ( ( ord_less @ int @ K @ J3 )
& ( ord_less_eq @ int @ J3 @ ( zero_zero @ int ) )
& ( N3
= ( plus_plus @ int @ ( times_times @ int @ K @ I3 ) @ J3 ) ) )
=> ( P @ J3 ) ) ) ) ) ).
% split_zmod
thf(fact_2568_int__mod__neg__eq,axiom,
! [A2: int,B2: int,Q2: int,R2: int] :
( ( A2
= ( plus_plus @ int @ ( times_times @ int @ B2 @ Q2 ) @ R2 ) )
=> ( ( ord_less_eq @ int @ R2 @ ( zero_zero @ int ) )
=> ( ( ord_less @ int @ B2 @ R2 )
=> ( ( modulo_modulo @ int @ A2 @ B2 )
= R2 ) ) ) ) ).
% int_mod_neg_eq
thf(fact_2569_int__mod__pos__eq,axiom,
! [A2: int,B2: int,Q2: int,R2: int] :
( ( A2
= ( plus_plus @ int @ ( times_times @ int @ B2 @ Q2 ) @ R2 ) )
=> ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ R2 )
=> ( ( ord_less @ int @ R2 @ B2 )
=> ( ( modulo_modulo @ int @ A2 @ B2 )
= R2 ) ) ) ) ).
% int_mod_pos_eq
thf(fact_2570_mod__add__if__z,axiom,
! [X: int,Z: int,Y: int] :
( ( ord_less @ int @ X @ Z )
=> ( ( ord_less @ int @ Y @ Z )
=> ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ Y )
=> ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ X )
=> ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ Z )
=> ( ( ( ord_less @ int @ ( plus_plus @ int @ X @ Y ) @ Z )
=> ( ( modulo_modulo @ int @ ( plus_plus @ int @ X @ Y ) @ Z )
= ( plus_plus @ int @ X @ Y ) ) )
& ( ~ ( ord_less @ int @ ( plus_plus @ int @ X @ Y ) @ Z )
=> ( ( modulo_modulo @ int @ ( plus_plus @ int @ X @ Y ) @ Z )
= ( minus_minus @ int @ ( plus_plus @ int @ X @ Y ) @ Z ) ) ) ) ) ) ) ) ) ).
% mod_add_if_z
thf(fact_2571_mod__sub__if__z,axiom,
! [X: int,Z: int,Y: int] :
( ( ord_less @ int @ X @ Z )
=> ( ( ord_less @ int @ Y @ Z )
=> ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ Y )
=> ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ X )
=> ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ Z )
=> ( ( ( ord_less_eq @ int @ Y @ X )
=> ( ( modulo_modulo @ int @ ( minus_minus @ int @ X @ Y ) @ Z )
= ( minus_minus @ int @ X @ Y ) ) )
& ( ~ ( ord_less_eq @ int @ Y @ X )
=> ( ( modulo_modulo @ int @ ( minus_minus @ int @ X @ Y ) @ Z )
= ( plus_plus @ int @ ( minus_minus @ int @ X @ Y ) @ Z ) ) ) ) ) ) ) ) ) ).
% mod_sub_if_z
thf(fact_2572_zmod__zmult2__eq,axiom,
! [C2: int,A2: int,B2: int] :
( ( ord_less_eq @ int @ ( zero_zero @ int ) @ C2 )
=> ( ( modulo_modulo @ int @ A2 @ ( times_times @ int @ B2 @ C2 ) )
= ( plus_plus @ int @ ( times_times @ int @ B2 @ ( modulo_modulo @ int @ ( divide_divide @ int @ A2 @ B2 ) @ C2 ) ) @ ( modulo_modulo @ int @ A2 @ B2 ) ) ) ) ).
% zmod_zmult2_eq
thf(fact_2573_axxmod2,axiom,
! [X: int] :
( ( ( modulo_modulo @ int @ ( plus_plus @ int @ ( plus_plus @ int @ ( one_one @ int ) @ X ) @ X ) @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) )
= ( one_one @ int ) )
& ( ( modulo_modulo @ int @ ( plus_plus @ int @ ( plus_plus @ int @ ( zero_zero @ int ) @ X ) @ X ) @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) )
= ( zero_zero @ int ) ) ) ).
% axxmod2
thf(fact_2574_z1pmod2,axiom,
! [B2: int] :
( ( modulo_modulo @ int @ ( plus_plus @ int @ ( times_times @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ B2 ) @ ( one_one @ int ) ) @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) )
= ( one_one @ int ) ) ).
% z1pmod2
thf(fact_2575_verit__le__mono__div__int,axiom,
! [A3: int,B6: int,N3: int] :
( ( ord_less @ int @ A3 @ B6 )
=> ( ( ord_less @ int @ ( zero_zero @ int ) @ N3 )
=> ( ord_less_eq @ int
@ ( plus_plus @ int @ ( divide_divide @ int @ A3 @ N3 )
@ ( if @ int
@ ( ( modulo_modulo @ int @ B6 @ N3 )
= ( zero_zero @ int ) )
@ ( one_one @ int )
@ ( zero_zero @ int ) ) )
@ ( divide_divide @ int @ B6 @ N3 ) ) ) ) ).
% verit_le_mono_div_int
thf(fact_2576_split__pos__lemma,axiom,
! [K: int,P: int > int > $o,N3: int] :
( ( ord_less @ int @ ( zero_zero @ int ) @ K )
=> ( ( P @ ( divide_divide @ int @ N3 @ K ) @ ( modulo_modulo @ int @ N3 @ K ) )
= ( ! [I3: int,J3: int] :
( ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ J3 )
& ( ord_less @ int @ J3 @ K )
& ( N3
= ( plus_plus @ int @ ( times_times @ int @ K @ I3 ) @ J3 ) ) )
=> ( P @ I3 @ J3 ) ) ) ) ) ).
% split_pos_lemma
thf(fact_2577_split__neg__lemma,axiom,
! [K: int,P: int > int > $o,N3: int] :
( ( ord_less @ int @ K @ ( zero_zero @ int ) )
=> ( ( P @ ( divide_divide @ int @ N3 @ K ) @ ( modulo_modulo @ int @ N3 @ K ) )
= ( ! [I3: int,J3: int] :
( ( ( ord_less @ int @ K @ J3 )
& ( ord_less_eq @ int @ J3 @ ( zero_zero @ int ) )
& ( N3
= ( plus_plus @ int @ ( times_times @ int @ K @ I3 ) @ J3 ) ) )
=> ( P @ I3 @ J3 ) ) ) ) ) ).
% split_neg_lemma
thf(fact_2578_list__decode_Ocases,axiom,
! [X: nat] :
( ( X
!= ( zero_zero @ nat ) )
=> ~ ! [N: nat] :
( X
!= ( suc @ N ) ) ) ).
% list_decode.cases
thf(fact_2579_dvd__productE,axiom,
! [A: $tType] :
( ( semiring_gcd @ A )
=> ! [P4: A,A2: A,B2: A] :
( ( dvd_dvd @ A @ P4 @ ( times_times @ A @ A2 @ B2 ) )
=> ~ ! [X3: A,Y3: A] :
( ( P4
= ( times_times @ A @ X3 @ Y3 ) )
=> ( ( dvd_dvd @ A @ X3 @ A2 )
=> ~ ( dvd_dvd @ A @ Y3 @ B2 ) ) ) ) ) ).
% dvd_productE
thf(fact_2580_division__decomp,axiom,
! [A: $tType] :
( ( semiring_gcd @ A )
=> ! [A2: A,B2: A,C2: A] :
( ( dvd_dvd @ A @ A2 @ ( times_times @ A @ B2 @ C2 ) )
=> ? [B8: A,C6: A] :
( ( A2
= ( times_times @ A @ B8 @ C6 ) )
& ( dvd_dvd @ A @ B8 @ B2 )
& ( dvd_dvd @ A @ C6 @ C2 ) ) ) ) ).
% division_decomp
thf(fact_2581_Euclid__induct,axiom,
! [P: nat > nat > $o,A2: nat,B2: nat] :
( ! [A4: nat,B3: nat] :
( ( P @ A4 @ B3 )
= ( P @ B3 @ A4 ) )
=> ( ! [A4: nat] : ( P @ A4 @ ( zero_zero @ nat ) )
=> ( ! [A4: nat,B3: nat] :
( ( P @ A4 @ B3 )
=> ( P @ A4 @ ( plus_plus @ nat @ A4 @ B3 ) ) )
=> ( P @ A2 @ B2 ) ) ) ) ).
% Euclid_induct
thf(fact_2582_p1mod22k_H,axiom,
! [B2: int,N3: nat] :
( ( modulo_modulo @ int @ ( plus_plus @ int @ ( one_one @ int ) @ ( times_times @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ B2 ) ) @ ( times_times @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ ( power_power @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ N3 ) ) )
= ( plus_plus @ int @ ( one_one @ int ) @ ( times_times @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ ( modulo_modulo @ int @ B2 @ ( power_power @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ N3 ) ) ) ) ) ).
% p1mod22k'
thf(fact_2583_p1mod22k,axiom,
! [B2: int,N3: nat] :
( ( modulo_modulo @ int @ ( plus_plus @ int @ ( times_times @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ B2 ) @ ( one_one @ int ) ) @ ( times_times @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ ( power_power @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ N3 ) ) )
= ( plus_plus @ int @ ( times_times @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ ( modulo_modulo @ int @ B2 @ ( power_power @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ N3 ) ) ) @ ( one_one @ int ) ) ) ).
% p1mod22k
thf(fact_2584_eq__diff__eq_H,axiom,
! [X: real,Y: real,Z: real] :
( ( X
= ( minus_minus @ real @ Y @ Z ) )
= ( Y
= ( plus_plus @ real @ X @ Z ) ) ) ).
% eq_diff_eq'
thf(fact_2585_pos__zmod__mult__2,axiom,
! [A2: int,B2: int] :
( ( ord_less_eq @ int @ ( zero_zero @ int ) @ A2 )
=> ( ( modulo_modulo @ int @ ( plus_plus @ int @ ( one_one @ int ) @ ( times_times @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ B2 ) ) @ ( times_times @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ A2 ) )
= ( plus_plus @ int @ ( one_one @ int ) @ ( times_times @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ ( modulo_modulo @ int @ B2 @ A2 ) ) ) ) ) ).
% pos_zmod_mult_2
thf(fact_2586_sb__inc__lem,axiom,
! [A2: int,K: nat] :
( ( ord_less @ int @ ( plus_plus @ int @ A2 @ ( power_power @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ K ) ) @ ( zero_zero @ int ) )
=> ( ord_less_eq @ int @ ( plus_plus @ int @ ( plus_plus @ int @ A2 @ ( power_power @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ K ) ) @ ( power_power @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ ( suc @ K ) ) ) @ ( modulo_modulo @ int @ ( plus_plus @ int @ A2 @ ( power_power @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ K ) ) @ ( power_power @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ ( suc @ K ) ) ) ) ) ).
% sb_inc_lem
thf(fact_2587_neg__zmod__mult__2,axiom,
! [A2: int,B2: int] :
( ( ord_less_eq @ int @ A2 @ ( zero_zero @ int ) )
=> ( ( modulo_modulo @ int @ ( plus_plus @ int @ ( one_one @ int ) @ ( times_times @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ B2 ) ) @ ( times_times @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ A2 ) )
= ( minus_minus @ int @ ( times_times @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ ( modulo_modulo @ int @ ( plus_plus @ int @ B2 @ ( one_one @ int ) ) @ A2 ) ) @ ( one_one @ int ) ) ) ) ).
% neg_zmod_mult_2
thf(fact_2588_dvd__pos__nat,axiom,
! [N3: nat,M: nat] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ N3 )
=> ( ( dvd_dvd @ nat @ M @ N3 )
=> ( ord_less @ nat @ ( zero_zero @ nat ) @ M ) ) ) ).
% dvd_pos_nat
thf(fact_2589_gcd__nat__induct,axiom,
! [P: nat > nat > $o,M: nat,N3: nat] :
( ! [M4: nat] : ( P @ M4 @ ( zero_zero @ nat ) )
=> ( ! [M4: nat,N: nat] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ N )
=> ( ( P @ N @ ( modulo_modulo @ nat @ M4 @ N ) )
=> ( P @ M4 @ N ) ) )
=> ( P @ M @ N3 ) ) ) ).
% gcd_nat_induct
thf(fact_2590_bezout__add__nat,axiom,
! [A2: nat,B2: nat] :
? [D3: nat,X3: nat,Y3: nat] :
( ( dvd_dvd @ nat @ D3 @ A2 )
& ( dvd_dvd @ nat @ D3 @ B2 )
& ( ( ( times_times @ nat @ A2 @ X3 )
= ( plus_plus @ nat @ ( times_times @ nat @ B2 @ Y3 ) @ D3 ) )
| ( ( times_times @ nat @ B2 @ X3 )
= ( plus_plus @ nat @ ( times_times @ nat @ A2 @ Y3 ) @ D3 ) ) ) ) ).
% bezout_add_nat
thf(fact_2591_bezout__lemma__nat,axiom,
! [D: nat,A2: nat,B2: nat,X: nat,Y: nat] :
( ( dvd_dvd @ nat @ D @ A2 )
=> ( ( dvd_dvd @ nat @ D @ B2 )
=> ( ( ( ( times_times @ nat @ A2 @ X )
= ( plus_plus @ nat @ ( times_times @ nat @ B2 @ Y ) @ D ) )
| ( ( times_times @ nat @ B2 @ X )
= ( plus_plus @ nat @ ( times_times @ nat @ A2 @ Y ) @ D ) ) )
=> ? [X3: nat,Y3: nat] :
( ( dvd_dvd @ nat @ D @ A2 )
& ( dvd_dvd @ nat @ D @ ( plus_plus @ nat @ A2 @ B2 ) )
& ( ( ( times_times @ nat @ A2 @ X3 )
= ( plus_plus @ nat @ ( times_times @ nat @ ( plus_plus @ nat @ A2 @ B2 ) @ Y3 ) @ D ) )
| ( ( times_times @ nat @ ( plus_plus @ nat @ A2 @ B2 ) @ X3 )
= ( plus_plus @ nat @ ( times_times @ nat @ A2 @ Y3 ) @ D ) ) ) ) ) ) ) ).
% bezout_lemma_nat
thf(fact_2592_bezout1__nat,axiom,
! [A2: nat,B2: nat] :
? [D3: nat,X3: nat,Y3: nat] :
( ( dvd_dvd @ nat @ D3 @ A2 )
& ( dvd_dvd @ nat @ D3 @ B2 )
& ( ( ( minus_minus @ nat @ ( times_times @ nat @ A2 @ X3 ) @ ( times_times @ nat @ B2 @ Y3 ) )
= D3 )
| ( ( minus_minus @ nat @ ( times_times @ nat @ B2 @ X3 ) @ ( times_times @ nat @ A2 @ Y3 ) )
= D3 ) ) ) ).
% bezout1_nat
thf(fact_2593_bezout__add__strong__nat,axiom,
! [A2: nat,B2: nat] :
( ( A2
!= ( zero_zero @ nat ) )
=> ? [D3: nat,X3: nat,Y3: nat] :
( ( dvd_dvd @ nat @ D3 @ A2 )
& ( dvd_dvd @ nat @ D3 @ B2 )
& ( ( times_times @ nat @ A2 @ X3 )
= ( plus_plus @ nat @ ( times_times @ nat @ B2 @ Y3 ) @ D3 ) ) ) ) ).
% bezout_add_strong_nat
thf(fact_2594_htt__vebt__buildupi_H__univ,axiom,
! [U: nat,N3: nat] :
( ( U
= ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N3 ) )
=> ( time_htt @ vEBT_VEBTi @ ( one_one @ assn ) @ ( vEBT_V739175172307565963ildupi @ N3 ) @ ( vEBT_vebt_assn_raw @ ( vEBT_vebt_buildup @ N3 ) ) @ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ ( bit1 @ ( bit0 @ one2 ) ) ) ) @ U ) ) ) ).
% htt_vebt_buildupi'_univ
thf(fact_2595_htt__vebt__buildupi__univ,axiom,
! [U: nat,N3: nat] :
( ( U
= ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N3 ) )
=> ( time_htt @ vEBT_VEBTi @ ( one_one @ assn ) @ ( vEBT_vebt_buildupi @ N3 ) @ ( vEBT_vebt_assn_raw @ ( vEBT_vebt_buildup @ N3 ) ) @ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ ( bit1 @ ( bit0 @ one2 ) ) ) ) @ U ) ) ) ).
% htt_vebt_buildupi_univ
thf(fact_2596_T__vebt__buildupi,axiom,
! [N3: nat,H2: heap_ext @ product_unit] : ( ord_less_eq @ nat @ ( time_time @ vEBT_VEBTi @ ( vEBT_V739175172307565963ildupi @ N3 ) @ H2 ) @ ( vEBT_V441764108873111860ildupi @ N3 ) ) ).
% T_vebt_buildupi
thf(fact_2597_htt__vebt__buildupi,axiom,
! [N3: nat] : ( time_htt @ vEBT_VEBTi @ ( one_one @ assn ) @ ( vEBT_vebt_buildupi @ N3 ) @ ( vEBT_vebt_assn_raw @ ( vEBT_vebt_buildup @ N3 ) ) @ ( vEBT_V441764108873111860ildupi @ N3 ) ) ).
% htt_vebt_buildupi
thf(fact_2598_htt__vebt__buildupi_H,axiom,
! [N3: nat] : ( time_htt @ vEBT_VEBTi @ ( one_one @ assn ) @ ( vEBT_V739175172307565963ildupi @ N3 ) @ ( vEBT_vebt_assn_raw @ ( vEBT_vebt_buildup @ N3 ) ) @ ( vEBT_V441764108873111860ildupi @ N3 ) ) ).
% htt_vebt_buildupi'
thf(fact_2599_TBOUND__vebt__inserti,axiom,
! [T2: vEBT_VEBT,Ti: vEBT_VEBTi,X: nat] : ( time_TBOUND @ vEBT_VEBTi @ ( vEBT_V3964819847710782039nserti @ T2 @ Ti @ X ) @ ( if @ nat @ ( vEBT_VEBT_minNull @ T2 ) @ ( one_one @ nat ) @ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit1 @ ( bit0 @ ( bit1 @ one2 ) ) ) ) @ ( plus_plus @ nat @ ( one_one @ nat ) @ ( vEBT_VEBT_height @ T2 ) ) ) ) ) ).
% TBOUND_vebt_inserti
thf(fact_2600_TBOUND__vebt__buildupi,axiom,
! [N3: nat] : ( time_TBOUND @ vEBT_VEBTi @ ( vEBT_V739175172307565963ildupi @ N3 ) @ ( vEBT_V441764108873111860ildupi @ N3 ) ) ).
% TBOUND_vebt_buildupi
thf(fact_2601_TBOUND__minNull,axiom,
! [T2: vEBT_VEBT,Ti: vEBT_VEBTi,X: nat] :
( ( vEBT_VEBT_minNull @ T2 )
=> ( time_TBOUND @ vEBT_VEBTi @ ( vEBT_V3964819847710782039nserti @ T2 @ Ti @ X ) @ ( one_one @ nat ) ) ) ).
% TBOUND_minNull
thf(fact_2602_TBOUND__buildupi,axiom,
! [N3: nat] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ N3 )
=> ( time_TBOUND @ vEBT_VEBTi @ ( vEBT_vebt_buildupi @ N3 ) @ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ ( bit1 @ ( bit0 @ one2 ) ) ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N3 ) ) ) ) ).
% TBOUND_buildupi
thf(fact_2603_htt__vebt__inserti,axiom,
! [T2: vEBT_VEBT,Ti: vEBT_VEBTi,X: nat] : ( time_htt @ vEBT_VEBTi @ ( vEBT_vebt_assn_raw @ T2 @ Ti ) @ ( vEBT_vebt_inserti @ Ti @ X ) @ ( vEBT_vebt_assn_raw @ ( vEBT_vebt_insert @ T2 @ X ) ) @ ( plus_plus @ nat @ ( numeral_numeral @ nat @ ( bit1 @ ( bit0 @ ( bit1 @ one2 ) ) ) ) @ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit1 @ ( bit0 @ ( bit1 @ one2 ) ) ) ) @ ( vEBT_VEBT_height @ T2 ) ) ) ) ).
% htt_vebt_inserti
thf(fact_2604_time__replicate,axiom,
! [A: $tType,X: heap_Time_Heap @ A,C2: nat,N3: nat,H2: heap_ext @ product_unit] :
( ! [H4: heap_ext @ product_unit] : ( ord_less_eq @ nat @ ( time_time @ A @ X @ H4 ) @ C2 )
=> ( ord_less_eq @ nat @ ( time_time @ ( list @ A ) @ ( vEBT_VEBT_replicatei @ A @ N3 @ X ) @ H2 ) @ ( plus_plus @ nat @ ( one_one @ nat ) @ ( times_times @ nat @ ( plus_plus @ nat @ ( one_one @ nat ) @ C2 ) @ N3 ) ) ) ) ).
% time_replicate
thf(fact_2605_TBOUND__replicate,axiom,
! [A: $tType,X: heap_Time_Heap @ A,C2: nat,N3: nat] :
( ( time_TBOUND @ A @ X @ C2 )
=> ( time_TBOUND @ ( list @ A ) @ ( vEBT_VEBT_replicatei @ A @ N3 @ X ) @ ( plus_plus @ nat @ ( one_one @ nat ) @ ( times_times @ nat @ ( plus_plus @ nat @ ( one_one @ nat ) @ C2 ) @ N3 ) ) ) ) ).
% TBOUND_replicate
thf(fact_2606_vebt__buildupi__rule,axiom,
! [N3: nat] : ( time_htt @ vEBT_VEBTi @ ( pure_assn @ ( ord_less @ nat @ ( zero_zero @ nat ) @ N3 ) ) @ ( vEBT_vebt_buildupi @ N3 ) @ ( vEBT_vebt_assn_raw @ ( vEBT_vebt_buildup @ N3 ) ) @ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ ( bit1 @ ( bit0 @ one2 ) ) ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N3 ) ) ) ).
% vebt_buildupi_rule
thf(fact_2607_htt__vebt__inserti__invar__vebt,axiom,
! [T2: vEBT_VEBT,N3: nat,Ti: vEBT_VEBTi,X: nat] :
( ( vEBT_invar_vebt @ T2 @ N3 )
=> ( time_htt @ vEBT_VEBTi @ ( vEBT_vebt_assn_raw @ T2 @ Ti ) @ ( vEBT_vebt_inserti @ Ti @ X ) @ ( vEBT_vebt_assn_raw @ ( vEBT_vebt_insert @ T2 @ X ) ) @ ( plus_plus @ nat @ ( numeral_numeral @ nat @ ( bit1 @ ( bit0 @ ( bit1 @ one2 ) ) ) ) @ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit1 @ ( bit0 @ ( bit1 @ one2 ) ) ) ) @ ( nat2 @ ( archimedean_ceiling @ real @ ( log @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) @ ( semiring_1_of_nat @ real @ N3 ) ) ) ) ) ) ) ) ).
% htt_vebt_inserti_invar_vebt
thf(fact_2608_neg__eucl__rel__int__mult__2,axiom,
! [B2: int,A2: int,Q2: int,R2: int] :
( ( ord_less_eq @ int @ B2 @ ( zero_zero @ int ) )
=> ( ( eucl_rel_int @ ( plus_plus @ int @ A2 @ ( one_one @ int ) ) @ B2 @ ( product_Pair @ int @ int @ Q2 @ R2 ) )
=> ( eucl_rel_int @ ( plus_plus @ int @ ( one_one @ int ) @ ( times_times @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ A2 ) ) @ ( times_times @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ B2 ) @ ( product_Pair @ int @ int @ Q2 @ ( minus_minus @ int @ ( times_times @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ R2 ) @ ( one_one @ int ) ) ) ) ) ) ).
% neg_eucl_rel_int_mult_2
thf(fact_2609_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_2610_nat__numeral,axiom,
! [K: num] :
( ( nat2 @ ( numeral_numeral @ int @ K ) )
= ( numeral_numeral @ nat @ K ) ) ).
% nat_numeral
thf(fact_2611_nat__1,axiom,
( ( nat2 @ ( one_one @ int ) )
= ( suc @ ( zero_zero @ nat ) ) ) ).
% nat_1
thf(fact_2612_nat__le__0,axiom,
! [Z: int] :
( ( ord_less_eq @ int @ Z @ ( zero_zero @ int ) )
=> ( ( nat2 @ Z )
= ( zero_zero @ nat ) ) ) ).
% nat_le_0
thf(fact_2613_nat__0__iff,axiom,
! [I: int] :
( ( ( nat2 @ I )
= ( zero_zero @ nat ) )
= ( ord_less_eq @ int @ I @ ( zero_zero @ int ) ) ) ).
% nat_0_iff
thf(fact_2614_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_2615_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_2616_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_2617_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_2618_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_2619_nat__eq__numeral__power__cancel__iff,axiom,
! [Y: int,X: num,N3: nat] :
( ( ( nat2 @ Y )
= ( power_power @ nat @ ( numeral_numeral @ nat @ X ) @ N3 ) )
= ( Y
= ( power_power @ int @ ( numeral_numeral @ int @ X ) @ N3 ) ) ) ).
% nat_eq_numeral_power_cancel_iff
thf(fact_2620_numeral__power__eq__nat__cancel__iff,axiom,
! [X: num,N3: nat,Y: int] :
( ( ( power_power @ nat @ ( numeral_numeral @ nat @ X ) @ N3 )
= ( nat2 @ Y ) )
= ( ( power_power @ int @ ( numeral_numeral @ int @ X ) @ N3 )
= Y ) ) ).
% numeral_power_eq_nat_cancel_iff
thf(fact_2621_nat__ceiling__le__eq,axiom,
! [X: real,A2: nat] :
( ( ord_less_eq @ nat @ ( nat2 @ ( archimedean_ceiling @ real @ X ) ) @ A2 )
= ( ord_less_eq @ real @ X @ ( semiring_1_of_nat @ real @ A2 ) ) ) ).
% nat_ceiling_le_eq
thf(fact_2622_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_2623_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_2624_nat__less__numeral__power__cancel__iff,axiom,
! [A2: int,X: num,N3: nat] :
( ( ord_less @ nat @ ( nat2 @ A2 ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ X ) @ N3 ) )
= ( ord_less @ int @ A2 @ ( power_power @ int @ ( numeral_numeral @ int @ X ) @ N3 ) ) ) ).
% nat_less_numeral_power_cancel_iff
thf(fact_2625_numeral__power__less__nat__cancel__iff,axiom,
! [X: num,N3: nat,A2: int] :
( ( ord_less @ nat @ ( power_power @ nat @ ( numeral_numeral @ nat @ X ) @ N3 ) @ ( nat2 @ A2 ) )
= ( ord_less @ int @ ( power_power @ int @ ( numeral_numeral @ int @ X ) @ N3 ) @ A2 ) ) ).
% numeral_power_less_nat_cancel_iff
thf(fact_2626_numeral__power__le__nat__cancel__iff,axiom,
! [X: num,N3: nat,A2: int] :
( ( ord_less_eq @ nat @ ( power_power @ nat @ ( numeral_numeral @ nat @ X ) @ N3 ) @ ( nat2 @ A2 ) )
= ( ord_less_eq @ int @ ( power_power @ int @ ( numeral_numeral @ int @ X ) @ N3 ) @ A2 ) ) ).
% numeral_power_le_nat_cancel_iff
thf(fact_2627_nat__le__numeral__power__cancel__iff,axiom,
! [A2: int,X: num,N3: nat] :
( ( ord_less_eq @ nat @ ( nat2 @ A2 ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ X ) @ N3 ) )
= ( ord_less_eq @ int @ A2 @ ( power_power @ int @ ( numeral_numeral @ int @ X ) @ N3 ) ) ) ).
% nat_le_numeral_power_cancel_iff
thf(fact_2628_TBOUND__vebt__memberi,axiom,
! [T2: vEBT_VEBT,Ti: vEBT_VEBTi,X: nat] : ( time_TBOUND @ $o @ ( vEBT_V854960066525838166emberi @ T2 @ Ti @ X ) @ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ ( bit0 @ one2 ) ) ) @ ( plus_plus @ nat @ ( one_one @ nat ) @ ( vEBT_VEBT_height @ T2 ) ) ) ) ).
% TBOUND_vebt_memberi
thf(fact_2629_TBOUND__vebt__predi,axiom,
! [T2: vEBT_VEBT,Ti: vEBT_VEBTi,X: nat] : ( time_TBOUND @ ( option @ nat ) @ ( vEBT_VEBT_vebt_predi @ T2 @ Ti @ X ) @ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit1 @ ( bit1 @ one2 ) ) ) @ ( plus_plus @ nat @ ( one_one @ nat ) @ ( vEBT_VEBT_height @ T2 ) ) ) ) ).
% TBOUND_vebt_predi
thf(fact_2630_TBOUND__vebt__succi,axiom,
! [T2: vEBT_VEBT,Ti: vEBT_VEBTi,X: nat] : ( time_TBOUND @ ( option @ nat ) @ ( vEBT_VEBT_vebt_succi @ T2 @ Ti @ X ) @ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit1 @ ( bit1 @ one2 ) ) ) @ ( plus_plus @ nat @ ( one_one @ nat ) @ ( vEBT_VEBT_height @ T2 ) ) ) ) ).
% TBOUND_vebt_succi
thf(fact_2631_unique__quotient,axiom,
! [A2: int,B2: int,Q2: int,R2: int,Q5: int,R4: int] :
( ( eucl_rel_int @ A2 @ B2 @ ( product_Pair @ int @ int @ Q2 @ R2 ) )
=> ( ( eucl_rel_int @ A2 @ B2 @ ( product_Pair @ int @ int @ Q5 @ R4 ) )
=> ( Q2 = Q5 ) ) ) ).
% unique_quotient
thf(fact_2632_unique__remainder,axiom,
! [A2: int,B2: int,Q2: int,R2: int,Q5: int,R4: int] :
( ( eucl_rel_int @ A2 @ B2 @ ( product_Pair @ int @ int @ Q2 @ R2 ) )
=> ( ( eucl_rel_int @ A2 @ B2 @ ( product_Pair @ int @ int @ Q5 @ R4 ) )
=> ( R2 = R4 ) ) ) ).
% unique_remainder
thf(fact_2633_nat__numeral__as__int,axiom,
( ( numeral_numeral @ nat )
= ( ^ [I3: num] : ( nat2 @ ( numeral_numeral @ int @ I3 ) ) ) ) ).
% nat_numeral_as_int
thf(fact_2634_nat__mono,axiom,
! [X: int,Y: int] :
( ( ord_less_eq @ int @ X @ Y )
=> ( ord_less_eq @ nat @ ( nat2 @ X ) @ ( nat2 @ Y ) ) ) ).
% nat_mono
thf(fact_2635_nat__one__as__int,axiom,
( ( one_one @ nat )
= ( nat2 @ ( one_one @ int ) ) ) ).
% nat_one_as_int
thf(fact_2636_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_2637_all__nat,axiom,
( ( ^ [P2: nat > $o] :
! [X4: nat] : ( P2 @ X4 ) )
= ( ^ [P3: nat > $o] :
! [X2: int] :
( ( ord_less_eq @ int @ ( zero_zero @ int ) @ X2 )
=> ( P3 @ ( nat2 @ X2 ) ) ) ) ) ).
% all_nat
thf(fact_2638_ex__nat,axiom,
( ( ^ [P2: nat > $o] :
? [X4: nat] : ( P2 @ X4 ) )
= ( ^ [P3: nat > $o] :
? [X2: int] :
( ( ord_less_eq @ int @ ( zero_zero @ int ) @ X2 )
& ( P3 @ ( nat2 @ X2 ) ) ) ) ) ).
% ex_nat
thf(fact_2639_unset__bit__nat__def,axiom,
( ( bit_se2638667681897837118et_bit @ nat )
= ( ^ [M5: nat,N2: nat] : ( nat2 @ ( bit_se2638667681897837118et_bit @ int @ M5 @ ( semiring_1_of_nat @ int @ N2 ) ) ) ) ) ).
% unset_bit_nat_def
thf(fact_2640_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_2641_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_2642_nat__le__iff,axiom,
! [X: int,N3: nat] :
( ( ord_less_eq @ nat @ ( nat2 @ X ) @ N3 )
= ( ord_less_eq @ int @ X @ ( semiring_1_of_nat @ int @ N3 ) ) ) ).
% nat_le_iff
thf(fact_2643_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_2644_nat__int__add,axiom,
! [A2: nat,B2: nat] :
( ( nat2 @ ( plus_plus @ int @ ( semiring_1_of_nat @ int @ A2 ) @ ( semiring_1_of_nat @ int @ B2 ) ) )
= ( plus_plus @ nat @ A2 @ B2 ) ) ).
% nat_int_add
thf(fact_2645_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_2646_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_2647_int__minus,axiom,
! [N3: nat,M: nat] :
( ( semiring_1_of_nat @ int @ ( minus_minus @ nat @ N3 @ M ) )
= ( semiring_1_of_nat @ int @ ( nat2 @ ( minus_minus @ int @ ( semiring_1_of_nat @ int @ N3 ) @ ( semiring_1_of_nat @ int @ M ) ) ) ) ) ).
% int_minus
thf(fact_2648_eucl__rel__int__by0,axiom,
! [K: int] : ( eucl_rel_int @ K @ ( zero_zero @ int ) @ ( product_Pair @ int @ int @ ( zero_zero @ int ) @ K ) ) ).
% eucl_rel_int_by0
thf(fact_2649_mod__int__unique,axiom,
! [K: int,L2: int,Q2: int,R2: int] :
( ( eucl_rel_int @ K @ L2 @ ( product_Pair @ int @ int @ Q2 @ R2 ) )
=> ( ( modulo_modulo @ int @ K @ L2 )
= R2 ) ) ).
% mod_int_unique
thf(fact_2650_div__int__unique,axiom,
! [K: int,L2: int,Q2: int,R2: int] :
( ( eucl_rel_int @ K @ L2 @ ( product_Pair @ int @ int @ Q2 @ R2 ) )
=> ( ( divide_divide @ int @ K @ L2 )
= Q2 ) ) ).
% div_int_unique
thf(fact_2651_real__nat__ceiling__ge,axiom,
! [X: real] : ( ord_less_eq @ real @ X @ ( semiring_1_of_nat @ real @ ( nat2 @ ( archimedean_ceiling @ real @ X ) ) ) ) ).
% real_nat_ceiling_ge
thf(fact_2652_nat__plus__as__int,axiom,
( ( plus_plus @ nat )
= ( ^ [A6: nat,B5: nat] : ( nat2 @ ( plus_plus @ int @ ( semiring_1_of_nat @ int @ A6 ) @ ( semiring_1_of_nat @ int @ B5 ) ) ) ) ) ).
% nat_plus_as_int
thf(fact_2653_nat__times__as__int,axiom,
( ( times_times @ nat )
= ( ^ [A6: nat,B5: nat] : ( nat2 @ ( times_times @ int @ ( semiring_1_of_nat @ int @ A6 ) @ ( semiring_1_of_nat @ int @ B5 ) ) ) ) ) ).
% nat_times_as_int
thf(fact_2654_nat__minus__as__int,axiom,
( ( minus_minus @ nat )
= ( ^ [A6: nat,B5: nat] : ( nat2 @ ( minus_minus @ int @ ( semiring_1_of_nat @ int @ A6 ) @ ( semiring_1_of_nat @ int @ B5 ) ) ) ) ) ).
% nat_minus_as_int
thf(fact_2655_nat__div__as__int,axiom,
( ( divide_divide @ nat )
= ( ^ [A6: nat,B5: nat] : ( nat2 @ ( divide_divide @ int @ ( semiring_1_of_nat @ int @ A6 ) @ ( semiring_1_of_nat @ int @ B5 ) ) ) ) ) ).
% nat_div_as_int
thf(fact_2656_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_2657_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_2658_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_2659_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_2660_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_2661_le__nat__iff,axiom,
! [K: int,N3: nat] :
( ( ord_less_eq @ int @ ( zero_zero @ int ) @ K )
=> ( ( ord_less_eq @ nat @ N3 @ ( nat2 @ K ) )
= ( ord_less_eq @ int @ ( semiring_1_of_nat @ int @ N3 ) @ K ) ) ) ).
% le_nat_iff
thf(fact_2662_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_2663_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_2664_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_2665_nat__diff__distrib_H,axiom,
! [X: int,Y: int] :
( ( ord_less_eq @ int @ ( zero_zero @ int ) @ X )
=> ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ Y )
=> ( ( nat2 @ ( minus_minus @ int @ X @ Y ) )
= ( minus_minus @ nat @ ( nat2 @ X ) @ ( nat2 @ Y ) ) ) ) ) ).
% nat_diff_distrib'
thf(fact_2666_nat__div__distrib,axiom,
! [X: int,Y: int] :
( ( ord_less_eq @ int @ ( zero_zero @ int ) @ X )
=> ( ( nat2 @ ( divide_divide @ int @ X @ Y ) )
= ( divide_divide @ nat @ ( nat2 @ X ) @ ( nat2 @ Y ) ) ) ) ).
% nat_div_distrib
thf(fact_2667_nat__div__distrib_H,axiom,
! [Y: int,X: int] :
( ( ord_less_eq @ int @ ( zero_zero @ int ) @ Y )
=> ( ( nat2 @ ( divide_divide @ int @ X @ Y ) )
= ( divide_divide @ nat @ ( nat2 @ X ) @ ( nat2 @ Y ) ) ) ) ).
% nat_div_distrib'
thf(fact_2668_nat__power__eq,axiom,
! [Z: int,N3: nat] :
( ( ord_less_eq @ int @ ( zero_zero @ int ) @ Z )
=> ( ( nat2 @ ( power_power @ int @ Z @ N3 ) )
= ( power_power @ nat @ ( nat2 @ Z ) @ N3 ) ) ) ).
% nat_power_eq
thf(fact_2669_nat__mod__distrib,axiom,
! [X: int,Y: int] :
( ( ord_less_eq @ int @ ( zero_zero @ int ) @ X )
=> ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ Y )
=> ( ( nat2 @ ( modulo_modulo @ int @ X @ Y ) )
= ( modulo_modulo @ nat @ ( nat2 @ X ) @ ( nat2 @ Y ) ) ) ) ) ).
% nat_mod_distrib
thf(fact_2670_word__of__int__nat,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [X: int] :
( ( ord_less_eq @ int @ ( zero_zero @ int ) @ X )
=> ( ( ring_1_of_int @ ( word @ A ) @ X )
= ( semiring_1_of_nat @ ( word @ A ) @ ( nat2 @ X ) ) ) ) ) ).
% word_of_int_nat
thf(fact_2671_eucl__rel__int__dividesI,axiom,
! [L2: int,K: int,Q2: int] :
( ( L2
!= ( zero_zero @ int ) )
=> ( ( K
= ( times_times @ int @ Q2 @ L2 ) )
=> ( eucl_rel_int @ K @ L2 @ ( product_Pair @ int @ int @ Q2 @ ( zero_zero @ int ) ) ) ) ) ).
% eucl_rel_int_dividesI
thf(fact_2672_eucl__rel__int,axiom,
! [K: int,L2: int] : ( eucl_rel_int @ K @ L2 @ ( product_Pair @ int @ int @ ( divide_divide @ int @ K @ L2 ) @ ( modulo_modulo @ int @ K @ L2 ) ) ) ).
% eucl_rel_int
thf(fact_2673_nat__2,axiom,
( ( nat2 @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) )
= ( suc @ ( suc @ ( zero_zero @ nat ) ) ) ) ).
% nat_2
thf(fact_2674_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_2675_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_2676_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_2677_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_2678_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_2679_eucl__rel__int__iff,axiom,
! [K: int,L2: int,Q2: int,R2: int] :
( ( eucl_rel_int @ K @ L2 @ ( product_Pair @ int @ int @ Q2 @ R2 ) )
= ( ( K
= ( plus_plus @ int @ ( times_times @ int @ L2 @ Q2 ) @ R2 ) )
& ( ( ord_less @ int @ ( zero_zero @ int ) @ L2 )
=> ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ R2 )
& ( ord_less @ int @ R2 @ L2 ) ) )
& ( ~ ( ord_less @ int @ ( zero_zero @ int ) @ L2 )
=> ( ( ( ord_less @ int @ L2 @ ( zero_zero @ int ) )
=> ( ( ord_less @ int @ L2 @ R2 )
& ( ord_less_eq @ int @ R2 @ ( zero_zero @ int ) ) ) )
& ( ~ ( ord_less @ int @ L2 @ ( zero_zero @ int ) )
=> ( Q2
= ( zero_zero @ int ) ) ) ) ) ) ) ).
% eucl_rel_int_iff
thf(fact_2680_pos__eucl__rel__int__mult__2,axiom,
! [B2: int,A2: int,Q2: int,R2: int] :
( ( ord_less_eq @ int @ ( zero_zero @ int ) @ B2 )
=> ( ( eucl_rel_int @ A2 @ B2 @ ( product_Pair @ int @ int @ Q2 @ R2 ) )
=> ( eucl_rel_int @ ( plus_plus @ int @ ( one_one @ int ) @ ( times_times @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ A2 ) ) @ ( times_times @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ B2 ) @ ( product_Pair @ int @ int @ Q2 @ ( plus_plus @ int @ ( one_one @ int ) @ ( times_times @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ R2 ) ) ) ) ) ) ).
% pos_eucl_rel_int_mult_2
thf(fact_2681_norm__pre__pure__iff__sng,axiom,
! [A: $tType,B2: $o,F2: heap_Time_Heap @ A,Q: A > assn] :
( ( hoare_hoare_triple @ A @ ( pure_assn @ B2 ) @ F2 @ Q )
= ( B2
=> ( hoare_hoare_triple @ A @ ( one_one @ assn ) @ F2 @ Q ) ) ) ).
% norm_pre_pure_iff_sng
thf(fact_2682_htt__vebt__memberi__invar__vebt,axiom,
! [T2: vEBT_VEBT,N3: nat,Ti: vEBT_VEBTi,X: nat] :
( ( vEBT_invar_vebt @ T2 @ N3 )
=> ( time_htt @ $o @ ( vEBT_vebt_assn_raw @ T2 @ Ti ) @ ( vEBT_vebt_memberi @ Ti @ X )
@ ^ [R5: $o] :
( times_times @ assn @ ( vEBT_vebt_assn_raw @ T2 @ Ti )
@ ( pure_assn
@ ( R5
= ( vEBT_vebt_member @ T2 @ X ) ) ) )
@ ( plus_plus @ nat @ ( numeral_numeral @ nat @ ( bit1 @ ( bit0 @ one2 ) ) ) @ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit1 @ ( bit0 @ one2 ) ) ) @ ( nat2 @ ( archimedean_ceiling @ real @ ( log @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) @ ( semiring_1_of_nat @ real @ N3 ) ) ) ) ) ) ) ) ).
% htt_vebt_memberi_invar_vebt
thf(fact_2683_htt__vebt__succi,axiom,
! [T2: vEBT_VEBT,N3: nat,Ti: vEBT_VEBTi,X: nat] :
( ( vEBT_invar_vebt @ T2 @ N3 )
=> ( time_htt @ ( option @ nat ) @ ( vEBT_vebt_assn_raw @ T2 @ Ti ) @ ( vEBT_vebt_succi @ Ti @ X )
@ ^ [R5: option @ nat] :
( times_times @ assn @ ( vEBT_vebt_assn_raw @ T2 @ Ti )
@ ( pure_assn
@ ( R5
= ( vEBT_vebt_succ @ T2 @ X ) ) ) )
@ ( plus_plus @ nat @ ( numeral_numeral @ nat @ ( bit1 @ ( bit1 @ one2 ) ) ) @ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit1 @ ( bit1 @ one2 ) ) ) @ ( nat2 @ ( archimedean_ceiling @ real @ ( log @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) @ ( semiring_1_of_nat @ real @ N3 ) ) ) ) ) ) ) ) ).
% htt_vebt_succi
thf(fact_2684_htt__vebt__predi,axiom,
! [T2: vEBT_VEBT,N3: nat,Ti: vEBT_VEBTi,X: nat] :
( ( vEBT_invar_vebt @ T2 @ N3 )
=> ( time_htt @ ( option @ nat ) @ ( vEBT_vebt_assn_raw @ T2 @ Ti ) @ ( vEBT_vebt_predi @ Ti @ X )
@ ^ [R5: option @ nat] :
( times_times @ assn @ ( vEBT_vebt_assn_raw @ T2 @ Ti )
@ ( pure_assn
@ ( R5
= ( vEBT_vebt_pred @ T2 @ X ) ) ) )
@ ( plus_plus @ nat @ ( numeral_numeral @ nat @ ( bit1 @ ( bit1 @ one2 ) ) ) @ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit1 @ ( bit1 @ one2 ) ) ) @ ( nat2 @ ( archimedean_ceiling @ real @ ( log @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) @ ( semiring_1_of_nat @ real @ N3 ) ) ) ) ) ) ) ) ).
% htt_vebt_predi
thf(fact_2685_pure__true,axiom,
( ( pure_assn @ $true )
= ( one_one @ assn ) ) ).
% pure_true
thf(fact_2686_pure__assn__eq__emp__iff,axiom,
! [P: $o] :
( ( ( pure_assn @ P )
= ( one_one @ assn ) )
= P ) ).
% pure_assn_eq_emp_iff
thf(fact_2687_vebt__memberi_H__rf__abstr,axiom,
! [T2: vEBT_VEBT,Ti: vEBT_VEBTi,X: nat] :
( hoare_hoare_triple @ $o @ ( vEBT_vebt_assn_raw @ T2 @ Ti ) @ ( vEBT_V854960066525838166emberi @ T2 @ Ti @ X )
@ ^ [R5: $o] :
( times_times @ assn @ ( vEBT_vebt_assn_raw @ T2 @ Ti )
@ ( pure_assn
@ ( R5
= ( vEBT_vebt_member @ T2 @ X ) ) ) ) ) ).
% vebt_memberi'_rf_abstr
thf(fact_2688_vebt__pred_H__rf__abstr,axiom,
! [T2: vEBT_VEBT,N3: nat,Ti: vEBT_VEBTi,X: nat] :
( ( vEBT_invar_vebt @ T2 @ N3 )
=> ( hoare_hoare_triple @ ( option @ nat ) @ ( vEBT_vebt_assn_raw @ T2 @ Ti ) @ ( vEBT_VEBT_vebt_predi @ T2 @ Ti @ X )
@ ^ [R5: option @ nat] :
( times_times @ assn @ ( vEBT_vebt_assn_raw @ T2 @ Ti )
@ ( pure_assn
@ ( R5
= ( vEBT_vebt_pred @ T2 @ X ) ) ) ) ) ) ).
% vebt_pred'_rf_abstr
thf(fact_2689_vebt__succi_H__rf__abstr,axiom,
! [T2: vEBT_VEBT,N3: nat,Ti: vEBT_VEBTi,X: nat] :
( ( vEBT_invar_vebt @ T2 @ N3 )
=> ( hoare_hoare_triple @ ( option @ nat ) @ ( vEBT_vebt_assn_raw @ T2 @ Ti ) @ ( vEBT_VEBT_vebt_succi @ T2 @ Ti @ X )
@ ^ [R5: option @ nat] :
( times_times @ assn @ ( vEBT_vebt_assn_raw @ T2 @ Ti )
@ ( pure_assn
@ ( R5
= ( vEBT_vebt_succ @ T2 @ X ) ) ) ) ) ) ).
% vebt_succi'_rf_abstr
thf(fact_2690_htt__vebt__memberi,axiom,
! [T2: vEBT_VEBT,Ti: vEBT_VEBTi,X: nat] :
( time_htt @ $o @ ( vEBT_vebt_assn_raw @ T2 @ Ti ) @ ( vEBT_vebt_memberi @ Ti @ X )
@ ^ [R5: $o] :
( times_times @ assn @ ( vEBT_vebt_assn_raw @ T2 @ Ti )
@ ( pure_assn
@ ( R5
= ( vEBT_vebt_member @ T2 @ X ) ) ) )
@ ( plus_plus @ nat @ ( numeral_numeral @ nat @ ( bit1 @ ( bit0 @ one2 ) ) ) @ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit1 @ ( bit0 @ one2 ) ) ) @ ( vEBT_VEBT_height @ T2 ) ) ) ) ).
% htt_vebt_memberi
thf(fact_2691_assn__one__left,axiom,
! [P: assn] :
( ( times_times @ assn @ ( one_one @ assn ) @ P )
= P ) ).
% assn_one_left
thf(fact_2692_norm__pre__pure__rule2,axiom,
! [A: $tType,B2: $o,F2: heap_Time_Heap @ A,Q: A > assn] :
( ( B2
=> ( hoare_hoare_triple @ A @ ( one_one @ assn ) @ F2 @ Q ) )
=> ( hoare_hoare_triple @ A @ ( pure_assn @ B2 ) @ F2 @ Q ) ) ).
% norm_pre_pure_rule2
thf(fact_2693_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_2694_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_2695_TBOUND__minNulli,axiom,
! [T2: vEBT_VEBTi] : ( time_TBOUND @ $o @ ( vEBT_VEBT_minNulli @ T2 ) @ ( one_one @ nat ) ) ).
% TBOUND_minNulli
thf(fact_2696_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_2697_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_2698_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_2699_TBOUND__vebt__maxti,axiom,
! [T2: vEBT_VEBTi] : ( time_TBOUND @ ( option @ nat ) @ ( vEBT_vebt_maxti @ T2 ) @ ( one_one @ nat ) ) ).
% TBOUND_vebt_maxti
thf(fact_2700_TBOUND__vebt__minti,axiom,
! [T2: vEBT_VEBTi] : ( time_TBOUND @ ( option @ nat ) @ ( vEBT_vebt_minti @ T2 ) @ ( one_one @ nat ) ) ).
% TBOUND_vebt_minti
thf(fact_2701_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_2702_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_2703_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_2704_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_2705_vebt__memberi__refines,axiom,
! [Ti: vEBT_VEBTi,X: nat,T2: vEBT_VEBT] : ( refine_Imp_refines @ $o @ ( vEBT_vebt_memberi @ Ti @ X ) @ ( vEBT_V854960066525838166emberi @ T2 @ Ti @ X ) ) ).
% vebt_memberi_refines
thf(fact_2706_vebt__succi__refines,axiom,
! [Ti: vEBT_VEBTi,X: nat,T2: vEBT_VEBT] : ( refine_Imp_refines @ ( option @ nat ) @ ( vEBT_vebt_succi @ Ti @ X ) @ ( vEBT_VEBT_vebt_succi @ T2 @ Ti @ X ) ) ).
% vebt_succi_refines
thf(fact_2707_vebt__predi__refines,axiom,
! [Ti: vEBT_VEBTi,X: nat,T2: vEBT_VEBT] : ( refine_Imp_refines @ ( option @ nat ) @ ( vEBT_vebt_predi @ Ti @ X ) @ ( vEBT_VEBT_vebt_predi @ T2 @ Ti @ X ) ) ).
% vebt_predi_refines
thf(fact_2708_vebt__inserti__refines,axiom,
! [Ti: vEBT_VEBTi,X: nat,T2: vEBT_VEBT] : ( refine_Imp_refines @ vEBT_VEBTi @ ( vEBT_vebt_inserti @ Ti @ X ) @ ( vEBT_V3964819847710782039nserti @ T2 @ Ti @ X ) ) ).
% vebt_inserti_refines
thf(fact_2709_refines__replicate,axiom,
! [A: $tType,F2: heap_Time_Heap @ A,F5: heap_Time_Heap @ A,N3: nat] :
( ( refine_Imp_refines @ A @ F2 @ F5 )
=> ( refine_Imp_refines @ ( list @ A ) @ ( vEBT_VEBT_replicatei @ A @ N3 @ F2 ) @ ( vEBT_VEBT_replicatei @ A @ N3 @ F5 ) ) ) ).
% refines_replicate
thf(fact_2710_vebt__buildupi__refines,axiom,
! [N3: nat] : ( refine_Imp_refines @ vEBT_VEBTi @ ( vEBT_vebt_buildupi @ N3 ) @ ( vEBT_V739175172307565963ildupi @ N3 ) ) ).
% vebt_buildupi_refines
thf(fact_2711_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,B3: $o] : ( refine_Imp_refines @ A @ ( F1 @ A4 @ B3 ) @ ( F12 @ A4 @ B3 ) )
=> ( ! [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_2712_flip__bit__Suc,axiom,
! [A: $tType] :
( ( bit_se359711467146920520ations @ A )
=> ! [N3: nat,A2: A] :
( ( bit_se8732182000553998342ip_bit @ A @ ( suc @ N3 ) @ A2 )
= ( plus_plus @ A @ ( modulo_modulo @ A @ A2 @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) @ ( times_times @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ ( bit_se8732182000553998342ip_bit @ A @ N3 @ ( divide_divide @ A @ A2 @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) ) ) ) ) ) ).
% flip_bit_Suc
thf(fact_2713_product__nth,axiom,
! [A: $tType,B: $tType,N3: nat,Xs2: list @ A,Ys: list @ B] :
( ( ord_less @ nat @ N3 @ ( times_times @ nat @ ( size_size @ ( list @ A ) @ Xs2 ) @ ( size_size @ ( list @ B ) @ Ys ) ) )
=> ( ( nth @ ( product_prod @ A @ B ) @ ( product @ A @ B @ Xs2 @ Ys ) @ N3 )
= ( product_Pair @ A @ B @ ( nth @ A @ Xs2 @ ( divide_divide @ nat @ N3 @ ( size_size @ ( list @ B ) @ Ys ) ) ) @ ( nth @ B @ Ys @ ( modulo_modulo @ nat @ N3 @ ( size_size @ ( list @ B ) @ Ys ) ) ) ) ) ) ).
% product_nth
thf(fact_2714_lowi__h,axiom,
! [X: nat,N3: nat] :
( hoare_hoare_triple @ nat @ ( one_one @ assn ) @ ( vEBT_VEBT_lowi @ X @ N3 )
@ ^ [R5: nat] :
( pure_assn
@ ( R5
= ( vEBT_VEBT_low @ X @ N3 ) ) ) ) ).
% lowi_h
thf(fact_2715_highi__h,axiom,
! [X: nat,N3: nat] :
( hoare_hoare_triple @ nat @ ( one_one @ assn ) @ ( vEBT_VEBT_highi @ X @ N3 )
@ ^ [R5: nat] :
( pure_assn
@ ( R5
= ( vEBT_VEBT_high @ X @ N3 ) ) ) ) ).
% highi_h
thf(fact_2716_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_2717_triangle__def,axiom,
( nat_triangle
= ( ^ [N2: nat] : ( divide_divide @ nat @ ( times_times @ nat @ N2 @ ( suc @ N2 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ).
% triangle_def
thf(fact_2718_TBOUND__highi,axiom,
! [X: nat,N3: nat] : ( time_TBOUND @ nat @ ( vEBT_VEBT_highi @ X @ N3 ) @ ( one_one @ nat ) ) ).
% TBOUND_highi
thf(fact_2719_TBOUND__lowi,axiom,
! [X: nat,N3: nat] : ( time_TBOUND @ nat @ ( vEBT_VEBT_lowi @ X @ N3 ) @ ( one_one @ nat ) ) ).
% TBOUND_lowi
thf(fact_2720_flip__bit__nonnegative__int__iff,axiom,
! [N3: nat,K: int] :
( ( ord_less_eq @ int @ ( zero_zero @ int ) @ ( bit_se8732182000553998342ip_bit @ int @ N3 @ K ) )
= ( ord_less_eq @ int @ ( zero_zero @ int ) @ K ) ) ).
% flip_bit_nonnegative_int_iff
thf(fact_2721_flip__bit__negative__int__iff,axiom,
! [N3: nat,K: int] :
( ( ord_less @ int @ ( bit_se8732182000553998342ip_bit @ int @ N3 @ K ) @ ( zero_zero @ int ) )
= ( ord_less @ int @ K @ ( zero_zero @ int ) ) ) ).
% flip_bit_negative_int_iff
thf(fact_2722_length__product,axiom,
! [A: $tType,B: $tType,Xs2: list @ A,Ys: list @ B] :
( ( size_size @ ( list @ ( product_prod @ A @ B ) ) @ ( product @ A @ B @ Xs2 @ Ys ) )
= ( times_times @ nat @ ( size_size @ ( list @ A ) @ Xs2 ) @ ( size_size @ ( list @ B ) @ Ys ) ) ) ).
% length_product
thf(fact_2723_triangle__Suc,axiom,
! [N3: nat] :
( ( nat_triangle @ ( suc @ N3 ) )
= ( plus_plus @ nat @ ( nat_triangle @ N3 ) @ ( suc @ N3 ) ) ) ).
% triangle_Suc
thf(fact_2724_TBOUND__VEBT__case,axiom,
! [A: $tType,Ti: vEBT_VEBTi,F2: $o > $o > ( heap_Time_Heap @ A ),Bnd: $o > $o > nat,F5: ( 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,B3: $o] :
( ( Ti
= ( vEBT_Leafi @ A4 @ B3 ) )
=> ( time_TBOUND @ A @ ( F2 @ A4 @ B3 ) @ ( Bnd @ A4 @ B3 ) ) )
=> ( ! [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 @ ( F5 @ Info2 @ Deg2 @ TreeArray @ Summary2 ) @ ( Bnd2 @ Info2 @ Deg2 @ TreeArray @ Summary2 ) ) )
=> ( time_TBOUND @ A @ ( vEBT_case_VEBTi @ ( heap_Time_Heap @ A ) @ F5 @ F2 @ Ti ) @ ( vEBT_case_VEBTi @ nat @ Bnd2 @ Bnd @ Ti ) ) ) ) ).
% TBOUND_VEBT_case
thf(fact_2725_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_2726_even__flip__bit__iff,axiom,
! [A: $tType] :
( ( bit_se359711467146920520ations @ A )
=> ! [M: nat,A2: A] :
( ( dvd_dvd @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ ( bit_se8732182000553998342ip_bit @ A @ M @ A2 ) )
= ( ( dvd_dvd @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ A2 )
!= ( M
= ( zero_zero @ nat ) ) ) ) ) ).
% even_flip_bit_iff
thf(fact_2727_lowi__hT,axiom,
! [X: nat,N3: nat] :
( time_htt @ nat @ ( one_one @ assn ) @ ( vEBT_VEBT_lowi @ X @ N3 )
@ ^ [R5: nat] :
( pure_assn
@ ( R5
= ( vEBT_VEBT_low @ X @ N3 ) ) )
@ ( one_one @ nat ) ) ).
% lowi_hT
thf(fact_2728_highi__hT,axiom,
! [X: nat,N3: nat] :
( time_htt @ nat @ ( one_one @ assn ) @ ( vEBT_VEBT_highi @ X @ N3 )
@ ^ [R5: nat] :
( pure_assn
@ ( R5
= ( vEBT_VEBT_high @ X @ N3 ) ) )
@ ( one_one @ nat ) ) ).
% highi_hT
thf(fact_2729_lowi__def,axiom,
( vEBT_VEBT_lowi
= ( ^ [X2: nat,N2: nat] : ( heap_Time_return @ nat @ ( modulo_modulo @ nat @ X2 @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N2 ) ) ) ) ) ).
% lowi_def
thf(fact_2730_highi__def,axiom,
( vEBT_VEBT_highi
= ( ^ [X2: nat,N2: nat] : ( heap_Time_return @ nat @ ( divide_divide @ nat @ X2 @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N2 ) ) ) ) ) ).
% highi_def
thf(fact_2731_obtain__set__pred,axiom,
! [Z: nat,X: nat,A3: set @ nat] :
( ( ord_less @ nat @ Z @ X )
=> ( ( vEBT_VEBT_min_in_set @ A3 @ Z )
=> ( ( finite_finite2 @ nat @ A3 )
=> ? [X_12: nat] : ( vEBT_is_pred_in_set @ A3 @ X @ X_12 ) ) ) ) ).
% obtain_set_pred
thf(fact_2732_obtain__set__succ,axiom,
! [X: nat,Z: nat,A3: set @ nat,B6: set @ nat] :
( ( ord_less @ nat @ X @ Z )
=> ( ( vEBT_VEBT_max_in_set @ A3 @ Z )
=> ( ( finite_finite2 @ nat @ B6 )
=> ( ( A3 = B6 )
=> ? [X_12: nat] : ( vEBT_is_succ_in_set @ A3 @ X @ X_12 ) ) ) ) ) ).
% obtain_set_succ
thf(fact_2733_highsimp,axiom,
! [X: nat,N3: nat] :
( ( heap_Time_return @ nat @ ( vEBT_VEBT_high @ X @ N3 ) )
= ( vEBT_VEBT_highi @ X @ N3 ) ) ).
% highsimp
thf(fact_2734_lowsimp,axiom,
! [X: nat,N3: nat] :
( ( heap_Time_return @ nat @ ( vEBT_VEBT_low @ X @ N3 ) )
= ( vEBT_VEBT_lowi @ X @ N3 ) ) ).
% lowsimp
thf(fact_2735_set__vebt__finite,axiom,
! [T2: vEBT_VEBT,N3: nat] :
( ( vEBT_invar_vebt @ T2 @ N3 )
=> ( finite_finite2 @ nat @ ( vEBT_VEBT_set_vebt @ T2 ) ) ) ).
% set_vebt_finite
thf(fact_2736_succ__none__empty,axiom,
! [Xs2: set @ nat,A2: nat] :
( ~ ? [X_12: nat] : ( vEBT_is_succ_in_set @ Xs2 @ A2 @ X_12 )
=> ( ( finite_finite2 @ nat @ Xs2 )
=> ~ ? [X5: nat] :
( ( member @ nat @ X5 @ Xs2 )
& ( ord_less @ nat @ A2 @ X5 ) ) ) ) ).
% succ_none_empty
thf(fact_2737_pred__none__empty,axiom,
! [Xs2: set @ nat,A2: nat] :
( ~ ? [X_12: nat] : ( vEBT_is_pred_in_set @ Xs2 @ A2 @ X_12 )
=> ( ( finite_finite2 @ nat @ Xs2 )
=> ~ ? [X5: nat] :
( ( member @ nat @ X5 @ Xs2 )
& ( ord_less @ nat @ X5 @ A2 ) ) ) ) ).
% pred_none_empty
thf(fact_2738_List_Ofinite__set,axiom,
! [A: $tType,Xs2: list @ A] : ( finite_finite2 @ A @ ( set2 @ A @ Xs2 ) ) ).
% List.finite_set
thf(fact_2739_infinite__Icc__iff,axiom,
! [A: $tType] :
( ( dense_linorder @ A )
=> ! [A2: A,B2: A] :
( ( ~ ( finite_finite2 @ A @ ( set_or1337092689740270186AtMost @ A @ A2 @ B2 ) ) )
= ( ord_less @ A @ A2 @ B2 ) ) ) ).
% infinite_Icc_iff
thf(fact_2740_finite__if__eq__beyond__finite,axiom,
! [A: $tType,S3: set @ A,S4: set @ A] :
( ( finite_finite2 @ A @ S3 )
=> ( finite_finite2 @ ( set @ A )
@ ( collect @ ( set @ A )
@ ^ [S5: set @ A] :
( ( minus_minus @ ( set @ A ) @ S5 @ S3 )
= ( minus_minus @ ( set @ A ) @ S4 @ S3 ) ) ) ) ) ).
% finite_if_eq_beyond_finite
thf(fact_2741_bounded__nat__set__is__finite,axiom,
! [N7: set @ nat,N3: nat] :
( ! [X3: nat] :
( ( member @ nat @ X3 @ N7 )
=> ( ord_less @ nat @ X3 @ N3 ) )
=> ( finite_finite2 @ nat @ N7 ) ) ).
% bounded_nat_set_is_finite
thf(fact_2742_finite__nat__set__iff__bounded,axiom,
( ( finite_finite2 @ nat )
= ( ^ [N8: set @ nat] :
? [M5: nat] :
! [X2: nat] :
( ( member @ nat @ X2 @ N8 )
=> ( ord_less @ nat @ X2 @ M5 ) ) ) ) ).
% finite_nat_set_iff_bounded
thf(fact_2743_finite__nat__set__iff__bounded__le,axiom,
( ( finite_finite2 @ nat )
= ( ^ [N8: set @ nat] :
? [M5: nat] :
! [X2: nat] :
( ( member @ nat @ X2 @ N8 )
=> ( ord_less_eq @ nat @ X2 @ M5 ) ) ) ) ).
% finite_nat_set_iff_bounded_le
thf(fact_2744_finite__list,axiom,
! [A: $tType,A3: set @ A] :
( ( finite_finite2 @ A @ A3 )
=> ? [Xs3: list @ A] :
( ( set2 @ A @ Xs3 )
= A3 ) ) ).
% finite_list
thf(fact_2745_finite__M__bounded__by__nat,axiom,
! [P: nat > $o,I: nat] :
( finite_finite2 @ nat
@ ( collect @ nat
@ ^ [K2: nat] :
( ( P @ K2 )
& ( ord_less @ nat @ K2 @ I ) ) ) ) ).
% finite_M_bounded_by_nat
thf(fact_2746_finite__less__ub,axiom,
! [F2: nat > nat,U: nat] :
( ! [N: nat] : ( ord_less_eq @ nat @ N @ ( F2 @ N ) )
=> ( finite_finite2 @ nat
@ ( collect @ nat
@ ^ [N2: nat] : ( ord_less_eq @ nat @ ( F2 @ N2 ) @ U ) ) ) ) ).
% finite_less_ub
thf(fact_2747_finite__lists__length__eq,axiom,
! [A: $tType,A3: set @ A,N3: nat] :
( ( finite_finite2 @ A @ A3 )
=> ( finite_finite2 @ ( list @ A )
@ ( collect @ ( list @ A )
@ ^ [Xs: list @ A] :
( ( ord_less_eq @ ( set @ A ) @ ( set2 @ A @ Xs ) @ A3 )
& ( ( size_size @ ( list @ A ) @ Xs )
= N3 ) ) ) ) ) ).
% finite_lists_length_eq
thf(fact_2748_infinite__Icc,axiom,
! [A: $tType] :
( ( dense_linorder @ A )
=> ! [A2: A,B2: A] :
( ( ord_less @ A @ A2 @ B2 )
=> ~ ( finite_finite2 @ A @ ( set_or1337092689740270186AtMost @ A @ A2 @ B2 ) ) ) ) ).
% infinite_Icc
thf(fact_2749_finite__lists__length__le,axiom,
! [A: $tType,A3: set @ A,N3: nat] :
( ( finite_finite2 @ A @ A3 )
=> ( finite_finite2 @ ( list @ A )
@ ( collect @ ( list @ A )
@ ^ [Xs: list @ A] :
( ( ord_less_eq @ ( set @ A ) @ ( set2 @ A @ Xs ) @ A3 )
& ( ord_less_eq @ nat @ ( size_size @ ( list @ A ) @ Xs ) @ N3 ) ) ) ) ) ).
% finite_lists_length_le
thf(fact_2750_finite__divisors__nat,axiom,
! [M: nat] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ M )
=> ( finite_finite2 @ nat
@ ( collect @ nat
@ ^ [D2: nat] : ( dvd_dvd @ nat @ D2 @ M ) ) ) ) ).
% finite_divisors_nat
thf(fact_2751_subset__eq__atLeast0__atMost__finite,axiom,
! [N7: set @ nat,N3: nat] :
( ( ord_less_eq @ ( set @ nat ) @ N7 @ ( set_or1337092689740270186AtMost @ nat @ ( zero_zero @ nat ) @ N3 ) )
=> ( finite_finite2 @ nat @ N7 ) ) ).
% subset_eq_atLeast0_atMost_finite
thf(fact_2752_finite__roots__unity,axiom,
! [A: $tType] :
( ( ( real_V8999393235501362500lgebra @ A )
& ( idom @ A ) )
=> ! [N3: nat] :
( ( ord_less_eq @ nat @ ( one_one @ nat ) @ N3 )
=> ( finite_finite2 @ A
@ ( collect @ A
@ ^ [Z3: A] :
( ( power_power @ A @ Z3 @ N3 )
= ( one_one @ A ) ) ) ) ) ) ).
% finite_roots_unity
thf(fact_2753_finite__Diff__insert,axiom,
! [A: $tType,A3: set @ A,A2: A,B6: set @ A] :
( ( finite_finite2 @ A @ ( minus_minus @ ( set @ A ) @ A3 @ ( insert @ A @ A2 @ B6 ) ) )
= ( finite_finite2 @ A @ ( minus_minus @ ( set @ A ) @ A3 @ B6 ) ) ) ).
% finite_Diff_insert
thf(fact_2754_finite__Collect__le__nat,axiom,
! [K: nat] :
( finite_finite2 @ nat
@ ( collect @ nat
@ ^ [N2: nat] : ( ord_less_eq @ nat @ N2 @ K ) ) ) ).
% finite_Collect_le_nat
thf(fact_2755_finite__Collect__less__nat,axiom,
! [K: nat] :
( finite_finite2 @ nat
@ ( collect @ nat
@ ^ [N2: nat] : ( ord_less @ nat @ N2 @ K ) ) ) ).
% finite_Collect_less_nat
thf(fact_2756_finite__Collect__subsets,axiom,
! [A: $tType,A3: set @ A] :
( ( finite_finite2 @ A @ A3 )
=> ( finite_finite2 @ ( set @ A )
@ ( collect @ ( set @ A )
@ ^ [B7: set @ A] : ( ord_less_eq @ ( set @ A ) @ B7 @ A3 ) ) ) ) ).
% finite_Collect_subsets
thf(fact_2757_finite__induct__select,axiom,
! [A: $tType,S3: set @ A,P: ( set @ A ) > $o] :
( ( finite_finite2 @ A @ S3 )
=> ( ( P @ ( bot_bot @ ( set @ A ) ) )
=> ( ! [T4: set @ A] :
( ( ord_less @ ( set @ A ) @ T4 @ S3 )
=> ( ( P @ T4 )
=> ? [X5: A] :
( ( member @ A @ X5 @ ( minus_minus @ ( set @ A ) @ S3 @ T4 ) )
& ( P @ ( insert @ A @ X5 @ T4 ) ) ) ) )
=> ( P @ S3 ) ) ) ) ).
% finite_induct_select
thf(fact_2758_finite__Diff2,axiom,
! [A: $tType,B6: set @ A,A3: set @ A] :
( ( finite_finite2 @ A @ B6 )
=> ( ( finite_finite2 @ A @ ( minus_minus @ ( set @ A ) @ A3 @ B6 ) )
= ( finite_finite2 @ A @ A3 ) ) ) ).
% finite_Diff2
thf(fact_2759_finite__interval__int1,axiom,
! [A2: int,B2: int] :
( finite_finite2 @ int
@ ( collect @ int
@ ^ [I3: int] :
( ( ord_less_eq @ int @ A2 @ I3 )
& ( ord_less_eq @ int @ I3 @ B2 ) ) ) ) ).
% finite_interval_int1
thf(fact_2760_finite__Diff,axiom,
! [A: $tType,A3: set @ A,B6: set @ A] :
( ( finite_finite2 @ A @ A3 )
=> ( finite_finite2 @ A @ ( minus_minus @ ( set @ A ) @ A3 @ B6 ) ) ) ).
% finite_Diff
thf(fact_2761_finite__interval__int2,axiom,
! [A2: int,B2: int] :
( finite_finite2 @ int
@ ( collect @ int
@ ^ [I3: int] :
( ( ord_less_eq @ int @ A2 @ I3 )
& ( ord_less @ int @ I3 @ B2 ) ) ) ) ).
% finite_interval_int2
thf(fact_2762_finite__interval__int3,axiom,
! [A2: int,B2: int] :
( finite_finite2 @ int
@ ( collect @ int
@ ^ [I3: int] :
( ( ord_less @ int @ A2 @ I3 )
& ( ord_less_eq @ int @ I3 @ B2 ) ) ) ) ).
% finite_interval_int3
thf(fact_2763_finite__maxlen,axiom,
! [A: $tType,M8: set @ ( list @ A )] :
( ( finite_finite2 @ ( list @ A ) @ M8 )
=> ? [N: nat] :
! [X5: list @ A] :
( ( member @ ( list @ A ) @ X5 @ M8 )
=> ( ord_less @ nat @ ( size_size @ ( list @ A ) @ X5 ) @ N ) ) ) ).
% finite_maxlen
thf(fact_2764_finite__has__maximal2,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [A3: set @ A,A2: A] :
( ( finite_finite2 @ A @ A3 )
=> ( ( member @ A @ A2 @ A3 )
=> ? [X3: A] :
( ( member @ A @ X3 @ A3 )
& ( ord_less_eq @ A @ A2 @ X3 )
& ! [Xa2: A] :
( ( member @ A @ Xa2 @ A3 )
=> ( ( ord_less_eq @ A @ X3 @ Xa2 )
=> ( X3 = Xa2 ) ) ) ) ) ) ) ).
% finite_has_maximal2
thf(fact_2765_finite__has__minimal2,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [A3: set @ A,A2: A] :
( ( finite_finite2 @ A @ A3 )
=> ( ( member @ A @ A2 @ A3 )
=> ? [X3: A] :
( ( member @ A @ X3 @ A3 )
& ( ord_less_eq @ A @ X3 @ A2 )
& ! [Xa2: A] :
( ( member @ A @ Xa2 @ A3 )
=> ( ( ord_less_eq @ A @ Xa2 @ X3 )
=> ( X3 = Xa2 ) ) ) ) ) ) ) ).
% finite_has_minimal2
thf(fact_2766_rev__finite__subset,axiom,
! [A: $tType,B6: set @ A,A3: set @ A] :
( ( finite_finite2 @ A @ B6 )
=> ( ( ord_less_eq @ ( set @ A ) @ A3 @ B6 )
=> ( finite_finite2 @ A @ A3 ) ) ) ).
% rev_finite_subset
thf(fact_2767_infinite__super,axiom,
! [A: $tType,S3: set @ A,T5: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ S3 @ T5 )
=> ( ~ ( finite_finite2 @ A @ S3 )
=> ~ ( finite_finite2 @ A @ T5 ) ) ) ).
% infinite_super
thf(fact_2768_finite__subset,axiom,
! [A: $tType,A3: set @ A,B6: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ A3 @ B6 )
=> ( ( finite_finite2 @ A @ B6 )
=> ( finite_finite2 @ A @ A3 ) ) ) ).
% finite_subset
thf(fact_2769_Diff__infinite__finite,axiom,
! [A: $tType,T5: set @ A,S3: set @ A] :
( ( finite_finite2 @ A @ T5 )
=> ( ~ ( finite_finite2 @ A @ S3 )
=> ~ ( finite_finite2 @ A @ ( minus_minus @ ( set @ A ) @ S3 @ T5 ) ) ) ) ).
% Diff_infinite_finite
thf(fact_2770_finite__has__maximal,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [A3: set @ A] :
( ( finite_finite2 @ A @ A3 )
=> ( ( A3
!= ( bot_bot @ ( set @ A ) ) )
=> ? [X3: A] :
( ( member @ A @ X3 @ A3 )
& ! [Xa2: A] :
( ( member @ A @ Xa2 @ A3 )
=> ( ( ord_less_eq @ A @ X3 @ Xa2 )
=> ( X3 = Xa2 ) ) ) ) ) ) ) ).
% finite_has_maximal
thf(fact_2771_finite__has__minimal,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [A3: set @ A] :
( ( finite_finite2 @ A @ A3 )
=> ( ( A3
!= ( bot_bot @ ( set @ A ) ) )
=> ? [X3: A] :
( ( member @ A @ X3 @ A3 )
& ! [Xa2: A] :
( ( member @ A @ Xa2 @ A3 )
=> ( ( ord_less_eq @ A @ Xa2 @ X3 )
=> ( X3 = Xa2 ) ) ) ) ) ) ) ).
% finite_has_minimal
thf(fact_2772_finite__subset__induct,axiom,
! [A: $tType,F4: set @ A,A3: set @ A,P: ( set @ A ) > $o] :
( ( finite_finite2 @ A @ F4 )
=> ( ( ord_less_eq @ ( set @ A ) @ F4 @ A3 )
=> ( ( P @ ( bot_bot @ ( set @ A ) ) )
=> ( ! [A4: A,F6: set @ A] :
( ( finite_finite2 @ A @ F6 )
=> ( ( member @ A @ A4 @ A3 )
=> ( ~ ( member @ A @ A4 @ F6 )
=> ( ( P @ F6 )
=> ( P @ ( insert @ A @ A4 @ F6 ) ) ) ) ) )
=> ( P @ F4 ) ) ) ) ) ).
% finite_subset_induct
thf(fact_2773_finite__subset__induct_H,axiom,
! [A: $tType,F4: set @ A,A3: set @ A,P: ( set @ A ) > $o] :
( ( finite_finite2 @ A @ F4 )
=> ( ( ord_less_eq @ ( set @ A ) @ F4 @ A3 )
=> ( ( P @ ( bot_bot @ ( set @ A ) ) )
=> ( ! [A4: A,F6: set @ A] :
( ( finite_finite2 @ A @ F6 )
=> ( ( member @ A @ A4 @ A3 )
=> ( ( ord_less_eq @ ( set @ A ) @ F6 @ A3 )
=> ( ~ ( member @ A @ A4 @ F6 )
=> ( ( P @ F6 )
=> ( P @ ( insert @ A @ A4 @ F6 ) ) ) ) ) ) )
=> ( P @ F4 ) ) ) ) ) ).
% finite_subset_induct'
thf(fact_2774_finite__empty__induct,axiom,
! [A: $tType,A3: set @ A,P: ( set @ A ) > $o] :
( ( finite_finite2 @ A @ A3 )
=> ( ( P @ A3 )
=> ( ! [A4: A,A8: set @ A] :
( ( finite_finite2 @ A @ A8 )
=> ( ( member @ A @ A4 @ A8 )
=> ( ( P @ A8 )
=> ( P @ ( minus_minus @ ( set @ A ) @ A8 @ ( insert @ A @ A4 @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ) )
=> ( P @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ).
% finite_empty_induct
thf(fact_2775_infinite__coinduct,axiom,
! [A: $tType,X8: ( set @ A ) > $o,A3: set @ A] :
( ( X8 @ A3 )
=> ( ! [A8: set @ A] :
( ( X8 @ A8 )
=> ? [X5: A] :
( ( member @ A @ X5 @ A8 )
& ( ( X8 @ ( minus_minus @ ( set @ A ) @ A8 @ ( insert @ A @ X5 @ ( bot_bot @ ( set @ A ) ) ) ) )
| ~ ( finite_finite2 @ A @ ( minus_minus @ ( set @ A ) @ A8 @ ( insert @ A @ X5 @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ) )
=> ~ ( finite_finite2 @ A @ A3 ) ) ) ).
% infinite_coinduct
thf(fact_2776_infinite__remove,axiom,
! [A: $tType,S3: set @ A,A2: A] :
( ~ ( finite_finite2 @ A @ S3 )
=> ~ ( finite_finite2 @ A @ ( minus_minus @ ( set @ A ) @ S3 @ ( insert @ A @ A2 @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ).
% infinite_remove
thf(fact_2777_remove__induct,axiom,
! [A: $tType,P: ( set @ A ) > $o,B6: set @ A] :
( ( P @ ( bot_bot @ ( set @ A ) ) )
=> ( ( ~ ( finite_finite2 @ A @ B6 )
=> ( P @ B6 ) )
=> ( ! [A8: set @ A] :
( ( finite_finite2 @ A @ A8 )
=> ( ( A8
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( ord_less_eq @ ( set @ A ) @ A8 @ B6 )
=> ( ! [X5: A] :
( ( member @ A @ X5 @ A8 )
=> ( P @ ( minus_minus @ ( set @ A ) @ A8 @ ( insert @ A @ X5 @ ( bot_bot @ ( set @ A ) ) ) ) ) )
=> ( P @ A8 ) ) ) ) )
=> ( P @ B6 ) ) ) ) ).
% remove_induct
thf(fact_2778_finite__remove__induct,axiom,
! [A: $tType,B6: set @ A,P: ( set @ A ) > $o] :
( ( finite_finite2 @ A @ B6 )
=> ( ( P @ ( bot_bot @ ( set @ A ) ) )
=> ( ! [A8: set @ A] :
( ( finite_finite2 @ A @ A8 )
=> ( ( A8
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( ord_less_eq @ ( set @ A ) @ A8 @ B6 )
=> ( ! [X5: A] :
( ( member @ A @ X5 @ A8 )
=> ( P @ ( minus_minus @ ( set @ A ) @ A8 @ ( insert @ A @ X5 @ ( bot_bot @ ( set @ A ) ) ) ) ) )
=> ( P @ A8 ) ) ) ) )
=> ( P @ B6 ) ) ) ) ).
% finite_remove_induct
thf(fact_2779_finite__nth__roots,axiom,
! [N3: nat,C2: complex] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ N3 )
=> ( finite_finite2 @ complex
@ ( collect @ complex
@ ^ [Z3: complex] :
( ( power_power @ complex @ Z3 @ N3 )
= C2 ) ) ) ) ).
% finite_nth_roots
thf(fact_2780_set__encode__insert,axiom,
! [A3: set @ nat,N3: nat] :
( ( finite_finite2 @ nat @ A3 )
=> ( ~ ( member @ nat @ N3 @ A3 )
=> ( ( nat_set_encode @ ( insert @ nat @ N3 @ A3 ) )
= ( plus_plus @ nat @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N3 ) @ ( nat_set_encode @ A3 ) ) ) ) ) ).
% set_encode_insert
thf(fact_2781_diff__preserves__multiset,axiom,
! [A: $tType,M8: A > nat,N7: A > nat] :
( ( finite_finite2 @ A
@ ( collect @ A
@ ^ [X2: A] : ( ord_less @ nat @ ( zero_zero @ nat ) @ ( M8 @ X2 ) ) ) )
=> ( finite_finite2 @ A
@ ( collect @ A
@ ^ [X2: A] : ( ord_less @ nat @ ( zero_zero @ nat ) @ ( minus_minus @ nat @ ( M8 @ X2 ) @ ( N7 @ X2 ) ) ) ) ) ) ).
% diff_preserves_multiset
thf(fact_2782_add__mset__in__multiset,axiom,
! [A: $tType,M8: A > nat,A2: A] :
( ( finite_finite2 @ A
@ ( collect @ A
@ ^ [X2: A] : ( ord_less @ nat @ ( zero_zero @ nat ) @ ( M8 @ X2 ) ) ) )
=> ( finite_finite2 @ A
@ ( collect @ A
@ ^ [X2: A] : ( ord_less @ nat @ ( zero_zero @ nat ) @ ( if @ nat @ ( X2 = A2 ) @ ( suc @ ( M8 @ X2 ) ) @ ( M8 @ X2 ) ) ) ) ) ) ).
% add_mset_in_multiset
thf(fact_2783_finite__linorder__max__induct,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A3: set @ A,P: ( set @ A ) > $o] :
( ( finite_finite2 @ A @ A3 )
=> ( ( P @ ( bot_bot @ ( set @ A ) ) )
=> ( ! [B3: A,A8: set @ A] :
( ( finite_finite2 @ A @ A8 )
=> ( ! [X5: A] :
( ( member @ A @ X5 @ A8 )
=> ( ord_less @ A @ X5 @ B3 ) )
=> ( ( P @ A8 )
=> ( P @ ( insert @ A @ B3 @ A8 ) ) ) ) )
=> ( P @ A3 ) ) ) ) ) ).
% finite_linorder_max_induct
thf(fact_2784_finite__linorder__min__induct,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A3: set @ A,P: ( set @ A ) > $o] :
( ( finite_finite2 @ A @ A3 )
=> ( ( P @ ( bot_bot @ ( set @ A ) ) )
=> ( ! [B3: A,A8: set @ A] :
( ( finite_finite2 @ A @ A8 )
=> ( ! [X5: A] :
( ( member @ A @ X5 @ A8 )
=> ( ord_less @ A @ B3 @ X5 ) )
=> ( ( P @ A8 )
=> ( P @ ( insert @ A @ B3 @ A8 ) ) ) ) )
=> ( P @ A3 ) ) ) ) ) ).
% finite_linorder_min_induct
thf(fact_2785_size__empty,axiom,
! [A: $tType] :
( ( size_size @ ( multiset @ A ) @ ( zero_zero @ ( multiset @ A ) ) )
= ( zero_zero @ nat ) ) ).
% size_empty
thf(fact_2786_size__eq__0__iff__empty,axiom,
! [A: $tType,M8: multiset @ A] :
( ( ( size_size @ ( multiset @ A ) @ M8 )
= ( zero_zero @ nat ) )
= ( M8
= ( zero_zero @ ( multiset @ A ) ) ) ) ).
% size_eq_0_iff_empty
thf(fact_2787_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_2788_ex__has__least__nat,axiom,
! [A: $tType,P: A > $o,K: A,M: A > nat] :
( ( P @ K )
=> ? [X3: A] :
( ( P @ X3 )
& ! [Y4: A] :
( ( P @ Y4 )
=> ( ord_less_eq @ nat @ ( M @ X3 ) @ ( M @ Y4 ) ) ) ) ) ).
% ex_has_least_nat
thf(fact_2789_nonempty__has__size,axiom,
! [A: $tType,S3: multiset @ A] :
( ( S3
!= ( zero_zero @ ( multiset @ A ) ) )
= ( ord_less @ nat @ ( zero_zero @ nat ) @ ( size_size @ ( multiset @ A ) @ S3 ) ) ) ).
% nonempty_has_size
thf(fact_2790_Lattices__Big_Oex__has__greatest__nat,axiom,
! [A: $tType,P: A > $o,K: A,F2: A > nat,B2: nat] :
( ( P @ K )
=> ( ! [Y3: A] :
( ( P @ Y3 )
=> ( ord_less @ nat @ ( F2 @ Y3 ) @ B2 ) )
=> ? [X3: A] :
( ( P @ X3 )
& ! [Y4: A] :
( ( P @ Y4 )
=> ( ord_less_eq @ nat @ ( F2 @ Y4 ) @ ( F2 @ X3 ) ) ) ) ) ) ).
% Lattices_Big.ex_has_greatest_nat
thf(fact_2791_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_2792_complex__mod__triangle__ineq2,axiom,
! [B2: complex,A2: complex] : ( ord_less_eq @ real @ ( minus_minus @ real @ ( real_V7770717601297561774m_norm @ complex @ ( plus_plus @ complex @ B2 @ A2 ) ) @ ( real_V7770717601297561774m_norm @ complex @ B2 ) ) @ ( real_V7770717601297561774m_norm @ complex @ A2 ) ) ).
% complex_mod_triangle_ineq2
thf(fact_2793_even__set__encode__iff,axiom,
! [A3: set @ nat] :
( ( finite_finite2 @ nat @ A3 )
=> ( ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( nat_set_encode @ A3 ) )
= ( ~ ( member @ nat @ ( zero_zero @ nat ) @ A3 ) ) ) ) ).
% even_set_encode_iff
thf(fact_2794_infinite__growing,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [X8: set @ A] :
( ( X8
!= ( bot_bot @ ( set @ A ) ) )
=> ( ! [X3: A] :
( ( member @ A @ X3 @ X8 )
=> ? [Xa2: A] :
( ( member @ A @ Xa2 @ X8 )
& ( ord_less @ A @ X3 @ Xa2 ) ) )
=> ~ ( finite_finite2 @ A @ X8 ) ) ) ) ).
% infinite_growing
thf(fact_2795_ex__min__if__finite,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [S3: set @ A] :
( ( finite_finite2 @ A @ S3 )
=> ( ( S3
!= ( bot_bot @ ( set @ A ) ) )
=> ? [X3: A] :
( ( member @ A @ X3 @ S3 )
& ~ ? [Xa2: A] :
( ( member @ A @ Xa2 @ S3 )
& ( ord_less @ A @ Xa2 @ X3 ) ) ) ) ) ) ).
% ex_min_if_finite
thf(fact_2796_ex__has__greatest__nat__lemma,axiom,
! [A: $tType,P: A > $o,K: A,F2: A > nat,N3: nat] :
( ( P @ K )
=> ( ! [X3: A] :
( ( P @ X3 )
=> ? [Y4: A] :
( ( P @ Y4 )
& ~ ( ord_less_eq @ nat @ ( F2 @ Y4 ) @ ( F2 @ X3 ) ) ) )
=> ? [Y3: A] :
( ( P @ Y3 )
& ~ ( ord_less @ nat @ ( F2 @ Y3 ) @ ( plus_plus @ nat @ ( F2 @ K ) @ N3 ) ) ) ) ) ).
% ex_has_greatest_nat_lemma
thf(fact_2797_filter__preserves__multiset,axiom,
! [A: $tType,M8: A > nat,P: A > $o] :
( ( finite_finite2 @ A
@ ( collect @ A
@ ^ [X2: A] : ( ord_less @ nat @ ( zero_zero @ nat ) @ ( M8 @ X2 ) ) ) )
=> ( finite_finite2 @ A
@ ( collect @ A
@ ^ [X2: A] : ( ord_less @ nat @ ( zero_zero @ nat ) @ ( if @ nat @ ( P @ X2 ) @ ( M8 @ X2 ) @ ( zero_zero @ nat ) ) ) ) ) ) ).
% filter_preserves_multiset
thf(fact_2798_finite__ranking__induct,axiom,
! [A: $tType,B: $tType] :
( ( linorder @ A )
=> ! [S3: set @ B,P: ( set @ B ) > $o,F2: B > A] :
( ( finite_finite2 @ B @ S3 )
=> ( ( P @ ( bot_bot @ ( set @ B ) ) )
=> ( ! [X3: B,S6: set @ B] :
( ( finite_finite2 @ B @ S6 )
=> ( ! [Y4: B] :
( ( member @ B @ Y4 @ S6 )
=> ( ord_less_eq @ A @ ( F2 @ Y4 ) @ ( F2 @ X3 ) ) )
=> ( ( P @ S6 )
=> ( P @ ( insert @ B @ X3 @ S6 ) ) ) ) )
=> ( P @ S3 ) ) ) ) ) ).
% finite_ranking_induct
thf(fact_2799_TBOUND__def,axiom,
! [A: $tType] :
( ( time_TBOUND @ A )
= ( ^ [M5: heap_Time_Heap @ A,T3: nat] :
! [H: heap_ext @ product_unit] : ( ord_less_eq @ nat @ ( time_time @ A @ M5 @ H ) @ T3 ) ) ) ).
% TBOUND_def
thf(fact_2800_TBOUNDI,axiom,
! [A: $tType,M: heap_Time_Heap @ A,T2: nat] :
( ! [H4: heap_ext @ product_unit] : ( ord_less_eq @ nat @ ( time_time @ A @ M @ H4 ) @ T2 )
=> ( time_TBOUND @ A @ M @ T2 ) ) ).
% TBOUNDI
thf(fact_2801_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_2802_vebt__buildup_Oelims,axiom,
! [X: nat,Y: vEBT_VEBT] :
( ( ( vEBT_vebt_buildup @ X )
= Y )
=> ( ( ( X
= ( zero_zero @ nat ) )
=> ( Y
!= ( vEBT_Leaf @ $false @ $false ) ) )
=> ( ( ( X
= ( suc @ ( zero_zero @ nat ) ) )
=> ( Y
!= ( vEBT_Leaf @ $false @ $false ) ) )
=> ~ ! [Va3: nat] :
( ( X
= ( suc @ ( suc @ Va3 ) ) )
=> ~ ( ( ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( suc @ ( suc @ Va3 ) ) )
=> ( Y
= ( 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 ) ) )
=> ( Y
= ( 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_2803_prod_Ofinite__Collect__op,axiom,
! [A: $tType,B: $tType] :
( ( comm_monoid_mult @ A )
=> ! [I5: set @ B,X: B > A,Y: B > A] :
( ( finite_finite2 @ B
@ ( collect @ B
@ ^ [I3: B] :
( ( member @ B @ I3 @ I5 )
& ( ( X @ I3 )
!= ( one_one @ A ) ) ) ) )
=> ( ( finite_finite2 @ B
@ ( collect @ B
@ ^ [I3: B] :
( ( member @ B @ I3 @ I5 )
& ( ( Y @ I3 )
!= ( one_one @ A ) ) ) ) )
=> ( finite_finite2 @ B
@ ( collect @ B
@ ^ [I3: B] :
( ( member @ B @ I3 @ I5 )
& ( ( times_times @ A @ ( X @ I3 ) @ ( Y @ I3 ) )
!= ( one_one @ A ) ) ) ) ) ) ) ) ).
% prod.finite_Collect_op
thf(fact_2804_sum_Ofinite__Collect__op,axiom,
! [A: $tType,B: $tType] :
( ( comm_monoid_add @ A )
=> ! [I5: set @ B,X: B > A,Y: B > A] :
( ( finite_finite2 @ B
@ ( collect @ B
@ ^ [I3: B] :
( ( member @ B @ I3 @ I5 )
& ( ( X @ I3 )
!= ( zero_zero @ A ) ) ) ) )
=> ( ( finite_finite2 @ B
@ ( collect @ B
@ ^ [I3: B] :
( ( member @ B @ I3 @ I5 )
& ( ( Y @ I3 )
!= ( zero_zero @ A ) ) ) ) )
=> ( finite_finite2 @ B
@ ( collect @ B
@ ^ [I3: B] :
( ( member @ B @ I3 @ I5 )
& ( ( plus_plus @ A @ ( X @ I3 ) @ ( Y @ I3 ) )
!= ( zero_zero @ A ) ) ) ) ) ) ) ) ).
% sum.finite_Collect_op
thf(fact_2805_intind,axiom,
! [A: $tType,I: nat,N3: nat,P: A > $o,X: A] :
( ( ord_less @ nat @ I @ N3 )
=> ( ( P @ X )
=> ( P @ ( nth @ A @ ( replicate @ A @ N3 @ X ) @ I ) ) ) ) ).
% intind
thf(fact_2806_repli__cons__repl,axiom,
! [B: $tType,A: $tType,Q: assn,X: heap_Time_Heap @ A,A3: B > A > assn,Y: B,N3: nat] :
( ( hoare_hoare_triple @ A @ Q @ X
@ ^ [R5: A] : ( times_times @ assn @ Q @ ( A3 @ Y @ R5 ) ) )
=> ( hoare_hoare_triple @ ( list @ A ) @ Q @ ( vEBT_VEBT_replicatei @ A @ N3 @ X )
@ ^ [R5: list @ A] : ( times_times @ assn @ Q @ ( vEBT_List_list_assn @ B @ A @ A3 @ ( replicate @ B @ N3 @ Y ) @ R5 ) ) ) ) ).
% repli_cons_repl
thf(fact_2807_repli__emp,axiom,
! [A: $tType,B: $tType,X: heap_Time_Heap @ A,A3: B > A > assn,Y: B,N3: nat] :
( ( hoare_hoare_triple @ A @ ( one_one @ assn ) @ X @ ( A3 @ Y ) )
=> ( hoare_hoare_triple @ ( list @ A ) @ ( one_one @ assn ) @ ( vEBT_VEBT_replicatei @ A @ N3 @ X ) @ ( vEBT_List_list_assn @ B @ A @ A3 @ ( replicate @ B @ N3 @ Y ) ) ) ) ).
% repli_emp
thf(fact_2808_length__replicate,axiom,
! [A: $tType,N3: nat,X: A] :
( ( size_size @ ( list @ A ) @ ( replicate @ A @ N3 @ X ) )
= N3 ) ).
% length_replicate
thf(fact_2809_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_2810_Ball__set__replicate,axiom,
! [A: $tType,N3: nat,A2: A,P: A > $o] :
( ( ! [X2: A] :
( ( member @ A @ X2 @ ( set2 @ A @ ( replicate @ A @ N3 @ A2 ) ) )
=> ( P @ X2 ) ) )
= ( ( P @ A2 )
| ( N3
= ( zero_zero @ nat ) ) ) ) ).
% Ball_set_replicate
thf(fact_2811_Bex__set__replicate,axiom,
! [A: $tType,N3: nat,A2: A,P: A > $o] :
( ( ? [X2: A] :
( ( member @ A @ X2 @ ( set2 @ A @ ( replicate @ A @ N3 @ A2 ) ) )
& ( P @ X2 ) ) )
= ( ( P @ A2 )
& ( N3
!= ( zero_zero @ nat ) ) ) ) ).
% Bex_set_replicate
thf(fact_2812_in__set__replicate,axiom,
! [A: $tType,X: A,N3: nat,Y: A] :
( ( member @ A @ X @ ( set2 @ A @ ( replicate @ A @ N3 @ Y ) ) )
= ( ( X = Y )
& ( N3
!= ( zero_zero @ nat ) ) ) ) ).
% in_set_replicate
thf(fact_2813_nth__replicate,axiom,
! [A: $tType,I: nat,N3: nat,X: A] :
( ( ord_less @ nat @ I @ N3 )
=> ( ( nth @ A @ ( replicate @ A @ N3 @ X ) @ I )
= X ) ) ).
% nth_replicate
thf(fact_2814_set__replicate,axiom,
! [A: $tType,N3: nat,X: A] :
( ( N3
!= ( zero_zero @ nat ) )
=> ( ( set2 @ A @ ( replicate @ A @ N3 @ X ) )
= ( insert @ A @ X @ ( bot_bot @ ( set @ A ) ) ) ) ) ).
% set_replicate
thf(fact_2815_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_2816_Multiset_Odiff__cancel,axiom,
! [A: $tType,A3: multiset @ A] :
( ( minus_minus @ ( multiset @ A ) @ A3 @ A3 )
= ( zero_zero @ ( multiset @ A ) ) ) ).
% Multiset.diff_cancel
thf(fact_2817_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_2818_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_2819_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_2820_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_2821_union__diff__assoc,axiom,
! [A: $tType,C5: multiset @ A,B6: multiset @ A,A3: multiset @ A] :
( ( ( minus_minus @ ( multiset @ A ) @ C5 @ B6 )
= ( zero_zero @ ( multiset @ A ) ) )
=> ( ( minus_minus @ ( multiset @ A ) @ ( plus_plus @ ( multiset @ A ) @ A3 @ B6 ) @ C5 )
= ( plus_plus @ ( multiset @ A ) @ A3 @ ( minus_minus @ ( multiset @ A ) @ B6 @ C5 ) ) ) ) ).
% union_diff_assoc
thf(fact_2822_replicate__eqI,axiom,
! [A: $tType,Xs2: list @ A,N3: nat,X: A] :
( ( ( size_size @ ( list @ A ) @ Xs2 )
= N3 )
=> ( ! [Y3: A] :
( ( member @ A @ Y3 @ ( set2 @ A @ Xs2 ) )
=> ( Y3 = X ) )
=> ( Xs2
= ( replicate @ A @ N3 @ X ) ) ) ) ).
% replicate_eqI
thf(fact_2823_replicate__length__same,axiom,
! [A: $tType,Xs2: list @ A,X: A] :
( ! [X3: A] :
( ( member @ A @ X3 @ ( set2 @ A @ Xs2 ) )
=> ( X3 = X ) )
=> ( ( replicate @ A @ ( size_size @ ( list @ A ) @ Xs2 ) @ X )
= Xs2 ) ) ).
% replicate_length_same
thf(fact_2824_map__replicate__const,axiom,
! [B: $tType,A: $tType,K: A,Lst: list @ B] :
( ( map @ B @ A
@ ^ [X2: B] : K
@ Lst )
= ( replicate @ A @ ( size_size @ ( list @ B ) @ Lst ) @ K ) ) ).
% map_replicate_const
thf(fact_2825_set__replicate__Suc,axiom,
! [A: $tType,N3: nat,X: A] :
( ( set2 @ A @ ( replicate @ A @ ( suc @ N3 ) @ X ) )
= ( insert @ A @ X @ ( bot_bot @ ( set @ A ) ) ) ) ).
% set_replicate_Suc
thf(fact_2826_set__replicate__conv__if,axiom,
! [A: $tType,N3: nat,X: A] :
( ( ( N3
= ( zero_zero @ nat ) )
=> ( ( set2 @ A @ ( replicate @ A @ N3 @ X ) )
= ( bot_bot @ ( set @ A ) ) ) )
& ( ( N3
!= ( zero_zero @ nat ) )
=> ( ( set2 @ A @ ( replicate @ A @ N3 @ X ) )
= ( insert @ A @ X @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ).
% set_replicate_conv_if
thf(fact_2827_TBOUND__mono,axiom,
! [A: $tType,C2: heap_Time_Heap @ A,T2: nat,T6: nat] :
( ( time_TBOUND @ A @ C2 @ T2 )
=> ( ( ord_less_eq @ nat @ T2 @ T6 )
=> ( time_TBOUND @ A @ C2 @ T6 ) ) ) ).
% TBOUND_mono
thf(fact_2828_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_2829_TBOUND__return,axiom,
! [A: $tType,X: A] : ( time_TBOUND @ A @ ( heap_Time_return @ A @ X ) @ ( one_one @ nat ) ) ).
% TBOUND_return
thf(fact_2830_time__return,axiom,
! [A: $tType,X: A,H2: heap_ext @ product_unit] :
( ( time_time @ A @ ( heap_Time_return @ A @ X ) @ H2 )
= ( one_one @ nat ) ) ).
% time_return
thf(fact_2831_ceiling__log__eq__powr__iff,axiom,
! [X: real,B2: real,K: nat] :
( ( ord_less @ real @ ( zero_zero @ real ) @ X )
=> ( ( ord_less @ real @ ( one_one @ real ) @ B2 )
=> ( ( ( archimedean_ceiling @ real @ ( log @ B2 @ X ) )
= ( plus_plus @ int @ ( semiring_1_of_nat @ int @ K ) @ ( one_one @ int ) ) )
= ( ( ord_less @ real @ ( powr @ real @ B2 @ ( semiring_1_of_nat @ real @ K ) ) @ X )
& ( ord_less_eq @ real @ X @ ( powr @ real @ B2 @ ( semiring_1_of_nat @ real @ ( plus_plus @ nat @ K @ ( one_one @ nat ) ) ) ) ) ) ) ) ) ).
% ceiling_log_eq_powr_iff
thf(fact_2832_sum__gp,axiom,
! [A: $tType] :
( ( ( division_ring @ A )
& ( comm_ring @ A ) )
=> ! [N3: nat,M: nat,X: A] :
( ( ( ord_less @ nat @ N3 @ M )
=> ( ( groups7311177749621191930dd_sum @ nat @ A @ ( power_power @ A @ X ) @ ( set_or1337092689740270186AtMost @ nat @ M @ N3 ) )
= ( zero_zero @ A ) ) )
& ( ~ ( ord_less @ nat @ N3 @ M )
=> ( ( ( X
= ( one_one @ A ) )
=> ( ( groups7311177749621191930dd_sum @ nat @ A @ ( power_power @ A @ X ) @ ( set_or1337092689740270186AtMost @ nat @ M @ N3 ) )
= ( semiring_1_of_nat @ A @ ( minus_minus @ nat @ ( plus_plus @ nat @ N3 @ ( one_one @ nat ) ) @ M ) ) ) )
& ( ( X
!= ( one_one @ A ) )
=> ( ( groups7311177749621191930dd_sum @ nat @ A @ ( power_power @ A @ X ) @ ( set_or1337092689740270186AtMost @ nat @ M @ N3 ) )
= ( divide_divide @ A @ ( minus_minus @ A @ ( power_power @ A @ X @ M ) @ ( power_power @ A @ X @ ( suc @ N3 ) ) ) @ ( minus_minus @ A @ ( one_one @ A ) @ X ) ) ) ) ) ) ) ) ).
% sum_gp
thf(fact_2833_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_2834_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
@ ^ [N2: nat] : ( times_times @ A @ ( semiring_1_of_nat @ A @ ( suc @ N2 ) ) @ ( power_power @ A @ Z @ N2 ) )
@ ( 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_2835_VEBT__internal_Oheight_Osimps_I1_J,axiom,
! [A2: $o,B2: $o] :
( ( vEBT_VEBT_height @ ( vEBT_Leaf @ A2 @ B2 ) )
= ( zero_zero @ nat ) ) ).
% VEBT_internal.height.simps(1)
thf(fact_2836_powr__one__eq__one,axiom,
! [A: $tType] :
( ( ln @ A )
=> ! [A2: A] :
( ( powr @ A @ ( one_one @ A ) @ A2 )
= ( one_one @ A ) ) ) ).
% powr_one_eq_one
thf(fact_2837_powr__zero__eq__one,axiom,
! [A: $tType] :
( ( ln @ A )
=> ! [X: A] :
( ( ( X
= ( zero_zero @ A ) )
=> ( ( powr @ A @ X @ ( zero_zero @ A ) )
= ( zero_zero @ A ) ) )
& ( ( X
!= ( zero_zero @ A ) )
=> ( ( powr @ A @ X @ ( zero_zero @ A ) )
= ( one_one @ A ) ) ) ) ) ).
% powr_zero_eq_one
thf(fact_2838_powr__nonneg__iff,axiom,
! [A2: real,X: real] :
( ( ord_less_eq @ real @ ( powr @ real @ A2 @ X ) @ ( zero_zero @ real ) )
= ( A2
= ( zero_zero @ real ) ) ) ).
% powr_nonneg_iff
thf(fact_2839_powr__less__cancel__iff,axiom,
! [X: real,A2: real,B2: real] :
( ( ord_less @ real @ ( one_one @ real ) @ X )
=> ( ( ord_less @ real @ ( powr @ real @ X @ A2 ) @ ( powr @ real @ X @ B2 ) )
= ( ord_less @ real @ A2 @ B2 ) ) ) ).
% powr_less_cancel_iff
thf(fact_2840_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_2841_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_2842_sum_Oinsert,axiom,
! [A: $tType,B: $tType] :
( ( comm_monoid_add @ A )
=> ! [A3: set @ B,X: B,G: B > A] :
( ( finite_finite2 @ B @ A3 )
=> ( ~ ( member @ B @ X @ A3 )
=> ( ( groups7311177749621191930dd_sum @ B @ A @ G @ ( insert @ B @ X @ A3 ) )
= ( plus_plus @ A @ ( G @ X ) @ ( groups7311177749621191930dd_sum @ B @ A @ G @ A3 ) ) ) ) ) ) ).
% sum.insert
thf(fact_2843_powr__eq__one__iff,axiom,
! [A2: real,X: real] :
( ( ord_less @ real @ ( one_one @ real ) @ A2 )
=> ( ( ( powr @ real @ A2 @ X )
= ( one_one @ real ) )
= ( X
= ( zero_zero @ real ) ) ) ) ).
% powr_eq_one_iff
thf(fact_2844_powr__one__gt__zero__iff,axiom,
! [X: real] :
( ( ( powr @ real @ X @ ( one_one @ real ) )
= X )
= ( ord_less_eq @ real @ ( zero_zero @ real ) @ X ) ) ).
% powr_one_gt_zero_iff
thf(fact_2845_powr__one,axiom,
! [X: real] :
( ( ord_less_eq @ real @ ( zero_zero @ real ) @ X )
=> ( ( powr @ real @ X @ ( one_one @ real ) )
= X ) ) ).
% powr_one
thf(fact_2846_powr__le__cancel__iff,axiom,
! [X: real,A2: real,B2: real] :
( ( ord_less @ real @ ( one_one @ real ) @ X )
=> ( ( ord_less_eq @ real @ ( powr @ real @ X @ A2 ) @ ( powr @ real @ X @ B2 ) )
= ( ord_less_eq @ real @ A2 @ B2 ) ) ) ).
% powr_le_cancel_iff
thf(fact_2847_numeral__powr__numeral__real,axiom,
! [M: num,N3: num] :
( ( powr @ real @ ( numeral_numeral @ real @ M ) @ ( numeral_numeral @ real @ N3 ) )
= ( power_power @ real @ ( numeral_numeral @ real @ M ) @ ( numeral_numeral @ nat @ N3 ) ) ) ).
% numeral_powr_numeral_real
thf(fact_2848_log__powr__cancel,axiom,
! [A2: real,Y: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ A2 )
=> ( ( A2
!= ( one_one @ real ) )
=> ( ( log @ A2 @ ( powr @ real @ A2 @ Y ) )
= Y ) ) ) ).
% log_powr_cancel
thf(fact_2849_powr__log__cancel,axiom,
! [A2: real,X: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ A2 )
=> ( ( A2
!= ( one_one @ real ) )
=> ( ( ord_less @ real @ ( zero_zero @ real ) @ X )
=> ( ( powr @ real @ A2 @ ( log @ A2 @ X ) )
= X ) ) ) ) ).
% powr_log_cancel
thf(fact_2850_powser__sums__zero__iff,axiom,
! [A: $tType] :
( ( real_V8999393235501362500lgebra @ A )
=> ! [A2: nat > A,X: A] :
( ( sums @ A
@ ^ [N2: nat] : ( times_times @ A @ ( A2 @ N2 ) @ ( power_power @ A @ ( zero_zero @ A ) @ N2 ) )
@ X )
= ( ( A2 @ ( zero_zero @ nat ) )
= X ) ) ) ).
% powser_sums_zero_iff
thf(fact_2851_sum_Ocl__ivl__Suc,axiom,
! [A: $tType] :
( ( comm_monoid_add @ A )
=> ! [N3: nat,M: nat,G: nat > A] :
( ( ( ord_less @ nat @ ( suc @ N3 ) @ M )
=> ( ( groups7311177749621191930dd_sum @ nat @ A @ G @ ( set_or1337092689740270186AtMost @ nat @ M @ ( suc @ N3 ) ) )
= ( zero_zero @ A ) ) )
& ( ~ ( ord_less @ nat @ ( suc @ N3 ) @ M )
=> ( ( groups7311177749621191930dd_sum @ nat @ A @ G @ ( set_or1337092689740270186AtMost @ nat @ M @ ( suc @ N3 ) ) )
= ( plus_plus @ A @ ( groups7311177749621191930dd_sum @ nat @ A @ G @ ( set_or1337092689740270186AtMost @ nat @ M @ N3 ) ) @ ( G @ ( suc @ N3 ) ) ) ) ) ) ) ).
% sum.cl_ivl_Suc
thf(fact_2852_powr__numeral,axiom,
! [X: real,N3: num] :
( ( ord_less_eq @ real @ ( zero_zero @ real ) @ X )
=> ( ( powr @ real @ X @ ( numeral_numeral @ real @ N3 ) )
= ( power_power @ real @ X @ ( numeral_numeral @ nat @ N3 ) ) ) ) ).
% powr_numeral
thf(fact_2853_sum__zero__power,axiom,
! [A: $tType] :
( ( division_ring @ A )
=> ! [A3: set @ nat,C2: nat > A] :
( ( ( ( finite_finite2 @ nat @ A3 )
& ( member @ nat @ ( zero_zero @ nat ) @ A3 ) )
=> ( ( groups7311177749621191930dd_sum @ nat @ A
@ ^ [I3: nat] : ( times_times @ A @ ( C2 @ I3 ) @ ( power_power @ A @ ( zero_zero @ A ) @ I3 ) )
@ A3 )
= ( C2 @ ( zero_zero @ nat ) ) ) )
& ( ~ ( ( finite_finite2 @ nat @ A3 )
& ( member @ nat @ ( zero_zero @ nat ) @ A3 ) )
=> ( ( groups7311177749621191930dd_sum @ nat @ A
@ ^ [I3: nat] : ( times_times @ A @ ( C2 @ I3 ) @ ( power_power @ A @ ( zero_zero @ A ) @ I3 ) )
@ A3 )
= ( zero_zero @ A ) ) ) ) ) ).
% sum_zero_power
thf(fact_2854_sum__zero__power_H,axiom,
! [A: $tType] :
( ( field @ A )
=> ! [A3: set @ nat,C2: nat > A,D: nat > A] :
( ( ( ( finite_finite2 @ nat @ A3 )
& ( member @ nat @ ( zero_zero @ nat ) @ A3 ) )
=> ( ( groups7311177749621191930dd_sum @ nat @ A
@ ^ [I3: nat] : ( divide_divide @ A @ ( times_times @ A @ ( C2 @ I3 ) @ ( power_power @ A @ ( zero_zero @ A ) @ I3 ) ) @ ( D @ I3 ) )
@ A3 )
= ( divide_divide @ A @ ( C2 @ ( zero_zero @ nat ) ) @ ( D @ ( zero_zero @ nat ) ) ) ) )
& ( ~ ( ( finite_finite2 @ nat @ A3 )
& ( member @ nat @ ( zero_zero @ nat ) @ A3 ) )
=> ( ( groups7311177749621191930dd_sum @ nat @ A
@ ^ [I3: nat] : ( divide_divide @ A @ ( times_times @ A @ ( C2 @ I3 ) @ ( power_power @ A @ ( zero_zero @ A ) @ I3 ) ) @ ( D @ I3 ) )
@ A3 )
= ( zero_zero @ A ) ) ) ) ) ).
% sum_zero_power'
thf(fact_2855_sum__norm__le,axiom,
! [A: $tType,B: $tType] :
( ( real_V822414075346904944vector @ A )
=> ! [S3: set @ B,F2: B > A,G: B > real] :
( ! [X3: B] :
( ( member @ B @ X3 @ S3 )
=> ( ord_less_eq @ real @ ( real_V7770717601297561774m_norm @ A @ ( F2 @ X3 ) ) @ ( G @ X3 ) ) )
=> ( ord_less_eq @ real @ ( real_V7770717601297561774m_norm @ A @ ( groups7311177749621191930dd_sum @ B @ A @ F2 @ S3 ) ) @ ( groups7311177749621191930dd_sum @ B @ real @ G @ S3 ) ) ) ) ).
% sum_norm_le
thf(fact_2856_norm__sum,axiom,
! [A: $tType,B: $tType] :
( ( real_V822414075346904944vector @ A )
=> ! [F2: B > A,A3: set @ B] :
( ord_less_eq @ real @ ( real_V7770717601297561774m_norm @ A @ ( groups7311177749621191930dd_sum @ B @ A @ F2 @ A3 ) )
@ ( groups7311177749621191930dd_sum @ B @ real
@ ^ [I3: B] : ( real_V7770717601297561774m_norm @ A @ ( F2 @ I3 ) )
@ A3 ) ) ) ).
% norm_sum
thf(fact_2857_powr__powr,axiom,
! [X: real,A2: real,B2: real] :
( ( powr @ real @ ( powr @ real @ X @ A2 ) @ B2 )
= ( powr @ real @ X @ ( times_times @ real @ A2 @ B2 ) ) ) ).
% powr_powr
thf(fact_2858_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_2859_sum_Odistrib,axiom,
! [A: $tType,B: $tType] :
( ( comm_monoid_add @ A )
=> ! [G: B > A,H2: B > A,A3: set @ B] :
( ( groups7311177749621191930dd_sum @ B @ A
@ ^ [X2: B] : ( plus_plus @ A @ ( G @ X2 ) @ ( H2 @ X2 ) )
@ A3 )
= ( plus_plus @ A @ ( groups7311177749621191930dd_sum @ B @ A @ G @ A3 ) @ ( groups7311177749621191930dd_sum @ B @ A @ H2 @ A3 ) ) ) ) ).
% sum.distrib
thf(fact_2860_sum__product,axiom,
! [B: $tType,C: $tType,A: $tType] :
( ( semiring_0 @ B )
=> ! [F2: A > B,A3: set @ A,G: C > B,B6: set @ C] :
( ( times_times @ B @ ( groups7311177749621191930dd_sum @ A @ B @ F2 @ A3 ) @ ( groups7311177749621191930dd_sum @ C @ B @ G @ B6 ) )
= ( groups7311177749621191930dd_sum @ A @ B
@ ^ [I3: A] :
( groups7311177749621191930dd_sum @ C @ B
@ ^ [J3: C] : ( times_times @ B @ ( F2 @ I3 ) @ ( G @ J3 ) )
@ B6 )
@ A3 ) ) ) ).
% sum_product
thf(fact_2861_sum__distrib__right,axiom,
! [A: $tType,B: $tType] :
( ( semiring_0 @ A )
=> ! [F2: B > A,A3: set @ B,R2: A] :
( ( times_times @ A @ ( groups7311177749621191930dd_sum @ B @ A @ F2 @ A3 ) @ R2 )
= ( groups7311177749621191930dd_sum @ B @ A
@ ^ [N2: B] : ( times_times @ A @ ( F2 @ N2 ) @ R2 )
@ A3 ) ) ) ).
% sum_distrib_right
thf(fact_2862_sum__distrib__left,axiom,
! [A: $tType,B: $tType] :
( ( semiring_0 @ A )
=> ! [R2: A,F2: B > A,A3: set @ B] :
( ( times_times @ A @ R2 @ ( groups7311177749621191930dd_sum @ B @ A @ F2 @ A3 ) )
= ( groups7311177749621191930dd_sum @ B @ A
@ ^ [N2: B] : ( times_times @ A @ R2 @ ( F2 @ N2 ) )
@ A3 ) ) ) ).
% sum_distrib_left
thf(fact_2863_sum__subtractf,axiom,
! [A: $tType,B: $tType] :
( ( ab_group_add @ A )
=> ! [F2: B > A,G: B > A,A3: set @ B] :
( ( groups7311177749621191930dd_sum @ B @ A
@ ^ [X2: B] : ( minus_minus @ A @ ( F2 @ X2 ) @ ( G @ X2 ) )
@ A3 )
= ( minus_minus @ A @ ( groups7311177749621191930dd_sum @ B @ A @ F2 @ A3 ) @ ( groups7311177749621191930dd_sum @ B @ A @ G @ A3 ) ) ) ) ).
% sum_subtractf
thf(fact_2864_sum__divide__distrib,axiom,
! [A: $tType,B: $tType] :
( ( field @ A )
=> ! [F2: B > A,A3: set @ B,R2: A] :
( ( divide_divide @ A @ ( groups7311177749621191930dd_sum @ B @ A @ F2 @ A3 ) @ R2 )
= ( groups7311177749621191930dd_sum @ B @ A
@ ^ [N2: B] : ( divide_divide @ A @ ( F2 @ N2 ) @ R2 )
@ A3 ) ) ) ).
% sum_divide_distrib
thf(fact_2865_mod__sum__eq,axiom,
! [B: $tType,A: $tType] :
( ( euclid4440199948858584721cancel @ A )
=> ! [F2: B > A,A2: A,A3: set @ B] :
( ( modulo_modulo @ A
@ ( groups7311177749621191930dd_sum @ B @ A
@ ^ [I3: B] : ( modulo_modulo @ A @ ( F2 @ I3 ) @ A2 )
@ A3 )
@ A2 )
= ( modulo_modulo @ A @ ( groups7311177749621191930dd_sum @ B @ A @ F2 @ A3 ) @ A2 ) ) ) ).
% mod_sum_eq
thf(fact_2866_sum__nonpos,axiom,
! [B: $tType,A: $tType] :
( ( ordere6911136660526730532id_add @ A )
=> ! [A3: set @ B,F2: B > A] :
( ! [X3: B] :
( ( member @ B @ X3 @ A3 )
=> ( ord_less_eq @ A @ ( F2 @ X3 ) @ ( zero_zero @ A ) ) )
=> ( ord_less_eq @ A @ ( groups7311177749621191930dd_sum @ B @ A @ F2 @ A3 ) @ ( zero_zero @ A ) ) ) ) ).
% sum_nonpos
thf(fact_2867_sum__nonneg,axiom,
! [A: $tType,B: $tType] :
( ( ordere6911136660526730532id_add @ A )
=> ! [A3: set @ B,F2: B > A] :
( ! [X3: B] :
( ( member @ B @ X3 @ A3 )
=> ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( F2 @ X3 ) ) )
=> ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( groups7311177749621191930dd_sum @ B @ A @ F2 @ A3 ) ) ) ) ).
% sum_nonneg
thf(fact_2868_sum__mono__inv,axiom,
! [A: $tType,I6: $tType] :
( ( ordere8940638589300402666id_add @ A )
=> ! [F2: I6 > A,I5: set @ I6,G: I6 > A,I: I6] :
( ( ( groups7311177749621191930dd_sum @ I6 @ A @ F2 @ I5 )
= ( groups7311177749621191930dd_sum @ I6 @ A @ G @ I5 ) )
=> ( ! [I2: I6] :
( ( member @ I6 @ I2 @ I5 )
=> ( ord_less_eq @ A @ ( F2 @ I2 ) @ ( G @ I2 ) ) )
=> ( ( member @ I6 @ I @ I5 )
=> ( ( finite_finite2 @ I6 @ I5 )
=> ( ( F2 @ I )
= ( G @ I ) ) ) ) ) ) ) ).
% sum_mono_inv
thf(fact_2869_powr__ge__pzero,axiom,
! [X: real,Y: real] : ( ord_less_eq @ real @ ( zero_zero @ real ) @ ( powr @ real @ X @ Y ) ) ).
% powr_ge_pzero
thf(fact_2870_powr__mono2,axiom,
! [A2: real,X: real,Y: real] :
( ( ord_less_eq @ real @ ( zero_zero @ real ) @ A2 )
=> ( ( ord_less_eq @ real @ ( zero_zero @ real ) @ X )
=> ( ( ord_less_eq @ real @ X @ Y )
=> ( ord_less_eq @ real @ ( powr @ real @ X @ A2 ) @ ( powr @ real @ Y @ A2 ) ) ) ) ) ).
% powr_mono2
thf(fact_2871_sum__cong__Suc,axiom,
! [A: $tType] :
( ( comm_monoid_add @ A )
=> ! [A3: set @ nat,F2: nat > A,G: nat > A] :
( ~ ( member @ nat @ ( zero_zero @ nat ) @ A3 )
=> ( ! [X3: nat] :
( ( member @ nat @ ( suc @ X3 ) @ A3 )
=> ( ( F2 @ ( suc @ X3 ) )
= ( G @ ( suc @ X3 ) ) ) )
=> ( ( groups7311177749621191930dd_sum @ nat @ A @ F2 @ A3 )
= ( groups7311177749621191930dd_sum @ nat @ A @ G @ A3 ) ) ) ) ) ).
% sum_cong_Suc
thf(fact_2872_powr__less__cancel,axiom,
! [X: real,A2: real,B2: real] :
( ( ord_less @ real @ ( powr @ real @ X @ A2 ) @ ( powr @ real @ X @ B2 ) )
=> ( ( ord_less @ real @ ( one_one @ real ) @ X )
=> ( ord_less @ real @ A2 @ B2 ) ) ) ).
% powr_less_cancel
thf(fact_2873_powr__less__mono,axiom,
! [A2: real,B2: real,X: real] :
( ( ord_less @ real @ A2 @ B2 )
=> ( ( ord_less @ real @ ( one_one @ real ) @ X )
=> ( ord_less @ real @ ( powr @ real @ X @ A2 ) @ ( powr @ real @ X @ B2 ) ) ) ) ).
% powr_less_mono
thf(fact_2874_powr__mono,axiom,
! [A2: real,B2: real,X: real] :
( ( ord_less_eq @ real @ A2 @ B2 )
=> ( ( ord_less_eq @ real @ ( one_one @ real ) @ X )
=> ( ord_less_eq @ real @ ( powr @ real @ X @ A2 ) @ ( powr @ real @ X @ B2 ) ) ) ) ).
% powr_mono
thf(fact_2875_sum_Oshift__bounds__cl__Suc__ivl,axiom,
! [A: $tType] :
( ( comm_monoid_add @ A )
=> ! [G: nat > A,M: nat,N3: nat] :
( ( groups7311177749621191930dd_sum @ nat @ A @ G @ ( set_or1337092689740270186AtMost @ nat @ ( suc @ M ) @ ( suc @ N3 ) ) )
= ( groups7311177749621191930dd_sum @ nat @ A
@ ^ [I3: nat] : ( G @ ( suc @ I3 ) )
@ ( set_or1337092689740270186AtMost @ nat @ M @ N3 ) ) ) ) ).
% sum.shift_bounds_cl_Suc_ivl
thf(fact_2876_sum_Oshift__bounds__cl__nat__ivl,axiom,
! [A: $tType] :
( ( comm_monoid_add @ A )
=> ! [G: nat > A,M: nat,K: nat,N3: nat] :
( ( groups7311177749621191930dd_sum @ nat @ A @ G @ ( set_or1337092689740270186AtMost @ nat @ ( plus_plus @ nat @ M @ K ) @ ( plus_plus @ nat @ N3 @ K ) ) )
= ( groups7311177749621191930dd_sum @ nat @ A
@ ^ [I3: nat] : ( G @ ( plus_plus @ nat @ I3 @ K ) )
@ ( set_or1337092689740270186AtMost @ nat @ M @ N3 ) ) ) ) ).
% sum.shift_bounds_cl_nat_ivl
thf(fact_2877_sum__nonneg__eq__0__iff,axiom,
! [A: $tType,B: $tType] :
( ( ordere6911136660526730532id_add @ A )
=> ! [A3: set @ B,F2: B > A] :
( ( finite_finite2 @ B @ A3 )
=> ( ! [X3: B] :
( ( member @ B @ X3 @ A3 )
=> ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( F2 @ X3 ) ) )
=> ( ( ( groups7311177749621191930dd_sum @ B @ A @ F2 @ A3 )
= ( zero_zero @ A ) )
= ( ! [X2: B] :
( ( member @ B @ X2 @ A3 )
=> ( ( F2 @ X2 )
= ( zero_zero @ A ) ) ) ) ) ) ) ) ).
% sum_nonneg_eq_0_iff
thf(fact_2878_sum__le__included,axiom,
! [B: $tType,A: $tType,C: $tType] :
( ( ordere6911136660526730532id_add @ A )
=> ! [S: set @ B,T2: set @ C,G: C > A,I: C > B,F2: B > A] :
( ( finite_finite2 @ B @ S )
=> ( ( finite_finite2 @ C @ T2 )
=> ( ! [X3: C] :
( ( member @ C @ X3 @ T2 )
=> ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( G @ X3 ) ) )
=> ( ! [X3: B] :
( ( member @ B @ X3 @ S )
=> ? [Xa2: C] :
( ( member @ C @ Xa2 @ T2 )
& ( ( I @ Xa2 )
= X3 )
& ( ord_less_eq @ A @ ( F2 @ X3 ) @ ( G @ Xa2 ) ) ) )
=> ( ord_less_eq @ A @ ( groups7311177749621191930dd_sum @ B @ A @ F2 @ S ) @ ( groups7311177749621191930dd_sum @ C @ A @ G @ T2 ) ) ) ) ) ) ) ).
% sum_le_included
thf(fact_2879_sum__strict__mono__ex1,axiom,
! [A: $tType,I6: $tType] :
( ( ordere8940638589300402666id_add @ A )
=> ! [A3: set @ I6,F2: I6 > A,G: I6 > A] :
( ( finite_finite2 @ I6 @ A3 )
=> ( ! [X3: I6] :
( ( member @ I6 @ X3 @ A3 )
=> ( ord_less_eq @ A @ ( F2 @ X3 ) @ ( G @ X3 ) ) )
=> ( ? [X5: I6] :
( ( member @ I6 @ X5 @ A3 )
& ( ord_less @ A @ ( F2 @ X5 ) @ ( G @ X5 ) ) )
=> ( ord_less @ A @ ( groups7311177749621191930dd_sum @ I6 @ A @ F2 @ A3 ) @ ( groups7311177749621191930dd_sum @ I6 @ A @ G @ A3 ) ) ) ) ) ) ).
% sum_strict_mono_ex1
thf(fact_2880_sum_Orelated,axiom,
! [A: $tType,B: $tType] :
( ( comm_monoid_add @ A )
=> ! [R: A > A > $o,S3: set @ B,H2: B > A,G: B > A] :
( ( R @ ( zero_zero @ A ) @ ( zero_zero @ A ) )
=> ( ! [X1: A,Y1: A,X23: A,Y23: A] :
( ( ( R @ X1 @ X23 )
& ( R @ Y1 @ Y23 ) )
=> ( R @ ( plus_plus @ A @ X1 @ Y1 ) @ ( plus_plus @ A @ X23 @ Y23 ) ) )
=> ( ( finite_finite2 @ B @ S3 )
=> ( ! [X3: B] :
( ( member @ B @ X3 @ S3 )
=> ( R @ ( H2 @ X3 ) @ ( G @ X3 ) ) )
=> ( R @ ( groups7311177749621191930dd_sum @ B @ A @ H2 @ S3 ) @ ( groups7311177749621191930dd_sum @ B @ A @ G @ S3 ) ) ) ) ) ) ) ).
% sum.related
thf(fact_2881_sum__strict__mono,axiom,
! [A: $tType,B: $tType] :
( ( strict7427464778891057005id_add @ A )
=> ! [A3: set @ B,F2: B > A,G: B > A] :
( ( finite_finite2 @ B @ A3 )
=> ( ( A3
!= ( bot_bot @ ( set @ B ) ) )
=> ( ! [X3: B] :
( ( member @ B @ X3 @ A3 )
=> ( ord_less @ A @ ( F2 @ X3 ) @ ( G @ X3 ) ) )
=> ( ord_less @ A @ ( groups7311177749621191930dd_sum @ B @ A @ F2 @ A3 ) @ ( groups7311177749621191930dd_sum @ B @ A @ G @ A3 ) ) ) ) ) ) ).
% sum_strict_mono
thf(fact_2882_sum_Oinsert__if,axiom,
! [A: $tType,B: $tType] :
( ( comm_monoid_add @ A )
=> ! [A3: set @ B,X: B,G: B > A] :
( ( finite_finite2 @ B @ A3 )
=> ( ( ( member @ B @ X @ A3 )
=> ( ( groups7311177749621191930dd_sum @ B @ A @ G @ ( insert @ B @ X @ A3 ) )
= ( groups7311177749621191930dd_sum @ B @ A @ G @ A3 ) ) )
& ( ~ ( member @ B @ X @ A3 )
=> ( ( groups7311177749621191930dd_sum @ B @ A @ G @ ( insert @ B @ X @ A3 ) )
= ( plus_plus @ A @ ( G @ X ) @ ( groups7311177749621191930dd_sum @ B @ A @ G @ A3 ) ) ) ) ) ) ) ).
% sum.insert_if
thf(fact_2883_sum_Oreindex__bij__witness__not__neutral,axiom,
! [B: $tType,A: $tType,C: $tType] :
( ( comm_monoid_add @ A )
=> ! [S7: set @ B,T7: set @ C,S3: set @ B,I: C > B,J: B > C,T5: 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 ) @ S3 @ S7 ) )
=> ( ( I @ ( J @ A4 ) )
= A4 ) )
=> ( ! [A4: B] :
( ( member @ B @ A4 @ ( minus_minus @ ( set @ B ) @ S3 @ S7 ) )
=> ( member @ C @ ( J @ A4 ) @ ( minus_minus @ ( set @ C ) @ T5 @ T7 ) ) )
=> ( ! [B3: C] :
( ( member @ C @ B3 @ ( minus_minus @ ( set @ C ) @ T5 @ T7 ) )
=> ( ( J @ ( I @ B3 ) )
= B3 ) )
=> ( ! [B3: C] :
( ( member @ C @ B3 @ ( minus_minus @ ( set @ C ) @ T5 @ T7 ) )
=> ( member @ B @ ( I @ B3 ) @ ( minus_minus @ ( set @ B ) @ S3 @ S7 ) ) )
=> ( ! [A4: B] :
( ( member @ B @ A4 @ S7 )
=> ( ( G @ A4 )
= ( zero_zero @ A ) ) )
=> ( ! [B3: C] :
( ( member @ C @ B3 @ T7 )
=> ( ( H2 @ B3 )
= ( zero_zero @ A ) ) )
=> ( ! [A4: B] :
( ( member @ B @ A4 @ S3 )
=> ( ( H2 @ ( J @ A4 ) )
= ( G @ A4 ) ) )
=> ( ( groups7311177749621191930dd_sum @ B @ A @ G @ S3 )
= ( groups7311177749621191930dd_sum @ C @ A @ H2 @ T5 ) ) ) ) ) ) ) ) ) ) ) ) ).
% sum.reindex_bij_witness_not_neutral
thf(fact_2884_powr__mono2_H,axiom,
! [A2: real,X: real,Y: real] :
( ( ord_less_eq @ real @ A2 @ ( zero_zero @ real ) )
=> ( ( ord_less @ real @ ( zero_zero @ real ) @ X )
=> ( ( ord_less_eq @ real @ X @ Y )
=> ( ord_less_eq @ real @ ( powr @ real @ Y @ A2 ) @ ( powr @ real @ X @ A2 ) ) ) ) ) ).
% powr_mono2'
thf(fact_2885_powr__less__mono2,axiom,
! [A2: real,X: real,Y: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ A2 )
=> ( ( ord_less_eq @ real @ ( zero_zero @ real ) @ X )
=> ( ( ord_less @ real @ X @ Y )
=> ( ord_less @ real @ ( powr @ real @ X @ A2 ) @ ( powr @ real @ Y @ A2 ) ) ) ) ) ).
% powr_less_mono2
thf(fact_2886_powr__inj,axiom,
! [A2: real,X: real,Y: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ A2 )
=> ( ( A2
!= ( one_one @ real ) )
=> ( ( ( powr @ real @ A2 @ X )
= ( powr @ real @ A2 @ Y ) )
= ( X = Y ) ) ) ) ).
% powr_inj
thf(fact_2887_gr__one__powr,axiom,
! [X: real,Y: real] :
( ( ord_less @ real @ ( one_one @ real ) @ X )
=> ( ( ord_less @ real @ ( zero_zero @ real ) @ Y )
=> ( ord_less @ real @ ( one_one @ real ) @ ( powr @ real @ X @ Y ) ) ) ) ).
% gr_one_powr
thf(fact_2888_ge__one__powr__ge__zero,axiom,
! [X: real,A2: real] :
( ( ord_less_eq @ real @ ( one_one @ real ) @ X )
=> ( ( ord_less_eq @ real @ ( zero_zero @ real ) @ A2 )
=> ( ord_less_eq @ real @ ( one_one @ real ) @ ( powr @ real @ X @ A2 ) ) ) ) ).
% ge_one_powr_ge_zero
thf(fact_2889_powr__mono__both,axiom,
! [A2: real,B2: real,X: real,Y: real] :
( ( ord_less_eq @ real @ ( zero_zero @ real ) @ A2 )
=> ( ( ord_less_eq @ real @ A2 @ B2 )
=> ( ( ord_less_eq @ real @ ( one_one @ real ) @ X )
=> ( ( ord_less_eq @ real @ X @ Y )
=> ( ord_less_eq @ real @ ( powr @ real @ X @ A2 ) @ ( powr @ real @ Y @ B2 ) ) ) ) ) ) ).
% powr_mono_both
thf(fact_2890_powr__le1,axiom,
! [A2: real,X: real] :
( ( ord_less_eq @ real @ ( zero_zero @ real ) @ A2 )
=> ( ( ord_less_eq @ real @ ( zero_zero @ real ) @ X )
=> ( ( ord_less_eq @ real @ X @ ( one_one @ real ) )
=> ( ord_less_eq @ real @ ( powr @ real @ X @ A2 ) @ ( one_one @ real ) ) ) ) ) ).
% powr_le1
thf(fact_2891_powr__divide,axiom,
! [X: real,Y: real,A2: real] :
( ( ord_less_eq @ real @ ( zero_zero @ real ) @ X )
=> ( ( ord_less_eq @ real @ ( zero_zero @ real ) @ Y )
=> ( ( powr @ real @ ( divide_divide @ real @ X @ Y ) @ A2 )
= ( divide_divide @ real @ ( powr @ real @ X @ A2 ) @ ( powr @ real @ Y @ A2 ) ) ) ) ) ).
% powr_divide
thf(fact_2892_powr__mult,axiom,
! [X: real,Y: real,A2: real] :
( ( ord_less_eq @ real @ ( zero_zero @ real ) @ X )
=> ( ( ord_less_eq @ real @ ( zero_zero @ real ) @ Y )
=> ( ( powr @ real @ ( times_times @ real @ X @ Y ) @ A2 )
= ( times_times @ real @ ( powr @ real @ X @ A2 ) @ ( powr @ real @ Y @ A2 ) ) ) ) ) ).
% powr_mult
thf(fact_2893_log__base__powr,axiom,
! [A2: real,B2: real,X: real] :
( ( A2
!= ( zero_zero @ real ) )
=> ( ( log @ ( powr @ real @ A2 @ B2 ) @ X )
= ( divide_divide @ real @ ( log @ A2 @ X ) @ B2 ) ) ) ).
% log_base_powr
thf(fact_2894_log__powr,axiom,
! [X: real,B2: real,Y: real] :
( ( X
!= ( zero_zero @ real ) )
=> ( ( log @ B2 @ ( powr @ real @ X @ Y ) )
= ( times_times @ real @ Y @ ( log @ B2 @ X ) ) ) ) ).
% log_powr
thf(fact_2895_powr__add,axiom,
! [A: $tType] :
( ( ( real_V3459762299906320749_field @ A )
& ( ln @ A ) )
=> ! [X: A,A2: A,B2: A] :
( ( powr @ A @ X @ ( plus_plus @ A @ A2 @ B2 ) )
= ( times_times @ A @ ( powr @ A @ X @ A2 ) @ ( powr @ A @ X @ B2 ) ) ) ) ).
% powr_add
thf(fact_2896_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_2897_sum__nonneg__leq__bound,axiom,
! [B: $tType,A: $tType] :
( ( ordere6911136660526730532id_add @ A )
=> ! [S: set @ B,F2: B > A,B6: A,I: B] :
( ( finite_finite2 @ B @ S )
=> ( ! [I2: B] :
( ( member @ B @ I2 @ S )
=> ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( F2 @ I2 ) ) )
=> ( ( ( groups7311177749621191930dd_sum @ B @ A @ F2 @ S )
= B6 )
=> ( ( member @ B @ I @ S )
=> ( ord_less_eq @ A @ ( F2 @ I ) @ B6 ) ) ) ) ) ) ).
% sum_nonneg_leq_bound
thf(fact_2898_sum__nonneg__0,axiom,
! [B: $tType,A: $tType] :
( ( ordere6911136660526730532id_add @ A )
=> ! [S: set @ B,F2: B > A,I: B] :
( ( finite_finite2 @ B @ S )
=> ( ! [I2: B] :
( ( member @ B @ I2 @ S )
=> ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( F2 @ I2 ) ) )
=> ( ( ( groups7311177749621191930dd_sum @ B @ A @ F2 @ S )
= ( zero_zero @ A ) )
=> ( ( member @ B @ I @ S )
=> ( ( F2 @ I )
= ( zero_zero @ A ) ) ) ) ) ) ) ).
% sum_nonneg_0
thf(fact_2899_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_2900_sum_Osetdiff__irrelevant,axiom,
! [A: $tType,B: $tType] :
( ( comm_monoid_add @ A )
=> ! [A3: set @ B,G: B > A] :
( ( finite_finite2 @ B @ A3 )
=> ( ( groups7311177749621191930dd_sum @ B @ A @ G
@ ( minus_minus @ ( set @ B ) @ A3
@ ( collect @ B
@ ^ [X2: B] :
( ( G @ X2 )
= ( zero_zero @ A ) ) ) ) )
= ( groups7311177749621191930dd_sum @ B @ A @ G @ A3 ) ) ) ) ).
% sum.setdiff_irrelevant
thf(fact_2901_VEBT__internal_OminNulli_Osimps_I4_J,axiom,
! [Uw2: nat,Ux: array @ vEBT_VEBTi,Uy: vEBT_VEBTi] :
( ( vEBT_VEBT_minNulli @ ( vEBT_Nodei @ ( none @ ( product_prod @ nat @ nat ) ) @ Uw2 @ Ux @ Uy ) )
= ( heap_Time_return @ $o @ $true ) ) ).
% VEBT_internal.minNulli.simps(4)
thf(fact_2902_sum__power__add,axiom,
! [A: $tType] :
( ( ( monoid_mult @ A )
& ( comm_ring @ A ) )
=> ! [X: A,M: nat,I5: set @ nat] :
( ( groups7311177749621191930dd_sum @ nat @ A
@ ^ [I3: nat] : ( power_power @ A @ X @ ( plus_plus @ nat @ M @ I3 ) )
@ I5 )
= ( times_times @ A @ ( power_power @ A @ X @ M ) @ ( groups7311177749621191930dd_sum @ nat @ A @ ( power_power @ A @ X ) @ I5 ) ) ) ) ).
% sum_power_add
thf(fact_2903_sum_OatLeastAtMost__rev,axiom,
! [A: $tType] :
( ( comm_monoid_add @ A )
=> ! [G: nat > A,N3: nat,M: nat] :
( ( groups7311177749621191930dd_sum @ nat @ A @ G @ ( set_or1337092689740270186AtMost @ nat @ N3 @ M ) )
= ( groups7311177749621191930dd_sum @ nat @ A
@ ^ [I3: nat] : ( G @ ( minus_minus @ nat @ ( plus_plus @ nat @ M @ N3 ) @ I3 ) )
@ ( set_or1337092689740270186AtMost @ nat @ N3 @ M ) ) ) ) ).
% sum.atLeastAtMost_rev
thf(fact_2904_sum__pos2,axiom,
! [A: $tType,B: $tType] :
( ( ordere6911136660526730532id_add @ A )
=> ! [I5: set @ B,I: B,F2: B > A] :
( ( finite_finite2 @ B @ I5 )
=> ( ( member @ B @ I @ I5 )
=> ( ( ord_less @ A @ ( zero_zero @ A ) @ ( F2 @ I ) )
=> ( ! [I2: B] :
( ( member @ B @ I2 @ I5 )
=> ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( F2 @ I2 ) ) )
=> ( ord_less @ A @ ( zero_zero @ A ) @ ( groups7311177749621191930dd_sum @ B @ A @ F2 @ I5 ) ) ) ) ) ) ) ).
% sum_pos2
thf(fact_2905_sum__pos,axiom,
! [A: $tType,B: $tType] :
( ( ordere6911136660526730532id_add @ A )
=> ! [I5: set @ B,F2: B > A] :
( ( finite_finite2 @ B @ I5 )
=> ( ( I5
!= ( bot_bot @ ( set @ B ) ) )
=> ( ! [I2: B] :
( ( member @ B @ I2 @ I5 )
=> ( ord_less @ A @ ( zero_zero @ A ) @ ( F2 @ I2 ) ) )
=> ( ord_less @ A @ ( zero_zero @ A ) @ ( groups7311177749621191930dd_sum @ B @ A @ F2 @ I5 ) ) ) ) ) ) ).
% sum_pos
thf(fact_2906_sum_Omono__neutral__cong__right,axiom,
! [A: $tType,B: $tType] :
( ( comm_monoid_add @ A )
=> ! [T5: set @ B,S3: set @ B,G: B > A,H2: B > A] :
( ( finite_finite2 @ B @ T5 )
=> ( ( ord_less_eq @ ( set @ B ) @ S3 @ T5 )
=> ( ! [X3: B] :
( ( member @ B @ X3 @ ( minus_minus @ ( set @ B ) @ T5 @ S3 ) )
=> ( ( G @ X3 )
= ( zero_zero @ A ) ) )
=> ( ! [X3: B] :
( ( member @ B @ X3 @ S3 )
=> ( ( G @ X3 )
= ( H2 @ X3 ) ) )
=> ( ( groups7311177749621191930dd_sum @ B @ A @ G @ T5 )
= ( groups7311177749621191930dd_sum @ B @ A @ H2 @ S3 ) ) ) ) ) ) ) ).
% sum.mono_neutral_cong_right
thf(fact_2907_sum_Omono__neutral__cong__left,axiom,
! [A: $tType,B: $tType] :
( ( comm_monoid_add @ A )
=> ! [T5: set @ B,S3: set @ B,H2: B > A,G: B > A] :
( ( finite_finite2 @ B @ T5 )
=> ( ( ord_less_eq @ ( set @ B ) @ S3 @ T5 )
=> ( ! [X3: B] :
( ( member @ B @ X3 @ ( minus_minus @ ( set @ B ) @ T5 @ S3 ) )
=> ( ( H2 @ X3 )
= ( zero_zero @ A ) ) )
=> ( ! [X3: B] :
( ( member @ B @ X3 @ S3 )
=> ( ( G @ X3 )
= ( H2 @ X3 ) ) )
=> ( ( groups7311177749621191930dd_sum @ B @ A @ G @ S3 )
= ( groups7311177749621191930dd_sum @ B @ A @ H2 @ T5 ) ) ) ) ) ) ) ).
% sum.mono_neutral_cong_left
thf(fact_2908_sum_Omono__neutral__right,axiom,
! [A: $tType,B: $tType] :
( ( comm_monoid_add @ A )
=> ! [T5: set @ B,S3: set @ B,G: B > A] :
( ( finite_finite2 @ B @ T5 )
=> ( ( ord_less_eq @ ( set @ B ) @ S3 @ T5 )
=> ( ! [X3: B] :
( ( member @ B @ X3 @ ( minus_minus @ ( set @ B ) @ T5 @ S3 ) )
=> ( ( G @ X3 )
= ( zero_zero @ A ) ) )
=> ( ( groups7311177749621191930dd_sum @ B @ A @ G @ T5 )
= ( groups7311177749621191930dd_sum @ B @ A @ G @ S3 ) ) ) ) ) ) ).
% sum.mono_neutral_right
thf(fact_2909_sum_Omono__neutral__left,axiom,
! [A: $tType,B: $tType] :
( ( comm_monoid_add @ A )
=> ! [T5: set @ B,S3: set @ B,G: B > A] :
( ( finite_finite2 @ B @ T5 )
=> ( ( ord_less_eq @ ( set @ B ) @ S3 @ T5 )
=> ( ! [X3: B] :
( ( member @ B @ X3 @ ( minus_minus @ ( set @ B ) @ T5 @ S3 ) )
=> ( ( G @ X3 )
= ( zero_zero @ A ) ) )
=> ( ( groups7311177749621191930dd_sum @ B @ A @ G @ S3 )
= ( groups7311177749621191930dd_sum @ B @ A @ G @ T5 ) ) ) ) ) ) ).
% sum.mono_neutral_left
thf(fact_2910_sum_Osame__carrierI,axiom,
! [A: $tType,B: $tType] :
( ( comm_monoid_add @ A )
=> ! [C5: set @ B,A3: set @ B,B6: set @ B,G: B > A,H2: B > A] :
( ( finite_finite2 @ B @ C5 )
=> ( ( ord_less_eq @ ( set @ B ) @ A3 @ C5 )
=> ( ( ord_less_eq @ ( set @ B ) @ B6 @ C5 )
=> ( ! [A4: B] :
( ( member @ B @ A4 @ ( minus_minus @ ( set @ B ) @ C5 @ A3 ) )
=> ( ( G @ A4 )
= ( zero_zero @ A ) ) )
=> ( ! [B3: B] :
( ( member @ B @ B3 @ ( minus_minus @ ( set @ B ) @ C5 @ B6 ) )
=> ( ( H2 @ B3 )
= ( zero_zero @ A ) ) )
=> ( ( ( groups7311177749621191930dd_sum @ B @ A @ G @ C5 )
= ( groups7311177749621191930dd_sum @ B @ A @ H2 @ C5 ) )
=> ( ( groups7311177749621191930dd_sum @ B @ A @ G @ A3 )
= ( groups7311177749621191930dd_sum @ B @ A @ H2 @ B6 ) ) ) ) ) ) ) ) ) ).
% sum.same_carrierI
thf(fact_2911_sum_Osame__carrier,axiom,
! [A: $tType,B: $tType] :
( ( comm_monoid_add @ A )
=> ! [C5: set @ B,A3: set @ B,B6: set @ B,G: B > A,H2: B > A] :
( ( finite_finite2 @ B @ C5 )
=> ( ( ord_less_eq @ ( set @ B ) @ A3 @ C5 )
=> ( ( ord_less_eq @ ( set @ B ) @ B6 @ C5 )
=> ( ! [A4: B] :
( ( member @ B @ A4 @ ( minus_minus @ ( set @ B ) @ C5 @ A3 ) )
=> ( ( G @ A4 )
= ( zero_zero @ A ) ) )
=> ( ! [B3: B] :
( ( member @ B @ B3 @ ( minus_minus @ ( set @ B ) @ C5 @ B6 ) )
=> ( ( H2 @ B3 )
= ( zero_zero @ A ) ) )
=> ( ( ( groups7311177749621191930dd_sum @ B @ A @ G @ A3 )
= ( groups7311177749621191930dd_sum @ B @ A @ H2 @ B6 ) )
= ( ( groups7311177749621191930dd_sum @ B @ A @ G @ C5 )
= ( groups7311177749621191930dd_sum @ B @ A @ H2 @ C5 ) ) ) ) ) ) ) ) ) ).
% sum.same_carrier
thf(fact_2912_sum_Osubset__diff,axiom,
! [A: $tType,B: $tType] :
( ( comm_monoid_add @ A )
=> ! [B6: set @ B,A3: set @ B,G: B > A] :
( ( ord_less_eq @ ( set @ B ) @ B6 @ A3 )
=> ( ( finite_finite2 @ B @ A3 )
=> ( ( groups7311177749621191930dd_sum @ B @ A @ G @ A3 )
= ( plus_plus @ A @ ( groups7311177749621191930dd_sum @ B @ A @ G @ ( minus_minus @ ( set @ B ) @ A3 @ B6 ) ) @ ( groups7311177749621191930dd_sum @ B @ A @ G @ B6 ) ) ) ) ) ) ).
% sum.subset_diff
thf(fact_2913_sum__diff,axiom,
! [A: $tType,B: $tType] :
( ( ab_group_add @ A )
=> ! [A3: set @ B,B6: set @ B,F2: B > A] :
( ( finite_finite2 @ B @ A3 )
=> ( ( ord_less_eq @ ( set @ B ) @ B6 @ A3 )
=> ( ( groups7311177749621191930dd_sum @ B @ A @ F2 @ ( minus_minus @ ( set @ B ) @ A3 @ B6 ) )
= ( minus_minus @ A @ ( groups7311177749621191930dd_sum @ B @ A @ F2 @ A3 ) @ ( groups7311177749621191930dd_sum @ B @ A @ F2 @ B6 ) ) ) ) ) ) ).
% sum_diff
thf(fact_2914_powser__sums__zero,axiom,
! [A: $tType] :
( ( real_V8999393235501362500lgebra @ A )
=> ! [A2: nat > A] :
( sums @ A
@ ^ [N2: nat] : ( times_times @ A @ ( A2 @ N2 ) @ ( power_power @ A @ ( zero_zero @ A ) @ N2 ) )
@ ( A2 @ ( zero_zero @ nat ) ) ) ) ).
% powser_sums_zero
thf(fact_2915_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_2916_sum_OatLeast0__atMost__Suc,axiom,
! [A: $tType] :
( ( comm_monoid_add @ A )
=> ! [G: nat > A,N3: nat] :
( ( groups7311177749621191930dd_sum @ nat @ A @ G @ ( set_or1337092689740270186AtMost @ nat @ ( zero_zero @ nat ) @ ( suc @ N3 ) ) )
= ( plus_plus @ A @ ( groups7311177749621191930dd_sum @ nat @ A @ G @ ( set_or1337092689740270186AtMost @ nat @ ( zero_zero @ nat ) @ N3 ) ) @ ( G @ ( suc @ N3 ) ) ) ) ) ).
% sum.atLeast0_atMost_Suc
thf(fact_2917_sum_OatLeast__Suc__atMost,axiom,
! [A: $tType] :
( ( comm_monoid_add @ A )
=> ! [M: nat,N3: nat,G: nat > A] :
( ( ord_less_eq @ nat @ M @ N3 )
=> ( ( groups7311177749621191930dd_sum @ nat @ A @ G @ ( set_or1337092689740270186AtMost @ nat @ M @ N3 ) )
= ( plus_plus @ A @ ( G @ M ) @ ( groups7311177749621191930dd_sum @ nat @ A @ G @ ( set_or1337092689740270186AtMost @ nat @ ( suc @ M ) @ N3 ) ) ) ) ) ) ).
% sum.atLeast_Suc_atMost
thf(fact_2918_sum_Onat__ivl__Suc_H,axiom,
! [A: $tType] :
( ( comm_monoid_add @ A )
=> ! [M: nat,N3: nat,G: nat > A] :
( ( ord_less_eq @ nat @ M @ ( suc @ N3 ) )
=> ( ( groups7311177749621191930dd_sum @ nat @ A @ G @ ( set_or1337092689740270186AtMost @ nat @ M @ ( suc @ N3 ) ) )
= ( plus_plus @ A @ ( G @ ( suc @ N3 ) ) @ ( groups7311177749621191930dd_sum @ nat @ A @ G @ ( set_or1337092689740270186AtMost @ nat @ M @ N3 ) ) ) ) ) ) ).
% sum.nat_ivl_Suc'
thf(fact_2919_powr__realpow,axiom,
! [X: real,N3: nat] :
( ( ord_less @ real @ ( zero_zero @ real ) @ X )
=> ( ( powr @ real @ X @ ( semiring_1_of_nat @ real @ N3 ) )
= ( power_power @ real @ X @ N3 ) ) ) ).
% powr_realpow
thf(fact_2920_less__log__iff,axiom,
! [B2: real,X: real,Y: real] :
( ( ord_less @ real @ ( one_one @ real ) @ B2 )
=> ( ( ord_less @ real @ ( zero_zero @ real ) @ X )
=> ( ( ord_less @ real @ Y @ ( log @ B2 @ X ) )
= ( ord_less @ real @ ( powr @ real @ B2 @ Y ) @ X ) ) ) ) ).
% less_log_iff
thf(fact_2921_log__less__iff,axiom,
! [B2: real,X: real,Y: real] :
( ( ord_less @ real @ ( one_one @ real ) @ B2 )
=> ( ( ord_less @ real @ ( zero_zero @ real ) @ X )
=> ( ( ord_less @ real @ ( log @ B2 @ X ) @ Y )
= ( ord_less @ real @ X @ ( powr @ real @ B2 @ Y ) ) ) ) ) ).
% log_less_iff
thf(fact_2922_less__powr__iff,axiom,
! [B2: real,X: real,Y: real] :
( ( ord_less @ real @ ( one_one @ real ) @ B2 )
=> ( ( ord_less @ real @ ( zero_zero @ real ) @ X )
=> ( ( ord_less @ real @ X @ ( powr @ real @ B2 @ Y ) )
= ( ord_less @ real @ ( log @ B2 @ X ) @ Y ) ) ) ) ).
% less_powr_iff
thf(fact_2923_powr__less__iff,axiom,
! [B2: real,X: real,Y: real] :
( ( ord_less @ real @ ( one_one @ real ) @ B2 )
=> ( ( ord_less @ real @ ( zero_zero @ real ) @ X )
=> ( ( ord_less @ real @ ( powr @ real @ B2 @ Y ) @ X )
= ( ord_less @ real @ Y @ ( log @ B2 @ X ) ) ) ) ) ).
% powr_less_iff
thf(fact_2924_dbl__inc__def,axiom,
! [A: $tType] :
( ( neg_numeral @ A )
=> ( ( neg_numeral_dbl_inc @ A )
= ( ^ [X2: A] : ( plus_plus @ A @ ( plus_plus @ A @ X2 @ X2 ) @ ( one_one @ A ) ) ) ) ) ).
% dbl_inc_def
thf(fact_2925_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_2926_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_2927_sum_OSuc__reindex__ivl,axiom,
! [A: $tType] :
( ( comm_monoid_add @ A )
=> ! [M: nat,N3: nat,G: nat > A] :
( ( ord_less_eq @ nat @ M @ N3 )
=> ( ( plus_plus @ A @ ( groups7311177749621191930dd_sum @ nat @ A @ G @ ( set_or1337092689740270186AtMost @ nat @ M @ N3 ) ) @ ( G @ ( suc @ N3 ) ) )
= ( plus_plus @ A @ ( G @ M )
@ ( groups7311177749621191930dd_sum @ nat @ A
@ ^ [I3: nat] : ( G @ ( suc @ I3 ) )
@ ( set_or1337092689740270186AtMost @ nat @ M @ N3 ) ) ) ) ) ) ).
% sum.Suc_reindex_ivl
thf(fact_2928_sum__Suc__diff,axiom,
! [A: $tType] :
( ( ab_group_add @ A )
=> ! [M: nat,N3: nat,F2: nat > A] :
( ( ord_less_eq @ nat @ M @ ( suc @ N3 ) )
=> ( ( groups7311177749621191930dd_sum @ nat @ A
@ ^ [I3: nat] : ( minus_minus @ A @ ( F2 @ ( suc @ I3 ) ) @ ( F2 @ I3 ) )
@ ( set_or1337092689740270186AtMost @ nat @ M @ N3 ) )
= ( minus_minus @ A @ ( F2 @ ( suc @ N3 ) ) @ ( F2 @ M ) ) ) ) ) ).
% sum_Suc_diff
thf(fact_2929_vebt__minti_Osimps_I2_J,axiom,
! [Uu2: nat,Uv2: array @ vEBT_VEBTi,Uw2: vEBT_VEBTi] :
( ( vEBT_vebt_minti @ ( vEBT_Nodei @ ( none @ ( product_prod @ nat @ nat ) ) @ Uu2 @ Uv2 @ Uw2 ) )
= ( heap_Time_return @ ( option @ nat ) @ ( none @ nat ) ) ) ).
% vebt_minti.simps(2)
thf(fact_2930_vebt__maxti_Osimps_I2_J,axiom,
! [Uu2: nat,Uv2: array @ vEBT_VEBTi,Uw2: vEBT_VEBTi] :
( ( vEBT_vebt_maxti @ ( vEBT_Nodei @ ( none @ ( product_prod @ nat @ nat ) ) @ Uu2 @ Uv2 @ Uw2 ) )
= ( heap_Time_return @ ( option @ nat ) @ ( none @ nat ) ) ) ).
% vebt_maxti.simps(2)
thf(fact_2931_sum__mono2,axiom,
! [A: $tType,B: $tType] :
( ( ordere6911136660526730532id_add @ A )
=> ! [B6: set @ B,A3: set @ B,F2: B > A] :
( ( finite_finite2 @ B @ B6 )
=> ( ( ord_less_eq @ ( set @ B ) @ A3 @ B6 )
=> ( ! [B3: B] :
( ( member @ B @ B3 @ ( minus_minus @ ( set @ B ) @ B6 @ A3 ) )
=> ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( F2 @ B3 ) ) )
=> ( ord_less_eq @ A @ ( groups7311177749621191930dd_sum @ B @ A @ F2 @ A3 ) @ ( groups7311177749621191930dd_sum @ B @ A @ F2 @ B6 ) ) ) ) ) ) ).
% sum_mono2
thf(fact_2932_sum_Oremove,axiom,
! [A: $tType,B: $tType] :
( ( comm_monoid_add @ A )
=> ! [A3: set @ B,X: B,G: B > A] :
( ( finite_finite2 @ B @ A3 )
=> ( ( member @ B @ X @ A3 )
=> ( ( groups7311177749621191930dd_sum @ B @ A @ G @ A3 )
= ( plus_plus @ A @ ( G @ X ) @ ( groups7311177749621191930dd_sum @ B @ A @ G @ ( minus_minus @ ( set @ B ) @ A3 @ ( insert @ B @ X @ ( bot_bot @ ( set @ B ) ) ) ) ) ) ) ) ) ) ).
% sum.remove
thf(fact_2933_sum_Oinsert__remove,axiom,
! [A: $tType,B: $tType] :
( ( comm_monoid_add @ A )
=> ! [A3: set @ B,G: B > A,X: B] :
( ( finite_finite2 @ B @ A3 )
=> ( ( groups7311177749621191930dd_sum @ B @ A @ G @ ( insert @ B @ X @ A3 ) )
= ( plus_plus @ A @ ( G @ X ) @ ( groups7311177749621191930dd_sum @ B @ A @ G @ ( minus_minus @ ( set @ B ) @ A3 @ ( insert @ B @ X @ ( bot_bot @ ( set @ B ) ) ) ) ) ) ) ) ) ).
% sum.insert_remove
thf(fact_2934_sum__diff1,axiom,
! [A: $tType,B: $tType] :
( ( ab_group_add @ A )
=> ! [A3: set @ B,A2: B,F2: B > A] :
( ( finite_finite2 @ B @ A3 )
=> ( ( ( member @ B @ A2 @ A3 )
=> ( ( groups7311177749621191930dd_sum @ B @ A @ F2 @ ( minus_minus @ ( set @ B ) @ A3 @ ( insert @ B @ A2 @ ( bot_bot @ ( set @ B ) ) ) ) )
= ( minus_minus @ A @ ( groups7311177749621191930dd_sum @ B @ A @ F2 @ A3 ) @ ( F2 @ A2 ) ) ) )
& ( ~ ( member @ B @ A2 @ A3 )
=> ( ( groups7311177749621191930dd_sum @ B @ A @ F2 @ ( minus_minus @ ( set @ B ) @ A3 @ ( insert @ B @ A2 @ ( bot_bot @ ( set @ B ) ) ) ) )
= ( groups7311177749621191930dd_sum @ B @ A @ F2 @ A3 ) ) ) ) ) ) ).
% sum_diff1
thf(fact_2935_powr__mult__base,axiom,
! [X: real,Y: real] :
( ( ord_less_eq @ real @ ( zero_zero @ real ) @ X )
=> ( ( times_times @ real @ X @ ( powr @ real @ X @ Y ) )
= ( powr @ real @ X @ ( plus_plus @ real @ ( one_one @ real ) @ Y ) ) ) ) ).
% powr_mult_base
thf(fact_2936_powr__le__iff,axiom,
! [B2: real,X: real,Y: real] :
( ( ord_less @ real @ ( one_one @ real ) @ B2 )
=> ( ( ord_less @ real @ ( zero_zero @ real ) @ X )
=> ( ( ord_less_eq @ real @ ( powr @ real @ B2 @ Y ) @ X )
= ( ord_less_eq @ real @ Y @ ( log @ B2 @ X ) ) ) ) ) ).
% powr_le_iff
thf(fact_2937_le__powr__iff,axiom,
! [B2: real,X: real,Y: real] :
( ( ord_less @ real @ ( one_one @ real ) @ B2 )
=> ( ( ord_less @ real @ ( zero_zero @ real ) @ X )
=> ( ( ord_less_eq @ real @ X @ ( powr @ real @ B2 @ Y ) )
= ( ord_less_eq @ real @ ( log @ B2 @ X ) @ Y ) ) ) ) ).
% le_powr_iff
thf(fact_2938_log__le__iff,axiom,
! [B2: real,X: real,Y: real] :
( ( ord_less @ real @ ( one_one @ real ) @ B2 )
=> ( ( ord_less @ real @ ( zero_zero @ real ) @ X )
=> ( ( ord_less_eq @ real @ ( log @ B2 @ X ) @ Y )
= ( ord_less_eq @ real @ X @ ( powr @ real @ B2 @ Y ) ) ) ) ) ).
% log_le_iff
thf(fact_2939_le__log__iff,axiom,
! [B2: real,X: real,Y: real] :
( ( ord_less @ real @ ( one_one @ real ) @ B2 )
=> ( ( ord_less @ real @ ( zero_zero @ real ) @ X )
=> ( ( ord_less_eq @ real @ Y @ ( log @ B2 @ X ) )
= ( ord_less_eq @ real @ ( powr @ real @ B2 @ Y ) @ X ) ) ) ) ).
% le_log_iff
thf(fact_2940_sum_Oub__add__nat,axiom,
! [A: $tType] :
( ( comm_monoid_add @ A )
=> ! [M: nat,N3: nat,G: nat > A,P4: nat] :
( ( ord_less_eq @ nat @ M @ ( plus_plus @ nat @ N3 @ ( one_one @ nat ) ) )
=> ( ( groups7311177749621191930dd_sum @ nat @ A @ G @ ( set_or1337092689740270186AtMost @ nat @ M @ ( plus_plus @ nat @ N3 @ P4 ) ) )
= ( plus_plus @ A @ ( groups7311177749621191930dd_sum @ nat @ A @ G @ ( set_or1337092689740270186AtMost @ nat @ M @ N3 ) ) @ ( groups7311177749621191930dd_sum @ nat @ A @ G @ ( set_or1337092689740270186AtMost @ nat @ ( plus_plus @ nat @ N3 @ ( one_one @ nat ) ) @ ( plus_plus @ nat @ N3 @ P4 ) ) ) ) ) ) ) ).
% sum.ub_add_nat
thf(fact_2941_sum_Odelta__remove,axiom,
! [A: $tType,B: $tType] :
( ( comm_monoid_add @ A )
=> ! [S3: set @ B,A2: B,B2: B > A,C2: B > A] :
( ( finite_finite2 @ B @ S3 )
=> ( ( ( member @ B @ A2 @ S3 )
=> ( ( groups7311177749621191930dd_sum @ B @ A
@ ^ [K2: B] : ( if @ A @ ( K2 = A2 ) @ ( B2 @ K2 ) @ ( C2 @ K2 ) )
@ S3 )
= ( plus_plus @ A @ ( B2 @ A2 ) @ ( groups7311177749621191930dd_sum @ B @ A @ C2 @ ( minus_minus @ ( set @ B ) @ S3 @ ( insert @ B @ A2 @ ( bot_bot @ ( set @ B ) ) ) ) ) ) ) )
& ( ~ ( member @ B @ A2 @ S3 )
=> ( ( groups7311177749621191930dd_sum @ B @ A
@ ^ [K2: B] : ( if @ A @ ( K2 = A2 ) @ ( B2 @ K2 ) @ ( C2 @ K2 ) )
@ S3 )
= ( groups7311177749621191930dd_sum @ B @ A @ C2 @ ( minus_minus @ ( set @ B ) @ S3 @ ( insert @ B @ A2 @ ( bot_bot @ ( set @ B ) ) ) ) ) ) ) ) ) ) ).
% sum.delta_remove
thf(fact_2942_set__encode__def,axiom,
( nat_set_encode
= ( groups7311177749621191930dd_sum @ nat @ nat @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ).
% set_encode_def
thf(fact_2943_VEBT__internal_OminNulli_Oelims,axiom,
! [X: vEBT_VEBTi,Y: heap_Time_Heap @ $o] :
( ( ( vEBT_VEBT_minNulli @ X )
= Y )
=> ( ( ( X
= ( vEBT_Leafi @ $false @ $false ) )
=> ( Y
!= ( heap_Time_return @ $o @ $true ) ) )
=> ( ( ? [Uv: $o] :
( X
= ( vEBT_Leafi @ $true @ Uv ) )
=> ( Y
!= ( heap_Time_return @ $o @ $false ) ) )
=> ( ( ? [Uu: $o] :
( X
= ( vEBT_Leafi @ Uu @ $true ) )
=> ( Y
!= ( heap_Time_return @ $o @ $false ) ) )
=> ( ( ? [Uw: nat,Ux2: array @ vEBT_VEBTi,Uy2: vEBT_VEBTi] :
( X
= ( vEBT_Nodei @ ( none @ ( product_prod @ nat @ nat ) ) @ Uw @ Ux2 @ Uy2 ) )
=> ( Y
!= ( heap_Time_return @ $o @ $true ) ) )
=> ~ ( ? [Uz2: product_prod @ nat @ nat,Va2: nat,Vb2: array @ vEBT_VEBTi,Vc2: vEBT_VEBTi] :
( X
= ( vEBT_Nodei @ ( some @ ( product_prod @ nat @ nat ) @ Uz2 ) @ Va2 @ Vb2 @ Vc2 ) )
=> ( Y
!= ( heap_Time_return @ $o @ $false ) ) ) ) ) ) ) ) ).
% VEBT_internal.minNulli.elims
thf(fact_2944_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_2945_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_2946_sum__strict__mono2,axiom,
! [B: $tType,A: $tType] :
( ( ordere8940638589300402666id_add @ B )
=> ! [B6: set @ A,A3: set @ A,B2: A,F2: A > B] :
( ( finite_finite2 @ A @ B6 )
=> ( ( ord_less_eq @ ( set @ A ) @ A3 @ B6 )
=> ( ( member @ A @ B2 @ ( minus_minus @ ( set @ A ) @ B6 @ A3 ) )
=> ( ( ord_less @ B @ ( zero_zero @ B ) @ ( F2 @ B2 ) )
=> ( ! [X3: A] :
( ( member @ A @ X3 @ B6 )
=> ( ord_less_eq @ B @ ( zero_zero @ B ) @ ( F2 @ X3 ) ) )
=> ( ord_less @ B @ ( groups7311177749621191930dd_sum @ A @ B @ F2 @ A3 ) @ ( groups7311177749621191930dd_sum @ A @ B @ F2 @ B6 ) ) ) ) ) ) ) ) ).
% sum_strict_mono2
thf(fact_2947_member__le__sum,axiom,
! [B: $tType,C: $tType] :
( ( ( ordere6911136660526730532id_add @ B )
& ( semiring_1 @ B ) )
=> ! [I: C,A3: set @ C,F2: C > B] :
( ( member @ C @ I @ A3 )
=> ( ! [X3: C] :
( ( member @ C @ X3 @ ( minus_minus @ ( set @ C ) @ A3 @ ( insert @ C @ I @ ( bot_bot @ ( set @ C ) ) ) ) )
=> ( ord_less_eq @ B @ ( zero_zero @ B ) @ ( F2 @ X3 ) ) )
=> ( ( finite_finite2 @ C @ A3 )
=> ( ord_less_eq @ B @ ( F2 @ I ) @ ( groups7311177749621191930dd_sum @ C @ B @ F2 @ A3 ) ) ) ) ) ) ).
% member_le_sum
thf(fact_2948_add__log__eq__powr,axiom,
! [B2: real,X: real,Y: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ B2 )
=> ( ( B2
!= ( one_one @ real ) )
=> ( ( ord_less @ real @ ( zero_zero @ real ) @ X )
=> ( ( plus_plus @ real @ Y @ ( log @ B2 @ X ) )
= ( log @ B2 @ ( times_times @ real @ ( powr @ real @ B2 @ Y ) @ X ) ) ) ) ) ) ).
% add_log_eq_powr
thf(fact_2949_log__add__eq__powr,axiom,
! [B2: real,X: real,Y: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ B2 )
=> ( ( B2
!= ( one_one @ real ) )
=> ( ( ord_less @ real @ ( zero_zero @ real ) @ X )
=> ( ( plus_plus @ real @ ( log @ B2 @ X ) @ Y )
= ( log @ B2 @ ( times_times @ real @ X @ ( powr @ real @ B2 @ Y ) ) ) ) ) ) ) ).
% log_add_eq_powr
thf(fact_2950_minus__log__eq__powr,axiom,
! [B2: real,X: real,Y: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ B2 )
=> ( ( B2
!= ( one_one @ real ) )
=> ( ( ord_less @ real @ ( zero_zero @ real ) @ X )
=> ( ( minus_minus @ real @ Y @ ( log @ B2 @ X ) )
= ( log @ B2 @ ( divide_divide @ real @ ( powr @ real @ B2 @ Y ) @ X ) ) ) ) ) ) ).
% minus_log_eq_powr
thf(fact_2951_sum__natinterval__diff,axiom,
! [A: $tType] :
( ( ab_group_add @ A )
=> ! [M: nat,N3: nat,F2: nat > A] :
( ( ( ord_less_eq @ nat @ M @ N3 )
=> ( ( groups7311177749621191930dd_sum @ nat @ A
@ ^ [K2: nat] : ( minus_minus @ A @ ( F2 @ K2 ) @ ( F2 @ ( plus_plus @ nat @ K2 @ ( one_one @ nat ) ) ) )
@ ( set_or1337092689740270186AtMost @ nat @ M @ N3 ) )
= ( minus_minus @ A @ ( F2 @ M ) @ ( F2 @ ( plus_plus @ nat @ N3 @ ( one_one @ nat ) ) ) ) ) )
& ( ~ ( ord_less_eq @ nat @ M @ N3 )
=> ( ( groups7311177749621191930dd_sum @ nat @ A
@ ^ [K2: nat] : ( minus_minus @ A @ ( F2 @ K2 ) @ ( F2 @ ( plus_plus @ nat @ K2 @ ( one_one @ nat ) ) ) )
@ ( set_or1337092689740270186AtMost @ nat @ M @ N3 ) )
= ( zero_zero @ A ) ) ) ) ) ).
% sum_natinterval_diff
thf(fact_2952_sum__telescope_H_H,axiom,
! [A: $tType] :
( ( ab_group_add @ A )
=> ! [M: nat,N3: nat,F2: nat > A] :
( ( ord_less_eq @ nat @ M @ N3 )
=> ( ( groups7311177749621191930dd_sum @ nat @ A
@ ^ [K2: nat] : ( minus_minus @ A @ ( F2 @ K2 ) @ ( F2 @ ( minus_minus @ nat @ K2 @ ( one_one @ nat ) ) ) )
@ ( set_or1337092689740270186AtMost @ nat @ ( suc @ M ) @ N3 ) )
= ( minus_minus @ A @ ( F2 @ N3 ) @ ( F2 @ M ) ) ) ) ) ).
% sum_telescope''
thf(fact_2953_vebt__minti_Osimps_I1_J,axiom,
! [A2: $o,B2: $o] :
( ( A2
=> ( ( vEBT_vebt_minti @ ( vEBT_Leafi @ A2 @ B2 ) )
= ( heap_Time_return @ ( option @ nat ) @ ( some @ nat @ ( zero_zero @ nat ) ) ) ) )
& ( ~ A2
=> ( ( B2
=> ( ( vEBT_vebt_minti @ ( vEBT_Leafi @ A2 @ B2 ) )
= ( heap_Time_return @ ( option @ nat ) @ ( some @ nat @ ( one_one @ nat ) ) ) ) )
& ( ~ B2
=> ( ( vEBT_vebt_minti @ ( vEBT_Leafi @ A2 @ B2 ) )
= ( heap_Time_return @ ( option @ nat ) @ ( none @ nat ) ) ) ) ) ) ) ).
% vebt_minti.simps(1)
thf(fact_2954_vebt__maxti_Osimps_I1_J,axiom,
! [B2: $o,A2: $o] :
( ( B2
=> ( ( vEBT_vebt_maxti @ ( vEBT_Leafi @ A2 @ B2 ) )
= ( heap_Time_return @ ( option @ nat ) @ ( some @ nat @ ( one_one @ nat ) ) ) ) )
& ( ~ B2
=> ( ( A2
=> ( ( vEBT_vebt_maxti @ ( vEBT_Leafi @ A2 @ B2 ) )
= ( heap_Time_return @ ( option @ nat ) @ ( some @ nat @ ( zero_zero @ nat ) ) ) ) )
& ( ~ A2
=> ( ( vEBT_vebt_maxti @ ( vEBT_Leafi @ A2 @ B2 ) )
= ( heap_Time_return @ ( option @ nat ) @ ( none @ nat ) ) ) ) ) ) ) ).
% vebt_maxti.simps(1)
thf(fact_2955_mask__eq__sum__exp,axiom,
! [A: $tType] :
( ( semiring_parity @ A )
=> ! [N3: nat] :
( ( minus_minus @ A @ ( power_power @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ N3 ) @ ( one_one @ A ) )
= ( groups7311177749621191930dd_sum @ nat @ A @ ( power_power @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) )
@ ( collect @ nat
@ ^ [Q4: nat] : ( ord_less @ nat @ Q4 @ N3 ) ) ) ) ) ).
% mask_eq_sum_exp
thf(fact_2956_sum__gp__multiplied,axiom,
! [A: $tType] :
( ( ( monoid_mult @ A )
& ( comm_ring @ A ) )
=> ! [M: nat,N3: nat,X: A] :
( ( ord_less_eq @ nat @ M @ N3 )
=> ( ( times_times @ A @ ( minus_minus @ A @ ( one_one @ A ) @ X ) @ ( groups7311177749621191930dd_sum @ nat @ A @ ( power_power @ A @ X ) @ ( set_or1337092689740270186AtMost @ nat @ M @ N3 ) ) )
= ( minus_minus @ A @ ( power_power @ A @ X @ M ) @ ( power_power @ A @ X @ ( suc @ N3 ) ) ) ) ) ) ).
% sum_gp_multiplied
thf(fact_2957_sum_Oin__pairs,axiom,
! [A: $tType] :
( ( comm_monoid_add @ A )
=> ! [G: nat > A,M: nat,N3: 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 ) ) @ N3 ) ) ) )
= ( groups7311177749621191930dd_sum @ nat @ A
@ ^ [I3: nat] : ( plus_plus @ A @ ( G @ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ I3 ) ) @ ( G @ ( suc @ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ I3 ) ) ) )
@ ( set_or1337092689740270186AtMost @ nat @ M @ N3 ) ) ) ) ).
% sum.in_pairs
thf(fact_2958_VEBT__internal_Oheight_Ocases,axiom,
! [X: vEBT_VEBT] :
( ! [A4: $o,B3: $o] :
( X
!= ( vEBT_Leaf @ A4 @ B3 ) )
=> ~ ! [Uu: option @ ( product_prod @ nat @ nat ),Deg2: nat,TreeList2: list @ vEBT_VEBT,Summary2: vEBT_VEBT] :
( X
!= ( vEBT_Node @ Uu @ Deg2 @ TreeList2 @ Summary2 ) ) ) ).
% VEBT_internal.height.cases
thf(fact_2959_sums__if_H,axiom,
! [G: nat > real,X: real] :
( ( sums @ real @ G @ X )
=> ( sums @ real
@ ^ [N2: nat] : ( if @ real @ ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N2 ) @ ( zero_zero @ real ) @ ( G @ ( divide_divide @ nat @ ( minus_minus @ nat @ N2 @ ( one_one @ nat ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) )
@ X ) ) ).
% sums_if'
thf(fact_2960_sums__if,axiom,
! [G: nat > real,X: real,F2: nat > real,Y: real] :
( ( sums @ real @ G @ X )
=> ( ( sums @ real @ F2 @ Y )
=> ( sums @ real
@ ^ [N2: nat] : ( if @ real @ ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N2 ) @ ( F2 @ ( divide_divide @ nat @ N2 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( G @ ( divide_divide @ nat @ ( minus_minus @ nat @ N2 @ ( one_one @ nat ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) )
@ ( plus_plus @ real @ X @ Y ) ) ) ) ).
% sums_if
thf(fact_2961_mask__eq__sum__exp__nat,axiom,
! [N3: nat] :
( ( minus_minus @ nat @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N3 ) @ ( suc @ ( zero_zero @ nat ) ) )
= ( groups7311177749621191930dd_sum @ nat @ nat @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) )
@ ( collect @ nat
@ ^ [Q4: nat] : ( ord_less @ nat @ Q4 @ N3 ) ) ) ) ).
% mask_eq_sum_exp_nat
thf(fact_2962_gauss__sum__nat,axiom,
! [N3: nat] :
( ( groups7311177749621191930dd_sum @ nat @ nat
@ ^ [X2: nat] : X2
@ ( set_or1337092689740270186AtMost @ nat @ ( zero_zero @ nat ) @ N3 ) )
= ( divide_divide @ nat @ ( times_times @ nat @ N3 @ ( suc @ N3 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ).
% gauss_sum_nat
thf(fact_2963_double__gauss__sum,axiom,
! [A: $tType] :
( ( comm_semiring_1 @ A )
=> ! [N3: nat] :
( ( times_times @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ ( groups7311177749621191930dd_sum @ nat @ A @ ( semiring_1_of_nat @ A ) @ ( set_or1337092689740270186AtMost @ nat @ ( zero_zero @ nat ) @ N3 ) ) )
= ( times_times @ A @ ( semiring_1_of_nat @ A @ N3 ) @ ( plus_plus @ A @ ( semiring_1_of_nat @ A @ N3 ) @ ( one_one @ A ) ) ) ) ) ).
% double_gauss_sum
thf(fact_2964_double__arith__series,axiom,
! [A: $tType] :
( ( comm_semiring_1 @ A )
=> ! [A2: A,D: A,N3: nat] :
( ( times_times @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) )
@ ( groups7311177749621191930dd_sum @ nat @ A
@ ^ [I3: nat] : ( plus_plus @ A @ A2 @ ( times_times @ A @ ( semiring_1_of_nat @ A @ I3 ) @ D ) )
@ ( set_or1337092689740270186AtMost @ nat @ ( zero_zero @ nat ) @ N3 ) ) )
= ( times_times @ A @ ( plus_plus @ A @ ( semiring_1_of_nat @ A @ N3 ) @ ( one_one @ A ) ) @ ( plus_plus @ A @ ( times_times @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ A2 ) @ ( times_times @ A @ ( semiring_1_of_nat @ A @ N3 ) @ D ) ) ) ) ) ).
% double_arith_series
thf(fact_2965_arith__series__nat,axiom,
! [A2: nat,D: nat,N3: nat] :
( ( groups7311177749621191930dd_sum @ nat @ nat
@ ^ [I3: nat] : ( plus_plus @ nat @ A2 @ ( times_times @ nat @ I3 @ D ) )
@ ( set_or1337092689740270186AtMost @ nat @ ( zero_zero @ nat ) @ N3 ) )
= ( divide_divide @ nat @ ( times_times @ nat @ ( suc @ N3 ) @ ( plus_plus @ nat @ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ A2 ) @ ( times_times @ nat @ N3 @ D ) ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ).
% arith_series_nat
thf(fact_2966_Sum__Icc__nat,axiom,
! [M: nat,N3: nat] :
( ( groups7311177749621191930dd_sum @ nat @ nat
@ ^ [X2: nat] : X2
@ ( set_or1337092689740270186AtMost @ nat @ M @ N3 ) )
= ( divide_divide @ nat @ ( minus_minus @ nat @ ( times_times @ nat @ N3 @ ( plus_plus @ nat @ N3 @ ( one_one @ nat ) ) ) @ ( times_times @ nat @ M @ ( minus_minus @ nat @ M @ ( one_one @ nat ) ) ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ).
% Sum_Icc_nat
thf(fact_2967_double__gauss__sum__from__Suc__0,axiom,
! [A: $tType] :
( ( comm_semiring_1 @ A )
=> ! [N3: 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 ) ) @ N3 ) ) )
= ( times_times @ A @ ( semiring_1_of_nat @ A @ N3 ) @ ( plus_plus @ A @ ( semiring_1_of_nat @ A @ N3 ) @ ( one_one @ A ) ) ) ) ) ).
% double_gauss_sum_from_Suc_0
thf(fact_2968_arith__series,axiom,
! [A: $tType] :
( ( euclid5411537665997757685th_nat @ A )
=> ! [A2: A,D: A,N3: nat] :
( ( groups7311177749621191930dd_sum @ nat @ A
@ ^ [I3: nat] : ( plus_plus @ A @ A2 @ ( times_times @ A @ ( semiring_1_of_nat @ A @ I3 ) @ D ) )
@ ( set_or1337092689740270186AtMost @ nat @ ( zero_zero @ nat ) @ N3 ) )
= ( divide_divide @ A @ ( times_times @ A @ ( plus_plus @ A @ ( semiring_1_of_nat @ A @ N3 ) @ ( one_one @ A ) ) @ ( plus_plus @ A @ ( times_times @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ A2 ) @ ( times_times @ A @ ( semiring_1_of_nat @ A @ N3 ) @ D ) ) ) @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) ) ) ).
% arith_series
thf(fact_2969_gauss__sum,axiom,
! [A: $tType] :
( ( euclid5411537665997757685th_nat @ A )
=> ! [N3: nat] :
( ( groups7311177749621191930dd_sum @ nat @ A @ ( semiring_1_of_nat @ A ) @ ( set_or1337092689740270186AtMost @ nat @ ( zero_zero @ nat ) @ N3 ) )
= ( divide_divide @ A @ ( times_times @ A @ ( semiring_1_of_nat @ A @ N3 ) @ ( plus_plus @ A @ ( semiring_1_of_nat @ A @ N3 ) @ ( one_one @ A ) ) ) @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) ) ) ).
% gauss_sum
thf(fact_2970_sum__gp__offset,axiom,
! [A: $tType] :
( ( ( division_ring @ A )
& ( comm_ring @ A ) )
=> ! [X: A,M: nat,N3: nat] :
( ( ( X
= ( one_one @ A ) )
=> ( ( groups7311177749621191930dd_sum @ nat @ A @ ( power_power @ A @ X ) @ ( set_or1337092689740270186AtMost @ nat @ M @ ( plus_plus @ nat @ M @ N3 ) ) )
= ( plus_plus @ A @ ( semiring_1_of_nat @ A @ N3 ) @ ( one_one @ A ) ) ) )
& ( ( X
!= ( one_one @ A ) )
=> ( ( groups7311177749621191930dd_sum @ nat @ A @ ( power_power @ A @ X ) @ ( set_or1337092689740270186AtMost @ nat @ M @ ( plus_plus @ nat @ M @ N3 ) ) )
= ( divide_divide @ A @ ( times_times @ A @ ( power_power @ A @ X @ M ) @ ( minus_minus @ A @ ( one_one @ A ) @ ( power_power @ A @ X @ ( suc @ N3 ) ) ) ) @ ( minus_minus @ A @ ( one_one @ A ) @ X ) ) ) ) ) ) ).
% sum_gp_offset
thf(fact_2971_vebt__minti_Oelims,axiom,
! [X: vEBT_VEBTi,Y: heap_Time_Heap @ ( option @ nat )] :
( ( ( vEBT_vebt_minti @ X )
= Y )
=> ( ! [A4: $o,B3: $o] :
( ( X
= ( vEBT_Leafi @ A4 @ B3 ) )
=> ~ ( ( A4
=> ( Y
= ( heap_Time_return @ ( option @ nat ) @ ( some @ nat @ ( zero_zero @ nat ) ) ) ) )
& ( ~ A4
=> ( ( B3
=> ( Y
= ( heap_Time_return @ ( option @ nat ) @ ( some @ nat @ ( one_one @ nat ) ) ) ) )
& ( ~ B3
=> ( Y
= ( heap_Time_return @ ( option @ nat ) @ ( none @ nat ) ) ) ) ) ) ) )
=> ( ( ? [Uu: nat,Uv: array @ vEBT_VEBTi,Uw: vEBT_VEBTi] :
( X
= ( vEBT_Nodei @ ( none @ ( product_prod @ nat @ nat ) ) @ Uu @ Uv @ Uw ) )
=> ( Y
!= ( heap_Time_return @ ( option @ nat ) @ ( none @ nat ) ) ) )
=> ~ ! [Mi2: nat] :
( ? [Ma2: nat,Ux2: nat,Uy2: array @ vEBT_VEBTi,Uz2: vEBT_VEBTi] :
( X
= ( vEBT_Nodei @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi2 @ Ma2 ) ) @ Ux2 @ Uy2 @ Uz2 ) )
=> ( Y
!= ( heap_Time_return @ ( option @ nat ) @ ( some @ nat @ Mi2 ) ) ) ) ) ) ) ).
% vebt_minti.elims
thf(fact_2972_vebt__maxti_Oelims,axiom,
! [X: vEBT_VEBTi,Y: heap_Time_Heap @ ( option @ nat )] :
( ( ( vEBT_vebt_maxti @ X )
= Y )
=> ( ! [A4: $o,B3: $o] :
( ( X
= ( vEBT_Leafi @ A4 @ B3 ) )
=> ~ ( ( B3
=> ( Y
= ( heap_Time_return @ ( option @ nat ) @ ( some @ nat @ ( one_one @ nat ) ) ) ) )
& ( ~ B3
=> ( ( A4
=> ( Y
= ( heap_Time_return @ ( option @ nat ) @ ( some @ nat @ ( zero_zero @ nat ) ) ) ) )
& ( ~ A4
=> ( Y
= ( heap_Time_return @ ( option @ nat ) @ ( none @ nat ) ) ) ) ) ) ) )
=> ( ( ? [Uu: nat,Uv: array @ vEBT_VEBTi,Uw: vEBT_VEBTi] :
( X
= ( vEBT_Nodei @ ( none @ ( product_prod @ nat @ nat ) ) @ Uu @ Uv @ Uw ) )
=> ( Y
!= ( heap_Time_return @ ( option @ nat ) @ ( none @ nat ) ) ) )
=> ~ ! [Mi2: nat,Ma2: nat] :
( ? [Ux2: nat,Uy2: array @ vEBT_VEBTi,Uz2: vEBT_VEBTi] :
( X
= ( vEBT_Nodei @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi2 @ Ma2 ) ) @ Ux2 @ Uy2 @ Uz2 ) )
=> ( Y
!= ( heap_Time_return @ ( option @ nat ) @ ( some @ nat @ Ma2 ) ) ) ) ) ) ) ).
% vebt_maxti.elims
thf(fact_2973_gauss__sum__from__Suc__0,axiom,
! [A: $tType] :
( ( euclid5411537665997757685th_nat @ A )
=> ! [N3: nat] :
( ( groups7311177749621191930dd_sum @ nat @ A @ ( semiring_1_of_nat @ A ) @ ( set_or1337092689740270186AtMost @ nat @ ( suc @ ( zero_zero @ nat ) ) @ N3 ) )
= ( divide_divide @ A @ ( times_times @ A @ ( semiring_1_of_nat @ A @ N3 ) @ ( plus_plus @ A @ ( semiring_1_of_nat @ A @ N3 ) @ ( one_one @ A ) ) ) @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) ) ) ).
% gauss_sum_from_Suc_0
thf(fact_2974_geometric__sums,axiom,
! [A: $tType] :
( ( real_V3459762299906320749_field @ A )
=> ! [C2: A] :
( ( ord_less @ real @ ( real_V7770717601297561774m_norm @ A @ C2 ) @ ( one_one @ real ) )
=> ( sums @ A @ ( power_power @ A @ C2 ) @ ( divide_divide @ A @ ( one_one @ A ) @ ( minus_minus @ A @ ( one_one @ A ) @ C2 ) ) ) ) ) ).
% geometric_sums
thf(fact_2975_power__half__series,axiom,
( sums @ real
@ ^ [N2: nat] : ( power_power @ real @ ( divide_divide @ real @ ( one_one @ real ) @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) @ ( suc @ N2 ) )
@ ( one_one @ real ) ) ).
% power_half_series
thf(fact_2976_sums__If__finite__set_H,axiom,
! [A: $tType] :
( ( ( topolo1287966508704411220up_add @ A )
& ( topological_t2_space @ A ) )
=> ! [G: nat > A,S3: A,A3: set @ nat,S7: A,F2: nat > A] :
( ( sums @ A @ G @ S3 )
=> ( ( finite_finite2 @ nat @ A3 )
=> ( ( S7
= ( plus_plus @ A @ S3
@ ( groups7311177749621191930dd_sum @ nat @ A
@ ^ [N2: nat] : ( minus_minus @ A @ ( F2 @ N2 ) @ ( G @ N2 ) )
@ A3 ) ) )
=> ( sums @ A
@ ^ [N2: nat] : ( if @ A @ ( member @ nat @ N2 @ A3 ) @ ( F2 @ N2 ) @ ( G @ N2 ) )
@ S7 ) ) ) ) ) ).
% sums_If_finite_set'
thf(fact_2977_powser__sums__if,axiom,
! [A: $tType] :
( ( ( ring_1 @ A )
& ( topolo4958980785337419405_space @ A ) )
=> ! [M: nat,Z: A] :
( sums @ A
@ ^ [N2: nat] : ( times_times @ A @ ( if @ A @ ( N2 = M ) @ ( one_one @ A ) @ ( zero_zero @ A ) ) @ ( power_power @ A @ Z @ N2 ) )
@ ( power_power @ A @ Z @ M ) ) ) ).
% powser_sums_if
thf(fact_2978_sums__zero__iff__shift,axiom,
! [A: $tType] :
( ( real_V822414075346904944vector @ A )
=> ! [N3: nat,F2: nat > A,S: A] :
( ! [I2: nat] :
( ( ord_less @ nat @ I2 @ N3 )
=> ( ( F2 @ I2 )
= ( zero_zero @ A ) ) )
=> ( ( sums @ A
@ ^ [I3: nat] : ( F2 @ ( plus_plus @ nat @ I3 @ N3 ) )
@ S )
= ( sums @ A @ F2 @ S ) ) ) ) ).
% sums_zero_iff_shift
thf(fact_2979_sum__subtractf__nat,axiom,
! [A: $tType,A3: set @ A,G: A > nat,F2: A > nat] :
( ! [X3: A] :
( ( member @ A @ X3 @ A3 )
=> ( ord_less_eq @ nat @ ( G @ X3 ) @ ( F2 @ X3 ) ) )
=> ( ( groups7311177749621191930dd_sum @ A @ nat
@ ^ [X2: A] : ( minus_minus @ nat @ ( F2 @ X2 ) @ ( G @ X2 ) )
@ A3 )
= ( minus_minus @ nat @ ( groups7311177749621191930dd_sum @ A @ nat @ F2 @ A3 ) @ ( groups7311177749621191930dd_sum @ A @ nat @ G @ A3 ) ) ) ) ).
% sum_subtractf_nat
thf(fact_2980_sum__SucD,axiom,
! [A: $tType,F2: A > nat,A3: set @ A,N3: nat] :
( ( ( groups7311177749621191930dd_sum @ A @ nat @ F2 @ A3 )
= ( suc @ N3 ) )
=> ? [X3: A] :
( ( member @ A @ X3 @ A3 )
& ( ord_less @ nat @ ( zero_zero @ nat ) @ ( F2 @ X3 ) ) ) ) ).
% sum_SucD
thf(fact_2981_sum__eq__Suc0__iff,axiom,
! [A: $tType,A3: set @ A,F2: A > nat] :
( ( finite_finite2 @ A @ A3 )
=> ( ( ( groups7311177749621191930dd_sum @ A @ nat @ F2 @ A3 )
= ( suc @ ( zero_zero @ nat ) ) )
= ( ? [X2: A] :
( ( member @ A @ X2 @ A3 )
& ( ( F2 @ X2 )
= ( suc @ ( zero_zero @ nat ) ) )
& ! [Y2: A] :
( ( member @ A @ Y2 @ A3 )
=> ( ( X2 != Y2 )
=> ( ( F2 @ Y2 )
= ( zero_zero @ nat ) ) ) ) ) ) ) ) ).
% sum_eq_Suc0_iff
thf(fact_2982_sum__eq__1__iff,axiom,
! [A: $tType,A3: set @ A,F2: A > nat] :
( ( finite_finite2 @ A @ A3 )
=> ( ( ( groups7311177749621191930dd_sum @ A @ nat @ F2 @ A3 )
= ( one_one @ nat ) )
= ( ? [X2: A] :
( ( member @ A @ X2 @ A3 )
& ( ( F2 @ X2 )
= ( one_one @ nat ) )
& ! [Y2: A] :
( ( member @ A @ Y2 @ A3 )
=> ( ( X2 != Y2 )
=> ( ( F2 @ Y2 )
= ( zero_zero @ nat ) ) ) ) ) ) ) ) ).
% sum_eq_1_iff
thf(fact_2983_sum__nth__roots,axiom,
! [N3: nat,C2: complex] :
( ( ord_less @ nat @ ( one_one @ nat ) @ N3 )
=> ( ( groups7311177749621191930dd_sum @ complex @ complex
@ ^ [X2: complex] : X2
@ ( collect @ complex
@ ^ [Z3: complex] :
( ( power_power @ complex @ Z3 @ N3 )
= C2 ) ) )
= ( zero_zero @ complex ) ) ) ).
% sum_nth_roots
thf(fact_2984_sum__roots__unity,axiom,
! [N3: nat] :
( ( ord_less @ nat @ ( one_one @ nat ) @ N3 )
=> ( ( groups7311177749621191930dd_sum @ complex @ complex
@ ^ [X2: complex] : X2
@ ( collect @ complex
@ ^ [Z3: complex] :
( ( power_power @ complex @ Z3 @ N3 )
= ( one_one @ complex ) ) ) )
= ( zero_zero @ complex ) ) ) ).
% sum_roots_unity
thf(fact_2985_sum__diff__nat,axiom,
! [A: $tType,B6: set @ A,A3: set @ A,F2: A > nat] :
( ( finite_finite2 @ A @ B6 )
=> ( ( ord_less_eq @ ( set @ A ) @ B6 @ A3 )
=> ( ( groups7311177749621191930dd_sum @ A @ nat @ F2 @ ( minus_minus @ ( set @ A ) @ A3 @ B6 ) )
= ( minus_minus @ nat @ ( groups7311177749621191930dd_sum @ A @ nat @ F2 @ A3 ) @ ( groups7311177749621191930dd_sum @ A @ nat @ F2 @ B6 ) ) ) ) ) ).
% sum_diff_nat
thf(fact_2986_sum__diff1__nat,axiom,
! [A: $tType,A2: A,A3: set @ A,F2: A > nat] :
( ( ( member @ A @ A2 @ A3 )
=> ( ( groups7311177749621191930dd_sum @ A @ nat @ F2 @ ( minus_minus @ ( set @ A ) @ A3 @ ( insert @ A @ A2 @ ( bot_bot @ ( set @ A ) ) ) ) )
= ( minus_minus @ nat @ ( groups7311177749621191930dd_sum @ A @ nat @ F2 @ A3 ) @ ( F2 @ A2 ) ) ) )
& ( ~ ( member @ A @ A2 @ A3 )
=> ( ( groups7311177749621191930dd_sum @ A @ nat @ F2 @ ( minus_minus @ ( set @ A ) @ A3 @ ( insert @ A @ A2 @ ( bot_bot @ ( set @ A ) ) ) ) )
= ( groups7311177749621191930dd_sum @ A @ nat @ F2 @ A3 ) ) ) ) ).
% sum_diff1_nat
thf(fact_2987_sums__le,axiom,
! [A: $tType] :
( ( ( ordere6911136660526730532id_add @ A )
& ( topolo1944317154257567458pology @ A ) )
=> ! [F2: nat > A,G: nat > A,S: A,T2: A] :
( ! [N: nat] : ( ord_less_eq @ A @ ( F2 @ N ) @ ( G @ N ) )
=> ( ( sums @ A @ F2 @ S )
=> ( ( sums @ A @ G @ T2 )
=> ( ord_less_eq @ A @ S @ T2 ) ) ) ) ) ).
% sums_le
thf(fact_2988_sums__add,axiom,
! [A: $tType] :
( ( ( topolo5987344860129210374id_add @ A )
& ( topological_t2_space @ A ) )
=> ! [F2: nat > A,A2: A,G: nat > A,B2: A] :
( ( sums @ A @ F2 @ A2 )
=> ( ( sums @ A @ G @ B2 )
=> ( sums @ A
@ ^ [N2: nat] : ( plus_plus @ A @ ( F2 @ N2 ) @ ( G @ N2 ) )
@ ( plus_plus @ A @ A2 @ B2 ) ) ) ) ) ).
% sums_add
thf(fact_2989_sums__mult,axiom,
! [A: $tType] :
( ( real_V4412858255891104859lgebra @ A )
=> ! [F2: nat > A,A2: A,C2: A] :
( ( sums @ A @ F2 @ A2 )
=> ( sums @ A
@ ^ [N2: nat] : ( times_times @ A @ C2 @ ( F2 @ N2 ) )
@ ( times_times @ A @ C2 @ A2 ) ) ) ) ).
% sums_mult
thf(fact_2990_sums__mult2,axiom,
! [A: $tType] :
( ( real_V4412858255891104859lgebra @ A )
=> ! [F2: nat > A,A2: A,C2: A] :
( ( sums @ A @ F2 @ A2 )
=> ( sums @ A
@ ^ [N2: nat] : ( times_times @ A @ ( F2 @ N2 ) @ C2 )
@ ( times_times @ A @ A2 @ C2 ) ) ) ) ).
% sums_mult2
thf(fact_2991_sums__diff,axiom,
! [A: $tType] :
( ( real_V822414075346904944vector @ A )
=> ! [F2: nat > A,A2: A,G: nat > A,B2: A] :
( ( sums @ A @ F2 @ A2 )
=> ( ( sums @ A @ G @ B2 )
=> ( sums @ A
@ ^ [N2: nat] : ( minus_minus @ A @ ( F2 @ N2 ) @ ( G @ N2 ) )
@ ( minus_minus @ A @ A2 @ B2 ) ) ) ) ) ).
% sums_diff
thf(fact_2992_sums__divide,axiom,
! [A: $tType] :
( ( real_V3459762299906320749_field @ A )
=> ! [F2: nat > A,A2: A,C2: A] :
( ( sums @ A @ F2 @ A2 )
=> ( sums @ A
@ ^ [N2: nat] : ( divide_divide @ A @ ( F2 @ N2 ) @ C2 )
@ ( divide_divide @ A @ A2 @ C2 ) ) ) ) ).
% sums_divide
thf(fact_2993_Sum__Icc__int,axiom,
! [M: int,N3: int] :
( ( ord_less_eq @ int @ M @ N3 )
=> ( ( groups7311177749621191930dd_sum @ int @ int
@ ^ [X2: int] : X2
@ ( set_or1337092689740270186AtMost @ int @ M @ N3 ) )
= ( divide_divide @ int @ ( minus_minus @ int @ ( times_times @ int @ N3 @ ( plus_plus @ int @ N3 @ ( one_one @ int ) ) ) @ ( times_times @ int @ M @ ( minus_minus @ int @ M @ ( one_one @ int ) ) ) ) @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) ) ) ) ).
% Sum_Icc_int
thf(fact_2994_sums__mult__iff,axiom,
! [A: $tType] :
( ( ( field @ A )
& ( real_V4412858255891104859lgebra @ A ) )
=> ! [C2: A,F2: nat > A,D: A] :
( ( C2
!= ( zero_zero @ A ) )
=> ( ( sums @ A
@ ^ [N2: nat] : ( times_times @ A @ C2 @ ( F2 @ N2 ) )
@ ( times_times @ A @ C2 @ D ) )
= ( sums @ A @ F2 @ D ) ) ) ) ).
% sums_mult_iff
thf(fact_2995_sums__mult2__iff,axiom,
! [A: $tType] :
( ( ( field @ A )
& ( real_V4412858255891104859lgebra @ A ) )
=> ! [C2: A,F2: nat > A,D: A] :
( ( C2
!= ( zero_zero @ A ) )
=> ( ( sums @ A
@ ^ [N2: nat] : ( times_times @ A @ ( F2 @ N2 ) @ C2 )
@ ( times_times @ A @ D @ C2 ) )
= ( sums @ A @ F2 @ D ) ) ) ) ).
% sums_mult2_iff
thf(fact_2996_sums__mult__D,axiom,
! [A: $tType] :
( ( real_V3459762299906320749_field @ A )
=> ! [C2: A,F2: nat > A,A2: A] :
( ( sums @ A
@ ^ [N2: nat] : ( times_times @ A @ C2 @ ( F2 @ N2 ) )
@ A2 )
=> ( ( C2
!= ( zero_zero @ A ) )
=> ( sums @ A @ F2 @ ( divide_divide @ A @ A2 @ C2 ) ) ) ) ) ).
% sums_mult_D
thf(fact_2997_sums__Suc__imp,axiom,
! [A: $tType] :
( ( real_V822414075346904944vector @ A )
=> ! [F2: nat > A,S: A] :
( ( ( F2 @ ( zero_zero @ nat ) )
= ( zero_zero @ A ) )
=> ( ( sums @ A
@ ^ [N2: nat] : ( F2 @ ( suc @ N2 ) )
@ S )
=> ( sums @ A @ F2 @ S ) ) ) ) ).
% sums_Suc_imp
thf(fact_2998_sums__Suc__iff,axiom,
! [A: $tType] :
( ( real_V822414075346904944vector @ A )
=> ! [F2: nat > A,S: A] :
( ( sums @ A
@ ^ [N2: nat] : ( F2 @ ( suc @ N2 ) )
@ S )
= ( sums @ A @ F2 @ ( plus_plus @ A @ S @ ( F2 @ ( zero_zero @ nat ) ) ) ) ) ) ).
% sums_Suc_iff
thf(fact_2999_sums__Suc,axiom,
! [A: $tType] :
( ( ( topolo5987344860129210374id_add @ A )
& ( topological_t2_space @ A ) )
=> ! [F2: nat > A,L2: A] :
( ( sums @ A
@ ^ [N2: nat] : ( F2 @ ( suc @ N2 ) )
@ L2 )
=> ( sums @ A @ F2 @ ( plus_plus @ A @ L2 @ ( F2 @ ( zero_zero @ nat ) ) ) ) ) ) ).
% sums_Suc
thf(fact_3000_lemma__termdiff2,axiom,
! [A: $tType] :
( ( field @ A )
=> ! [H2: A,Z: A,N3: nat] :
( ( H2
!= ( zero_zero @ A ) )
=> ( ( minus_minus @ A @ ( divide_divide @ A @ ( minus_minus @ A @ ( power_power @ A @ ( plus_plus @ A @ Z @ H2 ) @ N3 ) @ ( power_power @ A @ Z @ N3 ) ) @ H2 ) @ ( times_times @ A @ ( semiring_1_of_nat @ A @ N3 ) @ ( power_power @ A @ Z @ ( minus_minus @ nat @ N3 @ ( suc @ ( zero_zero @ nat ) ) ) ) ) )
= ( times_times @ A @ H2
@ ( groups7311177749621191930dd_sum @ nat @ A
@ ^ [P5: nat] :
( groups7311177749621191930dd_sum @ nat @ A
@ ^ [Q4: nat] : ( times_times @ A @ ( power_power @ A @ ( plus_plus @ A @ Z @ H2 ) @ Q4 ) @ ( power_power @ A @ Z @ ( minus_minus @ nat @ ( minus_minus @ nat @ N3 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ Q4 ) ) )
@ ( set_ord_lessThan @ nat @ ( minus_minus @ nat @ ( minus_minus @ nat @ N3 @ ( suc @ ( zero_zero @ nat ) ) ) @ P5 ) ) )
@ ( set_ord_lessThan @ nat @ ( minus_minus @ nat @ N3 @ ( suc @ ( zero_zero @ nat ) ) ) ) ) ) ) ) ) ).
% lemma_termdiff2
thf(fact_3001_pochhammer__double,axiom,
! [A: $tType] :
( ( field_char_0 @ A )
=> ! [Z: A,N3: nat] :
( ( comm_s3205402744901411588hammer @ A @ ( times_times @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ Z ) @ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N3 ) )
= ( 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 ) ) @ N3 ) ) ) @ ( comm_s3205402744901411588hammer @ A @ Z @ N3 ) ) @ ( comm_s3205402744901411588hammer @ A @ ( plus_plus @ A @ Z @ ( divide_divide @ A @ ( one_one @ A ) @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) ) @ N3 ) ) ) ) ).
% pochhammer_double
thf(fact_3002_of__nat__code,axiom,
! [A: $tType] :
( ( semiring_1 @ A )
=> ( ( semiring_1_of_nat @ A )
= ( ^ [N2: nat] :
( semiri8178284476397505188at_aux @ A
@ ^ [I3: A] : ( plus_plus @ A @ I3 @ ( one_one @ A ) )
@ N2
@ ( zero_zero @ A ) ) ) ) ) ).
% of_nat_code
thf(fact_3003_floor__log__nat__eq__powr__iff,axiom,
! [B2: nat,K: nat,N3: nat] :
( ( ord_less_eq @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ B2 )
=> ( ( ord_less @ nat @ ( zero_zero @ nat ) @ K )
=> ( ( ( archim6421214686448440834_floor @ real @ ( log @ ( semiring_1_of_nat @ real @ B2 ) @ ( semiring_1_of_nat @ real @ K ) ) )
= ( semiring_1_of_nat @ int @ N3 ) )
= ( ( ord_less_eq @ nat @ ( power_power @ nat @ B2 @ N3 ) @ K )
& ( ord_less @ nat @ K @ ( power_power @ nat @ B2 @ ( plus_plus @ nat @ N3 @ ( one_one @ nat ) ) ) ) ) ) ) ) ).
% floor_log_nat_eq_powr_iff
thf(fact_3004_prod__cases4,axiom,
! [A: $tType,B: $tType,C: $tType,D5: $tType,Y: product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ D5 ) )] :
~ ! [A4: A,B3: B,C4: C,D3: D5] :
( Y
!= ( product_Pair @ A @ ( product_prod @ B @ ( product_prod @ C @ D5 ) ) @ A4 @ ( product_Pair @ B @ ( product_prod @ C @ D5 ) @ B3 @ ( product_Pair @ C @ D5 @ C4 @ D3 ) ) ) ) ).
% prod_cases4
thf(fact_3005_old_Oprod_Oinject,axiom,
! [A: $tType,B: $tType,A2: A,B2: B,A5: A,B4: B] :
( ( ( product_Pair @ A @ B @ A2 @ B2 )
= ( product_Pair @ A @ B @ A5 @ B4 ) )
= ( ( A2 = A5 )
& ( B2 = B4 ) ) ) ).
% old.prod.inject
thf(fact_3006_prod_Oinject,axiom,
! [A: $tType,B: $tType,X15: A,X22: B,Y15: A,Y22: B] :
( ( ( product_Pair @ A @ B @ X15 @ X22 )
= ( product_Pair @ A @ B @ Y15 @ Y22 ) )
= ( ( X15 = Y15 )
& ( X22 = Y22 ) ) ) ).
% prod.inject
thf(fact_3007_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_3008_of__int__floor__cancel,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [X: A] :
( ( ( ring_1_of_int @ A @ ( archim6421214686448440834_floor @ A @ X ) )
= X )
= ( ? [N2: int] :
( X
= ( ring_1_of_int @ A @ N2 ) ) ) ) ) ).
% of_int_floor_cancel
thf(fact_3009_pochhammer__1,axiom,
! [A: $tType] :
( ( comm_semiring_1 @ A )
=> ! [A2: A] :
( ( comm_s3205402744901411588hammer @ A @ A2 @ ( one_one @ nat ) )
= A2 ) ) ).
% pochhammer_1
thf(fact_3010_lessThan__subset__iff,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [X: A,Y: A] :
( ( ord_less_eq @ ( set @ A ) @ ( set_ord_lessThan @ A @ X ) @ ( set_ord_lessThan @ A @ Y ) )
= ( ord_less_eq @ A @ X @ Y ) ) ) ).
% lessThan_subset_iff
thf(fact_3011_floor__numeral,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [V: num] :
( ( archim6421214686448440834_floor @ A @ ( numeral_numeral @ A @ V ) )
= ( numeral_numeral @ int @ V ) ) ) ).
% floor_numeral
thf(fact_3012_floor__one,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ( ( archim6421214686448440834_floor @ A @ ( one_one @ A ) )
= ( one_one @ int ) ) ) ).
% floor_one
thf(fact_3013_pochhammer__0,axiom,
! [A: $tType] :
( ( comm_semiring_1 @ A )
=> ! [A2: A] :
( ( comm_s3205402744901411588hammer @ A @ A2 @ ( zero_zero @ nat ) )
= ( one_one @ A ) ) ) ).
% pochhammer_0
thf(fact_3014_pochhammer__Suc0,axiom,
! [A: $tType] :
( ( comm_semiring_1 @ A )
=> ! [A2: A] :
( ( comm_s3205402744901411588hammer @ A @ A2 @ ( suc @ ( zero_zero @ nat ) ) )
= A2 ) ) ).
% pochhammer_Suc0
thf(fact_3015_sum_OlessThan__Suc,axiom,
! [A: $tType] :
( ( comm_monoid_add @ A )
=> ! [G: nat > A,N3: nat] :
( ( groups7311177749621191930dd_sum @ nat @ A @ G @ ( set_ord_lessThan @ nat @ ( suc @ N3 ) ) )
= ( plus_plus @ A @ ( groups7311177749621191930dd_sum @ nat @ A @ G @ ( set_ord_lessThan @ nat @ N3 ) ) @ ( G @ N3 ) ) ) ) ).
% sum.lessThan_Suc
thf(fact_3016_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_3017_floor__diff__of__int,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [X: A,Z: int] :
( ( archim6421214686448440834_floor @ A @ ( minus_minus @ A @ X @ ( ring_1_of_int @ A @ Z ) ) )
= ( minus_minus @ int @ ( archim6421214686448440834_floor @ A @ X ) @ Z ) ) ) ).
% floor_diff_of_int
thf(fact_3018_zero__le__floor,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [X: A] :
( ( ord_less_eq @ int @ ( zero_zero @ int ) @ ( archim6421214686448440834_floor @ A @ X ) )
= ( ord_less_eq @ A @ ( zero_zero @ A ) @ X ) ) ) ).
% zero_le_floor
thf(fact_3019_numeral__le__floor,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [V: num,X: A] :
( ( ord_less_eq @ int @ ( numeral_numeral @ int @ V ) @ ( archim6421214686448440834_floor @ A @ X ) )
= ( ord_less_eq @ A @ ( numeral_numeral @ A @ V ) @ X ) ) ) ).
% numeral_le_floor
thf(fact_3020_floor__less__zero,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [X: A] :
( ( ord_less @ int @ ( archim6421214686448440834_floor @ A @ X ) @ ( zero_zero @ int ) )
= ( ord_less @ A @ X @ ( zero_zero @ A ) ) ) ) ).
% floor_less_zero
thf(fact_3021_floor__less__numeral,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [X: A,V: num] :
( ( ord_less @ int @ ( archim6421214686448440834_floor @ A @ X ) @ ( numeral_numeral @ int @ V ) )
= ( ord_less @ A @ X @ ( numeral_numeral @ A @ V ) ) ) ) ).
% floor_less_numeral
thf(fact_3022_zero__less__floor,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [X: A] :
( ( ord_less @ int @ ( zero_zero @ int ) @ ( archim6421214686448440834_floor @ A @ X ) )
= ( ord_less_eq @ A @ ( one_one @ A ) @ X ) ) ) ).
% zero_less_floor
thf(fact_3023_floor__le__zero,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [X: A] :
( ( ord_less_eq @ int @ ( archim6421214686448440834_floor @ A @ X ) @ ( zero_zero @ int ) )
= ( ord_less @ A @ X @ ( one_one @ A ) ) ) ) ).
% floor_le_zero
thf(fact_3024_one__le__floor,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [X: A] :
( ( ord_less_eq @ int @ ( one_one @ int ) @ ( archim6421214686448440834_floor @ A @ X ) )
= ( ord_less_eq @ A @ ( one_one @ A ) @ X ) ) ) ).
% one_le_floor
thf(fact_3025_floor__less__one,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [X: A] :
( ( ord_less @ int @ ( archim6421214686448440834_floor @ A @ X ) @ ( one_one @ int ) )
= ( ord_less @ A @ X @ ( one_one @ A ) ) ) ) ).
% floor_less_one
thf(fact_3026_floor__diff__numeral,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [X: A,V: num] :
( ( archim6421214686448440834_floor @ A @ ( minus_minus @ A @ X @ ( numeral_numeral @ A @ V ) ) )
= ( minus_minus @ int @ ( archim6421214686448440834_floor @ A @ X ) @ ( numeral_numeral @ int @ V ) ) ) ) ).
% floor_diff_numeral
thf(fact_3027_floor__numeral__power,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [X: num,N3: nat] :
( ( archim6421214686448440834_floor @ A @ ( power_power @ A @ ( numeral_numeral @ A @ X ) @ N3 ) )
= ( power_power @ int @ ( numeral_numeral @ int @ X ) @ N3 ) ) ) ).
% floor_numeral_power
thf(fact_3028_floor__diff__one,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [X: A] :
( ( archim6421214686448440834_floor @ A @ ( minus_minus @ A @ X @ ( one_one @ A ) ) )
= ( minus_minus @ int @ ( archim6421214686448440834_floor @ A @ X ) @ ( one_one @ int ) ) ) ) ).
% floor_diff_one
thf(fact_3029_floor__divide__eq__div__numeral,axiom,
! [A2: num,B2: num] :
( ( archim6421214686448440834_floor @ real @ ( divide_divide @ real @ ( numeral_numeral @ real @ A2 ) @ ( numeral_numeral @ real @ B2 ) ) )
= ( divide_divide @ int @ ( numeral_numeral @ int @ A2 ) @ ( numeral_numeral @ int @ B2 ) ) ) ).
% floor_divide_eq_div_numeral
thf(fact_3030_numeral__less__floor,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [V: num,X: A] :
( ( ord_less @ int @ ( numeral_numeral @ int @ V ) @ ( archim6421214686448440834_floor @ A @ X ) )
= ( ord_less_eq @ A @ ( plus_plus @ A @ ( numeral_numeral @ A @ V ) @ ( one_one @ A ) ) @ X ) ) ) ).
% numeral_less_floor
thf(fact_3031_floor__le__numeral,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [X: A,V: num] :
( ( ord_less_eq @ int @ ( archim6421214686448440834_floor @ A @ X ) @ ( numeral_numeral @ int @ V ) )
= ( ord_less @ A @ X @ ( plus_plus @ A @ ( numeral_numeral @ A @ V ) @ ( one_one @ A ) ) ) ) ) ).
% floor_le_numeral
thf(fact_3032_one__less__floor,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [X: A] :
( ( ord_less @ int @ ( one_one @ int ) @ ( archim6421214686448440834_floor @ A @ X ) )
= ( ord_less_eq @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ X ) ) ) ).
% one_less_floor
thf(fact_3033_floor__le__one,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [X: A] :
( ( ord_less_eq @ int @ ( archim6421214686448440834_floor @ A @ X ) @ ( one_one @ int ) )
= ( ord_less @ A @ X @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) ) ) ).
% floor_le_one
thf(fact_3034_floor__one__divide__eq__div__numeral,axiom,
! [B2: num] :
( ( archim6421214686448440834_floor @ real @ ( divide_divide @ real @ ( one_one @ real ) @ ( numeral_numeral @ real @ B2 ) ) )
= ( divide_divide @ int @ ( one_one @ int ) @ ( numeral_numeral @ int @ B2 ) ) ) ).
% floor_one_divide_eq_div_numeral
thf(fact_3035_lessThan__def,axiom,
! [A: $tType] :
( ( ord @ A )
=> ( ( set_ord_lessThan @ A )
= ( ^ [U2: A] :
( collect @ A
@ ^ [X2: A] : ( ord_less @ A @ X2 @ U2 ) ) ) ) ) ).
% lessThan_def
thf(fact_3036_of__int__floor__le,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [X: A] : ( ord_less_eq @ A @ ( ring_1_of_int @ A @ ( archim6421214686448440834_floor @ A @ X ) ) @ X ) ) ).
% of_int_floor_le
thf(fact_3037_floor__mono,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [X: A,Y: A] :
( ( ord_less_eq @ A @ X @ Y )
=> ( ord_less_eq @ int @ ( archim6421214686448440834_floor @ A @ X ) @ ( archim6421214686448440834_floor @ A @ Y ) ) ) ) ).
% floor_mono
thf(fact_3038_floor__less__cancel,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [X: A,Y: A] :
( ( ord_less @ int @ ( archim6421214686448440834_floor @ A @ X ) @ ( archim6421214686448440834_floor @ A @ Y ) )
=> ( ord_less @ A @ X @ Y ) ) ) ).
% floor_less_cancel
thf(fact_3039_floor__le__ceiling,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [X: A] : ( ord_less_eq @ int @ ( archim6421214686448440834_floor @ A @ X ) @ ( archimedean_ceiling @ A @ X ) ) ) ).
% floor_le_ceiling
thf(fact_3040_lessThan__strict__subset__iff,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [M: A,N3: A] :
( ( ord_less @ ( set @ A ) @ ( set_ord_lessThan @ A @ M ) @ ( set_ord_lessThan @ A @ N3 ) )
= ( ord_less @ A @ M @ N3 ) ) ) ).
% lessThan_strict_subset_iff
thf(fact_3041_pochhammer__pos,axiom,
! [A: $tType] :
( ( linordered_semidom @ A )
=> ! [X: A,N3: nat] :
( ( ord_less @ A @ ( zero_zero @ A ) @ X )
=> ( ord_less @ A @ ( zero_zero @ A ) @ ( comm_s3205402744901411588hammer @ A @ X @ N3 ) ) ) ) ).
% pochhammer_pos
thf(fact_3042_lessThan__Suc,axiom,
! [K: nat] :
( ( set_ord_lessThan @ nat @ ( suc @ K ) )
= ( insert @ nat @ K @ ( set_ord_lessThan @ nat @ K ) ) ) ).
% lessThan_Suc
thf(fact_3043_pochhammer__neq__0__mono,axiom,
! [A: $tType] :
( ( field_char_0 @ A )
=> ! [A2: A,M: nat,N3: nat] :
( ( ( comm_s3205402744901411588hammer @ A @ A2 @ M )
!= ( zero_zero @ A ) )
=> ( ( ord_less_eq @ nat @ N3 @ M )
=> ( ( comm_s3205402744901411588hammer @ A @ A2 @ N3 )
!= ( zero_zero @ A ) ) ) ) ) ).
% pochhammer_neq_0_mono
thf(fact_3044_pochhammer__eq__0__mono,axiom,
! [A: $tType] :
( ( field_char_0 @ A )
=> ! [A2: A,N3: nat,M: nat] :
( ( ( comm_s3205402744901411588hammer @ A @ A2 @ N3 )
= ( zero_zero @ A ) )
=> ( ( ord_less_eq @ nat @ N3 @ M )
=> ( ( comm_s3205402744901411588hammer @ A @ A2 @ M )
= ( zero_zero @ A ) ) ) ) ) ).
% pochhammer_eq_0_mono
thf(fact_3045_floor__le__round,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [X: A] : ( ord_less_eq @ int @ ( archim6421214686448440834_floor @ A @ X ) @ ( archimedean_round @ A @ X ) ) ) ).
% floor_le_round
thf(fact_3046_le__floor__iff,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [Z: int,X: A] :
( ( ord_less_eq @ int @ Z @ ( archim6421214686448440834_floor @ A @ X ) )
= ( ord_less_eq @ A @ ( ring_1_of_int @ A @ Z ) @ X ) ) ) ).
% le_floor_iff
thf(fact_3047_floor__less__iff,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [X: A,Z: int] :
( ( ord_less @ int @ ( archim6421214686448440834_floor @ A @ X ) @ Z )
= ( ord_less @ A @ X @ ( ring_1_of_int @ A @ Z ) ) ) ) ).
% floor_less_iff
thf(fact_3048_int__add__floor,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [Z: int,X: A] :
( ( plus_plus @ int @ Z @ ( archim6421214686448440834_floor @ A @ X ) )
= ( archim6421214686448440834_floor @ A @ ( plus_plus @ A @ ( ring_1_of_int @ A @ Z ) @ X ) ) ) ) ).
% int_add_floor
thf(fact_3049_floor__add__int,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [X: A,Z: int] :
( ( plus_plus @ int @ ( archim6421214686448440834_floor @ A @ X ) @ Z )
= ( archim6421214686448440834_floor @ A @ ( plus_plus @ A @ X @ ( ring_1_of_int @ A @ Z ) ) ) ) ) ).
% floor_add_int
thf(fact_3050_le__floor__add,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [X: A,Y: A] : ( ord_less_eq @ int @ ( plus_plus @ int @ ( archim6421214686448440834_floor @ A @ X ) @ ( archim6421214686448440834_floor @ A @ Y ) ) @ ( archim6421214686448440834_floor @ A @ ( plus_plus @ A @ X @ Y ) ) ) ) ).
% le_floor_add
thf(fact_3051_floor__power,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [X: A,N3: nat] :
( ( X
= ( ring_1_of_int @ A @ ( archim6421214686448440834_floor @ A @ X ) ) )
=> ( ( archim6421214686448440834_floor @ A @ ( power_power @ A @ X @ N3 ) )
= ( power_power @ int @ ( archim6421214686448440834_floor @ A @ X ) @ N3 ) ) ) ) ).
% floor_power
thf(fact_3052_floor__divide__of__int__eq,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [K: int,L2: int] :
( ( archim6421214686448440834_floor @ A @ ( divide_divide @ A @ ( ring_1_of_int @ A @ K ) @ ( ring_1_of_int @ A @ L2 ) ) )
= ( divide_divide @ int @ K @ L2 ) ) ) ).
% floor_divide_of_int_eq
thf(fact_3053_pochhammer__nonneg,axiom,
! [A: $tType] :
( ( linordered_semidom @ A )
=> ! [X: A,N3: nat] :
( ( ord_less @ A @ ( zero_zero @ A ) @ X )
=> ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( comm_s3205402744901411588hammer @ A @ X @ N3 ) ) ) ) ).
% pochhammer_nonneg
thf(fact_3054_of__nat__aux_Osimps_I2_J,axiom,
! [A: $tType] :
( ( semiring_1 @ A )
=> ! [Inc: A > A,N3: nat,I: A] :
( ( semiri8178284476397505188at_aux @ A @ Inc @ ( suc @ N3 ) @ I )
= ( semiri8178284476397505188at_aux @ A @ Inc @ N3 @ ( Inc @ I ) ) ) ) ).
% of_nat_aux.simps(2)
thf(fact_3055_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_3056_pochhammer__0__left,axiom,
! [A: $tType] :
( ( comm_semiring_1 @ A )
=> ! [N3: nat] :
( ( ( N3
= ( zero_zero @ nat ) )
=> ( ( comm_s3205402744901411588hammer @ A @ ( zero_zero @ A ) @ N3 )
= ( one_one @ A ) ) )
& ( ( N3
!= ( zero_zero @ nat ) )
=> ( ( comm_s3205402744901411588hammer @ A @ ( zero_zero @ A ) @ N3 )
= ( zero_zero @ A ) ) ) ) ) ).
% pochhammer_0_left
thf(fact_3057_Pair__inject,axiom,
! [A: $tType,B: $tType,A2: A,B2: B,A5: A,B4: B] :
( ( ( product_Pair @ A @ B @ A2 @ B2 )
= ( product_Pair @ A @ B @ A5 @ B4 ) )
=> ~ ( ( A2 = A5 )
=> ( B2 != B4 ) ) ) ).
% Pair_inject
thf(fact_3058_prod__cases,axiom,
! [B: $tType,A: $tType,P: ( product_prod @ A @ B ) > $o,P4: product_prod @ A @ B] :
( ! [A4: A,B3: B] : ( P @ ( product_Pair @ A @ B @ A4 @ B3 ) )
=> ( P @ P4 ) ) ).
% prod_cases
thf(fact_3059_surj__pair,axiom,
! [A: $tType,B: $tType,P4: product_prod @ A @ B] :
? [X3: A,Y3: B] :
( P4
= ( product_Pair @ A @ B @ X3 @ Y3 ) ) ).
% surj_pair
thf(fact_3060_old_Oprod_Oexhaust,axiom,
! [A: $tType,B: $tType,Y: product_prod @ A @ B] :
~ ! [A4: A,B3: B] :
( Y
!= ( product_Pair @ A @ B @ A4 @ B3 ) ) ).
% old.prod.exhaust
thf(fact_3061_sum_Onat__diff__reindex,axiom,
! [A: $tType] :
( ( comm_monoid_add @ A )
=> ! [G: nat > A,N3: nat] :
( ( groups7311177749621191930dd_sum @ nat @ A
@ ^ [I3: nat] : ( G @ ( minus_minus @ nat @ N3 @ ( suc @ I3 ) ) )
@ ( set_ord_lessThan @ nat @ N3 ) )
= ( groups7311177749621191930dd_sum @ nat @ A @ G @ ( set_ord_lessThan @ nat @ N3 ) ) ) ) ).
% sum.nat_diff_reindex
thf(fact_3062_sum__diff__distrib,axiom,
! [A: $tType] :
( ( ord @ A )
=> ! [Q: A > nat,P: A > nat,N3: A] :
( ! [X3: A] : ( ord_less_eq @ nat @ ( Q @ X3 ) @ ( P @ X3 ) )
=> ( ( minus_minus @ nat @ ( groups7311177749621191930dd_sum @ A @ nat @ P @ ( set_ord_lessThan @ A @ N3 ) ) @ ( groups7311177749621191930dd_sum @ A @ nat @ Q @ ( set_ord_lessThan @ A @ N3 ) ) )
= ( groups7311177749621191930dd_sum @ A @ nat
@ ^ [X2: A] : ( minus_minus @ nat @ ( P @ X2 ) @ ( Q @ X2 ) )
@ ( set_ord_lessThan @ A @ N3 ) ) ) ) ) ).
% sum_diff_distrib
thf(fact_3063_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_3064_one__add__floor,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [X: A] :
( ( plus_plus @ int @ ( archim6421214686448440834_floor @ A @ X ) @ ( one_one @ int ) )
= ( archim6421214686448440834_floor @ A @ ( plus_plus @ A @ X @ ( one_one @ A ) ) ) ) ) ).
% one_add_floor
thf(fact_3065_floor__divide__of__nat__eq,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [M: nat,N3: nat] :
( ( archim6421214686448440834_floor @ A @ ( divide_divide @ A @ ( semiring_1_of_nat @ A @ M ) @ ( semiring_1_of_nat @ A @ N3 ) ) )
= ( semiring_1_of_nat @ int @ ( divide_divide @ nat @ M @ N3 ) ) ) ) ).
% floor_divide_of_nat_eq
thf(fact_3066_le__mult__nat__floor,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [A2: A,B2: A] : ( ord_less_eq @ nat @ ( times_times @ nat @ ( nat2 @ ( archim6421214686448440834_floor @ A @ A2 ) ) @ ( nat2 @ ( archim6421214686448440834_floor @ A @ B2 ) ) ) @ ( nat2 @ ( archim6421214686448440834_floor @ A @ ( times_times @ A @ A2 @ B2 ) ) ) ) ) ).
% le_mult_nat_floor
thf(fact_3067_nat__floor__neg,axiom,
! [X: real] :
( ( ord_less_eq @ real @ X @ ( zero_zero @ real ) )
=> ( ( nat2 @ ( archim6421214686448440834_floor @ real @ X ) )
= ( zero_zero @ nat ) ) ) ).
% nat_floor_neg
thf(fact_3068_floor__eq3,axiom,
! [N3: nat,X: real] :
( ( ord_less @ real @ ( semiring_1_of_nat @ real @ N3 ) @ X )
=> ( ( ord_less @ real @ X @ ( semiring_1_of_nat @ real @ ( suc @ N3 ) ) )
=> ( ( nat2 @ ( archim6421214686448440834_floor @ real @ X ) )
= N3 ) ) ) ).
% floor_eq3
thf(fact_3069_ceiling__altdef,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ( ( archimedean_ceiling @ A )
= ( ^ [X2: A] :
( if @ int
@ ( X2
= ( ring_1_of_int @ A @ ( archim6421214686448440834_floor @ A @ X2 ) ) )
@ ( archim6421214686448440834_floor @ A @ X2 )
@ ( plus_plus @ int @ ( archim6421214686448440834_floor @ A @ X2 ) @ ( one_one @ int ) ) ) ) ) ) ).
% ceiling_altdef
thf(fact_3070_le__nat__floor,axiom,
! [X: nat,A2: real] :
( ( ord_less_eq @ real @ ( semiring_1_of_nat @ real @ X ) @ A2 )
=> ( ord_less_eq @ nat @ X @ ( nat2 @ ( archim6421214686448440834_floor @ real @ A2 ) ) ) ) ).
% le_nat_floor
thf(fact_3071_ceiling__diff__floor__le__1,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [X: A] : ( ord_less_eq @ int @ ( minus_minus @ int @ ( archimedean_ceiling @ A @ X ) @ ( archim6421214686448440834_floor @ A @ X ) ) @ ( one_one @ int ) ) ) ).
% ceiling_diff_floor_le_1
thf(fact_3072_floor__eq,axiom,
! [N3: int,X: real] :
( ( ord_less @ real @ ( ring_1_of_int @ real @ N3 ) @ X )
=> ( ( ord_less @ real @ X @ ( plus_plus @ real @ ( ring_1_of_int @ real @ N3 ) @ ( one_one @ real ) ) )
=> ( ( archim6421214686448440834_floor @ real @ X )
= N3 ) ) ) ).
% floor_eq
thf(fact_3073_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_3074_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_3075_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_3076_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_3077_pochhammer__rec,axiom,
! [A: $tType] :
( ( comm_semiring_1 @ A )
=> ! [A2: A,N3: nat] :
( ( comm_s3205402744901411588hammer @ A @ A2 @ ( suc @ N3 ) )
= ( times_times @ A @ A2 @ ( comm_s3205402744901411588hammer @ A @ ( plus_plus @ A @ A2 @ ( one_one @ A ) ) @ N3 ) ) ) ) ).
% pochhammer_rec
thf(fact_3078_pochhammer__rec_H,axiom,
! [A: $tType] :
( ( comm_semiring_1 @ A )
=> ! [Z: A,N3: nat] :
( ( comm_s3205402744901411588hammer @ A @ Z @ ( suc @ N3 ) )
= ( times_times @ A @ ( plus_plus @ A @ Z @ ( semiring_1_of_nat @ A @ N3 ) ) @ ( comm_s3205402744901411588hammer @ A @ Z @ N3 ) ) ) ) ).
% pochhammer_rec'
thf(fact_3079_pochhammer__Suc,axiom,
! [A: $tType] :
( ( comm_semiring_1 @ A )
=> ! [A2: A,N3: nat] :
( ( comm_s3205402744901411588hammer @ A @ A2 @ ( suc @ N3 ) )
= ( times_times @ A @ ( comm_s3205402744901411588hammer @ A @ A2 @ N3 ) @ ( plus_plus @ A @ A2 @ ( semiring_1_of_nat @ A @ N3 ) ) ) ) ) ).
% pochhammer_Suc
thf(fact_3080_pochhammer__product_H,axiom,
! [A: $tType] :
( ( comm_semiring_1 @ A )
=> ! [Z: A,N3: nat,M: nat] :
( ( comm_s3205402744901411588hammer @ A @ Z @ ( plus_plus @ nat @ N3 @ M ) )
= ( times_times @ A @ ( comm_s3205402744901411588hammer @ A @ Z @ N3 ) @ ( comm_s3205402744901411588hammer @ A @ ( plus_plus @ A @ Z @ ( semiring_1_of_nat @ A @ N3 ) ) @ M ) ) ) ) ).
% pochhammer_product'
thf(fact_3081_sum_OlessThan__Suc__shift,axiom,
! [A: $tType] :
( ( comm_monoid_add @ A )
=> ! [G: nat > A,N3: nat] :
( ( groups7311177749621191930dd_sum @ nat @ A @ G @ ( set_ord_lessThan @ nat @ ( suc @ N3 ) ) )
= ( plus_plus @ A @ ( G @ ( zero_zero @ nat ) )
@ ( groups7311177749621191930dd_sum @ nat @ A
@ ^ [I3: nat] : ( G @ ( suc @ I3 ) )
@ ( set_ord_lessThan @ nat @ N3 ) ) ) ) ) ).
% sum.lessThan_Suc_shift
thf(fact_3082_sum__lessThan__telescope_H,axiom,
! [A: $tType] :
( ( ab_group_add @ A )
=> ! [F2: nat > A,M: nat] :
( ( groups7311177749621191930dd_sum @ nat @ A
@ ^ [N2: nat] : ( minus_minus @ A @ ( F2 @ N2 ) @ ( F2 @ ( suc @ N2 ) ) )
@ ( set_ord_lessThan @ nat @ M ) )
= ( minus_minus @ A @ ( F2 @ ( zero_zero @ nat ) ) @ ( F2 @ M ) ) ) ) ).
% sum_lessThan_telescope'
thf(fact_3083_sum__lessThan__telescope,axiom,
! [A: $tType] :
( ( ab_group_add @ A )
=> ! [F2: nat > A,M: nat] :
( ( groups7311177749621191930dd_sum @ nat @ A
@ ^ [N2: nat] : ( minus_minus @ A @ ( F2 @ ( suc @ N2 ) ) @ ( F2 @ N2 ) )
@ ( set_ord_lessThan @ nat @ M ) )
= ( minus_minus @ A @ ( F2 @ M ) @ ( F2 @ ( zero_zero @ nat ) ) ) ) ) ).
% sum_lessThan_telescope
thf(fact_3084_sumr__diff__mult__const2,axiom,
! [A: $tType] :
( ( ring_1 @ A )
=> ! [F2: nat > A,N3: nat,R2: A] :
( ( minus_minus @ A @ ( groups7311177749621191930dd_sum @ nat @ A @ F2 @ ( set_ord_lessThan @ nat @ N3 ) ) @ ( times_times @ A @ ( semiring_1_of_nat @ A @ N3 ) @ R2 ) )
= ( groups7311177749621191930dd_sum @ nat @ A
@ ^ [I3: nat] : ( minus_minus @ A @ ( F2 @ I3 ) @ R2 )
@ ( set_ord_lessThan @ nat @ N3 ) ) ) ) ).
% sumr_diff_mult_const2
thf(fact_3085_sum_OatLeast1__atMost__eq,axiom,
! [A: $tType] :
( ( comm_monoid_add @ A )
=> ! [G: nat > A,N3: nat] :
( ( groups7311177749621191930dd_sum @ nat @ A @ G @ ( set_or1337092689740270186AtMost @ nat @ ( suc @ ( zero_zero @ nat ) ) @ N3 ) )
= ( groups7311177749621191930dd_sum @ nat @ A
@ ^ [K2: nat] : ( G @ ( suc @ K2 ) )
@ ( set_ord_lessThan @ nat @ N3 ) ) ) ) ).
% sum.atLeast1_atMost_eq
thf(fact_3086_sums__iff__shift,axiom,
! [A: $tType] :
( ( real_V822414075346904944vector @ A )
=> ! [F2: nat > A,N3: nat,S: A] :
( ( sums @ A
@ ^ [I3: nat] : ( F2 @ ( plus_plus @ nat @ I3 @ N3 ) )
@ S )
= ( sums @ A @ F2 @ ( plus_plus @ A @ S @ ( groups7311177749621191930dd_sum @ nat @ A @ F2 @ ( set_ord_lessThan @ nat @ N3 ) ) ) ) ) ) ).
% sums_iff_shift
thf(fact_3087_sums__split__initial__segment,axiom,
! [A: $tType] :
( ( real_V822414075346904944vector @ A )
=> ! [F2: nat > A,S: A,N3: nat] :
( ( sums @ A @ F2 @ S )
=> ( sums @ A
@ ^ [I3: nat] : ( F2 @ ( plus_plus @ nat @ I3 @ N3 ) )
@ ( minus_minus @ A @ S @ ( groups7311177749621191930dd_sum @ nat @ A @ F2 @ ( set_ord_lessThan @ nat @ N3 ) ) ) ) ) ) ).
% sums_split_initial_segment
thf(fact_3088_sums__iff__shift_H,axiom,
! [A: $tType] :
( ( real_V822414075346904944vector @ A )
=> ! [F2: nat > A,N3: nat,S: A] :
( ( sums @ A
@ ^ [I3: nat] : ( F2 @ ( plus_plus @ nat @ I3 @ N3 ) )
@ ( minus_minus @ A @ S @ ( groups7311177749621191930dd_sum @ nat @ A @ F2 @ ( set_ord_lessThan @ nat @ N3 ) ) ) )
= ( sums @ A @ F2 @ S ) ) ) ).
% sums_iff_shift'
thf(fact_3089_floor__split,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [P: int > $o,T2: A] :
( ( P @ ( archim6421214686448440834_floor @ A @ T2 ) )
= ( ! [I3: int] :
( ( ( ord_less_eq @ A @ ( ring_1_of_int @ A @ I3 ) @ T2 )
& ( ord_less @ A @ T2 @ ( plus_plus @ A @ ( ring_1_of_int @ A @ I3 ) @ ( one_one @ A ) ) ) )
=> ( P @ I3 ) ) ) ) ) ).
% floor_split
thf(fact_3090_floor__eq__iff,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [X: A,A2: int] :
( ( ( archim6421214686448440834_floor @ A @ X )
= A2 )
= ( ( ord_less_eq @ A @ ( ring_1_of_int @ A @ A2 ) @ X )
& ( ord_less @ A @ X @ ( plus_plus @ A @ ( ring_1_of_int @ A @ A2 ) @ ( one_one @ A ) ) ) ) ) ) ).
% floor_eq_iff
thf(fact_3091_floor__unique,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [Z: int,X: A] :
( ( ord_less_eq @ A @ ( ring_1_of_int @ A @ Z ) @ X )
=> ( ( ord_less @ A @ X @ ( plus_plus @ A @ ( ring_1_of_int @ A @ Z ) @ ( one_one @ A ) ) )
=> ( ( archim6421214686448440834_floor @ A @ X )
= Z ) ) ) ) ).
% floor_unique
thf(fact_3092_less__floor__iff,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [Z: int,X: A] :
( ( ord_less @ int @ Z @ ( archim6421214686448440834_floor @ A @ X ) )
= ( ord_less_eq @ A @ ( plus_plus @ A @ ( ring_1_of_int @ A @ Z ) @ ( one_one @ A ) ) @ X ) ) ) ).
% less_floor_iff
thf(fact_3093_floor__le__iff,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [X: A,Z: int] :
( ( ord_less_eq @ int @ ( archim6421214686448440834_floor @ A @ X ) @ Z )
= ( ord_less @ A @ X @ ( plus_plus @ A @ ( ring_1_of_int @ A @ Z ) @ ( one_one @ A ) ) ) ) ) ).
% floor_le_iff
thf(fact_3094_le__mult__floor,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [A2: A,B2: A] :
( ( ord_less_eq @ A @ ( zero_zero @ A ) @ A2 )
=> ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ B2 )
=> ( ord_less_eq @ int @ ( times_times @ int @ ( archim6421214686448440834_floor @ A @ A2 ) @ ( archim6421214686448440834_floor @ A @ B2 ) ) @ ( archim6421214686448440834_floor @ A @ ( times_times @ A @ A2 @ B2 ) ) ) ) ) ) ).
% le_mult_floor
thf(fact_3095_floor__correct,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [X: A] :
( ( ord_less_eq @ A @ ( ring_1_of_int @ A @ ( archim6421214686448440834_floor @ A @ X ) ) @ X )
& ( ord_less @ A @ X @ ( ring_1_of_int @ A @ ( plus_plus @ int @ ( archim6421214686448440834_floor @ A @ X ) @ ( one_one @ int ) ) ) ) ) ) ).
% floor_correct
thf(fact_3096_floor__eq4,axiom,
! [N3: nat,X: real] :
( ( ord_less_eq @ real @ ( semiring_1_of_nat @ real @ N3 ) @ X )
=> ( ( ord_less @ real @ X @ ( semiring_1_of_nat @ real @ ( suc @ N3 ) ) )
=> ( ( nat2 @ ( archim6421214686448440834_floor @ real @ X ) )
= N3 ) ) ) ).
% floor_eq4
thf(fact_3097_floor__eq2,axiom,
! [N3: int,X: real] :
( ( ord_less_eq @ real @ ( ring_1_of_int @ real @ N3 ) @ X )
=> ( ( ord_less @ real @ X @ ( plus_plus @ real @ ( ring_1_of_int @ real @ N3 ) @ ( one_one @ real ) ) )
=> ( ( archim6421214686448440834_floor @ real @ X )
= N3 ) ) ) ).
% floor_eq2
thf(fact_3098_floor__divide__real__eq__div,axiom,
! [B2: int,A2: real] :
( ( ord_less_eq @ int @ ( zero_zero @ int ) @ B2 )
=> ( ( archim6421214686448440834_floor @ real @ ( divide_divide @ real @ A2 @ ( ring_1_of_int @ real @ B2 ) ) )
= ( divide_divide @ int @ ( archim6421214686448440834_floor @ real @ A2 ) @ B2 ) ) ) ).
% floor_divide_real_eq_div
thf(fact_3099_power__diff__1__eq,axiom,
! [A: $tType] :
( ( ( monoid_mult @ A )
& ( comm_ring @ A ) )
=> ! [X: A,N3: nat] :
( ( minus_minus @ A @ ( power_power @ A @ X @ N3 ) @ ( one_one @ A ) )
= ( times_times @ A @ ( minus_minus @ A @ X @ ( one_one @ A ) ) @ ( groups7311177749621191930dd_sum @ nat @ A @ ( power_power @ A @ X ) @ ( set_ord_lessThan @ nat @ N3 ) ) ) ) ) ).
% power_diff_1_eq
thf(fact_3100_one__diff__power__eq,axiom,
! [A: $tType] :
( ( ( monoid_mult @ A )
& ( comm_ring @ A ) )
=> ! [X: A,N3: nat] :
( ( minus_minus @ A @ ( one_one @ A ) @ ( power_power @ A @ X @ N3 ) )
= ( times_times @ A @ ( minus_minus @ A @ ( one_one @ A ) @ X ) @ ( groups7311177749621191930dd_sum @ nat @ A @ ( power_power @ A @ X ) @ ( set_ord_lessThan @ nat @ N3 ) ) ) ) ) ).
% one_diff_power_eq
thf(fact_3101_geometric__sum,axiom,
! [A: $tType] :
( ( field @ A )
=> ! [X: A,N3: nat] :
( ( X
!= ( one_one @ A ) )
=> ( ( groups7311177749621191930dd_sum @ nat @ A @ ( power_power @ A @ X ) @ ( set_ord_lessThan @ nat @ N3 ) )
= ( divide_divide @ A @ ( minus_minus @ A @ ( power_power @ A @ X @ N3 ) @ ( one_one @ A ) ) @ ( minus_minus @ A @ X @ ( one_one @ A ) ) ) ) ) ) ).
% geometric_sum
thf(fact_3102_pochhammer__product,axiom,
! [A: $tType] :
( ( comm_semiring_1 @ A )
=> ! [M: nat,N3: nat,Z: A] :
( ( ord_less_eq @ nat @ M @ N3 )
=> ( ( comm_s3205402744901411588hammer @ A @ Z @ N3 )
= ( times_times @ A @ ( comm_s3205402744901411588hammer @ A @ Z @ M ) @ ( comm_s3205402744901411588hammer @ A @ ( plus_plus @ A @ Z @ ( semiring_1_of_nat @ A @ M ) ) @ ( minus_minus @ nat @ N3 @ M ) ) ) ) ) ) ).
% pochhammer_product
thf(fact_3103_floor__divide__lower,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [Q2: A,P4: A] :
( ( ord_less @ A @ ( zero_zero @ A ) @ Q2 )
=> ( ord_less_eq @ A @ ( times_times @ A @ ( ring_1_of_int @ A @ ( archim6421214686448440834_floor @ A @ ( divide_divide @ A @ P4 @ Q2 ) ) ) @ Q2 ) @ P4 ) ) ) ).
% floor_divide_lower
thf(fact_3104_lemma__termdiff1,axiom,
! [A: $tType] :
( ( ( monoid_mult @ A )
& ( comm_ring @ A ) )
=> ! [Z: A,H2: A,M: nat] :
( ( groups7311177749621191930dd_sum @ nat @ A
@ ^ [P5: nat] : ( minus_minus @ A @ ( times_times @ A @ ( power_power @ A @ ( plus_plus @ A @ Z @ H2 ) @ ( minus_minus @ nat @ M @ P5 ) ) @ ( power_power @ A @ Z @ P5 ) ) @ ( power_power @ A @ Z @ M ) )
@ ( set_ord_lessThan @ nat @ M ) )
= ( groups7311177749621191930dd_sum @ nat @ A
@ ^ [P5: nat] : ( times_times @ A @ ( power_power @ A @ Z @ P5 ) @ ( minus_minus @ A @ ( power_power @ A @ ( plus_plus @ A @ Z @ H2 ) @ ( minus_minus @ nat @ M @ P5 ) ) @ ( power_power @ A @ Z @ ( minus_minus @ nat @ M @ P5 ) ) ) )
@ ( set_ord_lessThan @ nat @ M ) ) ) ) ).
% lemma_termdiff1
thf(fact_3105_sum__gp__strict,axiom,
! [A: $tType] :
( ( ( division_ring @ A )
& ( comm_ring @ A ) )
=> ! [X: A,N3: nat] :
( ( ( X
= ( one_one @ A ) )
=> ( ( groups7311177749621191930dd_sum @ nat @ A @ ( power_power @ A @ X ) @ ( set_ord_lessThan @ nat @ N3 ) )
= ( semiring_1_of_nat @ A @ N3 ) ) )
& ( ( X
!= ( one_one @ A ) )
=> ( ( groups7311177749621191930dd_sum @ nat @ A @ ( power_power @ A @ X ) @ ( set_ord_lessThan @ nat @ N3 ) )
= ( divide_divide @ A @ ( minus_minus @ A @ ( one_one @ A ) @ ( power_power @ A @ X @ N3 ) ) @ ( minus_minus @ A @ ( one_one @ A ) @ X ) ) ) ) ) ) ).
% sum_gp_strict
thf(fact_3106_power__diff__sumr2,axiom,
! [A: $tType] :
( ( ( monoid_mult @ A )
& ( comm_ring @ A ) )
=> ! [X: A,N3: nat,Y: A] :
( ( minus_minus @ A @ ( power_power @ A @ X @ N3 ) @ ( power_power @ A @ Y @ N3 ) )
= ( times_times @ A @ ( minus_minus @ A @ X @ Y )
@ ( groups7311177749621191930dd_sum @ nat @ A
@ ^ [I3: nat] : ( times_times @ A @ ( power_power @ A @ Y @ ( minus_minus @ nat @ N3 @ ( suc @ I3 ) ) ) @ ( power_power @ A @ X @ I3 ) )
@ ( set_ord_lessThan @ nat @ N3 ) ) ) ) ) ).
% power_diff_sumr2
thf(fact_3107_diff__power__eq__sum,axiom,
! [A: $tType] :
( ( ( monoid_mult @ A )
& ( comm_ring @ A ) )
=> ! [X: A,N3: nat,Y: A] :
( ( minus_minus @ A @ ( power_power @ A @ X @ ( suc @ N3 ) ) @ ( power_power @ A @ Y @ ( suc @ N3 ) ) )
= ( times_times @ A @ ( minus_minus @ A @ X @ Y )
@ ( groups7311177749621191930dd_sum @ nat @ A
@ ^ [P5: nat] : ( times_times @ A @ ( power_power @ A @ X @ P5 ) @ ( power_power @ A @ Y @ ( minus_minus @ nat @ N3 @ P5 ) ) )
@ ( set_ord_lessThan @ nat @ ( suc @ N3 ) ) ) ) ) ) ).
% diff_power_eq_sum
thf(fact_3108_floor__divide__upper,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [Q2: A,P4: A] :
( ( ord_less @ A @ ( zero_zero @ A ) @ Q2 )
=> ( ord_less @ A @ P4 @ ( times_times @ A @ ( plus_plus @ A @ ( ring_1_of_int @ A @ ( archim6421214686448440834_floor @ A @ ( divide_divide @ A @ P4 @ Q2 ) ) ) @ ( one_one @ A ) ) @ Q2 ) ) ) ) ).
% floor_divide_upper
thf(fact_3109_round__def,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ( ( archimedean_round @ A )
= ( ^ [X2: A] : ( archim6421214686448440834_floor @ A @ ( plus_plus @ A @ X2 @ ( divide_divide @ A @ ( one_one @ A ) @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) ) ) ) ) ) ).
% round_def
thf(fact_3110_real__sum__nat__ivl__bounded2,axiom,
! [A: $tType] :
( ( linordered_semidom @ A )
=> ! [N3: nat,F2: nat > A,K6: A,K: nat] :
( ! [P7: nat] :
( ( ord_less @ nat @ P7 @ N3 )
=> ( 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 @ N3 @ K ) ) ) @ ( times_times @ A @ ( semiring_1_of_nat @ A @ N3 ) @ K6 ) ) ) ) ) ).
% real_sum_nat_ivl_bounded2
thf(fact_3111_prod__induct3,axiom,
! [C: $tType,B: $tType,A: $tType,P: ( product_prod @ A @ ( product_prod @ B @ C ) ) > $o,X: product_prod @ A @ ( product_prod @ B @ C )] :
( ! [A4: A,B3: B,C4: C] : ( P @ ( product_Pair @ A @ ( product_prod @ B @ C ) @ A4 @ ( product_Pair @ B @ C @ B3 @ C4 ) ) )
=> ( P @ X ) ) ).
% prod_induct3
thf(fact_3112_prod__cases3,axiom,
! [A: $tType,B: $tType,C: $tType,Y: product_prod @ A @ ( product_prod @ B @ C )] :
~ ! [A4: A,B3: B,C4: C] :
( Y
!= ( product_Pair @ A @ ( product_prod @ B @ C ) @ A4 @ ( product_Pair @ B @ C @ B3 @ C4 ) ) ) ).
% prod_cases3
thf(fact_3113_one__diff__power__eq_H,axiom,
! [A: $tType] :
( ( ( monoid_mult @ A )
& ( comm_ring @ A ) )
=> ! [X: A,N3: nat] :
( ( minus_minus @ A @ ( one_one @ A ) @ ( power_power @ A @ X @ N3 ) )
= ( times_times @ A @ ( minus_minus @ A @ ( one_one @ A ) @ X )
@ ( groups7311177749621191930dd_sum @ nat @ A
@ ^ [I3: nat] : ( power_power @ A @ X @ ( minus_minus @ nat @ N3 @ ( suc @ I3 ) ) )
@ ( set_ord_lessThan @ nat @ N3 ) ) ) ) ) ).
% one_diff_power_eq'
thf(fact_3114_sum__split__even__odd,axiom,
! [F2: nat > real,G: nat > real,N3: nat] :
( ( groups7311177749621191930dd_sum @ nat @ real
@ ^ [I3: nat] : ( if @ real @ ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ I3 ) @ ( F2 @ I3 ) @ ( G @ I3 ) )
@ ( set_ord_lessThan @ nat @ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N3 ) ) )
= ( plus_plus @ real
@ ( groups7311177749621191930dd_sum @ nat @ real
@ ^ [I3: nat] : ( F2 @ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ I3 ) )
@ ( set_ord_lessThan @ nat @ N3 ) )
@ ( groups7311177749621191930dd_sum @ nat @ real
@ ^ [I3: nat] : ( G @ ( plus_plus @ nat @ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ I3 ) @ ( one_one @ nat ) ) )
@ ( set_ord_lessThan @ nat @ N3 ) ) ) ) ).
% sum_split_even_odd
thf(fact_3115_floor__log__eq__powr__iff,axiom,
! [X: real,B2: real,K: int] :
( ( ord_less @ real @ ( zero_zero @ real ) @ X )
=> ( ( ord_less @ real @ ( one_one @ real ) @ B2 )
=> ( ( ( archim6421214686448440834_floor @ real @ ( log @ B2 @ X ) )
= K )
= ( ( ord_less_eq @ real @ ( powr @ real @ B2 @ ( ring_1_of_int @ real @ K ) ) @ X )
& ( ord_less @ real @ X @ ( powr @ real @ B2 @ ( ring_1_of_int @ real @ ( plus_plus @ int @ K @ ( one_one @ int ) ) ) ) ) ) ) ) ) ).
% floor_log_eq_powr_iff
thf(fact_3116_floor__log2__div2,axiom,
! [N3: nat] :
( ( ord_less_eq @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N3 )
=> ( ( archim6421214686448440834_floor @ real @ ( log @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) @ ( semiring_1_of_nat @ real @ N3 ) ) )
= ( plus_plus @ int @ ( archim6421214686448440834_floor @ real @ ( log @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) @ ( semiring_1_of_nat @ real @ ( divide_divide @ nat @ N3 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) @ ( one_one @ int ) ) ) ) ).
% floor_log2_div2
thf(fact_3117_floor__log__nat__eq__if,axiom,
! [B2: nat,N3: nat,K: nat] :
( ( ord_less_eq @ nat @ ( power_power @ nat @ B2 @ N3 ) @ K )
=> ( ( ord_less @ nat @ K @ ( power_power @ nat @ B2 @ ( plus_plus @ nat @ N3 @ ( one_one @ nat ) ) ) )
=> ( ( ord_less_eq @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ B2 )
=> ( ( archim6421214686448440834_floor @ real @ ( log @ ( semiring_1_of_nat @ real @ B2 ) @ ( semiring_1_of_nat @ real @ K ) ) )
= ( semiring_1_of_nat @ int @ N3 ) ) ) ) ) ).
% floor_log_nat_eq_if
thf(fact_3118_prod__induct7,axiom,
! [G2: $tType,F: $tType,E3: $tType,D5: $tType,C: $tType,B: $tType,A: $tType,P: ( product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D5 @ ( product_prod @ E3 @ ( product_prod @ F @ G2 ) ) ) ) ) ) > $o,X: product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D5 @ ( product_prod @ E3 @ ( product_prod @ F @ G2 ) ) ) ) )] :
( ! [A4: A,B3: B,C4: C,D3: D5,E2: E3,F3: F,G3: G2] : ( P @ ( product_Pair @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D5 @ ( product_prod @ E3 @ ( product_prod @ F @ G2 ) ) ) ) ) @ A4 @ ( product_Pair @ B @ ( product_prod @ C @ ( product_prod @ D5 @ ( product_prod @ E3 @ ( product_prod @ F @ G2 ) ) ) ) @ B3 @ ( product_Pair @ C @ ( product_prod @ D5 @ ( product_prod @ E3 @ ( product_prod @ F @ G2 ) ) ) @ C4 @ ( product_Pair @ D5 @ ( product_prod @ E3 @ ( product_prod @ F @ G2 ) ) @ D3 @ ( product_Pair @ E3 @ ( product_prod @ F @ G2 ) @ E2 @ ( product_Pair @ F @ G2 @ F3 @ G3 ) ) ) ) ) ) )
=> ( P @ X ) ) ).
% prod_induct7
thf(fact_3119_prod__induct6,axiom,
! [F: $tType,E3: $tType,D5: $tType,C: $tType,B: $tType,A: $tType,P: ( product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D5 @ ( product_prod @ E3 @ F ) ) ) ) ) > $o,X: product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D5 @ ( product_prod @ E3 @ F ) ) ) )] :
( ! [A4: A,B3: B,C4: C,D3: D5,E2: E3,F3: F] : ( P @ ( product_Pair @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D5 @ ( product_prod @ E3 @ F ) ) ) ) @ A4 @ ( product_Pair @ B @ ( product_prod @ C @ ( product_prod @ D5 @ ( product_prod @ E3 @ F ) ) ) @ B3 @ ( product_Pair @ C @ ( product_prod @ D5 @ ( product_prod @ E3 @ F ) ) @ C4 @ ( product_Pair @ D5 @ ( product_prod @ E3 @ F ) @ D3 @ ( product_Pair @ E3 @ F @ E2 @ F3 ) ) ) ) ) )
=> ( P @ X ) ) ).
% prod_induct6
thf(fact_3120_prod__induct5,axiom,
! [E3: $tType,D5: $tType,C: $tType,B: $tType,A: $tType,P: ( product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D5 @ E3 ) ) ) ) > $o,X: product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D5 @ E3 ) ) )] :
( ! [A4: A,B3: B,C4: C,D3: D5,E2: E3] : ( P @ ( product_Pair @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D5 @ E3 ) ) ) @ A4 @ ( product_Pair @ B @ ( product_prod @ C @ ( product_prod @ D5 @ E3 ) ) @ B3 @ ( product_Pair @ C @ ( product_prod @ D5 @ E3 ) @ C4 @ ( product_Pair @ D5 @ E3 @ D3 @ E2 ) ) ) ) )
=> ( P @ X ) ) ).
% prod_induct5
thf(fact_3121_prod__induct4,axiom,
! [D5: $tType,C: $tType,B: $tType,A: $tType,P: ( product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ D5 ) ) ) > $o,X: product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ D5 ) )] :
( ! [A4: A,B3: B,C4: C,D3: D5] : ( P @ ( product_Pair @ A @ ( product_prod @ B @ ( product_prod @ C @ D5 ) ) @ A4 @ ( product_Pair @ B @ ( product_prod @ C @ D5 ) @ B3 @ ( product_Pair @ C @ D5 @ C4 @ D3 ) ) ) )
=> ( P @ X ) ) ).
% prod_induct4
thf(fact_3122_prod__cases7,axiom,
! [A: $tType,B: $tType,C: $tType,D5: $tType,E3: $tType,F: $tType,G2: $tType,Y: product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D5 @ ( product_prod @ E3 @ ( product_prod @ F @ G2 ) ) ) ) )] :
~ ! [A4: A,B3: B,C4: C,D3: D5,E2: E3,F3: F,G3: G2] :
( Y
!= ( product_Pair @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D5 @ ( product_prod @ E3 @ ( product_prod @ F @ G2 ) ) ) ) ) @ A4 @ ( product_Pair @ B @ ( product_prod @ C @ ( product_prod @ D5 @ ( product_prod @ E3 @ ( product_prod @ F @ G2 ) ) ) ) @ B3 @ ( product_Pair @ C @ ( product_prod @ D5 @ ( product_prod @ E3 @ ( product_prod @ F @ G2 ) ) ) @ C4 @ ( product_Pair @ D5 @ ( product_prod @ E3 @ ( product_prod @ F @ G2 ) ) @ D3 @ ( product_Pair @ E3 @ ( product_prod @ F @ G2 ) @ E2 @ ( product_Pair @ F @ G2 @ F3 @ G3 ) ) ) ) ) ) ) ).
% prod_cases7
thf(fact_3123_prod__cases6,axiom,
! [A: $tType,B: $tType,C: $tType,D5: $tType,E3: $tType,F: $tType,Y: product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D5 @ ( product_prod @ E3 @ F ) ) ) )] :
~ ! [A4: A,B3: B,C4: C,D3: D5,E2: E3,F3: F] :
( Y
!= ( product_Pair @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D5 @ ( product_prod @ E3 @ F ) ) ) ) @ A4 @ ( product_Pair @ B @ ( product_prod @ C @ ( product_prod @ D5 @ ( product_prod @ E3 @ F ) ) ) @ B3 @ ( product_Pair @ C @ ( product_prod @ D5 @ ( product_prod @ E3 @ F ) ) @ C4 @ ( product_Pair @ D5 @ ( product_prod @ E3 @ F ) @ D3 @ ( product_Pair @ E3 @ F @ E2 @ F3 ) ) ) ) ) ) ).
% prod_cases6
thf(fact_3124_prod__cases5,axiom,
! [A: $tType,B: $tType,C: $tType,D5: $tType,E3: $tType,Y: product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D5 @ E3 ) ) )] :
~ ! [A4: A,B3: B,C4: C,D3: D5,E2: E3] :
( Y
!= ( product_Pair @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D5 @ E3 ) ) ) @ A4 @ ( product_Pair @ B @ ( product_prod @ C @ ( product_prod @ D5 @ E3 ) ) @ B3 @ ( product_Pair @ C @ ( product_prod @ D5 @ E3 ) @ C4 @ ( product_Pair @ D5 @ E3 @ D3 @ E2 ) ) ) ) ) ).
% prod_cases5
thf(fact_3125_sum__bounds__lt__plus1,axiom,
! [A: $tType] :
( ( comm_monoid_add @ A )
=> ! [F2: nat > A,Mm: nat] :
( ( groups7311177749621191930dd_sum @ nat @ A
@ ^ [K2: nat] : ( F2 @ ( suc @ K2 ) )
@ ( set_ord_lessThan @ nat @ Mm ) )
= ( groups7311177749621191930dd_sum @ nat @ A @ F2 @ ( set_or1337092689740270186AtMost @ nat @ ( one_one @ nat ) @ Mm ) ) ) ) ).
% sum_bounds_lt_plus1
thf(fact_3126_pochhammer__times__pochhammer__half,axiom,
! [A: $tType] :
( ( field_char_0 @ A )
=> ! [Z: A,N3: nat] :
( ( times_times @ A @ ( comm_s3205402744901411588hammer @ A @ Z @ ( suc @ N3 ) ) @ ( comm_s3205402744901411588hammer @ A @ ( plus_plus @ A @ Z @ ( divide_divide @ A @ ( one_one @ A ) @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) ) @ ( suc @ N3 ) ) )
= ( groups7121269368397514597t_prod @ nat @ A
@ ^ [K2: nat] : ( plus_plus @ A @ Z @ ( divide_divide @ A @ ( semiring_1_of_nat @ A @ K2 ) @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) )
@ ( set_or1337092689740270186AtMost @ nat @ ( zero_zero @ nat ) @ ( plus_plus @ nat @ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N3 ) @ ( one_one @ nat ) ) ) ) ) ) ).
% pochhammer_times_pochhammer_half
thf(fact_3127_sumr__cos__zero__one,axiom,
! [N3: nat] :
( ( groups7311177749621191930dd_sum @ nat @ real
@ ^ [M5: nat] : ( times_times @ real @ ( cos_coeff @ M5 ) @ ( power_power @ real @ ( zero_zero @ real ) @ M5 ) )
@ ( set_ord_lessThan @ nat @ ( suc @ N3 ) ) )
= ( one_one @ real ) ) ).
% sumr_cos_zero_one
thf(fact_3128_pochhammer__code,axiom,
! [A: $tType] :
( ( comm_semiring_1 @ A )
=> ( ( comm_s3205402744901411588hammer @ A )
= ( ^ [A6: A,N2: nat] :
( if @ A
@ ( N2
= ( 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 @ N2 @ ( one_one @ nat ) )
@ ( one_one @ A ) ) ) ) ) ) ).
% pochhammer_code
thf(fact_3129_finite__nat__iff__bounded,axiom,
( ( finite_finite2 @ nat )
= ( ^ [S8: set @ nat] :
? [K2: nat] : ( ord_less_eq @ ( set @ nat ) @ S8 @ ( set_ord_lessThan @ nat @ K2 ) ) ) ) ).
% finite_nat_iff_bounded
thf(fact_3130_prod_Oneutral__const,axiom,
! [B: $tType,A: $tType] :
( ( comm_monoid_mult @ A )
=> ! [A3: set @ B] :
( ( groups7121269368397514597t_prod @ B @ A
@ ^ [Uu3: B] : ( one_one @ A )
@ A3 )
= ( one_one @ A ) ) ) ).
% prod.neutral_const
thf(fact_3131_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_3132_prod_Oinfinite,axiom,
! [B: $tType,A: $tType] :
( ( comm_monoid_mult @ A )
=> ! [A3: set @ B,G: B > A] :
( ~ ( finite_finite2 @ B @ A3 )
=> ( ( groups7121269368397514597t_prod @ B @ A @ G @ A3 )
= ( one_one @ A ) ) ) ) ).
% prod.infinite
thf(fact_3133_cos__coeff__0,axiom,
( ( cos_coeff @ ( zero_zero @ nat ) )
= ( one_one @ real ) ) ).
% cos_coeff_0
thf(fact_3134_prod_Odelta_H,axiom,
! [B: $tType,A: $tType] :
( ( comm_monoid_mult @ A )
=> ! [S3: set @ B,A2: B,B2: B > A] :
( ( finite_finite2 @ B @ S3 )
=> ( ( ( member @ B @ A2 @ S3 )
=> ( ( groups7121269368397514597t_prod @ B @ A
@ ^ [K2: B] : ( if @ A @ ( A2 = K2 ) @ ( B2 @ K2 ) @ ( one_one @ A ) )
@ S3 )
= ( B2 @ A2 ) ) )
& ( ~ ( member @ B @ A2 @ S3 )
=> ( ( groups7121269368397514597t_prod @ B @ A
@ ^ [K2: B] : ( if @ A @ ( A2 = K2 ) @ ( B2 @ K2 ) @ ( one_one @ A ) )
@ S3 )
= ( one_one @ A ) ) ) ) ) ) ).
% prod.delta'
thf(fact_3135_prod_Odelta,axiom,
! [B: $tType,A: $tType] :
( ( comm_monoid_mult @ A )
=> ! [S3: set @ B,A2: B,B2: B > A] :
( ( finite_finite2 @ B @ S3 )
=> ( ( ( member @ B @ A2 @ S3 )
=> ( ( groups7121269368397514597t_prod @ B @ A
@ ^ [K2: B] : ( if @ A @ ( K2 = A2 ) @ ( B2 @ K2 ) @ ( one_one @ A ) )
@ S3 )
= ( B2 @ A2 ) ) )
& ( ~ ( member @ B @ A2 @ S3 )
=> ( ( groups7121269368397514597t_prod @ B @ A
@ ^ [K2: B] : ( if @ A @ ( K2 = A2 ) @ ( B2 @ K2 ) @ ( one_one @ A ) )
@ S3 )
= ( one_one @ A ) ) ) ) ) ) ).
% prod.delta
thf(fact_3136_prod_Oinsert,axiom,
! [A: $tType,B: $tType] :
( ( comm_monoid_mult @ A )
=> ! [A3: set @ B,X: B,G: B > A] :
( ( finite_finite2 @ B @ A3 )
=> ( ~ ( member @ B @ X @ A3 )
=> ( ( groups7121269368397514597t_prod @ B @ A @ G @ ( insert @ B @ X @ A3 ) )
= ( times_times @ A @ ( G @ X ) @ ( groups7121269368397514597t_prod @ B @ A @ G @ A3 ) ) ) ) ) ) ).
% prod.insert
thf(fact_3137_prod_OlessThan__Suc,axiom,
! [A: $tType] :
( ( comm_monoid_mult @ A )
=> ! [G: nat > A,N3: nat] :
( ( groups7121269368397514597t_prod @ nat @ A @ G @ ( set_ord_lessThan @ nat @ ( suc @ N3 ) ) )
= ( times_times @ A @ ( groups7121269368397514597t_prod @ nat @ A @ G @ ( set_ord_lessThan @ nat @ N3 ) ) @ ( G @ N3 ) ) ) ) ).
% prod.lessThan_Suc
thf(fact_3138_prod_Ocl__ivl__Suc,axiom,
! [A: $tType] :
( ( comm_monoid_mult @ A )
=> ! [N3: nat,M: nat,G: nat > A] :
( ( ( ord_less @ nat @ ( suc @ N3 ) @ M )
=> ( ( groups7121269368397514597t_prod @ nat @ A @ G @ ( set_or1337092689740270186AtMost @ nat @ M @ ( suc @ N3 ) ) )
= ( one_one @ A ) ) )
& ( ~ ( ord_less @ nat @ ( suc @ N3 ) @ M )
=> ( ( groups7121269368397514597t_prod @ nat @ A @ G @ ( set_or1337092689740270186AtMost @ nat @ M @ ( suc @ N3 ) ) )
= ( times_times @ A @ ( groups7121269368397514597t_prod @ nat @ A @ G @ ( set_or1337092689740270186AtMost @ nat @ M @ N3 ) ) @ ( G @ ( suc @ N3 ) ) ) ) ) ) ) ).
% prod.cl_ivl_Suc
thf(fact_3139_prod_Oneutral,axiom,
! [B: $tType,A: $tType] :
( ( comm_monoid_mult @ A )
=> ! [A3: set @ B,G: B > A] :
( ! [X3: B] :
( ( member @ B @ X3 @ A3 )
=> ( ( G @ X3 )
= ( one_one @ A ) ) )
=> ( ( groups7121269368397514597t_prod @ B @ A @ G @ A3 )
= ( one_one @ A ) ) ) ) ).
% prod.neutral
thf(fact_3140_prod_Onot__neutral__contains__not__neutral,axiom,
! [B: $tType,A: $tType] :
( ( comm_monoid_mult @ A )
=> ! [G: B > A,A3: set @ B] :
( ( ( groups7121269368397514597t_prod @ B @ A @ G @ A3 )
!= ( one_one @ A ) )
=> ~ ! [A4: B] :
( ( member @ B @ A4 @ A3 )
=> ( ( G @ A4 )
= ( one_one @ A ) ) ) ) ) ).
% prod.not_neutral_contains_not_neutral
thf(fact_3141_norm__prod__le,axiom,
! [A: $tType,B: $tType] :
( ( ( comm_monoid_mult @ A )
& ( real_V2822296259951069270ebra_1 @ A ) )
=> ! [F2: B > A,A3: set @ B] :
( ord_less_eq @ real @ ( real_V7770717601297561774m_norm @ A @ ( groups7121269368397514597t_prod @ B @ A @ F2 @ A3 ) )
@ ( groups7121269368397514597t_prod @ B @ real
@ ^ [A6: B] : ( real_V7770717601297561774m_norm @ A @ ( F2 @ A6 ) )
@ A3 ) ) ) ).
% norm_prod_le
thf(fact_3142_prod_Odistrib,axiom,
! [A: $tType,B: $tType] :
( ( comm_monoid_mult @ A )
=> ! [G: B > A,H2: B > A,A3: set @ B] :
( ( groups7121269368397514597t_prod @ B @ A
@ ^ [X2: B] : ( times_times @ A @ ( G @ X2 ) @ ( H2 @ X2 ) )
@ A3 )
= ( times_times @ A @ ( groups7121269368397514597t_prod @ B @ A @ G @ A3 ) @ ( groups7121269368397514597t_prod @ B @ A @ H2 @ A3 ) ) ) ) ).
% prod.distrib
thf(fact_3143_prod__dividef,axiom,
! [A: $tType,B: $tType] :
( ( field @ A )
=> ! [F2: B > A,G: B > A,A3: set @ B] :
( ( groups7121269368397514597t_prod @ B @ A
@ ^ [X2: B] : ( divide_divide @ A @ ( F2 @ X2 ) @ ( G @ X2 ) )
@ A3 )
= ( divide_divide @ A @ ( groups7121269368397514597t_prod @ B @ A @ F2 @ A3 ) @ ( groups7121269368397514597t_prod @ B @ A @ G @ A3 ) ) ) ) ).
% prod_dividef
thf(fact_3144_prod__power__distrib,axiom,
! [B: $tType,A: $tType] :
( ( comm_semiring_1 @ B )
=> ! [F2: A > B,A3: set @ A,N3: nat] :
( ( power_power @ B @ ( groups7121269368397514597t_prod @ A @ B @ F2 @ A3 ) @ N3 )
= ( groups7121269368397514597t_prod @ A @ B
@ ^ [X2: A] : ( power_power @ B @ ( F2 @ X2 ) @ N3 )
@ A3 ) ) ) ).
% prod_power_distrib
thf(fact_3145_mod__prod__eq,axiom,
! [B: $tType,A: $tType] :
( ( euclid4440199948858584721cancel @ A )
=> ! [F2: B > A,A2: A,A3: set @ B] :
( ( modulo_modulo @ A
@ ( groups7121269368397514597t_prod @ B @ A
@ ^ [I3: B] : ( modulo_modulo @ A @ ( F2 @ I3 ) @ A2 )
@ A3 )
@ A2 )
= ( modulo_modulo @ A @ ( groups7121269368397514597t_prod @ B @ A @ F2 @ A3 ) @ A2 ) ) ) ).
% mod_prod_eq
thf(fact_3146_prod__mono,axiom,
! [A: $tType,B: $tType] :
( ( linordered_semidom @ A )
=> ! [A3: set @ B,F2: B > A,G: B > A] :
( ! [I2: B] :
( ( member @ B @ I2 @ A3 )
=> ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( F2 @ I2 ) )
& ( ord_less_eq @ A @ ( F2 @ I2 ) @ ( G @ I2 ) ) ) )
=> ( ord_less_eq @ A @ ( groups7121269368397514597t_prod @ B @ A @ F2 @ A3 ) @ ( groups7121269368397514597t_prod @ B @ A @ G @ A3 ) ) ) ) ).
% prod_mono
thf(fact_3147_prod__nonneg,axiom,
! [A: $tType,B: $tType] :
( ( linordered_semidom @ A )
=> ! [A3: set @ B,F2: B > A] :
( ! [X3: B] :
( ( member @ B @ X3 @ A3 )
=> ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( F2 @ X3 ) ) )
=> ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( groups7121269368397514597t_prod @ B @ A @ F2 @ A3 ) ) ) ) ).
% prod_nonneg
thf(fact_3148_prod__pos,axiom,
! [A: $tType,B: $tType] :
( ( linordered_semidom @ A )
=> ! [A3: set @ B,F2: B > A] :
( ! [X3: B] :
( ( member @ B @ X3 @ A3 )
=> ( ord_less @ A @ ( zero_zero @ A ) @ ( F2 @ X3 ) ) )
=> ( ord_less @ A @ ( zero_zero @ A ) @ ( groups7121269368397514597t_prod @ B @ A @ F2 @ A3 ) ) ) ) ).
% prod_pos
thf(fact_3149_prod__ge__1,axiom,
! [A: $tType,B: $tType] :
( ( linord181362715937106298miring @ A )
=> ! [A3: set @ B,F2: B > A] :
( ! [X3: B] :
( ( member @ B @ X3 @ A3 )
=> ( ord_less_eq @ A @ ( one_one @ A ) @ ( F2 @ X3 ) ) )
=> ( ord_less_eq @ A @ ( one_one @ A ) @ ( groups7121269368397514597t_prod @ B @ A @ F2 @ A3 ) ) ) ) ).
% prod_ge_1
thf(fact_3150_prod__atLeastAtMost__code,axiom,
! [A: $tType] :
( ( comm_monoid_mult @ A )
=> ! [F2: nat > A,A2: nat,B2: nat] :
( ( groups7121269368397514597t_prod @ nat @ A @ F2 @ ( set_or1337092689740270186AtMost @ nat @ A2 @ B2 ) )
= ( set_fo6178422350223883121st_nat @ A
@ ^ [A6: nat] : ( times_times @ A @ ( F2 @ A6 ) )
@ A2
@ B2
@ ( one_one @ A ) ) ) ) ).
% prod_atLeastAtMost_code
thf(fact_3151_prod_Ointer__filter,axiom,
! [A: $tType,B: $tType] :
( ( comm_monoid_mult @ A )
=> ! [A3: set @ B,G: B > A,P: B > $o] :
( ( finite_finite2 @ B @ A3 )
=> ( ( groups7121269368397514597t_prod @ B @ A @ G
@ ( collect @ B
@ ^ [X2: B] :
( ( member @ B @ X2 @ A3 )
& ( P @ X2 ) ) ) )
= ( groups7121269368397514597t_prod @ B @ A
@ ^ [X2: B] : ( if @ A @ ( P @ X2 ) @ ( G @ X2 ) @ ( one_one @ A ) )
@ A3 ) ) ) ) ).
% prod.inter_filter
thf(fact_3152_prod_Oshift__bounds__cl__Suc__ivl,axiom,
! [A: $tType] :
( ( comm_monoid_mult @ A )
=> ! [G: nat > A,M: nat,N3: nat] :
( ( groups7121269368397514597t_prod @ nat @ A @ G @ ( set_or1337092689740270186AtMost @ nat @ ( suc @ M ) @ ( suc @ N3 ) ) )
= ( groups7121269368397514597t_prod @ nat @ A
@ ^ [I3: nat] : ( G @ ( suc @ I3 ) )
@ ( set_or1337092689740270186AtMost @ nat @ M @ N3 ) ) ) ) ).
% prod.shift_bounds_cl_Suc_ivl
thf(fact_3153_power__sum,axiom,
! [A: $tType,B: $tType] :
( ( comm_monoid_mult @ A )
=> ! [C2: A,F2: B > nat,A3: set @ B] :
( ( power_power @ A @ C2 @ ( groups7311177749621191930dd_sum @ B @ nat @ F2 @ A3 ) )
= ( groups7121269368397514597t_prod @ B @ A
@ ^ [A6: B] : ( power_power @ A @ C2 @ ( F2 @ A6 ) )
@ A3 ) ) ) ).
% power_sum
thf(fact_3154_prod_Oshift__bounds__cl__nat__ivl,axiom,
! [A: $tType] :
( ( comm_monoid_mult @ A )
=> ! [G: nat > A,M: nat,K: nat,N3: nat] :
( ( groups7121269368397514597t_prod @ nat @ A @ G @ ( set_or1337092689740270186AtMost @ nat @ ( plus_plus @ nat @ M @ K ) @ ( plus_plus @ nat @ N3 @ K ) ) )
= ( groups7121269368397514597t_prod @ nat @ A
@ ^ [I3: nat] : ( G @ ( plus_plus @ nat @ I3 @ K ) )
@ ( set_or1337092689740270186AtMost @ nat @ M @ N3 ) ) ) ) ).
% prod.shift_bounds_cl_nat_ivl
thf(fact_3155_prod__le__1,axiom,
! [B: $tType,A: $tType] :
( ( linord181362715937106298miring @ A )
=> ! [A3: set @ B,F2: B > A] :
( ! [X3: B] :
( ( member @ B @ X3 @ A3 )
=> ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( F2 @ X3 ) )
& ( ord_less_eq @ A @ ( F2 @ X3 ) @ ( one_one @ A ) ) ) )
=> ( ord_less_eq @ A @ ( groups7121269368397514597t_prod @ B @ A @ F2 @ A3 ) @ ( one_one @ A ) ) ) ) ).
% prod_le_1
thf(fact_3156_prod_Orelated,axiom,
! [A: $tType,B: $tType] :
( ( comm_monoid_mult @ A )
=> ! [R: A > A > $o,S3: set @ B,H2: B > A,G: B > A] :
( ( R @ ( one_one @ A ) @ ( one_one @ A ) )
=> ( ! [X1: A,Y1: A,X23: A,Y23: A] :
( ( ( R @ X1 @ X23 )
& ( R @ Y1 @ Y23 ) )
=> ( R @ ( times_times @ A @ X1 @ Y1 ) @ ( times_times @ A @ X23 @ Y23 ) ) )
=> ( ( finite_finite2 @ B @ S3 )
=> ( ! [X3: B] :
( ( member @ B @ X3 @ S3 )
=> ( R @ ( H2 @ X3 ) @ ( G @ X3 ) ) )
=> ( R @ ( groups7121269368397514597t_prod @ B @ A @ H2 @ S3 ) @ ( groups7121269368397514597t_prod @ B @ A @ G @ S3 ) ) ) ) ) ) ) ).
% prod.related
thf(fact_3157_prod_Oinsert__if,axiom,
! [A: $tType,B: $tType] :
( ( comm_monoid_mult @ A )
=> ! [A3: set @ B,X: B,G: B > A] :
( ( finite_finite2 @ B @ A3 )
=> ( ( ( member @ B @ X @ A3 )
=> ( ( groups7121269368397514597t_prod @ B @ A @ G @ ( insert @ B @ X @ A3 ) )
= ( groups7121269368397514597t_prod @ B @ A @ G @ A3 ) ) )
& ( ~ ( member @ B @ X @ A3 )
=> ( ( groups7121269368397514597t_prod @ B @ A @ G @ ( insert @ B @ X @ A3 ) )
= ( times_times @ A @ ( G @ X ) @ ( groups7121269368397514597t_prod @ B @ A @ G @ A3 ) ) ) ) ) ) ) ).
% prod.insert_if
thf(fact_3158_prod_Oreindex__bij__witness__not__neutral,axiom,
! [B: $tType,A: $tType,C: $tType] :
( ( comm_monoid_mult @ A )
=> ! [S7: set @ B,T7: set @ C,S3: set @ B,I: C > B,J: B > C,T5: 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 ) @ S3 @ S7 ) )
=> ( ( I @ ( J @ A4 ) )
= A4 ) )
=> ( ! [A4: B] :
( ( member @ B @ A4 @ ( minus_minus @ ( set @ B ) @ S3 @ S7 ) )
=> ( member @ C @ ( J @ A4 ) @ ( minus_minus @ ( set @ C ) @ T5 @ T7 ) ) )
=> ( ! [B3: C] :
( ( member @ C @ B3 @ ( minus_minus @ ( set @ C ) @ T5 @ T7 ) )
=> ( ( J @ ( I @ B3 ) )
= B3 ) )
=> ( ! [B3: C] :
( ( member @ C @ B3 @ ( minus_minus @ ( set @ C ) @ T5 @ T7 ) )
=> ( member @ B @ ( I @ B3 ) @ ( minus_minus @ ( set @ B ) @ S3 @ S7 ) ) )
=> ( ! [A4: B] :
( ( member @ B @ A4 @ S7 )
=> ( ( G @ A4 )
= ( one_one @ A ) ) )
=> ( ! [B3: C] :
( ( member @ C @ B3 @ T7 )
=> ( ( H2 @ B3 )
= ( one_one @ A ) ) )
=> ( ! [A4: B] :
( ( member @ B @ A4 @ S3 )
=> ( ( H2 @ ( J @ A4 ) )
= ( G @ A4 ) ) )
=> ( ( groups7121269368397514597t_prod @ B @ A @ G @ S3 )
= ( groups7121269368397514597t_prod @ C @ A @ H2 @ T5 ) ) ) ) ) ) ) ) ) ) ) ) ).
% prod.reindex_bij_witness_not_neutral
thf(fact_3159_prod__dvd__prod__subset2,axiom,
! [A: $tType,B: $tType] :
( ( comm_semiring_1 @ A )
=> ! [B6: set @ B,A3: set @ B,F2: B > A,G: B > A] :
( ( finite_finite2 @ B @ B6 )
=> ( ( ord_less_eq @ ( set @ B ) @ A3 @ B6 )
=> ( ! [A4: B] :
( ( member @ B @ A4 @ A3 )
=> ( dvd_dvd @ A @ ( F2 @ A4 ) @ ( G @ A4 ) ) )
=> ( dvd_dvd @ A @ ( groups7121269368397514597t_prod @ B @ A @ F2 @ A3 ) @ ( groups7121269368397514597t_prod @ B @ A @ G @ B6 ) ) ) ) ) ) ).
% prod_dvd_prod_subset2
thf(fact_3160_prod__dvd__prod__subset,axiom,
! [A: $tType,B: $tType] :
( ( comm_monoid_mult @ A )
=> ! [B6: set @ B,A3: set @ B,F2: B > A] :
( ( finite_finite2 @ B @ B6 )
=> ( ( ord_less_eq @ ( set @ B ) @ A3 @ B6 )
=> ( dvd_dvd @ A @ ( groups7121269368397514597t_prod @ B @ A @ F2 @ A3 ) @ ( groups7121269368397514597t_prod @ B @ A @ F2 @ B6 ) ) ) ) ) ).
% prod_dvd_prod_subset
thf(fact_3161_prod_Osetdiff__irrelevant,axiom,
! [A: $tType,B: $tType] :
( ( comm_monoid_mult @ A )
=> ! [A3: set @ B,G: B > A] :
( ( finite_finite2 @ B @ A3 )
=> ( ( groups7121269368397514597t_prod @ B @ A @ G
@ ( minus_minus @ ( set @ B ) @ A3
@ ( collect @ B
@ ^ [X2: B] :
( ( G @ X2 )
= ( one_one @ A ) ) ) ) )
= ( groups7121269368397514597t_prod @ B @ A @ G @ A3 ) ) ) ) ).
% prod.setdiff_irrelevant
thf(fact_3162_prod_Onat__diff__reindex,axiom,
! [A: $tType] :
( ( comm_monoid_mult @ A )
=> ! [G: nat > A,N3: nat] :
( ( groups7121269368397514597t_prod @ nat @ A
@ ^ [I3: nat] : ( G @ ( minus_minus @ nat @ N3 @ ( suc @ I3 ) ) )
@ ( set_ord_lessThan @ nat @ N3 ) )
= ( groups7121269368397514597t_prod @ nat @ A @ G @ ( set_ord_lessThan @ nat @ N3 ) ) ) ) ).
% prod.nat_diff_reindex
thf(fact_3163_prod_OatLeastAtMost__rev,axiom,
! [A: $tType] :
( ( comm_monoid_mult @ A )
=> ! [G: nat > A,N3: nat,M: nat] :
( ( groups7121269368397514597t_prod @ nat @ A @ G @ ( set_or1337092689740270186AtMost @ nat @ N3 @ M ) )
= ( groups7121269368397514597t_prod @ nat @ A
@ ^ [I3: nat] : ( G @ ( minus_minus @ nat @ ( plus_plus @ nat @ M @ N3 ) @ I3 ) )
@ ( set_or1337092689740270186AtMost @ nat @ N3 @ M ) ) ) ) ).
% prod.atLeastAtMost_rev
thf(fact_3164_less__1__prod2,axiom,
! [B: $tType,A: $tType] :
( ( linordered_idom @ B )
=> ! [I5: set @ A,I: A,F2: A > B] :
( ( finite_finite2 @ A @ I5 )
=> ( ( member @ A @ I @ I5 )
=> ( ( ord_less @ B @ ( one_one @ B ) @ ( F2 @ I ) )
=> ( ! [I2: A] :
( ( member @ A @ I2 @ I5 )
=> ( ord_less_eq @ B @ ( one_one @ B ) @ ( F2 @ I2 ) ) )
=> ( ord_less @ B @ ( one_one @ B ) @ ( groups7121269368397514597t_prod @ A @ B @ F2 @ I5 ) ) ) ) ) ) ) ).
% less_1_prod2
thf(fact_3165_less__1__prod,axiom,
! [B: $tType,A: $tType] :
( ( linordered_idom @ B )
=> ! [I5: set @ A,F2: A > B] :
( ( finite_finite2 @ A @ I5 )
=> ( ( I5
!= ( bot_bot @ ( set @ A ) ) )
=> ( ! [I2: A] :
( ( member @ A @ I2 @ I5 )
=> ( ord_less @ B @ ( one_one @ B ) @ ( F2 @ I2 ) ) )
=> ( ord_less @ B @ ( one_one @ B ) @ ( groups7121269368397514597t_prod @ A @ B @ F2 @ I5 ) ) ) ) ) ) ).
% less_1_prod
thf(fact_3166_prod_Omono__neutral__cong__right,axiom,
! [A: $tType,B: $tType] :
( ( comm_monoid_mult @ A )
=> ! [T5: set @ B,S3: set @ B,G: B > A,H2: B > A] :
( ( finite_finite2 @ B @ T5 )
=> ( ( ord_less_eq @ ( set @ B ) @ S3 @ T5 )
=> ( ! [X3: B] :
( ( member @ B @ X3 @ ( minus_minus @ ( set @ B ) @ T5 @ S3 ) )
=> ( ( G @ X3 )
= ( one_one @ A ) ) )
=> ( ! [X3: B] :
( ( member @ B @ X3 @ S3 )
=> ( ( G @ X3 )
= ( H2 @ X3 ) ) )
=> ( ( groups7121269368397514597t_prod @ B @ A @ G @ T5 )
= ( groups7121269368397514597t_prod @ B @ A @ H2 @ S3 ) ) ) ) ) ) ) ).
% prod.mono_neutral_cong_right
thf(fact_3167_prod_Omono__neutral__cong__left,axiom,
! [A: $tType,B: $tType] :
( ( comm_monoid_mult @ A )
=> ! [T5: set @ B,S3: set @ B,H2: B > A,G: B > A] :
( ( finite_finite2 @ B @ T5 )
=> ( ( ord_less_eq @ ( set @ B ) @ S3 @ T5 )
=> ( ! [X3: B] :
( ( member @ B @ X3 @ ( minus_minus @ ( set @ B ) @ T5 @ S3 ) )
=> ( ( H2 @ X3 )
= ( one_one @ A ) ) )
=> ( ! [X3: B] :
( ( member @ B @ X3 @ S3 )
=> ( ( G @ X3 )
= ( H2 @ X3 ) ) )
=> ( ( groups7121269368397514597t_prod @ B @ A @ G @ S3 )
= ( groups7121269368397514597t_prod @ B @ A @ H2 @ T5 ) ) ) ) ) ) ) ).
% prod.mono_neutral_cong_left
thf(fact_3168_prod_Omono__neutral__right,axiom,
! [A: $tType,B: $tType] :
( ( comm_monoid_mult @ A )
=> ! [T5: set @ B,S3: set @ B,G: B > A] :
( ( finite_finite2 @ B @ T5 )
=> ( ( ord_less_eq @ ( set @ B ) @ S3 @ T5 )
=> ( ! [X3: B] :
( ( member @ B @ X3 @ ( minus_minus @ ( set @ B ) @ T5 @ S3 ) )
=> ( ( G @ X3 )
= ( one_one @ A ) ) )
=> ( ( groups7121269368397514597t_prod @ B @ A @ G @ T5 )
= ( groups7121269368397514597t_prod @ B @ A @ G @ S3 ) ) ) ) ) ) ).
% prod.mono_neutral_right
thf(fact_3169_prod_Omono__neutral__left,axiom,
! [A: $tType,B: $tType] :
( ( comm_monoid_mult @ A )
=> ! [T5: set @ B,S3: set @ B,G: B > A] :
( ( finite_finite2 @ B @ T5 )
=> ( ( ord_less_eq @ ( set @ B ) @ S3 @ T5 )
=> ( ! [X3: B] :
( ( member @ B @ X3 @ ( minus_minus @ ( set @ B ) @ T5 @ S3 ) )
=> ( ( G @ X3 )
= ( one_one @ A ) ) )
=> ( ( groups7121269368397514597t_prod @ B @ A @ G @ S3 )
= ( groups7121269368397514597t_prod @ B @ A @ G @ T5 ) ) ) ) ) ) ).
% prod.mono_neutral_left
thf(fact_3170_prod_Osame__carrierI,axiom,
! [A: $tType,B: $tType] :
( ( comm_monoid_mult @ A )
=> ! [C5: set @ B,A3: set @ B,B6: set @ B,G: B > A,H2: B > A] :
( ( finite_finite2 @ B @ C5 )
=> ( ( ord_less_eq @ ( set @ B ) @ A3 @ C5 )
=> ( ( ord_less_eq @ ( set @ B ) @ B6 @ C5 )
=> ( ! [A4: B] :
( ( member @ B @ A4 @ ( minus_minus @ ( set @ B ) @ C5 @ A3 ) )
=> ( ( G @ A4 )
= ( one_one @ A ) ) )
=> ( ! [B3: B] :
( ( member @ B @ B3 @ ( minus_minus @ ( set @ B ) @ C5 @ B6 ) )
=> ( ( H2 @ B3 )
= ( one_one @ A ) ) )
=> ( ( ( groups7121269368397514597t_prod @ B @ A @ G @ C5 )
= ( groups7121269368397514597t_prod @ B @ A @ H2 @ C5 ) )
=> ( ( groups7121269368397514597t_prod @ B @ A @ G @ A3 )
= ( groups7121269368397514597t_prod @ B @ A @ H2 @ B6 ) ) ) ) ) ) ) ) ) ).
% prod.same_carrierI
thf(fact_3171_prod_Osame__carrier,axiom,
! [A: $tType,B: $tType] :
( ( comm_monoid_mult @ A )
=> ! [C5: set @ B,A3: set @ B,B6: set @ B,G: B > A,H2: B > A] :
( ( finite_finite2 @ B @ C5 )
=> ( ( ord_less_eq @ ( set @ B ) @ A3 @ C5 )
=> ( ( ord_less_eq @ ( set @ B ) @ B6 @ C5 )
=> ( ! [A4: B] :
( ( member @ B @ A4 @ ( minus_minus @ ( set @ B ) @ C5 @ A3 ) )
=> ( ( G @ A4 )
= ( one_one @ A ) ) )
=> ( ! [B3: B] :
( ( member @ B @ B3 @ ( minus_minus @ ( set @ B ) @ C5 @ B6 ) )
=> ( ( H2 @ B3 )
= ( one_one @ A ) ) )
=> ( ( ( groups7121269368397514597t_prod @ B @ A @ G @ A3 )
= ( groups7121269368397514597t_prod @ B @ A @ H2 @ B6 ) )
= ( ( groups7121269368397514597t_prod @ B @ A @ G @ C5 )
= ( groups7121269368397514597t_prod @ B @ A @ H2 @ C5 ) ) ) ) ) ) ) ) ) ).
% prod.same_carrier
thf(fact_3172_prod_Osubset__diff,axiom,
! [A: $tType,B: $tType] :
( ( comm_monoid_mult @ A )
=> ! [B6: set @ B,A3: set @ B,G: B > A] :
( ( ord_less_eq @ ( set @ B ) @ B6 @ A3 )
=> ( ( finite_finite2 @ B @ A3 )
=> ( ( groups7121269368397514597t_prod @ B @ A @ G @ A3 )
= ( times_times @ A @ ( groups7121269368397514597t_prod @ B @ A @ G @ ( minus_minus @ ( set @ B ) @ A3 @ B6 ) ) @ ( groups7121269368397514597t_prod @ B @ A @ G @ B6 ) ) ) ) ) ) ).
% prod.subset_diff
thf(fact_3173_prod_OatLeast0__atMost__Suc,axiom,
! [A: $tType] :
( ( comm_monoid_mult @ A )
=> ! [G: nat > A,N3: nat] :
( ( groups7121269368397514597t_prod @ nat @ A @ G @ ( set_or1337092689740270186AtMost @ nat @ ( zero_zero @ nat ) @ ( suc @ N3 ) ) )
= ( times_times @ A @ ( groups7121269368397514597t_prod @ nat @ A @ G @ ( set_or1337092689740270186AtMost @ nat @ ( zero_zero @ nat ) @ N3 ) ) @ ( G @ ( suc @ N3 ) ) ) ) ) ).
% prod.atLeast0_atMost_Suc
thf(fact_3174_prod_Onat__ivl__Suc_H,axiom,
! [A: $tType] :
( ( comm_monoid_mult @ A )
=> ! [M: nat,N3: nat,G: nat > A] :
( ( ord_less_eq @ nat @ M @ ( suc @ N3 ) )
=> ( ( groups7121269368397514597t_prod @ nat @ A @ G @ ( set_or1337092689740270186AtMost @ nat @ M @ ( suc @ N3 ) ) )
= ( times_times @ A @ ( G @ ( suc @ N3 ) ) @ ( groups7121269368397514597t_prod @ nat @ A @ G @ ( set_or1337092689740270186AtMost @ nat @ M @ N3 ) ) ) ) ) ) ).
% prod.nat_ivl_Suc'
thf(fact_3175_prod_OatLeast__Suc__atMost,axiom,
! [A: $tType] :
( ( comm_monoid_mult @ A )
=> ! [M: nat,N3: nat,G: nat > A] :
( ( ord_less_eq @ nat @ M @ N3 )
=> ( ( groups7121269368397514597t_prod @ nat @ A @ G @ ( set_or1337092689740270186AtMost @ nat @ M @ N3 ) )
= ( times_times @ A @ ( G @ M ) @ ( groups7121269368397514597t_prod @ nat @ A @ G @ ( set_or1337092689740270186AtMost @ nat @ ( suc @ M ) @ N3 ) ) ) ) ) ) ).
% prod.atLeast_Suc_atMost
thf(fact_3176_prod_OlessThan__Suc__shift,axiom,
! [A: $tType] :
( ( comm_monoid_mult @ A )
=> ! [G: nat > A,N3: nat] :
( ( groups7121269368397514597t_prod @ nat @ A @ G @ ( set_ord_lessThan @ nat @ ( suc @ N3 ) ) )
= ( times_times @ A @ ( G @ ( zero_zero @ nat ) )
@ ( groups7121269368397514597t_prod @ nat @ A
@ ^ [I3: nat] : ( G @ ( suc @ I3 ) )
@ ( set_ord_lessThan @ nat @ N3 ) ) ) ) ) ).
% prod.lessThan_Suc_shift
thf(fact_3177_prod_OSuc__reindex__ivl,axiom,
! [A: $tType] :
( ( comm_monoid_mult @ A )
=> ! [M: nat,N3: nat,G: nat > A] :
( ( ord_less_eq @ nat @ M @ N3 )
=> ( ( times_times @ A @ ( groups7121269368397514597t_prod @ nat @ A @ G @ ( set_or1337092689740270186AtMost @ nat @ M @ N3 ) ) @ ( G @ ( suc @ N3 ) ) )
= ( times_times @ A @ ( G @ M )
@ ( groups7121269368397514597t_prod @ nat @ A
@ ^ [I3: nat] : ( G @ ( suc @ I3 ) )
@ ( set_or1337092689740270186AtMost @ nat @ M @ N3 ) ) ) ) ) ) ).
% prod.Suc_reindex_ivl
thf(fact_3178_prod_OatLeast1__atMost__eq,axiom,
! [A: $tType] :
( ( comm_monoid_mult @ A )
=> ! [G: nat > A,N3: nat] :
( ( groups7121269368397514597t_prod @ nat @ A @ G @ ( set_or1337092689740270186AtMost @ nat @ ( suc @ ( zero_zero @ nat ) ) @ N3 ) )
= ( groups7121269368397514597t_prod @ nat @ A
@ ^ [K2: nat] : ( G @ ( suc @ K2 ) )
@ ( set_ord_lessThan @ nat @ N3 ) ) ) ) ).
% prod.atLeast1_atMost_eq
thf(fact_3179_prod__mono__strict,axiom,
! [A: $tType,B: $tType] :
( ( linordered_semidom @ A )
=> ! [A3: set @ B,F2: B > A,G: B > A] :
( ( finite_finite2 @ B @ A3 )
=> ( ! [I2: B] :
( ( member @ B @ I2 @ A3 )
=> ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( F2 @ I2 ) )
& ( ord_less @ A @ ( F2 @ I2 ) @ ( G @ I2 ) ) ) )
=> ( ( A3
!= ( bot_bot @ ( set @ B ) ) )
=> ( ord_less @ A @ ( groups7121269368397514597t_prod @ B @ A @ F2 @ A3 ) @ ( groups7121269368397514597t_prod @ B @ A @ G @ A3 ) ) ) ) ) ) ).
% prod_mono_strict
thf(fact_3180_even__prod__iff,axiom,
! [A: $tType,B: $tType] :
( ( semiring_parity @ A )
=> ! [A3: set @ B,F2: B > A] :
( ( finite_finite2 @ B @ A3 )
=> ( ( dvd_dvd @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ ( groups7121269368397514597t_prod @ B @ A @ F2 @ A3 ) )
= ( ? [X2: B] :
( ( member @ B @ X2 @ A3 )
& ( dvd_dvd @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ ( F2 @ X2 ) ) ) ) ) ) ) ).
% even_prod_iff
thf(fact_3181_prod_Oinsert__remove,axiom,
! [A: $tType,B: $tType] :
( ( comm_monoid_mult @ A )
=> ! [A3: set @ B,G: B > A,X: B] :
( ( finite_finite2 @ B @ A3 )
=> ( ( groups7121269368397514597t_prod @ B @ A @ G @ ( insert @ B @ X @ A3 ) )
= ( times_times @ A @ ( G @ X ) @ ( groups7121269368397514597t_prod @ B @ A @ G @ ( minus_minus @ ( set @ B ) @ A3 @ ( insert @ B @ X @ ( bot_bot @ ( set @ B ) ) ) ) ) ) ) ) ) ).
% prod.insert_remove
thf(fact_3182_prod_Oremove,axiom,
! [A: $tType,B: $tType] :
( ( comm_monoid_mult @ A )
=> ! [A3: set @ B,X: B,G: B > A] :
( ( finite_finite2 @ B @ A3 )
=> ( ( member @ B @ X @ A3 )
=> ( ( groups7121269368397514597t_prod @ B @ A @ G @ A3 )
= ( times_times @ A @ ( G @ X ) @ ( groups7121269368397514597t_prod @ B @ A @ G @ ( minus_minus @ ( set @ B ) @ A3 @ ( insert @ B @ X @ ( bot_bot @ ( set @ B ) ) ) ) ) ) ) ) ) ) ).
% prod.remove
thf(fact_3183_prod_Oub__add__nat,axiom,
! [A: $tType] :
( ( comm_monoid_mult @ A )
=> ! [M: nat,N3: nat,G: nat > A,P4: nat] :
( ( ord_less_eq @ nat @ M @ ( plus_plus @ nat @ N3 @ ( one_one @ nat ) ) )
=> ( ( groups7121269368397514597t_prod @ nat @ A @ G @ ( set_or1337092689740270186AtMost @ nat @ M @ ( plus_plus @ nat @ N3 @ P4 ) ) )
= ( times_times @ A @ ( groups7121269368397514597t_prod @ nat @ A @ G @ ( set_or1337092689740270186AtMost @ nat @ M @ N3 ) ) @ ( groups7121269368397514597t_prod @ nat @ A @ G @ ( set_or1337092689740270186AtMost @ nat @ ( plus_plus @ nat @ N3 @ ( one_one @ nat ) ) @ ( plus_plus @ nat @ N3 @ P4 ) ) ) ) ) ) ) ).
% prod.ub_add_nat
thf(fact_3184_fold__atLeastAtMost__nat_Oelims,axiom,
! [A: $tType,X: nat > A > A,Xa: nat,Xb: nat,Xc: A,Y: A] :
( ( ( set_fo6178422350223883121st_nat @ A @ X @ Xa @ Xb @ Xc )
= Y )
=> ( ( ( ord_less @ nat @ Xb @ Xa )
=> ( Y = Xc ) )
& ( ~ ( ord_less @ nat @ Xb @ Xa )
=> ( Y
= ( set_fo6178422350223883121st_nat @ A @ X @ ( plus_plus @ nat @ Xa @ ( one_one @ nat ) ) @ Xb @ ( X @ Xa @ Xc ) ) ) ) ) ) ).
% fold_atLeastAtMost_nat.elims
thf(fact_3185_fold__atLeastAtMost__nat_Osimps,axiom,
! [A: $tType] :
( ( set_fo6178422350223883121st_nat @ A )
= ( ^ [F7: nat > A > A,A6: nat,B5: nat,Acc2: A] : ( if @ A @ ( ord_less @ nat @ B5 @ A6 ) @ Acc2 @ ( set_fo6178422350223883121st_nat @ A @ F7 @ ( plus_plus @ nat @ A6 @ ( one_one @ nat ) ) @ B5 @ ( F7 @ A6 @ Acc2 ) ) ) ) ) ).
% fold_atLeastAtMost_nat.simps
thf(fact_3186_prod_Odelta__remove,axiom,
! [A: $tType,B: $tType] :
( ( comm_monoid_mult @ A )
=> ! [S3: set @ B,A2: B,B2: B > A,C2: B > A] :
( ( finite_finite2 @ B @ S3 )
=> ( ( ( member @ B @ A2 @ S3 )
=> ( ( groups7121269368397514597t_prod @ B @ A
@ ^ [K2: B] : ( if @ A @ ( K2 = A2 ) @ ( B2 @ K2 ) @ ( C2 @ K2 ) )
@ S3 )
= ( times_times @ A @ ( B2 @ A2 ) @ ( groups7121269368397514597t_prod @ B @ A @ C2 @ ( minus_minus @ ( set @ B ) @ S3 @ ( insert @ B @ A2 @ ( bot_bot @ ( set @ B ) ) ) ) ) ) ) )
& ( ~ ( member @ B @ A2 @ S3 )
=> ( ( groups7121269368397514597t_prod @ B @ A
@ ^ [K2: B] : ( if @ A @ ( K2 = A2 ) @ ( B2 @ K2 ) @ ( C2 @ K2 ) )
@ S3 )
= ( groups7121269368397514597t_prod @ B @ A @ C2 @ ( minus_minus @ ( set @ B ) @ S3 @ ( insert @ B @ A2 @ ( bot_bot @ ( set @ B ) ) ) ) ) ) ) ) ) ) ).
% prod.delta_remove
thf(fact_3187_norm__prod__diff,axiom,
! [A: $tType,I6: $tType] :
( ( ( comm_monoid_mult @ A )
& ( real_V2822296259951069270ebra_1 @ A ) )
=> ! [I5: set @ I6,Z: I6 > A,W: I6 > A] :
( ! [I2: I6] :
( ( member @ I6 @ I2 @ I5 )
=> ( ord_less_eq @ real @ ( real_V7770717601297561774m_norm @ A @ ( Z @ I2 ) ) @ ( one_one @ real ) ) )
=> ( ! [I2: I6] :
( ( member @ I6 @ I2 @ I5 )
=> ( 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 @ I6 @ A @ Z @ I5 ) @ ( groups7121269368397514597t_prod @ I6 @ A @ W @ I5 ) ) )
@ ( groups7311177749621191930dd_sum @ I6 @ real
@ ^ [I3: I6] : ( real_V7770717601297561774m_norm @ A @ ( minus_minus @ A @ ( Z @ I3 ) @ ( W @ I3 ) ) )
@ I5 ) ) ) ) ) ).
% norm_prod_diff
thf(fact_3188_prod__mono2,axiom,
! [B: $tType,A: $tType] :
( ( linordered_idom @ B )
=> ! [B6: set @ A,A3: set @ A,F2: A > B] :
( ( finite_finite2 @ A @ B6 )
=> ( ( ord_less_eq @ ( set @ A ) @ A3 @ B6 )
=> ( ! [B3: A] :
( ( member @ A @ B3 @ ( minus_minus @ ( set @ A ) @ B6 @ A3 ) )
=> ( ord_less_eq @ B @ ( one_one @ B ) @ ( F2 @ B3 ) ) )
=> ( ! [A4: A] :
( ( member @ A @ A4 @ A3 )
=> ( ord_less_eq @ B @ ( zero_zero @ B ) @ ( F2 @ A4 ) ) )
=> ( ord_less_eq @ B @ ( groups7121269368397514597t_prod @ A @ B @ F2 @ A3 ) @ ( groups7121269368397514597t_prod @ A @ B @ F2 @ B6 ) ) ) ) ) ) ) ).
% prod_mono2
thf(fact_3189_prod__diff1,axiom,
! [A: $tType,B: $tType] :
( ( semidom_divide @ A )
=> ! [A3: set @ B,F2: B > A,A2: B] :
( ( finite_finite2 @ B @ A3 )
=> ( ( ( F2 @ A2 )
!= ( zero_zero @ A ) )
=> ( ( ( member @ B @ A2 @ A3 )
=> ( ( groups7121269368397514597t_prod @ B @ A @ F2 @ ( minus_minus @ ( set @ B ) @ A3 @ ( insert @ B @ A2 @ ( bot_bot @ ( set @ B ) ) ) ) )
= ( divide_divide @ A @ ( groups7121269368397514597t_prod @ B @ A @ F2 @ A3 ) @ ( F2 @ A2 ) ) ) )
& ( ~ ( member @ B @ A2 @ A3 )
=> ( ( groups7121269368397514597t_prod @ B @ A @ F2 @ ( minus_minus @ ( set @ B ) @ A3 @ ( insert @ B @ A2 @ ( bot_bot @ ( set @ B ) ) ) ) )
= ( groups7121269368397514597t_prod @ B @ A @ F2 @ A3 ) ) ) ) ) ) ) ).
% prod_diff1
thf(fact_3190_pochhammer__Suc__prod,axiom,
! [A: $tType] :
( ( comm_semiring_1 @ A )
=> ! [A2: A,N3: nat] :
( ( comm_s3205402744901411588hammer @ A @ A2 @ ( suc @ N3 ) )
= ( groups7121269368397514597t_prod @ nat @ A
@ ^ [I3: nat] : ( plus_plus @ A @ A2 @ ( semiring_1_of_nat @ A @ I3 ) )
@ ( set_or1337092689740270186AtMost @ nat @ ( zero_zero @ nat ) @ N3 ) ) ) ) ).
% pochhammer_Suc_prod
thf(fact_3191_pochhammer__prod__rev,axiom,
! [A: $tType] :
( ( comm_semiring_1 @ A )
=> ( ( comm_s3205402744901411588hammer @ A )
= ( ^ [A6: A,N2: nat] :
( groups7121269368397514597t_prod @ nat @ A
@ ^ [I3: nat] : ( plus_plus @ A @ A6 @ ( semiring_1_of_nat @ A @ ( minus_minus @ nat @ N2 @ I3 ) ) )
@ ( set_or1337092689740270186AtMost @ nat @ ( one_one @ nat ) @ N2 ) ) ) ) ) ).
% pochhammer_prod_rev
thf(fact_3192_prod_Oin__pairs,axiom,
! [A: $tType] :
( ( comm_monoid_mult @ A )
=> ! [G: nat > A,M: nat,N3: 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 ) ) @ N3 ) ) ) )
= ( groups7121269368397514597t_prod @ nat @ A
@ ^ [I3: nat] : ( times_times @ A @ ( G @ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ I3 ) ) @ ( G @ ( suc @ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ I3 ) ) ) )
@ ( set_or1337092689740270186AtMost @ nat @ M @ N3 ) ) ) ) ).
% prod.in_pairs
thf(fact_3193_sum__atLeastAtMost__code,axiom,
! [A: $tType] :
( ( comm_monoid_add @ A )
=> ! [F2: nat > A,A2: nat,B2: nat] :
( ( groups7311177749621191930dd_sum @ nat @ A @ F2 @ ( set_or1337092689740270186AtMost @ nat @ A2 @ B2 ) )
= ( set_fo6178422350223883121st_nat @ A
@ ^ [A6: nat] : ( plus_plus @ A @ ( F2 @ A6 ) )
@ A2
@ B2
@ ( zero_zero @ A ) ) ) ) ).
% sum_atLeastAtMost_code
thf(fact_3194_pochhammer__Suc__prod__rev,axiom,
! [A: $tType] :
( ( comm_semiring_1 @ A )
=> ! [A2: A,N3: nat] :
( ( comm_s3205402744901411588hammer @ A @ A2 @ ( suc @ N3 ) )
= ( groups7121269368397514597t_prod @ nat @ A
@ ^ [I3: nat] : ( plus_plus @ A @ A2 @ ( semiring_1_of_nat @ A @ ( minus_minus @ nat @ N3 @ I3 ) ) )
@ ( set_or1337092689740270186AtMost @ nat @ ( zero_zero @ nat ) @ N3 ) ) ) ) ).
% pochhammer_Suc_prod_rev
thf(fact_3195_unbounded__k__infinite,axiom,
! [K: nat,S3: set @ nat] :
( ! [M4: nat] :
( ( ord_less @ nat @ K @ M4 )
=> ? [N9: nat] :
( ( ord_less @ nat @ M4 @ N9 )
& ( member @ nat @ N9 @ S3 ) ) )
=> ~ ( finite_finite2 @ nat @ S3 ) ) ).
% unbounded_k_infinite
thf(fact_3196_infinite__nat__iff__unbounded,axiom,
! [S3: set @ nat] :
( ( ~ ( finite_finite2 @ nat @ S3 ) )
= ( ! [M5: nat] :
? [N2: nat] :
( ( ord_less @ nat @ M5 @ N2 )
& ( member @ nat @ N2 @ S3 ) ) ) ) ).
% infinite_nat_iff_unbounded
thf(fact_3197_infinite__nat__iff__unbounded__le,axiom,
! [S3: set @ nat] :
( ( ~ ( finite_finite2 @ nat @ S3 ) )
= ( ! [M5: nat] :
? [N2: nat] :
( ( ord_less_eq @ nat @ M5 @ N2 )
& ( member @ nat @ N2 @ S3 ) ) ) ) ).
% infinite_nat_iff_unbounded_le
thf(fact_3198_finite__nat__bounded,axiom,
! [S3: set @ nat] :
( ( finite_finite2 @ nat @ S3 )
=> ? [K3: nat] : ( ord_less_eq @ ( set @ nat ) @ S3 @ ( set_ord_lessThan @ nat @ K3 ) ) ) ).
% finite_nat_bounded
thf(fact_3199_round__altdef,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ( ( archimedean_round @ A )
= ( ^ [X2: A] : ( if @ int @ ( ord_less_eq @ A @ ( divide_divide @ A @ ( one_one @ A ) @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) @ ( archimedean_frac @ A @ X2 ) ) @ ( archimedean_ceiling @ A @ X2 ) @ ( archim6421214686448440834_floor @ A @ X2 ) ) ) ) ) ).
% round_altdef
thf(fact_3200_old_Oprod_Orec,axiom,
! [A: $tType,T: $tType,B: $tType,F1: A > B > T,A2: A,B2: B] :
( ( product_rec_prod @ A @ B @ T @ F1 @ ( product_Pair @ A @ B @ A2 @ B2 ) )
= ( F1 @ A2 @ B2 ) ) ).
% old.prod.rec
thf(fact_3201_gchoose__row__sum__weighted,axiom,
! [A: $tType] :
( ( field_char_0 @ A )
=> ! [R2: A,M: nat] :
( ( groups7311177749621191930dd_sum @ nat @ A
@ ^ [K2: nat] : ( times_times @ A @ ( gbinomial @ A @ R2 @ K2 ) @ ( minus_minus @ A @ ( divide_divide @ A @ R2 @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) @ ( semiring_1_of_nat @ A @ K2 ) ) )
@ ( 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_3202_central__binomial__lower__bound,axiom,
! [N3: nat] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ N3 )
=> ( ord_less_eq @ real @ ( divide_divide @ real @ ( power_power @ real @ ( numeral_numeral @ real @ ( bit0 @ ( bit0 @ one2 ) ) ) @ N3 ) @ ( times_times @ real @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) @ ( semiring_1_of_nat @ real @ N3 ) ) ) @ ( semiring_1_of_nat @ real @ ( binomial @ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N3 ) @ N3 ) ) ) ) ).
% central_binomial_lower_bound
thf(fact_3203_prod__eq__1__iff,axiom,
! [A: $tType,A3: set @ A,F2: A > nat] :
( ( finite_finite2 @ A @ A3 )
=> ( ( ( groups7121269368397514597t_prod @ A @ nat @ F2 @ A3 )
= ( one_one @ nat ) )
= ( ! [X2: A] :
( ( member @ A @ X2 @ A3 )
=> ( ( F2 @ X2 )
= ( one_one @ nat ) ) ) ) ) ) ).
% prod_eq_1_iff
thf(fact_3204_binomial__Suc__n,axiom,
! [N3: nat] :
( ( binomial @ ( suc @ N3 ) @ N3 )
= ( suc @ N3 ) ) ).
% binomial_Suc_n
thf(fact_3205_binomial__n__n,axiom,
! [N3: nat] :
( ( binomial @ N3 @ N3 )
= ( one_one @ nat ) ) ).
% binomial_n_n
thf(fact_3206_gbinomial__1,axiom,
! [A: $tType] :
( ( ( semiring_char_0 @ A )
& ( semidom_divide @ A ) )
=> ! [A2: A] :
( ( gbinomial @ A @ A2 @ ( one_one @ nat ) )
= A2 ) ) ).
% gbinomial_1
thf(fact_3207_prod__pos__nat__iff,axiom,
! [A: $tType,A3: set @ A,F2: A > nat] :
( ( finite_finite2 @ A @ A3 )
=> ( ( ord_less @ nat @ ( zero_zero @ nat ) @ ( groups7121269368397514597t_prod @ A @ nat @ F2 @ A3 ) )
= ( ! [X2: A] :
( ( member @ A @ X2 @ A3 )
=> ( ord_less @ nat @ ( zero_zero @ nat ) @ ( F2 @ X2 ) ) ) ) ) ) ).
% prod_pos_nat_iff
thf(fact_3208_binomial__1,axiom,
! [N3: nat] :
( ( binomial @ N3 @ ( suc @ ( zero_zero @ nat ) ) )
= N3 ) ).
% binomial_1
thf(fact_3209_binomial__0__Suc,axiom,
! [K: nat] :
( ( binomial @ ( zero_zero @ nat ) @ ( suc @ K ) )
= ( zero_zero @ nat ) ) ).
% binomial_0_Suc
thf(fact_3210_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_3211_binomial__eq__0__iff,axiom,
! [N3: nat,K: nat] :
( ( ( binomial @ N3 @ K )
= ( zero_zero @ nat ) )
= ( ord_less @ nat @ N3 @ K ) ) ).
% binomial_eq_0_iff
thf(fact_3212_binomial__Suc__Suc,axiom,
! [N3: nat,K: nat] :
( ( binomial @ ( suc @ N3 ) @ ( suc @ K ) )
= ( plus_plus @ nat @ ( binomial @ N3 @ K ) @ ( binomial @ N3 @ ( suc @ K ) ) ) ) ).
% binomial_Suc_Suc
thf(fact_3213_gbinomial__0_I1_J,axiom,
! [A: $tType] :
( ( ( semiring_char_0 @ A )
& ( semidom_divide @ A ) )
=> ! [A2: A] :
( ( gbinomial @ A @ A2 @ ( zero_zero @ nat ) )
= ( one_one @ A ) ) ) ).
% gbinomial_0(1)
thf(fact_3214_binomial__n__0,axiom,
! [N3: nat] :
( ( binomial @ N3 @ ( zero_zero @ nat ) )
= ( one_one @ nat ) ) ).
% binomial_n_0
thf(fact_3215_gbinomial__Suc0,axiom,
! [A: $tType] :
( ( ( semiring_char_0 @ A )
& ( semidom_divide @ A ) )
=> ! [A2: A] :
( ( gbinomial @ A @ A2 @ ( suc @ ( zero_zero @ nat ) ) )
= A2 ) ) ).
% gbinomial_Suc0
thf(fact_3216_zero__less__binomial__iff,axiom,
! [N3: nat,K: nat] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ ( binomial @ N3 @ K ) )
= ( ord_less_eq @ nat @ K @ N3 ) ) ).
% zero_less_binomial_iff
thf(fact_3217_choose__one,axiom,
! [N3: nat] :
( ( binomial @ N3 @ ( one_one @ nat ) )
= N3 ) ).
% choose_one
thf(fact_3218_binomial__eq__0,axiom,
! [N3: nat,K: nat] :
( ( ord_less @ nat @ N3 @ K )
=> ( ( binomial @ N3 @ K )
= ( zero_zero @ nat ) ) ) ).
% binomial_eq_0
thf(fact_3219_Suc__times__binomial__eq,axiom,
! [N3: nat,K: nat] :
( ( times_times @ nat @ ( suc @ N3 ) @ ( binomial @ N3 @ K ) )
= ( times_times @ nat @ ( binomial @ ( suc @ N3 ) @ ( suc @ K ) ) @ ( suc @ K ) ) ) ).
% Suc_times_binomial_eq
thf(fact_3220_Suc__times__binomial,axiom,
! [K: nat,N3: nat] :
( ( times_times @ nat @ ( suc @ K ) @ ( binomial @ ( suc @ N3 ) @ ( suc @ K ) ) )
= ( times_times @ nat @ ( suc @ N3 ) @ ( binomial @ N3 @ K ) ) ) ).
% Suc_times_binomial
thf(fact_3221_binomial__symmetric,axiom,
! [K: nat,N3: nat] :
( ( ord_less_eq @ nat @ K @ N3 )
=> ( ( binomial @ N3 @ K )
= ( binomial @ N3 @ ( minus_minus @ nat @ N3 @ K ) ) ) ) ).
% binomial_symmetric
thf(fact_3222_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_3223_binomial__le__pow,axiom,
! [R2: nat,N3: nat] :
( ( ord_less_eq @ nat @ R2 @ N3 )
=> ( ord_less_eq @ nat @ ( binomial @ N3 @ R2 ) @ ( power_power @ nat @ N3 @ R2 ) ) ) ).
% binomial_le_pow
thf(fact_3224_frac__ge__0,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [X: A] : ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( archimedean_frac @ A @ X ) ) ) ).
% frac_ge_0
thf(fact_3225_frac__lt__1,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [X: A] : ( ord_less @ A @ ( archimedean_frac @ A @ X ) @ ( one_one @ A ) ) ) ).
% frac_lt_1
thf(fact_3226_frac__1__eq,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [X: A] :
( ( archimedean_frac @ A @ ( plus_plus @ A @ X @ ( one_one @ A ) ) )
= ( archimedean_frac @ A @ X ) ) ) ).
% frac_1_eq
thf(fact_3227_zero__less__binomial,axiom,
! [K: nat,N3: nat] :
( ( ord_less_eq @ nat @ K @ N3 )
=> ( ord_less @ nat @ ( zero_zero @ nat ) @ ( binomial @ N3 @ K ) ) ) ).
% zero_less_binomial
thf(fact_3228_Suc__times__binomial__add,axiom,
! [A2: nat,B2: nat] :
( ( times_times @ nat @ ( suc @ A2 ) @ ( binomial @ ( suc @ ( plus_plus @ nat @ A2 @ B2 ) ) @ ( suc @ A2 ) ) )
= ( times_times @ nat @ ( suc @ B2 ) @ ( binomial @ ( suc @ ( plus_plus @ nat @ A2 @ B2 ) ) @ A2 ) ) ) ).
% Suc_times_binomial_add
thf(fact_3229_binomial__Suc__Suc__eq__times,axiom,
! [N3: nat,K: nat] :
( ( binomial @ ( suc @ N3 ) @ ( suc @ K ) )
= ( divide_divide @ nat @ ( times_times @ nat @ ( suc @ N3 ) @ ( binomial @ N3 @ K ) ) @ ( suc @ K ) ) ) ).
% binomial_Suc_Suc_eq_times
thf(fact_3230_choose__mult,axiom,
! [K: nat,M: nat,N3: nat] :
( ( ord_less_eq @ nat @ K @ M )
=> ( ( ord_less_eq @ nat @ M @ N3 )
=> ( ( times_times @ nat @ ( binomial @ N3 @ M ) @ ( binomial @ M @ K ) )
= ( times_times @ nat @ ( binomial @ N3 @ K ) @ ( binomial @ ( minus_minus @ nat @ N3 @ K ) @ ( minus_minus @ nat @ M @ K ) ) ) ) ) ) ).
% choose_mult
thf(fact_3231_binomial__absorb__comp,axiom,
! [N3: nat,K: nat] :
( ( times_times @ nat @ ( minus_minus @ nat @ N3 @ K ) @ ( binomial @ N3 @ K ) )
= ( times_times @ nat @ N3 @ ( binomial @ ( minus_minus @ nat @ N3 @ ( one_one @ nat ) ) @ K ) ) ) ).
% binomial_absorb_comp
thf(fact_3232_gbinomial__Suc__Suc,axiom,
! [A: $tType] :
( ( field_char_0 @ A )
=> ! [A2: A,K: nat] :
( ( gbinomial @ A @ ( plus_plus @ A @ A2 @ ( one_one @ A ) ) @ ( suc @ K ) )
= ( plus_plus @ A @ ( gbinomial @ A @ A2 @ K ) @ ( gbinomial @ A @ A2 @ ( suc @ K ) ) ) ) ) ).
% gbinomial_Suc_Suc
thf(fact_3233_gbinomial__of__nat__symmetric,axiom,
! [A: $tType] :
( ( field_char_0 @ A )
=> ! [K: nat,N3: nat] :
( ( ord_less_eq @ nat @ K @ N3 )
=> ( ( gbinomial @ A @ ( semiring_1_of_nat @ A @ N3 ) @ K )
= ( gbinomial @ A @ ( semiring_1_of_nat @ A @ N3 ) @ ( minus_minus @ nat @ N3 @ K ) ) ) ) ) ).
% gbinomial_of_nat_symmetric
thf(fact_3234_small__lazy_H_Ocases,axiom,
! [X: product_prod @ int @ int] :
~ ! [D3: int,I2: int] :
( X
!= ( product_Pair @ int @ int @ D3 @ I2 ) ) ).
% small_lazy'.cases
thf(fact_3235_binomial__absorption,axiom,
! [K: nat,N3: nat] :
( ( times_times @ nat @ ( suc @ K ) @ ( binomial @ N3 @ ( suc @ K ) ) )
= ( times_times @ nat @ N3 @ ( binomial @ ( minus_minus @ nat @ N3 @ ( one_one @ nat ) ) @ K ) ) ) ).
% binomial_absorption
thf(fact_3236_gbinomial__addition__formula,axiom,
! [A: $tType] :
( ( field_char_0 @ A )
=> ! [A2: A,K: nat] :
( ( gbinomial @ A @ A2 @ ( suc @ K ) )
= ( plus_plus @ A @ ( gbinomial @ A @ ( minus_minus @ A @ A2 @ ( one_one @ A ) ) @ ( suc @ K ) ) @ ( gbinomial @ A @ ( minus_minus @ A @ A2 @ ( one_one @ A ) ) @ K ) ) ) ) ).
% gbinomial_addition_formula
thf(fact_3237_gbinomial__mult__1_H,axiom,
! [A: $tType] :
( ( field_char_0 @ A )
=> ! [A2: A,K: nat] :
( ( times_times @ A @ ( gbinomial @ A @ A2 @ K ) @ A2 )
= ( plus_plus @ A @ ( times_times @ A @ ( semiring_1_of_nat @ A @ K ) @ ( gbinomial @ A @ A2 @ K ) ) @ ( times_times @ A @ ( semiring_1_of_nat @ A @ ( suc @ K ) ) @ ( gbinomial @ A @ A2 @ ( suc @ K ) ) ) ) ) ) ).
% gbinomial_mult_1'
thf(fact_3238_gbinomial__mult__1,axiom,
! [A: $tType] :
( ( field_char_0 @ A )
=> ! [A2: A,K: nat] :
( ( times_times @ A @ A2 @ ( gbinomial @ A @ A2 @ K ) )
= ( plus_plus @ A @ ( times_times @ A @ ( semiring_1_of_nat @ A @ K ) @ ( gbinomial @ A @ A2 @ K ) ) @ ( times_times @ A @ ( semiring_1_of_nat @ A @ ( suc @ K ) ) @ ( gbinomial @ A @ A2 @ ( suc @ K ) ) ) ) ) ) ).
% gbinomial_mult_1
thf(fact_3239_gbinomial__absorb__comp,axiom,
! [A: $tType] :
( ( field_char_0 @ A )
=> ! [A2: A,K: nat] :
( ( times_times @ A @ ( minus_minus @ A @ A2 @ ( semiring_1_of_nat @ A @ K ) ) @ ( gbinomial @ A @ A2 @ K ) )
= ( times_times @ A @ A2 @ ( gbinomial @ A @ ( minus_minus @ A @ A2 @ ( one_one @ A ) ) @ K ) ) ) ) ).
% gbinomial_absorb_comp
thf(fact_3240_gbinomial__ge__n__over__k__pow__k,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [K: nat,A2: A] :
( ( ord_less_eq @ A @ ( semiring_1_of_nat @ A @ K ) @ A2 )
=> ( ord_less_eq @ A @ ( power_power @ A @ ( divide_divide @ A @ A2 @ ( semiring_1_of_nat @ A @ K ) ) @ K ) @ ( gbinomial @ A @ A2 @ K ) ) ) ) ).
% gbinomial_ge_n_over_k_pow_k
thf(fact_3241_frac__def,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ( ( archimedean_frac @ A )
= ( ^ [X2: A] : ( minus_minus @ A @ X2 @ ( ring_1_of_int @ A @ ( archim6421214686448440834_floor @ A @ X2 ) ) ) ) ) ) ).
% frac_def
thf(fact_3242_prod__int__plus__eq,axiom,
! [I: nat,J: nat] :
( ( groups7121269368397514597t_prod @ nat @ int @ ( semiring_1_of_nat @ int ) @ ( set_or1337092689740270186AtMost @ nat @ I @ ( plus_plus @ nat @ I @ J ) ) )
= ( groups7121269368397514597t_prod @ int @ int
@ ^ [X2: int] : X2
@ ( set_or1337092689740270186AtMost @ int @ ( semiring_1_of_nat @ int @ I ) @ ( semiring_1_of_nat @ int @ ( plus_plus @ nat @ I @ J ) ) ) ) ) ).
% prod_int_plus_eq
thf(fact_3243_binomial__antimono,axiom,
! [K: nat,K4: nat,N3: nat] :
( ( ord_less_eq @ nat @ K @ K4 )
=> ( ( ord_less_eq @ nat @ ( divide_divide @ nat @ N3 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ K )
=> ( ( ord_less_eq @ nat @ K4 @ N3 )
=> ( ord_less_eq @ nat @ ( binomial @ N3 @ K4 ) @ ( binomial @ N3 @ K ) ) ) ) ) ).
% binomial_antimono
thf(fact_3244_binomial__maximum,axiom,
! [N3: nat,K: nat] : ( ord_less_eq @ nat @ ( binomial @ N3 @ K ) @ ( binomial @ N3 @ ( divide_divide @ nat @ N3 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ).
% binomial_maximum
thf(fact_3245_binomial__ge__n__over__k__pow__k,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [K: nat,N3: nat] :
( ( ord_less_eq @ nat @ K @ N3 )
=> ( ord_less_eq @ A @ ( power_power @ A @ ( divide_divide @ A @ ( semiring_1_of_nat @ A @ N3 ) @ ( semiring_1_of_nat @ A @ K ) ) @ K ) @ ( semiring_1_of_nat @ A @ ( binomial @ N3 @ K ) ) ) ) ) ).
% binomial_ge_n_over_k_pow_k
thf(fact_3246_binomial__maximum_H,axiom,
! [N3: nat,K: nat] : ( ord_less_eq @ nat @ ( binomial @ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N3 ) @ K ) @ ( binomial @ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N3 ) @ N3 ) ) ).
% binomial_maximum'
thf(fact_3247_binomial__mono,axiom,
! [K: nat,K4: nat,N3: nat] :
( ( ord_less_eq @ nat @ K @ K4 )
=> ( ( ord_less_eq @ nat @ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ K4 ) @ N3 )
=> ( ord_less_eq @ nat @ ( binomial @ N3 @ K ) @ ( binomial @ N3 @ K4 ) ) ) ) ).
% binomial_mono
thf(fact_3248_binomial__le__pow2,axiom,
! [N3: nat,K: nat] : ( ord_less_eq @ nat @ ( binomial @ N3 @ K ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N3 ) ) ).
% binomial_le_pow2
thf(fact_3249_choose__reduce__nat,axiom,
! [N3: nat,K: nat] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ N3 )
=> ( ( ord_less @ nat @ ( zero_zero @ nat ) @ K )
=> ( ( binomial @ N3 @ K )
= ( plus_plus @ nat @ ( binomial @ ( minus_minus @ nat @ N3 @ ( one_one @ nat ) ) @ ( minus_minus @ nat @ K @ ( one_one @ nat ) ) ) @ ( binomial @ ( minus_minus @ nat @ N3 @ ( one_one @ nat ) ) @ K ) ) ) ) ) ).
% choose_reduce_nat
thf(fact_3250_times__binomial__minus1__eq,axiom,
! [K: nat,N3: nat] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ K )
=> ( ( times_times @ nat @ K @ ( binomial @ N3 @ K ) )
= ( times_times @ nat @ N3 @ ( binomial @ ( minus_minus @ nat @ N3 @ ( one_one @ nat ) ) @ ( minus_minus @ nat @ K @ ( one_one @ nat ) ) ) ) ) ) ).
% times_binomial_minus1_eq
thf(fact_3251_Suc__times__gbinomial,axiom,
! [A: $tType] :
( ( field_char_0 @ A )
=> ! [K: nat,A2: A] :
( ( times_times @ A @ ( semiring_1_of_nat @ A @ ( suc @ K ) ) @ ( gbinomial @ A @ ( plus_plus @ A @ A2 @ ( one_one @ A ) ) @ ( suc @ K ) ) )
= ( times_times @ A @ ( plus_plus @ A @ A2 @ ( one_one @ A ) ) @ ( gbinomial @ A @ A2 @ K ) ) ) ) ).
% Suc_times_gbinomial
thf(fact_3252_gbinomial__absorption,axiom,
! [A: $tType] :
( ( field_char_0 @ A )
=> ! [K: nat,A2: A] :
( ( times_times @ A @ ( semiring_1_of_nat @ A @ ( suc @ K ) ) @ ( gbinomial @ A @ A2 @ ( suc @ K ) ) )
= ( times_times @ A @ A2 @ ( gbinomial @ A @ ( minus_minus @ A @ A2 @ ( one_one @ A ) ) @ K ) ) ) ) ).
% gbinomial_absorption
thf(fact_3253_gbinomial__trinomial__revision,axiom,
! [A: $tType] :
( ( field_char_0 @ A )
=> ! [K: nat,M: nat,A2: A] :
( ( ord_less_eq @ nat @ K @ M )
=> ( ( times_times @ A @ ( gbinomial @ A @ A2 @ M ) @ ( gbinomial @ A @ ( semiring_1_of_nat @ A @ M ) @ K ) )
= ( times_times @ A @ ( gbinomial @ A @ A2 @ K ) @ ( gbinomial @ A @ ( minus_minus @ A @ A2 @ ( semiring_1_of_nat @ A @ K ) ) @ ( minus_minus @ nat @ M @ K ) ) ) ) ) ) ).
% gbinomial_trinomial_revision
thf(fact_3254_frac__eq,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [X: A] :
( ( ( archimedean_frac @ A @ X )
= X )
= ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ X )
& ( ord_less @ A @ X @ ( one_one @ A ) ) ) ) ) ).
% frac_eq
thf(fact_3255_frac__add,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [X: A,Y: A] :
( ( ( ord_less @ A @ ( plus_plus @ A @ ( archimedean_frac @ A @ X ) @ ( archimedean_frac @ A @ Y ) ) @ ( one_one @ A ) )
=> ( ( archimedean_frac @ A @ ( plus_plus @ A @ X @ Y ) )
= ( plus_plus @ A @ ( archimedean_frac @ A @ X ) @ ( archimedean_frac @ A @ Y ) ) ) )
& ( ~ ( ord_less @ A @ ( plus_plus @ A @ ( archimedean_frac @ A @ X ) @ ( archimedean_frac @ A @ Y ) ) @ ( one_one @ A ) )
=> ( ( archimedean_frac @ A @ ( plus_plus @ A @ X @ Y ) )
= ( minus_minus @ A @ ( plus_plus @ A @ ( archimedean_frac @ A @ X ) @ ( archimedean_frac @ A @ Y ) ) @ ( one_one @ A ) ) ) ) ) ) ).
% frac_add
thf(fact_3256_binomial__less__binomial__Suc,axiom,
! [K: nat,N3: nat] :
( ( ord_less @ nat @ K @ ( divide_divide @ nat @ N3 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) )
=> ( ord_less @ nat @ ( binomial @ N3 @ K ) @ ( binomial @ N3 @ ( suc @ K ) ) ) ) ).
% binomial_less_binomial_Suc
thf(fact_3257_binomial__strict__antimono,axiom,
! [K: nat,K4: nat,N3: nat] :
( ( ord_less @ nat @ K @ K4 )
=> ( ( ord_less_eq @ nat @ N3 @ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ K ) )
=> ( ( ord_less_eq @ nat @ K4 @ N3 )
=> ( ord_less @ nat @ ( binomial @ N3 @ K4 ) @ ( binomial @ N3 @ K ) ) ) ) ) ).
% binomial_strict_antimono
thf(fact_3258_binomial__strict__mono,axiom,
! [K: nat,K4: nat,N3: nat] :
( ( ord_less @ nat @ K @ K4 )
=> ( ( ord_less_eq @ nat @ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ K4 ) @ N3 )
=> ( ord_less @ nat @ ( binomial @ N3 @ K ) @ ( binomial @ N3 @ K4 ) ) ) ) ).
% binomial_strict_mono
thf(fact_3259_central__binomial__odd,axiom,
! [N3: nat] :
( ~ ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N3 )
=> ( ( binomial @ N3 @ ( suc @ ( divide_divide @ nat @ N3 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) )
= ( binomial @ N3 @ ( divide_divide @ nat @ N3 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) ).
% central_binomial_odd
thf(fact_3260_binomial__addition__formula,axiom,
! [N3: nat,K: nat] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ N3 )
=> ( ( binomial @ N3 @ ( suc @ K ) )
= ( plus_plus @ nat @ ( binomial @ ( minus_minus @ nat @ N3 @ ( one_one @ nat ) ) @ ( suc @ K ) ) @ ( binomial @ ( minus_minus @ nat @ N3 @ ( one_one @ nat ) ) @ K ) ) ) ) ).
% binomial_addition_formula
thf(fact_3261_gbinomial__factors,axiom,
! [A: $tType] :
( ( field_char_0 @ A )
=> ! [A2: A,K: nat] :
( ( gbinomial @ A @ ( plus_plus @ A @ A2 @ ( one_one @ A ) ) @ ( suc @ K ) )
= ( times_times @ A @ ( divide_divide @ A @ ( plus_plus @ A @ A2 @ ( one_one @ A ) ) @ ( semiring_1_of_nat @ A @ ( suc @ K ) ) ) @ ( gbinomial @ A @ A2 @ K ) ) ) ) ).
% gbinomial_factors
thf(fact_3262_gbinomial__rec,axiom,
! [A: $tType] :
( ( field_char_0 @ A )
=> ! [A2: A,K: nat] :
( ( gbinomial @ A @ ( plus_plus @ A @ A2 @ ( one_one @ A ) ) @ ( suc @ K ) )
= ( times_times @ A @ ( gbinomial @ A @ A2 @ K ) @ ( divide_divide @ A @ ( plus_plus @ A @ A2 @ ( one_one @ A ) ) @ ( semiring_1_of_nat @ A @ ( suc @ K ) ) ) ) ) ) ).
% gbinomial_rec
thf(fact_3263_choose__two,axiom,
! [N3: nat] :
( ( binomial @ N3 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) )
= ( divide_divide @ nat @ ( times_times @ nat @ N3 @ ( minus_minus @ nat @ N3 @ ( one_one @ nat ) ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ).
% choose_two
thf(fact_3264_gbinomial__reduce__nat,axiom,
! [A: $tType] :
( ( field_char_0 @ A )
=> ! [K: nat,A2: A] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ K )
=> ( ( gbinomial @ A @ A2 @ K )
= ( plus_plus @ A @ ( gbinomial @ A @ ( minus_minus @ A @ A2 @ ( one_one @ A ) ) @ ( minus_minus @ nat @ K @ ( one_one @ nat ) ) ) @ ( gbinomial @ A @ ( minus_minus @ A @ A2 @ ( one_one @ A ) ) @ K ) ) ) ) ) ).
% gbinomial_reduce_nat
thf(fact_3265_floor__add,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [X: A,Y: A] :
( ( ( ord_less @ A @ ( plus_plus @ A @ ( archimedean_frac @ A @ X ) @ ( archimedean_frac @ A @ Y ) ) @ ( one_one @ A ) )
=> ( ( archim6421214686448440834_floor @ A @ ( plus_plus @ A @ X @ Y ) )
= ( plus_plus @ int @ ( archim6421214686448440834_floor @ A @ X ) @ ( archim6421214686448440834_floor @ A @ Y ) ) ) )
& ( ~ ( ord_less @ A @ ( plus_plus @ A @ ( archimedean_frac @ A @ X ) @ ( archimedean_frac @ A @ Y ) ) @ ( one_one @ A ) )
=> ( ( archim6421214686448440834_floor @ A @ ( plus_plus @ A @ X @ Y ) )
= ( plus_plus @ int @ ( plus_plus @ int @ ( archim6421214686448440834_floor @ A @ X ) @ ( archim6421214686448440834_floor @ A @ Y ) ) @ ( one_one @ int ) ) ) ) ) ) ).
% floor_add
thf(fact_3266_gbinomial__sum__up__index,axiom,
! [A: $tType] :
( ( field_char_0 @ A )
=> ! [K: nat,N3: nat] :
( ( groups7311177749621191930dd_sum @ nat @ A
@ ^ [J3: nat] : ( gbinomial @ A @ ( semiring_1_of_nat @ A @ J3 ) @ K )
@ ( set_or1337092689740270186AtMost @ nat @ ( zero_zero @ nat ) @ N3 ) )
= ( gbinomial @ A @ ( plus_plus @ A @ ( semiring_1_of_nat @ A @ N3 ) @ ( one_one @ A ) ) @ ( plus_plus @ nat @ K @ ( one_one @ nat ) ) ) ) ) ).
% gbinomial_sum_up_index
thf(fact_3267_gbinomial__absorption_H,axiom,
! [A: $tType] :
( ( field_char_0 @ A )
=> ! [K: nat,A2: A] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ K )
=> ( ( gbinomial @ A @ A2 @ K )
= ( times_times @ A @ ( divide_divide @ A @ A2 @ ( semiring_1_of_nat @ A @ K ) ) @ ( gbinomial @ A @ ( minus_minus @ A @ A2 @ ( one_one @ A ) ) @ ( minus_minus @ nat @ K @ ( one_one @ nat ) ) ) ) ) ) ) ).
% gbinomial_absorption'
thf(fact_3268_choose__even__sum,axiom,
! [A: $tType] :
( ( comm_ring_1 @ A )
=> ! [N3: nat] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ N3 )
=> ( ( times_times @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) )
@ ( groups7311177749621191930dd_sum @ nat @ A
@ ^ [I3: nat] : ( if @ A @ ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ I3 ) @ ( semiring_1_of_nat @ A @ ( binomial @ N3 @ I3 ) ) @ ( zero_zero @ A ) )
@ ( set_ord_atMost @ nat @ N3 ) ) )
= ( power_power @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ N3 ) ) ) ) ).
% choose_even_sum
thf(fact_3269_choose__odd__sum,axiom,
! [A: $tType] :
( ( comm_ring_1 @ A )
=> ! [N3: nat] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ N3 )
=> ( ( times_times @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) )
@ ( groups7311177749621191930dd_sum @ nat @ A
@ ^ [I3: nat] :
( if @ A
@ ~ ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ I3 )
@ ( semiring_1_of_nat @ A @ ( binomial @ N3 @ I3 ) )
@ ( zero_zero @ A ) )
@ ( set_ord_atMost @ nat @ N3 ) ) )
= ( power_power @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ N3 ) ) ) ) ).
% choose_odd_sum
thf(fact_3270_gbinomial__partial__row__sum,axiom,
! [A: $tType] :
( ( field_char_0 @ A )
=> ! [A2: A,M: nat] :
( ( groups7311177749621191930dd_sum @ nat @ A
@ ^ [K2: nat] : ( times_times @ A @ ( gbinomial @ A @ A2 @ K2 ) @ ( minus_minus @ A @ ( divide_divide @ A @ A2 @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) @ ( semiring_1_of_nat @ A @ K2 ) ) )
@ ( set_ord_atMost @ nat @ M ) )
= ( times_times @ A @ ( divide_divide @ A @ ( plus_plus @ A @ ( semiring_1_of_nat @ A @ M ) @ ( one_one @ A ) ) @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) @ ( gbinomial @ A @ A2 @ ( plus_plus @ nat @ M @ ( one_one @ nat ) ) ) ) ) ) ).
% gbinomial_partial_row_sum
thf(fact_3271_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_3272_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_3273_atMost__subset__iff,axiom,
! [A: $tType] :
( ( preorder @ A )
=> ! [X: A,Y: A] :
( ( ord_less_eq @ ( set @ A ) @ ( set_ord_atMost @ A @ X ) @ ( set_ord_atMost @ A @ Y ) )
= ( ord_less_eq @ A @ X @ Y ) ) ) ).
% atMost_subset_iff
thf(fact_3274_Icc__subset__Iic__iff,axiom,
! [A: $tType] :
( ( preorder @ A )
=> ! [L2: A,H2: A,H3: A] :
( ( ord_less_eq @ ( set @ A ) @ ( set_or1337092689740270186AtMost @ A @ L2 @ H2 ) @ ( set_ord_atMost @ A @ H3 ) )
= ( ~ ( ord_less_eq @ A @ L2 @ H2 )
| ( ord_less_eq @ A @ H2 @ H3 ) ) ) ) ).
% Icc_subset_Iic_iff
thf(fact_3275_sum_OatMost__Suc,axiom,
! [A: $tType] :
( ( comm_monoid_add @ A )
=> ! [G: nat > A,N3: nat] :
( ( groups7311177749621191930dd_sum @ nat @ A @ G @ ( set_ord_atMost @ nat @ ( suc @ N3 ) ) )
= ( plus_plus @ A @ ( groups7311177749621191930dd_sum @ nat @ A @ G @ ( set_ord_atMost @ nat @ N3 ) ) @ ( G @ ( suc @ N3 ) ) ) ) ) ).
% sum.atMost_Suc
thf(fact_3276_prod_OatMost__Suc,axiom,
! [A: $tType] :
( ( comm_monoid_mult @ A )
=> ! [G: nat > A,N3: nat] :
( ( groups7121269368397514597t_prod @ nat @ A @ G @ ( set_ord_atMost @ nat @ ( suc @ N3 ) ) )
= ( times_times @ A @ ( groups7121269368397514597t_prod @ nat @ A @ G @ ( set_ord_atMost @ nat @ N3 ) ) @ ( G @ ( suc @ N3 ) ) ) ) ) ).
% prod.atMost_Suc
thf(fact_3277_atMost__def,axiom,
! [A: $tType] :
( ( ord @ A )
=> ( ( set_ord_atMost @ A )
= ( ^ [U2: A] :
( collect @ A
@ ^ [X2: A] : ( ord_less_eq @ A @ X2 @ U2 ) ) ) ) ) ).
% atMost_def
thf(fact_3278_lessThan__Suc__atMost,axiom,
! [K: nat] :
( ( set_ord_lessThan @ nat @ ( suc @ K ) )
= ( set_ord_atMost @ nat @ K ) ) ).
% lessThan_Suc_atMost
thf(fact_3279_atMost__Suc,axiom,
! [K: nat] :
( ( set_ord_atMost @ nat @ ( suc @ K ) )
= ( insert @ nat @ ( suc @ K ) @ ( set_ord_atMost @ nat @ K ) ) ) ).
% atMost_Suc
thf(fact_3280_not__Iic__le__Icc,axiom,
! [A: $tType] :
( ( no_bot @ A )
=> ! [H2: A,L4: A,H3: A] :
~ ( ord_less_eq @ ( set @ A ) @ ( set_ord_atMost @ A @ H2 ) @ ( set_or1337092689740270186AtMost @ A @ L4 @ H3 ) ) ) ).
% not_Iic_le_Icc
thf(fact_3281_finite__nat__iff__bounded__le,axiom,
( ( finite_finite2 @ nat )
= ( ^ [S8: set @ nat] :
? [K2: nat] : ( ord_less_eq @ ( set @ nat ) @ S8 @ ( set_ord_atMost @ nat @ K2 ) ) ) ) ).
% finite_nat_iff_bounded_le
thf(fact_3282_Iic__subset__Iio__iff,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A2: A,B2: A] :
( ( ord_less_eq @ ( set @ A ) @ ( set_ord_atMost @ A @ A2 ) @ ( set_ord_lessThan @ A @ B2 ) )
= ( ord_less @ A @ A2 @ B2 ) ) ) ).
% Iic_subset_Iio_iff
thf(fact_3283_sum__choose__upper,axiom,
! [M: nat,N3: nat] :
( ( groups7311177749621191930dd_sum @ nat @ nat
@ ^ [K2: nat] : ( binomial @ K2 @ M )
@ ( set_ord_atMost @ nat @ N3 ) )
= ( binomial @ ( suc @ N3 ) @ ( suc @ M ) ) ) ).
% sum_choose_upper
thf(fact_3284_full__exhaustive__int_H_Ocases,axiom,
! [X: product_prod @ ( ( product_prod @ int @ ( product_unit > code_term ) ) > ( option @ ( product_prod @ $o @ ( list @ code_term ) ) ) ) @ ( product_prod @ int @ int )] :
~ ! [F3: ( product_prod @ int @ ( product_unit > code_term ) ) > ( option @ ( product_prod @ $o @ ( list @ code_term ) ) ),D3: int,I2: int] :
( X
!= ( product_Pair @ ( ( product_prod @ int @ ( product_unit > code_term ) ) > ( option @ ( product_prod @ $o @ ( list @ code_term ) ) ) ) @ ( product_prod @ int @ int ) @ F3 @ ( product_Pair @ int @ int @ D3 @ I2 ) ) ) ).
% full_exhaustive_int'.cases
thf(fact_3285_exhaustive__int_H_Ocases,axiom,
! [X: product_prod @ ( int > ( option @ ( product_prod @ $o @ ( list @ code_term ) ) ) ) @ ( product_prod @ int @ int )] :
~ ! [F3: int > ( option @ ( product_prod @ $o @ ( list @ code_term ) ) ),D3: int,I2: int] :
( X
!= ( product_Pair @ ( int > ( option @ ( product_prod @ $o @ ( list @ code_term ) ) ) ) @ ( product_prod @ int @ int ) @ F3 @ ( product_Pair @ int @ int @ D3 @ I2 ) ) ) ).
% exhaustive_int'.cases
thf(fact_3286_sum_OatMost__Suc__shift,axiom,
! [A: $tType] :
( ( comm_monoid_add @ A )
=> ! [G: nat > A,N3: nat] :
( ( groups7311177749621191930dd_sum @ nat @ A @ G @ ( set_ord_atMost @ nat @ ( suc @ N3 ) ) )
= ( plus_plus @ A @ ( G @ ( zero_zero @ nat ) )
@ ( groups7311177749621191930dd_sum @ nat @ A
@ ^ [I3: nat] : ( G @ ( suc @ I3 ) )
@ ( set_ord_atMost @ nat @ N3 ) ) ) ) ) ).
% sum.atMost_Suc_shift
thf(fact_3287_sum__telescope,axiom,
! [A: $tType] :
( ( ab_group_add @ A )
=> ! [F2: nat > A,I: nat] :
( ( groups7311177749621191930dd_sum @ nat @ A
@ ^ [I3: nat] : ( minus_minus @ A @ ( F2 @ I3 ) @ ( F2 @ ( suc @ I3 ) ) )
@ ( set_ord_atMost @ nat @ I ) )
= ( minus_minus @ A @ ( F2 @ ( zero_zero @ nat ) ) @ ( F2 @ ( suc @ I ) ) ) ) ) ).
% sum_telescope
thf(fact_3288_polyfun__eq__coeffs,axiom,
! [A: $tType] :
( ( ( real_V8999393235501362500lgebra @ A )
& ( idom @ A ) )
=> ! [C2: nat > A,N3: nat,D: nat > A] :
( ( ! [X2: A] :
( ( groups7311177749621191930dd_sum @ nat @ A
@ ^ [I3: nat] : ( times_times @ A @ ( C2 @ I3 ) @ ( power_power @ A @ X2 @ I3 ) )
@ ( set_ord_atMost @ nat @ N3 ) )
= ( groups7311177749621191930dd_sum @ nat @ A
@ ^ [I3: nat] : ( times_times @ A @ ( D @ I3 ) @ ( power_power @ A @ X2 @ I3 ) )
@ ( set_ord_atMost @ nat @ N3 ) ) ) )
= ( ! [I3: nat] :
( ( ord_less_eq @ nat @ I3 @ N3 )
=> ( ( C2 @ I3 )
= ( D @ I3 ) ) ) ) ) ) ).
% polyfun_eq_coeffs
thf(fact_3289_prod_OatMost__Suc__shift,axiom,
! [A: $tType] :
( ( comm_monoid_mult @ A )
=> ! [G: nat > A,N3: nat] :
( ( groups7121269368397514597t_prod @ nat @ A @ G @ ( set_ord_atMost @ nat @ ( suc @ N3 ) ) )
= ( times_times @ A @ ( G @ ( zero_zero @ nat ) )
@ ( groups7121269368397514597t_prod @ nat @ A
@ ^ [I3: nat] : ( G @ ( suc @ I3 ) )
@ ( set_ord_atMost @ nat @ N3 ) ) ) ) ) ).
% prod.atMost_Suc_shift
thf(fact_3290_sum_Onested__swap_H,axiom,
! [A: $tType] :
( ( comm_monoid_add @ A )
=> ! [A2: nat > nat > A,N3: nat] :
( ( groups7311177749621191930dd_sum @ nat @ A
@ ^ [I3: nat] : ( groups7311177749621191930dd_sum @ nat @ A @ ( A2 @ I3 ) @ ( set_ord_lessThan @ nat @ I3 ) )
@ ( set_ord_atMost @ nat @ N3 ) )
= ( groups7311177749621191930dd_sum @ nat @ A
@ ^ [J3: nat] :
( groups7311177749621191930dd_sum @ nat @ A
@ ^ [I3: nat] : ( A2 @ I3 @ J3 )
@ ( set_or1337092689740270186AtMost @ nat @ ( suc @ J3 ) @ N3 ) )
@ ( set_ord_lessThan @ nat @ N3 ) ) ) ) ).
% sum.nested_swap'
thf(fact_3291_prod_Onested__swap_H,axiom,
! [A: $tType] :
( ( comm_monoid_mult @ A )
=> ! [A2: nat > nat > A,N3: nat] :
( ( groups7121269368397514597t_prod @ nat @ A
@ ^ [I3: nat] : ( groups7121269368397514597t_prod @ nat @ A @ ( A2 @ I3 ) @ ( set_ord_lessThan @ nat @ I3 ) )
@ ( set_ord_atMost @ nat @ N3 ) )
= ( groups7121269368397514597t_prod @ nat @ A
@ ^ [J3: nat] :
( groups7121269368397514597t_prod @ nat @ A
@ ^ [I3: nat] : ( A2 @ I3 @ J3 )
@ ( set_or1337092689740270186AtMost @ nat @ ( suc @ J3 ) @ N3 ) )
@ ( set_ord_lessThan @ nat @ N3 ) ) ) ) ).
% prod.nested_swap'
thf(fact_3292_sum__choose__lower,axiom,
! [R2: nat,N3: nat] :
( ( groups7311177749621191930dd_sum @ nat @ nat
@ ^ [K2: nat] : ( binomial @ ( plus_plus @ nat @ R2 @ K2 ) @ K2 )
@ ( set_ord_atMost @ nat @ N3 ) )
= ( binomial @ ( suc @ ( plus_plus @ nat @ R2 @ N3 ) ) @ N3 ) ) ).
% sum_choose_lower
thf(fact_3293_choose__rising__sum_I2_J,axiom,
! [N3: nat,M: nat] :
( ( groups7311177749621191930dd_sum @ nat @ nat
@ ^ [J3: nat] : ( binomial @ ( plus_plus @ nat @ N3 @ J3 ) @ N3 )
@ ( set_ord_atMost @ nat @ M ) )
= ( binomial @ ( plus_plus @ nat @ ( plus_plus @ nat @ N3 @ M ) @ ( one_one @ nat ) ) @ M ) ) ).
% choose_rising_sum(2)
thf(fact_3294_choose__rising__sum_I1_J,axiom,
! [N3: nat,M: nat] :
( ( groups7311177749621191930dd_sum @ nat @ nat
@ ^ [J3: nat] : ( binomial @ ( plus_plus @ nat @ N3 @ J3 ) @ N3 )
@ ( set_ord_atMost @ nat @ M ) )
= ( binomial @ ( plus_plus @ nat @ ( plus_plus @ nat @ N3 @ M ) @ ( one_one @ nat ) ) @ ( plus_plus @ nat @ N3 @ ( one_one @ nat ) ) ) ) ).
% choose_rising_sum(1)
thf(fact_3295_zero__polynom__imp__zero__coeffs,axiom,
! [A: $tType] :
( ( ( ab_semigroup_mult @ A )
& ( real_V8999393235501362500lgebra @ A ) )
=> ! [C2: nat > A,N3: nat,K: nat] :
( ! [W3: A] :
( ( groups7311177749621191930dd_sum @ nat @ A
@ ^ [I3: nat] : ( times_times @ A @ ( C2 @ I3 ) @ ( power_power @ A @ W3 @ I3 ) )
@ ( set_ord_atMost @ nat @ N3 ) )
= ( zero_zero @ A ) )
=> ( ( ord_less_eq @ nat @ K @ N3 )
=> ( ( C2 @ K )
= ( zero_zero @ A ) ) ) ) ) ).
% zero_polynom_imp_zero_coeffs
thf(fact_3296_polyfun__eq__0,axiom,
! [A: $tType] :
( ( ( real_V8999393235501362500lgebra @ A )
& ( idom @ A ) )
=> ! [C2: nat > A,N3: nat] :
( ( ! [X2: A] :
( ( groups7311177749621191930dd_sum @ nat @ A
@ ^ [I3: nat] : ( times_times @ A @ ( C2 @ I3 ) @ ( power_power @ A @ X2 @ I3 ) )
@ ( set_ord_atMost @ nat @ N3 ) )
= ( zero_zero @ A ) ) )
= ( ! [I3: nat] :
( ( ord_less_eq @ nat @ I3 @ N3 )
=> ( ( C2 @ I3 )
= ( zero_zero @ A ) ) ) ) ) ) ).
% polyfun_eq_0
thf(fact_3297_sum_OatMost__shift,axiom,
! [A: $tType] :
( ( comm_monoid_add @ A )
=> ! [G: nat > A,N3: nat] :
( ( groups7311177749621191930dd_sum @ nat @ A @ G @ ( set_ord_atMost @ nat @ N3 ) )
= ( plus_plus @ A @ ( G @ ( zero_zero @ nat ) )
@ ( groups7311177749621191930dd_sum @ nat @ A
@ ^ [I3: nat] : ( G @ ( suc @ I3 ) )
@ ( set_ord_lessThan @ nat @ N3 ) ) ) ) ) ).
% sum.atMost_shift
thf(fact_3298_sum__up__index__split,axiom,
! [A: $tType] :
( ( comm_monoid_add @ A )
=> ! [F2: nat > A,M: nat,N3: nat] :
( ( groups7311177749621191930dd_sum @ nat @ A @ F2 @ ( set_ord_atMost @ nat @ ( plus_plus @ nat @ M @ N3 ) ) )
= ( 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 @ N3 ) ) ) ) ) ) ).
% sum_up_index_split
thf(fact_3299_prod_OatMost__shift,axiom,
! [A: $tType] :
( ( comm_monoid_mult @ A )
=> ! [G: nat > A,N3: nat] :
( ( groups7121269368397514597t_prod @ nat @ A @ G @ ( set_ord_atMost @ nat @ N3 ) )
= ( times_times @ A @ ( G @ ( zero_zero @ nat ) )
@ ( groups7121269368397514597t_prod @ nat @ A
@ ^ [I3: nat] : ( G @ ( suc @ I3 ) )
@ ( set_ord_lessThan @ nat @ N3 ) ) ) ) ) ).
% prod.atMost_shift
thf(fact_3300_gbinomial__parallel__sum,axiom,
! [A: $tType] :
( ( field_char_0 @ A )
=> ! [A2: A,N3: nat] :
( ( groups7311177749621191930dd_sum @ nat @ A
@ ^ [K2: nat] : ( gbinomial @ A @ ( plus_plus @ A @ A2 @ ( semiring_1_of_nat @ A @ K2 ) ) @ K2 )
@ ( set_ord_atMost @ nat @ N3 ) )
= ( gbinomial @ A @ ( plus_plus @ A @ ( plus_plus @ A @ A2 @ ( semiring_1_of_nat @ A @ N3 ) ) @ ( one_one @ A ) ) @ N3 ) ) ) ).
% gbinomial_parallel_sum
thf(fact_3301_sum__choose__diagonal,axiom,
! [M: nat,N3: nat] :
( ( ord_less_eq @ nat @ M @ N3 )
=> ( ( groups7311177749621191930dd_sum @ nat @ nat
@ ^ [K2: nat] : ( binomial @ ( minus_minus @ nat @ N3 @ K2 ) @ ( minus_minus @ nat @ M @ K2 ) )
@ ( set_ord_atMost @ nat @ M ) )
= ( binomial @ ( suc @ N3 ) @ M ) ) ) ).
% sum_choose_diagonal
thf(fact_3302_vandermonde,axiom,
! [M: nat,N3: nat,R2: nat] :
( ( groups7311177749621191930dd_sum @ nat @ nat
@ ^ [K2: nat] : ( times_times @ nat @ ( binomial @ M @ K2 ) @ ( binomial @ N3 @ ( minus_minus @ nat @ R2 @ K2 ) ) )
@ ( set_ord_atMost @ nat @ R2 ) )
= ( binomial @ ( plus_plus @ nat @ M @ N3 ) @ R2 ) ) ).
% vandermonde
thf(fact_3303_sum__gp__basic,axiom,
! [A: $tType] :
( ( ( monoid_mult @ A )
& ( comm_ring @ A ) )
=> ! [X: A,N3: nat] :
( ( times_times @ A @ ( minus_minus @ A @ ( one_one @ A ) @ X ) @ ( groups7311177749621191930dd_sum @ nat @ A @ ( power_power @ A @ X ) @ ( set_ord_atMost @ nat @ N3 ) ) )
= ( minus_minus @ A @ ( one_one @ A ) @ ( power_power @ A @ X @ ( suc @ N3 ) ) ) ) ) ).
% sum_gp_basic
thf(fact_3304_polyfun__roots__finite,axiom,
! [A: $tType] :
( ( ( real_V8999393235501362500lgebra @ A )
& ( idom @ A ) )
=> ! [C2: nat > A,K: nat,N3: nat] :
( ( ( C2 @ K )
!= ( zero_zero @ A ) )
=> ( ( ord_less_eq @ nat @ K @ N3 )
=> ( finite_finite2 @ A
@ ( collect @ A
@ ^ [Z3: A] :
( ( groups7311177749621191930dd_sum @ nat @ A
@ ^ [I3: nat] : ( times_times @ A @ ( C2 @ I3 ) @ ( power_power @ A @ Z3 @ I3 ) )
@ ( set_ord_atMost @ nat @ N3 ) )
= ( zero_zero @ A ) ) ) ) ) ) ) ).
% polyfun_roots_finite
thf(fact_3305_polyfun__finite__roots,axiom,
! [A: $tType] :
( ( ( real_V8999393235501362500lgebra @ A )
& ( idom @ A ) )
=> ! [C2: nat > A,N3: nat] :
( ( finite_finite2 @ A
@ ( collect @ A
@ ^ [X2: A] :
( ( groups7311177749621191930dd_sum @ nat @ A
@ ^ [I3: nat] : ( times_times @ A @ ( C2 @ I3 ) @ ( power_power @ A @ X2 @ I3 ) )
@ ( set_ord_atMost @ nat @ N3 ) )
= ( zero_zero @ A ) ) ) )
= ( ? [I3: nat] :
( ( ord_less_eq @ nat @ I3 @ N3 )
& ( ( C2 @ I3 )
!= ( zero_zero @ A ) ) ) ) ) ) ).
% polyfun_finite_roots
thf(fact_3306_polyfun__linear__factor__root,axiom,
! [A: $tType] :
( ( idom @ A )
=> ! [C2: nat > A,A2: A,N3: nat] :
( ( ( groups7311177749621191930dd_sum @ nat @ A
@ ^ [I3: nat] : ( times_times @ A @ ( C2 @ I3 ) @ ( power_power @ A @ A2 @ I3 ) )
@ ( set_ord_atMost @ nat @ N3 ) )
= ( zero_zero @ A ) )
=> ~ ! [B3: nat > A] :
~ ! [Z4: A] :
( ( groups7311177749621191930dd_sum @ nat @ A
@ ^ [I3: nat] : ( times_times @ A @ ( C2 @ I3 ) @ ( power_power @ A @ Z4 @ I3 ) )
@ ( set_ord_atMost @ nat @ N3 ) )
= ( times_times @ A @ ( minus_minus @ A @ Z4 @ A2 )
@ ( groups7311177749621191930dd_sum @ nat @ A
@ ^ [I3: nat] : ( times_times @ A @ ( B3 @ I3 ) @ ( power_power @ A @ Z4 @ I3 ) )
@ ( set_ord_lessThan @ nat @ N3 ) ) ) ) ) ) ).
% polyfun_linear_factor_root
thf(fact_3307_polyfun__linear__factor,axiom,
! [A: $tType] :
( ( idom @ A )
=> ! [C2: nat > A,N3: nat,A2: A] :
? [B3: nat > A] :
! [Z4: A] :
( ( groups7311177749621191930dd_sum @ nat @ A
@ ^ [I3: nat] : ( times_times @ A @ ( C2 @ I3 ) @ ( power_power @ A @ Z4 @ I3 ) )
@ ( set_ord_atMost @ nat @ N3 ) )
= ( plus_plus @ A
@ ( times_times @ A @ ( minus_minus @ A @ Z4 @ A2 )
@ ( groups7311177749621191930dd_sum @ nat @ A
@ ^ [I3: nat] : ( times_times @ A @ ( B3 @ I3 ) @ ( power_power @ A @ Z4 @ I3 ) )
@ ( set_ord_lessThan @ nat @ N3 ) ) )
@ ( groups7311177749621191930dd_sum @ nat @ A
@ ^ [I3: nat] : ( times_times @ A @ ( C2 @ I3 ) @ ( power_power @ A @ A2 @ I3 ) )
@ ( set_ord_atMost @ nat @ N3 ) ) ) ) ) ).
% polyfun_linear_factor
thf(fact_3308_sum__power__shift,axiom,
! [A: $tType] :
( ( ( monoid_mult @ A )
& ( comm_ring @ A ) )
=> ! [M: nat,N3: nat,X: A] :
( ( ord_less_eq @ nat @ M @ N3 )
=> ( ( groups7311177749621191930dd_sum @ nat @ A @ ( power_power @ A @ X ) @ ( set_or1337092689740270186AtMost @ nat @ M @ N3 ) )
= ( times_times @ A @ ( power_power @ A @ X @ M ) @ ( groups7311177749621191930dd_sum @ nat @ A @ ( power_power @ A @ X ) @ ( set_ord_atMost @ nat @ ( minus_minus @ nat @ N3 @ M ) ) ) ) ) ) ) ).
% sum_power_shift
thf(fact_3309_atLeast1__atMost__eq__remove0,axiom,
! [N3: nat] :
( ( set_or1337092689740270186AtMost @ nat @ ( suc @ ( zero_zero @ nat ) ) @ N3 )
= ( minus_minus @ ( set @ nat ) @ ( set_ord_atMost @ nat @ N3 ) @ ( insert @ nat @ ( zero_zero @ nat ) @ ( bot_bot @ ( set @ nat ) ) ) ) ) ).
% atLeast1_atMost_eq_remove0
thf(fact_3310_choose__row__sum,axiom,
! [N3: nat] :
( ( groups7311177749621191930dd_sum @ nat @ nat @ ( binomial @ N3 ) @ ( set_ord_atMost @ nat @ N3 ) )
= ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N3 ) ) ).
% choose_row_sum
thf(fact_3311_binomial,axiom,
! [A2: nat,B2: nat,N3: nat] :
( ( power_power @ nat @ ( plus_plus @ nat @ A2 @ B2 ) @ N3 )
= ( groups7311177749621191930dd_sum @ nat @ nat
@ ^ [K2: nat] : ( times_times @ nat @ ( times_times @ nat @ ( semiring_1_of_nat @ nat @ ( binomial @ N3 @ K2 ) ) @ ( power_power @ nat @ A2 @ K2 ) ) @ ( power_power @ nat @ B2 @ ( minus_minus @ nat @ N3 @ K2 ) ) )
@ ( set_ord_atMost @ nat @ N3 ) ) ) ).
% binomial
thf(fact_3312_sum_Oin__pairs__0,axiom,
! [A: $tType] :
( ( comm_monoid_add @ A )
=> ! [G: nat > A,N3: nat] :
( ( groups7311177749621191930dd_sum @ nat @ A @ G @ ( set_ord_atMost @ nat @ ( suc @ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N3 ) ) ) )
= ( groups7311177749621191930dd_sum @ nat @ A
@ ^ [I3: nat] : ( plus_plus @ A @ ( G @ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ I3 ) ) @ ( G @ ( suc @ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ I3 ) ) ) )
@ ( set_ord_atMost @ nat @ N3 ) ) ) ) ).
% sum.in_pairs_0
thf(fact_3313_polynomial__product,axiom,
! [A: $tType] :
( ( idom @ A )
=> ! [M: nat,A2: nat > A,N3: nat,B2: nat > A,X: A] :
( ! [I2: nat] :
( ( ord_less @ nat @ M @ I2 )
=> ( ( A2 @ I2 )
= ( zero_zero @ A ) ) )
=> ( ! [J2: nat] :
( ( ord_less @ nat @ N3 @ J2 )
=> ( ( B2 @ J2 )
= ( zero_zero @ A ) ) )
=> ( ( times_times @ A
@ ( groups7311177749621191930dd_sum @ nat @ A
@ ^ [I3: nat] : ( times_times @ A @ ( A2 @ I3 ) @ ( power_power @ A @ X @ I3 ) )
@ ( set_ord_atMost @ nat @ M ) )
@ ( groups7311177749621191930dd_sum @ nat @ A
@ ^ [J3: nat] : ( times_times @ A @ ( B2 @ J3 ) @ ( power_power @ A @ X @ J3 ) )
@ ( set_ord_atMost @ nat @ N3 ) ) )
= ( groups7311177749621191930dd_sum @ nat @ A
@ ^ [R5: nat] :
( times_times @ A
@ ( groups7311177749621191930dd_sum @ nat @ A
@ ^ [K2: nat] : ( times_times @ A @ ( A2 @ K2 ) @ ( B2 @ ( minus_minus @ nat @ R5 @ K2 ) ) )
@ ( set_ord_atMost @ nat @ R5 ) )
@ ( power_power @ A @ X @ R5 ) )
@ ( set_ord_atMost @ nat @ ( plus_plus @ nat @ M @ N3 ) ) ) ) ) ) ) ).
% polynomial_product
thf(fact_3314_prod_Oin__pairs__0,axiom,
! [A: $tType] :
( ( comm_monoid_mult @ A )
=> ! [G: nat > A,N3: nat] :
( ( groups7121269368397514597t_prod @ nat @ A @ G @ ( set_ord_atMost @ nat @ ( suc @ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N3 ) ) ) )
= ( groups7121269368397514597t_prod @ nat @ A
@ ^ [I3: nat] : ( times_times @ A @ ( G @ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ I3 ) ) @ ( G @ ( suc @ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ I3 ) ) ) )
@ ( set_ord_atMost @ nat @ N3 ) ) ) ) ).
% prod.in_pairs_0
thf(fact_3315_polyfun__eq__const,axiom,
! [A: $tType] :
( ( ( real_V8999393235501362500lgebra @ A )
& ( idom @ A ) )
=> ! [C2: nat > A,N3: nat,K: A] :
( ( ! [X2: A] :
( ( groups7311177749621191930dd_sum @ nat @ A
@ ^ [I3: nat] : ( times_times @ A @ ( C2 @ I3 ) @ ( power_power @ A @ X2 @ I3 ) )
@ ( set_ord_atMost @ nat @ N3 ) )
= K ) )
= ( ( ( C2 @ ( zero_zero @ nat ) )
= K )
& ! [X2: nat] :
( ( member @ nat @ X2 @ ( set_or1337092689740270186AtMost @ nat @ ( one_one @ nat ) @ N3 ) )
=> ( ( C2 @ X2 )
= ( zero_zero @ A ) ) ) ) ) ) ).
% polyfun_eq_const
thf(fact_3316_binomial__ring,axiom,
! [A: $tType] :
( ( comm_semiring_1 @ A )
=> ! [A2: A,B2: A,N3: nat] :
( ( power_power @ A @ ( plus_plus @ A @ A2 @ B2 ) @ N3 )
= ( groups7311177749621191930dd_sum @ nat @ A
@ ^ [K2: nat] : ( times_times @ A @ ( times_times @ A @ ( semiring_1_of_nat @ A @ ( binomial @ N3 @ K2 ) ) @ ( power_power @ A @ A2 @ K2 ) ) @ ( power_power @ A @ B2 @ ( minus_minus @ nat @ N3 @ K2 ) ) )
@ ( set_ord_atMost @ nat @ N3 ) ) ) ) ).
% binomial_ring
thf(fact_3317_polynomial__product__nat,axiom,
! [M: nat,A2: nat > nat,N3: nat,B2: nat > nat,X: nat] :
( ! [I2: nat] :
( ( ord_less @ nat @ M @ I2 )
=> ( ( A2 @ I2 )
= ( zero_zero @ nat ) ) )
=> ( ! [J2: nat] :
( ( ord_less @ nat @ N3 @ J2 )
=> ( ( B2 @ J2 )
= ( zero_zero @ nat ) ) )
=> ( ( times_times @ nat
@ ( groups7311177749621191930dd_sum @ nat @ nat
@ ^ [I3: nat] : ( times_times @ nat @ ( A2 @ I3 ) @ ( power_power @ nat @ X @ I3 ) )
@ ( set_ord_atMost @ nat @ M ) )
@ ( groups7311177749621191930dd_sum @ nat @ nat
@ ^ [J3: nat] : ( times_times @ nat @ ( B2 @ J3 ) @ ( power_power @ nat @ X @ J3 ) )
@ ( set_ord_atMost @ nat @ N3 ) ) )
= ( groups7311177749621191930dd_sum @ nat @ nat
@ ^ [R5: nat] :
( times_times @ nat
@ ( groups7311177749621191930dd_sum @ nat @ nat
@ ^ [K2: nat] : ( times_times @ nat @ ( A2 @ K2 ) @ ( B2 @ ( minus_minus @ nat @ R5 @ K2 ) ) )
@ ( set_ord_atMost @ nat @ R5 ) )
@ ( power_power @ nat @ X @ R5 ) )
@ ( set_ord_atMost @ nat @ ( plus_plus @ nat @ M @ N3 ) ) ) ) ) ) ).
