TPTP Problem File: ITP282^4.p
View Solutions
- Solve Problem
%------------------------------------------------------------------------------
% File : ITP282^4 : TPTP v8.2.0. Released v8.0.0.
% Domain : Interactive Theorem Proving
% Problem : Sledgehammer problem VEBT_BuildupMemImp 00337_015688
% 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 : 0093_VEBT_BuildupMemImp_00337_015688 [Des22]
% Status : Theorem
% Rating : 1.00 v8.1.0
% Syntax : Number of formulae : 10093 (3184 unt; 612 typ; 0 def)
% Number of atoms : 29510 (10628 equ; 0 cnn)
% Maximal formula atoms : 71 ( 3 avg)
% Number of connectives : 198190 (2477 ~; 326 |;2323 &;179966 @)
% ( 0 <=>;13098 =>; 0 <=; 0 <~>)
% Maximal formula depth : 40 ( 7 avg)
% Number of types : 21 ( 20 usr)
% Number of type conns : 2796 (2796 >; 0 *; 0 +; 0 <<)
% Number of symbols : 595 ( 592 usr; 10 con; 0-13 aty)
% Number of variables : 27764 (2071 ^;24331 !; 897 ?;27764 :)
% ( 465 !>; 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 18:08:53.153
%------------------------------------------------------------------------------
% Could-be-implicit typings (35)
thf(ty_t_VEBT__BuildupMemImp_OVEBTi,type,
vEBT_VEBTi: $tType ).
thf(ty_t_VEBT__Definitions_OVEBT,type,
vEBT_VEBT: $tType ).
thf(ty_t_Heap__Time__Monad_OHeap,type,
heap_Time_Heap: $tType > $tType ).
thf(ty_t_Code__Numeral_Ointeger,type,
code_integer: $tType ).
thf(ty_t_Heap_Oheap_Oheap__ext,type,
heap_ext: $tType > $tType ).
thf(ty_t_Product__Type_Ounit,type,
product_unit: $tType ).
thf(ty_t_Product__Type_Oprod,type,
product_prod: $tType > $tType > $tType ).
thf(ty_t_Numeral__Type_Onum1,type,
numeral_num1: $tType ).
thf(ty_t_Numeral__Type_Onum0,type,
numeral_num0: $tType ).
thf(ty_t_Numeral__Type_Obit1,type,
numeral_bit1: $tType > $tType ).
thf(ty_t_Numeral__Type_Obit0,type,
numeral_bit0: $tType > $tType ).
thf(ty_t_Extended__Nat_Oenat,type,
extended_enat: $tType ).
thf(ty_t_Multiset_Omultiset,type,
multiset: $tType > $tType ).
thf(ty_t_Complex_Ocomplex,type,
complex: $tType ).
thf(ty_t_Assertions_Oassn,type,
assn: $tType ).
thf(ty_t_String_Oliteral,type,
literal: $tType ).
thf(ty_t_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 (577)
thf(sy_cl_Complete__Lattices_Ocomplete__linorder,type,
comple5582772986160207858norder:
!>[A: $tType] : $o ).
thf(sy_cl_Enum_Oenum,type,
enum:
!>[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_Rings_Osemidom,type,
semidom:
!>[A: $tType] : $o ).
thf(sy_cl_Fields_Oinverse,type,
inverse:
!>[A: $tType] : $o ).
thf(sy_cl_Rings_Osemiring,type,
semiring:
!>[A: $tType] : $o ).
thf(sy_cl_Nat_Oring__char__0,type,
ring_char_0:
!>[A: $tType] : $o ).
thf(sy_cl_Num_Oneg__numeral,type,
neg_numeral:
!>[A: $tType] : $o ).
thf(sy_cl_Orderings_Oorder,type,
order:
!>[A: $tType] : $o ).
thf(sy_cl_Rings_Ocomm__ring,type,
comm_ring:
!>[A: $tType] : $o ).
thf(sy_cl_Rings_Omult__zero,type,
mult_zero:
!>[A: $tType] : $o ).
thf(sy_cl_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_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_Rings_Ocomm__semiring__1__cancel,type,
comm_s4317794764714335236cancel:
!>[A: $tType] : $o ).
thf(sy_cl_Rings_Olinordered__ring__strict,type,
linord4710134922213307826strict:
!>[A: $tType] : $o ).
thf(sy_cl_Groups_Ocancel__comm__monoid__add,type,
cancel1802427076303600483id_add:
!>[A: $tType] : $o ).
thf(sy_cl_Limits_Otopological__monoid__add,type,
topolo6943815403480290642id_add:
!>[A: $tType] : $o ).
thf(sy_cl_Real__Vector__Spaces_Oreal__field,type,
real_V7773925162809079976_field:
!>[A: $tType] : $o ).
thf(sy_cl_Rings_Oring__1__no__zero__divisors,type,
ring_15535105094025558882visors:
!>[A: $tType] : $o ).
thf(sy_cl_Groups_Ocancel__ab__semigroup__add,type,
cancel2418104881723323429up_add:
!>[A: $tType] : $o ).
thf(sy_cl_Groups_Olinordered__ab__group__add,type,
linord5086331880401160121up_add:
!>[A: $tType] : $o ).
thf(sy_cl_Groups_Oordered__comm__monoid__add,type,
ordere6911136660526730532id_add:
!>[A: $tType] : $o ).
thf(sy_cl_Limits_Otopological__monoid__mult,type,
topolo1898628316856586783d_mult:
!>[A: $tType] : $o ).
thf(sy_cl_Real__Vector__Spaces_Oreal__vector,type,
real_V4867850818363320053vector:
!>[A: $tType] : $o ).
thf(sy_cl_Archimedean__Field_Ofloor__ceiling,type,
archim2362893244070406136eiling:
!>[A: $tType] : $o ).
thf(sy_cl_Groups_Oordered__ab__group__add__abs,type,
ordere166539214618696060dd_abs:
!>[A: $tType] : $o ).
thf(sy_cl_Groups_Oordered__ab__semigroup__add,type,
ordere6658533253407199908up_add:
!>[A: $tType] : $o ).
thf(sy_cl_Limits_Otopological__ab__group__add,type,
topolo1287966508704411220up_add:
!>[A: $tType] : $o ).
thf(sy_cl_Real__Vector__Spaces_Ometric__space,type,
real_V7819770556892013058_space:
!>[A: $tType] : $o ).
thf(sy_cl_Real__Vector__Spaces_Oreal__algebra,type,
real_V6157519004096292374lgebra:
!>[A: $tType] : $o ).
thf(sy_cl_Rings_Osemiring__no__zero__divisors,type,
semiri3467727345109120633visors:
!>[A: $tType] : $o ).
thf(sy_cl_Boolean__Algebras_Oboolean__algebra,type,
boolea8198339166811842893lgebra:
!>[A: $tType] : $o ).
thf(sy_cl_Rings_Olinordered__semiring__strict,type,
linord8928482502909563296strict:
!>[A: $tType] : $o ).
thf(sy_cl_Topological__Spaces_Operfect__space,type,
topolo8386298272705272623_space:
!>[A: $tType] : $o ).
thf(sy_cl_Topological__Spaces_Ouniform__space,type,
topolo7287701948861334536_space:
!>[A: $tType] : $o ).
thf(sy_cl_Euclidean__Division_Oeuclidean__ring,type,
euclid5891614535332579305n_ring:
!>[A: $tType] : $o ).
thf(sy_cl_Limits_Otopological__semigroup__mult,type,
topolo4211221413907600880p_mult:
!>[A: $tType] : $o ).
thf(sy_cl_Real__Vector__Spaces_Ocomplete__space,type,
real_V8037385150606011577_space:
!>[A: $tType] : $o ).
thf(sy_cl_Real__Vector__Spaces_Oreal__algebra__1,type,
real_V2191834092415804123ebra_1:
!>[A: $tType] : $o ).
thf(sy_cl_Rings_Olinordered__nonzero__semiring,type,
linord181362715937106298miring:
!>[A: $tType] : $o ).
thf(sy_cl_Rings_Osemiring__1__no__zero__divisors,type,
semiri2026040879449505780visors:
!>[A: $tType] : $o ).
thf(sy_cl_Topological__Spaces_Oorder__topology,type,
topolo2564578578187576103pology:
!>[A: $tType] : $o ).
thf(sy_cl_Bit__Operations_Oring__bit__operations,type,
bit_ri3973907225187159222ations:
!>[A: $tType] : $o ).
thf(sy_cl_Groups_Olinordered__ab__semigroup__add,type,
linord4140545234300271783up_add:
!>[A: $tType] : $o ).
thf(sy_cl_Limits_Otopological__comm__monoid__add,type,
topolo5987344860129210374id_add:
!>[A: $tType] : $o ).
thf(sy_cl_Orderings_Ounbounded__dense__linorder,type,
unboun7993243217541854897norder:
!>[A: $tType] : $o ).
thf(sy_cl_Real__Vector__Spaces_Ouniformity__dist,type,
real_V768167426530841204y_dist:
!>[A: $tType] : $o ).
thf(sy_cl_Rings_Olinordered__semiring__1__strict,type,
linord715952674999750819strict:
!>[A: $tType] : $o ).
thf(sy_cl_Archimedean__Field_Oarchimedean__field,type,
archim462609752435547400_field:
!>[A: $tType] : $o ).
thf(sy_cl_Bit__Comprehension_Obit__comprehension,type,
bit_bi6583157726757044596ension:
!>[A: $tType] : $o ).
thf(sy_cl_Limits_Otopological__comm__monoid__mult,type,
topolo4987421752381908075d_mult:
!>[A: $tType] : $o ).
thf(sy_cl_Real__Vector__Spaces_Oreal__div__algebra,type,
real_V5047593784448816457lgebra:
!>[A: $tType] : $o ).
thf(sy_cl_Real__Vector__Spaces_Oreal__normed__field,type,
real_V3459762299906320749_field:
!>[A: $tType] : $o ).
thf(sy_cl_Topological__Spaces_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_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__linorder,type,
condit6923001295902523014norder:
!>[A: $tType] : $o ).
thf(sy_c_Archimedean__Field_Oceiling,type,
archimedean_ceiling:
!>[A: $tType] : ( A > int ) ).
thf(sy_c_Archimedean__Field_Ofloor__ceiling__class_Ofloor,type,
archim6421214686448440834_floor:
!>[A: $tType] : ( A > int ) ).
thf(sy_c_Archimedean__Field_Ofrac,type,
archimedean_frac:
!>[A: $tType] : ( A > A ) ).
thf(sy_c_Archimedean__Field_Oround,type,
archimedean_round:
!>[A: $tType] : ( A > int ) ).
thf(sy_c_Array__Time_Ofreeze,type,
array_freeze:
!>[A: $tType] : ( ( array @ A ) > ( heap_Time_Heap @ ( list @ A ) ) ) ).
thf(sy_c_Array__Time_Olen,type,
array_len:
!>[A: $tType] : ( ( array @ A ) > ( heap_Time_Heap @ nat ) ) ).
thf(sy_c_Array__Time_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_Automation_OFI__QUERY,type,
fI_QUERY: assn > assn > assn > $o ).
thf(sy_c_Binomial_Obinomial,type,
binomial: nat > nat > nat ).
thf(sy_c_Binomial_Ogbinomial,type,
gbinomial:
!>[A: $tType] : ( A > nat > A ) ).
thf(sy_c_Bit__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_Oand__not__num,type,
bit_and_not_num: num > num > ( option @ num ) ).
thf(sy_c_Bit__Operations_Oconcat__bit,type,
bit_concat_bit: nat > int > int > int ).
thf(sy_c_Bit__Operations_Oor__not__num__neg,type,
bit_or_not_num_neg: num > num > num ).
thf(sy_c_Bit__Operations_Oring__bit__operations__class_Onot,type,
bit_ri4277139882892585799ns_not:
!>[A: $tType] : ( A > A ) ).
thf(sy_c_Bit__Operations_Oring__bit__operations__class_Osigned__take__bit,type,
bit_ri4674362597316999326ke_bit:
!>[A: $tType] : ( nat > A > A ) ).
thf(sy_c_Bit__Operations_Osemiring__bit__operations__class_Oand,type,
bit_se5824344872417868541ns_and:
!>[A: $tType] : ( A > A > A ) ).
thf(sy_c_Bit__Operations_Osemiring__bit__operations__class_Odrop__bit,type,
bit_se4197421643247451524op_bit:
!>[A: $tType] : ( nat > A > A ) ).
thf(sy_c_Bit__Operations_Osemiring__bit__operations__class_Oflip__bit,type,
bit_se8732182000553998342ip_bit:
!>[A: $tType] : ( nat > A > A ) ).
thf(sy_c_Bit__Operations_Osemiring__bit__operations__class_Omask,type,
bit_se2239418461657761734s_mask:
!>[A: $tType] : ( nat > A ) ).
thf(sy_c_Bit__Operations_Osemiring__bit__operations__class_Oor,type,
bit_se1065995026697491101ons_or:
!>[A: $tType] : ( A > A > A ) ).
thf(sy_c_Bit__Operations_Osemiring__bit__operations__class_Opush__bit,type,
bit_se4730199178511100633sh_bit:
!>[A: $tType] : ( nat > A > A ) ).
thf(sy_c_Bit__Operations_Osemiring__bit__operations__class_Oset__bit,type,
bit_se5668285175392031749et_bit:
!>[A: $tType] : ( nat > A > A ) ).
thf(sy_c_Bit__Operations_Osemiring__bit__operations__class_Otake__bit,type,
bit_se2584673776208193580ke_bit:
!>[A: $tType] : ( nat > A > A ) ).
thf(sy_c_Bit__Operations_Osemiring__bit__operations__class_Ounset__bit,type,
bit_se2638667681897837118et_bit:
!>[A: $tType] : ( nat > A > A ) ).
thf(sy_c_Bit__Operations_Osemiring__bit__operations__class_Oxor,type,
bit_se5824344971392196577ns_xor:
!>[A: $tType] : ( A > A > A ) ).
thf(sy_c_Bit__Operations_Osemiring__bits__class_Obit,type,
bit_se5641148757651400278ts_bit:
!>[A: $tType] : ( A > nat > $o ) ).
thf(sy_c_Bit__Operations_Osemiring__bits__class_Opossible__bit,type,
bit_se6407376104438227557le_bit:
!>[A: $tType] : ( ( itself @ A ) > nat > $o ) ).
thf(sy_c_Bit__Operations_Otake__bit__num,type,
bit_take_bit_num: nat > num > ( option @ num ) ).
thf(sy_c_Bit__Shifts__Infix__Syntax_Osemiring__bit__operations__class_Oshiftl,type,
bit_Sh4282982442137083160shiftl:
!>[A: $tType] : ( A > nat > A ) ).
thf(sy_c_Bit__Shifts__Infix__Syntax_Osemiring__bit__operations__class_Oshiftr,type,
bit_Sh4282982442137083166shiftr:
!>[A: $tType] : ( A > nat > A ) ).
thf(sy_c_Bit__Shifts__Infix__Syntax_Osshiftr,type,
bit_Sh8784991116023147202shiftr:
!>[A: $tType] : ( ( word @ A ) > nat > ( word @ A ) ) ).
thf(sy_c_Bits__Integer_OBit__integer,type,
bits_Bit_integer: code_integer > $o > code_integer ).
thf(sy_c_Bits__Integer_Obin__last__integer,type,
bits_b8758750999018896077nteger: code_integer > $o ).
thf(sy_c_Bits__Integer_Obin__rest__integer,type,
bits_b2549910563261871055nteger: code_integer > code_integer ).
thf(sy_c_Boolean__Algebras_Oabstract__boolean__algebra__sym__diff,type,
boolea3799213064322606851m_diff:
!>[A: $tType] : ( ( A > A > A ) > ( A > A > A ) > ( A > A ) > A > A > ( A > A > A ) > $o ) ).
thf(sy_c_Code__Numeral_Odup,type,
code_dup: code_integer > code_integer ).
thf(sy_c_Code__Numeral_Ointeger_Ointeger__of__int,type,
code_integer_of_int: int > code_integer ).
thf(sy_c_Code__Target__Int_Onegative,type,
code_Target_negative: num > int ).
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_Complex_OArg,type,
arg: complex > real ).
thf(sy_c_Complex_Ocis,type,
cis: real > complex ).
thf(sy_c_Complex_Ocnj,type,
cnj: complex > complex ).
thf(sy_c_Complex_Ocomplex_OComplex,type,
complex2: real > real > complex ).
thf(sy_c_Complex_Ocomplex_OIm,type,
im: complex > real ).
thf(sy_c_Complex_Ocomplex_ORe,type,
re: complex > real ).
thf(sy_c_Complex_Ocsqrt,type,
csqrt: complex > complex ).
thf(sy_c_Complex_Oimaginary__unit,type,
imaginary_unit: complex ).
thf(sy_c_Deriv_Odifferentiable,type,
differentiable:
!>[A: $tType,B: $tType] : ( ( A > B ) > ( filter @ A ) > $o ) ).
thf(sy_c_Deriv_Ohas__derivative,type,
has_derivative:
!>[A: $tType,B: $tType] : ( ( A > B ) > ( A > B ) > ( filter @ A ) > $o ) ).
thf(sy_c_Deriv_Ohas__field__derivative,type,
has_field_derivative:
!>[A: $tType] : ( ( A > A ) > A > ( filter @ A ) > $o ) ).
thf(sy_c_Divides_Oadjust__div,type,
adjust_div: ( product_prod @ int @ int ) > int ).
thf(sy_c_Divides_Odivmod__nat,type,
divmod_nat: nat > nat > ( product_prod @ nat @ nat ) ).
thf(sy_c_Divides_Oeucl__rel__int,type,
eucl_rel_int: int > int > ( product_prod @ int @ int ) > $o ).
thf(sy_c_Divides_Ounique__euclidean__semiring__numeral__class_Odivides__aux,type,
unique5940410009612947441es_aux:
!>[A: $tType] : ( ( product_prod @ A @ A ) > $o ) ).
thf(sy_c_Divides_Ounique__euclidean__semiring__numeral__class_Odivmod,type,
unique8689654367752047608divmod:
!>[A: $tType] : ( num > num > ( product_prod @ A @ A ) ) ).
thf(sy_c_Divides_Ounique__euclidean__semiring__numeral__class_Odivmod__step,type,
unique1321980374590559556d_step:
!>[A: $tType] : ( num > ( product_prod @ A @ A ) > ( product_prod @ A @ A ) ) ).
thf(sy_c_Factorial_Ocomm__semiring__1__class_Opochhammer,type,
comm_s3205402744901411588hammer:
!>[A: $tType] : ( A > nat > A ) ).
thf(sy_c_Factorial_Osemiring__char__0__class_Ofact,type,
semiring_char_0_fact:
!>[A: $tType] : ( nat > A ) ).
thf(sy_c_Fields_Oinverse__class_Oinverse,type,
inverse_inverse:
!>[A: $tType] : ( A > A ) ).
thf(sy_c_Filter_Oat__bot,type,
at_bot:
!>[A: $tType] : ( filter @ A ) ).
thf(sy_c_Filter_Oat__top,type,
at_top:
!>[A: $tType] : ( filter @ A ) ).
thf(sy_c_Filter_Oeventually,type,
eventually:
!>[A: $tType] : ( ( A > $o ) > ( filter @ A ) > $o ) ).
thf(sy_c_Filter_Ofilterlim,type,
filterlim:
!>[A: $tType,B: $tType] : ( ( A > B ) > ( filter @ B ) > ( filter @ A ) > $o ) ).
thf(sy_c_Finite__Set_Ocard,type,
finite_card:
!>[B: $tType] : ( ( set @ B ) > nat ) ).
thf(sy_c_Finite__Set_Ofinite,type,
finite_finite2:
!>[A: $tType] : ( ( set @ A ) > $o ) ).
thf(sy_c_Finite__Set_Ofold,type,
finite_fold:
!>[A: $tType,B: $tType] : ( ( A > B > B ) > B > ( set @ A ) > B ) ).
thf(sy_c_Fun_Obij__betw,type,
bij_betw:
!>[A: $tType,B: $tType] : ( ( A > B ) > ( set @ A ) > ( set @ B ) > $o ) ).
thf(sy_c_Fun_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_HOL_Oundefined,type,
undefined:
!>[A: $tType] : 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_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_OBleast,type,
bleast:
!>[A: $tType] : ( ( set @ A ) > ( A > $o ) > A ) ).
thf(sy_c_List_Oabort__Bleast,type,
abort_Bleast:
!>[A: $tType] : ( ( set @ A ) > ( A > $o ) > 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_Oextract,type,
extract:
!>[A: $tType] : ( ( A > $o ) > ( list @ A ) > ( option @ ( product_prod @ ( list @ A ) @ ( product_prod @ A @ ( list @ A ) ) ) ) ) ).
thf(sy_c_List_Ofilter,type,
filter2:
!>[A: $tType] : ( ( A > $o ) > ( list @ A ) > ( list @ A ) ) ).
thf(sy_c_List_Ofoldr,type,
foldr:
!>[A: $tType,B: $tType] : ( ( A > B > B ) > ( list @ A ) > B > B ) ).
thf(sy_c_List_Olinorder__class_Osort__key,type,
linorder_sort_key:
!>[B: $tType,A: $tType] : ( ( B > A ) > ( list @ B ) > ( list @ B ) ) ).
thf(sy_c_List_Olinorder__class_Osorted__list__of__set,type,
linord4507533701916653071of_set:
!>[A: $tType] : ( ( set @ A ) > ( list @ A ) ) ).
thf(sy_c_List_Olist_OCons,type,
cons:
!>[A: $tType] : ( A > ( list @ A ) > ( list @ A ) ) ).
thf(sy_c_List_Olist_ONil,type,
nil:
!>[A: $tType] : ( list @ A ) ).
thf(sy_c_List_Olist_Ocase__list,type,
case_list:
!>[B: $tType,A: $tType] : ( B > ( A > ( list @ A ) > B ) > ( list @ A ) > B ) ).
thf(sy_c_List_Olist_Ohd,type,
hd:
!>[A: $tType] : ( ( list @ A ) > A ) ).
thf(sy_c_List_Olist_Omap,type,
map:
!>[A: $tType,Aa: $tType] : ( ( A > Aa ) > ( list @ A ) > ( list @ Aa ) ) ).
thf(sy_c_List_Olist_Oset,type,
set2:
!>[A: $tType] : ( ( list @ A ) > ( set @ A ) ) ).
thf(sy_c_List_Olist_Osize__list,type,
size_list:
!>[A: $tType] : ( ( A > nat ) > ( list @ A ) > nat ) ).
thf(sy_c_List_Olist_Otl,type,
tl:
!>[A: $tType] : ( ( list @ A ) > ( list @ A ) ) ).
thf(sy_c_List_Olist__update,type,
list_update:
!>[A: $tType] : ( ( list @ A ) > nat > A > ( list @ A ) ) ).
thf(sy_c_List_Olistset,type,
listset:
!>[A: $tType] : ( ( list @ ( set @ A ) ) > ( set @ ( list @ A ) ) ) ).
thf(sy_c_List_Omap__filter,type,
map_filter:
!>[A: $tType,B: $tType] : ( ( A > ( option @ B ) ) > ( list @ A ) > ( list @ B ) ) ).
thf(sy_c_List_Omin__list,type,
min_list:
!>[A: $tType] : ( ( list @ A ) > A ) ).
thf(sy_c_List_Omin__list__rel,type,
min_list_rel:
!>[A: $tType] : ( ( list @ A ) > ( list @ A ) > $o ) ).
thf(sy_c_List_On__lists,type,
n_lists:
!>[A: $tType] : ( nat > ( list @ A ) > ( list @ ( list @ A ) ) ) ).
thf(sy_c_List_Onth,type,
nth:
!>[A: $tType] : ( ( list @ A ) > nat > A ) ).
thf(sy_c_List_Oproduct,type,
product:
!>[A: $tType,B: $tType] : ( ( list @ A ) > ( list @ B ) > ( list @ ( product_prod @ A @ B ) ) ) ).
thf(sy_c_List_Oproduct__lists,type,
product_lists:
!>[A: $tType] : ( ( list @ ( list @ A ) ) > ( list @ ( list @ A ) ) ) ).
thf(sy_c_List_Oreplicate,type,
replicate:
!>[A: $tType] : ( nat > A > ( list @ A ) ) ).
thf(sy_c_List_Oset__Cons,type,
set_Cons:
!>[A: $tType] : ( ( set @ A ) > ( set @ ( list @ A ) ) > ( set @ ( 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_Otranspose__rel,type,
transpose_rel:
!>[A: $tType] : ( ( list @ ( list @ A ) ) > ( list @ ( list @ A ) ) > $o ) ).
thf(sy_c_List_Oupt,type,
upt: nat > nat > ( list @ nat ) ).
thf(sy_c_List_Oupto,type,
upto: int > int > ( list @ int ) ).
thf(sy_c_List_Oupto__aux,type,
upto_aux: int > int > ( list @ int ) > ( list @ int ) ).
thf(sy_c_List_Oupto__rel,type,
upto_rel: ( product_prod @ int @ int ) > ( product_prod @ int @ int ) > $o ).
thf(sy_c_Map_Oran,type,
ran:
!>[A: $tType,B: $tType] : ( ( A > ( option @ B ) ) > ( set @ B ) ) ).
thf(sy_c_Misc_OEps__Opt,type,
eps_Opt:
!>[A: $tType] : ( ( A > $o ) > ( option @ 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_Multiset_Ois__empty,type,
is_empty:
!>[A: $tType] : ( ( multiset @ A ) > $o ) ).
thf(sy_c_Multiset_Olinorder__class_Opart,type,
linorder_part:
!>[B: $tType,A: $tType] : ( ( B > A ) > A > ( list @ B ) > ( product_prod @ ( list @ B ) @ ( product_prod @ ( list @ B ) @ ( list @ B ) ) ) ) ).
thf(sy_c_Nat_OSuc,type,
suc: nat > nat ).
thf(sy_c_Nat_Onat_Ocase__nat,type,
case_nat:
!>[A: $tType] : ( A > ( nat > A ) > nat > A ) ).
thf(sy_c_Nat_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_Onum__of__nat,type,
num_of_nat: nat > num ).
thf(sy_c_Num_Onumeral__class_Onumeral,type,
numeral_numeral:
!>[A: $tType] : ( num > A ) ).
thf(sy_c_Num_Opow,type,
pow: num > num > num ).
thf(sy_c_Num_Opred__numeral,type,
pred_numeral: num > nat ).
thf(sy_c_Num_Oring__1__class_Oiszero,type,
ring_1_iszero:
!>[A: $tType] : ( A > $o ) ).
thf(sy_c_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_Ocase__option,type,
case_option:
!>[B: $tType,A: $tType] : ( B > ( A > B ) > ( option @ A ) > B ) ).
thf(sy_c_Option_Ooption_Osize__option,type,
size_option:
!>[A: $tType] : ( ( A > nat ) > ( option @ A ) > nat ) ).
thf(sy_c_Option_Ooption_Othe,type,
the2:
!>[A: $tType] : ( ( option @ A ) > A ) ).
thf(sy_c_Orderings_Obot__class_Obot,type,
bot_bot:
!>[A: $tType] : A ).
thf(sy_c_Orderings_Oord__class_OLeast,type,
ord_Least:
!>[A: $tType] : ( ( A > $o ) > A ) ).
thf(sy_c_Orderings_Oord__class_Oless,type,
ord_less:
!>[A: $tType] : ( A > A > $o ) ).
thf(sy_c_Orderings_Oord__class_Oless__eq,type,
ord_less_eq:
!>[A: $tType] : ( A > A > $o ) ).
thf(sy_c_Orderings_Oord__class_Omax,type,
ord_max:
!>[A: $tType] : ( A > A > A ) ).
thf(sy_c_Orderings_Oord__class_Omin,type,
ord_min:
!>[A: $tType] : ( A > A > 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_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_OReals,type,
real_Vector_Reals:
!>[A: $tType] : ( set @ A ) ).
thf(sy_c_Real__Vector__Spaces_Obounded__linear,type,
real_V3181309239436604168linear:
!>[A: $tType,B: $tType] : ( ( A > B ) > $o ) ).
thf(sy_c_Real__Vector__Spaces_Odist__class_Odist,type,
real_V557655796197034286t_dist:
!>[A: $tType] : ( A > A > real ) ).
thf(sy_c_Real__Vector__Spaces_Onorm__class_Onorm,type,
real_V7770717601297561774m_norm:
!>[A: $tType] : ( A > real ) ).
thf(sy_c_Real__Vector__Spaces_Oof__real,type,
real_Vector_of_real:
!>[A: $tType] : ( real > A ) ).
thf(sy_c_Real__Vector__Spaces_OscaleR__class_OscaleR,type,
real_V8093663219630862766scaleR:
!>[A: $tType] : ( real > A > A ) ).
thf(sy_c_Rings_Odivide__class_Odivide,type,
divide_divide:
!>[A: $tType] : ( A > A > A ) ).
thf(sy_c_Rings_Odvd__class_Odvd,type,
dvd_dvd:
!>[A: $tType] : ( A > A > $o ) ).
thf(sy_c_Rings_Omodulo__class_Omodulo,type,
modulo_modulo:
!>[A: $tType] : ( A > A > A ) ).
thf(sy_c_Rings_Ozero__neq__one__class_Oof__bool,type,
zero_neq_one_of_bool:
!>[A: $tType] : ( $o > A ) ).
thf(sy_c_Series_Osuminf,type,
suminf:
!>[A: $tType] : ( ( nat > A ) > A ) ).
thf(sy_c_Series_Osummable,type,
summable:
!>[A: $tType] : ( ( nat > A ) > $o ) ).
thf(sy_c_Series_Osums,type,
sums:
!>[A: $tType] : ( ( nat > A ) > A > $o ) ).
thf(sy_c_Set_OBall,type,
ball:
!>[A: $tType] : ( ( set @ A ) > ( A > $o ) > $o ) ).
thf(sy_c_Set_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_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_Ochar_Osize__char,type,
size_char: char > nat ).
thf(sy_c_String_Ocomm__semiring__1__class_Oof__char,type,
comm_s6883823935334413003f_char:
!>[A: $tType] : ( char > A ) ).
thf(sy_c_String_Ointeger__of__char,type,
integer_of_char: char > code_integer ).
thf(sy_c_String_Ounique__euclidean__semiring__with__bit__operations__class_Ochar__of,type,
unique5772411509450598832har_of:
!>[A: $tType] : ( A > char ) ).
thf(sy_c_Syntax__Match_Osyntax__fo__nomatch,type,
syntax7388354845996824322omatch:
!>[A: $tType,B: $tType] : ( A > B > $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_Ocauchy__filter,type,
topolo6773858410816713723filter:
!>[A: $tType] : ( ( filter @ 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_OUint32,type,
uint322: code_integer > uint32 ).
thf(sy_c_Uint32_OUint32__signed,type,
uint32_signed: code_integer > uint32 ).
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__rel,type,
vEBT_V2957053500504383685pi_rel: 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_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_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_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__Definitions_OVEBT_OLeaf,type,
vEBT_Leaf: $o > $o > vEBT_VEBT ).
thf(sy_c_VEBT__Definitions_OVEBT_ONode,type,
vEBT_Node: ( option @ ( product_prod @ nat @ nat ) ) > nat > ( list @ vEBT_VEBT ) > vEBT_VEBT > vEBT_VEBT ).
thf(sy_c_VEBT__Definitions_OVEBT_Osize__VEBT,type,
vEBT_size_VEBT: vEBT_VEBT > nat ).
thf(sy_c_VEBT__Definitions_OVEBT__internal_Oboth__member__options,type,
vEBT_V8194947554948674370ptions: vEBT_VEBT > nat > $o ).
thf(sy_c_VEBT__Definitions_OVEBT__internal_Ohigh,type,
vEBT_VEBT_high: nat > nat > nat ).
thf(sy_c_VEBT__Definitions_OVEBT__internal_Oin__children,type,
vEBT_V5917875025757280293ildren: nat > ( list @ vEBT_VEBT ) > nat > $o ).
thf(sy_c_VEBT__Definitions_OVEBT__internal_Olow,type,
vEBT_VEBT_low: nat > nat > nat ).
thf(sy_c_VEBT__Definitions_OVEBT__internal_Omembermima,type,
vEBT_VEBT_membermima: vEBT_VEBT > nat > $o ).
thf(sy_c_VEBT__Definitions_OVEBT__internal_Omembermima__rel,type,
vEBT_V4351362008482014158ma_rel: ( product_prod @ vEBT_VEBT @ nat ) > ( product_prod @ vEBT_VEBT @ nat ) > $o ).
thf(sy_c_VEBT__Definitions_OVEBT__internal_Onaive__member,type,
vEBT_V5719532721284313246member: vEBT_VEBT > nat > $o ).
thf(sy_c_VEBT__Definitions_OVEBT__internal_Onaive__member__rel,type,
vEBT_V5765760719290551771er_rel: ( product_prod @ vEBT_VEBT @ nat ) > ( product_prod @ vEBT_VEBT @ nat ) > $o ).
thf(sy_c_VEBT__Definitions_OVEBT__internal_Ovalid_H,type,
vEBT_VEBT_valid: vEBT_VEBT > nat > $o ).
thf(sy_c_VEBT__Definitions_OVEBT__internal_Ovalid_H__rel,type,
vEBT_VEBT_valid_rel: ( product_prod @ vEBT_VEBT @ nat ) > ( product_prod @ vEBT_VEBT @ nat ) > $o ).
thf(sy_c_VEBT__Definitions_Oinvar__vebt,type,
vEBT_invar_vebt: vEBT_VEBT > nat > $o ).
thf(sy_c_VEBT__Definitions_Oset__vebt,type,
vEBT_set_vebt: vEBT_VEBT > ( set @ nat ) ).
thf(sy_c_VEBT__Definitions_Ovebt__buildup,type,
vEBT_vebt_buildup: nat > vEBT_VEBT ).
thf(sy_c_VEBT__Definitions_Ovebt__buildup__rel,type,
vEBT_v4011308405150292612up_rel: nat > nat > $o ).
thf(sy_c_VEBT__DeleteBounds_OT_092_060_094sub_062d_092_060_094sub_062e_092_060_094sub_062l_092_060_094sub_062e_092_060_094sub_062t_092_060_094sub_062e,type,
vEBT_T_d_e_l_e_t_e: vEBT_VEBT > nat > nat ).
thf(sy_c_VEBT__DeleteBounds_OT_092_060_094sub_062d_092_060_094sub_062e_092_060_094sub_062l_092_060_094sub_062e_092_060_094sub_062t_092_060_094sub_062e__rel,type,
vEBT_T8441311223069195367_e_rel: ( product_prod @ vEBT_VEBT @ nat ) > ( product_prod @ vEBT_VEBT @ nat ) > $o ).
thf(sy_c_VEBT__DeleteBounds_OVEBT__internal_OT_092_060_094sub_062d_092_060_094sub_062e_092_060_094sub_062l_092_060_094sub_062e_092_060_094sub_062t_092_060_094sub_062e_H,type,
vEBT_V1232361888498592333_e_t_e: vEBT_VEBT > nat > nat ).
thf(sy_c_VEBT__DeleteBounds_OVEBT__internal_OT_092_060_094sub_062d_092_060_094sub_062e_092_060_094sub_062l_092_060_094sub_062e_092_060_094sub_062t_092_060_094sub_062e_H__rel,type,
vEBT_V6368547301243506412_e_rel: ( product_prod @ vEBT_VEBT @ nat ) > ( product_prod @ vEBT_VEBT @ nat ) > $o ).
thf(sy_c_VEBT__Delete_Ovebt__delete,type,
vEBT_vebt_delete: vEBT_VEBT > nat > vEBT_VEBT ).
thf(sy_c_VEBT__Delete_Ovebt__delete__rel,type,
vEBT_vebt_delete_rel: ( product_prod @ vEBT_VEBT @ nat ) > ( product_prod @ vEBT_VEBT @ nat ) > $o ).
thf(sy_c_VEBT__Height_OVEBT__internal_Oheight,type,
vEBT_VEBT_height: vEBT_VEBT > nat ).
thf(sy_c_VEBT__Height_OVEBT__internal_Oheight__rel,type,
vEBT_VEBT_height_rel: vEBT_VEBT > vEBT_VEBT > $o ).
thf(sy_c_VEBT__Insert_Ovebt__insert,type,
vEBT_vebt_insert: vEBT_VEBT > nat > vEBT_VEBT ).
thf(sy_c_VEBT__Insert_Ovebt__insert__rel,type,
vEBT_vebt_insert_rel: ( product_prod @ vEBT_VEBT @ nat ) > ( product_prod @ vEBT_VEBT @ nat ) > $o ).
thf(sy_c_VEBT__List__Assn_OlistI__assn,type,
vEBT_List_listI_assn:
!>[A: $tType,B: $tType] : ( ( set @ nat ) > ( A > B > assn ) > ( list @ A ) > ( list @ B ) > assn ) ).
thf(sy_c_VEBT__List__Assn_Olist__assn,type,
vEBT_List_list_assn:
!>[A: $tType,C: $tType] : ( ( A > C > assn ) > ( list @ A ) > ( list @ C ) > assn ) ).
thf(sy_c_VEBT__Member_OVEBT__internal_Obit__concat,type,
vEBT_VEBT_bit_concat: nat > nat > nat > nat ).
thf(sy_c_VEBT__Member_OVEBT__internal_OminNull,type,
vEBT_VEBT_minNull: vEBT_VEBT > $o ).
thf(sy_c_VEBT__Member_OVEBT__internal_OminNull__rel,type,
vEBT_V6963167321098673237ll_rel: vEBT_VEBT > vEBT_VEBT > $o ).
thf(sy_c_VEBT__Member_OVEBT__internal_Oset__vebt_H,type,
vEBT_VEBT_set_vebt: vEBT_VEBT > ( set @ nat ) ).
thf(sy_c_VEBT__Member_Ovebt__member,type,
vEBT_vebt_member: vEBT_VEBT > nat > $o ).
thf(sy_c_VEBT__Member_Ovebt__member__rel,type,
vEBT_vebt_member_rel: ( product_prod @ vEBT_VEBT @ nat ) > ( product_prod @ vEBT_VEBT @ nat ) > $o ).
thf(sy_c_VEBT__MinMax_OVEBT__internal_Oadd,type,
vEBT_VEBT_add: ( option @ nat ) > ( option @ nat ) > ( option @ nat ) ).
thf(sy_c_VEBT__MinMax_OVEBT__internal_Ogreater,type,
vEBT_VEBT_greater: ( option @ nat ) > ( option @ nat ) > $o ).
thf(sy_c_VEBT__MinMax_OVEBT__internal_Oless,type,
vEBT_VEBT_less: ( option @ nat ) > ( option @ nat ) > $o ).
thf(sy_c_VEBT__MinMax_OVEBT__internal_Olesseq,type,
vEBT_VEBT_lesseq: ( option @ nat ) > ( option @ nat ) > $o ).
thf(sy_c_VEBT__MinMax_OVEBT__internal_Omax__in__set,type,
vEBT_VEBT_max_in_set: ( set @ nat ) > nat > $o ).
thf(sy_c_VEBT__MinMax_OVEBT__internal_Omin__in__set,type,
vEBT_VEBT_min_in_set: ( set @ nat ) > nat > $o ).
thf(sy_c_VEBT__MinMax_OVEBT__internal_Omul,type,
vEBT_VEBT_mul: ( option @ nat ) > ( option @ nat ) > ( option @ nat ) ).
thf(sy_c_VEBT__MinMax_OVEBT__internal_Ooption__shift,type,
vEBT_V2048590022279873568_shift:
!>[A: $tType] : ( ( A > A > A ) > ( option @ A ) > ( option @ A ) > ( option @ A ) ) ).
thf(sy_c_VEBT__MinMax_OVEBT__internal_Opower,type,
vEBT_VEBT_power: ( option @ nat ) > ( option @ nat ) > ( option @ nat ) ).
thf(sy_c_VEBT__MinMax_Ovebt__maxt,type,
vEBT_vebt_maxt: vEBT_VEBT > ( option @ nat ) ).
thf(sy_c_VEBT__MinMax_Ovebt__maxt__rel,type,
vEBT_vebt_maxt_rel: vEBT_VEBT > vEBT_VEBT > $o ).
thf(sy_c_VEBT__MinMax_Ovebt__mint,type,
vEBT_vebt_mint: vEBT_VEBT > ( option @ nat ) ).
thf(sy_c_VEBT__MinMax_Ovebt__mint__rel,type,
vEBT_vebt_mint_rel: vEBT_VEBT > vEBT_VEBT > $o ).
thf(sy_c_VEBT__Pred_Ois__pred__in__set,type,
vEBT_is_pred_in_set: ( set @ nat ) > nat > nat > $o ).
thf(sy_c_VEBT__Pred_Ovebt__pred,type,
vEBT_vebt_pred: vEBT_VEBT > nat > ( option @ nat ) ).
thf(sy_c_VEBT__Pred_Ovebt__pred__rel,type,
vEBT_vebt_pred_rel: ( product_prod @ vEBT_VEBT @ nat ) > ( product_prod @ vEBT_VEBT @ nat ) > $o ).
thf(sy_c_VEBT__Space_OVEBT__internal_OT_092_060_094sub_062b_092_060_094sub_062u_092_060_094sub_062i_092_060_094sub_062l_092_060_094sub_062d,type,
vEBT_V8646137997579335489_i_l_d: nat > nat ).
thf(sy_c_VEBT__Space_OVEBT__internal_OT_092_060_094sub_062b_092_060_094sub_062u_092_060_094sub_062i_092_060_094sub_062l_092_060_094sub_062d_092_060_094sub_062u_092_060_094sub_062p,type,
vEBT_V8346862874174094_d_u_p: nat > nat ).
thf(sy_c_VEBT__Space_OVEBT__internal_OT_092_060_094sub_062b_092_060_094sub_062u_092_060_094sub_062i_092_060_094sub_062l_092_060_094sub_062d_092_060_094sub_062u_092_060_094sub_062p__rel,type,
vEBT_V1247956027447740395_p_rel: nat > nat > $o ).
thf(sy_c_VEBT__Space_OVEBT__internal_OT_092_060_094sub_062b_092_060_094sub_062u_092_060_094sub_062i_092_060_094sub_062l_092_060_094sub_062d__rel,type,
vEBT_V5144397997797733112_d_rel: nat > nat > $o ).
thf(sy_c_VEBT__Space_OVEBT__internal_Ocnt,type,
vEBT_VEBT_cnt: vEBT_VEBT > real ).
thf(sy_c_VEBT__Space_OVEBT__internal_Ocnt_H,type,
vEBT_VEBT_cnt2: vEBT_VEBT > nat ).
thf(sy_c_VEBT__Space_OVEBT__internal_Ocnt_H__rel,type,
vEBT_VEBT_cnt_rel: vEBT_VEBT > vEBT_VEBT > $o ).
thf(sy_c_VEBT__Space_OVEBT__internal_Ocnt__rel,type,
vEBT_VEBT_cnt_rel2: vEBT_VEBT > vEBT_VEBT > $o ).
thf(sy_c_VEBT__Space_OVEBT__internal_Ospace,type,
vEBT_VEBT_space: vEBT_VEBT > nat ).
thf(sy_c_VEBT__Space_OVEBT__internal_Ospace_H,type,
vEBT_VEBT_space2: vEBT_VEBT > nat ).
thf(sy_c_VEBT__Space_OVEBT__internal_Ospace_H__rel,type,
vEBT_VEBT_space_rel: vEBT_VEBT > vEBT_VEBT > $o ).
thf(sy_c_VEBT__Space_OVEBT__internal_Ospace__rel,type,
vEBT_VEBT_space_rel2: vEBT_VEBT > vEBT_VEBT > $o ).
thf(sy_c_VEBT__Succ_Ois__succ__in__set,type,
vEBT_is_succ_in_set: ( set @ nat ) > nat > nat > $o ).
thf(sy_c_VEBT__Succ_Ovebt__succ,type,
vEBT_vebt_succ: vEBT_VEBT > nat > ( option @ nat ) ).
thf(sy_c_VEBT__Succ_Ovebt__succ__rel,type,
vEBT_vebt_succ_rel: ( product_prod @ vEBT_VEBT @ nat ) > ( product_prod @ vEBT_VEBT @ nat ) > $o ).
thf(sy_c_Wellfounded_Oaccp,type,
accp:
!>[A: $tType] : ( ( A > A > $o ) > A > $o ) ).
thf(sy_c_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__bits__axioms,type,
word_T1405001541757447525axioms:
!>[A: $tType,B: $tType] : ( ( A > ( word @ B ) ) > ( nat > A > A ) > $o ) ).
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_fChoice,type,
fChoice:
!>[A: $tType] : ( ( A > $o ) > A ) ).
thf(sy_c_member,type,
member:
!>[A: $tType] : ( A > ( set @ A ) > $o ) ).
thf(sy_v_na____,type,
na: nat ).
thf(sy_v_x____,type,
x: list @ vEBT_VEBTi ).
thf(sy_v_xa____,type,
xa: array @ vEBT_VEBTi ).
thf(sy_v_xb____,type,
xb: vEBT_VEBTi ).
thf(sy_v_xc____,type,
xc: vEBT_VEBTi ).
% Relevant facts (8179)
thf(fact_0_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_1_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_2_False,axiom,
~ ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( suc @ ( suc @ na ) ) ) ).
% False
thf(fact_3_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_4_ent__pure__pre__iff,axiom,
! [P: assn,B2: $o,Q: assn] :
( ( entails @ ( times_times @ assn @ P @ ( pure_assn @ B2 ) ) @ Q )
= ( B2
=> ( entails @ P @ Q ) ) ) ).
% ent_pure_pre_iff
thf(fact_5_power__odd__eq,axiom,
! [A: $tType] :
( ( monoid_mult @ A )
=> ! [A2: A,N: nat] :
( ( power_power @ A @ A2 @ ( suc @ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N ) ) )
= ( times_times @ A @ A2 @ ( power_power @ A @ ( power_power @ A @ A2 @ N ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) ).
% power_odd_eq
thf(fact_6_merge__pure__star,axiom,
! [A2: $o,B2: $o] :
( ( times_times @ assn @ ( pure_assn @ A2 ) @ ( pure_assn @ B2 ) )
= ( pure_assn
@ ( A2
& B2 ) ) ) ).
% merge_pure_star
thf(fact_7_power__even__eq,axiom,
! [A: $tType] :
( ( monoid_mult @ A )
=> ! [A2: A,N: nat] :
( ( power_power @ A @ A2 @ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N ) )
= ( power_power @ A @ ( power_power @ A @ A2 @ N ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ).
% power_even_eq
thf(fact_8_Suc__double__not__eq__double,axiom,
! [M: nat,N: nat] :
( ( suc @ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ M ) )
!= ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N ) ) ).
% Suc_double_not_eq_double
thf(fact_9_double__not__eq__Suc__double,axiom,
! [M: nat,N: nat] :
( ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ M )
!= ( suc @ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N ) ) ) ).
% double_not_eq_Suc_double
thf(fact_10_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_11_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_12_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_13_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_14_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_15_pure__assn__eq__conv,axiom,
! [P: $o,Q: $o] :
( ( ( pure_assn @ P )
= ( pure_assn @ Q ) )
= ( P = Q ) ) ).
% pure_assn_eq_conv
thf(fact_16_power__mult__numeral,axiom,
! [A: $tType] :
( ( monoid_mult @ A )
=> ! [A2: A,M: num,N: num] :
( ( power_power @ A @ ( power_power @ A @ A2 @ ( numeral_numeral @ nat @ M ) ) @ ( numeral_numeral @ nat @ N ) )
= ( power_power @ A @ A2 @ ( numeral_numeral @ nat @ ( times_times @ num @ M @ N ) ) ) ) ) ).
% power_mult_numeral
thf(fact_17_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_18_even__Suc__Suc__iff,axiom,
! [N: nat] :
( ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( suc @ ( suc @ N ) ) )
= ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N ) ) ).
% even_Suc_Suc_iff
thf(fact_19_even__Suc,axiom,
! [N: nat] :
( ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( suc @ N ) )
= ( ~ ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N ) ) ) ).
% even_Suc
thf(fact_20_even__Suc__div__two,axiom,
! [N: nat] :
( ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N )
=> ( ( divide_divide @ nat @ ( suc @ N ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) )
= ( divide_divide @ nat @ N @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ).
% even_Suc_div_two
thf(fact_21_odd__Suc__div__two,axiom,
! [N: nat] :
( ~ ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N )
=> ( ( divide_divide @ nat @ ( suc @ N ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) )
= ( suc @ ( divide_divide @ nat @ N @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) ).
% odd_Suc_div_two
thf(fact_22_dvd__power__same,axiom,
! [A: $tType] :
( ( comm_semiring_1 @ A )
=> ! [X: A,Y: A,N: nat] :
( ( dvd_dvd @ A @ X @ Y )
=> ( dvd_dvd @ A @ ( power_power @ A @ X @ N ) @ ( power_power @ A @ Y @ N ) ) ) ) ).
% dvd_power_same
thf(fact_23_div__power,axiom,
! [A: $tType] :
( ( algebraic_semidom @ A )
=> ! [B2: A,A2: A,N: nat] :
( ( dvd_dvd @ A @ B2 @ A2 )
=> ( ( power_power @ A @ ( divide_divide @ A @ A2 @ B2 ) @ N )
= ( divide_divide @ A @ ( power_power @ A @ A2 @ N ) @ ( power_power @ A @ B2 @ N ) ) ) ) ) ).
% div_power
thf(fact_24_div__mult2__numeral__eq,axiom,
! [A: $tType] :
( ( euclid5411537665997757685th_nat @ A )
=> ! [A2: A,K: num,L: num] :
( ( divide_divide @ A @ ( divide_divide @ A @ A2 @ ( numeral_numeral @ A @ K ) ) @ ( numeral_numeral @ A @ L ) )
= ( divide_divide @ A @ A2 @ ( numeral_numeral @ A @ ( times_times @ num @ K @ L ) ) ) ) ) ).
% div_mult2_numeral_eq
thf(fact_25_even__numeral,axiom,
! [A: $tType] :
( ( semiring_parity @ A )
=> ! [N: num] : ( dvd_dvd @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ ( numeral_numeral @ A @ ( bit0 @ N ) ) ) ) ).
% even_numeral
thf(fact_26_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_27_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_28_assn__times__assoc,axiom,
! [P: assn,Q: assn,R: assn] :
( ( times_times @ assn @ ( times_times @ assn @ P @ Q ) @ R )
= ( times_times @ assn @ P @ ( times_times @ assn @ Q @ R ) ) ) ).
% assn_times_assoc
thf(fact_29_assn__times__comm,axiom,
( ( times_times @ assn )
= ( ^ [P2: assn,Q2: assn] : ( times_times @ assn @ Q2 @ P2 ) ) ) ).
% assn_times_comm
thf(fact_30_ent__trans,axiom,
! [P: assn,Q: assn,R: assn] :
( ( entails @ P @ Q )
=> ( ( entails @ Q @ R )
=> ( entails @ P @ R ) ) ) ).
% ent_trans
thf(fact_31_ent__refl,axiom,
! [P: assn] : ( entails @ P @ P ) ).
% ent_refl
thf(fact_32_ent__iffI,axiom,
! [A3: assn,B4: assn] :
( ( entails @ A3 @ B4 )
=> ( ( entails @ B4 @ A3 )
=> ( A3 = B4 ) ) ) ).
% ent_iffI
thf(fact_33_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_34_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_35_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_36_power__commuting__commutes,axiom,
! [A: $tType] :
( ( monoid_mult @ A )
=> ! [X: A,Y: A,N: nat] :
( ( ( times_times @ A @ X @ Y )
= ( times_times @ A @ Y @ X ) )
=> ( ( times_times @ A @ ( power_power @ A @ X @ N ) @ Y )
= ( times_times @ A @ Y @ ( power_power @ A @ X @ N ) ) ) ) ) ).
% power_commuting_commutes
thf(fact_37_power__mult__distrib,axiom,
! [A: $tType] :
( ( comm_monoid_mult @ A )
=> ! [A2: A,B2: A,N: nat] :
( ( power_power @ A @ ( times_times @ A @ A2 @ B2 ) @ N )
= ( times_times @ A @ ( power_power @ A @ A2 @ N ) @ ( power_power @ A @ B2 @ N ) ) ) ) ).
% power_mult_distrib
thf(fact_38_power__commutes,axiom,
! [A: $tType] :
( ( monoid_mult @ A )
=> ! [A2: A,N: nat] :
( ( times_times @ A @ ( power_power @ A @ A2 @ N ) @ A2 )
= ( times_times @ A @ A2 @ ( power_power @ A @ A2 @ N ) ) ) ) ).
% power_commutes
thf(fact_39_power__divide,axiom,
! [A: $tType] :
( ( field @ A )
=> ! [A2: A,B2: A,N: nat] :
( ( power_power @ A @ ( divide_divide @ A @ A2 @ B2 ) @ N )
= ( divide_divide @ A @ ( power_power @ A @ A2 @ N ) @ ( power_power @ A @ B2 @ N ) ) ) ) ).
% power_divide
thf(fact_40_power__mult,axiom,
! [A: $tType] :
( ( monoid_mult @ A )
=> ! [A2: A,M: nat,N: nat] :
( ( power_power @ A @ A2 @ ( times_times @ nat @ M @ N ) )
= ( power_power @ A @ ( power_power @ A @ A2 @ M ) @ N ) ) ) ).
% power_mult
thf(fact_41_div__mult2__eq,axiom,
! [M: nat,N: nat,Q3: nat] :
( ( divide_divide @ nat @ M @ ( times_times @ nat @ N @ Q3 ) )
= ( divide_divide @ nat @ ( divide_divide @ nat @ M @ N ) @ Q3 ) ) ).
% div_mult2_eq
thf(fact_42_ent__star__mono,axiom,
! [P: assn,P3: assn,Q: assn,Q4: assn] :
( ( entails @ P @ P3 )
=> ( ( entails @ Q @ Q4 )
=> ( entails @ ( times_times @ assn @ P @ Q ) @ ( times_times @ assn @ P3 @ Q4 ) ) ) ) ).
% ent_star_mono
thf(fact_43_power__Suc2,axiom,
! [A: $tType] :
( ( monoid_mult @ A )
=> ! [A2: A,N: nat] :
( ( power_power @ A @ A2 @ ( suc @ N ) )
= ( times_times @ A @ ( power_power @ A @ A2 @ N ) @ A2 ) ) ) ).
% power_Suc2
thf(fact_44_power__Suc,axiom,
! [A: $tType] :
( ( power @ A )
=> ! [A2: A,N: nat] :
( ( power_power @ A @ A2 @ ( suc @ N ) )
= ( times_times @ A @ A2 @ ( power_power @ A @ A2 @ N ) ) ) ) ).
% power_Suc
thf(fact_45_mem__Collect__eq,axiom,
! [A: $tType,A2: A,P: A > $o] :
( ( member @ A @ A2 @ ( collect @ A @ P ) )
= ( P @ A2 ) ) ).
% mem_Collect_eq
thf(fact_46_Collect__mem__eq,axiom,
! [A: $tType,A3: set @ A] :
( ( collect @ A
@ ^ [X2: A] : ( member @ A @ X2 @ A3 ) )
= A3 ) ).
% Collect_mem_eq
thf(fact_47_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_48_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_49_numeral__Bit0__div__2,axiom,
! [A: $tType] :
( ( euclid5411537665997757685th_nat @ A )
=> ! [N: num] :
( ( divide_divide @ A @ ( numeral_numeral @ A @ ( bit0 @ N ) ) @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) )
= ( numeral_numeral @ A @ N ) ) ) ).
% numeral_Bit0_div_2
thf(fact_50_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_51_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_52_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_53_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_54_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_55_bit__eq__rec,axiom,
! [A: $tType] :
( ( bit_semiring_bits @ A )
=> ( ( ^ [Y2: A,Z2: A] : Y2 = Z2 )
= ( ^ [A4: A,B5: A] :
( ( ( dvd_dvd @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ A4 )
= ( dvd_dvd @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ B5 ) )
& ( ( divide_divide @ A @ A4 @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) )
= ( divide_divide @ A @ B5 @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) ) ) ) ) ) ).
% bit_eq_rec
thf(fact_56_num__double,axiom,
! [N: num] :
( ( times_times @ num @ ( bit0 @ one2 ) @ N )
= ( bit0 @ N ) ) ).
% num_double
thf(fact_57_semiring__norm_I83_J,axiom,
! [N: num] :
( one2
!= ( bit0 @ N ) ) ).
% semiring_norm(83)
thf(fact_58_semiring__norm_I85_J,axiom,
! [M: num] :
( ( bit0 @ M )
!= one2 ) ).
% semiring_norm(85)
thf(fact_59_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_60_numeral__times__numeral,axiom,
! [A: $tType] :
( ( semiring_numeral @ A )
=> ! [M: num,N: num] :
( ( times_times @ A @ ( numeral_numeral @ A @ M ) @ ( numeral_numeral @ A @ N ) )
= ( numeral_numeral @ A @ ( times_times @ num @ M @ N ) ) ) ) ).
% numeral_times_numeral
thf(fact_61_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_62_buildup__nothing__in__leaf,axiom,
! [N: nat,X: nat] :
~ ( vEBT_V5719532721284313246member @ ( vEBT_vebt_buildup @ N ) @ X ) ).
% buildup_nothing_in_leaf
thf(fact_63_even__odd__cases,axiom,
! [X: nat] :
( ! [N2: nat] :
( X
!= ( plus_plus @ nat @ N2 @ N2 ) )
=> ~ ! [N2: nat] :
( X
!= ( plus_plus @ nat @ N2 @ ( suc @ N2 ) ) ) ) ).
% even_odd_cases
thf(fact_64_numeral__eq__iff,axiom,
! [A: $tType] :
( ( semiring_char_0 @ A )
=> ! [M: num,N: num] :
( ( ( numeral_numeral @ A @ M )
= ( numeral_numeral @ A @ N ) )
= ( M = N ) ) ) ).
% numeral_eq_iff
thf(fact_65_semiring__norm_I87_J,axiom,
! [M: num,N: num] :
( ( ( bit0 @ M )
= ( bit0 @ N ) )
= ( M = N ) ) ).
% semiring_norm(87)
thf(fact_66_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_67_numeral__plus__numeral,axiom,
! [A: $tType] :
( ( numeral @ A )
=> ! [M: num,N: num] :
( ( plus_plus @ A @ ( numeral_numeral @ A @ M ) @ ( numeral_numeral @ A @ N ) )
= ( numeral_numeral @ A @ ( plus_plus @ num @ M @ N ) ) ) ) ).
% numeral_plus_numeral
thf(fact_68_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_69_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_70_semiring__norm_I13_J,axiom,
! [M: num,N: num] :
( ( times_times @ num @ ( bit0 @ M ) @ ( bit0 @ N ) )
= ( bit0 @ ( bit0 @ ( times_times @ num @ M @ N ) ) ) ) ).
% semiring_norm(13)
thf(fact_71_semiring__norm_I12_J,axiom,
! [N: num] :
( ( times_times @ num @ one2 @ N )
= N ) ).
% semiring_norm(12)
thf(fact_72_semiring__norm_I11_J,axiom,
! [M: num] :
( ( times_times @ num @ M @ one2 )
= M ) ).
% semiring_norm(11)
thf(fact_73_bit__concat__def,axiom,
( vEBT_VEBT_bit_concat
= ( ^ [H: nat,L2: nat,D: nat] : ( plus_plus @ nat @ ( times_times @ nat @ H @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ D ) ) @ L2 ) ) ) ).
% bit_concat_def
thf(fact_74_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_75_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_76_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_77_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_78_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_79_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_80_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_81_add__2__eq__Suc,axiom,
! [N: nat] :
( ( plus_plus @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N )
= ( suc @ ( suc @ N ) ) ) ).
% add_2_eq_Suc
thf(fact_82_add__2__eq__Suc_H,axiom,
! [N: nat] :
( ( plus_plus @ nat @ N @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) )
= ( suc @ ( suc @ N ) ) ) ).
% add_2_eq_Suc'
thf(fact_83_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_84_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_85_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_86_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_87_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_88_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_89_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_90_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_91_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_92_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_93_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_94_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_95_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_96_numeral__Bit0,axiom,
! [A: $tType] :
( ( numeral @ A )
=> ! [N: num] :
( ( numeral_numeral @ A @ ( bit0 @ N ) )
= ( plus_plus @ A @ ( numeral_numeral @ A @ N ) @ ( numeral_numeral @ A @ N ) ) ) ) ).
% numeral_Bit0
thf(fact_97_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_98_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_99_power__add,axiom,
! [A: $tType] :
( ( monoid_mult @ A )
=> ! [A2: A,M: nat,N: nat] :
( ( power_power @ A @ A2 @ ( plus_plus @ nat @ M @ N ) )
= ( times_times @ A @ ( power_power @ A @ A2 @ M ) @ ( power_power @ A @ A2 @ N ) ) ) ) ).
% power_add
thf(fact_100_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_101_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_102_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_103_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_104_dvd__refl,axiom,
! [A: $tType] :
( ( comm_monoid_mult @ A )
=> ! [A2: A] : ( dvd_dvd @ A @ A2 @ A2 ) ) ).
% dvd_refl
thf(fact_105_div__exp__eq,axiom,
! [A: $tType] :
( ( bit_semiring_bits @ A )
=> ! [A2: A,M: nat,N: nat] :
( ( divide_divide @ A @ ( divide_divide @ A @ A2 @ ( power_power @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ M ) ) @ ( power_power @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ N ) )
= ( divide_divide @ A @ A2 @ ( power_power @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ ( plus_plus @ nat @ M @ N ) ) ) ) ) ).
% div_exp_eq
thf(fact_106_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_107_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_108_dvdE,axiom,
! [A: $tType] :
( ( dvd @ A )
=> ! [B2: A,A2: A] :
( ( dvd_dvd @ A @ B2 @ A2 )
=> ~ ! [K2: A] :
( A2
!= ( times_times @ A @ B2 @ K2 ) ) ) ) ).
% dvdE
thf(fact_109_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_110_dvd__def,axiom,
! [A: $tType] :
( ( dvd @ A )
=> ( ( dvd_dvd @ A )
= ( ^ [B5: A,A4: A] :
? [K3: A] :
( A4
= ( times_times @ A @ B5 @ K3 ) ) ) ) ) ).
% dvd_def
thf(fact_111_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_112_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_113_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_114_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_115_mult__dvd__mono,axiom,
! [A: $tType] :
( ( comm_monoid_mult @ A )
=> ! [A2: A,B2: A,C2: A,D2: A] :
( ( dvd_dvd @ A @ A2 @ B2 )
=> ( ( dvd_dvd @ A @ C2 @ D2 )
=> ( dvd_dvd @ A @ ( times_times @ A @ A2 @ C2 ) @ ( times_times @ A @ B2 @ D2 ) ) ) ) ) ).
% mult_dvd_mono
thf(fact_116_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_117_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_118_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_119_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_120_div__div__div__same,axiom,
! [A: $tType] :
( ( algebraic_semidom @ A )
=> ! [D2: A,B2: A,A2: A] :
( ( dvd_dvd @ A @ D2 @ B2 )
=> ( ( dvd_dvd @ A @ B2 @ A2 )
=> ( ( divide_divide @ A @ ( divide_divide @ A @ A2 @ D2 ) @ ( divide_divide @ A @ B2 @ D2 ) )
= ( divide_divide @ A @ A2 @ B2 ) ) ) ) ) ).
% div_div_div_same
thf(fact_121_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_122_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_123_divide__numeral__1,axiom,
! [A: $tType] :
( ( field @ A )
=> ! [A2: A] :
( ( divide_divide @ A @ A2 @ ( numeral_numeral @ A @ one2 ) )
= A2 ) ) ).
% divide_numeral_1
thf(fact_124_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_125_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_126_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_127_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_128_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_129_div__mult__div__if__dvd,axiom,
! [A: $tType] :
( ( algebraic_semidom @ A )
=> ! [B2: A,A2: A,D2: A,C2: A] :
( ( dvd_dvd @ A @ B2 @ A2 )
=> ( ( dvd_dvd @ A @ D2 @ C2 )
=> ( ( times_times @ A @ ( divide_divide @ A @ A2 @ B2 ) @ ( divide_divide @ A @ C2 @ D2 ) )
= ( divide_divide @ A @ ( times_times @ A @ A2 @ C2 ) @ ( times_times @ A @ B2 @ D2 ) ) ) ) ) ) ).
% div_mult_div_if_dvd
thf(fact_130_mult__Suc__right,axiom,
! [M: nat,N: nat] :
( ( times_times @ nat @ M @ ( suc @ N ) )
= ( plus_plus @ nat @ M @ ( times_times @ nat @ M @ N ) ) ) ).
% mult_Suc_right
thf(fact_131_add__Suc__right,axiom,
! [M: nat,N: nat] :
( ( plus_plus @ nat @ M @ ( suc @ N ) )
= ( suc @ ( plus_plus @ nat @ M @ N ) ) ) ).
% add_Suc_right
thf(fact_132_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_133_VEBT__internal_OT__vebt__buildupi_Osimps_I3_J,axiom,
! [N: nat] :
( ( ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N )
=> ( ( vEBT_V441764108873111860ildupi @ ( suc @ ( suc @ N ) ) )
= ( suc @ ( suc @ ( suc @ ( plus_plus @ nat @ ( vEBT_V441764108873111860ildupi @ ( suc @ ( divide_divide @ nat @ N @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( plus_plus @ nat @ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ ( bit0 @ one2 ) ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ N @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( times_times @ nat @ ( vEBT_V441764108873111860ildupi @ ( suc @ ( divide_divide @ nat @ N @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ N @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) ) ) ) ) ) ) )
& ( ~ ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N )
=> ( ( vEBT_V441764108873111860ildupi @ ( suc @ ( suc @ N ) ) )
= ( suc @ ( suc @ ( suc @ ( plus_plus @ nat @ ( vEBT_V441764108873111860ildupi @ ( suc @ ( suc @ ( divide_divide @ nat @ N @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) @ ( plus_plus @ nat @ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ ( bit0 @ ( bit0 @ one2 ) ) ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ N @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ ( bit0 @ one2 ) ) ) @ ( times_times @ nat @ ( vEBT_V441764108873111860ildupi @ ( suc @ ( divide_divide @ nat @ N @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ N @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).
% VEBT_internal.T_vebt_buildupi.simps(3)
thf(fact_134_triangle__def,axiom,
( nat_triangle
= ( ^ [N3: nat] : ( divide_divide @ nat @ ( times_times @ nat @ N3 @ ( suc @ N3 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ).
% triangle_def
thf(fact_135_high__def,axiom,
( vEBT_VEBT_high
= ( ^ [X2: nat,N3: nat] : ( divide_divide @ nat @ X2 @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N3 ) ) ) ) ).
% high_def
thf(fact_136_bezout__lemma__nat,axiom,
! [D2: nat,A2: nat,B2: nat,X: nat,Y: nat] :
( ( dvd_dvd @ nat @ D2 @ A2 )
=> ( ( dvd_dvd @ nat @ D2 @ B2 )
=> ( ( ( ( times_times @ nat @ A2 @ X )
= ( plus_plus @ nat @ ( times_times @ nat @ B2 @ Y ) @ D2 ) )
| ( ( times_times @ nat @ B2 @ X )
= ( plus_plus @ nat @ ( times_times @ nat @ A2 @ Y ) @ D2 ) ) )
=> ? [X3: nat,Y3: nat] :
( ( dvd_dvd @ nat @ D2 @ A2 )
& ( dvd_dvd @ nat @ D2 @ ( plus_plus @ nat @ A2 @ B2 ) )
& ( ( ( times_times @ nat @ A2 @ X3 )
= ( plus_plus @ nat @ ( times_times @ nat @ ( plus_plus @ nat @ A2 @ B2 ) @ Y3 ) @ D2 ) )
| ( ( times_times @ nat @ ( plus_plus @ nat @ A2 @ B2 ) @ X3 )
= ( plus_plus @ nat @ ( times_times @ nat @ A2 @ Y3 ) @ D2 ) ) ) ) ) ) ) ).
% bezout_lemma_nat
thf(fact_137_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_138_mult__Suc,axiom,
! [M: nat,N: nat] :
( ( times_times @ nat @ ( suc @ M ) @ N )
= ( plus_plus @ nat @ N @ ( times_times @ nat @ M @ N ) ) ) ).
% mult_Suc
thf(fact_139_highi__def,axiom,
( vEBT_VEBT_highi
= ( ^ [X2: nat,N3: nat] : ( heap_Time_return @ nat @ ( divide_divide @ nat @ X2 @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N3 ) ) ) ) ) ).
% highi_def
thf(fact_140_set__decode__Suc,axiom,
! [N: nat,X: nat] :
( ( member @ nat @ ( suc @ N ) @ ( nat_set_decode @ X ) )
= ( member @ nat @ N @ ( nat_set_decode @ ( divide_divide @ nat @ X @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) ).
% set_decode_Suc
thf(fact_141_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_142_old_Onat_Oinject,axiom,
! [Nat: nat,Nat2: nat] :
( ( ( suc @ Nat )
= ( suc @ Nat2 ) )
= ( Nat = Nat2 ) ) ).
% old.nat.inject
thf(fact_143_nat_Oinject,axiom,
! [X22: nat,Y22: nat] :
( ( ( suc @ X22 )
= ( suc @ Y22 ) )
= ( X22 = Y22 ) ) ).
% nat.inject
thf(fact_144_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_145_semiring__norm_I6_J,axiom,
! [M: num,N: num] :
( ( plus_plus @ num @ ( bit0 @ M ) @ ( bit0 @ N ) )
= ( bit0 @ ( plus_plus @ num @ M @ N ) ) ) ).
% semiring_norm(6)
thf(fact_146_semiring__norm_I2_J,axiom,
( ( plus_plus @ num @ one2 @ one2 )
= ( bit0 @ one2 ) ) ).
% semiring_norm(2)
thf(fact_147_triangle__Suc,axiom,
! [N: nat] :
( ( nat_triangle @ ( suc @ N ) )
= ( plus_plus @ nat @ ( nat_triangle @ N ) @ ( suc @ N ) ) ) ).
% triangle_Suc
thf(fact_148_Suc__numeral,axiom,
! [N: num] :
( ( suc @ ( numeral_numeral @ nat @ N ) )
= ( numeral_numeral @ nat @ ( plus_plus @ num @ N @ one2 ) ) ) ).
% Suc_numeral
thf(fact_149_power__add__numeral2,axiom,
! [A: $tType] :
( ( monoid_mult @ A )
=> ! [A2: A,M: num,N: num,B2: A] :
( ( times_times @ A @ ( power_power @ A @ A2 @ ( numeral_numeral @ nat @ M ) ) @ ( times_times @ A @ ( power_power @ A @ A2 @ ( numeral_numeral @ nat @ N ) ) @ B2 ) )
= ( times_times @ A @ ( power_power @ A @ A2 @ ( numeral_numeral @ nat @ ( plus_plus @ num @ M @ N ) ) ) @ B2 ) ) ) ).
% power_add_numeral2
thf(fact_150_power__add__numeral,axiom,
! [A: $tType] :
( ( monoid_mult @ A )
=> ! [A2: A,M: num,N: num] :
( ( times_times @ A @ ( power_power @ A @ A2 @ ( numeral_numeral @ nat @ M ) ) @ ( power_power @ A @ A2 @ ( numeral_numeral @ nat @ N ) ) )
= ( power_power @ A @ A2 @ ( numeral_numeral @ nat @ ( plus_plus @ num @ M @ N ) ) ) ) ) ).
% power_add_numeral
thf(fact_151_add__One__commute,axiom,
! [N: num] :
( ( plus_plus @ num @ one2 @ N )
= ( plus_plus @ num @ N @ one2 ) ) ).
% add_One_commute
thf(fact_152_n__not__Suc__n,axiom,
! [N: nat] :
( N
!= ( suc @ N ) ) ).
% n_not_Suc_n
thf(fact_153_Suc__inject,axiom,
! [X: nat,Y: nat] :
( ( ( suc @ X )
= ( suc @ Y ) )
=> ( X = Y ) ) ).
% Suc_inject
thf(fact_154_gcd__nat_Oasym,axiom,
! [A2: nat,B2: nat] :
( ( ( dvd_dvd @ nat @ A2 @ B2 )
& ( A2 != B2 ) )
=> ~ ( ( dvd_dvd @ nat @ B2 @ A2 )
& ( B2 != A2 ) ) ) ).
% gcd_nat.asym
thf(fact_155_gcd__nat_Orefl,axiom,
! [A2: nat] : ( dvd_dvd @ nat @ A2 @ A2 ) ).
% gcd_nat.refl
thf(fact_156_gcd__nat_Otrans,axiom,
! [A2: nat,B2: nat,C2: nat] :
( ( dvd_dvd @ nat @ A2 @ B2 )
=> ( ( dvd_dvd @ nat @ B2 @ C2 )
=> ( dvd_dvd @ nat @ A2 @ C2 ) ) ) ).
% gcd_nat.trans
thf(fact_157_gcd__nat_Oeq__iff,axiom,
( ( ^ [Y2: nat,Z2: nat] : Y2 = Z2 )
= ( ^ [A4: nat,B5: nat] :
( ( dvd_dvd @ nat @ A4 @ B5 )
& ( dvd_dvd @ nat @ B5 @ A4 ) ) ) ) ).
% gcd_nat.eq_iff
thf(fact_158_gcd__nat_Oirrefl,axiom,
! [A2: nat] :
~ ( ( dvd_dvd @ nat @ A2 @ A2 )
& ( A2 != A2 ) ) ).
% gcd_nat.irrefl
thf(fact_159_gcd__nat_Oantisym,axiom,
! [A2: nat,B2: nat] :
( ( dvd_dvd @ nat @ A2 @ B2 )
=> ( ( dvd_dvd @ nat @ B2 @ A2 )
=> ( A2 = B2 ) ) ) ).
% gcd_nat.antisym
thf(fact_160_gcd__nat_Ostrict__trans,axiom,
! [A2: nat,B2: nat,C2: nat] :
( ( ( dvd_dvd @ nat @ A2 @ B2 )
& ( A2 != B2 ) )
=> ( ( ( dvd_dvd @ nat @ B2 @ C2 )
& ( B2 != C2 ) )
=> ( ( dvd_dvd @ nat @ A2 @ C2 )
& ( A2 != C2 ) ) ) ) ).
% gcd_nat.strict_trans
thf(fact_161_gcd__nat_Ostrict__trans1,axiom,
! [A2: nat,B2: nat,C2: nat] :
( ( dvd_dvd @ nat @ A2 @ B2 )
=> ( ( ( dvd_dvd @ nat @ B2 @ C2 )
& ( B2 != C2 ) )
=> ( ( dvd_dvd @ nat @ A2 @ C2 )
& ( A2 != C2 ) ) ) ) ).
% gcd_nat.strict_trans1
thf(fact_162_gcd__nat_Ostrict__trans2,axiom,
! [A2: nat,B2: nat,C2: nat] :
( ( ( dvd_dvd @ nat @ A2 @ B2 )
& ( A2 != B2 ) )
=> ( ( dvd_dvd @ nat @ B2 @ C2 )
=> ( ( dvd_dvd @ nat @ A2 @ C2 )
& ( A2 != C2 ) ) ) ) ).
% gcd_nat.strict_trans2
thf(fact_163_gcd__nat_Ostrict__iff__not,axiom,
! [A2: nat,B2: nat] :
( ( ( dvd_dvd @ nat @ A2 @ B2 )
& ( A2 != B2 ) )
= ( ( dvd_dvd @ nat @ A2 @ B2 )
& ~ ( dvd_dvd @ nat @ B2 @ A2 ) ) ) ).
% gcd_nat.strict_iff_not
thf(fact_164_gcd__nat_Oorder__iff__strict,axiom,
( ( dvd_dvd @ nat )
= ( ^ [A4: nat,B5: nat] :
( ( ( dvd_dvd @ nat @ A4 @ B5 )
& ( A4 != B5 ) )
| ( A4 = B5 ) ) ) ) ).
% gcd_nat.order_iff_strict
thf(fact_165_gcd__nat_Ostrict__iff__order,axiom,
! [A2: nat,B2: nat] :
( ( ( dvd_dvd @ nat @ A2 @ B2 )
& ( A2 != B2 ) )
= ( ( dvd_dvd @ nat @ A2 @ B2 )
& ( A2 != B2 ) ) ) ).
% gcd_nat.strict_iff_order
thf(fact_166_gcd__nat_Ostrict__implies__order,axiom,
! [A2: nat,B2: nat] :
( ( ( dvd_dvd @ nat @ A2 @ B2 )
& ( A2 != B2 ) )
=> ( dvd_dvd @ nat @ A2 @ B2 ) ) ).
% gcd_nat.strict_implies_order
thf(fact_167_gcd__nat_Ostrict__implies__not__eq,axiom,
! [A2: nat,B2: nat] :
( ( ( dvd_dvd @ nat @ A2 @ B2 )
& ( A2 != B2 ) )
=> ( A2 != B2 ) ) ).
% gcd_nat.strict_implies_not_eq
thf(fact_168_gcd__nat_Onot__eq__order__implies__strict,axiom,
! [A2: nat,B2: nat] :
( ( A2 != B2 )
=> ( ( dvd_dvd @ nat @ A2 @ B2 )
=> ( ( dvd_dvd @ nat @ A2 @ B2 )
& ( A2 != B2 ) ) ) ) ).
% gcd_nat.not_eq_order_implies_strict
thf(fact_169_dvd__antisym,axiom,
! [M: nat,N: nat] :
( ( dvd_dvd @ nat @ M @ N )
=> ( ( dvd_dvd @ nat @ N @ M )
=> ( M = N ) ) ) ).
% dvd_antisym
thf(fact_170_Suc__nat__number__of__add,axiom,
! [V: num,N: nat] :
( ( suc @ ( plus_plus @ nat @ ( numeral_numeral @ nat @ V ) @ N ) )
= ( plus_plus @ nat @ ( numeral_numeral @ nat @ ( plus_plus @ num @ V @ one2 ) ) @ N ) ) ).
% Suc_nat_number_of_add
thf(fact_171_division__decomp,axiom,
! [A: $tType] :
( ( semiring_gcd @ A )
=> ! [A2: A,B2: A,C2: A] :
( ( dvd_dvd @ A @ A2 @ ( times_times @ A @ B2 @ C2 ) )
=> ? [B6: A,C3: A] :
( ( A2
= ( times_times @ A @ B6 @ C3 ) )
& ( dvd_dvd @ A @ B6 @ B2 )
& ( dvd_dvd @ A @ C3 @ C2 ) ) ) ) ).
% division_decomp
thf(fact_172_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_173_add__Suc__shift,axiom,
! [M: nat,N: nat] :
( ( plus_plus @ nat @ ( suc @ M ) @ N )
= ( plus_plus @ nat @ M @ ( suc @ N ) ) ) ).
% add_Suc_shift
thf(fact_174_add__Suc,axiom,
! [M: nat,N: nat] :
( ( plus_plus @ nat @ ( suc @ M ) @ N )
= ( suc @ ( plus_plus @ nat @ M @ N ) ) ) ).
% add_Suc
thf(fact_175_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_176_Suc__mult__cancel1,axiom,
! [K: nat,M: nat,N: nat] :
( ( ( times_times @ nat @ ( suc @ K ) @ M )
= ( times_times @ nat @ ( suc @ K ) @ N ) )
= ( M = N ) ) ).
% Suc_mult_cancel1
thf(fact_177_add__mult__distrib2,axiom,
! [K: nat,M: nat,N: nat] :
( ( times_times @ nat @ K @ ( plus_plus @ nat @ M @ N ) )
= ( plus_plus @ nat @ ( times_times @ nat @ K @ M ) @ ( times_times @ nat @ K @ N ) ) ) ).
% add_mult_distrib2
thf(fact_178_add__mult__distrib,axiom,
! [M: nat,N: nat,K: nat] :
( ( times_times @ nat @ ( plus_plus @ nat @ M @ N ) @ K )
= ( plus_plus @ nat @ ( times_times @ nat @ M @ K ) @ ( times_times @ nat @ N @ K ) ) ) ).
% add_mult_distrib
thf(fact_179_high__inv,axiom,
! [X: nat,N: nat,Y: nat] :
( ( ord_less @ nat @ X @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N ) )
=> ( ( vEBT_VEBT_high @ ( plus_plus @ nat @ ( times_times @ nat @ Y @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N ) ) @ X ) @ N )
= Y ) ) ).
% high_inv
thf(fact_180_high__bound__aux,axiom,
! [Ma: nat,N: nat,M: nat] :
( ( ord_less @ nat @ Ma @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( plus_plus @ nat @ N @ M ) ) )
=> ( ord_less @ nat @ ( vEBT_VEBT_high @ Ma @ N ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ M ) ) ) ).
% high_bound_aux
thf(fact_181_VEBT__internal_OT__vebt__buildupi_Oelims,axiom,
! [X: nat,Y: nat] :
( ( ( vEBT_V441764108873111860ildupi @ X )
= Y )
=> ( ( ( X
= ( zero_zero @ nat ) )
=> ( Y
!= ( suc @ ( zero_zero @ nat ) ) ) )
=> ( ( ( X
= ( suc @ ( zero_zero @ nat ) ) )
=> ( Y
!= ( suc @ ( zero_zero @ nat ) ) ) )
=> ~ ! [N2: nat] :
( ( X
= ( suc @ ( suc @ N2 ) ) )
=> ~ ( ( ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N2 )
=> ( Y
= ( suc @ ( suc @ ( suc @ ( plus_plus @ nat @ ( vEBT_V441764108873111860ildupi @ ( suc @ ( divide_divide @ nat @ N2 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( plus_plus @ nat @ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ ( bit0 @ one2 ) ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ N2 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( times_times @ nat @ ( vEBT_V441764108873111860ildupi @ ( suc @ ( divide_divide @ nat @ N2 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ N2 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) ) ) ) ) ) ) )
& ( ~ ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N2 )
=> ( Y
= ( suc @ ( suc @ ( suc @ ( plus_plus @ nat @ ( vEBT_V441764108873111860ildupi @ ( suc @ ( suc @ ( divide_divide @ nat @ N2 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) @ ( plus_plus @ nat @ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ ( bit0 @ ( bit0 @ one2 ) ) ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ N2 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ ( bit0 @ one2 ) ) ) @ ( times_times @ nat @ ( vEBT_V441764108873111860ildupi @ ( suc @ ( divide_divide @ nat @ N2 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ N2 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).
% VEBT_internal.T_vebt_buildupi.elims
thf(fact_182_power__numeral,axiom,
! [A: $tType] :
( ( semiring_numeral @ A )
=> ! [K: num,L: num] :
( ( power_power @ A @ ( numeral_numeral @ A @ K ) @ ( numeral_numeral @ nat @ L ) )
= ( numeral_numeral @ A @ ( pow @ K @ L ) ) ) ) ).
% power_numeral
thf(fact_183_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_184_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_185_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_186_verit__eq__simplify_I8_J,axiom,
! [X22: num,Y22: num] :
( ( ( bit0 @ X22 )
= ( bit0 @ Y22 ) )
= ( X22 = Y22 ) ) ).
% verit_eq_simplify(8)
thf(fact_187_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_188_lowi__def,axiom,
( vEBT_VEBT_lowi
= ( ^ [X2: nat,N3: nat] : ( heap_Time_return @ nat @ ( modulo_modulo @ nat @ X2 @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N3 ) ) ) ) ) ).
% lowi_def
thf(fact_189_list__assn__mono,axiom,
! [A: $tType,B: $tType,P: A > B > assn,P3: A > B > assn,L: list @ A,L3: list @ B] :
( ! [X3: A,X4: B] : ( entails @ ( P @ X3 @ X4 ) @ ( P3 @ X3 @ X4 ) )
=> ( entails @ ( vEBT_List_list_assn @ A @ B @ P @ L @ L3 ) @ ( vEBT_List_list_assn @ A @ B @ P3 @ L @ L3 ) ) ) ).
% list_assn_mono
thf(fact_190_buildup__nothing__in__min__max,axiom,
! [N: nat,X: nat] :
~ ( vEBT_VEBT_membermima @ ( vEBT_vebt_buildup @ N ) @ X ) ).
% buildup_nothing_in_min_max
thf(fact_191_power__one__right,axiom,
! [A: $tType] :
( ( monoid_mult @ A )
=> ! [A2: A] :
( ( power_power @ A @ A2 @ ( one_one @ nat ) )
= A2 ) ) ).
% power_one_right
thf(fact_192_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_193_nat__1__eq__mult__iff,axiom,
! [M: nat,N: nat] :
( ( ( one_one @ nat )
= ( times_times @ nat @ M @ N ) )
= ( ( M
= ( one_one @ nat ) )
& ( N
= ( one_one @ nat ) ) ) ) ).
% nat_1_eq_mult_iff
thf(fact_194_nat__mult__eq__1__iff,axiom,
! [M: nat,N: nat] :
( ( ( times_times @ nat @ M @ N )
= ( one_one @ nat ) )
= ( ( M
= ( one_one @ nat ) )
& ( N
= ( one_one @ nat ) ) ) ) ).
% nat_mult_eq_1_iff
thf(fact_195_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_196_pure__true,axiom,
( ( pure_assn @ $true )
= ( one_one @ assn ) ) ).
% pure_true
thf(fact_197_pure__assn__eq__emp__iff,axiom,
! [P: $o] :
( ( ( pure_assn @ P )
= ( one_one @ assn ) )
= P ) ).
% pure_assn_eq_emp_iff
thf(fact_198_not__gr__zero,axiom,
! [A: $tType] :
( ( canoni5634975068530333245id_add @ A )
=> ! [N: A] :
( ( ~ ( ord_less @ A @ ( zero_zero @ A ) @ N ) )
= ( N
= ( zero_zero @ A ) ) ) ) ).
% not_gr_zero
thf(fact_199_numeral__less__iff,axiom,
! [A: $tType] :
( ( linord181362715937106298miring @ A )
=> ! [M: num,N: num] :
( ( ord_less @ A @ ( numeral_numeral @ A @ M ) @ ( numeral_numeral @ A @ N ) )
= ( ord_less @ num @ M @ N ) ) ) ).
% numeral_less_iff
thf(fact_200_add__0,axiom,
! [A: $tType] :
( ( monoid_add @ A )
=> ! [A2: A] :
( ( plus_plus @ A @ ( zero_zero @ A ) @ A2 )
= A2 ) ) ).
% add_0
thf(fact_201_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_202_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_203_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_204_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_205_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_206_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_207_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_208_add_Oright__neutral,axiom,
! [A: $tType] :
( ( monoid_add @ A )
=> ! [A2: A] :
( ( plus_plus @ A @ A2 @ ( zero_zero @ A ) )
= A2 ) ) ).
% add.right_neutral
thf(fact_209_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_210_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_211_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_212_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_213_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_214_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_215_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_216_div__0,axiom,
! [A: $tType] :
( ( semidom_divide @ A )
=> ! [A2: A] :
( ( divide_divide @ A @ ( zero_zero @ A ) @ A2 )
= ( zero_zero @ A ) ) ) ).
% div_0
thf(fact_217_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_218_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_219_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_220_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_221_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_222_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_223_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_224_mult_Oright__neutral,axiom,
! [A: $tType] :
( ( monoid_mult @ A )
=> ! [A2: A] :
( ( times_times @ A @ A2 @ ( one_one @ A ) )
= A2 ) ) ).
% mult.right_neutral
thf(fact_225_mult__1,axiom,
! [A: $tType] :
( ( monoid_mult @ A )
=> ! [A2: A] :
( ( times_times @ A @ ( one_one @ A ) @ A2 )
= A2 ) ) ).
% mult_1
thf(fact_226_div__by__1,axiom,
! [A: $tType] :
( ( semidom_divide @ A )
=> ! [A2: A] :
( ( divide_divide @ A @ A2 @ ( one_one @ A ) )
= A2 ) ) ).
% div_by_1
thf(fact_227_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_228_power__one,axiom,
! [A: $tType] :
( ( monoid_mult @ A )
=> ! [N: nat] :
( ( power_power @ A @ ( one_one @ A ) @ N )
= ( one_one @ A ) ) ) ).
% power_one
thf(fact_229_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_230_dvd__0__right,axiom,
! [A: $tType] :
( ( comm_semiring_1 @ A )
=> ! [A2: A] : ( dvd_dvd @ A @ A2 @ ( zero_zero @ A ) ) ) ).
% dvd_0_right
thf(fact_231_lessI,axiom,
! [N: nat] : ( ord_less @ nat @ N @ ( suc @ N ) ) ).
% lessI
thf(fact_232_Suc__mono,axiom,
! [M: nat,N: nat] :
( ( ord_less @ nat @ M @ N )
=> ( ord_less @ nat @ ( suc @ M ) @ ( suc @ N ) ) ) ).
% Suc_mono
thf(fact_233_Suc__less__eq,axiom,
! [M: nat,N: nat] :
( ( ord_less @ nat @ ( suc @ M ) @ ( suc @ N ) )
= ( ord_less @ nat @ M @ N ) ) ).
% Suc_less_eq
thf(fact_234_mod__0,axiom,
! [A: $tType] :
( ( semidom_modulo @ A )
=> ! [A2: A] :
( ( modulo_modulo @ A @ ( zero_zero @ A ) @ A2 )
= ( zero_zero @ A ) ) ) ).
% mod_0
thf(fact_235_mod__by__0,axiom,
! [A: $tType] :
( ( semidom_modulo @ A )
=> ! [A2: A] :
( ( modulo_modulo @ A @ A2 @ ( zero_zero @ A ) )
= A2 ) ) ).
% mod_by_0
thf(fact_236_mod__self,axiom,
! [A: $tType] :
( ( semidom_modulo @ A )
=> ! [A2: A] :
( ( modulo_modulo @ A @ A2 @ A2 )
= ( zero_zero @ A ) ) ) ).
% mod_self
thf(fact_237_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_238_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_239_less__one,axiom,
! [N: nat] :
( ( ord_less @ nat @ N @ ( one_one @ nat ) )
= ( N
= ( zero_zero @ nat ) ) ) ).
% less_one
thf(fact_240_neq0__conv,axiom,
! [N: nat] :
( ( N
!= ( zero_zero @ nat ) )
= ( ord_less @ nat @ ( zero_zero @ nat ) @ N ) ) ).
% neq0_conv
thf(fact_241_less__nat__zero__code,axiom,
! [N: nat] :
~ ( ord_less @ nat @ N @ ( zero_zero @ nat ) ) ).
% less_nat_zero_code
thf(fact_242_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_243_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_244_Nat_Oadd__0__right,axiom,
! [M: nat] :
( ( plus_plus @ nat @ M @ ( zero_zero @ nat ) )
= M ) ).
% Nat.add_0_right
thf(fact_245_add__is__0,axiom,
! [M: nat,N: nat] :
( ( ( plus_plus @ nat @ M @ N )
= ( zero_zero @ nat ) )
= ( ( M
= ( zero_zero @ nat ) )
& ( N
= ( zero_zero @ nat ) ) ) ) ).
% add_is_0
thf(fact_246_nat__add__left__cancel__less,axiom,
! [K: nat,M: nat,N: nat] :
( ( ord_less @ nat @ ( plus_plus @ nat @ K @ M ) @ ( plus_plus @ nat @ K @ N ) )
= ( ord_less @ nat @ M @ N ) ) ).
% nat_add_left_cancel_less
thf(fact_247_mult__is__0,axiom,
! [M: nat,N: nat] :
( ( ( times_times @ nat @ M @ N )
= ( zero_zero @ nat ) )
= ( ( M
= ( zero_zero @ nat ) )
| ( N
= ( zero_zero @ nat ) ) ) ) ).
% mult_is_0
thf(fact_248_mult__0__right,axiom,
! [M: nat] :
( ( times_times @ nat @ M @ ( zero_zero @ nat ) )
= ( zero_zero @ nat ) ) ).
% mult_0_right
thf(fact_249_mult__cancel1,axiom,
! [K: nat,M: nat,N: nat] :
( ( ( times_times @ nat @ K @ M )
= ( times_times @ nat @ K @ N ) )
= ( ( M = N )
| ( K
= ( zero_zero @ nat ) ) ) ) ).
% mult_cancel1
thf(fact_250_mult__cancel2,axiom,
! [M: nat,K: nat,N: nat] :
( ( ( times_times @ nat @ M @ K )
= ( times_times @ nat @ N @ K ) )
= ( ( M = N )
| ( K
= ( zero_zero @ nat ) ) ) ) ).
% mult_cancel2
thf(fact_251_mod__less,axiom,
! [M: nat,N: nat] :
( ( ord_less @ nat @ M @ N )
=> ( ( modulo_modulo @ nat @ M @ N )
= M ) ) ).
% mod_less
thf(fact_252_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_253_triangle__0,axiom,
( ( nat_triangle @ ( zero_zero @ nat ) )
= ( zero_zero @ nat ) ) ).
% triangle_0
thf(fact_254_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_255_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_256_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_257_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_258_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_259_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_260_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_261_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_262_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_263_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_264_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_265_numeral__eq__one__iff,axiom,
! [A: $tType] :
( ( semiring_char_0 @ A )
=> ! [N: num] :
( ( ( numeral_numeral @ A @ N )
= ( one_one @ A ) )
= ( N = one2 ) ) ) ).
% numeral_eq_one_iff
thf(fact_266_one__eq__numeral__iff,axiom,
! [A: $tType] :
( ( semiring_char_0 @ A )
=> ! [N: num] :
( ( ( one_one @ A )
= ( numeral_numeral @ A @ N ) )
= ( one2 = N ) ) ) ).
% one_eq_numeral_iff
thf(fact_267_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_268_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_269_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_270_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_271_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_272_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_273_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_274_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_275_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_276_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_277_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_278_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_279_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_280_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_281_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_282_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_283_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_284_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_285_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_286_power__inject__exp,axiom,
! [A: $tType] :
( ( linordered_semidom @ A )
=> ! [A2: A,M: nat,N: nat] :
( ( ord_less @ A @ ( one_one @ A ) @ A2 )
=> ( ( ( power_power @ A @ A2 @ M )
= ( power_power @ A @ A2 @ N ) )
= ( M = N ) ) ) ) ).
% power_inject_exp
thf(fact_287_power__0__Suc,axiom,
! [A: $tType] :
( ( semiring_1 @ A )
=> ! [N: nat] :
( ( power_power @ A @ ( zero_zero @ A ) @ ( suc @ N ) )
= ( zero_zero @ A ) ) ) ).
% power_0_Suc
thf(fact_288_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_289_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_290_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_291_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_292_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_293_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_294_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_295_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_296_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_297_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_298_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_299_zero__less__Suc,axiom,
! [N: nat] : ( ord_less @ nat @ ( zero_zero @ nat ) @ ( suc @ N ) ) ).
% zero_less_Suc
thf(fact_300_less__Suc0,axiom,
! [N: nat] :
( ( ord_less @ nat @ N @ ( suc @ ( zero_zero @ nat ) ) )
= ( N
= ( zero_zero @ nat ) ) ) ).
% less_Suc0
thf(fact_301_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_302_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_303_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_304_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_305_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_306_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_307_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_308_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_309_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_310_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_311_add__gr__0,axiom,
! [M: nat,N: nat] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ ( plus_plus @ nat @ M @ N ) )
= ( ( ord_less @ nat @ ( zero_zero @ nat ) @ M )
| ( ord_less @ nat @ ( zero_zero @ nat ) @ N ) ) ) ).
% add_gr_0
thf(fact_312_div__by__Suc__0,axiom,
! [M: nat] :
( ( divide_divide @ nat @ M @ ( suc @ ( zero_zero @ nat ) ) )
= M ) ).
% div_by_Suc_0
thf(fact_313_mult__eq__1__iff,axiom,
! [M: nat,N: nat] :
( ( ( times_times @ nat @ M @ N )
= ( suc @ ( zero_zero @ nat ) ) )
= ( ( M
= ( suc @ ( zero_zero @ nat ) ) )
& ( N
= ( suc @ ( zero_zero @ nat ) ) ) ) ) ).
% mult_eq_1_iff
thf(fact_314_one__eq__mult__iff,axiom,
! [M: nat,N: nat] :
( ( ( suc @ ( zero_zero @ nat ) )
= ( times_times @ nat @ M @ N ) )
= ( ( M
= ( suc @ ( zero_zero @ nat ) ) )
& ( N
= ( suc @ ( zero_zero @ nat ) ) ) ) ) ).
% one_eq_mult_iff
thf(fact_315_power__Suc__0,axiom,
! [N: nat] :
( ( power_power @ nat @ ( suc @ ( zero_zero @ nat ) ) @ N )
= ( suc @ ( zero_zero @ nat ) ) ) ).
% power_Suc_0
thf(fact_316_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_317_div__eq__dividend__iff,axiom,
! [M: nat,N: nat] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ M )
=> ( ( ( divide_divide @ nat @ M @ N )
= M )
= ( N
= ( one_one @ nat ) ) ) ) ).
% div_eq_dividend_iff
thf(fact_318_div__less,axiom,
! [M: nat,N: nat] :
( ( ord_less @ nat @ M @ N )
=> ( ( divide_divide @ nat @ M @ N )
= ( zero_zero @ nat ) ) ) ).
% div_less
thf(fact_319_nat__mult__less__cancel__disj,axiom,
! [K: nat,M: nat,N: nat] :
( ( ord_less @ nat @ ( times_times @ nat @ K @ M ) @ ( times_times @ nat @ K @ N ) )
= ( ( ord_less @ nat @ ( zero_zero @ nat ) @ K )
& ( ord_less @ nat @ M @ N ) ) ) ).
% nat_mult_less_cancel_disj
thf(fact_320_nat__0__less__mult__iff,axiom,
! [M: nat,N: nat] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ ( times_times @ nat @ M @ N ) )
= ( ( ord_less @ nat @ ( zero_zero @ nat ) @ M )
& ( ord_less @ nat @ ( zero_zero @ nat ) @ N ) ) ) ).
% nat_0_less_mult_iff
thf(fact_321_mult__less__cancel2,axiom,
! [M: nat,K: nat,N: nat] :
( ( ord_less @ nat @ ( times_times @ nat @ M @ K ) @ ( times_times @ nat @ N @ K ) )
= ( ( ord_less @ nat @ ( zero_zero @ nat ) @ K )
& ( ord_less @ nat @ M @ N ) ) ) ).
% mult_less_cancel2
thf(fact_322_nat__zero__less__power__iff,axiom,
! [X: nat,N: nat] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ ( power_power @ nat @ X @ N ) )
= ( ( ord_less @ nat @ ( zero_zero @ nat ) @ X )
| ( N
= ( zero_zero @ nat ) ) ) ) ).
% nat_zero_less_power_iff
thf(fact_323_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_324_dvd__1__left,axiom,
! [K: nat] : ( dvd_dvd @ nat @ ( suc @ ( zero_zero @ nat ) ) @ K ) ).
% dvd_1_left
thf(fact_325_mod__by__Suc__0,axiom,
! [M: nat] :
( ( modulo_modulo @ nat @ M @ ( suc @ ( zero_zero @ nat ) ) )
= ( zero_zero @ nat ) ) ).
% mod_by_Suc_0
thf(fact_326_nat__mult__div__cancel__disj,axiom,
! [K: nat,M: nat,N: nat] :
( ( ( K
= ( zero_zero @ nat ) )
=> ( ( divide_divide @ nat @ ( times_times @ nat @ K @ M ) @ ( times_times @ nat @ K @ N ) )
= ( zero_zero @ nat ) ) )
& ( ( K
!= ( zero_zero @ nat ) )
=> ( ( divide_divide @ nat @ ( times_times @ nat @ K @ M ) @ ( times_times @ nat @ K @ N ) )
= ( divide_divide @ nat @ M @ N ) ) ) ) ).
% nat_mult_div_cancel_disj
thf(fact_327_nat__mult__dvd__cancel__disj,axiom,
! [K: nat,M: nat,N: nat] :
( ( dvd_dvd @ nat @ ( times_times @ nat @ K @ M ) @ ( times_times @ nat @ K @ N ) )
= ( ( K
= ( zero_zero @ nat ) )
| ( dvd_dvd @ nat @ M @ N ) ) ) ).
% nat_mult_dvd_cancel_disj
thf(fact_328_one__less__numeral__iff,axiom,
! [A: $tType] :
( ( linord181362715937106298miring @ A )
=> ! [N: num] :
( ( ord_less @ A @ ( one_one @ A ) @ ( numeral_numeral @ A @ N ) )
= ( ord_less @ num @ one2 @ N ) ) ) ).
% one_less_numeral_iff
thf(fact_329_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_330_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_331_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_332_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_333_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_334_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_335_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_336_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_337_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_338_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_339_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_340_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_341_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_342_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_343_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_344_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_345_Suc__1,axiom,
( ( suc @ ( one_one @ nat ) )
= ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ).
% Suc_1
thf(fact_346_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_347_power__eq__0__iff,axiom,
! [A: $tType] :
( ( semiri2026040879449505780visors @ A )
=> ! [A2: A,N: nat] :
( ( ( power_power @ A @ A2 @ N )
= ( zero_zero @ A ) )
= ( ( A2
= ( zero_zero @ A ) )
& ( ord_less @ nat @ ( zero_zero @ nat ) @ N ) ) ) ) ).
% power_eq_0_iff
thf(fact_348_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_349_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_350_pow__divides__pow__iff,axiom,
! [A: $tType] :
( ( semiring_gcd @ A )
=> ! [N: nat,A2: A,B2: A] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ N )
=> ( ( dvd_dvd @ A @ ( power_power @ A @ A2 @ N ) @ ( power_power @ A @ B2 @ N ) )
= ( dvd_dvd @ A @ A2 @ B2 ) ) ) ) ).
% pow_divides_pow_iff
thf(fact_351_div__mult__self__is__m,axiom,
! [N: nat,M: nat] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ N )
=> ( ( divide_divide @ nat @ ( times_times @ nat @ M @ N ) @ N )
= M ) ) ).