% polynomial_product_nat
thf(fact_3318_pochhammer__binomial__sum,axiom,
! [A: $tType] :
( ( comm_ring_1 @ A )
=> ! [A2: A,B2: A,N3: nat] :
( ( comm_s3205402744901411588hammer @ A @ ( plus_plus @ A @ A2 @ B2 ) @ N3 )
= ( groups7311177749621191930dd_sum @ nat @ A
@ ^ [K2: nat] : ( times_times @ A @ ( times_times @ A @ ( semiring_1_of_nat @ A @ ( binomial @ N3 @ K2 ) ) @ ( comm_s3205402744901411588hammer @ A @ A2 @ K2 ) ) @ ( comm_s3205402744901411588hammer @ A @ B2 @ ( minus_minus @ nat @ N3 @ K2 ) ) )
@ ( set_ord_atMost @ nat @ N3 ) ) ) ) ).
% pochhammer_binomial_sum
thf(fact_3319_choose__square__sum,axiom,
! [N3: nat] :
( ( groups7311177749621191930dd_sum @ nat @ nat
@ ^ [K2: nat] : ( power_power @ nat @ ( binomial @ N3 @ K2 ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) )
@ ( set_ord_atMost @ nat @ N3 ) )
= ( binomial @ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N3 ) @ N3 ) ) ).
% choose_square_sum
thf(fact_3320_sum_Ozero__middle,axiom,
! [A: $tType] :
( ( comm_monoid_add @ A )
=> ! [P4: nat,K: nat,G: nat > A,H2: nat > A] :
( ( ord_less_eq @ nat @ ( one_one @ nat ) @ P4 )
=> ( ( ord_less_eq @ nat @ K @ P4 )
=> ( ( 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 @ P4 ) )
= ( groups7311177749621191930dd_sum @ nat @ A
@ ^ [J3: nat] : ( if @ A @ ( ord_less @ nat @ J3 @ K ) @ ( G @ J3 ) @ ( H2 @ J3 ) )
@ ( set_ord_atMost @ nat @ ( minus_minus @ nat @ P4 @ ( suc @ ( zero_zero @ nat ) ) ) ) ) ) ) ) ) ).
% sum.zero_middle
thf(fact_3321_prod_Ozero__middle,axiom,
! [A: $tType] :
( ( comm_monoid_mult @ A )
=> ! [P4: nat,K: nat,G: nat > A,H2: nat > A] :
( ( ord_less_eq @ nat @ ( one_one @ nat ) @ P4 )
=> ( ( ord_less_eq @ nat @ K @ P4 )
=> ( ( 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 @ P4 ) )
= ( groups7121269368397514597t_prod @ nat @ A
@ ^ [J3: nat] : ( if @ A @ ( ord_less @ nat @ J3 @ K ) @ ( G @ J3 ) @ ( H2 @ J3 ) )
@ ( set_ord_atMost @ nat @ ( minus_minus @ nat @ P4 @ ( suc @ ( zero_zero @ nat ) ) ) ) ) ) ) ) ) ).
% prod.zero_middle
thf(fact_3322_sum__gp0,axiom,
! [A: $tType] :
( ( ( division_ring @ A )
& ( comm_ring @ A ) )
=> ! [X: A,N3: nat] :
( ( ( X
= ( one_one @ A ) )
=> ( ( groups7311177749621191930dd_sum @ nat @ A @ ( power_power @ A @ X ) @ ( set_ord_atMost @ nat @ N3 ) )
= ( semiring_1_of_nat @ A @ ( plus_plus @ nat @ N3 @ ( one_one @ nat ) ) ) ) )
& ( ( X
!= ( one_one @ A ) )
=> ( ( groups7311177749621191930dd_sum @ nat @ A @ ( power_power @ A @ X ) @ ( set_ord_atMost @ nat @ N3 ) )
= ( divide_divide @ A @ ( minus_minus @ A @ ( one_one @ A ) @ ( power_power @ A @ X @ ( suc @ N3 ) ) ) @ ( minus_minus @ A @ ( one_one @ A ) @ X ) ) ) ) ) ) ).
% sum_gp0
thf(fact_3323_gbinomial__sum__nat__pow2,axiom,
! [A: $tType] :
( ( field_char_0 @ A )
=> ! [M: nat] :
( ( groups7311177749621191930dd_sum @ nat @ A
@ ^ [K2: nat] : ( divide_divide @ A @ ( gbinomial @ A @ ( semiring_1_of_nat @ A @ ( plus_plus @ nat @ M @ K2 ) ) @ K2 ) @ ( power_power @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ K2 ) )
@ ( set_ord_atMost @ nat @ M ) )
= ( power_power @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ M ) ) ) ).
% gbinomial_sum_nat_pow2
thf(fact_3324_gbinomial__partial__sum__poly__xpos,axiom,
! [A: $tType] :
( ( field_char_0 @ A )
=> ! [M: nat,A2: A,X: A,Y: A] :
( ( groups7311177749621191930dd_sum @ nat @ A
@ ^ [K2: nat] : ( times_times @ A @ ( times_times @ A @ ( gbinomial @ A @ ( plus_plus @ A @ ( semiring_1_of_nat @ A @ M ) @ A2 ) @ K2 ) @ ( power_power @ A @ X @ K2 ) ) @ ( power_power @ A @ Y @ ( minus_minus @ nat @ M @ K2 ) ) )
@ ( set_ord_atMost @ nat @ M ) )
= ( groups7311177749621191930dd_sum @ nat @ A
@ ^ [K2: nat] : ( times_times @ A @ ( times_times @ A @ ( gbinomial @ A @ ( minus_minus @ A @ ( plus_plus @ A @ ( semiring_1_of_nat @ A @ K2 ) @ A2 ) @ ( one_one @ A ) ) @ K2 ) @ ( power_power @ A @ X @ K2 ) ) @ ( power_power @ A @ ( plus_plus @ A @ X @ Y ) @ ( minus_minus @ nat @ M @ K2 ) ) )
@ ( set_ord_atMost @ nat @ M ) ) ) ) ).
% gbinomial_partial_sum_poly_xpos
thf(fact_3325_polyfun__diff__alt,axiom,
! [A: $tType] :
( ( idom @ A )
=> ! [N3: nat,A2: nat > A,X: A,Y: A] :
( ( ord_less_eq @ nat @ ( one_one @ nat ) @ N3 )
=> ( ( minus_minus @ A
@ ( groups7311177749621191930dd_sum @ nat @ A
@ ^ [I3: nat] : ( times_times @ A @ ( A2 @ I3 ) @ ( power_power @ A @ X @ I3 ) )
@ ( set_ord_atMost @ nat @ N3 ) )
@ ( groups7311177749621191930dd_sum @ nat @ A
@ ^ [I3: nat] : ( times_times @ A @ ( A2 @ I3 ) @ ( power_power @ A @ Y @ I3 ) )
@ ( set_ord_atMost @ nat @ N3 ) ) )
= ( times_times @ A @ ( minus_minus @ A @ X @ Y )
@ ( groups7311177749621191930dd_sum @ nat @ A
@ ^ [J3: nat] :
( groups7311177749621191930dd_sum @ nat @ A
@ ^ [K2: nat] : ( times_times @ A @ ( times_times @ A @ ( A2 @ ( plus_plus @ nat @ ( plus_plus @ nat @ J3 @ K2 ) @ ( one_one @ nat ) ) ) @ ( power_power @ A @ Y @ K2 ) ) @ ( power_power @ A @ X @ J3 ) )
@ ( set_ord_lessThan @ nat @ ( minus_minus @ nat @ N3 @ J3 ) ) )
@ ( set_ord_lessThan @ nat @ N3 ) ) ) ) ) ) ).
% polyfun_diff_alt
thf(fact_3326_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_3327_choose__linear__sum,axiom,
! [N3: nat] :
( ( groups7311177749621191930dd_sum @ nat @ nat
@ ^ [I3: nat] : ( times_times @ nat @ I3 @ ( binomial @ N3 @ I3 ) )
@ ( set_ord_atMost @ nat @ N3 ) )
= ( times_times @ nat @ N3 @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( minus_minus @ nat @ N3 @ ( one_one @ nat ) ) ) ) ) ).
% choose_linear_sum
thf(fact_3328_polyfun__extremal__lemma,axiom,
! [A: $tType] :
( ( real_V8999393235501362500lgebra @ A )
=> ! [E: real,C2: nat > A,N3: nat] :
( ( ord_less @ real @ ( zero_zero @ real ) @ E )
=> ? [M10: real] :
! [Z4: A] :
( ( ord_less_eq @ real @ M10 @ ( real_V7770717601297561774m_norm @ A @ Z4 ) )
=> ( ord_less_eq @ real
@ ( real_V7770717601297561774m_norm @ A
@ ( groups7311177749621191930dd_sum @ nat @ A
@ ^ [I3: nat] : ( times_times @ A @ ( C2 @ I3 ) @ ( power_power @ A @ Z4 @ I3 ) )
@ ( set_ord_atMost @ nat @ N3 ) ) )
@ ( times_times @ real @ E @ ( power_power @ real @ ( real_V7770717601297561774m_norm @ A @ Z4 ) @ ( suc @ N3 ) ) ) ) ) ) ) ).
% polyfun_extremal_lemma
thf(fact_3329_polyfun__diff,axiom,
! [A: $tType] :
( ( idom @ A )
=> ! [N3: nat,A2: nat > A,X: A,Y: A] :
( ( ord_less_eq @ nat @ ( one_one @ nat ) @ N3 )
=> ( ( minus_minus @ A
@ ( groups7311177749621191930dd_sum @ nat @ A
@ ^ [I3: nat] : ( times_times @ A @ ( A2 @ I3 ) @ ( power_power @ A @ X @ I3 ) )
@ ( set_ord_atMost @ nat @ N3 ) )
@ ( groups7311177749621191930dd_sum @ nat @ A
@ ^ [I3: nat] : ( times_times @ A @ ( A2 @ I3 ) @ ( power_power @ A @ Y @ I3 ) )
@ ( set_ord_atMost @ nat @ N3 ) ) )
= ( times_times @ A @ ( minus_minus @ A @ X @ Y )
@ ( groups7311177749621191930dd_sum @ nat @ A
@ ^ [J3: nat] :
( times_times @ A
@ ( groups7311177749621191930dd_sum @ nat @ A
@ ^ [I3: nat] : ( times_times @ A @ ( A2 @ I3 ) @ ( power_power @ A @ Y @ ( minus_minus @ nat @ ( minus_minus @ nat @ I3 @ J3 ) @ ( one_one @ nat ) ) ) )
@ ( set_or1337092689740270186AtMost @ nat @ ( suc @ J3 ) @ N3 ) )
@ ( power_power @ A @ X @ J3 ) )
@ ( set_ord_lessThan @ nat @ N3 ) ) ) ) ) ) ).
% polyfun_diff
thf(fact_3330_gbinomial__code,axiom,
! [A: $tType] :
( ( field_char_0 @ A )
=> ( ( gbinomial @ A )
= ( ^ [A6: A,K2: nat] :
( if @ A
@ ( K2
= ( zero_zero @ nat ) )
@ ( one_one @ A )
@ ( divide_divide @ A
@ ( set_fo6178422350223883121st_nat @ A
@ ^ [L: nat] : ( times_times @ A @ ( minus_minus @ A @ A6 @ ( semiring_1_of_nat @ A @ L ) ) )
@ ( zero_zero @ nat )
@ ( minus_minus @ nat @ K2 @ ( one_one @ nat ) )
@ ( one_one @ A ) )
@ ( semiring_char_0_fact @ A @ K2 ) ) ) ) ) ) ).
% gbinomial_code
thf(fact_3331_powr__int,axiom,
! [X: real,I: int] :
( ( ord_less @ real @ ( zero_zero @ real ) @ X )
=> ( ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ I )
=> ( ( powr @ real @ X @ ( ring_1_of_int @ real @ I ) )
= ( power_power @ real @ X @ ( nat2 @ I ) ) ) )
& ( ~ ( ord_less_eq @ int @ ( zero_zero @ int ) @ I )
=> ( ( powr @ real @ X @ ( ring_1_of_int @ real @ I ) )
= ( divide_divide @ real @ ( one_one @ real ) @ ( power_power @ real @ X @ ( nat2 @ ( uminus_uminus @ int @ I ) ) ) ) ) ) ) ) ).
% powr_int
thf(fact_3332_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_3333_choose__alternating__sum,axiom,
! [A: $tType] :
( ( comm_ring_1 @ A )
=> ! [N3: nat] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ N3 )
=> ( ( groups7311177749621191930dd_sum @ nat @ A
@ ^ [I3: nat] : ( times_times @ A @ ( power_power @ A @ ( uminus_uminus @ A @ ( one_one @ A ) ) @ I3 ) @ ( semiring_1_of_nat @ A @ ( binomial @ N3 @ I3 ) ) )
@ ( set_ord_atMost @ nat @ N3 ) )
= ( zero_zero @ A ) ) ) ) ).
% choose_alternating_sum
thf(fact_3334_predicate1I,axiom,
! [A: $tType,P: A > $o,Q: A > $o] :
( ! [X3: A] :
( ( P @ X3 )
=> ( Q @ X3 ) )
=> ( ord_less_eq @ ( A > $o ) @ P @ Q ) ) ).
% predicate1I
thf(fact_3335_order__refl,axiom,
! [A: $tType] :
( ( preorder @ A )
=> ! [X: A] : ( ord_less_eq @ A @ X @ X ) ) ).
% order_refl
thf(fact_3336_dual__order_Orefl,axiom,
! [A: $tType] :
( ( preorder @ A )
=> ! [A2: A] : ( ord_less_eq @ A @ A2 @ A2 ) ) ).
% dual_order.refl
thf(fact_3337_Compl__subset__Compl__iff,axiom,
! [A: $tType,A3: set @ A,B6: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ ( uminus_uminus @ ( set @ A ) @ A3 ) @ ( uminus_uminus @ ( set @ A ) @ B6 ) )
= ( ord_less_eq @ ( set @ A ) @ B6 @ A3 ) ) ).
% Compl_subset_Compl_iff
thf(fact_3338_Compl__anti__mono,axiom,
! [A: $tType,A3: set @ A,B6: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ A3 @ B6 )
=> ( ord_less_eq @ ( set @ A ) @ ( uminus_uminus @ ( set @ A ) @ B6 ) @ ( uminus_uminus @ ( set @ A ) @ A3 ) ) ) ).
% Compl_anti_mono
thf(fact_3339_of__nat__id,axiom,
( ( semiring_1_of_nat @ nat )
= ( ^ [N2: nat] : N2 ) ) ).
% of_nat_id
thf(fact_3340_neg__le__iff__le,axiom,
! [A: $tType] :
( ( ordered_ab_group_add @ A )
=> ! [B2: A,A2: A] :
( ( ord_less_eq @ A @ ( uminus_uminus @ A @ B2 ) @ ( uminus_uminus @ A @ A2 ) )
= ( ord_less_eq @ A @ A2 @ B2 ) ) ) ).
% neg_le_iff_le
thf(fact_3341_neg__less__iff__less,axiom,
! [A: $tType] :
( ( ordered_ab_group_add @ A )
=> ! [B2: A,A2: A] :
( ( ord_less @ A @ ( uminus_uminus @ A @ B2 ) @ ( uminus_uminus @ A @ A2 ) )
= ( ord_less @ A @ A2 @ B2 ) ) ) ).
% neg_less_iff_less
thf(fact_3342_neg__numeral__eq__iff,axiom,
! [A: $tType] :
( ( ring_char_0 @ A )
=> ! [M: num,N3: num] :
( ( ( uminus_uminus @ A @ ( numeral_numeral @ A @ M ) )
= ( uminus_uminus @ A @ ( numeral_numeral @ A @ N3 ) ) )
= ( M = N3 ) ) ) ).
% neg_numeral_eq_iff
thf(fact_3343_minus__add__distrib,axiom,
! [A: $tType] :
( ( ab_group_add @ A )
=> ! [A2: A,B2: A] :
( ( uminus_uminus @ A @ ( plus_plus @ A @ A2 @ B2 ) )
= ( plus_plus @ A @ ( uminus_uminus @ A @ A2 ) @ ( uminus_uminus @ A @ B2 ) ) ) ) ).
% minus_add_distrib
thf(fact_3344_minus__add__cancel,axiom,
! [A: $tType] :
( ( group_add @ A )
=> ! [A2: A,B2: A] :
( ( plus_plus @ A @ ( uminus_uminus @ A @ A2 ) @ ( plus_plus @ A @ A2 @ B2 ) )
= B2 ) ) ).
% minus_add_cancel
thf(fact_3345_add__minus__cancel,axiom,
! [A: $tType] :
( ( group_add @ A )
=> ! [A2: A,B2: A] :
( ( plus_plus @ A @ A2 @ ( plus_plus @ A @ ( uminus_uminus @ A @ A2 ) @ B2 ) )
= B2 ) ) ).
% add_minus_cancel
thf(fact_3346_mult__minus__right,axiom,
! [A: $tType] :
( ( ring @ A )
=> ! [A2: A,B2: A] :
( ( times_times @ A @ A2 @ ( uminus_uminus @ A @ B2 ) )
= ( uminus_uminus @ A @ ( times_times @ A @ A2 @ B2 ) ) ) ) ).
% mult_minus_right
thf(fact_3347_minus__mult__minus,axiom,
! [A: $tType] :
( ( ring @ A )
=> ! [A2: A,B2: A] :
( ( times_times @ A @ ( uminus_uminus @ A @ A2 ) @ ( uminus_uminus @ A @ B2 ) )
= ( times_times @ A @ A2 @ B2 ) ) ) ).
% minus_mult_minus
thf(fact_3348_mult__minus__left,axiom,
! [A: $tType] :
( ( ring @ A )
=> ! [A2: A,B2: A] :
( ( times_times @ A @ ( uminus_uminus @ A @ A2 ) @ B2 )
= ( uminus_uminus @ A @ ( times_times @ A @ A2 @ B2 ) ) ) ) ).
% mult_minus_left
thf(fact_3349_minus__diff__eq,axiom,
! [A: $tType] :
( ( group_add @ A )
=> ! [A2: A,B2: A] :
( ( uminus_uminus @ A @ ( minus_minus @ A @ A2 @ B2 ) )
= ( minus_minus @ A @ B2 @ A2 ) ) ) ).
% minus_diff_eq
thf(fact_3350_div__minus__minus,axiom,
! [A: $tType] :
( ( euclid8851590272496341667cancel @ A )
=> ! [A2: A,B2: A] :
( ( divide_divide @ A @ ( uminus_uminus @ A @ A2 ) @ ( uminus_uminus @ A @ B2 ) )
= ( divide_divide @ A @ A2 @ B2 ) ) ) ).
% div_minus_minus
thf(fact_3351_minus__dvd__iff,axiom,
! [A: $tType] :
( ( comm_ring_1 @ A )
=> ! [X: A,Y: A] :
( ( dvd_dvd @ A @ ( uminus_uminus @ A @ X ) @ Y )
= ( dvd_dvd @ A @ X @ Y ) ) ) ).
% minus_dvd_iff
thf(fact_3352_dvd__minus__iff,axiom,
! [A: $tType] :
( ( comm_ring_1 @ A )
=> ! [X: A,Y: A] :
( ( dvd_dvd @ A @ X @ ( uminus_uminus @ A @ Y ) )
= ( dvd_dvd @ A @ X @ Y ) ) ) ).
% dvd_minus_iff
thf(fact_3353_mod__minus__minus,axiom,
! [A: $tType] :
( ( euclid8851590272496341667cancel @ A )
=> ! [A2: A,B2: A] :
( ( modulo_modulo @ A @ ( uminus_uminus @ A @ A2 ) @ ( uminus_uminus @ A @ B2 ) )
= ( uminus_uminus @ A @ ( modulo_modulo @ A @ A2 @ B2 ) ) ) ) ).
% mod_minus_minus
thf(fact_3354_real__add__minus__iff,axiom,
! [X: real,A2: real] :
( ( ( plus_plus @ real @ X @ ( uminus_uminus @ real @ A2 ) )
= ( zero_zero @ real ) )
= ( X = A2 ) ) ).
% real_add_minus_iff
thf(fact_3355_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_3356_word__minus__one__le,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [X: word @ A] :
( ( ord_less_eq @ ( word @ A ) @ ( uminus_uminus @ ( word @ A ) @ ( one_one @ ( word @ A ) ) ) @ X )
= ( X
= ( uminus_uminus @ ( word @ A ) @ ( one_one @ ( word @ A ) ) ) ) ) ) ).
% word_minus_one_le
thf(fact_3357_max__word__max,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [N3: word @ A] : ( ord_less_eq @ ( word @ A ) @ N3 @ ( uminus_uminus @ ( word @ A ) @ ( one_one @ ( word @ A ) ) ) ) ) ).
% max_word_max
thf(fact_3358_word__n1__ge,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [Y: word @ A] : ( ord_less_eq @ ( word @ A ) @ Y @ ( uminus_uminus @ ( word @ A ) @ ( one_one @ ( word @ A ) ) ) ) ) ).
% word_n1_ge
thf(fact_3359_word__order_Oextremum__unique,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [A2: word @ A] :
( ( ord_less_eq @ ( word @ A ) @ ( uminus_uminus @ ( word @ A ) @ ( one_one @ ( word @ A ) ) ) @ A2 )
= ( A2
= ( uminus_uminus @ ( word @ A ) @ ( one_one @ ( word @ A ) ) ) ) ) ) ).
% word_order.extremum_unique
thf(fact_3360_word__order_Oextremum,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [A2: word @ A] : ( ord_less_eq @ ( word @ A ) @ A2 @ ( uminus_uminus @ ( word @ A ) @ ( one_one @ ( word @ A ) ) ) ) ) ).
% word_order.extremum
thf(fact_3361_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_3362_neg__less__eq__nonneg,axiom,
! [A: $tType] :
( ( linord5086331880401160121up_add @ A )
=> ! [A2: A] :
( ( ord_less_eq @ A @ ( uminus_uminus @ A @ A2 ) @ A2 )
= ( ord_less_eq @ A @ ( zero_zero @ A ) @ A2 ) ) ) ).
% neg_less_eq_nonneg
thf(fact_3363_less__eq__neg__nonpos,axiom,
! [A: $tType] :
( ( linord5086331880401160121up_add @ A )
=> ! [A2: A] :
( ( ord_less_eq @ A @ A2 @ ( uminus_uminus @ A @ A2 ) )
= ( ord_less_eq @ A @ A2 @ ( zero_zero @ A ) ) ) ) ).
% less_eq_neg_nonpos
thf(fact_3364_neg__le__0__iff__le,axiom,
! [A: $tType] :
( ( ordered_ab_group_add @ A )
=> ! [A2: A] :
( ( ord_less_eq @ A @ ( uminus_uminus @ A @ A2 ) @ ( zero_zero @ A ) )
= ( ord_less_eq @ A @ ( zero_zero @ A ) @ A2 ) ) ) ).
% neg_le_0_iff_le
thf(fact_3365_neg__0__le__iff__le,axiom,
! [A: $tType] :
( ( ordered_ab_group_add @ A )
=> ! [A2: A] :
( ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( uminus_uminus @ A @ A2 ) )
= ( ord_less_eq @ A @ A2 @ ( zero_zero @ A ) ) ) ) ).
% neg_0_le_iff_le
thf(fact_3366_neg__less__0__iff__less,axiom,
! [A: $tType] :
( ( ordered_ab_group_add @ A )
=> ! [A2: A] :
( ( ord_less @ A @ ( uminus_uminus @ A @ A2 ) @ ( zero_zero @ A ) )
= ( ord_less @ A @ ( zero_zero @ A ) @ A2 ) ) ) ).
% neg_less_0_iff_less
thf(fact_3367_neg__0__less__iff__less,axiom,
! [A: $tType] :
( ( ordered_ab_group_add @ A )
=> ! [A2: A] :
( ( ord_less @ A @ ( zero_zero @ A ) @ ( uminus_uminus @ A @ A2 ) )
= ( ord_less @ A @ A2 @ ( zero_zero @ A ) ) ) ) ).
% neg_0_less_iff_less
thf(fact_3368_neg__less__pos,axiom,
! [A: $tType] :
( ( linord5086331880401160121up_add @ A )
=> ! [A2: A] :
( ( ord_less @ A @ ( uminus_uminus @ A @ A2 ) @ A2 )
= ( ord_less @ A @ ( zero_zero @ A ) @ A2 ) ) ) ).
% neg_less_pos
thf(fact_3369_less__neg__neg,axiom,
! [A: $tType] :
( ( linord5086331880401160121up_add @ A )
=> ! [A2: A] :
( ( ord_less @ A @ A2 @ ( uminus_uminus @ A @ A2 ) )
= ( ord_less @ A @ A2 @ ( zero_zero @ A ) ) ) ) ).
% less_neg_neg
thf(fact_3370_ab__left__minus,axiom,
! [A: $tType] :
( ( group_add @ A )
=> ! [A2: A] :
( ( plus_plus @ A @ ( uminus_uminus @ A @ A2 ) @ A2 )
= ( zero_zero @ A ) ) ) ).
% ab_left_minus
thf(fact_3371_add_Oright__inverse,axiom,
! [A: $tType] :
( ( group_add @ A )
=> ! [A2: A] :
( ( plus_plus @ A @ A2 @ ( uminus_uminus @ A @ A2 ) )
= ( zero_zero @ A ) ) ) ).
% add.right_inverse
thf(fact_3372_add__neg__numeral__simps_I3_J,axiom,
! [A: $tType] :
( ( neg_numeral @ A )
=> ! [M: num,N3: num] :
( ( plus_plus @ A @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ M ) ) @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ N3 ) ) )
= ( uminus_uminus @ A @ ( plus_plus @ A @ ( numeral_numeral @ A @ M ) @ ( numeral_numeral @ A @ N3 ) ) ) ) ) ).
% add_neg_numeral_simps(3)
thf(fact_3373_diff__0,axiom,
! [A: $tType] :
( ( group_add @ A )
=> ! [A2: A] :
( ( minus_minus @ A @ ( zero_zero @ A ) @ A2 )
= ( uminus_uminus @ A @ A2 ) ) ) ).
% diff_0
thf(fact_3374_verit__minus__simplify_I3_J,axiom,
! [B: $tType] :
( ( group_add @ B )
=> ! [B2: B] :
( ( minus_minus @ B @ ( zero_zero @ B ) @ B2 )
= ( uminus_uminus @ B @ B2 ) ) ) ).
% verit_minus_simplify(3)
thf(fact_3375_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_3376_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_3377_diff__minus__eq__add,axiom,
! [A: $tType] :
( ( group_add @ A )
=> ! [A2: A,B2: A] :
( ( minus_minus @ A @ A2 @ ( uminus_uminus @ A @ B2 ) )
= ( plus_plus @ A @ A2 @ B2 ) ) ) ).
% diff_minus_eq_add
thf(fact_3378_uminus__add__conv__diff,axiom,
! [A: $tType] :
( ( ab_group_add @ A )
=> ! [A2: A,B2: A] :
( ( plus_plus @ A @ ( uminus_uminus @ A @ A2 ) @ B2 )
= ( minus_minus @ A @ B2 @ A2 ) ) ) ).
% uminus_add_conv_diff
thf(fact_3379_divide__minus1,axiom,
! [A: $tType] :
( ( field @ A )
=> ! [X: A] :
( ( divide_divide @ A @ X @ ( uminus_uminus @ A @ ( one_one @ A ) ) )
= ( uminus_uminus @ A @ X ) ) ) ).
% divide_minus1
thf(fact_3380_div__minus1__right,axiom,
! [A: $tType] :
( ( euclid8851590272496341667cancel @ A )
=> ! [A2: A] :
( ( divide_divide @ A @ A2 @ ( uminus_uminus @ A @ ( one_one @ A ) ) )
= ( uminus_uminus @ A @ A2 ) ) ) ).
% div_minus1_right
thf(fact_3381_minus__mod__self1,axiom,
! [A: $tType] :
( ( euclid8851590272496341667cancel @ A )
=> ! [B2: A,A2: A] :
( ( modulo_modulo @ A @ ( minus_minus @ A @ B2 @ A2 ) @ B2 )
= ( modulo_modulo @ A @ ( uminus_uminus @ A @ A2 ) @ B2 ) ) ) ).
% minus_mod_self1
thf(fact_3382_subset__Compl__singleton,axiom,
! [A: $tType,A3: set @ A,B2: A] :
( ( ord_less_eq @ ( set @ A ) @ A3 @ ( uminus_uminus @ ( set @ A ) @ ( insert @ A @ B2 @ ( bot_bot @ ( set @ A ) ) ) ) )
= ( ~ ( member @ A @ B2 @ A3 ) ) ) ).
% subset_Compl_singleton
thf(fact_3383_fact__0,axiom,
! [A: $tType] :
( ( semiring_char_0 @ A )
=> ( ( semiring_char_0_fact @ A @ ( zero_zero @ nat ) )
= ( one_one @ A ) ) ) ).
% fact_0
thf(fact_3384_fact__1,axiom,
! [A: $tType] :
( ( semiring_char_0 @ A )
=> ( ( semiring_char_0_fact @ A @ ( one_one @ nat ) )
= ( one_one @ A ) ) ) ).
% fact_1
thf(fact_3385_negative__zle,axiom,
! [N3: nat,M: nat] : ( ord_less_eq @ int @ ( uminus_uminus @ int @ ( semiring_1_of_nat @ int @ N3 ) ) @ ( semiring_1_of_nat @ int @ M ) ) ).
% negative_zle
thf(fact_3386_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_3387_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_3388_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_3389_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_3390_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_3391_numeral__eq__neg__one__iff,axiom,
! [A: $tType] :
( ( ring_char_0 @ A )
=> ! [N3: num] :
( ( ( uminus_uminus @ A @ ( numeral_numeral @ A @ N3 ) )
= ( uminus_uminus @ A @ ( one_one @ A ) ) )
= ( N3 = one2 ) ) ) ).
% numeral_eq_neg_one_iff
thf(fact_3392_neg__one__eq__numeral__iff,axiom,
! [A: $tType] :
( ( ring_char_0 @ A )
=> ! [N3: num] :
( ( ( uminus_uminus @ A @ ( one_one @ A ) )
= ( uminus_uminus @ A @ ( numeral_numeral @ A @ N3 ) ) )
= ( N3 = one2 ) ) ) ).
% neg_one_eq_numeral_iff
thf(fact_3393_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_3394_left__minus__one__mult__self,axiom,
! [A: $tType] :
( ( comm_ring_1 @ A )
=> ! [N3: nat,A2: A] :
( ( times_times @ A @ ( power_power @ A @ ( uminus_uminus @ A @ ( one_one @ A ) ) @ N3 ) @ ( times_times @ A @ ( power_power @ A @ ( uminus_uminus @ A @ ( one_one @ A ) ) @ N3 ) @ A2 ) )
= A2 ) ) ).
% left_minus_one_mult_self
thf(fact_3395_minus__one__mult__self,axiom,
! [A: $tType] :
( ( comm_ring_1 @ A )
=> ! [N3: nat] :
( ( times_times @ A @ ( power_power @ A @ ( uminus_uminus @ A @ ( one_one @ A ) ) @ N3 ) @ ( power_power @ A @ ( uminus_uminus @ A @ ( one_one @ A ) ) @ N3 ) )
= ( one_one @ A ) ) ) ).
% minus_one_mult_self
thf(fact_3396_mod__minus1__right,axiom,
! [A: $tType] :
( ( euclid8851590272496341667cancel @ A )
=> ! [A2: A] :
( ( modulo_modulo @ A @ A2 @ ( uminus_uminus @ A @ ( one_one @ A ) ) )
= ( zero_zero @ A ) ) ) ).
% mod_minus1_right
thf(fact_3397_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_3398_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_3399_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_3400_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_3401_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_3402_fact__Suc,axiom,
! [A: $tType] :
( ( semiring_char_0 @ A )
=> ! [N3: nat] :
( ( semiring_char_0_fact @ A @ ( suc @ N3 ) )
= ( times_times @ A @ ( semiring_1_of_nat @ A @ ( suc @ N3 ) ) @ ( semiring_char_0_fact @ A @ N3 ) ) ) ) ).
% fact_Suc
thf(fact_3403_semiring__norm_I168_J,axiom,
! [A: $tType] :
( ( neg_numeral @ A )
=> ! [V: num,W: num,Y: A] :
( ( plus_plus @ A @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ V ) ) @ ( plus_plus @ A @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ W ) ) @ Y ) )
= ( plus_plus @ A @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ ( plus_plus @ num @ V @ W ) ) ) @ Y ) ) ) ).
% semiring_norm(168)
thf(fact_3404_diff__numeral__simps_I3_J,axiom,
! [A: $tType] :
( ( neg_numeral @ A )
=> ! [M: num,N3: num] :
( ( minus_minus @ A @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ M ) ) @ ( numeral_numeral @ A @ N3 ) )
= ( uminus_uminus @ A @ ( numeral_numeral @ A @ ( plus_plus @ num @ M @ N3 ) ) ) ) ) ).
% diff_numeral_simps(3)
thf(fact_3405_diff__numeral__simps_I2_J,axiom,
! [A: $tType] :
( ( neg_numeral @ A )
=> ! [M: num,N3: num] :
( ( minus_minus @ A @ ( numeral_numeral @ A @ M ) @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ N3 ) ) )
= ( numeral_numeral @ A @ ( plus_plus @ num @ M @ N3 ) ) ) ) ).
% diff_numeral_simps(2)
thf(fact_3406_negative__zless,axiom,
! [N3: nat,M: nat] : ( ord_less @ int @ ( uminus_uminus @ int @ ( semiring_1_of_nat @ int @ ( suc @ N3 ) ) ) @ ( semiring_1_of_nat @ int @ M ) ) ).
% negative_zless
thf(fact_3407_semiring__norm_I170_J,axiom,
! [A: $tType] :
( ( ring_1 @ A )
=> ! [V: num,W: num,Y: A] :
( ( times_times @ A @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ V ) ) @ ( times_times @ A @ ( numeral_numeral @ A @ W ) @ Y ) )
= ( times_times @ A @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ ( times_times @ num @ V @ W ) ) ) @ Y ) ) ) ).
% semiring_norm(170)
thf(fact_3408_semiring__norm_I171_J,axiom,
! [A: $tType] :
( ( ring_1 @ A )
=> ! [V: num,W: num,Y: A] :
( ( times_times @ A @ ( numeral_numeral @ A @ V ) @ ( times_times @ A @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ W ) ) @ Y ) )
= ( times_times @ A @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ ( times_times @ num @ V @ W ) ) ) @ Y ) ) ) ).
% semiring_norm(171)
thf(fact_3409_semiring__norm_I172_J,axiom,
! [A: $tType] :
( ( ring_1 @ A )
=> ! [V: num,W: num,Y: A] :
( ( times_times @ A @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ V ) ) @ ( times_times @ A @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ W ) ) @ Y ) )
= ( times_times @ A @ ( numeral_numeral @ A @ ( times_times @ num @ V @ W ) ) @ Y ) ) ) ).
% semiring_norm(172)
thf(fact_3410_mult__neg__numeral__simps_I1_J,axiom,
! [A: $tType] :
( ( ring_1 @ A )
=> ! [M: num,N3: num] :
( ( times_times @ A @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ M ) ) @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ N3 ) ) )
= ( numeral_numeral @ A @ ( times_times @ num @ M @ N3 ) ) ) ) ).
% mult_neg_numeral_simps(1)
thf(fact_3411_mult__neg__numeral__simps_I2_J,axiom,
! [A: $tType] :
( ( ring_1 @ A )
=> ! [M: num,N3: num] :
( ( times_times @ A @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ M ) ) @ ( numeral_numeral @ A @ N3 ) )
= ( uminus_uminus @ A @ ( numeral_numeral @ A @ ( times_times @ num @ M @ N3 ) ) ) ) ) ).
% mult_neg_numeral_simps(2)
thf(fact_3412_mult__neg__numeral__simps_I3_J,axiom,
! [A: $tType] :
( ( ring_1 @ A )
=> ! [M: num,N3: num] :
( ( times_times @ A @ ( numeral_numeral @ A @ M ) @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ N3 ) ) )
= ( uminus_uminus @ A @ ( numeral_numeral @ A @ ( times_times @ num @ M @ N3 ) ) ) ) ) ).
% mult_neg_numeral_simps(3)
thf(fact_3413_neg__numeral__le__iff,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [M: num,N3: num] :
( ( ord_less_eq @ A @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ M ) ) @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ N3 ) ) )
= ( ord_less_eq @ num @ N3 @ M ) ) ) ).
% neg_numeral_le_iff
thf(fact_3414_neg__numeral__less__iff,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [M: num,N3: num] :
( ( ord_less @ A @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ M ) ) @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ N3 ) ) )
= ( ord_less @ num @ N3 @ M ) ) ) ).
% neg_numeral_less_iff
thf(fact_3415_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_3416_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_3417_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_3418_eq__divide__eq__numeral1_I2_J,axiom,
! [A: $tType] :
( ( division_ring @ A )
=> ! [A2: A,B2: A,W: num] :
( ( A2
= ( divide_divide @ A @ B2 @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ W ) ) ) )
= ( ( ( ( uminus_uminus @ A @ ( numeral_numeral @ A @ W ) )
!= ( zero_zero @ A ) )
=> ( ( times_times @ A @ A2 @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ W ) ) )
= B2 ) )
& ( ( ( uminus_uminus @ A @ ( numeral_numeral @ A @ W ) )
= ( zero_zero @ A ) )
=> ( A2
= ( zero_zero @ A ) ) ) ) ) ) ).
% eq_divide_eq_numeral1(2)
thf(fact_3419_divide__eq__eq__numeral1_I2_J,axiom,
! [A: $tType] :
( ( division_ring @ A )
=> ! [B2: A,W: num,A2: A] :
( ( ( divide_divide @ A @ B2 @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ W ) ) )
= A2 )
= ( ( ( ( uminus_uminus @ A @ ( numeral_numeral @ A @ W ) )
!= ( zero_zero @ A ) )
=> ( B2
= ( times_times @ A @ A2 @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ W ) ) ) ) )
& ( ( ( uminus_uminus @ A @ ( numeral_numeral @ A @ W ) )
= ( zero_zero @ A ) )
=> ( A2
= ( zero_zero @ A ) ) ) ) ) ) ).
% divide_eq_eq_numeral1(2)
thf(fact_3420_le__divide__eq__numeral1_I2_J,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [A2: A,B2: A,W: num] :
( ( ord_less_eq @ A @ A2 @ ( divide_divide @ A @ B2 @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ W ) ) ) )
= ( ord_less_eq @ A @ B2 @ ( times_times @ A @ A2 @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ W ) ) ) ) ) ) ).
% le_divide_eq_numeral1(2)
thf(fact_3421_divide__le__eq__numeral1_I2_J,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [B2: A,W: num,A2: A] :
( ( ord_less_eq @ A @ ( divide_divide @ A @ B2 @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ W ) ) ) @ A2 )
= ( ord_less_eq @ A @ ( times_times @ A @ A2 @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ W ) ) ) @ B2 ) ) ) ).
% divide_le_eq_numeral1(2)
thf(fact_3422_less__divide__eq__numeral1_I2_J,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [A2: A,B2: A,W: num] :
( ( ord_less @ A @ A2 @ ( divide_divide @ A @ B2 @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ W ) ) ) )
= ( ord_less @ A @ B2 @ ( times_times @ A @ A2 @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ W ) ) ) ) ) ) ).
% less_divide_eq_numeral1(2)
thf(fact_3423_divide__less__eq__numeral1_I2_J,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [B2: A,W: num,A2: A] :
( ( ord_less @ A @ ( divide_divide @ A @ B2 @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ W ) ) ) @ A2 )
= ( ord_less @ A @ ( times_times @ A @ A2 @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ W ) ) ) @ B2 ) ) ) ).
% divide_less_eq_numeral1(2)
thf(fact_3424_power2__minus,axiom,
! [A: $tType] :
( ( ring_1 @ A )
=> ! [A2: A] :
( ( power_power @ A @ ( uminus_uminus @ A @ A2 ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) )
= ( power_power @ A @ A2 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ).
% power2_minus
thf(fact_3425_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_3426_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_3427_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_3428_int__div__minus__is__minus1,axiom,
! [A2: int,B2: int] :
( ( ord_less @ int @ A2 @ ( zero_zero @ int ) )
=> ( ( ( divide_divide @ int @ A2 @ B2 )
= ( uminus_uminus @ int @ A2 ) )
= ( B2
= ( uminus_uminus @ int @ ( one_one @ int ) ) ) ) ) ).
% int_div_minus_is_minus1
thf(fact_3429_round__neg__numeral,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [N3: num] :
( ( archimedean_round @ A @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ N3 ) ) )
= ( uminus_uminus @ int @ ( numeral_numeral @ int @ N3 ) ) ) ) ).
% round_neg_numeral
thf(fact_3430_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_3431_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_3432_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_3433_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_3434_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_3435_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_3436_Power_Oring__1__class_Opower__minus__even,axiom,
! [A: $tType] :
( ( ring_1 @ A )
=> ! [A2: A,N3: nat] :
( ( power_power @ A @ ( uminus_uminus @ A @ A2 ) @ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N3 ) )
= ( power_power @ A @ A2 @ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N3 ) ) ) ) ).
% Power.ring_1_class.power_minus_even
thf(fact_3437_Parity_Oring__1__class_Opower__minus__even,axiom,
! [A: $tType] :
( ( ring_1 @ A )
=> ! [N3: nat,A2: A] :
( ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N3 )
=> ( ( power_power @ A @ ( uminus_uminus @ A @ A2 ) @ N3 )
= ( power_power @ A @ A2 @ N3 ) ) ) ) ).
% Parity.ring_1_class.power_minus_even
thf(fact_3438_power__minus__odd,axiom,
! [A: $tType] :
( ( ring_1 @ A )
=> ! [N3: nat,A2: A] :
( ~ ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N3 )
=> ( ( power_power @ A @ ( uminus_uminus @ A @ A2 ) @ N3 )
= ( uminus_uminus @ A @ ( power_power @ A @ A2 @ N3 ) ) ) ) ) ).
% power_minus_odd
thf(fact_3439_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_3440_diff__numeral__special_I3_J,axiom,
! [A: $tType] :
( ( neg_numeral @ A )
=> ! [N3: num] :
( ( minus_minus @ A @ ( one_one @ A ) @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ N3 ) ) )
= ( numeral_numeral @ A @ ( plus_plus @ num @ one2 @ N3 ) ) ) ) ).
% diff_numeral_special(3)
thf(fact_3441_ceiling__less__zero,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [X: A] :
( ( ord_less @ int @ ( archimedean_ceiling @ A @ X ) @ ( zero_zero @ int ) )
= ( ord_less_eq @ A @ X @ ( uminus_uminus @ A @ ( one_one @ A ) ) ) ) ) ).
% ceiling_less_zero
thf(fact_3442_zero__le__ceiling,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [X: A] :
( ( ord_less_eq @ int @ ( zero_zero @ int ) @ ( archimedean_ceiling @ A @ X ) )
= ( ord_less @ A @ ( uminus_uminus @ A @ ( one_one @ A ) ) @ X ) ) ) ).
% zero_le_ceiling
thf(fact_3443_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_3444_floor__minus__divide__eq__div__numeral,axiom,
! [A2: num,B2: num] :
( ( archim6421214686448440834_floor @ real @ ( uminus_uminus @ real @ ( divide_divide @ real @ ( numeral_numeral @ real @ A2 ) @ ( numeral_numeral @ real @ B2 ) ) ) )
= ( divide_divide @ int @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ A2 ) ) @ ( numeral_numeral @ int @ B2 ) ) ) ).
% floor_minus_divide_eq_div_numeral
thf(fact_3445_ceiling__minus__divide__eq__div__numeral,axiom,
! [A2: num,B2: num] :
( ( archimedean_ceiling @ real @ ( uminus_uminus @ real @ ( divide_divide @ real @ ( numeral_numeral @ real @ A2 ) @ ( numeral_numeral @ real @ B2 ) ) ) )
= ( uminus_uminus @ int @ ( divide_divide @ int @ ( numeral_numeral @ int @ A2 ) @ ( numeral_numeral @ int @ B2 ) ) ) ) ).
% ceiling_minus_divide_eq_div_numeral
thf(fact_3446_ceiling__divide__eq__div__numeral,axiom,
! [A2: num,B2: num] :
( ( archimedean_ceiling @ real @ ( divide_divide @ real @ ( numeral_numeral @ real @ A2 ) @ ( numeral_numeral @ real @ B2 ) ) )
= ( uminus_uminus @ int @ ( divide_divide @ int @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ A2 ) ) @ ( numeral_numeral @ int @ B2 ) ) ) ) ).
% ceiling_divide_eq_div_numeral
thf(fact_3447_power__minus1__even,axiom,
! [A: $tType] :
( ( ring_1 @ A )
=> ! [N3: nat] :
( ( power_power @ A @ ( uminus_uminus @ A @ ( one_one @ A ) ) @ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N3 ) )
= ( one_one @ A ) ) ) ).
% power_minus1_even
thf(fact_3448_neg__one__even__power,axiom,
! [A: $tType] :
( ( ring_1 @ A )
=> ! [N3: nat] :
( ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N3 )
=> ( ( power_power @ A @ ( uminus_uminus @ A @ ( one_one @ A ) ) @ N3 )
= ( one_one @ A ) ) ) ) ).
% neg_one_even_power
thf(fact_3449_neg__one__odd__power,axiom,
! [A: $tType] :
( ( ring_1 @ A )
=> ! [N3: nat] :
( ~ ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N3 )
=> ( ( power_power @ A @ ( uminus_uminus @ A @ ( one_one @ A ) ) @ N3 )
= ( uminus_uminus @ A @ ( one_one @ A ) ) ) ) ) ).
% neg_one_odd_power
thf(fact_3450_of__int__eq__neg__numeral__power__cancel__iff,axiom,
! [A: $tType] :
( ( ring_char_0 @ A )
=> ! [Y: int,X: num,N3: nat] :
( ( ( ring_1_of_int @ A @ Y )
= ( power_power @ A @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ X ) ) @ N3 ) )
= ( Y
= ( power_power @ int @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ X ) ) @ N3 ) ) ) ) ).
% of_int_eq_neg_numeral_power_cancel_iff
thf(fact_3451_neg__numeral__power__eq__of__int__cancel__iff,axiom,
! [A: $tType] :
( ( ring_char_0 @ A )
=> ! [X: num,N3: nat,Y: int] :
( ( ( power_power @ A @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ X ) ) @ N3 )
= ( ring_1_of_int @ A @ Y ) )
= ( ( power_power @ int @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ X ) ) @ N3 )
= Y ) ) ) ).
% neg_numeral_power_eq_of_int_cancel_iff
thf(fact_3452_neg__numeral__le__floor,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [V: num,X: A] :
( ( ord_less_eq @ int @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ V ) ) @ ( archim6421214686448440834_floor @ A @ X ) )
= ( ord_less_eq @ A @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ V ) ) @ X ) ) ) ).
% neg_numeral_le_floor
thf(fact_3453_floor__less__neg__numeral,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [X: A,V: num] :
( ( ord_less @ int @ ( archim6421214686448440834_floor @ A @ X ) @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ V ) ) )
= ( ord_less @ A @ X @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ V ) ) ) ) ) ).
% floor_less_neg_numeral
thf(fact_3454_ceiling__le__neg__numeral,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [X: A,V: num] :
( ( ord_less_eq @ int @ ( archimedean_ceiling @ A @ X ) @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ V ) ) )
= ( ord_less_eq @ A @ X @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ V ) ) ) ) ) ).
% ceiling_le_neg_numeral
thf(fact_3455_neg__numeral__less__ceiling,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [V: num,X: A] :
( ( ord_less @ int @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ V ) ) @ ( archimedean_ceiling @ A @ X ) )
= ( ord_less @ A @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ V ) ) @ X ) ) ) ).
% neg_numeral_less_ceiling
thf(fact_3456_floor__minus__one__divide__eq__div__numeral,axiom,
! [B2: num] :
( ( archim6421214686448440834_floor @ real @ ( uminus_uminus @ real @ ( divide_divide @ real @ ( one_one @ real ) @ ( numeral_numeral @ real @ B2 ) ) ) )
= ( divide_divide @ int @ ( uminus_uminus @ int @ ( one_one @ int ) ) @ ( numeral_numeral @ int @ B2 ) ) ) ).
% floor_minus_one_divide_eq_div_numeral
thf(fact_3457_neg__numeral__less__floor,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [V: num,X: A] :
( ( ord_less @ int @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ V ) ) @ ( archim6421214686448440834_floor @ A @ X ) )
= ( ord_less_eq @ A @ ( plus_plus @ A @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ V ) ) @ ( one_one @ A ) ) @ X ) ) ) ).
% neg_numeral_less_floor
thf(fact_3458_floor__le__neg__numeral,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [X: A,V: num] :
( ( ord_less_eq @ int @ ( archim6421214686448440834_floor @ A @ X ) @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ V ) ) )
= ( ord_less @ A @ X @ ( plus_plus @ A @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ V ) ) @ ( one_one @ A ) ) ) ) ) ).
% floor_le_neg_numeral
thf(fact_3459_ceiling__less__neg__numeral,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [X: A,V: num] :
( ( ord_less @ int @ ( archimedean_ceiling @ A @ X ) @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ V ) ) )
= ( ord_less_eq @ A @ X @ ( minus_minus @ A @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ V ) ) @ ( one_one @ A ) ) ) ) ) ).
% ceiling_less_neg_numeral
thf(fact_3460_neg__numeral__le__ceiling,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [V: num,X: A] :
( ( ord_less_eq @ int @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ V ) ) @ ( archimedean_ceiling @ A @ X ) )
= ( ord_less @ A @ ( minus_minus @ A @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ V ) ) @ ( one_one @ A ) ) @ X ) ) ) ).
% neg_numeral_le_ceiling
thf(fact_3461_neg__numeral__power__le__of__int__cancel__iff,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [X: num,N3: nat,A2: int] :
( ( ord_less_eq @ A @ ( power_power @ A @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ X ) ) @ N3 ) @ ( ring_1_of_int @ A @ A2 ) )
= ( ord_less_eq @ int @ ( power_power @ int @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ X ) ) @ N3 ) @ A2 ) ) ) ).
% neg_numeral_power_le_of_int_cancel_iff
thf(fact_3462_of__int__le__neg__numeral__power__cancel__iff,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [A2: int,X: num,N3: nat] :
( ( ord_less_eq @ A @ ( ring_1_of_int @ A @ A2 ) @ ( power_power @ A @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ X ) ) @ N3 ) )
= ( ord_less_eq @ int @ A2 @ ( power_power @ int @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ X ) ) @ N3 ) ) ) ) ).
% of_int_le_neg_numeral_power_cancel_iff
thf(fact_3463_of__int__less__neg__numeral__power__cancel__iff,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [A2: int,X: num,N3: nat] :
( ( ord_less @ A @ ( ring_1_of_int @ A @ A2 ) @ ( power_power @ A @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ X ) ) @ N3 ) )
= ( ord_less @ int @ A2 @ ( power_power @ int @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ X ) ) @ N3 ) ) ) ) ).
% of_int_less_neg_numeral_power_cancel_iff
thf(fact_3464_neg__numeral__power__less__of__int__cancel__iff,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [X: num,N3: nat,A2: int] :
( ( ord_less @ A @ ( power_power @ A @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ X ) ) @ N3 ) @ ( ring_1_of_int @ A @ A2 ) )
= ( ord_less @ int @ ( power_power @ int @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ X ) ) @ N3 ) @ A2 ) ) ) ).
% neg_numeral_power_less_of_int_cancel_iff
thf(fact_3465_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_3466_is__num__normalize_I8_J,axiom,
! [A: $tType] :
( ( neg_numeral @ A )
=> ! [A2: A,B2: A] :
( ( uminus_uminus @ A @ ( plus_plus @ A @ A2 @ B2 ) )
= ( plus_plus @ A @ ( uminus_uminus @ A @ B2 ) @ ( uminus_uminus @ A @ A2 ) ) ) ) ).
% is_num_normalize(8)
thf(fact_3467_group__cancel_Oneg1,axiom,
! [A: $tType] :
( ( ab_group_add @ A )
=> ! [A3: A,K: A,A2: A] :
( ( A3
= ( plus_plus @ A @ K @ A2 ) )
=> ( ( uminus_uminus @ A @ A3 )
= ( plus_plus @ A @ ( uminus_uminus @ A @ K ) @ ( uminus_uminus @ A @ A2 ) ) ) ) ) ).
% group_cancel.neg1
thf(fact_3468_add_Oinverse__distrib__swap,axiom,
! [A: $tType] :
( ( group_add @ A )
=> ! [A2: A,B2: A] :
( ( uminus_uminus @ A @ ( plus_plus @ A @ A2 @ B2 ) )
= ( plus_plus @ A @ ( uminus_uminus @ A @ B2 ) @ ( uminus_uminus @ A @ A2 ) ) ) ) ).
% add.inverse_distrib_swap
thf(fact_3469_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_3470_square__eq__iff,axiom,
! [A: $tType] :
( ( idom @ A )
=> ! [A2: A,B2: A] :
( ( ( times_times @ A @ A2 @ A2 )
= ( times_times @ A @ B2 @ B2 ) )
= ( ( A2 = B2 )
| ( A2
= ( uminus_uminus @ A @ B2 ) ) ) ) ) ).
% square_eq_iff
thf(fact_3471_minus__mult__commute,axiom,
! [A: $tType] :
( ( ring @ A )
=> ! [A2: A,B2: A] :
( ( times_times @ A @ ( uminus_uminus @ A @ A2 ) @ B2 )
= ( times_times @ A @ A2 @ ( uminus_uminus @ A @ B2 ) ) ) ) ).
% minus_mult_commute
thf(fact_3472_minus__diff__minus,axiom,
! [A: $tType] :
( ( ab_group_add @ A )
=> ! [A2: A,B2: A] :
( ( minus_minus @ A @ ( uminus_uminus @ A @ A2 ) @ ( uminus_uminus @ A @ B2 ) )
= ( uminus_uminus @ A @ ( minus_minus @ A @ A2 @ B2 ) ) ) ) ).
% minus_diff_minus
thf(fact_3473_minus__diff__commute,axiom,
! [A: $tType] :
( ( ab_group_add @ A )
=> ! [B2: A,A2: A] :
( ( minus_minus @ A @ ( uminus_uminus @ A @ B2 ) @ A2 )
= ( minus_minus @ A @ ( uminus_uminus @ A @ A2 ) @ B2 ) ) ) ).
% minus_diff_commute
thf(fact_3474_div__minus__right,axiom,
! [A: $tType] :
( ( euclid8851590272496341667cancel @ A )
=> ! [A2: A,B2: A] :
( ( divide_divide @ A @ A2 @ ( uminus_uminus @ A @ B2 ) )
= ( divide_divide @ A @ ( uminus_uminus @ A @ A2 ) @ B2 ) ) ) ).
% div_minus_right
thf(fact_3475_minus__divide__right,axiom,
! [A: $tType] :
( ( field @ A )
=> ! [A2: A,B2: A] :
( ( uminus_uminus @ A @ ( divide_divide @ A @ A2 @ B2 ) )
= ( divide_divide @ A @ A2 @ ( uminus_uminus @ A @ B2 ) ) ) ) ).
% minus_divide_right
thf(fact_3476_minus__divide__divide,axiom,
! [A: $tType] :
( ( field @ A )
=> ! [A2: A,B2: A] :
( ( divide_divide @ A @ ( uminus_uminus @ A @ A2 ) @ ( uminus_uminus @ A @ B2 ) )
= ( divide_divide @ A @ A2 @ B2 ) ) ) ).
% minus_divide_divide
thf(fact_3477_minus__divide__left,axiom,
! [A: $tType] :
( ( division_ring @ A )
=> ! [A2: A,B2: A] :
( ( uminus_uminus @ A @ ( divide_divide @ A @ A2 @ B2 ) )
= ( divide_divide @ A @ ( uminus_uminus @ A @ A2 ) @ B2 ) ) ) ).
% minus_divide_left
thf(fact_3478_fact__mono__nat,axiom,
! [M: nat,N3: nat] :
( ( ord_less_eq @ nat @ M @ N3 )
=> ( ord_less_eq @ nat @ ( semiring_char_0_fact @ nat @ M ) @ ( semiring_char_0_fact @ nat @ N3 ) ) ) ).
% fact_mono_nat
thf(fact_3479_fact__ge__self,axiom,
! [N3: nat] : ( ord_less_eq @ nat @ N3 @ ( semiring_char_0_fact @ nat @ N3 ) ) ).
% fact_ge_self
thf(fact_3480_neg__numeral__neq__numeral,axiom,
! [A: $tType] :
( ( ring_char_0 @ A )
=> ! [M: num,N3: num] :
( ( uminus_uminus @ A @ ( numeral_numeral @ A @ M ) )
!= ( numeral_numeral @ A @ N3 ) ) ) ).
% neg_numeral_neq_numeral
thf(fact_3481_numeral__neq__neg__numeral,axiom,
! [A: $tType] :
( ( ring_char_0 @ A )
=> ! [M: num,N3: num] :
( ( numeral_numeral @ A @ M )
!= ( uminus_uminus @ A @ ( numeral_numeral @ A @ N3 ) ) ) ) ).
% numeral_neq_neg_numeral
thf(fact_3482_verit__negate__coefficient_I2_J,axiom,
! [A: $tType] :
( ( ordered_ab_group_add @ A )
=> ! [A2: A,B2: A] :
( ( ord_less @ A @ A2 @ B2 )
=> ( ord_less @ A @ ( uminus_uminus @ A @ B2 ) @ ( uminus_uminus @ A @ A2 ) ) ) ) ).
% verit_negate_coefficient(2)
thf(fact_3483_less__minus__iff,axiom,
! [A: $tType] :
( ( ordered_ab_group_add @ A )
=> ! [A2: A,B2: A] :
( ( ord_less @ A @ A2 @ ( uminus_uminus @ A @ B2 ) )
= ( ord_less @ A @ B2 @ ( uminus_uminus @ A @ A2 ) ) ) ) ).
% less_minus_iff
thf(fact_3484_minus__less__iff,axiom,
! [A: $tType] :
( ( ordered_ab_group_add @ A )
=> ! [A2: A,B2: A] :
( ( ord_less @ A @ ( uminus_uminus @ A @ A2 ) @ B2 )
= ( ord_less @ A @ ( uminus_uminus @ A @ B2 ) @ A2 ) ) ) ).
% minus_less_iff
thf(fact_3485_le__imp__neg__le,axiom,
! [A: $tType] :
( ( ordered_ab_group_add @ A )
=> ! [A2: A,B2: A] :
( ( ord_less_eq @ A @ A2 @ B2 )
=> ( ord_less_eq @ A @ ( uminus_uminus @ A @ B2 ) @ ( uminus_uminus @ A @ A2 ) ) ) ) ).
% le_imp_neg_le
thf(fact_3486_minus__le__iff,axiom,
! [A: $tType] :
( ( ordered_ab_group_add @ A )
=> ! [A2: A,B2: A] :
( ( ord_less_eq @ A @ ( uminus_uminus @ A @ A2 ) @ B2 )
= ( ord_less_eq @ A @ ( uminus_uminus @ A @ B2 ) @ A2 ) ) ) ).
% minus_le_iff
thf(fact_3487_le__minus__iff,axiom,
! [A: $tType] :
( ( ordered_ab_group_add @ A )
=> ! [A2: A,B2: A] :
( ( ord_less_eq @ A @ A2 @ ( uminus_uminus @ A @ B2 ) )
= ( ord_less_eq @ A @ B2 @ ( uminus_uminus @ A @ A2 ) ) ) ) ).
% le_minus_iff
thf(fact_3488_mod__minus__eq,axiom,
! [A: $tType] :
( ( euclid8851590272496341667cancel @ A )
=> ! [A2: A,B2: A] :
( ( modulo_modulo @ A @ ( uminus_uminus @ A @ ( modulo_modulo @ A @ A2 @ B2 ) ) @ B2 )
= ( modulo_modulo @ A @ ( uminus_uminus @ A @ A2 ) @ B2 ) ) ) ).
% mod_minus_eq
thf(fact_3489_euclidean__ring__cancel__class_Omod__minus__cong,axiom,
! [A: $tType] :
( ( euclid8851590272496341667cancel @ A )
=> ! [A2: A,B2: A,A5: A] :
( ( ( modulo_modulo @ A @ A2 @ B2 )
= ( modulo_modulo @ A @ A5 @ B2 ) )
=> ( ( modulo_modulo @ A @ ( uminus_uminus @ A @ A2 ) @ B2 )
= ( modulo_modulo @ A @ ( uminus_uminus @ A @ A5 ) @ B2 ) ) ) ) ).
% euclidean_ring_cancel_class.mod_minus_cong
thf(fact_3490_mod__minus__right,axiom,
! [A: $tType] :
( ( euclid8851590272496341667cancel @ A )
=> ! [A2: A,B2: A] :
( ( modulo_modulo @ A @ A2 @ ( uminus_uminus @ A @ B2 ) )
= ( uminus_uminus @ A @ ( modulo_modulo @ A @ ( uminus_uminus @ A @ A2 ) @ B2 ) ) ) ) ).
% mod_minus_right
thf(fact_3491_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_3492_fact__less__mono__nat,axiom,
! [M: nat,N3: nat] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ M )
=> ( ( ord_less @ nat @ M @ N3 )
=> ( ord_less @ nat @ ( semiring_char_0_fact @ nat @ M ) @ ( semiring_char_0_fact @ nat @ N3 ) ) ) ) ).
% fact_less_mono_nat
thf(fact_3493_fact__ge__zero,axiom,
! [A: $tType] :
( ( linordered_semidom @ A )
=> ! [N3: nat] : ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( semiring_char_0_fact @ A @ N3 ) ) ) ).
% fact_ge_zero
thf(fact_3494_fact__gt__zero,axiom,
! [A: $tType] :
( ( linordered_semidom @ A )
=> ! [N3: nat] : ( ord_less @ A @ ( zero_zero @ A ) @ ( semiring_char_0_fact @ A @ N3 ) ) ) ).
% fact_gt_zero
thf(fact_3495_fact__not__neg,axiom,
! [A: $tType] :
( ( linordered_semidom @ A )
=> ! [N3: nat] :
~ ( ord_less @ A @ ( semiring_char_0_fact @ A @ N3 ) @ ( zero_zero @ A ) ) ) ).
% fact_not_neg
thf(fact_3496_fact__ge__1,axiom,
! [A: $tType] :
( ( linordered_semidom @ A )
=> ! [N3: nat] : ( ord_less_eq @ A @ ( one_one @ A ) @ ( semiring_char_0_fact @ A @ N3 ) ) ) ).
% fact_ge_1
thf(fact_3497_zero__neq__neg__numeral,axiom,
! [A: $tType] :
( ( ring_char_0 @ A )
=> ! [N3: num] :
( ( zero_zero @ A )
!= ( uminus_uminus @ A @ ( numeral_numeral @ A @ N3 ) ) ) ) ).
% zero_neq_neg_numeral
thf(fact_3498_neg__numeral__le__numeral,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [M: num,N3: num] : ( ord_less_eq @ A @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ M ) ) @ ( numeral_numeral @ A @ N3 ) ) ) ).
% neg_numeral_le_numeral
thf(fact_3499_not__numeral__le__neg__numeral,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [M: num,N3: num] :
~ ( ord_less_eq @ A @ ( numeral_numeral @ A @ M ) @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ N3 ) ) ) ) ).
% not_numeral_le_neg_numeral
thf(fact_3500_not__numeral__less__neg__numeral,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [M: num,N3: num] :
~ ( ord_less @ A @ ( numeral_numeral @ A @ M ) @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ N3 ) ) ) ) ).
% not_numeral_less_neg_numeral
thf(fact_3501_neg__numeral__less__numeral,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [M: num,N3: num] : ( ord_less @ A @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ M ) ) @ ( numeral_numeral @ A @ N3 ) ) ) ).
% neg_numeral_less_numeral
thf(fact_3502_add__eq__0__iff,axiom,
! [A: $tType] :
( ( group_add @ A )
=> ! [A2: A,B2: A] :
( ( ( plus_plus @ A @ A2 @ B2 )
= ( zero_zero @ A ) )
= ( B2
= ( uminus_uminus @ A @ A2 ) ) ) ) ).
% add_eq_0_iff
thf(fact_3503_ab__group__add__class_Oab__left__minus,axiom,
! [A: $tType] :
( ( ab_group_add @ A )
=> ! [A2: A] :
( ( plus_plus @ A @ ( uminus_uminus @ A @ A2 ) @ A2 )
= ( zero_zero @ A ) ) ) ).
% ab_group_add_class.ab_left_minus
thf(fact_3504_add_Oinverse__unique,axiom,
! [A: $tType] :
( ( group_add @ A )
=> ! [A2: A,B2: A] :
( ( ( plus_plus @ A @ A2 @ B2 )
= ( zero_zero @ A ) )
=> ( ( uminus_uminus @ A @ A2 )
= B2 ) ) ) ).
% add.inverse_unique
thf(fact_3505_eq__neg__iff__add__eq__0,axiom,
! [A: $tType] :
( ( group_add @ A )
=> ! [A2: A,B2: A] :
( ( A2
= ( uminus_uminus @ A @ B2 ) )
= ( ( plus_plus @ A @ A2 @ B2 )
= ( zero_zero @ A ) ) ) ) ).
% eq_neg_iff_add_eq_0
thf(fact_3506_neg__eq__iff__add__eq__0,axiom,
! [A: $tType] :
( ( group_add @ A )
=> ! [A2: A,B2: A] :
( ( ( uminus_uminus @ A @ A2 )
= B2 )
= ( ( plus_plus @ A @ A2 @ B2 )
= ( zero_zero @ A ) ) ) ) ).
% neg_eq_iff_add_eq_0
thf(fact_3507_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_3508_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_3509_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_3510_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_3511_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_3512_numeral__neq__neg__one,axiom,
! [A: $tType] :
( ( ring_char_0 @ A )
=> ! [N3: num] :
( ( numeral_numeral @ A @ N3 )
!= ( uminus_uminus @ A @ ( one_one @ A ) ) ) ) ).
% numeral_neq_neg_one
thf(fact_3513_one__neq__neg__numeral,axiom,
! [A: $tType] :
( ( ring_char_0 @ A )
=> ! [N3: num] :
( ( one_one @ A )
!= ( uminus_uminus @ A @ ( numeral_numeral @ A @ N3 ) ) ) ) ).
% one_neq_neg_numeral
thf(fact_3514_numeral__times__minus__swap,axiom,
! [A: $tType] :
( ( comm_ring_1 @ A )
=> ! [W: num,X: A] :
( ( times_times @ A @ ( numeral_numeral @ A @ W ) @ ( uminus_uminus @ A @ X ) )
= ( times_times @ A @ X @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ W ) ) ) ) ) ).
% numeral_times_minus_swap
thf(fact_3515_nonzero__minus__divide__divide,axiom,
! [A: $tType] :
( ( division_ring @ A )
=> ! [B2: A,A2: A] :
( ( B2
!= ( zero_zero @ A ) )
=> ( ( divide_divide @ A @ ( uminus_uminus @ A @ A2 ) @ ( uminus_uminus @ A @ B2 ) )
= ( divide_divide @ A @ A2 @ B2 ) ) ) ) ).
% nonzero_minus_divide_divide
thf(fact_3516_nonzero__minus__divide__right,axiom,
! [A: $tType] :
( ( division_ring @ A )
=> ! [B2: A,A2: A] :
( ( B2
!= ( zero_zero @ A ) )
=> ( ( uminus_uminus @ A @ ( divide_divide @ A @ A2 @ B2 ) )
= ( divide_divide @ A @ A2 @ ( uminus_uminus @ A @ B2 ) ) ) ) ) ).
% nonzero_minus_divide_right
thf(fact_3517_square__eq__1__iff,axiom,
! [A: $tType] :
( ( ring_15535105094025558882visors @ A )
=> ! [X: A] :
( ( ( times_times @ A @ X @ X )
= ( one_one @ A ) )
= ( ( X
= ( one_one @ A ) )
| ( X
= ( uminus_uminus @ A @ ( one_one @ A ) ) ) ) ) ) ).
% square_eq_1_iff
thf(fact_3518_fact__mono,axiom,
! [A: $tType] :
( ( linordered_semidom @ A )
=> ! [M: nat,N3: nat] :
( ( ord_less_eq @ nat @ M @ N3 )
=> ( ord_less_eq @ A @ ( semiring_char_0_fact @ A @ M ) @ ( semiring_char_0_fact @ A @ N3 ) ) ) ) ).
% fact_mono
thf(fact_3519_group__cancel_Osub2,axiom,
! [A: $tType] :
( ( ab_group_add @ A )
=> ! [B6: A,K: A,B2: A,A2: A] :
( ( B6
= ( plus_plus @ A @ K @ B2 ) )
=> ( ( minus_minus @ A @ A2 @ B6 )
= ( plus_plus @ A @ ( uminus_uminus @ A @ K ) @ ( minus_minus @ A @ A2 @ B2 ) ) ) ) ) ).
% group_cancel.sub2
thf(fact_3520_diff__conv__add__uminus,axiom,
! [A: $tType] :
( ( group_add @ A )
=> ( ( minus_minus @ A )
= ( ^ [A6: A,B5: A] : ( plus_plus @ A @ A6 @ ( uminus_uminus @ A @ B5 ) ) ) ) ) ).
% diff_conv_add_uminus
thf(fact_3521_ab__group__add__class_Oab__diff__conv__add__uminus,axiom,
! [A: $tType] :
( ( ab_group_add @ A )
=> ( ( minus_minus @ A )
= ( ^ [A6: A,B5: A] : ( plus_plus @ A @ A6 @ ( uminus_uminus @ A @ B5 ) ) ) ) ) ).
% ab_group_add_class.ab_diff_conv_add_uminus
thf(fact_3522_dvd__neg__div,axiom,
! [A: $tType] :
( ( idom_divide @ A )
=> ! [B2: A,A2: A] :
( ( dvd_dvd @ A @ B2 @ A2 )
=> ( ( divide_divide @ A @ ( uminus_uminus @ A @ A2 ) @ B2 )
= ( uminus_uminus @ A @ ( divide_divide @ A @ A2 @ B2 ) ) ) ) ) ).
% dvd_neg_div
thf(fact_3523_dvd__div__neg,axiom,
! [A: $tType] :
( ( idom_divide @ A )
=> ! [B2: A,A2: A] :
( ( dvd_dvd @ A @ B2 @ A2 )
=> ( ( divide_divide @ A @ A2 @ ( uminus_uminus @ A @ B2 ) )
= ( uminus_uminus @ A @ ( divide_divide @ A @ A2 @ B2 ) ) ) ) ) ).
% dvd_div_neg
thf(fact_3524_fact__dvd,axiom,
! [A: $tType] :
( ( linordered_semidom @ A )
=> ! [N3: nat,M: nat] :
( ( ord_less_eq @ nat @ N3 @ M )
=> ( dvd_dvd @ A @ ( semiring_char_0_fact @ A @ N3 ) @ ( semiring_char_0_fact @ A @ M ) ) ) ) ).
% fact_dvd
thf(fact_3525_subset__Compl__self__eq,axiom,
! [A: $tType,A3: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ A3 @ ( uminus_uminus @ ( set @ A ) @ A3 ) )
= ( A3
= ( bot_bot @ ( set @ A ) ) ) ) ).
% subset_Compl_self_eq
thf(fact_3526_int__of__nat__induct,axiom,
! [P: int > $o,Z: int] :
( ! [N: nat] : ( P @ ( semiring_1_of_nat @ int @ N ) )
=> ( ! [N: nat] : ( P @ ( uminus_uminus @ int @ ( semiring_1_of_nat @ int @ ( suc @ N ) ) ) )
=> ( P @ Z ) ) ) ).
% int_of_nat_induct
thf(fact_3527_int__cases,axiom,
! [Z: int] :
( ! [N: nat] :
( Z
!= ( semiring_1_of_nat @ int @ N ) )
=> ~ ! [N: nat] :
( Z
!= ( uminus_uminus @ int @ ( semiring_1_of_nat @ int @ ( suc @ N ) ) ) ) ) ).
% int_cases
thf(fact_3528_dbl__def,axiom,
! [A: $tType] :
( ( neg_numeral @ A )
=> ( ( neg_numeral_dbl @ A )
= ( ^ [X2: A] : ( plus_plus @ A @ X2 @ X2 ) ) ) ) ).
% dbl_def
thf(fact_3529_real__minus__mult__self__le,axiom,
! [U: real,X: real] : ( ord_less_eq @ real @ ( uminus_uminus @ real @ ( times_times @ real @ U @ U ) ) @ ( times_times @ real @ X @ X ) ) ).
% real_minus_mult_self_le
thf(fact_3530_pos__zmult__eq__1__iff__lemma,axiom,
! [M: int,N3: int] :
( ( ( times_times @ int @ M @ N3 )
= ( one_one @ int ) )
=> ( ( M
= ( one_one @ int ) )
| ( M
= ( uminus_uminus @ int @ ( one_one @ int ) ) ) ) ) ).
% pos_zmult_eq_1_iff_lemma
thf(fact_3531_zmult__eq__1__iff,axiom,
! [M: int,N3: int] :
( ( ( times_times @ int @ M @ N3 )
= ( one_one @ int ) )
= ( ( ( M
= ( one_one @ int ) )
& ( N3
= ( one_one @ int ) ) )
| ( ( M
= ( uminus_uminus @ int @ ( one_one @ int ) ) )
& ( N3
= ( uminus_uminus @ int @ ( one_one @ int ) ) ) ) ) ) ).
% zmult_eq_1_iff
thf(fact_3532_minus__int__code_I2_J,axiom,
! [L2: int] :
( ( minus_minus @ int @ ( zero_zero @ int ) @ L2 )
= ( uminus_uminus @ int @ L2 ) ) ).
% minus_int_code(2)
thf(fact_3533_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_3534_word__order_Oextremum__uniqueI,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [A2: word @ A] :
( ( ord_less_eq @ ( word @ A ) @ ( uminus_uminus @ ( word @ A ) @ ( one_one @ ( word @ A ) ) ) @ A2 )
=> ( A2
= ( uminus_uminus @ ( word @ A ) @ ( one_one @ ( word @ A ) ) ) ) ) ) ).
% word_order.extremum_uniqueI
thf(fact_3535_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_3536_word__not__simps_I3_J,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [Y: word @ A] :
~ ( ord_less @ ( word @ A ) @ ( uminus_uminus @ ( word @ A ) @ ( one_one @ ( word @ A ) ) ) @ Y ) ) ).
% word_not_simps(3)
thf(fact_3537_word__order_Oextremum__strict,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [A2: word @ A] :
~ ( ord_less @ ( word @ A ) @ ( uminus_uminus @ ( word @ A ) @ ( one_one @ ( word @ A ) ) ) @ A2 ) ) ).
% word_order.extremum_strict
thf(fact_3538_word__order_Onot__eq__extremum,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [A2: word @ A] :
( ( A2
!= ( uminus_uminus @ ( word @ A ) @ ( one_one @ ( word @ A ) ) ) )
= ( ord_less @ ( word @ A ) @ A2 @ ( uminus_uminus @ ( word @ A ) @ ( one_one @ ( word @ A ) ) ) ) ) ) ).
% word_order.not_eq_extremum
thf(fact_3539_max__word__not__less,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [X: word @ A] :
~ ( ord_less @ ( word @ A ) @ ( uminus_uminus @ ( word @ A ) @ ( one_one @ ( word @ A ) ) ) @ X ) ) ).
% max_word_not_less
thf(fact_3540_minus__real__def,axiom,
( ( minus_minus @ real )
= ( ^ [X2: real,Y2: real] : ( plus_plus @ real @ X2 @ ( uminus_uminus @ real @ Y2 ) ) ) ) ).
% minus_real_def
thf(fact_3541_complex__mod__minus__le__complex__mod,axiom,
! [X: complex] : ( ord_less_eq @ real @ ( uminus_uminus @ real @ ( real_V7770717601297561774m_norm @ complex @ X ) ) @ ( real_V7770717601297561774m_norm @ complex @ X ) ) ).
% complex_mod_minus_le_complex_mod
thf(fact_3542_pochhammer__same,axiom,
! [A: $tType] :
( ( ( semiring_char_0 @ A )
& ( comm_ring_1 @ A )
& ( semiri3467727345109120633visors @ A ) )
=> ! [N3: nat] :
( ( comm_s3205402744901411588hammer @ A @ ( uminus_uminus @ A @ ( semiring_1_of_nat @ A @ N3 ) ) @ N3 )
= ( times_times @ A @ ( power_power @ A @ ( uminus_uminus @ A @ ( one_one @ A ) ) @ N3 ) @ ( semiring_char_0_fact @ A @ N3 ) ) ) ) ).
% pochhammer_same
thf(fact_3543_fact__ge__Suc__0__nat,axiom,
! [N3: nat] : ( ord_less_eq @ nat @ ( suc @ ( zero_zero @ nat ) ) @ ( semiring_char_0_fact @ nat @ N3 ) ) ).
% fact_ge_Suc_0_nat
thf(fact_3544_dvd__fact,axiom,
! [M: nat,N3: nat] :
( ( ord_less_eq @ nat @ ( one_one @ nat ) @ M )
=> ( ( ord_less_eq @ nat @ M @ N3 )
=> ( dvd_dvd @ nat @ M @ ( semiring_char_0_fact @ nat @ N3 ) ) ) ) ).
% dvd_fact
thf(fact_3545_neg__numeral__le__zero,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [N3: num] : ( ord_less_eq @ A @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ N3 ) ) @ ( zero_zero @ A ) ) ) ).
% neg_numeral_le_zero
thf(fact_3546_not__zero__le__neg__numeral,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [N3: num] :
~ ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ N3 ) ) ) ) ).
% not_zero_le_neg_numeral
thf(fact_3547_not__zero__less__neg__numeral,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [N3: num] :
~ ( ord_less @ A @ ( zero_zero @ A ) @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ N3 ) ) ) ) ).
% not_zero_less_neg_numeral
thf(fact_3548_neg__numeral__less__zero,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [N3: num] : ( ord_less @ A @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ N3 ) ) @ ( zero_zero @ A ) ) ) ).
% neg_numeral_less_zero
thf(fact_3549_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_3550_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_3551_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_3552_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_3553_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_3554_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_3555_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_3556_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_3557_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_3558_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_3559_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_3560_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_3561_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_3562_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_3563_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_3564_gbinomial__pochhammer,axiom,
! [A: $tType] :
( ( field_char_0 @ A )
=> ( ( gbinomial @ A )
= ( ^ [A6: A,K2: nat] : ( divide_divide @ A @ ( times_times @ A @ ( power_power @ A @ ( uminus_uminus @ A @ ( one_one @ A ) ) @ K2 ) @ ( comm_s3205402744901411588hammer @ A @ ( uminus_uminus @ A @ A6 ) @ K2 ) ) @ ( semiring_char_0_fact @ A @ K2 ) ) ) ) ) ).
% gbinomial_pochhammer
thf(fact_3565_mult__1s__ring__1_I1_J,axiom,
! [A: $tType] :
( ( ring_1 @ A )
=> ! [B2: A] :
( ( times_times @ A @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ one2 ) ) @ B2 )
= ( uminus_uminus @ A @ B2 ) ) ) ).
% mult_1s_ring_1(1)
thf(fact_3566_mult__1s__ring__1_I2_J,axiom,
! [A: $tType] :
( ( ring_1 @ A )
=> ! [B2: A] :
( ( times_times @ A @ B2 @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ one2 ) ) )
= ( uminus_uminus @ A @ B2 ) ) ) ).
% mult_1s_ring_1(2)
thf(fact_3567_divide__eq__minus__1__iff,axiom,
! [A: $tType] :
( ( field @ A )
=> ! [A2: A,B2: A] :
( ( ( divide_divide @ A @ A2 @ B2 )
= ( uminus_uminus @ A @ ( one_one @ A ) ) )
= ( ( B2
!= ( zero_zero @ A ) )
& ( A2
= ( uminus_uminus @ A @ B2 ) ) ) ) ) ).
% divide_eq_minus_1_iff
thf(fact_3568_nonzero__neg__divide__eq__eq2,axiom,
! [A: $tType] :
( ( division_ring @ A )
=> ! [B2: A,C2: A,A2: A] :
( ( B2
!= ( zero_zero @ A ) )
=> ( ( C2
= ( uminus_uminus @ A @ ( divide_divide @ A @ A2 @ B2 ) ) )
= ( ( times_times @ A @ C2 @ B2 )
= ( uminus_uminus @ A @ A2 ) ) ) ) ) ).
% nonzero_neg_divide_eq_eq2
thf(fact_3569_nonzero__neg__divide__eq__eq,axiom,
! [A: $tType] :
( ( division_ring @ A )
=> ! [B2: A,A2: A,C2: A] :
( ( B2
!= ( zero_zero @ A ) )
=> ( ( ( uminus_uminus @ A @ ( divide_divide @ A @ A2 @ B2 ) )
= C2 )
= ( ( uminus_uminus @ A @ A2 )
= ( times_times @ A @ C2 @ B2 ) ) ) ) ) ).
% nonzero_neg_divide_eq_eq
thf(fact_3570_minus__divide__eq__eq,axiom,
! [A: $tType] :
( ( division_ring @ A )
=> ! [B2: A,C2: A,A2: A] :
( ( ( uminus_uminus @ A @ ( divide_divide @ A @ B2 @ C2 ) )
= A2 )
= ( ( ( C2
!= ( zero_zero @ A ) )
=> ( ( uminus_uminus @ A @ B2 )
= ( times_times @ A @ A2 @ C2 ) ) )
& ( ( C2
= ( zero_zero @ A ) )
=> ( A2
= ( zero_zero @ A ) ) ) ) ) ) ).
% minus_divide_eq_eq
thf(fact_3571_eq__minus__divide__eq,axiom,
! [A: $tType] :
( ( division_ring @ A )
=> ! [A2: A,B2: A,C2: A] :
( ( A2
= ( uminus_uminus @ A @ ( divide_divide @ A @ B2 @ C2 ) ) )
= ( ( ( C2
!= ( zero_zero @ A ) )
=> ( ( times_times @ A @ A2 @ C2 )
= ( uminus_uminus @ A @ B2 ) ) )
& ( ( C2
= ( zero_zero @ A ) )
=> ( A2
= ( zero_zero @ A ) ) ) ) ) ) ).
% eq_minus_divide_eq
thf(fact_3572_fact__less__mono,axiom,
! [A: $tType] :
( ( linordered_semidom @ A )
=> ! [M: nat,N3: nat] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ M )
=> ( ( ord_less @ nat @ M @ N3 )
=> ( ord_less @ A @ ( semiring_char_0_fact @ A @ M ) @ ( semiring_char_0_fact @ A @ N3 ) ) ) ) ) ).
% fact_less_mono
thf(fact_3573_power__minus,axiom,
! [A: $tType] :
( ( ring_1 @ A )
=> ! [A2: A,N3: nat] :
( ( power_power @ A @ ( uminus_uminus @ A @ A2 ) @ N3 )
= ( times_times @ A @ ( power_power @ A @ ( uminus_uminus @ A @ ( one_one @ A ) ) @ N3 ) @ ( power_power @ A @ A2 @ N3 ) ) ) ) ).
% power_minus
thf(fact_3574_power__minus__Bit0,axiom,
! [A: $tType] :
( ( ring_1 @ A )
=> ! [X: A,K: num] :
( ( power_power @ A @ ( uminus_uminus @ A @ X ) @ ( numeral_numeral @ nat @ ( bit0 @ K ) ) )
= ( power_power @ A @ X @ ( numeral_numeral @ nat @ ( bit0 @ K ) ) ) ) ) ).
% power_minus_Bit0
thf(fact_3575_fact__mod,axiom,
! [A: $tType] :
( ( ( linordered_semidom @ A )
& ( semidom_modulo @ A ) )
=> ! [M: nat,N3: nat] :
( ( ord_less_eq @ nat @ M @ N3 )
=> ( ( modulo_modulo @ A @ ( semiring_char_0_fact @ A @ N3 ) @ ( semiring_char_0_fact @ A @ M ) )
= ( zero_zero @ A ) ) ) ) ).
% fact_mod
thf(fact_3576_fact__fact__dvd__fact,axiom,
! [A: $tType] :
( ( linordered_semidom @ A )
=> ! [K: nat,N3: nat] : ( dvd_dvd @ A @ ( times_times @ A @ ( semiring_char_0_fact @ A @ K ) @ ( semiring_char_0_fact @ A @ N3 ) ) @ ( semiring_char_0_fact @ A @ ( plus_plus @ nat @ K @ N3 ) ) ) ) ).
% fact_fact_dvd_fact
thf(fact_3577_power__minus__Bit1,axiom,
! [A: $tType] :
( ( ring_1 @ A )
=> ! [X: A,K: num] :
( ( power_power @ A @ ( uminus_uminus @ A @ X ) @ ( numeral_numeral @ nat @ ( bit1 @ K ) ) )
= ( uminus_uminus @ A @ ( power_power @ A @ X @ ( numeral_numeral @ nat @ ( bit1 @ K ) ) ) ) ) ) ).
% power_minus_Bit1
thf(fact_3578_fact__le__power,axiom,
! [A: $tType] :
( ( linordered_semidom @ A )
=> ! [N3: nat] : ( ord_less_eq @ A @ ( semiring_char_0_fact @ A @ N3 ) @ ( semiring_1_of_nat @ A @ ( power_power @ nat @ N3 @ N3 ) ) ) ) ).
% fact_le_power
thf(fact_3579_norm__uminus__minus,axiom,
! [A: $tType] :
( ( real_V822414075346904944vector @ A )
=> ! [X: A,Y: A] :
( ( real_V7770717601297561774m_norm @ A @ ( minus_minus @ A @ ( uminus_uminus @ A @ X ) @ Y ) )
= ( real_V7770717601297561774m_norm @ A @ ( plus_plus @ A @ X @ Y ) ) ) ) ).
% norm_uminus_minus
thf(fact_3580_Compl__insert,axiom,
! [A: $tType,X: A,A3: set @ A] :
( ( uminus_uminus @ ( set @ A ) @ ( insert @ A @ X @ A3 ) )
= ( minus_minus @ ( set @ A ) @ ( uminus_uminus @ ( set @ A ) @ A3 ) @ ( insert @ A @ X @ ( bot_bot @ ( set @ A ) ) ) ) ) ).
% Compl_insert
thf(fact_3581_powr__minus__divide,axiom,
! [A: $tType] :
( ( ( real_V3459762299906320749_field @ A )
& ( ln @ A ) )
=> ! [X: A,A2: A] :
( ( powr @ A @ X @ ( uminus_uminus @ A @ A2 ) )
= ( divide_divide @ A @ ( one_one @ A ) @ ( powr @ A @ X @ A2 ) ) ) ) ).
% powr_minus_divide
thf(fact_3582_int__cases4,axiom,
! [M: int] :
( ! [N: nat] :
( M
!= ( semiring_1_of_nat @ int @ N ) )
=> ~ ! [N: nat] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ N )
=> ( M
!= ( uminus_uminus @ int @ ( semiring_1_of_nat @ int @ N ) ) ) ) ) ).
% int_cases4
thf(fact_3583_nle__le,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A2: A,B2: A] :
( ( ~ ( ord_less_eq @ A @ A2 @ B2 ) )
= ( ( ord_less_eq @ A @ B2 @ A2 )
& ( B2 != A2 ) ) ) ) ).
% nle_le
thf(fact_3584_le__cases3,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [X: A,Y: A,Z: A] :
( ( ( ord_less_eq @ A @ X @ Y )
=> ~ ( ord_less_eq @ A @ Y @ Z ) )
=> ( ( ( ord_less_eq @ A @ Y @ X )
=> ~ ( ord_less_eq @ A @ X @ Z ) )
=> ( ( ( ord_less_eq @ A @ X @ Z )
=> ~ ( ord_less_eq @ A @ Z @ Y ) )
=> ( ( ( ord_less_eq @ A @ Z @ Y )
=> ~ ( ord_less_eq @ A @ Y @ X ) )
=> ( ( ( ord_less_eq @ A @ Y @ Z )
=> ~ ( ord_less_eq @ A @ Z @ X ) )
=> ~ ( ( ord_less_eq @ A @ Z @ X )
=> ~ ( ord_less_eq @ A @ X @ Y ) ) ) ) ) ) ) ) ).
% le_cases3
thf(fact_3585_order__class_Oorder__eq__iff,axiom,
! [A: $tType] :
( ( order @ A )
=> ( ( ^ [Y5: A,Z5: A] : Y5 = Z5 )
= ( ^ [X2: A,Y2: A] :
( ( ord_less_eq @ A @ X2 @ Y2 )
& ( ord_less_eq @ A @ Y2 @ X2 ) ) ) ) ) ).
% order_class.order_eq_iff
thf(fact_3586_ord__eq__le__trans,axiom,
! [A: $tType] :
( ( ord @ A )
=> ! [A2: A,B2: A,C2: A] :
( ( A2 = B2 )
=> ( ( ord_less_eq @ A @ B2 @ C2 )
=> ( ord_less_eq @ A @ A2 @ C2 ) ) ) ) ).
% ord_eq_le_trans
thf(fact_3587_ord__le__eq__trans,axiom,
! [A: $tType] :
( ( ord @ A )
=> ! [A2: A,B2: A,C2: A] :
( ( ord_less_eq @ A @ A2 @ B2 )
=> ( ( B2 = C2 )
=> ( ord_less_eq @ A @ A2 @ C2 ) ) ) ) ).
% ord_le_eq_trans
thf(fact_3588_order__antisym,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [X: A,Y: A] :
( ( ord_less_eq @ A @ X @ Y )
=> ( ( ord_less_eq @ A @ Y @ X )
=> ( X = Y ) ) ) ) ).
% order_antisym
thf(fact_3589_order_Otrans,axiom,
! [A: $tType] :
( ( preorder @ A )
=> ! [A2: A,B2: A,C2: A] :
( ( ord_less_eq @ A @ A2 @ B2 )
=> ( ( ord_less_eq @ A @ B2 @ C2 )
=> ( ord_less_eq @ A @ A2 @ C2 ) ) ) ) ).
% order.trans
thf(fact_3590_order__trans,axiom,
! [A: $tType] :
( ( preorder @ A )
=> ! [X: A,Y: A,Z: A] :
( ( ord_less_eq @ A @ X @ Y )
=> ( ( ord_less_eq @ A @ Y @ Z )
=> ( ord_less_eq @ A @ X @ Z ) ) ) ) ).
% order_trans
thf(fact_3591_linorder__wlog,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [P: A > A > $o,A2: A,B2: A] :
( ! [A4: A,B3: A] :
( ( ord_less_eq @ A @ A4 @ B3 )
=> ( P @ A4 @ B3 ) )
=> ( ! [A4: A,B3: A] :
( ( P @ B3 @ A4 )
=> ( P @ A4 @ B3 ) )
=> ( P @ A2 @ B2 ) ) ) ) ).
% linorder_wlog
thf(fact_3592_dual__order_Oeq__iff,axiom,
! [A: $tType] :
( ( order @ A )
=> ( ( ^ [Y5: A,Z5: A] : Y5 = Z5 )
= ( ^ [A6: A,B5: A] :
( ( ord_less_eq @ A @ B5 @ A6 )
& ( ord_less_eq @ A @ A6 @ B5 ) ) ) ) ) ).
% dual_order.eq_iff
thf(fact_3593_dual__order_Oantisym,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [B2: A,A2: A] :
( ( ord_less_eq @ A @ B2 @ A2 )
=> ( ( ord_less_eq @ A @ A2 @ B2 )
=> ( A2 = B2 ) ) ) ) ).
% dual_order.antisym
thf(fact_3594_dual__order_Otrans,axiom,
! [A: $tType] :
( ( preorder @ A )
=> ! [B2: A,A2: A,C2: A] :
( ( ord_less_eq @ A @ B2 @ A2 )
=> ( ( ord_less_eq @ A @ C2 @ B2 )
=> ( ord_less_eq @ A @ C2 @ A2 ) ) ) ) ).
% dual_order.trans
thf(fact_3595_antisym,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [A2: A,B2: A] :
( ( ord_less_eq @ A @ A2 @ B2 )
=> ( ( ord_less_eq @ A @ B2 @ A2 )
=> ( A2 = B2 ) ) ) ) ).
% antisym
thf(fact_3596_Orderings_Oorder__eq__iff,axiom,
! [A: $tType] :
( ( order @ A )
=> ( ( ^ [Y5: A,Z5: A] : Y5 = Z5 )
= ( ^ [A6: A,B5: A] :
( ( ord_less_eq @ A @ A6 @ B5 )
& ( ord_less_eq @ A @ B5 @ A6 ) ) ) ) ) ).
% Orderings.order_eq_iff
thf(fact_3597_order__subst1,axiom,
! [A: $tType,B: $tType] :
( ( ( order @ B )
& ( order @ A ) )
=> ! [A2: A,F2: B > A,B2: B,C2: B] :
( ( ord_less_eq @ A @ A2 @ ( F2 @ B2 ) )
=> ( ( ord_less_eq @ B @ B2 @ C2 )
=> ( ! [X3: B,Y3: B] :
( ( ord_less_eq @ B @ X3 @ Y3 )
=> ( ord_less_eq @ A @ ( F2 @ X3 ) @ ( F2 @ Y3 ) ) )
=> ( ord_less_eq @ A @ A2 @ ( F2 @ C2 ) ) ) ) ) ) ).
% order_subst1
thf(fact_3598_order__subst2,axiom,
! [A: $tType,C: $tType] :
( ( ( order @ C )
& ( order @ A ) )
=> ! [A2: A,B2: A,F2: A > C,C2: C] :
( ( ord_less_eq @ A @ A2 @ B2 )
=> ( ( ord_less_eq @ C @ ( F2 @ B2 ) @ C2 )
=> ( ! [X3: A,Y3: A] :
( ( ord_less_eq @ A @ X3 @ Y3 )
=> ( ord_less_eq @ C @ ( F2 @ X3 ) @ ( F2 @ Y3 ) ) )
=> ( ord_less_eq @ C @ ( F2 @ A2 ) @ C2 ) ) ) ) ) ).
% order_subst2
thf(fact_3599_order__eq__refl,axiom,
! [A: $tType] :
( ( preorder @ A )
=> ! [X: A,Y: A] :
( ( X = Y )
=> ( ord_less_eq @ A @ X @ Y ) ) ) ).
% order_eq_refl
thf(fact_3600_linorder__linear,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [X: A,Y: A] :
( ( ord_less_eq @ A @ X @ Y )
| ( ord_less_eq @ A @ Y @ X ) ) ) ).
% linorder_linear
thf(fact_3601_ord__eq__le__subst,axiom,
! [A: $tType,B: $tType] :
( ( ( ord @ B )
& ( ord @ A ) )
=> ! [A2: A,F2: B > A,B2: B,C2: B] :
( ( A2
= ( F2 @ B2 ) )
=> ( ( ord_less_eq @ B @ B2 @ C2 )
=> ( ! [X3: B,Y3: B] :
( ( ord_less_eq @ B @ X3 @ Y3 )
=> ( ord_less_eq @ A @ ( F2 @ X3 ) @ ( F2 @ Y3 ) ) )
=> ( ord_less_eq @ A @ A2 @ ( F2 @ C2 ) ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_3602_ord__le__eq__subst,axiom,
! [A: $tType,B: $tType] :
( ( ( ord @ B )
& ( ord @ A ) )
=> ! [A2: A,B2: A,F2: A > B,C2: B] :
( ( ord_less_eq @ A @ A2 @ B2 )
=> ( ( ( F2 @ B2 )
= C2 )
=> ( ! [X3: A,Y3: A] :
( ( ord_less_eq @ A @ X3 @ Y3 )
=> ( ord_less_eq @ B @ ( F2 @ X3 ) @ ( F2 @ Y3 ) ) )
=> ( ord_less_eq @ B @ ( F2 @ A2 ) @ C2 ) ) ) ) ) ).
% ord_le_eq_subst
thf(fact_3603_linorder__le__cases,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [X: A,Y: A] :
( ~ ( ord_less_eq @ A @ X @ Y )
=> ( ord_less_eq @ A @ Y @ X ) ) ) ).
% linorder_le_cases
thf(fact_3604_order__antisym__conv,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [Y: A,X: A] :
( ( ord_less_eq @ A @ Y @ X )
=> ( ( ord_less_eq @ A @ X @ Y )
= ( X = Y ) ) ) ) ).
% order_antisym_conv
thf(fact_3605_order__less__imp__not__less,axiom,
! [A: $tType] :
( ( preorder @ A )
=> ! [X: A,Y: A] :
( ( ord_less @ A @ X @ Y )
=> ~ ( ord_less @ A @ Y @ X ) ) ) ).
% order_less_imp_not_less
thf(fact_3606_order__less__imp__not__eq2,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [X: A,Y: A] :
( ( ord_less @ A @ X @ Y )
=> ( Y != X ) ) ) ).
% order_less_imp_not_eq2
thf(fact_3607_order__less__imp__not__eq,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [X: A,Y: A] :
( ( ord_less @ A @ X @ Y )
=> ( X != Y ) ) ) ).
% order_less_imp_not_eq
thf(fact_3608_linorder__less__linear,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [X: A,Y: A] :
( ( ord_less @ A @ X @ Y )
| ( X = Y )
| ( ord_less @ A @ Y @ X ) ) ) ).
% linorder_less_linear
thf(fact_3609_order__less__imp__triv,axiom,
! [A: $tType] :
( ( preorder @ A )
=> ! [X: A,Y: A,P: $o] :
( ( ord_less @ A @ X @ Y )
=> ( ( ord_less @ A @ Y @ X )
=> P ) ) ) ).
% order_less_imp_triv
thf(fact_3610_order__less__not__sym,axiom,
! [A: $tType] :
( ( preorder @ A )
=> ! [X: A,Y: A] :
( ( ord_less @ A @ X @ Y )
=> ~ ( ord_less @ A @ Y @ X ) ) ) ).
% order_less_not_sym
thf(fact_3611_order__less__subst2,axiom,
! [A: $tType,C: $tType] :
( ( ( order @ C )
& ( order @ A ) )
=> ! [A2: A,B2: A,F2: A > C,C2: C] :
( ( ord_less @ A @ A2 @ B2 )
=> ( ( ord_less @ C @ ( F2 @ B2 ) @ C2 )
=> ( ! [X3: A,Y3: A] :
( ( ord_less @ A @ X3 @ Y3 )
=> ( ord_less @ C @ ( F2 @ X3 ) @ ( F2 @ Y3 ) ) )
=> ( ord_less @ C @ ( F2 @ A2 ) @ C2 ) ) ) ) ) ).
% order_less_subst2
thf(fact_3612_order__less__subst1,axiom,
! [A: $tType,B: $tType] :
( ( ( order @ B )
& ( order @ A ) )
=> ! [A2: A,F2: B > A,B2: B,C2: B] :
( ( ord_less @ A @ A2 @ ( F2 @ B2 ) )
=> ( ( ord_less @ B @ B2 @ C2 )
=> ( ! [X3: B,Y3: B] :
( ( ord_less @ B @ X3 @ Y3 )
=> ( ord_less @ A @ ( F2 @ X3 ) @ ( F2 @ Y3 ) ) )
=> ( ord_less @ A @ A2 @ ( F2 @ C2 ) ) ) ) ) ) ).
% order_less_subst1
thf(fact_3613_order__less__irrefl,axiom,
! [A: $tType] :
( ( preorder @ A )
=> ! [X: A] :
~ ( ord_less @ A @ X @ X ) ) ).
% order_less_irrefl
thf(fact_3614_ord__less__eq__subst,axiom,
! [A: $tType,B: $tType] :
( ( ( ord @ B )
& ( ord @ A ) )
=> ! [A2: A,B2: A,F2: A > B,C2: B] :
( ( ord_less @ A @ A2 @ B2 )
=> ( ( ( F2 @ B2 )
= C2 )
=> ( ! [X3: A,Y3: A] :
( ( ord_less @ A @ X3 @ Y3 )
=> ( ord_less @ B @ ( F2 @ X3 ) @ ( F2 @ Y3 ) ) )
=> ( ord_less @ B @ ( F2 @ A2 ) @ C2 ) ) ) ) ) ).
% ord_less_eq_subst
thf(fact_3615_ord__eq__less__subst,axiom,
! [A: $tType,B: $tType] :
( ( ( ord @ B )
& ( ord @ A ) )
=> ! [A2: A,F2: B > A,B2: B,C2: B] :
( ( A2
= ( F2 @ B2 ) )
=> ( ( ord_less @ B @ B2 @ C2 )
=> ( ! [X3: B,Y3: B] :
( ( ord_less @ B @ X3 @ Y3 )
=> ( ord_less @ A @ ( F2 @ X3 ) @ ( F2 @ Y3 ) ) )
=> ( ord_less @ A @ A2 @ ( F2 @ C2 ) ) ) ) ) ) ).
% ord_eq_less_subst
thf(fact_3616_order__less__trans,axiom,
! [A: $tType] :
( ( preorder @ A )
=> ! [X: A,Y: A,Z: A] :
( ( ord_less @ A @ X @ Y )
=> ( ( ord_less @ A @ Y @ Z )
=> ( ord_less @ A @ X @ Z ) ) ) ) ).
% order_less_trans
thf(fact_3617_order__less__asym_H,axiom,
! [A: $tType] :
( ( preorder @ A )
=> ! [A2: A,B2: A] :
( ( ord_less @ A @ A2 @ B2 )
=> ~ ( ord_less @ A @ B2 @ A2 ) ) ) ).
% order_less_asym'
thf(fact_3618_linorder__neq__iff,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [X: A,Y: A] :
( ( X != Y )
= ( ( ord_less @ A @ X @ Y )
| ( ord_less @ A @ Y @ X ) ) ) ) ).
% linorder_neq_iff
thf(fact_3619_order__less__asym,axiom,
! [A: $tType] :
( ( preorder @ A )
=> ! [X: A,Y: A] :
( ( ord_less @ A @ X @ Y )
=> ~ ( ord_less @ A @ Y @ X ) ) ) ).
% order_less_asym
thf(fact_3620_linorder__neqE,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [X: A,Y: A] :
( ( X != Y )
=> ( ~ ( ord_less @ A @ X @ Y )
=> ( ord_less @ A @ Y @ X ) ) ) ) ).
% linorder_neqE
thf(fact_3621_dual__order_Ostrict__implies__not__eq,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [B2: A,A2: A] :
( ( ord_less @ A @ B2 @ A2 )
=> ( A2 != B2 ) ) ) ).
% dual_order.strict_implies_not_eq
thf(fact_3622_order_Ostrict__implies__not__eq,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [A2: A,B2: A] :
( ( ord_less @ A @ A2 @ B2 )
=> ( A2 != B2 ) ) ) ).
% order.strict_implies_not_eq
thf(fact_3623_dual__order_Ostrict__trans,axiom,
! [A: $tType] :
( ( preorder @ A )
=> ! [B2: A,A2: A,C2: A] :
( ( ord_less @ A @ B2 @ A2 )
=> ( ( ord_less @ A @ C2 @ B2 )
=> ( ord_less @ A @ C2 @ A2 ) ) ) ) ).
% dual_order.strict_trans
thf(fact_3624_not__less__iff__gr__or__eq,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [X: A,Y: A] :
( ( ~ ( ord_less @ A @ X @ Y ) )
= ( ( ord_less @ A @ Y @ X )
| ( X = Y ) ) ) ) ).
% not_less_iff_gr_or_eq
thf(fact_3625_order_Ostrict__trans,axiom,
! [A: $tType] :
( ( preorder @ A )
=> ! [A2: A,B2: A,C2: A] :
( ( ord_less @ A @ A2 @ B2 )
=> ( ( ord_less @ A @ B2 @ C2 )
=> ( ord_less @ A @ A2 @ C2 ) ) ) ) ).
% order.strict_trans
thf(fact_3626_linorder__less__wlog,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [P: A > A > $o,A2: A,B2: A] :
( ! [A4: A,B3: A] :
( ( ord_less @ A @ A4 @ B3 )
=> ( P @ A4 @ B3 ) )
=> ( ! [A4: A] : ( P @ A4 @ A4 )
=> ( ! [A4: A,B3: A] :
( ( P @ B3 @ A4 )
=> ( P @ A4 @ B3 ) )
=> ( P @ A2 @ B2 ) ) ) ) ) ).
% linorder_less_wlog
thf(fact_3627_exists__least__iff,axiom,
! [A: $tType] :
( ( wellorder @ A )
=> ( ( ^ [P2: A > $o] :
? [X4: A] : ( P2 @ X4 ) )
= ( ^ [P3: A > $o] :
? [N2: A] :
( ( P3 @ N2 )
& ! [M5: A] :
( ( ord_less @ A @ M5 @ N2 )
=> ~ ( P3 @ M5 ) ) ) ) ) ) ).
% exists_least_iff
thf(fact_3628_dual__order_Oirrefl,axiom,
! [A: $tType] :
( ( preorder @ A )
=> ! [A2: A] :
~ ( ord_less @ A @ A2 @ A2 ) ) ).
% dual_order.irrefl
thf(fact_3629_dual__order_Oasym,axiom,
! [A: $tType] :
( ( preorder @ A )
=> ! [B2: A,A2: A] :
( ( ord_less @ A @ B2 @ A2 )
=> ~ ( ord_less @ A @ A2 @ B2 ) ) ) ).
% dual_order.asym
thf(fact_3630_linorder__cases,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [X: A,Y: A] :
( ~ ( ord_less @ A @ X @ Y )
=> ( ( X != Y )
=> ( ord_less @ A @ Y @ X ) ) ) ) ).
% linorder_cases
thf(fact_3631_antisym__conv3,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [Y: A,X: A] :
( ~ ( ord_less @ A @ Y @ X )
=> ( ( ~ ( ord_less @ A @ X @ Y ) )
= ( X = Y ) ) ) ) ).
% antisym_conv3
thf(fact_3632_less__induct,axiom,
! [A: $tType] :
( ( wellorder @ A )
=> ! [P: A > $o,A2: A] :
( ! [X3: A] :
( ! [Y4: A] :
( ( ord_less @ A @ Y4 @ X3 )
=> ( P @ Y4 ) )
=> ( P @ X3 ) )
=> ( P @ A2 ) ) ) ).
% less_induct
thf(fact_3633_ord__less__eq__trans,axiom,
! [A: $tType] :
( ( ord @ A )
=> ! [A2: A,B2: A,C2: A] :
( ( ord_less @ A @ A2 @ B2 )
=> ( ( B2 = C2 )
=> ( ord_less @ A @ A2 @ C2 ) ) ) ) ).
% ord_less_eq_trans
thf(fact_3634_ord__eq__less__trans,axiom,
! [A: $tType] :
( ( ord @ A )
=> ! [A2: A,B2: A,C2: A] :
( ( A2 = B2 )
=> ( ( ord_less @ A @ B2 @ C2 )
=> ( ord_less @ A @ A2 @ C2 ) ) ) ) ).
% ord_eq_less_trans
thf(fact_3635_order_Oasym,axiom,
! [A: $tType] :
( ( preorder @ A )
=> ! [A2: A,B2: A] :
( ( ord_less @ A @ A2 @ B2 )
=> ~ ( ord_less @ A @ B2 @ A2 ) ) ) ).
% order.asym
thf(fact_3636_less__imp__neq,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [X: A,Y: A] :
( ( ord_less @ A @ X @ Y )
=> ( X != Y ) ) ) ).
% less_imp_neq
thf(fact_3637_dense,axiom,
! [A: $tType] :
( ( dense_order @ A )
=> ! [X: A,Y: A] :
( ( ord_less @ A @ X @ Y )
=> ? [Z2: A] :
( ( ord_less @ A @ X @ Z2 )
& ( ord_less @ A @ Z2 @ Y ) ) ) ) ).
% dense
thf(fact_3638_gt__ex,axiom,
! [A: $tType] :
( ( no_top @ A )
=> ! [X: A] :
? [X_12: A] : ( ord_less @ A @ X @ X_12 ) ) ).
% gt_ex
thf(fact_3639_lt__ex,axiom,
! [A: $tType] :
( ( no_bot @ A )
=> ! [X: A] :
? [Y3: A] : ( ord_less @ A @ Y3 @ X ) ) ).
% lt_ex
thf(fact_3640_real__add__less__0__iff,axiom,
! [X: real,Y: real] :
( ( ord_less @ real @ ( plus_plus @ real @ X @ Y ) @ ( zero_zero @ real ) )
= ( ord_less @ real @ Y @ ( uminus_uminus @ real @ X ) ) ) ).
% real_add_less_0_iff
thf(fact_3641_real__0__less__add__iff,axiom,
! [X: real,Y: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ ( plus_plus @ real @ X @ Y ) )
= ( ord_less @ real @ ( uminus_uminus @ real @ X ) @ Y ) ) ).
% real_0_less_add_iff
thf(fact_3642_int__zle__neg,axiom,
! [N3: nat,M: nat] :
( ( ord_less_eq @ int @ ( semiring_1_of_nat @ int @ N3 ) @ ( uminus_uminus @ int @ ( semiring_1_of_nat @ int @ M ) ) )
= ( ( N3
= ( zero_zero @ nat ) )
& ( M
= ( zero_zero @ nat ) ) ) ) ).
% int_zle_neg
thf(fact_3643_real__0__le__add__iff,axiom,
! [X: real,Y: real] :
( ( ord_less_eq @ real @ ( zero_zero @ real ) @ ( plus_plus @ real @ X @ Y ) )
= ( ord_less_eq @ real @ ( uminus_uminus @ real @ X ) @ Y ) ) ).
% real_0_le_add_iff
thf(fact_3644_real__add__le__0__iff,axiom,
! [X: real,Y: real] :
( ( ord_less_eq @ real @ ( plus_plus @ real @ X @ Y ) @ ( zero_zero @ real ) )
= ( ord_less_eq @ real @ Y @ ( uminus_uminus @ real @ X ) ) ) ).
% real_add_le_0_iff
thf(fact_3645_negative__zle__0,axiom,
! [N3: nat] : ( ord_less_eq @ int @ ( uminus_uminus @ int @ ( semiring_1_of_nat @ int @ N3 ) ) @ ( zero_zero @ int ) ) ).
% negative_zle_0
thf(fact_3646_nonpos__int__cases,axiom,
! [K: int] :
( ( ord_less_eq @ int @ K @ ( zero_zero @ int ) )
=> ~ ! [N: nat] :
( K
!= ( uminus_uminus @ int @ ( semiring_1_of_nat @ int @ N ) ) ) ) ).
% nonpos_int_cases
thf(fact_3647_overflow__plus__one__self,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [P4: word @ A] :
( ( ord_less_eq @ ( word @ A ) @ ( plus_plus @ ( word @ A ) @ ( one_one @ ( word @ A ) ) @ P4 ) @ P4 )
= ( P4
= ( uminus_uminus @ ( word @ A ) @ ( one_one @ ( word @ A ) ) ) ) ) ) ).
% overflow_plus_one_self
thf(fact_3648_plus__1__less,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [X: word @ A] :
( ( ord_less_eq @ ( word @ A ) @ ( plus_plus @ ( word @ A ) @ X @ ( one_one @ ( word @ A ) ) ) @ X )
= ( X
= ( uminus_uminus @ ( word @ A ) @ ( one_one @ ( word @ A ) ) ) ) ) ) ).
% plus_1_less
thf(fact_3649_max__word__wrap,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [X: word @ A] :
( ( ( plus_plus @ ( word @ A ) @ X @ ( one_one @ ( word @ A ) ) )
= ( zero_zero @ ( word @ A ) ) )
=> ( X
= ( uminus_uminus @ ( word @ A ) @ ( one_one @ ( word @ A ) ) ) ) ) ) ).
% max_word_wrap
thf(fact_3650_word__add__no__overflow,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [X: word @ A] :
( ( ord_less @ ( word @ A ) @ X @ ( uminus_uminus @ ( word @ A ) @ ( one_one @ ( word @ A ) ) ) )
=> ( ord_less @ ( word @ A ) @ X @ ( plus_plus @ ( word @ A ) @ X @ ( one_one @ ( word @ A ) ) ) ) ) ) ).
% word_add_no_overflow
thf(fact_3651_less__x__plus__1,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [X: word @ A,Y: word @ A] :
( ( X
!= ( uminus_uminus @ ( word @ A ) @ ( one_one @ ( word @ A ) ) ) )
=> ( ( ord_less @ ( word @ A ) @ Y @ ( plus_plus @ ( word @ A ) @ X @ ( one_one @ ( word @ A ) ) ) )
= ( ( ord_less @ ( word @ A ) @ Y @ X )
| ( Y = X ) ) ) ) ) ).
% less_x_plus_1
thf(fact_3652_zmod__zminus1__eq__if,axiom,
! [A2: int,B2: int] :
( ( ( ( modulo_modulo @ int @ A2 @ B2 )
= ( zero_zero @ int ) )
=> ( ( modulo_modulo @ int @ ( uminus_uminus @ int @ A2 ) @ B2 )
= ( zero_zero @ int ) ) )
& ( ( ( modulo_modulo @ int @ A2 @ B2 )
!= ( zero_zero @ int ) )
=> ( ( modulo_modulo @ int @ ( uminus_uminus @ int @ A2 ) @ B2 )
= ( minus_minus @ int @ B2 @ ( modulo_modulo @ int @ A2 @ B2 ) ) ) ) ) ).
% zmod_zminus1_eq_if
thf(fact_3653_zmod__zminus2__eq__if,axiom,
! [A2: int,B2: int] :
( ( ( ( modulo_modulo @ int @ A2 @ B2 )
= ( zero_zero @ int ) )
=> ( ( modulo_modulo @ int @ A2 @ ( uminus_uminus @ int @ B2 ) )
= ( zero_zero @ int ) ) )
& ( ( ( modulo_modulo @ int @ A2 @ B2 )
!= ( zero_zero @ int ) )
=> ( ( modulo_modulo @ int @ A2 @ ( uminus_uminus @ int @ B2 ) )
= ( minus_minus @ int @ ( modulo_modulo @ int @ A2 @ B2 ) @ B2 ) ) ) ) ).
% zmod_zminus2_eq_if
thf(fact_3654_no__plus__overflow__neg,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [X: word @ A,Y: word @ A] :
( ( ord_less @ ( word @ A ) @ X @ ( uminus_uminus @ ( word @ A ) @ Y ) )
=> ( ord_less_eq @ ( word @ A ) @ X @ ( plus_plus @ ( word @ A ) @ X @ Y ) ) ) ) ).
% no_plus_overflow_neg
thf(fact_3655_divide__powr__uminus,axiom,
! [A2: real,B2: real,C2: real] :
( ( divide_divide @ real @ A2 @ ( powr @ real @ B2 @ C2 ) )
= ( times_times @ real @ A2 @ ( powr @ real @ B2 @ ( uminus_uminus @ real @ C2 ) ) ) ) ).
% divide_powr_uminus
thf(fact_3656_fact__prod,axiom,
! [A: $tType] :
( ( semiring_char_0 @ A )
=> ( ( semiring_char_0_fact @ A )
= ( ^ [N2: nat] :
( semiring_1_of_nat @ A
@ ( groups7121269368397514597t_prod @ nat @ nat
@ ^ [X2: nat] : X2
@ ( set_or1337092689740270186AtMost @ nat @ ( one_one @ nat ) @ N2 ) ) ) ) ) ) ).
% fact_prod
thf(fact_3657_less__fun__def,axiom,
! [B: $tType,A: $tType] :
( ( ord @ B )
=> ( ( ord_less @ ( A > B ) )
= ( ^ [F7: A > B,G4: A > B] :
( ( ord_less_eq @ ( A > B ) @ F7 @ G4 )
& ~ ( ord_less_eq @ ( A > B ) @ G4 @ F7 ) ) ) ) ) ).
% less_fun_def
thf(fact_3658_fact__diff__Suc,axiom,
! [N3: nat,M: nat] :
( ( ord_less @ nat @ N3 @ ( suc @ M ) )
=> ( ( semiring_char_0_fact @ nat @ ( minus_minus @ nat @ ( suc @ M ) @ N3 ) )
= ( times_times @ nat @ ( minus_minus @ nat @ ( suc @ M ) @ N3 ) @ ( semiring_char_0_fact @ nat @ ( minus_minus @ nat @ M @ N3 ) ) ) ) ) ).
% fact_diff_Suc
thf(fact_3659_fact__div__fact__le__pow,axiom,
! [R2: nat,N3: nat] :
( ( ord_less_eq @ nat @ R2 @ N3 )
=> ( ord_less_eq @ nat @ ( divide_divide @ nat @ ( semiring_char_0_fact @ nat @ N3 ) @ ( semiring_char_0_fact @ nat @ ( minus_minus @ nat @ N3 @ R2 ) ) ) @ ( power_power @ nat @ N3 @ R2 ) ) ) ).
% fact_div_fact_le_pow
thf(fact_3660_binomial__fact__lemma,axiom,
! [K: nat,N3: nat] :
( ( ord_less_eq @ nat @ K @ N3 )
=> ( ( times_times @ nat @ ( times_times @ nat @ ( semiring_char_0_fact @ nat @ K ) @ ( semiring_char_0_fact @ nat @ ( minus_minus @ nat @ N3 @ K ) ) ) @ ( binomial @ N3 @ K ) )
= ( semiring_char_0_fact @ nat @ N3 ) ) ) ).
% binomial_fact_lemma
thf(fact_3661_pos__minus__divide__less__eq,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [C2: A,B2: A,A2: A] :
( ( ord_less @ A @ ( zero_zero @ A ) @ C2 )
=> ( ( ord_less @ A @ ( uminus_uminus @ A @ ( divide_divide @ A @ B2 @ C2 ) ) @ A2 )
= ( ord_less @ A @ ( uminus_uminus @ A @ B2 ) @ ( times_times @ A @ A2 @ C2 ) ) ) ) ) ).
% pos_minus_divide_less_eq
thf(fact_3662_pos__less__minus__divide__eq,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [C2: A,A2: A,B2: A] :
( ( ord_less @ A @ ( zero_zero @ A ) @ C2 )
=> ( ( ord_less @ A @ A2 @ ( uminus_uminus @ A @ ( divide_divide @ A @ B2 @ C2 ) ) )
= ( ord_less @ A @ ( times_times @ A @ A2 @ C2 ) @ ( uminus_uminus @ A @ B2 ) ) ) ) ) ).
% pos_less_minus_divide_eq
thf(fact_3663_neg__minus__divide__less__eq,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [C2: A,B2: A,A2: A] :
( ( ord_less @ A @ C2 @ ( zero_zero @ A ) )
=> ( ( ord_less @ A @ ( uminus_uminus @ A @ ( divide_divide @ A @ B2 @ C2 ) ) @ A2 )
= ( ord_less @ A @ ( times_times @ A @ A2 @ C2 ) @ ( uminus_uminus @ A @ B2 ) ) ) ) ) ).
% neg_minus_divide_less_eq
thf(fact_3664_neg__less__minus__divide__eq,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [C2: A,A2: A,B2: A] :
( ( ord_less @ A @ C2 @ ( zero_zero @ A ) )
=> ( ( ord_less @ A @ A2 @ ( uminus_uminus @ A @ ( divide_divide @ A @ B2 @ C2 ) ) )
= ( ord_less @ A @ ( uminus_uminus @ A @ B2 ) @ ( times_times @ A @ A2 @ C2 ) ) ) ) ) ).
% neg_less_minus_divide_eq
thf(fact_3665_minus__divide__less__eq,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [B2: A,C2: A,A2: A] :
( ( ord_less @ A @ ( uminus_uminus @ A @ ( divide_divide @ A @ B2 @ C2 ) ) @ A2 )
= ( ( ( ord_less @ A @ ( zero_zero @ A ) @ C2 )
=> ( ord_less @ A @ ( uminus_uminus @ A @ B2 ) @ ( times_times @ A @ A2 @ C2 ) ) )
& ( ~ ( ord_less @ A @ ( zero_zero @ A ) @ C2 )
=> ( ( ( ord_less @ A @ C2 @ ( zero_zero @ A ) )
=> ( ord_less @ A @ ( times_times @ A @ A2 @ C2 ) @ ( uminus_uminus @ A @ B2 ) ) )
& ( ~ ( ord_less @ A @ C2 @ ( zero_zero @ A ) )
=> ( ord_less @ A @ ( zero_zero @ A ) @ A2 ) ) ) ) ) ) ) ).
% minus_divide_less_eq
thf(fact_3666_less__minus__divide__eq,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [A2: A,B2: A,C2: A] :
( ( ord_less @ A @ A2 @ ( uminus_uminus @ A @ ( divide_divide @ A @ B2 @ C2 ) ) )
= ( ( ( ord_less @ A @ ( zero_zero @ A ) @ C2 )
=> ( ord_less @ A @ ( times_times @ A @ A2 @ C2 ) @ ( uminus_uminus @ A @ B2 ) ) )
& ( ~ ( ord_less @ A @ ( zero_zero @ A ) @ C2 )
=> ( ( ( ord_less @ A @ C2 @ ( zero_zero @ A ) )
=> ( ord_less @ A @ ( uminus_uminus @ A @ B2 ) @ ( times_times @ A @ A2 @ C2 ) ) )
& ( ~ ( ord_less @ A @ C2 @ ( zero_zero @ A ) )
=> ( ord_less @ A @ A2 @ ( zero_zero @ A ) ) ) ) ) ) ) ) ).
% less_minus_divide_eq
thf(fact_3667_eq__divide__eq__numeral_I2_J,axiom,
! [A: $tType] :
( ( division_ring @ A )
=> ! [W: num,B2: A,C2: A] :
( ( ( uminus_uminus @ A @ ( numeral_numeral @ A @ W ) )
= ( divide_divide @ A @ B2 @ C2 ) )
= ( ( ( C2
!= ( zero_zero @ A ) )
=> ( ( times_times @ A @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ W ) ) @ C2 )
= B2 ) )
& ( ( C2
= ( zero_zero @ A ) )
=> ( ( uminus_uminus @ A @ ( numeral_numeral @ A @ W ) )
= ( zero_zero @ A ) ) ) ) ) ) ).
% eq_divide_eq_numeral(2)
thf(fact_3668_divide__eq__eq__numeral_I2_J,axiom,
! [A: $tType] :
( ( division_ring @ A )
=> ! [B2: A,C2: A,W: num] :
( ( ( divide_divide @ A @ B2 @ C2 )
= ( uminus_uminus @ A @ ( numeral_numeral @ A @ W ) ) )
= ( ( ( C2
!= ( zero_zero @ A ) )
=> ( B2
= ( times_times @ A @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ W ) ) @ C2 ) ) )
& ( ( C2
= ( zero_zero @ A ) )
=> ( ( uminus_uminus @ A @ ( numeral_numeral @ A @ W ) )
= ( zero_zero @ A ) ) ) ) ) ) ).
% divide_eq_eq_numeral(2)
thf(fact_3669_minus__divide__add__eq__iff,axiom,
! [A: $tType] :
( ( division_ring @ A )
=> ! [Z: A,X: A,Y: A] :
( ( Z
!= ( zero_zero @ A ) )
=> ( ( plus_plus @ A @ ( uminus_uminus @ A @ ( divide_divide @ A @ X @ Z ) ) @ Y )
= ( divide_divide @ A @ ( plus_plus @ A @ ( uminus_uminus @ A @ X ) @ ( times_times @ A @ Y @ Z ) ) @ Z ) ) ) ) ).
% minus_divide_add_eq_iff
thf(fact_3670_add__divide__eq__if__simps_I3_J,axiom,
! [A: $tType] :
( ( division_ring @ A )
=> ! [Z: A,A2: A,B2: A] :
( ( ( Z
= ( zero_zero @ A ) )
=> ( ( plus_plus @ A @ ( uminus_uminus @ A @ ( divide_divide @ A @ A2 @ Z ) ) @ B2 )
= B2 ) )
& ( ( Z
!= ( zero_zero @ A ) )
=> ( ( plus_plus @ A @ ( uminus_uminus @ A @ ( divide_divide @ A @ A2 @ Z ) ) @ B2 )
= ( divide_divide @ A @ ( plus_plus @ A @ ( uminus_uminus @ A @ A2 ) @ ( times_times @ A @ B2 @ Z ) ) @ Z ) ) ) ) ) ).
% add_divide_eq_if_simps(3)
thf(fact_3671_add__divide__eq__if__simps_I6_J,axiom,
! [A: $tType] :
( ( division_ring @ A )
=> ! [Z: A,A2: A,B2: A] :
( ( ( Z
= ( zero_zero @ A ) )
=> ( ( minus_minus @ A @ ( uminus_uminus @ A @ ( divide_divide @ A @ A2 @ Z ) ) @ B2 )
= ( uminus_uminus @ A @ B2 ) ) )
& ( ( Z
!= ( zero_zero @ A ) )
=> ( ( minus_minus @ A @ ( uminus_uminus @ A @ ( divide_divide @ A @ A2 @ Z ) ) @ B2 )
= ( divide_divide @ A @ ( minus_minus @ A @ ( uminus_uminus @ A @ A2 ) @ ( times_times @ A @ B2 @ Z ) ) @ Z ) ) ) ) ) ).
% add_divide_eq_if_simps(6)
thf(fact_3672_add__divide__eq__if__simps_I5_J,axiom,
! [A: $tType] :
( ( division_ring @ A )
=> ! [Z: A,A2: A,B2: A] :
( ( ( Z
= ( zero_zero @ A ) )
=> ( ( minus_minus @ A @ ( divide_divide @ A @ A2 @ Z ) @ B2 )
= ( uminus_uminus @ A @ B2 ) ) )
& ( ( Z
!= ( zero_zero @ A ) )
=> ( ( minus_minus @ A @ ( divide_divide @ A @ A2 @ Z ) @ B2 )
= ( divide_divide @ A @ ( minus_minus @ A @ A2 @ ( times_times @ A @ B2 @ Z ) ) @ Z ) ) ) ) ) ).
% add_divide_eq_if_simps(5)
thf(fact_3673_minus__divide__diff__eq__iff,axiom,
! [A: $tType] :
( ( division_ring @ A )
=> ! [Z: A,X: A,Y: A] :
( ( Z
!= ( zero_zero @ A ) )
=> ( ( minus_minus @ A @ ( uminus_uminus @ A @ ( divide_divide @ A @ X @ Z ) ) @ Y )
= ( divide_divide @ A @ ( minus_minus @ A @ ( uminus_uminus @ A @ X ) @ ( times_times @ A @ Y @ Z ) ) @ Z ) ) ) ) ).
% minus_divide_diff_eq_iff
thf(fact_3674_even__minus,axiom,
! [A: $tType] :
( ( ring_parity @ A )
=> ! [A2: A] :
( ( dvd_dvd @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ ( uminus_uminus @ A @ A2 ) )
= ( dvd_dvd @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ A2 ) ) ) ).
% even_minus
thf(fact_3675_power2__eq__iff,axiom,
! [A: $tType] :
( ( idom @ A )
=> ! [X: A,Y: A] :
( ( ( power_power @ A @ X @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) )
= ( power_power @ A @ Y @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) )
= ( ( X = Y )
| ( X
= ( uminus_uminus @ A @ Y ) ) ) ) ) ).
% power2_eq_iff
thf(fact_3676_choose__dvd,axiom,
! [A: $tType] :
( ( linordered_semidom @ A )
=> ! [K: nat,N3: nat] :
( ( ord_less_eq @ nat @ K @ N3 )
=> ( dvd_dvd @ A @ ( times_times @ A @ ( semiring_char_0_fact @ A @ K ) @ ( semiring_char_0_fact @ A @ ( minus_minus @ nat @ N3 @ K ) ) ) @ ( semiring_char_0_fact @ A @ N3 ) ) ) ) ).
% choose_dvd
thf(fact_3677_predicate1D,axiom,
! [A: $tType,P: A > $o,Q: A > $o,X: A] :
( ( ord_less_eq @ ( A > $o ) @ P @ Q )
=> ( ( P @ X )
=> ( Q @ X ) ) ) ).
% predicate1D
thf(fact_3678_rev__predicate1D,axiom,
! [A: $tType,P: A > $o,X: A,Q: A > $o] :
( ( P @ X )
=> ( ( ord_less_eq @ ( A > $o ) @ P @ Q )
=> ( Q @ X ) ) ) ).
% rev_predicate1D
thf(fact_3679_int__cases3,axiom,
! [K: int] :
( ( K
!= ( zero_zero @ int ) )
=> ( ! [N: nat] :
( ( K
= ( semiring_1_of_nat @ int @ N ) )
=> ~ ( ord_less @ nat @ ( zero_zero @ nat ) @ N ) )
=> ~ ! [N: nat] :
( ( K
= ( uminus_uminus @ int @ ( semiring_1_of_nat @ int @ N ) ) )
=> ~ ( ord_less @ nat @ ( zero_zero @ nat ) @ N ) ) ) ) ).
% int_cases3
thf(fact_3680_fact__eq__fact__times,axiom,
! [N3: nat,M: nat] :
( ( ord_less_eq @ nat @ N3 @ M )
=> ( ( semiring_char_0_fact @ nat @ M )
= ( times_times @ nat @ ( semiring_char_0_fact @ nat @ N3 )
@ ( groups7121269368397514597t_prod @ nat @ nat
@ ^ [X2: nat] : X2
@ ( set_or1337092689740270186AtMost @ nat @ ( suc @ N3 ) @ M ) ) ) ) ) ).
% fact_eq_fact_times
thf(fact_3681_pochhammer__eq__0__iff,axiom,
! [A: $tType] :
( ( field_char_0 @ A )
=> ! [A2: A,N3: nat] :
( ( ( comm_s3205402744901411588hammer @ A @ A2 @ N3 )
= ( zero_zero @ A ) )
= ( ? [K2: nat] :
( ( ord_less @ nat @ K2 @ N3 )
& ( A2
= ( uminus_uminus @ A @ ( semiring_1_of_nat @ A @ K2 ) ) ) ) ) ) ) ).
% pochhammer_eq_0_iff
thf(fact_3682_pochhammer__of__nat__eq__0__iff,axiom,
! [A: $tType] :
( ( ( ring_char_0 @ A )
& ( idom @ A ) )
=> ! [N3: nat,K: nat] :
( ( ( comm_s3205402744901411588hammer @ A @ ( uminus_uminus @ A @ ( semiring_1_of_nat @ A @ N3 ) ) @ K )
= ( zero_zero @ A ) )
= ( ord_less @ nat @ N3 @ K ) ) ) ).
% pochhammer_of_nat_eq_0_iff
thf(fact_3683_pochhammer__of__nat__eq__0__lemma,axiom,
! [A: $tType] :
( ( idom @ A )
=> ! [N3: nat,K: nat] :
( ( ord_less @ nat @ N3 @ K )
=> ( ( comm_s3205402744901411588hammer @ A @ ( uminus_uminus @ A @ ( semiring_1_of_nat @ A @ N3 ) ) @ K )
= ( zero_zero @ A ) ) ) ) ).
% pochhammer_of_nat_eq_0_lemma
thf(fact_3684_pochhammer__of__nat__eq__0__lemma_H,axiom,
! [A: $tType] :
( ( ( ring_char_0 @ A )
& ( idom @ A ) )
=> ! [K: nat,N3: nat] :
( ( ord_less_eq @ nat @ K @ N3 )
=> ( ( comm_s3205402744901411588hammer @ A @ ( uminus_uminus @ A @ ( semiring_1_of_nat @ A @ N3 ) ) @ K )
!= ( zero_zero @ A ) ) ) ) ).
% pochhammer_of_nat_eq_0_lemma'
thf(fact_3685_not__zle__0__negative,axiom,
! [N3: nat] :
~ ( ord_less_eq @ int @ ( zero_zero @ int ) @ ( uminus_uminus @ int @ ( semiring_1_of_nat @ int @ ( suc @ N3 ) ) ) ) ).
% not_zle_0_negative
thf(fact_3686_negative__zless__0,axiom,
! [N3: nat] : ( ord_less @ int @ ( uminus_uminus @ int @ ( semiring_1_of_nat @ int @ ( suc @ N3 ) ) ) @ ( zero_zero @ int ) ) ).
% negative_zless_0
thf(fact_3687_negD,axiom,
! [X: int] :
( ( ord_less @ int @ X @ ( zero_zero @ int ) )
=> ? [N: nat] :
( X
= ( uminus_uminus @ int @ ( semiring_1_of_nat @ int @ ( suc @ N ) ) ) ) ) ).
% negD
thf(fact_3688_verit__less__mono__div__int2,axiom,
! [A3: int,B6: int,N3: int] :
( ( ord_less_eq @ int @ A3 @ B6 )
=> ( ( ord_less @ int @ ( zero_zero @ int ) @ ( uminus_uminus @ int @ N3 ) )
=> ( ord_less_eq @ int @ ( divide_divide @ int @ B6 @ N3 ) @ ( divide_divide @ int @ A3 @ N3 ) ) ) ) ).
% verit_less_mono_div_int2
thf(fact_3689_div__eq__minus1,axiom,
! [B2: int] :
( ( ord_less @ int @ ( zero_zero @ int ) @ B2 )
=> ( ( divide_divide @ int @ ( uminus_uminus @ int @ ( one_one @ int ) ) @ B2 )
= ( uminus_uminus @ int @ ( one_one @ int ) ) ) ) ).
% div_eq_minus1
thf(fact_3690_word__le__make__less,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [Y: word @ A,X: word @ A] :
( ( Y
!= ( uminus_uminus @ ( word @ A ) @ ( one_one @ ( word @ A ) ) ) )
=> ( ( ord_less_eq @ ( word @ A ) @ X @ Y )
= ( ord_less @ ( word @ A ) @ X @ ( plus_plus @ ( word @ A ) @ Y @ ( one_one @ ( word @ A ) ) ) ) ) ) ) ).
% word_le_make_less
thf(fact_3691_word__Suc__leq,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [K: word @ A,X: word @ A] :
( ( K
!= ( uminus_uminus @ ( word @ A ) @ ( one_one @ ( word @ A ) ) ) )
=> ( ( ord_less @ ( word @ A ) @ X @ ( plus_plus @ ( word @ A ) @ K @ ( one_one @ ( word @ A ) ) ) )
= ( ord_less_eq @ ( word @ A ) @ X @ K ) ) ) ) ).
% word_Suc_leq
thf(fact_3692_word__Suc__le,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [X: word @ A,K: word @ A] :
( ( X
!= ( uminus_uminus @ ( word @ A ) @ ( one_one @ ( word @ A ) ) ) )
=> ( ( ord_less_eq @ ( word @ A ) @ ( plus_plus @ ( word @ A ) @ X @ ( one_one @ ( word @ A ) ) ) @ K )
= ( ord_less @ ( word @ A ) @ X @ K ) ) ) ) ).
% word_Suc_le
thf(fact_3693_ceiling__divide__eq__div,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [A2: int,B2: int] :
( ( archimedean_ceiling @ A @ ( divide_divide @ A @ ( ring_1_of_int @ A @ A2 ) @ ( ring_1_of_int @ A @ B2 ) ) )
= ( uminus_uminus @ int @ ( divide_divide @ int @ ( uminus_uminus @ int @ A2 ) @ B2 ) ) ) ) ).
% ceiling_divide_eq_div
thf(fact_3694_binomial__altdef__nat,axiom,
! [K: nat,N3: nat] :
( ( ord_less_eq @ nat @ K @ N3 )
=> ( ( binomial @ N3 @ K )
= ( divide_divide @ nat @ ( semiring_char_0_fact @ nat @ N3 ) @ ( times_times @ nat @ ( semiring_char_0_fact @ nat @ K ) @ ( semiring_char_0_fact @ nat @ ( minus_minus @ nat @ N3 @ K ) ) ) ) ) ) ).
% binomial_altdef_nat
thf(fact_3695_cos__coeff__def,axiom,
( cos_coeff
= ( ^ [N2: nat] : ( if @ real @ ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N2 ) @ ( divide_divide @ real @ ( power_power @ real @ ( uminus_uminus @ real @ ( one_one @ real ) ) @ ( divide_divide @ nat @ N2 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( semiring_char_0_fact @ real @ N2 ) ) @ ( zero_zero @ real ) ) ) ) ).
% cos_coeff_def
thf(fact_3696_pos__minus__divide__le__eq,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [C2: A,B2: A,A2: A] :
( ( ord_less @ A @ ( zero_zero @ A ) @ C2 )
=> ( ( ord_less_eq @ A @ ( uminus_uminus @ A @ ( divide_divide @ A @ B2 @ C2 ) ) @ A2 )
= ( ord_less_eq @ A @ ( uminus_uminus @ A @ B2 ) @ ( times_times @ A @ A2 @ C2 ) ) ) ) ) ).
% pos_minus_divide_le_eq
thf(fact_3697_pos__le__minus__divide__eq,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [C2: A,A2: A,B2: A] :
( ( ord_less @ A @ ( zero_zero @ A ) @ C2 )
=> ( ( ord_less_eq @ A @ A2 @ ( uminus_uminus @ A @ ( divide_divide @ A @ B2 @ C2 ) ) )
= ( ord_less_eq @ A @ ( times_times @ A @ A2 @ C2 ) @ ( uminus_uminus @ A @ B2 ) ) ) ) ) ).
% pos_le_minus_divide_eq
thf(fact_3698_neg__minus__divide__le__eq,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [C2: A,B2: A,A2: A] :
( ( ord_less @ A @ C2 @ ( zero_zero @ A ) )
=> ( ( ord_less_eq @ A @ ( uminus_uminus @ A @ ( divide_divide @ A @ B2 @ C2 ) ) @ A2 )
= ( ord_less_eq @ A @ ( times_times @ A @ A2 @ C2 ) @ ( uminus_uminus @ A @ B2 ) ) ) ) ) ).
% neg_minus_divide_le_eq
thf(fact_3699_neg__le__minus__divide__eq,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [C2: A,A2: A,B2: A] :
( ( ord_less @ A @ C2 @ ( zero_zero @ A ) )
=> ( ( ord_less_eq @ A @ A2 @ ( uminus_uminus @ A @ ( divide_divide @ A @ B2 @ C2 ) ) )
= ( ord_less_eq @ A @ ( uminus_uminus @ A @ B2 ) @ ( times_times @ A @ A2 @ C2 ) ) ) ) ) ).
% neg_le_minus_divide_eq
thf(fact_3700_minus__divide__le__eq,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [B2: A,C2: A,A2: A] :
( ( ord_less_eq @ A @ ( uminus_uminus @ A @ ( divide_divide @ A @ B2 @ C2 ) ) @ A2 )
= ( ( ( ord_less @ A @ ( zero_zero @ A ) @ C2 )
=> ( ord_less_eq @ A @ ( uminus_uminus @ A @ B2 ) @ ( times_times @ A @ A2 @ C2 ) ) )
& ( ~ ( ord_less @ A @ ( zero_zero @ A ) @ C2 )
=> ( ( ( ord_less @ A @ C2 @ ( zero_zero @ A ) )
=> ( ord_less_eq @ A @ ( times_times @ A @ A2 @ C2 ) @ ( uminus_uminus @ A @ B2 ) ) )
& ( ~ ( ord_less @ A @ C2 @ ( zero_zero @ A ) )
=> ( ord_less_eq @ A @ ( zero_zero @ A ) @ A2 ) ) ) ) ) ) ) ).
% minus_divide_le_eq
thf(fact_3701_le__minus__divide__eq,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [A2: A,B2: A,C2: A] :
( ( ord_less_eq @ A @ A2 @ ( uminus_uminus @ A @ ( divide_divide @ A @ B2 @ C2 ) ) )
= ( ( ( ord_less @ A @ ( zero_zero @ A ) @ C2 )
=> ( ord_less_eq @ A @ ( times_times @ A @ A2 @ C2 ) @ ( uminus_uminus @ A @ B2 ) ) )
& ( ~ ( ord_less @ A @ ( zero_zero @ A ) @ C2 )
=> ( ( ( ord_less @ A @ C2 @ ( zero_zero @ A ) )
=> ( ord_less_eq @ A @ ( uminus_uminus @ A @ B2 ) @ ( times_times @ A @ A2 @ C2 ) ) )
& ( ~ ( ord_less @ A @ C2 @ ( zero_zero @ A ) )
=> ( ord_less_eq @ A @ A2 @ ( zero_zero @ A ) ) ) ) ) ) ) ) ).
% le_minus_divide_eq
thf(fact_3702_less__divide__eq__numeral_I2_J,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [W: num,B2: A,C2: A] :
( ( ord_less @ A @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ W ) ) @ ( divide_divide @ A @ B2 @ C2 ) )
= ( ( ( ord_less @ A @ ( zero_zero @ A ) @ C2 )
=> ( ord_less @ A @ ( times_times @ A @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ W ) ) @ C2 ) @ B2 ) )
& ( ~ ( ord_less @ A @ ( zero_zero @ A ) @ C2 )
=> ( ( ( ord_less @ A @ C2 @ ( zero_zero @ A ) )
=> ( ord_less @ A @ B2 @ ( times_times @ A @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ W ) ) @ C2 ) ) )
& ( ~ ( ord_less @ A @ C2 @ ( zero_zero @ A ) )
=> ( ord_less @ A @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ W ) ) @ ( zero_zero @ A ) ) ) ) ) ) ) ) ).
% less_divide_eq_numeral(2)
thf(fact_3703_divide__less__eq__numeral_I2_J,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [B2: A,C2: A,W: num] :
( ( ord_less @ A @ ( divide_divide @ A @ B2 @ C2 ) @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ W ) ) )
= ( ( ( ord_less @ A @ ( zero_zero @ A ) @ C2 )
=> ( ord_less @ A @ B2 @ ( times_times @ A @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ W ) ) @ C2 ) ) )
& ( ~ ( ord_less @ A @ ( zero_zero @ A ) @ C2 )
=> ( ( ( ord_less @ A @ C2 @ ( zero_zero @ A ) )
=> ( ord_less @ A @ ( times_times @ A @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ W ) ) @ C2 ) @ B2 ) )
& ( ~ ( ord_less @ A @ C2 @ ( zero_zero @ A ) )
=> ( ord_less @ A @ ( zero_zero @ A ) @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ W ) ) ) ) ) ) ) ) ) ).
% divide_less_eq_numeral(2)
thf(fact_3704_power2__eq__1__iff,axiom,
! [A: $tType] :
( ( ring_15535105094025558882visors @ A )
=> ! [A2: A] :
( ( ( power_power @ A @ A2 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) )
= ( one_one @ A ) )
= ( ( A2
= ( one_one @ A ) )
| ( A2
= ( uminus_uminus @ A @ ( one_one @ A ) ) ) ) ) ) ).
% power2_eq_1_iff
thf(fact_3705_uminus__power__if,axiom,
! [A: $tType] :
( ( ring_1 @ A )
=> ! [N3: nat,A2: A] :
( ( ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N3 )
=> ( ( power_power @ A @ ( uminus_uminus @ A @ A2 ) @ N3 )
= ( power_power @ A @ A2 @ N3 ) ) )
& ( ~ ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N3 )
=> ( ( power_power @ A @ ( uminus_uminus @ A @ A2 ) @ N3 )
= ( uminus_uminus @ A @ ( power_power @ A @ A2 @ N3 ) ) ) ) ) ) ).
% uminus_power_if
thf(fact_3706_neg__one__power__add__eq__neg__one__power__diff,axiom,
! [A: $tType] :
( ( ring_1 @ A )
=> ! [K: nat,N3: nat] :
( ( ord_less_eq @ nat @ K @ N3 )
=> ( ( power_power @ A @ ( uminus_uminus @ A @ ( one_one @ A ) ) @ ( plus_plus @ nat @ N3 @ K ) )
= ( power_power @ A @ ( uminus_uminus @ A @ ( one_one @ A ) ) @ ( minus_minus @ nat @ N3 @ K ) ) ) ) ) ).
% neg_one_power_add_eq_neg_one_power_diff
thf(fact_3707_realpow__square__minus__le,axiom,
! [U: real,X: real] : ( ord_less_eq @ real @ ( uminus_uminus @ real @ ( power_power @ real @ U @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( power_power @ real @ X @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ).
% realpow_square_minus_le
thf(fact_3708_fact__div__fact,axiom,
! [N3: nat,M: nat] :
( ( ord_less_eq @ nat @ N3 @ M )
=> ( ( divide_divide @ nat @ ( semiring_char_0_fact @ nat @ M ) @ ( semiring_char_0_fact @ nat @ N3 ) )
= ( groups7121269368397514597t_prod @ nat @ nat
@ ^ [X2: nat] : X2
@ ( set_or1337092689740270186AtMost @ nat @ ( plus_plus @ nat @ N3 @ ( one_one @ nat ) ) @ M ) ) ) ) ).
% fact_div_fact
thf(fact_3709_neg__int__cases,axiom,
! [K: int] :
( ( ord_less @ int @ K @ ( zero_zero @ int ) )
=> ~ ! [N: nat] :
( ( K
= ( uminus_uminus @ int @ ( semiring_1_of_nat @ int @ N ) ) )
=> ~ ( ord_less @ nat @ ( zero_zero @ nat ) @ N ) ) ) ).
% neg_int_cases
thf(fact_3710_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_3711_powr__neg__one,axiom,
! [X: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ X )
=> ( ( powr @ real @ X @ ( uminus_uminus @ real @ ( one_one @ real ) ) )
= ( divide_divide @ real @ ( one_one @ real ) @ X ) ) ) ).
% powr_neg_one
thf(fact_3712_minus__mod__int__eq,axiom,
! [L2: int,K: int] :
( ( ord_less_eq @ int @ ( zero_zero @ int ) @ L2 )
=> ( ( modulo_modulo @ int @ ( uminus_uminus @ int @ K ) @ L2 )
= ( minus_minus @ int @ ( minus_minus @ int @ L2 @ ( one_one @ int ) ) @ ( modulo_modulo @ int @ ( minus_minus @ int @ K @ ( one_one @ int ) ) @ L2 ) ) ) ) ).
% minus_mod_int_eq
thf(fact_3713_zmod__minus1,axiom,
! [B2: int] :
( ( ord_less @ int @ ( zero_zero @ int ) @ B2 )
=> ( ( modulo_modulo @ int @ ( uminus_uminus @ int @ ( one_one @ int ) ) @ B2 )
= ( minus_minus @ int @ B2 @ ( one_one @ int ) ) ) ) ).
% zmod_minus1
thf(fact_3714_zdiv__zminus1__eq__if,axiom,
! [B2: int,A2: int] :
( ( B2
!= ( zero_zero @ int ) )
=> ( ( ( ( modulo_modulo @ int @ A2 @ B2 )
= ( zero_zero @ int ) )
=> ( ( divide_divide @ int @ ( uminus_uminus @ int @ A2 ) @ B2 )
= ( uminus_uminus @ int @ ( divide_divide @ int @ A2 @ B2 ) ) ) )
& ( ( ( modulo_modulo @ int @ A2 @ B2 )
!= ( zero_zero @ int ) )
=> ( ( divide_divide @ int @ ( uminus_uminus @ int @ A2 ) @ B2 )
= ( minus_minus @ int @ ( uminus_uminus @ int @ ( divide_divide @ int @ A2 @ B2 ) ) @ ( one_one @ int ) ) ) ) ) ) ).
% zdiv_zminus1_eq_if
thf(fact_3715_zdiv__zminus2__eq__if,axiom,
! [B2: int,A2: int] :
( ( B2
!= ( zero_zero @ int ) )
=> ( ( ( ( modulo_modulo @ int @ A2 @ B2 )
= ( zero_zero @ int ) )
=> ( ( divide_divide @ int @ A2 @ ( uminus_uminus @ int @ B2 ) )
= ( uminus_uminus @ int @ ( divide_divide @ int @ A2 @ B2 ) ) ) )
& ( ( ( modulo_modulo @ int @ A2 @ B2 )
!= ( zero_zero @ int ) )
=> ( ( divide_divide @ int @ A2 @ ( uminus_uminus @ int @ B2 ) )
= ( minus_minus @ int @ ( uminus_uminus @ int @ ( divide_divide @ int @ A2 @ B2 ) ) @ ( one_one @ int ) ) ) ) ) ) ).
% zdiv_zminus2_eq_if
thf(fact_3716_square__fact__le__2__fact,axiom,
! [N3: nat] : ( ord_less_eq @ real @ ( times_times @ real @ ( semiring_char_0_fact @ real @ N3 ) @ ( semiring_char_0_fact @ real @ N3 ) ) @ ( semiring_char_0_fact @ real @ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N3 ) ) ) ).
% square_fact_le_2_fact
thf(fact_3717_zminus1__lemma,axiom,
! [A2: int,B2: int,Q2: int,R2: int] :
( ( eucl_rel_int @ A2 @ B2 @ ( product_Pair @ int @ int @ Q2 @ R2 ) )
=> ( ( B2
!= ( zero_zero @ int ) )
=> ( eucl_rel_int @ ( uminus_uminus @ int @ A2 ) @ B2
@ ( product_Pair @ int @ int
@ ( if @ int
@ ( R2
= ( zero_zero @ int ) )
@ ( uminus_uminus @ int @ Q2 )
@ ( minus_minus @ int @ ( uminus_uminus @ int @ Q2 ) @ ( one_one @ int ) ) )
@ ( if @ int
@ ( R2
= ( zero_zero @ int ) )
@ ( zero_zero @ int )
@ ( minus_minus @ int @ B2 @ R2 ) ) ) ) ) ) ).
% zminus1_lemma
thf(fact_3718_le__divide__eq__numeral_I2_J,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [W: num,B2: A,C2: A] :
( ( ord_less_eq @ A @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ W ) ) @ ( divide_divide @ A @ B2 @ C2 ) )
= ( ( ( ord_less @ A @ ( zero_zero @ A ) @ C2 )
=> ( ord_less_eq @ A @ ( times_times @ A @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ W ) ) @ C2 ) @ B2 ) )
& ( ~ ( ord_less @ A @ ( zero_zero @ A ) @ C2 )
=> ( ( ( ord_less @ A @ C2 @ ( zero_zero @ A ) )
=> ( ord_less_eq @ A @ B2 @ ( times_times @ A @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ W ) ) @ C2 ) ) )
& ( ~ ( ord_less @ A @ C2 @ ( zero_zero @ A ) )
=> ( ord_less_eq @ A @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ W ) ) @ ( zero_zero @ A ) ) ) ) ) ) ) ) ).
% le_divide_eq_numeral(2)
thf(fact_3719_divide__le__eq__numeral_I2_J,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [B2: A,C2: A,W: num] :
( ( ord_less_eq @ A @ ( divide_divide @ A @ B2 @ C2 ) @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ W ) ) )
= ( ( ( ord_less @ A @ ( zero_zero @ A ) @ C2 )
=> ( ord_less_eq @ A @ B2 @ ( times_times @ A @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ W ) ) @ C2 ) ) )
& ( ~ ( ord_less @ A @ ( zero_zero @ A ) @ C2 )
=> ( ( ( ord_less @ A @ C2 @ ( zero_zero @ A ) )
=> ( ord_less_eq @ A @ ( times_times @ A @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ W ) ) @ C2 ) @ B2 ) )
& ( ~ ( ord_less @ A @ C2 @ ( zero_zero @ A ) )
=> ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ W ) ) ) ) ) ) ) ) ) ).
% divide_le_eq_numeral(2)
thf(fact_3720_square__le__1,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [X: A] :
( ( ord_less_eq @ A @ ( uminus_uminus @ A @ ( one_one @ A ) ) @ X )
=> ( ( ord_less_eq @ A @ X @ ( one_one @ A ) )
=> ( ord_less_eq @ A @ ( power_power @ A @ X @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ ( one_one @ A ) ) ) ) ) ).
% square_le_1
thf(fact_3721_minus__power__mult__self,axiom,
! [A: $tType] :
( ( comm_ring_1 @ A )
=> ! [A2: A,N3: nat] :
( ( times_times @ A @ ( power_power @ A @ ( uminus_uminus @ A @ A2 ) @ N3 ) @ ( power_power @ A @ ( uminus_uminus @ A @ A2 ) @ N3 ) )
= ( power_power @ A @ A2 @ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N3 ) ) ) ) ).
% minus_power_mult_self
thf(fact_3722_fact__code,axiom,
! [A: $tType] :
( ( semiring_char_0 @ A )
=> ( ( semiring_char_0_fact @ A )
= ( ^ [N2: nat] : ( semiring_1_of_nat @ A @ ( set_fo6178422350223883121st_nat @ nat @ ( times_times @ nat ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N2 @ ( one_one @ nat ) ) ) ) ) ) ).
% fact_code
thf(fact_3723_fact__num__eq__if,axiom,
! [A: $tType] :
( ( semiring_char_0 @ A )
=> ( ( semiring_char_0_fact @ A )
= ( ^ [M5: nat] :
( if @ A
@ ( M5
= ( zero_zero @ nat ) )
@ ( one_one @ A )
@ ( times_times @ A @ ( semiring_1_of_nat @ A @ M5 ) @ ( semiring_char_0_fact @ A @ ( minus_minus @ nat @ M5 @ ( one_one @ nat ) ) ) ) ) ) ) ) ).
% fact_num_eq_if
thf(fact_3724_minus__one__power__iff,axiom,
! [A: $tType] :
( ( ring_1 @ A )
=> ! [N3: nat] :
( ( ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N3 )
=> ( ( power_power @ A @ ( uminus_uminus @ A @ ( one_one @ A ) ) @ N3 )
= ( one_one @ A ) ) )
& ( ~ ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N3 )
=> ( ( power_power @ A @ ( uminus_uminus @ A @ ( one_one @ A ) ) @ N3 )
= ( uminus_uminus @ A @ ( one_one @ A ) ) ) ) ) ) ).
% minus_one_power_iff
thf(fact_3725_fact__reduce,axiom,
! [A: $tType] :
( ( semiring_char_0 @ A )
=> ! [N3: nat] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ N3 )
=> ( ( semiring_char_0_fact @ A @ N3 )
= ( times_times @ A @ ( semiring_1_of_nat @ A @ N3 ) @ ( semiring_char_0_fact @ A @ ( minus_minus @ nat @ N3 @ ( one_one @ nat ) ) ) ) ) ) ) ).
% fact_reduce
thf(fact_3726_binomial__fact,axiom,
! [A: $tType] :
( ( field_char_0 @ A )
=> ! [K: nat,N3: nat] :
( ( ord_less_eq @ nat @ K @ N3 )
=> ( ( semiring_1_of_nat @ A @ ( binomial @ N3 @ K ) )
= ( divide_divide @ A @ ( semiring_char_0_fact @ A @ N3 ) @ ( times_times @ A @ ( semiring_char_0_fact @ A @ K ) @ ( semiring_char_0_fact @ A @ ( minus_minus @ nat @ N3 @ K ) ) ) ) ) ) ) ).
% binomial_fact
thf(fact_3727_fact__binomial,axiom,
! [A: $tType] :
( ( field_char_0 @ A )
=> ! [K: nat,N3: nat] :
( ( ord_less_eq @ nat @ K @ N3 )
=> ( ( times_times @ A @ ( semiring_char_0_fact @ A @ K ) @ ( semiring_1_of_nat @ A @ ( binomial @ N3 @ K ) ) )
= ( divide_divide @ A @ ( semiring_char_0_fact @ A @ N3 ) @ ( semiring_char_0_fact @ A @ ( minus_minus @ nat @ N3 @ K ) ) ) ) ) ) ).
% fact_binomial
thf(fact_3728_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_3729_gbinomial__index__swap,axiom,
! [A: $tType] :
( ( field_char_0 @ A )
=> ! [K: nat,N3: 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 @ N3 ) ) @ ( one_one @ A ) ) @ K ) )
= ( times_times @ A @ ( power_power @ A @ ( uminus_uminus @ A @ ( one_one @ A ) ) @ N3 ) @ ( gbinomial @ A @ ( minus_minus @ A @ ( uminus_uminus @ A @ ( semiring_1_of_nat @ A @ K ) ) @ ( one_one @ A ) ) @ N3 ) ) ) ) ).
% gbinomial_index_swap
thf(fact_3730_gbinomial__negated__upper,axiom,
! [A: $tType] :
( ( field_char_0 @ A )
=> ( ( gbinomial @ A )
= ( ^ [A6: A,K2: nat] : ( times_times @ A @ ( power_power @ A @ ( uminus_uminus @ A @ ( one_one @ A ) ) @ K2 ) @ ( gbinomial @ A @ ( minus_minus @ A @ ( minus_minus @ A @ ( semiring_1_of_nat @ A @ K2 ) @ A6 ) @ ( one_one @ A ) ) @ K2 ) ) ) ) ) ).
% gbinomial_negated_upper
thf(fact_3731_minus__1__div__exp__eq__int,axiom,
! [N3: nat] :
( ( divide_divide @ int @ ( uminus_uminus @ int @ ( one_one @ int ) ) @ ( power_power @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ N3 ) )
= ( uminus_uminus @ int @ ( one_one @ int ) ) ) ).
% minus_1_div_exp_eq_int
thf(fact_3732_div__pos__neg__trivial,axiom,
! [K: int,L2: int] :
( ( ord_less @ int @ ( zero_zero @ int ) @ K )
=> ( ( ord_less_eq @ int @ ( plus_plus @ int @ K @ L2 ) @ ( zero_zero @ int ) )
=> ( ( divide_divide @ int @ K @ L2 )
= ( uminus_uminus @ int @ ( one_one @ int ) ) ) ) ) ).
% div_pos_neg_trivial
thf(fact_3733_Bernoulli__inequality,axiom,
! [X: real,N3: nat] :
( ( ord_less_eq @ real @ ( uminus_uminus @ real @ ( one_one @ real ) ) @ X )
=> ( ord_less_eq @ real @ ( plus_plus @ real @ ( one_one @ real ) @ ( times_times @ real @ ( semiring_1_of_nat @ real @ N3 ) @ X ) ) @ ( power_power @ real @ ( plus_plus @ real @ ( one_one @ real ) @ X ) @ N3 ) ) ) ).
% Bernoulli_inequality
thf(fact_3734_order__le__imp__less__or__eq,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [X: A,Y: A] :
( ( ord_less_eq @ A @ X @ Y )
=> ( ( ord_less @ A @ X @ Y )
| ( X = Y ) ) ) ) ).
% order_le_imp_less_or_eq
thf(fact_3735_linorder__le__less__linear,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [X: A,Y: A] :
( ( ord_less_eq @ A @ X @ Y )
| ( ord_less @ A @ Y @ X ) ) ) ).
% linorder_le_less_linear
thf(fact_3736_order__less__le__subst2,axiom,
! [A: $tType,C: $tType] :
( ( ( order @ C )
& ( order @ A ) )
=> ! [A2: A,B2: A,F2: A > C,C2: C] :
( ( ord_less @ A @ A2 @ B2 )
=> ( ( ord_less_eq @ C @ ( F2 @ B2 ) @ C2 )
=> ( ! [X3: A,Y3: A] :
( ( ord_less @ A @ X3 @ Y3 )
=> ( ord_less @ C @ ( F2 @ X3 ) @ ( F2 @ Y3 ) ) )
=> ( ord_less @ C @ ( F2 @ A2 ) @ C2 ) ) ) ) ) ).
% order_less_le_subst2
thf(fact_3737_order__less__le__subst1,axiom,
! [A: $tType,B: $tType] :
( ( ( order @ B )
& ( order @ A ) )
=> ! [A2: A,F2: B > A,B2: B,C2: B] :
( ( ord_less @ A @ A2 @ ( F2 @ B2 ) )
=> ( ( ord_less_eq @ B @ B2 @ C2 )
=> ( ! [X3: B,Y3: B] :
( ( ord_less_eq @ B @ X3 @ Y3 )
=> ( ord_less_eq @ A @ ( F2 @ X3 ) @ ( F2 @ Y3 ) ) )
=> ( ord_less @ A @ A2 @ ( F2 @ C2 ) ) ) ) ) ) ).
% order_less_le_subst1
thf(fact_3738_order__le__less__subst2,axiom,
! [A: $tType,C: $tType] :
( ( ( order @ C )
& ( order @ A ) )
=> ! [A2: A,B2: A,F2: A > C,C2: C] :
( ( ord_less_eq @ A @ A2 @ B2 )
=> ( ( ord_less @ C @ ( F2 @ B2 ) @ C2 )
=> ( ! [X3: A,Y3: A] :
( ( ord_less_eq @ A @ X3 @ Y3 )
=> ( ord_less_eq @ C @ ( F2 @ X3 ) @ ( F2 @ Y3 ) ) )
=> ( ord_less @ C @ ( F2 @ A2 ) @ C2 ) ) ) ) ) ).
% order_le_less_subst2
thf(fact_3739_order__le__less__subst1,axiom,
! [A: $tType,B: $tType] :
( ( ( order @ B )
& ( order @ A ) )
=> ! [A2: A,F2: B > A,B2: B,C2: B] :
( ( ord_less_eq @ A @ A2 @ ( F2 @ B2 ) )
=> ( ( ord_less @ B @ B2 @ C2 )
=> ( ! [X3: B,Y3: B] :
( ( ord_less @ B @ X3 @ Y3 )
=> ( ord_less @ A @ ( F2 @ X3 ) @ ( F2 @ Y3 ) ) )
=> ( ord_less @ A @ A2 @ ( F2 @ C2 ) ) ) ) ) ) ).
% order_le_less_subst1
thf(fact_3740_order__less__le__trans,axiom,
! [A: $tType] :
( ( preorder @ A )
=> ! [X: A,Y: A,Z: A] :
( ( ord_less @ A @ X @ Y )
=> ( ( ord_less_eq @ A @ Y @ Z )
=> ( ord_less @ A @ X @ Z ) ) ) ) ).
% order_less_le_trans
thf(fact_3741_order__le__less__trans,axiom,
! [A: $tType] :
( ( preorder @ A )
=> ! [X: A,Y: A,Z: A] :
( ( ord_less_eq @ A @ X @ Y )
=> ( ( ord_less @ A @ Y @ Z )
=> ( ord_less @ A @ X @ Z ) ) ) ) ).
% order_le_less_trans
thf(fact_3742_order__neq__le__trans,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [A2: A,B2: A] :
( ( A2 != B2 )
=> ( ( ord_less_eq @ A @ A2 @ B2 )
=> ( ord_less @ A @ A2 @ B2 ) ) ) ) ).
% order_neq_le_trans
thf(fact_3743_order__le__neq__trans,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [A2: A,B2: A] :
( ( ord_less_eq @ A @ A2 @ B2 )
=> ( ( A2 != B2 )
=> ( ord_less @ A @ A2 @ B2 ) ) ) ) ).
% order_le_neq_trans
thf(fact_3744_order__less__imp__le,axiom,
! [A: $tType] :
( ( preorder @ A )
=> ! [X: A,Y: A] :
( ( ord_less @ A @ X @ Y )
=> ( ord_less_eq @ A @ X @ Y ) ) ) ).
% order_less_imp_le
thf(fact_3745_linorder__not__less,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [X: A,Y: A] :
( ( ~ ( ord_less @ A @ X @ Y ) )
= ( ord_less_eq @ A @ Y @ X ) ) ) ).
% linorder_not_less
thf(fact_3746_linorder__not__le,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [X: A,Y: A] :
( ( ~ ( ord_less_eq @ A @ X @ Y ) )
= ( ord_less @ A @ Y @ X ) ) ) ).
% linorder_not_le
thf(fact_3747_order__less__le,axiom,
! [A: $tType] :
( ( order @ A )
=> ( ( ord_less @ A )
= ( ^ [X2: A,Y2: A] :
( ( ord_less_eq @ A @ X2 @ Y2 )
& ( X2 != Y2 ) ) ) ) ) ).
% order_less_le
thf(fact_3748_order__le__less,axiom,
! [A: $tType] :
( ( order @ A )
=> ( ( ord_less_eq @ A )
= ( ^ [X2: A,Y2: A] :
( ( ord_less @ A @ X2 @ Y2 )
| ( X2 = Y2 ) ) ) ) ) ).
% order_le_less
thf(fact_3749_dual__order_Ostrict__implies__order,axiom,
! [A: $tType] :
( ( preorder @ A )
=> ! [B2: A,A2: A] :
( ( ord_less @ A @ B2 @ A2 )
=> ( ord_less_eq @ A @ B2 @ A2 ) ) ) ).
% dual_order.strict_implies_order
thf(fact_3750_order_Ostrict__implies__order,axiom,
! [A: $tType] :
( ( preorder @ A )
=> ! [A2: A,B2: A] :
( ( ord_less @ A @ A2 @ B2 )
=> ( ord_less_eq @ A @ A2 @ B2 ) ) ) ).
% order.strict_implies_order
thf(fact_3751_dual__order_Ostrict__iff__not,axiom,
! [A: $tType] :
( ( preorder @ A )
=> ( ( ord_less @ A )
= ( ^ [B5: A,A6: A] :
( ( ord_less_eq @ A @ B5 @ A6 )
& ~ ( ord_less_eq @ A @ A6 @ B5 ) ) ) ) ) ).
% dual_order.strict_iff_not
thf(fact_3752_dual__order_Ostrict__trans2,axiom,
! [A: $tType] :
( ( preorder @ A )
=> ! [B2: A,A2: A,C2: A] :
( ( ord_less @ A @ B2 @ A2 )
=> ( ( ord_less_eq @ A @ C2 @ B2 )
=> ( ord_less @ A @ C2 @ A2 ) ) ) ) ).
% dual_order.strict_trans2
thf(fact_3753_dual__order_Ostrict__trans1,axiom,
! [A: $tType] :
( ( preorder @ A )
=> ! [B2: A,A2: A,C2: A] :
( ( ord_less_eq @ A @ B2 @ A2 )
=> ( ( ord_less @ A @ C2 @ B2 )
=> ( ord_less @ A @ C2 @ A2 ) ) ) ) ).
% dual_order.strict_trans1
thf(fact_3754_dual__order_Ostrict__iff__order,axiom,
! [A: $tType] :
( ( order @ A )
=> ( ( ord_less @ A )
= ( ^ [B5: A,A6: A] :
( ( ord_less_eq @ A @ B5 @ A6 )
& ( A6 != B5 ) ) ) ) ) ).
% dual_order.strict_iff_order
thf(fact_3755_dual__order_Oorder__iff__strict,axiom,
! [A: $tType] :
( ( order @ A )
=> ( ( ord_less_eq @ A )
= ( ^ [B5: A,A6: A] :
( ( ord_less @ A @ B5 @ A6 )
| ( A6 = B5 ) ) ) ) ) ).
% dual_order.order_iff_strict
thf(fact_3756_dense__le__bounded,axiom,
! [A: $tType] :
( ( dense_linorder @ A )
=> ! [X: A,Y: A,Z: A] :
( ( ord_less @ A @ X @ Y )
=> ( ! [W3: A] :
( ( ord_less @ A @ X @ W3 )
=> ( ( ord_less @ A @ W3 @ Y )
=> ( ord_less_eq @ A @ W3 @ Z ) ) )
=> ( ord_less_eq @ A @ Y @ Z ) ) ) ) ).
% dense_le_bounded
thf(fact_3757_dense__ge__bounded,axiom,
! [A: $tType] :
( ( dense_linorder @ A )
=> ! [Z: A,X: A,Y: A] :
( ( ord_less @ A @ Z @ X )
=> ( ! [W3: A] :
( ( ord_less @ A @ Z @ W3 )
=> ( ( ord_less @ A @ W3 @ X )
=> ( ord_less_eq @ A @ Y @ W3 ) ) )
=> ( ord_less_eq @ A @ Y @ Z ) ) ) ) ).
% dense_ge_bounded
thf(fact_3758_order_Ostrict__iff__not,axiom,
! [A: $tType] :
( ( preorder @ A )
=> ( ( ord_less @ A )
= ( ^ [A6: A,B5: A] :
( ( ord_less_eq @ A @ A6 @ B5 )
& ~ ( ord_less_eq @ A @ B5 @ A6 ) ) ) ) ) ).
% order.strict_iff_not
thf(fact_3759_order_Ostrict__trans2,axiom,
! [A: $tType] :
( ( preorder @ A )
=> ! [A2: A,B2: A,C2: A] :
( ( ord_less @ A @ A2 @ B2 )
=> ( ( ord_less_eq @ A @ B2 @ C2 )
=> ( ord_less @ A @ A2 @ C2 ) ) ) ) ).
% order.strict_trans2
thf(fact_3760_order_Ostrict__trans1,axiom,
! [A: $tType] :
( ( preorder @ A )
=> ! [A2: A,B2: A,C2: A] :
( ( ord_less_eq @ A @ A2 @ B2 )
=> ( ( ord_less @ A @ B2 @ C2 )
=> ( ord_less @ A @ A2 @ C2 ) ) ) ) ).
% order.strict_trans1
thf(fact_3761_order_Ostrict__iff__order,axiom,
! [A: $tType] :
( ( order @ A )
=> ( ( ord_less @ A )
= ( ^ [A6: A,B5: A] :
( ( ord_less_eq @ A @ A6 @ B5 )
& ( A6 != B5 ) ) ) ) ) ).
% order.strict_iff_order
thf(fact_3762_order_Oorder__iff__strict,axiom,
! [A: $tType] :
( ( order @ A )
=> ( ( ord_less_eq @ A )
= ( ^ [A6: A,B5: A] :
( ( ord_less @ A @ A6 @ B5 )
| ( A6 = B5 ) ) ) ) ) ).
% order.order_iff_strict
thf(fact_3763_not__le__imp__less,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [Y: A,X: A] :
( ~ ( ord_less_eq @ A @ Y @ X )
=> ( ord_less @ A @ X @ Y ) ) ) ).
% not_le_imp_less
thf(fact_3764_less__le__not__le,axiom,
! [A: $tType] :
( ( preorder @ A )
=> ( ( ord_less @ A )
= ( ^ [X2: A,Y2: A] :
( ( ord_less_eq @ A @ X2 @ Y2 )
& ~ ( ord_less_eq @ A @ Y2 @ X2 ) ) ) ) ) ).
% less_le_not_le
thf(fact_3765_dense__le,axiom,
! [A: $tType] :
( ( dense_linorder @ A )
=> ! [Y: A,Z: A] :
( ! [X3: A] :
( ( ord_less @ A @ X3 @ Y )
=> ( ord_less_eq @ A @ X3 @ Z ) )
=> ( ord_less_eq @ A @ Y @ Z ) ) ) ).
% dense_le
thf(fact_3766_dense__ge,axiom,
! [A: $tType] :
( ( dense_linorder @ A )
=> ! [Z: A,Y: A] :
( ! [X3: A] :
( ( ord_less @ A @ Z @ X3 )
=> ( ord_less_eq @ A @ Y @ X3 ) )
=> ( ord_less_eq @ A @ Y @ Z ) ) ) ).
% dense_ge
thf(fact_3767_antisym__conv2,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [X: A,Y: A] :
( ( ord_less_eq @ A @ X @ Y )
=> ( ( ~ ( ord_less @ A @ X @ Y ) )
= ( X = Y ) ) ) ) ).
% antisym_conv2
thf(fact_3768_antisym__conv1,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [X: A,Y: A] :
( ~ ( ord_less @ A @ X @ Y )
=> ( ( ord_less_eq @ A @ X @ Y )
= ( X = Y ) ) ) ) ).
% antisym_conv1
thf(fact_3769_nless__le,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [A2: A,B2: A] :
( ( ~ ( ord_less @ A @ A2 @ B2 ) )
= ( ~ ( ord_less_eq @ A @ A2 @ B2 )
| ( A2 = B2 ) ) ) ) ).
% nless_le
thf(fact_3770_leI,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [X: A,Y: A] :
( ~ ( ord_less @ A @ X @ Y )
=> ( ord_less_eq @ A @ Y @ X ) ) ) ).
% leI
thf(fact_3771_leD,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [Y: A,X: A] :
( ( ord_less_eq @ A @ Y @ X )
=> ~ ( ord_less @ A @ X @ Y ) ) ) ).
% leD
thf(fact_3772_bot_Oextremum,axiom,
! [A: $tType] :
( ( order_bot @ A )
=> ! [A2: A] : ( ord_less_eq @ A @ ( bot_bot @ A ) @ A2 ) ) ).
% bot.extremum
thf(fact_3773_bot_Oextremum__unique,axiom,
! [A: $tType] :
( ( order_bot @ A )
=> ! [A2: A] :
( ( ord_less_eq @ A @ A2 @ ( bot_bot @ A ) )
= ( A2
= ( bot_bot @ A ) ) ) ) ).
% bot.extremum_unique
thf(fact_3774_bot_Oextremum__uniqueI,axiom,
! [A: $tType] :
( ( order_bot @ A )
=> ! [A2: A] :
( ( ord_less_eq @ A @ A2 @ ( bot_bot @ A ) )
=> ( A2
= ( bot_bot @ A ) ) ) ) ).
% bot.extremum_uniqueI
thf(fact_3775_bot_Oextremum__strict,axiom,
! [A: $tType] :
( ( order_bot @ A )
=> ! [A2: A] :
~ ( ord_less @ A @ A2 @ ( bot_bot @ A ) ) ) ).
% bot.extremum_strict
thf(fact_3776_bot_Onot__eq__extremum,axiom,
! [A: $tType] :
( ( order_bot @ A )
=> ! [A2: A] :
( ( A2
!= ( bot_bot @ A ) )
= ( ord_less @ A @ ( bot_bot @ A ) @ A2 ) ) ) ).
% bot.not_eq_extremum
thf(fact_3777_power__minus1__odd,axiom,
! [A: $tType] :
( ( ring_1 @ A )
=> ! [N3: nat] :
( ( power_power @ A @ ( uminus_uminus @ A @ ( one_one @ A ) ) @ ( suc @ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N3 ) ) )
= ( uminus_uminus @ A @ ( one_one @ A ) ) ) ) ).
% power_minus1_odd
thf(fact_3778_max__absorb2,axiom,
! [A: $tType] :
( ( ord @ A )
=> ! [X: A,Y: A] :
( ( ord_less_eq @ A @ X @ Y )
=> ( ( ord_max @ A @ X @ Y )
= Y ) ) ) ).
% max_absorb2
thf(fact_3779_max__absorb1,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [Y: A,X: A] :
( ( ord_less_eq @ A @ Y @ X )
=> ( ( ord_max @ A @ X @ Y )
= X ) ) ) ).
% max_absorb1
thf(fact_3780_max__def,axiom,
! [A: $tType] :
( ( ord @ A )
=> ( ( ord_max @ A )
= ( ^ [A6: A,B5: A] : ( if @ A @ ( ord_less_eq @ A @ A6 @ B5 ) @ B5 @ A6 ) ) ) ) ).
% max_def
thf(fact_3781_gbinomial__minus,axiom,
! [A: $tType] :
( ( field_char_0 @ A )
=> ! [A2: A,K: nat] :
( ( gbinomial @ A @ ( uminus_uminus @ A @ A2 ) @ K )
= ( times_times @ A @ ( power_power @ A @ ( uminus_uminus @ A @ ( one_one @ A ) ) @ K ) @ ( gbinomial @ A @ ( minus_minus @ A @ ( plus_plus @ A @ A2 @ ( semiring_1_of_nat @ A @ K ) ) @ ( one_one @ A ) ) @ K ) ) ) ) ).
% gbinomial_minus
thf(fact_3782_le__funD,axiom,
! [B: $tType,A: $tType] :
( ( ord @ B )
=> ! [F2: A > B,G: A > B,X: A] :
( ( ord_less_eq @ ( A > B ) @ F2 @ G )
=> ( ord_less_eq @ B @ ( F2 @ X ) @ ( G @ X ) ) ) ) ).
% le_funD
thf(fact_3783_le__funE,axiom,
! [B: $tType,A: $tType] :
( ( ord @ B )
=> ! [F2: A > B,G: A > B,X: A] :
( ( ord_less_eq @ ( A > B ) @ F2 @ G )
=> ( ord_less_eq @ B @ ( F2 @ X ) @ ( G @ X ) ) ) ) ).
% le_funE
thf(fact_3784_le__funI,axiom,
! [B: $tType,A: $tType] :
( ( ord @ B )
=> ! [F2: A > B,G: A > B] :
( ! [X3: A] : ( ord_less_eq @ B @ ( F2 @ X3 ) @ ( G @ X3 ) )
=> ( ord_less_eq @ ( A > B ) @ F2 @ G ) ) ) ).
% le_funI
thf(fact_3785_le__fun__def,axiom,
! [B: $tType,A: $tType] :
( ( ord @ B )
=> ( ( ord_less_eq @ ( A > B ) )
= ( ^ [F7: A > B,G4: A > B] :
! [X2: A] : ( ord_less_eq @ B @ ( F7 @ X2 ) @ ( G4 @ X2 ) ) ) ) ) ).
% le_fun_def
thf(fact_3786_pochhammer__minus,axiom,
! [A: $tType] :
( ( comm_ring_1 @ A )
=> ! [B2: A,K: nat] :
( ( comm_s3205402744901411588hammer @ A @ ( uminus_uminus @ A @ B2 ) @ K )
= ( times_times @ A @ ( power_power @ A @ ( uminus_uminus @ A @ ( one_one @ A ) ) @ K ) @ ( comm_s3205402744901411588hammer @ A @ ( plus_plus @ A @ ( minus_minus @ A @ B2 @ ( semiring_1_of_nat @ A @ K ) ) @ ( one_one @ A ) ) @ K ) ) ) ) ).
% pochhammer_minus
thf(fact_3787_pochhammer__minus_H,axiom,
! [A: $tType] :
( ( comm_ring_1 @ A )
=> ! [B2: A,K: nat] :
( ( comm_s3205402744901411588hammer @ A @ ( plus_plus @ A @ ( minus_minus @ A @ B2 @ ( semiring_1_of_nat @ A @ K ) ) @ ( one_one @ A ) ) @ K )
= ( times_times @ A @ ( power_power @ A @ ( uminus_uminus @ A @ ( one_one @ A ) ) @ K ) @ ( comm_s3205402744901411588hammer @ A @ ( uminus_uminus @ A @ B2 ) @ K ) ) ) ) ).
% pochhammer_minus'
thf(fact_3788_gbinomial__pochhammer_H,axiom,
! [A: $tType] :
( ( field_char_0 @ A )
=> ( ( gbinomial @ A )
= ( ^ [A6: A,K2: nat] : ( divide_divide @ A @ ( comm_s3205402744901411588hammer @ A @ ( plus_plus @ A @ ( minus_minus @ A @ A6 @ ( semiring_1_of_nat @ A @ K2 ) ) @ ( one_one @ A ) ) @ K2 ) @ ( semiring_char_0_fact @ A @ K2 ) ) ) ) ) ).
% gbinomial_pochhammer'
thf(fact_3789_int__bit__induct,axiom,
! [P: int > $o,K: int] :
( ( P @ ( zero_zero @ int ) )
=> ( ( P @ ( uminus_uminus @ int @ ( one_one @ int ) ) )
=> ( ! [K3: int] :
( ( P @ K3 )
=> ( ( K3
!= ( zero_zero @ int ) )
=> ( P @ ( times_times @ int @ K3 @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) ) ) ) )
=> ( ! [K3: int] :
( ( P @ K3 )
=> ( ( K3
!= ( uminus_uminus @ int @ ( one_one @ int ) ) )
=> ( P @ ( plus_plus @ int @ ( one_one @ int ) @ ( times_times @ int @ K3 @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) ) ) ) ) )
=> ( P @ K ) ) ) ) ) ).
% int_bit_induct
thf(fact_3790_m1mod2k,axiom,
! [N3: nat] :
( ( modulo_modulo @ int @ ( uminus_uminus @ int @ ( one_one @ int ) ) @ ( power_power @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ N3 ) )
= ( minus_minus @ int @ ( power_power @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ N3 ) @ ( one_one @ int ) ) ) ).
% m1mod2k
thf(fact_3791_log__minus__eq__powr,axiom,
! [B2: real,X: real,Y: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ B2 )
=> ( ( B2
!= ( one_one @ real ) )
=> ( ( ord_less @ real @ ( zero_zero @ real ) @ X )
=> ( ( minus_minus @ real @ ( log @ B2 @ X ) @ Y )
= ( log @ B2 @ ( times_times @ real @ X @ ( powr @ real @ B2 @ ( uminus_uminus @ real @ Y ) ) ) ) ) ) ) ) ).
% log_minus_eq_powr
thf(fact_3792_gbinomial__sum__lower__neg,axiom,
! [A: $tType] :
( ( field_char_0 @ A )
=> ! [A2: A,M: nat] :
( ( groups7311177749621191930dd_sum @ nat @ A
@ ^ [K2: nat] : ( times_times @ A @ ( gbinomial @ A @ A2 @ K2 ) @ ( power_power @ A @ ( uminus_uminus @ A @ ( one_one @ A ) ) @ K2 ) )
@ ( set_ord_atMost @ nat @ M ) )
= ( times_times @ A @ ( power_power @ A @ ( uminus_uminus @ A @ ( one_one @ A ) ) @ M ) @ ( gbinomial @ A @ ( minus_minus @ A @ A2 @ ( one_one @ A ) ) @ M ) ) ) ) ).
% gbinomial_sum_lower_neg
thf(fact_3793_Maclaurin__zero,axiom,
! [A: $tType] :
( ( zero @ A )
=> ! [X: real,N3: nat,Diff: nat > A > real] :
( ( X
= ( zero_zero @ real ) )
=> ( ( N3
!= ( zero_zero @ nat ) )
=> ( ( groups7311177749621191930dd_sum @ nat @ real
@ ^ [M5: nat] : ( times_times @ real @ ( divide_divide @ real @ ( Diff @ M5 @ ( zero_zero @ A ) ) @ ( semiring_char_0_fact @ real @ M5 ) ) @ ( power_power @ real @ X @ M5 ) )
@ ( set_ord_lessThan @ nat @ N3 ) )
= ( Diff @ ( zero_zero @ nat ) @ ( zero_zero @ A ) ) ) ) ) ) ).
% Maclaurin_zero
thf(fact_3794_Maclaurin__lemma,axiom,
! [H2: real,F2: real > real,J: nat > real,N3: nat] :
( ( ord_less @ real @ ( zero_zero @ real ) @ H2 )
=> ? [B9: real] :
( ( F2 @ H2 )
= ( plus_plus @ real
@ ( groups7311177749621191930dd_sum @ nat @ real
@ ^ [M5: nat] : ( times_times @ real @ ( divide_divide @ real @ ( J @ M5 ) @ ( semiring_char_0_fact @ real @ M5 ) ) @ ( power_power @ real @ H2 @ M5 ) )
@ ( set_ord_lessThan @ nat @ N3 ) )
@ ( times_times @ real @ B9 @ ( divide_divide @ real @ ( power_power @ real @ H2 @ N3 ) @ ( semiring_char_0_fact @ real @ N3 ) ) ) ) ) ) ).
% Maclaurin_lemma
thf(fact_3795_sb__dec__lem_H,axiom,
! [K: nat,A2: int] :
( ( ord_less_eq @ int @ ( power_power @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ K ) @ A2 )
=> ( ord_less_eq @ int @ ( modulo_modulo @ int @ ( plus_plus @ int @ A2 @ ( power_power @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ K ) ) @ ( times_times @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ ( power_power @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ K ) ) ) @ ( plus_plus @ int @ ( uminus_uminus @ int @ ( power_power @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ K ) ) @ A2 ) ) ) ).
% sb_dec_lem'
thf(fact_3796_m1mod22k,axiom,
! [N3: 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 ) ) @ N3 ) ) )
= ( minus_minus @ int @ ( times_times @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ ( power_power @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ N3 ) ) @ ( one_one @ int ) ) ) ).
% m1mod22k
thf(fact_3797_powr__neg__numeral,axiom,
! [X: real,N3: num] :
( ( ord_less @ real @ ( zero_zero @ real ) @ X )
=> ( ( powr @ real @ X @ ( uminus_uminus @ real @ ( numeral_numeral @ real @ N3 ) ) )
= ( divide_divide @ real @ ( one_one @ real ) @ ( power_power @ real @ X @ ( numeral_numeral @ nat @ N3 ) ) ) ) ) ).
% powr_neg_numeral
thf(fact_3798_gbinomial__Suc,axiom,
! [A: $tType] :
( ( ( semiring_char_0 @ A )
& ( semidom_divide @ A ) )
=> ! [A2: A,K: nat] :
( ( gbinomial @ A @ A2 @ ( suc @ K ) )
= ( divide_divide @ A
@ ( groups7121269368397514597t_prod @ nat @ A
@ ^ [I3: nat] : ( minus_minus @ A @ A2 @ ( semiring_1_of_nat @ A @ I3 ) )
@ ( set_or1337092689740270186AtMost @ nat @ ( zero_zero @ nat ) @ K ) )
@ ( semiring_char_0_fact @ A @ ( suc @ K ) ) ) ) ) ).
% gbinomial_Suc
thf(fact_3799_gbinomial__partial__sum__poly,axiom,
! [A: $tType] :
( ( field_char_0 @ A )
=> ! [M: nat,A2: A,X: A,Y: A] :
( ( groups7311177749621191930dd_sum @ nat @ A
@ ^ [K2: nat] : ( times_times @ A @ ( times_times @ A @ ( gbinomial @ A @ ( plus_plus @ A @ ( semiring_1_of_nat @ A @ M ) @ A2 ) @ K2 ) @ ( power_power @ A @ X @ K2 ) ) @ ( power_power @ A @ Y @ ( minus_minus @ nat @ M @ K2 ) ) )
@ ( set_ord_atMost @ nat @ M ) )
= ( groups7311177749621191930dd_sum @ nat @ A
@ ^ [K2: nat] : ( times_times @ A @ ( times_times @ A @ ( gbinomial @ A @ ( uminus_uminus @ A @ A2 ) @ K2 ) @ ( power_power @ A @ ( uminus_uminus @ A @ X ) @ K2 ) ) @ ( power_power @ A @ ( plus_plus @ A @ X @ Y ) @ ( minus_minus @ nat @ M @ K2 ) ) )
@ ( set_ord_atMost @ nat @ M ) ) ) ) ).
% gbinomial_partial_sum_poly
thf(fact_3800_sb__inc__lem_H,axiom,
! [A2: int,K: nat] :
( ( ord_less @ int @ A2 @ ( uminus_uminus @ int @ ( power_power @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ K ) ) )
=> ( ord_less_eq @ int @ ( plus_plus @ int @ ( plus_plus @ int @ A2 @ ( power_power @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ K ) ) @ ( power_power @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ ( suc @ K ) ) ) @ ( modulo_modulo @ int @ ( plus_plus @ int @ A2 @ ( power_power @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ K ) ) @ ( power_power @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ ( suc @ K ) ) ) ) ) ).
% sb_inc_lem'
thf(fact_3801_sb__dec__lem,axiom,
! [K: nat,A2: int] :
( ( ord_less_eq @ int @ ( zero_zero @ int ) @ ( plus_plus @ int @ ( uminus_uminus @ int @ ( power_power @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ K ) ) @ A2 ) )
=> ( ord_less_eq @ int @ ( modulo_modulo @ int @ ( plus_plus @ int @ A2 @ ( power_power @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ K ) ) @ ( times_times @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ ( power_power @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ K ) ) ) @ ( plus_plus @ int @ ( uminus_uminus @ int @ ( power_power @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ K ) ) @ A2 ) ) ) ).
% sb_dec_lem
thf(fact_3802_root__polyfun,axiom,
! [A: $tType] :
( ( idom @ A )
=> ! [N3: nat,Z: A,A2: A] :
( ( ord_less_eq @ nat @ ( one_one @ nat ) @ N3 )
=> ( ( ( power_power @ A @ Z @ N3 )
= A2 )
= ( ( groups7311177749621191930dd_sum @ nat @ A
@ ^ [I3: nat] :
( times_times @ A
@ ( if @ A
@ ( I3
= ( zero_zero @ nat ) )
@ ( uminus_uminus @ A @ A2 )
@ ( if @ A @ ( I3 = N3 ) @ ( one_one @ A ) @ ( zero_zero @ A ) ) )
@ ( power_power @ A @ Z @ I3 ) )
@ ( set_ord_atMost @ nat @ N3 ) )
= ( zero_zero @ A ) ) ) ) ) ).
% root_polyfun
thf(fact_3803_choose__alternating__linear__sum,axiom,
! [A: $tType] :
( ( comm_ring_1 @ A )
=> ! [N3: nat] :
( ( N3
!= ( one_one @ nat ) )
=> ( ( groups7311177749621191930dd_sum @ nat @ A
@ ^ [I3: nat] : ( times_times @ A @ ( times_times @ A @ ( power_power @ A @ ( uminus_uminus @ A @ ( one_one @ A ) ) @ I3 ) @ ( semiring_1_of_nat @ A @ I3 ) ) @ ( semiring_1_of_nat @ A @ ( binomial @ N3 @ I3 ) ) )
@ ( set_ord_atMost @ nat @ N3 ) )
= ( zero_zero @ A ) ) ) ) ).
% choose_alternating_linear_sum
thf(fact_3804_fact__double,axiom,
! [A: $tType] :
( ( field_char_0 @ A )
=> ! [N3: nat] :
( ( semiring_char_0_fact @ A @ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N3 ) )
= ( times_times @ A @ ( times_times @ A @ ( power_power @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N3 ) ) @ ( comm_s3205402744901411588hammer @ A @ ( divide_divide @ A @ ( one_one @ A ) @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) @ N3 ) ) @ ( semiring_char_0_fact @ A @ N3 ) ) ) ) ).
% fact_double
thf(fact_3805_binomial__code,axiom,
( binomial
= ( ^ [N2: nat,K2: nat] : ( if @ nat @ ( ord_less @ nat @ N2 @ K2 ) @ ( zero_zero @ nat ) @ ( if @ nat @ ( ord_less @ nat @ N2 @ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ K2 ) ) @ ( binomial @ N2 @ ( minus_minus @ nat @ N2 @ K2 ) ) @ ( divide_divide @ nat @ ( set_fo6178422350223883121st_nat @ nat @ ( times_times @ nat ) @ ( plus_plus @ nat @ ( minus_minus @ nat @ N2 @ K2 ) @ ( one_one @ nat ) ) @ N2 @ ( one_one @ nat ) ) @ ( semiring_char_0_fact @ nat @ K2 ) ) ) ) ) ) ).
% binomial_code
thf(fact_3806_of__int__code__if,axiom,
! [A: $tType] :
( ( ring_1 @ A )
=> ( ( ring_1_of_int @ A )
= ( ^ [K2: int] :
( if @ A
@ ( K2
= ( zero_zero @ int ) )
@ ( zero_zero @ A )
@ ( if @ A @ ( ord_less @ int @ K2 @ ( zero_zero @ int ) ) @ ( uminus_uminus @ A @ ( ring_1_of_int @ A @ ( uminus_uminus @ int @ K2 ) ) )
@ ( if @ A
@ ( ( modulo_modulo @ int @ K2 @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) )
= ( zero_zero @ int ) )
@ ( times_times @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ ( ring_1_of_int @ A @ ( divide_divide @ int @ K2 @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) ) ) )
@ ( plus_plus @ A @ ( times_times @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ ( ring_1_of_int @ A @ ( divide_divide @ int @ K2 @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) ) ) ) @ ( one_one @ A ) ) ) ) ) ) ) ) ).
% of_int_code_if
thf(fact_3807_one__div__minus__numeral,axiom,
! [N3: num] :
( ( divide_divide @ int @ ( one_one @ int ) @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ N3 ) ) )
= ( uminus_uminus @ int @ ( adjust_div @ ( unique8689654367752047608divmod @ int @ one2 @ N3 ) ) ) ) ).
% one_div_minus_numeral
thf(fact_3808_minus__one__div__numeral,axiom,
! [N3: num] :
( ( divide_divide @ int @ ( uminus_uminus @ int @ ( one_one @ int ) ) @ ( numeral_numeral @ int @ N3 ) )
= ( uminus_uminus @ int @ ( adjust_div @ ( unique8689654367752047608divmod @ int @ one2 @ N3 ) ) ) ) ).
% minus_one_div_numeral
thf(fact_3809_sin__coeff__def,axiom,
( sin_coeff
= ( ^ [N2: nat] : ( if @ real @ ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N2 ) @ ( zero_zero @ real ) @ ( divide_divide @ real @ ( power_power @ real @ ( uminus_uminus @ real @ ( one_one @ real ) ) @ ( divide_divide @ nat @ ( minus_minus @ nat @ N2 @ ( suc @ ( zero_zero @ nat ) ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( semiring_char_0_fact @ real @ N2 ) ) ) ) ) ).
% sin_coeff_def
thf(fact_3810_compl__less__compl__iff,axiom,
! [A: $tType] :
( ( boolea8198339166811842893lgebra @ A )
=> ! [X: A,Y: A] :
( ( ord_less @ A @ ( uminus_uminus @ A @ X ) @ ( uminus_uminus @ A @ Y ) )
= ( ord_less @ A @ Y @ X ) ) ) ).
% compl_less_compl_iff
thf(fact_3811_assn__basic__inequalities_I3_J,axiom,
( ( bot_bot @ assn )
!= ( one_one @ assn ) ) ).
% assn_basic_inequalities(3)
thf(fact_3812_compl__le__compl__iff,axiom,
! [A: $tType] :
( ( boolea8198339166811842893lgebra @ A )
=> ! [X: A,Y: A] :
( ( ord_less_eq @ A @ ( uminus_uminus @ A @ X ) @ ( uminus_uminus @ A @ Y ) )
= ( ord_less_eq @ A @ Y @ X ) ) ) ).
% compl_le_compl_iff
thf(fact_3813_bot__nat__def,axiom,
( ( bot_bot @ nat )
= ( zero_zero @ nat ) ) ).
% bot_nat_def
thf(fact_3814_bot__option__def,axiom,
! [A: $tType] :
( ( order @ A )
=> ( ( bot_bot @ ( option @ A ) )
= ( none @ A ) ) ) ).
% bot_option_def
thf(fact_3815_vebt__assn__raw_Osimps_I4_J,axiom,
! [Vd: $o,Ve: $o,V: option @ ( product_prod @ nat @ nat ),Va: nat,Vb: array @ vEBT_VEBTi,Vc: vEBT_VEBTi] :
( ( vEBT_vebt_assn_raw @ ( vEBT_Leaf @ Vd @ Ve ) @ ( vEBT_Nodei @ V @ Va @ Vb @ Vc ) )
= ( bot_bot @ assn ) ) ).
% vebt_assn_raw.simps(4)
thf(fact_3816_vebt__assn__raw_Osimps_I3_J,axiom,
! [V: option @ ( product_prod @ nat @ nat ),Va: nat,Vb: list @ vEBT_VEBT,Vc: vEBT_VEBT,Vd: $o,Ve: $o] :
( ( vEBT_vebt_assn_raw @ ( vEBT_Node @ V @ Va @ Vb @ Vc ) @ ( vEBT_Leafi @ Vd @ Ve ) )
= ( bot_bot @ assn ) ) ).
% vebt_assn_raw.simps(3)
thf(fact_3817_compl__mono,axiom,
! [A: $tType] :
( ( boolea8198339166811842893lgebra @ A )
=> ! [X: A,Y: A] :
( ( ord_less_eq @ A @ X @ Y )
=> ( ord_less_eq @ A @ ( uminus_uminus @ A @ Y ) @ ( uminus_uminus @ A @ X ) ) ) ) ).
% compl_mono
thf(fact_3818_compl__le__swap1,axiom,
! [A: $tType] :
( ( boolea8198339166811842893lgebra @ A )
=> ! [Y: A,X: A] :
( ( ord_less_eq @ A @ Y @ ( uminus_uminus @ A @ X ) )
=> ( ord_less_eq @ A @ X @ ( uminus_uminus @ A @ Y ) ) ) ) ).
% compl_le_swap1
thf(fact_3819_compl__le__swap2,axiom,
! [A: $tType] :
( ( boolea8198339166811842893lgebra @ A )
=> ! [Y: A,X: A] :
( ( ord_less_eq @ A @ ( uminus_uminus @ A @ Y ) @ X )
=> ( ord_less_eq @ A @ ( uminus_uminus @ A @ X ) @ Y ) ) ) ).
% compl_le_swap2
thf(fact_3820_compl__less__swap1,axiom,
! [A: $tType] :
( ( boolea8198339166811842893lgebra @ A )
=> ! [Y: A,X: A] :
( ( ord_less @ A @ Y @ ( uminus_uminus @ A @ X ) )
=> ( ord_less @ A @ X @ ( uminus_uminus @ A @ Y ) ) ) ) ).
% compl_less_swap1
thf(fact_3821_compl__less__swap2,axiom,
! [A: $tType] :
( ( boolea8198339166811842893lgebra @ A )
=> ! [Y: A,X: A] :
( ( ord_less @ A @ ( uminus_uminus @ A @ Y ) @ X )
=> ( ord_less @ A @ ( uminus_uminus @ A @ X ) @ Y ) ) ) ).
% compl_less_swap2
thf(fact_3822_sin__coeff__Suc,axiom,
! [N3: nat] :
( ( sin_coeff @ ( suc @ N3 ) )
= ( divide_divide @ real @ ( cos_coeff @ N3 ) @ ( semiring_1_of_nat @ real @ ( suc @ N3 ) ) ) ) ).
% sin_coeff_Suc
thf(fact_3823_cos__coeff__Suc,axiom,
! [N3: nat] :
( ( cos_coeff @ ( suc @ N3 ) )
= ( divide_divide @ real @ ( uminus_uminus @ real @ ( sin_coeff @ N3 ) ) @ ( semiring_1_of_nat @ real @ ( suc @ N3 ) ) ) ) ).
% cos_coeff_Suc
thf(fact_3824_diff__shunt__var,axiom,
! [A: $tType] :
( ( boolea8198339166811842893lgebra @ A )
=> ! [X: A,Y: A] :
( ( ( minus_minus @ A @ X @ Y )
= ( bot_bot @ A ) )
= ( ord_less_eq @ A @ X @ Y ) ) ) ).
% diff_shunt_var
thf(fact_3825_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_3826_signed__take__bit__rec,axiom,
! [A: $tType] :
( ( bit_ri3973907225187159222ations @ A )
=> ( ( bit_ri4674362597316999326ke_bit @ A )
= ( ^ [N2: nat,A6: A] :
( if @ A
@ ( N2
= ( 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 @ N2 @ ( one_one @ nat ) ) @ ( divide_divide @ A @ A6 @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) ) ) ) ) ) ) ) ).
% signed_take_bit_rec
thf(fact_3827_sin__paired,axiom,
! [X: real] :
( sums @ real
@ ^ [N2: nat] : ( times_times @ real @ ( divide_divide @ real @ ( power_power @ real @ ( uminus_uminus @ real @ ( one_one @ real ) ) @ N2 ) @ ( semiring_char_0_fact @ real @ ( plus_plus @ nat @ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N2 ) @ ( one_one @ nat ) ) ) ) @ ( power_power @ real @ X @ ( plus_plus @ nat @ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N2 ) @ ( one_one @ nat ) ) ) )
@ ( sin @ real @ X ) ) ).
% sin_paired
thf(fact_3828_signed__take__bit__Suc__minus__bit1,axiom,
! [N3: nat,K: num] :
( ( bit_ri4674362597316999326ke_bit @ int @ ( suc @ N3 ) @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ ( bit1 @ K ) ) ) )
= ( plus_plus @ int @ ( times_times @ int @ ( bit_ri4674362597316999326ke_bit @ int @ N3 @ ( 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_3829_signed__take__bit__of__0,axiom,
! [A: $tType] :
( ( bit_ri3973907225187159222ations @ A )
=> ! [N3: nat] :
( ( bit_ri4674362597316999326ke_bit @ A @ N3 @ ( zero_zero @ A ) )
= ( zero_zero @ A ) ) ) ).
% signed_take_bit_of_0
thf(fact_3830_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_3831_signed__take__bit__Suc__1,axiom,
! [A: $tType] :
( ( bit_ri3973907225187159222ations @ A )
=> ! [N3: nat] :
( ( bit_ri4674362597316999326ke_bit @ A @ ( suc @ N3 ) @ ( one_one @ A ) )
= ( one_one @ A ) ) ) ).
% signed_take_bit_Suc_1
thf(fact_3832_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_3833_signed__take__bit__of__minus__1,axiom,
! [A: $tType] :
( ( bit_ri3973907225187159222ations @ A )
=> ! [N3: nat] :
( ( bit_ri4674362597316999326ke_bit @ A @ N3 @ ( uminus_uminus @ A @ ( one_one @ A ) ) )
= ( uminus_uminus @ A @ ( one_one @ A ) ) ) ) ).
% signed_take_bit_of_minus_1
thf(fact_3834_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_3835_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_3836_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_3837_signed__take__bit__Suc__bit0,axiom,
! [N3: nat,K: num] :
( ( bit_ri4674362597316999326ke_bit @ int @ ( suc @ N3 ) @ ( numeral_numeral @ int @ ( bit0 @ K ) ) )
= ( times_times @ int @ ( bit_ri4674362597316999326ke_bit @ int @ N3 @ ( numeral_numeral @ int @ K ) ) @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) ) ) ).
% signed_take_bit_Suc_bit0
thf(fact_3838_signed__take__bit__0,axiom,
! [A: $tType] :
( ( bit_ri3973907225187159222ations @ A )
=> ! [A2: A] :
( ( bit_ri4674362597316999326ke_bit @ A @ ( zero_zero @ nat ) @ A2 )
= ( uminus_uminus @ A @ ( modulo_modulo @ A @ A2 @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) ) ) ) ).
% signed_take_bit_0
thf(fact_3839_signed__take__bit__Suc__minus__bit0,axiom,
! [N3: nat,K: num] :
( ( bit_ri4674362597316999326ke_bit @ int @ ( suc @ N3 ) @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ ( bit0 @ K ) ) ) )
= ( times_times @ int @ ( bit_ri4674362597316999326ke_bit @ int @ N3 @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ K ) ) ) @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) ) ) ).
% signed_take_bit_Suc_minus_bit0
thf(fact_3840_signed__take__bit__Suc__bit1,axiom,
! [N3: nat,K: num] :
( ( bit_ri4674362597316999326ke_bit @ int @ ( suc @ N3 ) @ ( numeral_numeral @ int @ ( bit1 @ K ) ) )
= ( plus_plus @ int @ ( times_times @ int @ ( bit_ri4674362597316999326ke_bit @ int @ N3 @ ( numeral_numeral @ int @ K ) ) @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) ) @ ( one_one @ int ) ) ) ).
% signed_take_bit_Suc_bit1
thf(fact_3841_signed__take__bit__minus,axiom,
! [N3: nat,K: int] :
( ( bit_ri4674362597316999326ke_bit @ int @ N3 @ ( uminus_uminus @ int @ ( bit_ri4674362597316999326ke_bit @ int @ N3 @ K ) ) )
= ( bit_ri4674362597316999326ke_bit @ int @ N3 @ ( uminus_uminus @ int @ K ) ) ) ).
% signed_take_bit_minus
thf(fact_3842_signed__take__bit__diff,axiom,
! [N3: nat,K: int,L2: int] :
( ( bit_ri4674362597316999326ke_bit @ int @ N3 @ ( minus_minus @ int @ ( bit_ri4674362597316999326ke_bit @ int @ N3 @ K ) @ ( bit_ri4674362597316999326ke_bit @ int @ N3 @ L2 ) ) )
= ( bit_ri4674362597316999326ke_bit @ int @ N3 @ ( minus_minus @ int @ K @ L2 ) ) ) ).
% signed_take_bit_diff
thf(fact_3843_signed__take__bit__mult,axiom,
! [N3: nat,K: int,L2: int] :
( ( bit_ri4674362597316999326ke_bit @ int @ N3 @ ( times_times @ int @ ( bit_ri4674362597316999326ke_bit @ int @ N3 @ K ) @ ( bit_ri4674362597316999326ke_bit @ int @ N3 @ L2 ) ) )
= ( bit_ri4674362597316999326ke_bit @ int @ N3 @ ( times_times @ int @ K @ L2 ) ) ) ).
% signed_take_bit_mult
thf(fact_3844_signed__take__bit__add,axiom,
! [N3: nat,K: int,L2: int] :
( ( bit_ri4674362597316999326ke_bit @ int @ N3 @ ( plus_plus @ int @ ( bit_ri4674362597316999326ke_bit @ int @ N3 @ K ) @ ( bit_ri4674362597316999326ke_bit @ int @ N3 @ L2 ) ) )
= ( bit_ri4674362597316999326ke_bit @ int @ N3 @ ( plus_plus @ int @ K @ L2 ) ) ) ).
% signed_take_bit_add
thf(fact_3845_sin__x__le__x,axiom,
! [X: real] :
( ( ord_less_eq @ real @ ( zero_zero @ real ) @ X )
=> ( ord_less_eq @ real @ ( sin @ real @ X ) @ X ) ) ).
% sin_x_le_x
thf(fact_3846_sin__le__one,axiom,
! [X: real] : ( ord_less_eq @ real @ ( sin @ real @ X ) @ ( one_one @ real ) ) ).
% sin_le_one
thf(fact_3847_sin__x__ge__neg__x,axiom,
! [X: real] :
( ( ord_less_eq @ real @ ( zero_zero @ real ) @ X )
=> ( ord_less_eq @ real @ ( uminus_uminus @ real @ X ) @ ( sin @ real @ X ) ) ) ).
% sin_x_ge_neg_x
thf(fact_3848_sin__ge__minus__one,axiom,
! [X: real] : ( ord_less_eq @ real @ ( uminus_uminus @ real @ ( one_one @ real ) ) @ ( sin @ real @ X ) ) ).
% sin_ge_minus_one
thf(fact_3849_sin__gt__zero__02,axiom,
! [X: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ X )
=> ( ( ord_less @ real @ X @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) )
=> ( ord_less @ real @ ( zero_zero @ real ) @ ( sin @ real @ X ) ) ) ) ).
% sin_gt_zero_02
thf(fact_3850_signed__take__bit__int__less__exp,axiom,
! [N3: nat,K: int] : ( ord_less @ int @ ( bit_ri4674362597316999326ke_bit @ int @ N3 @ K ) @ ( power_power @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ N3 ) ) ).
% signed_take_bit_int_less_exp
thf(fact_3851_even__signed__take__bit__iff,axiom,
! [A: $tType] :
( ( bit_ri3973907225187159222ations @ A )
=> ! [M: nat,A2: A] :
( ( dvd_dvd @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ ( bit_ri4674362597316999326ke_bit @ A @ M @ A2 ) )
= ( dvd_dvd @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ A2 ) ) ) ).
% even_signed_take_bit_iff
thf(fact_3852_signed__take__bit__int__less__self__iff,axiom,
! [N3: nat,K: int] :
( ( ord_less @ int @ ( bit_ri4674362597316999326ke_bit @ int @ N3 @ K ) @ K )
= ( ord_less_eq @ int @ ( power_power @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ N3 ) @ K ) ) ).
% signed_take_bit_int_less_self_iff
thf(fact_3853_signed__take__bit__int__greater__eq__self__iff,axiom,
! [K: int,N3: nat] :
( ( ord_less_eq @ int @ K @ ( bit_ri4674362597316999326ke_bit @ int @ N3 @ K ) )
= ( ord_less @ int @ K @ ( power_power @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ N3 ) ) ) ).
% signed_take_bit_int_greater_eq_self_iff
thf(fact_3854_signed__take__bit__int__greater__eq__minus__exp,axiom,
! [N3: nat,K: int] : ( ord_less_eq @ int @ ( uminus_uminus @ int @ ( power_power @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ N3 ) ) @ ( bit_ri4674362597316999326ke_bit @ int @ N3 @ K ) ) ).
% signed_take_bit_int_greater_eq_minus_exp
thf(fact_3855_signed__take__bit__int__less__eq__self__iff,axiom,
! [N3: nat,K: int] :
( ( ord_less_eq @ int @ ( bit_ri4674362597316999326ke_bit @ int @ N3 @ K ) @ K )
= ( ord_less_eq @ int @ ( uminus_uminus @ int @ ( power_power @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ N3 ) ) @ K ) ) ).
% signed_take_bit_int_less_eq_self_iff
thf(fact_3856_signed__take__bit__int__greater__self__iff,axiom,
! [K: int,N3: nat] :
( ( ord_less @ int @ K @ ( bit_ri4674362597316999326ke_bit @ int @ N3 @ K ) )
= ( ord_less @ int @ K @ ( uminus_uminus @ int @ ( power_power @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ N3 ) ) ) ) ).
% signed_take_bit_int_greater_self_iff
thf(fact_3857_dbl__dec__def,axiom,
! [A: $tType] :
( ( neg_numeral @ A )
=> ( ( neg_numeral_dbl_dec @ A )
= ( ^ [X2: A] : ( minus_minus @ A @ ( plus_plus @ A @ X2 @ X2 ) @ ( one_one @ A ) ) ) ) ) ).
% dbl_dec_def
thf(fact_3858_signed__take__bit__int__less__eq,axiom,
! [N3: nat,K: int] :
( ( ord_less_eq @ int @ ( power_power @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ N3 ) @ K )
=> ( ord_less_eq @ int @ ( bit_ri4674362597316999326ke_bit @ int @ N3 @ K ) @ ( minus_minus @ int @ K @ ( power_power @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ ( suc @ N3 ) ) ) ) ) ).
% signed_take_bit_int_less_eq
thf(fact_3859_signed__take__bit__int__eq__self__iff,axiom,
! [N3: nat,K: int] :
( ( ( bit_ri4674362597316999326ke_bit @ int @ N3 @ K )
= K )
= ( ( ord_less_eq @ int @ ( uminus_uminus @ int @ ( power_power @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ N3 ) ) @ K )
& ( ord_less @ int @ K @ ( power_power @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ N3 ) ) ) ) ).
% signed_take_bit_int_eq_self_iff
thf(fact_3860_signed__take__bit__int__eq__self,axiom,
! [N3: nat,K: int] :
( ( ord_less_eq @ int @ ( uminus_uminus @ int @ ( power_power @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ N3 ) ) @ K )
=> ( ( ord_less @ int @ K @ ( power_power @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ N3 ) )
=> ( ( bit_ri4674362597316999326ke_bit @ int @ N3 @ K )
= K ) ) ) ).
% signed_take_bit_int_eq_self
thf(fact_3861_signed__take__bit__int__greater__eq,axiom,
! [K: int,N3: nat] :
( ( ord_less @ int @ K @ ( uminus_uminus @ int @ ( power_power @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ N3 ) ) )
=> ( ord_less_eq @ int @ ( plus_plus @ int @ K @ ( power_power @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ ( suc @ N3 ) ) ) @ ( bit_ri4674362597316999326ke_bit @ int @ N3 @ K ) ) ) ).
% signed_take_bit_int_greater_eq
thf(fact_3862_signed__take__bit__Suc,axiom,
! [A: $tType] :
( ( bit_ri3973907225187159222ations @ A )
=> ! [N3: nat,A2: A] :
( ( bit_ri4674362597316999326ke_bit @ A @ ( suc @ N3 ) @ A2 )
= ( plus_plus @ A @ ( modulo_modulo @ A @ A2 @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) @ ( times_times @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ ( bit_ri4674362597316999326ke_bit @ A @ N3 @ ( divide_divide @ A @ A2 @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) ) ) ) ) ) ).
% signed_take_bit_Suc
thf(fact_3863_sin__x__sin__y,axiom,
! [A: $tType] :
( ( ( real_Vector_banach @ A )
& ( real_V3459762299906320749_field @ A ) )
=> ! [X: A,Y: A] :
( sums @ A
@ ^ [P5: nat] :
( groups7311177749621191930dd_sum @ nat @ A
@ ^ [N2: nat] :
( if @ A
@ ( ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ P5 )
& ~ ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N2 ) )
@ ( times_times @ A @ ( real_V8093663219630862766scaleR @ A @ ( uminus_uminus @ real @ ( divide_divide @ real @ ( ring_1_of_int @ real @ ( times_times @ int @ ( power_power @ int @ ( uminus_uminus @ int @ ( one_one @ int ) ) @ ( divide_divide @ nat @ P5 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( semiring_1_of_nat @ int @ ( binomial @ P5 @ N2 ) ) ) ) @ ( semiring_char_0_fact @ real @ P5 ) ) ) @ ( power_power @ A @ X @ N2 ) ) @ ( power_power @ A @ Y @ ( minus_minus @ nat @ P5 @ N2 ) ) )
@ ( zero_zero @ A ) )
@ ( set_ord_atMost @ nat @ P5 ) )
@ ( times_times @ A @ ( sin @ A @ X ) @ ( sin @ A @ Y ) ) ) ) ).
% sin_x_sin_y
thf(fact_3864_Maclaurin__sin__expansion3,axiom,
! [N3: nat,X: real] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ N3 )
=> ( ( ord_less @ real @ ( zero_zero @ real ) @ X )
=> ? [T8: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ T8 )
& ( ord_less @ real @ T8 @ X )
& ( ( sin @ real @ X )
= ( plus_plus @ real
@ ( groups7311177749621191930dd_sum @ nat @ real
@ ^ [M5: nat] : ( times_times @ real @ ( sin_coeff @ M5 ) @ ( power_power @ real @ X @ M5 ) )
@ ( set_ord_lessThan @ nat @ N3 ) )
@ ( 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 @ N3 ) ) @ pi ) ) ) @ ( semiring_char_0_fact @ real @ N3 ) ) @ ( power_power @ real @ X @ N3 ) ) ) ) ) ) ) ).
% Maclaurin_sin_expansion3
thf(fact_3865_Maclaurin__sin__expansion4,axiom,
! [X: real,N3: nat] :
( ( ord_less @ real @ ( zero_zero @ real ) @ X )
=> ? [T8: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ T8 )
& ( ord_less_eq @ real @ T8 @ X )
& ( ( sin @ real @ X )
= ( plus_plus @ real
@ ( groups7311177749621191930dd_sum @ nat @ real
@ ^ [M5: nat] : ( times_times @ real @ ( sin_coeff @ M5 ) @ ( power_power @ real @ X @ M5 ) )
@ ( set_ord_lessThan @ nat @ N3 ) )
@ ( 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 @ N3 ) ) @ pi ) ) ) @ ( semiring_char_0_fact @ real @ N3 ) ) @ ( power_power @ real @ X @ N3 ) ) ) ) ) ) ).
% Maclaurin_sin_expansion4
thf(fact_3866_sin__cos__npi,axiom,
! [N3: nat] :
( ( sin @ real @ ( divide_divide @ real @ ( times_times @ real @ ( semiring_1_of_nat @ real @ ( suc @ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N3 ) ) ) @ pi ) @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) )
= ( power_power @ real @ ( uminus_uminus @ real @ ( one_one @ real ) ) @ N3 ) ) ).
% sin_cos_npi
thf(fact_3867_mult__scaleR__right,axiom,
! [A: $tType] :
( ( real_V6157519004096292374lgebra @ A )
=> ! [X: A,A2: real,Y: A] :
( ( times_times @ A @ X @ ( real_V8093663219630862766scaleR @ A @ A2 @ Y ) )
= ( real_V8093663219630862766scaleR @ A @ A2 @ ( times_times @ A @ X @ Y ) ) ) ) ).
% mult_scaleR_right
thf(fact_3868_mult__scaleR__left,axiom,
! [A: $tType] :
( ( real_V6157519004096292374lgebra @ A )
=> ! [A2: real,X: A,Y: A] :
( ( times_times @ A @ ( real_V8093663219630862766scaleR @ A @ A2 @ X ) @ Y )
= ( real_V8093663219630862766scaleR @ A @ A2 @ ( times_times @ A @ X @ Y ) ) ) ) ).
% mult_scaleR_left
thf(fact_3869_scaleR__one,axiom,
! [A: $tType] :
( ( real_V4867850818363320053vector @ A )
=> ! [X: A] :
( ( real_V8093663219630862766scaleR @ A @ ( one_one @ real ) @ X )
= X ) ) ).
% scaleR_one
thf(fact_3870_scaleR__scaleR,axiom,
! [A: $tType] :
( ( real_V4867850818363320053vector @ A )
=> ! [A2: real,B2: real,X: A] :
( ( real_V8093663219630862766scaleR @ A @ A2 @ ( real_V8093663219630862766scaleR @ A @ B2 @ X ) )
= ( real_V8093663219630862766scaleR @ A @ ( times_times @ real @ A2 @ B2 ) @ X ) ) ) ).
% scaleR_scaleR
thf(fact_3871_scaleR__eq__iff,axiom,
! [A: $tType] :
( ( real_V4867850818363320053vector @ A )
=> ! [B2: A,U: real,A2: A] :
( ( ( plus_plus @ A @ B2 @ ( real_V8093663219630862766scaleR @ A @ U @ A2 ) )
= ( plus_plus @ A @ A2 @ ( real_V8093663219630862766scaleR @ A @ U @ B2 ) ) )
= ( ( A2 = B2 )
| ( U
= ( one_one @ real ) ) ) ) ) ).
% scaleR_eq_iff
thf(fact_3872_scaleR__power,axiom,
! [A: $tType] :
( ( real_V2822296259951069270ebra_1 @ A )
=> ! [X: real,Y: A,N3: nat] :
( ( power_power @ A @ ( real_V8093663219630862766scaleR @ A @ X @ Y ) @ N3 )
= ( real_V8093663219630862766scaleR @ A @ ( power_power @ real @ X @ N3 ) @ ( power_power @ A @ Y @ N3 ) ) ) ) ).
% scaleR_power
thf(fact_3873_sin__pi__minus,axiom,
! [X: real] :
( ( sin @ real @ ( minus_minus @ real @ pi @ X ) )
= ( sin @ real @ X ) ) ).
% sin_pi_minus
thf(fact_3874_scaleR__minus1__left,axiom,
! [A: $tType] :
( ( real_V4867850818363320053vector @ A )
=> ! [X: A] :
( ( real_V8093663219630862766scaleR @ A @ ( uminus_uminus @ real @ ( one_one @ real ) ) @ X )
= ( uminus_uminus @ A @ X ) ) ) ).
% scaleR_minus1_left
thf(fact_3875_scaleR__collapse,axiom,
! [A: $tType] :
( ( real_V4867850818363320053vector @ A )
=> ! [U: real,A2: A] :
( ( plus_plus @ A @ ( real_V8093663219630862766scaleR @ A @ ( minus_minus @ real @ ( one_one @ real ) @ U ) @ A2 ) @ ( real_V8093663219630862766scaleR @ A @ U @ A2 ) )
= A2 ) ) ).
% scaleR_collapse
thf(fact_3876_sin__periodic__pi,axiom,
! [X: real] :
( ( sin @ real @ ( plus_plus @ real @ X @ pi ) )
= ( uminus_uminus @ real @ ( sin @ real @ X ) ) ) ).
% sin_periodic_pi
thf(fact_3877_sin__periodic__pi2,axiom,
! [X: real] :
( ( sin @ real @ ( plus_plus @ real @ pi @ X ) )
= ( uminus_uminus @ real @ ( sin @ real @ X ) ) ) ).
% sin_periodic_pi2
thf(fact_3878_sin__minus__pi,axiom,
! [X: real] :
( ( sin @ real @ ( minus_minus @ real @ X @ pi ) )
= ( uminus_uminus @ real @ ( sin @ real @ X ) ) ) ).
% sin_minus_pi
thf(fact_3879_scaleR__times,axiom,
! [A: $tType] :
( ( real_V2191834092415804123ebra_1 @ A )
=> ! [U: num,W: num,A2: A] :
( ( real_V8093663219630862766scaleR @ A @ ( numeral_numeral @ real @ U ) @ ( times_times @ A @ ( numeral_numeral @ A @ W ) @ A2 ) )
= ( real_V8093663219630862766scaleR @ A @ ( times_times @ real @ ( numeral_numeral @ real @ U ) @ ( numeral_numeral @ real @ W ) ) @ A2 ) ) ) ).
% scaleR_times
thf(fact_3880_sin__npi,axiom,
! [N3: nat] :
( ( sin @ real @ ( times_times @ real @ ( semiring_1_of_nat @ real @ N3 ) @ pi ) )
= ( zero_zero @ real ) ) ).
% sin_npi
thf(fact_3881_sin__npi2,axiom,
! [N3: nat] :
( ( sin @ real @ ( times_times @ real @ pi @ ( semiring_1_of_nat @ real @ N3 ) ) )
= ( zero_zero @ real ) ) ).
% sin_npi2
thf(fact_3882_sin__npi__int,axiom,
! [N3: int] :
( ( sin @ real @ ( times_times @ real @ pi @ ( ring_1_of_int @ real @ N3 ) ) )
= ( zero_zero @ real ) ) ).
% sin_npi_int
thf(fact_3883_inverse__scaleR__times,axiom,
! [A: $tType] :
( ( real_V2191834092415804123ebra_1 @ A )
=> ! [V: num,W: num,A2: A] :
( ( real_V8093663219630862766scaleR @ A @ ( divide_divide @ real @ ( one_one @ real ) @ ( numeral_numeral @ real @ V ) ) @ ( times_times @ A @ ( numeral_numeral @ A @ W ) @ A2 ) )
= ( real_V8093663219630862766scaleR @ A @ ( divide_divide @ real @ ( numeral_numeral @ real @ W ) @ ( numeral_numeral @ real @ V ) ) @ A2 ) ) ) ).
% inverse_scaleR_times
thf(fact_3884_fraction__scaleR__times,axiom,
! [A: $tType] :
( ( real_V2191834092415804123ebra_1 @ A )
=> ! [U: num,V: num,W: num,A2: A] :
( ( real_V8093663219630862766scaleR @ A @ ( divide_divide @ real @ ( numeral_numeral @ real @ U ) @ ( numeral_numeral @ real @ V ) ) @ ( times_times @ A @ ( numeral_numeral @ A @ W ) @ A2 ) )
= ( real_V8093663219630862766scaleR @ A @ ( divide_divide @ real @ ( times_times @ real @ ( numeral_numeral @ real @ U ) @ ( numeral_numeral @ real @ W ) ) @ ( numeral_numeral @ real @ V ) ) @ A2 ) ) ) ).
% fraction_scaleR_times
thf(fact_3885_scaleR__half__double,axiom,
! [A: $tType] :
( ( real_V4867850818363320053vector @ A )
=> ! [A2: A] :
( ( real_V8093663219630862766scaleR @ A @ ( divide_divide @ real @ ( one_one @ real ) @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) @ ( plus_plus @ A @ A2 @ A2 ) )
= A2 ) ) ).
% scaleR_half_double
thf(fact_3886_sin__two__pi,axiom,
( ( sin @ real @ ( times_times @ real @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) @ pi ) )
= ( zero_zero @ real ) ) ).
% sin_two_pi
thf(fact_3887_sin__pi__half,axiom,
( ( sin @ real @ ( divide_divide @ real @ pi @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) )
= ( one_one @ real ) ) ).
% sin_pi_half
thf(fact_3888_sin__periodic,axiom,
! [X: real] :
( ( sin @ real @ ( plus_plus @ real @ X @ ( times_times @ real @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) @ pi ) ) )
= ( sin @ real @ X ) ) ).
% sin_periodic
thf(fact_3889_sin__2npi,axiom,
! [N3: nat] :
( ( sin @ real @ ( times_times @ real @ ( times_times @ real @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) @ ( semiring_1_of_nat @ real @ N3 ) ) @ pi ) )
= ( zero_zero @ real ) ) ).
% sin_2npi
thf(fact_3890_sin__2pi__minus,axiom,
! [X: real] :
( ( sin @ real @ ( minus_minus @ real @ ( times_times @ real @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) @ pi ) @ X ) )
= ( uminus_uminus @ real @ ( sin @ real @ X ) ) ) ).
% sin_2pi_minus
thf(fact_3891_sin__int__2pin,axiom,
! [N3: int] :
( ( sin @ real @ ( times_times @ real @ ( times_times @ real @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) @ pi ) @ ( ring_1_of_int @ real @ N3 ) ) )
= ( zero_zero @ real ) ) ).
% sin_int_2pin
thf(fact_3892_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_3893_scaleR__right__distrib,axiom,
! [A: $tType] :
( ( real_V4867850818363320053vector @ A )
=> ! [A2: real,X: A,Y: A] :
( ( real_V8093663219630862766scaleR @ A @ A2 @ ( plus_plus @ A @ X @ Y ) )
= ( plus_plus @ A @ ( real_V8093663219630862766scaleR @ A @ A2 @ X ) @ ( real_V8093663219630862766scaleR @ A @ A2 @ Y ) ) ) ) ).
% scaleR_right_distrib
thf(fact_3894_scaleR__right__diff__distrib,axiom,
! [A: $tType] :
( ( real_V4867850818363320053vector @ A )
=> ! [A2: real,X: A,Y: A] :
( ( real_V8093663219630862766scaleR @ A @ A2 @ ( minus_minus @ A @ X @ Y ) )
= ( minus_minus @ A @ ( real_V8093663219630862766scaleR @ A @ A2 @ X ) @ ( real_V8093663219630862766scaleR @ A @ A2 @ Y ) ) ) ) ).
% scaleR_right_diff_distrib
thf(fact_3895_real__scaleR__def,axiom,
( ( real_V8093663219630862766scaleR @ real )
= ( times_times @ real ) ) ).
% real_scaleR_def
thf(fact_3896_scaleR__left_Oadd,axiom,
! [A: $tType] :
( ( real_V4867850818363320053vector @ A )
=> ! [X: real,Y: real,Xa: A] :
( ( real_V8093663219630862766scaleR @ A @ ( plus_plus @ real @ X @ Y ) @ Xa )
= ( plus_plus @ A @ ( real_V8093663219630862766scaleR @ A @ X @ Xa ) @ ( real_V8093663219630862766scaleR @ A @ Y @ Xa ) ) ) ) ).
% scaleR_left.add
thf(fact_3897_scaleR__left__distrib,axiom,
! [A: $tType] :
( ( real_V4867850818363320053vector @ A )
=> ! [A2: real,B2: real,X: A] :
( ( real_V8093663219630862766scaleR @ A @ ( plus_plus @ real @ A2 @ B2 ) @ X )
= ( plus_plus @ A @ ( real_V8093663219630862766scaleR @ A @ A2 @ X ) @ ( real_V8093663219630862766scaleR @ A @ B2 @ X ) ) ) ) ).
% scaleR_left_distrib
thf(fact_3898_scaleR__left__diff__distrib,axiom,
! [A: $tType] :
( ( real_V4867850818363320053vector @ A )
=> ! [A2: real,B2: real,X: A] :
( ( real_V8093663219630862766scaleR @ A @ ( minus_minus @ real @ A2 @ B2 ) @ X )
= ( minus_minus @ A @ ( real_V8093663219630862766scaleR @ A @ A2 @ X ) @ ( real_V8093663219630862766scaleR @ A @ B2 @ X ) ) ) ) ).
% scaleR_left_diff_distrib
thf(fact_3899_scaleR__left_Odiff,axiom,
! [A: $tType] :
( ( real_V4867850818363320053vector @ A )
=> ! [X: real,Y: real,Xa: A] :
( ( real_V8093663219630862766scaleR @ A @ ( minus_minus @ real @ X @ Y ) @ Xa )
= ( minus_minus @ A @ ( real_V8093663219630862766scaleR @ A @ X @ Xa ) @ ( real_V8093663219630862766scaleR @ A @ Y @ Xa ) ) ) ) ).
% scaleR_left.diff
thf(fact_3900_pi__ge__zero,axiom,
ord_less_eq @ real @ ( zero_zero @ real ) @ pi ).
% pi_ge_zero
thf(fact_3901_scaleR__right__mono__neg,axiom,
! [A: $tType] :
( ( real_V5355595471888546746vector @ A )
=> ! [B2: real,A2: real,C2: A] :
( ( ord_less_eq @ real @ B2 @ A2 )
=> ( ( ord_less_eq @ A @ C2 @ ( zero_zero @ A ) )
=> ( ord_less_eq @ A @ ( real_V8093663219630862766scaleR @ A @ A2 @ C2 ) @ ( real_V8093663219630862766scaleR @ A @ B2 @ C2 ) ) ) ) ) ).
% scaleR_right_mono_neg
thf(fact_3902_scaleR__right__mono,axiom,
! [A: $tType] :
( ( real_V5355595471888546746vector @ A )
=> ! [A2: real,B2: real,X: A] :
( ( ord_less_eq @ real @ A2 @ B2 )
=> ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ X )
=> ( ord_less_eq @ A @ ( real_V8093663219630862766scaleR @ A @ A2 @ X ) @ ( real_V8093663219630862766scaleR @ A @ B2 @ X ) ) ) ) ) ).
% scaleR_right_mono
thf(fact_3903_scaleR__le__cancel__left__pos,axiom,
! [A: $tType] :
( ( real_V5355595471888546746vector @ A )
=> ! [C2: real,A2: A,B2: A] :
( ( ord_less @ real @ ( zero_zero @ real ) @ C2 )
=> ( ( ord_less_eq @ A @ ( real_V8093663219630862766scaleR @ A @ C2 @ A2 ) @ ( real_V8093663219630862766scaleR @ A @ C2 @ B2 ) )
= ( ord_less_eq @ A @ A2 @ B2 ) ) ) ) ).
% scaleR_le_cancel_left_pos
thf(fact_3904_scaleR__le__cancel__left__neg,axiom,
! [A: $tType] :
( ( real_V5355595471888546746vector @ A )
=> ! [C2: real,A2: A,B2: A] :
( ( ord_less @ real @ C2 @ ( zero_zero @ real ) )
=> ( ( ord_less_eq @ A @ ( real_V8093663219630862766scaleR @ A @ C2 @ A2 ) @ ( real_V8093663219630862766scaleR @ A @ C2 @ B2 ) )
= ( ord_less_eq @ A @ B2 @ A2 ) ) ) ) ).
% scaleR_le_cancel_left_neg
thf(fact_3905_scaleR__le__cancel__left,axiom,
! [A: $tType] :
( ( real_V5355595471888546746vector @ A )
=> ! [C2: real,A2: A,B2: A] :
( ( ord_less_eq @ A @ ( real_V8093663219630862766scaleR @ A @ C2 @ A2 ) @ ( real_V8093663219630862766scaleR @ A @ C2 @ B2 ) )
= ( ( ( ord_less @ real @ ( zero_zero @ real ) @ C2 )
=> ( ord_less_eq @ A @ A2 @ B2 ) )
& ( ( ord_less @ real @ C2 @ ( zero_zero @ real ) )
=> ( ord_less_eq @ A @ B2 @ A2 ) ) ) ) ) ).
% scaleR_le_cancel_left
thf(fact_3906_scaleR__left__mono__neg,axiom,
! [A: $tType] :
( ( real_V5355595471888546746vector @ A )
=> ! [B2: A,A2: A,C2: real] :
( ( ord_less_eq @ A @ B2 @ A2 )
=> ( ( ord_less_eq @ real @ C2 @ ( zero_zero @ real ) )
=> ( ord_less_eq @ A @ ( real_V8093663219630862766scaleR @ A @ C2 @ A2 ) @ ( real_V8093663219630862766scaleR @ A @ C2 @ B2 ) ) ) ) ) ).
% scaleR_left_mono_neg
thf(fact_3907_scaleR__left__mono,axiom,
! [A: $tType] :
( ( real_V5355595471888546746vector @ A )
=> ! [X: A,Y: A,A2: real] :
( ( ord_less_eq @ A @ X @ Y )
=> ( ( ord_less_eq @ real @ ( zero_zero @ real ) @ A2 )
=> ( ord_less_eq @ A @ ( real_V8093663219630862766scaleR @ A @ A2 @ X ) @ ( real_V8093663219630862766scaleR @ A @ A2 @ Y ) ) ) ) ) ).
% scaleR_left_mono
thf(fact_3908_eq__vector__fraction__iff,axiom,
! [A: $tType] :
( ( real_V4867850818363320053vector @ A )
=> ! [X: A,U: real,V: real,A2: A] :
( ( X
= ( real_V8093663219630862766scaleR @ A @ ( divide_divide @ real @ U @ V ) @ A2 ) )
= ( ( ( V
= ( zero_zero @ real ) )
=> ( X
= ( zero_zero @ A ) ) )
& ( ( V
!= ( zero_zero @ real ) )
=> ( ( real_V8093663219630862766scaleR @ A @ V @ X )
= ( real_V8093663219630862766scaleR @ A @ U @ A2 ) ) ) ) ) ) ).
% eq_vector_fraction_iff
thf(fact_3909_vector__fraction__eq__iff,axiom,
! [A: $tType] :
( ( real_V4867850818363320053vector @ A )
=> ! [U: real,V: real,A2: A,X: A] :
( ( ( real_V8093663219630862766scaleR @ A @ ( divide_divide @ real @ U @ V ) @ A2 )
= X )
= ( ( ( V
= ( zero_zero @ real ) )
=> ( X
= ( zero_zero @ A ) ) )
& ( ( V
!= ( zero_zero @ real ) )
=> ( ( real_V8093663219630862766scaleR @ A @ U @ A2 )
= ( real_V8093663219630862766scaleR @ A @ V @ X ) ) ) ) ) ) ).
% vector_fraction_eq_iff
thf(fact_3910_Real__Vector__Spaces_Ole__add__iff2,axiom,
! [A: $tType] :
( ( real_V5355595471888546746vector @ A )
=> ! [A2: real,E: A,C2: A,B2: real,D: A] :
( ( ord_less_eq @ A @ ( plus_plus @ A @ ( real_V8093663219630862766scaleR @ A @ A2 @ E ) @ C2 ) @ ( plus_plus @ A @ ( real_V8093663219630862766scaleR @ A @ B2 @ E ) @ D ) )
= ( ord_less_eq @ A @ C2 @ ( plus_plus @ A @ ( real_V8093663219630862766scaleR @ A @ ( minus_minus @ real @ B2 @ A2 ) @ E ) @ D ) ) ) ) ).
% Real_Vector_Spaces.le_add_iff2
thf(fact_3911_Real__Vector__Spaces_Ole__add__iff1,axiom,
! [A: $tType] :
( ( real_V5355595471888546746vector @ A )
=> ! [A2: real,E: A,C2: A,B2: real,D: A] :
( ( ord_less_eq @ A @ ( plus_plus @ A @ ( real_V8093663219630862766scaleR @ A @ A2 @ E ) @ C2 ) @ ( plus_plus @ A @ ( real_V8093663219630862766scaleR @ A @ B2 @ E ) @ D ) )
= ( ord_less_eq @ A @ ( plus_plus @ A @ ( real_V8093663219630862766scaleR @ A @ ( minus_minus @ real @ A2 @ B2 ) @ E ) @ C2 ) @ D ) ) ) ).
% Real_Vector_Spaces.le_add_iff1
thf(fact_3912_sin__ge__zero,axiom,
! [X: real] :
( ( ord_less_eq @ real @ ( zero_zero @ real ) @ X )
=> ( ( ord_less_eq @ real @ X @ pi )
=> ( ord_less_eq @ real @ ( zero_zero @ real ) @ ( sin @ real @ X ) ) ) ) ).
% sin_ge_zero
thf(fact_3913_scaleR__le__0__iff,axiom,
! [A: $tType] :
( ( real_V5355595471888546746vector @ A )
=> ! [A2: real,B2: A] :
( ( ord_less_eq @ A @ ( real_V8093663219630862766scaleR @ A @ A2 @ B2 ) @ ( zero_zero @ A ) )
= ( ( ( ord_less @ real @ ( zero_zero @ real ) @ A2 )
& ( ord_less_eq @ A @ B2 @ ( zero_zero @ A ) ) )
| ( ( ord_less @ real @ A2 @ ( zero_zero @ real ) )
& ( ord_less_eq @ A @ ( zero_zero @ A ) @ B2 ) )
| ( A2
= ( zero_zero @ real ) ) ) ) ) ).
% scaleR_le_0_iff
thf(fact_3914_zero__le__scaleR__iff,axiom,
! [A: $tType] :
( ( real_V5355595471888546746vector @ A )
=> ! [A2: real,B2: A] :
( ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( real_V8093663219630862766scaleR @ A @ A2 @ B2 ) )
= ( ( ( ord_less @ real @ ( zero_zero @ real ) @ A2 )
& ( ord_less_eq @ A @ ( zero_zero @ A ) @ B2 ) )
| ( ( ord_less @ real @ A2 @ ( zero_zero @ real ) )
& ( ord_less_eq @ A @ B2 @ ( zero_zero @ A ) ) )
| ( A2
= ( zero_zero @ real ) ) ) ) ) ).
% zero_le_scaleR_iff
thf(fact_3915_scaleR__mono,axiom,
! [A: $tType] :
( ( real_V5355595471888546746vector @ A )
=> ! [A2: real,B2: real,X: A,Y: A] :
( ( ord_less_eq @ real @ A2 @ B2 )
=> ( ( ord_less_eq @ A @ X @ Y )
=> ( ( ord_less_eq @ real @ ( zero_zero @ real ) @ B2 )
=> ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ X )
=> ( ord_less_eq @ A @ ( real_V8093663219630862766scaleR @ A @ A2 @ X ) @ ( real_V8093663219630862766scaleR @ A @ B2 @ Y ) ) ) ) ) ) ) ).
% scaleR_mono
thf(fact_3916_scaleR__mono_H,axiom,
! [A: $tType] :
( ( real_V5355595471888546746vector @ A )
=> ! [A2: real,B2: real,C2: A,D: A] :
( ( ord_less_eq @ real @ A2 @ B2 )
=> ( ( ord_less_eq @ A @ C2 @ D )
=> ( ( ord_less_eq @ real @ ( zero_zero @ real ) @ A2 )
=> ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ C2 )
=> ( ord_less_eq @ A @ ( real_V8093663219630862766scaleR @ A @ A2 @ C2 ) @ ( real_V8093663219630862766scaleR @ A @ B2 @ D ) ) ) ) ) ) ) ).
% scaleR_mono'
thf(fact_3917_split__scaleR__neg__le,axiom,
! [A: $tType] :
( ( real_V5355595471888546746vector @ A )
=> ! [A2: real,X: A] :
( ( ( ( ord_less_eq @ real @ ( zero_zero @ real ) @ A2 )
& ( ord_less_eq @ A @ X @ ( zero_zero @ A ) ) )
| ( ( ord_less_eq @ real @ A2 @ ( zero_zero @ real ) )
& ( ord_less_eq @ A @ ( zero_zero @ A ) @ X ) ) )
=> ( ord_less_eq @ A @ ( real_V8093663219630862766scaleR @ A @ A2 @ X ) @ ( zero_zero @ A ) ) ) ) ).
% split_scaleR_neg_le
thf(fact_3918_split__scaleR__pos__le,axiom,
! [A: $tType] :
( ( real_V5355595471888546746vector @ A )
=> ! [A2: real,B2: A] :
( ( ( ( ord_less_eq @ real @ ( zero_zero @ real ) @ A2 )
& ( ord_less_eq @ A @ ( zero_zero @ A ) @ B2 ) )
| ( ( ord_less_eq @ real @ A2 @ ( zero_zero @ real ) )
& ( ord_less_eq @ A @ B2 @ ( zero_zero @ A ) ) ) )
=> ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( real_V8093663219630862766scaleR @ A @ A2 @ B2 ) ) ) ) ).
% split_scaleR_pos_le
thf(fact_3919_scaleR__nonneg__nonneg,axiom,
! [A: $tType] :
( ( real_V5355595471888546746vector @ A )
=> ! [A2: real,X: A] :
( ( ord_less_eq @ real @ ( zero_zero @ real ) @ A2 )
=> ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ X )
=> ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( real_V8093663219630862766scaleR @ A @ A2 @ X ) ) ) ) ) ).
% scaleR_nonneg_nonneg
thf(fact_3920_scaleR__nonneg__nonpos,axiom,
! [A: $tType] :
( ( real_V5355595471888546746vector @ A )
=> ! [A2: real,X: A] :
( ( ord_less_eq @ real @ ( zero_zero @ real ) @ A2 )
=> ( ( ord_less_eq @ A @ X @ ( zero_zero @ A ) )
=> ( ord_less_eq @ A @ ( real_V8093663219630862766scaleR @ A @ A2 @ X ) @ ( zero_zero @ A ) ) ) ) ) ).
% scaleR_nonneg_nonpos
thf(fact_3921_scaleR__nonpos__nonneg,axiom,
! [A: $tType] :
( ( real_V5355595471888546746vector @ A )
=> ! [A2: real,X: A] :
( ( ord_less_eq @ real @ A2 @ ( zero_zero @ real ) )
=> ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ X )
=> ( ord_less_eq @ A @ ( real_V8093663219630862766scaleR @ A @ A2 @ X ) @ ( zero_zero @ A ) ) ) ) ) ).
% scaleR_nonpos_nonneg
thf(fact_3922_scaleR__nonpos__nonpos,axiom,
! [A: $tType] :
( ( real_V5355595471888546746vector @ A )
=> ! [A2: real,B2: A] :
( ( ord_less_eq @ real @ A2 @ ( zero_zero @ real ) )
=> ( ( ord_less_eq @ A @ B2 @ ( zero_zero @ A ) )
=> ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( real_V8093663219630862766scaleR @ A @ A2 @ B2 ) ) ) ) ) ).
% scaleR_nonpos_nonpos
thf(fact_3923_scaleR__2,axiom,
! [A: $tType] :
( ( real_V4867850818363320053vector @ A )
=> ! [X: A] :
( ( real_V8093663219630862766scaleR @ A @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) @ X )
= ( plus_plus @ A @ X @ X ) ) ) ).
% scaleR_2
thf(fact_3924_scaleR__left__le__one__le,axiom,
! [A: $tType] :
( ( real_V5355595471888546746vector @ A )
=> ! [X: A,A2: real] :
( ( ord_less_eq @ A @ ( zero_zero @ A ) @ X )
=> ( ( ord_less_eq @ real @ A2 @ ( one_one @ real ) )
=> ( ord_less_eq @ A @ ( real_V8093663219630862766scaleR @ A @ A2 @ X ) @ X ) ) ) ) ).
% scaleR_left_le_one_le
thf(fact_3925_pi__less__4,axiom,
ord_less @ real @ pi @ ( numeral_numeral @ real @ ( bit0 @ ( bit0 @ one2 ) ) ) ).
% pi_less_4
thf(fact_3926_pi__ge__two,axiom,
ord_less_eq @ real @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) @ pi ).
% pi_ge_two
thf(fact_3927_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_3928_sin__zero__iff__int2,axiom,
! [X: real] :
( ( ( sin @ real @ X )
= ( zero_zero @ real ) )
= ( ? [I3: int] :
( X
= ( times_times @ real @ ( ring_1_of_int @ real @ I3 ) @ pi ) ) ) ) ).
% sin_zero_iff_int2
thf(fact_3929_sin__converges,axiom,
! [A: $tType] :
( ( ( real_Vector_banach @ A )
& ( real_V2822296259951069270ebra_1 @ A ) )
=> ! [X: A] :
( sums @ A
@ ^ [N2: nat] : ( real_V8093663219630862766scaleR @ A @ ( sin_coeff @ N2 ) @ ( power_power @ A @ X @ N2 ) )
@ ( sin @ A @ X ) ) ) ).
% sin_converges
thf(fact_3930_pi__half__neq__zero,axiom,
( ( divide_divide @ real @ pi @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) )
!= ( zero_zero @ real ) ) ).
% pi_half_neq_zero
thf(fact_3931_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_3932_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_3933_sin__minus__converges,axiom,
! [A: $tType] :
( ( ( real_Vector_banach @ A )
& ( real_V2822296259951069270ebra_1 @ A ) )
=> ! [X: A] :
( sums @ A
@ ^ [N2: nat] : ( uminus_uminus @ A @ ( real_V8093663219630862766scaleR @ A @ ( sin_coeff @ N2 ) @ ( power_power @ A @ ( uminus_uminus @ A @ X ) @ N2 ) ) )
@ ( sin @ A @ X ) ) ) ).
% sin_minus_converges
thf(fact_3934_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_3935_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_3936_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_3937_sin__pi__divide__n__ge__0,axiom,
! [N3: nat] :
( ( N3
!= ( zero_zero @ nat ) )
=> ( ord_less_eq @ real @ ( zero_zero @ real ) @ ( sin @ real @ ( divide_divide @ real @ pi @ ( semiring_1_of_nat @ real @ N3 ) ) ) ) ) ).
% sin_pi_divide_n_ge_0
thf(fact_3938_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_3939_sin__gt__zero2,axiom,
! [X: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ X )
=> ( ( ord_less @ real @ X @ ( divide_divide @ real @ pi @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) )
=> ( ord_less @ real @ ( zero_zero @ real ) @ ( sin @ real @ X ) ) ) ) ).
% sin_gt_zero2
thf(fact_3940_sin__lt__zero,axiom,
! [X: real] :
( ( ord_less @ real @ pi @ X )
=> ( ( ord_less @ real @ X @ ( times_times @ real @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) @ pi ) )
=> ( ord_less @ real @ ( sin @ real @ X ) @ ( zero_zero @ real ) ) ) ) ).
% sin_lt_zero
thf(fact_3941_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_3942_sin__monotone__2pi__le,axiom,
! [Y: real,X: real] :
( ( ord_less_eq @ real @ ( uminus_uminus @ real @ ( divide_divide @ real @ pi @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) ) @ Y )
=> ( ( ord_less_eq @ real @ Y @ X )
=> ( ( ord_less_eq @ real @ X @ ( divide_divide @ real @ pi @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) )
=> ( ord_less_eq @ real @ ( sin @ real @ Y ) @ ( sin @ real @ X ) ) ) ) ) ).
% sin_monotone_2pi_le
thf(fact_3943_sin__mono__le__eq,axiom,
! [X: real,Y: real] :
( ( ord_less_eq @ real @ ( uminus_uminus @ real @ ( divide_divide @ real @ pi @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) ) @ X )
=> ( ( ord_less_eq @ real @ X @ ( divide_divide @ real @ pi @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) )
=> ( ( ord_less_eq @ real @ ( uminus_uminus @ real @ ( divide_divide @ real @ pi @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) ) @ Y )
=> ( ( ord_less_eq @ real @ Y @ ( divide_divide @ real @ pi @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) )
=> ( ( ord_less_eq @ real @ ( sin @ real @ X ) @ ( sin @ real @ Y ) )
= ( ord_less_eq @ real @ X @ Y ) ) ) ) ) ) ).
% sin_mono_le_eq
thf(fact_3944_sin__inj__pi,axiom,
! [X: real,Y: real] :
( ( ord_less_eq @ real @ ( uminus_uminus @ real @ ( divide_divide @ real @ pi @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) ) @ X )
=> ( ( ord_less_eq @ real @ X @ ( divide_divide @ real @ pi @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) )
=> ( ( ord_less_eq @ real @ ( uminus_uminus @ real @ ( divide_divide @ real @ pi @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) ) @ Y )
=> ( ( ord_less_eq @ real @ Y @ ( divide_divide @ real @ pi @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) )
=> ( ( ( sin @ real @ X )
= ( sin @ real @ Y ) )
=> ( X = Y ) ) ) ) ) ) ).
% sin_inj_pi
thf(fact_3945_sin__le__zero,axiom,
! [X: real] :
( ( ord_less_eq @ real @ pi @ X )
=> ( ( ord_less @ real @ X @ ( times_times @ real @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) @ pi ) )
=> ( ord_less_eq @ real @ ( sin @ real @ X ) @ ( zero_zero @ real ) ) ) ) ).
% sin_le_zero
thf(fact_3946_sin__less__zero,axiom,
! [X: real] :
( ( ord_less @ real @ ( divide_divide @ real @ ( uminus_uminus @ real @ pi ) @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) @ X )
=> ( ( ord_less @ real @ X @ ( zero_zero @ real ) )
=> ( ord_less @ real @ ( sin @ real @ X ) @ ( zero_zero @ real ) ) ) ) ).
% sin_less_zero
thf(fact_3947_sin__monotone__2pi,axiom,
! [Y: real,X: real] :
( ( ord_less_eq @ real @ ( uminus_uminus @ real @ ( divide_divide @ real @ pi @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) ) @ Y )
=> ( ( ord_less @ real @ Y @ X )
=> ( ( ord_less_eq @ real @ X @ ( divide_divide @ real @ pi @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) )
=> ( ord_less @ real @ ( sin @ real @ Y ) @ ( sin @ real @ X ) ) ) ) ) ).
% sin_monotone_2pi
thf(fact_3948_sin__mono__less__eq,axiom,
! [X: real,Y: real] :
( ( ord_less_eq @ real @ ( uminus_uminus @ real @ ( divide_divide @ real @ pi @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) ) @ X )
=> ( ( ord_less_eq @ real @ X @ ( divide_divide @ real @ pi @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) )
=> ( ( ord_less_eq @ real @ ( uminus_uminus @ real @ ( divide_divide @ real @ pi @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) ) @ Y )
=> ( ( ord_less_eq @ real @ Y @ ( divide_divide @ real @ pi @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) )
=> ( ( ord_less @ real @ ( sin @ real @ X ) @ ( sin @ real @ Y ) )
= ( ord_less @ real @ X @ Y ) ) ) ) ) ) ).
% sin_mono_less_eq
thf(fact_3949_sin__total,axiom,
! [Y: real] :
( ( ord_less_eq @ real @ ( uminus_uminus @ real @ ( one_one @ real ) ) @ Y )
=> ( ( ord_less_eq @ real @ Y @ ( one_one @ real ) )
=> ? [X3: real] :
( ( ord_less_eq @ real @ ( uminus_uminus @ real @ ( divide_divide @ real @ pi @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) ) @ X3 )
& ( ord_less_eq @ real @ X3 @ ( divide_divide @ real @ pi @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) )
& ( ( sin @ real @ X3 )
= Y )
& ! [Y4: real] :
( ( ( ord_less_eq @ real @ ( uminus_uminus @ real @ ( divide_divide @ real @ pi @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) ) @ Y4 )
& ( ord_less_eq @ real @ Y4 @ ( divide_divide @ real @ pi @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) )
& ( ( sin @ real @ Y4 )
= Y ) )
=> ( Y4 = X3 ) ) ) ) ) ).
% sin_total
thf(fact_3950_sin__pi__divide__n__gt__0,axiom,
! [N3: nat] :
( ( ord_less_eq @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N3 )
=> ( ord_less @ real @ ( zero_zero @ real ) @ ( sin @ real @ ( divide_divide @ real @ pi @ ( semiring_1_of_nat @ real @ N3 ) ) ) ) ) ).
% sin_pi_divide_n_gt_0
thf(fact_3951_sin__zero__iff__int,axiom,
! [X: real] :
( ( ( sin @ real @ X )
= ( zero_zero @ real ) )
= ( ? [I3: int] :
( ( dvd_dvd @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ I3 )
& ( X
= ( times_times @ real @ ( ring_1_of_int @ real @ I3 ) @ ( divide_divide @ real @ pi @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) ) ) ) ) ) ).
% sin_zero_iff_int
thf(fact_3952_sin__zero__lemma,axiom,
! [X: real] :
( ( ord_less_eq @ real @ ( zero_zero @ real ) @ X )
=> ( ( ( sin @ real @ X )
= ( zero_zero @ real ) )
=> ? [N: nat] :
( ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N )
& ( X
= ( times_times @ real @ ( semiring_1_of_nat @ real @ N ) @ ( divide_divide @ real @ pi @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) ) ) ) ) ) ).
% sin_zero_lemma
thf(fact_3953_sin__zero__iff,axiom,
! [X: real] :
( ( ( sin @ real @ X )
= ( zero_zero @ real ) )
= ( ? [N2: nat] :
( ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N2 )
& ( X
= ( times_times @ real @ ( semiring_1_of_nat @ real @ N2 ) @ ( divide_divide @ real @ pi @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) ) ) )
| ? [N2: nat] :
( ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N2 )
& ( X
= ( uminus_uminus @ real @ ( times_times @ real @ ( semiring_1_of_nat @ real @ N2 ) @ ( divide_divide @ real @ pi @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) ) ) ) ) ) ) ).
% sin_zero_iff
thf(fact_3954_Maclaurin__sin__expansion,axiom,
! [X: real,N3: nat] :
? [T8: real] :
( ( sin @ real @ X )
= ( plus_plus @ real
@ ( groups7311177749621191930dd_sum @ nat @ real
@ ^ [M5: nat] : ( times_times @ real @ ( sin_coeff @ M5 ) @ ( power_power @ real @ X @ M5 ) )
@ ( set_ord_lessThan @ nat @ N3 ) )
@ ( 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 @ N3 ) ) @ pi ) ) ) @ ( semiring_char_0_fact @ real @ N3 ) ) @ ( power_power @ real @ X @ N3 ) ) ) ) ).
% Maclaurin_sin_expansion
thf(fact_3955_sums__cos__x__plus__y,axiom,
! [A: $tType] :
( ( ( real_Vector_banach @ A )
& ( real_V3459762299906320749_field @ A ) )
=> ! [X: A,Y: A] :
( sums @ A
@ ^ [P5: nat] :
( groups7311177749621191930dd_sum @ nat @ A
@ ^ [N2: nat] : ( if @ A @ ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ P5 ) @ ( times_times @ A @ ( real_V8093663219630862766scaleR @ A @ ( divide_divide @ real @ ( ring_1_of_int @ real @ ( times_times @ int @ ( power_power @ int @ ( uminus_uminus @ int @ ( one_one @ int ) ) @ ( divide_divide @ nat @ P5 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( semiring_1_of_nat @ int @ ( binomial @ P5 @ N2 ) ) ) ) @ ( semiring_char_0_fact @ real @ P5 ) ) @ ( power_power @ A @ X @ N2 ) ) @ ( power_power @ A @ Y @ ( minus_minus @ nat @ P5 @ N2 ) ) ) @ ( zero_zero @ A ) )
@ ( set_ord_atMost @ nat @ P5 ) )
@ ( cos @ A @ ( plus_plus @ A @ X @ Y ) ) ) ) ).
% sums_cos_x_plus_y
thf(fact_3956_cos__x__cos__y,axiom,
! [A: $tType] :
( ( ( real_Vector_banach @ A )
& ( real_V3459762299906320749_field @ A ) )
=> ! [X: A,Y: A] :
( sums @ A
@ ^ [P5: nat] :
( groups7311177749621191930dd_sum @ nat @ A
@ ^ [N2: nat] :
( if @ A
@ ( ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ P5 )
& ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N2 ) )
@ ( times_times @ A @ ( real_V8093663219630862766scaleR @ A @ ( divide_divide @ real @ ( ring_1_of_int @ real @ ( times_times @ int @ ( power_power @ int @ ( uminus_uminus @ int @ ( one_one @ int ) ) @ ( divide_divide @ nat @ P5 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( semiring_1_of_nat @ int @ ( binomial @ P5 @ N2 ) ) ) ) @ ( semiring_char_0_fact @ real @ P5 ) ) @ ( power_power @ A @ X @ N2 ) ) @ ( power_power @ A @ Y @ ( minus_minus @ nat @ P5 @ N2 ) ) )
@ ( zero_zero @ A ) )
@ ( set_ord_atMost @ nat @ P5 ) )
@ ( times_times @ A @ ( cos @ A @ X ) @ ( cos @ A @ Y ) ) ) ) ).
% cos_x_cos_y
thf(fact_3957_Maclaurin__cos__expansion2,axiom,
! [X: real,N3: nat] :
( ( ord_less @ real @ ( zero_zero @ real ) @ X )
=> ( ( ord_less @ nat @ ( zero_zero @ nat ) @ N3 )
=> ? [T8: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ T8 )
& ( ord_less @ real @ T8 @ X )
& ( ( cos @ real @ X )
= ( plus_plus @ real
@ ( groups7311177749621191930dd_sum @ nat @ real
@ ^ [M5: nat] : ( times_times @ real @ ( cos_coeff @ M5 ) @ ( power_power @ real @ X @ M5 ) )
@ ( set_ord_lessThan @ nat @ N3 ) )
@ ( 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 @ N3 ) ) @ pi ) ) ) @ ( semiring_char_0_fact @ real @ N3 ) ) @ ( power_power @ real @ X @ N3 ) ) ) ) ) ) ) ).
% Maclaurin_cos_expansion2
thf(fact_3958_Maclaurin__minus__cos__expansion,axiom,
! [N3: nat,X: real] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ N3 )
=> ( ( ord_less @ real @ X @ ( zero_zero @ real ) )
=> ? [T8: real] :
( ( ord_less @ real @ X @ T8 )
& ( ord_less @ real @ T8 @ ( zero_zero @ real ) )
& ( ( cos @ real @ X )
= ( plus_plus @ real
@ ( groups7311177749621191930dd_sum @ nat @ real
@ ^ [M5: nat] : ( times_times @ real @ ( cos_coeff @ M5 ) @ ( power_power @ real @ X @ M5 ) )
@ ( set_ord_lessThan @ nat @ N3 ) )
@ ( 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 @ N3 ) ) @ pi ) ) ) @ ( semiring_char_0_fact @ real @ N3 ) ) @ ( power_power @ real @ X @ N3 ) ) ) ) ) ) ) ).
% Maclaurin_minus_cos_expansion
thf(fact_3959_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_3960_cos__pi,axiom,
( ( cos @ real @ pi )
= ( uminus_uminus @ real @ ( one_one @ real ) ) ) ).
% cos_pi
thf(fact_3961_cos__periodic__pi2,axiom,
! [X: real] :
( ( cos @ real @ ( plus_plus @ real @ pi @ X ) )
= ( uminus_uminus @ real @ ( cos @ real @ X ) ) ) ).
% cos_periodic_pi2
thf(fact_3962_cos__periodic__pi,axiom,
! [X: real] :
( ( cos @ real @ ( plus_plus @ real @ X @ pi ) )
= ( uminus_uminus @ real @ ( cos @ real @ X ) ) ) ).
% cos_periodic_pi
thf(fact_3963_cos__pi__minus,axiom,
! [X: real] :
( ( cos @ real @ ( minus_minus @ real @ pi @ X ) )
= ( uminus_uminus @ real @ ( cos @ real @ X ) ) ) ).
% cos_pi_minus
thf(fact_3964_cos__minus__pi,axiom,
! [X: real] :
( ( cos @ real @ ( minus_minus @ real @ X @ pi ) )
= ( uminus_uminus @ real @ ( cos @ real @ X ) ) ) ).
% cos_minus_pi
thf(fact_3965_sin__cos__squared__add3,axiom,
! [A: $tType] :
( ( ( real_Vector_banach @ A )
& ( real_V3459762299906320749_field @ A ) )
=> ! [X: A] :
( ( plus_plus @ A @ ( times_times @ A @ ( cos @ A @ X ) @ ( cos @ A @ X ) ) @ ( times_times @ A @ ( sin @ A @ X ) @ ( sin @ A @ X ) ) )
= ( one_one @ A ) ) ) ).
% sin_cos_squared_add3
thf(fact_3966_cos__pi__half,axiom,
( ( cos @ real @ ( divide_divide @ real @ pi @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) )
= ( zero_zero @ real ) ) ).
% cos_pi_half
thf(fact_3967_cos__two__pi,axiom,
( ( cos @ real @ ( times_times @ real @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) @ pi ) )
= ( one_one @ real ) ) ).
% cos_two_pi
thf(fact_3968_cos__periodic,axiom,
! [X: real] :
( ( cos @ real @ ( plus_plus @ real @ X @ ( times_times @ real @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) @ pi ) ) )
= ( cos @ real @ X ) ) ).
% cos_periodic
thf(fact_3969_cos__2pi__minus,axiom,
! [X: real] :
( ( cos @ real @ ( minus_minus @ real @ ( times_times @ real @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) @ pi ) @ X ) )
= ( cos @ real @ X ) ) ).
% cos_2pi_minus
thf(fact_3970_cos__npi2,axiom,
! [N3: nat] :
( ( cos @ real @ ( times_times @ real @ pi @ ( semiring_1_of_nat @ real @ N3 ) ) )
= ( power_power @ real @ ( uminus_uminus @ real @ ( one_one @ real ) ) @ N3 ) ) ).
% cos_npi2
thf(fact_3971_cos__npi,axiom,
! [N3: nat] :
( ( cos @ real @ ( times_times @ real @ ( semiring_1_of_nat @ real @ N3 ) @ pi ) )
= ( power_power @ real @ ( uminus_uminus @ real @ ( one_one @ real ) ) @ N3 ) ) ).
% cos_npi
thf(fact_3972_sin__cos__squared__add2,axiom,
! [A: $tType] :
( ( ( real_Vector_banach @ A )
& ( real_V3459762299906320749_field @ A ) )
=> ! [X: A] :
( ( plus_plus @ A @ ( power_power @ A @ ( cos @ A @ X ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ ( power_power @ A @ ( sin @ A @ X ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) )
= ( one_one @ A ) ) ) ).
% sin_cos_squared_add2
thf(fact_3973_sin__cos__squared__add,axiom,
! [A: $tType] :
( ( ( real_Vector_banach @ A )
& ( real_V3459762299906320749_field @ A ) )
=> ! [X: A] :
( ( plus_plus @ A @ ( power_power @ A @ ( sin @ A @ X ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ ( power_power @ A @ ( cos @ A @ X ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) )
= ( one_one @ A ) ) ) ).
% sin_cos_squared_add
thf(fact_3974_cos__2npi,axiom,
! [N3: nat] :
( ( cos @ real @ ( times_times @ real @ ( times_times @ real @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) @ ( semiring_1_of_nat @ real @ N3 ) ) @ pi ) )
= ( one_one @ real ) ) ).
% cos_2npi
thf(fact_3975_cos__int__2pin,axiom,
! [N3: int] :
( ( cos @ real @ ( times_times @ real @ ( times_times @ real @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) @ pi ) @ ( ring_1_of_int @ real @ N3 ) ) )
= ( one_one @ real ) ) ).
% cos_int_2pin
thf(fact_3976_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_3977_cos__npi__int,axiom,
! [N3: int] :
( ( ( dvd_dvd @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ N3 )
=> ( ( cos @ real @ ( times_times @ real @ pi @ ( ring_1_of_int @ real @ N3 ) ) )
= ( one_one @ real ) ) )
& ( ~ ( dvd_dvd @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ N3 )
=> ( ( cos @ real @ ( times_times @ real @ pi @ ( ring_1_of_int @ real @ N3 ) ) )
= ( uminus_uminus @ real @ ( one_one @ real ) ) ) ) ) ).
% cos_npi_int
thf(fact_3978_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_3979_cos__le__one,axiom,
! [X: real] : ( ord_less_eq @ real @ ( cos @ real @ X ) @ ( one_one @ real ) ) ).
% cos_le_one
thf(fact_3980_polar__Ex,axiom,
! [X: real,Y: real] :
? [R3: real,A4: real] :
( ( X
= ( times_times @ real @ R3 @ ( cos @ real @ A4 ) ) )
& ( Y
= ( times_times @ real @ R3 @ ( sin @ real @ A4 ) ) ) ) ).
% polar_Ex
thf(fact_3981_cos__one__sin__zero,axiom,
! [A: $tType] :
( ( ( real_Vector_banach @ A )
& ( real_V3459762299906320749_field @ A ) )
=> ! [X: A] :
( ( ( cos @ A @ X )
= ( one_one @ A ) )
=> ( ( sin @ A @ X )
= ( zero_zero @ A ) ) ) ) ).
% cos_one_sin_zero
thf(fact_3982_sin__add,axiom,
! [A: $tType] :
( ( ( real_Vector_banach @ A )
& ( real_V3459762299906320749_field @ A ) )
=> ! [X: A,Y: A] :
( ( sin @ A @ ( plus_plus @ A @ X @ Y ) )
= ( plus_plus @ A @ ( times_times @ A @ ( sin @ A @ X ) @ ( cos @ A @ Y ) ) @ ( times_times @ A @ ( cos @ A @ X ) @ ( sin @ A @ Y ) ) ) ) ) ).
% sin_add
thf(fact_3983_sin__diff,axiom,
! [A: $tType] :
( ( ( real_Vector_banach @ A )
& ( real_V3459762299906320749_field @ A ) )
=> ! [X: A,Y: A] :
( ( sin @ A @ ( minus_minus @ A @ X @ Y ) )
= ( minus_minus @ A @ ( times_times @ A @ ( sin @ A @ X ) @ ( cos @ A @ Y ) ) @ ( times_times @ A @ ( cos @ A @ X ) @ ( sin @ A @ Y ) ) ) ) ) ).
% sin_diff
thf(fact_3984_cos__monotone__0__pi__le,axiom,
! [Y: real,X: real] :
( ( ord_less_eq @ real @ ( zero_zero @ real ) @ Y )
=> ( ( ord_less_eq @ real @ Y @ X )
=> ( ( ord_less_eq @ real @ X @ pi )
=> ( ord_less_eq @ real @ ( cos @ real @ X ) @ ( cos @ real @ Y ) ) ) ) ) ).
% cos_monotone_0_pi_le
thf(fact_3985_cos__mono__le__eq,axiom,
! [X: real,Y: real] :
( ( ord_less_eq @ real @ ( zero_zero @ real ) @ X )
=> ( ( ord_less_eq @ real @ X @ pi )
=> ( ( ord_less_eq @ real @ ( zero_zero @ real ) @ Y )
=> ( ( ord_less_eq @ real @ Y @ pi )
=> ( ( ord_less_eq @ real @ ( cos @ real @ X ) @ ( cos @ real @ Y ) )
= ( ord_less_eq @ real @ Y @ X ) ) ) ) ) ) ).
% cos_mono_le_eq
thf(fact_3986_cos__inj__pi,axiom,
! [X: real,Y: real] :
( ( ord_less_eq @ real @ ( zero_zero @ real ) @ X )
=> ( ( ord_less_eq @ real @ X @ pi )
=> ( ( ord_less_eq @ real @ ( zero_zero @ real ) @ Y )
=> ( ( ord_less_eq @ real @ Y @ pi )
=> ( ( ( cos @ real @ X )
= ( cos @ real @ Y ) )
=> ( X = Y ) ) ) ) ) ) ).
% cos_inj_pi
thf(fact_3987_cos__ge__minus__one,axiom,
! [X: real] : ( ord_less_eq @ real @ ( uminus_uminus @ real @ ( one_one @ real ) ) @ ( cos @ real @ X ) ) ).
% cos_ge_minus_one
thf(fact_3988_cos__diff,axiom,
! [A: $tType] :
( ( ( real_Vector_banach @ A )
& ( real_V3459762299906320749_field @ A ) )
=> ! [X: A,Y: A] :
( ( cos @ A @ ( minus_minus @ A @ X @ Y ) )
= ( plus_plus @ A @ ( times_times @ A @ ( cos @ A @ X ) @ ( cos @ A @ Y ) ) @ ( times_times @ A @ ( sin @ A @ X ) @ ( sin @ A @ Y ) ) ) ) ) ).
% cos_diff
thf(fact_3989_cos__add,axiom,
! [A: $tType] :
( ( ( real_Vector_banach @ A )
& ( real_V3459762299906320749_field @ A ) )
=> ! [X: A,Y: A] :
( ( cos @ A @ ( plus_plus @ A @ X @ Y ) )
= ( minus_minus @ A @ ( times_times @ A @ ( cos @ A @ X ) @ ( cos @ A @ Y ) ) @ ( times_times @ A @ ( sin @ A @ X ) @ ( sin @ A @ Y ) ) ) ) ) ).
% cos_add
thf(fact_3990_sin__zero__norm__cos__one,axiom,
! [A: $tType] :
( ( ( real_Vector_banach @ A )
& ( real_V3459762299906320749_field @ A ) )
=> ! [X: A] :
( ( ( sin @ A @ X )
= ( zero_zero @ A ) )
=> ( ( real_V7770717601297561774m_norm @ A @ ( cos @ A @ X ) )
= ( one_one @ real ) ) ) ) ).
% sin_zero_norm_cos_one
thf(fact_3991_cos__two__neq__zero,axiom,
( ( cos @ real @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) )
!= ( zero_zero @ real ) ) ).
% cos_two_neq_zero
thf(fact_3992_cos__mono__less__eq,axiom,
! [X: real,Y: real] :
( ( ord_less_eq @ real @ ( zero_zero @ real ) @ X )
=> ( ( ord_less_eq @ real @ X @ pi )
=> ( ( ord_less_eq @ real @ ( zero_zero @ real ) @ Y )
=> ( ( ord_less_eq @ real @ Y @ pi )
=> ( ( ord_less @ real @ ( cos @ real @ X ) @ ( cos @ real @ Y ) )
= ( ord_less @ real @ Y @ X ) ) ) ) ) ) ).
% cos_mono_less_eq
thf(fact_3993_cos__monotone__0__pi,axiom,
! [Y: real,X: real] :
( ( ord_less_eq @ real @ ( zero_zero @ real ) @ Y )
=> ( ( ord_less @ real @ Y @ X )
=> ( ( ord_less_eq @ real @ X @ pi )
=> ( ord_less @ real @ ( cos @ real @ X ) @ ( cos @ real @ Y ) ) ) ) ) ).
% cos_monotone_0_pi
thf(fact_3994_cos__monotone__minus__pi__0_H,axiom,
! [Y: real,X: real] :
( ( ord_less_eq @ real @ ( uminus_uminus @ real @ pi ) @ Y )
=> ( ( ord_less_eq @ real @ Y @ X )
=> ( ( ord_less_eq @ real @ X @ ( zero_zero @ real ) )
=> ( ord_less_eq @ real @ ( cos @ real @ Y ) @ ( cos @ real @ X ) ) ) ) ) ).
% cos_monotone_minus_pi_0'
thf(fact_3995_cos__converges,axiom,
! [A: $tType] :
( ( ( real_Vector_banach @ A )
& ( real_V2822296259951069270ebra_1 @ A ) )
=> ! [X: A] :
( sums @ A
@ ^ [N2: nat] : ( real_V8093663219630862766scaleR @ A @ ( cos_coeff @ N2 ) @ ( power_power @ A @ X @ N2 ) )
@ ( cos @ A @ X ) ) ) ).
% cos_converges
thf(fact_3996_sin__double,axiom,
! [A: $tType] :
( ( ( real_Vector_banach @ A )
& ( real_V3459762299906320749_field @ A ) )
=> ! [X: A] :
( ( sin @ A @ ( times_times @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ X ) )
= ( times_times @ A @ ( times_times @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ ( sin @ A @ X ) ) @ ( cos @ A @ X ) ) ) ) ).
% sin_double
thf(fact_3997_cos__two__less__zero,axiom,
ord_less @ real @ ( cos @ real @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) @ ( zero_zero @ real ) ).
% cos_two_less_zero
thf(fact_3998_cos__is__zero,axiom,
? [X3: real] :
( ( ord_less_eq @ real @ ( zero_zero @ real ) @ X3 )
& ( ord_less_eq @ real @ X3 @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) )
& ( ( cos @ real @ X3 )
= ( zero_zero @ real ) )
& ! [Y4: real] :
( ( ( ord_less_eq @ real @ ( zero_zero @ real ) @ Y4 )
& ( ord_less_eq @ real @ Y4 @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) )
& ( ( cos @ real @ Y4 )
= ( zero_zero @ real ) ) )
=> ( Y4 = X3 ) ) ) ).
% cos_is_zero
thf(fact_3999_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_4000_cos__monotone__minus__pi__0,axiom,
! [Y: real,X: real] :
( ( ord_less_eq @ real @ ( uminus_uminus @ real @ pi ) @ Y )
=> ( ( ord_less @ real @ Y @ X )
=> ( ( ord_less_eq @ real @ X @ ( zero_zero @ real ) )
=> ( ord_less @ real @ ( cos @ real @ Y ) @ ( cos @ real @ X ) ) ) ) ) ).
% cos_monotone_minus_pi_0
thf(fact_4001_cos__total,axiom,
! [Y: real] :
( ( ord_less_eq @ real @ ( uminus_uminus @ real @ ( one_one @ real ) ) @ Y )
=> ( ( ord_less_eq @ real @ Y @ ( one_one @ real ) )
=> ? [X3: real] :
( ( ord_less_eq @ real @ ( zero_zero @ real ) @ X3 )
& ( ord_less_eq @ real @ X3 @ pi )
& ( ( cos @ real @ X3 )
= Y )
& ! [Y4: real] :
( ( ( ord_less_eq @ real @ ( zero_zero @ real ) @ Y4 )
& ( ord_less_eq @ real @ Y4 @ pi )
& ( ( cos @ real @ Y4 )
= Y ) )
=> ( Y4 = X3 ) ) ) ) ) ).
% cos_total
thf(fact_4002_sincos__principal__value,axiom,
! [X: real] :
? [Y3: real] :
( ( ord_less @ real @ ( uminus_uminus @ real @ pi ) @ Y3 )
& ( ord_less_eq @ real @ Y3 @ pi )
& ( ( sin @ real @ Y3 )
= ( sin @ real @ X ) )
& ( ( cos @ real @ Y3 )
= ( cos @ real @ X ) ) ) ).
% sincos_principal_value
thf(fact_4003_cos__minus__converges,axiom,
! [A: $tType] :
( ( ( real_Vector_banach @ A )
& ( real_V2822296259951069270ebra_1 @ A ) )
=> ! [X: A] :
( sums @ A
@ ^ [N2: nat] : ( real_V8093663219630862766scaleR @ A @ ( cos_coeff @ N2 ) @ ( power_power @ A @ ( uminus_uminus @ A @ X ) @ N2 ) )
@ ( cos @ A @ X ) ) ) ).
% cos_minus_converges
thf(fact_4004_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_4005_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_4006_sin__squared__eq,axiom,
! [A: $tType] :
( ( ( real_Vector_banach @ A )
& ( real_V3459762299906320749_field @ A ) )
=> ! [X: A] :
( ( power_power @ A @ ( sin @ A @ X ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) )
= ( minus_minus @ A @ ( one_one @ A ) @ ( power_power @ A @ ( cos @ A @ X ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) ).
% sin_squared_eq
thf(fact_4007_cos__squared__eq,axiom,
! [A: $tType] :
( ( ( real_Vector_banach @ A )
& ( real_V3459762299906320749_field @ A ) )
=> ! [X: A] :
( ( power_power @ A @ ( cos @ A @ X ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) )
= ( minus_minus @ A @ ( one_one @ A ) @ ( power_power @ A @ ( sin @ A @ X ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) ).
% cos_squared_eq
thf(fact_4008_cos__double__less__one,axiom,
! [X: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ X )
=> ( ( ord_less @ real @ X @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) )
=> ( ord_less @ real @ ( cos @ real @ ( times_times @ real @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) @ X ) ) @ ( one_one @ real ) ) ) ) ).
% cos_double_less_one
thf(fact_4009_cos__gt__zero,axiom,
! [X: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ X )
=> ( ( ord_less @ real @ X @ ( divide_divide @ real @ pi @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) )
=> ( ord_less @ real @ ( zero_zero @ real ) @ ( cos @ real @ X ) ) ) ) ).
% cos_gt_zero
thf(fact_4010_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_4011_cos__one__2pi__int,axiom,
! [X: real] :
( ( ( cos @ real @ X )
= ( one_one @ real ) )
= ( ? [X2: int] :
( X
= ( times_times @ real @ ( times_times @ real @ ( ring_1_of_int @ real @ X2 ) @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) @ pi ) ) ) ) ).
% cos_one_2pi_int
thf(fact_4012_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_4013_cos__treble__cos,axiom,
! [A: $tType] :
( ( ( real_Vector_banach @ A )
& ( real_V3459762299906320749_field @ A ) )
=> ! [X: A] :
( ( cos @ A @ ( times_times @ A @ ( numeral_numeral @ A @ ( bit1 @ one2 ) ) @ X ) )
= ( minus_minus @ A @ ( times_times @ A @ ( numeral_numeral @ A @ ( bit0 @ ( bit0 @ one2 ) ) ) @ ( power_power @ A @ ( cos @ A @ X ) @ ( numeral_numeral @ nat @ ( bit1 @ one2 ) ) ) ) @ ( times_times @ A @ ( numeral_numeral @ A @ ( bit1 @ one2 ) ) @ ( cos @ A @ X ) ) ) ) ) ).
% cos_treble_cos
thf(fact_4014_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_4015_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_4016_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_4017_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_4018_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_4019_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_4020_cos__double,axiom,
! [A: $tType] :
( ( ( real_Vector_banach @ A )
& ( real_V3459762299906320749_field @ A ) )
=> ! [X: A] :
( ( cos @ A @ ( times_times @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ X ) )
= ( minus_minus @ A @ ( power_power @ A @ ( cos @ A @ X ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ ( power_power @ A @ ( sin @ A @ X ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) ).
% cos_double
thf(fact_4021_cos__gt__zero__pi,axiom,
! [X: real] :
( ( ord_less @ real @ ( uminus_uminus @ real @ ( divide_divide @ real @ pi @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) ) @ X )
=> ( ( ord_less @ real @ X @ ( divide_divide @ real @ pi @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) )
=> ( ord_less @ real @ ( zero_zero @ real ) @ ( cos @ real @ X ) ) ) ) ).
% cos_gt_zero_pi
thf(fact_4022_cos__ge__zero,axiom,
! [X: real] :
( ( ord_less_eq @ real @ ( uminus_uminus @ real @ ( divide_divide @ real @ pi @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) ) @ X )
=> ( ( ord_less_eq @ real @ X @ ( divide_divide @ real @ pi @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) )
=> ( ord_less_eq @ real @ ( zero_zero @ real ) @ ( cos @ real @ X ) ) ) ) ).
% cos_ge_zero
thf(fact_4023_cos__one__2pi,axiom,
! [X: real] :
( ( ( cos @ real @ X )
= ( one_one @ real ) )
= ( ? [X2: nat] :
( X
= ( times_times @ real @ ( times_times @ real @ ( semiring_1_of_nat @ real @ X2 ) @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) @ pi ) )
| ? [X2: nat] :
( X
= ( uminus_uminus @ real @ ( times_times @ real @ ( times_times @ real @ ( semiring_1_of_nat @ real @ X2 ) @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) @ pi ) ) ) ) ) ).
% cos_one_2pi
thf(fact_4024_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_4025_sincos__total__pi,axiom,
! [Y: real,X: real] :
( ( ord_less_eq @ real @ ( zero_zero @ real ) @ Y )
=> ( ( ( plus_plus @ real @ ( power_power @ real @ X @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ ( power_power @ real @ Y @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) )
= ( one_one @ real ) )
=> ? [T8: real] :
( ( ord_less_eq @ real @ ( zero_zero @ real ) @ T8 )
& ( ord_less_eq @ real @ T8 @ pi )
& ( X
= ( cos @ real @ T8 ) )
& ( Y
= ( sin @ real @ T8 ) ) ) ) ) ).
% sincos_total_pi
thf(fact_4026_sin__expansion__lemma,axiom,
! [X: real,M: nat] :
( ( sin @ real @ ( plus_plus @ real @ X @ ( divide_divide @ real @ ( times_times @ real @ ( semiring_1_of_nat @ real @ ( suc @ M ) ) @ pi ) @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) ) )
= ( cos @ real @ ( plus_plus @ real @ X @ ( divide_divide @ real @ ( times_times @ real @ ( semiring_1_of_nat @ real @ M ) @ pi ) @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) ) ) ) ).
% sin_expansion_lemma
thf(fact_4027_cos__zero__iff__int,axiom,
! [X: real] :
( ( ( cos @ real @ X )
= ( zero_zero @ real ) )
= ( ? [I3: int] :
( ~ ( dvd_dvd @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ I3 )
& ( X
= ( times_times @ real @ ( ring_1_of_int @ real @ I3 ) @ ( divide_divide @ real @ pi @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) ) ) ) ) ) ).
% cos_zero_iff_int
thf(fact_4028_cos__zero__lemma,axiom,
! [X: real] :
( ( ord_less_eq @ real @ ( zero_zero @ real ) @ X )
=> ( ( ( cos @ real @ X )
= ( zero_zero @ real ) )
=> ? [N: nat] :
( ~ ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N )
& ( X
= ( times_times @ real @ ( semiring_1_of_nat @ real @ N ) @ ( divide_divide @ real @ pi @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) ) ) ) ) ) ).
% cos_zero_lemma
thf(fact_4029_cos__zero__iff,axiom,
! [X: real] :
( ( ( cos @ real @ X )
= ( zero_zero @ real ) )
= ( ? [N2: nat] :
( ~ ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N2 )
& ( X
= ( times_times @ real @ ( semiring_1_of_nat @ real @ N2 ) @ ( divide_divide @ real @ pi @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) ) ) )
| ? [N2: nat] :
( ~ ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N2 )
& ( X
= ( uminus_uminus @ real @ ( times_times @ real @ ( semiring_1_of_nat @ real @ N2 ) @ ( divide_divide @ real @ pi @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) ) ) ) ) ) ) ).
% cos_zero_iff
thf(fact_4030_cos__expansion__lemma,axiom,
! [X: real,M: nat] :
( ( cos @ real @ ( plus_plus @ real @ X @ ( divide_divide @ real @ ( times_times @ real @ ( semiring_1_of_nat @ real @ ( suc @ M ) ) @ pi ) @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) ) )
= ( uminus_uminus @ real @ ( sin @ real @ ( plus_plus @ real @ X @ ( divide_divide @ real @ ( times_times @ real @ ( semiring_1_of_nat @ real @ M ) @ pi ) @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) ) ) ) ) ).
% cos_expansion_lemma
thf(fact_4031_cos__paired,axiom,
! [X: real] :
( sums @ real
@ ^ [N2: nat] : ( times_times @ real @ ( divide_divide @ real @ ( power_power @ real @ ( uminus_uminus @ real @ ( one_one @ real ) ) @ N2 ) @ ( semiring_char_0_fact @ real @ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N2 ) ) ) @ ( power_power @ real @ X @ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N2 ) ) )
@ ( cos @ real @ X ) ) ).
% cos_paired
thf(fact_4032_sincos__total__pi__half,axiom,
! [X: real,Y: real] :
( ( ord_less_eq @ real @ ( zero_zero @ real ) @ X )
=> ( ( ord_less_eq @ real @ ( zero_zero @ real ) @ Y )
=> ( ( ( plus_plus @ real @ ( power_power @ real @ X @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ ( power_power @ real @ Y @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) )
= ( one_one @ real ) )
=> ? [T8: real] :
( ( ord_less_eq @ real @ ( zero_zero @ real ) @ T8 )
& ( ord_less_eq @ real @ T8 @ ( divide_divide @ real @ pi @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) )
& ( X
= ( cos @ real @ T8 ) )
& ( Y
= ( sin @ real @ T8 ) ) ) ) ) ) ).
% sincos_total_pi_half
thf(fact_4033_sincos__total__2pi__le,axiom,
! [X: real,Y: real] :
( ( ( plus_plus @ real @ ( power_power @ real @ X @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ ( power_power @ real @ Y @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) )
= ( one_one @ real ) )
=> ? [T8: real] :
( ( ord_less_eq @ real @ ( zero_zero @ real ) @ T8 )
& ( ord_less_eq @ real @ T8 @ ( times_times @ real @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) @ pi ) )
& ( X
= ( cos @ real @ T8 ) )
& ( Y
= ( sin @ real @ T8 ) ) ) ) ).
% sincos_total_2pi_le
thf(fact_4034_sincos__total__2pi,axiom,
! [X: real,Y: real] :
( ( ( plus_plus @ real @ ( power_power @ real @ X @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ ( power_power @ real @ Y @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) )
= ( one_one @ real ) )
=> ~ ! [T8: real] :
( ( ord_less_eq @ real @ ( zero_zero @ real ) @ T8 )
=> ( ( ord_less @ real @ T8 @ ( times_times @ real @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) @ pi ) )
=> ( ( X
= ( cos @ real @ T8 ) )
=> ( Y
!= ( sin @ real @ T8 ) ) ) ) ) ) ).
% sincos_total_2pi
thf(fact_4035_Maclaurin__cos__expansion,axiom,
! [X: real,N3: nat] :
? [T8: real] :
( ( ord_less_eq @ real @ ( abs_abs @ real @ T8 ) @ ( abs_abs @ real @ X ) )
& ( ( cos @ real @ X )
= ( plus_plus @ real
@ ( groups7311177749621191930dd_sum @ nat @ real
@ ^ [M5: nat] : ( times_times @ real @ ( cos_coeff @ M5 ) @ ( power_power @ real @ X @ M5 ) )
@ ( set_ord_lessThan @ nat @ N3 ) )
@ ( 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 @ N3 ) ) @ pi ) ) ) @ ( semiring_char_0_fact @ real @ N3 ) ) @ ( power_power @ real @ X @ N3 ) ) ) ) ) ).
% Maclaurin_cos_expansion
thf(fact_4036_Maclaurin__sin__expansion2,axiom,
! [X: real,N3: nat] :
? [T8: real] :
( ( ord_less_eq @ real @ ( abs_abs @ real @ T8 ) @ ( abs_abs @ real @ X ) )
& ( ( sin @ real @ X )
= ( plus_plus @ real
@ ( groups7311177749621191930dd_sum @ nat @ real
@ ^ [M5: nat] : ( times_times @ real @ ( sin_coeff @ M5 ) @ ( power_power @ real @ X @ M5 ) )
@ ( set_ord_lessThan @ nat @ N3 ) )
@ ( 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 @ N3 ) ) @ pi ) ) ) @ ( semiring_char_0_fact @ real @ N3 ) ) @ ( power_power @ real @ X @ N3 ) ) ) ) ) ).
% Maclaurin_sin_expansion2
thf(fact_4037_signed__take__bit__numeral__minus__bit1,axiom,
! [L2: num,K: num] :
( ( bit_ri4674362597316999326ke_bit @ int @ ( numeral_numeral @ nat @ L2 ) @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ ( bit1 @ K ) ) ) )
= ( plus_plus @ int @ ( times_times @ int @ ( bit_ri4674362597316999326ke_bit @ int @ ( pred_numeral @ L2 ) @ ( 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_4038_pi__series,axiom,
( ( divide_divide @ real @ pi @ ( numeral_numeral @ real @ ( bit0 @ ( bit0 @ one2 ) ) ) )
= ( suminf @ real
@ ^ [K2: nat] : ( divide_divide @ real @ ( times_times @ real @ ( power_power @ real @ ( uminus_uminus @ real @ ( one_one @ real ) ) @ K2 ) @ ( one_one @ real ) ) @ ( semiring_1_of_nat @ real @ ( plus_plus @ nat @ ( times_times @ nat @ K2 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ ( one_one @ nat ) ) ) ) ) ) ).
% pi_series
thf(fact_4039_abs__abs,axiom,
! [A: $tType] :
( ( idom_abs_sgn @ A )
=> ! [A2: A] :
( ( abs_abs @ A @ ( abs_abs @ A @ A2 ) )
= ( abs_abs @ A @ A2 ) ) ) ).
% abs_abs
thf(fact_4040_abs__0,axiom,
! [A: $tType] :
( ( idom_abs_sgn @ A )
=> ( ( abs_abs @ A @ ( zero_zero @ A ) )
= ( zero_zero @ A ) ) ) ).
% abs_0
thf(fact_4041_abs__numeral,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [N3: num] :
( ( abs_abs @ A @ ( numeral_numeral @ A @ N3 ) )
= ( numeral_numeral @ A @ N3 ) ) ) ).
% abs_numeral
thf(fact_4042_abs__add__abs,axiom,
! [A: $tType] :
( ( ordere166539214618696060dd_abs @ A )
=> ! [A2: A,B2: A] :
( ( abs_abs @ A @ ( plus_plus @ A @ ( abs_abs @ A @ A2 ) @ ( abs_abs @ A @ B2 ) ) )
= ( plus_plus @ A @ ( abs_abs @ A @ A2 ) @ ( abs_abs @ A @ B2 ) ) ) ) ).
% abs_add_abs
thf(fact_4043_abs__1,axiom,
! [A: $tType] :
( ( idom_abs_sgn @ A )
=> ( ( abs_abs @ A @ ( one_one @ A ) )
= ( one_one @ A ) ) ) ).
% abs_1
thf(fact_4044_abs__mult__self__eq,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [A2: A] :
( ( times_times @ A @ ( abs_abs @ A @ A2 ) @ ( abs_abs @ A @ A2 ) )
= ( times_times @ A @ A2 @ A2 ) ) ) ).
% abs_mult_self_eq
thf(fact_4045_abs__divide,axiom,
! [A: $tType] :
( ( field_abs_sgn @ A )
=> ! [A2: A,B2: A] :
( ( abs_abs @ A @ ( divide_divide @ A @ A2 @ B2 ) )
= ( divide_divide @ A @ ( abs_abs @ A @ A2 ) @ ( abs_abs @ A @ B2 ) ) ) ) ).
% abs_divide
thf(fact_4046_abs__minus,axiom,
! [A: $tType] :
( ( idom_abs_sgn @ A )
=> ! [A2: A] :
( ( abs_abs @ A @ ( uminus_uminus @ A @ A2 ) )
= ( abs_abs @ A @ A2 ) ) ) ).
% abs_minus
thf(fact_4047_dvd__abs__iff,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [M: A,K: A] :
( ( dvd_dvd @ A @ M @ ( abs_abs @ A @ K ) )
= ( dvd_dvd @ A @ M @ K ) ) ) ).
% dvd_abs_iff
thf(fact_4048_abs__dvd__iff,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [M: A,K: A] :
( ( dvd_dvd @ A @ ( abs_abs @ A @ M ) @ K )
= ( dvd_dvd @ A @ M @ K ) ) ) ).
% abs_dvd_iff
thf(fact_4049_abs__of__nat,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [N3: nat] :
( ( abs_abs @ A @ ( semiring_1_of_nat @ A @ N3 ) )
= ( semiring_1_of_nat @ A @ N3 ) ) ) ).
% abs_of_nat
thf(fact_4050_abs__of__nonneg,axiom,
! [A: $tType] :
( ( ordere166539214618696060dd_abs @ A )
=> ! [A2: A] :
( ( ord_less_eq @ A @ ( zero_zero @ A ) @ A2 )
=> ( ( abs_abs @ A @ A2 )
= A2 ) ) ) ).
% abs_of_nonneg
thf(fact_4051_abs__le__self__iff,axiom,
! [A: $tType] :
( ( ordere166539214618696060dd_abs @ A )
=> ! [A2: A] :
( ( ord_less_eq @ A @ ( abs_abs @ A @ A2 ) @ A2 )
= ( ord_less_eq @ A @ ( zero_zero @ A ) @ A2 ) ) ) ).
% abs_le_self_iff
thf(fact_4052_abs__le__zero__iff,axiom,
! [A: $tType] :
( ( ordere166539214618696060dd_abs @ A )
=> ! [A2: A] :
( ( ord_less_eq @ A @ ( abs_abs @ A @ A2 ) @ ( zero_zero @ A ) )
= ( A2
= ( zero_zero @ A ) ) ) ) ).
% abs_le_zero_iff
thf(fact_4053_zero__less__abs__iff,axiom,
! [A: $tType] :
( ( ordere166539214618696060dd_abs @ A )
=> ! [A2: A] :
( ( ord_less @ A @ ( zero_zero @ A ) @ ( abs_abs @ A @ A2 ) )
= ( A2
!= ( zero_zero @ A ) ) ) ) ).
% zero_less_abs_iff
thf(fact_4054_abs__neg__numeral,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [N3: num] :
( ( abs_abs @ A @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ N3 ) ) )
= ( numeral_numeral @ A @ N3 ) ) ) ).
% abs_neg_numeral
thf(fact_4055_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_4056_abs__power__minus,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [A2: A,N3: nat] :
( ( abs_abs @ A @ ( power_power @ A @ ( uminus_uminus @ A @ A2 ) @ N3 ) )
= ( abs_abs @ A @ ( power_power @ A @ A2 @ N3 ) ) ) ) ).
% abs_power_minus
thf(fact_4057_pred__numeral__simps_I1_J,axiom,
( ( pred_numeral @ one2 )
= ( zero_zero @ nat ) ) ).
% pred_numeral_simps(1)
thf(fact_4058_Suc__eq__numeral,axiom,
! [N3: nat,K: num] :
( ( ( suc @ N3 )
= ( numeral_numeral @ nat @ K ) )
= ( N3
= ( pred_numeral @ K ) ) ) ).
% Suc_eq_numeral
thf(fact_4059_eq__numeral__Suc,axiom,
! [K: num,N3: nat] :
( ( ( numeral_numeral @ nat @ K )
= ( suc @ N3 ) )
= ( ( pred_numeral @ K )
= N3 ) ) ).
% eq_numeral_Suc
thf(fact_4060_sum__abs,axiom,
! [B: $tType,A: $tType] :
( ( ordere166539214618696060dd_abs @ B )
=> ! [F2: A > B,A3: set @ A] :
( ord_less_eq @ B @ ( abs_abs @ B @ ( groups7311177749621191930dd_sum @ A @ B @ F2 @ A3 ) )
@ ( groups7311177749621191930dd_sum @ A @ B
@ ^ [I3: A] : ( abs_abs @ B @ ( F2 @ I3 ) )
@ A3 ) ) ) ).
% sum_abs
thf(fact_4061_zero__le__divide__abs__iff,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [A2: A,B2: A] :
( ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( divide_divide @ A @ A2 @ ( abs_abs @ A @ B2 ) ) )
= ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ A2 )
| ( B2
= ( zero_zero @ A ) ) ) ) ) ).
% zero_le_divide_abs_iff
thf(fact_4062_divide__le__0__abs__iff,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [A2: A,B2: A] :
( ( ord_less_eq @ A @ ( divide_divide @ A @ A2 @ ( abs_abs @ A @ B2 ) ) @ ( zero_zero @ A ) )
= ( ( ord_less_eq @ A @ A2 @ ( zero_zero @ A ) )
| ( B2
= ( zero_zero @ A ) ) ) ) ) ).
% divide_le_0_abs_iff
thf(fact_4063_abs__of__nonpos,axiom,
! [A: $tType] :
( ( ordere166539214618696060dd_abs @ A )
=> ! [A2: A] :
( ( ord_less_eq @ A @ A2 @ ( zero_zero @ A ) )
=> ( ( abs_abs @ A @ A2 )
= ( uminus_uminus @ A @ A2 ) ) ) ) ).
% abs_of_nonpos
thf(fact_4064_pred__numeral__simps_I3_J,axiom,
! [K: num] :
( ( pred_numeral @ ( bit1 @ K ) )
= ( numeral_numeral @ nat @ ( bit0 @ K ) ) ) ).
% pred_numeral_simps(3)
thf(fact_4065_less__Suc__numeral,axiom,
! [N3: nat,K: num] :
( ( ord_less @ nat @ ( suc @ N3 ) @ ( numeral_numeral @ nat @ K ) )
= ( ord_less @ nat @ N3 @ ( pred_numeral @ K ) ) ) ).
% less_Suc_numeral
thf(fact_4066_less__numeral__Suc,axiom,
! [K: num,N3: nat] :
( ( ord_less @ nat @ ( numeral_numeral @ nat @ K ) @ ( suc @ N3 ) )
= ( ord_less @ nat @ ( pred_numeral @ K ) @ N3 ) ) ).
% less_numeral_Suc
thf(fact_4067_le__Suc__numeral,axiom,
! [N3: nat,K: num] :
( ( ord_less_eq @ nat @ ( suc @ N3 ) @ ( numeral_numeral @ nat @ K ) )
= ( ord_less_eq @ nat @ N3 @ ( pred_numeral @ K ) ) ) ).
% le_Suc_numeral
thf(fact_4068_le__numeral__Suc,axiom,
! [K: num,N3: nat] :
( ( ord_less_eq @ nat @ ( numeral_numeral @ nat @ K ) @ ( suc @ N3 ) )
= ( ord_less_eq @ nat @ ( pred_numeral @ K ) @ N3 ) ) ).
% le_numeral_Suc
thf(fact_4069_diff__Suc__numeral,axiom,
! [N3: nat,K: num] :
( ( minus_minus @ nat @ ( suc @ N3 ) @ ( numeral_numeral @ nat @ K ) )
= ( minus_minus @ nat @ N3 @ ( pred_numeral @ K ) ) ) ).
% diff_Suc_numeral
thf(fact_4070_diff__numeral__Suc,axiom,
! [K: num,N3: nat] :
( ( minus_minus @ nat @ ( numeral_numeral @ nat @ K ) @ ( suc @ N3 ) )
= ( minus_minus @ nat @ ( pred_numeral @ K ) @ N3 ) ) ).
% diff_numeral_Suc
thf(fact_4071_norm__scaleR,axiom,
! [A: $tType] :
( ( real_V822414075346904944vector @ A )
=> ! [A2: real,X: A] :
( ( real_V7770717601297561774m_norm @ A @ ( real_V8093663219630862766scaleR @ A @ A2 @ X ) )
= ( times_times @ real @ ( abs_abs @ real @ A2 ) @ ( real_V7770717601297561774m_norm @ A @ X ) ) ) ) ).
% norm_scaleR
thf(fact_4072_max__Suc__numeral,axiom,
! [N3: nat,K: num] :
( ( ord_max @ nat @ ( suc @ N3 ) @ ( numeral_numeral @ nat @ K ) )
= ( suc @ ( ord_max @ nat @ N3 @ ( pred_numeral @ K ) ) ) ) ).
% max_Suc_numeral
thf(fact_4073_max__numeral__Suc,axiom,
! [K: num,N3: nat] :
( ( ord_max @ nat @ ( numeral_numeral @ nat @ K ) @ ( suc @ N3 ) )
= ( suc @ ( ord_max @ nat @ ( pred_numeral @ K ) @ N3 ) ) ) ).
% max_numeral_Suc
thf(fact_4074_sum__abs__ge__zero,axiom,
! [B: $tType,A: $tType] :
( ( ordere166539214618696060dd_abs @ B )
=> ! [F2: A > B,A3: set @ A] :
( ord_less_eq @ B @ ( zero_zero @ B )
@ ( groups7311177749621191930dd_sum @ A @ B
@ ^ [I3: A] : ( abs_abs @ B @ ( F2 @ I3 ) )
@ A3 ) ) ) ).
% sum_abs_ge_zero
thf(fact_4075_zero__less__power__abs__iff,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [A2: A,N3: nat] :
( ( ord_less @ A @ ( zero_zero @ A ) @ ( power_power @ A @ ( abs_abs @ A @ A2 ) @ N3 ) )
= ( ( A2
!= ( zero_zero @ A ) )
| ( N3
= ( zero_zero @ nat ) ) ) ) ) ).
% zero_less_power_abs_iff
thf(fact_4076_abs__power2,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [A2: A] :
( ( abs_abs @ A @ ( power_power @ A @ A2 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) )
= ( power_power @ A @ A2 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ).
% abs_power2
thf(fact_4077_power2__abs,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [A2: A] :
( ( power_power @ A @ ( abs_abs @ A @ A2 ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) )
= ( power_power @ A @ A2 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ).
% power2_abs
thf(fact_4078_powser__zero,axiom,
! [A: $tType] :
( ( real_V2822296259951069270ebra_1 @ A )
=> ! [F2: nat > A] :
( ( suminf @ A
@ ^ [N2: nat] : ( times_times @ A @ ( F2 @ N2 ) @ ( power_power @ A @ ( zero_zero @ A ) @ N2 ) ) )
= ( F2 @ ( zero_zero @ nat ) ) ) ) ).
% powser_zero
thf(fact_4079_power__even__abs__numeral,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [W: num,A2: A] :
( ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( numeral_numeral @ nat @ W ) )
=> ( ( power_power @ A @ ( abs_abs @ A @ A2 ) @ ( numeral_numeral @ nat @ W ) )
= ( power_power @ A @ A2 @ ( numeral_numeral @ nat @ W ) ) ) ) ) ).
% power_even_abs_numeral
thf(fact_4080_signed__take__bit__numeral__bit0,axiom,
! [L2: num,K: num] :
( ( bit_ri4674362597316999326ke_bit @ int @ ( numeral_numeral @ nat @ L2 ) @ ( numeral_numeral @ int @ ( bit0 @ K ) ) )
= ( times_times @ int @ ( bit_ri4674362597316999326ke_bit @ int @ ( pred_numeral @ L2 ) @ ( numeral_numeral @ int @ K ) ) @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) ) ) ).
% signed_take_bit_numeral_bit0
thf(fact_4081_signed__take__bit__numeral__minus__bit0,axiom,
! [L2: num,K: num] :
( ( bit_ri4674362597316999326ke_bit @ int @ ( numeral_numeral @ nat @ L2 ) @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ ( bit0 @ K ) ) ) )
= ( times_times @ int @ ( bit_ri4674362597316999326ke_bit @ int @ ( pred_numeral @ L2 ) @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ K ) ) ) @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) ) ) ).
% signed_take_bit_numeral_minus_bit0
thf(fact_4082_square__powr__half,axiom,
! [X: real] :
( ( powr @ real @ ( power_power @ real @ X @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ ( divide_divide @ real @ ( one_one @ real ) @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) )
= ( abs_abs @ real @ X ) ) ).
% square_powr_half
thf(fact_4083_signed__take__bit__numeral__bit1,axiom,
! [L2: num,K: num] :
( ( bit_ri4674362597316999326ke_bit @ int @ ( numeral_numeral @ nat @ L2 ) @ ( numeral_numeral @ int @ ( bit1 @ K ) ) )
= ( plus_plus @ int @ ( times_times @ int @ ( bit_ri4674362597316999326ke_bit @ int @ ( pred_numeral @ L2 ) @ ( numeral_numeral @ int @ K ) ) @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) ) @ ( one_one @ int ) ) ) ).
% signed_take_bit_numeral_bit1
thf(fact_4084_abs__eq__0__iff,axiom,
! [A: $tType] :
( ( idom_abs_sgn @ A )
=> ! [A2: A] :
( ( ( abs_abs @ A @ A2 )
= ( zero_zero @ A ) )
= ( A2
= ( zero_zero @ A ) ) ) ) ).
% abs_eq_0_iff
thf(fact_4085_abs__one,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ( ( abs_abs @ A @ ( one_one @ A ) )
= ( one_one @ A ) ) ) ).
% abs_one
thf(fact_4086_power__abs,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [A2: A,N3: nat] :
( ( abs_abs @ A @ ( power_power @ A @ A2 @ N3 ) )
= ( power_power @ A @ ( abs_abs @ A @ A2 ) @ N3 ) ) ) ).
% power_abs
thf(fact_4087_abs__mult,axiom,
! [A: $tType] :
( ( idom_abs_sgn @ A )
=> ! [A2: A,B2: A] :
( ( abs_abs @ A @ ( times_times @ A @ A2 @ B2 ) )
= ( times_times @ A @ ( abs_abs @ A @ A2 ) @ ( abs_abs @ A @ B2 ) ) ) ) ).
% abs_mult
thf(fact_4088_abs__minus__commute,axiom,
! [A: $tType] :
( ( ordere166539214618696060dd_abs @ A )
=> ! [A2: A,B2: A] :
( ( abs_abs @ A @ ( minus_minus @ A @ A2 @ B2 ) )
= ( abs_abs @ A @ ( minus_minus @ A @ B2 @ A2 ) ) ) ) ).
% abs_minus_commute
thf(fact_4089_dvd__if__abs__eq,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [L2: A,K: A] :
( ( ( abs_abs @ A @ L2 )
= ( abs_abs @ A @ K ) )
=> ( dvd_dvd @ A @ L2 @ K ) ) ) ).
% dvd_if_abs_eq
thf(fact_4090_abs__ge__self,axiom,
! [A: $tType] :
( ( ordere166539214618696060dd_abs @ A )
=> ! [A2: A] : ( ord_less_eq @ A @ A2 @ ( abs_abs @ A @ A2 ) ) ) ).
% abs_ge_self
thf(fact_4091_abs__le__D1,axiom,
! [A: $tType] :
( ( ordere166539214618696060dd_abs @ A )
=> ! [A2: A,B2: A] :
( ( ord_less_eq @ A @ ( abs_abs @ A @ A2 ) @ B2 )
=> ( ord_less_eq @ A @ A2 @ B2 ) ) ) ).
% abs_le_D1
thf(fact_4092_abs__eq__iff,axiom,
! [A: $tType] :
( ( linordered_ring @ A )
=> ! [X: A,Y: A] :
( ( ( abs_abs @ A @ X )
= ( abs_abs @ A @ Y ) )
= ( ( X = Y )
| ( X
= ( uminus_uminus @ A @ Y ) ) ) ) ) ).
% abs_eq_iff
thf(fact_4093_abs__ge__zero,axiom,
! [A: $tType] :
( ( ordere166539214618696060dd_abs @ A )
=> ! [A2: A] : ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( abs_abs @ A @ A2 ) ) ) ).
% abs_ge_zero
thf(fact_4094_abs__of__pos,axiom,
! [A: $tType] :
( ( ordere166539214618696060dd_abs @ A )
=> ! [A2: A] :
( ( ord_less @ A @ ( zero_zero @ A ) @ A2 )
=> ( ( abs_abs @ A @ A2 )
= A2 ) ) ) ).
% abs_of_pos
thf(fact_4095_abs__not__less__zero,axiom,
! [A: $tType] :
( ( ordere166539214618696060dd_abs @ A )
=> ! [A2: A] :
~ ( ord_less @ A @ ( abs_abs @ A @ A2 ) @ ( zero_zero @ A ) ) ) ).
% abs_not_less_zero
thf(fact_4096_abs__triangle__ineq,axiom,
! [A: $tType] :
( ( ordere166539214618696060dd_abs @ A )
=> ! [A2: A,B2: A] : ( ord_less_eq @ A @ ( abs_abs @ A @ ( plus_plus @ A @ A2 @ B2 ) ) @ ( plus_plus @ A @ ( abs_abs @ A @ A2 ) @ ( abs_abs @ A @ B2 ) ) ) ) ).
% abs_triangle_ineq
thf(fact_4097_abs__mult__less,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [A2: A,C2: A,B2: A,D: A] :
( ( ord_less @ A @ ( abs_abs @ A @ A2 ) @ C2 )
=> ( ( ord_less @ A @ ( abs_abs @ A @ B2 ) @ D )
=> ( ord_less @ A @ ( times_times @ A @ ( abs_abs @ A @ A2 ) @ ( abs_abs @ A @ B2 ) ) @ ( times_times @ A @ C2 @ D ) ) ) ) ) ).
% abs_mult_less
thf(fact_4098_abs__triangle__ineq2__sym,axiom,
! [A: $tType] :
( ( ordere166539214618696060dd_abs @ A )
=> ! [A2: A,B2: A] : ( ord_less_eq @ A @ ( minus_minus @ A @ ( abs_abs @ A @ A2 ) @ ( abs_abs @ A @ B2 ) ) @ ( abs_abs @ A @ ( minus_minus @ A @ B2 @ A2 ) ) ) ) ).
% abs_triangle_ineq2_sym
thf(fact_4099_abs__triangle__ineq3,axiom,
! [A: $tType] :
( ( ordere166539214618696060dd_abs @ A )
=> ! [A2: A,B2: A] : ( ord_less_eq @ A @ ( abs_abs @ A @ ( minus_minus @ A @ ( abs_abs @ A @ A2 ) @ ( abs_abs @ A @ B2 ) ) ) @ ( abs_abs @ A @ ( minus_minus @ A @ A2 @ B2 ) ) ) ) ).
% abs_triangle_ineq3
thf(fact_4100_abs__triangle__ineq2,axiom,
! [A: $tType] :
( ( ordere166539214618696060dd_abs @ A )
=> ! [A2: A,B2: A] : ( ord_less_eq @ A @ ( minus_minus @ A @ ( abs_abs @ A @ A2 ) @ ( abs_abs @ A @ B2 ) ) @ ( abs_abs @ A @ ( minus_minus @ A @ A2 @ B2 ) ) ) ) ).
% abs_triangle_ineq2
thf(fact_4101_nonzero__abs__divide,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [B2: A,A2: A] :
( ( B2
!= ( zero_zero @ A ) )
=> ( ( abs_abs @ A @ ( divide_divide @ A @ A2 @ B2 ) )
= ( divide_divide @ A @ ( abs_abs @ A @ A2 ) @ ( abs_abs @ A @ B2 ) ) ) ) ) ).
% nonzero_abs_divide
thf(fact_4102_abs__ge__minus__self,axiom,
! [A: $tType] :
( ( ordere166539214618696060dd_abs @ A )
=> ! [A2: A] : ( ord_less_eq @ A @ ( uminus_uminus @ A @ A2 ) @ ( abs_abs @ A @ A2 ) ) ) ).
% abs_ge_minus_self
thf(fact_4103_abs__le__iff,axiom,
! [A: $tType] :
( ( ordere166539214618696060dd_abs @ A )
=> ! [A2: A,B2: A] :
( ( ord_less_eq @ A @ ( abs_abs @ A @ A2 ) @ B2 )
= ( ( ord_less_eq @ A @ A2 @ B2 )
& ( ord_less_eq @ A @ ( uminus_uminus @ A @ A2 ) @ B2 ) ) ) ) ).
% abs_le_iff
thf(fact_4104_abs__le__D2,axiom,
! [A: $tType] :
( ( ordere166539214618696060dd_abs @ A )
=> ! [A2: A,B2: A] :
( ( ord_less_eq @ A @ ( abs_abs @ A @ A2 ) @ B2 )
=> ( ord_less_eq @ A @ ( uminus_uminus @ A @ A2 ) @ B2 ) ) ) ).
% abs_le_D2
thf(fact_4105_abs__leI,axiom,
! [A: $tType] :
( ( ordere166539214618696060dd_abs @ A )
=> ! [A2: A,B2: A] :
( ( ord_less_eq @ A @ A2 @ B2 )
=> ( ( ord_less_eq @ A @ ( uminus_uminus @ A @ A2 ) @ B2 )
=> ( ord_less_eq @ A @ ( abs_abs @ A @ A2 ) @ B2 ) ) ) ) ).
% abs_leI
thf(fact_4106_abs__less__iff,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [A2: A,B2: A] :
( ( ord_less @ A @ ( abs_abs @ A @ A2 ) @ B2 )
= ( ( ord_less @ A @ A2 @ B2 )
& ( ord_less @ A @ ( uminus_uminus @ A @ A2 ) @ B2 ) ) ) ) ).
% abs_less_iff
thf(fact_4107_abs__sin__x__le__abs__x,axiom,
! [X: real] : ( ord_less_eq @ real @ ( abs_abs @ real @ ( sin @ real @ X ) ) @ ( abs_abs @ real @ X ) ) ).
% abs_sin_x_le_abs_x
thf(fact_4108_numeral__eq__Suc,axiom,
( ( numeral_numeral @ nat )
= ( ^ [K2: num] : ( suc @ ( pred_numeral @ K2 ) ) ) ) ).
% numeral_eq_Suc
thf(fact_4109_dense__eq0__I,axiom,
! [A: $tType] :
( ( ( ordere166539214618696060dd_abs @ A )
& ( dense_linorder @ A ) )
=> ! [X: A] :
( ! [E2: A] :
( ( ord_less @ A @ ( zero_zero @ A ) @ E2 )
=> ( ord_less_eq @ A @ ( abs_abs @ A @ X ) @ E2 ) )
=> ( X
= ( zero_zero @ A ) ) ) ) ).
% dense_eq0_I
thf(fact_4110_abs__eq__mult,axiom,
! [A: $tType] :
( ( ordered_ring_abs @ A )
=> ! [A2: A,B2: A] :
( ( ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ A2 )
| ( ord_less_eq @ A @ A2 @ ( zero_zero @ A ) ) )
& ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ B2 )
| ( ord_less_eq @ A @ B2 @ ( zero_zero @ A ) ) ) )
=> ( ( abs_abs @ A @ ( times_times @ A @ A2 @ B2 ) )
= ( times_times @ A @ ( abs_abs @ A @ A2 ) @ ( abs_abs @ A @ B2 ) ) ) ) ) ).
% abs_eq_mult
thf(fact_4111_abs__mult__pos,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [X: A,Y: A] :
( ( ord_less_eq @ A @ ( zero_zero @ A ) @ X )
=> ( ( times_times @ A @ ( abs_abs @ A @ Y ) @ X )
= ( abs_abs @ A @ ( times_times @ A @ Y @ X ) ) ) ) ) ).
% abs_mult_pos
thf(fact_4112_abs__div__pos,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [Y: A,X: A] :
( ( ord_less @ A @ ( zero_zero @ A ) @ Y )
=> ( ( divide_divide @ A @ ( abs_abs @ A @ X ) @ Y )
= ( abs_abs @ A @ ( divide_divide @ A @ X @ Y ) ) ) ) ) ).
% abs_div_pos
thf(fact_4113_zero__le__power__abs,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [A2: A,N3: nat] : ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( power_power @ A @ ( abs_abs @ A @ A2 ) @ N3 ) ) ) ).
% zero_le_power_abs
thf(fact_4114_abs__minus__le__zero,axiom,
! [A: $tType] :
( ( ordere166539214618696060dd_abs @ A )
=> ! [A2: A] : ( ord_less_eq @ A @ ( uminus_uminus @ A @ ( abs_abs @ A @ A2 ) ) @ ( zero_zero @ A ) ) ) ).
% abs_minus_le_zero
thf(fact_4115_eq__abs__iff_H,axiom,
! [A: $tType] :
( ( linordered_ring @ A )
=> ! [A2: A,B2: A] :
( ( A2
= ( abs_abs @ A @ B2 ) )
= ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ A2 )
& ( ( B2 = A2 )
| ( B2
= ( uminus_uminus @ A @ A2 ) ) ) ) ) ) ).
% eq_abs_iff'
thf(fact_4116_abs__eq__iff_H,axiom,
! [A: $tType] :
( ( linordered_ring @ A )
=> ! [A2: A,B2: A] :
( ( ( abs_abs @ A @ A2 )
= B2 )
= ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ B2 )
& ( ( A2 = B2 )
| ( A2
= ( uminus_uminus @ A @ B2 ) ) ) ) ) ) ).
% abs_eq_iff'
thf(fact_4117_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_4118_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_4119_abs__of__neg,axiom,
! [A: $tType] :
( ( ordere166539214618696060dd_abs @ A )
=> ! [A2: A] :
( ( ord_less @ A @ A2 @ ( zero_zero @ A ) )
=> ( ( abs_abs @ A @ A2 )
= ( uminus_uminus @ A @ A2 ) ) ) ) ).
% abs_of_neg
thf(fact_4120_abs__diff__le__iff,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [X: A,A2: A,R2: A] :
( ( ord_less_eq @ A @ ( abs_abs @ A @ ( minus_minus @ A @ X @ A2 ) ) @ R2 )
= ( ( ord_less_eq @ A @ ( minus_minus @ A @ A2 @ R2 ) @ X )
& ( ord_less_eq @ A @ X @ ( plus_plus @ A @ A2 @ R2 ) ) ) ) ) ).
% abs_diff_le_iff
thf(fact_4121_abs__diff__triangle__ineq,axiom,
! [A: $tType] :
( ( ordere166539214618696060dd_abs @ A )
=> ! [A2: A,B2: A,C2: A,D: A] : ( ord_less_eq @ A @ ( abs_abs @ A @ ( minus_minus @ A @ ( plus_plus @ A @ A2 @ B2 ) @ ( plus_plus @ A @ C2 @ D ) ) ) @ ( plus_plus @ A @ ( abs_abs @ A @ ( minus_minus @ A @ A2 @ C2 ) ) @ ( abs_abs @ A @ ( minus_minus @ A @ B2 @ D ) ) ) ) ) ).
% abs_diff_triangle_ineq
thf(fact_4122_abs__triangle__ineq4,axiom,
! [A: $tType] :
( ( ordere166539214618696060dd_abs @ A )
=> ! [A2: A,B2: A] : ( ord_less_eq @ A @ ( abs_abs @ A @ ( minus_minus @ A @ A2 @ B2 ) ) @ ( plus_plus @ A @ ( abs_abs @ A @ A2 ) @ ( abs_abs @ A @ B2 ) ) ) ) ).
% abs_triangle_ineq4
thf(fact_4123_abs__diff__less__iff,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [X: A,A2: A,R2: A] :
( ( ord_less @ A @ ( abs_abs @ A @ ( minus_minus @ A @ X @ A2 ) ) @ R2 )
= ( ( ord_less @ A @ ( minus_minus @ A @ A2 @ R2 ) @ X )
& ( ord_less @ A @ X @ ( plus_plus @ A @ A2 @ R2 ) ) ) ) ) ).
% abs_diff_less_iff
thf(fact_4124_abs__real__def,axiom,
( ( abs_abs @ real )
= ( ^ [A6: real] : ( if @ real @ ( ord_less @ real @ A6 @ ( zero_zero @ real ) ) @ ( uminus_uminus @ real @ A6 ) @ A6 ) ) ) ).
% abs_real_def
thf(fact_4125_lemma__interval__lt,axiom,
! [A2: real,X: real,B2: real] :
( ( ord_less @ real @ A2 @ X )
=> ( ( ord_less @ real @ X @ B2 )
=> ? [D3: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ D3 )
& ! [Y4: real] :
( ( ord_less @ real @ ( abs_abs @ real @ ( minus_minus @ real @ X @ Y4 ) ) @ D3 )
=> ( ( ord_less @ real @ A2 @ Y4 )
& ( ord_less @ real @ Y4 @ B2 ) ) ) ) ) ) ).
% lemma_interval_lt
thf(fact_4126_abs__cos__le__one,axiom,
! [X: real] : ( ord_less_eq @ real @ ( abs_abs @ real @ ( cos @ real @ X ) ) @ ( one_one @ real ) ) ).
% abs_cos_le_one
thf(fact_4127_abs__sin__le__one,axiom,
! [X: real] : ( ord_less_eq @ real @ ( abs_abs @ real @ ( sin @ real @ X ) ) @ ( one_one @ real ) ) ).
% abs_sin_le_one
thf(fact_4128_sin__bound__lemma,axiom,
! [X: real,Y: real,U: real,V: real] :
( ( X = Y )
=> ( ( ord_less_eq @ real @ ( abs_abs @ real @ U ) @ V )
=> ( ord_less_eq @ real @ ( abs_abs @ real @ ( minus_minus @ real @ ( plus_plus @ real @ X @ U ) @ Y ) ) @ V ) ) ) ).
% sin_bound_lemma
thf(fact_4129_pred__numeral__def,axiom,
( pred_numeral
= ( ^ [K2: num] : ( minus_minus @ nat @ ( numeral_numeral @ nat @ K2 ) @ ( one_one @ nat ) ) ) ) ).
% pred_numeral_def
thf(fact_4130_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_4131_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_4132_abs__add__one__gt__zero,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [X: A] : ( ord_less @ A @ ( zero_zero @ A ) @ ( plus_plus @ A @ ( one_one @ A ) @ ( abs_abs @ A @ X ) ) ) ) ).
% abs_add_one_gt_zero
thf(fact_4133_of__int__leD,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [N3: int,X: A] :
( ( ord_less_eq @ A @ ( abs_abs @ A @ ( ring_1_of_int @ A @ N3 ) ) @ X )
=> ( ( N3
= ( zero_zero @ int ) )
| ( ord_less_eq @ A @ ( one_one @ A ) @ X ) ) ) ) ).
% of_int_leD
thf(fact_4134_of__int__lessD,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [N3: int,X: A] :
( ( ord_less @ A @ ( abs_abs @ A @ ( ring_1_of_int @ A @ N3 ) ) @ X )
=> ( ( N3
= ( zero_zero @ int ) )
| ( ord_less @ A @ ( one_one @ A ) @ X ) ) ) ) ).
% of_int_lessD
thf(fact_4135_lemma__interval,axiom,
! [A2: real,X: real,B2: real] :
( ( ord_less @ real @ A2 @ X )
=> ( ( ord_less @ real @ X @ B2 )
=> ? [D3: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ D3 )
& ! [Y4: real] :
( ( ord_less @ real @ ( abs_abs @ real @ ( minus_minus @ real @ X @ Y4 ) ) @ D3 )
=> ( ( ord_less_eq @ real @ A2 @ Y4 )
& ( ord_less_eq @ real @ Y4 @ B2 ) ) ) ) ) ) ).
% lemma_interval
thf(fact_4136_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_4137_norm__triangle__ineq3,axiom,
! [A: $tType] :
( ( real_V822414075346904944vector @ A )
=> ! [A2: A,B2: A] : ( ord_less_eq @ real @ ( abs_abs @ real @ ( minus_minus @ real @ ( real_V7770717601297561774m_norm @ A @ A2 ) @ ( real_V7770717601297561774m_norm @ A @ B2 ) ) ) @ ( real_V7770717601297561774m_norm @ A @ ( minus_minus @ A @ A2 @ B2 ) ) ) ) ).
% norm_triangle_ineq3
thf(fact_4138_sin__zero__abs__cos__one,axiom,
! [X: real] :
( ( ( sin @ real @ X )
= ( zero_zero @ real ) )
=> ( ( abs_abs @ real @ ( cos @ real @ X ) )
= ( one_one @ real ) ) ) ).
% sin_zero_abs_cos_one
thf(fact_4139_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_4140_sin__def,axiom,
! [A: $tType] :
( ( ( real_Vector_banach @ A )
& ( real_V2822296259951069270ebra_1 @ A ) )
=> ( ( sin @ A )
= ( ^ [X2: A] :
( suminf @ A
@ ^ [N2: nat] : ( real_V8093663219630862766scaleR @ A @ ( sin_coeff @ N2 ) @ ( power_power @ A @ X2 @ N2 ) ) ) ) ) ) ).
% sin_def
thf(fact_4141_cos__def,axiom,
! [A: $tType] :
( ( ( real_Vector_banach @ A )
& ( real_V2822296259951069270ebra_1 @ A ) )
=> ( ( cos @ A )
= ( ^ [X2: A] :
( suminf @ A
@ ^ [N2: nat] : ( real_V8093663219630862766scaleR @ A @ ( cos_coeff @ N2 ) @ ( power_power @ A @ X2 @ N2 ) ) ) ) ) ) ).
% cos_def
thf(fact_4142_abs__le__square__iff,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [X: A,Y: A] :
( ( ord_less_eq @ A @ ( abs_abs @ A @ X ) @ ( abs_abs @ A @ Y ) )
= ( ord_less_eq @ A @ ( power_power @ A @ X @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ ( power_power @ A @ Y @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) ).
% abs_le_square_iff
thf(fact_4143_abs__square__eq__1,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [X: A] :
( ( ( power_power @ A @ X @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) )
= ( one_one @ A ) )
= ( ( abs_abs @ A @ X )
= ( one_one @ A ) ) ) ) ).
% abs_square_eq_1
thf(fact_4144_power__even__abs,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [N3: nat,A2: A] :
( ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N3 )
=> ( ( power_power @ A @ ( abs_abs @ A @ A2 ) @ N3 )
= ( power_power @ A @ A2 @ N3 ) ) ) ) ).
% power_even_abs
thf(fact_4145_power2__le__iff__abs__le,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [Y: A,X: A] :
( ( ord_less_eq @ A @ ( zero_zero @ A ) @ Y )
=> ( ( ord_less_eq @ A @ ( power_power @ A @ X @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ ( power_power @ A @ Y @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) )
= ( ord_less_eq @ A @ ( abs_abs @ A @ X ) @ Y ) ) ) ) ).
% power2_le_iff_abs_le
thf(fact_4146_abs__sqrt__wlog,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [P: A > A > $o,X: A] :
( ! [X3: A] :
( ( ord_less_eq @ A @ ( zero_zero @ A ) @ X3 )
=> ( P @ X3 @ ( power_power @ A @ X3 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) )
=> ( P @ ( abs_abs @ A @ X ) @ ( power_power @ A @ X @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) ).
% abs_sqrt_wlog
thf(fact_4147_abs__square__le__1,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [X: A] :
( ( ord_less_eq @ A @ ( power_power @ A @ X @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ ( one_one @ A ) )
= ( ord_less_eq @ A @ ( abs_abs @ A @ X ) @ ( one_one @ A ) ) ) ) ).
% abs_square_le_1
thf(fact_4148_abs__square__less__1,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [X: A] :
( ( ord_less @ A @ ( power_power @ A @ X @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ ( one_one @ A ) )
= ( ord_less @ A @ ( abs_abs @ A @ X ) @ ( one_one @ A ) ) ) ) ).
% abs_square_less_1
thf(fact_4149_power__mono__even,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [N3: nat,A2: A,B2: A] :
( ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N3 )
=> ( ( ord_less_eq @ A @ ( abs_abs @ A @ A2 ) @ ( abs_abs @ A @ B2 ) )
=> ( ord_less_eq @ A @ ( power_power @ A @ A2 @ N3 ) @ ( power_power @ A @ B2 @ N3 ) ) ) ) ) ).
% power_mono_even
thf(fact_4150_convex__sum__bound__le,axiom,
! [A: $tType,B: $tType] :
( ( linordered_idom @ B )
=> ! [I5: set @ A,X: A > B,A2: A > B,B2: B,Delta: B] :
( ! [I2: A] :
( ( member @ A @ I2 @ I5 )
=> ( ord_less_eq @ B @ ( zero_zero @ B ) @ ( X @ I2 ) ) )
=> ( ( ( groups7311177749621191930dd_sum @ A @ B @ X @ I5 )
= ( one_one @ B ) )
=> ( ! [I2: A] :
( ( member @ A @ I2 @ I5 )
=> ( ord_less_eq @ B @ ( abs_abs @ B @ ( minus_minus @ B @ ( A2 @ I2 ) @ B2 ) ) @ Delta ) )
=> ( ord_less_eq @ B
@ ( abs_abs @ B
@ ( minus_minus @ B
@ ( groups7311177749621191930dd_sum @ A @ B
@ ^ [I3: A] : ( times_times @ B @ ( A2 @ I3 ) @ ( X @ I3 ) )
@ I5 )
@ B2 ) )
@ Delta ) ) ) ) ) ).
% convex_sum_bound_le
thf(fact_4151_sin__cos__le1,axiom,
! [X: real,Y: real] : ( ord_less_eq @ real @ ( abs_abs @ real @ ( plus_plus @ real @ ( times_times @ real @ ( sin @ real @ X ) @ ( sin @ real @ Y ) ) @ ( times_times @ real @ ( cos @ real @ X ) @ ( cos @ real @ Y ) ) ) ) @ ( one_one @ real ) ) ).
% sin_cos_le1
thf(fact_4152_suminf__geometric,axiom,
! [A: $tType] :
( ( real_V3459762299906320749_field @ A )
=> ! [C2: A] :
( ( ord_less @ real @ ( real_V7770717601297561774m_norm @ A @ C2 ) @ ( one_one @ real ) )
=> ( ( suminf @ A @ ( power_power @ A @ C2 ) )
= ( divide_divide @ A @ ( one_one @ A ) @ ( minus_minus @ A @ ( one_one @ A ) @ C2 ) ) ) ) ) ).
% suminf_geometric
thf(fact_4153_of__int__round__abs__le,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [X: A] : ( ord_less_eq @ A @ ( abs_abs @ A @ ( minus_minus @ A @ ( ring_1_of_int @ A @ ( archimedean_round @ A @ X ) ) @ X ) ) @ ( divide_divide @ A @ ( one_one @ A ) @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) ) ) ).
% of_int_round_abs_le
thf(fact_4154_round__unique_H,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [X: A,N3: int] :
( ( ord_less @ A @ ( abs_abs @ A @ ( minus_minus @ A @ X @ ( ring_1_of_int @ A @ N3 ) ) ) @ ( divide_divide @ A @ ( one_one @ A ) @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) )
=> ( ( archimedean_round @ A @ X )
= N3 ) ) ) ).
% round_unique'
thf(fact_4155_monoseq__arctan__series,axiom,
! [X: real] :
( ( ord_less_eq @ real @ ( abs_abs @ real @ X ) @ ( one_one @ real ) )
=> ( topological_monoseq @ real
@ ^ [N2: nat] : ( times_times @ real @ ( divide_divide @ real @ ( one_one @ real ) @ ( semiring_1_of_nat @ real @ ( plus_plus @ nat @ ( times_times @ nat @ N2 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ ( one_one @ nat ) ) ) ) @ ( power_power @ real @ X @ ( plus_plus @ nat @ ( times_times @ nat @ N2 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ ( one_one @ nat ) ) ) ) ) ) ).
% monoseq_arctan_series
thf(fact_4156_arctan__series,axiom,
! [X: real] :
( ( ord_less_eq @ real @ ( abs_abs @ real @ X ) @ ( one_one @ real ) )
=> ( ( arctan @ X )
= ( suminf @ real
@ ^ [K2: nat] : ( times_times @ real @ ( power_power @ real @ ( uminus_uminus @ real @ ( one_one @ real ) ) @ K2 ) @ ( times_times @ real @ ( divide_divide @ real @ ( one_one @ real ) @ ( semiring_1_of_nat @ real @ ( plus_plus @ nat @ ( times_times @ nat @ K2 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ ( one_one @ nat ) ) ) ) @ ( power_power @ real @ X @ ( plus_plus @ nat @ ( times_times @ nat @ K2 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ ( one_one @ nat ) ) ) ) ) ) ) ) ).
% arctan_series
thf(fact_4157_ln__series,axiom,
! [X: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ X )
=> ( ( ord_less @ real @ X @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) )
=> ( ( ln_ln @ real @ X )
= ( suminf @ real
@ ^ [N2: nat] : ( times_times @ real @ ( times_times @ real @ ( power_power @ real @ ( uminus_uminus @ real @ ( one_one @ real ) ) @ N2 ) @ ( divide_divide @ real @ ( one_one @ real ) @ ( semiring_1_of_nat @ real @ ( plus_plus @ nat @ N2 @ ( one_one @ nat ) ) ) ) ) @ ( power_power @ real @ ( minus_minus @ real @ X @ ( one_one @ real ) ) @ ( suc @ N2 ) ) ) ) ) ) ) ).
% ln_series
thf(fact_4158_summable__arctan__series,axiom,
! [X: real] :
( ( ord_less_eq @ real @ ( abs_abs @ real @ X ) @ ( one_one @ real ) )
=> ( summable @ real
@ ^ [K2: nat] : ( times_times @ real @ ( power_power @ real @ ( uminus_uminus @ real @ ( one_one @ real ) ) @ K2 ) @ ( times_times @ real @ ( divide_divide @ real @ ( one_one @ real ) @ ( semiring_1_of_nat @ real @ ( plus_plus @ nat @ ( times_times @ nat @ K2 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ ( one_one @ nat ) ) ) ) @ ( power_power @ real @ X @ ( plus_plus @ nat @ ( times_times @ nat @ K2 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ ( one_one @ nat ) ) ) ) ) ) ) ).
% summable_arctan_series
thf(fact_4159_zdvd1__eq,axiom,
! [X: int] :
( ( dvd_dvd @ int @ X @ ( one_one @ int ) )
= ( ( abs_abs @ int @ X )
= ( one_one @ int ) ) ) ).
% zdvd1_eq
thf(fact_4160_summable__iff__shift,axiom,
! [A: $tType] :
( ( real_V822414075346904944vector @ A )
=> ! [F2: nat > A,K: nat] :
( ( summable @ A
@ ^ [N2: nat] : ( F2 @ ( plus_plus @ nat @ N2 @ K ) ) )
= ( summable @ A @ F2 ) ) ) ).
% summable_iff_shift
thf(fact_4161_ln__one,axiom,
! [A: $tType] :
( ( ln @ A )
=> ( ( ln_ln @ A @ ( one_one @ A ) )
= ( zero_zero @ A ) ) ) ).
% ln_one
thf(fact_4162_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_4163_arctan__le__zero__iff,axiom,
! [X: real] :
( ( ord_less_eq @ real @ ( arctan @ X ) @ ( zero_zero @ real ) )
= ( ord_less_eq @ real @ X @ ( zero_zero @ real ) ) ) ).
% arctan_le_zero_iff
thf(fact_4164_zero__le__arctan__iff,axiom,
! [X: real] :
( ( ord_less_eq @ real @ ( zero_zero @ real ) @ ( arctan @ X ) )
= ( ord_less_eq @ real @ ( zero_zero @ real ) @ X ) ) ).
% zero_le_arctan_iff
thf(fact_4165_summable__cmult__iff,axiom,
! [A: $tType] :
( ( real_V3459762299906320749_field @ A )
=> ! [C2: A,F2: nat > A] :
( ( summable @ A
@ ^ [N2: nat] : ( times_times @ A @ C2 @ ( F2 @ N2 ) ) )
= ( ( C2
= ( zero_zero @ A ) )
| ( summable @ A @ F2 ) ) ) ) ).
% summable_cmult_iff
thf(fact_4166_summable__divide__iff,axiom,
! [A: $tType] :
( ( real_V3459762299906320749_field @ A )
=> ! [F2: nat > A,C2: A] :
( ( summable @ A
@ ^ [N2: nat] : ( divide_divide @ A @ ( F2 @ N2 ) @ C2 ) )
= ( ( C2
= ( zero_zero @ A ) )
| ( summable @ A @ F2 ) ) ) ) ).
% summable_divide_iff
thf(fact_4167_ln__le__cancel__iff,axiom,
! [X: real,Y: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ X )
=> ( ( ord_less @ real @ ( zero_zero @ real ) @ Y )
=> ( ( ord_less_eq @ real @ ( ln_ln @ real @ X ) @ ( ln_ln @ real @ Y ) )
= ( ord_less_eq @ real @ X @ Y ) ) ) ) ).
% ln_le_cancel_iff
thf(fact_4168_ln__eq__zero__iff,axiom,
! [X: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ X )
=> ( ( ( ln_ln @ real @ X )
= ( zero_zero @ real ) )
= ( X
= ( one_one @ real ) ) ) ) ).
% ln_eq_zero_iff
thf(fact_4169_ln__gt__zero__iff,axiom,
! [X: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ X )
=> ( ( ord_less @ real @ ( zero_zero @ real ) @ ( ln_ln @ real @ X ) )
= ( ord_less @ real @ ( one_one @ real ) @ X ) ) ) ).
% ln_gt_zero_iff
thf(fact_4170_ln__less__zero__iff,axiom,
! [X: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ X )
=> ( ( ord_less @ real @ ( ln_ln @ real @ X ) @ ( zero_zero @ real ) )
= ( ord_less @ real @ X @ ( one_one @ real ) ) ) ) ).
% ln_less_zero_iff
thf(fact_4171_ln__ge__zero__iff,axiom,
! [X: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ X )
=> ( ( ord_less_eq @ real @ ( zero_zero @ real ) @ ( ln_ln @ real @ X ) )
= ( ord_less_eq @ real @ ( one_one @ real ) @ X ) ) ) ).
% ln_ge_zero_iff
thf(fact_4172_ln__le__zero__iff,axiom,
! [X: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ X )
=> ( ( ord_less_eq @ real @ ( ln_ln @ real @ X ) @ ( zero_zero @ real ) )
= ( ord_less_eq @ real @ X @ ( one_one @ real ) ) ) ) ).
% ln_le_zero_iff
thf(fact_4173_summable__geometric__iff,axiom,
! [A: $tType] :
( ( real_V3459762299906320749_field @ A )
=> ! [C2: A] :
( ( summable @ A @ ( power_power @ A @ C2 ) )
= ( ord_less @ real @ ( real_V7770717601297561774m_norm @ A @ C2 ) @ ( one_one @ real ) ) ) ) ).
% summable_geometric_iff
thf(fact_4174_arctan__le__iff,axiom,
! [X: real,Y: real] :
( ( ord_less_eq @ real @ ( arctan @ X ) @ ( arctan @ Y ) )
= ( ord_less_eq @ real @ X @ Y ) ) ).
% arctan_le_iff
thf(fact_4175_arctan__monotone_H,axiom,
! [X: real,Y: real] :
( ( ord_less_eq @ real @ X @ Y )
=> ( ord_less_eq @ real @ ( arctan @ X ) @ ( arctan @ Y ) ) ) ).
% arctan_monotone'
thf(fact_4176_summable__comparison__test,axiom,
! [A: $tType] :
( ( real_Vector_banach @ A )
=> ! [F2: nat > A,G: nat > real] :
( ? [N10: nat] :
! [N: nat] :
( ( ord_less_eq @ nat @ N10 @ N )
=> ( ord_less_eq @ real @ ( real_V7770717601297561774m_norm @ A @ ( F2 @ N ) ) @ ( G @ N ) ) )
=> ( ( summable @ real @ G )
=> ( summable @ A @ F2 ) ) ) ) ).
% summable_comparison_test
thf(fact_4177_summable__comparison__test_H,axiom,
! [A: $tType] :
( ( real_Vector_banach @ A )
=> ! [G: nat > real,N7: nat,F2: nat > A] :
( ( summable @ real @ G )
=> ( ! [N: nat] :
( ( ord_less_eq @ nat @ N7 @ N )
=> ( ord_less_eq @ real @ ( real_V7770717601297561774m_norm @ A @ ( F2 @ N ) ) @ ( G @ N ) ) )
=> ( summable @ A @ F2 ) ) ) ) ).
% summable_comparison_test'
thf(fact_4178_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
@ ^ [N2: nat] : ( plus_plus @ A @ ( F2 @ N2 ) @ ( G @ N2 ) ) ) ) ) ) ).
% summable_add
thf(fact_4179_summable__mult,axiom,
! [A: $tType] :
( ( real_V4412858255891104859lgebra @ A )
=> ! [F2: nat > A,C2: A] :
( ( summable @ A @ F2 )
=> ( summable @ A
@ ^ [N2: nat] : ( times_times @ A @ C2 @ ( F2 @ N2 ) ) ) ) ) ).
% summable_mult
thf(fact_4180_summable__mult2,axiom,
! [A: $tType] :
( ( real_V4412858255891104859lgebra @ A )
=> ! [F2: nat > A,C2: A] :
( ( summable @ A @ F2 )
=> ( summable @ A
@ ^ [N2: nat] : ( times_times @ A @ ( F2 @ N2 ) @ C2 ) ) ) ) ).
% summable_mult2
thf(fact_4181_summable__diff,axiom,
! [A: $tType] :
( ( real_V822414075346904944vector @ A )
=> ! [F2: nat > A,G: nat > A] :
( ( summable @ A @ F2 )
=> ( ( summable @ A @ G )
=> ( summable @ A
@ ^ [N2: nat] : ( minus_minus @ A @ ( F2 @ N2 ) @ ( G @ N2 ) ) ) ) ) ) ).
% summable_diff
thf(fact_4182_summable__Suc__iff,axiom,
! [A: $tType] :
( ( real_V822414075346904944vector @ A )
=> ! [F2: nat > A] :
( ( summable @ A
@ ^ [N2: nat] : ( F2 @ ( suc @ N2 ) ) )
= ( summable @ A @ F2 ) ) ) ).
% summable_Suc_iff
thf(fact_4183_summable__divide,axiom,
! [A: $tType] :
( ( real_V3459762299906320749_field @ A )
=> ! [F2: nat > A,C2: A] :
( ( summable @ A @ F2 )
=> ( summable @ A
@ ^ [N2: nat] : ( divide_divide @ A @ ( F2 @ N2 ) @ C2 ) ) ) ) ).
% summable_divide
thf(fact_4184_summable__ignore__initial__segment,axiom,
! [A: $tType] :
( ( real_V822414075346904944vector @ A )
=> ! [F2: nat > A,K: nat] :
( ( summable @ A @ F2 )
=> ( summable @ A
@ ^ [N2: nat] : ( F2 @ ( plus_plus @ nat @ N2 @ K ) ) ) ) ) ).
% summable_ignore_initial_segment
thf(fact_4185_powser__insidea,axiom,
! [A: $tType] :
( ( real_V8999393235501362500lgebra @ A )
=> ! [F2: nat > A,X: A,Z: A] :
( ( summable @ A
@ ^ [N2: nat] : ( times_times @ A @ ( F2 @ N2 ) @ ( power_power @ A @ X @ N2 ) ) )
=> ( ( ord_less @ real @ ( real_V7770717601297561774m_norm @ A @ Z ) @ ( real_V7770717601297561774m_norm @ A @ X ) )
=> ( summable @ real
@ ^ [N2: nat] : ( real_V7770717601297561774m_norm @ A @ ( times_times @ A @ ( F2 @ N2 ) @ ( power_power @ A @ Z @ N2 ) ) ) ) ) ) ) ).
% powser_insidea
thf(fact_4186_suminf__le,axiom,
! [A: $tType] :
( ( ( ordere6911136660526730532id_add @ A )
& ( topolo1944317154257567458pology @ A ) )
=> ! [F2: nat > A,G: nat > A] :
( ! [N: nat] : ( ord_less_eq @ A @ ( F2 @ N ) @ ( G @ N ) )
=> ( ( summable @ A @ F2 )
=> ( ( summable @ A @ G )
=> ( ord_less_eq @ A @ ( suminf @ A @ F2 ) @ ( suminf @ A @ G ) ) ) ) ) ) ).
% suminf_le
thf(fact_4187_log__def,axiom,
( log
= ( ^ [A6: real,X2: real] : ( divide_divide @ real @ ( ln_ln @ real @ X2 ) @ ( ln_ln @ real @ A6 ) ) ) ) ).
% log_def
thf(fact_4188_abs__zmult__eq__1,axiom,
! [M: int,N3: int] :
( ( ( abs_abs @ int @ ( times_times @ int @ M @ N3 ) )
= ( one_one @ int ) )
=> ( ( abs_abs @ int @ M )
= ( one_one @ int ) ) ) ).
% abs_zmult_eq_1
thf(fact_4189_summable__mult__D,axiom,
! [A: $tType] :
( ( real_V3459762299906320749_field @ A )
=> ! [C2: A,F2: nat > A] :
( ( summable @ A
@ ^ [N2: nat] : ( times_times @ A @ C2 @ ( F2 @ N2 ) ) )
=> ( ( C2
!= ( zero_zero @ A ) )
=> ( summable @ A @ F2 ) ) ) ) ).
% summable_mult_D
thf(fact_4190_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_4191_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
@ ^ [N2: nat] : ( plus_plus @ A @ ( F2 @ N2 ) @ ( G @ N2 ) ) ) ) ) ) ) ).
% suminf_add
thf(fact_4192_suminf__mult,axiom,
! [A: $tType] :
( ( real_V4412858255891104859lgebra @ A )
=> ! [F2: nat > A,C2: A] :
( ( summable @ A @ F2 )
=> ( ( suminf @ A
@ ^ [N2: nat] : ( times_times @ A @ C2 @ ( F2 @ N2 ) ) )
= ( times_times @ A @ C2 @ ( suminf @ A @ F2 ) ) ) ) ) ).
% suminf_mult
thf(fact_4193_suminf__mult2,axiom,
! [A: $tType] :
( ( real_V4412858255891104859lgebra @ A )
=> ! [F2: nat > A,C2: A] :
( ( summable @ A @ F2 )
=> ( ( times_times @ A @ ( suminf @ A @ F2 ) @ C2 )
= ( suminf @ A
@ ^ [N2: nat] : ( times_times @ A @ ( F2 @ N2 ) @ C2 ) ) ) ) ) ).
% suminf_mult2
thf(fact_4194_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
@ ^ [N2: nat] : ( minus_minus @ A @ ( F2 @ N2 ) @ ( G @ N2 ) ) ) ) ) ) ) ).
% suminf_diff
thf(fact_4195_suminf__divide,axiom,
! [A: $tType] :
( ( real_V3459762299906320749_field @ A )
=> ! [F2: nat > A,C2: A] :
( ( summable @ A @ F2 )
=> ( ( suminf @ A
@ ^ [N2: nat] : ( divide_divide @ A @ ( F2 @ N2 ) @ C2 ) )
= ( divide_divide @ A @ ( suminf @ A @ F2 ) @ C2 ) ) ) ) ).
% suminf_divide
thf(fact_4196_summable__Cauchy__product,axiom,
! [A: $tType] :
( ( ( real_Vector_banach @ A )
& ( real_V4412858255891104859lgebra @ A ) )
=> ! [A2: nat > A,B2: nat > A] :
( ( summable @ real
@ ^ [K2: nat] : ( real_V7770717601297561774m_norm @ A @ ( A2 @ K2 ) ) )
=> ( ( summable @ real
@ ^ [K2: nat] : ( real_V7770717601297561774m_norm @ A @ ( B2 @ K2 ) ) )
=> ( summable @ A
@ ^ [K2: nat] :
( groups7311177749621191930dd_sum @ nat @ A
@ ^ [I3: nat] : ( times_times @ A @ ( A2 @ I3 ) @ ( B2 @ ( minus_minus @ nat @ K2 @ I3 ) ) )
@ ( set_ord_atMost @ nat @ K2 ) ) ) ) ) ) ).
% summable_Cauchy_product
thf(fact_4197_infinite__int__iff__unbounded__le,axiom,
! [S3: set @ int] :
( ( ~ ( finite_finite2 @ int @ S3 ) )
= ( ! [M5: int] :
? [N2: int] :
( ( ord_less_eq @ int @ M5 @ ( abs_abs @ int @ N2 ) )
& ( member @ int @ N2 @ S3 ) ) ) ) ).
% infinite_int_iff_unbounded_le
thf(fact_4198_suminf__nonneg,axiom,
! [A: $tType] :
( ( ( ordere6911136660526730532id_add @ A )
& ( topolo1944317154257567458pology @ A ) )
=> ! [F2: nat > A] :
( ( summable @ A @ F2 )
=> ( ! [N: nat] : ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( F2 @ N ) )
=> ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( suminf @ A @ F2 ) ) ) ) ) ).
% suminf_nonneg
thf(fact_4199_suminf__eq__zero__iff,axiom,
! [A: $tType] :
( ( ( ordere6911136660526730532id_add @ A )
& ( topolo1944317154257567458pology @ A ) )
=> ! [F2: nat > A] :
( ( summable @ A @ F2 )
=> ( ! [N: nat] : ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( F2 @ N ) )
=> ( ( ( suminf @ A @ F2 )
= ( zero_zero @ A ) )
= ( ! [N2: nat] :
( ( F2 @ N2 )
= ( zero_zero @ A ) ) ) ) ) ) ) ).
% suminf_eq_zero_iff
thf(fact_4200_suminf__pos,axiom,
! [A: $tType] :
( ( ( ordere6911136660526730532id_add @ A )
& ( topolo1944317154257567458pology @ A ) )
=> ! [F2: nat > A] :
( ( summable @ A @ F2 )
=> ( ! [N: nat] : ( ord_less @ A @ ( zero_zero @ A ) @ ( F2 @ N ) )
=> ( ord_less @ A @ ( zero_zero @ A ) @ ( suminf @ A @ F2 ) ) ) ) ) ).
% suminf_pos
thf(fact_4201_ln__bound,axiom,
! [X: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ X )
=> ( ord_less_eq @ real @ ( ln_ln @ real @ X ) @ X ) ) ).
% ln_bound
thf(fact_4202_ln__gt__zero,axiom,
! [X: real] :
( ( ord_less @ real @ ( one_one @ real ) @ X )
=> ( ord_less @ real @ ( zero_zero @ real ) @ ( ln_ln @ real @ X ) ) ) ).
% ln_gt_zero
thf(fact_4203_ln__less__zero,axiom,
! [X: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ X )
=> ( ( ord_less @ real @ X @ ( one_one @ real ) )
=> ( ord_less @ real @ ( ln_ln @ real @ X ) @ ( zero_zero @ real ) ) ) ) ).
% ln_less_zero
thf(fact_4204_ln__gt__zero__imp__gt__one,axiom,
! [X: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ ( ln_ln @ real @ X ) )
=> ( ( ord_less @ real @ ( zero_zero @ real ) @ X )
=> ( ord_less @ real @ ( one_one @ real ) @ X ) ) ) ).
% ln_gt_zero_imp_gt_one
thf(fact_4205_ln__ge__zero,axiom,
! [X: real] :
( ( ord_less_eq @ real @ ( one_one @ real ) @ X )
=> ( ord_less_eq @ real @ ( zero_zero @ real ) @ ( ln_ln @ real @ X ) ) ) ).
% ln_ge_zero
thf(fact_4206_ln__powr,axiom,
! [X: real,Y: real] :
( ( X
!= ( zero_zero @ real ) )
=> ( ( ln_ln @ real @ ( powr @ real @ X @ Y ) )
= ( times_times @ real @ Y @ ( ln_ln @ real @ X ) ) ) ) ).
% ln_powr
thf(fact_4207_summable__0__powser,axiom,
! [A: $tType] :
( ( real_V8999393235501362500lgebra @ A )
=> ! [F2: nat > A] :
( summable @ A
@ ^ [N2: nat] : ( times_times @ A @ ( F2 @ N2 ) @ ( power_power @ A @ ( zero_zero @ A ) @ N2 ) ) ) ) ).
% summable_0_powser
thf(fact_4208_summable__zero__power_H,axiom,
! [A: $tType] :
( ( ( ring_1 @ A )
& ( topolo4958980785337419405_space @ A ) )
=> ! [F2: nat > A] :
( summable @ A
@ ^ [N2: nat] : ( times_times @ A @ ( F2 @ N2 ) @ ( power_power @ A @ ( zero_zero @ A ) @ N2 ) ) ) ) ).
% summable_zero_power'
thf(fact_4209_summable__powser__split__head,axiom,
! [A: $tType] :
( ( real_V8999393235501362500lgebra @ A )
=> ! [F2: nat > A,Z: A] :
( ( summable @ A
@ ^ [N2: nat] : ( times_times @ A @ ( F2 @ ( suc @ N2 ) ) @ ( power_power @ A @ Z @ N2 ) ) )
= ( summable @ A
@ ^ [N2: nat] : ( times_times @ A @ ( F2 @ N2 ) @ ( power_power @ A @ Z @ N2 ) ) ) ) ) ).
% summable_powser_split_head
thf(fact_4210_powser__split__head_I3_J,axiom,
! [A: $tType] :
( ( ( real_Vector_banach @ A )
& ( real_V8999393235501362500lgebra @ A ) )
=> ! [F2: nat > A,Z: A] :
( ( summable @ A
@ ^ [N2: nat] : ( times_times @ A @ ( F2 @ N2 ) @ ( power_power @ A @ Z @ N2 ) ) )
=> ( summable @ A
@ ^ [N2: nat] : ( times_times @ A @ ( F2 @ ( suc @ N2 ) ) @ ( power_power @ A @ Z @ N2 ) ) ) ) ) ).
% powser_split_head(3)
thf(fact_4211_summable__powser__ignore__initial__segment,axiom,
! [A: $tType] :
( ( real_V8999393235501362500lgebra @ A )
=> ! [F2: nat > A,M: nat,Z: A] :
( ( summable @ A
@ ^ [N2: nat] : ( times_times @ A @ ( F2 @ ( plus_plus @ nat @ N2 @ M ) ) @ ( power_power @ A @ Z @ N2 ) ) )
= ( summable @ A
@ ^ [N2: nat] : ( times_times @ A @ ( F2 @ N2 ) @ ( power_power @ A @ Z @ N2 ) ) ) ) ) ).
% summable_powser_ignore_initial_segment
thf(fact_4212_abs__mod__less,axiom,
! [L2: int,K: int] :
( ( L2
!= ( zero_zero @ int ) )
=> ( ord_less @ int @ ( abs_abs @ int @ ( modulo_modulo @ int @ K @ L2 ) ) @ ( abs_abs @ int @ L2 ) ) ) ).
% abs_mod_less
thf(fact_4213_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_4214_dvd__imp__le__int,axiom,
! [I: int,D: int] :
( ( I
!= ( zero_zero @ int ) )
=> ( ( dvd_dvd @ int @ D @ I )
=> ( ord_less_eq @ int @ ( abs_abs @ int @ D ) @ ( abs_abs @ int @ I ) ) ) ) ).
% dvd_imp_le_int
thf(fact_4215_summable__norm__comparison__test,axiom,
! [A: $tType] :
( ( real_V822414075346904944vector @ A )
=> ! [F2: nat > A,G: nat > real] :
( ? [N10: nat] :
! [N: nat] :
( ( ord_less_eq @ nat @ N10 @ N )
=> ( ord_less_eq @ real @ ( real_V7770717601297561774m_norm @ A @ ( F2 @ N ) ) @ ( G @ N ) ) )
=> ( ( summable @ real @ G )
=> ( summable @ real
@ ^ [N2: nat] : ( real_V7770717601297561774m_norm @ A @ ( F2 @ N2 ) ) ) ) ) ) ).
% summable_norm_comparison_test
thf(fact_4216_summable__rabs__comparison__test,axiom,
! [F2: nat > real,G: nat > real] :
( ? [N10: nat] :
! [N: nat] :
( ( ord_less_eq @ nat @ N10 @ N )
=> ( ord_less_eq @ real @ ( abs_abs @ real @ ( F2 @ N ) ) @ ( G @ N ) ) )
=> ( ( summable @ real @ G )
=> ( summable @ real
@ ^ [N2: nat] : ( abs_abs @ real @ ( F2 @ N2 ) ) ) ) ) ).
% summable_rabs_comparison_test
thf(fact_4217_summable__rabs,axiom,
! [F2: nat > real] :
( ( summable @ real
@ ^ [N2: nat] : ( abs_abs @ real @ ( F2 @ N2 ) ) )
=> ( ord_less_eq @ real @ ( abs_abs @ real @ ( suminf @ real @ F2 ) )
@ ( suminf @ real
@ ^ [N2: nat] : ( abs_abs @ real @ ( F2 @ N2 ) ) ) ) ) ).
% summable_rabs
thf(fact_4218_suminf__pos2,axiom,
! [A: $tType] :
( ( ( ordere6911136660526730532id_add @ A )
& ( topolo1944317154257567458pology @ A ) )
=> ! [F2: nat > A,I: nat] :
( ( summable @ A @ F2 )
=> ( ! [N: nat] : ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( F2 @ N ) )
=> ( ( ord_less @ A @ ( zero_zero @ A ) @ ( F2 @ I ) )
=> ( ord_less @ A @ ( zero_zero @ A ) @ ( suminf @ A @ F2 ) ) ) ) ) ) ).
% suminf_pos2
thf(fact_4219_suminf__pos__iff,axiom,
! [A: $tType] :
( ( ( ordere6911136660526730532id_add @ A )
& ( topolo1944317154257567458pology @ A ) )
=> ! [F2: nat > A] :
( ( summable @ A @ F2 )
=> ( ! [N: nat] : ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( F2 @ N ) )
=> ( ( ord_less @ A @ ( zero_zero @ A ) @ ( suminf @ A @ F2 ) )
= ( ? [I3: nat] : ( ord_less @ A @ ( zero_zero @ A ) @ ( F2 @ I3 ) ) ) ) ) ) ) ).
% suminf_pos_iff
thf(fact_4220_ln__ge__zero__imp__ge__one,axiom,
! [X: real] :
( ( ord_less_eq @ real @ ( zero_zero @ real ) @ ( ln_ln @ real @ X ) )
=> ( ( ord_less @ real @ ( zero_zero @ real ) @ X )
=> ( ord_less_eq @ real @ ( one_one @ real ) @ X ) ) ) ).
% ln_ge_zero_imp_ge_one
thf(fact_4221_suminf__le__const,axiom,
! [A: $tType] :
( ( ( ordere6911136660526730532id_add @ A )
& ( topolo1944317154257567458pology @ A ) )
=> ! [F2: nat > A,X: A] :
( ( summable @ A @ F2 )
=> ( ! [N: nat] : ( ord_less_eq @ A @ ( groups7311177749621191930dd_sum @ nat @ A @ F2 @ ( set_ord_lessThan @ nat @ N ) ) @ X )
=> ( ord_less_eq @ A @ ( suminf @ A @ F2 ) @ X ) ) ) ) ).
% suminf_le_const
thf(fact_4222_ln__add__one__self__le__self,axiom,
! [X: real] :
( ( ord_less_eq @ real @ ( zero_zero @ real ) @ X )
=> ( ord_less_eq @ real @ ( ln_ln @ real @ ( plus_plus @ real @ ( one_one @ real ) @ X ) ) @ X ) ) ).
% ln_add_one_self_le_self
thf(fact_4223_ln__mult,axiom,
! [X: real,Y: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ X )
=> ( ( ord_less @ real @ ( zero_zero @ real ) @ Y )
=> ( ( ln_ln @ real @ ( times_times @ real @ X @ Y ) )
= ( plus_plus @ real @ ( ln_ln @ real @ X ) @ ( ln_ln @ real @ Y ) ) ) ) ) ).
% ln_mult
thf(fact_4224_ln__eq__minus__one,axiom,
! [X: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ X )
=> ( ( ( ln_ln @ real @ X )
= ( minus_minus @ real @ X @ ( one_one @ real ) ) )
=> ( X
= ( one_one @ real ) ) ) ) ).
% ln_eq_minus_one
thf(fact_4225_ln__div,axiom,
! [X: real,Y: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ X )
=> ( ( ord_less @ real @ ( zero_zero @ real ) @ Y )
=> ( ( ln_ln @ real @ ( divide_divide @ real @ X @ Y ) )
= ( minus_minus @ real @ ( ln_ln @ real @ X ) @ ( ln_ln @ real @ Y ) ) ) ) ) ).
% ln_div
thf(fact_4226_powser__inside,axiom,
! [A: $tType] :
( ( ( real_Vector_banach @ A )
& ( real_V8999393235501362500lgebra @ A ) )
=> ! [F2: nat > A,X: A,Z: A] :
( ( summable @ A
@ ^ [N2: nat] : ( times_times @ A @ ( F2 @ N2 ) @ ( power_power @ A @ X @ N2 ) ) )
=> ( ( ord_less @ real @ ( real_V7770717601297561774m_norm @ A @ Z ) @ ( real_V7770717601297561774m_norm @ A @ X ) )
=> ( summable @ A
@ ^ [N2: nat] : ( times_times @ A @ ( F2 @ N2 ) @ ( power_power @ A @ Z @ N2 ) ) ) ) ) ) ).
% powser_inside
thf(fact_4227_suminf__split__head,axiom,
! [A: $tType] :
( ( real_V822414075346904944vector @ A )
=> ! [F2: nat > A] :
( ( summable @ A @ F2 )
=> ( ( suminf @ A
@ ^ [N2: nat] : ( F2 @ ( suc @ N2 ) ) )
= ( minus_minus @ A @ ( suminf @ A @ F2 ) @ ( F2 @ ( zero_zero @ nat ) ) ) ) ) ) ).
% suminf_split_head
thf(fact_4228_summableI__nonneg__bounded,axiom,
! [A: $tType] :
( ( ( condit6923001295902523014norder @ A )
& ( ordere6911136660526730532id_add @ A )
& ( topolo1944317154257567458pology @ A ) )
=> ! [F2: nat > A,X: A] :
( ! [N: nat] : ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( F2 @ N ) )
=> ( ! [N: nat] : ( ord_less_eq @ A @ ( groups7311177749621191930dd_sum @ nat @ A @ F2 @ ( set_ord_lessThan @ nat @ N ) ) @ X )
=> ( summable @ A @ F2 ) ) ) ) ).
% summableI_nonneg_bounded
thf(fact_4229_bounded__imp__summable,axiom,
! [A: $tType] :
( ( ( condit6923001295902523014norder @ A )
& ( linord2810124833399127020strict @ A )
& ( topolo1944317154257567458pology @ A ) )
=> ! [A2: nat > A,B6: A] :
( ! [N: nat] : ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( A2 @ N ) )
=> ( ! [N: nat] : ( ord_less_eq @ A @ ( groups7311177749621191930dd_sum @ nat @ A @ A2 @ ( set_ord_atMost @ nat @ N ) ) @ B6 )
=> ( summable @ A @ A2 ) ) ) ) ).
% bounded_imp_summable
thf(fact_4230_complete__algebra__summable__geometric,axiom,
! [A: $tType] :
( ( ( real_Vector_banach @ A )
& ( real_V2822296259951069270ebra_1 @ A ) )
=> ! [X: A] :
( ( ord_less @ real @ ( real_V7770717601297561774m_norm @ A @ X ) @ ( one_one @ real ) )
=> ( summable @ A @ ( power_power @ A @ X ) ) ) ) ).
% complete_algebra_summable_geometric
thf(fact_4231_summable__geometric,axiom,
! [A: $tType] :
( ( real_V3459762299906320749_field @ A )
=> ! [C2: A] :
( ( ord_less @ real @ ( real_V7770717601297561774m_norm @ A @ C2 ) @ ( one_one @ real ) )
=> ( summable @ A @ ( power_power @ A @ C2 ) ) ) ) ).
% summable_geometric
thf(fact_4232_nat__abs__triangle__ineq,axiom,
! [K: int,L2: int] : ( ord_less_eq @ nat @ ( nat2 @ ( abs_abs @ int @ ( plus_plus @ int @ K @ L2 ) ) ) @ ( plus_plus @ nat @ ( nat2 @ ( abs_abs @ int @ K ) ) @ ( nat2 @ ( abs_abs @ int @ L2 ) ) ) ) ).
% nat_abs_triangle_ineq
thf(fact_4233_zdvd__mult__cancel1,axiom,
! [M: int,N3: int] :
( ( M
!= ( zero_zero @ int ) )
=> ( ( dvd_dvd @ int @ ( times_times @ int @ M @ N3 ) @ M )
= ( ( abs_abs @ int @ N3 )
= ( one_one @ int ) ) ) ) ).
% zdvd_mult_cancel1
thf(fact_4234_div__abs__eq__div__nat,axiom,
! [K: int,L2: int] :
( ( divide_divide @ int @ ( abs_abs @ int @ K ) @ ( abs_abs @ int @ L2 ) )
= ( semiring_1_of_nat @ int @ ( divide_divide @ nat @ ( nat2 @ ( abs_abs @ int @ K ) ) @ ( nat2 @ ( abs_abs @ int @ L2 ) ) ) ) ) ).
% div_abs_eq_div_nat
thf(fact_4235_summable__norm,axiom,
! [A: $tType] :
( ( real_Vector_banach @ A )
=> ! [F2: nat > A] :
( ( summable @ real
@ ^ [N2: nat] : ( real_V7770717601297561774m_norm @ A @ ( F2 @ N2 ) ) )
=> ( ord_less_eq @ real @ ( real_V7770717601297561774m_norm @ A @ ( suminf @ A @ F2 ) )
@ ( suminf @ real
@ ^ [N2: nat] : ( real_V7770717601297561774m_norm @ A @ ( F2 @ N2 ) ) ) ) ) ) ).
% summable_norm
thf(fact_4236_summable__norm__sin,axiom,
! [A: $tType] :
( ( ( real_Vector_banach @ A )
& ( real_V2822296259951069270ebra_1 @ A ) )
=> ! [X: A] :
( summable @ real
@ ^ [N2: nat] : ( real_V7770717601297561774m_norm @ A @ ( real_V8093663219630862766scaleR @ A @ ( sin_coeff @ N2 ) @ ( power_power @ A @ X @ N2 ) ) ) ) ) ).
% summable_norm_sin
thf(fact_4237_summable__norm__cos,axiom,
! [A: $tType] :
( ( ( real_Vector_banach @ A )
& ( real_V2822296259951069270ebra_1 @ A ) )
=> ! [X: A] :
( summable @ real
@ ^ [N2: nat] : ( real_V7770717601297561774m_norm @ A @ ( real_V8093663219630862766scaleR @ A @ ( cos_coeff @ N2 ) @ ( power_power @ A @ X @ N2 ) ) ) ) ) ).
% summable_norm_cos
thf(fact_4238_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_4239_sum__le__suminf,axiom,
! [A: $tType] :
( ( ( ordere6911136660526730532id_add @ A )
& ( topolo1944317154257567458pology @ A ) )
=> ! [F2: nat > A,I5: set @ nat] :
( ( summable @ A @ F2 )
=> ( ( finite_finite2 @ nat @ I5 )
=> ( ! [N: nat] :
( ( member @ nat @ N @ ( uminus_uminus @ ( set @ nat ) @ I5 ) )
=> ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( F2 @ N ) ) )
=> ( ord_less_eq @ A @ ( groups7311177749621191930dd_sum @ nat @ A @ F2 @ I5 ) @ ( suminf @ A @ F2 ) ) ) ) ) ) ).
% sum_le_suminf
thf(fact_4240_ln__le__minus__one,axiom,
! [X: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ X )
=> ( ord_less_eq @ real @ ( ln_ln @ real @ X ) @ ( minus_minus @ real @ X @ ( one_one @ real ) ) ) ) ).
% ln_le_minus_one
thf(fact_4241_ln__diff__le,axiom,
! [X: real,Y: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ X )
=> ( ( ord_less @ real @ ( zero_zero @ real ) @ Y )
=> ( ord_less_eq @ real @ ( minus_minus @ real @ ( ln_ln @ real @ X ) @ ( ln_ln @ real @ Y ) ) @ ( divide_divide @ real @ ( minus_minus @ real @ X @ Y ) @ Y ) ) ) ) ).
% ln_diff_le
thf(fact_4242_ln__add__one__self__le__self2,axiom,
! [X: real] :
( ( ord_less @ real @ ( uminus_uminus @ real @ ( one_one @ real ) ) @ X )
=> ( ord_less_eq @ real @ ( ln_ln @ real @ ( plus_plus @ real @ ( one_one @ real ) @ X ) ) @ X ) ) ).
% ln_add_one_self_le_self2
thf(fact_4243_ln__realpow,axiom,
! [X: real,N3: nat] :
( ( ord_less @ real @ ( zero_zero @ real ) @ X )
=> ( ( ln_ln @ real @ ( power_power @ real @ X @ N3 ) )
= ( times_times @ real @ ( semiring_1_of_nat @ real @ N3 ) @ ( ln_ln @ real @ X ) ) ) ) ).
% ln_realpow
thf(fact_4244_even__abs__add__iff,axiom,
! [K: int,L2: int] :
( ( dvd_dvd @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ ( plus_plus @ int @ ( abs_abs @ int @ K ) @ L2 ) )
= ( dvd_dvd @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ ( plus_plus @ int @ K @ L2 ) ) ) ).
% even_abs_add_iff
thf(fact_4245_even__add__abs__iff,axiom,
! [K: int,L2: int] :
( ( dvd_dvd @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ ( plus_plus @ int @ K @ ( abs_abs @ int @ L2 ) ) )
= ( dvd_dvd @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ ( plus_plus @ int @ K @ L2 ) ) ) ).
% even_add_abs_iff
thf(fact_4246_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
@ ^ [N2: nat] : ( F2 @ ( plus_plus @ nat @ N2 @ K ) ) )
@ ( groups7311177749621191930dd_sum @ nat @ A @ F2 @ ( set_ord_lessThan @ nat @ K ) ) ) ) ) ) ).
% suminf_split_initial_segment
thf(fact_4247_suminf__minus__initial__segment,axiom,
! [A: $tType] :
( ( real_V822414075346904944vector @ A )
=> ! [F2: nat > A,K: nat] :
( ( summable @ A @ F2 )
=> ( ( suminf @ A
@ ^ [N2: nat] : ( F2 @ ( plus_plus @ nat @ N2 @ K ) ) )
= ( minus_minus @ A @ ( suminf @ A @ F2 ) @ ( groups7311177749621191930dd_sum @ nat @ A @ F2 @ ( set_ord_lessThan @ nat @ K ) ) ) ) ) ) ).
% suminf_minus_initial_segment
thf(fact_4248_nat__abs__int__diff,axiom,
! [A2: nat,B2: nat] :
( ( ( ord_less_eq @ nat @ A2 @ B2 )
=> ( ( nat2 @ ( abs_abs @ int @ ( minus_minus @ int @ ( semiring_1_of_nat @ int @ A2 ) @ ( semiring_1_of_nat @ int @ B2 ) ) ) )
= ( minus_minus @ nat @ B2 @ A2 ) ) )
& ( ~ ( ord_less_eq @ nat @ A2 @ B2 )
=> ( ( nat2 @ ( abs_abs @ int @ ( minus_minus @ int @ ( semiring_1_of_nat @ int @ A2 ) @ ( semiring_1_of_nat @ int @ B2 ) ) ) )
= ( minus_minus @ nat @ A2 @ B2 ) ) ) ) ).
% nat_abs_int_diff
thf(fact_4249_sum__less__suminf,axiom,
! [A: $tType] :
( ( ( ordere8940638589300402666id_add @ A )
& ( topolo1944317154257567458pology @ A ) )
=> ! [F2: nat > A,N3: nat] :
( ( summable @ A @ F2 )
=> ( ! [M4: nat] :
( ( ord_less_eq @ nat @ N3 @ M4 )
=> ( ord_less @ A @ ( zero_zero @ A ) @ ( F2 @ M4 ) ) )
=> ( ord_less @ A @ ( groups7311177749621191930dd_sum @ nat @ A @ F2 @ ( set_ord_lessThan @ nat @ N3 ) ) @ ( suminf @ A @ F2 ) ) ) ) ) ).
% sum_less_suminf
thf(fact_4250_powser__split__head_I1_J,axiom,
! [A: $tType] :
( ( ( real_Vector_banach @ A )
& ( real_V8999393235501362500lgebra @ A ) )
=> ! [F2: nat > A,Z: A] :
( ( summable @ A
@ ^ [N2: nat] : ( times_times @ A @ ( F2 @ N2 ) @ ( power_power @ A @ Z @ N2 ) ) )
=> ( ( suminf @ A
@ ^ [N2: nat] : ( times_times @ A @ ( F2 @ N2 ) @ ( power_power @ A @ Z @ N2 ) ) )
= ( plus_plus @ A @ ( F2 @ ( zero_zero @ nat ) )
@ ( times_times @ A
@ ( suminf @ A
@ ^ [N2: nat] : ( times_times @ A @ ( F2 @ ( suc @ N2 ) ) @ ( power_power @ A @ Z @ N2 ) ) )
@ Z ) ) ) ) ) ).
% powser_split_head(1)
thf(fact_4251_powser__split__head_I2_J,axiom,
! [A: $tType] :
( ( ( real_Vector_banach @ A )
& ( real_V8999393235501362500lgebra @ A ) )
=> ! [F2: nat > A,Z: A] :
( ( summable @ A
@ ^ [N2: nat] : ( times_times @ A @ ( F2 @ N2 ) @ ( power_power @ A @ Z @ N2 ) ) )
=> ( ( times_times @ A
@ ( suminf @ A
@ ^ [N2: nat] : ( times_times @ A @ ( F2 @ ( suc @ N2 ) ) @ ( power_power @ A @ Z @ N2 ) ) )
@ Z )
= ( minus_minus @ A
@ ( suminf @ A
@ ^ [N2: nat] : ( times_times @ A @ ( F2 @ N2 ) @ ( power_power @ A @ Z @ N2 ) ) )
@ ( F2 @ ( zero_zero @ nat ) ) ) ) ) ) ).
% powser_split_head(2)
thf(fact_4252_ln__one__minus__pos__upper__bound,axiom,
! [X: real] :
( ( ord_less_eq @ real @ ( zero_zero @ real ) @ X )
=> ( ( ord_less @ real @ X @ ( one_one @ real ) )
=> ( ord_less_eq @ real @ ( ln_ln @ real @ ( minus_minus @ real @ ( one_one @ real ) @ X ) ) @ ( uminus_uminus @ real @ X ) ) ) ) ).
% ln_one_minus_pos_upper_bound
thf(fact_4253_ln__powr__bound,axiom,
! [X: real,A2: real] :
( ( ord_less_eq @ real @ ( one_one @ real ) @ X )
=> ( ( ord_less @ real @ ( zero_zero @ real ) @ A2 )
=> ( ord_less_eq @ real @ ( ln_ln @ real @ X ) @ ( divide_divide @ real @ ( powr @ real @ X @ A2 ) @ A2 ) ) ) ) ).
% ln_powr_bound
thf(fact_4254_ln__powr__bound2,axiom,
! [X: real,A2: real] :
( ( ord_less @ real @ ( one_one @ real ) @ X )
=> ( ( ord_less @ real @ ( zero_zero @ real ) @ A2 )
=> ( ord_less_eq @ real @ ( powr @ real @ ( ln_ln @ real @ X ) @ A2 ) @ ( times_times @ real @ ( powr @ real @ A2 @ A2 ) @ X ) ) ) ) ).
% ln_powr_bound2
thf(fact_4255_log__eq__div__ln__mult__log,axiom,
! [A2: real,B2: real,X: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ A2 )
=> ( ( A2
!= ( one_one @ real ) )
=> ( ( ord_less @ real @ ( zero_zero @ real ) @ B2 )
=> ( ( B2
!= ( one_one @ real ) )
=> ( ( ord_less @ real @ ( zero_zero @ real ) @ X )
=> ( ( log @ A2 @ X )
= ( times_times @ real @ ( divide_divide @ real @ ( ln_ln @ real @ B2 ) @ ( ln_ln @ real @ A2 ) ) @ ( log @ B2 @ X ) ) ) ) ) ) ) ) ).
% log_eq_div_ln_mult_log
thf(fact_4256_monoseq__realpow,axiom,
! [X: real] :
( ( ord_less_eq @ real @ ( zero_zero @ real ) @ X )
=> ( ( ord_less_eq @ real @ X @ ( one_one @ real ) )
=> ( topological_monoseq @ real @ ( power_power @ real @ X ) ) ) ) ).
% monoseq_realpow
thf(fact_4257_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
@ ^ [I3: nat] : ( F2 @ ( plus_plus @ nat @ I3 @ N9 ) ) ) )
@ R2 ) ) ) ) ) ).
% suminf_exist_split
thf(fact_4258_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] :
~ ! [M2: nat] :
( ( ord_less_eq @ nat @ N11 @ M2 )
=> ! [N9: nat] : ( ord_less @ real @ ( real_V7770717601297561774m_norm @ A @ ( groups7311177749621191930dd_sum @ nat @ A @ F2 @ ( set_or1337092689740270186AtMost @ nat @ M2 @ N9 ) ) ) @ E ) ) ) ) ) ).
% summable_partial_sum_bound
thf(fact_4259_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
@ ^ [I3: nat] : ( times_times @ real @ ( F2 @ I3 ) @ ( power_power @ real @ Z @ I3 ) ) ) ) ) ) ) ).
% summable_power_series
thf(fact_4260_Abel__lemma,axiom,
! [A: $tType] :
( ( real_V822414075346904944vector @ A )
=> ! [R2: real,R0: real,A2: nat > A,M8: real] :
( ( ord_less_eq @ real @ ( zero_zero @ real ) @ R2 )
=> ( ( ord_less @ real @ R2 @ R0 )
=> ( ! [N: nat] : ( ord_less_eq @ real @ ( times_times @ real @ ( real_V7770717601297561774m_norm @ A @ ( A2 @ N ) ) @ ( power_power @ real @ R0 @ N ) ) @ M8 )
=> ( summable @ real
@ ^ [N2: nat] : ( times_times @ real @ ( real_V7770717601297561774m_norm @ A @ ( A2 @ N2 ) ) @ ( power_power @ real @ R2 @ N2 ) ) ) ) ) ) ) ).
% Abel_lemma
thf(fact_4261_nat__intermed__int__val,axiom,
! [M: nat,N3: nat,F2: nat > int,K: int] :
( ! [I2: nat] :
( ( ( ord_less_eq @ nat @ M @ I2 )
& ( ord_less @ nat @ I2 @ N3 ) )
=> ( ord_less_eq @ int @ ( abs_abs @ int @ ( minus_minus @ int @ ( F2 @ ( suc @ I2 ) ) @ ( F2 @ I2 ) ) ) @ ( one_one @ int ) ) )
=> ( ( ord_less_eq @ nat @ M @ N3 )
=> ( ( ord_less_eq @ int @ ( F2 @ M ) @ K )
=> ( ( ord_less_eq @ int @ K @ ( F2 @ N3 ) )
=> ? [I2: nat] :
( ( ord_less_eq @ nat @ M @ I2 )
& ( ord_less_eq @ nat @ I2 @ N3 )
& ( ( F2 @ I2 )
= K ) ) ) ) ) ) ).