% div_mult_self_is_m
thf(fact_352_div__mult__self1__is__m,axiom,
! [N: nat,M: nat] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ N )
=> ( ( divide_divide @ nat @ ( times_times @ nat @ N @ M ) @ N )
= M ) ) ).
% div_mult_self1_is_m
thf(fact_353_Suc__times__numeral__mod__eq,axiom,
! [K: num,N: nat] :
( ( ( numeral_numeral @ nat @ K )
!= ( one_one @ nat ) )
=> ( ( modulo_modulo @ nat @ ( suc @ ( times_times @ nat @ ( numeral_numeral @ nat @ K ) @ N ) ) @ ( numeral_numeral @ nat @ K ) )
= ( one_one @ nat ) ) ) ).
% Suc_times_numeral_mod_eq
thf(fact_354_Suc__mod__mult__self1,axiom,
! [M: nat,K: nat,N: nat] :
( ( modulo_modulo @ nat @ ( suc @ ( plus_plus @ nat @ M @ ( times_times @ nat @ K @ N ) ) ) @ N )
= ( modulo_modulo @ nat @ ( suc @ M ) @ N ) ) ).
% Suc_mod_mult_self1
thf(fact_355_Suc__mod__mult__self2,axiom,
! [M: nat,N: nat,K: nat] :
( ( modulo_modulo @ nat @ ( suc @ ( plus_plus @ nat @ M @ ( times_times @ nat @ N @ K ) ) ) @ N )
= ( modulo_modulo @ nat @ ( suc @ M ) @ N ) ) ).
% Suc_mod_mult_self2
thf(fact_356_Suc__mod__mult__self3,axiom,
! [K: nat,N: nat,M: nat] :
( ( modulo_modulo @ nat @ ( suc @ ( plus_plus @ nat @ ( times_times @ nat @ K @ N ) @ M ) ) @ N )
= ( modulo_modulo @ nat @ ( suc @ M ) @ N ) ) ).
% Suc_mod_mult_self3
thf(fact_357_Suc__mod__mult__self4,axiom,
! [N: nat,K: nat,M: nat] :
( ( modulo_modulo @ nat @ ( suc @ ( plus_plus @ nat @ ( times_times @ nat @ N @ K ) @ M ) ) @ N )
= ( modulo_modulo @ nat @ ( suc @ M ) @ N ) ) ).
% Suc_mod_mult_self4
thf(fact_358_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_359_power__strict__decreasing__iff,axiom,
! [A: $tType] :
( ( linordered_semidom @ A )
=> ! [B2: A,M: nat,N: nat] :
( ( ord_less @ A @ ( zero_zero @ A ) @ B2 )
=> ( ( ord_less @ A @ B2 @ ( one_one @ A ) )
=> ( ( ord_less @ A @ ( power_power @ A @ B2 @ M ) @ ( power_power @ A @ B2 @ N ) )
= ( ord_less @ nat @ N @ M ) ) ) ) ) ).
% power_strict_decreasing_iff
thf(fact_360_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_361_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_362_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_363_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_364_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_365_numeral__plus__one,axiom,
! [A: $tType] :
( ( numeral @ A )
=> ! [N: num] :
( ( plus_plus @ A @ ( numeral_numeral @ A @ N ) @ ( one_one @ A ) )
= ( numeral_numeral @ A @ ( plus_plus @ num @ N @ one2 ) ) ) ) ).
% numeral_plus_one
thf(fact_366_one__plus__numeral,axiom,
! [A: $tType] :
( ( numeral @ A )
=> ! [N: num] :
( ( plus_plus @ A @ ( one_one @ A ) @ ( numeral_numeral @ A @ N ) )
= ( numeral_numeral @ A @ ( plus_plus @ num @ one2 @ N ) ) ) ) ).
% one_plus_numeral
thf(fact_367_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_368_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_369_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_370_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_371_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_372_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_373_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_374_not__mod2__eq__Suc__0__eq__0,axiom,
! [N: nat] :
( ( ( modulo_modulo @ nat @ N @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) )
!= ( suc @ ( zero_zero @ nat ) ) )
= ( ( modulo_modulo @ nat @ N @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) )
= ( zero_zero @ nat ) ) ) ).
% not_mod2_eq_Suc_0_eq_0
thf(fact_375_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_376_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_377_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_378_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_379_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_380_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_381_even__power,axiom,
! [A: $tType] :
( ( semiring_parity @ A )
=> ! [A2: A,N: nat] :
( ( dvd_dvd @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ ( power_power @ A @ A2 @ N ) )
= ( ( dvd_dvd @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ A2 )
& ( ord_less @ nat @ ( zero_zero @ nat ) @ N ) ) ) ) ).
% even_power
thf(fact_382_power__less__zero__eq,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [A2: A,N: nat] :
( ( ord_less @ A @ ( power_power @ A @ A2 @ N ) @ ( zero_zero @ A ) )
= ( ~ ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N )
& ( ord_less @ A @ A2 @ ( zero_zero @ A ) ) ) ) ) ).
% power_less_zero_eq
thf(fact_383_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_384_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_385_even__succ__div__exp,axiom,
! [A: $tType] :
( ( bit_semiring_bits @ A )
=> ! [A2: A,N: nat] :
( ( dvd_dvd @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ A2 )
=> ( ( ord_less @ nat @ ( zero_zero @ nat ) @ N )
=> ( ( divide_divide @ A @ ( plus_plus @ A @ ( one_one @ A ) @ A2 ) @ ( power_power @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ N ) )
= ( divide_divide @ A @ A2 @ ( power_power @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ N ) ) ) ) ) ) ).
% even_succ_div_exp
thf(fact_386_even__succ__mod__exp,axiom,
! [A: $tType] :
( ( bit_semiring_bits @ A )
=> ! [A2: A,N: nat] :
( ( dvd_dvd @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ A2 )
=> ( ( ord_less @ nat @ ( zero_zero @ nat ) @ N )
=> ( ( modulo_modulo @ A @ ( plus_plus @ A @ ( one_one @ A ) @ A2 ) @ ( power_power @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ N ) )
= ( plus_plus @ A @ ( one_one @ A ) @ ( modulo_modulo @ A @ A2 @ ( power_power @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ N ) ) ) ) ) ) ) ).
% even_succ_mod_exp
thf(fact_387_mod__eq__0D,axiom,
! [M: nat,D2: nat] :
( ( ( modulo_modulo @ nat @ M @ D2 )
= ( zero_zero @ nat ) )
=> ? [Q5: nat] :
( M
= ( times_times @ nat @ D2 @ Q5 ) ) ) ).
% mod_eq_0D
thf(fact_388_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_389_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_390_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_391_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_392_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_393_bot__nat__0_Oextremum__strict,axiom,
! [A2: nat] :
~ ( ord_less @ nat @ A2 @ ( zero_zero @ nat ) ) ).
% bot_nat_0.extremum_strict
thf(fact_394_gr0I,axiom,
! [N: nat] :
( ( N
!= ( zero_zero @ nat ) )
=> ( ord_less @ nat @ ( zero_zero @ nat ) @ N ) ) ).
% gr0I
thf(fact_395_not__gr0,axiom,
! [N: nat] :
( ( ~ ( ord_less @ nat @ ( zero_zero @ nat ) @ N ) )
= ( N
= ( zero_zero @ nat ) ) ) ).
% not_gr0
thf(fact_396_not__less0,axiom,
! [N: nat] :
~ ( ord_less @ nat @ N @ ( zero_zero @ nat ) ) ).
% not_less0
thf(fact_397_less__zeroE,axiom,
! [N: nat] :
~ ( ord_less @ nat @ N @ ( zero_zero @ nat ) ) ).
% less_zeroE
thf(fact_398_nat__neq__iff,axiom,
! [M: nat,N: nat] :
( ( M != N )
= ( ( ord_less @ nat @ M @ N )
| ( ord_less @ nat @ N @ M ) ) ) ).
% nat_neq_iff
thf(fact_399_less__not__refl,axiom,
! [N: nat] :
~ ( ord_less @ nat @ N @ N ) ).
% less_not_refl
thf(fact_400_gcd__nat__induct,axiom,
! [P: nat > nat > $o,M: nat,N: nat] :
( ! [M2: nat] : ( P @ M2 @ ( zero_zero @ nat ) )
=> ( ! [M2: nat,N2: nat] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ N2 )
=> ( ( P @ N2 @ ( modulo_modulo @ nat @ M2 @ N2 ) )
=> ( P @ M2 @ N2 ) ) )
=> ( P @ M @ N ) ) ) ).
% gcd_nat_induct
thf(fact_401_less__not__refl2,axiom,
! [N: nat,M: nat] :
( ( ord_less @ nat @ N @ M )
=> ( M != N ) ) ).
% less_not_refl2
thf(fact_402_less__not__refl3,axiom,
! [S: nat,T: nat] :
( ( ord_less @ nat @ S @ T )
=> ( S != T ) ) ).
% less_not_refl3
thf(fact_403_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_404_gr__implies__not0,axiom,
! [M: nat,N: nat] :
( ( ord_less @ nat @ M @ N )
=> ( N
!= ( zero_zero @ nat ) ) ) ).
% gr_implies_not0
thf(fact_405_less__irrefl__nat,axiom,
! [N: nat] :
~ ( ord_less @ nat @ N @ N ) ).
% less_irrefl_nat
thf(fact_406_nat__less__induct,axiom,
! [P: nat > $o,N: nat] :
( ! [N2: nat] :
( ! [M3: nat] :
( ( ord_less @ nat @ M3 @ N2 )
=> ( P @ M3 ) )
=> ( P @ N2 ) )
=> ( P @ N ) ) ).
% nat_less_induct
thf(fact_407_infinite__descent,axiom,
! [P: nat > $o,N: nat] :
( ! [N2: nat] :
( ~ ( P @ N2 )
=> ? [M3: nat] :
( ( ord_less @ nat @ M3 @ N2 )
& ~ ( P @ M3 ) ) )
=> ( P @ N ) ) ).
% infinite_descent
thf(fact_408_infinite__descent0,axiom,
! [P: nat > $o,N: nat] :
( ( P @ ( zero_zero @ nat ) )
=> ( ! [N2: nat] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ N2 )
=> ( ~ ( P @ N2 )
=> ? [M3: nat] :
( ( ord_less @ nat @ M3 @ N2 )
& ~ ( P @ M3 ) ) ) )
=> ( P @ N ) ) ) ).
% infinite_descent0
thf(fact_409_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_410_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_411_nat__induct__non__zero,axiom,
! [N: nat,P: nat > $o] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ N )
=> ( ( P @ ( one_one @ nat ) )
=> ( ! [N2: nat] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ N2 )
=> ( ( P @ N2 )
=> ( P @ ( suc @ N2 ) ) ) )
=> ( P @ N ) ) ) ) ).
% nat_induct_non_zero
thf(fact_412_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_413_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_414_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_415_power__strict__increasing,axiom,
! [A: $tType] :
( ( linordered_semidom @ A )
=> ! [N: nat,N4: nat,A2: A] :
( ( ord_less @ nat @ N @ N4 )
=> ( ( ord_less @ A @ ( one_one @ A ) @ A2 )
=> ( ord_less @ A @ ( power_power @ A @ A2 @ N ) @ ( power_power @ A @ A2 @ N4 ) ) ) ) ) ).
% power_strict_increasing
thf(fact_416_power__strict__decreasing,axiom,
! [A: $tType] :
( ( linordered_semidom @ A )
=> ! [N: nat,N4: nat,A2: A] :
( ( ord_less @ nat @ N @ N4 )
=> ( ( ord_less @ A @ ( zero_zero @ A ) @ A2 )
=> ( ( ord_less @ A @ A2 @ ( one_one @ A ) )
=> ( ord_less @ A @ ( power_power @ A @ A2 @ N4 ) @ ( power_power @ A @ A2 @ N ) ) ) ) ) ) ).
% power_strict_decreasing
thf(fact_417_power__less__imp__less__exp,axiom,
! [A: $tType] :
( ( linordered_semidom @ A )
=> ! [A2: A,M: nat,N: nat] :
( ( ord_less @ A @ ( one_one @ A ) @ A2 )
=> ( ( ord_less @ A @ ( power_power @ A @ A2 @ M ) @ ( power_power @ A @ A2 @ N ) )
=> ( ord_less @ nat @ M @ N ) ) ) ) ).
% power_less_imp_less_exp
thf(fact_418_one__less__power,axiom,
! [A: $tType] :
( ( linordered_semidom @ A )
=> ! [A2: A,N: nat] :
( ( ord_less @ A @ ( one_one @ A ) @ A2 )
=> ( ( ord_less @ nat @ ( zero_zero @ nat ) @ N )
=> ( ord_less @ A @ ( one_one @ A ) @ ( power_power @ A @ A2 @ N ) ) ) ) ) ).
% one_less_power
thf(fact_419_power__0__left,axiom,
! [A: $tType] :
( ( semiring_1 @ A )
=> ! [N: nat] :
( ( ( N
= ( zero_zero @ nat ) )
=> ( ( power_power @ A @ ( zero_zero @ A ) @ N )
= ( one_one @ A ) ) )
& ( ( N
!= ( zero_zero @ nat ) )
=> ( ( power_power @ A @ ( zero_zero @ A ) @ N )
= ( zero_zero @ A ) ) ) ) ) ).
% power_0_left
thf(fact_420_zero__power,axiom,
! [A: $tType] :
( ( semiring_1 @ A )
=> ! [N: nat] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ N )
=> ( ( power_power @ A @ ( zero_zero @ A ) @ N )
= ( zero_zero @ A ) ) ) ) ).
% zero_power
thf(fact_421_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_422_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_423_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_424_linordered__field__no__ub,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [X5: A] :
? [X_1: A] : ( ord_less @ A @ X5 @ X_1 ) ) ).
% linordered_field_no_ub
thf(fact_425_div__less__mono,axiom,
! [A3: nat,B4: nat,N: nat] :
( ( ord_less @ nat @ A3 @ B4 )
=> ( ( ord_less @ nat @ ( zero_zero @ nat ) @ N )
=> ( ( ( modulo_modulo @ nat @ A3 @ N )
= ( zero_zero @ nat ) )
=> ( ( ( modulo_modulo @ nat @ B4 @ N )
= ( zero_zero @ nat ) )
=> ( ord_less @ nat @ ( divide_divide @ nat @ A3 @ N ) @ ( divide_divide @ nat @ B4 @ N ) ) ) ) ) ) ).
% div_less_mono
thf(fact_426_mod__less__divisor,axiom,
! [N: nat,M: nat] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ N )
=> ( ord_less @ nat @ ( modulo_modulo @ nat @ M @ N ) @ N ) ) ).
% mod_less_divisor
thf(fact_427_div__less__dividend,axiom,
! [N: nat,M: nat] :
( ( ord_less @ nat @ ( one_one @ nat ) @ N )
=> ( ( ord_less @ nat @ ( zero_zero @ nat ) @ M )
=> ( ord_less @ nat @ ( divide_divide @ nat @ M @ N ) @ M ) ) ) ).
% div_less_dividend
thf(fact_428_zero__reorient,axiom,
! [A: $tType] :
( ( zero @ A )
=> ! [X: A] :
( ( ( zero_zero @ A )
= X )
= ( X
= ( zero_zero @ A ) ) ) ) ).
% zero_reorient
thf(fact_429_one__reorient,axiom,
! [A: $tType] :
( ( one @ A )
=> ! [X: A] :
( ( ( one_one @ A )
= X )
= ( X
= ( one_one @ A ) ) ) ) ).
% one_reorient
thf(fact_430_zero__less__iff__neq__zero,axiom,
! [A: $tType] :
( ( canoni5634975068530333245id_add @ A )
=> ! [N: A] :
( ( ord_less @ A @ ( zero_zero @ A ) @ N )
= ( N
!= ( zero_zero @ A ) ) ) ) ).
% zero_less_iff_neq_zero
thf(fact_431_gr__implies__not__zero,axiom,
! [A: $tType] :
( ( canoni5634975068530333245id_add @ A )
=> ! [M: A,N: A] :
( ( ord_less @ A @ M @ N )
=> ( N
!= ( zero_zero @ A ) ) ) ) ).
% gr_implies_not_zero
thf(fact_432_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_433_not__less__zero,axiom,
! [A: $tType] :
( ( canoni5634975068530333245id_add @ A )
=> ! [N: A] :
~ ( ord_less @ A @ N @ ( zero_zero @ A ) ) ) ).
% not_less_zero
thf(fact_434_gr__zeroI,axiom,
! [A: $tType] :
( ( canoni5634975068530333245id_add @ A )
=> ! [N: A] :
( ( N
!= ( zero_zero @ A ) )
=> ( ord_less @ A @ ( zero_zero @ A ) @ N ) ) ) ).
% gr_zeroI
thf(fact_435_verit__comp__simplify1_I1_J,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [A2: A] :
~ ( ord_less @ A @ A2 @ A2 ) ) ).
% verit_comp_simplify1(1)
thf(fact_436_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_437_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_438_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_439_zero__neq__one,axiom,
! [A: $tType] :
( ( zero_neq_one @ A )
=> ( ( zero_zero @ A )
!= ( one_one @ A ) ) ) ).
% zero_neq_one
thf(fact_440_zero__less__one,axiom,
! [A: $tType] :
( ( zero_less_one @ A )
=> ( ord_less @ A @ ( zero_zero @ A ) @ ( one_one @ A ) ) ) ).
% zero_less_one
thf(fact_441_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_442_not__one__less__zero,axiom,
! [A: $tType] :
( ( linord181362715937106298miring @ A )
=> ~ ( ord_less @ A @ ( one_one @ A ) @ ( zero_zero @ A ) ) ) ).
% not_one_less_zero
thf(fact_443_mod__greater__zero__iff__not__dvd,axiom,
! [M: nat,N: nat] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ ( modulo_modulo @ nat @ M @ N ) )
= ( ~ ( dvd_dvd @ nat @ N @ M ) ) ) ).
% mod_greater_zero_iff_not_dvd
thf(fact_444_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_445_Suc__times__mod__eq,axiom,
! [M: nat,N: nat] :
( ( ord_less @ nat @ ( suc @ ( zero_zero @ nat ) ) @ M )
=> ( ( modulo_modulo @ nat @ ( suc @ ( times_times @ nat @ M @ N ) ) @ M )
= ( one_one @ nat ) ) ) ).
% Suc_times_mod_eq
thf(fact_446_mod__induct,axiom,
! [P: nat > $o,N: nat,P4: nat,M: nat] :
( ( P @ N )
=> ( ( ord_less @ nat @ N @ P4 )
=> ( ( ord_less @ nat @ M @ P4 )
=> ( ! [N2: nat] :
( ( ord_less @ nat @ N2 @ P4 )
=> ( ( P @ N2 )
=> ( P @ ( modulo_modulo @ nat @ ( suc @ N2 ) @ P4 ) ) ) )
=> ( P @ M ) ) ) ) ) ).
% mod_induct
thf(fact_447_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_448_not__numeral__less__one,axiom,
! [A: $tType] :
( ( linord181362715937106298miring @ A )
=> ! [N: num] :
~ ( ord_less @ A @ ( numeral_numeral @ A @ N ) @ ( one_one @ A ) ) ) ).
% not_numeral_less_one
thf(fact_449_lift__Suc__mono__less__iff,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [F2: nat > A,N: nat,M: nat] :
( ! [N2: nat] : ( ord_less @ A @ ( F2 @ N2 ) @ ( F2 @ ( suc @ N2 ) ) )
=> ( ( ord_less @ A @ ( F2 @ N ) @ ( F2 @ M ) )
= ( ord_less @ nat @ N @ M ) ) ) ) ).
% lift_Suc_mono_less_iff
thf(fact_450_lift__Suc__mono__less,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [F2: nat > A,N: nat,N5: nat] :
( ! [N2: nat] : ( ord_less @ A @ ( F2 @ N2 ) @ ( F2 @ ( suc @ N2 ) ) )
=> ( ( ord_less @ nat @ N @ N5 )
=> ( ord_less @ A @ ( F2 @ N ) @ ( F2 @ N5 ) ) ) ) ) ).
% lift_Suc_mono_less
thf(fact_451_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_452_dvd__eq__mod__eq__0,axiom,
! [A: $tType] :
( ( semidom_modulo @ A )
=> ( ( dvd_dvd @ A )
= ( ^ [A4: A,B5: A] :
( ( modulo_modulo @ A @ B5 @ A4 )
= ( zero_zero @ A ) ) ) ) ) ).
% dvd_eq_mod_eq_0
thf(fact_453_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_454_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_455_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_456_less__1__mult,axiom,
! [A: $tType] :
( ( linordered_semidom @ A )
=> ! [M: A,N: A] :
( ( ord_less @ A @ ( one_one @ A ) @ M )
=> ( ( ord_less @ A @ ( one_one @ A ) @ N )
=> ( ord_less @ A @ ( one_one @ A ) @ ( times_times @ A @ M @ N ) ) ) ) ) ).
% less_1_mult
thf(fact_457_mod__Suc,axiom,
! [M: nat,N: nat] :
( ( ( ( suc @ ( modulo_modulo @ nat @ M @ N ) )
= N )
=> ( ( modulo_modulo @ nat @ ( suc @ M ) @ N )
= ( zero_zero @ nat ) ) )
& ( ( ( suc @ ( modulo_modulo @ nat @ M @ N ) )
!= N )
=> ( ( modulo_modulo @ nat @ ( suc @ M ) @ N )
= ( suc @ ( modulo_modulo @ nat @ M @ N ) ) ) ) ) ).
% mod_Suc
thf(fact_458_not__numeral__less__zero,axiom,
! [A: $tType] :
( ( linord181362715937106298miring @ A )
=> ! [N: num] :
~ ( ord_less @ A @ ( numeral_numeral @ A @ N ) @ ( zero_zero @ A ) ) ) ).
% not_numeral_less_zero
thf(fact_459_zero__less__numeral,axiom,
! [A: $tType] :
( ( linord181362715937106298miring @ A )
=> ! [N: num] : ( ord_less @ A @ ( zero_zero @ A ) @ ( numeral_numeral @ A @ N ) ) ) ).
% zero_less_numeral
thf(fact_460_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_461_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_462_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_463_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_464_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_465_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_466_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_467_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_468_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_469_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_470_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_471_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_472_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_473_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_474_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_475_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_476_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_477_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_478_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_479_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_480_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_481_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_482_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_483_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_484_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_485_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_486_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_487_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_488_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_489_zero__less__power,axiom,
! [A: $tType] :
( ( linordered_semidom @ A )
=> ! [A2: A,N: nat] :
( ( ord_less @ A @ ( zero_zero @ A ) @ A2 )
=> ( ord_less @ A @ ( zero_zero @ A ) @ ( power_power @ A @ A2 @ N ) ) ) ) ).
% zero_less_power
thf(fact_490_less__Suc__eq__0__disj,axiom,
! [M: nat,N: nat] :
( ( ord_less @ nat @ M @ ( suc @ N ) )
= ( ( M
= ( zero_zero @ nat ) )
| ? [J2: nat] :
( ( M
= ( suc @ J2 ) )
& ( ord_less @ nat @ J2 @ N ) ) ) ) ).
% less_Suc_eq_0_disj
thf(fact_491_gr0__implies__Suc,axiom,
! [N: nat] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ N )
=> ? [M2: nat] :
( N
= ( suc @ M2 ) ) ) ).
% gr0_implies_Suc
thf(fact_492_All__less__Suc2,axiom,
! [N: nat,P: nat > $o] :
( ( ! [I2: nat] :
( ( ord_less @ nat @ I2 @ ( suc @ N ) )
=> ( P @ I2 ) ) )
= ( ( P @ ( zero_zero @ nat ) )
& ! [I2: nat] :
( ( ord_less @ nat @ I2 @ N )
=> ( P @ ( suc @ I2 ) ) ) ) ) ).
% All_less_Suc2
thf(fact_493_gr0__conv__Suc,axiom,
! [N: nat] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ N )
= ( ? [M4: nat] :
( N
= ( suc @ M4 ) ) ) ) ).
% gr0_conv_Suc
thf(fact_494_Ex__less__Suc2,axiom,
! [N: nat,P: nat > $o] :
( ( ? [I2: nat] :
( ( ord_less @ nat @ I2 @ ( suc @ N ) )
& ( P @ I2 ) ) )
= ( ( P @ ( zero_zero @ nat ) )
| ? [I2: nat] :
( ( ord_less @ nat @ I2 @ N )
& ( P @ ( suc @ I2 ) ) ) ) ) ).
% Ex_less_Suc2
thf(fact_495_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_496_power__0,axiom,
! [A: $tType] :
( ( power @ A )
=> ! [A2: A] :
( ( power_power @ A @ A2 @ ( zero_zero @ nat ) )
= ( one_one @ A ) ) ) ).
% power_0
thf(fact_497_less__imp__add__positive,axiom,
! [I: nat,J: nat] :
( ( ord_less @ nat @ I @ J )
=> ? [K2: nat] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ K2 )
& ( ( plus_plus @ nat @ I @ K2 )
= J ) ) ) ).
% less_imp_add_positive
thf(fact_498_One__nat__def,axiom,
( ( one_one @ nat )
= ( suc @ ( zero_zero @ nat ) ) ) ).
% One_nat_def
thf(fact_499_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_500_Euclidean__Division_Odiv__eq__0__iff,axiom,
! [M: nat,N: nat] :
( ( ( divide_divide @ nat @ M @ N )
= ( zero_zero @ nat ) )
= ( ( ord_less @ nat @ M @ N )
| ( N
= ( zero_zero @ nat ) ) ) ) ).
% Euclidean_Division.div_eq_0_iff
thf(fact_501_nat__mult__eq__cancel1,axiom,
! [K: nat,M: nat,N: nat] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ K )
=> ( ( ( times_times @ nat @ K @ M )
= ( times_times @ nat @ K @ N ) )
= ( M = N ) ) ) ).
% nat_mult_eq_cancel1
thf(fact_502_nat__mult__less__cancel1,axiom,
! [K: nat,M: nat,N: nat] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ K )
=> ( ( ord_less @ nat @ ( times_times @ nat @ K @ M ) @ ( times_times @ nat @ K @ N ) )
= ( ord_less @ nat @ M @ N ) ) ) ).
% nat_mult_less_cancel1
thf(fact_503_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_504_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_505_nat__power__less__imp__less,axiom,
! [I: nat,M: nat,N: nat] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ I )
=> ( ( ord_less @ nat @ ( power_power @ nat @ I @ M ) @ ( power_power @ nat @ I @ N ) )
=> ( ord_less @ nat @ M @ N ) ) ) ).
% nat_power_less_imp_less
thf(fact_506_nat__dvd__not__less,axiom,
! [M: nat,N: nat] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ M )
=> ( ( ord_less @ nat @ M @ N )
=> ~ ( dvd_dvd @ nat @ N @ M ) ) ) ).
% nat_dvd_not_less
thf(fact_507_dvd__pos__nat,axiom,
! [N: nat,M: nat] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ N )
=> ( ( dvd_dvd @ nat @ M @ N )
=> ( ord_less @ nat @ ( zero_zero @ nat ) @ M ) ) ) ).
% dvd_pos_nat
thf(fact_508_mult__eq__self__implies__10,axiom,
! [M: nat,N: nat] :
( ( M
= ( times_times @ nat @ M @ N ) )
=> ( ( N
= ( one_one @ nat ) )
| ( M
= ( zero_zero @ nat ) ) ) ) ).
% mult_eq_self_implies_10
thf(fact_509_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_510_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_511_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_512_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_513_add__strict__mono,axiom,
! [A: $tType] :
( ( strict9044650504122735259up_add @ A )
=> ! [A2: A,B2: A,C2: A,D2: A] :
( ( ord_less @ A @ A2 @ B2 )
=> ( ( ord_less @ A @ C2 @ D2 )
=> ( ord_less @ A @ ( plus_plus @ A @ A2 @ C2 ) @ ( plus_plus @ A @ B2 @ D2 ) ) ) ) ) ).
% add_strict_mono
thf(fact_514_add__mono__thms__linordered__field_I1_J,axiom,
! [A: $tType] :
( ( ordere580206878836729694up_add @ A )
=> ! [I: A,J: A,K: A,L: A] :
( ( ( ord_less @ A @ I @ J )
& ( K = L ) )
=> ( ord_less @ A @ ( plus_plus @ A @ I @ K ) @ ( plus_plus @ A @ J @ L ) ) ) ) ).
% add_mono_thms_linordered_field(1)
thf(fact_515_add__mono__thms__linordered__field_I2_J,axiom,
! [A: $tType] :
( ( ordere580206878836729694up_add @ A )
=> ! [I: A,J: A,K: A,L: A] :
( ( ( I = J )
& ( ord_less @ A @ K @ L ) )
=> ( ord_less @ A @ ( plus_plus @ A @ I @ K ) @ ( plus_plus @ A @ J @ L ) ) ) ) ).
% add_mono_thms_linordered_field(2)
thf(fact_516_add__mono__thms__linordered__field_I5_J,axiom,
! [A: $tType] :
( ( ordere580206878836729694up_add @ A )
=> ! [I: A,J: A,K: A,L: A] :
( ( ( ord_less @ A @ I @ J )
& ( ord_less @ A @ K @ L ) )
=> ( ord_less @ A @ ( plus_plus @ A @ I @ K ) @ ( plus_plus @ A @ J @ L ) ) ) ) ).
% add_mono_thms_linordered_field(5)
thf(fact_517_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_518_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_519_verit__sum__simplify,axiom,
! [A: $tType] :
( ( cancel1802427076303600483id_add @ A )
=> ! [A2: A] :
( ( plus_plus @ A @ A2 @ ( zero_zero @ A ) )
= A2 ) ) ).
% verit_sum_simplify
thf(fact_520_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_521_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_522_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_523_power__Suc__less,axiom,
! [A: $tType] :
( ( linordered_semidom @ A )
=> ! [A2: A,N: nat] :
( ( ord_less @ A @ ( zero_zero @ A ) @ A2 )
=> ( ( ord_less @ A @ A2 @ ( one_one @ A ) )
=> ( ord_less @ A @ ( times_times @ A @ A2 @ ( power_power @ A @ A2 @ N ) ) @ ( power_power @ A @ A2 @ N ) ) ) ) ) ).
% power_Suc_less
thf(fact_524_power__Suc__less__one,axiom,
! [A: $tType] :
( ( linordered_semidom @ A )
=> ! [A2: A,N: nat] :
( ( ord_less @ A @ ( zero_zero @ A ) @ A2 )
=> ( ( ord_less @ A @ A2 @ ( one_one @ A ) )
=> ( ord_less @ A @ ( power_power @ A @ A2 @ ( suc @ N ) ) @ ( one_one @ A ) ) ) ) ) ).
% power_Suc_less_one
thf(fact_525_split__mod,axiom,
! [P: nat > $o,M: nat,N: nat] :
( ( P @ ( modulo_modulo @ nat @ M @ N ) )
= ( ( ( N
= ( zero_zero @ nat ) )
=> ( P @ M ) )
& ( ( N
!= ( zero_zero @ nat ) )
=> ! [I2: nat,J2: nat] :
( ( ord_less @ nat @ J2 @ N )
=> ( ( M
= ( plus_plus @ nat @ ( times_times @ nat @ N @ I2 ) @ J2 ) )
=> ( P @ J2 ) ) ) ) ) ) ).
% split_mod
thf(fact_526_dvd__power,axiom,
! [A: $tType] :
( ( comm_semiring_1 @ A )
=> ! [N: nat,X: A] :
( ( ( ord_less @ nat @ ( zero_zero @ nat ) @ N )
| ( X
= ( one_one @ A ) ) )
=> ( dvd_dvd @ A @ X @ ( power_power @ A @ X @ N ) ) ) ) ).
% dvd_power
thf(fact_527_cong__exp__iff__simps_I2_J,axiom,
! [A: $tType] :
( ( unique1627219031080169319umeral @ A )
=> ! [N: num,Q3: num] :
( ( ( modulo_modulo @ A @ ( numeral_numeral @ A @ ( bit0 @ N ) ) @ ( numeral_numeral @ A @ ( bit0 @ Q3 ) ) )
= ( zero_zero @ A ) )
= ( ( modulo_modulo @ A @ ( numeral_numeral @ A @ N ) @ ( numeral_numeral @ A @ Q3 ) )
= ( zero_zero @ A ) ) ) ) ).
% cong_exp_iff_simps(2)
thf(fact_528_cong__exp__iff__simps_I1_J,axiom,
! [A: $tType] :
( ( unique1627219031080169319umeral @ A )
=> ! [N: num] :
( ( modulo_modulo @ A @ ( numeral_numeral @ A @ N ) @ ( numeral_numeral @ A @ one2 ) )
= ( zero_zero @ A ) ) ) ).
% cong_exp_iff_simps(1)
thf(fact_529_dvd__mult__cancel2,axiom,
! [M: nat,N: nat] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ M )
=> ( ( dvd_dvd @ nat @ ( times_times @ nat @ N @ M ) @ M )
= ( N
= ( one_one @ nat ) ) ) ) ).
% dvd_mult_cancel2
thf(fact_530_dvd__mult__cancel1,axiom,
! [M: nat,N: nat] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ M )
=> ( ( dvd_dvd @ nat @ ( times_times @ nat @ M @ N ) @ M )
= ( N
= ( one_one @ nat ) ) ) ) ).
% dvd_mult_cancel1
thf(fact_531_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_532_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_533_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_534_power__gt1__lemma,axiom,
! [A: $tType] :
( ( linordered_semidom @ A )
=> ! [A2: A,N: nat] :
( ( ord_less @ A @ ( one_one @ A ) @ A2 )
=> ( ord_less @ A @ ( one_one @ A ) @ ( times_times @ A @ A2 @ ( power_power @ A @ A2 @ N ) ) ) ) ) ).
% power_gt1_lemma
thf(fact_535_power__less__power__Suc,axiom,
! [A: $tType] :
( ( linordered_semidom @ A )
=> ! [A2: A,N: nat] :
( ( ord_less @ A @ ( one_one @ A ) @ A2 )
=> ( ord_less @ A @ ( power_power @ A @ A2 @ N ) @ ( times_times @ A @ A2 @ ( power_power @ A @ A2 @ N ) ) ) ) ) ).
% power_less_power_Suc
thf(fact_536_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_537_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_538_power__gt1,axiom,
! [A: $tType] :
( ( linordered_semidom @ A )
=> ! [A2: A,N: nat] :
( ( ord_less @ A @ ( one_one @ A ) @ A2 )
=> ( ord_less @ A @ ( one_one @ A ) @ ( power_power @ A @ A2 @ ( suc @ N ) ) ) ) ) ).
% power_gt1
thf(fact_539_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_540_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_541_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_542_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_543_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_544_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_545_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_546_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_547_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_548_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_549_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_550_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_551_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_552_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_553_one__less__mult,axiom,
! [N: nat,M: nat] :
( ( ord_less @ nat @ ( suc @ ( zero_zero @ nat ) ) @ N )
=> ( ( ord_less @ nat @ ( suc @ ( zero_zero @ nat ) ) @ M )
=> ( ord_less @ nat @ ( suc @ ( zero_zero @ nat ) ) @ ( times_times @ nat @ M @ N ) ) ) ) ).
% one_less_mult
thf(fact_554_n__less__m__mult__n,axiom,
! [N: nat,M: nat] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ N )
=> ( ( ord_less @ nat @ ( suc @ ( zero_zero @ nat ) ) @ M )
=> ( ord_less @ nat @ N @ ( times_times @ nat @ M @ N ) ) ) ) ).
% n_less_m_mult_n
thf(fact_555_n__less__n__mult__m,axiom,
! [N: nat,M: nat] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ N )
=> ( ( ord_less @ nat @ ( suc @ ( zero_zero @ nat ) ) @ M )
=> ( ord_less @ nat @ N @ ( times_times @ nat @ N @ M ) ) ) ) ).
% n_less_n_mult_m
thf(fact_556_power__gt__expt,axiom,
! [N: nat,K: nat] :
( ( ord_less @ nat @ ( suc @ ( zero_zero @ nat ) ) @ N )
=> ( ord_less @ nat @ K @ ( power_power @ nat @ N @ K ) ) ) ).
% power_gt_expt
thf(fact_557_is__unit__power__iff,axiom,
! [A: $tType] :
( ( algebraic_semidom @ A )
=> ! [A2: A,N: nat] :
( ( dvd_dvd @ A @ ( power_power @ A @ A2 @ N ) @ ( one_one @ A ) )
= ( ( dvd_dvd @ A @ A2 @ ( one_one @ A ) )
| ( N
= ( zero_zero @ nat ) ) ) ) ) ).
% is_unit_power_iff
thf(fact_558_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_559_mod__add__cong,axiom,
! [A: $tType] :
( ( euclid4440199948858584721cancel @ A )
=> ! [A2: A,C2: A,A5: A,B2: A,B7: A] :
( ( ( modulo_modulo @ A @ A2 @ C2 )
= ( modulo_modulo @ A @ A5 @ C2 ) )
=> ( ( ( modulo_modulo @ A @ B2 @ C2 )
= ( modulo_modulo @ A @ B7 @ C2 ) )
=> ( ( modulo_modulo @ A @ ( plus_plus @ A @ A2 @ B2 ) @ C2 )
= ( modulo_modulo @ A @ ( plus_plus @ A @ A5 @ B7 ) @ C2 ) ) ) ) ) ).
% mod_add_cong
thf(fact_560_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_561_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_562_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_563_mod__mult__cong,axiom,
! [A: $tType] :
( ( euclid4440199948858584721cancel @ A )
=> ! [A2: A,C2: A,A5: A,B2: A,B7: A] :
( ( ( modulo_modulo @ A @ A2 @ C2 )
= ( modulo_modulo @ A @ A5 @ C2 ) )
=> ( ( ( modulo_modulo @ A @ B2 @ C2 )
= ( modulo_modulo @ A @ B7 @ C2 ) )
=> ( ( modulo_modulo @ A @ ( times_times @ A @ A2 @ B2 ) @ C2 )
= ( modulo_modulo @ A @ ( times_times @ A @ A5 @ B7 ) @ C2 ) ) ) ) ) ).
% mod_mult_cong
thf(fact_564_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_565_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_566_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_567_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_568_nat__mult__div__cancel1,axiom,
! [K: nat,M: nat,N: nat] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ K )
=> ( ( divide_divide @ nat @ ( times_times @ nat @ K @ M ) @ ( times_times @ nat @ K @ N ) )
= ( divide_divide @ nat @ M @ N ) ) ) ).
% nat_mult_div_cancel1
thf(fact_569_div__less__iff__less__mult,axiom,
! [Q3: nat,M: nat,N: nat] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ Q3 )
=> ( ( ord_less @ nat @ ( divide_divide @ nat @ M @ Q3 ) @ N )
= ( ord_less @ nat @ M @ ( times_times @ nat @ N @ Q3 ) ) ) ) ).
% div_less_iff_less_mult
thf(fact_570_power__mod,axiom,
! [A: $tType] :
( ( euclid4440199948858584721cancel @ A )
=> ! [A2: A,B2: A,N: nat] :
( ( modulo_modulo @ A @ ( power_power @ A @ ( modulo_modulo @ A @ A2 @ B2 ) @ N ) @ B2 )
= ( modulo_modulo @ A @ ( power_power @ A @ A2 @ N ) @ B2 ) ) ) ).
% power_mod
thf(fact_571_nat__mult__dvd__cancel1,axiom,
! [K: nat,M: nat,N: nat] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ K )
=> ( ( dvd_dvd @ nat @ ( times_times @ nat @ K @ M ) @ ( times_times @ nat @ K @ N ) )
= ( dvd_dvd @ nat @ M @ N ) ) ) ).
% nat_mult_dvd_cancel1
thf(fact_572_dvd__mult__cancel,axiom,
! [K: nat,M: nat,N: nat] :
( ( dvd_dvd @ nat @ ( times_times @ nat @ K @ M ) @ ( times_times @ nat @ K @ N ) )
=> ( ( ord_less @ nat @ ( zero_zero @ nat ) @ K )
=> ( dvd_dvd @ nat @ M @ N ) ) ) ).
% dvd_mult_cancel
thf(fact_573_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_574_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_575_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_576_dvd__mod,axiom,
! [A: $tType] :
( ( euclid4440199948858584721cancel @ A )
=> ! [K: A,M: A,N: A] :
( ( dvd_dvd @ A @ K @ M )
=> ( ( dvd_dvd @ A @ K @ N )
=> ( dvd_dvd @ A @ K @ ( modulo_modulo @ A @ M @ N ) ) ) ) ) ).
% dvd_mod
thf(fact_577_mod__Suc__eq,axiom,
! [M: nat,N: nat] :
( ( modulo_modulo @ nat @ ( suc @ ( modulo_modulo @ nat @ M @ N ) ) @ N )
= ( modulo_modulo @ nat @ ( suc @ M ) @ N ) ) ).
% mod_Suc_eq
thf(fact_578_mod__Suc__Suc__eq,axiom,
! [M: nat,N: nat] :
( ( modulo_modulo @ nat @ ( suc @ ( suc @ ( modulo_modulo @ nat @ M @ N ) ) ) @ N )
= ( modulo_modulo @ nat @ ( suc @ ( suc @ M ) ) @ N ) ) ).
% mod_Suc_Suc_eq
thf(fact_579_Nat_OlessE,axiom,
! [I: nat,K: nat] :
( ( ord_less @ nat @ I @ K )
=> ( ( K
!= ( suc @ I ) )
=> ~ ! [J3: nat] :
( ( ord_less @ nat @ I @ J3 )
=> ( K
!= ( suc @ J3 ) ) ) ) ) ).
% Nat.lessE
thf(fact_580_Suc__lessD,axiom,
! [M: nat,N: nat] :
( ( ord_less @ nat @ ( suc @ M ) @ N )
=> ( ord_less @ nat @ M @ N ) ) ).
% Suc_lessD
thf(fact_581_Suc__lessE,axiom,
! [I: nat,K: nat] :
( ( ord_less @ nat @ ( suc @ I ) @ K )
=> ~ ! [J3: nat] :
( ( ord_less @ nat @ I @ J3 )
=> ( K
!= ( suc @ J3 ) ) ) ) ).
% Suc_lessE
thf(fact_582_Suc__lessI,axiom,
! [M: nat,N: nat] :
( ( ord_less @ nat @ M @ N )
=> ( ( ( suc @ M )
!= N )
=> ( ord_less @ nat @ ( suc @ M ) @ N ) ) ) ).
% Suc_lessI
thf(fact_583_less__SucE,axiom,
! [M: nat,N: nat] :
( ( ord_less @ nat @ M @ ( suc @ N ) )
=> ( ~ ( ord_less @ nat @ M @ N )
=> ( M = N ) ) ) ).
% less_SucE
thf(fact_584_less__SucI,axiom,
! [M: nat,N: nat] :
( ( ord_less @ nat @ M @ N )
=> ( ord_less @ nat @ M @ ( suc @ N ) ) ) ).
% less_SucI
thf(fact_585_Ex__less__Suc,axiom,
! [N: nat,P: nat > $o] :
( ( ? [I2: nat] :
( ( ord_less @ nat @ I2 @ ( suc @ N ) )
& ( P @ I2 ) ) )
= ( ( P @ N )
| ? [I2: nat] :
( ( ord_less @ nat @ I2 @ N )
& ( P @ I2 ) ) ) ) ).
% Ex_less_Suc
thf(fact_586_less__Suc__eq,axiom,
! [M: nat,N: nat] :
( ( ord_less @ nat @ M @ ( suc @ N ) )
= ( ( ord_less @ nat @ M @ N )
| ( M = N ) ) ) ).
% less_Suc_eq
thf(fact_587_not__less__eq,axiom,
! [M: nat,N: nat] :
( ( ~ ( ord_less @ nat @ M @ N ) )
= ( ord_less @ nat @ N @ ( suc @ M ) ) ) ).
% not_less_eq
thf(fact_588_Nat_OAll__less__Suc,axiom,
! [N: nat,P: nat > $o] :
( ( ! [I2: nat] :
( ( ord_less @ nat @ I2 @ ( suc @ N ) )
=> ( P @ I2 ) ) )
= ( ( P @ N )
& ! [I2: nat] :
( ( ord_less @ nat @ I2 @ N )
=> ( P @ I2 ) ) ) ) ).
% Nat.All_less_Suc
thf(fact_589_Suc__less__eq2,axiom,
! [N: nat,M: nat] :
( ( ord_less @ nat @ ( suc @ N ) @ M )
= ( ? [M5: nat] :
( ( M
= ( suc @ M5 ) )
& ( ord_less @ nat @ N @ M5 ) ) ) ) ).
% Suc_less_eq2
thf(fact_590_less__antisym,axiom,
! [N: nat,M: nat] :
( ~ ( ord_less @ nat @ N @ M )
=> ( ( ord_less @ nat @ N @ ( suc @ M ) )
=> ( M = N ) ) ) ).
% less_antisym
thf(fact_591_Suc__less__SucD,axiom,
! [M: nat,N: nat] :
( ( ord_less @ nat @ ( suc @ M ) @ ( suc @ N ) )
=> ( ord_less @ nat @ M @ N ) ) ).
% Suc_less_SucD
thf(fact_592_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_593_less__Suc__induct,axiom,
! [I: nat,J: nat,P: nat > nat > $o] :
( ( ord_less @ nat @ I @ J )
=> ( ! [I3: nat] : ( P @ I3 @ ( suc @ I3 ) )
=> ( ! [I3: nat,J3: nat,K2: nat] :
( ( ord_less @ nat @ I3 @ J3 )
=> ( ( ord_less @ nat @ J3 @ K2 )
=> ( ( P @ I3 @ J3 )
=> ( ( P @ J3 @ K2 )
=> ( P @ I3 @ K2 ) ) ) ) )
=> ( P @ I @ J ) ) ) ) ).
% less_Suc_induct
thf(fact_594_strict__inc__induct,axiom,
! [I: nat,J: nat,P: nat > $o] :
( ( ord_less @ nat @ I @ J )
=> ( ! [I3: nat] :
( ( J
= ( suc @ I3 ) )
=> ( P @ I3 ) )
=> ( ! [I3: nat] :
( ( ord_less @ nat @ I3 @ J )
=> ( ( P @ ( suc @ I3 ) )
=> ( P @ I3 ) ) )
=> ( P @ I ) ) ) ) ).
% strict_inc_induct
thf(fact_595_not__less__less__Suc__eq,axiom,
! [N: nat,M: nat] :
( ~ ( ord_less @ nat @ N @ M )
=> ( ( ord_less @ nat @ N @ ( suc @ M ) )
= ( N = M ) ) ) ).
% not_less_less_Suc_eq
thf(fact_596_less__add__eq__less,axiom,
! [K: nat,L: nat,M: nat,N: nat] :
( ( ord_less @ nat @ K @ L )
=> ( ( ( plus_plus @ nat @ M @ L )
= ( plus_plus @ nat @ K @ N ) )
=> ( ord_less @ nat @ M @ N ) ) ) ).
% less_add_eq_less
thf(fact_597_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_598_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_599_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_600_not__add__less2,axiom,
! [J: nat,I: nat] :
~ ( ord_less @ nat @ ( plus_plus @ nat @ J @ I ) @ I ) ).
% not_add_less2
thf(fact_601_not__add__less1,axiom,
! [I: nat,J: nat] :
~ ( ord_less @ nat @ ( plus_plus @ nat @ I @ J ) @ I ) ).
% not_add_less1
thf(fact_602_add__less__mono,axiom,
! [I: nat,J: nat,K: nat,L: nat] :
( ( ord_less @ nat @ I @ J )
=> ( ( ord_less @ nat @ K @ L )
=> ( ord_less @ nat @ ( plus_plus @ nat @ I @ K ) @ ( plus_plus @ nat @ J @ L ) ) ) ) ).
% add_less_mono
thf(fact_603_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_604_cong__exp__iff__simps_I9_J,axiom,
! [A: $tType] :
( ( unique1627219031080169319umeral @ A )
=> ! [M: num,Q3: num,N: num] :
( ( ( modulo_modulo @ A @ ( numeral_numeral @ A @ ( bit0 @ M ) ) @ ( numeral_numeral @ A @ ( bit0 @ Q3 ) ) )
= ( modulo_modulo @ A @ ( numeral_numeral @ A @ ( bit0 @ N ) ) @ ( numeral_numeral @ A @ ( bit0 @ Q3 ) ) ) )
= ( ( modulo_modulo @ A @ ( numeral_numeral @ A @ M ) @ ( numeral_numeral @ A @ Q3 ) )
= ( modulo_modulo @ A @ ( numeral_numeral @ A @ N ) @ ( numeral_numeral @ A @ Q3 ) ) ) ) ) ).
% cong_exp_iff_simps(9)
thf(fact_605_cong__exp__iff__simps_I4_J,axiom,
! [A: $tType] :
( ( unique1627219031080169319umeral @ A )
=> ! [M: num,N: num] :
( ( modulo_modulo @ A @ ( numeral_numeral @ A @ M ) @ ( numeral_numeral @ A @ one2 ) )
= ( modulo_modulo @ A @ ( numeral_numeral @ A @ N ) @ ( numeral_numeral @ A @ one2 ) ) ) ) ).
% cong_exp_iff_simps(4)
thf(fact_606_zero__neq__numeral,axiom,
! [A: $tType] :
( ( semiring_char_0 @ A )
=> ! [N: num] :
( ( zero_zero @ A )
!= ( numeral_numeral @ A @ N ) ) ) ).
% zero_neq_numeral
thf(fact_607_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_608_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_609_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_610_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_611_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_612_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_613_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_614_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_615_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_616_one__dvd,axiom,
! [A: $tType] :
( ( comm_monoid_mult @ A )
=> ! [A2: A] : ( dvd_dvd @ A @ ( one_one @ A ) @ A2 ) ) ).
% one_dvd
thf(fact_617_semiring__1__no__zero__divisors__class_Opower__not__zero,axiom,
! [A: $tType] :
( ( semiri2026040879449505780visors @ A )
=> ! [A2: A,N: nat] :
( ( A2
!= ( zero_zero @ A ) )
=> ( ( power_power @ A @ A2 @ N )
!= ( zero_zero @ A ) ) ) ) ).
% semiring_1_no_zero_divisors_class.power_not_zero
thf(fact_618_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_619_list__decode_Ocases,axiom,
! [X: nat] :
( ( X
!= ( zero_zero @ nat ) )
=> ~ ! [N2: nat] :
( X
!= ( suc @ N2 ) ) ) ).
% list_decode.cases
thf(fact_620_nat_Odistinct_I1_J,axiom,
! [X22: nat] :
( ( zero_zero @ nat )
!= ( suc @ X22 ) ) ).
% nat.distinct(1)
thf(fact_621_old_Onat_Odistinct_I2_J,axiom,
! [Nat3: nat] :
( ( suc @ Nat3 )
!= ( zero_zero @ nat ) ) ).
% old.nat.distinct(2)
thf(fact_622_old_Onat_Odistinct_I1_J,axiom,
! [Nat2: nat] :
( ( zero_zero @ nat )
!= ( suc @ Nat2 ) ) ).
% old.nat.distinct(1)
thf(fact_623_nat_OdiscI,axiom,
! [Nat: nat,X22: nat] :
( ( Nat
= ( suc @ X22 ) )
=> ( Nat
!= ( zero_zero @ nat ) ) ) ).
% nat.discI
thf(fact_624_old_Onat_Oexhaust,axiom,
! [Y: nat] :
( ( Y
!= ( zero_zero @ nat ) )
=> ~ ! [Nat4: nat] :
( Y
!= ( suc @ Nat4 ) ) ) ).
% old.nat.exhaust
thf(fact_625_nat__induct,axiom,
! [P: nat > $o,N: nat] :
( ( P @ ( zero_zero @ nat ) )
=> ( ! [N2: nat] :
( ( P @ N2 )
=> ( P @ ( suc @ N2 ) ) )
=> ( P @ N ) ) ) ).
% nat_induct
thf(fact_626_diff__induct,axiom,
! [P: nat > nat > $o,M: nat,N: nat] :
( ! [X3: nat] : ( P @ X3 @ ( zero_zero @ nat ) )
=> ( ! [Y3: nat] : ( P @ ( zero_zero @ nat ) @ ( suc @ Y3 ) )
=> ( ! [X3: nat,Y3: nat] :
( ( P @ X3 @ Y3 )
=> ( P @ ( suc @ X3 ) @ ( suc @ Y3 ) ) )
=> ( P @ M @ N ) ) ) ) ).
% diff_induct
thf(fact_627_zero__induct,axiom,
! [P: nat > $o,K: nat] :
( ( P @ K )
=> ( ! [N2: nat] :
( ( P @ ( suc @ N2 ) )
=> ( P @ N2 ) )
=> ( P @ ( zero_zero @ nat ) ) ) ) ).
% zero_induct
thf(fact_628_Suc__neq__Zero,axiom,
! [M: nat] :
( ( suc @ M )
!= ( zero_zero @ nat ) ) ).
% Suc_neq_Zero
thf(fact_629_Zero__neq__Suc,axiom,
! [M: nat] :
( ( zero_zero @ nat )
!= ( suc @ M ) ) ).
% Zero_neq_Suc
thf(fact_630_Zero__not__Suc,axiom,
! [M: nat] :
( ( zero_zero @ nat )
!= ( suc @ M ) ) ).
% Zero_not_Suc
thf(fact_631_not0__implies__Suc,axiom,
! [N: nat] :
( ( N
!= ( zero_zero @ nat ) )
=> ? [M2: nat] :
( N
= ( suc @ M2 ) ) ) ).
% not0_implies_Suc
thf(fact_632_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_633_dvd__field__iff,axiom,
! [A: $tType] :
( ( field @ A )
=> ( ( dvd_dvd @ A )
= ( ^ [A4: A,B5: A] :
( ( A4
= ( zero_zero @ A ) )
=> ( B5
= ( zero_zero @ A ) ) ) ) ) ) ).
% dvd_field_iff
thf(fact_634_nat__mult__1,axiom,
! [N: nat] :
( ( times_times @ nat @ ( one_one @ nat ) @ N )
= N ) ).
% nat_mult_1
thf(fact_635_nat__mult__1__right,axiom,
! [N: nat] :
( ( times_times @ nat @ N @ ( one_one @ nat ) )
= N ) ).
% nat_mult_1_right
thf(fact_636_add__eq__self__zero,axiom,
! [M: nat,N: nat] :
( ( ( plus_plus @ nat @ M @ N )
= M )
=> ( N
= ( zero_zero @ nat ) ) ) ).
% add_eq_self_zero
thf(fact_637_Euclid__induct,axiom,
! [P: nat > nat > $o,A2: nat,B2: nat] :
( ! [A6: nat,B3: nat] :
( ( P @ A6 @ B3 )
= ( P @ B3 @ A6 ) )
=> ( ! [A6: nat] : ( P @ A6 @ ( zero_zero @ nat ) )
=> ( ! [A6: nat,B3: nat] :
( ( P @ A6 @ B3 )
=> ( P @ A6 @ ( plus_plus @ nat @ A6 @ B3 ) ) )
=> ( P @ A2 @ B2 ) ) ) ) ).
% Euclid_induct
thf(fact_638_plus__nat_Oadd__0,axiom,
! [N: nat] :
( ( plus_plus @ nat @ ( zero_zero @ nat ) @ N )
= N ) ).
% plus_nat.add_0
thf(fact_639_assn__one__left,axiom,
! [P: assn] :
( ( times_times @ assn @ ( one_one @ assn ) @ P )
= P ) ).
% assn_one_left
thf(fact_640_nat__mult__eq__cancel__disj,axiom,
! [K: nat,M: nat,N: nat] :
( ( ( times_times @ nat @ K @ M )
= ( times_times @ nat @ K @ N ) )
= ( ( K
= ( zero_zero @ nat ) )
| ( M = N ) ) ) ).
% nat_mult_eq_cancel_disj
thf(fact_641_mult__0,axiom,
! [N: nat] :
( ( times_times @ nat @ ( zero_zero @ nat ) @ N )
= ( zero_zero @ nat ) ) ).
% mult_0
thf(fact_642_nat__mod__eq__iff,axiom,
! [X: nat,N: nat,Y: nat] :
( ( ( modulo_modulo @ nat @ X @ N )
= ( modulo_modulo @ nat @ Y @ N ) )
= ( ? [Q1: nat,Q22: nat] :
( ( plus_plus @ nat @ X @ ( times_times @ nat @ N @ Q1 ) )
= ( plus_plus @ nat @ Y @ ( times_times @ nat @ N @ Q22 ) ) ) ) ) ).
% nat_mod_eq_iff
thf(fact_643_gcd__nat_Oextremum__uniqueI,axiom,
! [A2: nat] :
( ( dvd_dvd @ nat @ ( zero_zero @ nat ) @ A2 )
=> ( A2
= ( zero_zero @ nat ) ) ) ).
% gcd_nat.extremum_uniqueI
thf(fact_644_gcd__nat_Onot__eq__extremum,axiom,
! [A2: nat] :
( ( A2
!= ( zero_zero @ nat ) )
= ( ( dvd_dvd @ nat @ A2 @ ( zero_zero @ nat ) )
& ( A2
!= ( zero_zero @ nat ) ) ) ) ).
% gcd_nat.not_eq_extremum
thf(fact_645_gcd__nat_Oextremum__unique,axiom,
! [A2: nat] :
( ( dvd_dvd @ nat @ ( zero_zero @ nat ) @ A2 )
= ( A2
= ( zero_zero @ nat ) ) ) ).
% gcd_nat.extremum_unique
thf(fact_646_gcd__nat_Oextremum__strict,axiom,
! [A2: nat] :
~ ( ( dvd_dvd @ nat @ ( zero_zero @ nat ) @ A2 )
& ( ( zero_zero @ nat )
!= A2 ) ) ).
% gcd_nat.extremum_strict
thf(fact_647_gcd__nat_Oextremum,axiom,
! [A2: nat] : ( dvd_dvd @ nat @ A2 @ ( zero_zero @ nat ) ) ).
% gcd_nat.extremum
thf(fact_648_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_649_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_650_pos2,axiom,
ord_less @ nat @ ( zero_zero @ nat ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ).
% pos2
thf(fact_651_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_652_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_653_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_654_split__div,axiom,
! [P: nat > $o,M: nat,N: nat] :
( ( P @ ( divide_divide @ nat @ M @ N ) )
= ( ( ( N
= ( zero_zero @ nat ) )
=> ( P @ ( zero_zero @ nat ) ) )
& ( ( N
!= ( zero_zero @ nat ) )
=> ! [I2: nat,J2: nat] :
( ( ord_less @ nat @ J2 @ N )
=> ( ( M
= ( plus_plus @ nat @ ( times_times @ nat @ N @ I2 ) @ J2 ) )
=> ( P @ I2 ) ) ) ) ) ) ).
% split_div
thf(fact_655_dividend__less__div__times,axiom,
! [N: nat,M: nat] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ N )
=> ( ord_less @ nat @ M @ ( plus_plus @ nat @ N @ ( times_times @ nat @ ( divide_divide @ nat @ M @ N ) @ N ) ) ) ) ).
% dividend_less_div_times
thf(fact_656_dividend__less__times__div,axiom,
! [N: nat,M: nat] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ N )
=> ( ord_less @ nat @ M @ ( plus_plus @ nat @ N @ ( times_times @ nat @ N @ ( divide_divide @ nat @ M @ N ) ) ) ) ) ).
% dividend_less_times_div
thf(fact_657_zero__less__power__eq,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [A2: A,N: nat] :
( ( ord_less @ A @ ( zero_zero @ A ) @ ( power_power @ A @ A2 @ N ) )
= ( ( N
= ( zero_zero @ nat ) )
| ( ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N )
& ( A2
!= ( zero_zero @ A ) ) )
| ( ~ ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N )
& ( ord_less @ A @ ( zero_zero @ A ) @ A2 ) ) ) ) ) ).
% zero_less_power_eq
thf(fact_658_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_659_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_660_cong__exp__iff__simps_I8_J,axiom,
! [A: $tType] :
( ( unique1627219031080169319umeral @ A )
=> ! [M: num,Q3: num] :
( ( modulo_modulo @ A @ ( numeral_numeral @ A @ ( bit0 @ M ) ) @ ( numeral_numeral @ A @ ( bit0 @ Q3 ) ) )
!= ( modulo_modulo @ A @ ( numeral_numeral @ A @ one2 ) @ ( numeral_numeral @ A @ ( bit0 @ Q3 ) ) ) ) ) ).
% cong_exp_iff_simps(8)
thf(fact_661_cong__exp__iff__simps_I6_J,axiom,
! [A: $tType] :
( ( unique1627219031080169319umeral @ A )
=> ! [Q3: num,N: num] :
( ( modulo_modulo @ A @ ( numeral_numeral @ A @ one2 ) @ ( numeral_numeral @ A @ ( bit0 @ Q3 ) ) )
!= ( modulo_modulo @ A @ ( numeral_numeral @ A @ ( bit0 @ N ) ) @ ( numeral_numeral @ A @ ( bit0 @ Q3 ) ) ) ) ) ).
% cong_exp_iff_simps(6)
thf(fact_662_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_663_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_664_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_665_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_666_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_667_less__2__cases,axiom,
! [N: nat] :
( ( ord_less @ nat @ N @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) )
=> ( ( N
= ( zero_zero @ nat ) )
| ( N
= ( suc @ ( zero_zero @ nat ) ) ) ) ) ).
% less_2_cases
thf(fact_668_less__2__cases__iff,axiom,
! [N: nat] :
( ( ord_less @ nat @ N @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) )
= ( ( N
= ( zero_zero @ nat ) )
| ( N
= ( suc @ ( zero_zero @ nat ) ) ) ) ) ).
% less_2_cases_iff
thf(fact_669_odd__pos,axiom,
! [N: nat] :
( ~ ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N )
=> ( ord_less @ nat @ ( zero_zero @ nat ) @ N ) ) ).
% odd_pos
thf(fact_670_nat__induct2,axiom,
! [P: nat > $o,N: nat] :
( ( P @ ( zero_zero @ nat ) )
=> ( ( P @ ( one_one @ nat ) )
=> ( ! [N2: nat] :
( ( P @ N2 )
=> ( P @ ( plus_plus @ nat @ N2 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) )
=> ( P @ N ) ) ) ) ).
% nat_induct2
thf(fact_671_div__mod__decomp,axiom,
! [A3: nat,N: nat] :
( A3
= ( plus_plus @ nat @ ( times_times @ nat @ ( divide_divide @ nat @ A3 @ N ) @ N ) @ ( modulo_modulo @ nat @ A3 @ N ) ) ) ).
% div_mod_decomp
thf(fact_672_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_673_numeral__One,axiom,
! [A: $tType] :
( ( numeral @ A )
=> ( ( numeral_numeral @ A @ one2 )
= ( one_one @ A ) ) ) ).
% numeral_One
thf(fact_674_less__imp__Suc__add,axiom,
! [M: nat,N: nat] :
( ( ord_less @ nat @ M @ N )
=> ? [K2: nat] :
( N
= ( suc @ ( plus_plus @ nat @ M @ K2 ) ) ) ) ).
% less_imp_Suc_add
thf(fact_675_less__iff__Suc__add,axiom,
( ( ord_less @ nat )
= ( ^ [M4: nat,N3: nat] :
? [K3: nat] :
( N3
= ( suc @ ( plus_plus @ nat @ M4 @ K3 ) ) ) ) ) ).
% less_iff_Suc_add
thf(fact_676_less__add__Suc2,axiom,
! [I: nat,M: nat] : ( ord_less @ nat @ I @ ( suc @ ( plus_plus @ nat @ M @ I ) ) ) ).
% less_add_Suc2
thf(fact_677_less__add__Suc1,axiom,
! [I: nat,M: nat] : ( ord_less @ nat @ I @ ( suc @ ( plus_plus @ nat @ I @ M ) ) ) ).
% less_add_Suc1
thf(fact_678_less__natE,axiom,
! [M: nat,N: nat] :
( ( ord_less @ nat @ M @ N )
=> ~ ! [Q5: nat] :
( N
!= ( suc @ ( plus_plus @ nat @ M @ Q5 ) ) ) ) ).
% less_natE
thf(fact_679_left__right__inverse__power,axiom,
! [A: $tType] :
( ( monoid_mult @ A )
=> ! [X: A,Y: A,N: nat] :
( ( ( times_times @ A @ X @ Y )
= ( one_one @ A ) )
=> ( ( times_times @ A @ ( power_power @ A @ X @ N ) @ ( power_power @ A @ Y @ N ) )
= ( one_one @ A ) ) ) ) ).
% left_right_inverse_power
thf(fact_680_Suc__mult__less__cancel1,axiom,
! [K: nat,M: nat,N: nat] :
( ( ord_less @ nat @ ( times_times @ nat @ ( suc @ K ) @ M ) @ ( times_times @ nat @ ( suc @ K ) @ N ) )
= ( ord_less @ nat @ M @ N ) ) ).
% Suc_mult_less_cancel1
thf(fact_681_power__one__over,axiom,
! [A: $tType] :
( ( division_ring @ A )
=> ! [A2: A,N: nat] :
( ( power_power @ A @ ( divide_divide @ A @ ( one_one @ A ) @ A2 ) @ N )
= ( divide_divide @ A @ ( one_one @ A ) @ ( power_power @ A @ A2 @ N ) ) ) ) ).
% power_one_over
thf(fact_682_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_683_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_684_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_685_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_686_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_687_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_688_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_689_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_690_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_691_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_692_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_693_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_694_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_695_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_696_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_697_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_698_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_699_numerals_I1_J,axiom,
( ( numeral_numeral @ nat @ one2 )
= ( one_one @ nat ) ) ).
% numerals(1)
thf(fact_700_less__mult__imp__div__less,axiom,
! [M: nat,I: nat,N: nat] :
( ( ord_less @ nat @ M @ ( times_times @ nat @ I @ N ) )
=> ( ord_less @ nat @ ( divide_divide @ nat @ M @ N ) @ I ) ) ).
% less_mult_imp_div_less
thf(fact_701_Suc__eq__plus1__left,axiom,
( suc
= ( plus_plus @ nat @ ( one_one @ nat ) ) ) ).
% Suc_eq_plus1_left
thf(fact_702_plus__1__eq__Suc,axiom,
( ( plus_plus @ nat @ ( one_one @ nat ) )
= suc ) ).
% plus_1_eq_Suc
thf(fact_703_Suc__eq__plus1,axiom,
( suc
= ( ^ [N3: nat] : ( plus_plus @ nat @ N3 @ ( one_one @ nat ) ) ) ) ).
% Suc_eq_plus1
thf(fact_704_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_705_one__is__add,axiom,
! [M: nat,N: nat] :
( ( ( suc @ ( zero_zero @ nat ) )
= ( plus_plus @ nat @ M @ N ) )
= ( ( ( M
= ( suc @ ( zero_zero @ nat ) ) )
& ( N
= ( zero_zero @ nat ) ) )
| ( ( M
= ( zero_zero @ nat ) )
& ( N
= ( suc @ ( zero_zero @ nat ) ) ) ) ) ) ).
% one_is_add
thf(fact_706_add__is__1,axiom,
! [M: nat,N: nat] :
( ( ( plus_plus @ nat @ M @ N )
= ( suc @ ( zero_zero @ nat ) ) )
= ( ( ( M
= ( suc @ ( zero_zero @ nat ) ) )
& ( N
= ( zero_zero @ nat ) ) )
| ( ( M
= ( zero_zero @ nat ) )
& ( N
= ( suc @ ( zero_zero @ nat ) ) ) ) ) ) ).
% add_is_1
thf(fact_707_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_708_pow_Osimps_I1_J,axiom,
! [X: num] :
( ( pow @ X @ one2 )
= X ) ).
% pow.simps(1)
thf(fact_709_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_710_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_711_div__2__gt__zero,axiom,
! [N: nat] :
( ( ord_less @ nat @ ( suc @ ( zero_zero @ nat ) ) @ N )
=> ( ord_less @ nat @ ( zero_zero @ nat ) @ ( divide_divide @ nat @ N @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ).
% div_2_gt_zero
thf(fact_712_Suc__n__div__2__gt__zero,axiom,
! [N: nat] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ N )
=> ( ord_less @ nat @ ( zero_zero @ nat ) @ ( divide_divide @ nat @ ( suc @ N ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ).
% Suc_n_div_2_gt_zero
thf(fact_713_nat__bit__induct,axiom,
! [P: nat > $o,N: nat] :
( ( P @ ( zero_zero @ nat ) )
=> ( ! [N2: nat] :
( ( P @ N2 )
=> ( ( ord_less @ nat @ ( zero_zero @ nat ) @ N2 )
=> ( P @ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N2 ) ) ) )
=> ( ! [N2: nat] :
( ( P @ N2 )
=> ( P @ ( suc @ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N2 ) ) ) )
=> ( P @ N ) ) ) ) ).
% nat_bit_induct
thf(fact_714_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_715_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_716_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_717_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_718_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_719_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_720_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_721_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_722_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_723_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_724_odd__power__less__zero,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [A2: A,N: nat] :
( ( ord_less @ A @ A2 @ ( zero_zero @ A ) )
=> ( ord_less @ A @ ( power_power @ A @ A2 @ ( suc @ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N ) ) ) @ ( zero_zero @ A ) ) ) ) ).
% odd_power_less_zero
thf(fact_725_mod__mult2__eq,axiom,
! [M: nat,N: nat,Q3: nat] :
( ( modulo_modulo @ nat @ M @ ( times_times @ nat @ N @ Q3 ) )
= ( plus_plus @ nat @ ( times_times @ nat @ N @ ( modulo_modulo @ nat @ ( divide_divide @ nat @ M @ N ) @ Q3 ) ) @ ( modulo_modulo @ nat @ M @ N ) ) ) ).
% mod_mult2_eq
thf(fact_726_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_727_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_728_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_729_ab__semigroup__add__class_Oadd_Ocommute,axiom,
! [A: $tType] :
( ( ab_semigroup_add @ A )
=> ( ( plus_plus @ A )
= ( ^ [A4: A,B5: A] : ( plus_plus @ A @ B5 @ A4 ) ) ) ) ).
% ab_semigroup_add_class.add.commute
thf(fact_730_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_731_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_732_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_733_group__cancel_Oadd2,axiom,
! [A: $tType] :
( ( comm_monoid_add @ A )
=> ! [B4: A,K: A,B2: A,A2: A] :
( ( B4
= ( plus_plus @ A @ K @ B2 ) )
=> ( ( plus_plus @ A @ A2 @ B4 )
= ( plus_plus @ A @ K @ ( plus_plus @ A @ A2 @ B2 ) ) ) ) ) ).
% group_cancel.add2
thf(fact_734_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_735_add__mono__thms__linordered__semiring_I4_J,axiom,
! [A: $tType] :
( ( ordere6658533253407199908up_add @ A )
=> ! [I: A,J: A,K: A,L: A] :
( ( ( I = J )
& ( K = L ) )
=> ( ( plus_plus @ A @ I @ K )
= ( plus_plus @ A @ J @ L ) ) ) ) ).
% add_mono_thms_linordered_semiring(4)
thf(fact_736_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_737_ab__semigroup__mult__class_Omult_Ocommute,axiom,
! [A: $tType] :
( ( ab_semigroup_mult @ A )
=> ( ( times_times @ A )
= ( ^ [A4: A,B5: A] : ( times_times @ A @ B5 @ A4 ) ) ) ) ).
% ab_semigroup_mult_class.mult.commute
thf(fact_738_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_739_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_740_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_741_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_742_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_743_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_744_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_745_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_746_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_747_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_748_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_749_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_750_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_751_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_752_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_753_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_754_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_755_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_756_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_757_dvd__div__div__eq__mult,axiom,
! [A: $tType] :
( ( algebraic_semidom @ A )
=> ! [A2: A,C2: A,B2: A,D2: A] :
( ( A2
!= ( zero_zero @ A ) )
=> ( ( C2
!= ( zero_zero @ A ) )
=> ( ( dvd_dvd @ A @ A2 @ B2 )
=> ( ( dvd_dvd @ A @ C2 @ D2 )
=> ( ( ( divide_divide @ A @ B2 @ A2 )
= ( divide_divide @ A @ D2 @ C2 ) )
= ( ( times_times @ A @ B2 @ C2 )
= ( times_times @ A @ A2 @ D2 ) ) ) ) ) ) ) ) ).
% dvd_div_div_eq_mult
thf(fact_758_numeral__1__eq__Suc__0,axiom,
( ( numeral_numeral @ nat @ one2 )
= ( suc @ ( zero_zero @ nat ) ) ) ).
% numeral_1_eq_Suc_0
thf(fact_759_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_760_odd__one,axiom,
! [A: $tType] :
( ( semiring_parity @ A )
=> ~ ( dvd_dvd @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ ( one_one @ A ) ) ) ).
% odd_one
thf(fact_761_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_762_even__zero,axiom,
! [A: $tType] :
( ( semiring_parity @ A )
=> ( dvd_dvd @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ ( zero_zero @ A ) ) ) ).
% even_zero
thf(fact_763_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_764_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_765_numeral__2__eq__2,axiom,
( ( numeral_numeral @ nat @ ( bit0 @ one2 ) )
= ( suc @ ( suc @ ( zero_zero @ nat ) ) ) ) ).
% numeral_2_eq_2
thf(fact_766_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_767_exp__add__not__zero__imp__left,axiom,
! [A: $tType] :
( ( bit_semiring_bits @ A )
=> ! [M: nat,N: nat] :
( ( ( power_power @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ ( plus_plus @ nat @ M @ N ) )
!= ( zero_zero @ A ) )
=> ( ( power_power @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ M )
!= ( zero_zero @ A ) ) ) ) ).
% exp_add_not_zero_imp_left
thf(fact_768_exp__add__not__zero__imp__right,axiom,
! [A: $tType] :
( ( bit_semiring_bits @ A )
=> ! [M: nat,N: nat] :
( ( ( power_power @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ ( plus_plus @ nat @ M @ N ) )
!= ( zero_zero @ A ) )
=> ( ( power_power @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ N )
!= ( zero_zero @ A ) ) ) ) ).
% exp_add_not_zero_imp_right
thf(fact_769_div__exp__mod__exp__eq,axiom,
! [A: $tType] :
( ( bit_semiring_bits @ A )
=> ! [A2: A,N: nat,M: nat] :
( ( modulo_modulo @ A @ ( divide_divide @ A @ A2 @ ( power_power @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ N ) ) @ ( power_power @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ M ) )
= ( divide_divide @ A @ ( modulo_modulo @ A @ A2 @ ( power_power @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ ( plus_plus @ nat @ N @ M ) ) ) @ ( power_power @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ N ) ) ) ) ).
% div_exp_mod_exp_eq
thf(fact_770_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_771_verit__eq__simplify_I10_J,axiom,
! [X22: num] :
( one2
!= ( bit0 @ X22 ) ) ).
% verit_eq_simplify(10)
thf(fact_772_VEBT__internal_Oexp__split__high__low_I1_J,axiom,
! [X: nat,N: nat,M: nat] :
( ( ord_less @ nat @ X @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( plus_plus @ nat @ N @ M ) ) )
=> ( ( ord_less @ nat @ ( zero_zero @ nat ) @ N )
=> ( ( ord_less @ nat @ ( zero_zero @ nat ) @ M )
=> ( ord_less @ nat @ ( vEBT_VEBT_high @ X @ N ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ M ) ) ) ) ) ).
% VEBT_internal.exp_split_high_low(1)
thf(fact_773_nat__add__offset__less,axiom,
! [Y: nat,N: nat,X: nat,M: nat,Sz: nat] :
( ( ord_less @ nat @ Y @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N ) )
=> ( ( ord_less @ nat @ X @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ M ) )
=> ( ( Sz
= ( plus_plus @ nat @ M @ N ) )
=> ( ord_less @ nat @ ( plus_plus @ nat @ ( times_times @ nat @ X @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N ) ) @ Y ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ Sz ) ) ) ) ) ).
% nat_add_offset_less
thf(fact_774_replicate__eq__replicate,axiom,
! [A: $tType,M: nat,X: A,N: nat,Y: A] :
( ( ( replicate @ A @ M @ X )
= ( replicate @ A @ N @ Y ) )
= ( ( M = N )
& ( ( M
!= ( zero_zero @ nat ) )
=> ( X = Y ) ) ) ) ).
% replicate_eq_replicate
thf(fact_775_nat__mod__eq_H,axiom,
! [A2: nat,N: nat] :
( ( ord_less @ nat @ A2 @ N )
=> ( ( modulo_modulo @ nat @ A2 @ N )
= A2 ) ) ).
% nat_mod_eq'
thf(fact_776_mod__lemma,axiom,
! [C2: nat,R2: nat,B2: nat,Q3: 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 @ Q3 @ C2 ) ) @ R2 ) @ ( times_times @ nat @ B2 @ C2 ) ) ) ) ).
% mod_lemma
thf(fact_777_even__even__mod__4__iff,axiom,
! [N: nat] :
( ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N )
= ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( modulo_modulo @ nat @ N @ ( numeral_numeral @ nat @ ( bit0 @ ( bit0 @ one2 ) ) ) ) ) ) ).
% even_even_mod_4_iff
thf(fact_778_msrevs_I1_J,axiom,
! [N: nat,K: nat,M: nat] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ N )
=> ( ( divide_divide @ nat @ ( plus_plus @ nat @ ( times_times @ nat @ K @ N ) @ M ) @ N )
= ( plus_plus @ nat @ ( divide_divide @ nat @ M @ N ) @ K ) ) ) ).
% msrevs(1)
thf(fact_779_n__less__equal__power__2,axiom,
! [N: nat] : ( ord_less @ nat @ N @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N ) ) ).
% n_less_equal_power_2
thf(fact_780_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_781_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_782_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_783_semiring__norm_I78_J,axiom,
! [M: num,N: num] :
( ( ord_less @ num @ ( bit0 @ M ) @ ( bit0 @ N ) )
= ( ord_less @ num @ M @ N ) ) ).
% semiring_norm(78)
thf(fact_784_semiring__norm_I75_J,axiom,
! [M: num] :
~ ( ord_less @ num @ M @ one2 ) ).
% semiring_norm(75)
thf(fact_785_one__mod__exp__eq__one,axiom,
! [N: nat] :
( ( modulo_modulo @ int @ ( one_one @ int ) @ ( times_times @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ ( power_power @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ N ) ) )
= ( one_one @ int ) ) ).
% one_mod_exp_eq_one
thf(fact_786_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_787_semiring__norm_I76_J,axiom,
! [N: num] : ( ord_less @ num @ one2 @ ( bit0 @ N ) ) ).
% semiring_norm(76)
thf(fact_788_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_789_zdiv__mono__strict,axiom,
! [A3: int,B4: int,N: int] :
( ( ord_less @ int @ A3 @ B4 )
=> ( ( ord_less @ int @ ( zero_zero @ int ) @ N )
=> ( ( ( modulo_modulo @ int @ A3 @ N )
= ( zero_zero @ int ) )
=> ( ( ( modulo_modulo @ int @ B4 @ N )
= ( zero_zero @ int ) )
=> ( ord_less @ int @ ( divide_divide @ int @ A3 @ N ) @ ( divide_divide @ int @ B4 @ N ) ) ) ) ) ) ).
% zdiv_mono_strict
thf(fact_790_real__arch__pow__inv,axiom,
! [Y: real,X: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ Y )
=> ( ( ord_less @ real @ X @ ( one_one @ real ) )
=> ? [N2: nat] : ( ord_less @ real @ ( power_power @ real @ X @ N2 ) @ Y ) ) ) ).
% real_arch_pow_inv
thf(fact_791_pos__imp__zdiv__neg__iff,axiom,
! [B2: int,A2: int] :
( ( ord_less @ int @ ( zero_zero @ int ) @ B2 )
=> ( ( ord_less @ int @ ( divide_divide @ int @ A2 @ B2 ) @ ( zero_zero @ int ) )
= ( ord_less @ int @ A2 @ ( zero_zero @ int ) ) ) ) ).
% pos_imp_zdiv_neg_iff
thf(fact_792_neg__imp__zdiv__neg__iff,axiom,
! [B2: int,A2: int] :
( ( ord_less @ int @ B2 @ ( zero_zero @ int ) )
=> ( ( ord_less @ int @ ( divide_divide @ int @ A2 @ B2 ) @ ( zero_zero @ int ) )
= ( ord_less @ int @ ( zero_zero @ int ) @ A2 ) ) ) ).
% neg_imp_zdiv_neg_iff
thf(fact_793_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_794_div__neg__pos__less0,axiom,
! [A2: int,B2: int] :
( ( ord_less @ int @ A2 @ ( zero_zero @ int ) )
=> ( ( ord_less @ int @ ( zero_zero @ int ) @ B2 )
=> ( ord_less @ int @ ( divide_divide @ int @ A2 @ B2 ) @ ( zero_zero @ int ) ) ) ) ).
% div_neg_pos_less0
thf(fact_795_zdiv__mult__self,axiom,
! [M: int,A2: int,N: int] :
( ( M
!= ( zero_zero @ int ) )
=> ( ( divide_divide @ int @ ( plus_plus @ int @ A2 @ ( times_times @ int @ M @ N ) ) @ M )
= ( plus_plus @ int @ ( divide_divide @ int @ A2 @ M ) @ N ) ) ) ).
% zdiv_mult_self
thf(fact_796_realpow__pos__nth2,axiom,
! [A2: real,N: nat] :
( ( ord_less @ real @ ( zero_zero @ real ) @ A2 )
=> ? [R3: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ R3 )
& ( ( power_power @ real @ R3 @ ( suc @ N ) )
= A2 ) ) ) ).
% realpow_pos_nth2
thf(fact_797_mod__exp__less__eq__exp,axiom,
! [A2: int,N: nat] : ( ord_less @ int @ ( modulo_modulo @ int @ A2 @ ( power_power @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ N ) ) @ ( power_power @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ N ) ) ).
% mod_exp_less_eq_exp
thf(fact_798_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_799_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_800_nmod2,axiom,
! [N: int] :
( ( ( modulo_modulo @ int @ N @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) )
= ( zero_zero @ int ) )
| ( ( modulo_modulo @ int @ N @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) )
= ( one_one @ int ) ) ) ).
% nmod2
thf(fact_801_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_802_div__mod__decomp__int,axiom,
! [A3: int,N: int] :
( A3
= ( plus_plus @ int @ ( times_times @ int @ ( divide_divide @ int @ A3 @ N ) @ N ) @ ( modulo_modulo @ int @ A3 @ N ) ) ) ).
% div_mod_decomp_int
thf(fact_803_real__arch__pow,axiom,
! [X: real,Y: real] :
( ( ord_less @ real @ ( one_one @ real ) @ X )
=> ? [N2: nat] : ( ord_less @ real @ Y @ ( power_power @ real @ X @ N2 ) ) ) ).
% real_arch_pow
thf(fact_804_realpow__pos__nth,axiom,
! [N: nat,A2: real] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ N )
=> ( ( ord_less @ real @ ( zero_zero @ real ) @ A2 )
=> ? [R3: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ R3 )
& ( ( power_power @ real @ R3 @ N )
= A2 ) ) ) ) ).
% realpow_pos_nth
thf(fact_805_realpow__pos__nth__unique,axiom,
! [N: nat,A2: real] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ N )
=> ( ( ord_less @ real @ ( zero_zero @ real ) @ A2 )
=> ? [X3: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ X3 )
& ( ( power_power @ real @ X3 @ N )
= A2 )
& ! [Y4: real] :
( ( ( ord_less @ real @ ( zero_zero @ real ) @ Y4 )
& ( ( power_power @ real @ Y4 @ N )
= A2 ) )
=> ( Y4 = X3 ) ) ) ) ) ).
% realpow_pos_nth_unique
thf(fact_806_p1mod22k_H,axiom,
! [B2: int,N: 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 ) ) @ N ) ) )
= ( 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 ) ) @ N ) ) ) ) ) ).
% p1mod22k'
thf(fact_807_p1mod22k,axiom,
! [B2: int,N: 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 ) ) @ N ) ) )
= ( plus_plus @ int @ ( times_times @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ ( modulo_modulo @ int @ B2 @ ( power_power @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ N ) ) ) @ ( one_one @ int ) ) ) ).
% p1mod22k
thf(fact_808_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_809_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_810_minf_I11_J,axiom,
! [C: $tType,D4: $tType] :
( ( ord @ C )
=> ! [F3: D4] :
? [Z3: C] :
! [X5: C] :
( ( ord_less @ C @ X5 @ Z3 )
=> ( F3 = F3 ) ) ) ).
% minf(11)
thf(fact_811_minf_I7_J,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [T: A] :
? [Z3: A] :
! [X5: A] :
( ( ord_less @ A @ X5 @ Z3 )
=> ~ ( ord_less @ A @ T @ X5 ) ) ) ).
% minf(7)
thf(fact_812_minf_I5_J,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [T: A] :
? [Z3: A] :
! [X5: A] :
( ( ord_less @ A @ X5 @ Z3 )
=> ( ord_less @ A @ X5 @ T ) ) ) ).
% minf(5)
thf(fact_813_minf_I4_J,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [T: A] :
? [Z3: A] :
! [X5: A] :
( ( ord_less @ A @ X5 @ Z3 )
=> ( X5 != T ) ) ) ).
% minf(4)
thf(fact_814_minf_I3_J,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [T: A] :
? [Z3: A] :
! [X5: A] :
( ( ord_less @ A @ X5 @ Z3 )
=> ( X5 != T ) ) ) ).
% minf(3)
thf(fact_815_minf_I2_J,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [P: A > $o,P3: A > $o,Q: A > $o,Q4: A > $o] :
( ? [Z4: A] :
! [X3: A] :
( ( ord_less @ A @ X3 @ Z4 )
=> ( ( P @ X3 )
= ( P3 @ X3 ) ) )
=> ( ? [Z4: A] :
! [X3: A] :
( ( ord_less @ A @ X3 @ Z4 )
=> ( ( Q @ X3 )
= ( Q4 @ X3 ) ) )
=> ? [Z3: A] :
! [X5: A] :
( ( ord_less @ A @ X5 @ Z3 )
=> ( ( ( P @ X5 )
| ( Q @ X5 ) )
= ( ( P3 @ X5 )
| ( Q4 @ X5 ) ) ) ) ) ) ) ).
% minf(2)
thf(fact_816_minf_I1_J,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [P: A > $o,P3: A > $o,Q: A > $o,Q4: A > $o] :
( ? [Z4: A] :
! [X3: A] :
( ( ord_less @ A @ X3 @ Z4 )
=> ( ( P @ X3 )
= ( P3 @ X3 ) ) )
=> ( ? [Z4: A] :
! [X3: A] :
( ( ord_less @ A @ X3 @ Z4 )
=> ( ( Q @ X3 )
= ( Q4 @ X3 ) ) )
=> ? [Z3: A] :
! [X5: A] :
( ( ord_less @ A @ X5 @ Z3 )
=> ( ( ( P @ X5 )
& ( Q @ X5 ) )
= ( ( P3 @ X5 )
& ( Q4 @ X5 ) ) ) ) ) ) ) ).
% minf(1)
thf(fact_817_pinf_I11_J,axiom,
! [C: $tType,D4: $tType] :
( ( ord @ C )
=> ! [F3: D4] :
? [Z3: C] :
! [X5: C] :
( ( ord_less @ C @ Z3 @ X5 )
=> ( F3 = F3 ) ) ) ).
% pinf(11)
thf(fact_818_pinf_I7_J,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [T: A] :
? [Z3: A] :
! [X5: A] :
( ( ord_less @ A @ Z3 @ X5 )
=> ( ord_less @ A @ T @ X5 ) ) ) ).
% pinf(7)
thf(fact_819_pinf_I5_J,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [T: A] :
? [Z3: A] :
! [X5: A] :
( ( ord_less @ A @ Z3 @ X5 )
=> ~ ( ord_less @ A @ X5 @ T ) ) ) ).
% pinf(5)
thf(fact_820_pinf_I4_J,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [T: A] :
? [Z3: A] :
! [X5: A] :
( ( ord_less @ A @ Z3 @ X5 )
=> ( X5 != T ) ) ) ).
% pinf(4)
thf(fact_821_pinf_I3_J,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [T: A] :
? [Z3: A] :
! [X5: A] :
( ( ord_less @ A @ Z3 @ X5 )
=> ( X5 != T ) ) ) ).
% pinf(3)
thf(fact_822_pinf_I2_J,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [P: A > $o,P3: A > $o,Q: A > $o,Q4: A > $o] :
( ? [Z4: A] :
! [X3: A] :
( ( ord_less @ A @ Z4 @ X3 )
=> ( ( P @ X3 )
= ( P3 @ X3 ) ) )
=> ( ? [Z4: A] :
! [X3: A] :
( ( ord_less @ A @ Z4 @ X3 )
=> ( ( Q @ X3 )
= ( Q4 @ X3 ) ) )
=> ? [Z3: A] :
! [X5: A] :
( ( ord_less @ A @ Z3 @ X5 )
=> ( ( ( P @ X5 )
| ( Q @ X5 ) )
= ( ( P3 @ X5 )
| ( Q4 @ X5 ) ) ) ) ) ) ) ).
% pinf(2)
thf(fact_823_pinf_I1_J,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [P: A > $o,P3: A > $o,Q: A > $o,Q4: A > $o] :
( ? [Z4: A] :
! [X3: A] :
( ( ord_less @ A @ Z4 @ X3 )
=> ( ( P @ X3 )
= ( P3 @ X3 ) ) )
=> ( ? [Z4: A] :
! [X3: A] :
( ( ord_less @ A @ Z4 @ X3 )
=> ( ( Q @ X3 )
= ( Q4 @ X3 ) ) )
=> ? [Z3: A] :
! [X5: A] :
( ( ord_less @ A @ Z3 @ X5 )
=> ( ( ( P @ X5 )
& ( Q @ X5 ) )
= ( ( P3 @ X5 )
& ( Q4 @ X5 ) ) ) ) ) ) ) ).
% pinf(1)
thf(fact_824_VEBT__internal_OT_092_060_094sub_062b_092_060_094sub_062u_092_060_094sub_062i_092_060_094sub_062l_092_060_094sub_062d_092_060_094sub_062u_092_060_094sub_062p_Ocases,axiom,
! [X: nat] :
( ( X
!= ( zero_zero @ nat ) )
=> ( ( X
!= ( suc @ ( zero_zero @ nat ) ) )
=> ~ ! [Va: nat] :
( X
!= ( suc @ ( suc @ Va ) ) ) ) ) ).
% VEBT_internal.T\<^sub>b\<^sub>u\<^sub>i\<^sub>l\<^sub>d\<^sub>u\<^sub>p.cases
thf(fact_825_nat__mod__eq,axiom,
! [B2: nat,N: nat,A2: nat] :
( ( ord_less @ nat @ B2 @ N )
=> ( ( ( modulo_modulo @ nat @ A2 @ N )
= ( modulo_modulo @ nat @ B2 @ N ) )
=> ( ( modulo_modulo @ nat @ A2 @ N )
= B2 ) ) ) ).
% nat_mod_eq
thf(fact_826_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_827_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_828_pinf_I9_J,axiom,
! [B: $tType] :
( ( ( plus @ B )
& ( linorder @ B )
& ( dvd @ B ) )
=> ! [D2: B,S: B] :
? [Z3: B] :
! [X5: B] :
( ( ord_less @ B @ Z3 @ X5 )
=> ( ( dvd_dvd @ B @ D2 @ ( plus_plus @ B @ X5 @ S ) )
= ( dvd_dvd @ B @ D2 @ ( plus_plus @ B @ X5 @ S ) ) ) ) ) ).
% pinf(9)
thf(fact_829_pinf_I10_J,axiom,
! [B: $tType] :
( ( ( plus @ B )
& ( linorder @ B )
& ( dvd @ B ) )
=> ! [D2: B,S: B] :
? [Z3: B] :
! [X5: B] :
( ( ord_less @ B @ Z3 @ X5 )
=> ( ( ~ ( dvd_dvd @ B @ D2 @ ( plus_plus @ B @ X5 @ S ) ) )
= ( ~ ( dvd_dvd @ B @ D2 @ ( plus_plus @ B @ X5 @ S ) ) ) ) ) ) ).
% pinf(10)
thf(fact_830_minf_I9_J,axiom,
! [B: $tType] :
( ( ( plus @ B )
& ( linorder @ B )
& ( dvd @ B ) )
=> ! [D2: B,S: B] :
? [Z3: B] :
! [X5: B] :
( ( ord_less @ B @ X5 @ Z3 )
=> ( ( dvd_dvd @ B @ D2 @ ( plus_plus @ B @ X5 @ S ) )
= ( dvd_dvd @ B @ D2 @ ( plus_plus @ B @ X5 @ S ) ) ) ) ) ).
% minf(9)
thf(fact_831_minf_I10_J,axiom,
! [B: $tType] :
( ( ( plus @ B )
& ( linorder @ B )
& ( dvd @ B ) )
=> ! [D2: B,S: B] :
? [Z3: B] :
! [X5: B] :
( ( ord_less @ B @ X5 @ Z3 )
=> ( ( ~ ( dvd_dvd @ B @ D2 @ ( plus_plus @ B @ X5 @ S ) ) )
= ( ~ ( dvd_dvd @ B @ D2 @ ( plus_plus @ B @ X5 @ S ) ) ) ) ) ) ).
% minf(10)
thf(fact_832_nat__mod__lem,axiom,
! [N: nat,B2: nat] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ N )
=> ( ( ord_less @ nat @ B2 @ N )
= ( ( modulo_modulo @ nat @ B2 @ N )
= B2 ) ) ) ).
% nat_mod_lem
thf(fact_833_msrevs_I2_J,axiom,
! [K: nat,N: nat,M: nat] :
( ( modulo_modulo @ nat @ ( plus_plus @ nat @ ( times_times @ nat @ K @ N ) @ M ) @ N )
= ( modulo_modulo @ nat @ M @ N ) ) ).
% msrevs(2)
thf(fact_834_set__bit__Suc,axiom,
! [A: $tType] :
( ( bit_se359711467146920520ations @ A )
=> ! [N: nat,A2: A] :
( ( bit_se5668285175392031749et_bit @ A @ ( suc @ N ) @ A2 )
= ( plus_plus @ A @ ( modulo_modulo @ A @ A2 @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) @ ( times_times @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ ( bit_se5668285175392031749et_bit @ A @ N @ ( divide_divide @ A @ A2 @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) ) ) ) ) ) ).
% set_bit_Suc
thf(fact_835_unity__coeff__ex,axiom,
! [A: $tType] :
( ( ( dvd @ A )
& ( semiring_0 @ A ) )
=> ! [P: A > $o,L: A] :
( ( ? [X2: A] : ( P @ ( times_times @ A @ L @ X2 ) ) )
= ( ? [X2: A] :
( ( dvd_dvd @ A @ L @ ( plus_plus @ A @ X2 @ ( zero_zero @ A ) ) )
& ( P @ X2 ) ) ) ) ) ).
% unity_coeff_ex
thf(fact_836_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_837_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_838_enat__ord__number_I2_J,axiom,
! [M: num,N: num] :
( ( ord_less @ extended_enat @ ( numeral_numeral @ extended_enat @ M ) @ ( numeral_numeral @ extended_enat @ N ) )
= ( ord_less @ nat @ ( numeral_numeral @ nat @ M ) @ ( numeral_numeral @ nat @ N ) ) ) ).
% enat_ord_number(2)
thf(fact_839_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_840_low__inv,axiom,
! [X: nat,N: nat,Y: nat] :
( ( ord_less @ nat @ X @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N ) )
=> ( ( vEBT_VEBT_low @ ( plus_plus @ nat @ ( times_times @ nat @ Y @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N ) ) @ X ) @ N )
= X ) ) ).
% low_inv
thf(fact_841_flip__bit__Suc,axiom,
! [A: $tType] :
( ( bit_se359711467146920520ations @ A )
=> ! [N: nat,A2: A] :
( ( bit_se8732182000553998342ip_bit @ A @ ( suc @ N ) @ A2 )
= ( plus_plus @ A @ ( modulo_modulo @ A @ A2 @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) @ ( times_times @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ ( bit_se8732182000553998342ip_bit @ A @ N @ ( divide_divide @ A @ A2 @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) ) ) ) ) ) ).
% flip_bit_Suc
thf(fact_842_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_843_unset__bit__Suc,axiom,
! [A: $tType] :
( ( bit_se359711467146920520ations @ A )
=> ! [N: nat,A2: A] :
( ( bit_se2638667681897837118et_bit @ A @ ( suc @ N ) @ A2 )
= ( plus_plus @ A @ ( modulo_modulo @ A @ A2 @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) @ ( times_times @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ ( bit_se2638667681897837118et_bit @ A @ N @ ( divide_divide @ A @ A2 @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) ) ) ) ) ) ).
% unset_bit_Suc
thf(fact_844_low__def,axiom,
( vEBT_VEBT_low
= ( ^ [X2: nat,N3: nat] : ( modulo_modulo @ nat @ X2 @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N3 ) ) ) ) ).
% low_def
thf(fact_845_word__rot__lem,axiom,
! [L: nat,K: nat,D2: nat,N: nat] :
( ( ( plus_plus @ nat @ L @ K )
= ( plus_plus @ nat @ D2 @ ( modulo_modulo @ nat @ K @ L ) ) )
=> ( ( ord_less @ nat @ N @ L )
=> ( ( modulo_modulo @ nat @ ( plus_plus @ nat @ D2 @ N ) @ L )
= N ) ) ) ).
% word_rot_lem
thf(fact_846_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_847_mlex__snd__decrI,axiom,
! [A2: nat,A5: nat,B2: nat,B7: nat,N4: nat] :
( ( A2 = A5 )
=> ( ( ord_less @ nat @ B2 @ B7 )
=> ( ord_less @ nat @ ( plus_plus @ nat @ ( times_times @ nat @ A2 @ N4 ) @ B2 ) @ ( plus_plus @ nat @ ( times_times @ nat @ A5 @ N4 ) @ B7 ) ) ) ) ).
% mlex_snd_decrI
thf(fact_848_bit__split__inv,axiom,
! [X: nat,D2: nat] :
( ( vEBT_VEBT_bit_concat @ ( vEBT_VEBT_high @ X @ D2 ) @ ( vEBT_VEBT_low @ X @ D2 ) @ D2 )
= X ) ).
% bit_split_inv
thf(fact_849_i0__less,axiom,
! [N: extended_enat] :
( ( ord_less @ extended_enat @ ( zero_zero @ extended_enat ) @ N )
= ( N
!= ( zero_zero @ extended_enat ) ) ) ).
% i0_less
thf(fact_850_set__bit__negative__int__iff,axiom,
! [N: nat,K: int] :
( ( ord_less @ int @ ( bit_se5668285175392031749et_bit @ int @ N @ K ) @ ( zero_zero @ int ) )
= ( ord_less @ int @ K @ ( zero_zero @ int ) ) ) ).
% set_bit_negative_int_iff
thf(fact_851_flip__bit__negative__int__iff,axiom,
! [N: nat,K: int] :
( ( ord_less @ int @ ( bit_se8732182000553998342ip_bit @ int @ N @ K ) @ ( zero_zero @ int ) )
= ( ord_less @ int @ K @ ( zero_zero @ int ) ) ) ).
% flip_bit_negative_int_iff
thf(fact_852_unset__bit__negative__int__iff,axiom,
! [N: nat,K: int] :
( ( ord_less @ int @ ( bit_se2638667681897837118et_bit @ int @ N @ K ) @ ( zero_zero @ int ) )
= ( ord_less @ int @ K @ ( zero_zero @ int ) ) ) ).
% unset_bit_negative_int_iff
thf(fact_853_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_854_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_855_times__int__code_I2_J,axiom,
! [L: int] :
( ( times_times @ int @ ( zero_zero @ int ) @ L )
= ( zero_zero @ int ) ) ).
% times_int_code(2)
thf(fact_856_times__int__code_I1_J,axiom,
! [K: int] :
( ( times_times @ int @ K @ ( zero_zero @ int ) )
= ( zero_zero @ int ) ) ).
% times_int_code(1)
thf(fact_857_zmod__eq__0D,axiom,
! [M: int,D2: int] :
( ( ( modulo_modulo @ int @ M @ D2 )
= ( zero_zero @ int ) )
=> ? [Q5: int] :
( M
= ( times_times @ int @ D2 @ Q5 ) ) ) ).
% zmod_eq_0D
thf(fact_858_zdvd__mult__cancel,axiom,
! [K: int,M: int,N: int] :
( ( dvd_dvd @ int @ ( times_times @ int @ K @ M ) @ ( times_times @ int @ K @ N ) )
=> ( ( K
!= ( zero_zero @ int ) )
=> ( dvd_dvd @ int @ M @ N ) ) ) ).
% zdvd_mult_cancel
thf(fact_859_zdvd__mono,axiom,
! [K: int,M: int,T: int] :
( ( K
!= ( zero_zero @ int ) )
=> ( ( dvd_dvd @ int @ M @ T )
= ( dvd_dvd @ int @ ( times_times @ int @ K @ M ) @ ( times_times @ int @ K @ T ) ) ) ) ).
% zdvd_mono
thf(fact_860_zmod__eq__0__iff,axiom,
! [M: int,D2: int] :
( ( ( modulo_modulo @ int @ M @ D2 )
= ( zero_zero @ int ) )
= ( ? [Q6: int] :
( M
= ( times_times @ int @ D2 @ Q6 ) ) ) ) ).
% zmod_eq_0_iff
thf(fact_861_zmult__zless__mono2,axiom,
! [I: int,J: int,K: int] :
( ( ord_less @ int @ I @ J )
=> ( ( ord_less @ int @ ( zero_zero @ int ) @ K )
=> ( ord_less @ int @ ( times_times @ int @ K @ I ) @ ( times_times @ int @ K @ J ) ) ) ) ).
% zmult_zless_mono2
thf(fact_862_pos__zmult__eq__1__iff,axiom,
! [M: int,N: int] :
( ( ord_less @ int @ ( zero_zero @ int ) @ M )
=> ( ( ( times_times @ int @ M @ N )
= ( one_one @ int ) )
= ( ( M
= ( one_one @ int ) )
& ( N
= ( one_one @ int ) ) ) ) ) ).
% pos_zmult_eq_1_iff
thf(fact_863_enat__0__less__mult__iff,axiom,
! [M: extended_enat,N: extended_enat] :
( ( ord_less @ extended_enat @ ( zero_zero @ extended_enat ) @ ( times_times @ extended_enat @ M @ N ) )
= ( ( ord_less @ extended_enat @ ( zero_zero @ extended_enat ) @ M )
& ( ord_less @ extended_enat @ ( zero_zero @ extended_enat ) @ N ) ) ) ).
% enat_0_less_mult_iff
thf(fact_864_zmod__helper,axiom,
! [N: int,M: int,K: int,A2: int] :
( ( ( modulo_modulo @ int @ N @ M )
= K )
=> ( ( modulo_modulo @ int @ ( plus_plus @ int @ N @ A2 ) @ M )
= ( modulo_modulo @ int @ ( plus_plus @ int @ K @ A2 ) @ M ) ) ) ).
% zmod_helper
thf(fact_865_mod__plus__cong,axiom,
! [B2: int,B7: int,X: int,X6: int,Y: int,Y5: int,Z5: int] :
( ( B2 = B7 )
=> ( ( ( modulo_modulo @ int @ X @ B7 )
= ( modulo_modulo @ int @ X6 @ B7 ) )
=> ( ( ( modulo_modulo @ int @ Y @ B7 )
= ( modulo_modulo @ int @ Y5 @ B7 ) )
=> ( ( ( plus_plus @ int @ X6 @ Y5 )
= Z5 )
=> ( ( modulo_modulo @ int @ ( plus_plus @ int @ X @ Y ) @ B2 )
= ( modulo_modulo @ int @ Z5 @ B7 ) ) ) ) ) ) ).
% mod_plus_cong
thf(fact_866_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_867_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_868_int__gr__induct,axiom,
! [K: int,I: int,P: int > $o] :
( ( ord_less @ int @ K @ I )
=> ( ( P @ ( plus_plus @ int @ K @ ( one_one @ int ) ) )
=> ( ! [I3: int] :
( ( ord_less @ int @ K @ I3 )
=> ( ( P @ I3 )
=> ( P @ ( plus_plus @ int @ I3 @ ( one_one @ int ) ) ) ) )
=> ( P @ I ) ) ) ) ).
% int_gr_induct
thf(fact_869_odd__nonzero,axiom,
! [Z: int] :
( ( plus_plus @ int @ ( plus_plus @ int @ ( one_one @ int ) @ Z ) @ Z )
!= ( zero_zero @ int ) ) ).
% odd_nonzero
thf(fact_870_plus__int__code_I1_J,axiom,
! [K: int] :
( ( plus_plus @ int @ K @ ( zero_zero @ int ) )
= K ) ).
% plus_int_code(1)
thf(fact_871_plus__int__code_I2_J,axiom,
! [L: int] :
( ( plus_plus @ int @ ( zero_zero @ int ) @ L )
= L ) ).
% plus_int_code(2)
thf(fact_872_neg__mod__bound,axiom,
! [L: int,K: int] :
( ( ord_less @ int @ L @ ( zero_zero @ int ) )
=> ( ord_less @ int @ L @ ( modulo_modulo @ int @ K @ L ) ) ) ).
% neg_mod_bound
thf(fact_873_Euclidean__Division_Opos__mod__bound,axiom,
! [L: int,K: int] :
( ( ord_less @ int @ ( zero_zero @ int ) @ L )
=> ( ord_less @ int @ ( modulo_modulo @ int @ K @ L ) @ L ) ) ).
% Euclidean_Division.pos_mod_bound
thf(fact_874_enat__less__induct,axiom,
! [P: extended_enat > $o,N: extended_enat] :
( ! [N2: extended_enat] :
( ! [M3: extended_enat] :
( ( ord_less @ extended_enat @ M3 @ N2 )
=> ( P @ M3 ) )
=> ( P @ N2 ) )
=> ( P @ N ) ) ).
% enat_less_induct
thf(fact_875_not__iless0,axiom,
! [N: extended_enat] :
~ ( ord_less @ extended_enat @ N @ ( zero_zero @ extended_enat ) ) ).
% not_iless0
thf(fact_876_less__int__code_I1_J,axiom,
~ ( ord_less @ int @ ( zero_zero @ int ) @ ( zero_zero @ int ) ) ).
% less_int_code(1)
thf(fact_877_zdvd__not__zless,axiom,
! [M: int,N: int] :
( ( ord_less @ int @ ( zero_zero @ int ) @ M )
=> ( ( ord_less @ int @ M @ N )
=> ~ ( dvd_dvd @ int @ N @ M ) ) ) ).
% zdvd_not_zless
thf(fact_878_dbl__def,axiom,
! [A: $tType] :
( ( neg_numeral @ A )
=> ( ( neg_numeral_dbl @ A )
= ( ^ [X2: A] : ( plus_plus @ A @ X2 @ X2 ) ) ) ) ).
% dbl_def
thf(fact_879_VEBT__internal_Oexp__split__high__low_I2_J,axiom,
! [X: nat,N: nat,M: nat] :
( ( ord_less @ nat @ X @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( plus_plus @ nat @ N @ M ) ) )
=> ( ( ord_less @ nat @ ( zero_zero @ nat ) @ N )
=> ( ( ord_less @ nat @ ( zero_zero @ nat ) @ M )
=> ( ord_less @ nat @ ( vEBT_VEBT_low @ X @ N ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N ) ) ) ) ) ).
% VEBT_internal.exp_split_high_low(2)
thf(fact_880_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_881_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_882_mlex__bound,axiom,
! [A2: nat,A3: nat,B2: nat,N4: nat] :
( ( ord_less @ nat @ A2 @ A3 )
=> ( ( ord_less @ nat @ B2 @ N4 )
=> ( ord_less @ nat @ ( plus_plus @ nat @ ( times_times @ nat @ A2 @ N4 ) @ B2 ) @ ( times_times @ nat @ A3 @ N4 ) ) ) ) ).
% mlex_bound
thf(fact_883_mlex__fst__decrI,axiom,
! [A2: nat,A5: nat,B2: nat,N4: nat,B7: nat] :
( ( ord_less @ nat @ A2 @ A5 )
=> ( ( ord_less @ nat @ B2 @ N4 )
=> ( ( ord_less @ nat @ B7 @ N4 )
=> ( ord_less @ nat @ ( plus_plus @ nat @ ( times_times @ nat @ A2 @ N4 ) @ B2 ) @ ( plus_plus @ nat @ ( times_times @ nat @ A5 @ N4 ) @ B7 ) ) ) ) ) ).
% mlex_fst_decrI
thf(fact_884_Comparator__Generator_OAll__less__Suc,axiom,
! [X: nat,P: nat > $o] :
( ( ! [I2: nat] :
( ( ord_less @ nat @ I2 @ ( suc @ X ) )
=> ( P @ I2 ) ) )
= ( ( P @ ( zero_zero @ nat ) )
& ! [I2: nat] :
( ( ord_less @ nat @ I2 @ X )
=> ( P @ ( suc @ I2 ) ) ) ) ) ).
% Comparator_Generator.All_less_Suc
thf(fact_885_forall__finite_I2_J,axiom,
! [P: nat > $o] :
( ( ! [I2: nat] :
( ( ord_less @ nat @ I2 @ ( suc @ ( zero_zero @ nat ) ) )
=> ( P @ I2 ) ) )
= ( P @ ( zero_zero @ nat ) ) ) ).
% forall_finite(2)
thf(fact_886_forall__finite_I3_J,axiom,
! [X: nat,P: nat > $o] :
( ( ! [I2: nat] :
( ( ord_less @ nat @ I2 @ ( suc @ ( suc @ X ) ) )
=> ( P @ I2 ) ) )
= ( ( P @ ( zero_zero @ nat ) )
& ! [I2: nat] :
( ( ord_less @ nat @ I2 @ ( suc @ X ) )
=> ( P @ ( suc @ I2 ) ) ) ) ) ).
% forall_finite(3)
thf(fact_887_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_888_p2__eq__1,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [N: nat] :
( ( ( power_power @ ( word @ A ) @ ( numeral_numeral @ ( word @ A ) @ ( bit0 @ one2 ) ) @ N )
= ( one_one @ ( word @ A ) ) )
= ( N
= ( zero_zero @ nat ) ) ) ) ).
% p2_eq_1
thf(fact_889_add__scale__eq__noteq,axiom,
! [A: $tType] :
( ( semiri1453513574482234551roduct @ A )
=> ! [R2: A,A2: A,B2: A,C2: A,D2: A] :
( ( R2
!= ( zero_zero @ A ) )
=> ( ( ( A2 = B2 )
& ( C2 != D2 ) )
=> ( ( plus_plus @ A @ A2 @ ( times_times @ A @ R2 @ C2 ) )
!= ( plus_plus @ A @ B2 @ ( times_times @ A @ R2 @ D2 ) ) ) ) ) ) ).
% add_scale_eq_noteq
thf(fact_890_word__gt__0__no,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [Y: num] :
( ( ord_less @ ( word @ A ) @ ( zero_zero @ ( word @ A ) ) @ ( numeral_numeral @ ( word @ A ) @ Y ) )
= ( ( zero_zero @ ( word @ A ) )
!= ( numeral_numeral @ ( word @ A ) @ Y ) ) ) ) ).
% word_gt_0_no
thf(fact_891_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_892_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_893_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_894_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_895_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_896_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_897_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_898_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_899_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_900_Suc__0__mod__eq,axiom,
! [N: nat] :
( ( modulo_modulo @ nat @ ( suc @ ( zero_zero @ nat ) ) @ N )
= ( zero_neq_one_of_bool @ nat
@ ( N
!= ( suc @ ( zero_zero @ nat ) ) ) ) ) ).
% Suc_0_mod_eq
thf(fact_901_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_902_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_903_one__div__2__pow__eq,axiom,
! [A: $tType] :
( ( euclid5411537665997757685th_nat @ A )
=> ! [N: nat] :
( ( divide_divide @ A @ ( one_one @ A ) @ ( power_power @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ N ) )
= ( zero_neq_one_of_bool @ A
@ ( N
= ( zero_zero @ nat ) ) ) ) ) ).
% one_div_2_pow_eq
thf(fact_904_bits__1__div__exp,axiom,
! [A: $tType] :
( ( bit_semiring_bits @ A )
=> ! [N: nat] :
( ( divide_divide @ A @ ( one_one @ A ) @ ( power_power @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ N ) )
= ( zero_neq_one_of_bool @ A
@ ( N
= ( zero_zero @ nat ) ) ) ) ) ).
% bits_1_div_exp
thf(fact_905_one__mod__2__pow__eq,axiom,
! [A: $tType] :
( ( euclid5411537665997757685th_nat @ A )
=> ! [N: nat] :
( ( modulo_modulo @ A @ ( one_one @ A ) @ ( power_power @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ N ) )
= ( zero_neq_one_of_bool @ A @ ( ord_less @ nat @ ( zero_zero @ nat ) @ N ) ) ) ) ).
% one_mod_2_pow_eq
thf(fact_906_imult__is__0,axiom,
! [M: extended_enat,N: extended_enat] :
( ( ( times_times @ extended_enat @ M @ N )
= ( zero_zero @ extended_enat ) )
= ( ( M
= ( zero_zero @ extended_enat ) )
| ( N
= ( zero_zero @ extended_enat ) ) ) ) ).
% imult_is_0
thf(fact_907_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_908_word__gr0__conv__Suc,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [M: word @ A] :
( ( ord_less @ ( word @ A ) @ ( zero_zero @ ( word @ A ) ) @ M )
=> ? [N2: word @ A] :
( M
= ( plus_plus @ ( word @ A ) @ N2 @ ( one_one @ ( word @ A ) ) ) ) ) ) ).
% word_gr0_conv_Suc
thf(fact_909_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_910_iadd__is__0,axiom,
! [M: extended_enat,N: extended_enat] :
( ( ( plus_plus @ extended_enat @ M @ N )
= ( zero_zero @ extended_enat ) )
= ( ( M
= ( zero_zero @ extended_enat ) )
& ( N
= ( zero_zero @ extended_enat ) ) ) ) ).
% iadd_is_0
thf(fact_911_word__induct__less,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [P: ( word @ A ) > $o,M: word @ A] :
( ( P @ ( zero_zero @ ( word @ A ) ) )
=> ( ! [N2: word @ A] :
( ( ord_less @ ( word @ A ) @ N2 @ M )
=> ( ( P @ N2 )
=> ( P @ ( plus_plus @ ( word @ A ) @ ( one_one @ ( word @ A ) ) @ N2 ) ) ) )
=> ( P @ M ) ) ) ) ).
% word_induct_less
thf(fact_912_word__induct2,axiom,
! [B: $tType] :
( ( type_len @ B )
=> ! [P: ( word @ B ) > $o,N: word @ B] :
( ( P @ ( zero_zero @ ( word @ B ) ) )
=> ( ! [N2: word @ B] :
( ( ( plus_plus @ ( word @ B ) @ ( one_one @ ( word @ B ) ) @ N2 )
!= ( zero_zero @ ( word @ B ) ) )
=> ( ( P @ N2 )
=> ( P @ ( plus_plus @ ( word @ B ) @ ( one_one @ ( word @ B ) ) @ N2 ) ) ) )
=> ( P @ N ) ) ) ) ).
% word_induct2
thf(fact_913_word__induct,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [P: ( word @ A ) > $o,M: word @ A] :
( ( P @ ( zero_zero @ ( word @ A ) ) )
=> ( ! [N2: word @ A] :
( ( P @ N2 )
=> ( P @ ( plus_plus @ ( word @ A ) @ ( one_one @ ( word @ A ) ) @ N2 ) ) )
=> ( P @ M ) ) ) ) ).
% word_induct
thf(fact_914_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_915_word__div__1,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [N: word @ A] :
( ( divide_divide @ ( word @ A ) @ N @ ( one_one @ ( word @ A ) ) )
= N ) ) ).
% word_div_1
thf(fact_916_div__less__dividend__word,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [X: word @ A,N: word @ A] :
( ( X
!= ( zero_zero @ ( word @ A ) ) )
=> ( ( N
!= ( one_one @ ( word @ A ) ) )
=> ( ord_less @ ( word @ A ) @ ( divide_divide @ ( word @ A ) @ X @ N ) @ X ) ) ) ) ).
% div_less_dividend_word
thf(fact_917_zero__one__enat__neq_I1_J,axiom,
( ( zero_zero @ extended_enat )
!= ( one_one @ extended_enat ) ) ).
% zero_one_enat_neq(1)
thf(fact_918_of__bool__eq__iff,axiom,
! [A: $tType] :
( ( zero_neq_one @ A )
=> ! [P4: $o,Q3: $o] :
( ( ( zero_neq_one_of_bool @ A @ P4 )
= ( zero_neq_one_of_bool @ A @ Q3 ) )
= ( P4 = Q3 ) ) ) ).
% of_bool_eq_iff
thf(fact_919_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_920_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_921_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_922_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_923_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_924_bits__induct,axiom,
! [A: $tType] :
( ( bit_semiring_bits @ A )
=> ! [P: A > $o,A2: A] :
( ! [A6: A] :
( ( ( divide_divide @ A @ A6 @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) )
= A6 )
=> ( P @ A6 ) )
=> ( ! [A6: A,B3: $o] :
( ( P @ A6 )
=> ( ( ( divide_divide @ A @ ( plus_plus @ A @ ( zero_neq_one_of_bool @ A @ B3 ) @ ( times_times @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ A6 ) ) @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) )
= A6 )
=> ( P @ ( plus_plus @ A @ ( zero_neq_one_of_bool @ A @ B3 ) @ ( times_times @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ A6 ) ) ) ) )
=> ( P @ A2 ) ) ) ) ).
% bits_induct
thf(fact_925_exp__mod__exp,axiom,
! [A: $tType] :
( ( euclid5411537665997757685th_nat @ A )
=> ! [M: nat,N: nat] :
( ( modulo_modulo @ A @ ( power_power @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ M ) @ ( power_power @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ N ) )
= ( times_times @ A @ ( zero_neq_one_of_bool @ A @ ( ord_less @ nat @ M @ N ) ) @ ( power_power @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ M ) ) ) ) ).
% exp_mod_exp
thf(fact_926_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_927_crossproduct__noteq,axiom,
! [A: $tType] :
( ( semiri1453513574482234551roduct @ A )
=> ! [A2: A,B2: A,C2: A,D2: A] :
( ( ( A2 != B2 )
& ( C2 != D2 ) )
= ( ( plus_plus @ A @ ( times_times @ A @ A2 @ C2 ) @ ( times_times @ A @ B2 @ D2 ) )
!= ( plus_plus @ A @ ( times_times @ A @ A2 @ D2 ) @ ( times_times @ A @ B2 @ C2 ) ) ) ) ) ).
% crossproduct_noteq
thf(fact_928_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_929_forall__finite_I1_J,axiom,
! [P: nat > $o,I4: nat] :
( ( ord_less @ nat @ I4 @ ( zero_zero @ nat ) )
=> ( P @ I4 ) ) ).
% forall_finite(1)
thf(fact_930_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_931_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_932_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_933_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_934_signed__take__bit__Suc,axiom,
! [A: $tType] :
( ( bit_ri3973907225187159222ations @ A )
=> ! [N: nat,A2: A] :
( ( bit_ri4674362597316999326ke_bit @ A @ ( suc @ N ) @ A2 )
= ( plus_plus @ A @ ( modulo_modulo @ A @ A2 @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) @ ( times_times @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ ( bit_ri4674362597316999326ke_bit @ A @ N @ ( divide_divide @ A @ A2 @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) ) ) ) ) ) ).
% signed_take_bit_Suc
thf(fact_935_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_936_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_937_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_938_less__option__None,axiom,
! [A: $tType] :
( ( preorder @ A )
=> ! [X: option @ A] :
~ ( ord_less @ ( option @ A ) @ X @ ( none @ A ) ) ) ).
% less_option_None
thf(fact_939_min__in__set__def,axiom,
( vEBT_VEBT_min_in_set
= ( ^ [Xs: set @ nat,X2: nat] :
( ( member @ nat @ X2 @ Xs )
& ! [Y6: nat] :
( ( member @ nat @ Y6 @ Xs )
=> ( ord_less_eq @ nat @ X2 @ Y6 ) ) ) ) ) ).
% min_in_set_def
thf(fact_940_max__in__set__def,axiom,
( vEBT_VEBT_max_in_set
= ( ^ [Xs: set @ nat,X2: nat] :
( ( member @ nat @ X2 @ Xs )
& ! [Y6: nat] :
( ( member @ nat @ Y6 @ Xs )
=> ( ord_less_eq @ nat @ Y6 @ X2 ) ) ) ) ) ).
% max_in_set_def
thf(fact_941_semiring__norm_I71_J,axiom,
! [M: num,N: num] :
( ( ord_less_eq @ num @ ( bit0 @ M ) @ ( bit0 @ N ) )
= ( ord_less_eq @ num @ M @ N ) ) ).
% semiring_norm(71)
thf(fact_942_semiring__norm_I68_J,axiom,
! [N: num] : ( ord_less_eq @ num @ one2 @ N ) ).
% semiring_norm(68)
thf(fact_943_Suc__le__mono,axiom,
! [N: nat,M: nat] :
( ( ord_less_eq @ nat @ ( suc @ N ) @ ( suc @ M ) )
= ( ord_less_eq @ nat @ N @ M ) ) ).
% Suc_le_mono
thf(fact_944_bot__nat__0_Oextremum,axiom,
! [A2: nat] : ( ord_less_eq @ nat @ ( zero_zero @ nat ) @ A2 ) ).
% bot_nat_0.extremum
thf(fact_945_le0,axiom,
! [N: nat] : ( ord_less_eq @ nat @ ( zero_zero @ nat ) @ N ) ).
% le0
thf(fact_946_nat__add__left__cancel__le,axiom,
! [K: nat,M: nat,N: nat] :
( ( ord_less_eq @ nat @ ( plus_plus @ nat @ K @ M ) @ ( plus_plus @ nat @ K @ N ) )
= ( ord_less_eq @ nat @ M @ N ) ) ).
% nat_add_left_cancel_le
thf(fact_947_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_948_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_949_mod__word__self,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [W: word @ A] :
( ( modulo_modulo @ ( word @ A ) @ W @ W )
= ( zero_zero @ ( word @ A ) ) ) ) ).
% mod_word_self
thf(fact_950_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_951_le__zero__eq,axiom,
! [A: $tType] :
( ( canoni5634975068530333245id_add @ A )
=> ! [N: A] :
( ( ord_less_eq @ A @ N @ ( zero_zero @ A ) )
= ( N
= ( zero_zero @ A ) ) ) ) ).
% le_zero_eq
thf(fact_952_numeral__le__iff,axiom,
! [A: $tType] :
( ( linord181362715937106298miring @ A )
=> ! [M: num,N: num] :
( ( ord_less_eq @ A @ ( numeral_numeral @ A @ M ) @ ( numeral_numeral @ A @ N ) )
= ( ord_less_eq @ num @ M @ N ) ) ) ).
% numeral_le_iff
thf(fact_953_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_954_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_955_semiring__norm_I69_J,axiom,
! [M: num] :
~ ( ord_less_eq @ num @ ( bit0 @ M ) @ one2 ) ).
% semiring_norm(69)
thf(fact_956_signed__take__bit__of__0,axiom,
! [A: $tType] :
( ( bit_ri3973907225187159222ations @ A )
=> ! [N: nat] :
( ( bit_ri4674362597316999326ke_bit @ A @ N @ ( zero_zero @ A ) )
= ( zero_zero @ A ) ) ) ).
% signed_take_bit_of_0
thf(fact_957_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_958_div__of__0__id,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [N: word @ A] :
( ( divide_divide @ ( word @ A ) @ ( zero_zero @ ( word @ A ) ) @ N )
= ( zero_zero @ ( word @ A ) ) ) ) ).
% div_of_0_id
thf(fact_959_enat__ord__number_I1_J,axiom,
! [M: num,N: num] :
( ( ord_less_eq @ extended_enat @ ( numeral_numeral @ extended_enat @ M ) @ ( numeral_numeral @ extended_enat @ N ) )
= ( ord_less_eq @ nat @ ( numeral_numeral @ nat @ M ) @ ( numeral_numeral @ nat @ N ) ) ) ).
% enat_ord_number(1)
thf(fact_960_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_961_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_962_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_963_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_964_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_965_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_966_one__le__mult__iff,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq @ nat @ ( suc @ ( zero_zero @ nat ) ) @ ( times_times @ nat @ M @ N ) )
= ( ( ord_less_eq @ nat @ ( suc @ ( zero_zero @ nat ) ) @ M )
& ( ord_less_eq @ nat @ ( suc @ ( zero_zero @ nat ) ) @ N ) ) ) ).
% one_le_mult_iff
thf(fact_967_mult__le__cancel2,axiom,
! [M: nat,K: nat,N: nat] :
( ( ord_less_eq @ nat @ ( times_times @ nat @ M @ K ) @ ( times_times @ nat @ N @ K ) )
= ( ( ord_less @ nat @ ( zero_zero @ nat ) @ K )
=> ( ord_less_eq @ nat @ M @ N ) ) ) ).
% mult_le_cancel2
thf(fact_968_nat__mult__le__cancel__disj,axiom,
! [K: nat,M: nat,N: nat] :
( ( ord_less_eq @ nat @ ( times_times @ nat @ K @ M ) @ ( times_times @ nat @ K @ N ) )
= ( ( ord_less @ nat @ ( zero_zero @ nat ) @ K )
=> ( ord_less_eq @ nat @ M @ N ) ) ) ).
% nat_mult_le_cancel_disj
thf(fact_969_signed__take__bit__Suc__1,axiom,
! [A: $tType] :
( ( bit_ri3973907225187159222ations @ A )
=> ! [N: nat] :
( ( bit_ri4674362597316999326ke_bit @ A @ ( suc @ N ) @ ( one_one @ A ) )
= ( one_one @ A ) ) ) ).
% signed_take_bit_Suc_1
thf(fact_970_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_971_unset__bit__nonnegative__int__iff,axiom,
! [N: nat,K: int] :
( ( ord_less_eq @ int @ ( zero_zero @ int ) @ ( bit_se2638667681897837118et_bit @ int @ N @ K ) )
= ( ord_less_eq @ int @ ( zero_zero @ int ) @ K ) ) ).
% unset_bit_nonnegative_int_iff
thf(fact_972_set__bit__nonnegative__int__iff,axiom,
! [N: nat,K: int] :
( ( ord_less_eq @ int @ ( zero_zero @ int ) @ ( bit_se5668285175392031749et_bit @ int @ N @ K ) )
= ( ord_less_eq @ int @ ( zero_zero @ int ) @ K ) ) ).
% set_bit_nonnegative_int_iff
thf(fact_973_flip__bit__nonnegative__int__iff,axiom,
! [N: nat,K: int] :
( ( ord_less_eq @ int @ ( zero_zero @ int ) @ ( bit_se8732182000553998342ip_bit @ int @ N @ K ) )
= ( ord_less_eq @ int @ ( zero_zero @ int ) @ K ) ) ).
% flip_bit_nonnegative_int_iff
thf(fact_974_numeral__le__one__iff,axiom,
! [A: $tType] :
( ( linord181362715937106298miring @ A )
=> ! [N: num] :
( ( ord_less_eq @ A @ ( numeral_numeral @ A @ N ) @ ( one_one @ A ) )
= ( ord_less_eq @ num @ N @ one2 ) ) ) ).
% numeral_le_one_iff
thf(fact_975_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_976_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_977_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_978_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_979_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_980_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_981_mod__neg__neg__trivial,axiom,
! [K: int,L: int] :
( ( ord_less_eq @ int @ K @ ( zero_zero @ int ) )
=> ( ( ord_less @ int @ L @ K )
=> ( ( modulo_modulo @ int @ K @ L )
= K ) ) ) ).
% mod_neg_neg_trivial
thf(fact_982_mod__pos__pos__trivial,axiom,
! [K: int,L: int] :
( ( ord_less_eq @ int @ ( zero_zero @ int ) @ K )
=> ( ( ord_less @ int @ K @ L )
=> ( ( modulo_modulo @ int @ K @ L )
= K ) ) ) ).
% mod_pos_pos_trivial
thf(fact_983_div__pos__pos__trivial,axiom,
! [K: int,L: int] :
( ( ord_less_eq @ int @ ( zero_zero @ int ) @ K )
=> ( ( ord_less @ int @ K @ L )
=> ( ( divide_divide @ int @ K @ L )
= ( zero_zero @ int ) ) ) ) ).
% div_pos_pos_trivial
thf(fact_984_div__neg__neg__trivial,axiom,
! [K: int,L: int] :
( ( ord_less_eq @ int @ K @ ( zero_zero @ int ) )
=> ( ( ord_less @ int @ L @ K )
=> ( ( divide_divide @ int @ K @ L )
= ( zero_zero @ int ) ) ) ) ).
% div_neg_neg_trivial
thf(fact_985_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_986_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_987_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_988_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_989_power__decreasing__iff,axiom,
! [A: $tType] :
( ( linordered_semidom @ A )
=> ! [B2: A,M: nat,N: nat] :
( ( ord_less @ A @ ( zero_zero @ A ) @ B2 )
=> ( ( ord_less @ A @ B2 @ ( one_one @ A ) )
=> ( ( ord_less_eq @ A @ ( power_power @ A @ B2 @ M ) @ ( power_power @ A @ B2 @ N ) )
= ( ord_less_eq @ nat @ N @ M ) ) ) ) ) ).
% power_decreasing_iff
thf(fact_990_power__mono__iff,axiom,
! [A: $tType] :
( ( linordered_semidom @ A )
=> ! [A2: A,B2: A,N: nat] :
( ( ord_less_eq @ A @ ( zero_zero @ A ) @ A2 )
=> ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ B2 )
=> ( ( ord_less @ nat @ ( zero_zero @ nat ) @ N )
=> ( ( ord_less_eq @ A @ ( power_power @ A @ A2 @ N ) @ ( power_power @ A @ B2 @ N ) )
= ( ord_less_eq @ A @ A2 @ B2 ) ) ) ) ) ) ).
% power_mono_iff
thf(fact_991_signed__take__bit__Suc__bit0,axiom,
! [N: nat,K: num] :
( ( bit_ri4674362597316999326ke_bit @ int @ ( suc @ N ) @ ( numeral_numeral @ int @ ( bit0 @ K ) ) )
= ( times_times @ int @ ( bit_ri4674362597316999326ke_bit @ int @ N @ ( numeral_numeral @ int @ K ) ) @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) ) ) ).
% signed_take_bit_Suc_bit0
thf(fact_992_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_993_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_994_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_995_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_996_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_997_div__lt__mult,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [I: word @ A,K: word @ A,X: word @ A] :
( ( ord_less @ ( word @ A ) @ I @ ( divide_divide @ ( word @ A ) @ K @ X ) )
=> ( ( ord_less @ ( word @ A ) @ ( zero_zero @ ( word @ A ) ) @ X )
=> ( ord_less @ ( word @ A ) @ ( times_times @ ( word @ A ) @ I @ X ) @ K ) ) ) ) ).
% div_lt_mult
thf(fact_998_More__Word_Oword__div__mult,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [C2: word @ A,A2: word @ A,B2: word @ A] :
( ( ord_less @ ( word @ A ) @ ( zero_zero @ ( word @ A ) ) @ C2 )
=> ( ( ord_less @ ( word @ A ) @ A2 @ ( times_times @ ( word @ A ) @ B2 @ C2 ) )
=> ( ord_less @ ( word @ A ) @ ( divide_divide @ ( word @ A ) @ A2 @ C2 ) @ B2 ) ) ) ) ).
% More_Word.word_div_mult
thf(fact_999_word__mod__div__equality,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [N: word @ A,B2: word @ A] :
( ( plus_plus @ ( word @ A ) @ ( times_times @ ( word @ A ) @ ( divide_divide @ ( word @ A ) @ N @ B2 ) @ B2 ) @ ( modulo_modulo @ ( word @ A ) @ N @ B2 ) )
= N ) ) ).
% word_mod_div_equality
thf(fact_1000_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_1001_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_1002_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_1003_div__by__0__word,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [X: word @ A] :
( ( divide_divide @ ( word @ A ) @ X @ ( zero_zero @ ( word @ A ) ) )
= ( zero_zero @ ( word @ A ) ) ) ) ).
% div_by_0_word
thf(fact_1004_word__div__less,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [W: word @ A,V: word @ A] :
( ( ord_less @ ( word @ A ) @ W @ V )
=> ( ( divide_divide @ ( word @ A ) @ W @ V )
= ( zero_zero @ ( word @ A ) ) ) ) ) ).
% word_div_less
thf(fact_1005_word__less__div,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [X: word @ A,Y: word @ A] :
( ( ( divide_divide @ ( word @ A ) @ X @ Y )
= ( zero_zero @ ( word @ A ) ) )
=> ( ( Y
= ( zero_zero @ ( word @ A ) ) )
| ( ord_less @ ( word @ A ) @ X @ Y ) ) ) ) ).
% word_less_div
thf(fact_1006_word__div__lt__eq__0,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [X: word @ A,Y: word @ A] :
( ( ord_less @ ( word @ A ) @ X @ Y )
=> ( ( divide_divide @ ( word @ A ) @ X @ Y )
= ( zero_zero @ ( word @ A ) ) ) ) ) ).
% word_div_lt_eq_0
thf(fact_1007_word__mod__less__divisor,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [N: word @ A,M: word @ A] :
( ( ord_less @ ( word @ A ) @ ( zero_zero @ ( word @ A ) ) @ N )
=> ( ord_less @ ( word @ A ) @ ( modulo_modulo @ ( word @ A ) @ M @ N ) @ N ) ) ) ).
% word_mod_less_divisor
thf(fact_1008_word__greater__zero__iff,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [A2: word @ A] :
( ( ord_less @ ( word @ A ) @ ( zero_zero @ ( word @ A ) ) @ A2 )
= ( A2
!= ( zero_zero @ ( word @ A ) ) ) ) ) ).
% word_greater_zero_iff
thf(fact_1009_word__le__not__less,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ( ( ord_less_eq @ ( word @ A ) )
= ( ^ [B5: word @ A,A4: word @ A] :
~ ( ord_less @ ( word @ A ) @ A4 @ B5 ) ) ) ) ).
% word_le_not_less
thf(fact_1010_word__gt__a__gt__0,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [A2: word @ A,N: word @ A] :
( ( ord_less @ ( word @ A ) @ A2 @ N )
=> ( ord_less @ ( word @ A ) @ ( zero_zero @ ( word @ A ) ) @ N ) ) ) ).
% word_gt_a_gt_0
thf(fact_1011_word__mod__by__0,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [K: word @ A] :
( ( modulo_modulo @ ( word @ A ) @ K @ ( zero_zero @ ( word @ A ) ) )
= K ) ) ).
% word_mod_by_0
thf(fact_1012_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_1013_word__neq__0__conv,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [W: word @ A] :
( ( W
!= ( zero_zero @ ( word @ A ) ) )
= ( ord_less @ ( word @ A ) @ ( zero_zero @ ( word @ A ) ) @ W ) ) ) ).
% word_neq_0_conv
thf(fact_1014_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_1015_word__gt__0,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [Y: word @ A] :
( ( ord_less @ ( word @ A ) @ ( zero_zero @ ( word @ A ) ) @ Y )
= ( ( zero_zero @ ( word @ A ) )
!= Y ) ) ) ).
% word_gt_0
thf(fact_1016_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_1017_word__coorder_Oextremum__strict,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [A2: word @ A] :
~ ( ord_less @ ( word @ A ) @ A2 @ ( zero_zero @ ( word @ A ) ) ) ) ).
% word_coorder.extremum_strict
thf(fact_1018_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_1019_word__not__simps_I1_J,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [X: word @ A] :
~ ( ord_less @ ( word @ A ) @ X @ ( zero_zero @ ( word @ A ) ) ) ) ).
% word_not_simps(1)
thf(fact_1020_lift__Suc__mono__le,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [F2: nat > A,N: nat,N5: nat] :
( ! [N2: nat] : ( ord_less_eq @ A @ ( F2 @ N2 ) @ ( F2 @ ( suc @ N2 ) ) )
=> ( ( ord_less_eq @ nat @ N @ N5 )
=> ( ord_less_eq @ A @ ( F2 @ N ) @ ( F2 @ N5 ) ) ) ) ) ).
% lift_Suc_mono_le
thf(fact_1021_lift__Suc__antimono__le,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [F2: nat > A,N: nat,N5: nat] :
( ! [N2: nat] : ( ord_less_eq @ A @ ( F2 @ ( suc @ N2 ) ) @ ( F2 @ N2 ) )
=> ( ( ord_less_eq @ nat @ N @ N5 )
=> ( ord_less_eq @ A @ ( F2 @ N5 ) @ ( F2 @ N ) ) ) ) ) ).
% lift_Suc_antimono_le
thf(fact_1022_verit__comp__simplify1_I2_J,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [A2: A] : ( ord_less_eq @ A @ A2 @ A2 ) ) ).
% verit_comp_simplify1(2)
thf(fact_1023_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_1024_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_1025_mod__word__less,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [W: word @ A,V: word @ A] :
( ( ord_less @ ( word @ A ) @ W @ V )
=> ( ( modulo_modulo @ ( word @ A ) @ W @ V )
= W ) ) ) ).
% mod_word_less
thf(fact_1026_word__le__less__eq,axiom,
! [Z6: $tType] :
( ( type_len @ Z6 )
=> ( ( ord_less_eq @ ( word @ Z6 ) )
= ( ^ [X2: word @ Z6,Y6: word @ Z6] :
( ( X2 = Y6 )
| ( ord_less @ ( word @ Z6 ) @ X2 @ Y6 ) ) ) ) ) ).
% word_le_less_eq
thf(fact_1027_plus__le__left__cancel__wrap,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [X: word @ A,Y5: word @ A,Y: word @ A] :
( ( ord_less @ ( word @ A ) @ ( plus_plus @ ( word @ A ) @ X @ Y5 ) @ X )
=> ( ( ord_less @ ( word @ A ) @ ( plus_plus @ ( word @ A ) @ X @ Y ) @ X )
=> ( ( ord_less @ ( word @ A ) @ ( plus_plus @ ( word @ A ) @ X @ Y5 ) @ ( plus_plus @ ( word @ A ) @ X @ Y ) )
= ( ord_less @ ( word @ A ) @ Y5 @ Y ) ) ) ) ) ).
% plus_le_left_cancel_wrap
thf(fact_1028_plus__le__left__cancel__nowrap,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [X: word @ A,Y5: word @ A,Y: word @ A] :
( ( ord_less_eq @ ( word @ A ) @ X @ ( plus_plus @ ( word @ A ) @ X @ Y5 ) )
=> ( ( ord_less_eq @ ( word @ A ) @ X @ ( plus_plus @ ( word @ A ) @ X @ Y ) )
=> ( ( ord_less @ ( word @ A ) @ ( plus_plus @ ( word @ A ) @ X @ Y5 ) @ ( plus_plus @ ( word @ A ) @ X @ Y ) )
= ( ord_less @ ( word @ A ) @ Y5 @ Y ) ) ) ) ) ).
% plus_le_left_cancel_nowrap
thf(fact_1029_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_1030_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_1031_subset__decode__imp__le,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq @ ( set @ nat ) @ ( nat_set_decode @ M ) @ ( nat_set_decode @ N ) )
=> ( ord_less_eq @ nat @ M @ N ) ) ).
% subset_decode_imp_le
thf(fact_1032_power__increasing,axiom,
! [A: $tType] :
( ( linordered_semidom @ A )
=> ! [N: nat,N4: nat,A2: A] :
( ( ord_less_eq @ nat @ N @ N4 )
=> ( ( ord_less_eq @ A @ ( one_one @ A ) @ A2 )
=> ( ord_less_eq @ A @ ( power_power @ A @ A2 @ N ) @ ( power_power @ A @ A2 @ N4 ) ) ) ) ) ).
% power_increasing
thf(fact_1033_signed__take__bit__add,axiom,
! [N: nat,K: int,L: int] :
( ( bit_ri4674362597316999326ke_bit @ int @ N @ ( plus_plus @ int @ ( bit_ri4674362597316999326ke_bit @ int @ N @ K ) @ ( bit_ri4674362597316999326ke_bit @ int @ N @ L ) ) )
= ( bit_ri4674362597316999326ke_bit @ int @ N @ ( plus_plus @ int @ K @ L ) ) ) ).
% signed_take_bit_add
thf(fact_1034_signed__take__bit__mult,axiom,
! [N: nat,K: int,L: int] :
( ( bit_ri4674362597316999326ke_bit @ int @ N @ ( times_times @ int @ ( bit_ri4674362597316999326ke_bit @ int @ N @ K ) @ ( bit_ri4674362597316999326ke_bit @ int @ N @ L ) ) )
= ( bit_ri4674362597316999326ke_bit @ int @ N @ ( times_times @ int @ K @ L ) ) ) ).
% signed_take_bit_mult
thf(fact_1035_zero__le,axiom,
! [A: $tType] :
( ( canoni5634975068530333245id_add @ A )
=> ! [X: A] : ( ord_less_eq @ A @ ( zero_zero @ A ) @ X ) ) ).
% zero_le
thf(fact_1036_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_1037_verit__comp__simplify1_I3_J,axiom,
! [B: $tType] :
( ( linorder @ B )
=> ! [B7: B,A5: B] :
( ( ~ ( ord_less_eq @ B @ B7 @ A5 ) )
= ( ord_less @ B @ A5 @ B7 ) ) ) ).
% verit_comp_simplify1(3)
thf(fact_1038_pinf_I6_J,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [T: A] :
? [Z3: A] :
! [X5: A] :
( ( ord_less @ A @ Z3 @ X5 )
=> ~ ( ord_less_eq @ A @ X5 @ T ) ) ) ).
% pinf(6)
thf(fact_1039_pinf_I8_J,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [T: A] :
? [Z3: A] :
! [X5: A] :
( ( ord_less @ A @ Z3 @ X5 )
=> ( ord_less_eq @ A @ T @ X5 ) ) ) ).
% pinf(8)
thf(fact_1040_minf_I6_J,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [T: A] :
? [Z3: A] :
! [X5: A] :
( ( ord_less @ A @ X5 @ Z3 )
=> ( ord_less_eq @ A @ X5 @ T ) ) ) ).
% minf(6)
thf(fact_1041_minf_I8_J,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [T: A] :
? [Z3: A] :
! [X5: A] :
( ( ord_less @ A @ X5 @ Z3 )
=> ~ ( ord_less_eq @ A @ T @ X5 ) ) ) ).
% minf(8)
thf(fact_1042_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_1043_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_1044_le__iff__add,axiom,
! [A: $tType] :
( ( canoni5634975068530333245id_add @ A )
=> ( ( ord_less_eq @ A )
= ( ^ [A4: A,B5: A] :
? [C5: A] :
( B5
= ( plus_plus @ A @ A4 @ C5 ) ) ) ) ) ).
% le_iff_add
thf(fact_1045_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_1046_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_1047_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_1048_add__mono,axiom,
! [A: $tType] :
( ( ordere6658533253407199908up_add @ A )
=> ! [A2: A,B2: A,C2: A,D2: A] :
( ( ord_less_eq @ A @ A2 @ B2 )
=> ( ( ord_less_eq @ A @ C2 @ D2 )
=> ( ord_less_eq @ A @ ( plus_plus @ A @ A2 @ C2 ) @ ( plus_plus @ A @ B2 @ D2 ) ) ) ) ) ).
% add_mono
thf(fact_1049_add__mono__thms__linordered__semiring_I1_J,axiom,
! [A: $tType] :
( ( ordere6658533253407199908up_add @ A )
=> ! [I: A,J: A,K: A,L: A] :
( ( ( ord_less_eq @ A @ I @ J )
& ( ord_less_eq @ A @ K @ L ) )
=> ( ord_less_eq @ A @ ( plus_plus @ A @ I @ K ) @ ( plus_plus @ A @ J @ L ) ) ) ) ).
% add_mono_thms_linordered_semiring(1)
thf(fact_1050_add__mono__thms__linordered__semiring_I2_J,axiom,
! [A: $tType] :
( ( ordere6658533253407199908up_add @ A )
=> ! [I: A,J: A,K: A,L: A] :
( ( ( I = J )
& ( ord_less_eq @ A @ K @ L ) )
=> ( ord_less_eq @ A @ ( plus_plus @ A @ I @ K ) @ ( plus_plus @ A @ J @ L ) ) ) ) ).
% add_mono_thms_linordered_semiring(2)
thf(fact_1051_add__mono__thms__linordered__semiring_I3_J,axiom,
! [A: $tType] :
( ( ordere6658533253407199908up_add @ A )
=> ! [I: A,J: A,K: A,L: A] :
( ( ( ord_less_eq @ A @ I @ J )
& ( K = L ) )
=> ( ord_less_eq @ A @ ( plus_plus @ A @ I @ K ) @ ( plus_plus @ A @ J @ L ) ) ) ) ).
% add_mono_thms_linordered_semiring(3)
thf(fact_1052_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_1053_le__num__One__iff,axiom,
! [X: num] :
( ( ord_less_eq @ num @ X @ one2 )
= ( X = one2 ) ) ).
% le_num_One_iff
thf(fact_1054_Suc__leD,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq @ nat @ ( suc @ M ) @ N )
=> ( ord_less_eq @ nat @ M @ N ) ) ).
% Suc_leD
thf(fact_1055_le__SucE,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq @ nat @ M @ ( suc @ N ) )
=> ( ~ ( ord_less_eq @ nat @ M @ N )
=> ( M
= ( suc @ N ) ) ) ) ).
% le_SucE
thf(fact_1056_le__SucI,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq @ nat @ M @ N )
=> ( ord_less_eq @ nat @ M @ ( suc @ N ) ) ) ).
% le_SucI
thf(fact_1057_Suc__le__D,axiom,
! [N: nat,M6: nat] :
( ( ord_less_eq @ nat @ ( suc @ N ) @ M6 )
=> ? [M2: nat] :
( M6
= ( suc @ M2 ) ) ) ).
% Suc_le_D
thf(fact_1058_le__Suc__eq,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq @ nat @ M @ ( suc @ N ) )
= ( ( ord_less_eq @ nat @ M @ N )
| ( M
= ( suc @ N ) ) ) ) ).
% le_Suc_eq
thf(fact_1059_Suc__n__not__le__n,axiom,
! [N: nat] :
~ ( ord_less_eq @ nat @ ( suc @ N ) @ N ) ).
% Suc_n_not_le_n
thf(fact_1060_not__less__eq__eq,axiom,
! [M: nat,N: nat] :
( ( ~ ( ord_less_eq @ nat @ M @ N ) )
= ( ord_less_eq @ nat @ ( suc @ N ) @ M ) ) ).
% not_less_eq_eq
thf(fact_1061_full__nat__induct,axiom,
! [P: nat > $o,N: nat] :
( ! [N2: nat] :
( ! [M3: nat] :
( ( ord_less_eq @ nat @ ( suc @ M3 ) @ N2 )
=> ( P @ M3 ) )
=> ( P @ N2 ) )
=> ( P @ N ) ) ).
% full_nat_induct
thf(fact_1062_nat__induct__at__least,axiom,
! [M: nat,N: nat,P: nat > $o] :
( ( ord_less_eq @ nat @ M @ N )
=> ( ( P @ M )
=> ( ! [N2: nat] :
( ( ord_less_eq @ nat @ M @ N2 )
=> ( ( P @ N2 )
=> ( P @ ( suc @ N2 ) ) ) )
=> ( P @ N ) ) ) ) ).
% nat_induct_at_least
thf(fact_1063_transitive__stepwise__le,axiom,
! [M: nat,N: nat,R: nat > nat > $o] :
( ( ord_less_eq @ nat @ M @ N )
=> ( ! [X3: nat] : ( R @ X3 @ X3 )
=> ( ! [X3: nat,Y3: nat,Z3: nat] :
( ( R @ X3 @ Y3 )
=> ( ( R @ Y3 @ Z3 )
=> ( R @ X3 @ Z3 ) ) )
=> ( ! [N2: nat] : ( R @ N2 @ ( suc @ N2 ) )
=> ( R @ M @ N ) ) ) ) ) ).
% transitive_stepwise_le
thf(fact_1064_less__eq__nat_Osimps_I1_J,axiom,
! [N: nat] : ( ord_less_eq @ nat @ ( zero_zero @ nat ) @ N ) ).
% less_eq_nat.simps(1)
thf(fact_1065_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_1066_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_1067_le__0__eq,axiom,
! [N: nat] :
( ( ord_less_eq @ nat @ N @ ( zero_zero @ nat ) )
= ( N
= ( zero_zero @ nat ) ) ) ).
% le_0_eq
thf(fact_1068_exists__leI,axiom,
! [N: nat,P: nat > $o] :
( ( ! [N6: nat] :
( ( ord_less @ nat @ N6 @ N )
=> ~ ( P @ N6 ) )
=> ( P @ N ) )
=> ? [N7: nat] :
( ( ord_less_eq @ nat @ N7 @ N )
& ( P @ N7 ) ) ) ).
% exists_leI
thf(fact_1069_nat__less__le,axiom,
( ( ord_less @ nat )
= ( ^ [M4: nat,N3: nat] :
( ( ord_less_eq @ nat @ M4 @ N3 )
& ( M4 != N3 ) ) ) ) ).
% nat_less_le
thf(fact_1070_less__imp__le__nat,axiom,
! [M: nat,N: nat] :
( ( ord_less @ nat @ M @ N )
=> ( ord_less_eq @ nat @ M @ N ) ) ).
% less_imp_le_nat
thf(fact_1071_le__eq__less__or__eq,axiom,
( ( ord_less_eq @ nat )
= ( ^ [M4: nat,N3: nat] :
( ( ord_less @ nat @ M4 @ N3 )
| ( M4 = N3 ) ) ) ) ).
% le_eq_less_or_eq
thf(fact_1072_less__or__eq__imp__le,axiom,
! [M: nat,N: nat] :
( ( ( ord_less @ nat @ M @ N )
| ( M = N ) )
=> ( ord_less_eq @ nat @ M @ N ) ) ).
% less_or_eq_imp_le
thf(fact_1073_le__neq__implies__less,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq @ nat @ M @ N )
=> ( ( M != N )
=> ( ord_less @ nat @ M @ N ) ) ) ).
% le_neq_implies_less
thf(fact_1074_less__mono__imp__le__mono,axiom,
! [F2: nat > nat,I: nat,J: nat] :
( ! [I3: nat,J3: nat] :
( ( ord_less @ nat @ I3 @ J3 )
=> ( ord_less @ nat @ ( F2 @ I3 ) @ ( F2 @ J3 ) ) )
=> ( ( ord_less_eq @ nat @ I @ J )
=> ( ord_less_eq @ nat @ ( F2 @ I ) @ ( F2 @ J ) ) ) ) ).
% less_mono_imp_le_mono
thf(fact_1075_nat__le__iff__add,axiom,
( ( ord_less_eq @ nat )
= ( ^ [M4: nat,N3: nat] :
? [K3: nat] :
( N3
= ( plus_plus @ nat @ M4 @ K3 ) ) ) ) ).
% nat_le_iff_add
thf(fact_1076_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_1077_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_1078_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_1079_add__le__mono,axiom,
! [I: nat,J: nat,K: nat,L: nat] :
( ( ord_less_eq @ nat @ I @ J )
=> ( ( ord_less_eq @ nat @ K @ L )
=> ( ord_less_eq @ nat @ ( plus_plus @ nat @ I @ K ) @ ( plus_plus @ nat @ J @ L ) ) ) ) ).
% add_le_mono
thf(fact_1080_le__Suc__ex,axiom,
! [K: nat,L: nat] :
( ( ord_less_eq @ nat @ K @ L )
=> ? [N2: nat] :
( L
= ( plus_plus @ nat @ K @ N2 ) ) ) ).
% le_Suc_ex
thf(fact_1081_add__leD2,axiom,
! [M: nat,K: nat,N: nat] :
( ( ord_less_eq @ nat @ ( plus_plus @ nat @ M @ K ) @ N )
=> ( ord_less_eq @ nat @ K @ N ) ) ).
% add_leD2
thf(fact_1082_add__leD1,axiom,
! [M: nat,K: nat,N: nat] :
( ( ord_less_eq @ nat @ ( plus_plus @ nat @ M @ K ) @ N )
=> ( ord_less_eq @ nat @ M @ N ) ) ).
% add_leD1
thf(fact_1083_le__add2,axiom,
! [N: nat,M: nat] : ( ord_less_eq @ nat @ N @ ( plus_plus @ nat @ M @ N ) ) ).
% le_add2
thf(fact_1084_le__add1,axiom,
! [N: nat,M: nat] : ( ord_less_eq @ nat @ N @ ( plus_plus @ nat @ N @ M ) ) ).
% le_add1
thf(fact_1085_add__leE,axiom,
! [M: nat,K: nat,N: nat] :
( ( ord_less_eq @ nat @ ( plus_plus @ nat @ M @ K ) @ N )
=> ~ ( ( ord_less_eq @ nat @ M @ N )
=> ~ ( ord_less_eq @ nat @ K @ N ) ) ) ).
% add_leE
thf(fact_1086_less__eq__real__def,axiom,
( ( ord_less_eq @ real )
= ( ^ [X2: real,Y6: real] :
( ( ord_less @ real @ X2 @ Y6 )
| ( X2 = Y6 ) ) ) ) ).
% less_eq_real_def
thf(fact_1087_conj__le__cong,axiom,
! [X: int,X6: int,P: $o,P3: $o] :
( ( X = X6 )
=> ( ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ X6 )
=> ( P = P3 ) )
=> ( ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ X )
& P )
= ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ X6 )
& P3 ) ) ) ) ).
% conj_le_cong
thf(fact_1088_imp__le__cong,axiom,
! [X: int,X6: int,P: $o,P3: $o] :
( ( X = X6 )
=> ( ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ X6 )
=> ( P = P3 ) )
=> ( ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ X )
=> P )
= ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ X6 )
=> P3 ) ) ) ) ).
% imp_le_cong
thf(fact_1089_less__eq__int__code_I1_J,axiom,
ord_less_eq @ int @ ( zero_zero @ int ) @ ( zero_zero @ int ) ).
% less_eq_int_code(1)
thf(fact_1090_div__le__mono,axiom,
! [M: nat,N: nat,K: nat] :
( ( ord_less_eq @ nat @ M @ N )
=> ( ord_less_eq @ nat @ ( divide_divide @ nat @ M @ K ) @ ( divide_divide @ nat @ N @ K ) ) ) ).
% div_le_mono
thf(fact_1091_div__le__dividend,axiom,
! [M: nat,N: nat] : ( ord_less_eq @ nat @ ( divide_divide @ nat @ M @ N ) @ M ) ).
% div_le_dividend
thf(fact_1092_le__cube,axiom,
! [M: nat] : ( ord_less_eq @ nat @ M @ ( times_times @ nat @ M @ ( times_times @ nat @ M @ M ) ) ) ).
% le_cube
thf(fact_1093_le__square,axiom,
! [M: nat] : ( ord_less_eq @ nat @ M @ ( times_times @ nat @ M @ M ) ) ).
% le_square
thf(fact_1094_mult__le__mono,axiom,
! [I: nat,J: nat,K: nat,L: nat] :
( ( ord_less_eq @ nat @ I @ J )
=> ( ( ord_less_eq @ nat @ K @ L )
=> ( ord_less_eq @ nat @ ( times_times @ nat @ I @ K ) @ ( times_times @ nat @ J @ L ) ) ) ) ).
% mult_le_mono
thf(fact_1095_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_1096_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_1097_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_1098_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_1099_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_1100_word__le__plus__1,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [Y: word @ A,N: word @ A,A2: word @ A] :
( ( ord_less @ ( word @ A ) @ Y @ ( plus_plus @ ( word @ A ) @ Y @ N ) )
=> ( ( ord_less @ ( word @ A ) @ A2 @ N )
=> ( ord_less_eq @ ( word @ A ) @ ( plus_plus @ ( word @ A ) @ Y @ A2 ) @ ( plus_plus @ ( word @ A ) @ ( plus_plus @ ( word @ A ) @ Y @ A2 ) @ ( one_one @ ( word @ A ) ) ) ) ) ) ) ).
% word_le_plus_1
thf(fact_1101_plus__one__helper,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [X: word @ A,N: word @ A] :
( ( ord_less @ ( word @ A ) @ X @ ( plus_plus @ ( word @ A ) @ N @ ( one_one @ ( word @ A ) ) ) )
=> ( ord_less_eq @ ( word @ A ) @ X @ N ) ) ) ).
% plus_one_helper
thf(fact_1102_plus__one__helper2,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [X: word @ A,N: word @ A] :
( ( ord_less_eq @ ( word @ A ) @ X @ N )
=> ( ( ( plus_plus @ ( word @ A ) @ N @ ( one_one @ ( word @ A ) ) )
!= ( zero_zero @ ( word @ A ) ) )
=> ( ord_less @ ( word @ A ) @ X @ ( plus_plus @ ( word @ A ) @ N @ ( one_one @ ( word @ A ) ) ) ) ) ) ) ).
% plus_one_helper2
thf(fact_1103_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_1104_mod__less__eq__dividend,axiom,
! [M: nat,N: nat] : ( ord_less_eq @ nat @ ( modulo_modulo @ nat @ M @ N ) @ M ) ).
% mod_less_eq_dividend
thf(fact_1105_i0__lb,axiom,
! [N: extended_enat] : ( ord_less_eq @ extended_enat @ ( zero_zero @ extended_enat ) @ N ) ).
% i0_lb
thf(fact_1106_ile0__eq,axiom,
! [N: extended_enat] :
( ( ord_less_eq @ extended_enat @ N @ ( zero_zero @ extended_enat ) )
= ( N
= ( zero_zero @ extended_enat ) ) ) ).
% ile0_eq
thf(fact_1107_power__decreasing,axiom,
! [A: $tType] :
( ( linordered_semidom @ A )
=> ! [N: nat,N4: nat,A2: A] :
( ( ord_less_eq @ nat @ N @ N4 )
=> ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ A2 )
=> ( ( ord_less_eq @ A @ A2 @ ( one_one @ A ) )
=> ( ord_less_eq @ A @ ( power_power @ A @ A2 @ N4 ) @ ( power_power @ A @ A2 @ N ) ) ) ) ) ) ).
% power_decreasing
thf(fact_1108_power__le__imp__le__exp,axiom,
! [A: $tType] :
( ( linordered_semidom @ A )
=> ! [A2: A,M: nat,N: nat] :
( ( ord_less @ A @ ( one_one @ A ) @ A2 )
=> ( ( ord_less_eq @ A @ ( power_power @ A @ A2 @ M ) @ ( power_power @ A @ A2 @ N ) )
=> ( ord_less_eq @ nat @ M @ N ) ) ) ) ).
% power_le_imp_le_exp
thf(fact_1109_unset__bit__less__eq,axiom,
! [N: nat,K: int] : ( ord_less_eq @ int @ ( bit_se2638667681897837118et_bit @ int @ N @ K ) @ K ) ).
% unset_bit_less_eq
thf(fact_1110_set__bit__greater__eq,axiom,
! [K: int,N: nat] : ( ord_less_eq @ int @ K @ ( bit_se5668285175392031749et_bit @ int @ N @ K ) ) ).
% set_bit_greater_eq
thf(fact_1111_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_1112_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_1113_not__numeral__le__zero,axiom,
! [A: $tType] :
( ( linord181362715937106298miring @ A )
=> ! [N: num] :
~ ( ord_less_eq @ A @ ( numeral_numeral @ A @ N ) @ ( zero_zero @ A ) ) ) ).
% not_numeral_le_zero
thf(fact_1114_zero__le__numeral,axiom,
! [A: $tType] :
( ( linord181362715937106298miring @ A )
=> ! [N: num] : ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( numeral_numeral @ A @ N ) ) ) ).
% zero_le_numeral
thf(fact_1115_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_1116_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_1117_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_1118_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_1119_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_1120_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_1121_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_1122_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_1123_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_1124_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_1125_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_1126_mult__mono,axiom,
! [A: $tType] :
( ( ordered_semiring @ A )
=> ! [A2: A,B2: A,C2: A,D2: A] :
( ( ord_less_eq @ A @ A2 @ B2 )
=> ( ( ord_less_eq @ A @ C2 @ D2 )
=> ( ( 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 @ D2 ) ) ) ) ) ) ) ).
% mult_mono
thf(fact_1127_mult__mono_H,axiom,
! [A: $tType] :
( ( ordered_semiring @ A )
=> ! [A2: A,B2: A,C2: A,D2: A] :
( ( ord_less_eq @ A @ A2 @ B2 )
=> ( ( ord_less_eq @ A @ C2 @ D2 )
=> ( ( 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 @ D2 ) ) ) ) ) ) ) ).
% mult_mono'
thf(fact_1128_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_1129_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_1130_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_1131_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_1132_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_1133_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_1134_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_1135_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_1136_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_1137_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_1138_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_1139_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_1140_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_1141_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_1142_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_1143_add__less__le__mono,axiom,
! [A: $tType] :
( ( ordere580206878836729694up_add @ A )
=> ! [A2: A,B2: A,C2: A,D2: A] :
( ( ord_less @ A @ A2 @ B2 )
=> ( ( ord_less_eq @ A @ C2 @ D2 )
=> ( ord_less @ A @ ( plus_plus @ A @ A2 @ C2 ) @ ( plus_plus @ A @ B2 @ D2 ) ) ) ) ) ).
% add_less_le_mono
thf(fact_1144_add__le__less__mono,axiom,
! [A: $tType] :
( ( ordere580206878836729694up_add @ A )
=> ! [A2: A,B2: A,C2: A,D2: A] :
( ( ord_less_eq @ A @ A2 @ B2 )
=> ( ( ord_less @ A @ C2 @ D2 )
=> ( ord_less @ A @ ( plus_plus @ A @ A2 @ C2 ) @ ( plus_plus @ A @ B2 @ D2 ) ) ) ) ) ).
% add_le_less_mono
thf(fact_1145_add__mono__thms__linordered__field_I3_J,axiom,
! [A: $tType] :
( ( ordere580206878836729694up_add @ A )
=> ! [I: A,J: A,K: A,L: A] :
( ( ( ord_less @ A @ I @ J )
& ( ord_less_eq @ A @ K @ L ) )
=> ( ord_less @ A @ ( plus_plus @ A @ I @ K ) @ ( plus_plus @ A @ J @ L ) ) ) ) ).
% add_mono_thms_linordered_field(3)
thf(fact_1146_add__mono__thms__linordered__field_I4_J,axiom,
! [A: $tType] :
( ( ordere580206878836729694up_add @ A )
=> ! [I: A,J: A,K: A,L: A] :
( ( ( ord_less_eq @ A @ I @ J )
& ( ord_less @ A @ K @ L ) )
=> ( ord_less @ A @ ( plus_plus @ A @ I @ K ) @ ( plus_plus @ A @ J @ L ) ) ) ) ).
% add_mono_thms_linordered_field(4)
thf(fact_1147_one__le__numeral,axiom,
! [A: $tType] :
( ( linord181362715937106298miring @ A )
=> ! [N: num] : ( ord_less_eq @ A @ ( one_one @ A ) @ ( numeral_numeral @ A @ N ) ) ) ).
% one_le_numeral
thf(fact_1148_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_1149_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_1150_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_1151_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_1152_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_1153_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_1154_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_1155_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_1156_zero__le__power,axiom,
! [A: $tType] :
( ( linordered_semidom @ A )
=> ! [A2: A,N: nat] :
( ( ord_less_eq @ A @ ( zero_zero @ A ) @ A2 )
=> ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( power_power @ A @ A2 @ N ) ) ) ) ).
% zero_le_power
thf(fact_1157_power__mono,axiom,
! [A: $tType] :
( ( linordered_semidom @ A )
=> ! [A2: A,B2: A,N: nat] :
( ( ord_less_eq @ A @ A2 @ B2 )
=> ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ A2 )
=> ( ord_less_eq @ A @ ( power_power @ A @ A2 @ N ) @ ( power_power @ A @ B2 @ N ) ) ) ) ) ).
% power_mono
thf(fact_1158_one__le__power,axiom,
! [A: $tType] :
( ( linordered_semidom @ A )
=> ! [A2: A,N: nat] :
( ( ord_less_eq @ A @ ( one_one @ A ) @ A2 )
=> ( ord_less_eq @ A @ ( one_one @ A ) @ ( power_power @ A @ A2 @ N ) ) ) ) ).
% one_le_power
thf(fact_1159_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_1160_nat__compl__induct_H,axiom,
! [P: nat > $o,N: nat] :
( ( P @ ( zero_zero @ nat ) )
=> ( ! [N2: nat] :
( ! [Nn: nat] :
( ( ord_less_eq @ nat @ Nn @ N2 )
=> ( P @ Nn ) )
=> ( P @ ( suc @ N2 ) ) )
=> ( P @ N ) ) ) ).
% nat_compl_induct'
thf(fact_1161_nat__compl__induct,axiom,
! [P: nat > $o,N: nat] :
( ( P @ ( zero_zero @ nat ) )
=> ( ! [N2: nat] :
( ! [Nn: nat] :
( ( ord_less_eq @ nat @ Nn @ N2 )
=> ( P @ Nn ) )
=> ( P @ ( suc @ N2 ) ) )
=> ( P @ N ) ) ) ).
% nat_compl_induct
thf(fact_1162_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_1163_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_1164_le__imp__less__Suc,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq @ nat @ M @ N )
=> ( ord_less @ nat @ M @ ( suc @ N ) ) ) ).
% le_imp_less_Suc
thf(fact_1165_less__eq__Suc__le,axiom,
( ( ord_less @ nat )
= ( ^ [N3: nat] : ( ord_less_eq @ nat @ ( suc @ N3 ) ) ) ) ).
% less_eq_Suc_le
thf(fact_1166_less__Suc__eq__le,axiom,
! [M: nat,N: nat] :
( ( ord_less @ nat @ M @ ( suc @ N ) )
= ( ord_less_eq @ nat @ M @ N ) ) ).
% less_Suc_eq_le
thf(fact_1167_le__less__Suc__eq,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq @ nat @ M @ N )
=> ( ( ord_less @ nat @ N @ ( suc @ M ) )
= ( N = M ) ) ) ).
% le_less_Suc_eq
thf(fact_1168_Suc__le__lessD,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq @ nat @ ( suc @ M ) @ N )
=> ( ord_less @ nat @ M @ N ) ) ).
% Suc_le_lessD
thf(fact_1169_inc__induct,axiom,
! [I: nat,J: nat,P: nat > $o] :
( ( ord_less_eq @ nat @ I @ J )
=> ( ( P @ J )
=> ( ! [N2: nat] :
( ( ord_less_eq @ nat @ I @ N2 )
=> ( ( ord_less @ nat @ N2 @ J )
=> ( ( P @ ( suc @ N2 ) )
=> ( P @ N2 ) ) ) )
=> ( P @ I ) ) ) ) ).
% inc_induct
thf(fact_1170_dec__induct,axiom,
! [I: nat,J: nat,P: nat > $o] :
( ( ord_less_eq @ nat @ I @ J )
=> ( ( P @ I )
=> ( ! [N2: nat] :
( ( ord_less_eq @ nat @ I @ N2 )
=> ( ( ord_less @ nat @ N2 @ J )
=> ( ( P @ N2 )
=> ( P @ ( suc @ N2 ) ) ) ) )
=> ( P @ J ) ) ) ) ).
% dec_induct
thf(fact_1171_Suc__le__eq,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq @ nat @ ( suc @ M ) @ N )
= ( ord_less @ nat @ M @ N ) ) ).
% Suc_le_eq
thf(fact_1172_Suc__leI,axiom,
! [M: nat,N: nat] :
( ( ord_less @ nat @ M @ N )
=> ( ord_less_eq @ nat @ ( suc @ M ) @ N ) ) ).
% Suc_leI
thf(fact_1173_ex__least__nat__le,axiom,
! [P: nat > $o,N: nat] :
( ( P @ N )
=> ( ~ ( P @ ( zero_zero @ nat ) )
=> ? [K2: nat] :
( ( ord_less_eq @ nat @ K2 @ N )
& ! [I4: nat] :
( ( ord_less @ nat @ I4 @ K2 )
=> ~ ( P @ I4 ) )
& ( P @ K2 ) ) ) ) ).
% ex_least_nat_le
thf(fact_1174_mono__nat__linear__lb,axiom,
! [F2: nat > nat,M: nat,K: nat] :
( ! [M2: nat,N2: nat] :
( ( ord_less @ nat @ M2 @ N2 )
=> ( ord_less @ nat @ ( F2 @ M2 ) @ ( F2 @ N2 ) ) )
=> ( ord_less_eq @ nat @ ( plus_plus @ nat @ ( F2 @ M ) @ K ) @ ( F2 @ ( plus_plus @ nat @ M @ K ) ) ) ) ).
% mono_nat_linear_lb
thf(fact_1175_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_1176_le__imp__power__dvd,axiom,
! [A: $tType] :
( ( comm_semiring_1 @ A )
=> ! [M: nat,N: nat,A2: A] :
( ( ord_less_eq @ nat @ M @ N )
=> ( dvd_dvd @ A @ ( power_power @ A @ A2 @ M ) @ ( power_power @ A @ A2 @ N ) ) ) ) ).
% le_imp_power_dvd
thf(fact_1177_power__le__dvd,axiom,
! [A: $tType] :
( ( comm_semiring_1 @ A )
=> ! [A2: A,N: nat,B2: A,M: nat] :
( ( dvd_dvd @ A @ ( power_power @ A @ A2 @ N ) @ B2 )
=> ( ( ord_less_eq @ nat @ M @ N )
=> ( dvd_dvd @ A @ ( power_power @ A @ A2 @ M ) @ B2 ) ) ) ) ).
% power_le_dvd
thf(fact_1178_dvd__power__le,axiom,
! [A: $tType] :
( ( comm_semiring_1 @ A )
=> ! [X: A,Y: A,N: nat,M: nat] :
( ( dvd_dvd @ A @ X @ Y )
=> ( ( ord_less_eq @ nat @ N @ M )
=> ( dvd_dvd @ A @ ( power_power @ A @ X @ N ) @ ( power_power @ A @ Y @ M ) ) ) ) ) ).
% dvd_power_le
thf(fact_1179_Suc__div__le__mono,axiom,
! [M: nat,N: nat] : ( ord_less_eq @ nat @ ( divide_divide @ nat @ M @ N ) @ ( divide_divide @ nat @ ( suc @ M ) @ N ) ) ).
% Suc_div_le_mono
thf(fact_1180_Suc__mult__le__cancel1,axiom,
! [K: nat,M: nat,N: nat] :
( ( ord_less_eq @ nat @ ( times_times @ nat @ ( suc @ K ) @ M ) @ ( times_times @ nat @ ( suc @ K ) @ N ) )
= ( ord_less_eq @ nat @ M @ N ) ) ).
% Suc_mult_le_cancel1
thf(fact_1181_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_1182_mlex__leI,axiom,
! [A2: nat,A5: nat,B2: nat,B7: nat,N4: nat] :
( ( ord_less_eq @ nat @ A2 @ A5 )
=> ( ( ord_less_eq @ nat @ B2 @ B7 )
=> ( ord_less_eq @ nat @ ( plus_plus @ nat @ ( times_times @ nat @ A2 @ N4 ) @ B2 ) @ ( plus_plus @ nat @ ( times_times @ nat @ A5 @ N4 ) @ B7 ) ) ) ) ).
% mlex_leI
thf(fact_1183_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_1184_mod__Suc__le__divisor,axiom,
! [M: nat,N: nat] : ( ord_less_eq @ nat @ ( modulo_modulo @ nat @ M @ ( suc @ N ) ) @ N ) ).
% mod_Suc_le_divisor
thf(fact_1185_div__times__less__eq__dividend,axiom,
! [M: nat,N: nat] : ( ord_less_eq @ nat @ ( times_times @ nat @ ( divide_divide @ nat @ M @ N ) @ N ) @ M ) ).
% div_times_less_eq_dividend
thf(fact_1186_times__div__less__eq__dividend,axiom,
! [N: nat,M: nat] : ( ord_less_eq @ nat @ ( times_times @ nat @ N @ ( divide_divide @ nat @ M @ N ) ) @ M ) ).
% times_div_less_eq_dividend
thf(fact_1187_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_1188_int__ge__induct,axiom,
! [K: int,I: int,P: int > $o] :
( ( ord_less_eq @ int @ K @ I )
=> ( ( P @ K )
=> ( ! [I3: int] :
( ( ord_less_eq @ int @ K @ I3 )
=> ( ( P @ I3 )
=> ( P @ ( plus_plus @ int @ I3 @ ( one_one @ int ) ) ) ) )
=> ( P @ I ) ) ) ) ).
% int_ge_induct
thf(fact_1189_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_1190_zdvd__antisym__nonneg,axiom,
! [M: int,N: int] :
( ( ord_less_eq @ int @ ( zero_zero @ int ) @ M )
=> ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ N )
=> ( ( dvd_dvd @ int @ M @ N )
=> ( ( dvd_dvd @ int @ N @ M )
=> ( M = N ) ) ) ) ) ).
% zdvd_antisym_nonneg
thf(fact_1191_signed__take__bit__int__less__self__iff,axiom,
! [N: nat,K: int] :
( ( ord_less @ int @ ( bit_ri4674362597316999326ke_bit @ int @ N @ K ) @ K )
= ( ord_less_eq @ int @ ( power_power @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ N ) @ K ) ) ).
% signed_take_bit_int_less_self_iff
thf(fact_1192_signed__take__bit__int__greater__eq__self__iff,axiom,
! [K: int,N: nat] :
( ( ord_less_eq @ int @ K @ ( bit_ri4674362597316999326ke_bit @ int @ N @ K ) )
= ( ord_less @ int @ K @ ( power_power @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ N ) ) ) ).
% signed_take_bit_int_greater_eq_self_iff
thf(fact_1193_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_1194_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_1195_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_1196_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_1197_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_1198_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_1199_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_1200_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_1201_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_1202_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_1203_mult__strict__mono,axiom,
! [A: $tType] :
( ( linord8928482502909563296strict @ A )
=> ! [A2: A,B2: A,C2: A,D2: A] :
( ( ord_less @ A @ A2 @ B2 )
=> ( ( ord_less @ A @ C2 @ D2 )
=> ( ( 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 @ D2 ) ) ) ) ) ) ) ).
% mult_strict_mono
thf(fact_1204_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_1205_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_1206_mult__strict__mono_H,axiom,
! [A: $tType] :
( ( linord8928482502909563296strict @ A )
=> ! [A2: A,B2: A,C2: A,D2: A] :
( ( ord_less @ A @ A2 @ B2 )
=> ( ( ord_less @ A @ C2 @ D2 )
=> ( ( 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 @ D2 ) ) ) ) ) ) ) ).
% mult_strict_mono'
thf(fact_1207_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_1208_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_1209_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_1210_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_1211_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_1212_mult__le__less__imp__less,axiom,
! [A: $tType] :
( ( linord8928482502909563296strict @ A )
=> ! [A2: A,B2: A,C2: A,D2: A] :
( ( ord_less_eq @ A @ A2 @ B2 )
=> ( ( ord_less @ A @ C2 @ D2 )
=> ( ( 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 @ D2 ) ) ) ) ) ) ) ).
% mult_le_less_imp_less
thf(fact_1213_mult__less__le__imp__less,axiom,
! [A: $tType] :
( ( linord8928482502909563296strict @ A )
=> ! [A2: A,B2: A,C2: A,D2: A] :
( ( ord_less @ A @ A2 @ B2 )
=> ( ( ord_less_eq @ A @ C2 @ D2 )
=> ( ( 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 @ D2 ) ) ) ) ) ) ) ).
% mult_less_le_imp_less
thf(fact_1214_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_1215_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_1216_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_1217_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_1218_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_1219_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_1220_discrete,axiom,
! [A: $tType] :
( ( unique1627219031080169319umeral @ A )
=> ( ( ord_less @ A )
= ( ^ [A4: A] : ( ord_less_eq @ A @ ( plus_plus @ A @ A4 @ ( one_one @ A ) ) ) ) ) ) ).
% discrete
thf(fact_1221_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_1222_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_1223_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_1224_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_1225_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_1226_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_1227_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_1228_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_1229_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_1230_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_1231_power__less__imp__less__base,axiom,
! [A: $tType] :
( ( linordered_semidom @ A )
=> ! [A2: A,N: nat,B2: A] :
( ( ord_less @ A @ ( power_power @ A @ A2 @ N ) @ ( power_power @ A @ B2 @ N ) )
=> ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ B2 )
=> ( ord_less @ A @ A2 @ B2 ) ) ) ) ).
% power_less_imp_less_base
thf(fact_1232_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_1233_power__le__one,axiom,
! [A: $tType] :
( ( linordered_semidom @ A )
=> ! [A2: A,N: nat] :
( ( ord_less_eq @ A @ ( zero_zero @ A ) @ A2 )
=> ( ( ord_less_eq @ A @ A2 @ ( one_one @ A ) )
=> ( ord_less_eq @ A @ ( power_power @ A @ A2 @ N ) @ ( one_one @ A ) ) ) ) ) ).
% power_le_one
thf(fact_1234_power__inject__base,axiom,
! [A: $tType] :
( ( linordered_semidom @ A )
=> ! [A2: A,N: nat,B2: A] :
( ( ( power_power @ A @ A2 @ ( suc @ N ) )
= ( power_power @ A @ B2 @ ( suc @ N ) ) )
=> ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ A2 )
=> ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ B2 )
=> ( A2 = B2 ) ) ) ) ) ).
% power_inject_base
thf(fact_1235_power__le__imp__le__base,axiom,
! [A: $tType] :
( ( linordered_semidom @ A )
=> ! [A2: A,N: nat,B2: A] :
( ( ord_less_eq @ A @ ( power_power @ A @ A2 @ ( suc @ N ) ) @ ( power_power @ A @ B2 @ ( suc @ N ) ) )
=> ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ B2 )
=> ( ord_less_eq @ A @ A2 @ B2 ) ) ) ) ).
% power_le_imp_le_base
thf(fact_1236_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_1237_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_1238_ex__least__nat__less,axiom,
! [P: nat > $o,N: nat] :
( ( P @ N )
=> ( ~ ( P @ ( zero_zero @ nat ) )
=> ? [K2: nat] :
( ( ord_less @ nat @ K2 @ N )
& ! [I4: nat] :
( ( ord_less_eq @ nat @ I4 @ K2 )
=> ~ ( P @ I4 ) )
& ( P @ ( suc @ K2 ) ) ) ) ) ).
% ex_least_nat_less
thf(fact_1239_nat__one__le__power,axiom,
! [I: nat,N: nat] :
( ( ord_less_eq @ nat @ ( suc @ ( zero_zero @ nat ) ) @ I )
=> ( ord_less_eq @ nat @ ( suc @ ( zero_zero @ nat ) ) @ ( power_power @ nat @ I @ N ) ) ) ).
% nat_one_le_power
thf(fact_1240_div__le__mono2,axiom,
! [M: nat,N: nat,K: nat] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ M )
=> ( ( ord_less_eq @ nat @ M @ N )
=> ( ord_less_eq @ nat @ ( divide_divide @ nat @ K @ N ) @ ( divide_divide @ nat @ K @ M ) ) ) ) ).
% div_le_mono2
thf(fact_1241_div__greater__zero__iff,axiom,
! [M: nat,N: nat] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ ( divide_divide @ nat @ M @ N ) )
= ( ( ord_less_eq @ nat @ N @ M )
& ( ord_less @ nat @ ( zero_zero @ nat ) @ N ) ) ) ).
% div_greater_zero_iff
thf(fact_1242_nat__mult__le__cancel1,axiom,
! [K: nat,M: nat,N: nat] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ K )
=> ( ( ord_less_eq @ nat @ ( times_times @ nat @ K @ M ) @ ( times_times @ nat @ K @ N ) )
= ( ord_less_eq @ nat @ M @ N ) ) ) ).
% nat_mult_le_cancel1
thf(fact_1243_dvd__imp__le,axiom,
! [K: nat,N: nat] :
( ( dvd_dvd @ nat @ K @ N )
=> ( ( ord_less @ nat @ ( zero_zero @ nat ) @ N )
=> ( ord_less_eq @ nat @ K @ N ) ) ) ).
% dvd_imp_le
thf(fact_1244_mod__le__divisor,axiom,
! [N: nat,M: nat] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ N )
=> ( ord_less_eq @ nat @ ( modulo_modulo @ nat @ M @ N ) @ N ) ) ).
% mod_le_divisor
thf(fact_1245_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_1246_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_1247_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_1248_neg__mod__sign,axiom,
! [L: int,K: int] :
( ( ord_less @ int @ L @ ( zero_zero @ int ) )
=> ( ord_less_eq @ int @ ( modulo_modulo @ int @ K @ L ) @ ( zero_zero @ int ) ) ) ).
% neg_mod_sign
thf(fact_1249_Euclidean__Division_Opos__mod__sign,axiom,
! [L: int,K: int] :
( ( ord_less @ int @ ( zero_zero @ int ) @ L )
=> ( ord_less_eq @ int @ ( zero_zero @ int ) @ ( modulo_modulo @ int @ K @ L ) ) ) ).
% Euclidean_Division.pos_mod_sign
thf(fact_1250_int__mod__lem,axiom,
! [N: int,B2: int] :
( ( ord_less @ int @ ( zero_zero @ int ) @ N )
=> ( ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ B2 )
& ( ord_less @ int @ B2 @ N ) )
= ( ( modulo_modulo @ int @ B2 @ N )
= B2 ) ) ) ).
% int_mod_lem
thf(fact_1251_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_1252_int__mod__eq,axiom,
! [B2: int,N: int,A2: int] :
( ( ord_less_eq @ int @ ( zero_zero @ int ) @ B2 )
=> ( ( ord_less @ int @ B2 @ N )
=> ( ( ( modulo_modulo @ int @ A2 @ N )
= ( modulo_modulo @ int @ B2 @ N ) )
=> ( ( modulo_modulo @ int @ A2 @ N )
= B2 ) ) ) ) ).
% int_mod_eq
thf(fact_1253_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_1254_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_1255_int__mod__ge,axiom,
! [A2: int,N: int] :
( ( ord_less @ int @ A2 @ N )
=> ( ( ord_less @ int @ ( zero_zero @ int ) @ N )
=> ( ord_less_eq @ int @ A2 @ ( modulo_modulo @ int @ A2 @ N ) ) ) ) ).
% int_mod_ge
thf(fact_1256_mod__eq__nat1E,axiom,
! [M: nat,Q3: nat,N: nat] :
( ( ( modulo_modulo @ nat @ M @ Q3 )
= ( modulo_modulo @ nat @ N @ Q3 ) )
=> ( ( ord_less_eq @ nat @ N @ M )
=> ~ ! [S2: nat] :
( M
!= ( plus_plus @ nat @ N @ ( times_times @ nat @ Q3 @ S2 ) ) ) ) ) ).
% mod_eq_nat1E
thf(fact_1257_mod__eq__nat2E,axiom,
! [M: nat,Q3: nat,N: nat] :
( ( ( modulo_modulo @ nat @ M @ Q3 )
= ( modulo_modulo @ nat @ N @ Q3 ) )
=> ( ( ord_less_eq @ nat @ M @ N )
=> ~ ! [S2: nat] :
( N
!= ( plus_plus @ nat @ M @ ( times_times @ nat @ Q3 @ S2 ) ) ) ) ) ).
% mod_eq_nat2E
thf(fact_1258_nat__mod__eq__lemma,axiom,
! [X: nat,N: nat,Y: nat] :
( ( ( modulo_modulo @ nat @ X @ N )
= ( modulo_modulo @ nat @ Y @ N ) )
=> ( ( ord_less_eq @ nat @ Y @ X )
=> ? [Q5: nat] :
( X
= ( plus_plus @ nat @ Y @ ( times_times @ nat @ N @ Q5 ) ) ) ) ) ).
% nat_mod_eq_lemma
thf(fact_1259_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_1260_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_1261_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_1262_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_1263_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_1264_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_1265_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_1266_div__positive__int,axiom,
! [L: int,K: int] :
( ( ord_less_eq @ int @ L @ K )
=> ( ( ord_less @ int @ ( zero_zero @ int ) @ L )
=> ( ord_less @ int @ ( zero_zero @ int ) @ ( divide_divide @ int @ K @ L ) ) ) ) ).
% div_positive_int
thf(fact_1267_div__int__pos__iff,axiom,
! [K: int,L: int] :
( ( ord_less_eq @ int @ ( zero_zero @ int ) @ ( divide_divide @ int @ K @ L ) )
= ( ( K
= ( zero_zero @ int ) )
| ( L
= ( zero_zero @ int ) )
| ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ K )
& ( ord_less_eq @ int @ ( zero_zero @ int ) @ L ) )
| ( ( ord_less @ int @ K @ ( zero_zero @ int ) )
& ( ord_less @ int @ L @ ( zero_zero @ int ) ) ) ) ) ).
% div_int_pos_iff
thf(fact_1268_zdiv__mono2__neg,axiom,
! [A2: int,B7: int,B2: int] :
( ( ord_less @ int @ A2 @ ( zero_zero @ int ) )
=> ( ( ord_less @ int @ ( zero_zero @ int ) @ B7 )
=> ( ( ord_less_eq @ int @ B7 @ B2 )
=> ( ord_less_eq @ int @ ( divide_divide @ int @ A2 @ B7 ) @ ( divide_divide @ int @ A2 @ B2 ) ) ) ) ) ).
% zdiv_mono2_neg
thf(fact_1269_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_1270_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_1271_zdiv__mono2,axiom,
! [A2: int,B7: int,B2: int] :
( ( ord_less_eq @ int @ ( zero_zero @ int ) @ A2 )
=> ( ( ord_less @ int @ ( zero_zero @ int ) @ B7 )
=> ( ( ord_less_eq @ int @ B7 @ B2 )
=> ( ord_less_eq @ int @ ( divide_divide @ int @ A2 @ B2 ) @ ( divide_divide @ int @ A2 @ B7 ) ) ) ) ) ).
% zdiv_mono2
thf(fact_1272_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_1273_zdvd__imp__le,axiom,
! [Z: int,N: int] :
( ( dvd_dvd @ int @ Z @ N )
=> ( ( ord_less @ int @ ( zero_zero @ int ) @ N )
=> ( ord_less_eq @ int @ Z @ N ) ) ) ).
% zdvd_imp_le
thf(fact_1274_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_1275_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_1276_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_1277_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_1278_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_1279_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_1280_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_1281_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_1282_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_1283_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_1284_field__le__mult__one__interval,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [X: A,Y: A] :
( ! [Z3: A] :
( ( ord_less @ A @ ( zero_zero @ A ) @ Z3 )
=> ( ( ord_less @ A @ Z3 @ ( one_one @ A ) )
=> ( ord_less_eq @ A @ ( times_times @ A @ Z3 @ X ) @ Y ) ) )
=> ( ord_less_eq @ A @ X @ Y ) ) ) ).
% field_le_mult_one_interval
thf(fact_1285_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_1286_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_1287_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_1288_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_1289_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_1290_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_1291_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_1292_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_1293_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_1294_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_1295_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_1296_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_1297_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_1298_power__Suc__le__self,axiom,
! [A: $tType] :
( ( linordered_semidom @ A )
=> ! [A2: A,N: nat] :
( ( ord_less_eq @ A @ ( zero_zero @ A ) @ A2 )
=> ( ( ord_less_eq @ A @ A2 @ ( one_one @ A ) )
=> ( ord_less_eq @ A @ ( power_power @ A @ A2 @ ( suc @ N ) ) @ A2 ) ) ) ) ).
% power_Suc_le_self
thf(fact_1299_power__eq__imp__eq__base,axiom,
! [A: $tType] :
( ( linordered_semidom @ A )
=> ! [A2: A,N: nat,B2: A] :
( ( ( power_power @ A @ A2 @ N )
= ( power_power @ A @ B2 @ N ) )
=> ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ A2 )
=> ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ B2 )
=> ( ( ord_less @ nat @ ( zero_zero @ nat ) @ N )
=> ( A2 = B2 ) ) ) ) ) ) ).
% power_eq_imp_eq_base
thf(fact_1300_power__eq__iff__eq__base,axiom,
! [A: $tType] :
( ( linordered_semidom @ A )
=> ! [N: nat,A2: A,B2: A] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ N )
=> ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ A2 )
=> ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ B2 )
=> ( ( ( power_power @ A @ A2 @ N )
= ( power_power @ A @ B2 @ N ) )
= ( A2 = B2 ) ) ) ) ) ) ).
% power_eq_iff_eq_base
thf(fact_1301_self__le__power,axiom,
! [A: $tType] :
( ( linordered_semidom @ A )
=> ! [A2: A,N: nat] :
( ( ord_less_eq @ A @ ( one_one @ A ) @ A2 )
=> ( ( ord_less @ nat @ ( zero_zero @ nat ) @ N )
=> ( ord_less_eq @ A @ A2 @ ( power_power @ A @ A2 @ N ) ) ) ) ) ).
% self_le_power
thf(fact_1302_dvd__power__iff,axiom,
! [A: $tType] :
( ( algebraic_semidom @ A )
=> ! [X: A,M: nat,N: nat] :
( ( X
!= ( zero_zero @ A ) )
=> ( ( dvd_dvd @ A @ ( power_power @ A @ X @ M ) @ ( power_power @ A @ X @ N ) )
= ( ( dvd_dvd @ A @ X @ ( one_one @ A ) )
| ( ord_less_eq @ nat @ M @ N ) ) ) ) ) ).
% dvd_power_iff
thf(fact_1303_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_1304_power2__nat__le__eq__le,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq @ nat @ ( power_power @ nat @ M @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ ( power_power @ nat @ N @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) )
= ( ord_less_eq @ nat @ M @ N ) ) ).
% power2_nat_le_eq_le
thf(fact_1305_power2__nat__le__imp__le,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq @ nat @ ( power_power @ nat @ M @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ N )
=> ( ord_less_eq @ nat @ M @ N ) ) ).
% power2_nat_le_imp_le
thf(fact_1306_div__nat__eqI,axiom,
! [N: nat,Q3: nat,M: nat] :
( ( ord_less_eq @ nat @ ( times_times @ nat @ N @ Q3 ) @ M )
=> ( ( ord_less @ nat @ M @ ( times_times @ nat @ N @ ( suc @ Q3 ) ) )
=> ( ( divide_divide @ nat @ M @ N )
= Q3 ) ) ) ).
% div_nat_eqI
thf(fact_1307_less__eq__div__iff__mult__less__eq,axiom,
! [Q3: nat,M: nat,N: nat] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ Q3 )
=> ( ( ord_less_eq @ nat @ M @ ( divide_divide @ nat @ N @ Q3 ) )
= ( ord_less_eq @ nat @ ( times_times @ nat @ M @ Q3 ) @ N ) ) ) ).
% less_eq_div_iff_mult_less_eq
thf(fact_1308_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_1309_power__dvd__imp__le,axiom,
! [I: nat,M: nat,N: nat] :
( ( dvd_dvd @ nat @ ( power_power @ nat @ I @ M ) @ ( power_power @ nat @ I @ N ) )
=> ( ( ord_less @ nat @ ( one_one @ nat ) @ I )
=> ( ord_less_eq @ nat @ M @ N ) ) ) ).
% power_dvd_imp_le
thf(fact_1310_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_1311_pos__mult__pos__ge,axiom,
! [X: int,N: int] :
( ( ord_less @ int @ ( zero_zero @ int ) @ X )
=> ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ N )
=> ( ord_less_eq @ int @ ( times_times @ int @ N @ ( one_one @ int ) ) @ ( times_times @ int @ N @ X ) ) ) ) ).
% pos_mult_pos_ge
thf(fact_1312_q__pos__lemma,axiom,
! [B7: int,Q7: int,R4: int] :
( ( ord_less_eq @ int @ ( zero_zero @ int ) @ ( plus_plus @ int @ ( times_times @ int @ B7 @ Q7 ) @ R4 ) )
=> ( ( ord_less @ int @ R4 @ B7 )
=> ( ( ord_less @ int @ ( zero_zero @ int ) @ B7 )
=> ( ord_less_eq @ int @ ( zero_zero @ int ) @ Q7 ) ) ) ) ).
% q_pos_lemma
thf(fact_1313_zdiv__mono2__lemma,axiom,
! [B2: int,Q3: int,R2: int,B7: int,Q7: int,R4: int] :
( ( ( plus_plus @ int @ ( times_times @ int @ B2 @ Q3 ) @ R2 )
= ( plus_plus @ int @ ( times_times @ int @ B7 @ Q7 ) @ R4 ) )
=> ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ ( plus_plus @ int @ ( times_times @ int @ B7 @ Q7 ) @ R4 ) )
=> ( ( ord_less @ int @ R4 @ B7 )
=> ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ R2 )
=> ( ( ord_less @ int @ ( zero_zero @ int ) @ B7 )
=> ( ( ord_less_eq @ int @ B7 @ B2 )
=> ( ord_less_eq @ int @ Q3 @ Q7 ) ) ) ) ) ) ) ).
% zdiv_mono2_lemma
thf(fact_1314_incr__mult__lemma,axiom,
! [D2: int,P: int > $o,K: int] :
( ( ord_less @ int @ ( zero_zero @ int ) @ D2 )
=> ( ! [X3: int] :
( ( P @ X3 )
=> ( P @ ( plus_plus @ int @ X3 @ D2 ) ) )
=> ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ K )
=> ! [X5: int] :
( ( P @ X5 )
=> ( P @ ( plus_plus @ int @ X5 @ ( times_times @ int @ K @ D2 ) ) ) ) ) ) ) ).
% incr_mult_lemma
thf(fact_1315_zdiv__mono2__neg__lemma,axiom,
! [B2: int,Q3: int,R2: int,B7: int,Q7: int,R4: int] :
( ( ( plus_plus @ int @ ( times_times @ int @ B2 @ Q3 ) @ R2 )
= ( plus_plus @ int @ ( times_times @ int @ B7 @ Q7 ) @ R4 ) )
=> ( ( ord_less @ int @ ( plus_plus @ int @ ( times_times @ int @ B7 @ Q7 ) @ R4 ) @ ( zero_zero @ int ) )
=> ( ( ord_less @ int @ R2 @ B2 )
=> ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ R4 )
=> ( ( ord_less @ int @ ( zero_zero @ int ) @ B7 )
=> ( ( ord_less_eq @ int @ B7 @ B2 )
=> ( ord_less_eq @ int @ Q7 @ Q3 ) ) ) ) ) ) ) ).
% zdiv_mono2_neg_lemma
thf(fact_1316_unique__quotient__lemma,axiom,
! [B2: int,Q7: int,R4: int,Q3: int,R2: int] :
( ( ord_less_eq @ int @ ( plus_plus @ int @ ( times_times @ int @ B2 @ Q7 ) @ R4 ) @ ( plus_plus @ int @ ( times_times @ int @ B2 @ Q3 ) @ R2 ) )
=> ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ R4 )
=> ( ( ord_less @ int @ R4 @ B2 )
=> ( ( ord_less @ int @ R2 @ B2 )
=> ( ord_less_eq @ int @ Q7 @ Q3 ) ) ) ) ) ).
% unique_quotient_lemma
thf(fact_1317_unique__quotient__lemma__neg,axiom,
! [B2: int,Q7: int,R4: int,Q3: int,R2: int] :
( ( ord_less_eq @ int @ ( plus_plus @ int @ ( times_times @ int @ B2 @ Q7 ) @ R4 ) @ ( plus_plus @ int @ ( times_times @ int @ B2 @ Q3 ) @ R2 ) )
=> ( ( ord_less_eq @ int @ R2 @ ( zero_zero @ int ) )
=> ( ( ord_less @ int @ B2 @ R2 )
=> ( ( ord_less @ int @ B2 @ R4 )
=> ( ord_less_eq @ int @ Q3 @ Q7 ) ) ) ) ) ).
% unique_quotient_lemma_neg
thf(fact_1318_mod__pos__neg__trivial,axiom,
! [K: int,L: int] :
( ( ord_less @ int @ ( zero_zero @ int ) @ K )
=> ( ( ord_less_eq @ int @ ( plus_plus @ int @ K @ L ) @ ( zero_zero @ int ) )
=> ( ( modulo_modulo @ int @ K @ L )
= ( plus_plus @ int @ K @ L ) ) ) ) ).
% mod_pos_neg_trivial
thf(fact_1319_int__mod__ge_H,axiom,
! [B2: int,N: int] :
( ( ord_less @ int @ B2 @ ( zero_zero @ int ) )
=> ( ( ord_less @ int @ ( zero_zero @ int ) @ N )
=> ( ord_less_eq @ int @ ( plus_plus @ int @ B2 @ N ) @ ( modulo_modulo @ int @ B2 @ N ) ) ) ) ).
% int_mod_ge'
thf(fact_1320_mod__int__pos__iff,axiom,
! [K: int,L: int] :
( ( ord_less_eq @ int @ ( zero_zero @ int ) @ ( modulo_modulo @ int @ K @ L ) )
= ( ( dvd_dvd @ int @ L @ K )
| ( ( L
= ( zero_zero @ int ) )
& ( ord_less_eq @ int @ ( zero_zero @ int ) @ K ) )
| ( ord_less @ int @ ( zero_zero @ int ) @ L ) ) ) ).
% mod_int_pos_iff
thf(fact_1321_signed__take__bit__int__less__exp,axiom,
! [N: nat,K: int] : ( ord_less @ int @ ( bit_ri4674362597316999326ke_bit @ int @ N @ K ) @ ( power_power @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ N ) ) ).
% signed_take_bit_int_less_exp
thf(fact_1322_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_1323_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_1324_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_1325_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_1326_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_1327_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_1328_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_1329_power__strict__mono,axiom,
! [A: $tType] :
( ( linordered_semidom @ A )
=> ! [A2: A,B2: A,N: nat] :
( ( ord_less @ A @ A2 @ B2 )
=> ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ A2 )
=> ( ( ord_less @ nat @ ( zero_zero @ nat ) @ N )
=> ( ord_less @ A @ ( power_power @ A @ A2 @ N ) @ ( power_power @ A @ B2 @ N ) ) ) ) ) ) ).
% power_strict_mono
thf(fact_1330_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_1331_num_Osize__gen_I1_J,axiom,
( ( size_num @ one2 )
= ( zero_zero @ nat ) ) ).
% num.size_gen(1)
thf(fact_1332_power__mono__odd,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [N: nat,A2: A,B2: A] :
( ~ ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N )
=> ( ( ord_less_eq @ A @ A2 @ B2 )
=> ( ord_less_eq @ A @ ( power_power @ A @ A2 @ N ) @ ( power_power @ A @ B2 @ N ) ) ) ) ) ).
% power_mono_odd
thf(fact_1333_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_1334_split__div_H,axiom,
! [P: nat > $o,M: nat,N: nat] :
( ( P @ ( divide_divide @ nat @ M @ N ) )
= ( ( ( N
= ( zero_zero @ nat ) )
& ( P @ ( zero_zero @ nat ) ) )
| ? [Q6: nat] :
( ( ord_less_eq @ nat @ ( times_times @ nat @ N @ Q6 ) @ M )
& ( ord_less @ nat @ M @ ( times_times @ nat @ N @ ( suc @ Q6 ) ) )
& ( P @ Q6 ) ) ) ) ).
% split_div'
thf(fact_1335_not__exp__less__eq__0__int,axiom,
! [N: nat] :
~ ( ord_less_eq @ int @ ( power_power @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ N ) @ ( zero_zero @ int ) ) ).
% not_exp_less_eq_0_int
thf(fact_1336_dvd__power__iff__le,axiom,
! [K: nat,M: nat,N: nat] :
( ( ord_less_eq @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ K )
=> ( ( dvd_dvd @ nat @ ( power_power @ nat @ K @ M ) @ ( power_power @ nat @ K @ N ) )
= ( ord_less_eq @ nat @ M @ N ) ) ) ).
% dvd_power_iff_le
thf(fact_1337_two__realpow__ge__one,axiom,
! [N: nat] : ( ord_less_eq @ real @ ( one_one @ real ) @ ( power_power @ real @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) @ N ) ) ).
% two_realpow_ge_one
thf(fact_1338_split__zmod,axiom,
! [P: int > $o,N: int,K: int] :
( ( P @ ( modulo_modulo @ int @ N @ K ) )
= ( ( ( K
= ( zero_zero @ int ) )
=> ( P @ N ) )
& ( ( ord_less @ int @ ( zero_zero @ int ) @ K )
=> ! [I2: int,J2: int] :
( ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ J2 )
& ( ord_less @ int @ J2 @ K )
& ( N
= ( plus_plus @ int @ ( times_times @ int @ K @ I2 ) @ J2 ) ) )
=> ( P @ J2 ) ) )
& ( ( ord_less @ int @ K @ ( zero_zero @ int ) )
=> ! [I2: int,J2: int] :
( ( ( ord_less @ int @ K @ J2 )
& ( ord_less_eq @ int @ J2 @ ( zero_zero @ int ) )
& ( N
= ( plus_plus @ int @ ( times_times @ int @ K @ I2 ) @ J2 ) ) )
=> ( P @ J2 ) ) ) ) ) ).
% split_zmod
thf(fact_1339_int__mod__neg__eq,axiom,
! [A2: int,B2: int,Q3: int,R2: int] :
( ( A2
= ( plus_plus @ int @ ( times_times @ int @ B2 @ Q3 ) @ R2 ) )
=> ( ( ord_less_eq @ int @ R2 @ ( zero_zero @ int ) )
=> ( ( ord_less @ int @ B2 @ R2 )
=> ( ( modulo_modulo @ int @ A2 @ B2 )
= R2 ) ) ) ) ).
% int_mod_neg_eq
thf(fact_1340_int__mod__pos__eq,axiom,
! [A2: int,B2: int,Q3: int,R2: int] :
( ( A2
= ( plus_plus @ int @ ( times_times @ int @ B2 @ Q3 ) @ R2 ) )
=> ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ R2 )
=> ( ( ord_less @ int @ R2 @ B2 )
=> ( ( modulo_modulo @ int @ A2 @ B2 )
= R2 ) ) ) ) ).
% int_mod_pos_eq
thf(fact_1341_int__div__pos__eq,axiom,
! [A2: int,B2: int,Q3: int,R2: int] :
( ( A2
= ( plus_plus @ int @ ( times_times @ int @ B2 @ Q3 ) @ R2 ) )
=> ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ R2 )
=> ( ( ord_less @ int @ R2 @ B2 )
=> ( ( divide_divide @ int @ A2 @ B2 )
= Q3 ) ) ) ) ).
% int_div_pos_eq
thf(fact_1342_int__div__neg__eq,axiom,
! [A2: int,B2: int,Q3: int,R2: int] :
( ( A2
= ( plus_plus @ int @ ( times_times @ int @ B2 @ Q3 ) @ R2 ) )
=> ( ( ord_less_eq @ int @ R2 @ ( zero_zero @ int ) )
=> ( ( ord_less @ int @ B2 @ R2 )
=> ( ( divide_divide @ int @ A2 @ B2 )
= Q3 ) ) ) ) ).
% int_div_neg_eq
thf(fact_1343_split__zdiv,axiom,
! [P: int > $o,N: int,K: int] :
( ( P @ ( divide_divide @ int @ N @ K ) )
= ( ( ( K
= ( zero_zero @ int ) )
=> ( P @ ( zero_zero @ int ) ) )
& ( ( ord_less @ int @ ( zero_zero @ int ) @ K )
=> ! [I2: int,J2: int] :
( ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ J2 )
& ( ord_less @ int @ J2 @ K )
& ( N
= ( plus_plus @ int @ ( times_times @ int @ K @ I2 ) @ J2 ) ) )
=> ( P @ I2 ) ) )
& ( ( ord_less @ int @ K @ ( zero_zero @ int ) )
=> ! [I2: int,J2: int] :
( ( ( ord_less @ int @ K @ J2 )
& ( ord_less_eq @ int @ J2 @ ( zero_zero @ int ) )
& ( N
= ( plus_plus @ int @ ( times_times @ int @ K @ I2 ) @ J2 ) ) )
=> ( P @ I2 ) ) ) ) ) ).