% nat_intermed_int_val
thf(fact_4262_Cauchy__product,axiom,
! [A: $tType] :
( ( ( real_Vector_banach @ A )
& ( real_V4412858255891104859lgebra @ A ) )
=> ! [A2: nat > A,B2: nat > A] :
( ( summable @ real
@ ^ [K2: nat] : ( real_V7770717601297561774m_norm @ A @ ( A2 @ K2 ) ) )
=> ( ( summable @ real
@ ^ [K2: nat] : ( real_V7770717601297561774m_norm @ A @ ( B2 @ K2 ) ) )
=> ( ( times_times @ A @ ( suminf @ A @ A2 ) @ ( suminf @ A @ B2 ) )
= ( suminf @ A
@ ^ [K2: nat] :
( groups7311177749621191930dd_sum @ nat @ A
@ ^ [I3: nat] : ( times_times @ A @ ( A2 @ I3 ) @ ( B2 @ ( minus_minus @ nat @ K2 @ I3 ) ) )
@ ( set_ord_atMost @ nat @ K2 ) ) ) ) ) ) ) ).
% Cauchy_product
thf(fact_4263_decr__lemma,axiom,
! [D: int,X: int,Z: int] :
( ( ord_less @ int @ ( zero_zero @ int ) @ D )
=> ( ord_less @ int @ ( minus_minus @ int @ X @ ( times_times @ int @ ( plus_plus @ int @ ( abs_abs @ int @ ( minus_minus @ int @ X @ Z ) ) @ ( one_one @ int ) ) @ D ) ) @ Z ) ) ).
% decr_lemma
thf(fact_4264_incr__lemma,axiom,
! [D: int,Z: int,X: int] :
( ( ord_less @ int @ ( zero_zero @ int ) @ D )
=> ( ord_less @ int @ Z @ ( plus_plus @ int @ X @ ( times_times @ int @ ( plus_plus @ int @ ( abs_abs @ int @ ( minus_minus @ int @ X @ Z ) ) @ ( one_one @ int ) ) @ D ) ) ) ) ).
% incr_lemma
thf(fact_4265_summable__ratio__test,axiom,
! [A: $tType] :
( ( real_Vector_banach @ A )
=> ! [C2: real,N7: nat,F2: nat > A] :
( ( ord_less @ real @ C2 @ ( one_one @ real ) )
=> ( ! [N: nat] :
( ( ord_less_eq @ nat @ N7 @ N )
=> ( ord_less_eq @ real @ ( real_V7770717601297561774m_norm @ A @ ( F2 @ ( suc @ N ) ) ) @ ( times_times @ real @ C2 @ ( real_V7770717601297561774m_norm @ A @ ( F2 @ N ) ) ) ) )
=> ( summable @ A @ F2 ) ) ) ) ).
% summable_ratio_test
thf(fact_4266_sum__less__suminf2,axiom,
! [A: $tType] :
( ( ( ordere8940638589300402666id_add @ A )
& ( topolo1944317154257567458pology @ A ) )
=> ! [F2: nat > A,N3: nat,I: nat] :
( ( summable @ A @ F2 )
=> ( ! [M4: nat] :
( ( ord_less_eq @ nat @ N3 @ M4 )
=> ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( F2 @ M4 ) ) )
=> ( ( ord_less_eq @ nat @ N3 @ I )
=> ( ( ord_less @ A @ ( zero_zero @ A ) @ ( F2 @ I ) )
=> ( ord_less @ A @ ( groups7311177749621191930dd_sum @ nat @ A @ F2 @ ( set_ord_lessThan @ nat @ N3 ) ) @ ( suminf @ A @ F2 ) ) ) ) ) ) ) ).
% sum_less_suminf2
thf(fact_4267_arctan__ubound,axiom,
! [Y: real] : ( ord_less @ real @ ( arctan @ Y ) @ ( divide_divide @ real @ pi @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) ) ).
% arctan_ubound
thf(fact_4268_arctan__one,axiom,
( ( arctan @ ( one_one @ real ) )
= ( divide_divide @ real @ pi @ ( numeral_numeral @ real @ ( bit0 @ ( bit0 @ one2 ) ) ) ) ) ).
% arctan_one
thf(fact_4269_nat__ivt__aux,axiom,
! [N3: nat,F2: nat > int,K: int] :
( ! [I2: nat] :
( ( ord_less @ nat @ I2 @ N3 )
=> ( 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 @ N3 ) )
=> ? [I2: nat] :
( ( ord_less_eq @ nat @ I2 @ N3 )
& ( ( F2 @ I2 )
= K ) ) ) ) ) ).
% nat_ivt_aux
thf(fact_4270_Cauchy__product__sums,axiom,
! [A: $tType] :
( ( ( real_Vector_banach @ A )
& ( real_V4412858255891104859lgebra @ A ) )
=> ! [A2: nat > A,B2: nat > A] :
( ( summable @ real
@ ^ [K2: nat] : ( real_V7770717601297561774m_norm @ A @ ( A2 @ K2 ) ) )
=> ( ( summable @ real
@ ^ [K2: nat] : ( real_V7770717601297561774m_norm @ A @ ( B2 @ K2 ) ) )
=> ( sums @ A
@ ^ [K2: nat] :
( groups7311177749621191930dd_sum @ nat @ A
@ ^ [I3: nat] : ( times_times @ A @ ( A2 @ I3 ) @ ( B2 @ ( minus_minus @ nat @ K2 @ I3 ) ) )
@ ( set_ord_atMost @ nat @ K2 ) )
@ ( times_times @ A @ ( suminf @ A @ A2 ) @ ( suminf @ A @ B2 ) ) ) ) ) ) ).
% Cauchy_product_sums
thf(fact_4271_arctan__bounded,axiom,
! [Y: real] :
( ( ord_less @ real @ ( uminus_uminus @ real @ ( divide_divide @ real @ pi @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) ) @ ( arctan @ Y ) )
& ( ord_less @ real @ ( arctan @ Y ) @ ( divide_divide @ real @ pi @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) ) ) ).
% arctan_bounded
thf(fact_4272_arctan__lbound,axiom,
! [Y: real] : ( ord_less @ real @ ( uminus_uminus @ real @ ( divide_divide @ real @ pi @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) ) @ ( arctan @ Y ) ) ).
% arctan_lbound
thf(fact_4273_nat0__intermed__int__val,axiom,
! [N3: nat,F2: nat > int,K: int] :
( ! [I2: nat] :
( ( ord_less @ nat @ I2 @ N3 )
=> ( 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 @ N3 ) )
=> ? [I2: nat] :
( ( ord_less_eq @ nat @ I2 @ N3 )
& ( ( F2 @ I2 )
= K ) ) ) ) ) ).
% nat0_intermed_int_val
thf(fact_4274_ln__one__plus__pos__lower__bound,axiom,
! [X: real] :
( ( ord_less_eq @ real @ ( zero_zero @ real ) @ X )
=> ( ( ord_less_eq @ real @ X @ ( one_one @ real ) )
=> ( ord_less_eq @ real @ ( minus_minus @ real @ X @ ( power_power @ real @ X @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( ln_ln @ real @ ( plus_plus @ real @ ( one_one @ real ) @ X ) ) ) ) ) ).
% ln_one_plus_pos_lower_bound
thf(fact_4275_arctan__add,axiom,
! [X: real,Y: real] :
( ( ord_less_eq @ real @ ( abs_abs @ real @ X ) @ ( one_one @ real ) )
=> ( ( ord_less @ real @ ( abs_abs @ real @ Y ) @ ( one_one @ real ) )
=> ( ( plus_plus @ real @ ( arctan @ X ) @ ( arctan @ Y ) )
= ( arctan @ ( divide_divide @ real @ ( plus_plus @ real @ X @ Y ) @ ( minus_minus @ real @ ( one_one @ real ) @ ( times_times @ real @ X @ Y ) ) ) ) ) ) ) ).
% arctan_add
thf(fact_4276_abs__ln__one__plus__x__minus__x__bound__nonneg,axiom,
! [X: real] :
( ( ord_less_eq @ real @ ( zero_zero @ real ) @ X )
=> ( ( ord_less_eq @ real @ X @ ( one_one @ real ) )
=> ( ord_less_eq @ real @ ( abs_abs @ real @ ( minus_minus @ real @ ( ln_ln @ real @ ( plus_plus @ real @ ( one_one @ real ) @ X ) ) @ X ) ) @ ( power_power @ real @ X @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) ).
% abs_ln_one_plus_x_minus_x_bound_nonneg
thf(fact_4277_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_4278_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_4279_sum__pos__lt__pair,axiom,
! [F2: nat > real,K: nat] :
( ( summable @ real @ F2 )
=> ( ! [D3: nat] : ( ord_less @ real @ ( zero_zero @ real ) @ ( plus_plus @ real @ ( F2 @ ( plus_plus @ nat @ K @ ( times_times @ nat @ ( suc @ ( suc @ ( zero_zero @ nat ) ) ) @ D3 ) ) ) @ ( F2 @ ( plus_plus @ nat @ K @ ( plus_plus @ nat @ ( times_times @ nat @ ( suc @ ( suc @ ( zero_zero @ nat ) ) ) @ D3 ) @ ( one_one @ nat ) ) ) ) ) )
=> ( ord_less @ real @ ( groups7311177749621191930dd_sum @ nat @ real @ F2 @ ( set_ord_lessThan @ nat @ K ) ) @ ( suminf @ real @ F2 ) ) ) ) ).
% sum_pos_lt_pair
thf(fact_4280_ln__one__minus__pos__lower__bound,axiom,
! [X: real] :
( ( ord_less_eq @ real @ ( zero_zero @ real ) @ X )
=> ( ( ord_less_eq @ real @ X @ ( divide_divide @ real @ ( one_one @ real ) @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) )
=> ( ord_less_eq @ real @ ( minus_minus @ real @ ( uminus_uminus @ real @ X ) @ ( times_times @ real @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) @ ( power_power @ real @ X @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( ln_ln @ real @ ( minus_minus @ real @ ( one_one @ real ) @ X ) ) ) ) ) ).
% ln_one_minus_pos_lower_bound
thf(fact_4281_abs__ln__one__plus__x__minus__x__bound,axiom,
! [X: real] :
( ( ord_less_eq @ real @ ( abs_abs @ real @ X ) @ ( divide_divide @ real @ ( one_one @ real ) @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) )
=> ( ord_less_eq @ real @ ( abs_abs @ real @ ( minus_minus @ real @ ( ln_ln @ real @ ( plus_plus @ real @ ( one_one @ real ) @ X ) ) @ X ) ) @ ( times_times @ real @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) @ ( power_power @ real @ X @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) ).
% abs_ln_one_plus_x_minus_x_bound
thf(fact_4282_arctan__double,axiom,
! [X: real] :
( ( ord_less @ real @ ( abs_abs @ real @ X ) @ ( one_one @ real ) )
=> ( ( times_times @ real @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) @ ( arctan @ X ) )
= ( arctan @ ( divide_divide @ real @ ( times_times @ real @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) @ X ) @ ( minus_minus @ real @ ( one_one @ real ) @ ( power_power @ real @ X @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) ) ) ).
% arctan_double
thf(fact_4283_abs__ln__one__plus__x__minus__x__bound__nonpos,axiom,
! [X: real] :
( ( ord_less_eq @ real @ ( uminus_uminus @ real @ ( divide_divide @ real @ ( one_one @ real ) @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) ) @ X )
=> ( ( ord_less_eq @ real @ X @ ( zero_zero @ real ) )
=> ( ord_less_eq @ real @ ( abs_abs @ real @ ( minus_minus @ real @ ( ln_ln @ real @ ( plus_plus @ real @ ( one_one @ real ) @ X ) ) @ X ) ) @ ( times_times @ real @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) @ ( power_power @ real @ X @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) ) ).
% abs_ln_one_plus_x_minus_x_bound_nonpos
thf(fact_4284_artanh__def,axiom,
! [A: $tType] :
( ( ( real_V3459762299906320749_field @ A )
& ( ln @ A ) )
=> ( ( artanh @ A )
= ( ^ [X2: A] : ( divide_divide @ A @ ( ln_ln @ A @ ( divide_divide @ A @ ( plus_plus @ A @ ( one_one @ A ) @ X2 ) @ ( minus_minus @ A @ ( one_one @ A ) @ X2 ) ) ) @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) ) ) ) ).
% artanh_def
thf(fact_4285_diffs__equiv,axiom,
! [A: $tType] :
( ( ( real_V822414075346904944vector @ A )
& ( ring_1 @ A ) )
=> ! [C2: nat > A,X: A] :
( ( summable @ A
@ ^ [N2: nat] : ( times_times @ A @ ( diffs @ A @ C2 @ N2 ) @ ( power_power @ A @ X @ N2 ) ) )
=> ( sums @ A
@ ^ [N2: nat] : ( times_times @ A @ ( times_times @ A @ ( semiring_1_of_nat @ A @ N2 ) @ ( C2 @ N2 ) ) @ ( power_power @ A @ X @ ( minus_minus @ nat @ N2 @ ( suc @ ( zero_zero @ nat ) ) ) ) )
@ ( suminf @ A
@ ^ [N2: nat] : ( times_times @ A @ ( diffs @ A @ C2 @ N2 ) @ ( power_power @ A @ X @ N2 ) ) ) ) ) ) ).
% diffs_equiv
thf(fact_4286_tanh__ln__real,axiom,
! [X: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ X )
=> ( ( tanh @ real @ ( ln_ln @ real @ X ) )
= ( divide_divide @ real @ ( minus_minus @ real @ ( power_power @ real @ X @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ ( one_one @ real ) ) @ ( plus_plus @ real @ ( power_power @ real @ X @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ ( one_one @ real ) ) ) ) ) ).
% tanh_ln_real
thf(fact_4287_tan__double,axiom,
! [A: $tType] :
( ( ( real_Vector_banach @ A )
& ( real_V3459762299906320749_field @ A ) )
=> ! [X: A] :
( ( ( cos @ A @ X )
!= ( zero_zero @ A ) )
=> ( ( ( cos @ A @ ( times_times @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ X ) )
!= ( zero_zero @ A ) )
=> ( ( tan @ A @ ( times_times @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ X ) )
= ( divide_divide @ A @ ( times_times @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ ( tan @ A @ X ) ) @ ( minus_minus @ A @ ( one_one @ A ) @ ( power_power @ A @ ( tan @ A @ X ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) ) ) ) ).
% tan_double
thf(fact_4288_tan__periodic__pi,axiom,
! [X: real] :
( ( tan @ real @ ( plus_plus @ real @ X @ pi ) )
= ( tan @ real @ X ) ) ).
% tan_periodic_pi
thf(fact_4289_tanh__real__le__iff,axiom,
! [X: real,Y: real] :
( ( ord_less_eq @ real @ ( tanh @ real @ X ) @ ( tanh @ real @ Y ) )
= ( ord_less_eq @ real @ X @ Y ) ) ).
% tanh_real_le_iff
thf(fact_4290_tanh__real__nonpos__iff,axiom,
! [X: real] :
( ( ord_less_eq @ real @ ( tanh @ real @ X ) @ ( zero_zero @ real ) )
= ( ord_less_eq @ real @ X @ ( zero_zero @ real ) ) ) ).
% tanh_real_nonpos_iff
thf(fact_4291_tanh__real__nonneg__iff,axiom,
! [X: real] :
( ( ord_less_eq @ real @ ( zero_zero @ real ) @ ( tanh @ real @ X ) )
= ( ord_less_eq @ real @ ( zero_zero @ real ) @ X ) ) ).
% tanh_real_nonneg_iff
thf(fact_4292_tan__npi,axiom,
! [N3: nat] :
( ( tan @ real @ ( times_times @ real @ ( semiring_1_of_nat @ real @ N3 ) @ pi ) )
= ( zero_zero @ real ) ) ).
% tan_npi
thf(fact_4293_artanh__minus__real,axiom,
! [X: real] :
( ( ord_less @ real @ ( abs_abs @ real @ X ) @ ( one_one @ real ) )
=> ( ( artanh @ real @ ( uminus_uminus @ real @ X ) )
= ( uminus_uminus @ real @ ( artanh @ real @ X ) ) ) ) ).
% artanh_minus_real
thf(fact_4294_tan__periodic__n,axiom,
! [X: real,N3: num] :
( ( tan @ real @ ( plus_plus @ real @ X @ ( times_times @ real @ ( numeral_numeral @ real @ N3 ) @ pi ) ) )
= ( tan @ real @ X ) ) ).
% tan_periodic_n
thf(fact_4295_tan__periodic__nat,axiom,
! [X: real,N3: nat] :
( ( tan @ real @ ( plus_plus @ real @ X @ ( times_times @ real @ ( semiring_1_of_nat @ real @ N3 ) @ pi ) ) )
= ( tan @ real @ X ) ) ).
% tan_periodic_nat
thf(fact_4296_tan__periodic__int,axiom,
! [X: real,I: int] :
( ( tan @ real @ ( plus_plus @ real @ X @ ( times_times @ real @ ( ring_1_of_int @ real @ I ) @ pi ) ) )
= ( tan @ real @ X ) ) ).
% tan_periodic_int
thf(fact_4297_tan__periodic,axiom,
! [X: real] :
( ( tan @ real @ ( plus_plus @ real @ X @ ( times_times @ real @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) @ pi ) ) )
= ( tan @ real @ X ) ) ).
% tan_periodic
thf(fact_4298_tanh__real__lt__1,axiom,
! [X: real] : ( ord_less @ real @ ( tanh @ real @ X ) @ ( one_one @ real ) ) ).
% tanh_real_lt_1
thf(fact_4299_tanh__real__gt__neg1,axiom,
! [X: real] : ( ord_less @ real @ ( uminus_uminus @ real @ ( one_one @ real ) ) @ ( tanh @ real @ X ) ) ).
% tanh_real_gt_neg1
thf(fact_4300_tan__def,axiom,
! [A: $tType] :
( ( ( real_Vector_banach @ A )
& ( real_V3459762299906320749_field @ A ) )
=> ( ( tan @ A )
= ( ^ [X2: A] : ( divide_divide @ A @ ( sin @ A @ X2 ) @ ( cos @ A @ X2 ) ) ) ) ) ).
% tan_def
thf(fact_4301_diffs__def,axiom,
! [A: $tType] :
( ( ring_1 @ A )
=> ( ( diffs @ A )
= ( ^ [C3: nat > A,N2: nat] : ( times_times @ A @ ( semiring_1_of_nat @ A @ ( suc @ N2 ) ) @ ( C3 @ ( suc @ N2 ) ) ) ) ) ) ).
% diffs_def
thf(fact_4302_termdiff__converges__all,axiom,
! [A: $tType] :
( ( ( real_Vector_banach @ A )
& ( real_V3459762299906320749_field @ A ) )
=> ! [C2: nat > A,X: A] :
( ! [X3: A] :
( summable @ A
@ ^ [N2: nat] : ( times_times @ A @ ( C2 @ N2 ) @ ( power_power @ A @ X3 @ N2 ) ) )
=> ( summable @ A
@ ^ [N2: nat] : ( times_times @ A @ ( diffs @ A @ C2 @ N2 ) @ ( power_power @ A @ X @ N2 ) ) ) ) ) ).
% termdiff_converges_all
thf(fact_4303_tan__45,axiom,
( ( tan @ real @ ( divide_divide @ real @ pi @ ( numeral_numeral @ real @ ( bit0 @ ( bit0 @ one2 ) ) ) ) )
= ( one_one @ real ) ) ).
% tan_45
thf(fact_4304_tan__gt__zero,axiom,
! [X: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ X )
=> ( ( ord_less @ real @ X @ ( divide_divide @ real @ pi @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) )
=> ( ord_less @ real @ ( zero_zero @ real ) @ ( tan @ real @ X ) ) ) ) ).
% tan_gt_zero
thf(fact_4305_lemma__tan__total,axiom,
! [Y: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ Y )
=> ? [X3: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ X3 )
& ( ord_less @ real @ X3 @ ( divide_divide @ real @ pi @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) )
& ( ord_less @ real @ Y @ ( tan @ real @ X3 ) ) ) ) ).
% lemma_tan_total
thf(fact_4306_lemma__tan__total1,axiom,
! [Y: real] :
? [X3: real] :
( ( ord_less @ real @ ( uminus_uminus @ real @ ( divide_divide @ real @ pi @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) ) @ X3 )
& ( ord_less @ real @ X3 @ ( divide_divide @ real @ pi @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) )
& ( ( tan @ real @ X3 )
= Y ) ) ).
% lemma_tan_total1
thf(fact_4307_tan__mono__lt__eq,axiom,
! [X: real,Y: real] :
( ( ord_less @ real @ ( uminus_uminus @ real @ ( divide_divide @ real @ pi @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) ) @ X )
=> ( ( ord_less @ real @ X @ ( divide_divide @ real @ pi @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) )
=> ( ( ord_less @ real @ ( uminus_uminus @ real @ ( divide_divide @ real @ pi @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) ) @ Y )
=> ( ( ord_less @ real @ Y @ ( divide_divide @ real @ pi @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) )
=> ( ( ord_less @ real @ ( tan @ real @ X ) @ ( tan @ real @ Y ) )
= ( ord_less @ real @ X @ Y ) ) ) ) ) ) ).
% tan_mono_lt_eq
thf(fact_4308_tan__monotone_H,axiom,
! [Y: real,X: real] :
( ( ord_less @ real @ ( uminus_uminus @ real @ ( divide_divide @ real @ pi @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) ) @ Y )
=> ( ( ord_less @ real @ Y @ ( divide_divide @ real @ pi @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) )
=> ( ( ord_less @ real @ ( uminus_uminus @ real @ ( divide_divide @ real @ pi @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) ) @ X )
=> ( ( ord_less @ real @ X @ ( divide_divide @ real @ pi @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) )
=> ( ( ord_less @ real @ Y @ X )
= ( ord_less @ real @ ( tan @ real @ Y ) @ ( tan @ real @ X ) ) ) ) ) ) ) ).
% tan_monotone'
thf(fact_4309_tan__monotone,axiom,
! [Y: real,X: real] :
( ( ord_less @ real @ ( uminus_uminus @ real @ ( divide_divide @ real @ pi @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) ) @ Y )
=> ( ( ord_less @ real @ Y @ X )
=> ( ( ord_less @ real @ X @ ( divide_divide @ real @ pi @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) )
=> ( ord_less @ real @ ( tan @ real @ Y ) @ ( tan @ real @ X ) ) ) ) ) ).
% tan_monotone
thf(fact_4310_tan__total,axiom,
! [Y: real] :
? [X3: real] :
( ( ord_less @ real @ ( uminus_uminus @ real @ ( divide_divide @ real @ pi @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) ) @ X3 )
& ( ord_less @ real @ X3 @ ( divide_divide @ real @ pi @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) )
& ( ( tan @ real @ X3 )
= Y )
& ! [Y4: real] :
( ( ( ord_less @ real @ ( uminus_uminus @ real @ ( divide_divide @ real @ pi @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) ) @ Y4 )
& ( ord_less @ real @ Y4 @ ( divide_divide @ real @ pi @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) )
& ( ( tan @ real @ Y4 )
= Y ) )
=> ( Y4 = X3 ) ) ) ).
% tan_total
thf(fact_4311_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_4312_tan__inverse,axiom,
! [Y: real] :
( ( divide_divide @ real @ ( one_one @ real ) @ ( tan @ real @ Y ) )
= ( tan @ real @ ( minus_minus @ real @ ( divide_divide @ real @ pi @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) @ Y ) ) ) ).
% tan_inverse
thf(fact_4313_add__tan__eq,axiom,
! [A: $tType] :
( ( ( real_Vector_banach @ A )
& ( real_V3459762299906320749_field @ A ) )
=> ! [X: A,Y: A] :
( ( ( cos @ A @ X )
!= ( zero_zero @ A ) )
=> ( ( ( cos @ A @ Y )
!= ( zero_zero @ A ) )
=> ( ( plus_plus @ A @ ( tan @ A @ X ) @ ( tan @ A @ Y ) )
= ( divide_divide @ A @ ( sin @ A @ ( plus_plus @ A @ X @ Y ) ) @ ( times_times @ A @ ( cos @ A @ X ) @ ( cos @ A @ Y ) ) ) ) ) ) ) ).
% add_tan_eq
thf(fact_4314_termdiff__converges,axiom,
! [A: $tType] :
( ( ( real_Vector_banach @ A )
& ( real_V3459762299906320749_field @ A ) )
=> ! [X: A,K6: real,C2: nat > A] :
( ( ord_less @ real @ ( real_V7770717601297561774m_norm @ A @ X ) @ K6 )
=> ( ! [X3: A] :
( ( ord_less @ real @ ( real_V7770717601297561774m_norm @ A @ X3 ) @ K6 )
=> ( summable @ A
@ ^ [N2: nat] : ( times_times @ A @ ( C2 @ N2 ) @ ( power_power @ A @ X3 @ N2 ) ) ) )
=> ( summable @ A
@ ^ [N2: nat] : ( times_times @ A @ ( diffs @ A @ C2 @ N2 ) @ ( power_power @ A @ X @ N2 ) ) ) ) ) ) ).
% termdiff_converges
thf(fact_4315_tan__pos__pi2__le,axiom,
! [X: real] :
( ( ord_less_eq @ real @ ( zero_zero @ real ) @ X )
=> ( ( ord_less @ real @ X @ ( divide_divide @ real @ pi @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) )
=> ( ord_less_eq @ real @ ( zero_zero @ real ) @ ( tan @ real @ X ) ) ) ) ).
% tan_pos_pi2_le
thf(fact_4316_tan__total__pos,axiom,
! [Y: real] :
( ( ord_less_eq @ real @ ( zero_zero @ real ) @ Y )
=> ? [X3: real] :
( ( ord_less_eq @ real @ ( zero_zero @ real ) @ X3 )
& ( ord_less @ real @ X3 @ ( divide_divide @ real @ pi @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) )
& ( ( tan @ real @ X3 )
= Y ) ) ) ).
% tan_total_pos
thf(fact_4317_tan__less__zero,axiom,
! [X: real] :
( ( ord_less @ real @ ( divide_divide @ real @ ( uminus_uminus @ real @ pi ) @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) @ X )
=> ( ( ord_less @ real @ X @ ( zero_zero @ real ) )
=> ( ord_less @ real @ ( tan @ real @ X ) @ ( zero_zero @ real ) ) ) ) ).
% tan_less_zero
thf(fact_4318_tan__mono__le__eq,axiom,
! [X: real,Y: real] :
( ( ord_less @ real @ ( uminus_uminus @ real @ ( divide_divide @ real @ pi @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) ) @ X )
=> ( ( ord_less @ real @ X @ ( divide_divide @ real @ pi @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) )
=> ( ( ord_less @ real @ ( uminus_uminus @ real @ ( divide_divide @ real @ pi @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) ) @ Y )
=> ( ( ord_less @ real @ Y @ ( divide_divide @ real @ pi @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) )
=> ( ( ord_less_eq @ real @ ( tan @ real @ X ) @ ( tan @ real @ Y ) )
= ( ord_less_eq @ real @ X @ Y ) ) ) ) ) ) ).
% tan_mono_le_eq
thf(fact_4319_tan__mono__le,axiom,
! [X: real,Y: real] :
( ( ord_less @ real @ ( uminus_uminus @ real @ ( divide_divide @ real @ pi @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) ) @ X )
=> ( ( ord_less_eq @ real @ X @ Y )
=> ( ( ord_less @ real @ Y @ ( divide_divide @ real @ pi @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) )
=> ( ord_less_eq @ real @ ( tan @ real @ X ) @ ( tan @ real @ Y ) ) ) ) ) ).
% tan_mono_le
thf(fact_4320_tan__bound__pi2,axiom,
! [X: real] :
( ( ord_less @ real @ ( abs_abs @ real @ X ) @ ( divide_divide @ real @ pi @ ( numeral_numeral @ real @ ( bit0 @ ( bit0 @ one2 ) ) ) ) )
=> ( ord_less @ real @ ( abs_abs @ real @ ( tan @ real @ X ) ) @ ( one_one @ real ) ) ) ).
% tan_bound_pi2
thf(fact_4321_arctan,axiom,
! [Y: real] :
( ( ord_less @ real @ ( uminus_uminus @ real @ ( divide_divide @ real @ pi @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) ) @ ( arctan @ Y ) )
& ( ord_less @ real @ ( arctan @ Y ) @ ( divide_divide @ real @ pi @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) )
& ( ( tan @ real @ ( arctan @ Y ) )
= Y ) ) ).
% arctan
thf(fact_4322_arctan__tan,axiom,
! [X: real] :
( ( ord_less @ real @ ( uminus_uminus @ real @ ( divide_divide @ real @ pi @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) ) @ X )
=> ( ( ord_less @ real @ X @ ( divide_divide @ real @ pi @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) )
=> ( ( arctan @ ( tan @ real @ X ) )
= X ) ) ) ).
% arctan_tan
thf(fact_4323_arctan__unique,axiom,
! [X: real,Y: real] :
( ( ord_less @ real @ ( uminus_uminus @ real @ ( divide_divide @ real @ pi @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) ) @ X )
=> ( ( ord_less @ real @ X @ ( divide_divide @ real @ pi @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) )
=> ( ( ( tan @ real @ X )
= Y )
=> ( ( arctan @ Y )
= X ) ) ) ) ).
% arctan_unique
thf(fact_4324_lemma__tan__add1,axiom,
! [A: $tType] :
( ( ( real_Vector_banach @ A )
& ( real_V3459762299906320749_field @ A ) )
=> ! [X: A,Y: A] :
( ( ( cos @ A @ X )
!= ( zero_zero @ A ) )
=> ( ( ( cos @ A @ Y )
!= ( zero_zero @ A ) )
=> ( ( minus_minus @ A @ ( one_one @ A ) @ ( times_times @ A @ ( tan @ A @ X ) @ ( tan @ A @ Y ) ) )
= ( divide_divide @ A @ ( cos @ A @ ( plus_plus @ A @ X @ Y ) ) @ ( times_times @ A @ ( cos @ A @ X ) @ ( cos @ A @ Y ) ) ) ) ) ) ) ).
% lemma_tan_add1
thf(fact_4325_tan__diff,axiom,
! [A: $tType] :
( ( ( real_Vector_banach @ A )
& ( real_V3459762299906320749_field @ A ) )
=> ! [X: A,Y: A] :
( ( ( cos @ A @ X )
!= ( zero_zero @ A ) )
=> ( ( ( cos @ A @ Y )
!= ( zero_zero @ A ) )
=> ( ( ( cos @ A @ ( minus_minus @ A @ X @ Y ) )
!= ( zero_zero @ A ) )
=> ( ( tan @ A @ ( minus_minus @ A @ X @ Y ) )
= ( divide_divide @ A @ ( minus_minus @ A @ ( tan @ A @ X ) @ ( tan @ A @ Y ) ) @ ( plus_plus @ A @ ( one_one @ A ) @ ( times_times @ A @ ( tan @ A @ X ) @ ( tan @ A @ Y ) ) ) ) ) ) ) ) ) ).
% tan_diff
thf(fact_4326_tan__add,axiom,
! [A: $tType] :
( ( ( real_Vector_banach @ A )
& ( real_V3459762299906320749_field @ A ) )
=> ! [X: A,Y: A] :
( ( ( cos @ A @ X )
!= ( zero_zero @ A ) )
=> ( ( ( cos @ A @ Y )
!= ( zero_zero @ A ) )
=> ( ( ( cos @ A @ ( plus_plus @ A @ X @ Y ) )
!= ( zero_zero @ A ) )
=> ( ( tan @ A @ ( plus_plus @ A @ X @ Y ) )
= ( divide_divide @ A @ ( plus_plus @ A @ ( tan @ A @ X ) @ ( tan @ A @ Y ) ) @ ( minus_minus @ A @ ( one_one @ A ) @ ( times_times @ A @ ( tan @ A @ X ) @ ( tan @ A @ Y ) ) ) ) ) ) ) ) ) ).
% tan_add
thf(fact_4327_tan__total__pi4,axiom,
! [X: real] :
( ( ord_less @ real @ ( abs_abs @ real @ X ) @ ( 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 )
= X ) ) ) ).
% tan_total_pi4
thf(fact_4328_tan__half,axiom,
! [A: $tType] :
( ( ( real_Vector_banach @ A )
& ( real_V3459762299906320749_field @ A ) )
=> ( ( tan @ A )
= ( ^ [X2: A] : ( divide_divide @ A @ ( sin @ A @ ( times_times @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ X2 ) ) @ ( plus_plus @ A @ ( cos @ A @ ( times_times @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ X2 ) ) @ ( one_one @ A ) ) ) ) ) ) ).
% tan_half
thf(fact_4329_arcosh__def,axiom,
! [A: $tType] :
( ( ln @ A )
=> ( ( arcosh @ A )
= ( ^ [X2: A] : ( ln_ln @ A @ ( plus_plus @ A @ X2 @ ( powr @ A @ ( minus_minus @ A @ ( power_power @ A @ X2 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ ( one_one @ A ) ) @ ( real_Vector_of_real @ A @ ( divide_divide @ real @ ( one_one @ real ) @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) ) ) ) ) ) ) ) ).
% arcosh_def
thf(fact_4330_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_4331_Maclaurin__exp__lt,axiom,
! [X: real,N3: nat] :
( ( X
!= ( zero_zero @ real ) )
=> ( ( ord_less @ nat @ ( zero_zero @ nat ) @ N3 )
=> ? [T8: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ ( abs_abs @ real @ T8 ) )
& ( ord_less @ real @ ( abs_abs @ real @ T8 ) @ ( abs_abs @ real @ X ) )
& ( ( exp @ real @ X )
= ( plus_plus @ real
@ ( groups7311177749621191930dd_sum @ nat @ real
@ ^ [M5: nat] : ( divide_divide @ real @ ( power_power @ real @ X @ M5 ) @ ( semiring_char_0_fact @ real @ M5 ) )
@ ( set_ord_lessThan @ nat @ N3 ) )
@ ( times_times @ real @ ( divide_divide @ real @ ( exp @ real @ T8 ) @ ( semiring_char_0_fact @ real @ N3 ) ) @ ( power_power @ real @ X @ N3 ) ) ) ) ) ) ) ).
% Maclaurin_exp_lt
thf(fact_4332_list__assn__aux__ineq__len,axiom,
! [B: $tType,A: $tType,L2: list @ A,Li2: list @ B,A3: A > B > assn] :
( ( ( size_size @ ( list @ A ) @ L2 )
!= ( size_size @ ( list @ B ) @ Li2 ) )
=> ( ( vEBT_List_list_assn @ A @ B @ A3 @ L2 @ Li2 )
= ( bot_bot @ assn ) ) ) ).
% list_assn_aux_ineq_len
thf(fact_4333_exp__le__cancel__iff,axiom,
! [X: real,Y: real] :
( ( ord_less_eq @ real @ ( exp @ real @ X ) @ ( exp @ real @ Y ) )
= ( ord_less_eq @ real @ X @ Y ) ) ).
% exp_le_cancel_iff
thf(fact_4334_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_4335_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_4336_of__real__eq__1__iff,axiom,
! [A: $tType] :
( ( real_V2191834092415804123ebra_1 @ A )
=> ! [X: real] :
( ( ( real_Vector_of_real @ A @ X )
= ( one_one @ A ) )
= ( X
= ( one_one @ real ) ) ) ) ).
% of_real_eq_1_iff
thf(fact_4337_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_4338_of__real__mult,axiom,
! [A: $tType] :
( ( real_V2191834092415804123ebra_1 @ A )
=> ! [X: real,Y: real] :
( ( real_Vector_of_real @ A @ ( times_times @ real @ X @ Y ) )
= ( times_times @ A @ ( real_Vector_of_real @ A @ X ) @ ( real_Vector_of_real @ A @ Y ) ) ) ) ).
% of_real_mult
thf(fact_4339_of__real__divide,axiom,
! [A: $tType] :
( ( real_V5047593784448816457lgebra @ A )
=> ! [X: real,Y: real] :
( ( real_Vector_of_real @ A @ ( divide_divide @ real @ X @ Y ) )
= ( divide_divide @ A @ ( real_Vector_of_real @ A @ X ) @ ( real_Vector_of_real @ A @ Y ) ) ) ) ).
% of_real_divide
thf(fact_4340_of__real__add,axiom,
! [A: $tType] :
( ( real_V2191834092415804123ebra_1 @ A )
=> ! [X: real,Y: real] :
( ( real_Vector_of_real @ A @ ( plus_plus @ real @ X @ Y ) )
= ( plus_plus @ A @ ( real_Vector_of_real @ A @ X ) @ ( real_Vector_of_real @ A @ Y ) ) ) ) ).
% of_real_add
thf(fact_4341_of__real__power,axiom,
! [A: $tType] :
( ( real_V2191834092415804123ebra_1 @ A )
=> ! [X: real,N3: nat] :
( ( real_Vector_of_real @ A @ ( power_power @ real @ X @ N3 ) )
= ( power_power @ A @ ( real_Vector_of_real @ A @ X ) @ N3 ) ) ) ).
% of_real_power
thf(fact_4342_of__real__diff,axiom,
! [A: $tType] :
( ( real_V2191834092415804123ebra_1 @ A )
=> ! [X: real,Y: real] :
( ( real_Vector_of_real @ A @ ( minus_minus @ real @ X @ Y ) )
= ( minus_minus @ A @ ( real_Vector_of_real @ A @ X ) @ ( real_Vector_of_real @ A @ Y ) ) ) ) ).
% of_real_diff
thf(fact_4343_exp__eq__one__iff,axiom,
! [X: real] :
( ( ( exp @ real @ X )
= ( one_one @ real ) )
= ( X
= ( zero_zero @ real ) ) ) ).
% exp_eq_one_iff
thf(fact_4344_exp__less__one__iff,axiom,
! [X: real] :
( ( ord_less @ real @ ( exp @ real @ X ) @ ( one_one @ real ) )
= ( ord_less @ real @ X @ ( zero_zero @ real ) ) ) ).
% exp_less_one_iff
thf(fact_4345_one__less__exp__iff,axiom,
! [X: real] :
( ( ord_less @ real @ ( one_one @ real ) @ ( exp @ real @ X ) )
= ( ord_less @ real @ ( zero_zero @ real ) @ X ) ) ).
% one_less_exp_iff
thf(fact_4346_one__le__exp__iff,axiom,
! [X: real] :
( ( ord_less_eq @ real @ ( one_one @ real ) @ ( exp @ real @ X ) )
= ( ord_less_eq @ real @ ( zero_zero @ real ) @ X ) ) ).
% one_le_exp_iff
thf(fact_4347_exp__le__one__iff,axiom,
! [X: real] :
( ( ord_less_eq @ real @ ( exp @ real @ X ) @ ( one_one @ real ) )
= ( ord_less_eq @ real @ X @ ( zero_zero @ real ) ) ) ).
% exp_le_one_iff
thf(fact_4348_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_4349_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_4350_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_4351_norm__of__real__add1,axiom,
! [A: $tType] :
( ( real_V8999393235501362500lgebra @ A )
=> ! [X: real] :
( ( real_V7770717601297561774m_norm @ A @ ( plus_plus @ A @ ( real_Vector_of_real @ A @ X ) @ ( one_one @ A ) ) )
= ( abs_abs @ real @ ( plus_plus @ real @ X @ ( one_one @ real ) ) ) ) ) ).
% norm_of_real_add1
thf(fact_4352_norm__of__real__addn,axiom,
! [A: $tType] :
( ( real_V8999393235501362500lgebra @ A )
=> ! [X: real,B2: num] :
( ( real_V7770717601297561774m_norm @ A @ ( plus_plus @ A @ ( real_Vector_of_real @ A @ X ) @ ( numeral_numeral @ A @ B2 ) ) )
= ( abs_abs @ real @ ( plus_plus @ real @ X @ ( numeral_numeral @ real @ B2 ) ) ) ) ) ).
% norm_of_real_addn
thf(fact_4353_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_4354_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_4355_norm__exp,axiom,
! [A: $tType] :
( ( ( real_Vector_banach @ A )
& ( real_V2822296259951069270ebra_1 @ A ) )
=> ! [X: A] : ( ord_less_eq @ real @ ( real_V7770717601297561774m_norm @ A @ ( exp @ A @ X ) ) @ ( exp @ real @ ( real_V7770717601297561774m_norm @ A @ X ) ) ) ) ).
% norm_exp
thf(fact_4356_complex__diff,axiom,
! [A2: real,B2: real,C2: real,D: real] :
( ( minus_minus @ complex @ ( complex2 @ A2 @ B2 ) @ ( complex2 @ C2 @ D ) )
= ( complex2 @ ( minus_minus @ real @ A2 @ C2 ) @ ( minus_minus @ real @ B2 @ D ) ) ) ).
% complex_diff
thf(fact_4357_exp__times__arg__commute,axiom,
! [A: $tType] :
( ( ( real_Vector_banach @ A )
& ( real_V2822296259951069270ebra_1 @ A ) )
=> ! [A3: A] :
( ( times_times @ A @ ( exp @ A @ A3 ) @ A3 )
= ( times_times @ A @ A3 @ ( exp @ A @ A3 ) ) ) ) ).
% exp_times_arg_commute
thf(fact_4358_complex__of__real__mult__Complex,axiom,
! [R2: real,X: real,Y: real] :
( ( times_times @ complex @ ( real_Vector_of_real @ complex @ R2 ) @ ( complex2 @ X @ Y ) )
= ( complex2 @ ( times_times @ real @ R2 @ X ) @ ( times_times @ real @ R2 @ Y ) ) ) ).
% complex_of_real_mult_Complex
thf(fact_4359_Complex__mult__complex__of__real,axiom,
! [X: real,Y: real,R2: real] :
( ( times_times @ complex @ ( complex2 @ X @ Y ) @ ( real_Vector_of_real @ complex @ R2 ) )
= ( complex2 @ ( times_times @ real @ X @ R2 ) @ ( times_times @ real @ Y @ R2 ) ) ) ).
% Complex_mult_complex_of_real
thf(fact_4360_Complex__add__complex__of__real,axiom,
! [X: real,Y: real,R2: real] :
( ( plus_plus @ complex @ ( complex2 @ X @ Y ) @ ( real_Vector_of_real @ complex @ R2 ) )
= ( complex2 @ ( plus_plus @ real @ X @ R2 ) @ Y ) ) ).
% Complex_add_complex_of_real
thf(fact_4361_complex__of__real__add__Complex,axiom,
! [R2: real,X: real,Y: real] :
( ( plus_plus @ complex @ ( real_Vector_of_real @ complex @ R2 ) @ ( complex2 @ X @ Y ) )
= ( complex2 @ ( plus_plus @ real @ R2 @ X ) @ Y ) ) ).
% complex_of_real_add_Complex
thf(fact_4362_complex__scaleR,axiom,
! [R2: real,A2: real,B2: real] :
( ( real_V8093663219630862766scaleR @ complex @ R2 @ ( complex2 @ A2 @ B2 ) )
= ( complex2 @ ( times_times @ real @ R2 @ A2 ) @ ( times_times @ real @ R2 @ B2 ) ) ) ).
% complex_scaleR
thf(fact_4363_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_4364_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_4365_not__exp__le__zero,axiom,
! [X: real] :
~ ( ord_less_eq @ real @ ( exp @ real @ X ) @ ( zero_zero @ real ) ) ).
% not_exp_le_zero
thf(fact_4366_exp__ge__zero,axiom,
! [X: real] : ( ord_less_eq @ real @ ( zero_zero @ real ) @ ( exp @ real @ X ) ) ).
% exp_ge_zero
thf(fact_4367_exp__add__commuting,axiom,
! [A: $tType] :
( ( ( real_Vector_banach @ A )
& ( real_V2822296259951069270ebra_1 @ A ) )
=> ! [X: A,Y: A] :
( ( ( times_times @ A @ X @ Y )
= ( times_times @ A @ Y @ X ) )
=> ( ( exp @ A @ ( plus_plus @ A @ X @ Y ) )
= ( times_times @ A @ ( exp @ A @ X ) @ ( exp @ A @ Y ) ) ) ) ) ).
% exp_add_commuting
thf(fact_4368_mult__exp__exp,axiom,
! [A: $tType] :
( ( ( real_Vector_banach @ A )
& ( real_V3459762299906320749_field @ A ) )
=> ! [X: A,Y: A] :
( ( times_times @ A @ ( exp @ A @ X ) @ ( exp @ A @ Y ) )
= ( exp @ A @ ( plus_plus @ A @ X @ Y ) ) ) ) ).
% mult_exp_exp
thf(fact_4369_exp__diff,axiom,
! [A: $tType] :
( ( ( real_Vector_banach @ A )
& ( real_V3459762299906320749_field @ A ) )
=> ! [X: A,Y: A] :
( ( exp @ A @ ( minus_minus @ A @ X @ Y ) )
= ( divide_divide @ A @ ( exp @ A @ X ) @ ( exp @ A @ Y ) ) ) ) ).
% exp_diff
thf(fact_4370_Complex__eq__numeral,axiom,
! [A2: real,B2: real,W: num] :
( ( ( complex2 @ A2 @ B2 )
= ( numeral_numeral @ complex @ W ) )
= ( ( A2
= ( numeral_numeral @ real @ W ) )
& ( B2
= ( zero_zero @ real ) ) ) ) ).
% Complex_eq_numeral
thf(fact_4371_complex__add,axiom,
! [A2: real,B2: real,C2: real,D: real] :
( ( plus_plus @ complex @ ( complex2 @ A2 @ B2 ) @ ( complex2 @ C2 @ D ) )
= ( complex2 @ ( plus_plus @ real @ A2 @ C2 ) @ ( plus_plus @ real @ B2 @ D ) ) ) ).
% complex_add
thf(fact_4372_nonzero__of__real__divide,axiom,
! [A: $tType] :
( ( real_V7773925162809079976_field @ A )
=> ! [Y: real,X: real] :
( ( Y
!= ( zero_zero @ real ) )
=> ( ( real_Vector_of_real @ A @ ( divide_divide @ real @ X @ Y ) )
= ( divide_divide @ A @ ( real_Vector_of_real @ A @ X ) @ ( real_Vector_of_real @ A @ Y ) ) ) ) ) ).
% nonzero_of_real_divide
thf(fact_4373_exp__gt__one,axiom,
! [X: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ X )
=> ( ord_less @ real @ ( one_one @ real ) @ ( exp @ real @ X ) ) ) ).
% exp_gt_one
thf(fact_4374_exp__ge__add__one__self,axiom,
! [X: real] : ( ord_less_eq @ real @ ( plus_plus @ real @ ( one_one @ real ) @ X ) @ ( exp @ real @ X ) ) ).
% exp_ge_add_one_self
thf(fact_4375_exp__minus__inverse,axiom,
! [A: $tType] :
( ( ( real_Vector_banach @ A )
& ( real_V2822296259951069270ebra_1 @ A ) )
=> ! [X: A] :
( ( times_times @ A @ ( exp @ A @ X ) @ ( exp @ A @ ( uminus_uminus @ A @ X ) ) )
= ( one_one @ A ) ) ) ).
% exp_minus_inverse
thf(fact_4376_exp__of__nat2__mult,axiom,
! [A: $tType] :
( ( ( real_Vector_banach @ A )
& ( real_V3459762299906320749_field @ A ) )
=> ! [X: A,N3: nat] :
( ( exp @ A @ ( times_times @ A @ X @ ( semiring_1_of_nat @ A @ N3 ) ) )
= ( power_power @ A @ ( exp @ A @ X ) @ N3 ) ) ) ).
% exp_of_nat2_mult
thf(fact_4377_exp__of__nat__mult,axiom,
! [A: $tType] :
( ( ( real_Vector_banach @ A )
& ( real_V3459762299906320749_field @ A ) )
=> ! [N3: nat,X: A] :
( ( exp @ A @ ( times_times @ A @ ( semiring_1_of_nat @ A @ N3 ) @ X ) )
= ( power_power @ A @ ( exp @ A @ X ) @ N3 ) ) ) ).
% exp_of_nat_mult
thf(fact_4378_log__ln,axiom,
( ( ln_ln @ real )
= ( log @ ( exp @ real @ ( one_one @ real ) ) ) ) ).
% log_ln
thf(fact_4379_Complex__eq__neg__numeral,axiom,
! [A2: real,B2: real,W: num] :
( ( ( complex2 @ A2 @ B2 )
= ( uminus_uminus @ complex @ ( numeral_numeral @ complex @ W ) ) )
= ( ( A2
= ( uminus_uminus @ real @ ( numeral_numeral @ real @ W ) ) )
& ( B2
= ( zero_zero @ real ) ) ) ) ).
% Complex_eq_neg_numeral
thf(fact_4380_complex__mult,axiom,
! [A2: real,B2: real,C2: real,D: real] :
( ( times_times @ complex @ ( complex2 @ A2 @ B2 ) @ ( complex2 @ C2 @ D ) )
= ( complex2 @ ( minus_minus @ real @ ( times_times @ real @ A2 @ C2 ) @ ( times_times @ real @ B2 @ D ) ) @ ( plus_plus @ real @ ( times_times @ real @ A2 @ D ) @ ( times_times @ real @ B2 @ C2 ) ) ) ) ).
% complex_mult
thf(fact_4381_Complex__eq__1,axiom,
! [A2: real,B2: real] :
( ( ( complex2 @ A2 @ B2 )
= ( one_one @ complex ) )
= ( ( A2
= ( one_one @ real ) )
& ( B2
= ( zero_zero @ real ) ) ) ) ).
% Complex_eq_1
thf(fact_4382_one__complex_Ocode,axiom,
( ( one_one @ complex )
= ( complex2 @ ( one_one @ real ) @ ( zero_zero @ real ) ) ) ).
% one_complex.code
thf(fact_4383_norm__less__p1,axiom,
! [A: $tType] :
( ( real_V2822296259951069270ebra_1 @ A )
=> ! [X: A] : ( ord_less @ real @ ( real_V7770717601297561774m_norm @ A @ X ) @ ( real_V7770717601297561774m_norm @ A @ ( plus_plus @ A @ ( real_Vector_of_real @ A @ ( real_V7770717601297561774m_norm @ A @ X ) ) @ ( one_one @ A ) ) ) ) ) ).
% norm_less_p1
thf(fact_4384_exp__ge__add__one__self__aux,axiom,
! [X: real] :
( ( ord_less_eq @ real @ ( zero_zero @ real ) @ X )
=> ( ord_less_eq @ real @ ( plus_plus @ real @ ( one_one @ real ) @ X ) @ ( exp @ real @ X ) ) ) ).
% exp_ge_add_one_self_aux
thf(fact_4385_lemma__exp__total,axiom,
! [Y: real] :
( ( ord_less_eq @ real @ ( one_one @ real ) @ Y )
=> ? [X3: real] :
( ( ord_less_eq @ real @ ( zero_zero @ real ) @ X3 )
& ( ord_less_eq @ real @ X3 @ ( minus_minus @ real @ Y @ ( one_one @ real ) ) )
& ( ( exp @ real @ X3 )
= Y ) ) ) ).
% lemma_exp_total
thf(fact_4386_ln__ge__iff,axiom,
! [X: real,Y: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ X )
=> ( ( ord_less_eq @ real @ Y @ ( ln_ln @ real @ X ) )
= ( ord_less_eq @ real @ ( exp @ real @ Y ) @ X ) ) ) ).
% ln_ge_iff
thf(fact_4387_ln__x__over__x__mono,axiom,
! [X: real,Y: real] :
( ( ord_less_eq @ real @ ( exp @ real @ ( one_one @ real ) ) @ X )
=> ( ( ord_less_eq @ real @ X @ Y )
=> ( ord_less_eq @ real @ ( divide_divide @ real @ ( ln_ln @ real @ Y ) @ Y ) @ ( divide_divide @ real @ ( ln_ln @ real @ X ) @ X ) ) ) ) ).
% ln_x_over_x_mono
thf(fact_4388_powr__def,axiom,
! [A: $tType] :
( ( ln @ A )
=> ( ( powr @ A )
= ( ^ [X2: A,A6: A] :
( if @ A
@ ( X2
= ( zero_zero @ A ) )
@ ( zero_zero @ A )
@ ( exp @ A @ ( times_times @ A @ A6 @ ( ln_ln @ A @ X2 ) ) ) ) ) ) ) ).
% powr_def
thf(fact_4389_Complex__eq__neg__1,axiom,
! [A2: real,B2: real] :
( ( ( complex2 @ A2 @ B2 )
= ( uminus_uminus @ complex @ ( one_one @ complex ) ) )
= ( ( A2
= ( uminus_uminus @ real @ ( one_one @ real ) ) )
& ( B2
= ( zero_zero @ real ) ) ) ) ).
% Complex_eq_neg_1
thf(fact_4390_exp__le,axiom,
ord_less_eq @ real @ ( exp @ real @ ( one_one @ real ) ) @ ( numeral_numeral @ real @ ( bit1 @ one2 ) ) ).
% exp_le
thf(fact_4391_norm__of__real__diff,axiom,
! [A: $tType] :
( ( real_V2822296259951069270ebra_1 @ A )
=> ! [B2: real,A2: real] : ( ord_less_eq @ real @ ( real_V7770717601297561774m_norm @ A @ ( minus_minus @ A @ ( real_Vector_of_real @ A @ B2 ) @ ( real_Vector_of_real @ A @ A2 ) ) ) @ ( abs_abs @ real @ ( minus_minus @ real @ B2 @ A2 ) ) ) ) ).
% norm_of_real_diff
thf(fact_4392_exp__divide__power__eq,axiom,
! [A: $tType] :
( ( ( real_Vector_banach @ A )
& ( real_V3459762299906320749_field @ A ) )
=> ! [N3: nat,X: A] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ N3 )
=> ( ( power_power @ A @ ( exp @ A @ ( divide_divide @ A @ X @ ( semiring_1_of_nat @ A @ N3 ) ) ) @ N3 )
= ( exp @ A @ X ) ) ) ) ).
% exp_divide_power_eq
thf(fact_4393_tanh__altdef,axiom,
! [A: $tType] :
( ( ( real_Vector_banach @ A )
& ( real_V3459762299906320749_field @ A ) )
=> ( ( tanh @ A )
= ( ^ [X2: A] : ( divide_divide @ A @ ( minus_minus @ A @ ( exp @ A @ X2 ) @ ( exp @ A @ ( uminus_uminus @ A @ X2 ) ) ) @ ( plus_plus @ A @ ( exp @ A @ X2 ) @ ( exp @ A @ ( uminus_uminus @ A @ X2 ) ) ) ) ) ) ) ).
% tanh_altdef
thf(fact_4394_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_4395_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_4396_cos__int__times__real,axiom,
! [A: $tType] :
( ( ( real_Vector_banach @ A )
& ( real_V2822296259951069270ebra_1 @ A ) )
=> ! [M: int,X: real] :
( ( cos @ A @ ( times_times @ A @ ( ring_1_of_int @ A @ M ) @ ( real_Vector_of_real @ A @ X ) ) )
= ( real_Vector_of_real @ A @ ( cos @ real @ ( times_times @ real @ ( ring_1_of_int @ real @ M ) @ X ) ) ) ) ) ).
% cos_int_times_real
thf(fact_4397_sin__int__times__real,axiom,
! [A: $tType] :
( ( ( real_Vector_banach @ A )
& ( real_V2822296259951069270ebra_1 @ A ) )
=> ! [M: int,X: real] :
( ( sin @ A @ ( times_times @ A @ ( ring_1_of_int @ A @ M ) @ ( real_Vector_of_real @ A @ X ) ) )
= ( real_Vector_of_real @ A @ ( sin @ real @ ( times_times @ real @ ( ring_1_of_int @ real @ M ) @ X ) ) ) ) ) ).
% sin_int_times_real
thf(fact_4398_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_4399_sin__cos__eq,axiom,
! [A: $tType] :
( ( ( real_Vector_banach @ A )
& ( real_V3459762299906320749_field @ A ) )
=> ( ( sin @ A )
= ( ^ [X2: A] : ( cos @ A @ ( minus_minus @ A @ ( divide_divide @ A @ ( real_Vector_of_real @ A @ pi ) @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) @ X2 ) ) ) ) ) ).
% sin_cos_eq
thf(fact_4400_cos__sin__eq,axiom,
! [A: $tType] :
( ( ( real_Vector_banach @ A )
& ( real_V3459762299906320749_field @ A ) )
=> ( ( cos @ A )
= ( ^ [X2: A] : ( sin @ A @ ( minus_minus @ A @ ( divide_divide @ A @ ( real_Vector_of_real @ A @ pi ) @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) @ X2 ) ) ) ) ) ).
% cos_sin_eq
thf(fact_4401_exp__bound,axiom,
! [X: real] :
( ( ord_less_eq @ real @ ( zero_zero @ real ) @ X )
=> ( ( ord_less_eq @ real @ X @ ( one_one @ real ) )
=> ( ord_less_eq @ real @ ( exp @ real @ X ) @ ( plus_plus @ real @ ( plus_plus @ real @ ( one_one @ real ) @ X ) @ ( power_power @ real @ X @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) ) ).
% exp_bound
thf(fact_4402_list__assn__cong,axiom,
! [A: $tType,B: $tType,Xs2: list @ A,Xs4: list @ A,Xsi: list @ B,Xsi2: list @ B,A3: A > B > assn,A9: A > B > assn] :
( ( Xs2 = Xs4 )
=> ( ( Xsi = Xsi2 )
=> ( ! [X3: A,Xi: B] :
( ( member @ A @ X3 @ ( set2 @ A @ Xs4 ) )
=> ( ( member @ B @ Xi @ ( set2 @ B @ Xsi2 ) )
=> ( ( A3 @ X3 @ Xi )
= ( A9 @ X3 @ Xi ) ) ) )
=> ( ( vEBT_List_list_assn @ A @ B @ A3 @ Xs2 @ Xsi )
= ( vEBT_List_list_assn @ A @ B @ A9 @ Xs4 @ Xsi2 ) ) ) ) ) ).
% list_assn_cong
thf(fact_4403_minus__sin__cos__eq,axiom,
! [A: $tType] :
( ( ( real_Vector_banach @ A )
& ( real_V3459762299906320749_field @ A ) )
=> ! [X: A] :
( ( uminus_uminus @ A @ ( sin @ A @ X ) )
= ( cos @ A @ ( plus_plus @ A @ X @ ( divide_divide @ A @ ( real_Vector_of_real @ A @ pi ) @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) ) ) ) ) ).
% minus_sin_cos_eq
thf(fact_4404_real__exp__bound__lemma,axiom,
! [X: real] :
( ( ord_less_eq @ real @ ( zero_zero @ real ) @ X )
=> ( ( ord_less_eq @ real @ X @ ( divide_divide @ real @ ( one_one @ real ) @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) )
=> ( ord_less_eq @ real @ ( exp @ real @ X ) @ ( plus_plus @ real @ ( one_one @ real ) @ ( times_times @ real @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) @ X ) ) ) ) ) ).
% real_exp_bound_lemma
thf(fact_4405_exp__ge__one__plus__x__over__n__power__n,axiom,
! [N3: nat,X: real] :
( ( ord_less_eq @ real @ ( uminus_uminus @ real @ ( semiring_1_of_nat @ real @ N3 ) ) @ X )
=> ( ( ord_less @ nat @ ( zero_zero @ nat ) @ N3 )
=> ( ord_less_eq @ real @ ( power_power @ real @ ( plus_plus @ real @ ( one_one @ real ) @ ( divide_divide @ real @ X @ ( semiring_1_of_nat @ real @ N3 ) ) ) @ N3 ) @ ( exp @ real @ X ) ) ) ) ).
% exp_ge_one_plus_x_over_n_power_n
thf(fact_4406_exp__ge__one__minus__x__over__n__power__n,axiom,
! [X: real,N3: nat] :
( ( ord_less_eq @ real @ X @ ( semiring_1_of_nat @ real @ N3 ) )
=> ( ( ord_less @ nat @ ( zero_zero @ nat ) @ N3 )
=> ( ord_less_eq @ real @ ( power_power @ real @ ( minus_minus @ real @ ( one_one @ real ) @ ( divide_divide @ real @ X @ ( semiring_1_of_nat @ real @ N3 ) ) ) @ N3 ) @ ( exp @ real @ ( uminus_uminus @ real @ X ) ) ) ) ) ).
% exp_ge_one_minus_x_over_n_power_n
thf(fact_4407_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_4408_Maclaurin__exp__le,axiom,
! [X: real,N3: nat] :
? [T8: real] :
( ( ord_less_eq @ real @ ( abs_abs @ real @ T8 ) @ ( abs_abs @ real @ X ) )
& ( ( exp @ real @ X )
= ( plus_plus @ real
@ ( groups7311177749621191930dd_sum @ nat @ real
@ ^ [M5: nat] : ( divide_divide @ real @ ( power_power @ real @ X @ M5 ) @ ( semiring_char_0_fact @ real @ M5 ) )
@ ( set_ord_lessThan @ nat @ N3 ) )
@ ( times_times @ real @ ( divide_divide @ real @ ( exp @ real @ T8 ) @ ( semiring_char_0_fact @ real @ N3 ) ) @ ( power_power @ real @ X @ N3 ) ) ) ) ) ).
% Maclaurin_exp_le
thf(fact_4409_exp__lower__Taylor__quadratic,axiom,
! [X: real] :
( ( ord_less_eq @ real @ ( zero_zero @ real ) @ X )
=> ( ord_less_eq @ real @ ( plus_plus @ real @ ( plus_plus @ real @ ( one_one @ real ) @ X ) @ ( divide_divide @ real @ ( power_power @ real @ X @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) ) @ ( exp @ real @ X ) ) ) ).
% exp_lower_Taylor_quadratic
thf(fact_4410_log__base__10__eq2,axiom,
! [X: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ X )
=> ( ( log @ ( numeral_numeral @ real @ ( bit0 @ ( bit1 @ ( bit0 @ one2 ) ) ) ) @ X )
= ( times_times @ real @ ( log @ ( numeral_numeral @ real @ ( bit0 @ ( bit1 @ ( bit0 @ one2 ) ) ) ) @ ( exp @ real @ ( one_one @ real ) ) ) @ ( ln_ln @ real @ X ) ) ) ) ).
% log_base_10_eq2
thf(fact_4411_tanh__real__altdef,axiom,
( ( tanh @ real )
= ( ^ [X2: real] : ( divide_divide @ real @ ( minus_minus @ real @ ( one_one @ real ) @ ( exp @ real @ ( times_times @ real @ ( uminus_uminus @ real @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) @ X2 ) ) ) @ ( plus_plus @ real @ ( one_one @ real ) @ ( exp @ real @ ( times_times @ real @ ( uminus_uminus @ real @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) @ X2 ) ) ) ) ) ) ).
% tanh_real_altdef
thf(fact_4412_log__base__10__eq1,axiom,
! [X: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ X )
=> ( ( log @ ( numeral_numeral @ real @ ( bit0 @ ( bit1 @ ( bit0 @ one2 ) ) ) ) @ X )
= ( times_times @ real @ ( divide_divide @ real @ ( ln_ln @ real @ ( exp @ real @ ( one_one @ real ) ) ) @ ( ln_ln @ real @ ( numeral_numeral @ real @ ( bit0 @ ( bit1 @ ( bit0 @ one2 ) ) ) ) ) ) @ ( ln_ln @ real @ X ) ) ) ) ).
% log_base_10_eq1
thf(fact_4413_arsinh__def,axiom,
! [A: $tType] :
( ( ln @ A )
=> ( ( arsinh @ A )
= ( ^ [X2: A] : ( ln_ln @ A @ ( plus_plus @ A @ X2 @ ( powr @ A @ ( plus_plus @ A @ ( power_power @ A @ X2 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ ( one_one @ A ) ) @ ( real_Vector_of_real @ A @ ( divide_divide @ real @ ( one_one @ real ) @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) ) ) ) ) ) ) ) ).
% arsinh_def
thf(fact_4414_sin__tan,axiom,
! [X: real] :
( ( ord_less @ real @ ( abs_abs @ real @ X ) @ ( divide_divide @ real @ pi @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) )
=> ( ( sin @ real @ X )
= ( divide_divide @ real @ ( tan @ real @ X ) @ ( sqrt @ ( plus_plus @ real @ ( one_one @ real ) @ ( power_power @ real @ ( tan @ real @ X ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) ) ) ).
% sin_tan
thf(fact_4415_cos__tan,axiom,
! [X: real] :
( ( ord_less @ real @ ( abs_abs @ real @ X ) @ ( divide_divide @ real @ pi @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) )
=> ( ( cos @ real @ X )
= ( divide_divide @ real @ ( one_one @ real ) @ ( sqrt @ ( plus_plus @ real @ ( one_one @ real ) @ ( power_power @ real @ ( tan @ real @ X ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) ) ) ).
% cos_tan
thf(fact_4416_monoI1,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [X8: nat > A] :
( ! [M4: nat,N: nat] :
( ( ord_less_eq @ nat @ M4 @ N )
=> ( ord_less_eq @ A @ ( X8 @ M4 ) @ ( X8 @ N ) ) )
=> ( topological_monoseq @ A @ X8 ) ) ) ).
% monoI1
thf(fact_4417_real__sqrt__eq__iff,axiom,
! [X: real,Y: real] :
( ( ( sqrt @ X )
= ( sqrt @ Y ) )
= ( X = Y ) ) ).
% real_sqrt_eq_iff
thf(fact_4418_real__sqrt__eq__zero__cancel__iff,axiom,
! [X: real] :
( ( ( sqrt @ X )
= ( zero_zero @ real ) )
= ( X
= ( zero_zero @ real ) ) ) ).
% real_sqrt_eq_zero_cancel_iff
thf(fact_4419_real__sqrt__zero,axiom,
( ( sqrt @ ( zero_zero @ real ) )
= ( zero_zero @ real ) ) ).
% real_sqrt_zero
thf(fact_4420_real__sqrt__less__iff,axiom,
! [X: real,Y: real] :
( ( ord_less @ real @ ( sqrt @ X ) @ ( sqrt @ Y ) )
= ( ord_less @ real @ X @ Y ) ) ).
% real_sqrt_less_iff
thf(fact_4421_real__sqrt__le__iff,axiom,
! [X: real,Y: real] :
( ( ord_less_eq @ real @ ( sqrt @ X ) @ ( sqrt @ Y ) )
= ( ord_less_eq @ real @ X @ Y ) ) ).
% real_sqrt_le_iff
thf(fact_4422_real__sqrt__eq__1__iff,axiom,
! [X: real] :
( ( ( sqrt @ X )
= ( one_one @ real ) )
= ( X
= ( one_one @ real ) ) ) ).
% real_sqrt_eq_1_iff
thf(fact_4423_real__sqrt__one,axiom,
( ( sqrt @ ( one_one @ real ) )
= ( one_one @ real ) ) ).
% real_sqrt_one
thf(fact_4424_real__sqrt__gt__0__iff,axiom,
! [Y: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ ( sqrt @ Y ) )
= ( ord_less @ real @ ( zero_zero @ real ) @ Y ) ) ).
% real_sqrt_gt_0_iff
thf(fact_4425_real__sqrt__lt__0__iff,axiom,
! [X: real] :
( ( ord_less @ real @ ( sqrt @ X ) @ ( zero_zero @ real ) )
= ( ord_less @ real @ X @ ( zero_zero @ real ) ) ) ).
% real_sqrt_lt_0_iff
thf(fact_4426_real__sqrt__ge__0__iff,axiom,
! [Y: real] :
( ( ord_less_eq @ real @ ( zero_zero @ real ) @ ( sqrt @ Y ) )
= ( ord_less_eq @ real @ ( zero_zero @ real ) @ Y ) ) ).
% real_sqrt_ge_0_iff
thf(fact_4427_real__sqrt__le__0__iff,axiom,
! [X: real] :
( ( ord_less_eq @ real @ ( sqrt @ X ) @ ( zero_zero @ real ) )
= ( ord_less_eq @ real @ X @ ( zero_zero @ real ) ) ) ).
% real_sqrt_le_0_iff
thf(fact_4428_real__sqrt__gt__1__iff,axiom,
! [Y: real] :
( ( ord_less @ real @ ( one_one @ real ) @ ( sqrt @ Y ) )
= ( ord_less @ real @ ( one_one @ real ) @ Y ) ) ).
% real_sqrt_gt_1_iff
thf(fact_4429_real__sqrt__lt__1__iff,axiom,
! [X: real] :
( ( ord_less @ real @ ( sqrt @ X ) @ ( one_one @ real ) )
= ( ord_less @ real @ X @ ( one_one @ real ) ) ) ).
% real_sqrt_lt_1_iff
thf(fact_4430_real__sqrt__le__1__iff,axiom,
! [X: real] :
( ( ord_less_eq @ real @ ( sqrt @ X ) @ ( one_one @ real ) )
= ( ord_less_eq @ real @ X @ ( one_one @ real ) ) ) ).
% real_sqrt_le_1_iff
thf(fact_4431_real__sqrt__ge__1__iff,axiom,
! [Y: real] :
( ( ord_less_eq @ real @ ( one_one @ real ) @ ( sqrt @ Y ) )
= ( ord_less_eq @ real @ ( one_one @ real ) @ Y ) ) ).
% real_sqrt_ge_1_iff
thf(fact_4432_real__sqrt__abs2,axiom,
! [X: real] :
( ( sqrt @ ( times_times @ real @ X @ X ) )
= ( abs_abs @ real @ X ) ) ).
% real_sqrt_abs2
thf(fact_4433_real__sqrt__mult__self,axiom,
! [A2: real] :
( ( times_times @ real @ ( sqrt @ A2 ) @ ( sqrt @ A2 ) )
= ( abs_abs @ real @ A2 ) ) ).
% real_sqrt_mult_self
thf(fact_4434_real__sqrt__four,axiom,
( ( sqrt @ ( numeral_numeral @ real @ ( bit0 @ ( bit0 @ one2 ) ) ) )
= ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) ).
% real_sqrt_four
thf(fact_4435_real__sqrt__abs,axiom,
! [X: real] :
( ( sqrt @ ( power_power @ real @ X @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) )
= ( abs_abs @ real @ X ) ) ).
% real_sqrt_abs
thf(fact_4436_real__sqrt__pow2,axiom,
! [X: real] :
( ( ord_less_eq @ real @ ( zero_zero @ real ) @ X )
=> ( ( power_power @ real @ ( sqrt @ X ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) )
= X ) ) ).
% real_sqrt_pow2
thf(fact_4437_real__sqrt__pow2__iff,axiom,
! [X: real] :
( ( ( power_power @ real @ ( sqrt @ X ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) )
= X )
= ( ord_less_eq @ real @ ( zero_zero @ real ) @ X ) ) ).
% real_sqrt_pow2_iff
thf(fact_4438_real__sqrt__sum__squares__mult__squared__eq,axiom,
! [X: real,Y: real,Xa: real,Ya: real] :
( ( power_power @ real @ ( sqrt @ ( times_times @ real @ ( plus_plus @ real @ ( power_power @ real @ X @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ ( power_power @ real @ Y @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( plus_plus @ real @ ( power_power @ real @ Xa @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ ( power_power @ real @ Ya @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) )
= ( times_times @ real @ ( plus_plus @ real @ ( power_power @ real @ X @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ ( power_power @ real @ Y @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( plus_plus @ real @ ( power_power @ real @ Xa @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ ( power_power @ real @ Ya @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) ).
% real_sqrt_sum_squares_mult_squared_eq
thf(fact_4439_real__sqrt__power,axiom,
! [X: real,K: nat] :
( ( sqrt @ ( power_power @ real @ X @ K ) )
= ( power_power @ real @ ( sqrt @ X ) @ K ) ) ).
% real_sqrt_power
thf(fact_4440_real__sqrt__mult,axiom,
! [X: real,Y: real] :
( ( sqrt @ ( times_times @ real @ X @ Y ) )
= ( times_times @ real @ ( sqrt @ X ) @ ( sqrt @ Y ) ) ) ).
% real_sqrt_mult
thf(fact_4441_real__sqrt__divide,axiom,
! [X: real,Y: real] :
( ( sqrt @ ( divide_divide @ real @ X @ Y ) )
= ( divide_divide @ real @ ( sqrt @ X ) @ ( sqrt @ Y ) ) ) ).
% real_sqrt_divide
thf(fact_4442_real__sqrt__less__mono,axiom,
! [X: real,Y: real] :
( ( ord_less @ real @ X @ Y )
=> ( ord_less @ real @ ( sqrt @ X ) @ ( sqrt @ Y ) ) ) ).
% real_sqrt_less_mono
thf(fact_4443_real__sqrt__le__mono,axiom,
! [X: real,Y: real] :
( ( ord_less_eq @ real @ X @ Y )
=> ( ord_less_eq @ real @ ( sqrt @ X ) @ ( sqrt @ Y ) ) ) ).
% real_sqrt_le_mono
thf(fact_4444_real__sqrt__minus,axiom,
! [X: real] :
( ( sqrt @ ( uminus_uminus @ real @ X ) )
= ( uminus_uminus @ real @ ( sqrt @ X ) ) ) ).
% real_sqrt_minus
thf(fact_4445_real__sqrt__gt__zero,axiom,
! [X: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ X )
=> ( ord_less @ real @ ( zero_zero @ real ) @ ( sqrt @ X ) ) ) ).
% real_sqrt_gt_zero
thf(fact_4446_real__sqrt__ge__zero,axiom,
! [X: real] :
( ( ord_less_eq @ real @ ( zero_zero @ real ) @ X )
=> ( ord_less_eq @ real @ ( zero_zero @ real ) @ ( sqrt @ X ) ) ) ).
% real_sqrt_ge_zero
thf(fact_4447_real__sqrt__eq__zero__cancel,axiom,
! [X: real] :
( ( ord_less_eq @ real @ ( zero_zero @ real ) @ X )
=> ( ( ( sqrt @ X )
= ( zero_zero @ real ) )
=> ( X
= ( zero_zero @ real ) ) ) ) ).
% real_sqrt_eq_zero_cancel
thf(fact_4448_real__sqrt__ge__one,axiom,
! [X: real] :
( ( ord_less_eq @ real @ ( one_one @ real ) @ X )
=> ( ord_less_eq @ real @ ( one_one @ real ) @ ( sqrt @ X ) ) ) ).
% real_sqrt_ge_one
thf(fact_4449_real__div__sqrt,axiom,
! [X: real] :
( ( ord_less_eq @ real @ ( zero_zero @ real ) @ X )
=> ( ( divide_divide @ real @ X @ ( sqrt @ X ) )
= ( sqrt @ X ) ) ) ).
% real_div_sqrt
thf(fact_4450_sqrt__add__le__add__sqrt,axiom,
! [X: real,Y: real] :
( ( ord_less_eq @ real @ ( zero_zero @ real ) @ X )
=> ( ( ord_less_eq @ real @ ( zero_zero @ real ) @ Y )
=> ( ord_less_eq @ real @ ( sqrt @ ( plus_plus @ real @ X @ Y ) ) @ ( plus_plus @ real @ ( sqrt @ X ) @ ( sqrt @ Y ) ) ) ) ) ).
% sqrt_add_le_add_sqrt
thf(fact_4451_le__real__sqrt__sumsq,axiom,
! [X: real,Y: real] : ( ord_less_eq @ real @ X @ ( sqrt @ ( plus_plus @ real @ ( times_times @ real @ X @ X ) @ ( times_times @ real @ Y @ Y ) ) ) ) ).
% le_real_sqrt_sumsq
thf(fact_4452_sqrt2__less__2,axiom,
ord_less @ real @ ( sqrt @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ).
% sqrt2_less_2
thf(fact_4453_arsinh__real__def,axiom,
( ( arsinh @ real )
= ( ^ [X2: real] : ( ln_ln @ real @ ( plus_plus @ real @ X2 @ ( sqrt @ ( plus_plus @ real @ ( power_power @ real @ X2 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ ( one_one @ real ) ) ) ) ) ) ) ).
% arsinh_real_def
thf(fact_4454_real__less__rsqrt,axiom,
! [X: real,Y: real] :
( ( ord_less @ real @ ( power_power @ real @ X @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ Y )
=> ( ord_less @ real @ X @ ( sqrt @ Y ) ) ) ).
% real_less_rsqrt
thf(fact_4455_sqrt__le__D,axiom,
! [X: real,Y: real] :
( ( ord_less_eq @ real @ ( sqrt @ X ) @ Y )
=> ( ord_less_eq @ real @ X @ ( power_power @ real @ Y @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ).
% sqrt_le_D
thf(fact_4456_real__le__rsqrt,axiom,
! [X: real,Y: real] :
( ( ord_less_eq @ real @ ( power_power @ real @ X @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ Y )
=> ( ord_less_eq @ real @ X @ ( sqrt @ Y ) ) ) ).
% real_le_rsqrt
thf(fact_4457_real__le__lsqrt,axiom,
! [X: real,Y: real] :
( ( ord_less_eq @ real @ ( zero_zero @ real ) @ X )
=> ( ( ord_less_eq @ real @ ( zero_zero @ real ) @ Y )
=> ( ( ord_less_eq @ real @ X @ ( power_power @ real @ Y @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) )
=> ( ord_less_eq @ real @ ( sqrt @ X ) @ Y ) ) ) ) ).
% real_le_lsqrt
thf(fact_4458_real__sqrt__unique,axiom,
! [Y: real,X: real] :
( ( ( power_power @ real @ Y @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) )
= X )
=> ( ( ord_less_eq @ real @ ( zero_zero @ real ) @ Y )
=> ( ( sqrt @ X )
= Y ) ) ) ).
% real_sqrt_unique
thf(fact_4459_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_4460_real__sqrt__sum__squares__eq__cancel,axiom,
! [X: real,Y: real] :
( ( ( sqrt @ ( plus_plus @ real @ ( power_power @ real @ X @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ ( power_power @ real @ Y @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) )
= X )
=> ( Y
= ( zero_zero @ real ) ) ) ).
% real_sqrt_sum_squares_eq_cancel
thf(fact_4461_real__sqrt__sum__squares__eq__cancel2,axiom,
! [X: real,Y: real] :
( ( ( sqrt @ ( plus_plus @ real @ ( power_power @ real @ X @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ ( power_power @ real @ Y @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) )
= Y )
=> ( X
= ( zero_zero @ real ) ) ) ).
% real_sqrt_sum_squares_eq_cancel2
thf(fact_4462_real__sqrt__sum__squares__ge1,axiom,
! [X: real,Y: real] : ( ord_less_eq @ real @ X @ ( sqrt @ ( plus_plus @ real @ ( power_power @ real @ X @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ ( power_power @ real @ Y @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) ).
% real_sqrt_sum_squares_ge1
thf(fact_4463_real__sqrt__sum__squares__ge2,axiom,
! [Y: real,X: real] : ( ord_less_eq @ real @ Y @ ( sqrt @ ( plus_plus @ real @ ( power_power @ real @ X @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ ( power_power @ real @ Y @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) ).
% real_sqrt_sum_squares_ge2
thf(fact_4464_real__sqrt__sum__squares__triangle__ineq,axiom,
! [A2: real,C2: real,B2: real,D: real] : ( ord_less_eq @ real @ ( sqrt @ ( plus_plus @ real @ ( power_power @ real @ ( plus_plus @ real @ A2 @ C2 ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ ( power_power @ real @ ( plus_plus @ real @ B2 @ D ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( plus_plus @ real @ ( sqrt @ ( plus_plus @ real @ ( power_power @ real @ A2 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ ( power_power @ real @ B2 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( sqrt @ ( plus_plus @ real @ ( power_power @ real @ C2 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ ( power_power @ real @ D @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) ) ).
% real_sqrt_sum_squares_triangle_ineq
thf(fact_4465_sqrt__ge__absD,axiom,
! [X: real,Y: real] :
( ( ord_less_eq @ real @ ( abs_abs @ real @ X ) @ ( sqrt @ Y ) )
=> ( ord_less_eq @ real @ ( power_power @ real @ X @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ Y ) ) ).
% sqrt_ge_absD
thf(fact_4466_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_4467_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_4468_tan__60,axiom,
( ( tan @ real @ ( divide_divide @ real @ pi @ ( numeral_numeral @ real @ ( bit1 @ one2 ) ) ) )
= ( sqrt @ ( numeral_numeral @ real @ ( bit1 @ one2 ) ) ) ) ).
% tan_60
thf(fact_4469_real__less__lsqrt,axiom,
! [X: real,Y: real] :
( ( ord_less_eq @ real @ ( zero_zero @ real ) @ X )
=> ( ( ord_less_eq @ real @ ( zero_zero @ real ) @ Y )
=> ( ( ord_less @ real @ X @ ( power_power @ real @ Y @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) )
=> ( ord_less @ real @ ( sqrt @ X ) @ Y ) ) ) ) ).
% real_less_lsqrt
thf(fact_4470_sqrt__sum__squares__le__sum,axiom,
! [X: real,Y: real] :
( ( ord_less_eq @ real @ ( zero_zero @ real ) @ X )
=> ( ( ord_less_eq @ real @ ( zero_zero @ real ) @ Y )
=> ( ord_less_eq @ real @ ( sqrt @ ( plus_plus @ real @ ( power_power @ real @ X @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ ( power_power @ real @ Y @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( plus_plus @ real @ X @ Y ) ) ) ) ).
% sqrt_sum_squares_le_sum
thf(fact_4471_sqrt__even__pow2,axiom,
! [N3: nat] :
( ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N3 )
=> ( ( sqrt @ ( power_power @ real @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) @ N3 ) )
= ( power_power @ real @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ N3 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) ).
% sqrt_even_pow2
thf(fact_4472_sqrt__sum__squares__le__sum__abs,axiom,
! [X: real,Y: real] : ( ord_less_eq @ real @ ( sqrt @ ( plus_plus @ real @ ( power_power @ real @ X @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ ( power_power @ real @ Y @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( plus_plus @ real @ ( abs_abs @ real @ X ) @ ( abs_abs @ real @ Y ) ) ) ).
% sqrt_sum_squares_le_sum_abs
thf(fact_4473_real__sqrt__ge__abs2,axiom,
! [Y: real,X: real] : ( ord_less_eq @ real @ ( abs_abs @ real @ Y ) @ ( sqrt @ ( plus_plus @ real @ ( power_power @ real @ X @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ ( power_power @ real @ Y @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) ).
% real_sqrt_ge_abs2
thf(fact_4474_real__sqrt__ge__abs1,axiom,
! [X: real,Y: real] : ( ord_less_eq @ real @ ( abs_abs @ real @ X ) @ ( sqrt @ ( plus_plus @ real @ ( power_power @ real @ X @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ ( power_power @ real @ Y @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) ).
% real_sqrt_ge_abs1
thf(fact_4475_ln__sqrt,axiom,
! [X: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ X )
=> ( ( ln_ln @ real @ ( sqrt @ X ) )
= ( divide_divide @ real @ ( ln_ln @ real @ X ) @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) ) ) ).
% ln_sqrt
thf(fact_4476_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_4477_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_4478_complex__norm,axiom,
! [X: real,Y: real] :
( ( real_V7770717601297561774m_norm @ complex @ ( complex2 @ X @ Y ) )
= ( sqrt @ ( plus_plus @ real @ ( power_power @ real @ X @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ ( power_power @ real @ Y @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) ).
% complex_norm
thf(fact_4479_real__sqrt__power__even,axiom,
! [N3: nat,X: real] :
( ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N3 )
=> ( ( ord_less_eq @ real @ ( zero_zero @ real ) @ X )
=> ( ( power_power @ real @ ( sqrt @ X ) @ N3 )
= ( power_power @ real @ X @ ( divide_divide @ nat @ N3 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) ) ).
% real_sqrt_power_even
thf(fact_4480_arsinh__real__aux,axiom,
! [X: real] : ( ord_less @ real @ ( zero_zero @ real ) @ ( plus_plus @ real @ X @ ( sqrt @ ( plus_plus @ real @ ( power_power @ real @ X @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ ( one_one @ real ) ) ) ) ) ).
% arsinh_real_aux
thf(fact_4481_real__sqrt__sum__squares__mult__ge__zero,axiom,
! [X: real,Y: real,Xa: real,Ya: real] : ( ord_less_eq @ real @ ( zero_zero @ real ) @ ( sqrt @ ( times_times @ real @ ( plus_plus @ real @ ( power_power @ real @ X @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ ( power_power @ real @ Y @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( plus_plus @ real @ ( power_power @ real @ Xa @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ ( power_power @ real @ Ya @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) ) ).
% real_sqrt_sum_squares_mult_ge_zero
thf(fact_4482_arith__geo__mean__sqrt,axiom,
! [X: real,Y: real] :
( ( ord_less_eq @ real @ ( zero_zero @ real ) @ X )
=> ( ( ord_less_eq @ real @ ( zero_zero @ real ) @ Y )
=> ( ord_less_eq @ real @ ( sqrt @ ( times_times @ real @ X @ Y ) ) @ ( divide_divide @ real @ ( plus_plus @ real @ X @ Y ) @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) ) ) ) ).
% arith_geo_mean_sqrt
thf(fact_4483_powr__half__sqrt,axiom,
! [X: real] :
( ( ord_less_eq @ real @ ( zero_zero @ real ) @ X )
=> ( ( powr @ real @ X @ ( divide_divide @ real @ ( one_one @ real ) @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) )
= ( sqrt @ X ) ) ) ).
% powr_half_sqrt
thf(fact_4484_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_4485_cos__x__y__le__one,axiom,
! [X: real,Y: real] : ( ord_less_eq @ real @ ( abs_abs @ real @ ( divide_divide @ real @ X @ ( sqrt @ ( plus_plus @ real @ ( power_power @ real @ X @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ ( power_power @ real @ Y @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) ) @ ( one_one @ real ) ) ).
% cos_x_y_le_one
thf(fact_4486_real__sqrt__sum__squares__less,axiom,
! [X: real,U: real,Y: real] :
( ( ord_less @ real @ ( abs_abs @ real @ X ) @ ( divide_divide @ real @ U @ ( sqrt @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) ) )
=> ( ( ord_less @ real @ ( abs_abs @ real @ Y ) @ ( divide_divide @ real @ U @ ( sqrt @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) ) )
=> ( ord_less @ real @ ( sqrt @ ( plus_plus @ real @ ( power_power @ real @ X @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ ( power_power @ real @ Y @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ U ) ) ) ).
% real_sqrt_sum_squares_less
thf(fact_4487_arcosh__real__def,axiom,
! [X: real] :
( ( ord_less_eq @ real @ ( one_one @ real ) @ X )
=> ( ( arcosh @ real @ X )
= ( ln_ln @ real @ ( plus_plus @ real @ X @ ( sqrt @ ( minus_minus @ real @ ( power_power @ real @ X @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ ( one_one @ real ) ) ) ) ) ) ) ).
% arcosh_real_def
thf(fact_4488_cos__arctan,axiom,
! [X: real] :
( ( cos @ real @ ( arctan @ X ) )
= ( divide_divide @ real @ ( one_one @ real ) @ ( sqrt @ ( plus_plus @ real @ ( one_one @ real ) @ ( power_power @ real @ X @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) ) ).
% cos_arctan
thf(fact_4489_sin__arctan,axiom,
! [X: real] :
( ( sin @ real @ ( arctan @ X ) )
= ( divide_divide @ real @ X @ ( sqrt @ ( plus_plus @ real @ ( one_one @ real ) @ ( power_power @ real @ X @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) ) ).
% sin_arctan
thf(fact_4490_sqrt__sum__squares__half__less,axiom,
! [X: real,U: real,Y: real] :
( ( ord_less @ real @ X @ ( divide_divide @ real @ U @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) )
=> ( ( ord_less @ real @ Y @ ( divide_divide @ real @ U @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) )
=> ( ( ord_less_eq @ real @ ( zero_zero @ real ) @ X )
=> ( ( ord_less_eq @ real @ ( zero_zero @ real ) @ Y )
=> ( ord_less @ real @ ( sqrt @ ( plus_plus @ real @ ( power_power @ real @ X @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ ( power_power @ real @ Y @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ U ) ) ) ) ) ).
% sqrt_sum_squares_half_less
thf(fact_4491_sin__cos__sqrt,axiom,
! [X: real] :
( ( ord_less_eq @ real @ ( zero_zero @ real ) @ ( sin @ real @ X ) )
=> ( ( sin @ real @ X )
= ( sqrt @ ( minus_minus @ real @ ( one_one @ real ) @ ( power_power @ real @ ( cos @ real @ X ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) ) ).
% sin_cos_sqrt
thf(fact_4492_arctan__half,axiom,
( arctan
= ( ^ [X2: real] : ( times_times @ real @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) @ ( arctan @ ( divide_divide @ real @ X2 @ ( plus_plus @ real @ ( one_one @ real ) @ ( sqrt @ ( plus_plus @ real @ ( one_one @ real ) @ ( power_power @ real @ X2 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) ) ) ) ) ) ).
% arctan_half
thf(fact_4493_monoseq__Suc,axiom,
! [A: $tType] :
( ( order @ A )
=> ( ( topological_monoseq @ A )
= ( ^ [X6: nat > A] :
( ! [N2: nat] : ( ord_less_eq @ A @ ( X6 @ N2 ) @ ( X6 @ ( suc @ N2 ) ) )
| ! [N2: nat] : ( ord_less_eq @ A @ ( X6 @ ( suc @ N2 ) ) @ ( X6 @ N2 ) ) ) ) ) ) ).
% monoseq_Suc
thf(fact_4494_mono__SucI2,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [X8: nat > A] :
( ! [N: nat] : ( ord_less_eq @ A @ ( X8 @ ( suc @ N ) ) @ ( X8 @ N ) )
=> ( topological_monoseq @ A @ X8 ) ) ) ).
% mono_SucI2
thf(fact_4495_mono__SucI1,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [X8: nat > A] :
( ! [N: nat] : ( ord_less_eq @ A @ ( X8 @ N ) @ ( X8 @ ( suc @ N ) ) )
=> ( topological_monoseq @ A @ X8 ) ) ) ).
% mono_SucI1
thf(fact_4496_monoseq__def,axiom,
! [A: $tType] :
( ( order @ A )
=> ( ( topological_monoseq @ A )
= ( ^ [X6: nat > A] :
( ! [M5: nat,N2: nat] :
( ( ord_less_eq @ nat @ M5 @ N2 )
=> ( ord_less_eq @ A @ ( X6 @ M5 ) @ ( X6 @ N2 ) ) )
| ! [M5: nat,N2: nat] :
( ( ord_less_eq @ nat @ M5 @ N2 )
=> ( ord_less_eq @ A @ ( X6 @ N2 ) @ ( X6 @ M5 ) ) ) ) ) ) ) ).
% monoseq_def
thf(fact_4497_monoI2,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [X8: nat > A] :
( ! [M4: nat,N: nat] :
( ( ord_less_eq @ nat @ M4 @ N )
=> ( ord_less_eq @ A @ ( X8 @ N ) @ ( X8 @ M4 ) ) )
=> ( topological_monoseq @ A @ X8 ) ) ) ).
% monoI2
thf(fact_4498_cos__arcsin,axiom,
! [X: real] :
( ( ord_less_eq @ real @ ( uminus_uminus @ real @ ( one_one @ real ) ) @ X )
=> ( ( ord_less_eq @ real @ X @ ( one_one @ real ) )
=> ( ( cos @ real @ ( arcsin @ X ) )
= ( sqrt @ ( minus_minus @ real @ ( one_one @ real ) @ ( power_power @ real @ X @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) ) ) ).
% cos_arcsin
thf(fact_4499_sin__arccos__abs,axiom,
! [Y: real] :
( ( ord_less_eq @ real @ ( abs_abs @ real @ Y ) @ ( one_one @ real ) )
=> ( ( sin @ real @ ( arccos @ Y ) )
= ( sqrt @ ( minus_minus @ real @ ( one_one @ real ) @ ( power_power @ real @ Y @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) ) ).
% sin_arccos_abs
thf(fact_4500_sin__arccos,axiom,
! [X: real] :
( ( ord_less_eq @ real @ ( uminus_uminus @ real @ ( one_one @ real ) ) @ X )
=> ( ( ord_less_eq @ real @ X @ ( one_one @ real ) )
=> ( ( sin @ real @ ( arccos @ X ) )
= ( sqrt @ ( minus_minus @ real @ ( one_one @ real ) @ ( power_power @ real @ X @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) ) ) ).
% sin_arccos
thf(fact_4501_exp__first__two__terms,axiom,
! [A: $tType] :
( ( ( real_Vector_banach @ A )
& ( real_V2822296259951069270ebra_1 @ A ) )
=> ( ( exp @ A )
= ( ^ [X2: A] :
( plus_plus @ A @ ( plus_plus @ A @ ( one_one @ A ) @ X2 )
@ ( suminf @ A
@ ^ [N2: nat] : ( real_V8093663219630862766scaleR @ A @ ( inverse_inverse @ real @ ( semiring_char_0_fact @ real @ ( plus_plus @ nat @ N2 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( power_power @ A @ X2 @ ( plus_plus @ nat @ N2 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) ) ) ) ) ).
% exp_first_two_terms
thf(fact_4502_inverse__1,axiom,
! [A: $tType] :
( ( division_ring @ A )
=> ( ( inverse_inverse @ A @ ( one_one @ A ) )
= ( one_one @ A ) ) ) ).
% inverse_1
thf(fact_4503_inverse__eq__1__iff,axiom,
! [A: $tType] :
( ( field @ A )
=> ! [X: A] :
( ( ( inverse_inverse @ A @ X )
= ( one_one @ A ) )
= ( X
= ( one_one @ A ) ) ) ) ).
% inverse_eq_1_iff
thf(fact_4504_inverse__mult__distrib,axiom,
! [A: $tType] :
( ( field @ A )
=> ! [A2: A,B2: A] :
( ( inverse_inverse @ A @ ( times_times @ A @ A2 @ B2 ) )
= ( times_times @ A @ ( inverse_inverse @ A @ A2 ) @ ( inverse_inverse @ A @ B2 ) ) ) ) ).
% inverse_mult_distrib
thf(fact_4505_inverse__divide,axiom,
! [A: $tType] :
( ( field @ A )
=> ! [A2: A,B2: A] :
( ( inverse_inverse @ A @ ( divide_divide @ A @ A2 @ B2 ) )
= ( divide_divide @ A @ B2 @ A2 ) ) ) ).
% inverse_divide
thf(fact_4506_inverse__nonnegative__iff__nonnegative,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [A2: A] :
( ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( inverse_inverse @ A @ A2 ) )
= ( ord_less_eq @ A @ ( zero_zero @ A ) @ A2 ) ) ) ).
% inverse_nonnegative_iff_nonnegative
thf(fact_4507_inverse__nonpositive__iff__nonpositive,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [A2: A] :
( ( ord_less_eq @ A @ ( inverse_inverse @ A @ A2 ) @ ( zero_zero @ A ) )
= ( ord_less_eq @ A @ A2 @ ( zero_zero @ A ) ) ) ) ).
% inverse_nonpositive_iff_nonpositive
thf(fact_4508_inverse__less__iff__less,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [A2: A,B2: A] :
( ( ord_less @ A @ ( zero_zero @ A ) @ A2 )
=> ( ( ord_less @ A @ ( zero_zero @ A ) @ B2 )
=> ( ( ord_less @ A @ ( inverse_inverse @ A @ A2 ) @ ( inverse_inverse @ A @ B2 ) )
= ( ord_less @ A @ B2 @ A2 ) ) ) ) ) ).
% inverse_less_iff_less
thf(fact_4509_inverse__less__iff__less__neg,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [A2: A,B2: A] :
( ( ord_less @ A @ A2 @ ( zero_zero @ A ) )
=> ( ( ord_less @ A @ B2 @ ( zero_zero @ A ) )
=> ( ( ord_less @ A @ ( inverse_inverse @ A @ A2 ) @ ( inverse_inverse @ A @ B2 ) )
= ( ord_less @ A @ B2 @ A2 ) ) ) ) ) ).
% inverse_less_iff_less_neg
thf(fact_4510_inverse__negative__iff__negative,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [A2: A] :
( ( ord_less @ A @ ( inverse_inverse @ A @ A2 ) @ ( zero_zero @ A ) )
= ( ord_less @ A @ A2 @ ( zero_zero @ A ) ) ) ) ).
% inverse_negative_iff_negative
thf(fact_4511_inverse__positive__iff__positive,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [A2: A] :
( ( ord_less @ A @ ( zero_zero @ A ) @ ( inverse_inverse @ A @ A2 ) )
= ( ord_less @ A @ ( zero_zero @ A ) @ A2 ) ) ) ).
% inverse_positive_iff_positive
thf(fact_4512_arccos__1,axiom,
( ( arccos @ ( one_one @ real ) )
= ( zero_zero @ real ) ) ).
% arccos_1
thf(fact_4513_inverse__le__iff__le,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [A2: A,B2: A] :
( ( ord_less @ A @ ( zero_zero @ A ) @ A2 )
=> ( ( ord_less @ A @ ( zero_zero @ A ) @ B2 )
=> ( ( ord_less_eq @ A @ ( inverse_inverse @ A @ A2 ) @ ( inverse_inverse @ A @ B2 ) )
= ( ord_less_eq @ A @ B2 @ A2 ) ) ) ) ) ).
% inverse_le_iff_le
thf(fact_4514_inverse__le__iff__le__neg,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [A2: A,B2: A] :
( ( ord_less @ A @ A2 @ ( zero_zero @ A ) )
=> ( ( ord_less @ A @ B2 @ ( zero_zero @ A ) )
=> ( ( ord_less_eq @ A @ ( inverse_inverse @ A @ A2 ) @ ( inverse_inverse @ A @ B2 ) )
= ( ord_less_eq @ A @ B2 @ A2 ) ) ) ) ) ).
% inverse_le_iff_le_neg
thf(fact_4515_right__inverse,axiom,
! [A: $tType] :
( ( division_ring @ A )
=> ! [A2: A] :
( ( A2
!= ( zero_zero @ A ) )
=> ( ( times_times @ A @ A2 @ ( inverse_inverse @ A @ A2 ) )
= ( one_one @ A ) ) ) ) ).
% right_inverse
thf(fact_4516_left__inverse,axiom,
! [A: $tType] :
( ( division_ring @ A )
=> ! [A2: A] :
( ( A2
!= ( zero_zero @ A ) )
=> ( ( times_times @ A @ ( inverse_inverse @ A @ A2 ) @ A2 )
= ( one_one @ A ) ) ) ) ).
% left_inverse
thf(fact_4517_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_4518_arccos__minus__1,axiom,
( ( arccos @ ( uminus_uminus @ real @ ( one_one @ real ) ) )
= pi ) ).
% arccos_minus_1
thf(fact_4519_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_4520_cos__arccos,axiom,
! [Y: real] :
( ( ord_less_eq @ real @ ( uminus_uminus @ real @ ( one_one @ real ) ) @ Y )
=> ( ( ord_less_eq @ real @ Y @ ( one_one @ real ) )
=> ( ( cos @ real @ ( arccos @ Y ) )
= Y ) ) ) ).
% cos_arccos
thf(fact_4521_sin__arcsin,axiom,
! [Y: real] :
( ( ord_less_eq @ real @ ( uminus_uminus @ real @ ( one_one @ real ) ) @ Y )
=> ( ( ord_less_eq @ real @ Y @ ( one_one @ real ) )
=> ( ( sin @ real @ ( arcsin @ Y ) )
= Y ) ) ) ).
% sin_arcsin
thf(fact_4522_arccos__0,axiom,
( ( arccos @ ( zero_zero @ real ) )
= ( divide_divide @ real @ pi @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) ) ).
% arccos_0
thf(fact_4523_arcsin__1,axiom,
( ( arcsin @ ( one_one @ real ) )
= ( divide_divide @ real @ pi @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) ) ).
% arcsin_1
thf(fact_4524_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_4525_real__sqrt__inverse,axiom,
! [X: real] :
( ( sqrt @ ( inverse_inverse @ real @ X ) )
= ( inverse_inverse @ real @ ( sqrt @ X ) ) ) ).
% real_sqrt_inverse
thf(fact_4526_mult__commute__imp__mult__inverse__commute,axiom,
! [A: $tType] :
( ( division_ring @ A )
=> ! [Y: A,X: A] :
( ( ( times_times @ A @ Y @ X )
= ( times_times @ A @ X @ Y ) )
=> ( ( times_times @ A @ ( inverse_inverse @ A @ Y ) @ X )
= ( times_times @ A @ X @ ( inverse_inverse @ A @ Y ) ) ) ) ) ).
% mult_commute_imp_mult_inverse_commute
thf(fact_4527_power__inverse,axiom,
! [A: $tType] :
( ( division_ring @ A )
=> ! [A2: A,N3: nat] :
( ( power_power @ A @ ( inverse_inverse @ A @ A2 ) @ N3 )
= ( inverse_inverse @ A @ ( power_power @ A @ A2 @ N3 ) ) ) ) ).
% power_inverse
thf(fact_4528_norm__inverse__le__norm,axiom,
! [A: $tType] :
( ( real_V8999393235501362500lgebra @ A )
=> ! [R2: real,X: A] :
( ( ord_less_eq @ real @ R2 @ ( real_V7770717601297561774m_norm @ A @ X ) )
=> ( ( ord_less @ real @ ( zero_zero @ real ) @ R2 )
=> ( ord_less_eq @ real @ ( real_V7770717601297561774m_norm @ A @ ( inverse_inverse @ A @ X ) ) @ ( inverse_inverse @ real @ R2 ) ) ) ) ) ).
% norm_inverse_le_norm
thf(fact_4529_positive__imp__inverse__positive,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [A2: A] :
( ( ord_less @ A @ ( zero_zero @ A ) @ A2 )
=> ( ord_less @ A @ ( zero_zero @ A ) @ ( inverse_inverse @ A @ A2 ) ) ) ) ).
% positive_imp_inverse_positive
thf(fact_4530_negative__imp__inverse__negative,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [A2: A] :
( ( ord_less @ A @ A2 @ ( zero_zero @ A ) )
=> ( ord_less @ A @ ( inverse_inverse @ A @ A2 ) @ ( zero_zero @ A ) ) ) ) ).
% negative_imp_inverse_negative
thf(fact_4531_inverse__positive__imp__positive,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [A2: A] :
( ( ord_less @ A @ ( zero_zero @ A ) @ ( inverse_inverse @ A @ A2 ) )
=> ( ( A2
!= ( zero_zero @ A ) )
=> ( ord_less @ A @ ( zero_zero @ A ) @ A2 ) ) ) ) ).
% inverse_positive_imp_positive
thf(fact_4532_inverse__negative__imp__negative,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [A2: A] :
( ( ord_less @ A @ ( inverse_inverse @ A @ A2 ) @ ( zero_zero @ A ) )
=> ( ( A2
!= ( zero_zero @ A ) )
=> ( ord_less @ A @ A2 @ ( zero_zero @ A ) ) ) ) ) ).
% inverse_negative_imp_negative
thf(fact_4533_less__imp__inverse__less__neg,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [A2: A,B2: A] :
( ( ord_less @ A @ A2 @ B2 )
=> ( ( ord_less @ A @ B2 @ ( zero_zero @ A ) )
=> ( ord_less @ A @ ( inverse_inverse @ A @ B2 ) @ ( inverse_inverse @ A @ A2 ) ) ) ) ) ).
% less_imp_inverse_less_neg
thf(fact_4534_inverse__less__imp__less__neg,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [A2: A,B2: A] :
( ( ord_less @ A @ ( inverse_inverse @ A @ A2 ) @ ( inverse_inverse @ A @ B2 ) )
=> ( ( ord_less @ A @ B2 @ ( zero_zero @ A ) )
=> ( ord_less @ A @ B2 @ A2 ) ) ) ) ).
% inverse_less_imp_less_neg
thf(fact_4535_less__imp__inverse__less,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [A2: A,B2: A] :
( ( ord_less @ A @ A2 @ B2 )
=> ( ( ord_less @ A @ ( zero_zero @ A ) @ A2 )
=> ( ord_less @ A @ ( inverse_inverse @ A @ B2 ) @ ( inverse_inverse @ A @ A2 ) ) ) ) ) ).
% less_imp_inverse_less
thf(fact_4536_inverse__less__imp__less,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [A2: A,B2: A] :
( ( ord_less @ A @ ( inverse_inverse @ A @ A2 ) @ ( inverse_inverse @ A @ B2 ) )
=> ( ( ord_less @ A @ ( zero_zero @ A ) @ A2 )
=> ( ord_less @ A @ B2 @ A2 ) ) ) ) ).
% inverse_less_imp_less
thf(fact_4537_nonzero__inverse__mult__distrib,axiom,
! [A: $tType] :
( ( division_ring @ A )
=> ! [A2: A,B2: A] :
( ( A2
!= ( zero_zero @ A ) )
=> ( ( B2
!= ( zero_zero @ A ) )
=> ( ( inverse_inverse @ A @ ( times_times @ A @ A2 @ B2 ) )
= ( times_times @ A @ ( inverse_inverse @ A @ B2 ) @ ( inverse_inverse @ A @ A2 ) ) ) ) ) ) ).
% nonzero_inverse_mult_distrib
thf(fact_4538_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_4539_inverse__unique,axiom,
! [A: $tType] :
( ( division_ring @ A )
=> ! [A2: A,B2: A] :
( ( ( times_times @ A @ A2 @ B2 )
= ( one_one @ A ) )
=> ( ( inverse_inverse @ A @ A2 )
= B2 ) ) ) ).
% inverse_unique
thf(fact_4540_inverse__eq__divide,axiom,
! [A: $tType] :
( ( division_ring @ A )
=> ( ( inverse_inverse @ A )
= ( divide_divide @ A @ ( one_one @ A ) ) ) ) ).
% inverse_eq_divide
thf(fact_4541_divide__inverse__commute,axiom,
! [A: $tType] :
( ( field @ A )
=> ( ( divide_divide @ A )
= ( ^ [A6: A,B5: A] : ( times_times @ A @ ( inverse_inverse @ A @ B5 ) @ A6 ) ) ) ) ).
% divide_inverse_commute
thf(fact_4542_divide__inverse,axiom,
! [A: $tType] :
( ( division_ring @ A )
=> ( ( divide_divide @ A )
= ( ^ [A6: A,B5: A] : ( times_times @ A @ A6 @ ( inverse_inverse @ A @ B5 ) ) ) ) ) ).
% divide_inverse
thf(fact_4543_field__class_Ofield__divide__inverse,axiom,
! [A: $tType] :
( ( field @ A )
=> ( ( divide_divide @ A )
= ( ^ [A6: A,B5: A] : ( times_times @ A @ A6 @ ( inverse_inverse @ A @ B5 ) ) ) ) ) ).
% field_class.field_divide_inverse
thf(fact_4544_power__mult__power__inverse__commute,axiom,
! [A: $tType] :
( ( division_ring @ A )
=> ! [X: A,M: nat,N3: nat] :
( ( times_times @ A @ ( power_power @ A @ X @ M ) @ ( power_power @ A @ ( inverse_inverse @ A @ X ) @ N3 ) )
= ( times_times @ A @ ( power_power @ A @ ( inverse_inverse @ A @ X ) @ N3 ) @ ( power_power @ A @ X @ M ) ) ) ) ).
% power_mult_power_inverse_commute
thf(fact_4545_power__mult__inverse__distrib,axiom,
! [A: $tType] :
( ( division_ring @ A )
=> ! [X: A,M: nat] :
( ( times_times @ A @ ( power_power @ A @ X @ M ) @ ( inverse_inverse @ A @ X ) )
= ( times_times @ A @ ( inverse_inverse @ A @ X ) @ ( power_power @ A @ X @ M ) ) ) ) ).
% power_mult_inverse_distrib
thf(fact_4546_mult__inverse__of__nat__commute,axiom,
! [A: $tType] :
( ( division_ring @ A )
=> ! [Xa: nat,X: A] :
( ( times_times @ A @ ( inverse_inverse @ A @ ( semiring_1_of_nat @ A @ Xa ) ) @ X )
= ( times_times @ A @ X @ ( inverse_inverse @ A @ ( semiring_1_of_nat @ A @ Xa ) ) ) ) ) ).
% mult_inverse_of_nat_commute
thf(fact_4547_mult__inverse__of__int__commute,axiom,
! [A: $tType] :
( ( division_ring @ A )
=> ! [Xa: int,X: A] :
( ( times_times @ A @ ( inverse_inverse @ A @ ( ring_1_of_int @ A @ Xa ) ) @ X )
= ( times_times @ A @ X @ ( inverse_inverse @ A @ ( ring_1_of_int @ A @ Xa ) ) ) ) ) ).
% mult_inverse_of_int_commute
thf(fact_4548_divide__real__def,axiom,
( ( divide_divide @ real )
= ( ^ [X2: real,Y2: real] : ( times_times @ real @ X2 @ ( inverse_inverse @ real @ Y2 ) ) ) ) ).
% divide_real_def
thf(fact_4549_inverse__le__imp__le,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [A2: A,B2: A] :
( ( ord_less_eq @ A @ ( inverse_inverse @ A @ A2 ) @ ( inverse_inverse @ A @ B2 ) )
=> ( ( ord_less @ A @ ( zero_zero @ A ) @ A2 )
=> ( ord_less_eq @ A @ B2 @ A2 ) ) ) ) ).
% inverse_le_imp_le
thf(fact_4550_le__imp__inverse__le,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [A2: A,B2: A] :
( ( ord_less_eq @ A @ A2 @ B2 )
=> ( ( ord_less @ A @ ( zero_zero @ A ) @ A2 )
=> ( ord_less_eq @ A @ ( inverse_inverse @ A @ B2 ) @ ( inverse_inverse @ A @ A2 ) ) ) ) ) ).
% le_imp_inverse_le
thf(fact_4551_inverse__le__imp__le__neg,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [A2: A,B2: A] :
( ( ord_less_eq @ A @ ( inverse_inverse @ A @ A2 ) @ ( inverse_inverse @ A @ B2 ) )
=> ( ( ord_less @ A @ B2 @ ( zero_zero @ A ) )
=> ( ord_less_eq @ A @ B2 @ A2 ) ) ) ) ).
% inverse_le_imp_le_neg
thf(fact_4552_le__imp__inverse__le__neg,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [A2: A,B2: A] :
( ( ord_less_eq @ A @ A2 @ B2 )
=> ( ( ord_less @ A @ B2 @ ( zero_zero @ A ) )
=> ( ord_less_eq @ A @ ( inverse_inverse @ A @ B2 ) @ ( inverse_inverse @ A @ A2 ) ) ) ) ) ).
% le_imp_inverse_le_neg
thf(fact_4553_inverse__le__1__iff,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [X: A] :
( ( ord_less_eq @ A @ ( inverse_inverse @ A @ X ) @ ( one_one @ A ) )
= ( ( ord_less_eq @ A @ X @ ( zero_zero @ A ) )
| ( ord_less_eq @ A @ ( one_one @ A ) @ X ) ) ) ) ).
% inverse_le_1_iff
thf(fact_4554_one__less__inverse,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [A2: A] :
( ( ord_less @ A @ ( zero_zero @ A ) @ A2 )
=> ( ( ord_less @ A @ A2 @ ( one_one @ A ) )
=> ( ord_less @ A @ ( one_one @ A ) @ ( inverse_inverse @ A @ A2 ) ) ) ) ) ).
% one_less_inverse
thf(fact_4555_one__less__inverse__iff,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [X: A] :
( ( ord_less @ A @ ( one_one @ A ) @ ( inverse_inverse @ A @ X ) )
= ( ( ord_less @ A @ ( zero_zero @ A ) @ X )
& ( ord_less @ A @ X @ ( one_one @ A ) ) ) ) ) ).
% one_less_inverse_iff
thf(fact_4556_division__ring__inverse__add,axiom,
! [A: $tType] :
( ( division_ring @ A )
=> ! [A2: A,B2: A] :
( ( A2
!= ( zero_zero @ A ) )
=> ( ( B2
!= ( zero_zero @ A ) )
=> ( ( plus_plus @ A @ ( inverse_inverse @ A @ A2 ) @ ( inverse_inverse @ A @ B2 ) )
= ( times_times @ A @ ( times_times @ A @ ( inverse_inverse @ A @ A2 ) @ ( plus_plus @ A @ A2 @ B2 ) ) @ ( inverse_inverse @ A @ B2 ) ) ) ) ) ) ).
% division_ring_inverse_add
thf(fact_4557_inverse__add,axiom,
! [A: $tType] :
( ( field @ A )
=> ! [A2: A,B2: A] :
( ( A2
!= ( zero_zero @ A ) )
=> ( ( B2
!= ( zero_zero @ A ) )
=> ( ( plus_plus @ A @ ( inverse_inverse @ A @ A2 ) @ ( inverse_inverse @ A @ B2 ) )
= ( times_times @ A @ ( times_times @ A @ ( plus_plus @ A @ A2 @ B2 ) @ ( inverse_inverse @ A @ A2 ) ) @ ( inverse_inverse @ A @ B2 ) ) ) ) ) ) ).
% inverse_add
thf(fact_4558_field__class_Ofield__inverse,axiom,
! [A: $tType] :
( ( field @ A )
=> ! [A2: A] :
( ( A2
!= ( zero_zero @ A ) )
=> ( ( times_times @ A @ ( inverse_inverse @ A @ A2 ) @ A2 )
= ( one_one @ A ) ) ) ) ).
% field_class.field_inverse
thf(fact_4559_division__ring__inverse__diff,axiom,
! [A: $tType] :
( ( division_ring @ A )
=> ! [A2: A,B2: A] :
( ( A2
!= ( zero_zero @ A ) )
=> ( ( B2
!= ( zero_zero @ A ) )
=> ( ( minus_minus @ A @ ( inverse_inverse @ A @ A2 ) @ ( inverse_inverse @ A @ B2 ) )
= ( times_times @ A @ ( times_times @ A @ ( inverse_inverse @ A @ A2 ) @ ( minus_minus @ A @ B2 @ A2 ) ) @ ( inverse_inverse @ A @ B2 ) ) ) ) ) ) ).
% division_ring_inverse_diff
thf(fact_4560_nonzero__inverse__eq__divide,axiom,
! [A: $tType] :
( ( division_ring @ A )
=> ! [A2: A] :
( ( A2
!= ( zero_zero @ A ) )
=> ( ( inverse_inverse @ A @ A2 )
= ( divide_divide @ A @ ( one_one @ A ) @ A2 ) ) ) ) ).
% nonzero_inverse_eq_divide
thf(fact_4561_inverse__powr,axiom,
! [Y: real,A2: real] :
( ( ord_less_eq @ real @ ( zero_zero @ real ) @ Y )
=> ( ( powr @ real @ ( inverse_inverse @ real @ Y ) @ A2 )
= ( inverse_inverse @ real @ ( powr @ real @ Y @ A2 ) ) ) ) ).
% inverse_powr
thf(fact_4562_arccos__le__arccos,axiom,
! [X: real,Y: real] :
( ( ord_less_eq @ real @ ( uminus_uminus @ real @ ( one_one @ real ) ) @ X )
=> ( ( ord_less_eq @ real @ X @ Y )
=> ( ( ord_less_eq @ real @ Y @ ( one_one @ real ) )
=> ( ord_less_eq @ real @ ( arccos @ Y ) @ ( arccos @ X ) ) ) ) ) ).
% arccos_le_arccos
thf(fact_4563_arccos__eq__iff,axiom,
! [X: real,Y: real] :
( ( ( ord_less_eq @ real @ ( abs_abs @ real @ X ) @ ( one_one @ real ) )
& ( ord_less_eq @ real @ ( abs_abs @ real @ Y ) @ ( one_one @ real ) ) )
=> ( ( ( arccos @ X )
= ( arccos @ Y ) )
= ( X = Y ) ) ) ).
% arccos_eq_iff
thf(fact_4564_arccos__le__mono,axiom,
! [X: real,Y: real] :
( ( ord_less_eq @ real @ ( abs_abs @ real @ X ) @ ( one_one @ real ) )
=> ( ( ord_less_eq @ real @ ( abs_abs @ real @ Y ) @ ( one_one @ real ) )
=> ( ( ord_less_eq @ real @ ( arccos @ X ) @ ( arccos @ Y ) )
= ( ord_less_eq @ real @ Y @ X ) ) ) ) ).
% arccos_le_mono
thf(fact_4565_arcsin__le__arcsin,axiom,
! [X: real,Y: real] :
( ( ord_less_eq @ real @ ( uminus_uminus @ real @ ( one_one @ real ) ) @ X )
=> ( ( ord_less_eq @ real @ X @ Y )
=> ( ( ord_less_eq @ real @ Y @ ( one_one @ real ) )
=> ( ord_less_eq @ real @ ( arcsin @ X ) @ ( arcsin @ Y ) ) ) ) ) ).
% arcsin_le_arcsin
thf(fact_4566_arcsin__minus,axiom,
! [X: real] :
( ( ord_less_eq @ real @ ( uminus_uminus @ real @ ( one_one @ real ) ) @ X )
=> ( ( ord_less_eq @ real @ X @ ( one_one @ real ) )
=> ( ( arcsin @ ( uminus_uminus @ real @ X ) )
= ( uminus_uminus @ real @ ( arcsin @ X ) ) ) ) ) ).
% arcsin_minus
thf(fact_4567_arcsin__eq__iff,axiom,
! [X: real,Y: real] :
( ( ord_less_eq @ real @ ( abs_abs @ real @ X ) @ ( one_one @ real ) )
=> ( ( ord_less_eq @ real @ ( abs_abs @ real @ Y ) @ ( one_one @ real ) )
=> ( ( ( arcsin @ X )
= ( arcsin @ Y ) )
= ( X = Y ) ) ) ) ).
% arcsin_eq_iff
thf(fact_4568_arcsin__le__mono,axiom,
! [X: real,Y: real] :
( ( ord_less_eq @ real @ ( abs_abs @ real @ X ) @ ( one_one @ real ) )
=> ( ( ord_less_eq @ real @ ( abs_abs @ real @ Y ) @ ( one_one @ real ) )
=> ( ( ord_less_eq @ real @ ( arcsin @ X ) @ ( arcsin @ Y ) )
= ( ord_less_eq @ real @ X @ Y ) ) ) ) ).
% arcsin_le_mono
thf(fact_4569_one__le__inverse,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [A2: A] :
( ( ord_less @ A @ ( zero_zero @ A ) @ A2 )
=> ( ( ord_less_eq @ A @ A2 @ ( one_one @ A ) )
=> ( ord_less_eq @ A @ ( one_one @ A ) @ ( inverse_inverse @ A @ A2 ) ) ) ) ) ).
% one_le_inverse
thf(fact_4570_inverse__less__1__iff,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [X: A] :
( ( ord_less @ A @ ( inverse_inverse @ A @ X ) @ ( one_one @ A ) )
= ( ( ord_less_eq @ A @ X @ ( zero_zero @ A ) )
| ( ord_less @ A @ ( one_one @ A ) @ X ) ) ) ) ).
% inverse_less_1_iff
thf(fact_4571_one__le__inverse__iff,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [X: A] :
( ( ord_less_eq @ A @ ( one_one @ A ) @ ( inverse_inverse @ A @ X ) )
= ( ( ord_less @ A @ ( zero_zero @ A ) @ X )
& ( ord_less_eq @ A @ X @ ( one_one @ A ) ) ) ) ) ).
% one_le_inverse_iff
thf(fact_4572_inverse__less__iff,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [A2: A,B2: A] :
( ( ord_less @ A @ ( inverse_inverse @ A @ A2 ) @ ( inverse_inverse @ A @ B2 ) )
= ( ( ( ord_less @ A @ ( zero_zero @ A ) @ ( times_times @ A @ A2 @ B2 ) )
=> ( ord_less @ A @ B2 @ A2 ) )
& ( ( ord_less_eq @ A @ ( times_times @ A @ A2 @ B2 ) @ ( zero_zero @ A ) )
=> ( ord_less @ A @ A2 @ B2 ) ) ) ) ) ).
% inverse_less_iff
thf(fact_4573_inverse__le__iff,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [A2: A,B2: A] :
( ( ord_less_eq @ A @ ( inverse_inverse @ A @ A2 ) @ ( inverse_inverse @ A @ B2 ) )
= ( ( ( ord_less @ A @ ( zero_zero @ A ) @ ( times_times @ A @ A2 @ B2 ) )
=> ( ord_less_eq @ A @ B2 @ A2 ) )
& ( ( ord_less_eq @ A @ ( times_times @ A @ A2 @ B2 ) @ ( zero_zero @ A ) )
=> ( ord_less_eq @ A @ A2 @ B2 ) ) ) ) ) ).
% inverse_le_iff
thf(fact_4574_inverse__diff__inverse,axiom,
! [A: $tType] :
( ( division_ring @ A )
=> ! [A2: A,B2: A] :
( ( A2
!= ( zero_zero @ A ) )
=> ( ( B2
!= ( zero_zero @ A ) )
=> ( ( minus_minus @ A @ ( inverse_inverse @ A @ A2 ) @ ( inverse_inverse @ A @ B2 ) )
= ( uminus_uminus @ A @ ( times_times @ A @ ( times_times @ A @ ( inverse_inverse @ A @ A2 ) @ ( minus_minus @ A @ A2 @ B2 ) ) @ ( inverse_inverse @ A @ B2 ) ) ) ) ) ) ) ).
% inverse_diff_inverse
thf(fact_4575_reals__Archimedean,axiom,
! [A: $tType] :
( ( archim462609752435547400_field @ A )
=> ! [X: A] :
( ( ord_less @ A @ ( zero_zero @ A ) @ X )
=> ? [N: nat] : ( ord_less @ A @ ( inverse_inverse @ A @ ( semiring_1_of_nat @ A @ ( suc @ N ) ) ) @ X ) ) ) ).
% reals_Archimedean
thf(fact_4576_real__vector__eq__affinity,axiom,
! [A: $tType] :
( ( real_V4867850818363320053vector @ A )
=> ! [M: real,Y: A,X: A,C2: A] :
( ( M
!= ( zero_zero @ real ) )
=> ( ( Y
= ( plus_plus @ A @ ( real_V8093663219630862766scaleR @ A @ M @ X ) @ C2 ) )
= ( ( minus_minus @ A @ ( real_V8093663219630862766scaleR @ A @ ( inverse_inverse @ real @ M ) @ Y ) @ ( real_V8093663219630862766scaleR @ A @ ( inverse_inverse @ real @ M ) @ C2 ) )
= X ) ) ) ) ).
% real_vector_eq_affinity
thf(fact_4577_real__vector__affinity__eq,axiom,
! [A: $tType] :
( ( real_V4867850818363320053vector @ A )
=> ! [M: real,X: A,C2: A,Y: A] :
( ( M
!= ( zero_zero @ real ) )
=> ( ( ( plus_plus @ A @ ( real_V8093663219630862766scaleR @ A @ M @ X ) @ C2 )
= Y )
= ( X
= ( minus_minus @ A @ ( real_V8093663219630862766scaleR @ A @ ( inverse_inverse @ real @ M ) @ Y ) @ ( real_V8093663219630862766scaleR @ A @ ( inverse_inverse @ real @ M ) @ C2 ) ) ) ) ) ) ).
% real_vector_affinity_eq
thf(fact_4578_pos__divideR__le__eq,axiom,
! [A: $tType] :
( ( real_V5355595471888546746vector @ A )
=> ! [C2: real,B2: A,A2: A] :
( ( ord_less @ real @ ( zero_zero @ real ) @ C2 )
=> ( ( ord_less_eq @ A @ ( real_V8093663219630862766scaleR @ A @ ( inverse_inverse @ real @ C2 ) @ B2 ) @ A2 )
= ( ord_less_eq @ A @ B2 @ ( real_V8093663219630862766scaleR @ A @ C2 @ A2 ) ) ) ) ) ).
% pos_divideR_le_eq
thf(fact_4579_pos__le__divideR__eq,axiom,
! [A: $tType] :
( ( real_V5355595471888546746vector @ A )
=> ! [C2: real,A2: A,B2: A] :
( ( ord_less @ real @ ( zero_zero @ real ) @ C2 )
=> ( ( ord_less_eq @ A @ A2 @ ( real_V8093663219630862766scaleR @ A @ ( inverse_inverse @ real @ C2 ) @ B2 ) )
= ( ord_less_eq @ A @ ( real_V8093663219630862766scaleR @ A @ C2 @ A2 ) @ B2 ) ) ) ) ).
% pos_le_divideR_eq
thf(fact_4580_neg__divideR__le__eq,axiom,
! [A: $tType] :
( ( real_V5355595471888546746vector @ A )
=> ! [C2: real,B2: A,A2: A] :
( ( ord_less @ real @ C2 @ ( zero_zero @ real ) )
=> ( ( ord_less_eq @ A @ ( real_V8093663219630862766scaleR @ A @ ( inverse_inverse @ real @ C2 ) @ B2 ) @ A2 )
= ( ord_less_eq @ A @ ( real_V8093663219630862766scaleR @ A @ C2 @ A2 ) @ B2 ) ) ) ) ).
% neg_divideR_le_eq
thf(fact_4581_neg__le__divideR__eq,axiom,
! [A: $tType] :
( ( real_V5355595471888546746vector @ A )
=> ! [C2: real,A2: A,B2: A] :
( ( ord_less @ real @ C2 @ ( zero_zero @ real ) )
=> ( ( ord_less_eq @ A @ A2 @ ( real_V8093663219630862766scaleR @ A @ ( inverse_inverse @ real @ C2 ) @ B2 ) )
= ( ord_less_eq @ A @ B2 @ ( real_V8093663219630862766scaleR @ A @ C2 @ A2 ) ) ) ) ) ).
% neg_le_divideR_eq
thf(fact_4582_neg__less__divideR__eq,axiom,
! [A: $tType] :
( ( real_V5355595471888546746vector @ A )
=> ! [C2: real,A2: A,B2: A] :
( ( ord_less @ real @ C2 @ ( zero_zero @ real ) )
=> ( ( ord_less @ A @ A2 @ ( real_V8093663219630862766scaleR @ A @ ( inverse_inverse @ real @ C2 ) @ B2 ) )
= ( ord_less @ A @ B2 @ ( real_V8093663219630862766scaleR @ A @ C2 @ A2 ) ) ) ) ) ).
% neg_less_divideR_eq
thf(fact_4583_neg__divideR__less__eq,axiom,
! [A: $tType] :
( ( real_V5355595471888546746vector @ A )
=> ! [C2: real,B2: A,A2: A] :
( ( ord_less @ real @ C2 @ ( zero_zero @ real ) )
=> ( ( ord_less @ A @ ( real_V8093663219630862766scaleR @ A @ ( inverse_inverse @ real @ C2 ) @ B2 ) @ A2 )
= ( ord_less @ A @ ( real_V8093663219630862766scaleR @ A @ C2 @ A2 ) @ B2 ) ) ) ) ).
% neg_divideR_less_eq
thf(fact_4584_pos__less__divideR__eq,axiom,
! [A: $tType] :
( ( real_V5355595471888546746vector @ A )
=> ! [C2: real,A2: A,B2: A] :
( ( ord_less @ real @ ( zero_zero @ real ) @ C2 )
=> ( ( ord_less @ A @ A2 @ ( real_V8093663219630862766scaleR @ A @ ( inverse_inverse @ real @ C2 ) @ B2 ) )
= ( ord_less @ A @ ( real_V8093663219630862766scaleR @ A @ C2 @ A2 ) @ B2 ) ) ) ) ).
% pos_less_divideR_eq
thf(fact_4585_pos__divideR__less__eq,axiom,
! [A: $tType] :
( ( real_V5355595471888546746vector @ A )
=> ! [C2: real,B2: A,A2: A] :
( ( ord_less @ real @ ( zero_zero @ real ) @ C2 )
=> ( ( ord_less @ A @ ( real_V8093663219630862766scaleR @ A @ ( inverse_inverse @ real @ C2 ) @ B2 ) @ A2 )
= ( ord_less @ A @ B2 @ ( real_V8093663219630862766scaleR @ A @ C2 @ A2 ) ) ) ) ) ).
% pos_divideR_less_eq
thf(fact_4586_forall__pos__mono__1,axiom,
! [P: real > $o,E: real] :
( ! [D3: real,E2: real] :
( ( ord_less @ real @ D3 @ E2 )
=> ( ( P @ D3 )
=> ( P @ E2 ) ) )
=> ( ! [N: nat] : ( P @ ( inverse_inverse @ real @ ( semiring_1_of_nat @ real @ ( suc @ N ) ) ) )
=> ( ( ord_less @ real @ ( zero_zero @ real ) @ E )
=> ( P @ E ) ) ) ) ).
% forall_pos_mono_1
thf(fact_4587_forall__pos__mono,axiom,
! [P: real > $o,E: real] :
( ! [D3: real,E2: real] :
( ( ord_less @ real @ D3 @ E2 )
=> ( ( P @ D3 )
=> ( P @ E2 ) ) )
=> ( ! [N: nat] :
( ( N
!= ( zero_zero @ nat ) )
=> ( P @ ( inverse_inverse @ real @ ( semiring_1_of_nat @ real @ N ) ) ) )
=> ( ( ord_less @ real @ ( zero_zero @ real ) @ E )
=> ( P @ E ) ) ) ) ).
% forall_pos_mono
thf(fact_4588_real__arch__inverse,axiom,
! [E: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ E )
= ( ? [N2: nat] :
( ( N2
!= ( zero_zero @ nat ) )
& ( ord_less @ real @ ( zero_zero @ real ) @ ( inverse_inverse @ real @ ( semiring_1_of_nat @ real @ N2 ) ) )
& ( ord_less @ real @ ( inverse_inverse @ real @ ( semiring_1_of_nat @ real @ N2 ) ) @ E ) ) ) ) ).
% real_arch_inverse
thf(fact_4589_sqrt__divide__self__eq,axiom,
! [X: real] :
( ( ord_less_eq @ real @ ( zero_zero @ real ) @ X )
=> ( ( divide_divide @ real @ ( sqrt @ X ) @ X )
= ( inverse_inverse @ real @ ( sqrt @ X ) ) ) ) ).
% sqrt_divide_self_eq
thf(fact_4590_summable__exp,axiom,
! [A: $tType] :
( ( ( real_Vector_banach @ A )
& ( real_V3459762299906320749_field @ A ) )
=> ! [X: A] :
( summable @ A
@ ^ [N2: nat] : ( times_times @ A @ ( inverse_inverse @ A @ ( semiring_char_0_fact @ A @ N2 ) ) @ ( power_power @ A @ X @ N2 ) ) ) ) ).
% summable_exp
thf(fact_4591_summable__exp__generic,axiom,
! [A: $tType] :
( ( ( real_Vector_banach @ A )
& ( real_V2822296259951069270ebra_1 @ A ) )
=> ! [X: A] :
( summable @ A
@ ^ [N2: nat] : ( real_V8093663219630862766scaleR @ A @ ( inverse_inverse @ real @ ( semiring_char_0_fact @ real @ N2 ) ) @ ( power_power @ A @ X @ N2 ) ) ) ) ).
% summable_exp_generic
thf(fact_4592_ex__inverse__of__nat__less,axiom,
! [A: $tType] :
( ( archim462609752435547400_field @ A )
=> ! [X: A] :
( ( ord_less @ A @ ( zero_zero @ A ) @ X )
=> ? [N: nat] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ N )
& ( ord_less @ A @ ( inverse_inverse @ A @ ( semiring_1_of_nat @ A @ N ) ) @ X ) ) ) ) ).
% ex_inverse_of_nat_less
thf(fact_4593_arccos__lbound,axiom,
! [Y: real] :
( ( ord_less_eq @ real @ ( uminus_uminus @ real @ ( one_one @ real ) ) @ Y )
=> ( ( ord_less_eq @ real @ Y @ ( one_one @ real ) )
=> ( ord_less_eq @ real @ ( zero_zero @ real ) @ ( arccos @ Y ) ) ) ) ).
% arccos_lbound
thf(fact_4594_arccos__less__arccos,axiom,
! [X: real,Y: real] :
( ( ord_less_eq @ real @ ( uminus_uminus @ real @ ( one_one @ real ) ) @ X )
=> ( ( ord_less @ real @ X @ Y )
=> ( ( ord_less_eq @ real @ Y @ ( one_one @ real ) )
=> ( ord_less @ real @ ( arccos @ Y ) @ ( arccos @ X ) ) ) ) ) ).
% arccos_less_arccos
thf(fact_4595_power__diff__conv__inverse,axiom,
! [A: $tType] :
( ( division_ring @ A )
=> ! [X: A,M: nat,N3: nat] :
( ( X
!= ( zero_zero @ A ) )
=> ( ( ord_less_eq @ nat @ M @ N3 )
=> ( ( power_power @ A @ X @ ( minus_minus @ nat @ N3 @ M ) )
= ( times_times @ A @ ( power_power @ A @ X @ N3 ) @ ( power_power @ A @ ( inverse_inverse @ A @ X ) @ M ) ) ) ) ) ) ).
% power_diff_conv_inverse
thf(fact_4596_arccos__less__mono,axiom,
! [X: real,Y: real] :
( ( ord_less_eq @ real @ ( abs_abs @ real @ X ) @ ( one_one @ real ) )
=> ( ( ord_less_eq @ real @ ( abs_abs @ real @ Y ) @ ( one_one @ real ) )
=> ( ( ord_less @ real @ ( arccos @ X ) @ ( arccos @ Y ) )
= ( ord_less @ real @ Y @ X ) ) ) ) ).
% arccos_less_mono
thf(fact_4597_arccos__ubound,axiom,
! [Y: real] :
( ( ord_less_eq @ real @ ( uminus_uminus @ real @ ( one_one @ real ) ) @ Y )
=> ( ( ord_less_eq @ real @ Y @ ( one_one @ real ) )
=> ( ord_less_eq @ real @ ( arccos @ Y ) @ pi ) ) ) ).
% arccos_ubound
thf(fact_4598_arccos__cos,axiom,
! [X: real] :
( ( ord_less_eq @ real @ ( zero_zero @ real ) @ X )
=> ( ( ord_less_eq @ real @ X @ pi )
=> ( ( arccos @ ( cos @ real @ X ) )
= X ) ) ) ).
% arccos_cos
thf(fact_4599_arcsin__less__arcsin,axiom,
! [X: real,Y: real] :
( ( ord_less_eq @ real @ ( uminus_uminus @ real @ ( one_one @ real ) ) @ X )
=> ( ( ord_less @ real @ X @ Y )
=> ( ( ord_less_eq @ real @ Y @ ( one_one @ real ) )
=> ( ord_less @ real @ ( arcsin @ X ) @ ( arcsin @ Y ) ) ) ) ) ).
% arcsin_less_arcsin
thf(fact_4600_arcsin__less__mono,axiom,
! [X: real,Y: real] :
( ( ord_less_eq @ real @ ( abs_abs @ real @ X ) @ ( one_one @ real ) )
=> ( ( ord_less_eq @ real @ ( abs_abs @ real @ Y ) @ ( one_one @ real ) )
=> ( ( ord_less @ real @ ( arcsin @ X ) @ ( arcsin @ Y ) )
= ( ord_less @ real @ X @ Y ) ) ) ) ).
% arcsin_less_mono
thf(fact_4601_cos__arccos__abs,axiom,
! [Y: real] :
( ( ord_less_eq @ real @ ( abs_abs @ real @ Y ) @ ( one_one @ real ) )
=> ( ( cos @ real @ ( arccos @ Y ) )
= Y ) ) ).
% cos_arccos_abs
thf(fact_4602_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_4603_neg__minus__divideR__le__eq,axiom,
! [A: $tType] :
( ( real_V5355595471888546746vector @ A )
=> ! [C2: real,B2: A,A2: A] :
( ( ord_less @ real @ C2 @ ( zero_zero @ real ) )
=> ( ( ord_less_eq @ A @ ( uminus_uminus @ A @ ( real_V8093663219630862766scaleR @ A @ ( inverse_inverse @ real @ C2 ) @ B2 ) ) @ A2 )
= ( ord_less_eq @ A @ ( real_V8093663219630862766scaleR @ A @ C2 @ A2 ) @ ( uminus_uminus @ A @ B2 ) ) ) ) ) ).
% neg_minus_divideR_le_eq
thf(fact_4604_neg__le__minus__divideR__eq,axiom,
! [A: $tType] :
( ( real_V5355595471888546746vector @ A )
=> ! [C2: real,A2: A,B2: A] :
( ( ord_less @ real @ C2 @ ( zero_zero @ real ) )
=> ( ( ord_less_eq @ A @ A2 @ ( uminus_uminus @ A @ ( real_V8093663219630862766scaleR @ A @ ( inverse_inverse @ real @ C2 ) @ B2 ) ) )
= ( ord_less_eq @ A @ ( uminus_uminus @ A @ B2 ) @ ( real_V8093663219630862766scaleR @ A @ C2 @ A2 ) ) ) ) ) ).
% neg_le_minus_divideR_eq
thf(fact_4605_pos__minus__divideR__le__eq,axiom,
! [A: $tType] :
( ( real_V5355595471888546746vector @ A )
=> ! [C2: real,B2: A,A2: A] :
( ( ord_less @ real @ ( zero_zero @ real ) @ C2 )
=> ( ( ord_less_eq @ A @ ( uminus_uminus @ A @ ( real_V8093663219630862766scaleR @ A @ ( inverse_inverse @ real @ C2 ) @ B2 ) ) @ A2 )
= ( ord_less_eq @ A @ ( uminus_uminus @ A @ B2 ) @ ( real_V8093663219630862766scaleR @ A @ C2 @ A2 ) ) ) ) ) ).
% pos_minus_divideR_le_eq
thf(fact_4606_pos__le__minus__divideR__eq,axiom,
! [A: $tType] :
( ( real_V5355595471888546746vector @ A )
=> ! [C2: real,A2: A,B2: A] :
( ( ord_less @ real @ ( zero_zero @ real ) @ C2 )
=> ( ( ord_less_eq @ A @ A2 @ ( uminus_uminus @ A @ ( real_V8093663219630862766scaleR @ A @ ( inverse_inverse @ real @ C2 ) @ B2 ) ) )
= ( ord_less_eq @ A @ ( real_V8093663219630862766scaleR @ A @ C2 @ A2 ) @ ( uminus_uminus @ A @ B2 ) ) ) ) ) ).
% pos_le_minus_divideR_eq
thf(fact_4607_pos__less__minus__divideR__eq,axiom,
! [A: $tType] :
( ( real_V5355595471888546746vector @ A )
=> ! [C2: real,A2: A,B2: A] :
( ( ord_less @ real @ ( zero_zero @ real ) @ C2 )
=> ( ( ord_less @ A @ A2 @ ( uminus_uminus @ A @ ( real_V8093663219630862766scaleR @ A @ ( inverse_inverse @ real @ C2 ) @ B2 ) ) )
= ( ord_less @ A @ ( real_V8093663219630862766scaleR @ A @ C2 @ A2 ) @ ( uminus_uminus @ A @ B2 ) ) ) ) ) ).
% pos_less_minus_divideR_eq
thf(fact_4608_pos__minus__divideR__less__eq,axiom,
! [A: $tType] :
( ( real_V5355595471888546746vector @ A )
=> ! [C2: real,B2: A,A2: A] :
( ( ord_less @ real @ ( zero_zero @ real ) @ C2 )
=> ( ( ord_less @ A @ ( uminus_uminus @ A @ ( real_V8093663219630862766scaleR @ A @ ( inverse_inverse @ real @ C2 ) @ B2 ) ) @ A2 )
= ( ord_less @ A @ ( uminus_uminus @ A @ B2 ) @ ( real_V8093663219630862766scaleR @ A @ C2 @ A2 ) ) ) ) ) ).
% pos_minus_divideR_less_eq
thf(fact_4609_neg__less__minus__divideR__eq,axiom,
! [A: $tType] :
( ( real_V5355595471888546746vector @ A )
=> ! [C2: real,A2: A,B2: A] :
( ( ord_less @ real @ C2 @ ( zero_zero @ real ) )
=> ( ( ord_less @ A @ A2 @ ( uminus_uminus @ A @ ( real_V8093663219630862766scaleR @ A @ ( inverse_inverse @ real @ C2 ) @ B2 ) ) )
= ( ord_less @ A @ ( uminus_uminus @ A @ B2 ) @ ( real_V8093663219630862766scaleR @ A @ C2 @ A2 ) ) ) ) ) ).
% neg_less_minus_divideR_eq
thf(fact_4610_neg__minus__divideR__less__eq,axiom,
! [A: $tType] :
( ( real_V5355595471888546746vector @ A )
=> ! [C2: real,B2: A,A2: A] :
( ( ord_less @ real @ C2 @ ( zero_zero @ real ) )
=> ( ( ord_less @ A @ ( uminus_uminus @ A @ ( real_V8093663219630862766scaleR @ A @ ( inverse_inverse @ real @ C2 ) @ B2 ) ) @ A2 )
= ( ord_less @ A @ ( real_V8093663219630862766scaleR @ A @ C2 @ A2 ) @ ( uminus_uminus @ A @ B2 ) ) ) ) ) ).
% neg_minus_divideR_less_eq
thf(fact_4611_log__inverse,axiom,
! [A2: real,X: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ A2 )
=> ( ( A2
!= ( one_one @ real ) )
=> ( ( ord_less @ real @ ( zero_zero @ real ) @ X )
=> ( ( log @ A2 @ ( inverse_inverse @ real @ X ) )
= ( uminus_uminus @ real @ ( log @ A2 @ X ) ) ) ) ) ) ).
% log_inverse
thf(fact_4612_exp__converges,axiom,
! [A: $tType] :
( ( ( real_Vector_banach @ A )
& ( real_V2822296259951069270ebra_1 @ A ) )
=> ! [X: A] :
( sums @ A
@ ^ [N2: nat] : ( real_V8093663219630862766scaleR @ A @ ( inverse_inverse @ real @ ( semiring_char_0_fact @ real @ N2 ) ) @ ( power_power @ A @ X @ N2 ) )
@ ( exp @ A @ X ) ) ) ).
% exp_converges
thf(fact_4613_exp__def,axiom,
! [A: $tType] :
( ( ( real_Vector_banach @ A )
& ( real_V2822296259951069270ebra_1 @ A ) )
=> ( ( exp @ A )
= ( ^ [X2: A] :
( suminf @ A
@ ^ [N2: nat] : ( real_V8093663219630862766scaleR @ A @ ( inverse_inverse @ real @ ( semiring_char_0_fact @ real @ N2 ) ) @ ( power_power @ A @ X2 @ N2 ) ) ) ) ) ) ).
% exp_def
thf(fact_4614_summable__norm__exp,axiom,
! [A: $tType] :
( ( ( real_Vector_banach @ A )
& ( real_V2822296259951069270ebra_1 @ A ) )
=> ! [X: A] :
( summable @ real
@ ^ [N2: nat] : ( real_V7770717601297561774m_norm @ A @ ( real_V8093663219630862766scaleR @ A @ ( inverse_inverse @ real @ ( semiring_char_0_fact @ real @ N2 ) ) @ ( power_power @ A @ X @ N2 ) ) ) ) ) ).
% summable_norm_exp
thf(fact_4615_arccos__lt__bounded,axiom,
! [Y: real] :
( ( ord_less @ real @ ( uminus_uminus @ real @ ( one_one @ real ) ) @ Y )
=> ( ( ord_less @ real @ Y @ ( one_one @ real ) )
=> ( ( ord_less @ real @ ( zero_zero @ real ) @ ( arccos @ Y ) )
& ( ord_less @ real @ ( arccos @ Y ) @ pi ) ) ) ) ).
% arccos_lt_bounded
thf(fact_4616_arccos__bounded,axiom,
! [Y: real] :
( ( ord_less_eq @ real @ ( uminus_uminus @ real @ ( one_one @ real ) ) @ Y )
=> ( ( ord_less_eq @ real @ Y @ ( one_one @ real ) )
=> ( ( ord_less_eq @ real @ ( zero_zero @ real ) @ ( arccos @ Y ) )
& ( ord_less_eq @ real @ ( arccos @ Y ) @ pi ) ) ) ) ).
% arccos_bounded
thf(fact_4617_sin__arccos__nonzero,axiom,
! [X: real] :
( ( ord_less @ real @ ( uminus_uminus @ real @ ( one_one @ real ) ) @ X )
=> ( ( ord_less @ real @ X @ ( one_one @ real ) )
=> ( ( sin @ real @ ( arccos @ X ) )
!= ( zero_zero @ real ) ) ) ) ).
% sin_arccos_nonzero
thf(fact_4618_arccos__cos2,axiom,
! [X: real] :
( ( ord_less_eq @ real @ X @ ( zero_zero @ real ) )
=> ( ( ord_less_eq @ real @ ( uminus_uminus @ real @ pi ) @ X )
=> ( ( arccos @ ( cos @ real @ X ) )
= ( uminus_uminus @ real @ X ) ) ) ) ).
% arccos_cos2
thf(fact_4619_arccos__minus,axiom,
! [X: real] :
( ( ord_less_eq @ real @ ( uminus_uminus @ real @ ( one_one @ real ) ) @ X )
=> ( ( ord_less_eq @ real @ X @ ( one_one @ real ) )
=> ( ( arccos @ ( uminus_uminus @ real @ X ) )
= ( minus_minus @ real @ pi @ ( arccos @ X ) ) ) ) ) ).
% arccos_minus
thf(fact_4620_exp__plus__inverse__exp,axiom,
! [X: real] : ( ord_less_eq @ real @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) @ ( plus_plus @ real @ ( exp @ real @ X ) @ ( inverse_inverse @ real @ ( exp @ real @ X ) ) ) ) ).
% exp_plus_inverse_exp
thf(fact_4621_cos__arcsin__nonzero,axiom,
! [X: real] :
( ( ord_less @ real @ ( uminus_uminus @ real @ ( one_one @ real ) ) @ X )
=> ( ( ord_less @ real @ X @ ( one_one @ real ) )
=> ( ( cos @ real @ ( arcsin @ X ) )
!= ( zero_zero @ real ) ) ) ) ).
% cos_arcsin_nonzero
thf(fact_4622_plus__inverse__ge__2,axiom,
! [X: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ X )
=> ( ord_less_eq @ real @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) @ ( plus_plus @ real @ X @ ( inverse_inverse @ real @ X ) ) ) ) ).
% plus_inverse_ge_2
thf(fact_4623_real__inv__sqrt__pow2,axiom,
! [X: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ X )
=> ( ( power_power @ real @ ( inverse_inverse @ real @ ( sqrt @ X ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) )
= ( inverse_inverse @ real @ X ) ) ) ).
% real_inv_sqrt_pow2
thf(fact_4624_arccos,axiom,
! [Y: real] :
( ( ord_less_eq @ real @ ( uminus_uminus @ real @ ( one_one @ real ) ) @ Y )
=> ( ( ord_less_eq @ real @ Y @ ( one_one @ real ) )
=> ( ( ord_less_eq @ real @ ( zero_zero @ real ) @ ( arccos @ Y ) )
& ( ord_less_eq @ real @ ( arccos @ Y ) @ pi )
& ( ( cos @ real @ ( arccos @ Y ) )
= Y ) ) ) ) ).
% arccos
thf(fact_4625_tan__cot,axiom,
! [X: real] :
( ( tan @ real @ ( minus_minus @ real @ ( divide_divide @ real @ pi @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) @ X ) )
= ( inverse_inverse @ real @ ( tan @ real @ X ) ) ) ).
% tan_cot
thf(fact_4626_arccos__minus__abs,axiom,
! [X: real] :
( ( ord_less_eq @ real @ ( abs_abs @ real @ X ) @ ( one_one @ real ) )
=> ( ( arccos @ ( uminus_uminus @ real @ X ) )
= ( minus_minus @ real @ pi @ ( arccos @ X ) ) ) ) ).
% arccos_minus_abs
thf(fact_4627_real__le__x__sinh,axiom,
! [X: real] :
( ( ord_less_eq @ real @ ( zero_zero @ real ) @ X )
=> ( ord_less_eq @ real @ X @ ( divide_divide @ real @ ( minus_minus @ real @ ( exp @ real @ X ) @ ( inverse_inverse @ real @ ( exp @ real @ X ) ) ) @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) ) ) ).
% real_le_x_sinh
thf(fact_4628_real__le__abs__sinh,axiom,
! [X: real] : ( ord_less_eq @ real @ ( abs_abs @ real @ X ) @ ( abs_abs @ real @ ( divide_divide @ real @ ( minus_minus @ real @ ( exp @ real @ X ) @ ( inverse_inverse @ real @ ( exp @ real @ X ) ) ) @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) ) ) ).
% real_le_abs_sinh
thf(fact_4629_exp__series__add__commuting,axiom,
! [A: $tType] :
( ( ( real_Vector_banach @ A )
& ( real_V2822296259951069270ebra_1 @ A ) )
=> ! [X: A,Y: A,N3: nat] :
( ( ( times_times @ A @ X @ Y )
= ( times_times @ A @ Y @ X ) )
=> ( ( real_V8093663219630862766scaleR @ A @ ( inverse_inverse @ real @ ( semiring_char_0_fact @ real @ N3 ) ) @ ( power_power @ A @ ( plus_plus @ A @ X @ Y ) @ N3 ) )
= ( groups7311177749621191930dd_sum @ nat @ A
@ ^ [I3: nat] : ( times_times @ A @ ( real_V8093663219630862766scaleR @ A @ ( inverse_inverse @ real @ ( semiring_char_0_fact @ real @ I3 ) ) @ ( power_power @ A @ X @ I3 ) ) @ ( real_V8093663219630862766scaleR @ A @ ( inverse_inverse @ real @ ( semiring_char_0_fact @ real @ ( minus_minus @ nat @ N3 @ I3 ) ) ) @ ( power_power @ A @ Y @ ( minus_minus @ nat @ N3 @ I3 ) ) ) )
@ ( set_ord_atMost @ nat @ N3 ) ) ) ) ) ).
% exp_series_add_commuting
thf(fact_4630_exp__first__term,axiom,
! [A: $tType] :
( ( ( real_Vector_banach @ A )
& ( real_V2822296259951069270ebra_1 @ A ) )
=> ( ( exp @ A )
= ( ^ [X2: A] :
( plus_plus @ A @ ( one_one @ A )
@ ( suminf @ A
@ ^ [N2: nat] : ( real_V8093663219630862766scaleR @ A @ ( inverse_inverse @ real @ ( semiring_char_0_fact @ real @ ( suc @ N2 ) ) ) @ ( power_power @ A @ X2 @ ( suc @ N2 ) ) ) ) ) ) ) ) ).
% exp_first_term
thf(fact_4631_tan__sec,axiom,
! [A: $tType] :
( ( ( real_Vector_banach @ A )
& ( real_V3459762299906320749_field @ A ) )
=> ! [X: A] :
( ( ( cos @ A @ X )
!= ( zero_zero @ A ) )
=> ( ( plus_plus @ A @ ( one_one @ A ) @ ( power_power @ A @ ( tan @ A @ X ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) )
= ( power_power @ A @ ( inverse_inverse @ A @ ( cos @ A @ X ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) ).
% tan_sec
thf(fact_4632_arccos__le__pi2,axiom,
! [Y: real] :
( ( ord_less_eq @ real @ ( zero_zero @ real ) @ Y )
=> ( ( ord_less_eq @ real @ Y @ ( one_one @ real ) )
=> ( ord_less_eq @ real @ ( arccos @ Y ) @ ( divide_divide @ real @ pi @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) ) ) ) ).
% arccos_le_pi2
thf(fact_4633_powr__real__of__int,axiom,
! [X: real,N3: int] :
( ( ord_less @ real @ ( zero_zero @ real ) @ X )
=> ( ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ N3 )
=> ( ( powr @ real @ X @ ( ring_1_of_int @ real @ N3 ) )
= ( power_power @ real @ X @ ( nat2 @ N3 ) ) ) )
& ( ~ ( ord_less_eq @ int @ ( zero_zero @ int ) @ N3 )
=> ( ( powr @ real @ X @ ( ring_1_of_int @ real @ N3 ) )
= ( inverse_inverse @ real @ ( power_power @ real @ X @ ( nat2 @ ( uminus_uminus @ int @ N3 ) ) ) ) ) ) ) ) ).
% powr_real_of_int
thf(fact_4634_arcsin__lt__bounded,axiom,
! [Y: real] :
( ( ord_less @ real @ ( uminus_uminus @ real @ ( one_one @ real ) ) @ Y )
=> ( ( ord_less @ real @ Y @ ( one_one @ real ) )
=> ( ( ord_less @ real @ ( uminus_uminus @ real @ ( divide_divide @ real @ pi @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) ) @ ( arcsin @ Y ) )
& ( ord_less @ real @ ( arcsin @ Y ) @ ( divide_divide @ real @ pi @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) ) ) ) ) ).
% arcsin_lt_bounded
thf(fact_4635_arcsin__lbound,axiom,
! [Y: real] :
( ( ord_less_eq @ real @ ( uminus_uminus @ real @ ( one_one @ real ) ) @ Y )
=> ( ( ord_less_eq @ real @ Y @ ( one_one @ real ) )
=> ( ord_less_eq @ real @ ( uminus_uminus @ real @ ( divide_divide @ real @ pi @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) ) @ ( arcsin @ Y ) ) ) ) ).
% arcsin_lbound
thf(fact_4636_arcsin__ubound,axiom,
! [Y: real] :
( ( ord_less_eq @ real @ ( uminus_uminus @ real @ ( one_one @ real ) ) @ Y )
=> ( ( ord_less_eq @ real @ Y @ ( one_one @ real ) )
=> ( ord_less_eq @ real @ ( arcsin @ Y ) @ ( divide_divide @ real @ pi @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) ) ) ) ).
% arcsin_ubound
thf(fact_4637_arcsin__bounded,axiom,
! [Y: real] :
( ( ord_less_eq @ real @ ( uminus_uminus @ real @ ( one_one @ real ) ) @ Y )
=> ( ( ord_less_eq @ real @ Y @ ( one_one @ real ) )
=> ( ( ord_less_eq @ real @ ( uminus_uminus @ real @ ( divide_divide @ real @ pi @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) ) @ ( arcsin @ Y ) )
& ( ord_less_eq @ real @ ( arcsin @ Y ) @ ( divide_divide @ real @ pi @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) ) ) ) ) ).
% arcsin_bounded
thf(fact_4638_arcsin__sin,axiom,
! [X: real] :
( ( ord_less_eq @ real @ ( uminus_uminus @ real @ ( divide_divide @ real @ pi @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) ) @ X )
=> ( ( ord_less_eq @ real @ X @ ( divide_divide @ real @ pi @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) )
=> ( ( arcsin @ ( sin @ real @ X ) )
= X ) ) ) ).
% arcsin_sin
thf(fact_4639_exp__first__terms,axiom,
! [A: $tType] :
( ( ( real_Vector_banach @ A )
& ( real_V2822296259951069270ebra_1 @ A ) )
=> ! [K: nat] :
( ( exp @ A )
= ( ^ [X2: A] :
( plus_plus @ A
@ ( groups7311177749621191930dd_sum @ nat @ A
@ ^ [N2: nat] : ( real_V8093663219630862766scaleR @ A @ ( inverse_inverse @ real @ ( semiring_char_0_fact @ real @ N2 ) ) @ ( power_power @ A @ X2 @ N2 ) )
@ ( set_ord_lessThan @ nat @ K ) )
@ ( suminf @ A
@ ^ [N2: nat] : ( real_V8093663219630862766scaleR @ A @ ( inverse_inverse @ real @ ( semiring_char_0_fact @ real @ ( plus_plus @ nat @ N2 @ K ) ) ) @ ( power_power @ A @ X2 @ ( plus_plus @ nat @ N2 @ K ) ) ) ) ) ) ) ) ).
% exp_first_terms
thf(fact_4640_arcsin,axiom,
! [Y: real] :
( ( ord_less_eq @ real @ ( uminus_uminus @ real @ ( one_one @ real ) ) @ Y )
=> ( ( ord_less_eq @ real @ Y @ ( one_one @ real ) )
=> ( ( ord_less_eq @ real @ ( uminus_uminus @ real @ ( divide_divide @ real @ pi @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) ) @ ( arcsin @ Y ) )
& ( ord_less_eq @ real @ ( arcsin @ Y ) @ ( divide_divide @ real @ pi @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) )
& ( ( sin @ real @ ( arcsin @ Y ) )
= Y ) ) ) ) ).
% arcsin
thf(fact_4641_arcsin__pi,axiom,
! [Y: real] :
( ( ord_less_eq @ real @ ( uminus_uminus @ real @ ( one_one @ real ) ) @ Y )
=> ( ( ord_less_eq @ real @ Y @ ( one_one @ real ) )
=> ( ( ord_less_eq @ real @ ( uminus_uminus @ real @ ( divide_divide @ real @ pi @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) ) @ ( arcsin @ Y ) )
& ( ord_less_eq @ real @ ( arcsin @ Y ) @ pi )
& ( ( sin @ real @ ( arcsin @ Y ) )
= Y ) ) ) ) ).
% arcsin_pi
thf(fact_4642_arcsin__le__iff,axiom,
! [X: real,Y: real] :
( ( ord_less_eq @ real @ ( uminus_uminus @ real @ ( one_one @ real ) ) @ X )
=> ( ( ord_less_eq @ real @ X @ ( one_one @ real ) )
=> ( ( ord_less_eq @ real @ ( divide_divide @ real @ ( uminus_uminus @ real @ pi ) @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) @ Y )
=> ( ( ord_less_eq @ real @ Y @ ( divide_divide @ real @ pi @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) )
=> ( ( ord_less_eq @ real @ ( arcsin @ X ) @ Y )
= ( ord_less_eq @ real @ X @ ( sin @ real @ Y ) ) ) ) ) ) ) ).
% arcsin_le_iff
thf(fact_4643_le__arcsin__iff,axiom,
! [X: real,Y: real] :
( ( ord_less_eq @ real @ ( uminus_uminus @ real @ ( one_one @ real ) ) @ X )
=> ( ( ord_less_eq @ real @ X @ ( one_one @ real ) )
=> ( ( ord_less_eq @ real @ ( divide_divide @ real @ ( uminus_uminus @ real @ pi ) @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) @ Y )
=> ( ( ord_less_eq @ real @ Y @ ( divide_divide @ real @ pi @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) )
=> ( ( ord_less_eq @ real @ Y @ ( arcsin @ X ) )
= ( ord_less_eq @ real @ ( sin @ real @ Y ) @ X ) ) ) ) ) ) ).
% le_arcsin_iff
thf(fact_4644_arccos__cos__eq__abs__2pi,axiom,
! [Theta: real] :
~ ! [K3: int] :
( ( arccos @ ( cos @ real @ Theta ) )
!= ( abs_abs @ real @ ( minus_minus @ real @ Theta @ ( times_times @ real @ ( ring_1_of_int @ real @ K3 ) @ ( times_times @ real @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) @ pi ) ) ) ) ) ).
% arccos_cos_eq_abs_2pi
thf(fact_4645_Maclaurin__sin__bound,axiom,
! [X: real,N3: nat] :
( ord_less_eq @ real
@ ( abs_abs @ real
@ ( minus_minus @ real @ ( sin @ real @ X )
@ ( groups7311177749621191930dd_sum @ nat @ real
@ ^ [M5: nat] : ( times_times @ real @ ( sin_coeff @ M5 ) @ ( power_power @ real @ X @ M5 ) )
@ ( set_ord_lessThan @ nat @ N3 ) ) ) )
@ ( times_times @ real @ ( inverse_inverse @ real @ ( semiring_char_0_fact @ real @ N3 ) ) @ ( power_power @ real @ ( abs_abs @ real @ X ) @ N3 ) ) ) ).
% Maclaurin_sin_bound
thf(fact_4646_sinh__converges,axiom,
! [A: $tType] :
( ( ( real_Vector_banach @ A )
& ( real_V2822296259951069270ebra_1 @ A ) )
=> ! [X: A] :
( sums @ A
@ ^ [N2: nat] : ( if @ A @ ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N2 ) @ ( zero_zero @ A ) @ ( real_V8093663219630862766scaleR @ A @ ( inverse_inverse @ real @ ( semiring_char_0_fact @ real @ N2 ) ) @ ( power_power @ A @ X @ N2 ) ) )
@ ( sinh @ A @ X ) ) ) ).
% sinh_converges
thf(fact_4647_cosh__converges,axiom,
! [A: $tType] :
( ( ( real_Vector_banach @ A )
& ( real_V2822296259951069270ebra_1 @ A ) )
=> ! [X: A] :
( sums @ A
@ ^ [N2: nat] : ( if @ A @ ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N2 ) @ ( real_V8093663219630862766scaleR @ A @ ( inverse_inverse @ real @ ( semiring_char_0_fact @ real @ N2 ) ) @ ( power_power @ A @ X @ N2 ) ) @ ( zero_zero @ A ) )
@ ( cosh @ A @ X ) ) ) ).
% cosh_converges
thf(fact_4648_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_4649_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_4650_sinh__real__le__iff,axiom,
! [X: real,Y: real] :
( ( ord_less_eq @ real @ ( sinh @ real @ X ) @ ( sinh @ real @ Y ) )
= ( ord_less_eq @ real @ X @ Y ) ) ).
% sinh_real_le_iff
thf(fact_4651_sinh__real__nonpos__iff,axiom,
! [X: real] :
( ( ord_less_eq @ real @ ( sinh @ real @ X ) @ ( zero_zero @ real ) )
= ( ord_less_eq @ real @ X @ ( zero_zero @ real ) ) ) ).
% sinh_real_nonpos_iff
thf(fact_4652_sinh__real__nonneg__iff,axiom,
! [X: real] :
( ( ord_less_eq @ real @ ( zero_zero @ real ) @ ( sinh @ real @ X ) )
= ( ord_less_eq @ real @ ( zero_zero @ real ) @ X ) ) ).
% sinh_real_nonneg_iff
thf(fact_4653_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_4654_norm__ii,axiom,
( ( real_V7770717601297561774m_norm @ complex @ imaginary_unit )
= ( one_one @ real ) ) ).
% norm_ii
thf(fact_4655_i__squared,axiom,
( ( times_times @ complex @ imaginary_unit @ imaginary_unit )
= ( uminus_uminus @ complex @ ( one_one @ complex ) ) ) ).
% i_squared
thf(fact_4656_power2__i,axiom,
( ( power_power @ complex @ imaginary_unit @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) )
= ( uminus_uminus @ complex @ ( one_one @ complex ) ) ) ).
% power2_i
thf(fact_4657_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_4658_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_4659_i__even__power,axiom,
! [N3: nat] :
( ( power_power @ complex @ imaginary_unit @ ( times_times @ nat @ N3 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) )
= ( power_power @ complex @ ( uminus_uminus @ complex @ ( one_one @ complex ) ) @ N3 ) ) ).
% i_even_power
thf(fact_4660_sinh__le__cosh__real,axiom,
! [X: real] : ( ord_less_eq @ real @ ( sinh @ real @ X ) @ ( cosh @ real @ X ) ) ).
% sinh_le_cosh_real
thf(fact_4661_complex__i__not__one,axiom,
( imaginary_unit
!= ( one_one @ complex ) ) ).
% complex_i_not_one
thf(fact_4662_tanh__def,axiom,
! [A: $tType] :
( ( ( real_Vector_banach @ A )
& ( real_V3459762299906320749_field @ A ) )
=> ( ( tanh @ A )
= ( ^ [X2: A] : ( divide_divide @ A @ ( sinh @ A @ X2 ) @ ( cosh @ A @ X2 ) ) ) ) ) ).
% tanh_def
thf(fact_4663_sinh__diff,axiom,
! [A: $tType] :
( ( ( real_Vector_banach @ A )
& ( real_V3459762299906320749_field @ A ) )
=> ! [X: A,Y: A] :
( ( sinh @ A @ ( minus_minus @ A @ X @ Y ) )
= ( minus_minus @ A @ ( times_times @ A @ ( sinh @ A @ X ) @ ( cosh @ A @ Y ) ) @ ( times_times @ A @ ( cosh @ A @ X ) @ ( sinh @ A @ Y ) ) ) ) ) ).
% sinh_diff
thf(fact_4664_cosh__diff,axiom,
! [A: $tType] :
( ( ( real_Vector_banach @ A )
& ( real_V3459762299906320749_field @ A ) )
=> ! [X: A,Y: A] :
( ( cosh @ A @ ( minus_minus @ A @ X @ Y ) )
= ( minus_minus @ A @ ( times_times @ A @ ( cosh @ A @ X ) @ ( cosh @ A @ Y ) ) @ ( times_times @ A @ ( sinh @ A @ X ) @ ( sinh @ A @ Y ) ) ) ) ) ).
% cosh_diff
thf(fact_4665_sinh__add,axiom,
! [A: $tType] :
( ( ( real_Vector_banach @ A )
& ( real_V3459762299906320749_field @ A ) )
=> ! [X: A,Y: A] :
( ( sinh @ A @ ( plus_plus @ A @ X @ Y ) )
= ( plus_plus @ A @ ( times_times @ A @ ( sinh @ A @ X ) @ ( cosh @ A @ Y ) ) @ ( times_times @ A @ ( cosh @ A @ X ) @ ( sinh @ A @ Y ) ) ) ) ) ).
% sinh_add
thf(fact_4666_cosh__add,axiom,
! [A: $tType] :
( ( ( real_Vector_banach @ A )
& ( real_V3459762299906320749_field @ A ) )
=> ! [X: A,Y: A] :
( ( cosh @ A @ ( plus_plus @ A @ X @ Y ) )
= ( plus_plus @ A @ ( times_times @ A @ ( cosh @ A @ X ) @ ( cosh @ A @ Y ) ) @ ( times_times @ A @ ( sinh @ A @ X ) @ ( sinh @ A @ Y ) ) ) ) ) ).
% cosh_add
thf(fact_4667_sinh__plus__cosh,axiom,
! [A: $tType] :
( ( ( real_Vector_banach @ A )
& ( real_V2822296259951069270ebra_1 @ A ) )
=> ! [X: A] :
( ( plus_plus @ A @ ( sinh @ A @ X ) @ ( cosh @ A @ X ) )
= ( exp @ A @ X ) ) ) ).
% sinh_plus_cosh
thf(fact_4668_cosh__plus__sinh,axiom,
! [A: $tType] :
( ( ( real_Vector_banach @ A )
& ( real_V2822296259951069270ebra_1 @ A ) )
=> ! [X: A] :
( ( plus_plus @ A @ ( cosh @ A @ X ) @ ( sinh @ A @ X ) )
= ( exp @ A @ X ) ) ) ).
% cosh_plus_sinh
thf(fact_4669_sinh__minus__cosh,axiom,
! [A: $tType] :
( ( ( real_Vector_banach @ A )
& ( real_V2822296259951069270ebra_1 @ A ) )
=> ! [X: A] :
( ( minus_minus @ A @ ( sinh @ A @ X ) @ ( cosh @ A @ X ) )
= ( uminus_uminus @ A @ ( exp @ A @ ( uminus_uminus @ A @ X ) ) ) ) ) ).
% sinh_minus_cosh
thf(fact_4670_cosh__minus__sinh,axiom,
! [A: $tType] :
( ( ( real_Vector_banach @ A )
& ( real_V2822296259951069270ebra_1 @ A ) )
=> ! [X: A] :
( ( minus_minus @ A @ ( cosh @ A @ X ) @ ( sinh @ A @ X ) )
= ( exp @ A @ ( uminus_uminus @ A @ X ) ) ) ) ).
% cosh_minus_sinh
thf(fact_4671_cosh__real__nonneg,axiom,
! [X: real] : ( ord_less_eq @ real @ ( zero_zero @ real ) @ ( cosh @ real @ X ) ) ).
% cosh_real_nonneg
thf(fact_4672_cosh__real__nonneg__le__iff,axiom,
! [X: real,Y: real] :
( ( ord_less_eq @ real @ ( zero_zero @ real ) @ X )
=> ( ( ord_less_eq @ real @ ( zero_zero @ real ) @ Y )
=> ( ( ord_less_eq @ real @ ( cosh @ real @ X ) @ ( cosh @ real @ Y ) )
= ( ord_less_eq @ real @ X @ Y ) ) ) ) ).
% cosh_real_nonneg_le_iff
thf(fact_4673_cosh__real__nonpos__le__iff,axiom,
! [X: real,Y: real] :
( ( ord_less_eq @ real @ X @ ( zero_zero @ real ) )
=> ( ( ord_less_eq @ real @ Y @ ( zero_zero @ real ) )
=> ( ( ord_less_eq @ real @ ( cosh @ real @ X ) @ ( cosh @ real @ Y ) )
= ( ord_less_eq @ real @ Y @ X ) ) ) ) ).
% cosh_real_nonpos_le_iff
thf(fact_4674_sinh__double,axiom,
! [A: $tType] :
( ( ( real_Vector_banach @ A )
& ( real_V3459762299906320749_field @ A ) )
=> ! [X: A] :
( ( sinh @ A @ ( times_times @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ X ) )
= ( times_times @ A @ ( times_times @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ ( sinh @ A @ X ) ) @ ( cosh @ A @ X ) ) ) ) ).
% sinh_double
thf(fact_4675_cosh__real__ge__1,axiom,
! [X: real] : ( ord_less_eq @ real @ ( one_one @ real ) @ ( cosh @ real @ X ) ) ).
% cosh_real_ge_1
thf(fact_4676_cosh__real__nonpos__less__iff,axiom,
! [X: real,Y: real] :
( ( ord_less_eq @ real @ X @ ( zero_zero @ real ) )
=> ( ( ord_less_eq @ real @ Y @ ( zero_zero @ real ) )
=> ( ( ord_less @ real @ ( cosh @ real @ X ) @ ( cosh @ real @ Y ) )
= ( ord_less @ real @ Y @ X ) ) ) ) ).
% cosh_real_nonpos_less_iff
thf(fact_4677_cosh__real__nonneg__less__iff,axiom,
! [X: real,Y: real] :
( ( ord_less_eq @ real @ ( zero_zero @ real ) @ X )
=> ( ( ord_less_eq @ real @ ( zero_zero @ real ) @ Y )
=> ( ( ord_less @ real @ ( cosh @ real @ X ) @ ( cosh @ real @ Y ) )
= ( ord_less @ real @ X @ Y ) ) ) ) ).
% cosh_real_nonneg_less_iff
thf(fact_4678_cosh__real__strict__mono,axiom,
! [X: real,Y: real] :
( ( ord_less_eq @ real @ ( zero_zero @ real ) @ X )
=> ( ( ord_less @ real @ X @ Y )
=> ( ord_less @ real @ ( cosh @ real @ X ) @ ( cosh @ real @ Y ) ) ) ) ).
% cosh_real_strict_mono
thf(fact_4679_cosh__square__eq,axiom,
! [A: $tType] :
( ( ( real_Vector_banach @ A )
& ( real_V3459762299906320749_field @ A ) )
=> ! [X: A] :
( ( power_power @ A @ ( cosh @ A @ X ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) )
= ( plus_plus @ A @ ( power_power @ A @ ( sinh @ A @ X ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ ( one_one @ A ) ) ) ) ).
% cosh_square_eq
thf(fact_4680_sinh__square__eq,axiom,
! [A: $tType] :
( ( ( real_Vector_banach @ A )
& ( real_V3459762299906320749_field @ A ) )
=> ! [X: A] :
( ( power_power @ A @ ( sinh @ A @ X ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) )
= ( minus_minus @ A @ ( power_power @ A @ ( cosh @ A @ X ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ ( one_one @ A ) ) ) ) ).
% sinh_square_eq
thf(fact_4681_hyperbolic__pythagoras,axiom,
! [A: $tType] :
( ( ( real_Vector_banach @ A )
& ( real_V3459762299906320749_field @ A ) )
=> ! [X: A] :
( ( minus_minus @ A @ ( power_power @ A @ ( cosh @ A @ X ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ ( power_power @ A @ ( sinh @ A @ X ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) )
= ( one_one @ A ) ) ) ).
% hyperbolic_pythagoras
thf(fact_4682_arcosh__cosh__real,axiom,
! [X: real] :
( ( ord_less_eq @ real @ ( zero_zero @ real ) @ X )
=> ( ( arcosh @ real @ ( cosh @ real @ X ) )
= X ) ) ).
% arcosh_cosh_real
thf(fact_4683_Complex__eq__i,axiom,
! [X: real,Y: real] :
( ( ( complex2 @ X @ Y )
= imaginary_unit )
= ( ( X
= ( zero_zero @ real ) )
& ( Y
= ( one_one @ real ) ) ) ) ).
% Complex_eq_i
thf(fact_4684_imaginary__unit_Ocode,axiom,
( imaginary_unit
= ( complex2 @ ( zero_zero @ real ) @ ( one_one @ real ) ) ) ).
% imaginary_unit.code
thf(fact_4685_cosh__double,axiom,
! [A: $tType] :
( ( ( real_Vector_banach @ A )
& ( real_V3459762299906320749_field @ A ) )
=> ! [X: A] :
( ( cosh @ A @ ( times_times @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ X ) )
= ( plus_plus @ A @ ( power_power @ A @ ( cosh @ A @ X ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ ( power_power @ A @ ( sinh @ A @ X ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) ).
% cosh_double
thf(fact_4686_tanh__add,axiom,
! [A: $tType] :
( ( ( real_Vector_banach @ A )
& ( real_V3459762299906320749_field @ A ) )
=> ! [X: A,Y: A] :
( ( ( cosh @ A @ X )
!= ( zero_zero @ A ) )
=> ( ( ( cosh @ A @ Y )
!= ( zero_zero @ A ) )
=> ( ( tanh @ A @ ( plus_plus @ A @ X @ Y ) )
= ( divide_divide @ A @ ( plus_plus @ A @ ( tanh @ A @ X ) @ ( tanh @ A @ Y ) ) @ ( plus_plus @ A @ ( one_one @ A ) @ ( times_times @ A @ ( tanh @ A @ X ) @ ( tanh @ A @ Y ) ) ) ) ) ) ) ) ).
% tanh_add
thf(fact_4687_sinh__zero__iff,axiom,
! [A: $tType] :
( ( ( real_Vector_banach @ A )
& ( real_V3459762299906320749_field @ A ) )
=> ! [X: A] :
( ( ( sinh @ A @ X )
= ( zero_zero @ A ) )
= ( member @ A @ ( exp @ A @ X ) @ ( insert @ A @ ( one_one @ A ) @ ( insert @ A @ ( uminus_uminus @ A @ ( one_one @ A ) ) @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ) ).
% sinh_zero_iff
thf(fact_4688_cosh__field__def,axiom,
! [A: $tType] :
( ( ( real_Vector_banach @ A )
& ( real_V3459762299906320749_field @ A ) )
=> ( ( cosh @ A )
= ( ^ [Z3: A] : ( divide_divide @ A @ ( plus_plus @ A @ ( exp @ A @ Z3 ) @ ( exp @ A @ ( uminus_uminus @ A @ Z3 ) ) ) @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) ) ) ) ).
% cosh_field_def
thf(fact_4689_complex__inverse,axiom,
! [A2: real,B2: real] :
( ( inverse_inverse @ complex @ ( complex2 @ A2 @ B2 ) )
= ( complex2 @ ( divide_divide @ real @ A2 @ ( plus_plus @ real @ ( power_power @ real @ A2 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ ( power_power @ real @ B2 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( divide_divide @ real @ ( uminus_uminus @ real @ B2 ) @ ( plus_plus @ real @ ( power_power @ real @ A2 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ ( power_power @ real @ B2 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) ) ).
% complex_inverse
thf(fact_4690_sinh__field__def,axiom,
! [A: $tType] :
( ( ( real_Vector_banach @ A )
& ( real_V3459762299906320749_field @ A ) )
=> ( ( sinh @ A )
= ( ^ [Z3: A] : ( divide_divide @ A @ ( minus_minus @ A @ ( exp @ A @ Z3 ) @ ( exp @ A @ ( uminus_uminus @ A @ Z3 ) ) ) @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) ) ) ) ).
% sinh_field_def
thf(fact_4691_cosh__zero__iff,axiom,
! [A: $tType] :
( ( ( real_Vector_banach @ A )
& ( real_V3459762299906320749_field @ A ) )
=> ! [X: A] :
( ( ( cosh @ A @ X )
= ( zero_zero @ A ) )
= ( ( power_power @ A @ ( exp @ A @ X ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) )
= ( uminus_uminus @ A @ ( one_one @ A ) ) ) ) ) ).
% cosh_zero_iff
thf(fact_4692_cmod__unit__one,axiom,
! [A2: real] :
( ( real_V7770717601297561774m_norm @ complex @ ( plus_plus @ complex @ ( real_Vector_of_real @ complex @ ( cos @ real @ A2 ) ) @ ( times_times @ complex @ imaginary_unit @ ( real_Vector_of_real @ complex @ ( sin @ real @ A2 ) ) ) ) )
= ( one_one @ real ) ) ).
% cmod_unit_one
thf(fact_4693_cosh__def,axiom,
! [A: $tType] :
( ( ( real_Vector_banach @ A )
& ( real_V2822296259951069270ebra_1 @ A ) )
=> ( ( cosh @ A )
= ( ^ [X2: A] : ( real_V8093663219630862766scaleR @ A @ ( inverse_inverse @ real @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) @ ( plus_plus @ A @ ( exp @ A @ X2 ) @ ( exp @ A @ ( uminus_uminus @ A @ X2 ) ) ) ) ) ) ) ).
% cosh_def
thf(fact_4694_cosh__ln__real,axiom,
! [X: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ X )
=> ( ( cosh @ real @ ( ln_ln @ real @ X ) )
= ( divide_divide @ real @ ( plus_plus @ real @ X @ ( inverse_inverse @ real @ X ) ) @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) ) ) ).
% cosh_ln_real
thf(fact_4695_sinh__def,axiom,
! [A: $tType] :
( ( ( real_Vector_banach @ A )
& ( real_V2822296259951069270ebra_1 @ A ) )
=> ( ( sinh @ A )
= ( ^ [X2: A] : ( real_V8093663219630862766scaleR @ A @ ( inverse_inverse @ real @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) @ ( minus_minus @ A @ ( exp @ A @ X2 ) @ ( exp @ A @ ( uminus_uminus @ A @ X2 ) ) ) ) ) ) ) ).
% sinh_def
thf(fact_4696_sinh__ln__real,axiom,
! [X: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ X )
=> ( ( sinh @ real @ ( ln_ln @ real @ X ) )
= ( divide_divide @ real @ ( minus_minus @ real @ X @ ( inverse_inverse @ real @ X ) ) @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) ) ) ).
% sinh_ln_real
thf(fact_4697_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_4698_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_4699_Arg__ii,axiom,
( ( arg @ imaginary_unit )
= ( divide_divide @ real @ pi @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) ) ).
% Arg_ii
thf(fact_4700_csqrt__1,axiom,
( ( csqrt @ ( one_one @ complex ) )
= ( one_one @ complex ) ) ).
% csqrt_1
thf(fact_4701_csqrt__eq__1,axiom,
! [Z: complex] :
( ( ( csqrt @ Z )
= ( one_one @ complex ) )
= ( Z
= ( one_one @ complex ) ) ) ).
% csqrt_eq_1
thf(fact_4702_power2__csqrt,axiom,
! [Z: complex] :
( ( power_power @ complex @ ( csqrt @ Z ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) )
= Z ) ).
% power2_csqrt
thf(fact_4703_of__real__sqrt,axiom,
! [X: real] :
( ( ord_less_eq @ real @ ( zero_zero @ real ) @ X )
=> ( ( real_Vector_of_real @ complex @ ( sqrt @ X ) )
= ( csqrt @ ( real_Vector_of_real @ complex @ X ) ) ) ) ).
% of_real_sqrt
thf(fact_4704_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_4705_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_4706_cot__less__zero,axiom,
! [X: real] :
( ( ord_less @ real @ ( divide_divide @ real @ ( uminus_uminus @ real @ pi ) @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) @ X )
=> ( ( ord_less @ real @ X @ ( zero_zero @ real ) )
=> ( ord_less @ real @ ( cot @ real @ X ) @ ( zero_zero @ real ) ) ) ) ).
% cot_less_zero
thf(fact_4707_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_4708_norm__cis,axiom,
! [A2: real] :
( ( real_V7770717601297561774m_norm @ complex @ ( cis @ A2 ) )
= ( one_one @ real ) ) ).
% norm_cis
thf(fact_4709_cis__zero,axiom,
( ( cis @ ( zero_zero @ real ) )
= ( one_one @ complex ) ) ).
% cis_zero
thf(fact_4710_cis__pi,axiom,
( ( cis @ pi )
= ( uminus_uminus @ complex @ ( one_one @ complex ) ) ) ).
% cis_pi
thf(fact_4711_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_4712_sint__minus1,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [X: word @ A] :
( ( ( ring_1_signed @ A @ int @ X )
= ( uminus_uminus @ int @ ( one_one @ int ) ) )
= ( X
= ( uminus_uminus @ ( word @ A ) @ ( one_one @ ( word @ A ) ) ) ) ) ) ).
% sint_minus1
thf(fact_4713_cot__npi,axiom,
! [N3: nat] :
( ( cot @ real @ ( times_times @ real @ ( semiring_1_of_nat @ real @ N3 ) @ pi ) )
= ( zero_zero @ real ) ) ).
% cot_npi
thf(fact_4714_cis__pi__half,axiom,
( ( cis @ ( divide_divide @ real @ pi @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) )
= imaginary_unit ) ).
% cis_pi_half
thf(fact_4715_cis__2pi,axiom,
( ( cis @ ( times_times @ real @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) @ pi ) )
= ( one_one @ complex ) ) ).
% cis_2pi
thf(fact_4716_cot__periodic,axiom,
! [X: real] :
( ( cot @ real @ ( plus_plus @ real @ X @ ( times_times @ real @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) @ pi ) ) )
= ( cot @ real @ X ) ) ).
% cot_periodic
thf(fact_4717_cis__mult,axiom,
! [A2: real,B2: real] :
( ( times_times @ complex @ ( cis @ A2 ) @ ( cis @ B2 ) )
= ( cis @ ( plus_plus @ real @ A2 @ B2 ) ) ) ).
% cis_mult
thf(fact_4718_cis__divide,axiom,
! [A2: real,B2: real] :
( ( divide_divide @ complex @ ( cis @ A2 ) @ ( cis @ B2 ) )
= ( cis @ ( minus_minus @ real @ A2 @ B2 ) ) ) ).
% cis_divide
thf(fact_4719_DeMoivre,axiom,
! [A2: real,N3: nat] :
( ( power_power @ complex @ ( cis @ A2 ) @ N3 )
= ( cis @ ( times_times @ real @ ( semiring_1_of_nat @ real @ N3 ) @ A2 ) ) ) ).
% DeMoivre
thf(fact_4720_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_4721_cot__def,axiom,
! [A: $tType] :
( ( ( real_Vector_banach @ A )
& ( real_V3459762299906320749_field @ A ) )
=> ( ( cot @ A )
= ( ^ [X2: A] : ( divide_divide @ A @ ( cos @ A @ X2 ) @ ( sin @ A @ X2 ) ) ) ) ) ).
% cot_def
thf(fact_4722_sint__above__size,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [W: word @ A,X: int] :
( ( ord_less_eq @ int @ ( power_power @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ ( minus_minus @ nat @ ( size_size @ ( word @ A ) @ W ) @ ( one_one @ nat ) ) ) @ X )
=> ( ord_less @ int @ ( ring_1_signed @ A @ int @ W ) @ X ) ) ) ).
% sint_above_size
thf(fact_4723_sint__below__size,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [X: int,W: word @ A] :
( ( ord_less_eq @ int @ X @ ( uminus_uminus @ int @ ( power_power @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ ( minus_minus @ nat @ ( size_size @ ( word @ A ) @ W ) @ ( one_one @ nat ) ) ) ) )
=> ( ord_less_eq @ int @ X @ ( ring_1_signed @ A @ int @ W ) ) ) ) ).
% sint_below_size
thf(fact_4724_cot__gt__zero,axiom,
! [X: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ X )
=> ( ( ord_less @ real @ X @ ( divide_divide @ real @ pi @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) )
=> ( ord_less @ real @ ( zero_zero @ real ) @ ( cot @ real @ X ) ) ) ) ).
% cot_gt_zero
thf(fact_4725_tan__cot_H,axiom,
! [X: real] :
( ( tan @ real @ ( minus_minus @ real @ ( divide_divide @ real @ pi @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) @ X ) )
= ( cot @ real @ X ) ) ).
% tan_cot'
thf(fact_4726_bij__betw__roots__unity,axiom,
! [N3: nat] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ N3 )
=> ( bij_betw @ nat @ complex
@ ^ [K2: nat] : ( cis @ ( divide_divide @ real @ ( times_times @ real @ ( times_times @ real @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) @ pi ) @ ( semiring_1_of_nat @ real @ K2 ) ) @ ( semiring_1_of_nat @ real @ N3 ) ) )
@ ( set_ord_lessThan @ nat @ N3 )
@ ( collect @ complex
@ ^ [Z3: complex] :
( ( power_power @ complex @ Z3 @ N3 )
= ( one_one @ complex ) ) ) ) ) ).
% bij_betw_roots_unity
thf(fact_4727_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_4728_option_Osize__gen_I2_J,axiom,
! [A: $tType,X: A > nat,X22: A] :
( ( size_option @ A @ X @ ( some @ A @ X22 ) )
= ( plus_plus @ nat @ ( X @ X22 ) @ ( suc @ ( zero_zero @ nat ) ) ) ) ).
% option.size_gen(2)
thf(fact_4729_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_4730_pred__numeral__simps_I2_J,axiom,
! [K: num] :
( ( pred_numeral @ ( bit0 @ K ) )
= ( numeral_numeral @ nat @ ( bitM @ K ) ) ) ).
% pred_numeral_simps(2)
thf(fact_4731_semiring__norm_I26_J,axiom,
( ( bitM @ one2 )
= one2 ) ).
% semiring_norm(26)
thf(fact_4732_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_4733_semiring__norm_I28_J,axiom,
! [N3: num] :
( ( bitM @ ( bit1 @ N3 ) )
= ( bit1 @ ( bit0 @ N3 ) ) ) ).
% semiring_norm(28)
thf(fact_4734_semiring__norm_I27_J,axiom,
! [N3: num] :
( ( bitM @ ( bit0 @ N3 ) )
= ( bit1 @ ( bitM @ N3 ) ) ) ).
% semiring_norm(27)
thf(fact_4735_eval__nat__numeral_I2_J,axiom,
! [N3: num] :
( ( numeral_numeral @ nat @ ( bit0 @ N3 ) )
= ( suc @ ( numeral_numeral @ nat @ ( bitM @ N3 ) ) ) ) ).
% eval_nat_numeral(2)
thf(fact_4736_one__plus__BitM,axiom,
! [N3: num] :
( ( plus_plus @ num @ one2 @ ( bitM @ N3 ) )
= ( bit0 @ N3 ) ) ).
% one_plus_BitM
thf(fact_4737_BitM__plus__one,axiom,
! [N3: num] :
( ( plus_plus @ num @ ( bitM @ N3 ) @ one2 )
= ( bit0 @ N3 ) ) ).
% BitM_plus_one
thf(fact_4738_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,S3: set @ B,T5: set @ C,G: C > A] :
( ( finite_finite2 @ B @ S7 )
=> ( ( finite_finite2 @ C @ T7 )
=> ( ( bij_betw @ B @ C @ H2 @ ( minus_minus @ ( set @ B ) @ S3 @ S7 ) @ ( minus_minus @ ( set @ C ) @ T5 @ T7 ) )
=> ( ! [A4: B] :
( ( member @ B @ A4 @ S7 )
=> ( ( G @ ( H2 @ A4 ) )
= ( zero_zero @ A ) ) )
=> ( ! [B3: C] :
( ( member @ C @ B3 @ T7 )
=> ( ( G @ B3 )
= ( zero_zero @ A ) ) )
=> ( ( groups7311177749621191930dd_sum @ B @ A
@ ^ [X2: B] : ( G @ ( H2 @ X2 ) )
@ S3 )
= ( groups7311177749621191930dd_sum @ C @ A @ G @ T5 ) ) ) ) ) ) ) ) ).
% sum.reindex_bij_betw_not_neutral
thf(fact_4739_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,S3: set @ B,T5: set @ C,G: C > A] :
( ( finite_finite2 @ B @ S7 )
=> ( ( finite_finite2 @ C @ T7 )
=> ( ( bij_betw @ B @ C @ H2 @ ( minus_minus @ ( set @ B ) @ S3 @ S7 ) @ ( minus_minus @ ( set @ C ) @ T5 @ T7 ) )
=> ( ! [A4: B] :
( ( member @ B @ A4 @ S7 )
=> ( ( G @ ( H2 @ A4 ) )
= ( one_one @ A ) ) )
=> ( ! [B3: C] :
( ( member @ C @ B3 @ T7 )
=> ( ( G @ B3 )
= ( one_one @ A ) ) )
=> ( ( groups7121269368397514597t_prod @ B @ A
@ ^ [X2: B] : ( G @ ( H2 @ X2 ) )
@ S3 )
= ( groups7121269368397514597t_prod @ C @ A @ G @ T5 ) ) ) ) ) ) ) ) ).
% prod.reindex_bij_betw_not_neutral
thf(fact_4740_numeral__BitM,axiom,
! [A: $tType] :
( ( neg_numeral @ A )
=> ! [N3: num] :
( ( numeral_numeral @ A @ ( bitM @ N3 ) )
= ( minus_minus @ A @ ( numeral_numeral @ A @ ( bit0 @ N3 ) ) @ ( one_one @ A ) ) ) ) ).
% numeral_BitM
thf(fact_4741_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_4742_option_Osize__gen_I1_J,axiom,
! [A: $tType,X: A > nat] :
( ( size_option @ A @ X @ ( none @ A ) )
= ( suc @ ( zero_zero @ nat ) ) ) ).
% option.size_gen(1)
thf(fact_4743_infinite__imp__bij__betw,axiom,
! [A: $tType,A3: set @ A,A2: A] :
( ~ ( finite_finite2 @ A @ A3 )
=> ? [H4: A > A] : ( bij_betw @ A @ A @ H4 @ A3 @ ( minus_minus @ ( set @ A ) @ A3 @ ( insert @ A @ A2 @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ).
% infinite_imp_bij_betw
thf(fact_4744_set__decode__0,axiom,
! [X: nat] :
( ( member @ nat @ ( zero_zero @ nat ) @ ( nat_set_decode @ X ) )
= ( ~ ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ X ) ) ) ).
% set_decode_0
thf(fact_4745_flip__bit__0,axiom,
! [A: $tType] :
( ( bit_se359711467146920520ations @ A )
=> ! [A2: A] :
( ( bit_se8732182000553998342ip_bit @ A @ ( zero_zero @ nat ) @ A2 )
= ( plus_plus @ A @ ( zero_neq_one_of_bool @ A @ ( dvd_dvd @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ A2 ) ) @ ( times_times @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ ( divide_divide @ A @ A2 @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) ) ) ) ) ).
% flip_bit_0
thf(fact_4746_of__bool__eq__0__iff,axiom,
! [A: $tType] :
( ( zero_neq_one @ A )
=> ! [P: $o] :
( ( ( zero_neq_one_of_bool @ A @ P )
= ( zero_zero @ A ) )
= ~ P ) ) ).
% of_bool_eq_0_iff
thf(fact_4747_of__bool__eq_I1_J,axiom,
! [A: $tType] :
( ( zero_neq_one @ A )
=> ( ( zero_neq_one_of_bool @ A @ $false )
= ( zero_zero @ A ) ) ) ).
% of_bool_eq(1)
thf(fact_4748_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_4749_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_4750_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_4751_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_4752_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_4753_abs__bool__eq,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [P: $o] :
( ( abs_abs @ A @ ( zero_neq_one_of_bool @ A @ P ) )
= ( zero_neq_one_of_bool @ A @ P ) ) ) ).
% abs_bool_eq
thf(fact_4754_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_4755_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_4756_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_4757_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_4758_Suc__0__mod__eq,axiom,
! [N3: nat] :
( ( modulo_modulo @ nat @ ( suc @ ( zero_zero @ nat ) ) @ N3 )
= ( zero_neq_one_of_bool @ nat
@ ( N3
!= ( suc @ ( zero_zero @ nat ) ) ) ) ) ).
% Suc_0_mod_eq
thf(fact_4759_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_4760_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_4761_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_4762_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_4763_Divides_Oadjust__div__eq,axiom,
! [Q2: int,R2: int] :
( ( adjust_div @ ( product_Pair @ int @ int @ Q2 @ R2 ) )
= ( plus_plus @ int @ Q2
@ ( zero_neq_one_of_bool @ int
@ ( R2
!= ( zero_zero @ int ) ) ) ) ) ).
% Divides.adjust_div_eq
thf(fact_4764_odd__of__bool__self,axiom,
! [A: $tType] :
( ( semiring_parity @ A )
=> ! [P4: $o] :
( ( ~ ( dvd_dvd @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ ( zero_neq_one_of_bool @ A @ P4 ) ) )
= P4 ) ) ).
% odd_of_bool_self
thf(fact_4765_of__bool__half__eq__0,axiom,
! [A: $tType] :
( ( euclid5411537665997757685th_nat @ A )
=> ! [B2: $o] :
( ( divide_divide @ A @ ( zero_neq_one_of_bool @ A @ B2 ) @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) )
= ( zero_zero @ A ) ) ) ).
% of_bool_half_eq_0
thf(fact_4766_set__decode__Suc,axiom,
! [N3: nat,X: nat] :
( ( member @ nat @ ( suc @ N3 ) @ ( nat_set_decode @ X ) )
= ( member @ nat @ N3 @ ( nat_set_decode @ ( divide_divide @ nat @ X @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) ).
% set_decode_Suc
thf(fact_4767_bits__1__div__exp,axiom,
! [A: $tType] :
( ( bit_semiring_bits @ A )
=> ! [N3: nat] :
( ( divide_divide @ A @ ( one_one @ A ) @ ( power_power @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ N3 ) )
= ( zero_neq_one_of_bool @ A
@ ( N3
= ( zero_zero @ nat ) ) ) ) ) ).
% bits_1_div_exp
thf(fact_4768_one__div__2__pow__eq,axiom,
! [A: $tType] :
( ( euclid5411537665997757685th_nat @ A )
=> ! [N3: nat] :
( ( divide_divide @ A @ ( one_one @ A ) @ ( power_power @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ N3 ) )
= ( zero_neq_one_of_bool @ A
@ ( N3
= ( zero_zero @ nat ) ) ) ) ) ).
% one_div_2_pow_eq
thf(fact_4769_one__mod__2__pow__eq,axiom,
! [A: $tType] :
( ( euclid5411537665997757685th_nat @ A )
=> ! [N3: nat] :
( ( modulo_modulo @ A @ ( one_one @ A ) @ ( power_power @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ N3 ) )
= ( zero_neq_one_of_bool @ A @ ( ord_less @ nat @ ( zero_zero @ nat ) @ N3 ) ) ) ) ).
% one_mod_2_pow_eq
thf(fact_4770_of__bool__eq__iff,axiom,
! [A: $tType] :
( ( zero_neq_one @ A )
=> ! [P4: $o,Q2: $o] :
( ( ( zero_neq_one_of_bool @ A @ P4 )
= ( zero_neq_one_of_bool @ A @ Q2 ) )
= ( P4 = Q2 ) ) ) ).
% of_bool_eq_iff
thf(fact_4771_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_4772_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_4773_split__of__bool__asm,axiom,
! [A: $tType] :
( ( zero_neq_one @ A )
=> ! [P: A > $o,P4: $o] :
( ( P @ ( zero_neq_one_of_bool @ A @ P4 ) )
= ( ~ ( ( P4
& ~ ( P @ ( one_one @ A ) ) )
| ( ~ P4
& ~ ( P @ ( zero_zero @ A ) ) ) ) ) ) ) ).
% split_of_bool_asm
thf(fact_4774_split__of__bool,axiom,
! [A: $tType] :
( ( zero_neq_one @ A )
=> ! [P: A > $o,P4: $o] :
( ( P @ ( zero_neq_one_of_bool @ A @ P4 ) )
= ( ( P4
=> ( P @ ( one_one @ A ) ) )
& ( ~ P4
=> ( P @ ( zero_zero @ A ) ) ) ) ) ) ).
% split_of_bool
thf(fact_4775_of__bool__def,axiom,
! [A: $tType] :
( ( zero_neq_one @ A )
=> ( ( zero_neq_one_of_bool @ A )
= ( ^ [P5: $o] : ( if @ A @ P5 @ ( one_one @ A ) @ ( zero_zero @ A ) ) ) ) ) ).
% of_bool_def
thf(fact_4776_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_4777_subset__decode__imp__le,axiom,
! [M: nat,N3: nat] :
( ( ord_less_eq @ ( set @ nat ) @ ( nat_set_decode @ M ) @ ( nat_set_decode @ N3 ) )
=> ( ord_less_eq @ nat @ M @ N3 ) ) ).
% subset_decode_imp_le
thf(fact_4778_of__bool__odd__eq__mod__2,axiom,
! [A: $tType] :
( ( semiring_parity @ A )
=> ! [A2: A] :
( ( zero_neq_one_of_bool @ A
@ ~ ( dvd_dvd @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ A2 ) )
= ( modulo_modulo @ A @ A2 @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) ) ) ).
% of_bool_odd_eq_mod_2
thf(fact_4779_bits__induct,axiom,
! [A: $tType] :
( ( bit_semiring_bits @ A )
=> ! [P: A > $o,A2: A] :
( ! [A4: A] :
( ( ( divide_divide @ A @ A4 @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) )
= A4 )
=> ( P @ A4 ) )
=> ( ! [A4: A,B3: $o] :
( ( P @ A4 )
=> ( ( ( divide_divide @ A @ ( plus_plus @ A @ ( zero_neq_one_of_bool @ A @ B3 ) @ ( 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 @ B3 ) @ ( times_times @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ A4 ) ) ) ) )
=> ( P @ A2 ) ) ) ) ).
% bits_induct
thf(fact_4780_exp__mod__exp,axiom,
! [A: $tType] :
( ( euclid5411537665997757685th_nat @ A )
=> ! [M: nat,N3: nat] :
( ( modulo_modulo @ A @ ( power_power @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ M ) @ ( power_power @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ N3 ) )
= ( times_times @ A @ ( zero_neq_one_of_bool @ A @ ( ord_less @ nat @ M @ N3 ) ) @ ( power_power @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ M ) ) ) ) ).
% exp_mod_exp
thf(fact_4781_dependent__nat__choice,axiom,
! [A: $tType,P: nat > A > $o,Q: nat > A > A > $o] :
( ? [X_1: A] : ( P @ ( zero_zero @ nat ) @ X_1 )
=> ( ! [X3: A,N: nat] :
( ( P @ N @ X3 )
=> ? [Y4: A] :
( ( P @ ( suc @ N ) @ Y4 )
& ( Q @ N @ X3 @ Y4 ) ) )
=> ? [F3: nat > A] :
! [N9: nat] :
( ( P @ N9 @ ( F3 @ N9 ) )
& ( Q @ N9 @ ( F3 @ N9 ) @ ( F3 @ ( suc @ N9 ) ) ) ) ) ) ).
% dependent_nat_choice
thf(fact_4782_exp__div__exp__eq,axiom,
! [A: $tType] :
( ( bit_semiring_bits @ A )
=> ! [M: nat,N3: nat] :
( ( divide_divide @ A @ ( power_power @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ M ) @ ( power_power @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ N3 ) )
= ( 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 @ N3 @ M ) ) )
@ ( power_power @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ ( minus_minus @ nat @ M @ N3 ) ) ) ) ) ).
% exp_div_exp_eq
thf(fact_4783_set__decode__plus__power__2,axiom,
! [N3: nat,Z: nat] :
( ~ ( member @ nat @ N3 @ ( nat_set_decode @ Z ) )
=> ( ( nat_set_decode @ ( plus_plus @ nat @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N3 ) @ Z ) )
= ( insert @ nat @ N3 @ ( nat_set_decode @ Z ) ) ) ) ).
% set_decode_plus_power_2
thf(fact_4784_set__decode__def,axiom,
( nat_set_decode
= ( ^ [X2: nat] :
( collect @ nat
@ ^ [N2: nat] :
~ ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ X2 @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N2 ) ) ) ) ) ) ).
% set_decode_def
thf(fact_4785_bij__betw__nth__root__unity,axiom,
! [C2: complex,N3: nat] :
( ( C2
!= ( zero_zero @ complex ) )
=> ( ( ord_less @ nat @ ( zero_zero @ nat ) @ N3 )
=> ( bij_betw @ complex @ complex @ ( times_times @ complex @ ( times_times @ complex @ ( real_Vector_of_real @ complex @ ( root @ N3 @ ( real_V7770717601297561774m_norm @ complex @ C2 ) ) ) @ ( cis @ ( divide_divide @ real @ ( arg @ C2 ) @ ( semiring_1_of_nat @ real @ N3 ) ) ) ) )
@ ( collect @ complex
@ ^ [Z3: complex] :
( ( power_power @ complex @ Z3 @ N3 )
= ( one_one @ complex ) ) )
@ ( collect @ complex
@ ^ [Z3: complex] :
( ( power_power @ complex @ Z3 @ N3 )
= C2 ) ) ) ) ) ).
% bij_betw_nth_root_unity
thf(fact_4786_and__int_Osimps,axiom,
( ( bit_se5824344872417868541ns_and @ int )
= ( ^ [K2: int,L: int] :
( if @ int
@ ( ( member @ int @ K2 @ ( insert @ int @ ( zero_zero @ int ) @ ( insert @ int @ ( uminus_uminus @ int @ ( one_one @ int ) ) @ ( bot_bot @ ( set @ int ) ) ) ) )
& ( member @ int @ L @ ( insert @ int @ ( zero_zero @ int ) @ ( insert @ int @ ( uminus_uminus @ int @ ( one_one @ int ) ) @ ( bot_bot @ ( set @ int ) ) ) ) ) )
@ ( uminus_uminus @ int
@ ( zero_neq_one_of_bool @ int
@ ( ~ ( dvd_dvd @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ K2 )
& ~ ( dvd_dvd @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ L ) ) ) )
@ ( plus_plus @ int
@ ( zero_neq_one_of_bool @ int
@ ( ~ ( dvd_dvd @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ K2 )
& ~ ( dvd_dvd @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ L ) ) )
@ ( times_times @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ ( bit_se5824344872417868541ns_and @ int @ ( divide_divide @ int @ K2 @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) ) @ ( divide_divide @ int @ L @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) ) ) ) ) ) ) ) ).
% and_int.simps
thf(fact_4787_and__int_Oelims,axiom,
! [X: int,Xa: int,Y: int] :
( ( ( bit_se5824344872417868541ns_and @ int @ X @ Xa )
= Y )
=> ( ( ( ( member @ int @ X @ ( insert @ int @ ( zero_zero @ int ) @ ( insert @ int @ ( uminus_uminus @ int @ ( one_one @ int ) ) @ ( bot_bot @ ( set @ int ) ) ) ) )
& ( member @ int @ Xa @ ( insert @ int @ ( zero_zero @ int ) @ ( insert @ int @ ( uminus_uminus @ int @ ( one_one @ int ) ) @ ( bot_bot @ ( set @ int ) ) ) ) ) )
=> ( Y
= ( uminus_uminus @ int
@ ( zero_neq_one_of_bool @ int
@ ( ~ ( dvd_dvd @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ X )
& ~ ( dvd_dvd @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ Xa ) ) ) ) ) )
& ( ~ ( ( member @ int @ X @ ( insert @ int @ ( zero_zero @ int ) @ ( insert @ int @ ( uminus_uminus @ int @ ( one_one @ int ) ) @ ( bot_bot @ ( set @ int ) ) ) ) )
& ( member @ int @ Xa @ ( insert @ int @ ( zero_zero @ int ) @ ( insert @ int @ ( uminus_uminus @ int @ ( one_one @ int ) ) @ ( bot_bot @ ( set @ int ) ) ) ) ) )
=> ( Y
= ( plus_plus @ int
@ ( zero_neq_one_of_bool @ int
@ ( ~ ( dvd_dvd @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ X )
& ~ ( dvd_dvd @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ Xa ) ) )
@ ( times_times @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ ( bit_se5824344872417868541ns_and @ int @ ( divide_divide @ int @ X @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) ) @ ( divide_divide @ int @ Xa @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) ) ) ) ) ) ) ) ) ).
% and_int.elims
thf(fact_4788_and_Oright__idem,axiom,
! [A: $tType] :
( ( bit_se359711467146920520ations @ A )
=> ! [A2: A,B2: A] :
( ( bit_se5824344872417868541ns_and @ A @ ( bit_se5824344872417868541ns_and @ A @ A2 @ B2 ) @ B2 )
= ( bit_se5824344872417868541ns_and @ A @ A2 @ B2 ) ) ) ).
% and.right_idem
thf(fact_4789_and_Oleft__idem,axiom,
! [A: $tType] :
( ( bit_se359711467146920520ations @ A )
=> ! [A2: A,B2: A] :
( ( bit_se5824344872417868541ns_and @ A @ A2 @ ( bit_se5824344872417868541ns_and @ A @ A2 @ B2 ) )
= ( bit_se5824344872417868541ns_and @ A @ A2 @ B2 ) ) ) ).
% and.left_idem
thf(fact_4790_and_Oidem,axiom,
! [A: $tType] :
( ( bit_se359711467146920520ations @ A )
=> ! [A2: A] :
( ( bit_se5824344872417868541ns_and @ A @ A2 @ A2 )
= A2 ) ) ).
% and.idem
thf(fact_4791_and__zero__eq,axiom,
! [A: $tType] :
( ( bit_se359711467146920520ations @ A )
=> ! [A2: A] :
( ( bit_se5824344872417868541ns_and @ A @ A2 @ ( zero_zero @ A ) )
= ( zero_zero @ A ) ) ) ).
% and_zero_eq
thf(fact_4792_zero__and__eq,axiom,
! [A: $tType] :
( ( bit_se359711467146920520ations @ A )
=> ! [A2: A] :
( ( bit_se5824344872417868541ns_and @ A @ ( zero_zero @ A ) @ A2 )
= ( zero_zero @ A ) ) ) ).
% zero_and_eq
thf(fact_4793_bit_Oconj__zero__left,axiom,
! [A: $tType] :
( ( bit_ri3973907225187159222ations @ A )
=> ! [X: A] :
( ( bit_se5824344872417868541ns_and @ A @ ( zero_zero @ A ) @ X )
= ( zero_zero @ A ) ) ) ).
% bit.conj_zero_left
thf(fact_4794_bit_Oconj__zero__right,axiom,
! [A: $tType] :
( ( bit_ri3973907225187159222ations @ A )
=> ! [X: A] :
( ( bit_se5824344872417868541ns_and @ A @ X @ ( zero_zero @ A ) )
= ( zero_zero @ A ) ) ) ).
% bit.conj_zero_right
thf(fact_4795_real__root__zero,axiom,
! [N3: nat] :
( ( root @ N3 @ ( zero_zero @ real ) )
= ( zero_zero @ real ) ) ).
% real_root_zero
thf(fact_4796_bit_Oconj__one__right,axiom,
! [A: $tType] :
( ( bit_ri3973907225187159222ations @ A )
=> ! [X: A] :
( ( bit_se5824344872417868541ns_and @ A @ X @ ( uminus_uminus @ A @ ( one_one @ A ) ) )
= X ) ) ).
% bit.conj_one_right
thf(fact_4797_and_Oright__neutral,axiom,
! [A: $tType] :
( ( bit_ri3973907225187159222ations @ A )
=> ! [A2: A] :
( ( bit_se5824344872417868541ns_and @ A @ A2 @ ( uminus_uminus @ A @ ( one_one @ A ) ) )
= A2 ) ) ).
% and.right_neutral
thf(fact_4798_and_Oleft__neutral,axiom,
! [A: $tType] :
( ( bit_ri3973907225187159222ations @ A )
=> ! [A2: A] :
( ( bit_se5824344872417868541ns_and @ A @ ( uminus_uminus @ A @ ( one_one @ A ) ) @ A2 )
= A2 ) ) ).
% and.left_neutral
thf(fact_4799_real__root__Suc__0,axiom,
! [X: real] :
( ( root @ ( suc @ ( zero_zero @ nat ) ) @ X )
= X ) ).
% real_root_Suc_0
thf(fact_4800_real__root__eq__iff,axiom,
! [N3: nat,X: real,Y: real] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ N3 )
=> ( ( ( root @ N3 @ X )
= ( root @ N3 @ Y ) )
= ( X = Y ) ) ) ).
% real_root_eq_iff
thf(fact_4801_root__0,axiom,
! [X: real] :
( ( root @ ( zero_zero @ nat ) @ X )
= ( zero_zero @ real ) ) ).
% root_0
thf(fact_4802_and__nonnegative__int__iff,axiom,
! [K: int,L2: int] :
( ( ord_less_eq @ int @ ( zero_zero @ int ) @ ( bit_se5824344872417868541ns_and @ int @ K @ L2 ) )
= ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ K )
| ( ord_less_eq @ int @ ( zero_zero @ int ) @ L2 ) ) ) ).
% and_nonnegative_int_iff
thf(fact_4803_and__negative__int__iff,axiom,
! [K: int,L2: int] :
( ( ord_less @ int @ ( bit_se5824344872417868541ns_and @ int @ K @ L2 ) @ ( zero_zero @ int ) )
= ( ( ord_less @ int @ K @ ( zero_zero @ int ) )
& ( ord_less @ int @ L2 @ ( zero_zero @ int ) ) ) ) ).
% and_negative_int_iff
thf(fact_4804_and__numerals_I2_J,axiom,
! [A: $tType] :
( ( bit_un5681908812861735899ations @ A )
=> ! [Y: num] :
( ( bit_se5824344872417868541ns_and @ A @ ( one_one @ A ) @ ( numeral_numeral @ A @ ( bit1 @ Y ) ) )
= ( one_one @ A ) ) ) ).
% and_numerals(2)
thf(fact_4805_and__numerals_I8_J,axiom,
! [A: $tType] :
( ( bit_un5681908812861735899ations @ A )
=> ! [X: num] :
( ( bit_se5824344872417868541ns_and @ A @ ( numeral_numeral @ A @ ( bit1 @ X ) ) @ ( one_one @ A ) )
= ( one_one @ A ) ) ) ).
% and_numerals(8)
thf(fact_4806_real__root__eq__0__iff,axiom,
! [N3: nat,X: real] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ N3 )
=> ( ( ( root @ N3 @ X )
= ( zero_zero @ real ) )
= ( X
= ( zero_zero @ real ) ) ) ) ).
% real_root_eq_0_iff
thf(fact_4807_real__root__less__iff,axiom,
! [N3: nat,X: real,Y: real] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ N3 )
=> ( ( ord_less @ real @ ( root @ N3 @ X ) @ ( root @ N3 @ Y ) )
= ( ord_less @ real @ X @ Y ) ) ) ).
% real_root_less_iff
thf(fact_4808_real__root__le__iff,axiom,
! [N3: nat,X: real,Y: real] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ N3 )
=> ( ( ord_less_eq @ real @ ( root @ N3 @ X ) @ ( root @ N3 @ Y ) )
= ( ord_less_eq @ real @ X @ Y ) ) ) ).
% real_root_le_iff
thf(fact_4809_real__root__eq__1__iff,axiom,
! [N3: nat,X: real] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ N3 )
=> ( ( ( root @ N3 @ X )
= ( one_one @ real ) )
= ( X
= ( one_one @ real ) ) ) ) ).
% real_root_eq_1_iff
thf(fact_4810_real__root__one,axiom,
! [N3: nat] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ N3 )
=> ( ( root @ N3 @ ( one_one @ real ) )
= ( one_one @ real ) ) ) ).
% real_root_one
thf(fact_4811_and__numerals_I5_J,axiom,
! [A: $tType] :
( ( bit_un5681908812861735899ations @ A )
=> ! [X: num] :
( ( bit_se5824344872417868541ns_and @ A @ ( numeral_numeral @ A @ ( bit0 @ X ) ) @ ( one_one @ A ) )
= ( zero_zero @ A ) ) ) ).
% and_numerals(5)
thf(fact_4812_and__numerals_I1_J,axiom,
! [A: $tType] :
( ( bit_un5681908812861735899ations @ A )
=> ! [Y: num] :
( ( bit_se5824344872417868541ns_and @ A @ ( one_one @ A ) @ ( numeral_numeral @ A @ ( bit0 @ Y ) ) )
= ( zero_zero @ A ) ) ) ).
% and_numerals(1)
thf(fact_4813_and__numerals_I3_J,axiom,
! [A: $tType] :
( ( bit_un5681908812861735899ations @ A )
=> ! [X: num,Y: num] :
( ( bit_se5824344872417868541ns_and @ A @ ( numeral_numeral @ A @ ( bit0 @ X ) ) @ ( numeral_numeral @ A @ ( bit0 @ Y ) ) )
= ( times_times @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ ( bit_se5824344872417868541ns_and @ A @ ( numeral_numeral @ A @ X ) @ ( numeral_numeral @ A @ Y ) ) ) ) ) ).
% and_numerals(3)
thf(fact_4814_real__root__gt__0__iff,axiom,
! [N3: nat,Y: real] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ N3 )
=> ( ( ord_less @ real @ ( zero_zero @ real ) @ ( root @ N3 @ Y ) )
= ( ord_less @ real @ ( zero_zero @ real ) @ Y ) ) ) ).
% real_root_gt_0_iff
thf(fact_4815_real__root__lt__0__iff,axiom,
! [N3: nat,X: real] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ N3 )
=> ( ( ord_less @ real @ ( root @ N3 @ X ) @ ( zero_zero @ real ) )
= ( ord_less @ real @ X @ ( zero_zero @ real ) ) ) ) ).
% real_root_lt_0_iff
thf(fact_4816_real__root__ge__0__iff,axiom,
! [N3: nat,Y: real] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ N3 )
=> ( ( ord_less_eq @ real @ ( zero_zero @ real ) @ ( root @ N3 @ Y ) )
= ( ord_less_eq @ real @ ( zero_zero @ real ) @ Y ) ) ) ).
% real_root_ge_0_iff
thf(fact_4817_real__root__le__0__iff,axiom,
! [N3: nat,X: real] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ N3 )
=> ( ( ord_less_eq @ real @ ( root @ N3 @ X ) @ ( zero_zero @ real ) )
= ( ord_less_eq @ real @ X @ ( zero_zero @ real ) ) ) ) ).
% real_root_le_0_iff
thf(fact_4818_real__root__lt__1__iff,axiom,
! [N3: nat,X: real] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ N3 )
=> ( ( ord_less @ real @ ( root @ N3 @ X ) @ ( one_one @ real ) )
= ( ord_less @ real @ X @ ( one_one @ real ) ) ) ) ).
% real_root_lt_1_iff
thf(fact_4819_real__root__gt__1__iff,axiom,
! [N3: nat,Y: real] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ N3 )
=> ( ( ord_less @ real @ ( one_one @ real ) @ ( root @ N3 @ Y ) )
= ( ord_less @ real @ ( one_one @ real ) @ Y ) ) ) ).
% real_root_gt_1_iff
thf(fact_4820_real__root__ge__1__iff,axiom,
! [N3: nat,Y: real] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ N3 )
=> ( ( ord_less_eq @ real @ ( one_one @ real ) @ ( root @ N3 @ Y ) )
= ( ord_less_eq @ real @ ( one_one @ real ) @ Y ) ) ) ).
% real_root_ge_1_iff
thf(fact_4821_real__root__le__1__iff,axiom,
! [N3: nat,X: real] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ N3 )
=> ( ( ord_less_eq @ real @ ( root @ N3 @ X ) @ ( one_one @ real ) )
= ( ord_less_eq @ real @ X @ ( one_one @ real ) ) ) ) ).
% real_root_le_1_iff
thf(fact_4822_and__minus__numerals_I2_J,axiom,
! [N3: num] :
( ( bit_se5824344872417868541ns_and @ int @ ( one_one @ int ) @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ ( bit1 @ N3 ) ) ) )
= ( one_one @ int ) ) ).
% and_minus_numerals(2)
thf(fact_4823_and__minus__numerals_I6_J,axiom,
! [N3: num] :
( ( bit_se5824344872417868541ns_and @ int @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ ( bit1 @ N3 ) ) ) @ ( one_one @ int ) )
= ( one_one @ int ) ) ).
% and_minus_numerals(6)
thf(fact_4824_and__numerals_I4_J,axiom,
! [A: $tType] :
( ( bit_un5681908812861735899ations @ A )
=> ! [X: num,Y: num] :
( ( bit_se5824344872417868541ns_and @ A @ ( numeral_numeral @ A @ ( bit0 @ X ) ) @ ( numeral_numeral @ A @ ( bit1 @ Y ) ) )
= ( times_times @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ ( bit_se5824344872417868541ns_and @ A @ ( numeral_numeral @ A @ X ) @ ( numeral_numeral @ A @ Y ) ) ) ) ) ).
% and_numerals(4)
thf(fact_4825_and__numerals_I6_J,axiom,
! [A: $tType] :
( ( bit_un5681908812861735899ations @ A )
=> ! [X: num,Y: num] :
( ( bit_se5824344872417868541ns_and @ A @ ( numeral_numeral @ A @ ( bit1 @ X ) ) @ ( numeral_numeral @ A @ ( bit0 @ Y ) ) )
= ( times_times @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ ( bit_se5824344872417868541ns_and @ A @ ( numeral_numeral @ A @ X ) @ ( numeral_numeral @ A @ Y ) ) ) ) ) ).
% and_numerals(6)
thf(fact_4826_real__root__pow__pos2,axiom,
! [N3: nat,X: real] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ N3 )
=> ( ( ord_less_eq @ real @ ( zero_zero @ real ) @ X )
=> ( ( power_power @ real @ ( root @ N3 @ X ) @ N3 )
= X ) ) ) ).
% real_root_pow_pos2
thf(fact_4827_and__minus__numerals_I1_J,axiom,
! [N3: num] :
( ( bit_se5824344872417868541ns_and @ int @ ( one_one @ int ) @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ ( bit0 @ N3 ) ) ) )
= ( zero_zero @ int ) ) ).
% and_minus_numerals(1)
thf(fact_4828_and__minus__numerals_I5_J,axiom,
! [N3: num] :
( ( bit_se5824344872417868541ns_and @ int @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ ( bit0 @ N3 ) ) ) @ ( one_one @ int ) )
= ( zero_zero @ int ) ) ).
% and_minus_numerals(5)
thf(fact_4829_and__numerals_I7_J,axiom,
! [A: $tType] :
( ( bit_un5681908812861735899ations @ A )
=> ! [X: num,Y: num] :
( ( bit_se5824344872417868541ns_and @ A @ ( numeral_numeral @ A @ ( bit1 @ X ) ) @ ( numeral_numeral @ A @ ( bit1 @ Y ) ) )
= ( plus_plus @ A @ ( one_one @ A ) @ ( times_times @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ ( bit_se5824344872417868541ns_and @ A @ ( numeral_numeral @ A @ X ) @ ( numeral_numeral @ A @ Y ) ) ) ) ) ) ).
% and_numerals(7)
thf(fact_4830_real__root__mult__exp,axiom,
! [M: nat,N3: nat,X: real] :
( ( root @ ( times_times @ nat @ M @ N3 ) @ X )
= ( root @ M @ ( root @ N3 @ X ) ) ) ).
% real_root_mult_exp
thf(fact_4831_real__root__mult,axiom,
! [N3: nat,X: real,Y: real] :
( ( root @ N3 @ ( times_times @ real @ X @ Y ) )
= ( times_times @ real @ ( root @ N3 @ X ) @ ( root @ N3 @ Y ) ) ) ).
% real_root_mult
thf(fact_4832_real__root__minus,axiom,
! [N3: nat,X: real] :
( ( root @ N3 @ ( uminus_uminus @ real @ X ) )
= ( uminus_uminus @ real @ ( root @ N3 @ X ) ) ) ).
% real_root_minus
thf(fact_4833_and_Oleft__commute,axiom,
! [A: $tType] :
( ( bit_se359711467146920520ations @ A )
=> ! [B2: A,A2: A,C2: A] :
( ( bit_se5824344872417868541ns_and @ A @ B2 @ ( bit_se5824344872417868541ns_and @ A @ A2 @ C2 ) )
= ( bit_se5824344872417868541ns_and @ A @ A2 @ ( bit_se5824344872417868541ns_and @ A @ B2 @ C2 ) ) ) ) ).
% and.left_commute
thf(fact_4834_and_Ocommute,axiom,
! [A: $tType] :
( ( bit_se359711467146920520ations @ A )
=> ( ( bit_se5824344872417868541ns_and @ A )
= ( ^ [A6: A,B5: A] : ( bit_se5824344872417868541ns_and @ A @ B5 @ A6 ) ) ) ) ).
% and.commute
thf(fact_4835_and_Oassoc,axiom,
! [A: $tType] :
( ( bit_se359711467146920520ations @ A )
=> ! [A2: A,B2: A,C2: A] :
( ( bit_se5824344872417868541ns_and @ A @ ( bit_se5824344872417868541ns_and @ A @ A2 @ B2 ) @ C2 )
= ( bit_se5824344872417868541ns_and @ A @ A2 @ ( bit_se5824344872417868541ns_and @ A @ B2 @ C2 ) ) ) ) ).
% and.assoc
thf(fact_4836_real__root__commute,axiom,
! [M: nat,N3: nat,X: real] :
( ( root @ M @ ( root @ N3 @ X ) )
= ( root @ N3 @ ( root @ M @ X ) ) ) ).
% real_root_commute
thf(fact_4837_of__int__and__eq,axiom,
! [A: $tType] :
( ( bit_ri3973907225187159222ations @ A )
=> ! [K: int,L2: int] :
( ( ring_1_of_int @ A @ ( bit_se5824344872417868541ns_and @ int @ K @ L2 ) )
= ( bit_se5824344872417868541ns_and @ A @ ( ring_1_of_int @ A @ K ) @ ( ring_1_of_int @ A @ L2 ) ) ) ) ).
% of_int_and_eq
thf(fact_4838_of__nat__and__eq,axiom,
! [A: $tType] :
( ( bit_se359711467146920520ations @ A )
=> ! [M: nat,N3: nat] :
( ( semiring_1_of_nat @ A @ ( bit_se5824344872417868541ns_and @ nat @ M @ N3 ) )
= ( bit_se5824344872417868541ns_and @ A @ ( semiring_1_of_nat @ A @ M ) @ ( semiring_1_of_nat @ A @ N3 ) ) ) ) ).
% of_nat_and_eq
thf(fact_4839_real__root__divide,axiom,
! [N3: nat,X: real,Y: real] :
( ( root @ N3 @ ( divide_divide @ real @ X @ Y ) )
= ( divide_divide @ real @ ( root @ N3 @ X ) @ ( root @ N3 @ Y ) ) ) ).
% real_root_divide
thf(fact_4840_real__root__inverse,axiom,
! [N3: nat,X: real] :
( ( root @ N3 @ ( inverse_inverse @ real @ X ) )
= ( inverse_inverse @ real @ ( root @ N3 @ X ) ) ) ).
% real_root_inverse
thf(fact_4841_and__eq__minus__1__iff,axiom,
! [A: $tType] :
( ( bit_ri3973907225187159222ations @ A )
=> ! [A2: A,B2: A] :
( ( ( bit_se5824344872417868541ns_and @ A @ A2 @ B2 )
= ( uminus_uminus @ A @ ( one_one @ A ) ) )
= ( ( A2
= ( uminus_uminus @ A @ ( one_one @ A ) ) )
& ( B2
= ( uminus_uminus @ A @ ( one_one @ A ) ) ) ) ) ) ).
% and_eq_minus_1_iff
thf(fact_4842_real__root__pos__pos__le,axiom,
! [X: real,N3: nat] :
( ( ord_less_eq @ real @ ( zero_zero @ real ) @ X )
=> ( ord_less_eq @ real @ ( zero_zero @ real ) @ ( root @ N3 @ X ) ) ) ).
% real_root_pos_pos_le
thf(fact_4843_AND__upper2_H,axiom,
! [Y: int,Z: int,X: int] :
( ( ord_less_eq @ int @ ( zero_zero @ int ) @ Y )
=> ( ( ord_less_eq @ int @ Y @ Z )
=> ( ord_less_eq @ int @ ( bit_se5824344872417868541ns_and @ int @ X @ Y ) @ Z ) ) ) ).
% AND_upper2'
thf(fact_4844_AND__upper1_H,axiom,
! [Y: int,Z: int,Ya: int] :
( ( ord_less_eq @ int @ ( zero_zero @ int ) @ Y )
=> ( ( ord_less_eq @ int @ Y @ Z )
=> ( ord_less_eq @ int @ ( bit_se5824344872417868541ns_and @ int @ Y @ Ya ) @ Z ) ) ) ).
% AND_upper1'
thf(fact_4845_AND__upper2,axiom,
! [Y: int,X: int] :
( ( ord_less_eq @ int @ ( zero_zero @ int ) @ Y )
=> ( ord_less_eq @ int @ ( bit_se5824344872417868541ns_and @ int @ X @ Y ) @ Y ) ) ).
% AND_upper2
thf(fact_4846_AND__upper1,axiom,
! [X: int,Y: int] :
( ( ord_less_eq @ int @ ( zero_zero @ int ) @ X )
=> ( ord_less_eq @ int @ ( bit_se5824344872417868541ns_and @ int @ X @ Y ) @ X ) ) ).
% AND_upper1
thf(fact_4847_AND__lower,axiom,
! [X: int,Y: int] :
( ( ord_less_eq @ int @ ( zero_zero @ int ) @ X )
=> ( ord_less_eq @ int @ ( zero_zero @ int ) @ ( bit_se5824344872417868541ns_and @ int @ X @ Y ) ) ) ).
% AND_lower
thf(fact_4848_real__root__less__mono,axiom,
! [N3: nat,X: real,Y: real] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ N3 )
=> ( ( ord_less @ real @ X @ Y )
=> ( ord_less @ real @ ( root @ N3 @ X ) @ ( root @ N3 @ Y ) ) ) ) ).
% real_root_less_mono
thf(fact_4849_real__root__le__mono,axiom,
! [N3: nat,X: real,Y: real] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ N3 )
=> ( ( ord_less_eq @ real @ X @ Y )
=> ( ord_less_eq @ real @ ( root @ N3 @ X ) @ ( root @ N3 @ Y ) ) ) ) ).
% real_root_le_mono
thf(fact_4850_real__root__power,axiom,
! [N3: nat,X: real,K: nat] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ N3 )
=> ( ( root @ N3 @ ( power_power @ real @ X @ K ) )
= ( power_power @ real @ ( root @ N3 @ X ) @ K ) ) ) ).
% real_root_power
thf(fact_4851_real__root__abs,axiom,
! [N3: nat,X: real] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ N3 )
=> ( ( root @ N3 @ ( abs_abs @ real @ X ) )
= ( abs_abs @ real @ ( root @ N3 @ X ) ) ) ) ).
% real_root_abs
thf(fact_4852_and__less__eq,axiom,
! [L2: int,K: int] :
( ( ord_less @ int @ L2 @ ( zero_zero @ int ) )
=> ( ord_less_eq @ int @ ( bit_se5824344872417868541ns_and @ int @ K @ L2 ) @ K ) ) ).
% and_less_eq
thf(fact_4853_AND__upper1_H_H,axiom,
! [Y: int,Z: int,Ya: int] :
( ( ord_less_eq @ int @ ( zero_zero @ int ) @ Y )
=> ( ( ord_less @ int @ Y @ Z )
=> ( ord_less @ int @ ( bit_se5824344872417868541ns_and @ int @ Y @ Ya ) @ Z ) ) ) ).
% AND_upper1''
thf(fact_4854_AND__upper2_H_H,axiom,
! [Y: int,Z: int,X: int] :
( ( ord_less_eq @ int @ ( zero_zero @ int ) @ Y )
=> ( ( ord_less @ int @ Y @ Z )
=> ( ord_less @ int @ ( bit_se5824344872417868541ns_and @ int @ X @ Y ) @ Z ) ) ) ).
% AND_upper2''
thf(fact_4855_even__and__iff,axiom,
! [A: $tType] :
( ( bit_se359711467146920520ations @ A )
=> ! [A2: A,B2: A] :
( ( dvd_dvd @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ ( bit_se5824344872417868541ns_and @ A @ A2 @ B2 ) )
= ( ( dvd_dvd @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ A2 )
| ( dvd_dvd @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ B2 ) ) ) ) ).
% even_and_iff
thf(fact_4856_real__root__gt__zero,axiom,
! [N3: nat,X: real] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ N3 )
=> ( ( ord_less @ real @ ( zero_zero @ real ) @ X )
=> ( ord_less @ real @ ( zero_zero @ real ) @ ( root @ N3 @ X ) ) ) ) ).
% real_root_gt_zero
thf(fact_4857_real__root__strict__decreasing,axiom,
! [N3: nat,N7: nat,X: real] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ N3 )
=> ( ( ord_less @ nat @ N3 @ N7 )
=> ( ( ord_less @ real @ ( one_one @ real ) @ X )
=> ( ord_less @ real @ ( root @ N7 @ X ) @ ( root @ N3 @ X ) ) ) ) ) ).
% real_root_strict_decreasing
thf(fact_4858_sqrt__def,axiom,
( sqrt
= ( root @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ).
% sqrt_def
thf(fact_4859_root__abs__power,axiom,
! [N3: nat,Y: real] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ N3 )
=> ( ( abs_abs @ real @ ( root @ N3 @ ( power_power @ real @ Y @ N3 ) ) )
= ( abs_abs @ real @ Y ) ) ) ).
% root_abs_power
thf(fact_4860_even__and__iff__int,axiom,
! [K: int,L2: int] :
( ( dvd_dvd @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ ( bit_se5824344872417868541ns_and @ int @ K @ L2 ) )
= ( ( dvd_dvd @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ K )
| ( dvd_dvd @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ L2 ) ) ) ).
% even_and_iff_int
thf(fact_4861_one__and__eq,axiom,
! [A: $tType] :
( ( bit_se359711467146920520ations @ A )
=> ! [A2: A] :
( ( bit_se5824344872417868541ns_and @ A @ ( one_one @ A ) @ A2 )
= ( modulo_modulo @ A @ A2 @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) ) ) ).
% one_and_eq
thf(fact_4862_and__one__eq,axiom,
! [A: $tType] :
( ( bit_se359711467146920520ations @ A )
=> ! [A2: A] :
( ( bit_se5824344872417868541ns_and @ A @ A2 @ ( one_one @ A ) )
= ( modulo_modulo @ A @ A2 @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) ) ) ).
% and_one_eq
thf(fact_4863_real__root__pos__pos,axiom,
! [N3: nat,X: real] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ N3 )
=> ( ( ord_less @ real @ ( zero_zero @ real ) @ X )
=> ( ord_less_eq @ real @ ( zero_zero @ real ) @ ( root @ N3 @ X ) ) ) ) ).
% real_root_pos_pos
thf(fact_4864_real__root__strict__increasing,axiom,
! [N3: nat,N7: nat,X: real] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ N3 )
=> ( ( ord_less @ nat @ N3 @ N7 )
=> ( ( ord_less @ real @ ( zero_zero @ real ) @ X )
=> ( ( ord_less @ real @ X @ ( one_one @ real ) )
=> ( ord_less @ real @ ( root @ N3 @ X ) @ ( root @ N7 @ X ) ) ) ) ) ) ).
% real_root_strict_increasing
thf(fact_4865_real__root__decreasing,axiom,
! [N3: nat,N7: nat,X: real] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ N3 )
=> ( ( ord_less_eq @ nat @ N3 @ N7 )
=> ( ( ord_less_eq @ real @ ( one_one @ real ) @ X )
=> ( ord_less_eq @ real @ ( root @ N7 @ X ) @ ( root @ N3 @ X ) ) ) ) ) ).
% real_root_decreasing
thf(fact_4866_odd__real__root__power__cancel,axiom,
! [N3: nat,X: real] :
( ~ ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N3 )
=> ( ( root @ N3 @ ( power_power @ real @ X @ N3 ) )
= X ) ) ).
% odd_real_root_power_cancel
thf(fact_4867_odd__real__root__unique,axiom,
! [N3: nat,Y: real,X: real] :
( ~ ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N3 )
=> ( ( ( power_power @ real @ Y @ N3 )
= X )
=> ( ( root @ N3 @ X )
= Y ) ) ) ).
% odd_real_root_unique
thf(fact_4868_odd__real__root__pow,axiom,
! [N3: nat,X: real] :
( ~ ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N3 )
=> ( ( power_power @ real @ ( root @ N3 @ X ) @ N3 )
= X ) ) ).
% odd_real_root_pow
thf(fact_4869_real__root__pow__pos,axiom,
! [N3: nat,X: real] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ N3 )
=> ( ( ord_less @ real @ ( zero_zero @ real ) @ X )
=> ( ( power_power @ real @ ( root @ N3 @ X ) @ N3 )
= X ) ) ) ).
% real_root_pow_pos
thf(fact_4870_real__root__pos__unique,axiom,
! [N3: nat,Y: real,X: real] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ N3 )
=> ( ( ord_less_eq @ real @ ( zero_zero @ real ) @ Y )
=> ( ( ( power_power @ real @ Y @ N3 )
= X )
=> ( ( root @ N3 @ X )
= Y ) ) ) ) ).
% real_root_pos_unique
thf(fact_4871_real__root__power__cancel,axiom,
! [N3: nat,X: real] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ N3 )
=> ( ( ord_less_eq @ real @ ( zero_zero @ real ) @ X )
=> ( ( root @ N3 @ ( power_power @ real @ X @ N3 ) )
= X ) ) ) ).
% real_root_power_cancel
thf(fact_4872_real__root__increasing,axiom,
! [N3: nat,N7: nat,X: real] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ N3 )
=> ( ( ord_less_eq @ nat @ N3 @ N7 )
=> ( ( ord_less_eq @ real @ ( zero_zero @ real ) @ X )
=> ( ( ord_less_eq @ real @ X @ ( one_one @ real ) )
=> ( ord_less_eq @ real @ ( root @ N3 @ X ) @ ( root @ N7 @ X ) ) ) ) ) ) ).
% real_root_increasing
thf(fact_4873_log__root,axiom,
! [N3: nat,A2: real,B2: real] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ N3 )
=> ( ( ord_less @ real @ ( zero_zero @ real ) @ A2 )
=> ( ( log @ B2 @ ( root @ N3 @ A2 ) )
= ( divide_divide @ real @ ( log @ B2 @ A2 ) @ ( semiring_1_of_nat @ real @ N3 ) ) ) ) ) ).
% log_root
thf(fact_4874_log__base__root,axiom,
! [N3: nat,B2: real,X: real] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ N3 )
=> ( ( ord_less @ real @ ( zero_zero @ real ) @ B2 )
=> ( ( log @ ( root @ N3 @ B2 ) @ X )
= ( times_times @ real @ ( semiring_1_of_nat @ real @ N3 ) @ ( log @ B2 @ X ) ) ) ) ) ).
% log_base_root
thf(fact_4875_ln__root,axiom,
! [N3: nat,B2: real] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ N3 )
=> ( ( ord_less @ real @ ( zero_zero @ real ) @ B2 )
=> ( ( ln_ln @ real @ ( root @ N3 @ B2 ) )
= ( divide_divide @ real @ ( ln_ln @ real @ B2 ) @ ( semiring_1_of_nat @ real @ N3 ) ) ) ) ) ).
% ln_root
thf(fact_4876_and__int__rec,axiom,
( ( bit_se5824344872417868541ns_and @ int )
= ( ^ [K2: int,L: int] :
( plus_plus @ int
@ ( zero_neq_one_of_bool @ int
@ ( ~ ( dvd_dvd @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ K2 )
& ~ ( dvd_dvd @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ L ) ) )
@ ( times_times @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ ( bit_se5824344872417868541ns_and @ int @ ( divide_divide @ int @ K2 @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) ) @ ( divide_divide @ int @ L @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) ) ) ) ) ) ) ).
% and_int_rec
thf(fact_4877_root__powr__inverse,axiom,
! [N3: nat,X: real] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ N3 )
=> ( ( ord_less @ real @ ( zero_zero @ real ) @ X )
=> ( ( root @ N3 @ X )
= ( powr @ real @ X @ ( divide_divide @ real @ ( one_one @ real ) @ ( semiring_1_of_nat @ real @ N3 ) ) ) ) ) ) ).
% root_powr_inverse
thf(fact_4878_and__int__unfold,axiom,
( ( bit_se5824344872417868541ns_and @ int )
= ( ^ [K2: int,L: int] :
( if @ int
@ ( ( K2
= ( zero_zero @ int ) )
| ( L
= ( zero_zero @ int ) ) )
@ ( zero_zero @ int )
@ ( if @ int
@ ( K2
= ( uminus_uminus @ int @ ( one_one @ int ) ) )
@ L
@ ( if @ int
@ ( L
= ( uminus_uminus @ int @ ( one_one @ int ) ) )
@ K2
@ ( plus_plus @ int @ ( times_times @ int @ ( modulo_modulo @ int @ K2 @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) ) @ ( modulo_modulo @ int @ L @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) ) ) @ ( times_times @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ ( bit_se5824344872417868541ns_and @ int @ ( divide_divide @ int @ K2 @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) ) @ ( divide_divide @ int @ L @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) ) ) ) ) ) ) ) ) ) ).
% and_int_unfold
thf(fact_4879_and__int_Opelims,axiom,
! [X: int,Xa: int,Y: int] :
( ( ( bit_se5824344872417868541ns_and @ int @ X @ Xa )
= Y )
=> ( ( accp @ ( product_prod @ int @ int ) @ bit_and_int_rel @ ( product_Pair @ int @ int @ X @ Xa ) )
=> ~ ( ( ( ( ( member @ int @ X @ ( insert @ int @ ( zero_zero @ int ) @ ( insert @ int @ ( uminus_uminus @ int @ ( one_one @ int ) ) @ ( bot_bot @ ( set @ int ) ) ) ) )
& ( member @ int @ Xa @ ( insert @ int @ ( zero_zero @ int ) @ ( insert @ int @ ( uminus_uminus @ int @ ( one_one @ int ) ) @ ( bot_bot @ ( set @ int ) ) ) ) ) )
=> ( Y
= ( uminus_uminus @ int
@ ( zero_neq_one_of_bool @ int
@ ( ~ ( dvd_dvd @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ X )
& ~ ( dvd_dvd @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ Xa ) ) ) ) ) )
& ( ~ ( ( member @ int @ X @ ( insert @ int @ ( zero_zero @ int ) @ ( insert @ int @ ( uminus_uminus @ int @ ( one_one @ int ) ) @ ( bot_bot @ ( set @ int ) ) ) ) )
& ( member @ int @ Xa @ ( insert @ int @ ( zero_zero @ int ) @ ( insert @ int @ ( uminus_uminus @ int @ ( one_one @ int ) ) @ ( bot_bot @ ( set @ int ) ) ) ) ) )
=> ( Y
= ( plus_plus @ int
@ ( zero_neq_one_of_bool @ int
@ ( ~ ( dvd_dvd @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ X )
& ~ ( dvd_dvd @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ Xa ) ) )
@ ( times_times @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ ( bit_se5824344872417868541ns_and @ int @ ( divide_divide @ int @ X @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) ) @ ( divide_divide @ int @ Xa @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) ) ) ) ) ) ) )
=> ~ ( accp @ ( product_prod @ int @ int ) @ bit_and_int_rel @ ( product_Pair @ int @ int @ X @ Xa ) ) ) ) ) ).
% and_int.pelims
thf(fact_4880_and__int_Opsimps,axiom,
! [K: int,L2: int] :
( ( accp @ ( product_prod @ int @ int ) @ bit_and_int_rel @ ( product_Pair @ int @ int @ K @ L2 ) )
=> ( ( ( ( member @ int @ K @ ( 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 ) ) ) ) ) )
=> ( ( bit_se5824344872417868541ns_and @ int @ K @ L2 )
= ( 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 ) ) @ L2 ) ) ) ) ) )
& ( ~ ( ( member @ int @ K @ ( 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 ) ) ) ) ) )
=> ( ( bit_se5824344872417868541ns_and @ int @ K @ L2 )
= ( 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 ) ) @ L2 ) ) )
@ ( times_times @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ ( bit_se5824344872417868541ns_and @ int @ ( divide_divide @ int @ K @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) ) @ ( divide_divide @ int @ L2 @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) ) ) ) ) ) ) ) ) ).
% and_int.psimps
thf(fact_4881_Comparator__Generator_OAll__less__Suc,axiom,
! [X: nat,P: nat > $o] :
( ( ! [I3: nat] :
( ( ord_less @ nat @ I3 @ ( suc @ X ) )
=> ( P @ I3 ) ) )
= ( ( P @ ( zero_zero @ nat ) )
& ! [I3: nat] :
( ( ord_less @ nat @ I3 @ X )
=> ( P @ ( suc @ I3 ) ) ) ) ) ).
% Comparator_Generator.All_less_Suc
thf(fact_4882_word__bitwise__m1__simps_I2_J,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [X: word @ A] :
( ( bit_se5824344872417868541ns_and @ ( word @ A ) @ ( uminus_uminus @ ( word @ A ) @ ( one_one @ ( word @ A ) ) ) @ X )
= X ) ) ).
% word_bitwise_m1_simps(2)
thf(fact_4883_word__and__max,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [X: word @ A] :
( ( bit_se5824344872417868541ns_and @ ( word @ A ) @ X @ ( uminus_uminus @ ( word @ A ) @ ( one_one @ ( word @ A ) ) ) )
= X ) ) ).
% word_and_max
thf(fact_4884_and__nat__numerals_I3_J,axiom,
! [X: num] :
( ( bit_se5824344872417868541ns_and @ nat @ ( numeral_numeral @ nat @ ( bit0 @ X ) ) @ ( suc @ ( zero_zero @ nat ) ) )
= ( zero_zero @ nat ) ) ).
% and_nat_numerals(3)
thf(fact_4885_and__nat__numerals_I1_J,axiom,
! [Y: num] :
( ( bit_se5824344872417868541ns_and @ nat @ ( suc @ ( zero_zero @ nat ) ) @ ( numeral_numeral @ nat @ ( bit0 @ Y ) ) )
= ( zero_zero @ nat ) ) ).
% and_nat_numerals(1)
thf(fact_4886_and__nat__numerals_I4_J,axiom,
! [X: num] :
( ( bit_se5824344872417868541ns_and @ nat @ ( numeral_numeral @ nat @ ( bit1 @ X ) ) @ ( suc @ ( zero_zero @ nat ) ) )
= ( one_one @ nat ) ) ).
% and_nat_numerals(4)
thf(fact_4887_and__nat__numerals_I2_J,axiom,
! [Y: num] :
( ( bit_se5824344872417868541ns_and @ nat @ ( suc @ ( zero_zero @ nat ) ) @ ( numeral_numeral @ nat @ ( bit1 @ Y ) ) )
= ( one_one @ nat ) ) ).
% and_nat_numerals(2)
thf(fact_4888_word__bitwise__1__simps_I2_J,axiom,
! [B: $tType] :
( ( type_len @ B )
=> ! [B2: num] :
( ( bit_se5824344872417868541ns_and @ ( word @ B ) @ ( one_one @ ( word @ B ) ) @ ( numeral_numeral @ ( word @ B ) @ B2 ) )
= ( ring_1_of_int @ ( word @ B ) @ ( bit_se5824344872417868541ns_and @ int @ ( one_one @ int ) @ ( numeral_numeral @ int @ B2 ) ) ) ) ) ).
% word_bitwise_1_simps(2)
thf(fact_4889_word__bitwise__1__simps_I4_J,axiom,
! [D5: $tType] :
( ( type_len @ D5 )
=> ! [A2: num] :
( ( bit_se5824344872417868541ns_and @ ( word @ D5 ) @ ( numeral_numeral @ ( word @ D5 ) @ A2 ) @ ( one_one @ ( word @ D5 ) ) )
= ( ring_1_of_int @ ( word @ D5 ) @ ( bit_se5824344872417868541ns_and @ int @ ( numeral_numeral @ int @ A2 ) @ ( one_one @ int ) ) ) ) ) ).
% word_bitwise_1_simps(4)
thf(fact_4890_Suc__0__and__eq,axiom,
! [N3: nat] :
( ( bit_se5824344872417868541ns_and @ nat @ ( suc @ ( zero_zero @ nat ) ) @ N3 )
= ( modulo_modulo @ nat @ N3 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ).
% Suc_0_and_eq
thf(fact_4891_and__Suc__0__eq,axiom,
! [N3: nat] :
( ( bit_se5824344872417868541ns_and @ nat @ N3 @ ( suc @ ( zero_zero @ nat ) ) )
= ( modulo_modulo @ nat @ N3 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ).
% and_Suc_0_eq
thf(fact_4892_word__bitwise__1__simps_I3_J,axiom,
! [C: $tType] :
( ( type_len @ C )
=> ! [B2: num] :
( ( bit_se5824344872417868541ns_and @ ( word @ C ) @ ( one_one @ ( word @ C ) ) @ ( uminus_uminus @ ( word @ C ) @ ( numeral_numeral @ ( word @ C ) @ B2 ) ) )
= ( ring_1_of_int @ ( word @ C ) @ ( bit_se5824344872417868541ns_and @ int @ ( one_one @ int ) @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ B2 ) ) ) ) ) ) ).
% word_bitwise_1_simps(3)
thf(fact_4893_word__bitwise__1__simps_I5_J,axiom,
! [E3: $tType] :
( ( type_len @ E3 )
=> ! [A2: num] :
( ( bit_se5824344872417868541ns_and @ ( word @ E3 ) @ ( uminus_uminus @ ( word @ E3 ) @ ( numeral_numeral @ ( word @ E3 ) @ A2 ) ) @ ( one_one @ ( word @ E3 ) ) )
= ( ring_1_of_int @ ( word @ E3 ) @ ( bit_se5824344872417868541ns_and @ int @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ A2 ) ) @ ( one_one @ int ) ) ) ) ) ).
% word_bitwise_1_simps(5)
thf(fact_4894_word__and__le1,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [Y: word @ A,A2: word @ A] : ( ord_less_eq @ ( word @ A ) @ ( bit_se5824344872417868541ns_and @ ( word @ A ) @ Y @ A2 ) @ A2 ) ) ).
% word_and_le1
thf(fact_4895_word__and__le2,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [A2: word @ A,Y: word @ A] : ( ord_less_eq @ ( word @ A ) @ ( bit_se5824344872417868541ns_and @ ( word @ A ) @ A2 @ Y ) @ A2 ) ) ).
% word_and_le2
thf(fact_4896_word__and__max__word,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [X: word @ A,A2: word @ A] :
( ( X
= ( uminus_uminus @ ( word @ A ) @ ( one_one @ ( word @ A ) ) ) )
=> ( ( bit_se5824344872417868541ns_and @ ( word @ A ) @ A2 @ X )
= A2 ) ) ) ).
% word_and_max_word
thf(fact_4897_and__nat__def,axiom,
( ( bit_se5824344872417868541ns_and @ nat )
= ( ^ [M5: nat,N2: nat] : ( nat2 @ ( bit_se5824344872417868541ns_and @ int @ ( semiring_1_of_nat @ int @ M5 ) @ ( semiring_1_of_nat @ int @ N2 ) ) ) ) ) ).
% and_nat_def
thf(fact_4898_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_4899_forall__finite_I1_J,axiom,
! [P: nat > $o,I4: nat] :
( ( ord_less @ nat @ I4 @ ( zero_zero @ nat ) )
=> ( P @ I4 ) ) ).
% forall_finite(1)
thf(fact_4900_and__nat__unfold,axiom,
( ( bit_se5824344872417868541ns_and @ nat )
= ( ^ [M5: nat,N2: nat] :
( if @ nat
@ ( ( M5
= ( zero_zero @ nat ) )
| ( N2
= ( zero_zero @ nat ) ) )
@ ( zero_zero @ nat )
@ ( plus_plus @ nat @ ( times_times @ nat @ ( modulo_modulo @ nat @ M5 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ ( modulo_modulo @ nat @ N2 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( bit_se5824344872417868541ns_and @ nat @ ( divide_divide @ nat @ M5 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ ( divide_divide @ nat @ N2 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) ) ) ) ).
% and_nat_unfold
thf(fact_4901_and__nat__rec,axiom,
( ( bit_se5824344872417868541ns_and @ nat )
= ( ^ [M5: nat,N2: nat] :
( plus_plus @ nat
@ ( zero_neq_one_of_bool @ nat
@ ( ~ ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ M5 )
& ~ ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N2 ) ) )
@ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( bit_se5824344872417868541ns_and @ nat @ ( divide_divide @ nat @ M5 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ ( divide_divide @ nat @ N2 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) ) ) ).
% and_nat_rec
thf(fact_4902_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 ) )
=> ( ! [K3: int,L3: int] :
( ( accp @ ( product_prod @ int @ int ) @ bit_and_int_rel @ ( product_Pair @ int @ int @ K3 @ L3 ) )
=> ( ( ~ ( ( 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 ) ) ) ) ) )
=> ( P @ ( divide_divide @ int @ K3 @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) ) @ ( divide_divide @ int @ L3 @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) ) ) )
=> ( P @ K3 @ L3 ) ) )
=> ( P @ A0 @ A12 ) ) ) ).
% and_int.pinduct
thf(fact_4903_forall__finite_I3_J,axiom,
! [X: nat,P: nat > $o] :
( ( ! [I3: nat] :
( ( ord_less @ nat @ I3 @ ( suc @ ( suc @ X ) ) )
=> ( P @ I3 ) ) )
= ( ( P @ ( zero_zero @ nat ) )
& ! [I3: nat] :
( ( ord_less @ nat @ I3 @ ( suc @ X ) )
=> ( P @ ( suc @ I3 ) ) ) ) ) ).
% forall_finite(3)
thf(fact_4904_forall__finite_I2_J,axiom,
! [P: nat > $o] :
( ( ! [I3: nat] :
( ( ord_less @ nat @ I3 @ ( suc @ ( zero_zero @ nat ) ) )
=> ( P @ I3 ) ) )
= ( P @ ( zero_zero @ nat ) ) ) ).
% forall_finite(2)
thf(fact_4905_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_4906_take__bit__word__Bit1__eq,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [N3: num,M: num] :
( ( bit_se2584673776208193580ke_bit @ ( word @ A ) @ ( numeral_numeral @ nat @ N3 ) @ ( 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 @ N3 ) @ ( numeral_numeral @ ( word @ A ) @ M ) ) ) ) ) ) ).
% take_bit_word_Bit1_eq
thf(fact_4907_uint32_Osize__eq,axiom,
( ( size_size @ uint32 )
= ( ^ [P5: uint32] : ( numeral_numeral @ nat @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ one2 ) ) ) ) ) ) ) ) ).
% uint32.size_eq
thf(fact_4908_take__bit__of__0,axiom,
! [A: $tType] :
( ( bit_se359711467146920520ations @ A )
=> ! [N3: nat] :
( ( bit_se2584673776208193580ke_bit @ A @ N3 @ ( zero_zero @ A ) )
= ( zero_zero @ A ) ) ) ).
% take_bit_of_0
thf(fact_4909_take__bit__and,axiom,
! [A: $tType] :
( ( bit_se359711467146920520ations @ A )
=> ! [N3: nat,A2: A,B2: A] :
( ( bit_se2584673776208193580ke_bit @ A @ N3 @ ( bit_se5824344872417868541ns_and @ A @ A2 @ B2 ) )
= ( bit_se5824344872417868541ns_and @ A @ ( bit_se2584673776208193580ke_bit @ A @ N3 @ A2 ) @ ( bit_se2584673776208193580ke_bit @ A @ N3 @ B2 ) ) ) ) ).
% take_bit_and
thf(fact_4910_take__bit__0,axiom,
! [A: $tType] :
( ( bit_se359711467146920520ations @ A )
=> ! [A2: A] :
( ( bit_se2584673776208193580ke_bit @ A @ ( zero_zero @ nat ) @ A2 )
= ( zero_zero @ A ) ) ) ).
% take_bit_0
thf(fact_4911_take__bit__Suc__1,axiom,
! [A: $tType] :
( ( bit_un5681908812861735899ations @ A )
=> ! [N3: nat] :
( ( bit_se2584673776208193580ke_bit @ A @ ( suc @ N3 ) @ ( one_one @ A ) )
= ( one_one @ A ) ) ) ).
% take_bit_Suc_1
thf(fact_4912_take__bit__numeral__1,axiom,
! [A: $tType] :
( ( bit_un5681908812861735899ations @ A )
=> ! [L2: num] :
( ( bit_se2584673776208193580ke_bit @ A @ ( numeral_numeral @ nat @ L2 ) @ ( one_one @ A ) )
= ( one_one @ A ) ) ) ).
% take_bit_numeral_1
thf(fact_4913_take__bit__of__1__eq__0__iff,axiom,
! [A: $tType] :
( ( bit_un5681908812861735899ations @ A )
=> ! [N3: nat] :
( ( ( bit_se2584673776208193580ke_bit @ A @ N3 @ ( one_one @ A ) )
= ( zero_zero @ A ) )
= ( N3
= ( zero_zero @ nat ) ) ) ) ).
% take_bit_of_1_eq_0_iff
thf(fact_4914_take__bit__of__1,axiom,
! [A: $tType] :
( ( bit_se359711467146920520ations @ A )
=> ! [N3: nat] :
( ( bit_se2584673776208193580ke_bit @ A @ N3 @ ( one_one @ A ) )
= ( zero_neq_one_of_bool @ A @ ( ord_less @ nat @ ( zero_zero @ nat ) @ N3 ) ) ) ) ).
% take_bit_of_1
thf(fact_4915_even__take__bit__eq,axiom,
! [A: $tType] :
( ( bit_se359711467146920520ations @ A )
=> ! [N3: nat,A2: A] :
( ( dvd_dvd @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ ( bit_se2584673776208193580ke_bit @ A @ N3 @ A2 ) )
= ( ( N3
= ( zero_zero @ nat ) )
| ( dvd_dvd @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ A2 ) ) ) ) ).
% even_take_bit_eq
thf(fact_4916_take__bit__Suc__0,axiom,
! [A: $tType] :
( ( bit_se359711467146920520ations @ A )
=> ! [A2: A] :
( ( bit_se2584673776208193580ke_bit @ A @ ( suc @ ( zero_zero @ nat ) ) @ A2 )
= ( modulo_modulo @ A @ A2 @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) ) ) ).
% take_bit_Suc_0
thf(fact_4917_take__bit__word__Bit0__eq,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [N3: num,M: num] :
( ( bit_se2584673776208193580ke_bit @ ( word @ A ) @ ( numeral_numeral @ nat @ N3 ) @ ( numeral_numeral @ ( word @ A ) @ ( bit0 @ M ) ) )
= ( times_times @ ( word @ A ) @ ( numeral_numeral @ ( word @ A ) @ ( bit0 @ one2 ) ) @ ( bit_se2584673776208193580ke_bit @ ( word @ A ) @ ( pred_numeral @ N3 ) @ ( numeral_numeral @ ( word @ A ) @ M ) ) ) ) ) ).
% take_bit_word_Bit0_eq
thf(fact_4918_take__bit__of__exp,axiom,
! [A: $tType] :
( ( bit_un5681908812861735899ations @ A )
=> ! [M: nat,N3: nat] :
( ( bit_se2584673776208193580ke_bit @ A @ M @ ( power_power @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ N3 ) )
= ( times_times @ A @ ( zero_neq_one_of_bool @ A @ ( ord_less @ nat @ N3 @ M ) ) @ ( power_power @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ N3 ) ) ) ) ).
% take_bit_of_exp
thf(fact_4919_take__bit__of__2,axiom,
! [A: $tType] :
( ( bit_un5681908812861735899ations @ A )
=> ! [N3: nat] :
( ( bit_se2584673776208193580ke_bit @ A @ N3 @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) )
= ( times_times @ A @ ( zero_neq_one_of_bool @ A @ ( ord_less_eq @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N3 ) ) @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) ) ) ).
% take_bit_of_2
thf(fact_4920_take__bit__word__minus__Bit0__eq,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [N3: num,M: num] :
( ( bit_se2584673776208193580ke_bit @ ( word @ A ) @ ( numeral_numeral @ nat @ N3 ) @ ( 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 @ N3 ) @ ( uminus_uminus @ ( word @ A ) @ ( numeral_numeral @ ( word @ A ) @ M ) ) ) ) ) ) ).
% take_bit_word_minus_Bit0_eq
thf(fact_4921_take__bit__tightened,axiom,
! [A: $tType] :
( ( bit_se359711467146920520ations @ A )
=> ! [N3: nat,A2: A,B2: A,M: nat] :
( ( ( bit_se2584673776208193580ke_bit @ A @ N3 @ A2 )
= ( bit_se2584673776208193580ke_bit @ A @ N3 @ B2 ) )
=> ( ( ord_less_eq @ nat @ M @ N3 )
=> ( ( bit_se2584673776208193580ke_bit @ A @ M @ A2 )
= ( bit_se2584673776208193580ke_bit @ A @ M @ B2 ) ) ) ) ) ).
% take_bit_tightened
thf(fact_4922_take__bit__of__int,axiom,
! [A: $tType] :
( ( bit_ri3973907225187159222ations @ A )
=> ! [N3: nat,K: int] :
( ( bit_se2584673776208193580ke_bit @ A @ N3 @ ( ring_1_of_int @ A @ K ) )
= ( ring_1_of_int @ A @ ( bit_se2584673776208193580ke_bit @ int @ N3 @ K ) ) ) ) ).
% take_bit_of_int
thf(fact_4923_take__bit__of__nat,axiom,
! [A: $tType] :
( ( bit_se359711467146920520ations @ A )
=> ! [N3: nat,M: nat] :
( ( bit_se2584673776208193580ke_bit @ A @ N3 @ ( semiring_1_of_nat @ A @ M ) )
= ( semiring_1_of_nat @ A @ ( bit_se2584673776208193580ke_bit @ nat @ N3 @ M ) ) ) ) ).
% take_bit_of_nat
thf(fact_4924_take__bit__add,axiom,
! [A: $tType] :
( ( bit_un5681908812861735899ations @ A )
=> ! [N3: nat,A2: A,B2: A] :
( ( bit_se2584673776208193580ke_bit @ A @ N3 @ ( plus_plus @ A @ ( bit_se2584673776208193580ke_bit @ A @ N3 @ A2 ) @ ( bit_se2584673776208193580ke_bit @ A @ N3 @ B2 ) ) )
= ( bit_se2584673776208193580ke_bit @ A @ N3 @ ( plus_plus @ A @ A2 @ B2 ) ) ) ) ).
% take_bit_add
thf(fact_4925_signed__take__bit__eq__iff__take__bit__eq,axiom,
! [A: $tType] :
( ( bit_ri3973907225187159222ations @ A )
=> ! [N3: nat,A2: A,B2: A] :
( ( ( bit_ri4674362597316999326ke_bit @ A @ N3 @ A2 )
= ( bit_ri4674362597316999326ke_bit @ A @ N3 @ B2 ) )
= ( ( bit_se2584673776208193580ke_bit @ A @ ( suc @ N3 ) @ A2 )
= ( bit_se2584673776208193580ke_bit @ A @ ( suc @ N3 ) @ B2 ) ) ) ) ).
% signed_take_bit_eq_iff_take_bit_eq
thf(fact_4926_signed__take__bit__take__bit,axiom,
! [A: $tType] :
( ( bit_ri3973907225187159222ations @ A )
=> ! [M: nat,N3: nat,A2: A] :
( ( bit_ri4674362597316999326ke_bit @ A @ M @ ( bit_se2584673776208193580ke_bit @ A @ N3 @ A2 ) )
= ( if @ ( A > A ) @ ( ord_less_eq @ nat @ N3 @ M ) @ ( bit_se2584673776208193580ke_bit @ A @ N3 ) @ ( bit_ri4674362597316999326ke_bit @ A @ M ) @ A2 ) ) ) ).
% signed_take_bit_take_bit
thf(fact_4927_take__bit__unset__bit__eq,axiom,
! [A: $tType] :
( ( bit_se359711467146920520ations @ A )
=> ! [N3: nat,M: nat,A2: A] :
( ( ( ord_less_eq @ nat @ N3 @ M )
=> ( ( bit_se2584673776208193580ke_bit @ A @ N3 @ ( bit_se2638667681897837118et_bit @ A @ M @ A2 ) )
= ( bit_se2584673776208193580ke_bit @ A @ N3 @ A2 ) ) )
& ( ~ ( ord_less_eq @ nat @ N3 @ M )
=> ( ( bit_se2584673776208193580ke_bit @ A @ N3 @ ( bit_se2638667681897837118et_bit @ A @ M @ A2 ) )
= ( bit_se2638667681897837118et_bit @ A @ M @ ( bit_se2584673776208193580ke_bit @ A @ N3 @ A2 ) ) ) ) ) ) ).
% take_bit_unset_bit_eq
thf(fact_4928_take__bit__set__bit__eq,axiom,
! [A: $tType] :
( ( bit_se359711467146920520ations @ A )
=> ! [N3: nat,M: nat,A2: A] :
( ( ( ord_less_eq @ nat @ N3 @ M )
=> ( ( bit_se2584673776208193580ke_bit @ A @ N3 @ ( bit_se5668285175392031749et_bit @ A @ M @ A2 ) )
= ( bit_se2584673776208193580ke_bit @ A @ N3 @ A2 ) ) )
& ( ~ ( ord_less_eq @ nat @ N3 @ M )
=> ( ( bit_se2584673776208193580ke_bit @ A @ N3 @ ( bit_se5668285175392031749et_bit @ A @ M @ A2 ) )
= ( bit_se5668285175392031749et_bit @ A @ M @ ( bit_se2584673776208193580ke_bit @ A @ N3 @ A2 ) ) ) ) ) ) ).
% take_bit_set_bit_eq
thf(fact_4929_take__bit__flip__bit__eq,axiom,
! [A: $tType] :
( ( bit_se359711467146920520ations @ A )
=> ! [N3: nat,M: nat,A2: A] :
( ( ( ord_less_eq @ nat @ N3 @ M )
=> ( ( bit_se2584673776208193580ke_bit @ A @ N3 @ ( bit_se8732182000553998342ip_bit @ A @ M @ A2 ) )
= ( bit_se2584673776208193580ke_bit @ A @ N3 @ A2 ) ) )
& ( ~ ( ord_less_eq @ nat @ N3 @ M )
=> ( ( bit_se2584673776208193580ke_bit @ A @ N3 @ ( bit_se8732182000553998342ip_bit @ A @ M @ A2 ) )
= ( bit_se8732182000553998342ip_bit @ A @ M @ ( bit_se2584673776208193580ke_bit @ A @ N3 @ A2 ) ) ) ) ) ) ).
% take_bit_flip_bit_eq
thf(fact_4930_take__bit__signed__take__bit,axiom,
! [A: $tType] :
( ( bit_ri3973907225187159222ations @ A )
=> ! [M: nat,N3: nat,A2: A] :
( ( ord_less_eq @ nat @ M @ ( suc @ N3 ) )
=> ( ( bit_se2584673776208193580ke_bit @ A @ M @ ( bit_ri4674362597316999326ke_bit @ A @ N3 @ A2 ) )
= ( bit_se2584673776208193580ke_bit @ A @ M @ A2 ) ) ) ) ).
% take_bit_signed_take_bit
thf(fact_4931_take__bit__Suc__bit0,axiom,
! [A: $tType] :
( ( bit_un5681908812861735899ations @ A )
=> ! [N3: nat,K: num] :
( ( bit_se2584673776208193580ke_bit @ A @ ( suc @ N3 ) @ ( numeral_numeral @ A @ ( bit0 @ K ) ) )
= ( times_times @ A @ ( bit_se2584673776208193580ke_bit @ A @ N3 @ ( numeral_numeral @ A @ K ) ) @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) ) ) ).
% take_bit_Suc_bit0
thf(fact_4932_take__bit__eq__mod,axiom,
! [A: $tType] :
( ( bit_se359711467146920520ations @ A )
=> ( ( bit_se2584673776208193580ke_bit @ A )
= ( ^ [N2: nat,A6: A] : ( modulo_modulo @ A @ A6 @ ( power_power @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ N2 ) ) ) ) ) ).
% take_bit_eq_mod
thf(fact_4933_take__bit__eq__0__iff,axiom,
! [A: $tType] :
( ( bit_se359711467146920520ations @ A )
=> ! [N3: nat,A2: A] :
( ( ( bit_se2584673776208193580ke_bit @ A @ N3 @ A2 )
= ( zero_zero @ A ) )
= ( dvd_dvd @ A @ ( power_power @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ N3 ) @ A2 ) ) ) ).
% take_bit_eq_0_iff
thf(fact_4934_bin__last__bintrunc,axiom,
! [A: $tType] :
( ( bit_se359711467146920520ations @ A )
=> ! [L2: nat,N3: A] :
( ( ~ ( dvd_dvd @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ ( bit_se2584673776208193580ke_bit @ A @ L2 @ N3 ) ) )
= ( ( ord_less @ nat @ ( zero_zero @ nat ) @ L2 )
& ~ ( dvd_dvd @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ N3 ) ) ) ) ).
% bin_last_bintrunc
thf(fact_4935_take__bit__numeral__bit0,axiom,
! [A: $tType] :
( ( bit_un5681908812861735899ations @ A )
=> ! [L2: num,K: num] :
( ( bit_se2584673776208193580ke_bit @ A @ ( numeral_numeral @ nat @ L2 ) @ ( numeral_numeral @ A @ ( bit0 @ K ) ) )
= ( times_times @ A @ ( bit_se2584673776208193580ke_bit @ A @ ( pred_numeral @ L2 ) @ ( numeral_numeral @ A @ K ) ) @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) ) ) ).
% take_bit_numeral_bit0
thf(fact_4936_take__bit__Suc__bit1,axiom,
! [A: $tType] :
( ( bit_un5681908812861735899ations @ A )
=> ! [N3: nat,K: num] :
( ( bit_se2584673776208193580ke_bit @ A @ ( suc @ N3 ) @ ( numeral_numeral @ A @ ( bit1 @ K ) ) )
= ( plus_plus @ A @ ( times_times @ A @ ( bit_se2584673776208193580ke_bit @ A @ N3 @ ( numeral_numeral @ A @ K ) ) @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) @ ( one_one @ A ) ) ) ) ).
% take_bit_Suc_bit1
thf(fact_4937_take__bit__Suc__minus__1__eq,axiom,
! [A: $tType] :
( ( bit_ri3973907225187159222ations @ A )
=> ! [N3: nat] :
( ( bit_se2584673776208193580ke_bit @ A @ ( suc @ N3 ) @ ( uminus_uminus @ A @ ( one_one @ A ) ) )
= ( minus_minus @ A @ ( power_power @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ ( suc @ N3 ) ) @ ( one_one @ A ) ) ) ) ).
% take_bit_Suc_minus_1_eq
thf(fact_4938_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_4939_take__bit__Suc,axiom,
! [A: $tType] :
( ( bit_se359711467146920520ations @ A )
=> ! [N3: nat,A2: A] :
( ( bit_se2584673776208193580ke_bit @ A @ ( suc @ N3 ) @ A2 )
= ( plus_plus @ A @ ( times_times @ A @ ( bit_se2584673776208193580ke_bit @ A @ N3 @ ( divide_divide @ A @ A2 @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) ) @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) @ ( modulo_modulo @ A @ A2 @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) ) ) ) ).
% take_bit_Suc
thf(fact_4940_stable__imp__take__bit__eq,axiom,
! [A: $tType] :
( ( bit_se359711467146920520ations @ A )
=> ! [A2: A,N3: nat] :
( ( ( divide_divide @ A @ A2 @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) )
= A2 )
=> ( ( ( dvd_dvd @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ A2 )
=> ( ( bit_se2584673776208193580ke_bit @ A @ N3 @ A2 )
= ( zero_zero @ A ) ) )
& ( ~ ( dvd_dvd @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ A2 )
=> ( ( bit_se2584673776208193580ke_bit @ A @ N3 @ A2 )
= ( minus_minus @ A @ ( power_power @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ N3 ) @ ( one_one @ A ) ) ) ) ) ) ) ).
% stable_imp_take_bit_eq
thf(fact_4941_take__bit__numeral__bit1,axiom,
! [A: $tType] :
( ( bit_un5681908812861735899ations @ A )
=> ! [L2: num,K: num] :
( ( bit_se2584673776208193580ke_bit @ A @ ( numeral_numeral @ nat @ L2 ) @ ( numeral_numeral @ A @ ( bit1 @ K ) ) )
= ( plus_plus @ A @ ( times_times @ A @ ( bit_se2584673776208193580ke_bit @ A @ ( pred_numeral @ L2 ) @ ( numeral_numeral @ A @ K ) ) @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) @ ( one_one @ A ) ) ) ) ).
% take_bit_numeral_bit1
thf(fact_4942_take__bit__rec,axiom,
! [A: $tType] :
( ( bit_se359711467146920520ations @ A )
=> ( ( bit_se2584673776208193580ke_bit @ A )
= ( ^ [N2: nat,A6: A] :
( if @ A
@ ( N2
= ( zero_zero @ nat ) )
@ ( zero_zero @ A )
@ ( plus_plus @ A @ ( times_times @ A @ ( bit_se2584673776208193580ke_bit @ A @ ( minus_minus @ nat @ N2 @ ( 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_4943_take__bit__word__minus__Bit1__eq,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [N3: num,M: num] :
( ( bit_se2584673776208193580ke_bit @ ( word @ A ) @ ( numeral_numeral @ nat @ N3 ) @ ( 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 @ N3 ) @ ( uminus_uminus @ ( word @ A ) @ ( numeral_numeral @ ( word @ A ) @ ( inc @ M ) ) ) ) ) ) ) ) ).
% take_bit_word_minus_Bit1_eq
thf(fact_4944_and__mask__arith_H,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [N3: nat,W: word @ A] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ N3 )
=> ( ( bit_se5824344872417868541ns_and @ ( word @ A ) @ W @ ( bit_se2239418461657761734s_mask @ ( word @ A ) @ N3 ) )
= ( 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 ) @ N3 ) ) ) @ ( power_power @ ( word @ A ) @ ( numeral_numeral @ ( word @ A ) @ ( bit0 @ one2 ) ) @ ( minus_minus @ nat @ ( size_size @ ( word @ A ) @ W ) @ N3 ) ) ) ) ) ) ).
% and_mask_arith'
thf(fact_4945_divmod__algorithm__code_I6_J,axiom,
! [A: $tType] :
( ( unique1627219031080169319umeral @ A )
=> ! [M: num,N3: num] :
( ( unique8689654367752047608divmod @ A @ ( bit1 @ M ) @ ( bit0 @ N3 ) )
= ( product_case_prod @ A @ A @ ( product_prod @ A @ A )
@ ^ [Q4: A,R5: A] : ( product_Pair @ A @ A @ Q4 @ ( plus_plus @ A @ ( times_times @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ R5 ) @ ( one_one @ A ) ) )
@ ( unique8689654367752047608divmod @ A @ M @ N3 ) ) ) ) ).
% divmod_algorithm_code(6)
thf(fact_4946_case__prod__conv,axiom,
! [B: $tType,A: $tType,C: $tType,F2: B > C > A,A2: B,B2: C] :
( ( product_case_prod @ B @ C @ A @ F2 @ ( product_Pair @ B @ C @ A2 @ B2 ) )
= ( F2 @ A2 @ B2 ) ) ).
% case_prod_conv
thf(fact_4947_mask__0,axiom,
! [A: $tType] :
( ( bit_se359711467146920520ations @ A )
=> ( ( bit_se2239418461657761734s_mask @ A @ ( zero_zero @ nat ) )
= ( zero_zero @ A ) ) ) ).
% mask_0
thf(fact_4948_mask__eq__0__iff,axiom,
! [A: $tType] :
( ( bit_se359711467146920520ations @ A )
=> ! [N3: nat] :
( ( ( bit_se2239418461657761734s_mask @ A @ N3 )
= ( zero_zero @ A ) )
= ( N3
= ( zero_zero @ nat ) ) ) ) ).
% mask_eq_0_iff
thf(fact_4949_pred__numeral__inc,axiom,
! [K: num] :
( ( pred_numeral @ ( inc @ K ) )
= ( numeral_numeral @ nat @ K ) ) ).
% pred_numeral_inc
thf(fact_4950_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_4951_take__bit__minus__one__eq__mask,axiom,
! [A: $tType] :
( ( bit_ri3973907225187159222ations @ A )
=> ! [N3: nat] :
( ( bit_se2584673776208193580ke_bit @ A @ N3 @ ( uminus_uminus @ A @ ( one_one @ A ) ) )
= ( bit_se2239418461657761734s_mask @ A @ N3 ) ) ) ).
% take_bit_minus_one_eq_mask
thf(fact_4952_of__nat__nat__take__bit__eq,axiom,
! [A: $tType] :
( ( ring_1 @ A )
=> ! [N3: nat,K: int] :
( ( semiring_1_of_nat @ A @ ( nat2 @ ( bit_se2584673776208193580ke_bit @ int @ N3 @ K ) ) )
= ( ring_1_of_int @ A @ ( bit_se2584673776208193580ke_bit @ int @ N3 @ K ) ) ) ) ).
% of_nat_nat_take_bit_eq
thf(fact_4953_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_4954_add__neg__numeral__special_I5_J,axiom,
! [A: $tType] :
( ( neg_numeral @ A )
=> ! [N3: num] :
( ( plus_plus @ A @ ( uminus_uminus @ A @ ( one_one @ A ) ) @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ N3 ) ) )
= ( uminus_uminus @ A @ ( numeral_numeral @ A @ ( inc @ N3 ) ) ) ) ) ).
% add_neg_numeral_special(5)
thf(fact_4955_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_4956_diff__numeral__special_I5_J,axiom,
! [A: $tType] :
( ( neg_numeral @ A )
=> ! [N3: num] :
( ( minus_minus @ A @ ( uminus_uminus @ A @ ( one_one @ A ) ) @ ( numeral_numeral @ A @ N3 ) )
= ( uminus_uminus @ A @ ( numeral_numeral @ A @ ( inc @ N3 ) ) ) ) ) ).
% diff_numeral_special(5)
thf(fact_4957_take__bit__of__Suc__0,axiom,
! [N3: nat] :
( ( bit_se2584673776208193580ke_bit @ nat @ N3 @ ( suc @ ( zero_zero @ nat ) ) )
= ( zero_neq_one_of_bool @ nat @ ( ord_less @ nat @ ( zero_zero @ nat ) @ N3 ) ) ) ).
% take_bit_of_Suc_0
thf(fact_4958_divmod__algorithm__code_I5_J,axiom,
! [A: $tType] :
( ( unique1627219031080169319umeral @ A )
=> ! [M: num,N3: num] :
( ( unique8689654367752047608divmod @ A @ ( bit0 @ M ) @ ( bit0 @ N3 ) )
= ( product_case_prod @ A @ A @ ( product_prod @ A @ A )
@ ^ [Q4: A,R5: A] : ( product_Pair @ A @ A @ Q4 @ ( times_times @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ R5 ) )
@ ( unique8689654367752047608divmod @ A @ M @ N3 ) ) ) ) ).
% divmod_algorithm_code(5)
thf(fact_4959_take__bit__mult,axiom,
! [N3: nat,K: int,L2: int] :
( ( bit_se2584673776208193580ke_bit @ int @ N3 @ ( times_times @ int @ ( bit_se2584673776208193580ke_bit @ int @ N3 @ K ) @ ( bit_se2584673776208193580ke_bit @ int @ N3 @ L2 ) ) )
= ( bit_se2584673776208193580ke_bit @ int @ N3 @ ( times_times @ int @ K @ L2 ) ) ) ).
% take_bit_mult
thf(fact_4960_take__bit__diff,axiom,
! [N3: nat,K: int,L2: int] :
( ( bit_se2584673776208193580ke_bit @ int @ N3 @ ( minus_minus @ int @ ( bit_se2584673776208193580ke_bit @ int @ N3 @ K ) @ ( bit_se2584673776208193580ke_bit @ int @ N3 @ L2 ) ) )
= ( bit_se2584673776208193580ke_bit @ int @ N3 @ ( minus_minus @ int @ K @ L2 ) ) ) ).
% take_bit_diff
thf(fact_4961_mask__bin,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ( ( bit_se2239418461657761734s_mask @ ( word @ A ) )
= ( ^ [N2: nat] : ( ring_1_of_int @ ( word @ A ) @ ( bit_se2584673776208193580ke_bit @ int @ N2 @ ( uminus_uminus @ int @ ( one_one @ int ) ) ) ) ) ) ) ).
% mask_bin
thf(fact_4962_take__bit__minus,axiom,
! [N3: nat,K: int] :
( ( bit_se2584673776208193580ke_bit @ int @ N3 @ ( uminus_uminus @ int @ ( bit_se2584673776208193580ke_bit @ int @ N3 @ K ) ) )
= ( bit_se2584673776208193580ke_bit @ int @ N3 @ ( uminus_uminus @ int @ K ) ) ) ).
% take_bit_minus
thf(fact_4963_take__bit__nat__less__eq__self,axiom,
! [N3: nat,M: nat] : ( ord_less_eq @ nat @ ( bit_se2584673776208193580ke_bit @ nat @ N3 @ M ) @ M ) ).
% take_bit_nat_less_eq_self
thf(fact_4964_take__bit__tightened__less__eq__nat,axiom,
! [M: nat,N3: nat,Q2: nat] :
( ( ord_less_eq @ nat @ M @ N3 )
=> ( ord_less_eq @ nat @ ( bit_se2584673776208193580ke_bit @ nat @ M @ Q2 ) @ ( bit_se2584673776208193580ke_bit @ nat @ N3 @ Q2 ) ) ) ).
% take_bit_tightened_less_eq_nat
thf(fact_4965_nat__take__bit__eq,axiom,
! [K: int,N3: nat] :
( ( ord_less_eq @ int @ ( zero_zero @ int ) @ K )
=> ( ( nat2 @ ( bit_se2584673776208193580ke_bit @ int @ N3 @ K ) )
= ( bit_se2584673776208193580ke_bit @ nat @ N3 @ ( nat2 @ K ) ) ) ) ).
% nat_take_bit_eq
thf(fact_4966_take__bit__nat__eq,axiom,
! [K: int,N3: nat] :
( ( ord_less_eq @ int @ ( zero_zero @ int ) @ K )
=> ( ( bit_se2584673776208193580ke_bit @ nat @ N3 @ ( nat2 @ K ) )
= ( nat2 @ ( bit_se2584673776208193580ke_bit @ int @ N3 @ K ) ) ) ) ).
% take_bit_nat_eq
thf(fact_4967_old_Oprod_Ocase,axiom,
! [A: $tType,C: $tType,B: $tType,F2: A > B > C,X15: A,X22: B] :
( ( product_case_prod @ A @ B @ C @ F2 @ ( product_Pair @ A @ B @ X15 @ X22 ) )
= ( F2 @ X15 @ X22 ) ) ).
% old.prod.case
thf(fact_4968_of__nat__mask__eq,axiom,
! [A: $tType] :
( ( bit_se359711467146920520ations @ A )
=> ! [N3: nat] :
( ( semiring_1_of_nat @ A @ ( bit_se2239418461657761734s_mask @ nat @ N3 ) )
= ( bit_se2239418461657761734s_mask @ A @ N3 ) ) ) ).
% of_nat_mask_eq
thf(fact_4969_of__int__mask__eq,axiom,
! [A: $tType] :
( ( bit_ri3973907225187159222ations @ A )
=> ! [N3: nat] :
( ( ring_1_of_int @ A @ ( bit_se2239418461657761734s_mask @ int @ N3 ) )
= ( bit_se2239418461657761734s_mask @ A @ N3 ) ) ) ).
% of_int_mask_eq
thf(fact_4970_nested__case__prod__simp,axiom,
! [A: $tType,D5: $tType,C: $tType,B: $tType] :
( ( product_case_prod @ B @ C @ ( D5 > A ) )
= ( ^ [F7: B > C > D5 > A,X2: product_prod @ B @ C,Y2: D5] :
( product_case_prod @ B @ C @ A
@ ^ [A6: B,B5: C] : ( F7 @ A6 @ B5 @ Y2 )
@ X2 ) ) ) ).
% nested_case_prod_simp
thf(fact_4971_ex__mask__1,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ? [X3: nat] :
( ( bit_se2239418461657761734s_mask @ ( word @ A ) @ X3 )
= ( one_one @ ( word @ A ) ) ) ) ).
% ex_mask_1
thf(fact_4972_num__induct,axiom,
! [P: num > $o,X: num] :
( ( P @ one2 )
=> ( ! [X3: num] :
( ( P @ X3 )
=> ( P @ ( inc @ X3 ) ) )
=> ( P @ X ) ) ) ).
% num_induct
thf(fact_4973_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 ) )
=> ~ ! [X3: B,Y3: C] :
( ( Z
= ( product_Pair @ B @ C @ X3 @ Y3 ) )
=> ~ ( Q @ ( P @ X3 @ Y3 ) ) ) ) ).
% case_prodE2
thf(fact_4974_case__prod__eta,axiom,
! [C: $tType,B: $tType,A: $tType,F2: ( product_prod @ A @ B ) > C] :
( ( product_case_prod @ A @ B @ C
@ ^ [X2: A,Y2: B] : ( F2 @ ( product_Pair @ A @ B @ X2 @ Y2 ) ) )
= F2 ) ).
% case_prod_eta
thf(fact_4975_cond__case__prod__eta,axiom,
! [C: $tType,B: $tType,A: $tType,F2: A > B > C,G: ( product_prod @ A @ B ) > C] :
( ! [X3: A,Y3: B] :
( ( F2 @ X3 @ Y3 )
= ( G @ ( product_Pair @ A @ B @ X3 @ Y3 ) ) )
=> ( ( product_case_prod @ A @ B @ C @ F2 )
= G ) ) ).
% cond_case_prod_eta
thf(fact_4976_add__inc,axiom,
! [X: num,Y: num] :
( ( plus_plus @ num @ X @ ( inc @ Y ) )
= ( inc @ ( plus_plus @ num @ X @ Y ) ) ) ).
% add_inc
thf(fact_4977_mask__twice2,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [N3: nat,M: nat,X: word @ A] :
( ( ord_less_eq @ nat @ N3 @ M )
=> ( ( bit_se5824344872417868541ns_and @ ( word @ A ) @ ( bit_se5824344872417868541ns_and @ ( word @ A ) @ X @ ( bit_se2239418461657761734s_mask @ ( word @ A ) @ M ) ) @ ( bit_se2239418461657761734s_mask @ ( word @ A ) @ N3 ) )
= ( bit_se5824344872417868541ns_and @ ( word @ A ) @ X @ ( bit_se2239418461657761734s_mask @ ( word @ A ) @ N3 ) ) ) ) ) ).
% mask_twice2
thf(fact_4978_le__mask__imp__and__mask,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [X: word @ A,N3: nat] :
( ( ord_less_eq @ ( word @ A ) @ X @ ( bit_se2239418461657761734s_mask @ ( word @ A ) @ N3 ) )
=> ( ( bit_se5824344872417868541ns_and @ ( word @ A ) @ X @ ( bit_se2239418461657761734s_mask @ ( word @ A ) @ N3 ) )
= X ) ) ) ).
% le_mask_imp_and_mask
thf(fact_4979_and__mask__eq__iff__le__mask,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [W: word @ A,N3: nat] :
( ( ( bit_se5824344872417868541ns_and @ ( word @ A ) @ W @ ( bit_se2239418461657761734s_mask @ ( word @ A ) @ N3 ) )
= W )
= ( ord_less_eq @ ( word @ A ) @ W @ ( bit_se2239418461657761734s_mask @ ( word @ A ) @ N3 ) ) ) ) ).
% and_mask_eq_iff_le_mask
thf(fact_4980_mask__eqs_I3_J,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [A2: word @ A,N3: nat,B2: word @ A] :
( ( bit_se5824344872417868541ns_and @ ( word @ A ) @ ( minus_minus @ ( word @ A ) @ ( bit_se5824344872417868541ns_and @ ( word @ A ) @ A2 @ ( bit_se2239418461657761734s_mask @ ( word @ A ) @ N3 ) ) @ B2 ) @ ( bit_se2239418461657761734s_mask @ ( word @ A ) @ N3 ) )
= ( bit_se5824344872417868541ns_and @ ( word @ A ) @ ( minus_minus @ ( word @ A ) @ A2 @ B2 ) @ ( bit_se2239418461657761734s_mask @ ( word @ A ) @ N3 ) ) ) ) ).
% mask_eqs(3)
thf(fact_4981_mask__eqs_I4_J,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [A2: word @ A,B2: word @ A,N3: nat] :
( ( bit_se5824344872417868541ns_and @ ( word @ A ) @ ( minus_minus @ ( word @ A ) @ A2 @ ( bit_se5824344872417868541ns_and @ ( word @ A ) @ B2 @ ( bit_se2239418461657761734s_mask @ ( word @ A ) @ N3 ) ) ) @ ( bit_se2239418461657761734s_mask @ ( word @ A ) @ N3 ) )
= ( bit_se5824344872417868541ns_and @ ( word @ A ) @ ( minus_minus @ ( word @ A ) @ A2 @ B2 ) @ ( bit_se2239418461657761734s_mask @ ( word @ A ) @ N3 ) ) ) ) ).
% mask_eqs(4)
thf(fact_4982_mask__eqs_I8_J,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [A2: word @ A,N3: nat,B2: word @ A] :
( ( bit_se5824344872417868541ns_and @ ( word @ A ) @ ( minus_minus @ ( word @ A ) @ ( bit_se5824344872417868541ns_and @ ( word @ A ) @ A2 @ ( bit_se2239418461657761734s_mask @ ( word @ A ) @ N3 ) ) @ ( bit_se5824344872417868541ns_and @ ( word @ A ) @ B2 @ ( bit_se2239418461657761734s_mask @ ( word @ A ) @ N3 ) ) ) @ ( bit_se2239418461657761734s_mask @ ( word @ A ) @ N3 ) )
= ( bit_se5824344872417868541ns_and @ ( word @ A ) @ ( minus_minus @ ( word @ A ) @ A2 @ B2 ) @ ( bit_se2239418461657761734s_mask @ ( word @ A ) @ N3 ) ) ) ) ).
% mask_eqs(8)
thf(fact_4983_take__bit__tightened__less__eq__int,axiom,
! [M: nat,N3: nat,K: int] :
( ( ord_less_eq @ nat @ M @ N3 )
=> ( ord_less_eq @ int @ ( bit_se2584673776208193580ke_bit @ int @ M @ K ) @ ( bit_se2584673776208193580ke_bit @ int @ N3 @ K ) ) ) ).
% take_bit_tightened_less_eq_int
thf(fact_4984_take__bit__nonnegative,axiom,
! [N3: nat,K: int] : ( ord_less_eq @ int @ ( zero_zero @ int ) @ ( bit_se2584673776208193580ke_bit @ int @ N3 @ K ) ) ).
% take_bit_nonnegative
thf(fact_4985_take__bit__int__less__eq__self__iff,axiom,
! [N3: nat,K: int] :
( ( ord_less_eq @ int @ ( bit_se2584673776208193580ke_bit @ int @ N3 @ K ) @ K )
= ( ord_less_eq @ int @ ( zero_zero @ int ) @ K ) ) ).
% take_bit_int_less_eq_self_iff
thf(fact_4986_take__bit__int__greater__self__iff,axiom,
! [K: int,N3: nat] :
( ( ord_less @ int @ K @ ( bit_se2584673776208193580ke_bit @ int @ N3 @ K ) )
= ( ord_less @ int @ K @ ( zero_zero @ int ) ) ) ).
% take_bit_int_greater_self_iff
thf(fact_4987_not__take__bit__negative,axiom,
! [N3: nat,K: int] :
~ ( ord_less @ int @ ( bit_se2584673776208193580ke_bit @ int @ N3 @ K ) @ ( zero_zero @ int ) ) ).
% not_take_bit_negative
thf(fact_4988_less__eq__mask__iff__take__bit__eq__self,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [W: word @ A,N3: nat] :
( ( ord_less_eq @ ( word @ A ) @ W @ ( bit_se2239418461657761734s_mask @ ( word @ A ) @ N3 ) )
= ( ( bit_se2584673776208193580ke_bit @ ( word @ A ) @ N3 @ W )
= W ) ) ) ).
% less_eq_mask_iff_take_bit_eq_self
thf(fact_4989_mask__1,axiom,
! [A: $tType] :
( ( bit_se359711467146920520ations @ A )
=> ( ( bit_se2239418461657761734s_mask @ A @ ( one_one @ nat ) )
= ( one_one @ A ) ) ) ).
% mask_1
thf(fact_4990_inc_Osimps_I1_J,axiom,
( ( inc @ one2 )
= ( bit0 @ one2 ) ) ).
% inc.simps(1)
thf(fact_4991_inc_Osimps_I2_J,axiom,
! [X: num] :
( ( inc @ ( bit0 @ X ) )
= ( bit1 @ X ) ) ).
% inc.simps(2)
thf(fact_4992_inc_Osimps_I3_J,axiom,
! [X: num] :
( ( inc @ ( bit1 @ X ) )
= ( bit0 @ ( inc @ X ) ) ) ).
% inc.simps(3)
thf(fact_4993_take__bit__eq__mask,axiom,
! [A: $tType] :
( ( bit_se359711467146920520ations @ A )
=> ( ( bit_se2584673776208193580ke_bit @ A )
= ( ^ [N2: nat,A6: A] : ( bit_se5824344872417868541ns_and @ A @ A6 @ ( bit_se2239418461657761734s_mask @ A @ N2 ) ) ) ) ) ).
% take_bit_eq_mask
thf(fact_4994_add__One,axiom,
! [X: num] :
( ( plus_plus @ num @ X @ one2 )
= ( inc @ X ) ) ).
% add_One
thf(fact_4995_inc__BitM__eq,axiom,
! [N3: num] :
( ( inc @ ( bitM @ N3 ) )
= ( bit0 @ N3 ) ) ).
% inc_BitM_eq
thf(fact_4996_BitM__inc__eq,axiom,
! [N3: num] :
( ( bitM @ ( inc @ N3 ) )
= ( bit1 @ N3 ) ) ).
% BitM_inc_eq
thf(fact_4997_mult__inc,axiom,
! [X: num,Y: num] :
( ( times_times @ num @ X @ ( inc @ Y ) )
= ( plus_plus @ num @ ( times_times @ num @ X @ Y ) @ X ) ) ).
% mult_inc
thf(fact_4998_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_4999_take__bit__decr__eq,axiom,
! [N3: nat,K: int] :
( ( ( bit_se2584673776208193580ke_bit @ int @ N3 @ K )
!= ( zero_zero @ int ) )
=> ( ( bit_se2584673776208193580ke_bit @ int @ N3 @ ( minus_minus @ int @ K @ ( one_one @ int ) ) )
= ( minus_minus @ int @ ( bit_se2584673776208193580ke_bit @ int @ N3 @ K ) @ ( one_one @ int ) ) ) ) ).
% take_bit_decr_eq
thf(fact_5000_numeral__inc,axiom,
! [A: $tType] :
( ( numeral @ A )
=> ! [X: num] :
( ( numeral_numeral @ A @ ( inc @ X ) )
= ( plus_plus @ A @ ( numeral_numeral @ A @ X ) @ ( one_one @ A ) ) ) ) ).
% numeral_inc
thf(fact_5001_take__bit__nat__eq__self,axiom,
! [M: nat,N3: nat] :
( ( ord_less @ nat @ M @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N3 ) )
=> ( ( bit_se2584673776208193580ke_bit @ nat @ N3 @ M )
= M ) ) ).
% take_bit_nat_eq_self
thf(fact_5002_take__bit__nat__less__exp,axiom,
! [N3: nat,M: nat] : ( ord_less @ nat @ ( bit_se2584673776208193580ke_bit @ nat @ N3 @ M ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N3 ) ) ).
% take_bit_nat_less_exp
thf(fact_5003_take__bit__nat__eq__self__iff,axiom,
! [N3: nat,M: nat] :
( ( ( bit_se2584673776208193580ke_bit @ nat @ N3 @ M )
= M )
= ( ord_less @ nat @ M @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N3 ) ) ) ).
% take_bit_nat_eq_self_iff
thf(fact_5004_take__bit__nat__def,axiom,
( ( bit_se2584673776208193580ke_bit @ nat )
= ( ^ [N2: nat,M5: nat] : ( modulo_modulo @ nat @ M5 @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N2 ) ) ) ) ).
% take_bit_nat_def
thf(fact_5005_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_5006_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_5007_take__bit__Suc__minus__bit1,axiom,
! [N3: nat,K: num] :
( ( bit_se2584673776208193580ke_bit @ int @ ( suc @ N3 ) @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ ( bit1 @ K ) ) ) )
= ( plus_plus @ int @ ( times_times @ int @ ( bit_se2584673776208193580ke_bit @ int @ N3 @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ ( inc @ K ) ) ) ) @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) ) @ ( one_one @ int ) ) ) ).
% take_bit_Suc_minus_bit1
thf(fact_5008_take__bit__int__less__exp,axiom,
! [N3: nat,K: int] : ( ord_less @ int @ ( bit_se2584673776208193580ke_bit @ int @ N3 @ K ) @ ( power_power @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ N3 ) ) ).
% take_bit_int_less_exp
thf(fact_5009_take__bit__int__def,axiom,
( ( bit_se2584673776208193580ke_bit @ int )
= ( ^ [N2: nat,K2: int] : ( modulo_modulo @ int @ K2 @ ( power_power @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ N2 ) ) ) ) ).
% take_bit_int_def
thf(fact_5010_take__bit__numeral__minus__bit1,axiom,
! [L2: num,K: num] :
( ( bit_se2584673776208193580ke_bit @ int @ ( numeral_numeral @ nat @ L2 ) @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ ( bit1 @ K ) ) ) )
= ( plus_plus @ int @ ( times_times @ int @ ( bit_se2584673776208193580ke_bit @ int @ ( pred_numeral @ L2 ) @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ ( inc @ K ) ) ) ) @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) ) @ ( one_one @ int ) ) ) ).
% take_bit_numeral_minus_bit1
thf(fact_5011_take__bit__nat__less__self__iff,axiom,
! [N3: nat,M: nat] :
( ( ord_less @ nat @ ( bit_se2584673776208193580ke_bit @ nat @ N3 @ M ) @ M )
= ( ord_less_eq @ nat @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N3 ) @ M ) ) ).
% take_bit_nat_less_self_iff
thf(fact_5012_semiring__bit__operations__class_Oeven__mask__iff,axiom,
! [A: $tType] :
( ( bit_se359711467146920520ations @ A )
=> ! [N3: nat] :
( ( dvd_dvd @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ ( bit_se2239418461657761734s_mask @ A @ N3 ) )
= ( N3
= ( zero_zero @ nat ) ) ) ) ).
% semiring_bit_operations_class.even_mask_iff
thf(fact_5013_mask__plus__1,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [N3: nat] :
( ( plus_plus @ ( word @ A ) @ ( bit_se2239418461657761734s_mask @ ( word @ A ) @ N3 ) @ ( one_one @ ( word @ A ) ) )
= ( power_power @ ( word @ A ) @ ( numeral_numeral @ ( word @ A ) @ ( bit0 @ one2 ) ) @ N3 ) ) ) ).
% mask_plus_1
thf(fact_5014_less__mask__eq,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [X: word @ A,N3: nat] :
( ( ord_less @ ( word @ A ) @ X @ ( power_power @ ( word @ A ) @ ( numeral_numeral @ ( word @ A ) @ ( bit0 @ one2 ) ) @ N3 ) )
=> ( ( bit_se5824344872417868541ns_and @ ( word @ A ) @ X @ ( bit_se2239418461657761734s_mask @ ( word @ A ) @ N3 ) )
= X ) ) ) ).
% less_mask_eq
thf(fact_5015_mask__eq__decr__exp,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ( ( bit_se2239418461657761734s_mask @ ( word @ A ) )
= ( ^ [N2: nat] : ( minus_minus @ ( word @ A ) @ ( power_power @ ( word @ A ) @ ( numeral_numeral @ ( word @ A ) @ ( bit0 @ one2 ) ) @ N2 ) @ ( one_one @ ( word @ A ) ) ) ) ) ) ).
% mask_eq_decr_exp
thf(fact_5016_take__bit__Suc__minus__bit0,axiom,
! [N3: nat,K: num] :
( ( bit_se2584673776208193580ke_bit @ int @ ( suc @ N3 ) @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ ( bit0 @ K ) ) ) )
= ( times_times @ int @ ( bit_se2584673776208193580ke_bit @ int @ N3 @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ K ) ) ) @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) ) ) ).
% take_bit_Suc_minus_bit0
thf(fact_5017_take__bit__int__greater__eq__self__iff,axiom,
! [K: int,N3: nat] :
( ( ord_less_eq @ int @ K @ ( bit_se2584673776208193580ke_bit @ int @ N3 @ K ) )
= ( ord_less @ int @ K @ ( power_power @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ N3 ) ) ) ).
% take_bit_int_greater_eq_self_iff
thf(fact_5018_take__bit__int__less__self__iff,axiom,
! [N3: nat,K: int] :
( ( ord_less @ int @ ( bit_se2584673776208193580ke_bit @ int @ N3 @ K ) @ K )
= ( ord_less_eq @ int @ ( power_power @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ N3 ) @ K ) ) ).
% take_bit_int_less_self_iff
thf(fact_5019_mask__eq__exp__minus__1,axiom,
! [A: $tType] :
( ( bit_se359711467146920520ations @ A )
=> ( ( bit_se2239418461657761734s_mask @ A )
= ( ^ [N2: nat] : ( minus_minus @ A @ ( power_power @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ N2 ) @ ( one_one @ A ) ) ) ) ) ).
% mask_eq_exp_minus_1
thf(fact_5020_mask__Suc__rec,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [N3: nat] :
( ( bit_se2239418461657761734s_mask @ ( word @ A ) @ ( suc @ N3 ) )
= ( plus_plus @ ( word @ A ) @ ( times_times @ ( word @ A ) @ ( numeral_numeral @ ( word @ A ) @ ( bit0 @ one2 ) ) @ ( bit_se2239418461657761734s_mask @ ( word @ A ) @ N3 ) ) @ ( one_one @ ( word @ A ) ) ) ) ) ).
% mask_Suc_rec
thf(fact_5021_is__aligned__AND__less__0,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [U: word @ A,N3: nat,V: word @ A] :
( ( ( bit_se5824344872417868541ns_and @ ( word @ A ) @ U @ ( bit_se2239418461657761734s_mask @ ( word @ A ) @ N3 ) )
= ( zero_zero @ ( word @ A ) ) )
=> ( ( ord_less @ ( word @ A ) @ V @ ( power_power @ ( word @ A ) @ ( numeral_numeral @ ( word @ A ) @ ( bit0 @ one2 ) ) @ N3 ) )
=> ( ( bit_se5824344872417868541ns_and @ ( word @ A ) @ U @ V )
= ( zero_zero @ ( word @ A ) ) ) ) ) ) ).
% is_aligned_AND_less_0
thf(fact_5022_add__mask__fold,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [X: word @ A,N3: nat] :
( ( minus_minus @ ( word @ A ) @ ( plus_plus @ ( word @ A ) @ X @ ( power_power @ ( word @ A ) @ ( numeral_numeral @ ( word @ A ) @ ( bit0 @ one2 ) ) @ N3 ) ) @ ( one_one @ ( word @ A ) ) )
= ( plus_plus @ ( word @ A ) @ X @ ( bit_se2239418461657761734s_mask @ ( word @ A ) @ N3 ) ) ) ) ).
% add_mask_fold
thf(fact_5023_take__bit__int__eq__self,axiom,
! [K: int,N3: nat] :
( ( ord_less_eq @ int @ ( zero_zero @ int ) @ K )
=> ( ( ord_less @ int @ K @ ( power_power @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ N3 ) )
=> ( ( bit_se2584673776208193580ke_bit @ int @ N3 @ K )
= K ) ) ) ).
% take_bit_int_eq_self
thf(fact_5024_take__bit__int__eq__self__iff,axiom,
! [N3: nat,K: int] :
( ( ( bit_se2584673776208193580ke_bit @ int @ N3 @ K )
= K )
= ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ K )
& ( ord_less @ int @ K @ ( power_power @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ N3 ) ) ) ) ).
% take_bit_int_eq_self_iff
thf(fact_5025_take__bit__incr__eq,axiom,
! [N3: nat,K: int] :
( ( ( bit_se2584673776208193580ke_bit @ int @ N3 @ K )
!= ( minus_minus @ int @ ( power_power @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ N3 ) @ ( one_one @ int ) ) )
=> ( ( bit_se2584673776208193580ke_bit @ int @ N3 @ ( plus_plus @ int @ K @ ( one_one @ int ) ) )
= ( plus_plus @ int @ ( one_one @ int ) @ ( bit_se2584673776208193580ke_bit @ int @ N3 @ K ) ) ) ) ).
% take_bit_incr_eq
thf(fact_5026_take__bit__numeral__minus__bit0,axiom,
! [L2: num,K: num] :
( ( bit_se2584673776208193580ke_bit @ int @ ( numeral_numeral @ nat @ L2 ) @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ ( bit0 @ K ) ) ) )
= ( times_times @ int @ ( bit_se2584673776208193580ke_bit @ int @ ( pred_numeral @ L2 ) @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ K ) ) ) @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) ) ) ).
% take_bit_numeral_minus_bit0
thf(fact_5027_and__mask__less__size,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [N3: nat,X: word @ A] :
( ( ord_less @ nat @ N3 @ ( size_size @ ( word @ A ) @ X ) )
=> ( ord_less @ ( word @ A ) @ ( bit_se5824344872417868541ns_and @ ( word @ A ) @ X @ ( bit_se2239418461657761734s_mask @ ( word @ A ) @ N3 ) ) @ ( power_power @ ( word @ A ) @ ( numeral_numeral @ ( word @ A ) @ ( bit0 @ one2 ) ) @ N3 ) ) ) ) ).
% and_mask_less_size
thf(fact_5028_mask__eq__iff__w2p,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [N3: nat,W: word @ A] :
( ( ord_less @ nat @ N3 @ ( size_size @ ( word @ A ) @ W ) )
=> ( ( ( bit_se5824344872417868541ns_and @ ( word @ A ) @ W @ ( bit_se2239418461657761734s_mask @ ( word @ A ) @ N3 ) )
= W )
= ( ord_less @ ( word @ A ) @ W @ ( power_power @ ( word @ A ) @ ( numeral_numeral @ ( word @ A ) @ ( bit0 @ one2 ) ) @ N3 ) ) ) ) ) ).
% mask_eq_iff_w2p
thf(fact_5029_word__and__mask__le__2pm1,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [W: word @ A,N3: nat] : ( ord_less_eq @ ( word @ A ) @ ( bit_se5824344872417868541ns_and @ ( word @ A ) @ W @ ( bit_se2239418461657761734s_mask @ ( word @ A ) @ N3 ) ) @ ( minus_minus @ ( word @ A ) @ ( power_power @ ( word @ A ) @ ( numeral_numeral @ ( word @ A ) @ ( bit0 @ one2 ) ) @ N3 ) @ ( one_one @ ( word @ A ) ) ) ) ) ).
% word_and_mask_le_2pm1
thf(fact_5030_word__mod__2p__is__mask,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [N3: nat,X: word @ A] :
( ( ord_less @ ( word @ A ) @ ( zero_zero @ ( word @ A ) ) @ ( power_power @ ( word @ A ) @ ( numeral_numeral @ ( word @ A ) @ ( bit0 @ one2 ) ) @ N3 ) )
=> ( ( modulo_modulo @ ( word @ A ) @ X @ ( power_power @ ( word @ A ) @ ( numeral_numeral @ ( word @ A ) @ ( bit0 @ one2 ) ) @ N3 ) )
= ( bit_se5824344872417868541ns_and @ ( word @ A ) @ X @ ( bit_se2239418461657761734s_mask @ ( word @ A ) @ N3 ) ) ) ) ) ).
% word_mod_2p_is_mask
thf(fact_5031_take__bit__int__less__eq,axiom,
! [N3: nat,K: int] :
( ( ord_less_eq @ int @ ( power_power @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ N3 ) @ K )
=> ( ( ord_less @ nat @ ( zero_zero @ nat ) @ N3 )
=> ( ord_less_eq @ int @ ( bit_se2584673776208193580ke_bit @ int @ N3 @ K ) @ ( minus_minus @ int @ K @ ( power_power @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ N3 ) ) ) ) ) ).
% take_bit_int_less_eq
thf(fact_5032_take__bit__int__greater__eq,axiom,
! [K: int,N3: nat] :
( ( ord_less @ int @ K @ ( zero_zero @ int ) )
=> ( ord_less_eq @ int @ ( plus_plus @ int @ K @ ( power_power @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ N3 ) ) @ ( bit_se2584673776208193580ke_bit @ int @ N3 @ K ) ) ) ).
% take_bit_int_greater_eq
thf(fact_5033_divmod__step__nat__def,axiom,
( ( unique1321980374590559556d_step @ nat )
= ( ^ [L: num] :
( product_case_prod @ nat @ nat @ ( product_prod @ nat @ nat )
@ ^ [Q4: nat,R5: nat] : ( if @ ( product_prod @ nat @ nat ) @ ( ord_less_eq @ nat @ ( numeral_numeral @ nat @ L ) @ R5 ) @ ( product_Pair @ nat @ nat @ ( plus_plus @ nat @ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ Q4 ) @ ( one_one @ nat ) ) @ ( minus_minus @ nat @ R5 @ ( numeral_numeral @ nat @ L ) ) ) @ ( product_Pair @ nat @ nat @ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ Q4 ) @ R5 ) ) ) ) ) ).
% divmod_step_nat_def
thf(fact_5034_signed__take__bit__eq__take__bit__shift,axiom,
( ( bit_ri4674362597316999326ke_bit @ int )
= ( ^ [N2: nat,K2: int] : ( minus_minus @ int @ ( bit_se2584673776208193580ke_bit @ int @ ( suc @ N2 ) @ ( plus_plus @ int @ K2 @ ( power_power @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ N2 ) ) ) @ ( power_power @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ N2 ) ) ) ) ).
% signed_take_bit_eq_take_bit_shift
thf(fact_5035_divmod__step__int__def,axiom,
( ( unique1321980374590559556d_step @ int )
= ( ^ [L: num] :
( product_case_prod @ int @ int @ ( product_prod @ int @ int )
@ ^ [Q4: int,R5: int] : ( if @ ( product_prod @ int @ int ) @ ( ord_less_eq @ int @ ( numeral_numeral @ int @ L ) @ R5 ) @ ( product_Pair @ int @ int @ ( plus_plus @ int @ ( times_times @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ Q4 ) @ ( one_one @ int ) ) @ ( minus_minus @ int @ R5 @ ( numeral_numeral @ int @ L ) ) ) @ ( product_Pair @ int @ int @ ( times_times @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ Q4 ) @ R5 ) ) ) ) ) ).
% divmod_step_int_def
thf(fact_5036_and__mask__arith,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [W: word @ A,N3: nat] :
( ( bit_se5824344872417868541ns_and @ ( word @ A ) @ W @ ( bit_se2239418461657761734s_mask @ ( word @ A ) @ N3 ) )
= ( 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 ) @ N3 ) ) ) @ ( power_power @ ( word @ A ) @ ( numeral_numeral @ ( word @ A ) @ ( bit0 @ one2 ) ) @ ( minus_minus @ nat @ ( size_size @ ( word @ A ) @ W ) @ N3 ) ) ) ) ) ).
% and_mask_arith
thf(fact_5037_mask__numeral,axiom,
! [A: $tType] :
( ( bit_se359711467146920520ations @ A )
=> ! [N3: num] :
( ( bit_se2239418461657761734s_mask @ A @ ( numeral_numeral @ nat @ N3 ) )
= ( plus_plus @ A @ ( one_one @ A ) @ ( times_times @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ ( bit_se2239418461657761734s_mask @ A @ ( pred_numeral @ N3 ) ) ) ) ) ) ).
% mask_numeral
thf(fact_5038_take__bit__minus__small__eq,axiom,
! [K: int,N3: nat] :
( ( ord_less @ int @ ( zero_zero @ int ) @ K )
=> ( ( ord_less_eq @ int @ K @ ( power_power @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ N3 ) )
=> ( ( bit_se2584673776208193580ke_bit @ int @ N3 @ ( uminus_uminus @ int @ K ) )
= ( minus_minus @ int @ ( power_power @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ N3 ) @ K ) ) ) ) ).
% take_bit_minus_small_eq
thf(fact_5039_divmod__step__def,axiom,
! [A: $tType] :
( ( unique1627219031080169319umeral @ A )
=> ( ( unique1321980374590559556d_step @ A )
= ( ^ [L: num] :
( product_case_prod @ A @ A @ ( product_prod @ A @ A )
@ ^ [Q4: A,R5: A] : ( if @ ( product_prod @ A @ A ) @ ( ord_less_eq @ A @ ( numeral_numeral @ A @ L ) @ R5 ) @ ( product_Pair @ A @ A @ ( plus_plus @ A @ ( times_times @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ Q4 ) @ ( one_one @ A ) ) @ ( minus_minus @ A @ R5 @ ( numeral_numeral @ A @ L ) ) ) @ ( product_Pair @ A @ A @ ( times_times @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ Q4 ) @ R5 ) ) ) ) ) ) ).
% divmod_step_def
thf(fact_5040_divmod__step__integer__def,axiom,
( ( unique1321980374590559556d_step @ code_integer )
= ( ^ [L: num] :
( product_case_prod @ code_integer @ code_integer @ ( product_prod @ code_integer @ code_integer )
@ ^ [Q4: code_integer,R5: code_integer] : ( if @ ( product_prod @ code_integer @ code_integer ) @ ( ord_less_eq @ code_integer @ ( numeral_numeral @ code_integer @ L ) @ R5 ) @ ( product_Pair @ code_integer @ code_integer @ ( plus_plus @ code_integer @ ( times_times @ code_integer @ ( numeral_numeral @ code_integer @ ( bit0 @ one2 ) ) @ Q4 ) @ ( one_one @ code_integer ) ) @ ( minus_minus @ code_integer @ R5 @ ( numeral_numeral @ code_integer @ L ) ) ) @ ( product_Pair @ code_integer @ code_integer @ ( times_times @ code_integer @ ( numeral_numeral @ code_integer @ ( bit0 @ one2 ) ) @ Q4 ) @ R5 ) ) ) ) ) ).
% divmod_step_integer_def
thf(fact_5041_divmod__nat__if,axiom,
( divmod_nat
= ( ^ [M5: nat,N2: nat] :
( if @ ( product_prod @ nat @ nat )
@ ( ( N2
= ( zero_zero @ nat ) )
| ( ord_less @ nat @ M5 @ N2 ) )
@ ( product_Pair @ nat @ nat @ ( zero_zero @ nat ) @ M5 )
@ ( product_case_prod @ nat @ nat @ ( product_prod @ nat @ nat )
@ ^ [Q4: nat] : ( product_Pair @ nat @ nat @ ( suc @ Q4 ) )
@ ( divmod_nat @ ( minus_minus @ nat @ M5 @ N2 ) @ N2 ) ) ) ) ) ).
% divmod_nat_if
thf(fact_5042_add__scale__eq__noteq,axiom,
! [A: $tType] :
( ( semiri1453513574482234551roduct @ A )
=> ! [R2: A,A2: A,B2: A,C2: A,D: A] :
( ( R2
!= ( zero_zero @ A ) )
=> ( ( ( A2 = B2 )
& ( C2 != D ) )
=> ( ( plus_plus @ A @ A2 @ ( times_times @ A @ R2 @ C2 ) )
!= ( plus_plus @ A @ B2 @ ( times_times @ A @ R2 @ D ) ) ) ) ) ) ).
% add_scale_eq_noteq
thf(fact_5043_case__prodI2_H,axiom,
! [A: $tType,B: $tType,C: $tType,P4: product_prod @ A @ B,C2: A > B > C > $o,X: C] :
( ! [A4: A,B3: B] :
( ( ( product_Pair @ A @ B @ A4 @ B3 )
= P4 )
=> ( C2 @ A4 @ B3 @ X ) )
=> ( product_case_prod @ A @ B @ ( C > $o ) @ C2 @ P4 @ X ) ) ).
% case_prodI2'
thf(fact_5044_case__prodI,axiom,
! [A: $tType,B: $tType,F2: A > B > $o,A2: A,B2: B] :
( ( F2 @ A2 @ B2 )
=> ( product_case_prod @ A @ B @ $o @ F2 @ ( product_Pair @ A @ B @ A2 @ B2 ) ) ) ).
% case_prodI
thf(fact_5045_case__prodI2,axiom,
! [B: $tType,A: $tType,P4: product_prod @ A @ B,C2: A > B > $o] :
( ! [A4: A,B3: B] :
( ( P4
= ( product_Pair @ A @ B @ A4 @ B3 ) )
=> ( C2 @ A4 @ B3 ) )
=> ( product_case_prod @ A @ B @ $o @ C2 @ P4 ) ) ).
% case_prodI2
thf(fact_5046_mem__case__prodI,axiom,
! [A: $tType,B: $tType,C: $tType,Z: A,C2: B > C > ( set @ A ),A2: B,B2: C] :
( ( member @ A @ Z @ ( C2 @ A2 @ B2 ) )
=> ( member @ A @ Z @ ( product_case_prod @ B @ C @ ( set @ A ) @ C2 @ ( product_Pair @ B @ C @ A2 @ B2 ) ) ) ) ).
% mem_case_prodI
thf(fact_5047_mem__case__prodI2,axiom,
! [C: $tType,B: $tType,A: $tType,P4: product_prod @ A @ B,Z: C,C2: A > B > ( set @ C )] :
( ! [A4: A,B3: B] :
( ( P4
= ( product_Pair @ A @ B @ A4 @ B3 ) )
=> ( member @ C @ Z @ ( C2 @ A4 @ B3 ) ) )
=> ( member @ C @ Z @ ( product_case_prod @ A @ B @ ( set @ C ) @ C2 @ P4 ) ) ) ).
% mem_case_prodI2
thf(fact_5048_mask__nat__positive__iff,axiom,
! [N3: nat] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ ( bit_se2239418461657761734s_mask @ nat @ N3 ) )
= ( ord_less @ nat @ ( zero_zero @ nat ) @ N3 ) ) ).
% mask_nat_positive_iff
thf(fact_5049_case__prodD_H,axiom,
! [B: $tType,A: $tType,C: $tType,R: A > B > C > $o,A2: A,B2: B,C2: C] :
( ( product_case_prod @ A @ B @ ( C > $o ) @ R @ ( product_Pair @ A @ B @ A2 @ B2 ) @ C2 )
=> ( R @ A2 @ B2 @ C2 ) ) ).
% case_prodD'
thf(fact_5050_case__prodE_H,axiom,
! [B: $tType,A: $tType,C: $tType,C2: A > B > C > $o,P4: product_prod @ A @ B,Z: C] :
( ( product_case_prod @ A @ B @ ( C > $o ) @ C2 @ P4 @ Z )
=> ~ ! [X3: A,Y3: B] :
( ( P4
= ( product_Pair @ A @ B @ X3 @ Y3 ) )
=> ~ ( C2 @ X3 @ Y3 @ Z ) ) ) ).
% case_prodE'
thf(fact_5051_mem__case__prodE,axiom,
! [B: $tType,A: $tType,C: $tType,Z: A,C2: B > C > ( set @ A ),P4: product_prod @ B @ C] :
( ( member @ A @ Z @ ( product_case_prod @ B @ C @ ( set @ A ) @ C2 @ P4 ) )
=> ~ ! [X3: B,Y3: C] :
( ( P4
= ( product_Pair @ B @ C @ X3 @ Y3 ) )
=> ~ ( member @ A @ Z @ ( C2 @ X3 @ Y3 ) ) ) ) ).
% mem_case_prodE
thf(fact_5052_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,B3: B] :
( ( T2
= ( product_Pair @ A @ B @ A4 @ B3 ) )
=> ( time_TBOUND @ C @ ( F2 @ A4 @ B3 ) @ ( Bnd @ A4 @ B3 ) ) )
=> ( 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_5053_nat__mask__eq,axiom,
! [N3: nat] :
( ( nat2 @ ( bit_se2239418461657761734s_mask @ int @ N3 ) )
= ( bit_se2239418461657761734s_mask @ nat @ N3 ) ) ).
% nat_mask_eq
thf(fact_5054_case__prodD,axiom,
! [A: $tType,B: $tType,F2: A > B > $o,A2: A,B2: B] :
( ( product_case_prod @ A @ B @ $o @ F2 @ ( product_Pair @ A @ B @ A2 @ B2 ) )
=> ( F2 @ A2 @ B2 ) ) ).
% case_prodD
thf(fact_5055_case__prodE,axiom,
! [A: $tType,B: $tType,C2: A > B > $o,P4: product_prod @ A @ B] :
( ( product_case_prod @ A @ B @ $o @ C2 @ P4 )
=> ~ ! [X3: A,Y3: B] :
( ( P4
= ( product_Pair @ A @ B @ X3 @ Y3 ) )
=> ~ ( C2 @ X3 @ Y3 ) ) ) ).
% case_prodE
thf(fact_5056_less__eq__mask,axiom,
! [N3: nat] : ( ord_less_eq @ nat @ N3 @ ( bit_se2239418461657761734s_mask @ nat @ N3 ) ) ).
% less_eq_mask
thf(fact_5057_full__exhaustive__integer_H_Ocases,axiom,
! [X: product_prod @ ( ( product_prod @ code_integer @ ( product_unit > code_term ) ) > ( option @ ( product_prod @ $o @ ( list @ code_term ) ) ) ) @ ( product_prod @ code_integer @ code_integer )] :
~ ! [F3: ( product_prod @ code_integer @ ( product_unit > code_term ) ) > ( option @ ( product_prod @ $o @ ( list @ code_term ) ) ),D3: code_integer,I2: code_integer] :
( X
!= ( product_Pair @ ( ( product_prod @ code_integer @ ( product_unit > code_term ) ) > ( option @ ( product_prod @ $o @ ( list @ code_term ) ) ) ) @ ( product_prod @ code_integer @ code_integer ) @ F3 @ ( product_Pair @ code_integer @ code_integer @ D3 @ I2 ) ) ) ).
% full_exhaustive_integer'.cases
thf(fact_5058_exhaustive__integer_H_Ocases,axiom,
! [X: product_prod @ ( code_integer > ( option @ ( product_prod @ $o @ ( list @ code_term ) ) ) ) @ ( product_prod @ code_integer @ code_integer )] :
~ ! [F3: code_integer > ( option @ ( product_prod @ $o @ ( list @ code_term ) ) ),D3: code_integer,I2: code_integer] :
( X
!= ( product_Pair @ ( code_integer > ( option @ ( product_prod @ $o @ ( list @ code_term ) ) ) ) @ ( product_prod @ code_integer @ code_integer ) @ F3 @ ( product_Pair @ code_integer @ code_integer @ D3 @ I2 ) ) ) ).
% exhaustive_integer'.cases
thf(fact_5059_divmod__integer_H__def,axiom,
( ( unique8689654367752047608divmod @ code_integer )
= ( ^ [M5: num,N2: num] : ( product_Pair @ code_integer @ code_integer @ ( divide_divide @ code_integer @ ( numeral_numeral @ code_integer @ M5 ) @ ( numeral_numeral @ code_integer @ N2 ) ) @ ( modulo_modulo @ code_integer @ ( numeral_numeral @ code_integer @ M5 ) @ ( numeral_numeral @ code_integer @ N2 ) ) ) ) ) ).
% divmod_integer'_def
thf(fact_5060_mask__nonnegative__int,axiom,
! [N3: nat] : ( ord_less_eq @ int @ ( zero_zero @ int ) @ ( bit_se2239418461657761734s_mask @ int @ N3 ) ) ).
% mask_nonnegative_int
thf(fact_5061_not__mask__negative__int,axiom,
! [N3: nat] :
~ ( ord_less @ int @ ( bit_se2239418461657761734s_mask @ int @ N3 ) @ ( zero_zero @ int ) ) ).
% not_mask_negative_int
thf(fact_5062_sum_Otriangle__reindex__eq,axiom,
! [A: $tType] :
( ( comm_monoid_add @ A )
=> ! [G: nat > nat > A,N3: 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
@ ^ [I3: nat,J3: nat] : ( ord_less_eq @ nat @ ( plus_plus @ nat @ I3 @ J3 ) @ N3 ) ) ) )
= ( groups7311177749621191930dd_sum @ nat @ A
@ ^ [K2: nat] :
( groups7311177749621191930dd_sum @ nat @ A
@ ^ [I3: nat] : ( G @ I3 @ ( minus_minus @ nat @ K2 @ I3 ) )
@ ( set_ord_atMost @ nat @ K2 ) )
@ ( set_ord_atMost @ nat @ N3 ) ) ) ) ).
% sum.triangle_reindex_eq
thf(fact_5063_case__prod__rule,axiom,
! [A: $tType,B: $tType,C: $tType,X: product_prod @ A @ B,P: assn,F2: A > B > ( heap_Time_Heap @ C ),Q: C > assn] :
( ! [A4: A,B3: B] :
( ( X
= ( product_Pair @ A @ B @ A4 @ B3 ) )
=> ( hoare_hoare_triple @ C @ P @ ( F2 @ A4 @ B3 ) @ Q ) )
=> ( hoare_hoare_triple @ C @ P @ ( product_case_prod @ A @ B @ ( heap_Time_Heap @ C ) @ F2 @ X ) @ Q ) ) ).
% case_prod_rule
thf(fact_5064_prod_Otriangle__reindex__eq,axiom,
! [A: $tType] :
( ( comm_monoid_mult @ A )
=> ! [G: nat > nat > A,N3: 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
@ ^ [I3: nat,J3: nat] : ( ord_less_eq @ nat @ ( plus_plus @ nat @ I3 @ J3 ) @ N3 ) ) ) )
= ( groups7121269368397514597t_prod @ nat @ A
@ ^ [K2: nat] :
( groups7121269368397514597t_prod @ nat @ A
@ ^ [I3: nat] : ( G @ I3 @ ( minus_minus @ nat @ K2 @ I3 ) )
@ ( set_ord_atMost @ nat @ K2 ) )
@ ( set_ord_atMost @ nat @ N3 ) ) ) ) ).
% prod.triangle_reindex_eq
thf(fact_5065_sum_Otriangle__reindex,axiom,
! [A: $tType] :
( ( comm_monoid_add @ A )
=> ! [G: nat > nat > A,N3: 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
@ ^ [I3: nat,J3: nat] : ( ord_less @ nat @ ( plus_plus @ nat @ I3 @ J3 ) @ N3 ) ) ) )
= ( groups7311177749621191930dd_sum @ nat @ A
@ ^ [K2: nat] :
( groups7311177749621191930dd_sum @ nat @ A
@ ^ [I3: nat] : ( G @ I3 @ ( minus_minus @ nat @ K2 @ I3 ) )
@ ( set_ord_atMost @ nat @ K2 ) )
@ ( set_ord_lessThan @ nat @ N3 ) ) ) ) ).
% sum.triangle_reindex
thf(fact_5066_prod_Otriangle__reindex,axiom,
! [A: $tType] :
( ( comm_monoid_mult @ A )
=> ! [G: nat > nat > A,N3: 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
@ ^ [I3: nat,J3: nat] : ( ord_less @ nat @ ( plus_plus @ nat @ I3 @ J3 ) @ N3 ) ) ) )
= ( groups7121269368397514597t_prod @ nat @ A
@ ^ [K2: nat] :
( groups7121269368397514597t_prod @ nat @ A
@ ^ [I3: nat] : ( G @ I3 @ ( minus_minus @ nat @ K2 @ I3 ) )
@ ( set_ord_atMost @ nat @ K2 ) )
@ ( set_ord_lessThan @ nat @ N3 ) ) ) ) ).
% prod.triangle_reindex
thf(fact_5067_less__mask,axiom,
! [N3: nat] :
( ( ord_less @ nat @ ( suc @ ( zero_zero @ nat ) ) @ N3 )
=> ( ord_less @ nat @ N3 @ ( bit_se2239418461657761734s_mask @ nat @ N3 ) ) ) ).
% less_mask
thf(fact_5068_one__natural_Orsp,axiom,
( ( one_one @ nat )
= ( one_one @ nat ) ) ).
% one_natural.rsp
thf(fact_5069_one__integer_Orsp,axiom,
( ( one_one @ int )
= ( one_one @ int ) ) ).
% one_integer.rsp
thf(fact_5070_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_5071_rel__of__def,axiom,
! [B: $tType,A: $tType] :
( ( rel_of @ A @ B )
= ( ^ [M5: A > ( option @ B ),P3: ( product_prod @ A @ B ) > $o] :
( collect @ ( product_prod @ A @ B )
@ ( product_case_prod @ A @ B @ $o
@ ^ [K2: A,V5: B] :
( ( ( M5 @ K2 )
= ( some @ B @ V5 ) )
& ( P3 @ ( product_Pair @ A @ B @ K2 @ V5 ) ) ) ) ) ) ) ).
% rel_of_def
thf(fact_5072_take__bit__eq__mask__iff,axiom,
! [N3: nat,K: int] :
( ( ( bit_se2584673776208193580ke_bit @ int @ N3 @ K )
= ( bit_se2239418461657761734s_mask @ int @ N3 ) )
= ( ( bit_se2584673776208193580ke_bit @ int @ N3 @ ( plus_plus @ int @ K @ ( one_one @ int ) ) )
= ( zero_zero @ int ) ) ) ).
% take_bit_eq_mask_iff
thf(fact_5073_Suc__mask__eq__exp,axiom,
! [N3: nat] :
( ( suc @ ( bit_se2239418461657761734s_mask @ nat @ N3 ) )
= ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N3 ) ) ).
% Suc_mask_eq_exp
thf(fact_5074_mask__nat__less__exp,axiom,
! [N3: nat] : ( ord_less @ nat @ ( bit_se2239418461657761734s_mask @ nat @ N3 ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N3 ) ) ).
% mask_nat_less_exp
thf(fact_5075_mask__half__int,axiom,
! [N3: nat] :
( ( divide_divide @ int @ ( bit_se2239418461657761734s_mask @ int @ N3 ) @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) )
= ( bit_se2239418461657761734s_mask @ int @ ( minus_minus @ nat @ N3 @ ( one_one @ nat ) ) ) ) ).
% mask_half_int
thf(fact_5076_mask__int__def,axiom,
( ( bit_se2239418461657761734s_mask @ int )
= ( ^ [N2: nat] : ( minus_minus @ int @ ( power_power @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ N2 ) @ ( one_one @ int ) ) ) ) ).
% mask_int_def
thf(fact_5077_mask__nat__def,axiom,
( ( bit_se2239418461657761734s_mask @ nat )
= ( ^ [N2: nat] : ( minus_minus @ nat @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N2 ) @ ( one_one @ nat ) ) ) ) ).
% mask_nat_def
thf(fact_5078_add__0__iff,axiom,
! [A: $tType] :
( ( semiri1453513574482234551roduct @ A )
=> ! [B2: A,A2: A] :
( ( B2
= ( plus_plus @ A @ B2 @ A2 ) )
= ( A2
= ( zero_zero @ A ) ) ) ) ).
% add_0_iff
thf(fact_5079_crossproduct__noteq,axiom,
! [A: $tType] :
( ( semiri1453513574482234551roduct @ A )
=> ! [A2: A,B2: A,C2: A,D: A] :
( ( ( A2 != B2 )
& ( C2 != D ) )
= ( ( plus_plus @ A @ ( times_times @ A @ A2 @ C2 ) @ ( times_times @ A @ B2 @ D ) )
!= ( plus_plus @ A @ ( times_times @ A @ A2 @ D ) @ ( times_times @ A @ B2 @ C2 ) ) ) ) ) ).
% crossproduct_noteq
thf(fact_5080_crossproduct__eq,axiom,
! [A: $tType] :
( ( semiri1453513574482234551roduct @ A )
=> ! [W: A,Y: A,X: A,Z: A] :
( ( ( plus_plus @ A @ ( times_times @ A @ W @ Y ) @ ( times_times @ A @ X @ Z ) )
= ( plus_plus @ A @ ( times_times @ A @ W @ Z ) @ ( times_times @ A @ X @ Y ) ) )
= ( ( W = X )
| ( Y = Z ) ) ) ) ).
% crossproduct_eq
thf(fact_5081_divmod__nat__def,axiom,
( divmod_nat
= ( ^ [M5: nat,N2: nat] : ( product_Pair @ nat @ nat @ ( divide_divide @ nat @ M5 @ N2 ) @ ( modulo_modulo @ nat @ M5 @ N2 ) ) ) ) ).
% divmod_nat_def
thf(fact_5082_minus__integer__code_I2_J,axiom,
! [L2: code_integer] :
( ( minus_minus @ code_integer @ ( zero_zero @ code_integer ) @ L2 )
= ( uminus_uminus @ code_integer @ L2 ) ) ).
% minus_integer_code(2)
thf(fact_5083_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_5084_take__bit__eq__mask__iff__exp__dvd,axiom,
! [N3: nat,K: int] :
( ( ( bit_se2584673776208193580ke_bit @ int @ N3 @ K )
= ( bit_se2239418461657761734s_mask @ int @ N3 ) )
= ( dvd_dvd @ int @ ( power_power @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ N3 ) @ ( plus_plus @ int @ K @ ( one_one @ int ) ) ) ) ).
% take_bit_eq_mask_iff_exp_dvd
thf(fact_5085_of__nat__code__if,axiom,
! [A: $tType] :
( ( semiring_1 @ A )
=> ( ( semiring_1_of_nat @ A )
= ( ^ [N2: nat] :
( if @ A
@ ( N2
= ( zero_zero @ nat ) )
@ ( zero_zero @ A )
@ ( product_case_prod @ nat @ nat @ A
@ ^ [M5: nat,Q4: nat] :
( if @ A
@ ( Q4
= ( zero_zero @ nat ) )
@ ( times_times @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ ( semiring_1_of_nat @ A @ M5 ) )
@ ( plus_plus @ A @ ( times_times @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ ( semiring_1_of_nat @ A @ M5 ) ) @ ( one_one @ A ) ) )
@ ( divmod_nat @ N2 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) ) ) ).
% of_nat_code_if
thf(fact_5086_integer__of__int__code,axiom,
( code_integer_of_int
= ( ^ [K2: int] :
( if @ code_integer @ ( ord_less @ int @ K2 @ ( zero_zero @ int ) ) @ ( uminus_uminus @ code_integer @ ( code_integer_of_int @ ( uminus_uminus @ int @ K2 ) ) )
@ ( if @ code_integer
@ ( K2
= ( zero_zero @ int ) )
@ ( zero_zero @ code_integer )
@ ( if @ code_integer
@ ( ( modulo_modulo @ int @ K2 @ ( 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 @ K2 @ ( 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 @ K2 @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) ) ) ) @ ( one_one @ code_integer ) ) ) ) ) ) ) ).
% integer_of_int_code
thf(fact_5087_int__ge__less__than__def,axiom,
( int_ge_less_than
= ( ^ [D2: int] :
( collect @ ( product_prod @ int @ int )
@ ( product_case_prod @ int @ int @ $o
@ ^ [Z8: int,Z3: int] :
( ( ord_less_eq @ int @ D2 @ Z8 )
& ( ord_less @ int @ Z8 @ Z3 ) ) ) ) ) ) ).
% int_ge_less_than_def
thf(fact_5088_Collect__case__prod__mono,axiom,
! [B: $tType,A: $tType,A3: A > B > $o,B6: A > B > $o] :
( ( ord_less_eq @ ( A > B > $o ) @ A3 @ B6 )
=> ( ord_less_eq @ ( set @ ( product_prod @ A @ B ) ) @ ( collect @ ( product_prod @ A @ B ) @ ( product_case_prod @ A @ B @ $o @ A3 ) ) @ ( collect @ ( product_prod @ A @ B ) @ ( product_case_prod @ A @ B @ $o @ B6 ) ) ) ) ).
% Collect_case_prod_mono
thf(fact_5089_one__integer__def,axiom,
( ( one_one @ code_integer )
= ( code_integer_of_int @ ( one_one @ int ) ) ) ).
% one_integer_def
thf(fact_5090_less__eq__integer_Oabs__eq,axiom,
! [Xa: int,X: int] :
( ( ord_less_eq @ code_integer @ ( code_integer_of_int @ Xa ) @ ( code_integer_of_int @ X ) )
= ( ord_less_eq @ int @ Xa @ X ) ) ).
% less_eq_integer.abs_eq
thf(fact_5091_minus__integer_Oabs__eq,axiom,
! [Xa: int,X: int] :
( ( minus_minus @ code_integer @ ( code_integer_of_int @ Xa ) @ ( code_integer_of_int @ X ) )
= ( code_integer_of_int @ ( minus_minus @ int @ Xa @ X ) ) ) ).
% minus_integer.abs_eq
thf(fact_5092_int__ge__less__than2__def,axiom,
( int_ge_less_than2
= ( ^ [D2: int] :
( collect @ ( product_prod @ int @ int )
@ ( product_case_prod @ int @ int @ $o
@ ^ [Z8: int,Z3: int] :
( ( ord_less_eq @ int @ D2 @ Z3 )
& ( ord_less @ int @ Z8 @ Z3 ) ) ) ) ) ) ).
% int_ge_less_than2_def
thf(fact_5093_slice__nth,axiom,
! [A: $tType,From: nat,To: nat,Xs2: list @ A,I: nat] :
( ( ord_less @ nat @ From @ To )
=> ( ( ord_less_eq @ nat @ To @ ( size_size @ ( list @ A ) @ Xs2 ) )
=> ( ( ord_less @ nat @ I @ ( minus_minus @ nat @ To @ From ) )
=> ( ( nth @ A @ ( slice @ A @ From @ To @ Xs2 ) @ I )
= ( nth @ A @ Xs2 @ ( plus_plus @ nat @ From @ I ) ) ) ) ) ) ).
% slice_nth
thf(fact_5094_word__2p__lem,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [N3: nat,W: word @ A] :
( ( ord_less @ nat @ N3 @ ( size_size @ ( word @ A ) @ W ) )
=> ( ( ord_less @ ( word @ A ) @ W @ ( power_power @ ( word @ A ) @ ( numeral_numeral @ ( word @ A ) @ ( bit0 @ one2 ) ) @ N3 ) )
= ( ord_less @ int @ ( semiring_1_unsigned @ A @ int @ W ) @ ( power_power @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ N3 ) ) ) ) ) ).
% word_2p_lem
thf(fact_5095_predicate2I,axiom,
! [B: $tType,A: $tType,P: A > B > $o,Q: A > B > $o] :
( ! [X3: A,Y3: B] :
( ( P @ X3 @ Y3 )
=> ( Q @ X3 @ Y3 ) )
=> ( ord_less_eq @ ( A > B > $o ) @ P @ Q ) ) ).
% predicate2I
thf(fact_5096_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_5097_uint__ge__0,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [X: word @ A] : ( ord_less_eq @ int @ ( zero_zero @ int ) @ ( semiring_1_unsigned @ A @ int @ X ) ) ) ).
% uint_ge_0
thf(fact_5098_uint__le__0__iff,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [X: word @ A] :
( ( ord_less_eq @ int @ ( semiring_1_unsigned @ A @ int @ X ) @ ( zero_zero @ int ) )
= ( ( semiring_1_unsigned @ A @ int @ X )
= ( zero_zero @ int ) ) ) ) ).
% uint_le_0_iff
thf(fact_5099_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_5100_slice__complete,axiom,
! [A: $tType,Xs2: list @ A] :
( ( slice @ A @ ( zero_zero @ nat ) @ ( size_size @ ( list @ A ) @ Xs2 ) @ Xs2 )
= Xs2 ) ).
% slice_complete
thf(fact_5101_word__le__no,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [A2: num,B2: num] :
( ( ord_less_eq @ ( word @ A ) @ ( numeral_numeral @ ( word @ A ) @ A2 ) @ ( numeral_numeral @ ( word @ A ) @ B2 ) )
= ( ord_less_eq @ int @ ( semiring_1_unsigned @ A @ int @ ( numeral_numeral @ ( word @ A ) @ A2 ) ) @ ( semiring_1_unsigned @ A @ int @ ( numeral_numeral @ ( word @ A ) @ B2 ) ) ) ) ) ).
% word_le_no
thf(fact_5102_slice__len,axiom,
! [A: $tType,From: nat,To: nat,Xs2: list @ A] :
( ( ord_less_eq @ nat @ From @ To )
=> ( ( ord_less_eq @ nat @ To @ ( size_size @ ( list @ A ) @ Xs2 ) )
=> ( ( size_size @ ( list @ A ) @ ( slice @ A @ From @ To @ Xs2 ) )
= ( minus_minus @ nat @ To @ From ) ) ) ) ).
% slice_len
thf(fact_5103_predicate2D,axiom,
! [A: $tType,B: $tType,P: A > B > $o,Q: A > B > $o,X: A,Y: B] :
( ( ord_less_eq @ ( A > B > $o ) @ P @ Q )
=> ( ( P @ X @ Y )
=> ( Q @ X @ Y ) ) ) ).
% predicate2D
thf(fact_5104_rev__predicate2D,axiom,
! [A: $tType,B: $tType,P: A > B > $o,X: A,Y: B,Q: A > B > $o] :
( ( P @ X @ Y )
=> ( ( ord_less_eq @ ( A > B > $o ) @ P @ Q )
=> ( Q @ X @ Y ) ) ) ).
% rev_predicate2D
thf(fact_5105_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_5106_word__le__def,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ( ( ord_less_eq @ ( word @ A ) )
= ( ^ [A6: word @ A,B5: word @ A] : ( ord_less_eq @ int @ ( semiring_1_unsigned @ A @ int @ A6 ) @ ( semiring_1_unsigned @ A @ int @ B5 ) ) ) ) ) ).
% word_le_def
thf(fact_5107_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_5108_word__less__eq__iff__unsigned,axiom,
! [A: $tType,B: $tType] :
( ( ( type_len @ B )
& ( linordered_semidom @ A ) )
=> ( ( ord_less_eq @ ( word @ B ) )
= ( ^ [A6: word @ B,B5: word @ B] : ( ord_less_eq @ A @ ( semiring_1_unsigned @ B @ A @ A6 ) @ ( semiring_1_unsigned @ B @ A @ B5 ) ) ) ) ) ).
% word_less_eq_iff_unsigned
thf(fact_5109_word__less__iff__unsigned,axiom,
! [A: $tType,B: $tType] :
( ( ( type_len @ B )
& ( linordered_semidom @ A ) )
=> ( ( ord_less @ ( word @ B ) )
= ( ^ [A6: word @ B,B5: word @ B] : ( ord_less @ A @ ( semiring_1_unsigned @ B @ A @ A6 ) @ ( semiring_1_unsigned @ B @ A @ B5 ) ) ) ) ) ).
% word_less_iff_unsigned
thf(fact_5110_uint__add__ge0,axiom,
! [Aa: $tType,A: $tType] :
( ( ( type_len @ A )
& ( type_len @ Aa ) )
=> ! [Xa: word @ Aa,X: word @ A] : ( ord_less_eq @ int @ ( zero_zero @ int ) @ ( plus_plus @ int @ ( semiring_1_unsigned @ Aa @ int @ Xa ) @ ( semiring_1_unsigned @ A @ int @ X ) ) ) ) ).
% uint_add_ge0
thf(fact_5111_uint__mult__ge0,axiom,
! [Aa: $tType,A: $tType] :
( ( ( type_len @ A )
& ( type_len @ Aa ) )
=> ! [Xa: word @ Aa,X: word @ A] : ( ord_less_eq @ int @ ( zero_zero @ int ) @ ( times_times @ int @ ( semiring_1_unsigned @ Aa @ int @ Xa ) @ ( semiring_1_unsigned @ A @ int @ X ) ) ) ) ).
% uint_mult_ge0
thf(fact_5112_nat__uint__less__helper,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [Y: word @ A,Z: nat,X: word @ A] :
( ( ( nat2 @ ( semiring_1_unsigned @ A @ int @ Y ) )
= Z )
=> ( ( ord_less @ ( word @ A ) @ X @ Y )
=> ( ord_less @ nat @ ( nat2 @ ( semiring_1_unsigned @ A @ int @ X ) ) @ Z ) ) ) ) ).
% nat_uint_less_helper
thf(fact_5113_uint__add__le,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [X: word @ A,Y: word @ A] : ( ord_less_eq @ int @ ( semiring_1_unsigned @ A @ int @ ( plus_plus @ ( word @ A ) @ X @ Y ) ) @ ( plus_plus @ int @ ( semiring_1_unsigned @ A @ int @ X ) @ ( semiring_1_unsigned @ A @ int @ Y ) ) ) ) ).
% uint_add_le
thf(fact_5114_uint__plus__simple__iff,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [X: word @ A,Y: word @ A] :
( ( ord_less_eq @ ( word @ A ) @ X @ ( plus_plus @ ( word @ A ) @ X @ Y ) )
= ( ( semiring_1_unsigned @ A @ int @ ( plus_plus @ ( word @ A ) @ X @ Y ) )
= ( plus_plus @ int @ ( semiring_1_unsigned @ A @ int @ X ) @ ( semiring_1_unsigned @ A @ int @ Y ) ) ) ) ) ).
% uint_plus_simple_iff
thf(fact_5115_uint__plus__simple,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [X: word @ A,Y: word @ A] :
( ( ord_less_eq @ ( word @ A ) @ X @ ( plus_plus @ ( word @ A ) @ X @ Y ) )
=> ( ( semiring_1_unsigned @ A @ int @ ( plus_plus @ ( word @ A ) @ X @ Y ) )
= ( plus_plus @ int @ ( semiring_1_unsigned @ A @ int @ X ) @ ( semiring_1_unsigned @ A @ int @ Y ) ) ) ) ) ).
% uint_plus_simple
thf(fact_5116_no__ulen__sub,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [X: word @ A,Y: word @ A] :
( ( ord_less_eq @ ( word @ A ) @ ( minus_minus @ ( word @ A ) @ X @ Y ) @ X )
= ( ord_less_eq @ int @ ( semiring_1_unsigned @ A @ int @ Y ) @ ( semiring_1_unsigned @ A @ int @ X ) ) ) ) ).
% no_ulen_sub
thf(fact_5117_uint__sub__lem,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [Y: word @ A,X: word @ A] :
( ( ord_less_eq @ int @ ( semiring_1_unsigned @ A @ int @ Y ) @ ( semiring_1_unsigned @ A @ int @ X ) )
= ( ( semiring_1_unsigned @ A @ int @ ( minus_minus @ ( word @ A ) @ X @ Y ) )
= ( minus_minus @ int @ ( semiring_1_unsigned @ A @ int @ X ) @ ( semiring_1_unsigned @ A @ int @ Y ) ) ) ) ) ).
% uint_sub_lem
thf(fact_5118_uint__sub__ge,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [X: word @ A,Y: word @ A] : ( ord_less_eq @ int @ ( minus_minus @ int @ ( semiring_1_unsigned @ A @ int @ X ) @ ( semiring_1_unsigned @ A @ int @ Y ) ) @ ( semiring_1_unsigned @ A @ int @ ( minus_minus @ ( word @ A ) @ X @ Y ) ) ) ) ).
% uint_sub_ge
thf(fact_5119_uint__minus__simple__alt,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ( ( ord_less_eq @ ( word @ A ) )
= ( ^ [Y2: word @ A,X2: word @ A] :
( ( semiring_1_unsigned @ A @ int @ ( minus_minus @ ( word @ A ) @ X2 @ Y2 ) )
= ( minus_minus @ int @ ( semiring_1_unsigned @ A @ int @ X2 ) @ ( semiring_1_unsigned @ A @ int @ Y2 ) ) ) ) ) ) ).
% uint_minus_simple_alt
thf(fact_5120_uint__minus__simple__iff,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [X: word @ A,Y: word @ A] :
( ( ord_less_eq @ ( word @ A ) @ ( minus_minus @ ( word @ A ) @ X @ Y ) @ X )
= ( ( semiring_1_unsigned @ A @ int @ ( minus_minus @ ( word @ A ) @ X @ Y ) )
= ( minus_minus @ int @ ( semiring_1_unsigned @ A @ int @ X ) @ ( semiring_1_unsigned @ A @ int @ Y ) ) ) ) ) ).
% uint_minus_simple_iff
thf(fact_5121_word__sub__wi,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ( ( minus_minus @ ( word @ A ) )
= ( ^ [A6: word @ A,B5: word @ A] : ( ring_1_of_int @ ( word @ A ) @ ( minus_minus @ int @ ( semiring_1_unsigned @ A @ int @ A6 ) @ ( semiring_1_unsigned @ A @ int @ B5 ) ) ) ) ) ) ).
% word_sub_wi
thf(fact_5122_udvd__incr__lem0,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [Up: int,Uq: int,N3: int,K6: word @ A,N4: int] :
( ( ord_less @ int @ Up @ Uq )
=> ( ( Up
= ( times_times @ int @ N3 @ ( semiring_1_unsigned @ A @ int @ K6 ) ) )
=> ( ( Uq
= ( times_times @ int @ N4 @ ( 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_5123_udvd__incr__lem,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [Up: int,Uq: int,Ua: int,N3: int,K6: word @ A,N4: int] :
( ( ord_less @ int @ Up @ Uq )
=> ( ( Up
= ( plus_plus @ int @ Ua @ ( times_times @ int @ N3 @ ( semiring_1_unsigned @ A @ int @ K6 ) ) ) )
=> ( ( Uq
= ( plus_plus @ int @ Ua @ ( times_times @ int @ N4 @ ( 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_5124_udvd__incr0,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [P4: word @ A,Q2: word @ A,N3: int,K6: word @ A,N4: int] :
( ( ord_less @ ( word @ A ) @ P4 @ Q2 )
=> ( ( ( semiring_1_unsigned @ A @ int @ P4 )
= ( times_times @ int @ N3 @ ( semiring_1_unsigned @ A @ int @ K6 ) ) )
=> ( ( ( semiring_1_unsigned @ A @ int @ Q2 )
= ( times_times @ int @ N4 @ ( semiring_1_unsigned @ A @ int @ K6 ) ) )
=> ( ord_less_eq @ ( word @ A ) @ ( plus_plus @ ( word @ A ) @ P4 @ K6 ) @ Q2 ) ) ) ) ) ).
% udvd_incr0
thf(fact_5125_udvd__decr0,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [P4: word @ A,Q2: word @ A,N3: int,K6: word @ A,N4: int] :
( ( ord_less @ ( word @ A ) @ P4 @ Q2 )
=> ( ( ( semiring_1_unsigned @ A @ int @ P4 )
= ( times_times @ int @ N3 @ ( semiring_1_unsigned @ A @ int @ K6 ) ) )
=> ( ( ( semiring_1_unsigned @ A @ int @ Q2 )
= ( times_times @ int @ N4 @ ( semiring_1_unsigned @ A @ int @ K6 ) ) )
=> ( ( ( semiring_1_unsigned @ A @ int @ Q2 )
= ( times_times @ int @ N4 @ ( semiring_1_unsigned @ A @ int @ K6 ) ) )
=> ( ord_less_eq @ ( word @ A ) @ P4 @ ( minus_minus @ ( word @ A ) @ Q2 @ K6 ) ) ) ) ) ) ) ).
% udvd_decr0
thf(fact_5126_udvd__incr_H,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [P4: word @ A,Q2: word @ A,Ua: int,N3: int,K6: word @ A,N4: int] :
( ( ord_less @ ( word @ A ) @ P4 @ Q2 )
=> ( ( ( semiring_1_unsigned @ A @ int @ P4 )
= ( plus_plus @ int @ Ua @ ( times_times @ int @ N3 @ ( semiring_1_unsigned @ A @ int @ K6 ) ) ) )
=> ( ( ( semiring_1_unsigned @ A @ int @ Q2 )
= ( plus_plus @ int @ Ua @ ( times_times @ int @ N4 @ ( semiring_1_unsigned @ A @ int @ K6 ) ) ) )
=> ( ord_less_eq @ ( word @ A ) @ ( plus_plus @ ( word @ A ) @ P4 @ K6 ) @ Q2 ) ) ) ) ) ).
% udvd_incr'
thf(fact_5127_udvd__decr_H,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [P4: word @ A,Q2: word @ A,Ua: int,N3: int,K6: word @ A,N4: int] :
( ( ord_less @ ( word @ A ) @ P4 @ Q2 )
=> ( ( ( semiring_1_unsigned @ A @ int @ P4 )
= ( plus_plus @ int @ Ua @ ( times_times @ int @ N3 @ ( semiring_1_unsigned @ A @ int @ K6 ) ) ) )
=> ( ( ( semiring_1_unsigned @ A @ int @ Q2 )
= ( plus_plus @ int @ Ua @ ( times_times @ int @ N4 @ ( semiring_1_unsigned @ A @ int @ K6 ) ) ) )
=> ( ( ( semiring_1_unsigned @ A @ int @ Q2 )
= ( plus_plus @ int @ Ua @ ( times_times @ int @ N4 @ ( semiring_1_unsigned @ A @ int @ K6 ) ) ) )
=> ( ord_less_eq @ ( word @ A ) @ P4 @ ( minus_minus @ ( word @ A ) @ Q2 @ K6 ) ) ) ) ) ) ) ).
% udvd_decr'
thf(fact_5128_and__mask__lt__2p,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [W: word @ A,N3: nat] : ( ord_less @ int @ ( semiring_1_unsigned @ A @ int @ ( bit_se5824344872417868541ns_and @ ( word @ A ) @ W @ ( bit_se2239418461657761734s_mask @ ( word @ A ) @ N3 ) ) ) @ ( power_power @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ N3 ) ) ) ).
% and_mask_lt_2p
thf(fact_5129_mask__eq__iff,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [W: word @ A,N3: nat] :
( ( ( bit_se5824344872417868541ns_and @ ( word @ A ) @ W @ ( bit_se2239418461657761734s_mask @ ( word @ A ) @ N3 ) )
= W )
= ( ord_less @ int @ ( semiring_1_unsigned @ A @ int @ W ) @ ( power_power @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ N3 ) ) ) ) ).
% mask_eq_iff
thf(fact_5130_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_5131_uint__2p,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [N3: nat] :
( ( ord_less @ ( word @ A ) @ ( zero_zero @ ( word @ A ) ) @ ( power_power @ ( word @ A ) @ ( numeral_numeral @ ( word @ A ) @ ( bit0 @ one2 ) ) @ N3 ) )
=> ( ( semiring_1_unsigned @ A @ int @ ( power_power @ ( word @ A ) @ ( numeral_numeral @ ( word @ A ) @ ( bit0 @ one2 ) ) @ N3 ) )
= ( power_power @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ N3 ) ) ) ) ).
% uint_2p
thf(fact_5132_and__mask__mod__2p,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [W: word @ A,N3: nat] :
( ( bit_se5824344872417868541ns_and @ ( word @ A ) @ W @ ( bit_se2239418461657761734s_mask @ ( word @ A ) @ N3 ) )
= ( ring_1_of_int @ ( word @ A ) @ ( modulo_modulo @ int @ ( semiring_1_unsigned @ A @ int @ W ) @ ( power_power @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ N3 ) ) ) ) ) ).
% and_mask_mod_2p
thf(fact_5133_and__mask__dvd,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [N3: nat,W: word @ A] :
( ( dvd_dvd @ int @ ( power_power @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ N3 ) @ ( semiring_1_unsigned @ A @ int @ W ) )
= ( ( bit_se5824344872417868541ns_and @ ( word @ A ) @ W @ ( bit_se2239418461657761734s_mask @ ( word @ A ) @ N3 ) )
= ( zero_zero @ ( word @ A ) ) ) ) ) ).
% and_mask_dvd
thf(fact_5134_no__plus__overflow__uint__size,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [X: word @ A,Y: word @ A] :
( ( ord_less_eq @ ( word @ A ) @ X @ ( plus_plus @ ( word @ A ) @ X @ Y ) )
= ( ord_less @ int @ ( plus_plus @ int @ ( semiring_1_unsigned @ A @ int @ X ) @ ( semiring_1_unsigned @ A @ int @ Y ) ) @ ( power_power @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ ( size_size @ ( word @ A ) @ X ) ) ) ) ) ).
% no_plus_overflow_uint_size
thf(fact_5135_uint__plus__if__size,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [X: word @ A,Y: word @ A] :
( ( ( ord_less @ int @ ( plus_plus @ int @ ( semiring_1_unsigned @ A @ int @ X ) @ ( semiring_1_unsigned @ A @ int @ Y ) ) @ ( power_power @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ ( size_size @ ( word @ A ) @ X ) ) )
=> ( ( semiring_1_unsigned @ A @ int @ ( plus_plus @ ( word @ A ) @ X @ Y ) )
= ( plus_plus @ int @ ( semiring_1_unsigned @ A @ int @ X ) @ ( semiring_1_unsigned @ A @ int @ Y ) ) ) )
& ( ~ ( ord_less @ int @ ( plus_plus @ int @ ( semiring_1_unsigned @ A @ int @ X ) @ ( semiring_1_unsigned @ A @ int @ Y ) ) @ ( power_power @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ ( size_size @ ( word @ A ) @ X ) ) )
=> ( ( semiring_1_unsigned @ A @ int @ ( plus_plus @ ( word @ A ) @ X @ Y ) )
= ( minus_minus @ int @ ( plus_plus @ int @ ( semiring_1_unsigned @ A @ int @ X ) @ ( semiring_1_unsigned @ A @ int @ Y ) ) @ ( power_power @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ ( size_size @ ( word @ A ) @ X ) ) ) ) ) ) ) ).
% uint_plus_if_size
thf(fact_5136_uint__sub__if__size,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [Y: word @ A,X: word @ A] :
( ( ( ord_less_eq @ int @ ( semiring_1_unsigned @ A @ int @ Y ) @ ( semiring_1_unsigned @ A @ int @ X ) )
=> ( ( semiring_1_unsigned @ A @ int @ ( minus_minus @ ( word @ A ) @ X @ Y ) )
= ( minus_minus @ int @ ( semiring_1_unsigned @ A @ int @ X ) @ ( semiring_1_unsigned @ A @ int @ Y ) ) ) )
& ( ~ ( ord_less_eq @ int @ ( semiring_1_unsigned @ A @ int @ Y ) @ ( semiring_1_unsigned @ A @ int @ X ) )
=> ( ( semiring_1_unsigned @ A @ int @ ( minus_minus @ ( word @ A ) @ X @ Y ) )
= ( plus_plus @ int @ ( minus_minus @ int @ ( semiring_1_unsigned @ A @ int @ X ) @ ( semiring_1_unsigned @ A @ int @ Y ) ) @ ( power_power @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ ( size_size @ ( word @ A ) @ X ) ) ) ) ) ) ) ).
% uint_sub_if_size
thf(fact_5137_pred__subset__eq2,axiom,
! [B: $tType,A: $tType,R: set @ ( product_prod @ A @ B ),S3: set @ ( product_prod @ A @ B )] :
( ( ord_less_eq @ ( A > B > $o )
@ ^ [X2: A,Y2: B] : ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X2 @ Y2 ) @ R )
@ ^ [X2: A,Y2: B] : ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X2 @ Y2 ) @ S3 ) )
= ( ord_less_eq @ ( set @ ( product_prod @ A @ B ) ) @ R @ S3 ) ) ).
% pred_subset_eq2
thf(fact_5138_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_5139_arctan__def,axiom,
( arctan
= ( ^ [Y2: real] :
( the @ real
@ ^ [X2: real] :
( ( ord_less @ real @ ( uminus_uminus @ real @ ( divide_divide @ real @ pi @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) ) @ X2 )
& ( ord_less @ real @ X2 @ ( divide_divide @ real @ pi @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) )
& ( ( tan @ real @ X2 )
= Y2 ) ) ) ) ) ).
% arctan_def
thf(fact_5140_Suc__unat__minus__one,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [X: word @ A] :
( ( X
!= ( zero_zero @ ( word @ A ) ) )
=> ( ( suc @ ( semiring_1_unsigned @ A @ nat @ ( minus_minus @ ( word @ A ) @ X @ ( one_one @ ( word @ A ) ) ) ) )
= ( semiring_1_unsigned @ A @ nat @ X ) ) ) ) ).
% Suc_unat_minus_one
thf(fact_5141_word__unat__Rep__inject1,axiom,
! [B: $tType,A: $tType] :
( ( ( type_len @ A )
& ( type_len @ B ) )
=> ! [X: word @ A] :
( ( ( semiring_1_unsigned @ A @ nat @ X )
= ( semiring_1_unsigned @ B @ nat @ ( one_one @ ( word @ B ) ) ) )
= ( X
= ( one_one @ ( word @ A ) ) ) ) ) ).
% word_unat_Rep_inject1
thf(fact_5142_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_5143_le__ucast__ucast__le,axiom,
! [A: $tType,B: $tType] :
( ( ( type_len @ B )
& ( type_len @ A ) )
=> ! [X: word @ A,Y: word @ B] :
( ( ord_less_eq @ ( word @ A ) @ X @ ( semiring_1_unsigned @ B @ ( word @ A ) @ Y ) )
=> ( ord_less_eq @ ( word @ B ) @ ( semiring_1_unsigned @ A @ ( word @ B ) @ X ) @ Y ) ) ) ).
% le_ucast_ucast_le
thf(fact_5144_word__less__nat__alt,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ( ( ord_less @ ( word @ A ) )
= ( ^ [A6: word @ A,B5: word @ A] : ( ord_less @ nat @ ( semiring_1_unsigned @ A @ nat @ A6 ) @ ( semiring_1_unsigned @ A @ nat @ B5 ) ) ) ) ) ).
% word_less_nat_alt
thf(fact_5145_unat__mono,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [A2: word @ A,B2: word @ A] :
( ( ord_less @ ( word @ A ) @ A2 @ B2 )
=> ( ord_less @ nat @ ( semiring_1_unsigned @ A @ nat @ A2 ) @ ( semiring_1_unsigned @ A @ nat @ B2 ) ) ) ) ).
% unat_mono
thf(fact_5146_word__le__nat__alt,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ( ( ord_less_eq @ ( word @ A ) )
= ( ^ [A6: word @ A,B5: word @ A] : ( ord_less_eq @ nat @ ( semiring_1_unsigned @ A @ nat @ A6 ) @ ( semiring_1_unsigned @ A @ nat @ B5 ) ) ) ) ) ).
% word_le_nat_alt
thf(fact_5147_unat__1,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ( ( semiring_1_unsigned @ A @ nat @ ( one_one @ ( word @ A ) ) )
= ( one_one @ nat ) ) ) ).
% unat_1
thf(fact_5148_word__unat__and__lt,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [X: word @ A,N3: nat,Y: word @ A] :
( ( ( ord_less @ nat @ ( semiring_1_unsigned @ A @ nat @ X ) @ N3 )
| ( ord_less @ nat @ ( semiring_1_unsigned @ A @ nat @ Y ) @ N3 ) )
=> ( ord_less @ nat @ ( semiring_1_unsigned @ A @ nat @ ( bit_se5824344872417868541ns_and @ ( word @ A ) @ X @ Y ) ) @ N3 ) ) ) ).
% word_unat_and_lt
thf(fact_5149_le__unat__uoi,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [Y: nat,Z: word @ A] :
( ( ord_less_eq @ nat @ Y @ ( semiring_1_unsigned @ A @ nat @ Z ) )
=> ( ( semiring_1_unsigned @ A @ nat @ ( semiring_1_of_nat @ ( word @ A ) @ Y ) )
= Y ) ) ) ).
% le_unat_uoi
thf(fact_5150_uno__simps_I2_J,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [Z: word @ A,N3: nat] :
( ( semiring_1_unsigned @ A @ nat @ ( semiring_1_of_nat @ ( word @ A ) @ ( divide_divide @ nat @ ( semiring_1_unsigned @ A @ nat @ Z ) @ N3 ) ) )
= ( divide_divide @ nat @ ( semiring_1_unsigned @ A @ nat @ Z ) @ N3 ) ) ) ).
% uno_simps(2)
thf(fact_5151_unat__div,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [X: word @ A,Y: word @ A] :
( ( semiring_1_unsigned @ A @ nat @ ( divide_divide @ ( word @ A ) @ X @ Y ) )
= ( divide_divide @ nat @ ( semiring_1_unsigned @ A @ nat @ X ) @ ( semiring_1_unsigned @ A @ nat @ Y ) ) ) ) ).
% unat_div
thf(fact_5152_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_5153_max__lt,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [A2: word @ A,B2: word @ A,C2: word @ A] :
( ( semiring_1_unsigned @ A @ nat @ ( divide_divide @ ( word @ A ) @ ( ord_max @ ( word @ A ) @ A2 @ B2 ) @ C2 ) )
= ( divide_divide @ nat @ ( semiring_1_unsigned @ A @ nat @ ( ord_max @ ( word @ A ) @ A2 @ B2 ) ) @ ( semiring_1_unsigned @ A @ nat @ C2 ) ) ) ) ).
% max_lt
thf(fact_5154_unat__eq__1,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [X: word @ A] :
( ( ( semiring_1_unsigned @ A @ nat @ X )
= ( suc @ ( zero_zero @ nat ) ) )
= ( X
= ( one_one @ ( word @ A ) ) ) ) ) ).
% unat_eq_1
thf(fact_5155_unat__gt__0,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [X: word @ A] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ ( semiring_1_unsigned @ A @ nat @ X ) )
= ( X
!= ( zero_zero @ ( word @ A ) ) ) ) ) ).
% unat_gt_0
thf(fact_5156_un__ui__le,axiom,
! [A: $tType,B: $tType] :
( ( ( type_len @ B )
& ( type_len @ A ) )
=> ! [A2: word @ A,B2: word @ B] :
( ( ord_less_eq @ nat @ ( semiring_1_unsigned @ A @ nat @ A2 ) @ ( semiring_1_unsigned @ B @ nat @ B2 ) )
= ( ord_less_eq @ int @ ( semiring_1_unsigned @ A @ int @ A2 ) @ ( semiring_1_unsigned @ B @ int @ B2 ) ) ) ) ).
% un_ui_le
thf(fact_5157_unat__plus__simple,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [X: word @ A,Y: word @ A] :
( ( ord_less_eq @ ( word @ A ) @ X @ ( plus_plus @ ( word @ A ) @ X @ Y ) )
= ( ( semiring_1_unsigned @ A @ nat @ ( plus_plus @ ( word @ A ) @ X @ Y ) )
= ( plus_plus @ nat @ ( semiring_1_unsigned @ A @ nat @ X ) @ ( semiring_1_unsigned @ A @ nat @ Y ) ) ) ) ) ).
% unat_plus_simple
thf(fact_5158_unat__sub,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [B2: word @ A,A2: word @ A] :
( ( ord_less_eq @ ( word @ A ) @ B2 @ A2 )
=> ( ( semiring_1_unsigned @ A @ nat @ ( minus_minus @ ( word @ A ) @ A2 @ B2 ) )
= ( minus_minus @ nat @ ( semiring_1_unsigned @ A @ nat @ A2 ) @ ( semiring_1_unsigned @ A @ nat @ B2 ) ) ) ) ) ).
% unat_sub
thf(fact_5159_word__of__nat__less,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [N3: nat,X: word @ A] :
( ( ord_less @ nat @ N3 @ ( semiring_1_unsigned @ A @ nat @ X ) )
=> ( ord_less @ ( word @ A ) @ ( semiring_1_of_nat @ ( word @ A ) @ N3 ) @ X ) ) ) ).
% word_of_nat_less
thf(fact_5160_unat__less__helper,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [X: word @ A,N3: nat] :
( ( ord_less @ ( word @ A ) @ X @ ( semiring_1_of_nat @ ( word @ A ) @ N3 ) )
=> ( ord_less @ nat @ ( semiring_1_unsigned @ A @ nat @ X ) @ N3 ) ) ) ).
% unat_less_helper
thf(fact_5161_word__of__nat__le,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [N3: nat,X: word @ A] :
( ( ord_less_eq @ nat @ N3 @ ( semiring_1_unsigned @ A @ nat @ X ) )
=> ( ord_less_eq @ ( word @ A ) @ ( semiring_1_of_nat @ ( word @ A ) @ N3 ) @ X ) ) ) ).
% word_of_nat_le
thf(fact_5162_word__unat__less__le,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [A2: word @ A,B2: nat] :
( ( ord_less_eq @ ( word @ A ) @ A2 @ ( semiring_1_of_nat @ ( word @ A ) @ B2 ) )
=> ( ord_less_eq @ nat @ ( semiring_1_unsigned @ A @ nat @ A2 ) @ B2 ) ) ) ).
% word_unat_less_le
thf(fact_5163_word__arith__nat__add,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ( ( plus_plus @ ( word @ A ) )
= ( ^ [A6: word @ A,B5: word @ A] : ( semiring_1_of_nat @ ( word @ A ) @ ( plus_plus @ nat @ ( semiring_1_unsigned @ A @ nat @ A6 ) @ ( semiring_1_unsigned @ A @ nat @ B5 ) ) ) ) ) ) ).
% word_arith_nat_add
thf(fact_5164_word__arith__nat__mult,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ( ( times_times @ ( word @ A ) )
= ( ^ [A6: word @ A,B5: word @ A] : ( semiring_1_of_nat @ ( word @ A ) @ ( times_times @ nat @ ( semiring_1_unsigned @ A @ nat @ A6 ) @ ( semiring_1_unsigned @ A @ nat @ B5 ) ) ) ) ) ) ).
% word_arith_nat_mult
thf(fact_5165_word__arith__nat__div,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ( ( divide_divide @ ( word @ A ) )
= ( ^ [A6: word @ A,B5: word @ A] : ( semiring_1_of_nat @ ( word @ A ) @ ( divide_divide @ nat @ ( semiring_1_unsigned @ A @ nat @ A6 ) @ ( semiring_1_unsigned @ A @ nat @ B5 ) ) ) ) ) ) ).
% word_arith_nat_div
thf(fact_5166_ln__neg__is__const,axiom,
! [X: real] :
( ( ord_less_eq @ real @ X @ ( zero_zero @ real ) )
=> ( ( ln_ln @ real @ X )
= ( the @ real
@ ^ [X2: real] : $false ) ) ) ).
% ln_neg_is_const
thf(fact_5167_unat__1__0,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [X: word @ A] :
( ( ord_less_eq @ ( word @ A ) @ ( one_one @ ( word @ A ) ) @ X )
= ( ord_less @ nat @ ( zero_zero @ nat ) @ ( semiring_1_unsigned @ A @ nat @ X ) ) ) ) ).
% unat_1_0
thf(fact_5168_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_5169_unatSuc2,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [N3: word @ A] :
( ( ( plus_plus @ ( word @ A ) @ N3 @ ( one_one @ ( word @ A ) ) )
!= ( zero_zero @ ( word @ A ) ) )
=> ( ( semiring_1_unsigned @ A @ nat @ ( plus_plus @ ( word @ A ) @ N3 @ ( one_one @ ( word @ A ) ) ) )
= ( suc @ ( semiring_1_unsigned @ A @ nat @ N3 ) ) ) ) ) ).
% unatSuc2
thf(fact_5170_unatSuc,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [N3: word @ A] :
( ( ( plus_plus @ ( word @ A ) @ ( one_one @ ( word @ A ) ) @ N3 )
!= ( zero_zero @ ( word @ A ) ) )
=> ( ( semiring_1_unsigned @ A @ nat @ ( plus_plus @ ( word @ A ) @ ( one_one @ ( word @ A ) ) @ N3 ) )
= ( suc @ ( semiring_1_unsigned @ A @ nat @ N3 ) ) ) ) ) ).
% unatSuc
thf(fact_5171_Suc__unat__diff__1,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [X: word @ A] :
( ( ord_less_eq @ ( word @ A ) @ ( one_one @ ( word @ A ) ) @ X )
=> ( ( suc @ ( semiring_1_unsigned @ A @ nat @ ( minus_minus @ ( word @ A ) @ X @ ( one_one @ ( word @ A ) ) ) ) )
= ( semiring_1_unsigned @ A @ nat @ X ) ) ) ) ).
% Suc_unat_diff_1
thf(fact_5172_unat__Suc2,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [N3: word @ A] :
( ( N3
!= ( uminus_uminus @ ( word @ A ) @ ( one_one @ ( word @ A ) ) ) )
=> ( ( semiring_1_unsigned @ A @ nat @ ( plus_plus @ ( word @ A ) @ N3 @ ( one_one @ ( word @ A ) ) ) )
= ( suc @ ( semiring_1_unsigned @ A @ nat @ N3 ) ) ) ) ) ).
% unat_Suc2
thf(fact_5173_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_5174_measure__unat,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [P4: word @ A] :
( ( P4
!= ( zero_zero @ ( word @ A ) ) )
=> ( ord_less @ nat @ ( semiring_1_unsigned @ A @ nat @ ( minus_minus @ ( word @ A ) @ P4 @ ( one_one @ ( word @ A ) ) ) ) @ ( semiring_1_unsigned @ A @ nat @ P4 ) ) ) ) ).
% measure_unat
thf(fact_5175_pred__equals__eq2,axiom,
! [B: $tType,A: $tType,R: set @ ( product_prod @ A @ B ),S3: set @ ( product_prod @ A @ B )] :
( ( ( ^ [X2: A,Y2: B] : ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X2 @ Y2 ) @ R ) )
= ( ^ [X2: A,Y2: B] : ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X2 @ Y2 ) @ S3 ) ) )
= ( R = S3 ) ) ).
% pred_equals_eq2
thf(fact_5176_word__overflow__unat,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [X: word @ A] :
( ( ( semiring_1_unsigned @ A @ nat @ ( plus_plus @ ( word @ A ) @ X @ ( one_one @ ( word @ A ) ) ) )
= ( plus_plus @ nat @ ( semiring_1_unsigned @ A @ nat @ X ) @ ( one_one @ nat ) ) )
| ( ( plus_plus @ ( word @ A ) @ X @ ( one_one @ ( word @ A ) ) )
= ( zero_zero @ ( word @ A ) ) ) ) ) ).
% word_overflow_unat
thf(fact_5177_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_5178_arccos__def,axiom,
( arccos
= ( ^ [Y2: real] :
( the @ real
@ ^ [X2: real] :
( ( ord_less_eq @ real @ ( zero_zero @ real ) @ X2 )
& ( ord_less_eq @ real @ X2 @ pi )
& ( ( cos @ real @ X2 )
= Y2 ) ) ) ) ) ).
% arccos_def
thf(fact_5179_lt__plus__1__le__word,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [N3: nat,MaxBound: word @ A,X: word @ A] :
( ( ord_less @ nat @ N3 @ ( semiring_1_unsigned @ A @ nat @ MaxBound ) )
=> ( ( ord_less @ ( word @ A ) @ X @ ( plus_plus @ ( word @ A ) @ ( one_one @ ( word @ A ) ) @ ( semiring_1_of_nat @ ( word @ A ) @ N3 ) ) )
= ( ord_less_eq @ ( word @ A ) @ X @ ( semiring_1_of_nat @ ( word @ A ) @ N3 ) ) ) ) ) ).
% lt_plus_1_le_word
thf(fact_5180_odd__word__imp__even__next,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [X: word @ A] :
( ~ ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( semiring_1_unsigned @ A @ nat @ X ) )
=> ( ( ( plus_plus @ ( word @ A ) @ X @ ( 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 ) @ X @ ( one_one @ ( word @ A ) ) ) ) ) ) ) ) ).
% odd_word_imp_even_next
thf(fact_5181_even__word__imp__odd__next,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [X: word @ A] :
( ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( semiring_1_unsigned @ A @ nat @ X ) )
=> ( ( ( plus_plus @ ( word @ A ) @ X @ ( 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 ) @ X @ ( one_one @ ( word @ A ) ) ) ) ) ) ) ) ).
% even_word_imp_odd_next
thf(fact_5182_subrelI,axiom,
! [B: $tType,A: $tType,R2: set @ ( product_prod @ A @ B ),S: set @ ( product_prod @ A @ B )] :
( ! [X3: A,Y3: B] :
( ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X3 @ Y3 ) @ R2 )
=> ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X3 @ Y3 ) @ S ) )
=> ( ord_less_eq @ ( set @ ( product_prod @ A @ B ) ) @ R2 @ S ) ) ).
% subrelI
thf(fact_5183_accp__subset__induct,axiom,
! [A: $tType,D4: A > $o,R: A > A > $o,X: A,P: A > $o] :
( ( ord_less_eq @ ( A > $o ) @ D4 @ ( accp @ A @ R ) )
=> ( ! [X3: A,Z2: A] :
( ( D4 @ X3 )
=> ( ( R @ Z2 @ X3 )
=> ( D4 @ Z2 ) ) )
=> ( ( D4 @ X )
=> ( ! [X3: A] :
( ( D4 @ X3 )
=> ( ! [Z4: A] :
( ( R @ Z4 @ X3 )
=> ( P @ Z4 ) )
=> ( P @ X3 ) ) )
=> ( P @ X ) ) ) ) ) ).
% accp_subset_induct
thf(fact_5184_word__div__eq__1__iff,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [N3: word @ A,M: word @ A] :
( ( ( divide_divide @ ( word @ A ) @ N3 @ M )
= ( one_one @ ( word @ A ) ) )
= ( ( ord_less_eq @ ( word @ A ) @ M @ N3 )
& ( ord_less @ nat @ ( semiring_1_unsigned @ A @ nat @ N3 ) @ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( semiring_1_unsigned @ A @ nat @ M ) ) ) ) ) ) ).
% word_div_eq_1_iff
thf(fact_5185_and__mask__dvd__nat,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [N3: nat,W: word @ A] :
( ( dvd_dvd @ nat @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N3 ) @ ( semiring_1_unsigned @ A @ nat @ W ) )
= ( ( bit_se5824344872417868541ns_and @ ( word @ A ) @ W @ ( bit_se2239418461657761734s_mask @ ( word @ A ) @ N3 ) )
= ( zero_zero @ ( word @ A ) ) ) ) ) ).
% and_mask_dvd_nat
thf(fact_5186_of__nat__eq__size,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [N3: nat,W: word @ A] :
( ( ( semiring_1_of_nat @ ( word @ A ) @ N3 )
= W )
= ( ? [Q4: nat] :
( N3
= ( plus_plus @ nat @ ( semiring_1_unsigned @ A @ nat @ W ) @ ( times_times @ nat @ Q4 @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( size_size @ ( word @ A ) @ W ) ) ) ) ) ) ) ) ).
% of_nat_eq_size
thf(fact_5187_bot__empty__eq2,axiom,
! [B: $tType,A: $tType] :
( ( bot_bot @ ( A > B > $o ) )
= ( ^ [X2: A,Y2: B] : ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X2 @ Y2 ) @ ( bot_bot @ ( set @ ( product_prod @ A @ B ) ) ) ) ) ) ).