% split_zdiv
thf(fact_1344_mod__power__lem,axiom,
! [A2: int,M: nat,N: nat] :
( ( ord_less @ int @ ( one_one @ int ) @ A2 )
=> ( ( ( ord_less_eq @ nat @ M @ N )
=> ( ( modulo_modulo @ int @ ( power_power @ int @ A2 @ N ) @ ( power_power @ int @ A2 @ M ) )
= ( zero_zero @ int ) ) )
& ( ~ ( ord_less_eq @ nat @ M @ N )
=> ( ( modulo_modulo @ int @ ( power_power @ int @ A2 @ N ) @ ( power_power @ int @ A2 @ M ) )
= ( power_power @ int @ A2 @ N ) ) ) ) ) ).
% mod_power_lem
thf(fact_1345_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_1346_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_1347_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_1348_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_1349_zero__le__even__power_H,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [A2: A,N: nat] : ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( power_power @ A @ A2 @ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N ) ) ) ) ).
% zero_le_even_power'
thf(fact_1350_zero__le__power__eq,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [A2: A,N: nat] :
( ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( power_power @ A @ A2 @ N ) )
= ( ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N )
| ( ~ ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N )
& ( ord_less_eq @ A @ ( zero_zero @ A ) @ A2 ) ) ) ) ) ).
% zero_le_power_eq
thf(fact_1351_zero__le__odd__power,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [N: nat,A2: A] :
( ~ ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N )
=> ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( power_power @ A @ A2 @ N ) )
= ( ord_less_eq @ A @ ( zero_zero @ A ) @ A2 ) ) ) ) ).
% zero_le_odd_power
thf(fact_1352_zero__le__even__power,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [N: nat,A2: A] :
( ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N )
=> ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( power_power @ A @ A2 @ N ) ) ) ) ).
% zero_le_even_power
thf(fact_1353_two__pow__div__gt__le,axiom,
! [V: nat,N: nat,M: nat] :
( ( ord_less @ nat @ V @ ( divide_divide @ nat @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ M ) ) )
=> ( ord_less_eq @ nat @ M @ N ) ) ).
% two_pow_div_gt_le
thf(fact_1354_L2__set__mult__ineq__lemma,axiom,
! [A2: real,C2: real,B2: real,D2: 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 @ D2 ) ) @ ( plus_plus @ real @ ( times_times @ real @ ( power_power @ real @ A2 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ ( power_power @ real @ D2 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( times_times @ real @ ( power_power @ real @ B2 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ ( power_power @ real @ C2 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) ).
% L2_set_mult_ineq_lemma
thf(fact_1355_verit__le__mono__div,axiom,
! [A3: nat,B4: nat,N: nat] :
( ( ord_less @ nat @ A3 @ B4 )
=> ( ( ord_less @ nat @ ( zero_zero @ nat ) @ N )
=> ( ord_less_eq @ nat
@ ( plus_plus @ nat @ ( divide_divide @ nat @ A3 @ N )
@ ( if @ nat
@ ( ( modulo_modulo @ nat @ B4 @ N )
= ( zero_zero @ nat ) )
@ ( one_one @ nat )
@ ( zero_zero @ nat ) ) )
@ ( divide_divide @ nat @ B4 @ N ) ) ) ) ).
% verit_le_mono_div
thf(fact_1356_verit__le__mono__div__int,axiom,
! [A3: int,B4: int,N: int] :
( ( ord_less @ int @ A3 @ B4 )
=> ( ( ord_less @ int @ ( zero_zero @ int ) @ N )
=> ( ord_less_eq @ int
@ ( plus_plus @ int @ ( divide_divide @ int @ A3 @ N )
@ ( if @ int
@ ( ( modulo_modulo @ int @ B4 @ N )
= ( zero_zero @ int ) )
@ ( one_one @ int )
@ ( zero_zero @ int ) ) )
@ ( divide_divide @ int @ B4 @ N ) ) ) ) ).
% verit_le_mono_div_int
thf(fact_1357_split__pos__lemma,axiom,
! [K: int,P: int > int > $o,N: int] :
( ( ord_less @ int @ ( zero_zero @ int ) @ K )
=> ( ( P @ ( divide_divide @ int @ N @ K ) @ ( modulo_modulo @ int @ N @ K ) )
= ( ! [I2: int,J2: int] :
( ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ J2 )
& ( ord_less @ int @ J2 @ K )
& ( N
= ( plus_plus @ int @ ( times_times @ int @ K @ I2 ) @ J2 ) ) )
=> ( P @ I2 @ J2 ) ) ) ) ) ).
% split_pos_lemma
thf(fact_1358_split__neg__lemma,axiom,
! [K: int,P: int > int > $o,N: int] :
( ( ord_less @ int @ K @ ( zero_zero @ int ) )
=> ( ( P @ ( divide_divide @ int @ N @ K ) @ ( modulo_modulo @ int @ N @ K ) )
= ( ! [I2: int,J2: int] :
( ( ( ord_less @ int @ K @ J2 )
& ( ord_less_eq @ int @ J2 @ ( zero_zero @ int ) )
& ( N
= ( plus_plus @ int @ ( times_times @ int @ K @ I2 ) @ J2 ) ) )
=> ( P @ I2 @ J2 ) ) ) ) ) ).
% split_neg_lemma
thf(fact_1359_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_1360_odd__0__le__power__imp__0__le,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [A2: A,N: nat] :
( ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( power_power @ A @ A2 @ ( suc @ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N ) ) ) )
=> ( ord_less_eq @ A @ ( zero_zero @ A ) @ A2 ) ) ) ).
% odd_0_le_power_imp_0_le
thf(fact_1361_ex__power__ivl1,axiom,
! [B2: nat,K: nat] :
( ( ord_less_eq @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ B2 )
=> ( ( ord_less_eq @ nat @ ( one_one @ nat ) @ K )
=> ? [N2: nat] :
( ( ord_less_eq @ nat @ ( power_power @ nat @ B2 @ N2 ) @ K )
& ( ord_less @ nat @ K @ ( power_power @ nat @ B2 @ ( plus_plus @ nat @ N2 @ ( one_one @ nat ) ) ) ) ) ) ) ).
% ex_power_ivl1
thf(fact_1362_ex__power__ivl2,axiom,
! [B2: nat,K: nat] :
( ( ord_less_eq @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ B2 )
=> ( ( ord_less_eq @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ K )
=> ? [N2: nat] :
( ( ord_less @ nat @ ( power_power @ nat @ B2 @ N2 ) @ K )
& ( ord_less_eq @ nat @ K @ ( power_power @ nat @ B2 @ ( plus_plus @ nat @ N2 @ ( one_one @ nat ) ) ) ) ) ) ) ).
% ex_power_ivl2
thf(fact_1363_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_1364_power__le__zero__eq,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [A2: A,N: nat] :
( ( ord_less_eq @ A @ ( power_power @ A @ A2 @ N ) @ ( zero_zero @ A ) )
= ( ( ord_less @ nat @ ( zero_zero @ nat ) @ N )
& ( ( ~ ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N )
& ( ord_less_eq @ A @ A2 @ ( zero_zero @ A ) ) )
| ( ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N )
& ( A2
= ( zero_zero @ A ) ) ) ) ) ) ) ).
% power_le_zero_eq
thf(fact_1365_nat__div__eq__Suc__0__iff,axiom,
! [N: nat,M: nat] :
( ( ( divide_divide @ nat @ N @ M )
= ( suc @ ( zero_zero @ nat ) ) )
= ( ( ord_less_eq @ nat @ M @ N )
& ( ord_less @ nat @ N @ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ M ) ) ) ) ).
% nat_div_eq_Suc_0_iff
thf(fact_1366_power__2__mult__step__le,axiom,
! [N5: nat,N: nat,K4: nat,K: nat] :
( ( ord_less_eq @ nat @ N5 @ N )
=> ( ( ord_less @ nat @ ( times_times @ nat @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N5 ) @ K4 ) @ ( times_times @ nat @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N ) @ K ) )
=> ( ord_less_eq @ nat @ ( times_times @ nat @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N5 ) @ ( plus_plus @ nat @ K4 @ ( one_one @ nat ) ) ) @ ( times_times @ nat @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N ) @ K ) ) ) ) ).
% power_2_mult_step_le
thf(fact_1367_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_1368_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_1369_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_1370_eme1p,axiom,
! [N: int,D2: int] :
( ( dvd_dvd @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ N )
=> ( ( dvd_dvd @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ D2 )
=> ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ D2 )
=> ( ( modulo_modulo @ int @ ( plus_plus @ int @ ( one_one @ int ) @ N ) @ D2 )
= ( plus_plus @ int @ ( one_one @ int ) @ ( modulo_modulo @ int @ N @ D2 ) ) ) ) ) ) ).
% eme1p
thf(fact_1371_emep1,axiom,
! [N: int,D2: int] :
( ( dvd_dvd @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ N )
=> ( ( dvd_dvd @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ D2 )
=> ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ D2 )
=> ( ( modulo_modulo @ int @ ( plus_plus @ int @ N @ ( one_one @ int ) ) @ D2 )
= ( plus_plus @ int @ ( modulo_modulo @ int @ N @ D2 ) @ ( one_one @ int ) ) ) ) ) ) ).
% emep1
thf(fact_1372_count__buildup,axiom,
! [N: nat] : ( ord_less_eq @ real @ ( vEBT_VEBT_cnt @ ( vEBT_vebt_buildup @ N ) ) @ ( times_times @ real @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) @ ( power_power @ real @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) @ N ) ) ) ).
% count_buildup
thf(fact_1373_two__realpow__ge__two,axiom,
! [N: nat] :
( ( ord_less_eq @ real @ ( one_one @ real ) @ ( semiring_1_of_nat @ real @ N ) )
=> ( ord_less_eq @ real @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) @ ( power_power @ real @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) @ N ) ) ) ).
% two_realpow_ge_two
thf(fact_1374_cnt__non__neg,axiom,
! [T: vEBT_VEBT] : ( ord_less_eq @ real @ ( zero_zero @ real ) @ ( vEBT_VEBT_cnt @ T ) ) ).
% cnt_non_neg
thf(fact_1375_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_1376_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_1377_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_1378_even__mult__exp__div__exp__iff,axiom,
! [A: $tType] :
( ( bit_semiring_bits @ A )
=> ! [A2: A,M: nat,N: nat] :
( ( dvd_dvd @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ ( divide_divide @ A @ ( times_times @ A @ A2 @ ( power_power @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ M ) ) @ ( power_power @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ N ) ) )
= ( ( ord_less @ nat @ N @ M )
| ( ( power_power @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ N )
= ( zero_zero @ A ) )
| ( ( ord_less_eq @ nat @ M @ N )
& ( dvd_dvd @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ ( divide_divide @ A @ A2 @ ( power_power @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ ( minus_minus @ nat @ N @ M ) ) ) ) ) ) ) ) ).
% even_mult_exp_div_exp_iff
thf(fact_1379_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_1380_odd__two__times__div__two__nat,axiom,
! [N: nat] :
( ~ ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N )
=> ( ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ N @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) )
= ( minus_minus @ nat @ N @ ( one_one @ nat ) ) ) ) ).
% odd_two_times_div_two_nat
thf(fact_1381_neg__equal__iff__equal,axiom,
! [A: $tType] :
( ( group_add @ A )
=> ! [A2: A,B2: A] :
( ( ( uminus_uminus @ A @ A2 )
= ( uminus_uminus @ A @ B2 ) )
= ( A2 = B2 ) ) ) ).
% neg_equal_iff_equal
thf(fact_1382_add_Oinverse__inverse,axiom,
! [A: $tType] :
( ( group_add @ A )
=> ! [A2: A] :
( ( uminus_uminus @ A @ ( uminus_uminus @ A @ A2 ) )
= A2 ) ) ).
% add.inverse_inverse
thf(fact_1383_verit__minus__simplify_I4_J,axiom,
! [B: $tType] :
( ( group_add @ B )
=> ! [B2: B] :
( ( uminus_uminus @ B @ ( uminus_uminus @ B @ B2 ) )
= B2 ) ) ).
% verit_minus_simplify(4)
thf(fact_1384_verit__eq__simplify_I9_J,axiom,
! [X32: num,Y32: num] :
( ( ( bit1 @ X32 )
= ( bit1 @ Y32 ) )
= ( X32 = Y32 ) ) ).
% verit_eq_simplify(9)
thf(fact_1385_semiring__norm_I90_J,axiom,
! [M: num,N: num] :
( ( ( bit1 @ M )
= ( bit1 @ N ) )
= ( M = N ) ) ).
% semiring_norm(90)
thf(fact_1386_of__nat__eq__iff,axiom,
! [A: $tType] :
( ( semiring_char_0 @ A )
=> ! [M: nat,N: nat] :
( ( ( semiring_1_of_nat @ A @ M )
= ( semiring_1_of_nat @ A @ N ) )
= ( M = N ) ) ) ).
% of_nat_eq_iff
thf(fact_1387_diff__self,axiom,
! [A: $tType] :
( ( group_add @ A )
=> ! [A2: A] :
( ( minus_minus @ A @ A2 @ A2 )
= ( zero_zero @ A ) ) ) ).
% diff_self
thf(fact_1388_diff__0__right,axiom,
! [A: $tType] :
( ( group_add @ A )
=> ! [A2: A] :
( ( minus_minus @ A @ A2 @ ( zero_zero @ A ) )
= A2 ) ) ).
% diff_0_right
thf(fact_1389_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_1390_diff__zero,axiom,
! [A: $tType] :
( ( cancel1802427076303600483id_add @ A )
=> ! [A2: A] :
( ( minus_minus @ A @ A2 @ ( zero_zero @ A ) )
= A2 ) ) ).
% diff_zero
thf(fact_1391_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_1392_add_Oinverse__neutral,axiom,
! [A: $tType] :
( ( group_add @ A )
=> ( ( uminus_uminus @ A @ ( zero_zero @ A ) )
= ( zero_zero @ A ) ) ) ).
% add.inverse_neutral
thf(fact_1393_neg__0__equal__iff__equal,axiom,
! [A: $tType] :
( ( group_add @ A )
=> ! [A2: A] :
( ( ( zero_zero @ A )
= ( uminus_uminus @ A @ A2 ) )
= ( ( zero_zero @ A )
= A2 ) ) ) ).
% neg_0_equal_iff_equal
thf(fact_1394_neg__equal__0__iff__equal,axiom,
! [A: $tType] :
( ( group_add @ A )
=> ! [A2: A] :
( ( ( uminus_uminus @ A @ A2 )
= ( zero_zero @ A ) )
= ( A2
= ( zero_zero @ A ) ) ) ) ).
% neg_equal_0_iff_equal
thf(fact_1395_equal__neg__zero,axiom,
! [A: $tType] :
( ( linord5086331880401160121up_add @ A )
=> ! [A2: A] :
( ( A2
= ( uminus_uminus @ A @ A2 ) )
= ( A2
= ( zero_zero @ A ) ) ) ) ).
% equal_neg_zero
thf(fact_1396_neg__equal__zero,axiom,
! [A: $tType] :
( ( linord5086331880401160121up_add @ A )
=> ! [A2: A] :
( ( ( uminus_uminus @ A @ A2 )
= A2 )
= ( A2
= ( zero_zero @ A ) ) ) ) ).
% neg_equal_zero
thf(fact_1397_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_1398_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_1399_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_1400_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_1401_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_1402_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_1403_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_1404_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_1405_neg__numeral__eq__iff,axiom,
! [A: $tType] :
( ( ring_char_0 @ A )
=> ! [M: num,N: num] :
( ( ( uminus_uminus @ A @ ( numeral_numeral @ A @ M ) )
= ( uminus_uminus @ A @ ( numeral_numeral @ A @ N ) ) )
= ( M = N ) ) ) ).
% neg_numeral_eq_iff
thf(fact_1406_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_1407_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_1408_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_1409_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_1410_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_1411_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_1412_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_1413_semiring__norm_I88_J,axiom,
! [M: num,N: num] :
( ( bit0 @ M )
!= ( bit1 @ N ) ) ).
% semiring_norm(88)
thf(fact_1414_semiring__norm_I89_J,axiom,
! [M: num,N: num] :
( ( bit1 @ M )
!= ( bit0 @ N ) ) ).
% semiring_norm(89)
thf(fact_1415_semiring__norm_I84_J,axiom,
! [N: num] :
( one2
!= ( bit1 @ N ) ) ).
% semiring_norm(84)
thf(fact_1416_semiring__norm_I86_J,axiom,
! [M: num] :
( ( bit1 @ M )
!= one2 ) ).
% semiring_norm(86)
thf(fact_1417_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_1418_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_1419_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_1420_Suc__diff__diff,axiom,
! [M: nat,N: nat,K: nat] :
( ( minus_minus @ nat @ ( minus_minus @ nat @ ( suc @ M ) @ N ) @ ( suc @ K ) )
= ( minus_minus @ nat @ ( minus_minus @ nat @ M @ N ) @ K ) ) ).
% Suc_diff_diff
thf(fact_1421_diff__Suc__Suc,axiom,
! [M: nat,N: nat] :
( ( minus_minus @ nat @ ( suc @ M ) @ ( suc @ N ) )
= ( minus_minus @ nat @ M @ N ) ) ).
% diff_Suc_Suc
thf(fact_1422_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_1423_diff__self__eq__0,axiom,
! [M: nat] :
( ( minus_minus @ nat @ M @ M )
= ( zero_zero @ nat ) ) ).
% diff_self_eq_0
thf(fact_1424_diff__0__eq__0,axiom,
! [N: nat] :
( ( minus_minus @ nat @ ( zero_zero @ nat ) @ N )
= ( zero_zero @ nat ) ) ).
% diff_0_eq_0
thf(fact_1425_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_1426_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_1427_diff__diff__cancel,axiom,
! [I: nat,N: nat] :
( ( ord_less_eq @ nat @ I @ N )
=> ( ( minus_minus @ nat @ N @ ( minus_minus @ nat @ N @ I ) )
= I ) ) ).
% diff_diff_cancel
thf(fact_1428_semiring__norm_I73_J,axiom,
! [M: num,N: num] :
( ( ord_less_eq @ num @ ( bit1 @ M ) @ ( bit1 @ N ) )
= ( ord_less_eq @ num @ M @ N ) ) ).
% semiring_norm(73)
thf(fact_1429_semiring__norm_I80_J,axiom,
! [M: num,N: num] :
( ( ord_less @ num @ ( bit1 @ M ) @ ( bit1 @ N ) )
= ( ord_less @ num @ M @ N ) ) ).
% semiring_norm(80)
thf(fact_1430_count__buildup_H,axiom,
! [N: nat] : ( ord_less_eq @ real @ ( vEBT_VEBT_cnt @ ( vEBT_vebt_buildup @ N ) ) @ ( semiring_1_of_nat @ real @ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N ) ) ) ) ).
% count_buildup'
thf(fact_1431_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_1432_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_1433_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_1434_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_1435_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_1436_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_1437_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_1438_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_1439_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_1440_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_1441_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_1442_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_1443_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_1444_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_1445_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_1446_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_1447_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_1448_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_1449_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_1450_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_1451_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_1452_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_1453_add__neg__numeral__simps_I3_J,axiom,
! [A: $tType] :
( ( neg_numeral @ A )
=> ! [M: num,N: num] :
( ( plus_plus @ A @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ M ) ) @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ N ) ) )
= ( uminus_uminus @ A @ ( plus_plus @ A @ ( numeral_numeral @ A @ M ) @ ( numeral_numeral @ A @ N ) ) ) ) ) ).
% add_neg_numeral_simps(3)
thf(fact_1454_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_1455_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_1456_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_1457_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_1458_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_1459_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_1460_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_1461_of__nat__0__eq__iff,axiom,
! [A: $tType] :
( ( semiring_char_0 @ A )
=> ! [N: nat] :
( ( ( zero_zero @ A )
= ( semiring_1_of_nat @ A @ N ) )
= ( ( zero_zero @ nat )
= N ) ) ) ).
% of_nat_0_eq_iff
thf(fact_1462_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_1463_of__nat__numeral,axiom,
! [A: $tType] :
( ( semiring_1 @ A )
=> ! [N: num] :
( ( semiring_1_of_nat @ A @ ( numeral_numeral @ nat @ N ) )
= ( numeral_numeral @ A @ N ) ) ) ).
% of_nat_numeral
thf(fact_1464_of__nat__less__iff,axiom,
! [A: $tType] :
( ( linord181362715937106298miring @ A )
=> ! [M: nat,N: nat] :
( ( ord_less @ A @ ( semiring_1_of_nat @ A @ M ) @ ( semiring_1_of_nat @ A @ N ) )
= ( ord_less @ nat @ M @ N ) ) ) ).
% of_nat_less_iff
thf(fact_1465_of__nat__le__iff,axiom,
! [A: $tType] :
( ( linord181362715937106298miring @ A )
=> ! [M: nat,N: nat] :
( ( ord_less_eq @ A @ ( semiring_1_of_nat @ A @ M ) @ ( semiring_1_of_nat @ A @ N ) )
= ( ord_less_eq @ nat @ M @ N ) ) ) ).
% of_nat_le_iff
thf(fact_1466_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_1467_of__nat__add,axiom,
! [A: $tType] :
( ( semiring_1 @ A )
=> ! [M: nat,N: nat] :
( ( semiring_1_of_nat @ A @ ( plus_plus @ nat @ M @ N ) )
= ( plus_plus @ A @ ( semiring_1_of_nat @ A @ M ) @ ( semiring_1_of_nat @ A @ N ) ) ) ) ).
% of_nat_add
thf(fact_1468_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_1469_of__nat__eq__1__iff,axiom,
! [A: $tType] :
( ( semiring_char_0 @ A )
=> ! [N: nat] :
( ( ( semiring_1_of_nat @ A @ N )
= ( one_one @ A ) )
= ( N
= ( one_one @ nat ) ) ) ) ).
% of_nat_eq_1_iff
thf(fact_1470_of__nat__1__eq__iff,axiom,
! [A: $tType] :
( ( semiring_char_0 @ A )
=> ! [N: nat] :
( ( ( one_one @ A )
= ( semiring_1_of_nat @ A @ N ) )
= ( N
= ( one_one @ nat ) ) ) ) ).
% of_nat_1_eq_iff
thf(fact_1471_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_1472_zero__less__diff,axiom,
! [N: nat,M: nat] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ ( minus_minus @ nat @ N @ M ) )
= ( ord_less @ nat @ M @ N ) ) ).
% zero_less_diff
thf(fact_1473_diff__Suc__1,axiom,
! [N: nat] :
( ( minus_minus @ nat @ ( suc @ N ) @ ( one_one @ nat ) )
= N ) ).
% diff_Suc_1
thf(fact_1474_diff__is__0__eq_H,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq @ nat @ M @ N )
=> ( ( minus_minus @ nat @ M @ N )
= ( zero_zero @ nat ) ) ) ).
% diff_is_0_eq'
thf(fact_1475_diff__is__0__eq,axiom,
! [M: nat,N: nat] :
( ( ( minus_minus @ nat @ M @ N )
= ( zero_zero @ nat ) )
= ( ord_less_eq @ nat @ M @ N ) ) ).
% diff_is_0_eq
thf(fact_1476_of__nat__mult,axiom,
! [A: $tType] :
( ( semiring_1 @ A )
=> ! [M: nat,N: nat] :
( ( semiring_1_of_nat @ A @ ( times_times @ nat @ M @ N ) )
= ( times_times @ A @ ( semiring_1_of_nat @ A @ M ) @ ( semiring_1_of_nat @ A @ N ) ) ) ) ).
% of_nat_mult
thf(fact_1477_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_1478_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_1479_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_1480_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_1481_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_1482_semiring__1__class_Oof__nat__power,axiom,
! [A: $tType] :
( ( semiring_1 @ A )
=> ! [M: nat,N: nat] :
( ( semiring_1_of_nat @ A @ ( power_power @ nat @ M @ N ) )
= ( power_power @ A @ ( semiring_1_of_nat @ A @ M ) @ N ) ) ) ).
% semiring_1_class.of_nat_power
thf(fact_1483_signed__take__bit__of__minus__1,axiom,
! [A: $tType] :
( ( bit_ri3973907225187159222ations @ A )
=> ! [N: nat] :
( ( bit_ri4674362597316999326ke_bit @ A @ N @ ( uminus_uminus @ A @ ( one_one @ A ) ) )
= ( uminus_uminus @ A @ ( one_one @ A ) ) ) ) ).
% signed_take_bit_of_minus_1
thf(fact_1484_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_1485_semiring__norm_I9_J,axiom,
! [M: num,N: num] :
( ( plus_plus @ num @ ( bit1 @ M ) @ ( bit0 @ N ) )
= ( bit1 @ ( plus_plus @ num @ M @ N ) ) ) ).
% semiring_norm(9)
thf(fact_1486_semiring__norm_I7_J,axiom,
! [M: num,N: num] :
( ( plus_plus @ num @ ( bit0 @ M ) @ ( bit1 @ N ) )
= ( bit1 @ ( plus_plus @ num @ M @ N ) ) ) ).
% semiring_norm(7)
thf(fact_1487_semiring__norm_I15_J,axiom,
! [M: num,N: num] :
( ( times_times @ num @ ( bit1 @ M ) @ ( bit0 @ N ) )
= ( bit0 @ ( times_times @ num @ ( bit1 @ M ) @ N ) ) ) ).
% semiring_norm(15)
thf(fact_1488_semiring__norm_I14_J,axiom,
! [M: num,N: num] :
( ( times_times @ num @ ( bit0 @ M ) @ ( bit1 @ N ) )
= ( bit0 @ ( times_times @ num @ M @ ( bit1 @ N ) ) ) ) ).
% semiring_norm(14)
thf(fact_1489_semiring__norm_I72_J,axiom,
! [M: num,N: num] :
( ( ord_less_eq @ num @ ( bit0 @ M ) @ ( bit1 @ N ) )
= ( ord_less_eq @ num @ M @ N ) ) ).
% semiring_norm(72)
thf(fact_1490_semiring__norm_I81_J,axiom,
! [M: num,N: num] :
( ( ord_less @ num @ ( bit1 @ M ) @ ( bit0 @ N ) )
= ( ord_less @ num @ M @ N ) ) ).
% semiring_norm(81)
thf(fact_1491_semiring__norm_I70_J,axiom,
! [M: num] :
~ ( ord_less_eq @ num @ ( bit1 @ M ) @ one2 ) ).
% semiring_norm(70)
thf(fact_1492_semiring__norm_I77_J,axiom,
! [N: num] : ( ord_less @ num @ one2 @ ( bit1 @ N ) ) ).
% semiring_norm(77)
thf(fact_1493_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_1494_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_1495_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_1496_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_1497_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_1498_numeral__eq__neg__one__iff,axiom,
! [A: $tType] :
( ( ring_char_0 @ A )
=> ! [N: num] :
( ( ( uminus_uminus @ A @ ( numeral_numeral @ A @ N ) )
= ( uminus_uminus @ A @ ( one_one @ A ) ) )
= ( N = one2 ) ) ) ).
% numeral_eq_neg_one_iff
thf(fact_1499_neg__one__eq__numeral__iff,axiom,
! [A: $tType] :
( ( ring_char_0 @ A )
=> ! [N: num] :
( ( ( uminus_uminus @ A @ ( one_one @ A ) )
= ( uminus_uminus @ A @ ( numeral_numeral @ A @ N ) ) )
= ( N = one2 ) ) ) ).
% neg_one_eq_numeral_iff
thf(fact_1500_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_1501_left__minus__one__mult__self,axiom,
! [A: $tType] :
( ( comm_ring_1 @ A )
=> ! [N: nat,A2: A] :
( ( times_times @ A @ ( power_power @ A @ ( uminus_uminus @ A @ ( one_one @ A ) ) @ N ) @ ( times_times @ A @ ( power_power @ A @ ( uminus_uminus @ A @ ( one_one @ A ) ) @ N ) @ A2 ) )
= A2 ) ) ).
% left_minus_one_mult_self
thf(fact_1502_minus__one__mult__self,axiom,
! [A: $tType] :
( ( comm_ring_1 @ A )
=> ! [N: nat] :
( ( times_times @ A @ ( power_power @ A @ ( uminus_uminus @ A @ ( one_one @ A ) ) @ N ) @ ( power_power @ A @ ( uminus_uminus @ A @ ( one_one @ A ) ) @ N ) )
= ( one_one @ A ) ) ) ).
% minus_one_mult_self
thf(fact_1503_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_1504_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_1505_Suc__pred,axiom,
! [N: nat] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ N )
=> ( ( suc @ ( minus_minus @ nat @ N @ ( suc @ ( zero_zero @ nat ) ) ) )
= N ) ) ).
% Suc_pred
thf(fact_1506_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_1507_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_1508_Suc__diff,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq @ nat @ M @ N )
=> ( ( ord_less_eq @ nat @ ( one_one @ nat ) @ M )
=> ( ( suc @ ( minus_minus @ nat @ N @ M ) )
= ( minus_minus @ nat @ N @ ( minus_minus @ nat @ M @ ( one_one @ nat ) ) ) ) ) ) ).
% Suc_diff
thf(fact_1509_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_1510_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_1511_diff__numeral__simps_I2_J,axiom,
! [A: $tType] :
( ( neg_numeral @ A )
=> ! [M: num,N: num] :
( ( minus_minus @ A @ ( numeral_numeral @ A @ M ) @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ N ) ) )
= ( numeral_numeral @ A @ ( plus_plus @ num @ M @ N ) ) ) ) ).
% diff_numeral_simps(2)
thf(fact_1512_diff__numeral__simps_I3_J,axiom,
! [A: $tType] :
( ( neg_numeral @ A )
=> ! [M: num,N: num] :
( ( minus_minus @ A @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ M ) ) @ ( numeral_numeral @ A @ N ) )
= ( uminus_uminus @ A @ ( numeral_numeral @ A @ ( plus_plus @ num @ M @ N ) ) ) ) ) ).
% diff_numeral_simps(3)
thf(fact_1513_semiring__norm_I3_J,axiom,
! [N: num] :
( ( plus_plus @ num @ one2 @ ( bit0 @ N ) )
= ( bit1 @ N ) ) ).
% semiring_norm(3)
thf(fact_1514_semiring__norm_I4_J,axiom,
! [N: num] :
( ( plus_plus @ num @ one2 @ ( bit1 @ N ) )
= ( bit0 @ ( plus_plus @ num @ N @ one2 ) ) ) ).
% semiring_norm(4)
thf(fact_1515_semiring__norm_I5_J,axiom,
! [M: num] :
( ( plus_plus @ num @ ( bit0 @ M ) @ one2 )
= ( bit1 @ M ) ) ).
% semiring_norm(5)
thf(fact_1516_semiring__norm_I8_J,axiom,
! [M: num] :
( ( plus_plus @ num @ ( bit1 @ M ) @ one2 )
= ( bit0 @ ( plus_plus @ num @ M @ one2 ) ) ) ).
% semiring_norm(8)
thf(fact_1517_semiring__norm_I10_J,axiom,
! [M: num,N: num] :
( ( plus_plus @ num @ ( bit1 @ M ) @ ( bit1 @ N ) )
= ( bit0 @ ( plus_plus @ num @ ( plus_plus @ num @ M @ N ) @ one2 ) ) ) ).
% semiring_norm(10)
thf(fact_1518_mult__neg__numeral__simps_I1_J,axiom,
! [A: $tType] :
( ( ring_1 @ A )
=> ! [M: num,N: num] :
( ( times_times @ A @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ M ) ) @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ N ) ) )
= ( numeral_numeral @ A @ ( times_times @ num @ M @ N ) ) ) ) ).
% mult_neg_numeral_simps(1)
thf(fact_1519_mult__neg__numeral__simps_I2_J,axiom,
! [A: $tType] :
( ( ring_1 @ A )
=> ! [M: num,N: num] :
( ( times_times @ A @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ M ) ) @ ( numeral_numeral @ A @ N ) )
= ( uminus_uminus @ A @ ( numeral_numeral @ A @ ( times_times @ num @ M @ N ) ) ) ) ) ).
% mult_neg_numeral_simps(2)
thf(fact_1520_mult__neg__numeral__simps_I3_J,axiom,
! [A: $tType] :
( ( ring_1 @ A )
=> ! [M: num,N: num] :
( ( times_times @ A @ ( numeral_numeral @ A @ M ) @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ N ) ) )
= ( uminus_uminus @ A @ ( numeral_numeral @ A @ ( times_times @ num @ M @ N ) ) ) ) ) ).
% mult_neg_numeral_simps(3)
thf(fact_1521_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_1522_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_1523_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_1524_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_1525_neg__numeral__le__iff,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [M: num,N: num] :
( ( ord_less_eq @ A @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ M ) ) @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ N ) ) )
= ( ord_less_eq @ num @ N @ M ) ) ) ).
% neg_numeral_le_iff
thf(fact_1526_neg__numeral__less__iff,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [M: num,N: num] :
( ( ord_less @ A @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ M ) ) @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ N ) ) )
= ( ord_less @ num @ N @ M ) ) ) ).
% neg_numeral_less_iff
thf(fact_1527_semiring__norm_I16_J,axiom,
! [M: num,N: num] :
( ( times_times @ num @ ( bit1 @ M ) @ ( bit1 @ N ) )
= ( bit1 @ ( plus_plus @ num @ ( plus_plus @ num @ M @ N ) @ ( bit0 @ ( times_times @ num @ M @ N ) ) ) ) ) ).
% semiring_norm(16)
thf(fact_1528_semiring__norm_I79_J,axiom,
! [M: num,N: num] :
( ( ord_less @ num @ ( bit0 @ M ) @ ( bit1 @ N ) )
= ( ord_less_eq @ num @ M @ N ) ) ).
% semiring_norm(79)
thf(fact_1529_semiring__norm_I74_J,axiom,
! [M: num,N: num] :
( ( ord_less_eq @ num @ ( bit1 @ M ) @ ( bit0 @ N ) )
= ( ord_less @ num @ M @ N ) ) ).
% semiring_norm(74)
thf(fact_1530_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_1531_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_1532_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_1533_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_1534_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_1535_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_1536_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_1537_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_1538_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_1539_of__nat__0__less__iff,axiom,
! [A: $tType] :
( ( linord181362715937106298miring @ A )
=> ! [N: nat] :
( ( ord_less @ A @ ( zero_zero @ A ) @ ( semiring_1_of_nat @ A @ N ) )
= ( ord_less @ nat @ ( zero_zero @ nat ) @ N ) ) ) ).
% of_nat_0_less_iff
thf(fact_1540_Suc__diff__1,axiom,
! [N: nat] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ N )
=> ( ( suc @ ( minus_minus @ nat @ N @ ( one_one @ nat ) ) )
= N ) ) ).
% Suc_diff_1
thf(fact_1541_real__of__nat__eq__numeral__power__cancel__iff,axiom,
! [A: $tType] :
( ( semiring_char_0 @ A )
=> ! [Y: nat,X: num,N: nat] :
( ( ( semiring_1_of_nat @ A @ Y )
= ( power_power @ A @ ( numeral_numeral @ A @ X ) @ N ) )
= ( Y
= ( power_power @ nat @ ( numeral_numeral @ nat @ X ) @ N ) ) ) ) ).
% real_of_nat_eq_numeral_power_cancel_iff
thf(fact_1542_numeral__power__eq__of__nat__cancel__iff,axiom,
! [A: $tType] :
( ( semiring_char_0 @ A )
=> ! [X: num,N: nat,Y: nat] :
( ( ( power_power @ A @ ( numeral_numeral @ A @ X ) @ N )
= ( semiring_1_of_nat @ A @ Y ) )
= ( ( power_power @ nat @ ( numeral_numeral @ nat @ X ) @ N )
= Y ) ) ) ).
% numeral_power_eq_of_nat_cancel_iff
thf(fact_1543_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_1544_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_1545_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_1546_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_1547_real__of__nat__less__numeral__iff,axiom,
! [N: nat,W: num] :
( ( ord_less @ real @ ( semiring_1_of_nat @ real @ N ) @ ( numeral_numeral @ real @ W ) )
= ( ord_less @ nat @ N @ ( numeral_numeral @ nat @ W ) ) ) ).
% real_of_nat_less_numeral_iff
thf(fact_1548_numeral__less__real__of__nat__iff,axiom,
! [W: num,N: nat] :
( ( ord_less @ real @ ( numeral_numeral @ real @ W ) @ ( semiring_1_of_nat @ real @ N ) )
= ( ord_less @ nat @ ( numeral_numeral @ nat @ W ) @ N ) ) ).
% numeral_less_real_of_nat_iff
thf(fact_1549_numeral__le__real__of__nat__iff,axiom,
! [N: num,M: nat] :
( ( ord_less_eq @ real @ ( numeral_numeral @ real @ N ) @ ( semiring_1_of_nat @ real @ M ) )
= ( ord_less_eq @ nat @ ( numeral_numeral @ nat @ N ) @ M ) ) ).
% numeral_le_real_of_nat_iff
thf(fact_1550_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_1551_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_1552_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_1553_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_1554_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_1555_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_1556_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_1557_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_1558_of__nat__zero__less__power__iff,axiom,
! [A: $tType] :
( ( linordered_semidom @ A )
=> ! [X: nat,N: nat] :
( ( ord_less @ A @ ( zero_zero @ A ) @ ( power_power @ A @ ( semiring_1_of_nat @ A @ X ) @ N ) )
= ( ( ord_less @ nat @ ( zero_zero @ nat ) @ X )
| ( N
= ( zero_zero @ nat ) ) ) ) ) ).
% of_nat_zero_less_power_iff
thf(fact_1559_Power_Oring__1__class_Opower__minus__even,axiom,
! [A: $tType] :
( ( ring_1 @ A )
=> ! [A2: A,N: nat] :
( ( power_power @ A @ ( uminus_uminus @ A @ A2 ) @ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N ) )
= ( power_power @ A @ A2 @ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N ) ) ) ) ).
% Power.ring_1_class.power_minus_even
thf(fact_1560_Parity_Oring__1__class_Opower__minus__even,axiom,
! [A: $tType] :
( ( ring_1 @ A )
=> ! [N: nat,A2: A] :
( ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N )
=> ( ( power_power @ A @ ( uminus_uminus @ A @ A2 ) @ N )
= ( power_power @ A @ A2 @ N ) ) ) ) ).
% Parity.ring_1_class.power_minus_even
thf(fact_1561_power__minus__odd,axiom,
! [A: $tType] :
( ( ring_1 @ A )
=> ! [N: nat,A2: A] :
( ~ ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N )
=> ( ( power_power @ A @ ( uminus_uminus @ A @ A2 ) @ N )
= ( uminus_uminus @ A @ ( power_power @ A @ A2 @ N ) ) ) ) ) ).
% power_minus_odd
thf(fact_1562_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_1563_div__Suc__eq__div__add3,axiom,
! [M: nat,N: nat] :
( ( divide_divide @ nat @ M @ ( suc @ ( suc @ ( suc @ N ) ) ) )
= ( divide_divide @ nat @ M @ ( plus_plus @ nat @ ( numeral_numeral @ nat @ ( bit1 @ one2 ) ) @ N ) ) ) ).
% div_Suc_eq_div_add3
thf(fact_1564_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_1565_mod__Suc__eq__mod__add3,axiom,
! [M: nat,N: nat] :
( ( modulo_modulo @ nat @ M @ ( suc @ ( suc @ ( suc @ N ) ) ) )
= ( modulo_modulo @ nat @ M @ ( plus_plus @ nat @ ( numeral_numeral @ nat @ ( bit1 @ one2 ) ) @ N ) ) ) ).
% mod_Suc_eq_mod_add3
thf(fact_1566_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_1567_diff__numeral__special_I3_J,axiom,
! [A: $tType] :
( ( neg_numeral @ A )
=> ! [N: num] :
( ( minus_minus @ A @ ( one_one @ A ) @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ N ) ) )
= ( numeral_numeral @ A @ ( plus_plus @ num @ one2 @ N ) ) ) ) ).
% diff_numeral_special(3)
thf(fact_1568_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_1569_power__minus1__even,axiom,
! [A: $tType] :
( ( ring_1 @ A )
=> ! [N: nat] :
( ( power_power @ A @ ( uminus_uminus @ A @ ( one_one @ A ) ) @ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N ) )
= ( one_one @ A ) ) ) ).
% power_minus1_even
thf(fact_1570_neg__one__even__power,axiom,
! [A: $tType] :
( ( ring_1 @ A )
=> ! [N: nat] :
( ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N )
=> ( ( power_power @ A @ ( uminus_uminus @ A @ ( one_one @ A ) ) @ N )
= ( one_one @ A ) ) ) ) ).
% neg_one_even_power
thf(fact_1571_neg__one__odd__power,axiom,
! [A: $tType] :
( ( ring_1 @ A )
=> ! [N: nat] :
( ~ ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N )
=> ( ( power_power @ A @ ( uminus_uminus @ A @ ( one_one @ A ) ) @ N )
= ( uminus_uminus @ A @ ( one_one @ A ) ) ) ) ) ).
% neg_one_odd_power
thf(fact_1572_of__nat__less__numeral__power__cancel__iff,axiom,
! [A: $tType] :
( ( linordered_semidom @ A )
=> ! [X: nat,I: num,N: nat] :
( ( ord_less @ A @ ( semiring_1_of_nat @ A @ X ) @ ( power_power @ A @ ( numeral_numeral @ A @ I ) @ N ) )
= ( ord_less @ nat @ X @ ( power_power @ nat @ ( numeral_numeral @ nat @ I ) @ N ) ) ) ) ).
% of_nat_less_numeral_power_cancel_iff
thf(fact_1573_numeral__power__less__of__nat__cancel__iff,axiom,
! [A: $tType] :
( ( linordered_semidom @ A )
=> ! [I: num,N: nat,X: nat] :
( ( ord_less @ A @ ( power_power @ A @ ( numeral_numeral @ A @ I ) @ N ) @ ( semiring_1_of_nat @ A @ X ) )
= ( ord_less @ nat @ ( power_power @ nat @ ( numeral_numeral @ nat @ I ) @ N ) @ X ) ) ) ).
% numeral_power_less_of_nat_cancel_iff
thf(fact_1574_of__nat__le__numeral__power__cancel__iff,axiom,
! [A: $tType] :
( ( linordered_semidom @ A )
=> ! [X: nat,I: num,N: nat] :
( ( ord_less_eq @ A @ ( semiring_1_of_nat @ A @ X ) @ ( power_power @ A @ ( numeral_numeral @ A @ I ) @ N ) )
= ( ord_less_eq @ nat @ X @ ( power_power @ nat @ ( numeral_numeral @ nat @ I ) @ N ) ) ) ) ).
% of_nat_le_numeral_power_cancel_iff
thf(fact_1575_numeral__power__le__of__nat__cancel__iff,axiom,
! [A: $tType] :
( ( linordered_semidom @ A )
=> ! [I: num,N: nat,X: nat] :
( ( ord_less_eq @ A @ ( power_power @ A @ ( numeral_numeral @ A @ I ) @ N ) @ ( semiring_1_of_nat @ A @ X ) )
= ( ord_less_eq @ nat @ ( power_power @ nat @ ( numeral_numeral @ nat @ I ) @ N ) @ X ) ) ) ).
% numeral_power_le_of_nat_cancel_iff
thf(fact_1576_even__of__nat,axiom,
! [A: $tType] :
( ( semiring_parity @ A )
=> ! [N: nat] :
( ( dvd_dvd @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ ( semiring_1_of_nat @ A @ N ) )
= ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N ) ) ) ).
% even_of_nat
thf(fact_1577_odd__Suc__minus__one,axiom,
! [N: nat] :
( ~ ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N )
=> ( ( suc @ ( minus_minus @ nat @ N @ ( suc @ ( zero_zero @ nat ) ) ) )
= N ) ) ).
% odd_Suc_minus_one
thf(fact_1578_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_1579_even__diff__nat,axiom,
! [M: nat,N: nat] :
( ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( minus_minus @ nat @ M @ N ) )
= ( ( ord_less @ nat @ M @ N )
| ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( plus_plus @ nat @ M @ N ) ) ) ) ).
% even_diff_nat
thf(fact_1580_signed__take__bit__Suc__minus__bit0,axiom,
! [N: nat,K: num] :
( ( bit_ri4674362597316999326ke_bit @ int @ ( suc @ N ) @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ ( bit0 @ K ) ) ) )
= ( times_times @ int @ ( bit_ri4674362597316999326ke_bit @ int @ N @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ K ) ) ) @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) ) ) ).
% signed_take_bit_Suc_minus_bit0
thf(fact_1581_semiring__parity__class_Oeven__mask__iff,axiom,
! [A: $tType] :
( ( semiring_parity @ A )
=> ! [N: nat] :
( ( dvd_dvd @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ ( minus_minus @ A @ ( power_power @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ N ) @ ( one_one @ A ) ) )
= ( N
= ( zero_zero @ nat ) ) ) ) ).
% semiring_parity_class.even_mask_iff
thf(fact_1582_signed__take__bit__Suc__bit1,axiom,
! [N: nat,K: num] :
( ( bit_ri4674362597316999326ke_bit @ int @ ( suc @ N ) @ ( numeral_numeral @ int @ ( bit1 @ K ) ) )
= ( plus_plus @ int @ ( times_times @ int @ ( bit_ri4674362597316999326ke_bit @ int @ N @ ( numeral_numeral @ int @ K ) ) @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) ) @ ( one_one @ int ) ) ) ).
% signed_take_bit_Suc_bit1
thf(fact_1583_signed__take__bit__Suc__minus__bit1,axiom,
! [N: nat,K: num] :
( ( bit_ri4674362597316999326ke_bit @ int @ ( suc @ N ) @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ ( bit1 @ K ) ) ) )
= ( plus_plus @ int @ ( times_times @ int @ ( bit_ri4674362597316999326ke_bit @ int @ N @ ( minus_minus @ int @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ K ) ) @ ( one_one @ int ) ) ) @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) ) @ ( one_one @ int ) ) ) ).
% signed_take_bit_Suc_minus_bit1
thf(fact_1584_cnt__cnt__eq,axiom,
( vEBT_VEBT_cnt
= ( ^ [T2: vEBT_VEBT] : ( semiring_1_of_nat @ real @ ( vEBT_VEBT_cnt2 @ T2 ) ) ) ) ).
% cnt_cnt_eq
thf(fact_1585_minus__int__code_I2_J,axiom,
! [L: int] :
( ( minus_minus @ int @ ( zero_zero @ int ) @ L )
= ( uminus_uminus @ int @ L ) ) ).
% minus_int_code(2)
thf(fact_1586_of__nat__diff,axiom,
! [A: $tType] :
( ( semiring_1_cancel @ A )
=> ! [N: nat,M: nat] :
( ( ord_less_eq @ nat @ N @ M )
=> ( ( semiring_1_of_nat @ A @ ( minus_minus @ nat @ M @ N ) )
= ( minus_minus @ A @ ( semiring_1_of_nat @ A @ M ) @ ( semiring_1_of_nat @ A @ N ) ) ) ) ) ).
% of_nat_diff
thf(fact_1587_le__refl,axiom,
! [N: nat] : ( ord_less_eq @ nat @ N @ N ) ).
% le_refl
thf(fact_1588_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_1589_eq__imp__le,axiom,
! [M: nat,N: nat] :
( ( M = N )
=> ( ord_less_eq @ nat @ M @ N ) ) ).
% eq_imp_le
thf(fact_1590_le__antisym,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq @ nat @ M @ N )
=> ( ( ord_less_eq @ nat @ N @ M )
=> ( M = N ) ) ) ).
% le_antisym
thf(fact_1591_eq__diff__iff,axiom,
! [K: nat,M: nat,N: nat] :
( ( ord_less_eq @ nat @ K @ M )
=> ( ( ord_less_eq @ nat @ K @ N )
=> ( ( ( minus_minus @ nat @ M @ K )
= ( minus_minus @ nat @ N @ K ) )
= ( M = N ) ) ) ) ).
% eq_diff_iff
thf(fact_1592_le__diff__iff,axiom,
! [K: nat,M: nat,N: nat] :
( ( ord_less_eq @ nat @ K @ M )
=> ( ( ord_less_eq @ nat @ K @ N )
=> ( ( ord_less_eq @ nat @ ( minus_minus @ nat @ M @ K ) @ ( minus_minus @ nat @ N @ K ) )
= ( ord_less_eq @ nat @ M @ N ) ) ) ) ).
% le_diff_iff
thf(fact_1593_Nat_Odiff__diff__eq,axiom,
! [K: nat,M: nat,N: nat] :
( ( ord_less_eq @ nat @ K @ M )
=> ( ( ord_less_eq @ nat @ K @ N )
=> ( ( minus_minus @ nat @ ( minus_minus @ nat @ M @ K ) @ ( minus_minus @ nat @ N @ K ) )
= ( minus_minus @ nat @ M @ N ) ) ) ) ).
% Nat.diff_diff_eq
thf(fact_1594_diff__le__mono,axiom,
! [M: nat,N: nat,L: nat] :
( ( ord_less_eq @ nat @ M @ N )
=> ( ord_less_eq @ nat @ ( minus_minus @ nat @ M @ L ) @ ( minus_minus @ nat @ N @ L ) ) ) ).
% diff_le_mono
thf(fact_1595_diff__le__self,axiom,
! [M: nat,N: nat] : ( ord_less_eq @ nat @ ( minus_minus @ nat @ M @ N ) @ M ) ).
% diff_le_self
thf(fact_1596_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_1597_diff__le__mono2,axiom,
! [M: nat,N: nat,L: nat] :
( ( ord_less_eq @ nat @ M @ N )
=> ( ord_less_eq @ nat @ ( minus_minus @ nat @ L @ N ) @ ( minus_minus @ nat @ L @ M ) ) ) ).
% diff_le_mono2
thf(fact_1598_nat__le__linear,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq @ nat @ M @ N )
| ( ord_less_eq @ nat @ N @ M ) ) ).
% nat_le_linear
thf(fact_1599_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_1600_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_1601_signed__take__bit__diff,axiom,
! [N: nat,K: int,L: int] :
( ( bit_ri4674362597316999326ke_bit @ int @ N @ ( minus_minus @ int @ ( bit_ri4674362597316999326ke_bit @ int @ N @ K ) @ ( bit_ri4674362597316999326ke_bit @ int @ N @ L ) ) )
= ( bit_ri4674362597316999326ke_bit @ int @ N @ ( minus_minus @ int @ K @ L ) ) ) ).
% signed_take_bit_diff
thf(fact_1602_signed__take__bit__minus,axiom,
! [N: nat,K: int] :
( ( bit_ri4674362597316999326ke_bit @ int @ N @ ( uminus_uminus @ int @ ( bit_ri4674362597316999326ke_bit @ int @ N @ K ) ) )
= ( bit_ri4674362597316999326ke_bit @ int @ N @ ( uminus_uminus @ int @ K ) ) ) ).
% signed_take_bit_minus
thf(fact_1603_ab__group__add__class_Oab__diff__conv__add__uminus,axiom,
! [A: $tType] :
( ( ab_group_add @ A )
=> ( ( minus_minus @ A )
= ( ^ [A4: A,B5: A] : ( plus_plus @ A @ A4 @ ( uminus_uminus @ A @ B5 ) ) ) ) ) ).
% ab_group_add_class.ab_diff_conv_add_uminus
thf(fact_1604_diff__conv__add__uminus,axiom,
! [A: $tType] :
( ( group_add @ A )
=> ( ( minus_minus @ A )
= ( ^ [A4: A,B5: A] : ( plus_plus @ A @ A4 @ ( uminus_uminus @ A @ B5 ) ) ) ) ) ).
% diff_conv_add_uminus
thf(fact_1605_group__cancel_Osub2,axiom,
! [A: $tType] :
( ( ab_group_add @ A )
=> ! [B4: A,K: A,B2: A,A2: A] :
( ( B4
= ( plus_plus @ A @ K @ B2 ) )
=> ( ( minus_minus @ A @ A2 @ B4 )
= ( plus_plus @ A @ ( uminus_uminus @ A @ K ) @ ( minus_minus @ A @ A2 @ B2 ) ) ) ) ) ).
% group_cancel.sub2
thf(fact_1606_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_1607_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_1608_minus__equation__iff,axiom,
! [A: $tType] :
( ( group_add @ A )
=> ! [A2: A,B2: A] :
( ( ( uminus_uminus @ A @ A2 )
= B2 )
= ( ( uminus_uminus @ A @ B2 )
= A2 ) ) ) ).
% minus_equation_iff
thf(fact_1609_equation__minus__iff,axiom,
! [A: $tType] :
( ( group_add @ A )
=> ! [A2: A,B2: A] :
( ( A2
= ( uminus_uminus @ A @ B2 ) )
= ( B2
= ( uminus_uminus @ A @ A2 ) ) ) ) ).
% equation_minus_iff
thf(fact_1610_diff__eq__diff__eq,axiom,
! [A: $tType] :
( ( group_add @ A )
=> ! [A2: A,B2: A,C2: A,D2: A] :
( ( ( minus_minus @ A @ A2 @ B2 )
= ( minus_minus @ A @ C2 @ D2 ) )
=> ( ( A2 = B2 )
= ( C2 = D2 ) ) ) ) ).
% diff_eq_diff_eq
thf(fact_1611_verit__negate__coefficient_I3_J,axiom,
! [A: $tType] :
( ( ordered_ab_group_add @ A )
=> ! [A2: A,B2: A] :
( ( A2 = B2 )
=> ( ( uminus_uminus @ A @ A2 )
= ( uminus_uminus @ A @ B2 ) ) ) ) ).
% verit_negate_coefficient(3)
thf(fact_1612_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_1613_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_1614_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_1615_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_1616_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_1617_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_1618_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_1619_eq__iff__diff__eq__0,axiom,
! [A: $tType] :
( ( group_add @ A )
=> ( ( ^ [Y2: A,Z2: A] : Y2 = Z2 )
= ( ^ [A4: A,B5: A] :
( ( minus_minus @ A @ A4 @ B5 )
= ( zero_zero @ A ) ) ) ) ) ).
% eq_iff_diff_eq_0
thf(fact_1620_diff__eq__diff__less__eq,axiom,
! [A: $tType] :
( ( ordered_ab_group_add @ A )
=> ! [A2: A,B2: A,C2: A,D2: A] :
( ( ( minus_minus @ A @ A2 @ B2 )
= ( minus_minus @ A @ C2 @ D2 ) )
=> ( ( ord_less_eq @ A @ A2 @ B2 )
= ( ord_less_eq @ A @ C2 @ D2 ) ) ) ) ).
% diff_eq_diff_less_eq
thf(fact_1621_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_1622_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_1623_diff__mono,axiom,
! [A: $tType] :
( ( ordered_ab_group_add @ A )
=> ! [A2: A,B2: A,D2: A,C2: A] :
( ( ord_less_eq @ A @ A2 @ B2 )
=> ( ( ord_less_eq @ A @ D2 @ C2 )
=> ( ord_less_eq @ A @ ( minus_minus @ A @ A2 @ C2 ) @ ( minus_minus @ A @ B2 @ D2 ) ) ) ) ) ).
% diff_mono
thf(fact_1624_diff__strict__mono,axiom,
! [A: $tType] :
( ( ordered_ab_group_add @ A )
=> ! [A2: A,B2: A,D2: A,C2: A] :
( ( ord_less @ A @ A2 @ B2 )
=> ( ( ord_less @ A @ D2 @ C2 )
=> ( ord_less @ A @ ( minus_minus @ A @ A2 @ C2 ) @ ( minus_minus @ A @ B2 @ D2 ) ) ) ) ) ).
% diff_strict_mono
thf(fact_1625_diff__eq__diff__less,axiom,
! [A: $tType] :
( ( ordered_ab_group_add @ A )
=> ! [A2: A,B2: A,C2: A,D2: A] :
( ( ( minus_minus @ A @ A2 @ B2 )
= ( minus_minus @ A @ C2 @ D2 ) )
=> ( ( ord_less @ A @ A2 @ B2 )
= ( ord_less @ A @ C2 @ D2 ) ) ) ) ).
% diff_eq_diff_less
thf(fact_1626_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_1627_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_1628_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_1629_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_1630_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_1631_add__diff__add,axiom,
! [A: $tType] :
( ( ab_group_add @ A )
=> ! [A2: A,C2: A,B2: A,D2: A] :
( ( minus_minus @ A @ ( plus_plus @ A @ A2 @ C2 ) @ ( plus_plus @ A @ B2 @ D2 ) )
= ( plus_plus @ A @ ( minus_minus @ A @ A2 @ B2 ) @ ( minus_minus @ A @ C2 @ D2 ) ) ) ) ).
% add_diff_add
thf(fact_1632_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_1633_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_1634_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_1635_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_1636_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_1637_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_1638_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_1639_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_1640_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_1641_inf__period_I1_J,axiom,
! [A: $tType] :
( ( ( comm_ring @ A )
& ( dvd @ A ) )
=> ! [P: A > $o,D5: A,Q: A > $o] :
( ! [X3: A,K2: A] :
( ( P @ X3 )
= ( P @ ( minus_minus @ A @ X3 @ ( times_times @ A @ K2 @ D5 ) ) ) )
=> ( ! [X3: A,K2: A] :
( ( Q @ X3 )
= ( Q @ ( minus_minus @ A @ X3 @ ( times_times @ A @ K2 @ D5 ) ) ) )
=> ! [X5: A,K5: A] :
( ( ( P @ X5 )
& ( Q @ X5 ) )
= ( ( P @ ( minus_minus @ A @ X5 @ ( times_times @ A @ K5 @ D5 ) ) )
& ( Q @ ( minus_minus @ A @ X5 @ ( times_times @ A @ K5 @ D5 ) ) ) ) ) ) ) ) ).
% inf_period(1)
thf(fact_1642_inf__period_I2_J,axiom,
! [A: $tType] :
( ( ( comm_ring @ A )
& ( dvd @ A ) )
=> ! [P: A > $o,D5: A,Q: A > $o] :
( ! [X3: A,K2: A] :
( ( P @ X3 )
= ( P @ ( minus_minus @ A @ X3 @ ( times_times @ A @ K2 @ D5 ) ) ) )
=> ( ! [X3: A,K2: A] :
( ( Q @ X3 )
= ( Q @ ( minus_minus @ A @ X3 @ ( times_times @ A @ K2 @ D5 ) ) ) )
=> ! [X5: A,K5: A] :
( ( ( P @ X5 )
| ( Q @ X5 ) )
= ( ( P @ ( minus_minus @ A @ X5 @ ( times_times @ A @ K5 @ D5 ) ) )
| ( Q @ ( minus_minus @ A @ X5 @ ( times_times @ A @ K5 @ D5 ) ) ) ) ) ) ) ) ).
% inf_period(2)
thf(fact_1643_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_1644_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_1645_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_1646_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_1647_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_1648_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_1649_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_1650_numeral__neq__neg__numeral,axiom,
! [A: $tType] :
( ( ring_char_0 @ A )
=> ! [M: num,N: num] :
( ( numeral_numeral @ A @ M )
!= ( uminus_uminus @ A @ ( numeral_numeral @ A @ N ) ) ) ) ).
% numeral_neq_neg_numeral
thf(fact_1651_neg__numeral__neq__numeral,axiom,
! [A: $tType] :
( ( ring_char_0 @ A )
=> ! [M: num,N: num] :
( ( uminus_uminus @ A @ ( numeral_numeral @ A @ M ) )
!= ( numeral_numeral @ A @ N ) ) ) ).
% neg_numeral_neq_numeral
thf(fact_1652_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_1653_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_1654_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_1655_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_1656_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_1657_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_1658_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_1659_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_1660_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_1661_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_1662_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_1663_verit__eq__simplify_I14_J,axiom,
! [X22: num,X32: num] :
( ( bit0 @ X22 )
!= ( bit1 @ X32 ) ) ).
% verit_eq_simplify(14)
thf(fact_1664_verit__eq__simplify_I12_J,axiom,
! [X32: num] :
( one2
!= ( bit1 @ X32 ) ) ).
% verit_eq_simplify(12)
thf(fact_1665_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_1666_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_1667_zero__induct__lemma,axiom,
! [P: nat > $o,K: nat,I: nat] :
( ( P @ K )
=> ( ! [N2: nat] :
( ( P @ ( suc @ N2 ) )
=> ( P @ N2 ) )
=> ( P @ ( minus_minus @ nat @ K @ I ) ) ) ) ).
% zero_induct_lemma
thf(fact_1668_Suc__diff__le,axiom,
! [N: nat,M: nat] :
( ( ord_less_eq @ nat @ N @ M )
=> ( ( minus_minus @ nat @ ( suc @ M ) @ N )
= ( suc @ ( minus_minus @ nat @ M @ N ) ) ) ) ).
% Suc_diff_le
thf(fact_1669_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_1670_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_1671_mod__diff__cong,axiom,
! [A: $tType] :
( ( euclid8851590272496341667cancel @ A )
=> ! [A2: A,C2: A,A5: A,B2: A,B7: A] :
( ( ( modulo_modulo @ A @ A2 @ C2 )
= ( modulo_modulo @ A @ A5 @ C2 ) )
=> ( ( ( modulo_modulo @ A @ B2 @ C2 )
= ( modulo_modulo @ A @ B7 @ C2 ) )
=> ( ( modulo_modulo @ A @ ( minus_minus @ A @ A2 @ B2 ) @ C2 )
= ( modulo_modulo @ A @ ( minus_minus @ A @ A5 @ B7 ) @ C2 ) ) ) ) ) ).
% mod_diff_cong
thf(fact_1672_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_1673_diffs0__imp__equal,axiom,
! [M: nat,N: nat] :
( ( ( minus_minus @ nat @ M @ N )
= ( zero_zero @ nat ) )
=> ( ( ( minus_minus @ nat @ N @ M )
= ( zero_zero @ nat ) )
=> ( M = N ) ) ) ).
% diffs0_imp_equal
thf(fact_1674_minus__nat_Odiff__0,axiom,
! [M: nat] :
( ( minus_minus @ nat @ M @ ( zero_zero @ nat ) )
= M ) ).
% minus_nat.diff_0
thf(fact_1675_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_1676_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_1677_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_1678_diff__diff__less,axiom,
! [I: nat,M: nat,N: nat] :
( ( ord_less @ nat @ I @ ( minus_minus @ nat @ M @ ( minus_minus @ nat @ M @ N ) ) )
= ( ( ord_less @ nat @ I @ M )
& ( ord_less @ nat @ I @ N ) ) ) ).
% diff_diff_less
thf(fact_1679_less__imp__diff__less,axiom,
! [J: nat,K: nat,N: nat] :
( ( ord_less @ nat @ J @ K )
=> ( ord_less @ nat @ ( minus_minus @ nat @ J @ N ) @ K ) ) ).
% less_imp_diff_less
thf(fact_1680_diff__less__mono2,axiom,
! [M: nat,N: nat,L: nat] :
( ( ord_less @ nat @ M @ N )
=> ( ( ord_less @ nat @ M @ L )
=> ( ord_less @ nat @ ( minus_minus @ nat @ L @ N ) @ ( minus_minus @ nat @ L @ M ) ) ) ) ).
% diff_less_mono2
thf(fact_1681_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_1682_less__diff__iff,axiom,
! [K: nat,M: nat,N: nat] :
( ( ord_less_eq @ nat @ K @ M )
=> ( ( ord_less_eq @ nat @ K @ N )
=> ( ( ord_less @ nat @ ( minus_minus @ nat @ M @ K ) @ ( minus_minus @ nat @ N @ K ) )
= ( ord_less @ nat @ M @ N ) ) ) ) ).
% less_diff_iff
thf(fact_1683_minus__int__code_I1_J,axiom,
! [K: int] :
( ( minus_minus @ int @ K @ ( zero_zero @ int ) )
= K ) ).
% minus_int_code(1)
thf(fact_1684_Nat_Odiff__cancel,axiom,
! [K: nat,M: nat,N: nat] :
( ( minus_minus @ nat @ ( plus_plus @ nat @ K @ M ) @ ( plus_plus @ nat @ K @ N ) )
= ( minus_minus @ nat @ M @ N ) ) ).
% Nat.diff_cancel
thf(fact_1685_diff__cancel2,axiom,
! [M: nat,K: nat,N: nat] :
( ( minus_minus @ nat @ ( plus_plus @ nat @ M @ K ) @ ( plus_plus @ nat @ N @ K ) )
= ( minus_minus @ nat @ M @ N ) ) ).
% diff_cancel2
thf(fact_1686_diff__add__inverse,axiom,
! [N: nat,M: nat] :
( ( minus_minus @ nat @ ( plus_plus @ nat @ N @ M ) @ N )
= M ) ).
% diff_add_inverse
thf(fact_1687_diff__add__inverse2,axiom,
! [M: nat,N: nat] :
( ( minus_minus @ nat @ ( plus_plus @ nat @ M @ N ) @ N )
= M ) ).
% diff_add_inverse2
thf(fact_1688_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_1689_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_1690_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_1691_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_1692_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_1693_uminus__int__code_I1_J,axiom,
( ( uminus_uminus @ int @ ( zero_zero @ int ) )
= ( zero_zero @ int ) ) ).
% uminus_int_code(1)
thf(fact_1694_diff__mult__distrib2,axiom,
! [K: nat,M: nat,N: nat] :
( ( times_times @ nat @ K @ ( minus_minus @ nat @ M @ N ) )
= ( minus_minus @ nat @ ( times_times @ nat @ K @ M ) @ ( times_times @ nat @ K @ N ) ) ) ).
% diff_mult_distrib2
thf(fact_1695_diff__mult__distrib,axiom,
! [M: nat,N: nat,K: nat] :
( ( times_times @ nat @ ( minus_minus @ nat @ M @ N ) @ K )
= ( minus_minus @ nat @ ( times_times @ nat @ M @ K ) @ ( times_times @ nat @ N @ K ) ) ) ).
% diff_mult_distrib
thf(fact_1696_Word_Omod__minus__cong,axiom,
! [B2: int,B7: int,X: int,X6: int,Y: int,Y5: int,Z5: int] :
( ( B2 = B7 )
=> ( ( ( modulo_modulo @ int @ X @ B7 )
= ( modulo_modulo @ int @ X6 @ B7 ) )
=> ( ( ( modulo_modulo @ int @ Y @ B7 )
= ( modulo_modulo @ int @ Y5 @ B7 ) )
=> ( ( ( minus_minus @ int @ X6 @ Y5 )
= Z5 )
=> ( ( modulo_modulo @ int @ ( minus_minus @ int @ X @ Y ) @ B2 )
= ( modulo_modulo @ int @ Z5 @ B7 ) ) ) ) ) ) ).
% Word.mod_minus_cong
thf(fact_1697_dvd__diff__nat,axiom,
! [K: nat,M: nat,N: nat] :
( ( dvd_dvd @ nat @ K @ M )
=> ( ( dvd_dvd @ nat @ K @ N )
=> ( dvd_dvd @ nat @ K @ ( minus_minus @ nat @ M @ N ) ) ) ) ).
% dvd_diff_nat
thf(fact_1698_dvd__diffD,axiom,
! [K: nat,M: nat,N: nat] :
( ( dvd_dvd @ nat @ K @ ( minus_minus @ nat @ M @ N ) )
=> ( ( dvd_dvd @ nat @ K @ N )
=> ( ( ord_less_eq @ nat @ N @ M )
=> ( dvd_dvd @ nat @ K @ M ) ) ) ) ).
% dvd_diffD
thf(fact_1699_dvd__diffD1,axiom,
! [K: nat,M: nat,N: nat] :
( ( dvd_dvd @ nat @ K @ ( minus_minus @ nat @ M @ N ) )
=> ( ( dvd_dvd @ nat @ K @ M )
=> ( ( ord_less_eq @ nat @ N @ M )
=> ( dvd_dvd @ nat @ K @ N ) ) ) ) ).
% dvd_diffD1
thf(fact_1700_less__eq__dvd__minus,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq @ nat @ M @ N )
=> ( ( dvd_dvd @ nat @ M @ N )
= ( dvd_dvd @ nat @ M @ ( minus_minus @ nat @ N @ M ) ) ) ) ).
% less_eq_dvd_minus
thf(fact_1701_le__mod__geq,axiom,
! [N: nat,M: nat] :
( ( ord_less_eq @ nat @ N @ M )
=> ( ( modulo_modulo @ nat @ M @ N )
= ( modulo_modulo @ nat @ ( minus_minus @ nat @ M @ N ) @ N ) ) ) ).
% le_mod_geq
thf(fact_1702_uminus__dvd__conv_I1_J,axiom,
( ( dvd_dvd @ int )
= ( ^ [D: int] : ( dvd_dvd @ int @ ( uminus_uminus @ int @ D ) ) ) ) ).
% uminus_dvd_conv(1)
thf(fact_1703_uminus__dvd__conv_I2_J,axiom,
( ( dvd_dvd @ int )
= ( ^ [D: int,T2: int] : ( dvd_dvd @ int @ D @ ( uminus_uminus @ int @ T2 ) ) ) ) ).
% uminus_dvd_conv(2)
thf(fact_1704_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_1705_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_1706_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_1707_neg__one__power__add__eq__neg__one__power__diff,axiom,
! [A: $tType] :
( ( ring_1 @ A )
=> ! [K: nat,N: nat] :
( ( ord_less_eq @ nat @ K @ N )
=> ( ( power_power @ A @ ( uminus_uminus @ A @ ( one_one @ A ) ) @ ( plus_plus @ nat @ N @ K ) )
= ( power_power @ A @ ( uminus_uminus @ A @ ( one_one @ A ) ) @ ( minus_minus @ nat @ N @ K ) ) ) ) ) ).
% neg_one_power_add_eq_neg_one_power_diff
thf(fact_1708_field__char__0__class_Oof__nat__div,axiom,
! [A: $tType] :
( ( field_char_0 @ A )
=> ! [M: nat,N: nat] :
( ( semiring_1_of_nat @ A @ ( divide_divide @ nat @ M @ N ) )
= ( divide_divide @ A @ ( minus_minus @ A @ ( semiring_1_of_nat @ A @ M ) @ ( semiring_1_of_nat @ A @ ( modulo_modulo @ nat @ M @ N ) ) ) @ ( semiring_1_of_nat @ A @ N ) ) ) ) ).
% field_char_0_class.of_nat_div
thf(fact_1709_of__nat__0__le__iff,axiom,
! [A: $tType] :
( ( linord181362715937106298miring @ A )
=> ! [N: nat] : ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( semiring_1_of_nat @ A @ N ) ) ) ).
% of_nat_0_le_iff
thf(fact_1710_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_1711_semiring__char__0__class_Oof__nat__neq__0,axiom,
! [A: $tType] :
( ( semiring_char_0 @ A )
=> ! [N: nat] :
( ( semiring_1_of_nat @ A @ ( suc @ N ) )
!= ( zero_zero @ A ) ) ) ).
% semiring_char_0_class.of_nat_neq_0
thf(fact_1712_of__nat__less__imp__less,axiom,
! [A: $tType] :
( ( linord181362715937106298miring @ A )
=> ! [M: nat,N: nat] :
( ( ord_less @ A @ ( semiring_1_of_nat @ A @ M ) @ ( semiring_1_of_nat @ A @ N ) )
=> ( ord_less @ nat @ M @ N ) ) ) ).
% of_nat_less_imp_less
thf(fact_1713_less__imp__of__nat__less,axiom,
! [A: $tType] :
( ( linord181362715937106298miring @ A )
=> ! [M: nat,N: nat] :
( ( ord_less @ nat @ M @ N )
=> ( ord_less @ A @ ( semiring_1_of_nat @ A @ M ) @ ( semiring_1_of_nat @ A @ N ) ) ) ) ).
% less_imp_of_nat_less
thf(fact_1714_div__mult2__eq_H,axiom,
! [A: $tType] :
( ( euclid5411537665997757685th_nat @ A )
=> ! [A2: A,M: nat,N: nat] :
( ( divide_divide @ A @ A2 @ ( times_times @ A @ ( semiring_1_of_nat @ A @ M ) @ ( semiring_1_of_nat @ A @ N ) ) )
= ( divide_divide @ A @ ( divide_divide @ A @ A2 @ ( semiring_1_of_nat @ A @ M ) ) @ ( semiring_1_of_nat @ A @ N ) ) ) ) ).
% div_mult2_eq'
thf(fact_1715_minus__mod__int__eq,axiom,
! [L: int,K: int] :
( ( ord_less_eq @ int @ ( zero_zero @ int ) @ L )
=> ( ( modulo_modulo @ int @ ( uminus_uminus @ int @ K ) @ L )
= ( minus_minus @ int @ ( minus_minus @ int @ L @ ( one_one @ int ) ) @ ( modulo_modulo @ int @ ( minus_minus @ int @ K @ ( one_one @ int ) ) @ L ) ) ) ) ).
% minus_mod_int_eq
thf(fact_1716_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_1717_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_1718_le__iff__diff__le__0,axiom,
! [A: $tType] :
( ( ordered_ab_group_add @ A )
=> ( ( ord_less_eq @ A )
= ( ^ [A4: A,B5: A] : ( ord_less_eq @ A @ ( minus_minus @ A @ A4 @ B5 ) @ ( zero_zero @ A ) ) ) ) ) ).
% le_iff_diff_le_0
thf(fact_1719_less__iff__diff__less__0,axiom,
! [A: $tType] :
( ( ordered_ab_group_add @ A )
=> ( ( ord_less @ A )
= ( ^ [A4: A,B5: A] : ( ord_less @ A @ ( minus_minus @ A @ A4 @ B5 ) @ ( zero_zero @ A ) ) ) ) ) ).
% less_iff_diff_less_0
thf(fact_1720_add__le__imp__le__diff,axiom,
! [A: $tType] :
( ( linordered_semidom @ A )
=> ! [I: A,K: A,N: A] :
( ( ord_less_eq @ A @ ( plus_plus @ A @ I @ K ) @ N )
=> ( ord_less_eq @ A @ I @ ( minus_minus @ A @ N @ K ) ) ) ) ).
% add_le_imp_le_diff
thf(fact_1721_add__le__add__imp__diff__le,axiom,
! [A: $tType] :
( ( linordered_semidom @ A )
=> ! [I: A,K: A,N: A,J: A] :
( ( ord_less_eq @ A @ ( plus_plus @ A @ I @ K ) @ N )
=> ( ( ord_less_eq @ A @ N @ ( plus_plus @ A @ J @ K ) )
=> ( ( ord_less_eq @ A @ ( plus_plus @ A @ I @ K ) @ N )
=> ( ( ord_less_eq @ A @ N @ ( plus_plus @ A @ J @ K ) )
=> ( ord_less_eq @ A @ ( minus_minus @ A @ N @ K ) @ J ) ) ) ) ) ) ).
% add_le_add_imp_diff_le
thf(fact_1722_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_1723_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_1724_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_1725_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_1726_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_1727_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_1728_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_1729_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_1730_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_1731_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_1732_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_1733_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_1734_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_1735_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_1736_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_1737_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_1738_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_1739_zero__neq__neg__numeral,axiom,
! [A: $tType] :
( ( ring_char_0 @ A )
=> ! [N: num] :
( ( zero_zero @ A )
!= ( uminus_uminus @ A @ ( numeral_numeral @ A @ N ) ) ) ) ).
% zero_neq_neg_numeral
thf(fact_1740_not__numeral__le__neg__numeral,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [M: num,N: num] :
~ ( ord_less_eq @ A @ ( numeral_numeral @ A @ M ) @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ N ) ) ) ) ).
% not_numeral_le_neg_numeral
thf(fact_1741_neg__numeral__le__numeral,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [M: num,N: num] : ( ord_less_eq @ A @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ M ) ) @ ( numeral_numeral @ A @ N ) ) ) ).
% neg_numeral_le_numeral
thf(fact_1742_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_1743_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_1744_eq__add__iff2,axiom,
! [A: $tType] :
( ( ring @ A )
=> ! [A2: A,E: A,C2: A,B2: A,D2: A] :
( ( ( plus_plus @ A @ ( times_times @ A @ A2 @ E ) @ C2 )
= ( plus_plus @ A @ ( times_times @ A @ B2 @ E ) @ D2 ) )
= ( C2
= ( plus_plus @ A @ ( times_times @ A @ ( minus_minus @ A @ B2 @ A2 ) @ E ) @ D2 ) ) ) ) ).
% eq_add_iff2
thf(fact_1745_eq__add__iff1,axiom,
! [A: $tType] :
( ( ring @ A )
=> ! [A2: A,E: A,C2: A,B2: A,D2: A] :
( ( ( plus_plus @ A @ ( times_times @ A @ A2 @ E ) @ C2 )
= ( plus_plus @ A @ ( times_times @ A @ B2 @ E ) @ D2 ) )
= ( ( plus_plus @ A @ ( times_times @ A @ ( minus_minus @ A @ A2 @ B2 ) @ E ) @ C2 )
= D2 ) ) ) ).
% eq_add_iff1
thf(fact_1746_neg__numeral__less__numeral,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [M: num,N: num] : ( ord_less @ A @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ M ) ) @ ( numeral_numeral @ A @ N ) ) ) ).
% neg_numeral_less_numeral
thf(fact_1747_not__numeral__less__neg__numeral,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [M: num,N: num] :
~ ( ord_less @ A @ ( numeral_numeral @ A @ M ) @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ N ) ) ) ) ).
% not_numeral_less_neg_numeral
thf(fact_1748_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_1749_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_1750_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_1751_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_1752_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_1753_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_1754_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_1755_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_1756_unique__euclidean__semiring__with__nat__class_Oof__nat__div,axiom,
! [A: $tType] :
( ( euclid5411537665997757685th_nat @ A )
=> ! [M: nat,N: nat] :
( ( semiring_1_of_nat @ A @ ( divide_divide @ nat @ M @ N ) )
= ( divide_divide @ A @ ( semiring_1_of_nat @ A @ M ) @ ( semiring_1_of_nat @ A @ N ) ) ) ) ).
% unique_euclidean_semiring_with_nat_class.of_nat_div
thf(fact_1757_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_1758_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_1759_numeral__neq__neg__one,axiom,
! [A: $tType] :
( ( ring_char_0 @ A )
=> ! [N: num] :
( ( numeral_numeral @ A @ N )
!= ( uminus_uminus @ A @ ( one_one @ A ) ) ) ) ).
% numeral_neq_neg_one
thf(fact_1760_one__neq__neg__numeral,axiom,
! [A: $tType] :
( ( ring_char_0 @ A )
=> ! [N: num] :
( ( one_one @ A )
!= ( uminus_uminus @ A @ ( numeral_numeral @ A @ N ) ) ) ) ).
% one_neq_neg_numeral
thf(fact_1761_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_1762_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_1763_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_1764_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_1765_num_Oexhaust,axiom,
! [Y: num] :
( ( Y != one2 )
=> ( ! [X23: num] :
( Y
!= ( bit0 @ X23 ) )
=> ~ ! [X33: num] :
( Y
!= ( bit1 @ X33 ) ) ) ) ).
% num.exhaust
thf(fact_1766_of__nat__dvd__iff,axiom,
! [A: $tType] :
( ( euclid5411537665997757685th_nat @ A )
=> ! [M: nat,N: nat] :
( ( dvd_dvd @ A @ ( semiring_1_of_nat @ A @ M ) @ ( semiring_1_of_nat @ A @ N ) )
= ( dvd_dvd @ nat @ M @ N ) ) ) ).
% of_nat_dvd_iff
thf(fact_1767_nat__less__real__le,axiom,
( ( ord_less @ nat )
= ( ^ [N3: nat,M4: nat] : ( ord_less_eq @ real @ ( plus_plus @ real @ ( semiring_1_of_nat @ real @ N3 ) @ ( one_one @ real ) ) @ ( semiring_1_of_nat @ real @ M4 ) ) ) ) ).
% nat_less_real_le
thf(fact_1768_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_1769_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_1770_nat__le__real__less,axiom,
( ( ord_less_eq @ nat )
= ( ^ [N3: nat,M4: nat] : ( ord_less @ real @ ( semiring_1_of_nat @ real @ N3 ) @ ( plus_plus @ real @ ( semiring_1_of_nat @ real @ M4 ) @ ( one_one @ real ) ) ) ) ) ).
% nat_le_real_less
thf(fact_1771_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_1772_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_1773_of__nat__mod,axiom,
! [A: $tType] :
( ( euclid5411537665997757685th_nat @ A )
=> ! [M: nat,N: nat] :
( ( semiring_1_of_nat @ A @ ( modulo_modulo @ nat @ M @ N ) )
= ( modulo_modulo @ A @ ( semiring_1_of_nat @ A @ M ) @ ( semiring_1_of_nat @ A @ N ) ) ) ) ).
% of_nat_mod
thf(fact_1774_Suc__to__right,axiom,
! [N: nat,M: nat] :
( ( ( suc @ N )
= M )
=> ( N
= ( minus_minus @ nat @ M @ ( suc @ ( zero_zero @ nat ) ) ) ) ) ).
% Suc_to_right
thf(fact_1775_Suc__diff__Suc,axiom,
! [N: nat,M: nat] :
( ( ord_less @ nat @ N @ M )
=> ( ( suc @ ( minus_minus @ nat @ M @ ( suc @ N ) ) )
= ( minus_minus @ nat @ M @ N ) ) ) ).
% Suc_diff_Suc
thf(fact_1776_diff__less__Suc,axiom,
! [M: nat,N: nat] : ( ord_less @ nat @ ( minus_minus @ nat @ M @ N ) @ ( suc @ M ) ) ).
% diff_less_Suc
thf(fact_1777_diff__less,axiom,
! [N: nat,M: nat] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ N )
=> ( ( ord_less @ nat @ ( zero_zero @ nat ) @ M )
=> ( ord_less @ nat @ ( minus_minus @ nat @ M @ N ) @ M ) ) ) ).
% diff_less
thf(fact_1778_real__of__nat__div4,axiom,
! [N: nat,X: nat] : ( ord_less_eq @ real @ ( semiring_1_of_nat @ real @ ( divide_divide @ nat @ N @ X ) ) @ ( divide_divide @ real @ ( semiring_1_of_nat @ real @ N ) @ ( semiring_1_of_nat @ real @ X ) ) ) ).
% real_of_nat_div4
thf(fact_1779_diff__Suc__eq__diff__pred,axiom,
! [M: nat,N: nat] :
( ( minus_minus @ nat @ M @ ( suc @ N ) )
= ( minus_minus @ nat @ ( minus_minus @ nat @ M @ ( one_one @ nat ) ) @ N ) ) ).
% diff_Suc_eq_diff_pred
thf(fact_1780_diff__add__0,axiom,
! [N: nat,M: nat] :
( ( minus_minus @ nat @ N @ ( plus_plus @ nat @ N @ M ) )
= ( zero_zero @ nat ) ) ).
% diff_add_0
thf(fact_1781_add__diff__inverse__nat,axiom,
! [M: nat,N: nat] :
( ~ ( ord_less @ nat @ M @ N )
=> ( ( plus_plus @ nat @ N @ ( minus_minus @ nat @ M @ N ) )
= M ) ) ).
% add_diff_inverse_nat
thf(fact_1782_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_1783_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_1784_int__le__induct,axiom,
! [I: int,K: int,P: int > $o] :
( ( ord_less_eq @ int @ I @ K )
=> ( ( P @ K )
=> ( ! [I3: int] :
( ( ord_less_eq @ int @ I3 @ K )
=> ( ( P @ I3 )
=> ( P @ ( minus_minus @ int @ I3 @ ( one_one @ int ) ) ) ) )
=> ( P @ I ) ) ) ) ).
% int_le_induct
thf(fact_1785_int__less__induct,axiom,
! [I: int,K: int,P: int > $o] :
( ( ord_less @ int @ I @ K )
=> ( ( P @ ( minus_minus @ int @ K @ ( one_one @ int ) ) )
=> ( ! [I3: int] :
( ( ord_less @ int @ I3 @ K )
=> ( ( P @ I3 )
=> ( P @ ( minus_minus @ int @ I3 @ ( one_one @ int ) ) ) ) )
=> ( P @ I ) ) ) ) ).
% int_less_induct
thf(fact_1786_nat__diff__add__eq2,axiom,
! [I: nat,J: nat,U: nat,M: nat,N: nat] :
( ( ord_less_eq @ nat @ I @ J )
=> ( ( minus_minus @ nat @ ( plus_plus @ nat @ ( times_times @ nat @ I @ U ) @ M ) @ ( plus_plus @ nat @ ( times_times @ nat @ J @ U ) @ N ) )
= ( minus_minus @ nat @ M @ ( plus_plus @ nat @ ( times_times @ nat @ ( minus_minus @ nat @ J @ I ) @ U ) @ N ) ) ) ) ).
% nat_diff_add_eq2
thf(fact_1787_nat__diff__add__eq1,axiom,
! [J: nat,I: nat,U: nat,M: nat,N: nat] :
( ( ord_less_eq @ nat @ J @ I )
=> ( ( minus_minus @ nat @ ( plus_plus @ nat @ ( times_times @ nat @ I @ U ) @ M ) @ ( plus_plus @ nat @ ( times_times @ nat @ J @ U ) @ N ) )
= ( minus_minus @ nat @ ( plus_plus @ nat @ ( times_times @ nat @ ( minus_minus @ nat @ I @ J ) @ U ) @ M ) @ N ) ) ) ).
% nat_diff_add_eq1
thf(fact_1788_nat__le__add__iff2,axiom,
! [I: nat,J: nat,U: nat,M: nat,N: nat] :
( ( ord_less_eq @ nat @ I @ J )
=> ( ( ord_less_eq @ nat @ ( plus_plus @ nat @ ( times_times @ nat @ I @ U ) @ M ) @ ( plus_plus @ nat @ ( times_times @ nat @ J @ U ) @ N ) )
= ( ord_less_eq @ nat @ M @ ( plus_plus @ nat @ ( times_times @ nat @ ( minus_minus @ nat @ J @ I ) @ U ) @ N ) ) ) ) ).
% nat_le_add_iff2
thf(fact_1789_nat__le__add__iff1,axiom,
! [J: nat,I: nat,U: nat,M: nat,N: nat] :
( ( ord_less_eq @ nat @ J @ I )
=> ( ( ord_less_eq @ nat @ ( plus_plus @ nat @ ( times_times @ nat @ I @ U ) @ M ) @ ( plus_plus @ nat @ ( times_times @ nat @ J @ U ) @ N ) )
= ( ord_less_eq @ nat @ ( plus_plus @ nat @ ( times_times @ nat @ ( minus_minus @ nat @ I @ J ) @ U ) @ M ) @ N ) ) ) ).
% nat_le_add_iff1
thf(fact_1790_nat__eq__add__iff2,axiom,
! [I: nat,J: nat,U: nat,M: nat,N: nat] :
( ( ord_less_eq @ nat @ I @ J )
=> ( ( ( plus_plus @ nat @ ( times_times @ nat @ I @ U ) @ M )
= ( plus_plus @ nat @ ( times_times @ nat @ J @ U ) @ N ) )
= ( M
= ( plus_plus @ nat @ ( times_times @ nat @ ( minus_minus @ nat @ J @ I ) @ U ) @ N ) ) ) ) ).
% nat_eq_add_iff2
thf(fact_1791_nat__eq__add__iff1,axiom,
! [J: nat,I: nat,U: nat,M: nat,N: nat] :
( ( ord_less_eq @ nat @ J @ I )
=> ( ( ( plus_plus @ nat @ ( times_times @ nat @ I @ U ) @ M )
= ( plus_plus @ nat @ ( times_times @ nat @ J @ U ) @ N ) )
= ( ( plus_plus @ nat @ ( times_times @ nat @ ( minus_minus @ nat @ I @ J ) @ U ) @ M )
= N ) ) ) ).
% nat_eq_add_iff1
thf(fact_1792_dvd__minus__self,axiom,
! [M: nat,N: nat] :
( ( dvd_dvd @ nat @ M @ ( minus_minus @ nat @ N @ M ) )
= ( ( ord_less @ nat @ N @ M )
| ( dvd_dvd @ nat @ M @ N ) ) ) ).
% dvd_minus_self
thf(fact_1793_nat__minus__mod,axiom,
! [N: nat,M: nat] :
( ( modulo_modulo @ nat @ ( minus_minus @ nat @ N @ ( modulo_modulo @ nat @ N @ M ) ) @ M )
= ( zero_zero @ nat ) ) ).
% nat_minus_mod
thf(fact_1794_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_1795_mod__geq,axiom,
! [M: nat,N: nat] :
( ~ ( ord_less @ nat @ M @ N )
=> ( ( modulo_modulo @ nat @ M @ N )
= ( modulo_modulo @ nat @ ( minus_minus @ nat @ M @ N ) @ N ) ) ) ).
% mod_geq
thf(fact_1796_mod__if,axiom,
( ( modulo_modulo @ nat )
= ( ^ [M4: nat,N3: nat] : ( if @ nat @ ( ord_less @ nat @ M4 @ N3 ) @ M4 @ ( modulo_modulo @ nat @ ( minus_minus @ nat @ M4 @ N3 ) @ N3 ) ) ) ) ).
% mod_if
thf(fact_1797_pos__zmult__eq__1__iff__lemma,axiom,
! [M: int,N: int] :
( ( ( times_times @ int @ M @ N )
= ( one_one @ int ) )
=> ( ( M
= ( one_one @ int ) )
| ( M
= ( uminus_uminus @ int @ ( one_one @ int ) ) ) ) ) ).
% pos_zmult_eq_1_iff_lemma
thf(fact_1798_zmult__eq__1__iff,axiom,
! [M: int,N: int] :
( ( ( times_times @ int @ M @ N )
= ( one_one @ int ) )
= ( ( ( M
= ( one_one @ int ) )
& ( N
= ( one_one @ int ) ) )
| ( ( M
= ( uminus_uminus @ int @ ( one_one @ int ) ) )
& ( N
= ( uminus_uminus @ int @ ( one_one @ int ) ) ) ) ) ) ).
% zmult_eq_1_iff
thf(fact_1799_zmod__zminus1__not__zero,axiom,
! [K: int,L: int] :
( ( ( modulo_modulo @ int @ ( uminus_uminus @ int @ K ) @ L )
!= ( zero_zero @ int ) )
=> ( ( modulo_modulo @ int @ K @ L )
!= ( zero_zero @ int ) ) ) ).
% zmod_zminus1_not_zero
thf(fact_1800_zmod__zminus2__not__zero,axiom,
! [K: int,L: int] :
( ( ( modulo_modulo @ int @ K @ ( uminus_uminus @ int @ L ) )
!= ( zero_zero @ int ) )
=> ( ( modulo_modulo @ int @ K @ L )
!= ( zero_zero @ int ) ) ) ).
% zmod_zminus2_not_zero
thf(fact_1801_nat__minus__mod__plus__right,axiom,
! [N: nat,X: nat,M: nat] :
( ( modulo_modulo @ nat @ ( minus_minus @ nat @ ( plus_plus @ nat @ N @ X ) @ ( modulo_modulo @ nat @ N @ M ) ) @ M )
= ( modulo_modulo @ nat @ X @ M ) ) ).
% nat_minus_mod_plus_right
thf(fact_1802_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_1803_mod__eq__dvd__iff__nat,axiom,
! [N: nat,M: nat,Q3: nat] :
( ( ord_less_eq @ nat @ N @ M )
=> ( ( ( modulo_modulo @ nat @ M @ Q3 )
= ( modulo_modulo @ nat @ N @ Q3 ) )
= ( dvd_dvd @ nat @ Q3 @ ( minus_minus @ nat @ M @ N ) ) ) ) ).
% mod_eq_dvd_iff_nat
thf(fact_1804_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_1805_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_1806_ordered__ring__class_Ole__add__iff2,axiom,
! [A: $tType] :
( ( ordered_ring @ A )
=> ! [A2: A,E: A,C2: A,B2: A,D2: A] :
( ( ord_less_eq @ A @ ( plus_plus @ A @ ( times_times @ A @ A2 @ E ) @ C2 ) @ ( plus_plus @ A @ ( times_times @ A @ B2 @ E ) @ D2 ) )
= ( ord_less_eq @ A @ C2 @ ( plus_plus @ A @ ( times_times @ A @ ( minus_minus @ A @ B2 @ A2 ) @ E ) @ D2 ) ) ) ) ).
% ordered_ring_class.le_add_iff2
thf(fact_1807_ordered__ring__class_Ole__add__iff1,axiom,
! [A: $tType] :
( ( ordered_ring @ A )
=> ! [A2: A,E: A,C2: A,B2: A,D2: A] :
( ( ord_less_eq @ A @ ( plus_plus @ A @ ( times_times @ A @ A2 @ E ) @ C2 ) @ ( plus_plus @ A @ ( times_times @ A @ B2 @ E ) @ D2 ) )
= ( ord_less_eq @ A @ ( plus_plus @ A @ ( times_times @ A @ ( minus_minus @ A @ A2 @ B2 ) @ E ) @ C2 ) @ D2 ) ) ) ).
% ordered_ring_class.le_add_iff1
thf(fact_1808_neg__numeral__le__zero,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [N: num] : ( ord_less_eq @ A @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ N ) ) @ ( zero_zero @ A ) ) ) ).
% neg_numeral_le_zero
thf(fact_1809_not__zero__le__neg__numeral,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [N: num] :
~ ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ N ) ) ) ) ).
% not_zero_le_neg_numeral
thf(fact_1810_less__add__iff2,axiom,
! [A: $tType] :
( ( ordered_ring @ A )
=> ! [A2: A,E: A,C2: A,B2: A,D2: A] :
( ( ord_less @ A @ ( plus_plus @ A @ ( times_times @ A @ A2 @ E ) @ C2 ) @ ( plus_plus @ A @ ( times_times @ A @ B2 @ E ) @ D2 ) )
= ( ord_less @ A @ C2 @ ( plus_plus @ A @ ( times_times @ A @ ( minus_minus @ A @ B2 @ A2 ) @ E ) @ D2 ) ) ) ) ).
% less_add_iff2
thf(fact_1811_less__add__iff1,axiom,
! [A: $tType] :
( ( ordered_ring @ A )
=> ! [A2: A,E: A,C2: A,B2: A,D2: A] :
( ( ord_less @ A @ ( plus_plus @ A @ ( times_times @ A @ A2 @ E ) @ C2 ) @ ( plus_plus @ A @ ( times_times @ A @ B2 @ E ) @ D2 ) )
= ( ord_less @ A @ ( plus_plus @ A @ ( times_times @ A @ ( minus_minus @ A @ A2 @ B2 ) @ E ) @ C2 ) @ D2 ) ) ) ).
% less_add_iff1
thf(fact_1812_neg__numeral__less__zero,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [N: num] : ( ord_less @ A @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ N ) ) @ ( zero_zero @ A ) ) ) ).
% neg_numeral_less_zero
thf(fact_1813_not__zero__less__neg__numeral,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [N: num] :
~ ( ord_less @ A @ ( zero_zero @ A ) @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ N ) ) ) ) ).
% not_zero_less_neg_numeral
thf(fact_1814_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_1815_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_1816_m1mod2k,axiom,
! [N: nat] :
( ( modulo_modulo @ int @ ( uminus_uminus @ int @ ( one_one @ int ) ) @ ( power_power @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ N ) )
= ( minus_minus @ int @ ( power_power @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ N ) @ ( one_one @ int ) ) ) ).
% m1mod2k
thf(fact_1817_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_1818_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_1819_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_1820_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_1821_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_1822_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_1823_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_1824_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_1825_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_1826_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_1827_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_1828_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_1829_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_1830_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_1831_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_1832_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_1833_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_1834_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_1835_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_1836_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_1837_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_1838_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_1839_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_1840_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_1841_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_1842_inf__period_I4_J,axiom,
! [A: $tType] :
( ( ( comm_ring @ A )
& ( dvd @ A ) )
=> ! [D2: A,D5: A,T: A] :
( ( dvd_dvd @ A @ D2 @ D5 )
=> ! [X5: A,K5: A] :
( ( ~ ( dvd_dvd @ A @ D2 @ ( plus_plus @ A @ X5 @ T ) ) )
= ( ~ ( dvd_dvd @ A @ D2 @ ( plus_plus @ A @ ( minus_minus @ A @ X5 @ ( times_times @ A @ K5 @ D5 ) ) @ T ) ) ) ) ) ) ).
% inf_period(4)
thf(fact_1843_inf__period_I3_J,axiom,
! [A: $tType] :
( ( ( comm_ring @ A )
& ( dvd @ A ) )
=> ! [D2: A,D5: A,T: A] :
( ( dvd_dvd @ A @ D2 @ D5 )
=> ! [X5: A,K5: A] :
( ( dvd_dvd @ A @ D2 @ ( plus_plus @ A @ X5 @ T ) )
= ( dvd_dvd @ A @ D2 @ ( plus_plus @ A @ ( minus_minus @ A @ X5 @ ( times_times @ A @ K5 @ D5 ) ) @ T ) ) ) ) ) ).
% inf_period(3)
thf(fact_1844_power__minus,axiom,
! [A: $tType] :
( ( ring_1 @ A )
=> ! [A2: A,N: nat] :
( ( power_power @ A @ ( uminus_uminus @ A @ A2 ) @ N )
= ( times_times @ A @ ( power_power @ A @ ( uminus_uminus @ A @ ( one_one @ A ) ) @ N ) @ ( power_power @ A @ A2 @ N ) ) ) ) ).
% power_minus
thf(fact_1845_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_1846_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_1847_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_1848_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_1849_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_1850_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_1851_numeral__Bit1,axiom,
! [A: $tType] :
( ( numeral @ A )
=> ! [N: num] :
( ( numeral_numeral @ A @ ( bit1 @ N ) )
= ( plus_plus @ A @ ( plus_plus @ A @ ( numeral_numeral @ A @ N ) @ ( numeral_numeral @ A @ N ) ) @ ( one_one @ A ) ) ) ) ).
% numeral_Bit1
thf(fact_1852_eval__nat__numeral_I3_J,axiom,
! [N: num] :
( ( numeral_numeral @ nat @ ( bit1 @ N ) )
= ( suc @ ( numeral_numeral @ nat @ ( bit0 @ N ) ) ) ) ).
% eval_nat_numeral(3)
thf(fact_1853_power__diff,axiom,
! [A: $tType] :
( ( semidom_divide @ A )
=> ! [A2: A,N: nat,M: nat] :
( ( A2
!= ( zero_zero @ A ) )
=> ( ( ord_less_eq @ nat @ N @ M )
=> ( ( power_power @ A @ A2 @ ( minus_minus @ nat @ M @ N ) )
= ( divide_divide @ A @ ( power_power @ A @ A2 @ M ) @ ( power_power @ A @ A2 @ N ) ) ) ) ) ) ).
% power_diff
thf(fact_1854_cong__exp__iff__simps_I10_J,axiom,
! [A: $tType] :
( ( unique1627219031080169319umeral @ A )
=> ! [M: num,Q3: num,N: num] :
( ( modulo_modulo @ A @ ( numeral_numeral @ A @ ( bit0 @ M ) ) @ ( numeral_numeral @ A @ ( bit0 @ Q3 ) ) )
!= ( modulo_modulo @ A @ ( numeral_numeral @ A @ ( bit1 @ N ) ) @ ( numeral_numeral @ A @ ( bit0 @ Q3 ) ) ) ) ) ).
% cong_exp_iff_simps(10)
thf(fact_1855_cong__exp__iff__simps_I12_J,axiom,
! [A: $tType] :
( ( unique1627219031080169319umeral @ A )
=> ! [M: num,Q3: num,N: num] :
( ( modulo_modulo @ A @ ( numeral_numeral @ A @ ( bit1 @ M ) ) @ ( numeral_numeral @ A @ ( bit0 @ Q3 ) ) )
!= ( modulo_modulo @ A @ ( numeral_numeral @ A @ ( bit0 @ N ) ) @ ( numeral_numeral @ A @ ( bit0 @ Q3 ) ) ) ) ) ).
% cong_exp_iff_simps(12)
thf(fact_1856_cong__exp__iff__simps_I13_J,axiom,
! [A: $tType] :
( ( unique1627219031080169319umeral @ A )
=> ! [M: num,Q3: num,N: num] :
( ( ( modulo_modulo @ A @ ( numeral_numeral @ A @ ( bit1 @ M ) ) @ ( numeral_numeral @ A @ ( bit0 @ Q3 ) ) )
= ( modulo_modulo @ A @ ( numeral_numeral @ A @ ( bit1 @ N ) ) @ ( numeral_numeral @ A @ ( bit0 @ Q3 ) ) ) )
= ( ( modulo_modulo @ A @ ( numeral_numeral @ A @ M ) @ ( numeral_numeral @ A @ Q3 ) )
= ( modulo_modulo @ A @ ( numeral_numeral @ A @ N ) @ ( numeral_numeral @ A @ Q3 ) ) ) ) ) ).
% cong_exp_iff_simps(13)
thf(fact_1857_reals__Archimedean3,axiom,
! [X: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ X )
=> ! [Y4: real] :
? [N2: nat] : ( ord_less @ real @ Y4 @ ( times_times @ real @ ( semiring_1_of_nat @ real @ N2 ) @ X ) ) ) ).
% reals_Archimedean3
thf(fact_1858_diff__Suc__less,axiom,
! [N: nat,I: nat] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ N )
=> ( ord_less @ nat @ ( minus_minus @ nat @ N @ ( suc @ I ) ) @ N ) ) ).
% diff_Suc_less
thf(fact_1859_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_1860_Suc__n__minus__m__eq,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq @ nat @ M @ N )
=> ( ( ord_less @ nat @ ( one_one @ nat ) @ M )
=> ( ( suc @ ( minus_minus @ nat @ N @ M ) )
= ( minus_minus @ nat @ N @ ( minus_minus @ nat @ M @ ( one_one @ nat ) ) ) ) ) ) ).
% Suc_n_minus_m_eq
thf(fact_1861_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 ) ) )
& ! [D: nat] :
( ( A2
= ( plus_plus @ nat @ B2 @ D ) )
=> ( P @ D ) ) ) ) ).
% nat_diff_split
thf(fact_1862_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 ) ) )
| ? [D: nat] :
( ( A2
= ( plus_plus @ nat @ B2 @ D ) )
& ~ ( P @ D ) ) ) ) ) ).
% nat_diff_split_asm
thf(fact_1863_real__of__nat__div,axiom,
! [D2: nat,N: nat] :
( ( dvd_dvd @ nat @ D2 @ N )
=> ( ( semiring_1_of_nat @ real @ ( divide_divide @ nat @ N @ D2 ) )
= ( divide_divide @ real @ ( semiring_1_of_nat @ real @ N ) @ ( semiring_1_of_nat @ real @ D2 ) ) ) ) ).
% real_of_nat_div
thf(fact_1864_nat__less__add__iff2,axiom,
! [I: nat,J: nat,U: nat,M: nat,N: nat] :
( ( ord_less_eq @ nat @ I @ J )
=> ( ( ord_less @ nat @ ( plus_plus @ nat @ ( times_times @ nat @ I @ U ) @ M ) @ ( plus_plus @ nat @ ( times_times @ nat @ J @ U ) @ N ) )
= ( ord_less @ nat @ M @ ( plus_plus @ nat @ ( times_times @ nat @ ( minus_minus @ nat @ J @ I ) @ U ) @ N ) ) ) ) ).
% nat_less_add_iff2
thf(fact_1865_nat__less__add__iff1,axiom,
! [J: nat,I: nat,U: nat,M: nat,N: nat] :
( ( ord_less_eq @ nat @ J @ I )
=> ( ( ord_less @ nat @ ( plus_plus @ nat @ ( times_times @ nat @ I @ U ) @ M ) @ ( plus_plus @ nat @ ( times_times @ nat @ J @ U ) @ N ) )
= ( ord_less @ nat @ ( plus_plus @ nat @ ( times_times @ nat @ ( minus_minus @ nat @ I @ J ) @ U ) @ M ) @ N ) ) ) ).
% nat_less_add_iff1
thf(fact_1866_real__of__nat__div__aux,axiom,
! [X: nat,D2: nat] :
( ( divide_divide @ real @ ( semiring_1_of_nat @ real @ X ) @ ( semiring_1_of_nat @ real @ D2 ) )
= ( plus_plus @ real @ ( semiring_1_of_nat @ real @ ( divide_divide @ nat @ X @ D2 ) ) @ ( divide_divide @ real @ ( semiring_1_of_nat @ real @ ( modulo_modulo @ nat @ X @ D2 ) ) @ ( semiring_1_of_nat @ real @ D2 ) ) ) ) ).
% real_of_nat_div_aux
thf(fact_1867_minusinfinity,axiom,
! [D2: int,P1: int > $o,P: int > $o] :
( ( ord_less @ int @ ( zero_zero @ int ) @ D2 )
=> ( ! [X3: int,K2: int] :
( ( P1 @ X3 )
= ( P1 @ ( minus_minus @ int @ X3 @ ( times_times @ int @ K2 @ D2 ) ) ) )
=> ( ? [Z4: int] :
! [X3: int] :
( ( ord_less @ int @ X3 @ Z4 )
=> ( ( P @ X3 )
= ( P1 @ X3 ) ) )
=> ( ? [X_12: int] : ( P1 @ X_12 )
=> ? [X_1: int] : ( P @ X_1 ) ) ) ) ) ).
% minusinfinity
thf(fact_1868_plusinfinity,axiom,
! [D2: int,P3: int > $o,P: int > $o] :
( ( ord_less @ int @ ( zero_zero @ int ) @ D2 )
=> ( ! [X3: int,K2: int] :
( ( P3 @ X3 )
= ( P3 @ ( minus_minus @ int @ X3 @ ( times_times @ int @ K2 @ D2 ) ) ) )
=> ( ? [Z4: int] :
! [X3: int] :
( ( ord_less @ int @ Z4 @ X3 )
=> ( ( P @ X3 )
= ( P3 @ X3 ) ) )
=> ( ? [X_12: int] : ( P3 @ X_12 )
=> ? [X_1: int] : ( P @ X_1 ) ) ) ) ) ).
% plusinfinity
thf(fact_1869_less__1__helper,axiom,
! [N: int,M: int] :
( ( ord_less_eq @ int @ N @ M )
=> ( ord_less @ int @ ( minus_minus @ int @ N @ ( one_one @ int ) ) @ M ) ) ).
% less_1_helper
thf(fact_1870_int__induct,axiom,
! [P: int > $o,K: int,I: int] :
( ( P @ K )
=> ( ! [I3: int] :
( ( ord_less_eq @ int @ K @ I3 )
=> ( ( P @ I3 )
=> ( P @ ( plus_plus @ int @ I3 @ ( one_one @ int ) ) ) ) )
=> ( ! [I3: int] :
( ( ord_less_eq @ int @ I3 @ K )
=> ( ( P @ I3 )
=> ( P @ ( minus_minus @ int @ I3 @ ( one_one @ int ) ) ) ) )
=> ( P @ I ) ) ) ) ).
% int_induct
thf(fact_1871_int__mod__le_H,axiom,
! [B2: int,N: int] :
( ( ord_less_eq @ int @ ( zero_zero @ int ) @ ( minus_minus @ int @ B2 @ N ) )
=> ( ord_less_eq @ int @ ( modulo_modulo @ int @ B2 @ N ) @ ( minus_minus @ int @ B2 @ N ) ) ) ).
% int_mod_le'
thf(fact_1872_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_1873_dvd__minus__add,axiom,
! [Q3: nat,N: nat,R2: nat,M: nat] :
( ( ord_less_eq @ nat @ Q3 @ N )
=> ( ( ord_less_eq @ nat @ Q3 @ ( times_times @ nat @ R2 @ M ) )
=> ( ( dvd_dvd @ nat @ M @ ( minus_minus @ nat @ N @ Q3 ) )
= ( dvd_dvd @ nat @ M @ ( plus_plus @ nat @ N @ ( minus_minus @ nat @ ( times_times @ nat @ R2 @ M ) @ Q3 ) ) ) ) ) ) ).
% dvd_minus_add
thf(fact_1874_diff__mod__le,axiom,
! [A2: nat,D2: nat,B2: nat] :
( ( ord_less @ nat @ A2 @ D2 )
=> ( ( dvd_dvd @ nat @ B2 @ D2 )
=> ( ord_less_eq @ nat @ ( minus_minus @ nat @ A2 @ ( modulo_modulo @ nat @ A2 @ B2 ) ) @ ( minus_minus @ nat @ D2 @ B2 ) ) ) ) ).
% diff_mod_le
thf(fact_1875_mod__nat__eqI,axiom,
! [R2: nat,N: nat,M: nat] :
( ( ord_less @ nat @ R2 @ N )
=> ( ( ord_less_eq @ nat @ R2 @ M )
=> ( ( dvd_dvd @ nat @ N @ ( minus_minus @ nat @ M @ R2 ) )
=> ( ( modulo_modulo @ nat @ M @ N )
= R2 ) ) ) ) ).
% mod_nat_eqI
thf(fact_1876_modulo__nat__def,axiom,
( ( modulo_modulo @ nat )
= ( ^ [M4: nat,N3: nat] : ( minus_minus @ nat @ M4 @ ( times_times @ nat @ ( divide_divide @ nat @ M4 @ N3 ) @ N3 ) ) ) ) ).
% modulo_nat_def
thf(fact_1877_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_1878_m1mod22k,axiom,
! [N: nat] :
( ( modulo_modulo @ int @ ( uminus_uminus @ int @ ( one_one @ int ) ) @ ( times_times @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ ( power_power @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ N ) ) )
= ( minus_minus @ int @ ( times_times @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ ( power_power @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ N ) ) @ ( one_one @ int ) ) ) ).
% m1mod22k
thf(fact_1879_inverse__of__nat__le,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [N: nat,M: nat] :
( ( ord_less_eq @ nat @ N @ M )
=> ( ( N
!= ( zero_zero @ nat ) )
=> ( ord_less_eq @ A @ ( divide_divide @ A @ ( one_one @ A ) @ ( semiring_1_of_nat @ A @ M ) ) @ ( divide_divide @ A @ ( one_one @ A ) @ ( semiring_1_of_nat @ A @ N ) ) ) ) ) ) ).
% inverse_of_nat_le
thf(fact_1880_mod__mult2__eq_H,axiom,
! [A: $tType] :
( ( euclid5411537665997757685th_nat @ A )
=> ! [A2: A,M: nat,N: nat] :
( ( modulo_modulo @ A @ A2 @ ( times_times @ A @ ( semiring_1_of_nat @ A @ M ) @ ( semiring_1_of_nat @ A @ N ) ) )
= ( plus_plus @ A @ ( times_times @ A @ ( semiring_1_of_nat @ A @ M ) @ ( modulo_modulo @ A @ ( divide_divide @ A @ A2 @ ( semiring_1_of_nat @ A @ M ) ) @ ( semiring_1_of_nat @ A @ N ) ) ) @ ( modulo_modulo @ A @ A2 @ ( semiring_1_of_nat @ A @ M ) ) ) ) ) ).
% mod_mult2_eq'
thf(fact_1881_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_1882_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_1883_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_1884_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_1885_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_1886_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_1887_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_1888_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_1889_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_1890_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_1891_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_1892_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_1893_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_1894_odd__mod__4__div__2,axiom,
! [N: nat] :
( ( ( modulo_modulo @ nat @ N @ ( numeral_numeral @ nat @ ( bit0 @ ( bit0 @ one2 ) ) ) )
= ( numeral_numeral @ nat @ ( bit1 @ one2 ) ) )
=> ~ ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ ( minus_minus @ nat @ N @ ( suc @ ( zero_zero @ nat ) ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ).
% odd_mod_4_div_2
thf(fact_1895_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_1896_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_1897_numeral__Bit1__div__2,axiom,
! [A: $tType] :
( ( euclid5411537665997757685th_nat @ A )
=> ! [N: num] :
( ( divide_divide @ A @ ( numeral_numeral @ A @ ( bit1 @ N ) ) @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) )
= ( numeral_numeral @ A @ N ) ) ) ).
% numeral_Bit1_div_2
thf(fact_1898_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 )
=> ( ! [M2: nat] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ M2 )
=> ( ord_less_eq @ real @ ( times_times @ real @ ( semiring_1_of_nat @ real @ M2 ) @ X ) @ C2 ) )
=> ( X
= ( zero_zero @ real ) ) ) ) ) ).
% real_archimedian_rdiv_eq_0
thf(fact_1899_cong__exp__iff__simps_I3_J,axiom,
! [A: $tType] :
( ( unique1627219031080169319umeral @ A )
=> ! [N: num,Q3: num] :
( ( modulo_modulo @ A @ ( numeral_numeral @ A @ ( bit1 @ N ) ) @ ( numeral_numeral @ A @ ( bit0 @ Q3 ) ) )
!= ( zero_zero @ A ) ) ) ).
% cong_exp_iff_simps(3)
thf(fact_1900_odd__numeral,axiom,
! [A: $tType] :
( ( semiring_parity @ A )
=> ! [N: num] :
~ ( dvd_dvd @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ ( numeral_numeral @ A @ ( bit1 @ N ) ) ) ) ).
% odd_numeral
thf(fact_1901_power__diff__power__eq,axiom,
! [A: $tType] :
( ( euclid4440199948858584721cancel @ A )
=> ! [A2: A,N: nat,M: nat] :
( ( A2
!= ( zero_zero @ A ) )
=> ( ( ( ord_less_eq @ nat @ N @ M )
=> ( ( divide_divide @ A @ ( power_power @ A @ A2 @ M ) @ ( power_power @ A @ A2 @ N ) )
= ( power_power @ A @ A2 @ ( minus_minus @ nat @ M @ N ) ) ) )
& ( ~ ( ord_less_eq @ nat @ N @ M )
=> ( ( divide_divide @ A @ ( power_power @ A @ A2 @ M ) @ ( power_power @ A @ A2 @ N ) )
= ( divide_divide @ A @ ( one_one @ A ) @ ( power_power @ A @ A2 @ ( minus_minus @ nat @ N @ M ) ) ) ) ) ) ) ) ).
% power_diff_power_eq
thf(fact_1902_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_1903_numeral__3__eq__3,axiom,
( ( numeral_numeral @ nat @ ( bit1 @ one2 ) )
= ( suc @ ( suc @ ( suc @ ( zero_zero @ nat ) ) ) ) ) ).
% numeral_3_eq_3
thf(fact_1904_Suc3__eq__add__3,axiom,
! [N: nat] :
( ( suc @ ( suc @ ( suc @ N ) ) )
= ( plus_plus @ nat @ ( numeral_numeral @ nat @ ( bit1 @ one2 ) ) @ N ) ) ).
% Suc3_eq_add_3
thf(fact_1905_diff__le__diff__pow,axiom,
! [K: nat,M: nat,N: nat] :
( ( ord_less_eq @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ K )
=> ( ord_less_eq @ nat @ ( minus_minus @ nat @ M @ N ) @ ( minus_minus @ nat @ ( power_power @ nat @ K @ M ) @ ( power_power @ nat @ K @ N ) ) ) ) ).
% diff_le_diff_pow
thf(fact_1906_Suc__diff__eq__diff__pred,axiom,
! [N: nat,M: nat] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ N )
=> ( ( minus_minus @ nat @ ( suc @ M ) @ N )
= ( minus_minus @ nat @ M @ ( minus_minus @ nat @ N @ ( one_one @ nat ) ) ) ) ) ).
% Suc_diff_eq_diff_pred
thf(fact_1907_Suc__pred_H,axiom,
! [N: nat] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ N )
=> ( N
= ( suc @ ( minus_minus @ nat @ N @ ( one_one @ nat ) ) ) ) ) ).
% Suc_pred'
thf(fact_1908_add__eq__if,axiom,
( ( plus_plus @ nat )
= ( ^ [M4: nat,N3: nat] :
( if @ nat
@ ( M4
= ( zero_zero @ nat ) )
@ N3
@ ( suc @ ( plus_plus @ nat @ ( minus_minus @ nat @ M4 @ ( one_one @ nat ) ) @ N3 ) ) ) ) ) ).
% add_eq_if
thf(fact_1909_div__if,axiom,
( ( divide_divide @ nat )
= ( ^ [M4: nat,N3: nat] :
( if @ nat
@ ( ( ord_less @ nat @ M4 @ N3 )
| ( N3
= ( zero_zero @ nat ) ) )
@ ( zero_zero @ nat )
@ ( suc @ ( divide_divide @ nat @ ( minus_minus @ nat @ M4 @ N3 ) @ N3 ) ) ) ) ) ).
% div_if
thf(fact_1910_div__geq,axiom,
! [N: nat,M: nat] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ N )
=> ( ~ ( ord_less @ nat @ M @ N )
=> ( ( divide_divide @ nat @ M @ N )
= ( suc @ ( divide_divide @ nat @ ( minus_minus @ nat @ M @ N ) @ N ) ) ) ) ) ).
% div_geq
thf(fact_1911_le__div__geq,axiom,
! [N: nat,M: nat] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ N )
=> ( ( ord_less_eq @ nat @ N @ M )
=> ( ( divide_divide @ nat @ M @ N )
= ( suc @ ( divide_divide @ nat @ ( minus_minus @ nat @ M @ N ) @ N ) ) ) ) ) ).
% le_div_geq
thf(fact_1912_mult__eq__if,axiom,
( ( times_times @ nat )
= ( ^ [M4: nat,N3: nat] :
( if @ nat
@ ( M4
= ( zero_zero @ nat ) )
@ ( zero_zero @ nat )
@ ( plus_plus @ nat @ N3 @ ( times_times @ nat @ ( minus_minus @ nat @ M4 @ ( one_one @ nat ) ) @ N3 ) ) ) ) ) ).
% mult_eq_if
thf(fact_1913_power__sub,axiom,
! [N: nat,M: nat,A2: nat] :
( ( ord_less_eq @ nat @ N @ M )
=> ( ( ord_less @ nat @ ( zero_zero @ nat ) @ A2 )
=> ( ( power_power @ nat @ A2 @ ( minus_minus @ nat @ M @ N ) )
= ( divide_divide @ nat @ ( power_power @ nat @ A2 @ M ) @ ( power_power @ nat @ A2 @ N ) ) ) ) ) ).
% power_sub
thf(fact_1914_nat__mult__power__less__eq,axiom,
! [B2: nat,A2: nat,N: nat,M: nat] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ B2 )
=> ( ( ord_less @ nat @ ( times_times @ nat @ A2 @ ( power_power @ nat @ B2 @ N ) ) @ ( power_power @ nat @ B2 @ M ) )
= ( ord_less @ nat @ A2 @ ( power_power @ nat @ B2 @ ( minus_minus @ nat @ M @ N ) ) ) ) ) ).
% nat_mult_power_less_eq
thf(fact_1915_decr__mult__lemma,axiom,
! [D2: int,P: int > $o,K: int] :
( ( ord_less @ int @ ( zero_zero @ int ) @ D2 )
=> ( ! [X3: int] :
( ( P @ X3 )
=> ( P @ ( minus_minus @ int @ X3 @ D2 ) ) )
=> ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ K )
=> ! [X5: int] :
( ( P @ X5 )
=> ( P @ ( minus_minus @ int @ X5 @ ( times_times @ int @ K @ D2 ) ) ) ) ) ) ) ).
% decr_mult_lemma
thf(fact_1916_mod__pos__geq,axiom,
! [L: int,K: int] :
( ( ord_less @ int @ ( zero_zero @ int ) @ L )
=> ( ( ord_less_eq @ int @ L @ K )
=> ( ( modulo_modulo @ int @ K @ L )
= ( modulo_modulo @ int @ ( minus_minus @ int @ K @ L ) @ L ) ) ) ) ).
% mod_pos_geq
thf(fact_1917_verit__less__mono__div__int2,axiom,
! [A3: int,B4: int,N: int] :
( ( ord_less_eq @ int @ A3 @ B4 )
=> ( ( ord_less @ int @ ( zero_zero @ int ) @ ( uminus_uminus @ int @ N ) )
=> ( ord_less_eq @ int @ ( divide_divide @ int @ B4 @ N ) @ ( divide_divide @ int @ A3 @ N ) ) ) ) ).
% verit_less_mono_div_int2
thf(fact_1918_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_1919_int__div__sub__1,axiom,
! [M: int,N: int] :
( ( ord_less_eq @ int @ ( one_one @ int ) @ M )
=> ( ( ( dvd_dvd @ int @ M @ N )
=> ( ( divide_divide @ int @ ( minus_minus @ int @ N @ ( one_one @ int ) ) @ M )
= ( minus_minus @ int @ ( divide_divide @ int @ N @ M ) @ ( one_one @ int ) ) ) )
& ( ~ ( dvd_dvd @ int @ M @ N )
=> ( ( divide_divide @ int @ ( minus_minus @ int @ N @ ( one_one @ int ) ) @ M )
= ( divide_divide @ int @ N @ M ) ) ) ) ) ).
% int_div_sub_1
thf(fact_1920_of__nat__less__two__power,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [N: nat] : ( ord_less @ A @ ( semiring_1_of_nat @ A @ N ) @ ( power_power @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ N ) ) ) ).
% of_nat_less_two_power
thf(fact_1921_num_Osize__gen_I3_J,axiom,
! [X32: num] :
( ( size_num @ ( bit1 @ X32 ) )
= ( plus_plus @ nat @ ( size_num @ X32 ) @ ( suc @ ( zero_zero @ nat ) ) ) ) ).
% num.size_gen(3)
thf(fact_1922_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_1923_signed__take__bit__rec,axiom,
! [A: $tType] :
( ( bit_ri3973907225187159222ations @ A )
=> ( ( bit_ri4674362597316999326ke_bit @ A )
= ( ^ [N3: nat,A4: A] :
( if @ A
@ ( N3
= ( zero_zero @ nat ) )
@ ( uminus_uminus @ A @ ( modulo_modulo @ A @ A4 @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) )
@ ( plus_plus @ A @ ( modulo_modulo @ A @ A4 @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) @ ( times_times @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ ( bit_ri4674362597316999326ke_bit @ A @ ( minus_minus @ nat @ N3 @ ( one_one @ nat ) ) @ ( divide_divide @ A @ A4 @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) ) ) ) ) ) ) ) ).
% signed_take_bit_rec
thf(fact_1924_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_1925_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_1926_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_1927_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_1928_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_1929_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_1930_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_1931_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_1932_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_1933_exp__not__zero__imp__exp__diff__not__zero,axiom,
! [A: $tType] :
( ( bit_semiring_bits @ A )
=> ! [N: nat,M: nat] :
( ( ( power_power @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ N )
!= ( zero_zero @ A ) )
=> ( ( power_power @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ ( minus_minus @ nat @ N @ M ) )
!= ( zero_zero @ A ) ) ) ) ).
% exp_not_zero_imp_exp_diff_not_zero
thf(fact_1934_cong__exp__iff__simps_I7_J,axiom,
! [A: $tType] :
( ( unique1627219031080169319umeral @ A )
=> ! [Q3: num,N: num] :
( ( ( modulo_modulo @ A @ ( numeral_numeral @ A @ one2 ) @ ( numeral_numeral @ A @ ( bit0 @ Q3 ) ) )
= ( modulo_modulo @ A @ ( numeral_numeral @ A @ ( bit1 @ N ) ) @ ( numeral_numeral @ A @ ( bit0 @ Q3 ) ) ) )
= ( ( modulo_modulo @ A @ ( numeral_numeral @ A @ N ) @ ( numeral_numeral @ A @ Q3 ) )
= ( zero_zero @ A ) ) ) ) ).
% cong_exp_iff_simps(7)
thf(fact_1935_cong__exp__iff__simps_I11_J,axiom,
! [A: $tType] :
( ( unique1627219031080169319umeral @ A )
=> ! [M: num,Q3: num] :
( ( ( modulo_modulo @ A @ ( numeral_numeral @ A @ ( bit1 @ M ) ) @ ( numeral_numeral @ A @ ( bit0 @ Q3 ) ) )
= ( modulo_modulo @ A @ ( numeral_numeral @ A @ one2 ) @ ( numeral_numeral @ A @ ( bit0 @ Q3 ) ) ) )
= ( ( modulo_modulo @ A @ ( numeral_numeral @ A @ M ) @ ( numeral_numeral @ A @ Q3 ) )
= ( zero_zero @ A ) ) ) ) ).
% cong_exp_iff_simps(11)
thf(fact_1936_uminus__power__if,axiom,
! [A: $tType] :
( ( ring_1 @ A )
=> ! [N: nat,A2: A] :
( ( ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N )
=> ( ( power_power @ A @ ( uminus_uminus @ A @ A2 ) @ N )
= ( power_power @ A @ A2 @ N ) ) )
& ( ~ ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N )
=> ( ( power_power @ A @ ( uminus_uminus @ A @ A2 ) @ N )
= ( uminus_uminus @ A @ ( power_power @ A @ A2 @ N ) ) ) ) ) ) ).
% uminus_power_if
thf(fact_1937_power__eq__if,axiom,
! [A: $tType] :
( ( power @ A )
=> ( ( power_power @ A )
= ( ^ [P5: A,M4: nat] :
( if @ A
@ ( M4
= ( zero_zero @ nat ) )
@ ( one_one @ A )
@ ( times_times @ A @ P5 @ ( power_power @ A @ P5 @ ( minus_minus @ nat @ M4 @ ( one_one @ nat ) ) ) ) ) ) ) ) ).
% power_eq_if
thf(fact_1938_power__minus__mult,axiom,
! [A: $tType] :
( ( monoid_mult @ A )
=> ! [N: nat,A2: A] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ N )
=> ( ( times_times @ A @ ( power_power @ A @ A2 @ ( minus_minus @ nat @ N @ ( one_one @ nat ) ) ) @ A2 )
= ( power_power @ A @ A2 @ N ) ) ) ) ).
% power_minus_mult
thf(fact_1939_Suc__div__eq__add3__div,axiom,
! [M: nat,N: nat] :
( ( divide_divide @ nat @ ( suc @ ( suc @ ( suc @ M ) ) ) @ N )
= ( divide_divide @ nat @ ( plus_plus @ nat @ ( numeral_numeral @ nat @ ( bit1 @ one2 ) ) @ M ) @ N ) ) ).
% Suc_div_eq_add3_div
thf(fact_1940_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_1941_nat__le__power__trans,axiom,
! [N: nat,M: nat,K: nat] :
( ( ord_less_eq @ nat @ N @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( minus_minus @ nat @ M @ K ) ) )
=> ( ( ord_less_eq @ nat @ K @ M )
=> ( ord_less_eq @ nat @ ( times_times @ nat @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ K ) @ N ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ M ) ) ) ) ).
% nat_le_power_trans
thf(fact_1942_Suc__mod__eq__add3__mod,axiom,
! [M: nat,N: nat] :
( ( modulo_modulo @ nat @ ( suc @ ( suc @ ( suc @ M ) ) ) @ N )
= ( modulo_modulo @ nat @ ( plus_plus @ nat @ ( numeral_numeral @ nat @ ( bit1 @ one2 ) ) @ M ) @ N ) ) ).
% Suc_mod_eq_add3_mod
thf(fact_1943_even__diff__iff,axiom,
! [K: int,L: int] :
( ( dvd_dvd @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ ( minus_minus @ int @ K @ L ) )
= ( dvd_dvd @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ ( plus_plus @ int @ K @ L ) ) ) ).
% even_diff_iff
thf(fact_1944_signed__take__bit__int__greater__eq__minus__exp,axiom,
! [N: nat,K: int] : ( ord_less_eq @ int @ ( uminus_uminus @ int @ ( power_power @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ N ) ) @ ( bit_ri4674362597316999326ke_bit @ int @ N @ K ) ) ).
% signed_take_bit_int_greater_eq_minus_exp
thf(fact_1945_signed__take__bit__int__less__eq__self__iff,axiom,
! [N: nat,K: int] :
( ( ord_less_eq @ int @ ( bit_ri4674362597316999326ke_bit @ int @ N @ K ) @ K )
= ( ord_less_eq @ int @ ( uminus_uminus @ int @ ( power_power @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ N ) ) @ K ) ) ).
% signed_take_bit_int_less_eq_self_iff
thf(fact_1946_signed__take__bit__int__greater__self__iff,axiom,
! [K: int,N: nat] :
( ( ord_less @ int @ K @ ( bit_ri4674362597316999326ke_bit @ int @ N @ K ) )
= ( ord_less @ int @ K @ ( uminus_uminus @ int @ ( power_power @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ N ) ) ) ) ).
% signed_take_bit_int_greater_self_iff
thf(fact_1947_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_1948_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_1949_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_1950_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_1951_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_1952_minus__power__mult__self,axiom,
! [A: $tType] :
( ( comm_ring_1 @ A )
=> ! [A2: A,N: nat] :
( ( times_times @ A @ ( power_power @ A @ ( uminus_uminus @ A @ A2 ) @ N ) @ ( power_power @ A @ ( uminus_uminus @ A @ A2 ) @ N ) )
= ( power_power @ A @ A2 @ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N ) ) ) ) ).
% minus_power_mult_self
thf(fact_1953_minus__one__power__iff,axiom,
! [A: $tType] :
( ( ring_1 @ A )
=> ! [N: nat] :
( ( ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N )
=> ( ( power_power @ A @ ( uminus_uminus @ A @ ( one_one @ A ) ) @ N )
= ( one_one @ A ) ) )
& ( ~ ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N )
=> ( ( power_power @ A @ ( uminus_uminus @ A @ ( one_one @ A ) ) @ N )
= ( uminus_uminus @ A @ ( one_one @ A ) ) ) ) ) ) ).
% minus_one_power_iff
thf(fact_1954_mult__exp__mod__exp__eq,axiom,
! [A: $tType] :
( ( bit_semiring_bits @ A )
=> ! [M: nat,N: nat,A2: A] :
( ( ord_less_eq @ nat @ M @ N )
=> ( ( modulo_modulo @ A @ ( times_times @ A @ A2 @ ( power_power @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ M ) ) @ ( power_power @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ N ) )
= ( times_times @ A @ ( modulo_modulo @ A @ A2 @ ( power_power @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ ( minus_minus @ nat @ N @ M ) ) ) @ ( power_power @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ M ) ) ) ) ) ).
% mult_exp_mod_exp_eq
thf(fact_1955_signed__take__bit__int__less__eq,axiom,
! [N: nat,K: int] :
( ( ord_less_eq @ int @ ( power_power @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ N ) @ K )
=> ( ord_less_eq @ int @ ( bit_ri4674362597316999326ke_bit @ int @ N @ K ) @ ( minus_minus @ int @ K @ ( power_power @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ ( suc @ N ) ) ) ) ) ).
% signed_take_bit_int_less_eq
thf(fact_1956_less__two__pow__divI,axiom,
! [X: nat,N: nat,M: nat] :
( ( ord_less @ nat @ X @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( minus_minus @ nat @ N @ M ) ) )
=> ( ( ord_less_eq @ nat @ M @ N )
=> ( ord_less @ nat @ X @ ( divide_divide @ nat @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ M ) ) ) ) ) ).
% less_two_pow_divI
thf(fact_1957_less__two__pow__divD,axiom,
! [X: nat,N: nat,M: nat] :
( ( ord_less @ nat @ X @ ( divide_divide @ nat @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ M ) ) )
=> ( ( ord_less_eq @ nat @ M @ N )
& ( ord_less @ nat @ X @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( minus_minus @ nat @ N @ M ) ) ) ) ) ).
% less_two_pow_divD
thf(fact_1958_nat__power__less__diff,axiom,
! [N: nat,Q3: nat,M: nat] :
( ( ord_less @ nat @ ( times_times @ nat @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N ) @ Q3 ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ M ) )
=> ( ord_less @ nat @ Q3 @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( minus_minus @ nat @ M @ N ) ) ) ) ).
% nat_power_less_diff
thf(fact_1959_nat__less__power__trans,axiom,
! [N: nat,M: nat,K: nat] :
( ( ord_less @ nat @ N @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( minus_minus @ nat @ M @ K ) ) )
=> ( ( ord_less_eq @ nat @ K @ M )
=> ( ord_less @ nat @ ( times_times @ nat @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ K ) @ N ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ M ) ) ) ) ).
% nat_less_power_trans
thf(fact_1960_signed__take__bit__int__eq__self__iff,axiom,
! [N: nat,K: int] :
( ( ( bit_ri4674362597316999326ke_bit @ int @ N @ K )
= K )
= ( ( ord_less_eq @ int @ ( uminus_uminus @ int @ ( power_power @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ N ) ) @ K )
& ( ord_less @ int @ K @ ( power_power @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ N ) ) ) ) ).
% signed_take_bit_int_eq_self_iff
thf(fact_1961_signed__take__bit__int__eq__self,axiom,
! [N: nat,K: int] :
( ( ord_less_eq @ int @ ( uminus_uminus @ int @ ( power_power @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ N ) ) @ K )
=> ( ( ord_less @ int @ K @ ( power_power @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ N ) )
=> ( ( bit_ri4674362597316999326ke_bit @ int @ N @ K )
= K ) ) ) ).
% signed_take_bit_int_eq_self
thf(fact_1962_power__mod__div,axiom,
! [X: nat,N: nat,M: nat] :
( ( divide_divide @ nat @ ( modulo_modulo @ nat @ X @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N ) ) @ ( 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 @ N @ M ) ) ) ) ).
% power_mod_div
thf(fact_1963_minus__1__div__exp__eq__int,axiom,
! [N: nat] :
( ( divide_divide @ int @ ( uminus_uminus @ int @ ( one_one @ int ) ) @ ( power_power @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ N ) )
= ( uminus_uminus @ int @ ( one_one @ int ) ) ) ).
% minus_1_div_exp_eq_int
thf(fact_1964_div__pos__geq,axiom,
! [L: int,K: int] :
( ( ord_less @ int @ ( zero_zero @ int ) @ L )
=> ( ( ord_less_eq @ int @ L @ K )
=> ( ( divide_divide @ int @ K @ L )
= ( plus_plus @ int @ ( divide_divide @ int @ ( minus_minus @ int @ K @ L ) @ L ) @ ( one_one @ int ) ) ) ) ) ).
% div_pos_geq
thf(fact_1965_div__pos__neg__trivial,axiom,
! [K: int,L: int] :
( ( ord_less @ int @ ( zero_zero @ int ) @ K )
=> ( ( ord_less_eq @ int @ ( plus_plus @ int @ K @ L ) @ ( zero_zero @ int ) )
=> ( ( divide_divide @ int @ K @ L )
= ( uminus_uminus @ int @ ( one_one @ int ) ) ) ) ) ).
% div_pos_neg_trivial
thf(fact_1966_word__less__two__pow__divD,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [X: word @ A,N: nat,M: nat] :
( ( ord_less @ ( word @ A ) @ X @ ( divide_divide @ ( word @ A ) @ ( power_power @ ( word @ A ) @ ( numeral_numeral @ ( word @ A ) @ ( bit0 @ one2 ) ) @ N ) @ ( power_power @ ( word @ A ) @ ( numeral_numeral @ ( word @ A ) @ ( bit0 @ one2 ) ) @ M ) ) )
=> ( ( ord_less_eq @ nat @ M @ N )
& ( ord_less @ ( word @ A ) @ X @ ( power_power @ ( word @ A ) @ ( numeral_numeral @ ( word @ A ) @ ( bit0 @ one2 ) ) @ ( minus_minus @ nat @ N @ M ) ) ) ) ) ) ).
% word_less_two_pow_divD
thf(fact_1967_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_1968_even__mask__div__iff_H,axiom,
! [A: $tType] :
( ( euclid5411537665997757685th_nat @ A )
=> ! [M: nat,N: nat] :
( ( dvd_dvd @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ ( divide_divide @ A @ ( minus_minus @ A @ ( power_power @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ M ) @ ( one_one @ A ) ) @ ( power_power @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ N ) ) )
= ( ord_less_eq @ nat @ M @ N ) ) ) ).
% even_mask_div_iff'
thf(fact_1969_power__minus1__odd,axiom,
! [A: $tType] :
( ( ring_1 @ A )
=> ! [N: nat] :
( ( power_power @ A @ ( uminus_uminus @ A @ ( one_one @ A ) ) @ ( suc @ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N ) ) )
= ( uminus_uminus @ A @ ( one_one @ A ) ) ) ) ).
% power_minus1_odd
thf(fact_1970_Bernoulli__inequality__even,axiom,
! [N: nat,X: real] :
( ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N )
=> ( ord_less_eq @ real @ ( plus_plus @ real @ ( one_one @ real ) @ ( times_times @ real @ ( semiring_1_of_nat @ real @ N ) @ X ) ) @ ( power_power @ real @ ( plus_plus @ real @ ( one_one @ real ) @ X ) @ N ) ) ) ).
% Bernoulli_inequality_even
thf(fact_1971_int__bit__induct,axiom,
! [P: int > $o,K: int] :
( ( P @ ( zero_zero @ int ) )
=> ( ( P @ ( uminus_uminus @ int @ ( one_one @ int ) ) )
=> ( ! [K2: int] :
( ( P @ K2 )
=> ( ( K2
!= ( zero_zero @ int ) )
=> ( P @ ( times_times @ int @ K2 @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) ) ) ) )
=> ( ! [K2: int] :
( ( P @ K2 )
=> ( ( K2
!= ( uminus_uminus @ int @ ( one_one @ int ) ) )
=> ( P @ ( plus_plus @ int @ ( one_one @ int ) @ ( times_times @ int @ K2 @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) ) ) ) ) )
=> ( P @ K ) ) ) ) ) ).
% int_bit_induct
thf(fact_1972_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_1973_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_1974_even__mask__div__iff,axiom,
! [A: $tType] :
( ( bit_semiring_bits @ A )
=> ! [M: nat,N: nat] :
( ( dvd_dvd @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ ( divide_divide @ A @ ( minus_minus @ A @ ( power_power @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ M ) @ ( one_one @ A ) ) @ ( power_power @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ N ) ) )
= ( ( ( power_power @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ N )
= ( zero_zero @ A ) )
| ( ord_less_eq @ nat @ M @ N ) ) ) ) ).
% even_mask_div_iff
thf(fact_1975_exp__div__exp__eq,axiom,
! [A: $tType] :
( ( bit_semiring_bits @ A )
=> ! [M: nat,N: nat] :
( ( divide_divide @ A @ ( power_power @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ M ) @ ( power_power @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ N ) )
= ( times_times @ A
@ ( zero_neq_one_of_bool @ A
@ ( ( ( power_power @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ M )
!= ( zero_zero @ A ) )
& ( ord_less_eq @ nat @ N @ M ) ) )
@ ( power_power @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ ( minus_minus @ nat @ M @ N ) ) ) ) ) ).
% exp_div_exp_eq
thf(fact_1976_even__mod__4__div__2,axiom,
! [N: nat] :
( ( ( modulo_modulo @ nat @ N @ ( numeral_numeral @ nat @ ( bit0 @ ( bit0 @ one2 ) ) ) )
= ( suc @ ( zero_zero @ nat ) ) )
=> ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ ( minus_minus @ nat @ N @ ( suc @ ( zero_zero @ nat ) ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ).
% even_mod_4_div_2
thf(fact_1977_signed__take__bit__int__greater__eq,axiom,
! [K: int,N: nat] :
( ( ord_less @ int @ K @ ( uminus_uminus @ int @ ( power_power @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ N ) ) )
=> ( ord_less_eq @ int @ ( plus_plus @ int @ K @ ( power_power @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ ( suc @ N ) ) ) @ ( bit_ri4674362597316999326ke_bit @ int @ N @ K ) ) ) ).
% signed_take_bit_int_greater_eq
thf(fact_1978_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_1979_space__cnt,axiom,
! [T: vEBT_VEBT] : ( ord_less_eq @ real @ ( semiring_1_of_nat @ real @ ( vEBT_VEBT_space2 @ T ) ) @ ( times_times @ real @ ( numeral_numeral @ real @ ( bit0 @ ( bit1 @ one2 ) ) ) @ ( vEBT_VEBT_cnt @ T ) ) ) ).
% space_cnt
thf(fact_1980_t__build__cnt,axiom,
! [N: nat] : ( ord_less_eq @ real @ ( semiring_1_of_nat @ real @ ( vEBT_V8646137997579335489_i_l_d @ N ) ) @ ( times_times @ real @ ( vEBT_VEBT_cnt @ ( vEBT_vebt_buildup @ N ) ) @ ( numeral_numeral @ real @ ( bit1 @ ( bit0 @ ( bit1 @ one2 ) ) ) ) ) ) ).
% t_build_cnt
thf(fact_1981_t__buildup__cnt,axiom,
! [N: nat] : ( ord_less_eq @ real @ ( semiring_1_of_nat @ real @ ( vEBT_V8346862874174094_d_u_p @ N ) ) @ ( times_times @ real @ ( vEBT_VEBT_cnt @ ( vEBT_vebt_buildup @ N ) ) @ ( numeral_numeral @ real @ ( bit1 @ ( bit0 @ ( bit1 @ one2 ) ) ) ) ) ) ).
% t_buildup_cnt
thf(fact_1982_vebt__buildup__bound,axiom,
! [U: nat,N: nat] :
( ( U
= ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N ) )
=> ( ord_less_eq @ nat @ ( vEBT_V8346862874174094_d_u_p @ N ) @ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ ( bit1 @ ( bit0 @ ( bit1 @ one2 ) ) ) ) ) @ U ) ) ) ).
% vebt_buildup_bound
thf(fact_1983_linear__plus__1__le__power,axiom,
! [X: real,N: nat] :
( ( ord_less_eq @ real @ ( zero_zero @ real ) @ X )
=> ( ord_less_eq @ real @ ( plus_plus @ real @ ( times_times @ real @ ( semiring_1_of_nat @ real @ N ) @ X ) @ ( one_one @ real ) ) @ ( power_power @ real @ ( plus_plus @ real @ X @ ( one_one @ real ) ) @ N ) ) ) ).
% linear_plus_1_le_power
thf(fact_1984_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_1985_nat__approx__posE,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [E: A] :
( ( ord_less @ A @ ( zero_zero @ A ) @ E )
=> ~ ! [N2: nat] :
~ ( ord_less @ A @ ( divide_divide @ A @ ( one_one @ A ) @ ( semiring_1_of_nat @ A @ ( suc @ N2 ) ) ) @ E ) ) ) ).
% nat_approx_posE
thf(fact_1986_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_1987_buildup__build__time,axiom,
! [N: nat] : ( ord_less @ nat @ ( vEBT_V8346862874174094_d_u_p @ N ) @ ( vEBT_V8646137997579335489_i_l_d @ N ) ) ).
% buildup_build_time
thf(fact_1988_idiff__0,axiom,
! [N: extended_enat] :
( ( minus_minus @ extended_enat @ ( zero_zero @ extended_enat ) @ N )
= ( zero_zero @ extended_enat ) ) ).
% idiff_0
thf(fact_1989_idiff__0__right,axiom,
! [N: extended_enat] :
( ( minus_minus @ extended_enat @ N @ ( zero_zero @ extended_enat ) )
= N ) ).
% idiff_0_right
thf(fact_1990_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_1991_negative__eq__positive,axiom,
! [N: nat,M: nat] :
( ( ( uminus_uminus @ int @ ( semiring_1_of_nat @ int @ N ) )
= ( semiring_1_of_nat @ int @ M ) )
= ( ( N
= ( zero_zero @ nat ) )
& ( M
= ( zero_zero @ nat ) ) ) ) ).
% negative_eq_positive
thf(fact_1992_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_1993_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_1994_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_1995_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_1996_max__word__max,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [N: word @ A] : ( ord_less_eq @ ( word @ A ) @ N @ ( uminus_uminus @ ( word @ A ) @ ( one_one @ ( word @ A ) ) ) ) ) ).
% max_word_max
thf(fact_1997_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_1998_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_1999_int__dvd__int__iff,axiom,
! [M: nat,N: nat] :
( ( dvd_dvd @ int @ ( semiring_1_of_nat @ int @ M ) @ ( semiring_1_of_nat @ int @ N ) )
= ( dvd_dvd @ nat @ M @ N ) ) ).
% int_dvd_int_iff
thf(fact_2000_negative__zless,axiom,
! [N: nat,M: nat] : ( ord_less @ int @ ( uminus_uminus @ int @ ( semiring_1_of_nat @ int @ ( suc @ N ) ) ) @ ( semiring_1_of_nat @ int @ M ) ) ).
% negative_zless
thf(fact_2001_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_2002_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_2003_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_2004_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_2005_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_2006_minus__real__def,axiom,
( ( minus_minus @ real )
= ( ^ [X2: real,Y6: real] : ( plus_plus @ real @ X2 @ ( uminus_uminus @ real @ Y6 ) ) ) ) ).
% minus_real_def
thf(fact_2007_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_2008_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_2009_word__less__minus__mono,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [A2: word @ A,C2: word @ A,D2: word @ A,B2: word @ A] :
( ( ord_less @ ( word @ A ) @ A2 @ C2 )
=> ( ( ord_less @ ( word @ A ) @ D2 @ B2 )
=> ( ( ord_less @ ( word @ A ) @ ( minus_minus @ ( word @ A ) @ A2 @ B2 ) @ A2 )
=> ( ( ord_less @ ( word @ A ) @ ( minus_minus @ ( word @ A ) @ C2 @ D2 ) @ C2 )
=> ( ord_less @ ( word @ A ) @ ( minus_minus @ ( word @ A ) @ A2 @ B2 ) @ ( minus_minus @ ( word @ A ) @ C2 @ D2 ) ) ) ) ) ) ) ).
% word_less_minus_mono
thf(fact_2010_Bolzano,axiom,
! [A2: real,B2: real,P: real > real > $o] :
( ( ord_less_eq @ real @ A2 @ B2 )
=> ( ! [A6: real,B3: real,C4: real] :
( ( P @ A6 @ B3 )
=> ( ( P @ B3 @ C4 )
=> ( ( ord_less_eq @ real @ A6 @ B3 )
=> ( ( ord_less_eq @ real @ B3 @ C4 )
=> ( P @ A6 @ C4 ) ) ) ) )
=> ( ! [X3: real] :
( ( ord_less_eq @ real @ A2 @ X3 )
=> ( ( ord_less_eq @ real @ X3 @ B2 )
=> ? [D6: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ D6 )
& ! [A6: real,B3: real] :
( ( ( ord_less_eq @ real @ A6 @ X3 )
& ( ord_less_eq @ real @ X3 @ B3 )
& ( ord_less @ real @ ( minus_minus @ real @ B3 @ A6 ) @ D6 ) )
=> ( P @ A6 @ B3 ) ) ) ) )
=> ( P @ A2 @ B2 ) ) ) ) ).
% Bolzano
thf(fact_2011_int__ops_I3_J,axiom,
! [N: num] :
( ( semiring_1_of_nat @ int @ ( numeral_numeral @ nat @ N ) )
= ( numeral_numeral @ int @ N ) ) ).
% int_ops(3)
thf(fact_2012_Abs__fnat__hom__0,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ( ( zero_zero @ ( word @ A ) )
= ( semiring_1_of_nat @ ( word @ A ) @ ( zero_zero @ nat ) ) ) ) ).
% Abs_fnat_hom_0
thf(fact_2013_int__ops_I1_J,axiom,
( ( semiring_1_of_nat @ int @ ( zero_zero @ nat ) )
= ( zero_zero @ int ) ) ).
% int_ops(1)
thf(fact_2014_int__cases,axiom,
! [Z: int] :
( ! [N2: nat] :
( Z
!= ( semiring_1_of_nat @ int @ N2 ) )
=> ~ ! [N2: nat] :
( Z
!= ( uminus_uminus @ int @ ( semiring_1_of_nat @ int @ ( suc @ N2 ) ) ) ) ) ).
% int_cases
thf(fact_2015_int__of__nat__induct,axiom,
! [P: int > $o,Z: int] :
( ! [N2: nat] : ( P @ ( semiring_1_of_nat @ int @ N2 ) )
=> ( ! [N2: nat] : ( P @ ( uminus_uminus @ int @ ( semiring_1_of_nat @ int @ ( suc @ N2 ) ) ) )
=> ( P @ Z ) ) ) ).
% int_of_nat_induct
thf(fact_2016_nat__int__comparison_I2_J,axiom,
( ( ord_less @ nat )
= ( ^ [A4: nat,B5: nat] : ( ord_less @ int @ ( semiring_1_of_nat @ int @ A4 ) @ ( semiring_1_of_nat @ int @ B5 ) ) ) ) ).
% nat_int_comparison(2)
thf(fact_2017_nat__int__comparison_I3_J,axiom,
( ( ord_less_eq @ nat )
= ( ^ [A4: nat,B5: nat] : ( ord_less_eq @ int @ ( semiring_1_of_nat @ int @ A4 ) @ ( semiring_1_of_nat @ int @ B5 ) ) ) ) ).
% nat_int_comparison(3)
thf(fact_2018_int__ops_I2_J,axiom,
( ( semiring_1_of_nat @ int @ ( one_one @ nat ) )
= ( one_one @ int ) ) ).
% int_ops(2)
thf(fact_2019_zero__le__imp__eq__int,axiom,
! [K: int] :
( ( ord_less_eq @ int @ ( zero_zero @ int ) @ K )
=> ? [N2: nat] :
( K
= ( semiring_1_of_nat @ int @ N2 ) ) ) ).
% zero_le_imp_eq_int
thf(fact_2020_nonneg__int__cases,axiom,
! [K: int] :
( ( ord_less_eq @ int @ ( zero_zero @ int ) @ K )
=> ~ ! [N2: nat] :
( K
!= ( semiring_1_of_nat @ int @ N2 ) ) ) ).
% nonneg_int_cases
thf(fact_2021_zadd__int__left,axiom,
! [M: nat,N: nat,Z: int] :
( ( plus_plus @ int @ ( semiring_1_of_nat @ int @ M ) @ ( plus_plus @ int @ ( semiring_1_of_nat @ int @ N ) @ Z ) )
= ( plus_plus @ int @ ( semiring_1_of_nat @ int @ ( plus_plus @ nat @ M @ N ) ) @ Z ) ) ).
% zadd_int_left
thf(fact_2022_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_2023_int__plus,axiom,
! [N: nat,M: nat] :
( ( semiring_1_of_nat @ int @ ( plus_plus @ nat @ N @ M ) )
= ( plus_plus @ int @ ( semiring_1_of_nat @ int @ N ) @ ( semiring_1_of_nat @ int @ M ) ) ) ).
% int_plus
thf(fact_2024_not__int__zless__negative,axiom,
! [N: nat,M: nat] :
~ ( ord_less @ int @ ( semiring_1_of_nat @ int @ N ) @ ( uminus_uminus @ int @ ( semiring_1_of_nat @ int @ M ) ) ) ).
% not_int_zless_negative
thf(fact_2025_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_2026_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_2027_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_2028_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_2029_le__step__down__word,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [I: word @ A,N: word @ A] :
( ( ord_less_eq @ ( word @ A ) @ I @ N )
=> ( ( I != N )
=> ( ord_less_eq @ ( word @ A ) @ I @ ( minus_minus @ ( word @ A ) @ N @ ( one_one @ ( word @ A ) ) ) ) ) ) ) ).
% le_step_down_word
thf(fact_2030_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_2031_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_2032_word__less__imp__diff__less,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [K: word @ A,N: word @ A,M: word @ A] :
( ( ord_less_eq @ ( word @ A ) @ K @ N )
=> ( ( ord_less @ ( word @ A ) @ N @ M )
=> ( ord_less @ ( word @ A ) @ ( minus_minus @ ( word @ A ) @ N @ K ) @ M ) ) ) ) ).
% word_less_imp_diff_less
thf(fact_2033_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_2034_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_2035_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_2036_zmod__int,axiom,
! [A2: nat,B2: nat] :
( ( semiring_1_of_nat @ int @ ( modulo_modulo @ nat @ A2 @ B2 ) )
= ( modulo_modulo @ int @ ( semiring_1_of_nat @ int @ A2 ) @ ( semiring_1_of_nat @ int @ B2 ) ) ) ).
% zmod_int
thf(fact_2037_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_2038_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_2039_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_2040_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_2041_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_2042_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_2043_int__cases4,axiom,
! [M: int] :
( ! [N2: nat] :
( M
!= ( semiring_1_of_nat @ int @ N2 ) )
=> ~ ! [N2: nat] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ N2 )
=> ( M
!= ( uminus_uminus @ int @ ( semiring_1_of_nat @ int @ N2 ) ) ) ) ) ).
% int_cases4
thf(fact_2044_int__Suc,axiom,
! [N: nat] :
( ( semiring_1_of_nat @ int @ ( suc @ N ) )
= ( plus_plus @ int @ ( semiring_1_of_nat @ int @ N ) @ ( one_one @ int ) ) ) ).
% int_Suc
thf(fact_2045_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_2046_zless__iff__Suc__zadd,axiom,
( ( ord_less @ int )
= ( ^ [W2: int,Z7: int] :
? [N3: nat] :
( Z7
= ( plus_plus @ int @ W2 @ ( semiring_1_of_nat @ int @ ( suc @ N3 ) ) ) ) ) ) ).
% zless_iff_Suc_zadd
thf(fact_2047_int__zle__neg,axiom,
! [N: nat,M: nat] :
( ( ord_less_eq @ int @ ( semiring_1_of_nat @ int @ N ) @ ( uminus_uminus @ int @ ( semiring_1_of_nat @ int @ M ) ) )
= ( ( N
= ( zero_zero @ nat ) )
& ( M
= ( zero_zero @ nat ) ) ) ) ).
% int_zle_neg
thf(fact_2048_nonpos__int__cases,axiom,
! [K: int] :
( ( ord_less_eq @ int @ K @ ( zero_zero @ int ) )
=> ~ ! [N2: nat] :
( K
!= ( uminus_uminus @ int @ ( semiring_1_of_nat @ int @ N2 ) ) ) ) ).
% nonpos_int_cases
thf(fact_2049_negative__zle__0,axiom,
! [N: nat] : ( ord_less_eq @ int @ ( uminus_uminus @ int @ ( semiring_1_of_nat @ int @ N ) ) @ ( zero_zero @ int ) ) ).
% negative_zle_0
thf(fact_2050_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_2051_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_2052_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_2053_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_2054_word__must__wrap,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [X: word @ A,N: word @ A] :
( ( ord_less_eq @ ( word @ A ) @ X @ ( minus_minus @ ( word @ A ) @ N @ ( one_one @ ( word @ A ) ) ) )
=> ( ( ord_less_eq @ ( word @ A ) @ N @ X )
=> ( N
= ( zero_zero @ ( word @ A ) ) ) ) ) ) ).
% word_must_wrap
thf(fact_2055_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_2056_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_2057_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_2058_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_2059_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_2060_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_2061_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_2062_word__diff__less,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [N: word @ A,M: word @ A] :
( ( ord_less @ ( word @ A ) @ ( zero_zero @ ( word @ A ) ) @ N )
=> ( ( ord_less @ ( word @ A ) @ ( zero_zero @ ( word @ A ) ) @ M )
=> ( ( ord_less_eq @ ( word @ A ) @ N @ M )
=> ( ord_less @ ( word @ A ) @ ( minus_minus @ ( word @ A ) @ M @ N ) @ M ) ) ) ) ) ).
% word_diff_less
thf(fact_2063_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_2064_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_2065_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_2066_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_2067_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_2068_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_2069_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_2070_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_2071_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_2072_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_2073_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_2074_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_2075_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_2076_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_2077_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_2078_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_2079_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_2080_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_2081_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_2082_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_2083_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_2084_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_2085_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_2086_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_2087_zero__less__imp__eq__int,axiom,
! [K: int] :
( ( ord_less @ int @ ( zero_zero @ int ) @ K )
=> ? [N2: nat] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ N2 )
& ( K
= ( semiring_1_of_nat @ int @ N2 ) ) ) ) ).
% zero_less_imp_eq_int
thf(fact_2088_pos__int__cases,axiom,
! [K: int] :
( ( ord_less @ int @ ( zero_zero @ int ) @ K )
=> ~ ! [N2: nat] :
( ( K
= ( semiring_1_of_nat @ int @ N2 ) )
=> ~ ( ord_less @ nat @ ( zero_zero @ nat ) @ N2 ) ) ) ).
% pos_int_cases
thf(fact_2089_int__cases3,axiom,
! [K: int] :
( ( K
!= ( zero_zero @ int ) )
=> ( ! [N2: nat] :
( ( K
= ( semiring_1_of_nat @ int @ N2 ) )
=> ~ ( ord_less @ nat @ ( zero_zero @ nat ) @ N2 ) )
=> ~ ! [N2: nat] :
( ( K
= ( uminus_uminus @ int @ ( semiring_1_of_nat @ int @ N2 ) ) )
=> ~ ( ord_less @ nat @ ( zero_zero @ nat ) @ N2 ) ) ) ) ).
% int_cases3
thf(fact_2090_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_2091_not__zle__0__negative,axiom,
! [N: nat] :
~ ( ord_less_eq @ int @ ( zero_zero @ int ) @ ( uminus_uminus @ int @ ( semiring_1_of_nat @ int @ ( suc @ N ) ) ) ) ).
% not_zle_0_negative
thf(fact_2092_negative__zless__0,axiom,
! [N: nat] : ( ord_less @ int @ ( uminus_uminus @ int @ ( semiring_1_of_nat @ int @ ( suc @ N ) ) ) @ ( zero_zero @ int ) ) ).
% negative_zless_0
thf(fact_2093_negD,axiom,
! [X: int] :
( ( ord_less @ int @ X @ ( zero_zero @ int ) )
=> ? [N2: nat] :
( X
= ( uminus_uminus @ int @ ( semiring_1_of_nat @ int @ ( suc @ N2 ) ) ) ) ) ).
% negD
thf(fact_2094_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_2095_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_2096_word__sub__plus__one__nonzero,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [N5: word @ A,N: word @ A] :
( ( ord_less_eq @ ( word @ A ) @ N5 @ N )
=> ( ( N5
!= ( zero_zero @ ( word @ A ) ) )
=> ( ( plus_plus @ ( word @ A ) @ ( minus_minus @ ( word @ A ) @ N @ N5 ) @ ( one_one @ ( word @ A ) ) )
!= ( zero_zero @ ( word @ A ) ) ) ) ) ) ).
% word_sub_plus_one_nonzero
thf(fact_2097_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_2098_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_2099_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_2100_less__1__simp,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [N: word @ A,M: word @ A] :
( ( ord_less @ ( word @ A ) @ ( minus_minus @ ( word @ A ) @ N @ ( one_one @ ( word @ A ) ) ) @ M )
= ( ( ord_less_eq @ ( word @ A ) @ N @ M )
& ( N
!= ( zero_zero @ ( word @ A ) ) ) ) ) ) ).
% less_1_simp
thf(fact_2101_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_2102_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_2103_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_2104_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_2105_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_2106_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_2107_neg__int__cases,axiom,
! [K: int] :
( ( ord_less @ int @ K @ ( zero_zero @ int ) )
=> ~ ! [N2: nat] :
( ( K
= ( uminus_uminus @ int @ ( semiring_1_of_nat @ int @ N2 ) ) )
=> ~ ( ord_less @ nat @ ( zero_zero @ nat ) @ N2 ) ) ) ).
% neg_int_cases
thf(fact_2108_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_2109_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_2110_real__of__nat__div2,axiom,
! [N: nat,X: nat] : ( ord_less_eq @ real @ ( zero_zero @ real ) @ ( minus_minus @ real @ ( divide_divide @ real @ ( semiring_1_of_nat @ real @ N ) @ ( semiring_1_of_nat @ real @ X ) ) @ ( semiring_1_of_nat @ real @ ( divide_divide @ nat @ N @ X ) ) ) ) ).
% real_of_nat_div2
thf(fact_2111_real__of__nat__div3,axiom,
! [N: nat,X: nat] : ( ord_less_eq @ real @ ( minus_minus @ real @ ( divide_divide @ real @ ( semiring_1_of_nat @ real @ N ) @ ( semiring_1_of_nat @ real @ X ) ) @ ( semiring_1_of_nat @ real @ ( divide_divide @ nat @ N @ X ) ) ) @ ( one_one @ real ) ) ).
% real_of_nat_div3
thf(fact_2112_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_2113_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_2114_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_2115_word__unat__power,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [N: nat] :
( ( power_power @ ( word @ A ) @ ( numeral_numeral @ ( word @ A ) @ ( bit0 @ one2 ) ) @ N )
= ( semiring_1_of_nat @ ( word @ A ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N ) ) ) ) ).
% word_unat_power
thf(fact_2116_Bernoulli__inequality,axiom,
! [X: real,N: nat] :
( ( ord_less_eq @ real @ ( uminus_uminus @ real @ ( one_one @ real ) ) @ X )
=> ( ord_less_eq @ real @ ( plus_plus @ real @ ( one_one @ real ) @ ( times_times @ real @ ( semiring_1_of_nat @ real @ N ) @ X ) ) @ ( power_power @ real @ ( plus_plus @ real @ ( one_one @ real ) @ X ) @ N ) ) ) ).
% Bernoulli_inequality
thf(fact_2117_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_2118_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_2119_exists__least__lemma,axiom,
! [P: nat > $o] :
( ~ ( P @ ( zero_zero @ nat ) )
=> ( ? [X_12: nat] : ( P @ X_12 )
=> ? [N2: nat] :
( ~ ( P @ N2 )
& ( P @ ( suc @ N2 ) ) ) ) ) ).
% exists_least_lemma
thf(fact_2120_reals__Archimedean2,axiom,
! [A: $tType] :
( ( archim462609752435547400_field @ A )
=> ! [X: A] :
? [N2: nat] : ( ord_less @ A @ X @ ( semiring_1_of_nat @ A @ N2 ) ) ) ).
% reals_Archimedean2
thf(fact_2121_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_2122_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_2123_ex__less__of__nat__mult,axiom,
! [A: $tType] :
( ( archim462609752435547400_field @ A )
=> ! [X: A,Y: A] :
( ( ord_less @ A @ ( zero_zero @ A ) @ X )
=> ? [N2: nat] : ( ord_less @ A @ Y @ ( times_times @ A @ ( semiring_1_of_nat @ A @ N2 ) @ X ) ) ) ) ).
% ex_less_of_nat_mult
thf(fact_2124_space__space_H,axiom,
! [T: vEBT_VEBT] : ( ord_less @ nat @ ( vEBT_VEBT_space @ T ) @ ( vEBT_VEBT_space2 @ T ) ) ).
% space_space'
thf(fact_2125_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_2126_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_2127_space__2__pow__bound,axiom,
! [T: vEBT_VEBT,N: nat] :
( ( vEBT_invar_vebt @ T @ N )
=> ( ord_less_eq @ real @ ( semiring_1_of_nat @ real @ ( vEBT_VEBT_space2 @ T ) ) @ ( times_times @ real @ ( numeral_numeral @ real @ ( bit0 @ ( bit0 @ ( bit1 @ one2 ) ) ) ) @ ( minus_minus @ real @ ( power_power @ real @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) @ N ) @ ( one_one @ real ) ) ) ) ) ).
% space_2_pow_bound
thf(fact_2128_cnt__bound,axiom,
! [T: vEBT_VEBT,N: nat] :
( ( vEBT_invar_vebt @ T @ N )
=> ( ord_less_eq @ real @ ( vEBT_VEBT_cnt @ T ) @ ( times_times @ real @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) @ ( minus_minus @ real @ ( power_power @ real @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) @ N ) @ ( divide_divide @ real @ ( numeral_numeral @ real @ ( bit1 @ ( bit1 @ ( bit1 @ one2 ) ) ) ) @ ( numeral_numeral @ real @ ( bit0 @ ( bit1 @ ( bit0 @ one2 ) ) ) ) ) ) ) ) ) ).
% cnt_bound
thf(fact_2129_cnt__bound_H,axiom,
! [T: vEBT_VEBT,N: nat] :
( ( vEBT_invar_vebt @ T @ N )
=> ( ord_less_eq @ real @ ( vEBT_VEBT_cnt @ T ) @ ( times_times @ real @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) @ ( minus_minus @ real @ ( power_power @ real @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) @ N ) @ ( one_one @ real ) ) ) ) ) ).
% cnt_bound'
thf(fact_2130_space_H__bound,axiom,
! [T: vEBT_VEBT,N: nat,U: nat] :
( ( vEBT_invar_vebt @ T @ N )
=> ( ( U
= ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N ) )
=> ( ord_less_eq @ nat @ ( vEBT_VEBT_space2 @ T ) @ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ ( bit0 @ ( bit1 @ one2 ) ) ) ) @ U ) ) ) ) ).
% space'_bound
thf(fact_2131_valid__0__not,axiom,
! [T: vEBT_VEBT] :
~ ( vEBT_invar_vebt @ T @ ( zero_zero @ nat ) ) ).
% valid_0_not
thf(fact_2132_valid__tree__deg__neq__0,axiom,
! [T: vEBT_VEBT] :
~ ( vEBT_invar_vebt @ T @ ( zero_zero @ nat ) ) ).
% valid_tree_deg_neq_0
thf(fact_2133_deg__not__0,axiom,
! [T: vEBT_VEBT,N: nat] :
( ( vEBT_invar_vebt @ T @ N )
=> ( ord_less @ nat @ ( zero_zero @ nat ) @ N ) ) ).
% deg_not_0
thf(fact_2134_buildup__gives__valid,axiom,
! [N: nat] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ N )
=> ( vEBT_invar_vebt @ ( vEBT_vebt_buildup @ N ) @ N ) ) ).
% buildup_gives_valid
thf(fact_2135_space__bound,axiom,
! [T: vEBT_VEBT,N: nat,U: nat] :
( ( vEBT_invar_vebt @ T @ N )
=> ( ( U
= ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N ) )
=> ( ord_less_eq @ nat @ ( vEBT_VEBT_space @ T ) @ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ ( bit0 @ ( bit1 @ one2 ) ) ) ) @ U ) ) ) ) ).
% space_bound
thf(fact_2136_int__if,axiom,
! [P: $o,A2: nat,B2: nat] :
( ( P
=> ( ( semiring_1_of_nat @ int @ ( if @ nat @ P @ A2 @ B2 ) )
= ( semiring_1_of_nat @ int @ A2 ) ) )
& ( ~ P
=> ( ( semiring_1_of_nat @ int @ ( if @ nat @ P @ A2 @ B2 ) )
= ( semiring_1_of_nat @ int @ B2 ) ) ) ) ).
% int_if
thf(fact_2137_nat__int__comparison_I1_J,axiom,
( ( ^ [Y2: nat,Z2: nat] : Y2 = Z2 )
= ( ^ [A4: nat,B5: nat] :
( ( semiring_1_of_nat @ int @ A4 )
= ( semiring_1_of_nat @ int @ B5 ) ) ) ) ).
% nat_int_comparison(1)
thf(fact_2138_two__powr__height__bound__deg,axiom,
! [T: vEBT_VEBT,N: nat] :
( ( vEBT_invar_vebt @ T @ N )
=> ( ord_less_eq @ nat @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( vEBT_VEBT_height @ T ) ) @ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N ) ) ) ).
% two_powr_height_bound_deg
thf(fact_2139_member__bound,axiom,
! [Tree: vEBT_VEBT,X: nat,N: nat] :
( ( vEBT_vebt_member @ Tree @ X )
=> ( ( vEBT_invar_vebt @ Tree @ N )
=> ( ord_less @ nat @ X @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N ) ) ) ) ).
% member_bound
thf(fact_2140_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_2141_signed__take__bit__numeral__minus__bit1,axiom,
! [L: num,K: num] :
( ( bit_ri4674362597316999326ke_bit @ int @ ( numeral_numeral @ nat @ L ) @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ ( bit1 @ K ) ) ) )
= ( plus_plus @ int @ ( times_times @ int @ ( bit_ri4674362597316999326ke_bit @ int @ ( pred_numeral @ L ) @ ( minus_minus @ int @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ K ) ) @ ( one_one @ int ) ) ) @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) ) @ ( one_one @ int ) ) ) ).
% signed_take_bit_numeral_minus_bit1
thf(fact_2142_Suc__if__eq,axiom,
! [A: $tType,F2: nat > A,H2: nat > A,G: A,N: nat] :
( ! [N2: nat] :
( ( F2 @ ( suc @ N2 ) )
= ( H2 @ N2 ) )
=> ( ( ( F2 @ ( zero_zero @ nat ) )
= G )
=> ( ( ( N
= ( zero_zero @ nat ) )
=> ( ( F2 @ N )
= G ) )
& ( ( N
!= ( zero_zero @ nat ) )
=> ( ( F2 @ N )
= ( H2 @ ( minus_minus @ nat @ N @ ( one_one @ nat ) ) ) ) ) ) ) ) ).
% Suc_if_eq
thf(fact_2143_member__valid__both__member__options,axiom,
! [Tree: vEBT_VEBT,N: nat,X: nat] :
( ( vEBT_invar_vebt @ Tree @ N )
=> ( ( vEBT_vebt_member @ Tree @ X )
=> ( ( vEBT_V5719532721284313246member @ Tree @ X )
| ( vEBT_VEBT_membermima @ Tree @ X ) ) ) ) ).
% member_valid_both_member_options
thf(fact_2144_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_2145_member__correct,axiom,
! [T: vEBT_VEBT,N: nat,X: nat] :
( ( vEBT_invar_vebt @ T @ N )
=> ( ( vEBT_vebt_member @ T @ X )
= ( member @ nat @ X @ ( vEBT_set_vebt @ T ) ) ) ) ).
% member_correct
thf(fact_2146_pred__numeral__simps_I1_J,axiom,
( ( pred_numeral @ one2 )
= ( zero_zero @ nat ) ) ).
% pred_numeral_simps(1)
thf(fact_2147_Suc__eq__numeral,axiom,
! [N: nat,K: num] :
( ( ( suc @ N )
= ( numeral_numeral @ nat @ K ) )
= ( N
= ( pred_numeral @ K ) ) ) ).
% Suc_eq_numeral
thf(fact_2148_eq__numeral__Suc,axiom,
! [K: num,N: nat] :
( ( ( numeral_numeral @ nat @ K )
= ( suc @ N ) )
= ( ( pred_numeral @ K )
= N ) ) ).
% eq_numeral_Suc
thf(fact_2149_pred__numeral__simps_I3_J,axiom,
! [K: num] :
( ( pred_numeral @ ( bit1 @ K ) )
= ( numeral_numeral @ nat @ ( bit0 @ K ) ) ) ).
% pred_numeral_simps(3)
thf(fact_2150_less__Suc__numeral,axiom,
! [N: nat,K: num] :
( ( ord_less @ nat @ ( suc @ N ) @ ( numeral_numeral @ nat @ K ) )
= ( ord_less @ nat @ N @ ( pred_numeral @ K ) ) ) ).
% less_Suc_numeral
thf(fact_2151_less__numeral__Suc,axiom,
! [K: num,N: nat] :
( ( ord_less @ nat @ ( numeral_numeral @ nat @ K ) @ ( suc @ N ) )
= ( ord_less @ nat @ ( pred_numeral @ K ) @ N ) ) ).
% less_numeral_Suc
thf(fact_2152_le__Suc__numeral,axiom,
! [N: nat,K: num] :
( ( ord_less_eq @ nat @ ( suc @ N ) @ ( numeral_numeral @ nat @ K ) )
= ( ord_less_eq @ nat @ N @ ( pred_numeral @ K ) ) ) ).
% le_Suc_numeral
thf(fact_2153_le__numeral__Suc,axiom,
! [K: num,N: nat] :
( ( ord_less_eq @ nat @ ( numeral_numeral @ nat @ K ) @ ( suc @ N ) )
= ( ord_less_eq @ nat @ ( pred_numeral @ K ) @ N ) ) ).
% le_numeral_Suc
thf(fact_2154_diff__Suc__numeral,axiom,
! [N: nat,K: num] :
( ( minus_minus @ nat @ ( suc @ N ) @ ( numeral_numeral @ nat @ K ) )
= ( minus_minus @ nat @ N @ ( pred_numeral @ K ) ) ) ).
% diff_Suc_numeral
thf(fact_2155_diff__numeral__Suc,axiom,
! [K: num,N: nat] :
( ( minus_minus @ nat @ ( numeral_numeral @ nat @ K ) @ ( suc @ N ) )
= ( minus_minus @ nat @ ( pred_numeral @ K ) @ N ) ) ).
% diff_numeral_Suc
thf(fact_2156_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_2157_signed__take__bit__numeral__bit0,axiom,
! [L: num,K: num] :
( ( bit_ri4674362597316999326ke_bit @ int @ ( numeral_numeral @ nat @ L ) @ ( numeral_numeral @ int @ ( bit0 @ K ) ) )
= ( times_times @ int @ ( bit_ri4674362597316999326ke_bit @ int @ ( pred_numeral @ L ) @ ( numeral_numeral @ int @ K ) ) @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) ) ) ).
% signed_take_bit_numeral_bit0
thf(fact_2158_signed__take__bit__numeral__minus__bit0,axiom,
! [L: num,K: num] :
( ( bit_ri4674362597316999326ke_bit @ int @ ( numeral_numeral @ nat @ L ) @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ ( bit0 @ K ) ) ) )
= ( times_times @ int @ ( bit_ri4674362597316999326ke_bit @ int @ ( pred_numeral @ L ) @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ K ) ) ) @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) ) ) ).
% signed_take_bit_numeral_minus_bit0
thf(fact_2159_signed__take__bit__numeral__bit1,axiom,
! [L: num,K: num] :
( ( bit_ri4674362597316999326ke_bit @ int @ ( numeral_numeral @ nat @ L ) @ ( numeral_numeral @ int @ ( bit1 @ K ) ) )
= ( plus_plus @ int @ ( times_times @ int @ ( bit_ri4674362597316999326ke_bit @ int @ ( pred_numeral @ L ) @ ( numeral_numeral @ int @ K ) ) @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) ) @ ( one_one @ int ) ) ) ).
% signed_take_bit_numeral_bit1
thf(fact_2160_numeral__eq__Suc,axiom,
( ( numeral_numeral @ nat )
= ( ^ [K3: num] : ( suc @ ( pred_numeral @ K3 ) ) ) ) ).
% numeral_eq_Suc
thf(fact_2161_pred__numeral__def,axiom,
( pred_numeral
= ( ^ [K3: num] : ( minus_minus @ nat @ ( numeral_numeral @ nat @ K3 ) @ ( one_one @ nat ) ) ) ) ).
% pred_numeral_def
thf(fact_2162_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_2163_delete__bound__height,axiom,
! [T: vEBT_VEBT,N: nat,X: nat] :
( ( vEBT_invar_vebt @ T @ N )
=> ( ord_less_eq @ nat @ ( vEBT_T_d_e_l_e_t_e @ T @ X ) @ ( times_times @ nat @ ( plus_plus @ nat @ ( one_one @ nat ) @ ( vEBT_VEBT_height @ T ) ) @ ( numeral_numeral @ nat @ ( bit0 @ ( bit1 @ ( bit1 @ ( bit0 @ ( bit0 @ ( bit0 @ one2 ) ) ) ) ) ) ) ) ) ) ).
% delete_bound_height
thf(fact_2164_delete__bound__height_H,axiom,
! [T: vEBT_VEBT,N: nat,X: nat] :
( ( vEBT_invar_vebt @ T @ N )
=> ( ord_less_eq @ nat @ ( vEBT_V1232361888498592333_e_t_e @ T @ X ) @ ( plus_plus @ nat @ ( one_one @ nat ) @ ( vEBT_VEBT_height @ T ) ) ) ) ).
% delete_bound_height'
thf(fact_2165_post__member__pre__member,axiom,
! [T: vEBT_VEBT,N: nat,X: nat,Y: nat] :
( ( vEBT_invar_vebt @ T @ N )
=> ( ( ord_less @ nat @ X @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N ) )
=> ( ( ord_less @ nat @ Y @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N ) )
=> ( ( vEBT_vebt_member @ ( vEBT_vebt_insert @ T @ X ) @ Y )
=> ( ( vEBT_vebt_member @ T @ Y )
| ( X = Y ) ) ) ) ) ) ).
% post_member_pre_member
thf(fact_2166_height__double__log__univ__size,axiom,
! [U: real,Deg: nat,T: vEBT_VEBT] :
( ( U
= ( power_power @ real @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) @ Deg ) )
=> ( ( vEBT_invar_vebt @ T @ Deg )
=> ( ord_less_eq @ real @ ( semiring_1_of_nat @ real @ ( vEBT_VEBT_height @ T ) ) @ ( 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_2167_lemma__termdiff3,axiom,
! [A: $tType] :
( ( real_V3459762299906320749_field @ A )
=> ! [H2: A,Z: A,K6: real,N: nat] :
( ( H2
!= ( zero_zero @ A ) )
=> ( ( ord_less_eq @ real @ ( real_V7770717601297561774m_norm @ A @ Z ) @ K6 )
=> ( ( ord_less_eq @ real @ ( real_V7770717601297561774m_norm @ A @ ( plus_plus @ A @ Z @ H2 ) ) @ K6 )
=> ( ord_less_eq @ real @ ( real_V7770717601297561774m_norm @ A @ ( minus_minus @ A @ ( divide_divide @ A @ ( minus_minus @ A @ ( power_power @ A @ ( plus_plus @ A @ Z @ H2 ) @ N ) @ ( power_power @ A @ Z @ N ) ) @ H2 ) @ ( times_times @ A @ ( semiring_1_of_nat @ A @ N ) @ ( power_power @ A @ Z @ ( minus_minus @ nat @ N @ ( suc @ ( zero_zero @ nat ) ) ) ) ) ) ) @ ( times_times @ real @ ( times_times @ real @ ( times_times @ real @ ( semiring_1_of_nat @ real @ N ) @ ( semiring_1_of_nat @ real @ ( minus_minus @ nat @ N @ ( suc @ ( zero_zero @ nat ) ) ) ) ) @ ( power_power @ real @ K6 @ ( minus_minus @ nat @ N @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( real_V7770717601297561774m_norm @ A @ H2 ) ) ) ) ) ) ) ).
% lemma_termdiff3
thf(fact_2168_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_2169_pochhammer__double,axiom,
! [A: $tType] :
( ( field_char_0 @ A )
=> ! [Z: A,N: nat] :
( ( comm_s3205402744901411588hammer @ A @ ( times_times @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ Z ) @ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N ) )
= ( times_times @ A @ ( times_times @ A @ ( semiring_1_of_nat @ A @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N ) ) ) @ ( comm_s3205402744901411588hammer @ A @ Z @ N ) ) @ ( comm_s3205402744901411588hammer @ A @ ( plus_plus @ A @ Z @ ( divide_divide @ A @ ( one_one @ A ) @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) ) @ N ) ) ) ) ).
% pochhammer_double
thf(fact_2170_pochhammer__1,axiom,
! [A: $tType] :
( ( comm_semiring_1 @ A )
=> ! [A2: A] :
( ( comm_s3205402744901411588hammer @ A @ A2 @ ( one_one @ nat ) )
= A2 ) ) ).
% pochhammer_1
thf(fact_2171_delete__bound__size__univ_H,axiom,
! [T: vEBT_VEBT,N: nat,U: real,X: nat] :
( ( vEBT_invar_vebt @ T @ N )
=> ( ( U
= ( power_power @ real @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) @ N ) )
=> ( ord_less_eq @ real @ ( semiring_1_of_nat @ real @ ( vEBT_V1232361888498592333_e_t_e @ T @ 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_2172_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_2173_pochhammer__Suc0,axiom,
! [A: $tType] :
( ( comm_semiring_1 @ A )
=> ! [A2: A] :
( ( comm_s3205402744901411588hammer @ A @ A2 @ ( suc @ ( zero_zero @ nat ) ) )
= A2 ) ) ).
% pochhammer_Suc0
thf(fact_2174_log__one,axiom,
! [A2: real] :
( ( log @ A2 @ ( one_one @ real ) )
= ( zero_zero @ real ) ) ).
% log_one
thf(fact_2175_delete__bound__size__univ,axiom,
! [T: vEBT_VEBT,N: nat,U: real,X: nat] :
( ( vEBT_invar_vebt @ T @ N )
=> ( ( U
= ( power_power @ real @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) @ N ) )
=> ( ord_less_eq @ real @ ( semiring_1_of_nat @ real @ ( vEBT_T_d_e_l_e_t_e @ T @ 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_2176_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_2177_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_2178_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_2179_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_2180_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_2181_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_2182_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_2183_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_2184_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_2185_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_2186_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_2187_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_2188_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_2189_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_2190_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_2191_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_2192_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_2193_pochhammer__pos,axiom,
! [A: $tType] :
( ( linordered_semidom @ A )
=> ! [X: A,N: nat] :
( ( ord_less @ A @ ( zero_zero @ A ) @ X )
=> ( ord_less @ A @ ( zero_zero @ A ) @ ( comm_s3205402744901411588hammer @ A @ X @ N ) ) ) ) ).
% pochhammer_pos
thf(fact_2194_pochhammer__eq__0__mono,axiom,
! [A: $tType] :
( ( field_char_0 @ A )
=> ! [A2: A,N: nat,M: nat] :
( ( ( comm_s3205402744901411588hammer @ A @ A2 @ N )
= ( zero_zero @ A ) )
=> ( ( ord_less_eq @ nat @ N @ M )
=> ( ( comm_s3205402744901411588hammer @ A @ A2 @ M )
= ( zero_zero @ A ) ) ) ) ) ).
% pochhammer_eq_0_mono
thf(fact_2195_pochhammer__neq__0__mono,axiom,
! [A: $tType] :
( ( field_char_0 @ A )
=> ! [A2: A,M: nat,N: nat] :
( ( ( comm_s3205402744901411588hammer @ A @ A2 @ M )
!= ( zero_zero @ A ) )
=> ( ( ord_less_eq @ nat @ N @ M )
=> ( ( comm_s3205402744901411588hammer @ A @ A2 @ N )
!= ( zero_zero @ A ) ) ) ) ) ).
% pochhammer_neq_0_mono
thf(fact_2196_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_2197_log__of__power__eq,axiom,
! [M: nat,B2: real,N: nat] :
( ( ( semiring_1_of_nat @ real @ M )
= ( power_power @ real @ B2 @ N ) )
=> ( ( ord_less @ real @ ( one_one @ real ) @ B2 )
=> ( ( semiring_1_of_nat @ real @ N )
= ( log @ B2 @ ( semiring_1_of_nat @ real @ M ) ) ) ) ) ).
% log_of_power_eq
thf(fact_2198_less__log__of__power,axiom,
! [B2: real,N: nat,M: real] :
( ( ord_less @ real @ ( power_power @ real @ B2 @ N ) @ M )
=> ( ( ord_less @ real @ ( one_one @ real ) @ B2 )
=> ( ord_less @ real @ ( semiring_1_of_nat @ real @ N ) @ ( log @ B2 @ M ) ) ) ) ).
% less_log_of_power
thf(fact_2199_pochhammer__nonneg,axiom,
! [A: $tType] :
( ( linordered_semidom @ A )
=> ! [X: A,N: nat] :
( ( ord_less @ A @ ( zero_zero @ A ) @ X )
=> ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( comm_s3205402744901411588hammer @ A @ X @ N ) ) ) ) ).
% pochhammer_nonneg
thf(fact_2200_pochhammer__0__left,axiom,
! [A: $tType] :
( ( comm_semiring_1 @ A )
=> ! [N: nat] :
( ( ( N
= ( zero_zero @ nat ) )
=> ( ( comm_s3205402744901411588hammer @ A @ ( zero_zero @ A ) @ N )
= ( one_one @ A ) ) )
& ( ( N
!= ( zero_zero @ nat ) )
=> ( ( comm_s3205402744901411588hammer @ A @ ( zero_zero @ A ) @ N )
= ( zero_zero @ A ) ) ) ) ) ).
% pochhammer_0_left
thf(fact_2201_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_2202_le__log__of__power,axiom,
! [B2: real,N: nat,M: real] :
( ( ord_less_eq @ real @ ( power_power @ real @ B2 @ N ) @ M )
=> ( ( ord_less @ real @ ( one_one @ real ) @ B2 )
=> ( ord_less_eq @ real @ ( semiring_1_of_nat @ real @ N ) @ ( log @ B2 @ M ) ) ) ) ).
% le_log_of_power
thf(fact_2203_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_2204_log__base__pow,axiom,
! [A2: real,N: nat,X: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ A2 )
=> ( ( log @ ( power_power @ real @ A2 @ N ) @ X )
= ( divide_divide @ real @ ( log @ A2 @ X ) @ ( semiring_1_of_nat @ real @ N ) ) ) ) ).
% log_base_pow
thf(fact_2205_log__nat__power,axiom,
! [X: real,B2: real,N: nat] :
( ( ord_less @ real @ ( zero_zero @ real ) @ X )
=> ( ( log @ B2 @ ( power_power @ real @ X @ N ) )
= ( times_times @ real @ ( semiring_1_of_nat @ real @ N ) @ ( log @ B2 @ X ) ) ) ) ).
% log_nat_power
thf(fact_2206_pochhammer__rec,axiom,
! [A: $tType] :
( ( comm_semiring_1 @ A )
=> ! [A2: A,N: nat] :
( ( comm_s3205402744901411588hammer @ A @ A2 @ ( suc @ N ) )
= ( times_times @ A @ A2 @ ( comm_s3205402744901411588hammer @ A @ ( plus_plus @ A @ A2 @ ( one_one @ A ) ) @ N ) ) ) ) ).
% pochhammer_rec
thf(fact_2207_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_2208_log2__of__power__eq,axiom,
! [M: nat,N: nat] :
( ( M
= ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N ) )
=> ( ( semiring_1_of_nat @ real @ N )
= ( log @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) @ ( semiring_1_of_nat @ real @ M ) ) ) ) ).
% log2_of_power_eq
thf(fact_2209_pochhammer__rec_H,axiom,
! [A: $tType] :
( ( comm_semiring_1 @ A )
=> ! [Z: A,N: nat] :
( ( comm_s3205402744901411588hammer @ A @ Z @ ( suc @ N ) )
= ( times_times @ A @ ( plus_plus @ A @ Z @ ( semiring_1_of_nat @ A @ N ) ) @ ( comm_s3205402744901411588hammer @ A @ Z @ N ) ) ) ) ).
% pochhammer_rec'
thf(fact_2210_pochhammer__Suc,axiom,
! [A: $tType] :
( ( comm_semiring_1 @ A )
=> ! [A2: A,N: nat] :
( ( comm_s3205402744901411588hammer @ A @ A2 @ ( suc @ N ) )
= ( times_times @ A @ ( comm_s3205402744901411588hammer @ A @ A2 @ N ) @ ( plus_plus @ A @ A2 @ ( semiring_1_of_nat @ A @ N ) ) ) ) ) ).
% pochhammer_Suc
thf(fact_2211_pochhammer__of__nat__eq__0__lemma,axiom,
! [A: $tType] :
( ( idom @ A )
=> ! [N: nat,K: nat] :
( ( ord_less @ nat @ N @ K )
=> ( ( comm_s3205402744901411588hammer @ A @ ( uminus_uminus @ A @ ( semiring_1_of_nat @ A @ N ) ) @ K )
= ( zero_zero @ A ) ) ) ) ).
% pochhammer_of_nat_eq_0_lemma
thf(fact_2212_pochhammer__of__nat__eq__0__iff,axiom,
! [A: $tType] :
( ( ( ring_char_0 @ A )
& ( idom @ A ) )
=> ! [N: nat,K: nat] :
( ( ( comm_s3205402744901411588hammer @ A @ ( uminus_uminus @ A @ ( semiring_1_of_nat @ A @ N ) ) @ K )
= ( zero_zero @ A ) )
= ( ord_less @ nat @ N @ K ) ) ) ).
% pochhammer_of_nat_eq_0_iff
thf(fact_2213_pochhammer__eq__0__iff,axiom,
! [A: $tType] :
( ( field_char_0 @ A )
=> ! [A2: A,N: nat] :
( ( ( comm_s3205402744901411588hammer @ A @ A2 @ N )
= ( zero_zero @ A ) )
= ( ? [K3: nat] :
( ( ord_less @ nat @ K3 @ N )
& ( A2
= ( uminus_uminus @ A @ ( semiring_1_of_nat @ A @ K3 ) ) ) ) ) ) ) ).
% pochhammer_eq_0_iff
thf(fact_2214_log__of__power__less,axiom,
! [M: nat,B2: real,N: nat] :
( ( ord_less @ real @ ( semiring_1_of_nat @ real @ M ) @ ( power_power @ real @ B2 @ N ) )
=> ( ( ord_less @ real @ ( one_one @ real ) @ B2 )
=> ( ( ord_less @ nat @ ( zero_zero @ nat ) @ M )
=> ( ord_less @ real @ ( log @ B2 @ ( semiring_1_of_nat @ real @ M ) ) @ ( semiring_1_of_nat @ real @ N ) ) ) ) ) ).
% log_of_power_less
thf(fact_2215_pochhammer__product_H,axiom,
! [A: $tType] :
( ( comm_semiring_1 @ A )
=> ! [Z: A,N: nat,M: nat] :
( ( comm_s3205402744901411588hammer @ A @ Z @ ( plus_plus @ nat @ N @ M ) )
= ( times_times @ A @ ( comm_s3205402744901411588hammer @ A @ Z @ N ) @ ( comm_s3205402744901411588hammer @ A @ ( plus_plus @ A @ Z @ ( semiring_1_of_nat @ A @ N ) ) @ M ) ) ) ) ).
% pochhammer_product'
thf(fact_2216_pochhammer__of__nat__eq__0__lemma_H,axiom,
! [A: $tType] :
( ( ( ring_char_0 @ A )
& ( idom @ A ) )
=> ! [K: nat,N: nat] :
( ( ord_less_eq @ nat @ K @ N )
=> ( ( comm_s3205402744901411588hammer @ A @ ( uminus_uminus @ A @ ( semiring_1_of_nat @ A @ N ) ) @ K )
!= ( zero_zero @ A ) ) ) ) ).
% pochhammer_of_nat_eq_0_lemma'
thf(fact_2217_log__of__power__le,axiom,
! [M: nat,B2: real,N: nat] :
( ( ord_less_eq @ real @ ( semiring_1_of_nat @ real @ M ) @ ( power_power @ real @ B2 @ N ) )
=> ( ( ord_less @ real @ ( one_one @ real ) @ B2 )
=> ( ( ord_less @ nat @ ( zero_zero @ nat ) @ M )
=> ( ord_less_eq @ real @ ( log @ B2 @ ( semiring_1_of_nat @ real @ M ) ) @ ( semiring_1_of_nat @ real @ N ) ) ) ) ) ).
% log_of_power_le
thf(fact_2218_lemma__NBseq__def,axiom,
! [A: $tType,B: $tType] :
( ( real_V822414075346904944vector @ B )
=> ! [X7: A > B] :
( ( ? [K7: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ K7 )
& ! [N3: A] : ( ord_less_eq @ real @ ( real_V7770717601297561774m_norm @ B @ ( X7 @ N3 ) ) @ K7 ) ) )
= ( ? [N8: nat] :
! [N3: A] : ( ord_less_eq @ real @ ( real_V7770717601297561774m_norm @ B @ ( X7 @ N3 ) ) @ ( semiring_1_of_nat @ real @ ( suc @ N8 ) ) ) ) ) ) ).
% lemma_NBseq_def
thf(fact_2219_lemma__NBseq__def2,axiom,
! [A: $tType,B: $tType] :
( ( real_V822414075346904944vector @ B )
=> ! [X7: A > B] :
( ( ? [K7: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ K7 )
& ! [N3: A] : ( ord_less_eq @ real @ ( real_V7770717601297561774m_norm @ B @ ( X7 @ N3 ) ) @ K7 ) ) )
= ( ? [N8: nat] :
! [N3: A] : ( ord_less @ real @ ( real_V7770717601297561774m_norm @ B @ ( X7 @ N3 ) ) @ ( semiring_1_of_nat @ real @ ( suc @ N8 ) ) ) ) ) ) ).
% lemma_NBseq_def2
thf(fact_2220_pochhammer__product,axiom,
! [A: $tType] :
( ( comm_semiring_1 @ A )
=> ! [M: nat,N: nat,Z: A] :
( ( ord_less_eq @ nat @ M @ N )
=> ( ( comm_s3205402744901411588hammer @ A @ Z @ N )
= ( times_times @ A @ ( comm_s3205402744901411588hammer @ A @ Z @ M ) @ ( comm_s3205402744901411588hammer @ A @ ( plus_plus @ A @ Z @ ( semiring_1_of_nat @ A @ M ) ) @ ( minus_minus @ nat @ N @ M ) ) ) ) ) ) ).
% pochhammer_product
thf(fact_2221_less__log2__of__power,axiom,
! [N: nat,M: nat] :
( ( ord_less @ nat @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N ) @ M )
=> ( ord_less @ real @ ( semiring_1_of_nat @ real @ N ) @ ( log @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) @ ( semiring_1_of_nat @ real @ M ) ) ) ) ).
% less_log2_of_power
thf(fact_2222_le__log2__of__power,axiom,
! [N: nat,M: nat] :
( ( ord_less_eq @ nat @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N ) @ M )
=> ( ord_less_eq @ real @ ( semiring_1_of_nat @ real @ N ) @ ( log @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) @ ( semiring_1_of_nat @ real @ M ) ) ) ) ).
% le_log2_of_power
thf(fact_2223_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_2224_log2__of__power__less,axiom,
! [M: nat,N: nat] :
( ( ord_less @ nat @ M @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N ) )
=> ( ( ord_less @ nat @ ( zero_zero @ nat ) @ M )
=> ( ord_less @ real @ ( log @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) @ ( semiring_1_of_nat @ real @ M ) ) @ ( semiring_1_of_nat @ real @ N ) ) ) ) ).
% log2_of_power_less
thf(fact_2225_log2__of__power__le,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq @ nat @ M @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N ) )
=> ( ( ord_less @ nat @ ( zero_zero @ nat ) @ M )
=> ( ord_less_eq @ real @ ( log @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) @ ( semiring_1_of_nat @ real @ M ) ) @ ( semiring_1_of_nat @ real @ N ) ) ) ) ).
% log2_of_power_le
thf(fact_2226_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_2227_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_2228_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_2229_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_2230_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_2231_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_2232_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_2233_zero__less__norm__iff,axiom,
! [A: $tType] :
( ( real_V822414075346904944vector @ A )
=> ! [X: A] :
( ( ord_less @ real @ ( zero_zero @ real ) @ ( real_V7770717601297561774m_norm @ A @ X ) )
= ( X
!= ( zero_zero @ A ) ) ) ) ).
% zero_less_norm_iff
thf(fact_2234_set__vebt__set__vebt_H__valid,axiom,
! [T: vEBT_VEBT,N: nat] :
( ( vEBT_invar_vebt @ T @ N )
=> ( ( vEBT_set_vebt @ T )
= ( vEBT_VEBT_set_vebt @ T ) ) ) ).
% set_vebt_set_vebt'_valid
thf(fact_2235_norm__eq__zero,axiom,
! [A: $tType] :
( ( real_V822414075346904944vector @ A )
=> ! [X: A] :
( ( ( real_V7770717601297561774m_norm @ A @ X )
= ( zero_zero @ real ) )
= ( X
= ( zero_zero @ A ) ) ) ) ).
% norm_eq_zero
thf(fact_2236_norm__zero,axiom,
! [A: $tType] :
( ( real_V822414075346904944vector @ A )
=> ( ( real_V7770717601297561774m_norm @ A @ ( zero_zero @ A ) )
= ( zero_zero @ real ) ) ) ).
% norm_zero
thf(fact_2237_norm__one,axiom,
! [A: $tType] :
( ( real_V2822296259951069270ebra_1 @ A )
=> ( ( real_V7770717601297561774m_norm @ A @ ( one_one @ A ) )
= ( one_one @ real ) ) ) ).
% norm_one
thf(fact_2238_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_2239_pred__member,axiom,
! [T: vEBT_VEBT,X: nat,Y: nat] :
( ( vEBT_is_pred_in_set @ ( vEBT_VEBT_set_vebt @ T ) @ X @ Y )
= ( ( vEBT_vebt_member @ T @ Y )
& ( ord_less @ nat @ Y @ X )
& ! [Z7: nat] :
( ( ( vEBT_vebt_member @ T @ Z7 )
& ( ord_less @ nat @ Z7 @ X ) )
=> ( ord_less_eq @ nat @ Z7 @ Y ) ) ) ) ).
% pred_member
thf(fact_2240_succ__member,axiom,
! [T: vEBT_VEBT,X: nat,Y: nat] :
( ( vEBT_is_succ_in_set @ ( vEBT_VEBT_set_vebt @ T ) @ X @ Y )
= ( ( vEBT_vebt_member @ T @ Y )
& ( ord_less @ nat @ X @ Y )
& ! [Z7: nat] :
( ( ( vEBT_vebt_member @ T @ Z7 )
& ( ord_less @ nat @ X @ Z7 ) )
=> ( ord_less_eq @ nat @ Y @ Z7 ) ) ) ) ).
% succ_member
thf(fact_2241_norm__not__less__zero,axiom,
! [A: $tType] :
( ( real_V822414075346904944vector @ A )
=> ! [X: A] :
~ ( ord_less @ real @ ( real_V7770717601297561774m_norm @ A @ X ) @ ( zero_zero @ real ) ) ) ).
% norm_not_less_zero
thf(fact_2242_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_2243_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_2244_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_2245_norm__power,axiom,
! [A: $tType] :
( ( real_V8999393235501362500lgebra @ A )
=> ! [X: A,N: nat] :
( ( real_V7770717601297561774m_norm @ A @ ( power_power @ A @ X @ N ) )
= ( power_power @ real @ ( real_V7770717601297561774m_norm @ A @ X ) @ N ) ) ) ).
% norm_power
thf(fact_2246_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_2247_power__eq__imp__eq__norm,axiom,
! [A: $tType] :
( ( real_V8999393235501362500lgebra @ A )
=> ! [W: A,N: nat,Z: A] :
( ( ( power_power @ A @ W @ N )
= ( power_power @ A @ Z @ N ) )
=> ( ( ord_less @ nat @ ( zero_zero @ nat ) @ N )
=> ( ( real_V7770717601297561774m_norm @ A @ W )
= ( real_V7770717601297561774m_norm @ A @ Z ) ) ) ) ) ).
% power_eq_imp_eq_norm
thf(fact_2248_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_2249_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_2250_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_2251_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_2252_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_2253_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_2254_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_2255_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_2256_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_2257_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_2258_norm__power__ineq,axiom,
! [A: $tType] :
( ( real_V2822296259951069270ebra_1 @ A )
=> ! [X: A,N: nat] : ( ord_less_eq @ real @ ( real_V7770717601297561774m_norm @ A @ ( power_power @ A @ X @ N ) ) @ ( power_power @ real @ ( real_V7770717601297561774m_norm @ A @ X ) @ N ) ) ) ).
% norm_power_ineq
thf(fact_2259_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_2260_power__eq__1__iff,axiom,
! [A: $tType] :
( ( real_V8999393235501362500lgebra @ A )
=> ! [W: A,N: nat] :
( ( ( power_power @ A @ W @ N )
= ( one_one @ A ) )
=> ( ( ( real_V7770717601297561774m_norm @ A @ W )
= ( one_one @ real ) )
| ( N
= ( zero_zero @ nat ) ) ) ) ) ).
% power_eq_1_iff
thf(fact_2261_norm__diff__triangle__ineq,axiom,
! [A: $tType] :
( ( real_V822414075346904944vector @ A )
=> ! [A2: A,B2: A,C2: A,D2: A] : ( ord_less_eq @ real @ ( real_V7770717601297561774m_norm @ A @ ( minus_minus @ A @ ( plus_plus @ A @ A2 @ B2 ) @ ( plus_plus @ A @ C2 @ D2 ) ) ) @ ( plus_plus @ real @ ( real_V7770717601297561774m_norm @ A @ ( minus_minus @ A @ A2 @ C2 ) ) @ ( real_V7770717601297561774m_norm @ A @ ( minus_minus @ A @ B2 @ D2 ) ) ) ) ) ).
% norm_diff_triangle_ineq
thf(fact_2262_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_2263_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_2264_inrange,axiom,
! [T: vEBT_VEBT,N: nat] :
( ( vEBT_invar_vebt @ T @ N )
=> ( ord_less_eq @ ( set @ nat ) @ ( vEBT_VEBT_set_vebt @ T ) @ ( set_or1337092689740270186AtMost @ nat @ ( zero_zero @ nat ) @ ( minus_minus @ nat @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N ) @ ( one_one @ nat ) ) ) ) ) ).
% inrange
thf(fact_2265_arcosh__1,axiom,
! [A: $tType] :
( ( ln @ A )
=> ( ( arcosh @ A @ ( one_one @ A ) )
= ( zero_zero @ A ) ) ) ).
% arcosh_1
thf(fact_2266_heigt__uplog__rel,axiom,
! [T: vEBT_VEBT,N: nat] :
( ( vEBT_invar_vebt @ T @ N )
=> ( ( semiring_1_of_nat @ int @ ( vEBT_VEBT_height @ T ) )
= ( archimedean_ceiling @ real @ ( log @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) @ ( semiring_1_of_nat @ real @ N ) ) ) ) ) ).
% heigt_uplog_rel
thf(fact_2267_valid__insert__both__member__options__add,axiom,
! [T: vEBT_VEBT,N: nat,X: nat] :
( ( vEBT_invar_vebt @ T @ N )
=> ( ( ord_less @ nat @ X @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N ) )
=> ( vEBT_V8194947554948674370ptions @ ( vEBT_vebt_insert @ T @ X ) @ X ) ) ) ).
% valid_insert_both_member_options_add
thf(fact_2268_valid__insert__both__member__options__pres,axiom,
! [T: vEBT_VEBT,N: nat,X: nat,Y: nat] :
( ( vEBT_invar_vebt @ T @ N )
=> ( ( ord_less @ nat @ X @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N ) )
=> ( ( ord_less @ nat @ Y @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N ) )
=> ( ( vEBT_V8194947554948674370ptions @ T @ X )
=> ( vEBT_V8194947554948674370ptions @ ( vEBT_vebt_insert @ T @ Y ) @ X ) ) ) ) ) ).
% valid_insert_both_member_options_pres
thf(fact_2269_artanh__0,axiom,
! [A: $tType] :
( ( ( real_V3459762299906320749_field @ A )
& ( ln @ A ) )
=> ( ( artanh @ A @ ( zero_zero @ A ) )
= ( zero_zero @ A ) ) ) ).
% artanh_0
thf(fact_2270_valid__member__both__member__options,axiom,
! [T: vEBT_VEBT,N: nat,X: nat] :
( ( vEBT_invar_vebt @ T @ N )
=> ( ( vEBT_V8194947554948674370ptions @ T @ X )
=> ( vEBT_vebt_member @ T @ X ) ) ) ).
% valid_member_both_member_options
thf(fact_2271_both__member__options__equiv__member,axiom,
! [T: vEBT_VEBT,N: nat,X: nat] :
( ( vEBT_invar_vebt @ T @ N )
=> ( ( vEBT_V8194947554948674370ptions @ T @ X )
= ( vEBT_vebt_member @ T @ X ) ) ) ).
% both_member_options_equiv_member
thf(fact_2272_both__member__options__def,axiom,
( vEBT_V8194947554948674370ptions
= ( ^ [T2: vEBT_VEBT,X2: nat] :
( ( vEBT_V5719532721284313246member @ T2 @ X2 )
| ( vEBT_VEBT_membermima @ T2 @ X2 ) ) ) ) ).
% both_member_options_def
thf(fact_2273_ceiling__zero,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ( ( archimedean_ceiling @ A @ ( zero_zero @ A ) )
= ( zero_zero @ int ) ) ) ).
% ceiling_zero
thf(fact_2274_ceiling__numeral,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [V: num] :
( ( archimedean_ceiling @ A @ ( numeral_numeral @ A @ V ) )
= ( numeral_numeral @ int @ V ) ) ) ).