% bot_empty_eq2
thf(fact_5188_pi__half,axiom,
( ( divide_divide @ real @ pi @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) )
= ( the @ real
@ ^ [X2: real] :
( ( ord_less_eq @ real @ ( zero_zero @ real ) @ X2 )
& ( ord_less_eq @ real @ X2 @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) )
& ( ( cos @ real @ X2 )
= ( zero_zero @ real ) ) ) ) ) ).
% pi_half
thf(fact_5189_pi__def,axiom,
( pi
= ( times_times @ real @ ( numeral_numeral @ real @ ( bit0 @ one2 ) )
@ ( the @ real
@ ^ [X2: real] :
( ( ord_less_eq @ real @ ( zero_zero @ real ) @ X2 )
& ( ord_less_eq @ real @ X2 @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) )
& ( ( cos @ real @ X2 )
= ( zero_zero @ real ) ) ) ) ) ) ).
% pi_def
thf(fact_5190_unat__plus__if__size,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [X: word @ A,Y: word @ A] :
( ( ( ord_less @ nat @ ( plus_plus @ nat @ ( semiring_1_unsigned @ A @ nat @ X ) @ ( semiring_1_unsigned @ A @ nat @ Y ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( size_size @ ( word @ A ) @ X ) ) )
=> ( ( semiring_1_unsigned @ A @ nat @ ( plus_plus @ ( word @ A ) @ X @ Y ) )
= ( plus_plus @ nat @ ( semiring_1_unsigned @ A @ nat @ X ) @ ( semiring_1_unsigned @ A @ nat @ Y ) ) ) )
& ( ~ ( ord_less @ nat @ ( plus_plus @ nat @ ( semiring_1_unsigned @ A @ nat @ X ) @ ( semiring_1_unsigned @ A @ nat @ Y ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( size_size @ ( word @ A ) @ X ) ) )
=> ( ( semiring_1_unsigned @ A @ nat @ ( plus_plus @ ( word @ A ) @ X @ Y ) )
= ( minus_minus @ nat @ ( plus_plus @ nat @ ( semiring_1_unsigned @ A @ nat @ X ) @ ( semiring_1_unsigned @ A @ nat @ Y ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( size_size @ ( word @ A ) @ X ) ) ) ) ) ) ) ).
% unat_plus_if_size
thf(fact_5191_unat__sub__if__size,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [Y: word @ A,X: word @ A] :
( ( ( ord_less_eq @ nat @ ( semiring_1_unsigned @ A @ nat @ Y ) @ ( semiring_1_unsigned @ A @ nat @ X ) )
=> ( ( semiring_1_unsigned @ A @ nat @ ( minus_minus @ ( word @ A ) @ X @ Y ) )
= ( minus_minus @ nat @ ( semiring_1_unsigned @ A @ nat @ X ) @ ( semiring_1_unsigned @ A @ nat @ Y ) ) ) )
& ( ~ ( ord_less_eq @ nat @ ( semiring_1_unsigned @ A @ nat @ Y ) @ ( semiring_1_unsigned @ A @ nat @ X ) )
=> ( ( semiring_1_unsigned @ A @ nat @ ( minus_minus @ ( word @ A ) @ X @ Y ) )
= ( minus_minus @ nat @ ( plus_plus @ nat @ ( semiring_1_unsigned @ A @ nat @ X ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( size_size @ ( word @ A ) @ X ) ) ) @ ( semiring_1_unsigned @ A @ nat @ Y ) ) ) ) ) ) ).
% unat_sub_if_size
thf(fact_5192_no__plus__overflow__unat__size,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [X: word @ A,Y: word @ A] :
( ( ord_less_eq @ ( word @ A ) @ X @ ( plus_plus @ ( word @ A ) @ X @ Y ) )
= ( ord_less @ nat @ ( plus_plus @ nat @ ( semiring_1_unsigned @ A @ nat @ X ) @ ( semiring_1_unsigned @ A @ nat @ Y ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( size_size @ ( word @ A ) @ X ) ) ) ) ) ).
% no_plus_overflow_unat_size
thf(fact_5193_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_5194_pred__subset__eq,axiom,
! [A: $tType,R: set @ A,S3: set @ A] :
( ( ord_less_eq @ ( A > $o )
@ ^ [X2: A] : ( member @ A @ X2 @ R )
@ ^ [X2: A] : ( member @ A @ X2 @ S3 ) )
= ( ord_less_eq @ ( set @ A ) @ R @ S3 ) ) ).
% pred_subset_eq
thf(fact_5195_arcsin__def,axiom,
( arcsin
= ( ^ [Y2: real] :
( the @ real
@ ^ [X2: real] :
( ( ord_less_eq @ real @ ( uminus_uminus @ real @ ( divide_divide @ real @ pi @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) ) @ X2 )
& ( ord_less_eq @ real @ X2 @ ( divide_divide @ real @ pi @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) )
& ( ( sin @ real @ X2 )
= Y2 ) ) ) ) ) ).
% arcsin_def
thf(fact_5196_VEBT__internal_Ospace_Opelims,axiom,
! [X: vEBT_VEBT,Y: nat] :
( ( ( vEBT_VEBT_space @ X )
= Y )
=> ( ( accp @ vEBT_VEBT @ vEBT_VEBT_space_rel2 @ X )
=> ( ! [A4: $o,B3: $o] :
( ( X
= ( vEBT_Leaf @ A4 @ B3 ) )
=> ( ( Y
= ( numeral_numeral @ nat @ ( bit1 @ one2 ) ) )
=> ~ ( accp @ vEBT_VEBT @ vEBT_VEBT_space_rel2 @ ( vEBT_Leaf @ A4 @ B3 ) ) ) )
=> ~ ! [Info2: option @ ( product_prod @ nat @ nat ),Deg2: nat,TreeList2: list @ vEBT_VEBT,Summary2: vEBT_VEBT] :
( ( X
= ( vEBT_Node @ Info2 @ Deg2 @ TreeList2 @ Summary2 ) )
=> ( ( Y
= ( plus_plus @ nat @ ( plus_plus @ nat @ ( plus_plus @ nat @ ( numeral_numeral @ nat @ ( bit1 @ ( bit0 @ one2 ) ) ) @ ( vEBT_VEBT_space @ Summary2 ) ) @ ( size_size @ ( list @ vEBT_VEBT ) @ TreeList2 ) ) @ ( foldr @ nat @ nat @ ( plus_plus @ nat ) @ ( map @ vEBT_VEBT @ nat @ vEBT_VEBT_space @ TreeList2 ) @ ( zero_zero @ nat ) ) ) )
=> ~ ( accp @ vEBT_VEBT @ vEBT_VEBT_space_rel2 @ ( vEBT_Node @ Info2 @ Deg2 @ TreeList2 @ Summary2 ) ) ) ) ) ) ) ).
% VEBT_internal.space.pelims
thf(fact_5197_VEBT__internal_OT__vebt__buildupi_H_Opelims,axiom,
! [X: nat,Y: int] :
( ( ( vEBT_V9176841429113362141ildupi @ X )
= Y )
=> ( ( accp @ nat @ vEBT_V3352910403632780892pi_rel @ X )
=> ( ( ( X
= ( zero_zero @ nat ) )
=> ( ( Y
= ( one_one @ int ) )
=> ~ ( accp @ nat @ vEBT_V3352910403632780892pi_rel @ ( zero_zero @ nat ) ) ) )
=> ( ( ( X
= ( suc @ ( zero_zero @ nat ) ) )
=> ( ( Y
= ( one_one @ int ) )
=> ~ ( accp @ nat @ vEBT_V3352910403632780892pi_rel @ ( suc @ ( zero_zero @ nat ) ) ) ) )
=> ~ ! [N: nat] :
( ( X
= ( suc @ ( suc @ N ) ) )
=> ( ( ( ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N )
=> ( Y
= ( 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 )
=> ( Y
= ( 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 ) ) ) ) ) ) ) ) ) ) ) )
=> ~ ( accp @ nat @ vEBT_V3352910403632780892pi_rel @ ( suc @ ( suc @ N ) ) ) ) ) ) ) ) ) ).
% VEBT_internal.T_vebt_buildupi'.pelims
thf(fact_5198_vebt__buildup_Opelims,axiom,
! [X: nat,Y: vEBT_VEBT] :
( ( ( vEBT_vebt_buildup @ X )
= Y )
=> ( ( accp @ nat @ vEBT_v4011308405150292612up_rel @ X )
=> ( ( ( X
= ( zero_zero @ nat ) )
=> ( ( Y
= ( vEBT_Leaf @ $false @ $false ) )
=> ~ ( accp @ nat @ vEBT_v4011308405150292612up_rel @ ( zero_zero @ nat ) ) ) )
=> ( ( ( X
= ( suc @ ( zero_zero @ nat ) ) )
=> ( ( Y
= ( vEBT_Leaf @ $false @ $false ) )
=> ~ ( accp @ nat @ vEBT_v4011308405150292612up_rel @ ( suc @ ( zero_zero @ nat ) ) ) ) )
=> ~ ! [Va3: nat] :
( ( X
= ( suc @ ( suc @ Va3 ) ) )
=> ( ( ( ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( suc @ ( suc @ Va3 ) ) )
=> ( Y
= ( 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 ) ) )
=> ( Y
= ( 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_5199_VEBT__internal_Ospace_H_Opelims,axiom,
! [X: vEBT_VEBT,Y: nat] :
( ( ( vEBT_VEBT_space2 @ X )
= Y )
=> ( ( accp @ vEBT_VEBT @ vEBT_VEBT_space_rel @ X )
=> ( ! [A4: $o,B3: $o] :
( ( X
= ( vEBT_Leaf @ A4 @ B3 ) )
=> ( ( Y
= ( numeral_numeral @ nat @ ( bit0 @ ( bit0 @ one2 ) ) ) )
=> ~ ( accp @ vEBT_VEBT @ vEBT_VEBT_space_rel @ ( vEBT_Leaf @ A4 @ B3 ) ) ) )
=> ~ ! [Info2: option @ ( product_prod @ nat @ nat ),Deg2: nat,TreeList2: list @ vEBT_VEBT,Summary2: vEBT_VEBT] :
( ( X
= ( vEBT_Node @ Info2 @ Deg2 @ TreeList2 @ Summary2 ) )
=> ( ( Y
= ( plus_plus @ nat @ ( plus_plus @ nat @ ( numeral_numeral @ nat @ ( bit0 @ ( bit1 @ one2 ) ) ) @ ( vEBT_VEBT_space2 @ Summary2 ) ) @ ( foldr @ nat @ nat @ ( plus_plus @ nat ) @ ( map @ vEBT_VEBT @ nat @ vEBT_VEBT_space2 @ TreeList2 ) @ ( zero_zero @ nat ) ) ) )
=> ~ ( accp @ vEBT_VEBT @ vEBT_VEBT_space_rel @ ( vEBT_Node @ Info2 @ Deg2 @ TreeList2 @ Summary2 ) ) ) ) ) ) ) ).
% VEBT_internal.space'.pelims
thf(fact_5200_VEBT__internal_OT_092_060_094sub_062b_092_060_094sub_062u_092_060_094sub_062i_092_060_094sub_062l_092_060_094sub_062d_Opelims,axiom,
! [X: nat,Y: nat] :
( ( ( vEBT_V8646137997579335489_i_l_d @ X )
= Y )
=> ( ( accp @ nat @ vEBT_V5144397997797733112_d_rel @ X )
=> ( ( ( X
= ( zero_zero @ nat ) )
=> ( ( Y
= ( numeral_numeral @ nat @ ( bit0 @ ( bit0 @ one2 ) ) ) )
=> ~ ( accp @ nat @ vEBT_V5144397997797733112_d_rel @ ( zero_zero @ nat ) ) ) )
=> ( ( ( X
= ( suc @ ( zero_zero @ nat ) ) )
=> ( ( Y
= ( numeral_numeral @ nat @ ( bit0 @ ( bit0 @ one2 ) ) ) )
=> ~ ( accp @ nat @ vEBT_V5144397997797733112_d_rel @ ( suc @ ( zero_zero @ nat ) ) ) ) )
=> ~ ! [Va3: nat] :
( ( X
= ( suc @ ( suc @ Va3 ) ) )
=> ( ( ( ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( suc @ ( suc @ Va3 ) ) )
=> ( Y
= ( plus_plus @ nat @ ( one_one @ nat ) @ ( plus_plus @ nat @ ( plus_plus @ nat @ ( numeral_numeral @ nat @ ( bit0 @ ( bit1 @ ( bit0 @ one2 ) ) ) ) @ ( vEBT_V8646137997579335489_i_l_d @ ( divide_divide @ nat @ ( suc @ ( suc @ 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 ) ) )
=> ( Y
= ( plus_plus @ nat @ ( plus_plus @ nat @ ( numeral_numeral @ nat @ ( bit0 @ ( bit0 @ ( bit1 @ one2 ) ) ) ) @ ( vEBT_V8646137997579335489_i_l_d @ ( suc @ ( divide_divide @ nat @ ( suc @ ( suc @ 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_5201_VEBT__internal_OT_092_060_094sub_062b_092_060_094sub_062u_092_060_094sub_062i_092_060_094sub_062l_092_060_094sub_062d_092_060_094sub_062u_092_060_094sub_062p_Opelims,axiom,
! [X: nat,Y: nat] :
( ( ( vEBT_V8346862874174094_d_u_p @ X )
= Y )
=> ( ( accp @ nat @ vEBT_V1247956027447740395_p_rel @ X )
=> ( ( ( X
= ( zero_zero @ nat ) )
=> ( ( Y
= ( numeral_numeral @ nat @ ( bit1 @ one2 ) ) )
=> ~ ( accp @ nat @ vEBT_V1247956027447740395_p_rel @ ( zero_zero @ nat ) ) ) )
=> ( ( ( X
= ( suc @ ( zero_zero @ nat ) ) )
=> ( ( Y
= ( numeral_numeral @ nat @ ( bit1 @ one2 ) ) )
=> ~ ( accp @ nat @ vEBT_V1247956027447740395_p_rel @ ( suc @ ( zero_zero @ nat ) ) ) ) )
=> ~ ! [Va3: nat] :
( ( X
= ( suc @ ( suc @ Va3 ) ) )
=> ( ( ( ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( suc @ ( suc @ Va3 ) ) )
=> ( Y
= ( plus_plus @ nat @ ( one_one @ nat ) @ ( plus_plus @ nat @ ( plus_plus @ nat @ ( numeral_numeral @ nat @ ( bit1 @ ( bit0 @ ( bit0 @ one2 ) ) ) ) @ ( vEBT_V8346862874174094_d_u_p @ ( divide_divide @ nat @ ( suc @ ( suc @ 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 ) ) )
=> ( Y
= ( plus_plus @ nat @ ( plus_plus @ nat @ ( numeral_numeral @ nat @ ( bit1 @ ( bit1 @ ( bit0 @ one2 ) ) ) ) @ ( vEBT_V8346862874174094_d_u_p @ ( suc @ ( divide_divide @ nat @ ( suc @ ( suc @ 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_5202_VEBT__internal_OTb_Opelims,axiom,
! [X: nat,Y: int] :
( ( ( vEBT_VEBT_Tb @ X )
= Y )
=> ( ( accp @ nat @ vEBT_VEBT_Tb_rel2 @ X )
=> ( ( ( X
= ( zero_zero @ nat ) )
=> ( ( Y
= ( numeral_numeral @ int @ ( bit1 @ one2 ) ) )
=> ~ ( accp @ nat @ vEBT_VEBT_Tb_rel2 @ ( zero_zero @ nat ) ) ) )
=> ( ( ( X
= ( suc @ ( zero_zero @ nat ) ) )
=> ( ( Y
= ( numeral_numeral @ int @ ( bit1 @ one2 ) ) )
=> ~ ( accp @ nat @ vEBT_VEBT_Tb_rel2 @ ( suc @ ( zero_zero @ nat ) ) ) ) )
=> ~ ! [N: nat] :
( ( X
= ( suc @ ( suc @ N ) ) )
=> ( ( ( ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N )
=> ( Y
= ( 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 )
=> ( Y
= ( 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 ) ) ) ) ) ) ) ) ) ) )
=> ~ ( accp @ nat @ vEBT_VEBT_Tb_rel2 @ ( suc @ ( suc @ N ) ) ) ) ) ) ) ) ) ).
% VEBT_internal.Tb.pelims
thf(fact_5203_VEBT__internal_OTb_H_Opelims,axiom,
! [X: nat,Y: nat] :
( ( ( vEBT_VEBT_Tb2 @ X )
= Y )
=> ( ( accp @ nat @ vEBT_VEBT_Tb_rel @ X )
=> ( ( ( X
= ( zero_zero @ nat ) )
=> ( ( Y
= ( numeral_numeral @ nat @ ( bit1 @ one2 ) ) )
=> ~ ( accp @ nat @ vEBT_VEBT_Tb_rel @ ( zero_zero @ nat ) ) ) )
=> ( ( ( X
= ( suc @ ( zero_zero @ nat ) ) )
=> ( ( Y
= ( numeral_numeral @ nat @ ( bit1 @ one2 ) ) )
=> ~ ( accp @ nat @ vEBT_VEBT_Tb_rel @ ( suc @ ( zero_zero @ nat ) ) ) ) )
=> ~ ! [N: nat] :
( ( X
= ( suc @ ( suc @ N ) ) )
=> ( ( ( ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N )
=> ( Y
= ( 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 )
=> ( Y
= ( 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 ) ) ) ) ) ) ) ) ) ) )
=> ~ ( accp @ nat @ vEBT_VEBT_Tb_rel @ ( suc @ ( suc @ N ) ) ) ) ) ) ) ) ) ).
% VEBT_internal.Tb'.pelims
thf(fact_5204_VEBT__internal_Ocnt_Opelims,axiom,
! [X: vEBT_VEBT,Y: real] :
( ( ( vEBT_VEBT_cnt @ X )
= Y )
=> ( ( accp @ vEBT_VEBT @ vEBT_VEBT_cnt_rel2 @ X )
=> ( ! [A4: $o,B3: $o] :
( ( X
= ( vEBT_Leaf @ A4 @ B3 ) )
=> ( ( Y
= ( one_one @ real ) )
=> ~ ( accp @ vEBT_VEBT @ vEBT_VEBT_cnt_rel2 @ ( vEBT_Leaf @ A4 @ B3 ) ) ) )
=> ~ ! [Info2: option @ ( product_prod @ nat @ nat ),Deg2: nat,TreeList2: list @ vEBT_VEBT,Summary2: vEBT_VEBT] :
( ( X
= ( vEBT_Node @ Info2 @ Deg2 @ TreeList2 @ Summary2 ) )
=> ( ( Y
= ( plus_plus @ real @ ( plus_plus @ real @ ( one_one @ real ) @ ( vEBT_VEBT_cnt @ Summary2 ) ) @ ( foldr @ real @ real @ ( plus_plus @ real ) @ ( map @ vEBT_VEBT @ real @ vEBT_VEBT_cnt @ TreeList2 ) @ ( zero_zero @ real ) ) ) )
=> ~ ( accp @ vEBT_VEBT @ vEBT_VEBT_cnt_rel2 @ ( vEBT_Node @ Info2 @ Deg2 @ TreeList2 @ Summary2 ) ) ) ) ) ) ) ).
% VEBT_internal.cnt.pelims
thf(fact_5205_VEBT__internal_OT__vebt__buildupi_Opelims,axiom,
! [X: nat,Y: nat] :
( ( ( vEBT_V441764108873111860ildupi @ X )
= Y )
=> ( ( accp @ nat @ vEBT_V2957053500504383685pi_rel @ X )
=> ( ( ( X
= ( zero_zero @ nat ) )
=> ( ( Y
= ( suc @ ( zero_zero @ nat ) ) )
=> ~ ( accp @ nat @ vEBT_V2957053500504383685pi_rel @ ( zero_zero @ nat ) ) ) )
=> ( ( ( X
= ( suc @ ( zero_zero @ nat ) ) )
=> ( ( Y
= ( suc @ ( zero_zero @ nat ) ) )
=> ~ ( accp @ nat @ vEBT_V2957053500504383685pi_rel @ ( suc @ ( zero_zero @ nat ) ) ) ) )
=> ~ ! [N: nat] :
( ( X
= ( suc @ ( suc @ N ) ) )
=> ( ( ( ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N )
=> ( Y
= ( 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 )
=> ( Y
= ( 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 ) ) ) ) ) ) ) ) ) ) ) ) ) )
=> ~ ( accp @ nat @ vEBT_V2957053500504383685pi_rel @ ( suc @ ( suc @ N ) ) ) ) ) ) ) ) ) ).
% VEBT_internal.T_vebt_buildupi.pelims
thf(fact_5206_VEBT__internal_Ocnt_H_Opelims,axiom,
! [X: vEBT_VEBT,Y: nat] :
( ( ( vEBT_VEBT_cnt2 @ X )
= Y )
=> ( ( accp @ vEBT_VEBT @ vEBT_VEBT_cnt_rel @ X )
=> ( ! [A4: $o,B3: $o] :
( ( X
= ( vEBT_Leaf @ A4 @ B3 ) )
=> ( ( Y
= ( one_one @ nat ) )
=> ~ ( accp @ vEBT_VEBT @ vEBT_VEBT_cnt_rel @ ( vEBT_Leaf @ A4 @ B3 ) ) ) )
=> ~ ! [Info2: option @ ( product_prod @ nat @ nat ),Deg2: nat,TreeList2: list @ vEBT_VEBT,Summary2: vEBT_VEBT] :
( ( X
= ( vEBT_Node @ Info2 @ Deg2 @ TreeList2 @ Summary2 ) )
=> ( ( Y
= ( plus_plus @ nat @ ( plus_plus @ nat @ ( one_one @ nat ) @ ( vEBT_VEBT_cnt2 @ Summary2 ) ) @ ( foldr @ nat @ nat @ ( plus_plus @ nat ) @ ( map @ vEBT_VEBT @ nat @ vEBT_VEBT_cnt2 @ TreeList2 ) @ ( zero_zero @ nat ) ) ) )
=> ~ ( accp @ vEBT_VEBT @ vEBT_VEBT_cnt_rel @ ( vEBT_Node @ Info2 @ Deg2 @ TreeList2 @ Summary2 ) ) ) ) ) ) ) ).
% VEBT_internal.cnt'.pelims
thf(fact_5207_vebt__maxti_Opelims,axiom,
! [X: vEBT_VEBTi,Y: heap_Time_Heap @ ( option @ nat )] :
( ( ( vEBT_vebt_maxti @ X )
= Y )
=> ( ( accp @ vEBT_VEBTi @ vEBT_vebt_maxti_rel @ X )
=> ( ! [A4: $o,B3: $o] :
( ( X
= ( vEBT_Leafi @ A4 @ B3 ) )
=> ( ( ( B3
=> ( Y
= ( heap_Time_return @ ( option @ nat ) @ ( some @ nat @ ( one_one @ nat ) ) ) ) )
& ( ~ B3
=> ( ( A4
=> ( Y
= ( heap_Time_return @ ( option @ nat ) @ ( some @ nat @ ( zero_zero @ nat ) ) ) ) )
& ( ~ A4
=> ( Y
= ( heap_Time_return @ ( option @ nat ) @ ( none @ nat ) ) ) ) ) ) )
=> ~ ( accp @ vEBT_VEBTi @ vEBT_vebt_maxti_rel @ ( vEBT_Leafi @ A4 @ B3 ) ) ) )
=> ( ! [Uu: nat,Uv: array @ vEBT_VEBTi,Uw: vEBT_VEBTi] :
( ( X
= ( vEBT_Nodei @ ( none @ ( product_prod @ nat @ nat ) ) @ Uu @ Uv @ Uw ) )
=> ( ( Y
= ( heap_Time_return @ ( option @ nat ) @ ( none @ nat ) ) )
=> ~ ( accp @ vEBT_VEBTi @ vEBT_vebt_maxti_rel @ ( vEBT_Nodei @ ( none @ ( product_prod @ nat @ nat ) ) @ Uu @ Uv @ Uw ) ) ) )
=> ~ ! [Mi2: nat,Ma2: nat,Ux2: nat,Uy2: array @ vEBT_VEBTi,Uz2: vEBT_VEBTi] :
( ( X
= ( vEBT_Nodei @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi2 @ Ma2 ) ) @ Ux2 @ Uy2 @ Uz2 ) )
=> ( ( Y
= ( heap_Time_return @ ( option @ nat ) @ ( some @ nat @ Ma2 ) ) )
=> ~ ( accp @ vEBT_VEBTi @ vEBT_vebt_maxti_rel @ ( vEBT_Nodei @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi2 @ Ma2 ) ) @ Ux2 @ Uy2 @ Uz2 ) ) ) ) ) ) ) ) ).
% vebt_maxti.pelims
thf(fact_5208_vebt__minti_Opelims,axiom,
! [X: vEBT_VEBTi,Y: heap_Time_Heap @ ( option @ nat )] :
( ( ( vEBT_vebt_minti @ X )
= Y )
=> ( ( accp @ vEBT_VEBTi @ vEBT_vebt_minti_rel @ X )
=> ( ! [A4: $o,B3: $o] :
( ( X
= ( vEBT_Leafi @ A4 @ B3 ) )
=> ( ( ( A4
=> ( Y
= ( heap_Time_return @ ( option @ nat ) @ ( some @ nat @ ( zero_zero @ nat ) ) ) ) )
& ( ~ A4
=> ( ( B3
=> ( Y
= ( heap_Time_return @ ( option @ nat ) @ ( some @ nat @ ( one_one @ nat ) ) ) ) )
& ( ~ B3
=> ( Y
= ( heap_Time_return @ ( option @ nat ) @ ( none @ nat ) ) ) ) ) ) )
=> ~ ( accp @ vEBT_VEBTi @ vEBT_vebt_minti_rel @ ( vEBT_Leafi @ A4 @ B3 ) ) ) )
=> ( ! [Uu: nat,Uv: array @ vEBT_VEBTi,Uw: vEBT_VEBTi] :
( ( X
= ( vEBT_Nodei @ ( none @ ( product_prod @ nat @ nat ) ) @ Uu @ Uv @ Uw ) )
=> ( ( Y
= ( heap_Time_return @ ( option @ nat ) @ ( none @ nat ) ) )
=> ~ ( accp @ vEBT_VEBTi @ vEBT_vebt_minti_rel @ ( vEBT_Nodei @ ( none @ ( product_prod @ nat @ nat ) ) @ Uu @ Uv @ Uw ) ) ) )
=> ~ ! [Mi2: nat,Ma2: nat,Ux2: nat,Uy2: array @ vEBT_VEBTi,Uz2: vEBT_VEBTi] :
( ( X
= ( vEBT_Nodei @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi2 @ Ma2 ) ) @ Ux2 @ Uy2 @ Uz2 ) )
=> ( ( Y
= ( heap_Time_return @ ( option @ nat ) @ ( some @ nat @ Mi2 ) ) )
=> ~ ( accp @ vEBT_VEBTi @ vEBT_vebt_minti_rel @ ( vEBT_Nodei @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi2 @ Ma2 ) ) @ Ux2 @ Uy2 @ Uz2 ) ) ) ) ) ) ) ) ).
% vebt_minti.pelims
thf(fact_5209_vebt__maxt_Opelims,axiom,
! [X: vEBT_VEBT,Y: option @ nat] :
( ( ( vEBT_vebt_maxt @ X )
= Y )
=> ( ( accp @ vEBT_VEBT @ vEBT_vebt_maxt_rel @ X )
=> ( ! [A4: $o,B3: $o] :
( ( X
= ( vEBT_Leaf @ A4 @ B3 ) )
=> ( ( ( B3
=> ( Y
= ( some @ nat @ ( one_one @ nat ) ) ) )
& ( ~ B3
=> ( ( A4
=> ( Y
= ( some @ nat @ ( zero_zero @ nat ) ) ) )
& ( ~ A4
=> ( Y
= ( none @ nat ) ) ) ) ) )
=> ~ ( accp @ vEBT_VEBT @ vEBT_vebt_maxt_rel @ ( vEBT_Leaf @ A4 @ B3 ) ) ) )
=> ( ! [Uu: nat,Uv: list @ vEBT_VEBT,Uw: vEBT_VEBT] :
( ( X
= ( vEBT_Node @ ( none @ ( product_prod @ nat @ nat ) ) @ Uu @ Uv @ Uw ) )
=> ( ( Y
= ( none @ nat ) )
=> ~ ( accp @ vEBT_VEBT @ vEBT_vebt_maxt_rel @ ( vEBT_Node @ ( none @ ( product_prod @ nat @ nat ) ) @ Uu @ Uv @ Uw ) ) ) )
=> ~ ! [Mi2: nat,Ma2: nat,Ux2: nat,Uy2: list @ vEBT_VEBT,Uz2: vEBT_VEBT] :
( ( X
= ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi2 @ Ma2 ) ) @ Ux2 @ Uy2 @ Uz2 ) )
=> ( ( Y
= ( some @ nat @ Ma2 ) )
=> ~ ( accp @ vEBT_VEBT @ vEBT_vebt_maxt_rel @ ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi2 @ Ma2 ) ) @ Ux2 @ Uy2 @ Uz2 ) ) ) ) ) ) ) ) ).
% vebt_maxt.pelims
thf(fact_5210_vebt__mint_Opelims,axiom,
! [X: vEBT_VEBT,Y: option @ nat] :
( ( ( vEBT_vebt_mint @ X )
= Y )
=> ( ( accp @ vEBT_VEBT @ vEBT_vebt_mint_rel @ X )
=> ( ! [A4: $o,B3: $o] :
( ( X
= ( vEBT_Leaf @ A4 @ B3 ) )
=> ( ( ( A4
=> ( Y
= ( some @ nat @ ( zero_zero @ nat ) ) ) )
& ( ~ A4
=> ( ( B3
=> ( Y
= ( some @ nat @ ( one_one @ nat ) ) ) )
& ( ~ B3
=> ( Y
= ( none @ nat ) ) ) ) ) )
=> ~ ( accp @ vEBT_VEBT @ vEBT_vebt_mint_rel @ ( vEBT_Leaf @ A4 @ B3 ) ) ) )
=> ( ! [Uu: nat,Uv: list @ vEBT_VEBT,Uw: vEBT_VEBT] :
( ( X
= ( vEBT_Node @ ( none @ ( product_prod @ nat @ nat ) ) @ Uu @ Uv @ Uw ) )
=> ( ( Y
= ( none @ nat ) )
=> ~ ( accp @ vEBT_VEBT @ vEBT_vebt_mint_rel @ ( vEBT_Node @ ( none @ ( product_prod @ nat @ nat ) ) @ Uu @ Uv @ Uw ) ) ) )
=> ~ ! [Mi2: nat,Ma2: nat,Ux2: nat,Uy2: list @ vEBT_VEBT,Uz2: vEBT_VEBT] :
( ( X
= ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi2 @ Ma2 ) ) @ Ux2 @ Uy2 @ Uz2 ) )
=> ( ( Y
= ( some @ nat @ Mi2 ) )
=> ~ ( accp @ vEBT_VEBT @ vEBT_vebt_mint_rel @ ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi2 @ Ma2 ) ) @ Ux2 @ Uy2 @ Uz2 ) ) ) ) ) ) ) ) ).
% vebt_mint.pelims
thf(fact_5211_T_092_060_094sub_062m_092_060_094sub_062i_092_060_094sub_062n_092_060_094sub_062t_Opelims,axiom,
! [X: vEBT_VEBT,Y: nat] :
( ( ( vEBT_T_m_i_n_t @ X )
= Y )
=> ( ( accp @ vEBT_VEBT @ vEBT_T_m_i_n_t_rel @ X )
=> ( ! [A4: $o,B3: $o] :
( ( X
= ( vEBT_Leaf @ A4 @ B3 ) )
=> ( ( Y
= ( plus_plus @ nat @ ( one_one @ nat ) @ ( if @ nat @ A4 @ ( zero_zero @ nat ) @ ( plus_plus @ nat @ ( one_one @ nat ) @ ( one_one @ nat ) ) ) ) )
=> ~ ( accp @ vEBT_VEBT @ vEBT_T_m_i_n_t_rel @ ( vEBT_Leaf @ A4 @ B3 ) ) ) )
=> ( ! [Uu: nat,Uv: list @ vEBT_VEBT,Uw: vEBT_VEBT] :
( ( X
= ( vEBT_Node @ ( none @ ( product_prod @ nat @ nat ) ) @ Uu @ Uv @ Uw ) )
=> ( ( Y
= ( one_one @ nat ) )
=> ~ ( accp @ vEBT_VEBT @ vEBT_T_m_i_n_t_rel @ ( vEBT_Node @ ( none @ ( product_prod @ nat @ nat ) ) @ Uu @ Uv @ Uw ) ) ) )
=> ~ ! [Mi2: nat,Ma2: nat,Ux2: nat,Uy2: list @ vEBT_VEBT,Uz2: vEBT_VEBT] :
( ( X
= ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi2 @ Ma2 ) ) @ Ux2 @ Uy2 @ Uz2 ) )
=> ( ( Y
= ( one_one @ nat ) )
=> ~ ( accp @ vEBT_VEBT @ vEBT_T_m_i_n_t_rel @ ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi2 @ Ma2 ) ) @ Ux2 @ Uy2 @ Uz2 ) ) ) ) ) ) ) ) ).
% T\<^sub>m\<^sub>i\<^sub>n\<^sub>t.pelims
thf(fact_5212_T_092_060_094sub_062m_092_060_094sub_062a_092_060_094sub_062x_092_060_094sub_062t_Opelims,axiom,
! [X: vEBT_VEBT,Y: nat] :
( ( ( vEBT_T_m_a_x_t @ X )
= Y )
=> ( ( accp @ vEBT_VEBT @ vEBT_T_m_a_x_t_rel @ X )
=> ( ! [A4: $o,B3: $o] :
( ( X
= ( vEBT_Leaf @ A4 @ B3 ) )
=> ( ( Y
= ( plus_plus @ nat @ ( one_one @ nat ) @ ( if @ nat @ B3 @ ( 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 @ B3 ) ) ) )
=> ( ! [Uu: nat,Uv: list @ vEBT_VEBT,Uw: vEBT_VEBT] :
( ( X
= ( vEBT_Node @ ( none @ ( product_prod @ nat @ nat ) ) @ Uu @ Uv @ Uw ) )
=> ( ( Y
= ( one_one @ nat ) )
=> ~ ( accp @ vEBT_VEBT @ vEBT_T_m_a_x_t_rel @ ( vEBT_Node @ ( none @ ( product_prod @ nat @ nat ) ) @ Uu @ Uv @ Uw ) ) ) )
=> ~ ! [Mi2: nat,Ma2: nat,Ux2: nat,Uy2: list @ vEBT_VEBT,Uz2: vEBT_VEBT] :
( ( X
= ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi2 @ Ma2 ) ) @ Ux2 @ Uy2 @ Uz2 ) )
=> ( ( Y
= ( one_one @ nat ) )
=> ~ ( accp @ vEBT_VEBT @ vEBT_T_m_a_x_t_rel @ ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi2 @ Ma2 ) ) @ Ux2 @ Uy2 @ Uz2 ) ) ) ) ) ) ) ) ).
% T\<^sub>m\<^sub>a\<^sub>x\<^sub>t.pelims
thf(fact_5213_fold__atLeastAtMost__nat_Opsimps,axiom,
! [A: $tType,F2: nat > A > A,A2: nat,B2: nat,Acc3: A] :
( ( accp @ ( product_prod @ ( nat > A > A ) @ ( product_prod @ nat @ ( product_prod @ nat @ A ) ) ) @ ( set_fo1817059534552279752at_rel @ A ) @ ( product_Pair @ ( nat > A > A ) @ ( product_prod @ nat @ ( product_prod @ nat @ A ) ) @ F2 @ ( product_Pair @ nat @ ( product_prod @ nat @ A ) @ A2 @ ( product_Pair @ nat @ A @ B2 @ Acc3 ) ) ) )
=> ( ( ( ord_less @ nat @ B2 @ A2 )
=> ( ( set_fo6178422350223883121st_nat @ A @ F2 @ A2 @ B2 @ Acc3 )
= Acc3 ) )
& ( ~ ( ord_less @ nat @ B2 @ A2 )
=> ( ( set_fo6178422350223883121st_nat @ A @ F2 @ A2 @ B2 @ Acc3 )
= ( set_fo6178422350223883121st_nat @ A @ F2 @ ( plus_plus @ nat @ A2 @ ( one_one @ nat ) ) @ B2 @ ( F2 @ A2 @ Acc3 ) ) ) ) ) ) ).
% fold_atLeastAtMost_nat.psimps
thf(fact_5214_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,B3: 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 @ B3 @ Acc ) ) ) )
=> ( ( ~ ( ord_less @ nat @ B3 @ A4 )
=> ( P @ F3 @ ( plus_plus @ nat @ A4 @ ( one_one @ nat ) ) @ B3 @ ( F3 @ A4 @ Acc ) ) )
=> ( P @ F3 @ A4 @ B3 @ Acc ) ) )
=> ( P @ A0 @ A12 @ A23 @ A32 ) ) ) ).
% fold_atLeastAtMost_nat.pinduct
thf(fact_5215_fold__atLeastAtMost__nat_Opelims,axiom,
! [A: $tType,X: nat > A > A,Xa: nat,Xb: nat,Xc: A,Y: A] :
( ( ( set_fo6178422350223883121st_nat @ A @ X @ Xa @ Xb @ Xc )
= Y )
=> ( ( accp @ ( product_prod @ ( nat > A > A ) @ ( product_prod @ nat @ ( product_prod @ nat @ A ) ) ) @ ( set_fo1817059534552279752at_rel @ A ) @ ( product_Pair @ ( nat > A > A ) @ ( product_prod @ nat @ ( product_prod @ nat @ A ) ) @ X @ ( product_Pair @ nat @ ( product_prod @ nat @ A ) @ Xa @ ( product_Pair @ nat @ A @ Xb @ Xc ) ) ) )
=> ~ ( ( ( ( ord_less @ nat @ Xb @ Xa )
=> ( Y = Xc ) )
& ( ~ ( ord_less @ nat @ Xb @ Xa )
=> ( Y
= ( set_fo6178422350223883121st_nat @ A @ X @ ( plus_plus @ nat @ Xa @ ( one_one @ nat ) ) @ Xb @ ( X @ Xa @ Xc ) ) ) ) )
=> ~ ( accp @ ( product_prod @ ( nat > A > A ) @ ( product_prod @ nat @ ( product_prod @ nat @ A ) ) ) @ ( set_fo1817059534552279752at_rel @ A ) @ ( product_Pair @ ( nat > A > A ) @ ( product_prod @ nat @ ( product_prod @ nat @ A ) ) @ X @ ( product_Pair @ nat @ ( product_prod @ nat @ A ) @ Xa @ ( product_Pair @ nat @ A @ Xb @ Xc ) ) ) ) ) ) ) ).
% fold_atLeastAtMost_nat.pelims
thf(fact_5216_VEBT__internal_OminNulli_Opelims,axiom,
! [X: vEBT_VEBTi,Y: heap_Time_Heap @ $o] :
( ( ( vEBT_VEBT_minNulli @ X )
= Y )
=> ( ( accp @ vEBT_VEBTi @ vEBT_V5740978063120863272li_rel @ X )
=> ( ( ( X
= ( vEBT_Leafi @ $false @ $false ) )
=> ( ( Y
= ( heap_Time_return @ $o @ $true ) )
=> ~ ( accp @ vEBT_VEBTi @ vEBT_V5740978063120863272li_rel @ ( vEBT_Leafi @ $false @ $false ) ) ) )
=> ( ! [Uv: $o] :
( ( X
= ( vEBT_Leafi @ $true @ Uv ) )
=> ( ( Y
= ( heap_Time_return @ $o @ $false ) )
=> ~ ( accp @ vEBT_VEBTi @ vEBT_V5740978063120863272li_rel @ ( vEBT_Leafi @ $true @ Uv ) ) ) )
=> ( ! [Uu: $o] :
( ( X
= ( vEBT_Leafi @ Uu @ $true ) )
=> ( ( Y
= ( heap_Time_return @ $o @ $false ) )
=> ~ ( accp @ vEBT_VEBTi @ vEBT_V5740978063120863272li_rel @ ( vEBT_Leafi @ Uu @ $true ) ) ) )
=> ( ! [Uw: nat,Ux2: array @ vEBT_VEBTi,Uy2: vEBT_VEBTi] :
( ( X
= ( vEBT_Nodei @ ( none @ ( product_prod @ nat @ nat ) ) @ Uw @ Ux2 @ Uy2 ) )
=> ( ( Y
= ( heap_Time_return @ $o @ $true ) )
=> ~ ( accp @ vEBT_VEBTi @ vEBT_V5740978063120863272li_rel @ ( vEBT_Nodei @ ( none @ ( product_prod @ nat @ nat ) ) @ Uw @ Ux2 @ Uy2 ) ) ) )
=> ~ ! [Uz2: product_prod @ nat @ nat,Va2: nat,Vb2: array @ vEBT_VEBTi,Vc2: vEBT_VEBTi] :
( ( X
= ( vEBT_Nodei @ ( some @ ( product_prod @ nat @ nat ) @ Uz2 ) @ Va2 @ Vb2 @ Vc2 ) )
=> ( ( Y
= ( 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_5217_T_092_060_094sub_062m_092_060_094sub_062i_092_060_094sub_062n_092_060_094sub_062N_092_060_094sub_062u_092_060_094sub_062l_092_060_094sub_062l_Opelims,axiom,
! [X: vEBT_VEBT,Y: nat] :
( ( ( vEBT_T_m_i_n_N_u_l_l @ X )
= Y )
=> ( ( accp @ vEBT_VEBT @ vEBT_T5462971552011256508_l_rel @ X )
=> ( ( ( X
= ( vEBT_Leaf @ $false @ $false ) )
=> ( ( Y
= ( one_one @ nat ) )
=> ~ ( accp @ vEBT_VEBT @ vEBT_T5462971552011256508_l_rel @ ( vEBT_Leaf @ $false @ $false ) ) ) )
=> ( ! [Uv: $o] :
( ( X
= ( vEBT_Leaf @ $true @ Uv ) )
=> ( ( Y
= ( one_one @ nat ) )
=> ~ ( accp @ vEBT_VEBT @ vEBT_T5462971552011256508_l_rel @ ( vEBT_Leaf @ $true @ Uv ) ) ) )
=> ( ! [Uu: $o] :
( ( X
= ( vEBT_Leaf @ Uu @ $true ) )
=> ( ( Y
= ( one_one @ nat ) )
=> ~ ( accp @ vEBT_VEBT @ vEBT_T5462971552011256508_l_rel @ ( vEBT_Leaf @ Uu @ $true ) ) ) )
=> ( ! [Uw: nat,Ux2: list @ vEBT_VEBT,Uy2: vEBT_VEBT] :
( ( X
= ( vEBT_Node @ ( none @ ( product_prod @ nat @ nat ) ) @ Uw @ Ux2 @ Uy2 ) )
=> ( ( Y
= ( one_one @ nat ) )
=> ~ ( accp @ vEBT_VEBT @ vEBT_T5462971552011256508_l_rel @ ( vEBT_Node @ ( none @ ( product_prod @ nat @ nat ) ) @ Uw @ Ux2 @ Uy2 ) ) ) )
=> ~ ! [Uz2: product_prod @ nat @ nat,Va2: nat,Vb2: list @ vEBT_VEBT,Vc2: vEBT_VEBT] :
( ( X
= ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ Uz2 ) @ Va2 @ Vb2 @ Vc2 ) )
=> ( ( Y
= ( 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_5218_VEBT__internal_Ooption__shift_Opelims,axiom,
! [A: $tType,X: A > A > A,Xa: option @ A,Xb: option @ A,Y: option @ A] :
( ( ( vEBT_V2048590022279873568_shift @ A @ X @ Xa @ Xb )
= Y )
=> ( ( accp @ ( product_prod @ ( A > A > A ) @ ( product_prod @ ( option @ A ) @ ( option @ A ) ) ) @ ( vEBT_V459564278314245337ft_rel @ A ) @ ( product_Pair @ ( A > A > A ) @ ( product_prod @ ( option @ A ) @ ( option @ A ) ) @ X @ ( product_Pair @ ( option @ A ) @ ( option @ A ) @ Xa @ Xb ) ) )
=> ( ( ( Xa
= ( none @ A ) )
=> ( ( Y
= ( none @ A ) )
=> ~ ( accp @ ( product_prod @ ( A > A > A ) @ ( product_prod @ ( option @ A ) @ ( option @ A ) ) ) @ ( vEBT_V459564278314245337ft_rel @ A ) @ ( product_Pair @ ( A > A > A ) @ ( product_prod @ ( option @ A ) @ ( option @ A ) ) @ X @ ( product_Pair @ ( option @ A ) @ ( option @ A ) @ ( none @ A ) @ Xb ) ) ) ) )
=> ( ! [V3: A] :
( ( Xa
= ( some @ A @ V3 ) )
=> ( ( Xb
= ( none @ A ) )
=> ( ( Y
= ( none @ A ) )
=> ~ ( accp @ ( product_prod @ ( A > A > A ) @ ( product_prod @ ( option @ A ) @ ( option @ A ) ) ) @ ( vEBT_V459564278314245337ft_rel @ A ) @ ( product_Pair @ ( A > A > A ) @ ( product_prod @ ( option @ A ) @ ( option @ A ) ) @ X @ ( product_Pair @ ( option @ A ) @ ( option @ A ) @ ( some @ A @ V3 ) @ ( none @ A ) ) ) ) ) ) )
=> ~ ! [A4: A] :
( ( Xa
= ( some @ A @ A4 ) )
=> ! [B3: A] :
( ( Xb
= ( some @ A @ B3 ) )
=> ( ( Y
= ( some @ A @ ( X @ A4 @ B3 ) ) )
=> ~ ( accp @ ( product_prod @ ( A > A > A ) @ ( product_prod @ ( option @ A ) @ ( option @ A ) ) ) @ ( vEBT_V459564278314245337ft_rel @ A ) @ ( product_Pair @ ( A > A > A ) @ ( product_prod @ ( option @ A ) @ ( option @ A ) ) @ X @ ( product_Pair @ ( option @ A ) @ ( option @ A ) @ ( some @ A @ A4 ) @ ( some @ A @ B3 ) ) ) ) ) ) ) ) ) ) ) ).
% VEBT_internal.option_shift.pelims
thf(fact_5219_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 )
=> ( ! [X5: product_prod @ A @ A,Xa2: A,Y4: A] :
( ( ord_less @ code_integer @ ( zero_zero @ code_integer ) @ I2 )
=> ( ( X5
= ( A_of_integer @ I2 ) )
=> ( ( ( product_Pair @ A @ A @ Xa2 @ Y4 )
= X5 )
=> ( P @ ( minus_minus @ code_integer @ I2 @ ( one_one @ code_integer ) ) ) ) ) )
=> ( P @ I2 ) ) )
=> ( P @ A0 ) ) ) ) ).
% quickcheck_narrowing_samples.narrowing_samples.pinduct
thf(fact_5220_in__measure,axiom,
! [A: $tType,X: A,Y: A,F2: A > nat] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X @ Y ) @ ( measure @ A @ F2 ) )
= ( ord_less @ nat @ ( F2 @ X ) @ ( F2 @ Y ) ) ) ).
% in_measure
thf(fact_5221_in__finite__psubset,axiom,
! [A: $tType,A3: set @ A,B6: set @ A] :
( ( member @ ( product_prod @ ( set @ A ) @ ( set @ A ) ) @ ( product_Pair @ ( set @ A ) @ ( set @ A ) @ A3 @ B6 ) @ ( finite_psubset @ A ) )
= ( ( ord_less @ ( set @ A ) @ A3 @ B6 )
& ( finite_finite2 @ A @ B6 ) ) ) ).
% in_finite_psubset
thf(fact_5222_pure__assn__raw_Ocases,axiom,
! [A: $tType,B: $tType,X: product_prod @ $o @ ( product_prod @ A @ ( set @ B ) )] :
~ ! [B3: $o,H4: A,As: set @ B] :
( X
!= ( product_Pair @ $o @ ( product_prod @ A @ ( set @ B ) ) @ B3 @ ( product_Pair @ A @ ( set @ B ) @ H4 @ As ) ) ) ).
% pure_assn_raw.cases
thf(fact_5223_VEBT__internal_OminNull_Opelims_I1_J,axiom,
! [X: vEBT_VEBT,Y: $o] :
( ( ( vEBT_VEBT_minNull @ X )
= Y )
=> ( ( accp @ vEBT_VEBT @ vEBT_V6963167321098673237ll_rel @ X )
=> ( ( ( X
= ( vEBT_Leaf @ $false @ $false ) )
=> ( Y
=> ~ ( accp @ vEBT_VEBT @ vEBT_V6963167321098673237ll_rel @ ( vEBT_Leaf @ $false @ $false ) ) ) )
=> ( ! [Uv: $o] :
( ( X
= ( vEBT_Leaf @ $true @ Uv ) )
=> ( ~ Y
=> ~ ( accp @ vEBT_VEBT @ vEBT_V6963167321098673237ll_rel @ ( vEBT_Leaf @ $true @ Uv ) ) ) )
=> ( ! [Uu: $o] :
( ( X
= ( vEBT_Leaf @ Uu @ $true ) )
=> ( ~ Y
=> ~ ( accp @ vEBT_VEBT @ vEBT_V6963167321098673237ll_rel @ ( vEBT_Leaf @ Uu @ $true ) ) ) )
=> ( ! [Uw: nat,Ux2: list @ vEBT_VEBT,Uy2: vEBT_VEBT] :
( ( X
= ( vEBT_Node @ ( none @ ( product_prod @ nat @ nat ) ) @ Uw @ Ux2 @ Uy2 ) )
=> ( Y
=> ~ ( accp @ vEBT_VEBT @ vEBT_V6963167321098673237ll_rel @ ( vEBT_Node @ ( none @ ( product_prod @ nat @ nat ) ) @ Uw @ Ux2 @ Uy2 ) ) ) )
=> ~ ! [Uz2: product_prod @ nat @ nat,Va2: nat,Vb2: list @ vEBT_VEBT,Vc2: vEBT_VEBT] :
( ( X
= ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ Uz2 ) @ Va2 @ Vb2 @ Vc2 ) )
=> ( ~ Y
=> ~ ( accp @ vEBT_VEBT @ vEBT_V6963167321098673237ll_rel @ ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ Uz2 ) @ Va2 @ Vb2 @ Vc2 ) ) ) ) ) ) ) ) ) ) ).
% VEBT_internal.minNull.pelims(1)
thf(fact_5224_modulo__int__def,axiom,
( ( modulo_modulo @ int )
= ( ^ [K2: int,L: int] :
( if @ int
@ ( L
= ( zero_zero @ int ) )
@ K2
@ ( if @ int
@ ( ( sgn_sgn @ int @ K2 )
= ( sgn_sgn @ int @ L ) )
@ ( times_times @ int @ ( sgn_sgn @ int @ L ) @ ( semiring_1_of_nat @ int @ ( modulo_modulo @ nat @ ( nat2 @ ( abs_abs @ int @ K2 ) ) @ ( nat2 @ ( abs_abs @ int @ L ) ) ) ) )
@ ( times_times @ int @ ( sgn_sgn @ int @ L )
@ ( minus_minus @ int
@ ( times_times @ int @ ( abs_abs @ int @ L )
@ ( zero_neq_one_of_bool @ int
@ ~ ( dvd_dvd @ int @ L @ K2 ) ) )
@ ( semiring_1_of_nat @ int @ ( modulo_modulo @ nat @ ( nat2 @ ( abs_abs @ int @ K2 ) ) @ ( nat2 @ ( abs_abs @ int @ L ) ) ) ) ) ) ) ) ) ) ).
% modulo_int_def
thf(fact_5225_neg__mask__is__div_H,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [N3: nat,W: word @ A] :
( ( ord_less @ nat @ N3 @ ( size_size @ ( word @ A ) @ W ) )
=> ( ( bit_se5824344872417868541ns_and @ ( word @ A ) @ W @ ( bit_ri4277139882892585799ns_not @ ( word @ A ) @ ( bit_se2239418461657761734s_mask @ ( word @ A ) @ N3 ) ) )
= ( times_times @ ( word @ A ) @ ( divide_divide @ ( word @ A ) @ W @ ( power_power @ ( word @ A ) @ ( numeral_numeral @ ( word @ A ) @ ( bit0 @ one2 ) ) @ N3 ) ) @ ( power_power @ ( word @ A ) @ ( numeral_numeral @ ( word @ A ) @ ( bit0 @ one2 ) ) @ N3 ) ) ) ) ) ).
% neg_mask_is_div'
thf(fact_5226_bit_Ocompl__eq__compl__iff,axiom,
! [A: $tType] :
( ( bit_ri3973907225187159222ations @ A )
=> ! [X: A,Y: A] :
( ( ( bit_ri4277139882892585799ns_not @ A @ X )
= ( bit_ri4277139882892585799ns_not @ A @ Y ) )
= ( X = Y ) ) ) ).
% bit.compl_eq_compl_iff
thf(fact_5227_bit_Odouble__compl,axiom,
! [A: $tType] :
( ( bit_ri3973907225187159222ations @ A )
=> ! [X: A] :
( ( bit_ri4277139882892585799ns_not @ A @ ( bit_ri4277139882892585799ns_not @ A @ X ) )
= X ) ) ).
% bit.double_compl
thf(fact_5228_sgn__sgn,axiom,
! [A: $tType] :
( ( idom_abs_sgn @ A )
=> ! [A2: A] :
( ( sgn_sgn @ A @ ( sgn_sgn @ A @ A2 ) )
= ( sgn_sgn @ A @ A2 ) ) ) ).
% sgn_sgn
thf(fact_5229_sgn__0,axiom,
! [A: $tType] :
( ( idom_abs_sgn @ A )
=> ( ( sgn_sgn @ A @ ( zero_zero @ A ) )
= ( zero_zero @ A ) ) ) ).
% sgn_0
thf(fact_5230_sgn__1,axiom,
! [A: $tType] :
( ( idom_abs_sgn @ A )
=> ( ( sgn_sgn @ A @ ( one_one @ A ) )
= ( one_one @ A ) ) ) ).
% sgn_1
thf(fact_5231_sgn__one,axiom,
! [A: $tType] :
( ( real_V2822296259951069270ebra_1 @ A )
=> ( ( sgn_sgn @ A @ ( one_one @ A ) )
= ( one_one @ A ) ) ) ).
% sgn_one
thf(fact_5232_sgn__divide,axiom,
! [A: $tType] :
( ( field_abs_sgn @ A )
=> ! [A2: A,B2: A] :
( ( sgn_sgn @ A @ ( divide_divide @ A @ A2 @ B2 ) )
= ( divide_divide @ A @ ( sgn_sgn @ A @ A2 ) @ ( sgn_sgn @ A @ B2 ) ) ) ) ).
% sgn_divide
thf(fact_5233_power__sgn,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [A2: A,N3: nat] :
( ( sgn_sgn @ A @ ( power_power @ A @ A2 @ N3 ) )
= ( power_power @ A @ ( sgn_sgn @ A @ A2 ) @ N3 ) ) ) ).
% power_sgn
thf(fact_5234_idom__abs__sgn__class_Osgn__minus,axiom,
! [A: $tType] :
( ( idom_abs_sgn @ A )
=> ! [A2: A] :
( ( sgn_sgn @ A @ ( uminus_uminus @ A @ A2 ) )
= ( uminus_uminus @ A @ ( sgn_sgn @ A @ A2 ) ) ) ) ).
% idom_abs_sgn_class.sgn_minus
thf(fact_5235_sgn__greater,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [A2: A] :
( ( ord_less @ A @ ( zero_zero @ A ) @ ( sgn_sgn @ A @ A2 ) )
= ( ord_less @ A @ ( zero_zero @ A ) @ A2 ) ) ) ).
% sgn_greater
thf(fact_5236_sgn__less,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [A2: A] :
( ( ord_less @ A @ ( sgn_sgn @ A @ A2 ) @ ( zero_zero @ A ) )
= ( ord_less @ A @ A2 @ ( zero_zero @ A ) ) ) ) ).
% sgn_less
thf(fact_5237_divide__sgn,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [A2: A,B2: A] :
( ( divide_divide @ A @ A2 @ ( sgn_sgn @ A @ B2 ) )
= ( times_times @ A @ A2 @ ( sgn_sgn @ A @ B2 ) ) ) ) ).
% divide_sgn
thf(fact_5238_bit_Oconj__cancel__left,axiom,
! [A: $tType] :
( ( bit_ri3973907225187159222ations @ A )
=> ! [X: A] :
( ( bit_se5824344872417868541ns_and @ A @ ( bit_ri4277139882892585799ns_not @ A @ X ) @ X )
= ( zero_zero @ A ) ) ) ).
% bit.conj_cancel_left
thf(fact_5239_bit_Oconj__cancel__right,axiom,
! [A: $tType] :
( ( bit_ri3973907225187159222ations @ A )
=> ! [X: A] :
( ( bit_se5824344872417868541ns_and @ A @ X @ ( bit_ri4277139882892585799ns_not @ A @ X ) )
= ( zero_zero @ A ) ) ) ).
% bit.conj_cancel_right
thf(fact_5240_sgn__pos,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [A2: A] :
( ( ord_less @ A @ ( zero_zero @ A ) @ A2 )
=> ( ( sgn_sgn @ A @ A2 )
= ( one_one @ A ) ) ) ) ).
% sgn_pos
thf(fact_5241_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_5242_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_5243_abs__sgn__eq__1,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [A2: A] :
( ( A2
!= ( zero_zero @ A ) )
=> ( ( abs_abs @ A @ ( sgn_sgn @ A @ A2 ) )
= ( one_one @ A ) ) ) ) ).
% abs_sgn_eq_1
thf(fact_5244_sgn__mult__self__eq,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [A2: A] :
( ( times_times @ A @ ( sgn_sgn @ A @ A2 ) @ ( sgn_sgn @ A @ A2 ) )
= ( zero_neq_one_of_bool @ A
@ ( A2
!= ( zero_zero @ A ) ) ) ) ) ).
% sgn_mult_self_eq
thf(fact_5245_sgn__abs,axiom,
! [A: $tType] :
( ( idom_abs_sgn @ A )
=> ! [A2: A] :
( ( abs_abs @ A @ ( sgn_sgn @ A @ A2 ) )
= ( zero_neq_one_of_bool @ A
@ ( A2
!= ( zero_zero @ A ) ) ) ) ) ).
% sgn_abs
thf(fact_5246_idom__abs__sgn__class_Oabs__sgn,axiom,
! [A: $tType] :
( ( idom_abs_sgn @ A )
=> ! [A2: A] :
( ( sgn_sgn @ A @ ( abs_abs @ A @ A2 ) )
= ( zero_neq_one_of_bool @ A
@ ( A2
!= ( zero_zero @ A ) ) ) ) ) ).
% idom_abs_sgn_class.abs_sgn
thf(fact_5247_minus__not__numeral__eq,axiom,
! [A: $tType] :
( ( bit_ri3973907225187159222ations @ A )
=> ! [N3: num] :
( ( uminus_uminus @ A @ ( bit_ri4277139882892585799ns_not @ A @ ( numeral_numeral @ A @ N3 ) ) )
= ( numeral_numeral @ A @ ( inc @ N3 ) ) ) ) ).
% minus_not_numeral_eq
thf(fact_5248_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_5249_word__add__not,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [X: word @ A] :
( ( plus_plus @ ( word @ A ) @ X @ ( bit_ri4277139882892585799ns_not @ ( word @ A ) @ X ) )
= ( uminus_uminus @ ( word @ A ) @ ( one_one @ ( word @ A ) ) ) ) ) ).
% word_add_not
thf(fact_5250_sgn__neg,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [A2: A] :
( ( ord_less @ A @ A2 @ ( zero_zero @ A ) )
=> ( ( sgn_sgn @ A @ A2 )
= ( uminus_uminus @ A @ ( one_one @ A ) ) ) ) ) ).
% sgn_neg
thf(fact_5251_even__not__iff,axiom,
! [A: $tType] :
( ( bit_ri3973907225187159222ations @ A )
=> ! [A2: A] :
( ( dvd_dvd @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ ( bit_ri4277139882892585799ns_not @ A @ A2 ) )
= ( ~ ( dvd_dvd @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ A2 ) ) ) ) ).
% even_not_iff
thf(fact_5252_sgn__of__nat,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [N3: nat] :
( ( sgn_sgn @ A @ ( semiring_1_of_nat @ A @ N3 ) )
= ( zero_neq_one_of_bool @ A @ ( ord_less @ nat @ ( zero_zero @ nat ) @ N3 ) ) ) ) ).
% sgn_of_nat
thf(fact_5253_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_5254_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_5255_take__bit__not__take__bit,axiom,
! [A: $tType] :
( ( bit_ri3973907225187159222ations @ A )
=> ! [N3: nat,A2: A] :
( ( bit_se2584673776208193580ke_bit @ A @ N3 @ ( bit_ri4277139882892585799ns_not @ A @ ( bit_se2584673776208193580ke_bit @ A @ N3 @ A2 ) ) )
= ( bit_se2584673776208193580ke_bit @ A @ N3 @ ( bit_ri4277139882892585799ns_not @ A @ A2 ) ) ) ) ).
% take_bit_not_take_bit
thf(fact_5256_take__bit__not__iff,axiom,
! [A: $tType] :
( ( bit_ri3973907225187159222ations @ A )
=> ! [N3: nat,A2: A,B2: A] :
( ( ( bit_se2584673776208193580ke_bit @ A @ N3 @ ( bit_ri4277139882892585799ns_not @ A @ A2 ) )
= ( bit_se2584673776208193580ke_bit @ A @ N3 @ ( bit_ri4277139882892585799ns_not @ A @ B2 ) ) )
= ( ( bit_se2584673776208193580ke_bit @ A @ N3 @ A2 )
= ( bit_se2584673776208193580ke_bit @ A @ N3 @ B2 ) ) ) ) ).
% take_bit_not_iff
thf(fact_5257_of__int__not__eq,axiom,
! [A: $tType] :
( ( bit_ri3973907225187159222ations @ A )
=> ! [K: int] :
( ( ring_1_of_int @ A @ ( bit_ri4277139882892585799ns_not @ int @ K ) )
= ( bit_ri4277139882892585799ns_not @ A @ ( ring_1_of_int @ A @ K ) ) ) ) ).
% of_int_not_eq
thf(fact_5258_sgn__eq__0__iff,axiom,
! [A: $tType] :
( ( idom_abs_sgn @ A )
=> ! [A2: A] :
( ( ( sgn_sgn @ A @ A2 )
= ( zero_zero @ A ) )
= ( A2
= ( zero_zero @ A ) ) ) ) ).
% sgn_eq_0_iff
thf(fact_5259_sgn__0__0,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [A2: A] :
( ( ( sgn_sgn @ A @ A2 )
= ( zero_zero @ A ) )
= ( A2
= ( zero_zero @ A ) ) ) ) ).
% sgn_0_0
thf(fact_5260_same__sgn__sgn__add,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [B2: A,A2: A] :
( ( ( sgn_sgn @ A @ B2 )
= ( sgn_sgn @ A @ A2 ) )
=> ( ( sgn_sgn @ A @ ( plus_plus @ A @ A2 @ B2 ) )
= ( sgn_sgn @ A @ A2 ) ) ) ) ).
% same_sgn_sgn_add
thf(fact_5261_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_5262_sgn__mult,axiom,
! [A: $tType] :
( ( idom_abs_sgn @ A )
=> ! [A2: A,B2: A] :
( ( sgn_sgn @ A @ ( times_times @ A @ A2 @ B2 ) )
= ( times_times @ A @ ( sgn_sgn @ A @ A2 ) @ ( sgn_sgn @ A @ B2 ) ) ) ) ).
% sgn_mult
thf(fact_5263_Real__Vector__Spaces_Osgn__mult,axiom,
! [A: $tType] :
( ( real_V8999393235501362500lgebra @ A )
=> ! [X: A,Y: A] :
( ( sgn_sgn @ A @ ( times_times @ A @ X @ Y ) )
= ( times_times @ A @ ( sgn_sgn @ A @ X ) @ ( sgn_sgn @ A @ Y ) ) ) ) ).
% Real_Vector_Spaces.sgn_mult
thf(fact_5264_not__add__distrib,axiom,
! [A: $tType] :
( ( bit_ri3973907225187159222ations @ A )
=> ! [A2: A,B2: A] :
( ( bit_ri4277139882892585799ns_not @ A @ ( plus_plus @ A @ A2 @ B2 ) )
= ( minus_minus @ A @ ( bit_ri4277139882892585799ns_not @ A @ A2 ) @ B2 ) ) ) ).
% not_add_distrib
thf(fact_5265_not__diff__distrib,axiom,
! [A: $tType] :
( ( bit_ri3973907225187159222ations @ A )
=> ! [A2: A,B2: A] :
( ( bit_ri4277139882892585799ns_not @ A @ ( minus_minus @ A @ A2 @ B2 ) )
= ( plus_plus @ A @ ( bit_ri4277139882892585799ns_not @ A @ A2 ) @ B2 ) ) ) ).
% not_diff_distrib
thf(fact_5266_sgn__not__eq__imp,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [B2: A,A2: A] :
( ( ( sgn_sgn @ A @ B2 )
!= ( sgn_sgn @ A @ A2 ) )
=> ( ( ( sgn_sgn @ A @ A2 )
!= ( zero_zero @ A ) )
=> ( ( ( sgn_sgn @ A @ B2 )
!= ( zero_zero @ A ) )
=> ( ( sgn_sgn @ A @ A2 )
= ( uminus_uminus @ A @ ( sgn_sgn @ A @ B2 ) ) ) ) ) ) ) ).
% sgn_not_eq_imp
thf(fact_5267_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_5268_same__sgn__abs__add,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [B2: A,A2: A] :
( ( ( sgn_sgn @ A @ B2 )
= ( sgn_sgn @ A @ A2 ) )
=> ( ( abs_abs @ A @ ( plus_plus @ A @ A2 @ B2 ) )
= ( plus_plus @ A @ ( abs_abs @ A @ A2 ) @ ( abs_abs @ A @ B2 ) ) ) ) ) ).
% same_sgn_abs_add
thf(fact_5269_mult__sgn__abs,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [X: A] :
( ( times_times @ A @ ( sgn_sgn @ A @ X ) @ ( abs_abs @ A @ X ) )
= X ) ) ).
% mult_sgn_abs
thf(fact_5270_sgn__mult__abs,axiom,
! [A: $tType] :
( ( idom_abs_sgn @ A )
=> ! [A2: A] :
( ( times_times @ A @ ( sgn_sgn @ A @ A2 ) @ ( abs_abs @ A @ A2 ) )
= A2 ) ) ).
% sgn_mult_abs
thf(fact_5271_abs__mult__sgn,axiom,
! [A: $tType] :
( ( idom_abs_sgn @ A )
=> ! [A2: A] :
( ( times_times @ A @ ( abs_abs @ A @ A2 ) @ ( sgn_sgn @ A @ A2 ) )
= A2 ) ) ).
% abs_mult_sgn
thf(fact_5272_linordered__idom__class_Oabs__sgn,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ( ( abs_abs @ A )
= ( ^ [K2: A] : ( times_times @ A @ K2 @ ( sgn_sgn @ A @ K2 ) ) ) ) ) ).
% linordered_idom_class.abs_sgn
thf(fact_5273_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_5274_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_5275_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_5276_sgn__1__pos,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [A2: A] :
( ( ( sgn_sgn @ A @ A2 )
= ( one_one @ A ) )
= ( ord_less @ A @ ( zero_zero @ A ) @ A2 ) ) ) ).
% sgn_1_pos
thf(fact_5277_abs__sgn__eq,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [A2: A] :
( ( ( A2
= ( zero_zero @ A ) )
=> ( ( abs_abs @ A @ ( sgn_sgn @ A @ A2 ) )
= ( zero_zero @ A ) ) )
& ( ( A2
!= ( zero_zero @ A ) )
=> ( ( abs_abs @ A @ ( sgn_sgn @ A @ A2 ) )
= ( one_one @ A ) ) ) ) ) ).
% abs_sgn_eq
thf(fact_5278_take__bit__not__eq__mask__diff,axiom,
! [A: $tType] :
( ( bit_ri3973907225187159222ations @ A )
=> ! [N3: nat,A2: A] :
( ( bit_se2584673776208193580ke_bit @ A @ N3 @ ( bit_ri4277139882892585799ns_not @ A @ A2 ) )
= ( minus_minus @ A @ ( bit_se2239418461657761734s_mask @ A @ N3 ) @ ( bit_se2584673776208193580ke_bit @ A @ N3 @ A2 ) ) ) ) ).
% take_bit_not_eq_mask_diff
thf(fact_5279_sgn__mod,axiom,
! [L2: int,K: int] :
( ( L2
!= ( zero_zero @ int ) )
=> ( ~ ( dvd_dvd @ int @ L2 @ K )
=> ( ( sgn_sgn @ int @ ( modulo_modulo @ int @ K @ L2 ) )
= ( sgn_sgn @ int @ L2 ) ) ) ) ).
% sgn_mod
thf(fact_5280_mask__lower__twice,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [N3: nat,M: nat,X: word @ A] :
( ( ord_less_eq @ nat @ N3 @ M )
=> ( ( bit_se5824344872417868541ns_and @ ( word @ A ) @ ( bit_se5824344872417868541ns_and @ ( word @ A ) @ X @ ( bit_ri4277139882892585799ns_not @ ( word @ A ) @ ( bit_se2239418461657761734s_mask @ ( word @ A ) @ N3 ) ) ) @ ( bit_ri4277139882892585799ns_not @ ( word @ A ) @ ( bit_se2239418461657761734s_mask @ ( word @ A ) @ M ) ) )
= ( bit_se5824344872417868541ns_and @ ( word @ A ) @ X @ ( bit_ri4277139882892585799ns_not @ ( word @ A ) @ ( bit_se2239418461657761734s_mask @ ( word @ A ) @ M ) ) ) ) ) ) ).
% mask_lower_twice
thf(fact_5281_mask__out__first__mask__some,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [X: word @ A,N3: nat,Y: word @ A,M: nat] :
( ( ( bit_se5824344872417868541ns_and @ ( word @ A ) @ X @ ( bit_ri4277139882892585799ns_not @ ( word @ A ) @ ( bit_se2239418461657761734s_mask @ ( word @ A ) @ N3 ) ) )
= Y )
=> ( ( ord_less_eq @ nat @ N3 @ M )
=> ( ( bit_se5824344872417868541ns_and @ ( word @ A ) @ X @ ( bit_ri4277139882892585799ns_not @ ( word @ A ) @ ( bit_se2239418461657761734s_mask @ ( word @ A ) @ M ) ) )
= ( bit_se5824344872417868541ns_and @ ( word @ A ) @ Y @ ( bit_ri4277139882892585799ns_not @ ( word @ A ) @ ( bit_se2239418461657761734s_mask @ ( word @ A ) @ M ) ) ) ) ) ) ) ).
% mask_out_first_mask_some
thf(fact_5282_NOT__eq,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ( ( bit_ri4277139882892585799ns_not @ ( word @ A ) )
= ( ^ [X2: word @ A] : ( minus_minus @ ( word @ A ) @ ( uminus_uminus @ ( word @ A ) @ X2 ) @ ( one_one @ ( word @ A ) ) ) ) ) ) ).
% NOT_eq
thf(fact_5283_minus__numeral__inc__eq,axiom,
! [A: $tType] :
( ( bit_ri3973907225187159222ations @ A )
=> ! [N3: num] :
( ( uminus_uminus @ A @ ( numeral_numeral @ A @ ( inc @ N3 ) ) )
= ( bit_ri4277139882892585799ns_not @ A @ ( numeral_numeral @ A @ N3 ) ) ) ) ).
% minus_numeral_inc_eq
thf(fact_5284_mask__out__sub__mask,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [X: word @ A,N3: nat] :
( ( bit_se5824344872417868541ns_and @ ( word @ A ) @ X @ ( bit_ri4277139882892585799ns_not @ ( word @ A ) @ ( bit_se2239418461657761734s_mask @ ( word @ A ) @ N3 ) ) )
= ( minus_minus @ ( word @ A ) @ X @ ( bit_se5824344872417868541ns_and @ ( word @ A ) @ X @ ( bit_se2239418461657761734s_mask @ ( word @ A ) @ N3 ) ) ) ) ) ).
% mask_out_sub_mask
thf(fact_5285_subtract__mask_I1_J,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [P4: word @ A,N3: nat] :
( ( minus_minus @ ( word @ A ) @ P4 @ ( bit_se5824344872417868541ns_and @ ( word @ A ) @ P4 @ ( bit_se2239418461657761734s_mask @ ( word @ A ) @ N3 ) ) )
= ( bit_se5824344872417868541ns_and @ ( word @ A ) @ P4 @ ( bit_ri4277139882892585799ns_not @ ( word @ A ) @ ( bit_se2239418461657761734s_mask @ ( word @ A ) @ N3 ) ) ) ) ) ).
% subtract_mask(1)
thf(fact_5286_subtract__mask_I2_J,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [P4: word @ A,N3: nat] :
( ( minus_minus @ ( word @ A ) @ P4 @ ( bit_se5824344872417868541ns_and @ ( word @ A ) @ P4 @ ( bit_ri4277139882892585799ns_not @ ( word @ A ) @ ( bit_se2239418461657761734s_mask @ ( word @ A ) @ N3 ) ) ) )
= ( bit_se5824344872417868541ns_and @ ( word @ A ) @ P4 @ ( bit_se2239418461657761734s_mask @ ( word @ A ) @ N3 ) ) ) ) ).
% subtract_mask(2)
thf(fact_5287_not__numeral__Bit0__eq,axiom,
! [A: $tType] :
( ( bit_ri3973907225187159222ations @ A )
=> ! [N3: num] :
( ( bit_ri4277139882892585799ns_not @ A @ ( numeral_numeral @ A @ ( bit0 @ N3 ) ) )
= ( uminus_uminus @ A @ ( numeral_numeral @ A @ ( bit1 @ N3 ) ) ) ) ) ).
% not_numeral_Bit0_eq
thf(fact_5288_sgn__1__neg,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [A2: A] :
( ( ( sgn_sgn @ A @ A2 )
= ( uminus_uminus @ A @ ( one_one @ A ) ) )
= ( ord_less @ A @ A2 @ ( zero_zero @ A ) ) ) ) ).
% sgn_1_neg
thf(fact_5289_sgn__if,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ( ( sgn_sgn @ A )
= ( ^ [X2: A] :
( if @ A
@ ( X2
= ( zero_zero @ A ) )
@ ( zero_zero @ A )
@ ( if @ A @ ( ord_less @ A @ ( zero_zero @ A ) @ X2 ) @ ( one_one @ A ) @ ( uminus_uminus @ A @ ( one_one @ A ) ) ) ) ) ) ) ).
% sgn_if
thf(fact_5290_zsgn__def,axiom,
( ( sgn_sgn @ int )
= ( ^ [I3: int] :
( if @ int
@ ( I3
= ( zero_zero @ int ) )
@ ( zero_zero @ int )
@ ( if @ int @ ( ord_less @ int @ ( zero_zero @ int ) @ I3 ) @ ( one_one @ int ) @ ( uminus_uminus @ int @ ( one_one @ int ) ) ) ) ) ) ).
% zsgn_def
thf(fact_5291_take__bit__not__mask__eq__0,axiom,
! [A: $tType] :
( ( bit_ri3973907225187159222ations @ A )
=> ! [M: nat,N3: nat] :
( ( ord_less_eq @ nat @ M @ N3 )
=> ( ( bit_se2584673776208193580ke_bit @ A @ M @ ( bit_ri4277139882892585799ns_not @ A @ ( bit_se2239418461657761734s_mask @ A @ N3 ) ) )
= ( zero_zero @ A ) ) ) ) ).
% take_bit_not_mask_eq_0
thf(fact_5292_norm__sgn,axiom,
! [A: $tType] :
( ( real_V822414075346904944vector @ A )
=> ! [X: A] :
( ( ( X
= ( zero_zero @ A ) )
=> ( ( real_V7770717601297561774m_norm @ A @ ( sgn_sgn @ A @ X ) )
= ( zero_zero @ real ) ) )
& ( ( X
!= ( zero_zero @ A ) )
=> ( ( real_V7770717601297561774m_norm @ A @ ( sgn_sgn @ A @ X ) )
= ( one_one @ real ) ) ) ) ) ).
% norm_sgn
thf(fact_5293_not__numeral__BitM__eq,axiom,
! [A: $tType] :
( ( bit_ri3973907225187159222ations @ A )
=> ! [N3: num] :
( ( bit_ri4277139882892585799ns_not @ A @ ( numeral_numeral @ A @ ( bitM @ N3 ) ) )
= ( uminus_uminus @ A @ ( numeral_numeral @ A @ ( bit0 @ N3 ) ) ) ) ) ).
% not_numeral_BitM_eq
thf(fact_5294_multiple__mask__trivia,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [M: nat,N3: nat,X: word @ A] :
( ( ord_less_eq @ nat @ M @ N3 )
=> ( ( plus_plus @ ( word @ A ) @ ( bit_se5824344872417868541ns_and @ ( word @ A ) @ X @ ( bit_ri4277139882892585799ns_not @ ( word @ A ) @ ( bit_se2239418461657761734s_mask @ ( word @ A ) @ N3 ) ) ) @ ( bit_se5824344872417868541ns_and @ ( word @ A ) @ X @ ( bit_se5824344872417868541ns_and @ ( word @ A ) @ ( bit_se2239418461657761734s_mask @ ( word @ A ) @ N3 ) @ ( bit_ri4277139882892585799ns_not @ ( word @ A ) @ ( bit_se2239418461657761734s_mask @ ( word @ A ) @ M ) ) ) ) )
= ( bit_se5824344872417868541ns_and @ ( word @ A ) @ X @ ( bit_ri4277139882892585799ns_not @ ( word @ A ) @ ( bit_se2239418461657761734s_mask @ ( word @ A ) @ M ) ) ) ) ) ) ).
% multiple_mask_trivia
thf(fact_5295_and__mask__0__iff__le__mask,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [W: word @ A,N3: nat] :
( ( ( bit_se5824344872417868541ns_and @ ( word @ A ) @ W @ ( bit_ri4277139882892585799ns_not @ ( word @ A ) @ ( bit_se2239418461657761734s_mask @ ( word @ A ) @ N3 ) ) )
= ( zero_zero @ ( word @ A ) ) )
= ( ord_less_eq @ ( word @ A ) @ W @ ( bit_se2239418461657761734s_mask @ ( word @ A ) @ N3 ) ) ) ) ).
% and_mask_0_iff_le_mask
thf(fact_5296_minus__exp__eq__not__mask,axiom,
! [A: $tType] :
( ( bit_ri3973907225187159222ations @ A )
=> ! [N3: nat] :
( ( uminus_uminus @ A @ ( power_power @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ N3 ) )
= ( bit_ri4277139882892585799ns_not @ A @ ( bit_se2239418461657761734s_mask @ A @ N3 ) ) ) ) ).
% minus_exp_eq_not_mask
thf(fact_5297_NOT__mask,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [N3: nat] :
( ( bit_ri4277139882892585799ns_not @ ( word @ A ) @ ( bit_se2239418461657761734s_mask @ ( word @ A ) @ N3 ) )
= ( uminus_uminus @ ( word @ A ) @ ( power_power @ ( word @ A ) @ ( numeral_numeral @ ( word @ A ) @ ( bit0 @ one2 ) ) @ N3 ) ) ) ) ).
% NOT_mask
thf(fact_5298_eucl__rel__int__remainderI,axiom,
! [R2: int,L2: int,K: int,Q2: int] :
( ( ( sgn_sgn @ int @ R2 )
= ( sgn_sgn @ int @ L2 ) )
=> ( ( ord_less @ int @ ( abs_abs @ int @ R2 ) @ ( abs_abs @ int @ L2 ) )
=> ( ( K
= ( plus_plus @ int @ ( times_times @ int @ Q2 @ L2 ) @ R2 ) )
=> ( eucl_rel_int @ K @ L2 @ ( product_Pair @ int @ int @ Q2 @ R2 ) ) ) ) ) ).
% eucl_rel_int_remainderI
thf(fact_5299_div__noneq__sgn__abs,axiom,
! [L2: int,K: int] :
( ( L2
!= ( zero_zero @ int ) )
=> ( ( ( sgn_sgn @ int @ K )
!= ( sgn_sgn @ int @ L2 ) )
=> ( ( divide_divide @ int @ K @ L2 )
= ( minus_minus @ int @ ( uminus_uminus @ int @ ( divide_divide @ int @ ( abs_abs @ int @ K ) @ ( abs_abs @ int @ L2 ) ) )
@ ( zero_neq_one_of_bool @ int
@ ~ ( dvd_dvd @ int @ L2 @ K ) ) ) ) ) ) ).
% div_noneq_sgn_abs
thf(fact_5300_eucl__rel__int_Osimps,axiom,
( eucl_rel_int
= ( ^ [A1: int,A22: int,A33: product_prod @ int @ int] :
( ? [K2: int] :
( ( A1 = K2 )
& ( A22
= ( zero_zero @ int ) )
& ( A33
= ( product_Pair @ int @ int @ ( zero_zero @ int ) @ K2 ) ) )
| ? [L: int,K2: int,Q4: int] :
( ( A1 = K2 )
& ( A22 = L )
& ( A33
= ( product_Pair @ int @ int @ Q4 @ ( zero_zero @ int ) ) )
& ( L
!= ( zero_zero @ int ) )
& ( K2
= ( times_times @ int @ Q4 @ L ) ) )
| ? [R5: int,L: int,K2: int,Q4: int] :
( ( A1 = K2 )
& ( A22 = L )
& ( A33
= ( product_Pair @ int @ int @ Q4 @ R5 ) )
& ( ( sgn_sgn @ int @ R5 )
= ( sgn_sgn @ int @ L ) )
& ( ord_less @ int @ ( abs_abs @ int @ R5 ) @ ( abs_abs @ int @ L ) )
& ( K2
= ( plus_plus @ int @ ( times_times @ int @ Q4 @ L ) @ R5 ) ) ) ) ) ) ).
% eucl_rel_int.simps
thf(fact_5301_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 ) ) )
=> ( ! [Q3: int] :
( ( A32
= ( product_Pair @ int @ int @ Q3 @ ( zero_zero @ int ) ) )
=> ( ( A23
!= ( zero_zero @ int ) )
=> ( A12
!= ( times_times @ int @ Q3 @ A23 ) ) ) )
=> ~ ! [R3: int,Q3: int] :
( ( A32
= ( product_Pair @ int @ int @ Q3 @ 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 @ Q3 @ A23 ) @ R3 ) ) ) ) ) ) ) ) ).
% eucl_rel_int.cases
thf(fact_5302_neg__mask__is__div,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [W: word @ A,N3: nat] :
( ( bit_se5824344872417868541ns_and @ ( word @ A ) @ W @ ( bit_ri4277139882892585799ns_not @ ( word @ A ) @ ( bit_se2239418461657761734s_mask @ ( word @ A ) @ N3 ) ) )
= ( times_times @ ( word @ A ) @ ( divide_divide @ ( word @ A ) @ W @ ( power_power @ ( word @ A ) @ ( numeral_numeral @ ( word @ A ) @ ( bit0 @ one2 ) ) @ N3 ) ) @ ( power_power @ ( word @ A ) @ ( numeral_numeral @ ( word @ A ) @ ( bit0 @ one2 ) ) @ N3 ) ) ) ) ).
% neg_mask_is_div
thf(fact_5303_VEBT__internal_OminNull_Opelims_I3_J,axiom,
! [X: vEBT_VEBT] :
( ~ ( vEBT_VEBT_minNull @ X )
=> ( ( accp @ vEBT_VEBT @ vEBT_V6963167321098673237ll_rel @ X )
=> ( ! [Uv: $o] :
( ( X
= ( vEBT_Leaf @ $true @ Uv ) )
=> ~ ( accp @ vEBT_VEBT @ vEBT_V6963167321098673237ll_rel @ ( vEBT_Leaf @ $true @ Uv ) ) )
=> ( ! [Uu: $o] :
( ( X
= ( vEBT_Leaf @ Uu @ $true ) )
=> ~ ( accp @ vEBT_VEBT @ vEBT_V6963167321098673237ll_rel @ ( vEBT_Leaf @ Uu @ $true ) ) )
=> ~ ! [Uz2: product_prod @ nat @ nat,Va2: nat,Vb2: list @ vEBT_VEBT,Vc2: vEBT_VEBT] :
( ( X
= ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ Uz2 ) @ Va2 @ Vb2 @ Vc2 ) )
=> ~ ( accp @ vEBT_VEBT @ vEBT_V6963167321098673237ll_rel @ ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ Uz2 ) @ Va2 @ Vb2 @ Vc2 ) ) ) ) ) ) ) ).
% VEBT_internal.minNull.pelims(3)
thf(fact_5304_VEBT__internal_OminNull_Opelims_I2_J,axiom,
! [X: vEBT_VEBT] :
( ( vEBT_VEBT_minNull @ X )
=> ( ( accp @ vEBT_VEBT @ vEBT_V6963167321098673237ll_rel @ X )
=> ( ( ( X
= ( vEBT_Leaf @ $false @ $false ) )
=> ~ ( accp @ vEBT_VEBT @ vEBT_V6963167321098673237ll_rel @ ( vEBT_Leaf @ $false @ $false ) ) )
=> ~ ! [Uw: nat,Ux2: list @ vEBT_VEBT,Uy2: vEBT_VEBT] :
( ( X
= ( vEBT_Node @ ( none @ ( product_prod @ nat @ nat ) ) @ Uw @ Ux2 @ Uy2 ) )
=> ~ ( accp @ vEBT_VEBT @ vEBT_V6963167321098673237ll_rel @ ( vEBT_Node @ ( none @ ( product_prod @ nat @ nat ) ) @ Uw @ Ux2 @ Uy2 ) ) ) ) ) ) ).
% VEBT_internal.minNull.pelims(2)
thf(fact_5305_divide__int__unfold,axiom,
! [L2: int,K: int,N3: nat,M: nat] :
( ( ( ( ( sgn_sgn @ int @ L2 )
= ( zero_zero @ int ) )
| ( ( sgn_sgn @ int @ K )
= ( zero_zero @ int ) )
| ( N3
= ( 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 @ L2 ) @ ( semiring_1_of_nat @ int @ N3 ) ) )
= ( zero_zero @ int ) ) )
& ( ~ ( ( ( sgn_sgn @ int @ L2 )
= ( zero_zero @ int ) )
| ( ( sgn_sgn @ int @ K )
= ( zero_zero @ int ) )
| ( N3
= ( zero_zero @ nat ) ) )
=> ( ( ( ( sgn_sgn @ int @ K )
= ( sgn_sgn @ int @ L2 ) )
=> ( ( divide_divide @ int @ ( times_times @ int @ ( sgn_sgn @ int @ K ) @ ( semiring_1_of_nat @ int @ M ) ) @ ( times_times @ int @ ( sgn_sgn @ int @ L2 ) @ ( semiring_1_of_nat @ int @ N3 ) ) )
= ( semiring_1_of_nat @ int @ ( divide_divide @ nat @ M @ N3 ) ) ) )
& ( ( ( sgn_sgn @ int @ K )
!= ( sgn_sgn @ int @ L2 ) )
=> ( ( divide_divide @ int @ ( times_times @ int @ ( sgn_sgn @ int @ K ) @ ( semiring_1_of_nat @ int @ M ) ) @ ( times_times @ int @ ( sgn_sgn @ int @ L2 ) @ ( semiring_1_of_nat @ int @ N3 ) ) )
= ( uminus_uminus @ int
@ ( semiring_1_of_nat @ int
@ ( plus_plus @ nat @ ( divide_divide @ nat @ M @ N3 )
@ ( zero_neq_one_of_bool @ nat
@ ~ ( dvd_dvd @ nat @ N3 @ M ) ) ) ) ) ) ) ) ) ) ).
% divide_int_unfold
thf(fact_5306_modulo__int__unfold,axiom,
! [L2: int,K: int,N3: nat,M: nat] :
( ( ( ( ( sgn_sgn @ int @ L2 )
= ( zero_zero @ int ) )
| ( ( sgn_sgn @ int @ K )
= ( zero_zero @ int ) )
| ( N3
= ( 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 @ L2 ) @ ( semiring_1_of_nat @ int @ N3 ) ) )
= ( times_times @ int @ ( sgn_sgn @ int @ K ) @ ( semiring_1_of_nat @ int @ M ) ) ) )
& ( ~ ( ( ( sgn_sgn @ int @ L2 )
= ( zero_zero @ int ) )
| ( ( sgn_sgn @ int @ K )
= ( zero_zero @ int ) )
| ( N3
= ( zero_zero @ nat ) ) )
=> ( ( ( ( sgn_sgn @ int @ K )
= ( sgn_sgn @ int @ L2 ) )
=> ( ( modulo_modulo @ int @ ( times_times @ int @ ( sgn_sgn @ int @ K ) @ ( semiring_1_of_nat @ int @ M ) ) @ ( times_times @ int @ ( sgn_sgn @ int @ L2 ) @ ( semiring_1_of_nat @ int @ N3 ) ) )
= ( times_times @ int @ ( sgn_sgn @ int @ L2 ) @ ( semiring_1_of_nat @ int @ ( modulo_modulo @ nat @ M @ N3 ) ) ) ) )
& ( ( ( sgn_sgn @ int @ K )
!= ( sgn_sgn @ int @ L2 ) )
=> ( ( modulo_modulo @ int @ ( times_times @ int @ ( sgn_sgn @ int @ K ) @ ( semiring_1_of_nat @ int @ M ) ) @ ( times_times @ int @ ( sgn_sgn @ int @ L2 ) @ ( semiring_1_of_nat @ int @ N3 ) ) )
= ( times_times @ int @ ( sgn_sgn @ int @ L2 )
@ ( minus_minus @ int
@ ( semiring_1_of_nat @ int
@ ( times_times @ nat @ N3
@ ( zero_neq_one_of_bool @ nat
@ ~ ( dvd_dvd @ nat @ N3 @ M ) ) ) )
@ ( semiring_1_of_nat @ int @ ( modulo_modulo @ nat @ M @ N3 ) ) ) ) ) ) ) ) ) ).
% modulo_int_unfold
thf(fact_5307_divide__int__def,axiom,
( ( divide_divide @ int )
= ( ^ [K2: int,L: int] :
( if @ int
@ ( L
= ( zero_zero @ int ) )
@ ( zero_zero @ int )
@ ( if @ int
@ ( ( sgn_sgn @ int @ K2 )
= ( sgn_sgn @ int @ L ) )
@ ( semiring_1_of_nat @ int @ ( divide_divide @ nat @ ( nat2 @ ( abs_abs @ int @ K2 ) ) @ ( nat2 @ ( abs_abs @ int @ L ) ) ) )
@ ( uminus_uminus @ int
@ ( semiring_1_of_nat @ int
@ ( plus_plus @ nat @ ( divide_divide @ nat @ ( nat2 @ ( abs_abs @ int @ K2 ) ) @ ( nat2 @ ( abs_abs @ int @ L ) ) )
@ ( zero_neq_one_of_bool @ nat
@ ~ ( dvd_dvd @ int @ L @ K2 ) ) ) ) ) ) ) ) ) ).
% divide_int_def
thf(fact_5308_signed__take__bit__eq__take__bit__minus,axiom,
( ( bit_ri4674362597316999326ke_bit @ int )
= ( ^ [N2: nat,K2: int] : ( minus_minus @ int @ ( bit_se2584673776208193580ke_bit @ int @ ( suc @ N2 ) @ K2 ) @ ( times_times @ int @ ( power_power @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ ( suc @ N2 ) ) @ ( zero_neq_one_of_bool @ int @ ( bit_se5641148757651400278ts_bit @ int @ K2 @ N2 ) ) ) ) ) ) ).
% signed_take_bit_eq_take_bit_minus
thf(fact_5309_neg__mask__add__mask,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [X: word @ A,N3: nat] :
( ( plus_plus @ ( word @ A ) @ ( bit_se5824344872417868541ns_and @ ( word @ A ) @ X @ ( bit_ri4277139882892585799ns_not @ ( word @ A ) @ ( bit_se2239418461657761734s_mask @ ( word @ A ) @ N3 ) ) ) @ ( minus_minus @ ( word @ A ) @ ( power_power @ ( word @ A ) @ ( numeral_numeral @ ( word @ A ) @ ( bit0 @ one2 ) ) @ N3 ) @ ( one_one @ ( word @ A ) ) ) )
= ( bit_se1065995026697491101ons_or @ ( word @ A ) @ X @ ( bit_se2239418461657761734s_mask @ ( word @ A ) @ N3 ) ) ) ) ).
% neg_mask_add_mask
thf(fact_5310_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_5311_or_Oidem,axiom,
! [A: $tType] :
( ( bit_se359711467146920520ations @ A )
=> ! [A2: A] :
( ( bit_se1065995026697491101ons_or @ A @ A2 @ A2 )
= A2 ) ) ).
% or.idem
thf(fact_5312_or_Oleft__idem,axiom,
! [A: $tType] :
( ( bit_se359711467146920520ations @ A )
=> ! [A2: A,B2: A] :
( ( bit_se1065995026697491101ons_or @ A @ A2 @ ( bit_se1065995026697491101ons_or @ A @ A2 @ B2 ) )
= ( bit_se1065995026697491101ons_or @ A @ A2 @ B2 ) ) ) ).
% or.left_idem
thf(fact_5313_or_Oright__idem,axiom,
! [A: $tType] :
( ( bit_se359711467146920520ations @ A )
=> ! [A2: A,B2: A] :
( ( bit_se1065995026697491101ons_or @ A @ ( bit_se1065995026697491101ons_or @ A @ A2 @ B2 ) @ B2 )
= ( bit_se1065995026697491101ons_or @ A @ A2 @ B2 ) ) ) ).
% or.right_idem
thf(fact_5314_bit__0__eq,axiom,
! [A: $tType] :
( ( bit_semiring_bits @ A )
=> ( ( bit_se5641148757651400278ts_bit @ A @ ( zero_zero @ A ) )
= ( bot_bot @ ( nat > $o ) ) ) ) ).
% bit_0_eq
thf(fact_5315_or_Oright__neutral,axiom,
! [A: $tType] :
( ( bit_se359711467146920520ations @ A )
=> ! [A2: A] :
( ( bit_se1065995026697491101ons_or @ A @ A2 @ ( zero_zero @ A ) )
= A2 ) ) ).
% or.right_neutral
thf(fact_5316_or_Oleft__neutral,axiom,
! [A: $tType] :
( ( bit_se359711467146920520ations @ A )
=> ! [A2: A] :
( ( bit_se1065995026697491101ons_or @ A @ ( zero_zero @ A ) @ A2 )
= A2 ) ) ).
% or.left_neutral
thf(fact_5317_sgn__le__0__iff,axiom,
! [X: real] :
( ( ord_less_eq @ real @ ( sgn_sgn @ real @ X ) @ ( zero_zero @ real ) )
= ( ord_less_eq @ real @ X @ ( zero_zero @ real ) ) ) ).
% sgn_le_0_iff
thf(fact_5318_zero__le__sgn__iff,axiom,
! [X: real] :
( ( ord_less_eq @ real @ ( zero_zero @ real ) @ ( sgn_sgn @ real @ X ) )
= ( ord_less_eq @ real @ ( zero_zero @ real ) @ X ) ) ).
% zero_le_sgn_iff
thf(fact_5319_take__bit__or,axiom,
! [A: $tType] :
( ( bit_se359711467146920520ations @ A )
=> ! [N3: nat,A2: A,B2: A] :
( ( bit_se2584673776208193580ke_bit @ A @ N3 @ ( bit_se1065995026697491101ons_or @ A @ A2 @ B2 ) )
= ( bit_se1065995026697491101ons_or @ A @ ( bit_se2584673776208193580ke_bit @ A @ N3 @ A2 ) @ ( bit_se2584673776208193580ke_bit @ A @ N3 @ B2 ) ) ) ) ).
% take_bit_or
thf(fact_5320_bit_Odisj__one__left,axiom,
! [A: $tType] :
( ( bit_ri3973907225187159222ations @ A )
=> ! [X: A] :
( ( bit_se1065995026697491101ons_or @ A @ ( uminus_uminus @ A @ ( one_one @ A ) ) @ X )
= ( uminus_uminus @ A @ ( one_one @ A ) ) ) ) ).
% bit.disj_one_left
thf(fact_5321_bit_Odisj__one__right,axiom,
! [A: $tType] :
( ( bit_ri3973907225187159222ations @ A )
=> ! [X: A] :
( ( bit_se1065995026697491101ons_or @ A @ X @ ( uminus_uminus @ A @ ( one_one @ A ) ) )
= ( uminus_uminus @ A @ ( one_one @ A ) ) ) ) ).
% bit.disj_one_right
thf(fact_5322_bit_Ode__Morgan__disj,axiom,
! [A: $tType] :
( ( bit_ri3973907225187159222ations @ A )
=> ! [X: A,Y: A] :
( ( bit_ri4277139882892585799ns_not @ A @ ( bit_se1065995026697491101ons_or @ A @ X @ Y ) )
= ( bit_se5824344872417868541ns_and @ A @ ( bit_ri4277139882892585799ns_not @ A @ X ) @ ( bit_ri4277139882892585799ns_not @ A @ Y ) ) ) ) ).
% bit.de_Morgan_disj
thf(fact_5323_bit_Ode__Morgan__conj,axiom,
! [A: $tType] :
( ( bit_ri3973907225187159222ations @ A )
=> ! [X: A,Y: A] :
( ( bit_ri4277139882892585799ns_not @ A @ ( bit_se5824344872417868541ns_and @ A @ X @ Y ) )
= ( bit_se1065995026697491101ons_or @ A @ ( bit_ri4277139882892585799ns_not @ A @ X ) @ ( bit_ri4277139882892585799ns_not @ A @ Y ) ) ) ) ).
% bit.de_Morgan_conj
thf(fact_5324_word__bitwise__m1__simps_I4_J,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [X: word @ A] :
( ( bit_se1065995026697491101ons_or @ ( word @ A ) @ ( uminus_uminus @ ( word @ A ) @ ( one_one @ ( word @ A ) ) ) @ X )
= ( uminus_uminus @ ( word @ A ) @ ( one_one @ ( word @ A ) ) ) ) ) ).
% word_bitwise_m1_simps(4)
thf(fact_5325_word__or__max,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [X: word @ A] :
( ( bit_se1065995026697491101ons_or @ ( word @ A ) @ X @ ( uminus_uminus @ ( word @ A ) @ ( one_one @ ( word @ A ) ) ) )
= ( uminus_uminus @ ( word @ A ) @ ( one_one @ ( word @ A ) ) ) ) ) ).
% word_or_max
thf(fact_5326_bit__numeral__Bit0__Suc__iff,axiom,
! [A: $tType] :
( ( bit_un5681908812861735899ations @ A )
=> ! [M: num,N3: nat] :
( ( bit_se5641148757651400278ts_bit @ A @ ( numeral_numeral @ A @ ( bit0 @ M ) ) @ ( suc @ N3 ) )
= ( bit_se5641148757651400278ts_bit @ A @ ( numeral_numeral @ A @ M ) @ N3 ) ) ) ).
% bit_numeral_Bit0_Suc_iff
thf(fact_5327_bit__numeral__Bit1__Suc__iff,axiom,
! [A: $tType] :
( ( bit_un5681908812861735899ations @ A )
=> ! [M: num,N3: nat] :
( ( bit_se5641148757651400278ts_bit @ A @ ( numeral_numeral @ A @ ( bit1 @ M ) ) @ ( suc @ N3 ) )
= ( bit_se5641148757651400278ts_bit @ A @ ( numeral_numeral @ A @ M ) @ N3 ) ) ) ).
% bit_numeral_Bit1_Suc_iff
thf(fact_5328_or__numerals_I8_J,axiom,
! [A: $tType] :
( ( bit_un5681908812861735899ations @ A )
=> ! [X: num] :
( ( bit_se1065995026697491101ons_or @ A @ ( numeral_numeral @ A @ ( bit1 @ X ) ) @ ( one_one @ A ) )
= ( numeral_numeral @ A @ ( bit1 @ X ) ) ) ) ).
% or_numerals(8)
thf(fact_5329_or__numerals_I2_J,axiom,
! [A: $tType] :
( ( bit_un5681908812861735899ations @ A )
=> ! [Y: num] :
( ( bit_se1065995026697491101ons_or @ A @ ( one_one @ A ) @ ( numeral_numeral @ A @ ( bit1 @ Y ) ) )
= ( numeral_numeral @ A @ ( bit1 @ Y ) ) ) ) ).
% or_numerals(2)
thf(fact_5330_bit_Odisj__cancel__right,axiom,
! [A: $tType] :
( ( bit_ri3973907225187159222ations @ A )
=> ! [X: A] :
( ( bit_se1065995026697491101ons_or @ A @ X @ ( bit_ri4277139882892585799ns_not @ A @ X ) )
= ( uminus_uminus @ A @ ( one_one @ A ) ) ) ) ).
% bit.disj_cancel_right
thf(fact_5331_bit_Odisj__cancel__left,axiom,
! [A: $tType] :
( ( bit_ri3973907225187159222ations @ A )
=> ! [X: A] :
( ( bit_se1065995026697491101ons_or @ A @ ( bit_ri4277139882892585799ns_not @ A @ X ) @ X )
= ( uminus_uminus @ A @ ( one_one @ A ) ) ) ) ).
% bit.disj_cancel_left
thf(fact_5332_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_5333_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_5334_signed__take__bit__nonnegative__iff,axiom,
! [N3: nat,K: int] :
( ( ord_less_eq @ int @ ( zero_zero @ int ) @ ( bit_ri4674362597316999326ke_bit @ int @ N3 @ K ) )
= ( ~ ( bit_se5641148757651400278ts_bit @ int @ K @ N3 ) ) ) ).
% signed_take_bit_nonnegative_iff
thf(fact_5335_signed__take__bit__negative__iff,axiom,
! [N3: nat,K: int] :
( ( ord_less @ int @ ( bit_ri4674362597316999326ke_bit @ int @ N3 @ K ) @ ( zero_zero @ int ) )
= ( bit_se5641148757651400278ts_bit @ int @ K @ N3 ) ) ).
% signed_take_bit_negative_iff
thf(fact_5336_word__or__not,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [X: word @ A] :
( ( bit_se1065995026697491101ons_or @ ( word @ A ) @ X @ ( bit_ri4277139882892585799ns_not @ ( word @ A ) @ X ) )
= ( uminus_uminus @ ( word @ A ) @ ( one_one @ ( word @ A ) ) ) ) ) ).
% word_or_not
thf(fact_5337_or__numerals_I3_J,axiom,
! [A: $tType] :
( ( bit_un5681908812861735899ations @ A )
=> ! [X: num,Y: num] :
( ( bit_se1065995026697491101ons_or @ A @ ( numeral_numeral @ A @ ( bit0 @ X ) ) @ ( numeral_numeral @ A @ ( bit0 @ Y ) ) )
= ( times_times @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ ( bit_se1065995026697491101ons_or @ A @ ( numeral_numeral @ A @ X ) @ ( numeral_numeral @ A @ Y ) ) ) ) ) ).
% or_numerals(3)
thf(fact_5338_or__numerals_I5_J,axiom,
! [A: $tType] :
( ( bit_un5681908812861735899ations @ A )
=> ! [X: num] :
( ( bit_se1065995026697491101ons_or @ A @ ( numeral_numeral @ A @ ( bit0 @ X ) ) @ ( one_one @ A ) )
= ( numeral_numeral @ A @ ( bit1 @ X ) ) ) ) ).
% or_numerals(5)
thf(fact_5339_or__numerals_I1_J,axiom,
! [A: $tType] :
( ( bit_un5681908812861735899ations @ A )
=> ! [Y: num] :
( ( bit_se1065995026697491101ons_or @ A @ ( one_one @ A ) @ ( numeral_numeral @ A @ ( bit0 @ Y ) ) )
= ( numeral_numeral @ A @ ( bit1 @ Y ) ) ) ) ).
% or_numerals(1)
thf(fact_5340_bit__minus__numeral__Bit0__Suc__iff,axiom,
! [W: num,N3: nat] :
( ( bit_se5641148757651400278ts_bit @ int @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ ( bit0 @ W ) ) ) @ ( suc @ N3 ) )
= ( bit_se5641148757651400278ts_bit @ int @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ W ) ) @ N3 ) ) ).
% bit_minus_numeral_Bit0_Suc_iff
thf(fact_5341_bit__numeral__simps_I2_J,axiom,
! [A: $tType] :
( ( bit_un5681908812861735899ations @ A )
=> ! [W: num,N3: num] :
( ( bit_se5641148757651400278ts_bit @ A @ ( numeral_numeral @ A @ ( bit0 @ W ) ) @ ( numeral_numeral @ nat @ N3 ) )
= ( bit_se5641148757651400278ts_bit @ A @ ( numeral_numeral @ A @ W ) @ ( pred_numeral @ N3 ) ) ) ) ).
% bit_numeral_simps(2)
thf(fact_5342_bit__minus__numeral__Bit1__Suc__iff,axiom,
! [W: num,N3: nat] :
( ( bit_se5641148757651400278ts_bit @ int @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ ( bit1 @ W ) ) ) @ ( suc @ N3 ) )
= ( ~ ( bit_se5641148757651400278ts_bit @ int @ ( numeral_numeral @ int @ W ) @ N3 ) ) ) ).
% bit_minus_numeral_Bit1_Suc_iff
thf(fact_5343_bit__numeral__simps_I3_J,axiom,
! [A: $tType] :
( ( bit_un5681908812861735899ations @ A )
=> ! [W: num,N3: num] :
( ( bit_se5641148757651400278ts_bit @ A @ ( numeral_numeral @ A @ ( bit1 @ W ) ) @ ( numeral_numeral @ nat @ N3 ) )
= ( bit_se5641148757651400278ts_bit @ A @ ( numeral_numeral @ A @ W ) @ ( pred_numeral @ N3 ) ) ) ) ).
% bit_numeral_simps(3)
thf(fact_5344_bit__0,axiom,
! [A: $tType] :
( ( bit_semiring_bits @ A )
=> ! [A2: A] :
( ( bit_se5641148757651400278ts_bit @ A @ A2 @ ( zero_zero @ nat ) )
= ( ~ ( dvd_dvd @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ A2 ) ) ) ) ).
% bit_0
thf(fact_5345_bit__minus__numeral__int_I1_J,axiom,
! [W: num,N3: num] :
( ( bit_se5641148757651400278ts_bit @ int @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ ( bit0 @ W ) ) ) @ ( numeral_numeral @ nat @ N3 ) )
= ( bit_se5641148757651400278ts_bit @ int @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ W ) ) @ ( pred_numeral @ N3 ) ) ) ).
% bit_minus_numeral_int(1)
thf(fact_5346_bit__minus__numeral__int_I2_J,axiom,
! [W: num,N3: num] :
( ( bit_se5641148757651400278ts_bit @ int @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ ( bit1 @ W ) ) ) @ ( numeral_numeral @ nat @ N3 ) )
= ( ~ ( bit_se5641148757651400278ts_bit @ int @ ( numeral_numeral @ int @ W ) @ ( pred_numeral @ N3 ) ) ) ) ).
% bit_minus_numeral_int(2)
thf(fact_5347_bit__mod__2__iff,axiom,
! [A: $tType] :
( ( bit_semiring_bits @ A )
=> ! [A2: A,N3: nat] :
( ( bit_se5641148757651400278ts_bit @ A @ ( modulo_modulo @ A @ A2 @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) @ N3 )
= ( ( N3
= ( zero_zero @ nat ) )
& ~ ( dvd_dvd @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ A2 ) ) ) ) ).
% bit_mod_2_iff
thf(fact_5348_bin__nth__minus__Bit0,axiom,
! [N3: nat,W: num] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ N3 )
=> ( ( bit_se5641148757651400278ts_bit @ int @ ( numeral_numeral @ int @ ( bit0 @ W ) ) @ N3 )
= ( bit_se5641148757651400278ts_bit @ int @ ( numeral_numeral @ int @ W ) @ ( minus_minus @ nat @ N3 @ ( one_one @ nat ) ) ) ) ) ).
% bin_nth_minus_Bit0
thf(fact_5349_bin__nth__minus__Bit1,axiom,
! [N3: nat,W: num] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ N3 )
=> ( ( bit_se5641148757651400278ts_bit @ int @ ( numeral_numeral @ int @ ( bit1 @ W ) ) @ N3 )
= ( bit_se5641148757651400278ts_bit @ int @ ( numeral_numeral @ int @ W ) @ ( minus_minus @ nat @ N3 @ ( one_one @ nat ) ) ) ) ) ).
% bin_nth_minus_Bit1
thf(fact_5350_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_5351_or__numerals_I7_J,axiom,
! [A: $tType] :
( ( bit_un5681908812861735899ations @ A )
=> ! [X: num,Y: num] :
( ( bit_se1065995026697491101ons_or @ A @ ( numeral_numeral @ A @ ( bit1 @ X ) ) @ ( numeral_numeral @ A @ ( bit1 @ Y ) ) )
= ( plus_plus @ A @ ( one_one @ A ) @ ( times_times @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ ( bit_se1065995026697491101ons_or @ A @ ( numeral_numeral @ A @ X ) @ ( numeral_numeral @ A @ Y ) ) ) ) ) ) ).
% or_numerals(7)
thf(fact_5352_or__numerals_I6_J,axiom,
! [A: $tType] :
( ( bit_un5681908812861735899ations @ A )
=> ! [X: num,Y: num] :
( ( bit_se1065995026697491101ons_or @ A @ ( numeral_numeral @ A @ ( bit1 @ X ) ) @ ( numeral_numeral @ A @ ( bit0 @ Y ) ) )
= ( plus_plus @ A @ ( one_one @ A ) @ ( times_times @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ ( bit_se1065995026697491101ons_or @ A @ ( numeral_numeral @ A @ X ) @ ( numeral_numeral @ A @ Y ) ) ) ) ) ) ).
% or_numerals(6)
thf(fact_5353_or__numerals_I4_J,axiom,
! [A: $tType] :
( ( bit_un5681908812861735899ations @ A )
=> ! [X: num,Y: num] :
( ( bit_se1065995026697491101ons_or @ A @ ( numeral_numeral @ A @ ( bit0 @ X ) ) @ ( numeral_numeral @ A @ ( bit1 @ Y ) ) )
= ( plus_plus @ A @ ( one_one @ A ) @ ( times_times @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ ( bit_se1065995026697491101ons_or @ A @ ( numeral_numeral @ A @ X ) @ ( numeral_numeral @ A @ Y ) ) ) ) ) ) ).
% or_numerals(4)
thf(fact_5354_bit__minus__int__iff,axiom,
! [K: int,N3: nat] :
( ( bit_se5641148757651400278ts_bit @ int @ ( uminus_uminus @ int @ K ) @ N3 )
= ( bit_se5641148757651400278ts_bit @ int @ ( bit_ri4277139882892585799ns_not @ int @ ( minus_minus @ int @ K @ ( one_one @ int ) ) ) @ N3 ) ) ).
% bit_minus_int_iff
thf(fact_5355_bit__not__int__iff,axiom,
! [K: int,N3: nat] :
( ( bit_se5641148757651400278ts_bit @ int @ ( bit_ri4277139882892585799ns_not @ int @ K ) @ N3 )
= ( ~ ( bit_se5641148757651400278ts_bit @ int @ K @ N3 ) ) ) ).
% bit_not_int_iff
thf(fact_5356_signed__take__bit__eq__if__negative,axiom,
! [A: $tType] :
( ( bit_ri3973907225187159222ations @ A )
=> ! [A2: A,N3: nat] :
( ( bit_se5641148757651400278ts_bit @ A @ A2 @ N3 )
=> ( ( bit_ri4674362597316999326ke_bit @ A @ N3 @ A2 )
= ( bit_se1065995026697491101ons_or @ A @ ( bit_se2584673776208193580ke_bit @ A @ N3 @ A2 ) @ ( bit_ri4277139882892585799ns_not @ A @ ( bit_se2239418461657761734s_mask @ A @ N3 ) ) ) ) ) ) ).
% signed_take_bit_eq_if_negative
thf(fact_5357_le__word__or2,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [X: word @ A,Y: word @ A] : ( ord_less_eq @ ( word @ A ) @ X @ ( bit_se1065995026697491101ons_or @ ( word @ A ) @ X @ Y ) ) ) ).
% le_word_or2
thf(fact_5358_le__word__or1,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [C2: word @ A,Y: word @ A] : ( ord_less_eq @ ( word @ A ) @ C2 @ ( bit_se1065995026697491101ons_or @ ( word @ A ) @ Y @ C2 ) ) ) ).
% le_word_or1
thf(fact_5359_bit__numeral__simps_I1_J,axiom,
! [A: $tType] :
( ( bit_un5681908812861735899ations @ A )
=> ! [N3: num] :
~ ( bit_se5641148757651400278ts_bit @ A @ ( one_one @ A ) @ ( numeral_numeral @ nat @ N3 ) ) ) ).
% bit_numeral_simps(1)
thf(fact_5360_bit__numeral__iff,axiom,
! [A: $tType] :
( ( bit_un5681908812861735899ations @ A )
=> ! [M: num,N3: nat] :
( ( bit_se5641148757651400278ts_bit @ A @ ( numeral_numeral @ A @ M ) @ N3 )
= ( bit_se5641148757651400278ts_bit @ nat @ ( numeral_numeral @ nat @ M ) @ N3 ) ) ) ).
% bit_numeral_iff
thf(fact_5361_bit__and__int__iff,axiom,
! [K: int,L2: int,N3: nat] :
( ( bit_se5641148757651400278ts_bit @ int @ ( bit_se5824344872417868541ns_and @ int @ K @ L2 ) @ N3 )
= ( ( bit_se5641148757651400278ts_bit @ int @ K @ N3 )
& ( bit_se5641148757651400278ts_bit @ int @ L2 @ N3 ) ) ) ).
% bit_and_int_iff
thf(fact_5362_bit_Odisj__conj__distrib2,axiom,
! [A: $tType] :
( ( bit_ri3973907225187159222ations @ A )
=> ! [Y: A,Z: A,X: A] :
( ( bit_se1065995026697491101ons_or @ A @ ( bit_se5824344872417868541ns_and @ A @ Y @ Z ) @ X )
= ( bit_se5824344872417868541ns_and @ A @ ( bit_se1065995026697491101ons_or @ A @ Y @ X ) @ ( bit_se1065995026697491101ons_or @ A @ Z @ X ) ) ) ) ).
% bit.disj_conj_distrib2
thf(fact_5363_bit_Oconj__disj__distrib2,axiom,
! [A: $tType] :
( ( bit_ri3973907225187159222ations @ A )
=> ! [Y: A,Z: A,X: A] :
( ( bit_se5824344872417868541ns_and @ A @ ( bit_se1065995026697491101ons_or @ A @ Y @ Z ) @ X )
= ( bit_se1065995026697491101ons_or @ A @ ( bit_se5824344872417868541ns_and @ A @ Y @ X ) @ ( bit_se5824344872417868541ns_and @ A @ Z @ X ) ) ) ) ).
% bit.conj_disj_distrib2
thf(fact_5364_bit_Odisj__conj__distrib,axiom,
! [A: $tType] :
( ( bit_ri3973907225187159222ations @ A )
=> ! [X: A,Y: A,Z: A] :
( ( bit_se1065995026697491101ons_or @ A @ X @ ( bit_se5824344872417868541ns_and @ A @ Y @ Z ) )
= ( bit_se5824344872417868541ns_and @ A @ ( bit_se1065995026697491101ons_or @ A @ X @ Y ) @ ( bit_se1065995026697491101ons_or @ A @ X @ Z ) ) ) ) ).
% bit.disj_conj_distrib
thf(fact_5365_bit_Oconj__disj__distrib,axiom,
! [A: $tType] :
( ( bit_ri3973907225187159222ations @ A )
=> ! [X: A,Y: A,Z: A] :
( ( bit_se5824344872417868541ns_and @ A @ X @ ( bit_se1065995026697491101ons_or @ A @ Y @ Z ) )
= ( bit_se1065995026697491101ons_or @ A @ ( bit_se5824344872417868541ns_and @ A @ X @ Y ) @ ( bit_se5824344872417868541ns_and @ A @ X @ Z ) ) ) ) ).
% bit.conj_disj_distrib
thf(fact_5366_bit__and__iff,axiom,
! [A: $tType] :
( ( bit_se359711467146920520ations @ A )
=> ! [A2: A,B2: A,N3: nat] :
( ( bit_se5641148757651400278ts_bit @ A @ ( bit_se5824344872417868541ns_and @ A @ A2 @ B2 ) @ N3 )
= ( ( bit_se5641148757651400278ts_bit @ A @ A2 @ N3 )
& ( bit_se5641148757651400278ts_bit @ A @ B2 @ N3 ) ) ) ) ).
% bit_and_iff
thf(fact_5367_bit__disjunctive__add__iff,axiom,
! [A: $tType] :
( ( bit_semiring_bits @ A )
=> ! [A2: A,B2: A,N3: nat] :
( ! [N: nat] :
( ~ ( bit_se5641148757651400278ts_bit @ A @ A2 @ N )
| ~ ( bit_se5641148757651400278ts_bit @ A @ B2 @ N ) )
=> ( ( bit_se5641148757651400278ts_bit @ A @ ( plus_plus @ A @ A2 @ B2 ) @ N3 )
= ( ( bit_se5641148757651400278ts_bit @ A @ A2 @ N3 )
| ( bit_se5641148757651400278ts_bit @ A @ B2 @ N3 ) ) ) ) ) ).
% bit_disjunctive_add_iff
thf(fact_5368_disjunctive__add,axiom,
! [A: $tType] :
( ( bit_se359711467146920520ations @ A )
=> ! [A2: A,B2: A] :
( ! [N: nat] :
( ~ ( bit_se5641148757651400278ts_bit @ A @ A2 @ N )
| ~ ( bit_se5641148757651400278ts_bit @ A @ B2 @ N ) )
=> ( ( plus_plus @ A @ A2 @ B2 )
= ( bit_se1065995026697491101ons_or @ A @ A2 @ B2 ) ) ) ) ).
% disjunctive_add
thf(fact_5369_bit__of__nat__iff__bit,axiom,
! [A: $tType] :
( ( bit_un5681908812861735899ations @ A )
=> ! [M: nat,N3: nat] :
( ( bit_se5641148757651400278ts_bit @ A @ ( semiring_1_of_nat @ A @ M ) @ N3 )
= ( bit_se5641148757651400278ts_bit @ nat @ M @ N3 ) ) ) ).
% bit_of_nat_iff_bit
thf(fact_5370_of__nat__or__eq,axiom,
! [A: $tType] :
( ( bit_se359711467146920520ations @ A )
=> ! [M: nat,N3: nat] :
( ( semiring_1_of_nat @ A @ ( bit_se1065995026697491101ons_or @ nat @ M @ N3 ) )
= ( bit_se1065995026697491101ons_or @ A @ ( semiring_1_of_nat @ A @ M ) @ ( semiring_1_of_nat @ A @ N3 ) ) ) ) ).
% of_nat_or_eq
thf(fact_5371_bit__unset__bit__iff,axiom,
! [A: $tType] :
( ( bit_se359711467146920520ations @ A )
=> ! [M: nat,A2: A,N3: nat] :
( ( bit_se5641148757651400278ts_bit @ A @ ( bit_se2638667681897837118et_bit @ A @ M @ A2 ) @ N3 )
= ( ( bit_se5641148757651400278ts_bit @ A @ A2 @ N3 )
& ( M != N3 ) ) ) ) ).
% bit_unset_bit_iff
thf(fact_5372_bit_Odisj__zero__right,axiom,
! [A: $tType] :
( ( bit_ri3973907225187159222ations @ A )
=> ! [X: A] :
( ( bit_se1065995026697491101ons_or @ A @ X @ ( zero_zero @ A ) )
= X ) ) ).
% bit.disj_zero_right
thf(fact_5373_or__eq__0__iff,axiom,
! [A: $tType] :
( ( bit_se359711467146920520ations @ A )
=> ! [A2: A,B2: A] :
( ( ( bit_se1065995026697491101ons_or @ A @ A2 @ B2 )
= ( zero_zero @ A ) )
= ( ( A2
= ( zero_zero @ A ) )
& ( B2
= ( zero_zero @ A ) ) ) ) ) ).
% or_eq_0_iff
thf(fact_5374_of__int__or__eq,axiom,
! [A: $tType] :
( ( bit_ri3973907225187159222ations @ A )
=> ! [K: int,L2: int] :
( ( ring_1_of_int @ A @ ( bit_se1065995026697491101ons_or @ int @ K @ L2 ) )
= ( bit_se1065995026697491101ons_or @ A @ ( ring_1_of_int @ A @ K ) @ ( ring_1_of_int @ A @ L2 ) ) ) ) ).
% of_int_or_eq
thf(fact_5375_or_Oassoc,axiom,
! [A: $tType] :
( ( bit_se359711467146920520ations @ A )
=> ! [A2: A,B2: A,C2: A] :
( ( bit_se1065995026697491101ons_or @ A @ ( bit_se1065995026697491101ons_or @ A @ A2 @ B2 ) @ C2 )
= ( bit_se1065995026697491101ons_or @ A @ A2 @ ( bit_se1065995026697491101ons_or @ A @ B2 @ C2 ) ) ) ) ).
% or.assoc
thf(fact_5376_bit__or__iff,axiom,
! [A: $tType] :
( ( bit_se359711467146920520ations @ A )
=> ! [A2: A,B2: A,N3: nat] :
( ( bit_se5641148757651400278ts_bit @ A @ ( bit_se1065995026697491101ons_or @ A @ A2 @ B2 ) @ N3 )
= ( ( bit_se5641148757651400278ts_bit @ A @ A2 @ N3 )
| ( bit_se5641148757651400278ts_bit @ A @ B2 @ N3 ) ) ) ) ).
% bit_or_iff
thf(fact_5377_or_Ocommute,axiom,
! [A: $tType] :
( ( bit_se359711467146920520ations @ A )
=> ( ( bit_se1065995026697491101ons_or @ A )
= ( ^ [A6: A,B5: A] : ( bit_se1065995026697491101ons_or @ A @ B5 @ A6 ) ) ) ) ).
% or.commute
thf(fact_5378_or_Oleft__commute,axiom,
! [A: $tType] :
( ( bit_se359711467146920520ations @ A )
=> ! [B2: A,A2: A,C2: A] :
( ( bit_se1065995026697491101ons_or @ A @ B2 @ ( bit_se1065995026697491101ons_or @ A @ A2 @ C2 ) )
= ( bit_se1065995026697491101ons_or @ A @ A2 @ ( bit_se1065995026697491101ons_or @ A @ B2 @ C2 ) ) ) ) ).
% or.left_commute
thf(fact_5379_not__bit__1__Suc,axiom,
! [A: $tType] :
( ( bit_se359711467146920520ations @ A )
=> ! [N3: nat] :
~ ( bit_se5641148757651400278ts_bit @ A @ ( one_one @ A ) @ ( suc @ N3 ) ) ) ).
% not_bit_1_Suc
thf(fact_5380_bit__1__iff,axiom,
! [A: $tType] :
( ( bit_semiring_bits @ A )
=> ! [N3: nat] :
( ( bit_se5641148757651400278ts_bit @ A @ ( one_one @ A ) @ N3 )
= ( N3
= ( zero_zero @ nat ) ) ) ) ).
% bit_1_iff
thf(fact_5381_bit__take__bit__iff,axiom,
! [A: $tType] :
( ( bit_se359711467146920520ations @ A )
=> ! [M: nat,A2: A,N3: nat] :
( ( bit_se5641148757651400278ts_bit @ A @ ( bit_se2584673776208193580ke_bit @ A @ M @ A2 ) @ N3 )
= ( ( ord_less @ nat @ N3 @ M )
& ( bit_se5641148757651400278ts_bit @ A @ A2 @ N3 ) ) ) ) ).
% bit_take_bit_iff
thf(fact_5382_bit__of__bool__iff,axiom,
! [A: $tType] :
( ( bit_semiring_bits @ A )
=> ! [B2: $o,N3: nat] :
( ( bit_se5641148757651400278ts_bit @ A @ ( zero_neq_one_of_bool @ A @ B2 ) @ N3 )
= ( B2
& ( N3
= ( zero_zero @ nat ) ) ) ) ) ).
% bit_of_bool_iff
thf(fact_5383_signed__take__bit__eq__if__positive,axiom,
! [A: $tType] :
( ( bit_ri3973907225187159222ations @ A )
=> ! [A2: A,N3: nat] :
( ~ ( bit_se5641148757651400278ts_bit @ A @ A2 @ N3 )
=> ( ( bit_ri4674362597316999326ke_bit @ A @ N3 @ A2 )
= ( bit_se2584673776208193580ke_bit @ A @ N3 @ A2 ) ) ) ) ).
% signed_take_bit_eq_if_positive
thf(fact_5384_or__eq__not__not__and,axiom,
! [A: $tType] :
( ( bit_ri3973907225187159222ations @ A )
=> ( ( bit_se1065995026697491101ons_or @ A )
= ( ^ [A6: A,B5: A] : ( bit_ri4277139882892585799ns_not @ A @ ( bit_se5824344872417868541ns_and @ A @ ( bit_ri4277139882892585799ns_not @ A @ A6 ) @ ( bit_ri4277139882892585799ns_not @ A @ B5 ) ) ) ) ) ) ).
% or_eq_not_not_and
thf(fact_5385_and__eq__not__not__or,axiom,
! [A: $tType] :
( ( bit_ri3973907225187159222ations @ A )
=> ( ( bit_se5824344872417868541ns_and @ A )
= ( ^ [A6: A,B5: A] : ( bit_ri4277139882892585799ns_not @ A @ ( bit_se1065995026697491101ons_or @ A @ ( bit_ri4277139882892585799ns_not @ A @ A6 ) @ ( bit_ri4277139882892585799ns_not @ A @ B5 ) ) ) ) ) ) ).
% and_eq_not_not_or
thf(fact_5386_real__sgn__eq,axiom,
( ( sgn_sgn @ real )
= ( ^ [X2: real] : ( divide_divide @ real @ X2 @ ( abs_abs @ real @ X2 ) ) ) ) ).
% real_sgn_eq
thf(fact_5387_signed__take__bit__def,axiom,
! [A: $tType] :
( ( bit_ri3973907225187159222ations @ A )
=> ( ( bit_ri4674362597316999326ke_bit @ A )
= ( ^ [N2: nat,A6: A] : ( bit_se1065995026697491101ons_or @ A @ ( bit_se2584673776208193580ke_bit @ A @ N2 @ A6 ) @ ( times_times @ A @ ( zero_neq_one_of_bool @ A @ ( bit_se5641148757651400278ts_bit @ A @ A6 @ N2 ) ) @ ( bit_ri4277139882892585799ns_not @ A @ ( bit_se2239418461657761734s_mask @ A @ N2 ) ) ) ) ) ) ) ).
% signed_take_bit_def
thf(fact_5388_disjunctive__diff,axiom,
! [A: $tType] :
( ( bit_ri3973907225187159222ations @ A )
=> ! [B2: A,A2: A] :
( ! [N: nat] :
( ( bit_se5641148757651400278ts_bit @ A @ B2 @ N )
=> ( bit_se5641148757651400278ts_bit @ A @ A2 @ N ) )
=> ( ( minus_minus @ A @ A2 @ B2 )
= ( bit_se5824344872417868541ns_and @ A @ A2 @ ( bit_ri4277139882892585799ns_not @ A @ B2 ) ) ) ) ) ).
% disjunctive_diff
thf(fact_5389_sgn__root,axiom,
! [N3: nat,X: real] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ N3 )
=> ( ( sgn_sgn @ real @ ( root @ N3 @ X ) )
= ( sgn_sgn @ real @ X ) ) ) ).
% sgn_root
thf(fact_5390_not__int__def,axiom,
( ( bit_ri4277139882892585799ns_not @ int )
= ( ^ [K2: int] : ( minus_minus @ int @ ( uminus_uminus @ int @ K2 ) @ ( one_one @ int ) ) ) ) ).
% not_int_def
thf(fact_5391_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_5392_bit__not__int__iff_H,axiom,
! [K: int,N3: nat] :
( ( bit_se5641148757651400278ts_bit @ int @ ( minus_minus @ int @ ( uminus_uminus @ int @ K ) @ ( one_one @ int ) ) @ N3 )
= ( ~ ( bit_se5641148757651400278ts_bit @ int @ K @ N3 ) ) ) ).
% bit_not_int_iff'
thf(fact_5393_flip__bit__eq__if,axiom,
! [A: $tType] :
( ( bit_se359711467146920520ations @ A )
=> ( ( bit_se8732182000553998342ip_bit @ A )
= ( ^ [N2: nat,A6: A] : ( if @ ( nat > A > A ) @ ( bit_se5641148757651400278ts_bit @ A @ A6 @ N2 ) @ ( bit_se2638667681897837118et_bit @ A ) @ ( bit_se5668285175392031749et_bit @ A ) @ N2 @ A6 ) ) ) ) ).
% flip_bit_eq_if
thf(fact_5394_even__or__iff,axiom,
! [A: $tType] :
( ( bit_se359711467146920520ations @ A )
=> ! [A2: A,B2: A] :
( ( dvd_dvd @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ ( bit_se1065995026697491101ons_or @ A @ A2 @ B2 ) )
= ( ( dvd_dvd @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ A2 )
& ( dvd_dvd @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ B2 ) ) ) ) ).
% even_or_iff
thf(fact_5395_bit_Ocomplement__unique,axiom,
! [A: $tType] :
( ( bit_ri3973907225187159222ations @ A )
=> ! [A2: A,X: A,Y: A] :
( ( ( bit_se5824344872417868541ns_and @ A @ A2 @ X )
= ( zero_zero @ A ) )
=> ( ( ( bit_se1065995026697491101ons_or @ A @ A2 @ X )
= ( uminus_uminus @ A @ ( one_one @ A ) ) )
=> ( ( ( bit_se5824344872417868541ns_and @ A @ A2 @ Y )
= ( zero_zero @ A ) )
=> ( ( ( bit_se1065995026697491101ons_or @ A @ A2 @ Y )
= ( uminus_uminus @ A @ ( one_one @ A ) ) )
=> ( X = Y ) ) ) ) ) ) ).
% bit.complement_unique
thf(fact_5396_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_5397_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_5398_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_5399_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_5400_and__not__numerals_I2_J,axiom,
! [N3: num] :
( ( bit_se5824344872417868541ns_and @ int @ ( one_one @ int ) @ ( bit_ri4277139882892585799ns_not @ int @ ( numeral_numeral @ int @ ( bit0 @ N3 ) ) ) )
= ( one_one @ int ) ) ).
% and_not_numerals(2)
thf(fact_5401_bit__imp__take__bit__positive,axiom,
! [N3: nat,M: nat,K: int] :
( ( ord_less @ nat @ N3 @ M )
=> ( ( bit_se5641148757651400278ts_bit @ int @ K @ N3 )
=> ( ord_less @ int @ ( zero_zero @ int ) @ ( bit_se2584673776208193580ke_bit @ int @ M @ K ) ) ) ) ).
% bit_imp_take_bit_positive
thf(fact_5402_mask__subsume,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [N3: nat,M: nat,X: word @ A,Y: word @ A] :
( ( ord_less_eq @ nat @ N3 @ M )
=> ( ( bit_se5824344872417868541ns_and @ ( word @ A ) @ ( bit_se1065995026697491101ons_or @ ( word @ A ) @ X @ ( bit_se5824344872417868541ns_and @ ( word @ A ) @ Y @ ( bit_se2239418461657761734s_mask @ ( word @ A ) @ N3 ) ) ) @ ( bit_ri4277139882892585799ns_not @ ( word @ A ) @ ( bit_se2239418461657761734s_mask @ ( word @ A ) @ M ) ) )
= ( bit_se5824344872417868541ns_and @ ( word @ A ) @ X @ ( bit_ri4277139882892585799ns_not @ ( word @ A ) @ ( bit_se2239418461657761734s_mask @ ( word @ A ) @ M ) ) ) ) ) ) ).
% mask_subsume
thf(fact_5403_sgn__integer__code,axiom,
( ( sgn_sgn @ code_integer )
= ( ^ [K2: code_integer] :
( if @ code_integer
@ ( K2
= ( zero_zero @ code_integer ) )
@ ( zero_zero @ code_integer )
@ ( if @ code_integer @ ( ord_less @ code_integer @ K2 @ ( zero_zero @ code_integer ) ) @ ( uminus_uminus @ code_integer @ ( one_one @ code_integer ) ) @ ( one_one @ code_integer ) ) ) ) ) ).
% sgn_integer_code
thf(fact_5404_exp__eq__0__imp__not__bit,axiom,
! [A: $tType] :
( ( bit_semiring_bits @ A )
=> ! [N3: nat,A2: A] :
( ( ( power_power @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ N3 )
= ( zero_zero @ A ) )
=> ~ ( bit_se5641148757651400278ts_bit @ A @ A2 @ N3 ) ) ) ).
% exp_eq_0_imp_not_bit
thf(fact_5405_bit__Suc,axiom,
! [A: $tType] :
( ( bit_semiring_bits @ A )
=> ! [A2: A,N3: nat] :
( ( bit_se5641148757651400278ts_bit @ A @ A2 @ ( suc @ N3 ) )
= ( bit_se5641148757651400278ts_bit @ A @ ( divide_divide @ A @ A2 @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) @ N3 ) ) ) ).
% bit_Suc
thf(fact_5406_bit__iff__idd__imp__stable,axiom,
! [A: $tType] :
( ( bit_semiring_bits @ A )
=> ! [A2: A] :
( ! [N: nat] :
( ( bit_se5641148757651400278ts_bit @ A @ A2 @ N )
= ( ~ ( dvd_dvd @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ A2 ) ) )
=> ( ( divide_divide @ A @ A2 @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) )
= A2 ) ) ) ).
% bit_iff_idd_imp_stable
thf(fact_5407_stable__imp__bit__iff__odd,axiom,
! [A: $tType] :
( ( bit_semiring_bits @ A )
=> ! [A2: A,N3: nat] :
( ( ( divide_divide @ A @ A2 @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) )
= A2 )
=> ( ( bit_se5641148757651400278ts_bit @ A @ A2 @ N3 )
= ( ~ ( dvd_dvd @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ A2 ) ) ) ) ) ).
% stable_imp_bit_iff_odd
thf(fact_5408_bit_Ocompl__unique,axiom,
! [A: $tType] :
( ( bit_ri3973907225187159222ations @ A )
=> ! [X: A,Y: A] :
( ( ( bit_se5824344872417868541ns_and @ A @ X @ Y )
= ( zero_zero @ A ) )
=> ( ( ( bit_se1065995026697491101ons_or @ A @ X @ Y )
= ( uminus_uminus @ A @ ( one_one @ A ) ) )
=> ( ( bit_ri4277139882892585799ns_not @ A @ X )
= Y ) ) ) ) ).
% bit.compl_unique
thf(fact_5409_sgn__power__injE,axiom,
! [A2: real,N3: nat,X: real,B2: real] :
( ( ( times_times @ real @ ( sgn_sgn @ real @ A2 ) @ ( power_power @ real @ ( abs_abs @ real @ A2 ) @ N3 ) )
= X )
=> ( ( X
= ( times_times @ real @ ( sgn_sgn @ real @ B2 ) @ ( power_power @ real @ ( abs_abs @ real @ B2 ) @ N3 ) ) )
=> ( ( ord_less @ nat @ ( zero_zero @ nat ) @ N3 )
=> ( A2 = B2 ) ) ) ) ).
% sgn_power_injE
thf(fact_5410_int__bit__bound,axiom,
! [K: int] :
~ ! [N: nat] :
( ! [M2: nat] :
( ( ord_less_eq @ nat @ N @ M2 )
=> ( ( bit_se5641148757651400278ts_bit @ int @ K @ M2 )
= ( bit_se5641148757651400278ts_bit @ int @ K @ N ) ) )
=> ~ ( ( ord_less @ nat @ ( zero_zero @ nat ) @ N )
=> ( ( bit_se5641148757651400278ts_bit @ int @ K @ ( minus_minus @ nat @ N @ ( one_one @ nat ) ) )
= ( ~ ( bit_se5641148757651400278ts_bit @ int @ K @ N ) ) ) ) ) ).
% int_bit_bound
thf(fact_5411_and__not__numerals_I5_J,axiom,
! [M: num,N3: num] :
( ( bit_se5824344872417868541ns_and @ int @ ( numeral_numeral @ int @ ( bit0 @ M ) ) @ ( bit_ri4277139882892585799ns_not @ int @ ( numeral_numeral @ int @ ( bit0 @ N3 ) ) ) )
= ( times_times @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ ( bit_se5824344872417868541ns_and @ int @ ( numeral_numeral @ int @ M ) @ ( bit_ri4277139882892585799ns_not @ int @ ( numeral_numeral @ int @ N3 ) ) ) ) ) ).
% and_not_numerals(5)
thf(fact_5412_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_5413_and__not__numerals_I3_J,axiom,
! [N3: num] :
( ( bit_se5824344872417868541ns_and @ int @ ( one_one @ int ) @ ( bit_ri4277139882892585799ns_not @ int @ ( numeral_numeral @ int @ ( bit1 @ N3 ) ) ) )
= ( zero_zero @ int ) ) ).
% and_not_numerals(3)
thf(fact_5414_num_Osize__gen_I1_J,axiom,
( ( size_num @ one2 )
= ( zero_zero @ nat ) ) ).
% num.size_gen(1)
thf(fact_5415_bit__iff__odd,axiom,
! [A: $tType] :
( ( bit_semiring_bits @ A )
=> ( ( bit_se5641148757651400278ts_bit @ A )
= ( ^ [A6: A,N2: nat] :
~ ( dvd_dvd @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ ( divide_divide @ A @ A6 @ ( power_power @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ N2 ) ) ) ) ) ) ).
% bit_iff_odd
thf(fact_5416_and__exp__eq__0__iff__not__bit,axiom,
! [A: $tType] :
( ( bit_se359711467146920520ations @ A )
=> ! [A2: A,N3: nat] :
( ( ( bit_se5824344872417868541ns_and @ A @ A2 @ ( power_power @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ N3 ) )
= ( zero_zero @ A ) )
= ( ~ ( bit_se5641148757651400278ts_bit @ A @ A2 @ N3 ) ) ) ) ).
% and_exp_eq_0_iff_not_bit
thf(fact_5417_bit__not__iff__eq,axiom,
! [A: $tType] :
( ( bit_ri3973907225187159222ations @ A )
=> ! [A2: A,N3: nat] :
( ( bit_se5641148757651400278ts_bit @ A @ ( bit_ri4277139882892585799ns_not @ A @ A2 ) @ N3 )
= ( ( ( power_power @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ N3 )
!= ( zero_zero @ A ) )
& ~ ( bit_se5641148757651400278ts_bit @ A @ A2 @ N3 ) ) ) ) ).
% bit_not_iff_eq
thf(fact_5418_mask__Suc__exp,axiom,
! [A: $tType] :
( ( bit_se359711467146920520ations @ A )
=> ! [N3: nat] :
( ( bit_se2239418461657761734s_mask @ A @ ( suc @ N3 ) )
= ( bit_se1065995026697491101ons_or @ A @ ( power_power @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ N3 ) @ ( bit_se2239418461657761734s_mask @ A @ N3 ) ) ) ) ).
% mask_Suc_exp
thf(fact_5419_and__not__numerals_I9_J,axiom,
! [M: num,N3: num] :
( ( bit_se5824344872417868541ns_and @ int @ ( numeral_numeral @ int @ ( bit1 @ M ) ) @ ( bit_ri4277139882892585799ns_not @ int @ ( numeral_numeral @ int @ ( bit1 @ N3 ) ) ) )
= ( times_times @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ ( bit_se5824344872417868541ns_and @ int @ ( numeral_numeral @ int @ M ) @ ( bit_ri4277139882892585799ns_not @ int @ ( numeral_numeral @ int @ N3 ) ) ) ) ) ).
% and_not_numerals(9)
thf(fact_5420_and__not__numerals_I6_J,axiom,
! [M: num,N3: num] :
( ( bit_se5824344872417868541ns_and @ int @ ( numeral_numeral @ int @ ( bit0 @ M ) ) @ ( bit_ri4277139882892585799ns_not @ int @ ( numeral_numeral @ int @ ( bit1 @ N3 ) ) ) )
= ( times_times @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ ( bit_se5824344872417868541ns_and @ int @ ( numeral_numeral @ int @ M ) @ ( bit_ri4277139882892585799ns_not @ int @ ( numeral_numeral @ int @ N3 ) ) ) ) ) ).
% and_not_numerals(6)
thf(fact_5421_sgn__power__root,axiom,
! [N3: nat,X: real] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ N3 )
=> ( ( times_times @ real @ ( sgn_sgn @ real @ ( root @ N3 @ X ) ) @ ( power_power @ real @ ( abs_abs @ real @ ( root @ N3 @ X ) ) @ N3 ) )
= X ) ) ).
% sgn_power_root
thf(fact_5422_root__sgn__power,axiom,
! [N3: nat,Y: real] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ N3 )
=> ( ( root @ N3 @ ( times_times @ real @ ( sgn_sgn @ real @ Y ) @ ( power_power @ real @ ( abs_abs @ real @ Y ) @ N3 ) ) )
= Y ) ) ).
% root_sgn_power
thf(fact_5423_bit__int__def,axiom,
( ( bit_se5641148757651400278ts_bit @ int )
= ( ^ [K2: int,N2: nat] :
~ ( dvd_dvd @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ ( divide_divide @ int @ K2 @ ( power_power @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ N2 ) ) ) ) ) ).
% bit_int_def
thf(fact_5424_even__bit__succ__iff,axiom,
! [A: $tType] :
( ( bit_semiring_bits @ A )
=> ! [A2: A,N3: nat] :
( ( dvd_dvd @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ A2 )
=> ( ( bit_se5641148757651400278ts_bit @ A @ ( plus_plus @ A @ ( one_one @ A ) @ A2 ) @ N3 )
= ( ( bit_se5641148757651400278ts_bit @ A @ A2 @ N3 )
| ( N3
= ( zero_zero @ nat ) ) ) ) ) ) ).
% even_bit_succ_iff
thf(fact_5425_odd__bit__iff__bit__pred,axiom,
! [A: $tType] :
( ( bit_semiring_bits @ A )
=> ! [A2: A,N3: nat] :
( ~ ( dvd_dvd @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ A2 )
=> ( ( bit_se5641148757651400278ts_bit @ A @ A2 @ N3 )
= ( ( bit_se5641148757651400278ts_bit @ A @ ( minus_minus @ A @ A2 @ ( one_one @ A ) ) @ N3 )
| ( N3
= ( zero_zero @ nat ) ) ) ) ) ) ).
% odd_bit_iff_bit_pred
thf(fact_5426_cis__Arg__unique,axiom,
! [Z: complex,X: real] :
( ( ( sgn_sgn @ complex @ Z )
= ( cis @ X ) )
=> ( ( ord_less @ real @ ( uminus_uminus @ real @ pi ) @ X )
=> ( ( ord_less_eq @ real @ X @ pi )
=> ( ( arg @ Z )
= X ) ) ) ) ).
% cis_Arg_unique
thf(fact_5427_mask__Suc__double,axiom,
! [A: $tType] :
( ( bit_se359711467146920520ations @ A )
=> ! [N3: nat] :
( ( bit_se2239418461657761734s_mask @ A @ ( suc @ N3 ) )
= ( bit_se1065995026697491101ons_or @ A @ ( one_one @ A ) @ ( times_times @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ ( bit_se2239418461657761734s_mask @ A @ N3 ) ) ) ) ) ).
% mask_Suc_double
thf(fact_5428_one__or__eq,axiom,
! [A: $tType] :
( ( bit_se359711467146920520ations @ A )
=> ! [A2: A] :
( ( bit_se1065995026697491101ons_or @ A @ ( one_one @ A ) @ A2 )
= ( plus_plus @ A @ A2 @ ( zero_neq_one_of_bool @ A @ ( dvd_dvd @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ A2 ) ) ) ) ) ).
% one_or_eq
thf(fact_5429_or__one__eq,axiom,
! [A: $tType] :
( ( bit_se359711467146920520ations @ A )
=> ! [A2: A] :
( ( bit_se1065995026697491101ons_or @ A @ A2 @ ( one_one @ A ) )
= ( plus_plus @ A @ A2 @ ( zero_neq_one_of_bool @ A @ ( dvd_dvd @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ A2 ) ) ) ) ) ).
% or_one_eq
thf(fact_5430_split__root,axiom,
! [P: real > $o,N3: nat,X: real] :
( ( P @ ( root @ N3 @ X ) )
= ( ( ( N3
= ( zero_zero @ nat ) )
=> ( P @ ( zero_zero @ real ) ) )
& ( ( ord_less @ nat @ ( zero_zero @ nat ) @ N3 )
=> ! [Y2: real] :
( ( ( times_times @ real @ ( sgn_sgn @ real @ Y2 ) @ ( power_power @ real @ ( abs_abs @ real @ Y2 ) @ N3 ) )
= X )
=> ( P @ Y2 ) ) ) ) ) ).
% split_root
thf(fact_5431_bit__sum__mult__2__cases,axiom,
! [A: $tType] :
( ( bit_se359711467146920520ations @ A )
=> ! [A2: A,B2: A,N3: nat] :
( ! [J2: nat] :
~ ( bit_se5641148757651400278ts_bit @ A @ A2 @ ( suc @ J2 ) )
=> ( ( bit_se5641148757651400278ts_bit @ A @ ( plus_plus @ A @ A2 @ ( times_times @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ B2 ) ) @ N3 )
= ( ( ( N3
= ( zero_zero @ nat ) )
=> ~ ( dvd_dvd @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ A2 ) )
& ( ( N3
!= ( zero_zero @ nat ) )
=> ( bit_se5641148757651400278ts_bit @ A @ ( times_times @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ B2 ) @ N3 ) ) ) ) ) ) ).
% bit_sum_mult_2_cases
thf(fact_5432_bit__rec,axiom,
! [A: $tType] :
( ( bit_semiring_bits @ A )
=> ( ( bit_se5641148757651400278ts_bit @ A )
= ( ^ [A6: A,N2: nat] :
( ( ( N2
= ( zero_zero @ nat ) )
=> ~ ( dvd_dvd @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ A6 ) )
& ( ( N2
!= ( zero_zero @ nat ) )
=> ( bit_se5641148757651400278ts_bit @ A @ ( divide_divide @ A @ A6 @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) @ ( minus_minus @ nat @ N2 @ ( one_one @ nat ) ) ) ) ) ) ) ) ).
% bit_rec
thf(fact_5433_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_5434_and__not__numerals_I8_J,axiom,
! [M: num,N3: num] :
( ( bit_se5824344872417868541ns_and @ int @ ( numeral_numeral @ int @ ( bit1 @ M ) ) @ ( bit_ri4277139882892585799ns_not @ int @ ( numeral_numeral @ int @ ( bit0 @ N3 ) ) ) )
= ( 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 @ N3 ) ) ) ) ) ) ).
% and_not_numerals(8)
thf(fact_5435_not__int__rec,axiom,
( ( bit_ri4277139882892585799ns_not @ int )
= ( ^ [K2: int] : ( plus_plus @ int @ ( zero_neq_one_of_bool @ int @ ( dvd_dvd @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ K2 ) ) @ ( times_times @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ ( bit_ri4277139882892585799ns_not @ int @ ( divide_divide @ int @ K2 @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) ) ) ) ) ) ) ).
% not_int_rec
thf(fact_5436_Bit__Operations_Oset__bit__eq,axiom,
( ( bit_se5668285175392031749et_bit @ int )
= ( ^ [N2: nat,K2: int] :
( plus_plus @ int @ K2
@ ( times_times @ int
@ ( zero_neq_one_of_bool @ int
@ ~ ( bit_se5641148757651400278ts_bit @ int @ K2 @ N2 ) )
@ ( power_power @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ N2 ) ) ) ) ) ).
% Bit_Operations.set_bit_eq
thf(fact_5437_unset__bit__eq,axiom,
( ( bit_se2638667681897837118et_bit @ int )
= ( ^ [N2: nat,K2: int] : ( minus_minus @ int @ K2 @ ( times_times @ int @ ( zero_neq_one_of_bool @ int @ ( bit_se5641148757651400278ts_bit @ int @ K2 @ N2 ) ) @ ( power_power @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ N2 ) ) ) ) ) ).
% unset_bit_eq
thf(fact_5438_take__bit__Suc__from__most,axiom,
! [N3: nat,K: int] :
( ( bit_se2584673776208193580ke_bit @ int @ ( suc @ N3 ) @ K )
= ( plus_plus @ int @ ( times_times @ int @ ( power_power @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ N3 ) @ ( zero_neq_one_of_bool @ int @ ( bit_se5641148757651400278ts_bit @ int @ K @ N3 ) ) ) @ ( bit_se2584673776208193580ke_bit @ int @ N3 @ K ) ) ) ).
% take_bit_Suc_from_most
thf(fact_5439_arctan__inverse,axiom,
! [X: real] :
( ( X
!= ( zero_zero @ real ) )
=> ( ( arctan @ ( divide_divide @ real @ ( one_one @ real ) @ X ) )
= ( minus_minus @ real @ ( divide_divide @ real @ ( times_times @ real @ ( sgn_sgn @ real @ X ) @ pi ) @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) @ ( arctan @ X ) ) ) ) ).
% arctan_inverse
thf(fact_5440_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_5441_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_5442_bitNOT__integer__code,axiom,
( ( bit_ri4277139882892585799ns_not @ code_integer )
= ( ^ [I3: code_integer] : ( minus_minus @ code_integer @ ( uminus_uminus @ code_integer @ I3 ) @ ( one_one @ code_integer ) ) ) ) ).
% bitNOT_integer_code
thf(fact_5443_xor__int__unfold,axiom,
( ( bit_se5824344971392196577ns_xor @ int )
= ( ^ [K2: int,L: int] :
( if @ int
@ ( K2
= ( uminus_uminus @ int @ ( one_one @ int ) ) )
@ ( bit_ri4277139882892585799ns_not @ int @ L )
@ ( if @ int
@ ( L
= ( uminus_uminus @ int @ ( one_one @ int ) ) )
@ ( bit_ri4277139882892585799ns_not @ int @ K2 )
@ ( if @ int
@ ( K2
= ( zero_zero @ int ) )
@ L
@ ( if @ int
@ ( L
= ( zero_zero @ int ) )
@ K2
@ ( plus_plus @ int @ ( abs_abs @ int @ ( minus_minus @ int @ ( modulo_modulo @ int @ K2 @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) ) @ ( modulo_modulo @ int @ L @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) ) ) ) @ ( times_times @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ ( bit_se5824344971392196577ns_xor @ int @ ( divide_divide @ int @ K2 @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) ) @ ( divide_divide @ int @ L @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) ) ) ) ) ) ) ) ) ) ) ).
% xor_int_unfold
thf(fact_5444_bit_Oxor__left__self,axiom,
! [A: $tType] :
( ( bit_ri3973907225187159222ations @ A )
=> ! [X: A,Y: A] :
( ( bit_se5824344971392196577ns_xor @ A @ X @ ( bit_se5824344971392196577ns_xor @ A @ X @ Y ) )
= Y ) ) ).
% bit.xor_left_self
thf(fact_5445_bit_Oxor__self,axiom,
! [A: $tType] :
( ( bit_ri3973907225187159222ations @ A )
=> ! [X: A] :
( ( bit_se5824344971392196577ns_xor @ A @ X @ X )
= ( zero_zero @ A ) ) ) ).
% bit.xor_self
thf(fact_5446_xor__self__eq,axiom,
! [A: $tType] :
( ( bit_se359711467146920520ations @ A )
=> ! [A2: A] :
( ( bit_se5824344971392196577ns_xor @ A @ A2 @ A2 )
= ( zero_zero @ A ) ) ) ).
% xor_self_eq
thf(fact_5447_xor_Oleft__neutral,axiom,
! [A: $tType] :
( ( bit_se359711467146920520ations @ A )
=> ! [A2: A] :
( ( bit_se5824344971392196577ns_xor @ A @ ( zero_zero @ A ) @ A2 )
= A2 ) ) ).
% xor.left_neutral
thf(fact_5448_xor_Oright__neutral,axiom,
! [A: $tType] :
( ( bit_se359711467146920520ations @ A )
=> ! [A2: A] :
( ( bit_se5824344971392196577ns_xor @ A @ A2 @ ( zero_zero @ A ) )
= A2 ) ) ).
% xor.right_neutral
thf(fact_5449_test__bit__1,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [N3: nat] :
( ( bit_se5641148757651400278ts_bit @ ( word @ A ) @ ( one_one @ ( word @ A ) ) @ N3 )
= ( N3
= ( zero_zero @ nat ) ) ) ) ).
% test_bit_1
thf(fact_5450_take__bit__xor,axiom,
! [A: $tType] :
( ( bit_se359711467146920520ations @ A )
=> ! [N3: nat,A2: A,B2: A] :
( ( bit_se2584673776208193580ke_bit @ A @ N3 @ ( bit_se5824344971392196577ns_xor @ A @ A2 @ B2 ) )
= ( bit_se5824344971392196577ns_xor @ A @ ( bit_se2584673776208193580ke_bit @ A @ N3 @ A2 ) @ ( bit_se2584673776208193580ke_bit @ A @ N3 @ B2 ) ) ) ) ).
% take_bit_xor
thf(fact_5451_bit_Oxor__compl__right,axiom,
! [A: $tType] :
( ( bit_ri3973907225187159222ations @ A )
=> ! [X: A,Y: A] :
( ( bit_se5824344971392196577ns_xor @ A @ X @ ( bit_ri4277139882892585799ns_not @ A @ Y ) )
= ( bit_ri4277139882892585799ns_not @ A @ ( bit_se5824344971392196577ns_xor @ A @ X @ Y ) ) ) ) ).
% bit.xor_compl_right
thf(fact_5452_bit_Oxor__compl__left,axiom,
! [A: $tType] :
( ( bit_ri3973907225187159222ations @ A )
=> ! [X: A,Y: A] :
( ( bit_se5824344971392196577ns_xor @ A @ ( bit_ri4277139882892585799ns_not @ A @ X ) @ Y )
= ( bit_ri4277139882892585799ns_not @ A @ ( bit_se5824344971392196577ns_xor @ A @ X @ Y ) ) ) ) ).
% bit.xor_compl_left
thf(fact_5453_map__nth__0,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [Xs2: list @ nat] :
( ( map @ nat @ $o @ ( bit_se5641148757651400278ts_bit @ ( word @ A ) @ ( zero_zero @ ( word @ A ) ) ) @ Xs2 )
= ( replicate @ $o @ ( size_size @ ( list @ nat ) @ Xs2 ) @ $false ) ) ) ).
% map_nth_0
thf(fact_5454_or__nonnegative__int__iff,axiom,
! [K: int,L2: int] :
( ( ord_less_eq @ int @ ( zero_zero @ int ) @ ( bit_se1065995026697491101ons_or @ int @ K @ L2 ) )
= ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ K )
& ( ord_less_eq @ int @ ( zero_zero @ int ) @ L2 ) ) ) ).
% or_nonnegative_int_iff
thf(fact_5455_or__negative__int__iff,axiom,
! [K: int,L2: int] :
( ( ord_less @ int @ ( bit_se1065995026697491101ons_or @ int @ K @ L2 ) @ ( zero_zero @ int ) )
= ( ( ord_less @ int @ K @ ( zero_zero @ int ) )
| ( ord_less @ int @ L2 @ ( zero_zero @ int ) ) ) ) ).
% or_negative_int_iff
thf(fact_5456_xor__nonnegative__int__iff,axiom,
! [K: int,L2: int] :
( ( ord_less_eq @ int @ ( zero_zero @ int ) @ ( bit_se5824344971392196577ns_xor @ int @ K @ L2 ) )
= ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ K )
= ( ord_less_eq @ int @ ( zero_zero @ int ) @ L2 ) ) ) ).
% xor_nonnegative_int_iff
thf(fact_5457_xor__negative__int__iff,axiom,
! [K: int,L2: int] :
( ( ord_less @ int @ ( bit_se5824344971392196577ns_xor @ int @ K @ L2 ) @ ( zero_zero @ int ) )
= ( ( ord_less @ int @ K @ ( zero_zero @ int ) )
!= ( ord_less @ int @ L2 @ ( zero_zero @ int ) ) ) ) ).
% xor_negative_int_iff
thf(fact_5458_bit_Oxor__one__left,axiom,
! [A: $tType] :
( ( bit_ri3973907225187159222ations @ A )
=> ! [X: A] :
( ( bit_se5824344971392196577ns_xor @ A @ ( uminus_uminus @ A @ ( one_one @ A ) ) @ X )
= ( bit_ri4277139882892585799ns_not @ A @ X ) ) ) ).
% bit.xor_one_left
thf(fact_5459_bit_Oxor__one__right,axiom,
! [A: $tType] :
( ( bit_ri3973907225187159222ations @ A )
=> ! [X: A] :
( ( bit_se5824344971392196577ns_xor @ A @ X @ ( uminus_uminus @ A @ ( one_one @ A ) ) )
= ( bit_ri4277139882892585799ns_not @ A @ X ) ) ) ).
% bit.xor_one_right
thf(fact_5460_bit_Oxor__cancel__left,axiom,
! [A: $tType] :
( ( bit_ri3973907225187159222ations @ A )
=> ! [X: A] :
( ( bit_se5824344971392196577ns_xor @ A @ ( bit_ri4277139882892585799ns_not @ A @ X ) @ X )
= ( uminus_uminus @ A @ ( one_one @ A ) ) ) ) ).
% bit.xor_cancel_left
thf(fact_5461_bit_Oxor__cancel__right,axiom,
! [A: $tType] :
( ( bit_ri3973907225187159222ations @ A )
=> ! [X: A] :
( ( bit_se5824344971392196577ns_xor @ A @ X @ ( bit_ri4277139882892585799ns_not @ A @ X ) )
= ( uminus_uminus @ A @ ( one_one @ A ) ) ) ) ).
% bit.xor_cancel_right
thf(fact_5462_xor__numerals_I3_J,axiom,
! [A: $tType] :
( ( bit_un5681908812861735899ations @ A )
=> ! [X: num,Y: num] :
( ( bit_se5824344971392196577ns_xor @ A @ ( numeral_numeral @ A @ ( bit0 @ X ) ) @ ( numeral_numeral @ A @ ( bit0 @ Y ) ) )
= ( times_times @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ ( bit_se5824344971392196577ns_xor @ A @ ( numeral_numeral @ A @ X ) @ ( numeral_numeral @ A @ Y ) ) ) ) ) ).
% xor_numerals(3)
thf(fact_5463_xor__numerals_I1_J,axiom,
! [A: $tType] :
( ( bit_un5681908812861735899ations @ A )
=> ! [Y: num] :
( ( bit_se5824344971392196577ns_xor @ A @ ( one_one @ A ) @ ( numeral_numeral @ A @ ( bit0 @ Y ) ) )
= ( numeral_numeral @ A @ ( bit1 @ Y ) ) ) ) ).
% xor_numerals(1)
thf(fact_5464_xor__numerals_I2_J,axiom,
! [A: $tType] :
( ( bit_un5681908812861735899ations @ A )
=> ! [Y: num] :
( ( bit_se5824344971392196577ns_xor @ A @ ( one_one @ A ) @ ( numeral_numeral @ A @ ( bit1 @ Y ) ) )
= ( numeral_numeral @ A @ ( bit0 @ Y ) ) ) ) ).
% xor_numerals(2)
thf(fact_5465_xor__numerals_I5_J,axiom,
! [A: $tType] :
( ( bit_un5681908812861735899ations @ A )
=> ! [X: num] :
( ( bit_se5824344971392196577ns_xor @ A @ ( numeral_numeral @ A @ ( bit0 @ X ) ) @ ( one_one @ A ) )
= ( numeral_numeral @ A @ ( bit1 @ X ) ) ) ) ).
% xor_numerals(5)
thf(fact_5466_xor__numerals_I8_J,axiom,
! [A: $tType] :
( ( bit_un5681908812861735899ations @ A )
=> ! [X: num] :
( ( bit_se5824344971392196577ns_xor @ A @ ( numeral_numeral @ A @ ( bit1 @ X ) ) @ ( one_one @ A ) )
= ( numeral_numeral @ A @ ( bit0 @ X ) ) ) ) ).
% xor_numerals(8)
thf(fact_5467_or__minus__numerals_I2_J,axiom,
! [N3: num] :
( ( bit_se1065995026697491101ons_or @ int @ ( one_one @ int ) @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ ( bit1 @ N3 ) ) ) )
= ( uminus_uminus @ int @ ( numeral_numeral @ int @ ( bit1 @ N3 ) ) ) ) ).
% or_minus_numerals(2)
thf(fact_5468_or__minus__numerals_I6_J,axiom,
! [N3: num] :
( ( bit_se1065995026697491101ons_or @ int @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ ( bit1 @ N3 ) ) ) @ ( one_one @ int ) )
= ( uminus_uminus @ int @ ( numeral_numeral @ int @ ( bit1 @ N3 ) ) ) ) ).
% or_minus_numerals(6)
thf(fact_5469_or__nat__numerals_I2_J,axiom,
! [Y: num] :
( ( bit_se1065995026697491101ons_or @ nat @ ( suc @ ( zero_zero @ nat ) ) @ ( numeral_numeral @ nat @ ( bit1 @ Y ) ) )
= ( numeral_numeral @ nat @ ( bit1 @ Y ) ) ) ).
% or_nat_numerals(2)
thf(fact_5470_or__nat__numerals_I4_J,axiom,
! [X: num] :
( ( bit_se1065995026697491101ons_or @ nat @ ( numeral_numeral @ nat @ ( bit1 @ X ) ) @ ( suc @ ( zero_zero @ nat ) ) )
= ( numeral_numeral @ nat @ ( bit1 @ X ) ) ) ).
% or_nat_numerals(4)
thf(fact_5471_xor__numerals_I7_J,axiom,
! [A: $tType] :
( ( bit_un5681908812861735899ations @ A )
=> ! [X: num,Y: num] :
( ( bit_se5824344971392196577ns_xor @ A @ ( numeral_numeral @ A @ ( bit1 @ X ) ) @ ( numeral_numeral @ A @ ( bit1 @ Y ) ) )
= ( times_times @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ ( bit_se5824344971392196577ns_xor @ A @ ( numeral_numeral @ A @ X ) @ ( numeral_numeral @ A @ Y ) ) ) ) ) ).
% xor_numerals(7)
thf(fact_5472_or__nat__numerals_I1_J,axiom,
! [Y: num] :
( ( bit_se1065995026697491101ons_or @ nat @ ( suc @ ( zero_zero @ nat ) ) @ ( numeral_numeral @ nat @ ( bit0 @ Y ) ) )
= ( numeral_numeral @ nat @ ( bit1 @ Y ) ) ) ).
% or_nat_numerals(1)
thf(fact_5473_or__nat__numerals_I3_J,axiom,
! [X: num] :
( ( bit_se1065995026697491101ons_or @ nat @ ( numeral_numeral @ nat @ ( bit0 @ X ) ) @ ( suc @ ( zero_zero @ nat ) ) )
= ( numeral_numeral @ nat @ ( bit1 @ X ) ) ) ).
% or_nat_numerals(3)
thf(fact_5474_word__bitwise__1__simps_I8_J,axiom,
! [H5: $tType] :
( ( type_len @ H5 )
=> ! [A2: num] :
( ( bit_se1065995026697491101ons_or @ ( word @ H5 ) @ ( numeral_numeral @ ( word @ H5 ) @ A2 ) @ ( one_one @ ( word @ H5 ) ) )
= ( ring_1_of_int @ ( word @ H5 ) @ ( bit_se1065995026697491101ons_or @ int @ ( numeral_numeral @ int @ A2 ) @ ( one_one @ int ) ) ) ) ) ).
% word_bitwise_1_simps(8)
thf(fact_5475_word__bitwise__1__simps_I6_J,axiom,
! [F: $tType] :
( ( type_len @ F )
=> ! [B2: num] :
( ( bit_se1065995026697491101ons_or @ ( word @ F ) @ ( one_one @ ( word @ F ) ) @ ( numeral_numeral @ ( word @ F ) @ B2 ) )
= ( ring_1_of_int @ ( word @ F ) @ ( bit_se1065995026697491101ons_or @ int @ ( one_one @ int ) @ ( numeral_numeral @ int @ B2 ) ) ) ) ) ).
% word_bitwise_1_simps(6)
thf(fact_5476_and__minus__minus__numerals,axiom,
! [M: num,N3: num] :
( ( bit_se5824344872417868541ns_and @ int @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ M ) ) @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ N3 ) ) )
= ( bit_ri4277139882892585799ns_not @ int @ ( bit_se1065995026697491101ons_or @ int @ ( minus_minus @ int @ ( numeral_numeral @ int @ M ) @ ( one_one @ int ) ) @ ( minus_minus @ int @ ( numeral_numeral @ int @ N3 ) @ ( one_one @ int ) ) ) ) ) ).
% and_minus_minus_numerals
thf(fact_5477_or__minus__minus__numerals,axiom,
! [M: num,N3: num] :
( ( bit_se1065995026697491101ons_or @ int @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ M ) ) @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ N3 ) ) )
= ( bit_ri4277139882892585799ns_not @ int @ ( bit_se5824344872417868541ns_and @ int @ ( minus_minus @ int @ ( numeral_numeral @ int @ M ) @ ( one_one @ int ) ) @ ( minus_minus @ int @ ( numeral_numeral @ int @ N3 ) @ ( one_one @ int ) ) ) ) ) ).
% or_minus_minus_numerals
thf(fact_5478_xor__numerals_I4_J,axiom,
! [A: $tType] :
( ( bit_un5681908812861735899ations @ A )
=> ! [X: num,Y: num] :
( ( bit_se5824344971392196577ns_xor @ A @ ( numeral_numeral @ A @ ( bit0 @ X ) ) @ ( numeral_numeral @ A @ ( bit1 @ Y ) ) )
= ( plus_plus @ A @ ( one_one @ A ) @ ( times_times @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ ( bit_se5824344971392196577ns_xor @ A @ ( numeral_numeral @ A @ X ) @ ( numeral_numeral @ A @ Y ) ) ) ) ) ) ).
% xor_numerals(4)
thf(fact_5479_xor__numerals_I6_J,axiom,
! [A: $tType] :
( ( bit_un5681908812861735899ations @ A )
=> ! [X: num,Y: num] :
( ( bit_se5824344971392196577ns_xor @ A @ ( numeral_numeral @ A @ ( bit1 @ X ) ) @ ( numeral_numeral @ A @ ( bit0 @ Y ) ) )
= ( plus_plus @ A @ ( one_one @ A ) @ ( times_times @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ ( bit_se5824344971392196577ns_xor @ A @ ( numeral_numeral @ A @ X ) @ ( numeral_numeral @ A @ Y ) ) ) ) ) ) ).
% xor_numerals(6)
thf(fact_5480_word__bitwise__1__simps_I9_J,axiom,
! [I6: $tType] :
( ( type_len @ I6 )
=> ! [A2: num] :
( ( bit_se1065995026697491101ons_or @ ( word @ I6 ) @ ( uminus_uminus @ ( word @ I6 ) @ ( numeral_numeral @ ( word @ I6 ) @ A2 ) ) @ ( one_one @ ( word @ I6 ) ) )
= ( ring_1_of_int @ ( word @ I6 ) @ ( bit_se1065995026697491101ons_or @ int @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ A2 ) ) @ ( one_one @ int ) ) ) ) ) ).
% word_bitwise_1_simps(9)
thf(fact_5481_word__bitwise__1__simps_I7_J,axiom,
! [G2: $tType] :
( ( type_len @ G2 )
=> ! [B2: num] :
( ( bit_se1065995026697491101ons_or @ ( word @ G2 ) @ ( one_one @ ( word @ G2 ) ) @ ( uminus_uminus @ ( word @ G2 ) @ ( numeral_numeral @ ( word @ G2 ) @ B2 ) ) )
= ( ring_1_of_int @ ( word @ G2 ) @ ( bit_se1065995026697491101ons_or @ int @ ( one_one @ int ) @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ B2 ) ) ) ) ) ) ).
% word_bitwise_1_simps(7)
thf(fact_5482_xor__int__def,axiom,
( ( bit_se5824344971392196577ns_xor @ int )
= ( ^ [K2: int,L: int] : ( bit_se1065995026697491101ons_or @ int @ ( bit_se5824344872417868541ns_and @ int @ K2 @ ( bit_ri4277139882892585799ns_not @ int @ L ) ) @ ( bit_se5824344872417868541ns_and @ int @ ( bit_ri4277139882892585799ns_not @ int @ K2 ) @ L ) ) ) ) ).
% xor_int_def
thf(fact_5483_not__bit__Suc__0__Suc,axiom,
! [N3: nat] :
~ ( bit_se5641148757651400278ts_bit @ nat @ ( suc @ ( zero_zero @ nat ) ) @ ( suc @ N3 ) ) ).
% not_bit_Suc_0_Suc
thf(fact_5484_bit__Suc__0__iff,axiom,
! [N3: nat] :
( ( bit_se5641148757651400278ts_bit @ nat @ ( suc @ ( zero_zero @ nat ) ) @ N3 )
= ( N3
= ( zero_zero @ nat ) ) ) ).
% bit_Suc_0_iff
thf(fact_5485_or__nat__def,axiom,
( ( bit_se1065995026697491101ons_or @ nat )
= ( ^ [M5: nat,N2: nat] : ( nat2 @ ( bit_se1065995026697491101ons_or @ int @ ( semiring_1_of_nat @ int @ M5 ) @ ( semiring_1_of_nat @ int @ N2 ) ) ) ) ) ).
% or_nat_def
thf(fact_5486_bit__xor__iff,axiom,
! [A: $tType] :
( ( bit_se359711467146920520ations @ A )
=> ! [A2: A,B2: A,N3: nat] :
( ( bit_se5641148757651400278ts_bit @ A @ ( bit_se5824344971392196577ns_xor @ A @ A2 @ B2 ) @ N3 )
= ( ( bit_se5641148757651400278ts_bit @ A @ A2 @ N3 )
!= ( bit_se5641148757651400278ts_bit @ A @ B2 @ N3 ) ) ) ) ).
% bit_xor_iff
thf(fact_5487_bit__or__int__iff,axiom,
! [K: int,L2: int,N3: nat] :
( ( bit_se5641148757651400278ts_bit @ int @ ( bit_se1065995026697491101ons_or @ int @ K @ L2 ) @ N3 )
= ( ( bit_se5641148757651400278ts_bit @ int @ K @ N3 )
| ( bit_se5641148757651400278ts_bit @ int @ L2 @ N3 ) ) ) ).
% bit_or_int_iff
thf(fact_5488_bit__xor__int__iff,axiom,
! [K: int,L2: int,N3: nat] :
( ( bit_se5641148757651400278ts_bit @ int @ ( bit_se5824344971392196577ns_xor @ int @ K @ L2 ) @ N3 )
= ( ( bit_se5641148757651400278ts_bit @ int @ K @ N3 )
!= ( bit_se5641148757651400278ts_bit @ int @ L2 @ N3 ) ) ) ).
% bit_xor_int_iff
thf(fact_5489_XOR__lower,axiom,
! [X: int,Y: int] :
( ( ord_less_eq @ int @ ( zero_zero @ int ) @ X )
=> ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ Y )
=> ( ord_less_eq @ int @ ( zero_zero @ int ) @ ( bit_se5824344971392196577ns_xor @ int @ X @ Y ) ) ) ) ).
% XOR_lower
thf(fact_5490_xor_Oleft__commute,axiom,
! [A: $tType] :
( ( bit_se359711467146920520ations @ A )
=> ! [B2: A,A2: A,C2: A] :
( ( bit_se5824344971392196577ns_xor @ A @ B2 @ ( bit_se5824344971392196577ns_xor @ A @ A2 @ C2 ) )
= ( bit_se5824344971392196577ns_xor @ A @ A2 @ ( bit_se5824344971392196577ns_xor @ A @ B2 @ C2 ) ) ) ) ).
% xor.left_commute
thf(fact_5491_xor_Ocommute,axiom,
! [A: $tType] :
( ( bit_se359711467146920520ations @ A )
=> ( ( bit_se5824344971392196577ns_xor @ A )
= ( ^ [A6: A,B5: A] : ( bit_se5824344971392196577ns_xor @ A @ B5 @ A6 ) ) ) ) ).
% xor.commute
thf(fact_5492_xor_Oassoc,axiom,
! [A: $tType] :
( ( bit_se359711467146920520ations @ A )
=> ! [A2: A,B2: A,C2: A] :
( ( bit_se5824344971392196577ns_xor @ A @ ( bit_se5824344971392196577ns_xor @ A @ A2 @ B2 ) @ C2 )
= ( bit_se5824344971392196577ns_xor @ A @ A2 @ ( bit_se5824344971392196577ns_xor @ A @ B2 @ C2 ) ) ) ) ).
% xor.assoc
thf(fact_5493_of__int__xor__eq,axiom,
! [A: $tType] :
( ( bit_ri3973907225187159222ations @ A )
=> ! [K: int,L2: int] :
( ( ring_1_of_int @ A @ ( bit_se5824344971392196577ns_xor @ int @ K @ L2 ) )
= ( bit_se5824344971392196577ns_xor @ A @ ( ring_1_of_int @ A @ K ) @ ( ring_1_of_int @ A @ L2 ) ) ) ) ).
% of_int_xor_eq
thf(fact_5494_of__nat__xor__eq,axiom,
! [A: $tType] :
( ( bit_se359711467146920520ations @ A )
=> ! [M: nat,N3: nat] :
( ( semiring_1_of_nat @ A @ ( bit_se5824344971392196577ns_xor @ nat @ M @ N3 ) )
= ( bit_se5824344971392196577ns_xor @ A @ ( semiring_1_of_nat @ A @ M ) @ ( semiring_1_of_nat @ A @ N3 ) ) ) ) ).
% of_nat_xor_eq
thf(fact_5495_bit_Oconj__xor__distrib,axiom,
! [A: $tType] :
( ( bit_ri3973907225187159222ations @ A )
=> ! [X: A,Y: A,Z: A] :
( ( bit_se5824344872417868541ns_and @ A @ X @ ( bit_se5824344971392196577ns_xor @ A @ Y @ Z ) )
= ( bit_se5824344971392196577ns_xor @ A @ ( bit_se5824344872417868541ns_and @ A @ X @ Y ) @ ( bit_se5824344872417868541ns_and @ A @ X @ Z ) ) ) ) ).
% bit.conj_xor_distrib
thf(fact_5496_bit_Oconj__xor__distrib2,axiom,
! [A: $tType] :
( ( bit_ri3973907225187159222ations @ A )
=> ! [Y: A,Z: A,X: A] :
( ( bit_se5824344872417868541ns_and @ A @ ( bit_se5824344971392196577ns_xor @ A @ Y @ Z ) @ X )
= ( bit_se5824344971392196577ns_xor @ A @ ( bit_se5824344872417868541ns_and @ A @ Y @ X ) @ ( bit_se5824344872417868541ns_and @ A @ Z @ X ) ) ) ) ).
% bit.conj_xor_distrib2
thf(fact_5497_or__greater__eq,axiom,
! [L2: int,K: int] :
( ( ord_less_eq @ int @ ( zero_zero @ int ) @ L2 )
=> ( ord_less_eq @ int @ K @ ( bit_se1065995026697491101ons_or @ int @ K @ L2 ) ) ) ).
% or_greater_eq
thf(fact_5498_OR__lower,axiom,
! [X: int,Y: int] :
( ( ord_less_eq @ int @ ( zero_zero @ int ) @ X )
=> ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ Y )
=> ( ord_less_eq @ int @ ( zero_zero @ int ) @ ( bit_se1065995026697491101ons_or @ int @ X @ Y ) ) ) ) ).
% OR_lower
thf(fact_5499_plus__and__or,axiom,
! [X: int,Y: int] :
( ( plus_plus @ int @ ( bit_se5824344872417868541ns_and @ int @ X @ Y ) @ ( bit_se1065995026697491101ons_or @ int @ X @ Y ) )
= ( plus_plus @ int @ X @ Y ) ) ).
% plus_and_or
thf(fact_5500_test__bit__size,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [W: word @ A,N3: nat] :
( ( bit_se5641148757651400278ts_bit @ ( word @ A ) @ W @ N3 )
=> ( ord_less @ nat @ N3 @ ( size_size @ ( word @ A ) @ W ) ) ) ) ).
% test_bit_size
thf(fact_5501_word__eqI,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [U: word @ A,V: word @ A] :
( ! [N: nat] :
( ( ord_less @ nat @ N @ ( size_size @ ( word @ A ) @ U ) )
=> ( ( bit_se5641148757651400278ts_bit @ ( word @ A ) @ U @ N )
= ( bit_se5641148757651400278ts_bit @ ( word @ A ) @ V @ N ) ) )
=> ( U = V ) ) ) ).
% word_eqI
thf(fact_5502_test__bit__over,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [X: word @ A,N3: nat] :
( ( ord_less_eq @ nat @ ( size_size @ ( word @ A ) @ X ) @ N3 )
=> ~ ( bit_se5641148757651400278ts_bit @ ( word @ A ) @ X @ N3 ) ) ) ).
% test_bit_over
thf(fact_5503_or__int__def,axiom,
( ( bit_se1065995026697491101ons_or @ int )
= ( ^ [K2: int,L: int] : ( bit_ri4277139882892585799ns_not @ int @ ( bit_se5824344872417868541ns_and @ int @ ( bit_ri4277139882892585799ns_not @ int @ K2 ) @ ( bit_ri4277139882892585799ns_not @ int @ L ) ) ) ) ) ).
% or_int_def
thf(fact_5504_not__bit__Suc__0__numeral,axiom,
! [N3: num] :
~ ( bit_se5641148757651400278ts_bit @ nat @ ( suc @ ( zero_zero @ nat ) ) @ ( numeral_numeral @ nat @ N3 ) ) ).
% not_bit_Suc_0_numeral
thf(fact_5505_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_5506_lsb__this__or__next,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [X: word @ A] :
( ~ ( bit_se5641148757651400278ts_bit @ ( word @ A ) @ ( plus_plus @ ( word @ A ) @ X @ ( one_one @ ( word @ A ) ) ) @ ( zero_zero @ nat ) )
=> ( bit_se5641148757651400278ts_bit @ ( word @ A ) @ X @ ( zero_zero @ nat ) ) ) ) ).
% lsb_this_or_next
thf(fact_5507_word__leI,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [U: word @ A,V: word @ A] :
( ! [N: nat] :
( ( ord_less @ nat @ N @ ( size_size @ ( word @ A ) @ U ) )
=> ( ( bit_se5641148757651400278ts_bit @ ( word @ A ) @ U @ N )
=> ( bit_se5641148757651400278ts_bit @ ( word @ A ) @ V @ N ) ) )
=> ( ord_less_eq @ ( word @ A ) @ U @ V ) ) ) ).
% word_leI
thf(fact_5508_nth__mask,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [N3: nat,I: nat] :
( ( bit_se5641148757651400278ts_bit @ ( word @ A ) @ ( bit_se2239418461657761734s_mask @ ( word @ A ) @ N3 ) @ I )
= ( ( ord_less @ nat @ I @ N3 )
& ( ord_less @ nat @ I @ ( size_size @ ( word @ A ) @ ( bit_se2239418461657761734s_mask @ ( word @ A ) @ N3 ) ) ) ) ) ) ).
% nth_mask
thf(fact_5509_bit_Oxor__def2,axiom,
! [A: $tType] :
( ( bit_ri3973907225187159222ations @ A )
=> ( ( bit_se5824344971392196577ns_xor @ A )
= ( ^ [X2: A,Y2: A] : ( bit_se5824344872417868541ns_and @ A @ ( bit_se1065995026697491101ons_or @ A @ X2 @ Y2 ) @ ( bit_se1065995026697491101ons_or @ A @ ( bit_ri4277139882892585799ns_not @ A @ X2 ) @ ( bit_ri4277139882892585799ns_not @ A @ Y2 ) ) ) ) ) ) ).
% bit.xor_def2
thf(fact_5510_bit_Oxor__def,axiom,
! [A: $tType] :
( ( bit_ri3973907225187159222ations @ A )
=> ( ( bit_se5824344971392196577ns_xor @ A )
= ( ^ [X2: A,Y2: A] : ( bit_se1065995026697491101ons_or @ A @ ( bit_se5824344872417868541ns_and @ A @ X2 @ ( bit_ri4277139882892585799ns_not @ A @ Y2 ) ) @ ( bit_se5824344872417868541ns_and @ A @ ( bit_ri4277139882892585799ns_not @ A @ X2 ) @ Y2 ) ) ) ) ) ).
% bit.xor_def
thf(fact_5511_even__xor__iff,axiom,
! [A: $tType] :
( ( bit_se359711467146920520ations @ A )
=> ! [A2: A,B2: A] :
( ( dvd_dvd @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ ( bit_se5824344971392196577ns_xor @ A @ A2 @ B2 ) )
= ( ( dvd_dvd @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ A2 )
= ( dvd_dvd @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ B2 ) ) ) ) ).
% even_xor_iff
thf(fact_5512_or__not__numerals_I2_J,axiom,
! [N3: num] :
( ( bit_se1065995026697491101ons_or @ int @ ( one_one @ int ) @ ( bit_ri4277139882892585799ns_not @ int @ ( numeral_numeral @ int @ ( bit0 @ N3 ) ) ) )
= ( bit_ri4277139882892585799ns_not @ int @ ( numeral_numeral @ int @ ( bit0 @ N3 ) ) ) ) ).
% or_not_numerals(2)
thf(fact_5513_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_5514_bit__nat__iff,axiom,
! [K: int,N3: nat] :
( ( bit_se5641148757651400278ts_bit @ nat @ ( nat2 @ K ) @ N3 )
= ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ K )
& ( bit_se5641148757651400278ts_bit @ int @ K @ N3 ) ) ) ).
% bit_nat_iff
thf(fact_5515_overflow__imp__lsb,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [X: word @ A] :
( ( ( plus_plus @ ( word @ A ) @ X @ ( one_one @ ( word @ A ) ) )
= ( zero_zero @ ( word @ A ) ) )
=> ( bit_se5641148757651400278ts_bit @ ( word @ A ) @ X @ ( zero_zero @ nat ) ) ) ) ).
% overflow_imp_lsb
thf(fact_5516_word__and__1,axiom,
! [B: $tType] :
( ( type_len @ B )
=> ! [N3: word @ B] :
( ( ( bit_se5641148757651400278ts_bit @ ( word @ B ) @ N3 @ ( zero_zero @ nat ) )
=> ( ( bit_se5824344872417868541ns_and @ ( word @ B ) @ N3 @ ( one_one @ ( word @ B ) ) )
= ( one_one @ ( word @ B ) ) ) )
& ( ~ ( bit_se5641148757651400278ts_bit @ ( word @ B ) @ N3 @ ( zero_zero @ nat ) )
=> ( ( bit_se5824344872417868541ns_and @ ( word @ B ) @ N3 @ ( one_one @ ( word @ B ) ) )
= ( zero_zero @ ( word @ B ) ) ) ) ) ) ).
% word_and_1
thf(fact_5517_test__bit__bin_H,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ( ( bit_se5641148757651400278ts_bit @ ( word @ A ) )
= ( ^ [W2: word @ A,N2: nat] :
( ( ord_less @ nat @ N2 @ ( size_size @ ( word @ A ) @ W2 ) )
& ( bit_se5641148757651400278ts_bit @ int @ ( semiring_1_unsigned @ A @ int @ W2 ) @ N2 ) ) ) ) ) ).
% test_bit_bin'
thf(fact_5518_or__not__numerals_I3_J,axiom,
! [N3: num] :
( ( bit_se1065995026697491101ons_or @ int @ ( one_one @ int ) @ ( bit_ri4277139882892585799ns_not @ int @ ( numeral_numeral @ int @ ( bit1 @ N3 ) ) ) )
= ( bit_ri4277139882892585799ns_not @ int @ ( numeral_numeral @ int @ ( bit0 @ N3 ) ) ) ) ).
% or_not_numerals(3)
thf(fact_5519_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_5520_bang__is__le,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [X: word @ A,M: nat] :
( ( bit_se5641148757651400278ts_bit @ ( word @ A ) @ X @ M )
=> ( ord_less_eq @ ( word @ A ) @ ( power_power @ ( word @ A ) @ ( numeral_numeral @ ( word @ A ) @ ( bit0 @ one2 ) ) @ M ) @ X ) ) ) ).
% bang_is_le
thf(fact_5521_bit__nat__def,axiom,
( ( bit_se5641148757651400278ts_bit @ nat )
= ( ^ [M5: nat,N2: nat] :
~ ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ M5 @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N2 ) ) ) ) ) ).
% bit_nat_def
thf(fact_5522_or__not__numerals_I6_J,axiom,
! [M: num,N3: num] :
( ( bit_se1065995026697491101ons_or @ int @ ( numeral_numeral @ int @ ( bit0 @ M ) ) @ ( bit_ri4277139882892585799ns_not @ int @ ( numeral_numeral @ int @ ( bit1 @ N3 ) ) ) )
= ( times_times @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ ( bit_se1065995026697491101ons_or @ int @ ( numeral_numeral @ int @ M ) @ ( bit_ri4277139882892585799ns_not @ int @ ( numeral_numeral @ int @ N3 ) ) ) ) ) ).
% or_not_numerals(6)
thf(fact_5523_odd__iff__lsb,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [X: word @ A] :
( ( ~ ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( semiring_1_unsigned @ A @ nat @ X ) ) )
= ( bit_se5641148757651400278ts_bit @ ( word @ A ) @ X @ ( zero_zero @ nat ) ) ) ) ).
% odd_iff_lsb
thf(fact_5524_nth__is__and__neq__0,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ( ( bit_se5641148757651400278ts_bit @ ( word @ A ) )
= ( ^ [X2: word @ A,N2: nat] :
( ( bit_se5824344872417868541ns_and @ ( word @ A ) @ X2 @ ( power_power @ ( word @ A ) @ ( numeral_numeral @ ( word @ A ) @ ( bit0 @ one2 ) ) @ N2 ) )
!= ( zero_zero @ ( word @ A ) ) ) ) ) ) ).
% nth_is_and_neq_0
thf(fact_5525_and__neq__0__is__nth,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [Y: word @ A,N3: nat,X: word @ A] :
( ( Y
= ( power_power @ ( word @ A ) @ ( numeral_numeral @ ( word @ A ) @ ( bit0 @ one2 ) ) @ N3 ) )
=> ( ( ( bit_se5824344872417868541ns_and @ ( word @ A ) @ X @ Y )
!= ( zero_zero @ ( word @ A ) ) )
= ( bit_se5641148757651400278ts_bit @ ( word @ A ) @ X @ N3 ) ) ) ) ).
% and_neq_0_is_nth
thf(fact_5526_XOR__upper,axiom,
! [X: int,N3: nat,Y: int] :
( ( ord_less_eq @ int @ ( zero_zero @ int ) @ X )
=> ( ( ord_less @ int @ X @ ( power_power @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ N3 ) )
=> ( ( ord_less @ int @ Y @ ( power_power @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ N3 ) )
=> ( ord_less @ int @ ( bit_se5824344971392196577ns_xor @ int @ X @ Y ) @ ( power_power @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ N3 ) ) ) ) ) ).
% XOR_upper
thf(fact_5527_OR__upper,axiom,
! [X: int,N3: nat,Y: int] :
( ( ord_less_eq @ int @ ( zero_zero @ int ) @ X )
=> ( ( ord_less @ int @ X @ ( power_power @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ N3 ) )
=> ( ( ord_less @ int @ Y @ ( power_power @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ N3 ) )
=> ( ord_less @ int @ ( bit_se1065995026697491101ons_or @ int @ X @ Y ) @ ( power_power @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ N3 ) ) ) ) ) ).
% OR_upper
thf(fact_5528_or__not__numerals_I5_J,axiom,
! [M: num,N3: num] :
( ( bit_se1065995026697491101ons_or @ int @ ( numeral_numeral @ int @ ( bit0 @ M ) ) @ ( bit_ri4277139882892585799ns_not @ int @ ( numeral_numeral @ int @ ( bit0 @ N3 ) ) ) )
= ( 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 @ N3 ) ) ) ) ) ) ).
% or_not_numerals(5)
thf(fact_5529_or__Suc__0__eq,axiom,
! [N3: nat] :
( ( bit_se1065995026697491101ons_or @ nat @ N3 @ ( suc @ ( zero_zero @ nat ) ) )
= ( plus_plus @ nat @ N3 @ ( zero_neq_one_of_bool @ nat @ ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N3 ) ) ) ) ).
% or_Suc_0_eq
thf(fact_5530_Suc__0__or__eq,axiom,
! [N3: nat] :
( ( bit_se1065995026697491101ons_or @ nat @ ( suc @ ( zero_zero @ nat ) ) @ N3 )
= ( plus_plus @ nat @ N3 @ ( zero_neq_one_of_bool @ nat @ ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N3 ) ) ) ) ).
% Suc_0_or_eq
thf(fact_5531_or__nat__rec,axiom,
( ( bit_se1065995026697491101ons_or @ nat )
= ( ^ [M5: nat,N2: nat] :
( plus_plus @ nat
@ ( zero_neq_one_of_bool @ nat
@ ( ~ ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ M5 )
| ~ ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N2 ) ) )
@ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( bit_se1065995026697491101ons_or @ nat @ ( divide_divide @ nat @ M5 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ ( divide_divide @ nat @ N2 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) ) ) ).
% or_nat_rec
thf(fact_5532_or__not__numerals_I8_J,axiom,
! [M: num,N3: num] :
( ( bit_se1065995026697491101ons_or @ int @ ( numeral_numeral @ int @ ( bit1 @ M ) ) @ ( bit_ri4277139882892585799ns_not @ int @ ( numeral_numeral @ int @ ( bit0 @ N3 ) ) ) )
= ( 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 @ N3 ) ) ) ) ) ) ).
% or_not_numerals(8)
thf(fact_5533_or__not__numerals_I9_J,axiom,
! [M: num,N3: num] :
( ( bit_se1065995026697491101ons_or @ int @ ( numeral_numeral @ int @ ( bit1 @ M ) ) @ ( bit_ri4277139882892585799ns_not @ int @ ( numeral_numeral @ int @ ( bit1 @ N3 ) ) ) )
= ( 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 @ N3 ) ) ) ) ) ) ).
% or_not_numerals(9)
thf(fact_5534_xor__int__rec,axiom,
( ( bit_se5824344971392196577ns_xor @ int )
= ( ^ [K2: int,L: int] :
( plus_plus @ int
@ ( zero_neq_one_of_bool @ int
@ ( ( ~ ( dvd_dvd @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ K2 ) )
!= ( ~ ( dvd_dvd @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ L ) ) ) )
@ ( times_times @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ ( bit_se5824344971392196577ns_xor @ int @ ( divide_divide @ int @ K2 @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) ) @ ( divide_divide @ int @ L @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) ) ) ) ) ) ) ).
% xor_int_rec
thf(fact_5535_xor__one__eq,axiom,
! [A: $tType] :
( ( bit_se359711467146920520ations @ A )
=> ! [A2: A] :
( ( bit_se5824344971392196577ns_xor @ A @ A2 @ ( one_one @ A ) )
= ( minus_minus @ A @ ( plus_plus @ A @ A2 @ ( zero_neq_one_of_bool @ A @ ( dvd_dvd @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ A2 ) ) )
@ ( zero_neq_one_of_bool @ A
@ ~ ( dvd_dvd @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ A2 ) ) ) ) ) ).
% xor_one_eq
thf(fact_5536_one__xor__eq,axiom,
! [A: $tType] :
( ( bit_se359711467146920520ations @ A )
=> ! [A2: A] :
( ( bit_se5824344971392196577ns_xor @ A @ ( one_one @ A ) @ A2 )
= ( minus_minus @ A @ ( plus_plus @ A @ A2 @ ( zero_neq_one_of_bool @ A @ ( dvd_dvd @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ A2 ) ) )
@ ( zero_neq_one_of_bool @ A
@ ~ ( dvd_dvd @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ A2 ) ) ) ) ) ).
% one_xor_eq
thf(fact_5537_or__int__rec,axiom,
( ( bit_se1065995026697491101ons_or @ int )
= ( ^ [K2: int,L: int] :
( plus_plus @ int
@ ( zero_neq_one_of_bool @ int
@ ( ~ ( dvd_dvd @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ K2 )
| ~ ( dvd_dvd @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ L ) ) )
@ ( times_times @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ ( bit_se1065995026697491101ons_or @ int @ ( divide_divide @ int @ K2 @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) ) @ ( divide_divide @ int @ L @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) ) ) ) ) ) ) ).
% or_int_rec
thf(fact_5538_or__nat__unfold,axiom,
( ( bit_se1065995026697491101ons_or @ nat )
= ( ^ [M5: nat,N2: nat] :
( if @ nat
@ ( M5
= ( zero_zero @ nat ) )
@ N2
@ ( if @ nat
@ ( N2
= ( zero_zero @ nat ) )
@ M5
@ ( plus_plus @ nat @ ( ord_max @ nat @ ( modulo_modulo @ nat @ M5 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ ( modulo_modulo @ nat @ N2 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( bit_se1065995026697491101ons_or @ nat @ ( divide_divide @ nat @ M5 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ ( divide_divide @ nat @ N2 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) ) ) ) ) ).
% or_nat_unfold
thf(fact_5539_or__int__unfold,axiom,
( ( bit_se1065995026697491101ons_or @ int )
= ( ^ [K2: int,L: int] :
( if @ int
@ ( ( K2
= ( uminus_uminus @ int @ ( one_one @ int ) ) )
| ( L
= ( uminus_uminus @ int @ ( one_one @ int ) ) ) )
@ ( uminus_uminus @ int @ ( one_one @ int ) )
@ ( if @ int
@ ( K2
= ( zero_zero @ int ) )
@ L
@ ( if @ int
@ ( L
= ( zero_zero @ int ) )
@ K2
@ ( plus_plus @ int @ ( ord_max @ int @ ( modulo_modulo @ int @ K2 @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) ) @ ( modulo_modulo @ int @ L @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) ) ) @ ( times_times @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ ( bit_se1065995026697491101ons_or @ int @ ( divide_divide @ int @ K2 @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) ) @ ( divide_divide @ int @ L @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) ) ) ) ) ) ) ) ) ) ).
% or_int_unfold
thf(fact_5540_Bit__integer_Oabs__eq,axiom,
! [Xa: int,X: $o] :
( ( bits_Bit_integer @ ( code_integer_of_int @ Xa ) @ X )
= ( code_integer_of_int @ ( plus_plus @ int @ ( zero_neq_one_of_bool @ int @ X ) @ ( times_times @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ Xa ) ) ) ) ).
% Bit_integer.abs_eq
thf(fact_5541_or__minus__numerals_I5_J,axiom,
! [N3: num] :
( ( bit_se1065995026697491101ons_or @ int @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ ( bit0 @ N3 ) ) ) @ ( one_one @ int ) )
= ( uminus_uminus @ int @ ( numeral_numeral @ int @ ( bit_or_not_num_neg @ one2 @ ( bitM @ N3 ) ) ) ) ) ).
% or_minus_numerals(5)
thf(fact_5542_or__minus__numerals_I1_J,axiom,
! [N3: num] :
( ( bit_se1065995026697491101ons_or @ int @ ( one_one @ int ) @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ ( bit0 @ N3 ) ) ) )
= ( uminus_uminus @ int @ ( numeral_numeral @ int @ ( bit_or_not_num_neg @ one2 @ ( bitM @ N3 ) ) ) ) ) ).
% or_minus_numerals(1)
thf(fact_5543_word__bitwise__m1__simps_I6_J,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [X: word @ A] :
( ( bit_se5824344971392196577ns_xor @ ( word @ A ) @ ( uminus_uminus @ ( word @ A ) @ ( one_one @ ( word @ A ) ) ) @ X )
= ( bit_ri4277139882892585799ns_not @ ( word @ A ) @ X ) ) ) ).
% word_bitwise_m1_simps(6)
thf(fact_5544_word__bitwise__m1__simps_I7_J,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [X: word @ A] :
( ( bit_se5824344971392196577ns_xor @ ( word @ A ) @ X @ ( uminus_uminus @ ( word @ A ) @ ( one_one @ ( word @ A ) ) ) )
= ( bit_ri4277139882892585799ns_not @ ( word @ A ) @ X ) ) ) ).
% word_bitwise_m1_simps(7)
thf(fact_5545_xor__nat__numerals_I4_J,axiom,
! [X: num] :
( ( bit_se5824344971392196577ns_xor @ nat @ ( numeral_numeral @ nat @ ( bit1 @ X ) ) @ ( suc @ ( zero_zero @ nat ) ) )
= ( numeral_numeral @ nat @ ( bit0 @ X ) ) ) ).
% xor_nat_numerals(4)
thf(fact_5546_xor__nat__numerals_I3_J,axiom,
! [X: num] :
( ( bit_se5824344971392196577ns_xor @ nat @ ( numeral_numeral @ nat @ ( bit0 @ X ) ) @ ( suc @ ( zero_zero @ nat ) ) )
= ( numeral_numeral @ nat @ ( bit1 @ X ) ) ) ).
% xor_nat_numerals(3)
thf(fact_5547_xor__nat__numerals_I2_J,axiom,
! [Y: num] :
( ( bit_se5824344971392196577ns_xor @ nat @ ( suc @ ( zero_zero @ nat ) ) @ ( numeral_numeral @ nat @ ( bit1 @ Y ) ) )
= ( numeral_numeral @ nat @ ( bit0 @ Y ) ) ) ).
% xor_nat_numerals(2)
thf(fact_5548_xor__nat__numerals_I1_J,axiom,
! [Y: num] :
( ( bit_se5824344971392196577ns_xor @ nat @ ( suc @ ( zero_zero @ nat ) ) @ ( numeral_numeral @ nat @ ( bit0 @ Y ) ) )
= ( numeral_numeral @ nat @ ( bit1 @ Y ) ) ) ).
% xor_nat_numerals(1)
thf(fact_5549_or__minus__numerals_I4_J,axiom,
! [M: num,N3: num] :
( ( bit_se1065995026697491101ons_or @ int @ ( numeral_numeral @ int @ M ) @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ ( bit1 @ N3 ) ) ) )
= ( uminus_uminus @ int @ ( numeral_numeral @ int @ ( bit_or_not_num_neg @ M @ ( bit0 @ N3 ) ) ) ) ) ).
% or_minus_numerals(4)
thf(fact_5550_or__minus__numerals_I8_J,axiom,
! [N3: num,M: num] :
( ( bit_se1065995026697491101ons_or @ int @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ ( bit1 @ N3 ) ) ) @ ( numeral_numeral @ int @ M ) )
= ( uminus_uminus @ int @ ( numeral_numeral @ int @ ( bit_or_not_num_neg @ M @ ( bit0 @ N3 ) ) ) ) ) ).
% or_minus_numerals(8)
thf(fact_5551_word__bitwise__1__simps_I10_J,axiom,
! [J4: $tType] :
( ( type_len @ J4 )
=> ! [B2: num] :
( ( bit_se5824344971392196577ns_xor @ ( word @ J4 ) @ ( one_one @ ( word @ J4 ) ) @ ( numeral_numeral @ ( word @ J4 ) @ B2 ) )
= ( ring_1_of_int @ ( word @ J4 ) @ ( bit_se5824344971392196577ns_xor @ int @ ( one_one @ int ) @ ( numeral_numeral @ int @ B2 ) ) ) ) ) ).
% word_bitwise_1_simps(10)
thf(fact_5552_word__bitwise__1__simps_I12_J,axiom,
! [L5: $tType] :
( ( type_len @ L5 )
=> ! [A2: num] :
( ( bit_se5824344971392196577ns_xor @ ( word @ L5 ) @ ( numeral_numeral @ ( word @ L5 ) @ A2 ) @ ( one_one @ ( word @ L5 ) ) )
= ( ring_1_of_int @ ( word @ L5 ) @ ( bit_se5824344971392196577ns_xor @ int @ ( numeral_numeral @ int @ A2 ) @ ( one_one @ int ) ) ) ) ) ).
% word_bitwise_1_simps(12)
thf(fact_5553_or__minus__numerals_I3_J,axiom,
! [M: num,N3: num] :
( ( bit_se1065995026697491101ons_or @ int @ ( numeral_numeral @ int @ M ) @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ ( bit0 @ N3 ) ) ) )
= ( uminus_uminus @ int @ ( numeral_numeral @ int @ ( bit_or_not_num_neg @ M @ ( bitM @ N3 ) ) ) ) ) ).
% or_minus_numerals(3)
thf(fact_5554_or__minus__numerals_I7_J,axiom,
! [N3: num,M: num] :
( ( bit_se1065995026697491101ons_or @ int @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ ( bit0 @ N3 ) ) ) @ ( numeral_numeral @ int @ M ) )
= ( uminus_uminus @ int @ ( numeral_numeral @ int @ ( bit_or_not_num_neg @ M @ ( bitM @ N3 ) ) ) ) ) ).
% or_minus_numerals(7)
thf(fact_5555_word__bitwise__1__simps_I11_J,axiom,
! [K8: $tType] :
( ( type_len @ K8 )
=> ! [B2: num] :
( ( bit_se5824344971392196577ns_xor @ ( word @ K8 ) @ ( one_one @ ( word @ K8 ) ) @ ( uminus_uminus @ ( word @ K8 ) @ ( numeral_numeral @ ( word @ K8 ) @ B2 ) ) )
= ( ring_1_of_int @ ( word @ K8 ) @ ( bit_se5824344971392196577ns_xor @ int @ ( one_one @ int ) @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ B2 ) ) ) ) ) ) ).
% word_bitwise_1_simps(11)
thf(fact_5556_word__bitwise__1__simps_I13_J,axiom,
! [M11: $tType] :
( ( type_len @ M11 )
=> ! [A2: num] :
( ( bit_se5824344971392196577ns_xor @ ( word @ M11 ) @ ( uminus_uminus @ ( word @ M11 ) @ ( numeral_numeral @ ( word @ M11 ) @ A2 ) ) @ ( one_one @ ( word @ M11 ) ) )
= ( ring_1_of_int @ ( word @ M11 ) @ ( bit_se5824344971392196577ns_xor @ int @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ A2 ) ) @ ( one_one @ int ) ) ) ) ) ).
% word_bitwise_1_simps(13)
thf(fact_5557_or__not__num__neg_Osimps_I1_J,axiom,
( ( bit_or_not_num_neg @ one2 @ one2 )
= one2 ) ).
% or_not_num_neg.simps(1)
thf(fact_5558_or__not__num__neg_Osimps_I4_J,axiom,
! [N3: num] :
( ( bit_or_not_num_neg @ ( bit0 @ N3 ) @ one2 )
= ( bit0 @ one2 ) ) ).
% or_not_num_neg.simps(4)
thf(fact_5559_or__not__num__neg_Osimps_I6_J,axiom,
! [N3: num,M: num] :
( ( bit_or_not_num_neg @ ( bit0 @ N3 ) @ ( bit1 @ M ) )
= ( bit0 @ ( bit_or_not_num_neg @ N3 @ M ) ) ) ).
% or_not_num_neg.simps(6)
thf(fact_5560_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_5561_or__not__num__neg_Osimps_I7_J,axiom,
! [N3: num] :
( ( bit_or_not_num_neg @ ( bit1 @ N3 ) @ one2 )
= one2 ) ).
% or_not_num_neg.simps(7)
thf(fact_5562_or__not__num__neg_Osimps_I5_J,axiom,
! [N3: num,M: num] :
( ( bit_or_not_num_neg @ ( bit0 @ N3 ) @ ( bit0 @ M ) )
= ( bitM @ ( bit_or_not_num_neg @ N3 @ M ) ) ) ).
% or_not_num_neg.simps(5)
thf(fact_5563_or__not__num__neg_Osimps_I9_J,axiom,
! [N3: num,M: num] :
( ( bit_or_not_num_neg @ ( bit1 @ N3 ) @ ( bit1 @ M ) )
= ( bitM @ ( bit_or_not_num_neg @ N3 @ M ) ) ) ).
% or_not_num_neg.simps(9)
thf(fact_5564_xor__nat__def,axiom,
( ( bit_se5824344971392196577ns_xor @ nat )
= ( ^ [M5: nat,N2: nat] : ( nat2 @ ( bit_se5824344971392196577ns_xor @ int @ ( semiring_1_of_nat @ int @ M5 ) @ ( semiring_1_of_nat @ int @ N2 ) ) ) ) ) ).
% xor_nat_def
thf(fact_5565_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_5566_or__not__num__neg_Osimps_I8_J,axiom,
! [N3: num,M: num] :
( ( bit_or_not_num_neg @ ( bit1 @ N3 ) @ ( bit0 @ M ) )
= ( bitM @ ( bit_or_not_num_neg @ N3 @ M ) ) ) ).
% or_not_num_neg.simps(8)
thf(fact_5567_or__not__num__neg_Oelims,axiom,
! [X: num,Xa: num,Y: num] :
( ( ( bit_or_not_num_neg @ X @ Xa )
= Y )
=> ( ( ( X = one2 )
=> ( ( Xa = one2 )
=> ( Y != one2 ) ) )
=> ( ( ( X = one2 )
=> ! [M4: num] :
( ( Xa
= ( bit0 @ M4 ) )
=> ( Y
!= ( bit1 @ M4 ) ) ) )
=> ( ( ( X = one2 )
=> ! [M4: num] :
( ( Xa
= ( bit1 @ M4 ) )
=> ( Y
!= ( bit1 @ M4 ) ) ) )
=> ( ( ? [N: num] :
( X
= ( bit0 @ N ) )
=> ( ( Xa = one2 )
=> ( Y
!= ( bit0 @ one2 ) ) ) )
=> ( ! [N: num] :
( ( X
= ( bit0 @ N ) )
=> ! [M4: num] :
( ( Xa
= ( bit0 @ M4 ) )
=> ( Y
!= ( bitM @ ( bit_or_not_num_neg @ N @ M4 ) ) ) ) )
=> ( ! [N: num] :
( ( X
= ( bit0 @ N ) )
=> ! [M4: num] :
( ( Xa
= ( bit1 @ M4 ) )
=> ( Y
!= ( bit0 @ ( bit_or_not_num_neg @ N @ M4 ) ) ) ) )
=> ( ( ? [N: num] :
( X
= ( bit1 @ N ) )
=> ( ( Xa = one2 )
=> ( Y != one2 ) ) )
=> ( ! [N: num] :
( ( X
= ( bit1 @ N ) )
=> ! [M4: num] :
( ( Xa
= ( bit0 @ M4 ) )
=> ( Y
!= ( bitM @ ( bit_or_not_num_neg @ N @ M4 ) ) ) ) )
=> ~ ! [N: num] :
( ( X
= ( bit1 @ N ) )
=> ! [M4: num] :
( ( Xa
= ( bit1 @ M4 ) )
=> ( Y
!= ( bitM @ ( bit_or_not_num_neg @ N @ M4 ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).
% or_not_num_neg.elims
thf(fact_5568_word__ops__nth__size,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [N3: nat,X: word @ A,Y: word @ A] :
( ( ord_less @ nat @ N3 @ ( size_size @ ( word @ A ) @ X ) )
=> ( ( ( bit_se5641148757651400278ts_bit @ ( word @ A ) @ ( bit_se1065995026697491101ons_or @ ( word @ A ) @ X @ Y ) @ N3 )
= ( ( bit_se5641148757651400278ts_bit @ ( word @ A ) @ X @ N3 )
| ( bit_se5641148757651400278ts_bit @ ( word @ A ) @ Y @ N3 ) ) )
& ( ( bit_se5641148757651400278ts_bit @ ( word @ A ) @ ( bit_se5824344872417868541ns_and @ ( word @ A ) @ X @ Y ) @ N3 )
= ( ( bit_se5641148757651400278ts_bit @ ( word @ A ) @ X @ N3 )
& ( bit_se5641148757651400278ts_bit @ ( word @ A ) @ Y @ N3 ) ) )
& ( ( bit_se5641148757651400278ts_bit @ ( word @ A ) @ ( bit_se5824344971392196577ns_xor @ ( word @ A ) @ X @ Y ) @ N3 )
= ( ( bit_se5641148757651400278ts_bit @ ( word @ A ) @ X @ N3 )
!= ( bit_se5641148757651400278ts_bit @ ( word @ A ) @ Y @ N3 ) ) )
& ( ( bit_se5641148757651400278ts_bit @ ( word @ A ) @ ( bit_ri4277139882892585799ns_not @ ( word @ A ) @ X ) @ N3 )
= ( ~ ( bit_se5641148757651400278ts_bit @ ( word @ A ) @ X @ N3 ) ) ) ) ) ) ).
% word_ops_nth_size
thf(fact_5569_int__numeral__or__not__num__neg,axiom,
! [M: num,N3: num] :
( ( bit_se1065995026697491101ons_or @ int @ ( numeral_numeral @ int @ M ) @ ( bit_ri4277139882892585799ns_not @ int @ ( numeral_numeral @ int @ N3 ) ) )
= ( uminus_uminus @ int @ ( numeral_numeral @ int @ ( bit_or_not_num_neg @ M @ N3 ) ) ) ) ).
% int_numeral_or_not_num_neg
thf(fact_5570_int__numeral__not__or__num__neg,axiom,
! [M: num,N3: num] :
( ( bit_se1065995026697491101ons_or @ int @ ( bit_ri4277139882892585799ns_not @ int @ ( numeral_numeral @ int @ M ) ) @ ( numeral_numeral @ int @ N3 ) )
= ( uminus_uminus @ int @ ( numeral_numeral @ int @ ( bit_or_not_num_neg @ N3 @ M ) ) ) ) ).
% int_numeral_not_or_num_neg
thf(fact_5571_numeral__or__not__num__eq,axiom,
! [M: num,N3: num] :
( ( numeral_numeral @ int @ ( bit_or_not_num_neg @ M @ N3 ) )
= ( uminus_uminus @ int @ ( bit_se1065995026697491101ons_or @ int @ ( numeral_numeral @ int @ M ) @ ( bit_ri4277139882892585799ns_not @ int @ ( numeral_numeral @ int @ N3 ) ) ) ) ) ).
% numeral_or_not_num_eq
thf(fact_5572_bit__twiddle__max,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [X: word @ A,Y: word @ A] :
( ( bit_se5824344971392196577ns_xor @ ( word @ A ) @ X @ ( bit_se5824344872417868541ns_and @ ( word @ A ) @ ( bit_se5824344971392196577ns_xor @ ( word @ A ) @ X @ Y ) @ ( if @ ( word @ A ) @ ( ord_less @ ( word @ A ) @ X @ Y ) @ ( uminus_uminus @ ( word @ A ) @ ( one_one @ ( word @ A ) ) ) @ ( zero_zero @ ( word @ A ) ) ) ) )
= ( ord_max @ ( word @ A ) @ X @ Y ) ) ) ).
% bit_twiddle_max
thf(fact_5573_xor__nat__unfold,axiom,
( ( bit_se5824344971392196577ns_xor @ nat )
= ( ^ [M5: nat,N2: nat] :
( if @ nat
@ ( M5
= ( zero_zero @ nat ) )
@ N2
@ ( if @ nat
@ ( N2
= ( zero_zero @ nat ) )
@ M5
@ ( plus_plus @ nat @ ( modulo_modulo @ nat @ ( plus_plus @ nat @ ( modulo_modulo @ nat @ M5 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ ( modulo_modulo @ nat @ N2 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( bit_se5824344971392196577ns_xor @ nat @ ( divide_divide @ nat @ M5 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ ( divide_divide @ nat @ N2 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) ) ) ) ) ).
% xor_nat_unfold
thf(fact_5574_xor__nat__rec,axiom,
( ( bit_se5824344971392196577ns_xor @ nat )
= ( ^ [M5: nat,N2: nat] :
( plus_plus @ nat
@ ( zero_neq_one_of_bool @ nat
@ ( ( ~ ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ M5 ) )
!= ( ~ ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N2 ) ) ) )
@ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( bit_se5824344971392196577ns_xor @ nat @ ( divide_divide @ nat @ M5 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ ( divide_divide @ nat @ N2 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) ) ) ).
% xor_nat_rec
thf(fact_5575_Suc__0__xor__eq,axiom,
! [N3: nat] :
( ( bit_se5824344971392196577ns_xor @ nat @ ( suc @ ( zero_zero @ nat ) ) @ N3 )
= ( minus_minus @ nat @ ( plus_plus @ nat @ N3 @ ( zero_neq_one_of_bool @ nat @ ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N3 ) ) )
@ ( zero_neq_one_of_bool @ nat
@ ~ ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N3 ) ) ) ) ).
% Suc_0_xor_eq
thf(fact_5576_xor__Suc__0__eq,axiom,
! [N3: nat] :
( ( bit_se5824344971392196577ns_xor @ nat @ N3 @ ( suc @ ( zero_zero @ nat ) ) )
= ( minus_minus @ nat @ ( plus_plus @ nat @ N3 @ ( zero_neq_one_of_bool @ nat @ ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N3 ) ) )
@ ( zero_neq_one_of_bool @ nat
@ ~ ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N3 ) ) ) ) ).
% xor_Suc_0_eq
thf(fact_5577_or__not__num__neg_Opelims,axiom,
! [X: num,Xa: num,Y: num] :
( ( ( bit_or_not_num_neg @ X @ Xa )
= Y )
=> ( ( accp @ ( product_prod @ num @ num ) @ bit_or3848514188828904588eg_rel @ ( product_Pair @ num @ num @ X @ Xa ) )
=> ( ( ( X = one2 )
=> ( ( Xa = one2 )
=> ( ( Y = one2 )
=> ~ ( accp @ ( product_prod @ num @ num ) @ bit_or3848514188828904588eg_rel @ ( product_Pair @ num @ num @ one2 @ one2 ) ) ) ) )
=> ( ( ( X = one2 )
=> ! [M4: num] :
( ( Xa
= ( bit0 @ M4 ) )
=> ( ( Y
= ( bit1 @ M4 ) )
=> ~ ( accp @ ( product_prod @ num @ num ) @ bit_or3848514188828904588eg_rel @ ( product_Pair @ num @ num @ one2 @ ( bit0 @ M4 ) ) ) ) ) )
=> ( ( ( X = one2 )
=> ! [M4: num] :
( ( Xa
= ( bit1 @ M4 ) )
=> ( ( Y
= ( bit1 @ M4 ) )
=> ~ ( accp @ ( product_prod @ num @ num ) @ bit_or3848514188828904588eg_rel @ ( product_Pair @ num @ num @ one2 @ ( bit1 @ M4 ) ) ) ) ) )
=> ( ! [N: num] :
( ( X
= ( bit0 @ N ) )
=> ( ( Xa = one2 )
=> ( ( Y
= ( bit0 @ one2 ) )
=> ~ ( accp @ ( product_prod @ num @ num ) @ bit_or3848514188828904588eg_rel @ ( product_Pair @ num @ num @ ( bit0 @ N ) @ one2 ) ) ) ) )
=> ( ! [N: num] :
( ( X
= ( bit0 @ N ) )
=> ! [M4: num] :
( ( Xa
= ( bit0 @ M4 ) )
=> ( ( Y
= ( bitM @ ( bit_or_not_num_neg @ N @ M4 ) ) )
=> ~ ( accp @ ( product_prod @ num @ num ) @ bit_or3848514188828904588eg_rel @ ( product_Pair @ num @ num @ ( bit0 @ N ) @ ( bit0 @ M4 ) ) ) ) ) )
=> ( ! [N: num] :
( ( X
= ( bit0 @ N ) )
=> ! [M4: num] :
( ( Xa
= ( bit1 @ M4 ) )
=> ( ( Y
= ( bit0 @ ( bit_or_not_num_neg @ N @ M4 ) ) )
=> ~ ( accp @ ( product_prod @ num @ num ) @ bit_or3848514188828904588eg_rel @ ( product_Pair @ num @ num @ ( bit0 @ N ) @ ( bit1 @ M4 ) ) ) ) ) )
=> ( ! [N: num] :
( ( X
= ( bit1 @ N ) )
=> ( ( Xa = one2 )
=> ( ( Y = one2 )
=> ~ ( accp @ ( product_prod @ num @ num ) @ bit_or3848514188828904588eg_rel @ ( product_Pair @ num @ num @ ( bit1 @ N ) @ one2 ) ) ) ) )
=> ( ! [N: num] :
( ( X
= ( bit1 @ N ) )
=> ! [M4: num] :
( ( Xa
= ( bit0 @ M4 ) )
=> ( ( Y
= ( bitM @ ( bit_or_not_num_neg @ N @ M4 ) ) )
=> ~ ( accp @ ( product_prod @ num @ num ) @ bit_or3848514188828904588eg_rel @ ( product_Pair @ num @ num @ ( bit1 @ N ) @ ( bit0 @ M4 ) ) ) ) ) )
=> ~ ! [N: num] :
( ( X
= ( bit1 @ N ) )
=> ! [M4: num] :
( ( Xa
= ( bit1 @ M4 ) )
=> ( ( Y
= ( bitM @ ( bit_or_not_num_neg @ N @ M4 ) ) )
=> ~ ( accp @ ( product_prod @ num @ num ) @ bit_or3848514188828904588eg_rel @ ( product_Pair @ num @ num @ ( bit1 @ N ) @ ( bit1 @ M4 ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).
% or_not_num_neg.pelims
thf(fact_5578_Arg__def,axiom,
( arg
= ( ^ [Z3: complex] :
( if @ real
@ ( Z3
= ( zero_zero @ complex ) )
@ ( zero_zero @ real )
@ ( fChoice @ real
@ ^ [A6: real] :
( ( ( sgn_sgn @ complex @ Z3 )
= ( cis @ A6 ) )
& ( ord_less @ real @ ( uminus_uminus @ real @ pi ) @ A6 )
& ( ord_less_eq @ real @ A6 @ pi ) ) ) ) ) ) ).
% Arg_def
thf(fact_5579_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_5580_some__insert__self,axiom,
! [A: $tType,S3: set @ A] :
( ( S3
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( insert @ A
@ ( fChoice @ A
@ ^ [X2: A] : ( member @ A @ X2 @ S3 ) )
@ S3 )
= S3 ) ) ).
% some_insert_self
thf(fact_5581_some__elem,axiom,
! [A: $tType,S3: set @ A] :
( ( S3
!= ( bot_bot @ ( set @ A ) ) )
=> ( member @ A
@ ( fChoice @ A
@ ^ [X2: A] : ( member @ A @ X2 @ S3 ) )
@ S3 ) ) ).
% some_elem
thf(fact_5582_fun__of__rel__def,axiom,
! [A: $tType,B: $tType] :
( ( fun_of_rel @ B @ A )
= ( ^ [R6: set @ ( product_prod @ B @ A ),X2: B] :
( fChoice @ A
@ ^ [Y2: A] : ( member @ ( product_prod @ B @ A ) @ ( product_Pair @ B @ A @ X2 @ Y2 ) @ R6 ) ) ) ) ).
% fun_of_rel_def
thf(fact_5583_hash__code__prod__simps,axiom,
! [A: $tType,B: $tType,H_a: A > uint32,H_b: B > uint32,X: A,Xa: B] :
( ( hash_hash_code_prod @ A @ B @ H_a @ H_b @ ( product_Pair @ A @ B @ X @ Xa ) )
= ( plus_plus @ uint32 @ ( times_times @ uint32 @ ( H_a @ X ) @ ( numeral_numeral @ uint32 @ ( bit0 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit0 @ ( bit1 @ ( bit1 @ ( bit0 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit0 @ ( bit0 @ ( bit1 @ ( bit1 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit1 @ ( bit0 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit0 @ ( bit0 @ ( bit1 @ ( bit1 @ one2 ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) @ ( plus_plus @ uint32 @ ( times_times @ uint32 @ ( H_b @ 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_5584_sum__count__set,axiom,
! [A: $tType,Xs2: list @ A,X8: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ ( set2 @ A @ Xs2 ) @ X8 )
=> ( ( finite_finite2 @ A @ X8 )
=> ( ( groups7311177749621191930dd_sum @ A @ nat @ ( count_list @ A @ Xs2 ) @ X8 )
= ( size_size @ ( list @ A ) @ Xs2 ) ) ) ) ).
% sum_count_set
thf(fact_5585_Eps__case__prod__eq,axiom,
! [A: $tType,B: $tType,X: A,Y: B] :
( ( fChoice @ ( product_prod @ A @ B )
@ ( product_case_prod @ A @ B @ $o
@ ^ [X9: A,Y7: B] :
( ( X = X9 )
& ( Y = Y7 ) ) ) )
= ( product_Pair @ A @ B @ X @ Y ) ) ).
% Eps_case_prod_eq
thf(fact_5586_count__notin,axiom,
! [A: $tType,X: A,Xs2: list @ A] :
( ~ ( member @ A @ X @ ( set2 @ A @ Xs2 ) )
=> ( ( count_list @ A @ Xs2 @ X )
= ( zero_zero @ nat ) ) ) ).
% count_notin
thf(fact_5587_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,B5: B] : ( P3 @ ( product_Pair @ A @ B @ A6 @ B5 ) ) ) ) ) ) ).
% split_paired_Eps
thf(fact_5588_count__le__length,axiom,
! [A: $tType,Xs2: list @ A,X: A] : ( ord_less_eq @ nat @ ( count_list @ A @ Xs2 @ X ) @ ( size_size @ ( list @ A ) @ Xs2 ) ) ).
% count_le_length
thf(fact_5589_hash__code__option__simps_I2_J,axiom,
! [A: $tType,H_a: A > uint32,X: A] :
( ( hash_h1887023736457453652option @ A @ H_a @ ( some @ A @ X ) )
= ( plus_plus @ uint32 @ ( times_times @ uint32 @ ( H_a @ X ) @ ( 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_5590_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_5591_dup__1,axiom,
( ( code_dup @ ( one_one @ code_integer ) )
= ( numeral_numeral @ code_integer @ ( bit0 @ one2 ) ) ) ).
% dup_1
thf(fact_5592_power__numeral,axiom,
! [A: $tType] :
( ( semiring_numeral @ A )
=> ! [K: num,L2: num] :
( ( power_power @ A @ ( numeral_numeral @ A @ K ) @ ( numeral_numeral @ nat @ L2 ) )
= ( numeral_numeral @ A @ ( pow @ K @ L2 ) ) ) ) ).
% power_numeral
thf(fact_5593_cis__multiple__2pi,axiom,
! [N3: real] :
( ( member @ real @ N3 @ ( ring_1_Ints @ real ) )
=> ( ( cis @ ( times_times @ real @ ( times_times @ real @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) @ pi ) @ N3 ) )
= ( one_one @ complex ) ) ) ).
% cis_multiple_2pi
thf(fact_5594_ran__nth__set__encoding__conv,axiom,
! [A: $tType,L2: list @ A] :
( ( ran @ nat @ A
@ ^ [I3: nat] : ( if @ ( option @ A ) @ ( ord_less @ nat @ I3 @ ( size_size @ ( list @ A ) @ L2 ) ) @ ( some @ A @ ( nth @ A @ L2 @ I3 ) ) @ ( none @ A ) ) )
= ( set2 @ A @ L2 ) ) ).
% ran_nth_set_encoding_conv
thf(fact_5595_floor__add2,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [X: A,Y: A] :
( ( ( member @ A @ X @ ( ring_1_Ints @ A ) )
| ( member @ A @ Y @ ( ring_1_Ints @ A ) ) )
=> ( ( archim6421214686448440834_floor @ A @ ( plus_plus @ A @ X @ Y ) )
= ( plus_plus @ int @ ( archim6421214686448440834_floor @ A @ X ) @ ( archim6421214686448440834_floor @ A @ Y ) ) ) ) ) ).
% floor_add2
thf(fact_5596_frac__gt__0__iff,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [X: A] :
( ( ord_less @ A @ ( zero_zero @ A ) @ ( archimedean_frac @ A @ X ) )
= ( ~ ( member @ A @ X @ ( ring_1_Ints @ A ) ) ) ) ) ).
% frac_gt_0_iff
thf(fact_5597_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
@ ^ [X2: A] : ( if @ ( option @ B ) @ ( X2 = K ) @ ( some @ B @ V ) @ ( M @ X2 ) ) )
= ( insert @ B @ V @ ( ran @ A @ B @ M ) ) ) ) ).
% map_update_eta_repair(2)
thf(fact_5598_Ints__1,axiom,
! [A: $tType] :
( ( ring_1 @ A )
=> ( member @ A @ ( one_one @ A ) @ ( ring_1_Ints @ A ) ) ) ).
% Ints_1
thf(fact_5599_Ints__power,axiom,
! [A: $tType] :
( ( ring_1 @ A )
=> ! [A2: A,N3: nat] :
( ( member @ A @ A2 @ ( ring_1_Ints @ A ) )
=> ( member @ A @ ( power_power @ A @ A2 @ N3 ) @ ( ring_1_Ints @ A ) ) ) ) ).
% Ints_power
thf(fact_5600_Ints__diff,axiom,
! [A: $tType] :
( ( ring_1 @ A )
=> ! [A2: A,B2: A] :
( ( member @ A @ A2 @ ( ring_1_Ints @ A ) )
=> ( ( member @ A @ B2 @ ( ring_1_Ints @ A ) )
=> ( member @ A @ ( minus_minus @ A @ A2 @ B2 ) @ ( ring_1_Ints @ A ) ) ) ) ) ).
% Ints_diff
thf(fact_5601_Ints__add,axiom,
! [A: $tType] :
( ( ring_1 @ A )
=> ! [A2: A,B2: A] :
( ( member @ A @ A2 @ ( ring_1_Ints @ A ) )
=> ( ( member @ A @ B2 @ ( ring_1_Ints @ A ) )
=> ( member @ A @ ( plus_plus @ A @ A2 @ B2 ) @ ( ring_1_Ints @ A ) ) ) ) ) ).
% Ints_add
thf(fact_5602_Ints__mult,axiom,
! [A: $tType] :
( ( ring_1 @ A )
=> ! [A2: A,B2: A] :
( ( member @ A @ A2 @ ( ring_1_Ints @ A ) )
=> ( ( member @ A @ B2 @ ( ring_1_Ints @ A ) )
=> ( member @ A @ ( times_times @ A @ A2 @ B2 ) @ ( ring_1_Ints @ A ) ) ) ) ) ).
% Ints_mult
thf(fact_5603_Ints__numeral,axiom,
! [A: $tType] :
( ( ring_1 @ A )
=> ! [N3: num] : ( member @ A @ ( numeral_numeral @ A @ N3 ) @ ( ring_1_Ints @ A ) ) ) ).
% Ints_numeral
thf(fact_5604_Ints__double__eq__0__iff,axiom,
! [A: $tType] :
( ( ring_char_0 @ A )
=> ! [A2: A] :
( ( member @ A @ A2 @ ( ring_1_Ints @ A ) )
=> ( ( ( plus_plus @ A @ A2 @ A2 )
= ( zero_zero @ A ) )
= ( A2
= ( zero_zero @ A ) ) ) ) ) ).
% Ints_double_eq_0_iff
thf(fact_5605_pow_Osimps_I1_J,axiom,
! [X: num] :
( ( pow @ X @ one2 )
= X ) ).
% pow.simps(1)
thf(fact_5606_finite__int__segment,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [A2: A,B2: A] :
( finite_finite2 @ A
@ ( collect @ A
@ ^ [X2: A] :
( ( member @ A @ X2 @ ( ring_1_Ints @ A ) )
& ( ord_less_eq @ A @ A2 @ X2 )
& ( ord_less_eq @ A @ X2 @ B2 ) ) ) ) ) ).
% finite_int_segment
thf(fact_5607_Ints__odd__nonzero,axiom,
! [A: $tType] :
( ( ring_char_0 @ A )
=> ! [A2: A] :
( ( member @ A @ A2 @ ( ring_1_Ints @ A ) )
=> ( ( plus_plus @ A @ ( plus_plus @ A @ ( one_one @ A ) @ A2 ) @ A2 )
!= ( zero_zero @ A ) ) ) ) ).
% Ints_odd_nonzero
thf(fact_5608_of__int__divide__in__Ints,axiom,
! [A: $tType] :
( ( idom_divide @ A )
=> ! [B2: int,A2: int] :
( ( dvd_dvd @ int @ B2 @ A2 )
=> ( member @ A @ ( divide_divide @ A @ ( ring_1_of_int @ A @ A2 ) @ ( ring_1_of_int @ A @ B2 ) ) @ ( ring_1_Ints @ A ) ) ) ) ).
% of_int_divide_in_Ints
thf(fact_5609_finite__abs__int__segment,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [A2: A] :
( finite_finite2 @ A
@ ( collect @ A
@ ^ [K2: A] :
( ( member @ A @ K2 @ ( ring_1_Ints @ A ) )
& ( ord_less_eq @ A @ ( abs_abs @ A @ K2 ) @ A2 ) ) ) ) ) ).
% finite_abs_int_segment
thf(fact_5610_Ints__odd__less__0,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [A2: A] :
( ( member @ A @ A2 @ ( ring_1_Ints @ A ) )
=> ( ( ord_less @ A @ ( plus_plus @ A @ ( plus_plus @ A @ ( one_one @ A ) @ A2 ) @ A2 ) @ ( zero_zero @ A ) )
= ( ord_less @ A @ A2 @ ( zero_zero @ A ) ) ) ) ) ).
% Ints_odd_less_0
thf(fact_5611_Ints__nonzero__abs__ge1,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [X: A] :
( ( member @ A @ X @ ( ring_1_Ints @ A ) )
=> ( ( X
!= ( zero_zero @ A ) )
=> ( ord_less_eq @ A @ ( one_one @ A ) @ ( abs_abs @ A @ X ) ) ) ) ) ).
% Ints_nonzero_abs_ge1
thf(fact_5612_Ints__nonzero__abs__less1,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [X: A] :
( ( member @ A @ X @ ( ring_1_Ints @ A ) )
=> ( ( ord_less @ A @ ( abs_abs @ A @ X ) @ ( one_one @ A ) )
=> ( X
= ( zero_zero @ A ) ) ) ) ) ).
% Ints_nonzero_abs_less1
thf(fact_5613_Ints__eq__abs__less1,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [X: A,Y: A] :
( ( member @ A @ X @ ( ring_1_Ints @ A ) )
=> ( ( member @ A @ Y @ ( ring_1_Ints @ A ) )
=> ( ( X = Y )
= ( ord_less @ A @ ( abs_abs @ A @ ( minus_minus @ A @ X @ Y ) ) @ ( one_one @ A ) ) ) ) ) ) ).
% Ints_eq_abs_less1
thf(fact_5614_sin__times__pi__eq__0,axiom,
! [X: real] :
( ( ( sin @ real @ ( times_times @ real @ X @ pi ) )
= ( zero_zero @ real ) )
= ( member @ real @ X @ ( ring_1_Ints @ real ) ) ) ).
% sin_times_pi_eq_0
thf(fact_5615_frac__neg,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [X: A] :
( ( ( member @ A @ X @ ( ring_1_Ints @ A ) )
=> ( ( archimedean_frac @ A @ ( uminus_uminus @ A @ X ) )
= ( zero_zero @ A ) ) )
& ( ~ ( member @ A @ X @ ( ring_1_Ints @ A ) )
=> ( ( archimedean_frac @ A @ ( uminus_uminus @ A @ X ) )
= ( minus_minus @ A @ ( one_one @ A ) @ ( archimedean_frac @ A @ X ) ) ) ) ) ) ).
% frac_neg
thf(fact_5616_le__mult__floor__Ints,axiom,
! [A: $tType,B: $tType] :
( ( ( archim2362893244070406136eiling @ B )
& ( linordered_idom @ A ) )
=> ! [A2: B,B2: B] :
( ( ord_less_eq @ B @ ( zero_zero @ B ) @ A2 )
=> ( ( member @ B @ A2 @ ( ring_1_Ints @ B ) )
=> ( ord_less_eq @ A @ ( ring_1_of_int @ A @ ( times_times @ int @ ( archim6421214686448440834_floor @ B @ A2 ) @ ( archim6421214686448440834_floor @ B @ B2 ) ) ) @ ( ring_1_of_int @ A @ ( archim6421214686448440834_floor @ B @ ( times_times @ B @ A2 @ B2 ) ) ) ) ) ) ) ).
% le_mult_floor_Ints
thf(fact_5617_mult__ceiling__le__Ints,axiom,
! [A: $tType,B: $tType] :
( ( ( archim2362893244070406136eiling @ B )
& ( linordered_idom @ A ) )
=> ! [A2: B,B2: B] :
( ( ord_less_eq @ B @ ( zero_zero @ B ) @ A2 )
=> ( ( member @ B @ A2 @ ( ring_1_Ints @ B ) )
=> ( ord_less_eq @ A @ ( ring_1_of_int @ A @ ( archimedean_ceiling @ B @ ( times_times @ B @ A2 @ B2 ) ) ) @ ( ring_1_of_int @ A @ ( times_times @ int @ ( archimedean_ceiling @ B @ A2 ) @ ( archimedean_ceiling @ B @ B2 ) ) ) ) ) ) ) ).
% mult_ceiling_le_Ints
thf(fact_5618_frac__unique__iff,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [X: A,A2: A] :
( ( ( archimedean_frac @ A @ X )
= A2 )
= ( ( member @ A @ ( minus_minus @ A @ X @ A2 ) @ ( ring_1_Ints @ A ) )
& ( ord_less_eq @ A @ ( zero_zero @ A ) @ A2 )
& ( ord_less @ A @ A2 @ ( one_one @ A ) ) ) ) ) ).
% frac_unique_iff
thf(fact_5619_sin__integer__2pi,axiom,
! [N3: real] :
( ( member @ real @ N3 @ ( ring_1_Ints @ real ) )
=> ( ( sin @ real @ ( times_times @ real @ ( times_times @ real @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) @ pi ) @ N3 ) )
= ( zero_zero @ real ) ) ) ).
% sin_integer_2pi
thf(fact_5620_cos__integer__2pi,axiom,
! [N3: real] :
( ( member @ real @ N3 @ ( ring_1_Ints @ real ) )
=> ( ( cos @ real @ ( times_times @ real @ ( times_times @ real @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) @ pi ) @ N3 ) )
= ( one_one @ real ) ) ) ).
% cos_integer_2pi
thf(fact_5621_ran__empty,axiom,
! [B: $tType,A: $tType] :
( ( ran @ B @ A
@ ^ [X2: B] : ( none @ A ) )
= ( bot_bot @ ( set @ A ) ) ) ).
% ran_empty
thf(fact_5622_listI__assn__extract,axiom,
! [A: $tType,B: $tType,I: nat,I5: set @ nat,Xs2: list @ A,A3: A > B > assn,Xsi: list @ B] :
( ( member @ nat @ I @ I5 )
=> ( ( ord_less @ nat @ I @ ( size_size @ ( list @ A ) @ Xs2 ) )
=> ( ( vEBT_List_listI_assn @ A @ B @ I5 @ A3 @ Xs2 @ Xsi )
= ( times_times @ assn @ ( A3 @ ( nth @ A @ Xs2 @ I ) @ ( nth @ B @ Xsi @ I ) ) @ ( vEBT_List_listI_assn @ A @ B @ ( minus_minus @ ( set @ nat ) @ I5 @ ( insert @ nat @ I @ ( bot_bot @ ( set @ nat ) ) ) ) @ A3 @ Xs2 @ Xsi ) ) ) ) ) ).
% listI_assn_extract
thf(fact_5623_setceilmax,axiom,
! [S: vEBT_VEBT,M: nat,Listy: list @ vEBT_VEBT,N3: nat] :
( ( vEBT_invar_vebt @ S @ M )
=> ( ! [X3: vEBT_VEBT] :
( ( member @ vEBT_VEBT @ X3 @ ( set2 @ vEBT_VEBT @ Listy ) )
=> ( vEBT_invar_vebt @ X3 @ N3 ) )
=> ( ( M
= ( suc @ N3 ) )
=> ( ! [X3: vEBT_VEBT] :
( ( member @ vEBT_VEBT @ X3 @ ( set2 @ vEBT_VEBT @ Listy ) )
=> ( ( semiring_1_of_nat @ int @ ( vEBT_VEBT_height @ X3 ) )
= ( archimedean_ceiling @ real @ ( log @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) @ ( semiring_1_of_nat @ real @ N3 ) ) ) ) )
=> ( ( ( semiring_1_of_nat @ int @ ( vEBT_VEBT_height @ S ) )
= ( archimedean_ceiling @ real @ ( log @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) @ ( semiring_1_of_nat @ real @ M ) ) ) )
=> ( ( semiring_1_of_nat @ int @ ( lattic643756798349783984er_Max @ nat @ ( image @ vEBT_VEBT @ nat @ vEBT_VEBT_height @ ( insert @ vEBT_VEBT @ S @ ( set2 @ vEBT_VEBT @ Listy ) ) ) ) )
= ( archimedean_ceiling @ real @ ( log @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) @ ( semiring_1_of_nat @ real @ M ) ) ) ) ) ) ) ) ) ).
% setceilmax
thf(fact_5624_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_5625_max__ins__scaled,axiom,
! [N3: nat,X14: vEBT_VEBT,M: nat,X13: list @ vEBT_VEBT] : ( ord_less_eq @ nat @ ( times_times @ nat @ N3 @ ( vEBT_VEBT_height @ X14 ) ) @ ( plus_plus @ nat @ M @ ( times_times @ nat @ N3 @ ( 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_5626_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_5627_image__add__0,axiom,
! [A: $tType] :
( ( comm_monoid_add @ A )
=> ! [S3: set @ A] :
( ( image @ A @ A @ ( plus_plus @ A @ ( zero_zero @ A ) ) @ S3 )
= S3 ) ) ).
% image_add_0
thf(fact_5628_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_5629_image__diff__atLeastAtMost,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [D: A,A2: A,B2: A] :
( ( image @ A @ A @ ( minus_minus @ A @ D ) @ ( set_or1337092689740270186AtMost @ A @ A2 @ B2 ) )
= ( set_or1337092689740270186AtMost @ A @ ( minus_minus @ A @ D @ B2 ) @ ( minus_minus @ A @ D @ A2 ) ) ) ) ).
% image_diff_atLeastAtMost
thf(fact_5630_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_5631_image__add__atMost,axiom,
! [A: $tType] :
( ( ordered_ab_group_add @ A )
=> ! [C2: A,A2: A] :
( ( image @ A @ A @ ( plus_plus @ A @ C2 ) @ ( set_ord_atMost @ A @ A2 ) )
= ( set_ord_atMost @ A @ ( plus_plus @ A @ C2 @ A2 ) ) ) ) ).
% image_add_atMost
thf(fact_5632_max__idx__list,axiom,
! [I: nat,X13: list @ vEBT_VEBT,N3: nat,X14: vEBT_VEBT] :
( ( ord_less @ nat @ I @ ( size_size @ ( list @ vEBT_VEBT ) @ X13 ) )
=> ( ord_less_eq @ nat @ ( times_times @ nat @ N3 @ ( vEBT_VEBT_height @ ( nth @ vEBT_VEBT @ X13 @ I ) ) ) @ ( suc @ ( suc @ ( times_times @ nat @ N3 @ ( 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_5633_image__add__atLeastAtMost_H,axiom,
! [A: $tType] :
( ( linordered_semidom @ A )
=> ! [K: A,I: A,J: A] :
( ( image @ A @ A
@ ^ [N2: A] : ( plus_plus @ A @ N2 @ K )
@ ( set_or1337092689740270186AtMost @ A @ I @ J ) )
= ( set_or1337092689740270186AtMost @ A @ ( plus_plus @ A @ I @ K ) @ ( plus_plus @ A @ J @ K ) ) ) ) ).
% image_add_atLeastAtMost'
thf(fact_5634_image__minus__const__atLeastAtMost_H,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [D: A,A2: A,B2: A] :
( ( image @ A @ A
@ ^ [T3: A] : ( minus_minus @ A @ T3 @ D )
@ ( set_or1337092689740270186AtMost @ A @ A2 @ B2 ) )
= ( set_or1337092689740270186AtMost @ A @ ( minus_minus @ A @ A2 @ D ) @ ( minus_minus @ A @ B2 @ D ) ) ) ) ).
% image_minus_const_atLeastAtMost'
thf(fact_5635_Max_Obounded__iff,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A3: set @ A,X: A] :
( ( finite_finite2 @ A @ A3 )
=> ( ( A3
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( ord_less_eq @ A @ ( lattic643756798349783984er_Max @ A @ A3 ) @ X )
= ( ! [X2: A] :
( ( member @ A @ X2 @ A3 )
=> ( ord_less_eq @ A @ X2 @ X ) ) ) ) ) ) ) ).
% Max.bounded_iff
thf(fact_5636_Max__less__iff,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A3: set @ A,X: A] :
( ( finite_finite2 @ A @ A3 )
=> ( ( A3
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( ord_less @ A @ ( lattic643756798349783984er_Max @ A @ A3 ) @ X )
= ( ! [X2: A] :
( ( member @ A @ X2 @ A3 )
=> ( ord_less @ A @ X2 @ X ) ) ) ) ) ) ) ).
% Max_less_iff
thf(fact_5637_image__mult__atLeastAtMost,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [D: A,A2: A,B2: A] :
( ( ord_less @ A @ ( zero_zero @ A ) @ D )
=> ( ( image @ A @ A @ ( times_times @ A @ D ) @ ( set_or1337092689740270186AtMost @ A @ A2 @ B2 ) )
= ( set_or1337092689740270186AtMost @ A @ ( times_times @ A @ D @ A2 ) @ ( times_times @ A @ D @ B2 ) ) ) ) ) ).
% image_mult_atLeastAtMost
thf(fact_5638_image__divide__atLeastAtMost,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [D: A,A2: A,B2: A] :
( ( ord_less @ A @ ( zero_zero @ A ) @ D )
=> ( ( image @ A @ A
@ ^ [C3: A] : ( divide_divide @ A @ C3 @ D )
@ ( set_or1337092689740270186AtMost @ A @ A2 @ B2 ) )
= ( set_or1337092689740270186AtMost @ A @ ( divide_divide @ A @ A2 @ D ) @ ( divide_divide @ A @ B2 @ D ) ) ) ) ) ).
% image_divide_atLeastAtMost
thf(fact_5639_Max__add__commute,axiom,
! [B: $tType,A: $tType] :
( ( linord4140545234300271783up_add @ A )
=> ! [S3: set @ B,F2: B > A,K: A] :
( ( finite_finite2 @ B @ S3 )
=> ( ( S3
!= ( bot_bot @ ( set @ B ) ) )
=> ( ( lattic643756798349783984er_Max @ A
@ ( image @ B @ A
@ ^ [X2: B] : ( plus_plus @ A @ ( F2 @ X2 ) @ K )
@ S3 ) )
= ( plus_plus @ A @ ( lattic643756798349783984er_Max @ A @ ( image @ B @ A @ F2 @ S3 ) ) @ K ) ) ) ) ) ).
% Max_add_commute
thf(fact_5640_image__diff__subset,axiom,
! [A: $tType,B: $tType,F2: B > A,A3: set @ B,B6: set @ B] : ( ord_less_eq @ ( set @ A ) @ ( minus_minus @ ( set @ A ) @ ( image @ B @ A @ F2 @ A3 ) @ ( image @ B @ A @ F2 @ B6 ) ) @ ( image @ B @ A @ F2 @ ( minus_minus @ ( set @ B ) @ A3 @ B6 ) ) ) ).
% image_diff_subset
thf(fact_5641_image__set,axiom,
! [A: $tType,B: $tType,F2: B > A,Xs2: list @ B] :
( ( image @ B @ A @ F2 @ ( set2 @ B @ Xs2 ) )
= ( set2 @ A @ ( map @ B @ A @ F2 @ Xs2 ) ) ) ).
% image_set
thf(fact_5642_image__mono,axiom,
! [B: $tType,A: $tType,A3: set @ A,B6: set @ A,F2: A > B] :
( ( ord_less_eq @ ( set @ A ) @ A3 @ B6 )
=> ( ord_less_eq @ ( set @ B ) @ ( image @ A @ B @ F2 @ A3 ) @ ( image @ A @ B @ F2 @ B6 ) ) ) ).
% image_mono
thf(fact_5643_image__subsetI,axiom,
! [A: $tType,B: $tType,A3: set @ A,F2: A > B,B6: set @ B] :
( ! [X3: A] :
( ( member @ A @ X3 @ A3 )
=> ( member @ B @ ( F2 @ X3 ) @ B6 ) )
=> ( ord_less_eq @ ( set @ B ) @ ( image @ A @ B @ F2 @ A3 ) @ B6 ) ) ).
% image_subsetI
thf(fact_5644_subset__imageE,axiom,
! [A: $tType,B: $tType,B6: set @ A,F2: B > A,A3: set @ B] :
( ( ord_less_eq @ ( set @ A ) @ B6 @ ( image @ B @ A @ F2 @ A3 ) )
=> ~ ! [C7: set @ B] :
( ( ord_less_eq @ ( set @ B ) @ C7 @ A3 )
=> ( B6
!= ( image @ B @ A @ F2 @ C7 ) ) ) ) ).
% subset_imageE
thf(fact_5645_image__subset__iff,axiom,
! [A: $tType,B: $tType,F2: B > A,A3: set @ B,B6: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ ( image @ B @ A @ F2 @ A3 ) @ B6 )
= ( ! [X2: B] :
( ( member @ B @ X2 @ A3 )
=> ( member @ A @ ( F2 @ X2 ) @ B6 ) ) ) ) ).
% image_subset_iff
thf(fact_5646_subset__image__iff,axiom,
! [A: $tType,B: $tType,B6: set @ A,F2: B > A,A3: set @ B] :
( ( ord_less_eq @ ( set @ A ) @ B6 @ ( image @ B @ A @ F2 @ A3 ) )
= ( ? [AA: set @ B] :
( ( ord_less_eq @ ( set @ B ) @ AA @ A3 )
& ( B6
= ( image @ B @ A @ F2 @ AA ) ) ) ) ) ).
% subset_image_iff
thf(fact_5647_all__subset__image,axiom,
! [A: $tType,B: $tType,F2: B > A,A3: set @ B,P: ( set @ A ) > $o] :
( ( ! [B7: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ B7 @ ( image @ B @ A @ F2 @ A3 ) )
=> ( P @ B7 ) ) )
= ( ! [B7: set @ B] :
( ( ord_less_eq @ ( set @ B ) @ B7 @ A3 )
=> ( P @ ( image @ B @ A @ F2 @ B7 ) ) ) ) ) ).
% all_subset_image
thf(fact_5648_finite__surj,axiom,
! [A: $tType,B: $tType,A3: set @ A,B6: set @ B,F2: A > B] :
( ( finite_finite2 @ A @ A3 )
=> ( ( ord_less_eq @ ( set @ B ) @ B6 @ ( image @ A @ B @ F2 @ A3 ) )
=> ( finite_finite2 @ B @ B6 ) ) ) ).
% finite_surj
thf(fact_5649_finite__subset__image,axiom,
! [A: $tType,B: $tType,B6: set @ A,F2: B > A,A3: set @ B] :
( ( finite_finite2 @ A @ B6 )
=> ( ( ord_less_eq @ ( set @ A ) @ B6 @ ( image @ B @ A @ F2 @ A3 ) )
=> ? [C7: set @ B] :
( ( ord_less_eq @ ( set @ B ) @ C7 @ A3 )
& ( finite_finite2 @ B @ C7 )
& ( B6
= ( image @ B @ A @ F2 @ C7 ) ) ) ) ) ).
% finite_subset_image
thf(fact_5650_ex__finite__subset__image,axiom,
! [A: $tType,B: $tType,F2: B > A,A3: set @ B,P: ( set @ A ) > $o] :
( ( ? [B7: set @ A] :
( ( finite_finite2 @ A @ B7 )
& ( ord_less_eq @ ( set @ A ) @ B7 @ ( image @ B @ A @ F2 @ A3 ) )
& ( P @ B7 ) ) )
= ( ? [B7: set @ B] :
( ( finite_finite2 @ B @ B7 )
& ( ord_less_eq @ ( set @ B ) @ B7 @ A3 )
& ( P @ ( image @ B @ A @ F2 @ B7 ) ) ) ) ) ).
% ex_finite_subset_image
thf(fact_5651_all__finite__subset__image,axiom,
! [A: $tType,B: $tType,F2: B > A,A3: set @ B,P: ( set @ A ) > $o] :
( ( ! [B7: set @ A] :
( ( ( finite_finite2 @ A @ B7 )
& ( ord_less_eq @ ( set @ A ) @ B7 @ ( image @ B @ A @ F2 @ A3 ) ) )
=> ( P @ B7 ) ) )
= ( ! [B7: set @ B] :
( ( ( finite_finite2 @ B @ B7 )
& ( ord_less_eq @ ( set @ B ) @ B7 @ A3 ) )
=> ( P @ ( image @ B @ A @ F2 @ B7 ) ) ) ) ) ).
% all_finite_subset_image
thf(fact_5652_Max_OcoboundedI,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A3: set @ A,A2: A] :
( ( finite_finite2 @ A @ A3 )
=> ( ( member @ A @ A2 @ A3 )
=> ( ord_less_eq @ A @ A2 @ ( lattic643756798349783984er_Max @ A @ A3 ) ) ) ) ) ).
% Max.coboundedI
thf(fact_5653_Max__eq__if,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A3: set @ A,B6: set @ A] :
( ( finite_finite2 @ A @ A3 )
=> ( ( finite_finite2 @ A @ B6 )
=> ( ! [X3: A] :
( ( member @ A @ X3 @ A3 )
=> ? [Xa2: A] :
( ( member @ A @ Xa2 @ B6 )
& ( ord_less_eq @ A @ X3 @ Xa2 ) ) )
=> ( ! [X3: A] :
( ( member @ A @ X3 @ B6 )
=> ? [Xa2: A] :
( ( member @ A @ Xa2 @ A3 )
& ( ord_less_eq @ A @ X3 @ Xa2 ) ) )
=> ( ( lattic643756798349783984er_Max @ A @ A3 )
= ( lattic643756798349783984er_Max @ A @ B6 ) ) ) ) ) ) ) ).
% Max_eq_if
thf(fact_5654_Max__eqI,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A3: set @ A,X: A] :
( ( finite_finite2 @ A @ A3 )
=> ( ! [Y3: A] :
( ( member @ A @ Y3 @ A3 )
=> ( ord_less_eq @ A @ Y3 @ X ) )
=> ( ( member @ A @ X @ A3 )
=> ( ( lattic643756798349783984er_Max @ A @ A3 )
= X ) ) ) ) ) ).
% Max_eqI
thf(fact_5655_Max__ge,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A3: set @ A,X: A] :
( ( finite_finite2 @ A @ A3 )
=> ( ( member @ A @ X @ A3 )
=> ( ord_less_eq @ A @ X @ ( lattic643756798349783984er_Max @ A @ A3 ) ) ) ) ) ).
% Max_ge
thf(fact_5656_Max_OboundedI,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A3: set @ A,X: A] :
( ( finite_finite2 @ A @ A3 )
=> ( ( A3
!= ( bot_bot @ ( set @ A ) ) )
=> ( ! [A4: A] :
( ( member @ A @ A4 @ A3 )
=> ( ord_less_eq @ A @ A4 @ X ) )
=> ( ord_less_eq @ A @ ( lattic643756798349783984er_Max @ A @ A3 ) @ X ) ) ) ) ) ).
% Max.boundedI
thf(fact_5657_Max_OboundedE,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A3: set @ A,X: A] :
( ( finite_finite2 @ A @ A3 )
=> ( ( A3
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( ord_less_eq @ A @ ( lattic643756798349783984er_Max @ A @ A3 ) @ X )
=> ! [A10: A] :
( ( member @ A @ A10 @ A3 )
=> ( ord_less_eq @ A @ A10 @ X ) ) ) ) ) ) ).
% Max.boundedE
thf(fact_5658_eq__Max__iff,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A3: set @ A,M: A] :
( ( finite_finite2 @ A @ A3 )
=> ( ( A3
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( M
= ( lattic643756798349783984er_Max @ A @ A3 ) )
= ( ( member @ A @ M @ A3 )
& ! [X2: A] :
( ( member @ A @ X2 @ A3 )
=> ( ord_less_eq @ A @ X2 @ M ) ) ) ) ) ) ) ).
% eq_Max_iff
thf(fact_5659_Max__ge__iff,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A3: set @ A,X: A] :
( ( finite_finite2 @ A @ A3 )
=> ( ( A3
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( ord_less_eq @ A @ X @ ( lattic643756798349783984er_Max @ A @ A3 ) )
= ( ? [X2: A] :
( ( member @ A @ X2 @ A3 )
& ( ord_less_eq @ A @ X @ X2 ) ) ) ) ) ) ) ).
% Max_ge_iff
thf(fact_5660_Max__eq__iff,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A3: set @ A,M: A] :
( ( finite_finite2 @ A @ A3 )
=> ( ( A3
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( ( lattic643756798349783984er_Max @ A @ A3 )
= M )
= ( ( member @ A @ M @ A3 )
& ! [X2: A] :
( ( member @ A @ X2 @ A3 )
=> ( ord_less_eq @ A @ X2 @ M ) ) ) ) ) ) ) ).
% Max_eq_iff
thf(fact_5661_Max__gr__iff,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A3: set @ A,X: A] :
( ( finite_finite2 @ A @ A3 )
=> ( ( A3
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( ord_less @ A @ X @ ( lattic643756798349783984er_Max @ A @ A3 ) )
= ( ? [X2: A] :
( ( member @ A @ X2 @ A3 )
& ( ord_less @ A @ X @ X2 ) ) ) ) ) ) ) ).
% Max_gr_iff
thf(fact_5662_Max__insert2,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A3: set @ A,A2: A] :
( ( finite_finite2 @ A @ A3 )
=> ( ! [B3: A] :
( ( member @ A @ B3 @ A3 )
=> ( ord_less_eq @ A @ B3 @ A2 ) )
=> ( ( lattic643756798349783984er_Max @ A @ ( insert @ A @ A2 @ A3 ) )
= A2 ) ) ) ) ).
% Max_insert2
thf(fact_5663_VEBT__internal_Oheight_Osimps_I2_J,axiom,
! [Uu2: option @ ( product_prod @ nat @ nat ),Deg: nat,TreeList: list @ vEBT_VEBT,Summary: vEBT_VEBT] :
( ( vEBT_VEBT_height @ ( vEBT_Node @ Uu2 @ Deg @ TreeList @ Summary ) )
= ( plus_plus @ nat @ ( one_one @ nat ) @ ( lattic643756798349783984er_Max @ nat @ ( image @ vEBT_VEBT @ nat @ vEBT_VEBT_height @ ( insert @ vEBT_VEBT @ Summary @ ( set2 @ vEBT_VEBT @ TreeList ) ) ) ) ) ) ).
% VEBT_internal.height.simps(2)
thf(fact_5664_Max_Oinfinite,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A3: set @ A] :
( ~ ( finite_finite2 @ A @ A3 )
=> ( ( lattic643756798349783984er_Max @ A @ A3 )
= ( the2 @ A @ ( none @ A ) ) ) ) ) ).
% Max.infinite
thf(fact_5665_sum_Ogroup,axiom,
! [C: $tType,A: $tType,B: $tType] :
( ( comm_monoid_add @ A )
=> ! [S3: set @ B,T5: set @ C,G: B > C,H2: B > A] :
( ( finite_finite2 @ B @ S3 )
=> ( ( finite_finite2 @ C @ T5 )
=> ( ( ord_less_eq @ ( set @ C ) @ ( image @ B @ C @ G @ S3 ) @ T5 )
=> ( ( groups7311177749621191930dd_sum @ C @ A
@ ^ [Y2: C] :
( groups7311177749621191930dd_sum @ B @ A @ H2
@ ( collect @ B
@ ^ [X2: B] :
( ( member @ B @ X2 @ S3 )
& ( ( G @ X2 )
= Y2 ) ) ) )
@ T5 )
= ( groups7311177749621191930dd_sum @ B @ A @ H2 @ S3 ) ) ) ) ) ) ).
% sum.group
thf(fact_5666_prod_Ogroup,axiom,
! [C: $tType,A: $tType,B: $tType] :
( ( comm_monoid_mult @ A )
=> ! [S3: set @ B,T5: set @ C,G: B > C,H2: B > A] :
( ( finite_finite2 @ B @ S3 )
=> ( ( finite_finite2 @ C @ T5 )
=> ( ( ord_less_eq @ ( set @ C ) @ ( image @ B @ C @ G @ S3 ) @ T5 )
=> ( ( groups7121269368397514597t_prod @ C @ A
@ ^ [Y2: C] :
( groups7121269368397514597t_prod @ B @ A @ H2
@ ( collect @ B
@ ^ [X2: B] :
( ( member @ B @ X2 @ S3 )
& ( ( G @ X2 )
= Y2 ) ) ) )
@ T5 )
= ( groups7121269368397514597t_prod @ B @ A @ H2 @ S3 ) ) ) ) ) ) ).
% prod.group
thf(fact_5667_listI__assn__weak__cong,axiom,
! [A: $tType,B: $tType,I5: set @ nat,I7: set @ nat,A3: A > B > assn,A9: A > B > assn,Xs2: list @ A,Xs4: list @ A,Xsi: list @ B,Xsi2: list @ B] :
( ( I5 = I7 )
=> ( ( A3 = A9 )
=> ( ( ( size_size @ ( list @ A ) @ Xs2 )
= ( size_size @ ( list @ A ) @ Xs4 ) )
=> ( ( ( size_size @ ( list @ B ) @ Xsi )
= ( size_size @ ( list @ B ) @ Xsi2 ) )
=> ( ! [I2: nat] :
( ( member @ nat @ I2 @ I5 )
=> ( ( ord_less @ nat @ I2 @ ( size_size @ ( list @ A ) @ Xs2 ) )
=> ( ( ( size_size @ ( list @ A ) @ Xs2 )
= ( size_size @ ( list @ B ) @ Xsi ) )
=> ( ( ( nth @ A @ Xs2 @ I2 )
= ( nth @ A @ Xs4 @ I2 ) )
& ( ( nth @ B @ Xsi @ I2 )
= ( nth @ B @ Xsi2 @ I2 ) ) ) ) ) )
=> ( ( vEBT_List_listI_assn @ A @ B @ I5 @ A3 @ Xs2 @ Xsi )
= ( vEBT_List_listI_assn @ A @ B @ I7 @ A9 @ Xs4 @ Xsi2 ) ) ) ) ) ) ) ).
% listI_assn_weak_cong
thf(fact_5668_listI__assn__cong,axiom,
! [A: $tType,B: $tType,I5: set @ nat,I7: set @ nat,Xs2: list @ A,Xs4: list @ A,Xsi: list @ B,Xsi2: list @ B,A3: A > B > assn,A9: A > B > assn] :
( ( I5 = I7 )
=> ( ( ( size_size @ ( list @ A ) @ Xs2 )
= ( size_size @ ( list @ A ) @ Xs4 ) )
=> ( ( ( size_size @ ( list @ B ) @ Xsi )
= ( size_size @ ( list @ B ) @ Xsi2 ) )
=> ( ! [I2: nat] :
( ( member @ nat @ I2 @ I5 )
=> ( ( ord_less @ nat @ I2 @ ( size_size @ ( list @ A ) @ Xs2 ) )
=> ( ( ( size_size @ ( list @ A ) @ Xs2 )
= ( size_size @ ( list @ B ) @ Xsi ) )
=> ( ( ( nth @ A @ Xs2 @ I2 )
= ( nth @ A @ Xs4 @ I2 ) )
& ( ( nth @ B @ Xsi @ I2 )
= ( nth @ B @ Xsi2 @ I2 ) )
& ( ( A3 @ ( nth @ A @ Xs2 @ I2 ) @ ( nth @ B @ Xsi @ I2 ) )
= ( A9 @ ( nth @ A @ Xs4 @ I2 ) @ ( nth @ B @ Xsi2 @ I2 ) ) ) ) ) ) )
=> ( ( vEBT_List_listI_assn @ A @ B @ I5 @ A3 @ Xs2 @ Xsi )
= ( vEBT_List_listI_assn @ A @ B @ I7 @ A9 @ Xs4 @ Xsi2 ) ) ) ) ) ) ).
% listI_assn_cong
thf(fact_5669_subst__not__in,axiom,
! [A: $tType,B: $tType,I: nat,I5: set @ nat,Xs2: list @ A,A3: A > B > assn,X15: A,Xsi: list @ B,X22: B] :
( ~ ( member @ nat @ I @ I5 )
=> ( ( ord_less @ nat @ I @ ( size_size @ ( list @ A ) @ Xs2 ) )
=> ( ( vEBT_List_listI_assn @ A @ B @ I5 @ A3 @ ( list_update @ A @ Xs2 @ I @ X15 ) @ ( list_update @ B @ Xsi @ I @ X22 ) )
= ( vEBT_List_listI_assn @ A @ B @ I5 @ A3 @ Xs2 @ Xsi ) ) ) ) ).
% subst_not_in
thf(fact_5670_set__image__eq__pointwiseI,axiom,
! [B: $tType,A: $tType,L2: list @ A,L4: list @ A,F2: A > B] :
( ( ( size_size @ ( list @ A ) @ L2 )
= ( size_size @ ( list @ A ) @ L4 ) )
=> ( ! [I2: nat] :
( ( ord_less @ nat @ I2 @ ( size_size @ ( list @ A ) @ L2 ) )
=> ( ( F2 @ ( nth @ A @ L2 @ I2 ) )
= ( F2 @ ( nth @ A @ L4 @ I2 ) ) ) )
=> ( ( image @ A @ B @ F2 @ ( set2 @ A @ L2 ) )
= ( image @ A @ B @ F2 @ ( set2 @ A @ L4 ) ) ) ) ) ).
% set_image_eq_pointwiseI
thf(fact_5671_in__set__image__conv__nth,axiom,
! [B: $tType,A: $tType,F2: B > A,X: B,L2: list @ B] :
( ( member @ A @ ( F2 @ X ) @ ( image @ B @ A @ F2 @ ( set2 @ B @ L2 ) ) )
= ( ? [I3: nat] :
( ( ord_less @ nat @ I3 @ ( size_size @ ( list @ B ) @ L2 ) )
& ( ( F2 @ ( nth @ B @ L2 @ I3 ) )
= ( F2 @ X ) ) ) ) ) ).
% in_set_image_conv_nth
thf(fact_5672_Max_Osubset__imp,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A3: set @ A,B6: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ A3 @ B6 )
=> ( ( A3
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( finite_finite2 @ A @ B6 )
=> ( ord_less_eq @ A @ ( lattic643756798349783984er_Max @ A @ A3 ) @ ( lattic643756798349783984er_Max @ A @ B6 ) ) ) ) ) ) ).
% Max.subset_imp
thf(fact_5673_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_5674_VEBT__internal_Oheight_Oelims,axiom,
! [X: vEBT_VEBT,Y: nat] :
( ( ( vEBT_VEBT_height @ X )
= Y )
=> ( ( ? [A4: $o,B3: $o] :
( X
= ( vEBT_Leaf @ A4 @ B3 ) )
=> ( Y
!= ( zero_zero @ nat ) ) )
=> ~ ! [Uu: option @ ( product_prod @ nat @ nat ),Deg2: nat,TreeList2: list @ vEBT_VEBT,Summary2: vEBT_VEBT] :
( ( X
= ( vEBT_Node @ Uu @ Deg2 @ TreeList2 @ Summary2 ) )
=> ( Y
!= ( plus_plus @ nat @ ( one_one @ nat ) @ ( lattic643756798349783984er_Max @ nat @ ( image @ vEBT_VEBT @ nat @ vEBT_VEBT_height @ ( insert @ vEBT_VEBT @ Summary2 @ ( set2 @ vEBT_VEBT @ TreeList2 ) ) ) ) ) ) ) ) ) ).
% VEBT_internal.height.elims
thf(fact_5675_Max_Osubset,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A3: set @ A,B6: set @ A] :
( ( finite_finite2 @ A @ A3 )
=> ( ( B6
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( ord_less_eq @ ( set @ A ) @ B6 @ A3 )
=> ( ( ord_max @ A @ ( lattic643756798349783984er_Max @ A @ B6 ) @ ( lattic643756798349783984er_Max @ A @ A3 ) )
= ( lattic643756798349783984er_Max @ A @ A3 ) ) ) ) ) ) ).
% Max.subset
thf(fact_5676_divide__nat__def,axiom,
( ( divide_divide @ nat )
= ( ^ [M5: nat,N2: nat] :
( if @ nat
@ ( N2
= ( zero_zero @ nat ) )
@ ( zero_zero @ nat )
@ ( lattic643756798349783984er_Max @ nat
@ ( collect @ nat
@ ^ [K2: nat] : ( ord_less_eq @ nat @ ( times_times @ nat @ K2 @ N2 ) @ M5 ) ) ) ) ) ) ).
% divide_nat_def
thf(fact_5677_Max_Oremove,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A3: set @ A,X: A] :
( ( finite_finite2 @ A @ A3 )
=> ( ( member @ A @ X @ A3 )
=> ( ( ( ( minus_minus @ ( set @ A ) @ A3 @ ( insert @ A @ X @ ( bot_bot @ ( set @ A ) ) ) )
= ( bot_bot @ ( set @ A ) ) )
=> ( ( lattic643756798349783984er_Max @ A @ A3 )
= X ) )
& ( ( ( minus_minus @ ( set @ A ) @ A3 @ ( insert @ A @ X @ ( bot_bot @ ( set @ A ) ) ) )
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( lattic643756798349783984er_Max @ A @ A3 )
= ( ord_max @ A @ X @ ( lattic643756798349783984er_Max @ A @ ( minus_minus @ ( set @ A ) @ A3 @ ( insert @ A @ X @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ) ) ) ) ) ) ).
% Max.remove
thf(fact_5678_Max_Oinsert__remove,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A3: set @ A,X: A] :
( ( finite_finite2 @ A @ A3 )
=> ( ( ( ( minus_minus @ ( set @ A ) @ A3 @ ( insert @ A @ X @ ( bot_bot @ ( set @ A ) ) ) )
= ( bot_bot @ ( set @ A ) ) )
=> ( ( lattic643756798349783984er_Max @ A @ ( insert @ A @ X @ A3 ) )
= X ) )
& ( ( ( minus_minus @ ( set @ A ) @ A3 @ ( insert @ A @ X @ ( bot_bot @ ( set @ A ) ) ) )
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( lattic643756798349783984er_Max @ A @ ( insert @ A @ X @ A3 ) )
= ( ord_max @ A @ X @ ( lattic643756798349783984er_Max @ A @ ( minus_minus @ ( set @ A ) @ A3 @ ( insert @ A @ X @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ) ) ) ) ) ).
% Max.insert_remove
thf(fact_5679_image__mult__atLeastAtMost__if,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [C2: A,X: A,Y: A] :
( ( ( ord_less @ A @ ( zero_zero @ A ) @ C2 )
=> ( ( image @ A @ A @ ( times_times @ A @ C2 ) @ ( set_or1337092689740270186AtMost @ A @ X @ Y ) )
= ( set_or1337092689740270186AtMost @ A @ ( times_times @ A @ C2 @ X ) @ ( times_times @ A @ C2 @ Y ) ) ) )
& ( ~ ( ord_less @ A @ ( zero_zero @ A ) @ C2 )
=> ( ( ( ord_less_eq @ A @ X @ Y )
=> ( ( image @ A @ A @ ( times_times @ A @ C2 ) @ ( set_or1337092689740270186AtMost @ A @ X @ Y ) )
= ( set_or1337092689740270186AtMost @ A @ ( times_times @ A @ C2 @ Y ) @ ( times_times @ A @ C2 @ X ) ) ) )
& ( ~ ( ord_less_eq @ A @ X @ Y )
=> ( ( image @ A @ A @ ( times_times @ A @ C2 ) @ ( set_or1337092689740270186AtMost @ A @ X @ Y ) )
= ( bot_bot @ ( set @ A ) ) ) ) ) ) ) ) ).
% image_mult_atLeastAtMost_if
thf(fact_5680_image__mult__atLeastAtMost__if_H,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [X: A,Y: A,C2: A] :
( ( ( ord_less_eq @ A @ X @ Y )
=> ( ( ( ord_less @ A @ ( zero_zero @ A ) @ C2 )
=> ( ( image @ A @ A
@ ^ [X2: A] : ( times_times @ A @ X2 @ C2 )
@ ( set_or1337092689740270186AtMost @ A @ X @ Y ) )
= ( set_or1337092689740270186AtMost @ A @ ( times_times @ A @ X @ C2 ) @ ( times_times @ A @ Y @ C2 ) ) ) )
& ( ~ ( ord_less @ A @ ( zero_zero @ A ) @ C2 )
=> ( ( image @ A @ A
@ ^ [X2: A] : ( times_times @ A @ X2 @ C2 )
@ ( set_or1337092689740270186AtMost @ A @ X @ Y ) )
= ( set_or1337092689740270186AtMost @ A @ ( times_times @ A @ Y @ C2 ) @ ( times_times @ A @ X @ C2 ) ) ) ) ) )
& ( ~ ( ord_less_eq @ A @ X @ Y )
=> ( ( image @ A @ A
@ ^ [X2: A] : ( times_times @ A @ X2 @ C2 )
@ ( set_or1337092689740270186AtMost @ A @ X @ Y ) )
= ( bot_bot @ ( set @ A ) ) ) ) ) ) ).
% image_mult_atLeastAtMost_if'
thf(fact_5681_image__affinity__atLeastAtMost,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [A2: A,B2: A,M: A,C2: A] :
( ( ( ( set_or1337092689740270186AtMost @ A @ A2 @ B2 )
= ( bot_bot @ ( set @ A ) ) )
=> ( ( image @ A @ A
@ ^ [X2: A] : ( plus_plus @ A @ ( times_times @ A @ M @ X2 ) @ C2 )
@ ( set_or1337092689740270186AtMost @ A @ A2 @ B2 ) )
= ( bot_bot @ ( set @ A ) ) ) )
& ( ( ( set_or1337092689740270186AtMost @ A @ A2 @ B2 )
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ M )
=> ( ( image @ A @ A
@ ^ [X2: A] : ( plus_plus @ A @ ( times_times @ A @ M @ X2 ) @ C2 )
@ ( set_or1337092689740270186AtMost @ A @ A2 @ B2 ) )
= ( set_or1337092689740270186AtMost @ A @ ( plus_plus @ A @ ( times_times @ A @ M @ A2 ) @ C2 ) @ ( plus_plus @ A @ ( times_times @ A @ M @ B2 ) @ C2 ) ) ) )
& ( ~ ( ord_less_eq @ A @ ( zero_zero @ A ) @ M )
=> ( ( image @ A @ A
@ ^ [X2: A] : ( plus_plus @ A @ ( times_times @ A @ M @ X2 ) @ C2 )
@ ( set_or1337092689740270186AtMost @ A @ A2 @ B2 ) )
= ( set_or1337092689740270186AtMost @ A @ ( plus_plus @ A @ ( times_times @ A @ M @ B2 ) @ C2 ) @ ( plus_plus @ A @ ( times_times @ A @ M @ A2 ) @ C2 ) ) ) ) ) ) ) ) ).
% image_affinity_atLeastAtMost
thf(fact_5682_image__affinity__atLeastAtMost__diff,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [A2: A,B2: A,M: A,C2: A] :
( ( ( ( set_or1337092689740270186AtMost @ A @ A2 @ B2 )
= ( bot_bot @ ( set @ A ) ) )
=> ( ( image @ A @ A
@ ^ [X2: A] : ( minus_minus @ A @ ( times_times @ A @ M @ X2 ) @ C2 )
@ ( set_or1337092689740270186AtMost @ A @ A2 @ B2 ) )
= ( bot_bot @ ( set @ A ) ) ) )
& ( ( ( set_or1337092689740270186AtMost @ A @ A2 @ B2 )
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ M )
=> ( ( image @ A @ A
@ ^ [X2: A] : ( minus_minus @ A @ ( times_times @ A @ M @ X2 ) @ C2 )
@ ( set_or1337092689740270186AtMost @ A @ A2 @ B2 ) )
= ( set_or1337092689740270186AtMost @ A @ ( minus_minus @ A @ ( times_times @ A @ M @ A2 ) @ C2 ) @ ( minus_minus @ A @ ( times_times @ A @ M @ B2 ) @ C2 ) ) ) )
& ( ~ ( ord_less_eq @ A @ ( zero_zero @ A ) @ M )
=> ( ( image @ A @ A
@ ^ [X2: A] : ( minus_minus @ A @ ( times_times @ A @ M @ X2 ) @ C2 )
@ ( set_or1337092689740270186AtMost @ A @ A2 @ B2 ) )
= ( set_or1337092689740270186AtMost @ A @ ( minus_minus @ A @ ( times_times @ A @ M @ B2 ) @ C2 ) @ ( minus_minus @ A @ ( times_times @ A @ M @ A2 ) @ C2 ) ) ) ) ) ) ) ) ).
% image_affinity_atLeastAtMost_diff
thf(fact_5683_image__affinity__atLeastAtMost__div,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [A2: A,B2: A,M: A,C2: A] :
( ( ( ( set_or1337092689740270186AtMost @ A @ A2 @ B2 )
= ( bot_bot @ ( set @ A ) ) )
=> ( ( image @ A @ A
@ ^ [X2: A] : ( plus_plus @ A @ ( divide_divide @ A @ X2 @ M ) @ C2 )
@ ( set_or1337092689740270186AtMost @ A @ A2 @ B2 ) )
= ( bot_bot @ ( set @ A ) ) ) )
& ( ( ( set_or1337092689740270186AtMost @ A @ A2 @ B2 )
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ M )
=> ( ( image @ A @ A
@ ^ [X2: A] : ( plus_plus @ A @ ( divide_divide @ A @ X2 @ M ) @ C2 )
@ ( set_or1337092689740270186AtMost @ A @ A2 @ B2 ) )
= ( set_or1337092689740270186AtMost @ A @ ( plus_plus @ A @ ( divide_divide @ A @ A2 @ M ) @ C2 ) @ ( plus_plus @ A @ ( divide_divide @ A @ B2 @ M ) @ C2 ) ) ) )
& ( ~ ( ord_less_eq @ A @ ( zero_zero @ A ) @ M )
=> ( ( image @ A @ A
@ ^ [X2: A] : ( plus_plus @ A @ ( divide_divide @ A @ X2 @ M ) @ C2 )
@ ( set_or1337092689740270186AtMost @ A @ A2 @ B2 ) )
= ( set_or1337092689740270186AtMost @ A @ ( plus_plus @ A @ ( divide_divide @ A @ B2 @ M ) @ C2 ) @ ( plus_plus @ A @ ( divide_divide @ A @ A2 @ M ) @ C2 ) ) ) ) ) ) ) ) ).
% image_affinity_atLeastAtMost_div
thf(fact_5684_image__affinity__atLeastAtMost__div__diff,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [A2: A,B2: A,M: A,C2: A] :
( ( ( ( set_or1337092689740270186AtMost @ A @ A2 @ B2 )
= ( bot_bot @ ( set @ A ) ) )
=> ( ( image @ A @ A
@ ^ [X2: A] : ( minus_minus @ A @ ( divide_divide @ A @ X2 @ M ) @ C2 )
@ ( set_or1337092689740270186AtMost @ A @ A2 @ B2 ) )
= ( bot_bot @ ( set @ A ) ) ) )
& ( ( ( set_or1337092689740270186AtMost @ A @ A2 @ B2 )
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ M )
=> ( ( image @ A @ A
@ ^ [X2: A] : ( minus_minus @ A @ ( divide_divide @ A @ X2 @ M ) @ C2 )
@ ( set_or1337092689740270186AtMost @ A @ A2 @ B2 ) )
= ( set_or1337092689740270186AtMost @ A @ ( minus_minus @ A @ ( divide_divide @ A @ A2 @ M ) @ C2 ) @ ( minus_minus @ A @ ( divide_divide @ A @ B2 @ M ) @ C2 ) ) ) )
& ( ~ ( ord_less_eq @ A @ ( zero_zero @ A ) @ M )
=> ( ( image @ A @ A
@ ^ [X2: A] : ( minus_minus @ A @ ( divide_divide @ A @ X2 @ M ) @ C2 )
@ ( set_or1337092689740270186AtMost @ A @ A2 @ B2 ) )
= ( set_or1337092689740270186AtMost @ A @ ( minus_minus @ A @ ( divide_divide @ A @ B2 @ M ) @ C2 ) @ ( minus_minus @ A @ ( divide_divide @ A @ A2 @ M ) @ C2 ) ) ) ) ) ) ) ) ).
% image_affinity_atLeastAtMost_div_diff
thf(fact_5685_listI__assn__insert,axiom,
! [A: $tType,B: $tType,I: nat,I5: set @ nat,Xs2: list @ A,A3: A > B > assn,Xsi: list @ B] :
( ~ ( member @ nat @ I @ I5 )
=> ( ( ord_less @ nat @ I @ ( size_size @ ( list @ A ) @ Xs2 ) )
=> ( ( vEBT_List_listI_assn @ A @ B @ ( insert @ nat @ I @ I5 ) @ A3 @ Xs2 @ Xsi )
= ( times_times @ assn @ ( A3 @ ( nth @ A @ Xs2 @ I ) @ ( nth @ B @ Xsi @ I ) ) @ ( vEBT_List_listI_assn @ A @ B @ I5 @ A3 @ Xs2 @ Xsi ) ) ) ) ) ).
% listI_assn_insert
thf(fact_5686_listI__assn__subst,axiom,
! [A: $tType,B: $tType,I: nat,I5: set @ nat,Xs2: list @ A,A3: A > B > assn,X15: A,Xsi: list @ B,X22: B] :
( ~ ( member @ nat @ I @ I5 )
=> ( ( ord_less @ nat @ I @ ( size_size @ ( list @ A ) @ Xs2 ) )
=> ( ( vEBT_List_listI_assn @ A @ B @ ( insert @ nat @ I @ I5 ) @ A3 @ ( list_update @ A @ Xs2 @ I @ X15 ) @ ( list_update @ B @ Xsi @ I @ X22 ) )
= ( times_times @ assn @ ( A3 @ X15 @ X22 ) @ ( vEBT_List_listI_assn @ A @ B @ I5 @ A3 @ Xs2 @ Xsi ) ) ) ) ) ).
% listI_assn_subst
thf(fact_5687_ranI,axiom,
! [A: $tType,B: $tType,M: B > ( option @ A ),A2: B,B2: A] :
( ( ( M @ A2 )
= ( some @ A @ B2 ) )
=> ( member @ A @ B2 @ ( ran @ B @ A @ M ) ) ) ).
% ranI
thf(fact_5688_VEBT__internal_Oheight_Opelims,axiom,
! [X: vEBT_VEBT,Y: nat] :
( ( ( vEBT_VEBT_height @ X )
= Y )
=> ( ( accp @ vEBT_VEBT @ vEBT_VEBT_height_rel @ X )
=> ( ! [A4: $o,B3: $o] :
( ( X
= ( vEBT_Leaf @ A4 @ B3 ) )
=> ( ( Y
= ( zero_zero @ nat ) )
=> ~ ( accp @ vEBT_VEBT @ vEBT_VEBT_height_rel @ ( vEBT_Leaf @ A4 @ B3 ) ) ) )
=> ~ ! [Uu: option @ ( product_prod @ nat @ nat ),Deg2: nat,TreeList2: list @ vEBT_VEBT,Summary2: vEBT_VEBT] :
( ( X
= ( vEBT_Node @ Uu @ Deg2 @ TreeList2 @ Summary2 ) )
=> ( ( Y
= ( plus_plus @ nat @ ( one_one @ nat ) @ ( lattic643756798349783984er_Max @ nat @ ( image @ vEBT_VEBT @ nat @ vEBT_VEBT_height @ ( insert @ vEBT_VEBT @ Summary2 @ ( set2 @ vEBT_VEBT @ TreeList2 ) ) ) ) ) )
=> ~ ( accp @ vEBT_VEBT @ vEBT_VEBT_height_rel @ ( vEBT_Node @ Uu @ Deg2 @ TreeList2 @ Summary2 ) ) ) ) ) ) ) ).
% VEBT_internal.height.pelims
thf(fact_5689_bij__betw__add,axiom,
! [A: $tType] :
( ( cancel_semigroup_add @ A )
=> ! [A2: A,A3: set @ A,B6: set @ A] :
( ( bij_betw @ A @ A @ ( plus_plus @ A @ A2 ) @ A3 @ B6 )
= ( ( image @ A @ A @ ( plus_plus @ A @ A2 ) @ A3 )
= B6 ) ) ) ).
% bij_betw_add
thf(fact_5690_listI__assn__wrap__insert,axiom,
! [E3: $tType,P: assn,Uu2: vEBT_VEBT,Uua: nat,Xi2: vEBT_VEBTi,I5: set @ nat,I: nat,Xs2: list @ vEBT_VEBT,Xsi: list @ vEBT_VEBTi,F4: assn,C2: heap_Time_Heap @ E3,Q: E3 > assn] :
( ( entails @ P @ ( times_times @ assn @ ( times_times @ assn @ ( vEBT_vebt_assn_raw @ ( vEBT_vebt_insert @ Uu2 @ Uua ) @ Xi2 ) @ ( vEBT_List_listI_assn @ vEBT_VEBT @ vEBT_VEBTi @ ( minus_minus @ ( set @ nat ) @ I5 @ ( insert @ nat @ I @ ( bot_bot @ ( set @ nat ) ) ) ) @ vEBT_vebt_assn_raw @ Xs2 @ Xsi ) ) @ F4 ) )
=> ( ( ord_less @ nat @ I @ ( size_size @ ( list @ vEBT_VEBT ) @ Xs2 ) )
=> ( ( member @ nat @ I @ I5 )
=> ( ( hoare_hoare_triple @ E3 @ ( times_times @ assn @ ( vEBT_List_listI_assn @ vEBT_VEBT @ vEBT_VEBTi @ I5 @ vEBT_vebt_assn_raw @ ( list_update @ vEBT_VEBT @ Xs2 @ I @ ( vEBT_vebt_insert @ Uu2 @ Uua ) ) @ ( list_update @ vEBT_VEBTi @ Xsi @ I @ Xi2 ) ) @ F4 ) @ C2 @ Q )
=> ( hoare_hoare_triple @ E3 @ P @ C2 @ Q ) ) ) ) ) ).
% listI_assn_wrap_insert
thf(fact_5691_swappa,axiom,
! [B6: assn,A3: assn,C5: assn,X8: assn] :
( ( entails @ ( times_times @ assn @ ( times_times @ assn @ B6 @ A3 ) @ C5 ) @ X8 )
=> ( entails @ ( times_times @ assn @ ( times_times @ assn @ A3 @ B6 ) @ C5 ) @ X8 ) ) ).
% swappa
thf(fact_5692_midextr,axiom,
! [P: assn,Q: assn,Q6: assn,R: assn,X8: assn] :
( ( entails @ ( times_times @ assn @ ( times_times @ assn @ ( times_times @ assn @ P @ Q ) @ Q6 ) @ R ) @ X8 )
=> ( entails @ ( times_times @ assn @ ( times_times @ assn @ ( times_times @ assn @ P @ R ) @ Q ) @ Q6 ) @ X8 ) ) ).
% midextr
thf(fact_5693_groupy,axiom,
! [A3: assn,B6: assn,C5: assn,D4: assn,X8: assn] :
( ( entails @ ( times_times @ assn @ ( times_times @ assn @ A3 @ B6 ) @ ( times_times @ assn @ C5 @ D4 ) ) @ X8 )
=> ( entails @ ( times_times @ assn @ ( times_times @ assn @ ( times_times @ assn @ A3 @ B6 ) @ C5 ) @ D4 ) @ X8 ) ) ).
% groupy
thf(fact_5694_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_5695_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_5696_pair__imageI,axiom,
! [C: $tType,B: $tType,A: $tType,A2: A,B2: B,A3: set @ ( product_prod @ A @ B ),F2: A > B > C] :
( ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ A2 @ B2 ) @ A3 )
=> ( member @ C @ ( F2 @ A2 @ B2 ) @ ( image @ ( product_prod @ A @ B ) @ C @ ( product_case_prod @ A @ B @ C @ F2 ) @ A3 ) ) ) ).
% pair_imageI
thf(fact_5697_ent__pure__pre__iff__sng,axiom,
! [B2: $o,Q: assn] :
( ( entails @ ( pure_assn @ B2 ) @ Q )
= ( B2
=> ( entails @ ( one_one @ assn ) @ Q ) ) ) ).
% ent_pure_pre_iff_sng
thf(fact_5698_None__notin__image__Some,axiom,
! [A: $tType,A3: set @ A] :
~ ( member @ ( option @ A ) @ ( none @ A ) @ ( image @ A @ ( option @ A ) @ ( some @ A ) @ A3 ) ) ).
% None_notin_image_Some
thf(fact_5699_zero__notin__Suc__image,axiom,
! [A3: set @ nat] :
~ ( member @ nat @ ( zero_zero @ nat ) @ ( image @ nat @ nat @ suc @ A3 ) ) ).
% zero_notin_Suc_image
thf(fact_5700_finite__conv__nat__seg__image,axiom,
! [A: $tType] :
( ( finite_finite2 @ A )
= ( ^ [A7: set @ A] :
? [N2: nat,F7: nat > A] :
( A7
= ( image @ nat @ A @ F7
@ ( collect @ nat
@ ^ [I3: nat] : ( ord_less @ nat @ I3 @ N2 ) ) ) ) ) ) ).
% finite_conv_nat_seg_image
thf(fact_5701_nat__seg__image__imp__finite,axiom,
! [A: $tType,A3: set @ A,F2: nat > A,N3: nat] :
( ( A3
= ( image @ nat @ A @ F2
@ ( collect @ nat
@ ^ [I3: nat] : ( ord_less @ nat @ I3 @ N3 ) ) ) )
=> ( finite_finite2 @ A @ A3 ) ) ).
% nat_seg_image_imp_finite
thf(fact_5702_finite__int__iff__bounded,axiom,
( ( finite_finite2 @ int )
= ( ^ [S8: set @ int] :
? [K2: int] : ( ord_less_eq @ ( set @ int ) @ ( image @ int @ int @ ( abs_abs @ int ) @ S8 ) @ ( set_ord_lessThan @ int @ K2 ) ) ) ) ).
% finite_int_iff_bounded
thf(fact_5703_finite__int__iff__bounded__le,axiom,
( ( finite_finite2 @ int )
= ( ^ [S8: set @ int] :
? [K2: int] : ( ord_less_eq @ ( set @ int ) @ ( image @ int @ int @ ( abs_abs @ int ) @ S8 ) @ ( set_ord_atMost @ int @ K2 ) ) ) ) ).
% finite_int_iff_bounded_le
thf(fact_5704_htt__cons__rule,axiom,
! [A: $tType,P6: assn,C2: heap_Time_Heap @ A,Q6: A > assn,T6: nat,P: assn,Q: A > assn,T2: nat] :
( ( time_htt @ A @ P6 @ C2 @ Q6 @ T6 )
=> ( ( entails @ P @ P6 )
=> ( ! [X3: A] : ( entails @ ( Q6 @ X3 ) @ ( Q @ X3 ) )
=> ( ( ord_less_eq @ nat @ T6 @ T2 )
=> ( time_htt @ A @ P @ C2 @ Q @ T2 ) ) ) ) ) ).
% htt_cons_rule
thf(fact_5705_in__image__insert__iff,axiom,
! [A: $tType,B6: set @ ( set @ A ),X: A,A3: set @ A] :
( ! [C7: set @ A] :
( ( member @ ( set @ A ) @ C7 @ B6 )
=> ~ ( member @ A @ X @ C7 ) )
=> ( ( member @ ( set @ A ) @ A3 @ ( image @ ( set @ A ) @ ( set @ A ) @ ( insert @ A @ X ) @ B6 ) )
= ( ( member @ A @ X @ A3 )
& ( member @ ( set @ A ) @ ( minus_minus @ ( set @ A ) @ A3 @ ( insert @ A @ X @ ( bot_bot @ ( set @ A ) ) ) ) @ B6 ) ) ) ) ).
% in_image_insert_iff
thf(fact_5706_image__Suc__lessThan,axiom,
! [N3: nat] :
( ( image @ nat @ nat @ suc @ ( set_ord_lessThan @ nat @ N3 ) )
= ( set_or1337092689740270186AtMost @ nat @ ( one_one @ nat ) @ N3 ) ) ).
% image_Suc_lessThan
thf(fact_5707_image__Suc__atMost,axiom,
! [N3: nat] :
( ( image @ nat @ nat @ suc @ ( set_ord_atMost @ nat @ N3 ) )
= ( set_or1337092689740270186AtMost @ nat @ ( one_one @ nat ) @ ( suc @ N3 ) ) ) ).
% image_Suc_atMost
thf(fact_5708_atLeast0__atMost__Suc__eq__insert__0,axiom,
! [N3: nat] :
( ( set_or1337092689740270186AtMost @ nat @ ( zero_zero @ nat ) @ ( suc @ N3 ) )
= ( insert @ nat @ ( zero_zero @ nat ) @ ( image @ nat @ nat @ suc @ ( set_or1337092689740270186AtMost @ nat @ ( zero_zero @ nat ) @ N3 ) ) ) ) ).
% atLeast0_atMost_Suc_eq_insert_0
thf(fact_5709_lessThan__Suc__eq__insert__0,axiom,
! [N3: nat] :
( ( set_ord_lessThan @ nat @ ( suc @ N3 ) )
= ( insert @ nat @ ( zero_zero @ nat ) @ ( image @ nat @ nat @ suc @ ( set_ord_lessThan @ nat @ N3 ) ) ) ) ).
% lessThan_Suc_eq_insert_0
thf(fact_5710_atMost__Suc__eq__insert__0,axiom,
! [N3: nat] :
( ( set_ord_atMost @ nat @ ( suc @ N3 ) )
= ( insert @ nat @ ( zero_zero @ nat ) @ ( image @ nat @ nat @ suc @ ( set_ord_atMost @ nat @ N3 ) ) ) ) ).
% atMost_Suc_eq_insert_0
thf(fact_5711_listI__assn__reinsert,axiom,
! [B: $tType,A: $tType,P: assn,A3: A > B > assn,Xs2: list @ A,I: nat,Xsi: list @ B,I5: set @ nat,F4: assn,Q: assn] :
( ( entails @ P @ ( times_times @ assn @ ( times_times @ assn @ ( A3 @ ( nth @ A @ Xs2 @ I ) @ ( nth @ B @ Xsi @ I ) ) @ ( vEBT_List_listI_assn @ A @ B @ ( minus_minus @ ( set @ nat ) @ I5 @ ( insert @ nat @ I @ ( bot_bot @ ( set @ nat ) ) ) ) @ A3 @ Xs2 @ Xsi ) ) @ F4 ) )
=> ( ( ord_less @ nat @ I @ ( size_size @ ( list @ A ) @ Xs2 ) )
=> ( ( member @ nat @ I @ I5 )
=> ( ( entails @ ( times_times @ assn @ ( vEBT_List_listI_assn @ A @ B @ I5 @ A3 @ Xs2 @ Xsi ) @ F4 ) @ Q )
=> ( entails @ P @ Q ) ) ) ) ) ).
% listI_assn_reinsert
thf(fact_5712_listI__assn__reinsert__upd,axiom,
! [D5: $tType,C: $tType,P: assn,A3: C > D5 > assn,X: C,Xi2: D5,I5: set @ nat,I: nat,Xs2: list @ C,Xsi: list @ D5,F4: assn,Q: assn] :
( ( entails @ P @ ( times_times @ assn @ ( times_times @ assn @ ( A3 @ X @ Xi2 ) @ ( vEBT_List_listI_assn @ C @ D5 @ ( minus_minus @ ( set @ nat ) @ I5 @ ( insert @ nat @ I @ ( bot_bot @ ( set @ nat ) ) ) ) @ A3 @ Xs2 @ Xsi ) ) @ F4 ) )
=> ( ( ord_less @ nat @ I @ ( size_size @ ( list @ C ) @ Xs2 ) )
=> ( ( member @ nat @ I @ I5 )
=> ( ( entails @ ( times_times @ assn @ ( vEBT_List_listI_assn @ C @ D5 @ I5 @ A3 @ ( list_update @ C @ Xs2 @ I @ X ) @ ( list_update @ D5 @ Xsi @ I @ Xi2 ) ) @ F4 ) @ Q )
=> ( entails @ P @ Q ) ) ) ) ) ).
% listI_assn_reinsert_upd
thf(fact_5713_listI__assn__reinsert_H,axiom,
! [A: $tType,B: $tType,C: $tType,P: assn,A3: A > B > assn,Xs2: list @ A,I: nat,Xsi: list @ B,I5: set @ nat,F4: assn,C2: heap_Time_Heap @ C,Q: C > assn] :
( ( entails @ P @ ( times_times @ assn @ ( times_times @ assn @ ( A3 @ ( nth @ A @ Xs2 @ I ) @ ( nth @ B @ Xsi @ I ) ) @ ( vEBT_List_listI_assn @ A @ B @ ( minus_minus @ ( set @ nat ) @ I5 @ ( insert @ nat @ I @ ( bot_bot @ ( set @ nat ) ) ) ) @ A3 @ Xs2 @ Xsi ) ) @ F4 ) )
=> ( ( ord_less @ nat @ I @ ( size_size @ ( list @ A ) @ Xs2 ) )
=> ( ( member @ nat @ I @ I5 )
=> ( ( hoare_hoare_triple @ C @ ( times_times @ assn @ ( vEBT_List_listI_assn @ A @ B @ I5 @ A3 @ Xs2 @ Xsi ) @ F4 ) @ C2 @ Q )
=> ( hoare_hoare_triple @ C @ P @ C2 @ Q ) ) ) ) ) ).
% listI_assn_reinsert'
thf(fact_5714_listI__assn__reinsert__upd_H,axiom,
! [C: $tType,D5: $tType,E3: $tType,P: assn,A3: C > D5 > assn,X: C,Xi2: D5,I5: set @ nat,I: nat,Xs2: list @ C,Xsi: list @ D5,F4: assn,C2: heap_Time_Heap @ E3,Q: E3 > assn] :
( ( entails @ P @ ( times_times @ assn @ ( times_times @ assn @ ( A3 @ X @ Xi2 ) @ ( vEBT_List_listI_assn @ C @ D5 @ ( minus_minus @ ( set @ nat ) @ I5 @ ( insert @ nat @ I @ ( bot_bot @ ( set @ nat ) ) ) ) @ A3 @ Xs2 @ Xsi ) ) @ F4 ) )
=> ( ( ord_less @ nat @ I @ ( size_size @ ( list @ C ) @ Xs2 ) )
=> ( ( member @ nat @ I @ I5 )
=> ( ( hoare_hoare_triple @ E3 @ ( times_times @ assn @ ( vEBT_List_listI_assn @ C @ D5 @ I5 @ A3 @ ( list_update @ C @ Xs2 @ I @ X ) @ ( list_update @ D5 @ Xsi @ I @ Xi2 ) ) @ F4 ) @ C2 @ Q )
=> ( hoare_hoare_triple @ E3 @ P @ C2 @ Q ) ) ) ) ) ).
% listI_assn_reinsert_upd'
thf(fact_5715_translation__diff,axiom,
! [A: $tType] :
( ( ab_group_add @ A )
=> ! [A2: A,S: set @ A,T2: set @ A] :
( ( image @ A @ A @ ( plus_plus @ A @ A2 ) @ ( minus_minus @ ( set @ A ) @ S @ T2 ) )
= ( minus_minus @ ( set @ A ) @ ( image @ A @ A @ ( plus_plus @ A @ A2 ) @ S ) @ ( image @ A @ A @ ( plus_plus @ A @ A2 ) @ T2 ) ) ) ) ).
% translation_diff
thf(fact_5716_bij__betw__subset,axiom,
! [A: $tType,B: $tType,F2: A > B,A3: set @ A,A9: set @ B,B6: set @ A,B10: set @ B] :
( ( bij_betw @ A @ B @ F2 @ A3 @ A9 )
=> ( ( ord_less_eq @ ( set @ A ) @ B6 @ A3 )
=> ( ( ( image @ A @ B @ F2 @ B6 )
= B10 )
=> ( bij_betw @ A @ B @ F2 @ B6 @ B10 ) ) ) ) ).
% bij_betw_subset
thf(fact_5717_bij__betw__byWitness,axiom,
! [A: $tType,B: $tType,A3: set @ A,F5: B > A,F2: A > B,A9: set @ B] :
( ! [X3: A] :
( ( member @ A @ X3 @ A3 )
=> ( ( F5 @ ( F2 @ X3 ) )
= X3 ) )
=> ( ! [X3: B] :
( ( member @ B @ X3 @ A9 )
=> ( ( F2 @ ( F5 @ X3 ) )
= X3 ) )
=> ( ( ord_less_eq @ ( set @ B ) @ ( image @ A @ B @ F2 @ A3 ) @ A9 )
=> ( ( ord_less_eq @ ( set @ A ) @ ( image @ B @ A @ F5 @ A9 ) @ A3 )
=> ( bij_betw @ A @ B @ F2 @ A3 @ A9 ) ) ) ) ) ).
% bij_betw_byWitness
thf(fact_5718_translation__Compl,axiom,
! [A: $tType] :
( ( ab_group_add @ A )
=> ! [A2: A,T2: set @ A] :
( ( image @ A @ A @ ( plus_plus @ A @ A2 ) @ ( uminus_uminus @ ( set @ A ) @ T2 ) )
= ( uminus_uminus @ ( set @ A ) @ ( image @ A @ A @ ( plus_plus @ A @ A2 ) @ T2 ) ) ) ) ).
% translation_Compl
thf(fact_5719_translation__subtract__diff,axiom,
! [A: $tType] :
( ( ab_group_add @ A )
=> ! [A2: A,S: set @ A,T2: set @ A] :
( ( image @ A @ A
@ ^ [X2: A] : ( minus_minus @ A @ X2 @ A2 )
@ ( minus_minus @ ( set @ A ) @ S @ T2 ) )
= ( minus_minus @ ( set @ A )
@ ( image @ A @ A
@ ^ [X2: A] : ( minus_minus @ A @ X2 @ A2 )
@ S )
@ ( image @ A @ A
@ ^ [X2: A] : ( minus_minus @ A @ X2 @ A2 )
@ T2 ) ) ) ) ).
% translation_subtract_diff
thf(fact_5720_translation__subtract__Compl,axiom,
! [A: $tType] :
( ( ab_group_add @ A )
=> ! [A2: A,T2: set @ A] :
( ( image @ A @ A
@ ^ [X2: A] : ( minus_minus @ A @ X2 @ A2 )
@ ( uminus_uminus @ ( set @ A ) @ T2 ) )
= ( uminus_uminus @ ( set @ A )
@ ( image @ A @ A
@ ^ [X2: A] : ( minus_minus @ A @ X2 @ A2 )
@ T2 ) ) ) ) ).
% translation_subtract_Compl
thf(fact_5721_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_5722_big__assn__simp_H,axiom,
! [H2: nat,TreeList: list @ vEBT_VEBT,Xaa: vEBT_VEBT,L2: nat,X: vEBT_VEBTi,Xb: option @ nat,X13: array @ vEBT_VEBTi,Tree_is: list @ vEBT_VEBTi,Summary: vEBT_VEBT,X14: vEBT_VEBTi] :
( ( ord_less @ nat @ H2 @ ( size_size @ ( list @ vEBT_VEBT ) @ TreeList ) )
=> ( ( Xaa
= ( vEBT_vebt_delete @ ( nth @ vEBT_VEBT @ TreeList @ H2 ) @ L2 ) )
=> ( entails
@ ( times_times @ assn
@ ( times_times @ assn @ ( vEBT_vebt_assn_raw @ Xaa @ X )
@ ( pure_assn
@ ( Xb
= ( vEBT_vebt_mint @ Xaa ) ) ) )
@ ( times_times @ assn @ ( times_times @ assn @ ( snga_assn @ vEBT_VEBTi @ X13 @ ( list_update @ vEBT_VEBTi @ Tree_is @ H2 @ X ) ) @ ( vEBT_vebt_assn_raw @ Summary @ X14 ) ) @ ( vEBT_List_listI_assn @ vEBT_VEBT @ vEBT_VEBTi @ ( minus_minus @ ( set @ nat ) @ ( set_or7035219750837199246ssThan @ nat @ ( zero_zero @ nat ) @ ( size_size @ ( list @ vEBT_VEBT ) @ TreeList ) ) @ ( insert @ nat @ H2 @ ( bot_bot @ ( set @ nat ) ) ) ) @ vEBT_vebt_assn_raw @ TreeList @ Tree_is ) ) )
@ ( times_times @ assn
@ ( times_times @ assn @ ( times_times @ assn @ ( snga_assn @ vEBT_VEBTi @ X13 @ ( list_update @ vEBT_VEBTi @ Tree_is @ H2 @ X ) ) @ ( vEBT_vebt_assn_raw @ Summary @ X14 ) )
@ ( pure_assn
@ ( Xb
= ( vEBT_vebt_mint @ Xaa ) ) ) )
@ ( vEBT_List_list_assn @ vEBT_VEBT @ vEBT_VEBTi @ vEBT_vebt_assn_raw @ ( list_update @ vEBT_VEBT @ TreeList @ H2 @ Xaa ) @ ( list_update @ vEBT_VEBTi @ Tree_is @ H2 @ X ) ) ) ) ) ) ).
% big_assn_simp'
thf(fact_5723_big__assn__simp,axiom,
! [H2: nat,TreeList: list @ vEBT_VEBT,L2: nat,X: vEBT_VEBTi,Xaa: option @ nat,X13: array @ vEBT_VEBTi,Tree_is: list @ vEBT_VEBTi,Summary: vEBT_VEBT,X14: vEBT_VEBTi] :
( ( ord_less @ nat @ H2 @ ( size_size @ ( list @ vEBT_VEBT ) @ TreeList ) )
=> ( entails
@ ( times_times @ assn
@ ( times_times @ assn @ ( vEBT_vebt_assn_raw @ ( vEBT_vebt_delete @ ( nth @ vEBT_VEBT @ TreeList @ H2 ) @ L2 ) @ X )
@ ( pure_assn
@ ( Xaa
= ( vEBT_vebt_mint @ ( vEBT_vebt_delete @ ( nth @ vEBT_VEBT @ TreeList @ H2 ) @ L2 ) ) ) ) )
@ ( times_times @ assn @ ( times_times @ assn @ ( snga_assn @ vEBT_VEBTi @ X13 @ ( list_update @ vEBT_VEBTi @ Tree_is @ H2 @ X ) ) @ ( vEBT_vebt_assn_raw @ Summary @ X14 ) ) @ ( vEBT_List_listI_assn @ vEBT_VEBT @ vEBT_VEBTi @ ( minus_minus @ ( set @ nat ) @ ( set_or7035219750837199246ssThan @ nat @ ( zero_zero @ nat ) @ ( size_size @ ( list @ vEBT_VEBT ) @ TreeList ) ) @ ( insert @ nat @ H2 @ ( bot_bot @ ( set @ nat ) ) ) ) @ vEBT_vebt_assn_raw @ TreeList @ Tree_is ) ) )
@ ( times_times @ assn
@ ( times_times @ assn @ ( times_times @ assn @ ( snga_assn @ vEBT_VEBTi @ X13 @ ( list_update @ vEBT_VEBTi @ Tree_is @ H2 @ X ) ) @ ( vEBT_vebt_assn_raw @ Summary @ X14 ) )
@ ( pure_assn
@ ( Xaa
= ( vEBT_vebt_mint @ ( vEBT_vebt_delete @ ( nth @ vEBT_VEBT @ TreeList @ H2 ) @ L2 ) ) ) ) )
@ ( 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 ) @ L2 ) ) @ ( list_update @ vEBT_VEBTi @ Tree_is @ H2 @ X ) ) ) ) ) ).
% big_assn_simp
thf(fact_5724_atLeastLessThan__iff,axiom,
! [A: $tType] :
( ( ord @ A )
=> ! [I: A,L2: A,U: A] :
( ( member @ A @ I @ ( set_or7035219750837199246ssThan @ A @ L2 @ U ) )
= ( ( ord_less_eq @ A @ L2 @ I )
& ( ord_less @ A @ I @ U ) ) ) ) ).
% atLeastLessThan_iff
thf(fact_5725_atLeastLessThan__empty,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [B2: A,A2: A] :
( ( ord_less_eq @ A @ B2 @ A2 )
=> ( ( set_or7035219750837199246ssThan @ A @ A2 @ B2 )
= ( bot_bot @ ( set @ A ) ) ) ) ) ).
% atLeastLessThan_empty
thf(fact_5726_atLeastLessThan__empty__iff,axiom,
! [A: $tType] :
( ( preorder @ A )
=> ! [A2: A,B2: A] :
( ( ( set_or7035219750837199246ssThan @ A @ A2 @ B2 )
= ( bot_bot @ ( set @ A ) ) )
= ( ~ ( ord_less @ A @ A2 @ B2 ) ) ) ) ).
% atLeastLessThan_empty_iff
thf(fact_5727_atLeastLessThan__empty__iff2,axiom,
! [A: $tType] :
( ( preorder @ A )
=> ! [A2: A,B2: A] :
( ( ( bot_bot @ ( set @ A ) )
= ( set_or7035219750837199246ssThan @ A @ A2 @ B2 ) )
= ( ~ ( ord_less @ A @ A2 @ B2 ) ) ) ) ).
% atLeastLessThan_empty_iff2
thf(fact_5728_infinite__Ico__iff,axiom,
! [A: $tType] :
( ( dense_linorder @ A )
=> ! [A2: A,B2: A] :
( ( ~ ( finite_finite2 @ A @ ( set_or7035219750837199246ssThan @ A @ A2 @ B2 ) ) )
= ( ord_less @ A @ A2 @ B2 ) ) ) ).
% infinite_Ico_iff
thf(fact_5729_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_5730_ivl__subset,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [I: A,J: A,M: A,N3: A] :
( ( ord_less_eq @ ( set @ A ) @ ( set_or7035219750837199246ssThan @ A @ I @ J ) @ ( set_or7035219750837199246ssThan @ A @ M @ N3 ) )
= ( ( ord_less_eq @ A @ J @ I )
| ( ( ord_less_eq @ A @ M @ I )
& ( ord_less_eq @ A @ J @ N3 ) ) ) ) ) ).
% ivl_subset
thf(fact_5731_ivl__diff,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [I: A,N3: A,M: A] :
( ( ord_less_eq @ A @ I @ N3 )
=> ( ( minus_minus @ ( set @ A ) @ ( set_or7035219750837199246ssThan @ A @ I @ M ) @ ( set_or7035219750837199246ssThan @ A @ I @ N3 ) )
= ( set_or7035219750837199246ssThan @ A @ N3 @ M ) ) ) ) ).
% ivl_diff
thf(fact_5732_lessThan__minus__lessThan,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [N3: A,M: A] :
( ( minus_minus @ ( set @ A ) @ ( set_ord_lessThan @ A @ N3 ) @ ( set_ord_lessThan @ A @ M ) )
= ( set_or7035219750837199246ssThan @ A @ M @ N3 ) ) ) ).
% lessThan_minus_lessThan
thf(fact_5733_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_5734_image__add__atLeastLessThan_H,axiom,
! [A: $tType] :
( ( linordered_semidom @ A )
=> ! [K: A,I: A,J: A] :
( ( image @ A @ A
@ ^ [N2: A] : ( plus_plus @ A @ N2 @ K )
@ ( set_or7035219750837199246ssThan @ A @ I @ J ) )
= ( set_or7035219750837199246ssThan @ A @ ( plus_plus @ A @ I @ K ) @ ( plus_plus @ A @ J @ K ) ) ) ) ).
% image_add_atLeastLessThan'
thf(fact_5735_local_Oext,axiom,
! [Y: nat,TreeList: list @ vEBT_VEBT,X13: array @ vEBT_VEBTi,Tree_is: list @ vEBT_VEBTi,Summary: vEBT_VEBT,X14: vEBT_VEBTi] :
( ( ord_less @ nat @ Y @ ( size_size @ ( list @ vEBT_VEBT ) @ TreeList ) )
=> ( entails @ ( times_times @ assn @ ( snga_assn @ vEBT_VEBTi @ X13 @ Tree_is ) @ ( times_times @ assn @ ( vEBT_vebt_assn_raw @ Summary @ X14 ) @ ( times_times @ assn @ ( vEBT_vebt_assn_raw @ ( nth @ vEBT_VEBT @ TreeList @ Y ) @ ( nth @ vEBT_VEBTi @ Tree_is @ Y ) ) @ ( vEBT_List_listI_assn @ vEBT_VEBT @ vEBT_VEBTi @ ( minus_minus @ ( set @ nat ) @ ( set_or7035219750837199246ssThan @ nat @ ( zero_zero @ nat ) @ ( size_size @ ( list @ vEBT_VEBT ) @ TreeList ) ) @ ( insert @ nat @ Y @ ( bot_bot @ ( set @ nat ) ) ) ) @ vEBT_vebt_assn_raw @ TreeList @ Tree_is ) ) ) ) @ ( times_times @ assn @ ( times_times @ assn @ ( times_times @ assn @ ( snga_assn @ vEBT_VEBTi @ X13 @ Tree_is ) @ ( vEBT_vebt_assn_raw @ Summary @ X14 ) ) @ ( vEBT_List_listI_assn @ vEBT_VEBT @ vEBT_VEBTi @ ( minus_minus @ ( set @ nat ) @ ( set_or7035219750837199246ssThan @ nat @ ( zero_zero @ nat ) @ ( size_size @ ( list @ vEBT_VEBT ) @ TreeList ) ) @ ( insert @ nat @ Y @ ( bot_bot @ ( set @ nat ) ) ) ) @ vEBT_vebt_assn_raw @ TreeList @ Tree_is ) ) @ ( vEBT_vebt_assn_raw @ ( nth @ vEBT_VEBT @ TreeList @ Y ) @ ( nth @ vEBT_VEBTi @ Tree_is @ Y ) ) ) ) ) ).
% local.ext
thf(fact_5736_atLeastLessThan__singleton,axiom,
! [M: nat] :
( ( set_or7035219750837199246ssThan @ nat @ M @ ( suc @ M ) )
= ( insert @ nat @ M @ ( bot_bot @ ( set @ nat ) ) ) ) ).
% atLeastLessThan_singleton
thf(fact_5737_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_5738_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_5739_txe,axiom,
! [Y: nat,TreeList: list @ vEBT_VEBT,Tree_is: list @ vEBT_VEBTi,X13: array @ vEBT_VEBTi,Summary: vEBT_VEBT,X14: vEBT_VEBTi] :
( ( ord_less @ nat @ Y @ ( size_size @ ( list @ vEBT_VEBT ) @ TreeList ) )
=> ( entails @ ( times_times @ assn @ ( times_times @ assn @ ( times_times @ assn @ ( vEBT_vebt_assn_raw @ ( nth @ vEBT_VEBT @ TreeList @ Y ) @ ( nth @ vEBT_VEBTi @ Tree_is @ Y ) ) @ ( snga_assn @ vEBT_VEBTi @ X13 @ Tree_is ) ) @ ( vEBT_vebt_assn_raw @ Summary @ X14 ) ) @ ( vEBT_List_listI_assn @ vEBT_VEBT @ vEBT_VEBTi @ ( minus_minus @ ( set @ nat ) @ ( set_or7035219750837199246ssThan @ nat @ ( zero_zero @ nat ) @ ( size_size @ ( list @ vEBT_VEBT ) @ TreeList ) ) @ ( insert @ nat @ Y @ ( bot_bot @ ( set @ nat ) ) ) ) @ vEBT_vebt_assn_raw @ TreeList @ Tree_is ) ) @ ( times_times @ assn @ ( times_times @ assn @ ( vEBT_vebt_assn_raw @ Summary @ X14 ) @ ( snga_assn @ vEBT_VEBTi @ X13 @ Tree_is ) ) @ ( vEBT_List_list_assn @ vEBT_VEBT @ vEBT_VEBTi @ vEBT_vebt_assn_raw @ TreeList @ Tree_is ) ) ) ) ).
% txe
thf(fact_5740_tcd,axiom,
! [A: $tType,I: nat,TreeList: list @ vEBT_VEBT,TreeList4: list @ A,Y: vEBT_VEBT,X: vEBT_VEBTi,X13: array @ vEBT_VEBTi,Tree_is: list @ vEBT_VEBTi,Summary: vEBT_VEBT,X14: vEBT_VEBTi] :
( ( ord_less @ nat @ I @ ( size_size @ ( list @ vEBT_VEBT ) @ TreeList ) )
=> ( ( ( size_size @ ( list @ vEBT_VEBT ) @ TreeList )
= ( size_size @ ( list @ A ) @ TreeList4 ) )
=> ( entails @ ( times_times @ assn @ ( times_times @ assn @ ( times_times @ assn @ ( vEBT_vebt_assn_raw @ Y @ X ) @ ( snga_assn @ vEBT_VEBTi @ X13 @ ( list_update @ vEBT_VEBTi @ Tree_is @ I @ X ) ) ) @ ( vEBT_vebt_assn_raw @ Summary @ X14 ) ) @ ( vEBT_List_listI_assn @ vEBT_VEBT @ vEBT_VEBTi @ ( minus_minus @ ( set @ nat ) @ ( set_or7035219750837199246ssThan @ nat @ ( zero_zero @ nat ) @ ( size_size @ ( list @ vEBT_VEBT ) @ TreeList ) ) @ ( insert @ nat @ I @ ( bot_bot @ ( set @ nat ) ) ) ) @ vEBT_vebt_assn_raw @ ( list_update @ vEBT_VEBT @ TreeList @ I @ Y ) @ ( list_update @ vEBT_VEBTi @ Tree_is @ I @ X ) ) ) @ ( times_times @ assn @ ( times_times @ assn @ ( snga_assn @ vEBT_VEBTi @ X13 @ ( list_update @ vEBT_VEBTi @ Tree_is @ I @ X ) ) @ ( vEBT_vebt_assn_raw @ Summary @ X14 ) ) @ ( vEBT_List_list_assn @ vEBT_VEBT @ vEBT_VEBTi @ vEBT_vebt_assn_raw @ ( list_update @ vEBT_VEBT @ TreeList @ I @ Y ) @ ( list_update @ vEBT_VEBTi @ Tree_is @ I @ X ) ) ) ) ) ) ).
% tcd
thf(fact_5741_sum_Oop__ivl__Suc,axiom,
! [A: $tType] :
( ( comm_monoid_add @ A )
=> ! [N3: nat,M: nat,G: nat > A] :
( ( ( ord_less @ nat @ N3 @ M )
=> ( ( groups7311177749621191930dd_sum @ nat @ A @ G @ ( set_or7035219750837199246ssThan @ nat @ M @ ( suc @ N3 ) ) )
= ( zero_zero @ A ) ) )
& ( ~ ( ord_less @ nat @ N3 @ M )
=> ( ( groups7311177749621191930dd_sum @ nat @ A @ G @ ( set_or7035219750837199246ssThan @ nat @ M @ ( suc @ N3 ) ) )
= ( plus_plus @ A @ ( groups7311177749621191930dd_sum @ nat @ A @ G @ ( set_or7035219750837199246ssThan @ nat @ M @ N3 ) ) @ ( G @ N3 ) ) ) ) ) ) ).
% sum.op_ivl_Suc
thf(fact_5742_prod_Oop__ivl__Suc,axiom,
! [A: $tType] :
( ( comm_monoid_mult @ A )
=> ! [N3: nat,M: nat,G: nat > A] :
( ( ( ord_less @ nat @ N3 @ M )
=> ( ( groups7121269368397514597t_prod @ nat @ A @ G @ ( set_or7035219750837199246ssThan @ nat @ M @ ( suc @ N3 ) ) )
= ( one_one @ A ) ) )
& ( ~ ( ord_less @ nat @ N3 @ M )
=> ( ( groups7121269368397514597t_prod @ nat @ A @ G @ ( set_or7035219750837199246ssThan @ nat @ M @ ( suc @ N3 ) ) )
= ( times_times @ A @ ( groups7121269368397514597t_prod @ nat @ A @ G @ ( set_or7035219750837199246ssThan @ nat @ M @ N3 ) ) @ ( G @ N3 ) ) ) ) ) ) ).
% prod.op_ivl_Suc
thf(fact_5743_nth__image__indices,axiom,
! [A: $tType,L2: list @ A] :
( ( image @ nat @ A @ ( nth @ A @ L2 ) @ ( set_or7035219750837199246ssThan @ nat @ ( zero_zero @ nat ) @ ( size_size @ ( list @ A ) @ L2 ) ) )
= ( set2 @ A @ L2 ) ) ).
% nth_image_indices
thf(fact_5744_atLeastLessThanSuc__atLeastAtMost,axiom,
! [L2: nat,U: nat] :
( ( set_or7035219750837199246ssThan @ nat @ L2 @ ( suc @ U ) )
= ( set_or1337092689740270186AtMost @ nat @ L2 @ U ) ) ).
% atLeastLessThanSuc_atLeastAtMost
thf(fact_5745_infinite__Ico,axiom,
! [A: $tType] :
( ( dense_linorder @ A )
=> ! [A2: A,B2: A] :
( ( ord_less @ A @ A2 @ B2 )
=> ~ ( finite_finite2 @ A @ ( set_or7035219750837199246ssThan @ A @ A2 @ B2 ) ) ) ) ).
% infinite_Ico
thf(fact_5746_atLeastLessThan__subset__iff,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A2: A,B2: A,C2: A,D: A] :
( ( ord_less_eq @ ( set @ A ) @ ( set_or7035219750837199246ssThan @ A @ A2 @ B2 ) @ ( set_or7035219750837199246ssThan @ A @ C2 @ D ) )
=> ( ( ord_less_eq @ A @ B2 @ A2 )
| ( ( ord_less_eq @ A @ C2 @ A2 )
& ( ord_less_eq @ A @ B2 @ D ) ) ) ) ) ).
% atLeastLessThan_subset_iff
thf(fact_5747_atLeastLessThan__inj_I2_J,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A2: A,B2: A,C2: A,D: A] :
( ( ( set_or7035219750837199246ssThan @ A @ A2 @ B2 )
= ( set_or7035219750837199246ssThan @ A @ C2 @ D ) )
=> ( ( ord_less @ A @ A2 @ B2 )
=> ( ( ord_less @ A @ C2 @ D )
=> ( B2 = D ) ) ) ) ) ).
% atLeastLessThan_inj(2)
thf(fact_5748_atLeastLessThan__inj_I1_J,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A2: A,B2: A,C2: A,D: A] :
( ( ( set_or7035219750837199246ssThan @ A @ A2 @ B2 )
= ( set_or7035219750837199246ssThan @ A @ C2 @ D ) )
=> ( ( ord_less @ A @ A2 @ B2 )
=> ( ( ord_less @ A @ C2 @ D )
=> ( A2 = C2 ) ) ) ) ) ).
% atLeastLessThan_inj(1)
thf(fact_5749_atLeastLessThan__eq__iff,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A2: A,B2: A,C2: A,D: A] :
( ( ord_less @ A @ A2 @ B2 )
=> ( ( ord_less @ A @ C2 @ D )
=> ( ( ( set_or7035219750837199246ssThan @ A @ A2 @ B2 )
= ( set_or7035219750837199246ssThan @ A @ C2 @ D ) )
= ( ( A2 = C2 )
& ( B2 = D ) ) ) ) ) ) ).
% atLeastLessThan_eq_iff
thf(fact_5750_ex__nat__less__eq,axiom,
! [N3: nat,P: nat > $o] :
( ( ? [M5: nat] :
( ( ord_less @ nat @ M5 @ N3 )
& ( P @ M5 ) ) )
= ( ? [X2: nat] :
( ( member @ nat @ X2 @ ( set_or7035219750837199246ssThan @ nat @ ( zero_zero @ nat ) @ N3 ) )
& ( P @ X2 ) ) ) ) ).
% ex_nat_less_eq
thf(fact_5751_all__nat__less__eq,axiom,
! [N3: nat,P: nat > $o] :
( ( ! [M5: nat] :
( ( ord_less @ nat @ M5 @ N3 )
=> ( P @ M5 ) ) )
= ( ! [X2: nat] :
( ( member @ nat @ X2 @ ( set_or7035219750837199246ssThan @ nat @ ( zero_zero @ nat ) @ N3 ) )
=> ( P @ X2 ) ) ) ) ).
% all_nat_less_eq
thf(fact_5752_sum_Oshift__bounds__Suc__ivl,axiom,
! [A: $tType] :
( ( comm_monoid_add @ A )
=> ! [G: nat > A,M: nat,N3: nat] :
( ( groups7311177749621191930dd_sum @ nat @ A @ G @ ( set_or7035219750837199246ssThan @ nat @ ( suc @ M ) @ ( suc @ N3 ) ) )
= ( groups7311177749621191930dd_sum @ nat @ A
@ ^ [I3: nat] : ( G @ ( suc @ I3 ) )
@ ( set_or7035219750837199246ssThan @ nat @ M @ N3 ) ) ) ) ).
% sum.shift_bounds_Suc_ivl
thf(fact_5753_sum_Oshift__bounds__nat__ivl,axiom,
! [A: $tType] :
( ( comm_monoid_add @ A )
=> ! [G: nat > A,M: nat,K: nat,N3: nat] :
( ( groups7311177749621191930dd_sum @ nat @ A @ G @ ( set_or7035219750837199246ssThan @ nat @ ( plus_plus @ nat @ M @ K ) @ ( plus_plus @ nat @ N3 @ K ) ) )
= ( groups7311177749621191930dd_sum @ nat @ A
@ ^ [I3: nat] : ( G @ ( plus_plus @ nat @ I3 @ K ) )
@ ( set_or7035219750837199246ssThan @ nat @ M @ N3 ) ) ) ) ).
% sum.shift_bounds_nat_ivl
thf(fact_5754_prod_Oshift__bounds__Suc__ivl,axiom,
! [A: $tType] :
( ( comm_monoid_mult @ A )
=> ! [G: nat > A,M: nat,N3: nat] :
( ( groups7121269368397514597t_prod @ nat @ A @ G @ ( set_or7035219750837199246ssThan @ nat @ ( suc @ M ) @ ( suc @ N3 ) ) )
= ( groups7121269368397514597t_prod @ nat @ A
@ ^ [I3: nat] : ( G @ ( suc @ I3 ) )
@ ( set_or7035219750837199246ssThan @ nat @ M @ N3 ) ) ) ) ).
% prod.shift_bounds_Suc_ivl
thf(fact_5755_prod_Oshift__bounds__nat__ivl,axiom,
! [A: $tType] :
( ( comm_monoid_mult @ A )
=> ! [G: nat > A,M: nat,K: nat,N3: nat] :
( ( groups7121269368397514597t_prod @ nat @ A @ G @ ( set_or7035219750837199246ssThan @ nat @ ( plus_plus @ nat @ M @ K ) @ ( plus_plus @ nat @ N3 @ K ) ) )
= ( groups7121269368397514597t_prod @ nat @ A
@ ^ [I3: nat] : ( G @ ( plus_plus @ nat @ I3 @ K ) )
@ ( set_or7035219750837199246ssThan @ nat @ M @ N3 ) ) ) ) ).
% prod.shift_bounds_nat_ivl
thf(fact_5756_sum_Oivl__cong,axiom,
! [A: $tType,B: $tType] :
( ( ( ord @ B )
& ( comm_monoid_add @ A ) )
=> ! [A2: B,C2: B,B2: B,D: B,G: B > A,H2: B > A] :
( ( A2 = C2 )
=> ( ( B2 = D )
=> ( ! [X3: B] :
( ( ord_less_eq @ B @ C2 @ X3 )
=> ( ( ord_less @ B @ X3 @ D )
=> ( ( G @ X3 )
= ( H2 @ X3 ) ) ) )
=> ( ( groups7311177749621191930dd_sum @ B @ A @ G @ ( set_or7035219750837199246ssThan @ B @ A2 @ B2 ) )
= ( groups7311177749621191930dd_sum @ B @ A @ H2 @ ( set_or7035219750837199246ssThan @ B @ C2 @ D ) ) ) ) ) ) ) ).
% sum.ivl_cong
thf(fact_5757_prod_Oivl__cong,axiom,
! [A: $tType,B: $tType] :
( ( ( ord @ B )
& ( comm_monoid_mult @ A ) )
=> ! [A2: B,C2: B,B2: B,D: B,G: B > A,H2: B > A] :
( ( A2 = C2 )
=> ( ( B2 = D )
=> ( ! [X3: B] :
( ( ord_less_eq @ B @ C2 @ X3 )
=> ( ( ord_less @ B @ X3 @ D )
=> ( ( G @ X3 )
= ( H2 @ X3 ) ) ) )
=> ( ( groups7121269368397514597t_prod @ B @ A @ G @ ( set_or7035219750837199246ssThan @ B @ A2 @ B2 ) )
= ( groups7121269368397514597t_prod @ B @ A @ H2 @ ( set_or7035219750837199246ssThan @ B @ C2 @ D ) ) ) ) ) ) ) ).
% prod.ivl_cong
thf(fact_5758_sum_OatLeastLessThan__concat,axiom,
! [A: $tType] :
( ( comm_monoid_add @ A )
=> ! [M: nat,N3: nat,P4: nat,G: nat > A] :
( ( ord_less_eq @ nat @ M @ N3 )
=> ( ( ord_less_eq @ nat @ N3 @ P4 )
=> ( ( plus_plus @ A @ ( groups7311177749621191930dd_sum @ nat @ A @ G @ ( set_or7035219750837199246ssThan @ nat @ M @ N3 ) ) @ ( groups7311177749621191930dd_sum @ nat @ A @ G @ ( set_or7035219750837199246ssThan @ nat @ N3 @ P4 ) ) )
= ( groups7311177749621191930dd_sum @ nat @ A @ G @ ( set_or7035219750837199246ssThan @ nat @ M @ P4 ) ) ) ) ) ) ).
% sum.atLeastLessThan_concat
thf(fact_5759_sum__diff__nat__ivl,axiom,
! [A: $tType] :
( ( ab_group_add @ A )
=> ! [M: nat,N3: nat,P4: nat,F2: nat > A] :
( ( ord_less_eq @ nat @ M @ N3 )
=> ( ( ord_less_eq @ nat @ N3 @ P4 )
=> ( ( minus_minus @ A @ ( groups7311177749621191930dd_sum @ nat @ A @ F2 @ ( set_or7035219750837199246ssThan @ nat @ M @ P4 ) ) @ ( groups7311177749621191930dd_sum @ nat @ A @ F2 @ ( set_or7035219750837199246ssThan @ nat @ M @ N3 ) ) )
= ( groups7311177749621191930dd_sum @ nat @ A @ F2 @ ( set_or7035219750837199246ssThan @ nat @ N3 @ P4 ) ) ) ) ) ) ).
% sum_diff_nat_ivl
thf(fact_5760_prod_OatLeastLessThan__concat,axiom,
! [A: $tType] :
( ( comm_monoid_mult @ A )
=> ! [M: nat,N3: nat,P4: nat,G: nat > A] :
( ( ord_less_eq @ nat @ M @ N3 )
=> ( ( ord_less_eq @ nat @ N3 @ P4 )
=> ( ( times_times @ A @ ( groups7121269368397514597t_prod @ nat @ A @ G @ ( set_or7035219750837199246ssThan @ nat @ M @ N3 ) ) @ ( groups7121269368397514597t_prod @ nat @ A @ G @ ( set_or7035219750837199246ssThan @ nat @ N3 @ P4 ) ) )
= ( groups7121269368397514597t_prod @ nat @ A @ G @ ( set_or7035219750837199246ssThan @ nat @ M @ P4 ) ) ) ) ) ) ).
% prod.atLeastLessThan_concat
thf(fact_5761_atLeast0__lessThan__Suc,axiom,
! [N3: nat] :
( ( set_or7035219750837199246ssThan @ nat @ ( zero_zero @ nat ) @ ( suc @ N3 ) )
= ( insert @ nat @ N3 @ ( set_or7035219750837199246ssThan @ nat @ ( zero_zero @ nat ) @ N3 ) ) ) ).
% atLeast0_lessThan_Suc
thf(fact_5762_atLeast0__lessThan__Suc__eq__insert__0,axiom,
! [N3: nat] :
( ( set_or7035219750837199246ssThan @ nat @ ( zero_zero @ nat ) @ ( suc @ N3 ) )
= ( insert @ nat @ ( zero_zero @ nat ) @ ( image @ nat @ nat @ suc @ ( set_or7035219750837199246ssThan @ nat @ ( zero_zero @ nat ) @ N3 ) ) ) ) ).
% atLeast0_lessThan_Suc_eq_insert_0
thf(fact_5763_subset__eq__atLeast0__lessThan__finite,axiom,
! [N7: set @ nat,N3: nat] :
( ( ord_less_eq @ ( set @ nat ) @ N7 @ ( set_or7035219750837199246ssThan @ nat @ ( zero_zero @ nat ) @ N3 ) )
=> ( finite_finite2 @ nat @ N7 ) ) ).
% subset_eq_atLeast0_lessThan_finite
thf(fact_5764_atLeastAtMost__subseteq__atLeastLessThan__iff,axiom,
! [A: $tType] :
( ( preorder @ A )
=> ! [A2: A,B2: A,C2: A,D: A] :
( ( ord_less_eq @ ( set @ A ) @ ( set_or1337092689740270186AtMost @ A @ A2 @ B2 ) @ ( set_or7035219750837199246ssThan @ A @ C2 @ D ) )
= ( ( ord_less_eq @ A @ A2 @ B2 )
=> ( ( ord_less_eq @ A @ C2 @ A2 )
& ( ord_less @ A @ B2 @ D ) ) ) ) ) ).
% atLeastAtMost_subseteq_atLeastLessThan_iff
thf(fact_5765_atLeastLessThan__subseteq__atLeastAtMost__iff,axiom,
! [A: $tType] :
( ( dense_linorder @ A )
=> ! [A2: A,B2: A,C2: A,D: A] :
( ( ord_less_eq @ ( set @ A ) @ ( set_or7035219750837199246ssThan @ A @ A2 @ B2 ) @ ( set_or1337092689740270186AtMost @ A @ C2 @ D ) )
= ( ( ord_less @ A @ A2 @ B2 )
=> ( ( ord_less_eq @ A @ C2 @ A2 )
& ( ord_less_eq @ A @ B2 @ D ) ) ) ) ) ).
% atLeastLessThan_subseteq_atLeastAtMost_iff
thf(fact_5766_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_5767_sum_OatLeast0__lessThan__Suc,axiom,
! [A: $tType] :
( ( comm_monoid_add @ A )
=> ! [G: nat > A,N3: nat] :
( ( groups7311177749621191930dd_sum @ nat @ A @ G @ ( set_or7035219750837199246ssThan @ nat @ ( zero_zero @ nat ) @ ( suc @ N3 ) ) )
= ( plus_plus @ A @ ( groups7311177749621191930dd_sum @ nat @ A @ G @ ( set_or7035219750837199246ssThan @ nat @ ( zero_zero @ nat ) @ N3 ) ) @ ( G @ N3 ) ) ) ) ).
% sum.atLeast0_lessThan_Suc
thf(fact_5768_sum_OatLeast__Suc__lessThan,axiom,
! [A: $tType] :
( ( comm_monoid_add @ A )
=> ! [M: nat,N3: nat,G: nat > A] :
( ( ord_less @ nat @ M @ N3 )
=> ( ( groups7311177749621191930dd_sum @ nat @ A @ G @ ( set_or7035219750837199246ssThan @ nat @ M @ N3 ) )
= ( plus_plus @ A @ ( G @ M ) @ ( groups7311177749621191930dd_sum @ nat @ A @ G @ ( set_or7035219750837199246ssThan @ nat @ ( suc @ M ) @ N3 ) ) ) ) ) ) ).
% sum.atLeast_Suc_lessThan
thf(fact_5769_sum_OatLeastLessThan__Suc,axiom,
! [A: $tType] :
( ( comm_monoid_add @ A )
=> ! [A2: nat,B2: nat,G: nat > A] :
( ( ord_less_eq @ nat @ A2 @ B2 )
=> ( ( groups7311177749621191930dd_sum @ nat @ A @ G @ ( set_or7035219750837199246ssThan @ nat @ A2 @ ( suc @ B2 ) ) )
= ( plus_plus @ A @ ( groups7311177749621191930dd_sum @ nat @ A @ G @ ( set_or7035219750837199246ssThan @ nat @ A2 @ B2 ) ) @ ( G @ B2 ) ) ) ) ) ).
% sum.atLeastLessThan_Suc
thf(fact_5770_atLeastLessThan__eq__atLeastAtMost__diff,axiom,
! [A: $tType] :
( ( order @ A )
=> ( ( set_or7035219750837199246ssThan @ A )
= ( ^ [A6: A,B5: A] : ( minus_minus @ ( set @ A ) @ ( set_or1337092689740270186AtMost @ A @ A6 @ B5 ) @ ( insert @ A @ B5 @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ) ).
% atLeastLessThan_eq_atLeastAtMost_diff
thf(fact_5771_prod_OatLeast0__lessThan__Suc,axiom,
! [A: $tType] :
( ( comm_monoid_mult @ A )
=> ! [G: nat > A,N3: nat] :
( ( groups7121269368397514597t_prod @ nat @ A @ G @ ( set_or7035219750837199246ssThan @ nat @ ( zero_zero @ nat ) @ ( suc @ N3 ) ) )
= ( times_times @ A @ ( groups7121269368397514597t_prod @ nat @ A @ G @ ( set_or7035219750837199246ssThan @ nat @ ( zero_zero @ nat ) @ N3 ) ) @ ( G @ N3 ) ) ) ) ).
% prod.atLeast0_lessThan_Suc
thf(fact_5772_prod_OatLeast__Suc__lessThan,axiom,
! [A: $tType] :
( ( comm_monoid_mult @ A )
=> ! [M: nat,N3: nat,G: nat > A] :
( ( ord_less @ nat @ M @ N3 )
=> ( ( groups7121269368397514597t_prod @ nat @ A @ G @ ( set_or7035219750837199246ssThan @ nat @ M @ N3 ) )
= ( times_times @ A @ ( G @ M ) @ ( groups7121269368397514597t_prod @ nat @ A @ G @ ( set_or7035219750837199246ssThan @ nat @ ( suc @ M ) @ N3 ) ) ) ) ) ) ).
% prod.atLeast_Suc_lessThan
thf(fact_5773_prod_OatLeastLessThan__Suc,axiom,
! [A: $tType] :
( ( comm_monoid_mult @ A )
=> ! [A2: nat,B2: nat,G: nat > A] :
( ( ord_less_eq @ nat @ A2 @ B2 )
=> ( ( groups7121269368397514597t_prod @ nat @ A @ G @ ( set_or7035219750837199246ssThan @ nat @ A2 @ ( suc @ B2 ) ) )
= ( times_times @ A @ ( groups7121269368397514597t_prod @ nat @ A @ G @ ( set_or7035219750837199246ssThan @ nat @ A2 @ B2 ) ) @ ( G @ B2 ) ) ) ) ) ).
% prod.atLeastLessThan_Suc
thf(fact_5774_sum_Olast__plus,axiom,
! [A: $tType] :
( ( comm_monoid_add @ A )
=> ! [M: nat,N3: nat,G: nat > A] :
( ( ord_less_eq @ nat @ M @ N3 )
=> ( ( groups7311177749621191930dd_sum @ nat @ A @ G @ ( set_or1337092689740270186AtMost @ nat @ M @ N3 ) )
= ( plus_plus @ A @ ( G @ N3 ) @ ( groups7311177749621191930dd_sum @ nat @ A @ G @ ( set_or7035219750837199246ssThan @ nat @ M @ N3 ) ) ) ) ) ) ).
% sum.last_plus
thf(fact_5775_prod_Olast__plus,axiom,
! [A: $tType] :
( ( comm_monoid_mult @ A )
=> ! [M: nat,N3: nat,G: nat > A] :
( ( ord_less_eq @ nat @ M @ N3 )
=> ( ( groups7121269368397514597t_prod @ nat @ A @ G @ ( set_or1337092689740270186AtMost @ nat @ M @ N3 ) )
= ( times_times @ A @ ( G @ N3 ) @ ( groups7121269368397514597t_prod @ nat @ A @ G @ ( set_or7035219750837199246ssThan @ nat @ M @ N3 ) ) ) ) ) ) ).
% prod.last_plus
thf(fact_5776_atLeastLessThanSuc,axiom,
! [M: nat,N3: nat] :
( ( ( ord_less_eq @ nat @ M @ N3 )
=> ( ( set_or7035219750837199246ssThan @ nat @ M @ ( suc @ N3 ) )
= ( insert @ nat @ N3 @ ( set_or7035219750837199246ssThan @ nat @ M @ N3 ) ) ) )
& ( ~ ( ord_less_eq @ nat @ M @ N3 )
=> ( ( set_or7035219750837199246ssThan @ nat @ M @ ( suc @ N3 ) )
= ( bot_bot @ ( set @ nat ) ) ) ) ) ).
% atLeastLessThanSuc
thf(fact_5777_le__mask__high__bits,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [W: word @ A,N3: nat] :
( ( ord_less_eq @ ( word @ A ) @ W @ ( bit_se2239418461657761734s_mask @ ( word @ A ) @ N3 ) )
= ( ! [X2: nat] :
( ( member @ nat @ X2 @ ( set_or7035219750837199246ssThan @ nat @ N3 @ ( size_size @ ( word @ A ) @ W ) ) )
=> ~ ( bit_se5641148757651400278ts_bit @ ( word @ A ) @ W @ X2 ) ) ) ) ) ).
% le_mask_high_bits
thf(fact_5778_listI__assn__conv,axiom,
! [A: $tType,B: $tType,N3: nat,Xs2: list @ A,A3: A > B > assn,Xsi: list @ B] :
( ( N3
= ( size_size @ ( list @ A ) @ Xs2 ) )
=> ( ( vEBT_List_listI_assn @ A @ B @ ( set_or7035219750837199246ssThan @ nat @ ( zero_zero @ nat ) @ N3 ) @ A3 @ Xs2 @ Xsi )
= ( vEBT_List_list_assn @ A @ B @ A3 @ Xs2 @ Xsi ) ) ) ).
% listI_assn_conv
thf(fact_5779_list__assn__conv__idx,axiom,
! [B: $tType,A: $tType] :
( ( vEBT_List_list_assn @ A @ B )
= ( ^ [A7: A > B > assn,Xs: list @ A] : ( vEBT_List_listI_assn @ A @ B @ ( set_or7035219750837199246ssThan @ nat @ ( zero_zero @ nat ) @ ( size_size @ ( list @ A ) @ Xs ) ) @ A7 @ Xs ) ) ) ).
% list_assn_conv_idx
thf(fact_5780_sum__Suc__diff_H,axiom,
! [A: $tType] :
( ( ab_group_add @ A )
=> ! [M: nat,N3: nat,F2: nat > A] :
( ( ord_less_eq @ nat @ M @ N3 )
=> ( ( groups7311177749621191930dd_sum @ nat @ A
@ ^ [I3: nat] : ( minus_minus @ A @ ( F2 @ ( suc @ I3 ) ) @ ( F2 @ I3 ) )
@ ( set_or7035219750837199246ssThan @ nat @ M @ N3 ) )
= ( minus_minus @ A @ ( F2 @ N3 ) @ ( F2 @ M ) ) ) ) ) ).
% sum_Suc_diff'
thf(fact_5781_sum_OatLeastLessThan__rev,axiom,
! [A: $tType] :
( ( comm_monoid_add @ A )
=> ! [G: nat > A,N3: nat,M: nat] :
( ( groups7311177749621191930dd_sum @ nat @ A @ G @ ( set_or7035219750837199246ssThan @ nat @ N3 @ M ) )
= ( groups7311177749621191930dd_sum @ nat @ A
@ ^ [I3: nat] : ( G @ ( minus_minus @ nat @ ( plus_plus @ nat @ M @ N3 ) @ ( suc @ I3 ) ) )
@ ( set_or7035219750837199246ssThan @ nat @ N3 @ M ) ) ) ) ).
% sum.atLeastLessThan_rev
thf(fact_5782_sum_Onested__swap,axiom,
! [A: $tType] :
( ( comm_monoid_add @ A )
=> ! [A2: nat > nat > A,N3: nat] :
( ( groups7311177749621191930dd_sum @ nat @ A
@ ^ [I3: nat] : ( groups7311177749621191930dd_sum @ nat @ A @ ( A2 @ I3 ) @ ( set_or7035219750837199246ssThan @ nat @ ( zero_zero @ nat ) @ I3 ) )
@ ( set_or1337092689740270186AtMost @ nat @ ( zero_zero @ nat ) @ N3 ) )
= ( groups7311177749621191930dd_sum @ nat @ A
@ ^ [J3: nat] :
( groups7311177749621191930dd_sum @ nat @ A
@ ^ [I3: nat] : ( A2 @ I3 @ J3 )
@ ( set_or1337092689740270186AtMost @ nat @ ( suc @ J3 ) @ N3 ) )
@ ( set_or7035219750837199246ssThan @ nat @ ( zero_zero @ nat ) @ N3 ) ) ) ) ).
% sum.nested_swap
thf(fact_5783_prod_OatLeastLessThan__rev,axiom,
! [A: $tType] :
( ( comm_monoid_mult @ A )
=> ! [G: nat > A,N3: nat,M: nat] :
( ( groups7121269368397514597t_prod @ nat @ A @ G @ ( set_or7035219750837199246ssThan @ nat @ N3 @ M ) )
= ( groups7121269368397514597t_prod @ nat @ A
@ ^ [I3: nat] : ( G @ ( minus_minus @ nat @ ( plus_plus @ nat @ M @ N3 ) @ ( suc @ I3 ) ) )
@ ( set_or7035219750837199246ssThan @ nat @ N3 @ M ) ) ) ) ).
% prod.atLeastLessThan_rev
thf(fact_5784_prod_Onested__swap,axiom,
! [A: $tType] :
( ( comm_monoid_mult @ A )
=> ! [A2: nat > nat > A,N3: nat] :
( ( groups7121269368397514597t_prod @ nat @ A
@ ^ [I3: nat] : ( groups7121269368397514597t_prod @ nat @ A @ ( A2 @ I3 ) @ ( set_or7035219750837199246ssThan @ nat @ ( zero_zero @ nat ) @ I3 ) )
@ ( set_or1337092689740270186AtMost @ nat @ ( zero_zero @ nat ) @ N3 ) )
= ( groups7121269368397514597t_prod @ nat @ A
@ ^ [J3: nat] :
( groups7121269368397514597t_prod @ nat @ A
@ ^ [I3: nat] : ( A2 @ I3 @ J3 )
@ ( set_or1337092689740270186AtMost @ nat @ ( suc @ J3 ) @ N3 ) )
@ ( set_or7035219750837199246ssThan @ nat @ ( zero_zero @ nat ) @ N3 ) ) ) ) ).
% prod.nested_swap
thf(fact_5785_sum_Onat__group,axiom,
! [A: $tType] :
( ( comm_monoid_add @ A )
=> ! [G: nat > A,K: nat,N3: nat] :
( ( groups7311177749621191930dd_sum @ nat @ A
@ ^ [M5: nat] : ( groups7311177749621191930dd_sum @ nat @ A @ G @ ( set_or7035219750837199246ssThan @ nat @ ( times_times @ nat @ M5 @ K ) @ ( plus_plus @ nat @ ( times_times @ nat @ M5 @ K ) @ K ) ) )
@ ( set_ord_lessThan @ nat @ N3 ) )
= ( groups7311177749621191930dd_sum @ nat @ A @ G @ ( set_ord_lessThan @ nat @ ( times_times @ nat @ N3 @ K ) ) ) ) ) ).
% sum.nat_group
thf(fact_5786_prod_Onat__group,axiom,
! [A: $tType] :
( ( comm_monoid_mult @ A )
=> ! [G: nat > A,K: nat,N3: nat] :
( ( groups7121269368397514597t_prod @ nat @ A
@ ^ [M5: nat] : ( groups7121269368397514597t_prod @ nat @ A @ G @ ( set_or7035219750837199246ssThan @ nat @ ( times_times @ nat @ M5 @ K ) @ ( plus_plus @ nat @ ( times_times @ nat @ M5 @ K ) @ K ) ) )
@ ( set_ord_lessThan @ nat @ N3 ) )
= ( groups7121269368397514597t_prod @ nat @ A @ G @ ( set_ord_lessThan @ nat @ ( times_times @ nat @ N3 @ K ) ) ) ) ) ).
% prod.nat_group
thf(fact_5787_prod__Suc__Suc__fact,axiom,
! [N3: nat] :
( ( groups7121269368397514597t_prod @ nat @ nat @ suc @ ( set_or7035219750837199246ssThan @ nat @ ( suc @ ( zero_zero @ nat ) ) @ N3 ) )
= ( semiring_char_0_fact @ nat @ N3 ) ) ).
% prod_Suc_Suc_fact
thf(fact_5788_prod__Suc__fact,axiom,
! [N3: nat] :
( ( groups7121269368397514597t_prod @ nat @ nat @ suc @ ( set_or7035219750837199246ssThan @ nat @ ( zero_zero @ nat ) @ N3 ) )
= ( semiring_char_0_fact @ nat @ N3 ) ) ).
% prod_Suc_fact
thf(fact_5789_sum_Ohead__if,axiom,
! [A: $tType] :
( ( comm_monoid_add @ A )
=> ! [N3: nat,M: nat,G: nat > A] :
( ( ( ord_less @ nat @ N3 @ M )
=> ( ( groups7311177749621191930dd_sum @ nat @ A @ G @ ( set_or1337092689740270186AtMost @ nat @ M @ N3 ) )
= ( zero_zero @ A ) ) )
& ( ~ ( ord_less @ nat @ N3 @ M )
=> ( ( groups7311177749621191930dd_sum @ nat @ A @ G @ ( set_or1337092689740270186AtMost @ nat @ M @ N3 ) )
= ( plus_plus @ A @ ( groups7311177749621191930dd_sum @ nat @ A @ G @ ( set_or7035219750837199246ssThan @ nat @ M @ N3 ) ) @ ( G @ N3 ) ) ) ) ) ) ).
% sum.head_if
thf(fact_5790_prod_Ohead__if,axiom,
! [A: $tType] :
( ( comm_monoid_mult @ A )
=> ! [N3: nat,M: nat,G: nat > A] :
( ( ( ord_less @ nat @ N3 @ M )
=> ( ( groups7121269368397514597t_prod @ nat @ A @ G @ ( set_or1337092689740270186AtMost @ nat @ M @ N3 ) )
= ( one_one @ A ) ) )
& ( ~ ( ord_less @ nat @ N3 @ M )
=> ( ( groups7121269368397514597t_prod @ nat @ A @ G @ ( set_or1337092689740270186AtMost @ nat @ M @ N3 ) )
= ( times_times @ A @ ( groups7121269368397514597t_prod @ nat @ A @ G @ ( set_or7035219750837199246ssThan @ nat @ M @ N3 ) ) @ ( G @ N3 ) ) ) ) ) ) ).
% prod.head_if
thf(fact_5791_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_5792_listI__assn__conv_H,axiom,
! [B: $tType,A: $tType,N3: nat,Xs2: list @ A,A3: A > B > assn,Xsi: list @ B,F4: assn] :
( ( N3
= ( size_size @ ( list @ A ) @ Xs2 ) )
=> ( ( times_times @ assn @ ( vEBT_List_listI_assn @ A @ B @ ( set_or7035219750837199246ssThan @ nat @ ( zero_zero @ nat ) @ N3 ) @ A3 @ Xs2 @ Xsi ) @ F4 )
= ( times_times @ assn @ ( vEBT_List_list_assn @ A @ B @ A3 @ Xs2 @ Xsi ) @ F4 ) ) ) ).
% listI_assn_conv'
thf(fact_5793_fact__prod__Suc,axiom,
! [A: $tType] :
( ( semiring_char_0 @ A )
=> ( ( semiring_char_0_fact @ A )
= ( ^ [N2: nat] : ( semiring_1_of_nat @ A @ ( groups7121269368397514597t_prod @ nat @ nat @ suc @ ( set_or7035219750837199246ssThan @ nat @ ( zero_zero @ nat ) @ N2 ) ) ) ) ) ) ).
% fact_prod_Suc
thf(fact_5794_sum_OatLeastLessThan__rev__at__least__Suc__atMost,axiom,
! [A: $tType] :
( ( comm_monoid_add @ A )
=> ! [G: nat > A,N3: nat,M: nat] :
( ( groups7311177749621191930dd_sum @ nat @ A @ G @ ( set_or7035219750837199246ssThan @ nat @ N3 @ M ) )
= ( groups7311177749621191930dd_sum @ nat @ A
@ ^ [I3: nat] : ( G @ ( minus_minus @ nat @ ( plus_plus @ nat @ M @ N3 ) @ I3 ) )
@ ( set_or1337092689740270186AtMost @ nat @ ( suc @ N3 ) @ M ) ) ) ) ).
% sum.atLeastLessThan_rev_at_least_Suc_atMost
thf(fact_5795_prod_OatLeastLessThan__rev__at__least__Suc__atMost,axiom,
! [A: $tType] :
( ( comm_monoid_mult @ A )
=> ! [G: nat > A,N3: nat,M: nat] :
( ( groups7121269368397514597t_prod @ nat @ A @ G @ ( set_or7035219750837199246ssThan @ nat @ N3 @ M ) )
= ( groups7121269368397514597t_prod @ nat @ A
@ ^ [I3: nat] : ( G @ ( minus_minus @ nat @ ( plus_plus @ nat @ M @ N3 ) @ I3 ) )
@ ( set_or1337092689740270186AtMost @ nat @ ( suc @ N3 ) @ M ) ) ) ) ).
% prod.atLeastLessThan_rev_at_least_Suc_atMost
thf(fact_5796_pochhammer__prod,axiom,
! [A: $tType] :
( ( comm_semiring_1 @ A )
=> ( ( comm_s3205402744901411588hammer @ A )
= ( ^ [A6: A,N2: nat] :
( groups7121269368397514597t_prod @ nat @ A
@ ^ [I3: nat] : ( plus_plus @ A @ A6 @ ( semiring_1_of_nat @ A @ I3 ) )
@ ( set_or7035219750837199246ssThan @ nat @ ( zero_zero @ nat ) @ N2 ) ) ) ) ) ).
% pochhammer_prod
thf(fact_5797_image__minus__const__atLeastLessThan__nat,axiom,
! [C2: nat,Y: nat,X: nat] :
( ( ( ord_less @ nat @ C2 @ Y )
=> ( ( image @ nat @ nat
@ ^ [I3: nat] : ( minus_minus @ nat @ I3 @ C2 )
@ ( set_or7035219750837199246ssThan @ nat @ X @ Y ) )
= ( set_or7035219750837199246ssThan @ nat @ ( minus_minus @ nat @ X @ C2 ) @ ( minus_minus @ nat @ Y @ C2 ) ) ) )
& ( ~ ( ord_less @ nat @ C2 @ Y )
=> ( ( ( ord_less @ nat @ X @ Y )
=> ( ( image @ nat @ nat
@ ^ [I3: nat] : ( minus_minus @ nat @ I3 @ C2 )
@ ( set_or7035219750837199246ssThan @ nat @ X @ Y ) )
= ( insert @ nat @ ( zero_zero @ nat ) @ ( bot_bot @ ( set @ nat ) ) ) ) )
& ( ~ ( ord_less @ nat @ X @ Y )
=> ( ( image @ nat @ nat
@ ^ [I3: nat] : ( minus_minus @ nat @ I3 @ C2 )
@ ( set_or7035219750837199246ssThan @ nat @ X @ Y ) )
= ( bot_bot @ ( set @ nat ) ) ) ) ) ) ) ).
% image_minus_const_atLeastLessThan_nat
thf(fact_5798_fact__prod__rev,axiom,
! [A: $tType] :
( ( semiring_char_0 @ A )
=> ( ( semiring_char_0_fact @ A )
= ( ^ [N2: nat] : ( semiring_1_of_nat @ A @ ( groups7121269368397514597t_prod @ nat @ nat @ ( minus_minus @ nat @ N2 ) @ ( set_or7035219750837199246ssThan @ nat @ ( zero_zero @ nat ) @ N2 ) ) ) ) ) ) ).
% fact_prod_rev
thf(fact_5799_summable__Cauchy,axiom,
! [A: $tType] :
( ( real_Vector_banach @ A )
=> ( ( summable @ A )
= ( ^ [F7: nat > A] :
! [E4: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ E4 )
=> ? [N8: nat] :
! [M5: nat] :
( ( ord_less_eq @ nat @ N8 @ M5 )
=> ! [N2: nat] : ( ord_less @ real @ ( real_V7770717601297561774m_norm @ A @ ( groups7311177749621191930dd_sum @ nat @ A @ F7 @ ( set_or7035219750837199246ssThan @ nat @ M5 @ N2 ) ) ) @ E4 ) ) ) ) ) ) ).
% summable_Cauchy
thf(fact_5800_sums__group,axiom,
! [A: $tType] :
( ( ( comm_monoid_add @ A )
& ( topolo4958980785337419405_space @ A ) )
=> ! [F2: nat > A,S: A,K: nat] :
( ( sums @ A @ F2 @ S )
=> ( ( ord_less @ nat @ ( zero_zero @ nat ) @ K )
=> ( sums @ A
@ ^ [N2: nat] : ( groups7311177749621191930dd_sum @ nat @ A @ F2 @ ( set_or7035219750837199246ssThan @ nat @ ( times_times @ nat @ N2 @ K ) @ ( plus_plus @ nat @ ( times_times @ nat @ N2 @ K ) @ K ) ) )
@ S ) ) ) ) ).
% sums_group
thf(fact_5801_atLeast1__lessThan__eq__remove0,axiom,
! [N3: nat] :
( ( set_or7035219750837199246ssThan @ nat @ ( suc @ ( zero_zero @ nat ) ) @ N3 )
= ( minus_minus @ ( set @ nat ) @ ( set_ord_lessThan @ nat @ N3 ) @ ( insert @ nat @ ( zero_zero @ nat ) @ ( bot_bot @ ( set @ nat ) ) ) ) ) ).
% atLeast1_lessThan_eq_remove0
thf(fact_5802_fact__split,axiom,
! [A: $tType] :
( ( semiring_char_0 @ A )
=> ! [K: nat,N3: nat] :
( ( ord_less_eq @ nat @ K @ N3 )
=> ( ( semiring_char_0_fact @ A @ N3 )
= ( times_times @ A @ ( semiring_1_of_nat @ A @ ( groups7121269368397514597t_prod @ nat @ nat @ suc @ ( set_or7035219750837199246ssThan @ nat @ ( minus_minus @ nat @ N3 @ K ) @ N3 ) ) ) @ ( semiring_char_0_fact @ A @ ( minus_minus @ nat @ N3 @ K ) ) ) ) ) ) ).
% fact_split
thf(fact_5803_binomial__altdef__of__nat,axiom,
! [A: $tType] :
( ( field_char_0 @ A )
=> ! [K: nat,N3: nat] :
( ( ord_less_eq @ nat @ K @ N3 )
=> ( ( semiring_1_of_nat @ A @ ( binomial @ N3 @ K ) )
= ( groups7121269368397514597t_prod @ nat @ A
@ ^ [I3: nat] : ( divide_divide @ A @ ( semiring_1_of_nat @ A @ ( minus_minus @ nat @ N3 @ I3 ) ) @ ( semiring_1_of_nat @ A @ ( minus_minus @ nat @ K @ I3 ) ) )
@ ( set_or7035219750837199246ssThan @ nat @ ( zero_zero @ nat ) @ K ) ) ) ) ) ).
% binomial_altdef_of_nat
thf(fact_5804_gbinomial__altdef__of__nat,axiom,
! [A: $tType] :
( ( field_char_0 @ A )
=> ( ( gbinomial @ A )
= ( ^ [A6: A,K2: nat] :
( groups7121269368397514597t_prod @ nat @ A
@ ^ [I3: nat] : ( divide_divide @ A @ ( minus_minus @ A @ A6 @ ( semiring_1_of_nat @ A @ I3 ) ) @ ( semiring_1_of_nat @ A @ ( minus_minus @ nat @ K2 @ I3 ) ) )
@ ( set_or7035219750837199246ssThan @ nat @ ( zero_zero @ nat ) @ K2 ) ) ) ) ) ).
% gbinomial_altdef_of_nat
thf(fact_5805_gbinomial__mult__fact_H,axiom,
! [A: $tType] :
( ( field_char_0 @ A )
=> ! [A2: A,K: nat] :
( ( times_times @ A @ ( gbinomial @ A @ A2 @ K ) @ ( semiring_char_0_fact @ A @ K ) )
= ( groups7121269368397514597t_prod @ nat @ A
@ ^ [I3: nat] : ( minus_minus @ A @ A2 @ ( semiring_1_of_nat @ A @ I3 ) )
@ ( set_or7035219750837199246ssThan @ nat @ ( zero_zero @ nat ) @ K ) ) ) ) ).
% gbinomial_mult_fact'
thf(fact_5806_gbinomial__mult__fact,axiom,
! [A: $tType] :
( ( field_char_0 @ A )
=> ! [K: nat,A2: A] :
( ( times_times @ A @ ( semiring_char_0_fact @ A @ K ) @ ( gbinomial @ A @ A2 @ K ) )
= ( groups7121269368397514597t_prod @ nat @ A
@ ^ [I3: nat] : ( minus_minus @ A @ A2 @ ( semiring_1_of_nat @ A @ I3 ) )
@ ( set_or7035219750837199246ssThan @ nat @ ( zero_zero @ nat ) @ K ) ) ) ) ).
% gbinomial_mult_fact
thf(fact_5807_gbinomial__prod__rev,axiom,
! [A: $tType] :
( ( ( semiring_char_0 @ A )
& ( semidom_divide @ A ) )
=> ( ( gbinomial @ A )
= ( ^ [A6: A,K2: nat] :
( divide_divide @ A
@ ( groups7121269368397514597t_prod @ nat @ A
@ ^ [I3: nat] : ( minus_minus @ A @ A6 @ ( semiring_1_of_nat @ A @ I3 ) )
@ ( set_or7035219750837199246ssThan @ nat @ ( zero_zero @ nat ) @ K2 ) )
@ ( semiring_char_0_fact @ A @ K2 ) ) ) ) ) ).
% gbinomial_prod_rev
thf(fact_5808_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_5809_Sum__Ico__nat,axiom,
! [M: nat,N3: nat] :
( ( groups7311177749621191930dd_sum @ nat @ nat
@ ^ [X2: nat] : X2
@ ( set_or7035219750837199246ssThan @ nat @ M @ N3 ) )
= ( divide_divide @ nat @ ( minus_minus @ nat @ ( times_times @ nat @ N3 @ ( minus_minus @ nat @ N3 @ ( one_one @ nat ) ) ) @ ( times_times @ nat @ M @ ( minus_minus @ nat @ M @ ( one_one @ nat ) ) ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ).
% Sum_Ico_nat
thf(fact_5810_rule__at__index,axiom,
! [A: $tType,B: $tType,C: $tType,P: assn,A3: A > B > assn,Xs2: list @ A,Xsi: list @ B,F4: assn,I: nat,C2: heap_Time_Heap @ C,Q6: C > assn,F8: C > assn] :
( ( entails @ P @ ( times_times @ assn @ ( vEBT_List_list_assn @ A @ B @ A3 @ Xs2 @ Xsi ) @ F4 ) )
=> ( ( ord_less @ nat @ I @ ( size_size @ ( list @ A ) @ Xs2 ) )
=> ( ( hoare_hoare_triple @ C @ ( times_times @ assn @ ( times_times @ assn @ ( A3 @ ( nth @ A @ Xs2 @ 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 ) @ Xs2 ) ) @ ( insert @ nat @ I @ ( bot_bot @ ( set @ nat ) ) ) ) @ A3 @ Xs2 @ Xsi ) ) @ F4 ) @ C2 @ Q6 )
=> ( ! [R3: C] : ( entails @ ( Q6 @ R3 ) @ ( times_times @ assn @ ( times_times @ assn @ ( A3 @ ( nth @ A @ Xs2 @ 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 ) @ Xs2 ) ) @ ( insert @ nat @ I @ ( bot_bot @ ( set @ nat ) ) ) ) @ A3 @ Xs2 @ Xsi ) ) @ ( F8 @ R3 ) ) )
=> ( hoare_hoare_triple @ C @ P @ C2
@ ^ [R5: C] : ( times_times @ assn @ ( vEBT_List_list_assn @ A @ B @ A3 @ Xs2 @ Xsi ) @ ( F8 @ R5 ) ) ) ) ) ) ) ).
% rule_at_index
thf(fact_5811_Chebyshev__sum__upper,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [N3: nat,A2: nat > A,B2: nat > A] :
( ! [I2: nat,J2: nat] :
( ( ord_less_eq @ nat @ I2 @ J2 )
=> ( ( ord_less @ nat @ J2 @ N3 )
=> ( ord_less_eq @ A @ ( A2 @ I2 ) @ ( A2 @ J2 ) ) ) )
=> ( ! [I2: nat,J2: nat] :
( ( ord_less_eq @ nat @ I2 @ J2 )
=> ( ( ord_less @ nat @ J2 @ N3 )
=> ( ord_less_eq @ A @ ( B2 @ J2 ) @ ( B2 @ I2 ) ) ) )
=> ( ord_less_eq @ A
@ ( times_times @ A @ ( semiring_1_of_nat @ A @ N3 )
@ ( groups7311177749621191930dd_sum @ nat @ A
@ ^ [K2: nat] : ( times_times @ A @ ( A2 @ K2 ) @ ( B2 @ K2 ) )
@ ( set_or7035219750837199246ssThan @ nat @ ( zero_zero @ nat ) @ N3 ) ) )
@ ( times_times @ A @ ( groups7311177749621191930dd_sum @ nat @ A @ A2 @ ( set_or7035219750837199246ssThan @ nat @ ( zero_zero @ nat ) @ N3 ) ) @ ( groups7311177749621191930dd_sum @ nat @ A @ B2 @ ( set_or7035219750837199246ssThan @ nat @ ( zero_zero @ nat ) @ N3 ) ) ) ) ) ) ) ).
% Chebyshev_sum_upper
thf(fact_5812_Chebyshev__sum__upper__nat,axiom,
! [N3: nat,A2: nat > nat,B2: nat > nat] :
( ! [I2: nat,J2: nat] :
( ( ord_less_eq @ nat @ I2 @ J2 )
=> ( ( ord_less @ nat @ J2 @ N3 )
=> ( ord_less_eq @ nat @ ( A2 @ I2 ) @ ( A2 @ J2 ) ) ) )
=> ( ! [I2: nat,J2: nat] :
( ( ord_less_eq @ nat @ I2 @ J2 )
=> ( ( ord_less @ nat @ J2 @ N3 )
=> ( ord_less_eq @ nat @ ( B2 @ J2 ) @ ( B2 @ I2 ) ) ) )
=> ( ord_less_eq @ nat
@ ( times_times @ nat @ N3
@ ( groups7311177749621191930dd_sum @ nat @ nat
@ ^ [I3: nat] : ( times_times @ nat @ ( A2 @ I3 ) @ ( B2 @ I3 ) )
@ ( set_or7035219750837199246ssThan @ nat @ ( zero_zero @ nat ) @ N3 ) ) )
@ ( times_times @ nat @ ( groups7311177749621191930dd_sum @ nat @ nat @ A2 @ ( set_or7035219750837199246ssThan @ nat @ ( zero_zero @ nat ) @ N3 ) ) @ ( groups7311177749621191930dd_sum @ nat @ nat @ B2 @ ( set_or7035219750837199246ssThan @ nat @ ( zero_zero @ nat ) @ N3 ) ) ) ) ) ) ).
% Chebyshev_sum_upper_nat
thf(fact_5813_heaphelp,axiom,
! [A: $tType,Xa: array @ vEBT_VEBTi,Tree_is: list @ vEBT_VEBTi,TreeList: list @ vEBT_VEBT,Summary: vEBT_VEBT,Xb: vEBT_VEBTi,N3: 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 ) )
& ( N3 = N3 ) ) ) )
@ ( pure_assn
@ ( Xc
= ( vEBT_Nodei @ ( none @ ( product_prod @ nat @ nat ) ) @ N3 @ Xa @ Xb ) ) ) )
@ H2 )
=> ( rep_assn @ ( vEBT_vebt_assn_raw @ ( vEBT_Node @ ( none @ ( product_prod @ nat @ nat ) ) @ N3 @ TreeList @ Summary ) @ Xc ) @ H2 ) ) ).
% heaphelp
thf(fact_5814_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_5815_mod__h__bot__iff_I1_J,axiom,
! [B2: $o,H2: heap_ext @ product_unit] :
( ( rep_assn @ ( pure_assn @ B2 ) @ ( product_Pair @ ( heap_ext @ product_unit ) @ ( set @ nat ) @ H2 @ ( bot_bot @ ( set @ nat ) ) ) )
= B2 ) ).
% mod_h_bot_iff(1)
thf(fact_5816_mod__h__bot__iff_I4_J,axiom,
! [B: $tType] :
( ( heap @ B )
=> ! [Q2: array @ B,Y: list @ B,H2: heap_ext @ product_unit] :
~ ( rep_assn @ ( snga_assn @ B @ Q2 @ Y ) @ ( product_Pair @ ( heap_ext @ product_unit ) @ ( set @ nat ) @ H2 @ ( bot_bot @ ( set @ nat ) ) ) ) ) ).
% mod_h_bot_iff(4)
thf(fact_5817_ent__pure__post__iff__sng,axiom,
! [P: assn,B2: $o] :
( ( entails @ P @ ( pure_assn @ B2 ) )
= ( ! [H: product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat )] :
( ( rep_assn @ P @ H )
=> B2 )
& ( entails @ P @ ( one_one @ assn ) ) ) ) ).
% ent_pure_post_iff_sng
thf(fact_5818_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_5819_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_5820_mod__h__bot__indep,axiom,
! [P: assn,H2: heap_ext @ product_unit,H3: heap_ext @ product_unit] :
( ( rep_assn @ P @ ( product_Pair @ ( heap_ext @ product_unit ) @ ( set @ nat ) @ H2 @ ( bot_bot @ ( set @ nat ) ) ) )
= ( rep_assn @ P @ ( product_Pair @ ( heap_ext @ product_unit ) @ ( set @ nat ) @ H3 @ ( bot_bot @ ( set @ nat ) ) ) ) ) ).
% mod_h_bot_indep
thf(fact_5821_extract__pre__list__assn__lengthD,axiom,
! [B: $tType,A: $tType,A3: A > B > assn,Xs2: list @ A,Xsi: list @ B,H2: product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat )] :
( ( rep_assn @ ( vEBT_List_list_assn @ A @ B @ A3 @ Xs2 @ Xsi ) @ H2 )
=> ( ( size_size @ ( list @ B ) @ Xsi )
= ( size_size @ ( list @ A ) @ Xs2 ) ) ) ).
% extract_pre_list_assn_lengthD
thf(fact_5822_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_5823_atLeastLessThanPlusOne__atLeastAtMost__integer,axiom,
! [L2: code_integer,U: code_integer] :
( ( set_or7035219750837199246ssThan @ code_integer @ L2 @ ( plus_plus @ code_integer @ U @ ( one_one @ code_integer ) ) )
= ( set_or1337092689740270186AtMost @ code_integer @ L2 @ U ) ) ).
% atLeastLessThanPlusOne_atLeastAtMost_integer
thf(fact_5824_atLeastLessThanPlusOne__atLeastAtMost__int,axiom,
! [L2: int,U: int] :
( ( set_or7035219750837199246ssThan @ int @ L2 @ ( plus_plus @ int @ U @ ( one_one @ int ) ) )
= ( set_or1337092689740270186AtMost @ int @ L2 @ U ) ) ).
% atLeastLessThanPlusOne_atLeastAtMost_int
thf(fact_5825_image__add__integer__atLeastLessThan,axiom,
! [L2: code_integer,U: code_integer] :
( ( image @ code_integer @ code_integer
@ ^ [X2: code_integer] : ( plus_plus @ code_integer @ X2 @ L2 )
@ ( set_or7035219750837199246ssThan @ code_integer @ ( zero_zero @ code_integer ) @ ( minus_minus @ code_integer @ U @ L2 ) ) )
= ( set_or7035219750837199246ssThan @ code_integer @ L2 @ U ) ) ).
% image_add_integer_atLeastLessThan
thf(fact_5826_htt__def,axiom,
! [A: $tType] :
( ( time_htt @ A )
= ( ^ [P3: assn,C3: heap_Time_Heap @ A,Q7: A > assn,T3: nat] :
( ( hoare_hoare_triple @ A @ P3 @ C3 @ Q7 )
& ! [H: heap_ext @ product_unit,As2: set @ nat] :
( ( rep_assn @ P3 @ ( product_Pair @ ( heap_ext @ product_unit ) @ ( set @ nat ) @ H @ As2 ) )
=> ( ord_less_eq @ nat @ ( time_time @ A @ C3 @ H ) @ T3 ) ) ) ) ) ).
% htt_def
thf(fact_5827_httI,axiom,
! [A: $tType,P: assn,C2: heap_Time_Heap @ A,Q: A > assn,T2: nat] :
( ( hoare_hoare_triple @ A @ P @ C2 @ Q )
=> ( ! [H4: heap_ext @ product_unit,As: set @ nat] :
( ( rep_assn @ P @ ( product_Pair @ ( heap_ext @ product_unit ) @ ( set @ nat ) @ H4 @ As ) )
=> ( ord_less_eq @ nat @ ( time_time @ A @ C2 @ H4 ) @ T2 ) )
=> ( time_htt @ A @ P @ C2 @ Q @ T2 ) ) ) ).
% httI
thf(fact_5828_word__atLeastLessThan__Suc__atLeastAtMost,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [U: word @ A,L2: word @ A] :
( ( U
!= ( uminus_uminus @ ( word @ A ) @ ( one_one @ ( word @ A ) ) ) )
=> ( ( set_or7035219750837199246ssThan @ ( word @ A ) @ L2 @ ( plus_plus @ ( word @ A ) @ U @ ( one_one @ ( word @ A ) ) ) )
= ( set_or1337092689740270186AtMost @ ( word @ A ) @ L2 @ U ) ) ) ) ).
% word_atLeastLessThan_Suc_atLeastAtMost
thf(fact_5829_word__range__minus__1,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [B2: word @ A,A2: word @ A] :
( ( B2
!= ( zero_zero @ ( word @ A ) ) )
=> ( ( set_or1337092689740270186AtMost @ ( word @ A ) @ A2 @ ( minus_minus @ ( word @ A ) @ B2 @ ( one_one @ ( word @ A ) ) ) )
= ( set_or7035219750837199246ssThan @ ( word @ A ) @ A2 @ B2 ) ) ) ) ).
% word_range_minus_1
thf(fact_5830_image__add__int__atLeastLessThan,axiom,
! [L2: int,U: int] :
( ( image @ int @ int
@ ^ [X2: int] : ( plus_plus @ int @ X2 @ L2 )
@ ( set_or7035219750837199246ssThan @ int @ ( zero_zero @ int ) @ ( minus_minus @ int @ U @ L2 ) ) )
= ( set_or7035219750837199246ssThan @ int @ L2 @ U ) ) ).
% image_add_int_atLeastLessThan
thf(fact_5831_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_5832_nth__rule,axiom,
! [A: $tType] :
( ( heap @ A )
=> ! [I: nat,Xs2: list @ A,A2: array @ A] :
( ( ord_less @ nat @ I @ ( size_size @ ( list @ A ) @ Xs2 ) )
=> ( hoare_hoare_triple @ A @ ( snga_assn @ A @ A2 @ Xs2 ) @ ( array_nth @ A @ A2 @ I )
@ ^ [R5: A] :
( times_times @ assn @ ( snga_assn @ A @ A2 @ Xs2 )
@ ( pure_assn
@ ( R5
= ( nth @ A @ Xs2 @ I ) ) ) ) ) ) ) ).
% nth_rule
thf(fact_5833_vebt__assn__raw_Oelims,axiom,
! [X: vEBT_VEBT,Xa: vEBT_VEBTi,Y: assn] :
( ( ( vEBT_vebt_assn_raw @ X @ Xa )
= Y )
=> ( ! [A4: $o,B3: $o] :
( ( X
= ( vEBT_Leaf @ A4 @ B3 ) )
=> ! [Ai: $o,Bi: $o] :
( ( Xa
= ( vEBT_Leafi @ Ai @ Bi ) )
=> ( Y
!= ( pure_assn
@ ( ( Ai = A4 )
& ( Bi = B3 ) ) ) ) ) )
=> ( ! [Mmo: option @ ( product_prod @ nat @ nat ),Deg2: nat,Tree_list: list @ vEBT_VEBT,Summary2: vEBT_VEBT] :
( ( X
= ( 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 ) )
=> ( Y
!= ( 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] :
( X
= ( vEBT_Node @ V3 @ Va3 @ Vb3 @ Vc3 ) )
=> ( ? [Vd3: $o,Ve3: $o] :
( Xa
= ( vEBT_Leafi @ Vd3 @ Ve3 ) )
=> ( Y
!= ( bot_bot @ assn ) ) ) )
=> ~ ( ? [Vd3: $o,Ve3: $o] :
( X
= ( vEBT_Leaf @ Vd3 @ Ve3 ) )
=> ( ? [V3: option @ ( product_prod @ nat @ nat ),Va3: nat,Vb3: array @ vEBT_VEBTi,Vc3: vEBT_VEBTi] :
( Xa
= ( vEBT_Nodei @ V3 @ Va3 @ Vb3 @ Vc3 ) )
=> ( Y
!= ( bot_bot @ assn ) ) ) ) ) ) ) ) ).
% vebt_assn_raw.elims
thf(fact_5834_length__rule,axiom,
! [A: $tType] :
( ( heap @ A )
=> ! [A2: array @ A,Xs2: list @ A] :
( hoare_hoare_triple @ nat @ ( snga_assn @ A @ A2 @ Xs2 ) @ ( array_len @ A @ A2 )
@ ^ [R5: nat] :
( times_times @ assn @ ( snga_assn @ A @ A2 @ Xs2 )
@ ( pure_assn
@ ( R5
= ( size_size @ ( list @ A ) @ Xs2 ) ) ) ) ) ) ).
% length_rule
thf(fact_5835_finite__atLeastLessThan__integer,axiom,
! [L2: code_integer,U: code_integer] : ( finite_finite2 @ code_integer @ ( set_or7035219750837199246ssThan @ code_integer @ L2 @ U ) ) ).
% finite_atLeastLessThan_integer
thf(fact_5836_finite__atLeastAtMost__integer,axiom,
! [L2: code_integer,U: code_integer] : ( finite_finite2 @ code_integer @ ( set_or1337092689740270186AtMost @ code_integer @ L2 @ U ) ) ).