% ceiling_numeral
thf(fact_2275_ceiling__one,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ( ( archimedean_ceiling @ A @ ( one_one @ A ) )
= ( one_one @ int ) ) ) ).
% ceiling_one
thf(fact_2276_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_2277_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_2278_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_2279_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_2280_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_2281_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_2282_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_2283_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_2284_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_2285_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_2286_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_2287_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_2288_ceiling__numeral__power,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [X: num,N: nat] :
( ( archimedean_ceiling @ A @ ( power_power @ A @ ( numeral_numeral @ A @ X ) @ N ) )
= ( power_power @ int @ ( numeral_numeral @ int @ X ) @ N ) ) ) ).
% ceiling_numeral_power
thf(fact_2289_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_2290_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_2291_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_2292_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_2293_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_2294_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_2295_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_2296_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_2297_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_2298_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_2299_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_2300_atLeastatMost__psubset__iff,axiom,
! [A: $tType] :
( ( preorder @ A )
=> ! [A2: A,B2: A,C2: A,D2: A] :
( ( ord_less @ ( set @ A ) @ ( set_or1337092689740270186AtMost @ A @ A2 @ B2 ) @ ( set_or1337092689740270186AtMost @ A @ C2 @ D2 ) )
= ( ( ~ ( ord_less_eq @ A @ A2 @ B2 )
| ( ( ord_less_eq @ A @ C2 @ A2 )
& ( ord_less_eq @ A @ B2 @ D2 )
& ( ( ord_less @ A @ C2 @ A2 )
| ( ord_less @ A @ B2 @ D2 ) ) ) )
& ( ord_less_eq @ A @ C2 @ D2 ) ) ) ) ).
% atLeastatMost_psubset_iff
thf(fact_2301_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_2302_ex__nat__less,axiom,
! [N: nat,P: nat > $o] :
( ( ? [M4: nat] :
( ( ord_less_eq @ nat @ M4 @ N )
& ( P @ M4 ) ) )
= ( ? [X2: nat] :
( ( member @ nat @ X2 @ ( set_or1337092689740270186AtMost @ nat @ ( zero_zero @ nat ) @ N ) )
& ( P @ X2 ) ) ) ) ).
% ex_nat_less
thf(fact_2303_all__nat__less,axiom,
! [N: nat,P: nat > $o] :
( ( ! [M4: nat] :
( ( ord_less_eq @ nat @ M4 @ N )
=> ( P @ M4 ) ) )
= ( ! [X2: nat] :
( ( member @ nat @ X2 @ ( set_or1337092689740270186AtMost @ nat @ ( zero_zero @ nat ) @ N ) )
=> ( P @ X2 ) ) ) ) ).
% all_nat_less
thf(fact_2304_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_2305_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_2306_ceiling__log__nat__eq__if,axiom,
! [B2: nat,N: nat,K: nat] :
( ( ord_less @ nat @ ( power_power @ nat @ B2 @ N ) @ K )
=> ( ( ord_less_eq @ nat @ K @ ( power_power @ nat @ B2 @ ( plus_plus @ nat @ N @ ( one_one @ nat ) ) ) )
=> ( ( ord_less_eq @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ B2 )
=> ( ( archimedean_ceiling @ real @ ( log @ ( semiring_1_of_nat @ real @ B2 ) @ ( semiring_1_of_nat @ real @ K ) ) )
= ( plus_plus @ int @ ( semiring_1_of_nat @ int @ N ) @ ( one_one @ int ) ) ) ) ) ) ).
% ceiling_log_nat_eq_if
thf(fact_2307_ceiling__log2__div2,axiom,
! [N: nat] :
( ( ord_less_eq @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N )
=> ( ( archimedean_ceiling @ real @ ( log @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) @ ( semiring_1_of_nat @ real @ N ) ) )
= ( plus_plus @ int @ ( archimedean_ceiling @ real @ ( log @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) @ ( semiring_1_of_nat @ real @ ( plus_plus @ nat @ ( divide_divide @ nat @ ( minus_minus @ nat @ N @ ( one_one @ nat ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ ( one_one @ nat ) ) ) ) ) @ ( one_one @ int ) ) ) ) ).
% ceiling_log2_div2
thf(fact_2308_ceiling__log__nat__eq__powr__iff,axiom,
! [B2: nat,K: nat,N: nat] :
( ( ord_less_eq @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ B2 )
=> ( ( ord_less @ nat @ ( zero_zero @ nat ) @ K )
=> ( ( ( archimedean_ceiling @ real @ ( log @ ( semiring_1_of_nat @ real @ B2 ) @ ( semiring_1_of_nat @ real @ K ) ) )
= ( plus_plus @ int @ ( semiring_1_of_nat @ int @ N ) @ ( one_one @ int ) ) )
= ( ( ord_less @ nat @ ( power_power @ nat @ B2 @ N ) @ K )
& ( ord_less_eq @ nat @ K @ ( power_power @ nat @ B2 @ ( plus_plus @ nat @ N @ ( one_one @ nat ) ) ) ) ) ) ) ) ).
% ceiling_log_nat_eq_powr_iff
thf(fact_2309_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_2310_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_2311_arsinh__0,axiom,
! [A: $tType] :
( ( ln @ A )
=> ( ( arsinh @ A @ ( zero_zero @ A ) )
= ( zero_zero @ A ) ) ) ).
% arsinh_0
thf(fact_2312_is__succ__in__set__def,axiom,
( vEBT_is_succ_in_set
= ( ^ [Xs: set @ nat,X2: nat,Y6: nat] :
( ( member @ nat @ Y6 @ Xs )
& ( ord_less @ nat @ X2 @ Y6 )
& ! [Z7: nat] :
( ( member @ nat @ Z7 @ Xs )
=> ( ( ord_less @ nat @ X2 @ Z7 )
=> ( ord_less_eq @ nat @ Y6 @ Z7 ) ) ) ) ) ) ).
% is_succ_in_set_def
thf(fact_2313_is__pred__in__set__def,axiom,
( vEBT_is_pred_in_set
= ( ^ [Xs: set @ nat,X2: nat,Y6: nat] :
( ( member @ nat @ Y6 @ Xs )
& ( ord_less @ nat @ Y6 @ X2 )
& ! [Z7: nat] :
( ( member @ nat @ Z7 @ Xs )
=> ( ( ord_less @ nat @ Z7 @ X2 )
=> ( ord_less_eq @ nat @ Z7 @ Y6 ) ) ) ) ) ) ).
% is_pred_in_set_def
thf(fact_2314_member__bound__size__univ,axiom,
! [T: vEBT_VEBT,N: nat,U: real,X: nat] :
( ( vEBT_invar_vebt @ T @ N )
=> ( ( U
= ( power_power @ real @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) @ N ) )
=> ( ord_less_eq @ real @ ( semiring_1_of_nat @ real @ ( vEBT_T_m_e_m_b_e_r @ T @ 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_2315_ln__less__cancel__iff,axiom,
! [X: real,Y: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ X )
=> ( ( ord_less @ real @ ( zero_zero @ real ) @ Y )
=> ( ( ord_less @ real @ ( ln_ln @ real @ X ) @ ( ln_ln @ real @ Y ) )
= ( ord_less @ real @ X @ Y ) ) ) ) ).
% ln_less_cancel_iff
thf(fact_2316_ln__inj__iff,axiom,
! [X: real,Y: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ X )
=> ( ( ord_less @ real @ ( zero_zero @ real ) @ Y )
=> ( ( ( ln_ln @ real @ X )
= ( ln_ln @ real @ Y ) )
= ( X = Y ) ) ) ) ).
% ln_inj_iff
thf(fact_2317_of__int__ceiling__cancel,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [X: A] :
( ( ( ring_1_of_int @ A @ ( archimedean_ceiling @ A @ X ) )
= X )
= ( ? [N3: int] :
( X
= ( ring_1_of_int @ A @ N3 ) ) ) ) ) ).
% of_int_ceiling_cancel
thf(fact_2318_of__int__eq__0__iff,axiom,
! [A: $tType] :
( ( ring_char_0 @ A )
=> ! [Z: int] :
( ( ( ring_1_of_int @ A @ Z )
= ( zero_zero @ A ) )
= ( Z
= ( zero_zero @ int ) ) ) ) ).
% of_int_eq_0_iff
thf(fact_2319_of__int__0__eq__iff,axiom,
! [A: $tType] :
( ( ring_char_0 @ A )
=> ! [Z: int] :
( ( ( zero_zero @ A )
= ( ring_1_of_int @ A @ Z ) )
= ( Z
= ( zero_zero @ int ) ) ) ) ).
% of_int_0_eq_iff
thf(fact_2320_of__int__0,axiom,
! [A: $tType] :
( ( ring_1 @ A )
=> ( ( ring_1_of_int @ A @ ( zero_zero @ int ) )
= ( zero_zero @ A ) ) ) ).
% of_int_0
thf(fact_2321_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_2322_of__int__eq__numeral__iff,axiom,
! [A: $tType] :
( ( ring_char_0 @ A )
=> ! [Z: int,N: num] :
( ( ( ring_1_of_int @ A @ Z )
= ( numeral_numeral @ A @ N ) )
= ( Z
= ( numeral_numeral @ int @ N ) ) ) ) ).
% of_int_eq_numeral_iff
thf(fact_2323_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_2324_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_2325_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_2326_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_2327_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_2328_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_2329_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_2330_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_2331_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_2332_ln__one,axiom,
! [A: $tType] :
( ( ln @ A )
=> ( ( ln_ln @ A @ ( one_one @ A ) )
= ( zero_zero @ A ) ) ) ).
% ln_one
thf(fact_2333_of__int__power,axiom,
! [A: $tType] :
( ( ring_1 @ A )
=> ! [Z: int,N: nat] :
( ( ring_1_of_int @ A @ ( power_power @ int @ Z @ N ) )
= ( power_power @ A @ ( ring_1_of_int @ A @ Z ) @ N ) ) ) ).
% of_int_power
thf(fact_2334_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_2335_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_2336_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_2337_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_2338_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_2339_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_2340_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_2341_of__int__le__numeral__iff,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [Z: int,N: num] :
( ( ord_less_eq @ A @ ( ring_1_of_int @ A @ Z ) @ ( numeral_numeral @ A @ N ) )
= ( ord_less_eq @ int @ Z @ ( numeral_numeral @ int @ N ) ) ) ) ).
% of_int_le_numeral_iff
thf(fact_2342_of__int__numeral__le__iff,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [N: num,Z: int] :
( ( ord_less_eq @ A @ ( numeral_numeral @ A @ N ) @ ( ring_1_of_int @ A @ Z ) )
= ( ord_less_eq @ int @ ( numeral_numeral @ int @ N ) @ Z ) ) ) ).
% of_int_numeral_le_iff
thf(fact_2343_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_2344_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_2345_of__int__numeral__less__iff,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [N: num,Z: int] :
( ( ord_less @ A @ ( numeral_numeral @ A @ N ) @ ( ring_1_of_int @ A @ Z ) )
= ( ord_less @ int @ ( numeral_numeral @ int @ N ) @ Z ) ) ) ).
% of_int_numeral_less_iff
thf(fact_2346_of__int__less__numeral__iff,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [Z: int,N: num] :
( ( ord_less @ A @ ( ring_1_of_int @ A @ Z ) @ ( numeral_numeral @ A @ N ) )
= ( ord_less @ int @ Z @ ( numeral_numeral @ int @ N ) ) ) ) ).
% of_int_less_numeral_iff
thf(fact_2347_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_2348_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_2349_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_2350_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_2351_numeral__power__eq__of__int__cancel__iff,axiom,
! [A: $tType] :
( ( ring_char_0 @ A )
=> ! [X: num,N: nat,Y: int] :
( ( ( power_power @ A @ ( numeral_numeral @ A @ X ) @ N )
= ( ring_1_of_int @ A @ Y ) )
= ( ( power_power @ int @ ( numeral_numeral @ int @ X ) @ N )
= Y ) ) ) ).
% numeral_power_eq_of_int_cancel_iff
thf(fact_2352_of__int__eq__numeral__power__cancel__iff,axiom,
! [A: $tType] :
( ( ring_char_0 @ A )
=> ! [Y: int,X: num,N: nat] :
( ( ( ring_1_of_int @ A @ Y )
= ( power_power @ A @ ( numeral_numeral @ A @ X ) @ N ) )
= ( Y
= ( power_power @ int @ ( numeral_numeral @ int @ X ) @ N ) ) ) ) ).
% of_int_eq_numeral_power_cancel_iff
thf(fact_2353_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_2354_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_2355_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_2356_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_2357_numeral__power__le__of__int__cancel__iff,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [X: num,N: nat,A2: int] :
( ( ord_less_eq @ A @ ( power_power @ A @ ( numeral_numeral @ A @ X ) @ N ) @ ( ring_1_of_int @ A @ A2 ) )
= ( ord_less_eq @ int @ ( power_power @ int @ ( numeral_numeral @ int @ X ) @ N ) @ A2 ) ) ) ).
% numeral_power_le_of_int_cancel_iff
thf(fact_2358_of__int__le__numeral__power__cancel__iff,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [A2: int,X: num,N: nat] :
( ( ord_less_eq @ A @ ( ring_1_of_int @ A @ A2 ) @ ( power_power @ A @ ( numeral_numeral @ A @ X ) @ N ) )
= ( ord_less_eq @ int @ A2 @ ( power_power @ int @ ( numeral_numeral @ int @ X ) @ N ) ) ) ) ).
% of_int_le_numeral_power_cancel_iff
thf(fact_2359_numeral__power__less__of__int__cancel__iff,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [X: num,N: nat,A2: int] :
( ( ord_less @ A @ ( power_power @ A @ ( numeral_numeral @ A @ X ) @ N ) @ ( ring_1_of_int @ A @ A2 ) )
= ( ord_less @ int @ ( power_power @ int @ ( numeral_numeral @ int @ X ) @ N ) @ A2 ) ) ) ).
% numeral_power_less_of_int_cancel_iff
thf(fact_2360_of__int__less__numeral__power__cancel__iff,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [A2: int,X: num,N: nat] :
( ( ord_less @ A @ ( ring_1_of_int @ A @ A2 ) @ ( power_power @ A @ ( numeral_numeral @ A @ X ) @ N ) )
= ( ord_less @ int @ A2 @ ( power_power @ int @ ( numeral_numeral @ int @ X ) @ N ) ) ) ) ).
% of_int_less_numeral_power_cancel_iff
thf(fact_2361_neg__numeral__power__eq__of__int__cancel__iff,axiom,
! [A: $tType] :
( ( ring_char_0 @ A )
=> ! [X: num,N: nat,Y: int] :
( ( ( power_power @ A @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ X ) ) @ N )
= ( ring_1_of_int @ A @ Y ) )
= ( ( power_power @ int @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ X ) ) @ N )
= Y ) ) ) ).
% neg_numeral_power_eq_of_int_cancel_iff
thf(fact_2362_of__int__eq__neg__numeral__power__cancel__iff,axiom,
! [A: $tType] :
( ( ring_char_0 @ A )
=> ! [Y: int,X: num,N: nat] :
( ( ( ring_1_of_int @ A @ Y )
= ( power_power @ A @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ X ) ) @ N ) )
= ( Y
= ( power_power @ int @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ X ) ) @ N ) ) ) ) ).
% of_int_eq_neg_numeral_power_cancel_iff
thf(fact_2363_of__int__le__neg__numeral__power__cancel__iff,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [A2: int,X: num,N: nat] :
( ( ord_less_eq @ A @ ( ring_1_of_int @ A @ A2 ) @ ( power_power @ A @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ X ) ) @ N ) )
= ( ord_less_eq @ int @ A2 @ ( power_power @ int @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ X ) ) @ N ) ) ) ) ).
% of_int_le_neg_numeral_power_cancel_iff
thf(fact_2364_neg__numeral__power__le__of__int__cancel__iff,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [X: num,N: nat,A2: int] :
( ( ord_less_eq @ A @ ( power_power @ A @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ X ) ) @ N ) @ ( ring_1_of_int @ A @ A2 ) )
= ( ord_less_eq @ int @ ( power_power @ int @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ X ) ) @ N ) @ A2 ) ) ) ).
% neg_numeral_power_le_of_int_cancel_iff
thf(fact_2365_neg__numeral__power__less__of__int__cancel__iff,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [X: num,N: nat,A2: int] :
( ( ord_less @ A @ ( power_power @ A @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ X ) ) @ N ) @ ( ring_1_of_int @ A @ A2 ) )
= ( ord_less @ int @ ( power_power @ int @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ X ) ) @ N ) @ A2 ) ) ) ).
% neg_numeral_power_less_of_int_cancel_iff
thf(fact_2366_of__int__less__neg__numeral__power__cancel__iff,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [A2: int,X: num,N: nat] :
( ( ord_less @ A @ ( ring_1_of_int @ A @ A2 ) @ ( power_power @ A @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ X ) ) @ N ) )
= ( ord_less @ int @ A2 @ ( power_power @ int @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ X ) ) @ N ) ) ) ) ).
% of_int_less_neg_numeral_power_cancel_iff
thf(fact_2367_ex__less__of__int,axiom,
! [A: $tType] :
( ( archim462609752435547400_field @ A )
=> ! [X: A] :
? [Z3: int] : ( ord_less @ A @ X @ ( ring_1_of_int @ A @ Z3 ) ) ) ).
% ex_less_of_int
thf(fact_2368_ex__of__int__less,axiom,
! [A: $tType] :
( ( archim462609752435547400_field @ A )
=> ! [X: A] :
? [Z3: int] : ( ord_less @ A @ ( ring_1_of_int @ A @ Z3 ) @ X ) ) ).
% ex_of_int_less
thf(fact_2369_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_2370_ln__less__self,axiom,
! [X: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ X )
=> ( ord_less @ real @ ( ln_ln @ real @ X ) @ X ) ) ).
% ln_less_self
thf(fact_2371_log__def,axiom,
( log
= ( ^ [A4: real,X2: real] : ( divide_divide @ real @ ( ln_ln @ real @ X2 ) @ ( ln_ln @ real @ A4 ) ) ) ) ).
% log_def
thf(fact_2372_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_2373_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_2374_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_2375_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_2376_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_2377_bset_I1_J,axiom,
! [D5: int,B4: set @ int,P: int > $o,Q: int > $o] :
( ! [X3: int] :
( ! [Xa2: int] :
( ( member @ int @ Xa2 @ ( set_or1337092689740270186AtMost @ int @ ( one_one @ int ) @ D5 ) )
=> ! [Xb2: int] :
( ( member @ int @ Xb2 @ B4 )
=> ( X3
!= ( plus_plus @ int @ Xb2 @ Xa2 ) ) ) )
=> ( ( P @ X3 )
=> ( P @ ( minus_minus @ int @ X3 @ D5 ) ) ) )
=> ( ! [X3: int] :
( ! [Xa2: int] :
( ( member @ int @ Xa2 @ ( set_or1337092689740270186AtMost @ int @ ( one_one @ int ) @ D5 ) )
=> ! [Xb2: int] :
( ( member @ int @ Xb2 @ B4 )
=> ( X3
!= ( plus_plus @ int @ Xb2 @ Xa2 ) ) ) )
=> ( ( Q @ X3 )
=> ( Q @ ( minus_minus @ int @ X3 @ D5 ) ) ) )
=> ! [X5: int] :
( ! [Xa3: int] :
( ( member @ int @ Xa3 @ ( set_or1337092689740270186AtMost @ int @ ( one_one @ int ) @ D5 ) )
=> ! [Xb3: int] :
( ( member @ int @ Xb3 @ B4 )
=> ( X5
!= ( plus_plus @ int @ Xb3 @ Xa3 ) ) ) )
=> ( ( ( P @ X5 )
& ( Q @ X5 ) )
=> ( ( P @ ( minus_minus @ int @ X5 @ D5 ) )
& ( Q @ ( minus_minus @ int @ X5 @ D5 ) ) ) ) ) ) ) ).
% bset(1)
thf(fact_2378_bset_I2_J,axiom,
! [D5: int,B4: set @ int,P: int > $o,Q: int > $o] :
( ! [X3: int] :
( ! [Xa2: int] :
( ( member @ int @ Xa2 @ ( set_or1337092689740270186AtMost @ int @ ( one_one @ int ) @ D5 ) )
=> ! [Xb2: int] :
( ( member @ int @ Xb2 @ B4 )
=> ( X3
!= ( plus_plus @ int @ Xb2 @ Xa2 ) ) ) )
=> ( ( P @ X3 )
=> ( P @ ( minus_minus @ int @ X3 @ D5 ) ) ) )
=> ( ! [X3: int] :
( ! [Xa2: int] :
( ( member @ int @ Xa2 @ ( set_or1337092689740270186AtMost @ int @ ( one_one @ int ) @ D5 ) )
=> ! [Xb2: int] :
( ( member @ int @ Xb2 @ B4 )
=> ( X3
!= ( plus_plus @ int @ Xb2 @ Xa2 ) ) ) )
=> ( ( Q @ X3 )
=> ( Q @ ( minus_minus @ int @ X3 @ D5 ) ) ) )
=> ! [X5: int] :
( ! [Xa3: int] :
( ( member @ int @ Xa3 @ ( set_or1337092689740270186AtMost @ int @ ( one_one @ int ) @ D5 ) )
=> ! [Xb3: int] :
( ( member @ int @ Xb3 @ B4 )
=> ( X5
!= ( plus_plus @ int @ Xb3 @ Xa3 ) ) ) )
=> ( ( ( P @ X5 )
| ( Q @ X5 ) )
=> ( ( P @ ( minus_minus @ int @ X5 @ D5 ) )
| ( Q @ ( minus_minus @ int @ X5 @ D5 ) ) ) ) ) ) ) ).
% bset(2)
thf(fact_2379_aset_I1_J,axiom,
! [D5: int,A3: set @ int,P: int > $o,Q: int > $o] :
( ! [X3: int] :
( ! [Xa2: int] :
( ( member @ int @ Xa2 @ ( set_or1337092689740270186AtMost @ int @ ( one_one @ int ) @ D5 ) )
=> ! [Xb2: int] :
( ( member @ int @ Xb2 @ A3 )
=> ( X3
!= ( minus_minus @ int @ Xb2 @ Xa2 ) ) ) )
=> ( ( P @ X3 )
=> ( P @ ( plus_plus @ int @ X3 @ D5 ) ) ) )
=> ( ! [X3: int] :
( ! [Xa2: int] :
( ( member @ int @ Xa2 @ ( set_or1337092689740270186AtMost @ int @ ( one_one @ int ) @ D5 ) )
=> ! [Xb2: int] :
( ( member @ int @ Xb2 @ A3 )
=> ( X3
!= ( minus_minus @ int @ Xb2 @ Xa2 ) ) ) )
=> ( ( Q @ X3 )
=> ( Q @ ( plus_plus @ int @ X3 @ D5 ) ) ) )
=> ! [X5: int] :
( ! [Xa3: int] :
( ( member @ int @ Xa3 @ ( set_or1337092689740270186AtMost @ int @ ( one_one @ int ) @ D5 ) )
=> ! [Xb3: int] :
( ( member @ int @ Xb3 @ A3 )
=> ( X5
!= ( minus_minus @ int @ Xb3 @ Xa3 ) ) ) )
=> ( ( ( P @ X5 )
& ( Q @ X5 ) )
=> ( ( P @ ( plus_plus @ int @ X5 @ D5 ) )
& ( Q @ ( plus_plus @ int @ X5 @ D5 ) ) ) ) ) ) ) ).
% aset(1)
thf(fact_2380_aset_I2_J,axiom,
! [D5: int,A3: set @ int,P: int > $o,Q: int > $o] :
( ! [X3: int] :
( ! [Xa2: int] :
( ( member @ int @ Xa2 @ ( set_or1337092689740270186AtMost @ int @ ( one_one @ int ) @ D5 ) )
=> ! [Xb2: int] :
( ( member @ int @ Xb2 @ A3 )
=> ( X3
!= ( minus_minus @ int @ Xb2 @ Xa2 ) ) ) )
=> ( ( P @ X3 )
=> ( P @ ( plus_plus @ int @ X3 @ D5 ) ) ) )
=> ( ! [X3: int] :
( ! [Xa2: int] :
( ( member @ int @ Xa2 @ ( set_or1337092689740270186AtMost @ int @ ( one_one @ int ) @ D5 ) )
=> ! [Xb2: int] :
( ( member @ int @ Xb2 @ A3 )
=> ( X3
!= ( minus_minus @ int @ Xb2 @ Xa2 ) ) ) )
=> ( ( Q @ X3 )
=> ( Q @ ( plus_plus @ int @ X3 @ D5 ) ) ) )
=> ! [X5: int] :
( ! [Xa3: int] :
( ( member @ int @ Xa3 @ ( set_or1337092689740270186AtMost @ int @ ( one_one @ int ) @ D5 ) )
=> ! [Xb3: int] :
( ( member @ int @ Xb3 @ A3 )
=> ( X5
!= ( minus_minus @ int @ Xb3 @ Xa3 ) ) ) )
=> ( ( ( P @ X5 )
| ( Q @ X5 ) )
=> ( ( P @ ( plus_plus @ int @ X5 @ D5 ) )
| ( Q @ ( plus_plus @ int @ X5 @ D5 ) ) ) ) ) ) ) ).
% aset(2)
thf(fact_2381_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_2382_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_2383_real__of__int__div4,axiom,
! [N: int,X: int] : ( ord_less_eq @ real @ ( ring_1_of_int @ real @ ( divide_divide @ int @ N @ X ) ) @ ( divide_divide @ real @ ( ring_1_of_int @ real @ N ) @ ( ring_1_of_int @ real @ X ) ) ) ).
% real_of_int_div4
thf(fact_2384_real__of__int__div,axiom,
! [D2: int,N: int] :
( ( dvd_dvd @ int @ D2 @ N )
=> ( ( ring_1_of_int @ real @ ( divide_divide @ int @ N @ D2 ) )
= ( divide_divide @ real @ ( ring_1_of_int @ real @ N ) @ ( ring_1_of_int @ real @ D2 ) ) ) ) ).
% real_of_int_div
thf(fact_2385_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_2386_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_2387_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_2388_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_2389_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_2390_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_2391_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_2392_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_2393_floor__exists,axiom,
! [A: $tType] :
( ( archim462609752435547400_field @ A )
=> ! [X: A] :
? [Z3: int] :
( ( ord_less_eq @ A @ ( ring_1_of_int @ A @ Z3 ) @ X )
& ( ord_less @ A @ X @ ( ring_1_of_int @ A @ ( plus_plus @ int @ Z3 @ ( one_one @ int ) ) ) ) ) ) ).
% floor_exists
thf(fact_2394_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_2395_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_2396_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_2397_of__nat__less__of__int__iff,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [N: nat,X: int] :
( ( ord_less @ A @ ( semiring_1_of_nat @ A @ N ) @ ( ring_1_of_int @ A @ X ) )
= ( ord_less @ int @ ( semiring_1_of_nat @ int @ N ) @ X ) ) ) ).
% of_nat_less_of_int_iff
thf(fact_2398_aset_I10_J,axiom,
! [D2: int,D5: int,A3: set @ int,T: int] :
( ( dvd_dvd @ int @ D2 @ D5 )
=> ! [X5: int] :
( ! [Xa3: int] :
( ( member @ int @ Xa3 @ ( set_or1337092689740270186AtMost @ int @ ( one_one @ int ) @ D5 ) )
=> ! [Xb3: int] :
( ( member @ int @ Xb3 @ A3 )
=> ( X5
!= ( minus_minus @ int @ Xb3 @ Xa3 ) ) ) )
=> ( ~ ( dvd_dvd @ int @ D2 @ ( plus_plus @ int @ X5 @ T ) )
=> ~ ( dvd_dvd @ int @ D2 @ ( plus_plus @ int @ ( plus_plus @ int @ X5 @ D5 ) @ T ) ) ) ) ) ).
% aset(10)
thf(fact_2399_aset_I9_J,axiom,
! [D2: int,D5: int,A3: set @ int,T: int] :
( ( dvd_dvd @ int @ D2 @ D5 )
=> ! [X5: int] :
( ! [Xa3: int] :
( ( member @ int @ Xa3 @ ( set_or1337092689740270186AtMost @ int @ ( one_one @ int ) @ D5 ) )
=> ! [Xb3: int] :
( ( member @ int @ Xb3 @ A3 )
=> ( X5
!= ( minus_minus @ int @ Xb3 @ Xa3 ) ) ) )
=> ( ( dvd_dvd @ int @ D2 @ ( plus_plus @ int @ X5 @ T ) )
=> ( dvd_dvd @ int @ D2 @ ( plus_plus @ int @ ( plus_plus @ int @ X5 @ D5 ) @ T ) ) ) ) ) ).
% aset(9)
thf(fact_2400_bset_I10_J,axiom,
! [D2: int,D5: int,B4: set @ int,T: int] :
( ( dvd_dvd @ int @ D2 @ D5 )
=> ! [X5: int] :
( ! [Xa3: int] :
( ( member @ int @ Xa3 @ ( set_or1337092689740270186AtMost @ int @ ( one_one @ int ) @ D5 ) )
=> ! [Xb3: int] :
( ( member @ int @ Xb3 @ B4 )
=> ( X5
!= ( plus_plus @ int @ Xb3 @ Xa3 ) ) ) )
=> ( ~ ( dvd_dvd @ int @ D2 @ ( plus_plus @ int @ X5 @ T ) )
=> ~ ( dvd_dvd @ int @ D2 @ ( plus_plus @ int @ ( minus_minus @ int @ X5 @ D5 ) @ T ) ) ) ) ) ).
% bset(10)
thf(fact_2401_bset_I9_J,axiom,
! [D2: int,D5: int,B4: set @ int,T: int] :
( ( dvd_dvd @ int @ D2 @ D5 )
=> ! [X5: int] :
( ! [Xa3: int] :
( ( member @ int @ Xa3 @ ( set_or1337092689740270186AtMost @ int @ ( one_one @ int ) @ D5 ) )
=> ! [Xb3: int] :
( ( member @ int @ Xb3 @ B4 )
=> ( X5
!= ( plus_plus @ int @ Xb3 @ Xa3 ) ) ) )
=> ( ( dvd_dvd @ int @ D2 @ ( plus_plus @ int @ X5 @ T ) )
=> ( dvd_dvd @ int @ D2 @ ( plus_plus @ int @ ( minus_minus @ int @ X5 @ D5 ) @ T ) ) ) ) ) ).
% bset(9)
thf(fact_2402_int__le__real__less,axiom,
( ( ord_less_eq @ int )
= ( ^ [N3: int,M4: int] : ( ord_less @ real @ ( ring_1_of_int @ real @ N3 ) @ ( plus_plus @ real @ ( ring_1_of_int @ real @ M4 ) @ ( one_one @ real ) ) ) ) ) ).
% int_le_real_less
thf(fact_2403_int__less__real__le,axiom,
( ( ord_less @ int )
= ( ^ [N3: int,M4: int] : ( ord_less_eq @ real @ ( plus_plus @ real @ ( ring_1_of_int @ real @ N3 ) @ ( one_one @ real ) ) @ ( ring_1_of_int @ real @ M4 ) ) ) ) ).
% int_less_real_le
thf(fact_2404_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_2405_real__of__int__div__aux,axiom,
! [X: int,D2: int] :
( ( divide_divide @ real @ ( ring_1_of_int @ real @ X ) @ ( ring_1_of_int @ real @ D2 ) )
= ( plus_plus @ real @ ( ring_1_of_int @ real @ ( divide_divide @ int @ X @ D2 ) ) @ ( divide_divide @ real @ ( ring_1_of_int @ real @ ( modulo_modulo @ int @ X @ D2 ) ) @ ( ring_1_of_int @ real @ D2 ) ) ) ) ).
% real_of_int_div_aux
thf(fact_2406_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_2407_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_2408_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_2409_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_2410_ln__realpow,axiom,
! [X: real,N: nat] :
( ( ord_less @ real @ ( zero_zero @ real ) @ X )
=> ( ( ln_ln @ real @ ( power_power @ real @ X @ N ) )
= ( times_times @ real @ ( semiring_1_of_nat @ real @ N ) @ ( ln_ln @ real @ X ) ) ) ) ).
% ln_realpow
thf(fact_2411_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_2412_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_2413_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_2414_ceiling__split,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [P: int > $o,T: A] :
( ( P @ ( archimedean_ceiling @ A @ T ) )
= ( ! [I2: int] :
( ( ( ord_less @ A @ ( minus_minus @ A @ ( ring_1_of_int @ A @ I2 ) @ ( one_one @ A ) ) @ T )
& ( ord_less_eq @ A @ T @ ( ring_1_of_int @ A @ I2 ) ) )
=> ( P @ I2 ) ) ) ) ) ).
% ceiling_split
thf(fact_2415_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_2416_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_2417_bset_I3_J,axiom,
! [D5: int,T: int,B4: set @ int] :
( ( ord_less @ int @ ( zero_zero @ int ) @ D5 )
=> ( ( member @ int @ ( minus_minus @ int @ T @ ( one_one @ int ) ) @ B4 )
=> ! [X5: int] :
( ! [Xa3: int] :
( ( member @ int @ Xa3 @ ( set_or1337092689740270186AtMost @ int @ ( one_one @ int ) @ D5 ) )
=> ! [Xb3: int] :
( ( member @ int @ Xb3 @ B4 )
=> ( X5
!= ( plus_plus @ int @ Xb3 @ Xa3 ) ) ) )
=> ( ( X5 = T )
=> ( ( minus_minus @ int @ X5 @ D5 )
= T ) ) ) ) ) ).
% bset(3)
thf(fact_2418_bset_I4_J,axiom,
! [D5: int,T: int,B4: set @ int] :
( ( ord_less @ int @ ( zero_zero @ int ) @ D5 )
=> ( ( member @ int @ T @ B4 )
=> ! [X5: int] :
( ! [Xa3: int] :
( ( member @ int @ Xa3 @ ( set_or1337092689740270186AtMost @ int @ ( one_one @ int ) @ D5 ) )
=> ! [Xb3: int] :
( ( member @ int @ Xb3 @ B4 )
=> ( X5
!= ( plus_plus @ int @ Xb3 @ Xa3 ) ) ) )
=> ( ( X5 != T )
=> ( ( minus_minus @ int @ X5 @ D5 )
!= T ) ) ) ) ) ).
% bset(4)
thf(fact_2419_bset_I5_J,axiom,
! [D5: int,B4: set @ int,T: int] :
( ( ord_less @ int @ ( zero_zero @ int ) @ D5 )
=> ! [X5: int] :
( ! [Xa3: int] :
( ( member @ int @ Xa3 @ ( set_or1337092689740270186AtMost @ int @ ( one_one @ int ) @ D5 ) )
=> ! [Xb3: int] :
( ( member @ int @ Xb3 @ B4 )
=> ( X5
!= ( plus_plus @ int @ Xb3 @ Xa3 ) ) ) )
=> ( ( ord_less @ int @ X5 @ T )
=> ( ord_less @ int @ ( minus_minus @ int @ X5 @ D5 ) @ T ) ) ) ) ).
% bset(5)
thf(fact_2420_bset_I7_J,axiom,
! [D5: int,T: int,B4: set @ int] :
( ( ord_less @ int @ ( zero_zero @ int ) @ D5 )
=> ( ( member @ int @ T @ B4 )
=> ! [X5: int] :
( ! [Xa3: int] :
( ( member @ int @ Xa3 @ ( set_or1337092689740270186AtMost @ int @ ( one_one @ int ) @ D5 ) )
=> ! [Xb3: int] :
( ( member @ int @ Xb3 @ B4 )
=> ( X5
!= ( plus_plus @ int @ Xb3 @ Xa3 ) ) ) )
=> ( ( ord_less @ int @ T @ X5 )
=> ( ord_less @ int @ T @ ( minus_minus @ int @ X5 @ D5 ) ) ) ) ) ) ).
% bset(7)
thf(fact_2421_aset_I3_J,axiom,
! [D5: int,T: int,A3: set @ int] :
( ( ord_less @ int @ ( zero_zero @ int ) @ D5 )
=> ( ( member @ int @ ( plus_plus @ int @ T @ ( one_one @ int ) ) @ A3 )
=> ! [X5: int] :
( ! [Xa3: int] :
( ( member @ int @ Xa3 @ ( set_or1337092689740270186AtMost @ int @ ( one_one @ int ) @ D5 ) )
=> ! [Xb3: int] :
( ( member @ int @ Xb3 @ A3 )
=> ( X5
!= ( minus_minus @ int @ Xb3 @ Xa3 ) ) ) )
=> ( ( X5 = T )
=> ( ( plus_plus @ int @ X5 @ D5 )
= T ) ) ) ) ) ).
% aset(3)
thf(fact_2422_aset_I4_J,axiom,
! [D5: int,T: int,A3: set @ int] :
( ( ord_less @ int @ ( zero_zero @ int ) @ D5 )
=> ( ( member @ int @ T @ A3 )
=> ! [X5: int] :
( ! [Xa3: int] :
( ( member @ int @ Xa3 @ ( set_or1337092689740270186AtMost @ int @ ( one_one @ int ) @ D5 ) )
=> ! [Xb3: int] :
( ( member @ int @ Xb3 @ A3 )
=> ( X5
!= ( minus_minus @ int @ Xb3 @ Xa3 ) ) ) )
=> ( ( X5 != T )
=> ( ( plus_plus @ int @ X5 @ D5 )
!= T ) ) ) ) ) ).
% aset(4)
thf(fact_2423_aset_I5_J,axiom,
! [D5: int,T: int,A3: set @ int] :
( ( ord_less @ int @ ( zero_zero @ int ) @ D5 )
=> ( ( member @ int @ T @ A3 )
=> ! [X5: int] :
( ! [Xa3: int] :
( ( member @ int @ Xa3 @ ( set_or1337092689740270186AtMost @ int @ ( one_one @ int ) @ D5 ) )
=> ! [Xb3: int] :
( ( member @ int @ Xb3 @ A3 )
=> ( X5
!= ( minus_minus @ int @ Xb3 @ Xa3 ) ) ) )
=> ( ( ord_less @ int @ X5 @ T )
=> ( ord_less @ int @ ( plus_plus @ int @ X5 @ D5 ) @ T ) ) ) ) ) ).
% aset(5)
thf(fact_2424_aset_I7_J,axiom,
! [D5: int,A3: set @ int,T: int] :
( ( ord_less @ int @ ( zero_zero @ int ) @ D5 )
=> ! [X5: int] :
( ! [Xa3: int] :
( ( member @ int @ Xa3 @ ( set_or1337092689740270186AtMost @ int @ ( one_one @ int ) @ D5 ) )
=> ! [Xb3: int] :
( ( member @ int @ Xb3 @ A3 )
=> ( X5
!= ( minus_minus @ int @ Xb3 @ Xa3 ) ) ) )
=> ( ( ord_less @ int @ T @ X5 )
=> ( ord_less @ int @ T @ ( plus_plus @ int @ X5 @ D5 ) ) ) ) ) ).
% aset(7)
thf(fact_2425_periodic__finite__ex,axiom,
! [D2: int,P: int > $o] :
( ( ord_less @ int @ ( zero_zero @ int ) @ D2 )
=> ( ! [X3: int,K2: int] :
( ( P @ X3 )
= ( P @ ( minus_minus @ int @ X3 @ ( times_times @ int @ K2 @ D2 ) ) ) )
=> ( ( ? [X8: int] : ( P @ X8 ) )
= ( ? [X2: int] :
( ( member @ int @ X2 @ ( set_or1337092689740270186AtMost @ int @ ( one_one @ int ) @ D2 ) )
& ( P @ X2 ) ) ) ) ) ) ).
% periodic_finite_ex
thf(fact_2426_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_2427_real__of__int__div2,axiom,
! [N: int,X: int] : ( ord_less_eq @ real @ ( zero_zero @ real ) @ ( minus_minus @ real @ ( divide_divide @ real @ ( ring_1_of_int @ real @ N ) @ ( ring_1_of_int @ real @ X ) ) @ ( ring_1_of_int @ real @ ( divide_divide @ int @ N @ X ) ) ) ) ).
% real_of_int_div2
thf(fact_2428_real__of__int__div3,axiom,
! [N: int,X: int] : ( ord_less_eq @ real @ ( minus_minus @ real @ ( divide_divide @ real @ ( ring_1_of_int @ real @ N ) @ ( ring_1_of_int @ real @ X ) ) @ ( ring_1_of_int @ real @ ( divide_divide @ int @ N @ X ) ) ) @ ( one_one @ real ) ) ).
% real_of_int_div3
thf(fact_2429_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_2430_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_2431_ceiling__divide__upper,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [Q3: A,P4: A] :
( ( ord_less @ A @ ( zero_zero @ A ) @ Q3 )
=> ( ord_less_eq @ A @ P4 @ ( times_times @ A @ ( ring_1_of_int @ A @ ( archimedean_ceiling @ A @ ( divide_divide @ A @ P4 @ Q3 ) ) ) @ Q3 ) ) ) ) ).
% ceiling_divide_upper
thf(fact_2432_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_2433_bset_I6_J,axiom,
! [D5: int,B4: set @ int,T: int] :
( ( ord_less @ int @ ( zero_zero @ int ) @ D5 )
=> ! [X5: int] :
( ! [Xa3: int] :
( ( member @ int @ Xa3 @ ( set_or1337092689740270186AtMost @ int @ ( one_one @ int ) @ D5 ) )
=> ! [Xb3: int] :
( ( member @ int @ Xb3 @ B4 )
=> ( X5
!= ( plus_plus @ int @ Xb3 @ Xa3 ) ) ) )
=> ( ( ord_less_eq @ int @ X5 @ T )
=> ( ord_less_eq @ int @ ( minus_minus @ int @ X5 @ D5 ) @ T ) ) ) ) ).
% bset(6)
thf(fact_2434_bset_I8_J,axiom,
! [D5: int,T: int,B4: set @ int] :
( ( ord_less @ int @ ( zero_zero @ int ) @ D5 )
=> ( ( member @ int @ ( minus_minus @ int @ T @ ( one_one @ int ) ) @ B4 )
=> ! [X5: int] :
( ! [Xa3: int] :
( ( member @ int @ Xa3 @ ( set_or1337092689740270186AtMost @ int @ ( one_one @ int ) @ D5 ) )
=> ! [Xb3: int] :
( ( member @ int @ Xb3 @ B4 )
=> ( X5
!= ( plus_plus @ int @ Xb3 @ Xa3 ) ) ) )
=> ( ( ord_less_eq @ int @ T @ X5 )
=> ( ord_less_eq @ int @ T @ ( minus_minus @ int @ X5 @ D5 ) ) ) ) ) ) ).
% bset(8)
thf(fact_2435_aset_I6_J,axiom,
! [D5: int,T: int,A3: set @ int] :
( ( ord_less @ int @ ( zero_zero @ int ) @ D5 )
=> ( ( member @ int @ ( plus_plus @ int @ T @ ( one_one @ int ) ) @ A3 )
=> ! [X5: int] :
( ! [Xa3: int] :
( ( member @ int @ Xa3 @ ( set_or1337092689740270186AtMost @ int @ ( one_one @ int ) @ D5 ) )
=> ! [Xb3: int] :
( ( member @ int @ Xb3 @ A3 )
=> ( X5
!= ( minus_minus @ int @ Xb3 @ Xa3 ) ) ) )
=> ( ( ord_less_eq @ int @ X5 @ T )
=> ( ord_less_eq @ int @ ( plus_plus @ int @ X5 @ D5 ) @ T ) ) ) ) ) ).
% aset(6)
thf(fact_2436_aset_I8_J,axiom,
! [D5: int,A3: set @ int,T: int] :
( ( ord_less @ int @ ( zero_zero @ int ) @ D5 )
=> ! [X5: int] :
( ! [Xa3: int] :
( ( member @ int @ Xa3 @ ( set_or1337092689740270186AtMost @ int @ ( one_one @ int ) @ D5 ) )
=> ! [Xb3: int] :
( ( member @ int @ Xb3 @ A3 )
=> ( X5
!= ( minus_minus @ int @ Xb3 @ Xa3 ) ) ) )
=> ( ( ord_less_eq @ int @ T @ X5 )
=> ( ord_less_eq @ int @ T @ ( plus_plus @ int @ X5 @ D5 ) ) ) ) ) ).
% aset(8)
thf(fact_2437_cpmi,axiom,
! [D5: int,P: int > $o,P3: int > $o,B4: set @ int] :
( ( ord_less @ int @ ( zero_zero @ int ) @ D5 )
=> ( ? [Z4: int] :
! [X3: int] :
( ( ord_less @ int @ X3 @ Z4 )
=> ( ( P @ X3 )
= ( P3 @ X3 ) ) )
=> ( ! [X3: int] :
( ! [Xa2: int] :
( ( member @ int @ Xa2 @ ( set_or1337092689740270186AtMost @ int @ ( one_one @ int ) @ D5 ) )
=> ! [Xb2: int] :
( ( member @ int @ Xb2 @ B4 )
=> ( X3
!= ( plus_plus @ int @ Xb2 @ Xa2 ) ) ) )
=> ( ( P @ X3 )
=> ( P @ ( minus_minus @ int @ X3 @ D5 ) ) ) )
=> ( ! [X3: int,K2: int] :
( ( P3 @ X3 )
= ( P3 @ ( minus_minus @ int @ X3 @ ( times_times @ int @ K2 @ D5 ) ) ) )
=> ( ( ? [X8: int] : ( P @ X8 ) )
= ( ? [X2: int] :
( ( member @ int @ X2 @ ( set_or1337092689740270186AtMost @ int @ ( one_one @ int ) @ D5 ) )
& ( P3 @ X2 ) )
| ? [X2: int] :
( ( member @ int @ X2 @ ( set_or1337092689740270186AtMost @ int @ ( one_one @ int ) @ D5 ) )
& ? [Y6: int] :
( ( member @ int @ Y6 @ B4 )
& ( P @ ( plus_plus @ int @ Y6 @ X2 ) ) ) ) ) ) ) ) ) ) ).
% cpmi
thf(fact_2438_cppi,axiom,
! [D5: int,P: int > $o,P3: int > $o,A3: set @ int] :
( ( ord_less @ int @ ( zero_zero @ int ) @ D5 )
=> ( ? [Z4: int] :
! [X3: int] :
( ( ord_less @ int @ Z4 @ X3 )
=> ( ( P @ X3 )
= ( P3 @ X3 ) ) )
=> ( ! [X3: int] :
( ! [Xa2: int] :
( ( member @ int @ Xa2 @ ( set_or1337092689740270186AtMost @ int @ ( one_one @ int ) @ D5 ) )
=> ! [Xb2: int] :
( ( member @ int @ Xb2 @ A3 )
=> ( X3
!= ( minus_minus @ int @ Xb2 @ Xa2 ) ) ) )
=> ( ( P @ X3 )
=> ( P @ ( plus_plus @ int @ X3 @ D5 ) ) ) )
=> ( ! [X3: int,K2: int] :
( ( P3 @ X3 )
= ( P3 @ ( minus_minus @ int @ X3 @ ( times_times @ int @ K2 @ D5 ) ) ) )
=> ( ( ? [X8: int] : ( P @ X8 ) )
= ( ? [X2: int] :
( ( member @ int @ X2 @ ( set_or1337092689740270186AtMost @ int @ ( one_one @ int ) @ D5 ) )
& ( P3 @ X2 ) )
| ? [X2: int] :
( ( member @ int @ X2 @ ( set_or1337092689740270186AtMost @ int @ ( one_one @ int ) @ D5 ) )
& ? [Y6: int] :
( ( member @ int @ Y6 @ A3 )
& ( P @ ( minus_minus @ int @ Y6 @ X2 ) ) ) ) ) ) ) ) ) ) ).
% cppi
thf(fact_2439_ceiling__divide__lower,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [Q3: A,P4: A] :
( ( ord_less @ A @ ( zero_zero @ A ) @ Q3 )
=> ( ord_less @ A @ ( times_times @ A @ ( minus_minus @ A @ ( ring_1_of_int @ A @ ( archimedean_ceiling @ A @ ( divide_divide @ A @ P4 @ Q3 ) ) ) @ ( one_one @ A ) ) @ Q3 ) @ P4 ) ) ) ).
% ceiling_divide_lower
thf(fact_2440_ceiling__eq,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [N: int,X: A] :
( ( ord_less @ A @ ( ring_1_of_int @ A @ N ) @ X )
=> ( ( ord_less_eq @ A @ X @ ( plus_plus @ A @ ( ring_1_of_int @ A @ N ) @ ( one_one @ A ) ) )
=> ( ( archimedean_ceiling @ A @ X )
= ( plus_plus @ int @ N @ ( one_one @ int ) ) ) ) ) ) ).
% ceiling_eq
thf(fact_2441_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_2442_member__bound__height,axiom,
! [T: vEBT_VEBT,N: nat,X: nat] :
( ( vEBT_invar_vebt @ T @ N )
=> ( ord_less_eq @ nat @ ( vEBT_T_m_e_m_b_e_r @ T @ X ) @ ( times_times @ nat @ ( plus_plus @ nat @ ( one_one @ nat ) @ ( vEBT_VEBT_height @ T ) ) @ ( numeral_numeral @ nat @ ( bit1 @ ( bit1 @ ( bit1 @ one2 ) ) ) ) ) ) ) ).
% member_bound_height
thf(fact_2443_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_2444_psubsetI,axiom,
! [A: $tType,A3: set @ A,B4: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ A3 @ B4 )
=> ( ( A3 != B4 )
=> ( ord_less @ ( set @ A ) @ A3 @ B4 ) ) ) ).
% psubsetI
thf(fact_2445_succ__bound__size__univ,axiom,
! [T: vEBT_VEBT,N: nat,U: real,X: nat] :
( ( vEBT_invar_vebt @ T @ N )
=> ( ( U
= ( power_power @ real @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) @ N ) )
=> ( ord_less_eq @ real @ ( semiring_1_of_nat @ real @ ( vEBT_T_s_u_c_c @ T @ 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_2446_insert__bound__size__univ,axiom,
! [T: vEBT_VEBT,N: nat,U: real,X: nat] :
( ( vEBT_invar_vebt @ T @ N )
=> ( ( U
= ( power_power @ real @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) @ N ) )
=> ( ord_less_eq @ real @ ( semiring_1_of_nat @ real @ ( vEBT_T_i_n_s_e_r_t @ T @ 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_2447_pred__bound__size__univ,axiom,
! [T: vEBT_VEBT,N: nat,U: real,X: nat] :
( ( vEBT_invar_vebt @ T @ N )
=> ( ( U
= ( power_power @ real @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) @ N ) )
=> ( ord_less_eq @ real @ ( semiring_1_of_nat @ real @ ( vEBT_T_p_r_e_d @ T @ 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_2448_succ__bound__height,axiom,
! [T: vEBT_VEBT,N: nat,X: nat] :
( ( vEBT_invar_vebt @ T @ N )
=> ( ord_less_eq @ nat @ ( vEBT_T_s_u_c_c @ T @ X ) @ ( times_times @ nat @ ( plus_plus @ nat @ ( one_one @ nat ) @ ( vEBT_VEBT_height @ T ) ) @ ( numeral_numeral @ nat @ ( bit1 @ ( bit1 @ ( bit0 @ ( bit1 @ one2 ) ) ) ) ) ) ) ) ).
% succ_bound_height
thf(fact_2449_pred__bound__height,axiom,
! [T: vEBT_VEBT,N: nat,X: nat] :
( ( vEBT_invar_vebt @ T @ N )
=> ( ord_less_eq @ nat @ ( vEBT_T_p_r_e_d @ T @ X ) @ ( times_times @ nat @ ( plus_plus @ nat @ ( one_one @ nat ) @ ( vEBT_VEBT_height @ T ) ) @ ( numeral_numeral @ nat @ ( bit1 @ ( bit0 @ ( bit1 @ ( bit1 @ one2 ) ) ) ) ) ) ) ) ).
% pred_bound_height
thf(fact_2450_word__of__int__numeral,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [Bin: num] :
( ( ring_1_of_int @ ( word @ A ) @ ( numeral_numeral @ int @ Bin ) )
= ( numeral_numeral @ ( word @ A ) @ Bin ) ) ) ).
% word_of_int_numeral
thf(fact_2451_word__of__int__0,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ( ( ring_1_of_int @ ( word @ A ) @ ( zero_zero @ int ) )
= ( zero_zero @ ( word @ A ) ) ) ) ).
% word_of_int_0
thf(fact_2452_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_2453_word__of__int__neg__numeral,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [Bin: num] :
( ( ring_1_of_int @ ( word @ A ) @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ Bin ) ) )
= ( uminus_uminus @ ( word @ A ) @ ( numeral_numeral @ ( word @ A ) @ Bin ) ) ) ) ).
% word_of_int_neg_numeral
thf(fact_2454_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_2455_word__numeral__alt,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ( ( numeral_numeral @ ( word @ A ) )
= ( ^ [B5: num] : ( ring_1_of_int @ ( word @ A ) @ ( numeral_numeral @ int @ B5 ) ) ) ) ) ).
% word_numeral_alt
thf(fact_2456_psubsetD,axiom,
! [A: $tType,A3: set @ A,B4: set @ A,C2: A] :
( ( ord_less @ ( set @ A ) @ A3 @ B4 )
=> ( ( member @ A @ C2 @ A3 )
=> ( member @ A @ C2 @ B4 ) ) ) ).
% psubsetD
thf(fact_2457_psubset__trans,axiom,
! [A: $tType,A3: set @ A,B4: set @ A,C6: set @ A] :
( ( ord_less @ ( set @ A ) @ A3 @ B4 )
=> ( ( ord_less @ ( set @ A ) @ B4 @ C6 )
=> ( ord_less @ ( set @ A ) @ A3 @ C6 ) ) ) ).
% psubset_trans
thf(fact_2458_psubset__imp__ex__mem,axiom,
! [A: $tType,A3: set @ A,B4: set @ A] :
( ( ord_less @ ( set @ A ) @ A3 @ B4 )
=> ? [B3: A] : ( member @ A @ B3 @ ( minus_minus @ ( set @ A ) @ B4 @ A3 ) ) ) ).
% psubset_imp_ex_mem
thf(fact_2459_word__neg__numeral__alt,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [B2: num] :
( ( uminus_uminus @ ( word @ A ) @ ( numeral_numeral @ ( word @ A ) @ B2 ) )
= ( ring_1_of_int @ ( word @ A ) @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ B2 ) ) ) ) ) ).
% word_neg_numeral_alt
thf(fact_2460_word__of__int__2p,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [N: nat] :
( ( ring_1_of_int @ ( word @ A ) @ ( power_power @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ N ) )
= ( power_power @ ( word @ A ) @ ( numeral_numeral @ ( word @ A ) @ ( bit0 @ one2 ) ) @ N ) ) ) ).
% word_of_int_2p
thf(fact_2461_psubsetE,axiom,
! [A: $tType,A3: set @ A,B4: set @ A] :
( ( ord_less @ ( set @ A ) @ A3 @ B4 )
=> ~ ( ( ord_less_eq @ ( set @ A ) @ A3 @ B4 )
=> ( ord_less_eq @ ( set @ A ) @ B4 @ A3 ) ) ) ).
% psubsetE
thf(fact_2462_psubset__eq,axiom,
! [A: $tType] :
( ( ord_less @ ( set @ A ) )
= ( ^ [A7: set @ A,B8: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ A7 @ B8 )
& ( A7 != B8 ) ) ) ) ).
% psubset_eq
thf(fact_2463_psubset__imp__subset,axiom,
! [A: $tType,A3: set @ A,B4: set @ A] :
( ( ord_less @ ( set @ A ) @ A3 @ B4 )
=> ( ord_less_eq @ ( set @ A ) @ A3 @ B4 ) ) ).
% psubset_imp_subset
thf(fact_2464_psubset__subset__trans,axiom,
! [A: $tType,A3: set @ A,B4: set @ A,C6: set @ A] :
( ( ord_less @ ( set @ A ) @ A3 @ B4 )
=> ( ( ord_less_eq @ ( set @ A ) @ B4 @ C6 )
=> ( ord_less @ ( set @ A ) @ A3 @ C6 ) ) ) ).
% psubset_subset_trans
thf(fact_2465_subset__not__subset__eq,axiom,
! [A: $tType] :
( ( ord_less @ ( set @ A ) )
= ( ^ [A7: set @ A,B8: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ A7 @ B8 )
& ~ ( ord_less_eq @ ( set @ A ) @ B8 @ A7 ) ) ) ) ).
% subset_not_subset_eq
thf(fact_2466_subset__psubset__trans,axiom,
! [A: $tType,A3: set @ A,B4: set @ A,C6: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ A3 @ B4 )
=> ( ( ord_less @ ( set @ A ) @ B4 @ C6 )
=> ( ord_less @ ( set @ A ) @ A3 @ C6 ) ) ) ).
% subset_psubset_trans
thf(fact_2467_subset__iff__psubset__eq,axiom,
! [A: $tType] :
( ( ord_less_eq @ ( set @ A ) )
= ( ^ [A7: set @ A,B8: set @ A] :
( ( ord_less @ ( set @ A ) @ A7 @ B8 )
| ( A7 = B8 ) ) ) ) ).
% subset_iff_psubset_eq
thf(fact_2468_insersimp,axiom,
! [T: vEBT_VEBT,N: nat,Y: nat] :
( ( vEBT_invar_vebt @ T @ N )
=> ( ~ ? [X_1: nat] : ( vEBT_V8194947554948674370ptions @ T @ X_1 )
=> ( ord_less_eq @ nat @ ( vEBT_T_i_n_s_e_r_t @ T @ Y ) @ ( numeral_numeral @ nat @ ( bit1 @ one2 ) ) ) ) ) ).
% insersimp
thf(fact_2469_insert__bound__height,axiom,
! [T: vEBT_VEBT,N: nat,X: nat] :
( ( vEBT_invar_vebt @ T @ N )
=> ( ord_less_eq @ nat @ ( vEBT_T_i_n_s_e_r_t @ T @ X ) @ ( times_times @ nat @ ( plus_plus @ nat @ ( one_one @ nat ) @ ( vEBT_VEBT_height @ T ) ) @ ( numeral_numeral @ nat @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit0 @ one2 ) ) ) ) ) ) ) ) ).
% insert_bound_height
thf(fact_2470_succ__bound__size__univ_H,axiom,
! [T: vEBT_VEBT,N: nat,U: real,X: nat] :
( ( vEBT_invar_vebt @ T @ N )
=> ( ( U
= ( power_power @ real @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) @ N ) )
=> ( ord_less_eq @ real @ ( semiring_1_of_nat @ real @ ( vEBT_T_s_u_c_c2 @ T @ 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_2471_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_2472_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_2473_pred__bound__size__univ_H,axiom,
! [T: vEBT_VEBT,N: nat,U: real,X: nat] :
( ( vEBT_invar_vebt @ T @ N )
=> ( ( U
= ( power_power @ real @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) @ N ) )
=> ( ord_less_eq @ real @ ( semiring_1_of_nat @ real @ ( vEBT_T_p_r_e_d2 @ T @ 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_2474_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_2475_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_1: nat] : ( vEBT_is_pred_in_set @ A3 @ X @ X_1 ) ) ) ) ).
% obtain_set_pred
thf(fact_2476_set__vebt__finite,axiom,
! [T: vEBT_VEBT,N: nat] :
( ( vEBT_invar_vebt @ T @ N )
=> ( finite_finite2 @ nat @ ( vEBT_VEBT_set_vebt @ T ) ) ) ).
% set_vebt_finite
thf(fact_2477_succ__none__empty,axiom,
! [Xs2: set @ nat,A2: nat] :
( ~ ? [X_1: nat] : ( vEBT_is_succ_in_set @ Xs2 @ A2 @ X_1 )
=> ( ( finite_finite2 @ nat @ Xs2 )
=> ~ ? [X5: nat] :
( ( member @ nat @ X5 @ Xs2 )
& ( ord_less @ nat @ A2 @ X5 ) ) ) ) ).
% succ_none_empty
thf(fact_2478_pred__none__empty,axiom,
! [Xs2: set @ nat,A2: nat] :
( ~ ? [X_1: nat] : ( vEBT_is_pred_in_set @ Xs2 @ A2 @ X_1 )
=> ( ( finite_finite2 @ nat @ Xs2 )
=> ~ ? [X5: nat] :
( ( member @ nat @ X5 @ Xs2 )
& ( ord_less @ nat @ X5 @ A2 ) ) ) ) ).
% pred_none_empty
thf(fact_2479_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_2480_abs__idempotent,axiom,
! [A: $tType] :
( ( ordere166539214618696060dd_abs @ A )
=> ! [A2: A] :
( ( abs_abs @ A @ ( abs_abs @ A @ A2 ) )
= ( abs_abs @ A @ A2 ) ) ) ).
% abs_idempotent
thf(fact_2481_obtain__set__succ,axiom,
! [X: nat,Z: nat,A3: set @ nat,B4: set @ nat] :
( ( ord_less @ nat @ X @ Z )
=> ( ( vEBT_VEBT_max_in_set @ A3 @ Z )
=> ( ( finite_finite2 @ nat @ B4 )
=> ( ( A3 = B4 )
=> ? [X_1: nat] : ( vEBT_is_succ_in_set @ A3 @ X @ X_1 ) ) ) ) ) ).
% obtain_set_succ
thf(fact_2482_abs__zero,axiom,
! [A: $tType] :
( ( ordere166539214618696060dd_abs @ A )
=> ( ( abs_abs @ A @ ( zero_zero @ A ) )
= ( zero_zero @ A ) ) ) ).
% abs_zero
thf(fact_2483_abs__eq__0,axiom,
! [A: $tType] :
( ( ordere166539214618696060dd_abs @ A )
=> ! [A2: A] :
( ( ( abs_abs @ A @ A2 )
= ( zero_zero @ A ) )
= ( A2
= ( zero_zero @ A ) ) ) ) ).
% abs_eq_0
thf(fact_2484_abs__0__eq,axiom,
! [A: $tType] :
( ( ordere166539214618696060dd_abs @ A )
=> ! [A2: A] :
( ( ( zero_zero @ A )
= ( abs_abs @ A @ A2 ) )
= ( A2
= ( zero_zero @ A ) ) ) ) ).
% abs_0_eq
thf(fact_2485_abs__0,axiom,
! [A: $tType] :
( ( idom_abs_sgn @ A )
=> ( ( abs_abs @ A @ ( zero_zero @ A ) )
= ( zero_zero @ A ) ) ) ).
% abs_0
thf(fact_2486_abs__numeral,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [N: num] :
( ( abs_abs @ A @ ( numeral_numeral @ A @ N ) )
= ( numeral_numeral @ A @ N ) ) ) ).
% abs_numeral
thf(fact_2487_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_2488_abs__1,axiom,
! [A: $tType] :
( ( idom_abs_sgn @ A )
=> ( ( abs_abs @ A @ ( one_one @ A ) )
= ( one_one @ A ) ) ) ).
% abs_1
thf(fact_2489_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_2490_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_2491_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_2492_abs__minus__cancel,axiom,
! [A: $tType] :
( ( ordere166539214618696060dd_abs @ A )
=> ! [A2: A] :
( ( abs_abs @ A @ ( uminus_uminus @ A @ A2 ) )
= ( abs_abs @ A @ A2 ) ) ) ).
% abs_minus_cancel
thf(fact_2493_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_2494_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_2495_abs__of__nat,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [N: nat] :
( ( abs_abs @ A @ ( semiring_1_of_nat @ A @ N ) )
= ( semiring_1_of_nat @ A @ N ) ) ) ).
% abs_of_nat
thf(fact_2496_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_2497_tanh__0,axiom,
! [A: $tType] :
( ( ( real_Vector_banach @ A )
& ( real_V3459762299906320749_field @ A ) )
=> ( ( tanh @ A @ ( zero_zero @ A ) )
= ( zero_zero @ A ) ) ) ).
% tanh_0
thf(fact_2498_tanh__real__zero__iff,axiom,
! [X: real] :
( ( ( tanh @ real @ X )
= ( zero_zero @ real ) )
= ( X
= ( zero_zero @ real ) ) ) ).
% tanh_real_zero_iff
thf(fact_2499_tanh__real__less__iff,axiom,
! [X: real,Y: real] :
( ( ord_less @ real @ ( tanh @ real @ X ) @ ( tanh @ real @ Y ) )
= ( ord_less @ real @ X @ Y ) ) ).
% tanh_real_less_iff
thf(fact_2500_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_2501_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_2502_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_2503_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_2504_abs__neg__numeral,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [N: num] :
( ( abs_abs @ A @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ N ) ) )
= ( numeral_numeral @ A @ N ) ) ) ).
% abs_neg_numeral
thf(fact_2505_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_2506_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_2507_abs__power__minus,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [A2: A,N: nat] :
( ( abs_abs @ A @ ( power_power @ A @ ( uminus_uminus @ A @ A2 ) @ N ) )
= ( abs_abs @ A @ ( power_power @ A @ A2 @ N ) ) ) ) ).
% abs_power_minus
thf(fact_2508_round__0,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ( ( archimedean_round @ A @ ( zero_zero @ A ) )
= ( zero_zero @ int ) ) ) ).
% round_0
thf(fact_2509_round__numeral,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [N: num] :
( ( archimedean_round @ A @ ( numeral_numeral @ A @ N ) )
= ( numeral_numeral @ int @ N ) ) ) ).
% round_numeral
thf(fact_2510_tanh__real__pos__iff,axiom,
! [X: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ ( tanh @ real @ X ) )
= ( ord_less @ real @ ( zero_zero @ real ) @ X ) ) ).
% tanh_real_pos_iff
thf(fact_2511_tanh__real__neg__iff,axiom,
! [X: real] :
( ( ord_less @ real @ ( tanh @ real @ X ) @ ( zero_zero @ real ) )
= ( ord_less @ real @ X @ ( zero_zero @ real ) ) ) ).
% tanh_real_neg_iff
thf(fact_2512_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_2513_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_2514_round__1,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ( ( archimedean_round @ A @ ( one_one @ A ) )
= ( one_one @ int ) ) ) ).
% round_1
thf(fact_2515_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_2516_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_2517_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_2518_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_2519_zero__less__power__abs__iff,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [A2: A,N: nat] :
( ( ord_less @ A @ ( zero_zero @ A ) @ ( power_power @ A @ ( abs_abs @ A @ A2 ) @ N ) )
= ( ( A2
!= ( zero_zero @ A ) )
| ( N
= ( zero_zero @ nat ) ) ) ) ) ).
% zero_less_power_abs_iff
thf(fact_2520_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_2521_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_2522_round__neg__numeral,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [N: num] :
( ( archimedean_round @ A @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ N ) ) )
= ( uminus_uminus @ int @ ( numeral_numeral @ int @ N ) ) ) ) ).
% round_neg_numeral
thf(fact_2523_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_2524_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_2525_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_2526_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_2527_abs__one,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ( ( abs_abs @ A @ ( one_one @ A ) )
= ( one_one @ A ) ) ) ).
% abs_one
thf(fact_2528_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_2529_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_2530_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_2531_power__abs,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [A2: A,N: nat] :
( ( abs_abs @ A @ ( power_power @ A @ A2 @ N ) )
= ( power_power @ A @ ( abs_abs @ A @ A2 ) @ N ) ) ) ).
% power_abs
thf(fact_2532_dvd__if__abs__eq,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [L: A,K: A] :
( ( ( abs_abs @ A @ L )
= ( abs_abs @ A @ K ) )
=> ( dvd_dvd @ A @ L @ K ) ) ) ).
% dvd_if_abs_eq
thf(fact_2533_finite__nat__set__iff__bounded,axiom,
( ( finite_finite2 @ nat )
= ( ^ [N8: set @ nat] :
? [M4: nat] :
! [X2: nat] :
( ( member @ nat @ X2 @ N8 )
=> ( ord_less @ nat @ X2 @ M4 ) ) ) ) ).
% finite_nat_set_iff_bounded
thf(fact_2534_bounded__nat__set__is__finite,axiom,
! [N4: set @ nat,N: nat] :
( ! [X3: nat] :
( ( member @ nat @ X3 @ N4 )
=> ( ord_less @ nat @ X3 @ N ) )
=> ( finite_finite2 @ nat @ N4 ) ) ).
% bounded_nat_set_is_finite
thf(fact_2535_finite__set__decode,axiom,
! [N: nat] : ( finite_finite2 @ nat @ ( nat_set_decode @ N ) ) ).
% finite_set_decode
thf(fact_2536_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_2537_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_2538_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_2539_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_2540_abs__mult__less,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [A2: A,C2: A,B2: A,D2: A] :
( ( ord_less @ A @ ( abs_abs @ A @ A2 ) @ C2 )
=> ( ( ord_less @ A @ ( abs_abs @ A @ B2 ) @ D2 )
=> ( ord_less @ A @ ( times_times @ A @ ( abs_abs @ A @ A2 ) @ ( abs_abs @ A @ B2 ) ) @ ( times_times @ A @ C2 @ D2 ) ) ) ) ) ).
% abs_mult_less
thf(fact_2541_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_2542_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_2543_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_2544_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_2545_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_2546_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_2547_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_2548_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_2549_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_2550_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_2551_tanh__real__lt__1,axiom,
! [X: real] : ( ord_less @ real @ ( tanh @ real @ X ) @ ( one_one @ real ) ) ).
% tanh_real_lt_1
thf(fact_2552_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_2553_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_2554_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_2555_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_2556_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_2557_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_2558_zero__le__power__abs,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [A2: A,N: nat] : ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( power_power @ A @ ( abs_abs @ A @ A2 ) @ N ) ) ) ).
% zero_le_power_abs
thf(fact_2559_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_2560_abs__if__raw,axiom,
! [A: $tType] :
( ( abs_if @ A )
=> ( ( abs_abs @ A )
= ( ^ [A4: A] : ( if @ A @ ( ord_less @ A @ A4 @ ( zero_zero @ A ) ) @ ( uminus_uminus @ A @ A4 ) @ A4 ) ) ) ) ).
% abs_if_raw
thf(fact_2561_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_2562_abs__if,axiom,
! [A: $tType] :
( ( abs_if @ A )
=> ( ( abs_abs @ A )
= ( ^ [A4: A] : ( if @ A @ ( ord_less @ A @ A4 @ ( zero_zero @ A ) ) @ ( uminus_uminus @ A @ A4 ) @ A4 ) ) ) ) ).
% abs_if
thf(fact_2563_abs__diff__triangle__ineq,axiom,
! [A: $tType] :
( ( ordere166539214618696060dd_abs @ A )
=> ! [A2: A,B2: A,C2: A,D2: A] : ( ord_less_eq @ A @ ( abs_abs @ A @ ( minus_minus @ A @ ( plus_plus @ A @ A2 @ B2 ) @ ( plus_plus @ A @ C2 @ D2 ) ) ) @ ( plus_plus @ A @ ( abs_abs @ A @ ( minus_minus @ A @ A2 @ C2 ) ) @ ( abs_abs @ A @ ( minus_minus @ A @ B2 @ D2 ) ) ) ) ) ).
% abs_diff_triangle_ineq
thf(fact_2564_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_2565_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_2566_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_2567_abs__real__def,axiom,
( ( abs_abs @ real )
= ( ^ [A4: real] : ( if @ real @ ( ord_less @ real @ A4 @ ( zero_zero @ real ) ) @ ( uminus_uminus @ real @ A4 ) @ A4 ) ) ) ).
% abs_real_def
thf(fact_2568_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_2569_subset__eq__atLeast0__atMost__finite,axiom,
! [N4: set @ nat,N: nat] :
( ( ord_less_eq @ ( set @ nat ) @ N4 @ ( set_or1337092689740270186AtMost @ nat @ ( zero_zero @ nat ) @ N ) )
=> ( finite_finite2 @ nat @ N4 ) ) ).
% subset_eq_atLeast0_atMost_finite
thf(fact_2570_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_2571_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_2572_of__int__leD,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [N: int,X: A] :
( ( ord_less_eq @ A @ ( abs_abs @ A @ ( ring_1_of_int @ A @ N ) ) @ X )
=> ( ( N
= ( zero_zero @ int ) )
| ( ord_less_eq @ A @ ( one_one @ A ) @ X ) ) ) ) ).
% of_int_leD
thf(fact_2573_of__int__lessD,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [N: int,X: A] :
( ( ord_less @ A @ ( abs_abs @ A @ ( ring_1_of_int @ A @ N ) ) @ X )
=> ( ( N
= ( zero_zero @ int ) )
| ( ord_less @ A @ ( one_one @ A ) @ X ) ) ) ) ).
% of_int_lessD
thf(fact_2574_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_2575_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_2576_round__unique_H,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [X: A,N: int] :
( ( ord_less @ A @ ( abs_abs @ A @ ( minus_minus @ A @ X @ ( ring_1_of_int @ A @ N ) ) ) @ ( divide_divide @ A @ ( one_one @ A ) @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) )
=> ( ( archimedean_round @ A @ X )
= N ) ) ) ).
% round_unique'
thf(fact_2577_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_2578_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_2579_power__even__abs,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [N: nat,A2: A] :
( ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N )
=> ( ( power_power @ A @ ( abs_abs @ A @ A2 ) @ N )
= ( power_power @ A @ A2 @ N ) ) ) ) ).
% power_even_abs
thf(fact_2580_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_2581_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_2582_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_2583_power__mono__even,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [N: nat,A2: A,B2: A] :
( ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N )
=> ( ( ord_less_eq @ A @ ( abs_abs @ A @ A2 ) @ ( abs_abs @ A @ B2 ) )
=> ( ord_less_eq @ A @ ( power_power @ A @ A2 @ N ) @ ( power_power @ A @ B2 @ N ) ) ) ) ) ).
% power_mono_even
thf(fact_2584_pred__bound__height_H,axiom,
! [T: vEBT_VEBT,N: nat,X: nat] :
( ( vEBT_invar_vebt @ T @ N )
=> ( ord_less_eq @ nat @ ( vEBT_T_p_r_e_d2 @ T @ X ) @ ( plus_plus @ nat @ ( one_one @ nat ) @ ( vEBT_VEBT_height @ T ) ) ) ) ).
% pred_bound_height'
thf(fact_2585_succ_H__bound__height,axiom,
! [T: vEBT_VEBT,N: nat,X: nat] :
( ( vEBT_invar_vebt @ T @ N )
=> ( ord_less_eq @ nat @ ( vEBT_T_s_u_c_c2 @ T @ X ) @ ( plus_plus @ nat @ ( one_one @ nat ) @ ( vEBT_VEBT_height @ T ) ) ) ) ).
% succ'_bound_height
thf(fact_2586_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_2587_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_2588_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_2589_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_2590_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_2591_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_2592_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_2593_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_2594_floor__log__nat__eq__powr__iff,axiom,
! [B2: nat,K: nat,N: nat] :
( ( ord_less_eq @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ B2 )
=> ( ( ord_less @ nat @ ( zero_zero @ nat ) @ K )
=> ( ( ( archim6421214686448440834_floor @ real @ ( log @ ( semiring_1_of_nat @ real @ B2 ) @ ( semiring_1_of_nat @ real @ K ) ) )
= ( semiring_1_of_nat @ int @ N ) )
= ( ( ord_less_eq @ nat @ ( power_power @ nat @ B2 @ N ) @ K )
& ( ord_less @ nat @ K @ ( power_power @ nat @ B2 @ ( plus_plus @ nat @ N @ ( one_one @ nat ) ) ) ) ) ) ) ) ).
% floor_log_nat_eq_powr_iff
thf(fact_2595_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_2596_central__binomial__lower__bound,axiom,
! [N: nat] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ N )
=> ( ord_less_eq @ real @ ( divide_divide @ real @ ( power_power @ real @ ( numeral_numeral @ real @ ( bit0 @ ( bit0 @ one2 ) ) ) @ N ) @ ( times_times @ real @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) @ ( semiring_1_of_nat @ real @ N ) ) ) @ ( semiring_1_of_nat @ real @ ( binomial @ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N ) @ N ) ) ) ) ).
% central_binomial_lower_bound
thf(fact_2597_deg__deg__n,axiom,
! [Info: option @ ( product_prod @ nat @ nat ),Deg: nat,TreeList: list @ vEBT_VEBT,Summary: vEBT_VEBT,N: nat] :
( ( vEBT_invar_vebt @ ( vEBT_Node @ Info @ Deg @ TreeList @ Summary ) @ N )
=> ( Deg = N ) ) ).
% deg_deg_n
thf(fact_2598_deg__SUcn__Node,axiom,
! [Tree: vEBT_VEBT,N: nat] :
( ( vEBT_invar_vebt @ Tree @ ( suc @ ( suc @ N ) ) )
=> ? [Info2: option @ ( product_prod @ nat @ nat ),TreeList2: list @ vEBT_VEBT,S2: vEBT_VEBT] :
( Tree
= ( vEBT_Node @ Info2 @ ( suc @ ( suc @ N ) ) @ TreeList2 @ S2 ) ) ) ).
% deg_SUcn_Node
thf(fact_2599_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_2600_powr__0,axiom,
! [A: $tType] :
( ( ln @ A )
=> ! [Z: A] :
( ( powr @ A @ ( zero_zero @ A ) @ Z )
= ( zero_zero @ A ) ) ) ).
% powr_0
thf(fact_2601_powr__eq__0__iff,axiom,
! [A: $tType] :
( ( ln @ A )
=> ! [W: A,Z: A] :
( ( ( powr @ A @ W @ Z )
= ( zero_zero @ A ) )
= ( W
= ( zero_zero @ A ) ) ) ) ).
% powr_eq_0_iff
thf(fact_2602_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_2603_zdvd1__eq,axiom,
! [X: int] :
( ( dvd_dvd @ int @ X @ ( one_one @ int ) )
= ( ( abs_abs @ int @ X )
= ( one_one @ int ) ) ) ).
% zdvd1_eq
thf(fact_2604_of__int__floor__cancel,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [X: A] :
( ( ( ring_1_of_int @ A @ ( archim6421214686448440834_floor @ A @ X ) )
= X )
= ( ? [N3: int] :
( X
= ( ring_1_of_int @ A @ N3 ) ) ) ) ) ).
% of_int_floor_cancel
thf(fact_2605_arctan__eq__zero__iff,axiom,
! [X: real] :
( ( ( arctan @ X )
= ( zero_zero @ real ) )
= ( X
= ( zero_zero @ real ) ) ) ).
% arctan_eq_zero_iff
thf(fact_2606_arctan__zero__zero,axiom,
( ( arctan @ ( zero_zero @ real ) )
= ( zero_zero @ real ) ) ).
% arctan_zero_zero
thf(fact_2607_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_2608_floor__zero,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ( ( archim6421214686448440834_floor @ A @ ( zero_zero @ A ) )
= ( zero_zero @ int ) ) ) ).
% floor_zero
thf(fact_2609_floor__numeral,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [V: num] :
( ( archim6421214686448440834_floor @ A @ ( numeral_numeral @ A @ V ) )
= ( numeral_numeral @ int @ V ) ) ) ).
% floor_numeral
thf(fact_2610_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_2611_floor__one,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ( ( archim6421214686448440834_floor @ A @ ( one_one @ A ) )
= ( one_one @ int ) ) ) ).
% floor_one
thf(fact_2612_powr__gt__zero,axiom,
! [X: real,A2: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ ( powr @ real @ X @ A2 ) )
= ( X
!= ( zero_zero @ real ) ) ) ).
% powr_gt_zero
thf(fact_2613_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_2614_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_2615_arctan__less__zero__iff,axiom,
! [X: real] :
( ( ord_less @ real @ ( arctan @ X ) @ ( zero_zero @ real ) )
= ( ord_less @ real @ X @ ( zero_zero @ real ) ) ) ).
% arctan_less_zero_iff
thf(fact_2616_zero__less__arctan__iff,axiom,
! [X: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ ( arctan @ X ) )
= ( ord_less @ real @ ( zero_zero @ real ) @ X ) ) ).
% zero_less_arctan_iff
thf(fact_2617_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_2618_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_2619_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_2620_powr__one,axiom,
! [X: real] :
( ( ord_less_eq @ real @ ( zero_zero @ real ) @ X )
=> ( ( powr @ real @ X @ ( one_one @ real ) )
= X ) ) ).
% powr_one
thf(fact_2621_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_2622_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_2623_numeral__powr__numeral__real,axiom,
! [M: num,N: num] :
( ( powr @ real @ ( numeral_numeral @ real @ M ) @ ( numeral_numeral @ real @ N ) )
= ( power_power @ real @ ( numeral_numeral @ real @ M ) @ ( numeral_numeral @ nat @ N ) ) ) ).
% numeral_powr_numeral_real
thf(fact_2624_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_2625_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_2626_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_2627_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_2628_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_2629_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_2630_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_2631_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_2632_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_2633_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_2634_floor__numeral__power,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [X: num,N: nat] :
( ( archim6421214686448440834_floor @ A @ ( power_power @ A @ ( numeral_numeral @ A @ X ) @ N ) )
= ( power_power @ int @ ( numeral_numeral @ int @ X ) @ N ) ) ) ).
% floor_numeral_power
thf(fact_2635_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_2636_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_2637_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_2638_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_2639_powr__numeral,axiom,
! [X: real,N: num] :
( ( ord_less_eq @ real @ ( zero_zero @ real ) @ X )
=> ( ( powr @ real @ X @ ( numeral_numeral @ real @ N ) )
= ( power_power @ real @ X @ ( numeral_numeral @ nat @ N ) ) ) ) ).
% powr_numeral
thf(fact_2640_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_2641_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_2642_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_2643_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_2644_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_2645_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_2646_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_2647_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_2648_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_2649_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_2650_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_2651_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_2652_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_2653_arctan__monotone,axiom,
! [X: real,Y: real] :
( ( ord_less @ real @ X @ Y )
=> ( ord_less @ real @ ( arctan @ X ) @ ( arctan @ Y ) ) ) ).
% arctan_monotone
thf(fact_2654_arctan__less__iff,axiom,
! [X: real,Y: real] :
( ( ord_less @ real @ ( arctan @ X ) @ ( arctan @ Y ) )
= ( ord_less @ real @ X @ Y ) ) ).
% arctan_less_iff
thf(fact_2655_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_2656_powr__non__neg,axiom,
! [A2: real,X: real] :
~ ( ord_less @ real @ ( powr @ real @ A2 @ X ) @ ( zero_zero @ real ) ) ).
% powr_non_neg
thf(fact_2657_powr__less__mono2__neg,axiom,
! [A2: real,X: real,Y: real] :
( ( ord_less @ real @ A2 @ ( zero_zero @ real ) )
=> ( ( ord_less @ real @ ( zero_zero @ real ) @ X )
=> ( ( ord_less @ real @ X @ Y )
=> ( ord_less @ real @ ( powr @ real @ Y @ A2 ) @ ( powr @ real @ X @ A2 ) ) ) ) ) ).
% powr_less_mono2_neg
thf(fact_2658_powr__ge__pzero,axiom,
! [X: real,Y: real] : ( ord_less_eq @ real @ ( zero_zero @ real ) @ ( powr @ real @ X @ Y ) ) ).
% powr_ge_pzero
thf(fact_2659_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_2660_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_2661_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_2662_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_2663_abs__zmult__eq__1,axiom,
! [M: int,N: int] :
( ( ( abs_abs @ int @ ( times_times @ int @ M @ N ) )
= ( one_one @ int ) )
=> ( ( abs_abs @ int @ M )
= ( one_one @ int ) ) ) ).
% abs_zmult_eq_1
thf(fact_2664_abs__div,axiom,
! [Y: int,X: int] :
( ( dvd_dvd @ int @ Y @ X )
=> ( ( abs_abs @ int @ ( divide_divide @ int @ X @ Y ) )
= ( divide_divide @ int @ ( abs_abs @ int @ X ) @ ( abs_abs @ int @ Y ) ) ) ) ).
% abs_div
thf(fact_2665_VEBT__internal_Onaive__member_Osimps_I2_J,axiom,
! [Uu: option @ ( product_prod @ nat @ nat ),Uv: list @ vEBT_VEBT,Uw: vEBT_VEBT,Ux: nat] :
~ ( vEBT_V5719532721284313246member @ ( vEBT_Node @ Uu @ ( zero_zero @ nat ) @ Uv @ Uw ) @ Ux ) ).
% VEBT_internal.naive_member.simps(2)
thf(fact_2666_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_2667_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_2668_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_2669_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_2670_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_2671_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_2672_floor__power,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [X: A,N: nat] :
( ( X
= ( ring_1_of_int @ A @ ( archim6421214686448440834_floor @ A @ X ) ) )
=> ( ( archim6421214686448440834_floor @ A @ ( power_power @ A @ X @ N ) )
= ( power_power @ int @ ( archim6421214686448440834_floor @ A @ X ) @ N ) ) ) ) ).
% floor_power
thf(fact_2673_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_2674_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_2675_floor__divide__of__int__eq,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [K: int,L: int] :
( ( archim6421214686448440834_floor @ A @ ( divide_divide @ A @ ( ring_1_of_int @ A @ K ) @ ( ring_1_of_int @ A @ L ) ) )
= ( divide_divide @ int @ K @ L ) ) ) ).
% floor_divide_of_int_eq
thf(fact_2676_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_2677_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_2678_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_2679_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_2680_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_2681_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_2682_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_2683_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_2684_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_2685_zabs__def,axiom,
( ( abs_abs @ int )
= ( ^ [I2: int] : ( if @ int @ ( ord_less @ int @ I2 @ ( zero_zero @ int ) ) @ ( uminus_uminus @ int @ I2 ) @ I2 ) ) ) ).
% zabs_def
thf(fact_2686_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_2687_abs__mod__less,axiom,
! [L: int,K: int] :
( ( L
!= ( zero_zero @ int ) )
=> ( ord_less @ int @ ( abs_abs @ int @ ( modulo_modulo @ int @ K @ L ) ) @ ( abs_abs @ int @ L ) ) ) ).
% abs_mod_less
thf(fact_2688_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_2689_dvd__imp__le__int,axiom,
! [I: int,D2: int] :
( ( I
!= ( zero_zero @ int ) )
=> ( ( dvd_dvd @ int @ D2 @ I )
=> ( ord_less_eq @ int @ ( abs_abs @ int @ D2 ) @ ( abs_abs @ int @ I ) ) ) ) ).
% dvd_imp_le_int
thf(fact_2690_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_2691_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_2692_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,Va2: vEBT_VEBT,Vb: nat] :
( ( vEBT_T_p_r_e_d @ ( vEBT_Node @ ( none @ ( product_prod @ nat @ nat ) ) @ Uy @ Uz @ Va2 ) @ Vb )
= ( one_one @ nat ) ) ).
% T\<^sub>p\<^sub>r\<^sub>e\<^sub>d.simps(4)
thf(fact_2693_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,Va2: vEBT_VEBT,Vb: nat] :
( ( vEBT_T_p_r_e_d2 @ ( vEBT_Node @ ( none @ ( product_prod @ nat @ nat ) ) @ Uy @ Uz @ Va2 ) @ Vb )
= ( one_one @ nat ) ) ).
% T\<^sub>p\<^sub>r\<^sub>e\<^sub>d'.simps(4)
thf(fact_2694_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,Va2: nat] :
( ( vEBT_T_s_u_c_c @ ( vEBT_Node @ ( none @ ( product_prod @ nat @ nat ) ) @ Ux @ Uy @ Uz ) @ Va2 )
= ( one_one @ nat ) ) ).
% T\<^sub>s\<^sub>u\<^sub>c\<^sub>c.simps(3)
thf(fact_2695_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,Va2: nat] :
( ( vEBT_T_s_u_c_c2 @ ( vEBT_Node @ ( none @ ( product_prod @ nat @ nat ) ) @ Ux @ Uy @ Uz ) @ Va2 )
= ( one_one @ nat ) ) ).
% T\<^sub>s\<^sub>u\<^sub>c\<^sub>c'.simps(3)
thf(fact_2696_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_2697_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_2698_floor__divide__of__nat__eq,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [M: nat,N: nat] :
( ( archim6421214686448440834_floor @ A @ ( divide_divide @ A @ ( semiring_1_of_nat @ A @ M ) @ ( semiring_1_of_nat @ A @ N ) ) )
= ( semiring_1_of_nat @ int @ ( divide_divide @ nat @ M @ N ) ) ) ) ).
% floor_divide_of_nat_eq
thf(fact_2699_powr__realpow,axiom,
! [X: real,N: nat] :
( ( ord_less @ real @ ( zero_zero @ real ) @ X )
=> ( ( powr @ real @ X @ ( semiring_1_of_nat @ real @ N ) )
= ( power_power @ real @ X @ N ) ) ) ).
% powr_realpow
thf(fact_2700_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_2701_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_2702_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_2703_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_2704_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_2705_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_2706_floor__eq,axiom,
! [N: int,X: real] :
( ( ord_less @ real @ ( ring_1_of_int @ real @ N ) @ X )
=> ( ( ord_less @ real @ X @ ( plus_plus @ real @ ( ring_1_of_int @ real @ N ) @ ( one_one @ real ) ) )
=> ( ( archim6421214686448440834_floor @ real @ X )
= N ) ) ) ).
% floor_eq
thf(fact_2707_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_2708_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_2709_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_2710_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_2711_zdvd__mult__cancel1,axiom,
! [M: int,N: int] :
( ( M
!= ( zero_zero @ int ) )
=> ( ( dvd_dvd @ int @ ( times_times @ int @ M @ N ) @ M )
= ( ( abs_abs @ int @ N )
= ( one_one @ int ) ) ) ) ).
% zdvd_mult_cancel1
thf(fact_2712_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_2713_floor__split,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [P: int > $o,T: A] :
( ( P @ ( archim6421214686448440834_floor @ A @ T ) )
= ( ! [I2: int] :
( ( ( ord_less_eq @ A @ ( ring_1_of_int @ A @ I2 ) @ T )
& ( ord_less @ A @ T @ ( plus_plus @ A @ ( ring_1_of_int @ A @ I2 ) @ ( one_one @ A ) ) ) )
=> ( P @ I2 ) ) ) ) ) ).
% floor_split
thf(fact_2714_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_2715_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_2716_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_2717_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_2718_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_2719_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_2720_binomial__antimono,axiom,
! [K: nat,K4: nat,N: nat] :
( ( ord_less_eq @ nat @ K @ K4 )
=> ( ( ord_less_eq @ nat @ ( divide_divide @ nat @ N @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ K )
=> ( ( ord_less_eq @ nat @ K4 @ N )
=> ( ord_less_eq @ nat @ ( binomial @ N @ K4 ) @ ( binomial @ N @ K ) ) ) ) ) ).
% binomial_antimono
thf(fact_2721_binomial__maximum,axiom,
! [N: nat,K: nat] : ( ord_less_eq @ nat @ ( binomial @ N @ K ) @ ( binomial @ N @ ( divide_divide @ nat @ N @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ).
% binomial_maximum
thf(fact_2722_binomial__maximum_H,axiom,
! [N: nat,K: nat] : ( ord_less_eq @ nat @ ( binomial @ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N ) @ K ) @ ( binomial @ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N ) @ N ) ) ).
% binomial_maximum'
thf(fact_2723_binomial__mono,axiom,
! [K: nat,K4: nat,N: nat] :
( ( ord_less_eq @ nat @ K @ K4 )
=> ( ( ord_less_eq @ nat @ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ K4 ) @ N )
=> ( ord_less_eq @ nat @ ( binomial @ N @ K ) @ ( binomial @ N @ K4 ) ) ) ) ).
% binomial_mono
thf(fact_2724_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_2725_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_2726_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_2727_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_2728_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_2729_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_2730_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_2731_floor__eq2,axiom,
! [N: int,X: real] :
( ( ord_less_eq @ real @ ( ring_1_of_int @ real @ N ) @ X )
=> ( ( ord_less @ real @ X @ ( plus_plus @ real @ ( ring_1_of_int @ real @ N ) @ ( one_one @ real ) ) )
=> ( ( archim6421214686448440834_floor @ real @ X )
= N ) ) ) ).
% floor_eq2
thf(fact_2732_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_2733_even__abs__add__iff,axiom,
! [K: int,L: int] :
( ( dvd_dvd @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ ( plus_plus @ int @ ( abs_abs @ int @ K ) @ L ) )
= ( dvd_dvd @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ ( plus_plus @ int @ K @ L ) ) ) ).
% even_abs_add_iff
thf(fact_2734_even__add__abs__iff,axiom,
! [K: int,L: int] :
( ( dvd_dvd @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ ( plus_plus @ int @ K @ ( abs_abs @ int @ L ) ) )
= ( dvd_dvd @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ ( plus_plus @ int @ K @ L ) ) ) ).
% even_add_abs_iff
thf(fact_2735_floor__divide__lower,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [Q3: A,P4: A] :
( ( ord_less @ A @ ( zero_zero @ A ) @ Q3 )
=> ( ord_less_eq @ A @ ( times_times @ A @ ( ring_1_of_int @ A @ ( archim6421214686448440834_floor @ A @ ( divide_divide @ A @ P4 @ Q3 ) ) ) @ Q3 ) @ P4 ) ) ) ).
% floor_divide_lower
thf(fact_2736_binomial__less__binomial__Suc,axiom,
! [K: nat,N: nat] :
( ( ord_less @ nat @ K @ ( divide_divide @ nat @ N @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) )
=> ( ord_less @ nat @ ( binomial @ N @ K ) @ ( binomial @ N @ ( suc @ K ) ) ) ) ).
% binomial_less_binomial_Suc
thf(fact_2737_T_092_060_094sub_062m_092_060_094sub_062e_092_060_094sub_062m_092_060_094sub_062b_092_060_094sub_062e_092_060_094sub_062r_Osimps_I2_J,axiom,
! [Uu: nat,Uv: list @ vEBT_VEBT,Uw: vEBT_VEBT,X: nat] :
( ( vEBT_T_m_e_m_b_e_r @ ( vEBT_Node @ ( none @ ( product_prod @ nat @ nat ) ) @ Uu @ Uv @ Uw ) @ X )
= ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ).
% T\<^sub>m\<^sub>e\<^sub>m\<^sub>b\<^sub>e\<^sub>r.simps(2)
thf(fact_2738_binomial__strict__antimono,axiom,
! [K: nat,K4: nat,N: nat] :
( ( ord_less @ nat @ K @ K4 )
=> ( ( ord_less_eq @ nat @ N @ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ K ) )
=> ( ( ord_less_eq @ nat @ K4 @ N )
=> ( ord_less @ nat @ ( binomial @ N @ K4 ) @ ( binomial @ N @ K ) ) ) ) ) ).
% binomial_strict_antimono
thf(fact_2739_binomial__strict__mono,axiom,
! [K: nat,K4: nat,N: nat] :
( ( ord_less @ nat @ K @ K4 )
=> ( ( ord_less_eq @ nat @ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ K4 ) @ N )
=> ( ord_less @ nat @ ( binomial @ N @ K ) @ ( binomial @ N @ K4 ) ) ) ) ).
% binomial_strict_mono
thf(fact_2740_central__binomial__odd,axiom,
! [N: nat] :
( ~ ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N )
=> ( ( binomial @ N @ ( suc @ ( divide_divide @ nat @ N @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) )
= ( binomial @ N @ ( divide_divide @ nat @ N @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) ).
% central_binomial_odd
thf(fact_2741_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_2742_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_2743_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_2744_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_2745_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_2746_nat__intermed__int__val,axiom,
! [M: nat,N: nat,F2: nat > int,K: int] :
( ! [I3: nat] :
( ( ( ord_less_eq @ nat @ M @ I3 )
& ( ord_less @ nat @ I3 @ N ) )
=> ( ord_less_eq @ int @ ( abs_abs @ int @ ( minus_minus @ int @ ( F2 @ ( suc @ I3 ) ) @ ( F2 @ I3 ) ) ) @ ( one_one @ int ) ) )
=> ( ( ord_less_eq @ nat @ M @ N )
=> ( ( ord_less_eq @ int @ ( F2 @ M ) @ K )
=> ( ( ord_less_eq @ int @ K @ ( F2 @ N ) )
=> ? [I3: nat] :
( ( ord_less_eq @ nat @ M @ I3 )
& ( ord_less_eq @ nat @ I3 @ N )
& ( ( F2 @ I3 )
= K ) ) ) ) ) ) ).
% nat_intermed_int_val
thf(fact_2747_incr__lemma,axiom,
! [D2: int,Z: int,X: int] :
( ( ord_less @ int @ ( zero_zero @ int ) @ D2 )
=> ( ord_less @ int @ Z @ ( plus_plus @ int @ X @ ( times_times @ int @ ( plus_plus @ int @ ( abs_abs @ int @ ( minus_minus @ int @ X @ Z ) ) @ ( one_one @ int ) ) @ D2 ) ) ) ) ).
% incr_lemma
thf(fact_2748_decr__lemma,axiom,
! [D2: int,X: int,Z: int] :
( ( ord_less @ int @ ( zero_zero @ int ) @ D2 )
=> ( ord_less @ int @ ( minus_minus @ int @ X @ ( times_times @ int @ ( plus_plus @ int @ ( abs_abs @ int @ ( minus_minus @ int @ X @ Z ) ) @ ( one_one @ int ) ) @ D2 ) ) @ Z ) ) ).
% decr_lemma
thf(fact_2749_floor__divide__upper,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [Q3: A,P4: A] :
( ( ord_less @ A @ ( zero_zero @ A ) @ Q3 )
=> ( ord_less @ A @ P4 @ ( times_times @ A @ ( plus_plus @ A @ ( ring_1_of_int @ A @ ( archim6421214686448440834_floor @ A @ ( divide_divide @ A @ P4 @ Q3 ) ) ) @ ( one_one @ A ) ) @ Q3 ) ) ) ) ).
% floor_divide_upper
thf(fact_2750_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_2751_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_2752_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_2753_nat__ivt__aux,axiom,
! [N: nat,F2: nat > int,K: int] :
( ! [I3: nat] :
( ( ord_less @ nat @ I3 @ N )
=> ( ord_less_eq @ int @ ( abs_abs @ int @ ( minus_minus @ int @ ( F2 @ ( suc @ I3 ) ) @ ( F2 @ I3 ) ) ) @ ( one_one @ int ) ) )
=> ( ( ord_less_eq @ int @ ( F2 @ ( zero_zero @ nat ) ) @ K )
=> ( ( ord_less_eq @ int @ K @ ( F2 @ N ) )
=> ? [I3: nat] :
( ( ord_less_eq @ nat @ I3 @ N )
& ( ( F2 @ I3 )
= K ) ) ) ) ) ).
% nat_ivt_aux
thf(fact_2754_powr__neg__numeral,axiom,
! [X: real,N: num] :
( ( ord_less @ real @ ( zero_zero @ real ) @ X )
=> ( ( powr @ real @ X @ ( uminus_uminus @ real @ ( numeral_numeral @ real @ N ) ) )
= ( divide_divide @ real @ ( one_one @ real ) @ ( power_power @ real @ X @ ( numeral_numeral @ nat @ N ) ) ) ) ) ).
% powr_neg_numeral
thf(fact_2755_nat0__intermed__int__val,axiom,
! [N: nat,F2: nat > int,K: int] :
( ! [I3: nat] :
( ( ord_less @ nat @ I3 @ N )
=> ( ord_less_eq @ int @ ( abs_abs @ int @ ( minus_minus @ int @ ( F2 @ ( plus_plus @ nat @ I3 @ ( one_one @ nat ) ) ) @ ( F2 @ I3 ) ) ) @ ( one_one @ int ) ) )
=> ( ( ord_less_eq @ int @ ( F2 @ ( zero_zero @ nat ) ) @ K )
=> ( ( ord_less_eq @ int @ K @ ( F2 @ N ) )
=> ? [I3: nat] :
( ( ord_less_eq @ nat @ I3 @ N )
& ( ( F2 @ I3 )
= K ) ) ) ) ) ).
% nat0_intermed_int_val
thf(fact_2756_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_2757_floor__log2__div2,axiom,
! [N: nat] :
( ( ord_less_eq @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N )
=> ( ( archim6421214686448440834_floor @ real @ ( log @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) @ ( semiring_1_of_nat @ real @ N ) ) )
= ( plus_plus @ int @ ( archim6421214686448440834_floor @ real @ ( log @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) @ ( semiring_1_of_nat @ real @ ( divide_divide @ nat @ N @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) @ ( one_one @ int ) ) ) ) ).
% floor_log2_div2
thf(fact_2758_floor__log__nat__eq__if,axiom,
! [B2: nat,N: nat,K: nat] :
( ( ord_less_eq @ nat @ ( power_power @ nat @ B2 @ N ) @ K )
=> ( ( ord_less @ nat @ K @ ( power_power @ nat @ B2 @ ( plus_plus @ nat @ N @ ( one_one @ nat ) ) ) )
=> ( ( ord_less_eq @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ B2 )
=> ( ( archim6421214686448440834_floor @ real @ ( log @ ( semiring_1_of_nat @ real @ B2 ) @ ( semiring_1_of_nat @ real @ K ) ) )
= ( semiring_1_of_nat @ int @ N ) ) ) ) ) ).
% floor_log_nat_eq_if
thf(fact_2759_zero__less__binomial__iff,axiom,
! [N: nat,K: nat] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ ( binomial @ N @ K ) )
= ( ord_less_eq @ nat @ K @ N ) ) ).
% zero_less_binomial_iff
thf(fact_2760_choose__two,axiom,
! [N: nat] :
( ( binomial @ N @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) )
= ( divide_divide @ nat @ ( times_times @ nat @ N @ ( minus_minus @ nat @ N @ ( one_one @ nat ) ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ).
% choose_two
thf(fact_2761_both__member__options__ding,axiom,
! [Info: option @ ( product_prod @ nat @ nat ),Deg: nat,TreeList: list @ vEBT_VEBT,Summary: vEBT_VEBT,N: nat,X: nat] :
( ( vEBT_invar_vebt @ ( vEBT_Node @ Info @ Deg @ TreeList @ Summary ) @ N )
=> ( ( ord_less @ nat @ X @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ Deg ) )
=> ( ( vEBT_V8194947554948674370ptions @ ( nth @ vEBT_VEBT @ TreeList @ ( vEBT_VEBT_high @ X @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( vEBT_VEBT_low @ X @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) )
=> ( vEBT_V8194947554948674370ptions @ ( vEBT_Node @ Info @ Deg @ TreeList @ Summary ) @ X ) ) ) ) ).