% finite_atLeastAtMost_integer
thf(fact_5837_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 ) ) ) )
= ( ? [X2: C] : ( rep_assn @ ( R @ X2 ) @ ( product_Pair @ ( heap_ext @ product_unit ) @ ( set @ nat ) @ H2 @ ( bot_bot @ ( set @ nat ) ) ) ) ) ) ).
% mod_h_bot_iff(8)
thf(fact_5838_time__array__len,axiom,
! [A: $tType] :
( ( heap @ A )
=> ! [P4: array @ A,H2: heap_ext @ product_unit] :
( ( time_time @ nat @ ( array_len @ A @ P4 ) @ H2 )
= ( one_one @ nat ) ) ) ).
% time_array_len
thf(fact_5839_times__assn__raw_Ocases,axiom,
! [X: product_prod @ ( ( product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ) ) > $o ) @ ( product_prod @ ( ( product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ) ) > $o ) @ ( product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ) ) )] :
~ ! [P8: ( product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ) ) > $o,Q8: ( product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ) ) > $o,H4: heap_ext @ product_unit,As: set @ nat] :
( X
!= ( product_Pair @ ( ( product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ) ) > $o ) @ ( product_prod @ ( ( product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ) ) > $o ) @ ( product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ) ) ) @ P8 @ ( product_Pair @ ( ( product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ) ) > $o ) @ ( product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ) ) @ Q8 @ ( product_Pair @ ( heap_ext @ product_unit ) @ ( set @ nat ) @ H4 @ As ) ) ) ) ).
% times_assn_raw.cases
thf(fact_5840_sngr__assn__raw_Ocases,axiom,
! [A: $tType] :
( ( heap @ A )
=> ! [X: product_prod @ ( ref @ A ) @ ( product_prod @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ) ) )] :
~ ! [R3: ref @ A,X3: A,H4: heap_ext @ product_unit,As: set @ nat] :
( X
!= ( product_Pair @ ( ref @ A ) @ ( product_prod @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ) ) ) @ R3 @ ( product_Pair @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ) ) @ X3 @ ( product_Pair @ ( heap_ext @ product_unit ) @ ( set @ nat ) @ H4 @ As ) ) ) ) ) ).
% sngr_assn_raw.cases
thf(fact_5841_snga__assn__raw_Ocases,axiom,
! [A: $tType] :
( ( heap @ A )
=> ! [X: product_prod @ ( array @ A ) @ ( product_prod @ ( list @ A ) @ ( product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ) ) )] :
~ ! [R3: array @ A,X3: list @ A,H4: heap_ext @ product_unit,As: set @ nat] :
( X
!= ( product_Pair @ ( array @ A ) @ ( product_prod @ ( list @ A ) @ ( product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ) ) ) @ R3 @ ( product_Pair @ ( list @ A ) @ ( product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ) ) @ X3 @ ( product_Pair @ ( heap_ext @ product_unit ) @ ( set @ nat ) @ H4 @ As ) ) ) ) ) ).
% snga_assn_raw.cases
thf(fact_5842_one__assn__raw_Ocases,axiom,
! [X: product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat )] :
~ ! [H4: heap_ext @ product_unit,As: set @ nat] :
( X
!= ( product_Pair @ ( heap_ext @ product_unit ) @ ( set @ nat ) @ H4 @ As ) ) ).
% one_assn_raw.cases
thf(fact_5843_TBOUND__nth,axiom,
! [A: $tType] :
( ( heap @ A )
=> ! [Xs2: array @ A,I: nat] : ( time_TBOUND @ A @ ( array_nth @ A @ Xs2 @ I ) @ ( one_one @ nat ) ) ) ).
% TBOUND_nth
thf(fact_5844_TBOUND__len,axiom,
! [A: $tType] :
( ( heap @ A )
=> ! [Xs2: array @ A] : ( time_TBOUND @ nat @ ( array_len @ A @ Xs2 ) @ ( one_one @ nat ) ) ) ).
% TBOUND_len
thf(fact_5845_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_5846_vebt__assn__raw_Opelims,axiom,
! [X: vEBT_VEBT,Xa: vEBT_VEBTi,Y: assn] :
( ( ( vEBT_vebt_assn_raw @ X @ Xa )
= Y )
=> ( ( accp @ ( product_prod @ vEBT_VEBT @ vEBT_VEBTi ) @ vEBT_v8524038756793281170aw_rel @ ( product_Pair @ vEBT_VEBT @ vEBT_VEBTi @ X @ Xa ) )
=> ( ! [A4: $o,B3: $o] :
( ( X
= ( vEBT_Leaf @ A4 @ B3 ) )
=> ! [Ai: $o,Bi: $o] :
( ( Xa
= ( vEBT_Leafi @ Ai @ Bi ) )
=> ( ( Y
= ( pure_assn
@ ( ( Ai = A4 )
& ( Bi = B3 ) ) ) )
=> ~ ( accp @ ( product_prod @ vEBT_VEBT @ vEBT_VEBTi ) @ vEBT_v8524038756793281170aw_rel @ ( product_Pair @ vEBT_VEBT @ vEBT_VEBTi @ ( vEBT_Leaf @ A4 @ B3 ) @ ( vEBT_Leafi @ Ai @ Bi ) ) ) ) ) )
=> ( ! [Mmo: option @ ( product_prod @ nat @ nat ),Deg2: nat,Tree_list: list @ vEBT_VEBT,Summary2: vEBT_VEBT] :
( ( X
= ( 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 ) )
=> ( ( Y
= ( 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] :
( ( X
= ( vEBT_Node @ V3 @ Va3 @ Vb3 @ Vc3 ) )
=> ! [Vd3: $o,Ve3: $o] :
( ( Xa
= ( vEBT_Leafi @ Vd3 @ Ve3 ) )
=> ( ( Y
= ( bot_bot @ assn ) )
=> ~ ( accp @ ( product_prod @ vEBT_VEBT @ vEBT_VEBTi ) @ vEBT_v8524038756793281170aw_rel @ ( product_Pair @ vEBT_VEBT @ vEBT_VEBTi @ ( vEBT_Node @ V3 @ Va3 @ Vb3 @ Vc3 ) @ ( vEBT_Leafi @ Vd3 @ Ve3 ) ) ) ) ) )
=> ~ ! [Vd3: $o,Ve3: $o] :
( ( X
= ( vEBT_Leaf @ Vd3 @ Ve3 ) )
=> ! [V3: option @ ( product_prod @ nat @ nat ),Va3: nat,Vb3: array @ vEBT_VEBTi,Vc3: vEBT_VEBTi] :
( ( Xa
= ( vEBT_Nodei @ V3 @ Va3 @ Vb3 @ Vc3 ) )
=> ( ( Y
= ( bot_bot @ assn ) )
=> ~ ( accp @ ( product_prod @ vEBT_VEBT @ vEBT_VEBTi ) @ vEBT_v8524038756793281170aw_rel @ ( product_Pair @ vEBT_VEBT @ vEBT_VEBTi @ ( vEBT_Leaf @ Vd3 @ Ve3 ) @ ( vEBT_Nodei @ V3 @ Va3 @ Vb3 @ Vc3 ) ) ) ) ) ) ) ) ) ) ) ).
% vebt_assn_raw.pelims
thf(fact_5847_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_5848_upd__rule,axiom,
! [A: $tType] :
( ( heap @ A )
=> ! [I: nat,Xs2: list @ A,A2: array @ A,X: A] :
( ( ord_less @ nat @ I @ ( size_size @ ( list @ A ) @ Xs2 ) )
=> ( hoare_hoare_triple @ ( array @ A ) @ ( snga_assn @ A @ A2 @ Xs2 ) @ ( array_upd @ A @ I @ X @ A2 )
@ ^ [R5: array @ A] : ( times_times @ assn @ ( snga_assn @ A @ A2 @ ( list_update @ A @ Xs2 @ I @ X ) ) @ ( pure_assn @ ( R5 = A2 ) ) ) ) ) ) ).
% upd_rule
thf(fact_5849_assn__basic__inequalities_I1_J,axiom,
( ( top_top @ assn )
!= ( one_one @ assn ) ) ).
% assn_basic_inequalities(1)
thf(fact_5850_top__greatest,axiom,
! [A: $tType] :
( ( order_top @ A )
=> ! [A2: A] : ( ord_less_eq @ A @ A2 @ ( top_top @ A ) ) ) ).
% top_greatest
thf(fact_5851_top_Oextremum__unique,axiom,
! [A: $tType] :
( ( order_top @ A )
=> ! [A2: A] :
( ( ord_less_eq @ A @ ( top_top @ A ) @ A2 )
= ( A2
= ( top_top @ A ) ) ) ) ).
% top.extremum_unique
thf(fact_5852_top_Oextremum__uniqueI,axiom,
! [A: $tType] :
( ( order_top @ A )
=> ! [A2: A] :
( ( ord_less_eq @ A @ ( top_top @ A ) @ A2 )
=> ( A2
= ( top_top @ A ) ) ) ) ).
% top.extremum_uniqueI
thf(fact_5853_top__option__def,axiom,
! [A: $tType] :
( ( order_top @ A )
=> ( ( top_top @ ( option @ A ) )
= ( some @ A @ ( top_top @ A ) ) ) ) ).
% top_option_def
thf(fact_5854_top_Oextremum__strict,axiom,
! [A: $tType] :
( ( order_top @ A )
=> ! [A2: A] :
~ ( ord_less @ A @ ( top_top @ A ) @ A2 ) ) ).
% top.extremum_strict
thf(fact_5855_top_Onot__eq__extremum,axiom,
! [A: $tType] :
( ( order_top @ A )
=> ! [A2: A] :
( ( A2
!= ( top_top @ A ) )
= ( ord_less @ A @ A2 @ ( top_top @ A ) ) ) ) ).
% top.not_eq_extremum
thf(fact_5856_TBOUND__upd,axiom,
! [A: $tType] :
( ( heap @ A )
=> ! [Xs2: nat,I: A,X: array @ A] : ( time_TBOUND @ ( array @ A ) @ ( array_upd @ A @ Xs2 @ I @ X ) @ ( one_one @ nat ) ) ) ).
% TBOUND_upd
thf(fact_5857_mod__h__bot__iff_I2_J,axiom,
! [H2: heap_ext @ product_unit] : ( rep_assn @ ( top_top @ assn ) @ ( product_Pair @ ( heap_ext @ product_unit ) @ ( set @ nat ) @ H2 @ ( bot_bot @ ( set @ nat ) ) ) ) ).
% mod_h_bot_iff(2)
thf(fact_5858_listI__assn__def,axiom,
! [A: $tType,B: $tType] :
( ( vEBT_List_listI_assn @ A @ B )
= ( ^ [I8: set @ nat,A7: A > B > assn,Xs: list @ A,Xsi3: list @ B] :
( times_times @ assn
@ ( pure_assn
@ ( ( ( size_size @ ( list @ B ) @ Xsi3 )
= ( size_size @ ( list @ A ) @ Xs ) )
& ( ord_less_eq @ ( set @ nat ) @ I8 @ ( set_or7035219750837199246ssThan @ nat @ ( zero_zero @ nat ) @ ( size_size @ ( list @ A ) @ Xs ) ) ) ) )
@ ( finite_fold @ nat @ assn
@ ^ [I3: nat,A6: assn] : ( times_times @ assn @ A6 @ ( A7 @ ( nth @ A @ Xs @ I3 ) @ ( nth @ B @ Xsi3 @ I3 ) ) )
@ ( one_one @ assn )
@ I8 ) ) ) ) ).
% listI_assn_def
thf(fact_5859_new__rule,axiom,
! [A: $tType] :
( ( heap @ A )
=> ! [N3: nat,X: A] :
( hoare_hoare_triple @ ( array @ A ) @ ( one_one @ assn ) @ ( array_new @ A @ N3 @ X )
@ ^ [R5: array @ A] : ( snga_assn @ A @ R5 @ ( replicate @ A @ N3 @ X ) ) ) ) ).
% new_rule
thf(fact_5860_bin__last__integer__nbe,axiom,
( bits_b8758750999018896077nteger
= ( ^ [I3: code_integer] :
( ( modulo_modulo @ code_integer @ I3 @ ( numeral_numeral @ code_integer @ ( bit0 @ one2 ) ) )
!= ( zero_zero @ code_integer ) ) ) ) ).
% bin_last_integer_nbe
thf(fact_5861_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_5862_range__add,axiom,
! [A: $tType] :
( ( group_add @ A )
=> ! [A2: A] :
( ( image @ A @ A @ ( plus_plus @ A @ A2 ) @ ( top_top @ ( set @ A ) ) )
= ( top_top @ ( set @ A ) ) ) ) ).
% range_add
thf(fact_5863_surj__plus,axiom,
! [A: $tType] :
( ( ab_group_add @ A )
=> ! [A2: A] :
( ( image @ A @ A @ ( plus_plus @ A @ A2 ) @ ( top_top @ ( set @ A ) ) )
= ( top_top @ ( set @ A ) ) ) ) ).
% surj_plus
thf(fact_5864_range__diff,axiom,
! [A: $tType] :
( ( group_add @ A )
=> ! [A2: A] :
( ( image @ A @ A @ ( minus_minus @ A @ A2 ) @ ( top_top @ ( set @ A ) ) )
= ( top_top @ ( set @ A ) ) ) ) ).
% range_diff
thf(fact_5865_Diff__UNIV,axiom,
! [A: $tType,A3: set @ A] :
( ( minus_minus @ ( set @ A ) @ A3 @ ( top_top @ ( set @ A ) ) )
= ( bot_bot @ ( set @ A ) ) ) ).
% Diff_UNIV
thf(fact_5866_range__mult,axiom,
! [A2: real] :
( ( ( A2
= ( zero_zero @ real ) )
=> ( ( image @ real @ real @ ( times_times @ real @ A2 ) @ ( top_top @ ( set @ real ) ) )
= ( insert @ real @ ( zero_zero @ real ) @ ( bot_bot @ ( set @ real ) ) ) ) )
& ( ( A2
!= ( zero_zero @ real ) )
=> ( ( image @ real @ real @ ( times_times @ real @ A2 ) @ ( top_top @ ( set @ real ) ) )
= ( top_top @ ( set @ real ) ) ) ) ) ).
% range_mult
thf(fact_5867_surj__diff__right,axiom,
! [A: $tType] :
( ( ab_group_add @ A )
=> ! [A2: A] :
( ( image @ A @ A
@ ^ [X2: A] : ( minus_minus @ A @ X2 @ A2 )
@ ( top_top @ ( set @ A ) ) )
= ( top_top @ ( set @ A ) ) ) ) ).
% surj_diff_right
thf(fact_5868_top__empty__eq2,axiom,
! [B: $tType,A: $tType] :
( ( top_top @ ( A > B > $o ) )
= ( ^ [X2: A,Y2: B] : ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X2 @ Y2 ) @ ( top_top @ ( set @ ( product_prod @ A @ B ) ) ) ) ) ) ).
% top_empty_eq2
thf(fact_5869_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_5870_subset__UNIV,axiom,
! [A: $tType,A3: set @ A] : ( ord_less_eq @ ( set @ A ) @ A3 @ ( top_top @ ( set @ A ) ) ) ).
% subset_UNIV
thf(fact_5871_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_5872_range__subsetD,axiom,
! [B: $tType,A: $tType,F2: B > A,B6: set @ A,I: B] :
( ( ord_less_eq @ ( set @ A ) @ ( image @ B @ A @ F2 @ ( top_top @ ( set @ B ) ) ) @ B6 )
=> ( member @ A @ ( F2 @ I ) @ B6 ) ) ).
% range_subsetD
thf(fact_5873_not__UNIV__le__Icc,axiom,
! [A: $tType] :
( ( no_top @ A )
=> ! [L2: A,H2: A] :
~ ( ord_less_eq @ ( set @ A ) @ ( top_top @ ( set @ A ) ) @ ( set_or1337092689740270186AtMost @ A @ L2 @ H2 ) ) ) ).
% not_UNIV_le_Icc
thf(fact_5874_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_5875_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_5876_surj__Compl__image__subset,axiom,
! [A: $tType,B: $tType,F2: B > A,A3: set @ B] :
( ( ( image @ B @ A @ F2 @ ( top_top @ ( set @ B ) ) )
= ( top_top @ ( set @ A ) ) )
=> ( ord_less_eq @ ( set @ A ) @ ( uminus_uminus @ ( set @ A ) @ ( image @ B @ A @ F2 @ A3 ) ) @ ( image @ B @ A @ F2 @ ( uminus_uminus @ ( set @ B ) @ A3 ) ) ) ) ).
% surj_Compl_image_subset
thf(fact_5877_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_5878_notin__range__Some,axiom,
! [A: $tType,X: option @ A] :
( ( ~ ( member @ ( option @ A ) @ X @ ( image @ A @ ( option @ A ) @ ( some @ A ) @ ( top_top @ ( set @ A ) ) ) ) )
= ( X
= ( none @ A ) ) ) ).
% notin_range_Some
thf(fact_5879_time__array__new,axiom,
! [A: $tType] :
( ( heap @ A )
=> ! [N3: nat,X: A,H2: heap_ext @ product_unit] :
( ( time_time @ ( array @ A ) @ ( array_new @ A @ N3 @ X ) @ H2 )
= ( plus_plus @ nat @ N3 @ ( one_one @ nat ) ) ) ) ).
% time_array_new
thf(fact_5880_TBOUND__new,axiom,
! [A: $tType] :
( ( heap @ A )
=> ! [N3: nat,X: A] : ( time_TBOUND @ ( array @ A ) @ ( array_new @ A @ N3 @ X ) @ ( plus_plus @ nat @ N3 @ ( one_one @ nat ) ) ) ) ).
% TBOUND_new
thf(fact_5881_bin__last__integer__code,axiom,
( bits_b8758750999018896077nteger
= ( ^ [I3: code_integer] :
( ( bit_se5824344872417868541ns_and @ code_integer @ I3 @ ( one_one @ code_integer ) )
!= ( zero_zero @ code_integer ) ) ) ) ).
% bin_last_integer_code
thf(fact_5882_range__mod,axiom,
! [N3: nat] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ N3 )
=> ( ( image @ nat @ nat
@ ^ [M5: nat] : ( modulo_modulo @ nat @ M5 @ N3 )
@ ( top_top @ ( set @ nat ) ) )
= ( set_or7035219750837199246ssThan @ nat @ ( zero_zero @ nat ) @ N3 ) ) ) ).
% range_mod
thf(fact_5883_bin__last__integer_Oabs__eq,axiom,
! [X: int] :
( ( bits_b8758750999018896077nteger @ ( code_integer_of_int @ X ) )
= ( ~ ( dvd_dvd @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ X ) ) ) ).
% bin_last_integer.abs_eq
thf(fact_5884_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_5885_bitXOR__integer__unfold,axiom,
( ( bit_se5824344971392196577ns_xor @ code_integer )
= ( ^ [X2: code_integer,Y2: code_integer] :
( if @ code_integer
@ ( X2
= ( zero_zero @ code_integer ) )
@ Y2
@ ( if @ code_integer
@ ( X2
= ( uminus_uminus @ code_integer @ ( one_one @ code_integer ) ) )
@ ( bit_ri4277139882892585799ns_not @ code_integer @ Y2 )
@ ( bits_Bit_integer @ ( bit_se5824344971392196577ns_xor @ code_integer @ ( bits_b2549910563261871055nteger @ X2 ) @ ( bits_b2549910563261871055nteger @ Y2 ) )
@ ( ( ~ ( bits_b8758750999018896077nteger @ X2 ) )
= ( bits_b8758750999018896077nteger @ Y2 ) ) ) ) ) ) ) ).
% bitXOR_integer_unfold
thf(fact_5886_bitAND__integer__unfold,axiom,
( ( bit_se5824344872417868541ns_and @ code_integer )
= ( ^ [X2: code_integer,Y2: code_integer] :
( if @ code_integer
@ ( X2
= ( zero_zero @ code_integer ) )
@ ( zero_zero @ code_integer )
@ ( if @ code_integer
@ ( X2
= ( uminus_uminus @ code_integer @ ( one_one @ code_integer ) ) )
@ Y2
@ ( bits_Bit_integer @ ( bit_se5824344872417868541ns_and @ code_integer @ ( bits_b2549910563261871055nteger @ X2 ) @ ( bits_b2549910563261871055nteger @ Y2 ) )
@ ( ( bits_b8758750999018896077nteger @ X2 )
& ( bits_b8758750999018896077nteger @ Y2 ) ) ) ) ) ) ) ).
% bitAND_integer_unfold
thf(fact_5887_bin__rest__integer__code,axiom,
( bits_b2549910563261871055nteger
= ( ^ [I3: code_integer] : ( divide_divide @ code_integer @ I3 @ ( numeral_numeral @ code_integer @ ( bit0 @ one2 ) ) ) ) ) ).
% bin_rest_integer_code
thf(fact_5888_bin__rest__integer_Oabs__eq,axiom,
! [X: int] :
( ( bits_b2549910563261871055nteger @ ( code_integer_of_int @ X ) )
= ( code_integer_of_int @ ( divide_divide @ int @ X @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) ) ) ) ).
% bin_rest_integer.abs_eq
thf(fact_5889_bitOR__integer__unfold,axiom,
( ( bit_se1065995026697491101ons_or @ code_integer )
= ( ^ [X2: code_integer,Y2: code_integer] :
( if @ code_integer
@ ( X2
= ( zero_zero @ code_integer ) )
@ Y2
@ ( if @ code_integer
@ ( X2
= ( 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 @ X2 ) @ ( bits_b2549910563261871055nteger @ Y2 ) )
@ ( ( bits_b8758750999018896077nteger @ X2 )
| ( bits_b8758750999018896077nteger @ Y2 ) ) ) ) ) ) ) ).
% bitOR_integer_unfold
thf(fact_5890_root__def,axiom,
( root
= ( ^ [N2: nat,X2: real] :
( if @ real
@ ( N2
= ( zero_zero @ nat ) )
@ ( zero_zero @ real )
@ ( the_inv_into @ real @ real @ ( top_top @ ( set @ real ) )
@ ^ [Y2: real] : ( times_times @ real @ ( sgn_sgn @ real @ Y2 ) @ ( power_power @ real @ ( abs_abs @ real @ Y2 ) @ N2 ) )
@ X2 ) ) ) ) ).
% root_def
thf(fact_5891_aux,axiom,
! [B: $tType,A: $tType,P: A > B > assn,A2: A,As3: list @ A,C2: B,Cs: list @ B] :
( ( finite_fold @ nat @ assn
@ ^ [I3: nat,Aa2: assn] : ( times_times @ assn @ Aa2 @ ( P @ ( nth @ A @ ( cons @ A @ A2 @ As3 ) @ I3 ) @ ( nth @ B @ ( cons @ B @ C2 @ Cs ) @ I3 ) ) )
@ ( one_one @ assn )
@ ( set_or7035219750837199246ssThan @ nat @ ( zero_zero @ nat ) @ ( suc @ ( size_size @ ( list @ A ) @ As3 ) ) ) )
= ( times_times @ assn @ ( P @ A2 @ C2 )
@ ( finite_fold @ nat @ assn
@ ^ [I3: nat,Aa2: assn] : ( times_times @ assn @ Aa2 @ ( P @ ( nth @ A @ As3 @ I3 ) @ ( nth @ B @ Cs @ I3 ) ) )
@ ( one_one @ assn )
@ ( set_or7035219750837199246ssThan @ nat @ ( zero_zero @ nat ) @ ( size_size @ ( list @ A ) @ As3 ) ) ) ) ) ).
% aux
thf(fact_5892_sofl__test,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [X: word @ A,Y: word @ A] :
( ( ( plus_plus @ int @ ( ring_1_signed @ A @ int @ X ) @ ( ring_1_signed @ A @ int @ Y ) )
= ( ring_1_signed @ A @ int @ ( plus_plus @ ( word @ A ) @ X @ Y ) ) )
= ( ( bit_se4197421643247451524op_bit @ ( word @ A ) @ ( minus_minus @ nat @ ( size_size @ ( word @ A ) @ X ) @ ( one_one @ nat ) ) @ ( bit_se5824344872417868541ns_and @ ( word @ A ) @ ( bit_se5824344971392196577ns_xor @ ( word @ A ) @ ( plus_plus @ ( word @ A ) @ X @ Y ) @ X ) @ ( bit_se5824344971392196577ns_xor @ ( word @ A ) @ ( plus_plus @ ( word @ A ) @ X @ Y ) @ Y ) ) )
= ( zero_zero @ ( word @ A ) ) ) ) ) ).
% sofl_test
thf(fact_5893_drop__bit__of__0,axiom,
! [A: $tType] :
( ( bit_se359711467146920520ations @ A )
=> ! [N3: nat] :
( ( bit_se4197421643247451524op_bit @ A @ N3 @ ( zero_zero @ A ) )
= ( zero_zero @ A ) ) ) ).
% drop_bit_of_0
thf(fact_5894_drop__bit__drop__bit,axiom,
! [A: $tType] :
( ( bit_se359711467146920520ations @ A )
=> ! [M: nat,N3: nat,A2: A] :
( ( bit_se4197421643247451524op_bit @ A @ M @ ( bit_se4197421643247451524op_bit @ A @ N3 @ A2 ) )
= ( bit_se4197421643247451524op_bit @ A @ ( plus_plus @ nat @ M @ N3 ) @ A2 ) ) ) ).
% drop_bit_drop_bit
thf(fact_5895_drop__bit__and,axiom,
! [A: $tType] :
( ( bit_se359711467146920520ations @ A )
=> ! [N3: nat,A2: A,B2: A] :
( ( bit_se4197421643247451524op_bit @ A @ N3 @ ( bit_se5824344872417868541ns_and @ A @ A2 @ B2 ) )
= ( bit_se5824344872417868541ns_and @ A @ ( bit_se4197421643247451524op_bit @ A @ N3 @ A2 ) @ ( bit_se4197421643247451524op_bit @ A @ N3 @ B2 ) ) ) ) ).
% drop_bit_and
thf(fact_5896_drop__bit__or,axiom,
! [A: $tType] :
( ( bit_se359711467146920520ations @ A )
=> ! [N3: nat,A2: A,B2: A] :
( ( bit_se4197421643247451524op_bit @ A @ N3 @ ( bit_se1065995026697491101ons_or @ A @ A2 @ B2 ) )
= ( bit_se1065995026697491101ons_or @ A @ ( bit_se4197421643247451524op_bit @ A @ N3 @ A2 ) @ ( bit_se4197421643247451524op_bit @ A @ N3 @ B2 ) ) ) ) ).
% drop_bit_or
thf(fact_5897_drop__bit__xor,axiom,
! [A: $tType] :
( ( bit_se359711467146920520ations @ A )
=> ! [N3: nat,A2: A,B2: A] :
( ( bit_se4197421643247451524op_bit @ A @ N3 @ ( bit_se5824344971392196577ns_xor @ A @ A2 @ B2 ) )
= ( bit_se5824344971392196577ns_xor @ A @ ( bit_se4197421643247451524op_bit @ A @ N3 @ A2 ) @ ( bit_se4197421643247451524op_bit @ A @ N3 @ B2 ) ) ) ) ).
% drop_bit_xor
thf(fact_5898_nth__Cons__Suc,axiom,
! [A: $tType,X: A,Xs2: list @ A,N3: nat] :
( ( nth @ A @ ( cons @ A @ X @ Xs2 ) @ ( suc @ N3 ) )
= ( nth @ A @ Xs2 @ N3 ) ) ).
% nth_Cons_Suc
thf(fact_5899_length__nth__simps_I4_J,axiom,
! [B: $tType,X: B,Xs2: list @ B,N3: nat] :
( ( nth @ B @ ( cons @ B @ X @ Xs2 ) @ ( suc @ N3 ) )
= ( nth @ B @ Xs2 @ N3 ) ) ).
% length_nth_simps(4)
thf(fact_5900_nth__Cons__0,axiom,
! [A: $tType,X: A,Xs2: list @ A] :
( ( nth @ A @ ( cons @ A @ X @ Xs2 ) @ ( zero_zero @ nat ) )
= X ) ).
% nth_Cons_0
thf(fact_5901_length__nth__simps_I3_J,axiom,
! [B: $tType,X: B,Xs2: list @ B] :
( ( nth @ B @ ( cons @ B @ X @ Xs2 ) @ ( zero_zero @ nat ) )
= X ) ).
% length_nth_simps(3)
thf(fact_5902_list_Osimps_I15_J,axiom,
! [A: $tType,X21: A,X222: list @ A] :
( ( set2 @ A @ ( cons @ A @ X21 @ X222 ) )
= ( insert @ A @ X21 @ ( set2 @ A @ X222 ) ) ) ).
% list.simps(15)
thf(fact_5903_drop__bit__of__bool,axiom,
! [A: $tType] :
( ( bit_se359711467146920520ations @ A )
=> ! [N3: nat,B2: $o] :
( ( bit_se4197421643247451524op_bit @ A @ N3 @ ( zero_neq_one_of_bool @ A @ B2 ) )
= ( zero_neq_one_of_bool @ A
@ ( ( N3
= ( zero_zero @ nat ) )
& B2 ) ) ) ) ).
% drop_bit_of_bool
thf(fact_5904_drop__bit__Suc__bit0,axiom,
! [A: $tType] :
( ( bit_un5681908812861735899ations @ A )
=> ! [N3: nat,K: num] :
( ( bit_se4197421643247451524op_bit @ A @ ( suc @ N3 ) @ ( numeral_numeral @ A @ ( bit0 @ K ) ) )
= ( bit_se4197421643247451524op_bit @ A @ N3 @ ( numeral_numeral @ A @ K ) ) ) ) ).
% drop_bit_Suc_bit0
thf(fact_5905_drop__bit__Suc__bit1,axiom,
! [A: $tType] :
( ( bit_un5681908812861735899ations @ A )
=> ! [N3: nat,K: num] :
( ( bit_se4197421643247451524op_bit @ A @ ( suc @ N3 ) @ ( numeral_numeral @ A @ ( bit1 @ K ) ) )
= ( bit_se4197421643247451524op_bit @ A @ N3 @ ( numeral_numeral @ A @ K ) ) ) ) ).
% drop_bit_Suc_bit1
thf(fact_5906_drop__bit__of__1,axiom,
! [A: $tType] :
( ( bit_se359711467146920520ations @ A )
=> ! [N3: nat] :
( ( bit_se4197421643247451524op_bit @ A @ N3 @ ( one_one @ A ) )
= ( zero_neq_one_of_bool @ A
@ ( N3
= ( zero_zero @ nat ) ) ) ) ) ).
% drop_bit_of_1
thf(fact_5907_nth__Cons__numeral,axiom,
! [A: $tType,X: A,Xs2: list @ A,V: num] :
( ( nth @ A @ ( cons @ A @ X @ Xs2 ) @ ( numeral_numeral @ nat @ V ) )
= ( nth @ A @ Xs2 @ ( minus_minus @ nat @ ( numeral_numeral @ nat @ V ) @ ( one_one @ nat ) ) ) ) ).
% nth_Cons_numeral
thf(fact_5908_drop__bit__numeral__bit0,axiom,
! [A: $tType] :
( ( bit_un5681908812861735899ations @ A )
=> ! [L2: num,K: num] :
( ( bit_se4197421643247451524op_bit @ A @ ( numeral_numeral @ nat @ L2 ) @ ( numeral_numeral @ A @ ( bit0 @ K ) ) )
= ( bit_se4197421643247451524op_bit @ A @ ( pred_numeral @ L2 ) @ ( numeral_numeral @ A @ K ) ) ) ) ).
% drop_bit_numeral_bit0
thf(fact_5909_drop__bit__numeral__bit1,axiom,
! [A: $tType] :
( ( bit_un5681908812861735899ations @ A )
=> ! [L2: num,K: num] :
( ( bit_se4197421643247451524op_bit @ A @ ( numeral_numeral @ nat @ L2 ) @ ( numeral_numeral @ A @ ( bit1 @ K ) ) )
= ( bit_se4197421643247451524op_bit @ A @ ( pred_numeral @ L2 ) @ ( numeral_numeral @ A @ K ) ) ) ) ).
% drop_bit_numeral_bit1
thf(fact_5910_nth__Cons__pos,axiom,
! [A: $tType,N3: nat,X: A,Xs2: list @ A] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ N3 )
=> ( ( nth @ A @ ( cons @ A @ X @ Xs2 ) @ N3 )
= ( nth @ A @ Xs2 @ ( minus_minus @ nat @ N3 @ ( one_one @ nat ) ) ) ) ) ).
% nth_Cons_pos
thf(fact_5911_drop__bit__mask__eq,axiom,
! [A: $tType] :
( ( bit_un5681908812861735899ations @ A )
=> ! [M: nat,N3: nat] :
( ( bit_se4197421643247451524op_bit @ A @ M @ ( bit_se2239418461657761734s_mask @ A @ N3 ) )
= ( bit_se2239418461657761734s_mask @ A @ ( minus_minus @ nat @ N3 @ M ) ) ) ) ).
% drop_bit_mask_eq
thf(fact_5912_take__bit__drop__bit,axiom,
! [A: $tType] :
( ( bit_se359711467146920520ations @ A )
=> ! [M: nat,N3: nat,A2: A] :
( ( bit_se2584673776208193580ke_bit @ A @ M @ ( bit_se4197421643247451524op_bit @ A @ N3 @ A2 ) )
= ( bit_se4197421643247451524op_bit @ A @ N3 @ ( bit_se2584673776208193580ke_bit @ A @ ( plus_plus @ nat @ M @ N3 ) @ A2 ) ) ) ) ).
% take_bit_drop_bit
thf(fact_5913_drop__bit__take__bit,axiom,
! [A: $tType] :
( ( bit_se359711467146920520ations @ A )
=> ! [M: nat,N3: nat,A2: A] :
( ( bit_se4197421643247451524op_bit @ A @ M @ ( bit_se2584673776208193580ke_bit @ A @ N3 @ A2 ) )
= ( bit_se2584673776208193580ke_bit @ A @ ( minus_minus @ nat @ N3 @ M ) @ ( bit_se4197421643247451524op_bit @ A @ M @ A2 ) ) ) ) ).
% drop_bit_take_bit
thf(fact_5914_list__update__code_I3_J,axiom,
! [A: $tType,X: A,Xs2: list @ A,I: nat,Y: A] :
( ( list_update @ A @ ( cons @ A @ X @ Xs2 ) @ ( suc @ I ) @ Y )
= ( cons @ A @ X @ ( list_update @ A @ Xs2 @ I @ Y ) ) ) ).
% list_update_code(3)
thf(fact_5915_length__nth__simps_I2_J,axiom,
! [B: $tType,X: B,Xs2: list @ B] :
( ( size_size @ ( list @ B ) @ ( cons @ B @ X @ Xs2 ) )
= ( suc @ ( size_size @ ( list @ B ) @ Xs2 ) ) ) ).
% length_nth_simps(2)
thf(fact_5916_Suc__length__conv,axiom,
! [A: $tType,N3: nat,Xs2: list @ A] :
( ( ( suc @ N3 )
= ( size_size @ ( list @ A ) @ Xs2 ) )
= ( ? [Y2: A,Ys3: list @ A] :
( ( Xs2
= ( cons @ A @ Y2 @ Ys3 ) )
& ( ( size_size @ ( list @ A ) @ Ys3 )
= N3 ) ) ) ) ).
% Suc_length_conv
thf(fact_5917_length__Suc__conv,axiom,
! [A: $tType,Xs2: list @ A,N3: nat] :
( ( ( size_size @ ( list @ A ) @ Xs2 )
= ( suc @ N3 ) )
= ( ? [Y2: A,Ys3: list @ A] :
( ( Xs2
= ( cons @ A @ Y2 @ Ys3 ) )
& ( ( size_size @ ( list @ A ) @ Ys3 )
= N3 ) ) ) ) ).
% length_Suc_conv
thf(fact_5918_take__bit__eq__self__iff__drop__bit__eq__0,axiom,
! [A: $tType] :
( ( bit_se359711467146920520ations @ A )
=> ! [N3: nat,A2: A] :
( ( ( bit_se2584673776208193580ke_bit @ A @ N3 @ A2 )
= A2 )
= ( ( bit_se4197421643247451524op_bit @ A @ N3 @ A2 )
= ( zero_zero @ A ) ) ) ) ).
% take_bit_eq_self_iff_drop_bit_eq_0
thf(fact_5919_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,L3: list @ B] :
( ( Ls
= ( cons @ B @ A4 @ L3 ) )
=> ( ( ( F2 @ A4 )
= Fa )
=> ( ( map @ B @ A @ F2 @ L3 )
!= Fl ) ) ) ) ).
% map_eq_consE
thf(fact_5920_map__consI_I1_J,axiom,
! [A: $tType,B: $tType,W: list @ A,F2: B > A,Ww: list @ B,A2: B] :
( ( W
= ( map @ B @ A @ F2 @ Ww ) )
=> ( ( cons @ A @ ( F2 @ A2 ) @ W )
= ( map @ B @ A @ F2 @ ( cons @ B @ A2 @ Ww ) ) ) ) ).
% map_consI(1)
thf(fact_5921_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_5922_set__ConsD,axiom,
! [A: $tType,Y: A,X: A,Xs2: list @ A] :
( ( member @ A @ Y @ ( set2 @ A @ ( cons @ A @ X @ Xs2 ) ) )
=> ( ( Y = X )
| ( member @ A @ Y @ ( set2 @ A @ Xs2 ) ) ) ) ).
% set_ConsD
thf(fact_5923_list_Oset__cases,axiom,
! [A: $tType,E: A,A2: list @ A] :
( ( member @ A @ E @ ( set2 @ A @ A2 ) )
=> ( ! [Z23: list @ A] :
( A2
!= ( cons @ A @ E @ Z23 ) )
=> ~ ! [Z12: A,Z23: list @ A] :
( ( A2
= ( cons @ A @ Z12 @ Z23 ) )
=> ~ ( member @ A @ E @ ( set2 @ A @ Z23 ) ) ) ) ) ).
% list.set_cases
thf(fact_5924_list_Oset__intros_I1_J,axiom,
! [A: $tType,X21: A,X222: list @ A] : ( member @ A @ X21 @ ( set2 @ A @ ( cons @ A @ X21 @ X222 ) ) ) ).
% list.set_intros(1)
thf(fact_5925_list_Oset__intros_I2_J,axiom,
! [A: $tType,Y: A,X222: list @ A,X21: A] :
( ( member @ A @ Y @ ( set2 @ A @ X222 ) )
=> ( member @ A @ Y @ ( set2 @ A @ ( cons @ A @ X21 @ X222 ) ) ) ) ).
% list.set_intros(2)
thf(fact_5926_drop__bit__of__nat,axiom,
! [A: $tType] :
( ( bit_un5681908812861735899ations @ A )
=> ! [N3: nat,M: nat] :
( ( bit_se4197421643247451524op_bit @ A @ N3 @ ( semiring_1_of_nat @ A @ M ) )
= ( semiring_1_of_nat @ A @ ( bit_se4197421643247451524op_bit @ nat @ N3 @ M ) ) ) ) ).
% drop_bit_of_nat
thf(fact_5927_of__nat__drop__bit,axiom,
! [A: $tType] :
( ( bit_un5681908812861735899ations @ A )
=> ! [M: nat,N3: nat] :
( ( semiring_1_of_nat @ A @ ( bit_se4197421643247451524op_bit @ nat @ M @ N3 ) )
= ( bit_se4197421643247451524op_bit @ A @ M @ ( semiring_1_of_nat @ A @ N3 ) ) ) ) ).
% of_nat_drop_bit
thf(fact_5928_set__subset__Cons,axiom,
! [A: $tType,Xs2: list @ A,X: A] : ( ord_less_eq @ ( set @ A ) @ ( set2 @ A @ Xs2 ) @ ( set2 @ A @ ( cons @ A @ X @ Xs2 ) ) ) ).
% set_subset_Cons
thf(fact_5929_impossible__Cons,axiom,
! [A: $tType,Xs2: list @ A,Ys: list @ A,X: A] :
( ( ord_less_eq @ nat @ ( size_size @ ( list @ A ) @ Xs2 ) @ ( size_size @ ( list @ A ) @ Ys ) )
=> ( Xs2
!= ( cons @ A @ X @ Ys ) ) ) ).
% impossible_Cons
thf(fact_5930_replicate__Suc,axiom,
! [A: $tType,N3: nat,X: A] :
( ( replicate @ A @ ( suc @ N3 ) @ X )
= ( cons @ A @ X @ ( replicate @ A @ N3 @ X ) ) ) ).
% replicate_Suc
thf(fact_5931_bit__word__iff__drop__bit__and,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ( ( bit_se5641148757651400278ts_bit @ ( word @ A ) )
= ( ^ [A6: word @ A,N2: nat] :
( ( bit_se5824344872417868541ns_and @ ( word @ A ) @ ( bit_se4197421643247451524op_bit @ ( word @ A ) @ N2 @ A6 ) @ ( one_one @ ( word @ A ) ) )
= ( one_one @ ( word @ A ) ) ) ) ) ) ).
% bit_word_iff_drop_bit_and
thf(fact_5932_Suc__le__length__iff,axiom,
! [A: $tType,N3: nat,Xs2: list @ A] :
( ( ord_less_eq @ nat @ ( suc @ N3 ) @ ( size_size @ ( list @ A ) @ Xs2 ) )
= ( ? [X2: A,Ys3: list @ A] :
( ( Xs2
= ( cons @ A @ X2 @ Ys3 ) )
& ( ord_less_eq @ nat @ N3 @ ( size_size @ ( list @ A ) @ Ys3 ) ) ) ) ) ).
% Suc_le_length_iff
thf(fact_5933_bit__iff__and__drop__bit__eq__1,axiom,
! [A: $tType] :
( ( bit_se359711467146920520ations @ A )
=> ( ( bit_se5641148757651400278ts_bit @ A )
= ( ^ [A6: A,N2: nat] :
( ( bit_se5824344872417868541ns_and @ A @ ( bit_se4197421643247451524op_bit @ A @ N2 @ A6 ) @ ( one_one @ A ) )
= ( one_one @ A ) ) ) ) ) ).
% bit_iff_and_drop_bit_eq_1
thf(fact_5934_count__list_Osimps_I2_J,axiom,
! [A: $tType,X: A,Y: A,Xs2: list @ A] :
( ( ( X = Y )
=> ( ( count_list @ A @ ( cons @ A @ X @ Xs2 ) @ Y )
= ( plus_plus @ nat @ ( count_list @ A @ Xs2 @ Y ) @ ( one_one @ nat ) ) ) )
& ( ( X != Y )
=> ( ( count_list @ A @ ( cons @ A @ X @ Xs2 ) @ Y )
= ( count_list @ A @ Xs2 @ Y ) ) ) ) ).
% count_list.simps(2)
thf(fact_5935_stable__imp__drop__bit__eq,axiom,
! [A: $tType] :
( ( bit_se359711467146920520ations @ A )
=> ! [A2: A,N3: nat] :
( ( ( divide_divide @ A @ A2 @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) )
= A2 )
=> ( ( bit_se4197421643247451524op_bit @ A @ N3 @ A2 )
= A2 ) ) ) ).
% stable_imp_drop_bit_eq
thf(fact_5936_drop__bit__half,axiom,
! [A: $tType] :
( ( bit_se359711467146920520ations @ A )
=> ! [N3: nat,A2: A] :
( ( bit_se4197421643247451524op_bit @ A @ N3 @ ( divide_divide @ A @ A2 @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) )
= ( divide_divide @ A @ ( bit_se4197421643247451524op_bit @ A @ N3 @ A2 ) @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) ) ) ).
% drop_bit_half
thf(fact_5937_list_Osize_I4_J,axiom,
! [A: $tType,X21: A,X222: list @ A] :
( ( size_size @ ( list @ A ) @ ( cons @ A @ X21 @ X222 ) )
= ( plus_plus @ nat @ ( size_size @ ( list @ A ) @ X222 ) @ ( suc @ ( zero_zero @ nat ) ) ) ) ).
% list.size(4)
thf(fact_5938_nth__Cons_H,axiom,
! [A: $tType,N3: nat,X: A,Xs2: list @ A] :
( ( ( N3
= ( zero_zero @ nat ) )
=> ( ( nth @ A @ ( cons @ A @ X @ Xs2 ) @ N3 )
= X ) )
& ( ( N3
!= ( zero_zero @ nat ) )
=> ( ( nth @ A @ ( cons @ A @ X @ Xs2 ) @ N3 )
= ( nth @ A @ Xs2 @ ( minus_minus @ nat @ N3 @ ( one_one @ nat ) ) ) ) ) ) ).
% nth_Cons'
thf(fact_5939_drop__bit__Suc,axiom,
! [A: $tType] :
( ( bit_se359711467146920520ations @ A )
=> ! [N3: nat,A2: A] :
( ( bit_se4197421643247451524op_bit @ A @ ( suc @ N3 ) @ A2 )
= ( bit_se4197421643247451524op_bit @ A @ N3 @ ( divide_divide @ A @ A2 @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) ) ) ) ).
% drop_bit_Suc
thf(fact_5940_drop__bit__eq__div,axiom,
! [A: $tType] :
( ( bit_se359711467146920520ations @ A )
=> ( ( bit_se4197421643247451524op_bit @ A )
= ( ^ [N2: nat,A6: A] : ( divide_divide @ A @ A6 @ ( power_power @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ N2 ) ) ) ) ) ).
% drop_bit_eq_div
thf(fact_5941_nth__non__equal__first__eq,axiom,
! [A: $tType,X: A,Y: A,Xs2: list @ A,N3: nat] :
( ( X != Y )
=> ( ( ( nth @ A @ ( cons @ A @ X @ Xs2 ) @ N3 )
= Y )
= ( ( ( nth @ A @ Xs2 @ ( minus_minus @ nat @ N3 @ ( one_one @ nat ) ) )
= Y )
& ( ord_less @ nat @ ( zero_zero @ nat ) @ N3 ) ) ) ) ).
% nth_non_equal_first_eq
thf(fact_5942_bit__iff__odd__drop__bit,axiom,
! [A: $tType] :
( ( bit_se359711467146920520ations @ A )
=> ( ( bit_se5641148757651400278ts_bit @ A )
= ( ^ [A6: A,N2: nat] :
~ ( dvd_dvd @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ ( bit_se4197421643247451524op_bit @ A @ N2 @ A6 ) ) ) ) ) ).
% bit_iff_odd_drop_bit
thf(fact_5943_even__drop__bit__iff__not__bit,axiom,
! [A: $tType] :
( ( bit_se359711467146920520ations @ A )
=> ! [N3: nat,A2: A] :
( ( dvd_dvd @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ ( bit_se4197421643247451524op_bit @ A @ N3 @ A2 ) )
= ( ~ ( bit_se5641148757651400278ts_bit @ A @ A2 @ N3 ) ) ) ) ).
% even_drop_bit_iff_not_bit
thf(fact_5944_nth__equal__first__eq,axiom,
! [A: $tType,X: A,Xs2: list @ A,N3: nat] :
( ~ ( member @ A @ X @ ( set2 @ A @ Xs2 ) )
=> ( ( ord_less_eq @ nat @ N3 @ ( size_size @ ( list @ A ) @ Xs2 ) )
=> ( ( ( nth @ A @ ( cons @ A @ X @ Xs2 ) @ N3 )
= X )
= ( N3
= ( zero_zero @ nat ) ) ) ) ) ).
% nth_equal_first_eq
thf(fact_5945_Cons__replicate__eq,axiom,
! [A: $tType,X: A,Xs2: list @ A,N3: nat,Y: A] :
( ( ( cons @ A @ X @ Xs2 )
= ( replicate @ A @ N3 @ Y ) )
= ( ( X = Y )
& ( ord_less @ nat @ ( zero_zero @ nat ) @ N3 )
& ( Xs2
= ( replicate @ A @ ( minus_minus @ nat @ N3 @ ( one_one @ nat ) ) @ X ) ) ) ) ).
% Cons_replicate_eq
thf(fact_5946_slice__Cons,axiom,
! [A: $tType,Begin: nat,End: nat,X: A,Xs2: list @ A] :
( ( ( ( Begin
= ( zero_zero @ nat ) )
& ( ord_less @ nat @ ( zero_zero @ nat ) @ End ) )
=> ( ( slice @ A @ Begin @ End @ ( cons @ A @ X @ Xs2 ) )
= ( cons @ A @ X @ ( slice @ A @ Begin @ ( minus_minus @ nat @ End @ ( one_one @ nat ) ) @ Xs2 ) ) ) )
& ( ~ ( ( Begin
= ( zero_zero @ nat ) )
& ( ord_less @ nat @ ( zero_zero @ nat ) @ End ) )
=> ( ( slice @ A @ Begin @ End @ ( cons @ A @ X @ Xs2 ) )
= ( slice @ A @ ( minus_minus @ nat @ Begin @ ( one_one @ nat ) ) @ ( minus_minus @ nat @ End @ ( one_one @ nat ) ) @ Xs2 ) ) ) ) ).
% slice_Cons
thf(fact_5947_drop__bit__rec,axiom,
! [A: $tType] :
( ( bit_se359711467146920520ations @ A )
=> ( ( bit_se4197421643247451524op_bit @ A )
= ( ^ [N2: nat,A6: A] :
( if @ A
@ ( N2
= ( zero_zero @ nat ) )
@ A6
@ ( bit_se4197421643247451524op_bit @ A @ ( minus_minus @ nat @ N2 @ ( one_one @ nat ) ) @ ( divide_divide @ A @ A6 @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) ) ) ) ) ) ).
% drop_bit_rec
thf(fact_5948_hash__code__list__simps_I2_J,axiom,
! [A: $tType,H_a: A > uint32,X: A,Xa: list @ A] :
( ( hash_hash_code_list @ A @ H_a @ ( cons @ A @ X @ Xa ) )
= ( plus_plus @ uint32 @ ( times_times @ uint32 @ ( H_a @ X ) @ ( 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_5949_div__half__word,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [Y: word @ A,X: word @ A] :
( ( Y
!= ( zero_zero @ ( word @ A ) ) )
=> ( ( product_Pair @ ( word @ A ) @ ( word @ A ) @ ( divide_divide @ ( word @ A ) @ X @ Y ) @ ( modulo_modulo @ ( word @ A ) @ X @ Y ) )
= ( if @ ( product_prod @ ( word @ A ) @ ( word @ A ) ) @ ( ord_less_eq @ ( word @ A ) @ Y @ ( minus_minus @ ( word @ A ) @ X @ ( times_times @ ( word @ A ) @ ( bit_se4730199178511100633sh_bit @ ( word @ A ) @ ( one_one @ nat ) @ ( divide_divide @ ( word @ A ) @ ( bit_se4197421643247451524op_bit @ ( word @ A ) @ ( one_one @ nat ) @ X ) @ Y ) ) @ Y ) ) ) @ ( product_Pair @ ( word @ A ) @ ( word @ A ) @ ( plus_plus @ ( word @ A ) @ ( bit_se4730199178511100633sh_bit @ ( word @ A ) @ ( one_one @ nat ) @ ( divide_divide @ ( word @ A ) @ ( bit_se4197421643247451524op_bit @ ( word @ A ) @ ( one_one @ nat ) @ X ) @ Y ) ) @ ( one_one @ ( word @ A ) ) ) @ ( minus_minus @ ( word @ A ) @ ( minus_minus @ ( word @ A ) @ X @ ( times_times @ ( word @ A ) @ ( bit_se4730199178511100633sh_bit @ ( word @ A ) @ ( one_one @ nat ) @ ( divide_divide @ ( word @ A ) @ ( bit_se4197421643247451524op_bit @ ( word @ A ) @ ( one_one @ nat ) @ X ) @ Y ) ) @ Y ) ) @ Y ) ) @ ( product_Pair @ ( word @ A ) @ ( word @ A ) @ ( bit_se4730199178511100633sh_bit @ ( word @ A ) @ ( one_one @ nat ) @ ( divide_divide @ ( word @ A ) @ ( bit_se4197421643247451524op_bit @ ( word @ A ) @ ( one_one @ nat ) @ X ) @ Y ) ) @ ( minus_minus @ ( word @ A ) @ X @ ( times_times @ ( word @ A ) @ ( bit_se4730199178511100633sh_bit @ ( word @ A ) @ ( one_one @ nat ) @ ( divide_divide @ ( word @ A ) @ ( bit_se4197421643247451524op_bit @ ( word @ A ) @ ( one_one @ nat ) @ X ) @ Y ) ) @ Y ) ) ) ) ) ) ) ).
% div_half_word
thf(fact_5950_length__Cons,axiom,
! [A: $tType,X: A,Xs2: list @ A] :
( ( size_size @ ( list @ A ) @ ( cons @ A @ X @ Xs2 ) )
= ( suc @ ( size_size @ ( list @ A ) @ Xs2 ) ) ) ).
% length_Cons
thf(fact_5951_push__bit__eq__0__iff,axiom,
! [A: $tType] :
( ( bit_un5681908812861735899ations @ A )
=> ! [N3: nat,A2: A] :
( ( ( bit_se4730199178511100633sh_bit @ A @ N3 @ A2 )
= ( zero_zero @ A ) )
= ( A2
= ( zero_zero @ A ) ) ) ) ).
% push_bit_eq_0_iff
thf(fact_5952_push__bit__of__0,axiom,
! [A: $tType] :
( ( bit_se359711467146920520ations @ A )
=> ! [N3: nat] :
( ( bit_se4730199178511100633sh_bit @ A @ N3 @ ( zero_zero @ A ) )
= ( zero_zero @ A ) ) ) ).
% push_bit_of_0
thf(fact_5953_drop__bit__nonnegative__int__iff,axiom,
! [N3: nat,K: int] :
( ( ord_less_eq @ int @ ( zero_zero @ int ) @ ( bit_se4197421643247451524op_bit @ int @ N3 @ K ) )
= ( ord_less_eq @ int @ ( zero_zero @ int ) @ K ) ) ).
% drop_bit_nonnegative_int_iff
thf(fact_5954_drop__bit__negative__int__iff,axiom,
! [N3: nat,K: int] :
( ( ord_less @ int @ ( bit_se4197421643247451524op_bit @ int @ N3 @ K ) @ ( zero_zero @ int ) )
= ( ord_less @ int @ K @ ( zero_zero @ int ) ) ) ).
% drop_bit_negative_int_iff
thf(fact_5955_drop__bit__minus__one,axiom,
! [N3: nat] :
( ( bit_se4197421643247451524op_bit @ int @ N3 @ ( uminus_uminus @ int @ ( one_one @ int ) ) )
= ( uminus_uminus @ int @ ( one_one @ int ) ) ) ).
% drop_bit_minus_one
thf(fact_5956_push__bit__push__bit,axiom,
! [A: $tType] :
( ( bit_se359711467146920520ations @ A )
=> ! [M: nat,N3: nat,A2: A] :
( ( bit_se4730199178511100633sh_bit @ A @ M @ ( bit_se4730199178511100633sh_bit @ A @ N3 @ A2 ) )
= ( bit_se4730199178511100633sh_bit @ A @ ( plus_plus @ nat @ M @ N3 ) @ A2 ) ) ) ).
% push_bit_push_bit
thf(fact_5957_push__bit__and,axiom,
! [A: $tType] :
( ( bit_se359711467146920520ations @ A )
=> ! [N3: nat,A2: A,B2: A] :
( ( bit_se4730199178511100633sh_bit @ A @ N3 @ ( bit_se5824344872417868541ns_and @ A @ A2 @ B2 ) )
= ( bit_se5824344872417868541ns_and @ A @ ( bit_se4730199178511100633sh_bit @ A @ N3 @ A2 ) @ ( bit_se4730199178511100633sh_bit @ A @ N3 @ B2 ) ) ) ) ).
% push_bit_and
thf(fact_5958_push__bit__or,axiom,
! [A: $tType] :
( ( bit_se359711467146920520ations @ A )
=> ! [N3: nat,A2: A,B2: A] :
( ( bit_se4730199178511100633sh_bit @ A @ N3 @ ( bit_se1065995026697491101ons_or @ A @ A2 @ B2 ) )
= ( bit_se1065995026697491101ons_or @ A @ ( bit_se4730199178511100633sh_bit @ A @ N3 @ A2 ) @ ( bit_se4730199178511100633sh_bit @ A @ N3 @ B2 ) ) ) ) ).
% push_bit_or
thf(fact_5959_push__bit__xor,axiom,
! [A: $tType] :
( ( bit_se359711467146920520ations @ A )
=> ! [N3: nat,A2: A,B2: A] :
( ( bit_se4730199178511100633sh_bit @ A @ N3 @ ( bit_se5824344971392196577ns_xor @ A @ A2 @ B2 ) )
= ( bit_se5824344971392196577ns_xor @ A @ ( bit_se4730199178511100633sh_bit @ A @ N3 @ A2 ) @ ( bit_se4730199178511100633sh_bit @ A @ N3 @ B2 ) ) ) ) ).
% push_bit_xor
thf(fact_5960_push__bit__Suc__numeral,axiom,
! [A: $tType] :
( ( bit_se359711467146920520ations @ A )
=> ! [N3: nat,K: num] :
( ( bit_se4730199178511100633sh_bit @ A @ ( suc @ N3 ) @ ( numeral_numeral @ A @ K ) )
= ( bit_se4730199178511100633sh_bit @ A @ N3 @ ( numeral_numeral @ A @ ( bit0 @ K ) ) ) ) ) ).
% push_bit_Suc_numeral
thf(fact_5961_drop__bit__Suc__minus__bit0,axiom,
! [N3: nat,K: num] :
( ( bit_se4197421643247451524op_bit @ int @ ( suc @ N3 ) @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ ( bit0 @ K ) ) ) )
= ( bit_se4197421643247451524op_bit @ int @ N3 @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ K ) ) ) ) ).
% drop_bit_Suc_minus_bit0
thf(fact_5962_drop__bit__of__Suc__0,axiom,
! [N3: nat] :
( ( bit_se4197421643247451524op_bit @ nat @ N3 @ ( suc @ ( zero_zero @ nat ) ) )
= ( zero_neq_one_of_bool @ nat
@ ( N3
= ( zero_zero @ nat ) ) ) ) ).
% drop_bit_of_Suc_0
thf(fact_5963_push__bit__Suc__minus__numeral,axiom,
! [A: $tType] :
( ( bit_ri3973907225187159222ations @ A )
=> ! [N3: nat,K: num] :
( ( bit_se4730199178511100633sh_bit @ A @ ( suc @ N3 ) @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ K ) ) )
= ( bit_se4730199178511100633sh_bit @ A @ N3 @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ ( bit0 @ K ) ) ) ) ) ) ).
% push_bit_Suc_minus_numeral
thf(fact_5964_push__bit__minus__one__eq__not__mask,axiom,
! [A: $tType] :
( ( bit_ri3973907225187159222ations @ A )
=> ! [N3: nat] :
( ( bit_se4730199178511100633sh_bit @ A @ N3 @ ( uminus_uminus @ A @ ( one_one @ A ) ) )
= ( bit_ri4277139882892585799ns_not @ A @ ( bit_se2239418461657761734s_mask @ A @ N3 ) ) ) ) ).
% push_bit_minus_one_eq_not_mask
thf(fact_5965_push__bit__numeral,axiom,
! [A: $tType] :
( ( bit_se359711467146920520ations @ A )
=> ! [L2: num,K: num] :
( ( bit_se4730199178511100633sh_bit @ A @ ( numeral_numeral @ nat @ L2 ) @ ( numeral_numeral @ A @ K ) )
= ( bit_se4730199178511100633sh_bit @ A @ ( pred_numeral @ L2 ) @ ( numeral_numeral @ A @ ( bit0 @ K ) ) ) ) ) ).
% push_bit_numeral
thf(fact_5966_drop__bit__numeral__minus__bit0,axiom,
! [L2: num,K: num] :
( ( bit_se4197421643247451524op_bit @ int @ ( numeral_numeral @ nat @ L2 ) @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ ( bit0 @ K ) ) ) )
= ( bit_se4197421643247451524op_bit @ int @ ( pred_numeral @ L2 ) @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ K ) ) ) ) ).
% drop_bit_numeral_minus_bit0
thf(fact_5967_drop__bit__Suc__minus__bit1,axiom,
! [N3: nat,K: num] :
( ( bit_se4197421643247451524op_bit @ int @ ( suc @ N3 ) @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ ( bit1 @ K ) ) ) )
= ( bit_se4197421643247451524op_bit @ int @ N3 @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ ( inc @ K ) ) ) ) ) ).
% drop_bit_Suc_minus_bit1
thf(fact_5968_push__bit__Suc,axiom,
! [A: $tType] :
( ( bit_se359711467146920520ations @ A )
=> ! [N3: nat,A2: A] :
( ( bit_se4730199178511100633sh_bit @ A @ ( suc @ N3 ) @ A2 )
= ( bit_se4730199178511100633sh_bit @ A @ N3 @ ( times_times @ A @ A2 @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) ) ) ) ).
% push_bit_Suc
thf(fact_5969_push__bit__of__1,axiom,
! [A: $tType] :
( ( bit_se359711467146920520ations @ A )
=> ! [N3: nat] :
( ( bit_se4730199178511100633sh_bit @ A @ N3 @ ( one_one @ A ) )
= ( power_power @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ N3 ) ) ) ).
% push_bit_of_1
thf(fact_5970_even__push__bit__iff,axiom,
! [A: $tType] :
( ( bit_se359711467146920520ations @ A )
=> ! [N3: nat,A2: A] :
( ( dvd_dvd @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ ( bit_se4730199178511100633sh_bit @ A @ N3 @ A2 ) )
= ( ( N3
!= ( zero_zero @ nat ) )
| ( dvd_dvd @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ A2 ) ) ) ) ).
% even_push_bit_iff
thf(fact_5971_push__bit__minus__numeral,axiom,
! [A: $tType] :
( ( bit_ri3973907225187159222ations @ A )
=> ! [L2: num,K: num] :
( ( bit_se4730199178511100633sh_bit @ A @ ( numeral_numeral @ nat @ L2 ) @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ K ) ) )
= ( bit_se4730199178511100633sh_bit @ A @ ( pred_numeral @ L2 ) @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ ( bit0 @ K ) ) ) ) ) ) ).
% push_bit_minus_numeral
thf(fact_5972_drop__bit__numeral__minus__bit1,axiom,
! [L2: num,K: num] :
( ( bit_se4197421643247451524op_bit @ int @ ( numeral_numeral @ nat @ L2 ) @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ ( bit1 @ K ) ) ) )
= ( bit_se4197421643247451524op_bit @ int @ ( pred_numeral @ L2 ) @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ ( inc @ K ) ) ) ) ) ).
% drop_bit_numeral_minus_bit1
thf(fact_5973_push__bit__numeral__minus__1,axiom,
! [A: $tType] :
( ( bit_ri3973907225187159222ations @ A )
=> ! [N3: num] :
( ( bit_se4730199178511100633sh_bit @ A @ ( numeral_numeral @ nat @ N3 ) @ ( uminus_uminus @ A @ ( one_one @ A ) ) )
= ( uminus_uminus @ A @ ( power_power @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ ( numeral_numeral @ nat @ N3 ) ) ) ) ) ).
% push_bit_numeral_minus_1
thf(fact_5974_drop__bit__nat__eq,axiom,
! [N3: nat,K: int] :
( ( bit_se4197421643247451524op_bit @ nat @ N3 @ ( nat2 @ K ) )
= ( nat2 @ ( bit_se4197421643247451524op_bit @ int @ N3 @ K ) ) ) ).
% drop_bit_nat_eq
thf(fact_5975_drop__bit__int__code_I2_J,axiom,
! [N3: nat] :
( ( bit_se4197421643247451524op_bit @ int @ ( suc @ N3 ) @ ( zero_zero @ int ) )
= ( zero_zero @ int ) ) ).
% drop_bit_int_code(2)
thf(fact_5976_take__bit__push__bit,axiom,
! [A: $tType] :
( ( bit_se359711467146920520ations @ A )
=> ! [M: nat,N3: nat,A2: A] :
( ( bit_se2584673776208193580ke_bit @ A @ M @ ( bit_se4730199178511100633sh_bit @ A @ N3 @ A2 ) )
= ( bit_se4730199178511100633sh_bit @ A @ N3 @ ( bit_se2584673776208193580ke_bit @ A @ ( minus_minus @ nat @ M @ N3 ) @ A2 ) ) ) ) ).
% take_bit_push_bit
thf(fact_5977_push__bit__of__nat,axiom,
! [A: $tType] :
( ( bit_se359711467146920520ations @ A )
=> ! [N3: nat,M: nat] :
( ( bit_se4730199178511100633sh_bit @ A @ N3 @ ( semiring_1_of_nat @ A @ M ) )
= ( semiring_1_of_nat @ A @ ( bit_se4730199178511100633sh_bit @ nat @ N3 @ M ) ) ) ) ).
% push_bit_of_nat
thf(fact_5978_of__nat__push__bit,axiom,
! [A: $tType] :
( ( bit_se359711467146920520ations @ A )
=> ! [M: nat,N3: nat] :
( ( semiring_1_of_nat @ A @ ( bit_se4730199178511100633sh_bit @ nat @ M @ N3 ) )
= ( bit_se4730199178511100633sh_bit @ A @ M @ ( semiring_1_of_nat @ A @ N3 ) ) ) ) ).
% of_nat_push_bit
thf(fact_5979_push__bit__of__int,axiom,
! [A: $tType] :
( ( bit_ri3973907225187159222ations @ A )
=> ! [N3: nat,K: int] :
( ( bit_se4730199178511100633sh_bit @ A @ N3 @ ( ring_1_of_int @ A @ K ) )
= ( ring_1_of_int @ A @ ( bit_se4730199178511100633sh_bit @ int @ N3 @ K ) ) ) ) ).
% push_bit_of_int
thf(fact_5980_push__bit__minus,axiom,
! [A: $tType] :
( ( bit_ri3973907225187159222ations @ A )
=> ! [N3: nat,A2: A] :
( ( bit_se4730199178511100633sh_bit @ A @ N3 @ ( uminus_uminus @ A @ A2 ) )
= ( uminus_uminus @ A @ ( bit_se4730199178511100633sh_bit @ A @ N3 @ A2 ) ) ) ) ).
% push_bit_minus
thf(fact_5981_push__bit__add,axiom,
! [A: $tType] :
( ( bit_se359711467146920520ations @ A )
=> ! [N3: nat,A2: A,B2: A] :
( ( bit_se4730199178511100633sh_bit @ A @ N3 @ ( plus_plus @ A @ A2 @ B2 ) )
= ( plus_plus @ A @ ( bit_se4730199178511100633sh_bit @ A @ N3 @ A2 ) @ ( bit_se4730199178511100633sh_bit @ A @ N3 @ B2 ) ) ) ) ).
% push_bit_add
thf(fact_5982_push__bit__take__bit,axiom,
! [A: $tType] :
( ( bit_se359711467146920520ations @ A )
=> ! [M: nat,N3: nat,A2: A] :
( ( bit_se4730199178511100633sh_bit @ A @ M @ ( bit_se2584673776208193580ke_bit @ A @ N3 @ A2 ) )
= ( bit_se2584673776208193580ke_bit @ A @ ( plus_plus @ nat @ M @ N3 ) @ ( bit_se4730199178511100633sh_bit @ A @ M @ A2 ) ) ) ) ).
% push_bit_take_bit
thf(fact_5983_div__push__bit__of__1__eq__drop__bit,axiom,
! [A: $tType] :
( ( bit_se359711467146920520ations @ A )
=> ! [A2: A,N3: nat] :
( ( divide_divide @ A @ A2 @ ( bit_se4730199178511100633sh_bit @ A @ N3 @ ( one_one @ A ) ) )
= ( bit_se4197421643247451524op_bit @ A @ N3 @ A2 ) ) ) ).
% div_push_bit_of_1_eq_drop_bit
thf(fact_5984_bits__ident,axiom,
! [A: $tType] :
( ( bit_se359711467146920520ations @ A )
=> ! [N3: nat,A2: A] :
( ( plus_plus @ A @ ( bit_se4730199178511100633sh_bit @ A @ N3 @ ( bit_se4197421643247451524op_bit @ A @ N3 @ A2 ) ) @ ( bit_se2584673776208193580ke_bit @ A @ N3 @ A2 ) )
= A2 ) ) ).
% bits_ident
thf(fact_5985_set__bit__eq__or,axiom,
! [A: $tType] :
( ( bit_se359711467146920520ations @ A )
=> ( ( bit_se5668285175392031749et_bit @ A )
= ( ^ [N2: nat,A6: A] : ( bit_se1065995026697491101ons_or @ A @ A6 @ ( bit_se4730199178511100633sh_bit @ A @ N2 @ ( one_one @ A ) ) ) ) ) ) ).
% set_bit_eq_or
thf(fact_5986_flip__bit__eq__xor,axiom,
! [A: $tType] :
( ( bit_se359711467146920520ations @ A )
=> ( ( bit_se8732182000553998342ip_bit @ A )
= ( ^ [N2: nat,A6: A] : ( bit_se5824344971392196577ns_xor @ A @ A6 @ ( bit_se4730199178511100633sh_bit @ A @ N2 @ ( one_one @ A ) ) ) ) ) ) ).
% flip_bit_eq_xor
thf(fact_5987_push__bit__double,axiom,
! [A: $tType] :
( ( bit_se359711467146920520ations @ A )
=> ! [N3: nat,A2: A] :
( ( bit_se4730199178511100633sh_bit @ A @ N3 @ ( times_times @ A @ A2 @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) )
= ( times_times @ A @ ( bit_se4730199178511100633sh_bit @ A @ N3 @ A2 ) @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) ) ) ).
% push_bit_double
thf(fact_5988_bit__iff__and__push__bit__not__eq__0,axiom,
! [A: $tType] :
( ( bit_se359711467146920520ations @ A )
=> ( ( bit_se5641148757651400278ts_bit @ A )
= ( ^ [A6: A,N2: nat] :
( ( bit_se5824344872417868541ns_and @ A @ A6 @ ( bit_se4730199178511100633sh_bit @ A @ N2 @ ( one_one @ A ) ) )
!= ( zero_zero @ A ) ) ) ) ) ).
% bit_iff_and_push_bit_not_eq_0
thf(fact_5989_push__bit__mask__eq,axiom,
! [A: $tType] :
( ( bit_ri3973907225187159222ations @ A )
=> ! [M: nat,N3: nat] :
( ( bit_se4730199178511100633sh_bit @ A @ M @ ( bit_se2239418461657761734s_mask @ A @ N3 ) )
= ( bit_se5824344872417868541ns_and @ A @ ( bit_se2239418461657761734s_mask @ A @ ( plus_plus @ nat @ N3 @ M ) ) @ ( bit_ri4277139882892585799ns_not @ A @ ( bit_se2239418461657761734s_mask @ A @ M ) ) ) ) ) ).
% push_bit_mask_eq
thf(fact_5990_unset__bit__eq__and__not,axiom,
! [A: $tType] :
( ( bit_ri3973907225187159222ations @ A )
=> ( ( bit_se2638667681897837118et_bit @ A )
= ( ^ [N2: nat,A6: A] : ( bit_se5824344872417868541ns_and @ A @ A6 @ ( bit_ri4277139882892585799ns_not @ A @ ( bit_se4730199178511100633sh_bit @ A @ N2 @ ( one_one @ A ) ) ) ) ) ) ) ).
% unset_bit_eq_and_not
thf(fact_5991_shiftr__integer__conv__div__pow2,axiom,
( ( bit_se4197421643247451524op_bit @ code_integer )
= ( ^ [N2: nat,X2: code_integer] : ( divide_divide @ code_integer @ X2 @ ( power_power @ code_integer @ ( numeral_numeral @ code_integer @ ( bit0 @ one2 ) ) @ N2 ) ) ) ) ).
% shiftr_integer_conv_div_pow2
thf(fact_5992_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_5993_subset__Collect__iff,axiom,
! [A: $tType,B6: set @ A,A3: set @ A,P: A > $o] :
( ( ord_less_eq @ ( set @ A ) @ B6 @ A3 )
=> ( ( ord_less_eq @ ( set @ A ) @ B6
@ ( collect @ A
@ ^ [X2: A] :
( ( member @ A @ X2 @ A3 )
& ( P @ X2 ) ) ) )
= ( ! [X2: A] :
( ( member @ A @ X2 @ B6 )
=> ( P @ X2 ) ) ) ) ) ).
% subset_Collect_iff
thf(fact_5994_subset__CollectI,axiom,
! [A: $tType,B6: set @ A,A3: set @ A,Q: A > $o,P: A > $o] :
( ( ord_less_eq @ ( set @ A ) @ B6 @ A3 )
=> ( ! [X3: A] :
( ( member @ A @ X3 @ B6 )
=> ( ( Q @ X3 )
=> ( P @ X3 ) ) )
=> ( ord_less_eq @ ( set @ A )
@ ( collect @ A
@ ^ [X2: A] :
( ( member @ A @ X2 @ B6 )
& ( Q @ X2 ) ) )
@ ( collect @ A
@ ^ [X2: A] :
( ( member @ A @ X2 @ A3 )
& ( P @ X2 ) ) ) ) ) ) ).
% subset_CollectI
thf(fact_5995_drop__bit__int__def,axiom,
( ( bit_se4197421643247451524op_bit @ int )
= ( ^ [N2: nat,K2: int] : ( divide_divide @ int @ K2 @ ( power_power @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ N2 ) ) ) ) ).
% drop_bit_int_def
thf(fact_5996_push__bit__eq__mult,axiom,
! [A: $tType] :
( ( bit_se359711467146920520ations @ A )
=> ( ( bit_se4730199178511100633sh_bit @ A )
= ( ^ [N2: nat,A6: A] : ( times_times @ A @ A6 @ ( power_power @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ N2 ) ) ) ) ) ).
% push_bit_eq_mult
thf(fact_5997_exp__dvdE,axiom,
! [A: $tType] :
( ( bit_se359711467146920520ations @ A )
=> ! [N3: nat,A2: A] :
( ( dvd_dvd @ A @ ( power_power @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ N3 ) @ A2 )
=> ~ ! [B3: A] :
( A2
!= ( bit_se4730199178511100633sh_bit @ A @ N3 @ B3 ) ) ) ) ).
% exp_dvdE
thf(fact_5998_drop__bit__nat__def,axiom,
( ( bit_se4197421643247451524op_bit @ nat )
= ( ^ [N2: nat,M5: nat] : ( divide_divide @ nat @ M5 @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N2 ) ) ) ) ).
% drop_bit_nat_def
thf(fact_5999_slice__eq__mask,axiom,
! [A: $tType] :
( ( bit_ri3973907225187159222ations @ A )
=> ! [N3: nat,M: nat,A2: A] :
( ( bit_se4730199178511100633sh_bit @ A @ N3 @ ( bit_se2584673776208193580ke_bit @ A @ M @ ( bit_se4197421643247451524op_bit @ A @ N3 @ A2 ) ) )
= ( bit_se5824344872417868541ns_and @ A @ A2 @ ( bit_se5824344872417868541ns_and @ A @ ( bit_se2239418461657761734s_mask @ A @ ( plus_plus @ nat @ M @ N3 ) ) @ ( bit_ri4277139882892585799ns_not @ A @ ( bit_se2239418461657761734s_mask @ A @ N3 ) ) ) ) ) ) ).
% slice_eq_mask
thf(fact_6000_take__bit__sum,axiom,
! [A: $tType] :
( ( bit_un5681908812861735899ations @ A )
=> ( ( bit_se2584673776208193580ke_bit @ A )
= ( ^ [N2: nat,A6: A] :
( groups7311177749621191930dd_sum @ nat @ A
@ ^ [K2: nat] : ( bit_se4730199178511100633sh_bit @ A @ K2 @ ( zero_neq_one_of_bool @ A @ ( bit_se5641148757651400278ts_bit @ A @ A6 @ K2 ) ) )
@ ( set_or7035219750837199246ssThan @ nat @ ( zero_zero @ nat ) @ N2 ) ) ) ) ) ).
% take_bit_sum
thf(fact_6001_word__and__mask__or__conv__and__mask,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [N3: word @ A,Index: nat] :
( ( bit_se5641148757651400278ts_bit @ ( word @ A ) @ N3 @ Index )
=> ( ( bit_se1065995026697491101ons_or @ ( word @ A ) @ ( bit_se5824344872417868541ns_and @ ( word @ A ) @ N3 @ ( bit_se2239418461657761734s_mask @ ( word @ A ) @ Index ) ) @ ( bit_se4730199178511100633sh_bit @ ( word @ A ) @ Index @ ( one_one @ ( word @ A ) ) ) )
= ( bit_se5824344872417868541ns_and @ ( word @ A ) @ N3 @ ( bit_se2239418461657761734s_mask @ ( word @ A ) @ ( plus_plus @ nat @ Index @ ( one_one @ nat ) ) ) ) ) ) ) ).
% word_and_mask_or_conv_and_mask
thf(fact_6002_signed__take__bit__code,axiom,
! [A: $tType] :
( ( bit_ri3973907225187159222ations @ A )
=> ( ( bit_ri4674362597316999326ke_bit @ A )
= ( ^ [N2: nat,A6: A] : ( if @ A @ ( bit_se5641148757651400278ts_bit @ A @ ( bit_se2584673776208193580ke_bit @ A @ ( suc @ N2 ) @ A6 ) @ N2 ) @ ( plus_plus @ A @ ( bit_se2584673776208193580ke_bit @ A @ ( suc @ N2 ) @ A6 ) @ ( bit_se4730199178511100633sh_bit @ A @ ( suc @ N2 ) @ ( uminus_uminus @ A @ ( one_one @ A ) ) ) ) @ ( bit_se2584673776208193580ke_bit @ A @ ( suc @ N2 ) @ A6 ) ) ) ) ) ).
% signed_take_bit_code
thf(fact_6003_set__bits__aux__Suc,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [F2: nat > $o,N3: nat,W: word @ A] :
( ( code_T2661198915054445665ts_aux @ A @ F2 @ ( suc @ N3 ) @ W )
= ( code_T2661198915054445665ts_aux @ A @ F2 @ N3 @ ( bit_se1065995026697491101ons_or @ ( word @ A ) @ ( bit_se4730199178511100633sh_bit @ ( word @ A ) @ ( one_one @ nat ) @ W ) @ ( if @ ( word @ A ) @ ( F2 @ N3 ) @ ( one_one @ ( word @ A ) ) @ ( zero_zero @ ( word @ A ) ) ) ) ) ) ) ).
% set_bits_aux_Suc
thf(fact_6004_set__bits__aux__rec,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ( ( code_T2661198915054445665ts_aux @ A )
= ( ^ [F7: nat > $o,N2: nat,W2: word @ A] :
( if @ ( word @ A )
@ ( N2
= ( zero_zero @ nat ) )
@ W2
@ ( code_T2661198915054445665ts_aux @ A @ F7 @ ( minus_minus @ nat @ N2 @ ( one_one @ nat ) ) @ ( bit_se1065995026697491101ons_or @ ( word @ A ) @ ( bit_se4730199178511100633sh_bit @ ( word @ A ) @ ( one_one @ nat ) @ W2 ) @ ( if @ ( word @ A ) @ ( F7 @ ( minus_minus @ nat @ N2 @ ( one_one @ nat ) ) ) @ ( one_one @ ( word @ A ) ) @ ( zero_zero @ ( word @ A ) ) ) ) ) ) ) ) ) ).
% set_bits_aux_rec
thf(fact_6005_push__bit__nonnegative__int__iff,axiom,
! [N3: nat,K: int] :
( ( ord_less_eq @ int @ ( zero_zero @ int ) @ ( bit_se4730199178511100633sh_bit @ int @ N3 @ K ) )
= ( ord_less_eq @ int @ ( zero_zero @ int ) @ K ) ) ).
% push_bit_nonnegative_int_iff
thf(fact_6006_push__bit__negative__int__iff,axiom,
! [N3: nat,K: int] :
( ( ord_less @ int @ ( bit_se4730199178511100633sh_bit @ int @ N3 @ K ) @ ( zero_zero @ int ) )
= ( ord_less @ int @ K @ ( zero_zero @ int ) ) ) ).
% push_bit_negative_int_iff
thf(fact_6007_push__bit__of__Suc__0,axiom,
! [N3: nat] :
( ( bit_se4730199178511100633sh_bit @ nat @ N3 @ ( suc @ ( zero_zero @ nat ) ) )
= ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N3 ) ) ).
% push_bit_of_Suc_0
thf(fact_6008_drop__bit__push__bit__int,axiom,
! [M: nat,N3: nat,K: int] :
( ( bit_se4197421643247451524op_bit @ int @ M @ ( bit_se4730199178511100633sh_bit @ int @ N3 @ K ) )
= ( bit_se4197421643247451524op_bit @ int @ ( minus_minus @ nat @ M @ N3 ) @ ( bit_se4730199178511100633sh_bit @ int @ ( minus_minus @ nat @ N3 @ M ) @ K ) ) ) ).
% drop_bit_push_bit_int
thf(fact_6009_push__bit__nat__eq,axiom,
! [N3: nat,K: int] :
( ( bit_se4730199178511100633sh_bit @ nat @ N3 @ ( nat2 @ K ) )
= ( nat2 @ ( bit_se4730199178511100633sh_bit @ int @ N3 @ K ) ) ) ).
% push_bit_nat_eq
thf(fact_6010_flip__bit__nat__def,axiom,
( ( bit_se8732182000553998342ip_bit @ nat )
= ( ^ [M5: nat,N2: nat] : ( bit_se5824344971392196577ns_xor @ nat @ N2 @ ( bit_se4730199178511100633sh_bit @ nat @ M5 @ ( one_one @ nat ) ) ) ) ) ).
% flip_bit_nat_def
thf(fact_6011_set__bit__nat__def,axiom,
( ( bit_se5668285175392031749et_bit @ nat )
= ( ^ [M5: nat,N2: nat] : ( bit_se1065995026697491101ons_or @ nat @ N2 @ ( bit_se4730199178511100633sh_bit @ nat @ M5 @ ( one_one @ nat ) ) ) ) ) ).
% set_bit_nat_def
thf(fact_6012_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_6013_bit__push__bit__iff__int,axiom,
! [M: nat,K: int,N3: nat] :
( ( bit_se5641148757651400278ts_bit @ int @ ( bit_se4730199178511100633sh_bit @ int @ M @ K ) @ N3 )
= ( ( ord_less_eq @ nat @ M @ N3 )
& ( bit_se5641148757651400278ts_bit @ int @ K @ ( minus_minus @ nat @ N3 @ M ) ) ) ) ).
% bit_push_bit_iff_int
thf(fact_6014_Bit__Operations_Oset__bit__int__def,axiom,
( ( bit_se5668285175392031749et_bit @ int )
= ( ^ [N2: nat,K2: int] : ( bit_se1065995026697491101ons_or @ int @ K2 @ ( bit_se4730199178511100633sh_bit @ int @ N2 @ ( one_one @ int ) ) ) ) ) ).
% Bit_Operations.set_bit_int_def
thf(fact_6015_bit__push__bit__iff__nat,axiom,
! [M: nat,Q2: nat,N3: nat] :
( ( bit_se5641148757651400278ts_bit @ nat @ ( bit_se4730199178511100633sh_bit @ nat @ M @ Q2 ) @ N3 )
= ( ( ord_less_eq @ nat @ M @ N3 )
& ( bit_se5641148757651400278ts_bit @ nat @ Q2 @ ( minus_minus @ nat @ N3 @ M ) ) ) ) ).
% bit_push_bit_iff_nat
thf(fact_6016_flip__bit__int__def,axiom,
( ( bit_se8732182000553998342ip_bit @ int )
= ( ^ [N2: nat,K2: int] : ( bit_se5824344971392196577ns_xor @ int @ K2 @ ( bit_se4730199178511100633sh_bit @ int @ N2 @ ( one_one @ int ) ) ) ) ) ).
% flip_bit_int_def
thf(fact_6017_shiftl__integer__conv__mult__pow2,axiom,
( ( bit_se4730199178511100633sh_bit @ code_integer )
= ( ^ [N2: nat,X2: code_integer] : ( times_times @ code_integer @ X2 @ ( power_power @ code_integer @ ( numeral_numeral @ code_integer @ ( bit0 @ one2 ) ) @ N2 ) ) ) ) ).
% shiftl_integer_conv_mult_pow2
thf(fact_6018_unset__bit__int__def,axiom,
( ( bit_se2638667681897837118et_bit @ int )
= ( ^ [N2: nat,K2: int] : ( bit_se5824344872417868541ns_and @ int @ K2 @ ( bit_ri4277139882892585799ns_not @ int @ ( bit_se4730199178511100633sh_bit @ int @ N2 @ ( one_one @ int ) ) ) ) ) ) ).
% unset_bit_int_def
thf(fact_6019_push__bit__int__def,axiom,
( ( bit_se4730199178511100633sh_bit @ int )
= ( ^ [N2: nat,K2: int] : ( times_times @ int @ K2 @ ( power_power @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ N2 ) ) ) ) ).
% push_bit_int_def
thf(fact_6020_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_6021_push__bit__nat__def,axiom,
( ( bit_se4730199178511100633sh_bit @ nat )
= ( ^ [N2: nat,M5: nat] : ( times_times @ nat @ M5 @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N2 ) ) ) ) ).
% push_bit_nat_def
thf(fact_6022_push__bit__minus__one,axiom,
! [N3: nat] :
( ( bit_se4730199178511100633sh_bit @ int @ N3 @ ( uminus_uminus @ int @ ( one_one @ int ) ) )
= ( uminus_uminus @ int @ ( power_power @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ N3 ) ) ) ).
% push_bit_minus_one
thf(fact_6023_test__bit__split,axiom,
! [A: $tType,C: $tType,B: $tType] :
( ( ( type_len @ B )
& ( type_len @ C )
& ( type_len @ A ) )
=> ! [C2: word @ C,A2: word @ A,B2: word @ B] :
( ( ( word_split @ C @ A @ B @ C2 )
= ( product_Pair @ ( word @ A ) @ ( word @ B ) @ A2 @ B2 ) )
=> ( ! [N9: nat] :
( ( bit_se5641148757651400278ts_bit @ ( word @ B ) @ B2 @ N9 )
= ( ( ord_less @ nat @ N9 @ ( size_size @ ( word @ B ) @ B2 ) )
& ( bit_se5641148757651400278ts_bit @ ( word @ C ) @ C2 @ N9 ) ) )
& ! [M2: nat] :
( ( bit_se5641148757651400278ts_bit @ ( word @ A ) @ A2 @ M2 )
= ( ( ord_less @ nat @ M2 @ ( size_size @ ( word @ A ) @ A2 ) )
& ( bit_se5641148757651400278ts_bit @ ( word @ C ) @ C2 @ ( plus_plus @ nat @ M2 @ ( size_size @ ( word @ B ) @ B2 ) ) ) ) ) ) ) ) ).
% test_bit_split
thf(fact_6024_test__bit__split_H,axiom,
! [A: $tType,C: $tType,B: $tType] :
( ( ( type_len @ B )
& ( type_len @ C )
& ( type_len @ A ) )
=> ! [C2: word @ C,A2: word @ A,B2: word @ B] :
( ( ( word_split @ C @ A @ B @ C2 )
= ( product_Pair @ ( word @ A ) @ ( word @ B ) @ A2 @ B2 ) )
=> ! [N9: nat,M2: nat] :
( ( ( bit_se5641148757651400278ts_bit @ ( word @ B ) @ B2 @ N9 )
= ( ( ord_less @ nat @ N9 @ ( size_size @ ( word @ B ) @ B2 ) )
& ( bit_se5641148757651400278ts_bit @ ( word @ C ) @ C2 @ N9 ) ) )
& ( ( bit_se5641148757651400278ts_bit @ ( word @ A ) @ A2 @ M2 )
= ( ( ord_less @ nat @ M2 @ ( size_size @ ( word @ A ) @ A2 ) )
& ( bit_se5641148757651400278ts_bit @ ( word @ C ) @ C2 @ ( plus_plus @ nat @ M2 @ ( size_size @ ( word @ B ) @ B2 ) ) ) ) ) ) ) ) ).
% test_bit_split'
thf(fact_6025_test__bit__split__eq,axiom,
! [B: $tType,C: $tType,A: $tType] :
( ( ( type_len @ A )
& ( type_len @ C )
& ( type_len @ B ) )
=> ! [C2: word @ C,A2: word @ A,B2: word @ B] :
( ( ( word_split @ C @ A @ B @ C2 )
= ( product_Pair @ ( word @ A ) @ ( word @ B ) @ A2 @ B2 ) )
= ( ! [N2: nat] :
( ( bit_se5641148757651400278ts_bit @ ( word @ B ) @ B2 @ N2 )
= ( ( ord_less @ nat @ N2 @ ( size_size @ ( word @ B ) @ B2 ) )
& ( bit_se5641148757651400278ts_bit @ ( word @ C ) @ C2 @ N2 ) ) )
& ! [M5: nat] :
( ( bit_se5641148757651400278ts_bit @ ( word @ A ) @ A2 @ M5 )
= ( ( ord_less @ nat @ M5 @ ( size_size @ ( word @ A ) @ A2 ) )
& ( bit_se5641148757651400278ts_bit @ ( word @ C ) @ C2 @ ( plus_plus @ nat @ M5 @ ( size_size @ ( word @ B ) @ B2 ) ) ) ) ) ) ) ) ).
% test_bit_split_eq
thf(fact_6026_set__bit__integer__conv__masks,axiom,
( ( generi7602027413899671122et_bit @ code_integer )
= ( ^ [X2: code_integer,I3: nat,B5: $o] : ( if @ code_integer @ B5 @ ( bit_se1065995026697491101ons_or @ code_integer @ X2 @ ( bit_se4730199178511100633sh_bit @ code_integer @ I3 @ ( one_one @ code_integer ) ) ) @ ( bit_se5824344872417868541ns_and @ code_integer @ X2 @ ( bit_ri4277139882892585799ns_not @ code_integer @ ( bit_se4730199178511100633sh_bit @ code_integer @ I3 @ ( one_one @ code_integer ) ) ) ) ) ) ) ).
% set_bit_integer_conv_masks
thf(fact_6027_upto__aux__rec,axiom,
( upto_aux
= ( ^ [I3: int,J3: int,Js: list @ int] : ( if @ ( list @ int ) @ ( ord_less @ int @ J3 @ I3 ) @ Js @ ( upto_aux @ I3 @ ( minus_minus @ int @ J3 @ ( one_one @ int ) ) @ ( cons @ int @ J3 @ Js ) ) ) ) ) ).
% upto_aux_rec
thf(fact_6028_bit__set__bit__iff__2n,axiom,
! [A: $tType] :
( ( generic_set_set_bit @ A )
=> ! [A2: A,M: nat,B2: $o,N3: nat] :
( ( bit_se5641148757651400278ts_bit @ A @ ( generi7602027413899671122et_bit @ A @ A2 @ M @ B2 ) @ N3 )
= ( ( ( M = N3 )
=> B2 )
& ( ( M != N3 )
=> ( bit_se5641148757651400278ts_bit @ A @ A2 @ N3 ) )
& ( ( power_power @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ N3 )
!= ( zero_zero @ A ) ) ) ) ) ).
% bit_set_bit_iff_2n
thf(fact_6029_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_6030_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_6031_set__bit__beyond,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [X: word @ A,N3: nat,B2: $o] :
( ( ord_less_eq @ nat @ ( size_size @ ( word @ A ) @ X ) @ N3 )
=> ( ( generi7602027413899671122et_bit @ ( word @ A ) @ X @ N3 @ B2 )
= X ) ) ) ).
% set_bit_beyond
thf(fact_6032_word__set__ge,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [W: word @ A,N3: nat] : ( ord_less_eq @ ( word @ A ) @ W @ ( generi7602027413899671122et_bit @ ( word @ A ) @ W @ N3 @ $true ) ) ) ).
% word_set_ge
thf(fact_6033_word__clr__le,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [W: word @ A,N3: nat] : ( ord_less_eq @ ( word @ A ) @ ( generi7602027413899671122et_bit @ ( word @ A ) @ W @ N3 @ $false ) @ W ) ) ).
% word_clr_le
thf(fact_6034_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_6035_test__bit__set__gen,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [W: word @ A,N3: nat,X: $o,M: nat] :
( ( bit_se5641148757651400278ts_bit @ ( word @ A ) @ ( generi7602027413899671122et_bit @ ( word @ A ) @ W @ N3 @ X ) @ M )
= ( ( ( M = N3 )
=> ( ( ord_less @ nat @ N3 @ ( size_size @ ( word @ A ) @ W ) )
& X ) )
& ( ( M != N3 )
=> ( bit_se5641148757651400278ts_bit @ ( word @ A ) @ W @ M ) ) ) ) ) ).
% test_bit_set_gen
thf(fact_6036_test__bit__set,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [W: word @ A,N3: nat,X: $o] :
( ( bit_se5641148757651400278ts_bit @ ( word @ A ) @ ( generi7602027413899671122et_bit @ ( word @ A ) @ W @ N3 @ X ) @ N3 )
= ( ( ord_less @ nat @ N3 @ ( size_size @ ( word @ A ) @ W ) )
& X ) ) ) ).
% test_bit_set
thf(fact_6037_word__set__nth__iff,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [W: word @ A,N3: nat,B2: $o] :
( ( ( generi7602027413899671122et_bit @ ( word @ A ) @ W @ N3 @ B2 )
= W )
= ( ( ( bit_se5641148757651400278ts_bit @ ( word @ A ) @ W @ N3 )
= B2 )
| ( ord_less_eq @ nat @ ( size_size @ ( word @ A ) @ W ) @ N3 ) ) ) ) ).
% word_set_nth_iff
thf(fact_6038_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_6039_one__bit__pow,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [N3: nat] :
( ( generi7602027413899671122et_bit @ ( word @ A ) @ ( zero_zero @ ( word @ A ) ) @ N3 @ $true )
= ( power_power @ ( word @ A ) @ ( numeral_numeral @ ( word @ A ) @ ( bit0 @ one2 ) ) @ N3 ) ) ) ).
% one_bit_pow
thf(fact_6040_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_6041_int__set__bit__True__conv__OR,axiom,
! [I: int,N3: nat] :
( ( generi7602027413899671122et_bit @ int @ I @ N3 @ $true )
= ( bit_se1065995026697491101ons_or @ int @ I @ ( bit_se4730199178511100633sh_bit @ int @ N3 @ ( one_one @ int ) ) ) ) ).
% int_set_bit_True_conv_OR
thf(fact_6042_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_6043_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_6044_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_6045_int__set__bit__False__conv__NAND,axiom,
! [I: int,N3: nat] :
( ( generi7602027413899671122et_bit @ int @ I @ N3 @ $false )
= ( bit_se5824344872417868541ns_and @ int @ I @ ( bit_ri4277139882892585799ns_not @ int @ ( bit_se4730199178511100633sh_bit @ int @ N3 @ ( one_one @ int ) ) ) ) ) ).
% int_set_bit_False_conv_NAND
thf(fact_6046_int__set__bit__conv__ops,axiom,
( ( generi7602027413899671122et_bit @ int )
= ( ^ [I3: int,N2: nat,B5: $o] : ( if @ int @ B5 @ ( bit_se1065995026697491101ons_or @ int @ I3 @ ( bit_se4730199178511100633sh_bit @ int @ N2 @ ( one_one @ int ) ) ) @ ( bit_se5824344872417868541ns_and @ int @ I3 @ ( bit_ri4277139882892585799ns_not @ int @ ( bit_se4730199178511100633sh_bit @ int @ N2 @ ( one_one @ int ) ) ) ) ) ) ) ).
% int_set_bit_conv_ops
thf(fact_6047_concat__bit__Suc,axiom,
! [N3: nat,K: int,L2: int] :
( ( bit_concat_bit @ ( suc @ N3 ) @ K @ L2 )
= ( plus_plus @ int @ ( modulo_modulo @ int @ K @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) ) @ ( times_times @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ ( bit_concat_bit @ N3 @ ( divide_divide @ int @ K @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) ) @ L2 ) ) ) ) ).
% concat_bit_Suc
thf(fact_6048_case__prod__Pair__iden,axiom,
! [B: $tType,A: $tType,P4: product_prod @ A @ B] :
( ( product_case_prod @ A @ B @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B ) @ P4 )
= P4 ) ).
% case_prod_Pair_iden
thf(fact_6049_int__lsb__numeral_I2_J,axiom,
least_8051144512741203767sb_lsb @ int @ ( one_one @ int ) ).
% int_lsb_numeral(2)
thf(fact_6050_int__lsb__numeral_I6_J,axiom,
! [W: num] :
~ ( least_8051144512741203767sb_lsb @ int @ ( numeral_numeral @ int @ ( bit0 @ W ) ) ) ).
% int_lsb_numeral(6)
thf(fact_6051_int__lsb__numeral_I3_J,axiom,
least_8051144512741203767sb_lsb @ int @ ( numeral_numeral @ int @ one2 ) ).
% int_lsb_numeral(3)
thf(fact_6052_int__lsb__numeral_I4_J,axiom,
least_8051144512741203767sb_lsb @ int @ ( uminus_uminus @ int @ ( one_one @ int ) ) ).
% int_lsb_numeral(4)
thf(fact_6053_concat__bit__0,axiom,
! [K: int,L2: int] :
( ( bit_concat_bit @ ( zero_zero @ nat ) @ K @ L2 )
= L2 ) ).
% concat_bit_0
thf(fact_6054_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_6055_int__lsb__numeral_I5_J,axiom,
least_8051144512741203767sb_lsb @ int @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ one2 ) ) ).
% int_lsb_numeral(5)
thf(fact_6056_concat__bit__of__zero__2,axiom,
! [N3: nat,K: int] :
( ( bit_concat_bit @ N3 @ K @ ( zero_zero @ int ) )
= ( bit_se2584673776208193580ke_bit @ int @ N3 @ K ) ) ).
% concat_bit_of_zero_2
thf(fact_6057_concat__bit__nonnegative__iff,axiom,
! [N3: nat,K: int,L2: int] :
( ( ord_less_eq @ int @ ( zero_zero @ int ) @ ( bit_concat_bit @ N3 @ K @ L2 ) )
= ( ord_less_eq @ int @ ( zero_zero @ int ) @ L2 ) ) ).
% concat_bit_nonnegative_iff
thf(fact_6058_concat__bit__negative__iff,axiom,
! [N3: nat,K: int,L2: int] :
( ( ord_less @ int @ ( bit_concat_bit @ N3 @ K @ L2 ) @ ( zero_zero @ int ) )
= ( ord_less @ int @ L2 @ ( zero_zero @ int ) ) ) ).
% concat_bit_negative_iff
thf(fact_6059_concat__bit__of__zero__1,axiom,
! [N3: nat,L2: int] :
( ( bit_concat_bit @ N3 @ ( zero_zero @ int ) @ L2 )
= ( bit_se4730199178511100633sh_bit @ int @ N3 @ L2 ) ) ).
% concat_bit_of_zero_1
thf(fact_6060_concat__bit__assoc,axiom,
! [N3: nat,K: int,M: nat,L2: int,R2: int] :
( ( bit_concat_bit @ N3 @ K @ ( bit_concat_bit @ M @ L2 @ R2 ) )
= ( bit_concat_bit @ ( plus_plus @ nat @ M @ N3 ) @ ( bit_concat_bit @ N3 @ K @ L2 ) @ R2 ) ) ).
% concat_bit_assoc
thf(fact_6061_concat__bit__take__bit__eq,axiom,
! [N3: nat,B2: int] :
( ( bit_concat_bit @ N3 @ ( bit_se2584673776208193580ke_bit @ int @ N3 @ B2 ) )
= ( bit_concat_bit @ N3 @ B2 ) ) ).
% concat_bit_take_bit_eq
thf(fact_6062_concat__bit__eq__iff,axiom,
! [N3: nat,K: int,L2: int,R2: int,S: int] :
( ( ( bit_concat_bit @ N3 @ K @ L2 )
= ( bit_concat_bit @ N3 @ R2 @ S ) )
= ( ( ( bit_se2584673776208193580ke_bit @ int @ N3 @ K )
= ( bit_se2584673776208193580ke_bit @ int @ N3 @ R2 ) )
& ( L2 = S ) ) ) ).
% concat_bit_eq_iff
thf(fact_6063_concat__bit__eq,axiom,
( bit_concat_bit
= ( ^ [N2: nat,K2: int,L: int] : ( plus_plus @ int @ ( bit_se2584673776208193580ke_bit @ int @ N2 @ K2 ) @ ( bit_se4730199178511100633sh_bit @ int @ N2 @ L ) ) ) ) ).
% concat_bit_eq
thf(fact_6064_concat__bit__def,axiom,
( bit_concat_bit
= ( ^ [N2: nat,K2: int,L: int] : ( bit_se1065995026697491101ons_or @ int @ ( bit_se2584673776208193580ke_bit @ int @ N2 @ K2 ) @ ( bit_se4730199178511100633sh_bit @ int @ N2 @ L ) ) ) ) ).
% concat_bit_def
thf(fact_6065_predicate2D__conj,axiom,
! [A: $tType,B: $tType,P: A > B > $o,Q: A > B > $o,R: $o,X: A,Y: B] :
( ( ( ord_less_eq @ ( A > B > $o ) @ P @ Q )
& R )
=> ( R
& ( ( P @ X @ Y )
=> ( Q @ X @ Y ) ) ) ) ).
% predicate2D_conj
thf(fact_6066_bit__concat__bit__iff,axiom,
! [M: nat,K: int,L2: int,N3: nat] :
( ( bit_se5641148757651400278ts_bit @ int @ ( bit_concat_bit @ M @ K @ L2 ) @ N3 )
= ( ( ( ord_less @ nat @ N3 @ M )
& ( bit_se5641148757651400278ts_bit @ int @ K @ N3 ) )
| ( ( ord_less_eq @ nat @ M @ N3 )
& ( bit_se5641148757651400278ts_bit @ int @ L2 @ ( minus_minus @ nat @ N3 @ M ) ) ) ) ) ).
% bit_concat_bit_iff
thf(fact_6067_signed__take__bit__eq__concat__bit,axiom,
( ( bit_ri4674362597316999326ke_bit @ int )
= ( ^ [N2: nat,K2: int] : ( bit_concat_bit @ N2 @ K2 @ ( uminus_uminus @ int @ ( zero_neq_one_of_bool @ int @ ( bit_se5641148757651400278ts_bit @ int @ K2 @ N2 ) ) ) ) ) ) ).
% signed_take_bit_eq_concat_bit
thf(fact_6068_eq__subset,axiom,
! [A: $tType,P: A > A > $o] :
( ord_less_eq @ ( A > A > $o )
@ ^ [Y5: A,Z5: A] : Y5 = Z5
@ ^ [A6: A,B5: A] :
( ( P @ A6 @ B5 )
| ( A6 = B5 ) ) ) ).
% eq_subset
thf(fact_6069_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_6070_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_6071_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_6072_test__bit__cat,axiom,
! [A: $tType,B: $tType,C: $tType] :
( ( ( type_len @ C )
& ( type_len @ B )
& ( type_len @ A ) )
=> ! [A2: word @ B,B2: word @ C,N3: nat] :
( ( bit_se5641148757651400278ts_bit @ ( word @ A ) @ ( word_cat @ B @ C @ A @ A2 @ B2 ) @ N3 )
= ( ( ord_less @ nat @ N3 @ ( size_size @ ( word @ A ) @ ( word_cat @ B @ C @ A @ A2 @ B2 ) ) )
& ( ( ord_less @ nat @ N3 @ ( size_size @ ( word @ C ) @ B2 ) )
=> ( bit_se5641148757651400278ts_bit @ ( word @ C ) @ B2 @ N3 ) )
& ( ~ ( ord_less @ nat @ N3 @ ( size_size @ ( word @ C ) @ B2 ) )
=> ( bit_se5641148757651400278ts_bit @ ( word @ B ) @ A2 @ ( minus_minus @ nat @ N3 @ ( size_size @ ( word @ C ) @ B2 ) ) ) ) ) ) ) ).
% test_bit_cat
thf(fact_6073_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_6074_cat__slices,axiom,
! [C: $tType,A: $tType,B: $tType] :
( ( ( type_len @ B )
& ( type_len @ A )
& ( type_len @ C ) )
=> ! [A2: word @ A,N3: nat,C2: word @ B,B2: word @ C] :
( ( A2
= ( slice2 @ B @ A @ N3 @ C2 ) )
=> ( ( B2
= ( slice2 @ B @ C @ ( zero_zero @ nat ) @ C2 ) )
=> ( ( N3
= ( size_size @ ( word @ C ) @ B2 ) )
=> ( ( ord_less_eq @ nat @ ( size_size @ ( word @ B ) @ C2 ) @ ( plus_plus @ nat @ ( size_size @ ( word @ A ) @ A2 ) @ ( size_size @ ( word @ C ) @ B2 ) ) )
=> ( ( word_cat @ A @ C @ B @ A2 @ B2 )
= C2 ) ) ) ) ) ) ).
% cat_slices
thf(fact_6075_sum__diff1_H__aux,axiom,
! [B: $tType,A: $tType] :
( ( ab_group_add @ B )
=> ! [F4: set @ A,I5: set @ A,F2: A > B,I: A] :
( ( finite_finite2 @ A @ F4 )
=> ( ( ord_less_eq @ ( set @ A )
@ ( collect @ A
@ ^ [I3: A] :
( ( member @ A @ I3 @ I5 )
& ( ( F2 @ I3 )
!= ( zero_zero @ B ) ) ) )
@ F4 )
=> ( ( ( member @ A @ I @ I5 )
=> ( ( groups1027152243600224163dd_sum @ A @ B @ F2 @ ( minus_minus @ ( set @ A ) @ I5 @ ( insert @ A @ I @ ( bot_bot @ ( set @ A ) ) ) ) )
= ( minus_minus @ B @ ( groups1027152243600224163dd_sum @ A @ B @ F2 @ I5 ) @ ( F2 @ I ) ) ) )
& ( ~ ( member @ A @ I @ I5 )
=> ( ( groups1027152243600224163dd_sum @ A @ B @ F2 @ ( minus_minus @ ( set @ A ) @ I5 @ ( insert @ A @ I @ ( bot_bot @ ( set @ A ) ) ) ) )
= ( groups1027152243600224163dd_sum @ A @ B @ F2 @ I5 ) ) ) ) ) ) ) ).
% sum_diff1'_aux
thf(fact_6076_sum_Oinsert_H,axiom,
! [A: $tType,B: $tType] :
( ( comm_monoid_add @ A )
=> ! [I5: set @ B,P4: B > A,I: B] :
( ( finite_finite2 @ B
@ ( collect @ B
@ ^ [X2: B] :
( ( member @ B @ X2 @ I5 )
& ( ( P4 @ X2 )
!= ( zero_zero @ A ) ) ) ) )
=> ( ( ( member @ B @ I @ I5 )
=> ( ( groups1027152243600224163dd_sum @ B @ A @ P4 @ ( insert @ B @ I @ I5 ) )
= ( groups1027152243600224163dd_sum @ B @ A @ P4 @ I5 ) ) )
& ( ~ ( member @ B @ I @ I5 )
=> ( ( groups1027152243600224163dd_sum @ B @ A @ P4 @ ( insert @ B @ I @ I5 ) )
= ( plus_plus @ A @ ( P4 @ I ) @ ( groups1027152243600224163dd_sum @ B @ A @ P4 @ I5 ) ) ) ) ) ) ) ).
% sum.insert'
thf(fact_6077_sum_Odistrib__triv_H,axiom,
! [A: $tType,B: $tType] :
( ( comm_monoid_add @ A )
=> ! [I5: set @ B,G: B > A,H2: B > A] :
( ( finite_finite2 @ B @ I5 )
=> ( ( groups1027152243600224163dd_sum @ B @ A
@ ^ [I3: B] : ( plus_plus @ A @ ( G @ I3 ) @ ( H2 @ I3 ) )
@ I5 )
= ( plus_plus @ A @ ( groups1027152243600224163dd_sum @ B @ A @ G @ I5 ) @ ( groups1027152243600224163dd_sum @ B @ A @ H2 @ I5 ) ) ) ) ) ).
% sum.distrib_triv'
thf(fact_6078_sum_Omono__neutral__cong__right_H,axiom,
! [A: $tType,B: $tType] :
( ( comm_monoid_add @ A )
=> ! [S3: set @ B,T5: set @ B,G: B > A,H2: B > A] :
( ( ord_less_eq @ ( set @ B ) @ S3 @ T5 )
=> ( ! [X3: B] :
( ( member @ B @ X3 @ ( minus_minus @ ( set @ B ) @ T5 @ S3 ) )
=> ( ( G @ X3 )
= ( zero_zero @ A ) ) )
=> ( ! [X3: B] :
( ( member @ B @ X3 @ S3 )
=> ( ( G @ X3 )
= ( H2 @ X3 ) ) )
=> ( ( groups1027152243600224163dd_sum @ B @ A @ G @ T5 )
= ( groups1027152243600224163dd_sum @ B @ A @ H2 @ S3 ) ) ) ) ) ) ).
% sum.mono_neutral_cong_right'
thf(fact_6079_sum_Omono__neutral__cong__left_H,axiom,
! [A: $tType,B: $tType] :
( ( comm_monoid_add @ A )
=> ! [S3: set @ B,T5: set @ B,H2: B > A,G: B > A] :
( ( ord_less_eq @ ( set @ B ) @ S3 @ T5 )
=> ( ! [I2: B] :
( ( member @ B @ I2 @ ( minus_minus @ ( set @ B ) @ T5 @ S3 ) )
=> ( ( H2 @ I2 )
= ( zero_zero @ A ) ) )
=> ( ! [X3: B] :
( ( member @ B @ X3 @ S3 )
=> ( ( G @ X3 )
= ( H2 @ X3 ) ) )
=> ( ( groups1027152243600224163dd_sum @ B @ A @ G @ S3 )
= ( groups1027152243600224163dd_sum @ B @ A @ H2 @ T5 ) ) ) ) ) ) ).
% sum.mono_neutral_cong_left'
thf(fact_6080_sum_Omono__neutral__right_H,axiom,
! [A: $tType,B: $tType] :
( ( comm_monoid_add @ A )
=> ! [S3: set @ B,T5: set @ B,G: B > A] :
( ( ord_less_eq @ ( set @ B ) @ S3 @ T5 )
=> ( ! [X3: B] :
( ( member @ B @ X3 @ ( minus_minus @ ( set @ B ) @ T5 @ S3 ) )
=> ( ( G @ X3 )
= ( zero_zero @ A ) ) )
=> ( ( groups1027152243600224163dd_sum @ B @ A @ G @ T5 )
= ( groups1027152243600224163dd_sum @ B @ A @ G @ S3 ) ) ) ) ) ).
% sum.mono_neutral_right'
thf(fact_6081_sum_Omono__neutral__left_H,axiom,
! [A: $tType,B: $tType] :
( ( comm_monoid_add @ A )
=> ! [S3: set @ B,T5: set @ B,G: B > A] :
( ( ord_less_eq @ ( set @ B ) @ S3 @ T5 )
=> ( ! [X3: B] :
( ( member @ B @ X3 @ ( minus_minus @ ( set @ B ) @ T5 @ S3 ) )
=> ( ( G @ X3 )
= ( zero_zero @ A ) ) )
=> ( ( groups1027152243600224163dd_sum @ B @ A @ G @ S3 )
= ( groups1027152243600224163dd_sum @ B @ A @ G @ T5 ) ) ) ) ) ).
% sum.mono_neutral_left'
thf(fact_6082_sum_Odistrib_H,axiom,
! [A: $tType,B: $tType] :
( ( comm_monoid_add @ A )
=> ! [I5: set @ B,G: B > A,H2: B > A] :
( ( finite_finite2 @ B
@ ( collect @ B
@ ^ [X2: B] :
( ( member @ B @ X2 @ I5 )
& ( ( G @ X2 )
!= ( zero_zero @ A ) ) ) ) )
=> ( ( finite_finite2 @ B
@ ( collect @ B
@ ^ [X2: B] :
( ( member @ B @ X2 @ I5 )
& ( ( H2 @ X2 )
!= ( zero_zero @ A ) ) ) ) )
=> ( ( groups1027152243600224163dd_sum @ B @ A
@ ^ [I3: B] : ( plus_plus @ A @ ( G @ I3 ) @ ( H2 @ I3 ) )
@ I5 )
= ( plus_plus @ A @ ( groups1027152243600224163dd_sum @ B @ A @ G @ I5 ) @ ( groups1027152243600224163dd_sum @ B @ A @ H2 @ I5 ) ) ) ) ) ) ).
% sum.distrib'
thf(fact_6083_slice__cat1,axiom,
! [C: $tType,A: $tType,B: $tType] :
( ( ( type_len @ B )
& ( type_len @ A )
& ( type_len @ C ) )
=> ! [A2: word @ B,B2: word @ C] :
( ( ord_less_eq @ nat @ ( plus_plus @ nat @ ( size_size @ ( word @ B ) @ A2 ) @ ( size_size @ ( word @ C ) @ B2 ) ) @ ( size_size @ ( word @ A ) @ ( word_cat @ B @ C @ A @ A2 @ B2 ) ) )
=> ( ( slice2 @ A @ B @ ( size_size @ ( word @ C ) @ B2 ) @ ( word_cat @ B @ C @ A @ A2 @ B2 ) )
= A2 ) ) ) ).
% slice_cat1
thf(fact_6084_sum__diff1_H,axiom,
! [B: $tType,A: $tType] :
( ( ab_group_add @ B )
=> ! [I5: set @ A,F2: A > B,I: A] :
( ( finite_finite2 @ A
@ ( collect @ A
@ ^ [I3: A] :
( ( member @ A @ I3 @ I5 )
& ( ( F2 @ I3 )
!= ( zero_zero @ B ) ) ) ) )
=> ( ( ( member @ A @ I @ I5 )
=> ( ( groups1027152243600224163dd_sum @ A @ B @ F2 @ ( minus_minus @ ( set @ A ) @ I5 @ ( insert @ A @ I @ ( bot_bot @ ( set @ A ) ) ) ) )
= ( minus_minus @ B @ ( groups1027152243600224163dd_sum @ A @ B @ F2 @ I5 ) @ ( F2 @ I ) ) ) )
& ( ~ ( member @ A @ I @ I5 )
=> ( ( groups1027152243600224163dd_sum @ A @ B @ F2 @ ( minus_minus @ ( set @ A ) @ I5 @ ( insert @ A @ I @ ( bot_bot @ ( set @ A ) ) ) ) )
= ( groups1027152243600224163dd_sum @ A @ B @ F2 @ I5 ) ) ) ) ) ) ).
% sum_diff1'
thf(fact_6085_split__slices,axiom,
! [A: $tType,B: $tType,C: $tType] :
( ( ( type_len @ C )
& ( type_len @ B )
& ( type_len @ A ) )
=> ! [W: word @ C,U: word @ A,V: word @ B] :
( ( ( word_split @ C @ A @ B @ W )
= ( product_Pair @ ( word @ A ) @ ( word @ B ) @ U @ V ) )
=> ( ( U
= ( slice2 @ C @ A @ ( size_size @ ( word @ B ) @ V ) @ W ) )
& ( V
= ( slice2 @ C @ B @ ( zero_zero @ nat ) @ W ) ) ) ) ) ).
% split_slices
thf(fact_6086_bit__horner__sum__bit__iff,axiom,
! [A: $tType] :
( ( bit_un5681908812861735899ations @ A )
=> ! [Bs: list @ $o,N3: nat] :
( ( bit_se5641148757651400278ts_bit @ A @ ( groups4207007520872428315er_sum @ $o @ A @ ( zero_neq_one_of_bool @ A ) @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ Bs ) @ N3 )
= ( ( ord_less @ nat @ N3 @ ( size_size @ ( list @ $o ) @ Bs ) )
& ( nth @ $o @ Bs @ N3 ) ) ) ) ).
% bit_horner_sum_bit_iff
thf(fact_6087_time__array__of__list,axiom,
! [A: $tType] :
( ( heap @ A )
=> ! [Xs2: list @ A,H2: heap_ext @ product_unit] :
( ( time_time @ ( array @ A ) @ ( array_of_list @ A @ Xs2 ) @ H2 )
= ( plus_plus @ nat @ ( size_size @ ( list @ A ) @ Xs2 ) @ ( one_one @ nat ) ) ) ) ).
% time_array_of_list
thf(fact_6088_TBOUND__of__list,axiom,
! [A: $tType] :
( ( heap @ A )
=> ! [Xs2: list @ A] : ( time_TBOUND @ ( array @ A ) @ ( array_of_list @ A @ Xs2 ) @ ( suc @ ( size_size @ ( list @ A ) @ Xs2 ) ) ) ) ).
% TBOUND_of_list
thf(fact_6089_of__list__rule,axiom,
! [A: $tType] :
( ( heap @ A )
=> ! [Xs2: list @ A] :
( hoare_hoare_triple @ ( array @ A ) @ ( one_one @ assn ) @ ( array_of_list @ A @ Xs2 )
@ ^ [R5: array @ A] : ( snga_assn @ A @ R5 @ Xs2 ) ) ) ).
% of_list_rule
thf(fact_6090_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_6091_horner__sum__simps_I2_J,axiom,
! [A: $tType,B: $tType] :
( ( comm_semiring_0 @ A )
=> ! [F2: B > A,A2: A,X: B,Xs2: list @ B] :
( ( groups4207007520872428315er_sum @ B @ A @ F2 @ A2 @ ( cons @ B @ X @ Xs2 ) )
= ( plus_plus @ A @ ( F2 @ X ) @ ( times_times @ A @ A2 @ ( groups4207007520872428315er_sum @ B @ A @ F2 @ A2 @ Xs2 ) ) ) ) ) ).
% horner_sum_simps(2)
thf(fact_6092_horner__sum__eq__sum,axiom,
! [A: $tType,B: $tType] :
( ( comm_semiring_1 @ A )
=> ( ( groups4207007520872428315er_sum @ B @ A )
= ( ^ [F7: B > A,A6: A,Xs: list @ B] :
( groups7311177749621191930dd_sum @ nat @ A
@ ^ [N2: nat] : ( times_times @ A @ ( F7 @ ( nth @ B @ Xs @ N2 ) ) @ ( power_power @ A @ A6 @ N2 ) )
@ ( set_or7035219750837199246ssThan @ nat @ ( zero_zero @ nat ) @ ( size_size @ ( list @ B ) @ Xs ) ) ) ) ) ) ).
% horner_sum_eq_sum
thf(fact_6093_horner__sum__foldr,axiom,
! [A: $tType,B: $tType] :
( ( comm_semiring_0 @ A )
=> ( ( groups4207007520872428315er_sum @ B @ A )
= ( ^ [F7: B > A,A6: A,Xs: list @ B] :
( foldr @ B @ A
@ ^ [X2: B,B5: A] : ( plus_plus @ A @ ( F7 @ X2 ) @ ( times_times @ A @ A6 @ B5 ) )
@ Xs
@ ( zero_zero @ A ) ) ) ) ) ).
% horner_sum_foldr
thf(fact_6094_horner__sum__bit__eq__take__bit,axiom,
! [A: $tType] :
( ( bit_se359711467146920520ations @ A )
=> ! [A2: A,N3: nat] :
( ( groups4207007520872428315er_sum @ $o @ A @ ( zero_neq_one_of_bool @ A ) @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ ( map @ nat @ $o @ ( bit_se5641148757651400278ts_bit @ A @ A2 ) @ ( upt @ ( zero_zero @ nat ) @ N3 ) ) )
= ( bit_se2584673776208193580ke_bit @ A @ N3 @ A2 ) ) ) ).
% horner_sum_bit_eq_take_bit
thf(fact_6095_sdiv__word__min,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [A2: word @ A,B2: word @ A] : ( ord_less_eq @ int @ ( uminus_uminus @ int @ ( power_power @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ ( minus_minus @ nat @ ( size_size @ ( word @ A ) @ A2 ) @ ( one_one @ nat ) ) ) ) @ ( signed7115095781618012415divide @ int @ ( ring_1_signed @ A @ int @ A2 ) @ ( ring_1_signed @ A @ int @ B2 ) ) ) ) ).
% sdiv_word_min
thf(fact_6096_length__upt,axiom,
! [I: nat,J: nat] :
( ( size_size @ ( list @ nat ) @ ( upt @ I @ J ) )
= ( minus_minus @ nat @ J @ I ) ) ).
% length_upt
thf(fact_6097_nth__upt,axiom,
! [I: nat,K: nat,J: nat] :
( ( ord_less @ nat @ ( plus_plus @ nat @ I @ K ) @ J )
=> ( ( nth @ nat @ ( upt @ I @ J ) @ K )
= ( plus_plus @ nat @ I @ K ) ) ) ).
% nth_upt
thf(fact_6098_upt__conv__Cons__Cons,axiom,
! [M: nat,N3: nat,Ns: list @ nat,Q2: nat] :
( ( ( cons @ nat @ M @ ( cons @ nat @ N3 @ Ns ) )
= ( upt @ M @ Q2 ) )
= ( ( cons @ nat @ N3 @ Ns )
= ( upt @ ( suc @ M ) @ Q2 ) ) ) ).
% upt_conv_Cons_Cons
thf(fact_6099_upt__conv__Cons,axiom,
! [I: nat,J: nat] :
( ( ord_less @ nat @ I @ J )
=> ( ( upt @ I @ J )
= ( cons @ nat @ I @ ( upt @ ( suc @ I ) @ J ) ) ) ) ).
% upt_conv_Cons
thf(fact_6100_atLeastLessThan__upt,axiom,
( ( set_or7035219750837199246ssThan @ nat )
= ( ^ [I3: nat,J3: nat] : ( set2 @ nat @ ( upt @ I3 @ J3 ) ) ) ) ).
% atLeastLessThan_upt
thf(fact_6101_map__Suc__upt,axiom,
! [M: nat,N3: nat] :
( ( map @ nat @ nat @ suc @ ( upt @ M @ N3 ) )
= ( upt @ ( suc @ M ) @ ( suc @ N3 ) ) ) ).
% map_Suc_upt
thf(fact_6102_map__add__upt_H,axiom,
! [Ofs: nat,A2: nat,B2: nat] :
( ( map @ nat @ nat
@ ^ [I3: nat] : ( plus_plus @ nat @ I3 @ Ofs )
@ ( upt @ A2 @ B2 ) )
= ( upt @ ( plus_plus @ nat @ A2 @ Ofs ) @ ( plus_plus @ nat @ B2 @ Ofs ) ) ) ).
% map_add_upt'
thf(fact_6103_map__add__upt,axiom,
! [N3: nat,M: nat] :
( ( map @ nat @ nat
@ ^ [I3: nat] : ( plus_plus @ nat @ I3 @ N3 )
@ ( upt @ ( zero_zero @ nat ) @ M ) )
= ( upt @ N3 @ ( plus_plus @ nat @ M @ N3 ) ) ) ).
% map_add_upt
thf(fact_6104_upt__eq__Cons__conv,axiom,
! [I: nat,J: nat,X: nat,Xs2: list @ nat] :
( ( ( upt @ I @ J )
= ( cons @ nat @ X @ Xs2 ) )
= ( ( ord_less @ nat @ I @ J )
& ( I = X )
& ( ( upt @ ( plus_plus @ nat @ I @ ( one_one @ nat ) ) @ J )
= Xs2 ) ) ) ).
% upt_eq_Cons_conv
thf(fact_6105_map__decr__upt,axiom,
! [M: nat,N3: nat] :
( ( map @ nat @ nat
@ ^ [N2: nat] : ( minus_minus @ nat @ N2 @ ( suc @ ( zero_zero @ nat ) ) )
@ ( upt @ ( suc @ M ) @ ( suc @ N3 ) ) )
= ( upt @ M @ N3 ) ) ).
% map_decr_upt
thf(fact_6106_atLeastAtMost__upt,axiom,
( ( set_or1337092689740270186AtMost @ nat )
= ( ^ [N2: nat,M5: nat] : ( set2 @ nat @ ( upt @ N2 @ ( suc @ M5 ) ) ) ) ) ).
% atLeastAtMost_upt
thf(fact_6107_atLeast__upt,axiom,
( ( set_ord_lessThan @ nat )
= ( ^ [N2: nat] : ( set2 @ nat @ ( upt @ ( zero_zero @ nat ) @ N2 ) ) ) ) ).
% atLeast_upt
thf(fact_6108_atMost__upto,axiom,
( ( set_ord_atMost @ nat )
= ( ^ [N2: nat] : ( set2 @ nat @ ( upt @ ( zero_zero @ nat ) @ ( suc @ N2 ) ) ) ) ) ).
% atMost_upto
thf(fact_6109_map__upt__Suc,axiom,
! [A: $tType,F2: nat > A,N3: nat] :
( ( map @ nat @ A @ F2 @ ( upt @ ( zero_zero @ nat ) @ ( suc @ N3 ) ) )
= ( cons @ A @ ( F2 @ ( zero_zero @ nat ) )
@ ( map @ nat @ A
@ ^ [I3: nat] : ( F2 @ ( suc @ I3 ) )
@ ( upt @ ( zero_zero @ nat ) @ N3 ) ) ) ) ).
% map_upt_Suc
thf(fact_6110_map__nth,axiom,
! [A: $tType,Xs2: list @ A] :
( ( map @ nat @ A @ ( nth @ A @ Xs2 ) @ ( upt @ ( zero_zero @ nat ) @ ( size_size @ ( list @ A ) @ Xs2 ) ) )
= Xs2 ) ).
% map_nth
thf(fact_6111_nth__map__upt,axiom,
! [A: $tType,I: nat,N3: nat,M: nat,F2: nat > A] :
( ( ord_less @ nat @ I @ ( minus_minus @ nat @ N3 @ M ) )
=> ( ( nth @ A @ ( map @ nat @ A @ F2 @ ( upt @ M @ N3 ) ) @ I )
= ( F2 @ ( plus_plus @ nat @ M @ I ) ) ) ) ).
% nth_map_upt
thf(fact_6112_map__upt__eqI,axiom,
! [A: $tType,Xs2: list @ A,N3: nat,M: nat,F2: nat > A] :
( ( ( size_size @ ( list @ A ) @ Xs2 )
= ( minus_minus @ nat @ N3 @ M ) )
=> ( ! [I2: nat] :
( ( ord_less @ nat @ I2 @ ( size_size @ ( list @ A ) @ Xs2 ) )
=> ( ( nth @ A @ Xs2 @ I2 )
= ( F2 @ ( plus_plus @ nat @ M @ I2 ) ) ) )
=> ( ( map @ nat @ A @ F2 @ ( upt @ M @ N3 ) )
= Xs2 ) ) ) ).
% map_upt_eqI
thf(fact_6113_sdiv__word__max,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [A2: word @ A,B2: word @ A] : ( ord_less_eq @ int @ ( signed7115095781618012415divide @ int @ ( ring_1_signed @ A @ int @ A2 ) @ ( ring_1_signed @ A @ int @ B2 ) ) @ ( power_power @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ ( minus_minus @ nat @ ( size_size @ ( word @ A ) @ A2 ) @ ( suc @ ( zero_zero @ nat ) ) ) ) ) ) ).
% sdiv_word_max
thf(fact_6114_int__sdiv__negated__is__minus1,axiom,
! [A2: int,B2: int] :
( ( A2
!= ( zero_zero @ int ) )
=> ( ( ( signed7115095781618012415divide @ int @ A2 @ B2 )
= ( uminus_uminus @ int @ A2 ) )
= ( B2
= ( uminus_uminus @ int @ ( one_one @ int ) ) ) ) ) ).
% int_sdiv_negated_is_minus1
thf(fact_6115_int__sdiv__simps_I3_J,axiom,
! [A2: int] :
( ( signed7115095781618012415divide @ int @ A2 @ ( uminus_uminus @ int @ ( one_one @ int ) ) )
= ( uminus_uminus @ int @ A2 ) ) ).
% int_sdiv_simps(3)
thf(fact_6116_int__sdiv__simps_I1_J,axiom,
! [A2: int] :
( ( signed7115095781618012415divide @ int @ A2 @ ( one_one @ int ) )
= A2 ) ).
% int_sdiv_simps(1)
thf(fact_6117_int__sdiv__same__is__1,axiom,
! [A2: int,B2: int] :
( ( A2
!= ( zero_zero @ int ) )
=> ( ( ( signed7115095781618012415divide @ int @ A2 @ B2 )
= A2 )
= ( B2
= ( one_one @ int ) ) ) ) ).
% int_sdiv_same_is_1
thf(fact_6118_make__rule,axiom,
! [A: $tType] :
( ( heap @ A )
=> ! [N3: nat,F2: nat > A] :
( hoare_hoare_triple @ ( array @ A ) @ ( one_one @ assn ) @ ( array_make @ A @ N3 @ F2 )
@ ^ [R5: array @ A] : ( snga_assn @ A @ R5 @ ( map @ nat @ A @ F2 @ ( upt @ ( zero_zero @ nat ) @ N3 ) ) ) ) ) ).
% make_rule
thf(fact_6119_Cauchy__iff2,axiom,
( ( topolo3814608138187158403Cauchy @ real )
= ( ^ [X6: nat > real] :
! [J3: nat] :
? [M12: nat] :
! [M5: nat] :
( ( ord_less_eq @ nat @ M12 @ M5 )
=> ! [N2: nat] :
( ( ord_less_eq @ nat @ M12 @ N2 )
=> ( ord_less @ real @ ( abs_abs @ real @ ( minus_minus @ real @ ( X6 @ M5 ) @ ( X6 @ N2 ) ) ) @ ( inverse_inverse @ real @ ( semiring_1_of_nat @ real @ ( suc @ J3 ) ) ) ) ) ) ) ) ).
% Cauchy_iff2
thf(fact_6120_time__array__make,axiom,
! [A: $tType] :
( ( heap @ A )
=> ! [N3: nat,F2: nat > A,H2: heap_ext @ product_unit] :
( ( time_time @ ( array @ A ) @ ( array_make @ A @ N3 @ F2 ) @ H2 )
= ( plus_plus @ nat @ N3 @ ( one_one @ nat ) ) ) ) ).
% time_array_make
thf(fact_6121_TBOUND__make,axiom,
! [A: $tType] :
( ( heap @ A )
=> ! [N3: nat,F2: nat > A] : ( time_TBOUND @ ( array @ A ) @ ( array_make @ A @ N3 @ F2 ) @ ( plus_plus @ nat @ N3 @ ( one_one @ nat ) ) ) ) ).
% TBOUND_make
thf(fact_6122_Cauchy__iff,axiom,
! [A: $tType] :
( ( real_V822414075346904944vector @ A )
=> ( ( topolo3814608138187158403Cauchy @ A )
= ( ^ [X6: nat > A] :
! [E4: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ E4 )
=> ? [M12: nat] :
! [M5: nat] :
( ( ord_less_eq @ nat @ M12 @ M5 )
=> ! [N2: nat] :
( ( ord_less_eq @ nat @ M12 @ N2 )
=> ( ord_less @ real @ ( real_V7770717601297561774m_norm @ A @ ( minus_minus @ A @ ( X6 @ M5 ) @ ( X6 @ N2 ) ) ) @ E4 ) ) ) ) ) ) ) ).
% Cauchy_iff
thf(fact_6123_CauchyI,axiom,
! [A: $tType] :
( ( real_V822414075346904944vector @ A )
=> ! [X8: nat > A] :
( ! [E2: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ E2 )
=> ? [M13: nat] :
! [M4: nat] :
( ( ord_less_eq @ nat @ M13 @ M4 )
=> ! [N: nat] :
( ( ord_less_eq @ nat @ M13 @ N )
=> ( ord_less @ real @ ( real_V7770717601297561774m_norm @ A @ ( minus_minus @ A @ ( X8 @ M4 ) @ ( X8 @ N ) ) ) @ E2 ) ) ) )
=> ( topolo3814608138187158403Cauchy @ A @ X8 ) ) ) ).
% CauchyI
thf(fact_6124_CauchyD,axiom,
! [A: $tType] :
( ( real_V822414075346904944vector @ A )
=> ! [X8: nat > A,E: real] :
( ( topolo3814608138187158403Cauchy @ A @ X8 )
=> ( ( ord_less @ real @ ( zero_zero @ real ) @ E )
=> ? [M10: nat] :
! [M2: nat] :
( ( ord_less_eq @ nat @ M10 @ M2 )
=> ! [N9: nat] :
( ( ord_less_eq @ nat @ M10 @ N9 )
=> ( ord_less @ real @ ( real_V7770717601297561774m_norm @ A @ ( minus_minus @ A @ ( X8 @ M2 ) @ ( X8 @ N9 ) ) ) @ E ) ) ) ) ) ) ).
% CauchyD
thf(fact_6125_array__of__list__make,axiom,
! [A: $tType] :
( ( heap @ A )
=> ( ( array_of_list @ A )
= ( ^ [Xs: list @ A] : ( array_make @ A @ ( size_size @ ( list @ A ) @ Xs ) @ ( nth @ A @ Xs ) ) ) ) ) ).
% array_of_list_make
thf(fact_6126_VEBT_Osize_I3_J,axiom,
! [X11: option @ ( product_prod @ nat @ nat ),X12: nat,X13: list @ vEBT_VEBT,X14: vEBT_VEBT] :
( ( size_size @ vEBT_VEBT @ ( vEBT_Node @ X11 @ X12 @ X13 @ X14 ) )
= ( plus_plus @ nat @ ( plus_plus @ nat @ ( size_list @ vEBT_VEBT @ ( size_size @ vEBT_VEBT ) @ X13 ) @ ( size_size @ vEBT_VEBT @ X14 ) ) @ ( suc @ ( zero_zero @ nat ) ) ) ) ).
% VEBT.size(3)
thf(fact_6127_size__list__estimation,axiom,
! [A: $tType,X: A,Xs2: list @ A,Y: nat,F2: A > nat] :
( ( member @ A @ X @ ( set2 @ A @ Xs2 ) )
=> ( ( ord_less @ nat @ Y @ ( F2 @ X ) )
=> ( ord_less @ nat @ Y @ ( size_list @ A @ F2 @ Xs2 ) ) ) ) ).
% size_list_estimation
thf(fact_6128_size__list__estimation_H,axiom,
! [A: $tType,X: A,Xs2: list @ A,Y: nat,F2: A > nat] :
( ( member @ A @ X @ ( set2 @ A @ Xs2 ) )
=> ( ( ord_less_eq @ nat @ Y @ ( F2 @ X ) )
=> ( ord_less_eq @ nat @ Y @ ( size_list @ A @ F2 @ Xs2 ) ) ) ) ).
% size_list_estimation'
thf(fact_6129_size__list__pointwise,axiom,
! [A: $tType,Xs2: list @ A,F2: A > nat,G: A > nat] :
( ! [X3: A] :
( ( member @ A @ X3 @ ( set2 @ A @ Xs2 ) )
=> ( ord_less_eq @ nat @ ( F2 @ X3 ) @ ( G @ X3 ) ) )
=> ( ord_less_eq @ nat @ ( size_list @ A @ F2 @ Xs2 ) @ ( size_list @ A @ G @ Xs2 ) ) ) ).
% size_list_pointwise
thf(fact_6130_list_Osize__gen_I2_J,axiom,
! [A: $tType,X: A > nat,X21: A,X222: list @ A] :
( ( size_list @ A @ X @ ( cons @ A @ X21 @ X222 ) )
= ( plus_plus @ nat @ ( plus_plus @ nat @ ( X @ X21 ) @ ( size_list @ A @ X @ X222 ) ) @ ( suc @ ( zero_zero @ nat ) ) ) ) ).
% list.size_gen(2)
thf(fact_6131_VEBT_Osize__gen_I1_J,axiom,
! [X11: option @ ( product_prod @ nat @ nat ),X12: nat,X13: list @ vEBT_VEBT,X14: vEBT_VEBT] :
( ( vEBT_size_VEBT @ ( vEBT_Node @ X11 @ X12 @ X13 @ X14 ) )
= ( plus_plus @ nat @ ( plus_plus @ nat @ ( size_list @ vEBT_VEBT @ vEBT_size_VEBT @ X13 ) @ ( vEBT_size_VEBT @ X14 ) ) @ ( suc @ ( zero_zero @ nat ) ) ) ) ).
% VEBT.size_gen(1)
thf(fact_6132_image__Collect__subsetI,axiom,
! [A: $tType,B: $tType,P: A > $o,F2: A > B,B6: set @ B] :
( ! [X3: A] :
( ( P @ X3 )
=> ( member @ B @ ( F2 @ X3 ) @ B6 ) )
=> ( ord_less_eq @ ( set @ B ) @ ( image @ A @ B @ F2 @ ( collect @ A @ P ) ) @ B6 ) ) ).
% image_Collect_subsetI
thf(fact_6133_ssubst__Pair__rhs,axiom,
! [B: $tType,A: $tType,R2: A,S: B,R: set @ ( product_prod @ A @ B ),S4: B] :
( ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ R2 @ S ) @ R )
=> ( ( S4 = S )
=> ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ R2 @ S4 ) @ R ) ) ) ).
% ssubst_Pair_rhs
thf(fact_6134_Collect__restrict,axiom,
! [A: $tType,X8: set @ A,P: A > $o] :
( ord_less_eq @ ( set @ A )
@ ( collect @ A
@ ^ [X2: A] :
( ( member @ A @ X2 @ X8 )
& ( P @ X2 ) ) )
@ X8 ) ).
% Collect_restrict
thf(fact_6135_prop__restrict,axiom,
! [A: $tType,X: A,Z9: set @ A,X8: set @ A,P: A > $o] :
( ( member @ A @ X @ Z9 )
=> ( ( ord_less_eq @ ( set @ A ) @ Z9
@ ( collect @ A
@ ^ [X2: A] :
( ( member @ A @ X2 @ X8 )
& ( P @ X2 ) ) ) )
=> ( P @ X ) ) ) ).
% prop_restrict
thf(fact_6136_VEBT_Osize__gen_I2_J,axiom,
! [X21: $o,X222: $o] :
( ( vEBT_size_VEBT @ ( vEBT_Leaf @ X21 @ X222 ) )
= ( zero_zero @ nat ) ) ).
% VEBT.size_gen(2)
thf(fact_6137_subset__emptyI,axiom,
! [A: $tType,A3: set @ A] :
( ! [X3: A] :
~ ( member @ A @ X3 @ A3 )
=> ( ord_less_eq @ ( set @ A ) @ A3 @ ( bot_bot @ ( set @ A ) ) ) ) ).
% subset_emptyI
thf(fact_6138_insert__subsetI,axiom,
! [A: $tType,X: A,A3: set @ A,X8: set @ A] :
( ( member @ A @ X @ A3 )
=> ( ( ord_less_eq @ ( set @ A ) @ X8 @ A3 )
=> ( ord_less_eq @ ( set @ A ) @ ( insert @ A @ X @ X8 ) @ A3 ) ) ) ).
% insert_subsetI
thf(fact_6139_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_6140_smod__int__range,axiom,
! [B2: int,A2: int] :
( ( B2
!= ( zero_zero @ int ) )
=> ( member @ int @ ( signed6721504322012087516modulo @ int @ A2 @ B2 ) @ ( set_or1337092689740270186AtMost @ int @ ( plus_plus @ int @ ( uminus_uminus @ int @ ( abs_abs @ int @ B2 ) ) @ ( one_one @ int ) ) @ ( minus_minus @ int @ ( abs_abs @ int @ B2 ) @ ( one_one @ int ) ) ) ) ) ).
% smod_int_range
thf(fact_6141_in__set__product__lists__length,axiom,
! [A: $tType,Xs2: list @ A,Xss: list @ ( list @ A )] :
( ( member @ ( list @ A ) @ Xs2 @ ( set2 @ ( list @ A ) @ ( product_lists @ A @ Xss ) ) )
=> ( ( size_size @ ( list @ A ) @ Xs2 )
= ( size_size @ ( list @ ( list @ A ) ) @ Xss ) ) ) ).
% in_set_product_lists_length
thf(fact_6142_smod__int__compares_I8_J,axiom,
! [A2: int,B2: int] :
( ( ord_less_eq @ int @ A2 @ ( zero_zero @ int ) )
=> ( ( ord_less @ int @ B2 @ ( zero_zero @ int ) )
=> ( ord_less_eq @ int @ B2 @ ( signed6721504322012087516modulo @ int @ A2 @ B2 ) ) ) ) ).
% smod_int_compares(8)
thf(fact_6143_smod__int__compares_I7_J,axiom,
! [A2: int,B2: int] :
( ( ord_less_eq @ int @ A2 @ ( zero_zero @ int ) )
=> ( ( ord_less @ int @ B2 @ ( zero_zero @ int ) )
=> ( ord_less_eq @ int @ ( signed6721504322012087516modulo @ int @ A2 @ B2 ) @ ( zero_zero @ int ) ) ) ) ).
% smod_int_compares(7)
thf(fact_6144_smod__int__compares_I6_J,axiom,
! [A2: int,B2: int] :
( ( ord_less_eq @ int @ ( zero_zero @ int ) @ A2 )
=> ( ( ord_less @ int @ B2 @ ( zero_zero @ int ) )
=> ( ord_less_eq @ int @ ( zero_zero @ int ) @ ( signed6721504322012087516modulo @ int @ A2 @ B2 ) ) ) ) ).
% smod_int_compares(6)
thf(fact_6145_smod__int__compares_I4_J,axiom,
! [A2: int,B2: int] :
( ( ord_less_eq @ int @ A2 @ ( zero_zero @ int ) )
=> ( ( ord_less @ int @ ( zero_zero @ int ) @ B2 )
=> ( ord_less_eq @ int @ ( signed6721504322012087516modulo @ int @ A2 @ B2 ) @ ( zero_zero @ int ) ) ) ) ).
% smod_int_compares(4)
thf(fact_6146_smod__int__compares_I2_J,axiom,
! [A2: int,B2: int] :
( ( ord_less_eq @ int @ ( zero_zero @ int ) @ A2 )
=> ( ( ord_less @ int @ ( zero_zero @ int ) @ B2 )
=> ( ord_less_eq @ int @ ( zero_zero @ int ) @ ( signed6721504322012087516modulo @ int @ A2 @ B2 ) ) ) ) ).
% smod_int_compares(2)
thf(fact_6147_smod__int__compares_I1_J,axiom,
! [A2: int,B2: int] :
( ( ord_less_eq @ int @ ( zero_zero @ int ) @ A2 )
=> ( ( ord_less @ int @ ( zero_zero @ int ) @ B2 )
=> ( ord_less @ int @ ( signed6721504322012087516modulo @ int @ A2 @ B2 ) @ B2 ) ) ) ).
% smod_int_compares(1)
thf(fact_6148_smod__mod__positive,axiom,
! [A2: int,B2: int] :
( ( ord_less_eq @ int @ ( zero_zero @ int ) @ A2 )
=> ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ B2 )
=> ( ( signed6721504322012087516modulo @ int @ A2 @ B2 )
= ( modulo_modulo @ int @ A2 @ B2 ) ) ) ) ).
% smod_mod_positive
thf(fact_6149_signed__modulo__int__def,axiom,
( ( signed6721504322012087516modulo @ int )
= ( ^ [K2: int,L: int] : ( minus_minus @ int @ K2 @ ( times_times @ int @ ( signed7115095781618012415divide @ int @ K2 @ L ) @ L ) ) ) ) ).
% signed_modulo_int_def
thf(fact_6150_smod__int__compares_I3_J,axiom,
! [A2: int,B2: int] :
( ( ord_less_eq @ int @ A2 @ ( zero_zero @ int ) )
=> ( ( ord_less @ int @ ( zero_zero @ int ) @ B2 )
=> ( ord_less @ int @ ( uminus_uminus @ int @ B2 ) @ ( signed6721504322012087516modulo @ int @ A2 @ B2 ) ) ) ) ).
% smod_int_compares(3)
thf(fact_6151_smod__int__compares_I5_J,axiom,
! [A2: int,B2: int] :
( ( ord_less_eq @ int @ ( zero_zero @ int ) @ A2 )
=> ( ( ord_less @ int @ B2 @ ( zero_zero @ int ) )
=> ( ord_less @ int @ ( signed6721504322012087516modulo @ int @ A2 @ B2 ) @ ( uminus_uminus @ int @ B2 ) ) ) ) ).
% smod_int_compares(5)
thf(fact_6152_length__subseqs,axiom,
! [A: $tType,Xs2: list @ A] :
( ( size_size @ ( list @ ( list @ A ) ) @ ( subseqs @ A @ Xs2 ) )
= ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( size_size @ ( list @ A ) @ Xs2 ) ) ) ).
% length_subseqs
thf(fact_6153_length__mul__elem,axiom,
! [A: $tType,Xs2: list @ ( list @ A ),N3: nat] :
( ! [X3: list @ A] :
( ( member @ ( list @ A ) @ X3 @ ( set2 @ ( list @ A ) @ Xs2 ) )
=> ( ( size_size @ ( list @ A ) @ X3 )
= N3 ) )
=> ( ( size_size @ ( list @ A ) @ ( concat @ A @ Xs2 ) )
= ( times_times @ nat @ ( size_size @ ( list @ ( list @ A ) ) @ Xs2 ) @ N3 ) ) ) ).
% length_mul_elem
thf(fact_6154_Cons__in__subseqsD,axiom,
! [A: $tType,Y: A,Ys: list @ A,Xs2: list @ A] :
( ( member @ ( list @ A ) @ ( cons @ A @ Y @ Ys ) @ ( set2 @ ( list @ A ) @ ( subseqs @ A @ Xs2 ) ) )
=> ( member @ ( list @ A ) @ Ys @ ( set2 @ ( list @ A ) @ ( subseqs @ A @ Xs2 ) ) ) ) ).
% Cons_in_subseqsD
thf(fact_6155_subseqs__refl,axiom,
! [A: $tType,Xs2: list @ A] : ( member @ ( list @ A ) @ Xs2 @ ( set2 @ ( list @ A ) @ ( subseqs @ A @ Xs2 ) ) ) ).
% subseqs_refl
thf(fact_6156_product__concat__map,axiom,
! [B: $tType,A: $tType] :
( ( product @ A @ B )
= ( ^ [Xs: list @ A,Ys3: list @ B] :
( concat @ ( product_prod @ A @ B )
@ ( map @ A @ ( list @ ( product_prod @ A @ B ) )
@ ^ [X2: A] : ( map @ B @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X2 ) @ Ys3 )
@ Xs ) ) ) ) ).
% product_concat_map
thf(fact_6157_subset__subseqs,axiom,
! [A: $tType,X8: set @ A,Xs2: list @ A] :
( ( ord_less_eq @ ( set @ A ) @ X8 @ ( set2 @ A @ Xs2 ) )
=> ( member @ ( set @ A ) @ X8 @ ( image @ ( list @ A ) @ ( set @ A ) @ ( set2 @ A ) @ ( set2 @ ( list @ A ) @ ( subseqs @ A @ Xs2 ) ) ) ) ) ).
% subset_subseqs
thf(fact_6158_set__n__lists,axiom,
! [A: $tType,N3: nat,Xs2: list @ A] :
( ( set2 @ ( list @ A ) @ ( n_lists @ A @ N3 @ Xs2 ) )
= ( collect @ ( list @ A )
@ ^ [Ys3: list @ A] :
( ( ( size_size @ ( list @ A ) @ Ys3 )
= N3 )
& ( ord_less_eq @ ( set @ A ) @ ( set2 @ A @ Ys3 ) @ ( set2 @ A @ Xs2 ) ) ) ) ) ).
% set_n_lists
thf(fact_6159_mod__word__minus__1__minus__numeral,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [B2: num] :
( ( modulo_modulo @ ( word @ A ) @ ( uminus_uminus @ ( word @ A ) @ ( one_one @ ( word @ A ) ) ) @ ( uminus_uminus @ ( word @ A ) @ ( numeral_numeral @ ( word @ A ) @ B2 ) ) )
= ( 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 @ B2 ) ) ) ) ) ) ) ).
% mod_word_minus_1_minus_numeral
thf(fact_6160_drop__bit__word__beyond,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [N3: nat,W: word @ A] :
( ( ord_less_eq @ nat @ ( type_len0_len_of @ A @ ( type2 @ A ) ) @ N3 )
=> ( ( bit_se4197421643247451524op_bit @ ( word @ A ) @ N3 @ W )
= ( zero_zero @ ( word @ A ) ) ) ) ) ).
% drop_bit_word_beyond
thf(fact_6161_push__bit__word__beyond,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [N3: nat,W: word @ A] :
( ( ord_less_eq @ nat @ ( type_len0_len_of @ A @ ( type2 @ A ) ) @ N3 )
=> ( ( bit_se4730199178511100633sh_bit @ ( word @ A ) @ N3 @ W )
= ( zero_zero @ ( word @ A ) ) ) ) ) ).
% push_bit_word_beyond
thf(fact_6162_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_6163_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_6164_less__eq__word__numeral__numeral,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [A2: num,B2: num] :
( ( ord_less_eq @ ( word @ A ) @ ( numeral_numeral @ ( word @ A ) @ A2 ) @ ( numeral_numeral @ ( word @ A ) @ B2 ) )
= ( ord_less_eq @ int @ ( bit_se2584673776208193580ke_bit @ int @ ( type_len0_len_of @ A @ ( type2 @ A ) ) @ ( numeral_numeral @ int @ A2 ) ) @ ( bit_se2584673776208193580ke_bit @ int @ ( type_len0_len_of @ A @ ( type2 @ A ) ) @ ( numeral_numeral @ int @ B2 ) ) ) ) ) ).
% less_eq_word_numeral_numeral
thf(fact_6165_bit__numeral__word__iff,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [W: num,N3: nat] :
( ( bit_se5641148757651400278ts_bit @ ( word @ A ) @ ( numeral_numeral @ ( word @ A ) @ W ) @ N3 )
= ( ( ord_less @ nat @ N3 @ ( type_len0_len_of @ A @ ( type2 @ A ) ) )
& ( bit_se5641148757651400278ts_bit @ int @ ( numeral_numeral @ int @ W ) @ N3 ) ) ) ) ).
% bit_numeral_word_iff
thf(fact_6166_unsigned__numeral,axiom,
! [A: $tType,B: $tType] :
( ( ( type_len @ B )
& ( semiring_1 @ A ) )
=> ! [N3: num] :
( ( semiring_1_unsigned @ B @ A @ ( numeral_numeral @ ( word @ B ) @ N3 ) )
= ( semiring_1_of_nat @ A @ ( bit_se2584673776208193580ke_bit @ nat @ ( type_len0_len_of @ B @ ( type2 @ B ) ) @ ( numeral_numeral @ nat @ N3 ) ) ) ) ) ).
% unsigned_numeral
thf(fact_6167_unat__lt2p,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [X: word @ A] : ( ord_less @ nat @ ( semiring_1_unsigned @ A @ nat @ X ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( type_len0_len_of @ A @ ( type2 @ A ) ) ) ) ) ).
% unat_lt2p
thf(fact_6168_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_6169_uint__lt2p,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [X: word @ A] : ( ord_less @ int @ ( semiring_1_unsigned @ A @ int @ X ) @ ( power_power @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ ( type_len0_len_of @ A @ ( type2 @ A ) ) ) ) ) ).
% uint_lt2p
thf(fact_6170_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_6171_exp__eq__zero__iff,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [N3: nat] :
( ( ( power_power @ ( word @ A ) @ ( numeral_numeral @ ( word @ A ) @ ( bit0 @ one2 ) ) @ N3 )
= ( zero_zero @ ( word @ A ) ) )
= ( ord_less_eq @ nat @ ( type_len0_len_of @ A @ ( type2 @ A ) ) @ N3 ) ) ) ).
% exp_eq_zero_iff
thf(fact_6172_signed__take__bit__word__Suc__numeral,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [N3: nat,K: num] :
( ( bit_ri4674362597316999326ke_bit @ ( word @ A ) @ ( suc @ N3 ) @ ( numeral_numeral @ ( word @ A ) @ K ) )
= ( ring_1_of_int @ ( word @ A ) @ ( bit_ri4674362597316999326ke_bit @ int @ ( suc @ N3 ) @ ( bit_se2584673776208193580ke_bit @ int @ ( type_len0_len_of @ A @ ( type2 @ A ) ) @ ( numeral_numeral @ int @ K ) ) ) ) ) ) ).
% signed_take_bit_word_Suc_numeral
thf(fact_6173_signed__take__bit__word__numeral,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [N3: num,K: num] :
( ( bit_ri4674362597316999326ke_bit @ ( word @ A ) @ ( numeral_numeral @ nat @ N3 ) @ ( numeral_numeral @ ( word @ A ) @ K ) )
= ( ring_1_of_int @ ( word @ A ) @ ( bit_ri4674362597316999326ke_bit @ int @ ( numeral_numeral @ nat @ N3 ) @ ( bit_se2584673776208193580ke_bit @ int @ ( type_len0_len_of @ A @ ( type2 @ A ) ) @ ( numeral_numeral @ int @ K ) ) ) ) ) ) ).
% signed_take_bit_word_numeral
thf(fact_6174_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_6175_signed__numeral,axiom,
! [A: $tType,B: $tType] :
( ( ( type_len @ B )
& ( ring_1 @ A ) )
=> ! [N3: num] :
( ( ring_1_signed @ B @ A @ ( numeral_numeral @ ( word @ B ) @ N3 ) )
= ( 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 @ N3 ) ) ) ) ) ).
% signed_numeral
thf(fact_6176_less__eq__word__minus__numeral__minus__numeral,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [A2: num,B2: num] :
( ( ord_less_eq @ ( word @ A ) @ ( uminus_uminus @ ( word @ A ) @ ( numeral_numeral @ ( word @ A ) @ A2 ) ) @ ( uminus_uminus @ ( word @ A ) @ ( numeral_numeral @ ( word @ A ) @ B2 ) ) )
= ( ord_less_eq @ int @ ( bit_se2584673776208193580ke_bit @ int @ ( type_len0_len_of @ A @ ( type2 @ A ) ) @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ A2 ) ) ) @ ( bit_se2584673776208193580ke_bit @ int @ ( type_len0_len_of @ A @ ( type2 @ A ) ) @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ B2 ) ) ) ) ) ) ).
% less_eq_word_minus_numeral_minus_numeral
thf(fact_6177_less__eq__word__numeral__minus__numeral,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [A2: num,B2: num] :
( ( ord_less_eq @ ( word @ A ) @ ( numeral_numeral @ ( word @ A ) @ A2 ) @ ( uminus_uminus @ ( word @ A ) @ ( numeral_numeral @ ( word @ A ) @ B2 ) ) )
= ( ord_less_eq @ int @ ( bit_se2584673776208193580ke_bit @ int @ ( type_len0_len_of @ A @ ( type2 @ A ) ) @ ( numeral_numeral @ int @ A2 ) ) @ ( bit_se2584673776208193580ke_bit @ int @ ( type_len0_len_of @ A @ ( type2 @ A ) ) @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ B2 ) ) ) ) ) ) ).
% less_eq_word_numeral_minus_numeral
thf(fact_6178_less__eq__word__minus__numeral__numeral,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [A2: num,B2: num] :
( ( ord_less_eq @ ( word @ A ) @ ( uminus_uminus @ ( word @ A ) @ ( numeral_numeral @ ( word @ A ) @ A2 ) ) @ ( numeral_numeral @ ( word @ A ) @ B2 ) )
= ( ord_less_eq @ int @ ( bit_se2584673776208193580ke_bit @ int @ ( type_len0_len_of @ A @ ( type2 @ A ) ) @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ A2 ) ) ) @ ( bit_se2584673776208193580ke_bit @ int @ ( type_len0_len_of @ A @ ( type2 @ A ) ) @ ( numeral_numeral @ int @ B2 ) ) ) ) ) ).
% less_eq_word_minus_numeral_numeral
thf(fact_6179_bit__neg__numeral__word__iff,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [W: num,N3: nat] :
( ( bit_se5641148757651400278ts_bit @ ( word @ A ) @ ( uminus_uminus @ ( word @ A ) @ ( numeral_numeral @ ( word @ A ) @ W ) ) @ N3 )
= ( ( ord_less @ nat @ N3 @ ( type_len0_len_of @ A @ ( type2 @ A ) ) )
& ( bit_se5641148757651400278ts_bit @ int @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ W ) ) @ N3 ) ) ) ) ).
% bit_neg_numeral_word_iff
thf(fact_6180_drop__bit__word__Suc__numeral,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [N3: nat,K: num] :
( ( bit_se4197421643247451524op_bit @ ( word @ A ) @ ( suc @ N3 ) @ ( numeral_numeral @ ( word @ A ) @ K ) )
= ( ring_1_of_int @ ( word @ A ) @ ( bit_se4197421643247451524op_bit @ int @ ( suc @ N3 ) @ ( bit_se2584673776208193580ke_bit @ int @ ( type_len0_len_of @ A @ ( type2 @ A ) ) @ ( numeral_numeral @ int @ K ) ) ) ) ) ) ).
% drop_bit_word_Suc_numeral
thf(fact_6181_drop__bit__word__numeral,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [N3: num,K: num] :
( ( bit_se4197421643247451524op_bit @ ( word @ A ) @ ( numeral_numeral @ nat @ N3 ) @ ( numeral_numeral @ ( word @ A ) @ K ) )
= ( ring_1_of_int @ ( word @ A ) @ ( bit_se4197421643247451524op_bit @ int @ ( numeral_numeral @ nat @ N3 ) @ ( bit_se2584673776208193580ke_bit @ int @ ( type_len0_len_of @ A @ ( type2 @ A ) ) @ ( numeral_numeral @ int @ K ) ) ) ) ) ) ).
% drop_bit_word_numeral
thf(fact_6182_unat__power__lower,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [N3: nat] :
( ( ord_less @ nat @ N3 @ ( type_len0_len_of @ A @ ( type2 @ A ) ) )
=> ( ( semiring_1_unsigned @ A @ nat @ ( power_power @ ( word @ A ) @ ( numeral_numeral @ ( word @ A ) @ ( bit0 @ one2 ) ) @ N3 ) )
= ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N3 ) ) ) ) ).
% unat_power_lower
thf(fact_6183_signed__take__bit__word__Suc__minus__numeral,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [N3: nat,K: num] :
( ( bit_ri4674362597316999326ke_bit @ ( word @ A ) @ ( suc @ N3 ) @ ( uminus_uminus @ ( word @ A ) @ ( numeral_numeral @ ( word @ A ) @ K ) ) )
= ( ring_1_of_int @ ( word @ A ) @ ( bit_ri4674362597316999326ke_bit @ int @ ( suc @ N3 ) @ ( 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_6184_signed__take__bit__word__minus__numeral,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [N3: num,K: num] :
( ( bit_ri4674362597316999326ke_bit @ ( word @ A ) @ ( numeral_numeral @ nat @ N3 ) @ ( uminus_uminus @ ( word @ A ) @ ( numeral_numeral @ ( word @ A ) @ K ) ) )
= ( ring_1_of_int @ ( word @ A ) @ ( bit_ri4674362597316999326ke_bit @ int @ ( numeral_numeral @ nat @ N3 ) @ ( 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_6185_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_6186_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_6187_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_6188_word__less__sub__le,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [N3: nat,X: word @ A] :
( ( ord_less @ nat @ N3 @ ( type_len0_len_of @ A @ ( type2 @ A ) ) )
=> ( ( ord_less_eq @ ( word @ A ) @ X @ ( minus_minus @ ( word @ A ) @ ( power_power @ ( word @ A ) @ ( numeral_numeral @ ( word @ A ) @ ( bit0 @ one2 ) ) @ N3 ) @ ( one_one @ ( word @ A ) ) ) )
= ( ord_less @ ( word @ A ) @ X @ ( power_power @ ( word @ A ) @ ( numeral_numeral @ ( word @ A ) @ ( bit0 @ one2 ) ) @ N3 ) ) ) ) ) ).
% word_less_sub_le
thf(fact_6189_signed__neg__numeral,axiom,
! [A: $tType,B: $tType] :
( ( ( type_len @ B )
& ( ring_1 @ A ) )
=> ! [N3: num] :
( ( ring_1_signed @ B @ A @ ( uminus_uminus @ ( word @ B ) @ ( numeral_numeral @ ( word @ B ) @ N3 ) ) )
= ( 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 @ N3 ) ) ) ) ) ) ).
% signed_neg_numeral
thf(fact_6190_drop__bit__word__Suc__minus__numeral,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [N3: nat,K: num] :
( ( bit_se4197421643247451524op_bit @ ( word @ A ) @ ( suc @ N3 ) @ ( uminus_uminus @ ( word @ A ) @ ( numeral_numeral @ ( word @ A ) @ K ) ) )
= ( ring_1_of_int @ ( word @ A ) @ ( bit_se4197421643247451524op_bit @ int @ ( suc @ N3 ) @ ( 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_6191_drop__bit__word__minus__numeral,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [N3: num,K: num] :
( ( bit_se4197421643247451524op_bit @ ( word @ A ) @ ( numeral_numeral @ nat @ N3 ) @ ( uminus_uminus @ ( word @ A ) @ ( numeral_numeral @ ( word @ A ) @ K ) ) )
= ( ring_1_of_int @ ( word @ A ) @ ( bit_se4197421643247451524op_bit @ int @ ( numeral_numeral @ nat @ N3 ) @ ( 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_6192_less__word__numeral__minus__1,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [A2: num] :
( ( ord_less @ ( word @ A ) @ ( numeral_numeral @ ( word @ A ) @ A2 ) @ ( uminus_uminus @ ( word @ A ) @ ( one_one @ ( word @ A ) ) ) )
= ( ord_less @ int @ ( bit_se2584673776208193580ke_bit @ int @ ( type_len0_len_of @ A @ ( type2 @ A ) ) @ ( numeral_numeral @ int @ A2 ) ) @ ( minus_minus @ int @ ( power_power @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ ( type_len0_len_of @ A @ ( type2 @ A ) ) ) @ ( one_one @ int ) ) ) ) ) ).
% less_word_numeral_minus_1
thf(fact_6193_less__word__minus__numeral__minus__1,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [A2: num] :
( ( ord_less @ ( word @ A ) @ ( uminus_uminus @ ( word @ A ) @ ( numeral_numeral @ ( word @ A ) @ A2 ) ) @ ( uminus_uminus @ ( word @ A ) @ ( one_one @ ( word @ A ) ) ) )
= ( ord_less @ int @ ( bit_se2584673776208193580ke_bit @ int @ ( type_len0_len_of @ A @ ( type2 @ A ) ) @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ A2 ) ) ) @ ( minus_minus @ int @ ( power_power @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ ( type_len0_len_of @ A @ ( type2 @ A ) ) ) @ ( one_one @ int ) ) ) ) ) ).
% less_word_minus_numeral_minus_1
thf(fact_6194_div__word__minus__1__numeral,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [B2: num] :
( ( divide_divide @ ( word @ A ) @ ( uminus_uminus @ ( word @ A ) @ ( one_one @ ( word @ A ) ) ) @ ( numeral_numeral @ ( word @ A ) @ B2 ) )
= ( 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 @ B2 ) ) ) ) ) ) ).
% div_word_minus_1_numeral
thf(fact_6195_mod__word__minus__1__numeral,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [B2: num] :
( ( modulo_modulo @ ( word @ A ) @ ( uminus_uminus @ ( word @ A ) @ ( one_one @ ( word @ A ) ) ) @ ( numeral_numeral @ ( word @ A ) @ B2 ) )
= ( 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 @ B2 ) ) ) ) ) ) ).
% mod_word_minus_1_numeral
thf(fact_6196_div__word__minus__1__minus__numeral,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [B2: num] :
( ( divide_divide @ ( word @ A ) @ ( uminus_uminus @ ( word @ A ) @ ( one_one @ ( word @ A ) ) ) @ ( uminus_uminus @ ( word @ A ) @ ( numeral_numeral @ ( word @ A ) @ B2 ) ) )
= ( 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 @ B2 ) ) ) ) ) ) ) ).
% div_word_minus_1_minus_numeral
thf(fact_6197_word__cat__inj,axiom,
! [C: $tType,A: $tType,B: $tType] :
( ( ( type_len @ B )
& ( type_len @ A )
& ( type_len @ C ) )
=> ! [A2: word @ A,B2: word @ B,C2: word @ A,D: word @ B] :
( ( ord_less_eq @ nat @ ( plus_plus @ nat @ ( type_len0_len_of @ A @ ( type2 @ A ) ) @ ( type_len0_len_of @ B @ ( type2 @ B ) ) ) @ ( type_len0_len_of @ C @ ( type2 @ C ) ) )
=> ( ( ( word_cat @ A @ B @ C @ A2 @ B2 )
= ( word_cat @ A @ B @ C @ C2 @ D ) )
= ( ( A2 = C2 )
& ( B2 = D ) ) ) ) ) ).
% word_cat_inj
thf(fact_6198_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,N3: nat] :
( ( bit_se5641148757651400278ts_bit @ ( word @ C ) @ ( word_cat @ A @ B @ C @ V @ W ) @ N3 )
= ( ( ord_less @ nat @ N3 @ ( type_len0_len_of @ C @ ( type2 @ C ) ) )
& ( ( ord_less @ nat @ N3 @ ( type_len0_len_of @ B @ ( type2 @ B ) ) )
=> ( bit_se5641148757651400278ts_bit @ ( word @ B ) @ W @ N3 ) )
& ( ~ ( ord_less @ nat @ N3 @ ( type_len0_len_of @ B @ ( type2 @ B ) ) )
=> ( bit_se5641148757651400278ts_bit @ ( word @ A ) @ V @ ( minus_minus @ nat @ N3 @ ( type_len0_len_of @ B @ ( type2 @ B ) ) ) ) ) ) ) ) ).
% bit_word_cat_iff
thf(fact_6199_ucast__mask__drop,axiom,
! [A: $tType,B: $tType] :
( ( ( type_len @ B )
& ( type_len @ A ) )
=> ! [N3: nat,X: word @ B] :
( ( ord_less_eq @ nat @ ( type_len0_len_of @ A @ ( type2 @ A ) ) @ N3 )
=> ( ( semiring_1_unsigned @ B @ ( word @ A ) @ ( bit_se5824344872417868541ns_and @ ( word @ B ) @ X @ ( bit_se2239418461657761734s_mask @ ( word @ B ) @ N3 ) ) )
= ( semiring_1_unsigned @ B @ ( word @ A ) @ X ) ) ) ) ).
% ucast_mask_drop
thf(fact_6200_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_6201_word__of__nat__less__iff,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [M: nat,N3: nat] :
( ( ord_less @ ( word @ A ) @ ( semiring_1_of_nat @ ( word @ A ) @ M ) @ ( semiring_1_of_nat @ ( word @ A ) @ N3 ) )
= ( 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 ) ) @ N3 ) ) ) ) ).
% word_of_nat_less_iff
thf(fact_6202_word__of__nat__less__eq__iff,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [M: nat,N3: nat] :
( ( ord_less_eq @ ( word @ A ) @ ( semiring_1_of_nat @ ( word @ A ) @ M ) @ ( semiring_1_of_nat @ ( word @ A ) @ N3 ) )
= ( 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 ) ) @ N3 ) ) ) ) ).
% word_of_nat_less_eq_iff
thf(fact_6203_test__bit__wi,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [X: int,N3: nat] :
( ( bit_se5641148757651400278ts_bit @ ( word @ A ) @ ( ring_1_of_int @ ( word @ A ) @ X ) @ N3 )
= ( ( ord_less @ nat @ N3 @ ( type_len0_len_of @ A @ ( type2 @ A ) ) )
& ( bit_se5641148757651400278ts_bit @ int @ X @ N3 ) ) ) ) ).
% test_bit_wi
thf(fact_6204_bit__word__of__int__iff,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [K: int,N3: nat] :
( ( bit_se5641148757651400278ts_bit @ ( word @ A ) @ ( ring_1_of_int @ ( word @ A ) @ K ) @ N3 )
= ( ( ord_less @ nat @ N3 @ ( type_len0_len_of @ A @ ( type2 @ A ) ) )
& ( bit_se5641148757651400278ts_bit @ int @ K @ N3 ) ) ) ) ).
% bit_word_of_int_iff
thf(fact_6205_signed__take__bit__eq,axiom,
! [A: $tType,B: $tType] :
( ( ( type_len @ B )
& ( bit_ri3973907225187159222ations @ A ) )
=> ! [N3: nat,W: word @ B] :
( ( ( ord_less @ nat @ N3 @ ( type_len0_len_of @ B @ ( type2 @ B ) ) )
=> ( ( ring_1_signed @ B @ A @ ( bit_se2584673776208193580ke_bit @ ( word @ B ) @ N3 @ W ) )
= ( bit_se2584673776208193580ke_bit @ A @ N3 @ ( ring_1_signed @ B @ A @ W ) ) ) )
& ( ~ ( ord_less @ nat @ N3 @ ( type_len0_len_of @ B @ ( type2 @ B ) ) )
=> ( ( ring_1_signed @ B @ A @ ( bit_se2584673776208193580ke_bit @ ( word @ B ) @ N3 @ W ) )
= ( ring_1_signed @ B @ A @ W ) ) ) ) ) ).
% signed_take_bit_eq
thf(fact_6206_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_6207_neg__test__bit,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [X: word @ A,N3: nat] :
( ( bit_se5641148757651400278ts_bit @ ( word @ A ) @ ( bit_ri4277139882892585799ns_not @ ( word @ A ) @ X ) @ N3 )
= ( ~ ( bit_se5641148757651400278ts_bit @ ( word @ A ) @ X @ N3 )
& ( ord_less @ nat @ N3 @ ( type_len0_len_of @ A @ ( type2 @ A ) ) ) ) ) ) ).
% neg_test_bit
thf(fact_6208_bit__word__ucast__iff,axiom,
! [B: $tType,A: $tType] :
( ( ( type_len @ A )
& ( type_len @ B ) )
=> ! [W: word @ A,N3: nat] :
( ( bit_se5641148757651400278ts_bit @ ( word @ B ) @ ( semiring_1_unsigned @ A @ ( word @ B ) @ W ) @ N3 )
= ( ( ord_less @ nat @ N3 @ ( type_len0_len_of @ A @ ( type2 @ A ) ) )
& ( ord_less @ nat @ N3 @ ( type_len0_len_of @ B @ ( type2 @ B ) ) )
& ( bit_se5641148757651400278ts_bit @ ( word @ A ) @ W @ N3 ) ) ) ) ).
% bit_word_ucast_iff
thf(fact_6209_bin__nth__uint__imp,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [W: word @ A,N3: nat] :
( ( bit_se5641148757651400278ts_bit @ int @ ( semiring_1_unsigned @ A @ int @ W ) @ N3 )
=> ( ord_less @ nat @ N3 @ ( type_len0_len_of @ A @ ( type2 @ A ) ) ) ) ) ).
% bin_nth_uint_imp
thf(fact_6210_nth__ucast,axiom,
! [B: $tType,A: $tType] :
( ( ( type_len @ A )
& ( type_len @ B ) )
=> ! [W: word @ B,N3: nat] :
( ( bit_se5641148757651400278ts_bit @ ( word @ A ) @ ( semiring_1_unsigned @ B @ ( word @ A ) @ W ) @ N3 )
= ( ( bit_se5641148757651400278ts_bit @ ( word @ B ) @ W @ N3 )
& ( ord_less @ nat @ N3 @ ( type_len0_len_of @ A @ ( type2 @ A ) ) ) ) ) ) ).
% nth_ucast
thf(fact_6211_bit__ucast__iff,axiom,
! [A: $tType,B: $tType] :
( ( ( type_len @ B )
& ( type_len @ A ) )
=> ! [A2: word @ B,N3: nat] :
( ( bit_se5641148757651400278ts_bit @ ( word @ A ) @ ( semiring_1_unsigned @ B @ ( word @ A ) @ A2 ) @ N3 )
= ( ( ord_less @ nat @ N3 @ ( type_len0_len_of @ A @ ( type2 @ A ) ) )
& ( bit_se5641148757651400278ts_bit @ ( word @ B ) @ A2 @ N3 ) ) ) ) ).
% bit_ucast_iff
thf(fact_6212_bit__word__eqI,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [A2: word @ A,B2: word @ A] :
( ! [N: nat] :
( ( ord_less @ nat @ N @ ( type_len0_len_of @ A @ ( type2 @ A ) ) )
=> ( ( bit_se5641148757651400278ts_bit @ ( word @ A ) @ A2 @ N )
= ( bit_se5641148757651400278ts_bit @ ( word @ A ) @ B2 @ N ) ) )
=> ( A2 = B2 ) ) ) ).
% bit_word_eqI
thf(fact_6213_word__eq__iff,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ( ( ^ [Y5: word @ A,Z5: word @ A] : Y5 = Z5 )
= ( ^ [X2: word @ A,Y2: word @ A] :
! [N2: nat] :
( ( ord_less @ nat @ N2 @ ( type_len0_len_of @ A @ ( type2 @ A ) ) )
=> ( ( bit_se5641148757651400278ts_bit @ ( word @ A ) @ X2 @ N2 )
= ( bit_se5641148757651400278ts_bit @ ( word @ A ) @ Y2 @ N2 ) ) ) ) ) ) ).
% word_eq_iff
thf(fact_6214_bit__imp__le__length,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [W: word @ A,N3: nat] :
( ( bit_se5641148757651400278ts_bit @ ( word @ A ) @ W @ N3 )
=> ( ord_less @ nat @ N3 @ ( type_len0_len_of @ A @ ( type2 @ A ) ) ) ) ) ).
% bit_imp_le_length
thf(fact_6215_test__bit__conj__lt,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [X: word @ A,M: nat] :
( ( ( bit_se5641148757651400278ts_bit @ ( word @ A ) @ X @ M )
& ( ord_less @ nat @ M @ ( type_len0_len_of @ A @ ( type2 @ A ) ) ) )
= ( bit_se5641148757651400278ts_bit @ ( word @ A ) @ X @ M ) ) ) ).
% test_bit_conj_lt
thf(fact_6216_test__bit__1_H,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [N3: nat] :
( ( bit_se5641148757651400278ts_bit @ ( word @ A ) @ ( one_one @ ( word @ A ) ) @ N3 )
= ( ( ord_less @ nat @ ( zero_zero @ nat ) @ ( type_len0_len_of @ A @ ( type2 @ A ) ) )
& ( N3
= ( zero_zero @ nat ) ) ) ) ) ).
% test_bit_1'
thf(fact_6217_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_6218_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_6219_degenerate__word,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [X: word @ A] :
( ( ( type_len0_len_of @ A @ ( type2 @ A ) )
= ( one_one @ nat ) )
=> ( ( X
= ( zero_zero @ ( word @ A ) ) )
| ( X
= ( one_one @ ( word @ A ) ) ) ) ) ) ).
% degenerate_word
thf(fact_6220_ucast__ucast__id,axiom,
! [B: $tType,A: $tType] :
( ( ( type_len @ A )
& ( type_len @ B ) )
=> ! [X: word @ A] :
( ( ord_less @ nat @ ( type_len0_len_of @ A @ ( type2 @ A ) ) @ ( type_len0_len_of @ B @ ( type2 @ B ) ) )
=> ( ( semiring_1_unsigned @ B @ ( word @ A ) @ ( semiring_1_unsigned @ A @ ( word @ B ) @ X ) )
= X ) ) ) ).
% ucast_ucast_id
thf(fact_6221_ucast__less__ucast__weak,axiom,
! [B: $tType,A: $tType] :
( ( ( type_len @ A )
& ( type_len @ B ) )
=> ! [X: word @ A,Y: word @ A] :
( ( ord_less @ nat @ ( type_len0_len_of @ A @ ( type2 @ A ) ) @ ( type_len0_len_of @ B @ ( type2 @ B ) ) )
=> ( ( ord_less @ ( word @ B ) @ ( semiring_1_unsigned @ A @ ( word @ B ) @ X ) @ ( semiring_1_unsigned @ A @ ( word @ B ) @ Y ) )
= ( ord_less @ ( word @ A ) @ X @ Y ) ) ) ) ).
% ucast_less_ucast_weak
thf(fact_6222_max__test__bit,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [N3: nat] :
( ( bit_se5641148757651400278ts_bit @ ( word @ A ) @ ( uminus_uminus @ ( word @ A ) @ ( one_one @ ( word @ A ) ) ) @ N3 )
= ( ord_less @ nat @ N3 @ ( type_len0_len_of @ A @ ( type2 @ A ) ) ) ) ) ).
% max_test_bit
thf(fact_6223_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_6224_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_6225_test__bit__bin,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ( ( bit_se5641148757651400278ts_bit @ ( word @ A ) )
= ( ^ [W2: word @ A,N2: nat] :
( ( ord_less @ nat @ N2 @ ( type_len0_len_of @ A @ ( type2 @ A ) ) )
& ( bit_se5641148757651400278ts_bit @ int @ ( semiring_1_unsigned @ A @ int @ W2 ) @ N2 ) ) ) ) ) ).
% test_bit_bin
thf(fact_6226_bit__uint__iff,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [W: word @ A,N3: nat] :
( ( bit_se5641148757651400278ts_bit @ int @ ( semiring_1_unsigned @ A @ int @ W ) @ N3 )
= ( ( ord_less @ nat @ N3 @ ( type_len0_len_of @ A @ ( type2 @ A ) ) )
& ( bit_se5641148757651400278ts_bit @ ( word @ A ) @ W @ N3 ) ) ) ) ).
% bit_uint_iff
thf(fact_6227_uint__word__arith__bintrs_I2_J,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [A2: word @ A,B2: word @ A] :
( ( semiring_1_unsigned @ A @ int @ ( minus_minus @ ( word @ A ) @ A2 @ B2 ) )
= ( bit_se2584673776208193580ke_bit @ int @ ( type_len0_len_of @ A @ ( type2 @ A ) ) @ ( minus_minus @ int @ ( semiring_1_unsigned @ A @ int @ A2 ) @ ( semiring_1_unsigned @ A @ int @ B2 ) ) ) ) ) ).
% uint_word_arith_bintrs(2)
thf(fact_6228_mask__over__length,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [N3: nat] :
( ( ord_less_eq @ nat @ ( type_len0_len_of @ A @ ( type2 @ A ) ) @ N3 )
=> ( ( bit_se2239418461657761734s_mask @ ( word @ A ) @ N3 )
= ( uminus_uminus @ ( word @ A ) @ ( one_one @ ( word @ A ) ) ) ) ) ) ).
% mask_over_length
thf(fact_6229_ucast__sub__ucast,axiom,
! [B: $tType,A: $tType] :
( ( ( type_len @ A )
& ( type_len @ B ) )
=> ! [Y: word @ A,X: word @ A] :
( ( ord_less_eq @ ( word @ A ) @ Y @ X )
=> ( ( ord_less_eq @ nat @ ( type_len0_len_of @ A @ ( type2 @ A ) ) @ ( type_len0_len_of @ B @ ( type2 @ B ) ) )
=> ( ( semiring_1_unsigned @ A @ ( word @ B ) @ ( minus_minus @ ( word @ A ) @ X @ Y ) )
= ( minus_minus @ ( word @ B ) @ ( semiring_1_unsigned @ A @ ( word @ B ) @ X ) @ ( semiring_1_unsigned @ A @ ( word @ B ) @ Y ) ) ) ) ) ) ).
% ucast_sub_ucast
thf(fact_6230_word__of__int__less__eq__iff,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [K: int,L2: int] :
( ( ord_less_eq @ ( word @ A ) @ ( ring_1_of_int @ ( word @ A ) @ K ) @ ( ring_1_of_int @ ( word @ A ) @ L2 ) )
= ( 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 ) ) @ L2 ) ) ) ) ).
% word_of_int_less_eq_iff
thf(fact_6231_up__ucast__inj,axiom,
! [B: $tType,A: $tType] :
( ( ( type_len @ A )
& ( type_len @ B ) )
=> ! [X: word @ A,Y: word @ A] :
( ( ( semiring_1_unsigned @ A @ ( word @ B ) @ X )
= ( semiring_1_unsigned @ A @ ( word @ B ) @ Y ) )
=> ( ( ord_less_eq @ nat @ ( type_len0_len_of @ A @ ( type2 @ A ) ) @ ( type_len0_len_of @ B @ ( type2 @ B ) ) )
=> ( X = Y ) ) ) ) ).
% up_ucast_inj
thf(fact_6232_ucast__le__ucast,axiom,
! [B: $tType,A: $tType] :
( ( ( type_len @ A )
& ( type_len @ B ) )
=> ! [X: word @ A,Y: word @ A] :
( ( ord_less_eq @ nat @ ( type_len0_len_of @ A @ ( type2 @ A ) ) @ ( type_len0_len_of @ B @ ( type2 @ B ) ) )
=> ( ( ord_less_eq @ ( word @ B ) @ ( semiring_1_unsigned @ A @ ( word @ B ) @ X ) @ ( semiring_1_unsigned @ A @ ( word @ B ) @ Y ) )
= ( ord_less_eq @ ( word @ A ) @ X @ Y ) ) ) ) ).
% ucast_le_ucast
thf(fact_6233_ucast__ucast__eq,axiom,
! [C: $tType,A: $tType,B: $tType] :
( ( ( type_len @ B )
& ( type_len @ A )
& ( type_len @ C ) )
=> ! [X: word @ A,Y: word @ B] :
( ( ( semiring_1_unsigned @ A @ ( word @ C ) @ X )
= ( semiring_1_unsigned @ A @ ( word @ C ) @ ( semiring_1_unsigned @ B @ ( word @ A ) @ Y ) ) )
=> ( ( ord_less_eq @ nat @ ( type_len0_len_of @ A @ ( type2 @ A ) ) @ ( type_len0_len_of @ B @ ( type2 @ B ) ) )
=> ( ( ord_less_eq @ nat @ ( type_len0_len_of @ B @ ( type2 @ B ) ) @ ( type_len0_len_of @ C @ ( type2 @ C ) ) )
=> ( X
= ( semiring_1_unsigned @ B @ ( word @ A ) @ Y ) ) ) ) ) ) ).
% ucast_ucast_eq
thf(fact_6234_up__ucast__inj__eq,axiom,
! [B: $tType,A: $tType] :
( ( ( type_len @ A )
& ( type_len @ B ) )
=> ! [X: word @ A,Y: word @ A] :
( ( ord_less_eq @ nat @ ( type_len0_len_of @ A @ ( type2 @ A ) ) @ ( type_len0_len_of @ B @ ( type2 @ B ) ) )
=> ( ( ( semiring_1_unsigned @ A @ ( word @ B ) @ X )
= ( semiring_1_unsigned @ A @ ( word @ B ) @ Y ) )
= ( X = Y ) ) ) ) ).
% up_ucast_inj_eq
thf(fact_6235_ucast__up__mono__le,axiom,
! [B: $tType,A: $tType] :
( ( ( type_len @ A )
& ( type_len @ B ) )
=> ! [X: word @ A,Y: word @ A] :
( ( ord_less_eq @ nat @ ( type_len0_len_of @ A @ ( type2 @ A ) ) @ ( type_len0_len_of @ B @ ( type2 @ B ) ) )
=> ( ( ord_less_eq @ ( word @ A ) @ X @ Y )
=> ( ord_less_eq @ ( word @ B ) @ ( semiring_1_unsigned @ A @ ( word @ B ) @ X ) @ ( semiring_1_unsigned @ A @ ( word @ B ) @ Y ) ) ) ) ) ).
% ucast_up_mono_le
thf(fact_6236_eq__ucast__ucast__eq,axiom,
! [A: $tType,B: $tType] :
( ( ( type_len @ B )
& ( type_len @ A ) )
=> ! [X: word @ A,Y: word @ B] :
( ( ord_less_eq @ nat @ ( type_len0_len_of @ B @ ( type2 @ B ) ) @ ( type_len0_len_of @ A @ ( type2 @ A ) ) )
=> ( ( X
= ( semiring_1_unsigned @ B @ ( word @ A ) @ Y ) )
=> ( ( semiring_1_unsigned @ A @ ( word @ B ) @ X )
= Y ) ) ) ) ).
% eq_ucast_ucast_eq
thf(fact_6237_unat__ucast__up__simp,axiom,
! [B: $tType,A: $tType] :
( ( ( type_len @ A )
& ( type_len @ B ) )
=> ! [X: word @ A] :
( ( ord_less_eq @ nat @ ( type_len0_len_of @ A @ ( type2 @ A ) ) @ ( type_len0_len_of @ B @ ( type2 @ B ) ) )
=> ( ( semiring_1_unsigned @ B @ nat @ ( semiring_1_unsigned @ A @ ( word @ B ) @ X ) )
= ( semiring_1_unsigned @ A @ nat @ X ) ) ) ) ).
% unat_ucast_up_simp
thf(fact_6238_less__ucast__ucast__less,axiom,
! [A: $tType,B: $tType] :
( ( ( type_len @ B )
& ( type_len @ A ) )
=> ! [X: word @ A,Y: word @ B] :
( ( ord_less_eq @ nat @ ( type_len0_len_of @ B @ ( type2 @ B ) ) @ ( type_len0_len_of @ A @ ( type2 @ A ) ) )
=> ( ( ord_less @ ( word @ A ) @ X @ ( semiring_1_unsigned @ B @ ( word @ A ) @ Y ) )
=> ( ord_less @ ( word @ B ) @ ( semiring_1_unsigned @ A @ ( word @ B ) @ X ) @ Y ) ) ) ) ).
% less_ucast_ucast_less
thf(fact_6239_ucast__less__ucast,axiom,
! [B: $tType,A: $tType] :
( ( ( type_len @ A )
& ( type_len @ B ) )
=> ! [X: word @ A,Y: word @ A] :
( ( ord_less_eq @ nat @ ( type_len0_len_of @ A @ ( type2 @ A ) ) @ ( type_len0_len_of @ B @ ( type2 @ B ) ) )
=> ( ( ord_less @ ( word @ B ) @ ( semiring_1_unsigned @ A @ ( word @ B ) @ X ) @ ( semiring_1_unsigned @ A @ ( word @ B ) @ Y ) )
= ( ord_less @ ( word @ A ) @ X @ Y ) ) ) ) ).
% ucast_less_ucast
thf(fact_6240_ucast__up__mono,axiom,
! [B: $tType,A: $tType] :
( ( ( type_len @ A )
& ( type_len @ B ) )
=> ! [X: word @ A,Y: word @ A] :
( ( ord_less_eq @ nat @ ( type_len0_len_of @ A @ ( type2 @ A ) ) @ ( type_len0_len_of @ B @ ( type2 @ B ) ) )
=> ( ( ord_less @ ( word @ A ) @ X @ Y )
=> ( ord_less @ ( word @ B ) @ ( semiring_1_unsigned @ A @ ( word @ B ) @ X ) @ ( semiring_1_unsigned @ A @ ( word @ B ) @ Y ) ) ) ) ) ).
% ucast_up_mono
thf(fact_6241_bintr__uint,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [N3: nat,W: word @ A] :
( ( ord_less_eq @ nat @ ( type_len0_len_of @ A @ ( type2 @ A ) ) @ N3 )
=> ( ( bit_se2584673776208193580ke_bit @ int @ N3 @ ( semiring_1_unsigned @ A @ int @ W ) )
= ( semiring_1_unsigned @ A @ int @ W ) ) ) ) ).
% bintr_uint
thf(fact_6242_up__scast__inj__eq,axiom,
! [B: $tType,A: $tType] :
( ( ( type_len @ A )
& ( type_len @ B ) )
=> ! [X: word @ A,Y: word @ A] :
( ( ord_less_eq @ nat @ ( type_len0_len_of @ A @ ( type2 @ A ) ) @ ( type_len0_len_of @ B @ ( type2 @ B ) ) )
=> ( ( ( ring_1_signed @ A @ ( word @ B ) @ X )
= ( ring_1_signed @ A @ ( word @ B ) @ Y ) )
= ( X = Y ) ) ) ) ).
% up_scast_inj_eq
thf(fact_6243_take__bit__word__beyond__length__eq,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [N3: nat,W: word @ A] :
( ( ord_less_eq @ nat @ ( type_len0_len_of @ A @ ( type2 @ A ) ) @ N3 )
=> ( ( bit_se2584673776208193580ke_bit @ ( word @ A ) @ N3 @ W )
= W ) ) ) ).
% take_bit_word_beyond_length_eq
thf(fact_6244_wi__bintr,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [N3: nat,W: int] :
( ( ord_less_eq @ nat @ ( type_len0_len_of @ A @ ( type2 @ A ) ) @ N3 )
=> ( ( ring_1_of_int @ ( word @ A ) @ ( bit_se2584673776208193580ke_bit @ int @ N3 @ W ) )
= ( ring_1_of_int @ ( word @ A ) @ W ) ) ) ) ).
% wi_bintr
thf(fact_6245_ucast__ucast__add,axiom,
! [A: $tType,B: $tType] :
( ( ( type_len @ B )
& ( type_len @ A ) )
=> ! [X: word @ A,Y: word @ B] :
( ( ord_less_eq @ nat @ ( type_len0_len_of @ A @ ( type2 @ A ) ) @ ( type_len0_len_of @ B @ ( type2 @ B ) ) )
=> ( ( semiring_1_unsigned @ B @ ( word @ A ) @ ( plus_plus @ ( word @ B ) @ ( semiring_1_unsigned @ A @ ( word @ B ) @ X ) @ Y ) )
= ( plus_plus @ ( word @ A ) @ X @ ( semiring_1_unsigned @ B @ ( word @ A ) @ Y ) ) ) ) ) ).
% ucast_ucast_add
thf(fact_6246_bit__set__bit__aux,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [F2: nat > $o,N3: nat,W: word @ A,M: nat] :
( ( bit_se5641148757651400278ts_bit @ ( word @ A ) @ ( code_T2661198915054445665ts_aux @ A @ F2 @ N3 @ W ) @ M )
= ( ( ord_less @ nat @ M @ ( type_len0_len_of @ A @ ( type2 @ A ) ) )
& ( ( ord_less @ nat @ M @ N3 )
=> ( F2 @ M ) )
& ( ~ ( ord_less @ nat @ M @ N3 )
=> ( bit_se5641148757651400278ts_bit @ ( word @ A ) @ W @ ( minus_minus @ nat @ M @ N3 ) ) ) ) ) ) ).
% bit_set_bit_aux
thf(fact_6247_ucast__drop__bit__eq,axiom,
! [B: $tType,A: $tType] :
( ( ( type_len @ A )
& ( type_len @ B ) )
=> ! [N3: 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 ) @ N3 @ W ) )
= ( bit_se4197421643247451524op_bit @ ( word @ B ) @ N3 @ ( semiring_1_unsigned @ A @ ( word @ B ) @ W ) ) ) ) ) ).
% ucast_drop_bit_eq
thf(fact_6248_bit__set__bit__word__iff,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [W: word @ A,M: nat,B2: $o,N3: nat] :
( ( bit_se5641148757651400278ts_bit @ ( word @ A ) @ ( generi7602027413899671122et_bit @ ( word @ A ) @ W @ M @ B2 ) @ N3 )
= ( ( ( M = N3 )
=> ( ( ord_less @ nat @ N3 @ ( type_len0_len_of @ A @ ( type2 @ A ) ) )
& B2 ) )
& ( ( M != N3 )
=> ( bit_se5641148757651400278ts_bit @ ( word @ A ) @ W @ N3 ) ) ) ) ) ).
% bit_set_bit_word_iff
thf(fact_6249_nth__slice,axiom,
! [B: $tType,A: $tType] :
( ( ( type_len @ A )
& ( type_len @ B ) )
=> ! [N3: nat,W: word @ B,M: nat] :
( ( bit_se5641148757651400278ts_bit @ ( word @ A ) @ ( slice2 @ B @ A @ N3 @ W ) @ M )
= ( ( bit_se5641148757651400278ts_bit @ ( word @ B ) @ W @ ( plus_plus @ nat @ M @ N3 ) )
& ( ord_less @ nat @ M @ ( type_len0_len_of @ A @ ( type2 @ A ) ) ) ) ) ) ).
% nth_slice
thf(fact_6250_size__word_Orep__eq,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ( ( size_size @ ( word @ A ) )
= ( ^ [X2: word @ A] : ( type_len0_len_of @ A @ ( type2 @ A ) ) ) ) ) ).
% size_word.rep_eq
thf(fact_6251_word__size,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ( ( size_size @ ( word @ A ) )
= ( ^ [W2: word @ A] : ( type_len0_len_of @ A @ ( type2 @ A ) ) ) ) ) ).
% word_size
thf(fact_6252_up__scast__inj,axiom,
! [B: $tType,A: $tType] :
( ( ( type_len @ A )
& ( type_len @ B ) )
=> ! [X: word @ A,Y: word @ A] :
( ( ( ring_1_signed @ A @ ( word @ B ) @ X )
= ( ring_1_signed @ A @ ( word @ B ) @ Y ) )
=> ( ( ord_less_eq @ nat @ ( size_size @ ( word @ A ) @ X ) @ ( type_len0_len_of @ B @ ( type2 @ B ) ) )
=> ( X = Y ) ) ) ) ).
% up_scast_inj
thf(fact_6253_two__power__eq,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [N3: nat,M: nat] :
( ( ord_less @ nat @ N3 @ ( 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 ) ) @ N3 )
= ( power_power @ ( word @ A ) @ ( numeral_numeral @ ( word @ A ) @ ( bit0 @ one2 ) ) @ M ) )
= ( N3 = M ) ) ) ) ) ).
% two_power_eq
thf(fact_6254_signed__take__bit__decr__length__iff,axiom,
! [A: $tType,B: $tType] :
( ( ( bit_ri3973907225187159222ations @ B )
& ( type_len @ A ) )
=> ! [K: B,L2: 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 ) ) ) @ L2 ) )
= ( ( bit_se2584673776208193580ke_bit @ B @ ( type_len0_len_of @ A @ ( type2 @ A ) ) @ K )
= ( bit_se2584673776208193580ke_bit @ B @ ( type_len0_len_of @ A @ ( type2 @ A ) ) @ L2 ) ) ) ) ).
% signed_take_bit_decr_length_iff
thf(fact_6255_num__of__sbintr_H,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [A2: num,B2: num] :
( ( ( bit_ri4674362597316999326ke_bit @ int @ ( minus_minus @ nat @ ( type_len0_len_of @ A @ ( type2 @ A ) ) @ ( one_one @ nat ) ) @ ( numeral_numeral @ int @ A2 ) )
= ( numeral_numeral @ int @ B2 ) )
=> ( ( numeral_numeral @ ( word @ A ) @ A2 )
= ( numeral_numeral @ ( word @ A ) @ B2 ) ) ) ) ).
% num_of_sbintr'
thf(fact_6256_word__split__bin_H,axiom,
! [C: $tType,B: $tType,A: $tType] :
( ( ( type_len @ A )
& ( type_len @ B )
& ( type_len @ C ) )
=> ( ( word_split @ A @ B @ C )
= ( ^ [W2: word @ A] : ( product_Pair @ ( word @ B ) @ ( word @ C ) @ ( semiring_1_unsigned @ A @ ( word @ B ) @ ( bit_se4197421643247451524op_bit @ ( word @ A ) @ ( type_len0_len_of @ C @ ( type2 @ C ) ) @ W2 ) ) @ ( semiring_1_unsigned @ A @ ( word @ C ) @ W2 ) ) ) ) ) ).
% word_split_bin'
thf(fact_6257_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_6258_bin__nth__sint,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [N3: nat,W: word @ A] :
( ( ord_less_eq @ nat @ ( type_len0_len_of @ A @ ( type2 @ A ) ) @ N3 )
=> ( ( bit_se5641148757651400278ts_bit @ int @ ( ring_1_signed @ A @ int @ W ) @ N3 )
= ( 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_6259_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_6260_neg__mask__test__bit,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [N3: nat,M: nat] :
( ( bit_se5641148757651400278ts_bit @ ( word @ A ) @ ( bit_ri4277139882892585799ns_not @ ( word @ A ) @ ( bit_se2239418461657761734s_mask @ ( word @ A ) @ N3 ) ) @ M )
= ( ( ord_less_eq @ nat @ N3 @ M )
& ( ord_less @ nat @ M @ ( type_len0_len_of @ A @ ( type2 @ A ) ) ) ) ) ) ).
% neg_mask_test_bit
thf(fact_6261_word__of__int__bin__cat__eq__iff,axiom,
! [C: $tType,B: $tType,A: $tType] :
( ( ( type_len @ A )
& ( type_len @ B )
& ( type_len @ C ) )
=> ! [B2: word @ B,A2: word @ A,D: word @ B,C2: word @ A] :
( ( ord_less_eq @ nat @ ( plus_plus @ nat @ ( type_len0_len_of @ A @ ( type2 @ A ) ) @ ( type_len0_len_of @ B @ ( type2 @ B ) ) ) @ ( type_len0_len_of @ C @ ( type2 @ C ) ) )
=> ( ( ( ring_1_of_int @ ( word @ C ) @ ( bit_concat_bit @ ( type_len0_len_of @ B @ ( type2 @ B ) ) @ ( semiring_1_unsigned @ B @ int @ B2 ) @ ( semiring_1_unsigned @ A @ int @ A2 ) ) )
= ( ring_1_of_int @ ( word @ C ) @ ( bit_concat_bit @ ( type_len0_len_of @ B @ ( type2 @ B ) ) @ ( semiring_1_unsigned @ B @ int @ D ) @ ( semiring_1_unsigned @ A @ int @ C2 ) ) ) )
= ( ( B2 = D )
& ( A2 = C2 ) ) ) ) ) ).
% word_of_int_bin_cat_eq_iff
thf(fact_6262_mask__exceed,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [N3: nat,X: word @ A] :
( ( ord_less_eq @ nat @ ( type_len0_len_of @ A @ ( type2 @ A ) ) @ N3 )
=> ( ( bit_se5824344872417868541ns_and @ ( word @ A ) @ X @ ( bit_ri4277139882892585799ns_not @ ( word @ A ) @ ( bit_se2239418461657761734s_mask @ ( word @ A ) @ N3 ) ) )
= ( zero_zero @ ( word @ A ) ) ) ) ) ).
% mask_exceed
thf(fact_6263_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_6264_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_6265_word__of__nat__inj,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [X: nat,Y: nat] :
( ( ord_less @ nat @ X @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( type_len0_len_of @ A @ ( type2 @ A ) ) ) )
=> ( ( ord_less @ nat @ Y @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( type_len0_len_of @ A @ ( type2 @ A ) ) ) )
=> ( ( ( semiring_1_of_nat @ ( word @ A ) @ X )
= ( semiring_1_of_nat @ ( word @ A ) @ Y ) )
=> ( X = Y ) ) ) ) ) ).
% word_of_nat_inj
thf(fact_6266_of__nat__inj,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [X: nat,Y: nat] :
( ( ord_less @ nat @ X @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( type_len0_len_of @ A @ ( type2 @ A ) ) ) )
=> ( ( ord_less @ nat @ Y @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( type_len0_len_of @ A @ ( type2 @ A ) ) ) )
=> ( ( ( semiring_1_of_nat @ ( word @ A ) @ X )
= ( semiring_1_of_nat @ ( word @ A ) @ Y ) )
= ( X = Y ) ) ) ) ) ).
% of_nat_inj
thf(fact_6267_word__nat__cases,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [X: word @ A] :
~ ! [N: nat] :
( ( X
= ( semiring_1_of_nat @ ( word @ A ) @ N ) )
=> ~ ( ord_less @ nat @ N @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( type_len0_len_of @ A @ ( type2 @ A ) ) ) ) ) ) ).
% word_nat_cases
thf(fact_6268_word__nchotomy,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [W4: word @ A] :
? [N: nat] :
( ( W4
= ( semiring_1_of_nat @ ( word @ A ) @ N ) )
& ( ord_less @ nat @ N @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( type_len0_len_of @ A @ ( type2 @ A ) ) ) ) ) ) ).
% word_nchotomy
thf(fact_6269_More__Word_Opower__not__zero,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [N3: nat] :
( ( ord_less @ nat @ N3 @ ( type_len0_len_of @ A @ ( type2 @ A ) ) )
=> ( ( power_power @ ( word @ A ) @ ( numeral_numeral @ ( word @ A ) @ ( bit0 @ one2 ) ) @ N3 )
!= ( zero_zero @ ( word @ A ) ) ) ) ) ).
% More_Word.power_not_zero
thf(fact_6270_word__power__increasing,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [X: nat,Y: nat] :
( ( ord_less @ ( word @ A ) @ ( power_power @ ( word @ A ) @ ( numeral_numeral @ ( word @ A ) @ ( bit0 @ one2 ) ) @ X ) @ ( power_power @ ( word @ A ) @ ( numeral_numeral @ ( word @ A ) @ ( bit0 @ one2 ) ) @ Y ) )
=> ( ( ord_less @ nat @ X @ ( type_len0_len_of @ A @ ( type2 @ A ) ) )
=> ( ( ord_less @ nat @ Y @ ( type_len0_len_of @ A @ ( type2 @ A ) ) )
=> ( ord_less @ nat @ X @ Y ) ) ) ) ) ).
% word_power_increasing
thf(fact_6271_power__overflow,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [N3: nat] :
( ( ord_less_eq @ nat @ ( type_len0_len_of @ A @ ( type2 @ A ) ) @ N3 )
=> ( ( power_power @ ( word @ A ) @ ( numeral_numeral @ ( word @ A ) @ ( bit0 @ one2 ) ) @ N3 )
= ( zero_zero @ ( word @ A ) ) ) ) ) ).
% power_overflow
thf(fact_6272_nth__w2p__same,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [N3: nat] :
( ( bit_se5641148757651400278ts_bit @ ( word @ A ) @ ( power_power @ ( word @ A ) @ ( numeral_numeral @ ( word @ A ) @ ( bit0 @ one2 ) ) @ N3 ) @ N3 )
= ( ord_less @ nat @ N3 @ ( type_len0_len_of @ A @ ( type2 @ A ) ) ) ) ) ).
% nth_w2p_same
thf(fact_6273_nth__w2p,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [N3: nat,M: nat] :
( ( bit_se5641148757651400278ts_bit @ ( word @ A ) @ ( power_power @ ( word @ A ) @ ( numeral_numeral @ ( word @ A ) @ ( bit0 @ one2 ) ) @ N3 ) @ M )
= ( ( M = N3 )
& ( ord_less @ nat @ M @ ( type_len0_len_of @ A @ ( type2 @ A ) ) ) ) ) ) ).
% nth_w2p
thf(fact_6274_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_6275_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_6276_of__nat__neq__iff__word,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [X: nat,Y: nat] :
( ( ( modulo_modulo @ nat @ X @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( type_len0_len_of @ A @ ( type2 @ A ) ) ) )
!= ( modulo_modulo @ nat @ Y @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( type_len0_len_of @ A @ ( type2 @ A ) ) ) ) )
=> ( ( ( semiring_1_of_nat @ ( word @ A ) @ X )
!= ( semiring_1_of_nat @ ( word @ A ) @ Y ) )
= ( X != Y ) ) ) ) ).
% of_nat_neq_iff_word
thf(fact_6277_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_6278_num__abs__sbintr,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ( ( numeral_numeral @ ( word @ A ) )
= ( ^ [X2: 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 @ X2 ) ) ) ) ) ) ).
% num_abs_sbintr
thf(fact_6279_ucast__mono,axiom,
! [A: $tType,B: $tType] :
( ( ( type_len @ B )
& ( type_len @ A ) )
=> ! [X: word @ B,Y: word @ B] :
( ( ord_less @ ( word @ B ) @ X @ Y )
=> ( ( ord_less @ ( word @ B ) @ Y @ ( power_power @ ( word @ B ) @ ( numeral_numeral @ ( word @ B ) @ ( bit0 @ one2 ) ) @ ( type_len0_len_of @ A @ ( type2 @ A ) ) ) )
=> ( ord_less @ ( word @ A ) @ ( semiring_1_unsigned @ B @ ( word @ A ) @ X ) @ ( semiring_1_unsigned @ B @ ( word @ A ) @ Y ) ) ) ) ) ).
% ucast_mono
thf(fact_6280_ucast__ucast__len,axiom,
! [B: $tType,A: $tType] :
( ( ( type_len @ A )
& ( type_len @ B ) )
=> ! [X: word @ A] :
( ( ord_less @ ( word @ A ) @ X @ ( power_power @ ( word @ A ) @ ( numeral_numeral @ ( word @ A ) @ ( bit0 @ one2 ) ) @ ( type_len0_len_of @ B @ ( type2 @ B ) ) ) )
=> ( ( semiring_1_unsigned @ B @ ( word @ A ) @ ( semiring_1_unsigned @ A @ ( word @ B ) @ X ) )
= X ) ) ) ).
% ucast_ucast_len
thf(fact_6281_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_6282_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_6283_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_6284_sint__word__ariths_I1_J,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [A2: word @ A,B2: word @ A] :
( ( ring_1_signed @ A @ int @ ( plus_plus @ ( word @ A ) @ A2 @ B2 ) )
= ( 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 @ A2 ) @ ( ring_1_signed @ A @ int @ B2 ) ) ) ) ) ).
% sint_word_ariths(1)
thf(fact_6285_nth__sint,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [W: word @ A,N3: nat] :
( ( bit_se5641148757651400278ts_bit @ int @ ( ring_1_signed @ A @ int @ W ) @ N3 )
= ( ( ( ord_less @ nat @ N3 @ ( minus_minus @ nat @ ( type_len0_len_of @ A @ ( type2 @ A ) ) @ ( one_one @ nat ) ) )
=> ( bit_se5641148757651400278ts_bit @ ( word @ A ) @ W @ N3 ) )
& ( ~ ( ord_less @ nat @ N3 @ ( 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_6286_bit__sint__iff,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [W: word @ A,N3: nat] :
( ( bit_se5641148757651400278ts_bit @ int @ ( ring_1_signed @ A @ int @ W ) @ N3 )
= ( ( ( ord_less_eq @ nat @ ( type_len0_len_of @ A @ ( type2 @ A ) ) @ N3 )
& ( 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 @ N3 ) ) ) ) ).
% bit_sint_iff
thf(fact_6287_sint__word__ariths_I4_J,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [A2: word @ A] :
( ( ring_1_signed @ A @ int @ ( uminus_uminus @ ( word @ A ) @ A2 ) )
= ( 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 @ A2 ) ) ) ) ) ).
% sint_word_ariths(4)
thf(fact_6288_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_6289_sint__word__ariths_I2_J,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [A2: word @ A,B2: word @ A] :
( ( ring_1_signed @ A @ int @ ( minus_minus @ ( word @ A ) @ A2 @ B2 ) )
= ( 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 @ A2 ) @ ( ring_1_signed @ A @ int @ B2 ) ) ) ) ) ).
% sint_word_ariths(2)
thf(fact_6290_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_6291_sint__word__ariths_I3_J,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [A2: word @ A,B2: word @ A] :
( ( ring_1_signed @ A @ int @ ( times_times @ ( word @ A ) @ A2 @ B2 ) )
= ( 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 @ A2 ) @ ( ring_1_signed @ A @ int @ B2 ) ) ) ) ) ).
% sint_word_ariths(3)
thf(fact_6292_less__Suc__unat__less__bound,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [N3: nat,X: word @ A] :
( ( ord_less @ nat @ N3 @ ( suc @ ( semiring_1_unsigned @ A @ nat @ X ) ) )
=> ( ord_less @ nat @ N3 @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( type_len0_len_of @ A @ ( type2 @ A ) ) ) ) ) ) ).
% less_Suc_unat_less_bound
thf(fact_6293_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_6294_lt2p__lem,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [N3: nat,W: word @ A] :
( ( ord_less_eq @ nat @ ( type_len0_len_of @ A @ ( type2 @ A ) ) @ N3 )
=> ( ord_less @ int @ ( semiring_1_unsigned @ A @ int @ W ) @ ( power_power @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ N3 ) ) ) ) ).
% lt2p_lem
thf(fact_6295_power__le__mono,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [N3: nat,M: nat] :
( ( ord_less_eq @ ( word @ A ) @ ( power_power @ ( word @ A ) @ ( numeral_numeral @ ( word @ A ) @ ( bit0 @ one2 ) ) @ N3 ) @ ( power_power @ ( word @ A ) @ ( numeral_numeral @ ( word @ A ) @ ( bit0 @ one2 ) ) @ M ) )
=> ( ( ord_less @ nat @ N3 @ ( type_len0_len_of @ A @ ( type2 @ A ) ) )
=> ( ( ord_less @ nat @ M @ ( type_len0_len_of @ A @ ( type2 @ A ) ) )
=> ( ord_less_eq @ nat @ N3 @ M ) ) ) ) ) ).
% power_le_mono
thf(fact_6296_two__power__increasing,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [N3: nat,M: nat] :
( ( ord_less_eq @ nat @ N3 @ 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 ) ) @ N3 ) @ ( power_power @ ( word @ A ) @ ( numeral_numeral @ ( word @ A ) @ ( bit0 @ one2 ) ) @ M ) ) ) ) ) ).
% two_power_increasing
thf(fact_6297_unat__numeral,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [B2: num] :
( ( semiring_1_unsigned @ A @ nat @ ( numeral_numeral @ ( word @ A ) @ B2 ) )
= ( modulo_modulo @ nat @ ( numeral_numeral @ nat @ B2 ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( type_len0_len_of @ A @ ( type2 @ A ) ) ) ) ) ) ).
% unat_numeral
thf(fact_6298_of__nat__mono__maybe_H,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [X: nat,Y: nat] :
( ( ord_less @ nat @ X @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( type_len0_len_of @ A @ ( type2 @ A ) ) ) )
=> ( ( ord_less @ nat @ Y @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( type_len0_len_of @ A @ ( type2 @ A ) ) ) )
=> ( ( ord_less @ nat @ Y @ X )
= ( ord_less @ ( word @ A ) @ ( semiring_1_of_nat @ ( word @ A ) @ Y ) @ ( semiring_1_of_nat @ ( word @ A ) @ X ) ) ) ) ) ) ).
% of_nat_mono_maybe'
thf(fact_6299_of__nat__mono__maybe,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [X: nat,Y: nat] :
( ( ord_less @ nat @ X @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( type_len0_len_of @ A @ ( type2 @ A ) ) ) )
=> ( ( ord_less @ nat @ Y @ X )
=> ( ord_less @ ( word @ A ) @ ( semiring_1_of_nat @ ( word @ A ) @ Y ) @ ( semiring_1_of_nat @ ( word @ A ) @ X ) ) ) ) ) ).
% of_nat_mono_maybe
thf(fact_6300_unat__of__nat__len,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [X: nat] :
( ( ord_less @ nat @ X @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( type_len0_len_of @ A @ ( type2 @ A ) ) ) )
=> ( ( semiring_1_unsigned @ A @ nat @ ( semiring_1_of_nat @ ( word @ A ) @ X ) )
= X ) ) ) ).
% unat_of_nat_len
thf(fact_6301_unat__eq__of__nat,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [N3: nat,X: word @ A] :
( ( ord_less @ nat @ N3 @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( type_len0_len_of @ A @ ( type2 @ A ) ) ) )
=> ( ( ( semiring_1_unsigned @ A @ nat @ X )
= N3 )
= ( X
= ( semiring_1_of_nat @ ( word @ A ) @ N3 ) ) ) ) ) ).
% unat_eq_of_nat
thf(fact_6302_unat__split__asm,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [P: nat > $o,X: word @ A] :
( ( P @ ( semiring_1_unsigned @ A @ nat @ X ) )
= ( ~ ? [N2: nat] :
( ( ( semiring_1_of_nat @ ( word @ A ) @ N2 )
= X )
& ( ord_less @ nat @ N2 @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( type_len0_len_of @ A @ ( type2 @ A ) ) ) )
& ~ ( P @ N2 ) ) ) ) ) ).
% unat_split_asm
thf(fact_6303_of__nat__inverse,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [R2: nat,A2: word @ A] :
( ( ( semiring_1_of_nat @ ( word @ A ) @ R2 )
= A2 )
=> ( ( ord_less @ nat @ R2 @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( type_len0_len_of @ A @ ( type2 @ A ) ) ) )
=> ( ( semiring_1_unsigned @ A @ nat @ A2 )
= R2 ) ) ) ) ).
% of_nat_inverse
thf(fact_6304_unat__split,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [P: nat > $o,X: word @ A] :
( ( P @ ( semiring_1_unsigned @ A @ nat @ X ) )
= ( ! [N2: nat] :
( ( ( ( semiring_1_of_nat @ ( word @ A ) @ N2 )
= X )
& ( ord_less @ nat @ N2 @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( type_len0_len_of @ A @ ( type2 @ A ) ) ) ) )
=> ( P @ N2 ) ) ) ) ) ).
% unat_split
thf(fact_6305_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_6306_x__less__2__0__1_H,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [X: word @ A] :
( ( ( type_len0_len_of @ A @ ( type2 @ A ) )
!= ( one_one @ nat ) )
=> ( ( ord_less @ ( word @ A ) @ X @ ( numeral_numeral @ ( word @ A ) @ ( bit0 @ one2 ) ) )
=> ( ( X
= ( zero_zero @ ( word @ A ) ) )
| ( X
= ( one_one @ ( word @ A ) ) ) ) ) ) ) ).
% x_less_2_0_1'
thf(fact_6307_test__bit__2p,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [N3: nat,M: nat] :
( ( bit_se5641148757651400278ts_bit @ ( word @ A ) @ ( ring_1_of_int @ ( word @ A ) @ ( power_power @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ N3 ) ) @ M )
= ( ( M = N3 )
& ( ord_less @ nat @ M @ ( type_len0_len_of @ A @ ( type2 @ A ) ) ) ) ) ) ).
% test_bit_2p
thf(fact_6308_Word_Oof__nat__0,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [M: nat] :
( ( ( semiring_1_of_nat @ ( word @ A ) @ M )
= ( zero_zero @ ( word @ A ) ) )
= ( ? [Q4: nat] :
( M
= ( times_times @ nat @ Q4 @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( type_len0_len_of @ A @ ( type2 @ A ) ) ) ) ) ) ) ) ).
% Word.of_nat_0
thf(fact_6309_word__1__le__power,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [N3: nat] :
( ( ord_less @ nat @ N3 @ ( 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 ) ) @ N3 ) ) ) ) ).
% word_1_le_power
thf(fact_6310_ucast__of__nat__small,axiom,
! [A: $tType,B: $tType] :
( ( ( type_len @ B )
& ( type_len @ A ) )
=> ! [X: nat] :
( ( ord_less @ nat @ X @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( type_len0_len_of @ A @ ( type2 @ A ) ) ) )
=> ( ( semiring_1_unsigned @ A @ ( word @ B ) @ ( semiring_1_of_nat @ ( word @ A ) @ X ) )
= ( semiring_1_of_nat @ ( word @ B ) @ X ) ) ) ) ).
% ucast_of_nat_small
thf(fact_6311_uint__sub__lt2p,axiom,
! [B: $tType,A: $tType] :
( ( ( type_len @ A )
& ( type_len @ B ) )
=> ! [X: word @ A,Y: word @ B] : ( ord_less @ int @ ( minus_minus @ int @ ( semiring_1_unsigned @ A @ int @ X ) @ ( semiring_1_unsigned @ B @ int @ Y ) ) @ ( power_power @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ ( type_len0_len_of @ A @ ( type2 @ A ) ) ) ) ) ).
% uint_sub_lt2p
thf(fact_6312_uint__numeral,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [B2: num] :
( ( semiring_1_unsigned @ A @ int @ ( numeral_numeral @ ( word @ A ) @ B2 ) )
= ( modulo_modulo @ int @ ( numeral_numeral @ int @ B2 ) @ ( power_power @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ ( type_len0_len_of @ A @ ( type2 @ A ) ) ) ) ) ) ).
% uint_numeral
thf(fact_6313_p2__gt__0,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [N3: nat] :
( ( ord_less @ ( word @ A ) @ ( zero_zero @ ( word @ A ) ) @ ( power_power @ ( word @ A ) @ ( numeral_numeral @ ( word @ A ) @ ( bit0 @ one2 ) ) @ N3 ) )
= ( ord_less @ nat @ N3 @ ( type_len0_len_of @ A @ ( type2 @ A ) ) ) ) ) ).
% p2_gt_0
thf(fact_6314_word__of__nat__eq__0__iff,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [N3: nat] :
( ( ( semiring_1_of_nat @ ( word @ A ) @ N3 )
= ( zero_zero @ ( word @ A ) ) )
= ( dvd_dvd @ nat @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( type_len0_len_of @ A @ ( type2 @ A ) ) ) @ N3 ) ) ) ).
% word_of_nat_eq_0_iff
thf(fact_6315_unat__ucast,axiom,
! [B: $tType,A: $tType] :
( ( ( type_len @ A )
& ( type_len @ B ) )
=> ! [X: word @ B] :
( ( semiring_1_unsigned @ A @ nat @ ( semiring_1_unsigned @ B @ ( word @ A ) @ X ) )
= ( modulo_modulo @ nat @ ( semiring_1_unsigned @ B @ nat @ X ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( type_len0_len_of @ A @ ( type2 @ A ) ) ) ) ) ) ).
% unat_ucast
thf(fact_6316_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_6317_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_6318_mask__lt__2pn,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [N3: nat] :
( ( ord_less @ nat @ N3 @ ( type_len0_len_of @ A @ ( type2 @ A ) ) )
=> ( ord_less @ ( word @ A ) @ ( bit_se2239418461657761734s_mask @ ( word @ A ) @ N3 ) @ ( power_power @ ( word @ A ) @ ( numeral_numeral @ ( word @ A ) @ ( bit0 @ one2 ) ) @ N3 ) ) ) ) ).
% mask_lt_2pn
thf(fact_6319_unat__of__nat,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [X: nat] :
( ( semiring_1_unsigned @ A @ nat @ ( semiring_1_of_nat @ ( word @ A ) @ X ) )
= ( modulo_modulo @ nat @ X @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( type_len0_len_of @ A @ ( type2 @ A ) ) ) ) ) ) ).
% unat_of_nat
thf(fact_6320_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_6321_ucast__less,axiom,
! [A: $tType,B: $tType] :
( ( ( type_len @ B )
& ( type_len @ A ) )
=> ! [X: word @ B] :
( ( ord_less @ nat @ ( type_len0_len_of @ B @ ( type2 @ B ) ) @ ( type_len0_len_of @ A @ ( type2 @ A ) ) )
=> ( ord_less @ ( word @ A ) @ ( semiring_1_unsigned @ B @ ( word @ A ) @ X ) @ ( power_power @ ( word @ A ) @ ( numeral_numeral @ ( word @ A ) @ ( bit0 @ one2 ) ) @ ( type_len0_len_of @ B @ ( type2 @ B ) ) ) ) ) ) ).
% ucast_less
thf(fact_6322_signed__of__int,axiom,
! [A: $tType,B: $tType] :
( ( ( type_len @ B )
& ( ring_1 @ A ) )
=> ! [N3: int] :
( ( ring_1_signed @ B @ A @ ( ring_1_of_int @ ( word @ B ) @ N3 ) )
= ( ring_1_of_int @ A @ ( bit_ri4674362597316999326ke_bit @ int @ ( minus_minus @ nat @ ( type_len0_len_of @ B @ ( type2 @ B ) ) @ ( suc @ ( zero_zero @ nat ) ) ) @ N3 ) ) ) ) ).
% signed_of_int
thf(fact_6323_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_6324_of__nat__n__less__equal__power__2,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [N3: nat] :
( ( ord_less @ nat @ N3 @ ( type_len0_len_of @ A @ ( type2 @ A ) ) )
=> ( ord_less @ ( word @ A ) @ ( semiring_1_of_nat @ ( word @ A ) @ N3 ) @ ( power_power @ ( word @ A ) @ ( numeral_numeral @ ( word @ A ) @ ( bit0 @ one2 ) ) @ N3 ) ) ) ) ).
% of_nat_n_less_equal_power_2
thf(fact_6325_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_6326_complement__nth__w2p,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [N4: nat,N3: nat] :
( ( ord_less @ nat @ N4 @ ( 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 ) ) @ N3 ) ) @ N4 )
= ( N4 != N3 ) ) ) ) ).
% complement_nth_w2p
thf(fact_6327_upper__trivial,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [X: word @ A] :
( ( X
!= ( minus_minus @ ( word @ A ) @ ( power_power @ ( word @ A ) @ ( numeral_numeral @ ( word @ A ) @ ( bit0 @ one2 ) ) @ ( type_len0_len_of @ A @ ( type2 @ A ) ) ) @ ( one_one @ ( word @ A ) ) ) )
=> ( ord_less @ ( word @ A ) @ X @ ( minus_minus @ ( word @ A ) @ ( power_power @ ( word @ A ) @ ( numeral_numeral @ ( word @ A ) @ ( bit0 @ one2 ) ) @ ( type_len0_len_of @ A @ ( type2 @ A ) ) ) @ ( one_one @ ( word @ A ) ) ) ) ) ) ).
% upper_trivial
thf(fact_6328_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_6329_bit__word__scast__iff,axiom,
! [B: $tType,A: $tType] :
( ( ( type_len @ A )
& ( type_len @ B ) )
=> ! [W: word @ A,N3: nat] :
( ( bit_se5641148757651400278ts_bit @ ( word @ B ) @ ( ring_1_signed @ A @ ( word @ B ) @ W ) @ N3 )
= ( ( ord_less @ nat @ N3 @ ( type_len0_len_of @ B @ ( type2 @ B ) ) )
& ( ( bit_se5641148757651400278ts_bit @ ( word @ A ) @ W @ N3 )
| ( ( ord_less_eq @ nat @ ( type_len0_len_of @ A @ ( type2 @ A ) ) @ N3 )
& ( 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_6330_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_6331_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_6332_unat__word__ariths_I7_J,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [A2: word @ A,B2: word @ A] :
( ( semiring_1_unsigned @ A @ nat @ ( modulo_modulo @ ( word @ A ) @ A2 @ B2 ) )
= ( modulo_modulo @ nat @ ( modulo_modulo @ nat @ ( semiring_1_unsigned @ A @ nat @ A2 ) @ ( semiring_1_unsigned @ A @ nat @ B2 ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( type_len0_len_of @ A @ ( type2 @ A ) ) ) ) ) ) ).
% unat_word_ariths(7)
thf(fact_6333_word__power__less__diff,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [N3: nat,Q2: word @ A,M: nat] :
( ( ord_less @ ( word @ A ) @ ( times_times @ ( word @ A ) @ ( power_power @ ( word @ A ) @ ( numeral_numeral @ ( word @ A ) @ ( bit0 @ one2 ) ) @ N3 ) @ Q2 ) @ ( power_power @ ( word @ A ) @ ( numeral_numeral @ ( word @ A ) @ ( bit0 @ one2 ) ) @ M ) )
=> ( ( ord_less @ ( word @ A ) @ Q2 @ ( power_power @ ( word @ A ) @ ( numeral_numeral @ ( word @ A ) @ ( bit0 @ one2 ) ) @ ( minus_minus @ nat @ ( type_len0_len_of @ A @ ( type2 @ A ) ) @ N3 ) ) )
=> ( ord_less @ ( word @ A ) @ Q2 @ ( power_power @ ( word @ A ) @ ( numeral_numeral @ ( word @ A ) @ ( bit0 @ one2 ) ) @ ( minus_minus @ nat @ M @ N3 ) ) ) ) ) ) ).
% word_power_less_diff
thf(fact_6334_ucast__mono__le,axiom,
! [B: $tType,A: $tType] :
( ( ( type_len @ A )
& ( type_len @ B ) )
=> ! [X: word @ A,Y: word @ A] :
( ( ord_less_eq @ ( word @ A ) @ X @ Y )
=> ( ( ord_less @ ( word @ A ) @ Y @ ( power_power @ ( word @ A ) @ ( numeral_numeral @ ( word @ A ) @ ( bit0 @ one2 ) ) @ ( type_len0_len_of @ B @ ( type2 @ B ) ) ) )
=> ( ord_less_eq @ ( word @ B ) @ ( semiring_1_unsigned @ A @ ( word @ B ) @ X ) @ ( semiring_1_unsigned @ A @ ( word @ B ) @ Y ) ) ) ) ) ).
% ucast_mono_le
thf(fact_6335_take__bit__word__eq__self__iff,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [N3: nat,W: word @ A] :
( ( ( bit_se2584673776208193580ke_bit @ ( word @ A ) @ N3 @ W )
= W )
= ( ( ord_less_eq @ nat @ ( type_len0_len_of @ A @ ( type2 @ A ) ) @ N3 )
| ( ord_less @ ( word @ A ) @ W @ ( power_power @ ( word @ A ) @ ( numeral_numeral @ ( word @ A ) @ ( bit0 @ one2 ) ) @ N3 ) ) ) ) ) ).
% take_bit_word_eq_self_iff
thf(fact_6336_signed__push__bit__eq,axiom,
! [A: $tType,B: $tType] :
( ( ( type_len @ B )
& ( bit_ri3973907225187159222ations @ A ) )
=> ! [N3: nat,W: word @ B] :
( ( ring_1_signed @ B @ A @ ( bit_se4730199178511100633sh_bit @ ( word @ B ) @ N3 @ W ) )
= ( bit_ri4674362597316999326ke_bit @ A @ ( minus_minus @ nat @ ( type_len0_len_of @ B @ ( type2 @ B ) ) @ ( suc @ ( zero_zero @ nat ) ) ) @ ( bit_se4730199178511100633sh_bit @ A @ N3 @ ( ring_1_signed @ B @ A @ W ) ) ) ) ) ).
% signed_push_bit_eq
thf(fact_6337_msb0,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [X: word @ A,Y: word @ A,I: nat] :
( ( ( bit_se5641148757651400278ts_bit @ ( word @ A ) @ ( bit_se1065995026697491101ons_or @ ( word @ A ) @ X @ Y ) @ ( minus_minus @ nat @ ( type_len0_len_of @ A @ ( type2 @ A ) ) @ ( suc @ I ) ) )
= ( ( bit_se5641148757651400278ts_bit @ ( word @ A ) @ X @ ( minus_minus @ nat @ ( type_len0_len_of @ A @ ( type2 @ A ) ) @ ( suc @ I ) ) )
| ( bit_se5641148757651400278ts_bit @ ( word @ A ) @ Y @ ( minus_minus @ nat @ ( type_len0_len_of @ A @ ( type2 @ A ) ) @ ( suc @ I ) ) ) ) )
& ( ( bit_se5641148757651400278ts_bit @ ( word @ A ) @ ( bit_se5824344872417868541ns_and @ ( word @ A ) @ X @ Y ) @ ( minus_minus @ nat @ ( type_len0_len_of @ A @ ( type2 @ A ) ) @ ( suc @ I ) ) )
= ( ( bit_se5641148757651400278ts_bit @ ( word @ A ) @ X @ ( minus_minus @ nat @ ( type_len0_len_of @ A @ ( type2 @ A ) ) @ ( suc @ I ) ) )
& ( bit_se5641148757651400278ts_bit @ ( word @ A ) @ Y @ ( minus_minus @ nat @ ( type_len0_len_of @ A @ ( type2 @ A ) ) @ ( suc @ I ) ) ) ) )
& ( ( bit_se5641148757651400278ts_bit @ ( word @ A ) @ ( bit_se5824344971392196577ns_xor @ ( word @ A ) @ X @ Y ) @ ( minus_minus @ nat @ ( type_len0_len_of @ A @ ( type2 @ A ) ) @ ( suc @ I ) ) )
= ( ( bit_se5641148757651400278ts_bit @ ( word @ A ) @ X @ ( minus_minus @ nat @ ( type_len0_len_of @ A @ ( type2 @ A ) ) @ ( suc @ I ) ) )
!= ( bit_se5641148757651400278ts_bit @ ( word @ A ) @ Y @ ( minus_minus @ nat @ ( type_len0_len_of @ A @ ( type2 @ A ) ) @ ( suc @ I ) ) ) ) )
& ( ( bit_se5641148757651400278ts_bit @ ( word @ A ) @ ( bit_ri4277139882892585799ns_not @ ( word @ A ) @ X ) @ ( minus_minus @ nat @ ( type_len0_len_of @ A @ ( type2 @ A ) ) @ ( suc @ I ) ) )
= ( ~ ( bit_se5641148757651400278ts_bit @ ( word @ A ) @ X @ ( minus_minus @ nat @ ( type_len0_len_of @ A @ ( type2 @ A ) ) @ ( suc @ I ) ) ) ) ) ) ) ).
% msb0
thf(fact_6338_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 )
@ ^ [X2: word @ B] : ( ord_less @ ( word @ B ) @ X2 @ ( power_power @ ( word @ B ) @ ( numeral_numeral @ ( word @ B ) @ ( bit0 @ one2 ) ) @ ( type_len0_len_of @ A @ ( type2 @ A ) ) ) ) ) ) ) ) ).
% ucast_range_less
thf(fact_6339_unat__add__lem_H,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [X: word @ A,Y: word @ A] :
( ( ord_less @ nat @ ( plus_plus @ nat @ ( semiring_1_unsigned @ A @ nat @ X ) @ ( semiring_1_unsigned @ A @ nat @ Y ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( type_len0_len_of @ A @ ( type2 @ A ) ) ) )
=> ( ( semiring_1_unsigned @ A @ nat @ ( plus_plus @ ( word @ A ) @ X @ Y ) )
= ( plus_plus @ nat @ ( semiring_1_unsigned @ A @ nat @ X ) @ ( semiring_1_unsigned @ A @ nat @ Y ) ) ) ) ) ).
% unat_add_lem'
thf(fact_6340_unat__add__lem,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [X: word @ A,Y: word @ A] :
( ( ord_less @ nat @ ( plus_plus @ nat @ ( semiring_1_unsigned @ A @ nat @ X ) @ ( semiring_1_unsigned @ A @ nat @ Y ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( type_len0_len_of @ A @ ( type2 @ A ) ) ) )
= ( ( semiring_1_unsigned @ A @ nat @ ( plus_plus @ ( word @ A ) @ X @ Y ) )
= ( plus_plus @ nat @ ( semiring_1_unsigned @ A @ nat @ X ) @ ( semiring_1_unsigned @ A @ nat @ Y ) ) ) ) ) ).
% unat_add_lem
thf(fact_6341_More__Word_Oof__nat__0,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [N3: nat] :
( ( ( semiring_1_of_nat @ ( word @ A ) @ N3 )
= ( zero_zero @ ( word @ A ) ) )
=> ( ( ord_less @ nat @ N3 @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( type_len0_len_of @ A @ ( type2 @ A ) ) ) )
=> ( N3
= ( zero_zero @ nat ) ) ) ) ) ).
% More_Word.of_nat_0
thf(fact_6342_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_6343_of__nat__mono__maybe__le,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [X: nat,Y: nat] :
( ( ord_less @ nat @ X @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( type_len0_len_of @ A @ ( type2 @ A ) ) ) )
=> ( ( ord_less @ nat @ Y @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( type_len0_len_of @ A @ ( type2 @ A ) ) ) )
=> ( ( ord_less_eq @ nat @ Y @ X )
= ( ord_less_eq @ ( word @ A ) @ ( semiring_1_of_nat @ ( word @ A ) @ Y ) @ ( semiring_1_of_nat @ ( word @ A ) @ X ) ) ) ) ) ) ).
% of_nat_mono_maybe_le
thf(fact_6344_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_6345_unat__word__ariths_I1_J,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [A2: word @ A,B2: word @ A] :
( ( semiring_1_unsigned @ A @ nat @ ( plus_plus @ ( word @ A ) @ A2 @ B2 ) )
= ( modulo_modulo @ nat @ ( plus_plus @ nat @ ( semiring_1_unsigned @ A @ nat @ A2 ) @ ( semiring_1_unsigned @ A @ nat @ B2 ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( type_len0_len_of @ A @ ( type2 @ A ) ) ) ) ) ) ).
% unat_word_ariths(1)
thf(fact_6346_bool__mask_H,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [X: word @ A] :
( ( ord_less @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( type_len0_len_of @ A @ ( type2 @ A ) ) )
=> ( ( ord_less @ ( word @ A ) @ ( zero_zero @ ( word @ A ) ) @ ( bit_se5824344872417868541ns_and @ ( word @ A ) @ X @ ( one_one @ ( word @ A ) ) ) )
= ( ( bit_se5824344872417868541ns_and @ ( word @ A ) @ X @ ( one_one @ ( word @ A ) ) )
= ( one_one @ ( word @ A ) ) ) ) ) ) ).
% bool_mask'
thf(fact_6347_uint__range_H,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [X: word @ A] :
( ( ord_less_eq @ int @ ( zero_zero @ int ) @ ( semiring_1_unsigned @ A @ int @ X ) )
& ( ord_less @ int @ ( semiring_1_unsigned @ A @ int @ X ) @ ( power_power @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ ( type_len0_len_of @ A @ ( type2 @ A ) ) ) ) ) ) ).
% uint_range'
thf(fact_6348_sint__lt,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [X: word @ A] : ( ord_less @ int @ ( ring_1_signed @ A @ int @ X ) @ ( power_power @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ ( minus_minus @ nat @ ( type_len0_len_of @ A @ ( type2 @ A ) ) @ ( one_one @ nat ) ) ) ) ) ).
% sint_lt
thf(fact_6349_ucast__mono__le_H,axiom,
! [B: $tType,A: $tType] :
( ( ( type_len @ A )
& ( type_len @ B ) )
=> ! [Y: word @ A,X: word @ A] :
( ( ord_less @ nat @ ( semiring_1_unsigned @ A @ nat @ Y ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( type_len0_len_of @ B @ ( type2 @ B ) ) ) )
=> ( ( ord_less @ nat @ ( type_len0_len_of @ B @ ( type2 @ B ) ) @ ( type_len0_len_of @ A @ ( type2 @ A ) ) )
=> ( ( ord_less_eq @ ( word @ A ) @ X @ Y )
=> ( ord_less_eq @ ( word @ B ) @ ( semiring_1_unsigned @ A @ ( word @ B ) @ X ) @ ( semiring_1_unsigned @ A @ ( word @ B ) @ Y ) ) ) ) ) ) ).
% ucast_mono_le'
thf(fact_6350_of__nat__eq,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [N3: nat,W: word @ A] :
( ( ( semiring_1_of_nat @ ( word @ A ) @ N3 )
= W )
= ( ? [Q4: nat] :
( N3
= ( plus_plus @ nat @ ( semiring_1_unsigned @ A @ nat @ W ) @ ( times_times @ nat @ Q4 @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( type_len0_len_of @ A @ ( type2 @ A ) ) ) ) ) ) ) ) ) ).
% of_nat_eq
thf(fact_6351_unat__mult__lem,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [X: word @ A,Y: word @ A] :
( ( ord_less @ nat @ ( times_times @ nat @ ( semiring_1_unsigned @ A @ nat @ X ) @ ( semiring_1_unsigned @ A @ nat @ Y ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( type_len0_len_of @ A @ ( type2 @ A ) ) ) )
= ( ( semiring_1_unsigned @ A @ nat @ ( times_times @ ( word @ A ) @ X @ Y ) )
= ( times_times @ nat @ ( semiring_1_unsigned @ A @ nat @ X ) @ ( semiring_1_unsigned @ A @ nat @ Y ) ) ) ) ) ).
% unat_mult_lem
thf(fact_6352_word__of__int__inj,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [X: int,Y: int] :
( ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ X )
& ( ord_less @ int @ X @ ( power_power @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ ( type_len0_len_of @ A @ ( type2 @ A ) ) ) ) )
=> ( ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ Y )
& ( ord_less @ int @ Y @ ( power_power @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ ( type_len0_len_of @ A @ ( type2 @ A ) ) ) ) )
=> ( ( ( ring_1_of_int @ ( word @ A ) @ X )
= ( ring_1_of_int @ ( word @ A ) @ Y ) )
= ( X = Y ) ) ) ) ) ).
% word_of_int_inj
thf(fact_6353_word__int__cases,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [X: word @ A] :
~ ! [N: int] :
( ( X
= ( ring_1_of_int @ ( word @ A ) @ N ) )
=> ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ N )
=> ~ ( ord_less @ int @ N @ ( power_power @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ ( type_len0_len_of @ A @ ( type2 @ A ) ) ) ) ) ) ) ).
% word_int_cases
thf(fact_6354_unat__ucast__no__overflow__le,axiom,
! [A: $tType,B: $tType] :
( ( ( type_len @ B )
& ( type_len @ A ) )
=> ! [B2: word @ B,F2: word @ A] :
( ( ord_less @ nat @ ( semiring_1_unsigned @ B @ nat @ B2 ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( type_len0_len_of @ A @ ( type2 @ A ) ) ) )
=> ( ( ord_less @ nat @ ( type_len0_len_of @ A @ ( type2 @ A ) ) @ ( type_len0_len_of @ B @ ( type2 @ B ) ) )
=> ( ( ord_less @ ( word @ B ) @ ( semiring_1_unsigned @ A @ ( word @ B ) @ F2 ) @ B2 )
= ( ord_less @ nat @ ( semiring_1_unsigned @ A @ nat @ F2 ) @ ( semiring_1_unsigned @ B @ nat @ B2 ) ) ) ) ) ) ).
% unat_ucast_no_overflow_le
thf(fact_6355_uint__m2p__not__non__neg,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [X: word @ A] :
~ ( ord_less_eq @ int @ ( zero_zero @ int ) @ ( minus_minus @ int @ ( semiring_1_unsigned @ A @ int @ X ) @ ( power_power @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ ( type_len0_len_of @ A @ ( type2 @ A ) ) ) ) ) ) ).
% uint_m2p_not_non_neg
thf(fact_6356_unat__ucast__less__no__overflow__simp,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [N3: nat,F2: word @ A] :
( ( ord_less @ nat @ N3 @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( type_len0_len_of @ A @ ( type2 @ A ) ) ) )
=> ( ( ord_less @ nat @ ( semiring_1_unsigned @ A @ nat @ F2 ) @ N3 )
= ( ord_less @ ( word @ A ) @ F2 @ ( semiring_1_of_nat @ ( word @ A ) @ N3 ) ) ) ) ) ).
% unat_ucast_less_no_overflow_simp
thf(fact_6357_unat__ucast__less__no__overflow,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [N3: nat,F2: word @ A] :
( ( ord_less @ nat @ N3 @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( type_len0_len_of @ A @ ( type2 @ A ) ) ) )
=> ( ( ord_less @ nat @ ( semiring_1_unsigned @ A @ nat @ F2 ) @ N3 )
=> ( ord_less @ ( word @ A ) @ F2 @ ( semiring_1_of_nat @ ( word @ A ) @ N3 ) ) ) ) ) ).
% unat_ucast_less_no_overflow
thf(fact_6358_uint__m2p__neg,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [X: word @ A] : ( ord_less @ int @ ( minus_minus @ int @ ( semiring_1_unsigned @ A @ int @ X ) @ ( power_power @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ ( type_len0_len_of @ A @ ( type2 @ A ) ) ) ) @ ( zero_zero @ int ) ) ) ).
% uint_m2p_neg
thf(fact_6359_uint__power__lower,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [N3: nat] :
( ( ord_less @ nat @ N3 @ ( type_len0_len_of @ A @ ( type2 @ A ) ) )
=> ( ( semiring_1_unsigned @ A @ int @ ( power_power @ ( word @ A ) @ ( numeral_numeral @ ( word @ A ) @ ( bit0 @ one2 ) ) @ N3 ) )
= ( power_power @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ N3 ) ) ) ) ).
% uint_power_lower
thf(fact_6360_nth__bounded,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [X: word @ A,N3: nat,M: nat] :
( ( bit_se5641148757651400278ts_bit @ ( word @ A ) @ X @ N3 )
=> ( ( ord_less @ ( word @ A ) @ X @ ( power_power @ ( word @ A ) @ ( numeral_numeral @ ( word @ A ) @ ( bit0 @ one2 ) ) @ M ) )
=> ( ( ord_less_eq @ nat @ M @ ( type_len0_len_of @ A @ ( type2 @ A ) ) )
=> ( ord_less @ nat @ N3 @ M ) ) ) ) ) ).
% nth_bounded
thf(fact_6361_upper__bits__unset__is__l2p,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [N3: nat,P4: word @ A] :
( ( ord_less @ nat @ N3 @ ( type_len0_len_of @ A @ ( type2 @ A ) ) )
=> ( ( ! [N12: nat] :
( ( ord_less_eq @ nat @ N3 @ N12 )
=> ( ( ord_less @ nat @ N12 @ ( type_len0_len_of @ A @ ( type2 @ A ) ) )
=> ~ ( bit_se5641148757651400278ts_bit @ ( word @ A ) @ P4 @ N12 ) ) ) )
= ( ord_less @ ( word @ A ) @ P4 @ ( power_power @ ( word @ A ) @ ( numeral_numeral @ ( word @ A ) @ ( bit0 @ one2 ) ) @ N3 ) ) ) ) ) ).
% upper_bits_unset_is_l2p
thf(fact_6362_less__2p__is__upper__bits__unset,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [P4: word @ A,N3: nat] :
( ( ord_less @ ( word @ A ) @ P4 @ ( power_power @ ( word @ A ) @ ( numeral_numeral @ ( word @ A ) @ ( bit0 @ one2 ) ) @ N3 ) )
= ( ( ord_less @ nat @ N3 @ ( type_len0_len_of @ A @ ( type2 @ A ) ) )
& ! [N12: nat] :
( ( ord_less_eq @ nat @ N3 @ N12 )
=> ( ( ord_less @ nat @ N12 @ ( type_len0_len_of @ A @ ( type2 @ A ) ) )
=> ~ ( bit_se5641148757651400278ts_bit @ ( word @ A ) @ P4 @ N12 ) ) ) ) ) ) ).
% less_2p_is_upper_bits_unset
thf(fact_6363_uint__add__lem,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [X: word @ A,Y: word @ A] :
( ( ord_less @ int @ ( plus_plus @ int @ ( semiring_1_unsigned @ A @ int @ X ) @ ( semiring_1_unsigned @ A @ int @ Y ) ) @ ( power_power @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ ( type_len0_len_of @ A @ ( type2 @ A ) ) ) )
= ( ( semiring_1_unsigned @ A @ int @ ( plus_plus @ ( word @ A ) @ X @ Y ) )
= ( plus_plus @ int @ ( semiring_1_unsigned @ A @ int @ X ) @ ( semiring_1_unsigned @ A @ int @ Y ) ) ) ) ) ).
% uint_add_lem
thf(fact_6364_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_6365_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_6366_wi__le,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [N3: int,M: int] :
( ( ord_less_eq @ ( word @ A ) @ ( ring_1_of_int @ ( word @ A ) @ N3 ) @ ( ring_1_of_int @ ( word @ A ) @ M ) )
= ( ord_less_eq @ int @ ( modulo_modulo @ int @ N3 @ ( 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_6367_uint__word__ariths_I1_J,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [A2: word @ A,B2: word @ A] :
( ( semiring_1_unsigned @ A @ int @ ( plus_plus @ ( word @ A ) @ A2 @ B2 ) )
= ( modulo_modulo @ int @ ( plus_plus @ int @ ( semiring_1_unsigned @ A @ int @ A2 ) @ ( semiring_1_unsigned @ A @ int @ B2 ) ) @ ( power_power @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ ( type_len0_len_of @ A @ ( type2 @ A ) ) ) ) ) ) ).
% uint_word_ariths(1)
thf(fact_6368_word__2p__mult__inc,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [N3: 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 ) ) @ N3 ) ) @ ( 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 @ N3 ) @ ( type_len0_len_of @ A @ ( type2 @ A ) ) )
=> ( ord_less @ ( word @ A ) @ ( power_power @ ( word @ A ) @ ( numeral_numeral @ ( word @ A ) @ ( bit0 @ one2 ) ) @ N3 ) @ ( power_power @ ( word @ A ) @ ( numeral_numeral @ ( word @ A ) @ ( bit0 @ one2 ) ) @ M ) ) ) ) ) ).
% word_2p_mult_inc
thf(fact_6369_unat__word__ariths_I2_J,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [A2: word @ A,B2: word @ A] :
( ( semiring_1_unsigned @ A @ nat @ ( times_times @ ( word @ A ) @ A2 @ B2 ) )
= ( modulo_modulo @ nat @ ( times_times @ nat @ ( semiring_1_unsigned @ A @ nat @ A2 ) @ ( semiring_1_unsigned @ A @ nat @ B2 ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( type_len0_len_of @ A @ ( type2 @ A ) ) ) ) ) ) ).
% unat_word_ariths(2)
thf(fact_6370_unat__word__ariths_I6_J,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [A2: word @ A,B2: word @ A] :
( ( semiring_1_unsigned @ A @ nat @ ( divide_divide @ ( word @ A ) @ A2 @ B2 ) )
= ( modulo_modulo @ nat @ ( divide_divide @ nat @ ( semiring_1_unsigned @ A @ nat @ A2 ) @ ( semiring_1_unsigned @ A @ nat @ B2 ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( type_len0_len_of @ A @ ( type2 @ A ) ) ) ) ) ) ).
% unat_word_ariths(6)
thf(fact_6371_wi__less,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [N3: int,M: int] :
( ( ord_less @ ( word @ A ) @ ( ring_1_of_int @ ( word @ A ) @ N3 ) @ ( ring_1_of_int @ ( word @ A ) @ M ) )
= ( ord_less @ int @ ( modulo_modulo @ int @ N3 @ ( 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_6372_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_6373_power__2__ge__iff,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [N3: nat] :
( ( ord_less @ ( word @ A ) @ ( minus_minus @ ( word @ A ) @ ( power_power @ ( word @ A ) @ ( numeral_numeral @ ( word @ A ) @ ( bit0 @ one2 ) ) @ N3 ) @ ( one_one @ ( word @ A ) ) ) @ ( power_power @ ( word @ A ) @ ( numeral_numeral @ ( word @ A ) @ ( bit0 @ one2 ) ) @ N3 ) )
= ( ord_less @ nat @ N3 @ ( type_len0_len_of @ A @ ( type2 @ A ) ) ) ) ) ).
% power_2_ge_iff
thf(fact_6374_uint__word__ariths_I4_J,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [A2: word @ A] :
( ( semiring_1_unsigned @ A @ int @ ( uminus_uminus @ ( word @ A ) @ A2 ) )
= ( modulo_modulo @ int @ ( uminus_uminus @ int @ ( semiring_1_unsigned @ A @ int @ A2 ) ) @ ( power_power @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ ( type_len0_len_of @ A @ ( type2 @ A ) ) ) ) ) ) ).
% uint_word_ariths(4)
thf(fact_6375_le__mask__iff__lt__2n,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [N3: nat,W: word @ A] :
( ( ord_less @ nat @ N3 @ ( type_len0_len_of @ A @ ( type2 @ A ) ) )
= ( ( ord_less_eq @ ( word @ A ) @ W @ ( bit_se2239418461657761734s_mask @ ( word @ A ) @ N3 ) )
= ( ord_less @ ( word @ A ) @ W @ ( power_power @ ( word @ A ) @ ( numeral_numeral @ ( word @ A ) @ ( bit0 @ one2 ) ) @ N3 ) ) ) ) ) ).
% le_mask_iff_lt_2n
thf(fact_6376_eq__mask__less,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [W: word @ A,N3: nat] :
( ( W
= ( bit_se5824344872417868541ns_and @ ( word @ A ) @ W @ ( bit_se2239418461657761734s_mask @ ( word @ A ) @ N3 ) ) )
=> ( ( ord_less @ nat @ N3 @ ( type_len0_len_of @ A @ ( type2 @ A ) ) )
=> ( ord_less @ ( word @ A ) @ W @ ( power_power @ ( word @ A ) @ ( numeral_numeral @ ( word @ A ) @ ( bit0 @ one2 ) ) @ N3 ) ) ) ) ) ).
% eq_mask_less
thf(fact_6377_and__mask__less_H,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [N3: nat,W: word @ A] :
( ( ord_less @ nat @ N3 @ ( type_len0_len_of @ A @ ( type2 @ A ) ) )
=> ( ord_less @ ( word @ A ) @ ( bit_se5824344872417868541ns_and @ ( word @ A ) @ W @ ( bit_se2239418461657761734s_mask @ ( word @ A ) @ N3 ) ) @ ( power_power @ ( word @ A ) @ ( numeral_numeral @ ( word @ A ) @ ( bit0 @ one2 ) ) @ N3 ) ) ) ) ).
% and_mask_less'
thf(fact_6378_sint__1__cases,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [A2: word @ A] :
( ( ( ( type_len0_len_of @ A @ ( type2 @ A ) )
= ( one_one @ nat ) )
=> ( ( A2
= ( zero_zero @ ( word @ A ) ) )
=> ( ( ring_1_signed @ A @ int @ A2 )
!= ( zero_zero @ int ) ) ) )
=> ( ( ( ( type_len0_len_of @ A @ ( type2 @ A ) )
= ( one_one @ nat ) )
=> ( ( A2
= ( one_one @ ( word @ A ) ) )
=> ( ( ring_1_signed @ A @ int @ ( one_one @ ( word @ A ) ) )
!= ( uminus_uminus @ int @ ( one_one @ int ) ) ) ) )
=> ~ ( ( ord_less @ nat @ ( one_one @ nat ) @ ( type_len0_len_of @ A @ ( type2 @ A ) ) )
=> ( ( ring_1_signed @ A @ int @ ( one_one @ ( word @ A ) ) )
!= ( one_one @ int ) ) ) ) ) ) ).
% sint_1_cases
thf(fact_6379_uint__word__ariths_I2_J,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [A2: word @ A,B2: word @ A] :
( ( semiring_1_unsigned @ A @ int @ ( minus_minus @ ( word @ A ) @ A2 @ B2 ) )
= ( modulo_modulo @ int @ ( minus_minus @ int @ ( semiring_1_unsigned @ A @ int @ A2 ) @ ( semiring_1_unsigned @ A @ int @ B2 ) ) @ ( power_power @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ ( type_len0_len_of @ A @ ( type2 @ A ) ) ) ) ) ) ).
% uint_word_ariths(2)
thf(fact_6380_uint__mult__lem,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [X: word @ A,Y: word @ A] :
( ( ord_less @ int @ ( times_times @ int @ ( semiring_1_unsigned @ A @ int @ X ) @ ( semiring_1_unsigned @ A @ int @ Y ) ) @ ( power_power @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ ( type_len0_len_of @ A @ ( type2 @ A ) ) ) )
= ( ( semiring_1_unsigned @ A @ int @ ( times_times @ ( word @ A ) @ X @ Y ) )
= ( times_times @ int @ ( semiring_1_unsigned @ A @ int @ X ) @ ( semiring_1_unsigned @ A @ int @ Y ) ) ) ) ) ).
% uint_mult_lem
thf(fact_6381_uint__word__ariths_I3_J,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [A2: word @ A,B2: word @ A] :
( ( semiring_1_unsigned @ A @ int @ ( times_times @ ( word @ A ) @ A2 @ B2 ) )
= ( modulo_modulo @ int @ ( times_times @ int @ ( semiring_1_unsigned @ A @ int @ A2 ) @ ( semiring_1_unsigned @ A @ int @ B2 ) ) @ ( power_power @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ ( type_len0_len_of @ A @ ( type2 @ A ) ) ) ) ) ) ).
% uint_word_ariths(3)
thf(fact_6382_signed__of__nat,axiom,
! [A: $tType,B: $tType] :
( ( ( type_len @ B )
& ( ring_1 @ A ) )
=> ! [N3: nat] :
( ( ring_1_signed @ B @ A @ ( semiring_1_of_nat @ ( word @ B ) @ N3 ) )
= ( 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 @ N3 ) ) ) ) ) ).
% signed_of_nat
thf(fact_6383_word__power__mod__div,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [N3: nat,M: nat,X: word @ A] :
( ( ord_less @ nat @ N3 @ ( type_len0_len_of @ A @ ( type2 @ A ) ) )
=> ( ( ord_less @ nat @ M @ ( type_len0_len_of @ A @ ( type2 @ A ) ) )
=> ( ( divide_divide @ ( word @ A ) @ ( modulo_modulo @ ( word @ A ) @ X @ ( power_power @ ( word @ A ) @ ( numeral_numeral @ ( word @ A ) @ ( bit0 @ one2 ) ) @ N3 ) ) @ ( power_power @ ( word @ A ) @ ( numeral_numeral @ ( word @ A ) @ ( bit0 @ one2 ) ) @ M ) )
= ( modulo_modulo @ ( word @ A ) @ ( divide_divide @ ( word @ A ) @ X @ ( power_power @ ( word @ A ) @ ( numeral_numeral @ ( word @ A ) @ ( bit0 @ one2 ) ) @ M ) ) @ ( power_power @ ( word @ A ) @ ( numeral_numeral @ ( word @ A ) @ ( bit0 @ one2 ) ) @ ( minus_minus @ nat @ N3 @ M ) ) ) ) ) ) ) ).
% word_power_mod_div
thf(fact_6384_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_6385_msb1,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [X: word @ A,Y: word @ A] :
( ( ( bit_se5641148757651400278ts_bit @ ( word @ A ) @ ( bit_se1065995026697491101ons_or @ ( word @ A ) @ X @ Y ) @ ( minus_minus @ nat @ ( type_len0_len_of @ A @ ( type2 @ A ) ) @ ( suc @ ( zero_zero @ nat ) ) ) )
= ( ( bit_se5641148757651400278ts_bit @ ( word @ A ) @ X @ ( minus_minus @ nat @ ( type_len0_len_of @ A @ ( type2 @ A ) ) @ ( suc @ ( zero_zero @ nat ) ) ) )
| ( bit_se5641148757651400278ts_bit @ ( word @ A ) @ Y @ ( minus_minus @ nat @ ( type_len0_len_of @ A @ ( type2 @ A ) ) @ ( suc @ ( zero_zero @ nat ) ) ) ) ) )
& ( ( bit_se5641148757651400278ts_bit @ ( word @ A ) @ ( bit_se5824344872417868541ns_and @ ( word @ A ) @ X @ Y ) @ ( minus_minus @ nat @ ( type_len0_len_of @ A @ ( type2 @ A ) ) @ ( suc @ ( zero_zero @ nat ) ) ) )
= ( ( bit_se5641148757651400278ts_bit @ ( word @ A ) @ X @ ( minus_minus @ nat @ ( type_len0_len_of @ A @ ( type2 @ A ) ) @ ( suc @ ( zero_zero @ nat ) ) ) )
& ( bit_se5641148757651400278ts_bit @ ( word @ A ) @ Y @ ( minus_minus @ nat @ ( type_len0_len_of @ A @ ( type2 @ A ) ) @ ( suc @ ( zero_zero @ nat ) ) ) ) ) )
& ( ( bit_se5641148757651400278ts_bit @ ( word @ A ) @ ( bit_se5824344971392196577ns_xor @ ( word @ A ) @ X @ Y ) @ ( minus_minus @ nat @ ( type_len0_len_of @ A @ ( type2 @ A ) ) @ ( suc @ ( zero_zero @ nat ) ) ) )
= ( ( bit_se5641148757651400278ts_bit @ ( word @ A ) @ X @ ( minus_minus @ nat @ ( type_len0_len_of @ A @ ( type2 @ A ) ) @ ( suc @ ( zero_zero @ nat ) ) ) )
!= ( bit_se5641148757651400278ts_bit @ ( word @ A ) @ Y @ ( minus_minus @ nat @ ( type_len0_len_of @ A @ ( type2 @ A ) ) @ ( suc @ ( zero_zero @ nat ) ) ) ) ) )
& ( ( bit_se5641148757651400278ts_bit @ ( word @ A ) @ ( bit_ri4277139882892585799ns_not @ ( word @ A ) @ X ) @ ( minus_minus @ nat @ ( type_len0_len_of @ A @ ( type2 @ A ) ) @ ( suc @ ( zero_zero @ nat ) ) ) )
= ( ~ ( bit_se5641148757651400278ts_bit @ ( word @ A ) @ X @ ( minus_minus @ nat @ ( type_len0_len_of @ A @ ( type2 @ A ) ) @ ( suc @ ( zero_zero @ nat ) ) ) ) ) ) ) ) ).
% msb1
thf(fact_6386_unat__plus__if_H,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [A2: word @ A,B2: word @ A] :
( ( ( ord_less @ nat @ ( plus_plus @ nat @ ( semiring_1_unsigned @ A @ nat @ A2 ) @ ( semiring_1_unsigned @ A @ nat @ B2 ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( type_len0_len_of @ A @ ( type2 @ A ) ) ) )
=> ( ( semiring_1_unsigned @ A @ nat @ ( plus_plus @ ( word @ A ) @ A2 @ B2 ) )
= ( plus_plus @ nat @ ( semiring_1_unsigned @ A @ nat @ A2 ) @ ( semiring_1_unsigned @ A @ nat @ B2 ) ) ) )
& ( ~ ( ord_less @ nat @ ( plus_plus @ nat @ ( semiring_1_unsigned @ A @ nat @ A2 ) @ ( semiring_1_unsigned @ A @ nat @ B2 ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( type_len0_len_of @ A @ ( type2 @ A ) ) ) )
=> ( ( semiring_1_unsigned @ A @ nat @ ( plus_plus @ ( word @ A ) @ A2 @ B2 ) )
= ( minus_minus @ nat @ ( plus_plus @ nat @ ( semiring_1_unsigned @ A @ nat @ A2 ) @ ( semiring_1_unsigned @ A @ nat @ B2 ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( type_len0_len_of @ A @ ( type2 @ A ) ) ) ) ) ) ) ) ).
% unat_plus_if'
thf(fact_6387_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_6388_unat__sub__if_H,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [Y: word @ A,X: word @ A] :
( ( ( ord_less_eq @ nat @ ( semiring_1_unsigned @ A @ nat @ Y ) @ ( semiring_1_unsigned @ A @ nat @ X ) )
=> ( ( semiring_1_unsigned @ A @ nat @ ( minus_minus @ ( word @ A ) @ X @ Y ) )
= ( minus_minus @ nat @ ( semiring_1_unsigned @ A @ nat @ X ) @ ( semiring_1_unsigned @ A @ nat @ Y ) ) ) )
& ( ~ ( ord_less_eq @ nat @ ( semiring_1_unsigned @ A @ nat @ Y ) @ ( semiring_1_unsigned @ A @ nat @ X ) )
=> ( ( semiring_1_unsigned @ A @ nat @ ( minus_minus @ ( word @ A ) @ X @ Y ) )
= ( minus_minus @ nat @ ( plus_plus @ nat @ ( semiring_1_unsigned @ A @ nat @ X ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( type_len0_len_of @ A @ ( type2 @ A ) ) ) ) @ ( semiring_1_unsigned @ A @ nat @ Y ) ) ) ) ) ) ).
% unat_sub_if'
thf(fact_6389_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_6390_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_6391_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_6392_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_6393_no__olen__add__nat,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [X: word @ A,Y: word @ A] :
( ( ord_less_eq @ ( word @ A ) @ X @ ( plus_plus @ ( word @ A ) @ X @ Y ) )
= ( ord_less @ nat @ ( plus_plus @ nat @ ( semiring_1_unsigned @ A @ nat @ X ) @ ( semiring_1_unsigned @ A @ nat @ Y ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( type_len0_len_of @ A @ ( type2 @ A ) ) ) ) ) ) ).
% no_olen_add_nat
thf(fact_6394_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_6395_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_6396_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_6397_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_6398_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_6399_sint__ge,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [X: word @ A] : ( ord_less_eq @ int @ ( uminus_uminus @ int @ ( power_power @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ ( minus_minus @ nat @ ( type_len0_len_of @ A @ ( type2 @ A ) ) @ ( one_one @ nat ) ) ) ) @ ( ring_1_signed @ A @ int @ X ) ) ) ).
% sint_ge
thf(fact_6400_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_6401_uint__split,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [P: int > $o,X: word @ A] :
( ( P @ ( semiring_1_unsigned @ A @ int @ X ) )
= ( ! [I3: int] :
( ( ( ( ring_1_of_int @ ( word @ A ) @ I3 )
= X )
& ( ord_less_eq @ int @ ( zero_zero @ int ) @ I3 )
& ( ord_less @ int @ I3 @ ( power_power @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ ( type_len0_len_of @ A @ ( type2 @ A ) ) ) ) )
=> ( P @ I3 ) ) ) ) ) ).
% uint_split
thf(fact_6402_uint__split__asm,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [P: int > $o,X: word @ A] :
( ( P @ ( semiring_1_unsigned @ A @ int @ X ) )
= ( ~ ? [I3: int] :
( ( ( ring_1_of_int @ ( word @ A ) @ I3 )
= X )
& ( ord_less_eq @ int @ ( zero_zero @ int ) @ I3 )
& ( ord_less @ int @ I3 @ ( power_power @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ ( type_len0_len_of @ A @ ( type2 @ A ) ) ) )
& ~ ( P @ I3 ) ) ) ) ) ).
% uint_split_asm
thf(fact_6403_word__of__int__inverse,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [R2: int,A2: word @ A] :
( ( ( ring_1_of_int @ ( word @ A ) @ R2 )
= A2 )
=> ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ R2 )
=> ( ( ord_less @ int @ R2 @ ( power_power @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ ( type_len0_len_of @ A @ ( type2 @ A ) ) ) )
=> ( ( semiring_1_unsigned @ A @ int @ A2 )
= R2 ) ) ) ) ) ).
% word_of_int_inverse
thf(fact_6404_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_6405_div__lt_H,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [I: word @ A,K: word @ A,X: word @ A] :
( ( ord_less_eq @ ( word @ A ) @ I @ ( divide_divide @ ( word @ A ) @ K @ X ) )
=> ( ord_less @ nat @ ( times_times @ nat @ ( semiring_1_unsigned @ A @ nat @ I ) @ ( semiring_1_unsigned @ A @ nat @ X ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( type_len0_len_of @ A @ ( type2 @ A ) ) ) ) ) ) ).
% div_lt'
thf(fact_6406_div__lt_H_H,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [I: word @ A,K: word @ A,X: word @ A] :
( ( ord_less @ ( word @ A ) @ I @ ( divide_divide @ ( word @ A ) @ K @ X ) )
=> ( ord_less @ nat @ ( times_times @ nat @ ( semiring_1_unsigned @ A @ nat @ I ) @ ( semiring_1_unsigned @ A @ nat @ X ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( type_len0_len_of @ A @ ( type2 @ A ) ) ) ) ) ) ).
% div_lt''
thf(fact_6407_double__eq__zero__iff,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [A2: word @ A] :
( ( ( times_times @ ( word @ A ) @ ( numeral_numeral @ ( word @ A ) @ ( bit0 @ one2 ) ) @ A2 )
= ( zero_zero @ ( word @ A ) ) )
= ( ( A2
= ( zero_zero @ ( word @ A ) ) )
| ( A2
= ( power_power @ ( word @ A ) @ ( numeral_numeral @ ( word @ A ) @ ( bit0 @ one2 ) ) @ ( minus_minus @ nat @ ( type_len0_len_of @ A @ ( type2 @ A ) ) @ ( suc @ ( zero_zero @ nat ) ) ) ) ) ) ) ) ).
% double_eq_zero_iff
thf(fact_6408_word__le__exists_H,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [X: word @ A,Y: word @ A] :
( ( ord_less_eq @ ( word @ A ) @ X @ Y )
=> ? [Z2: word @ A] :
( ( Y
= ( plus_plus @ ( word @ A ) @ X @ Z2 ) )
& ( ord_less @ int @ ( plus_plus @ int @ ( semiring_1_unsigned @ A @ int @ X ) @ ( 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_6409_no__olen__add_H,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [X: word @ A,Y: word @ A] :
( ( ord_less_eq @ ( word @ A ) @ X @ ( plus_plus @ ( word @ A ) @ Y @ X ) )
= ( ord_less @ int @ ( plus_plus @ int @ ( semiring_1_unsigned @ A @ int @ Y ) @ ( semiring_1_unsigned @ A @ int @ X ) ) @ ( power_power @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ ( type_len0_len_of @ A @ ( type2 @ A ) ) ) ) ) ) ).
% no_olen_add'
thf(fact_6410_no__olen__add,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [X: word @ A,Y: word @ A] :
( ( ord_less_eq @ ( word @ A ) @ X @ ( plus_plus @ ( word @ A ) @ X @ Y ) )
= ( ord_less @ int @ ( plus_plus @ int @ ( semiring_1_unsigned @ A @ int @ X ) @ ( semiring_1_unsigned @ A @ int @ Y ) ) @ ( power_power @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ ( type_len0_len_of @ A @ ( type2 @ A ) ) ) ) ) ) ).
% no_olen_add
thf(fact_6411_More__Word_Oof__nat__power,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [P4: nat,X: nat] :
( ( ord_less @ nat @ P4 @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ X ) )
=> ( ( ord_less @ nat @ X @ ( type_len0_len_of @ A @ ( type2 @ A ) ) )
=> ( ord_less @ ( word @ A ) @ ( semiring_1_of_nat @ ( word @ A ) @ P4 ) @ ( power_power @ ( word @ A ) @ ( numeral_numeral @ ( word @ A ) @ ( bit0 @ one2 ) ) @ X ) ) ) ) ) ).
% More_Word.of_nat_power
thf(fact_6412_uint__plus__if_H,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [A2: word @ A,B2: word @ A] :
( ( ( ord_less @ int @ ( plus_plus @ int @ ( semiring_1_unsigned @ A @ int @ A2 ) @ ( semiring_1_unsigned @ A @ int @ B2 ) ) @ ( power_power @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ ( type_len0_len_of @ A @ ( type2 @ A ) ) ) )
=> ( ( semiring_1_unsigned @ A @ int @ ( plus_plus @ ( word @ A ) @ A2 @ B2 ) )
= ( plus_plus @ int @ ( semiring_1_unsigned @ A @ int @ A2 ) @ ( semiring_1_unsigned @ A @ int @ B2 ) ) ) )
& ( ~ ( ord_less @ int @ ( plus_plus @ int @ ( semiring_1_unsigned @ A @ int @ A2 ) @ ( semiring_1_unsigned @ A @ int @ B2 ) ) @ ( power_power @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ ( type_len0_len_of @ A @ ( type2 @ A ) ) ) )
=> ( ( semiring_1_unsigned @ A @ int @ ( plus_plus @ ( word @ A ) @ A2 @ B2 ) )
= ( minus_minus @ int @ ( plus_plus @ int @ ( semiring_1_unsigned @ A @ int @ A2 ) @ ( semiring_1_unsigned @ A @ int @ B2 ) ) @ ( power_power @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ ( type_len0_len_of @ A @ ( type2 @ A ) ) ) ) ) ) ) ) ).
% uint_plus_if'
thf(fact_6413_word__less__power__trans,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [N3: word @ A,M: nat,K: nat] :
( ( ord_less @ ( word @ A ) @ N3 @ ( 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 ) @ N3 ) @ ( power_power @ ( word @ A ) @ ( numeral_numeral @ ( word @ A ) @ ( bit0 @ one2 ) ) @ M ) ) ) ) ) ) ).
% word_less_power_trans
thf(fact_6414_word__less__power__trans2,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [N3: word @ A,M: nat,K: nat] :
( ( ord_less @ ( word @ A ) @ N3 @ ( 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 ) @ N3 @ ( 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_6415_length__n__lists__elem,axiom,
! [A: $tType,Ys: list @ A,N3: nat,Xs2: list @ A] :
( ( member @ ( list @ A ) @ Ys @ ( set2 @ ( list @ A ) @ ( n_lists @ A @ N3 @ Xs2 ) ) )
=> ( ( size_size @ ( list @ A ) @ Ys )
= N3 ) ) ).
% length_n_lists_elem
thf(fact_6416_uint__sub__if_H,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [B2: word @ A,A2: word @ A] :
( ( ( ord_less_eq @ int @ ( semiring_1_unsigned @ A @ int @ B2 ) @ ( semiring_1_unsigned @ A @ int @ A2 ) )
=> ( ( semiring_1_unsigned @ A @ int @ ( minus_minus @ ( word @ A ) @ A2 @ B2 ) )
= ( minus_minus @ int @ ( semiring_1_unsigned @ A @ int @ A2 ) @ ( semiring_1_unsigned @ A @ int @ B2 ) ) ) )
& ( ~ ( ord_less_eq @ int @ ( semiring_1_unsigned @ A @ int @ B2 ) @ ( semiring_1_unsigned @ A @ int @ A2 ) )
=> ( ( semiring_1_unsigned @ A @ int @ ( minus_minus @ ( word @ A ) @ A2 @ B2 ) )
= ( plus_plus @ int @ ( minus_minus @ int @ ( semiring_1_unsigned @ A @ int @ A2 ) @ ( semiring_1_unsigned @ A @ int @ B2 ) ) @ ( power_power @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ ( type_len0_len_of @ A @ ( type2 @ A ) ) ) ) ) ) ) ) ).
% uint_sub_if'
thf(fact_6417_word__less__two__pow__divI,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [X: word @ A,N3: nat,M: nat] :
( ( ord_less @ ( word @ A ) @ X @ ( power_power @ ( word @ A ) @ ( numeral_numeral @ ( word @ A ) @ ( bit0 @ one2 ) ) @ ( minus_minus @ nat @ N3 @ M ) ) )
=> ( ( ord_less_eq @ nat @ M @ N3 )
=> ( ( ord_less @ nat @ N3 @ ( type_len0_len_of @ A @ ( type2 @ A ) ) )
=> ( ord_less @ ( word @ A ) @ X @ ( divide_divide @ ( word @ A ) @ ( power_power @ ( word @ A ) @ ( numeral_numeral @ ( word @ A ) @ ( bit0 @ one2 ) ) @ N3 ) @ ( power_power @ ( word @ A ) @ ( numeral_numeral @ ( word @ A ) @ ( bit0 @ one2 ) ) @ M ) ) ) ) ) ) ) ).
% word_less_two_pow_divI
thf(fact_6418_uint__neg__numeral,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [B2: num] :
( ( semiring_1_unsigned @ A @ int @ ( uminus_uminus @ ( word @ A ) @ ( numeral_numeral @ ( word @ A ) @ B2 ) ) )
= ( modulo_modulo @ int @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ B2 ) ) @ ( power_power @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ ( type_len0_len_of @ A @ ( type2 @ A ) ) ) ) ) ) ).
% uint_neg_numeral
thf(fact_6419_word__power__nonzero,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [X: word @ A,N3: nat] :
( ( ord_less @ ( word @ A ) @ X @ ( power_power @ ( word @ A ) @ ( numeral_numeral @ ( word @ A ) @ ( bit0 @ one2 ) ) @ ( minus_minus @ nat @ ( type_len0_len_of @ A @ ( type2 @ A ) ) @ N3 ) ) )
=> ( ( ord_less @ nat @ N3 @ ( type_len0_len_of @ A @ ( type2 @ A ) ) )
=> ( ( X
!= ( zero_zero @ ( word @ A ) ) )
=> ( ( times_times @ ( word @ A ) @ X @ ( power_power @ ( word @ A ) @ ( numeral_numeral @ ( word @ A ) @ ( bit0 @ one2 ) ) @ N3 ) )
!= ( zero_zero @ ( word @ A ) ) ) ) ) ) ) ).
% word_power_nonzero
thf(fact_6420_mult__pow2__inj,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [M: nat,N3: nat,X: word @ A,Y: word @ A] :
( ( ord_less_eq @ nat @ ( plus_plus @ nat @ M @ N3 ) @ ( type_len0_len_of @ A @ ( type2 @ A ) ) )
=> ( ( ord_less_eq @ ( word @ A ) @ X @ ( bit_se2239418461657761734s_mask @ ( word @ A ) @ M ) )
=> ( ( ord_less_eq @ ( word @ A ) @ Y @ ( bit_se2239418461657761734s_mask @ ( word @ A ) @ M ) )
=> ( ( ( times_times @ ( word @ A ) @ X @ ( power_power @ ( word @ A ) @ ( numeral_numeral @ ( word @ A ) @ ( bit0 @ one2 ) ) @ N3 ) )
= ( times_times @ ( word @ A ) @ Y @ ( power_power @ ( word @ A ) @ ( numeral_numeral @ ( word @ A ) @ ( bit0 @ one2 ) ) @ N3 ) ) )
=> ( X = Y ) ) ) ) ) ) ).
% mult_pow2_inj
thf(fact_6421_div__lt__uint_H,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [I: word @ A,K: word @ A,X: word @ A] :
( ( ord_less_eq @ ( word @ A ) @ I @ ( divide_divide @ ( word @ A ) @ K @ X ) )
=> ( ord_less @ int @ ( times_times @ int @ ( semiring_1_unsigned @ A @ int @ I ) @ ( semiring_1_unsigned @ A @ int @ X ) ) @ ( power_power @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ ( type_len0_len_of @ A @ ( type2 @ A ) ) ) ) ) ) ).
% div_lt_uint'
thf(fact_6422_div__lt__uint_H_H,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [I: word @ A,K: word @ A,X: word @ A] :
( ( ord_less @ ( word @ A ) @ I @ ( divide_divide @ ( word @ A ) @ K @ X ) )
=> ( ord_less @ int @ ( times_times @ int @ ( semiring_1_unsigned @ A @ int @ I ) @ ( semiring_1_unsigned @ A @ int @ X ) ) @ ( power_power @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ ( type_len0_len_of @ A @ ( type2 @ A ) ) ) ) ) ) ).
% div_lt_uint''
thf(fact_6423_push__bit__word__eq__nonzero,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [W: word @ A,M: nat,N3: nat] :
( ( ord_less @ ( word @ A ) @ W @ ( power_power @ ( word @ A ) @ ( numeral_numeral @ ( word @ A ) @ ( bit0 @ one2 ) ) @ M ) )
=> ( ( ord_less @ nat @ ( plus_plus @ nat @ M @ N3 ) @ ( type_len0_len_of @ A @ ( type2 @ A ) ) )
=> ( ( W
!= ( zero_zero @ ( word @ A ) ) )
=> ( ( bit_se4730199178511100633sh_bit @ ( word @ A ) @ N3 @ W )
!= ( zero_zero @ ( word @ A ) ) ) ) ) ) ) ).
% push_bit_word_eq_nonzero
thf(fact_6424_n__lists_Osimps_I2_J,axiom,
! [A: $tType,N3: nat,Xs2: list @ A] :
( ( n_lists @ A @ ( suc @ N3 ) @ Xs2 )
= ( concat @ ( list @ A )
@ ( map @ ( list @ A ) @ ( list @ ( list @ A ) )
@ ^ [Ys3: list @ A] :
( map @ A @ ( list @ A )
@ ^ [Y2: A] : ( cons @ A @ Y2 @ Ys3 )
@ Xs2 )
@ ( n_lists @ A @ N3 @ Xs2 ) ) ) ) ).
% n_lists.simps(2)
thf(fact_6425_uint__and__mask__or__full,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [N3: word @ A,Mask1: word @ A,Mask2: int] :
( ( bit_se5641148757651400278ts_bit @ ( word @ A ) @ N3 @ ( 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 ) @ N3 @ Mask1 ) ) @ Mask2 )
= ( semiring_1_unsigned @ A @ int @ N3 ) ) ) ) ) ) ).
% uint_and_mask_or_full
thf(fact_6426_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_6427_int__eq__sint,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [X: nat] :
( ( ord_less @ nat @ X @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( minus_minus @ nat @ ( type_len0_len_of @ A @ ( type2 @ A ) ) @ ( one_one @ nat ) ) ) )
=> ( ( ring_1_signed @ A @ int @ ( semiring_1_of_nat @ ( word @ A ) @ X ) )
= ( semiring_1_of_nat @ int @ X ) ) ) ) ).
% int_eq_sint
thf(fact_6428_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_6429_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_6430_smod__word__max,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [A2: word @ A,B2: word @ A] : ( ord_less @ int @ ( signed6721504322012087516modulo @ int @ ( ring_1_signed @ A @ int @ A2 ) @ ( ring_1_signed @ A @ int @ B2 ) ) @ ( power_power @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ ( minus_minus @ nat @ ( type_len0_len_of @ A @ ( type2 @ A ) ) @ ( suc @ ( zero_zero @ nat ) ) ) ) ) ) ).
% smod_word_max
thf(fact_6431_le__2p__upper__bits,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [P4: word @ A,N3: nat] :
( ( ord_less_eq @ ( word @ A ) @ P4 @ ( minus_minus @ ( word @ A ) @ ( power_power @ ( word @ A ) @ ( numeral_numeral @ ( word @ A ) @ ( bit0 @ one2 ) ) @ N3 ) @ ( one_one @ ( word @ A ) ) ) )
=> ( ( ord_less @ nat @ N3 @ ( type_len0_len_of @ A @ ( type2 @ A ) ) )
=> ! [N5: nat] :
( ( ord_less_eq @ nat @ N3 @ N5 )
=> ( ( ord_less @ nat @ N5 @ ( type_len0_len_of @ A @ ( type2 @ A ) ) )
=> ~ ( bit_se5641148757651400278ts_bit @ ( word @ A ) @ P4 @ N5 ) ) ) ) ) ) ).
% le_2p_upper_bits
thf(fact_6432_le2p__bits__unset,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [P4: word @ A,N3: nat] :
( ( ord_less_eq @ ( word @ A ) @ P4 @ ( minus_minus @ ( word @ A ) @ ( power_power @ ( word @ A ) @ ( numeral_numeral @ ( word @ A ) @ ( bit0 @ one2 ) ) @ N3 ) @ ( one_one @ ( word @ A ) ) ) )
=> ! [N5: nat] :
( ( ord_less_eq @ nat @ N3 @ N5 )
=> ( ( ord_less @ nat @ N5 @ ( type_len0_len_of @ A @ ( type2 @ A ) ) )
=> ~ ( bit_se5641148757651400278ts_bit @ ( word @ A ) @ P4 @ N5 ) ) ) ) ) ).
% le2p_bits_unset
thf(fact_6433_word__add__offset__less,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [Y: word @ A,N3: nat,X: word @ A,M: nat,Sz: nat] :
( ( ord_less @ ( word @ A ) @ Y @ ( power_power @ ( word @ A ) @ ( numeral_numeral @ ( word @ A ) @ ( bit0 @ one2 ) ) @ N3 ) )
=> ( ( ord_less @ ( word @ A ) @ X @ ( power_power @ ( word @ A ) @ ( numeral_numeral @ ( word @ A ) @ ( bit0 @ one2 ) ) @ M ) )
=> ( ( ord_less @ nat @ Sz @ ( type_len0_len_of @ A @ ( type2 @ A ) ) )
=> ( ( ord_less @ ( word @ A ) @ X @ ( power_power @ ( word @ A ) @ ( numeral_numeral @ ( word @ A ) @ ( bit0 @ one2 ) ) @ ( minus_minus @ nat @ ( type_len0_len_of @ A @ ( type2 @ A ) ) @ N3 ) ) )
=> ( ( Sz
= ( plus_plus @ nat @ M @ N3 ) )
=> ( ord_less @ ( word @ A ) @ ( plus_plus @ ( word @ A ) @ ( times_times @ ( word @ A ) @ X @ ( power_power @ ( word @ A ) @ ( numeral_numeral @ ( word @ A ) @ ( bit0 @ one2 ) ) @ N3 ) ) @ Y ) @ ( power_power @ ( word @ A ) @ ( numeral_numeral @ ( word @ A ) @ ( bit0 @ one2 ) ) @ Sz ) ) ) ) ) ) ) ) ).
% word_add_offset_less
thf(fact_6434_bit__horner__sum__uint__exp__iff,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [Ws: list @ ( word @ A ),N3: 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 ) @ N3 )
= ( ( ord_less @ nat @ ( divide_divide @ nat @ N3 @ ( 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 @ N3 @ ( type_len0_len_of @ A @ ( type2 @ A ) ) ) ) @ ( modulo_modulo @ nat @ N3 @ ( type_len0_len_of @ A @ ( type2 @ A ) ) ) ) ) ) ) ).
% bit_horner_sum_uint_exp_iff
thf(fact_6435_div__power__helper,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [X: nat,Y: nat] :
( ( ord_less_eq @ nat @ X @ Y )
=> ( ( ord_less @ nat @ Y @ ( type_len0_len_of @ A @ ( type2 @ A ) ) )
=> ( ( divide_divide @ ( word @ A ) @ ( minus_minus @ ( word @ A ) @ ( power_power @ ( word @ A ) @ ( numeral_numeral @ ( word @ A ) @ ( bit0 @ one2 ) ) @ Y ) @ ( one_one @ ( word @ A ) ) ) @ ( power_power @ ( word @ A ) @ ( numeral_numeral @ ( word @ A ) @ ( bit0 @ one2 ) ) @ X ) )
= ( minus_minus @ ( word @ A ) @ ( power_power @ ( word @ A ) @ ( numeral_numeral @ ( word @ A ) @ ( bit0 @ one2 ) ) @ ( minus_minus @ nat @ Y @ X ) ) @ ( one_one @ ( word @ A ) ) ) ) ) ) ) ).
% div_power_helper
thf(fact_6436_even__mult__exp__div__word__iff,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [A2: word @ A,M: nat,N3: nat] :
( ( dvd_dvd @ ( word @ A ) @ ( numeral_numeral @ ( word @ A ) @ ( bit0 @ one2 ) ) @ ( divide_divide @ ( word @ A ) @ ( times_times @ ( word @ A ) @ A2 @ ( power_power @ ( word @ A ) @ ( numeral_numeral @ ( word @ A ) @ ( bit0 @ one2 ) ) @ M ) ) @ ( power_power @ ( word @ A ) @ ( numeral_numeral @ ( word @ A ) @ ( bit0 @ one2 ) ) @ N3 ) ) )
= ( ~ ( ( ord_less_eq @ nat @ M @ N3 )
& ( ord_less @ nat @ N3 @ ( type_len0_len_of @ A @ ( type2 @ A ) ) )
& ~ ( dvd_dvd @ ( word @ A ) @ ( numeral_numeral @ ( word @ A ) @ ( bit0 @ one2 ) ) @ ( divide_divide @ ( word @ A ) @ A2 @ ( power_power @ ( word @ A ) @ ( numeral_numeral @ ( word @ A ) @ ( bit0 @ one2 ) ) @ ( minus_minus @ nat @ N3 @ M ) ) ) ) ) ) ) ) ).
% even_mult_exp_div_word_iff
thf(fact_6437_Suc__2p__unat__mask,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [N3: nat,K: nat] :
( ( ord_less @ nat @ N3 @ ( type_len0_len_of @ A @ ( type2 @ A ) ) )
=> ( ( suc @ ( plus_plus @ nat @ ( times_times @ nat @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N3 ) @ K ) @ ( semiring_1_unsigned @ A @ nat @ ( bit_se2239418461657761734s_mask @ ( word @ A ) @ N3 ) ) ) )
= ( times_times @ nat @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N3 ) @ ( plus_plus @ nat @ K @ ( one_one @ nat ) ) ) ) ) ) ).
% Suc_2p_unat_mask
thf(fact_6438_sint__of__nat__le,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [B2: nat,A2: nat] :
( ( ord_less @ nat @ B2 @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( minus_minus @ nat @ ( type_len0_len_of @ A @ ( type2 @ A ) ) @ ( one_one @ nat ) ) ) )
=> ( ( ord_less_eq @ nat @ A2 @ B2 )
=> ( ord_less_eq @ int @ ( ring_1_signed @ A @ int @ ( semiring_1_of_nat @ ( word @ A ) @ A2 ) ) @ ( ring_1_signed @ A @ int @ ( semiring_1_of_nat @ ( word @ A ) @ B2 ) ) ) ) ) ) ).
% sint_of_nat_le
thf(fact_6439_sint__of__nat__ge__zero,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [X: nat] :
( ( ord_less @ nat @ X @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( minus_minus @ nat @ ( type_len0_len_of @ A @ ( type2 @ A ) ) @ ( one_one @ nat ) ) ) )
=> ( ord_less_eq @ int @ ( zero_zero @ int ) @ ( ring_1_signed @ A @ int @ ( semiring_1_of_nat @ ( word @ A ) @ X ) ) ) ) ) ).
% sint_of_nat_ge_zero
thf(fact_6440_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_6441_sint__of__int__eq,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [X: int] :
( ( ord_less_eq @ int @ ( uminus_uminus @ int @ ( power_power @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ ( minus_minus @ nat @ ( type_len0_len_of @ A @ ( type2 @ A ) ) @ ( one_one @ nat ) ) ) ) @ X )
=> ( ( ord_less @ int @ X @ ( power_power @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ ( minus_minus @ nat @ ( type_len0_len_of @ A @ ( type2 @ A ) ) @ ( one_one @ nat ) ) ) )
=> ( ( ring_1_signed @ A @ int @ ( ring_1_of_int @ ( word @ A ) @ X ) )
= X ) ) ) ) ).
% sint_of_int_eq
thf(fact_6442_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_6443_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_6444_sint__numeral,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [B2: num] :
( ( ring_1_signed @ A @ int @ ( numeral_numeral @ ( word @ A ) @ B2 ) )
= ( minus_minus @ int @ ( modulo_modulo @ int @ ( plus_plus @ int @ ( numeral_numeral @ int @ B2 ) @ ( 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_6445_smod__word__min,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [A2: word @ A,B2: word @ A] : ( ord_less_eq @ int @ ( uminus_uminus @ int @ ( power_power @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ ( minus_minus @ nat @ ( type_len0_len_of @ A @ ( type2 @ A ) ) @ ( suc @ ( zero_zero @ nat ) ) ) ) ) @ ( signed6721504322012087516modulo @ int @ ( ring_1_signed @ A @ int @ A2 ) @ ( ring_1_signed @ A @ int @ B2 ) ) ) ) ).
% smod_word_min
thf(fact_6446_int__word__sint,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [X: int] :
( ( ring_1_signed @ A @ int @ ( ring_1_of_int @ ( word @ A ) @ X ) )
= ( minus_minus @ int @ ( modulo_modulo @ int @ ( plus_plus @ int @ X @ ( 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_6447_Word_Oword__div__mult,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [Y: word @ A,X: word @ A] :
( ( ord_less @ ( word @ A ) @ ( zero_zero @ ( word @ A ) ) @ Y )
=> ( ( ord_less @ nat @ ( times_times @ nat @ ( semiring_1_unsigned @ A @ nat @ X ) @ ( semiring_1_unsigned @ A @ nat @ Y ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( type_len0_len_of @ A @ ( type2 @ A ) ) ) )
=> ( ( divide_divide @ ( word @ A ) @ ( times_times @ ( word @ A ) @ X @ Y ) @ Y )
= X ) ) ) ) ).
% Word.word_div_mult
thf(fact_6448_of__nat__less__two__pow__div__set,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [N3: nat,M: nat] :
( ( ord_less @ nat @ N3 @ ( type_len0_len_of @ A @ ( type2 @ A ) ) )
=> ( ( collect @ ( word @ A )
@ ^ [X2: word @ A] : ( ord_less @ ( word @ A ) @ X2 @ ( divide_divide @ ( word @ A ) @ ( power_power @ ( word @ A ) @ ( numeral_numeral @ ( word @ A ) @ ( bit0 @ one2 ) ) @ N3 ) @ ( power_power @ ( word @ A ) @ ( numeral_numeral @ ( word @ A ) @ ( bit0 @ one2 ) ) @ M ) ) ) )
= ( image @ nat @ ( word @ A ) @ ( semiring_1_of_nat @ ( word @ A ) )
@ ( collect @ nat
@ ^ [K2: nat] : ( ord_less @ nat @ K2 @ ( divide_divide @ nat @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N3 ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ M ) ) ) ) ) ) ) ) ).
% of_nat_less_two_pow_div_set
thf(fact_6449_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_6450_word__less__power__trans__ofnat,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [N3: nat,M: nat,K: nat] :
( ( ord_less @ nat @ N3 @ ( 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 ) @ N3 ) @ ( 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_6451_word__bit__induct,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [P: ( word @ A ) > $o,A2: 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 @ A2 ) ) ) ) ) ).
% word_bit_induct
thf(fact_6452_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_6453_bit__word__half__eq,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [A2: word @ A,B2: $o] :
( ( ord_less @ ( word @ A ) @ A2 @ ( power_power @ ( word @ A ) @ ( numeral_numeral @ ( word @ A ) @ ( bit0 @ one2 ) ) @ ( minus_minus @ nat @ ( type_len0_len_of @ A @ ( type2 @ A ) ) @ ( suc @ ( zero_zero @ nat ) ) ) ) )
=> ( ( divide_divide @ ( word @ A ) @ ( plus_plus @ ( word @ A ) @ ( zero_neq_one_of_bool @ ( word @ A ) @ B2 ) @ ( times_times @ ( word @ A ) @ A2 @ ( numeral_numeral @ ( word @ A ) @ ( bit0 @ one2 ) ) ) ) @ ( numeral_numeral @ ( word @ A ) @ ( bit0 @ one2 ) ) )
= A2 ) ) ) ).
% bit_word_half_eq
thf(fact_6454_length__n__lists,axiom,
! [A: $tType,N3: nat,Xs2: list @ A] :
( ( size_size @ ( list @ ( list @ A ) ) @ ( n_lists @ A @ N3 @ Xs2 ) )
= ( power_power @ nat @ ( size_size @ ( list @ A ) @ Xs2 ) @ N3 ) ) ).
% length_n_lists
thf(fact_6455_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_6456_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_6457_alignUp__div__helper,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [K: nat,N3: nat,X: word @ A,A2: word @ A] :
( ( ord_less @ nat @ K @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( minus_minus @ nat @ ( type_len0_len_of @ A @ ( type2 @ A ) ) @ N3 ) ) )
=> ( ( X
= ( times_times @ ( word @ A ) @ ( power_power @ ( word @ A ) @ ( numeral_numeral @ ( word @ A ) @ ( bit0 @ one2 ) ) @ N3 ) @ ( semiring_1_of_nat @ ( word @ A ) @ K ) ) )
=> ( ( ord_less_eq @ ( word @ A ) @ A2 @ X )
=> ( ( ord_less @ nat @ N3 @ ( type_len0_len_of @ A @ ( type2 @ A ) ) )
=> ( ( ( modulo_modulo @ ( word @ A ) @ A2 @ ( power_power @ ( word @ A ) @ ( numeral_numeral @ ( word @ A ) @ ( bit0 @ one2 ) ) @ N3 ) )
!= ( zero_zero @ ( word @ A ) ) )
=> ( ord_less @ ( word @ A ) @ ( divide_divide @ ( word @ A ) @ A2 @ ( power_power @ ( word @ A ) @ ( numeral_numeral @ ( word @ A ) @ ( bit0 @ one2 ) ) @ N3 ) ) @ ( semiring_1_of_nat @ ( word @ A ) @ K ) ) ) ) ) ) ) ) ).
% alignUp_div_helper
thf(fact_6458_less__eq__decr__length__iff,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [N3: nat] :
( ( ord_less_eq @ nat @ N3 @ ( minus_minus @ nat @ ( type_len0_len_of @ A @ ( type2 @ A ) ) @ ( suc @ ( zero_zero @ nat ) ) ) )
= ( ord_less @ nat @ N3 @ ( type_len0_len_of @ A @ ( type2 @ A ) ) ) ) ) ).
% less_eq_decr_length_iff
thf(fact_6459_decr__length__less__iff,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [N3: nat] :
( ( ord_less @ nat @ ( minus_minus @ nat @ ( type_len0_len_of @ A @ ( type2 @ A ) ) @ ( suc @ ( zero_zero @ nat ) ) ) @ N3 )
= ( ord_less_eq @ nat @ ( type_len0_len_of @ A @ ( type2 @ A ) ) @ N3 ) ) ) ).
% decr_length_less_iff
thf(fact_6460_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_6461_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_6462_len__of__finite__1__def,axiom,
( ( type_len0_len_of @ finite_1 )
= ( ^ [X2: itself @ finite_1] : ( one_one @ nat ) ) ) ).
% len_of_finite_1_def
thf(fact_6463_len__num1,axiom,
( ( type_len0_len_of @ numeral_num1 )
= ( ^ [Uu4: itself @ numeral_num1] : ( one_one @ nat ) ) ) ).
% len_num1
thf(fact_6464_len__of__finite__2__def,axiom,
( ( type_len0_len_of @ finite_2 )
= ( ^ [X2: itself @ finite_2] : ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ).
% len_of_finite_2_def
thf(fact_6465_len__of__finite__3__def,axiom,
( ( type_len0_len_of @ finite_3 )
= ( ^ [X2: itself @ finite_3] : ( numeral_numeral @ nat @ ( bit0 @ ( bit0 @ one2 ) ) ) ) ) ).
% len_of_finite_3_def
thf(fact_6466_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_6467_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_6468_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_6469_divmod__via__sdivmod,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [Y: word @ A,X: word @ A] :
( ( Y
!= ( zero_zero @ ( word @ A ) ) )
=> ( ( ( ord_less_eq @ ( word @ A ) @ ( bit_se4730199178511100633sh_bit @ ( word @ A ) @ ( minus_minus @ nat @ ( type_len0_len_of @ A @ ( type2 @ A ) ) @ ( one_one @ nat ) ) @ ( one_one @ ( word @ A ) ) ) @ Y )
=> ( ( ( ord_less @ ( word @ A ) @ X @ Y )
=> ( ( product_Pair @ ( word @ A ) @ ( word @ A ) @ ( divide_divide @ ( word @ A ) @ X @ Y ) @ ( modulo_modulo @ ( word @ A ) @ X @ Y ) )
= ( product_Pair @ ( word @ A ) @ ( word @ A ) @ ( zero_zero @ ( word @ A ) ) @ X ) ) )
& ( ~ ( ord_less @ ( word @ A ) @ X @ Y )
=> ( ( product_Pair @ ( word @ A ) @ ( word @ A ) @ ( divide_divide @ ( word @ A ) @ X @ Y ) @ ( modulo_modulo @ ( word @ A ) @ X @ Y ) )
= ( product_Pair @ ( word @ A ) @ ( word @ A ) @ ( one_one @ ( word @ A ) ) @ ( minus_minus @ ( word @ A ) @ X @ Y ) ) ) ) ) )
& ( ~ ( ord_less_eq @ ( word @ A ) @ ( bit_se4730199178511100633sh_bit @ ( word @ A ) @ ( minus_minus @ nat @ ( type_len0_len_of @ A @ ( type2 @ A ) ) @ ( one_one @ nat ) ) @ ( one_one @ ( word @ A ) ) ) @ Y )
=> ( ( product_Pair @ ( word @ A ) @ ( word @ A ) @ ( divide_divide @ ( word @ A ) @ X @ Y ) @ ( modulo_modulo @ ( word @ A ) @ X @ Y ) )
= ( if @ ( product_prod @ ( word @ A ) @ ( word @ A ) ) @ ( ord_less_eq @ ( word @ A ) @ Y @ ( minus_minus @ ( word @ A ) @ X @ ( times_times @ ( word @ A ) @ ( bit_se4730199178511100633sh_bit @ ( word @ A ) @ ( one_one @ nat ) @ ( signed7115095781618012415divide @ ( word @ A ) @ ( bit_se4197421643247451524op_bit @ ( word @ A ) @ ( one_one @ nat ) @ X ) @ Y ) ) @ Y ) ) ) @ ( product_Pair @ ( word @ A ) @ ( word @ A ) @ ( plus_plus @ ( word @ A ) @ ( bit_se4730199178511100633sh_bit @ ( word @ A ) @ ( one_one @ nat ) @ ( signed7115095781618012415divide @ ( word @ A ) @ ( bit_se4197421643247451524op_bit @ ( word @ A ) @ ( one_one @ nat ) @ X ) @ Y ) ) @ ( one_one @ ( word @ A ) ) ) @ ( minus_minus @ ( word @ A ) @ ( minus_minus @ ( word @ A ) @ X @ ( times_times @ ( word @ A ) @ ( bit_se4730199178511100633sh_bit @ ( word @ A ) @ ( one_one @ nat ) @ ( signed7115095781618012415divide @ ( word @ A ) @ ( bit_se4197421643247451524op_bit @ ( word @ A ) @ ( one_one @ nat ) @ X ) @ Y ) ) @ Y ) ) @ Y ) ) @ ( product_Pair @ ( word @ A ) @ ( word @ A ) @ ( bit_se4730199178511100633sh_bit @ ( word @ A ) @ ( one_one @ nat ) @ ( signed7115095781618012415divide @ ( word @ A ) @ ( bit_se4197421643247451524op_bit @ ( word @ A ) @ ( one_one @ nat ) @ X ) @ Y ) ) @ ( minus_minus @ ( word @ A ) @ X @ ( times_times @ ( word @ A ) @ ( bit_se4730199178511100633sh_bit @ ( word @ A ) @ ( one_one @ nat ) @ ( signed7115095781618012415divide @ ( word @ A ) @ ( bit_se4197421643247451524op_bit @ ( word @ A ) @ ( one_one @ nat ) @ X ) @ Y ) ) @ Y ) ) ) ) ) ) ) ) ) ).
% divmod_via_sdivmod
thf(fact_6470_product__code,axiom,
! [B: $tType,A: $tType,Xs2: list @ A,Ys: list @ B] :
( ( product_product @ A @ B @ ( set2 @ A @ Xs2 ) @ ( set2 @ B @ Ys ) )
= ( set2 @ ( product_prod @ A @ B )
@ ( concat @ ( product_prod @ A @ B )
@ ( map @ A @ ( list @ ( product_prod @ A @ B ) )
@ ^ [X2: A] : ( map @ B @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X2 ) @ Ys )
@ Xs2 ) ) ) ) ).
% product_code
thf(fact_6471_word__sdiv__div1,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [A2: word @ A] :
( ( signed7115095781618012415divide @ ( word @ A ) @ A2 @ ( one_one @ ( word @ A ) ) )
= A2 ) ) ).
% word_sdiv_div1
thf(fact_6472_word__sdiv__div__minus1,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [A2: word @ A] :
( ( signed7115095781618012415divide @ ( word @ A ) @ A2 @ ( uminus_uminus @ ( word @ A ) @ ( one_one @ ( word @ A ) ) ) )
= ( uminus_uminus @ ( word @ A ) @ A2 ) ) ) ).
% word_sdiv_div_minus1
thf(fact_6473_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_6474_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_6475_sdiv__word__numeral__numeral,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [A2: num,B2: num] :
( ( signed7115095781618012415divide @ ( word @ A ) @ ( numeral_numeral @ ( word @ A ) @ A2 ) @ ( numeral_numeral @ ( word @ A ) @ B2 ) )
= ( 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 @ A2 ) ) @ ( bit_ri4674362597316999326ke_bit @ int @ ( minus_minus @ nat @ ( type_len0_len_of @ A @ ( type2 @ A ) ) @ ( one_one @ nat ) ) @ ( numeral_numeral @ int @ B2 ) ) ) ) ) ) ).
% sdiv_word_numeral_numeral
thf(fact_6476_sdiv__word__minus__numeral__minus__numeral,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [A2: num,B2: num] :
( ( signed7115095781618012415divide @ ( word @ A ) @ ( uminus_uminus @ ( word @ A ) @ ( numeral_numeral @ ( word @ A ) @ A2 ) ) @ ( uminus_uminus @ ( word @ A ) @ ( numeral_numeral @ ( word @ A ) @ B2 ) ) )
= ( 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 @ A2 ) ) ) @ ( bit_ri4674362597316999326ke_bit @ int @ ( minus_minus @ nat @ ( type_len0_len_of @ A @ ( type2 @ A ) ) @ ( one_one @ nat ) ) @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ B2 ) ) ) ) ) ) ) ).
% sdiv_word_minus_numeral_minus_numeral
thf(fact_6477_sdiv__word__numeral__minus__numeral,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [A2: num,B2: num] :
( ( signed7115095781618012415divide @ ( word @ A ) @ ( numeral_numeral @ ( word @ A ) @ A2 ) @ ( uminus_uminus @ ( word @ A ) @ ( numeral_numeral @ ( word @ A ) @ B2 ) ) )
= ( 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 @ A2 ) ) @ ( bit_ri4674362597316999326ke_bit @ int @ ( minus_minus @ nat @ ( type_len0_len_of @ A @ ( type2 @ A ) ) @ ( one_one @ nat ) ) @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ B2 ) ) ) ) ) ) ) ).
% sdiv_word_numeral_minus_numeral
thf(fact_6478_sdiv__word__minus__numeral__numeral,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [A2: num,B2: num] :
( ( signed7115095781618012415divide @ ( word @ A ) @ ( uminus_uminus @ ( word @ A ) @ ( numeral_numeral @ ( word @ A ) @ A2 ) ) @ ( numeral_numeral @ ( word @ A ) @ B2 ) )
= ( 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 @ A2 ) ) ) @ ( bit_ri4674362597316999326ke_bit @ int @ ( minus_minus @ nat @ ( type_len0_len_of @ A @ ( type2 @ A ) ) @ ( one_one @ nat ) ) @ ( numeral_numeral @ int @ B2 ) ) ) ) ) ) ).
% sdiv_word_minus_numeral_numeral
thf(fact_6479_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_6480_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_6481_word__sdiv__numerals_I3_J,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [Y: num] :
( ( signed7115095781618012415divide @ ( word @ A ) @ ( one_one @ ( word @ A ) ) @ ( numeral_numeral @ ( word @ A ) @ Y ) )
= ( 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 ) @ Y ) ) ) ) ) ) ).
% word_sdiv_numerals(3)
thf(fact_6482_word__sdiv__numerals_I7_J,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [X: num] :
( ( signed7115095781618012415divide @ ( word @ A ) @ ( numeral_numeral @ ( word @ A ) @ X ) @ ( one_one @ ( word @ A ) ) )
= ( ring_1_of_int @ ( word @ A ) @ ( signed7115095781618012415divide @ int @ ( ring_1_signed @ A @ int @ ( numeral_numeral @ ( word @ A ) @ X ) ) @ ( ring_1_signed @ A @ int @ ( one_one @ ( word @ A ) ) ) ) ) ) ) ).
% word_sdiv_numerals(7)
thf(fact_6483_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_6484_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_6485_signed__div__arith,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [A2: word @ A,B2: word @ A] :
( ( ring_1_signed @ A @ int @ ( signed7115095781618012415divide @ ( word @ A ) @ A2 @ B2 ) )
= ( bit_ri4674362597316999326ke_bit @ int @ ( minus_minus @ nat @ ( type_len0_len_of @ A @ ( type2 @ A ) ) @ ( one_one @ nat ) ) @ ( signed7115095781618012415divide @ int @ ( ring_1_signed @ A @ int @ A2 ) @ ( ring_1_signed @ A @ int @ B2 ) ) ) ) ) ).
% signed_div_arith
thf(fact_6486_signed__drop__bit__word__minus__numeral,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [N3: num,K: num] :
( ( signed_drop_bit @ A @ ( numeral_numeral @ nat @ N3 ) @ ( uminus_uminus @ ( word @ A ) @ ( numeral_numeral @ ( word @ A ) @ K ) ) )
= ( ring_1_of_int @ ( word @ A ) @ ( bit_se4197421643247451524op_bit @ int @ ( numeral_numeral @ nat @ N3 ) @ ( 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_6487_signed__drop__bit__word__Suc__minus__numeral,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [N3: nat,K: num] :
( ( signed_drop_bit @ A @ ( suc @ N3 ) @ ( uminus_uminus @ ( word @ A ) @ ( numeral_numeral @ ( word @ A ) @ K ) ) )
= ( ring_1_of_int @ ( word @ A ) @ ( bit_se4197421643247451524op_bit @ int @ ( suc @ N3 ) @ ( 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_6488_signed__drop__bit__signed__drop__bit,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [M: nat,N3: nat,W: word @ A] :
( ( signed_drop_bit @ A @ M @ ( signed_drop_bit @ A @ N3 @ W ) )
= ( signed_drop_bit @ A @ ( plus_plus @ nat @ M @ N3 ) @ W ) ) ) ).
% signed_drop_bit_signed_drop_bit
thf(fact_6489_signed__drop__bit__of__minus__1,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [N3: nat] :
( ( signed_drop_bit @ A @ N3 @ ( uminus_uminus @ ( word @ A ) @ ( one_one @ ( word @ A ) ) ) )
= ( uminus_uminus @ ( word @ A ) @ ( one_one @ ( word @ A ) ) ) ) ) ).
% signed_drop_bit_of_minus_1
thf(fact_6490_signed__drop__bit__of__1,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [N3: nat] :
( ( signed_drop_bit @ A @ N3 @ ( one_one @ ( word @ A ) ) )
= ( zero_neq_one_of_bool @ ( word @ A )
@ ( ( ( type_len0_len_of @ A @ ( type2 @ A ) )
= ( one_one @ nat ) )
| ( N3
= ( zero_zero @ nat ) ) ) ) ) ) ).
% signed_drop_bit_of_1
thf(fact_6491_signed__drop__bit__word__Suc__numeral,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [N3: nat,K: num] :
( ( signed_drop_bit @ A @ ( suc @ N3 ) @ ( numeral_numeral @ ( word @ A ) @ K ) )
= ( ring_1_of_int @ ( word @ A ) @ ( bit_se4197421643247451524op_bit @ int @ ( suc @ N3 ) @ ( 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_6492_signed__drop__bit__word__numeral,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [N3: num,K: num] :
( ( signed_drop_bit @ A @ ( numeral_numeral @ nat @ N3 ) @ ( numeral_numeral @ ( word @ A ) @ K ) )
= ( ring_1_of_int @ ( word @ A ) @ ( bit_se4197421643247451524op_bit @ int @ ( numeral_numeral @ nat @ N3 ) @ ( 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_6493_bit__signed__drop__bit__iff,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [M: nat,W: word @ A,N3: nat] :
( ( bit_se5641148757651400278ts_bit @ ( word @ A ) @ ( signed_drop_bit @ A @ M @ W ) @ N3 )
= ( bit_se5641148757651400278ts_bit @ ( word @ A ) @ W
@ ( if @ nat
@ ( ( ord_less_eq @ nat @ ( minus_minus @ nat @ ( type_len0_len_of @ A @ ( type2 @ A ) ) @ M ) @ N3 )
& ( ord_less @ nat @ N3 @ ( type_len0_len_of @ A @ ( type2 @ A ) ) ) )
@ ( minus_minus @ nat @ ( type_len0_len_of @ A @ ( type2 @ A ) ) @ ( one_one @ nat ) )
@ ( plus_plus @ nat @ M @ N3 ) ) ) ) ) ).
% bit_signed_drop_bit_iff
thf(fact_6494_signed__drop__bit__beyond,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [N3: nat,W: word @ A] :
( ( ord_less_eq @ nat @ ( type_len0_len_of @ A @ ( type2 @ A ) ) @ N3 )
=> ( ( ( bit_se5641148757651400278ts_bit @ ( word @ A ) @ W @ ( minus_minus @ nat @ ( type_len0_len_of @ A @ ( type2 @ A ) ) @ ( suc @ ( zero_zero @ nat ) ) ) )
=> ( ( signed_drop_bit @ A @ N3 @ 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 @ N3 @ W )
= ( zero_zero @ ( word @ A ) ) ) ) ) ) ) ).
% signed_drop_bit_beyond
thf(fact_6495_word__int__split__asm,axiom,
! [A: $tType,B: $tType] :
( ( type_len @ B )
=> ! [P: A > $o,F2: int > A,X: word @ B] :
( ( P @ ( word_int_case @ A @ B @ F2 @ X ) )
= ( ~ ? [N2: int] :
( ( X
= ( ring_1_of_int @ ( word @ B ) @ N2 ) )
& ( ord_less_eq @ int @ ( zero_zero @ int ) @ N2 )
& ( ord_less @ int @ N2 @ ( power_power @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ ( type_len0_len_of @ B @ ( type2 @ B ) ) ) )
& ~ ( P @ ( F2 @ N2 ) ) ) ) ) ) ).
% word_int_split_asm
thf(fact_6496_word__int__split,axiom,
! [A: $tType,B: $tType] :
( ( type_len @ B )
=> ! [P: A > $o,F2: int > A,X: word @ B] :
( ( P @ ( word_int_case @ A @ B @ F2 @ X ) )
= ( ! [I3: int] :
( ( ( X
= ( ring_1_of_int @ ( word @ B ) @ I3 ) )
& ( ord_less_eq @ int @ ( zero_zero @ int ) @ I3 )
& ( ord_less @ int @ I3 @ ( power_power @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ ( type_len0_len_of @ B @ ( type2 @ B ) ) ) ) )
=> ( P @ ( F2 @ I3 ) ) ) ) ) ) ).
% word_int_split
thf(fact_6497_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_6498_smod__word__minus__numeral__numeral,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [A2: num,B2: num] :
( ( signed6721504322012087516modulo @ ( word @ A ) @ ( uminus_uminus @ ( word @ A ) @ ( numeral_numeral @ ( word @ A ) @ A2 ) ) @ ( numeral_numeral @ ( word @ A ) @ B2 ) )
= ( 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 @ A2 ) ) ) @ ( bit_ri4674362597316999326ke_bit @ int @ ( minus_minus @ nat @ ( type_len0_len_of @ A @ ( type2 @ A ) ) @ ( one_one @ nat ) ) @ ( numeral_numeral @ int @ B2 ) ) ) ) ) ) ).
% smod_word_minus_numeral_numeral
thf(fact_6499_smod__word__numeral__minus__numeral,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [A2: num,B2: num] :
( ( signed6721504322012087516modulo @ ( word @ A ) @ ( numeral_numeral @ ( word @ A ) @ A2 ) @ ( uminus_uminus @ ( word @ A ) @ ( numeral_numeral @ ( word @ A ) @ B2 ) ) )
= ( 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 @ A2 ) ) @ ( bit_ri4674362597316999326ke_bit @ int @ ( minus_minus @ nat @ ( type_len0_len_of @ A @ ( type2 @ A ) ) @ ( one_one @ nat ) ) @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ B2 ) ) ) ) ) ) ) ).
% smod_word_numeral_minus_numeral
thf(fact_6500_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_6501_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_6502_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_6503_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_6504_smod__word__numeral__numeral,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [A2: num,B2: num] :
( ( signed6721504322012087516modulo @ ( word @ A ) @ ( numeral_numeral @ ( word @ A ) @ A2 ) @ ( numeral_numeral @ ( word @ A ) @ B2 ) )
= ( 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 @ A2 ) ) @ ( bit_ri4674362597316999326ke_bit @ int @ ( minus_minus @ nat @ ( type_len0_len_of @ A @ ( type2 @ A ) ) @ ( one_one @ nat ) ) @ ( numeral_numeral @ int @ B2 ) ) ) ) ) ) ).
% smod_word_numeral_numeral
thf(fact_6505_smod__word__minus__numeral__minus__numeral,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [A2: num,B2: num] :
( ( signed6721504322012087516modulo @ ( word @ A ) @ ( uminus_uminus @ ( word @ A ) @ ( numeral_numeral @ ( word @ A ) @ A2 ) ) @ ( uminus_uminus @ ( word @ A ) @ ( numeral_numeral @ ( word @ A ) @ B2 ) ) )
= ( 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 @ A2 ) ) ) @ ( bit_ri4674362597316999326ke_bit @ int @ ( minus_minus @ nat @ ( type_len0_len_of @ A @ ( type2 @ A ) ) @ ( one_one @ nat ) ) @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ B2 ) ) ) ) ) ) ) ).
% smod_word_minus_numeral_minus_numeral
thf(fact_6506_smod__word__alt__def,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ( ( signed6721504322012087516modulo @ ( word @ A ) )
= ( ^ [A6: word @ A,B5: word @ A] : ( minus_minus @ ( word @ A ) @ A6 @ ( times_times @ ( word @ A ) @ ( signed7115095781618012415divide @ ( word @ A ) @ A6 @ B5 ) @ B5 ) ) ) ) ) ).
% smod_word_alt_def
thf(fact_6507_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_6508_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_6509_word__smod__numerals_I3_J,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [Y: num] :
( ( signed6721504322012087516modulo @ ( word @ A ) @ ( one_one @ ( word @ A ) ) @ ( numeral_numeral @ ( word @ A ) @ Y ) )
= ( 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 ) @ Y ) ) ) ) ) ) ).
% word_smod_numerals(3)
thf(fact_6510_word__smod__numerals_I7_J,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [X: num] :
( ( signed6721504322012087516modulo @ ( word @ A ) @ ( numeral_numeral @ ( word @ A ) @ X ) @ ( one_one @ ( word @ A ) ) )
= ( ring_1_of_int @ ( word @ A ) @ ( signed6721504322012087516modulo @ int @ ( ring_1_signed @ A @ int @ ( numeral_numeral @ ( word @ A ) @ X ) ) @ ( ring_1_signed @ A @ int @ ( one_one @ ( word @ A ) ) ) ) ) ) ) ).
% word_smod_numerals(7)
thf(fact_6511_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_6512_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_6513_signed__mod__arith,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [A2: word @ A,B2: word @ A] :
( ( ring_1_signed @ A @ int @ ( signed6721504322012087516modulo @ ( word @ A ) @ A2 @ B2 ) )
= ( bit_ri4674362597316999326ke_bit @ int @ ( minus_minus @ nat @ ( type_len0_len_of @ A @ ( type2 @ A ) ) @ ( one_one @ nat ) ) @ ( signed6721504322012087516modulo @ int @ ( ring_1_signed @ A @ int @ A2 ) @ ( ring_1_signed @ A @ int @ B2 ) ) ) ) ) ).
% signed_mod_arith
thf(fact_6514_uint__word__ariths_I6_J,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [A2: word @ A] :
( ( semiring_1_unsigned @ A @ int @ ( word_pred @ A @ A2 ) )
= ( modulo_modulo @ int @ ( minus_minus @ int @ ( semiring_1_unsigned @ A @ int @ A2 ) @ ( one_one @ int ) ) @ ( power_power @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ ( type_len0_len_of @ A @ ( type2 @ A ) ) ) ) ) ) ).
% uint_word_ariths(6)
thf(fact_6515_slice1__def,axiom,
! [A: $tType,B: $tType] :
( ( ( type_len @ B )
& ( type_len @ A ) )
=> ( ( slice1 @ A @ B )
= ( ^ [N2: nat,W2: word @ A] : ( if @ ( word @ B ) @ ( ord_less @ nat @ N2 @ ( 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 ) ) @ N2 ) @ W2 ) ) @ ( bit_se4730199178511100633sh_bit @ ( word @ B ) @ ( minus_minus @ nat @ N2 @ ( type_len0_len_of @ A @ ( type2 @ A ) ) ) @ ( semiring_1_unsigned @ A @ ( word @ B ) @ W2 ) ) ) ) ) ) ).
% slice1_def
thf(fact_6516_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_6517_succ__pred__no_I4_J,axiom,
! [D5: $tType] :
( ( type_len @ D5 )
=> ! [W: num] :
( ( word_pred @ D5 @ ( uminus_uminus @ ( word @ D5 ) @ ( numeral_numeral @ ( word @ D5 ) @ W ) ) )
= ( minus_minus @ ( word @ D5 ) @ ( uminus_uminus @ ( word @ D5 ) @ ( numeral_numeral @ ( word @ D5 ) @ W ) ) @ ( one_one @ ( word @ D5 ) ) ) ) ) ).
% succ_pred_no(4)
thf(fact_6518_word__m1__ge,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [Y: word @ A] : ( ord_less_eq @ ( word @ A ) @ Y @ ( word_pred @ A @ ( zero_zero @ ( word @ A ) ) ) ) ) ).
% word_m1_ge
thf(fact_6519_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_6520_ucast__slice1,axiom,
! [A: $tType,B: $tType] :
( ( ( type_len @ B )
& ( type_len @ A ) )
=> ( ( semiring_1_unsigned @ B @ ( word @ A ) )
= ( ^ [W2: word @ B] : ( slice1 @ B @ A @ ( size_size @ ( word @ B ) @ W2 ) @ W2 ) ) ) ) ).
% ucast_slice1
thf(fact_6521_wi__hom__pred,axiom,
! [F: $tType] :
( ( type_len @ F )
=> ! [A2: int] :
( ( word_pred @ F @ ( ring_1_of_int @ ( word @ F ) @ A2 ) )
= ( ring_1_of_int @ ( word @ F ) @ ( minus_minus @ int @ A2 @ ( one_one @ int ) ) ) ) ) ).
% wi_hom_pred
thf(fact_6522_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_6523_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_6524_Word_Oslice__def,axiom,
! [A: $tType,B: $tType] :
( ( ( type_len @ B )
& ( type_len @ A ) )
=> ( ( slice2 @ A @ B )
= ( ^ [N2: nat] : ( slice1 @ A @ B @ ( minus_minus @ nat @ ( type_len0_len_of @ A @ ( type2 @ A ) ) @ N2 ) ) ) ) ) ).
% Word.slice_def
thf(fact_6525_uint__word__arith__bintrs_I6_J,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [A2: word @ A] :
( ( semiring_1_unsigned @ A @ int @ ( word_pred @ A @ A2 ) )
= ( bit_se2584673776208193580ke_bit @ int @ ( type_len0_len_of @ A @ ( type2 @ A ) ) @ ( minus_minus @ int @ ( semiring_1_unsigned @ A @ int @ A2 ) @ ( one_one @ int ) ) ) ) ) ).
% uint_word_arith_bintrs(6)
thf(fact_6526_sint__word__ariths_I6_J,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [A2: word @ A] :
( ( ring_1_signed @ A @ int @ ( word_pred @ A @ A2 ) )
= ( 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 @ A2 ) @ ( one_one @ int ) ) ) ) ) ).
% sint_word_ariths(6)
thf(fact_6527_uint__word__ariths_I5_J,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [A2: word @ A] :
( ( semiring_1_unsigned @ A @ int @ ( word_succ @ A @ A2 ) )
= ( modulo_modulo @ int @ ( plus_plus @ int @ ( semiring_1_unsigned @ A @ int @ A2 ) @ ( one_one @ int ) ) @ ( power_power @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ ( type_len0_len_of @ A @ ( type2 @ A ) ) ) ) ) ) ).
% uint_word_ariths(5)
thf(fact_6528_bit__word__roti__iff,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [K: int,W: word @ A,N3: nat] :
( ( bit_se5641148757651400278ts_bit @ ( word @ A ) @ ( word_roti @ A @ K @ W ) @ N3 )
= ( ( ord_less @ nat @ N3 @ ( 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 @ N3 ) @ K ) @ ( semiring_1_of_nat @ int @ ( type_len0_len_of @ A @ ( type2 @ A ) ) ) ) ) ) ) ) ) ).
% bit_word_roti_iff
thf(fact_6529_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_6530_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_6531_Abs__fnat__hom__Suc,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [A2: nat] :
( ( word_succ @ A @ ( semiring_1_of_nat @ ( word @ A ) @ A2 ) )
= ( semiring_1_of_nat @ ( word @ A ) @ ( suc @ A2 ) ) ) ) ).
% Abs_fnat_hom_Suc
thf(fact_6532_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_6533_wi__hom__succ,axiom,
! [E3: $tType] :
( ( type_len @ E3 )
=> ! [A2: int] :
( ( word_succ @ E3 @ ( ring_1_of_int @ ( word @ E3 ) @ A2 ) )
= ( ring_1_of_int @ ( word @ E3 ) @ ( plus_plus @ int @ A2 @ ( one_one @ int ) ) ) ) ) ).
% wi_hom_succ
thf(fact_6534_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_6535_word__roti__conv__mod_H,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ( ( word_roti @ A )
= ( ^ [N2: int,W2: word @ A] : ( word_roti @ A @ ( modulo_modulo @ int @ N2 @ ( semiring_1_of_nat @ int @ ( size_size @ ( word @ A ) @ W2 ) ) ) @ W2 ) ) ) ) ).
% word_roti_conv_mod'
thf(fact_6536_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_6537_word__sp__01,axiom,
! [C: $tType,A: $tType,B: $tType,D5: $tType] :
( ( ( type_len @ D5 )
& ( 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 @ D5 @ ( one_one @ ( word @ D5 ) ) )
= ( zero_zero @ ( word @ D5 ) ) ) ) ) ).
% word_sp_01
thf(fact_6538_uint__word__arith__bintrs_I5_J,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [A2: word @ A] :
( ( semiring_1_unsigned @ A @ int @ ( word_succ @ A @ A2 ) )
= ( bit_se2584673776208193580ke_bit @ int @ ( type_len0_len_of @ A @ ( type2 @ A ) ) @ ( plus_plus @ int @ ( semiring_1_unsigned @ A @ int @ A2 ) @ ( one_one @ int ) ) ) ) ) ).
% uint_word_arith_bintrs(5)
thf(fact_6539_sint__word__ariths_I5_J,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [A2: word @ A] :
( ( ring_1_signed @ A @ int @ ( word_succ @ A @ A2 ) )
= ( 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 @ A2 ) @ ( one_one @ int ) ) ) ) ) ).
% sint_word_ariths(5)
thf(fact_6540_unat__word__ariths_I3_J,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [A2: word @ A] :
( ( semiring_1_unsigned @ A @ nat @ ( word_succ @ A @ A2 ) )
= ( modulo_modulo @ nat @ ( suc @ ( semiring_1_unsigned @ A @ nat @ A2 ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( type_len0_len_of @ A @ ( type2 @ A ) ) ) ) ) ) ).
% unat_word_ariths(3)
thf(fact_6541_sless__eq__word__minus__numeral__numeral,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [A2: num,B2: num] :
( ( word_sle @ A @ ( uminus_uminus @ ( word @ A ) @ ( numeral_numeral @ ( word @ A ) @ A2 ) ) @ ( numeral_numeral @ ( word @ A ) @ B2 ) )
= ( ord_less_eq @ int @ ( bit_ri4674362597316999326ke_bit @ int @ ( minus_minus @ nat @ ( type_len0_len_of @ A @ ( type2 @ A ) ) @ ( one_one @ nat ) ) @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ A2 ) ) ) @ ( bit_ri4674362597316999326ke_bit @ int @ ( minus_minus @ nat @ ( type_len0_len_of @ A @ ( type2 @ A ) ) @ ( one_one @ nat ) ) @ ( numeral_numeral @ int @ B2 ) ) ) ) ) ).
% sless_eq_word_minus_numeral_numeral
thf(fact_6542_sless__eq__word__numeral__minus__numeral,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [A2: num,B2: num] :
( ( word_sle @ A @ ( numeral_numeral @ ( word @ A ) @ A2 ) @ ( uminus_uminus @ ( word @ A ) @ ( numeral_numeral @ ( word @ A ) @ B2 ) ) )
= ( ord_less_eq @ int @ ( bit_ri4674362597316999326ke_bit @ int @ ( minus_minus @ nat @ ( type_len0_len_of @ A @ ( type2 @ A ) ) @ ( one_one @ nat ) ) @ ( numeral_numeral @ int @ A2 ) ) @ ( bit_ri4674362597316999326ke_bit @ int @ ( minus_minus @ nat @ ( type_len0_len_of @ A @ ( type2 @ A ) ) @ ( one_one @ nat ) ) @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ B2 ) ) ) ) ) ) ).
% sless_eq_word_numeral_minus_numeral
thf(fact_6543_word__sle__no,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [A2: num,B2: num] :
( ( word_sle @ A @ ( numeral_numeral @ ( word @ A ) @ A2 ) @ ( numeral_numeral @ ( word @ A ) @ B2 ) )
= ( ord_less_eq @ int @ ( ring_1_signed @ A @ int @ ( numeral_numeral @ ( word @ A ) @ A2 ) ) @ ( ring_1_signed @ A @ int @ ( numeral_numeral @ ( word @ A ) @ B2 ) ) ) ) ) ).
% word_sle_no
thf(fact_6544_extra__sle__sless__unfolds_I6_J,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [N3: num] :
( ( word_sle @ A @ ( numeral_numeral @ ( word @ A ) @ N3 ) @ ( one_one @ ( word @ A ) ) )
= ( ord_less_eq @ int @ ( ring_1_signed @ A @ int @ ( numeral_numeral @ ( word @ A ) @ N3 ) ) @ ( ring_1_signed @ A @ int @ ( one_one @ ( word @ A ) ) ) ) ) ) ).
% extra_sle_sless_unfolds(6)
thf(fact_6545_extra__sle__sless__unfolds_I4_J,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [N3: num] :
( ( word_sle @ A @ ( one_one @ ( word @ A ) ) @ ( numeral_numeral @ ( word @ A ) @ N3 ) )
= ( ord_less_eq @ int @ ( ring_1_signed @ A @ int @ ( one_one @ ( word @ A ) ) ) @ ( ring_1_signed @ A @ int @ ( numeral_numeral @ ( word @ A ) @ N3 ) ) ) ) ) ).
% extra_sle_sless_unfolds(4)
thf(fact_6546_extra__sle__sless__unfolds_I5_J,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [N3: num] :
( ( word_sle @ A @ ( numeral_numeral @ ( word @ A ) @ N3 ) @ ( zero_zero @ ( word @ A ) ) )
= ( ord_less_eq @ int @ ( ring_1_signed @ A @ int @ ( numeral_numeral @ ( word @ A ) @ N3 ) ) @ ( ring_1_signed @ A @ int @ ( zero_zero @ ( word @ A ) ) ) ) ) ) ).
% extra_sle_sless_unfolds(5)
thf(fact_6547_extra__sle__sless__unfolds_I2_J,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [N3: num] :
( ( word_sle @ A @ ( zero_zero @ ( word @ A ) ) @ ( numeral_numeral @ ( word @ A ) @ N3 ) )
= ( ord_less_eq @ int @ ( ring_1_signed @ A @ int @ ( zero_zero @ ( word @ A ) ) ) @ ( ring_1_signed @ A @ int @ ( numeral_numeral @ ( word @ A ) @ N3 ) ) ) ) ) ).
% extra_sle_sless_unfolds(2)
thf(fact_6548_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_6549_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_6550_sless__eq__word__numeral__numeral,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [A2: num,B2: num] :
( ( word_sle @ A @ ( numeral_numeral @ ( word @ A ) @ A2 ) @ ( numeral_numeral @ ( word @ A ) @ B2 ) )
= ( ord_less_eq @ int @ ( bit_ri4674362597316999326ke_bit @ int @ ( minus_minus @ nat @ ( type_len0_len_of @ A @ ( type2 @ A ) ) @ ( one_one @ nat ) ) @ ( numeral_numeral @ int @ A2 ) ) @ ( bit_ri4674362597316999326ke_bit @ int @ ( minus_minus @ nat @ ( type_len0_len_of @ A @ ( type2 @ A ) ) @ ( one_one @ nat ) ) @ ( numeral_numeral @ int @ B2 ) ) ) ) ) ).
% sless_eq_word_numeral_numeral
thf(fact_6551_sless__eq__word__minus__numeral__minus__numeral,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [A2: num,B2: num] :
( ( word_sle @ A @ ( uminus_uminus @ ( word @ A ) @ ( numeral_numeral @ ( word @ A ) @ A2 ) ) @ ( uminus_uminus @ ( word @ A ) @ ( numeral_numeral @ ( word @ A ) @ B2 ) ) )
= ( ord_less_eq @ int @ ( bit_ri4674362597316999326ke_bit @ int @ ( minus_minus @ nat @ ( type_len0_len_of @ A @ ( type2 @ A ) ) @ ( one_one @ nat ) ) @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ A2 ) ) ) @ ( bit_ri4674362597316999326ke_bit @ int @ ( minus_minus @ nat @ ( type_len0_len_of @ A @ ( type2 @ A ) ) @ ( one_one @ nat ) ) @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ B2 ) ) ) ) ) ) ).
% sless_eq_word_minus_numeral_minus_numeral
thf(fact_6552_word__sle__eq,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ( ( word_sle @ A )
= ( ^ [A6: word @ A,B5: word @ A] : ( ord_less_eq @ int @ ( ring_1_signed @ A @ int @ A6 ) @ ( ring_1_signed @ A @ int @ B5 ) ) ) ) ) ).
% word_sle_eq
thf(fact_6553_signed_Olift__Suc__mono__le,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [F2: nat > ( word @ A ),N3: nat,N4: nat] :
( ! [N: nat] : ( word_sle @ A @ ( F2 @ N ) @ ( F2 @ ( suc @ N ) ) )
=> ( ( ord_less_eq @ nat @ N3 @ N4 )
=> ( word_sle @ A @ ( F2 @ N3 ) @ ( F2 @ N4 ) ) ) ) ) ).
% signed.lift_Suc_mono_le
thf(fact_6554_signed_Olift__Suc__antimono__le,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [F2: nat > ( word @ A ),N3: nat,N4: nat] :
( ! [N: nat] : ( word_sle @ A @ ( F2 @ ( suc @ N ) ) @ ( F2 @ N ) )
=> ( ( ord_less_eq @ nat @ N3 @ N4 )
=> ( word_sle @ A @ ( F2 @ N4 ) @ ( F2 @ N3 ) ) ) ) ) ).
% signed.lift_Suc_antimono_le
thf(fact_6555_word__0__sle__from__less,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [X: word @ A] :
( ( ord_less @ ( word @ A ) @ X @ ( power_power @ ( word @ A ) @ ( numeral_numeral @ ( word @ A ) @ ( bit0 @ one2 ) ) @ ( minus_minus @ nat @ ( type_len0_len_of @ A @ ( type2 @ A ) ) @ ( one_one @ nat ) ) ) )
=> ( word_sle @ A @ ( zero_zero @ ( word @ A ) ) @ X ) ) ) ).
% word_0_sle_from_less
thf(fact_6556_sless__word__minus__numeral__minus__numeral,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [A2: num,B2: num] :
( ( word_sless @ A @ ( uminus_uminus @ ( word @ A ) @ ( numeral_numeral @ ( word @ A ) @ A2 ) ) @ ( uminus_uminus @ ( word @ A ) @ ( numeral_numeral @ ( word @ A ) @ B2 ) ) )
= ( ord_less @ int @ ( bit_ri4674362597316999326ke_bit @ int @ ( minus_minus @ nat @ ( type_len0_len_of @ A @ ( type2 @ A ) ) @ ( one_one @ nat ) ) @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ A2 ) ) ) @ ( bit_ri4674362597316999326ke_bit @ int @ ( minus_minus @ nat @ ( type_len0_len_of @ A @ ( type2 @ A ) ) @ ( one_one @ nat ) ) @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ B2 ) ) ) ) ) ) ).
% sless_word_minus_numeral_minus_numeral
thf(fact_6557_sless__word__numeral__minus__numeral,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [A2: num,B2: num] :
( ( word_sless @ A @ ( numeral_numeral @ ( word @ A ) @ A2 ) @ ( uminus_uminus @ ( word @ A ) @ ( numeral_numeral @ ( word @ A ) @ B2 ) ) )
= ( ord_less @ int @ ( bit_ri4674362597316999326ke_bit @ int @ ( minus_minus @ nat @ ( type_len0_len_of @ A @ ( type2 @ A ) ) @ ( one_one @ nat ) ) @ ( numeral_numeral @ int @ A2 ) ) @ ( bit_ri4674362597316999326ke_bit @ int @ ( minus_minus @ nat @ ( type_len0_len_of @ A @ ( type2 @ A ) ) @ ( one_one @ nat ) ) @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ B2 ) ) ) ) ) ) ).
% sless_word_numeral_minus_numeral
thf(fact_6558_extra__sle__sless__unfolds_I12_J,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [N3: num] :
( ( word_sless @ A @ ( numeral_numeral @ ( word @ A ) @ N3 ) @ ( one_one @ ( word @ A ) ) )
= ( ord_less @ int @ ( ring_1_signed @ A @ int @ ( numeral_numeral @ ( word @ A ) @ N3 ) ) @ ( ring_1_signed @ A @ int @ ( one_one @ ( word @ A ) ) ) ) ) ) ).
% extra_sle_sless_unfolds(12)
thf(fact_6559_extra__sle__sless__unfolds_I10_J,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [N3: num] :
( ( word_sless @ A @ ( one_one @ ( word @ A ) ) @ ( numeral_numeral @ ( word @ A ) @ N3 ) )
= ( ord_less @ int @ ( ring_1_signed @ A @ int @ ( one_one @ ( word @ A ) ) ) @ ( ring_1_signed @ A @ int @ ( numeral_numeral @ ( word @ A ) @ N3 ) ) ) ) ) ).
% extra_sle_sless_unfolds(10)
thf(fact_6560_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_6561_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_6562_sless__word__numeral__numeral,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [A2: num,B2: num] :
( ( word_sless @ A @ ( numeral_numeral @ ( word @ A ) @ A2 ) @ ( numeral_numeral @ ( word @ A ) @ B2 ) )
= ( ord_less @ int @ ( bit_ri4674362597316999326ke_bit @ int @ ( minus_minus @ nat @ ( type_len0_len_of @ A @ ( type2 @ A ) ) @ ( one_one @ nat ) ) @ ( numeral_numeral @ int @ A2 ) ) @ ( bit_ri4674362597316999326ke_bit @ int @ ( minus_minus @ nat @ ( type_len0_len_of @ A @ ( type2 @ A ) ) @ ( one_one @ nat ) ) @ ( numeral_numeral @ int @ B2 ) ) ) ) ) ).
% sless_word_numeral_numeral
thf(fact_6563_sless__word__minus__numeral__numeral,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [A2: num,B2: num] :
( ( word_sless @ A @ ( uminus_uminus @ ( word @ A ) @ ( numeral_numeral @ ( word @ A ) @ A2 ) ) @ ( numeral_numeral @ ( word @ A ) @ B2 ) )
= ( ord_less @ int @ ( bit_ri4674362597316999326ke_bit @ int @ ( minus_minus @ nat @ ( type_len0_len_of @ A @ ( type2 @ A ) ) @ ( one_one @ nat ) ) @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ A2 ) ) ) @ ( bit_ri4674362597316999326ke_bit @ int @ ( minus_minus @ nat @ ( type_len0_len_of @ A @ ( type2 @ A ) ) @ ( one_one @ nat ) ) @ ( numeral_numeral @ int @ B2 ) ) ) ) ) ).
% sless_word_minus_numeral_numeral
thf(fact_6564_signed_Olift__Suc__mono__less__iff,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [F2: nat > ( word @ A ),N3: nat,M: nat] :
( ! [N: nat] : ( word_sless @ A @ ( F2 @ N ) @ ( F2 @ ( suc @ N ) ) )
=> ( ( word_sless @ A @ ( F2 @ N3 ) @ ( F2 @ M ) )
= ( ord_less @ nat @ N3 @ M ) ) ) ) ).
% signed.lift_Suc_mono_less_iff
thf(fact_6565_signed_Olift__Suc__mono__less,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [F2: nat > ( word @ A ),N3: nat,N4: nat] :
( ! [N: nat] : ( word_sless @ A @ ( F2 @ N ) @ ( F2 @ ( suc @ N ) ) )
=> ( ( ord_less @ nat @ N3 @ N4 )
=> ( word_sless @ A @ ( F2 @ N3 ) @ ( F2 @ N4 ) ) ) ) ) ).
% signed.lift_Suc_mono_less
thf(fact_6566_word__sless__sint__le,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [X: word @ A,Y: word @ A] :
( ( word_sless @ A @ X @ Y )
=> ( ord_less_eq @ int @ ( ring_1_signed @ A @ int @ X ) @ ( minus_minus @ int @ ( ring_1_signed @ A @ int @ Y ) @ ( one_one @ int ) ) ) ) ) ).
% word_sless_sint_le
thf(fact_6567_take__bit__word__minus__numeral,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [N3: num,K: num] :
( ( bit_se2584673776208193580ke_bit @ ( word @ A ) @ ( numeral_numeral @ nat @ N3 ) @ ( 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 @ N3 ) ) @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ K ) ) ) ) ) ) ).
% take_bit_word_minus_numeral
thf(fact_6568_take__bit__word__Suc__minus__numeral,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [N3: nat,K: num] :
( ( bit_se2584673776208193580ke_bit @ ( word @ A ) @ ( suc @ N3 ) @ ( 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 @ N3 ) ) @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ K ) ) ) ) ) ) ).
% take_bit_word_Suc_minus_numeral
thf(fact_6569_min__arg__le_I2_J,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [M: A,N3: A] :
( ( ord_less_eq @ A @ M @ ( ord_min @ A @ M @ N3 ) )
= ( ( ord_min @ A @ M @ N3 )
= M ) ) ) ).
% min_arg_le(2)
thf(fact_6570_min__arg__le_I1_J,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [N3: A,M: A] :
( ( ord_less_eq @ A @ N3 @ ( ord_min @ A @ M @ N3 ) )
= ( ( ord_min @ A @ M @ N3 )
= N3 ) ) ) ).
% min_arg_le(1)
thf(fact_6571_min__eq__arg_I2_J,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [M: A,N3: A] :
( ( ( ord_min @ A @ M @ N3 )
= N3 )
= ( ord_less_eq @ A @ N3 @ M ) ) ) ).
% min_eq_arg(2)
thf(fact_6572_min__eq__arg_I1_J,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [M: A,N3: A] :
( ( ( ord_min @ A @ M @ N3 )
= M )
= ( ord_less_eq @ A @ M @ N3 ) ) ) ).
% min_eq_arg(1)
thf(fact_6573_min_Oabsorb1,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A2: A,B2: A] :
( ( ord_less_eq @ A @ A2 @ B2 )
=> ( ( ord_min @ A @ A2 @ B2 )
= A2 ) ) ) ).
% min.absorb1
thf(fact_6574_min_Oabsorb2,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [B2: A,A2: A] :
( ( ord_less_eq @ A @ B2 @ A2 )
=> ( ( ord_min @ A @ A2 @ B2 )
= B2 ) ) ) ).
% min.absorb2
thf(fact_6575_min_Obounded__iff,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A2: A,B2: A,C2: A] :
( ( ord_less_eq @ A @ A2 @ ( ord_min @ A @ B2 @ C2 ) )
= ( ( ord_less_eq @ A @ A2 @ B2 )
& ( ord_less_eq @ A @ A2 @ C2 ) ) ) ) ).
% min.bounded_iff
thf(fact_6576_min_Oabsorb3,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A2: A,B2: A] :
( ( ord_less @ A @ A2 @ B2 )
=> ( ( ord_min @ A @ A2 @ B2 )
= A2 ) ) ) ).
% min.absorb3
thf(fact_6577_min_Oabsorb4,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [B2: A,A2: A] :
( ( ord_less @ A @ B2 @ A2 )
=> ( ( ord_min @ A @ A2 @ B2 )
= B2 ) ) ) ).
% min.absorb4
thf(fact_6578_min__less__iff__conj,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [Z: A,X: A,Y: A] :
( ( ord_less @ A @ Z @ ( ord_min @ A @ X @ Y ) )
= ( ( ord_less @ A @ Z @ X )
& ( ord_less @ A @ Z @ Y ) ) ) ) ).
% min_less_iff_conj
thf(fact_6579_min__simps_I2_J,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [B2: A,A2: A] :
( ( ord_less @ A @ B2 @ A2 )
=> ( ( ord_min @ A @ A2 @ B2 )
= B2 ) ) ) ).
% min_simps(2)
thf(fact_6580_min__simps_I1_J,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [A2: A,B2: A] :
( ( ord_less @ A @ A2 @ B2 )
=> ( ( ord_min @ A @ A2 @ B2 )
= A2 ) ) ) ).
% min_simps(1)
thf(fact_6581_min__less__self__conv_I2_J,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A2: A,B2: A] :
( ( ord_less @ A @ ( ord_min @ A @ A2 @ B2 ) @ B2 )
= ( ord_less @ A @ A2 @ B2 ) ) ) ).
% min_less_self_conv(2)
thf(fact_6582_min__less__self__conv_I1_J,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A2: A,B2: A] :
( ( ord_less @ A @ ( ord_min @ A @ A2 @ B2 ) @ A2 )
= ( ord_less @ A @ B2 @ A2 ) ) ) ).
% min_less_self_conv(1)
thf(fact_6583_min__arg__not__ge_I2_J,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [M: A,N3: A] :
( ( ~ ( ord_less @ A @ ( ord_min @ A @ M @ N3 ) @ N3 ) )
= ( ( ord_min @ A @ M @ N3 )
= N3 ) ) ) ).
% min_arg_not_ge(2)
thf(fact_6584_min__arg__not__ge_I1_J,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [M: A,N3: A] :
( ( ~ ( ord_less @ A @ ( ord_min @ A @ M @ N3 ) @ M ) )
= ( ( ord_min @ A @ M @ N3 )
= M ) ) ) ).
% min_arg_not_ge(1)
thf(fact_6585_min__Suc__Suc,axiom,
! [M: nat,N3: nat] :
( ( ord_min @ nat @ ( suc @ M ) @ ( suc @ N3 ) )
= ( suc @ ( ord_min @ nat @ M @ N3 ) ) ) ).
% min_Suc_Suc
thf(fact_6586_min__0L,axiom,
! [N3: nat] :
( ( ord_min @ nat @ ( zero_zero @ nat ) @ N3 )
= ( zero_zero @ nat ) ) ).
% min_0L
thf(fact_6587_min__0R,axiom,
! [N3: nat] :
( ( ord_min @ nat @ N3 @ ( zero_zero @ nat ) )
= ( zero_zero @ nat ) ) ).
% min_0R
thf(fact_6588_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_6589_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_6590_take__bit__take__bit,axiom,
! [A: $tType] :
( ( bit_se359711467146920520ations @ A )
=> ! [M: nat,N3: nat,A2: A] :
( ( bit_se2584673776208193580ke_bit @ A @ M @ ( bit_se2584673776208193580ke_bit @ A @ N3 @ A2 ) )
= ( bit_se2584673776208193580ke_bit @ A @ ( ord_min @ nat @ M @ N3 ) @ A2 ) ) ) ).
% take_bit_take_bit
thf(fact_6591_signed__take__bit__signed__take__bit,axiom,
! [A: $tType] :
( ( bit_ri3973907225187159222ations @ A )
=> ! [M: nat,N3: nat,A2: A] :
( ( bit_ri4674362597316999326ke_bit @ A @ M @ ( bit_ri4674362597316999326ke_bit @ A @ N3 @ A2 ) )
= ( bit_ri4674362597316999326ke_bit @ A @ ( ord_min @ nat @ M @ N3 ) @ A2 ) ) ) ).
% signed_take_bit_signed_take_bit
thf(fact_6592_min__0__1_I4_J,axiom,
! [A: $tType] :
( ( linord181362715937106298miring @ A )
=> ! [X: num] :
( ( ord_min @ A @ ( numeral_numeral @ A @ X ) @ ( zero_zero @ A ) )
= ( zero_zero @ A ) ) ) ).
% min_0_1(4)
thf(fact_6593_min__0__1_I3_J,axiom,
! [A: $tType] :
( ( linord181362715937106298miring @ A )
=> ! [X: num] :
( ( ord_min @ A @ ( zero_zero @ A ) @ ( numeral_numeral @ A @ X ) )
= ( zero_zero @ A ) ) ) ).
% min_0_1(3)
thf(fact_6594_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_6595_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_6596_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_6597_min__0__1_I5_J,axiom,
! [A: $tType] :
( ( linord181362715937106298miring @ A )
=> ! [X: num] :
( ( ord_min @ A @ ( one_one @ A ) @ ( numeral_numeral @ A @ X ) )
= ( one_one @ A ) ) ) ).
% min_0_1(5)
thf(fact_6598_min__0__1_I6_J,axiom,
! [A: $tType] :
( ( linord181362715937106298miring @ A )
=> ! [X: num] :
( ( ord_min @ A @ ( numeral_numeral @ A @ X ) @ ( one_one @ A ) )
= ( one_one @ A ) ) ) ).
% min_0_1(6)
thf(fact_6599_min__Suc__gt_I2_J,axiom,
! [A2: nat,B2: nat] :
( ( ord_less @ nat @ A2 @ B2 )
=> ( ( ord_min @ nat @ B2 @ ( suc @ A2 ) )
= ( suc @ A2 ) ) ) ).
% min_Suc_gt(2)
thf(fact_6600_min__Suc__gt_I1_J,axiom,
! [A2: nat,B2: nat] :
( ( ord_less @ nat @ A2 @ B2 )
=> ( ( ord_min @ nat @ ( suc @ A2 ) @ B2 )
= ( suc @ A2 ) ) ) ).
% min_Suc_gt(1)
thf(fact_6601_min__pm,axiom,
! [A2: nat,B2: nat] :
( ( plus_plus @ nat @ ( ord_min @ nat @ A2 @ B2 ) @ ( minus_minus @ nat @ A2 @ B2 ) )
= A2 ) ).
% min_pm
thf(fact_6602_min__pm1,axiom,
! [A2: nat,B2: nat] :
( ( plus_plus @ nat @ ( minus_minus @ nat @ A2 @ B2 ) @ ( ord_min @ nat @ A2 @ B2 ) )
= A2 ) ).
% min_pm1
thf(fact_6603_rev__min__pm,axiom,
! [B2: nat,A2: nat] :
( ( plus_plus @ nat @ ( ord_min @ nat @ B2 @ A2 ) @ ( minus_minus @ nat @ A2 @ B2 ) )
= A2 ) ).
% rev_min_pm
thf(fact_6604_rev__min__pm1,axiom,
! [A2: nat,B2: nat] :
( ( plus_plus @ nat @ ( minus_minus @ nat @ A2 @ B2 ) @ ( ord_min @ nat @ B2 @ A2 ) )
= A2 ) ).
% rev_min_pm1
thf(fact_6605_take__bit__of__mask,axiom,
! [A: $tType] :
( ( bit_se359711467146920520ations @ A )
=> ! [M: nat,N3: nat] :
( ( bit_se2584673776208193580ke_bit @ A @ M @ ( bit_se2239418461657761734s_mask @ A @ N3 ) )
= ( bit_se2239418461657761734s_mask @ A @ ( ord_min @ nat @ M @ N3 ) ) ) ) ).
% take_bit_of_mask
thf(fact_6606_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_6607_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_6608_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_6609_min__Suc__numeral,axiom,
! [N3: nat,K: num] :
( ( ord_min @ nat @ ( suc @ N3 ) @ ( numeral_numeral @ nat @ K ) )
= ( suc @ ( ord_min @ nat @ N3 @ ( pred_numeral @ K ) ) ) ) ).
% min_Suc_numeral
thf(fact_6610_min__numeral__Suc,axiom,
! [K: num,N3: nat] :
( ( ord_min @ nat @ ( numeral_numeral @ nat @ K ) @ ( suc @ N3 ) )
= ( suc @ ( ord_min @ nat @ ( pred_numeral @ K ) @ N3 ) ) ) ).
% min_numeral_Suc
thf(fact_6611_take__bit__word__Suc__numeral,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [N3: nat,K: num] :
( ( bit_se2584673776208193580ke_bit @ ( word @ A ) @ ( suc @ N3 ) @ ( 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 @ N3 ) ) @ ( numeral_numeral @ int @ K ) ) ) ) ) ).
% take_bit_word_Suc_numeral
thf(fact_6612_take__bit__word__numeral,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [N3: num,K: num] :
( ( bit_se2584673776208193580ke_bit @ ( word @ A ) @ ( numeral_numeral @ nat @ N3 ) @ ( 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 @ N3 ) ) @ ( numeral_numeral @ int @ K ) ) ) ) ) ).
% take_bit_word_numeral
thf(fact_6613_nat__mult__min__right,axiom,
! [M: nat,N3: nat,Q2: nat] :
( ( times_times @ nat @ M @ ( ord_min @ nat @ N3 @ Q2 ) )
= ( ord_min @ nat @ ( times_times @ nat @ M @ N3 ) @ ( times_times @ nat @ M @ Q2 ) ) ) ).
% nat_mult_min_right
thf(fact_6614_nat__mult__min__left,axiom,
! [M: nat,N3: nat,Q2: nat] :
( ( times_times @ nat @ ( ord_min @ nat @ M @ N3 ) @ Q2 )
= ( ord_min @ nat @ ( times_times @ nat @ M @ Q2 ) @ ( times_times @ nat @ N3 @ Q2 ) ) ) ).
% nat_mult_min_left
thf(fact_6615_min__add__distrib__right,axiom,
! [A: $tType] :
( ( ordere2412721322843649153imp_le @ A )
=> ! [X: A,Y: A,Z: A] :
( ( plus_plus @ A @ X @ ( ord_min @ A @ Y @ Z ) )
= ( ord_min @ A @ ( plus_plus @ A @ X @ Y ) @ ( plus_plus @ A @ X @ Z ) ) ) ) ).
% min_add_distrib_right
thf(fact_6616_min__add__distrib__left,axiom,
! [A: $tType] :
( ( ordere2412721322843649153imp_le @ A )
=> ! [X: A,Y: A,Z: A] :
( ( plus_plus @ A @ ( ord_min @ A @ X @ Y ) @ Z )
= ( ord_min @ A @ ( plus_plus @ A @ X @ Z ) @ ( plus_plus @ A @ Y @ Z ) ) ) ) ).
% min_add_distrib_left
thf(fact_6617_min__diff__distrib__left,axiom,
! [A: $tType] :
( ( ordered_ab_group_add @ A )
=> ! [X: A,Y: A,Z: A] :
( ( minus_minus @ A @ ( ord_min @ A @ X @ Y ) @ Z )
= ( ord_min @ A @ ( minus_minus @ A @ X @ Z ) @ ( minus_minus @ A @ Y @ Z ) ) ) ) ).
% min_diff_distrib_left
thf(fact_6618_min__diff,axiom,
! [M: nat,I: nat,N3: nat] :
( ( ord_min @ nat @ ( minus_minus @ nat @ M @ I ) @ ( minus_minus @ nat @ N3 @ I ) )
= ( minus_minus @ nat @ ( ord_min @ nat @ M @ N3 ) @ I ) ) ).
% min_diff
thf(fact_6619_min__less__iff__disj,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [X: A,Y: A,Z: A] :
( ( ord_less @ A @ ( ord_min @ A @ X @ Y ) @ Z )
= ( ( ord_less @ A @ X @ Z )
| ( ord_less @ A @ Y @ Z ) ) ) ) ).
% min_less_iff_disj
thf(fact_6620_min_Ostrict__boundedE,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A2: A,B2: A,C2: A] :
( ( ord_less @ A @ A2 @ ( ord_min @ A @ B2 @ C2 ) )
=> ~ ( ( ord_less @ A @ A2 @ B2 )
=> ~ ( ord_less @ A @ A2 @ C2 ) ) ) ) ).
% min.strict_boundedE
thf(fact_6621_min_Ostrict__order__iff,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ( ( ord_less @ A )
= ( ^ [A6: A,B5: A] :
( ( A6
= ( ord_min @ A @ A6 @ B5 ) )
& ( A6 != B5 ) ) ) ) ) ).
% min.strict_order_iff
thf(fact_6622_min_Ostrict__coboundedI1,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A2: A,C2: A,B2: A] :
( ( ord_less @ A @ A2 @ C2 )
=> ( ord_less @ A @ ( ord_min @ A @ A2 @ B2 ) @ C2 ) ) ) ).
% min.strict_coboundedI1
thf(fact_6623_min_Ostrict__coboundedI2,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [B2: A,C2: A,A2: A] :
( ( ord_less @ A @ B2 @ C2 )
=> ( ord_less @ A @ ( ord_min @ A @ A2 @ B2 ) @ C2 ) ) ) ).
% min.strict_coboundedI2
thf(fact_6624_of__nat__min,axiom,
! [A: $tType] :
( ( linord181362715937106298miring @ A )
=> ! [X: nat,Y: nat] :
( ( semiring_1_of_nat @ A @ ( ord_min @ nat @ X @ Y ) )
= ( ord_min @ A @ ( semiring_1_of_nat @ A @ X ) @ ( semiring_1_of_nat @ A @ Y ) ) ) ) ).
% of_nat_min
thf(fact_6625_min__def__raw,axiom,
! [A: $tType] :
( ( ord @ A )
=> ( ( ord_min @ A )
= ( ^ [A6: A,B5: A] : ( if @ A @ ( ord_less_eq @ A @ A6 @ B5 ) @ A6 @ B5 ) ) ) ) ).
% min_def_raw
thf(fact_6626_min_Omono,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A2: A,C2: A,B2: A,D: A] :
( ( ord_less_eq @ A @ A2 @ C2 )
=> ( ( ord_less_eq @ A @ B2 @ D )
=> ( ord_less_eq @ A @ ( ord_min @ A @ A2 @ B2 ) @ ( ord_min @ A @ C2 @ D ) ) ) ) ) ).
% min.mono
thf(fact_6627_min_OorderE,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A2: A,B2: A] :
( ( ord_less_eq @ A @ A2 @ B2 )
=> ( A2
= ( ord_min @ A @ A2 @ B2 ) ) ) ) ).
% min.orderE
thf(fact_6628_min_OorderI,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A2: A,B2: A] :
( ( A2
= ( ord_min @ A @ A2 @ B2 ) )
=> ( ord_less_eq @ A @ A2 @ B2 ) ) ) ).
% min.orderI
thf(fact_6629_min_OboundedE,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A2: A,B2: A,C2: A] :
( ( ord_less_eq @ A @ A2 @ ( ord_min @ A @ B2 @ C2 ) )
=> ~ ( ( ord_less_eq @ A @ A2 @ B2 )
=> ~ ( ord_less_eq @ A @ A2 @ C2 ) ) ) ) ).
% min.boundedE
thf(fact_6630_min_OboundedI,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A2: A,B2: A,C2: A] :
( ( ord_less_eq @ A @ A2 @ B2 )
=> ( ( ord_less_eq @ A @ A2 @ C2 )
=> ( ord_less_eq @ A @ A2 @ ( ord_min @ A @ B2 @ C2 ) ) ) ) ) ).
% min.boundedI
thf(fact_6631_min_Oorder__iff,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ( ( ord_less_eq @ A )
= ( ^ [A6: A,B5: A] :
( A6
= ( ord_min @ A @ A6 @ B5 ) ) ) ) ) ).
% min.order_iff
thf(fact_6632_min_Ocobounded1,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A2: A,B2: A] : ( ord_less_eq @ A @ ( ord_min @ A @ A2 @ B2 ) @ A2 ) ) ).
% min.cobounded1
thf(fact_6633_min_Ocobounded2,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A2: A,B2: A] : ( ord_less_eq @ A @ ( ord_min @ A @ A2 @ B2 ) @ B2 ) ) ).
% min.cobounded2
thf(fact_6634_min_Oabsorb__iff1,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ( ( ord_less_eq @ A )
= ( ^ [A6: A,B5: A] :
( ( ord_min @ A @ A6 @ B5 )
= A6 ) ) ) ) ).
% min.absorb_iff1
thf(fact_6635_min_Oabsorb__iff2,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ( ( ord_less_eq @ A )
= ( ^ [B5: A,A6: A] :
( ( ord_min @ A @ A6 @ B5 )
= B5 ) ) ) ) ).
% min.absorb_iff2
thf(fact_6636_min_OcoboundedI1,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A2: A,C2: A,B2: A] :
( ( ord_less_eq @ A @ A2 @ C2 )
=> ( ord_less_eq @ A @ ( ord_min @ A @ A2 @ B2 ) @ C2 ) ) ) ).
% min.coboundedI1
thf(fact_6637_min_OcoboundedI2,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [B2: A,C2: A,A2: A] :
( ( ord_less_eq @ A @ B2 @ C2 )
=> ( ord_less_eq @ A @ ( ord_min @ A @ A2 @ B2 ) @ C2 ) ) ) ).
% min.coboundedI2
thf(fact_6638_min__le__iff__disj,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [X: A,Y: A,Z: A] :
( ( ord_less_eq @ A @ ( ord_min @ A @ X @ Y ) @ Z )
= ( ( ord_less_eq @ A @ X @ Z )
| ( ord_less_eq @ A @ Y @ Z ) ) ) ) ).
% min_le_iff_disj
thf(fact_6639_min__absorb2,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [Y: A,X: A] :
( ( ord_less_eq @ A @ Y @ X )
=> ( ( ord_min @ A @ X @ Y )
= Y ) ) ) ).
% min_absorb2
thf(fact_6640_min__absorb1,axiom,
! [A: $tType] :
( ( ord @ A )
=> ! [X: A,Y: A] :
( ( ord_less_eq @ A @ X @ Y )
=> ( ( ord_min @ A @ X @ Y )
= X ) ) ) ).
% min_absorb1
thf(fact_6641_min__def,axiom,
! [A: $tType] :
( ( ord @ A )
=> ( ( ord_min @ A )
= ( ^ [A6: A,B5: A] : ( if @ A @ ( ord_less_eq @ A @ A6 @ B5 ) @ A6 @ B5 ) ) ) ) ).
% min_def
thf(fact_6642_concat__bit__assoc__sym,axiom,
! [M: nat,N3: nat,K: int,L2: int,R2: int] :
( ( bit_concat_bit @ M @ ( bit_concat_bit @ N3 @ K @ L2 ) @ R2 )
= ( bit_concat_bit @ ( ord_min @ nat @ M @ N3 ) @ K @ ( bit_concat_bit @ ( minus_minus @ nat @ M @ N3 ) @ L2 @ R2 ) ) ) ).
% concat_bit_assoc_sym
thf(fact_6643_take__bit__concat__bit__eq,axiom,
! [M: nat,N3: nat,K: int,L2: int] :
( ( bit_se2584673776208193580ke_bit @ int @ M @ ( bit_concat_bit @ N3 @ K @ L2 ) )
= ( bit_concat_bit @ ( ord_min @ nat @ M @ N3 ) @ K @ ( bit_se2584673776208193580ke_bit @ int @ ( minus_minus @ nat @ M @ N3 ) @ L2 ) ) ) ).
% take_bit_concat_bit_eq
thf(fact_6644_min__mult__distrib__right,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [P4: A,X: A,Y: A] :
( ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ P4 )
=> ( ( times_times @ A @ ( ord_min @ A @ X @ Y ) @ P4 )
= ( ord_min @ A @ ( times_times @ A @ X @ P4 ) @ ( times_times @ A @ Y @ P4 ) ) ) )
& ( ~ ( ord_less_eq @ A @ ( zero_zero @ A ) @ P4 )
=> ( ( times_times @ A @ ( ord_min @ A @ X @ Y ) @ P4 )
= ( ord_max @ A @ ( times_times @ A @ X @ P4 ) @ ( times_times @ A @ Y @ P4 ) ) ) ) ) ) ).
% min_mult_distrib_right
thf(fact_6645_max__mult__distrib__right,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [P4: A,X: A,Y: A] :
( ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ P4 )
=> ( ( times_times @ A @ ( ord_max @ A @ X @ Y ) @ P4 )
= ( ord_max @ A @ ( times_times @ A @ X @ P4 ) @ ( times_times @ A @ Y @ P4 ) ) ) )
& ( ~ ( ord_less_eq @ A @ ( zero_zero @ A ) @ P4 )
=> ( ( times_times @ A @ ( ord_max @ A @ X @ Y ) @ P4 )
= ( ord_min @ A @ ( times_times @ A @ X @ P4 ) @ ( times_times @ A @ Y @ P4 ) ) ) ) ) ) ).
% max_mult_distrib_right
thf(fact_6646_min__mult__distrib__left,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [P4: A,X: A,Y: A] :
( ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ P4 )
=> ( ( times_times @ A @ P4 @ ( ord_min @ A @ X @ Y ) )
= ( ord_min @ A @ ( times_times @ A @ P4 @ X ) @ ( times_times @ A @ P4 @ Y ) ) ) )
& ( ~ ( ord_less_eq @ A @ ( zero_zero @ A ) @ P4 )
=> ( ( times_times @ A @ P4 @ ( ord_min @ A @ X @ Y ) )
= ( ord_max @ A @ ( times_times @ A @ P4 @ X ) @ ( times_times @ A @ P4 @ Y ) ) ) ) ) ) ).
% min_mult_distrib_left
thf(fact_6647_max__mult__distrib__left,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [P4: A,X: A,Y: A] :
( ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ P4 )
=> ( ( times_times @ A @ P4 @ ( ord_max @ A @ X @ Y ) )
= ( ord_max @ A @ ( times_times @ A @ P4 @ X ) @ ( times_times @ A @ P4 @ Y ) ) ) )
& ( ~ ( ord_less_eq @ A @ ( zero_zero @ A ) @ P4 )
=> ( ( times_times @ A @ P4 @ ( ord_max @ A @ X @ Y ) )
= ( ord_min @ A @ ( times_times @ A @ P4 @ X ) @ ( times_times @ A @ P4 @ Y ) ) ) ) ) ) ).
% max_mult_distrib_left
thf(fact_6648_max__divide__distrib__right,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [P4: A,X: A,Y: A] :
( ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ P4 )
=> ( ( divide_divide @ A @ ( ord_max @ A @ X @ Y ) @ P4 )
= ( ord_max @ A @ ( divide_divide @ A @ X @ P4 ) @ ( divide_divide @ A @ Y @ P4 ) ) ) )
& ( ~ ( ord_less_eq @ A @ ( zero_zero @ A ) @ P4 )
=> ( ( divide_divide @ A @ ( ord_max @ A @ X @ Y ) @ P4 )
= ( ord_min @ A @ ( divide_divide @ A @ X @ P4 ) @ ( divide_divide @ A @ Y @ P4 ) ) ) ) ) ) ).
% max_divide_distrib_right
thf(fact_6649_min__divide__distrib__right,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [P4: A,X: A,Y: A] :
( ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ P4 )
=> ( ( divide_divide @ A @ ( ord_min @ A @ X @ Y ) @ P4 )
= ( ord_min @ A @ ( divide_divide @ A @ X @ P4 ) @ ( divide_divide @ A @ Y @ P4 ) ) ) )
& ( ~ ( ord_less_eq @ A @ ( zero_zero @ A ) @ P4 )
=> ( ( divide_divide @ A @ ( ord_min @ A @ X @ Y ) @ P4 )
= ( ord_max @ A @ ( divide_divide @ A @ X @ P4 ) @ ( divide_divide @ A @ Y @ P4 ) ) ) ) ) ) ).
% min_divide_distrib_right
thf(fact_6650_mod__exp__eq,axiom,
! [A: $tType] :
( ( bit_semiring_bits @ A )
=> ! [A2: A,M: nat,N3: nat] :
( ( modulo_modulo @ A @ ( modulo_modulo @ A @ A2 @ ( power_power @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ M ) ) @ ( power_power @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ N3 ) )
= ( modulo_modulo @ A @ A2 @ ( power_power @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ ( ord_min @ nat @ M @ N3 ) ) ) ) ) ).
% mod_exp_eq
thf(fact_6651_mod__mod__power,axiom,
! [K: nat,M: nat,N3: 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 ) ) @ N3 ) )
= ( modulo_modulo @ nat @ K @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( ord_min @ nat @ M @ N3 ) ) ) ) ).
% mod_mod_power
thf(fact_6652_Word_Obit__mask__iff,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [M: nat,N3: nat] :
( ( bit_se5641148757651400278ts_bit @ ( word @ A ) @ ( bit_se2239418461657761734s_mask @ ( word @ A ) @ M ) @ N3 )
= ( ord_less @ nat @ N3 @ ( ord_min @ nat @ ( type_len0_len_of @ A @ ( type2 @ A ) ) @ M ) ) ) ) ).
% Word.bit_mask_iff
thf(fact_6653_mask__mod__exp,axiom,
! [A: $tType] :
( ( euclid5411537665997757685th_nat @ A )
=> ! [N3: nat,M: nat] :
( ( modulo_modulo @ A @ ( minus_minus @ A @ ( power_power @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ N3 ) @ ( 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 @ N3 ) ) @ ( one_one @ A ) ) ) ) ).
% mask_mod_exp
thf(fact_6654_bit__slice__iff,axiom,
! [B: $tType,A: $tType] :
( ( ( type_len @ A )
& ( type_len @ B ) )
=> ! [M: nat,W: word @ A,N3: nat] :
( ( bit_se5641148757651400278ts_bit @ ( word @ B ) @ ( slice2 @ A @ B @ M @ W ) @ N3 )
= ( ( ord_less @ nat @ N3 @ ( 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 @ N3 @ ( type_len0_len_of @ A @ ( type2 @ A ) ) ) @ ( minus_minus @ nat @ ( type_len0_len_of @ A @ ( type2 @ A ) ) @ M ) ) ) ) ) ) ).
% bit_slice_iff
thf(fact_6655_bit__slice1__iff,axiom,
! [B: $tType,A: $tType] :
( ( ( type_len @ A )
& ( type_len @ B ) )
=> ! [M: nat,W: word @ A,N3: nat] :
( ( bit_se5641148757651400278ts_bit @ ( word @ B ) @ ( slice1 @ A @ B @ M @ W ) @ N3 )
= ( ( ord_less_eq @ nat @ ( minus_minus @ nat @ M @ ( type_len0_len_of @ A @ ( type2 @ A ) ) ) @ N3 )
& ( ord_less @ nat @ N3 @ ( ord_min @ nat @ ( type_len0_len_of @ B @ ( type2 @ B ) ) @ M ) )
& ( bit_se5641148757651400278ts_bit @ ( word @ A ) @ W @ ( minus_minus @ nat @ ( plus_plus @ nat @ N3 @ ( 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_6656_unat__mask,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [N3: nat] :
( ( semiring_1_unsigned @ A @ nat @ ( bit_se2239418461657761734s_mask @ ( word @ A ) @ N3 ) )
= ( minus_minus @ nat @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( ord_min @ nat @ N3 @ ( type_len0_len_of @ A @ ( type2 @ A ) ) ) ) @ ( one_one @ nat ) ) ) ) ).
% unat_mask
thf(fact_6657_bit__horner__sum__bit__word__iff,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [Bs: list @ $o,N3: 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 ) @ N3 )
= ( ( ord_less @ nat @ N3 @ ( ord_min @ nat @ ( type_len0_len_of @ A @ ( type2 @ A ) ) @ ( size_size @ ( list @ $o ) @ Bs ) ) )
& ( nth @ $o @ Bs @ N3 ) ) ) ) ).
% bit_horner_sum_bit_word_iff
thf(fact_6658_drop__bit__exp__eq,axiom,
! [A: $tType] :
( ( bit_se359711467146920520ations @ A )
=> ! [M: nat,N3: nat] :
( ( bit_se4197421643247451524op_bit @ A @ M @ ( power_power @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ N3 ) )
= ( times_times @ A
@ ( zero_neq_one_of_bool @ A
@ ( ( ord_less_eq @ nat @ M @ N3 )
& ( bit_se6407376104438227557le_bit @ A @ ( type2 @ A ) @ N3 ) ) )
@ ( power_power @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ ( minus_minus @ nat @ N3 @ M ) ) ) ) ) ).
% drop_bit_exp_eq
thf(fact_6659_bit__minus__2__iff,axiom,
! [A: $tType] :
( ( bit_ri3973907225187159222ations @ A )
=> ! [N3: nat] :
( ( bit_se5641148757651400278ts_bit @ A @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) @ N3 )
= ( ( bit_se6407376104438227557le_bit @ A @ ( type2 @ A ) @ N3 )
& ( ord_less @ nat @ ( zero_zero @ nat ) @ N3 ) ) ) ) ).
% bit_minus_2_iff
thf(fact_6660_possible__bit__min,axiom,
! [A: $tType] :
( ( bit_semiring_bits @ A )
=> ! [Tyrep: itself @ A,I: nat,J: nat] :
( ( bit_se6407376104438227557le_bit @ A @ Tyrep @ ( ord_min @ nat @ I @ J ) )
= ( ( bit_se6407376104438227557le_bit @ A @ Tyrep @ I )
| ( bit_se6407376104438227557le_bit @ A @ Tyrep @ J ) ) ) ) ).
% possible_bit_min
thf(fact_6661_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_6662_bit__minus__1__iff,axiom,
! [A: $tType] :
( ( bit_ri3973907225187159222ations @ A )
=> ! [N3: nat] :
( ( bit_se5641148757651400278ts_bit @ A @ ( uminus_uminus @ A @ ( one_one @ A ) ) @ N3 )
= ( bit_se6407376104438227557le_bit @ A @ ( type2 @ A ) @ N3 ) ) ) ).
% bit_minus_1_iff
thf(fact_6663_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_6664_possible__bit__0,axiom,
! [A: $tType] :
( ( bit_semiring_bits @ A )
=> ! [Ty: itself @ A] : ( bit_se6407376104438227557le_bit @ A @ Ty @ ( zero_zero @ nat ) ) ) ).
% possible_bit_0
thf(fact_6665_bit__eqI,axiom,
! [A: $tType] :
( ( bit_semiring_bits @ A )
=> ! [A2: A,B2: A] :
( ! [N: nat] :
( ( bit_se6407376104438227557le_bit @ A @ ( type2 @ A ) @ N )
=> ( ( bit_se5641148757651400278ts_bit @ A @ A2 @ N )
= ( bit_se5641148757651400278ts_bit @ A @ B2 @ N ) ) )
=> ( A2 = B2 ) ) ) ).
% bit_eqI
thf(fact_6666_bit__eq__iff,axiom,
! [A: $tType] :
( ( bit_semiring_bits @ A )
=> ( ( ^ [Y5: A,Z5: A] : Y5 = Z5 )
= ( ^ [A6: A,B5: A] :
! [N2: nat] :
( ( bit_se6407376104438227557le_bit @ A @ ( type2 @ A ) @ N2 )
=> ( ( bit_se5641148757651400278ts_bit @ A @ A6 @ N2 )
= ( bit_se5641148757651400278ts_bit @ A @ B5 @ N2 ) ) ) ) ) ) ).
% bit_eq_iff
thf(fact_6667_impossible__bit,axiom,
! [A: $tType] :
( ( bit_semiring_bits @ A )
=> ! [N3: nat,A2: A] :
( ~ ( bit_se6407376104438227557le_bit @ A @ ( type2 @ A ) @ N3 )
=> ~ ( bit_se5641148757651400278ts_bit @ A @ A2 @ N3 ) ) ) ).
% impossible_bit
thf(fact_6668_bit__imp__possible__bit,axiom,
! [A: $tType] :
( ( bit_semiring_bits @ A )
=> ! [A2: A,N3: nat] :
( ( bit_se5641148757651400278ts_bit @ A @ A2 @ N3 )
=> ( bit_se6407376104438227557le_bit @ A @ ( type2 @ A ) @ N3 ) ) ) ).
% bit_imp_possible_bit
thf(fact_6669_bit__not__iff,axiom,
! [A: $tType] :
( ( bit_ri3973907225187159222ations @ A )
=> ! [A2: A,N3: nat] :
( ( bit_se5641148757651400278ts_bit @ A @ ( bit_ri4277139882892585799ns_not @ A @ A2 ) @ N3 )
= ( ~ ~ ( bit_se6407376104438227557le_bit @ A @ ( type2 @ A ) @ N3 )
& ~ ( bit_se5641148757651400278ts_bit @ A @ A2 @ N3 ) ) ) ) ).
% bit_not_iff
thf(fact_6670_semiring__bit__operations__class_Obit__set__bit__iff,axiom,
! [A: $tType] :
( ( bit_se359711467146920520ations @ A )
=> ! [M: nat,A2: A,N3: nat] :
( ( bit_se5641148757651400278ts_bit @ A @ ( bit_se5668285175392031749et_bit @ A @ M @ A2 ) @ N3 )
= ( ( bit_se5641148757651400278ts_bit @ A @ A2 @ N3 )
| ( ( M = N3 )
& ( bit_se6407376104438227557le_bit @ A @ ( type2 @ A ) @ N3 ) ) ) ) ) ).
% semiring_bit_operations_class.bit_set_bit_iff
thf(fact_6671_bit__flip__bit__iff,axiom,
! [A: $tType] :
( ( bit_se359711467146920520ations @ A )
=> ! [M: nat,A2: A,N3: nat] :
( ( bit_se5641148757651400278ts_bit @ A @ ( bit_se8732182000553998342ip_bit @ A @ M @ A2 ) @ N3 )
= ( ( ( M = N3 )
= ( ~ ( bit_se5641148757651400278ts_bit @ A @ A2 @ N3 ) ) )
& ( bit_se6407376104438227557le_bit @ A @ ( type2 @ A ) @ N3 ) ) ) ) ).
% bit_flip_bit_iff
thf(fact_6672_bit__mask__iff,axiom,
! [A: $tType] :
( ( bit_se359711467146920520ations @ A )
=> ! [M: nat,N3: nat] :
( ( bit_se5641148757651400278ts_bit @ A @ ( bit_se2239418461657761734s_mask @ A @ M ) @ N3 )
= ( ( bit_se6407376104438227557le_bit @ A @ ( type2 @ A ) @ N3 )
& ( ord_less @ nat @ N3 @ M ) ) ) ) ).
% bit_mask_iff
thf(fact_6673_bit__of__int__iff,axiom,
! [A: $tType] :
( ( bit_ri3973907225187159222ations @ A )
=> ! [K: int,N3: nat] :
( ( bit_se5641148757651400278ts_bit @ A @ ( ring_1_of_int @ A @ K ) @ N3 )
= ( ( bit_se6407376104438227557le_bit @ A @ ( type2 @ A ) @ N3 )
& ( bit_se5641148757651400278ts_bit @ int @ K @ N3 ) ) ) ) ).
% bit_of_int_iff
thf(fact_6674_bit__signed__take__bit__iff,axiom,
! [A: $tType] :
( ( bit_ri3973907225187159222ations @ A )
=> ! [M: nat,A2: A,N3: nat] :
( ( bit_se5641148757651400278ts_bit @ A @ ( bit_ri4674362597316999326ke_bit @ A @ M @ A2 ) @ N3 )
= ( ( bit_se6407376104438227557le_bit @ A @ ( type2 @ A ) @ N3 )
& ( bit_se5641148757651400278ts_bit @ A @ A2 @ ( ord_min @ nat @ M @ N3 ) ) ) ) ) ).
% bit_signed_take_bit_iff
thf(fact_6675_bit__of__nat__iff,axiom,
! [A: $tType] :
( ( bit_semiring_bits @ A )
=> ! [M: nat,N3: nat] :
( ( bit_se5641148757651400278ts_bit @ A @ ( semiring_1_of_nat @ A @ M ) @ N3 )
= ( ( bit_se6407376104438227557le_bit @ A @ ( type2 @ A ) @ N3 )
& ( bit_se5641148757651400278ts_bit @ nat @ M @ N3 ) ) ) ) ).
% bit_of_nat_iff
thf(fact_6676_possible__bit__def,axiom,
! [A: $tType] :
( ( bit_semiring_bits @ A )
=> ( ( bit_se6407376104438227557le_bit @ A )
= ( ^ [Tyrep2: itself @ A,N2: nat] :
( ( power_power @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ N2 )
!= ( zero_zero @ A ) ) ) ) ) ).
% possible_bit_def
thf(fact_6677_bit__minus__iff,axiom,
! [A: $tType] :
( ( bit_ri3973907225187159222ations @ A )
=> ! [A2: A,N3: nat] :
( ( bit_se5641148757651400278ts_bit @ A @ ( uminus_uminus @ A @ A2 ) @ N3 )
= ( ( bit_se6407376104438227557le_bit @ A @ ( type2 @ A ) @ N3 )
& ~ ( bit_se5641148757651400278ts_bit @ A @ ( minus_minus @ A @ A2 @ ( one_one @ A ) ) @ N3 ) ) ) ) ).
% bit_minus_iff
thf(fact_6678_bit__twiddle__min,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [Y: word @ A,X: word @ A] :
( ( bit_se5824344971392196577ns_xor @ ( word @ A ) @ Y @ ( bit_se5824344872417868541ns_and @ ( word @ A ) @ ( bit_se5824344971392196577ns_xor @ ( word @ A ) @ X @ Y ) @ ( if @ ( word @ A ) @ ( ord_less @ ( word @ A ) @ X @ Y ) @ ( uminus_uminus @ ( word @ A ) @ ( one_one @ ( word @ A ) ) ) @ ( zero_zero @ ( word @ A ) ) ) ) )
= ( ord_min @ ( word @ A ) @ X @ Y ) ) ) ).
% bit_twiddle_min
thf(fact_6679_bit__push__bit__iff,axiom,
! [A: $tType] :
( ( bit_se359711467146920520ations @ A )
=> ! [M: nat,A2: A,N3: nat] :
( ( bit_se5641148757651400278ts_bit @ A @ ( bit_se4730199178511100633sh_bit @ A @ M @ A2 ) @ N3 )
= ( ( ord_less_eq @ nat @ M @ N3 )
& ( bit_se6407376104438227557le_bit @ A @ ( type2 @ A ) @ N3 )
& ( bit_se5641148757651400278ts_bit @ A @ A2 @ ( minus_minus @ nat @ N3 @ M ) ) ) ) ) ).
% bit_push_bit_iff
thf(fact_6680_fold__possible__bit,axiom,
! [A: $tType] :
( ( bit_semiring_bits @ A )
=> ! [N3: nat] :
( ( ( power_power @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ N3 )
= ( zero_zero @ A ) )
= ( ~ ( bit_se6407376104438227557le_bit @ A @ ( type2 @ A ) @ N3 ) ) ) ) ).
% fold_possible_bit
thf(fact_6681_bit__exp__iff,axiom,
! [A: $tType] :
( ( bit_semiring_bits @ A )
=> ! [M: nat,N3: nat] :
( ( bit_se5641148757651400278ts_bit @ A @ ( power_power @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ M ) @ N3 )
= ( ( bit_se6407376104438227557le_bit @ A @ ( type2 @ A ) @ N3 )
& ( M = N3 ) ) ) ) ).
% bit_exp_iff
thf(fact_6682_bit__2__iff,axiom,
! [A: $tType] :
( ( bit_semiring_bits @ A )
=> ! [N3: nat] :
( ( bit_se5641148757651400278ts_bit @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ N3 )
= ( ( bit_se6407376104438227557le_bit @ A @ ( type2 @ A ) @ ( one_one @ nat ) )
& ( N3
= ( one_one @ nat ) ) ) ) ) ).
% bit_2_iff
thf(fact_6683_bit__not__exp__iff,axiom,
! [A: $tType] :
( ( bit_ri3973907225187159222ations @ A )
=> ! [M: nat,N3: nat] :
( ( bit_se5641148757651400278ts_bit @ A @ ( bit_ri4277139882892585799ns_not @ A @ ( power_power @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ M ) ) @ N3 )
= ( ( bit_se6407376104438227557le_bit @ A @ ( type2 @ A ) @ N3 )
& ( N3 != M ) ) ) ) ).
% bit_not_exp_iff
thf(fact_6684_bit__minus__exp__iff,axiom,
! [A: $tType] :
( ( bit_ri3973907225187159222ations @ A )
=> ! [M: nat,N3: nat] :
( ( bit_se5641148757651400278ts_bit @ A @ ( uminus_uminus @ A @ ( power_power @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ M ) ) @ N3 )
= ( ( bit_se6407376104438227557le_bit @ A @ ( type2 @ A ) @ N3 )
& ( ord_less_eq @ nat @ M @ N3 ) ) ) ) ).
% bit_minus_exp_iff
thf(fact_6685_bit__mask__sub__iff,axiom,
! [A: $tType] :
( ( bit_semiring_bits @ A )
=> ! [M: nat,N3: nat] :
( ( bit_se5641148757651400278ts_bit @ A @ ( minus_minus @ A @ ( power_power @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ M ) @ ( one_one @ A ) ) @ N3 )
= ( ( bit_se6407376104438227557le_bit @ A @ ( type2 @ A ) @ N3 )
& ( ord_less @ nat @ N3 @ M ) ) ) ) ).
% bit_mask_sub_iff
thf(fact_6686_bit__double__iff,axiom,
! [A: $tType] :
( ( bit_semiring_bits @ A )
=> ! [A2: A,N3: nat] :
( ( bit_se5641148757651400278ts_bit @ A @ ( times_times @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ A2 ) @ N3 )
= ( ( bit_se5641148757651400278ts_bit @ A @ A2 @ ( minus_minus @ nat @ N3 @ ( one_one @ nat ) ) )
& ( N3
!= ( zero_zero @ nat ) )
& ( bit_se6407376104438227557le_bit @ A @ ( type2 @ A ) @ N3 ) ) ) ) ).
% bit_double_iff
thf(fact_6687_bit__signed__iff,axiom,
! [A: $tType,B: $tType] :
( ( ( type_len @ B )
& ( bit_ri3973907225187159222ations @ A ) )
=> ! [W: word @ B,N3: nat] :
( ( bit_se5641148757651400278ts_bit @ A @ ( ring_1_signed @ B @ A @ W ) @ N3 )
= ( ( bit_se6407376104438227557le_bit @ A @ ( type2 @ A ) @ N3 )
& ( bit_se5641148757651400278ts_bit @ ( word @ B ) @ W @ ( ord_min @ nat @ ( minus_minus @ nat @ ( type_len0_len_of @ B @ ( type2 @ B ) ) @ ( suc @ ( zero_zero @ nat ) ) ) @ N3 ) ) ) ) ) ).
% bit_signed_iff
thf(fact_6688_word__set__bits__def,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ( ( bit_bi4170147762399595738t_bits @ ( word @ A ) )
= ( ^ [P3: nat > $o] : ( groups4207007520872428315er_sum @ $o @ ( word @ A ) @ ( zero_neq_one_of_bool @ ( word @ A ) ) @ ( numeral_numeral @ ( word @ A ) @ ( bit0 @ one2 ) ) @ ( map @ nat @ $o @ P3 @ ( upt @ ( zero_zero @ nat ) @ ( type_len0_len_of @ A @ ( type2 @ A ) ) ) ) ) ) ) ) ).
% word_set_bits_def
thf(fact_6689_uint__word__rotr__eq,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [N3: nat,W: word @ A] :
( ( semiring_1_unsigned @ A @ int @ ( word_rotr @ A @ N3 @ W ) )
= ( bit_concat_bit @ ( minus_minus @ nat @ ( type_len0_len_of @ A @ ( type2 @ A ) ) @ ( modulo_modulo @ nat @ N3 @ ( type_len0_len_of @ A @ ( type2 @ A ) ) ) ) @ ( bit_se4197421643247451524op_bit @ int @ ( modulo_modulo @ nat @ N3 @ ( 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 @ N3 @ ( type_len0_len_of @ A @ ( type2 @ A ) ) ) @ W ) ) ) ) ) ).
% uint_word_rotr_eq
thf(fact_6690_word__rotr__word__rotr__eq,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [M: nat,N3: nat,W: word @ A] :
( ( word_rotr @ A @ M @ ( word_rotr @ A @ N3 @ W ) )
= ( word_rotr @ A @ ( plus_plus @ nat @ M @ N3 ) @ W ) ) ) ).
% word_rotr_word_rotr_eq
thf(fact_6691_bit__set__bits__word__iff,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [P: nat > $o,N3: nat] :
( ( bit_se5641148757651400278ts_bit @ ( word @ A ) @ ( bit_bi4170147762399595738t_bits @ ( word @ A ) @ P ) @ N3 )
= ( ( ord_less @ nat @ N3 @ ( type_len0_len_of @ A @ ( type2 @ A ) ) )
& ( P @ N3 ) ) ) ) ).
% bit_set_bits_word_iff
thf(fact_6692_word__test__bit__set__bits,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [F2: nat > $o,N3: nat] :
( ( bit_se5641148757651400278ts_bit @ ( word @ A ) @ ( bit_bi4170147762399595738t_bits @ ( word @ A ) @ F2 ) @ N3 )
= ( ( ord_less @ nat @ N3 @ ( type_len0_len_of @ A @ ( type2 @ A ) ) )
& ( F2 @ N3 ) ) ) ) ).
% word_test_bit_set_bits
thf(fact_6693_bit__word__rotr__iff,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [M: nat,W: word @ A,N3: nat] :
( ( bit_se5641148757651400278ts_bit @ ( word @ A ) @ ( word_rotr @ A @ M @ W ) @ N3 )
= ( ( ord_less @ nat @ N3 @ ( type_len0_len_of @ A @ ( type2 @ A ) ) )
& ( bit_se5641148757651400278ts_bit @ ( word @ A ) @ W @ ( modulo_modulo @ nat @ ( plus_plus @ nat @ N3 @ M ) @ ( type_len0_len_of @ A @ ( type2 @ A ) ) ) ) ) ) ) ).
% bit_word_rotr_iff
thf(fact_6694_word__roti__eq__word__rotr__word__rotl,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ( ( word_roti @ A )
= ( ^ [I3: int,W2: word @ A] : ( if @ ( word @ A ) @ ( ord_less_eq @ int @ ( zero_zero @ int ) @ I3 ) @ ( word_rotr @ A @ ( nat2 @ I3 ) @ W2 ) @ ( word_rotl @ A @ ( nat2 @ ( uminus_uminus @ int @ I3 ) ) @ W2 ) ) ) ) ) ).
% word_roti_eq_word_rotr_word_rotl
thf(fact_6695_word__roti_Oabs__eq,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [Xa: int,X: int] :
( ( word_roti @ A @ Xa @ ( word2 @ A @ X ) )
= ( 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 ) ) @ X ) ) @ ( bit_se2584673776208193580ke_bit @ int @ ( nat2 @ ( modulo_modulo @ int @ Xa @ ( semiring_1_of_nat @ int @ ( type_len0_len_of @ A @ ( type2 @ A ) ) ) ) ) @ X ) ) ) ) ) ).
% word_roti.abs_eq
thf(fact_6696_one__word__def,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ( ( one_one @ ( word @ A ) )
= ( word2 @ A @ ( one_one @ int ) ) ) ) ).
% one_word_def
thf(fact_6697_minus__word_Oabs__eq,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [Xa: int,X: int] :
( ( minus_minus @ ( word @ A ) @ ( word2 @ A @ Xa ) @ ( word2 @ A @ X ) )
= ( word2 @ A @ ( minus_minus @ int @ Xa @ X ) ) ) ) ).
% minus_word.abs_eq
thf(fact_6698_size__word_Oabs__eq,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [X: int] :
( ( size_size @ ( word @ A ) @ ( word2 @ A @ X ) )
= ( type_len0_len_of @ A @ ( type2 @ A ) ) ) ) ).
% size_word.abs_eq
thf(fact_6699_word__succ_Oabs__eq,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [X: int] :
( ( word_succ @ A @ ( word2 @ A @ X ) )
= ( word2 @ A @ ( plus_plus @ int @ X @ ( one_one @ int ) ) ) ) ) ).
% word_succ.abs_eq
thf(fact_6700_word__pred_Oabs__eq,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [X: int] :
( ( word_pred @ A @ ( word2 @ A @ X ) )
= ( word2 @ A @ ( minus_minus @ int @ X @ ( one_one @ int ) ) ) ) ) ).
% word_pred.abs_eq
thf(fact_6701_word__rotl_Oabs__eq,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [Xa: nat,X: int] :
( ( word_rotl @ A @ Xa @ ( word2 @ A @ X ) )
= ( 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 ) ) @ X ) ) @ ( 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 ) ) ) ) @ X ) ) ) ) ) ).
% word_rotl.abs_eq
thf(fact_6702_less__eq__word_Oabs__eq,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [Xa: int,X: int] :
( ( ord_less_eq @ ( word @ A ) @ ( word2 @ A @ Xa ) @ ( word2 @ A @ X ) )
= ( 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 ) ) @ X ) ) ) ) ).
% less_eq_word.abs_eq
thf(fact_6703_bit__word_Oabs__eq,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [X: int] :
( ( bit_se5641148757651400278ts_bit @ ( word @ A ) @ ( word2 @ A @ X ) )
= ( ^ [N2: nat] :
( ( ord_less @ nat @ N2 @ ( type_len0_len_of @ A @ ( type2 @ A ) ) )
& ( bit_se5641148757651400278ts_bit @ int @ X @ N2 ) ) ) ) ) ).
% bit_word.abs_eq
thf(fact_6704_word__rotl__eq__word__rotr,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ( ( word_rotl @ A )
= ( ^ [N2: nat] : ( word_rotr @ A @ ( minus_minus @ nat @ ( type_len0_len_of @ A @ ( type2 @ A ) ) @ ( modulo_modulo @ nat @ N2 @ ( type_len0_len_of @ A @ ( type2 @ A ) ) ) ) ) ) ) ) ).
% word_rotl_eq_word_rotr
thf(fact_6705_word__sle_Oabs__eq,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [Xa: int,X: int] :
( ( word_sle @ A @ ( word2 @ A @ Xa ) @ ( word2 @ A @ X ) )
= ( ord_less_eq @ int @ ( bit_ri4674362597316999326ke_bit @ int @ ( minus_minus @ nat @ ( type_len0_len_of @ A @ ( type2 @ A ) ) @ ( suc @ ( zero_zero @ nat ) ) ) @ Xa ) @ ( bit_ri4674362597316999326ke_bit @ int @ ( minus_minus @ nat @ ( type_len0_len_of @ A @ ( type2 @ A ) ) @ ( suc @ ( zero_zero @ nat ) ) ) @ X ) ) ) ) ).
% word_sle.abs_eq
thf(fact_6706_word__sless_Oabs__eq,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [Xa: int,X: int] :
( ( word_sless @ A @ ( word2 @ A @ Xa ) @ ( word2 @ A @ X ) )
= ( 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 ) ) ) @ X ) ) ) ) ).
% word_sless.abs_eq
thf(fact_6707_signed_Oabs__eq,axiom,
! [A: $tType,B: $tType] :
( ( ( type_len @ B )
& ( ring_1 @ A ) )
=> ! [X: int] :
( ( ring_1_signed @ B @ A @ ( word2 @ B @ X ) )
= ( ring_1_of_int @ A @ ( bit_ri4674362597316999326ke_bit @ int @ ( minus_minus @ nat @ ( type_len0_len_of @ B @ ( type2 @ B ) ) @ ( suc @ ( zero_zero @ nat ) ) ) @ X ) ) ) ) ).
% signed.abs_eq
thf(fact_6708_signed__drop__bit_Oabs__eq,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [Xa: nat,X: int] :
( ( signed_drop_bit @ A @ Xa @ ( word2 @ A @ X ) )
= ( 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 ) ) ) @ X ) ) ) ) ) ).
% signed_drop_bit.abs_eq
thf(fact_6709_signed__modulo__word_Oabs__eq,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [Xa: int,X: int] :
( ( signed6721504322012087516modulo @ ( word @ A ) @ ( word2 @ A @ Xa ) @ ( word2 @ A @ X ) )
= ( word2 @ A @ ( signed6721504322012087516modulo @ int @ ( bit_ri4674362597316999326ke_bit @ int @ ( minus_minus @ nat @ ( type_len0_len_of @ A @ ( type2 @ A ) ) @ ( suc @ ( zero_zero @ nat ) ) ) @ Xa ) @ ( bit_ri4674362597316999326ke_bit @ int @ ( minus_minus @ nat @ ( type_len0_len_of @ A @ ( type2 @ A ) ) @ ( suc @ ( zero_zero @ nat ) ) ) @ X ) ) ) ) ) ).
% signed_modulo_word.abs_eq
thf(fact_6710_signed__divide__word_Oabs__eq,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [Xa: int,X: int] :
( ( signed7115095781618012415divide @ ( word @ A ) @ ( word2 @ A @ Xa ) @ ( word2 @ A @ X ) )
= ( word2 @ A @ ( signed7115095781618012415divide @ int @ ( bit_ri4674362597316999326ke_bit @ int @ ( minus_minus @ nat @ ( type_len0_len_of @ A @ ( type2 @ A ) ) @ ( suc @ ( zero_zero @ nat ) ) ) @ Xa ) @ ( bit_ri4674362597316999326ke_bit @ int @ ( minus_minus @ nat @ ( type_len0_len_of @ A @ ( type2 @ A ) ) @ ( suc @ ( zero_zero @ nat ) ) ) @ X ) ) ) ) ) ).
% signed_divide_word.abs_eq
thf(fact_6711_word__rotr_Oabs__eq,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [Xa: nat,X: int] :
( ( word_rotr @ A @ Xa @ ( word2 @ A @ X ) )
= ( 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 ) ) @ X ) ) @ ( bit_se2584673776208193580ke_bit @ int @ ( modulo_modulo @ nat @ Xa @ ( type_len0_len_of @ A @ ( type2 @ A ) ) ) @ X ) ) ) ) ) ).
% word_rotr.abs_eq
thf(fact_6712_bit__word__rotl__iff,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [M: nat,W: word @ A,N3: nat] :
( ( bit_se5641148757651400278ts_bit @ ( word @ A ) @ ( word_rotl @ A @ M @ W ) @ N3 )
= ( ( ord_less @ nat @ N3 @ ( type_len0_len_of @ A @ ( type2 @ A ) ) )
& ( bit_se5641148757651400278ts_bit @ ( word @ A ) @ W @ ( modulo_modulo @ nat @ ( plus_plus @ nat @ N3 @ ( 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_6713_signed__cast_Oabs__eq,axiom,
! [B: $tType,A: $tType] :
( ( ( type_len @ A )
& ( type_len @ B ) )
=> ! [X: int] :
( ( signed_cast @ A @ B @ ( word2 @ A @ X ) )
= ( word2 @ B @ ( bit_ri4674362597316999326ke_bit @ int @ ( minus_minus @ nat @ ( type_len0_len_of @ A @ ( type2 @ A ) ) @ ( suc @ ( zero_zero @ nat ) ) ) @ X ) ) ) ) ).
% signed_cast.abs_eq
thf(fact_6714_the__signed__int_Oabs__eq,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [X: int] :
( ( the_signed_int @ A @ ( word2 @ A @ X ) )
= ( bit_ri4674362597316999326ke_bit @ int @ ( minus_minus @ nat @ ( type_len0_len_of @ A @ ( type2 @ A ) ) @ ( suc @ ( zero_zero @ nat ) ) ) @ X ) ) ) ).
% the_signed_int.abs_eq
thf(fact_6715_CHAR__eq0__iff,axiom,
! [A: $tType] :
( ( semiring_1 @ A )
=> ( ( ( semiri4206861660011772517g_char @ A @ ( type2 @ A ) )
= ( zero_zero @ nat ) )
= ( ! [N2: nat] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ N2 )
=> ( ( semiring_1_of_nat @ A @ N2 )
!= ( zero_zero @ A ) ) ) ) ) ) ).
% CHAR_eq0_iff
thf(fact_6716_CHAR__eq__posI,axiom,
! [A: $tType] :
( ( semiring_1 @ A )
=> ! [C2: nat] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ C2 )
=> ( ( ( semiring_1_of_nat @ A @ C2 )
= ( zero_zero @ A ) )
=> ( ! [X3: nat] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ X3 )
=> ( ( ord_less @ nat @ X3 @ C2 )
=> ( ( semiring_1_of_nat @ A @ X3 )
!= ( zero_zero @ A ) ) ) )
=> ( ( semiri4206861660011772517g_char @ A @ ( type2 @ A ) )
= C2 ) ) ) ) ) ).
% CHAR_eq_posI
thf(fact_6717_CHAR__pos__iff,axiom,
! [A: $tType] :
( ( semiring_1 @ A )
=> ( ( ord_less @ nat @ ( zero_zero @ nat ) @ ( semiri4206861660011772517g_char @ A @ ( type2 @ A ) ) )
= ( ? [N2: nat] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ N2 )
& ( ( semiring_1_of_nat @ A @ N2 )
= ( zero_zero @ A ) ) ) ) ) ) ).
% CHAR_pos_iff
thf(fact_6718_bit__sshiftr__iff,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [W: word @ A,M: nat,N3: nat] :
( ( bit_se5641148757651400278ts_bit @ ( word @ A ) @ ( bit_Sh8784991116023147202shiftr @ A @ W @ M ) @ N3 )
= ( bit_se5641148757651400278ts_bit @ ( word @ A ) @ W
@ ( if @ nat
@ ( ( ord_less_eq @ nat @ ( minus_minus @ nat @ ( type_len0_len_of @ A @ ( type2 @ A ) ) @ M ) @ N3 )
& ( ord_less @ nat @ N3 @ ( type_len0_len_of @ A @ ( type2 @ A ) ) ) )
@ ( minus_minus @ nat @ ( type_len0_len_of @ A @ ( type2 @ A ) ) @ ( one_one @ nat ) )
@ ( plus_plus @ nat @ M @ N3 ) ) ) ) ) ).
% bit_sshiftr_iff
thf(fact_6719_sshiftr__1,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [N3: nat] :
( ( bit_Sh8784991116023147202shiftr @ A @ ( one_one @ ( word @ A ) ) @ N3 )
= ( zero_neq_one_of_bool @ ( word @ A )
@ ( ( ( type_len0_len_of @ A @ ( type2 @ A ) )
= ( one_one @ nat ) )
| ( N3
= ( zero_zero @ nat ) ) ) ) ) ) ).
% sshiftr_1
thf(fact_6720_sshiftr__numeral__Suc,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [M: num,N3: nat] :
( ( bit_Sh8784991116023147202shiftr @ A @ ( numeral_numeral @ ( word @ A ) @ M ) @ ( suc @ N3 ) )
= ( signed_drop_bit @ A @ ( suc @ N3 ) @ ( numeral_numeral @ ( word @ A ) @ M ) ) ) ) ).
% sshiftr_numeral_Suc
thf(fact_6721_sshiftr__numeral__numeral,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [M: num,N3: num] :
( ( bit_Sh8784991116023147202shiftr @ A @ ( numeral_numeral @ ( word @ A ) @ M ) @ ( numeral_numeral @ nat @ N3 ) )
= ( signed_drop_bit @ A @ ( numeral_numeral @ nat @ N3 ) @ ( numeral_numeral @ ( word @ A ) @ M ) ) ) ) ).
% sshiftr_numeral_numeral
thf(fact_6722_sshiftr__minus__numeral__Suc,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [M: num,N3: nat] :
( ( bit_Sh8784991116023147202shiftr @ A @ ( uminus_uminus @ ( word @ A ) @ ( numeral_numeral @ ( word @ A ) @ M ) ) @ ( suc @ N3 ) )
= ( signed_drop_bit @ A @ ( suc @ N3 ) @ ( uminus_uminus @ ( word @ A ) @ ( numeral_numeral @ ( word @ A ) @ M ) ) ) ) ) ).
% sshiftr_minus_numeral_Suc
thf(fact_6723_sshiftr__minus__numeral__numeral,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [M: num,N3: num] :
( ( bit_Sh8784991116023147202shiftr @ A @ ( uminus_uminus @ ( word @ A ) @ ( numeral_numeral @ ( word @ A ) @ M ) ) @ ( numeral_numeral @ nat @ N3 ) )
= ( signed_drop_bit @ A @ ( numeral_numeral @ nat @ N3 ) @ ( uminus_uminus @ ( word @ A ) @ ( numeral_numeral @ ( word @ A ) @ M ) ) ) ) ) ).
% sshiftr_minus_numeral_numeral
thf(fact_6724_shiftl__Suc__0,axiom,
! [N3: nat] :
( ( bit_Sh4282982442137083160shiftl @ nat @ ( suc @ ( zero_zero @ nat ) ) @ N3 )
= ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N3 ) ) ).
% shiftl_Suc_0
thf(fact_6725_shiftl__minus__1__numeral,axiom,
! [A: $tType] :
( ( bit_ri3973907225187159222ations @ A )
=> ! [N3: num] :
( ( bit_Sh4282982442137083160shiftl @ A @ ( uminus_uminus @ A @ ( one_one @ A ) ) @ ( numeral_numeral @ nat @ N3 ) )
= ( bit_ri4277139882892585799ns_not @ A @ ( bit_se2239418461657761734s_mask @ A @ ( numeral_numeral @ nat @ N3 ) ) ) ) ) ).
% shiftl_minus_1_numeral
thf(fact_6726_shiftl__numeral__Suc,axiom,
! [A: $tType] :
( ( bit_se359711467146920520ations @ A )
=> ! [M: num,N3: nat] :
( ( bit_Sh4282982442137083160shiftl @ A @ ( numeral_numeral @ A @ M ) @ ( suc @ N3 ) )
= ( bit_se4730199178511100633sh_bit @ A @ ( suc @ N3 ) @ ( numeral_numeral @ A @ M ) ) ) ) ).
% shiftl_numeral_Suc
thf(fact_6727_shiftl__numeral__numeral,axiom,
! [A: $tType] :
( ( bit_se359711467146920520ations @ A )
=> ! [M: num,N3: num] :
( ( bit_Sh4282982442137083160shiftl @ A @ ( numeral_numeral @ A @ M ) @ ( numeral_numeral @ nat @ N3 ) )
= ( bit_se4730199178511100633sh_bit @ A @ ( numeral_numeral @ nat @ N3 ) @ ( numeral_numeral @ A @ M ) ) ) ) ).
% shiftl_numeral_numeral
thf(fact_6728_shiftl__minus__numeral__Suc,axiom,
! [A: $tType] :
( ( bit_ri3973907225187159222ations @ A )
=> ! [M: num,N3: nat] :
( ( bit_Sh4282982442137083160shiftl @ A @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ M ) ) @ ( suc @ N3 ) )
= ( bit_se4730199178511100633sh_bit @ A @ ( suc @ N3 ) @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ M ) ) ) ) ) ).
% shiftl_minus_numeral_Suc
thf(fact_6729_shiftl__minus__numeral__numeral,axiom,
! [A: $tType] :
( ( bit_ri3973907225187159222ations @ A )
=> ! [M: num,N3: num] :
( ( bit_Sh4282982442137083160shiftl @ A @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ M ) ) @ ( numeral_numeral @ nat @ N3 ) )
= ( bit_se4730199178511100633sh_bit @ A @ ( numeral_numeral @ nat @ N3 ) @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ M ) ) ) ) ) ).
% shiftl_minus_numeral_numeral
thf(fact_6730_shiftl__of__Suc,axiom,
! [A: $tType] :
( ( bit_se359711467146920520ations @ A )
=> ! [A2: A,N3: nat] :
( ( bit_Sh4282982442137083160shiftl @ A @ A2 @ ( suc @ N3 ) )
= ( bit_Sh4282982442137083160shiftl @ A @ ( times_times @ A @ A2 @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) @ N3 ) ) ) ).
% shiftl_of_Suc
thf(fact_6731_shiftl__1,axiom,
! [A: $tType] :
( ( bit_se359711467146920520ations @ A )
=> ! [N3: nat] :
( ( bit_Sh4282982442137083160shiftl @ A @ ( one_one @ A ) @ N3 )
= ( power_power @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ N3 ) ) ) ).
% shiftl_1
thf(fact_6732_one__bit__shiftl,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [N3: nat] :
( ( generi7602027413899671122et_bit @ ( word @ A ) @ ( zero_zero @ ( word @ A ) ) @ N3 @ $true )
= ( bit_Sh4282982442137083160shiftl @ ( word @ A ) @ ( one_one @ ( word @ A ) ) @ N3 ) ) ) ).
% one_bit_shiftl
thf(fact_6733_shiftl__eq__mult,axiom,
! [A: $tType] :
( ( bit_se359711467146920520ations @ A )
=> ( ( bit_Sh4282982442137083160shiftl @ A )
= ( ^ [X2: A,N2: nat] : ( times_times @ A @ X2 @ ( power_power @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ N2 ) ) ) ) ) ).
% shiftl_eq_mult
thf(fact_6734_bit__shiftl__iff,axiom,
! [A: $tType] :
( ( bit_se359711467146920520ations @ A )
=> ! [A2: A,M: nat,N3: nat] :
( ( bit_se5641148757651400278ts_bit @ A @ ( bit_Sh4282982442137083160shiftl @ A @ A2 @ M ) @ N3 )
= ( ( ord_less_eq @ nat @ M @ N3 )
& ( bit_se6407376104438227557le_bit @ A @ ( type2 @ A ) @ N3 )
& ( bit_se5641148757651400278ts_bit @ A @ A2 @ ( minus_minus @ nat @ N3 @ M ) ) ) ) ) ).
% bit_shiftl_iff
thf(fact_6735_shiftr__minus__numeral__numeral,axiom,
! [A: $tType] :
( ( bit_ri3973907225187159222ations @ A )
=> ! [M: num,N3: num] :
( ( bit_Sh4282982442137083166shiftr @ A @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ M ) ) @ ( numeral_numeral @ nat @ N3 ) )
= ( bit_se4197421643247451524op_bit @ A @ ( numeral_numeral @ nat @ N3 ) @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ M ) ) ) ) ) ).
% shiftr_minus_numeral_numeral
thf(fact_6736_shiftr__minus__numeral__Suc,axiom,
! [A: $tType] :
( ( bit_ri3973907225187159222ations @ A )
=> ! [M: num,N3: nat] :
( ( bit_Sh4282982442137083166shiftr @ A @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ M ) ) @ ( suc @ N3 ) )
= ( bit_se4197421643247451524op_bit @ A @ ( suc @ N3 ) @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ M ) ) ) ) ) ).
% shiftr_minus_numeral_Suc
thf(fact_6737_shiftr__Suc__0,axiom,
! [N3: nat] :
( ( bit_Sh4282982442137083166shiftr @ nat @ ( suc @ ( zero_zero @ nat ) ) @ N3 )
= ( zero_neq_one_of_bool @ nat
@ ( N3
= ( zero_zero @ nat ) ) ) ) ).
% shiftr_Suc_0
thf(fact_6738_shiftr__numeral__Suc,axiom,
! [A: $tType] :
( ( bit_se359711467146920520ations @ A )
=> ! [M: num,N3: nat] :
( ( bit_Sh4282982442137083166shiftr @ A @ ( numeral_numeral @ A @ M ) @ ( suc @ N3 ) )
= ( bit_se4197421643247451524op_bit @ A @ ( suc @ N3 ) @ ( numeral_numeral @ A @ M ) ) ) ) ).
% shiftr_numeral_Suc
thf(fact_6739_shiftr__numeral__numeral,axiom,
! [A: $tType] :
( ( bit_se359711467146920520ations @ A )
=> ! [M: num,N3: num] :
( ( bit_Sh4282982442137083166shiftr @ A @ ( numeral_numeral @ A @ M ) @ ( numeral_numeral @ nat @ N3 ) )
= ( bit_se4197421643247451524op_bit @ A @ ( numeral_numeral @ nat @ N3 ) @ ( numeral_numeral @ A @ M ) ) ) ) ).
% shiftr_numeral_numeral
thf(fact_6740_shiftr__1,axiom,
! [A: $tType] :
( ( bit_se359711467146920520ations @ A )
=> ! [N3: nat] :
( ( bit_Sh4282982442137083166shiftr @ A @ ( one_one @ A ) @ N3 )
= ( zero_neq_one_of_bool @ A
@ ( N3
= ( zero_zero @ nat ) ) ) ) ) ).
% shiftr_1
thf(fact_6741_shiftr__eq__div,axiom,
! [A: $tType] :
( ( bit_se359711467146920520ations @ A )
=> ( ( bit_Sh4282982442137083166shiftr @ A )
= ( ^ [X2: A,N2: nat] : ( divide_divide @ A @ X2 @ ( power_power @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ N2 ) ) ) ) ) ).
% shiftr_eq_div
thf(fact_6742_bit__revcast__iff,axiom,
! [B: $tType,A: $tType] :
( ( ( type_len @ A )
& ( type_len @ B ) )
=> ! [W: word @ A,N3: nat] :
( ( bit_se5641148757651400278ts_bit @ ( word @ B ) @ ( revcast @ A @ B @ W ) @ N3 )
= ( ( ord_less_eq @ nat @ ( minus_minus @ nat @ ( type_len0_len_of @ B @ ( type2 @ B ) ) @ ( type_len0_len_of @ A @ ( type2 @ A ) ) ) @ N3 )
& ( ord_less @ nat @ N3 @ ( type_len0_len_of @ B @ ( type2 @ B ) ) )
& ( bit_se5641148757651400278ts_bit @ ( word @ A ) @ W @ ( minus_minus @ nat @ ( plus_plus @ nat @ N3 @ ( 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_6743_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_6744_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_6745_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_6746_word__sle__msb__le,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ( ( word_sle @ A )
= ( ^ [X2: word @ A,Y2: word @ A] :
( ( ( most_s684356279273892711sb_msb @ ( word @ A ) @ Y2 )
=> ( most_s684356279273892711sb_msb @ ( word @ A ) @ X2 ) )
& ( ( ( most_s684356279273892711sb_msb @ ( word @ A ) @ X2 )
& ~ ( most_s684356279273892711sb_msb @ ( word @ A ) @ Y2 ) )
| ( ord_less_eq @ ( word @ A ) @ X2 @ Y2 ) ) ) ) ) ) ).
% word_sle_msb_le
thf(fact_6747_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_6748_revcast__slice1,axiom,
! [A: $tType,B: $tType] :
( ( ( type_len @ B )
& ( type_len @ A ) )
=> ! [W: word @ B] :
( ( slice1 @ B @ A @ ( size_size @ ( word @ A ) @ ( revcast @ B @ A @ W ) ) @ W )
= ( revcast @ B @ A @ W ) ) ) ).
% revcast_slice1
thf(fact_6749_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_6750_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_6751_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_6752_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_6753_msb__word__of__int,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [X: int] :
( ( most_s684356279273892711sb_msb @ ( word @ A ) @ ( ring_1_of_int @ ( word @ A ) @ X ) )
= ( bit_se5641148757651400278ts_bit @ int @ X @ ( minus_minus @ nat @ ( type_len0_len_of @ A @ ( type2 @ A ) ) @ ( one_one @ nat ) ) ) ) ) ).
% msb_word_of_int
thf(fact_6754_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_6755_word__sint__msb__eq,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ( ( ring_1_signed @ A @ int )
= ( ^ [X2: word @ A] : ( minus_minus @ int @ ( semiring_1_unsigned @ A @ int @ X2 ) @ ( if @ int @ ( most_s684356279273892711sb_msb @ ( word @ A ) @ X2 ) @ ( power_power @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ ( size_size @ ( word @ A ) @ X2 ) ) @ ( zero_zero @ int ) ) ) ) ) ) ).
% word_sint_msb_eq
thf(fact_6756_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_6757_word__type__copy__misc__axioms_Ointro,axiom,
! [B: $tType,A: $tType] :
( ( ( bit_ri3973907225187159222ations @ A )
& ( cl_HOL_Oequal @ A )
& ( linorder @ A )
& ( type_len @ B ) )
=> ! [Size: nat,Word_of: A > ( word @ B ),Set_bits_aux: ( nat > $o ) > nat > A > A] :
( ( Size
= ( type_len0_len_of @ B @ ( type2 @ B ) ) )
=> ( ! [P7: A] :
( ( most_s684356279273892711sb_msb @ A @ P7 )
= ( most_s684356279273892711sb_msb @ ( word @ B ) @ ( Word_of @ P7 ) ) )
=> ( ! [P7: A] :
( ( least_8051144512741203767sb_lsb @ A @ P7 )
= ( least_8051144512741203767sb_lsb @ ( word @ B ) @ ( Word_of @ P7 ) ) )
=> ( ! [P7: A] :
( ( size_size @ A @ P7 )
= ( size_size @ ( word @ B ) @ ( Word_of @ P7 ) ) )
=> ( ! [P7: A,N: nat,B3: $o] :
( ( Word_of @ ( generi7602027413899671122et_bit @ A @ P7 @ N @ B3 ) )
= ( generi7602027413899671122et_bit @ ( word @ B ) @ ( Word_of @ P7 ) @ N @ B3 ) )
=> ( ! [P8: nat > $o] :
( ( Word_of @ ( bit_bi4170147762399595738t_bits @ A @ P8 ) )
= ( bit_bi4170147762399595738t_bits @ ( word @ B ) @ P8 ) )
=> ( ! [P8: nat > $o,N: nat,P7: A] :
( ( Word_of @ ( Set_bits_aux @ P8 @ N @ P7 ) )
= ( code_T2661198915054445665ts_aux @ B @ P8 @ N @ ( Word_of @ P7 ) ) )
=> ( word_T8964210463127689547axioms @ A @ B @ Word_of @ Size @ Set_bits_aux ) ) ) ) ) ) ) ) ) ).
% word_type_copy_misc_axioms.intro
thf(fact_6758_word__type__copy__misc__axioms__def,axiom,
! [B: $tType,A: $tType] :
( ( ( bit_ri3973907225187159222ations @ A )
& ( cl_HOL_Oequal @ A )
& ( linorder @ A )
& ( type_len @ B ) )
=> ( ( word_T8964210463127689547axioms @ A @ B )
= ( ^ [Word_of2: A > ( word @ B ),Size2: nat,Set_bits_aux2: ( nat > $o ) > nat > A > A] :
( ( Size2
= ( type_len0_len_of @ B @ ( type2 @ B ) ) )
& ! [P5: A] :
( ( most_s684356279273892711sb_msb @ A @ P5 )
= ( most_s684356279273892711sb_msb @ ( word @ B ) @ ( Word_of2 @ P5 ) ) )
& ! [P5: A] :
( ( least_8051144512741203767sb_lsb @ A @ P5 )
= ( least_8051144512741203767sb_lsb @ ( word @ B ) @ ( Word_of2 @ P5 ) ) )
& ! [P5: A] :
( ( size_size @ A @ P5 )
= ( size_size @ ( word @ B ) @ ( Word_of2 @ P5 ) ) )
& ! [P5: A,N2: nat,B5: $o] :
( ( Word_of2 @ ( generi7602027413899671122et_bit @ A @ P5 @ N2 @ B5 ) )
= ( generi7602027413899671122et_bit @ ( word @ B ) @ ( Word_of2 @ P5 ) @ N2 @ B5 ) )
& ! [P3: nat > $o] :
( ( Word_of2 @ ( bit_bi4170147762399595738t_bits @ A @ P3 ) )
= ( bit_bi4170147762399595738t_bits @ ( word @ B ) @ P3 ) )
& ! [P3: nat > $o,N2: nat,P5: A] :
( ( Word_of2 @ ( Set_bits_aux2 @ P3 @ N2 @ P5 ) )
= ( code_T2661198915054445665ts_aux @ B @ P3 @ N2 @ ( Word_of2 @ P5 ) ) ) ) ) ) ) ).
% word_type_copy_misc_axioms_def
thf(fact_6759_msb__1,axiom,
~ ( most_s684356279273892711sb_msb @ int @ ( one_one @ int ) ) ).
% msb_1
thf(fact_6760_int__set__bits__K__True,axiom,
( ( bit_bi4170147762399595738t_bits @ int
@ ^ [Uu3: nat] : $true )
= ( uminus_uminus @ int @ ( one_one @ int ) ) ) ).
% int_set_bits_K_True
thf(fact_6761_msb__bin__rest,axiom,
! [X: int] :
( ( most_s684356279273892711sb_msb @ int @ ( divide_divide @ int @ X @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) ) )
= ( most_s684356279273892711sb_msb @ int @ X ) ) ).
% msb_bin_rest
thf(fact_6762_uint32__msb__test__bit,axiom,
( ( most_s684356279273892711sb_msb @ uint32 )
= ( ^ [X2: uint32] : ( bit_se5641148757651400278ts_bit @ uint32 @ X2 @ ( numeral_numeral @ nat @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ one2 ) ) ) ) ) ) ) ) ).
% uint32_msb_test_bit
thf(fact_6763_word__type__copy__misc_Oset__bits__aux__code,axiom,
! [B: $tType,A: $tType] :
( ( ( bit_ri3973907225187159222ations @ A )
& ( cl_HOL_Oequal @ A )
& ( linorder @ A )
& ( type_len @ B ) )
=> ! [Of_word: ( word @ B ) > A,Word_of: A > ( word @ B ),Signed_drop_bit: nat > A > A,Of_nat: nat > A,Nat_of: A > nat,Of_int: int > A,Int_of: A > int,Of_integer: code_integer > A,Integer_of: A > code_integer,Size: nat,Set_bits_aux: ( nat > $o ) > nat > A > A,N3: nat,F2: nat > $o,W: A] :
( ( word_T1870391057261780392y_misc @ B @ A @ Of_word @ Word_of @ Signed_drop_bit @ Of_nat @ Nat_of @ Of_int @ Int_of @ Of_integer @ Integer_of @ Size @ Set_bits_aux )
=> ( ( ( N3
= ( zero_zero @ nat ) )
=> ( ( Set_bits_aux @ F2 @ N3 @ W )
= W ) )
& ( ( N3
!= ( zero_zero @ nat ) )
=> ( ( Set_bits_aux @ F2 @ N3 @ W )
= ( Set_bits_aux @ F2 @ ( minus_minus @ nat @ N3 @ ( one_one @ nat ) ) @ ( bit_se1065995026697491101ons_or @ A @ ( bit_se4730199178511100633sh_bit @ A @ ( one_one @ nat ) @ W ) @ ( if @ A @ ( F2 @ ( minus_minus @ nat @ N3 @ ( one_one @ nat ) ) ) @ ( one_one @ A ) @ ( zero_zero @ A ) ) ) ) ) ) ) ) ) ).
% word_type_copy_misc.set_bits_aux_code
thf(fact_6764_bin__last__set__bits,axiom,
! [F2: nat > $o] :
( ( bit_wf_set_bits_int @ F2 )
=> ( ( ~ ( dvd_dvd @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ ( bit_bi4170147762399595738t_bits @ int @ F2 ) ) )
= ( F2 @ ( zero_zero @ nat ) ) ) ) ).
% bin_last_set_bits
thf(fact_6765_wf__set__bits__int__Suc,axiom,
! [F2: nat > $o] :
( ( bit_wf_set_bits_int
@ ^ [N2: nat] : ( F2 @ ( suc @ N2 ) ) )
= ( bit_wf_set_bits_int @ F2 ) ) ).
% wf_set_bits_int_Suc
thf(fact_6766_word__type__copy__misc_Osize__eq,axiom,
! [B: $tType,A: $tType] :
( ( ( bit_ri3973907225187159222ations @ A )
& ( cl_HOL_Oequal @ A )
& ( linorder @ A )
& ( type_len @ B ) )
=> ! [Of_word: ( word @ B ) > A,Word_of: A > ( word @ B ),Signed_drop_bit: nat > A > A,Of_nat: nat > A,Nat_of: A > nat,Of_int: int > A,Int_of: A > int,Of_integer: code_integer > A,Integer_of: A > code_integer,Size: nat,Set_bits_aux: ( nat > $o ) > nat > A > A,P4: A] :
( ( word_T1870391057261780392y_misc @ B @ A @ Of_word @ Word_of @ Signed_drop_bit @ Of_nat @ Nat_of @ Of_int @ Int_of @ Of_integer @ Integer_of @ Size @ Set_bits_aux )
=> ( ( size_size @ A @ P4 )
= Size ) ) ) ).
% word_type_copy_misc.size_eq
thf(fact_6767_wf__set__bits__int__simps,axiom,
( bit_wf_set_bits_int
= ( ^ [F7: nat > $o] :
? [N2: nat] :
( ! [N12: nat] :
( ( ord_less_eq @ nat @ N2 @ N12 )
=> ~ ( F7 @ N12 ) )
| ! [N12: nat] :
( ( ord_less_eq @ nat @ N2 @ N12 )
=> ( F7 @ N12 ) ) ) ) ) ).
% wf_set_bits_int_simps
thf(fact_6768_zeros,axiom,
! [N3: nat,F2: nat > $o] :
( ! [N6: nat] :
( ( ord_less_eq @ nat @ N3 @ N6 )
=> ~ ( F2 @ N6 ) )
=> ( bit_wf_set_bits_int @ F2 ) ) ).
% zeros
thf(fact_6769_wf__set__bits__int_Osimps,axiom,
( bit_wf_set_bits_int
= ( ^ [F7: nat > $o] :
( ? [N2: nat] :
! [N12: nat] :
( ( ord_less_eq @ nat @ N2 @ N12 )
=> ~ ( F7 @ N12 ) )
| ? [N2: nat] :
! [N12: nat] :
( ( ord_less_eq @ nat @ N2 @ N12 )
=> ( F7 @ N12 ) ) ) ) ) ).
% wf_set_bits_int.simps
thf(fact_6770_wf__set__bits__int_Ocases,axiom,
! [F2: nat > $o] :
( ( bit_wf_set_bits_int @ F2 )
=> ( ! [N: nat] :
~ ! [N5: nat] :
( ( ord_less_eq @ nat @ N @ N5 )
=> ~ ( F2 @ N5 ) )
=> ~ ! [N: nat] :
~ ! [N5: nat] :
( ( ord_less_eq @ nat @ N @ N5 )
=> ( F2 @ N5 ) ) ) ) ).
% wf_set_bits_int.cases
thf(fact_6771_ones,axiom,
! [N3: nat,F2: nat > $o] :
( ! [N6: nat] :
( ( ord_less_eq @ nat @ N3 @ N6 )
=> ( F2 @ N6 ) )
=> ( bit_wf_set_bits_int @ F2 ) ) ).
% ones
thf(fact_6772_word__type__copy__misc_Osize__eq__word__of,axiom,
! [B: $tType,A: $tType] :
( ( ( bit_ri3973907225187159222ations @ A )
& ( cl_HOL_Oequal @ A )
& ( linorder @ A )
& ( type_len @ B ) )
=> ! [Of_word: ( word @ B ) > A,Word_of: A > ( word @ B ),Signed_drop_bit: nat > A > A,Of_nat: nat > A,Nat_of: A > nat,Of_int: int > A,Int_of: A > int,Of_integer: code_integer > A,Integer_of: A > code_integer,Size: nat,Set_bits_aux: ( nat > $o ) > nat > A > A,P4: A] :
( ( word_T1870391057261780392y_misc @ B @ A @ Of_word @ Word_of @ Signed_drop_bit @ Of_nat @ Nat_of @ Of_int @ Int_of @ Of_integer @ Integer_of @ Size @ Set_bits_aux )
=> ( ( size_size @ A @ P4 )
= ( size_size @ ( word @ B ) @ ( Word_of @ P4 ) ) ) ) ) ).
% word_type_copy_misc.size_eq_word_of
thf(fact_6773_msb__uint32__code,axiom,
( ( most_s684356279273892711sb_msb @ uint32 )
= ( ^ [X2: uint32] : ( uint32_test_bit @ X2 @ ( numeral_numeral @ code_integer @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ one2 ) ) ) ) ) ) ) ) ).
% msb_uint32_code
thf(fact_6774_int__set__bits__unfold__BIT,axiom,
! [F2: nat > $o] :
( ( bit_wf_set_bits_int @ F2 )
=> ( ( bit_bi4170147762399595738t_bits @ int @ F2 )
= ( plus_plus @ int @ ( zero_neq_one_of_bool @ int @ ( F2 @ ( zero_zero @ nat ) ) ) @ ( times_times @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ ( bit_bi4170147762399595738t_bits @ int @ ( comp @ nat @ $o @ nat @ F2 @ suc ) ) ) ) ) ) ).
% int_set_bits_unfold_BIT
thf(fact_6775_bin__rest__set__bits,axiom,
! [F2: nat > $o] :
( ( bit_wf_set_bits_int @ F2 )
=> ( ( divide_divide @ int @ ( bit_bi4170147762399595738t_bits @ int @ F2 ) @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) )
= ( bit_bi4170147762399595738t_bits @ int @ ( comp @ nat @ $o @ nat @ F2 @ suc ) ) ) ) ).
% bin_rest_set_bits
thf(fact_6776_fun__comp__eq__conv,axiom,
! [B: $tType,C: $tType,A: $tType,F2: C > B,G: A > C,Fg: A > B] :
( ( ( comp @ C @ B @ A @ F2 @ G )
= Fg )
= ( ! [X2: A] :
( ( F2 @ ( G @ X2 ) )
= ( Fg @ X2 ) ) ) ) ).
% fun_comp_eq_conv
thf(fact_6777_comp__cong__right,axiom,
! [C: $tType,B: $tType,A: $tType,X: A > B,Y: A > B,F2: B > C] :
( ( X = Y )
=> ( ( comp @ B @ C @ A @ F2 @ X )
= ( comp @ B @ C @ A @ F2 @ Y ) ) ) ).
% comp_cong_right
thf(fact_6778_comp__cong__left,axiom,
! [B: $tType,A: $tType,C: $tType,X: A > B,Y: A > B,F2: C > A] :
( ( X = Y )
=> ( ( comp @ A @ B @ C @ X @ F2 )
= ( comp @ A @ B @ C @ Y @ F2 ) ) ) ).
% comp_cong_left
thf(fact_6779_sum__comp__morphism,axiom,
! [A: $tType,B: $tType,C: $tType] :
( ( ( comm_monoid_add @ B )
& ( comm_monoid_add @ A ) )
=> ! [H2: B > A,G: C > B,A3: set @ C] :
( ( ( H2 @ ( zero_zero @ B ) )
= ( zero_zero @ A ) )
=> ( ! [X3: B,Y3: B] :
( ( H2 @ ( plus_plus @ B @ X3 @ Y3 ) )
= ( plus_plus @ A @ ( H2 @ X3 ) @ ( H2 @ Y3 ) ) )
=> ( ( groups7311177749621191930dd_sum @ C @ A @ ( comp @ B @ A @ C @ H2 @ G ) @ A3 )
= ( H2 @ ( groups7311177749621191930dd_sum @ C @ B @ G @ A3 ) ) ) ) ) ) ).
% sum_comp_morphism
thf(fact_6780_bij__betw__comp__iff2,axiom,
! [C: $tType,A: $tType,B: $tType,F5: A > B,A9: set @ A,A11: set @ B,F2: C > A,A3: set @ C] :
( ( bij_betw @ A @ B @ F5 @ A9 @ A11 )
=> ( ( ord_less_eq @ ( set @ A ) @ ( image @ C @ A @ F2 @ A3 ) @ A9 )
=> ( ( bij_betw @ C @ A @ F2 @ A3 @ A9 )
= ( bij_betw @ C @ B @ ( comp @ A @ B @ C @ F5 @ F2 ) @ A3 @ A11 ) ) ) ) ).
% bij_betw_comp_iff2
thf(fact_6781_sum_OatLeast__Suc__lessThan__Suc__shift,axiom,
! [A: $tType] :
( ( comm_monoid_add @ A )
=> ! [G: nat > A,M: nat,N3: nat] :
( ( groups7311177749621191930dd_sum @ nat @ A @ G @ ( set_or7035219750837199246ssThan @ nat @ ( suc @ M ) @ ( suc @ N3 ) ) )
= ( groups7311177749621191930dd_sum @ nat @ A @ ( comp @ nat @ A @ nat @ G @ suc ) @ ( set_or7035219750837199246ssThan @ nat @ M @ N3 ) ) ) ) ).
% sum.atLeast_Suc_lessThan_Suc_shift
thf(fact_6782_sum_OatLeast__Suc__atMost__Suc__shift,axiom,
! [A: $tType] :
( ( comm_monoid_add @ A )
=> ! [G: nat > A,M: nat,N3: nat] :
( ( groups7311177749621191930dd_sum @ nat @ A @ G @ ( set_or1337092689740270186AtMost @ nat @ ( suc @ M ) @ ( suc @ N3 ) ) )
= ( groups7311177749621191930dd_sum @ nat @ A @ ( comp @ nat @ A @ nat @ G @ suc ) @ ( set_or1337092689740270186AtMost @ nat @ M @ N3 ) ) ) ) ).
% sum.atLeast_Suc_atMost_Suc_shift
thf(fact_6783_bit__drop__bit__eq,axiom,
! [A: $tType] :
( ( bit_se359711467146920520ations @ A )
=> ! [N3: nat,A2: A] :
( ( bit_se5641148757651400278ts_bit @ A @ ( bit_se4197421643247451524op_bit @ A @ N3 @ A2 ) )
= ( comp @ nat @ $o @ nat @ ( bit_se5641148757651400278ts_bit @ A @ A2 ) @ ( plus_plus @ nat @ N3 ) ) ) ) ).
% bit_drop_bit_eq
thf(fact_6784_sum_OatLeastLessThan__shift__bounds,axiom,
! [A: $tType] :
( ( comm_monoid_add @ A )
=> ! [G: nat > A,M: nat,K: nat,N3: nat] :
( ( groups7311177749621191930dd_sum @ nat @ A @ G @ ( set_or7035219750837199246ssThan @ nat @ ( plus_plus @ nat @ M @ K ) @ ( plus_plus @ nat @ N3 @ K ) ) )
= ( groups7311177749621191930dd_sum @ nat @ A @ ( comp @ nat @ A @ nat @ G @ ( plus_plus @ nat @ K ) ) @ ( set_or7035219750837199246ssThan @ nat @ M @ N3 ) ) ) ) ).
% sum.atLeastLessThan_shift_bounds
thf(fact_6785_sum_OatLeastAtMost__shift__bounds,axiom,
! [A: $tType] :
( ( comm_monoid_add @ A )
=> ! [G: nat > A,M: nat,K: nat,N3: nat] :
( ( groups7311177749621191930dd_sum @ nat @ A @ G @ ( set_or1337092689740270186AtMost @ nat @ ( plus_plus @ nat @ M @ K ) @ ( plus_plus @ nat @ N3 @ K ) ) )
= ( groups7311177749621191930dd_sum @ nat @ A @ ( comp @ nat @ A @ nat @ G @ ( plus_plus @ nat @ K ) ) @ ( set_or1337092689740270186AtMost @ nat @ M @ N3 ) ) ) ) ).
% sum.atLeastAtMost_shift_bounds
thf(fact_6786_prod_OatLeast__Suc__lessThan__Suc__shift,axiom,
! [A: $tType] :
( ( comm_monoid_mult @ A )
=> ! [G: nat > A,M: nat,N3: nat] :
( ( groups7121269368397514597t_prod @ nat @ A @ G @ ( set_or7035219750837199246ssThan @ nat @ ( suc @ M ) @ ( suc @ N3 ) ) )
= ( groups7121269368397514597t_prod @ nat @ A @ ( comp @ nat @ A @ nat @ G @ suc ) @ ( set_or7035219750837199246ssThan @ nat @ M @ N3 ) ) ) ) ).
% prod.atLeast_Suc_lessThan_Suc_shift
thf(fact_6787_prod_OatLeast__Suc__atMost__Suc__shift,axiom,
! [A: $tType] :
( ( comm_monoid_mult @ A )
=> ! [G: nat > A,M: nat,N3: nat] :
( ( groups7121269368397514597t_prod @ nat @ A @ G @ ( set_or1337092689740270186AtMost @ nat @ ( suc @ M ) @ ( suc @ N3 ) ) )
= ( groups7121269368397514597t_prod @ nat @ A @ ( comp @ nat @ A @ nat @ G @ suc ) @ ( set_or1337092689740270186AtMost @ nat @ M @ N3 ) ) ) ) ).
% prod.atLeast_Suc_atMost_Suc_shift
thf(fact_6788_prod_OatLeastLessThan__shift__bounds,axiom,
! [A: $tType] :
( ( comm_monoid_mult @ A )
=> ! [G: nat > A,M: nat,K: nat,N3: nat] :
( ( groups7121269368397514597t_prod @ nat @ A @ G @ ( set_or7035219750837199246ssThan @ nat @ ( plus_plus @ nat @ M @ K ) @ ( plus_plus @ nat @ N3 @ K ) ) )
= ( groups7121269368397514597t_prod @ nat @ A @ ( comp @ nat @ A @ nat @ G @ ( plus_plus @ nat @ K ) ) @ ( set_or7035219750837199246ssThan @ nat @ M @ N3 ) ) ) ) ).
% prod.atLeastLessThan_shift_bounds
thf(fact_6789_prod_OatLeastAtMost__shift__bounds,axiom,
! [A: $tType] :
( ( comm_monoid_mult @ A )
=> ! [G: nat > A,M: nat,K: nat,N3: nat] :
( ( groups7121269368397514597t_prod @ nat @ A @ G @ ( set_or1337092689740270186AtMost @ nat @ ( plus_plus @ nat @ M @ K ) @ ( plus_plus @ nat @ N3 @ K ) ) )
= ( groups7121269368397514597t_prod @ nat @ A @ ( comp @ nat @ A @ nat @ G @ ( plus_plus @ nat @ K ) ) @ ( set_or1337092689740270186AtMost @ nat @ M @ N3 ) ) ) ) ).
% prod.atLeastAtMost_shift_bounds
thf(fact_6790_bit__shiftr__eq,axiom,
! [A: $tType] :
( ( bit_se359711467146920520ations @ A )
=> ! [A2: A,N3: nat] :
( ( bit_se5641148757651400278ts_bit @ A @ ( bit_Sh4282982442137083166shiftr @ A @ A2 @ N3 ) )
= ( comp @ nat @ $o @ nat @ ( bit_se5641148757651400278ts_bit @ A @ A2 ) @ ( plus_plus @ nat @ N3 ) ) ) ) ).
% bit_shiftr_eq
thf(fact_6791_summable__inverse__divide,axiom,
! [A: $tType] :
( ( real_V3459762299906320749_field @ A )
=> ! [F2: nat > A,C2: A] :
( ( summable @ A @ ( comp @ A @ A @ nat @ ( inverse_inverse @ A ) @ F2 ) )
=> ( summable @ A
@ ^ [N2: nat] : ( divide_divide @ A @ C2 @ ( F2 @ N2 ) ) ) ) ) ).
% summable_inverse_divide
thf(fact_6792_prod_Oreindex__nontrivial,axiom,
! [C: $tType,A: $tType,B: $tType] :
( ( comm_monoid_mult @ A )
=> ! [A3: set @ B,H2: B > C,G: C > A] :
( ( finite_finite2 @ B @ A3 )
=> ( ! [X3: B,Y3: B] :
( ( member @ B @ X3 @ A3 )
=> ( ( member @ B @ Y3 @ A3 )
=> ( ( X3 != Y3 )
=> ( ( ( H2 @ X3 )
= ( H2 @ Y3 ) )
=> ( ( G @ ( H2 @ X3 ) )
= ( one_one @ A ) ) ) ) ) )
=> ( ( groups7121269368397514597t_prod @ C @ A @ G @ ( image @ B @ C @ H2 @ A3 ) )
= ( groups7121269368397514597t_prod @ B @ A @ ( comp @ C @ A @ B @ G @ H2 ) @ A3 ) ) ) ) ) ).
% prod.reindex_nontrivial
thf(fact_6793_sum__image__le,axiom,
! [A: $tType,B: $tType,C: $tType] :
( ( ordere6911136660526730532id_add @ B )
=> ! [I5: set @ C,G: A > B,F2: C > A] :
( ( finite_finite2 @ C @ I5 )
=> ( ! [I2: C] :
( ( member @ C @ I2 @ I5 )
=> ( ord_less_eq @ B @ ( zero_zero @ B ) @ ( G @ ( F2 @ I2 ) ) ) )
=> ( ord_less_eq @ B @ ( groups7311177749621191930dd_sum @ A @ B @ G @ ( image @ C @ A @ F2 @ I5 ) ) @ ( groups7311177749621191930dd_sum @ C @ B @ ( comp @ A @ B @ C @ G @ F2 ) @ I5 ) ) ) ) ) ).
% sum_image_le
thf(fact_6794_sum_OatLeast0__lessThan__Suc__shift,axiom,
! [A: $tType] :
( ( comm_monoid_add @ A )
=> ! [G: nat > A,N3: nat] :
( ( groups7311177749621191930dd_sum @ nat @ A @ G @ ( set_or7035219750837199246ssThan @ nat @ ( zero_zero @ nat ) @ ( suc @ N3 ) ) )
= ( plus_plus @ A @ ( G @ ( zero_zero @ nat ) ) @ ( groups7311177749621191930dd_sum @ nat @ A @ ( comp @ nat @ A @ nat @ G @ suc ) @ ( set_or7035219750837199246ssThan @ nat @ ( zero_zero @ nat ) @ N3 ) ) ) ) ) ).
% sum.atLeast0_lessThan_Suc_shift
thf(fact_6795_sum_OatLeast0__atMost__Suc__shift,axiom,
! [A: $tType] :
( ( comm_monoid_add @ A )
=> ! [G: nat > A,N3: nat] :
( ( groups7311177749621191930dd_sum @ nat @ A @ G @ ( set_or1337092689740270186AtMost @ nat @ ( zero_zero @ nat ) @ ( suc @ N3 ) ) )
= ( plus_plus @ A @ ( G @ ( zero_zero @ nat ) ) @ ( groups7311177749621191930dd_sum @ nat @ A @ ( comp @ nat @ A @ nat @ G @ suc ) @ ( set_or1337092689740270186AtMost @ nat @ ( zero_zero @ nat ) @ N3 ) ) ) ) ) ).
% sum.atLeast0_atMost_Suc_shift
thf(fact_6796_prod_OatLeast0__lessThan__Suc__shift,axiom,
! [A: $tType] :
( ( comm_monoid_mult @ A )
=> ! [G: nat > A,N3: nat] :
( ( groups7121269368397514597t_prod @ nat @ A @ G @ ( set_or7035219750837199246ssThan @ nat @ ( zero_zero @ nat ) @ ( suc @ N3 ) ) )
= ( times_times @ A @ ( G @ ( zero_zero @ nat ) ) @ ( groups7121269368397514597t_prod @ nat @ A @ ( comp @ nat @ A @ nat @ G @ suc ) @ ( set_or7035219750837199246ssThan @ nat @ ( zero_zero @ nat ) @ N3 ) ) ) ) ) ).
% prod.atLeast0_lessThan_Suc_shift
thf(fact_6797_prod_OatLeast0__atMost__Suc__shift,axiom,
! [A: $tType] :
( ( comm_monoid_mult @ A )
=> ! [G: nat > A,N3: nat] :
( ( groups7121269368397514597t_prod @ nat @ A @ G @ ( set_or1337092689740270186AtMost @ nat @ ( zero_zero @ nat ) @ ( suc @ N3 ) ) )
= ( times_times @ A @ ( G @ ( zero_zero @ nat ) ) @ ( groups7121269368397514597t_prod @ nat @ A @ ( comp @ nat @ A @ nat @ G @ suc ) @ ( set_or1337092689740270186AtMost @ nat @ ( zero_zero @ nat ) @ N3 ) ) ) ) ) ).
% prod.atLeast0_atMost_Suc_shift
thf(fact_6798_sum_OatLeastLessThan__shift__0,axiom,
! [A: $tType] :
( ( comm_monoid_add @ A )
=> ! [G: nat > A,M: nat,N3: nat] :
( ( groups7311177749621191930dd_sum @ nat @ A @ G @ ( set_or7035219750837199246ssThan @ nat @ M @ N3 ) )
= ( groups7311177749621191930dd_sum @ nat @ A @ ( comp @ nat @ A @ nat @ G @ ( plus_plus @ nat @ M ) ) @ ( set_or7035219750837199246ssThan @ nat @ ( zero_zero @ nat ) @ ( minus_minus @ nat @ N3 @ M ) ) ) ) ) ).
% sum.atLeastLessThan_shift_0
thf(fact_6799_prod_OatLeastLessThan__shift__0,axiom,
! [A: $tType] :
( ( comm_monoid_mult @ A )
=> ! [G: nat > A,M: nat,N3: nat] :
( ( groups7121269368397514597t_prod @ nat @ A @ G @ ( set_or7035219750837199246ssThan @ nat @ M @ N3 ) )
= ( groups7121269368397514597t_prod @ nat @ A @ ( comp @ nat @ A @ nat @ G @ ( plus_plus @ nat @ M ) ) @ ( set_or7035219750837199246ssThan @ nat @ ( zero_zero @ nat ) @ ( minus_minus @ nat @ N3 @ M ) ) ) ) ) ).
% prod.atLeastLessThan_shift_0
thf(fact_6800_sum_OatLeast__lessThan__pred__shift,axiom,
! [A: $tType] :
( ( comm_monoid_add @ A )
=> ! [G: nat > A,M: nat,N3: nat] :
( ( groups7311177749621191930dd_sum @ nat @ A
@ ( comp @ nat @ A @ nat @ G
@ ^ [N2: nat] : ( minus_minus @ nat @ N2 @ ( suc @ ( zero_zero @ nat ) ) ) )
@ ( set_or7035219750837199246ssThan @ nat @ ( suc @ M ) @ ( suc @ N3 ) ) )
= ( groups7311177749621191930dd_sum @ nat @ A @ G @ ( set_or7035219750837199246ssThan @ nat @ M @ N3 ) ) ) ) ).
% sum.atLeast_lessThan_pred_shift
thf(fact_6801_sum_OatLeast__atMost__pred__shift,axiom,
! [A: $tType] :
( ( comm_monoid_add @ A )
=> ! [G: nat > A,M: nat,N3: nat] :
( ( groups7311177749621191930dd_sum @ nat @ A
@ ( comp @ nat @ A @ nat @ G
@ ^ [N2: nat] : ( minus_minus @ nat @ N2 @ ( suc @ ( zero_zero @ nat ) ) ) )
@ ( set_or1337092689740270186AtMost @ nat @ ( suc @ M ) @ ( suc @ N3 ) ) )
= ( groups7311177749621191930dd_sum @ nat @ A @ G @ ( set_or1337092689740270186AtMost @ nat @ M @ N3 ) ) ) ) ).
% sum.atLeast_atMost_pred_shift
thf(fact_6802_prod_OatLeast__lessThan__pred__shift,axiom,
! [A: $tType] :
( ( comm_monoid_mult @ A )
=> ! [G: nat > A,M: nat,N3: nat] :
( ( groups7121269368397514597t_prod @ nat @ A
@ ( comp @ nat @ A @ nat @ G
@ ^ [N2: nat] : ( minus_minus @ nat @ N2 @ ( suc @ ( zero_zero @ nat ) ) ) )
@ ( set_or7035219750837199246ssThan @ nat @ ( suc @ M ) @ ( suc @ N3 ) ) )
= ( groups7121269368397514597t_prod @ nat @ A @ G @ ( set_or7035219750837199246ssThan @ nat @ M @ N3 ) ) ) ) ).
% prod.atLeast_lessThan_pred_shift
thf(fact_6803_prod_OatLeast__atMost__pred__shift,axiom,
! [A: $tType] :
( ( comm_monoid_mult @ A )
=> ! [G: nat > A,M: nat,N3: nat] :
( ( groups7121269368397514597t_prod @ nat @ A
@ ( comp @ nat @ A @ nat @ G
@ ^ [N2: nat] : ( minus_minus @ nat @ N2 @ ( suc @ ( zero_zero @ nat ) ) ) )
@ ( set_or1337092689740270186AtMost @ nat @ ( suc @ M ) @ ( suc @ N3 ) ) )
= ( groups7121269368397514597t_prod @ nat @ A @ G @ ( set_or1337092689740270186AtMost @ nat @ M @ N3 ) ) ) ) ).
% prod.atLeast_atMost_pred_shift
thf(fact_6804_sum_OatLeastAtMost__shift__0,axiom,
! [A: $tType] :
( ( comm_monoid_add @ A )
=> ! [M: nat,N3: nat,G: nat > A] :
( ( ord_less_eq @ nat @ M @ N3 )
=> ( ( groups7311177749621191930dd_sum @ nat @ A @ G @ ( set_or1337092689740270186AtMost @ nat @ M @ N3 ) )
= ( groups7311177749621191930dd_sum @ nat @ A @ ( comp @ nat @ A @ nat @ G @ ( plus_plus @ nat @ M ) ) @ ( set_or1337092689740270186AtMost @ nat @ ( zero_zero @ nat ) @ ( minus_minus @ nat @ N3 @ M ) ) ) ) ) ) ).
% sum.atLeastAtMost_shift_0
thf(fact_6805_prod_OatLeastAtMost__shift__0,axiom,
! [A: $tType] :
( ( comm_monoid_mult @ A )
=> ! [M: nat,N3: nat,G: nat > A] :
( ( ord_less_eq @ nat @ M @ N3 )
=> ( ( groups7121269368397514597t_prod @ nat @ A @ G @ ( set_or1337092689740270186AtMost @ nat @ M @ N3 ) )
= ( groups7121269368397514597t_prod @ nat @ A @ ( comp @ nat @ A @ nat @ G @ ( plus_plus @ nat @ M ) ) @ ( set_or1337092689740270186AtMost @ nat @ ( zero_zero @ nat ) @ ( minus_minus @ nat @ N3 @ M ) ) ) ) ) ) ).
% prod.atLeastAtMost_shift_0
thf(fact_6806_test__bit__uint32__code,axiom,
( ( bit_se5641148757651400278ts_bit @ uint32 )
= ( ^ [X2: uint32,N2: nat] :
( ( ord_less @ nat @ N2 @ ( numeral_numeral @ nat @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ one2 ) ) ) ) ) ) )
& ( uint32_test_bit @ X2 @ ( code_integer_of_nat @ N2 ) ) ) ) ) ).
% test_bit_uint32_code
thf(fact_6807_integer__of__uint32__code,axiom,
( integer_of_uint32
= ( ^ [N2: uint32] : ( if @ code_integer @ ( bit_se5641148757651400278ts_bit @ uint32 @ N2 @ ( numeral_numeral @ nat @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ one2 ) ) ) ) ) ) @ ( bit_se1065995026697491101ons_or @ code_integer @ ( intege5370686899274169573signed @ ( bit_se5824344872417868541ns_and @ uint32 @ N2 @ ( numeral_numeral @ uint32 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ one2 ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) @ ( numeral_numeral @ code_integer @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ one2 ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) @ ( intege5370686899274169573signed @ N2 ) ) ) ) ).
% integer_of_uint32_code
thf(fact_6808_card_Ocomp__fun__commute__on,axiom,
( ( comp @ nat @ nat @ nat @ suc @ suc )
= ( comp @ nat @ nat @ nat @ suc @ suc ) ) ).
% card.comp_fun_commute_on
thf(fact_6809_set__oo__map__alt,axiom,
! [B: $tType,A: $tType,F2: A > B] :
( ( comp @ ( list @ B ) @ ( set @ B ) @ ( list @ A ) @ ( set2 @ B ) @ ( map @ A @ B @ F2 ) )
= ( ^ [L: list @ A] : ( image @ A @ B @ F2 @ ( set2 @ A @ L ) ) ) ) ).
% set_oo_map_alt
thf(fact_6810_integer__of__nat__numeral,axiom,
! [N3: num] :
( ( code_integer_of_nat @ ( numeral_numeral @ nat @ N3 ) )
= ( numeral_numeral @ code_integer @ N3 ) ) ).
% integer_of_nat_numeral
thf(fact_6811_integer__of__nat__1,axiom,
( ( code_integer_of_nat @ ( one_one @ nat ) )
= ( one_one @ code_integer ) ) ).
% integer_of_nat_1
thf(fact_6812_sum_Oeq__fold,axiom,
! [A: $tType,B: $tType] :
( ( comm_monoid_add @ A )
=> ( ( groups7311177749621191930dd_sum @ B @ A )
= ( ^ [G4: B > A] : ( finite_fold @ B @ A @ ( comp @ A @ ( A > A ) @ B @ ( plus_plus @ A ) @ G4 ) @ ( zero_zero @ A ) ) ) ) ) ).
% sum.eq_fold
thf(fact_6813_prod_Oeq__fold,axiom,
! [A: $tType,B: $tType] :
( ( comm_monoid_mult @ A )
=> ( ( groups7121269368397514597t_prod @ B @ A )
= ( ^ [G4: B > A] : ( finite_fold @ B @ A @ ( comp @ A @ ( A > A ) @ B @ ( times_times @ A ) @ G4 ) @ ( one_one @ A ) ) ) ) ) ).
% prod.eq_fold
thf(fact_6814_set__bit__uint32__code,axiom,
( ( generi7602027413899671122et_bit @ uint32 )
= ( ^ [X2: uint32,N2: nat,B5: $o] : ( if @ uint32 @ ( ord_less @ nat @ N2 @ ( numeral_numeral @ nat @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ one2 ) ) ) ) ) ) ) @ ( uint32_set_bit @ X2 @ ( code_integer_of_nat @ N2 ) @ B5 ) @ X2 ) ) ) ).
% set_bit_uint32_code
thf(fact_6815_shiftr__uint32__code,axiom,
( ( bit_se4197421643247451524op_bit @ uint32 )
= ( ^ [N2: nat,X2: uint32] : ( if @ uint32 @ ( ord_less @ nat @ N2 @ ( numeral_numeral @ nat @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ one2 ) ) ) ) ) ) ) @ ( uint32_shiftr @ X2 @ ( code_integer_of_nat @ N2 ) ) @ ( zero_zero @ uint32 ) ) ) ) ).
% shiftr_uint32_code
thf(fact_6816_shiftl__uint32__code,axiom,
( ( bit_se4730199178511100633sh_bit @ uint32 )
= ( ^ [N2: nat,X2: uint32] : ( if @ uint32 @ ( ord_less @ nat @ N2 @ ( numeral_numeral @ nat @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ one2 ) ) ) ) ) ) ) @ ( uint32_shiftl @ X2 @ ( code_integer_of_nat @ N2 ) ) @ ( zero_zero @ uint32 ) ) ) ) ).
% shiftl_uint32_code
thf(fact_6817_sshiftr__uint32__code,axiom,
( signed489701013188660438uint32
= ( ^ [N2: nat,X2: uint32] : ( if @ uint32 @ ( ord_less @ nat @ N2 @ ( numeral_numeral @ nat @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ one2 ) ) ) ) ) ) ) @ ( uint32_sshiftr @ X2 @ ( code_integer_of_nat @ N2 ) ) @ ( if @ uint32 @ ( bit_se5641148757651400278ts_bit @ uint32 @ X2 @ ( numeral_numeral @ nat @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ one2 ) ) ) ) ) ) @ ( uminus_uminus @ uint32 @ ( one_one @ uint32 ) ) @ ( zero_zero @ uint32 ) ) ) ) ) ).
% sshiftr_uint32_code
thf(fact_6818_uint32_Oset__bits__aux__code,axiom,
( set_bits_aux_uint32
= ( ^ [F7: nat > $o,N2: nat,W2: uint32] :
( if @ uint32
@ ( N2
= ( zero_zero @ nat ) )
@ W2
@ ( set_bits_aux_uint32 @ F7 @ ( minus_minus @ nat @ N2 @ ( one_one @ nat ) ) @ ( bit_se1065995026697491101ons_or @ uint32 @ ( bit_se4730199178511100633sh_bit @ uint32 @ ( one_one @ nat ) @ W2 ) @ ( if @ uint32 @ ( F7 @ ( minus_minus @ nat @ N2 @ ( one_one @ nat ) ) ) @ ( one_one @ uint32 ) @ ( zero_zero @ uint32 ) ) ) ) ) ) ) ).
% uint32.set_bits_aux_code
thf(fact_6819_uint32__divmod__code,axiom,
( uint32_divmod
= ( ^ [X2: uint32,Y2: uint32] :
( if @ ( product_prod @ uint32 @ uint32 ) @ ( ord_less_eq @ uint32 @ ( numeral_numeral @ uint32 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ one2 ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) @ Y2 ) @ ( if @ ( product_prod @ uint32 @ uint32 ) @ ( ord_less @ uint32 @ X2 @ Y2 ) @ ( product_Pair @ uint32 @ uint32 @ ( zero_zero @ uint32 ) @ X2 ) @ ( product_Pair @ uint32 @ uint32 @ ( one_one @ uint32 ) @ ( minus_minus @ uint32 @ X2 @ Y2 ) ) )
@ ( if @ ( product_prod @ uint32 @ uint32 )
@ ( Y2
= ( zero_zero @ uint32 ) )
@ ( product_Pair @ uint32 @ uint32 @ ( div0_uint32 @ X2 ) @ ( mod0_uint32 @ X2 ) )
@ ( if @ ( product_prod @ uint32 @ uint32 ) @ ( ord_less_eq @ uint32 @ Y2 @ ( minus_minus @ uint32 @ X2 @ ( times_times @ uint32 @ ( bit_se4730199178511100633sh_bit @ uint32 @ ( one_one @ nat ) @ ( uint32_sdiv @ ( bit_se4197421643247451524op_bit @ uint32 @ ( one_one @ nat ) @ X2 ) @ Y2 ) ) @ Y2 ) ) ) @ ( product_Pair @ uint32 @ uint32 @ ( plus_plus @ uint32 @ ( bit_se4730199178511100633sh_bit @ uint32 @ ( one_one @ nat ) @ ( uint32_sdiv @ ( bit_se4197421643247451524op_bit @ uint32 @ ( one_one @ nat ) @ X2 ) @ Y2 ) ) @ ( one_one @ uint32 ) ) @ ( minus_minus @ uint32 @ ( minus_minus @ uint32 @ X2 @ ( times_times @ uint32 @ ( bit_se4730199178511100633sh_bit @ uint32 @ ( one_one @ nat ) @ ( uint32_sdiv @ ( bit_se4197421643247451524op_bit @ uint32 @ ( one_one @ nat ) @ X2 ) @ Y2 ) ) @ Y2 ) ) @ Y2 ) ) @ ( product_Pair @ uint32 @ uint32 @ ( bit_se4730199178511100633sh_bit @ uint32 @ ( one_one @ nat ) @ ( uint32_sdiv @ ( bit_se4197421643247451524op_bit @ uint32 @ ( one_one @ nat ) @ X2 ) @ Y2 ) ) @ ( minus_minus @ uint32 @ X2 @ ( times_times @ uint32 @ ( bit_se4730199178511100633sh_bit @ uint32 @ ( one_one @ nat ) @ ( uint32_sdiv @ ( bit_se4197421643247451524op_bit @ uint32 @ ( one_one @ nat ) @ X2 ) @ Y2 ) ) @ Y2 ) ) ) ) ) ) ) ) ).
% uint32_divmod_code
thf(fact_6820_uint32_Oset__bits__code,axiom,
( ( bit_bi4170147762399595738t_bits @ uint32 )
= ( ^ [P3: nat > $o] : ( set_bits_aux_uint32 @ P3 @ ( numeral_numeral @ nat @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ one2 ) ) ) ) ) ) @ ( zero_zero @ uint32 ) ) ) ) ).
% uint32.set_bits_code
thf(fact_6821_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_6822_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_6823_inj__mult__left,axiom,
! [A: $tType] :
( ( idom @ A )
=> ! [A2: A] :
( ( inj_on @ A @ A @ ( times_times @ A @ A2 ) @ ( top_top @ ( set @ A ) ) )
= ( A2
!= ( zero_zero @ A ) ) ) ) ).
% inj_mult_left
thf(fact_6824_inj__divide__right,axiom,
! [A: $tType] :
( ( field @ A )
=> ! [A2: A] :
( ( inj_on @ A @ A
@ ^ [B5: A] : ( divide_divide @ A @ B5 @ A2 )
@ ( top_top @ ( set @ A ) ) )
= ( A2
!= ( zero_zero @ A ) ) ) ) ).
% inj_divide_right
thf(fact_6825_inj__on__insert,axiom,
! [B: $tType,A: $tType,F2: A > B,A2: A,A3: set @ A] :
( ( inj_on @ A @ B @ F2 @ ( insert @ A @ A2 @ A3 ) )
= ( ( inj_on @ A @ B @ F2 @ A3 )
& ~ ( member @ B @ ( F2 @ A2 ) @ ( image @ A @ B @ F2 @ ( minus_minus @ ( set @ A ) @ A3 @ ( insert @ A @ A2 @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ) ) ).
% inj_on_insert
thf(fact_6826_inj__on__image__mem__iff,axiom,
! [B: $tType,A: $tType,F2: A > B,B6: set @ A,A2: A,A3: set @ A] :
( ( inj_on @ A @ B @ F2 @ B6 )
=> ( ( member @ A @ A2 @ B6 )
=> ( ( ord_less_eq @ ( set @ A ) @ A3 @ B6 )
=> ( ( member @ B @ ( F2 @ A2 ) @ ( image @ A @ B @ F2 @ A3 ) )
= ( member @ A @ A2 @ A3 ) ) ) ) ) ).
% inj_on_image_mem_iff
thf(fact_6827_inj__on__image__eq__iff,axiom,
! [B: $tType,A: $tType,F2: A > B,C5: set @ A,A3: set @ A,B6: set @ A] :
( ( inj_on @ A @ B @ F2 @ C5 )
=> ( ( ord_less_eq @ ( set @ A ) @ A3 @ C5 )
=> ( ( ord_less_eq @ ( set @ A ) @ B6 @ C5 )
=> ( ( ( image @ A @ B @ F2 @ A3 )
= ( image @ A @ B @ F2 @ B6 ) )
= ( A3 = B6 ) ) ) ) ) ).
% inj_on_image_eq_iff
thf(fact_6828_subset__image__inj,axiom,
! [A: $tType,B: $tType,S3: set @ A,F2: B > A,T5: set @ B] :
( ( ord_less_eq @ ( set @ A ) @ S3 @ ( image @ B @ A @ F2 @ T5 ) )
= ( ? [U3: set @ B] :
( ( ord_less_eq @ ( set @ B ) @ U3 @ T5 )
& ( inj_on @ B @ A @ F2 @ U3 )
& ( S3
= ( image @ B @ A @ F2 @ U3 ) ) ) ) ) ).
% subset_image_inj
thf(fact_6829_linorder__injI,axiom,
! [B: $tType,A: $tType] :
( ( linorder @ A )
=> ! [F2: A > B] :
( ! [X3: A,Y3: A] :
( ( ord_less @ A @ X3 @ Y3 )
=> ( ( F2 @ X3 )
!= ( F2 @ Y3 ) ) )
=> ( inj_on @ A @ B @ F2 @ ( top_top @ ( set @ A ) ) ) ) ) ).
% linorder_injI
thf(fact_6830_inj__diff__right,axiom,
! [A: $tType] :
( ( ab_group_add @ A )
=> ! [A2: A] :
( inj_on @ A @ A
@ ^ [B5: A] : ( minus_minus @ A @ B5 @ A2 )
@ ( top_top @ ( set @ A ) ) ) ) ).
% inj_diff_right
thf(fact_6831_inj__on__diff,axiom,
! [B: $tType,A: $tType,F2: A > B,A3: set @ A,B6: set @ A] :
( ( inj_on @ A @ B @ F2 @ A3 )
=> ( inj_on @ A @ B @ F2 @ ( minus_minus @ ( set @ A ) @ A3 @ B6 ) ) ) ).
% inj_on_diff
thf(fact_6832_linorder__inj__onI,axiom,
! [B: $tType,A: $tType] :
( ( order @ A )
=> ! [A3: set @ A,F2: A > B] :
( ! [X3: A,Y3: A] :
( ( ord_less @ A @ X3 @ Y3 )
=> ( ( member @ A @ X3 @ A3 )
=> ( ( member @ A @ Y3 @ A3 )
=> ( ( F2 @ X3 )
!= ( F2 @ Y3 ) ) ) ) )
=> ( ! [X3: A,Y3: A] :
( ( member @ A @ X3 @ A3 )
=> ( ( member @ A @ Y3 @ A3 )
=> ( ( ord_less_eq @ A @ X3 @ Y3 )
| ( ord_less_eq @ A @ Y3 @ X3 ) ) ) )
=> ( inj_on @ A @ B @ F2 @ A3 ) ) ) ) ).
% linorder_inj_onI
thf(fact_6833_inj__on__subset,axiom,
! [B: $tType,A: $tType,F2: A > B,A3: set @ A,B6: set @ A] :
( ( inj_on @ A @ B @ F2 @ A3 )
=> ( ( ord_less_eq @ ( set @ A ) @ B6 @ A3 )
=> ( inj_on @ A @ B @ F2 @ B6 ) ) ) ).
% inj_on_subset
thf(fact_6834_subset__inj__on,axiom,
! [B: $tType,A: $tType,F2: A > B,B6: set @ A,A3: set @ A] :
( ( inj_on @ A @ B @ F2 @ B6 )
=> ( ( ord_less_eq @ ( set @ A ) @ A3 @ B6 )
=> ( inj_on @ A @ B @ F2 @ A3 ) ) ) ).
% subset_inj_on
thf(fact_6835_inj__on__mult,axiom,
! [A: $tType] :
( ( semidom_divide @ A )
=> ! [A2: A,A3: set @ A] :
( ( A2
!= ( zero_zero @ A ) )
=> ( inj_on @ A @ A @ ( times_times @ A @ A2 ) @ A3 ) ) ) ).
% inj_on_mult
thf(fact_6836_inj__on__add_H,axiom,
! [A: $tType] :
( ( cancel_semigroup_add @ A )
=> ! [A2: A,A3: set @ A] :
( inj_on @ A @ A
@ ^ [B5: A] : ( plus_plus @ A @ B5 @ A2 )
@ A3 ) ) ).
% inj_on_add'
thf(fact_6837_inj__on__add,axiom,
! [A: $tType] :
( ( cancel_semigroup_add @ A )
=> ! [A2: A,A3: set @ A] : ( inj_on @ A @ A @ ( plus_plus @ A @ A2 ) @ A3 ) ) ).
% inj_on_add
thf(fact_6838_finite__Collect,axiom,
! [A: $tType,B: $tType,S3: set @ A,F2: B > A] :
( ( finite_finite2 @ A @ S3 )
=> ( ( inj_on @ B @ A @ F2 @ ( top_top @ ( set @ B ) ) )
=> ( finite_finite2 @ B
@ ( collect @ B
@ ^ [A6: B] : ( member @ A @ ( F2 @ A6 ) @ S3 ) ) ) ) ) ).
% finite_Collect
thf(fact_6839_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_6840_inj__add__left,axiom,
! [A: $tType] :
( ( cancel_semigroup_add @ A )
=> ! [A2: A] : ( inj_on @ A @ A @ ( plus_plus @ A @ A2 ) @ ( top_top @ ( set @ A ) ) ) ) ).
% inj_add_left
thf(fact_6841_inj__on__iff__surj,axiom,
! [A: $tType,B: $tType,A3: set @ A,A9: set @ B] :
( ( A3
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( ? [F7: A > B] :
( ( inj_on @ A @ B @ F7 @ A3 )
& ( ord_less_eq @ ( set @ B ) @ ( image @ A @ B @ F7 @ A3 ) @ A9 ) ) )
= ( ? [G4: B > A] :
( ( image @ B @ A @ G4 @ A9 )
= A3 ) ) ) ) ).
% inj_on_iff_surj
thf(fact_6842_inj__image__subset__iff,axiom,
! [B: $tType,A: $tType,F2: A > B,A3: set @ A,B6: set @ A] :
( ( inj_on @ A @ B @ F2 @ ( top_top @ ( set @ A ) ) )
=> ( ( ord_less_eq @ ( set @ B ) @ ( image @ A @ B @ F2 @ A3 ) @ ( image @ A @ B @ F2 @ B6 ) )
= ( ord_less_eq @ ( set @ A ) @ A3 @ B6 ) ) ) ).
% inj_image_subset_iff
thf(fact_6843_endo__inj__surj,axiom,
! [A: $tType,A3: set @ A,F2: A > A] :
( ( finite_finite2 @ A @ A3 )
=> ( ( ord_less_eq @ ( set @ A ) @ ( image @ A @ A @ F2 @ A3 ) @ A3 )
=> ( ( inj_on @ A @ A @ F2 @ A3 )
=> ( ( image @ A @ A @ F2 @ A3 )
= A3 ) ) ) ) ).
% endo_inj_surj
thf(fact_6844_inj__on__finite,axiom,
! [B: $tType,A: $tType,F2: A > B,A3: set @ A,B6: set @ B] :
( ( inj_on @ A @ B @ F2 @ A3 )
=> ( ( ord_less_eq @ ( set @ B ) @ ( image @ A @ B @ F2 @ A3 ) @ B6 )
=> ( ( finite_finite2 @ B @ B6 )
=> ( finite_finite2 @ A @ A3 ) ) ) ) ).
% inj_on_finite
thf(fact_6845_finite__surj__inj,axiom,
! [A: $tType,A3: set @ A,F2: A > A] :
( ( finite_finite2 @ A @ A3 )
=> ( ( ord_less_eq @ ( set @ A ) @ A3 @ ( image @ A @ A @ F2 @ A3 ) )
=> ( inj_on @ A @ A @ F2 @ A3 ) ) ) ).
% finite_surj_inj
thf(fact_6846_image__set__diff,axiom,
! [B: $tType,A: $tType,F2: A > B,A3: set @ A,B6: set @ A] :
( ( inj_on @ A @ B @ F2 @ ( top_top @ ( set @ A ) ) )
=> ( ( image @ A @ B @ F2 @ ( minus_minus @ ( set @ A ) @ A3 @ B6 ) )
= ( minus_minus @ ( set @ B ) @ ( image @ A @ B @ F2 @ A3 ) @ ( image @ A @ B @ F2 @ B6 ) ) ) ) ).
% image_set_diff
thf(fact_6847_inj__on__image__set__diff,axiom,
! [B: $tType,A: $tType,F2: A > B,C5: set @ A,A3: set @ A,B6: set @ A] :
( ( inj_on @ A @ B @ F2 @ C5 )
=> ( ( ord_less_eq @ ( set @ A ) @ ( minus_minus @ ( set @ A ) @ A3 @ B6 ) @ C5 )
=> ( ( ord_less_eq @ ( set @ A ) @ B6 @ C5 )
=> ( ( image @ A @ B @ F2 @ ( minus_minus @ ( set @ A ) @ A3 @ B6 ) )
= ( minus_minus @ ( set @ B ) @ ( image @ A @ B @ F2 @ A3 ) @ ( image @ A @ B @ F2 @ B6 ) ) ) ) ) ) ).
% inj_on_image_set_diff
thf(fact_6848_the__inv__into__into,axiom,
! [B: $tType,A: $tType,F2: A > B,A3: set @ A,X: B,B6: set @ A] :
( ( inj_on @ A @ B @ F2 @ A3 )
=> ( ( member @ B @ X @ ( image @ A @ B @ F2 @ A3 ) )
=> ( ( ord_less_eq @ ( set @ A ) @ A3 @ B6 )
=> ( member @ A @ ( the_inv_into @ A @ B @ A3 @ F2 @ X ) @ B6 ) ) ) ) ).
% the_inv_into_into
thf(fact_6849_finite__imp__nat__seg__image__inj__on,axiom,
! [A: $tType,A3: set @ A] :
( ( finite_finite2 @ A @ A3 )
=> ? [N: nat,F3: nat > A] :
( ( A3
= ( image @ nat @ A @ F3
@ ( collect @ nat
@ ^ [I3: nat] : ( ord_less @ nat @ I3 @ N ) ) ) )
& ( inj_on @ nat @ A @ F3
@ ( collect @ nat
@ ^ [I3: nat] : ( ord_less @ nat @ I3 @ N ) ) ) ) ) ).
% finite_imp_nat_seg_image_inj_on
thf(fact_6850_inj__image__Compl__subset,axiom,
! [B: $tType,A: $tType,F2: A > B,A3: set @ A] :
( ( inj_on @ A @ B @ F2 @ ( top_top @ ( set @ A ) ) )
=> ( ord_less_eq @ ( set @ B ) @ ( image @ A @ B @ F2 @ ( uminus_uminus @ ( set @ A ) @ A3 ) ) @ ( uminus_uminus @ ( set @ B ) @ ( image @ A @ B @ F2 @ A3 ) ) ) ) ).
% inj_image_Compl_subset
thf(fact_6851_infinite__countable__subset,axiom,
! [A: $tType,S3: set @ A] :
( ~ ( finite_finite2 @ A @ S3 )
=> ? [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 ) ) ) @ S3 ) ) ) ).
% infinite_countable_subset
thf(fact_6852_infinite__iff__countable__subset,axiom,
! [A: $tType,S3: set @ A] :
( ( ~ ( finite_finite2 @ A @ S3 ) )
= ( ? [F7: nat > A] :
( ( inj_on @ nat @ A @ F7 @ ( top_top @ ( set @ nat ) ) )
& ( ord_less_eq @ ( set @ A ) @ ( image @ nat @ A @ F7 @ ( top_top @ ( set @ nat ) ) ) @ S3 ) ) ) ) ).
% infinite_iff_countable_subset
thf(fact_6853_inj__on__map__inv__f,axiom,
! [B: $tType,A: $tType,L2: list @ A,A3: set @ A,F2: A > B] :
( ( ord_less_eq @ ( set @ A ) @ ( set2 @ A @ L2 ) @ A3 )
=> ( ( inj_on @ A @ B @ F2 @ A3 )
=> ( ( map @ B @ A @ ( inv_on @ A @ B @ F2 @ A3 ) @ ( map @ A @ B @ F2 @ L2 ) )
= L2 ) ) ) ).
% inj_on_map_inv_f
thf(fact_6854_Schroeder__Bernstein,axiom,
! [A: $tType,B: $tType,F2: A > B,A3: set @ A,B6: set @ B,G: B > A] :
( ( inj_on @ A @ B @ F2 @ A3 )
=> ( ( ord_less_eq @ ( set @ B ) @ ( image @ A @ B @ F2 @ A3 ) @ B6 )
=> ( ( inj_on @ B @ A @ G @ B6 )
=> ( ( ord_less_eq @ ( set @ A ) @ ( image @ B @ A @ G @ B6 ) @ A3 )
=> ? [H4: A > B] : ( bij_betw @ A @ B @ H4 @ A3 @ B6 ) ) ) ) ) ).
% Schroeder_Bernstein
thf(fact_6855_inv__on__f__f,axiom,
! [B: $tType,A: $tType,F2: A > B,A3: set @ A,X: A] :
( ( inj_on @ A @ B @ F2 @ A3 )
=> ( ( member @ A @ X @ A3 )
=> ( ( inv_on @ A @ B @ F2 @ A3 @ ( F2 @ X ) )
= X ) ) ) ).
% inv_on_f_f
thf(fact_6856_inj__Pair_I2_J,axiom,
! [B: $tType,A: $tType,C2: A > B,S3: set @ A] :
( inj_on @ A @ ( product_prod @ B @ A )
@ ^ [X2: A] : ( product_Pair @ B @ A @ ( C2 @ X2 ) @ X2 )
@ S3 ) ).
% inj_Pair(2)
thf(fact_6857_inj__Pair_I1_J,axiom,
! [B: $tType,A: $tType,C2: A > B,S3: set @ A] :
( inj_on @ A @ ( product_prod @ A @ B )
@ ^ [X2: A] : ( product_Pair @ A @ B @ X2 @ ( C2 @ X2 ) )
@ S3 ) ).
% inj_Pair(1)
thf(fact_6858_inj__on__diff__nat,axiom,
! [N7: set @ nat,K: nat] :
( ! [N: nat] :
( ( member @ nat @ N @ N7 )
=> ( ord_less_eq @ nat @ K @ N ) )
=> ( inj_on @ nat @ nat
@ ^ [N2: nat] : ( minus_minus @ nat @ N2 @ K )
@ N7 ) ) ).
% inj_on_diff_nat
thf(fact_6859_inj__on__convol__ident,axiom,
! [B: $tType,A: $tType,F2: A > B,X8: set @ A] :
( inj_on @ A @ ( product_prod @ A @ B )
@ ^ [X2: A] : ( product_Pair @ A @ B @ X2 @ ( F2 @ X2 ) )
@ X8 ) ).
% inj_on_convol_ident
thf(fact_6860_swap__inj__on,axiom,
! [B: $tType,A: $tType,A3: set @ ( product_prod @ A @ B )] :
( inj_on @ ( product_prod @ A @ B ) @ ( product_prod @ B @ A )
@ ( product_case_prod @ A @ B @ ( product_prod @ B @ A )
@ ^ [I3: A,J3: B] : ( product_Pair @ B @ A @ J3 @ I3 ) )
@ A3 ) ).
% swap_inj_on
thf(fact_6861_inj__Suc,axiom,
! [N7: set @ nat] : ( inj_on @ nat @ nat @ suc @ N7 ) ).
% inj_Suc
thf(fact_6862_inj__Some,axiom,
! [A: $tType,A3: set @ A] : ( inj_on @ A @ ( option @ A ) @ ( some @ A ) @ A3 ) ).
% inj_Some
thf(fact_6863_inv__on__f__range,axiom,
! [A: $tType,B: $tType,Y: A,F2: B > A,A3: set @ B] :
( ( member @ A @ Y @ ( image @ B @ A @ F2 @ A3 ) )
=> ( member @ B @ ( inv_on @ B @ A @ F2 @ A3 @ Y ) @ A3 ) ) ).
% inv_on_f_range
thf(fact_6864_f__inv__on__f,axiom,
! [B: $tType,A: $tType,Y: A,F2: B > A,A3: set @ B] :
( ( member @ A @ Y @ ( image @ B @ A @ F2 @ A3 ) )
=> ( ( F2 @ ( inv_on @ B @ A @ F2 @ A3 @ Y ) )
= Y ) ) ).
% f_inv_on_f
thf(fact_6865_le__rel__bool__arg__iff,axiom,
! [A: $tType] :
( ( ord @ A )
=> ( ( ord_less_eq @ ( $o > A ) )
= ( ^ [X6: $o > A,Y8: $o > A] :
( ( ord_less_eq @ A @ ( X6 @ $false ) @ ( Y8 @ $false ) )
& ( ord_less_eq @ A @ ( X6 @ $true ) @ ( Y8 @ $true ) ) ) ) ) ) ).
% le_rel_bool_arg_iff
thf(fact_6866_finite__imp__inj__to__nat__seg_H,axiom,
! [A: $tType,A3: set @ A] :
( ( finite_finite2 @ A @ A3 )
=> ~ ! [F3: A > nat] :
( ? [N: nat] :
( ( image @ A @ nat @ F3 @ A3 )
= ( collect @ nat
@ ^ [I3: nat] : ( ord_less @ nat @ I3 @ N ) ) )
=> ~ ( inj_on @ A @ nat @ F3 @ A3 ) ) ) ).
% finite_imp_inj_to_nat_seg'
thf(fact_6867_finite__imp__inj__to__nat__seg,axiom,
! [A: $tType,A3: set @ A] :
( ( finite_finite2 @ A @ A3 )
=> ? [F3: A > nat,N: nat] :
( ( ( image @ A @ nat @ F3 @ A3 )
= ( collect @ nat
@ ^ [I3: nat] : ( ord_less @ nat @ I3 @ N ) ) )
& ( inj_on @ A @ nat @ F3 @ A3 ) ) ) ).
% finite_imp_inj_to_nat_seg
thf(fact_6868_inv__on__def,axiom,
! [B: $tType,A: $tType] :
( ( inv_on @ A @ B )
= ( ^ [F7: A > B,A7: set @ A,X2: B] :
( fChoice @ A
@ ^ [Y2: A] :
( ( member @ A @ Y2 @ A7 )
& ( ( F7 @ Y2 )
= X2 ) ) ) ) ) ).
% inv_on_def
thf(fact_6869_summable__reindex,axiom,
! [F2: nat > real,G: nat > nat] :
( ( summable @ real @ F2 )
=> ( ( inj_on @ nat @ nat @ G @ ( top_top @ ( set @ nat ) ) )
=> ( ! [X3: nat] : ( ord_less_eq @ real @ ( zero_zero @ real ) @ ( F2 @ X3 ) )
=> ( summable @ real @ ( comp @ nat @ real @ nat @ F2 @ G ) ) ) ) ) ).
% summable_reindex
thf(fact_6870_suminf__reindex__mono,axiom,
! [F2: nat > real,G: nat > nat] :
( ( summable @ real @ F2 )
=> ( ( inj_on @ nat @ nat @ G @ ( top_top @ ( set @ nat ) ) )
=> ( ! [X3: nat] : ( ord_less_eq @ real @ ( zero_zero @ real ) @ ( F2 @ X3 ) )
=> ( ord_less_eq @ real @ ( suminf @ real @ ( comp @ nat @ real @ nat @ F2 @ G ) ) @ ( suminf @ real @ F2 ) ) ) ) ) ).
% suminf_reindex_mono
thf(fact_6871_suminf__reindex,axiom,
! [F2: nat > real,G: nat > nat] :
( ( summable @ real @ F2 )
=> ( ( inj_on @ nat @ nat @ G @ ( top_top @ ( set @ nat ) ) )
=> ( ! [X3: nat] : ( ord_less_eq @ real @ ( zero_zero @ real ) @ ( F2 @ X3 ) )
=> ( ! [X3: nat] :
( ~ ( member @ nat @ X3 @ ( image @ nat @ nat @ G @ ( top_top @ ( set @ nat ) ) ) )
=> ( ( F2 @ X3 )
= ( zero_zero @ real ) ) )
=> ( ( suminf @ real @ ( comp @ nat @ real @ nat @ F2 @ G ) )
= ( suminf @ real @ F2 ) ) ) ) ) ) ).
% suminf_reindex
thf(fact_6872_inj__sgn__power,axiom,
! [N3: nat] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ N3 )
=> ( inj_on @ real @ real
@ ^ [Y2: real] : ( times_times @ real @ ( sgn_sgn @ real @ Y2 ) @ ( power_power @ real @ ( abs_abs @ real @ Y2 ) @ N3 ) )
@ ( top_top @ ( set @ real ) ) ) ) ).
% inj_sgn_power
thf(fact_6873_ex__subset__image__inj,axiom,
! [A: $tType,B: $tType,F2: B > A,S3: set @ B,P: ( set @ A ) > $o] :
( ( ? [T9: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ T9 @ ( image @ B @ A @ F2 @ S3 ) )
& ( P @ T9 ) ) )
= ( ? [T9: set @ B] :
( ( ord_less_eq @ ( set @ B ) @ T9 @ S3 )
& ( inj_on @ B @ A @ F2 @ T9 )
& ( P @ ( image @ B @ A @ F2 @ T9 ) ) ) ) ) ).
% ex_subset_image_inj
thf(fact_6874_all__subset__image__inj,axiom,
! [A: $tType,B: $tType,F2: B > A,S3: set @ B,P: ( set @ A ) > $o] :
( ( ! [T9: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ T9 @ ( image @ B @ A @ F2 @ S3 ) )
=> ( P @ T9 ) ) )
= ( ! [T9: set @ B] :
( ( ( ord_less_eq @ ( set @ B ) @ T9 @ S3 )
& ( inj_on @ B @ A @ F2 @ T9 ) )
=> ( P @ ( image @ B @ A @ F2 @ T9 ) ) ) ) ) ).
% all_subset_image_inj
thf(fact_6875_valid__eq2,axiom,
! [T2: vEBT_VEBT,D: nat] :
( ( vEBT_VEBT_valid @ T2 @ D )
=> ( vEBT_invar_vebt @ T2 @ D ) ) ).
% valid_eq2
thf(fact_6876_valid__eq1,axiom,
! [T2: vEBT_VEBT,D: nat] :
( ( vEBT_invar_vebt @ T2 @ D )
=> ( vEBT_VEBT_valid @ T2 @ D ) ) ).
% valid_eq1
thf(fact_6877_valid__eq,axiom,
vEBT_VEBT_valid = vEBT_invar_vebt ).
% valid_eq
thf(fact_6878_VEBT__internal_Ovalid_H_Osimps_I1_J,axiom,
! [Uu2: $o,Uv2: $o,D: nat] :
( ( vEBT_VEBT_valid @ ( vEBT_Leaf @ Uu2 @ Uv2 ) @ D )
= ( D
= ( one_one @ nat ) ) ) ).
% VEBT_internal.valid'.simps(1)
thf(fact_6879_DERIV__even__real__root,axiom,
! [N3: nat,X: real] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ N3 )
=> ( ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N3 )
=> ( ( ord_less @ real @ X @ ( zero_zero @ real ) )
=> ( has_field_derivative @ real @ ( root @ N3 ) @ ( inverse_inverse @ real @ ( times_times @ real @ ( uminus_uminus @ real @ ( semiring_1_of_nat @ real @ N3 ) ) @ ( power_power @ real @ ( root @ N3 @ X ) @ ( minus_minus @ nat @ N3 @ ( suc @ ( zero_zero @ nat ) ) ) ) ) ) @ ( topolo174197925503356063within @ real @ X @ ( top_top @ ( set @ real ) ) ) ) ) ) ) ).
% DERIV_even_real_root
thf(fact_6880_DERIV__real__root__generic,axiom,
! [N3: nat,X: real,D4: real] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ N3 )
=> ( ( X
!= ( zero_zero @ real ) )
=> ( ( ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N3 )
=> ( ( ord_less @ real @ ( zero_zero @ real ) @ X )
=> ( D4
= ( inverse_inverse @ real @ ( times_times @ real @ ( semiring_1_of_nat @ real @ N3 ) @ ( power_power @ real @ ( root @ N3 @ X ) @ ( minus_minus @ nat @ N3 @ ( suc @ ( zero_zero @ nat ) ) ) ) ) ) ) ) )
=> ( ( ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N3 )
=> ( ( ord_less @ real @ X @ ( zero_zero @ real ) )
=> ( D4
= ( uminus_uminus @ real @ ( inverse_inverse @ real @ ( times_times @ real @ ( semiring_1_of_nat @ real @ N3 ) @ ( power_power @ real @ ( root @ N3 @ X ) @ ( minus_minus @ nat @ N3 @ ( suc @ ( zero_zero @ nat ) ) ) ) ) ) ) ) ) )
=> ( ( ~ ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N3 )
=> ( D4
= ( inverse_inverse @ real @ ( times_times @ real @ ( semiring_1_of_nat @ real @ N3 ) @ ( power_power @ real @ ( root @ N3 @ X ) @ ( minus_minus @ nat @ N3 @ ( suc @ ( zero_zero @ nat ) ) ) ) ) ) ) )
=> ( has_field_derivative @ real @ ( root @ N3 ) @ D4 @ ( topolo174197925503356063within @ real @ X @ ( top_top @ ( set @ real ) ) ) ) ) ) ) ) ) ).
% DERIV_real_root_generic
thf(fact_6881_DERIV__const__ratio__const,axiom,
! [A2: real,B2: real,F2: real > real,K: real] :
( ( A2 != B2 )
=> ( ! [X3: real] : ( has_field_derivative @ real @ F2 @ K @ ( topolo174197925503356063within @ real @ X3 @ ( top_top @ ( set @ real ) ) ) )
=> ( ( minus_minus @ real @ ( F2 @ B2 ) @ ( F2 @ A2 ) )
= ( times_times @ real @ ( minus_minus @ real @ B2 @ A2 ) @ K ) ) ) ) ).
% DERIV_const_ratio_const
thf(fact_6882_DERIV__const__ratio__const2,axiom,
! [A2: real,B2: real,F2: real > real,K: real] :
( ( A2 != B2 )
=> ( ! [X3: real] : ( has_field_derivative @ real @ F2 @ K @ ( topolo174197925503356063within @ real @ X3 @ ( top_top @ ( set @ real ) ) ) )
=> ( ( divide_divide @ real @ ( minus_minus @ real @ ( F2 @ B2 ) @ ( F2 @ A2 ) ) @ ( minus_minus @ real @ B2 @ A2 ) )
= K ) ) ) ).
% DERIV_const_ratio_const2
thf(fact_6883_DERIV__diff,axiom,
! [A: $tType] :
( ( real_V3459762299906320749_field @ A )
=> ! [F2: A > A,D4: A,X: A,S: set @ A,G: A > A,E5: A] :
( ( has_field_derivative @ A @ F2 @ D4 @ ( topolo174197925503356063within @ A @ X @ S ) )
=> ( ( has_field_derivative @ A @ G @ E5 @ ( topolo174197925503356063within @ A @ X @ S ) )
=> ( has_field_derivative @ A
@ ^ [X2: A] : ( minus_minus @ A @ ( F2 @ X2 ) @ ( G @ X2 ) )
@ ( minus_minus @ A @ D4 @ E5 )
@ ( topolo174197925503356063within @ A @ X @ S ) ) ) ) ) ).
% DERIV_diff
thf(fact_6884_field__differentiable__diff,axiom,
! [A: $tType] :
( ( real_V3459762299906320749_field @ A )
=> ! [F2: A > A,F5: A,F4: filter @ A,G: A > A,G5: A] :
( ( has_field_derivative @ A @ F2 @ F5 @ F4 )
=> ( ( has_field_derivative @ A @ G @ G5 @ F4 )
=> ( has_field_derivative @ A
@ ^ [Z3: A] : ( minus_minus @ A @ ( F2 @ Z3 ) @ ( G @ Z3 ) )
@ ( minus_minus @ A @ F5 @ G5 )
@ F4 ) ) ) ) ).
% field_differentiable_diff
thf(fact_6885_has__real__derivative__neg__dec__left,axiom,
! [F2: real > real,L2: real,X: real,S3: set @ real] :
( ( has_field_derivative @ real @ F2 @ L2 @ ( topolo174197925503356063within @ real @ X @ S3 ) )
=> ( ( ord_less @ real @ L2 @ ( zero_zero @ real ) )
=> ? [D3: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ D3 )
& ! [H6: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ H6 )
=> ( ( member @ real @ ( minus_minus @ real @ X @ H6 ) @ S3 )
=> ( ( ord_less @ real @ H6 @ D3 )
=> ( ord_less @ real @ ( F2 @ X ) @ ( F2 @ ( minus_minus @ real @ X @ H6 ) ) ) ) ) ) ) ) ) ).
% has_real_derivative_neg_dec_left
thf(fact_6886_has__real__derivative__pos__inc__left,axiom,
! [F2: real > real,L2: real,X: real,S3: set @ real] :
( ( has_field_derivative @ real @ F2 @ L2 @ ( topolo174197925503356063within @ real @ X @ S3 ) )
=> ( ( ord_less @ real @ ( zero_zero @ real ) @ L2 )
=> ? [D3: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ D3 )
& ! [H6: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ H6 )
=> ( ( member @ real @ ( minus_minus @ real @ X @ H6 ) @ S3 )
=> ( ( ord_less @ real @ H6 @ D3 )
=> ( ord_less @ real @ ( F2 @ ( minus_minus @ real @ X @ H6 ) ) @ ( F2 @ X ) ) ) ) ) ) ) ) ).
% has_real_derivative_pos_inc_left
thf(fact_6887_DERIV__divide,axiom,
! [A: $tType] :
( ( real_V3459762299906320749_field @ A )
=> ! [F2: A > A,D4: A,X: A,S: set @ A,G: A > A,E5: A] :
( ( has_field_derivative @ A @ F2 @ D4 @ ( topolo174197925503356063within @ A @ X @ S ) )
=> ( ( has_field_derivative @ A @ G @ E5 @ ( topolo174197925503356063within @ A @ X @ S ) )
=> ( ( ( G @ X )
!= ( zero_zero @ A ) )
=> ( has_field_derivative @ A
@ ^ [X2: A] : ( divide_divide @ A @ ( F2 @ X2 ) @ ( G @ X2 ) )
@ ( divide_divide @ A @ ( minus_minus @ A @ ( times_times @ A @ D4 @ ( G @ X ) ) @ ( times_times @ A @ ( F2 @ X ) @ E5 ) ) @ ( times_times @ A @ ( G @ X ) @ ( G @ X ) ) )
@ ( topolo174197925503356063within @ A @ X @ S ) ) ) ) ) ) ).
% DERIV_divide
thf(fact_6888_has__real__derivative__pos__inc__right,axiom,
! [F2: real > real,L2: real,X: real,S3: set @ real] :
( ( has_field_derivative @ real @ F2 @ L2 @ ( topolo174197925503356063within @ real @ X @ S3 ) )
=> ( ( ord_less @ real @ ( zero_zero @ real ) @ L2 )
=> ? [D3: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ D3 )
& ! [H6: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ H6 )
=> ( ( member @ real @ ( plus_plus @ real @ X @ H6 ) @ S3 )
=> ( ( ord_less @ real @ H6 @ D3 )
=> ( ord_less @ real @ ( F2 @ X ) @ ( F2 @ ( plus_plus @ real @ X @ H6 ) ) ) ) ) ) ) ) ) ).
% has_real_derivative_pos_inc_right
thf(fact_6889_has__real__derivative__neg__dec__right,axiom,
! [F2: real > real,L2: real,X: real,S3: set @ real] :
( ( has_field_derivative @ real @ F2 @ L2 @ ( topolo174197925503356063within @ real @ X @ S3 ) )
=> ( ( ord_less @ real @ L2 @ ( zero_zero @ real ) )
=> ? [D3: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ D3 )
& ! [H6: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ H6 )
=> ( ( member @ real @ ( plus_plus @ real @ X @ H6 ) @ S3 )
=> ( ( ord_less @ real @ H6 @ D3 )
=> ( ord_less @ real @ ( F2 @ ( plus_plus @ real @ X @ H6 ) ) @ ( F2 @ X ) ) ) ) ) ) ) ) ).
% has_real_derivative_neg_dec_right
thf(fact_6890_DERIV__local__const,axiom,
! [F2: real > real,L2: real,X: real,D: real] :
( ( has_field_derivative @ real @ F2 @ L2 @ ( topolo174197925503356063within @ real @ X @ ( top_top @ ( set @ real ) ) ) )
=> ( ( ord_less @ real @ ( zero_zero @ real ) @ D )
=> ( ! [Y3: real] :
( ( ord_less @ real @ ( abs_abs @ real @ ( minus_minus @ real @ X @ Y3 ) ) @ D )
=> ( ( F2 @ X )
= ( F2 @ Y3 ) ) )
=> ( L2
= ( zero_zero @ real ) ) ) ) ) ).
% DERIV_local_const
thf(fact_6891_DERIV__pos__inc__left,axiom,
! [F2: real > real,L2: real,X: real] :
( ( has_field_derivative @ real @ F2 @ L2 @ ( topolo174197925503356063within @ real @ X @ ( top_top @ ( set @ real ) ) ) )
=> ( ( ord_less @ real @ ( zero_zero @ real ) @ L2 )
=> ? [D3: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ D3 )
& ! [H6: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ H6 )
=> ( ( ord_less @ real @ H6 @ D3 )
=> ( ord_less @ real @ ( F2 @ ( minus_minus @ real @ X @ H6 ) ) @ ( F2 @ X ) ) ) ) ) ) ) ).
% DERIV_pos_inc_left
thf(fact_6892_DERIV__neg__dec__left,axiom,
! [F2: real > real,L2: real,X: real] :
( ( has_field_derivative @ real @ F2 @ L2 @ ( topolo174197925503356063within @ real @ X @ ( top_top @ ( set @ real ) ) ) )
=> ( ( ord_less @ real @ L2 @ ( zero_zero @ real ) )
=> ? [D3: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ D3 )
& ! [H6: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ H6 )
=> ( ( ord_less @ real @ H6 @ D3 )
=> ( ord_less @ real @ ( F2 @ X ) @ ( F2 @ ( minus_minus @ real @ X @ H6 ) ) ) ) ) ) ) ) ).
% DERIV_neg_dec_left
thf(fact_6893_DERIV__pos__inc__right,axiom,
! [F2: real > real,L2: real,X: real] :
( ( has_field_derivative @ real @ F2 @ L2 @ ( topolo174197925503356063within @ real @ X @ ( top_top @ ( set @ real ) ) ) )
=> ( ( ord_less @ real @ ( zero_zero @ real ) @ L2 )
=> ? [D3: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ D3 )
& ! [H6: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ H6 )
=> ( ( ord_less @ real @ H6 @ D3 )
=> ( ord_less @ real @ ( F2 @ X ) @ ( F2 @ ( plus_plus @ real @ X @ H6 ) ) ) ) ) ) ) ) ).
% DERIV_pos_inc_right
thf(fact_6894_DERIV__neg__dec__right,axiom,
! [F2: real > real,L2: real,X: real] :
( ( has_field_derivative @ real @ F2 @ L2 @ ( topolo174197925503356063within @ real @ X @ ( top_top @ ( set @ real ) ) ) )
=> ( ( ord_less @ real @ L2 @ ( zero_zero @ real ) )
=> ? [D3: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ D3 )
& ! [H6: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ H6 )
=> ( ( ord_less @ real @ H6 @ D3 )
=> ( ord_less @ real @ ( F2 @ ( plus_plus @ real @ X @ H6 ) ) @ ( F2 @ X ) ) ) ) ) ) ) ).
% DERIV_neg_dec_right
thf(fact_6895_DERIV__isconst__all,axiom,
! [F2: real > real,X: real,Y: real] :
( ! [X3: real] : ( has_field_derivative @ real @ F2 @ ( zero_zero @ real ) @ ( topolo174197925503356063within @ real @ X3 @ ( top_top @ ( set @ real ) ) ) )
=> ( ( F2 @ X )
= ( F2 @ Y ) ) ) ).
% DERIV_isconst_all
thf(fact_6896_DERIV__const,axiom,
! [A: $tType] :
( ( real_V3459762299906320749_field @ A )
=> ! [K: A,F4: filter @ A] :
( has_field_derivative @ A
@ ^ [X2: A] : K
@ ( zero_zero @ A )
@ F4 ) ) ).
% DERIV_const
thf(fact_6897_DERIV__cong,axiom,
! [A: $tType] :
( ( real_V3459762299906320749_field @ A )
=> ! [F2: A > A,X8: A,F4: filter @ A,Y9: A] :
( ( has_field_derivative @ A @ F2 @ X8 @ F4 )
=> ( ( X8 = Y9 )
=> ( has_field_derivative @ A @ F2 @ Y9 @ F4 ) ) ) ) ).
% DERIV_cong
thf(fact_6898_DERIV__sum,axiom,
! [A: $tType,C: $tType,B: $tType] :
( ( real_V3459762299906320749_field @ B )
=> ! [S3: set @ A,F2: B > A > B,F5: C > A > B,X: C,F4: filter @ B] :
( ! [N: A] :
( ( member @ A @ N @ S3 )
=> ( has_field_derivative @ B
@ ^ [X2: B] : ( F2 @ X2 @ N )
@ ( F5 @ X @ N )
@ F4 ) )
=> ( has_field_derivative @ B
@ ^ [X2: B] : ( groups7311177749621191930dd_sum @ A @ B @ ( F2 @ X2 ) @ S3 )
@ ( groups7311177749621191930dd_sum @ A @ B @ ( F5 @ X ) @ S3 )
@ F4 ) ) ) ).
% DERIV_sum
thf(fact_6899_DERIV__cdivide,axiom,
! [A: $tType] :
( ( real_V3459762299906320749_field @ A )
=> ! [F2: A > A,D4: A,X: A,S: set @ A,C2: A] :
( ( has_field_derivative @ A @ F2 @ D4 @ ( topolo174197925503356063within @ A @ X @ S ) )
=> ( has_field_derivative @ A
@ ^ [X2: A] : ( divide_divide @ A @ ( F2 @ X2 ) @ C2 )
@ ( divide_divide @ A @ D4 @ C2 )
@ ( topolo174197925503356063within @ A @ X @ S ) ) ) ) ).
% DERIV_cdivide
thf(fact_6900_field__differentiable__add,axiom,
! [A: $tType] :
( ( real_V3459762299906320749_field @ A )
=> ! [F2: A > A,F5: A,F4: filter @ A,G: A > A,G5: A] :
( ( has_field_derivative @ A @ F2 @ F5 @ F4 )
=> ( ( has_field_derivative @ A @ G @ G5 @ F4 )
=> ( has_field_derivative @ A
@ ^ [Z3: A] : ( plus_plus @ A @ ( F2 @ Z3 ) @ ( G @ Z3 ) )
@ ( plus_plus @ A @ F5 @ G5 )
@ F4 ) ) ) ) ).
% field_differentiable_add
thf(fact_6901_DERIV__add,axiom,
! [A: $tType] :
( ( real_V3459762299906320749_field @ A )
=> ! [F2: A > A,D4: A,X: A,S: set @ A,G: A > A,E5: A] :
( ( has_field_derivative @ A @ F2 @ D4 @ ( topolo174197925503356063within @ A @ X @ S ) )
=> ( ( has_field_derivative @ A @ G @ E5 @ ( topolo174197925503356063within @ A @ X @ S ) )
=> ( has_field_derivative @ A
@ ^ [X2: A] : ( plus_plus @ A @ ( F2 @ X2 ) @ ( G @ X2 ) )
@ ( plus_plus @ A @ D4 @ E5 )
@ ( topolo174197925503356063within @ A @ X @ S ) ) ) ) ) ).
% DERIV_add
thf(fact_6902_has__field__derivative__cosh,axiom,
! [A13: $tType] :
( ( ( real_Vector_banach @ A13 )
& ( real_V3459762299906320749_field @ A13 ) )
=> ! [G: A13 > A13,Db: A13,X: A13,S: set @ A13] :
( ( has_field_derivative @ A13 @ G @ Db @ ( topolo174197925503356063within @ A13 @ X @ S ) )
=> ( has_field_derivative @ A13
@ ^ [X2: A13] : ( cosh @ A13 @ ( G @ X2 ) )
@ ( times_times @ A13 @ ( sinh @ A13 @ ( G @ X ) ) @ Db )
@ ( topolo174197925503356063within @ A13 @ X @ S ) ) ) ) ).
% has_field_derivative_cosh
thf(fact_6903_has__field__derivative__sinh,axiom,
! [A13: $tType] :
( ( ( real_Vector_banach @ A13 )
& ( real_V3459762299906320749_field @ A13 ) )
=> ! [G: A13 > A13,Db: A13,X: A13,S: set @ A13] :
( ( has_field_derivative @ A13 @ G @ Db @ ( topolo174197925503356063within @ A13 @ X @ S ) )
=> ( has_field_derivative @ A13
@ ^ [X2: A13] : ( sinh @ A13 @ ( G @ X2 ) )
@ ( times_times @ A13 @ ( cosh @ A13 @ ( G @ X ) ) @ Db )
@ ( topolo174197925503356063within @ A13 @ X @ S ) ) ) ) ).
% has_field_derivative_sinh
thf(fact_6904_DERIV__cmult__Id,axiom,
! [A: $tType] :
( ( real_V3459762299906320749_field @ A )
=> ! [C2: A,X: A,S: set @ A] : ( has_field_derivative @ A @ ( times_times @ A @ C2 ) @ C2 @ ( topolo174197925503356063within @ A @ X @ S ) ) ) ).
% DERIV_cmult_Id
thf(fact_6905_DERIV__cmult,axiom,
! [A: $tType] :
( ( real_V3459762299906320749_field @ A )
=> ! [F2: A > A,D4: A,X: A,S: set @ A,C2: A] :
( ( has_field_derivative @ A @ F2 @ D4 @ ( topolo174197925503356063within @ A @ X @ S ) )
=> ( has_field_derivative @ A
@ ^ [X2: A] : ( times_times @ A @ C2 @ ( F2 @ X2 ) )
@ ( times_times @ A @ C2 @ D4 )
@ ( topolo174197925503356063within @ A @ X @ S ) ) ) ) ).
% DERIV_cmult
thf(fact_6906_DERIV__cmult__right,axiom,
! [A: $tType] :
( ( real_V3459762299906320749_field @ A )
=> ! [F2: A > A,D4: A,X: A,S: set @ A,C2: A] :
( ( has_field_derivative @ A @ F2 @ D4 @ ( topolo174197925503356063within @ A @ X @ S ) )
=> ( has_field_derivative @ A
@ ^ [X2: A] : ( times_times @ A @ ( F2 @ X2 ) @ C2 )
@ ( times_times @ A @ D4 @ C2 )
@ ( topolo174197925503356063within @ A @ X @ S ) ) ) ) ).
% DERIV_cmult_right
thf(fact_6907_DERIV__mult,axiom,
! [A: $tType] :
( ( real_V3459762299906320749_field @ A )
=> ! [F2: A > A,Da: A,X: A,S: set @ A,G: A > A,Db: A] :
( ( has_field_derivative @ A @ F2 @ Da @ ( topolo174197925503356063within @ A @ X @ S ) )
=> ( ( has_field_derivative @ A @ G @ Db @ ( topolo174197925503356063within @ A @ X @ S ) )
=> ( has_field_derivative @ A
@ ^ [X2: A] : ( times_times @ A @ ( F2 @ X2 ) @ ( G @ X2 ) )
@ ( plus_plus @ A @ ( times_times @ A @ Da @ ( G @ X ) ) @ ( times_times @ A @ Db @ ( F2 @ X ) ) )
@ ( topolo174197925503356063within @ A @ X @ S ) ) ) ) ) ).
% DERIV_mult
thf(fact_6908_DERIV__mult_H,axiom,
! [A: $tType] :
( ( real_V3459762299906320749_field @ A )
=> ! [F2: A > A,D4: A,X: A,S: set @ A,G: A > A,E5: A] :
( ( has_field_derivative @ A @ F2 @ D4 @ ( topolo174197925503356063within @ A @ X @ S ) )
=> ( ( has_field_derivative @ A @ G @ E5 @ ( topolo174197925503356063within @ A @ X @ S ) )
=> ( has_field_derivative @ A
@ ^ [X2: A] : ( times_times @ A @ ( F2 @ X2 ) @ ( G @ X2 ) )
@ ( plus_plus @ A @ ( times_times @ A @ ( F2 @ X ) @ E5 ) @ ( times_times @ A @ D4 @ ( G @ X ) ) )
@ ( topolo174197925503356063within @ A @ X @ S ) ) ) ) ) ).
% DERIV_mult'
thf(fact_6909_DERIV__unique,axiom,
! [A: $tType] :
( ( real_V3459762299906320749_field @ A )
=> ! [F2: A > A,D4: A,X: A,E5: A] :
( ( has_field_derivative @ A @ F2 @ D4 @ ( topolo174197925503356063within @ A @ X @ ( top_top @ ( set @ A ) ) ) )
=> ( ( has_field_derivative @ A @ F2 @ E5 @ ( topolo174197925503356063within @ A @ X @ ( top_top @ ( set @ A ) ) ) )
=> ( D4 = E5 ) ) ) ) ).
% DERIV_unique
thf(fact_6910_has__field__derivative__at__within,axiom,
! [A: $tType] :
( ( real_V3459762299906320749_field @ A )
=> ! [F2: A > A,F5: A,X: A,S: set @ A] :
( ( has_field_derivative @ A @ F2 @ F5 @ ( topolo174197925503356063within @ A @ X @ ( top_top @ ( set @ A ) ) ) )
=> ( has_field_derivative @ A @ F2 @ F5 @ ( topolo174197925503356063within @ A @ X @ S ) ) ) ) ).
% has_field_derivative_at_within
thf(fact_6911_DERIV__shift,axiom,
! [A: $tType] :
( ( real_V3459762299906320749_field @ A )
=> ! [F2: A > A,Y: A,X: A,Z: A] :
( ( has_field_derivative @ A @ F2 @ Y @ ( topolo174197925503356063within @ A @ ( plus_plus @ A @ X @ Z ) @ ( top_top @ ( set @ A ) ) ) )
= ( has_field_derivative @ A
@ ^ [X2: A] : ( F2 @ ( plus_plus @ A @ X2 @ Z ) )
@ Y
@ ( topolo174197925503356063within @ A @ X @ ( top_top @ ( set @ A ) ) ) ) ) ) ).
% DERIV_shift
thf(fact_6912_DERIV__chain_H,axiom,
! [A: $tType] :
( ( real_V3459762299906320749_field @ A )
=> ! [F2: A > A,D4: A,X: A,S: set @ A,G: A > A,E5: A] :
( ( has_field_derivative @ A @ F2 @ D4 @ ( topolo174197925503356063within @ A @ X @ S ) )
=> ( ( has_field_derivative @ A @ G @ E5 @ ( topolo174197925503356063within @ A @ ( F2 @ X ) @ ( top_top @ ( set @ A ) ) ) )
=> ( has_field_derivative @ A
@ ^ [X2: A] : ( G @ ( F2 @ X2 ) )
@ ( times_times @ A @ E5 @ D4 )
@ ( topolo174197925503356063within @ A @ X @ S ) ) ) ) ) ).
% DERIV_chain'
thf(fact_6913_DERIV__chain2,axiom,
! [A: $tType] :
( ( real_V3459762299906320749_field @ A )
=> ! [F2: A > A,Da: A,G: A > A,X: A,Db: A,S: set @ A] :
( ( has_field_derivative @ A @ F2 @ Da @ ( topolo174197925503356063within @ A @ ( G @ X ) @ ( top_top @ ( set @ A ) ) ) )
=> ( ( has_field_derivative @ A @ G @ Db @ ( topolo174197925503356063within @ A @ X @ S ) )
=> ( has_field_derivative @ A
@ ^ [X2: A] : ( F2 @ ( G @ X2 ) )
@ ( times_times @ A @ Da @ Db )
@ ( topolo174197925503356063within @ A @ X @ S ) ) ) ) ) ).
% DERIV_chain2
thf(fact_6914_DERIV__chain3,axiom,
! [A: $tType] :
( ( real_V3459762299906320749_field @ A )
=> ! [G: A > A,G5: A > A,F2: A > A,F5: A,X: A] :
( ! [X3: A] : ( has_field_derivative @ A @ G @ ( G5 @ X3 ) @ ( topolo174197925503356063within @ A @ X3 @ ( top_top @ ( set @ A ) ) ) )
=> ( ( has_field_derivative @ A @ F2 @ F5 @ ( topolo174197925503356063within @ A @ X @ ( top_top @ ( set @ A ) ) ) )
=> ( has_field_derivative @ A
@ ^ [X2: A] : ( G @ ( F2 @ X2 ) )
@ ( times_times @ A @ F5 @ ( G5 @ ( F2 @ X ) ) )
@ ( topolo174197925503356063within @ A @ X @ ( top_top @ ( set @ A ) ) ) ) ) ) ) ).
% DERIV_chain3
thf(fact_6915_DERIV__chain__s,axiom,
! [A: $tType] :
( ( real_V3459762299906320749_field @ A )
=> ! [S: set @ A,G: A > A,G5: A > A,F2: A > A,F5: A,X: A] :
( ! [X3: A] :
( ( member @ A @ X3 @ S )
=> ( has_field_derivative @ A @ G @ ( G5 @ X3 ) @ ( topolo174197925503356063within @ A @ X3 @ ( top_top @ ( set @ A ) ) ) ) )
=> ( ( has_field_derivative @ A @ F2 @ F5 @ ( topolo174197925503356063within @ A @ X @ ( top_top @ ( set @ A ) ) ) )
=> ( ( member @ A @ ( F2 @ X ) @ S )
=> ( has_field_derivative @ A
@ ^ [X2: A] : ( G @ ( F2 @ X2 ) )
@ ( times_times @ A @ F5 @ ( G5 @ ( F2 @ X ) ) )
@ ( topolo174197925503356063within @ A @ X @ ( top_top @ ( set @ A ) ) ) ) ) ) ) ) ).
% DERIV_chain_s
thf(fact_6916_DERIV__fun__sin,axiom,
! [A: $tType] :
( ( ( real_Vector_banach @ A )
& ( real_V3459762299906320749_field @ A ) )
=> ! [G: A > A,M: A,X: A] :
( ( has_field_derivative @ A @ G @ M @ ( topolo174197925503356063within @ A @ X @ ( top_top @ ( set @ A ) ) ) )
=> ( has_field_derivative @ A
@ ^ [X2: A] : ( sin @ A @ ( G @ X2 ) )
@ ( times_times @ A @ ( cos @ A @ ( G @ X ) ) @ M )
@ ( topolo174197925503356063within @ A @ X @ ( top_top @ ( set @ A ) ) ) ) ) ) ).
% DERIV_fun_sin
thf(fact_6917_DERIV__fun__exp,axiom,
! [A: $tType] :
( ( ( real_Vector_banach @ A )
& ( real_V3459762299906320749_field @ A ) )
=> ! [G: A > A,M: A,X: A] :
( ( has_field_derivative @ A @ G @ M @ ( topolo174197925503356063within @ A @ X @ ( top_top @ ( set @ A ) ) ) )
=> ( has_field_derivative @ A
@ ^ [X2: A] : ( exp @ A @ ( G @ X2 ) )
@ ( times_times @ A @ ( exp @ A @ ( G @ X ) ) @ M )
@ ( topolo174197925503356063within @ A @ X @ ( top_top @ ( set @ A ) ) ) ) ) ) ).
% DERIV_fun_exp
thf(fact_6918_DERIV__mirror,axiom,
! [F2: real > real,Y: real,X: real] :
( ( has_field_derivative @ real @ F2 @ Y @ ( topolo174197925503356063within @ real @ ( uminus_uminus @ real @ X ) @ ( top_top @ ( set @ real ) ) ) )
= ( has_field_derivative @ real
@ ^ [X2: real] : ( F2 @ ( uminus_uminus @ real @ X2 ) )
@ ( uminus_uminus @ real @ Y )
@ ( topolo174197925503356063within @ real @ X @ ( top_top @ ( set @ real ) ) ) ) ) ).
% DERIV_mirror
thf(fact_6919_DERIV__cos__add,axiom,
! [A: $tType] :
( ( ( real_Vector_banach @ A )
& ( real_V3459762299906320749_field @ A ) )
=> ! [K: A,Xa: A] :
( has_field_derivative @ A
@ ^ [X2: A] : ( cos @ A @ ( plus_plus @ A @ X2 @ K ) )
@ ( uminus_uminus @ A @ ( sin @ A @ ( plus_plus @ A @ Xa @ K ) ) )
@ ( topolo174197925503356063within @ A @ Xa @ ( top_top @ ( set @ A ) ) ) ) ) ).
% DERIV_cos_add
thf(fact_6920_DERIV__fun__cos,axiom,
! [A: $tType] :
( ( ( real_Vector_banach @ A )
& ( real_V3459762299906320749_field @ A ) )
=> ! [G: A > A,M: A,X: A] :
( ( has_field_derivative @ A @ G @ M @ ( topolo174197925503356063within @ A @ X @ ( top_top @ ( set @ A ) ) ) )
=> ( has_field_derivative @ A
@ ^ [X2: A] : ( cos @ A @ ( G @ X2 ) )
@ ( times_times @ A @ ( uminus_uminus @ A @ ( sin @ A @ ( G @ X ) ) ) @ M )
@ ( topolo174197925503356063within @ A @ X @ ( top_top @ ( set @ A ) ) ) ) ) ) ).
% DERIV_fun_cos
thf(fact_6921_field__differentiable__minus,axiom,
! [A: $tType] :
( ( real_V3459762299906320749_field @ A )
=> ! [F2: A > A,F5: A,F4: filter @ A] :
( ( has_field_derivative @ A @ F2 @ F5 @ F4 )
=> ( has_field_derivative @ A
@ ^ [Z3: A] : ( uminus_uminus @ A @ ( F2 @ Z3 ) )
@ ( uminus_uminus @ A @ F5 )
@ F4 ) ) ) ).
% field_differentiable_minus
thf(fact_6922_DERIV__minus,axiom,
! [A: $tType] :
( ( real_V3459762299906320749_field @ A )
=> ! [F2: A > A,D4: A,X: A,S: set @ A] :
( ( has_field_derivative @ A @ F2 @ D4 @ ( topolo174197925503356063within @ A @ X @ S ) )
=> ( has_field_derivative @ A
@ ^ [X2: A] : ( uminus_uminus @ A @ ( F2 @ X2 ) )
@ ( uminus_uminus @ A @ D4 )
@ ( topolo174197925503356063within @ A @ X @ S ) ) ) ) ).
% DERIV_minus
thf(fact_6923_DERIV__ident,axiom,
! [A: $tType] :
( ( real_V3459762299906320749_field @ A )
=> ! [F4: filter @ A] :
( has_field_derivative @ A
@ ^ [X2: A] : X2
@ ( one_one @ A )
@ F4 ) ) ).
% DERIV_ident
thf(fact_6924_DERIV__inverse_H,axiom,
! [A: $tType] :
( ( real_V3459762299906320749_field @ A )
=> ! [F2: A > A,D4: A,X: A,S: set @ A] :
( ( has_field_derivative @ A @ F2 @ D4 @ ( topolo174197925503356063within @ A @ X @ S ) )
=> ( ( ( F2 @ X )
!= ( zero_zero @ A ) )
=> ( has_field_derivative @ A
@ ^ [X2: A] : ( inverse_inverse @ A @ ( F2 @ X2 ) )
@ ( uminus_uminus @ A @ ( times_times @ A @ ( times_times @ A @ ( inverse_inverse @ A @ ( F2 @ X ) ) @ D4 ) @ ( inverse_inverse @ A @ ( F2 @ X ) ) ) )
@ ( topolo174197925503356063within @ A @ X @ S ) ) ) ) ) ).
% DERIV_inverse'
thf(fact_6925_DERIV__at__within__shift,axiom,
! [A: $tType] :
( ( real_V3459762299906320749_field @ A )
=> ! [F2: A > A,Y: A,Z: A,X: A,S3: set @ A] :
( ( has_field_derivative @ A @ F2 @ Y @ ( topolo174197925503356063within @ A @ ( plus_plus @ A @ Z @ X ) @ ( image @ A @ A @ ( plus_plus @ A @ Z ) @ S3 ) ) )
= ( has_field_derivative @ A
@ ^ [X2: A] : ( F2 @ ( plus_plus @ A @ Z @ X2 ) )
@ Y
@ ( topolo174197925503356063within @ A @ X @ S3 ) ) ) ) ).
% DERIV_at_within_shift
thf(fact_6926_DERIV__chain,axiom,
! [A: $tType] :
( ( real_V3459762299906320749_field @ A )
=> ! [F2: A > A,Da: A,G: A > A,X: A,Db: A,S: set @ A] :
( ( has_field_derivative @ A @ F2 @ Da @ ( topolo174197925503356063within @ A @ ( G @ X ) @ ( top_top @ ( set @ A ) ) ) )
=> ( ( has_field_derivative @ A @ G @ Db @ ( topolo174197925503356063within @ A @ X @ S ) )
=> ( has_field_derivative @ A @ ( comp @ A @ A @ A @ F2 @ G ) @ ( times_times @ A @ Da @ Db ) @ ( topolo174197925503356063within @ A @ X @ S ) ) ) ) ) ).
% DERIV_chain
thf(fact_6927_DERIV__image__chain,axiom,
! [A: $tType] :
( ( real_V3459762299906320749_field @ A )
=> ! [F2: A > A,Da: A,G: A > A,X: A,S: set @ A,Db: A] :
( ( has_field_derivative @ A @ F2 @ Da @ ( topolo174197925503356063within @ A @ ( G @ X ) @ ( image @ A @ A @ G @ S ) ) )
=> ( ( has_field_derivative @ A @ G @ Db @ ( topolo174197925503356063within @ A @ X @ S ) )
=> ( has_field_derivative @ A @ ( comp @ A @ A @ A @ F2 @ G ) @ ( times_times @ A @ Da @ Db ) @ ( topolo174197925503356063within @ A @ X @ S ) ) ) ) ) ).
% DERIV_image_chain
thf(fact_6928_DERIV__at__within__shift__lemma,axiom,
! [A: $tType] :
( ( real_V3459762299906320749_field @ A )
=> ! [F2: A > A,Y: A,Z: A,X: A,S3: set @ A] :
( ( has_field_derivative @ A @ F2 @ Y @ ( topolo174197925503356063within @ A @ ( plus_plus @ A @ Z @ X ) @ ( image @ A @ A @ ( plus_plus @ A @ Z ) @ S3 ) ) )
=> ( has_field_derivative @ A @ ( comp @ A @ A @ A @ F2 @ ( plus_plus @ A @ Z ) ) @ Y @ ( topolo174197925503356063within @ A @ X @ S3 ) ) ) ) ).
% DERIV_at_within_shift_lemma
thf(fact_6929_DERIV__nonneg__imp__nondecreasing,axiom,
! [A2: real,B2: real,F2: real > real] :
( ( ord_less_eq @ real @ A2 @ B2 )
=> ( ! [X3: real] :
( ( ord_less_eq @ real @ A2 @ X3 )
=> ( ( ord_less_eq @ real @ X3 @ B2 )
=> ? [Y4: real] :
( ( has_field_derivative @ real @ F2 @ Y4 @ ( topolo174197925503356063within @ real @ X3 @ ( top_top @ ( set @ real ) ) ) )
& ( ord_less_eq @ real @ ( zero_zero @ real ) @ Y4 ) ) ) )
=> ( ord_less_eq @ real @ ( F2 @ A2 ) @ ( F2 @ B2 ) ) ) ) ).
% DERIV_nonneg_imp_nondecreasing
thf(fact_6930_DERIV__nonpos__imp__nonincreasing,axiom,
! [A2: real,B2: real,F2: real > real] :
( ( ord_less_eq @ real @ A2 @ B2 )
=> ( ! [X3: real] :
( ( ord_less_eq @ real @ A2 @ X3 )
=> ( ( ord_less_eq @ real @ X3 @ B2 )
=> ? [Y4: real] :
( ( has_field_derivative @ real @ F2 @ Y4 @ ( topolo174197925503356063within @ real @ X3 @ ( top_top @ ( set @ real ) ) ) )
& ( ord_less_eq @ real @ Y4 @ ( zero_zero @ real ) ) ) ) )
=> ( ord_less_eq @ real @ ( F2 @ B2 ) @ ( F2 @ A2 ) ) ) ) ).
% DERIV_nonpos_imp_nonincreasing
thf(fact_6931_DERIV__pos__imp__increasing,axiom,
! [A2: real,B2: real,F2: real > real] :
( ( ord_less @ real @ A2 @ B2 )
=> ( ! [X3: real] :
( ( ord_less_eq @ real @ A2 @ X3 )
=> ( ( ord_less_eq @ real @ X3 @ B2 )
=> ? [Y4: real] :
( ( has_field_derivative @ real @ F2 @ Y4 @ ( topolo174197925503356063within @ real @ X3 @ ( top_top @ ( set @ real ) ) ) )
& ( ord_less @ real @ ( zero_zero @ real ) @ Y4 ) ) ) )
=> ( ord_less @ real @ ( F2 @ A2 ) @ ( F2 @ B2 ) ) ) ) ).
% DERIV_pos_imp_increasing
thf(fact_6932_DERIV__neg__imp__decreasing,axiom,
! [A2: real,B2: real,F2: real > real] :
( ( ord_less @ real @ A2 @ B2 )
=> ( ! [X3: real] :
( ( ord_less_eq @ real @ A2 @ X3 )
=> ( ( ord_less_eq @ real @ X3 @ B2 )
=> ? [Y4: real] :
( ( has_field_derivative @ real @ F2 @ Y4 @ ( topolo174197925503356063within @ real @ X3 @ ( top_top @ ( set @ real ) ) ) )
& ( ord_less @ real @ Y4 @ ( zero_zero @ real ) ) ) ) )
=> ( ord_less @ real @ ( F2 @ B2 ) @ ( F2 @ A2 ) ) ) ) ).
% DERIV_neg_imp_decreasing
thf(fact_6933_deriv__nonneg__imp__mono,axiom,
! [A2: real,B2: real,G: real > real,G5: real > real] :
( ! [X3: real] :
( ( member @ real @ X3 @ ( set_or1337092689740270186AtMost @ real @ A2 @ B2 ) )
=> ( has_field_derivative @ real @ G @ ( G5 @ X3 ) @ ( topolo174197925503356063within @ real @ X3 @ ( top_top @ ( set @ real ) ) ) ) )
=> ( ! [X3: real] :
( ( member @ real @ X3 @ ( set_or1337092689740270186AtMost @ real @ A2 @ B2 ) )
=> ( ord_less_eq @ real @ ( zero_zero @ real ) @ ( G5 @ X3 ) ) )
=> ( ( ord_less_eq @ real @ A2 @ B2 )
=> ( ord_less_eq @ real @ ( G @ A2 ) @ ( G @ B2 ) ) ) ) ) ).
% deriv_nonneg_imp_mono
thf(fact_6934_at__le,axiom,
! [A: $tType] :
( ( topolo4958980785337419405_space @ A )
=> ! [S: set @ A,T2: set @ A,X: A] :
( ( ord_less_eq @ ( set @ A ) @ S @ T2 )
=> ( ord_less_eq @ ( filter @ A ) @ ( topolo174197925503356063within @ A @ X @ S ) @ ( topolo174197925503356063within @ A @ X @ T2 ) ) ) ) ).
% at_le
thf(fact_6935_DERIV__subset,axiom,
! [A: $tType] :
( ( real_V3459762299906320749_field @ A )
=> ! [F2: A > A,F5: A,X: A,S: set @ A,T2: set @ A] :
( ( has_field_derivative @ A @ F2 @ F5 @ ( topolo174197925503356063within @ A @ X @ S ) )
=> ( ( ord_less_eq @ ( set @ A ) @ T2 @ S )
=> ( has_field_derivative @ A @ F2 @ F5 @ ( topolo174197925503356063within @ A @ X @ T2 ) ) ) ) ) ).
% DERIV_subset
thf(fact_6936_has__field__derivative__subset,axiom,
! [A: $tType] :
( ( real_V3459762299906320749_field @ A )
=> ! [F2: A > A,Y: A,X: A,S: set @ A,T2: set @ A] :
( ( has_field_derivative @ A @ F2 @ Y @ ( topolo174197925503356063within @ A @ X @ S ) )
=> ( ( ord_less_eq @ ( set @ A ) @ T2 @ S )
=> ( has_field_derivative @ A @ F2 @ Y @ ( topolo174197925503356063within @ A @ X @ T2 ) ) ) ) ) ).
% has_field_derivative_subset
thf(fact_6937_MVT2,axiom,
! [A2: real,B2: real,F2: real > real,F5: real > real] :
( ( ord_less @ real @ A2 @ B2 )
=> ( ! [X3: real] :
( ( ord_less_eq @ real @ A2 @ X3 )
=> ( ( ord_less_eq @ real @ X3 @ B2 )
=> ( has_field_derivative @ real @ F2 @ ( F5 @ X3 ) @ ( topolo174197925503356063within @ real @ X3 @ ( top_top @ ( set @ real ) ) ) ) ) )
=> ? [Z2: real] :
( ( ord_less @ real @ A2 @ Z2 )
& ( ord_less @ real @ Z2 @ B2 )
& ( ( minus_minus @ real @ ( F2 @ B2 ) @ ( F2 @ A2 ) )
= ( times_times @ real @ ( minus_minus @ real @ B2 @ A2 ) @ ( F5 @ Z2 ) ) ) ) ) ) ).
% MVT2
thf(fact_6938_DERIV__power__Suc,axiom,
! [A: $tType] :
( ( real_V3459762299906320749_field @ A )
=> ! [F2: A > A,D4: A,X: A,S: set @ A,N3: nat] :
( ( has_field_derivative @ A @ F2 @ D4 @ ( topolo174197925503356063within @ A @ X @ S ) )
=> ( has_field_derivative @ A
@ ^ [X2: A] : ( power_power @ A @ ( F2 @ X2 ) @ ( suc @ N3 ) )
@ ( times_times @ A @ ( plus_plus @ A @ ( one_one @ A ) @ ( semiring_1_of_nat @ A @ N3 ) ) @ ( times_times @ A @ D4 @ ( power_power @ A @ ( F2 @ X ) @ N3 ) ) )
@ ( topolo174197925503356063within @ A @ X @ S ) ) ) ) ).
% DERIV_power_Suc
thf(fact_6939_DERIV__const__average,axiom,
! [A2: real,B2: real,V: real > real,K: real] :
( ( A2 != B2 )
=> ( ! [X3: real] : ( has_field_derivative @ real @ V @ K @ ( topolo174197925503356063within @ real @ X3 @ ( top_top @ ( set @ real ) ) ) )
=> ( ( V @ ( divide_divide @ real @ ( plus_plus @ real @ A2 @ B2 ) @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) )
= ( divide_divide @ real @ ( plus_plus @ real @ ( V @ A2 ) @ ( V @ B2 ) ) @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) ) ) ) ).
% DERIV_const_average
thf(fact_6940_DERIV__power,axiom,
! [A: $tType] :
( ( real_V3459762299906320749_field @ A )
=> ! [F2: A > A,D4: A,X: A,S: set @ A,N3: nat] :
( ( has_field_derivative @ A @ F2 @ D4 @ ( topolo174197925503356063within @ A @ X @ S ) )
=> ( has_field_derivative @ A
@ ^ [X2: A] : ( power_power @ A @ ( F2 @ X2 ) @ N3 )
@ ( times_times @ A @ ( semiring_1_of_nat @ A @ N3 ) @ ( times_times @ A @ D4 @ ( power_power @ A @ ( F2 @ X ) @ ( minus_minus @ nat @ N3 @ ( suc @ ( zero_zero @ nat ) ) ) ) ) )
@ ( topolo174197925503356063within @ A @ X @ S ) ) ) ) ).
% DERIV_power
thf(fact_6941_DERIV__inverse,axiom,
! [A: $tType] :
( ( real_V3459762299906320749_field @ A )
=> ! [X: A,S: set @ A] :
( ( X
!= ( zero_zero @ A ) )
=> ( has_field_derivative @ A @ ( inverse_inverse @ A ) @ ( uminus_uminus @ A @ ( power_power @ A @ ( inverse_inverse @ A @ X ) @ ( suc @ ( suc @ ( zero_zero @ nat ) ) ) ) ) @ ( topolo174197925503356063within @ A @ X @ S ) ) ) ) ).
% DERIV_inverse
thf(fact_6942_DERIV__local__max,axiom,
! [F2: real > real,L2: real,X: real,D: real] :
( ( has_field_derivative @ real @ F2 @ L2 @ ( topolo174197925503356063within @ real @ X @ ( top_top @ ( set @ real ) ) ) )
=> ( ( ord_less @ real @ ( zero_zero @ real ) @ D )
=> ( ! [Y3: real] :
( ( ord_less @ real @ ( abs_abs @ real @ ( minus_minus @ real @ X @ Y3 ) ) @ D )
=> ( ord_less_eq @ real @ ( F2 @ Y3 ) @ ( F2 @ X ) ) )
=> ( L2
= ( zero_zero @ real ) ) ) ) ) ).
% DERIV_local_max
thf(fact_6943_DERIV__local__min,axiom,
! [F2: real > real,L2: real,X: real,D: real] :
( ( has_field_derivative @ real @ F2 @ L2 @ ( topolo174197925503356063within @ real @ X @ ( top_top @ ( set @ real ) ) ) )
=> ( ( ord_less @ real @ ( zero_zero @ real ) @ D )
=> ( ! [Y3: real] :
( ( ord_less @ real @ ( abs_abs @ real @ ( minus_minus @ real @ X @ Y3 ) ) @ D )
=> ( ord_less_eq @ real @ ( F2 @ X ) @ ( F2 @ Y3 ) ) )
=> ( L2
= ( zero_zero @ real ) ) ) ) ) ).
% DERIV_local_min
thf(fact_6944_DERIV__ln__divide,axiom,
! [X: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ X )
=> ( has_field_derivative @ real @ ( ln_ln @ real ) @ ( divide_divide @ real @ ( one_one @ real ) @ X ) @ ( topolo174197925503356063within @ real @ X @ ( top_top @ ( set @ real ) ) ) ) ) ).
% DERIV_ln_divide
thf(fact_6945_DERIV__pow,axiom,
! [N3: nat,X: real,S: set @ real] :
( has_field_derivative @ real
@ ^ [X2: real] : ( power_power @ real @ X2 @ N3 )
@ ( times_times @ real @ ( semiring_1_of_nat @ real @ N3 ) @ ( power_power @ real @ X @ ( minus_minus @ nat @ N3 @ ( suc @ ( zero_zero @ nat ) ) ) ) )
@ ( topolo174197925503356063within @ real @ X @ S ) ) ).
% DERIV_pow
thf(fact_6946_termdiffs__strong__converges__everywhere,axiom,
! [A: $tType] :
( ( ( real_Vector_banach @ A )
& ( real_V3459762299906320749_field @ A ) )
=> ! [C2: nat > A,X: A] :
( ! [Y3: A] :
( summable @ A
@ ^ [N2: nat] : ( times_times @ A @ ( C2 @ N2 ) @ ( power_power @ A @ Y3 @ N2 ) ) )
=> ( has_field_derivative @ A
@ ^ [X2: A] :
( suminf @ A
@ ^ [N2: nat] : ( times_times @ A @ ( C2 @ N2 ) @ ( power_power @ A @ X2 @ N2 ) ) )
@ ( suminf @ A
@ ^ [N2: nat] : ( times_times @ A @ ( diffs @ A @ C2 @ N2 ) @ ( power_power @ A @ X @ N2 ) ) )
@ ( topolo174197925503356063within @ A @ X @ ( top_top @ ( set @ A ) ) ) ) ) ) ).
% termdiffs_strong_converges_everywhere
thf(fact_6947_at__within__Icc__at,axiom,
! [A: $tType] :
( ( topolo2564578578187576103pology @ A )
=> ! [A2: A,X: A,B2: A] :
( ( ord_less @ A @ A2 @ X )
=> ( ( ord_less @ A @ X @ B2 )
=> ( ( topolo174197925503356063within @ A @ X @ ( set_or1337092689740270186AtMost @ A @ A2 @ B2 ) )
= ( topolo174197925503356063within @ A @ X @ ( top_top @ ( set @ A ) ) ) ) ) ) ) ).
% at_within_Icc_at
thf(fact_6948_DERIV__fun__pow,axiom,
! [G: real > real,M: real,X: real,N3: nat] :
( ( has_field_derivative @ real @ G @ M @ ( topolo174197925503356063within @ real @ X @ ( top_top @ ( set @ real ) ) ) )
=> ( has_field_derivative @ real
@ ^ [X2: real] : ( power_power @ real @ ( G @ X2 ) @ N3 )
@ ( times_times @ real @ ( times_times @ real @ ( semiring_1_of_nat @ real @ N3 ) @ ( power_power @ real @ ( G @ X ) @ ( minus_minus @ nat @ N3 @ ( one_one @ nat ) ) ) ) @ M )
@ ( topolo174197925503356063within @ real @ X @ ( top_top @ ( set @ real ) ) ) ) ) ).
% DERIV_fun_pow
thf(fact_6949_at__within__Icc__at__left,axiom,
! [A: $tType] :
( ( topolo2564578578187576103pology @ A )
=> ! [A2: A,B2: A] :
( ( ord_less @ A @ A2 @ B2 )
=> ( ( topolo174197925503356063within @ A @ B2 @ ( set_or1337092689740270186AtMost @ A @ A2 @ B2 ) )
= ( topolo174197925503356063within @ A @ B2 @ ( set_ord_lessThan @ A @ B2 ) ) ) ) ) ).
% at_within_Icc_at_left
thf(fact_6950_DERIV__quotient,axiom,
! [A: $tType] :
( ( real_V3459762299906320749_field @ A )
=> ! [F2: A > A,D: A,X: A,S: set @ A,G: A > A,E: A] :
( ( has_field_derivative @ A @ F2 @ D @ ( topolo174197925503356063within @ A @ X @ S ) )
=> ( ( has_field_derivative @ A @ G @ E @ ( topolo174197925503356063within @ A @ X @ S ) )
=> ( ( ( G @ X )
!= ( zero_zero @ A ) )
=> ( has_field_derivative @ A
@ ^ [Y2: A] : ( divide_divide @ A @ ( F2 @ Y2 ) @ ( G @ Y2 ) )
@ ( divide_divide @ A @ ( minus_minus @ A @ ( times_times @ A @ D @ ( G @ X ) ) @ ( times_times @ A @ E @ ( F2 @ X ) ) ) @ ( power_power @ A @ ( G @ X ) @ ( suc @ ( suc @ ( zero_zero @ nat ) ) ) ) )
@ ( topolo174197925503356063within @ A @ X @ S ) ) ) ) ) ) ).
% DERIV_quotient
thf(fact_6951_DERIV__inverse__fun,axiom,
! [A: $tType] :
( ( real_V3459762299906320749_field @ A )
=> ! [F2: A > A,D: A,X: A,S: set @ A] :
( ( has_field_derivative @ A @ F2 @ D @ ( topolo174197925503356063within @ A @ X @ S ) )
=> ( ( ( F2 @ X )
!= ( zero_zero @ A ) )
=> ( has_field_derivative @ A
@ ^ [X2: A] : ( inverse_inverse @ A @ ( F2 @ X2 ) )
@ ( uminus_uminus @ A @ ( times_times @ A @ D @ ( inverse_inverse @ A @ ( power_power @ A @ ( F2 @ X ) @ ( suc @ ( suc @ ( zero_zero @ nat ) ) ) ) ) ) )
@ ( topolo174197925503356063within @ A @ X @ S ) ) ) ) ) ).
% DERIV_inverse_fun
thf(fact_6952_termdiffs__sums__strong,axiom,
! [A: $tType] :
( ( ( real_Vector_banach @ A )
& ( real_V3459762299906320749_field @ A ) )
=> ! [K6: real,C2: nat > A,F2: A > A,F5: A,Z: A] :
( ! [Z2: A] :
( ( ord_less @ real @ ( real_V7770717601297561774m_norm @ A @ Z2 ) @ K6 )
=> ( sums @ A
@ ^ [N2: nat] : ( times_times @ A @ ( C2 @ N2 ) @ ( power_power @ A @ Z2 @ N2 ) )
@ ( F2 @ Z2 ) ) )
=> ( ( has_field_derivative @ A @ F2 @ F5 @ ( topolo174197925503356063within @ A @ Z @ ( top_top @ ( set @ A ) ) ) )
=> ( ( ord_less @ real @ ( real_V7770717601297561774m_norm @ A @ Z ) @ K6 )
=> ( sums @ A
@ ^ [N2: nat] : ( times_times @ A @ ( diffs @ A @ C2 @ N2 ) @ ( power_power @ A @ Z @ N2 ) )
@ F5 ) ) ) ) ) ).
% termdiffs_sums_strong
thf(fact_6953_has__real__derivative__powr,axiom,
! [Z: real,R2: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ Z )
=> ( has_field_derivative @ real
@ ^ [Z3: real] : ( powr @ real @ Z3 @ 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_6954_termdiffs__strong_H,axiom,
! [A: $tType] :
( ( ( real_Vector_banach @ A )
& ( real_V3459762299906320749_field @ A ) )
=> ! [K6: real,C2: nat > A,Z: A] :
( ! [Z2: A] :
( ( ord_less @ real @ ( real_V7770717601297561774m_norm @ A @ Z2 ) @ K6 )
=> ( summable @ A
@ ^ [N2: nat] : ( times_times @ A @ ( C2 @ N2 ) @ ( power_power @ A @ Z2 @ N2 ) ) ) )
=> ( ( ord_less @ real @ ( real_V7770717601297561774m_norm @ A @ Z ) @ K6 )
=> ( has_field_derivative @ A
@ ^ [Z3: A] :
( suminf @ A
@ ^ [N2: nat] : ( times_times @ A @ ( C2 @ N2 ) @ ( power_power @ A @ Z3 @ N2 ) ) )
@ ( suminf @ A
@ ^ [N2: nat] : ( times_times @ A @ ( diffs @ A @ C2 @ N2 ) @ ( power_power @ A @ Z @ N2 ) ) )
@ ( topolo174197925503356063within @ A @ Z @ ( top_top @ ( set @ A ) ) ) ) ) ) ) ).
% termdiffs_strong'
thf(fact_6955_termdiffs__strong,axiom,
! [A: $tType] :
( ( ( real_Vector_banach @ A )
& ( real_V3459762299906320749_field @ A ) )
=> ! [C2: nat > A,K6: A,X: A] :
( ( summable @ A
@ ^ [N2: nat] : ( times_times @ A @ ( C2 @ N2 ) @ ( power_power @ A @ K6 @ N2 ) ) )
=> ( ( ord_less @ real @ ( real_V7770717601297561774m_norm @ A @ X ) @ ( real_V7770717601297561774m_norm @ A @ K6 ) )
=> ( has_field_derivative @ A
@ ^ [X2: A] :
( suminf @ A
@ ^ [N2: nat] : ( times_times @ A @ ( C2 @ N2 ) @ ( power_power @ A @ X2 @ N2 ) ) )
@ ( suminf @ A
@ ^ [N2: nat] : ( times_times @ A @ ( diffs @ A @ C2 @ N2 ) @ ( power_power @ A @ X @ N2 ) ) )
@ ( topolo174197925503356063within @ A @ X @ ( top_top @ ( set @ A ) ) ) ) ) ) ) ).
% termdiffs_strong
thf(fact_6956_termdiffs,axiom,
! [A: $tType] :
( ( ( real_Vector_banach @ A )
& ( real_V3459762299906320749_field @ A ) )
=> ! [C2: nat > A,K6: A,X: A] :
( ( summable @ A
@ ^ [N2: nat] : ( times_times @ A @ ( C2 @ N2 ) @ ( power_power @ A @ K6 @ N2 ) ) )
=> ( ( summable @ A
@ ^ [N2: nat] : ( times_times @ A @ ( diffs @ A @ C2 @ N2 ) @ ( power_power @ A @ K6 @ N2 ) ) )
=> ( ( summable @ A
@ ^ [N2: nat] : ( times_times @ A @ ( diffs @ A @ ( diffs @ A @ C2 ) @ N2 ) @ ( power_power @ A @ K6 @ N2 ) ) )
=> ( ( ord_less @ real @ ( real_V7770717601297561774m_norm @ A @ X ) @ ( real_V7770717601297561774m_norm @ A @ K6 ) )
=> ( has_field_derivative @ A
@ ^ [X2: A] :
( suminf @ A
@ ^ [N2: nat] : ( times_times @ A @ ( C2 @ N2 ) @ ( power_power @ A @ X2 @ N2 ) ) )
@ ( suminf @ A
@ ^ [N2: nat] : ( times_times @ A @ ( diffs @ A @ C2 @ N2 ) @ ( power_power @ A @ X @ N2 ) ) )
@ ( topolo174197925503356063within @ A @ X @ ( top_top @ ( set @ A ) ) ) ) ) ) ) ) ) ).
% termdiffs
thf(fact_6957_DERIV__fun__powr,axiom,
! [G: real > real,M: real,X: real,R2: real] :
( ( has_field_derivative @ real @ G @ M @ ( topolo174197925503356063within @ real @ X @ ( top_top @ ( set @ real ) ) ) )
=> ( ( ord_less @ real @ ( zero_zero @ real ) @ ( G @ X ) )
=> ( has_field_derivative @ real
@ ^ [X2: real] : ( powr @ real @ ( G @ X2 ) @ R2 )
@ ( times_times @ real @ ( times_times @ real @ R2 @ ( powr @ real @ ( G @ X ) @ ( minus_minus @ real @ R2 @ ( semiring_1_of_nat @ real @ ( one_one @ nat ) ) ) ) ) @ M )
@ ( topolo174197925503356063within @ real @ X @ ( top_top @ ( set @ real ) ) ) ) ) ) ).
% DERIV_fun_powr
thf(fact_6958_DERIV__log,axiom,
! [X: real,B2: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ X )
=> ( has_field_derivative @ real @ ( log @ B2 ) @ ( divide_divide @ real @ ( one_one @ real ) @ ( times_times @ real @ ( ln_ln @ real @ B2 ) @ X ) ) @ ( topolo174197925503356063within @ real @ X @ ( top_top @ ( set @ real ) ) ) ) ) ).
% DERIV_log
thf(fact_6959_DERIV__powr,axiom,
! [G: real > real,M: real,X: real,F2: real > real,R2: real] :
( ( has_field_derivative @ real @ G @ M @ ( topolo174197925503356063within @ real @ X @ ( top_top @ ( set @ real ) ) ) )
=> ( ( ord_less @ real @ ( zero_zero @ real ) @ ( G @ X ) )
=> ( ( has_field_derivative @ real @ F2 @ R2 @ ( topolo174197925503356063within @ real @ X @ ( top_top @ ( set @ real ) ) ) )
=> ( has_field_derivative @ real
@ ^ [X2: real] : ( powr @ real @ ( G @ X2 ) @ ( F2 @ X2 ) )
@ ( times_times @ real @ ( powr @ real @ ( G @ X ) @ ( F2 @ X ) ) @ ( plus_plus @ real @ ( times_times @ real @ R2 @ ( ln_ln @ real @ ( G @ X ) ) ) @ ( divide_divide @ real @ ( times_times @ real @ M @ ( F2 @ X ) ) @ ( G @ X ) ) ) )
@ ( topolo174197925503356063within @ real @ X @ ( top_top @ ( set @ real ) ) ) ) ) ) ) ).
% DERIV_powr
thf(fact_6960_DERIV__tan,axiom,
! [A: $tType] :
( ( ( real_Vector_banach @ A )
& ( real_V3459762299906320749_field @ A ) )
=> ! [X: A] :
( ( ( cos @ A @ X )
!= ( zero_zero @ A ) )
=> ( has_field_derivative @ A @ ( tan @ A ) @ ( inverse_inverse @ A @ ( power_power @ A @ ( cos @ A @ X ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( topolo174197925503356063within @ A @ X @ ( top_top @ ( set @ A ) ) ) ) ) ) ).
% DERIV_tan
thf(fact_6961_DERIV__real__sqrt,axiom,
! [X: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ X )
=> ( has_field_derivative @ real @ sqrt @ ( divide_divide @ real @ ( inverse_inverse @ real @ ( sqrt @ X ) ) @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) @ ( topolo174197925503356063within @ real @ X @ ( top_top @ ( set @ real ) ) ) ) ) ).
% DERIV_real_sqrt
thf(fact_6962_DERIV__arctan,axiom,
! [X: real] : ( has_field_derivative @ real @ arctan @ ( inverse_inverse @ real @ ( plus_plus @ real @ ( one_one @ real ) @ ( power_power @ real @ X @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( topolo174197925503356063within @ real @ X @ ( top_top @ ( set @ real ) ) ) ) ).
% DERIV_arctan
thf(fact_6963_arsinh__real__has__field__derivative,axiom,
! [X: real,A3: set @ real] : ( has_field_derivative @ real @ ( arsinh @ real ) @ ( divide_divide @ real @ ( one_one @ real ) @ ( sqrt @ ( plus_plus @ real @ ( power_power @ real @ X @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ ( one_one @ real ) ) ) ) @ ( topolo174197925503356063within @ real @ X @ A3 ) ) ).
% arsinh_real_has_field_derivative
thf(fact_6964_DERIV__cot,axiom,
! [A: $tType] :
( ( ( real_Vector_banach @ A )
& ( real_V3459762299906320749_field @ A ) )
=> ! [X: A] :
( ( ( sin @ A @ X )
!= ( zero_zero @ A ) )
=> ( has_field_derivative @ A @ ( cot @ A ) @ ( uminus_uminus @ A @ ( inverse_inverse @ A @ ( power_power @ A @ ( sin @ A @ X ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( topolo174197925503356063within @ A @ X @ ( top_top @ ( set @ A ) ) ) ) ) ) ).
% DERIV_cot
thf(fact_6965_has__field__derivative__tanh,axiom,
! [A13: $tType] :
( ( ( real_Vector_banach @ A13 )
& ( real_V3459762299906320749_field @ A13 ) )
=> ! [G: A13 > A13,X: A13,Db: A13,S: set @ A13] :
( ( ( cosh @ A13 @ ( G @ X ) )
!= ( zero_zero @ A13 ) )
=> ( ( has_field_derivative @ A13 @ G @ Db @ ( topolo174197925503356063within @ A13 @ X @ S ) )
=> ( has_field_derivative @ A13
@ ^ [X2: A13] : ( tanh @ A13 @ ( G @ X2 ) )
@ ( times_times @ A13 @ ( minus_minus @ A13 @ ( one_one @ A13 ) @ ( power_power @ A13 @ ( tanh @ A13 @ ( G @ X ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ Db )
@ ( topolo174197925503356063within @ A13 @ X @ S ) ) ) ) ) ).
% has_field_derivative_tanh
thf(fact_6966_DERIV__real__sqrt__generic,axiom,
! [X: real,D4: real] :
( ( X
!= ( zero_zero @ real ) )
=> ( ( ( ord_less @ real @ ( zero_zero @ real ) @ X )
=> ( D4
= ( divide_divide @ real @ ( inverse_inverse @ real @ ( sqrt @ X ) ) @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) ) )
=> ( ( ( ord_less @ real @ X @ ( zero_zero @ real ) )
=> ( D4
= ( divide_divide @ real @ ( uminus_uminus @ real @ ( inverse_inverse @ real @ ( sqrt @ X ) ) ) @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) ) )
=> ( has_field_derivative @ real @ sqrt @ D4 @ ( topolo174197925503356063within @ real @ X @ ( top_top @ ( set @ real ) ) ) ) ) ) ) ).
% DERIV_real_sqrt_generic
thf(fact_6967_arcosh__real__has__field__derivative,axiom,
! [X: real,A3: set @ real] :
( ( ord_less @ real @ ( one_one @ real ) @ X )
=> ( has_field_derivative @ real @ ( arcosh @ real ) @ ( divide_divide @ real @ ( one_one @ real ) @ ( sqrt @ ( minus_minus @ real @ ( power_power @ real @ X @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ ( one_one @ real ) ) ) ) @ ( topolo174197925503356063within @ real @ X @ A3 ) ) ) ).
% arcosh_real_has_field_derivative
thf(fact_6968_artanh__real__has__field__derivative,axiom,
! [X: real,A3: set @ real] :
( ( ord_less @ real @ ( abs_abs @ real @ X ) @ ( one_one @ real ) )
=> ( has_field_derivative @ real @ ( artanh @ real ) @ ( divide_divide @ real @ ( one_one @ real ) @ ( minus_minus @ real @ ( one_one @ real ) @ ( power_power @ real @ X @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( topolo174197925503356063within @ real @ X @ A3 ) ) ) ).
% artanh_real_has_field_derivative
thf(fact_6969_DERIV__real__root,axiom,
! [N3: nat,X: real] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ N3 )
=> ( ( ord_less @ real @ ( zero_zero @ real ) @ X )
=> ( has_field_derivative @ real @ ( root @ N3 ) @ ( inverse_inverse @ real @ ( times_times @ real @ ( semiring_1_of_nat @ real @ N3 ) @ ( power_power @ real @ ( root @ N3 @ X ) @ ( minus_minus @ nat @ N3 @ ( suc @ ( zero_zero @ nat ) ) ) ) ) ) @ ( topolo174197925503356063within @ real @ X @ ( top_top @ ( set @ real ) ) ) ) ) ) ).
% DERIV_real_root
thf(fact_6970_DERIV__arccos,axiom,
! [X: real] :
( ( ord_less @ real @ ( uminus_uminus @ real @ ( one_one @ real ) ) @ X )
=> ( ( ord_less @ real @ X @ ( one_one @ real ) )
=> ( has_field_derivative @ real @ arccos @ ( inverse_inverse @ real @ ( uminus_uminus @ real @ ( sqrt @ ( minus_minus @ real @ ( one_one @ real ) @ ( power_power @ real @ X @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) ) @ ( topolo174197925503356063within @ real @ X @ ( top_top @ ( set @ real ) ) ) ) ) ) ).
% DERIV_arccos
thf(fact_6971_DERIV__arcsin,axiom,
! [X: real] :
( ( ord_less @ real @ ( uminus_uminus @ real @ ( one_one @ real ) ) @ X )
=> ( ( ord_less @ real @ X @ ( one_one @ real ) )
=> ( has_field_derivative @ real @ arcsin @ ( inverse_inverse @ real @ ( sqrt @ ( minus_minus @ real @ ( one_one @ real ) @ ( power_power @ real @ X @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) @ ( topolo174197925503356063within @ real @ X @ ( top_top @ ( set @ real ) ) ) ) ) ) ).
% DERIV_arcsin
thf(fact_6972_Maclaurin__all__le__objl,axiom,
! [Diff: nat > real > real,F2: real > real,X: real,N3: nat] :
( ( ( ( Diff @ ( zero_zero @ nat ) )
= F2 )
& ! [M4: nat,X3: real] : ( has_field_derivative @ real @ ( Diff @ M4 ) @ ( Diff @ ( suc @ M4 ) @ X3 ) @ ( topolo174197925503356063within @ real @ X3 @ ( top_top @ ( set @ real ) ) ) ) )
=> ? [T8: real] :
( ( ord_less_eq @ real @ ( abs_abs @ real @ T8 ) @ ( abs_abs @ real @ X ) )
& ( ( F2 @ X )
= ( plus_plus @ real
@ ( groups7311177749621191930dd_sum @ nat @ real
@ ^ [M5: nat] : ( times_times @ real @ ( divide_divide @ real @ ( Diff @ M5 @ ( zero_zero @ real ) ) @ ( semiring_char_0_fact @ real @ M5 ) ) @ ( power_power @ real @ X @ M5 ) )
@ ( set_ord_lessThan @ nat @ N3 ) )
@ ( times_times @ real @ ( divide_divide @ real @ ( Diff @ N3 @ T8 ) @ ( semiring_char_0_fact @ real @ N3 ) ) @ ( power_power @ real @ X @ N3 ) ) ) ) ) ) ).
% Maclaurin_all_le_objl
thf(fact_6973_Maclaurin__all__le,axiom,
! [Diff: nat > real > real,F2: real > real,X: real,N3: nat] :
( ( ( Diff @ ( zero_zero @ nat ) )
= F2 )
=> ( ! [M4: nat,X3: real] : ( has_field_derivative @ real @ ( Diff @ M4 ) @ ( Diff @ ( suc @ M4 ) @ X3 ) @ ( topolo174197925503356063within @ real @ X3 @ ( top_top @ ( set @ real ) ) ) )
=> ? [T8: real] :
( ( ord_less_eq @ real @ ( abs_abs @ real @ T8 ) @ ( abs_abs @ real @ X ) )
& ( ( F2 @ X )
= ( plus_plus @ real
@ ( groups7311177749621191930dd_sum @ nat @ real
@ ^ [M5: nat] : ( times_times @ real @ ( divide_divide @ real @ ( Diff @ M5 @ ( zero_zero @ real ) ) @ ( semiring_char_0_fact @ real @ M5 ) ) @ ( power_power @ real @ X @ M5 ) )
@ ( set_ord_lessThan @ nat @ N3 ) )
@ ( times_times @ real @ ( divide_divide @ real @ ( Diff @ N3 @ T8 ) @ ( semiring_char_0_fact @ real @ N3 ) ) @ ( power_power @ real @ X @ N3 ) ) ) ) ) ) ) ).
% Maclaurin_all_le
thf(fact_6974_DERIV__odd__real__root,axiom,
! [N3: nat,X: real] :
( ~ ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N3 )
=> ( ( X
!= ( zero_zero @ real ) )
=> ( has_field_derivative @ real @ ( root @ N3 ) @ ( inverse_inverse @ real @ ( times_times @ real @ ( semiring_1_of_nat @ real @ N3 ) @ ( power_power @ real @ ( root @ N3 @ X ) @ ( minus_minus @ nat @ N3 @ ( suc @ ( zero_zero @ nat ) ) ) ) ) ) @ ( topolo174197925503356063within @ real @ X @ ( top_top @ ( set @ real ) ) ) ) ) ) ).
% DERIV_odd_real_root
thf(fact_6975_Maclaurin,axiom,
! [H2: real,N3: nat,Diff: nat > real > real,F2: real > real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ H2 )
=> ( ( ord_less @ nat @ ( zero_zero @ nat ) @ N3 )
=> ( ( ( Diff @ ( zero_zero @ nat ) )
= F2 )
=> ( ! [M4: nat,T8: real] :
( ( ( ord_less @ nat @ M4 @ N3 )
& ( ord_less_eq @ real @ ( zero_zero @ real ) @ T8 )
& ( ord_less_eq @ real @ T8 @ H2 ) )
=> ( has_field_derivative @ real @ ( Diff @ M4 ) @ ( Diff @ ( suc @ M4 ) @ 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
@ ^ [M5: nat] : ( times_times @ real @ ( divide_divide @ real @ ( Diff @ M5 @ ( zero_zero @ real ) ) @ ( semiring_char_0_fact @ real @ M5 ) ) @ ( power_power @ real @ H2 @ M5 ) )
@ ( set_ord_lessThan @ nat @ N3 ) )
@ ( times_times @ real @ ( divide_divide @ real @ ( Diff @ N3 @ T8 ) @ ( semiring_char_0_fact @ real @ N3 ) ) @ ( power_power @ real @ H2 @ N3 ) ) ) ) ) ) ) ) ) ).
% Maclaurin
thf(fact_6976_Maclaurin2,axiom,
! [H2: real,Diff: nat > real > real,F2: real > real,N3: nat] :
( ( ord_less @ real @ ( zero_zero @ real ) @ H2 )
=> ( ( ( Diff @ ( zero_zero @ nat ) )
= F2 )
=> ( ! [M4: nat,T8: real] :
( ( ( ord_less @ nat @ M4 @ N3 )
& ( ord_less_eq @ real @ ( zero_zero @ real ) @ T8 )
& ( ord_less_eq @ real @ T8 @ H2 ) )
=> ( has_field_derivative @ real @ ( Diff @ M4 ) @ ( Diff @ ( suc @ M4 ) @ 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
@ ^ [M5: nat] : ( times_times @ real @ ( divide_divide @ real @ ( Diff @ M5 @ ( zero_zero @ real ) ) @ ( semiring_char_0_fact @ real @ M5 ) ) @ ( power_power @ real @ H2 @ M5 ) )
@ ( set_ord_lessThan @ nat @ N3 ) )
@ ( times_times @ real @ ( divide_divide @ real @ ( Diff @ N3 @ T8 ) @ ( semiring_char_0_fact @ real @ N3 ) ) @ ( power_power @ real @ H2 @ N3 ) ) ) ) ) ) ) ) ).
% Maclaurin2
thf(fact_6977_Maclaurin__minus,axiom,
! [H2: real,N3: nat,Diff: nat > real > real,F2: real > real] :
( ( ord_less @ real @ H2 @ ( zero_zero @ real ) )
=> ( ( ord_less @ nat @ ( zero_zero @ nat ) @ N3 )
=> ( ( ( Diff @ ( zero_zero @ nat ) )
= F2 )
=> ( ! [M4: nat,T8: real] :
( ( ( ord_less @ nat @ M4 @ N3 )
& ( ord_less_eq @ real @ H2 @ T8 )
& ( ord_less_eq @ real @ T8 @ ( zero_zero @ real ) ) )
=> ( has_field_derivative @ real @ ( Diff @ M4 ) @ ( Diff @ ( suc @ M4 ) @ 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
@ ^ [M5: nat] : ( times_times @ real @ ( divide_divide @ real @ ( Diff @ M5 @ ( zero_zero @ real ) ) @ ( semiring_char_0_fact @ real @ M5 ) ) @ ( power_power @ real @ H2 @ M5 ) )
@ ( set_ord_lessThan @ nat @ N3 ) )
@ ( times_times @ real @ ( divide_divide @ real @ ( Diff @ N3 @ T8 ) @ ( semiring_char_0_fact @ real @ N3 ) ) @ ( power_power @ real @ H2 @ N3 ) ) ) ) ) ) ) ) ) ).
% Maclaurin_minus
thf(fact_6978_Maclaurin__all__lt,axiom,
! [Diff: nat > real > real,F2: real > real,N3: nat,X: real] :
( ( ( Diff @ ( zero_zero @ nat ) )
= F2 )
=> ( ( ord_less @ nat @ ( zero_zero @ nat ) @ N3 )
=> ( ( X
!= ( zero_zero @ real ) )
=> ( ! [M4: nat,X3: real] : ( has_field_derivative @ real @ ( Diff @ M4 ) @ ( Diff @ ( suc @ M4 ) @ X3 ) @ ( topolo174197925503356063within @ real @ X3 @ ( top_top @ ( set @ real ) ) ) )
=> ? [T8: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ ( abs_abs @ real @ T8 ) )
& ( ord_less @ real @ ( abs_abs @ real @ T8 ) @ ( abs_abs @ real @ X ) )
& ( ( F2 @ X )
= ( plus_plus @ real
@ ( groups7311177749621191930dd_sum @ nat @ real
@ ^ [M5: nat] : ( times_times @ real @ ( divide_divide @ real @ ( Diff @ M5 @ ( zero_zero @ real ) ) @ ( semiring_char_0_fact @ real @ M5 ) ) @ ( power_power @ real @ X @ M5 ) )
@ ( set_ord_lessThan @ nat @ N3 ) )
@ ( times_times @ real @ ( divide_divide @ real @ ( Diff @ N3 @ T8 ) @ ( semiring_char_0_fact @ real @ N3 ) ) @ ( power_power @ real @ X @ N3 ) ) ) ) ) ) ) ) ) ).
% Maclaurin_all_lt
thf(fact_6979_Maclaurin__bi__le,axiom,
! [Diff: nat > real > real,F2: real > real,N3: nat,X: real] :
( ( ( Diff @ ( zero_zero @ nat ) )
= F2 )
=> ( ! [M4: nat,T8: real] :
( ( ( ord_less @ nat @ M4 @ N3 )
& ( ord_less_eq @ real @ ( abs_abs @ real @ T8 ) @ ( abs_abs @ real @ X ) ) )
=> ( has_field_derivative @ real @ ( Diff @ M4 ) @ ( Diff @ ( suc @ M4 ) @ T8 ) @ ( topolo174197925503356063within @ real @ T8 @ ( top_top @ ( set @ real ) ) ) ) )
=> ? [T8: real] :
( ( ord_less_eq @ real @ ( abs_abs @ real @ T8 ) @ ( abs_abs @ real @ X ) )
& ( ( F2 @ X )
= ( plus_plus @ real
@ ( groups7311177749621191930dd_sum @ nat @ real
@ ^ [M5: nat] : ( times_times @ real @ ( divide_divide @ real @ ( Diff @ M5 @ ( zero_zero @ real ) ) @ ( semiring_char_0_fact @ real @ M5 ) ) @ ( power_power @ real @ X @ M5 ) )
@ ( set_ord_lessThan @ nat @ N3 ) )
@ ( times_times @ real @ ( divide_divide @ real @ ( Diff @ N3 @ T8 ) @ ( semiring_char_0_fact @ real @ N3 ) ) @ ( power_power @ real @ X @ N3 ) ) ) ) ) ) ) ).
% Maclaurin_bi_le
thf(fact_6980_Taylor,axiom,
! [N3: nat,Diff: nat > real > real,F2: real > real,A2: real,B2: real,C2: real,X: real] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ N3 )
=> ( ( ( Diff @ ( zero_zero @ nat ) )
= F2 )
=> ( ! [M4: nat,T8: real] :
( ( ( ord_less @ nat @ M4 @ N3 )
& ( ord_less_eq @ real @ A2 @ T8 )
& ( ord_less_eq @ real @ T8 @ B2 ) )
=> ( has_field_derivative @ real @ ( Diff @ M4 ) @ ( Diff @ ( suc @ M4 ) @ T8 ) @ ( topolo174197925503356063within @ real @ T8 @ ( top_top @ ( set @ real ) ) ) ) )
=> ( ( ord_less_eq @ real @ A2 @ C2 )
=> ( ( ord_less_eq @ real @ C2 @ B2 )
=> ( ( ord_less_eq @ real @ A2 @ X )
=> ( ( ord_less_eq @ real @ X @ B2 )
=> ( ( X != C2 )
=> ? [T8: real] :
( ( ( ord_less @ real @ X @ C2 )
=> ( ( ord_less @ real @ X @ T8 )
& ( ord_less @ real @ T8 @ C2 ) ) )
& ( ~ ( ord_less @ real @ X @ C2 )
=> ( ( ord_less @ real @ C2 @ T8 )
& ( ord_less @ real @ T8 @ X ) ) )
& ( ( F2 @ X )
= ( plus_plus @ real
@ ( groups7311177749621191930dd_sum @ nat @ real
@ ^ [M5: nat] : ( times_times @ real @ ( divide_divide @ real @ ( Diff @ M5 @ C2 ) @ ( semiring_char_0_fact @ real @ M5 ) ) @ ( power_power @ real @ ( minus_minus @ real @ X @ C2 ) @ M5 ) )
@ ( set_ord_lessThan @ nat @ N3 ) )
@ ( times_times @ real @ ( divide_divide @ real @ ( Diff @ N3 @ T8 ) @ ( semiring_char_0_fact @ real @ N3 ) ) @ ( power_power @ real @ ( minus_minus @ real @ X @ C2 ) @ N3 ) ) ) ) ) ) ) ) ) ) ) ) ) ).
% Taylor
thf(fact_6981_Taylor__up,axiom,
! [N3: nat,Diff: nat > real > real,F2: real > real,A2: real,B2: real,C2: real] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ N3 )
=> ( ( ( Diff @ ( zero_zero @ nat ) )
= F2 )
=> ( ! [M4: nat,T8: real] :
( ( ( ord_less @ nat @ M4 @ N3 )
& ( ord_less_eq @ real @ A2 @ T8 )
& ( ord_less_eq @ real @ T8 @ B2 ) )
=> ( has_field_derivative @ real @ ( Diff @ M4 ) @ ( Diff @ ( suc @ M4 ) @ T8 ) @ ( topolo174197925503356063within @ real @ T8 @ ( top_top @ ( set @ real ) ) ) ) )
=> ( ( ord_less_eq @ real @ A2 @ C2 )
=> ( ( ord_less @ real @ C2 @ B2 )
=> ? [T8: real] :
( ( ord_less @ real @ C2 @ T8 )
& ( ord_less @ real @ T8 @ B2 )
& ( ( F2 @ B2 )
= ( plus_plus @ real
@ ( groups7311177749621191930dd_sum @ nat @ real
@ ^ [M5: nat] : ( times_times @ real @ ( divide_divide @ real @ ( Diff @ M5 @ C2 ) @ ( semiring_char_0_fact @ real @ M5 ) ) @ ( power_power @ real @ ( minus_minus @ real @ B2 @ C2 ) @ M5 ) )
@ ( set_ord_lessThan @ nat @ N3 ) )
@ ( times_times @ real @ ( divide_divide @ real @ ( Diff @ N3 @ T8 ) @ ( semiring_char_0_fact @ real @ N3 ) ) @ ( power_power @ real @ ( minus_minus @ real @ B2 @ C2 ) @ N3 ) ) ) ) ) ) ) ) ) ) ).
% Taylor_up
thf(fact_6982_Taylor__down,axiom,
! [N3: nat,Diff: nat > real > real,F2: real > real,A2: real,B2: real,C2: real] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ N3 )
=> ( ( ( Diff @ ( zero_zero @ nat ) )
= F2 )
=> ( ! [M4: nat,T8: real] :
( ( ( ord_less @ nat @ M4 @ N3 )
& ( ord_less_eq @ real @ A2 @ T8 )
& ( ord_less_eq @ real @ T8 @ B2 ) )
=> ( has_field_derivative @ real @ ( Diff @ M4 ) @ ( Diff @ ( suc @ M4 ) @ T8 ) @ ( topolo174197925503356063within @ real @ T8 @ ( top_top @ ( set @ real ) ) ) ) )
=> ( ( ord_less @ real @ A2 @ C2 )
=> ( ( ord_less_eq @ real @ C2 @ B2 )
=> ? [T8: real] :
( ( ord_less @ real @ A2 @ T8 )
& ( ord_less @ real @ T8 @ C2 )
& ( ( F2 @ A2 )
= ( plus_plus @ real
@ ( groups7311177749621191930dd_sum @ nat @ real
@ ^ [M5: nat] : ( times_times @ real @ ( divide_divide @ real @ ( Diff @ M5 @ C2 ) @ ( semiring_char_0_fact @ real @ M5 ) ) @ ( power_power @ real @ ( minus_minus @ real @ A2 @ C2 ) @ M5 ) )
@ ( set_ord_lessThan @ nat @ N3 ) )
@ ( times_times @ real @ ( divide_divide @ real @ ( Diff @ N3 @ T8 ) @ ( semiring_char_0_fact @ real @ N3 ) ) @ ( power_power @ real @ ( minus_minus @ real @ A2 @ C2 ) @ N3 ) ) ) ) ) ) ) ) ) ) ).
% Taylor_down
thf(fact_6983_Maclaurin__lemma2,axiom,
! [N3: nat,H2: real,Diff: nat > real > real,K: nat,B6: real] :
( ! [M4: nat,T8: real] :
( ( ( ord_less @ nat @ M4 @ N3 )
& ( ord_less_eq @ real @ ( zero_zero @ real ) @ T8 )
& ( ord_less_eq @ real @ T8 @ H2 ) )
=> ( has_field_derivative @ real @ ( Diff @ M4 ) @ ( Diff @ ( suc @ M4 ) @ T8 ) @ ( topolo174197925503356063within @ real @ T8 @ ( top_top @ ( set @ real ) ) ) ) )
=> ( ( N3
= ( suc @ K ) )
=> ! [M2: nat,T10: real] :
( ( ( ord_less @ nat @ M2 @ N3 )
& ( ord_less_eq @ real @ ( zero_zero @ real ) @ T10 )
& ( ord_less_eq @ real @ T10 @ H2 ) )
=> ( has_field_derivative @ real
@ ^ [U2: real] :
( minus_minus @ real @ ( Diff @ M2 @ U2 )
@ ( plus_plus @ real
@ ( groups7311177749621191930dd_sum @ nat @ real
@ ^ [P5: nat] : ( times_times @ real @ ( divide_divide @ real @ ( Diff @ ( plus_plus @ nat @ M2 @ P5 ) @ ( zero_zero @ real ) ) @ ( semiring_char_0_fact @ real @ P5 ) ) @ ( power_power @ real @ U2 @ P5 ) )
@ ( set_ord_lessThan @ nat @ ( minus_minus @ nat @ N3 @ M2 ) ) )
@ ( times_times @ real @ B6 @ ( divide_divide @ real @ ( power_power @ real @ U2 @ ( minus_minus @ nat @ N3 @ M2 ) ) @ ( semiring_char_0_fact @ real @ ( minus_minus @ nat @ N3 @ M2 ) ) ) ) ) )
@ ( minus_minus @ real @ ( Diff @ ( suc @ M2 ) @ T10 )
@ ( plus_plus @ real
@ ( groups7311177749621191930dd_sum @ nat @ real
@ ^ [P5: nat] : ( times_times @ real @ ( divide_divide @ real @ ( Diff @ ( plus_plus @ nat @ ( suc @ M2 ) @ P5 ) @ ( zero_zero @ real ) ) @ ( semiring_char_0_fact @ real @ P5 ) ) @ ( power_power @ real @ T10 @ P5 ) )
@ ( set_ord_lessThan @ nat @ ( minus_minus @ nat @ N3 @ ( suc @ M2 ) ) ) )
@ ( times_times @ real @ B6 @ ( divide_divide @ real @ ( power_power @ real @ T10 @ ( minus_minus @ nat @ N3 @ ( suc @ M2 ) ) ) @ ( semiring_char_0_fact @ real @ ( minus_minus @ nat @ N3 @ ( suc @ M2 ) ) ) ) ) ) )
@ ( topolo174197925503356063within @ real @ T10 @ ( top_top @ ( set @ real ) ) ) ) ) ) ) ).
% Maclaurin_lemma2
thf(fact_6984_DERIV__arctan__series,axiom,
! [X: real] :
( ( ord_less @ real @ ( abs_abs @ real @ X ) @ ( one_one @ real ) )
=> ( has_field_derivative @ real
@ ^ [X9: real] :
( suminf @ real
@ ^ [K2: nat] : ( times_times @ real @ ( power_power @ real @ ( uminus_uminus @ real @ ( one_one @ real ) ) @ K2 ) @ ( times_times @ real @ ( divide_divide @ real @ ( one_one @ real ) @ ( semiring_1_of_nat @ real @ ( plus_plus @ nat @ ( times_times @ nat @ K2 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ ( one_one @ nat ) ) ) ) @ ( power_power @ real @ X9 @ ( plus_plus @ nat @ ( times_times @ nat @ K2 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ ( one_one @ nat ) ) ) ) ) )
@ ( suminf @ real
@ ^ [K2: nat] : ( times_times @ real @ ( power_power @ real @ ( uminus_uminus @ real @ ( one_one @ real ) ) @ K2 ) @ ( power_power @ real @ X @ ( times_times @ nat @ K2 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) )
@ ( topolo174197925503356063within @ real @ X @ ( top_top @ ( set @ real ) ) ) ) ) ).
% DERIV_arctan_series
thf(fact_6985_DERIV__power__series_H,axiom,
! [R: real,F2: nat > real,X0: real] :
( ! [X3: real] :
( ( member @ real @ X3 @ ( set_or5935395276787703475ssThan @ real @ ( uminus_uminus @ real @ R ) @ R ) )
=> ( summable @ real
@ ^ [N2: nat] : ( times_times @ real @ ( times_times @ real @ ( F2 @ N2 ) @ ( semiring_1_of_nat @ real @ ( suc @ N2 ) ) ) @ ( power_power @ real @ X3 @ N2 ) ) ) )
=> ( ( member @ real @ X0 @ ( set_or5935395276787703475ssThan @ real @ ( uminus_uminus @ real @ R ) @ R ) )
=> ( ( ord_less @ real @ ( zero_zero @ real ) @ R )
=> ( has_field_derivative @ real
@ ^ [X2: real] :
( suminf @ real
@ ^ [N2: nat] : ( times_times @ real @ ( F2 @ N2 ) @ ( power_power @ real @ X2 @ ( suc @ N2 ) ) ) )
@ ( suminf @ real
@ ^ [N2: nat] : ( times_times @ real @ ( times_times @ real @ ( F2 @ N2 ) @ ( semiring_1_of_nat @ real @ ( suc @ N2 ) ) ) @ ( power_power @ real @ X0 @ N2 ) ) )
@ ( topolo174197925503356063within @ real @ X0 @ ( top_top @ ( set @ real ) ) ) ) ) ) ) ).
% DERIV_power_series'
thf(fact_6986_has__derivative__arcsin,axiom,
! [A: $tType] :
( ( real_V822414075346904944vector @ A )
=> ! [G: A > real,X: A,G5: A > real,S: set @ A] :
( ( ord_less @ real @ ( uminus_uminus @ real @ ( one_one @ real ) ) @ ( G @ X ) )
=> ( ( ord_less @ real @ ( G @ X ) @ ( one_one @ real ) )
=> ( ( has_derivative @ A @ real @ G @ G5 @ ( topolo174197925503356063within @ A @ X @ S ) )
=> ( has_derivative @ A @ real
@ ^ [X2: A] : ( arcsin @ ( G @ X2 ) )
@ ^ [X2: A] : ( times_times @ real @ ( G5 @ X2 ) @ ( inverse_inverse @ real @ ( sqrt @ ( minus_minus @ real @ ( one_one @ real ) @ ( power_power @ real @ ( G @ X ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) )
@ ( topolo174197925503356063within @ A @ X @ S ) ) ) ) ) ) ).
% has_derivative_arcsin
thf(fact_6987_greaterThanLessThan__iff,axiom,
! [A: $tType] :
( ( ord @ A )
=> ! [I: A,L2: A,U: A] :
( ( member @ A @ I @ ( set_or5935395276787703475ssThan @ A @ L2 @ U ) )
= ( ( ord_less @ A @ L2 @ I )
& ( ord_less @ A @ I @ U ) ) ) ) ).
% greaterThanLessThan_iff
thf(fact_6988_greaterThanLessThan__empty,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [L2: A,K: A] :
( ( ord_less_eq @ A @ L2 @ K )
=> ( ( set_or5935395276787703475ssThan @ A @ K @ L2 )
= ( bot_bot @ ( set @ A ) ) ) ) ) ).
% greaterThanLessThan_empty
thf(fact_6989_greaterThanLessThan__empty__iff,axiom,
! [A: $tType] :
( ( dense_linorder @ A )
=> ! [A2: A,B2: A] :
( ( ( set_or5935395276787703475ssThan @ A @ A2 @ B2 )
= ( bot_bot @ ( set @ A ) ) )
= ( ord_less_eq @ A @ B2 @ A2 ) ) ) ).
% greaterThanLessThan_empty_iff
thf(fact_6990_greaterThanLessThan__empty__iff2,axiom,
! [A: $tType] :
( ( dense_linorder @ A )
=> ! [A2: A,B2: A] :
( ( ( bot_bot @ ( set @ A ) )
= ( set_or5935395276787703475ssThan @ A @ A2 @ B2 ) )
= ( ord_less_eq @ A @ B2 @ A2 ) ) ) ).
% greaterThanLessThan_empty_iff2
thf(fact_6991_infinite__Ioo__iff,axiom,
! [A: $tType] :
( ( dense_linorder @ A )
=> ! [A2: A,B2: A] :
( ( ~ ( finite_finite2 @ A @ ( set_or5935395276787703475ssThan @ A @ A2 @ B2 ) ) )
= ( ord_less @ A @ A2 @ B2 ) ) ) ).
% infinite_Ioo_iff
thf(fact_6992_has__derivative__transform,axiom,
! [B: $tType,A: $tType] :
( ( ( real_V822414075346904944vector @ A )
& ( real_V822414075346904944vector @ B ) )
=> ! [X: A,S: set @ A,G: A > B,F2: A > B,F5: A > B] :
( ( member @ A @ X @ S )
=> ( ! [X3: A] :
( ( member @ A @ X3 @ S )
=> ( ( G @ X3 )
= ( F2 @ X3 ) ) )
=> ( ( has_derivative @ A @ B @ F2 @ F5 @ ( topolo174197925503356063within @ A @ X @ S ) )
=> ( has_derivative @ A @ B @ G @ F5 @ ( topolo174197925503356063within @ A @ X @ S ) ) ) ) ) ) ).
% has_derivative_transform
thf(fact_6993_has__derivative__scaleR,axiom,
! [C: $tType,D5: $tType] :
( ( ( real_V822414075346904944vector @ D5 )
& ( real_V822414075346904944vector @ C ) )
=> ! [F2: D5 > real,F5: D5 > real,X: D5,S: set @ D5,G: D5 > C,G5: D5 > C] :
( ( has_derivative @ D5 @ real @ F2 @ F5 @ ( topolo174197925503356063within @ D5 @ X @ S ) )
=> ( ( has_derivative @ D5 @ C @ G @ G5 @ ( topolo174197925503356063within @ D5 @ X @ S ) )
=> ( has_derivative @ D5 @ C
@ ^ [X2: D5] : ( real_V8093663219630862766scaleR @ C @ ( F2 @ X2 ) @ ( G @ X2 ) )
@ ^ [H: D5] : ( plus_plus @ C @ ( real_V8093663219630862766scaleR @ C @ ( F2 @ X ) @ ( G5 @ H ) ) @ ( real_V8093663219630862766scaleR @ C @ ( F5 @ H ) @ ( G @ X ) ) )
@ ( topolo174197925503356063within @ D5 @ X @ S ) ) ) ) ) ).
% has_derivative_scaleR
thf(fact_6994_has__field__derivative__imp__has__derivative,axiom,
! [A: $tType] :
( ( real_V3459762299906320749_field @ A )
=> ! [F2: A > A,D4: A,F4: filter @ A] :
( ( has_field_derivative @ A @ F2 @ D4 @ F4 )
=> ( has_derivative @ A @ A @ F2 @ ( times_times @ A @ D4 ) @ F4 ) ) ) ).
% has_field_derivative_imp_has_derivative
thf(fact_6995_has__derivative__imp__has__field__derivative,axiom,
! [A: $tType] :
( ( real_V3459762299906320749_field @ A )
=> ! [F2: A > A,D4: A > A,F4: filter @ A,D7: A] :
( ( has_derivative @ A @ A @ F2 @ D4 @ F4 )
=> ( ! [X3: A] :
( ( times_times @ A @ X3 @ D7 )
= ( D4 @ X3 ) )
=> ( has_field_derivative @ A @ F2 @ D7 @ F4 ) ) ) ) ).
% has_derivative_imp_has_field_derivative
thf(fact_6996_has__field__derivative__def,axiom,
! [A: $tType] :
( ( real_V3459762299906320749_field @ A )
=> ( ( has_field_derivative @ A )
= ( ^ [F7: A > A,D8: A] : ( has_derivative @ A @ A @ F7 @ ( times_times @ A @ D8 ) ) ) ) ) ).
% has_field_derivative_def
thf(fact_6997_has__derivative__unique,axiom,
! [B: $tType,A: $tType] :
( ( ( real_V822414075346904944vector @ A )
& ( real_V822414075346904944vector @ B ) )
=> ! [F2: A > B,F4: A > B,X: A,F8: A > B] :
( ( has_derivative @ A @ B @ F2 @ F4 @ ( topolo174197925503356063within @ A @ X @ ( top_top @ ( set @ A ) ) ) )
=> ( ( has_derivative @ A @ B @ F2 @ F8 @ ( topolo174197925503356063within @ A @ X @ ( top_top @ ( set @ A ) ) ) )
=> ( F4 = F8 ) ) ) ) ).
% has_derivative_unique
thf(fact_6998_has__derivative__compose,axiom,
! [C: $tType,B: $tType,A: $tType] :
( ( ( real_V822414075346904944vector @ A )
& ( real_V822414075346904944vector @ B )
& ( real_V822414075346904944vector @ C ) )
=> ! [F2: A > B,F5: A > B,X: A,S: set @ A,G: B > C,G5: B > C] :
( ( has_derivative @ A @ B @ F2 @ F5 @ ( topolo174197925503356063within @ A @ X @ S ) )
=> ( ( has_derivative @ B @ C @ G @ G5 @ ( topolo174197925503356063within @ B @ ( F2 @ X ) @ ( top_top @ ( set @ B ) ) ) )
=> ( has_derivative @ A @ C
@ ^ [X2: A] : ( G @ ( F2 @ X2 ) )
@ ^ [X2: A] : ( G5 @ ( F5 @ X2 ) )
@ ( topolo174197925503356063within @ A @ X @ S ) ) ) ) ) ).
% has_derivative_compose
thf(fact_6999_has__derivative__in__compose,axiom,
! [C: $tType,B: $tType,A: $tType] :
( ( ( real_V822414075346904944vector @ A )
& ( real_V822414075346904944vector @ B )
& ( real_V822414075346904944vector @ C ) )
=> ! [F2: A > B,F5: A > B,X: A,S: set @ A,G: B > C,G5: B > C] :
( ( has_derivative @ A @ B @ F2 @ F5 @ ( topolo174197925503356063within @ A @ X @ S ) )
=> ( ( has_derivative @ B @ C @ G @ G5 @ ( topolo174197925503356063within @ B @ ( F2 @ X ) @ ( image @ A @ B @ F2 @ S ) ) )
=> ( has_derivative @ A @ C
@ ^ [X2: A] : ( G @ ( F2 @ X2 ) )
@ ^ [X2: A] : ( G5 @ ( F5 @ X2 ) )
@ ( topolo174197925503356063within @ A @ X @ S ) ) ) ) ) ).
% has_derivative_in_compose
thf(fact_7000_has__derivative__subset,axiom,
! [B: $tType,A: $tType] :
( ( ( real_V822414075346904944vector @ A )
& ( real_V822414075346904944vector @ B ) )
=> ! [F2: A > B,F5: A > B,X: A,S: set @ A,T2: set @ A] :
( ( has_derivative @ A @ B @ F2 @ F5 @ ( topolo174197925503356063within @ A @ X @ S ) )
=> ( ( ord_less_eq @ ( set @ A ) @ T2 @ S )
=> ( has_derivative @ A @ B @ F2 @ F5 @ ( topolo174197925503356063within @ A @ X @ T2 ) ) ) ) ) ).
% has_derivative_subset
thf(fact_7001_has__derivative__minus,axiom,
! [B: $tType,A: $tType] :
( ( ( real_V822414075346904944vector @ A )
& ( real_V822414075346904944vector @ B ) )
=> ! [F2: A > B,F5: A > B,F4: filter @ A] :
( ( has_derivative @ A @ B @ F2 @ F5 @ F4 )
=> ( has_derivative @ A @ B
@ ^ [X2: A] : ( uminus_uminus @ B @ ( F2 @ X2 ) )
@ ^ [X2: A] : ( uminus_uminus @ B @ ( F5 @ X2 ) )
@ F4 ) ) ) ).
% has_derivative_minus
thf(fact_7002_has__derivative__mult__right,axiom,
! [A: $tType,C: $tType] :
( ( ( real_V822414075346904944vector @ C )
& ( real_V4412858255891104859lgebra @ A ) )
=> ! [G: C > A,G5: C > A,F4: filter @ C,X: A] :
( ( has_derivative @ C @ A @ G @ G5 @ F4 )
=> ( has_derivative @ C @ A
@ ^ [X2: C] : ( times_times @ A @ X @ ( G @ X2 ) )
@ ^ [X2: C] : ( times_times @ A @ X @ ( G5 @ X2 ) )
@ F4 ) ) ) ).
% has_derivative_mult_right
thf(fact_7003_has__derivative__mult__left,axiom,
! [A: $tType,C: $tType] :
( ( ( real_V822414075346904944vector @ C )
& ( real_V4412858255891104859lgebra @ A ) )
=> ! [G: C > A,G5: C > A,F4: filter @ C,Y: A] :
( ( has_derivative @ C @ A @ G @ G5 @ F4 )
=> ( has_derivative @ C @ A
@ ^ [X2: C] : ( times_times @ A @ ( G @ X2 ) @ Y )
@ ^ [X2: C] : ( times_times @ A @ ( G5 @ X2 ) @ Y )
@ F4 ) ) ) ).
% has_derivative_mult_left
thf(fact_7004_has__derivative__add,axiom,
! [B: $tType,A: $tType] :
( ( ( real_V822414075346904944vector @ A )
& ( real_V822414075346904944vector @ B ) )
=> ! [F2: A > B,F5: A > B,F4: filter @ A,G: A > B,G5: A > B] :
( ( has_derivative @ A @ B @ F2 @ F5 @ F4 )
=> ( ( has_derivative @ A @ B @ G @ G5 @ F4 )
=> ( has_derivative @ A @ B
@ ^ [X2: A] : ( plus_plus @ B @ ( F2 @ X2 ) @ ( G @ X2 ) )
@ ^ [X2: A] : ( plus_plus @ B @ ( F5 @ X2 ) @ ( G5 @ X2 ) )
@ F4 ) ) ) ) ).
% has_derivative_add
thf(fact_7005_has__derivative__sum,axiom,
! [C: $tType,A: $tType,B: $tType] :
( ( ( real_V822414075346904944vector @ B )
& ( real_V822414075346904944vector @ C ) )
=> ! [I5: set @ A,F2: A > B > C,F5: A > B > C,F4: filter @ B] :
( ! [I2: A] :
( ( member @ A @ I2 @ I5 )
=> ( has_derivative @ B @ C @ ( F2 @ I2 ) @ ( F5 @ I2 ) @ F4 ) )
=> ( has_derivative @ B @ C
@ ^ [X2: B] :
( groups7311177749621191930dd_sum @ A @ C
@ ^ [I3: A] : ( F2 @ I3 @ X2 )
@ I5 )
@ ^ [X2: B] :
( groups7311177749621191930dd_sum @ A @ C
@ ^ [I3: A] : ( F5 @ I3 @ X2 )
@ I5 )
@ F4 ) ) ) ).
% has_derivative_sum
thf(fact_7006_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
@ ^ [X2: C] : ( real_Vector_of_real @ A @ ( G @ X2 ) )
@ ^ [X2: C] : ( real_Vector_of_real @ A @ ( G5 @ X2 ) )
@ F4 ) ) ) ).
% has_derivative_of_real
thf(fact_7007_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
@ ^ [X2: C] : ( real_V8093663219630862766scaleR @ B @ R2 @ ( G @ X2 ) )
@ ^ [X2: C] : ( real_V8093663219630862766scaleR @ B @ R2 @ ( G5 @ X2 ) )
@ F4 ) ) ) ).
% has_derivative_scaleR_right
thf(fact_7008_has__derivative__scaleR__left,axiom,
! [B: $tType,C: $tType] :
( ( ( real_V822414075346904944vector @ C )
& ( real_V822414075346904944vector @ B ) )
=> ! [G: C > real,G5: C > real,F4: filter @ C,X: B] :
( ( has_derivative @ C @ real @ G @ G5 @ F4 )
=> ( has_derivative @ C @ B
@ ^ [X2: C] : ( real_V8093663219630862766scaleR @ B @ ( G @ X2 ) @ X )
@ ^ [X2: C] : ( real_V8093663219630862766scaleR @ B @ ( G5 @ X2 ) @ X )
@ F4 ) ) ) ).
% has_derivative_scaleR_left
thf(fact_7009_has__derivative__ident,axiom,
! [A: $tType] :
( ( real_V822414075346904944vector @ A )
=> ! [F4: filter @ A] :
( has_derivative @ A @ A
@ ^ [X2: A] : X2
@ ^ [X2: A] : X2
@ F4 ) ) ).
% has_derivative_ident
thf(fact_7010_has__derivative__eq__rhs,axiom,
! [B: $tType,A: $tType] :
( ( ( real_V822414075346904944vector @ A )
& ( real_V822414075346904944vector @ B ) )
=> ! [F2: A > B,F5: A > B,F4: filter @ A,G5: A > B] :
( ( has_derivative @ A @ B @ F2 @ F5 @ F4 )
=> ( ( F5 = G5 )
=> ( has_derivative @ A @ B @ F2 @ G5 @ F4 ) ) ) ) ).
% has_derivative_eq_rhs
thf(fact_7011_has__derivative__const,axiom,
! [B: $tType,A: $tType] :
( ( ( real_V822414075346904944vector @ A )
& ( real_V822414075346904944vector @ B ) )
=> ! [C2: B,F4: filter @ A] :
( has_derivative @ A @ B
@ ^ [X2: A] : C2
@ ^ [X2: A] : ( zero_zero @ B )
@ F4 ) ) ).
% has_derivative_const
thf(fact_7012_has__derivative__diff,axiom,
! [B: $tType,A: $tType] :
( ( ( real_V822414075346904944vector @ A )
& ( real_V822414075346904944vector @ B ) )
=> ! [F2: A > B,F5: A > B,F4: filter @ A,G: A > B,G5: A > B] :
( ( has_derivative @ A @ B @ F2 @ F5 @ F4 )
=> ( ( has_derivative @ A @ B @ G @ G5 @ F4 )
=> ( has_derivative @ A @ B
@ ^ [X2: A] : ( minus_minus @ B @ ( F2 @ X2 ) @ ( G @ X2 ) )
@ ^ [X2: A] : ( minus_minus @ B @ ( F5 @ X2 ) @ ( G5 @ X2 ) )
@ F4 ) ) ) ) ).
% has_derivative_diff
thf(fact_7013_infinite__Ioo,axiom,
! [A: $tType] :
( ( dense_linorder @ A )
=> ! [A2: A,B2: A] :
( ( ord_less @ A @ A2 @ B2 )
=> ~ ( finite_finite2 @ A @ ( set_or5935395276787703475ssThan @ A @ A2 @ B2 ) ) ) ) ).
% infinite_Ioo
thf(fact_7014_has__derivative__mult,axiom,
! [A: $tType,D5: $tType] :
( ( ( real_V822414075346904944vector @ D5 )
& ( real_V4412858255891104859lgebra @ A ) )
=> ! [F2: D5 > A,F5: D5 > A,X: D5,S: set @ D5,G: D5 > A,G5: D5 > A] :
( ( has_derivative @ D5 @ A @ F2 @ F5 @ ( topolo174197925503356063within @ D5 @ X @ S ) )
=> ( ( has_derivative @ D5 @ A @ G @ G5 @ ( topolo174197925503356063within @ D5 @ X @ S ) )
=> ( has_derivative @ D5 @ A
@ ^ [X2: D5] : ( times_times @ A @ ( F2 @ X2 ) @ ( G @ X2 ) )
@ ^ [H: D5] : ( plus_plus @ A @ ( times_times @ A @ ( F2 @ X ) @ ( G5 @ H ) ) @ ( times_times @ A @ ( F5 @ H ) @ ( G @ X ) ) )
@ ( topolo174197925503356063within @ D5 @ X @ S ) ) ) ) ) ).
% has_derivative_mult
thf(fact_7015_has__derivative__zero__unique,axiom,
! [B: $tType,A: $tType] :
( ( ( real_V822414075346904944vector @ A )
& ( real_V822414075346904944vector @ B ) )
=> ! [F4: A > B,X: A] :
( ( has_derivative @ A @ B
@ ^ [X2: A] : ( zero_zero @ B )
@ F4
@ ( topolo174197925503356063within @ A @ X @ ( top_top @ ( set @ A ) ) ) )
=> ( F4
= ( ^ [H: A] : ( zero_zero @ B ) ) ) ) ) ).
% has_derivative_zero_unique
thf(fact_7016_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,S: set @ C,X: C,F5: C > A] :
( ! [X3: A] :
( ( member @ A @ X3 @ T2 )
=> ( has_derivative @ A @ B @ G @ ( G5 @ X3 ) @ ( topolo174197925503356063within @ A @ X3 @ T2 ) ) )
=> ( ( ord_less_eq @ ( set @ A ) @ ( image @ C @ A @ F2 @ S ) @ T2 )
=> ( ( member @ C @ X @ S )
=> ( ( has_derivative @ C @ A @ F2 @ F5 @ ( topolo174197925503356063within @ C @ X @ S ) )
=> ( has_derivative @ C @ B
@ ^ [X2: C] : ( G @ ( F2 @ X2 ) )
@ ^ [Y2: C] : ( G5 @ ( F2 @ X ) @ ( F5 @ Y2 ) )
@ ( topolo174197925503356063within @ C @ X @ S ) ) ) ) ) ) ) ).
% has_derivative_in_compose2
thf(fact_7017_has__derivative__exp,axiom,
! [A: $tType] :
( ( real_V822414075346904944vector @ A )
=> ! [G: A > real,G5: A > real,X: A,S: set @ A] :
( ( has_derivative @ A @ real @ G @ G5 @ ( topolo174197925503356063within @ A @ X @ S ) )
=> ( has_derivative @ A @ real
@ ^ [X2: A] : ( exp @ real @ ( G @ X2 ) )
@ ^ [X2: A] : ( times_times @ real @ ( G5 @ X2 ) @ ( exp @ real @ ( G @ X ) ) )
@ ( topolo174197925503356063within @ A @ X @ S ) ) ) ) ).
% has_derivative_exp
thf(fact_7018_greaterThanLessThan__subseteq__greaterThanLessThan,axiom,
! [A: $tType] :
( ( dense_linorder @ A )
=> ! [A2: A,B2: A,C2: A,D: A] :
( ( ord_less_eq @ ( set @ A ) @ ( set_or5935395276787703475ssThan @ A @ A2 @ B2 ) @ ( set_or5935395276787703475ssThan @ A @ C2 @ D ) )
= ( ( ord_less @ A @ A2 @ B2 )
=> ( ( ord_less_eq @ A @ C2 @ A2 )
& ( ord_less_eq @ A @ B2 @ D ) ) ) ) ) ).
% greaterThanLessThan_subseteq_greaterThanLessThan
thf(fact_7019_tanh__real__bounds,axiom,
! [X: real] : ( member @ real @ ( tanh @ real @ X ) @ ( set_or5935395276787703475ssThan @ real @ ( uminus_uminus @ real @ ( one_one @ real ) ) @ ( one_one @ real ) ) ) ).
% tanh_real_bounds
thf(fact_7020_has__derivative__sinh,axiom,
! [A: $tType] :
( ( ( real_Vector_banach @ A )
& ( real_V3459762299906320749_field @ A ) )
=> ! [G: A > A,Db: A,X: A,S: set @ A] :
( ( has_derivative @ A @ A @ G @ ( times_times @ A @ Db ) @ ( topolo174197925503356063within @ A @ X @ S ) )
=> ( has_derivative @ A @ A
@ ^ [X2: A] : ( sinh @ A @ ( G @ X2 ) )
@ ( times_times @ A @ ( times_times @ A @ ( cosh @ A @ ( G @ X ) ) @ Db ) )
@ ( topolo174197925503356063within @ A @ X @ S ) ) ) ) ).
% has_derivative_sinh
thf(fact_7021_has__derivative__cosh,axiom,
! [A: $tType] :
( ( ( real_Vector_banach @ A )
& ( real_V3459762299906320749_field @ A ) )
=> ! [G: A > A,Db: A,X: A,S: set @ A] :
( ( has_derivative @ A @ A @ G @ ( times_times @ A @ Db ) @ ( topolo174197925503356063within @ A @ X @ S ) )
=> ( has_derivative @ A @ A
@ ^ [X2: A] : ( cosh @ A @ ( G @ X2 ) )
@ ( times_times @ A @ ( times_times @ A @ ( sinh @ A @ ( G @ X ) ) @ Db ) )
@ ( topolo174197925503356063within @ A @ X @ S ) ) ) ) ).
% has_derivative_cosh
thf(fact_7022_has__derivative__sin,axiom,
! [A: $tType] :
( ( real_V822414075346904944vector @ A )
=> ! [G: A > real,G5: A > real,X: A,S: set @ A] :
( ( has_derivative @ A @ real @ G @ G5 @ ( topolo174197925503356063within @ A @ X @ S ) )
=> ( has_derivative @ A @ real
@ ^ [X2: A] : ( sin @ real @ ( G @ X2 ) )
@ ^ [X2: A] : ( times_times @ real @ ( G5 @ X2 ) @ ( cos @ real @ ( G @ X ) ) )
@ ( topolo174197925503356063within @ A @ X @ S ) ) ) ) ).
% has_derivative_sin
thf(fact_7023_greaterThanLessThan__subseteq__atLeastAtMost__iff,axiom,
! [A: $tType] :
( ( dense_linorder @ A )
=> ! [A2: A,B2: A,C2: A,D: A] :
( ( ord_less_eq @ ( set @ A ) @ ( set_or5935395276787703475ssThan @ A @ A2 @ B2 ) @ ( set_or1337092689740270186AtMost @ A @ C2 @ D ) )
= ( ( ord_less @ A @ A2 @ B2 )
=> ( ( ord_less_eq @ A @ C2 @ A2 )
& ( ord_less_eq @ A @ B2 @ D ) ) ) ) ) ).
% greaterThanLessThan_subseteq_atLeastAtMost_iff
thf(fact_7024_greaterThanLessThan__subseteq__atLeastLessThan__iff,axiom,
! [A: $tType] :
( ( dense_linorder @ A )
=> ! [A2: A,B2: A,C2: A,D: A] :
( ( ord_less_eq @ ( set @ A ) @ ( set_or5935395276787703475ssThan @ A @ A2 @ B2 ) @ ( set_or7035219750837199246ssThan @ A @ C2 @ D ) )
= ( ( ord_less @ A @ A2 @ B2 )
=> ( ( ord_less_eq @ A @ C2 @ A2 )
& ( ord_less_eq @ A @ B2 @ D ) ) ) ) ) ).
% greaterThanLessThan_subseteq_atLeastLessThan_iff
thf(fact_7025_atLeastAtMost__diff__ends,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A2: A,B2: A] :
( ( minus_minus @ ( set @ A ) @ ( set_or1337092689740270186AtMost @ A @ A2 @ B2 ) @ ( insert @ A @ A2 @ ( insert @ A @ B2 @ ( bot_bot @ ( set @ A ) ) ) ) )
= ( set_or5935395276787703475ssThan @ A @ A2 @ B2 ) ) ) ).
% atLeastAtMost_diff_ends
thf(fact_7026_has__derivative__divide_H,axiom,
! [A: $tType,C: $tType] :
( ( ( real_V822414075346904944vector @ C )
& ( real_V3459762299906320749_field @ A ) )
=> ! [F2: C > A,F5: C > A,X: C,S3: set @ C,G: C > A,G5: C > A] :
( ( has_derivative @ C @ A @ F2 @ F5 @ ( topolo174197925503356063within @ C @ X @ S3 ) )
=> ( ( has_derivative @ C @ A @ G @ G5 @ ( topolo174197925503356063within @ C @ X @ S3 ) )
=> ( ( ( G @ X )
!= ( zero_zero @ A ) )
=> ( has_derivative @ C @ A
@ ^ [X2: C] : ( divide_divide @ A @ ( F2 @ X2 ) @ ( G @ X2 ) )
@ ^ [H: C] : ( divide_divide @ A @ ( minus_minus @ A @ ( times_times @ A @ ( F5 @ H ) @ ( G @ X ) ) @ ( times_times @ A @ ( F2 @ X ) @ ( G5 @ H ) ) ) @ ( times_times @ A @ ( G @ X ) @ ( G @ X ) ) )
@ ( topolo174197925503356063within @ C @ X @ S3 ) ) ) ) ) ) ).
% has_derivative_divide'
thf(fact_7027_has__derivative__inverse_H,axiom,
! [A: $tType] :
( ( real_V8999393235501362500lgebra @ A )
=> ! [X: A,S3: set @ A] :
( ( X
!= ( zero_zero @ A ) )
=> ( has_derivative @ A @ A @ ( inverse_inverse @ A )
@ ^ [H: A] : ( uminus_uminus @ A @ ( times_times @ A @ ( times_times @ A @ ( inverse_inverse @ A @ X ) @ H ) @ ( inverse_inverse @ A @ X ) ) )
@ ( topolo174197925503356063within @ A @ X @ S3 ) ) ) ) ).
% has_derivative_inverse'
thf(fact_7028_has__derivative__inverse,axiom,
! [A: $tType,C: $tType] :
( ( ( real_V822414075346904944vector @ C )
& ( real_V8999393235501362500lgebra @ A ) )
=> ! [F2: C > A,X: C,F5: C > A,S3: set @ C] :
( ( ( F2 @ X )
!= ( zero_zero @ A ) )
=> ( ( has_derivative @ C @ A @ F2 @ F5 @ ( topolo174197925503356063within @ C @ X @ S3 ) )
=> ( has_derivative @ C @ A
@ ^ [X2: C] : ( inverse_inverse @ A @ ( F2 @ X2 ) )
@ ^ [H: C] : ( uminus_uminus @ A @ ( times_times @ A @ ( times_times @ A @ ( inverse_inverse @ A @ ( F2 @ X ) ) @ ( F5 @ H ) ) @ ( inverse_inverse @ A @ ( F2 @ X ) ) ) )
@ ( topolo174197925503356063within @ C @ X @ S3 ) ) ) ) ) ).
% has_derivative_inverse
thf(fact_7029_DERIV__compose__FDERIV,axiom,
! [A: $tType] :
( ( real_V822414075346904944vector @ A )
=> ! [F2: real > real,F5: real,G: A > real,X: A,G5: A > real,S: set @ A] :
( ( has_field_derivative @ real @ F2 @ F5 @ ( topolo174197925503356063within @ real @ ( G @ X ) @ ( top_top @ ( set @ real ) ) ) )
=> ( ( has_derivative @ A @ real @ G @ G5 @ ( topolo174197925503356063within @ A @ X @ S ) )
=> ( has_derivative @ A @ real
@ ^ [X2: A] : ( F2 @ ( G @ X2 ) )
@ ^ [X2: A] : ( times_times @ real @ ( G5 @ X2 ) @ F5 )
@ ( topolo174197925503356063within @ A @ X @ S ) ) ) ) ) ).
% DERIV_compose_FDERIV
thf(fact_7030_has__derivative__cos,axiom,
! [A: $tType] :
( ( real_V822414075346904944vector @ A )
=> ! [G: A > real,G5: A > real,X: A,S: set @ A] :
( ( has_derivative @ A @ real @ G @ G5 @ ( topolo174197925503356063within @ A @ X @ S ) )
=> ( has_derivative @ A @ real
@ ^ [X2: A] : ( cos @ real @ ( G @ X2 ) )
@ ^ [X2: A] : ( times_times @ real @ ( G5 @ X2 ) @ ( uminus_uminus @ real @ ( sin @ real @ ( G @ X ) ) ) )
@ ( topolo174197925503356063within @ A @ X @ S ) ) ) ) ).
% has_derivative_cos
thf(fact_7031_DERIV__isconst3,axiom,
! [A2: real,B2: real,X: real,Y: real,F2: real > real] :
( ( ord_less @ real @ A2 @ B2 )
=> ( ( member @ real @ X @ ( set_or5935395276787703475ssThan @ real @ A2 @ B2 ) )
=> ( ( member @ real @ Y @ ( set_or5935395276787703475ssThan @ real @ A2 @ B2 ) )
=> ( ! [X3: real] :
( ( member @ real @ X3 @ ( set_or5935395276787703475ssThan @ real @ A2 @ B2 ) )
=> ( has_field_derivative @ real @ F2 @ ( zero_zero @ real ) @ ( topolo174197925503356063within @ real @ X3 @ ( top_top @ ( set @ real ) ) ) ) )
=> ( ( F2 @ X )
= ( F2 @ Y ) ) ) ) ) ) ).
% DERIV_isconst3
thf(fact_7032_has__derivative__power,axiom,
! [B: $tType,A: $tType] :
( ( ( real_V822414075346904944vector @ A )
& ( real_V3459762299906320749_field @ B ) )
=> ! [F2: A > B,F5: A > B,X: A,S3: set @ A,N3: nat] :
( ( has_derivative @ A @ B @ F2 @ F5 @ ( topolo174197925503356063within @ A @ X @ S3 ) )
=> ( has_derivative @ A @ B
@ ^ [X2: A] : ( power_power @ B @ ( F2 @ X2 ) @ N3 )
@ ^ [Y2: A] : ( times_times @ B @ ( times_times @ B @ ( semiring_1_of_nat @ B @ N3 ) @ ( F5 @ Y2 ) ) @ ( power_power @ B @ ( F2 @ X ) @ ( minus_minus @ nat @ N3 @ ( one_one @ nat ) ) ) )
@ ( topolo174197925503356063within @ A @ X @ S3 ) ) ) ) ).
% has_derivative_power
thf(fact_7033_has__derivative__ln,axiom,
! [A: $tType] :
( ( real_V822414075346904944vector @ A )
=> ! [G: A > real,X: A,G5: A > real,S: set @ A] :
( ( ord_less @ real @ ( zero_zero @ real ) @ ( G @ X ) )
=> ( ( has_derivative @ A @ real @ G @ G5 @ ( topolo174197925503356063within @ A @ X @ S ) )
=> ( has_derivative @ A @ real
@ ^ [X2: A] : ( ln_ln @ real @ ( G @ X2 ) )
@ ^ [X2: A] : ( times_times @ real @ ( G5 @ X2 ) @ ( inverse_inverse @ real @ ( G @ X ) ) )
@ ( topolo174197925503356063within @ A @ X @ S ) ) ) ) ) ).
% has_derivative_ln
thf(fact_7034_has__derivative__divide,axiom,
! [A: $tType,C: $tType] :
( ( ( real_V822414075346904944vector @ C )
& ( real_V8999393235501362500lgebra @ A ) )
=> ! [F2: C > A,F5: C > A,X: C,S3: set @ C,G: C > A,G5: C > A] :
( ( has_derivative @ C @ A @ F2 @ F5 @ ( topolo174197925503356063within @ C @ X @ S3 ) )
=> ( ( has_derivative @ C @ A @ G @ G5 @ ( topolo174197925503356063within @ C @ X @ S3 ) )
=> ( ( ( G @ X )
!= ( zero_zero @ A ) )
=> ( has_derivative @ C @ A
@ ^ [X2: C] : ( divide_divide @ A @ ( F2 @ X2 ) @ ( G @ X2 ) )
@ ^ [H: C] : ( plus_plus @ A @ ( times_times @ A @ ( uminus_uminus @ A @ ( F2 @ X ) ) @ ( times_times @ A @ ( times_times @ A @ ( inverse_inverse @ A @ ( G @ X ) ) @ ( G5 @ H ) ) @ ( inverse_inverse @ A @ ( G @ X ) ) ) ) @ ( divide_divide @ A @ ( F5 @ H ) @ ( G @ X ) ) )
@ ( topolo174197925503356063within @ C @ X @ S3 ) ) ) ) ) ) ).
% has_derivative_divide
thf(fact_7035_has__derivative__prod,axiom,
! [B: $tType,I6: $tType,A: $tType] :
( ( ( real_V822414075346904944vector @ A )
& ( real_V3459762299906320749_field @ B ) )
=> ! [I5: set @ I6,F2: I6 > A > B,F5: I6 > A > B,X: A,S3: set @ A] :
( ! [I2: I6] :
( ( member @ I6 @ I2 @ I5 )
=> ( has_derivative @ A @ B @ ( F2 @ I2 ) @ ( F5 @ I2 ) @ ( topolo174197925503356063within @ A @ X @ S3 ) ) )
=> ( has_derivative @ A @ B
@ ^ [X2: A] :
( groups7121269368397514597t_prod @ I6 @ B
@ ^ [I3: I6] : ( F2 @ I3 @ X2 )
@ I5 )
@ ^ [Y2: A] :
( groups7311177749621191930dd_sum @ I6 @ B
@ ^ [I3: I6] :
( times_times @ B @ ( F5 @ I3 @ Y2 )
@ ( groups7121269368397514597t_prod @ I6 @ B
@ ^ [J3: I6] : ( F2 @ J3 @ X )
@ ( minus_minus @ ( set @ I6 ) @ I5 @ ( insert @ I6 @ I3 @ ( bot_bot @ ( set @ I6 ) ) ) ) ) )
@ I5 )
@ ( topolo174197925503356063within @ A @ X @ S3 ) ) ) ) ).
% has_derivative_prod
thf(fact_7036_has__derivative__powr,axiom,
! [A: $tType] :
( ( real_V822414075346904944vector @ A )
=> ! [G: A > real,G5: A > real,X: A,X8: set @ A,F2: A > real,F5: A > real] :
( ( has_derivative @ A @ real @ G @ G5 @ ( topolo174197925503356063within @ A @ X @ X8 ) )
=> ( ( has_derivative @ A @ real @ F2 @ F5 @ ( topolo174197925503356063within @ A @ X @ X8 ) )
=> ( ( ord_less @ real @ ( zero_zero @ real ) @ ( G @ X ) )
=> ( ( member @ A @ X @ X8 )
=> ( has_derivative @ A @ real
@ ^ [X2: A] : ( powr @ real @ ( G @ X2 ) @ ( F2 @ X2 ) )
@ ^ [H: A] : ( times_times @ real @ ( powr @ real @ ( G @ X ) @ ( F2 @ X ) ) @ ( plus_plus @ real @ ( times_times @ real @ ( F5 @ H ) @ ( ln_ln @ real @ ( G @ X ) ) ) @ ( divide_divide @ real @ ( times_times @ real @ ( G5 @ H ) @ ( F2 @ X ) ) @ ( G @ X ) ) ) )
@ ( topolo174197925503356063within @ A @ X @ X8 ) ) ) ) ) ) ) ).
% has_derivative_powr
thf(fact_7037_has__derivative__real__sqrt,axiom,
! [A: $tType] :
( ( real_V822414075346904944vector @ A )
=> ! [G: A > real,X: A,G5: A > real,S: set @ A] :
( ( ord_less @ real @ ( zero_zero @ real ) @ ( G @ X ) )
=> ( ( has_derivative @ A @ real @ G @ G5 @ ( topolo174197925503356063within @ A @ X @ S ) )
=> ( has_derivative @ A @ real
@ ^ [X2: A] : ( sqrt @ ( G @ X2 ) )
@ ^ [X2: A] : ( times_times @ real @ ( G5 @ X2 ) @ ( divide_divide @ real @ ( inverse_inverse @ real @ ( sqrt @ ( G @ X ) ) ) @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) )
@ ( topolo174197925503356063within @ A @ X @ S ) ) ) ) ) ).
% has_derivative_real_sqrt
thf(fact_7038_has__derivative__arctan,axiom,
! [A: $tType] :
( ( real_V822414075346904944vector @ A )
=> ! [G: A > real,G5: A > real,X: A,S: set @ A] :
( ( has_derivative @ A @ real @ G @ G5 @ ( topolo174197925503356063within @ A @ X @ S ) )
=> ( has_derivative @ A @ real
@ ^ [X2: A] : ( arctan @ ( G @ X2 ) )
@ ^ [X2: A] : ( times_times @ real @ ( G5 @ X2 ) @ ( inverse_inverse @ real @ ( plus_plus @ real @ ( one_one @ real ) @ ( power_power @ real @ ( G @ X ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) )
@ ( topolo174197925503356063within @ A @ X @ S ) ) ) ) ).
% has_derivative_arctan
thf(fact_7039_has__derivative__tan,axiom,
! [A: $tType] :
( ( real_V822414075346904944vector @ A )
=> ! [G: A > real,X: A,G5: A > real,S: set @ A] :
( ( ( cos @ real @ ( G @ X ) )
!= ( zero_zero @ real ) )
=> ( ( has_derivative @ A @ real @ G @ G5 @ ( topolo174197925503356063within @ A @ X @ S ) )
=> ( has_derivative @ A @ real
@ ^ [X2: A] : ( tan @ real @ ( G @ X2 ) )
@ ^ [X2: A] : ( times_times @ real @ ( G5 @ X2 ) @ ( inverse_inverse @ real @ ( power_power @ real @ ( cos @ real @ ( G @ X ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) )
@ ( topolo174197925503356063within @ A @ X @ S ) ) ) ) ) ).
% has_derivative_tan
thf(fact_7040_DERIV__series_H,axiom,
! [F2: real > nat > real,F5: real > nat > real,X0: real,A2: real,B2: real,L6: nat > real] :
( ! [N: nat] :
( has_field_derivative @ real
@ ^ [X2: real] : ( F2 @ X2 @ N )
@ ( F5 @ X0 @ N )
@ ( topolo174197925503356063within @ real @ X0 @ ( top_top @ ( set @ real ) ) ) )
=> ( ! [X3: real] :
( ( member @ real @ X3 @ ( set_or5935395276787703475ssThan @ real @ A2 @ B2 ) )
=> ( summable @ real @ ( F2 @ X3 ) ) )
=> ( ( member @ real @ X0 @ ( set_or5935395276787703475ssThan @ real @ A2 @ B2 ) )
=> ( ( summable @ real @ ( F5 @ X0 ) )
=> ( ( summable @ real @ L6 )
=> ( ! [N: nat,X3: real,Y3: real] :
( ( member @ real @ X3 @ ( set_or5935395276787703475ssThan @ real @ A2 @ B2 ) )
=> ( ( member @ real @ Y3 @ ( set_or5935395276787703475ssThan @ real @ A2 @ B2 ) )
=> ( ord_less_eq @ real @ ( abs_abs @ real @ ( minus_minus @ real @ ( F2 @ X3 @ N ) @ ( F2 @ Y3 @ N ) ) ) @ ( times_times @ real @ ( L6 @ N ) @ ( abs_abs @ real @ ( minus_minus @ real @ X3 @ Y3 ) ) ) ) ) )
=> ( has_field_derivative @ real
@ ^ [X2: real] : ( suminf @ real @ ( F2 @ X2 ) )
@ ( suminf @ real @ ( F5 @ X0 ) )
@ ( topolo174197925503356063within @ real @ X0 @ ( top_top @ ( set @ real ) ) ) ) ) ) ) ) ) ) ).
% DERIV_series'
thf(fact_7041_has__derivative__arccos,axiom,
! [A: $tType] :
( ( real_V822414075346904944vector @ A )
=> ! [G: A > real,X: A,G5: A > real,S: set @ A] :
( ( ord_less @ real @ ( uminus_uminus @ real @ ( one_one @ real ) ) @ ( G @ X ) )
=> ( ( ord_less @ real @ ( G @ X ) @ ( one_one @ real ) )
=> ( ( has_derivative @ A @ real @ G @ G5 @ ( topolo174197925503356063within @ A @ X @ S ) )
=> ( has_derivative @ A @ real
@ ^ [X2: A] : ( arccos @ ( G @ X2 ) )
@ ^ [X2: A] : ( times_times @ real @ ( G5 @ X2 ) @ ( inverse_inverse @ real @ ( uminus_uminus @ real @ ( sqrt @ ( minus_minus @ real @ ( one_one @ real ) @ ( power_power @ real @ ( G @ X ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) ) )
@ ( topolo174197925503356063within @ A @ X @ S ) ) ) ) ) ) ).
% has_derivative_arccos
thf(fact_7042_has__derivative__floor,axiom,
! [Aa: $tType,A: $tType] :
( ( ( real_V822414075346904944vector @ A )
& ( archim2362893244070406136eiling @ Aa )
& ( topolo2564578578187576103pology @ Aa ) )
=> ! [G: A > real,X: A,F2: real > Aa,G5: A > real,S: set @ A] :
( ( topolo3448309680560233919inuous @ real @ Aa @ ( topolo174197925503356063within @ real @ ( G @ X ) @ ( top_top @ ( set @ real ) ) ) @ F2 )
=> ( ~ ( member @ Aa @ ( F2 @ ( G @ X ) ) @ ( ring_1_Ints @ Aa ) )
=> ( ( has_derivative @ A @ real @ G @ G5 @ ( topolo174197925503356063within @ A @ X @ S ) )
=> ( has_derivative @ A @ real
@ ^ [X2: A] : ( ring_1_of_int @ real @ ( archim6421214686448440834_floor @ Aa @ ( F2 @ ( G @ X2 ) ) ) )
@ ^ [X2: A] : ( times_times @ real @ ( G5 @ X2 ) @ ( zero_zero @ real ) )
@ ( topolo174197925503356063within @ A @ X @ S ) ) ) ) ) ) ).
% has_derivative_floor
thf(fact_7043_termdiffs__aux,axiom,
! [A: $tType] :
( ( ( real_Vector_banach @ A )
& ( real_V3459762299906320749_field @ A ) )
=> ! [C2: nat > A,K6: A,X: A] :
( ( summable @ A
@ ^ [N2: nat] : ( times_times @ A @ ( diffs @ A @ ( diffs @ A @ C2 ) @ N2 ) @ ( power_power @ A @ K6 @ N2 ) ) )
=> ( ( ord_less @ real @ ( real_V7770717601297561774m_norm @ A @ X ) @ ( real_V7770717601297561774m_norm @ A @ K6 ) )
=> ( filterlim @ A @ A
@ ^ [H: A] :
( suminf @ A
@ ^ [N2: nat] : ( times_times @ A @ ( C2 @ N2 ) @ ( minus_minus @ A @ ( divide_divide @ A @ ( minus_minus @ A @ ( power_power @ A @ ( plus_plus @ A @ X @ H ) @ N2 ) @ ( power_power @ A @ X @ N2 ) ) @ H ) @ ( times_times @ A @ ( semiring_1_of_nat @ A @ N2 ) @ ( power_power @ A @ X @ ( minus_minus @ nat @ N2 @ ( suc @ ( zero_zero @ nat ) ) ) ) ) ) ) )
@ ( topolo7230453075368039082e_nhds @ A @ ( zero_zero @ A ) )
@ ( topolo174197925503356063within @ A @ ( zero_zero @ A ) @ ( top_top @ ( set @ A ) ) ) ) ) ) ) ).
% termdiffs_aux
thf(fact_7044_finite__greaterThanLessThan__integer,axiom,
! [L2: code_integer,U: code_integer] : ( finite_finite2 @ code_integer @ ( set_or5935395276787703475ssThan @ code_integer @ L2 @ U ) ) ).
% finite_greaterThanLessThan_integer
thf(fact_7045_tendsto__mult__right__iff,axiom,
! [A: $tType,B: $tType] :
( ( ( field @ A )
& ( topolo4211221413907600880p_mult @ A ) )
=> ! [C2: A,F2: B > A,L2: A,F4: filter @ B] :
( ( C2
!= ( zero_zero @ A ) )
=> ( ( filterlim @ B @ A
@ ^ [X2: B] : ( times_times @ A @ ( F2 @ X2 ) @ C2 )
@ ( topolo7230453075368039082e_nhds @ A @ ( times_times @ A @ L2 @ C2 ) )
@ F4 )
= ( filterlim @ B @ A @ F2 @ ( topolo7230453075368039082e_nhds @ A @ L2 ) @ F4 ) ) ) ) ).
% tendsto_mult_right_iff
thf(fact_7046_tendsto__mult__left__iff,axiom,
! [A: $tType,B: $tType] :
( ( ( field @ A )
& ( topolo4211221413907600880p_mult @ A ) )
=> ! [C2: A,F2: B > A,L2: A,F4: filter @ B] :
( ( C2
!= ( zero_zero @ A ) )
=> ( ( filterlim @ B @ A
@ ^ [X2: B] : ( times_times @ A @ C2 @ ( F2 @ X2 ) )
@ ( topolo7230453075368039082e_nhds @ A @ ( times_times @ A @ C2 @ L2 ) )
@ F4 )
= ( filterlim @ B @ A @ F2 @ ( topolo7230453075368039082e_nhds @ A @ L2 ) @ F4 ) ) ) ) ).
% tendsto_mult_left_iff
thf(fact_7047_power__tendsto__0__iff,axiom,
! [A: $tType,N3: nat,F2: A > real,F4: filter @ A] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ N3 )
=> ( ( filterlim @ A @ real
@ ^ [X2: A] : ( power_power @ real @ ( F2 @ X2 ) @ N3 )
@ ( topolo7230453075368039082e_nhds @ real @ ( zero_zero @ real ) )
@ F4 )
= ( filterlim @ A @ real @ F2 @ ( topolo7230453075368039082e_nhds @ real @ ( zero_zero @ real ) ) @ F4 ) ) ) ).
% power_tendsto_0_iff
thf(fact_7048_LIM__offset,axiom,
! [B: $tType,A: $tType] :
( ( ( real_V822414075346904944vector @ A )
& ( topolo4958980785337419405_space @ B ) )
=> ! [F2: A > B,L6: B,A2: A,K: A] :
( ( filterlim @ A @ B @ F2 @ ( topolo7230453075368039082e_nhds @ B @ L6 ) @ ( topolo174197925503356063within @ A @ A2 @ ( top_top @ ( set @ A ) ) ) )
=> ( filterlim @ A @ B
@ ^ [X2: A] : ( F2 @ ( plus_plus @ A @ X2 @ K ) )
@ ( topolo7230453075368039082e_nhds @ B @ L6 )
@ ( topolo174197925503356063within @ A @ ( minus_minus @ A @ A2 @ K ) @ ( top_top @ ( set @ A ) ) ) ) ) ) ).
% LIM_offset
thf(fact_7049_isCont__Pair,axiom,
! [C: $tType,B: $tType,A: $tType] :
( ( ( topological_t2_space @ A )
& ( topolo4958980785337419405_space @ B )
& ( topolo4958980785337419405_space @ C ) )
=> ! [A2: A,F2: A > B,G: A > C] :
( ( topolo3448309680560233919inuous @ A @ B @ ( topolo174197925503356063within @ A @ A2 @ ( top_top @ ( set @ A ) ) ) @ F2 )
=> ( ( topolo3448309680560233919inuous @ A @ C @ ( topolo174197925503356063within @ A @ A2 @ ( top_top @ ( set @ A ) ) ) @ G )
=> ( topolo3448309680560233919inuous @ A @ ( product_prod @ B @ C ) @ ( topolo174197925503356063within @ A @ A2 @ ( top_top @ ( set @ A ) ) )
@ ^ [X2: A] : ( product_Pair @ B @ C @ ( F2 @ X2 ) @ ( G @ X2 ) ) ) ) ) ) ).
% isCont_Pair
thf(fact_7050_has__field__derivative__iff,axiom,
! [A: $tType] :
( ( real_V3459762299906320749_field @ A )
=> ! [F2: A > A,D4: A,X: A,S3: set @ A] :
( ( has_field_derivative @ A @ F2 @ D4 @ ( topolo174197925503356063within @ A @ X @ S3 ) )
= ( filterlim @ A @ A
@ ^ [Y2: A] : ( divide_divide @ A @ ( minus_minus @ A @ ( F2 @ Y2 ) @ ( F2 @ X ) ) @ ( minus_minus @ A @ Y2 @ X ) )
@ ( topolo7230453075368039082e_nhds @ A @ D4 )
@ ( topolo174197925503356063within @ A @ X @ S3 ) ) ) ) ).
% has_field_derivative_iff
thf(fact_7051_has__field__derivativeD,axiom,
! [A: $tType] :
( ( real_V3459762299906320749_field @ A )
=> ! [F2: A > A,D4: A,X: A,S3: set @ A] :
( ( has_field_derivative @ A @ F2 @ D4 @ ( topolo174197925503356063within @ A @ X @ S3 ) )
=> ( filterlim @ A @ A
@ ^ [Y2: A] : ( divide_divide @ A @ ( minus_minus @ A @ ( F2 @ Y2 ) @ ( F2 @ X ) ) @ ( minus_minus @ A @ Y2 @ X ) )
@ ( topolo7230453075368039082e_nhds @ A @ D4 )
@ ( topolo174197925503356063within @ A @ X @ S3 ) ) ) ) ).
% has_field_derivativeD
thf(fact_7052_isCont__LIM__compose2,axiom,
! [C: $tType,B: $tType,A: $tType] :
( ( ( real_V822414075346904944vector @ A )
& ( topolo4958980785337419405_space @ B )
& ( topolo4958980785337419405_space @ C ) )
=> ! [A2: A,F2: A > B,G: B > C,L2: C] :
( ( topolo3448309680560233919inuous @ A @ B @ ( topolo174197925503356063within @ A @ A2 @ ( top_top @ ( set @ A ) ) ) @ F2 )
=> ( ( filterlim @ B @ C @ G @ ( topolo7230453075368039082e_nhds @ C @ L2 ) @ ( topolo174197925503356063within @ B @ ( F2 @ A2 ) @ ( top_top @ ( set @ B ) ) ) )
=> ( ? [D6: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ D6 )
& ! [X3: A] :
( ( ( X3 != A2 )
& ( ord_less @ real @ ( real_V7770717601297561774m_norm @ A @ ( minus_minus @ A @ X3 @ A2 ) ) @ D6 ) )
=> ( ( F2 @ X3 )
!= ( F2 @ A2 ) ) ) )
=> ( filterlim @ A @ C
@ ^ [X2: A] : ( G @ ( F2 @ X2 ) )
@ ( topolo7230453075368039082e_nhds @ C @ L2 )
@ ( topolo174197925503356063within @ A @ A2 @ ( top_top @ ( set @ A ) ) ) ) ) ) ) ) ).
% isCont_LIM_compose2
thf(fact_7053_LIM__equal2,axiom,
! [B: $tType,A: $tType] :
( ( ( real_V822414075346904944vector @ A )
& ( topolo4958980785337419405_space @ B ) )
=> ! [R: real,A2: A,F2: A > B,G: A > B,L2: B] :
( ( ord_less @ real @ ( zero_zero @ real ) @ R )
=> ( ! [X3: A] :
( ( X3 != A2 )
=> ( ( ord_less @ real @ ( real_V7770717601297561774m_norm @ A @ ( minus_minus @ A @ X3 @ A2 ) ) @ R )
=> ( ( F2 @ X3 )
= ( G @ X3 ) ) ) )
=> ( ( filterlim @ A @ B @ G @ ( topolo7230453075368039082e_nhds @ B @ L2 ) @ ( topolo174197925503356063within @ A @ A2 @ ( top_top @ ( set @ A ) ) ) )
=> ( filterlim @ A @ B @ F2 @ ( topolo7230453075368039082e_nhds @ B @ L2 ) @ ( topolo174197925503356063within @ A @ A2 @ ( top_top @ ( set @ A ) ) ) ) ) ) ) ) ).
% LIM_equal2
thf(fact_7054_LIM__eq,axiom,
! [B: $tType,A: $tType] :
( ( ( real_V822414075346904944vector @ A )
& ( real_V822414075346904944vector @ B ) )
=> ! [F2: A > B,L6: B,A2: A] :
( ( filterlim @ A @ B @ F2 @ ( topolo7230453075368039082e_nhds @ B @ L6 ) @ ( topolo174197925503356063within @ A @ A2 @ ( top_top @ ( set @ A ) ) ) )
= ( ! [R5: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ R5 )
=> ? [S5: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ S5 )
& ! [X2: A] :
( ( ( X2 != A2 )
& ( ord_less @ real @ ( real_V7770717601297561774m_norm @ A @ ( minus_minus @ A @ X2 @ A2 ) ) @ S5 ) )
=> ( ord_less @ real @ ( real_V7770717601297561774m_norm @ B @ ( minus_minus @ B @ ( F2 @ X2 ) @ L6 ) ) @ R5 ) ) ) ) ) ) ) ).
% LIM_eq
thf(fact_7055_LIM__I,axiom,
! [B: $tType,A: $tType] :
( ( ( real_V822414075346904944vector @ A )
& ( real_V822414075346904944vector @ B ) )
=> ! [A2: A,F2: A > B,L6: B] :
( ! [R3: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ R3 )
=> ? [S9: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ S9 )
& ! [X3: A] :
( ( ( X3 != A2 )
& ( ord_less @ real @ ( real_V7770717601297561774m_norm @ A @ ( minus_minus @ A @ X3 @ A2 ) ) @ S9 ) )
=> ( ord_less @ real @ ( real_V7770717601297561774m_norm @ B @ ( minus_minus @ B @ ( F2 @ X3 ) @ L6 ) ) @ R3 ) ) ) )
=> ( filterlim @ A @ B @ F2 @ ( topolo7230453075368039082e_nhds @ B @ L6 ) @ ( topolo174197925503356063within @ A @ A2 @ ( top_top @ ( set @ A ) ) ) ) ) ) ).
% LIM_I
thf(fact_7056_LIM__D,axiom,
! [B: $tType,A: $tType] :
( ( ( real_V822414075346904944vector @ A )
& ( real_V822414075346904944vector @ B ) )
=> ! [F2: A > B,L6: B,A2: A,R2: real] :
( ( filterlim @ A @ B @ F2 @ ( topolo7230453075368039082e_nhds @ B @ L6 ) @ ( topolo174197925503356063within @ A @ A2 @ ( top_top @ ( set @ A ) ) ) )
=> ( ( ord_less @ real @ ( zero_zero @ real ) @ R2 )
=> ? [S2: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ S2 )
& ! [X5: A] :
( ( ( X5 != A2 )
& ( ord_less @ real @ ( real_V7770717601297561774m_norm @ A @ ( minus_minus @ A @ X5 @ A2 ) ) @ S2 ) )
=> ( ord_less @ real @ ( real_V7770717601297561774m_norm @ B @ ( minus_minus @ B @ ( F2 @ X5 ) @ L6 ) ) @ R2 ) ) ) ) ) ) ).
% LIM_D
thf(fact_7057_isCont__iff,axiom,
! [B: $tType,A: $tType] :
( ( ( real_V822414075346904944vector @ A )
& ( topolo4958980785337419405_space @ B ) )
=> ! [X: A,F2: A > B] :
( ( topolo3448309680560233919inuous @ A @ B @ ( topolo174197925503356063within @ A @ X @ ( top_top @ ( set @ A ) ) ) @ F2 )
= ( filterlim @ A @ B
@ ^ [H: A] : ( F2 @ ( plus_plus @ A @ X @ H ) )
@ ( topolo7230453075368039082e_nhds @ B @ ( F2 @ X ) )
@ ( topolo174197925503356063within @ A @ ( zero_zero @ A ) @ ( top_top @ ( set @ A ) ) ) ) ) ) ).
% isCont_iff
thf(fact_7058_LIM__isCont__iff,axiom,
! [B: $tType,A: $tType] :
( ( ( real_V822414075346904944vector @ A )
& ( topolo4958980785337419405_space @ B ) )
=> ! [F2: A > B,A2: A] :
( ( filterlim @ A @ B @ F2 @ ( topolo7230453075368039082e_nhds @ B @ ( F2 @ A2 ) ) @ ( topolo174197925503356063within @ A @ A2 @ ( top_top @ ( set @ A ) ) ) )
= ( filterlim @ A @ B
@ ^ [H: A] : ( F2 @ ( plus_plus @ A @ A2 @ H ) )
@ ( topolo7230453075368039082e_nhds @ B @ ( F2 @ A2 ) )
@ ( topolo174197925503356063within @ A @ ( zero_zero @ A ) @ ( top_top @ ( set @ A ) ) ) ) ) ) ).
% LIM_isCont_iff
thf(fact_7059_LIM__offset__zero,axiom,
! [B: $tType,A: $tType] :
( ( ( real_V822414075346904944vector @ A )
& ( topolo4958980785337419405_space @ B ) )
=> ! [F2: A > B,L6: B,A2: A] :
( ( filterlim @ A @ B @ F2 @ ( topolo7230453075368039082e_nhds @ B @ L6 ) @ ( topolo174197925503356063within @ A @ A2 @ ( top_top @ ( set @ A ) ) ) )
=> ( filterlim @ A @ B
@ ^ [H: A] : ( F2 @ ( plus_plus @ A @ A2 @ H ) )
@ ( topolo7230453075368039082e_nhds @ B @ L6 )
@ ( topolo174197925503356063within @ A @ ( zero_zero @ A ) @ ( top_top @ ( set @ A ) ) ) ) ) ) ).
% LIM_offset_zero
thf(fact_7060_LIM__offset__zero__cancel,axiom,
! [B: $tType,A: $tType] :
( ( ( real_V822414075346904944vector @ A )
& ( topolo4958980785337419405_space @ B ) )
=> ! [F2: A > B,A2: A,L6: B] :
( ( filterlim @ A @ B
@ ^ [H: A] : ( F2 @ ( plus_plus @ A @ A2 @ H ) )
@ ( topolo7230453075368039082e_nhds @ B @ L6 )
@ ( topolo174197925503356063within @ A @ ( zero_zero @ A ) @ ( top_top @ ( set @ A ) ) ) )
=> ( filterlim @ A @ B @ F2 @ ( topolo7230453075368039082e_nhds @ B @ L6 ) @ ( topolo174197925503356063within @ A @ A2 @ ( top_top @ ( set @ A ) ) ) ) ) ) ).
% LIM_offset_zero_cancel
thf(fact_7061_DERIV__continuous,axiom,
! [A: $tType] :
( ( real_V3459762299906320749_field @ A )
=> ! [F2: A > A,D4: A,X: A,S: set @ A] :
( ( has_field_derivative @ A @ F2 @ D4 @ ( topolo174197925503356063within @ A @ X @ S ) )
=> ( topolo3448309680560233919inuous @ A @ A @ ( topolo174197925503356063within @ A @ X @ S ) @ F2 ) ) ) ).
% DERIV_continuous
thf(fact_7062_has__derivative__continuous,axiom,
! [B: $tType,A: $tType] :
( ( ( real_V822414075346904944vector @ A )
& ( real_V822414075346904944vector @ B ) )
=> ! [F2: A > B,F5: A > B,X: A,S: set @ A] :
( ( has_derivative @ A @ B @ F2 @ F5 @ ( topolo174197925503356063within @ A @ X @ S ) )
=> ( topolo3448309680560233919inuous @ A @ B @ ( topolo174197925503356063within @ A @ X @ S ) @ F2 ) ) ) ).
% has_derivative_continuous
thf(fact_7063_real__LIM__sandwich__zero,axiom,
! [A: $tType] :
( ( topolo4958980785337419405_space @ A )
=> ! [F2: A > real,A2: A,G: A > real] :
( ( filterlim @ A @ real @ F2 @ ( topolo7230453075368039082e_nhds @ real @ ( zero_zero @ real ) ) @ ( topolo174197925503356063within @ A @ A2 @ ( top_top @ ( set @ A ) ) ) )
=> ( ! [X3: A] :
( ( X3 != A2 )
=> ( ord_less_eq @ real @ ( zero_zero @ real ) @ ( G @ X3 ) ) )
=> ( ! [X3: A] :
( ( X3 != A2 )
=> ( ord_less_eq @ real @ ( G @ X3 ) @ ( F2 @ X3 ) ) )
=> ( filterlim @ A @ real @ G @ ( topolo7230453075368039082e_nhds @ real @ ( zero_zero @ real ) ) @ ( topolo174197925503356063within @ A @ A2 @ ( top_top @ ( set @ A ) ) ) ) ) ) ) ) ).
% real_LIM_sandwich_zero
thf(fact_7064_tendsto__within__subset,axiom,
! [B: $tType,A: $tType] :
( ( topolo4958980785337419405_space @ A )
=> ! [F2: A > B,L2: filter @ B,X: A,S3: set @ A,T5: set @ A] :
( ( filterlim @ A @ B @ F2 @ L2 @ ( topolo174197925503356063within @ A @ X @ S3 ) )
=> ( ( ord_less_eq @ ( set @ A ) @ T5 @ S3 )
=> ( filterlim @ A @ B @ F2 @ L2 @ ( topolo174197925503356063within @ A @ X @ T5 ) ) ) ) ) ).
% tendsto_within_subset
thf(fact_7065_LIM__imp__LIM,axiom,
! [B: $tType,C: $tType,A: $tType] :
( ( ( topolo4958980785337419405_space @ A )
& ( real_V822414075346904944vector @ C )
& ( real_V822414075346904944vector @ B ) )
=> ! [F2: A > B,L2: B,A2: A,G: A > C,M: C] :
( ( filterlim @ A @ B @ F2 @ ( topolo7230453075368039082e_nhds @ B @ L2 ) @ ( topolo174197925503356063within @ A @ A2 @ ( top_top @ ( set @ A ) ) ) )
=> ( ! [X3: A] :
( ( X3 != A2 )
=> ( ord_less_eq @ real @ ( real_V7770717601297561774m_norm @ C @ ( minus_minus @ C @ ( G @ X3 ) @ M ) ) @ ( real_V7770717601297561774m_norm @ B @ ( minus_minus @ B @ ( F2 @ X3 ) @ L2 ) ) ) )
=> ( filterlim @ A @ C @ G @ ( topolo7230453075368039082e_nhds @ C @ M ) @ ( topolo174197925503356063within @ A @ A2 @ ( top_top @ ( set @ A ) ) ) ) ) ) ) ).
% LIM_imp_LIM
thf(fact_7066_IVT2,axiom,
! [A: $tType,B: $tType] :
( ( ( topolo1944317154257567458pology @ B )
& ( topolo8458572112393995274pology @ A ) )
=> ! [F2: A > B,B2: A,Y: B,A2: A] :
( ( ord_less_eq @ B @ ( F2 @ B2 ) @ Y )
=> ( ( ord_less_eq @ B @ Y @ ( F2 @ A2 ) )
=> ( ( ord_less_eq @ A @ A2 @ B2 )
=> ( ! [X3: A] :
( ( ( ord_less_eq @ A @ A2 @ X3 )
& ( ord_less_eq @ A @ X3 @ B2 ) )
=> ( topolo3448309680560233919inuous @ A @ B @ ( topolo174197925503356063within @ A @ X3 @ ( top_top @ ( set @ A ) ) ) @ F2 ) )
=> ? [X3: A] :
( ( ord_less_eq @ A @ A2 @ X3 )
& ( ord_less_eq @ A @ X3 @ B2 )
& ( ( F2 @ X3 )
= Y ) ) ) ) ) ) ) ).
% IVT2
thf(fact_7067_IVT,axiom,
! [A: $tType,B: $tType] :
( ( ( topolo1944317154257567458pology @ B )
& ( topolo8458572112393995274pology @ A ) )
=> ! [F2: A > B,A2: A,Y: B,B2: A] :
( ( ord_less_eq @ B @ ( F2 @ A2 ) @ Y )
=> ( ( ord_less_eq @ B @ Y @ ( F2 @ B2 ) )
=> ( ( ord_less_eq @ A @ A2 @ B2 )
=> ( ! [X3: A] :
( ( ( ord_less_eq @ A @ A2 @ X3 )
& ( ord_less_eq @ A @ X3 @ B2 ) )
=> ( topolo3448309680560233919inuous @ A @ B @ ( topolo174197925503356063within @ A @ X3 @ ( top_top @ ( set @ A ) ) ) @ F2 ) )
=> ? [X3: A] :
( ( ord_less_eq @ A @ A2 @ X3 )
& ( ord_less_eq @ A @ X3 @ B2 )
& ( ( F2 @ X3 )
= Y ) ) ) ) ) ) ) ).
% IVT
thf(fact_7068_tendsto__mono,axiom,
! [A: $tType,B: $tType] :
( ( topolo4958980785337419405_space @ A )
=> ! [F4: filter @ B,F8: filter @ B,F2: B > A,L2: A] :
( ( ord_less_eq @ ( filter @ B ) @ F4 @ F8 )
=> ( ( filterlim @ B @ A @ F2 @ ( topolo7230453075368039082e_nhds @ A @ L2 ) @ F8 )
=> ( filterlim @ B @ A @ F2 @ ( topolo7230453075368039082e_nhds @ A @ L2 ) @ F4 ) ) ) ) ).
% tendsto_mono
thf(fact_7069_atLeastSucLessThan__greaterThanLessThan,axiom,
! [L2: nat,U: nat] :
( ( set_or7035219750837199246ssThan @ nat @ ( suc @ L2 ) @ U )
= ( set_or5935395276787703475ssThan @ nat @ L2 @ U ) ) ).
% atLeastSucLessThan_greaterThanLessThan
thf(fact_7070_DERIV__isCont,axiom,
! [A: $tType] :
( ( real_V3459762299906320749_field @ A )
=> ! [F2: A > A,D4: A,X: A] :
( ( has_field_derivative @ A @ F2 @ D4 @ ( topolo174197925503356063within @ A @ X @ ( top_top @ ( set @ A ) ) ) )
=> ( topolo3448309680560233919inuous @ A @ A @ ( topolo174197925503356063within @ A @ X @ ( top_top @ ( set @ A ) ) ) @ F2 ) ) ) ).
% DERIV_isCont
thf(fact_7071_tendsto__log,axiom,
! [A: $tType,F2: A > real,A2: real,F4: filter @ A,G: A > real,B2: real] :
( ( filterlim @ A @ real @ F2 @ ( topolo7230453075368039082e_nhds @ real @ A2 ) @ F4 )
=> ( ( filterlim @ A @ real @ G @ ( topolo7230453075368039082e_nhds @ real @ B2 ) @ F4 )
=> ( ( ord_less @ real @ ( zero_zero @ real ) @ A2 )
=> ( ( A2
!= ( one_one @ real ) )
=> ( ( ord_less @ real @ ( zero_zero @ real ) @ B2 )
=> ( filterlim @ A @ real
@ ^ [X2: A] : ( log @ ( F2 @ X2 ) @ ( G @ X2 ) )
@ ( topolo7230453075368039082e_nhds @ real @ ( log @ A2 @ B2 ) )
@ F4 ) ) ) ) ) ) ).
% tendsto_log
thf(fact_7072_tendsto__arcosh,axiom,
! [B: $tType,F2: B > real,A2: real,F4: filter @ B] :
( ( filterlim @ B @ real @ F2 @ ( topolo7230453075368039082e_nhds @ real @ A2 ) @ F4 )
=> ( ( ord_less @ real @ ( one_one @ real ) @ A2 )
=> ( filterlim @ B @ real
@ ^ [X2: B] : ( arcosh @ real @ ( F2 @ X2 ) )
@ ( topolo7230453075368039082e_nhds @ real @ ( arcosh @ real @ A2 ) )
@ F4 ) ) ) ).
% tendsto_arcosh
thf(fact_7073_tendsto__artanh,axiom,
! [A: $tType,F2: A > real,A2: real,F4: filter @ A] :
( ( filterlim @ A @ real @ F2 @ ( topolo7230453075368039082e_nhds @ real @ A2 ) @ F4 )
=> ( ( ord_less @ real @ ( uminus_uminus @ real @ ( one_one @ real ) ) @ A2 )
=> ( ( ord_less @ real @ A2 @ ( one_one @ real ) )
=> ( filterlim @ A @ real
@ ^ [X2: A] : ( artanh @ real @ ( F2 @ X2 ) )
@ ( topolo7230453075368039082e_nhds @ real @ ( artanh @ real @ A2 ) )
@ F4 ) ) ) ) ).
% tendsto_artanh
thf(fact_7074_continuous__Pair,axiom,
! [C: $tType,B: $tType,A: $tType] :
( ( ( topological_t2_space @ A )
& ( topolo4958980785337419405_space @ B )
& ( topolo4958980785337419405_space @ C ) )
=> ! [F4: filter @ A,F2: A > B,G: A > C] :
( ( topolo3448309680560233919inuous @ A @ B @ F4 @ F2 )
=> ( ( topolo3448309680560233919inuous @ A @ C @ F4 @ G )
=> ( topolo3448309680560233919inuous @ A @ ( product_prod @ B @ C ) @ F4
@ ^ [X2: A] : ( product_Pair @ B @ C @ ( F2 @ X2 ) @ ( G @ X2 ) ) ) ) ) ) ).
% continuous_Pair
thf(fact_7075_tendsto__Pair,axiom,
! [C: $tType,B: $tType,A: $tType] :
( ( ( topolo4958980785337419405_space @ B )
& ( topolo4958980785337419405_space @ C ) )
=> ! [F2: A > B,A2: B,F4: filter @ A,G: A > C,B2: C] :
( ( filterlim @ A @ B @ F2 @ ( topolo7230453075368039082e_nhds @ B @ A2 ) @ F4 )
=> ( ( filterlim @ A @ C @ G @ ( topolo7230453075368039082e_nhds @ C @ B2 ) @ F4 )
=> ( filterlim @ A @ ( product_prod @ B @ C )
@ ^ [X2: A] : ( product_Pair @ B @ C @ ( F2 @ X2 ) @ ( G @ X2 ) )
@ ( topolo7230453075368039082e_nhds @ ( product_prod @ B @ C ) @ ( product_Pair @ B @ C @ A2 @ B2 ) )
@ F4 ) ) ) ) ).
% tendsto_Pair
thf(fact_7076_tendsto__diff,axiom,
! [A: $tType,B: $tType] :
( ( topolo1633459387980952147up_add @ A )
=> ! [F2: B > A,A2: A,F4: filter @ B,G: B > A,B2: A] :
( ( filterlim @ B @ A @ F2 @ ( topolo7230453075368039082e_nhds @ A @ A2 ) @ F4 )
=> ( ( filterlim @ B @ A @ G @ ( topolo7230453075368039082e_nhds @ A @ B2 ) @ F4 )
=> ( filterlim @ B @ A
@ ^ [X2: B] : ( minus_minus @ A @ ( F2 @ X2 ) @ ( G @ X2 ) )
@ ( topolo7230453075368039082e_nhds @ A @ ( minus_minus @ A @ A2 @ B2 ) )
@ F4 ) ) ) ) ).
% tendsto_diff
thf(fact_7077_continuous__diff,axiom,
! [B: $tType,A: $tType] :
( ( ( topological_t2_space @ A )
& ( topolo1633459387980952147up_add @ B ) )
=> ! [F4: filter @ A,F2: A > B,G: A > B] :
( ( topolo3448309680560233919inuous @ A @ B @ F4 @ F2 )
=> ( ( topolo3448309680560233919inuous @ A @ B @ F4 @ G )
=> ( topolo3448309680560233919inuous @ A @ B @ F4
@ ^ [X2: A] : ( minus_minus @ B @ ( F2 @ X2 ) @ ( G @ X2 ) ) ) ) ) ) ).
% continuous_diff
thf(fact_7078_tendsto__null__power,axiom,
! [B: $tType,A: $tType] :
( ( real_V2822296259951069270ebra_1 @ B )
=> ! [F2: A > B,F4: filter @ A,N3: nat] :
( ( filterlim @ A @ B @ F2 @ ( topolo7230453075368039082e_nhds @ B @ ( zero_zero @ B ) ) @ F4 )
=> ( ( ord_less @ nat @ ( zero_zero @ nat ) @ N3 )
=> ( filterlim @ A @ B
@ ^ [X2: A] : ( power_power @ B @ ( F2 @ X2 ) @ N3 )
@ ( topolo7230453075368039082e_nhds @ B @ ( zero_zero @ B ) )
@ F4 ) ) ) ) ).
% tendsto_null_power
thf(fact_7079_Lim__transform__eq,axiom,
! [A: $tType,B: $tType] :
( ( real_V822414075346904944vector @ A )
=> ! [F2: B > A,G: B > A,F4: filter @ B,A2: A] :
( ( filterlim @ B @ A
@ ^ [X2: B] : ( minus_minus @ A @ ( F2 @ X2 ) @ ( G @ X2 ) )
@ ( topolo7230453075368039082e_nhds @ A @ ( zero_zero @ A ) )
@ F4 )
=> ( ( filterlim @ B @ A @ F2 @ ( topolo7230453075368039082e_nhds @ A @ A2 ) @ F4 )
= ( filterlim @ B @ A @ G @ ( topolo7230453075368039082e_nhds @ A @ A2 ) @ F4 ) ) ) ) ).
% Lim_transform_eq
thf(fact_7080_LIM__zero__cancel,axiom,
! [B: $tType,A: $tType] :
( ( real_V822414075346904944vector @ B )
=> ! [F2: A > B,L2: B,F4: filter @ A] :
( ( filterlim @ A @ B
@ ^ [X2: A] : ( minus_minus @ B @ ( F2 @ X2 ) @ L2 )
@ ( topolo7230453075368039082e_nhds @ B @ ( zero_zero @ B ) )
@ F4 )
=> ( filterlim @ A @ B @ F2 @ ( topolo7230453075368039082e_nhds @ B @ L2 ) @ F4 ) ) ) ).
% LIM_zero_cancel
thf(fact_7081_Lim__transform2,axiom,
! [A: $tType,B: $tType] :
( ( real_V822414075346904944vector @ A )
=> ! [F2: B > A,A2: A,F4: filter @ B,G: B > A] :
( ( filterlim @ B @ A @ F2 @ ( topolo7230453075368039082e_nhds @ A @ A2 ) @ F4 )
=> ( ( filterlim @ B @ A
@ ^ [X2: B] : ( minus_minus @ A @ ( F2 @ X2 ) @ ( G @ X2 ) )
@ ( topolo7230453075368039082e_nhds @ A @ ( zero_zero @ A ) )
@ F4 )
=> ( filterlim @ B @ A @ G @ ( topolo7230453075368039082e_nhds @ A @ A2 ) @ F4 ) ) ) ) ).
% Lim_transform2
thf(fact_7082_Lim__transform,axiom,
! [A: $tType,B: $tType] :
( ( real_V822414075346904944vector @ A )
=> ! [G: B > A,A2: A,F4: filter @ B,F2: B > A] :
( ( filterlim @ B @ A @ G @ ( topolo7230453075368039082e_nhds @ A @ A2 ) @ F4 )
=> ( ( filterlim @ B @ A
@ ^ [X2: B] : ( minus_minus @ A @ ( F2 @ X2 ) @ ( G @ X2 ) )
@ ( topolo7230453075368039082e_nhds @ A @ ( zero_zero @ A ) )
@ F4 )
=> ( filterlim @ B @ A @ F2 @ ( topolo7230453075368039082e_nhds @ A @ A2 ) @ F4 ) ) ) ) ).
% Lim_transform
thf(fact_7083_LIM__zero__iff,axiom,
! [B: $tType,A: $tType] :
( ( real_V822414075346904944vector @ B )
=> ! [F2: A > B,L2: B,F4: filter @ A] :
( ( filterlim @ A @ B
@ ^ [X2: A] : ( minus_minus @ B @ ( F2 @ X2 ) @ L2 )
@ ( topolo7230453075368039082e_nhds @ B @ ( zero_zero @ B ) )
@ F4 )
= ( filterlim @ A @ B @ F2 @ ( topolo7230453075368039082e_nhds @ B @ L2 ) @ F4 ) ) ) ).
% LIM_zero_iff
thf(fact_7084_LIM__zero,axiom,
! [B: $tType,A: $tType] :
( ( real_V822414075346904944vector @ B )
=> ! [F2: A > B,L2: B,F4: filter @ A] :
( ( filterlim @ A @ B @ F2 @ ( topolo7230453075368039082e_nhds @ B @ L2 ) @ F4 )
=> ( filterlim @ A @ B
@ ^ [X2: A] : ( minus_minus @ B @ ( F2 @ X2 ) @ L2 )
@ ( topolo7230453075368039082e_nhds @ B @ ( zero_zero @ B ) )
@ F4 ) ) ) ).
% LIM_zero
thf(fact_7085_tendsto__mult__zero,axiom,
! [A: $tType,D5: $tType] :
( ( real_V4412858255891104859lgebra @ A )
=> ! [F2: D5 > A,F4: filter @ D5,G: D5 > A] :
( ( filterlim @ D5 @ A @ F2 @ ( topolo7230453075368039082e_nhds @ A @ ( zero_zero @ A ) ) @ F4 )
=> ( ( filterlim @ D5 @ A @ G @ ( topolo7230453075368039082e_nhds @ A @ ( zero_zero @ A ) ) @ F4 )
=> ( filterlim @ D5 @ A
@ ^ [X2: D5] : ( times_times @ A @ ( F2 @ X2 ) @ ( G @ X2 ) )
@ ( topolo7230453075368039082e_nhds @ A @ ( zero_zero @ A ) )
@ F4 ) ) ) ) ).
% tendsto_mult_zero
thf(fact_7086_tendsto__mult__left__zero,axiom,
! [A: $tType,D5: $tType] :
( ( real_V4412858255891104859lgebra @ A )
=> ! [F2: D5 > A,F4: filter @ D5,C2: A] :
( ( filterlim @ D5 @ A @ F2 @ ( topolo7230453075368039082e_nhds @ A @ ( zero_zero @ A ) ) @ F4 )
=> ( filterlim @ D5 @ A
@ ^ [X2: D5] : ( times_times @ A @ ( F2 @ X2 ) @ C2 )
@ ( topolo7230453075368039082e_nhds @ A @ ( zero_zero @ A ) )
@ F4 ) ) ) ).
% tendsto_mult_left_zero
thf(fact_7087_tendsto__mult__right__zero,axiom,
! [A: $tType,D5: $tType] :
( ( real_V4412858255891104859lgebra @ A )
=> ! [F2: D5 > A,F4: filter @ D5,C2: A] :
( ( filterlim @ D5 @ A @ F2 @ ( topolo7230453075368039082e_nhds @ A @ ( zero_zero @ A ) ) @ F4 )
=> ( filterlim @ D5 @ A
@ ^ [X2: D5] : ( times_times @ A @ C2 @ ( F2 @ X2 ) )
@ ( topolo7230453075368039082e_nhds @ A @ ( zero_zero @ A ) )
@ F4 ) ) ) ).
% tendsto_mult_right_zero
thf(fact_7088_tendsto__add__zero,axiom,
! [B: $tType,D5: $tType] :
( ( topolo6943815403480290642id_add @ B )
=> ! [F2: D5 > B,F4: filter @ D5,G: D5 > B] :
( ( filterlim @ D5 @ B @ F2 @ ( topolo7230453075368039082e_nhds @ B @ ( zero_zero @ B ) ) @ F4 )
=> ( ( filterlim @ D5 @ B @ G @ ( topolo7230453075368039082e_nhds @ B @ ( zero_zero @ B ) ) @ F4 )
=> ( filterlim @ D5 @ B
@ ^ [X2: D5] : ( plus_plus @ B @ ( F2 @ X2 ) @ ( G @ X2 ) )
@ ( topolo7230453075368039082e_nhds @ B @ ( zero_zero @ B ) )
@ F4 ) ) ) ) ).
% tendsto_add_zero
thf(fact_7089_tendsto__divide,axiom,
! [A: $tType,B: $tType] :
( ( real_V3459762299906320749_field @ A )
=> ! [F2: B > A,A2: A,F4: filter @ B,G: B > A,B2: A] :
( ( filterlim @ B @ A @ F2 @ ( topolo7230453075368039082e_nhds @ A @ A2 ) @ F4 )
=> ( ( filterlim @ B @ A @ G @ ( topolo7230453075368039082e_nhds @ A @ B2 ) @ F4 )
=> ( ( B2
!= ( zero_zero @ A ) )
=> ( filterlim @ B @ A
@ ^ [X2: B] : ( divide_divide @ A @ ( F2 @ X2 ) @ ( G @ X2 ) )
@ ( topolo7230453075368039082e_nhds @ A @ ( divide_divide @ A @ A2 @ B2 ) )
@ F4 ) ) ) ) ) ).
% tendsto_divide
thf(fact_7090_tendsto__divide__zero,axiom,
! [A: $tType,B: $tType] :
( ( real_V3459762299906320749_field @ A )
=> ! [F2: B > A,F4: filter @ B,C2: A] :
( ( filterlim @ B @ A @ F2 @ ( topolo7230453075368039082e_nhds @ A @ ( zero_zero @ A ) ) @ F4 )
=> ( filterlim @ B @ A
@ ^ [X2: B] : ( divide_divide @ A @ ( F2 @ X2 ) @ C2 )
@ ( topolo7230453075368039082e_nhds @ A @ ( zero_zero @ A ) )
@ F4 ) ) ) ).
% tendsto_divide_zero
thf(fact_7091_continuous__power,axiom,
! [B: $tType,A: $tType] :
( ( ( topological_t2_space @ A )
& ( power @ B )
& ( real_V4412858255891104859lgebra @ B ) )
=> ! [F4: filter @ A,F2: A > B,N3: nat] :
( ( topolo3448309680560233919inuous @ A @ B @ F4 @ F2 )
=> ( topolo3448309680560233919inuous @ A @ B @ F4
@ ^ [X2: A] : ( power_power @ B @ ( F2 @ X2 ) @ N3 ) ) ) ) ).
% continuous_power
thf(fact_7092_tendsto__power__strong,axiom,
! [B: $tType,C: $tType] :
( ( topolo1898628316856586783d_mult @ B )
=> ! [F2: C > B,A2: B,F4: filter @ C,G: C > nat,B2: nat] :
( ( filterlim @ C @ B @ F2 @ ( topolo7230453075368039082e_nhds @ B @ A2 ) @ F4 )
=> ( ( filterlim @ C @ nat @ G @ ( topolo7230453075368039082e_nhds @ nat @ B2 ) @ F4 )
=> ( filterlim @ C @ B
@ ^ [X2: C] : ( power_power @ B @ ( F2 @ X2 ) @ ( G @ X2 ) )
@ ( topolo7230453075368039082e_nhds @ B @ ( power_power @ B @ A2 @ B2 ) )
@ F4 ) ) ) ) ).
% tendsto_power_strong
thf(fact_7093_continuous__power_H,axiom,
! [B: $tType,C: $tType] :
( ( ( topological_t2_space @ C )
& ( topolo1898628316856586783d_mult @ B ) )
=> ! [F4: filter @ C,F2: C > B,G: C > nat] :
( ( topolo3448309680560233919inuous @ C @ B @ F4 @ F2 )
=> ( ( topolo3448309680560233919inuous @ C @ nat @ F4 @ G )
=> ( topolo3448309680560233919inuous @ C @ B @ F4
@ ^ [X2: C] : ( power_power @ B @ ( F2 @ X2 ) @ ( G @ X2 ) ) ) ) ) ) ).
% continuous_power'
thf(fact_7094_tendsto__power,axiom,
! [B: $tType,A: $tType] :
( ( ( power @ B )
& ( real_V4412858255891104859lgebra @ B ) )
=> ! [F2: A > B,A2: B,F4: filter @ A,N3: nat] :
( ( filterlim @ A @ B @ F2 @ ( topolo7230453075368039082e_nhds @ B @ A2 ) @ F4 )
=> ( filterlim @ A @ B
@ ^ [X2: A] : ( power_power @ B @ ( F2 @ X2 ) @ N3 )
@ ( topolo7230453075368039082e_nhds @ B @ ( power_power @ B @ A2 @ N3 ) )
@ F4 ) ) ) ).
% tendsto_power
thf(fact_7095_tendsto__real__root,axiom,
! [A: $tType,F2: A > real,X: real,F4: filter @ A,N3: nat] :
( ( filterlim @ A @ real @ F2 @ ( topolo7230453075368039082e_nhds @ real @ X ) @ F4 )
=> ( filterlim @ A @ real
@ ^ [X2: A] : ( root @ N3 @ ( F2 @ X2 ) )
@ ( topolo7230453075368039082e_nhds @ real @ ( root @ N3 @ X ) )
@ F4 ) ) ).
% tendsto_real_root
thf(fact_7096_tendsto__real__sqrt,axiom,
! [A: $tType,F2: A > real,X: real,F4: filter @ A] :
( ( filterlim @ A @ real @ F2 @ ( topolo7230453075368039082e_nhds @ real @ X ) @ F4 )
=> ( filterlim @ A @ real
@ ^ [X2: A] : ( sqrt @ ( F2 @ X2 ) )
@ ( topolo7230453075368039082e_nhds @ real @ ( sqrt @ X ) )
@ F4 ) ) ).
% tendsto_real_sqrt
thf(fact_7097_continuous__add,axiom,
! [B: $tType,D5: $tType] :
( ( ( topological_t2_space @ D5 )
& ( topolo6943815403480290642id_add @ B ) )
=> ! [F4: filter @ D5,F2: D5 > B,G: D5 > B] :
( ( topolo3448309680560233919inuous @ D5 @ B @ F4 @ F2 )
=> ( ( topolo3448309680560233919inuous @ D5 @ B @ F4 @ G )
=> ( topolo3448309680560233919inuous @ D5 @ B @ F4
@ ^ [X2: D5] : ( plus_plus @ B @ ( F2 @ X2 ) @ ( G @ X2 ) ) ) ) ) ) ).
% continuous_add
thf(fact_7098_tendsto__add__const__iff,axiom,
! [A: $tType,B: $tType] :
( ( topolo1633459387980952147up_add @ A )
=> ! [C2: A,F2: B > A,D: A,F4: filter @ B] :
( ( filterlim @ B @ A
@ ^ [X2: B] : ( plus_plus @ A @ C2 @ ( F2 @ X2 ) )
@ ( topolo7230453075368039082e_nhds @ A @ ( plus_plus @ A @ C2 @ D ) )
@ F4 )
= ( filterlim @ B @ A @ F2 @ ( topolo7230453075368039082e_nhds @ A @ D ) @ F4 ) ) ) ).
% tendsto_add_const_iff
thf(fact_7099_tendsto__add,axiom,
! [A: $tType,B: $tType] :
( ( topolo6943815403480290642id_add @ A )
=> ! [F2: B > A,A2: A,F4: filter @ B,G: B > A,B2: A] :
( ( filterlim @ B @ A @ F2 @ ( topolo7230453075368039082e_nhds @ A @ A2 ) @ F4 )
=> ( ( filterlim @ B @ A @ G @ ( topolo7230453075368039082e_nhds @ A @ B2 ) @ F4 )
=> ( filterlim @ B @ A
@ ^ [X2: B] : ( plus_plus @ A @ ( F2 @ X2 ) @ ( G @ X2 ) )
@ ( topolo7230453075368039082e_nhds @ A @ ( plus_plus @ A @ A2 @ B2 ) )
@ F4 ) ) ) ) ).
% tendsto_add
thf(fact_7100_continuous__mult__right,axiom,
! [A: $tType,B: $tType] :
( ( ( topological_t2_space @ B )
& ( real_V4412858255891104859lgebra @ A ) )
=> ! [F4: filter @ B,F2: B > A,C2: A] :
( ( topolo3448309680560233919inuous @ B @ A @ F4 @ F2 )
=> ( topolo3448309680560233919inuous @ B @ A @ F4
@ ^ [X2: B] : ( times_times @ A @ ( F2 @ X2 ) @ C2 ) ) ) ) ).
% continuous_mult_right
thf(fact_7101_continuous__mult__left,axiom,
! [A: $tType,B: $tType] :
( ( ( topological_t2_space @ B )
& ( real_V4412858255891104859lgebra @ A ) )
=> ! [F4: filter @ B,F2: B > A,C2: A] :
( ( topolo3448309680560233919inuous @ B @ A @ F4 @ F2 )
=> ( topolo3448309680560233919inuous @ B @ A @ F4
@ ^ [X2: B] : ( times_times @ A @ C2 @ ( F2 @ X2 ) ) ) ) ) ).
% continuous_mult_left
thf(fact_7102_continuous__mult_H,axiom,
! [B: $tType,D5: $tType] :
( ( ( topological_t2_space @ D5 )
& ( topolo4211221413907600880p_mult @ B ) )
=> ! [F4: filter @ D5,F2: D5 > B,G: D5 > B] :
( ( topolo3448309680560233919inuous @ D5 @ B @ F4 @ F2 )
=> ( ( topolo3448309680560233919inuous @ D5 @ B @ F4 @ G )
=> ( topolo3448309680560233919inuous @ D5 @ B @ F4
@ ^ [X2: D5] : ( times_times @ B @ ( F2 @ X2 ) @ ( G @ X2 ) ) ) ) ) ) ).
% continuous_mult'
thf(fact_7103_continuous__mult,axiom,
! [A: $tType,D5: $tType] :
( ( ( topological_t2_space @ D5 )
& ( real_V4412858255891104859lgebra @ A ) )
=> ! [F4: filter @ D5,F2: D5 > A,G: D5 > A] :
( ( topolo3448309680560233919inuous @ D5 @ A @ F4 @ F2 )
=> ( ( topolo3448309680560233919inuous @ D5 @ A @ F4 @ G )
=> ( topolo3448309680560233919inuous @ D5 @ A @ F4
@ ^ [X2: D5] : ( times_times @ A @ ( F2 @ X2 ) @ ( G @ X2 ) ) ) ) ) ) ).
% continuous_mult
thf(fact_7104_tendsto__mult,axiom,
! [A: $tType,B: $tType] :
( ( topolo4211221413907600880p_mult @ A )
=> ! [F2: B > A,A2: A,F4: filter @ B,G: B > A,B2: A] :
( ( filterlim @ B @ A @ F2 @ ( topolo7230453075368039082e_nhds @ A @ A2 ) @ F4 )
=> ( ( filterlim @ B @ A @ G @ ( topolo7230453075368039082e_nhds @ A @ B2 ) @ F4 )
=> ( filterlim @ B @ A
@ ^ [X2: B] : ( times_times @ A @ ( F2 @ X2 ) @ ( G @ X2 ) )
@ ( topolo7230453075368039082e_nhds @ A @ ( times_times @ A @ A2 @ B2 ) )
@ F4 ) ) ) ) ).
% tendsto_mult
thf(fact_7105_tendsto__mult__left,axiom,
! [A: $tType,B: $tType] :
( ( topolo4211221413907600880p_mult @ A )
=> ! [F2: B > A,L2: A,F4: filter @ B,C2: A] :
( ( filterlim @ B @ A @ F2 @ ( topolo7230453075368039082e_nhds @ A @ L2 ) @ F4 )
=> ( filterlim @ B @ A
@ ^ [X2: B] : ( times_times @ A @ C2 @ ( F2 @ X2 ) )
@ ( topolo7230453075368039082e_nhds @ A @ ( times_times @ A @ C2 @ L2 ) )
@ F4 ) ) ) ).
% tendsto_mult_left
thf(fact_7106_tendsto__mult__right,axiom,
! [A: $tType,B: $tType] :
( ( topolo4211221413907600880p_mult @ A )
=> ! [F2: B > A,L2: A,F4: filter @ B,C2: A] :
( ( filterlim @ B @ A @ F2 @ ( topolo7230453075368039082e_nhds @ A @ L2 ) @ F4 )
=> ( filterlim @ B @ A
@ ^ [X2: B] : ( times_times @ A @ ( F2 @ X2 ) @ C2 )
@ ( topolo7230453075368039082e_nhds @ A @ ( times_times @ A @ L2 @ C2 ) )
@ F4 ) ) ) ).
% tendsto_mult_right
thf(fact_7107_tendsto__mult__one,axiom,
! [B: $tType,D5: $tType] :
( ( topolo1898628316856586783d_mult @ B )
=> ! [F2: D5 > B,F4: filter @ D5,G: D5 > B] :
( ( filterlim @ D5 @ B @ F2 @ ( topolo7230453075368039082e_nhds @ B @ ( one_one @ B ) ) @ F4 )
=> ( ( filterlim @ D5 @ B @ G @ ( topolo7230453075368039082e_nhds @ B @ ( one_one @ B ) ) @ F4 )
=> ( filterlim @ D5 @ B
@ ^ [X2: D5] : ( times_times @ B @ ( F2 @ X2 ) @ ( G @ X2 ) )
@ ( topolo7230453075368039082e_nhds @ B @ ( one_one @ B ) )
@ F4 ) ) ) ) ).
% tendsto_mult_one
thf(fact_7108_tendsto__one__prod_H,axiom,
! [C: $tType,B: $tType,A: $tType] :
( ( topolo4987421752381908075d_mult @ C )
=> ! [I5: set @ B,F2: A > B > C,F4: filter @ A] :
( ! [I2: B] :
( ( member @ B @ I2 @ I5 )
=> ( filterlim @ A @ C
@ ^ [X2: A] : ( F2 @ X2 @ I2 )
@ ( topolo7230453075368039082e_nhds @ C @ ( one_one @ C ) )
@ F4 ) )
=> ( filterlim @ A @ C
@ ^ [I3: A] : ( groups7121269368397514597t_prod @ B @ C @ ( F2 @ I3 ) @ I5 )
@ ( topolo7230453075368039082e_nhds @ C @ ( one_one @ C ) )
@ F4 ) ) ) ).
% tendsto_one_prod'
thf(fact_7109_DERIV__LIM__iff,axiom,
! [A: $tType] :
( ( ( inverse @ A )
& ( real_V822414075346904944vector @ A ) )
=> ! [F2: A > A,A2: A,D4: A] :
( ( filterlim @ A @ A
@ ^ [H: A] : ( divide_divide @ A @ ( minus_minus @ A @ ( F2 @ ( plus_plus @ A @ A2 @ H ) ) @ ( F2 @ A2 ) ) @ H )
@ ( topolo7230453075368039082e_nhds @ A @ D4 )
@ ( topolo174197925503356063within @ A @ ( zero_zero @ A ) @ ( top_top @ ( set @ A ) ) ) )
= ( filterlim @ A @ A
@ ^ [X2: A] : ( divide_divide @ A @ ( minus_minus @ A @ ( F2 @ X2 ) @ ( F2 @ A2 ) ) @ ( minus_minus @ A @ X2 @ A2 ) )
@ ( topolo7230453075368039082e_nhds @ A @ D4 )
@ ( topolo174197925503356063within @ A @ A2 @ ( top_top @ ( set @ A ) ) ) ) ) ) ).
% DERIV_LIM_iff
thf(fact_7110_isCont__Lb__Ub,axiom,
! [A2: real,B2: real,F2: real > real] :
( ( ord_less_eq @ real @ A2 @ B2 )
=> ( ! [X3: real] :
( ( ( ord_less_eq @ real @ A2 @ X3 )
& ( ord_less_eq @ real @ X3 @ B2 ) )
=> ( topolo3448309680560233919inuous @ real @ real @ ( topolo174197925503356063within @ real @ X3 @ ( top_top @ ( set @ real ) ) ) @ F2 ) )
=> ? [L7: real,M10: real] :
( ! [X5: real] :
( ( ( ord_less_eq @ real @ A2 @ X5 )
& ( ord_less_eq @ real @ X5 @ B2 ) )
=> ( ( ord_less_eq @ real @ L7 @ ( F2 @ X5 ) )
& ( ord_less_eq @ real @ ( F2 @ X5 ) @ M10 ) ) )
& ! [Y4: real] :
( ( ( ord_less_eq @ real @ L7 @ Y4 )
& ( ord_less_eq @ real @ Y4 @ M10 ) )
=> ? [X3: real] :
( ( ord_less_eq @ real @ A2 @ X3 )
& ( ord_less_eq @ real @ X3 @ B2 )
& ( ( F2 @ X3 )
= Y4 ) ) ) ) ) ) ).
% isCont_Lb_Ub
thf(fact_7111_LIM__fun__gt__zero,axiom,
! [F2: real > real,L2: real,C2: real] :
( ( filterlim @ real @ real @ F2 @ ( topolo7230453075368039082e_nhds @ real @ L2 ) @ ( topolo174197925503356063within @ real @ C2 @ ( top_top @ ( set @ real ) ) ) )
=> ( ( ord_less @ real @ ( zero_zero @ real ) @ L2 )
=> ? [R3: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ R3 )
& ! [X5: real] :
( ( ( X5 != C2 )
& ( ord_less @ real @ ( abs_abs @ real @ ( minus_minus @ real @ C2 @ X5 ) ) @ R3 ) )
=> ( ord_less @ real @ ( zero_zero @ real ) @ ( F2 @ X5 ) ) ) ) ) ) ).
% LIM_fun_gt_zero
thf(fact_7112_LIM__fun__not__zero,axiom,
! [F2: real > real,L2: real,C2: real] :
( ( filterlim @ real @ real @ F2 @ ( topolo7230453075368039082e_nhds @ real @ L2 ) @ ( topolo174197925503356063within @ real @ C2 @ ( top_top @ ( set @ real ) ) ) )
=> ( ( L2
!= ( zero_zero @ real ) )
=> ? [R3: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ R3 )
& ! [X5: real] :
( ( ( X5 != C2 )
& ( ord_less @ real @ ( abs_abs @ real @ ( minus_minus @ real @ C2 @ X5 ) ) @ R3 ) )
=> ( ( F2 @ X5 )
!= ( zero_zero @ real ) ) ) ) ) ) ).
% LIM_fun_not_zero
thf(fact_7113_LIM__fun__less__zero,axiom,
! [F2: real > real,L2: real,C2: real] :
( ( filterlim @ real @ real @ F2 @ ( topolo7230453075368039082e_nhds @ real @ L2 ) @ ( topolo174197925503356063within @ real @ C2 @ ( top_top @ ( set @ real ) ) ) )
=> ( ( ord_less @ real @ L2 @ ( zero_zero @ real ) )
=> ? [R3: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ R3 )
& ! [X5: real] :
( ( ( X5 != C2 )
& ( ord_less @ real @ ( abs_abs @ real @ ( minus_minus @ real @ C2 @ X5 ) ) @ R3 ) )
=> ( ord_less @ real @ ( F2 @ X5 ) @ ( zero_zero @ real ) ) ) ) ) ) ).
% LIM_fun_less_zero
thf(fact_7114_LIM__compose2,axiom,
! [C: $tType,B: $tType,A: $tType] :
( ( ( real_V822414075346904944vector @ A )
& ( topolo4958980785337419405_space @ B )
& ( topolo4958980785337419405_space @ C ) )
=> ! [F2: A > B,B2: B,A2: A,G: B > C,C2: C] :
( ( filterlim @ A @ B @ F2 @ ( topolo7230453075368039082e_nhds @ B @ B2 ) @ ( topolo174197925503356063within @ A @ A2 @ ( top_top @ ( set @ A ) ) ) )
=> ( ( filterlim @ B @ C @ G @ ( topolo7230453075368039082e_nhds @ C @ C2 ) @ ( topolo174197925503356063within @ B @ B2 @ ( top_top @ ( set @ B ) ) ) )
=> ( ? [D6: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ D6 )
& ! [X3: A] :
( ( ( X3 != A2 )
& ( ord_less @ real @ ( real_V7770717601297561774m_norm @ A @ ( minus_minus @ A @ X3 @ A2 ) ) @ D6 ) )
=> ( ( F2 @ X3 )
!= B2 ) ) )
=> ( filterlim @ A @ C
@ ^ [X2: A] : ( G @ ( F2 @ X2 ) )
@ ( topolo7230453075368039082e_nhds @ C @ C2 )
@ ( topolo174197925503356063within @ A @ A2 @ ( top_top @ ( set @ A ) ) ) ) ) ) ) ) ).
% LIM_compose2
thf(fact_7115_isCont__real__sqrt,axiom,
! [X: real] : ( topolo3448309680560233919inuous @ real @ real @ ( topolo174197925503356063within @ real @ X @ ( top_top @ ( set @ real ) ) ) @ sqrt ) ).
% isCont_real_sqrt
thf(fact_7116_isCont__real__root,axiom,
! [X: real,N3: nat] : ( topolo3448309680560233919inuous @ real @ real @ ( topolo174197925503356063within @ real @ X @ ( top_top @ ( set @ real ) ) ) @ ( root @ N3 ) ) ).
% isCont_real_root
thf(fact_7117_continuous__at__within__divide,axiom,
! [B: $tType,A: $tType] :
( ( ( topological_t2_space @ A )
& ( real_V3459762299906320749_field @ B ) )
=> ! [A2: A,S: set @ A,F2: A > B,G: A > B] :
( ( topolo3448309680560233919inuous @ A @ B @ ( topolo174197925503356063within @ A @ A2 @ S ) @ F2 )
=> ( ( topolo3448309680560233919inuous @ A @ B @ ( topolo174197925503356063within @ A @ A2 @ S ) @ G )
=> ( ( ( G @ A2 )
!= ( zero_zero @ B ) )
=> ( topolo3448309680560233919inuous @ A @ B @ ( topolo174197925503356063within @ A @ A2 @ S )
@ ^ [X2: A] : ( divide_divide @ B @ ( F2 @ X2 ) @ ( G @ X2 ) ) ) ) ) ) ) ).
% continuous_at_within_divide
thf(fact_7118_isCont__add,axiom,
! [B: $tType,A: $tType] :
( ( ( topological_t2_space @ A )
& ( topolo6943815403480290642id_add @ B ) )
=> ! [A2: A,F2: A > B,G: A > B] :
( ( topolo3448309680560233919inuous @ A @ B @ ( topolo174197925503356063within @ A @ A2 @ ( top_top @ ( set @ A ) ) ) @ F2 )
=> ( ( topolo3448309680560233919inuous @ A @ B @ ( topolo174197925503356063within @ A @ A2 @ ( top_top @ ( set @ A ) ) ) @ G )
=> ( topolo3448309680560233919inuous @ A @ B @ ( topolo174197925503356063within @ A @ A2 @ ( top_top @ ( set @ A ) ) )
@ ^ [X2: A] : ( plus_plus @ B @ ( F2 @ X2 ) @ ( G @ X2 ) ) ) ) ) ) ).
% isCont_add
thf(fact_7119_isCont__mult,axiom,
! [B: $tType,A: $tType] :
( ( ( topological_t2_space @ A )
& ( real_V4412858255891104859lgebra @ B ) )
=> ! [A2: A,F2: A > B,G: A > B] :
( ( topolo3448309680560233919inuous @ A @ B @ ( topolo174197925503356063within @ A @ A2 @ ( top_top @ ( set @ A ) ) ) @ F2 )
=> ( ( topolo3448309680560233919inuous @ A @ B @ ( topolo174197925503356063within @ A @ A2 @ ( top_top @ ( set @ A ) ) ) @ G )
=> ( topolo3448309680560233919inuous @ A @ B @ ( topolo174197925503356063within @ A @ A2 @ ( top_top @ ( set @ A ) ) )
@ ^ [X2: A] : ( times_times @ B @ ( F2 @ X2 ) @ ( G @ X2 ) ) ) ) ) ) ).
% isCont_mult
thf(fact_7120_isCont__diff,axiom,
! [B: $tType,A: $tType] :
( ( ( topological_t2_space @ A )
& ( real_V822414075346904944vector @ B ) )
=> ! [A2: A,F2: A > B,G: A > B] :
( ( topolo3448309680560233919inuous @ A @ B @ ( topolo174197925503356063within @ A @ A2 @ ( top_top @ ( set @ A ) ) ) @ F2 )
=> ( ( topolo3448309680560233919inuous @ A @ B @ ( topolo174197925503356063within @ A @ A2 @ ( top_top @ ( set @ A ) ) ) @ G )
=> ( topolo3448309680560233919inuous @ A @ B @ ( topolo174197925503356063within @ A @ A2 @ ( top_top @ ( set @ A ) ) )
@ ^ [X2: A] : ( minus_minus @ B @ ( F2 @ X2 ) @ ( G @ X2 ) ) ) ) ) ) ).
% isCont_diff
thf(fact_7121_isCont__power,axiom,
! [B: $tType,A: $tType] :
( ( ( topological_t2_space @ A )
& ( power @ B )
& ( real_V4412858255891104859lgebra @ B ) )
=> ! [A2: A,F2: A > B,N3: nat] :
( ( topolo3448309680560233919inuous @ A @ B @ ( topolo174197925503356063within @ A @ A2 @ ( top_top @ ( set @ A ) ) ) @ F2 )
=> ( topolo3448309680560233919inuous @ A @ B @ ( topolo174197925503356063within @ A @ A2 @ ( top_top @ ( set @ A ) ) )
@ ^ [X2: A] : ( power_power @ B @ ( F2 @ X2 ) @ N3 ) ) ) ) ).
% isCont_power
thf(fact_7122_continuous__frac,axiom,
! [X: real] :
( ~ ( member @ real @ X @ ( ring_1_Ints @ real ) )
=> ( topolo3448309680560233919inuous @ real @ real @ ( topolo174197925503356063within @ real @ X @ ( top_top @ ( set @ real ) ) ) @ ( archimedean_frac @ real ) ) ) ).
% continuous_frac
thf(fact_7123_DERIV__D,axiom,
! [A: $tType] :
( ( real_V3459762299906320749_field @ A )
=> ! [F2: A > A,D4: A,X: A] :
( ( has_field_derivative @ A @ F2 @ D4 @ ( topolo174197925503356063within @ A @ X @ ( top_top @ ( set @ A ) ) ) )
=> ( filterlim @ A @ A
@ ^ [H: A] : ( divide_divide @ A @ ( minus_minus @ A @ ( F2 @ ( plus_plus @ A @ X @ H ) ) @ ( F2 @ X ) ) @ H )
@ ( topolo7230453075368039082e_nhds @ A @ D4 )
@ ( topolo174197925503356063within @ A @ ( zero_zero @ A ) @ ( top_top @ ( set @ A ) ) ) ) ) ) ).
% DERIV_D
thf(fact_7124_DERIV__def,axiom,
! [A: $tType] :
( ( real_V3459762299906320749_field @ A )
=> ! [F2: A > A,D4: A,X: A] :
( ( has_field_derivative @ A @ F2 @ D4 @ ( topolo174197925503356063within @ A @ X @ ( top_top @ ( set @ A ) ) ) )
= ( filterlim @ A @ A
@ ^ [H: A] : ( divide_divide @ A @ ( minus_minus @ A @ ( F2 @ ( plus_plus @ A @ X @ H ) ) @ ( F2 @ X ) ) @ H )
@ ( topolo7230453075368039082e_nhds @ A @ D4 )
@ ( topolo174197925503356063within @ A @ ( zero_zero @ A ) @ ( top_top @ ( set @ A ) ) ) ) ) ) ).
% DERIV_def
thf(fact_7125_lim__exp__minus__1,axiom,
! [A: $tType] :
( ( ( real_Vector_banach @ A )
& ( real_V3459762299906320749_field @ A ) )
=> ( filterlim @ A @ A
@ ^ [Z3: A] : ( divide_divide @ A @ ( minus_minus @ A @ ( exp @ A @ Z3 ) @ ( one_one @ A ) ) @ Z3 )
@ ( topolo7230453075368039082e_nhds @ A @ ( one_one @ A ) )
@ ( topolo174197925503356063within @ A @ ( zero_zero @ A ) @ ( top_top @ ( set @ A ) ) ) ) ) ).
% lim_exp_minus_1
thf(fact_7126_atLeastPlusOneLessThan__greaterThanLessThan__int,axiom,
! [L2: int,U: int] :
( ( set_or7035219750837199246ssThan @ int @ ( plus_plus @ int @ L2 @ ( one_one @ int ) ) @ U )
= ( set_or5935395276787703475ssThan @ int @ L2 @ U ) ) ).
% atLeastPlusOneLessThan_greaterThanLessThan_int
thf(fact_7127_greaterThanLessThan__upt,axiom,
( ( set_or5935395276787703475ssThan @ nat )
= ( ^ [N2: nat,M5: nat] : ( set2 @ nat @ ( upt @ ( suc @ N2 ) @ M5 ) ) ) ) ).
% greaterThanLessThan_upt
thf(fact_7128_lemma__termdiff4,axiom,
! [B: $tType,A: $tType] :
( ( ( real_V822414075346904944vector @ A )
& ( real_V822414075346904944vector @ B ) )
=> ! [K: real,F2: A > B,K6: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ K )
=> ( ! [H4: A] :
( ( H4
!= ( zero_zero @ A ) )
=> ( ( ord_less @ real @ ( real_V7770717601297561774m_norm @ A @ H4 ) @ K )
=> ( ord_less_eq @ real @ ( real_V7770717601297561774m_norm @ B @ ( F2 @ H4 ) ) @ ( times_times @ real @ K6 @ ( real_V7770717601297561774m_norm @ A @ H4 ) ) ) ) )
=> ( filterlim @ A @ B @ F2 @ ( topolo7230453075368039082e_nhds @ B @ ( zero_zero @ B ) ) @ ( topolo174197925503356063within @ A @ ( zero_zero @ A ) @ ( top_top @ ( set @ A ) ) ) ) ) ) ) ).
% lemma_termdiff4
thf(fact_7129_isCont__eq__Lb,axiom,
! [A: $tType] :
( ( topolo1944317154257567458pology @ A )
=> ! [A2: real,B2: real,F2: real > A] :
( ( ord_less_eq @ real @ A2 @ B2 )
=> ( ! [X3: real] :
( ( ( ord_less_eq @ real @ A2 @ X3 )
& ( ord_less_eq @ real @ X3 @ B2 ) )
=> ( topolo3448309680560233919inuous @ real @ A @ ( topolo174197925503356063within @ real @ X3 @ ( top_top @ ( set @ real ) ) ) @ F2 ) )
=> ? [M10: A] :
( ! [X5: real] :
( ( ( ord_less_eq @ real @ A2 @ X5 )
& ( ord_less_eq @ real @ X5 @ B2 ) )
=> ( ord_less_eq @ A @ M10 @ ( F2 @ X5 ) ) )
& ? [X3: real] :
( ( ord_less_eq @ real @ A2 @ X3 )
& ( ord_less_eq @ real @ X3 @ B2 )
& ( ( F2 @ X3 )
= M10 ) ) ) ) ) ) ).
% isCont_eq_Lb
thf(fact_7130_isCont__eq__Ub,axiom,
! [A: $tType] :
( ( topolo1944317154257567458pology @ A )
=> ! [A2: real,B2: real,F2: real > A] :
( ( ord_less_eq @ real @ A2 @ B2 )
=> ( ! [X3: real] :
( ( ( ord_less_eq @ real @ A2 @ X3 )
& ( ord_less_eq @ real @ X3 @ B2 ) )
=> ( topolo3448309680560233919inuous @ real @ A @ ( topolo174197925503356063within @ real @ X3 @ ( top_top @ ( set @ real ) ) ) @ F2 ) )
=> ? [M10: A] :
( ! [X5: real] :
( ( ( ord_less_eq @ real @ A2 @ X5 )
& ( ord_less_eq @ real @ X5 @ B2 ) )
=> ( ord_less_eq @ A @ ( F2 @ X5 ) @ M10 ) )
& ? [X3: real] :
( ( ord_less_eq @ real @ A2 @ X3 )
& ( ord_less_eq @ real @ X3 @ B2 )
& ( ( F2 @ X3 )
= M10 ) ) ) ) ) ) ).
% isCont_eq_Ub
thf(fact_7131_isCont__bounded,axiom,
! [A: $tType] :
( ( topolo1944317154257567458pology @ A )
=> ! [A2: real,B2: real,F2: real > A] :
( ( ord_less_eq @ real @ A2 @ B2 )
=> ( ! [X3: real] :
( ( ( ord_less_eq @ real @ A2 @ X3 )
& ( ord_less_eq @ real @ X3 @ B2 ) )
=> ( topolo3448309680560233919inuous @ real @ A @ ( topolo174197925503356063within @ real @ X3 @ ( top_top @ ( set @ real ) ) ) @ F2 ) )
=> ? [M10: A] :
! [X5: real] :
( ( ( ord_less_eq @ real @ A2 @ X5 )
& ( ord_less_eq @ real @ X5 @ B2 ) )
=> ( ord_less_eq @ A @ ( F2 @ X5 ) @ M10 ) ) ) ) ) ).
% isCont_bounded
thf(fact_7132_isCont__inverse__function2,axiom,
! [A2: real,X: real,B2: real,G: real > real,F2: real > real] :
( ( ord_less @ real @ A2 @ X )
=> ( ( ord_less @ real @ X @ B2 )
=> ( ! [Z2: real] :
( ( ord_less_eq @ real @ A2 @ Z2 )
=> ( ( ord_less_eq @ real @ Z2 @ B2 )
=> ( ( G @ ( F2 @ Z2 ) )
= Z2 ) ) )
=> ( ! [Z2: real] :
( ( ord_less_eq @ real @ A2 @ Z2 )
=> ( ( ord_less_eq @ real @ Z2 @ B2 )
=> ( topolo3448309680560233919inuous @ real @ real @ ( topolo174197925503356063within @ real @ Z2 @ ( top_top @ ( set @ real ) ) ) @ F2 ) ) )
=> ( topolo3448309680560233919inuous @ real @ real @ ( topolo174197925503356063within @ real @ ( F2 @ X ) @ ( top_top @ ( set @ real ) ) ) @ G ) ) ) ) ) ).
% isCont_inverse_function2
thf(fact_7133_field__has__derivative__at,axiom,
! [A: $tType] :
( ( real_V3459762299906320749_field @ A )
=> ! [F2: A > A,D4: A,X: A] :
( ( has_derivative @ A @ A @ F2 @ ( times_times @ A @ D4 ) @ ( topolo174197925503356063within @ A @ X @ ( top_top @ ( set @ A ) ) ) )
= ( filterlim @ A @ A
@ ^ [H: A] : ( divide_divide @ A @ ( minus_minus @ A @ ( F2 @ ( plus_plus @ A @ X @ H ) ) @ ( F2 @ X ) ) @ H )
@ ( topolo7230453075368039082e_nhds @ A @ D4 )
@ ( topolo174197925503356063within @ A @ ( zero_zero @ A ) @ ( top_top @ ( set @ A ) ) ) ) ) ) ).
% field_has_derivative_at
thf(fact_7134_isCont__divide,axiom,
! [B: $tType,A: $tType] :
( ( ( topological_t2_space @ A )
& ( real_V3459762299906320749_field @ B ) )
=> ! [A2: A,F2: A > B,G: A > B] :
( ( topolo3448309680560233919inuous @ A @ B @ ( topolo174197925503356063within @ A @ A2 @ ( top_top @ ( set @ A ) ) ) @ F2 )
=> ( ( topolo3448309680560233919inuous @ A @ B @ ( topolo174197925503356063within @ A @ A2 @ ( top_top @ ( set @ A ) ) ) @ G )
=> ( ( ( G @ A2 )
!= ( zero_zero @ B ) )
=> ( topolo3448309680560233919inuous @ A @ B @ ( topolo174197925503356063within @ A @ A2 @ ( top_top @ ( set @ A ) ) )
@ ^ [X2: A] : ( divide_divide @ B @ ( F2 @ X2 ) @ ( G @ X2 ) ) ) ) ) ) ) ).
% isCont_divide
thf(fact_7135_filterlim__at__to__0,axiom,
! [B: $tType,A: $tType] :
( ( real_V822414075346904944vector @ A )
=> ! [F2: A > B,F4: filter @ B,A2: A] :
( ( filterlim @ A @ B @ F2 @ F4 @ ( topolo174197925503356063within @ A @ A2 @ ( top_top @ ( set @ A ) ) ) )
= ( filterlim @ A @ B
@ ^ [X2: A] : ( F2 @ ( plus_plus @ A @ X2 @ A2 ) )
@ F4
@ ( topolo174197925503356063within @ A @ ( zero_zero @ A ) @ ( top_top @ ( set @ A ) ) ) ) ) ) ).
% filterlim_at_to_0
thf(fact_7136_atLeastPlusOneLessThan__greaterThanLessThan__integer,axiom,
! [L2: code_integer,U: code_integer] :
( ( set_or7035219750837199246ssThan @ code_integer @ ( plus_plus @ code_integer @ L2 @ ( one_one @ code_integer ) ) @ U )
= ( set_or5935395276787703475ssThan @ code_integer @ L2 @ U ) ) ).
% atLeastPlusOneLessThan_greaterThanLessThan_integer
thf(fact_7137_CARAT__DERIV,axiom,
! [A: $tType] :
( ( real_V3459762299906320749_field @ A )
=> ! [F2: A > A,L2: A,X: A] :
( ( has_field_derivative @ A @ F2 @ L2 @ ( topolo174197925503356063within @ A @ X @ ( top_top @ ( set @ A ) ) ) )
= ( ? [G4: A > A] :
( ! [Z3: A] :
( ( minus_minus @ A @ ( F2 @ Z3 ) @ ( F2 @ X ) )
= ( times_times @ A @ ( G4 @ Z3 ) @ ( minus_minus @ A @ Z3 @ X ) ) )
& ( topolo3448309680560233919inuous @ A @ A @ ( topolo174197925503356063within @ A @ X @ ( top_top @ ( set @ A ) ) ) @ G4 )
& ( ( G4 @ X )
= L2 ) ) ) ) ) ).
% CARAT_DERIV
thf(fact_7138_isCont__has__Ub,axiom,
! [A: $tType] :
( ( topolo1944317154257567458pology @ A )
=> ! [A2: real,B2: real,F2: real > A] :
( ( ord_less_eq @ real @ A2 @ B2 )
=> ( ! [X3: real] :
( ( ( ord_less_eq @ real @ A2 @ X3 )
& ( ord_less_eq @ real @ X3 @ B2 ) )
=> ( topolo3448309680560233919inuous @ real @ A @ ( topolo174197925503356063within @ real @ X3 @ ( top_top @ ( set @ real ) ) ) @ F2 ) )
=> ? [M10: A] :
( ! [X5: real] :
( ( ( ord_less_eq @ real @ A2 @ X5 )
& ( ord_less_eq @ real @ X5 @ B2 ) )
=> ( ord_less_eq @ A @ ( F2 @ X5 ) @ M10 ) )
& ! [N10: A] :
( ( ord_less @ A @ N10 @ M10 )
=> ? [X3: real] :
( ( ord_less_eq @ real @ A2 @ X3 )
& ( ord_less_eq @ real @ X3 @ B2 )
& ( ord_less @ A @ N10 @ ( F2 @ X3 ) ) ) ) ) ) ) ) ).
% isCont_has_Ub
thf(fact_7139_filterlim__shift__iff,axiom,
! [B: $tType,A: $tType] :
( ( real_V822414075346904944vector @ A )
=> ! [F2: A > B,D: A,F4: filter @ B,A2: A] :
( ( filterlim @ A @ B @ ( comp @ A @ B @ A @ F2 @ ( plus_plus @ A @ D ) ) @ F4 @ ( topolo174197925503356063within @ A @ ( minus_minus @ A @ A2 @ D ) @ ( top_top @ ( set @ A ) ) ) )
= ( filterlim @ A @ B @ F2 @ F4 @ ( topolo174197925503356063within @ A @ A2 @ ( top_top @ ( set @ A ) ) ) ) ) ) ).
% filterlim_shift_iff
thf(fact_7140_filterlim__shift,axiom,
! [B: $tType,A: $tType] :
( ( real_V822414075346904944vector @ A )
=> ! [F2: A > B,F4: filter @ B,A2: A,D: A] :
( ( filterlim @ A @ B @ F2 @ F4 @ ( topolo174197925503356063within @ A @ A2 @ ( top_top @ ( set @ A ) ) ) )
=> ( filterlim @ A @ B @ ( comp @ A @ B @ A @ F2 @ ( plus_plus @ A @ D ) ) @ F4 @ ( topolo174197925503356063within @ A @ ( minus_minus @ A @ A2 @ D ) @ ( top_top @ ( set @ A ) ) ) ) ) ) ).
% filterlim_shift
thf(fact_7141_powser__limit__0,axiom,
! [A: $tType] :
( ( ( real_Vector_banach @ A )
& ( real_V3459762299906320749_field @ A ) )
=> ! [S: real,A2: nat > A,F2: A > A] :
( ( ord_less @ real @ ( zero_zero @ real ) @ S )
=> ( ! [X3: A] :
( ( ord_less @ real @ ( real_V7770717601297561774m_norm @ A @ X3 ) @ S )
=> ( sums @ A
@ ^ [N2: nat] : ( times_times @ A @ ( A2 @ N2 ) @ ( power_power @ A @ X3 @ N2 ) )
@ ( F2 @ X3 ) ) )
=> ( filterlim @ A @ A @ F2 @ ( topolo7230453075368039082e_nhds @ A @ ( A2 @ ( zero_zero @ nat ) ) ) @ ( topolo174197925503356063within @ A @ ( zero_zero @ A ) @ ( top_top @ ( set @ A ) ) ) ) ) ) ) ).
% powser_limit_0
thf(fact_7142_powser__limit__0__strong,axiom,
! [A: $tType] :
( ( ( real_Vector_banach @ A )
& ( real_V3459762299906320749_field @ A ) )
=> ! [S: real,A2: nat > A,F2: A > A] :
( ( ord_less @ real @ ( zero_zero @ real ) @ S )
=> ( ! [X3: A] :
( ( X3
!= ( zero_zero @ A ) )
=> ( ( ord_less @ real @ ( real_V7770717601297561774m_norm @ A @ X3 ) @ S )
=> ( sums @ A
@ ^ [N2: nat] : ( times_times @ A @ ( A2 @ N2 ) @ ( power_power @ A @ X3 @ N2 ) )
@ ( F2 @ X3 ) ) ) )
=> ( filterlim @ A @ A @ F2 @ ( topolo7230453075368039082e_nhds @ A @ ( A2 @ ( zero_zero @ nat ) ) ) @ ( topolo174197925503356063within @ A @ ( zero_zero @ A ) @ ( top_top @ ( set @ A ) ) ) ) ) ) ) ).
% powser_limit_0_strong
thf(fact_7143_lemma__termdiff5,axiom,
! [B: $tType,A: $tType] :
( ( ( real_V822414075346904944vector @ A )
& ( real_Vector_banach @ B ) )
=> ! [K: real,F2: nat > real,G: A > nat > B] :
( ( ord_less @ real @ ( zero_zero @ real ) @ K )
=> ( ( summable @ real @ F2 )
=> ( ! [H4: A,N: nat] :
( ( H4
!= ( zero_zero @ A ) )
=> ( ( ord_less @ real @ ( real_V7770717601297561774m_norm @ A @ H4 ) @ K )
=> ( ord_less_eq @ real @ ( real_V7770717601297561774m_norm @ B @ ( G @ H4 @ N ) ) @ ( times_times @ real @ ( F2 @ N ) @ ( real_V7770717601297561774m_norm @ A @ H4 ) ) ) ) )
=> ( filterlim @ A @ B
@ ^ [H: A] : ( suminf @ B @ ( G @ H ) )
@ ( topolo7230453075368039082e_nhds @ B @ ( zero_zero @ B ) )
@ ( topolo174197925503356063within @ A @ ( zero_zero @ A ) @ ( top_top @ ( set @ A ) ) ) ) ) ) ) ) ).
% lemma_termdiff5
thf(fact_7144_isCont__arcosh,axiom,
! [X: real] :
( ( ord_less @ real @ ( one_one @ real ) @ X )
=> ( topolo3448309680560233919inuous @ real @ real @ ( topolo174197925503356063within @ real @ X @ ( top_top @ ( set @ real ) ) ) @ ( arcosh @ real ) ) ) ).
% isCont_arcosh
thf(fact_7145_LIM__cos__div__sin,axiom,
( filterlim @ real @ real
@ ^ [X2: real] : ( divide_divide @ real @ ( cos @ real @ X2 ) @ ( sin @ real @ X2 ) )
@ ( topolo7230453075368039082e_nhds @ real @ ( zero_zero @ real ) )
@ ( topolo174197925503356063within @ real @ ( divide_divide @ real @ pi @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) @ ( top_top @ ( set @ real ) ) ) ) ).
% LIM_cos_div_sin
thf(fact_7146_continuous__floor,axiom,
! [X: real] :
( ~ ( member @ real @ X @ ( ring_1_Ints @ real ) )
=> ( topolo3448309680560233919inuous @ real @ real @ ( topolo174197925503356063within @ real @ X @ ( top_top @ ( set @ real ) ) ) @ ( comp @ int @ real @ real @ ( ring_1_of_int @ real ) @ ( archim6421214686448440834_floor @ real ) ) ) ) ).
% continuous_floor
thf(fact_7147_DERIV__inverse__function,axiom,
! [F2: real > real,D4: real,G: real > real,X: real,A2: real,B2: real] :
( ( has_field_derivative @ real @ F2 @ D4 @ ( topolo174197925503356063within @ real @ ( G @ X ) @ ( top_top @ ( set @ real ) ) ) )
=> ( ( D4
!= ( zero_zero @ real ) )
=> ( ( ord_less @ real @ A2 @ X )
=> ( ( ord_less @ real @ X @ B2 )
=> ( ! [Y3: real] :
( ( ord_less @ real @ A2 @ Y3 )
=> ( ( ord_less @ real @ Y3 @ B2 )
=> ( ( F2 @ ( G @ Y3 ) )
= Y3 ) ) )
=> ( ( topolo3448309680560233919inuous @ real @ real @ ( topolo174197925503356063within @ real @ X @ ( top_top @ ( set @ real ) ) ) @ G )
=> ( has_field_derivative @ real @ G @ ( inverse_inverse @ real @ D4 ) @ ( topolo174197925503356063within @ real @ X @ ( top_top @ ( set @ real ) ) ) ) ) ) ) ) ) ) ).
% DERIV_inverse_function
thf(fact_7148_isCont__polynom,axiom,
! [A: $tType] :
( ( real_V8999393235501362500lgebra @ A )
=> ! [A2: A,C2: nat > A,N3: nat] :
( topolo3448309680560233919inuous @ A @ A @ ( topolo174197925503356063within @ A @ A2 @ ( top_top @ ( set @ A ) ) )
@ ^ [W2: A] :
( groups7311177749621191930dd_sum @ nat @ A
@ ^ [I3: nat] : ( times_times @ A @ ( C2 @ I3 ) @ ( power_power @ A @ W2 @ I3 ) )
@ ( set_ord_atMost @ nat @ N3 ) ) ) ) ).
% isCont_polynom
thf(fact_7149_isCont__powser__converges__everywhere,axiom,
! [A: $tType] :
( ( ( real_Vector_banach @ A )
& ( real_V3459762299906320749_field @ A ) )
=> ! [C2: nat > A,X: A] :
( ! [Y3: A] :
( summable @ A
@ ^ [N2: nat] : ( times_times @ A @ ( C2 @ N2 ) @ ( power_power @ A @ Y3 @ N2 ) ) )
=> ( topolo3448309680560233919inuous @ A @ A @ ( topolo174197925503356063within @ A @ X @ ( top_top @ ( set @ A ) ) )
@ ^ [X2: A] :
( suminf @ A
@ ^ [N2: nat] : ( times_times @ A @ ( C2 @ N2 ) @ ( power_power @ A @ X2 @ N2 ) ) ) ) ) ) ).
% isCont_powser_converges_everywhere
thf(fact_7150_isCont__arccos,axiom,
! [X: real] :
( ( ord_less @ real @ ( uminus_uminus @ real @ ( one_one @ real ) ) @ X )
=> ( ( ord_less @ real @ X @ ( one_one @ real ) )
=> ( topolo3448309680560233919inuous @ real @ real @ ( topolo174197925503356063within @ real @ X @ ( top_top @ ( set @ real ) ) ) @ arccos ) ) ) ).
% isCont_arccos
thf(fact_7151_isCont__arcsin,axiom,
! [X: real] :
( ( ord_less @ real @ ( uminus_uminus @ real @ ( one_one @ real ) ) @ X )
=> ( ( ord_less @ real @ X @ ( one_one @ real ) )
=> ( topolo3448309680560233919inuous @ real @ real @ ( topolo174197925503356063within @ real @ X @ ( top_top @ ( set @ real ) ) ) @ arcsin ) ) ) ).
% isCont_arcsin
thf(fact_7152_LIM__less__bound,axiom,
! [B2: real,X: real,F2: real > real] :
( ( ord_less @ real @ B2 @ X )
=> ( ! [X3: real] :
( ( member @ real @ X3 @ ( set_or5935395276787703475ssThan @ real @ B2 @ X ) )
=> ( ord_less_eq @ real @ ( zero_zero @ real ) @ ( F2 @ X3 ) ) )
=> ( ( topolo3448309680560233919inuous @ real @ real @ ( topolo174197925503356063within @ real @ X @ ( top_top @ ( set @ real ) ) ) @ F2 )
=> ( ord_less_eq @ real @ ( zero_zero @ real ) @ ( F2 @ X ) ) ) ) ) ).
% LIM_less_bound
thf(fact_7153_isCont__artanh,axiom,
! [X: real] :
( ( ord_less @ real @ ( uminus_uminus @ real @ ( one_one @ real ) ) @ X )
=> ( ( ord_less @ real @ X @ ( one_one @ real ) )
=> ( topolo3448309680560233919inuous @ real @ real @ ( topolo174197925503356063within @ real @ X @ ( top_top @ ( set @ real ) ) ) @ ( artanh @ real ) ) ) ) ).
% isCont_artanh
thf(fact_7154_isCont__inverse__function,axiom,
! [D: real,X: real,G: real > real,F2: real > real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ D )
=> ( ! [Z2: real] :
( ( ord_less_eq @ real @ ( abs_abs @ real @ ( minus_minus @ real @ Z2 @ X ) ) @ D )
=> ( ( G @ ( F2 @ Z2 ) )
= Z2 ) )
=> ( ! [Z2: real] :
( ( ord_less_eq @ real @ ( abs_abs @ real @ ( minus_minus @ real @ Z2 @ X ) ) @ D )
=> ( topolo3448309680560233919inuous @ real @ real @ ( topolo174197925503356063within @ real @ Z2 @ ( top_top @ ( set @ real ) ) ) @ F2 ) )
=> ( topolo3448309680560233919inuous @ real @ real @ ( topolo174197925503356063within @ real @ ( F2 @ X ) @ ( top_top @ ( set @ real ) ) ) @ G ) ) ) ) ).
% isCont_inverse_function
thf(fact_7155_GMVT_H,axiom,
! [A2: real,B2: real,F2: real > real,G: real > real,G5: real > real,F5: real > real] :
( ( ord_less @ real @ A2 @ B2 )
=> ( ! [Z2: real] :
( ( ord_less_eq @ real @ A2 @ Z2 )
=> ( ( ord_less_eq @ real @ Z2 @ B2 )
=> ( topolo3448309680560233919inuous @ real @ real @ ( topolo174197925503356063within @ real @ Z2 @ ( top_top @ ( set @ real ) ) ) @ F2 ) ) )
=> ( ! [Z2: real] :
( ( ord_less_eq @ real @ A2 @ Z2 )
=> ( ( ord_less_eq @ real @ Z2 @ B2 )
=> ( topolo3448309680560233919inuous @ real @ real @ ( topolo174197925503356063within @ real @ Z2 @ ( top_top @ ( set @ real ) ) ) @ G ) ) )
=> ( ! [Z2: real] :
( ( ord_less @ real @ A2 @ Z2 )
=> ( ( ord_less @ real @ Z2 @ B2 )
=> ( has_field_derivative @ real @ G @ ( G5 @ Z2 ) @ ( topolo174197925503356063within @ real @ Z2 @ ( top_top @ ( set @ real ) ) ) ) ) )
=> ( ! [Z2: real] :
( ( ord_less @ real @ A2 @ Z2 )
=> ( ( ord_less @ real @ Z2 @ B2 )
=> ( has_field_derivative @ real @ F2 @ ( F5 @ Z2 ) @ ( topolo174197925503356063within @ real @ Z2 @ ( top_top @ ( set @ real ) ) ) ) ) )
=> ? [C4: real] :
( ( ord_less @ real @ A2 @ C4 )
& ( ord_less @ real @ C4 @ B2 )
& ( ( times_times @ real @ ( minus_minus @ real @ ( F2 @ B2 ) @ ( F2 @ A2 ) ) @ ( G5 @ C4 ) )
= ( times_times @ real @ ( minus_minus @ real @ ( G @ B2 ) @ ( G @ A2 ) ) @ ( F5 @ C4 ) ) ) ) ) ) ) ) ) ).
% GMVT'
thf(fact_7156_floor__has__real__derivative,axiom,
! [A: $tType] :
( ( ( archim2362893244070406136eiling @ A )
& ( topolo2564578578187576103pology @ A ) )
=> ! [X: real,F2: real > A] :
( ( topolo3448309680560233919inuous @ real @ A @ ( topolo174197925503356063within @ real @ X @ ( top_top @ ( set @ real ) ) ) @ F2 )
=> ( ~ ( member @ A @ ( F2 @ X ) @ ( ring_1_Ints @ A ) )
=> ( has_field_derivative @ real
@ ^ [X2: real] : ( ring_1_of_int @ real @ ( archim6421214686448440834_floor @ A @ ( F2 @ X2 ) ) )
@ ( zero_zero @ real )
@ ( topolo174197925503356063within @ real @ X @ ( top_top @ ( set @ real ) ) ) ) ) ) ) ).
% floor_has_real_derivative
thf(fact_7157_isCont__powser,axiom,
! [A: $tType] :
( ( ( real_Vector_banach @ A )
& ( real_V3459762299906320749_field @ A ) )
=> ! [C2: nat > A,K6: A,X: A] :
( ( summable @ A
@ ^ [N2: nat] : ( times_times @ A @ ( C2 @ N2 ) @ ( power_power @ A @ K6 @ N2 ) ) )
=> ( ( ord_less @ real @ ( real_V7770717601297561774m_norm @ A @ X ) @ ( real_V7770717601297561774m_norm @ A @ K6 ) )
=> ( topolo3448309680560233919inuous @ A @ A @ ( topolo174197925503356063within @ A @ X @ ( top_top @ ( set @ A ) ) )
@ ^ [X2: A] :
( suminf @ A
@ ^ [N2: nat] : ( times_times @ A @ ( C2 @ N2 ) @ ( power_power @ A @ X2 @ N2 ) ) ) ) ) ) ) ).
% isCont_powser
thf(fact_7158_isCont__powser_H,axiom,
! [Aa: $tType,A: $tType] :
( ( ( topological_t2_space @ A )
& ( real_Vector_banach @ Aa )
& ( real_V3459762299906320749_field @ Aa ) )
=> ! [A2: A,F2: A > Aa,C2: nat > Aa,K6: Aa] :
( ( topolo3448309680560233919inuous @ A @ Aa @ ( topolo174197925503356063within @ A @ A2 @ ( top_top @ ( set @ A ) ) ) @ F2 )
=> ( ( summable @ Aa
@ ^ [N2: nat] : ( times_times @ Aa @ ( C2 @ N2 ) @ ( power_power @ Aa @ K6 @ N2 ) ) )
=> ( ( ord_less @ real @ ( real_V7770717601297561774m_norm @ Aa @ ( F2 @ A2 ) ) @ ( real_V7770717601297561774m_norm @ Aa @ K6 ) )
=> ( topolo3448309680560233919inuous @ A @ Aa @ ( topolo174197925503356063within @ A @ A2 @ ( top_top @ ( set @ A ) ) )
@ ^ [X2: A] :
( suminf @ Aa
@ ^ [N2: nat] : ( times_times @ Aa @ ( C2 @ N2 ) @ ( power_power @ Aa @ ( F2 @ X2 ) @ N2 ) ) ) ) ) ) ) ) ).
% isCont_powser'
thf(fact_7159_summable__Leibniz_I3_J,axiom,
! [A2: nat > real] :
( ( filterlim @ nat @ real @ A2 @ ( topolo7230453075368039082e_nhds @ real @ ( zero_zero @ real ) ) @ ( at_top @ nat ) )
=> ( ( topological_monoseq @ real @ A2 )
=> ( ( ord_less @ real @ ( A2 @ ( zero_zero @ nat ) ) @ ( zero_zero @ real ) )
=> ! [N9: nat] :
( member @ real
@ ( suminf @ real
@ ^ [I3: nat] : ( times_times @ real @ ( power_power @ real @ ( uminus_uminus @ real @ ( one_one @ real ) ) @ I3 ) @ ( A2 @ I3 ) ) )
@ ( set_or1337092689740270186AtMost @ real
@ ( groups7311177749621191930dd_sum @ nat @ real
@ ^ [I3: nat] : ( times_times @ real @ ( power_power @ real @ ( uminus_uminus @ real @ ( one_one @ real ) ) @ I3 ) @ ( A2 @ I3 ) )
@ ( set_ord_lessThan @ nat @ ( plus_plus @ nat @ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N9 ) @ ( one_one @ nat ) ) ) )
@ ( groups7311177749621191930dd_sum @ nat @ real
@ ^ [I3: nat] : ( times_times @ real @ ( power_power @ real @ ( uminus_uminus @ real @ ( one_one @ real ) ) @ I3 ) @ ( A2 @ I3 ) )
@ ( set_ord_lessThan @ nat @ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N9 ) ) ) ) ) ) ) ) ).
% summable_Leibniz(3)
thf(fact_7160_summable__Leibniz_I2_J,axiom,
! [A2: nat > real] :
( ( filterlim @ nat @ real @ A2 @ ( topolo7230453075368039082e_nhds @ real @ ( zero_zero @ real ) ) @ ( at_top @ nat ) )
=> ( ( topological_monoseq @ real @ A2 )
=> ( ( ord_less @ real @ ( zero_zero @ real ) @ ( A2 @ ( zero_zero @ nat ) ) )
=> ! [N9: nat] :
( member @ real
@ ( suminf @ real
@ ^ [I3: nat] : ( times_times @ real @ ( power_power @ real @ ( uminus_uminus @ real @ ( one_one @ real ) ) @ I3 ) @ ( A2 @ I3 ) ) )
@ ( set_or1337092689740270186AtMost @ real
@ ( groups7311177749621191930dd_sum @ nat @ real
@ ^ [I3: nat] : ( times_times @ real @ ( power_power @ real @ ( uminus_uminus @ real @ ( one_one @ real ) ) @ I3 ) @ ( A2 @ I3 ) )
@ ( set_ord_lessThan @ nat @ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N9 ) ) )
@ ( groups7311177749621191930dd_sum @ nat @ real
@ ^ [I3: nat] : ( times_times @ real @ ( power_power @ real @ ( uminus_uminus @ real @ ( one_one @ real ) ) @ I3 ) @ ( A2 @ I3 ) )
@ ( set_ord_lessThan @ nat @ ( plus_plus @ nat @ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N9 ) @ ( one_one @ nat ) ) ) ) ) ) ) ) ) ).
% summable_Leibniz(2)
thf(fact_7161_tendsto__zero__mult__right__iff,axiom,
! [A: $tType] :
( ( ( field @ A )
& ( topolo4211221413907600880p_mult @ A ) )
=> ! [C2: A,A2: nat > A] :
( ( C2
!= ( zero_zero @ A ) )
=> ( ( filterlim @ nat @ A
@ ^ [N2: nat] : ( times_times @ A @ ( A2 @ N2 ) @ C2 )
@ ( topolo7230453075368039082e_nhds @ A @ ( zero_zero @ A ) )
@ ( at_top @ nat ) )
= ( filterlim @ nat @ A @ A2 @ ( topolo7230453075368039082e_nhds @ A @ ( zero_zero @ A ) ) @ ( at_top @ nat ) ) ) ) ) ).
% tendsto_zero_mult_right_iff
thf(fact_7162_tendsto__zero__mult__left__iff,axiom,
! [A: $tType] :
( ( ( field @ A )
& ( topolo4211221413907600880p_mult @ A ) )
=> ! [C2: A,A2: nat > A] :
( ( C2
!= ( zero_zero @ A ) )
=> ( ( filterlim @ nat @ A
@ ^ [N2: nat] : ( times_times @ A @ C2 @ ( A2 @ N2 ) )
@ ( topolo7230453075368039082e_nhds @ A @ ( zero_zero @ A ) )
@ ( at_top @ nat ) )
= ( filterlim @ nat @ A @ A2 @ ( topolo7230453075368039082e_nhds @ A @ ( zero_zero @ A ) ) @ ( at_top @ nat ) ) ) ) ) ).
% tendsto_zero_mult_left_iff
thf(fact_7163_tendsto__zero__divide__iff,axiom,
! [A: $tType] :
( ( ( field @ A )
& ( topolo4211221413907600880p_mult @ A ) )
=> ! [C2: A,A2: nat > A] :
( ( C2
!= ( zero_zero @ A ) )
=> ( ( filterlim @ nat @ A
@ ^ [N2: nat] : ( divide_divide @ A @ ( A2 @ N2 ) @ C2 )
@ ( topolo7230453075368039082e_nhds @ A @ ( zero_zero @ A ) )
@ ( at_top @ nat ) )
= ( filterlim @ nat @ A @ A2 @ ( topolo7230453075368039082e_nhds @ A @ ( zero_zero @ A ) ) @ ( at_top @ nat ) ) ) ) ) ).
% tendsto_zero_divide_iff
thf(fact_7164_seq__offset__neg,axiom,
! [A: $tType] :
( ( topolo4958980785337419405_space @ A )
=> ! [F2: nat > A,L2: A,K: nat] :
( ( filterlim @ nat @ A @ F2 @ ( topolo7230453075368039082e_nhds @ A @ L2 ) @ ( at_top @ nat ) )
=> ( filterlim @ nat @ A
@ ^ [I3: nat] : ( F2 @ ( minus_minus @ nat @ I3 @ K ) )
@ ( topolo7230453075368039082e_nhds @ A @ L2 )
@ ( at_top @ nat ) ) ) ) ).
% seq_offset_neg
thf(fact_7165_approx__from__below__dense__linorder,axiom,
! [A: $tType] :
( ( ( dense_linorder @ A )
& ( topolo3112930676232923870pology @ A )
& ( topolo1944317154257567458pology @ A ) )
=> ! [Y: A,X: A] :
( ( ord_less @ A @ Y @ X )
=> ? [U4: nat > A] :
( ! [N9: nat] : ( ord_less @ A @ ( U4 @ N9 ) @ X )
& ( filterlim @ nat @ A @ U4 @ ( topolo7230453075368039082e_nhds @ A @ X ) @ ( at_top @ nat ) ) ) ) ) ).
% approx_from_below_dense_linorder
thf(fact_7166_approx__from__above__dense__linorder,axiom,
! [A: $tType] :
( ( ( dense_linorder @ A )
& ( topolo3112930676232923870pology @ A )
& ( topolo1944317154257567458pology @ A ) )
=> ! [X: A,Y: A] :
( ( ord_less @ A @ X @ Y )
=> ? [U4: nat > A] :
( ! [N9: nat] : ( ord_less @ A @ X @ ( U4 @ N9 ) )
& ( filterlim @ nat @ A @ U4 @ ( topolo7230453075368039082e_nhds @ A @ X ) @ ( at_top @ nat ) ) ) ) ) ).
% approx_from_above_dense_linorder
thf(fact_7167_continuous__real__root,axiom,
! [A: $tType] :
( ( topological_t2_space @ A )
=> ! [F4: filter @ A,F2: A > real,N3: nat] :
( ( topolo3448309680560233919inuous @ A @ real @ F4 @ F2 )
=> ( topolo3448309680560233919inuous @ A @ real @ F4
@ ^ [X2: A] : ( root @ N3 @ ( F2 @ X2 ) ) ) ) ) ).
% continuous_real_root
thf(fact_7168_LIMSEQ__imp__Suc,axiom,
! [A: $tType] :
( ( topolo4958980785337419405_space @ A )
=> ! [F2: nat > A,L2: A] :
( ( filterlim @ nat @ A
@ ^ [N2: nat] : ( F2 @ ( suc @ N2 ) )
@ ( topolo7230453075368039082e_nhds @ A @ L2 )
@ ( at_top @ nat ) )
=> ( filterlim @ nat @ A @ F2 @ ( topolo7230453075368039082e_nhds @ A @ L2 ) @ ( at_top @ nat ) ) ) ) ).
% LIMSEQ_imp_Suc
thf(fact_7169_LIMSEQ__Suc,axiom,
! [A: $tType] :
( ( topolo4958980785337419405_space @ A )
=> ! [F2: nat > A,L2: A] :
( ( filterlim @ nat @ A @ F2 @ ( topolo7230453075368039082e_nhds @ A @ L2 ) @ ( at_top @ nat ) )
=> ( filterlim @ nat @ A
@ ^ [N2: nat] : ( F2 @ ( suc @ N2 ) )
@ ( topolo7230453075368039082e_nhds @ A @ L2 )
@ ( at_top @ nat ) ) ) ) ).
% LIMSEQ_Suc
thf(fact_7170_continuous__real__sqrt,axiom,
! [A: $tType] :
( ( topological_t2_space @ A )
=> ! [F4: filter @ A,F2: A > real] :
( ( topolo3448309680560233919inuous @ A @ real @ F4 @ F2 )
=> ( topolo3448309680560233919inuous @ A @ real @ F4
@ ^ [X2: A] : ( sqrt @ ( F2 @ X2 ) ) ) ) ) ).
% continuous_real_sqrt
thf(fact_7171_LIMSEQ__ignore__initial__segment,axiom,
! [A: $tType] :
( ( topolo4958980785337419405_space @ A )
=> ! [F2: nat > A,A2: A,K: nat] :
( ( filterlim @ nat @ A @ F2 @ ( topolo7230453075368039082e_nhds @ A @ A2 ) @ ( at_top @ nat ) )
=> ( filterlim @ nat @ A
@ ^ [N2: nat] : ( F2 @ ( plus_plus @ nat @ N2 @ K ) )
@ ( topolo7230453075368039082e_nhds @ A @ A2 )
@ ( at_top @ nat ) ) ) ) ).
% LIMSEQ_ignore_initial_segment
thf(fact_7172_LIMSEQ__offset,axiom,
! [A: $tType] :
( ( topolo4958980785337419405_space @ A )
=> ! [F2: nat > A,K: nat,A2: A] :
( ( filterlim @ nat @ A
@ ^ [N2: nat] : ( F2 @ ( plus_plus @ nat @ N2 @ K ) )
@ ( topolo7230453075368039082e_nhds @ A @ A2 )
@ ( at_top @ nat ) )
=> ( filterlim @ nat @ A @ F2 @ ( topolo7230453075368039082e_nhds @ A @ A2 ) @ ( at_top @ nat ) ) ) ) ).
% LIMSEQ_offset
thf(fact_7173_lim__mono,axiom,
! [A: $tType] :
( ( topolo1944317154257567458pology @ A )
=> ! [N7: nat,X8: nat > A,Y9: nat > A,X: A,Y: A] :
( ! [N: nat] :
( ( ord_less_eq @ nat @ N7 @ N )
=> ( ord_less_eq @ A @ ( X8 @ N ) @ ( Y9 @ N ) ) )
=> ( ( filterlim @ nat @ A @ X8 @ ( topolo7230453075368039082e_nhds @ A @ X ) @ ( at_top @ nat ) )
=> ( ( filterlim @ nat @ A @ Y9 @ ( topolo7230453075368039082e_nhds @ A @ Y ) @ ( at_top @ nat ) )
=> ( ord_less_eq @ A @ X @ Y ) ) ) ) ) ).
% lim_mono
thf(fact_7174_LIMSEQ__le,axiom,
! [A: $tType] :
( ( topolo1944317154257567458pology @ A )
=> ! [X8: nat > A,X: A,Y9: nat > A,Y: A] :
( ( filterlim @ nat @ A @ X8 @ ( topolo7230453075368039082e_nhds @ A @ X ) @ ( at_top @ nat ) )
=> ( ( filterlim @ nat @ A @ Y9 @ ( topolo7230453075368039082e_nhds @ A @ Y ) @ ( at_top @ nat ) )
=> ( ? [N10: nat] :
! [N: nat] :
( ( ord_less_eq @ nat @ N10 @ N )
=> ( ord_less_eq @ A @ ( X8 @ N ) @ ( Y9 @ N ) ) )
=> ( ord_less_eq @ A @ X @ Y ) ) ) ) ) ).
% LIMSEQ_le
thf(fact_7175_Lim__bounded,axiom,
! [A: $tType] :
( ( topolo1944317154257567458pology @ A )
=> ! [F2: nat > A,L2: A,M8: nat,C5: A] :
( ( filterlim @ nat @ A @ F2 @ ( topolo7230453075368039082e_nhds @ A @ L2 ) @ ( at_top @ nat ) )
=> ( ! [N: nat] :
( ( ord_less_eq @ nat @ M8 @ N )
=> ( ord_less_eq @ A @ ( F2 @ N ) @ C5 ) )
=> ( ord_less_eq @ A @ L2 @ C5 ) ) ) ) ).
% Lim_bounded
thf(fact_7176_Lim__bounded2,axiom,
! [A: $tType] :
( ( topolo1944317154257567458pology @ A )
=> ! [F2: nat > A,L2: A,N7: nat,C5: A] :
( ( filterlim @ nat @ A @ F2 @ ( topolo7230453075368039082e_nhds @ A @ L2 ) @ ( at_top @ nat ) )
=> ( ! [N: nat] :
( ( ord_less_eq @ nat @ N7 @ N )
=> ( ord_less_eq @ A @ C5 @ ( F2 @ N ) ) )
=> ( ord_less_eq @ A @ C5 @ L2 ) ) ) ) ).
% Lim_bounded2
thf(fact_7177_LIMSEQ__le__const,axiom,
! [A: $tType] :
( ( topolo1944317154257567458pology @ A )
=> ! [X8: nat > A,X: A,A2: A] :
( ( filterlim @ nat @ A @ X8 @ ( topolo7230453075368039082e_nhds @ A @ X ) @ ( at_top @ nat ) )
=> ( ? [N10: nat] :
! [N: nat] :
( ( ord_less_eq @ nat @ N10 @ N )
=> ( ord_less_eq @ A @ A2 @ ( X8 @ N ) ) )
=> ( ord_less_eq @ A @ A2 @ X ) ) ) ) ).
% LIMSEQ_le_const
thf(fact_7178_LIMSEQ__le__const2,axiom,
! [A: $tType] :
( ( topolo1944317154257567458pology @ A )
=> ! [X8: nat > A,X: A,A2: A] :
( ( filterlim @ nat @ A @ X8 @ ( topolo7230453075368039082e_nhds @ A @ X ) @ ( at_top @ nat ) )
=> ( ? [N10: nat] :
! [N: nat] :
( ( ord_less_eq @ nat @ N10 @ N )
=> ( ord_less_eq @ A @ ( X8 @ N ) @ A2 ) )
=> ( ord_less_eq @ A @ X @ A2 ) ) ) ) ).
% LIMSEQ_le_const2
thf(fact_7179_mult__nat__left__at__top,axiom,
! [C2: nat] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ C2 )
=> ( filterlim @ nat @ nat @ ( times_times @ nat @ C2 ) @ ( at_top @ nat ) @ ( at_top @ nat ) ) ) ).
% mult_nat_left_at_top
thf(fact_7180_mult__nat__right__at__top,axiom,
! [C2: nat] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ C2 )
=> ( filterlim @ nat @ nat
@ ^ [X2: nat] : ( times_times @ nat @ X2 @ C2 )
@ ( at_top @ nat )
@ ( at_top @ nat ) ) ) ).
% mult_nat_right_at_top
thf(fact_7181_monoseq__convergent,axiom,
! [X8: nat > real,B6: real] :
( ( topological_monoseq @ real @ X8 )
=> ( ! [I2: nat] : ( ord_less_eq @ real @ ( abs_abs @ real @ ( X8 @ I2 ) ) @ B6 )
=> ~ ! [L7: real] :
~ ( filterlim @ nat @ real @ X8 @ ( topolo7230453075368039082e_nhds @ real @ L7 ) @ ( at_top @ nat ) ) ) ) ).
% monoseq_convergent
thf(fact_7182_LIMSEQ__root,axiom,
( filterlim @ nat @ real
@ ^ [N2: nat] : ( root @ N2 @ ( semiring_1_of_nat @ real @ N2 ) )
@ ( topolo7230453075368039082e_nhds @ real @ ( one_one @ real ) )
@ ( at_top @ nat ) ) ).
% LIMSEQ_root
thf(fact_7183_monoseq__le,axiom,
! [A: $tType] :
( ( topolo1944317154257567458pology @ A )
=> ! [A2: nat > A,X: A] :
( ( topological_monoseq @ A @ A2 )
=> ( ( filterlim @ nat @ A @ A2 @ ( topolo7230453075368039082e_nhds @ A @ X ) @ ( at_top @ nat ) )
=> ( ( ! [N9: nat] : ( ord_less_eq @ A @ ( A2 @ N9 ) @ X )
& ! [M2: nat,N9: nat] :
( ( ord_less_eq @ nat @ M2 @ N9 )
=> ( ord_less_eq @ A @ ( A2 @ M2 ) @ ( A2 @ N9 ) ) ) )
| ( ! [N9: nat] : ( ord_less_eq @ A @ X @ ( A2 @ N9 ) )
& ! [M2: nat,N9: nat] :
( ( ord_less_eq @ nat @ M2 @ N9 )
=> ( ord_less_eq @ A @ ( A2 @ N9 ) @ ( A2 @ M2 ) ) ) ) ) ) ) ) ).
% monoseq_le
thf(fact_7184_lim__const__over__n,axiom,
! [A: $tType] :
( ( real_V3459762299906320749_field @ A )
=> ! [A2: A] :
( filterlim @ nat @ A
@ ^ [N2: nat] : ( divide_divide @ A @ A2 @ ( semiring_1_of_nat @ A @ N2 ) )
@ ( topolo7230453075368039082e_nhds @ A @ ( zero_zero @ A ) )
@ ( at_top @ nat ) ) ) ).
% lim_const_over_n
thf(fact_7185_LIMSEQ__linear,axiom,
! [A: $tType] :
( ( topolo4958980785337419405_space @ A )
=> ! [X8: nat > A,X: A,L2: nat] :
( ( filterlim @ nat @ A @ X8 @ ( topolo7230453075368039082e_nhds @ A @ X ) @ ( at_top @ nat ) )
=> ( ( ord_less @ nat @ ( zero_zero @ nat ) @ L2 )
=> ( filterlim @ nat @ A
@ ^ [N2: nat] : ( X8 @ ( times_times @ nat @ N2 @ L2 ) )
@ ( topolo7230453075368039082e_nhds @ A @ X )
@ ( at_top @ nat ) ) ) ) ) ).
% LIMSEQ_linear
thf(fact_7186_telescope__summable,axiom,
! [A: $tType] :
( ( real_V822414075346904944vector @ A )
=> ! [F2: nat > A,C2: A] :
( ( filterlim @ nat @ A @ F2 @ ( topolo7230453075368039082e_nhds @ A @ C2 ) @ ( at_top @ nat ) )
=> ( summable @ A
@ ^ [N2: nat] : ( minus_minus @ A @ ( F2 @ ( suc @ N2 ) ) @ ( F2 @ N2 ) ) ) ) ) ).
% telescope_summable
thf(fact_7187_telescope__summable_H,axiom,
! [A: $tType] :
( ( real_V822414075346904944vector @ A )
=> ! [F2: nat > A,C2: A] :
( ( filterlim @ nat @ A @ F2 @ ( topolo7230453075368039082e_nhds @ A @ C2 ) @ ( at_top @ nat ) )
=> ( summable @ A
@ ^ [N2: nat] : ( minus_minus @ A @ ( F2 @ N2 ) @ ( F2 @ ( suc @ N2 ) ) ) ) ) ) ).
% telescope_summable'
thf(fact_7188_nested__sequence__unique,axiom,
! [F2: nat > real,G: nat > real] :
( ! [N: nat] : ( ord_less_eq @ real @ ( F2 @ N ) @ ( F2 @ ( suc @ N ) ) )
=> ( ! [N: nat] : ( ord_less_eq @ real @ ( G @ ( suc @ N ) ) @ ( G @ N ) )
=> ( ! [N: nat] : ( ord_less_eq @ real @ ( F2 @ N ) @ ( G @ N ) )
=> ( ( filterlim @ nat @ real
@ ^ [N2: nat] : ( minus_minus @ real @ ( F2 @ N2 ) @ ( G @ N2 ) )
@ ( topolo7230453075368039082e_nhds @ real @ ( zero_zero @ real ) )
@ ( at_top @ nat ) )
=> ? [L3: real] :
( ! [N9: nat] : ( ord_less_eq @ real @ ( F2 @ N9 ) @ L3 )
& ( filterlim @ nat @ real @ F2 @ ( topolo7230453075368039082e_nhds @ real @ L3 ) @ ( at_top @ nat ) )
& ! [N9: nat] : ( ord_less_eq @ real @ L3 @ ( G @ N9 ) )
& ( filterlim @ nat @ real @ G @ ( topolo7230453075368039082e_nhds @ real @ L3 ) @ ( at_top @ nat ) ) ) ) ) ) ) ).
% nested_sequence_unique
thf(fact_7189_LIMSEQ__inverse__zero,axiom,
! [X8: nat > real] :
( ! [R3: real] :
? [N10: nat] :
! [N: nat] :
( ( ord_less_eq @ nat @ N10 @ N )
=> ( ord_less @ real @ R3 @ ( X8 @ N ) ) )
=> ( filterlim @ nat @ real
@ ^ [N2: nat] : ( inverse_inverse @ real @ ( X8 @ N2 ) )
@ ( topolo7230453075368039082e_nhds @ real @ ( zero_zero @ real ) )
@ ( at_top @ nat ) ) ) ).
% LIMSEQ_inverse_zero
thf(fact_7190_lim__inverse__n_H,axiom,
( filterlim @ nat @ real
@ ^ [N2: nat] : ( divide_divide @ real @ ( one_one @ real ) @ ( semiring_1_of_nat @ real @ N2 ) )
@ ( topolo7230453075368039082e_nhds @ real @ ( zero_zero @ real ) )
@ ( at_top @ nat ) ) ).
% lim_inverse_n'
thf(fact_7191_LIMSEQ__inverse__real__of__nat,axiom,
( filterlim @ nat @ real
@ ^ [N2: nat] : ( inverse_inverse @ real @ ( semiring_1_of_nat @ real @ ( suc @ N2 ) ) )
@ ( topolo7230453075368039082e_nhds @ real @ ( zero_zero @ real ) )
@ ( at_top @ nat ) ) ).
% LIMSEQ_inverse_real_of_nat
thf(fact_7192_LIMSEQ__root__const,axiom,
! [C2: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ C2 )
=> ( filterlim @ nat @ real
@ ^ [N2: nat] : ( root @ N2 @ C2 )
@ ( topolo7230453075368039082e_nhds @ real @ ( one_one @ real ) )
@ ( at_top @ nat ) ) ) ).
% LIMSEQ_root_const
thf(fact_7193_LIMSEQ__inverse__real__of__nat__add,axiom,
! [R2: real] :
( filterlim @ nat @ real
@ ^ [N2: nat] : ( plus_plus @ real @ R2 @ ( inverse_inverse @ real @ ( semiring_1_of_nat @ real @ ( suc @ N2 ) ) ) )
@ ( topolo7230453075368039082e_nhds @ real @ R2 )
@ ( at_top @ nat ) ) ).
% LIMSEQ_inverse_real_of_nat_add
thf(fact_7194_continuous__at__within__log,axiom,
! [A: $tType] :
( ( topological_t2_space @ A )
=> ! [A2: A,S: set @ A,F2: A > real,G: A > real] :
( ( topolo3448309680560233919inuous @ A @ real @ ( topolo174197925503356063within @ A @ A2 @ S ) @ F2 )
=> ( ( topolo3448309680560233919inuous @ A @ real @ ( topolo174197925503356063within @ A @ A2 @ S ) @ G )
=> ( ( ord_less @ real @ ( zero_zero @ real ) @ ( F2 @ A2 ) )
=> ( ( ( F2 @ A2 )
!= ( one_one @ real ) )
=> ( ( ord_less @ real @ ( zero_zero @ real ) @ ( G @ A2 ) )
=> ( topolo3448309680560233919inuous @ A @ real @ ( topolo174197925503356063within @ A @ A2 @ S )
@ ^ [X2: A] : ( log @ ( F2 @ X2 ) @ ( G @ X2 ) ) ) ) ) ) ) ) ) ).
% continuous_at_within_log
thf(fact_7195_increasing__LIMSEQ,axiom,
! [F2: nat > real,L2: real] :
( ! [N: nat] : ( ord_less_eq @ real @ ( F2 @ N ) @ ( F2 @ ( suc @ N ) ) )
=> ( ! [N: nat] : ( ord_less_eq @ real @ ( F2 @ N ) @ L2 )
=> ( ! [E2: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ E2 )
=> ? [N9: nat] : ( ord_less_eq @ real @ L2 @ ( plus_plus @ real @ ( F2 @ N9 ) @ E2 ) ) )
=> ( filterlim @ nat @ real @ F2 @ ( topolo7230453075368039082e_nhds @ real @ L2 ) @ ( at_top @ nat ) ) ) ) ) ).
% increasing_LIMSEQ
thf(fact_7196_lim__1__over__n,axiom,
! [A: $tType] :
( ( real_V3459762299906320749_field @ A )
=> ( filterlim @ nat @ A
@ ^ [N2: nat] : ( divide_divide @ A @ ( one_one @ A ) @ ( semiring_1_of_nat @ A @ N2 ) )
@ ( topolo7230453075368039082e_nhds @ A @ ( zero_zero @ A ) )
@ ( at_top @ nat ) ) ) ).
% lim_1_over_n
thf(fact_7197_LIMSEQ__n__over__Suc__n,axiom,
! [A: $tType] :
( ( real_V3459762299906320749_field @ A )
=> ( filterlim @ nat @ A
@ ^ [N2: nat] : ( divide_divide @ A @ ( semiring_1_of_nat @ A @ N2 ) @ ( semiring_1_of_nat @ A @ ( suc @ N2 ) ) )
@ ( topolo7230453075368039082e_nhds @ A @ ( one_one @ A ) )
@ ( at_top @ nat ) ) ) ).
% LIMSEQ_n_over_Suc_n
thf(fact_7198_LIMSEQ__Suc__n__over__n,axiom,
! [A: $tType] :
( ( real_V3459762299906320749_field @ A )
=> ( filterlim @ nat @ A
@ ^ [N2: nat] : ( divide_divide @ A @ ( semiring_1_of_nat @ A @ ( suc @ N2 ) ) @ ( semiring_1_of_nat @ A @ N2 ) )
@ ( topolo7230453075368039082e_nhds @ A @ ( one_one @ A ) )
@ ( at_top @ nat ) ) ) ).
% LIMSEQ_Suc_n_over_n
thf(fact_7199_telescope__sums,axiom,
! [A: $tType] :
( ( real_V822414075346904944vector @ A )
=> ! [F2: nat > A,C2: A] :
( ( filterlim @ nat @ A @ F2 @ ( topolo7230453075368039082e_nhds @ A @ C2 ) @ ( at_top @ nat ) )
=> ( sums @ A
@ ^ [N2: nat] : ( minus_minus @ A @ ( F2 @ ( suc @ N2 ) ) @ ( F2 @ N2 ) )
@ ( minus_minus @ A @ C2 @ ( F2 @ ( zero_zero @ nat ) ) ) ) ) ) ).
% telescope_sums
thf(fact_7200_telescope__sums_H,axiom,
! [A: $tType] :
( ( real_V822414075346904944vector @ A )
=> ! [F2: nat > A,C2: A] :
( ( filterlim @ nat @ A @ F2 @ ( topolo7230453075368039082e_nhds @ A @ C2 ) @ ( at_top @ nat ) )
=> ( sums @ A
@ ^ [N2: nat] : ( minus_minus @ A @ ( F2 @ N2 ) @ ( F2 @ ( suc @ N2 ) ) )
@ ( minus_minus @ A @ ( F2 @ ( zero_zero @ nat ) ) @ C2 ) ) ) ) ).
% telescope_sums'
thf(fact_7201_LIMSEQ__realpow__zero,axiom,
! [X: real] :
( ( ord_less_eq @ real @ ( zero_zero @ real ) @ X )
=> ( ( ord_less @ real @ X @ ( one_one @ real ) )
=> ( filterlim @ nat @ real @ ( power_power @ real @ X ) @ ( topolo7230453075368039082e_nhds @ real @ ( zero_zero @ real ) ) @ ( at_top @ nat ) ) ) ) ).
% LIMSEQ_realpow_zero
thf(fact_7202_LIMSEQ__divide__realpow__zero,axiom,
! [X: real,A2: real] :
( ( ord_less @ real @ ( one_one @ real ) @ X )
=> ( filterlim @ nat @ real
@ ^ [N2: nat] : ( divide_divide @ real @ A2 @ ( power_power @ real @ X @ N2 ) )
@ ( topolo7230453075368039082e_nhds @ real @ ( zero_zero @ real ) )
@ ( at_top @ nat ) ) ) ).
% LIMSEQ_divide_realpow_zero
thf(fact_7203_LIMSEQ__abs__realpow__zero,axiom,
! [C2: real] :
( ( ord_less @ real @ ( abs_abs @ real @ C2 ) @ ( one_one @ real ) )
=> ( filterlim @ nat @ real @ ( power_power @ real @ ( abs_abs @ real @ C2 ) ) @ ( topolo7230453075368039082e_nhds @ real @ ( zero_zero @ real ) ) @ ( at_top @ nat ) ) ) ).
% LIMSEQ_abs_realpow_zero
thf(fact_7204_LIMSEQ__abs__realpow__zero2,axiom,
! [C2: real] :
( ( ord_less @ real @ ( abs_abs @ real @ C2 ) @ ( one_one @ real ) )
=> ( filterlim @ nat @ real @ ( power_power @ real @ C2 ) @ ( topolo7230453075368039082e_nhds @ real @ ( zero_zero @ real ) ) @ ( at_top @ nat ) ) ) ).
% LIMSEQ_abs_realpow_zero2
thf(fact_7205_LIMSEQ__inverse__realpow__zero,axiom,
! [X: real] :
( ( ord_less @ real @ ( one_one @ real ) @ X )
=> ( filterlim @ nat @ real
@ ^ [N2: nat] : ( inverse_inverse @ real @ ( power_power @ real @ X @ N2 ) )
@ ( topolo7230453075368039082e_nhds @ real @ ( zero_zero @ real ) )
@ ( at_top @ nat ) ) ) ).
% LIMSEQ_inverse_realpow_zero
thf(fact_7206_LIMSEQ__inverse__real__of__nat__add__minus,axiom,
! [R2: real] :
( filterlim @ nat @ real
@ ^ [N2: nat] : ( plus_plus @ real @ R2 @ ( uminus_uminus @ real @ ( inverse_inverse @ real @ ( semiring_1_of_nat @ real @ ( suc @ N2 ) ) ) ) )
@ ( topolo7230453075368039082e_nhds @ real @ R2 )
@ ( at_top @ nat ) ) ).
% LIMSEQ_inverse_real_of_nat_add_minus
thf(fact_7207_root__test__convergence,axiom,
! [A: $tType] :
( ( real_Vector_banach @ A )
=> ! [F2: nat > A,X: real] :
( ( filterlim @ nat @ real
@ ^ [N2: nat] : ( root @ N2 @ ( real_V7770717601297561774m_norm @ A @ ( F2 @ N2 ) ) )
@ ( topolo7230453075368039082e_nhds @ real @ X )
@ ( at_top @ nat ) )
=> ( ( ord_less @ real @ X @ ( one_one @ real ) )
=> ( summable @ A @ F2 ) ) ) ) ).
% root_test_convergence
thf(fact_7208_isCont__log,axiom,
! [A: $tType] :
( ( topological_t2_space @ A )
=> ! [A2: A,F2: A > real,G: A > real] :
( ( topolo3448309680560233919inuous @ A @ real @ ( topolo174197925503356063within @ A @ A2 @ ( top_top @ ( set @ A ) ) ) @ F2 )
=> ( ( topolo3448309680560233919inuous @ A @ real @ ( topolo174197925503356063within @ A @ A2 @ ( top_top @ ( set @ A ) ) ) @ G )
=> ( ( ord_less @ real @ ( zero_zero @ real ) @ ( F2 @ A2 ) )
=> ( ( ( F2 @ A2 )
!= ( one_one @ real ) )
=> ( ( ord_less @ real @ ( zero_zero @ real ) @ ( G @ A2 ) )
=> ( topolo3448309680560233919inuous @ A @ real @ ( topolo174197925503356063within @ A @ A2 @ ( top_top @ ( set @ A ) ) )
@ ^ [X2: A] : ( log @ ( F2 @ X2 ) @ ( G @ X2 ) ) ) ) ) ) ) ) ) ).
% isCont_log
thf(fact_7209_LIMSEQ__iff,axiom,
! [A: $tType] :
( ( real_V822414075346904944vector @ A )
=> ! [X8: nat > A,L6: A] :
( ( filterlim @ nat @ A @ X8 @ ( topolo7230453075368039082e_nhds @ A @ L6 ) @ ( at_top @ nat ) )
= ( ! [R5: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ R5 )
=> ? [No: nat] :
! [N2: nat] :
( ( ord_less_eq @ nat @ No @ N2 )
=> ( ord_less @ real @ ( real_V7770717601297561774m_norm @ A @ ( minus_minus @ A @ ( X8 @ N2 ) @ L6 ) ) @ R5 ) ) ) ) ) ) ).
% LIMSEQ_iff
thf(fact_7210_LIMSEQ__I,axiom,
! [A: $tType] :
( ( real_V822414075346904944vector @ A )
=> ! [X8: nat > A,L6: A] :
( ! [R3: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ R3 )
=> ? [No2: nat] :
! [N: nat] :
( ( ord_less_eq @ nat @ No2 @ N )
=> ( ord_less @ real @ ( real_V7770717601297561774m_norm @ A @ ( minus_minus @ A @ ( X8 @ N ) @ L6 ) ) @ R3 ) ) )
=> ( filterlim @ nat @ A @ X8 @ ( topolo7230453075368039082e_nhds @ A @ L6 ) @ ( at_top @ nat ) ) ) ) ).
% LIMSEQ_I
thf(fact_7211_LIMSEQ__D,axiom,
! [A: $tType] :
( ( real_V822414075346904944vector @ A )
=> ! [X8: nat > A,L6: A,R2: real] :
( ( filterlim @ nat @ A @ X8 @ ( topolo7230453075368039082e_nhds @ A @ L6 ) @ ( at_top @ nat ) )
=> ( ( ord_less @ real @ ( zero_zero @ real ) @ R2 )
=> ? [No3: nat] :
! [N9: nat] :
( ( ord_less_eq @ nat @ No3 @ N9 )
=> ( ord_less @ real @ ( real_V7770717601297561774m_norm @ A @ ( minus_minus @ A @ ( X8 @ N9 ) @ L6 ) ) @ R2 ) ) ) ) ) ).
% LIMSEQ_D
thf(fact_7212_LIMSEQ__power__zero,axiom,
! [A: $tType] :
( ( real_V2822296259951069270ebra_1 @ A )
=> ! [X: A] :
( ( ord_less @ real @ ( real_V7770717601297561774m_norm @ A @ X ) @ ( one_one @ real ) )
=> ( filterlim @ nat @ A @ ( power_power @ A @ X ) @ ( topolo7230453075368039082e_nhds @ A @ ( zero_zero @ A ) ) @ ( at_top @ nat ) ) ) ) ).
% LIMSEQ_power_zero
thf(fact_7213_tendsto__exp__limit__sequentially,axiom,
! [X: real] :
( filterlim @ nat @ real
@ ^ [N2: nat] : ( power_power @ real @ ( plus_plus @ real @ ( one_one @ real ) @ ( divide_divide @ real @ X @ ( semiring_1_of_nat @ real @ N2 ) ) ) @ N2 )
@ ( topolo7230453075368039082e_nhds @ real @ ( exp @ real @ X ) )
@ ( at_top @ nat ) ) ).
% tendsto_exp_limit_sequentially
thf(fact_7214_tendsto__at__iff__sequentially,axiom,
! [C: $tType,A: $tType] :
( ( ( topolo3112930676232923870pology @ A )
& ( topolo4958980785337419405_space @ C ) )
=> ! [F2: A > C,A2: C,X: A,S: set @ A] :
( ( filterlim @ A @ C @ F2 @ ( topolo7230453075368039082e_nhds @ C @ A2 ) @ ( topolo174197925503356063within @ A @ X @ S ) )
= ( ! [X6: nat > A] :
( ! [I3: nat] : ( member @ A @ ( X6 @ I3 ) @ ( minus_minus @ ( set @ A ) @ S @ ( insert @ A @ X @ ( bot_bot @ ( set @ A ) ) ) ) )
=> ( ( filterlim @ nat @ A @ X6 @ ( topolo7230453075368039082e_nhds @ A @ X ) @ ( at_top @ nat ) )
=> ( filterlim @ nat @ C @ ( comp @ A @ C @ nat @ F2 @ X6 ) @ ( topolo7230453075368039082e_nhds @ C @ A2 ) @ ( at_top @ nat ) ) ) ) ) ) ) ).
% tendsto_at_iff_sequentially
thf(fact_7215_tendsto__power__zero,axiom,
! [A: $tType,B: $tType] :
( ( real_V2822296259951069270ebra_1 @ A )
=> ! [F2: B > nat,F4: filter @ B,X: A] :
( ( filterlim @ B @ nat @ F2 @ ( at_top @ nat ) @ F4 )
=> ( ( ord_less @ real @ ( real_V7770717601297561774m_norm @ A @ X ) @ ( one_one @ real ) )
=> ( filterlim @ B @ A
@ ^ [Y2: B] : ( power_power @ A @ X @ ( F2 @ Y2 ) )
@ ( topolo7230453075368039082e_nhds @ A @ ( zero_zero @ A ) )
@ F4 ) ) ) ) ).
% tendsto_power_zero
thf(fact_7216_LIMSEQ__inverse__real__of__nat__add__minus__mult,axiom,
! [R2: real] :
( filterlim @ nat @ real
@ ^ [N2: nat] : ( times_times @ real @ R2 @ ( plus_plus @ real @ ( one_one @ real ) @ ( uminus_uminus @ real @ ( inverse_inverse @ real @ ( semiring_1_of_nat @ real @ ( suc @ N2 ) ) ) ) ) )
@ ( topolo7230453075368039082e_nhds @ real @ R2 )
@ ( at_top @ nat ) ) ).
% LIMSEQ_inverse_real_of_nat_add_minus_mult
thf(fact_7217_LIMSEQ__norm__0,axiom,
! [A: $tType] :
( ( real_V822414075346904944vector @ A )
=> ! [F2: nat > A] :
( ! [N: nat] : ( ord_less @ real @ ( real_V7770717601297561774m_norm @ A @ ( F2 @ N ) ) @ ( divide_divide @ real @ ( one_one @ real ) @ ( semiring_1_of_nat @ real @ ( suc @ N ) ) ) )
=> ( filterlim @ nat @ A @ F2 @ ( topolo7230453075368039082e_nhds @ A @ ( zero_zero @ A ) ) @ ( at_top @ nat ) ) ) ) ).
% LIMSEQ_norm_0
thf(fact_7218_summable__Leibniz_I1_J,axiom,
! [A2: nat > real] :
( ( filterlim @ nat @ real @ A2 @ ( topolo7230453075368039082e_nhds @ real @ ( zero_zero @ real ) ) @ ( at_top @ nat ) )
=> ( ( topological_monoseq @ real @ A2 )
=> ( summable @ real
@ ^ [N2: nat] : ( times_times @ real @ ( power_power @ real @ ( uminus_uminus @ real @ ( one_one @ real ) ) @ N2 ) @ ( A2 @ N2 ) ) ) ) ) ).
% summable_Leibniz(1)
thf(fact_7219_field__derivative__lim__unique,axiom,
! [A: $tType] :
( ( real_V3459762299906320749_field @ A )
=> ! [F2: A > A,Df: A,Z: A,S: nat > A,A2: A] :
( ( has_field_derivative @ A @ F2 @ Df @ ( topolo174197925503356063within @ A @ Z @ ( top_top @ ( set @ A ) ) ) )
=> ( ( filterlim @ nat @ A @ S @ ( topolo7230453075368039082e_nhds @ A @ ( zero_zero @ A ) ) @ ( at_top @ nat ) )
=> ( ! [N: nat] :
( ( S @ N )
!= ( zero_zero @ A ) )
=> ( ( filterlim @ nat @ A
@ ^ [N2: nat] : ( divide_divide @ A @ ( minus_minus @ A @ ( F2 @ ( plus_plus @ A @ Z @ ( S @ N2 ) ) ) @ ( F2 @ Z ) ) @ ( S @ N2 ) )
@ ( topolo7230453075368039082e_nhds @ A @ A2 )
@ ( at_top @ nat ) )
=> ( Df = A2 ) ) ) ) ) ) ).
% field_derivative_lim_unique
thf(fact_7220_powser__times__n__limit__0,axiom,
! [A: $tType] :
( ( ( real_Vector_banach @ A )
& ( real_V8999393235501362500lgebra @ A ) )
=> ! [X: A] :
( ( ord_less @ real @ ( real_V7770717601297561774m_norm @ A @ X ) @ ( one_one @ real ) )
=> ( filterlim @ nat @ A
@ ^ [N2: nat] : ( times_times @ A @ ( semiring_1_of_nat @ A @ N2 ) @ ( power_power @ A @ X @ N2 ) )
@ ( topolo7230453075368039082e_nhds @ A @ ( zero_zero @ A ) )
@ ( at_top @ nat ) ) ) ) ).
% powser_times_n_limit_0
thf(fact_7221_lim__n__over__pown,axiom,
! [A: $tType] :
( ( ( real_Vector_banach @ A )
& ( real_V3459762299906320749_field @ A ) )
=> ! [X: A] :
( ( ord_less @ real @ ( one_one @ real ) @ ( real_V7770717601297561774m_norm @ A @ X ) )
=> ( filterlim @ nat @ A
@ ^ [N2: nat] : ( divide_divide @ A @ ( semiring_1_of_nat @ A @ N2 ) @ ( power_power @ A @ X @ N2 ) )
@ ( topolo7230453075368039082e_nhds @ A @ ( zero_zero @ A ) )
@ ( at_top @ nat ) ) ) ) ).
% lim_n_over_pown
thf(fact_7222_summable,axiom,
! [A2: nat > real] :
( ( filterlim @ nat @ real @ A2 @ ( topolo7230453075368039082e_nhds @ real @ ( zero_zero @ real ) ) @ ( at_top @ nat ) )
=> ( ! [N: nat] : ( ord_less_eq @ real @ ( zero_zero @ real ) @ ( A2 @ N ) )
=> ( ! [N: nat] : ( ord_less_eq @ real @ ( A2 @ ( suc @ N ) ) @ ( A2 @ N ) )
=> ( summable @ real
@ ^ [N2: nat] : ( times_times @ real @ ( power_power @ real @ ( uminus_uminus @ real @ ( one_one @ real ) ) @ N2 ) @ ( A2 @ N2 ) ) ) ) ) ) ).
% summable
thf(fact_7223_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 ) )
=> ~ ! [K3: nat > int] :
~ ( filterlim @ nat @ real
@ ^ [J3: nat] : ( minus_minus @ real @ ( Theta @ J3 ) @ ( times_times @ real @ ( ring_1_of_int @ real @ ( K3 @ J3 ) ) @ ( times_times @ real @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) @ pi ) ) )
@ ( topolo7230453075368039082e_nhds @ real @ Theta2 )
@ ( at_top @ nat ) ) ) ).
% cos_diff_limit_1
thf(fact_7224_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 ) )
=> ? [K3: nat > int] :
( filterlim @ nat @ real
@ ^ [J3: nat] : ( minus_minus @ real @ ( Theta @ J3 ) @ ( times_times @ real @ ( ring_1_of_int @ real @ ( K3 @ J3 ) ) @ ( times_times @ real @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) @ pi ) ) )
@ ( topolo7230453075368039082e_nhds @ real @ ( zero_zero @ real ) )
@ ( at_top @ nat ) ) ) ).
% cos_limit_1
thf(fact_7225_summable__Leibniz_I4_J,axiom,
! [A2: nat > real] :
( ( filterlim @ nat @ real @ A2 @ ( topolo7230453075368039082e_nhds @ real @ ( zero_zero @ real ) ) @ ( at_top @ nat ) )
=> ( ( topological_monoseq @ real @ A2 )
=> ( filterlim @ nat @ real
@ ^ [N2: nat] :
( groups7311177749621191930dd_sum @ nat @ real
@ ^ [I3: nat] : ( times_times @ real @ ( power_power @ real @ ( uminus_uminus @ real @ ( one_one @ real ) ) @ I3 ) @ ( A2 @ I3 ) )
@ ( set_ord_lessThan @ nat @ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N2 ) ) )
@ ( topolo7230453075368039082e_nhds @ real
@ ( suminf @ real
@ ^ [I3: nat] : ( times_times @ real @ ( power_power @ real @ ( uminus_uminus @ real @ ( one_one @ real ) ) @ I3 ) @ ( A2 @ I3 ) ) ) )
@ ( at_top @ nat ) ) ) ) ).