% both_member_options_ding
thf(fact_2762_binomial__n__0,axiom,
! [N: nat] :
( ( binomial @ N @ ( zero_zero @ nat ) )
= ( one_one @ nat ) ) ).
% binomial_n_0
thf(fact_2763_binomial__Suc__Suc,axiom,
! [N: nat,K: nat] :
( ( binomial @ ( suc @ N ) @ ( suc @ K ) )
= ( plus_plus @ nat @ ( binomial @ N @ K ) @ ( binomial @ N @ ( suc @ K ) ) ) ) ).
% binomial_Suc_Suc
thf(fact_2764_binomial__eq__0__iff,axiom,
! [N: nat,K: nat] :
( ( ( binomial @ N @ K )
= ( zero_zero @ nat ) )
= ( ord_less @ nat @ N @ K ) ) ).
% binomial_eq_0_iff
thf(fact_2765_intind,axiom,
! [A: $tType,I: nat,N: nat,P: A > $o,X: A] :
( ( ord_less @ nat @ I @ N )
=> ( ( P @ X )
=> ( P @ ( nth @ A @ ( replicate @ A @ N @ X ) @ I ) ) ) ) ).
% intind
thf(fact_2766_binomial__Suc__n,axiom,
! [N: nat] :
( ( binomial @ ( suc @ N ) @ N )
= ( suc @ N ) ) ).
% binomial_Suc_n
thf(fact_2767_binomial__n__n,axiom,
! [N: nat] :
( ( binomial @ N @ N )
= ( one_one @ nat ) ) ).
% binomial_n_n
thf(fact_2768_binomial__0__Suc,axiom,
! [K: nat] :
( ( binomial @ ( zero_zero @ nat ) @ ( suc @ K ) )
= ( zero_zero @ nat ) ) ).
% binomial_0_Suc
thf(fact_2769_binomial__1,axiom,
! [N: nat] :
( ( binomial @ N @ ( suc @ ( zero_zero @ nat ) ) )
= N ) ).
% binomial_1
thf(fact_2770_nth__replicate,axiom,
! [A: $tType,I: nat,N: nat,X: A] :
( ( ord_less @ nat @ I @ N )
=> ( ( nth @ A @ ( replicate @ A @ N @ X ) @ I )
= X ) ) ).
% nth_replicate
thf(fact_2771_choose__one,axiom,
! [N: nat] :
( ( binomial @ N @ ( one_one @ nat ) )
= N ) ).
% choose_one
thf(fact_2772_binomial__eq__0,axiom,
! [N: nat,K: nat] :
( ( ord_less @ nat @ N @ K )
=> ( ( binomial @ N @ K )
= ( zero_zero @ nat ) ) ) ).
% binomial_eq_0
thf(fact_2773_Suc__times__binomial,axiom,
! [K: nat,N: nat] :
( ( times_times @ nat @ ( suc @ K ) @ ( binomial @ ( suc @ N ) @ ( suc @ K ) ) )
= ( times_times @ nat @ ( suc @ N ) @ ( binomial @ N @ K ) ) ) ).
% Suc_times_binomial
thf(fact_2774_Suc__times__binomial__eq,axiom,
! [N: nat,K: nat] :
( ( times_times @ nat @ ( suc @ N ) @ ( binomial @ N @ K ) )
= ( times_times @ nat @ ( binomial @ ( suc @ N ) @ ( suc @ K ) ) @ ( suc @ K ) ) ) ).
% Suc_times_binomial_eq
thf(fact_2775_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_2776_binomial__le__pow,axiom,
! [R2: nat,N: nat] :
( ( ord_less_eq @ nat @ R2 @ N )
=> ( ord_less_eq @ nat @ ( binomial @ N @ R2 ) @ ( power_power @ nat @ N @ R2 ) ) ) ).
% binomial_le_pow
thf(fact_2777_zero__less__binomial,axiom,
! [K: nat,N: nat] :
( ( ord_less_eq @ nat @ K @ N )
=> ( ord_less @ nat @ ( zero_zero @ nat ) @ ( binomial @ N @ K ) ) ) ).
% zero_less_binomial
thf(fact_2778_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_2779_binomial__Suc__Suc__eq__times,axiom,
! [N: nat,K: nat] :
( ( binomial @ ( suc @ N ) @ ( suc @ K ) )
= ( divide_divide @ nat @ ( times_times @ nat @ ( suc @ N ) @ ( binomial @ N @ K ) ) @ ( suc @ K ) ) ) ).
% binomial_Suc_Suc_eq_times
thf(fact_2780_choose__mult,axiom,
! [K: nat,M: nat,N: nat] :
( ( ord_less_eq @ nat @ K @ M )
=> ( ( ord_less_eq @ nat @ M @ N )
=> ( ( times_times @ nat @ ( binomial @ N @ M ) @ ( binomial @ M @ K ) )
= ( times_times @ nat @ ( binomial @ N @ K ) @ ( binomial @ ( minus_minus @ nat @ N @ K ) @ ( minus_minus @ nat @ M @ K ) ) ) ) ) ) ).
% choose_mult
thf(fact_2781_binomial__absorb__comp,axiom,
! [N: nat,K: nat] :
( ( times_times @ nat @ ( minus_minus @ nat @ N @ K ) @ ( binomial @ N @ K ) )
= ( times_times @ nat @ N @ ( binomial @ ( minus_minus @ nat @ N @ ( one_one @ nat ) ) @ K ) ) ) ).
% binomial_absorb_comp
thf(fact_2782_binomial__absorption,axiom,
! [K: nat,N: nat] :
( ( times_times @ nat @ ( suc @ K ) @ ( binomial @ N @ ( suc @ K ) ) )
= ( times_times @ nat @ N @ ( binomial @ ( minus_minus @ nat @ N @ ( one_one @ nat ) ) @ K ) ) ) ).
% binomial_absorption
thf(fact_2783_binomial__ge__n__over__k__pow__k,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [K: nat,N: nat] :
( ( ord_less_eq @ nat @ K @ N )
=> ( ord_less_eq @ A @ ( power_power @ A @ ( divide_divide @ A @ ( semiring_1_of_nat @ A @ N ) @ ( semiring_1_of_nat @ A @ K ) ) @ K ) @ ( semiring_1_of_nat @ A @ ( binomial @ N @ K ) ) ) ) ) ).
% binomial_ge_n_over_k_pow_k
thf(fact_2784_binomial__le__pow2,axiom,
! [N: nat,K: nat] : ( ord_less_eq @ nat @ ( binomial @ N @ K ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N ) ) ).
% binomial_le_pow2
thf(fact_2785_choose__reduce__nat,axiom,
! [N: nat,K: nat] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ N )
=> ( ( ord_less @ nat @ ( zero_zero @ nat ) @ K )
=> ( ( binomial @ N @ K )
= ( plus_plus @ nat @ ( binomial @ ( minus_minus @ nat @ N @ ( one_one @ nat ) ) @ ( minus_minus @ nat @ K @ ( one_one @ nat ) ) ) @ ( binomial @ ( minus_minus @ nat @ N @ ( one_one @ nat ) ) @ K ) ) ) ) ) ).
% choose_reduce_nat
thf(fact_2786_times__binomial__minus1__eq,axiom,
! [K: nat,N: nat] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ K )
=> ( ( times_times @ nat @ K @ ( binomial @ N @ K ) )
= ( times_times @ nat @ N @ ( binomial @ ( minus_minus @ nat @ N @ ( one_one @ nat ) ) @ ( minus_minus @ nat @ K @ ( one_one @ nat ) ) ) ) ) ) ).
% times_binomial_minus1_eq
thf(fact_2787_binomial__addition__formula,axiom,
! [N: nat,K: nat] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ N )
=> ( ( binomial @ N @ ( suc @ K ) )
= ( plus_plus @ nat @ ( binomial @ ( minus_minus @ nat @ N @ ( one_one @ nat ) ) @ ( suc @ K ) ) @ ( binomial @ ( minus_minus @ nat @ N @ ( one_one @ nat ) ) @ K ) ) ) ) ).
% binomial_addition_formula
thf(fact_2788_in__children__def,axiom,
( vEBT_V5917875025757280293ildren
= ( ^ [N3: nat,TreeList3: list @ vEBT_VEBT,X2: nat] : ( vEBT_V8194947554948674370ptions @ ( nth @ vEBT_VEBT @ TreeList3 @ ( vEBT_VEBT_high @ X2 @ N3 ) ) @ ( vEBT_VEBT_low @ X2 @ N3 ) ) ) ) ).
% in_children_def
thf(fact_2789_heaphelp,axiom,
! [A: $tType,Xa: array @ vEBT_VEBTi,Tree_is: list @ vEBT_VEBTi,TreeList: list @ vEBT_VEBT,Summary: vEBT_VEBT,Xb: vEBT_VEBTi,N: nat,Xc: vEBT_VEBTi,H2: product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat )] :
( ( rep_assn
@ ( times_times @ assn
@ ( times_times @ assn @ ( times_times @ assn @ ( times_times @ assn @ ( snga_assn @ vEBT_VEBTi @ Xa @ Tree_is ) @ ( vEBT_List_list_assn @ vEBT_VEBT @ vEBT_VEBTi @ vEBT_vebt_assn_raw @ TreeList @ Tree_is ) ) @ ( vEBT_vebt_assn_raw @ Summary @ Xb ) )
@ ( pure_assn
@ ( ( ( none @ A )
= ( none @ A ) )
& ( N = N ) ) ) )
@ ( pure_assn
@ ( Xc
= ( vEBT_Nodei @ ( none @ ( product_prod @ nat @ nat ) ) @ N @ Xa @ Xb ) ) ) )
@ H2 )
=> ( rep_assn @ ( vEBT_vebt_assn_raw @ ( vEBT_Node @ ( none @ ( product_prod @ nat @ nat ) ) @ N @ TreeList @ Summary ) @ Xc ) @ H2 ) ) ).
% heaphelp
thf(fact_2790_VEBT__internal_OT_092_060_094sub_062d_092_060_094sub_062e_092_060_094sub_062l_092_060_094sub_062e_092_060_094sub_062t_092_060_094sub_062e_H_Osimps_I4_J,axiom,
! [Deg: nat,TreeList: list @ vEBT_VEBT,Summary: vEBT_VEBT,Uu: nat] :
( ( vEBT_V1232361888498592333_e_t_e @ ( vEBT_Node @ ( none @ ( product_prod @ nat @ nat ) ) @ Deg @ TreeList @ Summary ) @ Uu )
= ( one_one @ nat ) ) ).
% VEBT_internal.T\<^sub>d\<^sub>e\<^sub>l\<^sub>e\<^sub>t\<^sub>e'.simps(4)
thf(fact_2791_T_092_060_094sub_062d_092_060_094sub_062e_092_060_094sub_062l_092_060_094sub_062e_092_060_094sub_062t_092_060_094sub_062e_Osimps_I4_J,axiom,
! [Deg: nat,TreeList: list @ vEBT_VEBT,Summary: vEBT_VEBT,Uu: nat] :
( ( vEBT_T_d_e_l_e_t_e @ ( vEBT_Node @ ( none @ ( product_prod @ nat @ nat ) ) @ Deg @ TreeList @ Summary ) @ Uu )
= ( one_one @ nat ) ) ).
% T\<^sub>d\<^sub>e\<^sub>l\<^sub>e\<^sub>t\<^sub>e.simps(4)
thf(fact_2792_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_2793_height__compose__child,axiom,
! [T: vEBT_VEBT,TreeList: list @ vEBT_VEBT,Info: option @ ( product_prod @ nat @ nat ),Deg: nat,Summary: vEBT_VEBT] :
( ( member @ vEBT_VEBT @ T @ ( set2 @ vEBT_VEBT @ TreeList ) )
=> ( ord_less_eq @ nat @ ( plus_plus @ nat @ ( one_one @ nat ) @ ( vEBT_VEBT_height @ T ) ) @ ( vEBT_VEBT_height @ ( vEBT_Node @ Info @ Deg @ TreeList @ Summary ) ) ) ) ).
% height_compose_child
thf(fact_2794_Rep__assn__inject,axiom,
! [X: assn,Y: assn] :
( ( ( rep_assn @ X )
= ( rep_assn @ Y ) )
= ( X = Y ) ) ).
% Rep_assn_inject
thf(fact_2795_List_Ofinite__set,axiom,
! [A: $tType,Xs2: list @ A] : ( finite_finite2 @ A @ ( set2 @ A @ Xs2 ) ) ).
% List.finite_set
thf(fact_2796_in__set__replicate,axiom,
! [A: $tType,X: A,N: nat,Y: A] :
( ( member @ A @ X @ ( set2 @ A @ ( replicate @ A @ N @ Y ) ) )
= ( ( X = Y )
& ( N
!= ( zero_zero @ nat ) ) ) ) ).
% in_set_replicate
thf(fact_2797_Bex__set__replicate,axiom,
! [A: $tType,N: nat,A2: A,P: A > $o] :
( ( ? [X2: A] :
( ( member @ A @ X2 @ ( set2 @ A @ ( replicate @ A @ N @ A2 ) ) )
& ( P @ X2 ) ) )
= ( ( P @ A2 )
& ( N
!= ( zero_zero @ nat ) ) ) ) ).
% Bex_set_replicate
thf(fact_2798_Ball__set__replicate,axiom,
! [A: $tType,N: nat,A2: A,P: A > $o] :
( ( ! [X2: A] :
( ( member @ A @ X2 @ ( set2 @ A @ ( replicate @ A @ N @ A2 ) ) )
=> ( P @ X2 ) ) )
= ( ( P @ A2 )
| ( N
= ( zero_zero @ nat ) ) ) ) ).
% Ball_set_replicate
thf(fact_2799_mod__pure__star__dist,axiom,
! [P: assn,B2: $o,H2: product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat )] :
( ( rep_assn @ ( times_times @ assn @ P @ ( pure_assn @ B2 ) ) @ H2 )
= ( ( rep_assn @ P @ H2 )
& B2 ) ) ).
% mod_pure_star_dist
thf(fact_2800_ent__pure__post__iff,axiom,
! [P: assn,Q: assn,B2: $o] :
( ( entails @ P @ ( times_times @ assn @ Q @ ( pure_assn @ B2 ) ) )
= ( ! [H: product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat )] :
( ( rep_assn @ P @ H )
=> B2 )
& ( entails @ P @ Q ) ) ) ).
% ent_pure_post_iff
thf(fact_2801_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_2802_mod__starE,axiom,
! [A2: assn,B2: assn,H2: product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat )] :
( ( rep_assn @ ( times_times @ assn @ A2 @ B2 ) @ H2 )
=> ~ ( ? [X_1: product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat )] : ( rep_assn @ A2 @ X_1 )
=> ! [H_2: product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat )] :
~ ( rep_assn @ B2 @ H_2 ) ) ) ).
% mod_starE
thf(fact_2803_mod__starD,axiom,
! [A3: assn,B4: assn,H2: product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat )] :
( ( rep_assn @ ( times_times @ assn @ A3 @ B4 ) @ H2 )
=> ? [H1: product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ),H22: product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat )] :
( ( rep_assn @ A3 @ H1 )
& ( rep_assn @ B4 @ H22 ) ) ) ).
% mod_starD
thf(fact_2804_entails__def,axiom,
( entails
= ( ^ [P2: assn,Q2: assn] :
! [H: product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat )] :
( ( rep_assn @ P2 @ H )
=> ( rep_assn @ Q2 @ H ) ) ) ) ).
% entails_def
thf(fact_2805_entailsI,axiom,
! [P: assn,Q: assn] :
( ! [H3: product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat )] :
( ( rep_assn @ P @ H3 )
=> ( rep_assn @ Q @ H3 ) )
=> ( entails @ P @ Q ) ) ).
% entailsI
thf(fact_2806_entailsD,axiom,
! [P: assn,Q: assn,H2: product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat )] :
( ( entails @ P @ Q )
=> ( ( rep_assn @ P @ H2 )
=> ( rep_assn @ Q @ H2 ) ) ) ).
% entailsD
thf(fact_2807_ent__fwd,axiom,
! [P: assn,H2: product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ),Q: assn] :
( ( rep_assn @ P @ H2 )
=> ( ( entails @ P @ Q )
=> ( rep_assn @ Q @ H2 ) ) ) ).
% ent_fwd
thf(fact_2808_finite__list,axiom,
! [A: $tType,A3: set @ A] :
( ( finite_finite2 @ A @ A3 )
=> ? [Xs3: list @ A] :
( ( set2 @ A @ Xs3 )
= A3 ) ) ).
% finite_list
thf(fact_2809_subset__code_I1_J,axiom,
! [A: $tType,Xs2: list @ A,B4: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ ( set2 @ A @ Xs2 ) @ B4 )
= ( ! [X2: A] :
( ( member @ A @ X2 @ ( set2 @ A @ Xs2 ) )
=> ( member @ A @ X2 @ B4 ) ) ) ) ).
% subset_code(1)
thf(fact_2810_list__assn__cong,axiom,
! [A: $tType,B: $tType,Xs2: list @ A,Xs4: list @ A,Xsi: list @ B,Xsi2: list @ B,A3: A > B > assn,A8: 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 )
= ( A8 @ X3 @ Xi ) ) ) )
=> ( ( vEBT_List_list_assn @ A @ B @ A3 @ Xs2 @ Xsi )
= ( vEBT_List_list_assn @ A @ B @ A8 @ Xs4 @ Xsi2 ) ) ) ) ) ).
% list_assn_cong
thf(fact_2811_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_2812_set__n__deg__not__0,axiom,
! [TreeList: list @ vEBT_VEBT,N: nat,M: nat] :
( ! [X3: vEBT_VEBT] :
( ( member @ vEBT_VEBT @ X3 @ ( set2 @ vEBT_VEBT @ TreeList ) )
=> ( vEBT_invar_vebt @ X3 @ N ) )
=> ( ( ( size_size @ ( list @ vEBT_VEBT ) @ TreeList )
= ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ M ) )
=> ( ord_less_eq @ nat @ ( one_one @ nat ) @ N ) ) ) ).
% set_n_deg_not_0
thf(fact_2813_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_2814_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_2815_vebt__member_Osimps_I2_J,axiom,
! [Uu: nat,Uv: list @ vEBT_VEBT,Uw: vEBT_VEBT,X: nat] :
~ ( vEBT_vebt_member @ ( vEBT_Node @ ( none @ ( product_prod @ nat @ nat ) ) @ Uu @ Uv @ Uw ) @ X ) ).
% vebt_member.simps(2)
thf(fact_2816_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_2817_invar__vebt_Ointros_I3_J,axiom,
! [TreeList: list @ vEBT_VEBT,N: nat,Summary: vEBT_VEBT,M: nat,Deg: nat] :
( ! [X3: vEBT_VEBT] :
( ( member @ vEBT_VEBT @ X3 @ ( set2 @ vEBT_VEBT @ TreeList ) )
=> ( vEBT_invar_vebt @ X3 @ N ) )
=> ( ( vEBT_invar_vebt @ Summary @ M )
=> ( ( ( size_size @ ( list @ vEBT_VEBT ) @ TreeList )
= ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ M ) )
=> ( ( M
= ( suc @ N ) )
=> ( ( Deg
= ( plus_plus @ nat @ N @ M ) )
=> ( ~ ? [X_1: nat] : ( vEBT_V8194947554948674370ptions @ Summary @ X_1 )
=> ( ! [X3: vEBT_VEBT] :
( ( member @ vEBT_VEBT @ X3 @ ( set2 @ vEBT_VEBT @ TreeList ) )
=> ~ ? [X_1: nat] : ( vEBT_V8194947554948674370ptions @ X3 @ X_1 ) )
=> ( vEBT_invar_vebt @ ( vEBT_Node @ ( none @ ( product_prod @ nat @ nat ) ) @ Deg @ TreeList @ Summary ) @ Deg ) ) ) ) ) ) ) ) ).
% invar_vebt.intros(3)
thf(fact_2818_inthall,axiom,
! [A: $tType,Xs2: list @ A,P: A > $o,N: nat] :
( ! [X3: A] :
( ( member @ A @ X3 @ ( set2 @ A @ Xs2 ) )
=> ( P @ X3 ) )
=> ( ( ord_less @ nat @ N @ ( size_size @ ( list @ A ) @ Xs2 ) )
=> ( P @ ( nth @ A @ Xs2 @ N ) ) ) ) ).
% inthall
thf(fact_2819_length__replicate,axiom,
! [A: $tType,N: nat,X: A] :
( ( size_size @ ( list @ A ) @ ( replicate @ A @ N @ X ) )
= N ) ).
% length_replicate
thf(fact_2820_of__real__eq__0__iff,axiom,
! [A: $tType] :
( ( real_V2191834092415804123ebra_1 @ A )
=> ! [X: real] :
( ( ( real_Vector_of_real @ A @ X )
= ( zero_zero @ A ) )
= ( X
= ( zero_zero @ real ) ) ) ) ).
% of_real_eq_0_iff
thf(fact_2821_of__real__0,axiom,
! [A: $tType] :
( ( real_V2191834092415804123ebra_1 @ A )
=> ( ( real_Vector_of_real @ A @ ( zero_zero @ real ) )
= ( zero_zero @ A ) ) ) ).
% of_real_0
thf(fact_2822_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_2823_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_2824_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_2825_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_2826_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_2827_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_2828_of__real__power,axiom,
! [A: $tType] :
( ( real_V2191834092415804123ebra_1 @ A )
=> ! [X: real,N: nat] :
( ( real_Vector_of_real @ A @ ( power_power @ real @ X @ N ) )
= ( power_power @ A @ ( real_Vector_of_real @ A @ X ) @ N ) ) ) ).
% of_real_power
thf(fact_2829_frac__of__int,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [Z: int] :
( ( archimedean_frac @ A @ ( ring_1_of_int @ A @ Z ) )
= ( zero_zero @ A ) ) ) ).
% frac_of_int
thf(fact_2830_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_2831_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_2832_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_2833_Ex__list__of__length,axiom,
! [A: $tType,N: nat] :
? [Xs3: list @ A] :
( ( size_size @ ( list @ A ) @ Xs3 )
= N ) ).
% Ex_list_of_length
thf(fact_2834_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_2835_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_2836_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_2837_finite__maxlen,axiom,
! [A: $tType,M7: set @ ( list @ A )] :
( ( finite_finite2 @ ( list @ A ) @ M7 )
=> ? [N2: nat] :
! [X5: list @ A] :
( ( member @ ( list @ A ) @ X5 @ M7 )
=> ( ord_less @ nat @ ( size_size @ ( list @ A ) @ X5 ) @ N2 ) ) ) ).
% finite_maxlen
thf(fact_2838_nth__equalityI,axiom,
! [A: $tType,Xs2: list @ A,Ys: list @ A] :
( ( ( size_size @ ( list @ A ) @ Xs2 )
= ( size_size @ ( list @ A ) @ Ys ) )
=> ( ! [I3: nat] :
( ( ord_less @ nat @ I3 @ ( size_size @ ( list @ A ) @ Xs2 ) )
=> ( ( nth @ A @ Xs2 @ I3 )
= ( nth @ A @ Ys @ I3 ) ) )
=> ( Xs2 = Ys ) ) ) ).
% nth_equalityI
thf(fact_2839_Skolem__list__nth,axiom,
! [A: $tType,K: nat,P: nat > A > $o] :
( ( ! [I2: nat] :
( ( ord_less @ nat @ I2 @ K )
=> ? [X8: A] : ( P @ I2 @ X8 ) ) )
= ( ? [Xs: list @ A] :
( ( ( size_size @ ( list @ A ) @ Xs )
= K )
& ! [I2: nat] :
( ( ord_less @ nat @ I2 @ K )
=> ( P @ I2 @ ( nth @ A @ Xs @ I2 ) ) ) ) ) ) ).
% Skolem_list_nth
thf(fact_2840_list__eq__iff__nth__eq,axiom,
! [A: $tType] :
( ( ^ [Y2: list @ A,Z2: list @ A] : Y2 = Z2 )
= ( ^ [Xs: list @ A,Ys3: list @ A] :
( ( ( size_size @ ( list @ A ) @ Xs )
= ( size_size @ ( list @ A ) @ Ys3 ) )
& ! [I2: nat] :
( ( ord_less @ nat @ I2 @ ( size_size @ ( list @ A ) @ Xs ) )
=> ( ( nth @ A @ Xs @ I2 )
= ( nth @ A @ Ys3 @ I2 ) ) ) ) ) ) ).
% list_eq_iff_nth_eq
thf(fact_2841_obtain__list__from__elements,axiom,
! [A: $tType,N: nat,P: A > nat > $o] :
( ! [I3: nat] :
( ( ord_less @ nat @ I3 @ N )
=> ? [Li: A] : ( P @ Li @ I3 ) )
=> ~ ! [L4: list @ A] :
( ( ( size_size @ ( list @ A ) @ L4 )
= N )
=> ~ ! [I4: nat] :
( ( ord_less @ nat @ I4 @ N )
=> ( P @ ( nth @ A @ L4 @ I4 ) @ I4 ) ) ) ) ).
% obtain_list_from_elements
thf(fact_2842_replicate__eqI,axiom,
! [A: $tType,Xs2: list @ A,N: nat,X: A] :
( ( ( size_size @ ( list @ A ) @ Xs2 )
= N )
=> ( ! [Y3: A] :
( ( member @ A @ Y3 @ ( set2 @ A @ Xs2 ) )
=> ( Y3 = X ) )
=> ( Xs2
= ( replicate @ A @ N @ X ) ) ) ) ).
% replicate_eqI
thf(fact_2843_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_2844_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_2845_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_2846_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_2847_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_2848_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_2849_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_2850_all__set__conv__all__nth,axiom,
! [A: $tType,Xs2: list @ A,P: A > $o] :
( ( ! [X2: A] :
( ( member @ A @ X2 @ ( set2 @ A @ Xs2 ) )
=> ( P @ X2 ) ) )
= ( ! [I2: nat] :
( ( ord_less @ nat @ I2 @ ( size_size @ ( list @ A ) @ Xs2 ) )
=> ( P @ ( nth @ A @ Xs2 @ I2 ) ) ) ) ) ).
% all_set_conv_all_nth
thf(fact_2851_all__nth__imp__all__set,axiom,
! [A: $tType,Xs2: list @ A,P: A > $o,X: A] :
( ! [I3: nat] :
( ( ord_less @ nat @ I3 @ ( size_size @ ( list @ A ) @ Xs2 ) )
=> ( P @ ( nth @ A @ Xs2 @ I3 ) ) )
=> ( ( member @ A @ X @ ( set2 @ A @ Xs2 ) )
=> ( P @ X ) ) ) ).
% all_nth_imp_all_set
thf(fact_2852_all__set__conv__nth,axiom,
! [A: $tType,L: list @ A,P: A > $o] :
( ( ! [X2: A] :
( ( member @ A @ X2 @ ( set2 @ A @ L ) )
=> ( P @ X2 ) ) )
= ( ! [I2: nat] :
( ( ord_less @ nat @ I2 @ ( size_size @ ( list @ A ) @ L ) )
=> ( P @ ( nth @ A @ L @ I2 ) ) ) ) ) ).
% all_set_conv_nth
thf(fact_2853_in__set__conv__nth,axiom,
! [A: $tType,X: A,Xs2: list @ A] :
( ( member @ A @ X @ ( set2 @ A @ Xs2 ) )
= ( ? [I2: nat] :
( ( ord_less @ nat @ I2 @ ( size_size @ ( list @ A ) @ Xs2 ) )
& ( ( nth @ A @ Xs2 @ I2 )
= X ) ) ) ) ).
% in_set_conv_nth
thf(fact_2854_list__ball__nth,axiom,
! [A: $tType,N: nat,Xs2: list @ A,P: A > $o] :
( ( ord_less @ nat @ N @ ( size_size @ ( list @ A ) @ Xs2 ) )
=> ( ! [X3: A] :
( ( member @ A @ X3 @ ( set2 @ A @ Xs2 ) )
=> ( P @ X3 ) )
=> ( P @ ( nth @ A @ Xs2 @ N ) ) ) ) ).
% list_ball_nth
thf(fact_2855_nth__mem,axiom,
! [A: $tType,N: nat,Xs2: list @ A] :
( ( ord_less @ nat @ N @ ( size_size @ ( list @ A ) @ Xs2 ) )
=> ( member @ A @ ( nth @ A @ Xs2 @ N ) @ ( set2 @ A @ Xs2 ) ) ) ).
% nth_mem
thf(fact_2856_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_2857_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_2858_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_2859_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_2860_invar__vebt_Ointros_I2_J,axiom,
! [TreeList: list @ vEBT_VEBT,N: nat,Summary: vEBT_VEBT,M: nat,Deg: nat] :
( ! [X3: vEBT_VEBT] :
( ( member @ vEBT_VEBT @ X3 @ ( set2 @ vEBT_VEBT @ TreeList ) )
=> ( vEBT_invar_vebt @ X3 @ N ) )
=> ( ( vEBT_invar_vebt @ Summary @ M )
=> ( ( ( size_size @ ( list @ vEBT_VEBT ) @ TreeList )
= ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ M ) )
=> ( ( M = N )
=> ( ( Deg
= ( plus_plus @ nat @ N @ M ) )
=> ( ~ ? [X_1: nat] : ( vEBT_V8194947554948674370ptions @ Summary @ X_1 )
=> ( ! [X3: vEBT_VEBT] :
( ( member @ vEBT_VEBT @ X3 @ ( set2 @ vEBT_VEBT @ TreeList ) )
=> ~ ? [X_1: nat] : ( vEBT_V8194947554948674370ptions @ X3 @ X_1 ) )
=> ( vEBT_invar_vebt @ ( vEBT_Node @ ( none @ ( product_prod @ nat @ nat ) ) @ Deg @ TreeList @ Summary ) @ Deg ) ) ) ) ) ) ) ) ).
% invar_vebt.intros(2)
thf(fact_2861_mod__frame__fwd,axiom,
! [Ps: assn,H2: product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ),P: assn,R: assn,F3: assn] :
( ( rep_assn @ Ps @ H2 )
=> ( ( entails @ P @ R )
=> ( ( entails @ Ps @ ( times_times @ assn @ P @ F3 ) )
=> ( rep_assn @ ( times_times @ assn @ R @ F3 ) @ H2 ) ) ) ) ).
% mod_frame_fwd
thf(fact_2862_ex__has__greatest__nat__lemma,axiom,
! [A: $tType,P: A > $o,K: A,F2: A > nat,N: 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 ) @ N ) ) ) ) ) ).
% ex_has_greatest_nat_lemma
thf(fact_2863_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_2864_take__bit__word__Bit1__eq,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [N: num,M: num] :
( ( bit_se2584673776208193580ke_bit @ ( word @ A ) @ ( numeral_numeral @ nat @ N ) @ ( numeral_numeral @ ( word @ A ) @ ( bit1 @ M ) ) )
= ( plus_plus @ ( word @ A ) @ ( one_one @ ( word @ A ) ) @ ( times_times @ ( word @ A ) @ ( numeral_numeral @ ( word @ A ) @ ( bit0 @ one2 ) ) @ ( bit_se2584673776208193580ke_bit @ ( word @ A ) @ ( pred_numeral @ N ) @ ( numeral_numeral @ ( word @ A ) @ M ) ) ) ) ) ) ).
% take_bit_word_Bit1_eq
thf(fact_2865_Leaf__0__not,axiom,
! [A2: $o,B2: $o] :
~ ( vEBT_invar_vebt @ ( vEBT_Leaf @ A2 @ B2 ) @ ( zero_zero @ nat ) ) ).
% Leaf_0_not
thf(fact_2866_deg1Leaf,axiom,
! [T: vEBT_VEBT] :
( ( vEBT_invar_vebt @ T @ ( one_one @ nat ) )
= ( ? [A4: $o,B5: $o] :
( T
= ( vEBT_Leaf @ A4 @ B5 ) ) ) ) ).
% deg1Leaf
thf(fact_2867_deg__1__Leaf,axiom,
! [T: vEBT_VEBT] :
( ( vEBT_invar_vebt @ T @ ( one_one @ nat ) )
=> ? [A6: $o,B3: $o] :
( T
= ( vEBT_Leaf @ A6 @ B3 ) ) ) ).
% deg_1_Leaf
thf(fact_2868_deg__1__Leafy,axiom,
! [T: vEBT_VEBT,N: nat] :
( ( vEBT_invar_vebt @ T @ N )
=> ( ( N
= ( one_one @ nat ) )
=> ? [A6: $o,B3: $o] :
( T
= ( vEBT_Leaf @ A6 @ B3 ) ) ) ) ).
% deg_1_Leafy
thf(fact_2869_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_2870_take__bit__of__0,axiom,
! [A: $tType] :
( ( bit_se359711467146920520ations @ A )
=> ! [N: nat] :
( ( bit_se2584673776208193580ke_bit @ A @ N @ ( zero_zero @ A ) )
= ( zero_zero @ A ) ) ) ).
% take_bit_of_0
thf(fact_2871_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_2872_exp__less__cancel__iff,axiom,
! [X: real,Y: real] :
( ( ord_less @ real @ ( exp @ real @ X ) @ ( exp @ real @ Y ) )
= ( ord_less @ real @ X @ Y ) ) ).
% exp_less_cancel_iff
thf(fact_2873_exp__less__mono,axiom,
! [X: real,Y: real] :
( ( ord_less @ real @ X @ Y )
=> ( ord_less @ real @ ( exp @ real @ X ) @ ( exp @ real @ Y ) ) ) ).
% exp_less_mono
thf(fact_2874_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_2875_take__bit__Suc__1,axiom,
! [A: $tType] :
( ( bit_un5681908812861735899ations @ A )
=> ! [N: nat] :
( ( bit_se2584673776208193580ke_bit @ A @ ( suc @ N ) @ ( one_one @ A ) )
= ( one_one @ A ) ) ) ).
% take_bit_Suc_1
thf(fact_2876_take__bit__numeral__1,axiom,
! [A: $tType] :
( ( bit_un5681908812861735899ations @ A )
=> ! [L: num] :
( ( bit_se2584673776208193580ke_bit @ A @ ( numeral_numeral @ nat @ L ) @ ( one_one @ A ) )
= ( one_one @ A ) ) ) ).
% take_bit_numeral_1
thf(fact_2877_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_2878_exp__eq__one__iff,axiom,
! [X: real] :
( ( ( exp @ real @ X )
= ( one_one @ real ) )
= ( X
= ( zero_zero @ real ) ) ) ).
% exp_eq_one_iff
thf(fact_2879_take__bit__of__1__eq__0__iff,axiom,
! [A: $tType] :
( ( bit_un5681908812861735899ations @ A )
=> ! [N: nat] :
( ( ( bit_se2584673776208193580ke_bit @ A @ N @ ( one_one @ A ) )
= ( zero_zero @ A ) )
= ( N
= ( zero_zero @ nat ) ) ) ) ).
% take_bit_of_1_eq_0_iff
thf(fact_2880_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_2881_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_2882_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_2883_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_2884_exp__ln__iff,axiom,
! [X: real] :
( ( ( exp @ real @ ( ln_ln @ real @ X ) )
= X )
= ( ord_less @ real @ ( zero_zero @ real ) @ X ) ) ).
% exp_ln_iff
thf(fact_2885_exp__ln,axiom,
! [X: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ X )
=> ( ( exp @ real @ ( ln_ln @ real @ X ) )
= X ) ) ).
% exp_ln
thf(fact_2886_take__bit__of__1,axiom,
! [A: $tType] :
( ( bit_se359711467146920520ations @ A )
=> ! [N: nat] :
( ( bit_se2584673776208193580ke_bit @ A @ N @ ( one_one @ A ) )
= ( zero_neq_one_of_bool @ A @ ( ord_less @ nat @ ( zero_zero @ nat ) @ N ) ) ) ) ).
% take_bit_of_1
thf(fact_2887_even__take__bit__eq,axiom,
! [A: $tType] :
( ( bit_se359711467146920520ations @ A )
=> ! [N: nat,A2: A] :
( ( dvd_dvd @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ ( bit_se2584673776208193580ke_bit @ A @ N @ A2 ) )
= ( ( N
= ( zero_zero @ nat ) )
| ( dvd_dvd @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ A2 ) ) ) ) ).
% even_take_bit_eq
thf(fact_2888_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_2889_take__bit__word__Bit0__eq,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [N: num,M: num] :
( ( bit_se2584673776208193580ke_bit @ ( word @ A ) @ ( numeral_numeral @ nat @ N ) @ ( numeral_numeral @ ( word @ A ) @ ( bit0 @ M ) ) )
= ( times_times @ ( word @ A ) @ ( numeral_numeral @ ( word @ A ) @ ( bit0 @ one2 ) ) @ ( bit_se2584673776208193580ke_bit @ ( word @ A ) @ ( pred_numeral @ N ) @ ( numeral_numeral @ ( word @ A ) @ M ) ) ) ) ) ).
% take_bit_word_Bit0_eq
thf(fact_2890_take__bit__of__exp,axiom,
! [A: $tType] :
( ( bit_un5681908812861735899ations @ A )
=> ! [M: nat,N: nat] :
( ( bit_se2584673776208193580ke_bit @ A @ M @ ( power_power @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ N ) )
= ( times_times @ A @ ( zero_neq_one_of_bool @ A @ ( ord_less @ nat @ N @ M ) ) @ ( power_power @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ N ) ) ) ) ).
% take_bit_of_exp
thf(fact_2891_take__bit__of__2,axiom,
! [A: $tType] :
( ( bit_un5681908812861735899ations @ A )
=> ! [N: nat] :
( ( bit_se2584673776208193580ke_bit @ A @ N @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) )
= ( times_times @ A @ ( zero_neq_one_of_bool @ A @ ( ord_less_eq @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N ) ) @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) ) ) ).
% take_bit_of_2
thf(fact_2892_take__bit__word__minus__Bit0__eq,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [N: num,M: num] :
( ( bit_se2584673776208193580ke_bit @ ( word @ A ) @ ( numeral_numeral @ nat @ N ) @ ( uminus_uminus @ ( word @ A ) @ ( numeral_numeral @ ( word @ A ) @ ( bit0 @ M ) ) ) )
= ( times_times @ ( word @ A ) @ ( numeral_numeral @ ( word @ A ) @ ( bit0 @ one2 ) ) @ ( bit_se2584673776208193580ke_bit @ ( word @ A ) @ ( pred_numeral @ N ) @ ( uminus_uminus @ ( word @ A ) @ ( numeral_numeral @ ( word @ A ) @ M ) ) ) ) ) ) ).
% take_bit_word_minus_Bit0_eq
thf(fact_2893_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_2894_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_2895_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_2896_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_2897_exp__less__cancel,axiom,
! [X: real,Y: real] :
( ( ord_less @ real @ ( exp @ real @ X ) @ ( exp @ real @ Y ) )
=> ( ord_less @ real @ X @ Y ) ) ).
% exp_less_cancel
thf(fact_2898_take__bit__add,axiom,
! [A: $tType] :
( ( bit_un5681908812861735899ations @ A )
=> ! [N: nat,A2: A,B2: A] :
( ( bit_se2584673776208193580ke_bit @ A @ N @ ( plus_plus @ A @ ( bit_se2584673776208193580ke_bit @ A @ N @ A2 ) @ ( bit_se2584673776208193580ke_bit @ A @ N @ B2 ) ) )
= ( bit_se2584673776208193580ke_bit @ A @ N @ ( plus_plus @ A @ A2 @ B2 ) ) ) ) ).
% take_bit_add
thf(fact_2899_take__bit__tightened,axiom,
! [A: $tType] :
( ( bit_se359711467146920520ations @ A )
=> ! [N: nat,A2: A,B2: A,M: nat] :
( ( ( bit_se2584673776208193580ke_bit @ A @ N @ A2 )
= ( bit_se2584673776208193580ke_bit @ A @ N @ B2 ) )
=> ( ( ord_less_eq @ nat @ M @ N )
=> ( ( bit_se2584673776208193580ke_bit @ A @ M @ A2 )
= ( bit_se2584673776208193580ke_bit @ A @ M @ B2 ) ) ) ) ) ).
% take_bit_tightened
thf(fact_2900_take__bit__of__int,axiom,
! [A: $tType] :
( ( bit_ri3973907225187159222ations @ A )
=> ! [N: nat,K: int] :
( ( bit_se2584673776208193580ke_bit @ A @ N @ ( ring_1_of_int @ A @ K ) )
= ( ring_1_of_int @ A @ ( bit_se2584673776208193580ke_bit @ int @ N @ K ) ) ) ) ).
% take_bit_of_int
thf(fact_2901_take__bit__of__nat,axiom,
! [A: $tType] :
( ( bit_se359711467146920520ations @ A )
=> ! [N: nat,M: nat] :
( ( bit_se2584673776208193580ke_bit @ A @ N @ ( semiring_1_of_nat @ A @ M ) )
= ( semiring_1_of_nat @ A @ ( bit_se2584673776208193580ke_bit @ nat @ N @ M ) ) ) ) ).
% take_bit_of_nat
thf(fact_2902_exp__not__eq__zero,axiom,
! [A: $tType] :
( ( ( real_Vector_banach @ A )
& ( real_V2822296259951069270ebra_1 @ A ) )
=> ! [X: A] :
( ( exp @ A @ X )
!= ( zero_zero @ A ) ) ) ).
% exp_not_eq_zero
thf(fact_2903_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_2904_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_2905_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_2906_VEBT__internal_Omembermima_Osimps_I1_J,axiom,
! [Uu: $o,Uv: $o,Uw: nat] :
~ ( vEBT_VEBT_membermima @ ( vEBT_Leaf @ Uu @ Uv ) @ Uw ) ).
% VEBT_internal.membermima.simps(1)
thf(fact_2907_exp__total,axiom,
! [Y: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ Y )
=> ? [X3: real] :
( ( exp @ real @ X3 )
= Y ) ) ).
% exp_total
thf(fact_2908_exp__gt__zero,axiom,
! [X: real] : ( ord_less @ real @ ( zero_zero @ real ) @ ( exp @ real @ X ) ) ).
% exp_gt_zero
thf(fact_2909_not__exp__less__zero,axiom,
! [X: real] :
~ ( ord_less @ real @ ( exp @ real @ X ) @ ( zero_zero @ real ) ) ).
% not_exp_less_zero
thf(fact_2910_not__exp__le__zero,axiom,
! [X: real] :
~ ( ord_less_eq @ real @ ( exp @ real @ X ) @ ( zero_zero @ real ) ) ).
% not_exp_le_zero
thf(fact_2911_exp__ge__zero,axiom,
! [X: real] : ( ord_less_eq @ real @ ( zero_zero @ real ) @ ( exp @ real @ X ) ) ).
% exp_ge_zero
thf(fact_2912_signed__take__bit__eq__iff__take__bit__eq,axiom,
! [A: $tType] :
( ( bit_ri3973907225187159222ations @ A )
=> ! [N: nat,A2: A,B2: A] :
( ( ( bit_ri4674362597316999326ke_bit @ A @ N @ A2 )
= ( bit_ri4674362597316999326ke_bit @ A @ N @ B2 ) )
= ( ( bit_se2584673776208193580ke_bit @ A @ ( suc @ N ) @ A2 )
= ( bit_se2584673776208193580ke_bit @ A @ ( suc @ N ) @ B2 ) ) ) ) ).
% signed_take_bit_eq_iff_take_bit_eq
thf(fact_2913_signed__take__bit__take__bit,axiom,
! [A: $tType] :
( ( bit_ri3973907225187159222ations @ A )
=> ! [M: nat,N: nat,A2: A] :
( ( bit_ri4674362597316999326ke_bit @ A @ M @ ( bit_se2584673776208193580ke_bit @ A @ N @ A2 ) )
= ( if @ ( A > A ) @ ( ord_less_eq @ nat @ N @ M ) @ ( bit_se2584673776208193580ke_bit @ A @ N ) @ ( bit_ri4674362597316999326ke_bit @ A @ M ) @ A2 ) ) ) ).
% signed_take_bit_take_bit
thf(fact_2914_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_2915_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_2916_take__bit__unset__bit__eq,axiom,
! [A: $tType] :
( ( bit_se359711467146920520ations @ A )
=> ! [N: nat,M: nat,A2: A] :
( ( ( ord_less_eq @ nat @ N @ M )
=> ( ( bit_se2584673776208193580ke_bit @ A @ N @ ( bit_se2638667681897837118et_bit @ A @ M @ A2 ) )
= ( bit_se2584673776208193580ke_bit @ A @ N @ A2 ) ) )
& ( ~ ( ord_less_eq @ nat @ N @ M )
=> ( ( bit_se2584673776208193580ke_bit @ A @ N @ ( bit_se2638667681897837118et_bit @ A @ M @ A2 ) )
= ( bit_se2638667681897837118et_bit @ A @ M @ ( bit_se2584673776208193580ke_bit @ A @ N @ A2 ) ) ) ) ) ) ).
% take_bit_unset_bit_eq
thf(fact_2917_take__bit__set__bit__eq,axiom,
! [A: $tType] :
( ( bit_se359711467146920520ations @ A )
=> ! [N: nat,M: nat,A2: A] :
( ( ( ord_less_eq @ nat @ N @ M )
=> ( ( bit_se2584673776208193580ke_bit @ A @ N @ ( bit_se5668285175392031749et_bit @ A @ M @ A2 ) )
= ( bit_se2584673776208193580ke_bit @ A @ N @ A2 ) ) )
& ( ~ ( ord_less_eq @ nat @ N @ M )
=> ( ( bit_se2584673776208193580ke_bit @ A @ N @ ( bit_se5668285175392031749et_bit @ A @ M @ A2 ) )
= ( bit_se5668285175392031749et_bit @ A @ M @ ( bit_se2584673776208193580ke_bit @ A @ N @ A2 ) ) ) ) ) ) ).
% take_bit_set_bit_eq
thf(fact_2918_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_2919_take__bit__flip__bit__eq,axiom,
! [A: $tType] :
( ( bit_se359711467146920520ations @ A )
=> ! [N: nat,M: nat,A2: A] :
( ( ( ord_less_eq @ nat @ N @ M )
=> ( ( bit_se2584673776208193580ke_bit @ A @ N @ ( bit_se8732182000553998342ip_bit @ A @ M @ A2 ) )
= ( bit_se2584673776208193580ke_bit @ A @ N @ A2 ) ) )
& ( ~ ( ord_less_eq @ nat @ N @ M )
=> ( ( bit_se2584673776208193580ke_bit @ A @ N @ ( bit_se8732182000553998342ip_bit @ A @ M @ A2 ) )
= ( bit_se8732182000553998342ip_bit @ A @ M @ ( bit_se2584673776208193580ke_bit @ A @ N @ A2 ) ) ) ) ) ) ).
% take_bit_flip_bit_eq
thf(fact_2920_vebt__buildup_Osimps_I1_J,axiom,
( ( vEBT_vebt_buildup @ ( zero_zero @ nat ) )
= ( vEBT_Leaf @ $false @ $false ) ) ).
% vebt_buildup.simps(1)
thf(fact_2921_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_2922_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_2923_take__bit__signed__take__bit,axiom,
! [A: $tType] :
( ( bit_ri3973907225187159222ations @ A )
=> ! [M: nat,N: nat,A2: A] :
( ( ord_less_eq @ nat @ M @ ( suc @ N ) )
=> ( ( bit_se2584673776208193580ke_bit @ A @ M @ ( bit_ri4674362597316999326ke_bit @ A @ N @ A2 ) )
= ( bit_se2584673776208193580ke_bit @ A @ M @ A2 ) ) ) ) ).
% take_bit_signed_take_bit
thf(fact_2924_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_2925_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_2926_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_2927_exp__of__nat2__mult,axiom,
! [A: $tType] :
( ( ( real_Vector_banach @ A )
& ( real_V3459762299906320749_field @ A ) )
=> ! [X: A,N: nat] :
( ( exp @ A @ ( times_times @ A @ X @ ( semiring_1_of_nat @ A @ N ) ) )
= ( power_power @ A @ ( exp @ A @ X ) @ N ) ) ) ).
% exp_of_nat2_mult
thf(fact_2928_exp__of__nat__mult,axiom,
! [A: $tType] :
( ( ( real_Vector_banach @ A )
& ( real_V3459762299906320749_field @ A ) )
=> ! [N: nat,X: A] :
( ( exp @ A @ ( times_times @ A @ ( semiring_1_of_nat @ A @ N ) @ X ) )
= ( power_power @ A @ ( exp @ A @ X ) @ N ) ) ) ).
% exp_of_nat_mult
thf(fact_2929_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_2930_log__ln,axiom,
( ( ln_ln @ real )
= ( log @ ( exp @ real @ ( one_one @ real ) ) ) ) ).
% log_ln
thf(fact_2931_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_2932_vebt__buildup_Osimps_I2_J,axiom,
( ( vEBT_vebt_buildup @ ( suc @ ( zero_zero @ nat ) ) )
= ( vEBT_Leaf @ $false @ $false ) ) ).
% vebt_buildup.simps(2)
thf(fact_2933_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_2934_T_092_060_094sub_062d_092_060_094sub_062e_092_060_094sub_062l_092_060_094sub_062e_092_060_094sub_062t_092_060_094sub_062e_Osimps_I3_J,axiom,
! [A2: $o,B2: $o,N: nat] :
( ( vEBT_T_d_e_l_e_t_e @ ( vEBT_Leaf @ A2 @ B2 ) @ ( suc @ ( suc @ N ) ) )
= ( one_one @ nat ) ) ).
% T\<^sub>d\<^sub>e\<^sub>l\<^sub>e\<^sub>t\<^sub>e.simps(3)
thf(fact_2935_VEBT__internal_OT_092_060_094sub_062d_092_060_094sub_062e_092_060_094sub_062l_092_060_094sub_062e_092_060_094sub_062t_092_060_094sub_062e_H_Osimps_I3_J,axiom,
! [A2: $o,B2: $o,N: nat] :
( ( vEBT_V1232361888498592333_e_t_e @ ( vEBT_Leaf @ A2 @ B2 ) @ ( suc @ ( suc @ N ) ) )
= ( one_one @ nat ) ) ).
% VEBT_internal.T\<^sub>d\<^sub>e\<^sub>l\<^sub>e\<^sub>t\<^sub>e'.simps(3)
thf(fact_2936_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_2937_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_2938_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_2939_T_092_060_094sub_062p_092_060_094sub_062r_092_060_094sub_062e_092_060_094sub_062d_H_Osimps_I3_J,axiom,
! [A2: $o,B2: $o,Va2: nat] :
( ( vEBT_T_p_r_e_d2 @ ( vEBT_Leaf @ A2 @ B2 ) @ ( suc @ ( suc @ Va2 ) ) )
= ( one_one @ nat ) ) ).
% T\<^sub>p\<^sub>r\<^sub>e\<^sub>d'.simps(3)
thf(fact_2940_T_092_060_094sub_062p_092_060_094sub_062r_092_060_094sub_062e_092_060_094sub_062d_Osimps_I1_J,axiom,
! [Uu: $o,Uv: $o] :
( ( vEBT_T_p_r_e_d @ ( vEBT_Leaf @ Uu @ Uv ) @ ( zero_zero @ nat ) )
= ( one_one @ nat ) ) ).
% T\<^sub>p\<^sub>r\<^sub>e\<^sub>d.simps(1)
thf(fact_2941_T_092_060_094sub_062p_092_060_094sub_062r_092_060_094sub_062e_092_060_094sub_062d_H_Osimps_I1_J,axiom,
! [Uu: $o,Uv: $o] :
( ( vEBT_T_p_r_e_d2 @ ( vEBT_Leaf @ Uu @ Uv ) @ ( zero_zero @ nat ) )
= ( one_one @ nat ) ) ).
% T\<^sub>p\<^sub>r\<^sub>e\<^sub>d'.simps(1)
thf(fact_2942_T_092_060_094sub_062s_092_060_094sub_062u_092_060_094sub_062c_092_060_094sub_062c_Osimps_I2_J,axiom,
! [Uv: $o,Uw: $o,N: nat] :
( ( vEBT_T_s_u_c_c @ ( vEBT_Leaf @ Uv @ Uw ) @ ( suc @ N ) )
= ( one_one @ nat ) ) ).
% T\<^sub>s\<^sub>u\<^sub>c\<^sub>c.simps(2)
thf(fact_2943_T_092_060_094sub_062s_092_060_094sub_062u_092_060_094sub_062c_092_060_094sub_062c_H_Osimps_I2_J,axiom,
! [Uv: $o,Uw: $o,N: nat] :
( ( vEBT_T_s_u_c_c2 @ ( vEBT_Leaf @ Uv @ Uw ) @ ( suc @ N ) )
= ( one_one @ nat ) ) ).
% T\<^sub>s\<^sub>u\<^sub>c\<^sub>c'.simps(2)
thf(fact_2944_T_092_060_094sub_062s_092_060_094sub_062u_092_060_094sub_062c_092_060_094sub_062c_H_Osimps_I1_J,axiom,
! [Uu: $o,B2: $o] :
( ( vEBT_T_s_u_c_c2 @ ( vEBT_Leaf @ Uu @ B2 ) @ ( zero_zero @ nat ) )
= ( one_one @ nat ) ) ).
% T\<^sub>s\<^sub>u\<^sub>c\<^sub>c'.simps(1)
thf(fact_2945_num_Osize_I4_J,axiom,
( ( size_size @ num @ one2 )
= ( zero_zero @ nat ) ) ).
% num.size(4)
thf(fact_2946_star__assoc,axiom,
! [A2: assn,B2: assn,C2: assn] :
( ( times_times @ assn @ ( times_times @ assn @ A2 @ B2 ) @ C2 )
= ( times_times @ assn @ A2 @ ( times_times @ assn @ B2 @ C2 ) ) ) ).
% star_assoc
thf(fact_2947_star__aci_I2_J,axiom,
( ( times_times @ assn )
= ( ^ [A4: assn,B5: assn] : ( times_times @ assn @ B5 @ A4 ) ) ) ).
% star_aci(2)
thf(fact_2948_star__aci_I3_J,axiom,
! [A2: assn,B2: assn,C2: assn] :
( ( times_times @ assn @ A2 @ ( times_times @ assn @ B2 @ C2 ) )
= ( times_times @ assn @ B2 @ ( times_times @ assn @ A2 @ C2 ) ) ) ).
% star_aci(3)
thf(fact_2949_assn__aci_I10_J,axiom,
! [A2: assn,B2: assn,C2: assn] :
( ( times_times @ assn @ ( times_times @ assn @ A2 @ B2 ) @ C2 )
= ( times_times @ assn @ ( times_times @ assn @ A2 @ C2 ) @ B2 ) ) ).
% assn_aci(10)
thf(fact_2950_is__entails,axiom,
! [P: assn,Q: assn] :
( ( entails @ P @ Q )
=> ( entails @ P @ Q ) ) ).
% is_entails
thf(fact_2951_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_2952_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_2953_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_2954_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_2955_powr__def,axiom,
! [A: $tType] :
( ( ln @ A )
=> ( ( powr @ A )
= ( ^ [X2: A,A4: A] :
( if @ A
@ ( X2
= ( zero_zero @ A ) )
@ ( zero_zero @ A )
@ ( exp @ A @ ( times_times @ A @ A4 @ ( ln_ln @ A @ X2 ) ) ) ) ) ) ) ).
% powr_def
thf(fact_2956_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_2957_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_2958_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_2959_T_092_060_094sub_062p_092_060_094sub_062r_092_060_094sub_062e_092_060_094sub_062d_H_Osimps_I2_J,axiom,
! [A2: $o,Uw: $o] :
( ( vEBT_T_p_r_e_d2 @ ( vEBT_Leaf @ A2 @ Uw ) @ ( suc @ ( zero_zero @ nat ) ) )
= ( one_one @ nat ) ) ).
% T\<^sub>p\<^sub>r\<^sub>e\<^sub>d'.simps(2)
thf(fact_2960_T_092_060_094sub_062p_092_060_094sub_062r_092_060_094sub_062e_092_060_094sub_062d_Osimps_I3_J,axiom,
! [A2: $o,B2: $o,Va2: nat] :
( ( vEBT_T_p_r_e_d @ ( vEBT_Leaf @ A2 @ B2 ) @ ( suc @ ( suc @ Va2 ) ) )
= ( plus_plus @ nat @ ( one_one @ nat ) @ ( if @ nat @ B2 @ ( one_one @ nat ) @ ( plus_plus @ nat @ ( one_one @ nat ) @ ( one_one @ nat ) ) ) ) ) ).
% T\<^sub>p\<^sub>r\<^sub>e\<^sub>d.simps(3)
thf(fact_2961_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_2962_T_092_060_094sub_062s_092_060_094sub_062u_092_060_094sub_062c_092_060_094sub_062c_Osimps_I1_J,axiom,
! [Uu: $o,B2: $o] :
( ( vEBT_T_s_u_c_c @ ( vEBT_Leaf @ Uu @ 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_2963_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_2964_take__bit__Suc__bit0,axiom,
! [A: $tType] :
( ( bit_un5681908812861735899ations @ A )
=> ! [N: nat,K: num] :
( ( bit_se2584673776208193580ke_bit @ A @ ( suc @ N ) @ ( numeral_numeral @ A @ ( bit0 @ K ) ) )
= ( times_times @ A @ ( bit_se2584673776208193580ke_bit @ A @ N @ ( numeral_numeral @ A @ K ) ) @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) ) ) ).
% take_bit_Suc_bit0
thf(fact_2965_take__bit__eq__mod,axiom,
! [A: $tType] :
( ( bit_se359711467146920520ations @ A )
=> ( ( bit_se2584673776208193580ke_bit @ A )
= ( ^ [N3: nat,A4: A] : ( modulo_modulo @ A @ A4 @ ( power_power @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ N3 ) ) ) ) ) ).
% take_bit_eq_mod
thf(fact_2966_exp__le,axiom,
ord_less_eq @ real @ ( exp @ real @ ( one_one @ real ) ) @ ( numeral_numeral @ real @ ( bit1 @ one2 ) ) ).
% exp_le
thf(fact_2967_exp__divide__power__eq,axiom,
! [A: $tType] :
( ( ( real_Vector_banach @ A )
& ( real_V3459762299906320749_field @ A ) )
=> ! [N: nat,X: A] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ N )
=> ( ( power_power @ A @ ( exp @ A @ ( divide_divide @ A @ X @ ( semiring_1_of_nat @ A @ N ) ) ) @ N )
= ( exp @ A @ X ) ) ) ) ).
% exp_divide_power_eq
thf(fact_2968_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_2969_T_092_060_094sub_062p_092_060_094sub_062r_092_060_094sub_062e_092_060_094sub_062d_Osimps_I2_J,axiom,
! [A2: $o,Uw: $o] :
( ( vEBT_T_p_r_e_d @ ( vEBT_Leaf @ A2 @ Uw ) @ ( suc @ ( zero_zero @ nat ) ) )
= ( plus_plus @ nat @ ( one_one @ nat ) @ ( one_one @ nat ) ) ) ).
% T\<^sub>p\<^sub>r\<^sub>e\<^sub>d.simps(2)
thf(fact_2970_take__bit__eq__0__iff,axiom,
! [A: $tType] :
( ( bit_se359711467146920520ations @ A )
=> ! [N: nat,A2: A] :
( ( ( bit_se2584673776208193580ke_bit @ A @ N @ A2 )
= ( zero_zero @ A ) )
= ( dvd_dvd @ A @ ( power_power @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ N ) @ A2 ) ) ) ).
% take_bit_eq_0_iff
thf(fact_2971_bin__last__bintrunc,axiom,
! [A: $tType] :
( ( bit_se359711467146920520ations @ A )
=> ! [L: nat,N: A] :
( ( ~ ( dvd_dvd @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ ( bit_se2584673776208193580ke_bit @ A @ L @ N ) ) )
= ( ( ord_less @ nat @ ( zero_zero @ nat ) @ L )
& ~ ( dvd_dvd @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ N ) ) ) ) ).
% bin_last_bintrunc
thf(fact_2972_take__bit__numeral__bit0,axiom,
! [A: $tType] :
( ( bit_un5681908812861735899ations @ A )
=> ! [L: num,K: num] :
( ( bit_se2584673776208193580ke_bit @ A @ ( numeral_numeral @ nat @ L ) @ ( numeral_numeral @ A @ ( bit0 @ K ) ) )
= ( times_times @ A @ ( bit_se2584673776208193580ke_bit @ A @ ( pred_numeral @ L ) @ ( numeral_numeral @ A @ K ) ) @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) ) ) ).
% take_bit_numeral_bit0
thf(fact_2973_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_2974_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_2975_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_2976_num_Osize_I6_J,axiom,
! [X32: num] :
( ( size_size @ num @ ( bit1 @ X32 ) )
= ( plus_plus @ nat @ ( size_size @ num @ X32 ) @ ( suc @ ( zero_zero @ nat ) ) ) ) ).
% num.size(6)
thf(fact_2977_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_2978_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_2979_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_2980_take__bit__Suc__bit1,axiom,
! [A: $tType] :
( ( bit_un5681908812861735899ations @ A )
=> ! [N: nat,K: num] :
( ( bit_se2584673776208193580ke_bit @ A @ ( suc @ N ) @ ( numeral_numeral @ A @ ( bit1 @ K ) ) )
= ( plus_plus @ A @ ( times_times @ A @ ( bit_se2584673776208193580ke_bit @ A @ N @ ( numeral_numeral @ A @ K ) ) @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) @ ( one_one @ A ) ) ) ) ).
% take_bit_Suc_bit1
thf(fact_2981_take__bit__Suc__minus__1__eq,axiom,
! [A: $tType] :
( ( bit_ri3973907225187159222ations @ A )
=> ! [N: nat] :
( ( bit_se2584673776208193580ke_bit @ A @ ( suc @ N ) @ ( uminus_uminus @ A @ ( one_one @ A ) ) )
= ( minus_minus @ A @ ( power_power @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ ( suc @ N ) ) @ ( one_one @ A ) ) ) ) ).
% take_bit_Suc_minus_1_eq
thf(fact_2982_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_2983_take__bit__Suc,axiom,
! [A: $tType] :
( ( bit_se359711467146920520ations @ A )
=> ! [N: nat,A2: A] :
( ( bit_se2584673776208193580ke_bit @ A @ ( suc @ N ) @ A2 )
= ( plus_plus @ A @ ( times_times @ A @ ( bit_se2584673776208193580ke_bit @ A @ N @ ( divide_divide @ A @ A2 @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) ) @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) @ ( modulo_modulo @ A @ A2 @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) ) ) ) ).
% take_bit_Suc
thf(fact_2984_ent__frame__fwd,axiom,
! [P: assn,R: assn,Ps: assn,F3: assn,Q: assn] :
( ( entails @ P @ R )
=> ( ( entails @ Ps @ ( times_times @ assn @ P @ F3 ) )
=> ( ( entails @ ( times_times @ assn @ R @ F3 ) @ Q )
=> ( entails @ Ps @ Q ) ) ) ) ).
% ent_frame_fwd
thf(fact_2985_fr__rot__rhs,axiom,
! [A3: assn,B4: assn,C6: assn] :
( ( entails @ A3 @ ( times_times @ assn @ B4 @ C6 ) )
=> ( entails @ A3 @ ( times_times @ assn @ C6 @ B4 ) ) ) ).
% fr_rot_rhs
thf(fact_2986_fr__refl,axiom,
! [A3: assn,B4: assn,C6: assn] :
( ( entails @ A3 @ B4 )
=> ( entails @ ( times_times @ assn @ A3 @ C6 ) @ ( times_times @ assn @ B4 @ C6 ) ) ) ).
% fr_refl
thf(fact_2987_fr__rot,axiom,
! [A3: assn,B4: assn,C6: assn] :
( ( entails @ ( times_times @ assn @ A3 @ B4 ) @ C6 )
=> ( entails @ ( times_times @ assn @ B4 @ A3 ) @ C6 ) ) ).
% fr_rot
thf(fact_2988_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_2989_norm__assertion__simps_I1_J,axiom,
! [A2: assn] :
( ( times_times @ assn @ ( one_one @ assn ) @ A2 )
= A2 ) ).
% norm_assertion_simps(1)
thf(fact_2990_norm__assertion__simps_I2_J,axiom,
! [A2: assn] :
( ( times_times @ assn @ A2 @ ( one_one @ assn ) )
= A2 ) ).
% norm_assertion_simps(2)
thf(fact_2991_stable__imp__take__bit__eq,axiom,
! [A: $tType] :
( ( bit_se359711467146920520ations @ A )
=> ! [A2: A,N: nat] :
( ( ( divide_divide @ A @ A2 @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) )
= A2 )
=> ( ( ( dvd_dvd @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ A2 )
=> ( ( bit_se2584673776208193580ke_bit @ A @ N @ A2 )
= ( zero_zero @ A ) ) )
& ( ~ ( dvd_dvd @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ A2 )
=> ( ( bit_se2584673776208193580ke_bit @ A @ N @ A2 )
= ( minus_minus @ A @ ( power_power @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ N ) @ ( one_one @ A ) ) ) ) ) ) ) ).
% stable_imp_take_bit_eq
thf(fact_2992_take__bit__numeral__bit1,axiom,
! [A: $tType] :
( ( bit_un5681908812861735899ations @ A )
=> ! [L: num,K: num] :
( ( bit_se2584673776208193580ke_bit @ A @ ( numeral_numeral @ nat @ L ) @ ( numeral_numeral @ A @ ( bit1 @ K ) ) )
= ( plus_plus @ A @ ( times_times @ A @ ( bit_se2584673776208193580ke_bit @ A @ ( pred_numeral @ L ) @ ( numeral_numeral @ A @ K ) ) @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) @ ( one_one @ A ) ) ) ) ).
% take_bit_numeral_bit1
thf(fact_2993_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_2994_exp__ge__one__plus__x__over__n__power__n,axiom,
! [N: nat,X: real] :
( ( ord_less_eq @ real @ ( uminus_uminus @ real @ ( semiring_1_of_nat @ real @ N ) ) @ X )
=> ( ( ord_less @ nat @ ( zero_zero @ nat ) @ N )
=> ( ord_less_eq @ real @ ( power_power @ real @ ( plus_plus @ real @ ( one_one @ real ) @ ( divide_divide @ real @ X @ ( semiring_1_of_nat @ real @ N ) ) ) @ N ) @ ( exp @ real @ X ) ) ) ) ).
% exp_ge_one_plus_x_over_n_power_n
thf(fact_2995_exp__ge__one__minus__x__over__n__power__n,axiom,
! [X: real,N: nat] :
( ( ord_less_eq @ real @ X @ ( semiring_1_of_nat @ real @ N ) )
=> ( ( ord_less @ nat @ ( zero_zero @ nat ) @ N )
=> ( ord_less_eq @ real @ ( power_power @ real @ ( minus_minus @ real @ ( one_one @ real ) @ ( divide_divide @ real @ X @ ( semiring_1_of_nat @ real @ N ) ) ) @ N ) @ ( exp @ real @ ( uminus_uminus @ real @ X ) ) ) ) ) ).
% exp_ge_one_minus_x_over_n_power_n
thf(fact_2996_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_2997_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_2998_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_2999_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_3000_take__bit__rec,axiom,
! [A: $tType] :
( ( bit_se359711467146920520ations @ A )
=> ( ( bit_se2584673776208193580ke_bit @ A )
= ( ^ [N3: nat,A4: A] :
( if @ A
@ ( N3
= ( zero_zero @ nat ) )
@ ( zero_zero @ A )
@ ( plus_plus @ A @ ( times_times @ A @ ( bit_se2584673776208193580ke_bit @ A @ ( minus_minus @ nat @ N3 @ ( one_one @ nat ) ) @ ( divide_divide @ A @ A4 @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) ) @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) @ ( modulo_modulo @ A @ A4 @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) ) ) ) ) ) ).
% take_bit_rec
thf(fact_3001_take__bit__word__minus__Bit1__eq,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [N: num,M: num] :
( ( bit_se2584673776208193580ke_bit @ ( word @ A ) @ ( numeral_numeral @ nat @ N ) @ ( uminus_uminus @ ( word @ A ) @ ( numeral_numeral @ ( word @ A ) @ ( bit1 @ M ) ) ) )
= ( plus_plus @ ( word @ A ) @ ( one_one @ ( word @ A ) ) @ ( times_times @ ( word @ A ) @ ( numeral_numeral @ ( word @ A ) @ ( bit0 @ one2 ) ) @ ( bit_se2584673776208193580ke_bit @ ( word @ A ) @ ( pred_numeral @ N ) @ ( uminus_uminus @ ( word @ A ) @ ( numeral_numeral @ ( word @ A ) @ ( inc @ M ) ) ) ) ) ) ) ) ).
% take_bit_word_minus_Bit1_eq
thf(fact_3002_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_3003_option_Osize_I3_J,axiom,
! [A: $tType] :
( ( size_size @ ( option @ A ) @ ( none @ A ) )
= ( suc @ ( zero_zero @ nat ) ) ) ).
% option.size(3)
thf(fact_3004_uint32_Osize__eq,axiom,
( ( size_size @ uint32 )
= ( ^ [P5: uint32] : ( numeral_numeral @ nat @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ one2 ) ) ) ) ) ) ) ) ).
% uint32.size_eq
thf(fact_3005_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_3006_pred__numeral__inc,axiom,
! [K: num] :
( ( pred_numeral @ ( inc @ K ) )
= ( numeral_numeral @ nat @ K ) ) ).
% pred_numeral_inc
thf(fact_3007_slice__complete,axiom,
! [A: $tType,Xs2: list @ A] :
( ( slice @ A @ ( zero_zero @ nat ) @ ( size_size @ ( list @ A ) @ Xs2 ) @ Xs2 )
= Xs2 ) ).
% slice_complete
thf(fact_3008_add__neg__numeral__special_I5_J,axiom,
! [A: $tType] :
( ( neg_numeral @ A )
=> ! [N: num] :
( ( plus_plus @ A @ ( uminus_uminus @ A @ ( one_one @ A ) ) @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ N ) ) )
= ( uminus_uminus @ A @ ( numeral_numeral @ A @ ( inc @ N ) ) ) ) ) ).
% add_neg_numeral_special(5)
thf(fact_3009_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_3010_diff__numeral__special_I5_J,axiom,
! [A: $tType] :
( ( neg_numeral @ A )
=> ! [N: num] :
( ( minus_minus @ A @ ( uminus_uminus @ A @ ( one_one @ A ) ) @ ( numeral_numeral @ A @ N ) )
= ( uminus_uminus @ A @ ( numeral_numeral @ A @ ( inc @ N ) ) ) ) ) ).
% diff_numeral_special(5)
thf(fact_3011_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_3012_take__bit__of__Suc__0,axiom,
! [N: nat] :
( ( bit_se2584673776208193580ke_bit @ nat @ N @ ( suc @ ( zero_zero @ nat ) ) )
= ( zero_neq_one_of_bool @ nat @ ( ord_less @ nat @ ( zero_zero @ nat ) @ N ) ) ) ).
% take_bit_of_Suc_0
thf(fact_3013_take__bit__tightened__less__eq__nat,axiom,
! [M: nat,N: nat,Q3: nat] :
( ( ord_less_eq @ nat @ M @ N )
=> ( ord_less_eq @ nat @ ( bit_se2584673776208193580ke_bit @ nat @ M @ Q3 ) @ ( bit_se2584673776208193580ke_bit @ nat @ N @ Q3 ) ) ) ).
% take_bit_tightened_less_eq_nat
thf(fact_3014_take__bit__nat__less__eq__self,axiom,
! [N: nat,M: nat] : ( ord_less_eq @ nat @ ( bit_se2584673776208193580ke_bit @ nat @ N @ M ) @ M ) ).
% take_bit_nat_less_eq_self
thf(fact_3015_take__bit__minus,axiom,
! [N: nat,K: int] :
( ( bit_se2584673776208193580ke_bit @ int @ N @ ( uminus_uminus @ int @ ( bit_se2584673776208193580ke_bit @ int @ N @ K ) ) )
= ( bit_se2584673776208193580ke_bit @ int @ N @ ( uminus_uminus @ int @ K ) ) ) ).
% take_bit_minus
thf(fact_3016_take__bit__mult,axiom,
! [N: nat,K: int,L: int] :
( ( bit_se2584673776208193580ke_bit @ int @ N @ ( times_times @ int @ ( bit_se2584673776208193580ke_bit @ int @ N @ K ) @ ( bit_se2584673776208193580ke_bit @ int @ N @ L ) ) )
= ( bit_se2584673776208193580ke_bit @ int @ N @ ( times_times @ int @ K @ L ) ) ) ).
% take_bit_mult
thf(fact_3017_take__bit__diff,axiom,
! [N: nat,K: int,L: int] :
( ( bit_se2584673776208193580ke_bit @ int @ N @ ( minus_minus @ int @ ( bit_se2584673776208193580ke_bit @ int @ N @ K ) @ ( bit_se2584673776208193580ke_bit @ int @ N @ L ) ) )
= ( bit_se2584673776208193580ke_bit @ int @ N @ ( minus_minus @ int @ K @ L ) ) ) ).
% take_bit_diff
thf(fact_3018_num__induct,axiom,
! [P: num > $o,X: num] :
( ( P @ one2 )
=> ( ! [X3: num] :
( ( P @ X3 )
=> ( P @ ( inc @ X3 ) ) )
=> ( P @ X ) ) ) ).
% num_induct
thf(fact_3019_add__inc,axiom,
! [X: num,Y: num] :
( ( plus_plus @ num @ X @ ( inc @ Y ) )
= ( inc @ ( plus_plus @ num @ X @ Y ) ) ) ).
% add_inc
thf(fact_3020_take__bit__tightened__less__eq__int,axiom,
! [M: nat,N: nat,K: int] :
( ( ord_less_eq @ nat @ M @ N )
=> ( ord_less_eq @ int @ ( bit_se2584673776208193580ke_bit @ int @ M @ K ) @ ( bit_se2584673776208193580ke_bit @ int @ N @ K ) ) ) ).
% take_bit_tightened_less_eq_int
thf(fact_3021_take__bit__nonnegative,axiom,
! [N: nat,K: int] : ( ord_less_eq @ int @ ( zero_zero @ int ) @ ( bit_se2584673776208193580ke_bit @ int @ N @ K ) ) ).
% take_bit_nonnegative
thf(fact_3022_take__bit__int__less__eq__self__iff,axiom,
! [N: nat,K: int] :
( ( ord_less_eq @ int @ ( bit_se2584673776208193580ke_bit @ int @ N @ K ) @ K )
= ( ord_less_eq @ int @ ( zero_zero @ int ) @ K ) ) ).
% take_bit_int_less_eq_self_iff
thf(fact_3023_not__take__bit__negative,axiom,
! [N: nat,K: int] :
~ ( ord_less @ int @ ( bit_se2584673776208193580ke_bit @ int @ N @ K ) @ ( zero_zero @ int ) ) ).
% not_take_bit_negative
thf(fact_3024_take__bit__int__greater__self__iff,axiom,
! [K: int,N: nat] :
( ( ord_less @ int @ K @ ( bit_se2584673776208193580ke_bit @ int @ N @ K ) )
= ( ord_less @ int @ K @ ( zero_zero @ int ) ) ) ).
% take_bit_int_greater_self_iff
thf(fact_3025_inc_Osimps_I1_J,axiom,
( ( inc @ one2 )
= ( bit0 @ one2 ) ) ).
% inc.simps(1)
thf(fact_3026_inc_Osimps_I3_J,axiom,
! [X: num] :
( ( inc @ ( bit1 @ X ) )
= ( bit0 @ ( inc @ X ) ) ) ).
% inc.simps(3)
thf(fact_3027_inc_Osimps_I2_J,axiom,
! [X: num] :
( ( inc @ ( bit0 @ X ) )
= ( bit1 @ X ) ) ).
% inc.simps(2)
thf(fact_3028_add__One,axiom,
! [X: num] :
( ( plus_plus @ num @ X @ one2 )
= ( inc @ X ) ) ).
% add_One
thf(fact_3029_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_3030_take__bit__decr__eq,axiom,
! [N: nat,K: int] :
( ( ( bit_se2584673776208193580ke_bit @ int @ N @ K )
!= ( zero_zero @ int ) )
=> ( ( bit_se2584673776208193580ke_bit @ int @ N @ ( minus_minus @ int @ K @ ( one_one @ int ) ) )
= ( minus_minus @ int @ ( bit_se2584673776208193580ke_bit @ int @ N @ K ) @ ( one_one @ int ) ) ) ) ).
% take_bit_decr_eq
thf(fact_3031_option_Osize__neq,axiom,
! [A: $tType,X: option @ A] :
( ( size_size @ ( option @ A ) @ X )
!= ( zero_zero @ nat ) ) ).
% option.size_neq
thf(fact_3032_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_3033_take__bit__nat__eq__self,axiom,
! [M: nat,N: nat] :
( ( ord_less @ nat @ M @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N ) )
=> ( ( bit_se2584673776208193580ke_bit @ nat @ N @ M )
= M ) ) ).
% take_bit_nat_eq_self
thf(fact_3034_take__bit__nat__less__exp,axiom,
! [N: nat,M: nat] : ( ord_less @ nat @ ( bit_se2584673776208193580ke_bit @ nat @ N @ M ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N ) ) ).
% take_bit_nat_less_exp
thf(fact_3035_take__bit__nat__eq__self__iff,axiom,
! [N: nat,M: nat] :
( ( ( bit_se2584673776208193580ke_bit @ nat @ N @ M )
= M )
= ( ord_less @ nat @ M @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N ) ) ) ).
% take_bit_nat_eq_self_iff
thf(fact_3036_take__bit__nat__def,axiom,
( ( bit_se2584673776208193580ke_bit @ nat )
= ( ^ [N3: nat,M4: nat] : ( modulo_modulo @ nat @ M4 @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N3 ) ) ) ) ).
% take_bit_nat_def
thf(fact_3037_take__bit__Suc__minus__bit1,axiom,
! [N: nat,K: num] :
( ( bit_se2584673776208193580ke_bit @ int @ ( suc @ N ) @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ ( bit1 @ K ) ) ) )
= ( plus_plus @ int @ ( times_times @ int @ ( bit_se2584673776208193580ke_bit @ int @ N @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ ( inc @ K ) ) ) ) @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) ) @ ( one_one @ int ) ) ) ).
% take_bit_Suc_minus_bit1
thf(fact_3038_take__bit__int__less__exp,axiom,
! [N: nat,K: int] : ( ord_less @ int @ ( bit_se2584673776208193580ke_bit @ int @ N @ K ) @ ( power_power @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ N ) ) ).
% take_bit_int_less_exp
thf(fact_3039_take__bit__int__def,axiom,
( ( bit_se2584673776208193580ke_bit @ int )
= ( ^ [N3: nat,K3: int] : ( modulo_modulo @ int @ K3 @ ( power_power @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ N3 ) ) ) ) ).
% take_bit_int_def
thf(fact_3040_take__bit__numeral__minus__bit1,axiom,
! [L: num,K: num] :
( ( bit_se2584673776208193580ke_bit @ int @ ( numeral_numeral @ nat @ L ) @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ ( bit1 @ K ) ) ) )
= ( plus_plus @ int @ ( times_times @ int @ ( bit_se2584673776208193580ke_bit @ int @ ( pred_numeral @ L ) @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ ( inc @ K ) ) ) ) @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) ) @ ( one_one @ int ) ) ) ).
% take_bit_numeral_minus_bit1
thf(fact_3041_take__bit__nat__less__self__iff,axiom,
! [N: nat,M: nat] :
( ( ord_less @ nat @ ( bit_se2584673776208193580ke_bit @ nat @ N @ M ) @ M )
= ( ord_less_eq @ nat @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N ) @ M ) ) ).
% take_bit_nat_less_self_iff
thf(fact_3042_take__bit__Suc__minus__bit0,axiom,
! [N: nat,K: num] :
( ( bit_se2584673776208193580ke_bit @ int @ ( suc @ N ) @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ ( bit0 @ K ) ) ) )
= ( times_times @ int @ ( bit_se2584673776208193580ke_bit @ int @ N @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ K ) ) ) @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) ) ) ).
% take_bit_Suc_minus_bit0
thf(fact_3043_take__bit__int__less__self__iff,axiom,
! [N: nat,K: int] :
( ( ord_less @ int @ ( bit_se2584673776208193580ke_bit @ int @ N @ K ) @ K )
= ( ord_less_eq @ int @ ( power_power @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ N ) @ K ) ) ).
% take_bit_int_less_self_iff
thf(fact_3044_take__bit__int__greater__eq__self__iff,axiom,
! [K: int,N: nat] :
( ( ord_less_eq @ int @ K @ ( bit_se2584673776208193580ke_bit @ int @ N @ K ) )
= ( ord_less @ int @ K @ ( power_power @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ N ) ) ) ).
% take_bit_int_greater_eq_self_iff
thf(fact_3045_take__bit__int__eq__self__iff,axiom,
! [N: nat,K: int] :
( ( ( bit_se2584673776208193580ke_bit @ int @ N @ K )
= K )
= ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ K )
& ( ord_less @ int @ K @ ( power_power @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ N ) ) ) ) ).
% take_bit_int_eq_self_iff
thf(fact_3046_take__bit__int__eq__self,axiom,
! [K: int,N: nat] :
( ( ord_less_eq @ int @ ( zero_zero @ int ) @ K )
=> ( ( ord_less @ int @ K @ ( power_power @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ N ) )
=> ( ( bit_se2584673776208193580ke_bit @ int @ N @ K )
= K ) ) ) ).
% take_bit_int_eq_self
thf(fact_3047_take__bit__incr__eq,axiom,
! [N: nat,K: int] :
( ( ( bit_se2584673776208193580ke_bit @ int @ N @ K )
!= ( minus_minus @ int @ ( power_power @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ N ) @ ( one_one @ int ) ) )
=> ( ( bit_se2584673776208193580ke_bit @ int @ N @ ( plus_plus @ int @ K @ ( one_one @ int ) ) )
= ( plus_plus @ int @ ( one_one @ int ) @ ( bit_se2584673776208193580ke_bit @ int @ N @ K ) ) ) ) ).
% take_bit_incr_eq
thf(fact_3048_take__bit__numeral__minus__bit0,axiom,
! [L: num,K: num] :
( ( bit_se2584673776208193580ke_bit @ int @ ( numeral_numeral @ nat @ L ) @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ ( bit0 @ K ) ) ) )
= ( times_times @ int @ ( bit_se2584673776208193580ke_bit @ int @ ( pred_numeral @ L ) @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ K ) ) ) @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) ) ) ).
% take_bit_numeral_minus_bit0
thf(fact_3049_take__bit__int__less__eq,axiom,
! [N: nat,K: int] :
( ( ord_less_eq @ int @ ( power_power @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ N ) @ K )
=> ( ( ord_less @ nat @ ( zero_zero @ nat ) @ N )
=> ( ord_less_eq @ int @ ( bit_se2584673776208193580ke_bit @ int @ N @ K ) @ ( minus_minus @ int @ K @ ( power_power @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ N ) ) ) ) ) ).
% take_bit_int_less_eq
thf(fact_3050_take__bit__int__greater__eq,axiom,
! [K: int,N: nat] :
( ( ord_less @ int @ K @ ( zero_zero @ int ) )
=> ( ord_less_eq @ int @ ( plus_plus @ int @ K @ ( power_power @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ N ) ) @ ( bit_se2584673776208193580ke_bit @ int @ N @ K ) ) ) ).
% take_bit_int_greater_eq
thf(fact_3051_signed__take__bit__eq__take__bit__shift,axiom,
( ( bit_ri4674362597316999326ke_bit @ int )
= ( ^ [N3: nat,K3: int] : ( minus_minus @ int @ ( bit_se2584673776208193580ke_bit @ int @ ( suc @ N3 ) @ ( plus_plus @ int @ K3 @ ( power_power @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ N3 ) ) ) @ ( power_power @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ N3 ) ) ) ) ).
% signed_take_bit_eq_take_bit_shift
thf(fact_3052_take__bit__minus__small__eq,axiom,
! [K: int,N: nat] :
( ( ord_less @ int @ ( zero_zero @ int ) @ K )
=> ( ( ord_less_eq @ int @ K @ ( power_power @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ N ) )
=> ( ( bit_se2584673776208193580ke_bit @ int @ N @ ( uminus_uminus @ int @ K ) )
= ( minus_minus @ int @ ( power_power @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ N ) @ K ) ) ) ) ).
% take_bit_minus_small_eq
thf(fact_3053_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_3054_word__2p__lem,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [N: nat,W: word @ A] :
( ( ord_less @ nat @ N @ ( size_size @ ( word @ A ) @ W ) )
=> ( ( ord_less @ ( word @ A ) @ W @ ( power_power @ ( word @ A ) @ ( numeral_numeral @ ( word @ A ) @ ( bit0 @ one2 ) ) @ N ) )
= ( ord_less @ int @ ( semiring_1_unsigned @ A @ int @ W ) @ ( power_power @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ N ) ) ) ) ) ).
% word_2p_lem
thf(fact_3055_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_3056_fact__double,axiom,
! [A: $tType] :
( ( field_char_0 @ A )
=> ! [N: nat] :
( ( semiring_char_0_fact @ A @ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N ) )
= ( times_times @ A @ ( times_times @ A @ ( power_power @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N ) ) @ ( comm_s3205402744901411588hammer @ A @ ( divide_divide @ A @ ( one_one @ A ) @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) @ N ) ) @ ( semiring_char_0_fact @ A @ N ) ) ) ) ).
% fact_double
thf(fact_3057_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_3058_set__decode__inverse,axiom,
! [N: nat] :
( ( nat_set_encode @ ( nat_set_decode @ N ) )
= N ) ).
% set_decode_inverse
thf(fact_3059_unsigned__0,axiom,
! [B: $tType,A: $tType] :
( ( ( semiring_1 @ A )
& ( type_len @ B ) )
=> ( ( semiring_1_unsigned @ B @ A @ ( zero_zero @ ( word @ B ) ) )
= ( zero_zero @ A ) ) ) ).
% unsigned_0
thf(fact_3060_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_3061_fact__0,axiom,
! [A: $tType] :
( ( semiring_char_0 @ A )
=> ( ( semiring_char_0_fact @ A @ ( zero_zero @ nat ) )
= ( one_one @ A ) ) ) ).
% fact_0
thf(fact_3062_nat__numeral,axiom,
! [K: num] :
( ( nat2 @ ( numeral_numeral @ int @ K ) )
= ( numeral_numeral @ nat @ K ) ) ).
% nat_numeral
thf(fact_3063_fact__1,axiom,
! [A: $tType] :
( ( semiring_char_0 @ A )
=> ( ( semiring_char_0_fact @ A @ ( one_one @ nat ) )
= ( one_one @ A ) ) ) ).
% fact_1
thf(fact_3064_signed__0,axiom,
! [B: $tType,A: $tType] :
( ( ( ring_1 @ A )
& ( type_len @ B ) )
=> ( ( ring_1_signed @ B @ A @ ( zero_zero @ ( word @ B ) ) )
= ( zero_zero @ A ) ) ) ).
% signed_0
thf(fact_3065_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_3066_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_3067_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_3068_uint__lt__0,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [X: word @ A] :
~ ( ord_less @ int @ ( semiring_1_unsigned @ A @ int @ X ) @ ( zero_zero @ int ) ) ) ).
% uint_lt_0
thf(fact_3069_More__Word_Osint__0,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [X: word @ A] :
( ( ( ring_1_signed @ A @ int @ X )
= ( zero_zero @ int ) )
= ( X
= ( zero_zero @ ( word @ A ) ) ) ) ) ).
% More_Word.sint_0
thf(fact_3070_set__encode__inverse,axiom,
! [A3: set @ nat] :
( ( finite_finite2 @ nat @ A3 )
=> ( ( nat_set_decode @ ( nat_set_encode @ A3 ) )
= A3 ) ) ).
% set_encode_inverse
thf(fact_3071_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_3072_fact__Suc,axiom,
! [A: $tType] :
( ( semiring_char_0 @ A )
=> ! [N: nat] :
( ( semiring_char_0_fact @ A @ ( suc @ N ) )
= ( times_times @ A @ ( semiring_1_of_nat @ A @ ( suc @ N ) ) @ ( semiring_char_0_fact @ A @ N ) ) ) ) ).
% fact_Suc
thf(fact_3073_nat__1,axiom,
( ( nat2 @ ( one_one @ int ) )
= ( suc @ ( zero_zero @ nat ) ) ) ).
% nat_1
thf(fact_3074_nat__le__0,axiom,
! [Z: int] :
( ( ord_less_eq @ int @ Z @ ( zero_zero @ int ) )
=> ( ( nat2 @ Z )
= ( zero_zero @ nat ) ) ) ).
% nat_le_0
thf(fact_3075_nat__0__iff,axiom,
! [I: int] :
( ( ( nat2 @ I )
= ( zero_zero @ nat ) )
= ( ord_less_eq @ int @ I @ ( zero_zero @ int ) ) ) ).
% nat_0_iff
thf(fact_3076_nat__neg__numeral,axiom,
! [K: num] :
( ( nat2 @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ K ) ) )
= ( zero_zero @ nat ) ) ).
% nat_neg_numeral
thf(fact_3077_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_3078_of__nat__nat__take__bit__eq,axiom,
! [A: $tType] :
( ( ring_1 @ A )
=> ! [N: nat,K: int] :
( ( semiring_1_of_nat @ A @ ( nat2 @ ( bit_se2584673776208193580ke_bit @ int @ N @ K ) ) )
= ( ring_1_of_int @ A @ ( bit_se2584673776208193580ke_bit @ int @ N @ K ) ) ) ) ).
% of_nat_nat_take_bit_eq
thf(fact_3079_nat__zminus__int,axiom,
! [N: nat] :
( ( nat2 @ ( uminus_uminus @ int @ ( semiring_1_of_nat @ int @ N ) ) )
= ( zero_zero @ nat ) ) ).
% nat_zminus_int
thf(fact_3080_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_3081_word__less__no,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [A2: num,B2: num] :
( ( ord_less @ ( word @ A ) @ ( numeral_numeral @ ( word @ A ) @ A2 ) @ ( numeral_numeral @ ( word @ A ) @ B2 ) )
= ( ord_less @ int @ ( semiring_1_unsigned @ A @ int @ ( numeral_numeral @ ( word @ A ) @ A2 ) ) @ ( semiring_1_unsigned @ A @ int @ ( numeral_numeral @ ( word @ A ) @ B2 ) ) ) ) ) ).
% word_less_no
thf(fact_3082_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_3083_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_3084_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_3085_diff__nat__numeral,axiom,
! [V: num,V3: num] :
( ( minus_minus @ nat @ ( numeral_numeral @ nat @ V ) @ ( numeral_numeral @ nat @ V3 ) )
= ( nat2 @ ( minus_minus @ int @ ( numeral_numeral @ int @ V ) @ ( numeral_numeral @ int @ V3 ) ) ) ) ).
% diff_nat_numeral
thf(fact_3086_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_3087_numeral__power__eq__nat__cancel__iff,axiom,
! [X: num,N: nat,Y: int] :
( ( ( power_power @ nat @ ( numeral_numeral @ nat @ X ) @ N )
= ( nat2 @ Y ) )
= ( ( power_power @ int @ ( numeral_numeral @ int @ X ) @ N )
= Y ) ) ).
% numeral_power_eq_nat_cancel_iff
thf(fact_3088_nat__eq__numeral__power__cancel__iff,axiom,
! [Y: int,X: num,N: nat] :
( ( ( nat2 @ Y )
= ( power_power @ nat @ ( numeral_numeral @ nat @ X ) @ N ) )
= ( Y
= ( power_power @ int @ ( numeral_numeral @ int @ X ) @ N ) ) ) ).
% nat_eq_numeral_power_cancel_iff
thf(fact_3089_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_3090_dvd__nat__abs__iff,axiom,
! [N: nat,K: int] :
( ( dvd_dvd @ nat @ N @ ( nat2 @ ( abs_abs @ int @ K ) ) )
= ( dvd_dvd @ int @ ( semiring_1_of_nat @ int @ N ) @ K ) ) ).
% dvd_nat_abs_iff
thf(fact_3091_nat__abs__dvd__iff,axiom,
! [K: int,N: nat] :
( ( dvd_dvd @ nat @ ( nat2 @ ( abs_abs @ int @ K ) ) @ N )
= ( dvd_dvd @ int @ K @ ( semiring_1_of_nat @ int @ N ) ) ) ).
% nat_abs_dvd_iff
thf(fact_3092_word__div__no,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [A2: num,B2: num] :
( ( divide_divide @ ( word @ A ) @ ( numeral_numeral @ ( word @ A ) @ A2 ) @ ( numeral_numeral @ ( word @ A ) @ B2 ) )
= ( ring_1_of_int @ ( word @ A ) @ ( divide_divide @ int @ ( semiring_1_unsigned @ A @ int @ ( numeral_numeral @ ( word @ A ) @ A2 ) ) @ ( semiring_1_unsigned @ A @ int @ ( numeral_numeral @ ( word @ A ) @ B2 ) ) ) ) ) ) ).
% word_div_no
thf(fact_3093_word__mod__no,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [A2: num,B2: num] :
( ( modulo_modulo @ ( word @ A ) @ ( numeral_numeral @ ( word @ A ) @ A2 ) @ ( numeral_numeral @ ( word @ A ) @ B2 ) )
= ( ring_1_of_int @ ( word @ A ) @ ( modulo_modulo @ int @ ( semiring_1_unsigned @ A @ int @ ( numeral_numeral @ ( word @ A ) @ A2 ) ) @ ( semiring_1_unsigned @ A @ int @ ( numeral_numeral @ ( word @ A ) @ B2 ) ) ) ) ) ) ).
% word_mod_no
thf(fact_3094_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_3095_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_3096_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_3097_numeral__power__less__nat__cancel__iff,axiom,
! [X: num,N: nat,A2: int] :
( ( ord_less @ nat @ ( power_power @ nat @ ( numeral_numeral @ nat @ X ) @ N ) @ ( nat2 @ A2 ) )
= ( ord_less @ int @ ( power_power @ int @ ( numeral_numeral @ int @ X ) @ N ) @ A2 ) ) ).
% numeral_power_less_nat_cancel_iff
thf(fact_3098_nat__less__numeral__power__cancel__iff,axiom,
! [A2: int,X: num,N: nat] :
( ( ord_less @ nat @ ( nat2 @ A2 ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ X ) @ N ) )
= ( ord_less @ int @ A2 @ ( power_power @ int @ ( numeral_numeral @ int @ X ) @ N ) ) ) ).
% nat_less_numeral_power_cancel_iff
thf(fact_3099_numeral__power__le__nat__cancel__iff,axiom,
! [X: num,N: nat,A2: int] :
( ( ord_less_eq @ nat @ ( power_power @ nat @ ( numeral_numeral @ nat @ X ) @ N ) @ ( nat2 @ A2 ) )
= ( ord_less_eq @ int @ ( power_power @ int @ ( numeral_numeral @ int @ X ) @ N ) @ A2 ) ) ).
% numeral_power_le_nat_cancel_iff
thf(fact_3100_nat__le__numeral__power__cancel__iff,axiom,
! [A2: int,X: num,N: nat] :
( ( ord_less_eq @ nat @ ( nat2 @ A2 ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ X ) @ N ) )
= ( ord_less_eq @ int @ A2 @ ( power_power @ int @ ( numeral_numeral @ int @ X ) @ N ) ) ) ).
% nat_le_numeral_power_cancel_iff
thf(fact_3101_fact__nonzero,axiom,
! [A: $tType] :
( ( ( semiring_char_0 @ A )
& ( semiri3467727345109120633visors @ A ) )
=> ! [N: nat] :
( ( semiring_char_0_fact @ A @ N )
!= ( zero_zero @ A ) ) ) ).
% fact_nonzero
thf(fact_3102_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_3103_fact__less__mono__nat,axiom,
! [M: nat,N: nat] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ M )
=> ( ( ord_less @ nat @ M @ N )
=> ( ord_less @ nat @ ( semiring_char_0_fact @ nat @ M ) @ ( semiring_char_0_fact @ nat @ N ) ) ) ) ).
% fact_less_mono_nat
thf(fact_3104_nat__numeral__as__int,axiom,
( ( numeral_numeral @ nat )
= ( ^ [I2: num] : ( nat2 @ ( numeral_numeral @ int @ I2 ) ) ) ) ).
% nat_numeral_as_int
thf(fact_3105_nat__zero__as__int,axiom,
( ( zero_zero @ nat )
= ( nat2 @ ( zero_zero @ int ) ) ) ).
% nat_zero_as_int
thf(fact_3106_fact__ge__zero,axiom,
! [A: $tType] :
( ( linordered_semidom @ A )
=> ! [N: nat] : ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( semiring_char_0_fact @ A @ N ) ) ) ).
% fact_ge_zero
thf(fact_3107_fact__gt__zero,axiom,
! [A: $tType] :
( ( linordered_semidom @ A )
=> ! [N: nat] : ( ord_less @ A @ ( zero_zero @ A ) @ ( semiring_char_0_fact @ A @ N ) ) ) ).
% fact_gt_zero
thf(fact_3108_fact__not__neg,axiom,
! [A: $tType] :
( ( linordered_semidom @ A )
=> ! [N: nat] :
~ ( ord_less @ A @ ( semiring_char_0_fact @ A @ N ) @ ( zero_zero @ A ) ) ) ).
% fact_not_neg
thf(fact_3109_fact__ge__1,axiom,
! [A: $tType] :
( ( linordered_semidom @ A )
=> ! [N: nat] : ( ord_less_eq @ A @ ( one_one @ A ) @ ( semiring_char_0_fact @ A @ N ) ) ) ).
% fact_ge_1
thf(fact_3110_nat__one__as__int,axiom,
( ( one_one @ nat )
= ( nat2 @ ( one_one @ int ) ) ) ).
% nat_one_as_int
thf(fact_3111_Word_Osint__0,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ( ( ring_1_signed @ A @ int @ ( zero_zero @ ( word @ A ) ) )
= ( zero_zero @ int ) ) ) ).
% Word.sint_0
thf(fact_3112_ex__nat,axiom,
( ( ^ [P6: nat > $o] :
? [X9: nat] : ( P6 @ X9 ) )
= ( ^ [P2: nat > $o] :
? [X2: int] :
( ( ord_less_eq @ int @ ( zero_zero @ int ) @ X2 )
& ( P2 @ ( nat2 @ X2 ) ) ) ) ) ).
% ex_nat
thf(fact_3113_all__nat,axiom,
( ( ^ [P6: nat > $o] :
! [X9: nat] : ( P6 @ X9 ) )
= ( ^ [P2: nat > $o] :
! [X2: int] :
( ( ord_less_eq @ int @ ( zero_zero @ int ) @ X2 )
=> ( P2 @ ( nat2 @ X2 ) ) ) ) ) ).
% all_nat
thf(fact_3114_eq__nat__nat__iff,axiom,
! [Z: int,Z5: int] :
( ( ord_less_eq @ int @ ( zero_zero @ int ) @ Z )
=> ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ Z5 )
=> ( ( ( nat2 @ Z )
= ( nat2 @ Z5 ) )
= ( Z = Z5 ) ) ) ) ).
% eq_nat_nat_iff
thf(fact_3115_uint__0__eq,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ( ( semiring_1_unsigned @ A @ int @ ( zero_zero @ ( word @ A ) ) )
= ( zero_zero @ int ) ) ) ).
% uint_0_eq
thf(fact_3116_uint__0__iff,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [X: word @ A] :
( ( ( semiring_1_unsigned @ A @ int @ X )
= ( zero_zero @ int ) )
= ( X
= ( zero_zero @ ( word @ A ) ) ) ) ) ).
% uint_0_iff
thf(fact_3117_fact__dvd,axiom,
! [A: $tType] :
( ( linordered_semidom @ A )
=> ! [N: nat,M: nat] :
( ( ord_less_eq @ nat @ N @ M )
=> ( dvd_dvd @ A @ ( semiring_char_0_fact @ A @ N ) @ ( semiring_char_0_fact @ A @ M ) ) ) ) ).
% fact_dvd
thf(fact_3118_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_3119_word__less__def,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ( ( ord_less @ ( word @ A ) )
= ( ^ [A4: word @ A,B5: word @ A] : ( ord_less @ int @ ( semiring_1_unsigned @ A @ int @ A4 ) @ ( semiring_1_unsigned @ A @ int @ B5 ) ) ) ) ) ).
% word_less_def
thf(fact_3120_set__encode__eq,axiom,
! [A3: set @ nat,B4: set @ nat] :
( ( finite_finite2 @ nat @ A3 )
=> ( ( finite_finite2 @ nat @ B4 )
=> ( ( ( nat_set_encode @ A3 )
= ( nat_set_encode @ B4 ) )
= ( A3 = B4 ) ) ) ) ).
% set_encode_eq
thf(fact_3121_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_3122_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_3123_uint__div__distrib,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [V: word @ A,W: word @ A] :
( ( semiring_1_unsigned @ A @ int @ ( divide_divide @ ( word @ A ) @ V @ W ) )
= ( divide_divide @ int @ ( semiring_1_unsigned @ A @ int @ V ) @ ( semiring_1_unsigned @ A @ int @ W ) ) ) ) ).
% uint_div_distrib
thf(fact_3124_uint__div,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [X: word @ A,Y: word @ A] :
( ( semiring_1_unsigned @ A @ int @ ( divide_divide @ ( word @ A ) @ X @ Y ) )
= ( divide_divide @ int @ ( semiring_1_unsigned @ A @ int @ X ) @ ( semiring_1_unsigned @ A @ int @ Y ) ) ) ) ).
% uint_div
thf(fact_3125_uint__mod,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [X: word @ A,Y: word @ A] :
( ( semiring_1_unsigned @ A @ int @ ( modulo_modulo @ ( word @ A ) @ X @ Y ) )
= ( modulo_modulo @ int @ ( semiring_1_unsigned @ A @ int @ X ) @ ( semiring_1_unsigned @ A @ int @ Y ) ) ) ) ).