% summable_Leibniz(4)
thf(fact_7226_zeroseq__arctan__series,axiom,
! [X: real] :
( ( ord_less_eq @ real @ ( abs_abs @ real @ X ) @ ( one_one @ real ) )
=> ( filterlim @ nat @ real
@ ^ [N2: nat] : ( times_times @ real @ ( divide_divide @ real @ ( one_one @ real ) @ ( semiring_1_of_nat @ real @ ( plus_plus @ nat @ ( times_times @ nat @ N2 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ ( one_one @ nat ) ) ) ) @ ( power_power @ real @ X @ ( plus_plus @ nat @ ( times_times @ nat @ N2 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ ( one_one @ nat ) ) ) )
@ ( topolo7230453075368039082e_nhds @ real @ ( zero_zero @ real ) )
@ ( at_top @ nat ) ) ) ).
% zeroseq_arctan_series
thf(fact_7227_summable__Leibniz_H_I3_J,axiom,
! [A2: nat > real] :
( ( filterlim @ nat @ real @ A2 @ ( topolo7230453075368039082e_nhds @ real @ ( zero_zero @ real ) ) @ ( at_top @ nat ) )
=> ( ! [N: nat] : ( ord_less_eq @ real @ ( zero_zero @ real ) @ ( A2 @ N ) )
=> ( ! [N: nat] : ( ord_less_eq @ real @ ( A2 @ ( suc @ N ) ) @ ( A2 @ N ) )
=> ( filterlim @ nat @ real
@ ^ [N2: nat] :
( groups7311177749621191930dd_sum @ nat @ real
@ ^ [I3: nat] : ( times_times @ real @ ( power_power @ real @ ( uminus_uminus @ real @ ( one_one @ real ) ) @ I3 ) @ ( A2 @ I3 ) )
@ ( set_ord_lessThan @ nat @ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N2 ) ) )
@ ( topolo7230453075368039082e_nhds @ real
@ ( suminf @ real
@ ^ [I3: nat] : ( times_times @ real @ ( power_power @ real @ ( uminus_uminus @ real @ ( one_one @ real ) ) @ I3 ) @ ( A2 @ I3 ) ) ) )
@ ( at_top @ nat ) ) ) ) ) ).
% summable_Leibniz'(3)
thf(fact_7228_summable__Leibniz_H_I2_J,axiom,
! [A2: nat > real,N3: nat] :
( ( filterlim @ nat @ real @ A2 @ ( topolo7230453075368039082e_nhds @ real @ ( zero_zero @ real ) ) @ ( at_top @ nat ) )
=> ( ! [N: nat] : ( ord_less_eq @ real @ ( zero_zero @ real ) @ ( A2 @ N ) )
=> ( ! [N: nat] : ( ord_less_eq @ real @ ( A2 @ ( suc @ N ) ) @ ( A2 @ N ) )
=> ( ord_less_eq @ real
@ ( groups7311177749621191930dd_sum @ nat @ real
@ ^ [I3: nat] : ( times_times @ real @ ( power_power @ real @ ( uminus_uminus @ real @ ( one_one @ real ) ) @ I3 ) @ ( A2 @ I3 ) )
@ ( set_ord_lessThan @ nat @ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N3 ) ) )
@ ( suminf @ real
@ ^ [I3: nat] : ( times_times @ real @ ( power_power @ real @ ( uminus_uminus @ real @ ( one_one @ real ) ) @ I3 ) @ ( A2 @ I3 ) ) ) ) ) ) ) ).
% summable_Leibniz'(2)
thf(fact_7229_sums__alternating__upper__lower,axiom,
! [A2: nat > real] :
( ! [N: nat] : ( ord_less_eq @ real @ ( A2 @ ( suc @ N ) ) @ ( A2 @ N ) )
=> ( ! [N: nat] : ( ord_less_eq @ real @ ( zero_zero @ real ) @ ( A2 @ N ) )
=> ( ( filterlim @ nat @ real @ A2 @ ( topolo7230453075368039082e_nhds @ real @ ( zero_zero @ real ) ) @ ( at_top @ nat ) )
=> ? [L3: real] :
( ! [N9: nat] :
( ord_less_eq @ real
@ ( groups7311177749621191930dd_sum @ nat @ real
@ ^ [I3: nat] : ( times_times @ real @ ( power_power @ real @ ( uminus_uminus @ real @ ( one_one @ real ) ) @ I3 ) @ ( A2 @ I3 ) )
@ ( set_ord_lessThan @ nat @ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N9 ) ) )
@ L3 )
& ( filterlim @ nat @ real
@ ^ [N2: nat] :
( groups7311177749621191930dd_sum @ nat @ real
@ ^ [I3: nat] : ( times_times @ real @ ( power_power @ real @ ( uminus_uminus @ real @ ( one_one @ real ) ) @ I3 ) @ ( A2 @ I3 ) )
@ ( set_ord_lessThan @ nat @ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N2 ) ) )
@ ( topolo7230453075368039082e_nhds @ real @ L3 )
@ ( at_top @ nat ) )
& ! [N9: nat] :
( ord_less_eq @ real @ L3
@ ( groups7311177749621191930dd_sum @ nat @ real
@ ^ [I3: nat] : ( times_times @ real @ ( power_power @ real @ ( uminus_uminus @ real @ ( one_one @ real ) ) @ I3 ) @ ( A2 @ I3 ) )
@ ( set_ord_lessThan @ nat @ ( plus_plus @ nat @ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N9 ) @ ( one_one @ nat ) ) ) ) )
& ( filterlim @ nat @ real
@ ^ [N2: nat] :
( groups7311177749621191930dd_sum @ nat @ real
@ ^ [I3: nat] : ( times_times @ real @ ( power_power @ real @ ( uminus_uminus @ real @ ( one_one @ real ) ) @ I3 ) @ ( A2 @ I3 ) )
@ ( set_ord_lessThan @ nat @ ( plus_plus @ nat @ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N2 ) @ ( one_one @ nat ) ) ) )
@ ( topolo7230453075368039082e_nhds @ real @ L3 )
@ ( at_top @ nat ) ) ) ) ) ) ).
% sums_alternating_upper_lower
thf(fact_7230_summable__Leibniz_I5_J,axiom,
! [A2: nat > real] :
( ( filterlim @ nat @ real @ A2 @ ( topolo7230453075368039082e_nhds @ real @ ( zero_zero @ real ) ) @ ( at_top @ nat ) )
=> ( ( topological_monoseq @ real @ A2 )
=> ( filterlim @ nat @ real
@ ^ [N2: nat] :
( groups7311177749621191930dd_sum @ nat @ real
@ ^ [I3: nat] : ( times_times @ real @ ( power_power @ real @ ( uminus_uminus @ real @ ( one_one @ real ) ) @ I3 ) @ ( A2 @ I3 ) )
@ ( set_ord_lessThan @ nat @ ( plus_plus @ nat @ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N2 ) @ ( one_one @ nat ) ) ) )
@ ( topolo7230453075368039082e_nhds @ real
@ ( suminf @ real
@ ^ [I3: nat] : ( times_times @ real @ ( power_power @ real @ ( uminus_uminus @ real @ ( one_one @ real ) ) @ I3 ) @ ( A2 @ I3 ) ) ) )
@ ( at_top @ nat ) ) ) ) ).
% summable_Leibniz(5)
thf(fact_7231_summable__Leibniz_H_I5_J,axiom,
! [A2: nat > real] :
( ( filterlim @ nat @ real @ A2 @ ( topolo7230453075368039082e_nhds @ real @ ( zero_zero @ real ) ) @ ( at_top @ nat ) )
=> ( ! [N: nat] : ( ord_less_eq @ real @ ( zero_zero @ real ) @ ( A2 @ N ) )
=> ( ! [N: nat] : ( ord_less_eq @ real @ ( A2 @ ( suc @ N ) ) @ ( A2 @ N ) )
=> ( filterlim @ nat @ real
@ ^ [N2: nat] :
( groups7311177749621191930dd_sum @ nat @ real
@ ^ [I3: nat] : ( times_times @ real @ ( power_power @ real @ ( uminus_uminus @ real @ ( one_one @ real ) ) @ I3 ) @ ( A2 @ I3 ) )
@ ( set_ord_lessThan @ nat @ ( plus_plus @ nat @ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N2 ) @ ( one_one @ nat ) ) ) )
@ ( topolo7230453075368039082e_nhds @ real
@ ( suminf @ real
@ ^ [I3: nat] : ( times_times @ real @ ( power_power @ real @ ( uminus_uminus @ real @ ( one_one @ real ) ) @ I3 ) @ ( A2 @ I3 ) ) ) )
@ ( at_top @ nat ) ) ) ) ) ).
% summable_Leibniz'(5)
thf(fact_7232_summable__Leibniz_H_I4_J,axiom,
! [A2: nat > real,N3: nat] :
( ( filterlim @ nat @ real @ A2 @ ( topolo7230453075368039082e_nhds @ real @ ( zero_zero @ real ) ) @ ( at_top @ nat ) )
=> ( ! [N: nat] : ( ord_less_eq @ real @ ( zero_zero @ real ) @ ( A2 @ N ) )
=> ( ! [N: nat] : ( ord_less_eq @ real @ ( A2 @ ( suc @ N ) ) @ ( A2 @ N ) )
=> ( ord_less_eq @ real
@ ( suminf @ real
@ ^ [I3: nat] : ( times_times @ real @ ( power_power @ real @ ( uminus_uminus @ real @ ( one_one @ real ) ) @ I3 ) @ ( A2 @ I3 ) ) )
@ ( groups7311177749621191930dd_sum @ nat @ real
@ ^ [I3: nat] : ( times_times @ real @ ( power_power @ real @ ( uminus_uminus @ real @ ( one_one @ real ) ) @ I3 ) @ ( A2 @ I3 ) )
@ ( set_ord_lessThan @ nat @ ( plus_plus @ nat @ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N3 ) @ ( one_one @ nat ) ) ) ) ) ) ) ) ).
% summable_Leibniz'(4)
thf(fact_7233_filterlim__sequentially__Suc,axiom,
! [A: $tType,F2: nat > A,F4: filter @ A] :
( ( filterlim @ nat @ A
@ ^ [X2: nat] : ( F2 @ ( suc @ X2 ) )
@ F4
@ ( at_top @ nat ) )
= ( filterlim @ nat @ A @ F2 @ F4 @ ( at_top @ nat ) ) ) ).
% filterlim_sequentially_Suc
thf(fact_7234_has__derivative__at2,axiom,
! [B: $tType,A: $tType] :
( ( ( real_V822414075346904944vector @ A )
& ( real_V822414075346904944vector @ B ) )
=> ! [F2: A > B,F5: A > B,X: A] :
( ( has_derivative @ A @ B @ F2 @ F5 @ ( topolo174197925503356063within @ A @ X @ ( top_top @ ( set @ A ) ) ) )
= ( ( real_V3181309239436604168linear @ A @ B @ F5 )
& ( filterlim @ A @ B
@ ^ [Y2: A] : ( real_V8093663219630862766scaleR @ B @ ( divide_divide @ real @ ( one_one @ real ) @ ( real_V7770717601297561774m_norm @ A @ ( minus_minus @ A @ Y2 @ X ) ) ) @ ( minus_minus @ B @ ( F2 @ Y2 ) @ ( plus_plus @ B @ ( F2 @ X ) @ ( F5 @ ( minus_minus @ A @ Y2 @ X ) ) ) ) )
@ ( topolo7230453075368039082e_nhds @ B @ ( zero_zero @ B ) )
@ ( topolo174197925503356063within @ A @ X @ ( top_top @ ( set @ A ) ) ) ) ) ) ) ).
% has_derivative_at2
thf(fact_7235_bounded__linear__sub,axiom,
! [B: $tType,A: $tType] :
( ( ( real_V822414075346904944vector @ A )
& ( real_V822414075346904944vector @ B ) )
=> ! [F2: A > B,G: A > B] :
( ( real_V3181309239436604168linear @ A @ B @ F2 )
=> ( ( real_V3181309239436604168linear @ A @ B @ G )
=> ( real_V3181309239436604168linear @ A @ B
@ ^ [X2: A] : ( minus_minus @ B @ ( F2 @ X2 ) @ ( G @ X2 ) ) ) ) ) ) ).
% bounded_linear_sub
thf(fact_7236_bounded__linear_Obounded__linear,axiom,
! [B: $tType,A: $tType] :
( ( ( real_V822414075346904944vector @ A )
& ( real_V822414075346904944vector @ B ) )
=> ! [F2: A > B] :
( ( real_V3181309239436604168linear @ A @ B @ F2 )
=> ( real_V3181309239436604168linear @ A @ B @ F2 ) ) ) ).
% bounded_linear.bounded_linear
thf(fact_7237_bounded__linear__divide,axiom,
! [A: $tType] :
( ( real_V3459762299906320749_field @ A )
=> ! [Y: A] :
( real_V3181309239436604168linear @ A @ A
@ ^ [X2: A] : ( divide_divide @ A @ X2 @ Y ) ) ) ).
% bounded_linear_divide
thf(fact_7238_bounded__linear__add,axiom,
! [B: $tType,A: $tType] :
( ( ( real_V822414075346904944vector @ A )
& ( real_V822414075346904944vector @ B ) )
=> ! [F2: A > B,G: A > B] :
( ( real_V3181309239436604168linear @ A @ B @ F2 )
=> ( ( real_V3181309239436604168linear @ A @ B @ G )
=> ( real_V3181309239436604168linear @ A @ B
@ ^ [X2: A] : ( plus_plus @ B @ ( F2 @ X2 ) @ ( G @ X2 ) ) ) ) ) ) ).
% bounded_linear_add
thf(fact_7239_bounded__linear__mult__left,axiom,
! [A: $tType] :
( ( real_V4412858255891104859lgebra @ A )
=> ! [Y: A] :
( real_V3181309239436604168linear @ A @ A
@ ^ [X2: A] : ( times_times @ A @ X2 @ Y ) ) ) ).
% bounded_linear_mult_left
thf(fact_7240_bounded__linear__const__mult,axiom,
! [A: $tType,C: $tType] :
( ( ( real_V822414075346904944vector @ C )
& ( real_V4412858255891104859lgebra @ A ) )
=> ! [G: C > A,X: A] :
( ( real_V3181309239436604168linear @ C @ A @ G )
=> ( real_V3181309239436604168linear @ C @ A
@ ^ [X2: C] : ( times_times @ A @ X @ ( G @ X2 ) ) ) ) ) ).
% bounded_linear_const_mult
thf(fact_7241_bounded__linear__mult__const,axiom,
! [A: $tType,C: $tType] :
( ( ( real_V822414075346904944vector @ C )
& ( real_V4412858255891104859lgebra @ A ) )
=> ! [G: C > A,Y: A] :
( ( real_V3181309239436604168linear @ C @ A @ G )
=> ( real_V3181309239436604168linear @ C @ A
@ ^ [X2: C] : ( times_times @ A @ ( G @ X2 ) @ Y ) ) ) ) ).
% bounded_linear_mult_const
thf(fact_7242_bounded__linear__mult__right,axiom,
! [A: $tType] :
( ( real_V4412858255891104859lgebra @ A )
=> ! [X: A] : ( real_V3181309239436604168linear @ A @ A @ ( times_times @ A @ X ) ) ) ).
% bounded_linear_mult_right
thf(fact_7243_real__bounded__linear,axiom,
( ( real_V3181309239436604168linear @ real @ real )
= ( ^ [F7: real > real] :
? [C3: real] :
( F7
= ( ^ [X2: real] : ( times_times @ real @ X2 @ C3 ) ) ) ) ) ).
% real_bounded_linear
thf(fact_7244_has__derivative__bounded__linear,axiom,
! [B: $tType,A: $tType] :
( ( ( real_V822414075346904944vector @ A )
& ( real_V822414075346904944vector @ B ) )
=> ! [F2: A > B,F5: A > B,F4: filter @ A] :
( ( has_derivative @ A @ B @ F2 @ F5 @ F4 )
=> ( real_V3181309239436604168linear @ A @ B @ F5 ) ) ) ).
% has_derivative_bounded_linear
thf(fact_7245_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
@ ^ [X2: C] : ( F2 @ ( G @ X2 ) )
@ ^ [X2: C] : ( F2 @ ( G5 @ X2 ) )
@ F4 ) ) ) ) ).
% bounded_linear.has_derivative
thf(fact_7246_bounded__linear_Obounded,axiom,
! [B: $tType,A: $tType] :
( ( ( real_V822414075346904944vector @ A )
& ( real_V822414075346904944vector @ B ) )
=> ! [F2: A > B] :
( ( real_V3181309239436604168linear @ A @ B @ F2 )
=> ? [K9: real] :
! [X5: A] : ( ord_less_eq @ real @ ( real_V7770717601297561774m_norm @ B @ ( F2 @ X5 ) ) @ ( times_times @ real @ ( real_V7770717601297561774m_norm @ A @ X5 ) @ K9 ) ) ) ) ).
% bounded_linear.bounded
thf(fact_7247_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 )
& ! [X5: A] : ( ord_less_eq @ real @ ( real_V7770717601297561774m_norm @ B @ ( F2 @ X5 ) ) @ ( times_times @ real @ ( real_V7770717601297561774m_norm @ A @ X5 ) @ K9 ) ) ) ) ) ).
% bounded_linear.nonneg_bounded
thf(fact_7248_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_7249_has__derivative__within__singleton__iff,axiom,
! [B: $tType,A: $tType] :
( ( ( real_V822414075346904944vector @ A )
& ( real_V822414075346904944vector @ B ) )
=> ! [F2: A > B,G: A > B,X: A] :
( ( has_derivative @ A @ B @ F2 @ G @ ( topolo174197925503356063within @ A @ X @ ( insert @ A @ X @ ( bot_bot @ ( set @ A ) ) ) ) )
= ( real_V3181309239436604168linear @ A @ B @ G ) ) ) ).
% has_derivative_within_singleton_iff
thf(fact_7250_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 )
& ! [X5: A] : ( ord_less_eq @ real @ ( real_V7770717601297561774m_norm @ B @ ( F2 @ X5 ) ) @ ( times_times @ real @ ( real_V7770717601297561774m_norm @ A @ X5 ) @ K9 ) ) ) ) ) ).
% bounded_linear.pos_bounded
thf(fact_7251_bounded__linear__intro,axiom,
! [B: $tType,A: $tType] :
( ( ( real_V822414075346904944vector @ A )
& ( real_V822414075346904944vector @ B ) )
=> ! [F2: A > B,K6: real] :
( ! [X3: A,Y3: A] :
( ( F2 @ ( plus_plus @ A @ X3 @ Y3 ) )
= ( plus_plus @ B @ ( F2 @ X3 ) @ ( F2 @ Y3 ) ) )
=> ( ! [R3: real,X3: A] :
( ( F2 @ ( real_V8093663219630862766scaleR @ A @ R3 @ X3 ) )
= ( real_V8093663219630862766scaleR @ B @ R3 @ ( F2 @ X3 ) ) )
=> ( ! [X3: A] : ( ord_less_eq @ real @ ( real_V7770717601297561774m_norm @ B @ ( F2 @ X3 ) ) @ ( times_times @ real @ ( real_V7770717601297561774m_norm @ A @ X3 ) @ K6 ) )
=> ( real_V3181309239436604168linear @ A @ B @ F2 ) ) ) ) ) ).
% bounded_linear_intro
thf(fact_7252_filterlim__Suc,axiom,
filterlim @ nat @ nat @ suc @ ( at_top @ nat ) @ ( at_top @ nat ) ).
% filterlim_Suc
thf(fact_7253_filterlim__mono,axiom,
! [B: $tType,A: $tType,F2: A > B,F24: filter @ B,F13: filter @ A,F25: filter @ B,F14: filter @ A] :
( ( filterlim @ A @ B @ F2 @ F24 @ F13 )
=> ( ( ord_less_eq @ ( filter @ B ) @ F24 @ F25 )
=> ( ( ord_less_eq @ ( filter @ A ) @ F14 @ F13 )
=> ( filterlim @ A @ B @ F2 @ F25 @ F14 ) ) ) ) ).
% filterlim_mono
thf(fact_7254_has__derivative__iff__norm,axiom,
! [B: $tType,A: $tType] :
( ( ( real_V822414075346904944vector @ A )
& ( real_V822414075346904944vector @ B ) )
=> ! [F2: A > B,F5: A > B,X: A,S: set @ A] :
( ( has_derivative @ A @ B @ F2 @ F5 @ ( topolo174197925503356063within @ A @ X @ S ) )
= ( ( real_V3181309239436604168linear @ A @ B @ F5 )
& ( filterlim @ A @ real
@ ^ [Y2: A] : ( divide_divide @ real @ ( real_V7770717601297561774m_norm @ B @ ( minus_minus @ B @ ( minus_minus @ B @ ( F2 @ Y2 ) @ ( F2 @ X ) ) @ ( F5 @ ( minus_minus @ A @ Y2 @ X ) ) ) ) @ ( real_V7770717601297561774m_norm @ A @ ( minus_minus @ A @ Y2 @ X ) ) )
@ ( topolo7230453075368039082e_nhds @ real @ ( zero_zero @ real ) )
@ ( topolo174197925503356063within @ A @ X @ S ) ) ) ) ) ).
% has_derivative_iff_norm
thf(fact_7255_has__derivativeI,axiom,
! [B: $tType,A: $tType] :
( ( ( real_V822414075346904944vector @ A )
& ( real_V822414075346904944vector @ B ) )
=> ! [F5: A > B,X: A,F2: A > B,S: set @ A] :
( ( real_V3181309239436604168linear @ A @ B @ F5 )
=> ( ( filterlim @ A @ B
@ ^ [Y2: A] : ( real_V8093663219630862766scaleR @ B @ ( inverse_inverse @ real @ ( real_V7770717601297561774m_norm @ A @ ( minus_minus @ A @ Y2 @ X ) ) ) @ ( minus_minus @ B @ ( minus_minus @ B @ ( F2 @ Y2 ) @ ( F2 @ X ) ) @ ( F5 @ ( minus_minus @ A @ Y2 @ X ) ) ) )
@ ( topolo7230453075368039082e_nhds @ B @ ( zero_zero @ B ) )
@ ( topolo174197925503356063within @ A @ X @ S ) )
=> ( has_derivative @ A @ B @ F2 @ F5 @ ( topolo174197925503356063within @ A @ X @ S ) ) ) ) ) ).
% has_derivativeI
thf(fact_7256_has__derivative__at__within,axiom,
! [B: $tType,A: $tType] :
( ( ( real_V822414075346904944vector @ A )
& ( real_V822414075346904944vector @ B ) )
=> ! [F2: A > B,F5: A > B,X: A,S: set @ A] :
( ( has_derivative @ A @ B @ F2 @ F5 @ ( topolo174197925503356063within @ A @ X @ S ) )
= ( ( real_V3181309239436604168linear @ A @ B @ F5 )
& ( filterlim @ A @ B
@ ^ [Y2: A] : ( real_V8093663219630862766scaleR @ B @ ( inverse_inverse @ real @ ( real_V7770717601297561774m_norm @ A @ ( minus_minus @ A @ Y2 @ X ) ) ) @ ( minus_minus @ B @ ( minus_minus @ B @ ( F2 @ Y2 ) @ ( F2 @ X ) ) @ ( F5 @ ( minus_minus @ A @ Y2 @ X ) ) ) )
@ ( topolo7230453075368039082e_nhds @ B @ ( zero_zero @ B ) )
@ ( topolo174197925503356063within @ A @ X @ S ) ) ) ) ) ).
% has_derivative_at_within
thf(fact_7257_has__derivative__iff__Ex,axiom,
! [B: $tType,A: $tType] :
( ( ( real_V822414075346904944vector @ A )
& ( real_V822414075346904944vector @ B ) )
=> ! [F2: A > B,F5: A > B,X: A] :
( ( has_derivative @ A @ B @ F2 @ F5 @ ( topolo174197925503356063within @ A @ X @ ( top_top @ ( set @ A ) ) ) )
= ( ( real_V3181309239436604168linear @ A @ B @ F5 )
& ? [E4: A > B] :
( ! [H: A] :
( ( F2 @ ( plus_plus @ A @ X @ H ) )
= ( plus_plus @ B @ ( plus_plus @ B @ ( F2 @ X ) @ ( F5 @ 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_7258_has__derivative__within,axiom,
! [B: $tType,A: $tType] :
( ( ( real_V822414075346904944vector @ A )
& ( real_V822414075346904944vector @ B ) )
=> ! [F2: A > B,F5: A > B,X: A,S: set @ A] :
( ( has_derivative @ A @ B @ F2 @ F5 @ ( topolo174197925503356063within @ A @ X @ S ) )
= ( ( real_V3181309239436604168linear @ A @ B @ F5 )
& ( filterlim @ A @ B
@ ^ [Y2: A] : ( real_V8093663219630862766scaleR @ B @ ( divide_divide @ real @ ( one_one @ real ) @ ( real_V7770717601297561774m_norm @ A @ ( minus_minus @ A @ Y2 @ X ) ) ) @ ( minus_minus @ B @ ( F2 @ Y2 ) @ ( plus_plus @ B @ ( F2 @ X ) @ ( F5 @ ( minus_minus @ A @ Y2 @ X ) ) ) ) )
@ ( topolo7230453075368039082e_nhds @ B @ ( zero_zero @ B ) )
@ ( topolo174197925503356063within @ A @ X @ S ) ) ) ) ) ).
% has_derivative_within
thf(fact_7259_has__derivative__at,axiom,
! [B: $tType,A: $tType] :
( ( ( real_V822414075346904944vector @ A )
& ( real_V822414075346904944vector @ B ) )
=> ! [F2: A > B,D4: A > B,X: A] :
( ( has_derivative @ A @ B @ F2 @ D4 @ ( topolo174197925503356063within @ A @ X @ ( top_top @ ( set @ A ) ) ) )
= ( ( real_V3181309239436604168linear @ A @ B @ D4 )
& ( filterlim @ A @ real
@ ^ [H: A] : ( divide_divide @ real @ ( real_V7770717601297561774m_norm @ B @ ( minus_minus @ B @ ( minus_minus @ B @ ( F2 @ ( plus_plus @ A @ X @ H ) ) @ ( F2 @ X ) ) @ ( D4 @ 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_7260_has__derivative__def,axiom,
! [B: $tType,A: $tType] :
( ( ( real_V822414075346904944vector @ A )
& ( real_V822414075346904944vector @ B ) )
=> ( ( has_derivative @ A @ B )
= ( ^ [F7: A > B,F11: A > B,F9: filter @ A] :
( ( real_V3181309239436604168linear @ A @ B @ F11 )
& ( filterlim @ A @ B
@ ^ [Y2: A] :
( real_V8093663219630862766scaleR @ B
@ ( inverse_inverse @ real
@ ( real_V7770717601297561774m_norm @ A
@ ( minus_minus @ A @ Y2
@ ( topolo3827282254853284352ce_Lim @ A @ A @ F9
@ ^ [X2: A] : X2 ) ) ) )
@ ( minus_minus @ B
@ ( minus_minus @ B @ ( F7 @ Y2 )
@ ( F7
@ ( topolo3827282254853284352ce_Lim @ A @ A @ F9
@ ^ [X2: A] : X2 ) ) )
@ ( F11
@ ( minus_minus @ A @ Y2
@ ( topolo3827282254853284352ce_Lim @ A @ A @ F9
@ ^ [X2: A] : X2 ) ) ) ) )
@ ( topolo7230453075368039082e_nhds @ B @ ( zero_zero @ B ) )
@ F9 ) ) ) ) ) ).
% has_derivative_def
thf(fact_7261_has__derivative__at__within__iff__Ex,axiom,
! [B: $tType,A: $tType] :
( ( ( real_V822414075346904944vector @ A )
& ( real_V822414075346904944vector @ B ) )
=> ! [X: A,S3: set @ A,F2: A > B,F5: A > B] :
( ( member @ A @ X @ S3 )
=> ( ( topolo1002775350975398744n_open @ A @ S3 )
=> ( ( has_derivative @ A @ B @ F2 @ F5 @ ( topolo174197925503356063within @ A @ X @ S3 ) )
= ( ( real_V3181309239436604168linear @ A @ B @ F5 )
& ? [E4: A > B] :
( ! [H: A] :
( ( member @ A @ ( plus_plus @ A @ X @ H ) @ S3 )
=> ( ( F2 @ ( plus_plus @ A @ X @ H ) )
= ( plus_plus @ B @ ( plus_plus @ B @ ( F2 @ X ) @ ( F5 @ 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_7262_has__derivative__transform__within__open,axiom,
! [B: $tType,A: $tType] :
( ( ( real_V822414075346904944vector @ A )
& ( real_V822414075346904944vector @ B ) )
=> ! [F2: A > B,F5: A > B,X: A,T2: set @ A,S: set @ A,G: A > B] :
( ( has_derivative @ A @ B @ F2 @ F5 @ ( topolo174197925503356063within @ A @ X @ T2 ) )
=> ( ( topolo1002775350975398744n_open @ A @ S )
=> ( ( member @ A @ X @ S )
=> ( ! [X3: A] :
( ( member @ A @ X3 @ S )
=> ( ( F2 @ X3 )
= ( G @ X3 ) ) )
=> ( has_derivative @ A @ B @ G @ F5 @ ( topolo174197925503356063within @ A @ X @ T2 ) ) ) ) ) ) ) ).
% has_derivative_transform_within_open
thf(fact_7263_at__within__open__subset,axiom,
! [A: $tType] :
( ( topolo4958980785337419405_space @ A )
=> ! [A2: A,S3: set @ A,T5: set @ A] :
( ( member @ A @ A2 @ S3 )
=> ( ( topolo1002775350975398744n_open @ A @ S3 )
=> ( ( ord_less_eq @ ( set @ A ) @ S3 @ T5 )
=> ( ( topolo174197925503356063within @ A @ A2 @ T5 )
= ( topolo174197925503356063within @ A @ A2 @ ( top_top @ ( set @ A ) ) ) ) ) ) ) ) ).
% at_within_open_subset
thf(fact_7264_open__right,axiom,
! [A: $tType] :
( ( topolo1944317154257567458pology @ A )
=> ! [S3: set @ A,X: A,Y: A] :
( ( topolo1002775350975398744n_open @ A @ S3 )
=> ( ( member @ A @ X @ S3 )
=> ( ( ord_less @ A @ X @ Y )
=> ? [B3: A] :
( ( ord_less @ A @ X @ B3 )
& ( ord_less_eq @ ( set @ A ) @ ( set_or7035219750837199246ssThan @ A @ X @ B3 ) @ S3 ) ) ) ) ) ) ).
% open_right
thf(fact_7265_first__countable__basis,axiom,
! [A: $tType] :
( ( topolo3112930676232923870pology @ A )
=> ! [X: A] :
? [A8: nat > ( set @ A )] :
( ! [I4: nat] :
( ( member @ A @ X @ ( A8 @ I4 ) )
& ( topolo1002775350975398744n_open @ A @ ( A8 @ I4 ) ) )
& ! [S10: set @ A] :
( ( ( topolo1002775350975398744n_open @ A @ S10 )
& ( member @ A @ X @ S10 ) )
=> ? [I2: nat] : ( ord_less_eq @ ( set @ A ) @ ( A8 @ I2 ) @ S10 ) ) ) ) ).
% first_countable_basis
thf(fact_7266_open__subopen,axiom,
! [A: $tType] :
( ( topolo4958980785337419405_space @ A )
=> ( ( topolo1002775350975398744n_open @ A )
= ( ^ [S8: set @ A] :
! [X2: A] :
( ( member @ A @ X2 @ S8 )
=> ? [T9: set @ A] :
( ( topolo1002775350975398744n_open @ A @ T9 )
& ( member @ A @ X2 @ T9 )
& ( ord_less_eq @ ( set @ A ) @ T9 @ S8 ) ) ) ) ) ) ).
% open_subopen
thf(fact_7267_openI,axiom,
! [A: $tType] :
( ( topolo4958980785337419405_space @ A )
=> ! [S3: set @ A] :
( ! [X3: A] :
( ( member @ A @ X3 @ S3 )
=> ? [T11: set @ A] :
( ( topolo1002775350975398744n_open @ A @ T11 )
& ( member @ A @ X3 @ T11 )
& ( ord_less_eq @ ( set @ A ) @ T11 @ S3 ) ) )
=> ( topolo1002775350975398744n_open @ A @ S3 ) ) ) ).
% openI
thf(fact_7268_has__field__derivative__transform__within__open,axiom,
! [A: $tType] :
( ( real_V3459762299906320749_field @ A )
=> ! [F2: A > A,F5: A,A2: A,S3: set @ A,G: A > A] :
( ( has_field_derivative @ A @ F2 @ F5 @ ( topolo174197925503356063within @ A @ A2 @ ( top_top @ ( set @ A ) ) ) )
=> ( ( topolo1002775350975398744n_open @ A @ S3 )
=> ( ( member @ A @ A2 @ S3 )
=> ( ! [X3: A] :
( ( member @ A @ X3 @ S3 )
=> ( ( F2 @ X3 )
= ( G @ X3 ) ) )
=> ( has_field_derivative @ A @ G @ F5 @ ( topolo174197925503356063within @ A @ A2 @ ( top_top @ ( set @ A ) ) ) ) ) ) ) ) ) ).
% has_field_derivative_transform_within_open
thf(fact_7269_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] :
! [N2: nat] :
( ( ord_less_eq @ nat @ N8 @ N2 )
=> ( member @ A @ ( F2 @ N2 ) @ S8 ) ) ) ) ) ) ) ).
% lim_explicit
thf(fact_7270_continuous__divide,axiom,
! [B: $tType,A: $tType] :
( ( ( topological_t2_space @ A )
& ( real_V3459762299906320749_field @ B ) )
=> ! [F4: filter @ A,F2: A > B,G: A > B] :
( ( topolo3448309680560233919inuous @ A @ B @ F4 @ F2 )
=> ( ( topolo3448309680560233919inuous @ A @ B @ F4 @ G )
=> ( ( ( G
@ ( topolo3827282254853284352ce_Lim @ A @ A @ F4
@ ^ [X2: A] : X2 ) )
!= ( zero_zero @ B ) )
=> ( topolo3448309680560233919inuous @ A @ B @ F4
@ ^ [X2: A] : ( divide_divide @ B @ ( F2 @ X2 ) @ ( G @ X2 ) ) ) ) ) ) ) ).
% continuous_divide
thf(fact_7271_continuous__arcosh,axiom,
! [A: $tType] :
( ( topological_t2_space @ A )
=> ! [F4: filter @ A,F2: A > real] :
( ( topolo3448309680560233919inuous @ A @ real @ F4 @ F2 )
=> ( ( ord_less @ real @ ( one_one @ real )
@ ( F2
@ ( topolo3827282254853284352ce_Lim @ A @ A @ F4
@ ^ [X2: A] : X2 ) ) )
=> ( topolo3448309680560233919inuous @ A @ real @ F4
@ ^ [X2: A] : ( arcosh @ real @ ( F2 @ X2 ) ) ) ) ) ) ).
% continuous_arcosh
thf(fact_7272_continuous__log,axiom,
! [A: $tType] :
( ( topological_t2_space @ A )
=> ! [F4: filter @ A,F2: A > real,G: A > real] :
( ( topolo3448309680560233919inuous @ A @ real @ F4 @ F2 )
=> ( ( topolo3448309680560233919inuous @ A @ real @ F4 @ G )
=> ( ( ord_less @ real @ ( zero_zero @ real )
@ ( F2
@ ( topolo3827282254853284352ce_Lim @ A @ A @ F4
@ ^ [X2: A] : X2 ) ) )
=> ( ( ( F2
@ ( topolo3827282254853284352ce_Lim @ A @ A @ F4
@ ^ [X2: A] : X2 ) )
!= ( one_one @ real ) )
=> ( ( ord_less @ real @ ( zero_zero @ real )
@ ( G
@ ( topolo3827282254853284352ce_Lim @ A @ A @ F4
@ ^ [X2: A] : X2 ) ) )
=> ( topolo3448309680560233919inuous @ A @ real @ F4
@ ^ [X2: A] : ( log @ ( F2 @ X2 ) @ ( G @ X2 ) ) ) ) ) ) ) ) ) ).
% continuous_log
thf(fact_7273_continuous__artanh,axiom,
! [A: $tType] :
( ( topological_t2_space @ A )
=> ! [F4: filter @ A,F2: A > real] :
( ( topolo3448309680560233919inuous @ A @ real @ F4 @ F2 )
=> ( ( member @ real
@ ( F2
@ ( topolo3827282254853284352ce_Lim @ A @ A @ F4
@ ^ [X2: A] : X2 ) )
@ ( set_or5935395276787703475ssThan @ real @ ( uminus_uminus @ real @ ( one_one @ real ) ) @ ( one_one @ real ) ) )
=> ( topolo3448309680560233919inuous @ A @ real @ F4
@ ^ [X2: A] : ( artanh @ real @ ( F2 @ X2 ) ) ) ) ) ) ).
% continuous_artanh
thf(fact_7274_tendsto__offset__zero__iff,axiom,
! [C: $tType,D5: $tType,A: $tType] :
( ( ( real_V822414075346904944vector @ A )
& ( topolo4958980785337419405_space @ D5 )
& ( zero @ C ) )
=> ! [A2: A,S3: set @ A,F2: A > D5,L6: D5] :
( ( nO_MATCH @ C @ A @ ( zero_zero @ C ) @ A2 )
=> ( ( member @ A @ A2 @ S3 )
=> ( ( topolo1002775350975398744n_open @ A @ S3 )
=> ( ( filterlim @ A @ D5 @ F2 @ ( topolo7230453075368039082e_nhds @ D5 @ L6 ) @ ( topolo174197925503356063within @ A @ A2 @ S3 ) )
= ( filterlim @ A @ D5
@ ^ [H: A] : ( F2 @ ( plus_plus @ A @ A2 @ H ) )
@ ( topolo7230453075368039082e_nhds @ D5 @ L6 )
@ ( topolo174197925503356063within @ A @ ( zero_zero @ A ) @ ( top_top @ ( set @ A ) ) ) ) ) ) ) ) ) ).
% tendsto_offset_zero_iff
thf(fact_7275_has__derivativeI__sandwich,axiom,
! [B: $tType,A: $tType] :
( ( ( real_V822414075346904944vector @ A )
& ( real_V822414075346904944vector @ B ) )
=> ! [E: real,F5: A > B,S: set @ A,X: A,F2: A > B,H7: A > real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ E )
=> ( ( real_V3181309239436604168linear @ A @ B @ F5 )
=> ( ! [Y3: A] :
( ( member @ A @ Y3 @ S )
=> ( ( Y3 != X )
=> ( ( ord_less @ real @ ( real_V557655796197034286t_dist @ A @ Y3 @ X ) @ E )
=> ( ord_less_eq @ real @ ( divide_divide @ real @ ( real_V7770717601297561774m_norm @ B @ ( minus_minus @ B @ ( minus_minus @ B @ ( F2 @ Y3 ) @ ( F2 @ X ) ) @ ( F5 @ ( minus_minus @ A @ Y3 @ X ) ) ) ) @ ( real_V7770717601297561774m_norm @ A @ ( minus_minus @ A @ Y3 @ X ) ) ) @ ( H7 @ Y3 ) ) ) ) )
=> ( ( filterlim @ A @ real @ H7 @ ( topolo7230453075368039082e_nhds @ real @ ( zero_zero @ real ) ) @ ( topolo174197925503356063within @ A @ X @ S ) )
=> ( has_derivative @ A @ B @ F2 @ F5 @ ( topolo174197925503356063within @ A @ X @ S ) ) ) ) ) ) ) ).
% has_derivativeI_sandwich
thf(fact_7276_dist__add__cancel,axiom,
! [A: $tType] :
( ( real_V822414075346904944vector @ A )
=> ! [A2: A,B2: A,C2: A] :
( ( real_V557655796197034286t_dist @ A @ ( plus_plus @ A @ A2 @ B2 ) @ ( plus_plus @ A @ A2 @ C2 ) )
= ( real_V557655796197034286t_dist @ A @ B2 @ C2 ) ) ) ).
% dist_add_cancel
thf(fact_7277_dist__add__cancel2,axiom,
! [A: $tType] :
( ( real_V822414075346904944vector @ A )
=> ! [B2: A,A2: A,C2: A] :
( ( real_V557655796197034286t_dist @ A @ ( plus_plus @ A @ B2 @ A2 ) @ ( plus_plus @ A @ C2 @ A2 ) )
= ( real_V557655796197034286t_dist @ A @ B2 @ C2 ) ) ) ).
% dist_add_cancel2
thf(fact_7278_dist__le__zero__iff,axiom,
! [A: $tType] :
( ( real_V7819770556892013058_space @ A )
=> ! [X: A,Y: A] :
( ( ord_less_eq @ real @ ( real_V557655796197034286t_dist @ A @ X @ Y ) @ ( zero_zero @ real ) )
= ( X = Y ) ) ) ).
% dist_le_zero_iff
thf(fact_7279_dist__diff_I2_J,axiom,
! [A: $tType] :
( ( real_V822414075346904944vector @ A )
=> ! [A2: A,B2: A] :
( ( real_V557655796197034286t_dist @ A @ ( minus_minus @ A @ A2 @ B2 ) @ A2 )
= ( real_V7770717601297561774m_norm @ A @ B2 ) ) ) ).
% dist_diff(2)
thf(fact_7280_dist__diff_I1_J,axiom,
! [A: $tType] :
( ( real_V822414075346904944vector @ A )
=> ! [A2: A,B2: A] :
( ( real_V557655796197034286t_dist @ A @ A2 @ ( minus_minus @ A @ A2 @ B2 ) )
= ( real_V7770717601297561774m_norm @ A @ B2 ) ) ) ).
% dist_diff(1)
thf(fact_7281_dist__scaleR,axiom,
! [A: $tType] :
( ( real_V822414075346904944vector @ A )
=> ! [X: real,A2: A,Y: real] :
( ( real_V557655796197034286t_dist @ A @ ( real_V8093663219630862766scaleR @ A @ X @ A2 ) @ ( real_V8093663219630862766scaleR @ A @ Y @ A2 ) )
= ( times_times @ real @ ( abs_abs @ real @ ( minus_minus @ real @ X @ Y ) ) @ ( real_V7770717601297561774m_norm @ A @ A2 ) ) ) ) ).
% dist_scaleR
thf(fact_7282_dist__triangle__less__add,axiom,
! [A: $tType] :
( ( real_V7819770556892013058_space @ A )
=> ! [X15: A,Y: A,E1: real,X22: A,E22: real] :
( ( ord_less @ real @ ( real_V557655796197034286t_dist @ A @ X15 @ Y ) @ E1 )
=> ( ( ord_less @ real @ ( real_V557655796197034286t_dist @ A @ X22 @ Y ) @ E22 )
=> ( ord_less @ real @ ( real_V557655796197034286t_dist @ A @ X15 @ X22 ) @ ( plus_plus @ real @ E1 @ E22 ) ) ) ) ) ).
% dist_triangle_less_add
thf(fact_7283_dist__triangle__lt,axiom,
! [A: $tType] :
( ( real_V7819770556892013058_space @ A )
=> ! [X: A,Z: A,Y: A,E: real] :
( ( ord_less @ real @ ( plus_plus @ real @ ( real_V557655796197034286t_dist @ A @ X @ Z ) @ ( real_V557655796197034286t_dist @ A @ Y @ Z ) ) @ E )
=> ( ord_less @ real @ ( real_V557655796197034286t_dist @ A @ X @ Y ) @ E ) ) ) ).
% dist_triangle_lt
thf(fact_7284_dist__triangle__le,axiom,
! [A: $tType] :
( ( real_V7819770556892013058_space @ A )
=> ! [X: A,Z: A,Y: A,E: real] :
( ( ord_less_eq @ real @ ( plus_plus @ real @ ( real_V557655796197034286t_dist @ A @ X @ Z ) @ ( real_V557655796197034286t_dist @ A @ Y @ Z ) ) @ E )
=> ( ord_less_eq @ real @ ( real_V557655796197034286t_dist @ A @ X @ Y ) @ E ) ) ) ).
% dist_triangle_le
thf(fact_7285_dist__triangle3,axiom,
! [A: $tType] :
( ( real_V7819770556892013058_space @ A )
=> ! [X: A,Y: A,A2: A] : ( ord_less_eq @ real @ ( real_V557655796197034286t_dist @ A @ X @ Y ) @ ( plus_plus @ real @ ( real_V557655796197034286t_dist @ A @ A2 @ X ) @ ( real_V557655796197034286t_dist @ A @ A2 @ Y ) ) ) ) ).
% dist_triangle3
thf(fact_7286_dist__triangle2,axiom,
! [A: $tType] :
( ( real_V7819770556892013058_space @ A )
=> ! [X: A,Y: A,Z: A] : ( ord_less_eq @ real @ ( real_V557655796197034286t_dist @ A @ X @ Y ) @ ( plus_plus @ real @ ( real_V557655796197034286t_dist @ A @ X @ Z ) @ ( real_V557655796197034286t_dist @ A @ Y @ Z ) ) ) ) ).
% dist_triangle2
thf(fact_7287_dist__triangle,axiom,
! [A: $tType] :
( ( real_V7819770556892013058_space @ A )
=> ! [X: A,Z: A,Y: A] : ( ord_less_eq @ real @ ( real_V557655796197034286t_dist @ A @ X @ Z ) @ ( plus_plus @ real @ ( real_V557655796197034286t_dist @ A @ X @ Y ) @ ( real_V557655796197034286t_dist @ A @ Y @ Z ) ) ) ) ).
% dist_triangle
thf(fact_7288_zero__le__dist,axiom,
! [A: $tType] :
( ( real_V7819770556892013058_space @ A )
=> ! [X: A,Y: A] : ( ord_less_eq @ real @ ( zero_zero @ real ) @ ( real_V557655796197034286t_dist @ A @ X @ Y ) ) ) ).
% zero_le_dist
thf(fact_7289_dist__complex__def,axiom,
( ( real_V557655796197034286t_dist @ complex )
= ( ^ [X2: complex,Y2: complex] : ( real_V7770717601297561774m_norm @ complex @ ( minus_minus @ complex @ X2 @ Y2 ) ) ) ) ).
% dist_complex_def
thf(fact_7290_dist__real__def,axiom,
( ( real_V557655796197034286t_dist @ real )
= ( ^ [X2: real,Y2: real] : ( abs_abs @ real @ ( minus_minus @ real @ X2 @ Y2 ) ) ) ) ).
% dist_real_def
thf(fact_7291_dist__norm,axiom,
! [A: $tType] :
( ( real_V6936659425649961206t_norm @ A )
=> ( ( real_V557655796197034286t_dist @ A )
= ( ^ [X2: A,Y2: A] : ( real_V7770717601297561774m_norm @ A @ ( minus_minus @ A @ X2 @ Y2 ) ) ) ) ) ).
% dist_norm
thf(fact_7292_abs__dist__diff__le,axiom,
! [A: $tType] :
( ( real_V7819770556892013058_space @ A )
=> ! [A2: A,B2: A,C2: A] : ( ord_less_eq @ real @ ( abs_abs @ real @ ( minus_minus @ real @ ( real_V557655796197034286t_dist @ A @ A2 @ B2 ) @ ( real_V557655796197034286t_dist @ A @ B2 @ C2 ) ) ) @ ( real_V557655796197034286t_dist @ A @ A2 @ C2 ) ) ) ).
% abs_dist_diff_le
thf(fact_7293_has__field__derivative__transform__within,axiom,
! [A: $tType] :
( ( real_V3459762299906320749_field @ A )
=> ! [F2: A > A,F5: A,A2: A,S3: set @ A,D: real,G: A > A] :
( ( has_field_derivative @ A @ F2 @ F5 @ ( topolo174197925503356063within @ A @ A2 @ S3 ) )
=> ( ( ord_less @ real @ ( zero_zero @ real ) @ D )
=> ( ( member @ A @ A2 @ S3 )
=> ( ! [X3: A] :
( ( member @ A @ X3 @ S3 )
=> ( ( ord_less @ real @ ( real_V557655796197034286t_dist @ A @ X3 @ A2 ) @ D )
=> ( ( F2 @ X3 )
= ( G @ X3 ) ) ) )
=> ( has_field_derivative @ A @ G @ F5 @ ( topolo174197925503356063within @ A @ A2 @ S3 ) ) ) ) ) ) ) ).
% has_field_derivative_transform_within
thf(fact_7294_has__derivative__transform__within,axiom,
! [B: $tType,A: $tType] :
( ( ( real_V822414075346904944vector @ A )
& ( real_V822414075346904944vector @ B ) )
=> ! [F2: A > B,F5: A > B,X: A,S: set @ A,D: real,G: A > B] :
( ( has_derivative @ A @ B @ F2 @ F5 @ ( topolo174197925503356063within @ A @ X @ S ) )
=> ( ( ord_less @ real @ ( zero_zero @ real ) @ D )
=> ( ( member @ A @ X @ S )
=> ( ! [X10: A] :
( ( member @ A @ X10 @ S )
=> ( ( ord_less @ real @ ( real_V557655796197034286t_dist @ A @ X10 @ X ) @ D )
=> ( ( F2 @ X10 )
= ( G @ X10 ) ) ) )
=> ( has_derivative @ A @ B @ G @ F5 @ ( topolo174197925503356063within @ A @ X @ S ) ) ) ) ) ) ) ).
% has_derivative_transform_within
thf(fact_7295_Cauchy__def,axiom,
! [A: $tType] :
( ( real_V7819770556892013058_space @ A )
=> ( ( topolo3814608138187158403Cauchy @ A )
= ( ^ [X6: nat > A] :
! [E4: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ E4 )
=> ? [M12: nat] :
! [M5: nat] :
( ( ord_less_eq @ nat @ M12 @ M5 )
=> ! [N2: nat] :
( ( ord_less_eq @ nat @ M12 @ N2 )
=> ( ord_less @ real @ ( real_V557655796197034286t_dist @ A @ ( X6 @ M5 ) @ ( X6 @ N2 ) ) @ E4 ) ) ) ) ) ) ) ).
% Cauchy_def
thf(fact_7296_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] :
! [N2: nat] :
( ( ord_less_eq @ nat @ N8 @ N2 )
=> ( ord_less @ real @ ( real_V557655796197034286t_dist @ A @ ( S5 @ N2 ) @ ( S5 @ N8 ) ) @ E4 ) ) ) ) ) ) ).
% Cauchy_altdef2
thf(fact_7297_metric__CauchyD,axiom,
! [A: $tType] :
( ( real_V7819770556892013058_space @ A )
=> ! [X8: nat > A,E: real] :
( ( topolo3814608138187158403Cauchy @ A @ X8 )
=> ( ( ord_less @ real @ ( zero_zero @ real ) @ E )
=> ? [M10: nat] :
! [M2: nat] :
( ( ord_less_eq @ nat @ M10 @ M2 )
=> ! [N9: nat] :
( ( ord_less_eq @ nat @ M10 @ N9 )
=> ( ord_less @ real @ ( real_V557655796197034286t_dist @ A @ ( X8 @ M2 ) @ ( X8 @ N9 ) ) @ E ) ) ) ) ) ) ).
% metric_CauchyD
thf(fact_7298_metric__CauchyI,axiom,
! [A: $tType] :
( ( real_V7819770556892013058_space @ A )
=> ! [X8: nat > A] :
( ! [E2: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ E2 )
=> ? [M13: nat] :
! [M4: nat] :
( ( ord_less_eq @ nat @ M13 @ M4 )
=> ! [N: nat] :
( ( ord_less_eq @ nat @ M13 @ N )
=> ( ord_less @ real @ ( real_V557655796197034286t_dist @ A @ ( X8 @ M4 ) @ ( X8 @ N ) ) @ E2 ) ) ) )
=> ( topolo3814608138187158403Cauchy @ A @ X8 ) ) ) ).
% metric_CauchyI
thf(fact_7299_dist__of__int,axiom,
! [A: $tType] :
( ( real_V2822296259951069270ebra_1 @ A )
=> ! [M: int,N3: int] :
( ( real_V557655796197034286t_dist @ A @ ( ring_1_of_int @ A @ M ) @ ( ring_1_of_int @ A @ N3 ) )
= ( ring_1_of_int @ real @ ( abs_abs @ int @ ( minus_minus @ int @ M @ N3 ) ) ) ) ) ).
% dist_of_int
thf(fact_7300_scale__right__distrib__NO__MATCH,axiom,
! [A: $tType] :
( ( real_V4867850818363320053vector @ A )
=> ! [X: A,Y: A,A2: real] :
( ( nO_MATCH @ A @ real @ ( divide_divide @ A @ X @ Y ) @ A2 )
=> ( ( real_V8093663219630862766scaleR @ A @ A2 @ ( plus_plus @ A @ X @ Y ) )
= ( plus_plus @ A @ ( real_V8093663219630862766scaleR @ A @ A2 @ X ) @ ( real_V8093663219630862766scaleR @ A @ A2 @ Y ) ) ) ) ) ).
% scale_right_distrib_NO_MATCH
thf(fact_7301_scale__right__diff__distrib__NO__MATCH,axiom,
! [A: $tType] :
( ( real_V4867850818363320053vector @ A )
=> ! [X: A,Y: A,A2: real] :
( ( nO_MATCH @ A @ real @ ( divide_divide @ A @ X @ Y ) @ A2 )
=> ( ( real_V8093663219630862766scaleR @ A @ A2 @ ( minus_minus @ A @ X @ Y ) )
= ( minus_minus @ A @ ( real_V8093663219630862766scaleR @ A @ A2 @ X ) @ ( real_V8093663219630862766scaleR @ A @ A2 @ Y ) ) ) ) ) ).
% scale_right_diff_distrib_NO_MATCH
thf(fact_7302_distrib__left__NO__MATCH,axiom,
! [B: $tType,A: $tType] :
( ( semiring @ A )
=> ! [X: B,Y: B,A2: A,B2: A,C2: A] :
( ( nO_MATCH @ B @ A @ ( divide_divide @ B @ X @ Y ) @ A2 )
=> ( ( times_times @ A @ A2 @ ( plus_plus @ A @ B2 @ C2 ) )
= ( plus_plus @ A @ ( times_times @ A @ A2 @ B2 ) @ ( times_times @ A @ A2 @ C2 ) ) ) ) ) ).
% distrib_left_NO_MATCH
thf(fact_7303_distrib__right__NO__MATCH,axiom,
! [B: $tType,A: $tType] :
( ( semiring @ A )
=> ! [X: B,Y: B,C2: A,A2: A,B2: A] :
( ( nO_MATCH @ B @ A @ ( divide_divide @ B @ X @ Y ) @ C2 )
=> ( ( times_times @ A @ ( plus_plus @ A @ A2 @ B2 ) @ C2 )
= ( plus_plus @ A @ ( times_times @ A @ A2 @ C2 ) @ ( times_times @ A @ B2 @ C2 ) ) ) ) ) ).
% distrib_right_NO_MATCH
thf(fact_7304_left__diff__distrib__NO__MATCH,axiom,
! [B: $tType,A: $tType] :
( ( ring @ A )
=> ! [X: B,Y: B,C2: A,A2: A,B2: A] :
( ( nO_MATCH @ B @ A @ ( divide_divide @ B @ X @ Y ) @ C2 )
=> ( ( times_times @ A @ ( minus_minus @ A @ A2 @ B2 ) @ C2 )
= ( minus_minus @ A @ ( times_times @ A @ A2 @ C2 ) @ ( times_times @ A @ B2 @ C2 ) ) ) ) ) ).
% left_diff_distrib_NO_MATCH
thf(fact_7305_right__diff__distrib__NO__MATCH,axiom,
! [B: $tType,A: $tType] :
( ( ring @ A )
=> ! [X: B,Y: B,A2: A,B2: A,C2: A] :
( ( nO_MATCH @ B @ A @ ( divide_divide @ B @ X @ Y ) @ A2 )
=> ( ( times_times @ A @ A2 @ ( minus_minus @ A @ B2 @ C2 ) )
= ( minus_minus @ A @ ( times_times @ A @ A2 @ B2 ) @ ( times_times @ A @ A2 @ C2 ) ) ) ) ) ).
% right_diff_distrib_NO_MATCH
thf(fact_7306_dist__triangle__half__r,axiom,
! [A: $tType] :
( ( real_V7819770556892013058_space @ A )
=> ! [Y: A,X15: A,E: real,X22: A] :
( ( ord_less @ real @ ( real_V557655796197034286t_dist @ A @ Y @ X15 ) @ ( divide_divide @ real @ E @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) )
=> ( ( ord_less @ real @ ( real_V557655796197034286t_dist @ A @ Y @ X22 ) @ ( divide_divide @ real @ E @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) )
=> ( ord_less @ real @ ( real_V557655796197034286t_dist @ A @ X15 @ X22 ) @ E ) ) ) ) ).
% dist_triangle_half_r
thf(fact_7307_dist__triangle__half__l,axiom,
! [A: $tType] :
( ( real_V7819770556892013058_space @ A )
=> ! [X15: A,Y: A,E: real,X22: A] :
( ( ord_less @ real @ ( real_V557655796197034286t_dist @ A @ X15 @ Y ) @ ( divide_divide @ real @ E @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) )
=> ( ( ord_less @ real @ ( real_V557655796197034286t_dist @ A @ X22 @ Y ) @ ( divide_divide @ real @ E @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) )
=> ( ord_less @ real @ ( real_V557655796197034286t_dist @ A @ X15 @ X22 ) @ E ) ) ) ) ).
% dist_triangle_half_l
thf(fact_7308_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,L2: A,A2: C,G: C > B,M: B] :
( ( filterlim @ C @ A @ F2 @ ( topolo7230453075368039082e_nhds @ A @ L2 ) @ ( topolo174197925503356063within @ C @ A2 @ ( top_top @ ( set @ C ) ) ) )
=> ( ! [X3: C] :
( ( X3 != A2 )
=> ( ord_less_eq @ real @ ( real_V557655796197034286t_dist @ B @ ( G @ X3 ) @ M ) @ ( real_V557655796197034286t_dist @ A @ ( F2 @ X3 ) @ L2 ) ) )
=> ( filterlim @ C @ B @ G @ ( topolo7230453075368039082e_nhds @ B @ M ) @ ( topolo174197925503356063within @ C @ A2 @ ( top_top @ ( set @ C ) ) ) ) ) ) ) ).
% metric_LIM_imp_LIM
thf(fact_7309_dist__triangle__third,axiom,
! [A: $tType] :
( ( real_V7819770556892013058_space @ A )
=> ! [X15: A,X22: A,E: real,X33: A,X42: A] :
( ( ord_less @ real @ ( real_V557655796197034286t_dist @ A @ X15 @ 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 @ X42 ) @ ( divide_divide @ real @ E @ ( numeral_numeral @ real @ ( bit1 @ one2 ) ) ) )
=> ( ord_less @ real @ ( real_V557655796197034286t_dist @ A @ X15 @ X42 ) @ E ) ) ) ) ) ).
% dist_triangle_third
thf(fact_7310_filterlim__transform__within,axiom,
! [B: $tType,A: $tType] :
( ( real_V7819770556892013058_space @ A )
=> ! [G: A > B,G6: filter @ B,X: A,S3: set @ A,F4: filter @ B,D: real,F2: A > B] :
( ( filterlim @ A @ B @ G @ G6 @ ( topolo174197925503356063within @ A @ X @ S3 ) )
=> ( ( ord_less_eq @ ( filter @ B ) @ G6 @ F4 )
=> ( ( ord_less @ real @ ( zero_zero @ real ) @ D )
=> ( ! [X10: A] :
( ( member @ A @ X10 @ S3 )
=> ( ( ord_less @ real @ ( zero_zero @ real ) @ ( real_V557655796197034286t_dist @ A @ X10 @ X ) )
=> ( ( ord_less @ real @ ( real_V557655796197034286t_dist @ A @ X10 @ X ) @ D )
=> ( ( F2 @ X10 )
= ( G @ X10 ) ) ) ) )
=> ( filterlim @ A @ B @ F2 @ F4 @ ( topolo174197925503356063within @ A @ X @ S3 ) ) ) ) ) ) ) ).
% filterlim_transform_within
thf(fact_7311_Cauchy__altdef,axiom,
! [A: $tType] :
( ( real_V7819770556892013058_space @ A )
=> ( ( topolo3814608138187158403Cauchy @ A )
= ( ^ [F7: nat > A] :
! [E4: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ E4 )
=> ? [M12: nat] :
! [M5: nat] :
( ( ord_less_eq @ nat @ M12 @ M5 )
=> ! [N2: nat] :
( ( ord_less @ nat @ M5 @ N2 )
=> ( ord_less @ real @ ( real_V557655796197034286t_dist @ A @ ( F7 @ M5 ) @ ( F7 @ N2 ) ) @ E4 ) ) ) ) ) ) ) ).
% Cauchy_altdef
thf(fact_7312_CauchyI_H,axiom,
! [A: $tType] :
( ( real_V7819770556892013058_space @ A )
=> ! [X8: nat > A] :
( ! [E2: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ E2 )
=> ? [M13: nat] :
! [M4: nat] :
( ( ord_less_eq @ nat @ M13 @ M4 )
=> ! [N: nat] :
( ( ord_less @ nat @ M4 @ N )
=> ( ord_less @ real @ ( real_V557655796197034286t_dist @ A @ ( X8 @ M4 ) @ ( X8 @ N ) ) @ E2 ) ) ) )
=> ( topolo3814608138187158403Cauchy @ A @ X8 ) ) ) ).
% CauchyI'
thf(fact_7313_dist__of__nat,axiom,
! [A: $tType] :
( ( real_V2822296259951069270ebra_1 @ A )
=> ! [M: nat,N3: nat] :
( ( real_V557655796197034286t_dist @ A @ ( semiring_1_of_nat @ A @ M ) @ ( semiring_1_of_nat @ A @ N3 ) )
= ( ring_1_of_int @ real @ ( abs_abs @ int @ ( minus_minus @ int @ ( semiring_1_of_nat @ int @ M ) @ ( semiring_1_of_nat @ int @ N3 ) ) ) ) ) ) ).
% dist_of_nat
thf(fact_7314_lim__sequentially,axiom,
! [A: $tType] :
( ( real_V7819770556892013058_space @ A )
=> ! [X8: nat > A,L6: A] :
( ( filterlim @ nat @ A @ X8 @ ( topolo7230453075368039082e_nhds @ A @ L6 ) @ ( at_top @ nat ) )
= ( ! [R5: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ R5 )
=> ? [No: nat] :
! [N2: nat] :
( ( ord_less_eq @ nat @ No @ N2 )
=> ( ord_less @ real @ ( real_V557655796197034286t_dist @ A @ ( X8 @ N2 ) @ L6 ) @ R5 ) ) ) ) ) ) ).
% lim_sequentially
thf(fact_7315_metric__LIMSEQ__I,axiom,
! [A: $tType] :
( ( real_V7819770556892013058_space @ A )
=> ! [X8: nat > A,L6: A] :
( ! [R3: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ R3 )
=> ? [No2: nat] :
! [N: nat] :
( ( ord_less_eq @ nat @ No2 @ N )
=> ( ord_less @ real @ ( real_V557655796197034286t_dist @ A @ ( X8 @ N ) @ L6 ) @ R3 ) ) )
=> ( filterlim @ nat @ A @ X8 @ ( topolo7230453075368039082e_nhds @ A @ L6 ) @ ( at_top @ nat ) ) ) ) ).
% metric_LIMSEQ_I
thf(fact_7316_metric__LIMSEQ__D,axiom,
! [A: $tType] :
( ( real_V7819770556892013058_space @ A )
=> ! [X8: nat > A,L6: A,R2: real] :
( ( filterlim @ nat @ A @ X8 @ ( topolo7230453075368039082e_nhds @ A @ L6 ) @ ( at_top @ nat ) )
=> ( ( ord_less @ real @ ( zero_zero @ real ) @ R2 )
=> ? [No3: nat] :
! [N9: nat] :
( ( ord_less_eq @ nat @ No3 @ N9 )
=> ( ord_less @ real @ ( real_V557655796197034286t_dist @ A @ ( X8 @ N9 ) @ L6 ) @ R2 ) ) ) ) ) ).
% metric_LIMSEQ_D
thf(fact_7317_power__minus_H,axiom,
! [A: $tType] :
( ( ring_1 @ A )
=> ! [X: A,N3: nat] :
( ( nO_MATCH @ A @ A @ ( one_one @ A ) @ X )
=> ( ( power_power @ A @ ( uminus_uminus @ A @ X ) @ N3 )
= ( times_times @ A @ ( power_power @ A @ ( uminus_uminus @ A @ ( one_one @ A ) ) @ N3 ) @ ( power_power @ A @ X @ N3 ) ) ) ) ) ).
% power_minus'
thf(fact_7318_metric__Cauchy__iff2,axiom,
! [A: $tType] :
( ( real_V7819770556892013058_space @ A )
=> ( ( topolo3814608138187158403Cauchy @ A )
= ( ^ [X6: nat > A] :
! [J3: nat] :
? [M12: nat] :
! [M5: nat] :
( ( ord_less_eq @ nat @ M12 @ M5 )
=> ! [N2: nat] :
( ( ord_less_eq @ nat @ M12 @ N2 )
=> ( ord_less @ real @ ( real_V557655796197034286t_dist @ A @ ( X6 @ M5 ) @ ( X6 @ N2 ) ) @ ( inverse_inverse @ real @ ( semiring_1_of_nat @ real @ ( suc @ J3 ) ) ) ) ) ) ) ) ) ).
% metric_Cauchy_iff2
thf(fact_7319_scale__left__distrib__NO__MATCH,axiom,
! [C: $tType,A: $tType] :
( ( real_V4867850818363320053vector @ A )
=> ! [X: A,Y: A,C2: C,A2: real,B2: real] :
( ( nO_MATCH @ A @ C @ ( divide_divide @ A @ X @ Y ) @ C2 )
=> ( ( real_V8093663219630862766scaleR @ A @ ( plus_plus @ real @ A2 @ B2 ) @ X )
= ( plus_plus @ A @ ( real_V8093663219630862766scaleR @ A @ A2 @ X ) @ ( real_V8093663219630862766scaleR @ A @ B2 @ X ) ) ) ) ) ).
% scale_left_distrib_NO_MATCH
thf(fact_7320_scale__left__diff__distrib__NO__MATCH,axiom,
! [C: $tType,A: $tType] :
( ( real_V4867850818363320053vector @ A )
=> ! [X: A,Y: A,C2: C,A2: real,B2: real] :
( ( nO_MATCH @ A @ C @ ( divide_divide @ A @ X @ Y ) @ C2 )
=> ( ( real_V8093663219630862766scaleR @ A @ ( minus_minus @ real @ A2 @ B2 ) @ X )
= ( minus_minus @ A @ ( real_V8093663219630862766scaleR @ A @ A2 @ X ) @ ( real_V8093663219630862766scaleR @ A @ B2 @ X ) ) ) ) ) ).
% scale_left_diff_distrib_NO_MATCH
thf(fact_7321_LIMSEQ__iff__nz,axiom,
! [A: $tType] :
( ( real_V7819770556892013058_space @ A )
=> ! [X8: nat > A,L6: A] :
( ( filterlim @ nat @ A @ X8 @ ( topolo7230453075368039082e_nhds @ A @ L6 ) @ ( at_top @ nat ) )
= ( ! [R5: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ R5 )
=> ? [No: nat] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ No )
& ! [N2: nat] :
( ( ord_less_eq @ nat @ No @ N2 )
=> ( ord_less @ real @ ( real_V557655796197034286t_dist @ A @ ( X8 @ N2 ) @ L6 ) @ R5 ) ) ) ) ) ) ) ).
% LIMSEQ_iff_nz
thf(fact_7322_LIM__offset__zero__iff,axiom,
! [C: $tType,D5: $tType,A: $tType] :
( ( ( real_V822414075346904944vector @ A )
& ( topolo4958980785337419405_space @ D5 )
& ( zero @ C ) )
=> ! [A2: A,F2: A > D5,L6: D5] :
( ( nO_MATCH @ C @ A @ ( zero_zero @ C ) @ A2 )
=> ( ( filterlim @ A @ D5 @ F2 @ ( topolo7230453075368039082e_nhds @ D5 @ L6 ) @ ( topolo174197925503356063within @ A @ A2 @ ( top_top @ ( set @ A ) ) ) )
= ( filterlim @ A @ D5
@ ^ [H: A] : ( F2 @ ( plus_plus @ A @ A2 @ H ) )
@ ( topolo7230453075368039082e_nhds @ D5 @ L6 )
@ ( topolo174197925503356063within @ A @ ( zero_zero @ A ) @ ( top_top @ ( set @ A ) ) ) ) ) ) ) ).
% LIM_offset_zero_iff
thf(fact_7323_div__add__self1__no__field,axiom,
! [B: $tType,A: $tType] :
( ( ( euclid4440199948858584721cancel @ A )
& ( field @ B ) )
=> ! [X: B,B2: A,A2: A] :
( ( nO_MATCH @ B @ A @ X @ B2 )
=> ( ( B2
!= ( zero_zero @ A ) )
=> ( ( divide_divide @ A @ ( plus_plus @ A @ B2 @ A2 ) @ B2 )
= ( plus_plus @ A @ ( divide_divide @ A @ A2 @ B2 ) @ ( one_one @ A ) ) ) ) ) ) ).
% div_add_self1_no_field
thf(fact_7324_div__add__self2__no__field,axiom,
! [B: $tType,A: $tType] :
( ( ( euclid4440199948858584721cancel @ A )
& ( field @ B ) )
=> ! [X: B,B2: A,A2: A] :
( ( nO_MATCH @ B @ A @ X @ B2 )
=> ( ( B2
!= ( zero_zero @ A ) )
=> ( ( divide_divide @ A @ ( plus_plus @ A @ A2 @ B2 ) @ B2 )
= ( plus_plus @ A @ ( divide_divide @ A @ A2 @ B2 ) @ ( one_one @ A ) ) ) ) ) ) ).
% div_add_self2_no_field
thf(fact_7325_tendsto__exp__limit__at__right,axiom,
! [X: real] :
( filterlim @ real @ real
@ ^ [Y2: real] : ( powr @ real @ ( plus_plus @ real @ ( one_one @ real ) @ ( times_times @ real @ X @ Y2 ) ) @ ( divide_divide @ real @ ( one_one @ real ) @ Y2 ) )
@ ( topolo7230453075368039082e_nhds @ real @ ( exp @ real @ X ) )
@ ( topolo174197925503356063within @ real @ ( zero_zero @ real ) @ ( set_ord_greaterThan @ real @ ( zero_zero @ real ) ) ) ) ).
% tendsto_exp_limit_at_right
thf(fact_7326_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_7327_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_7328_greaterThan__subset__iff,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [X: A,Y: A] :
( ( ord_less_eq @ ( set @ A ) @ ( set_ord_greaterThan @ A @ X ) @ ( set_ord_greaterThan @ A @ Y ) )
= ( ord_less_eq @ A @ Y @ X ) ) ) ).
% greaterThan_subset_iff
thf(fact_7329_at__within__Icc__at__right,axiom,
! [A: $tType] :
( ( topolo2564578578187576103pology @ A )
=> ! [A2: A,B2: A] :
( ( ord_less @ A @ A2 @ B2 )
=> ( ( topolo174197925503356063within @ A @ A2 @ ( set_or1337092689740270186AtMost @ A @ A2 @ B2 ) )
= ( topolo174197925503356063within @ A @ A2 @ ( set_ord_greaterThan @ A @ A2 ) ) ) ) ) ).
% at_within_Icc_at_right
thf(fact_7330_greaterThan__def,axiom,
! [A: $tType] :
( ( ord @ A )
=> ( ( set_ord_greaterThan @ A )
= ( ^ [L: A] : ( collect @ A @ ( ord_less @ A @ L ) ) ) ) ) ).
% greaterThan_def
thf(fact_7331_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_7332_filterlim__at__right__to__0,axiom,
! [A: $tType,F2: real > A,F4: filter @ A,A2: real] :
( ( filterlim @ real @ A @ F2 @ F4 @ ( topolo174197925503356063within @ real @ A2 @ ( set_ord_greaterThan @ real @ A2 ) ) )
= ( filterlim @ real @ A
@ ^ [X2: real] : ( F2 @ ( plus_plus @ real @ X2 @ A2 ) )
@ F4
@ ( topolo174197925503356063within @ real @ ( zero_zero @ real ) @ ( set_ord_greaterThan @ real @ ( zero_zero @ real ) ) ) ) ) ).
% filterlim_at_right_to_0
thf(fact_7333_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_7334_filterlim__times__pos,axiom,
! [A: $tType,B: $tType] :
( ( ( linordered_field @ A )
& ( topolo1944317154257567458pology @ A ) )
=> ! [F2: B > A,P4: A,F13: filter @ B,C2: A,L2: A] :
( ( filterlim @ B @ A @ F2 @ ( topolo174197925503356063within @ A @ P4 @ ( set_ord_greaterThan @ A @ P4 ) ) @ F13 )
=> ( ( ord_less @ A @ ( zero_zero @ A ) @ C2 )
=> ( ( L2
= ( times_times @ A @ C2 @ P4 ) )
=> ( filterlim @ B @ A
@ ^ [X2: B] : ( times_times @ A @ C2 @ ( F2 @ X2 ) )
@ ( topolo174197925503356063within @ A @ L2 @ ( set_ord_greaterThan @ A @ L2 ) )
@ F13 ) ) ) ) ) ).
% filterlim_times_pos
thf(fact_7335_log__inj,axiom,
! [B2: real] :
( ( ord_less @ real @ ( one_one @ real ) @ B2 )
=> ( inj_on @ real @ real @ ( log @ B2 ) @ ( set_ord_greaterThan @ real @ ( zero_zero @ real ) ) ) ) ).
% log_inj
thf(fact_7336_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_7337_filterlim__tendsto__pos__mult__at__bot,axiom,
! [A: $tType,F2: A > real,C2: real,F4: filter @ A,G: A > real] :
( ( filterlim @ A @ real @ F2 @ ( topolo7230453075368039082e_nhds @ real @ C2 ) @ F4 )
=> ( ( ord_less @ real @ ( zero_zero @ real ) @ C2 )
=> ( ( filterlim @ A @ real @ G @ ( at_bot @ real ) @ F4 )
=> ( filterlim @ A @ real
@ ^ [X2: A] : ( times_times @ real @ ( F2 @ X2 ) @ ( G @ X2 ) )
@ ( at_bot @ real )
@ F4 ) ) ) ) ).
% filterlim_tendsto_pos_mult_at_bot
thf(fact_7338_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_7339_isCont__If__ge,axiom,
! [B: $tType,A: $tType] :
( ( ( topolo1944317154257567458pology @ A )
& ( topolo4958980785337419405_space @ B ) )
=> ! [A2: A,G: A > B,F2: A > B] :
( ( topolo3448309680560233919inuous @ A @ B @ ( topolo174197925503356063within @ A @ A2 @ ( set_ord_lessThan @ A @ A2 ) ) @ G )
=> ( ( filterlim @ A @ B @ F2 @ ( topolo7230453075368039082e_nhds @ B @ ( G @ A2 ) ) @ ( topolo174197925503356063within @ A @ A2 @ ( set_ord_greaterThan @ A @ A2 ) ) )
=> ( topolo3448309680560233919inuous @ A @ B @ ( topolo174197925503356063within @ A @ A2 @ ( top_top @ ( set @ A ) ) )
@ ^ [X2: A] : ( if @ B @ ( ord_less_eq @ A @ X2 @ A2 ) @ ( G @ X2 ) @ ( F2 @ X2 ) ) ) ) ) ) ).
% isCont_If_ge
thf(fact_7340_DERIV__pos__imp__increasing__at__bot,axiom,
! [B2: real,F2: real > real,Flim: real] :
( ! [X3: real] :
( ( ord_less_eq @ real @ X3 @ B2 )
=> ? [Y4: real] :
( ( has_field_derivative @ real @ F2 @ Y4 @ ( topolo174197925503356063within @ real @ X3 @ ( top_top @ ( set @ real ) ) ) )
& ( ord_less @ real @ ( zero_zero @ real ) @ Y4 ) ) )
=> ( ( filterlim @ real @ real @ F2 @ ( topolo7230453075368039082e_nhds @ real @ Flim ) @ ( at_bot @ real ) )
=> ( ord_less @ real @ Flim @ ( F2 @ B2 ) ) ) ) ).
% DERIV_pos_imp_increasing_at_bot
thf(fact_7341_filterlim__pow__at__bot__odd,axiom,
! [N3: nat,F2: real > real,F4: filter @ real] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ N3 )
=> ( ( filterlim @ real @ real @ F2 @ ( at_bot @ real ) @ F4 )
=> ( ~ ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N3 )
=> ( filterlim @ real @ real
@ ^ [X2: real] : ( power_power @ real @ ( F2 @ X2 ) @ N3 )
@ ( at_bot @ real )
@ F4 ) ) ) ) ).
% filterlim_pow_at_bot_odd
thf(fact_7342_filterlim__pow__at__bot__even,axiom,
! [N3: nat,F2: real > real,F4: filter @ real] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ N3 )
=> ( ( filterlim @ real @ real @ F2 @ ( at_bot @ real ) @ F4 )
=> ( ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N3 )
=> ( filterlim @ real @ real
@ ^ [X2: real] : ( power_power @ real @ ( F2 @ X2 ) @ N3 )
@ ( at_top @ real )
@ F4 ) ) ) ) ).
% filterlim_pow_at_bot_even
thf(fact_7343_lim__zero__infinity,axiom,
! [A: $tType] :
( ( real_V3459762299906320749_field @ A )
=> ! [F2: A > A,L2: A] :
( ( filterlim @ A @ A
@ ^ [X2: A] : ( F2 @ ( divide_divide @ A @ ( one_one @ A ) @ X2 ) )
@ ( topolo7230453075368039082e_nhds @ A @ L2 )
@ ( topolo174197925503356063within @ A @ ( zero_zero @ A ) @ ( top_top @ ( set @ A ) ) ) )
=> ( filterlim @ A @ A @ F2 @ ( topolo7230453075368039082e_nhds @ A @ L2 ) @ ( at_infinity @ A ) ) ) ) ).
% lim_zero_infinity
thf(fact_7344_at__bot__le__at__infinity,axiom,
ord_less_eq @ ( filter @ real ) @ ( at_bot @ real ) @ ( at_infinity @ real ) ).
% at_bot_le_at_infinity
thf(fact_7345_sqrt__at__top,axiom,
filterlim @ real @ real @ sqrt @ ( at_top @ real ) @ ( at_top @ real ) ).
% sqrt_at_top
thf(fact_7346_filterlim__at__top__add__at__top,axiom,
! [A: $tType,F2: A > real,F4: filter @ A,G: A > real] :
( ( filterlim @ A @ real @ F2 @ ( at_top @ real ) @ F4 )
=> ( ( filterlim @ A @ real @ G @ ( at_top @ real ) @ F4 )
=> ( filterlim @ A @ real
@ ^ [X2: A] : ( plus_plus @ real @ ( F2 @ X2 ) @ ( G @ X2 ) )
@ ( at_top @ real )
@ F4 ) ) ) ).
% filterlim_at_top_add_at_top
thf(fact_7347_filterlim__at__top__mult__at__top,axiom,
! [A: $tType,F2: A > real,F4: filter @ A,G: A > real] :
( ( filterlim @ A @ real @ F2 @ ( at_top @ real ) @ F4 )
=> ( ( filterlim @ A @ real @ G @ ( at_top @ real ) @ F4 )
=> ( filterlim @ A @ real
@ ^ [X2: A] : ( times_times @ real @ ( F2 @ X2 ) @ ( G @ X2 ) )
@ ( at_top @ real )
@ F4 ) ) ) ).
% filterlim_at_top_mult_at_top
thf(fact_7348_at__top__le__at__infinity,axiom,
ord_less_eq @ ( filter @ real ) @ ( at_top @ real ) @ ( at_infinity @ real ) ).
% at_top_le_at_infinity
thf(fact_7349_filterlim__tendsto__add__at__top,axiom,
! [A: $tType,F2: A > real,C2: real,F4: filter @ A,G: A > real] :
( ( filterlim @ A @ real @ F2 @ ( topolo7230453075368039082e_nhds @ real @ C2 ) @ F4 )
=> ( ( filterlim @ A @ real @ G @ ( at_top @ real ) @ F4 )
=> ( filterlim @ A @ real
@ ^ [X2: A] : ( plus_plus @ real @ ( F2 @ X2 ) @ ( G @ X2 ) )
@ ( at_top @ real )
@ F4 ) ) ) ).
% filterlim_tendsto_add_at_top
thf(fact_7350_greaterThan__0,axiom,
( ( set_ord_greaterThan @ nat @ ( zero_zero @ nat ) )
= ( image @ nat @ nat @ suc @ ( top_top @ ( set @ nat ) ) ) ) ).
% greaterThan_0
thf(fact_7351_filterlim__pow__at__top,axiom,
! [A: $tType,N3: nat,F2: A > real,F4: filter @ A] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ N3 )
=> ( ( filterlim @ A @ real @ F2 @ ( at_top @ real ) @ F4 )
=> ( filterlim @ A @ real
@ ^ [X2: A] : ( power_power @ real @ ( F2 @ X2 ) @ N3 )
@ ( at_top @ real )
@ F4 ) ) ) ).
% filterlim_pow_at_top
thf(fact_7352_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_7353_tendsto__add__filterlim__at__infinity,axiom,
! [B: $tType,A: $tType] :
( ( real_V822414075346904944vector @ B )
=> ! [F2: A > B,C2: B,F4: filter @ A,G: A > B] :
( ( filterlim @ A @ B @ F2 @ ( topolo7230453075368039082e_nhds @ B @ C2 ) @ F4 )
=> ( ( filterlim @ A @ B @ G @ ( at_infinity @ B ) @ F4 )
=> ( filterlim @ A @ B
@ ^ [X2: A] : ( plus_plus @ B @ ( F2 @ X2 ) @ ( G @ X2 ) )
@ ( at_infinity @ B )
@ F4 ) ) ) ) ).
% tendsto_add_filterlim_at_infinity
thf(fact_7354_tendsto__add__filterlim__at__infinity_H,axiom,
! [B: $tType,A: $tType] :
( ( real_V822414075346904944vector @ B )
=> ! [F2: A > B,F4: filter @ A,G: A > B,C2: B] :
( ( filterlim @ A @ B @ F2 @ ( at_infinity @ B ) @ F4 )
=> ( ( filterlim @ A @ B @ G @ ( topolo7230453075368039082e_nhds @ B @ C2 ) @ F4 )
=> ( filterlim @ A @ B
@ ^ [X2: A] : ( plus_plus @ B @ ( F2 @ X2 ) @ ( G @ X2 ) )
@ ( at_infinity @ B )
@ F4 ) ) ) ) ).
% tendsto_add_filterlim_at_infinity'
thf(fact_7355_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_7356_real__tendsto__divide__at__top,axiom,
! [A: $tType,F2: A > real,C2: real,F4: filter @ A,G: A > real] :
( ( filterlim @ A @ real @ F2 @ ( topolo7230453075368039082e_nhds @ real @ C2 ) @ F4 )
=> ( ( filterlim @ A @ real @ G @ ( at_top @ real ) @ F4 )
=> ( filterlim @ A @ real
@ ^ [X2: A] : ( divide_divide @ real @ ( F2 @ X2 ) @ ( G @ X2 ) )
@ ( topolo7230453075368039082e_nhds @ real @ ( zero_zero @ real ) )
@ F4 ) ) ) ).
% real_tendsto_divide_at_top
thf(fact_7357_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_7358_filterlim__tendsto__pos__mult__at__top,axiom,
! [A: $tType,F2: A > real,C2: real,F4: filter @ A,G: A > real] :
( ( filterlim @ A @ real @ F2 @ ( topolo7230453075368039082e_nhds @ real @ C2 ) @ F4 )
=> ( ( ord_less @ real @ ( zero_zero @ real ) @ C2 )
=> ( ( filterlim @ A @ real @ G @ ( at_top @ real ) @ F4 )
=> ( filterlim @ A @ real
@ ^ [X2: A] : ( times_times @ real @ ( F2 @ X2 ) @ ( G @ X2 ) )
@ ( at_top @ real )
@ F4 ) ) ) ) ).
% filterlim_tendsto_pos_mult_at_top
thf(fact_7359_filterlim__at__top__mult__tendsto__pos,axiom,
! [A: $tType,F2: A > real,C2: real,F4: filter @ A,G: A > real] :
( ( filterlim @ A @ real @ F2 @ ( topolo7230453075368039082e_nhds @ real @ C2 ) @ F4 )
=> ( ( ord_less @ real @ ( zero_zero @ real ) @ C2 )
=> ( ( filterlim @ A @ real @ G @ ( at_top @ real ) @ F4 )
=> ( filterlim @ A @ real
@ ^ [X2: A] : ( times_times @ real @ ( G @ X2 ) @ ( F2 @ X2 ) )
@ ( at_top @ real )
@ F4 ) ) ) ) ).
% filterlim_at_top_mult_tendsto_pos
thf(fact_7360_tendsto__mult__filterlim__at__infinity,axiom,
! [A: $tType,B: $tType] :
( ( real_V3459762299906320749_field @ A )
=> ! [F2: B > A,C2: A,F4: filter @ B,G: B > A] :
( ( filterlim @ B @ A @ F2 @ ( topolo7230453075368039082e_nhds @ A @ C2 ) @ F4 )
=> ( ( C2
!= ( zero_zero @ A ) )
=> ( ( filterlim @ B @ A @ G @ ( at_infinity @ A ) @ F4 )
=> ( filterlim @ B @ A
@ ^ [X2: B] : ( times_times @ A @ ( F2 @ X2 ) @ ( G @ X2 ) )
@ ( at_infinity @ A )
@ F4 ) ) ) ) ) ).
% tendsto_mult_filterlim_at_infinity
thf(fact_7361_filterlim__power__at__infinity,axiom,
! [B: $tType,A: $tType] :
( ( real_V8999393235501362500lgebra @ B )
=> ! [F2: A > B,F4: filter @ A,N3: nat] :
( ( filterlim @ A @ B @ F2 @ ( at_infinity @ B ) @ F4 )
=> ( ( ord_less @ nat @ ( zero_zero @ nat ) @ N3 )
=> ( filterlim @ A @ B
@ ^ [X2: A] : ( power_power @ B @ ( F2 @ X2 ) @ N3 )
@ ( at_infinity @ B )
@ F4 ) ) ) ) ).
% filterlim_power_at_infinity
thf(fact_7362_tendsto__divide__0,axiom,
! [A: $tType,C: $tType] :
( ( real_V8999393235501362500lgebra @ A )
=> ! [F2: C > A,C2: A,F4: filter @ C,G: C > A] :
( ( filterlim @ C @ A @ F2 @ ( topolo7230453075368039082e_nhds @ A @ C2 ) @ F4 )
=> ( ( filterlim @ C @ A @ G @ ( at_infinity @ A ) @ F4 )
=> ( filterlim @ C @ A
@ ^ [X2: C] : ( divide_divide @ A @ ( F2 @ X2 ) @ ( G @ X2 ) )
@ ( topolo7230453075368039082e_nhds @ A @ ( zero_zero @ A ) )
@ F4 ) ) ) ) ).
% tendsto_divide_0
thf(fact_7363_ln__x__over__x__tendsto__0,axiom,
( filterlim @ real @ real
@ ^ [X2: real] : ( divide_divide @ real @ ( ln_ln @ real @ X2 ) @ X2 )
@ ( topolo7230453075368039082e_nhds @ real @ ( zero_zero @ real ) )
@ ( at_top @ real ) ) ).
% ln_x_over_x_tendsto_0
thf(fact_7364_filterlim__tendsto__neg__mult__at__bot,axiom,
! [A: $tType,F2: A > real,C2: real,F4: filter @ A,G: A > real] :
( ( filterlim @ A @ real @ F2 @ ( topolo7230453075368039082e_nhds @ real @ C2 ) @ F4 )
=> ( ( ord_less @ real @ C2 @ ( zero_zero @ real ) )
=> ( ( filterlim @ A @ real @ G @ ( at_top @ real ) @ F4 )
=> ( filterlim @ A @ real
@ ^ [X2: A] : ( times_times @ real @ ( F2 @ X2 ) @ ( G @ X2 ) )
@ ( at_bot @ real )
@ F4 ) ) ) ) ).
% filterlim_tendsto_neg_mult_at_bot
thf(fact_7365_tendsto__power__div__exp__0,axiom,
! [K: nat] :
( filterlim @ real @ real
@ ^ [X2: real] : ( divide_divide @ real @ ( power_power @ real @ X2 @ K ) @ ( exp @ real @ X2 ) )
@ ( topolo7230453075368039082e_nhds @ real @ ( zero_zero @ real ) )
@ ( at_top @ real ) ) ).
% tendsto_power_div_exp_0
thf(fact_7366_tendsto__exp__limit__at__top,axiom,
! [X: real] :
( filterlim @ real @ real
@ ^ [Y2: real] : ( powr @ real @ ( plus_plus @ real @ ( one_one @ real ) @ ( divide_divide @ real @ X @ Y2 ) ) @ Y2 )
@ ( topolo7230453075368039082e_nhds @ real @ ( exp @ real @ X ) )
@ ( at_top @ real ) ) ).
% tendsto_exp_limit_at_top
thf(fact_7367_filterlim__divide__at__infinity,axiom,
! [A: $tType] :
( ( real_V3459762299906320749_field @ A )
=> ! [F2: A > A,C2: A,F4: filter @ A,G: A > A] :
( ( filterlim @ A @ A @ F2 @ ( topolo7230453075368039082e_nhds @ A @ C2 ) @ F4 )
=> ( ( filterlim @ A @ A @ G @ ( topolo174197925503356063within @ A @ ( zero_zero @ A ) @ ( top_top @ ( set @ A ) ) ) @ F4 )
=> ( ( C2
!= ( zero_zero @ A ) )
=> ( filterlim @ A @ A
@ ^ [X2: A] : ( divide_divide @ A @ ( F2 @ X2 ) @ ( G @ X2 ) )
@ ( at_infinity @ A )
@ F4 ) ) ) ) ) ).
% filterlim_divide_at_infinity
thf(fact_7368_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_7369_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_7370_DERIV__neg__imp__decreasing__at__top,axiom,
! [B2: real,F2: real > real,Flim: real] :
( ! [X3: real] :
( ( ord_less_eq @ real @ B2 @ X3 )
=> ? [Y4: real] :
( ( has_field_derivative @ real @ F2 @ Y4 @ ( topolo174197925503356063within @ real @ X3 @ ( top_top @ ( set @ real ) ) ) )
& ( ord_less @ real @ Y4 @ ( zero_zero @ real ) ) ) )
=> ( ( filterlim @ real @ real @ F2 @ ( topolo7230453075368039082e_nhds @ real @ Flim ) @ ( at_top @ real ) )
=> ( ord_less @ real @ Flim @ ( F2 @ B2 ) ) ) ) ).
% DERIV_neg_imp_decreasing_at_top
thf(fact_7371_filterlim__realpow__sequentially__gt1,axiom,
! [A: $tType] :
( ( real_V8999393235501362500lgebra @ A )
=> ! [X: A] :
( ( ord_less @ real @ ( one_one @ real ) @ ( real_V7770717601297561774m_norm @ A @ X ) )
=> ( filterlim @ nat @ A @ ( power_power @ A @ X ) @ ( at_infinity @ A ) @ ( at_top @ nat ) ) ) ) ).
% filterlim_realpow_sequentially_gt1
thf(fact_7372_polyfun__extremal,axiom,
! [A: $tType] :
( ( real_V8999393235501362500lgebra @ A )
=> ! [C2: nat > A,K: nat,N3: nat,B6: real] :
( ( ( C2 @ K )
!= ( zero_zero @ A ) )
=> ( ( ord_less_eq @ nat @ ( one_one @ nat ) @ K )
=> ( ( ord_less_eq @ nat @ K @ N3 )
=> ( eventually @ A
@ ^ [Z3: A] :
( ord_less_eq @ real @ B6
@ ( real_V7770717601297561774m_norm @ A
@ ( groups7311177749621191930dd_sum @ nat @ A
@ ^ [I3: nat] : ( times_times @ A @ ( C2 @ I3 ) @ ( power_power @ A @ Z3 @ I3 ) )
@ ( set_ord_atMost @ nat @ N3 ) ) ) )
@ ( at_infinity @ A ) ) ) ) ) ) ).
% polyfun_extremal
thf(fact_7373_lhopital__right__0__at__top,axiom,
! [G: real > real,G5: real > real,F2: real > real,F5: real > real,X: real] :
( ( filterlim @ real @ real @ G @ ( at_top @ real ) @ ( topolo174197925503356063within @ real @ ( zero_zero @ real ) @ ( set_ord_greaterThan @ real @ ( zero_zero @ real ) ) ) )
=> ( ( eventually @ real
@ ^ [X2: real] :
( ( G5 @ X2 )
!= ( zero_zero @ real ) )
@ ( topolo174197925503356063within @ real @ ( zero_zero @ real ) @ ( set_ord_greaterThan @ real @ ( zero_zero @ real ) ) ) )
=> ( ( eventually @ real
@ ^ [X2: real] : ( has_field_derivative @ real @ F2 @ ( F5 @ X2 ) @ ( topolo174197925503356063within @ real @ X2 @ ( top_top @ ( set @ real ) ) ) )
@ ( topolo174197925503356063within @ real @ ( zero_zero @ real ) @ ( set_ord_greaterThan @ real @ ( zero_zero @ real ) ) ) )
=> ( ( eventually @ real
@ ^ [X2: real] : ( has_field_derivative @ real @ G @ ( G5 @ X2 ) @ ( topolo174197925503356063within @ real @ X2 @ ( top_top @ ( set @ real ) ) ) )
@ ( topolo174197925503356063within @ real @ ( zero_zero @ real ) @ ( set_ord_greaterThan @ real @ ( zero_zero @ real ) ) ) )
=> ( ( filterlim @ real @ real
@ ^ [X2: real] : ( divide_divide @ real @ ( F5 @ X2 ) @ ( G5 @ X2 ) )
@ ( topolo7230453075368039082e_nhds @ real @ X )
@ ( topolo174197925503356063within @ real @ ( zero_zero @ real ) @ ( set_ord_greaterThan @ real @ ( zero_zero @ real ) ) ) )
=> ( filterlim @ real @ real
@ ^ [X2: real] : ( divide_divide @ real @ ( F2 @ X2 ) @ ( G @ X2 ) )
@ ( topolo7230453075368039082e_nhds @ real @ X )
@ ( topolo174197925503356063within @ real @ ( zero_zero @ real ) @ ( set_ord_greaterThan @ real @ ( zero_zero @ real ) ) ) ) ) ) ) ) ) ).
% lhopital_right_0_at_top
thf(fact_7374_eventually__sequentially__Suc,axiom,
! [P: nat > $o] :
( ( eventually @ nat
@ ^ [I3: nat] : ( P @ ( suc @ I3 ) )
@ ( at_top @ nat ) )
= ( eventually @ nat @ P @ ( at_top @ nat ) ) ) ).
% eventually_sequentially_Suc
thf(fact_7375_eventually__sequentially__seg,axiom,
! [P: nat > $o,K: nat] :
( ( eventually @ nat
@ ^ [N2: nat] : ( P @ ( plus_plus @ nat @ N2 @ K ) )
@ ( at_top @ nat ) )
= ( eventually @ nat @ P @ ( at_top @ nat ) ) ) ).
% eventually_sequentially_seg
thf(fact_7376_eventually__at__right,axiom,
! [A: $tType] :
( ( topolo1944317154257567458pology @ A )
=> ! [X: A,Y: A,P: A > $o] :
( ( ord_less @ A @ X @ Y )
=> ( ( eventually @ A @ P @ ( topolo174197925503356063within @ A @ X @ ( set_ord_greaterThan @ A @ X ) ) )
= ( ? [B5: A] :
( ( ord_less @ A @ X @ B5 )
& ! [Y2: A] :
( ( ord_less @ A @ X @ Y2 )
=> ( ( ord_less @ A @ Y2 @ B5 )
=> ( P @ Y2 ) ) ) ) ) ) ) ) ).
% eventually_at_right
thf(fact_7377_eventually__at__right__field,axiom,
! [A: $tType] :
( ( ( linordered_field @ A )
& ( topolo1944317154257567458pology @ A ) )
=> ! [P: A > $o,X: A] :
( ( eventually @ A @ P @ ( topolo174197925503356063within @ A @ X @ ( set_ord_greaterThan @ A @ X ) ) )
= ( ? [B5: A] :
( ( ord_less @ A @ X @ B5 )
& ! [Y2: A] :
( ( ord_less @ A @ X @ Y2 )
=> ( ( ord_less @ A @ Y2 @ B5 )
=> ( P @ Y2 ) ) ) ) ) ) ) ).
% eventually_at_right_field
thf(fact_7378_filterlim__mono__eventually,axiom,
! [B: $tType,A: $tType,F2: A > B,F4: filter @ B,G6: filter @ A,F8: filter @ B,G7: filter @ A,F5: A > B] :
( ( filterlim @ A @ B @ F2 @ F4 @ G6 )
=> ( ( ord_less_eq @ ( filter @ B ) @ F4 @ F8 )
=> ( ( ord_less_eq @ ( filter @ A ) @ G7 @ G6 )
=> ( ( eventually @ A
@ ^ [X2: A] :
( ( F2 @ X2 )
= ( F5 @ X2 ) )
@ G7 )
=> ( filterlim @ A @ B @ F5 @ F8 @ G7 ) ) ) ) ) ).
% filterlim_mono_eventually
thf(fact_7379_filterlim__at__top__at__top,axiom,
! [B: $tType,A: $tType] :
( ( ( linorder @ A )
& ( linorder @ B ) )
=> ! [Q: A > $o,F2: A > B,P: B > $o,G: B > A] :
( ! [X3: A,Y3: A] :
( ( Q @ X3 )
=> ( ( Q @ Y3 )
=> ( ( ord_less_eq @ A @ X3 @ Y3 )
=> ( ord_less_eq @ B @ ( F2 @ X3 ) @ ( F2 @ Y3 ) ) ) ) )
=> ( ! [X3: B] :
( ( P @ X3 )
=> ( ( F2 @ ( G @ X3 ) )
= X3 ) )
=> ( ! [X3: B] :
( ( P @ X3 )
=> ( Q @ ( G @ X3 ) ) )
=> ( ( eventually @ A @ Q @ ( at_top @ A ) )
=> ( ( eventually @ B @ P @ ( at_top @ B ) )
=> ( filterlim @ A @ B @ F2 @ ( at_top @ B ) @ ( at_top @ A ) ) ) ) ) ) ) ) ).
% filterlim_at_top_at_top
thf(fact_7380_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_7381_eventually__ge__at__top,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [C2: A] : ( eventually @ A @ ( ord_less_eq @ A @ C2 ) @ ( at_top @ A ) ) ) ).
% eventually_ge_at_top
thf(fact_7382_eventually__sequentially,axiom,
! [P: nat > $o] :
( ( eventually @ nat @ P @ ( at_top @ nat ) )
= ( ? [N8: nat] :
! [N2: nat] :
( ( ord_less_eq @ nat @ N8 @ N2 )
=> ( P @ N2 ) ) ) ) ).
% eventually_sequentially
thf(fact_7383_eventually__sequentiallyI,axiom,
! [C2: nat,P: nat > $o] :
( ! [X3: nat] :
( ( ord_less_eq @ nat @ C2 @ X3 )
=> ( P @ X3 ) )
=> ( eventually @ nat @ P @ ( at_top @ nat ) ) ) ).
% eventually_sequentiallyI
thf(fact_7384_eventually__at__top__linorder,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [P: A > $o] :
( ( eventually @ A @ P @ ( at_top @ A ) )
= ( ? [N8: A] :
! [N2: A] :
( ( ord_less_eq @ A @ N8 @ N2 )
=> ( P @ N2 ) ) ) ) ) ).
% eventually_at_top_linorder
thf(fact_7385_eventually__at__top__linorderI,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [C2: A,P: A > $o] :
( ! [X3: A] :
( ( ord_less_eq @ A @ C2 @ X3 )
=> ( P @ X3 ) )
=> ( eventually @ A @ P @ ( at_top @ A ) ) ) ) ).
% eventually_at_top_linorderI
thf(fact_7386_eventually__at__infinity,axiom,
! [A: $tType] :
( ( real_V822414075346904944vector @ A )
=> ! [P: A > $o] :
( ( eventually @ A @ P @ ( at_infinity @ A ) )
= ( ? [B5: real] :
! [X2: A] :
( ( ord_less_eq @ real @ B5 @ ( real_V7770717601297561774m_norm @ A @ X2 ) )
=> ( P @ X2 ) ) ) ) ) ).
% eventually_at_infinity
thf(fact_7387_sequentially__offset,axiom,
! [P: nat > $o,K: nat] :
( ( eventually @ nat @ P @ ( at_top @ nat ) )
=> ( eventually @ nat
@ ^ [I3: nat] : ( P @ ( plus_plus @ nat @ I3 @ K ) )
@ ( at_top @ nat ) ) ) ).
% sequentially_offset
thf(fact_7388_eventually__gt__at__top,axiom,
! [A: $tType] :
( ( ( linorder @ A )
& ( no_top @ A ) )
=> ! [C2: A] : ( eventually @ A @ ( ord_less @ A @ C2 ) @ ( at_top @ A ) ) ) ).
% eventually_gt_at_top
thf(fact_7389_eventually__at__top__dense,axiom,
! [A: $tType] :
( ( ( linorder @ A )
& ( no_top @ A ) )
=> ! [P: A > $o] :
( ( eventually @ A @ P @ ( at_top @ A ) )
= ( ? [N8: A] :
! [N2: A] :
( ( ord_less @ A @ N8 @ N2 )
=> ( P @ N2 ) ) ) ) ) ).
% eventually_at_top_dense
thf(fact_7390_eventually__at__left,axiom,
! [A: $tType] :
( ( topolo1944317154257567458pology @ A )
=> ! [Y: A,X: A,P: A > $o] :
( ( ord_less @ A @ Y @ X )
=> ( ( eventually @ A @ P @ ( topolo174197925503356063within @ A @ X @ ( set_ord_lessThan @ A @ X ) ) )
= ( ? [B5: A] :
( ( ord_less @ A @ B5 @ X )
& ! [Y2: A] :
( ( ord_less @ A @ B5 @ Y2 )
=> ( ( ord_less @ A @ Y2 @ X )
=> ( P @ Y2 ) ) ) ) ) ) ) ) ).
% eventually_at_left
thf(fact_7391_eventually__at__left__field,axiom,
! [A: $tType] :
( ( ( linordered_field @ A )
& ( topolo1944317154257567458pology @ A ) )
=> ! [P: A > $o,X: A] :
( ( eventually @ A @ P @ ( topolo174197925503356063within @ A @ X @ ( set_ord_lessThan @ A @ X ) ) )
= ( ? [B5: A] :
( ( ord_less @ A @ B5 @ X )
& ! [Y2: A] :
( ( ord_less @ A @ B5 @ Y2 )
=> ( ( ord_less @ A @ Y2 @ X )
=> ( P @ Y2 ) ) ) ) ) ) ) ).
% eventually_at_left_field
thf(fact_7392_eventually__nhds__top,axiom,
! [A: $tType] :
( ( ( order_top @ A )
& ( topolo1944317154257567458pology @ A ) )
=> ! [B2: A,P: A > $o] :
( ( ord_less @ A @ B2 @ ( top_top @ A ) )
=> ( ( eventually @ A @ P @ ( topolo7230453075368039082e_nhds @ A @ ( top_top @ A ) ) )
= ( ? [B5: A] :
( ( ord_less @ A @ B5 @ ( top_top @ A ) )
& ! [Z3: A] :
( ( ord_less @ A @ B5 @ Z3 )
=> ( P @ Z3 ) ) ) ) ) ) ) ).
% eventually_nhds_top
thf(fact_7393_has__field__derivative__cong__eventually,axiom,
! [A: $tType] :
( ( real_V3459762299906320749_field @ A )
=> ! [F2: A > A,G: A > A,X: A,S3: set @ A,U: A] :
( ( eventually @ A
@ ^ [X2: A] :
( ( F2 @ X2 )
= ( G @ X2 ) )
@ ( topolo174197925503356063within @ A @ X @ S3 ) )
=> ( ( ( F2 @ X )
= ( G @ X ) )
=> ( ( has_field_derivative @ A @ F2 @ U @ ( topolo174197925503356063within @ A @ X @ S3 ) )
= ( has_field_derivative @ A @ G @ U @ ( topolo174197925503356063within @ A @ X @ S3 ) ) ) ) ) ) ).
% has_field_derivative_cong_eventually
thf(fact_7394_has__derivative__transform__eventually,axiom,
! [B: $tType,A: $tType] :
( ( ( real_V822414075346904944vector @ A )
& ( real_V822414075346904944vector @ B ) )
=> ! [F2: A > B,F5: A > B,X: A,S: set @ A,G: A > B] :
( ( has_derivative @ A @ B @ F2 @ F5 @ ( topolo174197925503356063within @ A @ X @ S ) )
=> ( ( eventually @ A
@ ^ [X9: A] :
( ( F2 @ X9 )
= ( G @ X9 ) )
@ ( topolo174197925503356063within @ A @ X @ S ) )
=> ( ( ( F2 @ X )
= ( G @ X ) )
=> ( ( member @ A @ X @ S )
=> ( has_derivative @ A @ B @ G @ F5 @ ( topolo174197925503356063within @ A @ X @ S ) ) ) ) ) ) ) ).
% has_derivative_transform_eventually
thf(fact_7395_filter__leD,axiom,
! [A: $tType,F4: filter @ A,F8: filter @ A,P: A > $o] :
( ( ord_less_eq @ ( filter @ A ) @ F4 @ F8 )
=> ( ( eventually @ A @ P @ F8 )
=> ( eventually @ A @ P @ F4 ) ) ) ).
% filter_leD
thf(fact_7396_filter__leI,axiom,
! [A: $tType,F8: filter @ A,F4: filter @ A] :
( ! [P8: A > $o] :
( ( eventually @ A @ P8 @ F8 )
=> ( eventually @ A @ P8 @ F4 ) )
=> ( ord_less_eq @ ( filter @ A ) @ F4 @ F8 ) ) ).
% filter_leI
thf(fact_7397_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_7398_eventually__at__bot__linorder,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [P: A > $o] :
( ( eventually @ A @ P @ ( at_bot @ A ) )
= ( ? [N8: A] :
! [N2: A] :
( ( ord_less_eq @ A @ N2 @ N8 )
=> ( P @ N2 ) ) ) ) ) ).
% eventually_at_bot_linorder
thf(fact_7399_eventually__le__at__bot,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [C2: A] :
( eventually @ A
@ ^ [X2: A] : ( ord_less_eq @ A @ X2 @ C2 )
@ ( at_bot @ A ) ) ) ).
% eventually_le_at_bot
thf(fact_7400_eventually__at__bot__dense,axiom,
! [A: $tType] :
( ( ( linorder @ A )
& ( no_bot @ A ) )
=> ! [P: A > $o] :
( ( eventually @ A @ P @ ( at_bot @ A ) )
= ( ? [N8: A] :
! [N2: A] :
( ( ord_less @ A @ N2 @ N8 )
=> ( P @ N2 ) ) ) ) ) ).
% eventually_at_bot_dense
thf(fact_7401_eventually__gt__at__bot,axiom,
! [A: $tType] :
( ( unboun7993243217541854897norder @ A )
=> ! [C2: A] :
( eventually @ A
@ ^ [X2: A] : ( ord_less @ A @ X2 @ C2 )
@ ( at_bot @ A ) ) ) ).
% eventually_gt_at_bot
thf(fact_7402_tendsto__le,axiom,
! [B: $tType,A: $tType] :
( ( topolo1944317154257567458pology @ A )
=> ! [F4: filter @ B,F2: B > A,X: A,G: B > A,Y: A] :
( ( F4
!= ( bot_bot @ ( filter @ B ) ) )
=> ( ( filterlim @ B @ A @ F2 @ ( topolo7230453075368039082e_nhds @ A @ X ) @ F4 )
=> ( ( filterlim @ B @ A @ G @ ( topolo7230453075368039082e_nhds @ A @ Y ) @ F4 )
=> ( ( eventually @ B
@ ^ [X2: B] : ( ord_less_eq @ A @ ( G @ X2 ) @ ( F2 @ X2 ) )
@ F4 )
=> ( ord_less_eq @ A @ Y @ X ) ) ) ) ) ) ).
% tendsto_le
thf(fact_7403_tendsto__lowerbound,axiom,
! [B: $tType,A: $tType] :
( ( topolo1944317154257567458pology @ A )
=> ! [F2: B > A,X: A,F4: filter @ B,A2: A] :
( ( filterlim @ B @ A @ F2 @ ( topolo7230453075368039082e_nhds @ A @ X ) @ F4 )
=> ( ( eventually @ B
@ ^ [I3: B] : ( ord_less_eq @ A @ A2 @ ( F2 @ I3 ) )
@ F4 )
=> ( ( F4
!= ( bot_bot @ ( filter @ B ) ) )
=> ( ord_less_eq @ A @ A2 @ X ) ) ) ) ) ).
% tendsto_lowerbound
thf(fact_7404_tendsto__upperbound,axiom,
! [B: $tType,A: $tType] :
( ( topolo1944317154257567458pology @ A )
=> ! [F2: B > A,X: A,F4: filter @ B,A2: A] :
( ( filterlim @ B @ A @ F2 @ ( topolo7230453075368039082e_nhds @ A @ X ) @ F4 )
=> ( ( eventually @ B
@ ^ [I3: B] : ( ord_less_eq @ A @ ( F2 @ I3 ) @ A2 )
@ F4 )
=> ( ( F4
!= ( bot_bot @ ( filter @ B ) ) )
=> ( ord_less_eq @ A @ X @ A2 ) ) ) ) ) ).
% tendsto_upperbound
thf(fact_7405_tendsto__sandwich,axiom,
! [A: $tType,B: $tType] :
( ( topolo2564578578187576103pology @ A )
=> ! [F2: B > A,G: B > A,Net: filter @ B,H2: B > A,C2: A] :
( ( eventually @ B
@ ^ [N2: B] : ( ord_less_eq @ A @ ( F2 @ N2 ) @ ( G @ N2 ) )
@ Net )
=> ( ( eventually @ B
@ ^ [N2: B] : ( ord_less_eq @ A @ ( G @ N2 ) @ ( H2 @ N2 ) )
@ Net )
=> ( ( filterlim @ B @ A @ F2 @ ( topolo7230453075368039082e_nhds @ A @ C2 ) @ Net )
=> ( ( filterlim @ B @ A @ H2 @ ( topolo7230453075368039082e_nhds @ A @ C2 ) @ Net )
=> ( filterlim @ B @ A @ G @ ( topolo7230453075368039082e_nhds @ A @ C2 ) @ Net ) ) ) ) ) ) ).
% tendsto_sandwich
thf(fact_7406_order__tendstoD_I2_J,axiom,
! [A: $tType,B: $tType] :
( ( topolo2564578578187576103pology @ A )
=> ! [F2: B > A,Y: A,F4: filter @ B,A2: A] :
( ( filterlim @ B @ A @ F2 @ ( topolo7230453075368039082e_nhds @ A @ Y ) @ F4 )
=> ( ( ord_less @ A @ Y @ A2 )
=> ( eventually @ B
@ ^ [X2: B] : ( ord_less @ A @ ( F2 @ X2 ) @ A2 )
@ F4 ) ) ) ) ).
% order_tendstoD(2)
thf(fact_7407_order__tendstoD_I1_J,axiom,
! [A: $tType,B: $tType] :
( ( topolo2564578578187576103pology @ A )
=> ! [F2: B > A,Y: A,F4: filter @ B,A2: A] :
( ( filterlim @ B @ A @ F2 @ ( topolo7230453075368039082e_nhds @ A @ Y ) @ F4 )
=> ( ( ord_less @ A @ A2 @ Y )
=> ( eventually @ B
@ ^ [X2: B] : ( ord_less @ A @ A2 @ ( F2 @ X2 ) )
@ F4 ) ) ) ) ).
% order_tendstoD(1)
thf(fact_7408_order__tendstoI,axiom,
! [A: $tType,B: $tType] :
( ( topolo2564578578187576103pology @ A )
=> ! [Y: A,F2: B > A,F4: filter @ B] :
( ! [A4: A] :
( ( ord_less @ A @ A4 @ Y )
=> ( eventually @ B
@ ^ [X2: B] : ( ord_less @ A @ A4 @ ( F2 @ X2 ) )
@ F4 ) )
=> ( ! [A4: A] :
( ( ord_less @ A @ Y @ A4 )
=> ( eventually @ B
@ ^ [X2: B] : ( ord_less @ A @ ( F2 @ X2 ) @ A4 )
@ F4 ) )
=> ( filterlim @ B @ A @ F2 @ ( topolo7230453075368039082e_nhds @ A @ Y ) @ F4 ) ) ) ) ).
% order_tendstoI
thf(fact_7409_order__tendsto__iff,axiom,
! [B: $tType,A: $tType] :
( ( topolo2564578578187576103pology @ A )
=> ! [F2: B > A,X: A,F4: filter @ B] :
( ( filterlim @ B @ A @ F2 @ ( topolo7230453075368039082e_nhds @ A @ X ) @ F4 )
= ( ! [L: A] :
( ( ord_less @ A @ L @ X )
=> ( eventually @ B
@ ^ [X2: B] : ( ord_less @ A @ L @ ( F2 @ X2 ) )
@ F4 ) )
& ! [U2: A] :
( ( ord_less @ A @ X @ U2 )
=> ( eventually @ B
@ ^ [X2: B] : ( ord_less @ A @ ( F2 @ X2 ) @ U2 )
@ F4 ) ) ) ) ) ).
% order_tendsto_iff
thf(fact_7410_filterlim__at__top__mono,axiom,
! [A: $tType,B: $tType] :
( ( linorder @ A )
=> ! [F2: B > A,F4: filter @ B,G: B > A] :
( ( filterlim @ B @ A @ F2 @ ( at_top @ A ) @ F4 )
=> ( ( eventually @ B
@ ^ [X2: B] : ( ord_less_eq @ A @ ( F2 @ X2 ) @ ( G @ X2 ) )
@ F4 )
=> ( filterlim @ B @ A @ G @ ( at_top @ A ) @ F4 ) ) ) ) ).
% filterlim_at_top_mono
thf(fact_7411_filterlim__at__top__ge,axiom,
! [A: $tType,B: $tType] :
( ( linorder @ B )
=> ! [F2: A > B,F4: filter @ A,C2: B] :
( ( filterlim @ A @ B @ F2 @ ( at_top @ B ) @ F4 )
= ( ! [Z10: B] :
( ( ord_less_eq @ B @ C2 @ Z10 )
=> ( eventually @ A
@ ^ [X2: A] : ( ord_less_eq @ B @ Z10 @ ( F2 @ X2 ) )
@ F4 ) ) ) ) ) ).
% filterlim_at_top_ge
thf(fact_7412_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
@ ^ [X2: A] : ( ord_less_eq @ B @ Z10 @ ( F2 @ X2 ) )
@ F4 ) ) ) ) ).
% filterlim_at_top
thf(fact_7413_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
@ ^ [X2: A] : ( ord_less @ B @ Z10 @ ( F2 @ X2 ) )
@ F4 ) ) ) ) ).
% filterlim_at_top_dense
thf(fact_7414_eventually__at__right__less,axiom,
! [A: $tType] :
( ( ( no_top @ A )
& ( topolo1944317154257567458pology @ A ) )
=> ! [X: A] : ( eventually @ A @ ( ord_less @ A @ X ) @ ( topolo174197925503356063within @ A @ X @ ( set_ord_greaterThan @ A @ X ) ) ) ) ).
% eventually_at_right_less
thf(fact_7415_has__field__derivative__cong__ev,axiom,
! [A: $tType] :
( ( real_V3459762299906320749_field @ A )
=> ! [X: A,Y: A,S3: set @ A,F2: A > A,G: A > A,U: A,V: A,T2: set @ A] :
( ( X = Y )
=> ( ( eventually @ A
@ ^ [X2: A] :
( ( member @ A @ X2 @ S3 )
=> ( ( F2 @ X2 )
= ( G @ X2 ) ) )
@ ( topolo7230453075368039082e_nhds @ A @ X ) )
=> ( ( U = V )
=> ( ( S3 = T2 )
=> ( ( member @ A @ X @ S3 )
=> ( ( has_field_derivative @ A @ F2 @ U @ ( topolo174197925503356063within @ A @ X @ S3 ) )
= ( has_field_derivative @ A @ G @ V @ ( topolo174197925503356063within @ A @ Y @ T2 ) ) ) ) ) ) ) ) ) ).
% has_field_derivative_cong_ev
thf(fact_7416_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
@ ^ [X2: A] : ( ord_less_eq @ B @ ( F2 @ X2 ) @ Z10 )
@ F4 ) ) ) ) ).
% filterlim_at_bot
thf(fact_7417_filterlim__at__bot__le,axiom,
! [A: $tType,B: $tType] :
( ( linorder @ B )
=> ! [F2: A > B,F4: filter @ A,C2: B] :
( ( filterlim @ A @ B @ F2 @ ( at_bot @ B ) @ F4 )
= ( ! [Z10: B] :
( ( ord_less_eq @ B @ Z10 @ C2 )
=> ( eventually @ A
@ ^ [X2: A] : ( ord_less_eq @ B @ ( F2 @ X2 ) @ Z10 )
@ F4 ) ) ) ) ) ).
% filterlim_at_bot_le
thf(fact_7418_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
@ ^ [X2: A] : ( ord_less @ B @ ( F2 @ X2 ) @ Z10 )
@ F4 ) ) ) ) ).
% filterlim_at_bot_dense
thf(fact_7419_real__tendsto__sandwich,axiom,
! [B: $tType,F2: B > real,G: B > real,Net: filter @ B,H2: B > real,C2: real] :
( ( eventually @ B
@ ^ [N2: B] : ( ord_less_eq @ real @ ( F2 @ N2 ) @ ( G @ N2 ) )
@ Net )
=> ( ( eventually @ B
@ ^ [N2: B] : ( ord_less_eq @ real @ ( G @ N2 ) @ ( H2 @ N2 ) )
@ Net )
=> ( ( filterlim @ B @ real @ F2 @ ( topolo7230453075368039082e_nhds @ real @ C2 ) @ Net )
=> ( ( filterlim @ B @ real @ H2 @ ( topolo7230453075368039082e_nhds @ real @ C2 ) @ Net )
=> ( filterlim @ B @ real @ G @ ( topolo7230453075368039082e_nhds @ real @ C2 ) @ Net ) ) ) ) ) ).
% real_tendsto_sandwich
thf(fact_7420_countable__basis__at__decseq,axiom,
! [A: $tType] :
( ( topolo3112930676232923870pology @ A )
=> ! [X: A] :
~ ! [A8: nat > ( set @ A )] :
( ! [I4: nat] : ( topolo1002775350975398744n_open @ A @ ( A8 @ I4 ) )
=> ( ! [I4: nat] : ( member @ A @ X @ ( A8 @ I4 ) )
=> ~ ! [S10: set @ A] :
( ( topolo1002775350975398744n_open @ A @ S10 )
=> ( ( member @ A @ X @ S10 )
=> ( eventually @ nat
@ ^ [I3: nat] : ( ord_less_eq @ ( set @ A ) @ ( A8 @ I3 ) @ S10 )
@ ( at_top @ nat ) ) ) ) ) ) ) ).
% countable_basis_at_decseq
thf(fact_7421_eventually__at__leftI,axiom,
! [A: $tType] :
( ( topolo2564578578187576103pology @ A )
=> ! [A2: A,B2: A,P: A > $o] :
( ! [X3: A] :
( ( member @ A @ X3 @ ( set_or5935395276787703475ssThan @ A @ A2 @ B2 ) )
=> ( P @ X3 ) )
=> ( ( ord_less @ A @ A2 @ B2 )
=> ( eventually @ A @ P @ ( topolo174197925503356063within @ A @ B2 @ ( set_ord_lessThan @ A @ B2 ) ) ) ) ) ) ).
% eventually_at_leftI
thf(fact_7422_eventually__at__rightI,axiom,
! [A: $tType] :
( ( topolo2564578578187576103pology @ A )
=> ! [A2: A,B2: A,P: A > $o] :
( ! [X3: A] :
( ( member @ A @ X3 @ ( set_or5935395276787703475ssThan @ A @ A2 @ B2 ) )
=> ( P @ X3 ) )
=> ( ( ord_less @ A @ A2 @ B2 )
=> ( eventually @ A @ P @ ( topolo174197925503356063within @ A @ A2 @ ( set_ord_greaterThan @ A @ A2 ) ) ) ) ) ) ).
% eventually_at_rightI
thf(fact_7423_eventually__at__to__0,axiom,
! [A: $tType] :
( ( real_V822414075346904944vector @ A )
=> ! [P: A > $o,A2: A] :
( ( eventually @ A @ P @ ( topolo174197925503356063within @ A @ A2 @ ( top_top @ ( set @ A ) ) ) )
= ( eventually @ A
@ ^ [X2: A] : ( P @ ( plus_plus @ A @ X2 @ A2 ) )
@ ( topolo174197925503356063within @ A @ ( zero_zero @ A ) @ ( top_top @ ( set @ A ) ) ) ) ) ) ).
% eventually_at_to_0
thf(fact_7424_increasing__tendsto,axiom,
! [A: $tType,B: $tType] :
( ( topolo2564578578187576103pology @ A )
=> ! [F2: B > A,L2: A,F4: filter @ B] :
( ( eventually @ B
@ ^ [N2: B] : ( ord_less_eq @ A @ ( F2 @ N2 ) @ L2 )
@ F4 )
=> ( ! [X3: A] :
( ( ord_less @ A @ X3 @ L2 )
=> ( eventually @ B
@ ^ [N2: B] : ( ord_less @ A @ X3 @ ( F2 @ N2 ) )
@ F4 ) )
=> ( filterlim @ B @ A @ F2 @ ( topolo7230453075368039082e_nhds @ A @ L2 ) @ F4 ) ) ) ) ).
% increasing_tendsto
thf(fact_7425_decreasing__tendsto,axiom,
! [A: $tType,B: $tType] :
( ( topolo2564578578187576103pology @ A )
=> ! [L2: A,F2: B > A,F4: filter @ B] :
( ( eventually @ B
@ ^ [N2: B] : ( ord_less_eq @ A @ L2 @ ( F2 @ N2 ) )
@ F4 )
=> ( ! [X3: A] :
( ( ord_less @ A @ L2 @ X3 )
=> ( eventually @ B
@ ^ [N2: B] : ( ord_less @ A @ ( F2 @ N2 ) @ X3 )
@ F4 ) )
=> ( filterlim @ B @ A @ F2 @ ( topolo7230453075368039082e_nhds @ A @ L2 ) @ F4 ) ) ) ) ).
% decreasing_tendsto
thf(fact_7426_filterlim__at__top__gt,axiom,
! [A: $tType,B: $tType] :
( ( unboun7993243217541854897norder @ B )
=> ! [F2: A > B,F4: filter @ A,C2: B] :
( ( filterlim @ A @ B @ F2 @ ( at_top @ B ) @ F4 )
= ( ! [Z10: B] :
( ( ord_less @ B @ C2 @ Z10 )
=> ( eventually @ A
@ ^ [X2: A] : ( ord_less_eq @ B @ Z10 @ ( F2 @ X2 ) )
@ F4 ) ) ) ) ) ).
% filterlim_at_top_gt
thf(fact_7427_DERIV__cong__ev,axiom,
! [A: $tType] :
( ( real_V3459762299906320749_field @ A )
=> ! [X: A,Y: A,F2: A > A,G: A > A,U: A,V: A] :
( ( X = Y )
=> ( ( eventually @ A
@ ^ [X2: A] :
( ( F2 @ X2 )
= ( G @ X2 ) )
@ ( topolo7230453075368039082e_nhds @ A @ X ) )
=> ( ( U = V )
=> ( ( has_field_derivative @ A @ F2 @ U @ ( topolo174197925503356063within @ A @ X @ ( top_top @ ( set @ A ) ) ) )
= ( has_field_derivative @ A @ G @ V @ ( topolo174197925503356063within @ A @ Y @ ( top_top @ ( set @ A ) ) ) ) ) ) ) ) ) ).
% DERIV_cong_ev
thf(fact_7428_filterlim__at__bot__lt,axiom,
! [A: $tType,B: $tType] :
( ( unboun7993243217541854897norder @ B )
=> ! [F2: A > B,F4: filter @ A,C2: B] :
( ( filterlim @ A @ B @ F2 @ ( at_bot @ B ) @ F4 )
= ( ! [Z10: B] :
( ( ord_less @ B @ Z10 @ C2 )
=> ( eventually @ A
@ ^ [X2: A] : ( ord_less_eq @ B @ ( F2 @ X2 ) @ Z10 )
@ F4 ) ) ) ) ) ).
% filterlim_at_bot_lt
thf(fact_7429_metric__tendsto__imp__tendsto,axiom,
! [A: $tType,B: $tType,C: $tType] :
( ( ( real_V7819770556892013058_space @ B )
& ( real_V7819770556892013058_space @ A ) )
=> ! [F2: C > A,A2: A,F4: filter @ C,G: C > B,B2: B] :
( ( filterlim @ C @ A @ F2 @ ( topolo7230453075368039082e_nhds @ A @ A2 ) @ F4 )
=> ( ( eventually @ C
@ ^ [X2: C] : ( ord_less_eq @ real @ ( real_V557655796197034286t_dist @ B @ ( G @ X2 ) @ B2 ) @ ( real_V557655796197034286t_dist @ A @ ( F2 @ X2 ) @ A2 ) )
@ F4 )
=> ( filterlim @ C @ B @ G @ ( topolo7230453075368039082e_nhds @ B @ B2 ) @ F4 ) ) ) ) ).
% metric_tendsto_imp_tendsto
thf(fact_7430_eventually__at__right__to__0,axiom,
! [P: real > $o,A2: real] :
( ( eventually @ real @ P @ ( topolo174197925503356063within @ real @ A2 @ ( set_ord_greaterThan @ real @ A2 ) ) )
= ( eventually @ real
@ ^ [X2: real] : ( P @ ( plus_plus @ real @ X2 @ A2 ) )
@ ( topolo174197925503356063within @ real @ ( zero_zero @ real ) @ ( set_ord_greaterThan @ real @ ( zero_zero @ real ) ) ) ) ) ).
% eventually_at_right_to_0
thf(fact_7431_continuous__arcosh__strong,axiom,
! [A: $tType] :
( ( topological_t2_space @ A )
=> ! [F4: filter @ A,F2: A > real] :
( ( topolo3448309680560233919inuous @ A @ real @ F4 @ F2 )
=> ( ( eventually @ A
@ ^ [X2: A] : ( ord_less_eq @ real @ ( one_one @ real ) @ ( F2 @ X2 ) )
@ F4 )
=> ( topolo3448309680560233919inuous @ A @ real @ F4
@ ^ [X2: A] : ( arcosh @ real @ ( F2 @ X2 ) ) ) ) ) ) ).
% continuous_arcosh_strong
thf(fact_7432_eventually__at__le,axiom,
! [A: $tType] :
( ( real_V7819770556892013058_space @ A )
=> ! [P: A > $o,A2: A,S3: set @ A] :
( ( eventually @ A @ P @ ( topolo174197925503356063within @ A @ A2 @ S3 ) )
= ( ? [D2: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ D2 )
& ! [X2: A] :
( ( member @ A @ X2 @ S3 )
=> ( ( ( X2 != A2 )
& ( ord_less_eq @ real @ ( real_V557655796197034286t_dist @ A @ X2 @ A2 ) @ D2 ) )
=> ( P @ X2 ) ) ) ) ) ) ) ).
% eventually_at_le
thf(fact_7433_eventually__at__infinity__pos,axiom,
! [A: $tType] :
( ( real_V822414075346904944vector @ A )
=> ! [P4: A > $o] :
( ( eventually @ A @ P4 @ ( at_infinity @ A ) )
= ( ? [B5: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ B5 )
& ! [X2: A] :
( ( ord_less_eq @ real @ B5 @ ( real_V7770717601297561774m_norm @ A @ X2 ) )
=> ( P4 @ X2 ) ) ) ) ) ) ).
% eventually_at_infinity_pos
thf(fact_7434_tendsto__imp__filterlim__at__left,axiom,
! [B: $tType,A: $tType] :
( ( topolo2564578578187576103pology @ B )
=> ! [F2: A > B,L6: B,F4: filter @ A] :
( ( filterlim @ A @ B @ F2 @ ( topolo7230453075368039082e_nhds @ B @ L6 ) @ F4 )
=> ( ( eventually @ A
@ ^ [X2: A] : ( ord_less @ B @ ( F2 @ X2 ) @ L6 )
@ F4 )
=> ( filterlim @ A @ B @ F2 @ ( topolo174197925503356063within @ B @ L6 @ ( set_ord_lessThan @ B @ L6 ) ) @ F4 ) ) ) ) ).
% tendsto_imp_filterlim_at_left
thf(fact_7435_tendsto__imp__filterlim__at__right,axiom,
! [B: $tType,A: $tType] :
( ( topolo2564578578187576103pology @ B )
=> ! [F2: A > B,L6: B,F4: filter @ A] :
( ( filterlim @ A @ B @ F2 @ ( topolo7230453075368039082e_nhds @ B @ L6 ) @ F4 )
=> ( ( eventually @ A
@ ^ [X2: A] : ( ord_less @ B @ L6 @ ( F2 @ X2 ) )
@ F4 )
=> ( filterlim @ A @ B @ F2 @ ( topolo174197925503356063within @ B @ L6 @ ( set_ord_greaterThan @ B @ L6 ) ) @ F4 ) ) ) ) ).
% tendsto_imp_filterlim_at_right
thf(fact_7436_summable__comparison__test__ev,axiom,
! [A: $tType] :
( ( real_Vector_banach @ A )
=> ! [F2: nat > A,G: nat > real] :
( ( eventually @ nat
@ ^ [N2: nat] : ( ord_less_eq @ real @ ( real_V7770717601297561774m_norm @ A @ ( F2 @ N2 ) ) @ ( G @ N2 ) )
@ ( at_top @ nat ) )
=> ( ( summable @ real @ G )
=> ( summable @ A @ F2 ) ) ) ) ).
% summable_comparison_test_ev
thf(fact_7437_tendsto__arcosh__strong,axiom,
! [B: $tType,F2: B > real,A2: real,F4: filter @ B] :
( ( filterlim @ B @ real @ F2 @ ( topolo7230453075368039082e_nhds @ real @ A2 ) @ F4 )
=> ( ( ord_less_eq @ real @ ( one_one @ real ) @ A2 )
=> ( ( eventually @ B
@ ^ [X2: B] : ( ord_less_eq @ real @ ( one_one @ real ) @ ( F2 @ X2 ) )
@ F4 )
=> ( filterlim @ B @ real
@ ^ [X2: B] : ( arcosh @ real @ ( F2 @ X2 ) )
@ ( topolo7230453075368039082e_nhds @ real @ ( arcosh @ real @ A2 ) )
@ F4 ) ) ) ) ).
% tendsto_arcosh_strong
thf(fact_7438_filterlim__at__top__at__left,axiom,
! [B: $tType,A: $tType] :
( ( ( topolo1944317154257567458pology @ A )
& ( linorder @ B ) )
=> ! [Q: A > $o,F2: A > B,P: B > $o,G: B > A,A2: A] :
( ! [X3: A,Y3: A] :
( ( Q @ X3 )
=> ( ( Q @ Y3 )
=> ( ( ord_less_eq @ A @ X3 @ Y3 )
=> ( ord_less_eq @ B @ ( F2 @ X3 ) @ ( F2 @ Y3 ) ) ) ) )
=> ( ! [X3: B] :
( ( P @ X3 )
=> ( ( F2 @ ( G @ X3 ) )
= X3 ) )
=> ( ! [X3: B] :
( ( P @ X3 )
=> ( Q @ ( G @ X3 ) ) )
=> ( ( eventually @ A @ Q @ ( topolo174197925503356063within @ A @ A2 @ ( set_ord_lessThan @ A @ A2 ) ) )
=> ( ! [B3: A] :
( ( Q @ B3 )
=> ( ord_less @ A @ B3 @ A2 ) )
=> ( ( eventually @ B @ P @ ( at_top @ B ) )
=> ( filterlim @ A @ B @ F2 @ ( at_top @ B ) @ ( topolo174197925503356063within @ A @ A2 @ ( set_ord_lessThan @ A @ A2 ) ) ) ) ) ) ) ) ) ) ).
% filterlim_at_top_at_left
thf(fact_7439_filterlim__at__bot__at__right,axiom,
! [B: $tType,A: $tType] :
( ( ( topolo1944317154257567458pology @ A )
& ( linorder @ B ) )
=> ! [Q: A > $o,F2: A > B,P: B > $o,G: B > A,A2: A] :
( ! [X3: A,Y3: A] :
( ( Q @ X3 )
=> ( ( Q @ Y3 )
=> ( ( ord_less_eq @ A @ X3 @ Y3 )
=> ( ord_less_eq @ B @ ( F2 @ X3 ) @ ( F2 @ Y3 ) ) ) ) )
=> ( ! [X3: B] :
( ( P @ X3 )
=> ( ( F2 @ ( G @ X3 ) )
= X3 ) )
=> ( ! [X3: B] :
( ( P @ X3 )
=> ( Q @ ( G @ X3 ) ) )
=> ( ( eventually @ A @ Q @ ( topolo174197925503356063within @ A @ A2 @ ( set_ord_greaterThan @ A @ A2 ) ) )
=> ( ! [B3: A] :
( ( Q @ B3 )
=> ( ord_less @ A @ A2 @ B3 ) )
=> ( ( eventually @ B @ P @ ( at_bot @ B ) )
=> ( filterlim @ A @ B @ F2 @ ( at_bot @ B ) @ ( topolo174197925503356063within @ A @ A2 @ ( set_ord_greaterThan @ A @ A2 ) ) ) ) ) ) ) ) ) ) ).
% filterlim_at_bot_at_right
thf(fact_7440_filterlim__at__withinI,axiom,
! [A: $tType,B: $tType] :
( ( topolo4958980785337419405_space @ A )
=> ! [F2: B > A,C2: A,F4: filter @ B,A3: set @ A] :
( ( filterlim @ B @ A @ F2 @ ( topolo7230453075368039082e_nhds @ A @ C2 ) @ F4 )
=> ( ( eventually @ B
@ ^ [X2: B] : ( member @ A @ ( F2 @ X2 ) @ ( minus_minus @ ( set @ A ) @ A3 @ ( insert @ A @ C2 @ ( bot_bot @ ( set @ A ) ) ) ) )
@ F4 )
=> ( filterlim @ B @ A @ F2 @ ( topolo174197925503356063within @ A @ C2 @ A3 ) @ F4 ) ) ) ) ).
% filterlim_at_withinI
thf(fact_7441_tendsto__0__le,axiom,
! [B: $tType,C: $tType,A: $tType] :
( ( ( real_V822414075346904944vector @ C )
& ( real_V822414075346904944vector @ B ) )
=> ! [F2: A > B,F4: filter @ A,G: A > C,K6: real] :
( ( filterlim @ A @ B @ F2 @ ( topolo7230453075368039082e_nhds @ B @ ( zero_zero @ B ) ) @ F4 )
=> ( ( eventually @ A
@ ^ [X2: A] : ( ord_less_eq @ real @ ( real_V7770717601297561774m_norm @ C @ ( G @ X2 ) ) @ ( times_times @ real @ ( real_V7770717601297561774m_norm @ B @ ( F2 @ X2 ) ) @ K6 ) )
@ F4 )
=> ( filterlim @ A @ C @ G @ ( topolo7230453075368039082e_nhds @ C @ ( zero_zero @ C ) ) @ F4 ) ) ) ) ).
% tendsto_0_le
thf(fact_7442_eventually__floor__less,axiom,
! [B: $tType,A: $tType] :
( ( ( archim2362893244070406136eiling @ B )
& ( topolo2564578578187576103pology @ B ) )
=> ! [F2: A > B,L2: B,F4: filter @ A] :
( ( filterlim @ A @ B @ F2 @ ( topolo7230453075368039082e_nhds @ B @ L2 ) @ F4 )
=> ( ~ ( member @ B @ L2 @ ( ring_1_Ints @ B ) )
=> ( eventually @ A
@ ^ [X2: A] : ( ord_less @ B @ ( ring_1_of_int @ B @ ( archim6421214686448440834_floor @ B @ L2 ) ) @ ( F2 @ X2 ) )
@ F4 ) ) ) ) ).
% eventually_floor_less
thf(fact_7443_eventually__less__ceiling,axiom,
! [B: $tType,A: $tType] :
( ( ( archim2362893244070406136eiling @ B )
& ( topolo2564578578187576103pology @ B ) )
=> ! [F2: A > B,L2: B,F4: filter @ A] :
( ( filterlim @ A @ B @ F2 @ ( topolo7230453075368039082e_nhds @ B @ L2 ) @ F4 )
=> ( ~ ( member @ B @ L2 @ ( ring_1_Ints @ B ) )
=> ( eventually @ A
@ ^ [X2: A] : ( ord_less @ B @ ( F2 @ X2 ) @ ( ring_1_of_int @ B @ ( archimedean_ceiling @ B @ L2 ) ) )
@ F4 ) ) ) ) ).
% eventually_less_ceiling
thf(fact_7444_filterlim__at__infinity,axiom,
! [C: $tType,A: $tType] :
( ( real_V822414075346904944vector @ A )
=> ! [C2: real,F2: C > A,F4: filter @ C] :
( ( ord_less_eq @ real @ ( zero_zero @ real ) @ C2 )
=> ( ( filterlim @ C @ A @ F2 @ ( at_infinity @ A ) @ F4 )
= ( ! [R5: real] :
( ( ord_less @ real @ C2 @ R5 )
=> ( eventually @ C
@ ^ [X2: C] : ( ord_less_eq @ real @ R5 @ ( real_V7770717601297561774m_norm @ A @ ( F2 @ X2 ) ) )
@ F4 ) ) ) ) ) ) ).
% filterlim_at_infinity
thf(fact_7445_tendsto__zero__powrI,axiom,
! [A: $tType,F2: A > real,F4: filter @ A,G: A > real,B2: real] :
( ( filterlim @ A @ real @ F2 @ ( topolo7230453075368039082e_nhds @ real @ ( zero_zero @ real ) ) @ F4 )
=> ( ( filterlim @ A @ real @ G @ ( topolo7230453075368039082e_nhds @ real @ B2 ) @ F4 )
=> ( ( eventually @ A
@ ^ [X2: A] : ( ord_less_eq @ real @ ( zero_zero @ real ) @ ( F2 @ X2 ) )
@ F4 )
=> ( ( ord_less @ real @ ( zero_zero @ real ) @ B2 )
=> ( filterlim @ A @ real
@ ^ [X2: A] : ( powr @ real @ ( F2 @ X2 ) @ ( G @ X2 ) )
@ ( topolo7230453075368039082e_nhds @ real @ ( zero_zero @ real ) )
@ F4 ) ) ) ) ) ).
% tendsto_zero_powrI
thf(fact_7446_tendsto__powr2,axiom,
! [A: $tType,F2: A > real,A2: real,F4: filter @ A,G: A > real,B2: real] :
( ( filterlim @ A @ real @ F2 @ ( topolo7230453075368039082e_nhds @ real @ A2 ) @ F4 )
=> ( ( filterlim @ A @ real @ G @ ( topolo7230453075368039082e_nhds @ real @ B2 ) @ F4 )
=> ( ( eventually @ A
@ ^ [X2: A] : ( ord_less_eq @ real @ ( zero_zero @ real ) @ ( F2 @ X2 ) )
@ F4 )
=> ( ( ord_less @ real @ ( zero_zero @ real ) @ B2 )
=> ( filterlim @ A @ real
@ ^ [X2: A] : ( powr @ real @ ( F2 @ X2 ) @ ( G @ X2 ) )
@ ( topolo7230453075368039082e_nhds @ real @ ( powr @ real @ A2 @ B2 ) )
@ F4 ) ) ) ) ) ).
% tendsto_powr2
thf(fact_7447_tendsto__powr_H,axiom,
! [A: $tType,F2: A > real,A2: real,F4: filter @ A,G: A > real,B2: real] :
( ( filterlim @ A @ real @ F2 @ ( topolo7230453075368039082e_nhds @ real @ A2 ) @ F4 )
=> ( ( filterlim @ A @ real @ G @ ( topolo7230453075368039082e_nhds @ real @ B2 ) @ F4 )
=> ( ( ( A2
!= ( zero_zero @ real ) )
| ( ( ord_less @ real @ ( zero_zero @ real ) @ B2 )
& ( eventually @ A
@ ^ [X2: A] : ( ord_less_eq @ real @ ( zero_zero @ real ) @ ( F2 @ X2 ) )
@ F4 ) ) )
=> ( filterlim @ A @ real
@ ^ [X2: A] : ( powr @ real @ ( F2 @ X2 ) @ ( G @ X2 ) )
@ ( topolo7230453075368039082e_nhds @ real @ ( powr @ real @ A2 @ B2 ) )
@ F4 ) ) ) ) ).
% tendsto_powr'
thf(fact_7448_LIM__at__top__divide,axiom,
! [A: $tType,F2: A > real,A2: real,F4: filter @ A,G: A > real] :
( ( filterlim @ A @ real @ F2 @ ( topolo7230453075368039082e_nhds @ real @ A2 ) @ F4 )
=> ( ( ord_less @ real @ ( zero_zero @ real ) @ A2 )
=> ( ( filterlim @ A @ real @ G @ ( topolo7230453075368039082e_nhds @ real @ ( zero_zero @ real ) ) @ F4 )
=> ( ( eventually @ A
@ ^ [X2: A] : ( ord_less @ real @ ( zero_zero @ real ) @ ( G @ X2 ) )
@ F4 )
=> ( filterlim @ A @ real
@ ^ [X2: A] : ( divide_divide @ real @ ( F2 @ X2 ) @ ( G @ X2 ) )
@ ( at_top @ real )
@ F4 ) ) ) ) ) ).
% LIM_at_top_divide
thf(fact_7449_lhopital__at__top__at__top,axiom,
! [F2: real > real,A2: real,G: real > real,F5: real > real,G5: real > real] :
( ( filterlim @ real @ real @ F2 @ ( at_top @ real ) @ ( topolo174197925503356063within @ real @ A2 @ ( top_top @ ( set @ real ) ) ) )
=> ( ( filterlim @ real @ real @ G @ ( at_top @ real ) @ ( topolo174197925503356063within @ real @ A2 @ ( top_top @ ( set @ real ) ) ) )
=> ( ( eventually @ real
@ ^ [X2: real] : ( has_field_derivative @ real @ F2 @ ( F5 @ X2 ) @ ( topolo174197925503356063within @ real @ X2 @ ( top_top @ ( set @ real ) ) ) )
@ ( topolo174197925503356063within @ real @ A2 @ ( top_top @ ( set @ real ) ) ) )
=> ( ( eventually @ real
@ ^ [X2: real] : ( has_field_derivative @ real @ G @ ( G5 @ X2 ) @ ( topolo174197925503356063within @ real @ X2 @ ( top_top @ ( set @ real ) ) ) )
@ ( topolo174197925503356063within @ real @ A2 @ ( top_top @ ( set @ real ) ) ) )
=> ( ( filterlim @ real @ real
@ ^ [X2: real] : ( divide_divide @ real @ ( F5 @ X2 ) @ ( G5 @ X2 ) )
@ ( at_top @ real )
@ ( topolo174197925503356063within @ real @ A2 @ ( top_top @ ( set @ real ) ) ) )
=> ( filterlim @ real @ real
@ ^ [X2: real] : ( divide_divide @ real @ ( F2 @ X2 ) @ ( G @ X2 ) )
@ ( at_top @ real )
@ ( topolo174197925503356063within @ real @ A2 @ ( top_top @ ( set @ real ) ) ) ) ) ) ) ) ) ).
% lhopital_at_top_at_top
thf(fact_7450_lhopital__left__at__top__at__top,axiom,
! [F2: real > real,A2: real,G: real > real,F5: real > real,G5: real > real] :
( ( filterlim @ real @ real @ F2 @ ( at_top @ real ) @ ( topolo174197925503356063within @ real @ A2 @ ( set_ord_lessThan @ real @ A2 ) ) )
=> ( ( filterlim @ real @ real @ G @ ( at_top @ real ) @ ( topolo174197925503356063within @ real @ A2 @ ( set_ord_lessThan @ real @ A2 ) ) )
=> ( ( eventually @ real
@ ^ [X2: real] : ( has_field_derivative @ real @ F2 @ ( F5 @ X2 ) @ ( topolo174197925503356063within @ real @ X2 @ ( top_top @ ( set @ real ) ) ) )
@ ( topolo174197925503356063within @ real @ A2 @ ( set_ord_lessThan @ real @ A2 ) ) )
=> ( ( eventually @ real
@ ^ [X2: real] : ( has_field_derivative @ real @ G @ ( G5 @ X2 ) @ ( topolo174197925503356063within @ real @ X2 @ ( top_top @ ( set @ real ) ) ) )
@ ( topolo174197925503356063within @ real @ A2 @ ( set_ord_lessThan @ real @ A2 ) ) )
=> ( ( filterlim @ real @ real
@ ^ [X2: real] : ( divide_divide @ real @ ( F5 @ X2 ) @ ( G5 @ X2 ) )
@ ( at_top @ real )
@ ( topolo174197925503356063within @ real @ A2 @ ( set_ord_lessThan @ real @ A2 ) ) )
=> ( filterlim @ real @ real
@ ^ [X2: real] : ( divide_divide @ real @ ( F2 @ X2 ) @ ( G @ X2 ) )
@ ( at_top @ real )
@ ( topolo174197925503356063within @ real @ A2 @ ( set_ord_lessThan @ real @ A2 ) ) ) ) ) ) ) ) ).
% lhopital_left_at_top_at_top
thf(fact_7451_lhopital,axiom,
! [F2: real > real,X: real,G: real > real,G5: real > real,F5: real > real,F4: filter @ real] :
( ( filterlim @ real @ real @ F2 @ ( topolo7230453075368039082e_nhds @ real @ ( zero_zero @ real ) ) @ ( topolo174197925503356063within @ real @ X @ ( top_top @ ( set @ real ) ) ) )
=> ( ( filterlim @ real @ real @ G @ ( topolo7230453075368039082e_nhds @ real @ ( zero_zero @ real ) ) @ ( topolo174197925503356063within @ real @ X @ ( top_top @ ( set @ real ) ) ) )
=> ( ( eventually @ real
@ ^ [X2: real] :
( ( G @ X2 )
!= ( zero_zero @ real ) )
@ ( topolo174197925503356063within @ real @ X @ ( top_top @ ( set @ real ) ) ) )
=> ( ( eventually @ real
@ ^ [X2: real] :
( ( G5 @ X2 )
!= ( zero_zero @ real ) )
@ ( topolo174197925503356063within @ real @ X @ ( top_top @ ( set @ real ) ) ) )
=> ( ( eventually @ real
@ ^ [X2: real] : ( has_field_derivative @ real @ F2 @ ( F5 @ X2 ) @ ( topolo174197925503356063within @ real @ X2 @ ( top_top @ ( set @ real ) ) ) )
@ ( topolo174197925503356063within @ real @ X @ ( top_top @ ( set @ real ) ) ) )
=> ( ( eventually @ real
@ ^ [X2: real] : ( has_field_derivative @ real @ G @ ( G5 @ X2 ) @ ( topolo174197925503356063within @ real @ X2 @ ( top_top @ ( set @ real ) ) ) )
@ ( topolo174197925503356063within @ real @ X @ ( top_top @ ( set @ real ) ) ) )
=> ( ( filterlim @ real @ real
@ ^ [X2: real] : ( divide_divide @ real @ ( F5 @ X2 ) @ ( G5 @ X2 ) )
@ F4
@ ( topolo174197925503356063within @ real @ X @ ( top_top @ ( set @ real ) ) ) )
=> ( filterlim @ real @ real
@ ^ [X2: real] : ( divide_divide @ real @ ( F2 @ X2 ) @ ( G @ X2 ) )
@ F4
@ ( topolo174197925503356063within @ real @ X @ ( top_top @ ( set @ real ) ) ) ) ) ) ) ) ) ) ) ).
% lhopital
thf(fact_7452_lhopital__left,axiom,
! [F2: real > real,X: real,G: real > real,G5: real > real,F5: real > real,F4: filter @ real] :
( ( filterlim @ real @ real @ F2 @ ( topolo7230453075368039082e_nhds @ real @ ( zero_zero @ real ) ) @ ( topolo174197925503356063within @ real @ X @ ( set_ord_lessThan @ real @ X ) ) )
=> ( ( filterlim @ real @ real @ G @ ( topolo7230453075368039082e_nhds @ real @ ( zero_zero @ real ) ) @ ( topolo174197925503356063within @ real @ X @ ( set_ord_lessThan @ real @ X ) ) )
=> ( ( eventually @ real
@ ^ [X2: real] :
( ( G @ X2 )
!= ( zero_zero @ real ) )
@ ( topolo174197925503356063within @ real @ X @ ( set_ord_lessThan @ real @ X ) ) )
=> ( ( eventually @ real
@ ^ [X2: real] :
( ( G5 @ X2 )
!= ( zero_zero @ real ) )
@ ( topolo174197925503356063within @ real @ X @ ( set_ord_lessThan @ real @ X ) ) )
=> ( ( eventually @ real
@ ^ [X2: real] : ( has_field_derivative @ real @ F2 @ ( F5 @ X2 ) @ ( topolo174197925503356063within @ real @ X2 @ ( top_top @ ( set @ real ) ) ) )
@ ( topolo174197925503356063within @ real @ X @ ( set_ord_lessThan @ real @ X ) ) )
=> ( ( eventually @ real
@ ^ [X2: real] : ( has_field_derivative @ real @ G @ ( G5 @ X2 ) @ ( topolo174197925503356063within @ real @ X2 @ ( top_top @ ( set @ real ) ) ) )
@ ( topolo174197925503356063within @ real @ X @ ( set_ord_lessThan @ real @ X ) ) )
=> ( ( filterlim @ real @ real
@ ^ [X2: real] : ( divide_divide @ real @ ( F5 @ X2 ) @ ( G5 @ X2 ) )
@ F4
@ ( topolo174197925503356063within @ real @ X @ ( set_ord_lessThan @ real @ X ) ) )
=> ( filterlim @ real @ real
@ ^ [X2: real] : ( divide_divide @ real @ ( F2 @ X2 ) @ ( G @ X2 ) )
@ F4
@ ( topolo174197925503356063within @ real @ X @ ( set_ord_lessThan @ real @ X ) ) ) ) ) ) ) ) ) ) ).
% lhopital_left
thf(fact_7453_lhopital__right__at__top__at__top,axiom,
! [F2: real > real,A2: real,G: real > real,F5: real > real,G5: real > real] :
( ( filterlim @ real @ real @ F2 @ ( at_top @ real ) @ ( topolo174197925503356063within @ real @ A2 @ ( set_ord_greaterThan @ real @ A2 ) ) )
=> ( ( filterlim @ real @ real @ G @ ( at_top @ real ) @ ( topolo174197925503356063within @ real @ A2 @ ( set_ord_greaterThan @ real @ A2 ) ) )
=> ( ( eventually @ real
@ ^ [X2: real] : ( has_field_derivative @ real @ F2 @ ( F5 @ X2 ) @ ( topolo174197925503356063within @ real @ X2 @ ( top_top @ ( set @ real ) ) ) )
@ ( topolo174197925503356063within @ real @ A2 @ ( set_ord_greaterThan @ real @ A2 ) ) )
=> ( ( eventually @ real
@ ^ [X2: real] : ( has_field_derivative @ real @ G @ ( G5 @ X2 ) @ ( topolo174197925503356063within @ real @ X2 @ ( top_top @ ( set @ real ) ) ) )
@ ( topolo174197925503356063within @ real @ A2 @ ( set_ord_greaterThan @ real @ A2 ) ) )
=> ( ( filterlim @ real @ real
@ ^ [X2: real] : ( divide_divide @ real @ ( F5 @ X2 ) @ ( G5 @ X2 ) )
@ ( at_top @ real )
@ ( topolo174197925503356063within @ real @ A2 @ ( set_ord_greaterThan @ real @ A2 ) ) )
=> ( filterlim @ real @ real
@ ^ [X2: real] : ( divide_divide @ real @ ( F2 @ X2 ) @ ( G @ X2 ) )
@ ( at_top @ real )
@ ( topolo174197925503356063within @ real @ A2 @ ( set_ord_greaterThan @ real @ A2 ) ) ) ) ) ) ) ) ).
% lhopital_right_at_top_at_top
thf(fact_7454_lhopital__at__top__at__bot,axiom,
! [F2: real > real,A2: real,G: real > real,F5: real > real,G5: real > real] :
( ( filterlim @ real @ real @ F2 @ ( at_top @ real ) @ ( topolo174197925503356063within @ real @ A2 @ ( top_top @ ( set @ real ) ) ) )
=> ( ( filterlim @ real @ real @ G @ ( at_bot @ real ) @ ( topolo174197925503356063within @ real @ A2 @ ( top_top @ ( set @ real ) ) ) )
=> ( ( eventually @ real
@ ^ [X2: real] : ( has_field_derivative @ real @ F2 @ ( F5 @ X2 ) @ ( topolo174197925503356063within @ real @ X2 @ ( top_top @ ( set @ real ) ) ) )
@ ( topolo174197925503356063within @ real @ A2 @ ( top_top @ ( set @ real ) ) ) )
=> ( ( eventually @ real
@ ^ [X2: real] : ( has_field_derivative @ real @ G @ ( G5 @ X2 ) @ ( topolo174197925503356063within @ real @ X2 @ ( top_top @ ( set @ real ) ) ) )
@ ( topolo174197925503356063within @ real @ A2 @ ( top_top @ ( set @ real ) ) ) )
=> ( ( filterlim @ real @ real
@ ^ [X2: real] : ( divide_divide @ real @ ( F5 @ X2 ) @ ( G5 @ X2 ) )
@ ( at_bot @ real )
@ ( topolo174197925503356063within @ real @ A2 @ ( top_top @ ( set @ real ) ) ) )
=> ( filterlim @ real @ real
@ ^ [X2: real] : ( divide_divide @ real @ ( F2 @ X2 ) @ ( G @ X2 ) )
@ ( at_bot @ real )
@ ( topolo174197925503356063within @ real @ A2 @ ( top_top @ ( set @ real ) ) ) ) ) ) ) ) ) ).
% lhopital_at_top_at_bot
thf(fact_7455_lhopital__left__at__top__at__bot,axiom,
! [F2: real > real,A2: real,G: real > real,F5: real > real,G5: real > real] :
( ( filterlim @ real @ real @ F2 @ ( at_top @ real ) @ ( topolo174197925503356063within @ real @ A2 @ ( set_ord_lessThan @ real @ A2 ) ) )
=> ( ( filterlim @ real @ real @ G @ ( at_bot @ real ) @ ( topolo174197925503356063within @ real @ A2 @ ( set_ord_lessThan @ real @ A2 ) ) )
=> ( ( eventually @ real
@ ^ [X2: real] : ( has_field_derivative @ real @ F2 @ ( F5 @ X2 ) @ ( topolo174197925503356063within @ real @ X2 @ ( top_top @ ( set @ real ) ) ) )
@ ( topolo174197925503356063within @ real @ A2 @ ( set_ord_lessThan @ real @ A2 ) ) )
=> ( ( eventually @ real
@ ^ [X2: real] : ( has_field_derivative @ real @ G @ ( G5 @ X2 ) @ ( topolo174197925503356063within @ real @ X2 @ ( top_top @ ( set @ real ) ) ) )
@ ( topolo174197925503356063within @ real @ A2 @ ( set_ord_lessThan @ real @ A2 ) ) )
=> ( ( filterlim @ real @ real
@ ^ [X2: real] : ( divide_divide @ real @ ( F5 @ X2 ) @ ( G5 @ X2 ) )
@ ( at_bot @ real )
@ ( topolo174197925503356063within @ real @ A2 @ ( set_ord_lessThan @ real @ A2 ) ) )
=> ( filterlim @ real @ real
@ ^ [X2: real] : ( divide_divide @ real @ ( F2 @ X2 ) @ ( G @ X2 ) )
@ ( at_bot @ real )
@ ( topolo174197925503356063within @ real @ A2 @ ( set_ord_lessThan @ real @ A2 ) ) ) ) ) ) ) ) ).
% lhopital_left_at_top_at_bot
thf(fact_7456_lhopital__left__at__top,axiom,
! [G: real > real,X: real,G5: real > real,F2: real > real,F5: real > real,Y: real] :
( ( filterlim @ real @ real @ G @ ( at_top @ real ) @ ( topolo174197925503356063within @ real @ X @ ( set_ord_lessThan @ real @ X ) ) )
=> ( ( eventually @ real
@ ^ [X2: real] :
( ( G5 @ X2 )
!= ( zero_zero @ real ) )
@ ( topolo174197925503356063within @ real @ X @ ( set_ord_lessThan @ real @ X ) ) )
=> ( ( eventually @ real
@ ^ [X2: real] : ( has_field_derivative @ real @ F2 @ ( F5 @ X2 ) @ ( topolo174197925503356063within @ real @ X2 @ ( top_top @ ( set @ real ) ) ) )
@ ( topolo174197925503356063within @ real @ X @ ( set_ord_lessThan @ real @ X ) ) )
=> ( ( eventually @ real
@ ^ [X2: real] : ( has_field_derivative @ real @ G @ ( G5 @ X2 ) @ ( topolo174197925503356063within @ real @ X2 @ ( top_top @ ( set @ real ) ) ) )
@ ( topolo174197925503356063within @ real @ X @ ( set_ord_lessThan @ real @ X ) ) )
=> ( ( filterlim @ real @ real
@ ^ [X2: real] : ( divide_divide @ real @ ( F5 @ X2 ) @ ( G5 @ X2 ) )
@ ( topolo7230453075368039082e_nhds @ real @ Y )
@ ( topolo174197925503356063within @ real @ X @ ( set_ord_lessThan @ real @ X ) ) )
=> ( filterlim @ real @ real
@ ^ [X2: real] : ( divide_divide @ real @ ( F2 @ X2 ) @ ( G @ X2 ) )
@ ( topolo7230453075368039082e_nhds @ real @ Y )
@ ( topolo174197925503356063within @ real @ X @ ( set_ord_lessThan @ real @ X ) ) ) ) ) ) ) ) ).
% lhopital_left_at_top
thf(fact_7457_lhopital__at__top,axiom,
! [G: real > real,X: real,G5: real > real,F2: real > real,F5: real > real,Y: real] :
( ( filterlim @ real @ real @ G @ ( at_top @ real ) @ ( topolo174197925503356063within @ real @ X @ ( top_top @ ( set @ real ) ) ) )
=> ( ( eventually @ real
@ ^ [X2: real] :
( ( G5 @ X2 )
!= ( zero_zero @ real ) )
@ ( topolo174197925503356063within @ real @ X @ ( top_top @ ( set @ real ) ) ) )
=> ( ( eventually @ real
@ ^ [X2: real] : ( has_field_derivative @ real @ F2 @ ( F5 @ X2 ) @ ( topolo174197925503356063within @ real @ X2 @ ( top_top @ ( set @ real ) ) ) )
@ ( topolo174197925503356063within @ real @ X @ ( top_top @ ( set @ real ) ) ) )
=> ( ( eventually @ real
@ ^ [X2: real] : ( has_field_derivative @ real @ G @ ( G5 @ X2 ) @ ( topolo174197925503356063within @ real @ X2 @ ( top_top @ ( set @ real ) ) ) )
@ ( topolo174197925503356063within @ real @ X @ ( top_top @ ( set @ real ) ) ) )
=> ( ( filterlim @ real @ real
@ ^ [X2: real] : ( divide_divide @ real @ ( F5 @ X2 ) @ ( G5 @ X2 ) )
@ ( topolo7230453075368039082e_nhds @ real @ Y )
@ ( topolo174197925503356063within @ real @ X @ ( top_top @ ( set @ real ) ) ) )
=> ( filterlim @ real @ real
@ ^ [X2: real] : ( divide_divide @ real @ ( F2 @ X2 ) @ ( G @ X2 ) )
@ ( topolo7230453075368039082e_nhds @ real @ Y )
@ ( topolo174197925503356063within @ real @ X @ ( top_top @ ( set @ real ) ) ) ) ) ) ) ) ) ).
% lhopital_at_top
thf(fact_7458_lhospital__at__top__at__top,axiom,
! [G: real > real,G5: real > real,F2: real > real,F5: real > real,X: real] :
( ( filterlim @ real @ real @ G @ ( at_top @ real ) @ ( at_top @ real ) )
=> ( ( eventually @ real
@ ^ [X2: real] :
( ( G5 @ X2 )
!= ( zero_zero @ real ) )
@ ( at_top @ real ) )
=> ( ( eventually @ real
@ ^ [X2: real] : ( has_field_derivative @ real @ F2 @ ( F5 @ X2 ) @ ( topolo174197925503356063within @ real @ X2 @ ( top_top @ ( set @ real ) ) ) )
@ ( at_top @ real ) )
=> ( ( eventually @ real
@ ^ [X2: real] : ( has_field_derivative @ real @ G @ ( G5 @ X2 ) @ ( topolo174197925503356063within @ real @ X2 @ ( top_top @ ( set @ real ) ) ) )
@ ( at_top @ real ) )
=> ( ( filterlim @ real @ real
@ ^ [X2: real] : ( divide_divide @ real @ ( F5 @ X2 ) @ ( G5 @ X2 ) )
@ ( topolo7230453075368039082e_nhds @ real @ X )
@ ( at_top @ real ) )
=> ( filterlim @ real @ real
@ ^ [X2: real] : ( divide_divide @ real @ ( F2 @ X2 ) @ ( G @ X2 ) )
@ ( topolo7230453075368039082e_nhds @ real @ X )
@ ( at_top @ real ) ) ) ) ) ) ) ).
% lhospital_at_top_at_top
thf(fact_7459_lhopital__right,axiom,
! [F2: real > real,X: real,G: real > real,G5: real > real,F5: real > real,F4: filter @ real] :
( ( filterlim @ real @ real @ F2 @ ( topolo7230453075368039082e_nhds @ real @ ( zero_zero @ real ) ) @ ( topolo174197925503356063within @ real @ X @ ( set_ord_greaterThan @ real @ X ) ) )
=> ( ( filterlim @ real @ real @ G @ ( topolo7230453075368039082e_nhds @ real @ ( zero_zero @ real ) ) @ ( topolo174197925503356063within @ real @ X @ ( set_ord_greaterThan @ real @ X ) ) )
=> ( ( eventually @ real
@ ^ [X2: real] :
( ( G @ X2 )
!= ( zero_zero @ real ) )
@ ( topolo174197925503356063within @ real @ X @ ( set_ord_greaterThan @ real @ X ) ) )
=> ( ( eventually @ real
@ ^ [X2: real] :
( ( G5 @ X2 )
!= ( zero_zero @ real ) )
@ ( topolo174197925503356063within @ real @ X @ ( set_ord_greaterThan @ real @ X ) ) )
=> ( ( eventually @ real
@ ^ [X2: real] : ( has_field_derivative @ real @ F2 @ ( F5 @ X2 ) @ ( topolo174197925503356063within @ real @ X2 @ ( top_top @ ( set @ real ) ) ) )
@ ( topolo174197925503356063within @ real @ X @ ( set_ord_greaterThan @ real @ X ) ) )
=> ( ( eventually @ real
@ ^ [X2: real] : ( has_field_derivative @ real @ G @ ( G5 @ X2 ) @ ( topolo174197925503356063within @ real @ X2 @ ( top_top @ ( set @ real ) ) ) )
@ ( topolo174197925503356063within @ real @ X @ ( set_ord_greaterThan @ real @ X ) ) )
=> ( ( filterlim @ real @ real
@ ^ [X2: real] : ( divide_divide @ real @ ( F5 @ X2 ) @ ( G5 @ X2 ) )
@ F4
@ ( topolo174197925503356063within @ real @ X @ ( set_ord_greaterThan @ real @ X ) ) )
=> ( filterlim @ real @ real
@ ^ [X2: real] : ( divide_divide @ real @ ( F2 @ X2 ) @ ( G @ X2 ) )
@ F4
@ ( topolo174197925503356063within @ real @ X @ ( set_ord_greaterThan @ real @ X ) ) ) ) ) ) ) ) ) ) ).
% lhopital_right
thf(fact_7460_lhopital__right__0,axiom,
! [F0: real > real,G0: real > real,G5: real > real,F5: real > real,F4: filter @ real] :
( ( filterlim @ real @ real @ F0 @ ( topolo7230453075368039082e_nhds @ real @ ( zero_zero @ real ) ) @ ( topolo174197925503356063within @ real @ ( zero_zero @ real ) @ ( set_ord_greaterThan @ real @ ( zero_zero @ real ) ) ) )
=> ( ( filterlim @ real @ real @ G0 @ ( topolo7230453075368039082e_nhds @ real @ ( zero_zero @ real ) ) @ ( topolo174197925503356063within @ real @ ( zero_zero @ real ) @ ( set_ord_greaterThan @ real @ ( zero_zero @ real ) ) ) )
=> ( ( eventually @ real
@ ^ [X2: real] :
( ( G0 @ X2 )
!= ( zero_zero @ real ) )
@ ( topolo174197925503356063within @ real @ ( zero_zero @ real ) @ ( set_ord_greaterThan @ real @ ( zero_zero @ real ) ) ) )
=> ( ( eventually @ real
@ ^ [X2: real] :
( ( G5 @ X2 )
!= ( zero_zero @ real ) )
@ ( topolo174197925503356063within @ real @ ( zero_zero @ real ) @ ( set_ord_greaterThan @ real @ ( zero_zero @ real ) ) ) )
=> ( ( eventually @ real
@ ^ [X2: real] : ( has_field_derivative @ real @ F0 @ ( F5 @ X2 ) @ ( topolo174197925503356063within @ real @ X2 @ ( top_top @ ( set @ real ) ) ) )
@ ( topolo174197925503356063within @ real @ ( zero_zero @ real ) @ ( set_ord_greaterThan @ real @ ( zero_zero @ real ) ) ) )
=> ( ( eventually @ real
@ ^ [X2: real] : ( has_field_derivative @ real @ G0 @ ( G5 @ X2 ) @ ( topolo174197925503356063within @ real @ X2 @ ( top_top @ ( set @ real ) ) ) )
@ ( topolo174197925503356063within @ real @ ( zero_zero @ real ) @ ( set_ord_greaterThan @ real @ ( zero_zero @ real ) ) ) )
=> ( ( filterlim @ real @ real
@ ^ [X2: real] : ( divide_divide @ real @ ( F5 @ X2 ) @ ( G5 @ X2 ) )
@ F4
@ ( topolo174197925503356063within @ real @ ( zero_zero @ real ) @ ( set_ord_greaterThan @ real @ ( zero_zero @ real ) ) ) )
=> ( filterlim @ real @ real
@ ^ [X2: real] : ( divide_divide @ real @ ( F0 @ X2 ) @ ( G0 @ X2 ) )
@ F4
@ ( topolo174197925503356063within @ real @ ( zero_zero @ real ) @ ( set_ord_greaterThan @ real @ ( zero_zero @ real ) ) ) ) ) ) ) ) ) ) ) ).
% lhopital_right_0
thf(fact_7461_lhopital__right__at__top__at__bot,axiom,
! [F2: real > real,A2: real,G: real > real,F5: real > real,G5: real > real] :
( ( filterlim @ real @ real @ F2 @ ( at_top @ real ) @ ( topolo174197925503356063within @ real @ A2 @ ( set_ord_greaterThan @ real @ A2 ) ) )
=> ( ( filterlim @ real @ real @ G @ ( at_bot @ real ) @ ( topolo174197925503356063within @ real @ A2 @ ( set_ord_greaterThan @ real @ A2 ) ) )
=> ( ( eventually @ real
@ ^ [X2: real] : ( has_field_derivative @ real @ F2 @ ( F5 @ X2 ) @ ( topolo174197925503356063within @ real @ X2 @ ( top_top @ ( set @ real ) ) ) )
@ ( topolo174197925503356063within @ real @ A2 @ ( set_ord_greaterThan @ real @ A2 ) ) )
=> ( ( eventually @ real
@ ^ [X2: real] : ( has_field_derivative @ real @ G @ ( G5 @ X2 ) @ ( topolo174197925503356063within @ real @ X2 @ ( top_top @ ( set @ real ) ) ) )
@ ( topolo174197925503356063within @ real @ A2 @ ( set_ord_greaterThan @ real @ A2 ) ) )
=> ( ( filterlim @ real @ real
@ ^ [X2: real] : ( divide_divide @ real @ ( F5 @ X2 ) @ ( G5 @ X2 ) )
@ ( at_bot @ real )
@ ( topolo174197925503356063within @ real @ A2 @ ( set_ord_greaterThan @ real @ A2 ) ) )
=> ( filterlim @ real @ real
@ ^ [X2: real] : ( divide_divide @ real @ ( F2 @ X2 ) @ ( G @ X2 ) )
@ ( at_bot @ real )
@ ( topolo174197925503356063within @ real @ A2 @ ( set_ord_greaterThan @ real @ A2 ) ) ) ) ) ) ) ) ).
% lhopital_right_at_top_at_bot
thf(fact_7462_lhopital__right__at__top,axiom,
! [G: real > real,X: real,G5: real > real,F2: real > real,F5: real > real,Y: real] :
( ( filterlim @ real @ real @ G @ ( at_top @ real ) @ ( topolo174197925503356063within @ real @ X @ ( set_ord_greaterThan @ real @ X ) ) )
=> ( ( eventually @ real
@ ^ [X2: real] :
( ( G5 @ X2 )
!= ( zero_zero @ real ) )
@ ( topolo174197925503356063within @ real @ X @ ( set_ord_greaterThan @ real @ X ) ) )
=> ( ( eventually @ real
@ ^ [X2: real] : ( has_field_derivative @ real @ F2 @ ( F5 @ X2 ) @ ( topolo174197925503356063within @ real @ X2 @ ( top_top @ ( set @ real ) ) ) )
@ ( topolo174197925503356063within @ real @ X @ ( set_ord_greaterThan @ real @ X ) ) )
=> ( ( eventually @ real
@ ^ [X2: real] : ( has_field_derivative @ real @ G @ ( G5 @ X2 ) @ ( topolo174197925503356063within @ real @ X2 @ ( top_top @ ( set @ real ) ) ) )
@ ( topolo174197925503356063within @ real @ X @ ( set_ord_greaterThan @ real @ X ) ) )
=> ( ( filterlim @ real @ real
@ ^ [X2: real] : ( divide_divide @ real @ ( F5 @ X2 ) @ ( G5 @ X2 ) )
@ ( topolo7230453075368039082e_nhds @ real @ Y )
@ ( topolo174197925503356063within @ real @ X @ ( set_ord_greaterThan @ real @ X ) ) )
=> ( filterlim @ real @ real
@ ^ [X2: real] : ( divide_divide @ real @ ( F2 @ X2 ) @ ( G @ X2 ) )
@ ( topolo7230453075368039082e_nhds @ real @ Y )
@ ( topolo174197925503356063within @ real @ X @ ( set_ord_greaterThan @ real @ X ) ) ) ) ) ) ) ) ).
% lhopital_right_at_top
thf(fact_7463_summable__Cauchy_H,axiom,
! [A: $tType] :
( ( real_Vector_banach @ A )
=> ! [F2: nat > A,G: nat > real] :
( ( eventually @ nat
@ ^ [M5: nat] :
! [N2: nat] :
( ( ord_less_eq @ nat @ M5 @ N2 )
=> ( ord_less_eq @ real @ ( real_V7770717601297561774m_norm @ A @ ( groups7311177749621191930dd_sum @ nat @ A @ F2 @ ( set_or7035219750837199246ssThan @ nat @ M5 @ N2 ) ) ) @ ( G @ M5 ) ) )
@ ( at_top @ nat ) )
=> ( ( filterlim @ nat @ real @ G @ ( topolo7230453075368039082e_nhds @ real @ ( zero_zero @ real ) ) @ ( at_top @ nat ) )
=> ( summable @ A @ F2 ) ) ) ) ).
% summable_Cauchy'
thf(fact_7464_Bfun__metric__def,axiom,
! [B: $tType,A: $tType] :
( ( real_V7819770556892013058_space @ B )
=> ( ( bfun @ A @ B )
= ( ^ [F7: A > B,F9: filter @ A] :
? [Y2: B,K7: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ K7 )
& ( eventually @ A
@ ^ [X2: A] : ( ord_less_eq @ real @ ( real_V557655796197034286t_dist @ B @ ( F7 @ X2 ) @ Y2 ) @ K7 )
@ F9 ) ) ) ) ) ).
% Bfun_metric_def
thf(fact_7465_Bseq__mult,axiom,
! [A: $tType] :
( ( real_V3459762299906320749_field @ A )
=> ! [F2: nat > A,G: nat > A] :
( ( bfun @ nat @ A @ F2 @ ( at_top @ nat ) )
=> ( ( bfun @ nat @ A @ G @ ( at_top @ nat ) )
=> ( bfun @ nat @ A
@ ^ [X2: nat] : ( times_times @ A @ ( F2 @ X2 ) @ ( G @ X2 ) )
@ ( at_top @ nat ) ) ) ) ) ).
% Bseq_mult
thf(fact_7466_Bseq__add,axiom,
! [A: $tType] :
( ( real_V822414075346904944vector @ A )
=> ! [F2: nat > A,C2: A] :
( ( bfun @ nat @ A @ F2 @ ( at_top @ nat ) )
=> ( bfun @ nat @ A
@ ^ [X2: nat] : ( plus_plus @ A @ ( F2 @ X2 ) @ C2 )
@ ( at_top @ nat ) ) ) ) ).
% Bseq_add
thf(fact_7467_Bseq__offset,axiom,
! [A: $tType] :
( ( real_V7819770556892013058_space @ A )
=> ! [X8: nat > A,K: nat] :
( ( bfun @ nat @ A
@ ^ [N2: nat] : ( X8 @ ( plus_plus @ nat @ N2 @ K ) )
@ ( at_top @ nat ) )
=> ( bfun @ nat @ A @ X8 @ ( at_top @ nat ) ) ) ) ).
% Bseq_offset
thf(fact_7468_Bseq__add__iff,axiom,
! [A: $tType] :
( ( real_V822414075346904944vector @ A )
=> ! [F2: nat > A,C2: A] :
( ( bfun @ nat @ A
@ ^ [X2: nat] : ( plus_plus @ A @ ( F2 @ X2 ) @ C2 )
@ ( at_top @ nat ) )
= ( bfun @ nat @ A @ F2 @ ( at_top @ nat ) ) ) ) ).
% Bseq_add_iff
thf(fact_7469_Bseq__ignore__initial__segment,axiom,
! [A: $tType] :
( ( real_V7819770556892013058_space @ A )
=> ! [X8: nat > A,K: nat] :
( ( bfun @ nat @ A @ X8 @ ( at_top @ nat ) )
=> ( bfun @ nat @ A
@ ^ [N2: nat] : ( X8 @ ( plus_plus @ nat @ N2 @ K ) )
@ ( at_top @ nat ) ) ) ) ).
% Bseq_ignore_initial_segment
thf(fact_7470_Bseq__Suc__iff,axiom,
! [A: $tType] :
( ( real_V822414075346904944vector @ A )
=> ! [F2: nat > A] :
( ( bfun @ nat @ A
@ ^ [N2: nat] : ( F2 @ ( suc @ N2 ) )
@ ( at_top @ nat ) )
= ( bfun @ nat @ A @ F2 @ ( at_top @ nat ) ) ) ) ).
% Bseq_Suc_iff
thf(fact_7471_BseqI_H,axiom,
! [A: $tType] :
( ( real_V822414075346904944vector @ A )
=> ! [X8: nat > A,K6: real] :
( ! [N: nat] : ( ord_less_eq @ real @ ( real_V7770717601297561774m_norm @ A @ ( X8 @ N ) ) @ K6 )
=> ( bfun @ nat @ A @ X8 @ ( at_top @ nat ) ) ) ) ).
% BseqI'
thf(fact_7472_eventually__all__ge__at__top,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [P: A > $o] :
( ( eventually @ A @ P @ ( at_top @ A ) )
=> ( eventually @ A
@ ^ [X2: A] :
! [Y2: A] :
( ( ord_less_eq @ A @ X2 @ Y2 )
=> ( P @ Y2 ) )
@ ( at_top @ A ) ) ) ) ).
% eventually_all_ge_at_top
thf(fact_7473_Bseq__eq__bounded,axiom,
! [F2: nat > real,A2: real,B2: real] :
( ( ord_less_eq @ ( set @ real ) @ ( image @ nat @ real @ F2 @ ( top_top @ ( set @ nat ) ) ) @ ( set_or1337092689740270186AtMost @ real @ A2 @ B2 ) )
=> ( bfun @ nat @ real @ F2 @ ( at_top @ nat ) ) ) ).
% Bseq_eq_bounded
thf(fact_7474_Bseq__cmult__iff,axiom,
! [A: $tType] :
( ( real_V3459762299906320749_field @ A )
=> ! [C2: A,F2: nat > A] :
( ( C2
!= ( zero_zero @ A ) )
=> ( ( bfun @ nat @ A
@ ^ [X2: nat] : ( times_times @ A @ C2 @ ( F2 @ X2 ) )
@ ( at_top @ nat ) )
= ( bfun @ nat @ A @ F2 @ ( at_top @ nat ) ) ) ) ) ).
% Bseq_cmult_iff
thf(fact_7475_Bseq__eventually__mono,axiom,
! [B: $tType,A: $tType] :
( ( ( real_V822414075346904944vector @ A )
& ( real_V822414075346904944vector @ B ) )
=> ! [F2: nat > A,G: nat > B] :
( ( eventually @ nat
@ ^ [N2: nat] : ( ord_less_eq @ real @ ( real_V7770717601297561774m_norm @ A @ ( F2 @ N2 ) ) @ ( real_V7770717601297561774m_norm @ B @ ( G @ N2 ) ) )
@ ( at_top @ nat ) )
=> ( ( bfun @ nat @ B @ G @ ( at_top @ nat ) )
=> ( bfun @ nat @ A @ F2 @ ( at_top @ nat ) ) ) ) ) ).
% Bseq_eventually_mono
thf(fact_7476_BseqD,axiom,
! [A: $tType] :
( ( real_V822414075346904944vector @ A )
=> ! [X8: nat > A] :
( ( bfun @ nat @ A @ X8 @ ( at_top @ nat ) )
=> ? [K9: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ K9 )
& ! [N9: nat] : ( ord_less_eq @ real @ ( real_V7770717601297561774m_norm @ A @ ( X8 @ N9 ) ) @ K9 ) ) ) ) ).
% BseqD
thf(fact_7477_BseqE,axiom,
! [A: $tType] :
( ( real_V822414075346904944vector @ A )
=> ! [X8: nat > A] :
( ( bfun @ nat @ A @ X8 @ ( at_top @ nat ) )
=> ~ ! [K9: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ K9 )
=> ~ ! [N9: nat] : ( ord_less_eq @ real @ ( real_V7770717601297561774m_norm @ A @ ( X8 @ N9 ) ) @ K9 ) ) ) ) ).
% BseqE
thf(fact_7478_BseqI,axiom,
! [A: $tType] :
( ( real_V822414075346904944vector @ A )
=> ! [K6: real,X8: nat > A] :
( ( ord_less @ real @ ( zero_zero @ real ) @ K6 )
=> ( ! [N: nat] : ( ord_less_eq @ real @ ( real_V7770717601297561774m_norm @ A @ ( X8 @ N ) ) @ K6 )
=> ( bfun @ nat @ A @ X8 @ ( at_top @ nat ) ) ) ) ) ).
% BseqI
thf(fact_7479_Bseq__def,axiom,
! [A: $tType] :
( ( real_V822414075346904944vector @ A )
=> ! [X8: nat > A] :
( ( bfun @ nat @ A @ X8 @ ( at_top @ nat ) )
= ( ? [K7: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ K7 )
& ! [N2: nat] : ( ord_less_eq @ real @ ( real_V7770717601297561774m_norm @ A @ ( X8 @ N2 ) ) @ K7 ) ) ) ) ) ).
% Bseq_def
thf(fact_7480_Bseq__iff1a,axiom,
! [A: $tType] :
( ( real_V822414075346904944vector @ A )
=> ! [X8: nat > A] :
( ( bfun @ nat @ A @ X8 @ ( at_top @ nat ) )
= ( ? [N8: nat] :
! [N2: nat] : ( ord_less @ real @ ( real_V7770717601297561774m_norm @ A @ ( X8 @ N2 ) ) @ ( semiring_1_of_nat @ real @ ( suc @ N8 ) ) ) ) ) ) ).
% Bseq_iff1a
thf(fact_7481_Bseq__iff,axiom,
! [A: $tType] :
( ( real_V822414075346904944vector @ A )
=> ! [X8: nat > A] :
( ( bfun @ nat @ A @ X8 @ ( at_top @ nat ) )
= ( ? [N8: nat] :
! [N2: nat] : ( ord_less_eq @ real @ ( real_V7770717601297561774m_norm @ A @ ( X8 @ N2 ) ) @ ( semiring_1_of_nat @ real @ ( suc @ N8 ) ) ) ) ) ) ).
% Bseq_iff
thf(fact_7482_Bseq__realpow,axiom,
! [X: real] :
( ( ord_less_eq @ real @ ( zero_zero @ real ) @ X )
=> ( ( ord_less_eq @ real @ X @ ( one_one @ real ) )
=> ( bfun @ nat @ real @ ( power_power @ real @ X ) @ ( at_top @ nat ) ) ) ) ).
% Bseq_realpow
thf(fact_7483_BfunI,axiom,
! [B: $tType,A: $tType] :
( ( real_V822414075346904944vector @ B )
=> ! [F2: A > B,K6: real,F4: filter @ A] :
( ( eventually @ A
@ ^ [X2: A] : ( ord_less_eq @ real @ ( real_V7770717601297561774m_norm @ B @ ( F2 @ X2 ) ) @ K6 )
@ F4 )
=> ( bfun @ A @ B @ F2 @ F4 ) ) ) ).
% BfunI
thf(fact_7484_Bseq__iff3,axiom,
! [A: $tType] :
( ( real_V822414075346904944vector @ A )
=> ! [X8: nat > A] :
( ( bfun @ nat @ A @ X8 @ ( at_top @ nat ) )
= ( ? [K2: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ K2 )
& ? [N8: nat] :
! [N2: nat] : ( ord_less_eq @ real @ ( real_V7770717601297561774m_norm @ A @ ( plus_plus @ A @ ( X8 @ N2 ) @ ( uminus_uminus @ A @ ( X8 @ N8 ) ) ) ) @ K2 ) ) ) ) ) ).
% Bseq_iff3
thf(fact_7485_Bseq__iff2,axiom,
! [A: $tType] :
( ( real_V822414075346904944vector @ A )
=> ! [X8: nat > A] :
( ( bfun @ nat @ A @ X8 @ ( at_top @ nat ) )
= ( ? [K2: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ K2 )
& ? [X2: A] :
! [N2: nat] : ( ord_less_eq @ real @ ( real_V7770717601297561774m_norm @ A @ ( plus_plus @ A @ ( X8 @ N2 ) @ ( uminus_uminus @ A @ X2 ) ) ) @ K2 ) ) ) ) ) ).
% Bseq_iff2
thf(fact_7486_Bfun__def,axiom,
! [B: $tType,A: $tType] :
( ( real_V822414075346904944vector @ B )
=> ( ( bfun @ A @ B )
= ( ^ [F7: A > B,F9: filter @ A] :
? [K7: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ K7 )
& ( eventually @ A
@ ^ [X2: A] : ( ord_less_eq @ real @ ( real_V7770717601297561774m_norm @ B @ ( F7 @ X2 ) ) @ K7 )
@ F9 ) ) ) ) ) ).
% Bfun_def
thf(fact_7487_BfunE,axiom,
! [B: $tType,A: $tType] :
( ( real_V822414075346904944vector @ B )
=> ! [F2: A > B,F4: filter @ A] :
( ( bfun @ A @ B @ F2 @ F4 )
=> ~ ! [B9: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ B9 )
=> ~ ( eventually @ A
@ ^ [X2: A] : ( ord_less_eq @ real @ ( real_V7770717601297561774m_norm @ B @ ( F2 @ X2 ) ) @ B9 )
@ F4 ) ) ) ) ).
% BfunE
thf(fact_7488_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 )
=> ! [B5: nat] :
( ( ord_less @ nat @ A6 @ B5 )
=> ( ord_less_eq @ real @ ( real_V7770717601297561774m_norm @ A @ ( groups7311177749621191930dd_sum @ nat @ A @ F2 @ ( set_or3652927894154168847AtMost @ nat @ A6 @ B5 ) ) ) @ ( 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_7489_Greatest__def,axiom,
! [A: $tType] :
( ( order @ A )
=> ( ( order_Greatest @ A )
= ( ^ [P3: A > $o] :
( the @ A
@ ^ [X2: A] :
( ( P3 @ X2 )
& ! [Y2: A] :
( ( P3 @ Y2 )
=> ( ord_less_eq @ A @ Y2 @ X2 ) ) ) ) ) ) ) ).
% Greatest_def
thf(fact_7490_The__split__eq,axiom,
! [A: $tType,B: $tType,X: A,Y: B] :
( ( the @ ( product_prod @ A @ B )
@ ( product_case_prod @ A @ B @ $o
@ ^ [X9: A,Y7: B] :
( ( X = X9 )
& ( Y = Y7 ) ) ) )
= ( product_Pair @ A @ B @ X @ Y ) ) ).
% The_split_eq
thf(fact_7491_greaterThanAtMost__iff,axiom,
! [A: $tType] :
( ( ord @ A )
=> ! [I: A,L2: A,U: A] :
( ( member @ A @ I @ ( set_or3652927894154168847AtMost @ A @ L2 @ U ) )
= ( ( ord_less @ A @ L2 @ I )
& ( ord_less_eq @ A @ I @ U ) ) ) ) ).
% greaterThanAtMost_iff
thf(fact_7492_greaterThanAtMost__empty,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [L2: A,K: A] :
( ( ord_less_eq @ A @ L2 @ K )
=> ( ( set_or3652927894154168847AtMost @ A @ K @ L2 )
= ( bot_bot @ ( set @ A ) ) ) ) ) ).
% greaterThanAtMost_empty
thf(fact_7493_greaterThanAtMost__empty__iff2,axiom,
! [A: $tType] :
( ( preorder @ A )
=> ! [K: A,L2: A] :
( ( ( bot_bot @ ( set @ A ) )
= ( set_or3652927894154168847AtMost @ A @ K @ L2 ) )
= ( ~ ( ord_less @ A @ K @ L2 ) ) ) ) ).
% greaterThanAtMost_empty_iff2
thf(fact_7494_greaterThanAtMost__empty__iff,axiom,
! [A: $tType] :
( ( preorder @ A )
=> ! [K: A,L2: A] :
( ( ( set_or3652927894154168847AtMost @ A @ K @ L2 )
= ( bot_bot @ ( set @ A ) ) )
= ( ~ ( ord_less @ A @ K @ L2 ) ) ) ) ).
% greaterThanAtMost_empty_iff
thf(fact_7495_infinite__Ioc__iff,axiom,
! [A: $tType] :
( ( dense_linorder @ A )
=> ! [A2: A,B2: A] :
( ( ~ ( finite_finite2 @ A @ ( set_or3652927894154168847AtMost @ A @ A2 @ B2 ) ) )
= ( ord_less @ A @ A2 @ B2 ) ) ) ).
% infinite_Ioc_iff
thf(fact_7496_image__add__greaterThanAtMost,axiom,
! [A: $tType] :
( ( linordered_semidom @ A )
=> ! [C2: A,A2: A,B2: A] :
( ( image @ A @ A @ ( plus_plus @ A @ C2 ) @ ( set_or3652927894154168847AtMost @ A @ A2 @ B2 ) )
= ( set_or3652927894154168847AtMost @ A @ ( plus_plus @ A @ C2 @ A2 ) @ ( plus_plus @ A @ C2 @ B2 ) ) ) ) ).
% image_add_greaterThanAtMost
thf(fact_7497_image__minus__const__greaterThanAtMost,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [C2: A,A2: A,B2: A] :
( ( image @ A @ A @ ( minus_minus @ A @ C2 ) @ ( set_or3652927894154168847AtMost @ A @ A2 @ B2 ) )
= ( set_or7035219750837199246ssThan @ A @ ( minus_minus @ A @ C2 @ B2 ) @ ( minus_minus @ A @ C2 @ A2 ) ) ) ) ).
% image_minus_const_greaterThanAtMost
thf(fact_7498_image__diff__atLeastLessThan,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [C2: A,A2: A,B2: A] :
( ( image @ A @ A @ ( minus_minus @ A @ C2 ) @ ( set_or7035219750837199246ssThan @ A @ A2 @ B2 ) )
= ( set_or3652927894154168847AtMost @ A @ ( minus_minus @ A @ C2 @ B2 ) @ ( minus_minus @ A @ C2 @ A2 ) ) ) ) ).
% image_diff_atLeastLessThan
thf(fact_7499_floor__rat__def,axiom,
( ( archim6421214686448440834_floor @ rat )
= ( ^ [X2: rat] :
( the @ int
@ ^ [Z3: int] :
( ( ord_less_eq @ rat @ ( ring_1_of_int @ rat @ Z3 ) @ X2 )
& ( ord_less @ rat @ X2 @ ( ring_1_of_int @ rat @ ( plus_plus @ int @ Z3 @ ( one_one @ int ) ) ) ) ) ) ) ) ).
% floor_rat_def
thf(fact_7500_Ioc__inj,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A2: A,B2: A,C2: A,D: A] :
( ( ( set_or3652927894154168847AtMost @ A @ A2 @ B2 )
= ( set_or3652927894154168847AtMost @ A @ C2 @ D ) )
= ( ( ( ord_less_eq @ A @ B2 @ A2 )
& ( ord_less_eq @ A @ D @ C2 ) )
| ( ( A2 = C2 )
& ( B2 = D ) ) ) ) ) ).
% Ioc_inj
thf(fact_7501_Ioc__subset__iff,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A2: A,B2: A,C2: A,D: A] :
( ( ord_less_eq @ ( set @ A ) @ ( set_or3652927894154168847AtMost @ A @ A2 @ B2 ) @ ( set_or3652927894154168847AtMost @ A @ C2 @ D ) )
= ( ( ord_less_eq @ A @ B2 @ A2 )
| ( ( ord_less_eq @ A @ C2 @ A2 )
& ( ord_less_eq @ A @ B2 @ D ) ) ) ) ) ).
% Ioc_subset_iff
thf(fact_7502_GreatestI__ex__nat,axiom,
! [P: nat > $o,B2: nat] :
( ? [X_1: nat] : ( P @ X_1 )
=> ( ! [Y3: nat] :
( ( P @ Y3 )
=> ( ord_less_eq @ nat @ Y3 @ B2 ) )
=> ( P @ ( order_Greatest @ nat @ P ) ) ) ) ).
% GreatestI_ex_nat
thf(fact_7503_Greatest__le__nat,axiom,
! [P: nat > $o,K: nat,B2: nat] :
( ( P @ K )
=> ( ! [Y3: nat] :
( ( P @ Y3 )
=> ( ord_less_eq @ nat @ Y3 @ B2 ) )
=> ( ord_less_eq @ nat @ K @ ( order_Greatest @ nat @ P ) ) ) ) ).
% Greatest_le_nat
thf(fact_7504_GreatestI__nat,axiom,
! [P: nat > $o,K: nat,B2: nat] :
( ( P @ K )
=> ( ! [Y3: nat] :
( ( P @ Y3 )
=> ( ord_less_eq @ nat @ Y3 @ B2 ) )
=> ( P @ ( order_Greatest @ nat @ P ) ) ) ) ).
% GreatestI_nat
thf(fact_7505_infinite__Ioc,axiom,
! [A: $tType] :
( ( dense_linorder @ A )
=> ! [A2: A,B2: A] :
( ( ord_less @ A @ A2 @ B2 )
=> ~ ( finite_finite2 @ A @ ( set_or3652927894154168847AtMost @ A @ A2 @ B2 ) ) ) ) ).
% infinite_Ioc
thf(fact_7506_atLeastSucAtMost__greaterThanAtMost,axiom,
! [L2: nat,U: nat] :
( ( set_or1337092689740270186AtMost @ nat @ ( suc @ L2 ) @ U )
= ( set_or3652927894154168847AtMost @ nat @ L2 @ U ) ) ).
% atLeastSucAtMost_greaterThanAtMost
thf(fact_7507_open__left,axiom,
! [A: $tType] :
( ( topolo1944317154257567458pology @ A )
=> ! [S3: set @ A,X: A,Y: A] :
( ( topolo1002775350975398744n_open @ A @ S3 )
=> ( ( member @ A @ X @ S3 )
=> ( ( ord_less @ A @ Y @ X )
=> ? [B3: A] :
( ( ord_less @ A @ B3 @ X )
& ( ord_less_eq @ ( set @ A ) @ ( set_or3652927894154168847AtMost @ A @ B3 @ X ) @ S3 ) ) ) ) ) ) ).
% open_left
thf(fact_7508_greaterThanAtMost__upt,axiom,
( ( set_or3652927894154168847AtMost @ nat )
= ( ^ [N2: nat,M5: nat] : ( set2 @ nat @ ( upt @ ( suc @ N2 ) @ ( suc @ M5 ) ) ) ) ) ).
% greaterThanAtMost_upt
thf(fact_7509_Greatest__equality,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [P: A > $o,X: A] :
( ( P @ X )
=> ( ! [Y3: A] :
( ( P @ Y3 )
=> ( ord_less_eq @ A @ Y3 @ X ) )
=> ( ( order_Greatest @ A @ P )
= X ) ) ) ) ).
% Greatest_equality
thf(fact_7510_GreatestI2__order,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [P: A > $o,X: A,Q: A > $o] :
( ( P @ X )
=> ( ! [Y3: A] :
( ( P @ Y3 )
=> ( ord_less_eq @ A @ Y3 @ X ) )
=> ( ! [X3: A] :
( ( P @ X3 )
=> ( ! [Y4: A] :
( ( P @ Y4 )
=> ( ord_less_eq @ A @ Y4 @ X3 ) )
=> ( Q @ X3 ) ) )
=> ( Q @ ( order_Greatest @ A @ P ) ) ) ) ) ) ).
% GreatestI2_order
thf(fact_7511_sum_Ohead,axiom,
! [A: $tType] :
( ( comm_monoid_add @ A )
=> ! [M: nat,N3: nat,G: nat > A] :
( ( ord_less_eq @ nat @ M @ N3 )
=> ( ( groups7311177749621191930dd_sum @ nat @ A @ G @ ( set_or1337092689740270186AtMost @ nat @ M @ N3 ) )
= ( plus_plus @ A @ ( G @ M ) @ ( groups7311177749621191930dd_sum @ nat @ A @ G @ ( set_or3652927894154168847AtMost @ nat @ M @ N3 ) ) ) ) ) ) ).
% sum.head
thf(fact_7512_prod_Ohead,axiom,
! [A: $tType] :
( ( comm_monoid_mult @ A )
=> ! [M: nat,N3: nat,G: nat > A] :
( ( ord_less_eq @ nat @ M @ N3 )
=> ( ( groups7121269368397514597t_prod @ nat @ A @ G @ ( set_or1337092689740270186AtMost @ nat @ M @ N3 ) )
= ( times_times @ A @ ( G @ M ) @ ( groups7121269368397514597t_prod @ nat @ A @ G @ ( set_or3652927894154168847AtMost @ nat @ M @ N3 ) ) ) ) ) ) ).
% prod.head
thf(fact_7513_greaterThanAtMost__subseteq__atLeastAtMost__iff,axiom,
! [A: $tType] :
( ( dense_linorder @ A )
=> ! [A2: A,B2: A,C2: A,D: A] :
( ( ord_less_eq @ ( set @ A ) @ ( set_or3652927894154168847AtMost @ A @ A2 @ B2 ) @ ( set_or1337092689740270186AtMost @ A @ C2 @ D ) )
= ( ( ord_less @ A @ A2 @ B2 )
=> ( ( ord_less_eq @ A @ C2 @ A2 )
& ( ord_less_eq @ A @ B2 @ D ) ) ) ) ) ).
% greaterThanAtMost_subseteq_atLeastAtMost_iff
thf(fact_7514_greaterThanAtMost__subseteq__atLeastLessThan__iff,axiom,
! [A: $tType] :
( ( dense_linorder @ A )
=> ! [A2: A,B2: A,C2: A,D: A] :
( ( ord_less_eq @ ( set @ A ) @ ( set_or3652927894154168847AtMost @ A @ A2 @ B2 ) @ ( set_or7035219750837199246ssThan @ A @ C2 @ D ) )
= ( ( ord_less @ A @ A2 @ B2 )
=> ( ( ord_less_eq @ A @ C2 @ A2 )
& ( ord_less @ A @ B2 @ D ) ) ) ) ) ).
% greaterThanAtMost_subseteq_atLeastLessThan_iff
thf(fact_7515_greaterThanLessThan__subseteq__greaterThanAtMost__iff,axiom,
! [A: $tType] :
( ( dense_linorder @ A )
=> ! [A2: A,B2: A,C2: A,D: A] :
( ( ord_less_eq @ ( set @ A ) @ ( set_or5935395276787703475ssThan @ A @ A2 @ B2 ) @ ( set_or3652927894154168847AtMost @ A @ C2 @ D ) )
= ( ( ord_less @ A @ A2 @ B2 )
=> ( ( ord_less_eq @ A @ C2 @ A2 )
& ( ord_less_eq @ A @ B2 @ D ) ) ) ) ) ).
% greaterThanLessThan_subseteq_greaterThanAtMost_iff
thf(fact_7516_greaterThanAtMost__eq__atLeastAtMost__diff,axiom,
! [A: $tType] :
( ( order @ A )
=> ( ( set_or3652927894154168847AtMost @ A )
= ( ^ [A6: A,B5: A] : ( minus_minus @ ( set @ A ) @ ( set_or1337092689740270186AtMost @ A @ A6 @ B5 ) @ ( insert @ A @ A6 @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ) ).
% greaterThanAtMost_eq_atLeastAtMost_diff
thf(fact_7517_floor__real__def,axiom,
( ( archim6421214686448440834_floor @ real )
= ( ^ [X2: real] :
( the @ int
@ ^ [Z3: int] :
( ( ord_less_eq @ real @ ( ring_1_of_int @ real @ Z3 ) @ X2 )
& ( ord_less @ real @ X2 @ ( ring_1_of_int @ real @ ( plus_plus @ int @ Z3 @ ( one_one @ int ) ) ) ) ) ) ) ) ).
% floor_real_def
thf(fact_7518_sequentially__imp__eventually__at__right,axiom,
! [A: $tType] :
( ( ( topolo3112930676232923870pology @ A )
& ( topolo1944317154257567458pology @ A ) )
=> ! [A2: A,B2: A,P: A > $o] :
( ( ord_less @ A @ A2 @ B2 )
=> ( ! [F3: nat > A] :
( ! [N9: nat] : ( ord_less @ A @ A2 @ ( F3 @ N9 ) )
=> ( ! [N9: nat] : ( ord_less @ A @ ( F3 @ N9 ) @ B2 )
=> ( ( order_antimono @ nat @ A @ F3 )
=> ( ( filterlim @ nat @ A @ F3 @ ( topolo7230453075368039082e_nhds @ A @ A2 ) @ ( at_top @ nat ) )
=> ( eventually @ nat
@ ^ [N2: nat] : ( P @ ( F3 @ N2 ) )
@ ( at_top @ nat ) ) ) ) ) )
=> ( eventually @ A @ P @ ( topolo174197925503356063within @ A @ A2 @ ( set_ord_greaterThan @ A @ A2 ) ) ) ) ) ) ).
% sequentially_imp_eventually_at_right
thf(fact_7519_suminf__eq__SUP__real,axiom,
! [X8: nat > real] :
( ( summable @ real @ X8 )
=> ( ! [I2: nat] : ( ord_less_eq @ real @ ( zero_zero @ real ) @ ( X8 @ I2 ) )
=> ( ( suminf @ real @ X8 )
= ( complete_Sup_Sup @ real
@ ( image @ nat @ real
@ ^ [I3: nat] : ( groups7311177749621191930dd_sum @ nat @ real @ X8 @ ( set_ord_lessThan @ nat @ I3 ) )
@ ( top_top @ ( set @ nat ) ) ) ) ) ) ) ).
% suminf_eq_SUP_real
thf(fact_7520_finite__greaterThanAtMost__integer,axiom,
! [L2: code_integer,U: code_integer] : ( finite_finite2 @ code_integer @ ( set_or3652927894154168847AtMost @ code_integer @ L2 @ U ) ) ).
% finite_greaterThanAtMost_integer
thf(fact_7521_Sup__atLeastAtMost,axiom,
! [A: $tType] :
( ( comple6319245703460814977attice @ A )
=> ! [X: A,Y: A] :
( ( ord_less_eq @ A @ X @ Y )
=> ( ( complete_Sup_Sup @ A @ ( set_or1337092689740270186AtMost @ A @ X @ Y ) )
= Y ) ) ) ).
% Sup_atLeastAtMost
thf(fact_7522_Sup__atLeastLessThan,axiom,
! [A: $tType] :
( ( ( comple6319245703460814977attice @ A )
& ( dense_linorder @ A ) )
=> ! [X: A,Y: A] :
( ( ord_less @ A @ X @ Y )
=> ( ( complete_Sup_Sup @ A @ ( set_or7035219750837199246ssThan @ A @ X @ Y ) )
= Y ) ) ) ).
% Sup_atLeastLessThan
thf(fact_7523_Sup__greaterThanLessThan,axiom,
! [A: $tType] :
( ( ( comple6319245703460814977attice @ A )
& ( dense_linorder @ A ) )
=> ! [X: A,Y: A] :
( ( ord_less @ A @ X @ Y )
=> ( ( complete_Sup_Sup @ A @ ( set_or5935395276787703475ssThan @ A @ X @ Y ) )
= Y ) ) ) ).
% Sup_greaterThanLessThan
thf(fact_7524_Sup__greaterThanAtMost,axiom,
! [A: $tType] :
( ( comple6319245703460814977attice @ A )
=> ! [X: A,Y: A] :
( ( ord_less @ A @ X @ Y )
=> ( ( complete_Sup_Sup @ A @ ( set_or3652927894154168847AtMost @ A @ X @ Y ) )
= Y ) ) ) ).
% Sup_greaterThanAtMost
thf(fact_7525_Sup__greaterThanAtLeast,axiom,
! [A: $tType] :
( ( comple6319245703460814977attice @ A )
=> ! [X: A] :
( ( ord_less @ A @ X @ ( top_top @ A ) )
=> ( ( complete_Sup_Sup @ A @ ( set_ord_greaterThan @ A @ X ) )
= ( top_top @ A ) ) ) ) ).
% Sup_greaterThanAtLeast
thf(fact_7526_less__eq__rat__def,axiom,
( ( ord_less_eq @ rat )
= ( ^ [X2: rat,Y2: rat] :
( ( ord_less @ rat @ X2 @ Y2 )
| ( X2 = Y2 ) ) ) ) ).
% less_eq_rat_def
thf(fact_7527_decseq__SucD,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [A3: nat > A,I: nat] :
( ( order_antimono @ nat @ A @ A3 )
=> ( ord_less_eq @ A @ ( A3 @ ( suc @ I ) ) @ ( A3 @ I ) ) ) ) ).
% decseq_SucD
thf(fact_7528_decseq__SucI,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [X8: nat > A] :
( ! [N: nat] : ( ord_less_eq @ A @ ( X8 @ ( suc @ N ) ) @ ( X8 @ N ) )
=> ( order_antimono @ nat @ A @ X8 ) ) ) ).
% decseq_SucI
thf(fact_7529_decseq__Suc__iff,axiom,
! [A: $tType] :
( ( order @ A )
=> ( ( order_antimono @ nat @ A )
= ( ^ [F7: nat > A] :
! [N2: nat] : ( ord_less_eq @ A @ ( F7 @ ( suc @ N2 ) ) @ ( F7 @ N2 ) ) ) ) ) ).
% decseq_Suc_iff
thf(fact_7530_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_7531_decseq__def,axiom,
! [A: $tType] :
( ( order @ A )
=> ( ( order_antimono @ nat @ A )
= ( ^ [X6: nat > A] :
! [M5: nat,N2: nat] :
( ( ord_less_eq @ nat @ M5 @ N2 )
=> ( ord_less_eq @ A @ ( X6 @ N2 ) @ ( X6 @ M5 ) ) ) ) ) ) ).
% decseq_def
thf(fact_7532_antimono__def,axiom,
! [B: $tType,A: $tType] :
( ( ( order @ A )
& ( order @ B ) )
=> ( ( order_antimono @ A @ B )
= ( ^ [F7: A > B] :
! [X2: A,Y2: A] :
( ( ord_less_eq @ A @ X2 @ Y2 )
=> ( ord_less_eq @ B @ ( F7 @ Y2 ) @ ( F7 @ X2 ) ) ) ) ) ) ).
% antimono_def
thf(fact_7533_antimonoI,axiom,
! [B: $tType,A: $tType] :
( ( ( order @ A )
& ( order @ B ) )
=> ! [F2: A > B] :
( ! [X3: A,Y3: A] :
( ( ord_less_eq @ A @ X3 @ Y3 )
=> ( ord_less_eq @ B @ ( F2 @ Y3 ) @ ( F2 @ X3 ) ) )
=> ( order_antimono @ A @ B @ F2 ) ) ) ).
% antimonoI
thf(fact_7534_antimonoE,axiom,
! [B: $tType,A: $tType] :
( ( ( order @ A )
& ( order @ B ) )
=> ! [F2: A > B,X: A,Y: A] :
( ( order_antimono @ A @ B @ F2 )
=> ( ( ord_less_eq @ A @ X @ Y )
=> ( ord_less_eq @ B @ ( F2 @ Y ) @ ( F2 @ X ) ) ) ) ) ).
% antimonoE
thf(fact_7535_antimonoD,axiom,
! [B: $tType,A: $tType] :
( ( ( order @ A )
& ( order @ B ) )
=> ! [F2: A > B,X: A,Y: A] :
( ( order_antimono @ A @ B @ F2 )
=> ( ( ord_less_eq @ A @ X @ Y )
=> ( ord_less_eq @ B @ ( F2 @ Y ) @ ( F2 @ X ) ) ) ) ) ).
% antimonoD
thf(fact_7536_Some__SUP,axiom,
! [B: $tType,A: $tType] :
( ( comple6319245703460814977attice @ B )
=> ! [A3: set @ A,F2: A > B] :
( ( A3
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( some @ B @ ( complete_Sup_Sup @ B @ ( image @ A @ B @ F2 @ A3 ) ) )
= ( complete_Sup_Sup @ ( option @ B )
@ ( image @ A @ ( option @ B )
@ ^ [X2: A] : ( some @ B @ ( F2 @ X2 ) )
@ A3 ) ) ) ) ) ).
% Some_SUP
thf(fact_7537_Some__Sup,axiom,
! [A: $tType] :
( ( comple6319245703460814977attice @ A )
=> ! [A3: set @ A] :
( ( A3
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( some @ A @ ( complete_Sup_Sup @ A @ A3 ) )
= ( complete_Sup_Sup @ ( option @ A ) @ ( image @ A @ ( option @ A ) @ ( some @ A ) @ A3 ) ) ) ) ) ).
% Some_Sup
thf(fact_7538_Sup__notin__open,axiom,
! [A: $tType] :
( ( topolo8458572112393995274pology @ A )
=> ! [A3: set @ A,X: A] :
( ( topolo1002775350975398744n_open @ A @ A3 )
=> ( ! [X3: A] :
( ( member @ A @ X3 @ A3 )
=> ( ord_less @ A @ X3 @ X ) )
=> ~ ( member @ A @ ( complete_Sup_Sup @ A @ A3 ) @ A3 ) ) ) ) ).
% Sup_notin_open
thf(fact_7539_obtain__pos__sum,axiom,
! [R2: rat] :
( ( ord_less @ rat @ ( zero_zero @ rat ) @ R2 )
=> ~ ! [S2: rat] :
( ( ord_less @ rat @ ( zero_zero @ rat ) @ S2 )
=> ! [T8: rat] :
( ( ord_less @ rat @ ( zero_zero @ rat ) @ T8 )
=> ( R2
!= ( plus_plus @ rat @ S2 @ T8 ) ) ) ) ) ).
% obtain_pos_sum
thf(fact_7540_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_7541_atLeastPlusOneAtMost__greaterThanAtMost__int,axiom,
! [L2: int,U: int] :
( ( set_or1337092689740270186AtMost @ int @ ( plus_plus @ int @ L2 @ ( one_one @ int ) ) @ U )
= ( set_or3652927894154168847AtMost @ int @ L2 @ U ) ) ).
% atLeastPlusOneAtMost_greaterThanAtMost_int
thf(fact_7542_cSup__asclose,axiom,
! [A: $tType] :
( ( ( condit6923001295902523014norder @ A )
& ( linordered_idom @ A ) )
=> ! [S3: set @ A,L2: A,E: A] :
( ( S3
!= ( bot_bot @ ( set @ A ) ) )
=> ( ! [X3: A] :
( ( member @ A @ X3 @ S3 )
=> ( ord_less_eq @ A @ ( abs_abs @ A @ ( minus_minus @ A @ X3 @ L2 ) ) @ E ) )
=> ( ord_less_eq @ A @ ( abs_abs @ A @ ( minus_minus @ A @ ( complete_Sup_Sup @ A @ S3 ) @ L2 ) ) @ E ) ) ) ) ).
% cSup_asclose
thf(fact_7543_decseq__bounded,axiom,
! [X8: nat > real,B6: real] :
( ( order_antimono @ nat @ real @ X8 )
=> ( ! [I2: nat] : ( ord_less_eq @ real @ B6 @ ( X8 @ I2 ) )
=> ( bfun @ nat @ real @ X8 @ ( at_top @ nat ) ) ) ) ).
% decseq_bounded
thf(fact_7544_Sup__lim,axiom,
! [A: $tType] :
( ( ( comple5582772986160207858norder @ A )
& ( topolo1944317154257567458pology @ A ) )
=> ! [B2: nat > A,S: set @ A,A2: A] :
( ! [N: nat] : ( member @ A @ ( B2 @ N ) @ S )
=> ( ( filterlim @ nat @ A @ B2 @ ( topolo7230453075368039082e_nhds @ A @ A2 ) @ ( at_top @ nat ) )
=> ( ord_less_eq @ A @ A2 @ ( complete_Sup_Sup @ A @ S ) ) ) ) ) ).
% Sup_lim
thf(fact_7545_atLeastPlusOneAtMost__greaterThanAtMost__integer,axiom,
! [L2: code_integer,U: code_integer] :
( ( set_or1337092689740270186AtMost @ code_integer @ ( plus_plus @ code_integer @ L2 @ ( one_one @ code_integer ) ) @ U )
= ( set_or3652927894154168847AtMost @ code_integer @ L2 @ U ) ) ).
% atLeastPlusOneAtMost_greaterThanAtMost_integer
thf(fact_7546_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_7547_word__range__minus__1_H,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [A2: word @ A,B2: word @ A] :
( ( A2
!= ( zero_zero @ ( word @ A ) ) )
=> ( ( set_or3652927894154168847AtMost @ ( word @ A ) @ ( minus_minus @ ( word @ A ) @ A2 @ ( one_one @ ( word @ A ) ) ) @ B2 )
= ( set_or1337092689740270186AtMost @ ( word @ A ) @ A2 @ B2 ) ) ) ) ).
% word_range_minus_1'
thf(fact_7548_decseq__ge,axiom,
! [A: $tType] :
( ( topolo1944317154257567458pology @ A )
=> ! [X8: nat > A,L6: A,N3: nat] :
( ( order_antimono @ nat @ A @ X8 )
=> ( ( filterlim @ nat @ A @ X8 @ ( topolo7230453075368039082e_nhds @ A @ L6 ) @ ( at_top @ nat ) )
=> ( ord_less_eq @ A @ L6 @ ( X8 @ N3 ) ) ) ) ) ).
% decseq_ge
thf(fact_7549_decseq__convergent,axiom,
! [X8: nat > real,B6: real] :
( ( order_antimono @ nat @ real @ X8 )
=> ( ! [I2: nat] : ( ord_less_eq @ real @ B6 @ ( X8 @ I2 ) )
=> ~ ! [L7: real] :
( ( filterlim @ nat @ real @ X8 @ ( topolo7230453075368039082e_nhds @ real @ L7 ) @ ( at_top @ nat ) )
=> ~ ! [I4: nat] : ( ord_less_eq @ real @ L7 @ ( X8 @ I4 ) ) ) ) ) ).
% decseq_convergent
thf(fact_7550_tendsto__at__right__sequentially,axiom,
! [C: $tType,B: $tType] :
( ( ( topolo3112930676232923870pology @ B )
& ( topolo1944317154257567458pology @ B )
& ( topolo4958980785337419405_space @ C ) )
=> ! [A2: B,B2: B,X8: B > C,L6: C] :
( ( ord_less @ B @ A2 @ B2 )
=> ( ! [S6: nat > B] :
( ! [N9: nat] : ( ord_less @ B @ A2 @ ( S6 @ N9 ) )
=> ( ! [N9: nat] : ( ord_less @ B @ ( S6 @ N9 ) @ B2 )
=> ( ( order_antimono @ nat @ B @ S6 )
=> ( ( filterlim @ nat @ B @ S6 @ ( topolo7230453075368039082e_nhds @ B @ A2 ) @ ( at_top @ nat ) )
=> ( filterlim @ nat @ C
@ ^ [N2: nat] : ( X8 @ ( S6 @ N2 ) )
@ ( topolo7230453075368039082e_nhds @ C @ L6 )
@ ( at_top @ nat ) ) ) ) ) )
=> ( filterlim @ B @ C @ X8 @ ( topolo7230453075368039082e_nhds @ C @ L6 ) @ ( topolo174197925503356063within @ B @ A2 @ ( set_ord_greaterThan @ B @ A2 ) ) ) ) ) ) ).
% tendsto_at_right_sequentially
thf(fact_7551_SUP__eq__top__iff,axiom,
! [B: $tType,A: $tType] :
( ( comple5582772986160207858norder @ A )
=> ! [F2: B > A,A3: set @ B] :
( ( ( complete_Sup_Sup @ A @ ( image @ B @ A @ F2 @ A3 ) )
= ( top_top @ A ) )
= ( ! [X2: A] :
( ( ord_less @ A @ X2 @ ( top_top @ A ) )
=> ? [Y2: B] :
( ( member @ B @ Y2 @ A3 )
& ( ord_less @ A @ X2 @ ( F2 @ Y2 ) ) ) ) ) ) ) ).
% SUP_eq_top_iff
thf(fact_7552_cSup__greaterThanAtMost,axiom,
! [A: $tType] :
( ( condit1219197933456340205attice @ A )
=> ! [Y: A,X: A] :
( ( ord_less @ A @ Y @ X )
=> ( ( complete_Sup_Sup @ A @ ( set_or3652927894154168847AtMost @ A @ Y @ X ) )
= X ) ) ) ).
% cSup_greaterThanAtMost
thf(fact_7553_empty__Sup,axiom,
! [A: $tType] :
( ( comple6319245703460814977attice @ A )
=> ( ( complete_Sup_Sup @ ( option @ A ) @ ( bot_bot @ ( set @ ( option @ A ) ) ) )
= ( none @ A ) ) ) ).
% empty_Sup
thf(fact_7554_Sup__eq__top__iff,axiom,
! [A: $tType] :
( ( comple5582772986160207858norder @ A )
=> ! [A3: set @ A] :
( ( ( complete_Sup_Sup @ A @ A3 )
= ( top_top @ A ) )
= ( ! [X2: A] :
( ( ord_less @ A @ X2 @ ( top_top @ A ) )
=> ? [Y2: A] :
( ( member @ A @ Y2 @ A3 )
& ( ord_less @ A @ X2 @ Y2 ) ) ) ) ) ) ).
% Sup_eq_top_iff
thf(fact_7555_cSup__atLeastAtMost,axiom,
! [A: $tType] :
( ( condit1219197933456340205attice @ A )
=> ! [Y: A,X: A] :
( ( ord_less_eq @ A @ Y @ X )
=> ( ( complete_Sup_Sup @ A @ ( set_or1337092689740270186AtMost @ A @ Y @ X ) )
= X ) ) ) ).
% cSup_atLeastAtMost
thf(fact_7556_cSup__atLeastLessThan,axiom,
! [A: $tType] :
( ( ( condit6923001295902523014norder @ A )
& ( dense_linorder @ A ) )
=> ! [Y: A,X: A] :
( ( ord_less @ A @ Y @ X )
=> ( ( complete_Sup_Sup @ A @ ( set_or7035219750837199246ssThan @ A @ Y @ X ) )
= X ) ) ) ).
% cSup_atLeastLessThan
thf(fact_7557_cSup__greaterThanLessThan,axiom,
! [A: $tType] :
( ( ( condit6923001295902523014norder @ A )
& ( dense_linorder @ A ) )
=> ! [Y: A,X: A] :
( ( ord_less @ A @ Y @ X )
=> ( ( complete_Sup_Sup @ A @ ( set_or5935395276787703475ssThan @ A @ Y @ X ) )
= X ) ) ) ).
% cSup_greaterThanLessThan
thf(fact_7558_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_7559_set__concat,axiom,
! [A: $tType,Xs2: list @ ( list @ A )] :
( ( set2 @ A @ ( concat @ A @ Xs2 ) )
= ( complete_Sup_Sup @ ( set @ A ) @ ( image @ ( list @ A ) @ ( set @ A ) @ ( set2 @ A ) @ ( set2 @ ( list @ A ) @ Xs2 ) ) ) ) ).
% set_concat
thf(fact_7560_inj__on__UNION__chain,axiom,
! [C: $tType,B: $tType,A: $tType,I5: set @ A,A3: A > ( set @ B ),F2: B > C] :
( ! [I2: A,J2: A] :
( ( member @ A @ I2 @ I5 )
=> ( ( member @ A @ J2 @ I5 )
=> ( ( ord_less_eq @ ( set @ B ) @ ( A3 @ I2 ) @ ( A3 @ J2 ) )
| ( ord_less_eq @ ( set @ B ) @ ( A3 @ J2 ) @ ( A3 @ I2 ) ) ) ) )
=> ( ! [I2: A] :
( ( member @ A @ I2 @ I5 )
=> ( inj_on @ B @ C @ F2 @ ( A3 @ I2 ) ) )
=> ( inj_on @ B @ C @ F2 @ ( complete_Sup_Sup @ ( set @ B ) @ ( image @ A @ ( set @ B ) @ A3 @ I5 ) ) ) ) ) ).
% inj_on_UNION_chain
thf(fact_7561_UN__subset__iff,axiom,
! [A: $tType,B: $tType,A3: B > ( set @ A ),I5: set @ B,B6: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ ( complete_Sup_Sup @ ( set @ A ) @ ( image @ B @ ( set @ A ) @ A3 @ I5 ) ) @ B6 )
= ( ! [X2: B] :
( ( member @ B @ X2 @ I5 )
=> ( ord_less_eq @ ( set @ A ) @ ( A3 @ X2 ) @ B6 ) ) ) ) ).
% UN_subset_iff
thf(fact_7562_UN__upper,axiom,
! [B: $tType,A: $tType,A2: A,A3: set @ A,B6: A > ( set @ B )] :
( ( member @ A @ A2 @ A3 )
=> ( ord_less_eq @ ( set @ B ) @ ( B6 @ A2 ) @ ( complete_Sup_Sup @ ( set @ B ) @ ( image @ A @ ( set @ B ) @ B6 @ A3 ) ) ) ) ).
% UN_upper
thf(fact_7563_UN__least,axiom,
! [A: $tType,B: $tType,A3: set @ A,B6: A > ( set @ B ),C5: set @ B] :
( ! [X3: A] :
( ( member @ A @ X3 @ A3 )
=> ( ord_less_eq @ ( set @ B ) @ ( B6 @ X3 ) @ C5 ) )
=> ( ord_less_eq @ ( set @ B ) @ ( complete_Sup_Sup @ ( set @ B ) @ ( image @ A @ ( set @ B ) @ B6 @ A3 ) ) @ C5 ) ) ).
% UN_least
thf(fact_7564_UN__mono,axiom,
! [B: $tType,A: $tType,A3: set @ A,B6: set @ A,F2: A > ( set @ B ),G: A > ( set @ B )] :
( ( ord_less_eq @ ( set @ A ) @ A3 @ B6 )
=> ( ! [X3: A] :
( ( member @ A @ X3 @ A3 )
=> ( ord_less_eq @ ( set @ B ) @ ( F2 @ X3 ) @ ( G @ X3 ) ) )
=> ( ord_less_eq @ ( set @ B ) @ ( complete_Sup_Sup @ ( set @ B ) @ ( image @ A @ ( set @ B ) @ F2 @ A3 ) ) @ ( complete_Sup_Sup @ ( set @ B ) @ ( image @ A @ ( set @ B ) @ G @ B6 ) ) ) ) ) ).
% UN_mono
thf(fact_7565_bij__betw__UNION__chain,axiom,
! [B: $tType,C: $tType,A: $tType,I5: set @ A,A3: A > ( set @ B ),F2: B > C,A9: A > ( set @ C )] :
( ! [I2: A,J2: A] :
( ( member @ A @ I2 @ I5 )
=> ( ( member @ A @ J2 @ I5 )
=> ( ( ord_less_eq @ ( set @ B ) @ ( A3 @ I2 ) @ ( A3 @ J2 ) )
| ( ord_less_eq @ ( set @ B ) @ ( A3 @ J2 ) @ ( A3 @ I2 ) ) ) ) )
=> ( ! [I2: A] :
( ( member @ A @ I2 @ I5 )
=> ( bij_betw @ B @ C @ F2 @ ( A3 @ I2 ) @ ( A9 @ I2 ) ) )
=> ( bij_betw @ B @ C @ F2 @ ( complete_Sup_Sup @ ( set @ B ) @ ( image @ A @ ( set @ B ) @ A3 @ I5 ) ) @ ( complete_Sup_Sup @ ( set @ C ) @ ( image @ A @ ( set @ C ) @ A9 @ I5 ) ) ) ) ) ).
% bij_betw_UNION_chain
thf(fact_7566_Union__mono,axiom,
! [A: $tType,A3: set @ ( set @ A ),B6: set @ ( set @ A )] :
( ( ord_less_eq @ ( set @ ( set @ A ) ) @ A3 @ B6 )
=> ( ord_less_eq @ ( set @ A ) @ ( complete_Sup_Sup @ ( set @ A ) @ A3 ) @ ( complete_Sup_Sup @ ( set @ A ) @ B6 ) ) ) ).
% Union_mono
thf(fact_7567_Union__least,axiom,
! [A: $tType,A3: set @ ( set @ A ),C5: set @ A] :
( ! [X16: set @ A] :
( ( member @ ( set @ A ) @ X16 @ A3 )
=> ( ord_less_eq @ ( set @ A ) @ X16 @ C5 ) )
=> ( ord_less_eq @ ( set @ A ) @ ( complete_Sup_Sup @ ( set @ A ) @ A3 ) @ C5 ) ) ).
% Union_least
thf(fact_7568_Union__upper,axiom,
! [A: $tType,B6: set @ A,A3: set @ ( set @ A )] :
( ( member @ ( set @ A ) @ B6 @ A3 )
=> ( ord_less_eq @ ( set @ A ) @ B6 @ ( complete_Sup_Sup @ ( set @ A ) @ A3 ) ) ) ).
% Union_upper
thf(fact_7569_Union__subsetI,axiom,
! [A: $tType,A3: set @ ( set @ A ),B6: set @ ( set @ A )] :
( ! [X3: set @ A] :
( ( member @ ( set @ A ) @ X3 @ A3 )
=> ? [Y4: set @ A] :
( ( member @ ( set @ A ) @ Y4 @ B6 )
& ( ord_less_eq @ ( set @ A ) @ X3 @ Y4 ) ) )
=> ( ord_less_eq @ ( set @ A ) @ ( complete_Sup_Sup @ ( set @ A ) @ A3 ) @ ( complete_Sup_Sup @ ( set @ A ) @ B6 ) ) ) ).
% Union_subsetI
thf(fact_7570_finite__UNION__then__finite,axiom,
! [A: $tType,B: $tType,B6: B > ( set @ A ),A3: set @ B,A2: B] :
( ( finite_finite2 @ A @ ( complete_Sup_Sup @ ( set @ A ) @ ( image @ B @ ( set @ A ) @ B6 @ A3 ) ) )
=> ( ( member @ B @ A2 @ A3 )
=> ( finite_finite2 @ A @ ( B6 @ A2 ) ) ) ) ).
% finite_UNION_then_finite
thf(fact_7571_UN__extend__simps_I6_J,axiom,
! [L5: $tType,K8: $tType,A3: K8 > ( set @ L5 ),C5: set @ K8,B6: set @ L5] :
( ( minus_minus @ ( set @ L5 ) @ ( complete_Sup_Sup @ ( set @ L5 ) @ ( image @ K8 @ ( set @ L5 ) @ A3 @ C5 ) ) @ B6 )
= ( complete_Sup_Sup @ ( set @ L5 )
@ ( image @ K8 @ ( set @ L5 )
@ ^ [X2: K8] : ( minus_minus @ ( set @ L5 ) @ ( A3 @ X2 ) @ B6 )
@ C5 ) ) ) ).
% UN_extend_simps(6)
thf(fact_7572_SUP__UN__eq2,axiom,
! [B: $tType,C: $tType,A: $tType,R2: C > ( set @ ( product_prod @ A @ B ) ),S3: set @ C] :
( ( complete_Sup_Sup @ ( A > B > $o )
@ ( image @ C @ ( A > B > $o )
@ ^ [I3: C,X2: A,Y2: B] : ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X2 @ Y2 ) @ ( R2 @ I3 ) )
@ S3 ) )
= ( ^ [X2: A,Y2: B] : ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X2 @ Y2 ) @ ( complete_Sup_Sup @ ( set @ ( product_prod @ A @ B ) ) @ ( image @ C @ ( set @ ( product_prod @ A @ B ) ) @ R2 @ S3 ) ) ) ) ) ).
% SUP_UN_eq2
thf(fact_7573_SUP__Sup__eq2,axiom,
! [B: $tType,A: $tType,S3: set @ ( set @ ( product_prod @ A @ B ) )] :
( ( complete_Sup_Sup @ ( A > B > $o )
@ ( image @ ( set @ ( product_prod @ A @ B ) ) @ ( A > B > $o )
@ ^ [I3: set @ ( product_prod @ A @ B ),X2: A,Y2: B] : ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X2 @ Y2 ) @ I3 )
@ S3 ) )
= ( ^ [X2: A,Y2: B] : ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X2 @ Y2 ) @ ( complete_Sup_Sup @ ( set @ ( product_prod @ A @ B ) ) @ S3 ) ) ) ) ).
% SUP_Sup_eq2
thf(fact_7574_Sup__SUP__eq2,axiom,
! [B: $tType,A: $tType] :
( ( complete_Sup_Sup @ ( A > B > $o ) )
= ( ^ [S8: set @ ( A > B > $o ),X2: A,Y2: B] : ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X2 @ Y2 ) @ ( 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_7575_prod_OUnion__comp,axiom,
! [A: $tType,B: $tType] :
( ( comm_monoid_mult @ A )
=> ! [B6: set @ ( set @ B ),G: B > A] :
( ! [X3: set @ B] :
( ( member @ ( set @ B ) @ X3 @ B6 )
=> ( finite_finite2 @ B @ X3 ) )
=> ( ! [A14: set @ B] :
( ( member @ ( set @ B ) @ A14 @ B6 )
=> ! [A24: set @ B] :
( ( member @ ( set @ B ) @ A24 @ B6 )
=> ( ( A14 != A24 )
=> ! [X3: B] :
( ( member @ B @ X3 @ A14 )
=> ( ( member @ B @ X3 @ A24 )
=> ( ( G @ X3 )
= ( one_one @ A ) ) ) ) ) ) )
=> ( ( groups7121269368397514597t_prod @ B @ A @ G @ ( complete_Sup_Sup @ ( set @ B ) @ B6 ) )
= ( comp @ ( ( set @ B ) > A ) @ ( ( set @ ( set @ B ) ) > A ) @ ( B > A ) @ ( groups7121269368397514597t_prod @ ( set @ B ) @ A ) @ ( groups7121269368397514597t_prod @ B @ A ) @ G @ B6 ) ) ) ) ) ).
% prod.Union_comp
thf(fact_7576_diff__rat__def,axiom,
( ( minus_minus @ rat )
= ( ^ [Q4: rat,R5: rat] : ( plus_plus @ rat @ Q4 @ ( uminus_uminus @ rat @ R5 ) ) ) ) ).
% diff_rat_def
thf(fact_7577_ex__gt__or__lt,axiom,
! [A: $tType] :
( ( condit5016429287641298734tinuum @ A )
=> ! [A2: A] :
? [B3: A] :
( ( ord_less @ A @ A2 @ B3 )
| ( ord_less @ A @ B3 @ A2 ) ) ) ).
% ex_gt_or_lt
thf(fact_7578_finite__subset__Union,axiom,
! [A: $tType,A3: set @ A,B11: set @ ( set @ A )] :
( ( finite_finite2 @ A @ A3 )
=> ( ( ord_less_eq @ ( set @ A ) @ A3 @ ( complete_Sup_Sup @ ( set @ A ) @ B11 ) )
=> ~ ! [F15: set @ ( set @ A )] :
( ( finite_finite2 @ ( set @ A ) @ F15 )
=> ( ( ord_less_eq @ ( set @ ( set @ A ) ) @ F15 @ B11 )
=> ~ ( ord_less_eq @ ( set @ A ) @ A3 @ ( complete_Sup_Sup @ ( set @ A ) @ F15 ) ) ) ) ) ) ).
% finite_subset_Union
thf(fact_7579_UN__le__add__shift__strict,axiom,
! [A: $tType,M8: nat > ( set @ A ),K: nat,N3: nat] :
( ( complete_Sup_Sup @ ( set @ A )
@ ( image @ nat @ ( set @ A )
@ ^ [I3: nat] : ( M8 @ ( plus_plus @ nat @ I3 @ K ) )
@ ( set_ord_lessThan @ nat @ N3 ) ) )
= ( complete_Sup_Sup @ ( set @ A ) @ ( image @ nat @ ( set @ A ) @ M8 @ ( set_or7035219750837199246ssThan @ nat @ K @ ( plus_plus @ nat @ N3 @ K ) ) ) ) ) ).
% UN_le_add_shift_strict
thf(fact_7580_UN__le__add__shift,axiom,
! [A: $tType,M8: nat > ( set @ A ),K: nat,N3: nat] :
( ( complete_Sup_Sup @ ( set @ A )
@ ( image @ nat @ ( set @ A )
@ ^ [I3: nat] : ( M8 @ ( plus_plus @ nat @ I3 @ K ) )
@ ( set_ord_atMost @ nat @ N3 ) ) )
= ( complete_Sup_Sup @ ( set @ A ) @ ( image @ nat @ ( set @ A ) @ M8 @ ( set_or1337092689740270186AtMost @ nat @ K @ ( plus_plus @ nat @ N3 @ K ) ) ) ) ) ).
% UN_le_add_shift
thf(fact_7581_UN__finite__subset,axiom,
! [A: $tType,A3: nat > ( set @ A ),C5: set @ A] :
( ! [N: nat] : ( ord_less_eq @ ( set @ A ) @ ( complete_Sup_Sup @ ( set @ A ) @ ( image @ nat @ ( set @ A ) @ A3 @ ( set_or7035219750837199246ssThan @ nat @ ( zero_zero @ nat ) @ N ) ) ) @ C5 )
=> ( ord_less_eq @ ( set @ A ) @ ( complete_Sup_Sup @ ( set @ A ) @ ( image @ nat @ ( set @ A ) @ A3 @ ( top_top @ ( set @ nat ) ) ) ) @ C5 ) ) ).
% UN_finite_subset
thf(fact_7582_UN__finite2__eq,axiom,
! [A: $tType,A3: nat > ( set @ A ),B6: nat > ( set @ A ),K: nat] :
( ! [N: nat] :
( ( complete_Sup_Sup @ ( set @ A ) @ ( image @ nat @ ( set @ A ) @ A3 @ ( set_or7035219750837199246ssThan @ nat @ ( zero_zero @ nat ) @ N ) ) )
= ( complete_Sup_Sup @ ( set @ A ) @ ( image @ nat @ ( set @ A ) @ B6 @ ( set_or7035219750837199246ssThan @ nat @ ( zero_zero @ nat ) @ ( plus_plus @ nat @ N @ K ) ) ) ) )
=> ( ( complete_Sup_Sup @ ( set @ A ) @ ( image @ nat @ ( set @ A ) @ A3 @ ( top_top @ ( set @ nat ) ) ) )
= ( complete_Sup_Sup @ ( set @ A ) @ ( image @ nat @ ( set @ A ) @ B6 @ ( top_top @ ( set @ nat ) ) ) ) ) ) ).
% UN_finite2_eq
thf(fact_7583_inj__on__mapI,axiom,
! [B: $tType,A: $tType,F2: A > B,A3: set @ ( list @ A )] :
( ( inj_on @ A @ B @ F2 @ ( complete_Sup_Sup @ ( set @ A ) @ ( image @ ( list @ A ) @ ( set @ A ) @ ( set2 @ A ) @ A3 ) ) )
=> ( inj_on @ ( list @ A ) @ ( list @ B ) @ ( map @ A @ B @ F2 ) @ A3 ) ) ).
% inj_on_mapI
thf(fact_7584_UN__finite2__subset,axiom,
! [A: $tType,A3: nat > ( set @ A ),B6: nat > ( set @ A ),K: nat] :
( ! [N: nat] : ( ord_less_eq @ ( set @ A ) @ ( complete_Sup_Sup @ ( set @ A ) @ ( image @ nat @ ( set @ A ) @ A3 @ ( set_or7035219750837199246ssThan @ nat @ ( zero_zero @ nat ) @ N ) ) ) @ ( complete_Sup_Sup @ ( set @ A ) @ ( image @ nat @ ( set @ A ) @ B6 @ ( set_or7035219750837199246ssThan @ nat @ ( zero_zero @ nat ) @ ( plus_plus @ nat @ N @ K ) ) ) ) )
=> ( ord_less_eq @ ( set @ A ) @ ( complete_Sup_Sup @ ( set @ A ) @ ( image @ nat @ ( set @ A ) @ A3 @ ( top_top @ ( set @ nat ) ) ) ) @ ( complete_Sup_Sup @ ( set @ A ) @ ( image @ nat @ ( set @ A ) @ B6 @ ( top_top @ ( set @ nat ) ) ) ) ) ) ).
% UN_finite2_subset
thf(fact_7585_complete__interval,axiom,
! [A: $tType] :
( ( condit6923001295902523014norder @ A )
=> ! [A2: A,B2: A,P: A > $o] :
( ( ord_less @ A @ A2 @ B2 )
=> ( ( P @ A2 )
=> ( ~ ( P @ B2 )
=> ? [C4: A] :
( ( ord_less_eq @ A @ A2 @ C4 )
& ( ord_less_eq @ A @ C4 @ B2 )
& ! [X5: A] :
( ( ( ord_less_eq @ A @ A2 @ X5 )
& ( ord_less @ A @ X5 @ C4 ) )
=> ( P @ X5 ) )
& ! [D6: A] :
( ! [X3: A] :
( ( ( ord_less_eq @ A @ A2 @ X3 )
& ( ord_less @ A @ X3 @ D6 ) )
=> ( P @ X3 ) )
=> ( ord_less_eq @ A @ D6 @ C4 ) ) ) ) ) ) ) ).
% complete_interval
thf(fact_7586_cSup__eq,axiom,
! [A: $tType] :
( ( ( condit1219197933456340205attice @ A )
& ( no_bot @ A ) )
=> ! [X8: set @ A,A2: A] :
( ! [X3: A] :
( ( member @ A @ X3 @ X8 )
=> ( ord_less_eq @ A @ X3 @ A2 ) )
=> ( ! [Y3: A] :
( ! [X5: A] :
( ( member @ A @ X5 @ X8 )
=> ( ord_less_eq @ A @ X5 @ Y3 ) )
=> ( ord_less_eq @ A @ A2 @ Y3 ) )
=> ( ( complete_Sup_Sup @ A @ X8 )
= A2 ) ) ) ) ).
% cSup_eq
thf(fact_7587_cSup__eq__maximum,axiom,
! [A: $tType] :
( ( condit1219197933456340205attice @ A )
=> ! [Z: A,X8: set @ A] :
( ( member @ A @ Z @ X8 )
=> ( ! [X3: A] :
( ( member @ A @ X3 @ X8 )
=> ( ord_less_eq @ A @ X3 @ Z ) )
=> ( ( complete_Sup_Sup @ A @ X8 )
= Z ) ) ) ) ).
% cSup_eq_maximum
thf(fact_7588_Sup__upper2,axiom,
! [A: $tType] :
( ( comple6319245703460814977attice @ A )
=> ! [U: A,A3: set @ A,V: A] :
( ( member @ A @ U @ A3 )
=> ( ( ord_less_eq @ A @ V @ U )
=> ( ord_less_eq @ A @ V @ ( complete_Sup_Sup @ A @ A3 ) ) ) ) ) ).
% Sup_upper2
thf(fact_7589_Sup__le__iff,axiom,
! [A: $tType] :
( ( comple6319245703460814977attice @ A )
=> ! [A3: set @ A,B2: A] :
( ( ord_less_eq @ A @ ( complete_Sup_Sup @ A @ A3 ) @ B2 )
= ( ! [X2: A] :
( ( member @ A @ X2 @ A3 )
=> ( ord_less_eq @ A @ X2 @ B2 ) ) ) ) ) ).
% Sup_le_iff
thf(fact_7590_Sup__upper,axiom,
! [A: $tType] :
( ( comple6319245703460814977attice @ A )
=> ! [X: A,A3: set @ A] :
( ( member @ A @ X @ A3 )
=> ( ord_less_eq @ A @ X @ ( complete_Sup_Sup @ A @ A3 ) ) ) ) ).
% Sup_upper
thf(fact_7591_Sup__least,axiom,
! [A: $tType] :
( ( comple6319245703460814977attice @ A )
=> ! [A3: set @ A,Z: A] :
( ! [X3: A] :
( ( member @ A @ X3 @ A3 )
=> ( ord_less_eq @ A @ X3 @ Z ) )
=> ( ord_less_eq @ A @ ( complete_Sup_Sup @ A @ A3 ) @ Z ) ) ) ).
% Sup_least
thf(fact_7592_Sup__mono,axiom,
! [A: $tType] :
( ( comple6319245703460814977attice @ A )
=> ! [A3: set @ A,B6: set @ A] :
( ! [A4: A] :
( ( member @ A @ A4 @ A3 )
=> ? [X5: A] :
( ( member @ A @ X5 @ B6 )
& ( ord_less_eq @ A @ A4 @ X5 ) ) )
=> ( ord_less_eq @ A @ ( complete_Sup_Sup @ A @ A3 ) @ ( complete_Sup_Sup @ A @ B6 ) ) ) ) ).
% Sup_mono
thf(fact_7593_Sup__eqI,axiom,
! [A: $tType] :
( ( comple6319245703460814977attice @ A )
=> ! [A3: set @ A,X: A] :
( ! [Y3: A] :
( ( member @ A @ Y3 @ A3 )
=> ( ord_less_eq @ A @ Y3 @ X ) )
=> ( ! [Y3: A] :
( ! [Z4: A] :
( ( member @ A @ Z4 @ A3 )
=> ( ord_less_eq @ A @ Z4 @ Y3 ) )
=> ( ord_less_eq @ A @ X @ Y3 ) )
=> ( ( complete_Sup_Sup @ A @ A3 )
= X ) ) ) ) ).
% Sup_eqI
thf(fact_7594_less__Sup__iff,axiom,
! [A: $tType] :
( ( comple5582772986160207858norder @ A )
=> ! [A2: A,S3: set @ A] :
( ( ord_less @ A @ A2 @ ( complete_Sup_Sup @ A @ S3 ) )
= ( ? [X2: A] :
( ( member @ A @ X2 @ S3 )
& ( ord_less @ A @ A2 @ X2 ) ) ) ) ) ).
% less_Sup_iff
thf(fact_7595_le__Sup__iff,axiom,
! [A: $tType] :
( ( comple5582772986160207858norder @ A )
=> ! [X: A,A3: set @ A] :
( ( ord_less_eq @ A @ X @ ( complete_Sup_Sup @ A @ A3 ) )
= ( ! [Y2: A] :
( ( ord_less @ A @ Y2 @ X )
=> ? [X2: A] :
( ( member @ A @ X2 @ A3 )
& ( ord_less @ A @ Y2 @ X2 ) ) ) ) ) ) ).
% le_Sup_iff
thf(fact_7596_SUP__eq,axiom,
! [B: $tType,A: $tType,C: $tType] :
( ( comple6319245703460814977attice @ A )
=> ! [A3: set @ B,B6: set @ C,F2: B > A,G: C > A] :
( ! [I2: B] :
( ( member @ B @ I2 @ A3 )
=> ? [X5: C] :
( ( member @ C @ X5 @ B6 )
& ( ord_less_eq @ A @ ( F2 @ I2 ) @ ( G @ X5 ) ) ) )
=> ( ! [J2: C] :
( ( member @ C @ J2 @ B6 )
=> ? [X5: B] :
( ( member @ B @ X5 @ A3 )
& ( ord_less_eq @ A @ ( G @ J2 ) @ ( F2 @ X5 ) ) ) )
=> ( ( complete_Sup_Sup @ A @ ( image @ B @ A @ F2 @ A3 ) )
= ( complete_Sup_Sup @ A @ ( image @ C @ A @ G @ B6 ) ) ) ) ) ) ).
% SUP_eq
thf(fact_7597_cSup__eq__non__empty,axiom,
! [A: $tType] :
( ( condit1219197933456340205attice @ A )
=> ! [X8: set @ A,A2: A] :
( ( X8
!= ( bot_bot @ ( set @ A ) ) )
=> ( ! [X3: A] :
( ( member @ A @ X3 @ X8 )
=> ( ord_less_eq @ A @ X3 @ A2 ) )
=> ( ! [Y3: A] :
( ! [X5: A] :
( ( member @ A @ X5 @ X8 )
=> ( ord_less_eq @ A @ X5 @ Y3 ) )
=> ( ord_less_eq @ A @ A2 @ Y3 ) )
=> ( ( complete_Sup_Sup @ A @ X8 )
= A2 ) ) ) ) ) ).
% cSup_eq_non_empty
thf(fact_7598_cSup__least,axiom,
! [A: $tType] :
( ( condit1219197933456340205attice @ A )
=> ! [X8: set @ A,Z: A] :
( ( X8
!= ( bot_bot @ ( set @ A ) ) )
=> ( ! [X3: A] :
( ( member @ A @ X3 @ X8 )
=> ( ord_less_eq @ A @ X3 @ Z ) )
=> ( ord_less_eq @ A @ ( complete_Sup_Sup @ A @ X8 ) @ Z ) ) ) ) ).
% cSup_least
thf(fact_7599_less__eq__Sup,axiom,
! [A: $tType] :
( ( comple6319245703460814977attice @ A )
=> ! [A3: set @ A,U: A] :
( ! [V3: A] :
( ( member @ A @ V3 @ A3 )
=> ( ord_less_eq @ A @ U @ V3 ) )
=> ( ( A3
!= ( bot_bot @ ( set @ A ) ) )
=> ( ord_less_eq @ A @ U @ ( complete_Sup_Sup @ A @ A3 ) ) ) ) ) ).
% less_eq_Sup
thf(fact_7600_le__cSup__finite,axiom,
! [A: $tType] :
( ( condit1219197933456340205attice @ A )
=> ! [X8: set @ A,X: A] :
( ( finite_finite2 @ A @ X8 )
=> ( ( member @ A @ X @ X8 )
=> ( ord_less_eq @ A @ X @ ( complete_Sup_Sup @ A @ X8 ) ) ) ) ) ).
% le_cSup_finite
thf(fact_7601_less__cSupD,axiom,
! [A: $tType] :
( ( condit6923001295902523014norder @ A )
=> ! [X8: set @ A,Z: A] :
( ( X8
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( ord_less @ A @ Z @ ( complete_Sup_Sup @ A @ X8 ) )
=> ? [X3: A] :
( ( member @ A @ X3 @ X8 )
& ( ord_less @ A @ Z @ X3 ) ) ) ) ) ).
% less_cSupD
thf(fact_7602_less__cSupE,axiom,
! [A: $tType] :
( ( condit6923001295902523014norder @ A )
=> ! [Y: A,X8: set @ A] :
( ( ord_less @ A @ Y @ ( complete_Sup_Sup @ A @ X8 ) )
=> ( ( X8
!= ( bot_bot @ ( set @ A ) ) )
=> ~ ! [X3: A] :
( ( member @ A @ X3 @ X8 )
=> ~ ( ord_less @ A @ Y @ X3 ) ) ) ) ) ).
% less_cSupE
thf(fact_7603_finite__imp__Sup__less,axiom,
! [A: $tType] :
( ( condit6923001295902523014norder @ A )
=> ! [X8: set @ A,X: A,A2: A] :
( ( finite_finite2 @ A @ X8 )
=> ( ( member @ A @ X @ X8 )
=> ( ! [X3: A] :
( ( member @ A @ X3 @ X8 )
=> ( ord_less @ A @ X3 @ A2 ) )
=> ( ord_less @ A @ ( complete_Sup_Sup @ A @ X8 ) @ A2 ) ) ) ) ) ).
% finite_imp_Sup_less
thf(fact_7604_Sup__subset__mono,axiom,
! [A: $tType] :
( ( comple6319245703460814977attice @ A )
=> ! [A3: set @ A,B6: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ A3 @ B6 )
=> ( ord_less_eq @ A @ ( complete_Sup_Sup @ A @ A3 ) @ ( complete_Sup_Sup @ A @ B6 ) ) ) ) ).
% Sup_subset_mono
thf(fact_7605_SUP__eqI,axiom,
! [B: $tType,A: $tType] :
( ( comple6319245703460814977attice @ A )
=> ! [A3: set @ B,F2: B > A,X: A] :
( ! [I2: B] :
( ( member @ B @ I2 @ A3 )
=> ( ord_less_eq @ A @ ( F2 @ I2 ) @ X ) )
=> ( ! [Y3: A] :
( ! [I4: B] :
( ( member @ B @ I4 @ A3 )
=> ( ord_less_eq @ A @ ( F2 @ I4 ) @ Y3 ) )
=> ( ord_less_eq @ A @ X @ Y3 ) )
=> ( ( complete_Sup_Sup @ A @ ( image @ B @ A @ F2 @ A3 ) )
= X ) ) ) ) ).
% SUP_eqI
thf(fact_7606_SUP__mono,axiom,
! [B: $tType,A: $tType,C: $tType] :
( ( comple6319245703460814977attice @ A )
=> ! [A3: set @ B,B6: set @ C,F2: B > A,G: C > A] :
( ! [N: B] :
( ( member @ B @ N @ A3 )
=> ? [X5: C] :
( ( member @ C @ X5 @ B6 )
& ( ord_less_eq @ A @ ( F2 @ N ) @ ( G @ X5 ) ) ) )
=> ( ord_less_eq @ A @ ( complete_Sup_Sup @ A @ ( image @ B @ A @ F2 @ A3 ) ) @ ( complete_Sup_Sup @ A @ ( image @ C @ A @ G @ B6 ) ) ) ) ) ).
% SUP_mono
thf(fact_7607_SUP__least,axiom,
! [B: $tType,A: $tType] :
( ( comple6319245703460814977attice @ A )
=> ! [A3: set @ B,F2: B > A,U: A] :
( ! [I2: B] :
( ( member @ B @ I2 @ A3 )
=> ( ord_less_eq @ A @ ( F2 @ I2 ) @ U ) )
=> ( ord_less_eq @ A @ ( complete_Sup_Sup @ A @ ( image @ B @ A @ F2 @ A3 ) ) @ U ) ) ) ).
% SUP_least
thf(fact_7608_SUP__mono_H,axiom,
! [A: $tType,B: $tType] :
( ( comple6319245703460814977attice @ A )
=> ! [F2: B > A,G: B > A,A3: set @ B] :
( ! [X3: B] : ( ord_less_eq @ A @ ( F2 @ X3 ) @ ( G @ X3 ) )
=> ( ord_less_eq @ A @ ( complete_Sup_Sup @ A @ ( image @ B @ A @ F2 @ A3 ) ) @ ( complete_Sup_Sup @ A @ ( image @ B @ A @ G @ A3 ) ) ) ) ) ).
% SUP_mono'
thf(fact_7609_SUP__upper,axiom,
! [A: $tType,B: $tType] :
( ( comple6319245703460814977attice @ A )
=> ! [I: B,A3: set @ B,F2: B > A] :
( ( member @ B @ I @ A3 )
=> ( ord_less_eq @ A @ ( F2 @ I ) @ ( complete_Sup_Sup @ A @ ( image @ B @ A @ F2 @ A3 ) ) ) ) ) ).
% SUP_upper
thf(fact_7610_SUP__le__iff,axiom,
! [A: $tType,B: $tType] :
( ( comple6319245703460814977attice @ A )
=> ! [F2: B > A,A3: set @ B,U: A] :
( ( ord_less_eq @ A @ ( complete_Sup_Sup @ A @ ( image @ B @ A @ F2 @ A3 ) ) @ U )
= ( ! [X2: B] :
( ( member @ B @ X2 @ A3 )
=> ( ord_less_eq @ A @ ( F2 @ X2 ) @ U ) ) ) ) ) ).
% SUP_le_iff
thf(fact_7611_SUP__upper2,axiom,
! [A: $tType,B: $tType] :
( ( comple6319245703460814977attice @ A )
=> ! [I: B,A3: set @ B,U: A,F2: B > A] :
( ( member @ B @ I @ A3 )
=> ( ( ord_less_eq @ A @ U @ ( F2 @ I ) )
=> ( ord_less_eq @ A @ U @ ( complete_Sup_Sup @ A @ ( image @ B @ A @ F2 @ A3 ) ) ) ) ) ) ).
% SUP_upper2
thf(fact_7612_less__SUP__iff,axiom,
! [A: $tType,B: $tType] :
( ( comple5582772986160207858norder @ A )
=> ! [A2: A,F2: B > A,A3: set @ B] :
( ( ord_less @ A @ A2 @ ( complete_Sup_Sup @ A @ ( image @ B @ A @ F2 @ A3 ) ) )
= ( ? [X2: B] :
( ( member @ B @ X2 @ A3 )
& ( ord_less @ A @ A2 @ ( F2 @ X2 ) ) ) ) ) ) ).
% less_SUP_iff
thf(fact_7613_SUP__lessD,axiom,
! [B: $tType,A: $tType] :
( ( comple6319245703460814977attice @ A )
=> ! [F2: B > A,A3: set @ B,Y: A,I: B] :
( ( ord_less @ A @ ( complete_Sup_Sup @ A @ ( image @ B @ A @ F2 @ A3 ) ) @ Y )
=> ( ( member @ B @ I @ A3 )
=> ( ord_less @ A @ ( F2 @ I ) @ Y ) ) ) ) ).
% SUP_lessD
thf(fact_7614_le__SUP__iff,axiom,
! [B: $tType,A: $tType] :
( ( comple5582772986160207858norder @ A )
=> ! [X: A,F2: B > A,A3: set @ B] :
( ( ord_less_eq @ A @ X @ ( complete_Sup_Sup @ A @ ( image @ B @ A @ F2 @ A3 ) ) )
= ( ! [Y2: A] :
( ( ord_less @ A @ Y2 @ X )
=> ? [X2: B] :
( ( member @ B @ X2 @ A3 )
& ( ord_less @ A @ Y2 @ ( F2 @ X2 ) ) ) ) ) ) ) ).
% le_SUP_iff
thf(fact_7615_cSUP__least,axiom,
! [B: $tType,A: $tType] :
( ( condit1219197933456340205attice @ A )
=> ! [A3: set @ B,F2: B > A,M8: A] :
( ( A3
!= ( bot_bot @ ( set @ B ) ) )
=> ( ! [X3: B] :
( ( member @ B @ X3 @ A3 )
=> ( ord_less_eq @ A @ ( F2 @ X3 ) @ M8 ) )
=> ( ord_less_eq @ A @ ( complete_Sup_Sup @ A @ ( image @ B @ A @ F2 @ A3 ) ) @ M8 ) ) ) ) ).
% cSUP_least
thf(fact_7616_SUP__eq__iff,axiom,
! [A: $tType,B: $tType] :
( ( comple6319245703460814977attice @ A )
=> ! [I5: set @ B,C2: A,F2: B > A] :
( ( I5
!= ( bot_bot @ ( set @ B ) ) )
=> ( ! [I2: B] :
( ( member @ B @ I2 @ I5 )
=> ( ord_less_eq @ A @ C2 @ ( F2 @ I2 ) ) )
=> ( ( ( complete_Sup_Sup @ A @ ( image @ B @ A @ F2 @ I5 ) )
= C2 )
= ( ! [X2: B] :
( ( member @ B @ X2 @ I5 )
=> ( ( F2 @ X2 )
= C2 ) ) ) ) ) ) ) ).
% SUP_eq_iff
thf(fact_7617_finite__Sup__less__iff,axiom,
! [A: $tType] :
( ( condit6923001295902523014norder @ A )
=> ! [X8: set @ A,A2: A] :
( ( finite_finite2 @ A @ X8 )
=> ( ( X8
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( ord_less @ A @ ( complete_Sup_Sup @ A @ X8 ) @ A2 )
= ( ! [X2: A] :
( ( member @ A @ X2 @ X8 )
=> ( ord_less @ A @ X2 @ A2 ) ) ) ) ) ) ) ).
% finite_Sup_less_iff
thf(fact_7618_cSup__abs__le,axiom,
! [A: $tType] :
( ( ( condit6923001295902523014norder @ A )
& ( linordered_idom @ A ) )
=> ! [S3: set @ A,A2: A] :
( ( S3
!= ( bot_bot @ ( set @ A ) ) )
=> ( ! [X3: A] :
( ( member @ A @ X3 @ S3 )
=> ( ord_less_eq @ A @ ( abs_abs @ A @ X3 ) @ A2 ) )
=> ( ord_less_eq @ A @ ( abs_abs @ A @ ( complete_Sup_Sup @ A @ S3 ) ) @ A2 ) ) ) ) ).
% cSup_abs_le
thf(fact_7619_SUP__subset__mono,axiom,
! [A: $tType,B: $tType] :
( ( comple6319245703460814977attice @ A )
=> ! [A3: set @ B,B6: set @ B,F2: B > A,G: B > A] :
( ( ord_less_eq @ ( set @ B ) @ A3 @ B6 )
=> ( ! [X3: B] :
( ( member @ B @ X3 @ A3 )
=> ( ord_less_eq @ A @ ( F2 @ X3 ) @ ( G @ X3 ) ) )
=> ( ord_less_eq @ A @ ( complete_Sup_Sup @ A @ ( image @ B @ A @ F2 @ A3 ) ) @ ( complete_Sup_Sup @ A @ ( image @ B @ A @ G @ B6 ) ) ) ) ) ) ).
% SUP_subset_mono
thf(fact_7620_totally__bounded__metric,axiom,
! [A: $tType] :
( ( real_V7819770556892013058_space @ A )
=> ( ( topolo6688025880775521714ounded @ A )
= ( ^ [S8: set @ A] :
! [E4: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ E4 )
=> ? [K2: set @ A] :
( ( finite_finite2 @ A @ K2 )
& ( ord_less_eq @ ( set @ A ) @ S8
@ ( complete_Sup_Sup @ ( set @ A )
@ ( image @ A @ ( set @ A )
@ ^ [X2: A] :
( collect @ A
@ ^ [Y2: A] : ( ord_less @ real @ ( real_V557655796197034286t_dist @ A @ X2 @ Y2 ) @ E4 ) )
@ K2 ) ) ) ) ) ) ) ) ).
% totally_bounded_metric
thf(fact_7621_interval__cases,axiom,
! [A: $tType] :
( ( condit6923001295902523014norder @ A )
=> ! [S3: set @ A] :
( ! [A4: A,B3: A,X3: A] :
( ( member @ A @ A4 @ S3 )
=> ( ( member @ A @ B3 @ S3 )
=> ( ( ord_less_eq @ A @ A4 @ X3 )
=> ( ( ord_less_eq @ A @ X3 @ B3 )
=> ( member @ A @ X3 @ S3 ) ) ) ) )
=> ? [A4: A,B3: A] :
( ( S3
= ( bot_bot @ ( set @ A ) ) )
| ( S3
= ( top_top @ ( set @ A ) ) )
| ( S3
= ( set_ord_lessThan @ A @ B3 ) )
| ( S3
= ( set_ord_atMost @ A @ B3 ) )
| ( S3
= ( set_ord_greaterThan @ A @ A4 ) )
| ( S3
= ( set_ord_atLeast @ A @ A4 ) )
| ( S3
= ( set_or5935395276787703475ssThan @ A @ A4 @ B3 ) )
| ( S3
= ( set_or3652927894154168847AtMost @ A @ A4 @ B3 ) )
| ( S3
= ( set_or7035219750837199246ssThan @ A @ A4 @ B3 ) )
| ( S3
= ( set_or1337092689740270186AtMost @ A @ A4 @ B3 ) ) ) ) ) ).
% interval_cases
thf(fact_7622_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_7623_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_7624_atLeast__subset__iff,axiom,
! [A: $tType] :
( ( preorder @ A )
=> ! [X: A,Y: A] :
( ( ord_less_eq @ ( set @ A ) @ ( set_ord_atLeast @ A @ X ) @ ( set_ord_atLeast @ A @ Y ) )
= ( ord_less_eq @ A @ Y @ X ) ) ) ).
% atLeast_subset_iff
thf(fact_7625_Icc__subset__Ici__iff,axiom,
! [A: $tType] :
( ( preorder @ A )
=> ! [L2: A,H2: A,L4: A] :
( ( ord_less_eq @ ( set @ A ) @ ( set_or1337092689740270186AtMost @ A @ L2 @ H2 ) @ ( set_ord_atLeast @ A @ L4 ) )
= ( ~ ( ord_less_eq @ A @ L2 @ H2 )
| ( ord_less_eq @ A @ L4 @ L2 ) ) ) ) ).
% Icc_subset_Ici_iff
thf(fact_7626_image__minus__const__atLeast,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [C2: A,A2: A] :
( ( image @ A @ A @ ( minus_minus @ A @ C2 ) @ ( set_ord_atLeast @ A @ A2 ) )
= ( set_ord_atMost @ A @ ( minus_minus @ A @ C2 @ A2 ) ) ) ) ).
% image_minus_const_atLeast
thf(fact_7627_image__minus__const__AtMost,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [C2: A,B2: A] :
( ( image @ A @ A @ ( minus_minus @ A @ C2 ) @ ( set_ord_atMost @ A @ B2 ) )
= ( set_ord_atLeast @ A @ ( minus_minus @ A @ C2 @ B2 ) ) ) ) ).
% image_minus_const_AtMost
thf(fact_7628_atLeast__Suc__greaterThan,axiom,
! [K: nat] :
( ( set_ord_atLeast @ nat @ ( suc @ K ) )
= ( set_ord_greaterThan @ nat @ K ) ) ).
% atLeast_Suc_greaterThan
thf(fact_7629_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_7630_not__Ici__le__Iic,axiom,
! [A: $tType] :
( ( no_top @ A )
=> ! [L2: A,H3: A] :
~ ( ord_less_eq @ ( set @ A ) @ ( set_ord_atLeast @ A @ L2 ) @ ( set_ord_atMost @ A @ H3 ) ) ) ).
% not_Ici_le_Iic
thf(fact_7631_not__Ici__le__Icc,axiom,
! [A: $tType] :
( ( no_top @ A )
=> ! [L2: A,L4: A,H3: A] :
~ ( ord_less_eq @ ( set @ A ) @ ( set_ord_atLeast @ A @ L2 ) @ ( set_or1337092689740270186AtMost @ A @ L4 @ H3 ) ) ) ).
% not_Ici_le_Icc
thf(fact_7632_atLeast__def,axiom,
! [A: $tType] :
( ( ord @ A )
=> ( ( set_ord_atLeast @ A )
= ( ^ [L: A] : ( collect @ A @ ( ord_less_eq @ A @ L ) ) ) ) ) ).
% atLeast_def
thf(fact_7633_totally__bounded__subset,axiom,
! [A: $tType] :
( ( topolo7287701948861334536_space @ A )
=> ! [S3: set @ A,T5: set @ A] :
( ( topolo6688025880775521714ounded @ A @ S3 )
=> ( ( ord_less_eq @ ( set @ A ) @ T5 @ S3 )
=> ( topolo6688025880775521714ounded @ A @ T5 ) ) ) ) ).
% totally_bounded_subset
thf(fact_7634_not__UNIV__le__Ici,axiom,
! [A: $tType] :
( ( no_bot @ A )
=> ! [L2: A] :
~ ( ord_less_eq @ ( set @ A ) @ ( top_top @ ( set @ A ) ) @ ( set_ord_atLeast @ A @ L2 ) ) ) ).
% not_UNIV_le_Ici
thf(fact_7635_Ioi__le__Ico,axiom,
! [A: $tType] :
( ( preorder @ A )
=> ! [A2: A] : ( ord_less_eq @ ( set @ A ) @ ( set_ord_greaterThan @ A @ A2 ) @ ( set_ord_atLeast @ A @ A2 ) ) ) ).
% Ioi_le_Ico
thf(fact_7636_Ici__subset__Ioi__iff,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A2: A,B2: A] :
( ( ord_less_eq @ ( set @ A ) @ ( set_ord_atLeast @ A @ A2 ) @ ( set_ord_greaterThan @ A @ B2 ) )
= ( ord_less @ A @ B2 @ A2 ) ) ) ).
% Ici_subset_Ioi_iff
thf(fact_7637_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_7638_UN__image__subset,axiom,
! [C: $tType,A: $tType,B: $tType,F2: B > ( set @ A ),G: C > ( set @ B ),X: C,X8: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ ( complete_Sup_Sup @ ( set @ A ) @ ( image @ B @ ( set @ A ) @ F2 @ ( G @ X ) ) ) @ X8 )
= ( ord_less_eq @ ( set @ B ) @ ( G @ X )
@ ( collect @ B
@ ^ [X2: B] : ( ord_less_eq @ ( set @ A ) @ ( F2 @ X2 ) @ X8 ) ) ) ) ).
% UN_image_subset
thf(fact_7639_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_7640_these__image__Some__eq,axiom,
! [A: $tType,A3: set @ A] :
( ( these @ A @ ( image @ A @ ( option @ A ) @ ( some @ A ) @ A3 ) )
= A3 ) ).
% these_image_Some_eq
thf(fact_7641_these__empty,axiom,
! [A: $tType] :
( ( these @ A @ ( bot_bot @ ( set @ ( option @ A ) ) ) )
= ( bot_bot @ ( set @ A ) ) ) ).
% these_empty
thf(fact_7642_these__insert__None,axiom,
! [A: $tType,A3: set @ ( option @ A )] :
( ( these @ A @ ( insert @ ( option @ A ) @ ( none @ A ) @ A3 ) )
= ( these @ A @ A3 ) ) ).
% these_insert_None
thf(fact_7643_these__insert__Some,axiom,
! [A: $tType,X: A,A3: set @ ( option @ A )] :
( ( these @ A @ ( insert @ ( option @ A ) @ ( some @ A @ X ) @ A3 ) )
= ( insert @ A @ X @ ( these @ A @ A3 ) ) ) ).
% these_insert_Some
thf(fact_7644_in__these__eq,axiom,
! [A: $tType,X: A,A3: set @ ( option @ A )] :
( ( member @ A @ X @ ( these @ A @ A3 ) )
= ( member @ ( option @ A ) @ ( some @ A @ X ) @ A3 ) ) ).
% in_these_eq
thf(fact_7645_Option_Othese__def,axiom,
! [A: $tType] :
( ( these @ A )
= ( ^ [A7: set @ ( option @ A )] :
( image @ ( option @ A ) @ A @ ( the2 @ A )
@ ( collect @ ( option @ A )
@ ^ [X2: option @ A] :
( ( member @ ( option @ A ) @ X2 @ A7 )
& ( X2
!= ( none @ A ) ) ) ) ) ) ) ).
% Option.these_def
thf(fact_7646_conj__subset__def,axiom,
! [A: $tType,A3: set @ A,P: A > $o,Q: A > $o] :
( ( ord_less_eq @ ( set @ A ) @ A3
@ ( collect @ A
@ ^ [X2: A] :
( ( P @ X2 )
& ( Q @ X2 ) ) ) )
= ( ( ord_less_eq @ ( set @ A ) @ A3 @ ( collect @ A @ P ) )
& ( ord_less_eq @ ( set @ A ) @ A3 @ ( collect @ A @ Q ) ) ) ) ).
% conj_subset_def
thf(fact_7647_Some__image__these__eq,axiom,
! [A: $tType,A3: set @ ( option @ A )] :
( ( image @ A @ ( option @ A ) @ ( some @ A ) @ ( these @ A @ A3 ) )
= ( collect @ ( option @ A )
@ ^ [X2: option @ A] :
( ( member @ ( option @ A ) @ X2 @ A3 )
& ( X2
!= ( none @ A ) ) ) ) ) ).
% Some_image_these_eq
thf(fact_7648_these__empty__eq,axiom,
! [A: $tType,B6: set @ ( option @ A )] :
( ( ( these @ A @ B6 )
= ( bot_bot @ ( set @ A ) ) )
= ( ( B6
= ( bot_bot @ ( set @ ( option @ A ) ) ) )
| ( B6
= ( insert @ ( option @ A ) @ ( none @ A ) @ ( bot_bot @ ( set @ ( option @ A ) ) ) ) ) ) ) ).
% these_empty_eq
thf(fact_7649_these__not__empty__eq,axiom,
! [A: $tType,B6: set @ ( option @ A )] :
( ( ( these @ A @ B6 )
!= ( bot_bot @ ( set @ A ) ) )
= ( ( B6
!= ( bot_bot @ ( set @ ( option @ A ) ) ) )
& ( B6
!= ( insert @ ( option @ A ) @ ( none @ A ) @ ( bot_bot @ ( set @ ( option @ A ) ) ) ) ) ) ) ).
% these_not_empty_eq
thf(fact_7650_map__filter__on__comp,axiom,
! [A: $tType,C: $tType,B: $tType,G: B > A,Y9: set @ B,X8: set @ A,F4: filter @ B,F2: A > C] :
( ( ord_less_eq @ ( set @ A ) @ ( image @ B @ A @ G @ Y9 ) @ X8 )
=> ( ( eventually @ B
@ ^ [X2: B] : ( member @ B @ X2 @ Y9 )
@ F4 )
=> ( ( map_filter_on @ A @ C @ X8 @ F2 @ ( map_filter_on @ B @ A @ Y9 @ G @ F4 ) )
= ( map_filter_on @ B @ C @ Y9 @ ( comp @ A @ C @ B @ F2 @ G ) @ F4 ) ) ) ) ).
% map_filter_on_comp
thf(fact_7651_rat__inverse__code,axiom,
! [P4: rat] :
( ( quotient_of @ ( inverse_inverse @ rat @ P4 ) )
= ( product_case_prod @ int @ int @ ( product_prod @ int @ int )
@ ^ [A6: int,B5: 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 ) @ B5 ) @ ( abs_abs @ int @ A6 ) ) )
@ ( quotient_of @ P4 ) ) ) ).
% rat_inverse_code
thf(fact_7652_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_7653_rat__one__code,axiom,
( ( quotient_of @ ( one_one @ rat ) )
= ( product_Pair @ int @ int @ ( one_one @ int ) @ ( one_one @ int ) ) ) ).
% rat_one_code
thf(fact_7654_rat__zero__code,axiom,
( ( quotient_of @ ( zero_zero @ rat ) )
= ( product_Pair @ int @ int @ ( zero_zero @ int ) @ ( one_one @ int ) ) ) ).
% rat_zero_code
thf(fact_7655_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_7656_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_7657_quotient__of__div,axiom,
! [R2: rat,N3: int,D: int] :
( ( ( quotient_of @ R2 )
= ( product_Pair @ int @ int @ N3 @ D ) )
=> ( R2
= ( divide_divide @ rat @ ( ring_1_of_int @ rat @ N3 ) @ ( ring_1_of_int @ rat @ D ) ) ) ) ).
% quotient_of_div
thf(fact_7658_quotient__of__denom__pos,axiom,
! [R2: rat,P4: int,Q2: int] :
( ( ( quotient_of @ R2 )
= ( product_Pair @ int @ int @ P4 @ Q2 ) )
=> ( ord_less @ int @ ( zero_zero @ int ) @ Q2 ) ) ).
% quotient_of_denom_pos
thf(fact_7659_rat__abs__code,axiom,
! [P4: rat] :
( ( quotient_of @ ( abs_abs @ rat @ P4 ) )
= ( product_case_prod @ int @ int @ ( product_prod @ int @ int )
@ ^ [A6: int] : ( product_Pair @ int @ int @ ( abs_abs @ int @ A6 ) )
@ ( quotient_of @ P4 ) ) ) ).
% rat_abs_code
thf(fact_7660_rat__uminus__code,axiom,
! [P4: rat] :
( ( quotient_of @ ( uminus_uminus @ rat @ P4 ) )
= ( product_case_prod @ int @ int @ ( product_prod @ int @ int )
@ ^ [A6: int] : ( product_Pair @ int @ int @ ( uminus_uminus @ int @ A6 ) )
@ ( quotient_of @ P4 ) ) ) ).
% rat_uminus_code
thf(fact_7661_rat__less__eq__code,axiom,
( ( ord_less_eq @ rat )
= ( ^ [P5: rat,Q4: rat] :
( product_case_prod @ int @ int @ $o
@ ^ [A6: int,C3: int] :
( product_case_prod @ int @ int @ $o
@ ^ [B5: int,D2: int] : ( ord_less_eq @ int @ ( times_times @ int @ A6 @ D2 ) @ ( times_times @ int @ C3 @ B5 ) )
@ ( quotient_of @ Q4 ) )
@ ( quotient_of @ P5 ) ) ) ) ).
% rat_less_eq_code
thf(fact_7662_quotient__of__int,axiom,
! [A2: int] :
( ( quotient_of @ ( of_int @ A2 ) )
= ( product_Pair @ int @ int @ A2 @ ( one_one @ int ) ) ) ).
% quotient_of_int
thf(fact_7663_rat__minus__code,axiom,
! [P4: rat,Q2: rat] :
( ( quotient_of @ ( minus_minus @ rat @ P4 @ Q2 ) )
= ( product_case_prod @ int @ int @ ( product_prod @ int @ int )
@ ^ [A6: int,C3: int] :
( product_case_prod @ int @ int @ ( product_prod @ int @ int )
@ ^ [B5: int,D2: int] : ( normalize @ ( product_Pair @ int @ int @ ( minus_minus @ int @ ( times_times @ int @ A6 @ D2 ) @ ( times_times @ int @ B5 @ C3 ) ) @ ( times_times @ int @ C3 @ D2 ) ) )
@ ( quotient_of @ Q2 ) )
@ ( quotient_of @ P4 ) ) ) ).
% rat_minus_code
thf(fact_7664_normalize__denom__zero,axiom,
! [P4: int] :
( ( normalize @ ( product_Pair @ int @ int @ P4 @ ( zero_zero @ int ) ) )
= ( product_Pair @ int @ int @ ( zero_zero @ int ) @ ( one_one @ int ) ) ) ).
% normalize_denom_zero
thf(fact_7665_normalize__negative,axiom,
! [Q2: int,P4: int] :
( ( ord_less @ int @ Q2 @ ( zero_zero @ int ) )
=> ( ( normalize @ ( product_Pair @ int @ int @ P4 @ Q2 ) )
= ( normalize @ ( product_Pair @ int @ int @ ( uminus_uminus @ int @ P4 ) @ ( uminus_uminus @ int @ Q2 ) ) ) ) ) ).
% normalize_negative
thf(fact_7666_normalize__denom__pos,axiom,
! [R2: product_prod @ int @ int,P4: int,Q2: int] :
( ( ( normalize @ R2 )
= ( product_Pair @ int @ int @ P4 @ Q2 ) )
=> ( ord_less @ int @ ( zero_zero @ int ) @ Q2 ) ) ).
% normalize_denom_pos
thf(fact_7667_normalize__crossproduct,axiom,
! [Q2: int,S: int,P4: int,R2: int] :
( ( Q2
!= ( zero_zero @ int ) )
=> ( ( S
!= ( zero_zero @ int ) )
=> ( ( ( normalize @ ( product_Pair @ int @ int @ P4 @ Q2 ) )
= ( normalize @ ( product_Pair @ int @ int @ R2 @ S ) ) )
=> ( ( times_times @ int @ P4 @ S )
= ( times_times @ int @ R2 @ Q2 ) ) ) ) ) ).
% normalize_crossproduct
thf(fact_7668_rat__times__code,axiom,
! [P4: rat,Q2: rat] :
( ( quotient_of @ ( times_times @ rat @ P4 @ Q2 ) )
= ( product_case_prod @ int @ int @ ( product_prod @ int @ int )
@ ^ [A6: int,C3: int] :
( product_case_prod @ int @ int @ ( product_prod @ int @ int )
@ ^ [B5: int,D2: int] : ( normalize @ ( product_Pair @ int @ int @ ( times_times @ int @ A6 @ B5 ) @ ( times_times @ int @ C3 @ D2 ) ) )
@ ( quotient_of @ Q2 ) )
@ ( quotient_of @ P4 ) ) ) ).
% rat_times_code
thf(fact_7669_rat__divide__code,axiom,
! [P4: rat,Q2: rat] :
( ( quotient_of @ ( divide_divide @ rat @ P4 @ Q2 ) )
= ( product_case_prod @ int @ int @ ( product_prod @ int @ int )
@ ^ [A6: int,C3: int] :
( product_case_prod @ int @ int @ ( product_prod @ int @ int )
@ ^ [B5: int,D2: int] : ( normalize @ ( product_Pair @ int @ int @ ( times_times @ int @ A6 @ D2 ) @ ( times_times @ int @ C3 @ B5 ) ) )
@ ( quotient_of @ Q2 ) )
@ ( quotient_of @ P4 ) ) ) ).
% rat_divide_code
thf(fact_7670_rat__plus__code,axiom,
! [P4: rat,Q2: rat] :
( ( quotient_of @ ( plus_plus @ rat @ P4 @ Q2 ) )
= ( product_case_prod @ int @ int @ ( product_prod @ int @ int )
@ ^ [A6: int,C3: int] :
( product_case_prod @ int @ int @ ( product_prod @ int @ int )
@ ^ [B5: int,D2: int] : ( normalize @ ( product_Pair @ int @ int @ ( plus_plus @ int @ ( times_times @ int @ A6 @ D2 ) @ ( times_times @ int @ B5 @ C3 ) ) @ ( times_times @ int @ C3 @ D2 ) ) )
@ ( quotient_of @ Q2 ) )
@ ( quotient_of @ P4 ) ) ) ).
% rat_plus_code
thf(fact_7671_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_7672_GMVT,axiom,
! [A2: real,B2: real,F2: real > real,G: real > real] :
( ( ord_less @ real @ A2 @ B2 )
=> ( ! [X3: real] :
( ( ( ord_less_eq @ real @ A2 @ X3 )
& ( ord_less_eq @ real @ X3 @ B2 ) )
=> ( topolo3448309680560233919inuous @ real @ real @ ( topolo174197925503356063within @ real @ X3 @ ( top_top @ ( set @ real ) ) ) @ F2 ) )
=> ( ! [X3: real] :
( ( ( ord_less @ real @ A2 @ X3 )
& ( ord_less @ real @ X3 @ B2 ) )
=> ( differentiable @ real @ real @ F2 @ ( topolo174197925503356063within @ real @ X3 @ ( top_top @ ( set @ real ) ) ) ) )
=> ( ! [X3: real] :
( ( ( ord_less_eq @ real @ A2 @ X3 )
& ( ord_less_eq @ real @ X3 @ B2 ) )
=> ( topolo3448309680560233919inuous @ real @ real @ ( topolo174197925503356063within @ real @ X3 @ ( top_top @ ( set @ real ) ) ) @ G ) )
=> ( ! [X3: real] :
( ( ( ord_less @ real @ A2 @ X3 )
& ( ord_less @ real @ X3 @ B2 ) )
=> ( differentiable @ real @ real @ G @ ( topolo174197925503356063within @ real @ X3 @ ( 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 @ A2 @ C4 )
& ( ord_less @ real @ C4 @ B2 )
& ( ( times_times @ real @ ( minus_minus @ real @ ( F2 @ B2 ) @ ( F2 @ A2 ) ) @ G_c )
= ( times_times @ real @ ( minus_minus @ real @ ( G @ B2 ) @ ( G @ A2 ) ) @ F_c ) ) ) ) ) ) ) ) ).
% GMVT
thf(fact_7673_differentiable__cmult__right__iff,axiom,
! [A: $tType,B: $tType] :
( ( ( real_V822414075346904944vector @ B )
& ( real_V3459762299906320749_field @ A ) )
=> ! [Q2: B > A,C2: A,T2: B] :
( ( differentiable @ B @ A
@ ^ [T3: B] : ( times_times @ A @ ( Q2 @ T3 ) @ C2 )
@ ( topolo174197925503356063within @ B @ T2 @ ( top_top @ ( set @ B ) ) ) )
= ( ( C2
= ( zero_zero @ A ) )
| ( differentiable @ B @ A @ Q2 @ ( topolo174197925503356063within @ B @ T2 @ ( top_top @ ( set @ B ) ) ) ) ) ) ) ).
% differentiable_cmult_right_iff
thf(fact_7674_differentiable__cmult__left__iff,axiom,
! [A: $tType,B: $tType] :
( ( ( real_V822414075346904944vector @ B )
& ( real_V3459762299906320749_field @ A ) )
=> ! [C2: A,Q2: B > A,T2: B] :
( ( differentiable @ B @ A
@ ^ [T3: B] : ( times_times @ A @ C2 @ ( Q2 @ T3 ) )
@ ( topolo174197925503356063within @ B @ T2 @ ( top_top @ ( set @ B ) ) ) )
= ( ( C2
= ( zero_zero @ A ) )
| ( differentiable @ B @ A @ Q2 @ ( topolo174197925503356063within @ B @ T2 @ ( top_top @ ( set @ B ) ) ) ) ) ) ) ).
% differentiable_cmult_left_iff
thf(fact_7675_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
@ ^ [X2: A] : ( plus_plus @ B @ ( F2 @ X2 ) @ ( G @ X2 ) )
@ F4 ) ) ) ) ).
% differentiable_add
thf(fact_7676_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
@ ^ [X2: A] : ( uminus_uminus @ B @ ( F2 @ X2 ) )
@ F4 ) ) ) ).
% differentiable_minus
thf(fact_7677_differentiable__def,axiom,
! [B: $tType,A: $tType] :
( ( ( real_V822414075346904944vector @ A )
& ( real_V822414075346904944vector @ B ) )
=> ( ( differentiable @ A @ B )
= ( ^ [F7: A > B,F9: filter @ A] :
? [D8: A > B] : ( has_derivative @ A @ B @ F7 @ D8 @ F9 ) ) ) ) ).
% differentiable_def
thf(fact_7678_differentiable__sum,axiom,
! [C: $tType,A: $tType,B: $tType] :
( ( ( real_V822414075346904944vector @ B )
& ( real_V822414075346904944vector @ C ) )
=> ! [S: set @ A,F2: A > B > C,Net: filter @ B] :
( ( finite_finite2 @ A @ S )
=> ( ! [X3: A] :
( ( member @ A @ X3 @ S )
=> ( differentiable @ B @ C @ ( F2 @ X3 ) @ Net ) )
=> ( differentiable @ B @ C
@ ^ [X2: B] :
( groups7311177749621191930dd_sum @ A @ C
@ ^ [A6: A] : ( F2 @ A6 @ X2 )
@ S )
@ Net ) ) ) ) ).
% differentiable_sum
thf(fact_7679_differentiable__const,axiom,
! [B: $tType,A: $tType] :
( ( ( real_V822414075346904944vector @ A )
& ( real_V822414075346904944vector @ B ) )
=> ! [A2: B,F4: filter @ A] :
( differentiable @ A @ B
@ ^ [Z3: A] : A2
@ F4 ) ) ).
% differentiable_const
thf(fact_7680_differentiable__ident,axiom,
! [A: $tType] :
( ( real_V822414075346904944vector @ A )
=> ! [F4: filter @ A] :
( differentiable @ A @ A
@ ^ [X2: A] : X2
@ F4 ) ) ).
% differentiable_ident
thf(fact_7681_differentiable__compose,axiom,
! [B: $tType,A: $tType,C: $tType] :
( ( ( real_V822414075346904944vector @ C )
& ( real_V822414075346904944vector @ A )
& ( real_V822414075346904944vector @ B ) )
=> ! [F2: A > B,G: C > A,X: C,S: set @ C] :
( ( differentiable @ A @ B @ F2 @ ( topolo174197925503356063within @ A @ ( G @ X ) @ ( top_top @ ( set @ A ) ) ) )
=> ( ( differentiable @ C @ A @ G @ ( topolo174197925503356063within @ C @ X @ S ) )
=> ( differentiable @ C @ B
@ ^ [X2: C] : ( F2 @ ( G @ X2 ) )
@ ( topolo174197925503356063within @ C @ X @ S ) ) ) ) ) ).
% differentiable_compose
thf(fact_7682_differentiable__mult,axiom,
! [B: $tType,A: $tType] :
( ( ( real_V822414075346904944vector @ A )
& ( real_V4412858255891104859lgebra @ B ) )
=> ! [F2: A > B,X: A,S: set @ A,G: A > B] :
( ( differentiable @ A @ B @ F2 @ ( topolo174197925503356063within @ A @ X @ S ) )
=> ( ( differentiable @ A @ B @ G @ ( topolo174197925503356063within @ A @ X @ S ) )
=> ( differentiable @ A @ B
@ ^ [X2: A] : ( times_times @ B @ ( F2 @ X2 ) @ ( G @ X2 ) )
@ ( topolo174197925503356063within @ A @ X @ S ) ) ) ) ) ).
% differentiable_mult
thf(fact_7683_differentiable__scaleR,axiom,
! [B: $tType,A: $tType] :
( ( ( real_V822414075346904944vector @ A )
& ( real_V822414075346904944vector @ B ) )
=> ! [F2: A > real,X: A,S: set @ A,G: A > B] :
( ( differentiable @ A @ real @ F2 @ ( topolo174197925503356063within @ A @ X @ S ) )
=> ( ( differentiable @ A @ B @ G @ ( topolo174197925503356063within @ A @ X @ S ) )
=> ( differentiable @ A @ B
@ ^ [X2: A] : ( real_V8093663219630862766scaleR @ B @ ( F2 @ X2 ) @ ( G @ X2 ) )
@ ( topolo174197925503356063within @ A @ X @ S ) ) ) ) ) ).
% differentiable_scaleR
thf(fact_7684_real__differentiable__def,axiom,
! [F2: real > real,X: real,S: set @ real] :
( ( differentiable @ real @ real @ F2 @ ( topolo174197925503356063within @ real @ X @ S ) )
= ( ? [D8: real] : ( has_field_derivative @ real @ F2 @ D8 @ ( topolo174197925503356063within @ real @ X @ S ) ) ) ) ).
% real_differentiable_def
thf(fact_7685_real__differentiableE,axiom,
! [F2: real > real,X: real,S: set @ real] :
( ( differentiable @ real @ real @ F2 @ ( topolo174197925503356063within @ real @ X @ S ) )
=> ~ ! [Df2: real] :
~ ( has_field_derivative @ real @ F2 @ Df2 @ ( topolo174197925503356063within @ real @ X @ S ) ) ) ).
% real_differentiableE
thf(fact_7686_differentiable__power,axiom,
! [B: $tType,A: $tType] :
( ( ( real_V822414075346904944vector @ A )
& ( real_V3459762299906320749_field @ B ) )
=> ! [F2: A > B,X: A,S: set @ A,N3: nat] :
( ( differentiable @ A @ B @ F2 @ ( topolo174197925503356063within @ A @ X @ S ) )
=> ( differentiable @ A @ B
@ ^ [X2: A] : ( power_power @ B @ ( F2 @ X2 ) @ N3 )
@ ( topolo174197925503356063within @ A @ X @ S ) ) ) ) ).
% differentiable_power
thf(fact_7687_differentiable__within__subset,axiom,
! [B: $tType,A: $tType] :
( ( ( real_V822414075346904944vector @ A )
& ( real_V822414075346904944vector @ B ) )
=> ! [F2: A > B,X: A,S: set @ A,T2: set @ A] :
( ( differentiable @ A @ B @ F2 @ ( topolo174197925503356063within @ A @ X @ S ) )
=> ( ( ord_less_eq @ ( set @ A ) @ T2 @ S )
=> ( differentiable @ A @ B @ F2 @ ( topolo174197925503356063within @ A @ X @ T2 ) ) ) ) ) ).
% differentiable_within_subset
thf(fact_7688_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
@ ^ [X2: A] : ( minus_minus @ B @ ( F2 @ X2 ) @ ( G @ X2 ) )
@ F4 ) ) ) ) ).
% differentiable_diff
thf(fact_7689_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,X: C,S: set @ C] :
( ( differentiable @ A @ B @ F2 @ ( topolo174197925503356063within @ A @ ( G @ X ) @ ( image @ C @ A @ G @ S ) ) )
=> ( ( differentiable @ C @ A @ G @ ( topolo174197925503356063within @ C @ X @ S ) )
=> ( differentiable @ C @ B
@ ^ [X2: C] : ( F2 @ ( G @ X2 ) )
@ ( topolo174197925503356063within @ C @ X @ S ) ) ) ) ) ).
% differentiable_in_compose
thf(fact_7690_differentiable__divide,axiom,
! [B: $tType,A: $tType] :
( ( ( real_V822414075346904944vector @ A )
& ( real_V3459762299906320749_field @ B ) )
=> ! [F2: A > B,X: A,S: set @ A,G: A > B] :
( ( differentiable @ A @ B @ F2 @ ( topolo174197925503356063within @ A @ X @ S ) )
=> ( ( differentiable @ A @ B @ G @ ( topolo174197925503356063within @ A @ X @ S ) )
=> ( ( ( G @ X )
!= ( zero_zero @ B ) )
=> ( differentiable @ A @ B
@ ^ [X2: A] : ( divide_divide @ B @ ( F2 @ X2 ) @ ( G @ X2 ) )
@ ( topolo174197925503356063within @ A @ X @ S ) ) ) ) ) ) ).
% differentiable_divide
thf(fact_7691_differentiable__inverse,axiom,
! [B: $tType,A: $tType] :
( ( ( real_V822414075346904944vector @ A )
& ( real_V3459762299906320749_field @ B ) )
=> ! [F2: A > B,X: A,S: set @ A] :
( ( differentiable @ A @ B @ F2 @ ( topolo174197925503356063within @ A @ X @ S ) )
=> ( ( ( F2 @ X )
!= ( zero_zero @ B ) )
=> ( differentiable @ A @ B
@ ^ [X2: A] : ( inverse_inverse @ B @ ( F2 @ X2 ) )
@ ( topolo174197925503356063within @ A @ X @ S ) ) ) ) ) ).
% differentiable_inverse
thf(fact_7692_Frct__code__post_I2_J,axiom,
! [A2: int] :
( ( frct @ ( product_Pair @ int @ int @ A2 @ ( zero_zero @ int ) ) )
= ( zero_zero @ rat ) ) ).
% Frct_code_post(2)
thf(fact_7693_Frct__code__post_I1_J,axiom,
! [A2: int] :
( ( frct @ ( product_Pair @ int @ int @ ( zero_zero @ int ) @ A2 ) )
= ( zero_zero @ rat ) ) ).
% Frct_code_post(1)
thf(fact_7694_Frct__code__post_I7_J,axiom,
! [A2: int,B2: int] :
( ( frct @ ( product_Pair @ int @ int @ ( uminus_uminus @ int @ A2 ) @ B2 ) )
= ( uminus_uminus @ rat @ ( frct @ ( product_Pair @ int @ int @ A2 @ B2 ) ) ) ) ).
% Frct_code_post(7)
thf(fact_7695_Frct__code__post_I8_J,axiom,
! [A2: int,B2: int] :
( ( frct @ ( product_Pair @ int @ int @ A2 @ ( uminus_uminus @ int @ B2 ) ) )
= ( uminus_uminus @ rat @ ( frct @ ( product_Pair @ int @ int @ A2 @ B2 ) ) ) ) ).
% Frct_code_post(8)
thf(fact_7696_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_7697_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_7698_Frct__code__post_I6_J,axiom,
! [K: num,L2: num] :
( ( frct @ ( product_Pair @ int @ int @ ( numeral_numeral @ int @ K ) @ ( numeral_numeral @ int @ L2 ) ) )
= ( divide_divide @ rat @ ( numeral_numeral @ rat @ K ) @ ( numeral_numeral @ rat @ L2 ) ) ) ).
% Frct_code_post(6)
thf(fact_7699_finite__mono__strict__prefix__implies__finite__fixpoint,axiom,
! [A: $tType,F2: nat > ( set @ A ),S3: set @ A] :
( ! [I2: nat] : ( ord_less_eq @ ( set @ A ) @ ( F2 @ I2 ) @ S3 )
=> ( ( finite_finite2 @ A @ S3 )
=> ( ? [N10: nat] :
( ! [N: nat] :
( ( ord_less_eq @ nat @ N @ N10 )
=> ! [M4: nat] :
( ( ord_less_eq @ nat @ M4 @ N10 )
=> ( ( ord_less @ nat @ M4 @ N )
=> ( ord_less @ ( set @ A ) @ ( F2 @ M4 ) @ ( F2 @ N ) ) ) ) )
& ! [N: nat] :
( ( ord_less_eq @ nat @ N10 @ N )
=> ( ( F2 @ N10 )
= ( F2 @ N ) ) ) )
=> ( ( F2 @ ( finite_card @ A @ S3 ) )
= ( complete_Sup_Sup @ ( set @ A ) @ ( image @ nat @ ( set @ A ) @ F2 @ ( top_top @ ( set @ nat ) ) ) ) ) ) ) ) ).
% finite_mono_strict_prefix_implies_finite_fixpoint
thf(fact_7700_MVT,axiom,
! [A2: real,B2: real,F2: real > real] :
( ( ord_less @ real @ A2 @ B2 )
=> ( ( topolo81223032696312382ous_on @ real @ real @ ( set_or1337092689740270186AtMost @ real @ A2 @ B2 ) @ F2 )
=> ( ! [X3: real] :
( ( ord_less @ real @ A2 @ X3 )
=> ( ( ord_less @ real @ X3 @ B2 )
=> ( differentiable @ real @ real @ F2 @ ( topolo174197925503356063within @ real @ X3 @ ( top_top @ ( set @ real ) ) ) ) ) )
=> ? [L3: real,Z2: real] :
( ( ord_less @ real @ A2 @ Z2 )
& ( ord_less @ real @ Z2 @ B2 )
& ( has_field_derivative @ real @ F2 @ L3 @ ( topolo174197925503356063within @ real @ Z2 @ ( top_top @ ( set @ real ) ) ) )
& ( ( minus_minus @ real @ ( F2 @ B2 ) @ ( F2 @ A2 ) )
= ( times_times @ real @ ( minus_minus @ real @ B2 @ A2 ) @ L3 ) ) ) ) ) ) ).
% MVT
thf(fact_7701_card__UNIV__unit,axiom,
( ( finite_card @ product_unit @ ( top_top @ ( set @ product_unit ) ) )
= ( one_one @ nat ) ) ).
% card_UNIV_unit
thf(fact_7702_card__Collect__less__nat,axiom,
! [N3: nat] :
( ( finite_card @ nat
@ ( collect @ nat
@ ^ [I3: nat] : ( ord_less @ nat @ I3 @ N3 ) ) )
= N3 ) ).
% card_Collect_less_nat
thf(fact_7703_card__eq__UNIV,axiom,
! [A: $tType] :
( ( finite_finite @ A )
=> ! [S3: set @ A] :
( ( ( finite_card @ A @ S3 )
= ( finite_card @ A @ ( top_top @ ( set @ A ) ) ) )
= ( S3
= ( top_top @ ( set @ A ) ) ) ) ) ).
% card_eq_UNIV
thf(fact_7704_card__eq__UNIV2,axiom,
! [A: $tType] :
( ( finite_finite @ A )
=> ! [S3: set @ A] :
( ( ( finite_card @ A @ ( top_top @ ( set @ A ) ) )
= ( finite_card @ A @ S3 ) )
= ( S3
= ( top_top @ ( set @ A ) ) ) ) ) ).
% card_eq_UNIV2
thf(fact_7705_card__atMost,axiom,
! [U: nat] :
( ( finite_card @ nat @ ( set_ord_atMost @ nat @ U ) )
= ( suc @ U ) ) ).
% card_atMost
thf(fact_7706_card__atLeastLessThan,axiom,
! [L2: nat,U: nat] :
( ( finite_card @ nat @ ( set_or7035219750837199246ssThan @ nat @ L2 @ U ) )
= ( minus_minus @ nat @ U @ L2 ) ) ).
% card_atLeastLessThan
thf(fact_7707_card__Collect__le__nat,axiom,
! [N3: nat] :
( ( finite_card @ nat
@ ( collect @ nat
@ ^ [I3: nat] : ( ord_less_eq @ nat @ I3 @ N3 ) ) )
= ( suc @ N3 ) ) ).
% card_Collect_le_nat
thf(fact_7708_card__greaterThanAtMost,axiom,
! [L2: nat,U: nat] :
( ( finite_card @ nat @ ( set_or3652927894154168847AtMost @ nat @ L2 @ U ) )
= ( minus_minus @ nat @ U @ L2 ) ) ).
% card_greaterThanAtMost
thf(fact_7709_card__UNIV__bool,axiom,
( ( finite_card @ $o @ ( top_top @ ( set @ $o ) ) )
= ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ).
% card_UNIV_bool
thf(fact_7710_card__ge__UNIV,axiom,
! [A: $tType] :
( ( finite_finite @ A )
=> ! [S3: set @ A] :
( ( ord_less_eq @ nat @ ( finite_card @ A @ ( top_top @ ( set @ A ) ) ) @ ( finite_card @ A @ S3 ) )
= ( S3
= ( top_top @ ( set @ A ) ) ) ) ) ).
% card_ge_UNIV
thf(fact_7711_card__atLeastAtMost,axiom,
! [L2: nat,U: nat] :
( ( finite_card @ nat @ ( set_or1337092689740270186AtMost @ nat @ L2 @ U ) )
= ( minus_minus @ nat @ ( suc @ U ) @ L2 ) ) ).
% card_atLeastAtMost
thf(fact_7712_card__atLeastLessThan__int,axiom,
! [L2: int,U: int] :
( ( finite_card @ int @ ( set_or7035219750837199246ssThan @ int @ L2 @ U ) )
= ( nat2 @ ( minus_minus @ int @ U @ L2 ) ) ) ).
% card_atLeastLessThan_int
thf(fact_7713_prod__constant,axiom,
! [A: $tType,B: $tType] :
( ( comm_monoid_mult @ A )
=> ! [Y: A,A3: set @ B] :
( ( groups7121269368397514597t_prod @ B @ A
@ ^ [X2: B] : Y
@ A3 )
= ( power_power @ A @ Y @ ( finite_card @ B @ A3 ) ) ) ) ).
% prod_constant
thf(fact_7714_card__greaterThanLessThan,axiom,
! [L2: nat,U: nat] :
( ( finite_card @ nat @ ( set_or5935395276787703475ssThan @ nat @ L2 @ U ) )
= ( minus_minus @ nat @ U @ ( suc @ L2 ) ) ) ).
% card_greaterThanLessThan
thf(fact_7715_card__greaterThanAtMost__int,axiom,
! [L2: int,U: int] :
( ( finite_card @ int @ ( set_or3652927894154168847AtMost @ int @ L2 @ U ) )
= ( nat2 @ ( minus_minus @ int @ U @ L2 ) ) ) ).
% card_greaterThanAtMost_int
thf(fact_7716_card__insert__disjoint,axiom,
! [A: $tType,A3: set @ A,X: A] :
( ( finite_finite2 @ A @ A3 )
=> ( ~ ( member @ A @ X @ A3 )
=> ( ( finite_card @ A @ ( insert @ A @ X @ A3 ) )
= ( suc @ ( finite_card @ A @ A3 ) ) ) ) ) ).
% card_insert_disjoint
thf(fact_7717_sum__constant,axiom,
! [B: $tType,A: $tType] :
( ( semiring_1 @ A )
=> ! [Y: A,A3: set @ B] :
( ( groups7311177749621191930dd_sum @ B @ A
@ ^ [X2: B] : Y
@ A3 )
= ( times_times @ A @ ( semiring_1_of_nat @ A @ ( finite_card @ B @ A3 ) ) @ Y ) ) ) ).
% sum_constant
thf(fact_7718_card__Diff__insert,axiom,
! [A: $tType,A2: A,A3: set @ A,B6: set @ A] :
( ( member @ A @ A2 @ A3 )
=> ( ~ ( member @ A @ A2 @ B6 )
=> ( ( finite_card @ A @ ( minus_minus @ ( set @ A ) @ A3 @ ( insert @ A @ A2 @ B6 ) ) )
= ( minus_minus @ nat @ ( finite_card @ A @ ( minus_minus @ ( set @ A ) @ A3 @ B6 ) ) @ ( one_one @ nat ) ) ) ) ) ).
% card_Diff_insert
thf(fact_7719_card__atLeastAtMost__int,axiom,
! [L2: int,U: int] :
( ( finite_card @ int @ ( set_or1337092689740270186AtMost @ int @ L2 @ U ) )
= ( nat2 @ ( plus_plus @ int @ ( minus_minus @ int @ U @ L2 ) @ ( one_one @ int ) ) ) ) ).
% card_atLeastAtMost_int
thf(fact_7720_card__doubleton__eq__2__iff,axiom,
! [A: $tType,A2: A,B2: A] :
( ( ( finite_card @ A @ ( insert @ A @ A2 @ ( insert @ A @ B2 @ ( bot_bot @ ( set @ A ) ) ) ) )
= ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) )
= ( A2 != B2 ) ) ).
% card_doubleton_eq_2_iff
thf(fact_7721_card__greaterThanLessThan__int,axiom,
! [L2: int,U: int] :
( ( finite_card @ int @ ( set_or5935395276787703475ssThan @ int @ L2 @ U ) )
= ( nat2 @ ( minus_minus @ int @ U @ ( plus_plus @ int @ L2 @ ( one_one @ int ) ) ) ) ) ).
% card_greaterThanLessThan_int
thf(fact_7722_continuous__on__mult__const,axiom,
! [A: $tType] :
( ( real_V4412858255891104859lgebra @ A )
=> ! [S: set @ A,C2: A] : ( topolo81223032696312382ous_on @ A @ A @ S @ ( times_times @ A @ C2 ) ) ) ).
% continuous_on_mult_const
thf(fact_7723_continuous__on__mult__right,axiom,
! [A: $tType,B: $tType] :
( ( ( topolo4958980785337419405_space @ B )
& ( real_V4412858255891104859lgebra @ A ) )
=> ! [S: set @ B,F2: B > A,C2: A] :
( ( topolo81223032696312382ous_on @ B @ A @ S @ F2 )
=> ( topolo81223032696312382ous_on @ B @ A @ S
@ ^ [X2: B] : ( times_times @ A @ ( F2 @ X2 ) @ C2 ) ) ) ) ).
% continuous_on_mult_right
thf(fact_7724_continuous__on__mult__left,axiom,
! [A: $tType,B: $tType] :
( ( ( topolo4958980785337419405_space @ B )
& ( real_V4412858255891104859lgebra @ A ) )
=> ! [S: set @ B,F2: B > A,C2: A] :
( ( topolo81223032696312382ous_on @ B @ A @ S @ F2 )
=> ( topolo81223032696312382ous_on @ B @ A @ S
@ ^ [X2: B] : ( times_times @ A @ C2 @ ( F2 @ X2 ) ) ) ) ) ).
% continuous_on_mult_left
thf(fact_7725_continuous__on__mult_H,axiom,
! [B: $tType,D5: $tType] :
( ( ( topolo4958980785337419405_space @ D5 )
& ( topolo4211221413907600880p_mult @ B ) )
=> ! [A3: set @ D5,F2: D5 > B,G: D5 > B] :
( ( topolo81223032696312382ous_on @ D5 @ B @ A3 @ F2 )
=> ( ( topolo81223032696312382ous_on @ D5 @ B @ A3 @ G )
=> ( topolo81223032696312382ous_on @ D5 @ B @ A3
@ ^ [X2: D5] : ( times_times @ B @ ( F2 @ X2 ) @ ( G @ X2 ) ) ) ) ) ) ).
% continuous_on_mult'
thf(fact_7726_continuous__on__mult,axiom,
! [A: $tType,D5: $tType] :
( ( ( topolo4958980785337419405_space @ D5 )
& ( real_V4412858255891104859lgebra @ A ) )
=> ! [S: set @ D5,F2: D5 > A,G: D5 > A] :
( ( topolo81223032696312382ous_on @ D5 @ A @ S @ F2 )
=> ( ( topolo81223032696312382ous_on @ D5 @ A @ S @ G )
=> ( topolo81223032696312382ous_on @ D5 @ A @ S
@ ^ [X2: D5] : ( times_times @ A @ ( F2 @ X2 ) @ ( G @ X2 ) ) ) ) ) ) ).
% continuous_on_mult
thf(fact_7727_continuous__onI__mono,axiom,
! [B: $tType,A: $tType] :
( ( ( topolo1944317154257567458pology @ A )
& ( dense_order @ B )
& ( topolo1944317154257567458pology @ B ) )
=> ! [F2: A > B,A3: set @ A] :
( ( topolo1002775350975398744n_open @ B @ ( image @ A @ B @ F2 @ A3 ) )
=> ( ! [X3: A,Y3: A] :
( ( member @ A @ X3 @ A3 )
=> ( ( member @ A @ Y3 @ A3 )
=> ( ( ord_less_eq @ A @ X3 @ Y3 )
=> ( ord_less_eq @ B @ ( F2 @ X3 ) @ ( F2 @ Y3 ) ) ) ) )
=> ( topolo81223032696312382ous_on @ A @ B @ A3 @ F2 ) ) ) ) ).
% continuous_onI_mono
thf(fact_7728_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,S: set @ C,F2: C > A] :
( ( topolo81223032696312382ous_on @ A @ B @ T2 @ G )
=> ( ( topolo81223032696312382ous_on @ C @ A @ S @ F2 )
=> ( ( ord_less_eq @ ( set @ A ) @ ( image @ C @ A @ F2 @ S ) @ T2 )
=> ( topolo81223032696312382ous_on @ C @ B @ S
@ ^ [X2: C] : ( G @ ( F2 @ X2 ) ) ) ) ) ) ) ).
% continuous_on_compose2
thf(fact_7729_card__image__le,axiom,
! [B: $tType,A: $tType,A3: set @ A,F2: A > B] :
( ( finite_finite2 @ A @ A3 )
=> ( ord_less_eq @ nat @ ( finite_card @ B @ ( image @ A @ B @ F2 @ A3 ) ) @ ( finite_card @ A @ A3 ) ) ) ).
% card_image_le
thf(fact_7730_pigeonhole,axiom,
! [A: $tType,B: $tType,F2: B > A,A3: set @ B] :
( ( ord_less @ nat @ ( finite_card @ A @ ( image @ B @ A @ F2 @ A3 ) ) @ ( finite_card @ B @ A3 ) )
=> ~ ( inj_on @ B @ A @ F2 @ A3 ) ) ).
% pigeonhole
thf(fact_7731_card__Union__le__sum__card__weak,axiom,
! [A: $tType,U5: set @ ( set @ A )] :
( ! [X3: set @ A] :
( ( member @ ( set @ A ) @ X3 @ U5 )
=> ( finite_finite2 @ A @ X3 ) )
=> ( ord_less_eq @ nat @ ( finite_card @ A @ ( complete_Sup_Sup @ ( set @ A ) @ U5 ) ) @ ( groups7311177749621191930dd_sum @ ( set @ A ) @ nat @ ( finite_card @ A ) @ U5 ) ) ) ).
% card_Union_le_sum_card_weak
thf(fact_7732_card__Union__le__sum__card,axiom,
! [A: $tType,U5: set @ ( set @ A )] : ( ord_less_eq @ nat @ ( finite_card @ A @ ( complete_Sup_Sup @ ( set @ A ) @ U5 ) ) @ ( groups7311177749621191930dd_sum @ ( set @ A ) @ nat @ ( finite_card @ A ) @ U5 ) ) ).
% card_Union_le_sum_card
thf(fact_7733_card__1__singletonE,axiom,
! [A: $tType,A3: set @ A] :
( ( ( finite_card @ A @ A3 )
= ( one_one @ nat ) )
=> ~ ! [X3: A] :
( A3
!= ( insert @ A @ X3 @ ( bot_bot @ ( set @ A ) ) ) ) ) ).
% card_1_singletonE
thf(fact_7734_continuous__on__diff,axiom,
! [B: $tType,D5: $tType] :
( ( ( topolo4958980785337419405_space @ D5 )
& ( topolo1633459387980952147up_add @ B ) )
=> ! [S: set @ D5,F2: D5 > B,G: D5 > B] :
( ( topolo81223032696312382ous_on @ D5 @ B @ S @ F2 )
=> ( ( topolo81223032696312382ous_on @ D5 @ B @ S @ G )
=> ( topolo81223032696312382ous_on @ D5 @ B @ S
@ ^ [X2: D5] : ( minus_minus @ B @ ( F2 @ X2 ) @ ( G @ X2 ) ) ) ) ) ) ).
% continuous_on_diff
thf(fact_7735_continuous__on__op__minus,axiom,
! [A: $tType] :
( ( topolo1633459387980952147up_add @ A )
=> ! [S: set @ A,X: A] : ( topolo81223032696312382ous_on @ A @ A @ S @ ( minus_minus @ A @ X ) ) ) ).
% continuous_on_op_minus
thf(fact_7736_continuous__inj__imp__mono,axiom,
! [B: $tType,A: $tType] :
( ( ( topolo8458572112393995274pology @ A )
& ( topolo1944317154257567458pology @ B ) )
=> ! [A2: A,X: A,B2: A,F2: A > B] :
( ( ord_less @ A @ A2 @ X )
=> ( ( ord_less @ A @ X @ B2 )
=> ( ( topolo81223032696312382ous_on @ A @ B @ ( set_or1337092689740270186AtMost @ A @ A2 @ B2 ) @ F2 )
=> ( ( inj_on @ A @ B @ F2 @ ( set_or1337092689740270186AtMost @ A @ A2 @ B2 ) )
=> ( ( ( ord_less @ B @ ( F2 @ A2 ) @ ( F2 @ X ) )
& ( ord_less @ B @ ( F2 @ X ) @ ( F2 @ B2 ) ) )
| ( ( ord_less @ B @ ( F2 @ B2 ) @ ( F2 @ X ) )
& ( ord_less @ B @ ( F2 @ X ) @ ( F2 @ A2 ) ) ) ) ) ) ) ) ) ).
% continuous_inj_imp_mono
thf(fact_7737_card__less__sym__Diff,axiom,
! [A: $tType,A3: set @ A,B6: set @ A] :
( ( finite_finite2 @ A @ A3 )
=> ( ( finite_finite2 @ A @ B6 )
=> ( ( ord_less @ nat @ ( finite_card @ A @ A3 ) @ ( finite_card @ A @ B6 ) )
=> ( ord_less @ nat @ ( finite_card @ A @ ( minus_minus @ ( set @ A ) @ A3 @ B6 ) ) @ ( finite_card @ A @ ( minus_minus @ ( set @ A ) @ B6 @ A3 ) ) ) ) ) ) ).
% card_less_sym_Diff
thf(fact_7738_card__le__sym__Diff,axiom,
! [A: $tType,A3: set @ A,B6: set @ A] :
( ( finite_finite2 @ A @ A3 )
=> ( ( finite_finite2 @ A @ B6 )
=> ( ( ord_less_eq @ nat @ ( finite_card @ A @ A3 ) @ ( finite_card @ A @ B6 ) )
=> ( ord_less_eq @ nat @ ( finite_card @ A @ ( minus_minus @ ( set @ A ) @ A3 @ B6 ) ) @ ( finite_card @ A @ ( minus_minus @ ( set @ A ) @ B6 @ A3 ) ) ) ) ) ) ).
% card_le_sym_Diff
thf(fact_7739_finite__if__finite__subsets__card__bdd,axiom,
! [A: $tType,F4: set @ A,C5: nat] :
( ! [G8: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ G8 @ F4 )
=> ( ( finite_finite2 @ A @ G8 )
=> ( ord_less_eq @ nat @ ( finite_card @ A @ G8 ) @ C5 ) ) )
=> ( ( finite_finite2 @ A @ F4 )
& ( ord_less_eq @ nat @ ( finite_card @ A @ F4 ) @ C5 ) ) ) ).
% finite_if_finite_subsets_card_bdd
thf(fact_7740_card__seteq,axiom,
! [A: $tType,B6: set @ A,A3: set @ A] :
( ( finite_finite2 @ A @ B6 )
=> ( ( ord_less_eq @ ( set @ A ) @ A3 @ B6 )
=> ( ( ord_less_eq @ nat @ ( finite_card @ A @ B6 ) @ ( finite_card @ A @ A3 ) )
=> ( A3 = B6 ) ) ) ) ).
% card_seteq
thf(fact_7741_card__mono,axiom,
! [A: $tType,B6: set @ A,A3: set @ A] :
( ( finite_finite2 @ A @ B6 )
=> ( ( ord_less_eq @ ( set @ A ) @ A3 @ B6 )
=> ( ord_less_eq @ nat @ ( finite_card @ A @ A3 ) @ ( finite_card @ A @ B6 ) ) ) ) ).
% card_mono
thf(fact_7742_obtain__subset__with__card__n,axiom,
! [A: $tType,N3: nat,S3: set @ A] :
( ( ord_less_eq @ nat @ N3 @ ( finite_card @ A @ S3 ) )
=> ~ ! [T4: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ T4 @ S3 )
=> ( ( ( finite_card @ A @ T4 )
= N3 )
=> ~ ( finite_finite2 @ A @ T4 ) ) ) ) ).
% obtain_subset_with_card_n
thf(fact_7743_card__lists__length__eq,axiom,
! [A: $tType,A3: set @ A,N3: nat] :
( ( finite_finite2 @ A @ A3 )
=> ( ( finite_card @ ( list @ A )
@ ( collect @ ( list @ A )
@ ^ [Xs: list @ A] :
( ( ord_less_eq @ ( set @ A ) @ ( set2 @ A @ Xs ) @ A3 )
& ( ( size_size @ ( list @ A ) @ Xs )
= N3 ) ) ) )
= ( power_power @ nat @ ( finite_card @ A @ A3 ) @ N3 ) ) ) ).
% card_lists_length_eq
thf(fact_7744_IVT_H,axiom,
! [A: $tType,B: $tType] :
( ( ( topolo1944317154257567458pology @ B )
& ( topolo8458572112393995274pology @ A ) )
=> ! [F2: A > B,A2: A,Y: B,B2: A] :
( ( ord_less_eq @ B @ ( F2 @ A2 ) @ Y )
=> ( ( ord_less_eq @ B @ Y @ ( F2 @ B2 ) )
=> ( ( ord_less_eq @ A @ A2 @ B2 )
=> ( ( topolo81223032696312382ous_on @ A @ B @ ( set_or1337092689740270186AtMost @ A @ A2 @ B2 ) @ F2 )
=> ? [X3: A] :
( ( ord_less_eq @ A @ A2 @ X3 )
& ( ord_less_eq @ A @ X3 @ B2 )
& ( ( F2 @ X3 )
= Y ) ) ) ) ) ) ) ).
% IVT'
thf(fact_7745_IVT2_H,axiom,
! [A: $tType,B: $tType] :
( ( ( topolo1944317154257567458pology @ B )
& ( topolo8458572112393995274pology @ A ) )
=> ! [F2: A > B,B2: A,Y: B,A2: A] :
( ( ord_less_eq @ B @ ( F2 @ B2 ) @ Y )
=> ( ( ord_less_eq @ B @ Y @ ( F2 @ A2 ) )
=> ( ( ord_less_eq @ A @ A2 @ B2 )
=> ( ( topolo81223032696312382ous_on @ A @ B @ ( set_or1337092689740270186AtMost @ A @ A2 @ B2 ) @ F2 )
=> ? [X3: A] :
( ( ord_less_eq @ A @ A2 @ X3 )
& ( ord_less_eq @ A @ X3 @ B2 )
& ( ( F2 @ X3 )
= Y ) ) ) ) ) ) ) ).
% IVT2'
thf(fact_7746_continuous__on__subset,axiom,
! [B: $tType,A: $tType] :
( ( ( topolo4958980785337419405_space @ A )
& ( topolo4958980785337419405_space @ B ) )
=> ! [S: set @ A,F2: A > B,T2: set @ A] :
( ( topolo81223032696312382ous_on @ A @ B @ S @ F2 )
=> ( ( ord_less_eq @ ( set @ A ) @ T2 @ S )
=> ( topolo81223032696312382ous_on @ A @ B @ T2 @ F2 ) ) ) ) ).
% continuous_on_subset
thf(fact_7747_card__insert__le,axiom,
! [A: $tType,A3: set @ A,X: A] : ( ord_less_eq @ nat @ ( finite_card @ A @ A3 ) @ ( finite_card @ A @ ( insert @ A @ X @ A3 ) ) ) ).
% card_insert_le
thf(fact_7748_n__subsets,axiom,
! [A: $tType,A3: set @ A,K: nat] :
( ( finite_finite2 @ A @ A3 )
=> ( ( finite_card @ ( set @ A )
@ ( collect @ ( set @ A )
@ ^ [B7: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ B7 @ A3 )
& ( ( finite_card @ A @ B7 )
= K ) ) ) )
= ( binomial @ ( finite_card @ A @ A3 ) @ K ) ) ) ).
% n_subsets
thf(fact_7749_card__le__if__inj__on__rel,axiom,
! [B: $tType,A: $tType,B6: set @ A,A3: set @ B,R2: B > A > $o] :
( ( finite_finite2 @ A @ B6 )
=> ( ! [A4: B] :
( ( member @ B @ A4 @ A3 )
=> ? [B12: A] :
( ( member @ A @ B12 @ B6 )
& ( R2 @ A4 @ B12 ) ) )
=> ( ! [A15: B,A25: B,B3: A] :
( ( member @ B @ A15 @ A3 )
=> ( ( member @ B @ A25 @ A3 )
=> ( ( member @ A @ B3 @ B6 )
=> ( ( R2 @ A15 @ B3 )
=> ( ( R2 @ A25 @ B3 )
=> ( A15 = A25 ) ) ) ) ) )
=> ( ord_less_eq @ nat @ ( finite_card @ B @ A3 ) @ ( finite_card @ A @ B6 ) ) ) ) ) ).
% card_le_if_inj_on_rel
thf(fact_7750_card__subset__eq,axiom,
! [A: $tType,B6: set @ A,A3: set @ A] :
( ( finite_finite2 @ A @ B6 )
=> ( ( ord_less_eq @ ( set @ A ) @ A3 @ B6 )
=> ( ( ( finite_card @ A @ A3 )
= ( finite_card @ A @ B6 ) )
=> ( A3 = B6 ) ) ) ) ).
% card_subset_eq
thf(fact_7751_infinite__arbitrarily__large,axiom,
! [A: $tType,A3: set @ A,N3: nat] :
( ~ ( finite_finite2 @ A @ A3 )
=> ? [B9: set @ A] :
( ( finite_finite2 @ A @ B9 )
& ( ( finite_card @ A @ B9 )
= N3 )
& ( ord_less_eq @ ( set @ A ) @ B9 @ A3 ) ) ) ).
% infinite_arbitrarily_large
thf(fact_7752_card__length,axiom,
! [A: $tType,Xs2: list @ A] : ( ord_less_eq @ nat @ ( finite_card @ A @ ( set2 @ A @ Xs2 ) ) @ ( size_size @ ( list @ A ) @ Xs2 ) ) ).
% card_length
thf(fact_7753_psubset__card__mono,axiom,
! [A: $tType,B6: set @ A,A3: set @ A] :
( ( finite_finite2 @ A @ B6 )
=> ( ( ord_less @ ( set @ A ) @ A3 @ B6 )
=> ( ord_less @ nat @ ( finite_card @ A @ A3 ) @ ( finite_card @ A @ B6 ) ) ) ) ).
% psubset_card_mono
thf(fact_7754_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
@ ^ [X2: A] : ( ord_less @ B @ ( F2 @ X2 ) @ ( G @ X2 ) ) ) ) ) ) ) ).
% open_Collect_less
thf(fact_7755_DERIV__continuous__on,axiom,
! [A: $tType] :
( ( real_V3459762299906320749_field @ A )
=> ! [S: set @ A,F2: A > A,D4: A > A] :
( ! [X3: A] :
( ( member @ A @ X3 @ S )
=> ( has_field_derivative @ A @ F2 @ ( D4 @ X3 ) @ ( topolo174197925503356063within @ A @ X3 @ S ) ) )
=> ( topolo81223032696312382ous_on @ A @ A @ S @ F2 ) ) ) ).
% DERIV_continuous_on
thf(fact_7756_continuous__on__Pair,axiom,
! [C: $tType,B: $tType,A: $tType] :
( ( ( topolo4958980785337419405_space @ A )
& ( topolo4958980785337419405_space @ B )
& ( topolo4958980785337419405_space @ C ) )
=> ! [S: set @ A,F2: A > B,G: A > C] :
( ( topolo81223032696312382ous_on @ A @ B @ S @ F2 )
=> ( ( topolo81223032696312382ous_on @ A @ C @ S @ G )
=> ( topolo81223032696312382ous_on @ A @ ( product_prod @ B @ C ) @ S
@ ^ [X2: A] : ( product_Pair @ B @ C @ ( F2 @ X2 ) @ ( G @ X2 ) ) ) ) ) ) ).
% continuous_on_Pair
thf(fact_7757_continuous__on__power,axiom,
! [B: $tType,C: $tType] :
( ( ( topolo4958980785337419405_space @ C )
& ( power @ B )
& ( real_V4412858255891104859lgebra @ B ) )
=> ! [S: set @ C,F2: C > B,N3: nat] :
( ( topolo81223032696312382ous_on @ C @ B @ S @ F2 )
=> ( topolo81223032696312382ous_on @ C @ B @ S
@ ^ [X2: C] : ( power_power @ B @ ( F2 @ X2 ) @ N3 ) ) ) ) ).
% continuous_on_power
thf(fact_7758_continuous__on__power_H,axiom,
! [B: $tType,C: $tType] :
( ( ( topolo4958980785337419405_space @ C )
& ( topolo1898628316856586783d_mult @ B ) )
=> ! [A3: set @ C,F2: C > B,G: C > nat] :
( ( topolo81223032696312382ous_on @ C @ B @ A3 @ F2 )
=> ( ( topolo81223032696312382ous_on @ C @ nat @ A3 @ G )
=> ( topolo81223032696312382ous_on @ C @ B @ A3
@ ^ [X2: C] : ( power_power @ B @ ( F2 @ X2 ) @ ( G @ X2 ) ) ) ) ) ) ).
% continuous_on_power'
thf(fact_7759_continuous__on__real__root,axiom,
! [A: $tType] :
( ( topolo4958980785337419405_space @ A )
=> ! [S: set @ A,F2: A > real,N3: nat] :
( ( topolo81223032696312382ous_on @ A @ real @ S @ F2 )
=> ( topolo81223032696312382ous_on @ A @ real @ S
@ ^ [X2: A] : ( root @ N3 @ ( F2 @ X2 ) ) ) ) ) ).
% continuous_on_real_root
thf(fact_7760_continuous__on__real__sqrt,axiom,
! [A: $tType] :
( ( topolo4958980785337419405_space @ A )
=> ! [S: set @ A,F2: A > real] :
( ( topolo81223032696312382ous_on @ A @ real @ S @ F2 )
=> ( topolo81223032696312382ous_on @ A @ real @ S
@ ^ [X2: A] : ( sqrt @ ( F2 @ X2 ) ) ) ) ) ).
% continuous_on_real_sqrt
thf(fact_7761_card__Suc__eq__finite,axiom,
! [A: $tType,A3: set @ A,K: nat] :
( ( ( finite_card @ A @ A3 )
= ( suc @ K ) )
= ( ? [B5: A,B7: set @ A] :
( ( A3
= ( insert @ A @ B5 @ B7 ) )
& ~ ( member @ A @ B5 @ B7 )
& ( ( finite_card @ A @ B7 )
= K )
& ( finite_finite2 @ A @ B7 ) ) ) ) ).
% card_Suc_eq_finite
thf(fact_7762_card__insert__if,axiom,
! [A: $tType,A3: set @ A,X: A] :
( ( finite_finite2 @ A @ A3 )
=> ( ( ( member @ A @ X @ A3 )
=> ( ( finite_card @ A @ ( insert @ A @ X @ A3 ) )
= ( finite_card @ A @ A3 ) ) )
& ( ~ ( member @ A @ X @ A3 )
=> ( ( finite_card @ A @ ( insert @ A @ X @ A3 ) )
= ( suc @ ( finite_card @ A @ A3 ) ) ) ) ) ) ).
% card_insert_if
thf(fact_7763_card__eq__sum,axiom,
! [A: $tType] :
( ( finite_card @ A )
= ( groups7311177749621191930dd_sum @ A @ nat
@ ^ [X2: A] : ( one_one @ nat ) ) ) ).
% card_eq_sum
thf(fact_7764_continuous__on__add,axiom,
! [B: $tType,D5: $tType] :
( ( ( topolo4958980785337419405_space @ D5 )
& ( topolo6943815403480290642id_add @ B ) )
=> ! [S: set @ D5,F2: D5 > B,G: D5 > B] :
( ( topolo81223032696312382ous_on @ D5 @ B @ S @ F2 )
=> ( ( topolo81223032696312382ous_on @ D5 @ B @ S @ G )
=> ( topolo81223032696312382ous_on @ D5 @ B @ S
@ ^ [X2: D5] : ( plus_plus @ B @ ( F2 @ X2 ) @ ( G @ X2 ) ) ) ) ) ) ).
% continuous_on_add
thf(fact_7765_card_Oeq__fold,axiom,
! [A: $tType] :
( ( finite_card @ A )
= ( finite_fold @ A @ nat
@ ^ [Uu3: A] : suc
@ ( zero_zero @ nat ) ) ) ).
% card.eq_fold
thf(fact_7766_card__ge__0__finite,axiom,
! [A: $tType,A3: set @ A] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ ( finite_card @ A @ A3 ) )
=> ( finite_finite2 @ A @ A3 ) ) ).
% card_ge_0_finite
thf(fact_7767_continuous__on__divide,axiom,
! [B: $tType,A: $tType] :
( ( ( topolo4958980785337419405_space @ A )
& ( real_V3459762299906320749_field @ B ) )
=> ! [S: set @ A,F2: A > B,G: A > B] :
( ( topolo81223032696312382ous_on @ A @ B @ S @ F2 )
=> ( ( topolo81223032696312382ous_on @ A @ B @ S @ G )
=> ( ! [X3: A] :
( ( member @ A @ X3 @ S )
=> ( ( G @ X3 )
!= ( zero_zero @ B ) ) )
=> ( topolo81223032696312382ous_on @ A @ B @ S
@ ^ [X2: A] : ( divide_divide @ B @ ( F2 @ X2 ) @ ( G @ X2 ) ) ) ) ) ) ) ).
% continuous_on_divide
thf(fact_7768_card__2__iff_H,axiom,
! [A: $tType,S3: set @ A] :
( ( ( finite_card @ A @ S3 )
= ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) )
= ( ? [X2: A] :
( ( member @ A @ X2 @ S3 )
& ? [Y2: A] :
( ( member @ A @ Y2 @ S3 )
& ( X2 != Y2 )
& ! [Z3: A] :
( ( member @ A @ Z3 @ S3 )
=> ( ( Z3 = X2 )
| ( Z3 = Y2 ) ) ) ) ) ) ) ).
% card_2_iff'
thf(fact_7769_card__less__Suc2,axiom,
! [M8: set @ nat,I: nat] :
( ~ ( member @ nat @ ( zero_zero @ nat ) @ M8 )
=> ( ( finite_card @ nat
@ ( collect @ nat
@ ^ [K2: nat] :
( ( member @ nat @ ( suc @ K2 ) @ M8 )
& ( ord_less @ nat @ K2 @ I ) ) ) )
= ( finite_card @ nat
@ ( collect @ nat
@ ^ [K2: nat] :
( ( member @ nat @ K2 @ M8 )
& ( ord_less @ nat @ K2 @ ( suc @ I ) ) ) ) ) ) ) ).
% card_less_Suc2
thf(fact_7770_card__less__Suc,axiom,
! [M8: set @ nat,I: nat] :
( ( member @ nat @ ( zero_zero @ nat ) @ M8 )
=> ( ( suc
@ ( finite_card @ nat
@ ( collect @ nat
@ ^ [K2: nat] :
( ( member @ nat @ ( suc @ K2 ) @ M8 )
& ( ord_less @ nat @ K2 @ I ) ) ) ) )
= ( finite_card @ nat
@ ( collect @ nat
@ ^ [K2: nat] :
( ( member @ nat @ K2 @ M8 )
& ( ord_less @ nat @ K2 @ ( suc @ I ) ) ) ) ) ) ) ).
% card_less_Suc
thf(fact_7771_card__less,axiom,
! [M8: set @ nat,I: nat] :
( ( member @ nat @ ( zero_zero @ nat ) @ M8 )
=> ( ( finite_card @ nat
@ ( collect @ nat
@ ^ [K2: nat] :
( ( member @ nat @ K2 @ M8 )
& ( ord_less @ nat @ K2 @ ( suc @ I ) ) ) ) )
!= ( zero_zero @ nat ) ) ) ).
% card_less
thf(fact_7772_card__map__elide,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [N3: nat] :
( ( ord_less_eq @ nat @ N3 @ ( 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 ) @ N3 ) ) )
= ( finite_card @ nat @ ( set_or7035219750837199246ssThan @ nat @ ( zero_zero @ nat ) @ N3 ) ) ) ) ) ).
% card_map_elide
thf(fact_7773_subset__card__intvl__is__intvl,axiom,
! [A3: set @ nat,K: nat] :
( ( ord_less_eq @ ( set @ nat ) @ A3 @ ( set_or7035219750837199246ssThan @ nat @ K @ ( plus_plus @ nat @ K @ ( finite_card @ nat @ A3 ) ) ) )
=> ( A3
= ( set_or7035219750837199246ssThan @ nat @ K @ ( plus_plus @ nat @ K @ ( finite_card @ nat @ A3 ) ) ) ) ) ).
% subset_card_intvl_is_intvl
thf(fact_7774_continuous__on__arcosh_H,axiom,
! [A3: set @ real,F2: real > real] :
( ( topolo81223032696312382ous_on @ real @ real @ A3 @ F2 )
=> ( ! [X3: real] :
( ( member @ real @ X3 @ A3 )
=> ( ord_less_eq @ real @ ( one_one @ real ) @ ( F2 @ X3 ) ) )
=> ( topolo81223032696312382ous_on @ real @ real @ A3
@ ^ [X2: real] : ( arcosh @ real @ ( F2 @ X2 ) ) ) ) ) ).
% continuous_on_arcosh'
thf(fact_7775_sum__Suc,axiom,
! [A: $tType,F2: A > nat,A3: set @ A] :
( ( groups7311177749621191930dd_sum @ A @ nat
@ ^ [X2: A] : ( suc @ ( F2 @ X2 ) )
@ A3 )
= ( plus_plus @ nat @ ( groups7311177749621191930dd_sum @ A @ nat @ F2 @ A3 ) @ ( finite_card @ A @ A3 ) ) ) ).
% sum_Suc
thf(fact_7776_sum__multicount,axiom,
! [A: $tType,B: $tType,S3: set @ A,T5: set @ B,R: A > B > $o,K: nat] :
( ( finite_finite2 @ A @ S3 )
=> ( ( finite_finite2 @ B @ T5 )
=> ( ! [X3: B] :
( ( member @ B @ X3 @ T5 )
=> ( ( finite_card @ A
@ ( collect @ A
@ ^ [I3: A] :
( ( member @ A @ I3 @ S3 )
& ( R @ I3 @ X3 ) ) ) )
= K ) )
=> ( ( groups7311177749621191930dd_sum @ A @ nat
@ ^ [I3: A] :
( finite_card @ B
@ ( collect @ B
@ ^ [J3: B] :
( ( member @ B @ J3 @ T5 )
& ( R @ I3 @ J3 ) ) ) )
@ S3 )
= ( times_times @ nat @ K @ ( finite_card @ B @ T5 ) ) ) ) ) ) ).
% sum_multicount
thf(fact_7777_real__of__card,axiom,
! [A: $tType,A3: set @ A] :
( ( semiring_1_of_nat @ real @ ( finite_card @ A @ A3 ) )
= ( groups7311177749621191930dd_sum @ A @ real
@ ^ [X2: A] : ( one_one @ real )
@ A3 ) ) ).
% real_of_card
thf(fact_7778_continuous__image__closed__interval,axiom,
! [A2: real,B2: real,F2: real > real] :
( ( ord_less_eq @ real @ A2 @ B2 )
=> ( ( topolo81223032696312382ous_on @ real @ real @ ( set_or1337092689740270186AtMost @ real @ A2 @ B2 ) @ F2 )
=> ? [C4: real,D3: real] :
( ( ( image @ real @ real @ F2 @ ( set_or1337092689740270186AtMost @ real @ A2 @ B2 ) )
= ( set_or1337092689740270186AtMost @ real @ C4 @ D3 ) )
& ( ord_less_eq @ real @ C4 @ D3 ) ) ) ) ).
% continuous_image_closed_interval
thf(fact_7779_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_7780_sum__bounded__below,axiom,
! [A: $tType,B: $tType] :
( ( ( ordere6911136660526730532id_add @ A )
& ( semiring_1 @ A ) )
=> ! [A3: set @ B,K6: A,F2: B > A] :
( ! [I2: B] :
( ( member @ B @ I2 @ A3 )
=> ( ord_less_eq @ A @ K6 @ ( F2 @ I2 ) ) )
=> ( ord_less_eq @ A @ ( times_times @ A @ ( semiring_1_of_nat @ A @ ( finite_card @ B @ A3 ) ) @ K6 ) @ ( groups7311177749621191930dd_sum @ B @ A @ F2 @ A3 ) ) ) ) ).
% sum_bounded_below
thf(fact_7781_sum__bounded__above,axiom,
! [B: $tType,A: $tType] :
( ( ( ordere6911136660526730532id_add @ A )
& ( semiring_1 @ A ) )
=> ! [A3: set @ B,F2: B > A,K6: A] :
( ! [I2: B] :
( ( member @ B @ I2 @ A3 )
=> ( ord_less_eq @ A @ ( F2 @ I2 ) @ K6 ) )
=> ( ord_less_eq @ A @ ( groups7311177749621191930dd_sum @ B @ A @ F2 @ A3 ) @ ( times_times @ A @ ( semiring_1_of_nat @ A @ ( finite_card @ B @ A3 ) ) @ K6 ) ) ) ) ).
% sum_bounded_above
thf(fact_7782_card__le__Suc0__iff__eq,axiom,
! [A: $tType,A3: set @ A] :
( ( finite_finite2 @ A @ A3 )
=> ( ( ord_less_eq @ nat @ ( finite_card @ A @ A3 ) @ ( suc @ ( zero_zero @ nat ) ) )
= ( ! [X2: A] :
( ( member @ A @ X2 @ A3 )
=> ! [Y2: A] :
( ( member @ A @ Y2 @ A3 )
=> ( X2 = Y2 ) ) ) ) ) ) ).
% card_le_Suc0_iff_eq
thf(fact_7783_card__gt__0__iff,axiom,
! [A: $tType,A3: set @ A] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ ( finite_card @ A @ A3 ) )
= ( ( A3
!= ( bot_bot @ ( set @ A ) ) )
& ( finite_finite2 @ A @ A3 ) ) ) ).
% card_gt_0_iff
thf(fact_7784_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_7785_card__1__singleton__iff,axiom,
! [A: $tType,A3: set @ A] :
( ( ( finite_card @ A @ A3 )
= ( suc @ ( zero_zero @ nat ) ) )
= ( ? [X2: A] :
( A3
= ( insert @ A @ X2 @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ).
% card_1_singleton_iff
thf(fact_7786_card__eq__SucD,axiom,
! [A: $tType,A3: set @ A,K: nat] :
( ( ( finite_card @ A @ A3 )
= ( suc @ K ) )
=> ? [B3: A,B9: set @ A] :
( ( A3
= ( insert @ A @ B3 @ B9 ) )
& ~ ( member @ A @ B3 @ B9 )
& ( ( finite_card @ A @ B9 )
= K )
& ( ( K
= ( zero_zero @ nat ) )
=> ( B9
= ( bot_bot @ ( set @ A ) ) ) ) ) ) ).
% card_eq_SucD
thf(fact_7787_card__Suc__eq,axiom,
! [A: $tType,A3: set @ A,K: nat] :
( ( ( finite_card @ A @ A3 )
= ( suc @ K ) )
= ( ? [B5: A,B7: set @ A] :
( ( A3
= ( insert @ A @ B5 @ B7 ) )
& ~ ( member @ A @ B5 @ B7 )
& ( ( finite_card @ A @ B7 )
= K )
& ( ( K
= ( zero_zero @ nat ) )
=> ( B7
= ( bot_bot @ ( set @ A ) ) ) ) ) ) ) ).
% card_Suc_eq
thf(fact_7788_continuous__on__arcosh,axiom,
! [A3: set @ real] :
( ( ord_less_eq @ ( set @ real ) @ A3 @ ( set_ord_atLeast @ real @ ( one_one @ real ) ) )
=> ( topolo81223032696312382ous_on @ real @ real @ A3 @ ( arcosh @ real ) ) ) ).
% continuous_on_arcosh
thf(fact_7789_card__le__Suc__iff,axiom,
! [A: $tType,N3: nat,A3: set @ A] :
( ( ord_less_eq @ nat @ ( suc @ N3 ) @ ( finite_card @ A @ A3 ) )
= ( ? [A6: A,B7: set @ A] :
( ( A3
= ( insert @ A @ A6 @ B7 ) )
& ~ ( member @ A @ A6 @ B7 )
& ( ord_less_eq @ nat @ N3 @ ( finite_card @ A @ B7 ) )
& ( finite_finite2 @ A @ B7 ) ) ) ) ).
% card_le_Suc_iff
thf(fact_7790_surj__card__le,axiom,
! [B: $tType,A: $tType,A3: set @ A,B6: set @ B,F2: A > B] :
( ( finite_finite2 @ A @ A3 )
=> ( ( ord_less_eq @ ( set @ B ) @ B6 @ ( image @ A @ B @ F2 @ A3 ) )
=> ( ord_less_eq @ nat @ ( finite_card @ B @ B6 ) @ ( finite_card @ A @ A3 ) ) ) ) ).
% surj_card_le
thf(fact_7791_card__1__singletonI,axiom,
! [A: $tType,S3: set @ A,X: A] :
( ( finite_finite2 @ A @ S3 )
=> ( ( ( finite_card @ A @ S3 )
= ( one_one @ nat ) )
=> ( ( member @ A @ X @ S3 )
=> ( S3
= ( insert @ A @ X @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ) ).
% card_1_singletonI
thf(fact_7792_card__bij__eq,axiom,
! [A: $tType,B: $tType,F2: A > B,A3: set @ A,B6: set @ B,G: B > A] :
( ( inj_on @ A @ B @ F2 @ A3 )
=> ( ( ord_less_eq @ ( set @ B ) @ ( image @ A @ B @ F2 @ A3 ) @ B6 )
=> ( ( inj_on @ B @ A @ G @ B6 )
=> ( ( ord_less_eq @ ( set @ A ) @ ( image @ B @ A @ G @ B6 ) @ A3 )
=> ( ( finite_finite2 @ A @ A3 )
=> ( ( finite_finite2 @ B @ B6 )
=> ( ( finite_card @ A @ A3 )
= ( finite_card @ B @ B6 ) ) ) ) ) ) ) ) ).
% card_bij_eq
thf(fact_7793_surjective__iff__injective__gen,axiom,
! [B: $tType,A: $tType,S3: set @ A,T5: set @ B,F2: A > B] :
( ( finite_finite2 @ A @ S3 )
=> ( ( finite_finite2 @ B @ T5 )
=> ( ( ( finite_card @ A @ S3 )
= ( finite_card @ B @ T5 ) )
=> ( ( ord_less_eq @ ( set @ B ) @ ( image @ A @ B @ F2 @ S3 ) @ T5 )
=> ( ( ! [X2: B] :
( ( member @ B @ X2 @ T5 )
=> ? [Y2: A] :
( ( member @ A @ Y2 @ S3 )
& ( ( F2 @ Y2 )
= X2 ) ) ) )
= ( inj_on @ A @ B @ F2 @ S3 ) ) ) ) ) ) ).
% surjective_iff_injective_gen
thf(fact_7794_card__Diff1__le,axiom,
! [A: $tType,A3: set @ A,X: A] : ( ord_less_eq @ nat @ ( finite_card @ A @ ( minus_minus @ ( set @ A ) @ A3 @ ( insert @ A @ X @ ( bot_bot @ ( set @ A ) ) ) ) ) @ ( finite_card @ A @ A3 ) ) ).
% card_Diff1_le
thf(fact_7795_card__Diff__subset,axiom,
! [A: $tType,B6: set @ A,A3: set @ A] :
( ( finite_finite2 @ A @ B6 )
=> ( ( ord_less_eq @ ( set @ A ) @ B6 @ A3 )
=> ( ( finite_card @ A @ ( minus_minus @ ( set @ A ) @ A3 @ B6 ) )
= ( minus_minus @ nat @ ( finite_card @ A @ A3 ) @ ( finite_card @ A @ B6 ) ) ) ) ) ).
% card_Diff_subset
thf(fact_7796_diff__card__le__card__Diff,axiom,
! [A: $tType,B6: set @ A,A3: set @ A] :
( ( finite_finite2 @ A @ B6 )
=> ( ord_less_eq @ nat @ ( minus_minus @ nat @ ( finite_card @ A @ A3 ) @ ( finite_card @ A @ B6 ) ) @ ( finite_card @ A @ ( minus_minus @ ( set @ A ) @ A3 @ B6 ) ) ) ) ).
% diff_card_le_card_Diff
thf(fact_7797_card__psubset,axiom,
! [A: $tType,B6: set @ A,A3: set @ A] :
( ( finite_finite2 @ A @ B6 )
=> ( ( ord_less_eq @ ( set @ A ) @ A3 @ B6 )
=> ( ( ord_less @ nat @ ( finite_card @ A @ A3 ) @ ( finite_card @ A @ B6 ) )
=> ( ord_less @ ( set @ A ) @ A3 @ B6 ) ) ) ) ).
% card_psubset
thf(fact_7798_continuous__on__powr_H,axiom,
! [C: $tType] :
( ( topolo4958980785337419405_space @ C )
=> ! [S: set @ C,F2: C > real,G: C > real] :
( ( topolo81223032696312382ous_on @ C @ real @ S @ F2 )
=> ( ( topolo81223032696312382ous_on @ C @ real @ S @ G )
=> ( ! [X3: C] :
( ( member @ C @ X3 @ S )
=> ( ( ord_less_eq @ real @ ( zero_zero @ real ) @ ( F2 @ X3 ) )
& ( ( ( F2 @ X3 )
= ( zero_zero @ real ) )
=> ( ord_less @ real @ ( zero_zero @ real ) @ ( G @ X3 ) ) ) ) )
=> ( topolo81223032696312382ous_on @ C @ real @ S
@ ^ [X2: C] : ( powr @ real @ ( F2 @ X2 ) @ ( G @ X2 ) ) ) ) ) ) ) ).
% continuous_on_powr'
thf(fact_7799_continuous__on__log,axiom,
! [A: $tType] :
( ( topolo4958980785337419405_space @ A )
=> ! [S: set @ A,F2: A > real,G: A > real] :
( ( topolo81223032696312382ous_on @ A @ real @ S @ F2 )
=> ( ( topolo81223032696312382ous_on @ A @ real @ S @ G )
=> ( ! [X3: A] :
( ( member @ A @ X3 @ S )
=> ( ord_less @ real @ ( zero_zero @ real ) @ ( F2 @ X3 ) ) )
=> ( ! [X3: A] :
( ( member @ A @ X3 @ S )
=> ( ( F2 @ X3 )
!= ( one_one @ real ) ) )
=> ( ! [X3: A] :
( ( member @ A @ X3 @ S )
=> ( ord_less @ real @ ( zero_zero @ real ) @ ( G @ X3 ) ) )
=> ( topolo81223032696312382ous_on @ A @ real @ S
@ ^ [X2: A] : ( log @ ( F2 @ X2 ) @ ( G @ X2 ) ) ) ) ) ) ) ) ) ).
% continuous_on_log
thf(fact_7800_card__lists__length__le,axiom,
! [A: $tType,A3: set @ A,N3: nat] :
( ( finite_finite2 @ A @ A3 )
=> ( ( finite_card @ ( list @ A )
@ ( collect @ ( list @ A )
@ ^ [Xs: list @ A] :
( ( ord_less_eq @ ( set @ A ) @ ( set2 @ A @ Xs ) @ A3 )
& ( ord_less_eq @ nat @ ( size_size @ ( list @ A ) @ Xs ) @ N3 ) ) ) )
= ( groups7311177749621191930dd_sum @ nat @ nat @ ( power_power @ nat @ ( finite_card @ A @ A3 ) ) @ ( set_ord_atMost @ nat @ N3 ) ) ) ) ).
% card_lists_length_le
thf(fact_7801_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_7802_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_7803_continuous__on__arccos,axiom,
! [A: $tType] :
( ( topolo4958980785337419405_space @ A )
=> ! [S: set @ A,F2: A > real] :
( ( topolo81223032696312382ous_on @ A @ real @ S @ F2 )
=> ( ! [X3: A] :
( ( member @ A @ X3 @ S )
=> ( ( ord_less_eq @ real @ ( uminus_uminus @ real @ ( one_one @ real ) ) @ ( F2 @ X3 ) )
& ( ord_less_eq @ real @ ( F2 @ X3 ) @ ( one_one @ real ) ) ) )
=> ( topolo81223032696312382ous_on @ A @ real @ S
@ ^ [X2: A] : ( arccos @ ( F2 @ X2 ) ) ) ) ) ) ).
% continuous_on_arccos
thf(fact_7804_continuous__on__arcsin,axiom,
! [A: $tType] :
( ( topolo4958980785337419405_space @ A )
=> ! [S: set @ A,F2: A > real] :
( ( topolo81223032696312382ous_on @ A @ real @ S @ F2 )
=> ( ! [X3: A] :
( ( member @ A @ X3 @ S )
=> ( ( ord_less_eq @ real @ ( uminus_uminus @ real @ ( one_one @ real ) ) @ ( F2 @ X3 ) )
& ( ord_less_eq @ real @ ( F2 @ X3 ) @ ( one_one @ real ) ) ) )
=> ( topolo81223032696312382ous_on @ A @ real @ S
@ ^ [X2: A] : ( arcsin @ ( F2 @ X2 ) ) ) ) ) ) ).
% continuous_on_arcsin
thf(fact_7805_ex__bij__betw__nat__finite__1,axiom,
! [A: $tType,M8: set @ A] :
( ( finite_finite2 @ A @ M8 )
=> ? [H4: nat > A] : ( bij_betw @ nat @ A @ H4 @ ( set_or1337092689740270186AtMost @ nat @ ( one_one @ nat ) @ ( finite_card @ A @ M8 ) ) @ M8 ) ) ).
% ex_bij_betw_nat_finite_1
thf(fact_7806_card__roots__unity,axiom,
! [A: $tType] :
( ( ( real_V8999393235501362500lgebra @ A )
& ( idom @ A ) )
=> ! [N3: nat] :
( ( ord_less_eq @ nat @ ( one_one @ nat ) @ N3 )
=> ( ord_less_eq @ nat
@ ( finite_card @ A
@ ( collect @ A
@ ^ [Z3: A] :
( ( power_power @ A @ Z3 @ N3 )
= ( one_one @ A ) ) ) )
@ N3 ) ) ) ).
% card_roots_unity
thf(fact_7807_subset__eq__atLeast0__lessThan__card,axiom,
! [N7: set @ nat,N3: nat] :
( ( ord_less_eq @ ( set @ nat ) @ N7 @ ( set_or7035219750837199246ssThan @ nat @ ( zero_zero @ nat ) @ N3 ) )
=> ( ord_less_eq @ nat @ ( finite_card @ nat @ N7 ) @ N3 ) ) ).
% subset_eq_atLeast0_lessThan_card
thf(fact_7808_card__le__Suc__Max,axiom,
! [S3: set @ nat] :
( ( finite_finite2 @ nat @ S3 )
=> ( ord_less_eq @ nat @ ( finite_card @ nat @ S3 ) @ ( suc @ ( lattic643756798349783984er_Max @ nat @ S3 ) ) ) ) ).
% card_le_Suc_Max
thf(fact_7809_DERIV__atLeastAtMost__imp__continuous__on,axiom,
! [A: $tType] :
( ( ( ord @ A )
& ( real_V3459762299906320749_field @ A ) )
=> ! [A2: A,B2: A,F2: A > A] :
( ! [X3: A] :
( ( ord_less_eq @ A @ A2 @ X3 )
=> ( ( ord_less_eq @ A @ X3 @ B2 )
=> ? [Y4: A] : ( has_field_derivative @ A @ F2 @ Y4 @ ( topolo174197925503356063within @ A @ X3 @ ( top_top @ ( set @ A ) ) ) ) ) )
=> ( topolo81223032696312382ous_on @ A @ A @ ( set_or1337092689740270186AtMost @ A @ A2 @ B2 ) @ F2 ) ) ) ).
% DERIV_atLeastAtMost_imp_continuous_on
thf(fact_7810_card__sum__le__nat__sum,axiom,
! [S3: set @ nat] :
( ord_less_eq @ nat
@ ( groups7311177749621191930dd_sum @ nat @ nat
@ ^ [X2: nat] : X2
@ ( set_or7035219750837199246ssThan @ nat @ ( zero_zero @ nat ) @ ( finite_card @ nat @ S3 ) ) )
@ ( groups7311177749621191930dd_sum @ nat @ nat
@ ^ [X2: nat] : X2
@ S3 ) ) ).
% card_sum_le_nat_sum
thf(fact_7811_continuous__on__artanh_H,axiom,
! [A3: set @ real,F2: real > real] :
( ( topolo81223032696312382ous_on @ real @ real @ A3 @ F2 )
=> ( ! [X3: real] :
( ( member @ real @ X3 @ A3 )
=> ( member @ real @ ( F2 @ X3 ) @ ( set_or5935395276787703475ssThan @ real @ ( uminus_uminus @ real @ ( one_one @ real ) ) @ ( one_one @ real ) ) ) )
=> ( topolo81223032696312382ous_on @ real @ real @ A3
@ ^ [X2: real] : ( artanh @ real @ ( F2 @ X2 ) ) ) ) ) ).
% continuous_on_artanh'
thf(fact_7812_card__nth__roots,axiom,
! [C2: complex,N3: nat] :
( ( C2
!= ( zero_zero @ complex ) )
=> ( ( ord_less @ nat @ ( zero_zero @ nat ) @ N3 )
=> ( ( finite_card @ complex
@ ( collect @ complex
@ ^ [Z3: complex] :
( ( power_power @ complex @ Z3 @ N3 )
= C2 ) ) )
= N3 ) ) ) ).
% card_nth_roots
thf(fact_7813_card__roots__unity__eq,axiom,
! [N3: nat] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ N3 )
=> ( ( finite_card @ complex
@ ( collect @ complex
@ ^ [Z3: complex] :
( ( power_power @ complex @ Z3 @ N3 )
= ( one_one @ complex ) ) ) )
= N3 ) ) ).
% card_roots_unity_eq
thf(fact_7814_card__2__iff,axiom,
! [A: $tType,S3: set @ A] :
( ( ( finite_card @ A @ S3 )
= ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) )
= ( ? [X2: A,Y2: A] :
( ( S3
= ( insert @ A @ X2 @ ( insert @ A @ Y2 @ ( bot_bot @ ( set @ A ) ) ) ) )
& ( X2 != Y2 ) ) ) ) ).
% card_2_iff
thf(fact_7815_card__3__iff,axiom,
! [A: $tType,S3: set @ A] :
( ( ( finite_card @ A @ S3 )
= ( numeral_numeral @ nat @ ( bit1 @ one2 ) ) )
= ( ? [X2: A,Y2: A,Z3: A] :
( ( S3
= ( insert @ A @ X2 @ ( insert @ A @ Y2 @ ( insert @ A @ Z3 @ ( bot_bot @ ( set @ A ) ) ) ) ) )
& ( X2 != Y2 )
& ( Y2 != Z3 )
& ( X2 != Z3 ) ) ) ) ).
% card_3_iff
thf(fact_7816_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_7817_odd__card__imp__not__empty,axiom,
! [A: $tType,A3: set @ A] :
( ~ ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( finite_card @ A @ A3 ) )
=> ( A3
!= ( bot_bot @ ( set @ A ) ) ) ) ).
% odd_card_imp_not_empty
thf(fact_7818_card__insert__disjoint_H,axiom,
! [A: $tType,A3: set @ A,X: A] :
( ( finite_finite2 @ A @ A3 )
=> ( ~ ( member @ A @ X @ A3 )
=> ( ( minus_minus @ nat @ ( finite_card @ A @ ( insert @ A @ X @ A3 ) ) @ ( suc @ ( zero_zero @ nat ) ) )
= ( finite_card @ A @ A3 ) ) ) ) ).
% card_insert_disjoint'
thf(fact_7819_card__Suc__Diff1,axiom,
! [A: $tType,A3: set @ A,X: A] :
( ( finite_finite2 @ A @ A3 )
=> ( ( member @ A @ X @ A3 )
=> ( ( suc @ ( finite_card @ A @ ( minus_minus @ ( set @ A ) @ A3 @ ( insert @ A @ X @ ( bot_bot @ ( set @ A ) ) ) ) ) )
= ( finite_card @ A @ A3 ) ) ) ) ).
% card_Suc_Diff1
thf(fact_7820_card_Oinsert__remove,axiom,
! [A: $tType,A3: set @ A,X: A] :
( ( finite_finite2 @ A @ A3 )
=> ( ( finite_card @ A @ ( insert @ A @ X @ A3 ) )
= ( suc @ ( finite_card @ A @ ( minus_minus @ ( set @ A ) @ A3 @ ( insert @ A @ X @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ) ) ).
% card.insert_remove
thf(fact_7821_card_Oremove,axiom,
! [A: $tType,A3: set @ A,X: A] :
( ( finite_finite2 @ A @ A3 )
=> ( ( member @ A @ X @ A3 )
=> ( ( finite_card @ A @ A3 )
= ( suc @ ( finite_card @ A @ ( minus_minus @ ( set @ A ) @ A3 @ ( insert @ A @ X @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ) ) ) ).
% card.remove
thf(fact_7822_card__Diff1__less__iff,axiom,
! [A: $tType,A3: set @ A,X: A] :
( ( ord_less @ nat @ ( finite_card @ A @ ( minus_minus @ ( set @ A ) @ A3 @ ( insert @ A @ X @ ( bot_bot @ ( set @ A ) ) ) ) ) @ ( finite_card @ A @ A3 ) )
= ( ( finite_finite2 @ A @ A3 )
& ( member @ A @ X @ A3 ) ) ) ).
% card_Diff1_less_iff
thf(fact_7823_card__Diff2__less,axiom,
! [A: $tType,A3: set @ A,X: A,Y: A] :
( ( finite_finite2 @ A @ A3 )
=> ( ( member @ A @ X @ A3 )
=> ( ( member @ A @ Y @ A3 )
=> ( ord_less @ nat @ ( finite_card @ A @ ( minus_minus @ ( set @ A ) @ ( minus_minus @ ( set @ A ) @ A3 @ ( insert @ A @ X @ ( bot_bot @ ( set @ A ) ) ) ) @ ( insert @ A @ Y @ ( bot_bot @ ( set @ A ) ) ) ) ) @ ( finite_card @ A @ A3 ) ) ) ) ) ).
% card_Diff2_less
thf(fact_7824_card__Diff1__less,axiom,
! [A: $tType,A3: set @ A,X: A] :
( ( finite_finite2 @ A @ A3 )
=> ( ( member @ A @ X @ A3 )
=> ( ord_less @ nat @ ( finite_card @ A @ ( minus_minus @ ( set @ A ) @ A3 @ ( insert @ A @ X @ ( bot_bot @ ( set @ A ) ) ) ) ) @ ( finite_card @ A @ A3 ) ) ) ) ).
% card_Diff1_less
thf(fact_7825_card__le__inj,axiom,
! [B: $tType,A: $tType,A3: set @ A,B6: set @ B] :
( ( finite_finite2 @ A @ A3 )
=> ( ( finite_finite2 @ B @ B6 )
=> ( ( ord_less_eq @ nat @ ( finite_card @ A @ A3 ) @ ( finite_card @ B @ B6 ) )
=> ? [F3: A > B] :
( ( ord_less_eq @ ( set @ B ) @ ( image @ A @ B @ F3 @ A3 ) @ B6 )
& ( inj_on @ A @ B @ F3 @ A3 ) ) ) ) ) ).
% card_le_inj
thf(fact_7826_card__inj__on__le,axiom,
! [A: $tType,B: $tType,F2: A > B,A3: set @ A,B6: set @ B] :
( ( inj_on @ A @ B @ F2 @ A3 )
=> ( ( ord_less_eq @ ( set @ B ) @ ( image @ A @ B @ F2 @ A3 ) @ B6 )
=> ( ( finite_finite2 @ B @ B6 )
=> ( ord_less_eq @ nat @ ( finite_card @ A @ A3 ) @ ( finite_card @ B @ B6 ) ) ) ) ) ).
% card_inj_on_le
thf(fact_7827_inj__on__iff__card__le,axiom,
! [A: $tType,B: $tType,A3: set @ A,B6: set @ B] :
( ( finite_finite2 @ A @ A3 )
=> ( ( finite_finite2 @ B @ B6 )
=> ( ( ? [F7: A > B] :
( ( inj_on @ A @ B @ F7 @ A3 )
& ( ord_less_eq @ ( set @ B ) @ ( image @ A @ B @ F7 @ A3 ) @ B6 ) ) )
= ( ord_less_eq @ nat @ ( finite_card @ A @ A3 ) @ ( finite_card @ B @ B6 ) ) ) ) ) ).
% inj_on_iff_card_le
thf(fact_7828_Rolle__deriv,axiom,
! [A2: real,B2: real,F2: real > real,F5: real > real > real] :
( ( ord_less @ real @ A2 @ B2 )
=> ( ( ( F2 @ A2 )
= ( F2 @ B2 ) )
=> ( ( topolo81223032696312382ous_on @ real @ real @ ( set_or1337092689740270186AtMost @ real @ A2 @ B2 ) @ F2 )
=> ( ! [X3: real] :
( ( ord_less @ real @ A2 @ X3 )
=> ( ( ord_less @ real @ X3 @ B2 )
=> ( has_derivative @ real @ real @ F2 @ ( F5 @ X3 ) @ ( topolo174197925503356063within @ real @ X3 @ ( top_top @ ( set @ real ) ) ) ) ) )
=> ? [Z2: real] :
( ( ord_less @ real @ A2 @ Z2 )
& ( ord_less @ real @ Z2 @ B2 )
& ( ( F5 @ Z2 )
= ( ^ [V5: real] : ( zero_zero @ real ) ) ) ) ) ) ) ) ).
% Rolle_deriv
thf(fact_7829_card__Diff__singleton,axiom,
! [A: $tType,X: A,A3: set @ A] :
( ( member @ A @ X @ A3 )
=> ( ( finite_card @ A @ ( minus_minus @ ( set @ A ) @ A3 @ ( insert @ A @ X @ ( bot_bot @ ( set @ A ) ) ) ) )
= ( minus_minus @ nat @ ( finite_card @ A @ A3 ) @ ( one_one @ nat ) ) ) ) ).
% card_Diff_singleton
thf(fact_7830_card__Diff__singleton__if,axiom,
! [A: $tType,X: A,A3: set @ A] :
( ( ( member @ A @ X @ A3 )
=> ( ( finite_card @ A @ ( minus_minus @ ( set @ A ) @ A3 @ ( insert @ A @ X @ ( bot_bot @ ( set @ A ) ) ) ) )
= ( minus_minus @ nat @ ( finite_card @ A @ A3 ) @ ( one_one @ nat ) ) ) )
& ( ~ ( member @ A @ X @ A3 )
=> ( ( finite_card @ A @ ( minus_minus @ ( set @ A ) @ A3 @ ( insert @ A @ X @ ( bot_bot @ ( set @ A ) ) ) ) )
= ( finite_card @ A @ A3 ) ) ) ) ).
% card_Diff_singleton_if
thf(fact_7831_mvt,axiom,
! [A2: real,B2: real,F2: real > real,F5: real > real > real] :
( ( ord_less @ real @ A2 @ B2 )
=> ( ( topolo81223032696312382ous_on @ real @ real @ ( set_or1337092689740270186AtMost @ real @ A2 @ B2 ) @ F2 )
=> ( ! [X3: real] :
( ( ord_less @ real @ A2 @ X3 )
=> ( ( ord_less @ real @ X3 @ B2 )
=> ( has_derivative @ real @ real @ F2 @ ( F5 @ X3 ) @ ( topolo174197925503356063within @ real @ X3 @ ( top_top @ ( set @ real ) ) ) ) ) )
=> ~ ! [Xi3: real] :
( ( ord_less @ real @ A2 @ Xi3 )
=> ( ( ord_less @ real @ Xi3 @ B2 )
=> ( ( minus_minus @ real @ ( F2 @ B2 ) @ ( F2 @ A2 ) )
!= ( F5 @ Xi3 @ ( minus_minus @ real @ B2 @ A2 ) ) ) ) ) ) ) ) ).
% mvt
thf(fact_7832_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 ) )
@ ^ [X2: A] : ( ring_1_of_int @ B @ ( archim6421214686448440834_floor @ A @ X2 ) ) ) ) ).
% continuous_on_of_int_floor
thf(fact_7833_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 ) )
@ ^ [X2: A] : ( ring_1_of_int @ B @ ( archimedean_ceiling @ A @ X2 ) ) ) ) ).
% continuous_on_of_int_ceiling
thf(fact_7834_sum__norm__bound,axiom,
! [A: $tType,B: $tType] :
( ( real_V822414075346904944vector @ A )
=> ! [S3: set @ B,F2: B > A,K6: real] :
( ! [X3: B] :
( ( member @ B @ X3 @ S3 )
=> ( ord_less_eq @ real @ ( real_V7770717601297561774m_norm @ A @ ( F2 @ X3 ) ) @ K6 ) )
=> ( ord_less_eq @ real @ ( real_V7770717601297561774m_norm @ A @ ( groups7311177749621191930dd_sum @ B @ A @ F2 @ S3 ) ) @ ( times_times @ real @ ( semiring_1_of_nat @ real @ ( finite_card @ B @ S3 ) ) @ K6 ) ) ) ) ).
% sum_norm_bound
thf(fact_7835_continuous__on__Icc__at__leftD,axiom,
! [B: $tType,A: $tType] :
( ( ( topolo1944317154257567458pology @ A )
& ( topolo4958980785337419405_space @ B ) )
=> ! [A2: A,B2: A,F2: A > B] :
( ( topolo81223032696312382ous_on @ A @ B @ ( set_or1337092689740270186AtMost @ A @ A2 @ B2 ) @ F2 )
=> ( ( ord_less @ A @ A2 @ B2 )
=> ( filterlim @ A @ B @ F2 @ ( topolo7230453075368039082e_nhds @ B @ ( F2 @ B2 ) ) @ ( topolo174197925503356063within @ A @ B2 @ ( set_ord_lessThan @ A @ B2 ) ) ) ) ) ) ).
% continuous_on_Icc_at_leftD
thf(fact_7836_continuous__on__Icc__at__rightD,axiom,
! [B: $tType,A: $tType] :
( ( ( topolo1944317154257567458pology @ A )
& ( topolo4958980785337419405_space @ B ) )
=> ! [A2: A,B2: A,F2: A > B] :
( ( topolo81223032696312382ous_on @ A @ B @ ( set_or1337092689740270186AtMost @ A @ A2 @ B2 ) @ F2 )
=> ( ( ord_less @ A @ A2 @ B2 )
=> ( filterlim @ A @ B @ F2 @ ( topolo7230453075368039082e_nhds @ B @ ( F2 @ A2 ) ) @ ( topolo174197925503356063within @ A @ A2 @ ( set_ord_greaterThan @ A @ A2 ) ) ) ) ) ) ).
% continuous_on_Icc_at_rightD
thf(fact_7837_card__UN__le,axiom,
! [B: $tType,A: $tType,I5: set @ A,A3: A > ( set @ B )] :
( ( finite_finite2 @ A @ I5 )
=> ( ord_less_eq @ nat @ ( finite_card @ B @ ( complete_Sup_Sup @ ( set @ B ) @ ( image @ A @ ( set @ B ) @ A3 @ I5 ) ) )
@ ( groups7311177749621191930dd_sum @ A @ nat
@ ^ [I3: A] : ( finite_card @ B @ ( A3 @ I3 ) )
@ I5 ) ) ) ).
% card_UN_le
thf(fact_7838_prod__le__power,axiom,
! [B: $tType,A: $tType] :
( ( linordered_semidom @ A )
=> ! [A3: set @ B,F2: B > A,N3: A,K: nat] :
( ! [I2: B] :
( ( member @ B @ I2 @ A3 )
=> ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( F2 @ I2 ) )
& ( ord_less_eq @ A @ ( F2 @ I2 ) @ N3 ) ) )
=> ( ( ord_less_eq @ nat @ ( finite_card @ B @ A3 ) @ K )
=> ( ( ord_less_eq @ A @ ( one_one @ A ) @ N3 )
=> ( ord_less_eq @ A @ ( groups7121269368397514597t_prod @ B @ A @ F2 @ A3 ) @ ( power_power @ A @ N3 @ K ) ) ) ) ) ) ).
% prod_le_power
thf(fact_7839_sum__bounded__above__strict,axiom,
! [B: $tType,A: $tType] :
( ( ( ordere8940638589300402666id_add @ A )
& ( semiring_1 @ A ) )
=> ! [A3: set @ B,F2: B > A,K6: A] :
( ! [I2: B] :
( ( member @ B @ I2 @ A3 )
=> ( ord_less @ A @ ( F2 @ I2 ) @ K6 ) )
=> ( ( ord_less @ nat @ ( zero_zero @ nat ) @ ( finite_card @ B @ A3 ) )
=> ( ord_less @ A @ ( groups7311177749621191930dd_sum @ B @ A @ F2 @ A3 ) @ ( times_times @ A @ ( semiring_1_of_nat @ A @ ( finite_card @ B @ A3 ) ) @ K6 ) ) ) ) ) ).
% sum_bounded_above_strict
thf(fact_7840_sum__bounded__above__divide,axiom,
! [B: $tType,A: $tType] :
( ( linordered_field @ A )
=> ! [A3: set @ B,F2: B > A,K6: A] :
( ! [I2: B] :
( ( member @ B @ I2 @ A3 )
=> ( ord_less_eq @ A @ ( F2 @ I2 ) @ ( divide_divide @ A @ K6 @ ( semiring_1_of_nat @ A @ ( finite_card @ B @ A3 ) ) ) ) )
=> ( ( finite_finite2 @ B @ A3 )
=> ( ( A3
!= ( bot_bot @ ( set @ B ) ) )
=> ( ord_less_eq @ A @ ( groups7311177749621191930dd_sum @ B @ A @ F2 @ A3 ) @ K6 ) ) ) ) ) ).
% sum_bounded_above_divide
thf(fact_7841_card__insert__le__m1,axiom,
! [A: $tType,N3: nat,Y: set @ A,X: A] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ N3 )
=> ( ( ord_less_eq @ nat @ ( finite_card @ A @ Y ) @ ( minus_minus @ nat @ N3 @ ( one_one @ nat ) ) )
=> ( ord_less_eq @ nat @ ( finite_card @ A @ ( insert @ A @ X @ Y ) ) @ N3 ) ) ) ).
% card_insert_le_m1
thf(fact_7842_DERIV__pos__imp__increasing__open,axiom,
! [A2: real,B2: real,F2: real > real] :
( ( ord_less @ real @ A2 @ B2 )
=> ( ! [X3: real] :
( ( ord_less @ real @ A2 @ X3 )
=> ( ( ord_less @ real @ X3 @ B2 )
=> ? [Y4: real] :
( ( has_field_derivative @ real @ F2 @ Y4 @ ( topolo174197925503356063within @ real @ X3 @ ( top_top @ ( set @ real ) ) ) )
& ( ord_less @ real @ ( zero_zero @ real ) @ Y4 ) ) ) )
=> ( ( topolo81223032696312382ous_on @ real @ real @ ( set_or1337092689740270186AtMost @ real @ A2 @ B2 ) @ F2 )
=> ( ord_less @ real @ ( F2 @ A2 ) @ ( F2 @ B2 ) ) ) ) ) ).
% DERIV_pos_imp_increasing_open
thf(fact_7843_DERIV__neg__imp__decreasing__open,axiom,
! [A2: real,B2: real,F2: real > real] :
( ( ord_less @ real @ A2 @ B2 )
=> ( ! [X3: real] :
( ( ord_less @ real @ A2 @ X3 )
=> ( ( ord_less @ real @ X3 @ B2 )
=> ? [Y4: real] :
( ( has_field_derivative @ real @ F2 @ Y4 @ ( topolo174197925503356063within @ real @ X3 @ ( top_top @ ( set @ real ) ) ) )
& ( ord_less @ real @ Y4 @ ( zero_zero @ real ) ) ) ) )
=> ( ( topolo81223032696312382ous_on @ real @ real @ ( set_or1337092689740270186AtMost @ real @ A2 @ B2 ) @ F2 )
=> ( ord_less @ real @ ( F2 @ B2 ) @ ( F2 @ A2 ) ) ) ) ) ).
% DERIV_neg_imp_decreasing_open
thf(fact_7844_DERIV__isconst__end,axiom,
! [A2: real,B2: real,F2: real > real] :
( ( ord_less @ real @ A2 @ B2 )
=> ( ( topolo81223032696312382ous_on @ real @ real @ ( set_or1337092689740270186AtMost @ real @ A2 @ B2 ) @ F2 )
=> ( ! [X3: real] :
( ( ord_less @ real @ A2 @ X3 )
=> ( ( ord_less @ real @ X3 @ B2 )
=> ( has_field_derivative @ real @ F2 @ ( zero_zero @ real ) @ ( topolo174197925503356063within @ real @ X3 @ ( top_top @ ( set @ real ) ) ) ) ) )
=> ( ( F2 @ B2 )
= ( F2 @ A2 ) ) ) ) ) ).
% DERIV_isconst_end
thf(fact_7845_continuous__on__artanh,axiom,
! [A3: set @ real] :
( ( ord_less_eq @ ( set @ real ) @ A3 @ ( set_or5935395276787703475ssThan @ real @ ( uminus_uminus @ real @ ( one_one @ real ) ) @ ( one_one @ real ) ) )
=> ( topolo81223032696312382ous_on @ real @ real @ A3 @ ( artanh @ real ) ) ) ).
% continuous_on_artanh
thf(fact_7846_card__word__size,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [X: word @ A] :
( ( finite_card @ ( word @ A ) @ ( top_top @ ( set @ ( word @ A ) ) ) )
= ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( size_size @ ( word @ A ) @ X ) ) ) ) ).
% card_word_size
thf(fact_7847_sum__fun__comp,axiom,
! [A: $tType,C: $tType,B: $tType] :
( ( semiring_1 @ C )
=> ! [S3: set @ A,R: set @ B,G: A > B,F2: B > C] :
( ( finite_finite2 @ A @ S3 )
=> ( ( finite_finite2 @ B @ R )
=> ( ( ord_less_eq @ ( set @ B ) @ ( image @ A @ B @ G @ S3 ) @ R )
=> ( ( groups7311177749621191930dd_sum @ A @ C
@ ^ [X2: A] : ( F2 @ ( G @ X2 ) )
@ S3 )
= ( groups7311177749621191930dd_sum @ B @ C
@ ^ [Y2: B] :
( times_times @ C
@ ( semiring_1_of_nat @ C
@ ( finite_card @ A
@ ( collect @ A
@ ^ [X2: A] :
( ( member @ A @ X2 @ S3 )
& ( ( G @ X2 )
= Y2 ) ) ) ) )
@ ( F2 @ Y2 ) )
@ R ) ) ) ) ) ) ).
% sum_fun_comp
thf(fact_7848_prod__gen__delta,axiom,
! [A: $tType,B: $tType] :
( ( comm_monoid_mult @ A )
=> ! [S3: set @ B,A2: B,B2: B > A,C2: A] :
( ( finite_finite2 @ B @ S3 )
=> ( ( ( member @ B @ A2 @ S3 )
=> ( ( groups7121269368397514597t_prod @ B @ A
@ ^ [K2: B] : ( if @ A @ ( K2 = A2 ) @ ( B2 @ K2 ) @ C2 )
@ S3 )
= ( times_times @ A @ ( B2 @ A2 ) @ ( power_power @ A @ C2 @ ( minus_minus @ nat @ ( finite_card @ B @ S3 ) @ ( one_one @ nat ) ) ) ) ) )
& ( ~ ( member @ B @ A2 @ S3 )
=> ( ( groups7121269368397514597t_prod @ B @ A
@ ^ [K2: B] : ( if @ A @ ( K2 = A2 ) @ ( B2 @ K2 ) @ C2 )
@ S3 )
= ( power_power @ A @ C2 @ ( finite_card @ B @ S3 ) ) ) ) ) ) ) ).
% prod_gen_delta
thf(fact_7849_polyfun__roots__card,axiom,
! [A: $tType] :
( ( ( real_V8999393235501362500lgebra @ A )
& ( idom @ A ) )
=> ! [C2: nat > A,K: nat,N3: nat] :
( ( ( C2 @ K )
!= ( zero_zero @ A ) )
=> ( ( ord_less_eq @ nat @ K @ N3 )
=> ( ord_less_eq @ nat
@ ( finite_card @ A
@ ( collect @ A
@ ^ [Z3: A] :
( ( groups7311177749621191930dd_sum @ nat @ A
@ ^ [I3: nat] : ( times_times @ A @ ( C2 @ I3 ) @ ( power_power @ A @ Z3 @ I3 ) )
@ ( set_ord_atMost @ nat @ N3 ) )
= ( zero_zero @ A ) ) ) )
@ N3 ) ) ) ) ).
% polyfun_roots_card
thf(fact_7850_sum__le__card__Max,axiom,
! [A: $tType,A3: set @ A,F2: A > nat] :
( ( finite_finite2 @ A @ A3 )
=> ( ord_less_eq @ nat @ ( groups7311177749621191930dd_sum @ A @ nat @ F2 @ A3 ) @ ( times_times @ nat @ ( finite_card @ A @ A3 ) @ ( lattic643756798349783984er_Max @ nat @ ( image @ A @ nat @ F2 @ A3 ) ) ) ) ) ).
% sum_le_card_Max
thf(fact_7851_DERIV__isconst2,axiom,
! [A2: real,B2: real,F2: real > real,X: real] :
( ( ord_less @ real @ A2 @ B2 )
=> ( ( topolo81223032696312382ous_on @ real @ real @ ( set_or1337092689740270186AtMost @ real @ A2 @ B2 ) @ F2 )
=> ( ! [X3: real] :
( ( ord_less @ real @ A2 @ X3 )
=> ( ( ord_less @ real @ X3 @ B2 )
=> ( has_field_derivative @ real @ F2 @ ( zero_zero @ real ) @ ( topolo174197925503356063within @ real @ X3 @ ( top_top @ ( set @ real ) ) ) ) ) )
=> ( ( ord_less_eq @ real @ A2 @ X )
=> ( ( ord_less_eq @ real @ X @ B2 )
=> ( ( F2 @ X )
= ( F2 @ A2 ) ) ) ) ) ) ) ).
% DERIV_isconst2
thf(fact_7852_card__map__elide2,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [N3: nat] :
( ( ord_less_eq @ nat @ N3 @ ( 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 ) @ N3 ) ) )
= N3 ) ) ) ).
% card_map_elide2
thf(fact_7853_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_7854_polyfun__rootbound,axiom,
! [A: $tType] :
( ( ( real_V8999393235501362500lgebra @ A )
& ( idom @ A ) )
=> ! [C2: nat > A,K: nat,N3: nat] :
( ( ( C2 @ K )
!= ( zero_zero @ A ) )
=> ( ( ord_less_eq @ nat @ K @ N3 )
=> ( ( finite_finite2 @ A
@ ( collect @ A
@ ^ [Z3: A] :
( ( groups7311177749621191930dd_sum @ nat @ A
@ ^ [I3: nat] : ( times_times @ A @ ( C2 @ I3 ) @ ( power_power @ A @ Z3 @ I3 ) )
@ ( set_ord_atMost @ nat @ N3 ) )
= ( zero_zero @ A ) ) ) )
& ( ord_less_eq @ nat
@ ( finite_card @ A
@ ( collect @ A
@ ^ [Z3: A] :
( ( groups7311177749621191930dd_sum @ nat @ A
@ ^ [I3: nat] : ( times_times @ A @ ( C2 @ I3 ) @ ( power_power @ A @ Z3 @ I3 ) )
@ ( set_ord_atMost @ nat @ N3 ) )
= ( zero_zero @ A ) ) ) )
@ N3 ) ) ) ) ) ).
% polyfun_rootbound
thf(fact_7855_continuous__on__IccI,axiom,
! [B: $tType,A: $tType] :
( ( ( topolo1944317154257567458pology @ A )
& ( topolo4958980785337419405_space @ B ) )
=> ! [F2: A > B,A2: A,B2: A] :
( ( filterlim @ A @ B @ F2 @ ( topolo7230453075368039082e_nhds @ B @ ( F2 @ A2 ) ) @ ( topolo174197925503356063within @ A @ A2 @ ( set_ord_greaterThan @ A @ A2 ) ) )
=> ( ( filterlim @ A @ B @ F2 @ ( topolo7230453075368039082e_nhds @ B @ ( F2 @ B2 ) ) @ ( topolo174197925503356063within @ A @ B2 @ ( set_ord_lessThan @ A @ B2 ) ) )
=> ( ! [X3: A] :
( ( ord_less @ A @ A2 @ X3 )
=> ( ( ord_less @ A @ X3 @ B2 )
=> ( filterlim @ A @ B @ F2 @ ( topolo7230453075368039082e_nhds @ B @ ( F2 @ X3 ) ) @ ( topolo174197925503356063within @ A @ X3 @ ( top_top @ ( set @ A ) ) ) ) ) )
=> ( ( ord_less @ A @ A2 @ B2 )
=> ( topolo81223032696312382ous_on @ A @ B @ ( set_or1337092689740270186AtMost @ A @ A2 @ B2 ) @ F2 ) ) ) ) ) ) ).
% continuous_on_IccI
thf(fact_7856_Rolle,axiom,
! [A2: real,B2: real,F2: real > real] :
( ( ord_less @ real @ A2 @ B2 )
=> ( ( ( F2 @ A2 )
= ( F2 @ B2 ) )
=> ( ( topolo81223032696312382ous_on @ real @ real @ ( set_or1337092689740270186AtMost @ real @ A2 @ B2 ) @ F2 )
=> ( ! [X3: real] :
( ( ord_less @ real @ A2 @ X3 )
=> ( ( ord_less @ real @ X3 @ B2 )
=> ( differentiable @ real @ real @ F2 @ ( topolo174197925503356063within @ real @ X3 @ ( top_top @ ( set @ real ) ) ) ) ) )
=> ? [Z2: real] :
( ( ord_less @ real @ A2 @ Z2 )
& ( ord_less @ real @ Z2 @ B2 )
& ( has_field_derivative @ real @ F2 @ ( zero_zero @ real ) @ ( topolo174197925503356063within @ real @ Z2 @ ( top_top @ ( set @ real ) ) ) ) ) ) ) ) ) ).
% Rolle
thf(fact_7857_even__sum__iff,axiom,
! [A: $tType,B: $tType] :
( ( semiring_parity @ A )
=> ! [A3: set @ B,F2: B > A] :
( ( finite_finite2 @ B @ A3 )
=> ( ( dvd_dvd @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ ( groups7311177749621191930dd_sum @ B @ A @ F2 @ A3 ) )
= ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) )
@ ( finite_card @ B
@ ( collect @ B
@ ^ [A6: B] :
( ( member @ B @ A6 @ A3 )
& ~ ( dvd_dvd @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ ( F2 @ A6 ) ) ) ) ) ) ) ) ) ).
% even_sum_iff
thf(fact_7858_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_7859_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_7860_card__num1,axiom,
( ( finite_card @ numeral_num1 @ ( top_top @ ( set @ numeral_num1 ) ) )
= ( one_one @ nat ) ) ).
% card_num1
thf(fact_7861_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_7862_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_7863_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_7864_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_7865_CARD__1,axiom,
! [A: $tType] :
( ( cARD_1 @ A )
=> ( ( finite_card @ A @ ( top_top @ ( set @ A ) ) )
= ( one_one @ nat ) ) ) ).
% CARD_1
thf(fact_7866_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_7867_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_7868_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_7869_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_7870_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_7871_bit0__cases,axiom,
! [A: $tType] :
( ( finite_finite @ A )
=> ! [X: numeral_bit0 @ A] :
~ ! [Z2: int] :
( ( X
= ( 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_7872_bit1__cases,axiom,
! [A: $tType] :
( ( finite_finite @ A )
=> ! [X: numeral_bit1 @ A] :
~ ! [Z2: int] :
( ( X
= ( 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_7873_bit0__induct,axiom,
! [A: $tType] :
( ( finite_finite @ A )
=> ! [P: ( numeral_bit0 @ A ) > $o,X: 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 @ X ) ) ) ).
% bit0_induct
thf(fact_7874_bit1__induct,axiom,
! [A: $tType] :
( ( finite_finite @ A )
=> ! [P: ( numeral_bit1 @ A ) > $o,X: 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 @ X ) ) ) ).
% bit1_induct
thf(fact_7875_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_7876_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_7877_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_7878_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_7879_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_7880_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_7881_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_7882_card__UNION,axiom,
! [A: $tType,A3: set @ ( set @ A )] :
( ( finite_finite2 @ ( set @ A ) @ A3 )
=> ( ! [X3: set @ A] :
( ( member @ ( set @ A ) @ X3 @ A3 )
=> ( finite_finite2 @ A @ X3 ) )
=> ( ( finite_card @ A @ ( complete_Sup_Sup @ ( set @ A ) @ A3 ) )
= ( nat2
@ ( groups7311177749621191930dd_sum @ ( set @ ( set @ A ) ) @ int
@ ^ [I8: set @ ( set @ A )] : ( times_times @ int @ ( power_power @ int @ ( uminus_uminus @ int @ ( one_one @ int ) ) @ ( plus_plus @ nat @ ( finite_card @ ( set @ A ) @ I8 ) @ ( one_one @ nat ) ) ) @ ( semiring_1_of_nat @ int @ ( finite_card @ A @ ( complete_Inf_Inf @ ( set @ A ) @ I8 ) ) ) )
@ ( collect @ ( set @ ( set @ A ) )
@ ^ [I8: set @ ( set @ A )] :
( ( ord_less_eq @ ( set @ ( set @ A ) ) @ I8 @ A3 )
& ( I8
!= ( bot_bot @ ( set @ ( set @ A ) ) ) ) ) ) ) ) ) ) ) ).
% card_UNION
thf(fact_7883_inj__on__Abs__bit1,axiom,
! [A: $tType] :
( ( finite_finite @ A )
=> ( inj_on @ int @ ( numeral_bit1 @ A ) @ ( numeral_Abs_bit12 @ 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_7884_Inf__eq__bot__iff,axiom,
! [A: $tType] :
( ( comple5582772986160207858norder @ A )
=> ! [A3: set @ A] :
( ( ( complete_Inf_Inf @ A @ A3 )
= ( bot_bot @ A ) )
= ( ! [X2: A] :
( ( ord_less @ A @ ( bot_bot @ A ) @ X2 )
=> ? [Y2: A] :
( ( member @ A @ Y2 @ A3 )
& ( ord_less @ A @ Y2 @ X2 ) ) ) ) ) ) ).
% Inf_eq_bot_iff
thf(fact_7885_cInf__atLeastAtMost,axiom,
! [A: $tType] :
( ( condit1219197933456340205attice @ A )
=> ! [Y: A,X: A] :
( ( ord_less_eq @ A @ Y @ X )
=> ( ( complete_Inf_Inf @ A @ ( set_or1337092689740270186AtMost @ A @ Y @ X ) )
= Y ) ) ) ).
% cInf_atLeastAtMost
thf(fact_7886_Inf__atLeastAtMost,axiom,
! [A: $tType] :
( ( comple6319245703460814977attice @ A )
=> ! [X: A,Y: A] :
( ( ord_less_eq @ A @ X @ Y )
=> ( ( complete_Inf_Inf @ A @ ( set_or1337092689740270186AtMost @ A @ X @ Y ) )
= X ) ) ) ).
% Inf_atLeastAtMost
thf(fact_7887_Inf__atLeastLessThan,axiom,
! [A: $tType] :
( ( comple6319245703460814977attice @ A )
=> ! [X: A,Y: A] :
( ( ord_less @ A @ X @ Y )
=> ( ( complete_Inf_Inf @ A @ ( set_or7035219750837199246ssThan @ A @ X @ Y ) )
= X ) ) ) ).
% Inf_atLeastLessThan
thf(fact_7888_cInf__atLeastLessThan,axiom,
! [A: $tType] :
( ( condit1219197933456340205attice @ A )
=> ! [Y: A,X: A] :
( ( ord_less @ A @ Y @ X )
=> ( ( complete_Inf_Inf @ A @ ( set_or7035219750837199246ssThan @ A @ Y @ X ) )
= Y ) ) ) ).
% cInf_atLeastLessThan
thf(fact_7889_Inf__greaterThanLessThan,axiom,
! [A: $tType] :
( ( ( comple6319245703460814977attice @ A )
& ( dense_linorder @ A ) )
=> ! [X: A,Y: A] :
( ( ord_less @ A @ X @ Y )
=> ( ( complete_Inf_Inf @ A @ ( set_or5935395276787703475ssThan @ A @ X @ Y ) )
= X ) ) ) ).
% Inf_greaterThanLessThan
thf(fact_7890_cInf__greaterThanLessThan,axiom,
! [A: $tType] :
( ( ( condit6923001295902523014norder @ A )
& ( dense_linorder @ A ) )
=> ! [Y: A,X: A] :
( ( ord_less @ A @ Y @ X )
=> ( ( complete_Inf_Inf @ A @ ( set_or5935395276787703475ssThan @ A @ Y @ X ) )
= Y ) ) ) ).
% cInf_greaterThanLessThan
thf(fact_7891_Inf__greaterThanAtMost,axiom,
! [A: $tType] :
( ( ( comple6319245703460814977attice @ A )
& ( dense_linorder @ A ) )
=> ! [X: A,Y: A] :
( ( ord_less @ A @ X @ Y )
=> ( ( complete_Inf_Inf @ A @ ( set_or3652927894154168847AtMost @ A @ X @ Y ) )
= X ) ) ) ).
% Inf_greaterThanAtMost
thf(fact_7892_cInf__greaterThanAtMost,axiom,
! [A: $tType] :
( ( ( condit6923001295902523014norder @ A )
& ( dense_linorder @ A ) )
=> ! [Y: A,X: A] :
( ( ord_less @ A @ Y @ X )
=> ( ( complete_Inf_Inf @ A @ ( set_or3652927894154168847AtMost @ A @ Y @ X ) )
= Y ) ) ) ).
% cInf_greaterThanAtMost
thf(fact_7893_INF__eq__bot__iff,axiom,
! [B: $tType,A: $tType] :
( ( comple5582772986160207858norder @ A )
=> ! [F2: B > A,A3: set @ B] :
( ( ( complete_Inf_Inf @ A @ ( image @ B @ A @ F2 @ A3 ) )
= ( bot_bot @ A ) )
= ( ! [X2: A] :
( ( ord_less @ A @ ( bot_bot @ A ) @ X2 )
=> ? [Y2: B] :
( ( member @ B @ Y2 @ A3 )
& ( ord_less @ A @ ( F2 @ Y2 ) @ X2 ) ) ) ) ) ) ).
% INF_eq_bot_iff
thf(fact_7894_Inf__atMostLessThan,axiom,
! [A: $tType] :
( ( comple6319245703460814977attice @ A )
=> ! [X: A] :
( ( ord_less @ A @ ( top_top @ A ) @ X )
=> ( ( complete_Inf_Inf @ A @ ( set_ord_lessThan @ A @ X ) )
= ( bot_bot @ A ) ) ) ) ).
% Inf_atMostLessThan
thf(fact_7895_INT__simps_I3_J,axiom,
! [E3: $tType,F: $tType,C5: set @ E3,A3: E3 > ( set @ F ),B6: set @ F] :
( ( ( C5
= ( bot_bot @ ( set @ E3 ) ) )
=> ( ( complete_Inf_Inf @ ( set @ F )
@ ( image @ E3 @ ( set @ F )
@ ^ [X2: E3] : ( minus_minus @ ( set @ F ) @ ( A3 @ X2 ) @ B6 )
@ C5 ) )
= ( top_top @ ( set @ F ) ) ) )
& ( ( C5
!= ( bot_bot @ ( set @ E3 ) ) )
=> ( ( complete_Inf_Inf @ ( set @ F )
@ ( image @ E3 @ ( set @ F )
@ ^ [X2: E3] : ( minus_minus @ ( set @ F ) @ ( A3 @ X2 ) @ B6 )
@ C5 ) )
= ( minus_minus @ ( set @ F ) @ ( complete_Inf_Inf @ ( set @ F ) @ ( image @ E3 @ ( set @ F ) @ A3 @ C5 ) ) @ B6 ) ) ) ) ).
% INT_simps(3)
thf(fact_7896_INT__simps_I4_J,axiom,
! [G2: $tType,H5: $tType,C5: set @ H5,A3: set @ G2,B6: H5 > ( set @ G2 )] :
( ( ( C5
= ( bot_bot @ ( set @ H5 ) ) )
=> ( ( complete_Inf_Inf @ ( set @ G2 )
@ ( image @ H5 @ ( set @ G2 )
@ ^ [X2: H5] : ( minus_minus @ ( set @ G2 ) @ A3 @ ( B6 @ X2 ) )
@ C5 ) )
= ( top_top @ ( set @ G2 ) ) ) )
& ( ( C5
!= ( bot_bot @ ( set @ H5 ) ) )
=> ( ( complete_Inf_Inf @ ( set @ G2 )
@ ( image @ H5 @ ( set @ G2 )
@ ^ [X2: H5] : ( minus_minus @ ( set @ G2 ) @ A3 @ ( B6 @ X2 ) )
@ C5 ) )
= ( minus_minus @ ( set @ G2 ) @ A3 @ ( complete_Sup_Sup @ ( set @ G2 ) @ ( image @ H5 @ ( set @ G2 ) @ B6 @ C5 ) ) ) ) ) ) ).
% INT_simps(4)
thf(fact_7897_finite__imp__less__Inf,axiom,
! [A: $tType] :
( ( condit6923001295902523014norder @ A )
=> ! [X8: set @ A,X: A,A2: A] :
( ( finite_finite2 @ A @ X8 )
=> ( ( member @ A @ X @ X8 )
=> ( ! [X3: A] :
( ( member @ A @ X3 @ X8 )
=> ( ord_less @ A @ A2 @ X3 ) )
=> ( ord_less @ A @ A2 @ ( complete_Inf_Inf @ A @ X8 ) ) ) ) ) ) ).
% finite_imp_less_Inf
thf(fact_7898_one__bit1__def,axiom,
! [A: $tType] :
( ( finite_finite @ A )
=> ( ( one_one @ ( numeral_bit1 @ A ) )
= ( numeral_Abs_bit12 @ A @ ( one_one @ int ) ) ) ) ).
% one_bit1_def
thf(fact_7899_finite__less__Inf__iff,axiom,
! [A: $tType] :
( ( condit6923001295902523014norder @ A )
=> ! [X8: set @ A,A2: A] :
( ( finite_finite2 @ A @ X8 )
=> ( ( X8
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( ord_less @ A @ A2 @ ( complete_Inf_Inf @ A @ X8 ) )
= ( ! [X2: A] :
( ( member @ A @ X2 @ X8 )
=> ( ord_less @ A @ A2 @ X2 ) ) ) ) ) ) ) ).
% finite_less_Inf_iff
thf(fact_7900_cInf__le__finite,axiom,
! [A: $tType] :
( ( condit1219197933456340205attice @ A )
=> ! [X8: set @ A,X: A] :
( ( finite_finite2 @ A @ X8 )
=> ( ( member @ A @ X @ X8 )
=> ( ord_less_eq @ A @ ( complete_Inf_Inf @ A @ X8 ) @ X ) ) ) ) ).
% cInf_le_finite
thf(fact_7901_Some__INF,axiom,
! [A: $tType,B: $tType] :
( ( comple6319245703460814977attice @ A )
=> ! [F2: B > A,A3: set @ B] :
( ( some @ A @ ( complete_Inf_Inf @ A @ ( image @ B @ A @ F2 @ A3 ) ) )
= ( complete_Inf_Inf @ ( option @ A )
@ ( image @ B @ ( option @ A )
@ ^ [X2: B] : ( some @ A @ ( F2 @ X2 ) )
@ A3 ) ) ) ) ).
% Some_INF
thf(fact_7902_Some__Inf,axiom,
! [A: $tType] :
( ( comple6319245703460814977attice @ A )
=> ! [A3: set @ A] :
( ( some @ A @ ( complete_Inf_Inf @ A @ A3 ) )
= ( complete_Inf_Inf @ ( option @ A ) @ ( image @ A @ ( option @ A ) @ ( some @ A ) @ A3 ) ) ) ) ).
% Some_Inf
thf(fact_7903_INF__less__iff,axiom,
! [A: $tType,B: $tType] :
( ( comple5582772986160207858norder @ A )
=> ! [F2: B > A,A3: set @ B,A2: A] :
( ( ord_less @ A @ ( complete_Inf_Inf @ A @ ( image @ B @ A @ F2 @ A3 ) ) @ A2 )
= ( ? [X2: B] :
( ( member @ B @ X2 @ A3 )
& ( ord_less @ A @ ( F2 @ X2 ) @ A2 ) ) ) ) ) ).
% INF_less_iff
thf(fact_7904_less__INF__D,axiom,
! [A: $tType,B: $tType] :
( ( comple6319245703460814977attice @ A )
=> ! [Y: A,F2: B > A,A3: set @ B,I: B] :
( ( ord_less @ A @ Y @ ( complete_Inf_Inf @ A @ ( image @ B @ A @ F2 @ A3 ) ) )
=> ( ( member @ B @ I @ A3 )
=> ( ord_less @ A @ Y @ ( F2 @ I ) ) ) ) ) ).
% less_INF_D
thf(fact_7905_INF__eq,axiom,
! [B: $tType,A: $tType,C: $tType] :
( ( comple6319245703460814977attice @ A )
=> ! [A3: set @ B,B6: set @ C,G: C > A,F2: B > A] :
( ! [I2: B] :
( ( member @ B @ I2 @ A3 )
=> ? [X5: C] :
( ( member @ C @ X5 @ B6 )
& ( ord_less_eq @ A @ ( G @ X5 ) @ ( F2 @ I2 ) ) ) )
=> ( ! [J2: C] :
( ( member @ C @ J2 @ B6 )
=> ? [X5: B] :
( ( member @ B @ X5 @ A3 )
& ( ord_less_eq @ A @ ( F2 @ X5 ) @ ( G @ J2 ) ) ) )
=> ( ( complete_Inf_Inf @ A @ ( image @ B @ A @ F2 @ A3 ) )
= ( complete_Inf_Inf @ A @ ( image @ C @ A @ G @ B6 ) ) ) ) ) ) ).
% INF_eq
thf(fact_7906_INF__eqI,axiom,
! [B: $tType,A: $tType] :
( ( comple6319245703460814977attice @ A )
=> ! [A3: set @ B,X: A,F2: B > A] :
( ! [I2: B] :
( ( member @ B @ I2 @ A3 )
=> ( ord_less_eq @ A @ X @ ( F2 @ I2 ) ) )
=> ( ! [Y3: A] :
( ! [I4: B] :
( ( member @ B @ I4 @ A3 )
=> ( ord_less_eq @ A @ Y3 @ ( F2 @ I4 ) ) )
=> ( ord_less_eq @ A @ Y3 @ X ) )
=> ( ( complete_Inf_Inf @ A @ ( image @ B @ A @ F2 @ A3 ) )
= X ) ) ) ) ).
% INF_eqI
thf(fact_7907_INF__mono,axiom,
! [C: $tType,A: $tType,B: $tType] :
( ( comple6319245703460814977attice @ A )
=> ! [B6: set @ B,A3: set @ C,F2: C > A,G: B > A] :
( ! [M4: B] :
( ( member @ B @ M4 @ B6 )
=> ? [X5: C] :
( ( member @ C @ X5 @ A3 )
& ( ord_less_eq @ A @ ( F2 @ X5 ) @ ( G @ M4 ) ) ) )
=> ( ord_less_eq @ A @ ( complete_Inf_Inf @ A @ ( image @ C @ A @ F2 @ A3 ) ) @ ( complete_Inf_Inf @ A @ ( image @ B @ A @ G @ B6 ) ) ) ) ) ).
% INF_mono
thf(fact_7908_INF__lower,axiom,
! [A: $tType,B: $tType] :
( ( comple6319245703460814977attice @ A )
=> ! [I: B,A3: set @ B,F2: B > A] :
( ( member @ B @ I @ A3 )
=> ( ord_less_eq @ A @ ( complete_Inf_Inf @ A @ ( image @ B @ A @ F2 @ A3 ) ) @ ( F2 @ I ) ) ) ) ).
% INF_lower
thf(fact_7909_INF__mono_H,axiom,
! [A: $tType,B: $tType] :
( ( comple6319245703460814977attice @ A )
=> ! [F2: B > A,G: B > A,A3: set @ B] :
( ! [X3: B] : ( ord_less_eq @ A @ ( F2 @ X3 ) @ ( G @ X3 ) )
=> ( ord_less_eq @ A @ ( complete_Inf_Inf @ A @ ( image @ B @ A @ F2 @ A3 ) ) @ ( complete_Inf_Inf @ A @ ( image @ B @ A @ G @ A3 ) ) ) ) ) ).
% INF_mono'
thf(fact_7910_INF__lower2,axiom,
! [B: $tType,A: $tType] :
( ( comple6319245703460814977attice @ A )
=> ! [I: B,A3: set @ B,F2: B > A,U: A] :
( ( member @ B @ I @ A3 )
=> ( ( ord_less_eq @ A @ ( F2 @ I ) @ U )
=> ( ord_less_eq @ A @ ( complete_Inf_Inf @ A @ ( image @ B @ A @ F2 @ A3 ) ) @ U ) ) ) ) ).
% INF_lower2
thf(fact_7911_le__INF__iff,axiom,
! [A: $tType,B: $tType] :
( ( comple6319245703460814977attice @ A )
=> ! [U: A,F2: B > A,A3: set @ B] :
( ( ord_less_eq @ A @ U @ ( complete_Inf_Inf @ A @ ( image @ B @ A @ F2 @ A3 ) ) )
= ( ! [X2: B] :
( ( member @ B @ X2 @ A3 )
=> ( ord_less_eq @ A @ U @ ( F2 @ X2 ) ) ) ) ) ) ).
% le_INF_iff
thf(fact_7912_INF__greatest,axiom,
! [A: $tType,B: $tType] :
( ( comple6319245703460814977attice @ A )
=> ! [A3: set @ B,U: A,F2: B > A] :
( ! [I2: B] :
( ( member @ B @ I2 @ A3 )
=> ( ord_less_eq @ A @ U @ ( F2 @ I2 ) ) )
=> ( ord_less_eq @ A @ U @ ( complete_Inf_Inf @ A @ ( image @ B @ A @ F2 @ A3 ) ) ) ) ) ).
% INF_greatest
thf(fact_7913_INT__lower,axiom,
! [B: $tType,A: $tType,A2: A,A3: set @ A,B6: A > ( set @ B )] :
( ( member @ A @ A2 @ A3 )
=> ( ord_less_eq @ ( set @ B ) @ ( complete_Inf_Inf @ ( set @ B ) @ ( image @ A @ ( set @ B ) @ B6 @ A3 ) ) @ ( B6 @ A2 ) ) ) ).
% INT_lower
thf(fact_7914_INT__greatest,axiom,
! [B: $tType,A: $tType,A3: set @ A,C5: set @ B,B6: A > ( set @ B )] :
( ! [X3: A] :
( ( member @ A @ X3 @ A3 )
=> ( ord_less_eq @ ( set @ B ) @ C5 @ ( B6 @ X3 ) ) )
=> ( ord_less_eq @ ( set @ B ) @ C5 @ ( complete_Inf_Inf @ ( set @ B ) @ ( image @ A @ ( set @ B ) @ B6 @ A3 ) ) ) ) ).
% INT_greatest
thf(fact_7915_INT__anti__mono,axiom,
! [B: $tType,A: $tType,A3: set @ A,B6: set @ A,F2: A > ( set @ B ),G: A > ( set @ B )] :
( ( ord_less_eq @ ( set @ A ) @ A3 @ B6 )
=> ( ! [X3: A] :
( ( member @ A @ X3 @ A3 )
=> ( ord_less_eq @ ( set @ B ) @ ( F2 @ X3 ) @ ( G @ X3 ) ) )
=> ( ord_less_eq @ ( set @ B ) @ ( complete_Inf_Inf @ ( set @ B ) @ ( image @ A @ ( set @ B ) @ F2 @ B6 ) ) @ ( complete_Inf_Inf @ ( set @ B ) @ ( image @ A @ ( set @ B ) @ G @ A3 ) ) ) ) ) ).
% INT_anti_mono
thf(fact_7916_INT__subset__iff,axiom,
! [A: $tType,B: $tType,B6: set @ A,A3: B > ( set @ A ),I5: set @ B] :
( ( ord_less_eq @ ( set @ A ) @ B6 @ ( complete_Inf_Inf @ ( set @ A ) @ ( image @ B @ ( set @ A ) @ A3 @ I5 ) ) )
= ( ! [X2: B] :
( ( member @ B @ X2 @ I5 )
=> ( ord_less_eq @ ( set @ A ) @ B6 @ ( A3 @ X2 ) ) ) ) ) ).
% INT_subset_iff
thf(fact_7917_cINF__greatest,axiom,
! [A: $tType,B: $tType] :
( ( condit1219197933456340205attice @ A )
=> ! [A3: set @ B,M: A,F2: B > A] :
( ( A3
!= ( bot_bot @ ( set @ B ) ) )
=> ( ! [X3: B] :
( ( member @ B @ X3 @ A3 )
=> ( ord_less_eq @ A @ M @ ( F2 @ X3 ) ) )
=> ( ord_less_eq @ A @ M @ ( complete_Inf_Inf @ A @ ( image @ B @ A @ F2 @ A3 ) ) ) ) ) ) ).
% cINF_greatest
thf(fact_7918_INF__eq__iff,axiom,
! [A: $tType,B: $tType] :
( ( comple6319245703460814977attice @ A )
=> ! [I5: set @ B,F2: B > A,C2: A] :
( ( I5
!= ( bot_bot @ ( set @ B ) ) )
=> ( ! [I2: B] :
( ( member @ B @ I2 @ I5 )
=> ( ord_less_eq @ A @ ( F2 @ I2 ) @ C2 ) )
=> ( ( ( complete_Inf_Inf @ A @ ( image @ B @ A @ F2 @ I5 ) )
= C2 )
= ( ! [X2: B] :
( ( member @ B @ X2 @ I5 )
=> ( ( F2 @ X2 )
= C2 ) ) ) ) ) ) ) ).
% INF_eq_iff
thf(fact_7919_INF__le__iff,axiom,
! [B: $tType,A: $tType] :
( ( comple5582772986160207858norder @ A )
=> ! [F2: B > A,A3: set @ B,X: A] :
( ( ord_less_eq @ A @ ( complete_Inf_Inf @ A @ ( image @ B @ A @ F2 @ A3 ) ) @ X )
= ( ! [Y2: A] :
( ( ord_less @ A @ X @ Y2 )
=> ? [X2: B] :
( ( member @ B @ X2 @ A3 )
& ( ord_less @ A @ ( F2 @ X2 ) @ Y2 ) ) ) ) ) ) ).
% INF_le_iff
thf(fact_7920_Inf__le__iff,axiom,
! [A: $tType] :
( ( comple5582772986160207858norder @ A )
=> ! [A3: set @ A,X: A] :
( ( ord_less_eq @ A @ ( complete_Inf_Inf @ A @ A3 ) @ X )
= ( ! [Y2: A] :
( ( ord_less @ A @ X @ Y2 )
=> ? [X2: A] :
( ( member @ A @ X2 @ A3 )
& ( ord_less @ A @ X2 @ Y2 ) ) ) ) ) ) ).
% Inf_le_iff
thf(fact_7921_Inf__le__Sup,axiom,
! [A: $tType] :
( ( comple6319245703460814977attice @ A )
=> ! [A3: set @ A] :
( ( A3
!= ( bot_bot @ ( set @ A ) ) )
=> ( ord_less_eq @ A @ ( complete_Inf_Inf @ A @ A3 ) @ ( complete_Sup_Sup @ A @ A3 ) ) ) ) ).
% Inf_le_Sup
thf(fact_7922_Inter__subset,axiom,
! [A: $tType,A3: set @ ( set @ A ),B6: set @ A] :
( ! [X16: set @ A] :
( ( member @ ( set @ A ) @ X16 @ A3 )
=> ( ord_less_eq @ ( set @ A ) @ X16 @ B6 ) )
=> ( ( A3
!= ( bot_bot @ ( set @ ( set @ A ) ) ) )
=> ( ord_less_eq @ ( set @ A ) @ ( complete_Inf_Inf @ ( set @ A ) @ A3 ) @ B6 ) ) ) ).
% Inter_subset
thf(fact_7923_Inf__less__eq,axiom,
! [A: $tType] :
( ( comple6319245703460814977attice @ A )
=> ! [A3: set @ A,U: A] :
( ! [V3: A] :
( ( member @ A @ V3 @ A3 )
=> ( ord_less_eq @ A @ V3 @ U ) )
=> ( ( A3
!= ( bot_bot @ ( set @ A ) ) )
=> ( ord_less_eq @ A @ ( complete_Inf_Inf @ A @ A3 ) @ U ) ) ) ) ).
% Inf_less_eq
thf(fact_7924_cInf__greatest,axiom,
! [A: $tType] :
( ( condit1219197933456340205attice @ A )
=> ! [X8: set @ A,Z: A] :
( ( X8
!= ( bot_bot @ ( set @ A ) ) )
=> ( ! [X3: A] :
( ( member @ A @ X3 @ X8 )
=> ( ord_less_eq @ A @ Z @ X3 ) )
=> ( ord_less_eq @ A @ Z @ ( complete_Inf_Inf @ A @ X8 ) ) ) ) ) ).
% cInf_greatest
thf(fact_7925_cInf__eq__non__empty,axiom,
! [A: $tType] :
( ( condit1219197933456340205attice @ A )
=> ! [X8: set @ A,A2: A] :
( ( X8
!= ( bot_bot @ ( set @ A ) ) )
=> ( ! [X3: A] :
( ( member @ A @ X3 @ X8 )
=> ( ord_less_eq @ A @ A2 @ X3 ) )
=> ( ! [Y3: A] :
( ! [X5: A] :
( ( member @ A @ X5 @ X8 )
=> ( ord_less_eq @ A @ Y3 @ X5 ) )
=> ( ord_less_eq @ A @ Y3 @ A2 ) )
=> ( ( complete_Inf_Inf @ A @ X8 )
= A2 ) ) ) ) ) ).
% cInf_eq_non_empty
thf(fact_7926_Inf__superset__mono,axiom,
! [A: $tType] :
( ( comple6319245703460814977attice @ A )
=> ! [B6: set @ A,A3: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ B6 @ A3 )
=> ( ord_less_eq @ A @ ( complete_Inf_Inf @ A @ A3 ) @ ( complete_Inf_Inf @ A @ B6 ) ) ) ) ).
% Inf_superset_mono
thf(fact_7927_Inf__eqI,axiom,
! [A: $tType] :
( ( comple6319245703460814977attice @ A )
=> ! [A3: set @ A,X: A] :
( ! [I2: A] :
( ( member @ A @ I2 @ A3 )
=> ( ord_less_eq @ A @ X @ I2 ) )
=> ( ! [Y3: A] :
( ! [I4: A] :
( ( member @ A @ I4 @ A3 )
=> ( ord_less_eq @ A @ Y3 @ I4 ) )
=> ( ord_less_eq @ A @ Y3 @ X ) )
=> ( ( complete_Inf_Inf @ A @ A3 )
= X ) ) ) ) ).
% Inf_eqI
thf(fact_7928_Inf__mono,axiom,
! [A: $tType] :
( ( comple6319245703460814977attice @ A )
=> ! [B6: set @ A,A3: set @ A] :
( ! [B3: A] :
( ( member @ A @ B3 @ B6 )
=> ? [X5: A] :
( ( member @ A @ X5 @ A3 )
& ( ord_less_eq @ A @ X5 @ B3 ) ) )
=> ( ord_less_eq @ A @ ( complete_Inf_Inf @ A @ A3 ) @ ( complete_Inf_Inf @ A @ B6 ) ) ) ) ).
% Inf_mono
thf(fact_7929_Inf__lower,axiom,
! [A: $tType] :
( ( comple6319245703460814977attice @ A )
=> ! [X: A,A3: set @ A] :
( ( member @ A @ X @ A3 )
=> ( ord_less_eq @ A @ ( complete_Inf_Inf @ A @ A3 ) @ X ) ) ) ).
% Inf_lower
thf(fact_7930_Inf__lower2,axiom,
! [A: $tType] :
( ( comple6319245703460814977attice @ A )
=> ! [U: A,A3: set @ A,V: A] :
( ( member @ A @ U @ A3 )
=> ( ( ord_less_eq @ A @ U @ V )
=> ( ord_less_eq @ A @ ( complete_Inf_Inf @ A @ A3 ) @ V ) ) ) ) ).
% Inf_lower2
thf(fact_7931_le__Inf__iff,axiom,
! [A: $tType] :
( ( comple6319245703460814977attice @ A )
=> ! [B2: A,A3: set @ A] :
( ( ord_less_eq @ A @ B2 @ ( complete_Inf_Inf @ A @ A3 ) )
= ( ! [X2: A] :
( ( member @ A @ X2 @ A3 )
=> ( ord_less_eq @ A @ B2 @ X2 ) ) ) ) ) ).
% le_Inf_iff
thf(fact_7932_Inf__greatest,axiom,
! [A: $tType] :
( ( comple6319245703460814977attice @ A )
=> ! [A3: set @ A,Z: A] :
( ! [X3: A] :
( ( member @ A @ X3 @ A3 )
=> ( ord_less_eq @ A @ Z @ X3 ) )
=> ( ord_less_eq @ A @ Z @ ( complete_Inf_Inf @ A @ A3 ) ) ) ) ).
% Inf_greatest
thf(fact_7933_cInf__eq__minimum,axiom,
! [A: $tType] :
( ( condit1219197933456340205attice @ A )
=> ! [Z: A,X8: set @ A] :
( ( member @ A @ Z @ X8 )
=> ( ! [X3: A] :
( ( member @ A @ X3 @ X8 )
=> ( ord_less_eq @ A @ Z @ X3 ) )
=> ( ( complete_Inf_Inf @ A @ X8 )
= Z ) ) ) ) ).
% cInf_eq_minimum
thf(fact_7934_cInf__eq,axiom,
! [A: $tType] :
( ( ( condit1219197933456340205attice @ A )
& ( no_top @ A ) )
=> ! [X8: set @ A,A2: A] :
( ! [X3: A] :
( ( member @ A @ X3 @ X8 )
=> ( ord_less_eq @ A @ A2 @ X3 ) )
=> ( ! [Y3: A] :
( ! [X5: A] :
( ( member @ A @ X5 @ X8 )
=> ( ord_less_eq @ A @ Y3 @ X5 ) )
=> ( ord_less_eq @ A @ Y3 @ A2 ) )
=> ( ( complete_Inf_Inf @ A @ X8 )
= A2 ) ) ) ) ).
% cInf_eq
thf(fact_7935_Inter__lower,axiom,
! [A: $tType,B6: set @ A,A3: set @ ( set @ A )] :
( ( member @ ( set @ A ) @ B6 @ A3 )
=> ( ord_less_eq @ ( set @ A ) @ ( complete_Inf_Inf @ ( set @ A ) @ A3 ) @ B6 ) ) ).
% Inter_lower
thf(fact_7936_Inter__greatest,axiom,
! [A: $tType,A3: set @ ( set @ A ),C5: set @ A] :
( ! [X16: set @ A] :
( ( member @ ( set @ A ) @ X16 @ A3 )
=> ( ord_less_eq @ ( set @ A ) @ C5 @ X16 ) )
=> ( ord_less_eq @ ( set @ A ) @ C5 @ ( complete_Inf_Inf @ ( set @ A ) @ A3 ) ) ) ).
% Inter_greatest
thf(fact_7937_Inter__anti__mono,axiom,
! [A: $tType,B6: set @ ( set @ A ),A3: set @ ( set @ A )] :
( ( ord_less_eq @ ( set @ ( set @ A ) ) @ B6 @ A3 )
=> ( ord_less_eq @ ( set @ A ) @ ( complete_Inf_Inf @ ( set @ A ) @ A3 ) @ ( complete_Inf_Inf @ ( set @ A ) @ B6 ) ) ) ).
% Inter_anti_mono
thf(fact_7938_cInf__lessD,axiom,
! [A: $tType] :
( ( condit6923001295902523014norder @ A )
=> ! [X8: set @ A,Z: A] :
( ( X8
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( ord_less @ A @ ( complete_Inf_Inf @ A @ X8 ) @ Z )
=> ? [X3: A] :
( ( member @ A @ X3 @ X8 )
& ( ord_less @ A @ X3 @ Z ) ) ) ) ) ).
% cInf_lessD
thf(fact_7939_Inf__notin__open,axiom,
! [A: $tType] :
( ( topolo8458572112393995274pology @ A )
=> ! [A3: set @ A,X: A] :
( ( topolo1002775350975398744n_open @ A @ A3 )
=> ( ! [X3: A] :
( ( member @ A @ X3 @ A3 )
=> ( ord_less @ A @ X @ X3 ) )
=> ~ ( member @ A @ ( complete_Inf_Inf @ A @ A3 ) @ A3 ) ) ) ) ).
% Inf_notin_open
thf(fact_7940_Inf__less__iff,axiom,
! [A: $tType] :
( ( comple5582772986160207858norder @ A )
=> ! [S3: set @ A,A2: A] :
( ( ord_less @ A @ ( complete_Inf_Inf @ A @ S3 ) @ A2 )
= ( ? [X2: A] :
( ( member @ A @ X2 @ S3 )
& ( ord_less @ A @ X2 @ A2 ) ) ) ) ) ).
% Inf_less_iff
thf(fact_7941_Inf__option__def,axiom,
! [A: $tType] :
( ( comple6319245703460814977attice @ A )
=> ( ( complete_Inf_Inf @ ( option @ A ) )
= ( ^ [A7: set @ ( option @ A )] : ( if @ ( option @ A ) @ ( member @ ( option @ A ) @ ( none @ A ) @ A7 ) @ ( none @ A ) @ ( some @ A @ ( complete_Inf_Inf @ A @ ( these @ A @ A7 ) ) ) ) ) ) ) ).
% Inf_option_def
thf(fact_7942_cInf__abs__ge,axiom,
! [A: $tType] :
( ( ( condit6923001295902523014norder @ A )
& ( linordered_idom @ A ) )
=> ! [S3: set @ A,A2: A] :
( ( S3
!= ( bot_bot @ ( set @ A ) ) )
=> ( ! [X3: A] :
( ( member @ A @ X3 @ S3 )
=> ( ord_less_eq @ A @ ( abs_abs @ A @ X3 ) @ A2 ) )
=> ( ord_less_eq @ A @ ( abs_abs @ A @ ( complete_Inf_Inf @ A @ S3 ) ) @ A2 ) ) ) ) ).
% cInf_abs_ge
thf(fact_7943_INF__superset__mono,axiom,
! [A: $tType,B: $tType] :
( ( comple6319245703460814977attice @ A )
=> ! [B6: set @ B,A3: set @ B,F2: B > A,G: B > A] :
( ( ord_less_eq @ ( set @ B ) @ B6 @ A3 )
=> ( ! [X3: B] :
( ( member @ B @ X3 @ B6 )
=> ( ord_less_eq @ A @ ( F2 @ X3 ) @ ( G @ X3 ) ) )
=> ( ord_less_eq @ A @ ( complete_Inf_Inf @ A @ ( image @ B @ A @ F2 @ A3 ) ) @ ( complete_Inf_Inf @ A @ ( image @ B @ A @ G @ B6 ) ) ) ) ) ) ).
% INF_superset_mono
thf(fact_7944_UN__extend__simps_I7_J,axiom,
! [M11: $tType,N13: $tType,A3: set @ M11,B6: N13 > ( set @ M11 ),C5: set @ N13] :
( ( minus_minus @ ( set @ M11 ) @ A3 @ ( complete_Inf_Inf @ ( set @ M11 ) @ ( image @ N13 @ ( set @ M11 ) @ B6 @ C5 ) ) )
= ( complete_Sup_Sup @ ( set @ M11 )
@ ( image @ N13 @ ( set @ M11 )
@ ^ [X2: N13] : ( minus_minus @ ( set @ M11 ) @ A3 @ ( B6 @ X2 ) )
@ C5 ) ) ) ).
% UN_extend_simps(7)
thf(fact_7945_INF__le__SUP,axiom,
! [A: $tType,B: $tType] :
( ( comple6319245703460814977attice @ A )
=> ! [A3: set @ B,F2: B > A] :
( ( A3
!= ( bot_bot @ ( set @ B ) ) )
=> ( ord_less_eq @ A @ ( complete_Inf_Inf @ A @ ( image @ B @ A @ F2 @ A3 ) ) @ ( complete_Sup_Sup @ A @ ( image @ B @ A @ F2 @ A3 ) ) ) ) ) ).
% INF_le_SUP
thf(fact_7946_cInf__asclose,axiom,
! [A: $tType] :
( ( ( condit6923001295902523014norder @ A )
& ( linordered_idom @ A ) )
=> ! [S3: set @ A,L2: A,E: A] :
( ( S3
!= ( bot_bot @ ( set @ A ) ) )
=> ( ! [X3: A] :
( ( member @ A @ X3 @ S3 )
=> ( ord_less_eq @ A @ ( abs_abs @ A @ ( minus_minus @ A @ X3 @ L2 ) ) @ E ) )
=> ( ord_less_eq @ A @ ( abs_abs @ A @ ( minus_minus @ A @ ( complete_Inf_Inf @ A @ S3 ) @ L2 ) ) @ E ) ) ) ) ).
% cInf_asclose
thf(fact_7947_Inf__lim,axiom,
! [A: $tType] :
( ( ( comple5582772986160207858norder @ A )
& ( topolo1944317154257567458pology @ A ) )
=> ! [B2: nat > A,S: set @ A,A2: A] :
( ! [N: nat] : ( member @ A @ ( B2 @ N ) @ S )
=> ( ( filterlim @ nat @ A @ B2 @ ( topolo7230453075368039082e_nhds @ A @ A2 ) @ ( at_top @ nat ) )
=> ( ord_less_eq @ A @ ( complete_Inf_Inf @ A @ S ) @ A2 ) ) ) ) ).
% Inf_lim
thf(fact_7948_INT__extend__simps_I3_J,axiom,
! [F: $tType,E3: $tType,C5: set @ E3,A3: E3 > ( set @ F ),B6: set @ F] :
( ( ( C5
= ( bot_bot @ ( set @ E3 ) ) )
=> ( ( minus_minus @ ( set @ F ) @ ( complete_Inf_Inf @ ( set @ F ) @ ( image @ E3 @ ( set @ F ) @ A3 @ C5 ) ) @ B6 )
= ( minus_minus @ ( set @ F ) @ ( top_top @ ( set @ F ) ) @ B6 ) ) )
& ( ( C5
!= ( bot_bot @ ( set @ E3 ) ) )
=> ( ( minus_minus @ ( set @ F ) @ ( complete_Inf_Inf @ ( set @ F ) @ ( image @ E3 @ ( set @ F ) @ A3 @ C5 ) ) @ B6 )
= ( complete_Inf_Inf @ ( set @ F )
@ ( image @ E3 @ ( set @ F )
@ ^ [X2: E3] : ( minus_minus @ ( set @ F ) @ ( A3 @ X2 ) @ B6 )
@ C5 ) ) ) ) ) ).
% INT_extend_simps(3)
thf(fact_7949_image__INT,axiom,
! [B: $tType,A: $tType,C: $tType,F2: A > B,C5: set @ A,A3: set @ C,B6: C > ( set @ A ),J: C] :
( ( inj_on @ A @ B @ F2 @ C5 )
=> ( ! [X3: C] :
( ( member @ C @ X3 @ A3 )
=> ( ord_less_eq @ ( set @ A ) @ ( B6 @ X3 ) @ C5 ) )
=> ( ( member @ C @ J @ A3 )
=> ( ( image @ A @ B @ F2 @ ( complete_Inf_Inf @ ( set @ A ) @ ( image @ C @ ( set @ A ) @ B6 @ A3 ) ) )
= ( complete_Inf_Inf @ ( set @ B )
@ ( image @ C @ ( set @ B )
@ ^ [X2: C] : ( image @ A @ B @ F2 @ ( B6 @ X2 ) )
@ A3 ) ) ) ) ) ) ).
% image_INT
thf(fact_7950_INT__extend__simps_I4_J,axiom,
! [G2: $tType,H5: $tType,C5: set @ H5,A3: set @ G2,B6: H5 > ( set @ G2 )] :
( ( ( C5
= ( bot_bot @ ( set @ H5 ) ) )
=> ( ( minus_minus @ ( set @ G2 ) @ A3 @ ( complete_Sup_Sup @ ( set @ G2 ) @ ( image @ H5 @ ( set @ G2 ) @ B6 @ C5 ) ) )
= A3 ) )
& ( ( C5
!= ( bot_bot @ ( set @ H5 ) ) )
=> ( ( minus_minus @ ( set @ G2 ) @ A3 @ ( complete_Sup_Sup @ ( set @ G2 ) @ ( image @ H5 @ ( set @ G2 ) @ B6 @ C5 ) ) )
= ( complete_Inf_Inf @ ( set @ G2 )
@ ( image @ H5 @ ( set @ G2 )
@ ^ [X2: H5] : ( minus_minus @ ( set @ G2 ) @ A3 @ ( B6 @ X2 ) )
@ C5 ) ) ) ) ) ).
% INT_extend_simps(4)
thf(fact_7951_Abs__bit1__cases,axiom,
! [A: $tType] :
( ( finite_finite @ A )
=> ! [X: numeral_bit1 @ A] :
~ ! [Y3: int] :
( ( X
= ( numeral_Abs_bit12 @ A @ Y3 ) )
=> ~ ( member @ int @ Y3 @ ( 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 ) ) ) ) ) ) ) ) ) ) ).
% Abs_bit1_cases
thf(fact_7952_Abs__bit1__induct,axiom,
! [A: $tType] :
( ( finite_finite @ A )
=> ! [P: ( numeral_bit1 @ A ) > $o,X: numeral_bit1 @ A] :
( ! [Y3: int] :
( ( member @ int @ Y3 @ ( 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 ) ) ) ) ) ) ) )
=> ( P @ ( numeral_Abs_bit12 @ A @ Y3 ) ) )
=> ( P @ X ) ) ) ).
% Abs_bit1_induct
thf(fact_7953_Abs__bit1__inject,axiom,
! [A: $tType] :
( ( finite_finite @ A )
=> ! [X: int,Y: int] :
( ( member @ int @ X @ ( 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 ) ) ) ) ) ) ) )
=> ( ( member @ int @ Y @ ( 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 ) ) ) ) ) ) ) )
=> ( ( ( numeral_Abs_bit12 @ A @ X )
= ( numeral_Abs_bit12 @ A @ Y ) )
= ( X = Y ) ) ) ) ) ).
% Abs_bit1_inject
thf(fact_7954_Abs__bit1__inverse,axiom,
! [A: $tType] :
( ( finite_finite @ A )
=> ! [Y: int] :
( ( member @ int @ Y @ ( 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 ) ) ) ) ) ) ) )
=> ( ( numeral_Rep_bit1 @ A @ ( numeral_Abs_bit12 @ A @ Y ) )
= Y ) ) ) ).
% Abs_bit1_inverse
thf(fact_7955_Rep__bit1,axiom,
! [A: $tType] :
( ( finite_finite @ A )
=> ! [X: numeral_bit1 @ A] : ( member @ int @ ( numeral_Rep_bit1 @ A @ X ) @ ( 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 ) ) ) ) ) ) ) ) ) ).
% Rep_bit1
thf(fact_7956_None__in__Inf,axiom,
! [A: $tType] :
( ( comple6319245703460814977attice @ A )
=> ! [A3: set @ ( option @ A )] :
( ( member @ ( option @ A ) @ ( none @ A ) @ A3 )
=> ( ( complete_Inf_Inf @ ( option @ A ) @ A3 )
= ( none @ A ) ) ) ) ).
% None_in_Inf
thf(fact_7957_bit1_ORep__Abs__1,axiom,
! [A: $tType] :
( ( finite_finite @ A )
=> ( ( numeral_Rep_bit1 @ A @ ( numeral_Abs_bit12 @ A @ ( one_one @ int ) ) )
= ( one_one @ int ) ) ) ).
% bit1.Rep_Abs_1
thf(fact_7958_eventually__Inf,axiom,
! [A: $tType,P: A > $o,B6: set @ ( filter @ A )] :
( ( eventually @ A @ P @ ( complete_Inf_Inf @ ( filter @ A ) @ B6 ) )
= ( ? [X6: set @ ( filter @ A )] :
( ( ord_less_eq @ ( set @ ( filter @ A ) ) @ X6 @ B6 )
& ( finite_finite2 @ ( filter @ A ) @ X6 )
& ( eventually @ A @ P @ ( complete_Inf_Inf @ ( filter @ A ) @ X6 ) ) ) ) ) ).
% eventually_Inf
thf(fact_7959_Inf__filter__not__bot,axiom,
! [A: $tType,B6: set @ ( filter @ A )] :
( ! [X16: set @ ( filter @ A )] :
( ( ord_less_eq @ ( set @ ( filter @ A ) ) @ X16 @ B6 )
=> ( ( finite_finite2 @ ( filter @ A ) @ X16 )
=> ( ( complete_Inf_Inf @ ( filter @ A ) @ X16 )
!= ( bot_bot @ ( filter @ A ) ) ) ) )
=> ( ( complete_Inf_Inf @ ( filter @ A ) @ B6 )
!= ( bot_bot @ ( filter @ A ) ) ) ) ).
% Inf_filter_not_bot
thf(fact_7960_eventually__INF,axiom,
! [A: $tType,B: $tType,P: A > $o,F4: B > ( filter @ A ),B6: set @ B] :
( ( eventually @ A @ P @ ( complete_Inf_Inf @ ( filter @ A ) @ ( image @ B @ ( filter @ A ) @ F4 @ B6 ) ) )
= ( ? [X6: set @ B] :
( ( ord_less_eq @ ( set @ B ) @ X6 @ B6 )
& ( finite_finite2 @ B @ X6 )
& ( eventually @ A @ P @ ( complete_Inf_Inf @ ( filter @ A ) @ ( image @ B @ ( filter @ A ) @ F4 @ X6 ) ) ) ) ) ) ).
% eventually_INF
thf(fact_7961_INF__Int__eq2,axiom,
! [B: $tType,A: $tType,S3: set @ ( set @ ( product_prod @ A @ B ) )] :
( ( complete_Inf_Inf @ ( A > B > $o )
@ ( image @ ( set @ ( product_prod @ A @ B ) ) @ ( A > B > $o )
@ ^ [I3: set @ ( product_prod @ A @ B ),X2: A,Y2: B] : ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X2 @ Y2 ) @ I3 )
@ S3 ) )
= ( ^ [X2: A,Y2: B] : ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X2 @ Y2 ) @ ( complete_Inf_Inf @ ( set @ ( product_prod @ A @ B ) ) @ S3 ) ) ) ) ).
% INF_Int_eq2
thf(fact_7962_less__eq__bit1__def,axiom,
! [A: $tType] :
( ( finite_finite @ A )
=> ( ( ord_less_eq @ ( numeral_bit1 @ A ) )
= ( ^ [A6: numeral_bit1 @ A,B5: numeral_bit1 @ A] : ( ord_less_eq @ int @ ( numeral_Rep_bit1 @ A @ A6 ) @ ( numeral_Rep_bit1 @ A @ B5 ) ) ) ) ) ).
% less_eq_bit1_def
thf(fact_7963_bit1_ORep__1,axiom,
! [A: $tType] :
( ( finite_finite @ A )
=> ( ( numeral_Rep_bit1 @ A @ ( one_one @ ( numeral_bit1 @ A ) ) )
= ( one_one @ int ) ) ) ).
% bit1.Rep_1
thf(fact_7964_INF__filter__not__bot,axiom,
! [I6: $tType,A: $tType,B6: set @ I6,F4: I6 > ( filter @ A )] :
( ! [X16: set @ I6] :
( ( ord_less_eq @ ( set @ I6 ) @ X16 @ B6 )
=> ( ( finite_finite2 @ I6 @ X16 )
=> ( ( complete_Inf_Inf @ ( filter @ A ) @ ( image @ I6 @ ( filter @ A ) @ F4 @ X16 ) )
!= ( bot_bot @ ( filter @ A ) ) ) ) )
=> ( ( complete_Inf_Inf @ ( filter @ A ) @ ( image @ I6 @ ( filter @ A ) @ F4 @ B6 ) )
!= ( bot_bot @ ( filter @ A ) ) ) ) ).
% INF_filter_not_bot
thf(fact_7965_INF__INT__eq2,axiom,
! [B: $tType,C: $tType,A: $tType,R2: C > ( set @ ( product_prod @ A @ B ) ),S3: set @ C] :
( ( complete_Inf_Inf @ ( A > B > $o )
@ ( image @ C @ ( A > B > $o )
@ ^ [I3: C,X2: A,Y2: B] : ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X2 @ Y2 ) @ ( R2 @ I3 ) )
@ S3 ) )
= ( ^ [X2: A,Y2: B] : ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X2 @ Y2 ) @ ( complete_Inf_Inf @ ( set @ ( product_prod @ A @ B ) ) @ ( image @ C @ ( set @ ( product_prod @ A @ B ) ) @ R2 @ S3 ) ) ) ) ) ).
% INF_INT_eq2
thf(fact_7966_Inf__real__def,axiom,
( ( complete_Inf_Inf @ real )
= ( ^ [X6: set @ real] : ( uminus_uminus @ real @ ( complete_Sup_Sup @ real @ ( image @ real @ real @ ( uminus_uminus @ real ) @ X6 ) ) ) ) ) ).
% Inf_real_def
thf(fact_7967_bit1_ORep__le__n,axiom,
! [A: $tType] :
( ( finite_finite @ A )
=> ! [X: numeral_bit1 @ A] : ( ord_less_eq @ int @ ( numeral_Rep_bit1 @ A @ X ) @ ( semiring_1_of_nat @ int @ ( finite_card @ ( numeral_bit1 @ A ) @ ( top_top @ ( set @ ( numeral_bit1 @ A ) ) ) ) ) ) ) ).
% bit1.Rep_le_n
thf(fact_7968_Inf__INT__eq2,axiom,
! [B: $tType,A: $tType] :
( ( complete_Inf_Inf @ ( A > B > $o ) )
= ( ^ [S8: set @ ( A > B > $o ),X2: A,Y2: B] : ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X2 @ Y2 ) @ ( complete_Inf_Inf @ ( set @ ( product_prod @ A @ B ) ) @ ( image @ ( ( product_prod @ A @ B ) > $o ) @ ( set @ ( product_prod @ A @ B ) ) @ ( collect @ ( product_prod @ A @ B ) ) @ ( image @ ( A > B > $o ) @ ( ( product_prod @ A @ B ) > $o ) @ ( product_case_prod @ A @ B @ $o ) @ S8 ) ) ) ) ) ) ).
% Inf_INT_eq2
thf(fact_7969_bit1_Odiff__def,axiom,
! [A: $tType] :
( ( finite_finite @ A )
=> ( ( minus_minus @ ( numeral_bit1 @ A ) )
= ( ^ [X2: numeral_bit1 @ A,Y2: numeral_bit1 @ A] : ( numeral_Abs_bit12 @ A @ ( modulo_modulo @ int @ ( minus_minus @ int @ ( numeral_Rep_bit1 @ A @ X2 ) @ ( numeral_Rep_bit1 @ A @ Y2 ) ) @ ( semiring_1_of_nat @ int @ ( finite_card @ ( numeral_bit1 @ A ) @ ( top_top @ ( set @ ( numeral_bit1 @ A ) ) ) ) ) ) ) ) ) ) ).
% bit1.diff_def
thf(fact_7970_Rep__bit1__induct,axiom,
! [A: $tType] :
( ( finite_finite @ A )
=> ! [Y: int,P: int > $o] :
( ( member @ int @ Y @ ( 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 ) ) ) ) ) ) ) )
=> ( ! [X3: numeral_bit1 @ A] : ( P @ ( numeral_Rep_bit1 @ A @ X3 ) )
=> ( P @ Y ) ) ) ) ).
% Rep_bit1_induct
thf(fact_7971_Rep__bit1__cases,axiom,
! [A: $tType] :
( ( finite_finite @ A )
=> ! [Y: int] :
( ( member @ int @ Y @ ( 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 ) ) ) ) ) ) ) )
=> ~ ! [X3: numeral_bit1 @ A] :
( Y
!= ( numeral_Rep_bit1 @ A @ X3 ) ) ) ) ).
% Rep_bit1_cases
thf(fact_7972_type__definition__bit1,axiom,
! [A: $tType] :
( ( finite_finite @ A )
=> ( type_definition @ ( numeral_bit1 @ A ) @ int @ ( numeral_Rep_bit1 @ A ) @ ( numeral_Abs_bit12 @ 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 ) ) ) ) ) ) ) ) ) ).
% type_definition_bit1
thf(fact_7973_card__lists__distinct__length__eq,axiom,
! [A: $tType,A3: set @ A,K: nat] :
( ( finite_finite2 @ A @ A3 )
=> ( ( ord_less_eq @ nat @ K @ ( finite_card @ A @ A3 ) )
=> ( ( finite_card @ ( list @ A )
@ ( collect @ ( list @ A )
@ ^ [Xs: list @ A] :
( ( ( size_size @ ( list @ A ) @ Xs )
= K )
& ( distinct @ A @ Xs )
& ( ord_less_eq @ ( set @ A ) @ ( set2 @ A @ Xs ) @ A3 ) ) ) )
= ( groups7121269368397514597t_prod @ nat @ nat
@ ^ [X2: nat] : X2
@ ( set_or1337092689740270186AtMost @ nat @ ( plus_plus @ nat @ ( minus_minus @ nat @ ( finite_card @ A @ A3 ) @ K ) @ ( one_one @ nat ) ) @ ( finite_card @ A @ A3 ) ) ) ) ) ) ).
% card_lists_distinct_length_eq
thf(fact_7974_distinct__swap,axiom,
! [A: $tType,I: nat,Xs2: list @ A,J: nat] :
( ( ord_less @ nat @ I @ ( size_size @ ( list @ A ) @ Xs2 ) )
=> ( ( ord_less @ nat @ J @ ( size_size @ ( list @ A ) @ Xs2 ) )
=> ( ( distinct @ A @ ( list_update @ A @ ( list_update @ A @ Xs2 @ I @ ( nth @ A @ Xs2 @ J ) ) @ J @ ( nth @ A @ Xs2 @ I ) ) )
= ( distinct @ A @ Xs2 ) ) ) ) ).
% distinct_swap
thf(fact_7975_finite__lists__distinct__length__eq,axiom,
! [A: $tType,A3: set @ A,N3: nat] :
( ( finite_finite2 @ A @ A3 )
=> ( finite_finite2 @ ( list @ A )
@ ( collect @ ( list @ A )
@ ^ [Xs: list @ A] :
( ( ( size_size @ ( list @ A ) @ Xs )
= N3 )
& ( distinct @ A @ Xs )
& ( ord_less_eq @ ( set @ A ) @ ( set2 @ A @ Xs ) @ A3 ) ) ) ) ) ).
% finite_lists_distinct_length_eq
thf(fact_7976_distinct__finite__set,axiom,
! [A: $tType,X: set @ A] :
( finite_finite2 @ ( list @ A )
@ ( collect @ ( list @ A )
@ ^ [Ys3: list @ A] :
( ( ( set2 @ A @ Ys3 )
= X )
& ( distinct @ A @ Ys3 ) ) ) ) ).
% distinct_finite_set
thf(fact_7977_finite__distinct__list,axiom,
! [A: $tType,A3: set @ A] :
( ( finite_finite2 @ A @ A3 )
=> ? [Xs3: list @ A] :
( ( ( set2 @ A @ Xs3 )
= A3 )
& ( distinct @ A @ Xs3 ) ) ) ).
% finite_distinct_list
thf(fact_7978_distinct__length__le,axiom,
! [A: $tType,Ys: list @ A,Xs2: list @ A] :
( ( distinct @ A @ Ys )
=> ( ( ( set2 @ A @ Ys )
= ( set2 @ A @ Xs2 ) )
=> ( ord_less_eq @ nat @ ( size_size @ ( list @ A ) @ Ys ) @ ( size_size @ ( list @ A ) @ Xs2 ) ) ) ) ).
% distinct_length_le
thf(fact_7979_nth__eq__iff__index__eq,axiom,
! [A: $tType,Xs2: list @ A,I: nat,J: nat] :
( ( distinct @ A @ Xs2 )
=> ( ( ord_less @ nat @ I @ ( size_size @ ( list @ A ) @ Xs2 ) )
=> ( ( ord_less @ nat @ J @ ( size_size @ ( list @ A ) @ Xs2 ) )
=> ( ( ( nth @ A @ Xs2 @ I )
= ( nth @ A @ Xs2 @ J ) )
= ( I = J ) ) ) ) ) ).
% nth_eq_iff_index_eq
thf(fact_7980_distinct__conv__nth,axiom,
! [A: $tType] :
( ( distinct @ A )
= ( ^ [Xs: list @ A] :
! [I3: nat] :
( ( ord_less @ nat @ I3 @ ( size_size @ ( list @ A ) @ Xs ) )
=> ! [J3: nat] :
( ( ord_less @ nat @ J3 @ ( size_size @ ( list @ A ) @ Xs ) )
=> ( ( I3 != J3 )
=> ( ( nth @ A @ Xs @ I3 )
!= ( nth @ A @ Xs @ J3 ) ) ) ) ) ) ) ).
% distinct_conv_nth
thf(fact_7981_distinct__set__subseqs,axiom,
! [A: $tType,Xs2: list @ A] :
( ( distinct @ A @ Xs2 )
=> ( distinct @ ( set @ A ) @ ( map @ ( list @ A ) @ ( set @ A ) @ ( set2 @ A ) @ ( subseqs @ A @ Xs2 ) ) ) ) ).
% distinct_set_subseqs
thf(fact_7982_distinct__map__eq,axiom,
! [A: $tType,B: $tType,F2: B > A,L2: list @ B,X: B,Y: B] :
( ( distinct @ A @ ( map @ B @ A @ F2 @ L2 ) )
=> ( ( ( F2 @ X )
= ( F2 @ Y ) )
=> ( ( member @ B @ X @ ( set2 @ B @ L2 ) )
=> ( ( member @ B @ Y @ ( set2 @ B @ L2 ) )
=> ( X = Y ) ) ) ) ) ).
% distinct_map_eq
thf(fact_7983_distinct_Osimps_I2_J,axiom,
! [A: $tType,X: A,Xs2: list @ A] :
( ( distinct @ A @ ( cons @ A @ X @ Xs2 ) )
= ( ~ ( member @ A @ X @ ( set2 @ A @ Xs2 ) )
& ( distinct @ A @ Xs2 ) ) ) ).
% distinct.simps(2)
thf(fact_7984_subseqs__distinctD,axiom,
! [A: $tType,Ys: list @ A,Xs2: list @ A] :
( ( member @ ( list @ A ) @ Ys @ ( set2 @ ( list @ A ) @ ( subseqs @ A @ Xs2 ) ) )
=> ( ( distinct @ A @ Xs2 )
=> ( distinct @ A @ Ys ) ) ) ).
% subseqs_distinctD
thf(fact_7985_distinct__product__lists,axiom,
! [A: $tType,Xss: list @ ( list @ A )] :
( ! [X3: list @ A] :
( ( member @ ( list @ A ) @ X3 @ ( set2 @ ( list @ A ) @ Xss ) )
=> ( distinct @ A @ X3 ) )
=> ( distinct @ ( list @ A ) @ ( product_lists @ A @ Xss ) ) ) ).
% distinct_product_lists
thf(fact_7986_distinct__mapI,axiom,
! [A: $tType,B: $tType,F2: B > A,L2: list @ B] :
( ( distinct @ A @ ( map @ B @ A @ F2 @ L2 ) )
=> ( distinct @ B @ L2 ) ) ).
% distinct_mapI
thf(fact_7987_distinct__map,axiom,
! [A: $tType,B: $tType,F2: B > A,Xs2: list @ B] :
( ( distinct @ A @ ( map @ B @ A @ F2 @ Xs2 ) )
= ( ( distinct @ B @ Xs2 )
& ( inj_on @ B @ A @ F2 @ ( set2 @ B @ Xs2 ) ) ) ) ).
% distinct_map
thf(fact_7988_distinct__card,axiom,
! [A: $tType,Xs2: list @ A] :
( ( distinct @ A @ Xs2 )
=> ( ( finite_card @ A @ ( set2 @ A @ Xs2 ) )
= ( size_size @ ( list @ A ) @ Xs2 ) ) ) ).
% distinct_card
thf(fact_7989_card__distinct,axiom,
! [A: $tType,Xs2: list @ A] :
( ( ( finite_card @ A @ ( set2 @ A @ Xs2 ) )
= ( size_size @ ( list @ A ) @ Xs2 ) )
=> ( distinct @ A @ Xs2 ) ) ).
% card_distinct
thf(fact_7990_distinct__Ex1,axiom,
! [A: $tType,Xs2: list @ A,X: A] :
( ( distinct @ A @ Xs2 )
=> ( ( member @ A @ X @ ( set2 @ A @ Xs2 ) )
=> ? [X3: nat] :
( ( ord_less @ nat @ X3 @ ( size_size @ ( list @ A ) @ Xs2 ) )
& ( ( nth @ A @ Xs2 @ X3 )
= X )
& ! [Y4: nat] :
( ( ( ord_less @ nat @ Y4 @ ( size_size @ ( list @ A ) @ Xs2 ) )
& ( ( nth @ A @ Xs2 @ Y4 )
= X ) )
=> ( Y4 = X3 ) ) ) ) ) ).
% distinct_Ex1
thf(fact_7991_distinct__idx,axiom,
! [B: $tType,A: $tType,F2: B > A,L2: list @ B,I: nat,J: nat] :
( ( distinct @ A @ ( map @ B @ A @ F2 @ L2 ) )
=> ( ( ord_less @ nat @ I @ ( size_size @ ( list @ B ) @ L2 ) )
=> ( ( ord_less @ nat @ J @ ( size_size @ ( list @ B ) @ L2 ) )
=> ( ( ( F2 @ ( nth @ B @ L2 @ I ) )
= ( F2 @ ( nth @ B @ L2 @ J ) ) )
=> ( I = J ) ) ) ) ) ).
% distinct_idx
thf(fact_7992_inj__on__nth,axiom,
! [A: $tType,Xs2: list @ A,I5: set @ nat] :
( ( distinct @ A @ Xs2 )
=> ( ! [X3: nat] :
( ( member @ nat @ X3 @ I5 )
=> ( ord_less @ nat @ X3 @ ( size_size @ ( list @ A ) @ Xs2 ) ) )
=> ( inj_on @ nat @ A @ ( nth @ A @ Xs2 ) @ I5 ) ) ) ).
% inj_on_nth
thf(fact_7993_distinct__finite__subset,axiom,
! [A: $tType,X: set @ A] :
( ( finite_finite2 @ A @ X )
=> ( finite_finite2 @ ( list @ A )
@ ( collect @ ( list @ A )
@ ^ [Ys3: list @ A] :
( ( ord_less_eq @ ( set @ A ) @ ( set2 @ A @ Ys3 ) @ X )
& ( distinct @ A @ Ys3 ) ) ) ) ) ).
% distinct_finite_subset
thf(fact_7994_finite__set__image,axiom,
! [A: $tType,A3: set @ ( list @ A )] :
( ( finite_finite2 @ ( set @ A ) @ ( image @ ( list @ A ) @ ( set @ A ) @ ( set2 @ A ) @ A3 ) )
=> ( ! [Xs3: list @ A] :
( ( member @ ( list @ A ) @ Xs3 @ A3 )
=> ( distinct @ A @ Xs3 ) )
=> ( finite_finite2 @ ( list @ A ) @ A3 ) ) ) ).
% finite_set_image
thf(fact_7995_bij__betw__nth,axiom,
! [A: $tType,Xs2: list @ A,A3: set @ nat,B6: set @ A] :
( ( distinct @ A @ Xs2 )
=> ( ( A3
= ( set_ord_lessThan @ nat @ ( size_size @ ( list @ A ) @ Xs2 ) ) )
=> ( ( B6
= ( set2 @ A @ Xs2 ) )
=> ( bij_betw @ nat @ A @ ( nth @ A @ Xs2 ) @ A3 @ B6 ) ) ) ) ).
% bij_betw_nth
thf(fact_7996_distinct__list__update,axiom,
! [A: $tType,Xs2: list @ A,A2: A,I: nat] :
( ( distinct @ A @ Xs2 )
=> ( ~ ( member @ A @ A2 @ ( minus_minus @ ( set @ A ) @ ( set2 @ A @ Xs2 ) @ ( insert @ A @ ( nth @ A @ Xs2 @ I ) @ ( bot_bot @ ( set @ A ) ) ) ) )
=> ( distinct @ A @ ( list_update @ A @ Xs2 @ I @ A2 ) ) ) ) ).
% distinct_list_update
thf(fact_7997_set__update__distinct,axiom,
! [A: $tType,Xs2: list @ A,N3: nat,X: A] :
( ( distinct @ A @ Xs2 )
=> ( ( ord_less @ nat @ N3 @ ( size_size @ ( list @ A ) @ Xs2 ) )
=> ( ( set2 @ A @ ( list_update @ A @ Xs2 @ N3 @ X ) )
= ( insert @ A @ X @ ( minus_minus @ ( set @ A ) @ ( set2 @ A @ Xs2 ) @ ( insert @ A @ ( nth @ A @ Xs2 @ N3 ) @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ) ) ).
% set_update_distinct
thf(fact_7998_card__lists__distinct__length__eq_H,axiom,
! [A: $tType,K: nat,A3: set @ A] :
( ( ord_less @ nat @ K @ ( finite_card @ A @ A3 ) )
=> ( ( finite_card @ ( list @ A )
@ ( collect @ ( list @ A )
@ ^ [Xs: list @ A] :
( ( ( size_size @ ( list @ A ) @ Xs )
= K )
& ( distinct @ A @ Xs )
& ( ord_less_eq @ ( set @ A ) @ ( set2 @ A @ Xs ) @ A3 ) ) ) )
= ( groups7121269368397514597t_prod @ nat @ nat
@ ^ [X2: nat] : X2
@ ( set_or1337092689740270186AtMost @ nat @ ( plus_plus @ nat @ ( minus_minus @ nat @ ( finite_card @ A @ A3 ) @ K ) @ ( one_one @ nat ) ) @ ( finite_card @ A @ A3 ) ) ) ) ) ).
% card_lists_distinct_length_eq'
thf(fact_7999_mod__type__def,axiom,
! [A: $tType] :
( ( ( minus @ A )
& ( one @ A )
& ( plus @ A )
& ( times @ A )
& ( uminus @ A )
& ( zero @ A ) )
=> ( ( numeral_mod_type @ A )
= ( ^ [N2: int,Rep: A > int,Abs: int > A] :
( ( type_definition @ A @ int @ Rep @ Abs @ ( set_or7035219750837199246ssThan @ int @ ( zero_zero @ int ) @ N2 ) )
& ( ord_less @ int @ ( one_one @ int ) @ N2 )
& ( ( zero_zero @ A )
= ( Abs @ ( zero_zero @ int ) ) )
& ( ( one_one @ A )
= ( Abs @ ( one_one @ int ) ) )
& ! [X2: A,Y2: A] :
( ( plus_plus @ A @ X2 @ Y2 )
= ( Abs @ ( modulo_modulo @ int @ ( plus_plus @ int @ ( Rep @ X2 ) @ ( Rep @ Y2 ) ) @ N2 ) ) )
& ! [X2: A,Y2: A] :
( ( times_times @ A @ X2 @ Y2 )
= ( Abs @ ( modulo_modulo @ int @ ( times_times @ int @ ( Rep @ X2 ) @ ( Rep @ Y2 ) ) @ N2 ) ) )
& ! [X2: A,Y2: A] :
( ( minus_minus @ A @ X2 @ Y2 )
= ( Abs @ ( modulo_modulo @ int @ ( minus_minus @ int @ ( Rep @ X2 ) @ ( Rep @ Y2 ) ) @ N2 ) ) )
& ! [X2: A] :
( ( uminus_uminus @ A @ X2 )
= ( Abs @ ( modulo_modulo @ int @ ( uminus_uminus @ int @ ( Rep @ X2 ) ) @ N2 ) ) ) ) ) ) ) ).
% mod_type_def
thf(fact_8000_mod__type_Ointro,axiom,
! [A: $tType] :
( ( ( minus @ A )
& ( one @ A )
& ( plus @ A )
& ( times @ A )
& ( uminus @ A )
& ( zero @ A ) )
=> ! [Rep2: A > int,Abs2: int > A,N3: int] :
( ( type_definition @ A @ int @ Rep2 @ Abs2 @ ( set_or7035219750837199246ssThan @ int @ ( zero_zero @ int ) @ N3 ) )
=> ( ( ord_less @ int @ ( one_one @ int ) @ N3 )
=> ( ( ( zero_zero @ A )
= ( Abs2 @ ( zero_zero @ int ) ) )
=> ( ( ( one_one @ A )
= ( Abs2 @ ( one_one @ int ) ) )
=> ( ! [X3: A,Y3: A] :
( ( plus_plus @ A @ X3 @ Y3 )
= ( Abs2 @ ( modulo_modulo @ int @ ( plus_plus @ int @ ( Rep2 @ X3 ) @ ( Rep2 @ Y3 ) ) @ N3 ) ) )
=> ( ! [X3: A,Y3: A] :
( ( times_times @ A @ X3 @ Y3 )
= ( Abs2 @ ( modulo_modulo @ int @ ( times_times @ int @ ( Rep2 @ X3 ) @ ( Rep2 @ Y3 ) ) @ N3 ) ) )
=> ( ! [X3: A,Y3: A] :
( ( minus_minus @ A @ X3 @ Y3 )
= ( Abs2 @ ( modulo_modulo @ int @ ( minus_minus @ int @ ( Rep2 @ X3 ) @ ( Rep2 @ Y3 ) ) @ N3 ) ) )
=> ( ! [X3: A] :
( ( uminus_uminus @ A @ X3 )
= ( Abs2 @ ( modulo_modulo @ int @ ( uminus_uminus @ int @ ( Rep2 @ X3 ) ) @ N3 ) ) )
=> ( numeral_mod_type @ A @ N3 @ Rep2 @ Abs2 ) ) ) ) ) ) ) ) ) ) ).
% mod_type.intro
thf(fact_8001_mod__type_ORep__le__n,axiom,
! [A: $tType] :
( ( ( minus @ A )
& ( one @ A )
& ( plus @ A )
& ( times @ A )
& ( uminus @ A )
& ( zero @ A ) )
=> ! [N3: int,Rep2: A > int,Abs2: int > A,X: A] :
( ( numeral_mod_type @ A @ N3 @ Rep2 @ Abs2 )
=> ( ord_less_eq @ int @ ( Rep2 @ X ) @ N3 ) ) ) ).
% mod_type.Rep_le_n
thf(fact_8002_mod__type_Osize1,axiom,
! [A: $tType] :
( ( ( minus @ A )
& ( one @ A )
& ( plus @ A )
& ( times @ A )
& ( uminus @ A )
& ( zero @ A ) )
=> ! [N3: int,Rep2: A > int,Abs2: int > A] :
( ( numeral_mod_type @ A @ N3 @ Rep2 @ Abs2 )
=> ( ord_less @ int @ ( one_one @ int ) @ N3 ) ) ) ).
% mod_type.size1
thf(fact_8003_mod__type_ORep__Abs__1,axiom,
! [A: $tType] :
( ( ( minus @ A )
& ( one @ A )
& ( plus @ A )
& ( times @ A )
& ( uminus @ A )
& ( zero @ A ) )
=> ! [N3: int,Rep2: A > int,Abs2: int > A] :
( ( numeral_mod_type @ A @ N3 @ Rep2 @ Abs2 )
=> ( ( Rep2 @ ( Abs2 @ ( one_one @ int ) ) )
= ( one_one @ int ) ) ) ) ).
% mod_type.Rep_Abs_1
thf(fact_8004_mod__type_Oone__def,axiom,
! [A: $tType] :
( ( ( minus @ A )
& ( one @ A )
& ( plus @ A )
& ( times @ A )
& ( uminus @ A )
& ( zero @ A ) )
=> ! [N3: int,Rep2: A > int,Abs2: int > A] :
( ( numeral_mod_type @ A @ N3 @ Rep2 @ Abs2 )
=> ( ( one_one @ A )
= ( Abs2 @ ( one_one @ int ) ) ) ) ) ).
% mod_type.one_def
thf(fact_8005_mod__type_ORep__1,axiom,
! [A: $tType] :
( ( ( minus @ A )
& ( one @ A )
& ( plus @ A )
& ( times @ A )
& ( uminus @ A )
& ( zero @ A ) )
=> ! [N3: int,Rep2: A > int,Abs2: int > A] :
( ( numeral_mod_type @ A @ N3 @ Rep2 @ Abs2 )
=> ( ( Rep2 @ ( one_one @ A ) )
= ( one_one @ int ) ) ) ) ).
% mod_type.Rep_1
thf(fact_8006_mod__type_Oadd__def,axiom,
! [A: $tType] :
( ( ( minus @ A )
& ( one @ A )
& ( plus @ A )
& ( times @ A )
& ( uminus @ A )
& ( zero @ A ) )
=> ! [N3: int,Rep2: A > int,Abs2: int > A,X: A,Y: A] :
( ( numeral_mod_type @ A @ N3 @ Rep2 @ Abs2 )
=> ( ( plus_plus @ A @ X @ Y )
= ( Abs2 @ ( modulo_modulo @ int @ ( plus_plus @ int @ ( Rep2 @ X ) @ ( Rep2 @ Y ) ) @ N3 ) ) ) ) ) ).
% mod_type.add_def
thf(fact_8007_mod__type_Omult__def,axiom,
! [A: $tType] :
( ( ( minus @ A )
& ( one @ A )
& ( plus @ A )
& ( times @ A )
& ( uminus @ A )
& ( zero @ A ) )
=> ! [N3: int,Rep2: A > int,Abs2: int > A,X: A,Y: A] :
( ( numeral_mod_type @ A @ N3 @ Rep2 @ Abs2 )
=> ( ( times_times @ A @ X @ Y )
= ( Abs2 @ ( modulo_modulo @ int @ ( times_times @ int @ ( Rep2 @ X ) @ ( Rep2 @ Y ) ) @ N3 ) ) ) ) ) ).
% mod_type.mult_def
thf(fact_8008_mod__type_Odiff__def,axiom,
! [A: $tType] :
( ( ( minus @ A )
& ( one @ A )
& ( plus @ A )
& ( times @ A )
& ( uminus @ A )
& ( zero @ A ) )
=> ! [N3: int,Rep2: A > int,Abs2: int > A,X: A,Y: A] :
( ( numeral_mod_type @ A @ N3 @ Rep2 @ Abs2 )
=> ( ( minus_minus @ A @ X @ Y )
= ( Abs2 @ ( modulo_modulo @ int @ ( minus_minus @ int @ ( Rep2 @ X ) @ ( Rep2 @ Y ) ) @ N3 ) ) ) ) ) ).
% mod_type.diff_def
thf(fact_8009_inj__on__Abs__bit0,axiom,
! [A: $tType] :
( ( finite_finite @ A )
=> ( inj_on @ int @ ( numeral_bit0 @ A ) @ ( numeral_Abs_bit02 @ A ) @ ( set_or7035219750837199246ssThan @ int @ ( zero_zero @ int ) @ ( times_times @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ ( semiring_1_of_nat @ int @ ( finite_card @ A @ ( top_top @ ( set @ A ) ) ) ) ) ) ) ) ).
% inj_on_Abs_bit0
thf(fact_8010_Abs__bit0__cases,axiom,
! [A: $tType] :
( ( finite_finite @ A )
=> ! [X: numeral_bit0 @ A] :
~ ! [Y3: int] :
( ( X
= ( numeral_Abs_bit02 @ A @ Y3 ) )
=> ~ ( member @ int @ Y3 @ ( set_or7035219750837199246ssThan @ int @ ( zero_zero @ int ) @ ( times_times @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ ( semiring_1_of_nat @ int @ ( finite_card @ A @ ( top_top @ ( set @ A ) ) ) ) ) ) ) ) ) ).
% Abs_bit0_cases
thf(fact_8011_one__bit0__def,axiom,
! [A: $tType] :
( ( finite_finite @ A )
=> ( ( one_one @ ( numeral_bit0 @ A ) )
= ( numeral_Abs_bit02 @ A @ ( one_one @ int ) ) ) ) ).
% one_bit0_def
thf(fact_8012_Abs__bit0__inject,axiom,
! [A: $tType] :
( ( finite_finite @ A )
=> ! [X: int,Y: int] :
( ( member @ int @ X @ ( set_or7035219750837199246ssThan @ int @ ( zero_zero @ int ) @ ( times_times @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ ( semiring_1_of_nat @ int @ ( finite_card @ A @ ( top_top @ ( set @ A ) ) ) ) ) ) )
=> ( ( member @ int @ Y @ ( set_or7035219750837199246ssThan @ int @ ( zero_zero @ int ) @ ( times_times @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ ( semiring_1_of_nat @ int @ ( finite_card @ A @ ( top_top @ ( set @ A ) ) ) ) ) ) )
=> ( ( ( numeral_Abs_bit02 @ A @ X )
= ( numeral_Abs_bit02 @ A @ Y ) )
= ( X = Y ) ) ) ) ) ).
% Abs_bit0_inject
thf(fact_8013_Abs__bit0__induct,axiom,
! [A: $tType] :
( ( finite_finite @ A )
=> ! [P: ( numeral_bit0 @ A ) > $o,X: numeral_bit0 @ A] :
( ! [Y3: int] :
( ( member @ int @ Y3 @ ( set_or7035219750837199246ssThan @ int @ ( zero_zero @ int ) @ ( times_times @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ ( semiring_1_of_nat @ int @ ( finite_card @ A @ ( top_top @ ( set @ A ) ) ) ) ) ) )
=> ( P @ ( numeral_Abs_bit02 @ A @ Y3 ) ) )
=> ( P @ X ) ) ) ).
% Abs_bit0_induct
thf(fact_8014_Abs__bit0__inverse,axiom,
! [A: $tType] :
( ( finite_finite @ A )
=> ! [Y: int] :
( ( member @ int @ Y @ ( set_or7035219750837199246ssThan @ int @ ( zero_zero @ int ) @ ( times_times @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ ( semiring_1_of_nat @ int @ ( finite_card @ A @ ( top_top @ ( set @ A ) ) ) ) ) ) )
=> ( ( numeral_Rep_bit0 @ A @ ( numeral_Abs_bit02 @ A @ Y ) )
= Y ) ) ) ).
% Abs_bit0_inverse
thf(fact_8015_type__definition__bit0,axiom,
! [A: $tType] :
( ( finite_finite @ A )
=> ( type_definition @ ( numeral_bit0 @ A ) @ int @ ( numeral_Rep_bit0 @ A ) @ ( numeral_Abs_bit02 @ A ) @ ( set_or7035219750837199246ssThan @ int @ ( zero_zero @ int ) @ ( times_times @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ ( semiring_1_of_nat @ int @ ( finite_card @ A @ ( top_top @ ( set @ A ) ) ) ) ) ) ) ) ).
% type_definition_bit0
thf(fact_8016_bit0_ORep__Abs__1,axiom,
! [A: $tType] :
( ( finite_finite @ A )
=> ( ( numeral_Rep_bit0 @ A @ ( numeral_Abs_bit02 @ A @ ( one_one @ int ) ) )
= ( one_one @ int ) ) ) ).
% bit0.Rep_Abs_1
thf(fact_8017_less__eq__bit0__def,axiom,
! [A: $tType] :
( ( finite_finite @ A )
=> ( ( ord_less_eq @ ( numeral_bit0 @ A ) )
= ( ^ [A6: numeral_bit0 @ A,B5: numeral_bit0 @ A] : ( ord_less_eq @ int @ ( numeral_Rep_bit0 @ A @ A6 ) @ ( numeral_Rep_bit0 @ A @ B5 ) ) ) ) ) ).
% less_eq_bit0_def
thf(fact_8018_bit0_ORep__1,axiom,
! [A: $tType] :
( ( finite_finite @ A )
=> ( ( numeral_Rep_bit0 @ A @ ( one_one @ ( numeral_bit0 @ A ) ) )
= ( one_one @ int ) ) ) ).
% bit0.Rep_1
thf(fact_8019_bit0_ORep__le__n,axiom,
! [A: $tType] :
( ( finite_finite @ A )
=> ! [X: numeral_bit0 @ A] : ( ord_less_eq @ int @ ( numeral_Rep_bit0 @ A @ X ) @ ( semiring_1_of_nat @ int @ ( finite_card @ ( numeral_bit0 @ A ) @ ( top_top @ ( set @ ( numeral_bit0 @ A ) ) ) ) ) ) ) ).
% bit0.Rep_le_n
thf(fact_8020_bit0_Odiff__def,axiom,
! [A: $tType] :
( ( finite_finite @ A )
=> ( ( minus_minus @ ( numeral_bit0 @ A ) )
= ( ^ [X2: numeral_bit0 @ A,Y2: numeral_bit0 @ A] : ( numeral_Abs_bit02 @ A @ ( modulo_modulo @ int @ ( minus_minus @ int @ ( numeral_Rep_bit0 @ A @ X2 ) @ ( numeral_Rep_bit0 @ A @ Y2 ) ) @ ( semiring_1_of_nat @ int @ ( finite_card @ ( numeral_bit0 @ A ) @ ( top_top @ ( set @ ( numeral_bit0 @ A ) ) ) ) ) ) ) ) ) ) ).
% bit0.diff_def
thf(fact_8021_Rep__bit0__induct,axiom,
! [A: $tType] :
( ( finite_finite @ A )
=> ! [Y: int,P: int > $o] :
( ( member @ int @ Y @ ( set_or7035219750837199246ssThan @ int @ ( zero_zero @ int ) @ ( times_times @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ ( semiring_1_of_nat @ int @ ( finite_card @ A @ ( top_top @ ( set @ A ) ) ) ) ) ) )
=> ( ! [X3: numeral_bit0 @ A] : ( P @ ( numeral_Rep_bit0 @ A @ X3 ) )
=> ( P @ Y ) ) ) ) ).
% Rep_bit0_induct
thf(fact_8022_Rep__bit0__cases,axiom,
! [A: $tType] :
( ( finite_finite @ A )
=> ! [Y: int] :
( ( member @ int @ Y @ ( set_or7035219750837199246ssThan @ int @ ( zero_zero @ int ) @ ( times_times @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ ( semiring_1_of_nat @ int @ ( finite_card @ A @ ( top_top @ ( set @ A ) ) ) ) ) ) )
=> ~ ! [X3: numeral_bit0 @ A] :
( Y
!= ( numeral_Rep_bit0 @ A @ X3 ) ) ) ) ).
% Rep_bit0_cases
thf(fact_8023_Rep__bit0,axiom,
! [A: $tType] :
( ( finite_finite @ A )
=> ! [X: numeral_bit0 @ A] : ( member @ int @ ( numeral_Rep_bit0 @ A @ X ) @ ( set_or7035219750837199246ssThan @ int @ ( zero_zero @ int ) @ ( times_times @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ ( semiring_1_of_nat @ int @ ( finite_card @ A @ ( top_top @ ( set @ A ) ) ) ) ) ) ) ) ).
% Rep_bit0
thf(fact_8024_card__UNIV__char,axiom,
( ( finite_card @ char @ ( top_top @ ( set @ char ) ) )
= ( numeral_numeral @ nat @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ one2 ) ) ) ) ) ) ) ) ) ) ).
% card_UNIV_char
thf(fact_8025_minus__bit1__def,axiom,
! [A: $tType] :
( ( finite_finite @ A )
=> ( ( minus_minus @ ( numeral_bit1 @ A ) )
= ( ^ [X2: numeral_bit1 @ A,Y2: numeral_bit1 @ A] : ( numeral_Abs_bit1 @ A @ ( minus_minus @ int @ ( numeral_Rep_bit1 @ A @ X2 ) @ ( numeral_Rep_bit1 @ A @ Y2 ) ) ) ) ) ) ).
% minus_bit1_def
thf(fact_8026_UNIV__char__of__nat,axiom,
( ( top_top @ ( set @ char ) )
= ( image @ nat @ char @ ( unique5772411509450598832har_of @ nat ) @ ( set_or7035219750837199246ssThan @ nat @ ( zero_zero @ nat ) @ ( numeral_numeral @ nat @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ one2 ) ) ) ) ) ) ) ) ) ) ) ) ).
% UNIV_char_of_nat
thf(fact_8027_minus__bit0__def,axiom,
! [A: $tType] :
( ( finite_finite @ A )
=> ( ( minus_minus @ ( numeral_bit0 @ A ) )
= ( ^ [X2: numeral_bit0 @ A,Y2: numeral_bit0 @ A] : ( numeral_Abs_bit0 @ A @ ( minus_minus @ int @ ( numeral_Rep_bit0 @ A @ X2 ) @ ( numeral_Rep_bit0 @ A @ Y2 ) ) ) ) ) ) ).
% minus_bit0_def
thf(fact_8028_char__of__quasi__inj,axiom,
! [A: $tType] :
( ( bit_un5681908812861735899ations @ A )
=> ! [M: A,N3: A] :
( ( ( unique5772411509450598832har_of @ A @ M )
= ( unique5772411509450598832har_of @ A @ N3 ) )
= ( ( modulo_modulo @ A @ M @ ( numeral_numeral @ A @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ one2 ) ) ) ) ) ) ) ) ) )
= ( modulo_modulo @ A @ N3 @ ( numeral_numeral @ A @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ one2 ) ) ) ) ) ) ) ) ) ) ) ) ) ).
% char_of_quasi_inj
thf(fact_8029_char__of__mod__256,axiom,
! [A: $tType] :
( ( bit_un5681908812861735899ations @ A )
=> ! [N3: A] :
( ( unique5772411509450598832har_of @ A @ ( modulo_modulo @ A @ N3 @ ( numeral_numeral @ A @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ one2 ) ) ) ) ) ) ) ) ) ) )
= ( unique5772411509450598832har_of @ A @ N3 ) ) ) ).
% char_of_mod_256
thf(fact_8030_inj__on__char__of__nat,axiom,
inj_on @ nat @ char @ ( unique5772411509450598832har_of @ nat ) @ ( set_or7035219750837199246ssThan @ nat @ ( zero_zero @ nat ) @ ( numeral_numeral @ nat @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ one2 ) ) ) ) ) ) ) ) ) ) ).
% inj_on_char_of_nat
thf(fact_8031_char__of__take__bit__eq,axiom,
! [A: $tType] :
( ( bit_un5681908812861735899ations @ A )
=> ! [N3: nat,M: A] :
( ( ord_less_eq @ nat @ ( numeral_numeral @ nat @ ( bit0 @ ( bit0 @ ( bit0 @ one2 ) ) ) ) @ N3 )
=> ( ( unique5772411509450598832har_of @ A @ ( bit_se2584673776208193580ke_bit @ A @ N3 @ M ) )
= ( unique5772411509450598832har_of @ A @ M ) ) ) ) ).
% char_of_take_bit_eq
thf(fact_8032_of__char__of,axiom,
! [A: $tType] :
( ( bit_un5681908812861735899ations @ A )
=> ! [A2: A] :
( ( comm_s6883823935334413003f_char @ A @ ( unique5772411509450598832har_of @ A @ A2 ) )
= ( modulo_modulo @ A @ A2 @ ( numeral_numeral @ A @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ one2 ) ) ) ) ) ) ) ) ) ) ) ) ).
% of_char_of
thf(fact_8033_char__of__def,axiom,
! [A: $tType] :
( ( bit_un5681908812861735899ations @ A )
=> ( ( unique5772411509450598832har_of @ A )
= ( ^ [N2: A] :
( char2
@ ~ ( dvd_dvd @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ N2 )
@ ( bit_se5641148757651400278ts_bit @ A @ N2 @ ( one_one @ nat ) )
@ ( bit_se5641148757651400278ts_bit @ A @ N2 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) )
@ ( bit_se5641148757651400278ts_bit @ A @ N2 @ ( numeral_numeral @ nat @ ( bit1 @ one2 ) ) )
@ ( bit_se5641148757651400278ts_bit @ A @ N2 @ ( numeral_numeral @ nat @ ( bit0 @ ( bit0 @ one2 ) ) ) )
@ ( bit_se5641148757651400278ts_bit @ A @ N2 @ ( numeral_numeral @ nat @ ( bit1 @ ( bit0 @ one2 ) ) ) )
@ ( bit_se5641148757651400278ts_bit @ A @ N2 @ ( numeral_numeral @ nat @ ( bit0 @ ( bit1 @ one2 ) ) ) )
@ ( bit_se5641148757651400278ts_bit @ A @ N2 @ ( numeral_numeral @ nat @ ( bit1 @ ( bit1 @ one2 ) ) ) ) ) ) ) ) ).
% char_of_def
thf(fact_8034_of__char__mod__256,axiom,
! [A: $tType] :
( ( bit_un5681908812861735899ations @ A )
=> ! [C2: char] :
( ( modulo_modulo @ A @ ( comm_s6883823935334413003f_char @ A @ C2 ) @ ( numeral_numeral @ A @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ one2 ) ) ) ) ) ) ) ) ) )
= ( comm_s6883823935334413003f_char @ A @ C2 ) ) ) ).
% of_char_mod_256
thf(fact_8035_char_Osize_I2_J,axiom,
! [X15: $o,X22: $o,X33: $o,X42: $o,X52: $o,X62: $o,X72: $o,X82: $o] :
( ( size_size @ char @ ( char2 @ X15 @ X22 @ X33 @ X42 @ X52 @ X62 @ X72 @ X82 ) )
= ( zero_zero @ nat ) ) ).
% char.size(2)
thf(fact_8036_nat__of__char__less__256,axiom,
! [C2: char] : ( ord_less @ nat @ ( comm_s6883823935334413003f_char @ nat @ C2 ) @ ( numeral_numeral @ nat @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ one2 ) ) ) ) ) ) ) ) ) ) ).
% nat_of_char_less_256
thf(fact_8037_range__nat__of__char,axiom,
( ( image @ char @ nat @ ( comm_s6883823935334413003f_char @ nat ) @ ( top_top @ ( set @ char ) ) )
= ( set_or7035219750837199246ssThan @ nat @ ( zero_zero @ nat ) @ ( numeral_numeral @ nat @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ one2 ) ) ) ) ) ) ) ) ) ) ) ).
% range_nat_of_char
thf(fact_8038_char__of__eq__iff,axiom,
! [A: $tType] :
( ( bit_un5681908812861735899ations @ A )
=> ! [N3: A,C2: char] :
( ( ( unique5772411509450598832har_of @ A @ N3 )
= C2 )
= ( ( bit_se2584673776208193580ke_bit @ A @ ( numeral_numeral @ nat @ ( bit0 @ ( bit0 @ ( bit0 @ one2 ) ) ) ) @ N3 )
= ( comm_s6883823935334413003f_char @ A @ C2 ) ) ) ) ).
% char_of_eq_iff
thf(fact_8039_integer__of__char__code,axiom,
! [B0: $o,B1: $o,B22: $o,B32: $o,B42: $o,B52: $o,B62: $o,B72: $o] :
( ( integer_of_char @ ( char2 @ B0 @ B1 @ B22 @ B32 @ B42 @ B52 @ B62 @ B72 ) )
= ( plus_plus @ code_integer @ ( times_times @ code_integer @ ( plus_plus @ code_integer @ ( times_times @ code_integer @ ( plus_plus @ code_integer @ ( times_times @ code_integer @ ( plus_plus @ code_integer @ ( times_times @ code_integer @ ( plus_plus @ code_integer @ ( times_times @ code_integer @ ( plus_plus @ code_integer @ ( times_times @ code_integer @ ( plus_plus @ code_integer @ ( times_times @ code_integer @ ( zero_neq_one_of_bool @ code_integer @ B72 ) @ ( numeral_numeral @ code_integer @ ( bit0 @ one2 ) ) ) @ ( zero_neq_one_of_bool @ code_integer @ B62 ) ) @ ( numeral_numeral @ code_integer @ ( bit0 @ one2 ) ) ) @ ( zero_neq_one_of_bool @ code_integer @ B52 ) ) @ ( numeral_numeral @ code_integer @ ( bit0 @ one2 ) ) ) @ ( zero_neq_one_of_bool @ code_integer @ B42 ) ) @ ( numeral_numeral @ code_integer @ ( bit0 @ one2 ) ) ) @ ( zero_neq_one_of_bool @ code_integer @ B32 ) ) @ ( numeral_numeral @ code_integer @ ( bit0 @ one2 ) ) ) @ ( zero_neq_one_of_bool @ code_integer @ B22 ) ) @ ( numeral_numeral @ code_integer @ ( bit0 @ one2 ) ) ) @ ( zero_neq_one_of_bool @ code_integer @ B1 ) ) @ ( numeral_numeral @ code_integer @ ( bit0 @ one2 ) ) ) @ ( zero_neq_one_of_bool @ code_integer @ B0 ) ) ) ).
% integer_of_char_code
thf(fact_8040_of__char__Char,axiom,
! [A: $tType] :
( ( comm_semiring_1 @ A )
=> ! [B0: $o,B1: $o,B22: $o,B32: $o,B42: $o,B52: $o,B62: $o,B72: $o] :
( ( comm_s6883823935334413003f_char @ A @ ( char2 @ B0 @ B1 @ B22 @ B32 @ B42 @ B52 @ B62 @ B72 ) )
= ( groups4207007520872428315er_sum @ $o @ A @ ( zero_neq_one_of_bool @ A ) @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ ( cons @ $o @ B0 @ ( cons @ $o @ B1 @ ( cons @ $o @ B22 @ ( cons @ $o @ B32 @ ( cons @ $o @ B42 @ ( cons @ $o @ B52 @ ( cons @ $o @ B62 @ ( cons @ $o @ B72 @ ( nil @ $o ) ) ) ) ) ) ) ) ) ) ) ) ).
% of_char_Char
thf(fact_8041_slice__Nil,axiom,
! [A: $tType,Begin: nat,End: nat] :
( ( slice @ A @ Begin @ End @ ( nil @ A ) )
= ( nil @ A ) ) ).
% slice_Nil
thf(fact_8042_slice__eq__bounds__empty,axiom,
! [A: $tType,I: nat,Xs2: list @ A] :
( ( slice @ A @ I @ I @ Xs2 )
= ( nil @ A ) ) ).
% slice_eq_bounds_empty
thf(fact_8043_length__0__conv,axiom,
! [A: $tType,Xs2: list @ A] :
( ( ( size_size @ ( list @ A ) @ Xs2 )
= ( zero_zero @ nat ) )
= ( Xs2
= ( nil @ A ) ) ) ).
% length_0_conv
thf(fact_8044_set__empty,axiom,
! [A: $tType,Xs2: list @ A] :
( ( ( set2 @ A @ Xs2 )
= ( bot_bot @ ( set @ A ) ) )
= ( Xs2
= ( nil @ A ) ) ) ).
% set_empty
thf(fact_8045_set__empty2,axiom,
! [A: $tType,Xs2: list @ A] :
( ( ( bot_bot @ ( set @ A ) )
= ( set2 @ A @ Xs2 ) )
= ( Xs2
= ( nil @ A ) ) ) ).
% set_empty2
thf(fact_8046_concat__eq__Nil__conv,axiom,
! [A: $tType,Xss: list @ ( list @ A )] :
( ( ( concat @ A @ Xss )
= ( nil @ A ) )
= ( ! [X2: list @ A] :
( ( member @ ( list @ A ) @ X2 @ ( set2 @ ( list @ A ) @ Xss ) )
=> ( X2
= ( nil @ A ) ) ) ) ) ).
% concat_eq_Nil_conv
thf(fact_8047_Nil__eq__concat__conv,axiom,
! [A: $tType,Xss: list @ ( list @ A )] :
( ( ( nil @ A )
= ( concat @ A @ Xss ) )
= ( ! [X2: list @ A] :
( ( member @ ( list @ A ) @ X2 @ ( set2 @ ( list @ A ) @ Xss ) )
=> ( X2
= ( nil @ A ) ) ) ) ) ).
% Nil_eq_concat_conv
thf(fact_8048_length__greater__0__conv,axiom,
! [A: $tType,Xs2: list @ A] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ ( size_size @ ( list @ A ) @ Xs2 ) )
= ( Xs2
!= ( nil @ A ) ) ) ).
% length_greater_0_conv
thf(fact_8049_length__ge__1__conv,axiom,
! [A: $tType,L2: list @ A] :
( ( ord_less_eq @ nat @ ( suc @ ( zero_zero @ nat ) ) @ ( size_size @ ( list @ A ) @ L2 ) )
= ( L2
!= ( nil @ A ) ) ) ).
% length_ge_1_conv
thf(fact_8050_hash__code__list__simps_I1_J,axiom,
! [A: $tType,H_a: A > uint32] :
( ( hash_hash_code_list @ A @ H_a @ ( nil @ A ) )
= ( numeral_numeral @ uint32 @ ( bit0 @ ( bit0 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit1 @ ( bit0 @ ( bit1 @ ( bit1 @ ( bit0 @ ( bit0 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit1 @ ( bit0 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit0 @ ( bit0 @ one2 ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).
% hash_code_list_simps(1)
thf(fact_8051_length__compl__induct,axiom,
! [A: $tType,P: ( list @ A ) > $o,L2: list @ A] :
( ( P @ ( nil @ A ) )
=> ( ! [E2: A,L3: list @ A] :
( ! [Ll: list @ A] :
( ( ord_less_eq @ nat @ ( size_size @ ( list @ A ) @ Ll ) @ ( size_size @ ( list @ A ) @ L3 ) )
=> ( P @ Ll ) )
=> ( P @ ( cons @ A @ E2 @ L3 ) ) )
=> ( P @ L2 ) ) ) ).
% length_compl_induct
thf(fact_8052_empty__set,axiom,
! [A: $tType] :
( ( bot_bot @ ( set @ A ) )
= ( set2 @ A @ ( nil @ A ) ) ) ).
% empty_set
thf(fact_8053_length__nth__simps_I1_J,axiom,
! [A: $tType] :
( ( size_size @ ( list @ A ) @ ( nil @ A ) )
= ( zero_zero @ nat ) ) ).
% length_nth_simps(1)
thf(fact_8054_list__induct4,axiom,
! [C: $tType,A: $tType,B: $tType,D5: $tType,Xs2: list @ A,Ys: list @ B,Zs: list @ C,Ws: list @ D5,P: ( list @ A ) > ( list @ B ) > ( list @ C ) > ( list @ D5 ) > $o] :
( ( ( size_size @ ( list @ A ) @ Xs2 )
= ( size_size @ ( list @ B ) @ Ys ) )
=> ( ( ( size_size @ ( list @ B ) @ Ys )
= ( size_size @ ( list @ C ) @ Zs ) )
=> ( ( ( size_size @ ( list @ C ) @ Zs )
= ( size_size @ ( list @ D5 ) @ Ws ) )
=> ( ( P @ ( nil @ A ) @ ( nil @ B ) @ ( nil @ C ) @ ( nil @ D5 ) )
=> ( ! [X3: A,Xs3: list @ A,Y3: B,Ys4: list @ B,Z2: C,Zs2: list @ C,W3: D5,Ws2: list @ D5] :
( ( ( size_size @ ( list @ A ) @ Xs3 )
= ( size_size @ ( list @ B ) @ Ys4 ) )
=> ( ( ( size_size @ ( list @ B ) @ Ys4 )
= ( size_size @ ( list @ C ) @ Zs2 ) )
=> ( ( ( size_size @ ( list @ C ) @ Zs2 )
= ( size_size @ ( list @ D5 ) @ Ws2 ) )
=> ( ( P @ Xs3 @ Ys4 @ Zs2 @ Ws2 )
=> ( P @ ( cons @ A @ X3 @ Xs3 ) @ ( cons @ B @ Y3 @ Ys4 ) @ ( cons @ C @ Z2 @ Zs2 ) @ ( cons @ D5 @ W3 @ Ws2 ) ) ) ) ) )
=> ( P @ Xs2 @ Ys @ Zs @ Ws ) ) ) ) ) ) ).
% list_induct4
thf(fact_8055_list__induct3,axiom,
! [B: $tType,A: $tType,C: $tType,Xs2: list @ A,Ys: list @ B,Zs: list @ C,P: ( list @ A ) > ( list @ B ) > ( list @ C ) > $o] :
( ( ( size_size @ ( list @ A ) @ Xs2 )
= ( size_size @ ( list @ B ) @ Ys ) )
=> ( ( ( size_size @ ( list @ B ) @ Ys )
= ( size_size @ ( list @ C ) @ Zs ) )
=> ( ( P @ ( nil @ A ) @ ( nil @ B ) @ ( nil @ C ) )
=> ( ! [X3: A,Xs3: list @ A,Y3: B,Ys4: list @ B,Z2: C,Zs2: list @ C] :
( ( ( size_size @ ( list @ A ) @ Xs3 )
= ( size_size @ ( list @ B ) @ Ys4 ) )
=> ( ( ( size_size @ ( list @ B ) @ Ys4 )
= ( size_size @ ( list @ C ) @ Zs2 ) )
=> ( ( P @ Xs3 @ Ys4 @ Zs2 )
=> ( P @ ( cons @ A @ X3 @ Xs3 ) @ ( cons @ B @ Y3 @ Ys4 ) @ ( cons @ C @ Z2 @ Zs2 ) ) ) ) )
=> ( P @ Xs2 @ Ys @ Zs ) ) ) ) ) ).
% list_induct3
thf(fact_8056_list__induct2,axiom,
! [A: $tType,B: $tType,Xs2: list @ A,Ys: list @ B,P: ( list @ A ) > ( list @ B ) > $o] :
( ( ( size_size @ ( list @ A ) @ Xs2 )
= ( size_size @ ( list @ B ) @ Ys ) )
=> ( ( P @ ( nil @ A ) @ ( nil @ B ) )
=> ( ! [X3: A,Xs3: list @ A,Y3: B,Ys4: list @ B] :
( ( ( size_size @ ( list @ A ) @ Xs3 )
= ( size_size @ ( list @ B ) @ Ys4 ) )
=> ( ( P @ Xs3 @ Ys4 )
=> ( P @ ( cons @ A @ X3 @ Xs3 ) @ ( cons @ B @ Y3 @ Ys4 ) ) ) )
=> ( P @ Xs2 @ Ys ) ) ) ) ).
% list_induct2
thf(fact_8057_mergesort__by__rel__merge__induct,axiom,
! [A: $tType,B: $tType,P: ( list @ A ) > ( list @ B ) > $o,R: A > B > $o,Xs2: list @ A,Ys: list @ B] :
( ! [Xs3: list @ A] : ( P @ Xs3 @ ( nil @ B ) )
=> ( ! [X_12: list @ B] : ( P @ ( nil @ A ) @ X_12 )
=> ( ! [X3: A,Xs3: list @ A,Y3: B,Ys4: list @ B] :
( ( R @ X3 @ Y3 )
=> ( ( P @ Xs3 @ ( cons @ B @ Y3 @ Ys4 ) )
=> ( P @ ( cons @ A @ X3 @ Xs3 ) @ ( cons @ B @ Y3 @ Ys4 ) ) ) )
=> ( ! [X3: A,Xs3: list @ A,Y3: B,Ys4: list @ B] :
( ~ ( R @ X3 @ Y3 )
=> ( ( P @ ( cons @ A @ X3 @ Xs3 ) @ Ys4 )
=> ( P @ ( cons @ A @ X3 @ Xs3 ) @ ( cons @ B @ Y3 @ Ys4 ) ) ) )
=> ( P @ Xs2 @ Ys ) ) ) ) ) ).
% mergesort_by_rel_merge_induct
thf(fact_8058_list__induct__first2,axiom,
! [A: $tType,P: ( list @ A ) > $o,Xs2: list @ A] :
( ( P @ ( nil @ A ) )
=> ( ! [X3: A] : ( P @ ( cons @ A @ X3 @ ( nil @ A ) ) )
=> ( ! [X1: A,X23: A,Xs3: list @ A] :
( ( P @ Xs3 )
=> ( P @ ( cons @ A @ X1 @ ( cons @ A @ X23 @ Xs3 ) ) ) )
=> ( P @ Xs2 ) ) ) ) ).
% list_induct_first2
thf(fact_8059_list__2pre__induct,axiom,
! [A: $tType,B: $tType,P: ( list @ A ) > ( list @ B ) > $o,W1: list @ A,W22: list @ B] :
( ( P @ ( nil @ A ) @ ( nil @ B ) )
=> ( ! [E2: A,W12: list @ A,W23: list @ B] :
( ( P @ W12 @ W23 )
=> ( P @ ( cons @ A @ E2 @ W12 ) @ W23 ) )
=> ( ! [E2: B,W13: list @ A,W24: list @ B] :
( ( P @ W13 @ W24 )
=> ( P @ W13 @ ( cons @ B @ E2 @ W24 ) ) )
=> ( P @ W1 @ W22 ) ) ) ) ).
% list_2pre_induct
thf(fact_8060_neq__NilE,axiom,
! [A: $tType,L2: list @ A] :
( ( L2
!= ( nil @ A ) )
=> ~ ! [X3: A,Xs3: list @ A] :
( L2
!= ( cons @ A @ X3 @ Xs3 ) ) ) ).
% neq_NilE
thf(fact_8061_Misc_Ofoldr__Cons,axiom,
! [A: $tType,Xs2: list @ A] :
( ( foldr @ A @ ( list @ A ) @ ( cons @ A ) @ Xs2 @ ( nil @ A ) )
= Xs2 ) ).
% Misc.foldr_Cons
thf(fact_8062_subset__eq__mset__impl_Ocases,axiom,
! [A: $tType,X: product_prod @ ( list @ A ) @ ( list @ A )] :
( ! [Ys4: list @ A] :
( X
!= ( product_Pair @ ( list @ A ) @ ( list @ A ) @ ( nil @ A ) @ Ys4 ) )
=> ~ ! [X3: A,Xs3: list @ A,Ys4: list @ A] :
( X
!= ( product_Pair @ ( list @ A ) @ ( list @ A ) @ ( cons @ A @ X3 @ Xs3 ) @ Ys4 ) ) ) ).
% subset_eq_mset_impl.cases
thf(fact_8063_zipf_Ocases,axiom,
! [C: $tType,A: $tType,B: $tType,X: product_prod @ ( A > B > C ) @ ( product_prod @ ( list @ A ) @ ( list @ B ) )] :
( ! [F3: A > B > C] :
( X
!= ( product_Pair @ ( A > B > C ) @ ( product_prod @ ( list @ A ) @ ( list @ B ) ) @ F3 @ ( product_Pair @ ( list @ A ) @ ( list @ B ) @ ( nil @ A ) @ ( nil @ B ) ) ) )
=> ( ! [F3: A > B > C,A4: A,As: list @ A,B3: B,Bs2: list @ B] :
( X
!= ( product_Pair @ ( A > B > C ) @ ( product_prod @ ( list @ A ) @ ( list @ B ) ) @ F3 @ ( product_Pair @ ( list @ A ) @ ( list @ B ) @ ( cons @ A @ A4 @ As ) @ ( cons @ B @ B3 @ Bs2 ) ) ) )
=> ( ! [A4: A > B > C,V3: A,Va3: list @ A] :
( X
!= ( product_Pair @ ( A > B > C ) @ ( product_prod @ ( list @ A ) @ ( list @ B ) ) @ A4 @ ( product_Pair @ ( list @ A ) @ ( list @ B ) @ ( cons @ A @ V3 @ Va3 ) @ ( nil @ B ) ) ) )
=> ~ ! [A4: A > B > C,V3: B,Va3: list @ B] :
( X
!= ( product_Pair @ ( A > B > C ) @ ( product_prod @ ( list @ A ) @ ( list @ B ) ) @ A4 @ ( product_Pair @ ( list @ A ) @ ( list @ B ) @ ( nil @ A ) @ ( cons @ B @ V3 @ Va3 ) ) ) ) ) ) ) ).
% zipf.cases
thf(fact_8064_merge_Ocases,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [X: product_prod @ ( list @ A ) @ ( list @ A )] :
( ! [L22: list @ A] :
( X
!= ( product_Pair @ ( list @ A ) @ ( list @ A ) @ ( nil @ A ) @ L22 ) )
=> ( ! [V3: A,Va3: list @ A] :
( X
!= ( product_Pair @ ( list @ A ) @ ( list @ A ) @ ( cons @ A @ V3 @ Va3 ) @ ( nil @ A ) ) )
=> ~ ! [X1: A,L1: list @ A,X23: A,L22: list @ A] :
( X
!= ( product_Pair @ ( list @ A ) @ ( list @ A ) @ ( cons @ A @ X1 @ L1 ) @ ( cons @ A @ X23 @ L22 ) ) ) ) ) ) ).
% merge.cases
thf(fact_8065_list__all__zip_Ocases,axiom,
! [A: $tType,B: $tType,X: product_prod @ ( A > B > $o ) @ ( product_prod @ ( list @ A ) @ ( list @ B ) )] :
( ! [P8: A > B > $o] :
( X
!= ( product_Pair @ ( A > B > $o ) @ ( product_prod @ ( list @ A ) @ ( list @ B ) ) @ P8 @ ( product_Pair @ ( list @ A ) @ ( list @ B ) @ ( nil @ A ) @ ( nil @ B ) ) ) )
=> ( ! [P8: A > B > $o,A4: A,As: list @ A,B3: B,Bs2: list @ B] :
( X
!= ( product_Pair @ ( A > B > $o ) @ ( product_prod @ ( list @ A ) @ ( list @ B ) ) @ P8 @ ( product_Pair @ ( list @ A ) @ ( list @ B ) @ ( cons @ A @ A4 @ As ) @ ( cons @ B @ B3 @ Bs2 ) ) ) )
=> ( ! [P8: A > B > $o,V3: A,Va3: list @ A] :
( X
!= ( product_Pair @ ( A > B > $o ) @ ( product_prod @ ( list @ A ) @ ( list @ B ) ) @ P8 @ ( product_Pair @ ( list @ A ) @ ( list @ B ) @ ( cons @ A @ V3 @ Va3 ) @ ( nil @ B ) ) ) )
=> ~ ! [P8: A > B > $o,V3: B,Va3: list @ B] :
( X
!= ( product_Pair @ ( A > B > $o ) @ ( product_prod @ ( list @ A ) @ ( list @ B ) ) @ P8 @ ( product_Pair @ ( list @ A ) @ ( list @ B ) @ ( nil @ A ) @ ( cons @ B @ V3 @ Va3 ) ) ) ) ) ) ) ).
% list_all_zip.cases
thf(fact_8066_partition__rev_Ocases,axiom,
! [A: $tType,X: product_prod @ ( A > $o ) @ ( product_prod @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( list @ A ) )] :
( ! [P8: A > $o,Yes: list @ A,No3: list @ A] :
( X
!= ( product_Pair @ ( A > $o ) @ ( product_prod @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( list @ A ) ) @ P8 @ ( product_Pair @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( list @ A ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ Yes @ No3 ) @ ( nil @ A ) ) ) )
=> ~ ! [P8: A > $o,Yes: list @ A,No3: list @ A,X3: A,Xs3: list @ A] :
( X
!= ( product_Pair @ ( A > $o ) @ ( product_prod @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( list @ A ) ) @ P8 @ ( product_Pair @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( list @ A ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ Yes @ No3 ) @ ( cons @ A @ X3 @ Xs3 ) ) ) ) ) ).
% partition_rev.cases
thf(fact_8067_quicksort__by__rel_Ocases,axiom,
! [A: $tType,X: product_prod @ ( A > A > $o ) @ ( product_prod @ ( list @ A ) @ ( list @ A ) )] :
( ! [R7: A > A > $o,Sl: list @ A] :
( X
!= ( product_Pair @ ( A > A > $o ) @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ R7 @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ Sl @ ( nil @ A ) ) ) )
=> ~ ! [R7: A > A > $o,Sl: list @ A,X3: A,Xs3: list @ A] :
( X
!= ( product_Pair @ ( A > A > $o ) @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ R7 @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ Sl @ ( cons @ A @ X3 @ Xs3 ) ) ) ) ) ).
% quicksort_by_rel.cases
thf(fact_8068_mergesort__by__rel__merge_Ocases,axiom,
! [A: $tType,X: product_prod @ ( A > A > $o ) @ ( product_prod @ ( list @ A ) @ ( list @ A ) )] :
( ! [R7: A > A > $o,X3: A,Xs3: list @ A,Y3: A,Ys4: list @ A] :
( X
!= ( product_Pair @ ( A > A > $o ) @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ R7 @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ ( cons @ A @ X3 @ Xs3 ) @ ( cons @ A @ Y3 @ Ys4 ) ) ) )
=> ( ! [R7: A > A > $o,Xs3: list @ A] :
( X
!= ( product_Pair @ ( A > A > $o ) @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ R7 @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ Xs3 @ ( nil @ A ) ) ) )
=> ~ ! [R7: A > A > $o,V3: A,Va3: list @ A] :
( X
!= ( product_Pair @ ( A > A > $o ) @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ R7 @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ ( nil @ A ) @ ( cons @ A @ V3 @ Va3 ) ) ) ) ) ) ).
% mergesort_by_rel_merge.cases
thf(fact_8069_mergesort__by__rel__split_Ocases,axiom,
! [A: $tType,X: product_prod @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( list @ A )] :
( ! [Xs1: list @ A,Xs22: list @ A] :
( X
!= ( product_Pair @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( list @ A ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ Xs1 @ Xs22 ) @ ( nil @ A ) ) )
=> ( ! [Xs1: list @ A,Xs22: list @ A,X3: A] :
( X
!= ( product_Pair @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( list @ A ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ Xs1 @ Xs22 ) @ ( cons @ A @ X3 @ ( nil @ A ) ) ) )
=> ~ ! [Xs1: list @ A,Xs22: list @ A,X1: A,X23: A,Xs3: list @ A] :
( X
!= ( product_Pair @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( list @ A ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ Xs1 @ Xs22 ) @ ( cons @ A @ X1 @ ( cons @ A @ X23 @ Xs3 ) ) ) ) ) ) ).
% mergesort_by_rel_split.cases
thf(fact_8070_map__tailrec__rev_Ocases,axiom,
! [A: $tType,B: $tType,X: product_prod @ ( A > B ) @ ( product_prod @ ( list @ A ) @ ( list @ B ) )] :
( ! [F3: A > B,Bs2: list @ B] :
( X
!= ( product_Pair @ ( A > B ) @ ( product_prod @ ( list @ A ) @ ( list @ B ) ) @ F3 @ ( product_Pair @ ( list @ A ) @ ( list @ B ) @ ( nil @ A ) @ Bs2 ) ) )
=> ~ ! [F3: A > B,A4: A,As: list @ A,Bs2: list @ B] :
( X
!= ( product_Pair @ ( A > B ) @ ( product_prod @ ( list @ A ) @ ( list @ B ) ) @ F3 @ ( product_Pair @ ( list @ A ) @ ( list @ B ) @ ( cons @ A @ A4 @ As ) @ Bs2 ) ) ) ) ).
% map_tailrec_rev.cases
thf(fact_8071_successively_Ocases,axiom,
! [A: $tType,X: product_prod @ ( A > A > $o ) @ ( list @ A )] :
( ! [P8: A > A > $o] :
( X
!= ( product_Pair @ ( A > A > $o ) @ ( list @ A ) @ P8 @ ( nil @ A ) ) )
=> ( ! [P8: A > A > $o,X3: A] :
( X
!= ( product_Pair @ ( A > A > $o ) @ ( list @ A ) @ P8 @ ( cons @ A @ X3 @ ( nil @ A ) ) ) )
=> ~ ! [P8: A > A > $o,X3: A,Y3: A,Xs3: list @ A] :
( X
!= ( product_Pair @ ( A > A > $o ) @ ( list @ A ) @ P8 @ ( cons @ A @ X3 @ ( cons @ A @ Y3 @ Xs3 ) ) ) ) ) ) ).
% successively.cases
thf(fact_8072_arg__min__list_Ocases,axiom,
! [B: $tType,A: $tType] :
( ( linorder @ B )
=> ! [X: product_prod @ ( A > B ) @ ( list @ A )] :
( ! [F3: A > B,X3: A] :
( X
!= ( product_Pair @ ( A > B ) @ ( list @ A ) @ F3 @ ( cons @ A @ X3 @ ( nil @ A ) ) ) )
=> ( ! [F3: A > B,X3: A,Y3: A,Zs2: list @ A] :
( X
!= ( product_Pair @ ( A > B ) @ ( list @ A ) @ F3 @ ( cons @ A @ X3 @ ( cons @ A @ Y3 @ Zs2 ) ) ) )
=> ~ ! [A4: A > B] :
( X
!= ( product_Pair @ ( A > B ) @ ( list @ A ) @ A4 @ ( nil @ A ) ) ) ) ) ) ).
% arg_min_list.cases
thf(fact_8073_sorted__wrt_Ocases,axiom,
! [A: $tType,X: product_prod @ ( A > A > $o ) @ ( list @ A )] :
( ! [P8: A > A > $o] :
( X
!= ( product_Pair @ ( A > A > $o ) @ ( list @ A ) @ P8 @ ( nil @ A ) ) )
=> ~ ! [P8: A > A > $o,X3: A,Ys4: list @ A] :
( X
!= ( product_Pair @ ( A > A > $o ) @ ( list @ A ) @ P8 @ ( cons @ A @ X3 @ Ys4 ) ) ) ) ).
% sorted_wrt.cases
thf(fact_8074_shuffles_Ocases,axiom,
! [A: $tType,X: product_prod @ ( list @ A ) @ ( list @ A )] :
( ! [Ys4: list @ A] :
( X
!= ( product_Pair @ ( list @ A ) @ ( list @ A ) @ ( nil @ A ) @ Ys4 ) )
=> ( ! [Xs3: list @ A] :
( X
!= ( product_Pair @ ( list @ A ) @ ( list @ A ) @ Xs3 @ ( nil @ A ) ) )
=> ~ ! [X3: A,Xs3: list @ A,Y3: A,Ys4: list @ A] :
( X
!= ( product_Pair @ ( list @ A ) @ ( list @ A ) @ ( cons @ A @ X3 @ Xs3 ) @ ( cons @ A @ Y3 @ Ys4 ) ) ) ) ) ).
% shuffles.cases
thf(fact_8075_list__assn_Ocases,axiom,
! [A: $tType,C: $tType,X: product_prod @ ( A > C > assn ) @ ( product_prod @ ( list @ A ) @ ( list @ C ) )] :
( ! [P8: A > C > assn] :
( X
!= ( product_Pair @ ( A > C > assn ) @ ( product_prod @ ( list @ A ) @ ( list @ C ) ) @ P8 @ ( product_Pair @ ( list @ A ) @ ( list @ C ) @ ( nil @ A ) @ ( nil @ C ) ) ) )
=> ( ! [P8: A > C > assn,A4: A,As: list @ A,C4: C,Cs2: list @ C] :
( X
!= ( product_Pair @ ( A > C > assn ) @ ( product_prod @ ( list @ A ) @ ( list @ C ) ) @ P8 @ ( product_Pair @ ( list @ A ) @ ( list @ C ) @ ( cons @ A @ A4 @ As ) @ ( cons @ C @ C4 @ Cs2 ) ) ) )
=> ( ! [Uu: A > C > assn,V3: A,Va3: list @ A] :
( X
!= ( product_Pair @ ( A > C > assn ) @ ( product_prod @ ( list @ A ) @ ( list @ C ) ) @ Uu @ ( product_Pair @ ( list @ A ) @ ( list @ C ) @ ( cons @ A @ V3 @ Va3 ) @ ( nil @ C ) ) ) )
=> ~ ! [Uu: A > C > assn,V3: C,Va3: list @ C] :
( X
!= ( product_Pair @ ( A > C > assn ) @ ( product_prod @ ( list @ A ) @ ( list @ C ) ) @ Uu @ ( product_Pair @ ( list @ A ) @ ( list @ C ) @ ( nil @ A ) @ ( cons @ C @ V3 @ Va3 ) ) ) ) ) ) ) ).
% list_assn.cases
thf(fact_8076_len__greater__imp__nonempty,axiom,
! [A: $tType,X: nat,L2: list @ A] :
( ( ord_less @ nat @ X @ ( size_size @ ( list @ A ) @ L2 ) )
=> ( L2
!= ( nil @ A ) ) ) ).
% len_greater_imp_nonempty
thf(fact_8077_list__decomp__1,axiom,
! [A: $tType,L2: list @ A] :
( ( ( size_size @ ( list @ A ) @ L2 )
= ( one_one @ nat ) )
=> ? [A4: A] :
( L2
= ( cons @ A @ A4 @ ( nil @ A ) ) ) ) ).
% list_decomp_1
thf(fact_8078_list__assn_Osimps_I1_J,axiom,
! [C: $tType,A: $tType,P: A > C > assn] :
( ( vEBT_List_list_assn @ A @ C @ P @ ( nil @ A ) @ ( nil @ C ) )
= ( one_one @ assn ) ) ).
% list_assn.simps(1)
thf(fact_8079_list__assn__simps_I1_J,axiom,
! [B: $tType,A: $tType,P: A > B > assn] :
( ( vEBT_List_list_assn @ A @ B @ P @ ( nil @ A ) @ ( nil @ B ) )
= ( one_one @ assn ) ) ).
% list_assn_simps(1)
thf(fact_8080_list__decomp__2,axiom,
! [A: $tType,L2: list @ A] :
( ( ( size_size @ ( list @ A ) @ L2 )
= ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) )
=> ? [A4: A,B3: A] :
( L2
= ( cons @ A @ A4 @ ( cons @ A @ B3 @ ( nil @ A ) ) ) ) ) ).
% list_decomp_2
thf(fact_8081_list__assn_Oelims,axiom,
! [C: $tType,A: $tType,X: A > C > assn,Xa: list @ A,Xb: list @ C,Y: assn] :
( ( ( vEBT_List_list_assn @ A @ C @ X @ Xa @ Xb )
= Y )
=> ( ( ( Xa
= ( nil @ A ) )
=> ( ( Xb
= ( nil @ C ) )
=> ( Y
!= ( one_one @ assn ) ) ) )
=> ( ! [A4: A,As: list @ A] :
( ( Xa
= ( cons @ A @ A4 @ As ) )
=> ! [C4: C,Cs2: list @ C] :
( ( Xb
= ( cons @ C @ C4 @ Cs2 ) )
=> ( Y
!= ( times_times @ assn @ ( X @ A4 @ C4 ) @ ( vEBT_List_list_assn @ A @ C @ X @ As @ Cs2 ) ) ) ) )
=> ( ( ? [V3: A,Va3: list @ A] :
( Xa
= ( cons @ A @ V3 @ Va3 ) )
=> ( ( Xb
= ( nil @ C ) )
=> ( Y
!= ( bot_bot @ assn ) ) ) )
=> ~ ( ( Xa
= ( nil @ A ) )
=> ( ? [V3: C,Va3: list @ C] :
( Xb
= ( cons @ C @ V3 @ Va3 ) )
=> ( Y
!= ( bot_bot @ assn ) ) ) ) ) ) ) ) ).
% list_assn.elims
thf(fact_8082_less__eq__char__simp,axiom,
! [B0: $o,B1: $o,B22: $o,B32: $o,B42: $o,B52: $o,B62: $o,B72: $o,C0: $o,C1: $o,C22: $o,C32: $o,C42: $o,C52: $o,C62: $o,C72: $o] :
( ( ord_less_eq @ char @ ( char2 @ B0 @ B1 @ B22 @ B32 @ B42 @ B52 @ B62 @ B72 ) @ ( char2 @ C0 @ C1 @ C22 @ C32 @ C42 @ C52 @ C62 @ C72 ) )
= ( ord_less_eq @ nat
@ ( foldr @ $o @ nat
@ ^ [B5: $o,K2: nat] : ( plus_plus @ nat @ ( zero_neq_one_of_bool @ nat @ B5 ) @ ( times_times @ nat @ K2 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) )
@ ( cons @ $o @ B0 @ ( cons @ $o @ B1 @ ( cons @ $o @ B22 @ ( cons @ $o @ B32 @ ( cons @ $o @ B42 @ ( cons @ $o @ B52 @ ( cons @ $o @ B62 @ ( cons @ $o @ B72 @ ( nil @ $o ) ) ) ) ) ) ) ) )
@ ( zero_zero @ nat ) )
@ ( foldr @ $o @ nat
@ ^ [B5: $o,K2: nat] : ( plus_plus @ nat @ ( zero_neq_one_of_bool @ nat @ B5 ) @ ( times_times @ nat @ K2 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) )
@ ( cons @ $o @ C0 @ ( cons @ $o @ C1 @ ( cons @ $o @ C22 @ ( cons @ $o @ C32 @ ( cons @ $o @ C42 @ ( cons @ $o @ C52 @ ( cons @ $o @ C62 @ ( cons @ $o @ C72 @ ( nil @ $o ) ) ) ) ) ) ) ) )
@ ( zero_zero @ nat ) ) ) ) ).
% less_eq_char_simp
thf(fact_8083_less__char__simp,axiom,
! [B0: $o,B1: $o,B22: $o,B32: $o,B42: $o,B52: $o,B62: $o,B72: $o,C0: $o,C1: $o,C22: $o,C32: $o,C42: $o,C52: $o,C62: $o,C72: $o] :
( ( ord_less @ char @ ( char2 @ B0 @ B1 @ B22 @ B32 @ B42 @ B52 @ B62 @ B72 ) @ ( char2 @ C0 @ C1 @ C22 @ C32 @ C42 @ C52 @ C62 @ C72 ) )
= ( ord_less @ nat
@ ( foldr @ $o @ nat
@ ^ [B5: $o,K2: nat] : ( plus_plus @ nat @ ( zero_neq_one_of_bool @ nat @ B5 ) @ ( times_times @ nat @ K2 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) )
@ ( cons @ $o @ B0 @ ( cons @ $o @ B1 @ ( cons @ $o @ B22 @ ( cons @ $o @ B32 @ ( cons @ $o @ B42 @ ( cons @ $o @ B52 @ ( cons @ $o @ B62 @ ( cons @ $o @ B72 @ ( nil @ $o ) ) ) ) ) ) ) ) )
@ ( zero_zero @ nat ) )
@ ( foldr @ $o @ nat
@ ^ [B5: $o,K2: nat] : ( plus_plus @ nat @ ( zero_neq_one_of_bool @ nat @ B5 ) @ ( times_times @ nat @ K2 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) )
@ ( cons @ $o @ C0 @ ( cons @ $o @ C1 @ ( cons @ $o @ C22 @ ( cons @ $o @ C32 @ ( cons @ $o @ C42 @ ( cons @ $o @ C52 @ ( cons @ $o @ C62 @ ( cons @ $o @ C72 @ ( nil @ $o ) ) ) ) ) ) ) ) )
@ ( zero_zero @ nat ) ) ) ) ).
% less_char_simp
thf(fact_8084_upt__0__eq__Nil__conv,axiom,
! [J: nat] :
( ( ( upt @ ( zero_zero @ nat ) @ J )
= ( nil @ nat ) )
= ( J
= ( zero_zero @ nat ) ) ) ).
% upt_0_eq_Nil_conv
thf(fact_8085_upt__conv__Nil,axiom,
! [J: nat,I: nat] :
( ( ord_less_eq @ nat @ J @ I )
=> ( ( upt @ I @ J )
= ( nil @ nat ) ) ) ).
% upt_conv_Nil
thf(fact_8086_upt__eq__Nil__conv,axiom,
! [I: nat,J: nat] :
( ( ( upt @ I @ J )
= ( nil @ nat ) )
= ( ( J
= ( zero_zero @ nat ) )
| ( ord_less_eq @ nat @ J @ I ) ) ) ).
% upt_eq_Nil_conv
thf(fact_8087_upt__rec__numeral,axiom,
! [M: num,N3: num] :
( ( ( ord_less @ nat @ ( numeral_numeral @ nat @ M ) @ ( numeral_numeral @ nat @ N3 ) )
=> ( ( upt @ ( numeral_numeral @ nat @ M ) @ ( numeral_numeral @ nat @ N3 ) )
= ( cons @ nat @ ( numeral_numeral @ nat @ M ) @ ( upt @ ( suc @ ( numeral_numeral @ nat @ M ) ) @ ( numeral_numeral @ nat @ N3 ) ) ) ) )
& ( ~ ( ord_less @ nat @ ( numeral_numeral @ nat @ M ) @ ( numeral_numeral @ nat @ N3 ) )
=> ( ( upt @ ( numeral_numeral @ nat @ M ) @ ( numeral_numeral @ nat @ N3 ) )
= ( nil @ nat ) ) ) ) ).
% upt_rec_numeral
thf(fact_8088_mergesort__by__rel_Ocases,axiom,
! [A: $tType,X: product_prod @ ( A > A > $o ) @ ( list @ A )] :
~ ! [R7: A > A > $o,Xs3: list @ A] :
( X
!= ( product_Pair @ ( A > A > $o ) @ ( list @ A ) @ R7 @ Xs3 ) ) ).
% mergesort_by_rel.cases
thf(fact_8089_upt__rec,axiom,
( upt
= ( ^ [I3: nat,J3: nat] : ( if @ ( list @ nat ) @ ( ord_less @ nat @ I3 @ J3 ) @ ( cons @ nat @ I3 @ ( upt @ ( suc @ I3 ) @ J3 ) ) @ ( nil @ nat ) ) ) ) ).
% upt_rec
thf(fact_8090_less__char__def,axiom,
( ( ord_less @ char )
= ( ^ [C12: char,C23: char] : ( ord_less @ nat @ ( comm_s6883823935334413003f_char @ nat @ C12 ) @ ( comm_s6883823935334413003f_char @ nat @ C23 ) ) ) ) ).
% less_char_def
thf(fact_8091_less__eq__char__def,axiom,
( ( ord_less_eq @ char )
= ( ^ [C12: char,C23: char] : ( ord_less_eq @ nat @ ( comm_s6883823935334413003f_char @ nat @ C12 ) @ ( comm_s6883823935334413003f_char @ nat @ C23 ) ) ) ) ).
% less_eq_char_def
thf(fact_8092_list__assn_Opelims,axiom,
! [A: $tType,C: $tType,X: A > C > assn,Xa: list @ A,Xb: list @ C,Y: assn] :
( ( ( vEBT_List_list_assn @ A @ C @ X @ Xa @ Xb )
= Y )
=> ( ( accp @ ( product_prod @ ( A > C > assn ) @ ( product_prod @ ( list @ A ) @ ( list @ C ) ) ) @ ( vEBT_L4249061453398456502sn_rel @ A @ C ) @ ( product_Pair @ ( A > C > assn ) @ ( product_prod @ ( list @ A ) @ ( list @ C ) ) @ X @ ( product_Pair @ ( list @ A ) @ ( list @ C ) @ Xa @ Xb ) ) )
=> ( ( ( Xa
= ( nil @ A ) )
=> ( ( Xb
= ( nil @ C ) )
=> ( ( Y
= ( one_one @ assn ) )
=> ~ ( accp @ ( product_prod @ ( A > C > assn ) @ ( product_prod @ ( list @ A ) @ ( list @ C ) ) ) @ ( vEBT_L4249061453398456502sn_rel @ A @ C ) @ ( product_Pair @ ( A > C > assn ) @ ( product_prod @ ( list @ A ) @ ( list @ C ) ) @ X @ ( product_Pair @ ( list @ A ) @ ( list @ C ) @ ( nil @ A ) @ ( nil @ C ) ) ) ) ) ) )
=> ( ! [A4: A,As: list @ A] :
( ( Xa
= ( cons @ A @ A4 @ As ) )
=> ! [C4: C,Cs2: list @ C] :
( ( Xb
= ( cons @ C @ C4 @ Cs2 ) )
=> ( ( Y
= ( times_times @ assn @ ( X @ A4 @ C4 ) @ ( vEBT_List_list_assn @ A @ C @ X @ As @ Cs2 ) ) )
=> ~ ( accp @ ( product_prod @ ( A > C > assn ) @ ( product_prod @ ( list @ A ) @ ( list @ C ) ) ) @ ( vEBT_L4249061453398456502sn_rel @ A @ C ) @ ( product_Pair @ ( A > C > assn ) @ ( product_prod @ ( list @ A ) @ ( list @ C ) ) @ X @ ( product_Pair @ ( list @ A ) @ ( list @ C ) @ ( cons @ A @ A4 @ As ) @ ( cons @ C @ C4 @ Cs2 ) ) ) ) ) ) )
=> ( ! [V3: A,Va3: list @ A] :
( ( Xa
= ( cons @ A @ V3 @ Va3 ) )
=> ( ( Xb
= ( nil @ C ) )
=> ( ( Y
= ( bot_bot @ assn ) )
=> ~ ( accp @ ( product_prod @ ( A > C > assn ) @ ( product_prod @ ( list @ A ) @ ( list @ C ) ) ) @ ( vEBT_L4249061453398456502sn_rel @ A @ C ) @ ( product_Pair @ ( A > C > assn ) @ ( product_prod @ ( list @ A ) @ ( list @ C ) ) @ X @ ( product_Pair @ ( list @ A ) @ ( list @ C ) @ ( cons @ A @ V3 @ Va3 ) @ ( nil @ C ) ) ) ) ) ) )
=> ~ ( ( Xa
= ( nil @ A ) )
=> ! [V3: C,Va3: list @ C] :
( ( Xb
= ( cons @ C @ V3 @ Va3 ) )
=> ( ( Y
= ( bot_bot @ assn ) )
=> ~ ( accp @ ( product_prod @ ( A > C > assn ) @ ( product_prod @ ( list @ A ) @ ( list @ C ) ) ) @ ( vEBT_L4249061453398456502sn_rel @ A @ C ) @ ( product_Pair @ ( A > C > assn ) @ ( product_prod @ ( list @ A ) @ ( list @ C ) ) @ X @ ( product_Pair @ ( list @ A ) @ ( list @ C ) @ ( nil @ A ) @ ( cons @ C @ V3 @ Va3 ) ) ) ) ) ) ) ) ) ) ) ) ).
% list_assn.pelims
thf(fact_8093_in__set__simps_I2_J,axiom,
! [A: $tType,X: A,Y: A] :
( ( member @ A @ X @ ( set2 @ A @ ( cons @ A @ Y @ ( nil @ A ) ) ) )
= ( X = Y ) ) ).
% in_set_simps(2)
thf(fact_8094_in__set__simps_I1_J,axiom,
! [A: $tType,X: A,Y: A,Z: A,Ys: list @ A] :
( ( member @ A @ X @ ( set2 @ A @ ( cons @ A @ Y @ ( cons @ A @ Z @ Ys ) ) ) )
= ( ( X = Y )
| ( member @ A @ X @ ( set2 @ A @ ( cons @ A @ Z @ Ys ) ) ) ) ) ).
% in_set_simps(1)
thf(fact_8095_in__set__simps_I3_J,axiom,
! [A: $tType,X: A] :
~ ( member @ A @ X @ ( set2 @ A @ ( nil @ A ) ) ) ).
% in_set_simps(3)
thf(fact_8096_String_Ochar__of__ascii__of,axiom,
! [C2: char] :
( ( comm_s6883823935334413003f_char @ nat @ ( ascii_of @ C2 ) )
= ( bit_se2584673776208193580ke_bit @ nat @ ( numeral_numeral @ nat @ ( bit1 @ ( bit1 @ one2 ) ) ) @ ( comm_s6883823935334413003f_char @ nat @ C2 ) ) ) ).
% String.char_of_ascii_of
thf(fact_8097_concat__inth,axiom,
! [A: $tType,Xs2: list @ A,X: A,Ys: list @ A] :
( ( nth @ A @ ( append @ A @ Xs2 @ ( append @ A @ ( cons @ A @ X @ ( nil @ A ) ) @ Ys ) ) @ ( size_size @ ( list @ A ) @ Xs2 ) )
= X ) ).
% concat_inth
thf(fact_8098_empty__append__eq__id,axiom,
! [A: $tType] :
( ( append @ A @ ( nil @ A ) )
= ( ^ [X2: list @ A] : X2 ) ) ).
% empty_append_eq_id
thf(fact_8099_op__conc__empty__img__id,axiom,
! [A: $tType,L6: set @ ( list @ A )] :
( ( image @ ( list @ A ) @ ( list @ A ) @ ( append @ A @ ( nil @ A ) ) @ L6 )
= L6 ) ).
% op_conc_empty_img_id
thf(fact_8100_append__eq__append__conv,axiom,
! [A: $tType,Xs2: list @ A,Ys: list @ A,Us: list @ A,Vs: list @ A] :
( ( ( ( size_size @ ( list @ A ) @ Xs2 )
= ( size_size @ ( list @ A ) @ Ys ) )
| ( ( size_size @ ( list @ A ) @ Us )
= ( size_size @ ( list @ A ) @ Vs ) ) )
=> ( ( ( append @ A @ Xs2 @ Us )
= ( append @ A @ Ys @ Vs ) )
= ( ( Xs2 = Ys )
& ( Us = Vs ) ) ) ) ).
% append_eq_append_conv
thf(fact_8101_upt__merge,axiom,
! [I: nat,J: nat,K: nat] :
( ( ( ord_less_eq @ nat @ I @ J )
& ( ord_less_eq @ nat @ J @ K ) )
=> ( ( append @ nat @ ( upt @ I @ J ) @ ( upt @ J @ K ) )
= ( upt @ I @ K ) ) ) ).
% upt_merge
thf(fact_8102_list__ee__eq__leel_I1_J,axiom,
! [A: $tType,E1: A,E22: A,L12: list @ A,E12: A,E23: A,L23: list @ A] :
( ( ( cons @ A @ E1 @ ( cons @ A @ E22 @ ( nil @ A ) ) )
= ( append @ A @ L12 @ ( cons @ A @ E12 @ ( cons @ A @ E23 @ L23 ) ) ) )
= ( ( L12
= ( nil @ A ) )
& ( E1 = E12 )
& ( E22 = E23 )
& ( L23
= ( nil @ A ) ) ) ) ).
% list_ee_eq_leel(1)
thf(fact_8103_list__ee__eq__leel_I2_J,axiom,
! [A: $tType,L12: list @ A,E12: A,E23: A,L23: list @ A,E1: A,E22: A] :
( ( ( append @ A @ L12 @ ( cons @ A @ E12 @ ( cons @ A @ E23 @ L23 ) ) )
= ( cons @ A @ E1 @ ( cons @ A @ E22 @ ( nil @ A ) ) ) )
= ( ( L12
= ( nil @ A ) )
& ( E1 = E12 )
& ( E22 = E23 )
& ( L23
= ( nil @ A ) ) ) ) ).
% list_ee_eq_leel(2)
thf(fact_8104_list__se__match_I1_J,axiom,
! [A: $tType,L12: list @ A,L23: list @ A,A2: A] :
( ( L12
!= ( nil @ A ) )
=> ( ( ( append @ A @ L12 @ L23 )
= ( cons @ A @ A2 @ ( nil @ A ) ) )
= ( ( L12
= ( cons @ A @ A2 @ ( nil @ A ) ) )
& ( L23
= ( nil @ A ) ) ) ) ) ).
% list_se_match(1)
thf(fact_8105_list__se__match_I2_J,axiom,
! [A: $tType,L23: list @ A,L12: list @ A,A2: A] :
( ( L23
!= ( nil @ A ) )
=> ( ( ( append @ A @ L12 @ L23 )
= ( cons @ A @ A2 @ ( nil @ A ) ) )
= ( ( L12
= ( nil @ A ) )
& ( L23
= ( cons @ A @ A2 @ ( nil @ A ) ) ) ) ) ) ).
% list_se_match(2)
thf(fact_8106_list__se__match_I3_J,axiom,
! [A: $tType,L12: list @ A,A2: A,L23: list @ A] :
( ( L12
!= ( nil @ A ) )
=> ( ( ( cons @ A @ A2 @ ( nil @ A ) )
= ( append @ A @ L12 @ L23 ) )
= ( ( L12
= ( cons @ A @ A2 @ ( nil @ A ) ) )
& ( L23
= ( nil @ A ) ) ) ) ) ).
% list_se_match(3)
thf(fact_8107_list__se__match_I4_J,axiom,
! [A: $tType,L23: list @ A,A2: A,L12: list @ A] :
( ( L23
!= ( nil @ A ) )
=> ( ( ( cons @ A @ A2 @ ( nil @ A ) )
= ( append @ A @ L12 @ L23 ) )
= ( ( L12
= ( nil @ A ) )
& ( L23
= ( cons @ A @ A2 @ ( nil @ A ) ) ) ) ) ) ).
% list_se_match(4)
thf(fact_8108_list__e__eq__lel_I1_J,axiom,
! [A: $tType,E: A,L12: list @ A,E6: A,L23: list @ A] :
( ( ( cons @ A @ E @ ( nil @ A ) )
= ( append @ A @ L12 @ ( cons @ A @ E6 @ L23 ) ) )
= ( ( L12
= ( nil @ A ) )
& ( E6 = E )
& ( L23
= ( nil @ A ) ) ) ) ).
% list_e_eq_lel(1)
thf(fact_8109_list__e__eq__lel_I2_J,axiom,
! [A: $tType,L12: list @ A,E6: A,L23: list @ A,E: A] :
( ( ( append @ A @ L12 @ ( cons @ A @ E6 @ L23 ) )
= ( cons @ A @ E @ ( nil @ A ) ) )
= ( ( L12
= ( nil @ A ) )
& ( E6 = E )
& ( L23
= ( nil @ A ) ) ) ) ).
% list_e_eq_lel(2)
thf(fact_8110_length__append,axiom,
! [A: $tType,Xs2: list @ A,Ys: list @ A] :
( ( size_size @ ( list @ A ) @ ( append @ A @ Xs2 @ Ys ) )
= ( plus_plus @ nat @ ( size_size @ ( list @ A ) @ Xs2 ) @ ( size_size @ ( list @ A ) @ Ys ) ) ) ).
% length_append
thf(fact_8111_size__list__append,axiom,
! [A: $tType,F2: A > nat,Xs2: list @ A,Ys: list @ A] :
( ( size_list @ A @ F2 @ ( append @ A @ Xs2 @ Ys ) )
= ( plus_plus @ nat @ ( size_list @ A @ F2 @ Xs2 ) @ ( size_list @ A @ F2 @ Ys ) ) ) ).
% size_list_append
thf(fact_8112_nth__append__first,axiom,
! [A: $tType,I: nat,L2: list @ A,L4: list @ A] :
( ( ord_less @ nat @ I @ ( size_size @ ( list @ A ) @ L2 ) )
=> ( ( nth @ A @ ( append @ A @ L2 @ L4 ) @ I )
= ( nth @ A @ L2 @ I ) ) ) ).
% nth_append_first
thf(fact_8113_nth__append__length,axiom,
! [A: $tType,Xs2: list @ A,X: A,Ys: list @ A] :
( ( nth @ A @ ( append @ A @ Xs2 @ ( cons @ A @ X @ Ys ) ) @ ( size_size @ ( list @ A ) @ Xs2 ) )
= X ) ).
% nth_append_length
thf(fact_8114_nth__append__length__plus,axiom,
! [A: $tType,Xs2: list @ A,Ys: list @ A,N3: nat] :
( ( nth @ A @ ( append @ A @ Xs2 @ Ys ) @ ( plus_plus @ nat @ ( size_size @ ( list @ A ) @ Xs2 ) @ N3 ) )
= ( nth @ A @ Ys @ N3 ) ) ).
% nth_append_length_plus
thf(fact_8115_list__update__length,axiom,
! [A: $tType,Xs2: list @ A,X: A,Ys: list @ A,Y: A] :
( ( list_update @ A @ ( append @ A @ Xs2 @ ( cons @ A @ X @ Ys ) ) @ ( size_size @ ( list @ A ) @ Xs2 ) @ Y )
= ( append @ A @ Xs2 @ ( cons @ A @ Y @ Ys ) ) ) ).
% list_update_length
thf(fact_8116_list__assn__aux__append2,axiom,
! [A: $tType,B: $tType,L23: list @ A,L24: list @ B,P: A > B > assn,L12: list @ A,L13: list @ B] :
( ( ( size_size @ ( list @ A ) @ L23 )
= ( size_size @ ( list @ B ) @ L24 ) )
=> ( ( vEBT_List_list_assn @ A @ B @ P @ ( append @ A @ L12 @ L23 ) @ ( append @ B @ L13 @ L24 ) )
= ( times_times @ assn @ ( vEBT_List_list_assn @ A @ B @ P @ L12 @ L13 ) @ ( vEBT_List_list_assn @ A @ B @ P @ L23 @ L24 ) ) ) ) ).
% list_assn_aux_append2
thf(fact_8117_list__assn__aux__append,axiom,
! [A: $tType,B: $tType,L12: list @ A,L13: list @ B,P: A > B > assn,L23: list @ A,L24: list @ B] :
( ( ( size_size @ ( list @ A ) @ L12 )
= ( size_size @ ( list @ B ) @ L13 ) )
=> ( ( vEBT_List_list_assn @ A @ B @ P @ ( append @ A @ L12 @ L23 ) @ ( append @ B @ L13 @ L24 ) )
= ( times_times @ assn @ ( vEBT_List_list_assn @ A @ B @ P @ L12 @ L13 ) @ ( vEBT_List_list_assn @ A @ B @ P @ L23 @ L24 ) ) ) ) ).
% list_assn_aux_append
thf(fact_8118_same__length__different,axiom,
! [A: $tType,Xs2: list @ A,Ys: list @ A] :
( ( Xs2 != Ys )
=> ( ( ( size_size @ ( list @ A ) @ Xs2 )
= ( size_size @ ( list @ A ) @ Ys ) )
=> ? [Pre: list @ A,X3: A,Xs5: list @ A,Y3: A,Ys5: list @ A] :
( ( X3 != Y3 )
& ( Xs2
= ( append @ A @ Pre @ ( append @ A @ ( cons @ A @ X3 @ ( nil @ A ) ) @ Xs5 ) ) )
& ( Ys
= ( append @ A @ Pre @ ( append @ A @ ( cons @ A @ Y3 @ ( nil @ A ) ) @ Ys5 ) ) ) ) ) ) ).
% same_length_different
thf(fact_8119_list__append__eq__Cons__cases,axiom,
! [A: $tType,Ys: list @ A,Zs: list @ A,X: A,Xs2: list @ A] :
( ( ( append @ A @ Ys @ Zs )
= ( cons @ A @ X @ Xs2 ) )
=> ( ( ( Ys
= ( nil @ A ) )
=> ( Zs
!= ( cons @ A @ X @ Xs2 ) ) )
=> ~ ! [Ys5: list @ A] :
( ( Ys
= ( cons @ A @ X @ Ys5 ) )
=> ( ( append @ A @ Ys5 @ Zs )
!= Xs2 ) ) ) ) ).
% list_append_eq_Cons_cases
thf(fact_8120_list__Cons__eq__append__cases,axiom,
! [A: $tType,X: A,Xs2: list @ A,Ys: list @ A,Zs: list @ A] :
( ( ( cons @ A @ X @ Xs2 )
= ( append @ A @ Ys @ Zs ) )
=> ( ( ( Ys
= ( nil @ A ) )
=> ( Zs
!= ( cons @ A @ X @ Xs2 ) ) )
=> ~ ! [Ys5: list @ A] :
( ( Ys
= ( cons @ A @ X @ Ys5 ) )
=> ( ( append @ A @ Ys5 @ Zs )
!= Xs2 ) ) ) ) ).
% list_Cons_eq_append_cases
thf(fact_8121_rev__nonempty__induct2_H,axiom,
! [A: $tType,B: $tType,Xs2: list @ A,Ys: list @ B,P: ( list @ A ) > ( list @ B ) > $o] :
( ( Xs2
!= ( nil @ A ) )
=> ( ( Ys
!= ( nil @ B ) )
=> ( ! [X3: A,Y3: B] : ( P @ ( cons @ A @ X3 @ ( nil @ A ) ) @ ( cons @ B @ Y3 @ ( nil @ B ) ) )
=> ( ! [X3: A,Xs3: list @ A,Y3: B] :
( ( Xs3
!= ( nil @ A ) )
=> ( P @ ( append @ A @ Xs3 @ ( cons @ A @ X3 @ ( nil @ A ) ) ) @ ( cons @ B @ Y3 @ ( nil @ B ) ) ) )
=> ( ! [X3: A,Y3: B,Ys4: list @ B] :
( ( Ys4
!= ( nil @ B ) )
=> ( P @ ( cons @ A @ X3 @ ( nil @ A ) ) @ ( append @ B @ Ys4 @ ( cons @ B @ Y3 @ ( nil @ B ) ) ) ) )
=> ( ! [X3: A,Xs3: list @ A,Y3: B,Ys4: list @ B] :
( ( P @ Xs3 @ Ys4 )
=> ( ( Xs3
!= ( nil @ A ) )
=> ( ( Ys4
!= ( nil @ B ) )
=> ( P @ ( append @ A @ Xs3 @ ( cons @ A @ X3 @ ( nil @ A ) ) ) @ ( append @ B @ Ys4 @ ( cons @ B @ Y3 @ ( nil @ B ) ) ) ) ) ) )
=> ( P @ Xs2 @ Ys ) ) ) ) ) ) ) ).
% rev_nonempty_induct2'
thf(fact_8122_neq__Nil__rev__conv,axiom,
! [A: $tType,L2: list @ A] :
( ( L2
!= ( nil @ A ) )
= ( ? [Xs: list @ A,X2: A] :
( L2
= ( append @ A @ Xs @ ( cons @ A @ X2 @ ( nil @ A ) ) ) ) ) ) ).
% neq_Nil_rev_conv
thf(fact_8123_rev__induct2_H,axiom,
! [A: $tType,B: $tType,P: ( list @ A ) > ( list @ B ) > $o,Xs2: list @ A,Ys: list @ B] :
( ( P @ ( nil @ A ) @ ( nil @ B ) )
=> ( ! [X3: A,Xs3: list @ A] : ( P @ ( append @ A @ Xs3 @ ( cons @ A @ X3 @ ( nil @ A ) ) ) @ ( nil @ B ) )
=> ( ! [Y3: B,Ys4: list @ B] : ( P @ ( nil @ A ) @ ( append @ B @ Ys4 @ ( cons @ B @ Y3 @ ( nil @ B ) ) ) )
=> ( ! [X3: A,Xs3: list @ A,Y3: B,Ys4: list @ B] :
( ( P @ Xs3 @ Ys4 )
=> ( P @ ( append @ A @ Xs3 @ ( cons @ A @ X3 @ ( nil @ A ) ) ) @ ( append @ B @ Ys4 @ ( cons @ B @ Y3 @ ( nil @ B ) ) ) ) )
=> ( P @ Xs2 @ Ys ) ) ) ) ) ).
% rev_induct2'
thf(fact_8124_neq__Nil__revE,axiom,
! [A: $tType,L2: list @ A] :
( ( L2
!= ( nil @ A ) )
=> ~ ! [Ll2: list @ A,E2: A] :
( L2
!= ( append @ A @ Ll2 @ ( cons @ A @ E2 @ ( nil @ A ) ) ) ) ) ).
% neq_Nil_revE
thf(fact_8125_list__match__lel__lel,axiom,
! [A: $tType,C1: list @ A,Qs: A,C22: list @ A,C13: list @ A,Qs2: A,C24: list @ A] :
( ( ( append @ A @ C1 @ ( cons @ A @ Qs @ C22 ) )
= ( append @ A @ C13 @ ( cons @ A @ Qs2 @ C24 ) ) )
=> ( ! [C21: list @ A] :
( ( C1
= ( append @ A @ C13 @ ( cons @ A @ Qs2 @ C21 ) ) )
=> ( C24
!= ( append @ A @ C21 @ ( cons @ A @ Qs @ C22 ) ) ) )
=> ( ( ( C13 = C1 )
=> ( ( Qs2 = Qs )
=> ( C24 != C22 ) ) )
=> ~ ! [C212: list @ A] :
( ( C13
= ( append @ A @ C1 @ ( cons @ A @ Qs @ C212 ) ) )
=> ( C22
!= ( append @ A @ C212 @ ( cons @ A @ Qs2 @ C24 ) ) ) ) ) ) ) ).
% list_match_lel_lel
thf(fact_8126_split__list,axiom,
! [A: $tType,X: A,Xs2: list @ A] :
( ( member @ A @ X @ ( set2 @ A @ Xs2 ) )
=> ? [Ys4: list @ A,Zs2: list @ A] :
( Xs2
= ( append @ A @ Ys4 @ ( cons @ A @ X @ Zs2 ) ) ) ) ).
% split_list
thf(fact_8127_split__list__last,axiom,
! [A: $tType,X: A,Xs2: list @ A] :
( ( member @ A @ X @ ( set2 @ A @ Xs2 ) )
=> ? [Ys4: list @ A,Zs2: list @ A] :
( ( Xs2
= ( append @ A @ Ys4 @ ( cons @ A @ X @ Zs2 ) ) )
& ~ ( member @ A @ X @ ( set2 @ A @ Zs2 ) ) ) ) ).
% split_list_last
thf(fact_8128_split__list__prop,axiom,
! [A: $tType,Xs2: list @ A,P: A > $o] :
( ? [X5: A] :
( ( member @ A @ X5 @ ( set2 @ A @ Xs2 ) )
& ( P @ X5 ) )
=> ? [Ys4: list @ A,X3: A] :
( ? [Zs2: list @ A] :
( Xs2
= ( append @ A @ Ys4 @ ( cons @ A @ X3 @ Zs2 ) ) )
& ( P @ X3 ) ) ) ).
% split_list_prop
thf(fact_8129_split__list__first,axiom,
! [A: $tType,X: A,Xs2: list @ A] :
( ( member @ A @ X @ ( set2 @ A @ Xs2 ) )
=> ? [Ys4: list @ A,Zs2: list @ A] :
( ( Xs2
= ( append @ A @ Ys4 @ ( cons @ A @ X @ Zs2 ) ) )
& ~ ( member @ A @ X @ ( set2 @ A @ Ys4 ) ) ) ) ).
% split_list_first
thf(fact_8130_split__list__propE,axiom,
! [A: $tType,Xs2: list @ A,P: A > $o] :
( ? [X5: A] :
( ( member @ A @ X5 @ ( set2 @ A @ Xs2 ) )
& ( P @ X5 ) )
=> ~ ! [Ys4: list @ A,X3: A] :
( ? [Zs2: list @ A] :
( Xs2
= ( append @ A @ Ys4 @ ( cons @ A @ X3 @ Zs2 ) ) )
=> ~ ( P @ X3 ) ) ) ).
% split_list_propE
thf(fact_8131_append__Cons__eq__iff,axiom,
! [A: $tType,X: A,Xs2: list @ A,Ys: list @ A,Xs4: list @ A,Ys6: list @ A] :
( ~ ( member @ A @ X @ ( set2 @ A @ Xs2 ) )
=> ( ~ ( member @ A @ X @ ( set2 @ A @ Ys ) )
=> ( ( ( append @ A @ Xs2 @ ( cons @ A @ X @ Ys ) )
= ( append @ A @ Xs4 @ ( cons @ A @ X @ Ys6 ) ) )
= ( ( Xs2 = Xs4 )
& ( Ys = Ys6 ) ) ) ) ) ).
% append_Cons_eq_iff
thf(fact_8132_in__set__conv__decomp,axiom,
! [A: $tType,X: A,Xs2: list @ A] :
( ( member @ A @ X @ ( set2 @ A @ Xs2 ) )
= ( ? [Ys3: list @ A,Zs3: list @ A] :
( Xs2
= ( append @ A @ Ys3 @ ( cons @ A @ X @ Zs3 ) ) ) ) ) ).
% in_set_conv_decomp
thf(fact_8133_split__list__last__prop,axiom,
! [A: $tType,Xs2: list @ A,P: A > $o] :
( ? [X5: A] :
( ( member @ A @ X5 @ ( set2 @ A @ Xs2 ) )
& ( P @ X5 ) )
=> ? [Ys4: list @ A,X3: A,Zs2: list @ A] :
( ( Xs2
= ( append @ A @ Ys4 @ ( cons @ A @ X3 @ Zs2 ) ) )
& ( P @ X3 )
& ! [Xa2: A] :
( ( member @ A @ Xa2 @ ( set2 @ A @ Zs2 ) )
=> ~ ( P @ Xa2 ) ) ) ) ).
% split_list_last_prop
thf(fact_8134_split__list__first__prop,axiom,
! [A: $tType,Xs2: list @ A,P: A > $o] :
( ? [X5: A] :
( ( member @ A @ X5 @ ( set2 @ A @ Xs2 ) )
& ( P @ X5 ) )
=> ? [Ys4: list @ A,X3: A] :
( ? [Zs2: list @ A] :
( Xs2
= ( append @ A @ Ys4 @ ( cons @ A @ X3 @ Zs2 ) ) )
& ( P @ X3 )
& ! [Xa2: A] :
( ( member @ A @ Xa2 @ ( set2 @ A @ Ys4 ) )
=> ~ ( P @ Xa2 ) ) ) ) ).
% split_list_first_prop
thf(fact_8135_split__list__last__propE,axiom,
! [A: $tType,Xs2: list @ A,P: A > $o] :
( ? [X5: A] :
( ( member @ A @ X5 @ ( set2 @ A @ Xs2 ) )
& ( P @ X5 ) )
=> ~ ! [Ys4: list @ A,X3: A,Zs2: list @ A] :
( ( Xs2
= ( append @ A @ Ys4 @ ( cons @ A @ X3 @ Zs2 ) ) )
=> ( ( P @ X3 )
=> ~ ! [Xa2: A] :
( ( member @ A @ Xa2 @ ( set2 @ A @ Zs2 ) )
=> ~ ( P @ Xa2 ) ) ) ) ) ).
% split_list_last_propE
thf(fact_8136_split__list__first__propE,axiom,
! [A: $tType,Xs2: list @ A,P: A > $o] :
( ? [X5: A] :
( ( member @ A @ X5 @ ( set2 @ A @ Xs2 ) )
& ( P @ X5 ) )
=> ~ ! [Ys4: list @ A,X3: A] :
( ? [Zs2: list @ A] :
( Xs2
= ( append @ A @ Ys4 @ ( cons @ A @ X3 @ Zs2 ) ) )
=> ( ( P @ X3 )
=> ~ ! [Xa2: A] :
( ( member @ A @ Xa2 @ ( set2 @ A @ Ys4 ) )
=> ~ ( P @ Xa2 ) ) ) ) ) ).
% split_list_first_propE
thf(fact_8137_in__set__conv__decomp__last,axiom,
! [A: $tType,X: A,Xs2: list @ A] :
( ( member @ A @ X @ ( set2 @ A @ Xs2 ) )
= ( ? [Ys3: list @ A,Zs3: list @ A] :
( ( Xs2
= ( append @ A @ Ys3 @ ( cons @ A @ X @ Zs3 ) ) )
& ~ ( member @ A @ X @ ( set2 @ A @ Zs3 ) ) ) ) ) ).
% in_set_conv_decomp_last
thf(fact_8138_in__set__conv__decomp__first,axiom,
! [A: $tType,X: A,Xs2: list @ A] :
( ( member @ A @ X @ ( set2 @ A @ Xs2 ) )
= ( ? [Ys3: list @ A,Zs3: list @ A] :
( ( Xs2
= ( append @ A @ Ys3 @ ( cons @ A @ X @ Zs3 ) ) )
& ~ ( member @ A @ X @ ( set2 @ A @ Ys3 ) ) ) ) ) ).
% in_set_conv_decomp_first
thf(fact_8139_split__list__last__prop__iff,axiom,
! [A: $tType,Xs2: list @ A,P: A > $o] :
( ( ? [X2: A] :
( ( member @ A @ X2 @ ( set2 @ A @ Xs2 ) )
& ( P @ X2 ) ) )
= ( ? [Ys3: list @ A,X2: A,Zs3: list @ A] :
( ( Xs2
= ( append @ A @ Ys3 @ ( cons @ A @ X2 @ Zs3 ) ) )
& ( P @ X2 )
& ! [Y2: A] :
( ( member @ A @ Y2 @ ( set2 @ A @ Zs3 ) )
=> ~ ( P @ Y2 ) ) ) ) ) ).
% split_list_last_prop_iff
thf(fact_8140_split__list__first__prop__iff,axiom,
! [A: $tType,Xs2: list @ A,P: A > $o] :
( ( ? [X2: A] :
( ( member @ A @ X2 @ ( set2 @ A @ Xs2 ) )
& ( P @ X2 ) ) )
= ( ? [Ys3: list @ A,X2: A] :
( ? [Zs3: list @ A] :
( Xs2
= ( append @ A @ Ys3 @ ( cons @ A @ X2 @ Zs3 ) ) )
& ( P @ X2 )
& ! [Y2: A] :
( ( member @ A @ Y2 @ ( set2 @ A @ Ys3 ) )
=> ~ ( P @ Y2 ) ) ) ) ) ).
% split_list_first_prop_iff
thf(fact_8141_xy__in__set__cases,axiom,
! [A: $tType,X: A,L2: list @ A,Y: A] :
( ( member @ A @ X @ ( set2 @ A @ L2 ) )
=> ( ( member @ A @ Y @ ( set2 @ A @ L2 ) )
=> ( ( ( X = Y )
=> ! [L1: list @ A,L22: list @ A] :
( L2
!= ( append @ A @ L1 @ ( cons @ A @ Y @ L22 ) ) ) )
=> ( ( ( X != Y )
=> ! [L1: list @ A,L22: list @ A,L32: list @ A] :
( L2
!= ( append @ A @ L1 @ ( cons @ A @ X @ ( append @ A @ L22 @ ( cons @ A @ Y @ L32 ) ) ) ) ) )
=> ~ ( ( X != Y )
=> ! [L1: list @ A,L22: list @ A,L32: list @ A] :
( L2
!= ( append @ A @ L1 @ ( cons @ A @ Y @ ( append @ A @ L22 @ ( cons @ A @ X @ L32 ) ) ) ) ) ) ) ) ) ) ).
% xy_in_set_cases
thf(fact_8142_in__set__list__format,axiom,
! [A: $tType,E: A,L2: list @ A] :
( ( member @ A @ E @ ( set2 @ A @ L2 ) )
=> ~ ! [L1: list @ A,L22: list @ A] :
( L2
!= ( append @ A @ L1 @ ( cons @ A @ E @ L22 ) ) ) ) ).
% in_set_list_format
thf(fact_8143_map__consI_I2_J,axiom,
! [B: $tType,A: $tType,W: list @ A,L2: list @ A,F2: B > A,Ww: list @ B,A2: B] :
( ( ( append @ A @ W @ L2 )
= ( append @ A @ ( map @ B @ A @ F2 @ Ww ) @ L2 ) )
=> ( ( cons @ A @ ( F2 @ A2 ) @ ( append @ A @ W @ L2 ) )
= ( append @ A @ ( map @ B @ A @ F2 @ ( cons @ B @ A2 @ Ww ) ) @ L2 ) ) ) ).
% map_consI(2)
thf(fact_8144_distinct__match,axiom,
! [A: $tType,Al: list @ A,E: A,Bl: list @ A,Al2: list @ A,Bl2: list @ A] :
( ( distinct @ A @ ( append @ A @ Al @ ( cons @ A @ E @ Bl ) ) )
=> ( ( ( append @ A @ Al @ ( cons @ A @ E @ Bl ) )
= ( append @ A @ Al2 @ ( cons @ A @ E @ Bl2 ) ) )
= ( ( Al = Al2 )
& ( Bl = Bl2 ) ) ) ) ).
% distinct_match
thf(fact_8145_not__distinct__conv__prefix,axiom,
! [A: $tType,As3: list @ A] :
( ( ~ ( distinct @ A @ As3 ) )
= ( ? [Xs: list @ A,Y2: A,Ys3: list @ A] :
( ( member @ A @ Y2 @ ( set2 @ A @ Xs ) )
& ( distinct @ A @ Xs )
& ( As3
= ( append @ A @ Xs @ ( cons @ A @ Y2 @ Ys3 ) ) ) ) ) ) ).
% not_distinct_conv_prefix
thf(fact_8146_Misc_Omap__eq__append__conv,axiom,
! [A: $tType,B: $tType,F2: B > A,Ls: list @ B,Fl: list @ A,Fl2: list @ A] :
( ( ( map @ B @ A @ F2 @ Ls )
= ( append @ A @ Fl @ Fl2 ) )
= ( ? [L: list @ B,L8: list @ B] :
( ( Ls
= ( append @ B @ L @ L8 ) )
& ( ( map @ B @ A @ F2 @ L )
= Fl )
& ( ( map @ B @ A @ F2 @ L8 )
= Fl2 ) ) ) ) ).
% Misc.map_eq_append_conv
thf(fact_8147_Misc_Oappend__eq__map__conv,axiom,
! [A: $tType,B: $tType,Fl: list @ A,Fl2: list @ A,F2: B > A,Ls: list @ B] :
( ( ( append @ A @ Fl @ Fl2 )
= ( map @ B @ A @ F2 @ Ls ) )
= ( ? [L: list @ B,L8: list @ B] :
( ( Ls
= ( append @ B @ L @ L8 ) )
& ( ( map @ B @ A @ F2 @ L )
= Fl )
& ( ( map @ B @ A @ F2 @ L8 )
= Fl2 ) ) ) ) ).
% Misc.append_eq_map_conv
thf(fact_8148_map__eq__appendE,axiom,
! [B: $tType,A: $tType,F2: B > A,Ls: list @ B,Fl: list @ A,Fl2: list @ A] :
( ( ( map @ B @ A @ F2 @ Ls )
= ( append @ A @ Fl @ Fl2 ) )
=> ~ ! [L3: list @ B,L9: list @ B] :
( ( Ls
= ( append @ B @ L3 @ L9 ) )
=> ( ( ( map @ B @ A @ F2 @ L3 )
= Fl )
=> ( ( map @ B @ A @ F2 @ L9 )
!= Fl2 ) ) ) ) ).
% map_eq_appendE
thf(fact_8149_append__eq__mapE,axiom,
! [B: $tType,A: $tType,Fl: list @ A,Fl2: list @ A,F2: B > A,Ls: list @ B] :
( ( ( append @ A @ Fl @ Fl2 )
= ( map @ B @ A @ F2 @ Ls ) )
=> ~ ! [L3: list @ B,L9: list @ B] :
( ( Ls
= ( append @ B @ L3 @ L9 ) )
=> ( ( ( map @ B @ A @ F2 @ L3 )
= Fl )
=> ( ( map @ B @ A @ F2 @ L9 )
!= Fl2 ) ) ) ) ).
% append_eq_mapE
thf(fact_8150_replicate__add,axiom,
! [A: $tType,N3: nat,M: nat,X: A] :
( ( replicate @ A @ ( plus_plus @ nat @ N3 @ M ) @ X )
= ( append @ A @ ( replicate @ A @ N3 @ X ) @ ( replicate @ A @ M @ X ) ) ) ).
% replicate_add
thf(fact_8151_upt__eq__append__conv,axiom,
! [I: nat,J: nat,Xs2: list @ nat,Ys: list @ nat] :
( ( ord_less_eq @ nat @ I @ J )
=> ( ( ( upt @ I @ J )
= ( append @ nat @ Xs2 @ Ys ) )
= ( ? [K2: nat] :
( ( ord_less_eq @ nat @ I @ K2 )
& ( ord_less_eq @ nat @ K2 @ J )
& ( ( upt @ I @ K2 )
= Xs2 )
& ( ( upt @ K2 @ J )
= Ys ) ) ) ) ) ).
% upt_eq_append_conv
thf(fact_8152_upt__add__eq__append,axiom,
! [I: nat,J: nat,K: nat] :
( ( ord_less_eq @ nat @ I @ J )
=> ( ( upt @ I @ ( plus_plus @ nat @ J @ K ) )
= ( append @ nat @ ( upt @ I @ J ) @ ( upt @ J @ ( plus_plus @ nat @ J @ K ) ) ) ) ) ).
% upt_add_eq_append
thf(fact_8153_upt__append,axiom,
! [I: nat,J: nat] :
( ( ord_less @ nat @ I @ J )
=> ( ( append @ nat @ ( upt @ ( zero_zero @ nat ) @ I ) @ ( upt @ I @ J ) )
= ( upt @ ( zero_zero @ nat ) @ J ) ) ) ).
% upt_append
thf(fact_8154_list__update__append1,axiom,
! [A: $tType,I: nat,Xs2: list @ A,Ys: list @ A,X: A] :
( ( ord_less @ nat @ I @ ( size_size @ ( list @ A ) @ Xs2 ) )
=> ( ( list_update @ A @ ( append @ A @ Xs2 @ Ys ) @ I @ X )
= ( append @ A @ ( list_update @ A @ Xs2 @ I @ X ) @ Ys ) ) ) ).
% list_update_append1
thf(fact_8155_list__rest__coinc,axiom,
! [A: $tType,S22: list @ A,S1: list @ A,R12: list @ A,R23: list @ A] :
( ( ord_less_eq @ nat @ ( size_size @ ( list @ A ) @ S22 ) @ ( size_size @ ( list @ A ) @ S1 ) )
=> ( ( ( append @ A @ S1 @ R12 )
= ( append @ A @ S22 @ R23 ) )
=> ? [R1p: list @ A] :
( R23
= ( append @ A @ R1p @ R12 ) ) ) ) ).
% list_rest_coinc
thf(fact_8156_length__append__singleton,axiom,
! [A: $tType,Xs2: list @ A,X: A] :
( ( size_size @ ( list @ A ) @ ( append @ A @ Xs2 @ ( cons @ A @ X @ ( nil @ A ) ) ) )
= ( suc @ ( size_size @ ( list @ A ) @ Xs2 ) ) ) ).
% length_append_singleton
thf(fact_8157_length__Suc__rev__conv,axiom,
! [A: $tType,Xs2: list @ A,N3: nat] :
( ( ( size_size @ ( list @ A ) @ Xs2 )
= ( suc @ N3 ) )
= ( ? [Ys3: list @ A,Y2: A] :
( ( Xs2
= ( append @ A @ Ys3 @ ( cons @ A @ Y2 @ ( nil @ A ) ) ) )
& ( ( size_size @ ( list @ A ) @ Ys3 )
= N3 ) ) ) ) ).
% length_Suc_rev_conv
thf(fact_8158_length__Suc__conv__rev,axiom,
! [A: $tType,Xs2: list @ A,N3: nat] :
( ( ( size_size @ ( list @ A ) @ Xs2 )
= ( suc @ N3 ) )
= ( ? [Y2: A,Ys3: list @ A] :
( ( Xs2
= ( append @ A @ Ys3 @ ( cons @ A @ Y2 @ ( nil @ A ) ) ) )
& ( ( size_size @ ( list @ A ) @ Ys3 )
= N3 ) ) ) ) ).
% length_Suc_conv_rev
thf(fact_8159_length__compl__rev__induct,axiom,
! [A: $tType,P: ( list @ A ) > $o,L2: list @ A] :
( ( P @ ( nil @ A ) )
=> ( ! [L3: list @ A,E2: A] :
( ! [Ll: list @ A] :
( ( ord_less_eq @ nat @ ( size_size @ ( list @ A ) @ Ll ) @ ( size_size @ ( list @ A ) @ L3 ) )
=> ( P @ Ll ) )
=> ( P @ ( append @ A @ L3 @ ( cons @ A @ E2 @ ( nil @ A ) ) ) ) )
=> ( P @ L2 ) ) ) ).
% length_compl_rev_induct
thf(fact_8160_nth__append,axiom,
! [A: $tType,N3: nat,Xs2: list @ A,Ys: list @ A] :
( ( ( ord_less @ nat @ N3 @ ( size_size @ ( list @ A ) @ Xs2 ) )
=> ( ( nth @ A @ ( append @ A @ Xs2 @ Ys ) @ N3 )
= ( nth @ A @ Xs2 @ N3 ) ) )
& ( ~ ( ord_less @ nat @ N3 @ ( size_size @ ( list @ A ) @ Xs2 ) )
=> ( ( nth @ A @ ( append @ A @ Xs2 @ Ys ) @ N3 )
= ( nth @ A @ Ys @ ( minus_minus @ nat @ N3 @ ( size_size @ ( list @ A ) @ Xs2 ) ) ) ) ) ) ).
% nth_append
thf(fact_8161_not__distinct__split__distinct,axiom,
! [A: $tType,Xs2: list @ A] :
( ~ ( distinct @ A @ Xs2 )
=> ~ ! [Y3: A,Ys4: list @ A] :
( ( distinct @ A @ Ys4 )
=> ( ( member @ A @ Y3 @ ( set2 @ A @ Ys4 ) )
=> ! [Zs2: list @ A] :
( Xs2
!= ( append @ A @ Ys4 @ ( append @ A @ ( cons @ A @ Y3 @ ( nil @ A ) ) @ Zs2 ) ) ) ) ) ) ).
% not_distinct_split_distinct
thf(fact_8162_replicate__Suc__conv__snoc,axiom,
! [A: $tType,N3: nat,X: A] :
( ( replicate @ A @ ( suc @ N3 ) @ X )
= ( append @ A @ ( replicate @ A @ N3 @ X ) @ ( cons @ A @ X @ ( nil @ A ) ) ) ) ).
% replicate_Suc_conv_snoc
thf(fact_8163_list__update__append,axiom,
! [A: $tType,N3: nat,Xs2: list @ A,Ys: list @ A,X: A] :
( ( ( ord_less @ nat @ N3 @ ( size_size @ ( list @ A ) @ Xs2 ) )
=> ( ( list_update @ A @ ( append @ A @ Xs2 @ Ys ) @ N3 @ X )
= ( append @ A @ ( list_update @ A @ Xs2 @ N3 @ X ) @ Ys ) ) )
& ( ~ ( ord_less @ nat @ N3 @ ( size_size @ ( list @ A ) @ Xs2 ) )
=> ( ( list_update @ A @ ( append @ A @ Xs2 @ Ys ) @ N3 @ X )
= ( append @ A @ Xs2 @ ( list_update @ A @ Ys @ ( minus_minus @ nat @ N3 @ ( size_size @ ( list @ A ) @ Xs2 ) ) @ X ) ) ) ) ) ).
% list_update_append
thf(fact_8164_comm__append__is__replicate,axiom,
! [A: $tType,Xs2: list @ A,Ys: list @ A] :
( ( Xs2
!= ( nil @ A ) )
=> ( ( Ys
!= ( nil @ A ) )
=> ( ( ( append @ A @ Xs2 @ Ys )
= ( append @ A @ Ys @ Xs2 ) )
=> ? [N: nat,Zs2: list @ A] :
( ( ord_less @ nat @ ( one_one @ nat ) @ N )
& ( ( concat @ A @ ( replicate @ ( list @ A ) @ N @ Zs2 ) )
= ( append @ A @ Xs2 @ Ys ) ) ) ) ) ) ).
% comm_append_is_replicate
thf(fact_8165_product_Osimps_I2_J,axiom,
! [A: $tType,B: $tType,X: A,Xs2: list @ A,Ys: list @ B] :
( ( product @ A @ B @ ( cons @ A @ X @ Xs2 ) @ Ys )
= ( append @ ( product_prod @ A @ B ) @ ( map @ B @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X ) @ Ys ) @ ( product @ A @ B @ Xs2 @ Ys ) ) ) ).
% product.simps(2)
thf(fact_8166_slice__prepend,axiom,
! [A: $tType,I: nat,K: nat,Xs2: list @ A,Ys: list @ A] :
( ( ord_less_eq @ nat @ I @ K )
=> ( ( ord_less_eq @ nat @ K @ ( size_size @ ( list @ A ) @ Xs2 ) )
=> ( ( slice @ A @ I @ K @ Xs2 )
= ( slice @ A @ ( plus_plus @ nat @ I @ ( size_size @ ( list @ A ) @ Ys ) ) @ ( plus_plus @ nat @ K @ ( size_size @ ( list @ A ) @ Ys ) ) @ ( append @ A @ Ys @ Xs2 ) ) ) ) ) ).
% slice_prepend
thf(fact_8167_horner__sum__append,axiom,
! [A: $tType,B: $tType] :
( ( comm_semiring_1 @ A )
=> ! [F2: B > A,A2: A,Xs2: list @ B,Ys: list @ B] :
( ( groups4207007520872428315er_sum @ B @ A @ F2 @ A2 @ ( append @ B @ Xs2 @ Ys ) )
= ( plus_plus @ A @ ( groups4207007520872428315er_sum @ B @ A @ F2 @ A2 @ Xs2 ) @ ( times_times @ A @ ( power_power @ A @ A2 @ ( size_size @ ( list @ B ) @ Xs2 ) ) @ ( groups4207007520872428315er_sum @ B @ A @ F2 @ A2 @ Ys ) ) ) ) ) ).
% horner_sum_append
thf(fact_8168_upt__eq__lel__conv,axiom,
! [L2: nat,H2: nat,Is1: list @ nat,I: nat,Is2: list @ nat] :
( ( ( upt @ L2 @ H2 )
= ( append @ nat @ Is1 @ ( cons @ nat @ I @ Is2 ) ) )
= ( ( Is1
= ( upt @ L2 @ I ) )
& ( Is2
= ( upt @ ( suc @ I ) @ H2 ) )
& ( ord_less_eq @ nat @ L2 @ I )
& ( ord_less @ nat @ I @ H2 ) ) ) ).
% upt_eq_lel_conv
thf(fact_8169_upt__Suc__append,axiom,
! [I: nat,J: nat] :
( ( ord_less_eq @ nat @ I @ J )
=> ( ( upt @ I @ ( suc @ J ) )
= ( append @ nat @ ( upt @ I @ J ) @ ( cons @ nat @ J @ ( nil @ nat ) ) ) ) ) ).
% upt_Suc_append
thf(fact_8170_upt__Suc,axiom,
! [I: nat,J: nat] :
( ( ( ord_less_eq @ nat @ I @ J )
=> ( ( upt @ I @ ( suc @ J ) )
= ( append @ nat @ ( upt @ I @ J ) @ ( cons @ nat @ J @ ( nil @ nat ) ) ) ) )
& ( ~ ( ord_less_eq @ nat @ I @ J )
=> ( ( upt @ I @ ( suc @ J ) )
= ( nil @ nat ) ) ) ) ).
% upt_Suc
thf(fact_8171_quickcheck__narrowing__samples_Onarrowing__samples_Opelims,axiom,
! [A: $tType] :
( ( ( code_term_of @ A )
& ( quickc6926020345158392990erm_of @ A ) )
=> ! [A_of_integer: code_integer > ( product_prod @ A @ A ),Zero: A,X: code_integer,Y: list @ A] :
( ( ( code_T4080844693773952564amples @ A @ A_of_integer @ Zero @ X )
= Y )
=> ( ( accp @ code_integer @ ( code_T1710151556404007877es_rel @ A @ A_of_integer ) @ X )
=> ~ ( ( ( ( ord_less @ code_integer @ ( zero_zero @ code_integer ) @ X )
=> ( Y
= ( product_case_prod @ A @ A @ ( list @ A )
@ ^ [A6: A,A16: A] : ( append @ A @ ( code_T4080844693773952564amples @ A @ A_of_integer @ Zero @ ( minus_minus @ code_integer @ X @ ( one_one @ code_integer ) ) ) @ ( cons @ A @ A6 @ ( cons @ A @ A16 @ ( nil @ A ) ) ) )
@ ( A_of_integer @ X ) ) ) )
& ( ~ ( ord_less @ code_integer @ ( zero_zero @ code_integer ) @ X )
=> ( Y
= ( cons @ A @ Zero @ ( nil @ A ) ) ) ) )
=> ~ ( accp @ code_integer @ ( code_T1710151556404007877es_rel @ A @ A_of_integer ) @ X ) ) ) ) ) ).
% quickcheck_narrowing_samples.narrowing_samples.pelims
thf(fact_8172_quickcheck__narrowing__samples_Onarrowing__samples_Opsimps,axiom,
! [A: $tType] :
( ( ( code_term_of @ A )
& ( quickc6926020345158392990erm_of @ A ) )
=> ! [A_of_integer: code_integer > ( product_prod @ A @ A ),I: code_integer,Zero: A] :
( ( accp @ code_integer @ ( code_T1710151556404007877es_rel @ A @ A_of_integer ) @ I )
=> ( ( ( ord_less @ code_integer @ ( zero_zero @ code_integer ) @ I )
=> ( ( code_T4080844693773952564amples @ A @ A_of_integer @ Zero @ I )
= ( product_case_prod @ A @ A @ ( list @ A )
@ ^ [A6: A,A16: A] : ( append @ A @ ( code_T4080844693773952564amples @ A @ A_of_integer @ Zero @ ( minus_minus @ code_integer @ I @ ( one_one @ code_integer ) ) ) @ ( cons @ A @ A6 @ ( cons @ A @ A16 @ ( nil @ A ) ) ) )
@ ( A_of_integer @ I ) ) ) )
& ( ~ ( ord_less @ code_integer @ ( zero_zero @ code_integer ) @ I )
=> ( ( code_T4080844693773952564amples @ A @ A_of_integer @ Zero @ I )
= ( cons @ A @ Zero @ ( nil @ A ) ) ) ) ) ) ) ).
% quickcheck_narrowing_samples.narrowing_samples.psimps
thf(fact_8173_quickcheck__narrowing__samples_Onarrowing__samples_Oelims,axiom,
! [A: $tType] :
( ( ( code_term_of @ A )
& ( quickc6926020345158392990erm_of @ A ) )
=> ! [A_of_integer: code_integer > ( product_prod @ A @ A ),Zero: A,X: code_integer,Y: list @ A] :
( ( ( code_T4080844693773952564amples @ A @ A_of_integer @ Zero @ X )
= Y )
=> ( ( ( ord_less @ code_integer @ ( zero_zero @ code_integer ) @ X )
=> ( Y
= ( product_case_prod @ A @ A @ ( list @ A )
@ ^ [A6: A,A16: A] : ( append @ A @ ( code_T4080844693773952564amples @ A @ A_of_integer @ Zero @ ( minus_minus @ code_integer @ X @ ( one_one @ code_integer ) ) ) @ ( cons @ A @ A6 @ ( cons @ A @ A16 @ ( nil @ A ) ) ) )
@ ( A_of_integer @ X ) ) ) )
& ( ~ ( ord_less @ code_integer @ ( zero_zero @ code_integer ) @ X )
=> ( Y
= ( cons @ A @ Zero @ ( nil @ A ) ) ) ) ) ) ) ).
% quickcheck_narrowing_samples.narrowing_samples.elims
thf(fact_8174_quickcheck__narrowing__samples_Onarrowing__samples_Osimps,axiom,
! [A: $tType] :
( ( ( code_term_of @ A )
& ( quickc6926020345158392990erm_of @ A ) )
=> ( ( code_T4080844693773952564amples @ A )
= ( ^ [A_of_integer2: code_integer > ( product_prod @ A @ A ),Zero2: A,I3: code_integer] :
( if @ ( list @ A ) @ ( ord_less @ code_integer @ ( zero_zero @ code_integer ) @ I3 )
@ ( product_case_prod @ A @ A @ ( list @ A )
@ ^ [A6: A,A16: A] : ( append @ A @ ( code_T4080844693773952564amples @ A @ A_of_integer2 @ Zero2 @ ( minus_minus @ code_integer @ I3 @ ( one_one @ code_integer ) ) ) @ ( cons @ A @ A6 @ ( cons @ A @ A16 @ ( nil @ A ) ) ) )
@ ( A_of_integer2 @ I3 ) )
@ ( cons @ A @ Zero2 @ ( nil @ A ) ) ) ) ) ) ).
% quickcheck_narrowing_samples.narrowing_samples.simps
thf(fact_8175_transpose__rectangle,axiom,
! [A: $tType,Xs2: list @ ( list @ A ),N3: nat] :
( ( ( Xs2
= ( nil @ ( list @ A ) ) )
=> ( N3
= ( zero_zero @ nat ) ) )
=> ( ! [I2: nat] :
( ( ord_less @ nat @ I2 @ ( size_size @ ( list @ ( list @ A ) ) @ Xs2 ) )
=> ( ( size_size @ ( list @ A ) @ ( nth @ ( list @ A ) @ Xs2 @ I2 ) )
= N3 ) )
=> ( ( transpose @ A @ Xs2 )
= ( map @ nat @ ( list @ A )
@ ^ [I3: nat] :
( map @ nat @ A
@ ^ [J3: nat] : ( nth @ A @ ( nth @ ( list @ A ) @ Xs2 @ J3 ) @ I3 )
@ ( upt @ ( zero_zero @ nat ) @ ( size_size @ ( list @ ( list @ A ) ) @ Xs2 ) ) )
@ ( upt @ ( zero_zero @ nat ) @ N3 ) ) ) ) ) ).
% transpose_rectangle
thf(fact_8176_length__remdups__concat,axiom,
! [A: $tType,Xss: list @ ( list @ A )] :
( ( size_size @ ( list @ A ) @ ( remdups @ A @ ( concat @ A @ Xss ) ) )
= ( finite_card @ A @ ( complete_Sup_Sup @ ( set @ A ) @ ( image @ ( list @ A ) @ ( set @ A ) @ ( set2 @ A ) @ ( set2 @ ( list @ A ) @ Xss ) ) ) ) ) ).
% length_remdups_concat
thf(fact_8177_length__remdups__eq,axiom,
! [A: $tType,Xs2: list @ A] :
( ( ( size_size @ ( list @ A ) @ ( remdups @ A @ Xs2 ) )
= ( size_size @ ( list @ A ) @ Xs2 ) )
= ( ( remdups @ A @ Xs2 )
= Xs2 ) ) ).
% length_remdups_eq
thf(fact_8178_set__remdups,axiom,
! [A: $tType,Xs2: list @ A] :
( ( set2 @ A @ ( remdups @ A @ Xs2 ) )
= ( set2 @ A @ Xs2 ) ) ).
% set_remdups
% Type constructors (1395)
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_Code__Evaluation_Oterm___Typerep_Otyperep_4,axiom,
typerep @ code_term ).
thf(tcon_Heap_Oheap_Oheap__ext___Typerep_Otyperep_5,axiom,
! [A13: $tType] :
( ( typerep @ A13 )
=> ( typerep @ ( heap_ext @ A13 ) ) ) ).
thf(tcon_Product__Type_Ounit___Typerep_Otyperep_6,axiom,
typerep @ product_unit ).
thf(tcon_Product__Type_Ounit___Enum_Oenum,axiom,
enum @ product_unit ).
thf(tcon_Product__Type_Oprod___Typerep_Otyperep_7,axiom,
! [A13: $tType,A17: $tType] :
( ( ( typerep @ A13 )
& ( typerep @ A17 ) )
=> ( typerep @ ( product_prod @ A13 @ A17 ) ) ) ).
thf(tcon_Product__Type_Oprod___Enum_Oenum_8,axiom,
! [A13: $tType,A17: $tType] :
( ( ( enum @ A13 )
& ( enum @ A17 ) )
=> ( enum @ ( product_prod @ A13 @ A17 ) ) ) ).
thf(tcon_Numeral__Type_Onum1___Typerep_Otyperep_9,axiom,
typerep @ numeral_num1 ).
thf(tcon_Numeral__Type_Onum1___Enum_Oenum_10,axiom,
enum @ numeral_num1 ).
thf(tcon_Numeral__Type_Obit1___Typerep_Otyperep_11,axiom,
! [A13: $tType] :
( ( typerep @ A13 )
=> ( typerep @ ( numeral_bit1 @ A13 ) ) ) ).
thf(tcon_Numeral__Type_Obit1___Enum_Oenum_12,axiom,
! [A13: $tType] :
( ( finite_finite @ A13 )
=> ( enum @ ( numeral_bit1 @ A13 ) ) ) ).
thf(tcon_Numeral__Type_Obit0___Typerep_Otyperep_13,axiom,
! [A13: $tType] :
( ( typerep @ A13 )
=> ( typerep @ ( numeral_bit0 @ A13 ) ) ) ).
thf(tcon_Numeral__Type_Obit0___Enum_Oenum_14,axiom,
! [A13: $tType] :
( ( finite_finite @ A13 )
=> ( enum @ ( numeral_bit0 @ A13 ) ) ) ).
thf(tcon_Multiset_Omultiset___Typerep_Otyperep_15,axiom,
! [A13: $tType] :
( ( typerep @ A13 )
=> ( typerep @ ( multiset @ A13 ) ) ) ).
thf(tcon_Extended__Nat_Oenat___Typerep_Otyperep_16,axiom,
typerep @ extended_enat ).
thf(tcon_Complex_Ocomplex___Typerep_Otyperep_17,axiom,
typerep @ complex ).
thf(tcon_Assertions_Oassn___Typerep_Otyperep_18,axiom,
typerep @ assn ).
thf(tcon_Uint32_Ouint32___Typerep_Otyperep_19,axiom,
typerep @ uint32 ).
thf(tcon_Option_Ooption___Typerep_Otyperep_20,axiom,
! [A13: $tType] :
( ( typerep @ A13 )
=> ( typerep @ ( option @ A13 ) ) ) ).
thf(tcon_Option_Ooption___Enum_Oenum_21,axiom,
! [A13: $tType] :
( ( enum @ A13 )
=> ( enum @ ( option @ A13 ) ) ) ).
thf(tcon_Filter_Ofilter___Typerep_Otyperep_22,axiom,
! [A13: $tType] :
( ( typerep @ A13 )
=> ( typerep @ ( filter @ A13 ) ) ) ).
thf(tcon_Enum_Ofinite__3___Typerep_Otyperep_23,axiom,
typerep @ finite_3 ).
thf(tcon_Enum_Ofinite__3___Enum_Oenum_24,axiom,
enum @ finite_3 ).
thf(tcon_Enum_Ofinite__2___Typerep_Otyperep_25,axiom,
typerep @ finite_2 ).
thf(tcon_Enum_Ofinite__2___Enum_Oenum_26,axiom,
enum @ finite_2 ).
thf(tcon_Enum_Ofinite__1___Typerep_Otyperep_27,axiom,
typerep @ finite_1 ).
thf(tcon_Enum_Ofinite__1___Enum_Oenum_28,axiom,
enum @ finite_1 ).
thf(tcon_Sum__Type_Osum___Typerep_Otyperep_29,axiom,
! [A13: $tType,A17: $tType] :
( ( ( typerep @ A13 )
& ( typerep @ A17 ) )
=> ( typerep @ ( sum_sum @ A13 @ A17 ) ) ) ).
thf(tcon_Sum__Type_Osum___Enum_Oenum_30,axiom,
! [A13: $tType,A17: $tType] :
( ( ( enum @ A13 )
& ( enum @ A17 ) )
=> ( enum @ ( sum_sum @ A13 @ A17 ) ) ) ).
thf(tcon_String_Ochar___Typerep_Otyperep_31,axiom,
typerep @ char ).
thf(tcon_String_Ochar___Enum_Oenum_32,axiom,
enum @ char ).
thf(tcon_Heap_Oarray___Typerep_Otyperep_33,axiom,
! [A13: $tType] : ( typerep @ ( array @ A13 ) ) ).
thf(tcon_Word_Oword___Typerep_Otyperep_34,axiom,
! [A13: $tType] :
( ( typerep @ A13 )
=> ( typerep @ ( word @ A13 ) ) ) ).
thf(tcon_Word_Oword___Enum_Oenum_35,axiom,
! [A13: $tType] :
( ( type_len @ A13 )
=> ( enum @ ( word @ A13 ) ) ) ).
thf(tcon_Real_Oreal___Typerep_Otyperep_36,axiom,
typerep @ real ).
thf(tcon_List_Olist___Typerep_Otyperep_37,axiom,
! [A13: $tType] :
( ( typerep @ A13 )
=> ( typerep @ ( list @ A13 ) ) ) ).
thf(tcon_Heap_Oref___Typerep_Otyperep_38,axiom,
! [A13: $tType] : ( typerep @ ( ref @ A13 ) ) ).
thf(tcon_HOL_Obool___Typerep_Otyperep_39,axiom,
typerep @ $o ).
thf(tcon_HOL_Obool___Enum_Oenum_40,axiom,
enum @ $o ).
thf(tcon_Set_Oset___Typerep_Otyperep_41,axiom,
! [A13: $tType] :
( ( typerep @ A13 )
=> ( typerep @ ( set @ A13 ) ) ) ).
thf(tcon_Set_Oset___Enum_Oenum_42,axiom,
! [A13: $tType] :
( ( enum @ A13 )
=> ( enum @ ( set @ A13 ) ) ) ).
thf(tcon_Rat_Orat___Typerep_Otyperep_43,axiom,
typerep @ rat ).
thf(tcon_Num_Onum___Typerep_Otyperep_44,axiom,
typerep @ num ).
thf(tcon_Nat_Onat___Typerep_Otyperep_45,axiom,
typerep @ nat ).
thf(tcon_Int_Oint___Typerep_Otyperep_46,axiom,
typerep @ int ).
thf(tcon_fun___Typerep_Otyperep_47,axiom,
! [A13: $tType,A17: $tType] :
( ( ( typerep @ A13 )
& ( typerep @ A17 ) )
=> ( typerep @ ( A13 > A17 ) ) ) ).
thf(tcon_fun___Enum_Oenum_48,axiom,
! [A13: $tType,A17: $tType] :
( ( ( enum @ A13 )
& ( enum @ A17 ) )
=> ( enum @ ( A13 > A17 ) ) ) ).
thf(tcon_fun___Conditionally__Complete__Lattices_Oconditionally__complete__lattice,axiom,
! [A13: $tType,A17: $tType] :
( ( comple6319245703460814977attice @ A17 )
=> ( condit1219197933456340205attice @ ( A13 > A17 ) ) ) ).
thf(tcon_fun___Complete__Lattices_Ocomplete__lattice,axiom,
! [A13: $tType,A17: $tType] :
( ( comple6319245703460814977attice @ A17 )
=> ( comple6319245703460814977attice @ ( A13 > A17 ) ) ) ).
thf(tcon_fun___Boolean__Algebras_Oboolean__algebra,axiom,
! [A13: $tType,A17: $tType] :
( ( boolea8198339166811842893lgebra @ A17 )
=> ( boolea8198339166811842893lgebra @ ( A13 > A17 ) ) ) ).
thf(tcon_fun___Code__Evaluation_Oterm__of,axiom,
! [A13: $tType,A17: $tType] :
( ( ( typerep @ A13 )
& ( typerep @ A17 ) )
=> ( code_term_of @ ( A13 > A17 ) ) ) ).
thf(tcon_fun___Orderings_Oorder__top,axiom,
! [A13: $tType,A17: $tType] :
( ( order_top @ A17 )
=> ( order_top @ ( A13 > A17 ) ) ) ).
thf(tcon_fun___Orderings_Oorder__bot,axiom,
! [A13: $tType,A17: $tType] :
( ( order_bot @ A17 )
=> ( order_bot @ ( A13 > A17 ) ) ) ).
thf(tcon_fun___Orderings_Opreorder,axiom,
! [A13: $tType,A17: $tType] :
( ( preorder @ A17 )
=> ( preorder @ ( A13 > A17 ) ) ) ).
thf(tcon_fun___Finite__Set_Ofinite,axiom,
! [A13: $tType,A17: $tType] :
( ( ( finite_finite @ A13 )
& ( finite_finite @ A17 ) )
=> ( finite_finite @ ( A13 > A17 ) ) ) ).
thf(tcon_fun___Orderings_Oorder,axiom,
! [A13: $tType,A17: $tType] :
( ( order @ A17 )
=> ( order @ ( A13 > A17 ) ) ) ).
thf(tcon_fun___Orderings_Oord,axiom,
! [A13: $tType,A17: $tType] :
( ( ord @ A17 )
=> ( ord @ ( A13 > A17 ) ) ) ).
thf(tcon_fun___Groups_Ouminus,axiom,
! [A13: $tType,A17: $tType] :
( ( uminus @ A17 )
=> ( uminus @ ( A13 > A17 ) ) ) ).
thf(tcon_fun___Groups_Ominus,axiom,
! [A13: $tType,A17: $tType] :
( ( minus @ A17 )
=> ( minus @ ( A13 > A17 ) ) ) ).
thf(tcon_fun___HOL_Oequal,axiom,
! [A13: $tType,A17: $tType] :
( ( ( enum @ A13 )
& ( cl_HOL_Oequal @ A17 ) )
=> ( cl_HOL_Oequal @ ( A13 > A17 ) ) ) ).
thf(tcon_Int_Oint___Conditionally__Complete__Lattices_Oconditionally__complete__linorder,axiom,
condit6923001295902523014norder @ int ).
thf(tcon_Int_Oint___Conditionally__Complete__Lattices_Oconditionally__complete__lattice_49,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___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___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_50,axiom,
code_term_of @ int ).
thf(tcon_Int_Oint___Rings_Oordered__semiring,axiom,
ordered_semiring @ int ).
thf(tcon_Int_Oint___Rings_Oordered__ring__abs,axiom,
ordered_ring_abs @ int ).
thf(tcon_Int_Oint___Parity_Osemiring__parity,axiom,
semiring_parity @ int ).
thf(tcon_Int_Oint___Groups_Ocomm__monoid__add,axiom,
comm_monoid_add @ int ).
thf(tcon_Int_Oint___Rings_Osemiring__modulo,axiom,
semiring_modulo @ int ).
thf(tcon_Int_Oint___Rings_Olinordered__ring,axiom,
linordered_ring @ int ).
thf(tcon_Int_Oint___Rings_Olinordered__idom,axiom,
linordered_idom @ int ).
thf(tcon_Int_Oint___Rings_Ocomm__semiring__1,axiom,
comm_semiring_1 @ int ).
thf(tcon_Int_Oint___Rings_Ocomm__semiring__0,axiom,
comm_semiring_0 @ int ).
thf(tcon_Int_Oint___Groups_Osemigroup__mult,axiom,
semigroup_mult @ int ).
thf(tcon_Int_Oint___Rings_Osemidom__modulo,axiom,
semidom_modulo @ int ).
thf(tcon_Int_Oint___Rings_Osemidom__divide,axiom,
semidom_divide @ int ).
thf(tcon_Int_Oint___Num_Osemiring__numeral,axiom,
semiring_numeral @ int ).
thf(tcon_Int_Oint___Groups_Osemigroup__add,axiom,
semigroup_add @ int ).
thf(tcon_Int_Oint___Rings_Ozero__less__one,axiom,
zero_less_one @ int ).
thf(tcon_Int_Oint___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_51,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_52,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_53,axiom,
ord @ int ).
thf(tcon_Int_Oint___Groups_Ouminus_54,axiom,
uminus @ int ).
thf(tcon_Int_Oint___Rings_Oring__1,axiom,
ring_1 @ int ).
thf(tcon_Int_Oint___Rings_Oabs__if,axiom,
abs_if @ int ).
thf(tcon_Int_Oint___Groups_Otimes,axiom,
times @ int ).
thf(tcon_Int_Oint___Groups_Ominus_55,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_Int_Oint___HOL_Oequal_56,axiom,
cl_HOL_Oequal @ int ).
thf(tcon_Nat_Onat___Conditionally__Complete__Lattices_Oconditionally__complete__linorder_57,axiom,
condit6923001295902523014norder @ nat ).
thf(tcon_Nat_Onat___Conditionally__Complete__Lattices_Oconditionally__complete__lattice_58,axiom,
condit1219197933456340205attice @ nat ).
thf(tcon_Nat_Onat___Bit__Operations_Ounique__euclidean__semiring__with__bit__operations_59,axiom,
bit_un5681908812861735899ations @ nat ).
thf(tcon_Nat_Onat___Semiring__Normalization_Ocomm__semiring__1__cancel__crossproduct_60,axiom,
semiri1453513574482234551roduct @ nat ).
thf(tcon_Nat_Onat___Euclidean__Division_Ounique__euclidean__semiring__with__nat_61,axiom,
euclid5411537665997757685th_nat @ nat ).
thf(tcon_Nat_Onat___Groups_Oordered__ab__semigroup__monoid__add__imp__le_62,axiom,
ordere1937475149494474687imp_le @ nat ).
thf(tcon_Nat_Onat___Euclidean__Division_Ounique__euclidean__semiring_63,axiom,
euclid3128863361964157862miring @ nat ).
thf(tcon_Nat_Onat___Euclidean__Division_Oeuclidean__semiring__cancel_64,axiom,
euclid4440199948858584721cancel @ nat ).
thf(tcon_Nat_Onat___Divides_Ounique__euclidean__semiring__numeral_65,axiom,
unique1627219031080169319umeral @ nat ).
thf(tcon_Nat_Onat___Rings_Osemiring__no__zero__divisors__cancel_66,axiom,
semiri6575147826004484403cancel @ nat ).
thf(tcon_Nat_Onat___Groups_Ostrict__ordered__ab__semigroup__add_67,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_68,axiom,
ordere580206878836729694up_add @ nat ).
thf(tcon_Nat_Onat___Groups_Oordered__ab__semigroup__add__imp__le_69,axiom,
ordere2412721322843649153imp_le @ nat ).
thf(tcon_Nat_Onat___Bit__Operations_Osemiring__bit__operations_70,axiom,
bit_se359711467146920520ations @ nat ).
thf(tcon_Nat_Onat___Rings_Olinordered__comm__semiring__strict_71,axiom,
linord2810124833399127020strict @ nat ).
thf(tcon_Nat_Onat___Groups_Ostrict__ordered__comm__monoid__add_72,axiom,
strict7427464778891057005id_add @ nat ).
thf(tcon_Nat_Onat___Groups_Oordered__cancel__comm__monoid__add_73,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_74,axiom,
euclid3725896446679973847miring @ nat ).
thf(tcon_Nat_Onat___Topological__Spaces_Otopological__space_75,axiom,
topolo4958980785337419405_space @ nat ).
thf(tcon_Nat_Onat___Topological__Spaces_Olinorder__topology_76,axiom,
topolo1944317154257567458pology @ nat ).
thf(tcon_Nat_Onat___Quickcheck__Narrowing_Opartial__term__of_77,axiom,
quickc6926020345158392990erm_of @ nat ).
thf(tcon_Nat_Onat___Limits_Otopological__comm__monoid__mult_78,axiom,
topolo4987421752381908075d_mult @ nat ).
thf(tcon_Nat_Onat___Limits_Otopological__comm__monoid__add_79,axiom,
topolo5987344860129210374id_add @ nat ).
thf(tcon_Nat_Onat___Groups_Olinordered__ab__semigroup__add_80,axiom,
linord4140545234300271783up_add @ nat ).
thf(tcon_Nat_Onat___Topological__Spaces_Oorder__topology_81,axiom,
topolo2564578578187576103pology @ nat ).
thf(tcon_Nat_Onat___Rings_Osemiring__1__no__zero__divisors_82,axiom,
semiri2026040879449505780visors @ nat ).
thf(tcon_Nat_Onat___Rings_Olinordered__nonzero__semiring_83,axiom,
linord181362715937106298miring @ nat ).
thf(tcon_Nat_Onat___Limits_Otopological__semigroup__mult_84,axiom,
topolo4211221413907600880p_mult @ nat ).
thf(tcon_Nat_Onat___Rings_Olinordered__semiring__strict_85,axiom,
linord8928482502909563296strict @ nat ).
thf(tcon_Nat_Onat___Rings_Osemiring__no__zero__divisors_86,axiom,
semiri3467727345109120633visors @ nat ).
thf(tcon_Nat_Onat___Groups_Oordered__ab__semigroup__add_87,axiom,
ordere6658533253407199908up_add @ nat ).
thf(tcon_Nat_Onat___Limits_Otopological__monoid__mult_88,axiom,
topolo1898628316856586783d_mult @ nat ).
thf(tcon_Nat_Onat___Groups_Oordered__comm__monoid__add_89,axiom,
ordere6911136660526730532id_add @ nat ).
thf(tcon_Nat_Onat___Groups_Ocancel__ab__semigroup__add_90,axiom,
cancel2418104881723323429up_add @ nat ).
thf(tcon_Nat_Onat___Limits_Otopological__monoid__add_91,axiom,
topolo6943815403480290642id_add @ nat ).
thf(tcon_Nat_Onat___Groups_Ocancel__comm__monoid__add_92,axiom,
cancel1802427076303600483id_add @ nat ).
thf(tcon_Nat_Onat___Rings_Ocomm__semiring__1__cancel_93,axiom,
comm_s4317794764714335236cancel @ nat ).
thf(tcon_Nat_Onat___Bit__Operations_Osemiring__bits_94,axiom,
bit_semiring_bits @ nat ).
thf(tcon_Nat_Onat___Topological__Spaces_Ot2__space_95,axiom,
topological_t2_space @ nat ).
thf(tcon_Nat_Onat___Rings_Oordered__comm__semiring_96,axiom,
ordere2520102378445227354miring @ nat ).
thf(tcon_Nat_Onat___Groups_Ocancel__semigroup__add_97,axiom,
cancel_semigroup_add @ nat ).
thf(tcon_Nat_Onat___Rings_Olinordered__semiring_98,axiom,
linordered_semiring @ nat ).
thf(tcon_Nat_Onat___Rings_Oordered__semiring__0_99,axiom,
ordered_semiring_0 @ nat ).
thf(tcon_Nat_Onat___Rings_Olinordered__semidom_100,axiom,
linordered_semidom @ nat ).
thf(tcon_Nat_Onat___Groups_Oab__semigroup__mult_101,axiom,
ab_semigroup_mult @ nat ).
thf(tcon_Nat_Onat___Rings_Osemiring__1__cancel_102,axiom,
semiring_1_cancel @ nat ).
thf(tcon_Nat_Onat___Rings_Oalgebraic__semidom_103,axiom,
algebraic_semidom @ nat ).
thf(tcon_Nat_Onat___Groups_Ocomm__monoid__mult_104,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_105,axiom,
ab_semigroup_add @ nat ).
thf(tcon_Nat_Onat___Code__Evaluation_Oterm__of_106,axiom,
code_term_of @ nat ).
thf(tcon_Nat_Onat___Rings_Oordered__semiring_107,axiom,
ordered_semiring @ nat ).
thf(tcon_Nat_Onat___Parity_Osemiring__parity_108,axiom,
semiring_parity @ nat ).
thf(tcon_Nat_Onat___Groups_Ocomm__monoid__add_109,axiom,
comm_monoid_add @ nat ).
thf(tcon_Nat_Onat___Rings_Osemiring__modulo_110,axiom,
semiring_modulo @ nat ).
thf(tcon_Nat_Onat___Rings_Ocomm__semiring__1_111,axiom,
comm_semiring_1 @ nat ).
thf(tcon_Nat_Onat___Rings_Ocomm__semiring__0_112,axiom,
comm_semiring_0 @ nat ).
thf(tcon_Nat_Onat___Groups_Osemigroup__mult_113,axiom,
semigroup_mult @ nat ).
thf(tcon_Nat_Onat___Rings_Osemidom__modulo_114,axiom,
semidom_modulo @ nat ).
thf(tcon_Nat_Onat___Rings_Osemidom__divide_115,axiom,
semidom_divide @ nat ).
thf(tcon_Nat_Onat___Num_Osemiring__numeral_116,axiom,
semiring_numeral @ nat ).
thf(tcon_Nat_Onat___Groups_Osemigroup__add_117,axiom,
semigroup_add @ nat ).
thf(tcon_Nat_Onat___Rings_Ozero__less__one_118,axiom,
zero_less_one @ nat ).
thf(tcon_Nat_Onat___Rings_Ocomm__semiring_119,axiom,
comm_semiring @ nat ).
thf(tcon_Nat_Onat___Orderings_Owellorder,axiom,
wellorder @ nat ).
thf(tcon_Nat_Onat___Orderings_Oorder__bot_120,axiom,
order_bot @ nat ).
thf(tcon_Nat_Onat___Nat_Osemiring__char__0_121,axiom,
semiring_char_0 @ nat ).
thf(tcon_Nat_Onat___Rings_Ozero__neq__one_122,axiom,
zero_neq_one @ nat ).
thf(tcon_Nat_Onat___Orderings_Opreorder_123,axiom,
preorder @ nat ).
thf(tcon_Nat_Onat___Orderings_Olinorder_124,axiom,
linorder @ nat ).
thf(tcon_Nat_Onat___Groups_Omonoid__mult_125,axiom,
monoid_mult @ nat ).
thf(tcon_Nat_Onat___Groups_Omonoid__add_126,axiom,
monoid_add @ nat ).
thf(tcon_Nat_Onat___Rings_Osemiring__1_127,axiom,
semiring_1 @ nat ).
thf(tcon_Nat_Onat___Rings_Osemiring__0_128,axiom,
semiring_0 @ nat ).
thf(tcon_Nat_Onat___Orderings_Ono__top_129,axiom,
no_top @ nat ).
thf(tcon_Nat_Onat___GCD_Osemiring__gcd_130,axiom,
semiring_gcd @ nat ).
thf(tcon_Nat_Onat___Rings_Omult__zero_131,axiom,
mult_zero @ nat ).
thf(tcon_Nat_Onat___Orderings_Oorder_132,axiom,
order @ nat ).
thf(tcon_Nat_Onat___Rings_Osemiring_133,axiom,
semiring @ nat ).
thf(tcon_Nat_Onat___Orderings_Oord_134,axiom,
ord @ nat ).
thf(tcon_Nat_Onat___Groups_Otimes_135,axiom,
times @ nat ).
thf(tcon_Nat_Onat___Groups_Ominus_136,axiom,
minus @ nat ).
thf(tcon_Nat_Onat___Power_Opower_137,axiom,
power @ nat ).
thf(tcon_Nat_Onat___Num_Onumeral_138,axiom,
numeral @ nat ).
thf(tcon_Nat_Onat___Groups_Ozero_139,axiom,
zero @ nat ).
thf(tcon_Nat_Onat___Groups_Oplus_140,axiom,
plus @ nat ).
thf(tcon_Nat_Onat___Groups_Oone_141,axiom,
one @ nat ).
thf(tcon_Nat_Onat___Rings_Odvd_142,axiom,
dvd @ nat ).
thf(tcon_Nat_Onat___Heap_Oheap_143,axiom,
heap @ nat ).
thf(tcon_Nat_Onat___HOL_Oequal_144,axiom,
cl_HOL_Oequal @ nat ).
thf(tcon_Nat_Onat___Nat_Osize,axiom,
size @ nat ).
thf(tcon_Num_Onum___Quickcheck__Narrowing_Opartial__term__of_145,axiom,
quickc6926020345158392990erm_of @ num ).
thf(tcon_Num_Onum___Code__Evaluation_Oterm__of_146,axiom,
code_term_of @ num ).
thf(tcon_Num_Onum___Orderings_Opreorder_147,axiom,
preorder @ num ).
thf(tcon_Num_Onum___Orderings_Olinorder_148,axiom,
linorder @ num ).
thf(tcon_Num_Onum___Orderings_Oorder_149,axiom,
order @ num ).
thf(tcon_Num_Onum___Orderings_Oord_150,axiom,
ord @ num ).
thf(tcon_Num_Onum___Groups_Otimes_151,axiom,
times @ num ).
thf(tcon_Num_Onum___Groups_Oplus_152,axiom,
plus @ num ).
thf(tcon_Num_Onum___HOL_Oequal_153,axiom,
cl_HOL_Oequal @ num ).
thf(tcon_Num_Onum___Nat_Osize_154,axiom,
size @ num ).
thf(tcon_Rat_Orat___Semiring__Normalization_Ocomm__semiring__1__cancel__crossproduct_155,axiom,
semiri1453513574482234551roduct @ rat ).
thf(tcon_Rat_Orat___Groups_Oordered__ab__semigroup__monoid__add__imp__le_156,axiom,
ordere1937475149494474687imp_le @ rat ).
thf(tcon_Rat_Orat___Rings_Osemiring__no__zero__divisors__cancel_157,axiom,
semiri6575147826004484403cancel @ rat ).
thf(tcon_Rat_Orat___Groups_Ostrict__ordered__ab__semigroup__add_158,axiom,
strict9044650504122735259up_add @ rat ).
thf(tcon_Rat_Orat___Groups_Oordered__cancel__ab__semigroup__add_159,axiom,
ordere580206878836729694up_add @ rat ).
thf(tcon_Rat_Orat___Groups_Oordered__ab__semigroup__add__imp__le_160,axiom,
ordere2412721322843649153imp_le @ rat ).
thf(tcon_Rat_Orat___Rings_Olinordered__comm__semiring__strict_161,axiom,
linord2810124833399127020strict @ rat ).
thf(tcon_Rat_Orat___Groups_Ostrict__ordered__comm__monoid__add_162,axiom,
strict7427464778891057005id_add @ rat ).
thf(tcon_Rat_Orat___Groups_Oordered__cancel__comm__monoid__add_163,axiom,
ordere8940638589300402666id_add @ rat ).
thf(tcon_Rat_Orat___Quickcheck__Narrowing_Opartial__term__of_164,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_165,axiom,
linord715952674999750819strict @ rat ).
thf(tcon_Rat_Orat___Orderings_Ounbounded__dense__linorder,axiom,
unboun7993243217541854897norder @ rat ).
thf(tcon_Rat_Orat___Groups_Olinordered__ab__semigroup__add_166,axiom,
linord4140545234300271783up_add @ rat ).
thf(tcon_Rat_Orat___Rings_Osemiring__1__no__zero__divisors_167,axiom,
semiri2026040879449505780visors @ rat ).
thf(tcon_Rat_Orat___Rings_Olinordered__nonzero__semiring_168,axiom,
linord181362715937106298miring @ rat ).
thf(tcon_Rat_Orat___Rings_Olinordered__semiring__strict_169,axiom,
linord8928482502909563296strict @ rat ).
thf(tcon_Rat_Orat___Rings_Osemiring__no__zero__divisors_170,axiom,
semiri3467727345109120633visors @ rat ).
thf(tcon_Rat_Orat___Groups_Oordered__ab__semigroup__add_171,axiom,
ordere6658533253407199908up_add @ rat ).
thf(tcon_Rat_Orat___Groups_Oordered__ab__group__add__abs_172,axiom,
ordere166539214618696060dd_abs @ rat ).
thf(tcon_Rat_Orat___Archimedean__Field_Ofloor__ceiling,axiom,
archim2362893244070406136eiling @ rat ).
thf(tcon_Rat_Orat___Groups_Oordered__comm__monoid__add_173,axiom,
ordere6911136660526730532id_add @ rat ).
thf(tcon_Rat_Orat___Groups_Olinordered__ab__group__add_174,axiom,
linord5086331880401160121up_add @ rat ).
thf(tcon_Rat_Orat___Groups_Ocancel__ab__semigroup__add_175,axiom,
cancel2418104881723323429up_add @ rat ).
thf(tcon_Rat_Orat___Rings_Oring__1__no__zero__divisors_176,axiom,
ring_15535105094025558882visors @ rat ).
thf(tcon_Rat_Orat___Groups_Ocancel__comm__monoid__add_177,axiom,
cancel1802427076303600483id_add @ rat ).
thf(tcon_Rat_Orat___Rings_Olinordered__ring__strict_178,axiom,
linord4710134922213307826strict @ rat ).
thf(tcon_Rat_Orat___Rings_Ocomm__semiring__1__cancel_179,axiom,
comm_s4317794764714335236cancel @ rat ).
thf(tcon_Rat_Orat___Rings_Oordered__comm__semiring_180,axiom,
ordere2520102378445227354miring @ rat ).
thf(tcon_Rat_Orat___Rings_Olinordered__semiring__1_181,axiom,
linord6961819062388156250ring_1 @ rat ).
thf(tcon_Rat_Orat___Groups_Oordered__ab__group__add_182,axiom,
ordered_ab_group_add @ rat ).
thf(tcon_Rat_Orat___Groups_Ocancel__semigroup__add_183,axiom,
cancel_semigroup_add @ rat ).
thf(tcon_Rat_Orat___Rings_Olinordered__semiring_184,axiom,
linordered_semiring @ rat ).
thf(tcon_Rat_Orat___Rings_Oordered__semiring__0_185,axiom,
ordered_semiring_0 @ rat ).
thf(tcon_Rat_Orat___Rings_Olinordered__semidom_186,axiom,
linordered_semidom @ rat ).
thf(tcon_Rat_Orat___Orderings_Odense__linorder,axiom,
dense_linorder @ rat ).
thf(tcon_Rat_Orat___Groups_Oab__semigroup__mult_187,axiom,
ab_semigroup_mult @ rat ).
thf(tcon_Rat_Orat___Rings_Osemiring__1__cancel_188,axiom,
semiring_1_cancel @ rat ).
thf(tcon_Rat_Orat___Groups_Ocomm__monoid__mult_189,axiom,
comm_monoid_mult @ rat ).
thf(tcon_Rat_Orat___Groups_Oab__semigroup__add_190,axiom,
ab_semigroup_add @ rat ).
thf(tcon_Rat_Orat___Fields_Olinordered__field,axiom,
linordered_field @ rat ).
thf(tcon_Rat_Orat___Code__Evaluation_Oterm__of_191,axiom,
code_term_of @ rat ).
thf(tcon_Rat_Orat___Rings_Oordered__semiring_192,axiom,
ordered_semiring @ rat ).
thf(tcon_Rat_Orat___Rings_Oordered__ring__abs_193,axiom,
ordered_ring_abs @ rat ).
thf(tcon_Rat_Orat___Groups_Ocomm__monoid__add_194,axiom,
comm_monoid_add @ rat ).
thf(tcon_Rat_Orat___Rings_Olinordered__ring_195,axiom,
linordered_ring @ rat ).
thf(tcon_Rat_Orat___Rings_Olinordered__idom_196,axiom,
linordered_idom @ rat ).
thf(tcon_Rat_Orat___Rings_Ocomm__semiring__1_197,axiom,
comm_semiring_1 @ rat ).
thf(tcon_Rat_Orat___Rings_Ocomm__semiring__0_198,axiom,
comm_semiring_0 @ rat ).
thf(tcon_Rat_Orat___Orderings_Odense__order,axiom,
dense_order @ rat ).
thf(tcon_Rat_Orat___Groups_Osemigroup__mult_199,axiom,
semigroup_mult @ rat ).
thf(tcon_Rat_Orat___Rings_Osemidom__divide_200,axiom,
semidom_divide @ rat ).
thf(tcon_Rat_Orat___Num_Osemiring__numeral_201,axiom,
semiring_numeral @ rat ).
thf(tcon_Rat_Orat___Groups_Osemigroup__add_202,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_203,axiom,
zero_less_one @ rat ).
thf(tcon_Rat_Orat___Rings_Ocomm__semiring_204,axiom,
comm_semiring @ rat ).
thf(tcon_Rat_Orat___Nat_Osemiring__char__0_205,axiom,
semiring_char_0 @ rat ).
thf(tcon_Rat_Orat___Groups_Oab__group__add_206,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_207,axiom,
zero_neq_one @ rat ).
thf(tcon_Rat_Orat___Rings_Oordered__ring_208,axiom,
ordered_ring @ rat ).
thf(tcon_Rat_Orat___Rings_Oidom__abs__sgn_209,axiom,
idom_abs_sgn @ rat ).
thf(tcon_Rat_Orat___Orderings_Opreorder_210,axiom,
preorder @ rat ).
thf(tcon_Rat_Orat___Orderings_Olinorder_211,axiom,
linorder @ rat ).
thf(tcon_Rat_Orat___Groups_Omonoid__mult_212,axiom,
monoid_mult @ rat ).
thf(tcon_Rat_Orat___Rings_Oidom__divide_213,axiom,
idom_divide @ rat ).
thf(tcon_Rat_Orat___Rings_Ocomm__ring__1_214,axiom,
comm_ring_1 @ rat ).
thf(tcon_Rat_Orat___Groups_Omonoid__add_215,axiom,
monoid_add @ rat ).
thf(tcon_Rat_Orat___Rings_Osemiring__1_216,axiom,
semiring_1 @ rat ).
thf(tcon_Rat_Orat___Rings_Osemiring__0_217,axiom,
semiring_0 @ rat ).
thf(tcon_Rat_Orat___Orderings_Ono__top_218,axiom,
no_top @ rat ).
thf(tcon_Rat_Orat___Orderings_Ono__bot_219,axiom,
no_bot @ rat ).
thf(tcon_Rat_Orat___Groups_Ogroup__add_220,axiom,
group_add @ rat ).
thf(tcon_Rat_Orat___Rings_Omult__zero_221,axiom,
mult_zero @ rat ).
thf(tcon_Rat_Orat___Rings_Ocomm__ring_222,axiom,
comm_ring @ rat ).
thf(tcon_Rat_Orat___Orderings_Oorder_223,axiom,
order @ rat ).
thf(tcon_Rat_Orat___Num_Oneg__numeral_224,axiom,
neg_numeral @ rat ).
thf(tcon_Rat_Orat___Nat_Oring__char__0_225,axiom,
ring_char_0 @ rat ).
thf(tcon_Rat_Orat___Rings_Osemiring_226,axiom,
semiring @ rat ).
thf(tcon_Rat_Orat___Fields_Oinverse,axiom,
inverse @ rat ).
thf(tcon_Rat_Orat___Orderings_Oord_227,axiom,
ord @ rat ).
thf(tcon_Rat_Orat___Groups_Ouminus_228,axiom,
uminus @ rat ).
thf(tcon_Rat_Orat___Rings_Oring__1_229,axiom,
ring_1 @ rat ).
thf(tcon_Rat_Orat___Rings_Oabs__if_230,axiom,
abs_if @ rat ).
thf(tcon_Rat_Orat___Groups_Otimes_231,axiom,
times @ rat ).
thf(tcon_Rat_Orat___Groups_Ominus_232,axiom,
minus @ rat ).
thf(tcon_Rat_Orat___Fields_Ofield,axiom,
field @ rat ).
thf(tcon_Rat_Orat___Power_Opower_233,axiom,
power @ rat ).
thf(tcon_Rat_Orat___Num_Onumeral_234,axiom,
numeral @ rat ).
thf(tcon_Rat_Orat___Groups_Ozero_235,axiom,
zero @ rat ).
thf(tcon_Rat_Orat___Groups_Oplus_236,axiom,
plus @ rat ).
thf(tcon_Rat_Orat___Rings_Oring_237,axiom,
ring @ rat ).
thf(tcon_Rat_Orat___Rings_Oidom_238,axiom,
idom @ rat ).
thf(tcon_Rat_Orat___Groups_Oone_239,axiom,
one @ rat ).
thf(tcon_Rat_Orat___Rings_Odvd_240,axiom,
dvd @ rat ).
thf(tcon_Rat_Orat___HOL_Oequal_241,axiom,
cl_HOL_Oequal @ rat ).
thf(tcon_Set_Oset___Conditionally__Complete__Lattices_Oconditionally__complete__lattice_242,axiom,
! [A13: $tType] : ( condit1219197933456340205attice @ ( set @ A13 ) ) ).
thf(tcon_Set_Oset___Quickcheck__Narrowing_Opartial__term__of_243,axiom,
! [A13: $tType] :
( ( typerep @ A13 )
=> ( quickc6926020345158392990erm_of @ ( set @ A13 ) ) ) ).
thf(tcon_Set_Oset___Complete__Lattices_Ocomplete__lattice_244,axiom,
! [A13: $tType] : ( comple6319245703460814977attice @ ( set @ A13 ) ) ).
thf(tcon_Set_Oset___Boolean__Algebras_Oboolean__algebra_245,axiom,
! [A13: $tType] : ( boolea8198339166811842893lgebra @ ( set @ A13 ) ) ).
thf(tcon_Set_Oset___Code__Evaluation_Oterm__of_246,axiom,
! [A13: $tType] :
( ( typerep @ A13 )
=> ( code_term_of @ ( set @ A13 ) ) ) ).
thf(tcon_Set_Oset___Orderings_Oorder__top_247,axiom,
! [A13: $tType] : ( order_top @ ( set @ A13 ) ) ).
thf(tcon_Set_Oset___Orderings_Oorder__bot_248,axiom,
! [A13: $tType] : ( order_bot @ ( set @ A13 ) ) ).
thf(tcon_Set_Oset___Orderings_Opreorder_249,axiom,
! [A13: $tType] : ( preorder @ ( set @ A13 ) ) ).
thf(tcon_Set_Oset___Finite__Set_Ofinite_250,axiom,
! [A13: $tType] :
( ( finite_finite @ A13 )
=> ( finite_finite @ ( set @ A13 ) ) ) ).
thf(tcon_Set_Oset___Orderings_Oorder_251,axiom,
! [A13: $tType] : ( order @ ( set @ A13 ) ) ).
thf(tcon_Set_Oset___Orderings_Oord_252,axiom,
! [A13: $tType] : ( ord @ ( set @ A13 ) ) ).
thf(tcon_Set_Oset___Groups_Ouminus_253,axiom,
! [A13: $tType] : ( uminus @ ( set @ A13 ) ) ).
thf(tcon_Set_Oset___Groups_Ominus_254,axiom,
! [A13: $tType] : ( minus @ ( set @ A13 ) ) ).
thf(tcon_Set_Oset___HOL_Oequal_255,axiom,
! [A13: $tType] :
( ( cl_HOL_Oequal @ A13 )
=> ( cl_HOL_Oequal @ ( set @ A13 ) ) ) ).
thf(tcon_HOL_Obool___Conditionally__Complete__Lattices_Oconditionally__complete__lattice_256,axiom,
condit1219197933456340205attice @ $o ).
thf(tcon_HOL_Obool___Topological__Spaces_Otopological__space_257,axiom,
topolo4958980785337419405_space @ $o ).
thf(tcon_HOL_Obool___Topological__Spaces_Olinorder__topology_258,axiom,
topolo1944317154257567458pology @ $o ).
thf(tcon_HOL_Obool___Quickcheck__Narrowing_Opartial__term__of_259,axiom,
quickc6926020345158392990erm_of @ $o ).
thf(tcon_HOL_Obool___Complete__Lattices_Ocomplete__lattice_260,axiom,
comple6319245703460814977attice @ $o ).
thf(tcon_HOL_Obool___Topological__Spaces_Oorder__topology_261,axiom,
topolo2564578578187576103pology @ $o ).
thf(tcon_HOL_Obool___Boolean__Algebras_Oboolean__algebra_262,axiom,
boolea8198339166811842893lgebra @ $o ).
thf(tcon_HOL_Obool___Topological__Spaces_Ot2__space_263,axiom,
topological_t2_space @ $o ).
thf(tcon_HOL_Obool___Code__Evaluation_Oterm__of_264,axiom,
code_term_of @ $o ).
thf(tcon_HOL_Obool___Orderings_Oorder__top_265,axiom,
order_top @ $o ).
thf(tcon_HOL_Obool___Orderings_Oorder__bot_266,axiom,
order_bot @ $o ).
thf(tcon_HOL_Obool___Orderings_Opreorder_267,axiom,
preorder @ $o ).
thf(tcon_HOL_Obool___Orderings_Olinorder_268,axiom,
linorder @ $o ).
thf(tcon_HOL_Obool___Finite__Set_Ofinite_269,axiom,
finite_finite @ $o ).
thf(tcon_HOL_Obool___Orderings_Oorder_270,axiom,
order @ $o ).
thf(tcon_HOL_Obool___Orderings_Oord_271,axiom,
ord @ $o ).
thf(tcon_HOL_Obool___Groups_Ouminus_272,axiom,
uminus @ $o ).
thf(tcon_HOL_Obool___Groups_Ominus_273,axiom,
minus @ $o ).
thf(tcon_HOL_Obool___Heap_Oheap_274,axiom,
heap @ $o ).
thf(tcon_HOL_Obool___HOL_Oequal_275,axiom,
cl_HOL_Oequal @ $o ).
thf(tcon_Heap_Oref___Quickcheck__Narrowing_Opartial__term__of_276,axiom,
! [A13: $tType] :
( ( typerep @ A13 )
=> ( quickc6926020345158392990erm_of @ ( ref @ A13 ) ) ) ).
thf(tcon_Heap_Oref___Code__Evaluation_Oterm__of_277,axiom,
! [A13: $tType] :
( ( typerep @ A13 )
=> ( code_term_of @ ( ref @ A13 ) ) ) ).
thf(tcon_Heap_Oref___Heap_Oheap_278,axiom,
! [A13: $tType] : ( heap @ ( ref @ A13 ) ) ).
thf(tcon_Heap_Oref___HOL_Oequal_279,axiom,
! [A13: $tType] : ( cl_HOL_Oequal @ ( ref @ A13 ) ) ).
thf(tcon_Heap_Oref___Nat_Osize_280,axiom,
! [A13: $tType] : ( size @ ( ref @ A13 ) ) ).
thf(tcon_List_Olist___Quickcheck__Narrowing_Opartial__term__of_281,axiom,
! [A13: $tType] :
( ( typerep @ A13 )
=> ( quickc6926020345158392990erm_of @ ( list @ A13 ) ) ) ).
thf(tcon_List_Olist___Code__Evaluation_Oterm__of_282,axiom,
! [A13: $tType] :
( ( typerep @ A13 )
=> ( code_term_of @ ( list @ A13 ) ) ) ).
thf(tcon_List_Olist___Heap_Oheap_283,axiom,
! [A13: $tType] :
( ( heap @ A13 )
=> ( heap @ ( list @ A13 ) ) ) ).
thf(tcon_List_Olist___HOL_Oequal_284,axiom,
! [A13: $tType] : ( cl_HOL_Oequal @ ( list @ A13 ) ) ).
thf(tcon_List_Olist___Nat_Osize_285,axiom,
! [A13: $tType] : ( size @ ( list @ A13 ) ) ).
thf(tcon_Real_Oreal___Conditionally__Complete__Lattices_Oconditionally__complete__linorder_286,axiom,
condit6923001295902523014norder @ real ).
thf(tcon_Real_Oreal___Conditionally__Complete__Lattices_Oconditionally__complete__lattice_287,axiom,
condit1219197933456340205attice @ real ).
thf(tcon_Real_Oreal___Semiring__Normalization_Ocomm__semiring__1__cancel__crossproduct_288,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_289,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_290,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_291,axiom,
strict9044650504122735259up_add @ real ).
thf(tcon_Real_Oreal___Groups_Oordered__cancel__ab__semigroup__add_292,axiom,
ordere580206878836729694up_add @ real ).
thf(tcon_Real_Oreal___Groups_Oordered__ab__semigroup__add__imp__le_293,axiom,
ordere2412721322843649153imp_le @ real ).
thf(tcon_Real_Oreal___Rings_Olinordered__comm__semiring__strict_294,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_295,axiom,
strict7427464778891057005id_add @ real ).
thf(tcon_Real_Oreal___Groups_Oordered__cancel__comm__monoid__add_296,axiom,
ordere8940638589300402666id_add @ real ).
thf(tcon_Real_Oreal___Topological__Spaces_Otopological__space_297,axiom,
topolo4958980785337419405_space @ real ).
thf(tcon_Real_Oreal___Topological__Spaces_Olinorder__topology_298,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_299,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_300,axiom,
archim462609752435547400_field @ real ).
thf(tcon_Real_Oreal___Rings_Olinordered__semiring__1__strict_301,axiom,
linord715952674999750819strict @ real ).
thf(tcon_Real_Oreal___Orderings_Ounbounded__dense__linorder_302,axiom,
unboun7993243217541854897norder @ real ).
thf(tcon_Real_Oreal___Limits_Otopological__comm__monoid__add_303,axiom,
topolo5987344860129210374id_add @ real ).
thf(tcon_Real_Oreal___Groups_Olinordered__ab__semigroup__add_304,axiom,
linord4140545234300271783up_add @ real ).
thf(tcon_Real_Oreal___Topological__Spaces_Oorder__topology_305,axiom,
topolo2564578578187576103pology @ real ).
thf(tcon_Real_Oreal___Rings_Osemiring__1__no__zero__divisors_306,axiom,
semiri2026040879449505780visors @ real ).
thf(tcon_Real_Oreal___Rings_Olinordered__nonzero__semiring_307,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_308,axiom,
topolo4211221413907600880p_mult @ real ).
thf(tcon_Real_Oreal___Topological__Spaces_Ouniform__space,axiom,
topolo7287701948861334536_space @ real ).
thf(tcon_Real_Oreal___Rings_Olinordered__semiring__strict_309,axiom,
linord8928482502909563296strict @ real ).
thf(tcon_Real_Oreal___Rings_Osemiring__no__zero__divisors_310,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_311,axiom,
ordere6658533253407199908up_add @ real ).
thf(tcon_Real_Oreal___Groups_Oordered__ab__group__add__abs_312,axiom,
ordere166539214618696060dd_abs @ real ).
thf(tcon_Real_Oreal___Archimedean__Field_Ofloor__ceiling_313,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_314,axiom,
ordere6911136660526730532id_add @ real ).
thf(tcon_Real_Oreal___Groups_Olinordered__ab__group__add_315,axiom,
linord5086331880401160121up_add @ real ).
thf(tcon_Real_Oreal___Groups_Ocancel__ab__semigroup__add_316,axiom,
cancel2418104881723323429up_add @ real ).
thf(tcon_Real_Oreal___Rings_Oring__1__no__zero__divisors_317,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_318,axiom,
topolo6943815403480290642id_add @ real ).
thf(tcon_Real_Oreal___Groups_Ocancel__comm__monoid__add_319,axiom,
cancel1802427076303600483id_add @ real ).
thf(tcon_Real_Oreal___Rings_Olinordered__ring__strict_320,axiom,
linord4710134922213307826strict @ real ).
thf(tcon_Real_Oreal___Rings_Ocomm__semiring__1__cancel_321,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_322,axiom,
topological_t2_space @ real ).
thf(tcon_Real_Oreal___Rings_Oordered__comm__semiring_323,axiom,
ordere2520102378445227354miring @ real ).
thf(tcon_Real_Oreal___Rings_Olinordered__semiring__1_324,axiom,
linord6961819062388156250ring_1 @ real ).
thf(tcon_Real_Oreal___Groups_Oordered__ab__group__add_325,axiom,
ordered_ab_group_add @ real ).
thf(tcon_Real_Oreal___Groups_Ocancel__semigroup__add_326,axiom,
cancel_semigroup_add @ real ).
thf(tcon_Real_Oreal___Rings_Olinordered__semiring_327,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_328,axiom,
ordered_semiring_0 @ real ).
thf(tcon_Real_Oreal___Rings_Olinordered__semidom_329,axiom,
linordered_semidom @ real ).
thf(tcon_Real_Oreal___Orderings_Odense__linorder_330,axiom,
dense_linorder @ real ).
thf(tcon_Real_Oreal___Groups_Oab__semigroup__mult_331,axiom,
ab_semigroup_mult @ real ).
thf(tcon_Real_Oreal___Rings_Osemiring__1__cancel_332,axiom,
semiring_1_cancel @ real ).
thf(tcon_Real_Oreal___Groups_Ocomm__monoid__mult_333,axiom,
comm_monoid_mult @ real ).
thf(tcon_Real_Oreal___Groups_Oab__semigroup__add_334,axiom,
ab_semigroup_add @ real ).
thf(tcon_Real_Oreal___Fields_Olinordered__field_335,axiom,
linordered_field @ real ).
thf(tcon_Real_Oreal___Code__Evaluation_Oterm__of_336,axiom,
code_term_of @ real ).
thf(tcon_Real_Oreal___Rings_Oordered__semiring_337,axiom,
ordered_semiring @ real ).
thf(tcon_Real_Oreal___Rings_Oordered__ring__abs_338,axiom,
ordered_ring_abs @ real ).
thf(tcon_Real_Oreal___Groups_Ocomm__monoid__add_339,axiom,
comm_monoid_add @ real ).
thf(tcon_Real_Oreal___Rings_Olinordered__ring_340,axiom,
linordered_ring @ real ).
thf(tcon_Real_Oreal___Rings_Olinordered__idom_341,axiom,
linordered_idom @ real ).
thf(tcon_Real_Oreal___Rings_Ocomm__semiring__1_342,axiom,
comm_semiring_1 @ real ).
thf(tcon_Real_Oreal___Rings_Ocomm__semiring__0_343,axiom,
comm_semiring_0 @ real ).
thf(tcon_Real_Oreal___Orderings_Odense__order_344,axiom,
dense_order @ real ).
thf(tcon_Real_Oreal___Groups_Osemigroup__mult_345,axiom,
semigroup_mult @ real ).
thf(tcon_Real_Oreal___Rings_Osemidom__divide_346,axiom,
semidom_divide @ real ).
thf(tcon_Real_Oreal___Num_Osemiring__numeral_347,axiom,
semiring_numeral @ real ).
thf(tcon_Real_Oreal___Groups_Osemigroup__add_348,axiom,
semigroup_add @ real ).
thf(tcon_Real_Oreal___Fields_Ofield__abs__sgn_349,axiom,
field_abs_sgn @ real ).
thf(tcon_Real_Oreal___Fields_Odivision__ring_350,axiom,
division_ring @ real ).
thf(tcon_Real_Oreal___Rings_Ozero__less__one_351,axiom,
zero_less_one @ real ).
thf(tcon_Real_Oreal___Rings_Ocomm__semiring_352,axiom,
comm_semiring @ real ).
thf(tcon_Real_Oreal___Nat_Osemiring__char__0_353,axiom,
semiring_char_0 @ real ).
thf(tcon_Real_Oreal___Groups_Oab__group__add_354,axiom,
ab_group_add @ real ).
thf(tcon_Real_Oreal___Fields_Ofield__char__0_355,axiom,
field_char_0 @ real ).
thf(tcon_Real_Oreal___Rings_Ozero__neq__one_356,axiom,
zero_neq_one @ real ).
thf(tcon_Real_Oreal___Rings_Oordered__ring_357,axiom,
ordered_ring @ real ).
thf(tcon_Real_Oreal___Rings_Oidom__abs__sgn_358,axiom,
idom_abs_sgn @ real ).
thf(tcon_Real_Oreal___Orderings_Opreorder_359,axiom,
preorder @ real ).
thf(tcon_Real_Oreal___Orderings_Olinorder_360,axiom,
linorder @ real ).
thf(tcon_Real_Oreal___Groups_Omonoid__mult_361,axiom,
monoid_mult @ real ).
thf(tcon_Real_Oreal___Transcendental_Oln,axiom,
ln @ real ).
thf(tcon_Real_Oreal___Rings_Oidom__divide_362,axiom,
idom_divide @ real ).
thf(tcon_Real_Oreal___Rings_Ocomm__ring__1_363,axiom,
comm_ring_1 @ real ).
thf(tcon_Real_Oreal___Groups_Omonoid__add_364,axiom,
monoid_add @ real ).
thf(tcon_Real_Oreal___Rings_Osemiring__1_365,axiom,
semiring_1 @ real ).
thf(tcon_Real_Oreal___Rings_Osemiring__0_366,axiom,
semiring_0 @ real ).
thf(tcon_Real_Oreal___Orderings_Ono__top_367,axiom,
no_top @ real ).
thf(tcon_Real_Oreal___Orderings_Ono__bot_368,axiom,
no_bot @ real ).
thf(tcon_Real_Oreal___Groups_Ogroup__add_369,axiom,
group_add @ real ).
thf(tcon_Real_Oreal___Rings_Omult__zero_370,axiom,
mult_zero @ real ).
thf(tcon_Real_Oreal___Rings_Ocomm__ring_371,axiom,
comm_ring @ real ).
thf(tcon_Real_Oreal___Orderings_Oorder_372,axiom,
order @ real ).
thf(tcon_Real_Oreal___Num_Oneg__numeral_373,axiom,
neg_numeral @ real ).
thf(tcon_Real_Oreal___Nat_Oring__char__0_374,axiom,
ring_char_0 @ real ).
thf(tcon_Real_Oreal___Rings_Osemiring_375,axiom,
semiring @ real ).
thf(tcon_Real_Oreal___Fields_Oinverse_376,axiom,
inverse @ real ).
thf(tcon_Real_Oreal___Orderings_Oord_377,axiom,
ord @ real ).
thf(tcon_Real_Oreal___Groups_Ouminus_378,axiom,
uminus @ real ).
thf(tcon_Real_Oreal___Rings_Oring__1_379,axiom,
ring_1 @ real ).
thf(tcon_Real_Oreal___Rings_Oabs__if_380,axiom,
abs_if @ real ).
thf(tcon_Real_Oreal___Groups_Otimes_381,axiom,
times @ real ).
thf(tcon_Real_Oreal___Groups_Ominus_382,axiom,
minus @ real ).
thf(tcon_Real_Oreal___Fields_Ofield_383,axiom,
field @ real ).
thf(tcon_Real_Oreal___Power_Opower_384,axiom,
power @ real ).
thf(tcon_Real_Oreal___Num_Onumeral_385,axiom,
numeral @ real ).
thf(tcon_Real_Oreal___Groups_Ozero_386,axiom,
zero @ real ).
thf(tcon_Real_Oreal___Groups_Oplus_387,axiom,
plus @ real ).
thf(tcon_Real_Oreal___Rings_Oring_388,axiom,
ring @ real ).
thf(tcon_Real_Oreal___Rings_Oidom_389,axiom,
idom @ real ).
thf(tcon_Real_Oreal___Groups_Oone_390,axiom,
one @ real ).
thf(tcon_Real_Oreal___Rings_Odvd_391,axiom,
dvd @ real ).
thf(tcon_Real_Oreal___HOL_Oequal_392,axiom,
cl_HOL_Oequal @ real ).
thf(tcon_Word_Oword___Bit__Operations_Osemiring__bit__operations_393,axiom,
! [A13: $tType] :
( ( type_len @ A13 )
=> ( bit_se359711467146920520ations @ ( word @ A13 ) ) ) ).
thf(tcon_Word_Oword___Quickcheck__Narrowing_Opartial__term__of_394,axiom,
! [A13: $tType] :
( ( typerep @ A13 )
=> ( quickc6926020345158392990erm_of @ ( word @ A13 ) ) ) ).
thf(tcon_Word_Oword___Bit__Operations_Oring__bit__operations_395,axiom,
! [A13: $tType] :
( ( type_len @ A13 )
=> ( bit_ri3973907225187159222ations @ ( word @ A13 ) ) ) ).
thf(tcon_Word_Oword___Groups_Ocancel__ab__semigroup__add_396,axiom,
! [A13: $tType] :
( ( type_len @ A13 )
=> ( cancel2418104881723323429up_add @ ( word @ A13 ) ) ) ).
thf(tcon_Word_Oword___Groups_Ocancel__comm__monoid__add_397,axiom,
! [A13: $tType] :
( ( type_len @ A13 )
=> ( cancel1802427076303600483id_add @ ( word @ A13 ) ) ) ).
thf(tcon_Word_Oword___Rings_Ocomm__semiring__1__cancel_398,axiom,
! [A13: $tType] :
( ( type_len @ A13 )
=> ( comm_s4317794764714335236cancel @ ( word @ A13 ) ) ) ).
thf(tcon_Word_Oword___Bit__Operations_Osemiring__bits_399,axiom,
! [A13: $tType] :
( ( type_len @ A13 )
=> ( bit_semiring_bits @ ( word @ A13 ) ) ) ).
thf(tcon_Word_Oword___Groups_Ocancel__semigroup__add_400,axiom,
! [A13: $tType] :
( ( type_len @ A13 )
=> ( cancel_semigroup_add @ ( word @ A13 ) ) ) ).
thf(tcon_Word_Oword___Least__significant__bit_Olsb_401,axiom,
! [A13: $tType] :
( ( type_len @ A13 )
=> ( least_6119777620449941438nt_lsb @ ( word @ A13 ) ) ) ).
thf(tcon_Word_Oword___Groups_Oab__semigroup__mult_402,axiom,
! [A13: $tType] :
( ( type_len @ A13 )
=> ( ab_semigroup_mult @ ( word @ A13 ) ) ) ).
thf(tcon_Word_Oword___Rings_Osemiring__1__cancel_403,axiom,
! [A13: $tType] :
( ( type_len @ A13 )
=> ( semiring_1_cancel @ ( word @ A13 ) ) ) ).
thf(tcon_Word_Oword___Groups_Ocomm__monoid__mult_404,axiom,
! [A13: $tType] :
( ( type_len @ A13 )
=> ( comm_monoid_mult @ ( word @ A13 ) ) ) ).
thf(tcon_Word_Oword___Groups_Oab__semigroup__add_405,axiom,
! [A13: $tType] :
( ( type_len @ A13 )
=> ( ab_semigroup_add @ ( word @ A13 ) ) ) ).
thf(tcon_Word_Oword___Generic__set__bit_Oset__bit_406,axiom,
! [A13: $tType] :
( ( type_len @ A13 )
=> ( generic_set_set_bit @ ( word @ A13 ) ) ) ).
thf(tcon_Word_Oword___Code__Evaluation_Oterm__of_407,axiom,
! [A13: $tType] :
( ( typerep @ A13 )
=> ( code_term_of @ ( word @ A13 ) ) ) ).
thf(tcon_Word_Oword___Parity_Osemiring__parity_408,axiom,
! [A13: $tType] :
( ( type_len @ A13 )
=> ( semiring_parity @ ( word @ A13 ) ) ) ).
thf(tcon_Word_Oword___Groups_Ocomm__monoid__add_409,axiom,
! [A13: $tType] :
( ( type_len @ A13 )
=> ( comm_monoid_add @ ( word @ A13 ) ) ) ).
thf(tcon_Word_Oword___Rings_Osemiring__modulo_410,axiom,
! [A13: $tType] :
( ( type_len @ A13 )
=> ( semiring_modulo @ ( word @ A13 ) ) ) ).
thf(tcon_Word_Oword___Rings_Ocomm__semiring__1_411,axiom,
! [A13: $tType] :
( ( type_len @ A13 )
=> ( comm_semiring_1 @ ( word @ A13 ) ) ) ).
thf(tcon_Word_Oword___Rings_Ocomm__semiring__0_412,axiom,
! [A13: $tType] :
( ( type_len @ A13 )
=> ( comm_semiring_0 @ ( word @ A13 ) ) ) ).
thf(tcon_Word_Oword___Groups_Osemigroup__mult_413,axiom,
! [A13: $tType] :
( ( type_len @ A13 )
=> ( semigroup_mult @ ( word @ A13 ) ) ) ).
thf(tcon_Word_Oword___Num_Osemiring__numeral_414,axiom,
! [A13: $tType] :
( ( type_len @ A13 )
=> ( semiring_numeral @ ( word @ A13 ) ) ) ).
thf(tcon_Word_Oword___Groups_Osemigroup__add_415,axiom,
! [A13: $tType] :
( ( type_len @ A13 )
=> ( semigroup_add @ ( word @ A13 ) ) ) ).
thf(tcon_Word_Oword___Rings_Ocomm__semiring_416,axiom,
! [A13: $tType] :
( ( type_len @ A13 )
=> ( comm_semiring @ ( word @ A13 ) ) ) ).
thf(tcon_Word_Oword___Orderings_Owellorder_417,axiom,
! [A13: $tType] :
( ( type_len @ A13 )
=> ( wellorder @ ( word @ A13 ) ) ) ).
thf(tcon_Word_Oword___Groups_Oab__group__add_418,axiom,
! [A13: $tType] :
( ( type_len @ A13 )
=> ( ab_group_add @ ( word @ A13 ) ) ) ).
thf(tcon_Word_Oword___Rings_Ozero__neq__one_419,axiom,
! [A13: $tType] :
( ( type_len @ A13 )
=> ( zero_neq_one @ ( word @ A13 ) ) ) ).
thf(tcon_Word_Oword___Parity_Oring__parity_420,axiom,
! [A13: $tType] :
( ( type_len @ A13 )
=> ( ring_parity @ ( word @ A13 ) ) ) ).
thf(tcon_Word_Oword___Orderings_Opreorder_421,axiom,
! [A13: $tType] :
( ( type_len @ A13 )
=> ( preorder @ ( word @ A13 ) ) ) ).
thf(tcon_Word_Oword___Orderings_Olinorder_422,axiom,
! [A13: $tType] :
( ( type_len @ A13 )
=> ( linorder @ ( word @ A13 ) ) ) ).
thf(tcon_Word_Oword___Groups_Omonoid__mult_423,axiom,
! [A13: $tType] :
( ( type_len @ A13 )
=> ( monoid_mult @ ( word @ A13 ) ) ) ).
thf(tcon_Word_Oword___Rings_Ocomm__ring__1_424,axiom,
! [A13: $tType] :
( ( type_len @ A13 )
=> ( comm_ring_1 @ ( word @ A13 ) ) ) ).
thf(tcon_Word_Oword___Groups_Omonoid__add_425,axiom,
! [A13: $tType] :
( ( type_len @ A13 )
=> ( monoid_add @ ( word @ A13 ) ) ) ).
thf(tcon_Word_Oword___Finite__Set_Ofinite_426,axiom,
! [A13: $tType] :
( ( type_len @ A13 )
=> ( finite_finite @ ( word @ A13 ) ) ) ).
thf(tcon_Word_Oword___Rings_Osemiring__1_427,axiom,
! [A13: $tType] :
( ( type_len @ A13 )
=> ( semiring_1 @ ( word @ A13 ) ) ) ).
thf(tcon_Word_Oword___Rings_Osemiring__0_428,axiom,
! [A13: $tType] :
( ( type_len @ A13 )
=> ( semiring_0 @ ( word @ A13 ) ) ) ).
thf(tcon_Word_Oword___Groups_Ogroup__add_429,axiom,
! [A13: $tType] :
( ( type_len @ A13 )
=> ( group_add @ ( word @ A13 ) ) ) ).
thf(tcon_Word_Oword___Rings_Omult__zero_430,axiom,
! [A13: $tType] :
( ( type_len @ A13 )
=> ( mult_zero @ ( word @ A13 ) ) ) ).
thf(tcon_Word_Oword___Rings_Ocomm__ring_431,axiom,
! [A13: $tType] :
( ( type_len @ A13 )
=> ( comm_ring @ ( word @ A13 ) ) ) ).
thf(tcon_Word_Oword___Orderings_Oorder_432,axiom,
! [A13: $tType] :
( ( type_len @ A13 )
=> ( order @ ( word @ A13 ) ) ) ).
thf(tcon_Word_Oword___Num_Oneg__numeral_433,axiom,
! [A13: $tType] :
( ( type_len @ A13 )
=> ( neg_numeral @ ( word @ A13 ) ) ) ).
thf(tcon_Word_Oword___Rings_Osemiring_434,axiom,
! [A13: $tType] :
( ( type_len @ A13 )
=> ( semiring @ ( word @ A13 ) ) ) ).
thf(tcon_Word_Oword___Orderings_Oord_435,axiom,
! [A13: $tType] :
( ( type_len @ A13 )
=> ( ord @ ( word @ A13 ) ) ) ).
thf(tcon_Word_Oword___Groups_Ouminus_436,axiom,
! [A13: $tType] :
( ( type_len @ A13 )
=> ( uminus @ ( word @ A13 ) ) ) ).
thf(tcon_Word_Oword___Rings_Oring__1_437,axiom,
! [A13: $tType] :
( ( type_len @ A13 )
=> ( ring_1 @ ( word @ A13 ) ) ) ).
thf(tcon_Word_Oword___Groups_Otimes_438,axiom,
! [A13: $tType] :
( ( type_len @ A13 )
=> ( times @ ( word @ A13 ) ) ) ).
thf(tcon_Word_Oword___Groups_Ominus_439,axiom,
! [A13: $tType] :
( ( type_len @ A13 )
=> ( minus @ ( word @ A13 ) ) ) ).
thf(tcon_Word_Oword___Power_Opower_440,axiom,
! [A13: $tType] :
( ( type_len @ A13 )
=> ( power @ ( word @ A13 ) ) ) ).
thf(tcon_Word_Oword___Num_Onumeral_441,axiom,
! [A13: $tType] :
( ( type_len @ A13 )
=> ( numeral @ ( word @ A13 ) ) ) ).
thf(tcon_Word_Oword___Groups_Ozero_442,axiom,
! [A13: $tType] :
( ( type_len @ A13 )
=> ( zero @ ( word @ A13 ) ) ) ).
thf(tcon_Word_Oword___Groups_Oplus_443,axiom,
! [A13: $tType] :
( ( type_len @ A13 )
=> ( plus @ ( word @ A13 ) ) ) ).
thf(tcon_Word_Oword___Rings_Oring_444,axiom,
! [A13: $tType] :
( ( type_len @ A13 )
=> ( ring @ ( word @ A13 ) ) ) ).
thf(tcon_Word_Oword___Groups_Oone_445,axiom,
! [A13: $tType] :
( ( type_len @ A13 )
=> ( one @ ( word @ A13 ) ) ) ).
thf(tcon_Word_Oword___Rings_Odvd_446,axiom,
! [A13: $tType] :
( ( type_len @ A13 )
=> ( dvd @ ( word @ A13 ) ) ) ).
thf(tcon_Word_Oword___HOL_Oequal_447,axiom,
! [A13: $tType] :
( ( type_len @ A13 )
=> ( cl_HOL_Oequal @ ( word @ A13 ) ) ) ).
thf(tcon_Word_Oword___Nat_Osize_448,axiom,
! [A13: $tType] :
( ( type_len @ A13 )
=> ( size @ ( word @ A13 ) ) ) ).
thf(tcon_Heap_Oarray___Quickcheck__Narrowing_Opartial__term__of_449,axiom,
! [A13: $tType] :
( ( typerep @ A13 )
=> ( quickc6926020345158392990erm_of @ ( array @ A13 ) ) ) ).
thf(tcon_Heap_Oarray___Code__Evaluation_Oterm__of_450,axiom,
! [A13: $tType] :
( ( typerep @ A13 )
=> ( code_term_of @ ( array @ A13 ) ) ) ).
thf(tcon_Heap_Oarray___Heap_Oheap_451,axiom,
! [A13: $tType] : ( heap @ ( array @ A13 ) ) ).
thf(tcon_Heap_Oarray___HOL_Oequal_452,axiom,
! [A13: $tType] : ( cl_HOL_Oequal @ ( array @ A13 ) ) ).
thf(tcon_Heap_Oarray___Nat_Osize_453,axiom,
! [A13: $tType] : ( size @ ( array @ A13 ) ) ).
thf(tcon_String_Ochar___Quickcheck__Narrowing_Opartial__term__of_454,axiom,
quickc6926020345158392990erm_of @ char ).
thf(tcon_String_Ochar___Code__Evaluation_Oterm__of_455,axiom,
code_term_of @ char ).
thf(tcon_String_Ochar___Orderings_Opreorder_456,axiom,
preorder @ char ).
thf(tcon_String_Ochar___Orderings_Olinorder_457,axiom,
linorder @ char ).
thf(tcon_String_Ochar___Finite__Set_Ofinite_458,axiom,
finite_finite @ char ).
thf(tcon_String_Ochar___Orderings_Oorder_459,axiom,
order @ char ).
thf(tcon_String_Ochar___Orderings_Oord_460,axiom,
ord @ char ).
thf(tcon_String_Ochar___Heap_Oheap_461,axiom,
heap @ char ).
thf(tcon_String_Ochar___HOL_Oequal_462,axiom,
cl_HOL_Oequal @ char ).
thf(tcon_String_Ochar___Nat_Osize_463,axiom,
size @ char ).
thf(tcon_Sum__Type_Osum___Quickcheck__Narrowing_Opartial__term__of_464,axiom,
! [A13: $tType,A17: $tType] :
( ( ( typerep @ A13 )
& ( typerep @ A17 ) )
=> ( quickc6926020345158392990erm_of @ ( sum_sum @ A13 @ A17 ) ) ) ).
thf(tcon_Sum__Type_Osum___Code__Evaluation_Oterm__of_465,axiom,
! [A13: $tType,A17: $tType] :
( ( ( typerep @ A13 )
& ( typerep @ A17 ) )
=> ( code_term_of @ ( sum_sum @ A13 @ A17 ) ) ) ).
thf(tcon_Sum__Type_Osum___Finite__Set_Ofinite_466,axiom,
! [A13: $tType,A17: $tType] :
( ( ( finite_finite @ A13 )
& ( finite_finite @ A17 ) )
=> ( finite_finite @ ( sum_sum @ A13 @ A17 ) ) ) ).
thf(tcon_Sum__Type_Osum___Heap_Oheap_467,axiom,
! [A13: $tType,A17: $tType] :
( ( ( heap @ A13 )
& ( heap @ A17 ) )
=> ( heap @ ( sum_sum @ A13 @ A17 ) ) ) ).
thf(tcon_Sum__Type_Osum___HOL_Oequal_468,axiom,
! [A13: $tType,A17: $tType] : ( cl_HOL_Oequal @ ( sum_sum @ A13 @ A17 ) ) ).
thf(tcon_Sum__Type_Osum___Nat_Osize_469,axiom,
! [A13: $tType,A17: $tType] : ( size @ ( sum_sum @ A13 @ A17 ) ) ).
thf(tcon_Enum_Ofinite__1___Conditionally__Complete__Lattices_Oconditionally__complete__linorder_470,axiom,
condit6923001295902523014norder @ finite_1 ).
thf(tcon_Enum_Ofinite__1___Conditionally__Complete__Lattices_Oconditionally__complete__lattice_471,axiom,
condit1219197933456340205attice @ finite_1 ).
thf(tcon_Enum_Ofinite__1___Groups_Oordered__ab__semigroup__monoid__add__imp__le_472,axiom,
ordere1937475149494474687imp_le @ finite_1 ).
thf(tcon_Enum_Ofinite__1___Rings_Osemiring__no__zero__divisors__cancel_473,axiom,
semiri6575147826004484403cancel @ finite_1 ).
thf(tcon_Enum_Ofinite__1___Groups_Ostrict__ordered__ab__semigroup__add_474,axiom,
strict9044650504122735259up_add @ finite_1 ).
thf(tcon_Enum_Ofinite__1___Groups_Oordered__cancel__comm__monoid__diff_475,axiom,
ordere1170586879665033532d_diff @ finite_1 ).
thf(tcon_Enum_Ofinite__1___Groups_Oordered__cancel__ab__semigroup__add_476,axiom,
ordere580206878836729694up_add @ finite_1 ).
thf(tcon_Enum_Ofinite__1___Groups_Oordered__ab__semigroup__add__imp__le_477,axiom,
ordere2412721322843649153imp_le @ finite_1 ).
thf(tcon_Enum_Ofinite__1___Rings_Olinordered__comm__semiring__strict_478,axiom,
linord2810124833399127020strict @ finite_1 ).
thf(tcon_Enum_Ofinite__1___Groups_Ostrict__ordered__comm__monoid__add_479,axiom,
strict7427464778891057005id_add @ finite_1 ).
thf(tcon_Enum_Ofinite__1___Groups_Oordered__cancel__comm__monoid__add_480,axiom,
ordere8940638589300402666id_add @ finite_1 ).
thf(tcon_Enum_Ofinite__1___Groups_Ocanonically__ordered__monoid__add_481,axiom,
canoni5634975068530333245id_add @ finite_1 ).
thf(tcon_Enum_Ofinite__1___Quickcheck__Narrowing_Opartial__term__of_482,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_483,axiom,
linord4140545234300271783up_add @ finite_1 ).
thf(tcon_Enum_Ofinite__1___Complete__Lattices_Ocomplete__lattice_484,axiom,
comple6319245703460814977attice @ finite_1 ).
thf(tcon_Enum_Ofinite__1___Rings_Olinordered__semiring__strict_485,axiom,
linord8928482502909563296strict @ finite_1 ).
thf(tcon_Enum_Ofinite__1___Boolean__Algebras_Oboolean__algebra_486,axiom,
boolea8198339166811842893lgebra @ finite_1 ).
thf(tcon_Enum_Ofinite__1___Rings_Osemiring__no__zero__divisors_487,axiom,
semiri3467727345109120633visors @ finite_1 ).
thf(tcon_Enum_Ofinite__1___Groups_Oordered__ab__semigroup__add_488,axiom,
ordere6658533253407199908up_add @ finite_1 ).
thf(tcon_Enum_Ofinite__1___Groups_Oordered__ab__group__add__abs_489,axiom,
ordere166539214618696060dd_abs @ finite_1 ).
thf(tcon_Enum_Ofinite__1___Groups_Oordered__comm__monoid__add_490,axiom,
ordere6911136660526730532id_add @ finite_1 ).
thf(tcon_Enum_Ofinite__1___Groups_Olinordered__ab__group__add_491,axiom,
linord5086331880401160121up_add @ finite_1 ).
thf(tcon_Enum_Ofinite__1___Groups_Ocancel__ab__semigroup__add_492,axiom,
cancel2418104881723323429up_add @ finite_1 ).
thf(tcon_Enum_Ofinite__1___Groups_Ocancel__comm__monoid__add_493,axiom,
cancel1802427076303600483id_add @ finite_1 ).
thf(tcon_Enum_Ofinite__1___Rings_Olinordered__ring__strict_494,axiom,
linord4710134922213307826strict @ finite_1 ).
thf(tcon_Enum_Ofinite__1___Rings_Oordered__comm__semiring_495,axiom,
ordere2520102378445227354miring @ finite_1 ).
thf(tcon_Enum_Ofinite__1___Groups_Oordered__ab__group__add_496,axiom,
ordered_ab_group_add @ finite_1 ).
thf(tcon_Enum_Ofinite__1___Groups_Ocancel__semigroup__add_497,axiom,
cancel_semigroup_add @ finite_1 ).
thf(tcon_Enum_Ofinite__1___Rings_Olinordered__semiring_498,axiom,
linordered_semiring @ finite_1 ).
thf(tcon_Enum_Ofinite__1___Rings_Oordered__semiring__0_499,axiom,
ordered_semiring_0 @ finite_1 ).
thf(tcon_Enum_Ofinite__1___Orderings_Odense__linorder_500,axiom,
dense_linorder @ finite_1 ).
thf(tcon_Enum_Ofinite__1___Groups_Oab__semigroup__mult_501,axiom,
ab_semigroup_mult @ finite_1 ).
thf(tcon_Enum_Ofinite__1___Groups_Ocomm__monoid__mult_502,axiom,
comm_monoid_mult @ finite_1 ).
thf(tcon_Enum_Ofinite__1___Groups_Ocomm__monoid__diff_503,axiom,
comm_monoid_diff @ finite_1 ).
thf(tcon_Enum_Ofinite__1___Groups_Oab__semigroup__add_504,axiom,
ab_semigroup_add @ finite_1 ).
thf(tcon_Enum_Ofinite__1___Code__Evaluation_Oterm__of_505,axiom,
code_term_of @ finite_1 ).
thf(tcon_Enum_Ofinite__1___Rings_Oordered__semiring_506,axiom,
ordered_semiring @ finite_1 ).
thf(tcon_Enum_Ofinite__1___Rings_Oordered__ring__abs_507,axiom,
ordered_ring_abs @ finite_1 ).
thf(tcon_Enum_Ofinite__1___Groups_Ocomm__monoid__add_508,axiom,
comm_monoid_add @ finite_1 ).
thf(tcon_Enum_Ofinite__1___Rings_Olinordered__ring_509,axiom,
linordered_ring @ finite_1 ).
thf(tcon_Enum_Ofinite__1___Rings_Ocomm__semiring__0_510,axiom,
comm_semiring_0 @ finite_1 ).
thf(tcon_Enum_Ofinite__1___Orderings_Odense__order_511,axiom,
dense_order @ finite_1 ).
thf(tcon_Enum_Ofinite__1___Groups_Osemigroup__mult_512,axiom,
semigroup_mult @ finite_1 ).
thf(tcon_Enum_Ofinite__1___Groups_Osemigroup__add_513,axiom,
semigroup_add @ finite_1 ).
thf(tcon_Enum_Ofinite__1___Rings_Ocomm__semiring_514,axiom,
comm_semiring @ finite_1 ).
thf(tcon_Enum_Ofinite__1___Orderings_Owellorder_515,axiom,
wellorder @ finite_1 ).
thf(tcon_Enum_Ofinite__1___Orderings_Oorder__top_516,axiom,
order_top @ finite_1 ).
thf(tcon_Enum_Ofinite__1___Orderings_Oorder__bot_517,axiom,
order_bot @ finite_1 ).
thf(tcon_Enum_Ofinite__1___Groups_Oab__group__add_518,axiom,
ab_group_add @ finite_1 ).
thf(tcon_Enum_Ofinite__1___Rings_Oordered__ring_519,axiom,
ordered_ring @ finite_1 ).
thf(tcon_Enum_Ofinite__1___Orderings_Opreorder_520,axiom,
preorder @ finite_1 ).
thf(tcon_Enum_Ofinite__1___Orderings_Olinorder_521,axiom,
linorder @ finite_1 ).
thf(tcon_Enum_Ofinite__1___Groups_Omonoid__mult_522,axiom,
monoid_mult @ finite_1 ).
thf(tcon_Enum_Ofinite__1___Groups_Omonoid__add_523,axiom,
monoid_add @ finite_1 ).
thf(tcon_Enum_Ofinite__1___Finite__Set_Ofinite_524,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_525,axiom,
semiring_0 @ finite_1 ).
thf(tcon_Enum_Ofinite__1___Groups_Ogroup__add_526,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_527,axiom,
mult_zero @ finite_1 ).
thf(tcon_Enum_Ofinite__1___Rings_Ocomm__ring_528,axiom,
comm_ring @ finite_1 ).
thf(tcon_Enum_Ofinite__1___Orderings_Oorder_529,axiom,
order @ finite_1 ).
thf(tcon_Enum_Ofinite__1___Rings_Osemiring_530,axiom,
semiring @ finite_1 ).
thf(tcon_Enum_Ofinite__1___Fields_Oinverse_531,axiom,
inverse @ finite_1 ).
thf(tcon_Enum_Ofinite__1___Orderings_Oord_532,axiom,
ord @ finite_1 ).
thf(tcon_Enum_Ofinite__1___Groups_Ouminus_533,axiom,
uminus @ finite_1 ).
thf(tcon_Enum_Ofinite__1___Rings_Oabs__if_534,axiom,
abs_if @ finite_1 ).
thf(tcon_Enum_Ofinite__1___Groups_Otimes_535,axiom,
times @ finite_1 ).
thf(tcon_Enum_Ofinite__1___Groups_Ominus_536,axiom,
minus @ finite_1 ).
thf(tcon_Enum_Ofinite__1___Power_Opower_537,axiom,
power @ finite_1 ).
thf(tcon_Enum_Ofinite__1___Groups_Ozero_538,axiom,
zero @ finite_1 ).
thf(tcon_Enum_Ofinite__1___Groups_Oplus_539,axiom,
plus @ finite_1 ).
thf(tcon_Enum_Ofinite__1___Rings_Oring_540,axiom,
ring @ finite_1 ).
thf(tcon_Enum_Ofinite__1___Groups_Oone_541,axiom,
one @ finite_1 ).
thf(tcon_Enum_Ofinite__1___Rings_Odvd_542,axiom,
dvd @ finite_1 ).
thf(tcon_Enum_Ofinite__1___HOL_Oequal_543,axiom,
cl_HOL_Oequal @ finite_1 ).
thf(tcon_Enum_Ofinite__2___Conditionally__Complete__Lattices_Oconditionally__complete__linorder_544,axiom,
condit6923001295902523014norder @ finite_2 ).
thf(tcon_Enum_Ofinite__2___Conditionally__Complete__Lattices_Oconditionally__complete__lattice_545,axiom,
condit1219197933456340205attice @ finite_2 ).
thf(tcon_Enum_Ofinite__2___Semiring__Normalization_Ocomm__semiring__1__cancel__crossproduct_546,axiom,
semiri1453513574482234551roduct @ finite_2 ).
thf(tcon_Enum_Ofinite__2___Euclidean__Division_Ounique__euclidean__semiring_547,axiom,
euclid3128863361964157862miring @ finite_2 ).
thf(tcon_Enum_Ofinite__2___Euclidean__Division_Oeuclidean__semiring__cancel_548,axiom,
euclid4440199948858584721cancel @ finite_2 ).
thf(tcon_Enum_Ofinite__2___Rings_Osemiring__no__zero__divisors__cancel_549,axiom,
semiri6575147826004484403cancel @ finite_2 ).
thf(tcon_Enum_Ofinite__2___Euclidean__Division_Oeuclidean__semiring_550,axiom,
euclid3725896446679973847miring @ finite_2 ).
thf(tcon_Enum_Ofinite__2___Quickcheck__Narrowing_Opartial__term__of_551,axiom,
quickc6926020345158392990erm_of @ finite_2 ).
thf(tcon_Enum_Ofinite__2___Complete__Lattices_Ocomplete__linorder_552,axiom,
comple5582772986160207858norder @ finite_2 ).
thf(tcon_Enum_Ofinite__2___Complete__Lattices_Ocomplete__lattice_553,axiom,
comple6319245703460814977attice @ finite_2 ).
thf(tcon_Enum_Ofinite__2___Rings_Osemiring__1__no__zero__divisors_554,axiom,
semiri2026040879449505780visors @ finite_2 ).
thf(tcon_Enum_Ofinite__2___Rings_Osemiring__no__zero__divisors_555,axiom,
semiri3467727345109120633visors @ finite_2 ).
thf(tcon_Enum_Ofinite__2___Groups_Ocancel__ab__semigroup__add_556,axiom,
cancel2418104881723323429up_add @ finite_2 ).
thf(tcon_Enum_Ofinite__2___Rings_Oring__1__no__zero__divisors_557,axiom,
ring_15535105094025558882visors @ finite_2 ).
thf(tcon_Enum_Ofinite__2___Groups_Ocancel__comm__monoid__add_558,axiom,
cancel1802427076303600483id_add @ finite_2 ).
thf(tcon_Enum_Ofinite__2___Rings_Ocomm__semiring__1__cancel_559,axiom,
comm_s4317794764714335236cancel @ finite_2 ).
thf(tcon_Enum_Ofinite__2___Groups_Ocancel__semigroup__add_560,axiom,
cancel_semigroup_add @ finite_2 ).
thf(tcon_Enum_Ofinite__2___Groups_Oab__semigroup__mult_561,axiom,
ab_semigroup_mult @ finite_2 ).
thf(tcon_Enum_Ofinite__2___Rings_Osemiring__1__cancel_562,axiom,
semiring_1_cancel @ finite_2 ).
thf(tcon_Enum_Ofinite__2___Rings_Oalgebraic__semidom_563,axiom,
algebraic_semidom @ finite_2 ).
thf(tcon_Enum_Ofinite__2___Groups_Ocomm__monoid__mult_564,axiom,
comm_monoid_mult @ finite_2 ).
thf(tcon_Enum_Ofinite__2___Groups_Oab__semigroup__add_565,axiom,
ab_semigroup_add @ finite_2 ).
thf(tcon_Enum_Ofinite__2___Code__Evaluation_Oterm__of_566,axiom,
code_term_of @ finite_2 ).
thf(tcon_Enum_Ofinite__2___Groups_Ocomm__monoid__add_567,axiom,
comm_monoid_add @ finite_2 ).
thf(tcon_Enum_Ofinite__2___Rings_Osemiring__modulo_568,axiom,
semiring_modulo @ finite_2 ).
thf(tcon_Enum_Ofinite__2___Rings_Ocomm__semiring__1_569,axiom,
comm_semiring_1 @ finite_2 ).
thf(tcon_Enum_Ofinite__2___Rings_Ocomm__semiring__0_570,axiom,
comm_semiring_0 @ finite_2 ).
thf(tcon_Enum_Ofinite__2___Groups_Osemigroup__mult_571,axiom,
semigroup_mult @ finite_2 ).
thf(tcon_Enum_Ofinite__2___Rings_Osemidom__modulo_572,axiom,
semidom_modulo @ finite_2 ).
thf(tcon_Enum_Ofinite__2___Rings_Osemidom__divide_573,axiom,
semidom_divide @ finite_2 ).
thf(tcon_Enum_Ofinite__2___Num_Osemiring__numeral_574,axiom,
semiring_numeral @ finite_2 ).
thf(tcon_Enum_Ofinite__2___Groups_Osemigroup__add_575,axiom,
semigroup_add @ finite_2 ).
thf(tcon_Enum_Ofinite__2___Fields_Odivision__ring_576,axiom,
division_ring @ finite_2 ).
thf(tcon_Enum_Ofinite__2___Rings_Ocomm__semiring_577,axiom,
comm_semiring @ finite_2 ).
thf(tcon_Enum_Ofinite__2___Orderings_Owellorder_578,axiom,
wellorder @ finite_2 ).
thf(tcon_Enum_Ofinite__2___Orderings_Oorder__top_579,axiom,
order_top @ finite_2 ).
thf(tcon_Enum_Ofinite__2___Orderings_Oorder__bot_580,axiom,
order_bot @ finite_2 ).
thf(tcon_Enum_Ofinite__2___Groups_Oab__group__add_581,axiom,
ab_group_add @ finite_2 ).
thf(tcon_Enum_Ofinite__2___Rings_Ozero__neq__one_582,axiom,
zero_neq_one @ finite_2 ).
thf(tcon_Enum_Ofinite__2___Rings_Oidom__abs__sgn_583,axiom,
idom_abs_sgn @ finite_2 ).
thf(tcon_Enum_Ofinite__2___Orderings_Opreorder_584,axiom,
preorder @ finite_2 ).
thf(tcon_Enum_Ofinite__2___Orderings_Olinorder_585,axiom,
linorder @ finite_2 ).
thf(tcon_Enum_Ofinite__2___Groups_Omonoid__mult_586,axiom,
monoid_mult @ finite_2 ).
thf(tcon_Enum_Ofinite__2___Rings_Oidom__modulo_587,axiom,
idom_modulo @ finite_2 ).
thf(tcon_Enum_Ofinite__2___Rings_Oidom__divide_588,axiom,
idom_divide @ finite_2 ).
thf(tcon_Enum_Ofinite__2___Rings_Ocomm__ring__1_589,axiom,
comm_ring_1 @ finite_2 ).
thf(tcon_Enum_Ofinite__2___Groups_Omonoid__add_590,axiom,
monoid_add @ finite_2 ).
thf(tcon_Enum_Ofinite__2___Finite__Set_Ofinite_591,axiom,
finite_finite @ finite_2 ).
thf(tcon_Enum_Ofinite__2___Type__Length_Olen0_592,axiom,
type_len0 @ finite_2 ).
thf(tcon_Enum_Ofinite__2___Rings_Osemiring__1_593,axiom,
semiring_1 @ finite_2 ).
thf(tcon_Enum_Ofinite__2___Rings_Osemiring__0_594,axiom,
semiring_0 @ finite_2 ).
thf(tcon_Enum_Ofinite__2___Groups_Ogroup__add_595,axiom,
group_add @ finite_2 ).
thf(tcon_Enum_Ofinite__2___Type__Length_Olen_596,axiom,
type_len @ finite_2 ).
thf(tcon_Enum_Ofinite__2___Rings_Omult__zero_597,axiom,
mult_zero @ finite_2 ).
thf(tcon_Enum_Ofinite__2___Rings_Ocomm__ring_598,axiom,
comm_ring @ finite_2 ).
thf(tcon_Enum_Ofinite__2___Orderings_Oorder_599,axiom,
order @ finite_2 ).
thf(tcon_Enum_Ofinite__2___Num_Oneg__numeral_600,axiom,
neg_numeral @ finite_2 ).
thf(tcon_Enum_Ofinite__2___Rings_Osemiring_601,axiom,
semiring @ finite_2 ).
thf(tcon_Enum_Ofinite__2___Fields_Oinverse_602,axiom,
inverse @ finite_2 ).
thf(tcon_Enum_Ofinite__2___Orderings_Oord_603,axiom,
ord @ finite_2 ).
thf(tcon_Enum_Ofinite__2___Groups_Ouminus_604,axiom,
uminus @ finite_2 ).
thf(tcon_Enum_Ofinite__2___Rings_Oring__1_605,axiom,
ring_1 @ finite_2 ).
thf(tcon_Enum_Ofinite__2___Groups_Otimes_606,axiom,
times @ finite_2 ).
thf(tcon_Enum_Ofinite__2___Groups_Ominus_607,axiom,
minus @ finite_2 ).
thf(tcon_Enum_Ofinite__2___Fields_Ofield_608,axiom,
field @ finite_2 ).
thf(tcon_Enum_Ofinite__2___Power_Opower_609,axiom,
power @ finite_2 ).
thf(tcon_Enum_Ofinite__2___Num_Onumeral_610,axiom,
numeral @ finite_2 ).
thf(tcon_Enum_Ofinite__2___Groups_Ozero_611,axiom,
zero @ finite_2 ).
thf(tcon_Enum_Ofinite__2___Groups_Oplus_612,axiom,
plus @ finite_2 ).
thf(tcon_Enum_Ofinite__2___Rings_Oring_613,axiom,
ring @ finite_2 ).
thf(tcon_Enum_Ofinite__2___Rings_Oidom_614,axiom,
idom @ finite_2 ).
thf(tcon_Enum_Ofinite__2___Groups_Oone_615,axiom,
one @ finite_2 ).
thf(tcon_Enum_Ofinite__2___Rings_Odvd_616,axiom,
dvd @ finite_2 ).
thf(tcon_Enum_Ofinite__2___HOL_Oequal_617,axiom,
cl_HOL_Oequal @ finite_2 ).
thf(tcon_Enum_Ofinite__3___Conditionally__Complete__Lattices_Oconditionally__complete__linorder_618,axiom,
condit6923001295902523014norder @ finite_3 ).
thf(tcon_Enum_Ofinite__3___Conditionally__Complete__Lattices_Oconditionally__complete__lattice_619,axiom,
condit1219197933456340205attice @ finite_3 ).
thf(tcon_Enum_Ofinite__3___Semiring__Normalization_Ocomm__semiring__1__cancel__crossproduct_620,axiom,
semiri1453513574482234551roduct @ finite_3 ).
thf(tcon_Enum_Ofinite__3___Euclidean__Division_Ounique__euclidean__semiring_621,axiom,
euclid3128863361964157862miring @ finite_3 ).
thf(tcon_Enum_Ofinite__3___Euclidean__Division_Oeuclidean__semiring__cancel_622,axiom,
euclid4440199948858584721cancel @ finite_3 ).
thf(tcon_Enum_Ofinite__3___Rings_Osemiring__no__zero__divisors__cancel_623,axiom,
semiri6575147826004484403cancel @ finite_3 ).
thf(tcon_Enum_Ofinite__3___Euclidean__Division_Oeuclidean__semiring_624,axiom,
euclid3725896446679973847miring @ finite_3 ).
thf(tcon_Enum_Ofinite__3___Quickcheck__Narrowing_Opartial__term__of_625,axiom,
quickc6926020345158392990erm_of @ finite_3 ).
thf(tcon_Enum_Ofinite__3___Complete__Lattices_Ocomplete__linorder_626,axiom,
comple5582772986160207858norder @ finite_3 ).
thf(tcon_Enum_Ofinite__3___Complete__Lattices_Ocomplete__lattice_627,axiom,
comple6319245703460814977attice @ finite_3 ).
thf(tcon_Enum_Ofinite__3___Rings_Osemiring__1__no__zero__divisors_628,axiom,
semiri2026040879449505780visors @ finite_3 ).
thf(tcon_Enum_Ofinite__3___Rings_Osemiring__no__zero__divisors_629,axiom,
semiri3467727345109120633visors @ finite_3 ).
thf(tcon_Enum_Ofinite__3___Groups_Ocancel__ab__semigroup__add_630,axiom,
cancel2418104881723323429up_add @ finite_3 ).
thf(tcon_Enum_Ofinite__3___Rings_Oring__1__no__zero__divisors_631,axiom,
ring_15535105094025558882visors @ finite_3 ).
thf(tcon_Enum_Ofinite__3___Groups_Ocancel__comm__monoid__add_632,axiom,
cancel1802427076303600483id_add @ finite_3 ).
thf(tcon_Enum_Ofinite__3___Rings_Ocomm__semiring__1__cancel_633,axiom,
comm_s4317794764714335236cancel @ finite_3 ).
thf(tcon_Enum_Ofinite__3___Groups_Ocancel__semigroup__add_634,axiom,
cancel_semigroup_add @ finite_3 ).
thf(tcon_Enum_Ofinite__3___Groups_Oab__semigroup__mult_635,axiom,
ab_semigroup_mult @ finite_3 ).
thf(tcon_Enum_Ofinite__3___Rings_Osemiring__1__cancel_636,axiom,
semiring_1_cancel @ finite_3 ).
thf(tcon_Enum_Ofinite__3___Rings_Oalgebraic__semidom_637,axiom,
algebraic_semidom @ finite_3 ).
thf(tcon_Enum_Ofinite__3___Groups_Ocomm__monoid__mult_638,axiom,
comm_monoid_mult @ finite_3 ).
thf(tcon_Enum_Ofinite__3___Groups_Oab__semigroup__add_639,axiom,
ab_semigroup_add @ finite_3 ).
thf(tcon_Enum_Ofinite__3___Code__Evaluation_Oterm__of_640,axiom,
code_term_of @ finite_3 ).
thf(tcon_Enum_Ofinite__3___Groups_Ocomm__monoid__add_641,axiom,
comm_monoid_add @ finite_3 ).
thf(tcon_Enum_Ofinite__3___Rings_Osemiring__modulo_642,axiom,
semiring_modulo @ finite_3 ).
thf(tcon_Enum_Ofinite__3___Rings_Ocomm__semiring__1_643,axiom,
comm_semiring_1 @ finite_3 ).
thf(tcon_Enum_Ofinite__3___Rings_Ocomm__semiring__0_644,axiom,
comm_semiring_0 @ finite_3 ).
thf(tcon_Enum_Ofinite__3___Groups_Osemigroup__mult_645,axiom,
semigroup_mult @ finite_3 ).
thf(tcon_Enum_Ofinite__3___Rings_Osemidom__modulo_646,axiom,
semidom_modulo @ finite_3 ).
thf(tcon_Enum_Ofinite__3___Rings_Osemidom__divide_647,axiom,
semidom_divide @ finite_3 ).
thf(tcon_Enum_Ofinite__3___Num_Osemiring__numeral_648,axiom,
semiring_numeral @ finite_3 ).
thf(tcon_Enum_Ofinite__3___Groups_Osemigroup__add_649,axiom,
semigroup_add @ finite_3 ).
thf(tcon_Enum_Ofinite__3___Fields_Odivision__ring_650,axiom,
division_ring @ finite_3 ).
thf(tcon_Enum_Ofinite__3___Rings_Ocomm__semiring_651,axiom,
comm_semiring @ finite_3 ).
thf(tcon_Enum_Ofinite__3___Orderings_Owellorder_652,axiom,
wellorder @ finite_3 ).
thf(tcon_Enum_Ofinite__3___Orderings_Oorder__top_653,axiom,
order_top @ finite_3 ).
thf(tcon_Enum_Ofinite__3___Orderings_Oorder__bot_654,axiom,
order_bot @ finite_3 ).
thf(tcon_Enum_Ofinite__3___Groups_Oab__group__add_655,axiom,
ab_group_add @ finite_3 ).
thf(tcon_Enum_Ofinite__3___Rings_Ozero__neq__one_656,axiom,
zero_neq_one @ finite_3 ).
thf(tcon_Enum_Ofinite__3___Rings_Oidom__abs__sgn_657,axiom,
idom_abs_sgn @ finite_3 ).
thf(tcon_Enum_Ofinite__3___Orderings_Opreorder_658,axiom,
preorder @ finite_3 ).
thf(tcon_Enum_Ofinite__3___Orderings_Olinorder_659,axiom,
linorder @ finite_3 ).
thf(tcon_Enum_Ofinite__3___Groups_Omonoid__mult_660,axiom,
monoid_mult @ finite_3 ).
thf(tcon_Enum_Ofinite__3___Rings_Oidom__modulo_661,axiom,
idom_modulo @ finite_3 ).
thf(tcon_Enum_Ofinite__3___Rings_Oidom__divide_662,axiom,
idom_divide @ finite_3 ).
thf(tcon_Enum_Ofinite__3___Rings_Ocomm__ring__1_663,axiom,
comm_ring_1 @ finite_3 ).
thf(tcon_Enum_Ofinite__3___Groups_Omonoid__add_664,axiom,
monoid_add @ finite_3 ).
thf(tcon_Enum_Ofinite__3___Finite__Set_Ofinite_665,axiom,
finite_finite @ finite_3 ).
thf(tcon_Enum_Ofinite__3___Type__Length_Olen0_666,axiom,
type_len0 @ finite_3 ).
thf(tcon_Enum_Ofinite__3___Rings_Osemiring__1_667,axiom,
semiring_1 @ finite_3 ).
thf(tcon_Enum_Ofinite__3___Rings_Osemiring__0_668,axiom,
semiring_0 @ finite_3 ).
thf(tcon_Enum_Ofinite__3___Groups_Ogroup__add_669,axiom,
group_add @ finite_3 ).
thf(tcon_Enum_Ofinite__3___Type__Length_Olen_670,axiom,
type_len @ finite_3 ).
thf(tcon_Enum_Ofinite__3___Rings_Omult__zero_671,axiom,
mult_zero @ finite_3 ).
thf(tcon_Enum_Ofinite__3___Rings_Ocomm__ring_672,axiom,
comm_ring @ finite_3 ).
thf(tcon_Enum_Ofinite__3___Orderings_Oorder_673,axiom,
order @ finite_3 ).
thf(tcon_Enum_Ofinite__3___Num_Oneg__numeral_674,axiom,
neg_numeral @ finite_3 ).
thf(tcon_Enum_Ofinite__3___Rings_Osemiring_675,axiom,
semiring @ finite_3 ).
thf(tcon_Enum_Ofinite__3___Fields_Oinverse_676,axiom,
inverse @ finite_3 ).
thf(tcon_Enum_Ofinite__3___Orderings_Oord_677,axiom,
ord @ finite_3 ).
thf(tcon_Enum_Ofinite__3___Groups_Ouminus_678,axiom,
uminus @ finite_3 ).
thf(tcon_Enum_Ofinite__3___Rings_Oring__1_679,axiom,
ring_1 @ finite_3 ).
thf(tcon_Enum_Ofinite__3___Groups_Otimes_680,axiom,
times @ finite_3 ).
thf(tcon_Enum_Ofinite__3___Groups_Ominus_681,axiom,
minus @ finite_3 ).
thf(tcon_Enum_Ofinite__3___Fields_Ofield_682,axiom,
field @ finite_3 ).
thf(tcon_Enum_Ofinite__3___Power_Opower_683,axiom,
power @ finite_3 ).
thf(tcon_Enum_Ofinite__3___Num_Onumeral_684,axiom,
numeral @ finite_3 ).
thf(tcon_Enum_Ofinite__3___Groups_Ozero_685,axiom,
zero @ finite_3 ).
thf(tcon_Enum_Ofinite__3___Groups_Oplus_686,axiom,
plus @ finite_3 ).
thf(tcon_Enum_Ofinite__3___Rings_Oring_687,axiom,
ring @ finite_3 ).
thf(tcon_Enum_Ofinite__3___Rings_Oidom_688,axiom,
idom @ finite_3 ).
thf(tcon_Enum_Ofinite__3___Groups_Oone_689,axiom,
one @ finite_3 ).
thf(tcon_Enum_Ofinite__3___Rings_Odvd_690,axiom,
dvd @ finite_3 ).
thf(tcon_Enum_Ofinite__3___HOL_Oequal_691,axiom,
cl_HOL_Oequal @ finite_3 ).
thf(tcon_Filter_Ofilter___Conditionally__Complete__Lattices_Oconditionally__complete__lattice_692,axiom,
! [A13: $tType] : ( condit1219197933456340205attice @ ( filter @ A13 ) ) ).
thf(tcon_Filter_Ofilter___Quickcheck__Narrowing_Opartial__term__of_693,axiom,
! [A13: $tType] :
( ( typerep @ A13 )
=> ( quickc6926020345158392990erm_of @ ( filter @ A13 ) ) ) ).
thf(tcon_Filter_Ofilter___Complete__Lattices_Ocomplete__lattice_694,axiom,
! [A13: $tType] : ( comple6319245703460814977attice @ ( filter @ A13 ) ) ).
thf(tcon_Filter_Ofilter___Code__Evaluation_Oterm__of_695,axiom,
! [A13: $tType] :
( ( typerep @ A13 )
=> ( code_term_of @ ( filter @ A13 ) ) ) ).
thf(tcon_Filter_Ofilter___Orderings_Oorder__top_696,axiom,
! [A13: $tType] : ( order_top @ ( filter @ A13 ) ) ).
thf(tcon_Filter_Ofilter___Orderings_Oorder__bot_697,axiom,
! [A13: $tType] : ( order_bot @ ( filter @ A13 ) ) ).
thf(tcon_Filter_Ofilter___Orderings_Opreorder_698,axiom,
! [A13: $tType] : ( preorder @ ( filter @ A13 ) ) ).
thf(tcon_Filter_Ofilter___Orderings_Oorder_699,axiom,
! [A13: $tType] : ( order @ ( filter @ A13 ) ) ).
thf(tcon_Filter_Ofilter___Orderings_Oord_700,axiom,
! [A13: $tType] : ( ord @ ( filter @ A13 ) ) ).
thf(tcon_Filter_Ofilter___HOL_Oequal_701,axiom,
! [A13: $tType] :
( ( cl_HOL_Oequal @ A13 )
=> ( cl_HOL_Oequal @ ( filter @ A13 ) ) ) ).
thf(tcon_Option_Ooption___Conditionally__Complete__Lattices_Oconditionally__complete__linorder_702,axiom,
! [A13: $tType] :
( ( comple5582772986160207858norder @ A13 )
=> ( condit6923001295902523014norder @ ( option @ A13 ) ) ) ).
thf(tcon_Option_Ooption___Conditionally__Complete__Lattices_Oconditionally__complete__lattice_703,axiom,
! [A13: $tType] :
( ( comple6319245703460814977attice @ A13 )
=> ( condit1219197933456340205attice @ ( option @ A13 ) ) ) ).
thf(tcon_Option_Ooption___Quickcheck__Narrowing_Opartial__term__of_704,axiom,
! [A13: $tType] :
( ( typerep @ A13 )
=> ( quickc6926020345158392990erm_of @ ( option @ A13 ) ) ) ).
thf(tcon_Option_Ooption___Complete__Lattices_Ocomplete__linorder_705,axiom,
! [A13: $tType] :
( ( comple5582772986160207858norder @ A13 )
=> ( comple5582772986160207858norder @ ( option @ A13 ) ) ) ).
thf(tcon_Option_Ooption___Complete__Lattices_Ocomplete__lattice_706,axiom,
! [A13: $tType] :
( ( comple6319245703460814977attice @ A13 )
=> ( comple6319245703460814977attice @ ( option @ A13 ) ) ) ).
thf(tcon_Option_Ooption___Code__Evaluation_Oterm__of_707,axiom,
! [A13: $tType] :
( ( typerep @ A13 )
=> ( code_term_of @ ( option @ A13 ) ) ) ).
thf(tcon_Option_Ooption___Orderings_Owellorder_708,axiom,
! [A13: $tType] :
( ( wellorder @ A13 )
=> ( wellorder @ ( option @ A13 ) ) ) ).
thf(tcon_Option_Ooption___Orderings_Oorder__top_709,axiom,
! [A13: $tType] :
( ( order_top @ A13 )
=> ( order_top @ ( option @ A13 ) ) ) ).
thf(tcon_Option_Ooption___Orderings_Oorder__bot_710,axiom,
! [A13: $tType] :
( ( order @ A13 )
=> ( order_bot @ ( option @ A13 ) ) ) ).
thf(tcon_Option_Ooption___Orderings_Opreorder_711,axiom,
! [A13: $tType] :
( ( preorder @ A13 )
=> ( preorder @ ( option @ A13 ) ) ) ).
thf(tcon_Option_Ooption___Orderings_Olinorder_712,axiom,
! [A13: $tType] :
( ( linorder @ A13 )
=> ( linorder @ ( option @ A13 ) ) ) ).
thf(tcon_Option_Ooption___Finite__Set_Ofinite_713,axiom,
! [A13: $tType] :
( ( finite_finite @ A13 )
=> ( finite_finite @ ( option @ A13 ) ) ) ).
thf(tcon_Option_Ooption___Orderings_Oorder_714,axiom,
! [A13: $tType] :
( ( order @ A13 )
=> ( order @ ( option @ A13 ) ) ) ).
thf(tcon_Option_Ooption___Orderings_Oord_715,axiom,
! [A13: $tType] :
( ( preorder @ A13 )
=> ( ord @ ( option @ A13 ) ) ) ).
thf(tcon_Option_Ooption___Heap_Oheap_716,axiom,
! [A13: $tType] :
( ( heap @ A13 )
=> ( heap @ ( option @ A13 ) ) ) ).
thf(tcon_Option_Ooption___HOL_Oequal_717,axiom,
! [A13: $tType] : ( cl_HOL_Oequal @ ( option @ A13 ) ) ).
thf(tcon_Option_Ooption___Nat_Osize_718,axiom,
! [A13: $tType] : ( size @ ( option @ A13 ) ) ).
thf(tcon_Uint32_Ouint32___Bit__Operations_Osemiring__bit__operations_719,axiom,
bit_se359711467146920520ations @ uint32 ).
thf(tcon_Uint32_Ouint32___Quickcheck__Narrowing_Opartial__term__of_720,axiom,
quickc6926020345158392990erm_of @ uint32 ).
thf(tcon_Uint32_Ouint32___Bit__Operations_Oring__bit__operations_721,axiom,
bit_ri3973907225187159222ations @ uint32 ).
thf(tcon_Uint32_Ouint32___Groups_Ocancel__ab__semigroup__add_722,axiom,
cancel2418104881723323429up_add @ uint32 ).
thf(tcon_Uint32_Ouint32___Groups_Ocancel__comm__monoid__add_723,axiom,
cancel1802427076303600483id_add @ uint32 ).
thf(tcon_Uint32_Ouint32___Rings_Ocomm__semiring__1__cancel_724,axiom,
comm_s4317794764714335236cancel @ uint32 ).
thf(tcon_Uint32_Ouint32___Bit__Operations_Osemiring__bits_725,axiom,
bit_semiring_bits @ uint32 ).
thf(tcon_Uint32_Ouint32___Groups_Ocancel__semigroup__add_726,axiom,
cancel_semigroup_add @ uint32 ).
thf(tcon_Uint32_Ouint32___Least__significant__bit_Olsb_727,axiom,
least_6119777620449941438nt_lsb @ uint32 ).
thf(tcon_Uint32_Ouint32___Groups_Oab__semigroup__mult_728,axiom,
ab_semigroup_mult @ uint32 ).
thf(tcon_Uint32_Ouint32___Rings_Osemiring__1__cancel_729,axiom,
semiring_1_cancel @ uint32 ).
thf(tcon_Uint32_Ouint32___Groups_Ocomm__monoid__mult_730,axiom,
comm_monoid_mult @ uint32 ).
thf(tcon_Uint32_Ouint32___Groups_Oab__semigroup__add_731,axiom,
ab_semigroup_add @ uint32 ).
thf(tcon_Uint32_Ouint32___Generic__set__bit_Oset__bit_732,axiom,
generic_set_set_bit @ uint32 ).
thf(tcon_Uint32_Ouint32___Code__Evaluation_Oterm__of_733,axiom,
code_term_of @ uint32 ).
thf(tcon_Uint32_Ouint32___Parity_Osemiring__parity_734,axiom,
semiring_parity @ uint32 ).
thf(tcon_Uint32_Ouint32___Groups_Ocomm__monoid__add_735,axiom,
comm_monoid_add @ uint32 ).
thf(tcon_Uint32_Ouint32___Rings_Osemiring__modulo_736,axiom,
semiring_modulo @ uint32 ).
thf(tcon_Uint32_Ouint32___Rings_Ocomm__semiring__1_737,axiom,
comm_semiring_1 @ uint32 ).
thf(tcon_Uint32_Ouint32___Rings_Ocomm__semiring__0_738,axiom,
comm_semiring_0 @ uint32 ).
thf(tcon_Uint32_Ouint32___Groups_Osemigroup__mult_739,axiom,
semigroup_mult @ uint32 ).
thf(tcon_Uint32_Ouint32___Num_Osemiring__numeral_740,axiom,
semiring_numeral @ uint32 ).
thf(tcon_Uint32_Ouint32___Groups_Osemigroup__add_741,axiom,
semigroup_add @ uint32 ).
thf(tcon_Uint32_Ouint32___Rings_Ocomm__semiring_742,axiom,
comm_semiring @ uint32 ).
thf(tcon_Uint32_Ouint32___Groups_Oab__group__add_743,axiom,
ab_group_add @ uint32 ).
thf(tcon_Uint32_Ouint32___Rings_Ozero__neq__one_744,axiom,
zero_neq_one @ uint32 ).
thf(tcon_Uint32_Ouint32___Parity_Oring__parity_745,axiom,
ring_parity @ uint32 ).
thf(tcon_Uint32_Ouint32___Orderings_Opreorder_746,axiom,
preorder @ uint32 ).
thf(tcon_Uint32_Ouint32___Orderings_Olinorder_747,axiom,
linorder @ uint32 ).
thf(tcon_Uint32_Ouint32___Groups_Omonoid__mult_748,axiom,
monoid_mult @ uint32 ).
thf(tcon_Uint32_Ouint32___Rings_Ocomm__ring__1_749,axiom,
comm_ring_1 @ uint32 ).
thf(tcon_Uint32_Ouint32___Groups_Omonoid__add_750,axiom,
monoid_add @ uint32 ).
thf(tcon_Uint32_Ouint32___Rings_Osemiring__1_751,axiom,
semiring_1 @ uint32 ).
thf(tcon_Uint32_Ouint32___Rings_Osemiring__0_752,axiom,
semiring_0 @ uint32 ).
thf(tcon_Uint32_Ouint32___Groups_Ogroup__add_753,axiom,
group_add @ uint32 ).
thf(tcon_Uint32_Ouint32___Rings_Omult__zero_754,axiom,
mult_zero @ uint32 ).
thf(tcon_Uint32_Ouint32___Rings_Ocomm__ring_755,axiom,
comm_ring @ uint32 ).
thf(tcon_Uint32_Ouint32___Orderings_Oorder_756,axiom,
order @ uint32 ).
thf(tcon_Uint32_Ouint32___Num_Oneg__numeral_757,axiom,
neg_numeral @ uint32 ).
thf(tcon_Uint32_Ouint32___Rings_Osemiring_758,axiom,
semiring @ uint32 ).
thf(tcon_Uint32_Ouint32___Orderings_Oord_759,axiom,
ord @ uint32 ).
thf(tcon_Uint32_Ouint32___Groups_Ouminus_760,axiom,
uminus @ uint32 ).
thf(tcon_Uint32_Ouint32___Rings_Oring__1_761,axiom,
ring_1 @ uint32 ).
thf(tcon_Uint32_Ouint32___Groups_Otimes_762,axiom,
times @ uint32 ).
thf(tcon_Uint32_Ouint32___Groups_Ominus_763,axiom,
minus @ uint32 ).
thf(tcon_Uint32_Ouint32___Power_Opower_764,axiom,
power @ uint32 ).
thf(tcon_Uint32_Ouint32___Num_Onumeral_765,axiom,
numeral @ uint32 ).
thf(tcon_Uint32_Ouint32___Groups_Ozero_766,axiom,
zero @ uint32 ).
thf(tcon_Uint32_Ouint32___Groups_Oplus_767,axiom,
plus @ uint32 ).
thf(tcon_Uint32_Ouint32___Rings_Oring_768,axiom,
ring @ uint32 ).
thf(tcon_Uint32_Ouint32___Groups_Oone_769,axiom,
one @ uint32 ).
thf(tcon_Uint32_Ouint32___Rings_Odvd_770,axiom,
dvd @ uint32 ).
thf(tcon_Uint32_Ouint32___HOL_Oequal_771,axiom,
cl_HOL_Oequal @ uint32 ).
thf(tcon_Uint32_Ouint32___Nat_Osize_772,axiom,
size @ uint32 ).
thf(tcon_Assertions_Oassn___Boolean__Algebras_Oboolean__algebra_773,axiom,
boolea8198339166811842893lgebra @ assn ).
thf(tcon_Assertions_Oassn___Groups_Oab__semigroup__mult_774,axiom,
ab_semigroup_mult @ assn ).
thf(tcon_Assertions_Oassn___Groups_Ocomm__monoid__mult_775,axiom,
comm_monoid_mult @ assn ).
thf(tcon_Assertions_Oassn___Groups_Osemigroup__mult_776,axiom,
semigroup_mult @ assn ).
thf(tcon_Assertions_Oassn___Orderings_Oorder__top_777,axiom,
order_top @ assn ).
thf(tcon_Assertions_Oassn___Orderings_Oorder__bot_778,axiom,
order_bot @ assn ).
thf(tcon_Assertions_Oassn___Orderings_Opreorder_779,axiom,
preorder @ assn ).
thf(tcon_Assertions_Oassn___Groups_Omonoid__mult_780,axiom,
monoid_mult @ assn ).
thf(tcon_Assertions_Oassn___Orderings_Oorder_781,axiom,
order @ assn ).
thf(tcon_Assertions_Oassn___Orderings_Oord_782,axiom,
ord @ assn ).
thf(tcon_Assertions_Oassn___Groups_Ouminus_783,axiom,
uminus @ assn ).
thf(tcon_Assertions_Oassn___Groups_Otimes_784,axiom,
times @ assn ).
thf(tcon_Assertions_Oassn___Groups_Ominus_785,axiom,
minus @ assn ).
thf(tcon_Assertions_Oassn___Power_Opower_786,axiom,
power @ assn ).
thf(tcon_Assertions_Oassn___Groups_Oone_787,axiom,
one @ assn ).
thf(tcon_Assertions_Oassn___Rings_Odvd_788,axiom,
dvd @ assn ).
thf(tcon_Complex_Ocomplex___Semiring__Normalization_Ocomm__semiring__1__cancel__crossproduct_789,axiom,
semiri1453513574482234551roduct @ complex ).
thf(tcon_Complex_Ocomplex___Topological__Spaces_Ofirst__countable__topology_790,axiom,
topolo3112930676232923870pology @ complex ).
thf(tcon_Complex_Ocomplex___Real__Vector__Spaces_Oreal__normed__div__algebra_791,axiom,
real_V8999393235501362500lgebra @ complex ).
thf(tcon_Complex_Ocomplex___Real__Vector__Spaces_Oreal__normed__algebra__1_792,axiom,
real_V2822296259951069270ebra_1 @ complex ).
thf(tcon_Complex_Ocomplex___Rings_Osemiring__no__zero__divisors__cancel_793,axiom,
semiri6575147826004484403cancel @ complex ).
thf(tcon_Complex_Ocomplex___Real__Vector__Spaces_Oreal__normed__algebra_794,axiom,
real_V4412858255891104859lgebra @ complex ).
thf(tcon_Complex_Ocomplex___Real__Vector__Spaces_Oreal__normed__vector_795,axiom,
real_V822414075346904944vector @ complex ).
thf(tcon_Complex_Ocomplex___Topological__Spaces_Otopological__space_796,axiom,
topolo4958980785337419405_space @ complex ).
thf(tcon_Complex_Ocomplex___Real__Vector__Spaces_Oreal__normed__field_797,axiom,
real_V3459762299906320749_field @ complex ).
thf(tcon_Complex_Ocomplex___Quickcheck__Narrowing_Opartial__term__of_798,axiom,
quickc6926020345158392990erm_of @ complex ).
thf(tcon_Complex_Ocomplex___Real__Vector__Spaces_Oreal__div__algebra_799,axiom,
real_V5047593784448816457lgebra @ complex ).
thf(tcon_Complex_Ocomplex___Limits_Otopological__comm__monoid__add_800,axiom,
topolo5987344860129210374id_add @ complex ).
thf(tcon_Complex_Ocomplex___Rings_Osemiring__1__no__zero__divisors_801,axiom,
semiri2026040879449505780visors @ complex ).
thf(tcon_Complex_Ocomplex___Real__Vector__Spaces_Oreal__algebra__1_802,axiom,
real_V2191834092415804123ebra_1 @ complex ).
thf(tcon_Complex_Ocomplex___Real__Vector__Spaces_Ocomplete__space_803,axiom,
real_V8037385150606011577_space @ complex ).
thf(tcon_Complex_Ocomplex___Limits_Otopological__semigroup__mult_804,axiom,
topolo4211221413907600880p_mult @ complex ).
thf(tcon_Complex_Ocomplex___Topological__Spaces_Ouniform__space_805,axiom,
topolo7287701948861334536_space @ complex ).
thf(tcon_Complex_Ocomplex___Rings_Osemiring__no__zero__divisors_806,axiom,
semiri3467727345109120633visors @ complex ).
thf(tcon_Complex_Ocomplex___Real__Vector__Spaces_Oreal__algebra_807,axiom,
real_V6157519004096292374lgebra @ complex ).
thf(tcon_Complex_Ocomplex___Real__Vector__Spaces_Ometric__space_808,axiom,
real_V7819770556892013058_space @ complex ).
thf(tcon_Complex_Ocomplex___Limits_Otopological__ab__group__add_809,axiom,
topolo1287966508704411220up_add @ complex ).
thf(tcon_Complex_Ocomplex___Real__Vector__Spaces_Oreal__vector_810,axiom,
real_V4867850818363320053vector @ complex ).
thf(tcon_Complex_Ocomplex___Groups_Ocancel__ab__semigroup__add_811,axiom,
cancel2418104881723323429up_add @ complex ).
thf(tcon_Complex_Ocomplex___Rings_Oring__1__no__zero__divisors_812,axiom,
ring_15535105094025558882visors @ complex ).
thf(tcon_Complex_Ocomplex___Real__Vector__Spaces_Oreal__field_813,axiom,
real_V7773925162809079976_field @ complex ).
thf(tcon_Complex_Ocomplex___Limits_Otopological__monoid__add_814,axiom,
topolo6943815403480290642id_add @ complex ).
thf(tcon_Complex_Ocomplex___Groups_Ocancel__comm__monoid__add_815,axiom,
cancel1802427076303600483id_add @ complex ).
thf(tcon_Complex_Ocomplex___Rings_Ocomm__semiring__1__cancel_816,axiom,
comm_s4317794764714335236cancel @ complex ).
thf(tcon_Complex_Ocomplex___Real__Vector__Spaces_Odist__norm_817,axiom,
real_V6936659425649961206t_norm @ complex ).
thf(tcon_Complex_Ocomplex___Limits_Otopological__group__add_818,axiom,
topolo1633459387980952147up_add @ complex ).
thf(tcon_Complex_Ocomplex___Topological__Spaces_Ot2__space_819,axiom,
topological_t2_space @ complex ).
thf(tcon_Complex_Ocomplex___Groups_Ocancel__semigroup__add_820,axiom,
cancel_semigroup_add @ complex ).
thf(tcon_Complex_Ocomplex___Real__Vector__Spaces_Obanach_821,axiom,
real_Vector_banach @ complex ).
thf(tcon_Complex_Ocomplex___Groups_Oab__semigroup__mult_822,axiom,
ab_semigroup_mult @ complex ).
thf(tcon_Complex_Ocomplex___Rings_Osemiring__1__cancel_823,axiom,
semiring_1_cancel @ complex ).
thf(tcon_Complex_Ocomplex___Groups_Ocomm__monoid__mult_824,axiom,
comm_monoid_mult @ complex ).
thf(tcon_Complex_Ocomplex___Groups_Oab__semigroup__add_825,axiom,
ab_semigroup_add @ complex ).
thf(tcon_Complex_Ocomplex___Code__Evaluation_Oterm__of_826,axiom,
code_term_of @ complex ).
thf(tcon_Complex_Ocomplex___Groups_Ocomm__monoid__add_827,axiom,
comm_monoid_add @ complex ).
thf(tcon_Complex_Ocomplex___Rings_Ocomm__semiring__1_828,axiom,
comm_semiring_1 @ complex ).
thf(tcon_Complex_Ocomplex___Rings_Ocomm__semiring__0_829,axiom,
comm_semiring_0 @ complex ).
thf(tcon_Complex_Ocomplex___Groups_Osemigroup__mult_830,axiom,
semigroup_mult @ complex ).
thf(tcon_Complex_Ocomplex___Rings_Osemidom__divide_831,axiom,
semidom_divide @ complex ).
thf(tcon_Complex_Ocomplex___Num_Osemiring__numeral_832,axiom,
semiring_numeral @ complex ).
thf(tcon_Complex_Ocomplex___Groups_Osemigroup__add_833,axiom,
semigroup_add @ complex ).
thf(tcon_Complex_Ocomplex___Fields_Ofield__abs__sgn_834,axiom,
field_abs_sgn @ complex ).
thf(tcon_Complex_Ocomplex___Fields_Odivision__ring_835,axiom,
division_ring @ complex ).
thf(tcon_Complex_Ocomplex___Rings_Ocomm__semiring_836,axiom,
comm_semiring @ complex ).
thf(tcon_Complex_Ocomplex___Nat_Osemiring__char__0_837,axiom,
semiring_char_0 @ complex ).
thf(tcon_Complex_Ocomplex___Groups_Oab__group__add_838,axiom,
ab_group_add @ complex ).
thf(tcon_Complex_Ocomplex___Fields_Ofield__char__0_839,axiom,
field_char_0 @ complex ).
thf(tcon_Complex_Ocomplex___Rings_Ozero__neq__one_840,axiom,
zero_neq_one @ complex ).
thf(tcon_Complex_Ocomplex___Rings_Oidom__abs__sgn_841,axiom,
idom_abs_sgn @ complex ).
thf(tcon_Complex_Ocomplex___Groups_Omonoid__mult_842,axiom,
monoid_mult @ complex ).
thf(tcon_Complex_Ocomplex___Rings_Oidom__divide_843,axiom,
idom_divide @ complex ).
thf(tcon_Complex_Ocomplex___Rings_Ocomm__ring__1_844,axiom,
comm_ring_1 @ complex ).
thf(tcon_Complex_Ocomplex___Groups_Omonoid__add_845,axiom,
monoid_add @ complex ).
thf(tcon_Complex_Ocomplex___Rings_Osemiring__1_846,axiom,
semiring_1 @ complex ).
thf(tcon_Complex_Ocomplex___Rings_Osemiring__0_847,axiom,
semiring_0 @ complex ).
thf(tcon_Complex_Ocomplex___Groups_Ogroup__add_848,axiom,
group_add @ complex ).
thf(tcon_Complex_Ocomplex___Rings_Omult__zero_849,axiom,
mult_zero @ complex ).
thf(tcon_Complex_Ocomplex___Rings_Ocomm__ring_850,axiom,
comm_ring @ complex ).
thf(tcon_Complex_Ocomplex___Num_Oneg__numeral_851,axiom,
neg_numeral @ complex ).
thf(tcon_Complex_Ocomplex___Nat_Oring__char__0_852,axiom,
ring_char_0 @ complex ).
thf(tcon_Complex_Ocomplex___Rings_Osemiring_853,axiom,
semiring @ complex ).
thf(tcon_Complex_Ocomplex___Fields_Oinverse_854,axiom,
inverse @ complex ).
thf(tcon_Complex_Ocomplex___Groups_Ouminus_855,axiom,
uminus @ complex ).
thf(tcon_Complex_Ocomplex___Rings_Oring__1_856,axiom,
ring_1 @ complex ).
thf(tcon_Complex_Ocomplex___Groups_Otimes_857,axiom,
times @ complex ).
thf(tcon_Complex_Ocomplex___Groups_Ominus_858,axiom,
minus @ complex ).
thf(tcon_Complex_Ocomplex___Fields_Ofield_859,axiom,
field @ complex ).
thf(tcon_Complex_Ocomplex___Power_Opower_860,axiom,
power @ complex ).
thf(tcon_Complex_Ocomplex___Num_Onumeral_861,axiom,
numeral @ complex ).
thf(tcon_Complex_Ocomplex___Groups_Ozero_862,axiom,
zero @ complex ).
thf(tcon_Complex_Ocomplex___Groups_Oplus_863,axiom,
plus @ complex ).
thf(tcon_Complex_Ocomplex___Rings_Oring_864,axiom,
ring @ complex ).
thf(tcon_Complex_Ocomplex___Rings_Oidom_865,axiom,
idom @ complex ).
thf(tcon_Complex_Ocomplex___Groups_Oone_866,axiom,
one @ complex ).
thf(tcon_Complex_Ocomplex___Rings_Odvd_867,axiom,
dvd @ complex ).
thf(tcon_Complex_Ocomplex___HOL_Oequal_868,axiom,
cl_HOL_Oequal @ complex ).
thf(tcon_Extended__Nat_Oenat___Conditionally__Complete__Lattices_Oconditionally__complete__linorder_869,axiom,
condit6923001295902523014norder @ extended_enat ).
thf(tcon_Extended__Nat_Oenat___Conditionally__Complete__Lattices_Oconditionally__complete__lattice_870,axiom,
condit1219197933456340205attice @ extended_enat ).
thf(tcon_Extended__Nat_Oenat___Groups_Ostrict__ordered__ab__semigroup__add_871,axiom,
strict9044650504122735259up_add @ extended_enat ).
thf(tcon_Extended__Nat_Oenat___Groups_Ostrict__ordered__comm__monoid__add_872,axiom,
strict7427464778891057005id_add @ extended_enat ).
thf(tcon_Extended__Nat_Oenat___Groups_Ocanonically__ordered__monoid__add_873,axiom,
canoni5634975068530333245id_add @ extended_enat ).
thf(tcon_Extended__Nat_Oenat___Quickcheck__Narrowing_Opartial__term__of_874,axiom,
quickc6926020345158392990erm_of @ extended_enat ).
thf(tcon_Extended__Nat_Oenat___Complete__Lattices_Ocomplete__linorder_875,axiom,
comple5582772986160207858norder @ extended_enat ).
thf(tcon_Extended__Nat_Oenat___Groups_Olinordered__ab__semigroup__add_876,axiom,
linord4140545234300271783up_add @ extended_enat ).
thf(tcon_Extended__Nat_Oenat___Complete__Lattices_Ocomplete__lattice_877,axiom,
comple6319245703460814977attice @ extended_enat ).
thf(tcon_Extended__Nat_Oenat___Rings_Olinordered__nonzero__semiring_878,axiom,
linord181362715937106298miring @ extended_enat ).
thf(tcon_Extended__Nat_Oenat___Rings_Osemiring__no__zero__divisors_879,axiom,
semiri3467727345109120633visors @ extended_enat ).
thf(tcon_Extended__Nat_Oenat___Groups_Oordered__ab__semigroup__add_880,axiom,
ordere6658533253407199908up_add @ extended_enat ).
thf(tcon_Extended__Nat_Oenat___Groups_Oordered__comm__monoid__add_881,axiom,
ordere6911136660526730532id_add @ extended_enat ).
thf(tcon_Extended__Nat_Oenat___Rings_Oordered__comm__semiring_882,axiom,
ordere2520102378445227354miring @ extended_enat ).
thf(tcon_Extended__Nat_Oenat___Groups_Oab__semigroup__mult_883,axiom,
ab_semigroup_mult @ extended_enat ).
thf(tcon_Extended__Nat_Oenat___Groups_Ocomm__monoid__mult_884,axiom,
comm_monoid_mult @ extended_enat ).
thf(tcon_Extended__Nat_Oenat___Groups_Oab__semigroup__add_885,axiom,
ab_semigroup_add @ extended_enat ).
thf(tcon_Extended__Nat_Oenat___Code__Evaluation_Oterm__of_886,axiom,
code_term_of @ extended_enat ).
thf(tcon_Extended__Nat_Oenat___Rings_Oordered__semiring_887,axiom,
ordered_semiring @ extended_enat ).
thf(tcon_Extended__Nat_Oenat___Groups_Ocomm__monoid__add_888,axiom,
comm_monoid_add @ extended_enat ).
thf(tcon_Extended__Nat_Oenat___Rings_Ocomm__semiring__1_889,axiom,
comm_semiring_1 @ extended_enat ).
thf(tcon_Extended__Nat_Oenat___Rings_Ocomm__semiring__0_890,axiom,
comm_semiring_0 @ extended_enat ).
thf(tcon_Extended__Nat_Oenat___Groups_Osemigroup__mult_891,axiom,
semigroup_mult @ extended_enat ).
thf(tcon_Extended__Nat_Oenat___Num_Osemiring__numeral_892,axiom,
semiring_numeral @ extended_enat ).
thf(tcon_Extended__Nat_Oenat___Groups_Osemigroup__add_893,axiom,
semigroup_add @ extended_enat ).
thf(tcon_Extended__Nat_Oenat___Rings_Ozero__less__one_894,axiom,
zero_less_one @ extended_enat ).
thf(tcon_Extended__Nat_Oenat___Rings_Ocomm__semiring_895,axiom,
comm_semiring @ extended_enat ).
thf(tcon_Extended__Nat_Oenat___Orderings_Owellorder_896,axiom,
wellorder @ extended_enat ).
thf(tcon_Extended__Nat_Oenat___Orderings_Oorder__top_897,axiom,
order_top @ extended_enat ).
thf(tcon_Extended__Nat_Oenat___Orderings_Oorder__bot_898,axiom,
order_bot @ extended_enat ).
thf(tcon_Extended__Nat_Oenat___Nat_Osemiring__char__0_899,axiom,
semiring_char_0 @ extended_enat ).
thf(tcon_Extended__Nat_Oenat___Rings_Ozero__neq__one_900,axiom,
zero_neq_one @ extended_enat ).
thf(tcon_Extended__Nat_Oenat___Orderings_Opreorder_901,axiom,
preorder @ extended_enat ).
thf(tcon_Extended__Nat_Oenat___Orderings_Olinorder_902,axiom,
linorder @ extended_enat ).
thf(tcon_Extended__Nat_Oenat___Groups_Omonoid__mult_903,axiom,
monoid_mult @ extended_enat ).
thf(tcon_Extended__Nat_Oenat___Groups_Omonoid__add_904,axiom,
monoid_add @ extended_enat ).
thf(tcon_Extended__Nat_Oenat___Rings_Osemiring__1_905,axiom,
semiring_1 @ extended_enat ).
thf(tcon_Extended__Nat_Oenat___Rings_Osemiring__0_906,axiom,
semiring_0 @ extended_enat ).
thf(tcon_Extended__Nat_Oenat___Rings_Omult__zero_907,axiom,
mult_zero @ extended_enat ).
thf(tcon_Extended__Nat_Oenat___Orderings_Oorder_908,axiom,
order @ extended_enat ).
thf(tcon_Extended__Nat_Oenat___Rings_Osemiring_909,axiom,
semiring @ extended_enat ).
thf(tcon_Extended__Nat_Oenat___Orderings_Oord_910,axiom,
ord @ extended_enat ).
thf(tcon_Extended__Nat_Oenat___Groups_Otimes_911,axiom,
times @ extended_enat ).
thf(tcon_Extended__Nat_Oenat___Groups_Ominus_912,axiom,
minus @ extended_enat ).
thf(tcon_Extended__Nat_Oenat___Power_Opower_913,axiom,
power @ extended_enat ).
thf(tcon_Extended__Nat_Oenat___Num_Onumeral_914,axiom,
numeral @ extended_enat ).
thf(tcon_Extended__Nat_Oenat___Groups_Ozero_915,axiom,
zero @ extended_enat ).
thf(tcon_Extended__Nat_Oenat___Groups_Oplus_916,axiom,
plus @ extended_enat ).
thf(tcon_Extended__Nat_Oenat___Groups_Oone_917,axiom,
one @ extended_enat ).
thf(tcon_Extended__Nat_Oenat___Rings_Odvd_918,axiom,
dvd @ extended_enat ).
thf(tcon_Extended__Nat_Oenat___HOL_Oequal_919,axiom,
cl_HOL_Oequal @ extended_enat ).
thf(tcon_Multiset_Omultiset___Quickcheck__Narrowing_Opartial__term__of_920,axiom,
! [A13: $tType] :
( ( typerep @ A13 )
=> ( quickc6926020345158392990erm_of @ ( multiset @ A13 ) ) ) ).
thf(tcon_Multiset_Omultiset___Groups_Oordered__ab__semigroup__add_921,axiom,
! [A13: $tType] :
( ( preorder @ A13 )
=> ( ordere6658533253407199908up_add @ ( multiset @ A13 ) ) ) ).
thf(tcon_Multiset_Omultiset___Groups_Ocancel__ab__semigroup__add_922,axiom,
! [A13: $tType] : ( cancel2418104881723323429up_add @ ( multiset @ A13 ) ) ).
thf(tcon_Multiset_Omultiset___Groups_Ocancel__comm__monoid__add_923,axiom,
! [A13: $tType] : ( cancel1802427076303600483id_add @ ( multiset @ A13 ) ) ).
thf(tcon_Multiset_Omultiset___Groups_Ocancel__semigroup__add_924,axiom,
! [A13: $tType] : ( cancel_semigroup_add @ ( multiset @ A13 ) ) ).
thf(tcon_Multiset_Omultiset___Groups_Ocomm__monoid__diff_925,axiom,
! [A13: $tType] : ( comm_monoid_diff @ ( multiset @ A13 ) ) ).
thf(tcon_Multiset_Omultiset___Groups_Oab__semigroup__add_926,axiom,
! [A13: $tType] : ( ab_semigroup_add @ ( multiset @ A13 ) ) ).
thf(tcon_Multiset_Omultiset___Code__Evaluation_Oterm__of_927,axiom,
! [A13: $tType] :
( ( typerep @ A13 )
=> ( code_term_of @ ( multiset @ A13 ) ) ) ).
thf(tcon_Multiset_Omultiset___Groups_Ocomm__monoid__add_928,axiom,
! [A13: $tType] : ( comm_monoid_add @ ( multiset @ A13 ) ) ).
thf(tcon_Multiset_Omultiset___Groups_Osemigroup__add_929,axiom,
! [A13: $tType] : ( semigroup_add @ ( multiset @ A13 ) ) ).
thf(tcon_Multiset_Omultiset___Orderings_Opreorder_930,axiom,
! [A13: $tType] :
( ( preorder @ A13 )
=> ( preorder @ ( multiset @ A13 ) ) ) ).
thf(tcon_Multiset_Omultiset___Groups_Omonoid__add_931,axiom,
! [A13: $tType] : ( monoid_add @ ( multiset @ A13 ) ) ).
thf(tcon_Multiset_Omultiset___Orderings_Oorder_932,axiom,
! [A13: $tType] :
( ( preorder @ A13 )
=> ( order @ ( multiset @ A13 ) ) ) ).
thf(tcon_Multiset_Omultiset___Orderings_Oord_933,axiom,
! [A13: $tType] :
( ( preorder @ A13 )
=> ( ord @ ( multiset @ A13 ) ) ) ).
thf(tcon_Multiset_Omultiset___Groups_Ominus_934,axiom,
! [A13: $tType] : ( minus @ ( multiset @ A13 ) ) ).
thf(tcon_Multiset_Omultiset___Groups_Ozero_935,axiom,
! [A13: $tType] : ( zero @ ( multiset @ A13 ) ) ).
thf(tcon_Multiset_Omultiset___Groups_Oplus_936,axiom,
! [A13: $tType] : ( plus @ ( multiset @ A13 ) ) ).
thf(tcon_Multiset_Omultiset___HOL_Oequal_937,axiom,
! [A13: $tType] :
( ( cl_HOL_Oequal @ A13 )
=> ( cl_HOL_Oequal @ ( multiset @ A13 ) ) ) ).
thf(tcon_Multiset_Omultiset___Nat_Osize_938,axiom,
! [A13: $tType] : ( size @ ( multiset @ A13 ) ) ).
thf(tcon_Numeral__Type_Obit0___Groups_Ocancel__ab__semigroup__add_939,axiom,
! [A13: $tType] :
( ( finite_finite @ A13 )
=> ( cancel2418104881723323429up_add @ ( numeral_bit0 @ A13 ) ) ) ).
thf(tcon_Numeral__Type_Obit0___Groups_Ocancel__comm__monoid__add_940,axiom,
! [A13: $tType] :
( ( finite_finite @ A13 )
=> ( cancel1802427076303600483id_add @ ( numeral_bit0 @ A13 ) ) ) ).
thf(tcon_Numeral__Type_Obit0___Rings_Ocomm__semiring__1__cancel_941,axiom,
! [A13: $tType] :
( ( finite_finite @ A13 )
=> ( comm_s4317794764714335236cancel @ ( numeral_bit0 @ A13 ) ) ) ).
thf(tcon_Numeral__Type_Obit0___Groups_Ocancel__semigroup__add_942,axiom,
! [A13: $tType] :
( ( finite_finite @ A13 )
=> ( cancel_semigroup_add @ ( numeral_bit0 @ A13 ) ) ) ).
thf(tcon_Numeral__Type_Obit0___Groups_Oab__semigroup__mult_943,axiom,
! [A13: $tType] :
( ( finite_finite @ A13 )
=> ( ab_semigroup_mult @ ( numeral_bit0 @ A13 ) ) ) ).
thf(tcon_Numeral__Type_Obit0___Rings_Osemiring__1__cancel_944,axiom,
! [A13: $tType] :
( ( finite_finite @ A13 )
=> ( semiring_1_cancel @ ( numeral_bit0 @ A13 ) ) ) ).
thf(tcon_Numeral__Type_Obit0___Groups_Ocomm__monoid__mult_945,axiom,
! [A13: $tType] :
( ( finite_finite @ A13 )
=> ( comm_monoid_mult @ ( numeral_bit0 @ A13 ) ) ) ).
thf(tcon_Numeral__Type_Obit0___Groups_Oab__semigroup__add_946,axiom,
! [A13: $tType] :
( ( finite_finite @ A13 )
=> ( ab_semigroup_add @ ( numeral_bit0 @ A13 ) ) ) ).
thf(tcon_Numeral__Type_Obit0___Code__Evaluation_Oterm__of_947,axiom,
! [A13: $tType] :
( ( typerep @ A13 )
=> ( code_term_of @ ( numeral_bit0 @ A13 ) ) ) ).
thf(tcon_Numeral__Type_Obit0___Groups_Ocomm__monoid__add_948,axiom,
! [A13: $tType] :
( ( finite_finite @ A13 )
=> ( comm_monoid_add @ ( numeral_bit0 @ A13 ) ) ) ).
thf(tcon_Numeral__Type_Obit0___Rings_Ocomm__semiring__1_949,axiom,
! [A13: $tType] :
( ( finite_finite @ A13 )
=> ( comm_semiring_1 @ ( numeral_bit0 @ A13 ) ) ) ).
thf(tcon_Numeral__Type_Obit0___Rings_Ocomm__semiring__0_950,axiom,
! [A13: $tType] :
( ( finite_finite @ A13 )
=> ( comm_semiring_0 @ ( numeral_bit0 @ A13 ) ) ) ).
thf(tcon_Numeral__Type_Obit0___Groups_Osemigroup__mult_951,axiom,
! [A13: $tType] :
( ( finite_finite @ A13 )
=> ( semigroup_mult @ ( numeral_bit0 @ A13 ) ) ) ).
thf(tcon_Numeral__Type_Obit0___Num_Osemiring__numeral_952,axiom,
! [A13: $tType] :
( ( finite_finite @ A13 )
=> ( semiring_numeral @ ( numeral_bit0 @ A13 ) ) ) ).
thf(tcon_Numeral__Type_Obit0___Groups_Osemigroup__add_953,axiom,
! [A13: $tType] :
( ( finite_finite @ A13 )
=> ( semigroup_add @ ( numeral_bit0 @ A13 ) ) ) ).
thf(tcon_Numeral__Type_Obit0___Rings_Ocomm__semiring_954,axiom,
! [A13: $tType] :
( ( finite_finite @ A13 )
=> ( comm_semiring @ ( numeral_bit0 @ A13 ) ) ) ).
thf(tcon_Numeral__Type_Obit0___Orderings_Owellorder_955,axiom,
! [A13: $tType] :
( ( finite_finite @ A13 )
=> ( wellorder @ ( numeral_bit0 @ A13 ) ) ) ).
thf(tcon_Numeral__Type_Obit0___Groups_Oab__group__add_956,axiom,
! [A13: $tType] :
( ( finite_finite @ A13 )
=> ( ab_group_add @ ( numeral_bit0 @ A13 ) ) ) ).
thf(tcon_Numeral__Type_Obit0___Rings_Ozero__neq__one_957,axiom,
! [A13: $tType] :
( ( finite_finite @ A13 )
=> ( zero_neq_one @ ( numeral_bit0 @ A13 ) ) ) ).
thf(tcon_Numeral__Type_Obit0___Orderings_Opreorder_958,axiom,
! [A13: $tType] :
( ( finite_finite @ A13 )
=> ( preorder @ ( numeral_bit0 @ A13 ) ) ) ).
thf(tcon_Numeral__Type_Obit0___Orderings_Olinorder_959,axiom,
! [A13: $tType] :
( ( finite_finite @ A13 )
=> ( linorder @ ( numeral_bit0 @ A13 ) ) ) ).
thf(tcon_Numeral__Type_Obit0___Groups_Omonoid__mult_960,axiom,
! [A13: $tType] :
( ( finite_finite @ A13 )
=> ( monoid_mult @ ( numeral_bit0 @ A13 ) ) ) ).
thf(tcon_Numeral__Type_Obit0___Rings_Ocomm__ring__1_961,axiom,
! [A13: $tType] :
( ( finite_finite @ A13 )
=> ( comm_ring_1 @ ( numeral_bit0 @ A13 ) ) ) ).
thf(tcon_Numeral__Type_Obit0___Groups_Omonoid__add_962,axiom,
! [A13: $tType] :
( ( finite_finite @ A13 )
=> ( monoid_add @ ( numeral_bit0 @ A13 ) ) ) ).
thf(tcon_Numeral__Type_Obit0___Finite__Set_Ofinite_963,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_964,axiom,
! [A13: $tType] :
( ( type_len0 @ A13 )
=> ( type_len0 @ ( numeral_bit0 @ A13 ) ) ) ).
thf(tcon_Numeral__Type_Obit0___Rings_Osemiring__1_965,axiom,
! [A13: $tType] :
( ( finite_finite @ A13 )
=> ( semiring_1 @ ( numeral_bit0 @ A13 ) ) ) ).
thf(tcon_Numeral__Type_Obit0___Rings_Osemiring__0_966,axiom,
! [A13: $tType] :
( ( finite_finite @ A13 )
=> ( semiring_0 @ ( numeral_bit0 @ A13 ) ) ) ).
thf(tcon_Numeral__Type_Obit0___Groups_Ogroup__add_967,axiom,
! [A13: $tType] :
( ( finite_finite @ A13 )
=> ( group_add @ ( numeral_bit0 @ A13 ) ) ) ).
thf(tcon_Numeral__Type_Obit0___Type__Length_Olen_968,axiom,
! [A13: $tType] :
( ( type_len @ A13 )
=> ( type_len @ ( numeral_bit0 @ A13 ) ) ) ).
thf(tcon_Numeral__Type_Obit0___Rings_Omult__zero_969,axiom,
! [A13: $tType] :
( ( finite_finite @ A13 )
=> ( mult_zero @ ( numeral_bit0 @ A13 ) ) ) ).
thf(tcon_Numeral__Type_Obit0___Rings_Ocomm__ring_970,axiom,
! [A13: $tType] :
( ( finite_finite @ A13 )
=> ( comm_ring @ ( numeral_bit0 @ A13 ) ) ) ).
thf(tcon_Numeral__Type_Obit0___Orderings_Oorder_971,axiom,
! [A13: $tType] :
( ( finite_finite @ A13 )
=> ( order @ ( numeral_bit0 @ A13 ) ) ) ).
thf(tcon_Numeral__Type_Obit0___Num_Oneg__numeral_972,axiom,
! [A13: $tType] :
( ( finite_finite @ A13 )
=> ( neg_numeral @ ( numeral_bit0 @ A13 ) ) ) ).
thf(tcon_Numeral__Type_Obit0___Rings_Osemiring_973,axiom,
! [A13: $tType] :
( ( finite_finite @ A13 )
=> ( semiring @ ( numeral_bit0 @ A13 ) ) ) ).
thf(tcon_Numeral__Type_Obit0___Orderings_Oord_974,axiom,
! [A13: $tType] :
( ( finite_finite @ A13 )
=> ( ord @ ( numeral_bit0 @ A13 ) ) ) ).
thf(tcon_Numeral__Type_Obit0___Groups_Ouminus_975,axiom,
! [A13: $tType] :
( ( finite_finite @ A13 )
=> ( uminus @ ( numeral_bit0 @ A13 ) ) ) ).
thf(tcon_Numeral__Type_Obit0___Rings_Oring__1_976,axiom,
! [A13: $tType] :
( ( finite_finite @ A13 )
=> ( ring_1 @ ( numeral_bit0 @ A13 ) ) ) ).
thf(tcon_Numeral__Type_Obit0___Groups_Otimes_977,axiom,
! [A13: $tType] :
( ( finite_finite @ A13 )
=> ( times @ ( numeral_bit0 @ A13 ) ) ) ).
thf(tcon_Numeral__Type_Obit0___Groups_Ominus_978,axiom,
! [A13: $tType] :
( ( finite_finite @ A13 )
=> ( minus @ ( numeral_bit0 @ A13 ) ) ) ).
thf(tcon_Numeral__Type_Obit0___Power_Opower_979,axiom,
! [A13: $tType] :
( ( finite_finite @ A13 )
=> ( power @ ( numeral_bit0 @ A13 ) ) ) ).
thf(tcon_Numeral__Type_Obit0___Num_Onumeral_980,axiom,
! [A13: $tType] :
( ( finite_finite @ A13 )
=> ( numeral @ ( numeral_bit0 @ A13 ) ) ) ).
thf(tcon_Numeral__Type_Obit0___Groups_Ozero_981,axiom,
! [A13: $tType] :
( ( finite_finite @ A13 )
=> ( zero @ ( numeral_bit0 @ A13 ) ) ) ).
thf(tcon_Numeral__Type_Obit0___Groups_Oplus_982,axiom,
! [A13: $tType] :
( ( finite_finite @ A13 )
=> ( plus @ ( numeral_bit0 @ A13 ) ) ) ).
thf(tcon_Numeral__Type_Obit0___Rings_Oring_983,axiom,
! [A13: $tType] :
( ( finite_finite @ A13 )
=> ( ring @ ( numeral_bit0 @ A13 ) ) ) ).
thf(tcon_Numeral__Type_Obit0___Groups_Oone_984,axiom,
! [A13: $tType] :
( ( finite_finite @ A13 )
=> ( one @ ( numeral_bit0 @ A13 ) ) ) ).
thf(tcon_Numeral__Type_Obit0___Rings_Odvd_985,axiom,
! [A13: $tType] :
( ( finite_finite @ A13 )
=> ( dvd @ ( numeral_bit0 @ A13 ) ) ) ).
thf(tcon_Numeral__Type_Obit0___HOL_Oequal_986,axiom,
! [A13: $tType] :
( ( finite_finite @ A13 )
=> ( cl_HOL_Oequal @ ( numeral_bit0 @ A13 ) ) ) ).
thf(tcon_Numeral__Type_Obit1___Groups_Ocancel__ab__semigroup__add_987,axiom,
! [A13: $tType] :
( ( finite_finite @ A13 )
=> ( cancel2418104881723323429up_add @ ( numeral_bit1 @ A13 ) ) ) ).
thf(tcon_Numeral__Type_Obit1___Groups_Ocancel__comm__monoid__add_988,axiom,
! [A13: $tType] :
( ( finite_finite @ A13 )
=> ( cancel1802427076303600483id_add @ ( numeral_bit1 @ A13 ) ) ) ).
thf(tcon_Numeral__Type_Obit1___Rings_Ocomm__semiring__1__cancel_989,axiom,
! [A13: $tType] :
( ( finite_finite @ A13 )
=> ( comm_s4317794764714335236cancel @ ( numeral_bit1 @ A13 ) ) ) ).
thf(tcon_Numeral__Type_Obit1___Groups_Ocancel__semigroup__add_990,axiom,
! [A13: $tType] :
( ( finite_finite @ A13 )
=> ( cancel_semigroup_add @ ( numeral_bit1 @ A13 ) ) ) ).
thf(tcon_Numeral__Type_Obit1___Groups_Oab__semigroup__mult_991,axiom,
! [A13: $tType] :
( ( finite_finite @ A13 )
=> ( ab_semigroup_mult @ ( numeral_bit1 @ A13 ) ) ) ).
thf(tcon_Numeral__Type_Obit1___Rings_Osemiring__1__cancel_992,axiom,
! [A13: $tType] :
( ( finite_finite @ A13 )
=> ( semiring_1_cancel @ ( numeral_bit1 @ A13 ) ) ) ).
thf(tcon_Numeral__Type_Obit1___Groups_Ocomm__monoid__mult_993,axiom,
! [A13: $tType] :
( ( finite_finite @ A13 )
=> ( comm_monoid_mult @ ( numeral_bit1 @ A13 ) ) ) ).
thf(tcon_Numeral__Type_Obit1___Groups_Oab__semigroup__add_994,axiom,
! [A13: $tType] :
( ( finite_finite @ A13 )
=> ( ab_semigroup_add @ ( numeral_bit1 @ A13 ) ) ) ).
thf(tcon_Numeral__Type_Obit1___Code__Evaluation_Oterm__of_995,axiom,
! [A13: $tType] :
( ( typerep @ A13 )
=> ( code_term_of @ ( numeral_bit1 @ A13 ) ) ) ).
thf(tcon_Numeral__Type_Obit1___Groups_Ocomm__monoid__add_996,axiom,
! [A13: $tType] :
( ( finite_finite @ A13 )
=> ( comm_monoid_add @ ( numeral_bit1 @ A13 ) ) ) ).
thf(tcon_Numeral__Type_Obit1___Rings_Ocomm__semiring__1_997,axiom,
! [A13: $tType] :
( ( finite_finite @ A13 )
=> ( comm_semiring_1 @ ( numeral_bit1 @ A13 ) ) ) ).
thf(tcon_Numeral__Type_Obit1___Rings_Ocomm__semiring__0_998,axiom,
! [A13: $tType] :
( ( finite_finite @ A13 )
=> ( comm_semiring_0 @ ( numeral_bit1 @ A13 ) ) ) ).
thf(tcon_Numeral__Type_Obit1___Groups_Osemigroup__mult_999,axiom,
! [A13: $tType] :
( ( finite_finite @ A13 )
=> ( semigroup_mult @ ( numeral_bit1 @ A13 ) ) ) ).
thf(tcon_Numeral__Type_Obit1___Num_Osemiring__numeral_1000,axiom,
! [A13: $tType] :
( ( finite_finite @ A13 )
=> ( semiring_numeral @ ( numeral_bit1 @ A13 ) ) ) ).
thf(tcon_Numeral__Type_Obit1___Groups_Osemigroup__add_1001,axiom,
! [A13: $tType] :
( ( finite_finite @ A13 )
=> ( semigroup_add @ ( numeral_bit1 @ A13 ) ) ) ).
thf(tcon_Numeral__Type_Obit1___Rings_Ocomm__semiring_1002,axiom,
! [A13: $tType] :
( ( finite_finite @ A13 )
=> ( comm_semiring @ ( numeral_bit1 @ A13 ) ) ) ).
thf(tcon_Numeral__Type_Obit1___Orderings_Owellorder_1003,axiom,
! [A13: $tType] :
( ( finite_finite @ A13 )
=> ( wellorder @ ( numeral_bit1 @ A13 ) ) ) ).
thf(tcon_Numeral__Type_Obit1___Groups_Oab__group__add_1004,axiom,
! [A13: $tType] :
( ( finite_finite @ A13 )
=> ( ab_group_add @ ( numeral_bit1 @ A13 ) ) ) ).
thf(tcon_Numeral__Type_Obit1___Rings_Ozero__neq__one_1005,axiom,
! [A13: $tType] :
( ( finite_finite @ A13 )
=> ( zero_neq_one @ ( numeral_bit1 @ A13 ) ) ) ).
thf(tcon_Numeral__Type_Obit1___Orderings_Opreorder_1006,axiom,
! [A13: $tType] :
( ( finite_finite @ A13 )
=> ( preorder @ ( numeral_bit1 @ A13 ) ) ) ).
thf(tcon_Numeral__Type_Obit1___Orderings_Olinorder_1007,axiom,
! [A13: $tType] :
( ( finite_finite @ A13 )
=> ( linorder @ ( numeral_bit1 @ A13 ) ) ) ).
thf(tcon_Numeral__Type_Obit1___Groups_Omonoid__mult_1008,axiom,
! [A13: $tType] :
( ( finite_finite @ A13 )
=> ( monoid_mult @ ( numeral_bit1 @ A13 ) ) ) ).
thf(tcon_Numeral__Type_Obit1___Rings_Ocomm__ring__1_1009,axiom,
! [A13: $tType] :
( ( finite_finite @ A13 )
=> ( comm_ring_1 @ ( numeral_bit1 @ A13 ) ) ) ).
thf(tcon_Numeral__Type_Obit1___Groups_Omonoid__add_1010,axiom,
! [A13: $tType] :
( ( finite_finite @ A13 )
=> ( monoid_add @ ( numeral_bit1 @ A13 ) ) ) ).
thf(tcon_Numeral__Type_Obit1___Finite__Set_Ofinite_1011,axiom,
! [A13: $tType] :
( ( finite_finite @ A13 )
=> ( finite_finite @ ( numeral_bit1 @ A13 ) ) ) ).
thf(tcon_Numeral__Type_Obit1___Cardinality_Ocard2_1012,axiom,
! [A13: $tType] :
( ( finite_finite @ A13 )
=> ( card2 @ ( numeral_bit1 @ A13 ) ) ) ).
thf(tcon_Numeral__Type_Obit1___Type__Length_Olen0_1013,axiom,
! [A13: $tType] :
( ( type_len0 @ A13 )
=> ( type_len0 @ ( numeral_bit1 @ A13 ) ) ) ).
thf(tcon_Numeral__Type_Obit1___Rings_Osemiring__1_1014,axiom,
! [A13: $tType] :
( ( finite_finite @ A13 )
=> ( semiring_1 @ ( numeral_bit1 @ A13 ) ) ) ).
thf(tcon_Numeral__Type_Obit1___Rings_Osemiring__0_1015,axiom,
! [A13: $tType] :
( ( finite_finite @ A13 )
=> ( semiring_0 @ ( numeral_bit1 @ A13 ) ) ) ).
thf(tcon_Numeral__Type_Obit1___Groups_Ogroup__add_1016,axiom,
! [A13: $tType] :
( ( finite_finite @ A13 )
=> ( group_add @ ( numeral_bit1 @ A13 ) ) ) ).
thf(tcon_Numeral__Type_Obit1___Type__Length_Olen_1017,axiom,
! [A13: $tType] :
( ( type_len0 @ A13 )
=> ( type_len @ ( numeral_bit1 @ A13 ) ) ) ).
thf(tcon_Numeral__Type_Obit1___Rings_Omult__zero_1018,axiom,
! [A13: $tType] :
( ( finite_finite @ A13 )
=> ( mult_zero @ ( numeral_bit1 @ A13 ) ) ) ).
thf(tcon_Numeral__Type_Obit1___Rings_Ocomm__ring_1019,axiom,
! [A13: $tType] :
( ( finite_finite @ A13 )
=> ( comm_ring @ ( numeral_bit1 @ A13 ) ) ) ).
thf(tcon_Numeral__Type_Obit1___Orderings_Oorder_1020,axiom,
! [A13: $tType] :
( ( finite_finite @ A13 )
=> ( order @ ( numeral_bit1 @ A13 ) ) ) ).
thf(tcon_Numeral__Type_Obit1___Num_Oneg__numeral_1021,axiom,
! [A13: $tType] :
( ( finite_finite @ A13 )
=> ( neg_numeral @ ( numeral_bit1 @ A13 ) ) ) ).
thf(tcon_Numeral__Type_Obit1___Rings_Osemiring_1022,axiom,
! [A13: $tType] :
( ( finite_finite @ A13 )
=> ( semiring @ ( numeral_bit1 @ A13 ) ) ) ).
thf(tcon_Numeral__Type_Obit1___Orderings_Oord_1023,axiom,
! [A13: $tType] :
( ( finite_finite @ A13 )
=> ( ord @ ( numeral_bit1 @ A13 ) ) ) ).
thf(tcon_Numeral__Type_Obit1___Groups_Ouminus_1024,axiom,
! [A13: $tType] :
( ( finite_finite @ A13 )
=> ( uminus @ ( numeral_bit1 @ A13 ) ) ) ).
thf(tcon_Numeral__Type_Obit1___Rings_Oring__1_1025,axiom,
! [A13: $tType] :
( ( finite_finite @ A13 )
=> ( ring_1 @ ( numeral_bit1 @ A13 ) ) ) ).
thf(tcon_Numeral__Type_Obit1___Groups_Otimes_1026,axiom,
! [A13: $tType] :
( ( finite_finite @ A13 )
=> ( times @ ( numeral_bit1 @ A13 ) ) ) ).
thf(tcon_Numeral__Type_Obit1___Groups_Ominus_1027,axiom,
! [A13: $tType] :
( ( finite_finite @ A13 )
=> ( minus @ ( numeral_bit1 @ A13 ) ) ) ).
thf(tcon_Numeral__Type_Obit1___Power_Opower_1028,axiom,
! [A13: $tType] :
( ( finite_finite @ A13 )
=> ( power @ ( numeral_bit1 @ A13 ) ) ) ).
thf(tcon_Numeral__Type_Obit1___Num_Onumeral_1029,axiom,
! [A13: $tType] :
( ( finite_finite @ A13 )
=> ( numeral @ ( numeral_bit1 @ A13 ) ) ) ).
thf(tcon_Numeral__Type_Obit1___Groups_Ozero_1030,axiom,
! [A13: $tType] :
( ( finite_finite @ A13 )
=> ( zero @ ( numeral_bit1 @ A13 ) ) ) ).
thf(tcon_Numeral__Type_Obit1___Groups_Oplus_1031,axiom,
! [A13: $tType] :
( ( finite_finite @ A13 )
=> ( plus @ ( numeral_bit1 @ A13 ) ) ) ).
thf(tcon_Numeral__Type_Obit1___Rings_Oring_1032,axiom,
! [A13: $tType] :
( ( finite_finite @ A13 )
=> ( ring @ ( numeral_bit1 @ A13 ) ) ) ).
thf(tcon_Numeral__Type_Obit1___Groups_Oone_1033,axiom,
! [A13: $tType] :
( ( finite_finite @ A13 )
=> ( one @ ( numeral_bit1 @ A13 ) ) ) ).
thf(tcon_Numeral__Type_Obit1___Rings_Odvd_1034,axiom,
! [A13: $tType] :
( ( finite_finite @ A13 )
=> ( dvd @ ( numeral_bit1 @ A13 ) ) ) ).
thf(tcon_Numeral__Type_Obit1___HOL_Oequal_1035,axiom,
! [A13: $tType] :
( ( finite_finite @ A13 )
=> ( cl_HOL_Oequal @ ( numeral_bit1 @ A13 ) ) ) ).
thf(tcon_Numeral__Type_Onum1___Quickcheck__Narrowing_Opartial__term__of_1036,axiom,
quickc6926020345158392990erm_of @ numeral_num1 ).
thf(tcon_Numeral__Type_Onum1___Groups_Ocancel__ab__semigroup__add_1037,axiom,
cancel2418104881723323429up_add @ numeral_num1 ).
thf(tcon_Numeral__Type_Onum1___Groups_Ocancel__comm__monoid__add_1038,axiom,
cancel1802427076303600483id_add @ numeral_num1 ).
thf(tcon_Numeral__Type_Onum1___Groups_Ocancel__semigroup__add_1039,axiom,
cancel_semigroup_add @ numeral_num1 ).
thf(tcon_Numeral__Type_Onum1___Groups_Oab__semigroup__mult_1040,axiom,
ab_semigroup_mult @ numeral_num1 ).
thf(tcon_Numeral__Type_Onum1___Groups_Ocomm__monoid__mult_1041,axiom,
comm_monoid_mult @ numeral_num1 ).
thf(tcon_Numeral__Type_Onum1___Groups_Oab__semigroup__add_1042,axiom,
ab_semigroup_add @ numeral_num1 ).
thf(tcon_Numeral__Type_Onum1___Code__Evaluation_Oterm__of_1043,axiom,
code_term_of @ numeral_num1 ).
thf(tcon_Numeral__Type_Onum1___Groups_Ocomm__monoid__add_1044,axiom,
comm_monoid_add @ numeral_num1 ).
thf(tcon_Numeral__Type_Onum1___Rings_Ocomm__semiring__0_1045,axiom,
comm_semiring_0 @ numeral_num1 ).
thf(tcon_Numeral__Type_Onum1___Groups_Osemigroup__mult_1046,axiom,
semigroup_mult @ numeral_num1 ).
thf(tcon_Numeral__Type_Onum1___Groups_Osemigroup__add_1047,axiom,
semigroup_add @ numeral_num1 ).
thf(tcon_Numeral__Type_Onum1___Rings_Ocomm__semiring_1048,axiom,
comm_semiring @ numeral_num1 ).
thf(tcon_Numeral__Type_Onum1___Orderings_Owellorder_1049,axiom,
wellorder @ numeral_num1 ).
thf(tcon_Numeral__Type_Onum1___Groups_Oab__group__add_1050,axiom,
ab_group_add @ numeral_num1 ).
thf(tcon_Numeral__Type_Onum1___Orderings_Opreorder_1051,axiom,
preorder @ numeral_num1 ).
thf(tcon_Numeral__Type_Onum1___Orderings_Olinorder_1052,axiom,
linorder @ numeral_num1 ).
thf(tcon_Numeral__Type_Onum1___Groups_Omonoid__mult_1053,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_1054,axiom,
monoid_add @ numeral_num1 ).
thf(tcon_Numeral__Type_Onum1___Finite__Set_Ofinite_1055,axiom,
finite_finite @ numeral_num1 ).
thf(tcon_Numeral__Type_Onum1___Type__Length_Olen0_1056,axiom,
type_len0 @ numeral_num1 ).
thf(tcon_Numeral__Type_Onum1___Rings_Osemiring__0_1057,axiom,
semiring_0 @ numeral_num1 ).
thf(tcon_Numeral__Type_Onum1___Groups_Ogroup__add_1058,axiom,
group_add @ numeral_num1 ).
thf(tcon_Numeral__Type_Onum1___Type__Length_Olen_1059,axiom,
type_len @ numeral_num1 ).
thf(tcon_Numeral__Type_Onum1___Rings_Omult__zero_1060,axiom,
mult_zero @ numeral_num1 ).
thf(tcon_Numeral__Type_Onum1___Rings_Ocomm__ring_1061,axiom,
comm_ring @ numeral_num1 ).
thf(tcon_Numeral__Type_Onum1___Orderings_Oorder_1062,axiom,
order @ numeral_num1 ).
thf(tcon_Numeral__Type_Onum1___Rings_Osemiring_1063,axiom,
semiring @ numeral_num1 ).
thf(tcon_Numeral__Type_Onum1___Orderings_Oord_1064,axiom,
ord @ numeral_num1 ).
thf(tcon_Numeral__Type_Onum1___Groups_Ouminus_1065,axiom,
uminus @ numeral_num1 ).
thf(tcon_Numeral__Type_Onum1___Groups_Otimes_1066,axiom,
times @ numeral_num1 ).
thf(tcon_Numeral__Type_Onum1___Groups_Ominus_1067,axiom,
minus @ numeral_num1 ).
thf(tcon_Numeral__Type_Onum1___Power_Opower_1068,axiom,
power @ numeral_num1 ).
thf(tcon_Numeral__Type_Onum1___Num_Onumeral_1069,axiom,
numeral @ numeral_num1 ).
thf(tcon_Numeral__Type_Onum1___Groups_Ozero_1070,axiom,
zero @ numeral_num1 ).
thf(tcon_Numeral__Type_Onum1___Groups_Oplus_1071,axiom,
plus @ numeral_num1 ).
thf(tcon_Numeral__Type_Onum1___Rings_Oring_1072,axiom,
ring @ numeral_num1 ).
thf(tcon_Numeral__Type_Onum1___Groups_Oone_1073,axiom,
one @ numeral_num1 ).
thf(tcon_Numeral__Type_Onum1___Rings_Odvd_1074,axiom,
dvd @ numeral_num1 ).
thf(tcon_Numeral__Type_Onum1___HOL_Oequal_1075,axiom,
cl_HOL_Oequal @ numeral_num1 ).
thf(tcon_Product__Type_Oprod___Topological__Spaces_Otopological__space_1076,axiom,
! [A13: $tType,A17: $tType] :
( ( ( topolo4958980785337419405_space @ A13 )
& ( topolo4958980785337419405_space @ A17 ) )
=> ( topolo4958980785337419405_space @ ( product_prod @ A13 @ A17 ) ) ) ).
thf(tcon_Product__Type_Oprod___Quickcheck__Narrowing_Opartial__term__of_1077,axiom,
! [A13: $tType,A17: $tType] :
( ( ( typerep @ A13 )
& ( typerep @ A17 ) )
=> ( quickc6926020345158392990erm_of @ ( product_prod @ A13 @ A17 ) ) ) ).
thf(tcon_Product__Type_Oprod___Topological__Spaces_Ot2__space_1078,axiom,
! [A13: $tType,A17: $tType] :
( ( ( topological_t2_space @ A13 )
& ( topological_t2_space @ A17 ) )
=> ( topological_t2_space @ ( product_prod @ A13 @ A17 ) ) ) ).
thf(tcon_Product__Type_Oprod___Code__Evaluation_Oterm__of_1079,axiom,
! [A13: $tType,A17: $tType] :
( ( ( typerep @ A13 )
& ( typerep @ A17 ) )
=> ( code_term_of @ ( product_prod @ A13 @ A17 ) ) ) ).
thf(tcon_Product__Type_Oprod___Finite__Set_Ofinite_1080,axiom,
! [A13: $tType,A17: $tType] :
( ( ( finite_finite @ A13 )
& ( finite_finite @ A17 ) )
=> ( finite_finite @ ( product_prod @ A13 @ A17 ) ) ) ).
thf(tcon_Product__Type_Oprod___Heap_Oheap_1081,axiom,
! [A13: $tType,A17: $tType] :
( ( ( heap @ A13 )
& ( heap @ A17 ) )
=> ( heap @ ( product_prod @ A13 @ A17 ) ) ) ).
thf(tcon_Product__Type_Oprod___HOL_Oequal_1082,axiom,
! [A13: $tType,A17: $tType] : ( cl_HOL_Oequal @ ( product_prod @ A13 @ A17 ) ) ).
thf(tcon_Product__Type_Oprod___Nat_Osize_1083,axiom,
! [A13: $tType,A17: $tType] : ( size @ ( product_prod @ A13 @ A17 ) ) ).
thf(tcon_Product__Type_Ounit___Conditionally__Complete__Lattices_Oconditionally__complete__linorder_1084,axiom,
condit6923001295902523014norder @ product_unit ).
thf(tcon_Product__Type_Ounit___Conditionally__Complete__Lattices_Oconditionally__complete__lattice_1085,axiom,
condit1219197933456340205attice @ product_unit ).
thf(tcon_Product__Type_Ounit___Quickcheck__Narrowing_Opartial__term__of_1086,axiom,
quickc6926020345158392990erm_of @ product_unit ).
thf(tcon_Product__Type_Ounit___Complete__Lattices_Ocomplete__linorder_1087,axiom,
comple5582772986160207858norder @ product_unit ).
thf(tcon_Product__Type_Ounit___Complete__Lattices_Ocomplete__lattice_1088,axiom,
comple6319245703460814977attice @ product_unit ).
thf(tcon_Product__Type_Ounit___Boolean__Algebras_Oboolean__algebra_1089,axiom,
boolea8198339166811842893lgebra @ product_unit ).
thf(tcon_Product__Type_Ounit___Code__Evaluation_Oterm__of_1090,axiom,
code_term_of @ product_unit ).
thf(tcon_Product__Type_Ounit___Orderings_Owellorder_1091,axiom,
wellorder @ product_unit ).
thf(tcon_Product__Type_Ounit___Orderings_Oorder__top_1092,axiom,
order_top @ product_unit ).
thf(tcon_Product__Type_Ounit___Orderings_Oorder__bot_1093,axiom,
order_bot @ product_unit ).
thf(tcon_Product__Type_Ounit___Orderings_Opreorder_1094,axiom,
preorder @ product_unit ).
thf(tcon_Product__Type_Ounit___Orderings_Olinorder_1095,axiom,
linorder @ product_unit ).
thf(tcon_Product__Type_Ounit___Finite__Set_Ofinite_1096,axiom,
finite_finite @ product_unit ).
thf(tcon_Product__Type_Ounit___Orderings_Oorder_1097,axiom,
order @ product_unit ).
thf(tcon_Product__Type_Ounit___Orderings_Oord_1098,axiom,
ord @ product_unit ).
thf(tcon_Product__Type_Ounit___Groups_Ouminus_1099,axiom,
uminus @ product_unit ).
thf(tcon_Product__Type_Ounit___Groups_Ominus_1100,axiom,
minus @ product_unit ).
thf(tcon_Product__Type_Ounit___Heap_Oheap_1101,axiom,
heap @ product_unit ).
thf(tcon_Product__Type_Ounit___HOL_Oequal_1102,axiom,
cl_HOL_Oequal @ product_unit ).
thf(tcon_Heap_Oheap_Oheap__ext___Quickcheck__Narrowing_Opartial__term__of_1103,axiom,
! [A13: $tType] :
( ( typerep @ A13 )
=> ( quickc6926020345158392990erm_of @ ( heap_ext @ A13 ) ) ) ).
thf(tcon_Heap_Oheap_Oheap__ext___Code__Evaluation_Oterm__of_1104,axiom,
! [A13: $tType] :
( ( typerep @ A13 )
=> ( code_term_of @ ( heap_ext @ A13 ) ) ) ).
thf(tcon_Heap_Oheap_Oheap__ext___HOL_Oequal_1105,axiom,
! [A13: $tType] : ( cl_HOL_Oequal @ ( heap_ext @ A13 ) ) ).
thf(tcon_Code__Evaluation_Oterm___Quickcheck__Narrowing_Opartial__term__of_1106,axiom,
quickc6926020345158392990erm_of @ code_term ).
thf(tcon_Code__Evaluation_Oterm___Code__Evaluation_Oterm__of_1107,axiom,
code_term_of @ code_term ).
thf(tcon_Code__Numeral_Ointeger___Bit__Operations_Ounique__euclidean__semiring__with__bit__operations_1108,axiom,
bit_un5681908812861735899ations @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Semiring__Normalization_Ocomm__semiring__1__cancel__crossproduct_1109,axiom,
semiri1453513574482234551roduct @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Euclidean__Division_Ounique__euclidean__semiring__with__nat_1110,axiom,
euclid5411537665997757685th_nat @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Euclidean__Division_Ounique__euclidean__ring__with__nat_1111,axiom,
euclid8789492081693882211th_nat @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Groups_Oordered__ab__semigroup__monoid__add__imp__le_1112,axiom,
ordere1937475149494474687imp_le @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Euclidean__Division_Ounique__euclidean__semiring_1113,axiom,
euclid3128863361964157862miring @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Euclidean__Division_Oeuclidean__semiring__cancel_1114,axiom,
euclid4440199948858584721cancel @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Divides_Ounique__euclidean__semiring__numeral_1115,axiom,
unique1627219031080169319umeral @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Euclidean__Division_Oeuclidean__ring__cancel_1116,axiom,
euclid8851590272496341667cancel @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Rings_Osemiring__no__zero__divisors__cancel_1117,axiom,
semiri6575147826004484403cancel @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Groups_Ostrict__ordered__ab__semigroup__add_1118,axiom,
strict9044650504122735259up_add @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Groups_Oordered__cancel__ab__semigroup__add_1119,axiom,
ordere580206878836729694up_add @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Groups_Oordered__ab__semigroup__add__imp__le_1120,axiom,
ordere2412721322843649153imp_le @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Bit__Operations_Osemiring__bit__operations_1121,axiom,
bit_se359711467146920520ations @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Rings_Olinordered__comm__semiring__strict_1122,axiom,
linord2810124833399127020strict @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Groups_Ostrict__ordered__comm__monoid__add_1123,axiom,
strict7427464778891057005id_add @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Groups_Oordered__cancel__comm__monoid__add_1124,axiom,
ordere8940638589300402666id_add @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Euclidean__Division_Oeuclidean__semiring_1125,axiom,
euclid3725896446679973847miring @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Quickcheck__Narrowing_Opartial__term__of_1126,axiom,
quickc6926020345158392990erm_of @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Rings_Olinordered__semiring__1__strict_1127,axiom,
linord715952674999750819strict @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Groups_Olinordered__ab__semigroup__add_1128,axiom,
linord4140545234300271783up_add @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Bit__Operations_Oring__bit__operations_1129,axiom,
bit_ri3973907225187159222ations @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Rings_Osemiring__1__no__zero__divisors_1130,axiom,
semiri2026040879449505780visors @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Rings_Olinordered__nonzero__semiring_1131,axiom,
linord181362715937106298miring @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Euclidean__Division_Oeuclidean__ring_1132,axiom,
euclid5891614535332579305n_ring @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Rings_Olinordered__semiring__strict_1133,axiom,
linord8928482502909563296strict @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Rings_Osemiring__no__zero__divisors_1134,axiom,
semiri3467727345109120633visors @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Groups_Oordered__ab__semigroup__add_1135,axiom,
ordere6658533253407199908up_add @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Groups_Oordered__ab__group__add__abs_1136,axiom,
ordere166539214618696060dd_abs @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Groups_Oordered__comm__monoid__add_1137,axiom,
ordere6911136660526730532id_add @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Groups_Olinordered__ab__group__add_1138,axiom,
linord5086331880401160121up_add @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Groups_Ocancel__ab__semigroup__add_1139,axiom,
cancel2418104881723323429up_add @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Rings_Oring__1__no__zero__divisors_1140,axiom,
ring_15535105094025558882visors @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Groups_Ocancel__comm__monoid__add_1141,axiom,
cancel1802427076303600483id_add @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Rings_Olinordered__ring__strict_1142,axiom,
linord4710134922213307826strict @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Rings_Ocomm__semiring__1__cancel_1143,axiom,
comm_s4317794764714335236cancel @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Bit__Operations_Osemiring__bits_1144,axiom,
bit_semiring_bits @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Rings_Oordered__comm__semiring_1145,axiom,
ordere2520102378445227354miring @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Rings_Olinordered__semiring__1_1146,axiom,
linord6961819062388156250ring_1 @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Groups_Oordered__ab__group__add_1147,axiom,
ordered_ab_group_add @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Groups_Ocancel__semigroup__add_1148,axiom,
cancel_semigroup_add @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Rings_Olinordered__semiring_1149,axiom,
linordered_semiring @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Least__significant__bit_Olsb_1150,axiom,
least_6119777620449941438nt_lsb @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Rings_Oordered__semiring__0_1151,axiom,
ordered_semiring_0 @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Rings_Olinordered__semidom_1152,axiom,
linordered_semidom @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Groups_Oab__semigroup__mult_1153,axiom,
ab_semigroup_mult @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Rings_Osemiring__1__cancel_1154,axiom,
semiring_1_cancel @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Rings_Oalgebraic__semidom_1155,axiom,
algebraic_semidom @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Groups_Ocomm__monoid__mult_1156,axiom,
comm_monoid_mult @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Groups_Oab__semigroup__add_1157,axiom,
ab_semigroup_add @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Generic__set__bit_Oset__bit_1158,axiom,
generic_set_set_bit @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Code__Evaluation_Oterm__of_1159,axiom,
code_term_of @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Rings_Oordered__semiring_1160,axiom,
ordered_semiring @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Rings_Oordered__ring__abs_1161,axiom,
ordered_ring_abs @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Parity_Osemiring__parity_1162,axiom,
semiring_parity @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Groups_Ocomm__monoid__add_1163,axiom,
comm_monoid_add @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Rings_Osemiring__modulo_1164,axiom,
semiring_modulo @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Rings_Olinordered__ring_1165,axiom,
linordered_ring @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Rings_Olinordered__idom_1166,axiom,
linordered_idom @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Rings_Ocomm__semiring__1_1167,axiom,
comm_semiring_1 @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Rings_Ocomm__semiring__0_1168,axiom,
comm_semiring_0 @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Groups_Osemigroup__mult_1169,axiom,
semigroup_mult @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Rings_Osemidom__modulo_1170,axiom,
semidom_modulo @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Rings_Osemidom__divide_1171,axiom,
semidom_divide @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Num_Osemiring__numeral_1172,axiom,
semiring_numeral @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Groups_Osemigroup__add_1173,axiom,
semigroup_add @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Rings_Ozero__less__one_1174,axiom,
zero_less_one @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Rings_Ocomm__semiring_1175,axiom,
comm_semiring @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Nat_Osemiring__char__0_1176,axiom,
semiring_char_0 @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Groups_Oab__group__add_1177,axiom,
ab_group_add @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Rings_Ozero__neq__one_1178,axiom,
zero_neq_one @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Rings_Oordered__ring_1179,axiom,
ordered_ring @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Rings_Oidom__abs__sgn_1180,axiom,
idom_abs_sgn @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Parity_Oring__parity_1181,axiom,
ring_parity @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Orderings_Opreorder_1182,axiom,
preorder @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Orderings_Olinorder_1183,axiom,
linorder @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Groups_Omonoid__mult_1184,axiom,
monoid_mult @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Rings_Oidom__modulo_1185,axiom,
idom_modulo @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Rings_Oidom__divide_1186,axiom,
idom_divide @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Rings_Ocomm__ring__1_1187,axiom,
comm_ring_1 @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Groups_Omonoid__add_1188,axiom,
monoid_add @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Rings_Osemiring__1_1189,axiom,
semiring_1 @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Rings_Osemiring__0_1190,axiom,
semiring_0 @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Groups_Ogroup__add_1191,axiom,
group_add @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Rings_Omult__zero_1192,axiom,
mult_zero @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Rings_Ocomm__ring_1193,axiom,
comm_ring @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Orderings_Oorder_1194,axiom,
order @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Num_Oneg__numeral_1195,axiom,
neg_numeral @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Nat_Oring__char__0_1196,axiom,
ring_char_0 @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Rings_Osemiring_1197,axiom,
semiring @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Orderings_Oord_1198,axiom,
ord @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Groups_Ouminus_1199,axiom,
uminus @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Rings_Oring__1_1200,axiom,
ring_1 @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Rings_Oabs__if_1201,axiom,
abs_if @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Groups_Otimes_1202,axiom,
times @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Groups_Ominus_1203,axiom,
minus @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Power_Opower_1204,axiom,
power @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Num_Onumeral_1205,axiom,
numeral @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Groups_Ozero_1206,axiom,
zero @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Groups_Oplus_1207,axiom,
plus @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Rings_Oring_1208,axiom,
ring @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Rings_Oidom_1209,axiom,
idom @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Groups_Oone_1210,axiom,
one @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Rings_Odvd_1211,axiom,
dvd @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___HOL_Oequal_1212,axiom,
cl_HOL_Oequal @ code_integer ).
thf(tcon_Heap__Time__Monad_OHeap___Quickcheck__Narrowing_Opartial__term__of_1213,axiom,
! [A13: $tType] :
( ( typerep @ A13 )
=> ( quickc6926020345158392990erm_of @ ( heap_Time_Heap @ A13 ) ) ) ).
thf(tcon_Heap__Time__Monad_OHeap___Code__Evaluation_Oterm__of_1214,axiom,
! [A13: $tType] :
( ( typerep @ A13 )
=> ( code_term_of @ ( heap_Time_Heap @ A13 ) ) ) ).
thf(tcon_Heap__Time__Monad_OHeap___HOL_Oequal_1215,axiom,
! [A13: $tType] : ( cl_HOL_Oequal @ ( heap_Time_Heap @ A13 ) ) ).
thf(tcon_Heap__Time__Monad_OHeap___Nat_Osize_1216,axiom,
! [A13: $tType] : ( size @ ( heap_Time_Heap @ A13 ) ) ).
thf(tcon_VEBT__Definitions_OVEBT___Quickcheck__Narrowing_Opartial__term__of_1217,axiom,
quickc6926020345158392990erm_of @ vEBT_VEBT ).
thf(tcon_VEBT__Definitions_OVEBT___Code__Evaluation_Oterm__of_1218,axiom,
code_term_of @ vEBT_VEBT ).
thf(tcon_VEBT__Definitions_OVEBT___HOL_Oequal_1219,axiom,
cl_HOL_Oequal @ vEBT_VEBT ).
thf(tcon_VEBT__Definitions_OVEBT___Nat_Osize_1220,axiom,
size @ vEBT_VEBT ).
thf(tcon_VEBT__BuildupMemImp_OVEBTi___Quickcheck__Narrowing_Opartial__term__of_1221,axiom,
quickc6926020345158392990erm_of @ vEBT_VEBTi ).
thf(tcon_VEBT__BuildupMemImp_OVEBTi___Code__Evaluation_Oterm__of_1222,axiom,
code_term_of @ vEBT_VEBTi ).
thf(tcon_VEBT__BuildupMemImp_OVEBTi___Heap_Oheap_1223,axiom,
heap @ vEBT_VEBTi ).
thf(tcon_VEBT__BuildupMemImp_OVEBTi___HOL_Oequal_1224,axiom,
cl_HOL_Oequal @ vEBT_VEBTi ).
thf(tcon_VEBT__BuildupMemImp_OVEBTi___Nat_Osize_1225,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,X: A,Y: A] :
( ( if @ A @ $false @ X @ Y )
= Y ) ).
thf(help_If_1_1_T,axiom,
! [A: $tType,X: A,Y: A] :
( ( if @ A @ $true @ X @ Y )
= X ) ).
thf(help_fChoice_1_1_T,axiom,
! [A: $tType,P: A > $o] :
( ( P @ ( fChoice @ A @ P ) )
= ( ? [X6: A] : ( P @ X6 ) ) ) ).
% Conjectures (9)
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,
( x11
= ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ mi @ ma ) ) ) ).
thf(conj_2,hypothesis,
~ ( ord_less @ nat @ ma @ mi ) ).
thf(conj_3,hypothesis,
( ( size_size @ ( list @ vEBT_VEBTi ) @ tree_is )
= ( size_size @ ( list @ vEBT_VEBT ) @ treeList ) ) ).
thf(conj_4,hypothesis,
ma != mi ).
thf(conj_5,hypothesis,
xa = mi ).
thf(conj_6,hypothesis,
( ( vEBT_vebt_mint @ ( 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 @ ( 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 ) ) ) ) ) ) )
= ( none @ nat ) ) ).
thf(conj_7,hypothesis,
( xaa
= ( none @ nat ) ) ).
thf(conj_8,conjecture,
( xe_7_ATP
= ( 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 ) ) @ ( 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 ) ) ) ) ) ) ) ).
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