% uint_mod
thf(fact_3126_uint__mod__distrib,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [V: word @ A,W: word @ A] :
( ( semiring_1_unsigned @ A @ int @ ( modulo_modulo @ ( word @ A ) @ V @ W ) )
= ( modulo_modulo @ int @ ( semiring_1_unsigned @ A @ int @ V ) @ ( semiring_1_unsigned @ A @ int @ W ) ) ) ) ).
% uint_mod_distrib
thf(fact_3127_unsigned__eq__0__iff,axiom,
! [A: $tType,B: $tType] :
( ( ( type_len @ B )
& ( semiring_char_0 @ A ) )
=> ! [W: word @ B] :
( ( ( semiring_1_unsigned @ B @ A @ W )
= ( zero_zero @ A ) )
= ( W
= ( zero_zero @ ( word @ B ) ) ) ) ) ).
% unsigned_eq_0_iff
thf(fact_3128_word__less__iff__unsigned,axiom,
! [A: $tType,B: $tType] :
( ( ( type_len @ B )
& ( linordered_semidom @ A ) )
=> ( ( ord_less @ ( word @ B ) )
= ( ^ [A4: word @ B,B5: word @ B] : ( ord_less @ A @ ( semiring_1_unsigned @ B @ A @ A4 ) @ ( semiring_1_unsigned @ B @ A @ B5 ) ) ) ) ) ).
% word_less_iff_unsigned
thf(fact_3129_unset__bit__nat__def,axiom,
( ( bit_se2638667681897837118et_bit @ nat )
= ( ^ [M4: nat,N3: nat] : ( nat2 @ ( bit_se2638667681897837118et_bit @ int @ M4 @ ( semiring_1_of_nat @ int @ N3 ) ) ) ) ) ).
% unset_bit_nat_def
thf(fact_3130_signed__eq__0__iff,axiom,
! [A: $tType,B: $tType] :
( ( ( type_len @ B )
& ( ring_char_0 @ A ) )
=> ! [W: word @ B] :
( ( ( ring_1_signed @ B @ A @ W )
= ( zero_zero @ A ) )
= ( W
= ( zero_zero @ ( word @ B ) ) ) ) ) ).
% signed_eq_0_iff
thf(fact_3131_fact__ge__Suc__0__nat,axiom,
! [N: nat] : ( ord_less_eq @ nat @ ( suc @ ( zero_zero @ nat ) ) @ ( semiring_char_0_fact @ nat @ N ) ) ).
% fact_ge_Suc_0_nat
thf(fact_3132_dvd__fact,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq @ nat @ ( one_one @ nat ) @ M )
=> ( ( ord_less_eq @ nat @ M @ N )
=> ( dvd_dvd @ nat @ M @ ( semiring_char_0_fact @ nat @ N ) ) ) ) ).
% dvd_fact
thf(fact_3133_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_3134_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_3135_fact__less__mono,axiom,
! [A: $tType] :
( ( linordered_semidom @ A )
=> ! [M: nat,N: nat] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ M )
=> ( ( ord_less @ nat @ M @ N )
=> ( ord_less @ A @ ( semiring_char_0_fact @ A @ M ) @ ( semiring_char_0_fact @ A @ N ) ) ) ) ) ).
% fact_less_mono
thf(fact_3136_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_3137_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_3138_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_3139_int__minus,axiom,
! [N: nat,M: nat] :
( ( semiring_1_of_nat @ int @ ( minus_minus @ nat @ N @ M ) )
= ( semiring_1_of_nat @ int @ ( nat2 @ ( minus_minus @ int @ ( semiring_1_of_nat @ int @ N ) @ ( semiring_1_of_nat @ int @ M ) ) ) ) ) ).
% int_minus
thf(fact_3140_fact__mod,axiom,
! [A: $tType] :
( ( ( linordered_semidom @ A )
& ( semidom_modulo @ A ) )
=> ! [M: nat,N: nat] :
( ( ord_less_eq @ nat @ M @ N )
=> ( ( modulo_modulo @ A @ ( semiring_char_0_fact @ A @ N ) @ ( semiring_char_0_fact @ A @ M ) )
= ( zero_zero @ A ) ) ) ) ).
% fact_mod
thf(fact_3141_fact__fact__dvd__fact,axiom,
! [A: $tType] :
( ( linordered_semidom @ A )
=> ! [K: nat,N: nat] : ( dvd_dvd @ A @ ( times_times @ A @ ( semiring_char_0_fact @ A @ K ) @ ( semiring_char_0_fact @ A @ N ) ) @ ( semiring_char_0_fact @ A @ ( plus_plus @ nat @ K @ N ) ) ) ) ).
% fact_fact_dvd_fact
thf(fact_3142_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_3143_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_3144_fact__le__power,axiom,
! [A: $tType] :
( ( linordered_semidom @ A )
=> ! [N: nat] : ( ord_less_eq @ A @ ( semiring_char_0_fact @ A @ N ) @ ( semiring_1_of_nat @ A @ ( power_power @ nat @ N @ N ) ) ) ) ).
% fact_le_power
thf(fact_3145_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_3146_word__div__def,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ( ( divide_divide @ ( word @ A ) )
= ( ^ [A4: word @ A,B5: word @ A] : ( ring_1_of_int @ ( word @ A ) @ ( divide_divide @ int @ ( semiring_1_unsigned @ A @ int @ A4 ) @ ( semiring_1_unsigned @ A @ int @ B5 ) ) ) ) ) ) ).
% word_div_def
thf(fact_3147_word__mod__def,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ( ( modulo_modulo @ ( word @ A ) )
= ( ^ [A4: word @ A,B5: word @ A] : ( ring_1_of_int @ ( word @ A ) @ ( modulo_modulo @ int @ ( semiring_1_unsigned @ A @ int @ A4 ) @ ( semiring_1_unsigned @ A @ int @ B5 ) ) ) ) ) ) ).
% word_mod_def
thf(fact_3148_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_3149_set__encode__inf,axiom,
! [A3: set @ nat] :
( ~ ( finite_finite2 @ nat @ A3 )
=> ( ( nat_set_encode @ A3 )
= ( zero_zero @ nat ) ) ) ).
% set_encode_inf
thf(fact_3150_fact__diff__Suc,axiom,
! [N: nat,M: nat] :
( ( ord_less @ nat @ N @ ( suc @ M ) )
=> ( ( semiring_char_0_fact @ nat @ ( minus_minus @ nat @ ( suc @ M ) @ N ) )
= ( times_times @ nat @ ( minus_minus @ nat @ ( suc @ M ) @ N ) @ ( semiring_char_0_fact @ nat @ ( minus_minus @ nat @ M @ N ) ) ) ) ) ).
% fact_diff_Suc
thf(fact_3151_fact__div__fact__le__pow,axiom,
! [R2: nat,N: nat] :
( ( ord_less_eq @ nat @ R2 @ N )
=> ( ord_less_eq @ nat @ ( divide_divide @ nat @ ( semiring_char_0_fact @ nat @ N ) @ ( semiring_char_0_fact @ nat @ ( minus_minus @ nat @ N @ R2 ) ) ) @ ( power_power @ nat @ N @ R2 ) ) ) ).
% fact_div_fact_le_pow
thf(fact_3152_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_3153_binomial__fact__lemma,axiom,
! [K: nat,N: nat] :
( ( ord_less_eq @ nat @ K @ N )
=> ( ( times_times @ nat @ ( times_times @ nat @ ( semiring_char_0_fact @ nat @ K ) @ ( semiring_char_0_fact @ nat @ ( minus_minus @ nat @ N @ K ) ) ) @ ( binomial @ N @ K ) )
= ( semiring_char_0_fact @ nat @ N ) ) ) ).
% binomial_fact_lemma
thf(fact_3154_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_3155_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_3156_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_3157_split__nat,axiom,
! [P: nat > $o,I: int] :
( ( P @ ( nat2 @ I ) )
= ( ! [N3: nat] :
( ( I
= ( semiring_1_of_nat @ int @ N3 ) )
=> ( P @ N3 ) )
& ( ( ord_less @ int @ I @ ( zero_zero @ int ) )
=> ( P @ ( zero_zero @ nat ) ) ) ) ) ).
% split_nat
thf(fact_3158_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_3159_nat__add__distrib,axiom,
! [Z: int,Z5: int] :
( ( ord_less_eq @ int @ ( zero_zero @ int ) @ Z )
=> ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ Z5 )
=> ( ( nat2 @ ( plus_plus @ int @ Z @ Z5 ) )
= ( plus_plus @ nat @ ( nat2 @ Z ) @ ( nat2 @ Z5 ) ) ) ) ) ).
% nat_add_distrib
thf(fact_3160_le__nat__iff,axiom,
! [K: int,N: nat] :
( ( ord_less_eq @ int @ ( zero_zero @ int ) @ K )
=> ( ( ord_less_eq @ nat @ N @ ( nat2 @ K ) )
= ( ord_less_eq @ int @ ( semiring_1_of_nat @ int @ N ) @ K ) ) ) ).
% le_nat_iff
thf(fact_3161_Suc__as__int,axiom,
( suc
= ( ^ [A4: nat] : ( nat2 @ ( plus_plus @ int @ ( semiring_1_of_nat @ int @ A4 ) @ ( one_one @ int ) ) ) ) ) ).
% Suc_as_int
thf(fact_3162_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_3163_nat__mult__distrib,axiom,
! [Z: int,Z5: int] :
( ( ord_less_eq @ int @ ( zero_zero @ int ) @ Z )
=> ( ( nat2 @ ( times_times @ int @ Z @ Z5 ) )
= ( times_times @ nat @ ( nat2 @ Z ) @ ( nat2 @ Z5 ) ) ) ) ).
% nat_mult_distrib
thf(fact_3164_nat__diff__distrib,axiom,
! [Z5: int,Z: int] :
( ( ord_less_eq @ int @ ( zero_zero @ int ) @ Z5 )
=> ( ( ord_less_eq @ int @ Z5 @ Z )
=> ( ( nat2 @ ( minus_minus @ int @ Z @ Z5 ) )
= ( minus_minus @ nat @ ( nat2 @ Z ) @ ( nat2 @ Z5 ) ) ) ) ) ).
% nat_diff_distrib
thf(fact_3165_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_3166_nat__abs__triangle__ineq,axiom,
! [K: int,L: int] : ( ord_less_eq @ nat @ ( nat2 @ ( abs_abs @ int @ ( plus_plus @ int @ K @ L ) ) ) @ ( plus_plus @ nat @ ( nat2 @ ( abs_abs @ int @ K ) ) @ ( nat2 @ ( abs_abs @ int @ L ) ) ) ) ).
% nat_abs_triangle_ineq
thf(fact_3167_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_3168_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_3169_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_3170_nat__power__eq,axiom,
! [Z: int,N: nat] :
( ( ord_less_eq @ int @ ( zero_zero @ int ) @ Z )
=> ( ( nat2 @ ( power_power @ int @ Z @ N ) )
= ( power_power @ nat @ ( nat2 @ Z ) @ N ) ) ) ).
% nat_power_eq
thf(fact_3171_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_3172_choose__dvd,axiom,
! [A: $tType] :
( ( linordered_semidom @ A )
=> ! [K: nat,N: nat] :
( ( ord_less_eq @ nat @ K @ N )
=> ( dvd_dvd @ A @ ( times_times @ A @ ( semiring_char_0_fact @ A @ K ) @ ( semiring_char_0_fact @ A @ ( minus_minus @ nat @ N @ K ) ) ) @ ( semiring_char_0_fact @ A @ N ) ) ) ) ).
% choose_dvd
thf(fact_3173_udvd__incr__lem0,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [Up: int,Uq: int,N: int,K6: word @ A,N5: int] :
( ( ord_less @ int @ Up @ Uq )
=> ( ( Up
= ( times_times @ int @ N @ ( semiring_1_unsigned @ A @ int @ K6 ) ) )
=> ( ( Uq
= ( times_times @ int @ N5 @ ( 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_3174_udvd__incr__lem,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [Up: int,Uq: int,Ua: int,N: int,K6: word @ A,N5: int] :
( ( ord_less @ int @ Up @ Uq )
=> ( ( Up
= ( plus_plus @ int @ Ua @ ( times_times @ int @ N @ ( semiring_1_unsigned @ A @ int @ K6 ) ) ) )
=> ( ( Uq
= ( plus_plus @ int @ Ua @ ( times_times @ int @ N5 @ ( 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_3175_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_3176_div__abs__eq__div__nat,axiom,
! [K: int,L: int] :
( ( divide_divide @ int @ ( abs_abs @ int @ K ) @ ( abs_abs @ int @ L ) )
= ( semiring_1_of_nat @ int @ ( divide_divide @ nat @ ( nat2 @ ( abs_abs @ int @ K ) ) @ ( nat2 @ ( abs_abs @ int @ L ) ) ) ) ) ).
% div_abs_eq_div_nat
thf(fact_3177_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_3178_mod__abs__eq__div__nat,axiom,
! [K: int,L: int] :
( ( modulo_modulo @ int @ ( abs_abs @ int @ K ) @ ( abs_abs @ int @ L ) )
= ( semiring_1_of_nat @ int @ ( modulo_modulo @ nat @ ( nat2 @ ( abs_abs @ int @ K ) ) @ ( nat2 @ ( abs_abs @ int @ L ) ) ) ) ) ).
% mod_abs_eq_div_nat
thf(fact_3179_udvd__incr0,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [P4: word @ A,Q3: word @ A,N: int,K6: word @ A,N5: int] :
( ( ord_less @ ( word @ A ) @ P4 @ Q3 )
=> ( ( ( semiring_1_unsigned @ A @ int @ P4 )
= ( times_times @ int @ N @ ( semiring_1_unsigned @ A @ int @ K6 ) ) )
=> ( ( ( semiring_1_unsigned @ A @ int @ Q3 )
= ( times_times @ int @ N5 @ ( semiring_1_unsigned @ A @ int @ K6 ) ) )
=> ( ord_less_eq @ ( word @ A ) @ ( plus_plus @ ( word @ A ) @ P4 @ K6 ) @ Q3 ) ) ) ) ) ).
% udvd_incr0
thf(fact_3180_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_3181_take__bit__nat__eq,axiom,
! [K: int,N: nat] :
( ( ord_less_eq @ int @ ( zero_zero @ int ) @ K )
=> ( ( bit_se2584673776208193580ke_bit @ nat @ N @ ( nat2 @ K ) )
= ( nat2 @ ( bit_se2584673776208193580ke_bit @ int @ N @ K ) ) ) ) ).
% take_bit_nat_eq
thf(fact_3182_nat__take__bit__eq,axiom,
! [K: int,N: nat] :
( ( ord_less_eq @ int @ ( zero_zero @ int ) @ K )
=> ( ( nat2 @ ( bit_se2584673776208193580ke_bit @ int @ N @ K ) )
= ( bit_se2584673776208193580ke_bit @ nat @ N @ ( nat2 @ K ) ) ) ) ).
% nat_take_bit_eq
thf(fact_3183_udvd__decr0,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [P4: word @ A,Q3: word @ A,N: int,K6: word @ A,N5: int] :
( ( ord_less @ ( word @ A ) @ P4 @ Q3 )
=> ( ( ( semiring_1_unsigned @ A @ int @ P4 )
= ( times_times @ int @ N @ ( semiring_1_unsigned @ A @ int @ K6 ) ) )
=> ( ( ( semiring_1_unsigned @ A @ int @ Q3 )
= ( times_times @ int @ N5 @ ( semiring_1_unsigned @ A @ int @ K6 ) ) )
=> ( ( ( semiring_1_unsigned @ A @ int @ Q3 )
= ( times_times @ int @ N5 @ ( semiring_1_unsigned @ A @ int @ K6 ) ) )
=> ( ord_less_eq @ ( word @ A ) @ P4 @ ( minus_minus @ ( word @ A ) @ Q3 @ K6 ) ) ) ) ) ) ) ).
% udvd_decr0
thf(fact_3184_floor__eq3,axiom,
! [N: nat,X: real] :
( ( ord_less @ real @ ( semiring_1_of_nat @ real @ N ) @ X )
=> ( ( ord_less @ real @ X @ ( semiring_1_of_nat @ real @ ( suc @ N ) ) )
=> ( ( nat2 @ ( archim6421214686448440834_floor @ real @ X ) )
= N ) ) ) ).
% floor_eq3
thf(fact_3185_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_3186_nat__2,axiom,
( ( nat2 @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) )
= ( suc @ ( suc @ ( zero_zero @ nat ) ) ) ) ).
% nat_2
thf(fact_3187_binomial__altdef__nat,axiom,
! [K: nat,N: nat] :
( ( ord_less_eq @ nat @ K @ N )
=> ( ( binomial @ N @ K )
= ( divide_divide @ nat @ ( semiring_char_0_fact @ nat @ N ) @ ( times_times @ nat @ ( semiring_char_0_fact @ nat @ K ) @ ( semiring_char_0_fact @ nat @ ( minus_minus @ nat @ N @ K ) ) ) ) ) ) ).
% binomial_altdef_nat
thf(fact_3188_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_3189_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_3190_nat__mult__distrib__neg,axiom,
! [Z: int,Z5: int] :
( ( ord_less_eq @ int @ Z @ ( zero_zero @ int ) )
=> ( ( nat2 @ ( times_times @ int @ Z @ Z5 ) )
= ( times_times @ nat @ ( nat2 @ ( uminus_uminus @ int @ Z ) ) @ ( nat2 @ ( uminus_uminus @ int @ Z5 ) ) ) ) ) ).
% nat_mult_distrib_neg
thf(fact_3191_floor__eq4,axiom,
! [N: nat,X: real] :
( ( ord_less_eq @ real @ ( semiring_1_of_nat @ real @ N ) @ X )
=> ( ( ord_less @ real @ X @ ( semiring_1_of_nat @ real @ ( suc @ N ) ) )
=> ( ( nat2 @ ( archim6421214686448440834_floor @ real @ X ) )
= N ) ) ) ).
% floor_eq4
thf(fact_3192_udvd__incr_H,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [P4: word @ A,Q3: word @ A,Ua: int,N: int,K6: word @ A,N5: int] :
( ( ord_less @ ( word @ A ) @ P4 @ Q3 )
=> ( ( ( semiring_1_unsigned @ A @ int @ P4 )
= ( plus_plus @ int @ Ua @ ( times_times @ int @ N @ ( semiring_1_unsigned @ A @ int @ K6 ) ) ) )
=> ( ( ( semiring_1_unsigned @ A @ int @ Q3 )
= ( plus_plus @ int @ Ua @ ( times_times @ int @ N5 @ ( semiring_1_unsigned @ A @ int @ K6 ) ) ) )
=> ( ord_less_eq @ ( word @ A ) @ ( plus_plus @ ( word @ A ) @ P4 @ K6 ) @ Q3 ) ) ) ) ) ).
% udvd_incr'
thf(fact_3193_udvd__decr_H,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [P4: word @ A,Q3: word @ A,Ua: int,N: int,K6: word @ A,N5: int] :
( ( ord_less @ ( word @ A ) @ P4 @ Q3 )
=> ( ( ( semiring_1_unsigned @ A @ int @ P4 )
= ( plus_plus @ int @ Ua @ ( times_times @ int @ N @ ( semiring_1_unsigned @ A @ int @ K6 ) ) ) )
=> ( ( ( semiring_1_unsigned @ A @ int @ Q3 )
= ( plus_plus @ int @ Ua @ ( times_times @ int @ N5 @ ( semiring_1_unsigned @ A @ int @ K6 ) ) ) )
=> ( ( ( semiring_1_unsigned @ A @ int @ Q3 )
= ( plus_plus @ int @ Ua @ ( times_times @ int @ N5 @ ( semiring_1_unsigned @ A @ int @ K6 ) ) ) )
=> ( ord_less_eq @ ( word @ A ) @ P4 @ ( minus_minus @ ( word @ A ) @ Q3 @ K6 ) ) ) ) ) ) ) ).
% udvd_decr'
thf(fact_3194_square__fact__le__2__fact,axiom,
! [N: nat] : ( ord_less_eq @ real @ ( times_times @ real @ ( semiring_char_0_fact @ real @ N ) @ ( semiring_char_0_fact @ real @ N ) ) @ ( semiring_char_0_fact @ real @ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N ) ) ) ).
% square_fact_le_2_fact
thf(fact_3195_of__int__of__nat,axiom,
! [A: $tType] :
( ( ring_1 @ A )
=> ( ( ring_1_of_int @ A )
= ( ^ [K3: int] : ( if @ A @ ( ord_less @ int @ K3 @ ( zero_zero @ int ) ) @ ( uminus_uminus @ A @ ( semiring_1_of_nat @ A @ ( nat2 @ ( uminus_uminus @ int @ K3 ) ) ) ) @ ( semiring_1_of_nat @ A @ ( nat2 @ K3 ) ) ) ) ) ) ).
% of_int_of_nat
thf(fact_3196_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_3197_fact__num__eq__if,axiom,
! [A: $tType] :
( ( semiring_char_0 @ A )
=> ( ( semiring_char_0_fact @ A )
= ( ^ [M4: nat] :
( if @ A
@ ( M4
= ( zero_zero @ nat ) )
@ ( one_one @ A )
@ ( times_times @ A @ ( semiring_1_of_nat @ A @ M4 ) @ ( semiring_char_0_fact @ A @ ( minus_minus @ nat @ M4 @ ( one_one @ nat ) ) ) ) ) ) ) ) ).
% fact_num_eq_if
thf(fact_3198_fact__reduce,axiom,
! [A: $tType] :
( ( semiring_char_0 @ A )
=> ! [N: nat] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ N )
=> ( ( semiring_char_0_fact @ A @ N )
= ( times_times @ A @ ( semiring_1_of_nat @ A @ N ) @ ( semiring_char_0_fact @ A @ ( minus_minus @ nat @ N @ ( one_one @ nat ) ) ) ) ) ) ) ).
% fact_reduce
thf(fact_3199_pochhammer__same,axiom,
! [A: $tType] :
( ( ( semiring_char_0 @ A )
& ( comm_ring_1 @ A )
& ( semiri3467727345109120633visors @ A ) )
=> ! [N: nat] :
( ( comm_s3205402744901411588hammer @ A @ ( uminus_uminus @ A @ ( semiring_1_of_nat @ A @ N ) ) @ N )
= ( times_times @ A @ ( power_power @ A @ ( uminus_uminus @ A @ ( one_one @ A ) ) @ N ) @ ( semiring_char_0_fact @ A @ N ) ) ) ) ).
% pochhammer_same
thf(fact_3200_fact__binomial,axiom,
! [A: $tType] :
( ( field_char_0 @ A )
=> ! [K: nat,N: nat] :
( ( ord_less_eq @ nat @ K @ N )
=> ( ( times_times @ A @ ( semiring_char_0_fact @ A @ K ) @ ( semiring_1_of_nat @ A @ ( binomial @ N @ K ) ) )
= ( divide_divide @ A @ ( semiring_char_0_fact @ A @ N ) @ ( semiring_char_0_fact @ A @ ( minus_minus @ nat @ N @ K ) ) ) ) ) ) ).
% fact_binomial
thf(fact_3201_binomial__fact,axiom,
! [A: $tType] :
( ( field_char_0 @ A )
=> ! [K: nat,N: nat] :
( ( ord_less_eq @ nat @ K @ N )
=> ( ( semiring_1_of_nat @ A @ ( binomial @ N @ K ) )
= ( divide_divide @ A @ ( semiring_char_0_fact @ A @ N ) @ ( times_times @ A @ ( semiring_char_0_fact @ A @ K ) @ ( semiring_char_0_fact @ A @ ( minus_minus @ nat @ N @ K ) ) ) ) ) ) ) ).
% binomial_fact
thf(fact_3202_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_3203_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_3204_uint__2p,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [N: nat] :
( ( ord_less @ ( word @ A ) @ ( zero_zero @ ( word @ A ) ) @ ( power_power @ ( word @ A ) @ ( numeral_numeral @ ( word @ A ) @ ( bit0 @ one2 ) ) @ N ) )
=> ( ( semiring_1_unsigned @ A @ int @ ( power_power @ ( word @ A ) @ ( numeral_numeral @ ( word @ A ) @ ( bit0 @ one2 ) ) @ N ) )
= ( power_power @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ N ) ) ) ) ).
% uint_2p
thf(fact_3205_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_3206_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_3207_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_3208_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_3209_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_3210_sin__coeff__def,axiom,
( sin_coeff
= ( ^ [N3: nat] : ( if @ real @ ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N3 ) @ ( zero_zero @ real ) @ ( divide_divide @ real @ ( power_power @ real @ ( uminus_uminus @ real @ ( one_one @ real ) ) @ ( divide_divide @ nat @ ( minus_minus @ nat @ N3 @ ( suc @ ( zero_zero @ nat ) ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( semiring_char_0_fact @ real @ N3 ) ) ) ) ) ).
% sin_coeff_def
thf(fact_3211_binomial__code,axiom,
( binomial
= ( ^ [N3: nat,K3: nat] : ( if @ nat @ ( ord_less @ nat @ N3 @ K3 ) @ ( zero_zero @ nat ) @ ( if @ nat @ ( ord_less @ nat @ N3 @ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ K3 ) ) @ ( binomial @ N3 @ ( minus_minus @ nat @ N3 @ K3 ) ) @ ( divide_divide @ nat @ ( set_fo6178422350223883121st_nat @ nat @ ( times_times @ nat ) @ ( plus_plus @ nat @ ( minus_minus @ nat @ N3 @ K3 ) @ ( one_one @ nat ) ) @ N3 @ ( one_one @ nat ) ) @ ( semiring_char_0_fact @ nat @ K3 ) ) ) ) ) ) ).
% binomial_code
thf(fact_3212_cos__coeff__def,axiom,
( cos_coeff
= ( ^ [N3: nat] : ( if @ real @ ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N3 ) @ ( divide_divide @ real @ ( power_power @ real @ ( uminus_uminus @ real @ ( one_one @ real ) ) @ ( divide_divide @ nat @ N3 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( semiring_char_0_fact @ real @ N3 ) ) @ ( zero_zero @ real ) ) ) ) ).
% cos_coeff_def
thf(fact_3213_fact__code,axiom,
! [A: $tType] :
( ( semiring_char_0 @ A )
=> ( ( semiring_char_0_fact @ A )
= ( ^ [N3: nat] : ( semiring_1_of_nat @ A @ ( set_fo6178422350223883121st_nat @ nat @ ( times_times @ nat ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N3 @ ( one_one @ nat ) ) ) ) ) ) ).
% fact_code
thf(fact_3214_divide__int__def,axiom,
( ( divide_divide @ int )
= ( ^ [K3: int,L2: int] :
( if @ int
@ ( L2
= ( zero_zero @ int ) )
@ ( zero_zero @ int )
@ ( if @ int
@ ( ( sgn_sgn @ int @ K3 )
= ( sgn_sgn @ int @ L2 ) )
@ ( semiring_1_of_nat @ int @ ( divide_divide @ nat @ ( nat2 @ ( abs_abs @ int @ K3 ) ) @ ( nat2 @ ( abs_abs @ int @ L2 ) ) ) )
@ ( uminus_uminus @ int
@ ( semiring_1_of_nat @ int
@ ( plus_plus @ nat @ ( divide_divide @ nat @ ( nat2 @ ( abs_abs @ int @ K3 ) ) @ ( nat2 @ ( abs_abs @ int @ L2 ) ) )
@ ( zero_neq_one_of_bool @ nat
@ ~ ( dvd_dvd @ int @ L2 @ K3 ) ) ) ) ) ) ) ) ) ).
% divide_int_def
thf(fact_3215_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_3216_sgn__zero,axiom,
! [A: $tType] :
( ( real_V822414075346904944vector @ A )
=> ( ( sgn_sgn @ A @ ( zero_zero @ A ) )
= ( zero_zero @ A ) ) ) ).
% sgn_zero
thf(fact_3217_sgn__0,axiom,
! [A: $tType] :
( ( idom_abs_sgn @ A )
=> ( ( sgn_sgn @ A @ ( zero_zero @ A ) )
= ( zero_zero @ A ) ) ) ).
% sgn_0
thf(fact_3218_sgn__one,axiom,
! [A: $tType] :
( ( real_V2822296259951069270ebra_1 @ A )
=> ( ( sgn_sgn @ A @ ( one_one @ A ) )
= ( one_one @ A ) ) ) ).
% sgn_one
thf(fact_3219_sgn__1,axiom,
! [A: $tType] :
( ( idom_abs_sgn @ A )
=> ( ( sgn_sgn @ A @ ( one_one @ A ) )
= ( one_one @ A ) ) ) ).
% sgn_1
thf(fact_3220_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_3221_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_3222_power__sgn,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [A2: A,N: nat] :
( ( sgn_sgn @ A @ ( power_power @ A @ A2 @ N ) )
= ( power_power @ A @ ( sgn_sgn @ A @ A2 ) @ N ) ) ) ).
% power_sgn
thf(fact_3223_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_3224_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_3225_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_3226_sgn__uint__eq,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [W: word @ A] :
( ( sgn_sgn @ int @ ( semiring_1_unsigned @ A @ int @ W ) )
= ( zero_neq_one_of_bool @ int
@ ( W
!= ( zero_zero @ ( word @ A ) ) ) ) ) ) ).
% sgn_uint_eq
thf(fact_3227_sin__coeff__0,axiom,
( ( sin_coeff @ ( zero_zero @ nat ) )
= ( zero_zero @ real ) ) ).
% sin_coeff_0
thf(fact_3228_cos__coeff__0,axiom,
( ( cos_coeff @ ( zero_zero @ nat ) )
= ( one_one @ real ) ) ).
% cos_coeff_0
thf(fact_3229_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_3230_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_3231_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_3232_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_3233_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_3234_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_3235_dvd__mult__sgn__iff,axiom,
! [L: int,K: int,R2: int] :
( ( dvd_dvd @ int @ L @ ( times_times @ int @ K @ ( sgn_sgn @ int @ R2 ) ) )
= ( ( dvd_dvd @ int @ L @ K )
| ( R2
= ( zero_zero @ int ) ) ) ) ).
% dvd_mult_sgn_iff
thf(fact_3236_dvd__sgn__mult__iff,axiom,
! [L: int,R2: int,K: int] :
( ( dvd_dvd @ int @ L @ ( times_times @ int @ ( sgn_sgn @ int @ R2 ) @ K ) )
= ( ( dvd_dvd @ int @ L @ K )
| ( R2
= ( zero_zero @ int ) ) ) ) ).
% dvd_sgn_mult_iff
thf(fact_3237_mult__sgn__dvd__iff,axiom,
! [L: int,R2: int,K: int] :
( ( dvd_dvd @ int @ ( times_times @ int @ L @ ( sgn_sgn @ int @ R2 ) ) @ K )
= ( ( dvd_dvd @ int @ L @ K )
& ( ( R2
= ( zero_zero @ int ) )
=> ( K
= ( zero_zero @ int ) ) ) ) ) ).
% mult_sgn_dvd_iff
thf(fact_3238_sgn__mult__dvd__iff,axiom,
! [R2: int,L: int,K: int] :
( ( dvd_dvd @ int @ ( times_times @ int @ ( sgn_sgn @ int @ R2 ) @ L ) @ K )
= ( ( dvd_dvd @ int @ L @ K )
& ( ( R2
= ( zero_zero @ int ) )
=> ( K
= ( zero_zero @ int ) ) ) ) ) ).
% sgn_mult_dvd_iff
thf(fact_3239_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_3240_sgn__of__nat,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [N: nat] :
( ( sgn_sgn @ A @ ( semiring_1_of_nat @ A @ N ) )
= ( zero_neq_one_of_bool @ A @ ( ord_less @ nat @ ( zero_zero @ nat ) @ N ) ) ) ) ).
% sgn_of_nat
thf(fact_3241_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_3242_ucast__0__I,axiom,
! [A: $tType,B: $tType] :
( ( ( type_len @ B )
& ( type_len @ A ) )
=> ! [X: word @ A] :
( ( X
= ( zero_zero @ ( word @ A ) ) )
=> ( ( semiring_1_unsigned @ A @ ( word @ B ) @ X )
= ( zero_zero @ ( word @ B ) ) ) ) ) ).
% ucast_0_I
thf(fact_3243_ucast__0,axiom,
! [B: $tType,A: $tType] :
( ( ( type_len @ A )
& ( type_len @ B ) )
=> ( ( semiring_1_unsigned @ B @ ( word @ A ) @ ( zero_zero @ ( word @ B ) ) )
= ( zero_zero @ ( word @ A ) ) ) ) ).
% ucast_0
thf(fact_3244_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_3245_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_3246_sgn__zero__iff,axiom,
! [A: $tType] :
( ( real_V822414075346904944vector @ A )
=> ! [X: A] :
( ( ( sgn_sgn @ A @ X )
= ( zero_zero @ A ) )
= ( X
= ( zero_zero @ A ) ) ) ) ).
% sgn_zero_iff
thf(fact_3247_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_3248_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_3249_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_3250_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_3251_scast__0,axiom,
! [B: $tType,A: $tType] :
( ( ( type_len @ A )
& ( type_len @ B ) )
=> ( ( ring_1_signed @ B @ ( word @ A ) @ ( zero_zero @ ( word @ B ) ) )
= ( zero_zero @ ( word @ A ) ) ) ) ).
% scast_0
thf(fact_3252_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_3253_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_3254_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_3255_linordered__idom__class_Oabs__sgn,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ( ( abs_abs @ A )
= ( ^ [K3: A] : ( times_times @ A @ K3 @ ( sgn_sgn @ A @ K3 ) ) ) ) ) ).
% linordered_idom_class.abs_sgn
thf(fact_3256_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_3257_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_3258_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_3259_unat__0,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ( ( semiring_1_unsigned @ A @ nat @ ( zero_zero @ ( word @ A ) ) )
= ( zero_zero @ nat ) ) ) ).
% unat_0
thf(fact_3260_unat__eq__zero,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [X: word @ A] :
( ( ( semiring_1_unsigned @ A @ nat @ X )
= ( zero_zero @ nat ) )
= ( X
= ( zero_zero @ ( word @ A ) ) ) ) ) ).
% unat_eq_zero
thf(fact_3261_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_3262_word__less__nat__alt,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ( ( ord_less @ ( word @ A ) )
= ( ^ [A4: word @ A,B5: word @ A] : ( ord_less @ nat @ ( semiring_1_unsigned @ A @ nat @ A4 ) @ ( semiring_1_unsigned @ A @ nat @ B5 ) ) ) ) ) ).
% word_less_nat_alt
thf(fact_3263_unat__1,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ( ( semiring_1_unsigned @ A @ nat @ ( one_one @ ( word @ A ) ) )
= ( one_one @ nat ) ) ) ).
% unat_1
thf(fact_3264_div__eq__sgn__abs,axiom,
! [K: int,L: int] :
( ( ( sgn_sgn @ int @ K )
= ( sgn_sgn @ int @ L ) )
=> ( ( divide_divide @ int @ K @ L )
= ( divide_divide @ int @ ( abs_abs @ int @ K ) @ ( abs_abs @ int @ L ) ) ) ) ).
% div_eq_sgn_abs
thf(fact_3265_uno__simps_I2_J,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [Z: word @ A,N: nat] :
( ( semiring_1_unsigned @ A @ nat @ ( semiring_1_of_nat @ ( word @ A ) @ ( divide_divide @ nat @ ( semiring_1_unsigned @ A @ nat @ Z ) @ N ) ) )
= ( divide_divide @ nat @ ( semiring_1_unsigned @ A @ nat @ Z ) @ N ) ) ) ).
% uno_simps(2)
thf(fact_3266_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_3267_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_3268_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_3269_unat__of__nat__less,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [A2: word @ A,B2: word @ A,C2: nat] :
( ( ord_less @ ( word @ A ) @ A2 @ B2 )
=> ( ( ( semiring_1_unsigned @ A @ nat @ B2 )
= C2 )
=> ( ord_less @ ( word @ A ) @ A2 @ ( semiring_1_of_nat @ ( word @ A ) @ C2 ) ) ) ) ) ).
% unat_of_nat_less
thf(fact_3270_unat__mod,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [X: word @ A,Y: word @ A] :
( ( semiring_1_unsigned @ A @ nat @ ( modulo_modulo @ ( word @ A ) @ X @ Y ) )
= ( modulo_modulo @ nat @ ( semiring_1_unsigned @ A @ nat @ X ) @ ( semiring_1_unsigned @ A @ nat @ Y ) ) ) ) ).
% unat_mod
thf(fact_3271_unat__mod__distrib,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [V: word @ A,W: word @ A] :
( ( semiring_1_unsigned @ A @ nat @ ( modulo_modulo @ ( word @ A ) @ V @ W ) )
= ( modulo_modulo @ nat @ ( semiring_1_unsigned @ A @ nat @ V ) @ ( semiring_1_unsigned @ A @ nat @ W ) ) ) ) ).
% unat_mod_distrib
thf(fact_3272_sin__coeff__Suc,axiom,
! [N: nat] :
( ( sin_coeff @ ( suc @ N ) )
= ( divide_divide @ real @ ( cos_coeff @ N ) @ ( semiring_1_of_nat @ real @ ( suc @ N ) ) ) ) ).
% sin_coeff_Suc
thf(fact_3273_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_3274_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_3275_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_3276_sgn__mod,axiom,
! [L: int,K: int] :
( ( L
!= ( zero_zero @ int ) )
=> ( ~ ( dvd_dvd @ int @ L @ K )
=> ( ( sgn_sgn @ int @ ( modulo_modulo @ int @ K @ L ) )
= ( sgn_sgn @ int @ L ) ) ) ) ).
% sgn_mod
thf(fact_3277_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_3278_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_3279_word__of__nat__less,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [N: nat,X: word @ A] :
( ( ord_less @ nat @ N @ ( semiring_1_unsigned @ A @ nat @ X ) )
=> ( ord_less @ ( word @ A ) @ ( semiring_1_of_nat @ ( word @ A ) @ N ) @ X ) ) ) ).
% word_of_nat_less
thf(fact_3280_unat__less__helper,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [X: word @ A,N: nat] :
( ( ord_less @ ( word @ A ) @ X @ ( semiring_1_of_nat @ ( word @ A ) @ N ) )
=> ( ord_less @ nat @ ( semiring_1_unsigned @ A @ nat @ X ) @ N ) ) ) ).
% unat_less_helper
thf(fact_3281_word__arith__nat__add,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ( ( plus_plus @ ( word @ A ) )
= ( ^ [A4: word @ A,B5: word @ A] : ( semiring_1_of_nat @ ( word @ A ) @ ( plus_plus @ nat @ ( semiring_1_unsigned @ A @ nat @ A4 ) @ ( semiring_1_unsigned @ A @ nat @ B5 ) ) ) ) ) ) ).
% word_arith_nat_add
thf(fact_3282_cos__coeff__Suc,axiom,
! [N: nat] :
( ( cos_coeff @ ( suc @ N ) )
= ( divide_divide @ real @ ( uminus_uminus @ real @ ( sin_coeff @ N ) ) @ ( semiring_1_of_nat @ real @ ( suc @ N ) ) ) ) ).
% cos_coeff_Suc
thf(fact_3283_word__arith__nat__mult,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ( ( times_times @ ( word @ A ) )
= ( ^ [A4: word @ A,B5: word @ A] : ( semiring_1_of_nat @ ( word @ A ) @ ( times_times @ nat @ ( semiring_1_unsigned @ A @ nat @ A4 ) @ ( semiring_1_unsigned @ A @ nat @ B5 ) ) ) ) ) ) ).
% word_arith_nat_mult
thf(fact_3284_word__arith__nat__div,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ( ( divide_divide @ ( word @ A ) )
= ( ^ [A4: word @ A,B5: word @ A] : ( semiring_1_of_nat @ ( word @ A ) @ ( divide_divide @ nat @ ( semiring_1_unsigned @ A @ nat @ A4 ) @ ( semiring_1_unsigned @ A @ nat @ B5 ) ) ) ) ) ) ).
% word_arith_nat_div
thf(fact_3285_word__arith__nat__mod,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ( ( modulo_modulo @ ( word @ A ) )
= ( ^ [A4: word @ A,B5: word @ A] : ( semiring_1_of_nat @ ( word @ A ) @ ( modulo_modulo @ nat @ ( semiring_1_unsigned @ A @ nat @ A4 ) @ ( semiring_1_unsigned @ A @ nat @ B5 ) ) ) ) ) ) ).
% word_arith_nat_mod
thf(fact_3286_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_3287_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_3288_zsgn__def,axiom,
( ( sgn_sgn @ int )
= ( ^ [I2: int] :
( if @ int
@ ( I2
= ( zero_zero @ int ) )
@ ( zero_zero @ int )
@ ( if @ int @ ( ord_less @ int @ ( zero_zero @ int ) @ I2 ) @ ( one_one @ int ) @ ( uminus_uminus @ int @ ( one_one @ int ) ) ) ) ) ) ).
% zsgn_def
thf(fact_3289_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_3290_div__sgn__abs__cancel,axiom,
! [V: int,K: int,L: int] :
( ( V
!= ( zero_zero @ int ) )
=> ( ( divide_divide @ int @ ( times_times @ int @ ( sgn_sgn @ int @ V ) @ ( abs_abs @ int @ K ) ) @ ( times_times @ int @ ( sgn_sgn @ int @ V ) @ ( abs_abs @ int @ L ) ) )
= ( divide_divide @ int @ ( abs_abs @ int @ K ) @ ( abs_abs @ int @ L ) ) ) ) ).
% div_sgn_abs_cancel
thf(fact_3291_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_3292_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_3293_div__dvd__sgn__abs,axiom,
! [L: int,K: int] :
( ( dvd_dvd @ int @ L @ K )
=> ( ( divide_divide @ int @ K @ L )
= ( times_times @ int @ ( times_times @ int @ ( sgn_sgn @ int @ K ) @ ( sgn_sgn @ int @ L ) ) @ ( divide_divide @ int @ ( abs_abs @ int @ K ) @ ( abs_abs @ int @ L ) ) ) ) ) ).
% div_dvd_sgn_abs
thf(fact_3294_unatSuc,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [N: word @ A] :
( ( ( plus_plus @ ( word @ A ) @ ( one_one @ ( word @ A ) ) @ N )
!= ( zero_zero @ ( word @ A ) ) )
=> ( ( semiring_1_unsigned @ A @ nat @ ( plus_plus @ ( word @ A ) @ ( one_one @ ( word @ A ) ) @ N ) )
= ( suc @ ( semiring_1_unsigned @ A @ nat @ N ) ) ) ) ) ).
% unatSuc
thf(fact_3295_unatSuc2,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [N: word @ A] :
( ( ( plus_plus @ ( word @ A ) @ N @ ( one_one @ ( word @ A ) ) )
!= ( zero_zero @ ( word @ A ) ) )
=> ( ( semiring_1_unsigned @ A @ nat @ ( plus_plus @ ( word @ A ) @ N @ ( one_one @ ( word @ A ) ) ) )
= ( suc @ ( semiring_1_unsigned @ A @ nat @ N ) ) ) ) ) ).
% unatSuc2
thf(fact_3296_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_3297_unat__Suc2,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [N: word @ A] :
( ( N
!= ( uminus_uminus @ ( word @ A ) @ ( one_one @ ( word @ A ) ) ) )
=> ( ( semiring_1_unsigned @ A @ nat @ ( plus_plus @ ( word @ A ) @ N @ ( one_one @ ( word @ A ) ) ) )
= ( suc @ ( semiring_1_unsigned @ A @ nat @ N ) ) ) ) ) ).
% unat_Suc2
thf(fact_3298_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_3299_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_3300_fold__atLeastAtMost__nat_Osimps,axiom,
! [A: $tType] :
( ( set_fo6178422350223883121st_nat @ A )
= ( ^ [F4: nat > A > A,A4: nat,B5: nat,Acc: A] : ( if @ A @ ( ord_less @ nat @ B5 @ A4 ) @ Acc @ ( set_fo6178422350223883121st_nat @ A @ F4 @ ( plus_plus @ nat @ A4 @ ( one_one @ nat ) ) @ B5 @ ( F4 @ A4 @ Acc ) ) ) ) ) ).
% fold_atLeastAtMost_nat.simps
thf(fact_3301_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_3302_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_3303_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_3304_div__noneq__sgn__abs,axiom,
! [L: int,K: int] :
( ( L
!= ( zero_zero @ int ) )
=> ( ( ( sgn_sgn @ int @ K )
!= ( sgn_sgn @ int @ L ) )
=> ( ( divide_divide @ int @ K @ L )
= ( minus_minus @ int @ ( uminus_uminus @ int @ ( divide_divide @ int @ ( abs_abs @ int @ K ) @ ( abs_abs @ int @ L ) ) )
@ ( zero_neq_one_of_bool @ int
@ ~ ( dvd_dvd @ int @ L @ K ) ) ) ) ) ) ).
% div_noneq_sgn_abs
thf(fact_3305_lt__plus__1__le__word,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [N: nat,MaxBound: word @ A,X: word @ A] :
( ( ord_less @ nat @ N @ ( semiring_1_unsigned @ A @ nat @ MaxBound ) )
=> ( ( ord_less @ ( word @ A ) @ X @ ( plus_plus @ ( word @ A ) @ ( one_one @ ( word @ A ) ) @ ( semiring_1_of_nat @ ( word @ A ) @ N ) ) )
= ( ord_less_eq @ ( word @ A ) @ X @ ( semiring_1_of_nat @ ( word @ A ) @ N ) ) ) ) ) ).
% lt_plus_1_le_word
thf(fact_3306_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_3307_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_3308_word__div__eq__1__iff,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [N: word @ A,M: word @ A] :
( ( ( divide_divide @ ( word @ A ) @ N @ M )
= ( one_one @ ( word @ A ) ) )
= ( ( ord_less_eq @ ( word @ A ) @ M @ N )
& ( ord_less @ nat @ ( semiring_1_unsigned @ A @ nat @ N ) @ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( semiring_1_unsigned @ A @ nat @ M ) ) ) ) ) ) ).
% word_div_eq_1_iff
thf(fact_3309_of__nat__eq__size,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [N: nat,W: word @ A] :
( ( ( semiring_1_of_nat @ ( word @ A ) @ N )
= W )
= ( ? [Q6: nat] :
( N
= ( plus_plus @ nat @ ( semiring_1_unsigned @ A @ nat @ W ) @ ( times_times @ nat @ Q6 @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( size_size @ ( word @ A ) @ W ) ) ) ) ) ) ) ) ).
% of_nat_eq_size
thf(fact_3310_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_3311_modulo__int__def,axiom,
( ( modulo_modulo @ int )
= ( ^ [K3: int,L2: int] :
( if @ int
@ ( L2
= ( zero_zero @ int ) )
@ K3
@ ( if @ int
@ ( ( sgn_sgn @ int @ K3 )
= ( sgn_sgn @ int @ L2 ) )
@ ( times_times @ int @ ( sgn_sgn @ int @ L2 ) @ ( semiring_1_of_nat @ int @ ( modulo_modulo @ nat @ ( nat2 @ ( abs_abs @ int @ K3 ) ) @ ( nat2 @ ( abs_abs @ int @ L2 ) ) ) ) )
@ ( times_times @ int @ ( sgn_sgn @ int @ L2 )
@ ( minus_minus @ int
@ ( times_times @ int @ ( abs_abs @ int @ L2 )
@ ( zero_neq_one_of_bool @ int
@ ~ ( dvd_dvd @ int @ L2 @ K3 ) ) )
@ ( semiring_1_of_nat @ int @ ( modulo_modulo @ nat @ ( nat2 @ ( abs_abs @ int @ K3 ) ) @ ( nat2 @ ( abs_abs @ int @ L2 ) ) ) ) ) ) ) ) ) ) ).
% modulo_int_def
thf(fact_3312_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_3313_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_3314_divide__int__unfold,axiom,
! [L: int,K: int,N: nat,M: nat] :
( ( ( ( ( sgn_sgn @ int @ L )
= ( zero_zero @ int ) )
| ( ( sgn_sgn @ int @ K )
= ( zero_zero @ int ) )
| ( N
= ( zero_zero @ nat ) ) )
=> ( ( divide_divide @ int @ ( times_times @ int @ ( sgn_sgn @ int @ K ) @ ( semiring_1_of_nat @ int @ M ) ) @ ( times_times @ int @ ( sgn_sgn @ int @ L ) @ ( semiring_1_of_nat @ int @ N ) ) )
= ( zero_zero @ int ) ) )
& ( ~ ( ( ( sgn_sgn @ int @ L )
= ( zero_zero @ int ) )
| ( ( sgn_sgn @ int @ K )
= ( zero_zero @ int ) )
| ( N
= ( zero_zero @ nat ) ) )
=> ( ( ( ( sgn_sgn @ int @ K )
= ( sgn_sgn @ int @ L ) )
=> ( ( divide_divide @ int @ ( times_times @ int @ ( sgn_sgn @ int @ K ) @ ( semiring_1_of_nat @ int @ M ) ) @ ( times_times @ int @ ( sgn_sgn @ int @ L ) @ ( semiring_1_of_nat @ int @ N ) ) )
= ( semiring_1_of_nat @ int @ ( divide_divide @ nat @ M @ N ) ) ) )
& ( ( ( sgn_sgn @ int @ K )
!= ( sgn_sgn @ int @ L ) )
=> ( ( divide_divide @ int @ ( times_times @ int @ ( sgn_sgn @ int @ K ) @ ( semiring_1_of_nat @ int @ M ) ) @ ( times_times @ int @ ( sgn_sgn @ int @ L ) @ ( semiring_1_of_nat @ int @ N ) ) )
= ( uminus_uminus @ int
@ ( semiring_1_of_nat @ int
@ ( plus_plus @ nat @ ( divide_divide @ nat @ M @ N )
@ ( zero_neq_one_of_bool @ nat
@ ~ ( dvd_dvd @ nat @ N @ M ) ) ) ) ) ) ) ) ) ) ).
% divide_int_unfold
thf(fact_3315_modulo__int__unfold,axiom,
! [L: int,K: int,N: nat,M: nat] :
( ( ( ( ( sgn_sgn @ int @ L )
= ( zero_zero @ int ) )
| ( ( sgn_sgn @ int @ K )
= ( zero_zero @ int ) )
| ( N
= ( zero_zero @ nat ) ) )
=> ( ( modulo_modulo @ int @ ( times_times @ int @ ( sgn_sgn @ int @ K ) @ ( semiring_1_of_nat @ int @ M ) ) @ ( times_times @ int @ ( sgn_sgn @ int @ L ) @ ( semiring_1_of_nat @ int @ N ) ) )
= ( times_times @ int @ ( sgn_sgn @ int @ K ) @ ( semiring_1_of_nat @ int @ M ) ) ) )
& ( ~ ( ( ( sgn_sgn @ int @ L )
= ( zero_zero @ int ) )
| ( ( sgn_sgn @ int @ K )
= ( zero_zero @ int ) )
| ( N
= ( zero_zero @ nat ) ) )
=> ( ( ( ( sgn_sgn @ int @ K )
= ( sgn_sgn @ int @ L ) )
=> ( ( modulo_modulo @ int @ ( times_times @ int @ ( sgn_sgn @ int @ K ) @ ( semiring_1_of_nat @ int @ M ) ) @ ( times_times @ int @ ( sgn_sgn @ int @ L ) @ ( semiring_1_of_nat @ int @ N ) ) )
= ( times_times @ int @ ( sgn_sgn @ int @ L ) @ ( semiring_1_of_nat @ int @ ( modulo_modulo @ nat @ M @ N ) ) ) ) )
& ( ( ( sgn_sgn @ int @ K )
!= ( sgn_sgn @ int @ L ) )
=> ( ( modulo_modulo @ int @ ( times_times @ int @ ( sgn_sgn @ int @ K ) @ ( semiring_1_of_nat @ int @ M ) ) @ ( times_times @ int @ ( sgn_sgn @ int @ L ) @ ( semiring_1_of_nat @ int @ N ) ) )
= ( times_times @ int @ ( sgn_sgn @ int @ L )
@ ( minus_minus @ int
@ ( semiring_1_of_nat @ int
@ ( times_times @ nat @ N
@ ( zero_neq_one_of_bool @ nat
@ ~ ( dvd_dvd @ nat @ N @ M ) ) ) )
@ ( semiring_1_of_nat @ int @ ( modulo_modulo @ nat @ M @ N ) ) ) ) ) ) ) ) ) ).
% modulo_int_unfold
thf(fact_3316_sgn__div__eq__sgn__mult,axiom,
! [A2: int,B2: int] :
( ( ( divide_divide @ int @ A2 @ B2 )
!= ( zero_zero @ int ) )
=> ( ( sgn_sgn @ int @ ( divide_divide @ int @ A2 @ B2 ) )
= ( sgn_sgn @ int @ ( times_times @ int @ A2 @ B2 ) ) ) ) ).
% sgn_div_eq_sgn_mult
thf(fact_3317_invar__vebt_Ocases,axiom,
! [A1: vEBT_VEBT,A22: nat] :
( ( vEBT_invar_vebt @ A1 @ A22 )
=> ( ( ? [A6: $o,B3: $o] :
( A1
= ( vEBT_Leaf @ A6 @ B3 ) )
=> ( A22
!= ( suc @ ( zero_zero @ nat ) ) ) )
=> ( ! [TreeList2: list @ vEBT_VEBT,N2: nat,Summary2: vEBT_VEBT,M2: nat,Deg2: nat] :
( ( A1
= ( vEBT_Node @ ( none @ ( product_prod @ nat @ nat ) ) @ Deg2 @ TreeList2 @ Summary2 ) )
=> ( ( A22 = Deg2 )
=> ( ! [X5: vEBT_VEBT] :
( ( member @ vEBT_VEBT @ X5 @ ( set2 @ vEBT_VEBT @ TreeList2 ) )
=> ( vEBT_invar_vebt @ X5 @ N2 ) )
=> ( ( vEBT_invar_vebt @ Summary2 @ M2 )
=> ( ( ( size_size @ ( list @ vEBT_VEBT ) @ TreeList2 )
= ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ M2 ) )
=> ( ( M2 = N2 )
=> ( ( Deg2
= ( plus_plus @ nat @ N2 @ M2 ) )
=> ( ~ ? [X_12: nat] : ( vEBT_V8194947554948674370ptions @ Summary2 @ X_12 )
=> ~ ! [X5: vEBT_VEBT] :
( ( member @ vEBT_VEBT @ X5 @ ( set2 @ vEBT_VEBT @ TreeList2 ) )
=> ~ ? [X_12: nat] : ( vEBT_V8194947554948674370ptions @ X5 @ X_12 ) ) ) ) ) ) ) ) ) )
=> ( ! [TreeList2: list @ vEBT_VEBT,N2: nat,Summary2: vEBT_VEBT,M2: nat,Deg2: nat] :
( ( A1
= ( vEBT_Node @ ( none @ ( product_prod @ nat @ nat ) ) @ Deg2 @ TreeList2 @ Summary2 ) )
=> ( ( A22 = Deg2 )
=> ( ! [X5: vEBT_VEBT] :
( ( member @ vEBT_VEBT @ X5 @ ( set2 @ vEBT_VEBT @ TreeList2 ) )
=> ( vEBT_invar_vebt @ X5 @ N2 ) )
=> ( ( vEBT_invar_vebt @ Summary2 @ M2 )
=> ( ( ( size_size @ ( list @ vEBT_VEBT ) @ TreeList2 )
= ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ M2 ) )
=> ( ( M2
= ( suc @ N2 ) )
=> ( ( Deg2
= ( plus_plus @ nat @ N2 @ M2 ) )
=> ( ~ ? [X_12: nat] : ( vEBT_V8194947554948674370ptions @ Summary2 @ X_12 )
=> ~ ! [X5: vEBT_VEBT] :
( ( member @ vEBT_VEBT @ X5 @ ( set2 @ vEBT_VEBT @ TreeList2 ) )
=> ~ ? [X_12: nat] : ( vEBT_V8194947554948674370ptions @ X5 @ X_12 ) ) ) ) ) ) ) ) ) )
=> ( ! [TreeList2: list @ vEBT_VEBT,N2: nat,Summary2: vEBT_VEBT,M2: nat,Deg2: nat,Mi: nat,Ma2: nat] :
( ( A1
= ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi @ Ma2 ) ) @ Deg2 @ TreeList2 @ Summary2 ) )
=> ( ( A22 = Deg2 )
=> ( ! [X5: vEBT_VEBT] :
( ( member @ vEBT_VEBT @ X5 @ ( set2 @ vEBT_VEBT @ TreeList2 ) )
=> ( vEBT_invar_vebt @ X5 @ N2 ) )
=> ( ( vEBT_invar_vebt @ Summary2 @ M2 )
=> ( ( ( size_size @ ( list @ vEBT_VEBT ) @ TreeList2 )
= ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ M2 ) )
=> ( ( M2 = N2 )
=> ( ( Deg2
= ( plus_plus @ nat @ N2 @ M2 ) )
=> ( ! [I4: nat] :
( ( ord_less @ nat @ I4 @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ M2 ) )
=> ( ( ? [X8: nat] : ( vEBT_V8194947554948674370ptions @ ( nth @ vEBT_VEBT @ TreeList2 @ I4 ) @ X8 ) )
= ( vEBT_V8194947554948674370ptions @ Summary2 @ I4 ) ) )
=> ( ( ( Mi = Ma2 )
=> ! [X5: vEBT_VEBT] :
( ( member @ vEBT_VEBT @ X5 @ ( set2 @ vEBT_VEBT @ TreeList2 ) )
=> ~ ? [X_12: nat] : ( vEBT_V8194947554948674370ptions @ X5 @ X_12 ) ) )
=> ( ( ord_less_eq @ nat @ Mi @ Ma2 )
=> ( ( ord_less @ nat @ Ma2 @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ Deg2 ) )
=> ~ ( ( Mi != Ma2 )
=> ! [I4: nat] :
( ( ord_less @ nat @ I4 @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ M2 ) )
=> ( ( ( ( vEBT_VEBT_high @ Ma2 @ N2 )
= I4 )
=> ( vEBT_V8194947554948674370ptions @ ( nth @ vEBT_VEBT @ TreeList2 @ I4 ) @ ( vEBT_VEBT_low @ Ma2 @ N2 ) ) )
& ! [X5: nat] :
( ( ( ( vEBT_VEBT_high @ X5 @ N2 )
= I4 )
& ( vEBT_V8194947554948674370ptions @ ( nth @ vEBT_VEBT @ TreeList2 @ I4 ) @ ( vEBT_VEBT_low @ X5 @ N2 ) ) )
=> ( ( ord_less @ nat @ Mi @ X5 )
& ( ord_less_eq @ nat @ X5 @ Ma2 ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) )
=> ~ ! [TreeList2: list @ vEBT_VEBT,N2: nat,Summary2: vEBT_VEBT,M2: nat,Deg2: nat,Mi: nat,Ma2: nat] :
( ( A1
= ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi @ Ma2 ) ) @ Deg2 @ TreeList2 @ Summary2 ) )
=> ( ( A22 = Deg2 )
=> ( ! [X5: vEBT_VEBT] :
( ( member @ vEBT_VEBT @ X5 @ ( set2 @ vEBT_VEBT @ TreeList2 ) )
=> ( vEBT_invar_vebt @ X5 @ N2 ) )
=> ( ( vEBT_invar_vebt @ Summary2 @ M2 )
=> ( ( ( size_size @ ( list @ vEBT_VEBT ) @ TreeList2 )
= ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ M2 ) )
=> ( ( M2
= ( suc @ N2 ) )
=> ( ( Deg2
= ( plus_plus @ nat @ N2 @ M2 ) )
=> ( ! [I4: nat] :
( ( ord_less @ nat @ I4 @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ M2 ) )
=> ( ( ? [X8: nat] : ( vEBT_V8194947554948674370ptions @ ( nth @ vEBT_VEBT @ TreeList2 @ I4 ) @ X8 ) )
= ( vEBT_V8194947554948674370ptions @ Summary2 @ I4 ) ) )
=> ( ( ( Mi = Ma2 )
=> ! [X5: vEBT_VEBT] :
( ( member @ vEBT_VEBT @ X5 @ ( set2 @ vEBT_VEBT @ TreeList2 ) )
=> ~ ? [X_12: nat] : ( vEBT_V8194947554948674370ptions @ X5 @ X_12 ) ) )
=> ( ( ord_less_eq @ nat @ Mi @ Ma2 )
=> ( ( ord_less @ nat @ Ma2 @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ Deg2 ) )
=> ~ ( ( Mi != Ma2 )
=> ! [I4: nat] :
( ( ord_less @ nat @ I4 @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ M2 ) )
=> ( ( ( ( vEBT_VEBT_high @ Ma2 @ N2 )
= I4 )
=> ( vEBT_V8194947554948674370ptions @ ( nth @ vEBT_VEBT @ TreeList2 @ I4 ) @ ( vEBT_VEBT_low @ Ma2 @ N2 ) ) )
& ! [X5: nat] :
( ( ( ( vEBT_VEBT_high @ X5 @ N2 )
= I4 )
& ( vEBT_V8194947554948674370ptions @ ( nth @ vEBT_VEBT @ TreeList2 @ I4 ) @ ( vEBT_VEBT_low @ X5 @ N2 ) ) )
=> ( ( ord_less @ nat @ Mi @ X5 )
& ( ord_less_eq @ nat @ X5 @ Ma2 ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).
% invar_vebt.cases
thf(fact_3318_invar__vebt_Osimps,axiom,
( vEBT_invar_vebt
= ( ^ [A12: vEBT_VEBT,A23: nat] :
( ( ? [A4: $o,B5: $o] :
( A12
= ( vEBT_Leaf @ A4 @ B5 ) )
& ( A23
= ( suc @ ( zero_zero @ nat ) ) ) )
| ? [TreeList3: list @ vEBT_VEBT,N3: nat,Summary3: vEBT_VEBT] :
( ( A12
= ( vEBT_Node @ ( none @ ( product_prod @ nat @ nat ) ) @ A23 @ TreeList3 @ Summary3 ) )
& ! [X2: vEBT_VEBT] :
( ( member @ vEBT_VEBT @ X2 @ ( set2 @ vEBT_VEBT @ TreeList3 ) )
=> ( vEBT_invar_vebt @ X2 @ N3 ) )
& ( vEBT_invar_vebt @ Summary3 @ N3 )
& ( ( size_size @ ( list @ vEBT_VEBT ) @ TreeList3 )
= ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N3 ) )
& ( A23
= ( plus_plus @ nat @ N3 @ N3 ) )
& ~ ? [X8: nat] : ( vEBT_V8194947554948674370ptions @ Summary3 @ X8 )
& ! [X2: vEBT_VEBT] :
( ( member @ vEBT_VEBT @ X2 @ ( set2 @ vEBT_VEBT @ TreeList3 ) )
=> ~ ? [X8: nat] : ( vEBT_V8194947554948674370ptions @ X2 @ X8 ) ) )
| ? [TreeList3: list @ vEBT_VEBT,N3: nat,Summary3: vEBT_VEBT] :
( ( A12
= ( vEBT_Node @ ( none @ ( product_prod @ nat @ nat ) ) @ A23 @ TreeList3 @ Summary3 ) )
& ! [X2: vEBT_VEBT] :
( ( member @ vEBT_VEBT @ X2 @ ( set2 @ vEBT_VEBT @ TreeList3 ) )
=> ( vEBT_invar_vebt @ X2 @ N3 ) )
& ( vEBT_invar_vebt @ Summary3 @ ( suc @ N3 ) )
& ( ( size_size @ ( list @ vEBT_VEBT ) @ TreeList3 )
= ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( suc @ N3 ) ) )
& ( A23
= ( plus_plus @ nat @ N3 @ ( suc @ N3 ) ) )
& ~ ? [X8: nat] : ( vEBT_V8194947554948674370ptions @ Summary3 @ X8 )
& ! [X2: vEBT_VEBT] :
( ( member @ vEBT_VEBT @ X2 @ ( set2 @ vEBT_VEBT @ TreeList3 ) )
=> ~ ? [X8: nat] : ( vEBT_V8194947554948674370ptions @ X2 @ X8 ) ) )
| ? [TreeList3: list @ vEBT_VEBT,N3: nat,Summary3: vEBT_VEBT,Mi2: nat,Ma3: nat] :
( ( A12
= ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi2 @ Ma3 ) ) @ A23 @ TreeList3 @ Summary3 ) )
& ! [X2: vEBT_VEBT] :
( ( member @ vEBT_VEBT @ X2 @ ( set2 @ vEBT_VEBT @ TreeList3 ) )
=> ( vEBT_invar_vebt @ X2 @ N3 ) )
& ( vEBT_invar_vebt @ Summary3 @ N3 )
& ( ( size_size @ ( list @ vEBT_VEBT ) @ TreeList3 )
= ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N3 ) )
& ( A23
= ( plus_plus @ nat @ N3 @ N3 ) )
& ! [I2: nat] :
( ( ord_less @ nat @ I2 @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N3 ) )
=> ( ( ? [X8: nat] : ( vEBT_V8194947554948674370ptions @ ( nth @ vEBT_VEBT @ TreeList3 @ I2 ) @ X8 ) )
= ( vEBT_V8194947554948674370ptions @ Summary3 @ I2 ) ) )
& ( ( Mi2 = Ma3 )
=> ! [X2: vEBT_VEBT] :
( ( member @ vEBT_VEBT @ X2 @ ( set2 @ vEBT_VEBT @ TreeList3 ) )
=> ~ ? [X8: nat] : ( vEBT_V8194947554948674370ptions @ X2 @ X8 ) ) )
& ( ord_less_eq @ nat @ Mi2 @ Ma3 )
& ( ord_less @ nat @ Ma3 @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ A23 ) )
& ( ( Mi2 != Ma3 )
=> ! [I2: nat] :
( ( ord_less @ nat @ I2 @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N3 ) )
=> ( ( ( ( vEBT_VEBT_high @ Ma3 @ N3 )
= I2 )
=> ( vEBT_V8194947554948674370ptions @ ( nth @ vEBT_VEBT @ TreeList3 @ I2 ) @ ( vEBT_VEBT_low @ Ma3 @ N3 ) ) )
& ! [X2: nat] :
( ( ( ( vEBT_VEBT_high @ X2 @ N3 )
= I2 )
& ( vEBT_V8194947554948674370ptions @ ( nth @ vEBT_VEBT @ TreeList3 @ I2 ) @ ( vEBT_VEBT_low @ X2 @ N3 ) ) )
=> ( ( ord_less @ nat @ Mi2 @ X2 )
& ( ord_less_eq @ nat @ X2 @ Ma3 ) ) ) ) ) ) )
| ? [TreeList3: list @ vEBT_VEBT,N3: nat,Summary3: vEBT_VEBT,Mi2: nat,Ma3: nat] :
( ( A12
= ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi2 @ Ma3 ) ) @ A23 @ TreeList3 @ Summary3 ) )
& ! [X2: vEBT_VEBT] :
( ( member @ vEBT_VEBT @ X2 @ ( set2 @ vEBT_VEBT @ TreeList3 ) )
=> ( vEBT_invar_vebt @ X2 @ N3 ) )
& ( vEBT_invar_vebt @ Summary3 @ ( suc @ N3 ) )
& ( ( size_size @ ( list @ vEBT_VEBT ) @ TreeList3 )
= ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( suc @ N3 ) ) )
& ( A23
= ( plus_plus @ nat @ N3 @ ( suc @ N3 ) ) )
& ! [I2: nat] :
( ( ord_less @ nat @ I2 @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( suc @ N3 ) ) )
=> ( ( ? [X8: nat] : ( vEBT_V8194947554948674370ptions @ ( nth @ vEBT_VEBT @ TreeList3 @ I2 ) @ X8 ) )
= ( vEBT_V8194947554948674370ptions @ Summary3 @ I2 ) ) )
& ( ( Mi2 = Ma3 )
=> ! [X2: vEBT_VEBT] :
( ( member @ vEBT_VEBT @ X2 @ ( set2 @ vEBT_VEBT @ TreeList3 ) )
=> ~ ? [X8: nat] : ( vEBT_V8194947554948674370ptions @ X2 @ X8 ) ) )
& ( ord_less_eq @ nat @ Mi2 @ Ma3 )
& ( ord_less @ nat @ Ma3 @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ A23 ) )
& ( ( Mi2 != Ma3 )
=> ! [I2: nat] :
( ( ord_less @ nat @ I2 @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( suc @ N3 ) ) )
=> ( ( ( ( vEBT_VEBT_high @ Ma3 @ N3 )
= I2 )
=> ( vEBT_V8194947554948674370ptions @ ( nth @ vEBT_VEBT @ TreeList3 @ I2 ) @ ( vEBT_VEBT_low @ Ma3 @ N3 ) ) )
& ! [X2: nat] :
( ( ( ( vEBT_VEBT_high @ X2 @ N3 )
= I2 )
& ( vEBT_V8194947554948674370ptions @ ( nth @ vEBT_VEBT @ TreeList3 @ I2 ) @ ( vEBT_VEBT_low @ X2 @ N3 ) ) )
=> ( ( ord_less @ nat @ Mi2 @ X2 )
& ( ord_less_eq @ nat @ X2 @ Ma3 ) ) ) ) ) ) ) ) ) ) ).
% invar_vebt.simps
thf(fact_3319_and__int__unfold,axiom,
( ( bit_se5824344872417868541ns_and @ int )
= ( ^ [K3: int,L2: int] :
( if @ int
@ ( ( K3
= ( zero_zero @ int ) )
| ( L2
= ( zero_zero @ int ) ) )
@ ( zero_zero @ int )
@ ( if @ int
@ ( K3
= ( uminus_uminus @ int @ ( one_one @ int ) ) )
@ L2
@ ( if @ int
@ ( L2
= ( uminus_uminus @ int @ ( one_one @ int ) ) )
@ K3
@ ( plus_plus @ int @ ( times_times @ int @ ( modulo_modulo @ int @ K3 @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) ) @ ( modulo_modulo @ int @ L2 @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) ) ) @ ( times_times @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ ( bit_se5824344872417868541ns_and @ int @ ( divide_divide @ int @ K3 @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) ) @ ( divide_divide @ int @ L2 @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) ) ) ) ) ) ) ) ) ) ).
% and_int_unfold
thf(fact_3320_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_3321_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_3322_and_Oidem,axiom,
! [A: $tType] :
( ( bit_se359711467146920520ations @ A )
=> ! [A2: A] :
( ( bit_se5824344872417868541ns_and @ A @ A2 @ A2 )
= A2 ) ) ).
% and.idem
thf(fact_3323_mi__eq__ma__no__ch,axiom,
! [Mi3: 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 @ Mi3 @ Ma ) ) @ Deg @ TreeList @ Summary ) @ Deg )
=> ( ( Mi3 = Ma )
=> ( ! [X5: vEBT_VEBT] :
( ( member @ vEBT_VEBT @ X5 @ ( set2 @ vEBT_VEBT @ TreeList ) )
=> ~ ? [X_12: nat] : ( vEBT_V8194947554948674370ptions @ X5 @ X_12 ) )
& ~ ? [X_12: nat] : ( vEBT_V8194947554948674370ptions @ Summary @ X_12 ) ) ) ) ).
% mi_eq_ma_no_ch
thf(fact_3324_insert__simp__mima,axiom,
! [X: nat,Mi3: nat,Ma: nat,Deg: nat,TreeList: list @ vEBT_VEBT,Summary: vEBT_VEBT] :
( ( ( X = Mi3 )
| ( 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 @ Mi3 @ Ma ) ) @ Deg @ TreeList @ Summary ) @ X )
= ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi3 @ Ma ) ) @ Deg @ TreeList @ Summary ) ) ) ) ).
% insert_simp_mima
thf(fact_3325_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_3326_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_3327_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_3328_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_3329_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_3330_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_3331_not__None__eq,axiom,
! [A: $tType,X: option @ A] :
( ( X
!= ( none @ A ) )
= ( ? [Y6: A] :
( X
= ( some @ A @ Y6 ) ) ) ) ).
% not_None_eq
thf(fact_3332_not__Some__eq,axiom,
! [A: $tType,X: option @ A] :
( ( ! [Y6: A] :
( X
!= ( some @ A @ Y6 ) ) )
= ( X
= ( none @ A ) ) ) ).
% not_Some_eq
thf(fact_3333_take__bit__and,axiom,
! [A: $tType] :
( ( bit_se359711467146920520ations @ A )
=> ! [N: nat,A2: A,B2: A] :
( ( bit_se2584673776208193580ke_bit @ A @ N @ ( bit_se5824344872417868541ns_and @ A @ A2 @ B2 ) )
= ( bit_se5824344872417868541ns_and @ A @ ( bit_se2584673776208193580ke_bit @ A @ N @ A2 ) @ ( bit_se2584673776208193580ke_bit @ A @ N @ B2 ) ) ) ) ).
% take_bit_and
thf(fact_3334_tdeletemimi,axiom,
! [Deg: nat,Mi3: 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 @ Mi3 @ Mi3 ) ) @ Deg @ TreeList @ Summary ) @ X ) @ ( numeral_numeral @ nat @ ( bit1 @ ( bit0 @ ( bit0 @ one2 ) ) ) ) ) ) ).
% tdeletemimi
thf(fact_3335_tdeletemimi_H,axiom,
! [Deg: nat,Mi3: 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 @ Mi3 @ Mi3 ) ) @ Deg @ TreeList @ Summary ) @ X ) @ ( one_one @ nat ) ) ) ).
% tdeletemimi'
thf(fact_3336_mi__ma__2__deg,axiom,
! [Mi3: nat,Ma: nat,Deg: nat,TreeList: list @ vEBT_VEBT,Summary: vEBT_VEBT,N: nat] :
( ( vEBT_invar_vebt @ ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi3 @ Ma ) ) @ Deg @ TreeList @ Summary ) @ N )
=> ( ( ord_less_eq @ nat @ Mi3 @ Ma )
& ( ord_less @ nat @ Ma @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ Deg ) ) ) ) ).
% mi_ma_2_deg
thf(fact_3337_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_3338_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_3339_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_3340_not__Some__eq2,axiom,
! [B: $tType,A: $tType,V: option @ ( product_prod @ A @ B )] :
( ( ! [X2: A,Y6: B] :
( V
!= ( some @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X2 @ Y6 ) ) ) )
= ( V
= ( none @ ( product_prod @ A @ B ) ) ) ) ).
% not_Some_eq2
thf(fact_3341_and__nonnegative__int__iff,axiom,
! [K: int,L: int] :
( ( ord_less_eq @ int @ ( zero_zero @ int ) @ ( bit_se5824344872417868541ns_and @ int @ K @ L ) )
= ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ K )
| ( ord_less_eq @ int @ ( zero_zero @ int ) @ L ) ) ) ).
% and_nonnegative_int_iff
thf(fact_3342_and__negative__int__iff,axiom,
! [K: int,L: int] :
( ( ord_less @ int @ ( bit_se5824344872417868541ns_and @ int @ K @ L ) @ ( zero_zero @ int ) )
= ( ( ord_less @ int @ K @ ( zero_zero @ int ) )
& ( ord_less @ int @ L @ ( zero_zero @ int ) ) ) ) ).
% and_negative_int_iff
thf(fact_3343_both__member__options__from__complete__tree__to__child,axiom,
! [Deg: nat,Mi3: 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 @ Mi3 @ 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 = Mi3 )
| ( X = Ma ) ) ) ) ).
% both_member_options_from_complete_tree_to_child
thf(fact_3344_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_3345_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_3346_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_3347_member__inv,axiom,
! [Mi3: 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 @ Mi3 @ Ma ) ) @ Deg @ TreeList @ Summary ) @ X )
=> ( ( ord_less_eq @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ Deg )
& ( ( X = Mi3 )
| ( X = Ma )
| ( ( ord_less @ nat @ X @ Ma )
& ( ord_less @ nat @ Mi3 @ 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_3348_both__member__options__from__chilf__to__complete__tree,axiom,
! [X: nat,Deg: nat,TreeList: list @ vEBT_VEBT,Mi3: 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 @ Mi3 @ Ma ) ) @ Deg @ TreeList @ Summary ) @ X ) ) ) ) ).
% both_member_options_from_chilf_to_complete_tree
thf(fact_3349_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_3350_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_3351_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_3352_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_3353_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_3354_and__minus__numerals_I6_J,axiom,
! [N: num] :
( ( bit_se5824344872417868541ns_and @ int @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ ( bit1 @ N ) ) ) @ ( one_one @ int ) )
= ( one_one @ int ) ) ).
% and_minus_numerals(6)
thf(fact_3355_and__minus__numerals_I2_J,axiom,
! [N: num] :
( ( bit_se5824344872417868541ns_and @ int @ ( one_one @ int ) @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ ( bit1 @ N ) ) ) )
= ( one_one @ int ) ) ).
% and_minus_numerals(2)
thf(fact_3356_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_3357_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_3358_and__minus__numerals_I5_J,axiom,
! [N: num] :
( ( bit_se5824344872417868541ns_and @ int @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ ( bit0 @ N ) ) ) @ ( one_one @ int ) )
= ( zero_zero @ int ) ) ).
% and_minus_numerals(5)
thf(fact_3359_and__minus__numerals_I1_J,axiom,
! [N: num] :
( ( bit_se5824344872417868541ns_and @ int @ ( one_one @ int ) @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ ( bit0 @ N ) ) ) )
= ( zero_zero @ int ) ) ).
% and_minus_numerals(1)
thf(fact_3360_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_3361_of__int__and__eq,axiom,
! [A: $tType] :
( ( bit_ri3973907225187159222ations @ A )
=> ! [K: int,L: int] :
( ( ring_1_of_int @ A @ ( bit_se5824344872417868541ns_and @ int @ K @ L ) )
= ( bit_se5824344872417868541ns_and @ A @ ( ring_1_of_int @ A @ K ) @ ( ring_1_of_int @ A @ L ) ) ) ) ).
% of_int_and_eq
thf(fact_3362_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_3363_and_Ocommute,axiom,
! [A: $tType] :
( ( bit_se359711467146920520ations @ A )
=> ( ( bit_se5824344872417868541ns_and @ A )
= ( ^ [A4: A,B5: A] : ( bit_se5824344872417868541ns_and @ A @ B5 @ A4 ) ) ) ) ).
% and.commute
thf(fact_3364_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_3365_prod__decode__aux_Ocases,axiom,
! [X: product_prod @ nat @ nat] :
~ ! [K2: nat,M2: nat] :
( X
!= ( product_Pair @ nat @ nat @ K2 @ M2 ) ) ).
% prod_decode_aux.cases
thf(fact_3366_of__nat__and__eq,axiom,
! [A: $tType] :
( ( bit_se359711467146920520ations @ A )
=> ! [M: nat,N: nat] :
( ( semiring_1_of_nat @ A @ ( bit_se5824344872417868541ns_and @ nat @ M @ N ) )
= ( bit_se5824344872417868541ns_and @ A @ ( semiring_1_of_nat @ A @ M ) @ ( semiring_1_of_nat @ A @ N ) ) ) ) ).
% of_nat_and_eq
thf(fact_3367_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 ) )
=> ( ! [A6: A,B3: B] :
( ( X
= ( some @ A @ A6 ) )
=> ( ( Y
= ( some @ B @ B3 ) )
=> ( P @ X @ Y ) ) )
=> ( P @ X @ Y ) ) ) ) ).
% combine_options_cases
thf(fact_3368_split__option__all,axiom,
! [A: $tType] :
( ( ^ [P6: ( option @ A ) > $o] :
! [X9: option @ A] : ( P6 @ X9 ) )
= ( ^ [P2: ( option @ A ) > $o] :
( ( P2 @ ( none @ A ) )
& ! [X2: A] : ( P2 @ ( some @ A @ X2 ) ) ) ) ) ).
% split_option_all
thf(fact_3369_split__option__ex,axiom,
! [A: $tType] :
( ( ^ [P6: ( option @ A ) > $o] :
? [X9: option @ A] : ( P6 @ X9 ) )
= ( ^ [P2: ( option @ A ) > $o] :
( ( P2 @ ( none @ A ) )
| ? [X2: A] : ( P2 @ ( some @ A @ X2 ) ) ) ) ) ).
% split_option_ex
thf(fact_3370_option_Oexhaust,axiom,
! [A: $tType,Y: option @ A] :
( ( Y
!= ( none @ A ) )
=> ~ ! [X23: A] :
( Y
!= ( some @ A @ X23 ) ) ) ).
% option.exhaust
thf(fact_3371_option_OdiscI,axiom,
! [A: $tType,Option: option @ A,X22: A] :
( ( Option
= ( some @ A @ X22 ) )
=> ( Option
!= ( none @ A ) ) ) ).
% option.discI
thf(fact_3372_option_Odistinct_I1_J,axiom,
! [A: $tType,X22: A] :
( ( none @ A )
!= ( some @ A @ X22 ) ) ).
% option.distinct(1)
thf(fact_3373_T_092_060_094sub_062m_092_060_094sub_062i_092_060_094sub_062n_092_060_094sub_062t_Ocases,axiom,
! [X: vEBT_VEBT] :
( ! [A6: $o,B3: $o] :
( X
!= ( vEBT_Leaf @ A6 @ B3 ) )
=> ( ! [Uu2: nat,Uv2: list @ vEBT_VEBT,Uw2: vEBT_VEBT] :
( X
!= ( vEBT_Node @ ( none @ ( product_prod @ nat @ nat ) ) @ Uu2 @ Uv2 @ Uw2 ) )
=> ~ ! [Mi: nat,Ma2: nat,Ux2: nat,Uy2: list @ vEBT_VEBT,Uz2: vEBT_VEBT] :
( X
!= ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi @ Ma2 ) ) @ Ux2 @ Uy2 @ Uz2 ) ) ) ) ).
% T\<^sub>m\<^sub>i\<^sub>n\<^sub>t.cases
thf(fact_3374_VEBT__internal_Omembermima_Osimps_I3_J,axiom,
! [Mi3: nat,Ma: nat,Va2: list @ vEBT_VEBT,Vb: vEBT_VEBT,X: nat] :
( ( vEBT_VEBT_membermima @ ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi3 @ Ma ) ) @ ( zero_zero @ nat ) @ Va2 @ Vb ) @ X )
= ( ( X = Mi3 )
| ( X = Ma ) ) ) ).
% VEBT_internal.membermima.simps(3)
thf(fact_3375_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_3376_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_3377_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_3378_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_3379_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_3380_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_3381_real__sgn__eq,axiom,
( ( sgn_sgn @ real )
= ( ^ [X2: real] : ( divide_divide @ real @ X2 @ ( abs_abs @ real @ X2 ) ) ) ) ).
% real_sgn_eq
thf(fact_3382_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,
! [Mi3: 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 @ Mi3 @ 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_3383_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,
! [Mi3: 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 @ Mi3 @ 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_3384_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_3385_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,
! [Mi3: 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 @ Mi3 @ 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_3386_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,
! [Mi3: 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 @ Mi3 @ 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_3387_less__option__None__is__Some,axiom,
! [A: $tType] :
( ( preorder @ A )
=> ! [X: option @ A] :
( ( ord_less @ ( option @ A ) @ ( none @ A ) @ X )
=> ? [Z3: A] :
( X
= ( some @ A @ Z3 ) ) ) ) ).
% less_option_None_is_Some
thf(fact_3388_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_3389_height__node,axiom,
! [Mi3: nat,Ma: nat,Deg: nat,TreeList: list @ vEBT_VEBT,Summary: vEBT_VEBT,N: nat] :
( ( vEBT_invar_vebt @ ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi3 @ Ma ) ) @ Deg @ TreeList @ Summary ) @ N )
=> ( ord_less_eq @ nat @ ( one_one @ nat ) @ ( vEBT_VEBT_height @ ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi3 @ Ma ) ) @ Deg @ TreeList @ Summary ) ) ) ) ).
% height_node
thf(fact_3390_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_3391_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_3392_and__less__eq,axiom,
! [L: int,K: int] :
( ( ord_less @ int @ L @ ( zero_zero @ int ) )
=> ( ord_less_eq @ int @ ( bit_se5824344872417868541ns_and @ int @ K @ L ) @ K ) ) ).
% and_less_eq
thf(fact_3393_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 ) )
=> ( ! [Uv2: $o] :
( X
!= ( vEBT_Leaf @ $true @ Uv2 ) )
=> ( ! [Uu2: $o] :
( X
!= ( vEBT_Leaf @ Uu2 @ $true ) )
=> ( ! [Uw2: nat,Ux2: list @ vEBT_VEBT,Uy2: vEBT_VEBT] :
( X
!= ( vEBT_Node @ ( none @ ( product_prod @ nat @ nat ) ) @ Uw2 @ Ux2 @ Uy2 ) )
=> ~ ! [Uz2: product_prod @ nat @ nat,Va3: nat,Vb2: list @ vEBT_VEBT,Vc: vEBT_VEBT] :
( X
!= ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ Uz2 ) @ Va3 @ Vb2 @ Vc ) ) ) ) ) ) ).
% T\<^sub>m\<^sub>i\<^sub>n\<^sub>N\<^sub>u\<^sub>l\<^sub>l.cases
thf(fact_3394_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_3395_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_3396_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_3397_sgn__real__def,axiom,
( ( sgn_sgn @ real )
= ( ^ [A4: real] :
( if @ real
@ ( A4
= ( zero_zero @ real ) )
@ ( zero_zero @ real )
@ ( if @ real @ ( ord_less @ real @ ( zero_zero @ real ) @ A4 ) @ ( one_one @ real ) @ ( uminus_uminus @ real @ ( one_one @ real ) ) ) ) ) ) ).
% sgn_real_def
thf(fact_3398_even__and__iff__int,axiom,
! [K: int,L: int] :
( ( dvd_dvd @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ ( bit_se5824344872417868541ns_and @ int @ K @ L ) )
= ( ( dvd_dvd @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ K )
| ( dvd_dvd @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ L ) ) ) ).
% even_and_iff_int
thf(fact_3399_vebt__member_Osimps_I4_J,axiom,
! [V: product_prod @ nat @ nat,Vb: list @ vEBT_VEBT,Vc2: vEBT_VEBT,X: nat] :
~ ( vEBT_vebt_member @ ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ V ) @ ( suc @ ( zero_zero @ nat ) ) @ Vb @ Vc2 ) @ X ) ).
% vebt_member.simps(4)
thf(fact_3400_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_3401_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_3402_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,Vc2: 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 ) @ Vc2 @ Vd ) @ Ve )
= ( one_one @ nat ) ) ).
% T\<^sub>s\<^sub>u\<^sub>c\<^sub>c.simps(4)
thf(fact_3403_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,Vc2: 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 ) @ Vc2 @ Vd ) @ Ve )
= ( one_one @ nat ) ) ).
% T\<^sub>s\<^sub>u\<^sub>c\<^sub>c'.simps(4)
thf(fact_3404_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_3405_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_3406_sgn__power__injE,axiom,
! [A2: real,N: nat,X: real,B2: real] :
( ( ( times_times @ real @ ( sgn_sgn @ real @ A2 ) @ ( power_power @ real @ ( abs_abs @ real @ A2 ) @ N ) )
= X )
=> ( ( X
= ( times_times @ real @ ( sgn_sgn @ real @ B2 ) @ ( power_power @ real @ ( abs_abs @ real @ B2 ) @ N ) ) )
=> ( ( ord_less @ nat @ ( zero_zero @ nat ) @ N )
=> ( A2 = B2 ) ) ) ) ).
% sgn_power_injE
thf(fact_3407_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_3408_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_3409_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_3410_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_3411_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_3412_and__int__rec,axiom,
( ( bit_se5824344872417868541ns_and @ int )
= ( ^ [K3: int,L2: int] :
( plus_plus @ int
@ ( zero_neq_one_of_bool @ int
@ ( ~ ( dvd_dvd @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ K3 )
& ~ ( dvd_dvd @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ L2 ) ) )
@ ( times_times @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ ( bit_se5824344872417868541ns_and @ int @ ( divide_divide @ int @ K3 @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) ) @ ( divide_divide @ int @ L2 @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) ) ) ) ) ) ) ).
% and_int_rec
thf(fact_3413_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,Vc2: 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 @ Vc2 ) @ X )
= ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ).
% T\<^sub>m\<^sub>e\<^sub>m\<^sub>b\<^sub>e\<^sub>r.simps(4)
thf(fact_3414_invar__vebt_Ointros_I4_J,axiom,
! [TreeList: list @ vEBT_VEBT,N: nat,Summary: vEBT_VEBT,M: nat,Deg: nat,Mi3: nat,Ma: nat] :
( ! [X3: vEBT_VEBT] :
( ( member @ vEBT_VEBT @ X3 @ ( set2 @ vEBT_VEBT @ TreeList ) )
=> ( vEBT_invar_vebt @ X3 @ N ) )
=> ( ( vEBT_invar_vebt @ Summary @ M )
=> ( ( ( size_size @ ( list @ vEBT_VEBT ) @ TreeList )
= ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ M ) )
=> ( ( M = N )
=> ( ( Deg
= ( plus_plus @ nat @ N @ M ) )
=> ( ! [I3: nat] :
( ( ord_less @ nat @ I3 @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ M ) )
=> ( ( ? [X8: nat] : ( vEBT_V8194947554948674370ptions @ ( nth @ vEBT_VEBT @ TreeList @ I3 ) @ X8 ) )
= ( vEBT_V8194947554948674370ptions @ Summary @ I3 ) ) )
=> ( ( ( Mi3 = Ma )
=> ! [X3: vEBT_VEBT] :
( ( member @ vEBT_VEBT @ X3 @ ( set2 @ vEBT_VEBT @ TreeList ) )
=> ~ ? [X_1: nat] : ( vEBT_V8194947554948674370ptions @ X3 @ X_1 ) ) )
=> ( ( ord_less_eq @ nat @ Mi3 @ Ma )
=> ( ( ord_less @ nat @ Ma @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ Deg ) )
=> ( ( ( Mi3 != Ma )
=> ! [I3: nat] :
( ( ord_less @ nat @ I3 @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ M ) )
=> ( ( ( ( vEBT_VEBT_high @ Ma @ N )
= I3 )
=> ( vEBT_V8194947554948674370ptions @ ( nth @ vEBT_VEBT @ TreeList @ I3 ) @ ( vEBT_VEBT_low @ Ma @ N ) ) )
& ! [X3: nat] :
( ( ( ( vEBT_VEBT_high @ X3 @ N )
= I3 )
& ( vEBT_V8194947554948674370ptions @ ( nth @ vEBT_VEBT @ TreeList @ I3 ) @ ( vEBT_VEBT_low @ X3 @ N ) ) )
=> ( ( ord_less @ nat @ Mi3 @ X3 )
& ( ord_less_eq @ nat @ X3 @ Ma ) ) ) ) ) )
=> ( vEBT_invar_vebt @ ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi3 @ Ma ) ) @ Deg @ TreeList @ Summary ) @ Deg ) ) ) ) ) ) ) ) ) ) ) ).
% invar_vebt.intros(4)
thf(fact_3415_invar__vebt_Ointros_I5_J,axiom,
! [TreeList: list @ vEBT_VEBT,N: nat,Summary: vEBT_VEBT,M: nat,Deg: nat,Mi3: nat,Ma: nat] :
( ! [X3: vEBT_VEBT] :
( ( member @ vEBT_VEBT @ X3 @ ( set2 @ vEBT_VEBT @ TreeList ) )
=> ( vEBT_invar_vebt @ X3 @ N ) )
=> ( ( vEBT_invar_vebt @ Summary @ M )
=> ( ( ( size_size @ ( list @ vEBT_VEBT ) @ TreeList )
= ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ M ) )
=> ( ( M
= ( suc @ N ) )
=> ( ( Deg
= ( plus_plus @ nat @ N @ M ) )
=> ( ! [I3: nat] :
( ( ord_less @ nat @ I3 @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ M ) )
=> ( ( ? [X8: nat] : ( vEBT_V8194947554948674370ptions @ ( nth @ vEBT_VEBT @ TreeList @ I3 ) @ X8 ) )
= ( vEBT_V8194947554948674370ptions @ Summary @ I3 ) ) )
=> ( ( ( Mi3 = Ma )
=> ! [X3: vEBT_VEBT] :
( ( member @ vEBT_VEBT @ X3 @ ( set2 @ vEBT_VEBT @ TreeList ) )
=> ~ ? [X_1: nat] : ( vEBT_V8194947554948674370ptions @ X3 @ X_1 ) ) )
=> ( ( ord_less_eq @ nat @ Mi3 @ Ma )
=> ( ( ord_less @ nat @ Ma @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ Deg ) )
=> ( ( ( Mi3 != Ma )
=> ! [I3: nat] :
( ( ord_less @ nat @ I3 @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ M ) )
=> ( ( ( ( vEBT_VEBT_high @ Ma @ N )
= I3 )
=> ( vEBT_V8194947554948674370ptions @ ( nth @ vEBT_VEBT @ TreeList @ I3 ) @ ( vEBT_VEBT_low @ Ma @ N ) ) )
& ! [X3: nat] :
( ( ( ( vEBT_VEBT_high @ X3 @ N )
= I3 )
& ( vEBT_V8194947554948674370ptions @ ( nth @ vEBT_VEBT @ TreeList @ I3 ) @ ( vEBT_VEBT_low @ X3 @ N ) ) )
=> ( ( ord_less @ nat @ Mi3 @ X3 )
& ( ord_less_eq @ nat @ X3 @ Ma ) ) ) ) ) )
=> ( vEBT_invar_vebt @ ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi3 @ Ma ) ) @ Deg @ TreeList @ Summary ) @ Deg ) ) ) ) ) ) ) ) ) ) ) ).
% invar_vebt.intros(5)
thf(fact_3416_succ__list__to__short,axiom,
! [Deg: nat,Mi3: 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 @ Mi3 @ 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 @ Mi3 @ Ma ) ) @ Deg @ TreeList @ Summary ) @ X )
= ( none @ nat ) ) ) ) ) ).
% succ_list_to_short
thf(fact_3417_pred__list__to__short,axiom,
! [Deg: nat,X: nat,Ma: nat,TreeList: list @ vEBT_VEBT,Mi3: 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 @ Mi3 @ Ma ) ) @ Deg @ TreeList @ Summary ) @ X )
= ( none @ nat ) ) ) ) ) ).
% pred_list_to_short
thf(fact_3418_divmod__step__eq,axiom,
! [A: $tType] :
( ( unique1627219031080169319umeral @ A )
=> ! [L: num,R2: A,Q3: A] :
( ( ( ord_less_eq @ A @ ( numeral_numeral @ A @ L ) @ R2 )
=> ( ( unique1321980374590559556d_step @ A @ L @ ( product_Pair @ A @ A @ Q3 @ R2 ) )
= ( product_Pair @ A @ A @ ( plus_plus @ A @ ( times_times @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ Q3 ) @ ( one_one @ A ) ) @ ( minus_minus @ A @ R2 @ ( numeral_numeral @ A @ L ) ) ) ) )
& ( ~ ( ord_less_eq @ A @ ( numeral_numeral @ A @ L ) @ R2 )
=> ( ( unique1321980374590559556d_step @ A @ L @ ( product_Pair @ A @ A @ Q3 @ R2 ) )
= ( product_Pair @ A @ A @ ( times_times @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ Q3 ) @ R2 ) ) ) ) ) ).
% divmod_step_eq
thf(fact_3419_geqmaxNone,axiom,
! [Mi3: nat,Ma: nat,Deg: nat,TreeList: list @ vEBT_VEBT,Summary: vEBT_VEBT,N: nat,X: nat] :
( ( vEBT_invar_vebt @ ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi3 @ Ma ) ) @ Deg @ TreeList @ Summary ) @ N )
=> ( ( ord_less_eq @ nat @ Ma @ X )
=> ( ( vEBT_vebt_succ @ ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi3 @ Ma ) ) @ Deg @ TreeList @ Summary ) @ X )
= ( none @ nat ) ) ) ) ).
% geqmaxNone
thf(fact_3420_del__single__cont,axiom,
! [X: nat,Mi3: nat,Ma: nat,Deg: nat,TreeList: list @ vEBT_VEBT,Summary: vEBT_VEBT] :
( ( ( X = Mi3 )
& ( 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 @ Mi3 @ Ma ) ) @ Deg @ TreeList @ Summary ) @ X )
= ( vEBT_Node @ ( none @ ( product_prod @ nat @ nat ) ) @ Deg @ TreeList @ Summary ) ) ) ) ).
% del_single_cont
thf(fact_3421_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_3422_delete__pres__valid,axiom,
! [T: vEBT_VEBT,N: nat,X: nat] :
( ( vEBT_invar_vebt @ T @ N )
=> ( vEBT_invar_vebt @ ( vEBT_vebt_delete @ T @ X ) @ N ) ) ).
% delete_pres_valid
thf(fact_3423_dele__bmo__cont__corr,axiom,
! [T: vEBT_VEBT,N: nat,X: nat,Y: nat] :
( ( vEBT_invar_vebt @ T @ N )
=> ( ( vEBT_V8194947554948674370ptions @ ( vEBT_vebt_delete @ T @ X ) @ Y )
= ( ( X != Y )
& ( vEBT_V8194947554948674370ptions @ T @ Y ) ) ) ) ).
% dele_bmo_cont_corr
thf(fact_3424_dele__member__cont__corr,axiom,
! [T: vEBT_VEBT,N: nat,X: nat,Y: nat] :
( ( vEBT_invar_vebt @ T @ N )
=> ( ( vEBT_vebt_member @ ( vEBT_vebt_delete @ T @ X ) @ Y )
= ( ( X != Y )
& ( vEBT_vebt_member @ T @ Y ) ) ) ) ).
% dele_member_cont_corr
thf(fact_3425_pred__corr,axiom,
! [T: vEBT_VEBT,N: nat,X: nat,Px: nat] :
( ( vEBT_invar_vebt @ T @ N )
=> ( ( ( vEBT_vebt_pred @ T @ X )
= ( some @ nat @ Px ) )
= ( vEBT_is_pred_in_set @ ( vEBT_VEBT_set_vebt @ T ) @ X @ Px ) ) ) ).
% pred_corr
thf(fact_3426_succ__corr,axiom,
! [T: vEBT_VEBT,N: nat,X: nat,Sx: nat] :
( ( vEBT_invar_vebt @ T @ N )
=> ( ( ( vEBT_vebt_succ @ T @ X )
= ( some @ nat @ Sx ) )
= ( vEBT_is_succ_in_set @ ( vEBT_VEBT_set_vebt @ T ) @ X @ Sx ) ) ) ).
% succ_corr
thf(fact_3427_pred__correct,axiom,
! [T: vEBT_VEBT,N: nat,X: nat,Sx: nat] :
( ( vEBT_invar_vebt @ T @ N )
=> ( ( ( vEBT_vebt_pred @ T @ X )
= ( some @ nat @ Sx ) )
= ( vEBT_is_pred_in_set @ ( vEBT_set_vebt @ T ) @ X @ Sx ) ) ) ).
% pred_correct
thf(fact_3428_succ__correct,axiom,
! [T: vEBT_VEBT,N: nat,X: nat,Sx: nat] :
( ( vEBT_invar_vebt @ T @ N )
=> ( ( ( vEBT_vebt_succ @ T @ X )
= ( some @ nat @ Sx ) )
= ( vEBT_is_succ_in_set @ ( vEBT_set_vebt @ T ) @ X @ Sx ) ) ) ).
% succ_correct
thf(fact_3429_helpyd,axiom,
! [T: vEBT_VEBT,N: nat,X: nat,Y: nat] :
( ( vEBT_invar_vebt @ T @ N )
=> ( ( ( vEBT_vebt_succ @ T @ X )
= ( some @ nat @ Y ) )
=> ( ord_less @ nat @ Y @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N ) ) ) ) ).
% helpyd
thf(fact_3430_helpypredd,axiom,
! [T: vEBT_VEBT,N: nat,X: nat,Y: nat] :
( ( vEBT_invar_vebt @ T @ N )
=> ( ( ( vEBT_vebt_pred @ T @ X )
= ( some @ nat @ Y ) )
=> ( ord_less @ nat @ Y @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N ) ) ) ) ).
% helpypredd
thf(fact_3431_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_3432_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_3433_delt__out__of__range,axiom,
! [X: nat,Mi3: nat,Ma: nat,Deg: nat,TreeList: list @ vEBT_VEBT,Summary: vEBT_VEBT] :
( ( ( ord_less @ nat @ X @ Mi3 )
| ( 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 @ Mi3 @ Ma ) ) @ Deg @ TreeList @ Summary ) @ X )
= ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi3 @ Ma ) ) @ Deg @ TreeList @ Summary ) ) ) ) ).
% delt_out_of_range
thf(fact_3434_succ__min,axiom,
! [Deg: nat,X: nat,Mi3: nat,Ma: nat,TreeList: list @ vEBT_VEBT,Summary: vEBT_VEBT] :
( ( ord_less_eq @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ Deg )
=> ( ( ord_less @ nat @ X @ Mi3 )
=> ( ( vEBT_vebt_succ @ ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi3 @ Ma ) ) @ Deg @ TreeList @ Summary ) @ X )
= ( some @ nat @ Mi3 ) ) ) ) ).
% succ_min
thf(fact_3435_pred__max,axiom,
! [Deg: nat,Ma: nat,X: nat,Mi3: 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 @ Mi3 @ Ma ) ) @ Deg @ TreeList @ Summary ) @ X )
= ( some @ nat @ Ma ) ) ) ) ).
% pred_max
thf(fact_3436_lesseq__shift,axiom,
( ( ord_less_eq @ nat )
= ( ^ [X2: nat,Y6: nat] : ( vEBT_VEBT_lesseq @ ( some @ nat @ X2 ) @ ( some @ nat @ Y6 ) ) ) ) ).
% lesseq_shift
thf(fact_3437_word__no__log__defs_I3_J,axiom,
! [C: $tType] :
( ( type_len @ C )
=> ! [A2: num,B2: num] :
( ( bit_se5824344872417868541ns_and @ ( word @ C ) @ ( numeral_numeral @ ( word @ C ) @ A2 ) @ ( numeral_numeral @ ( word @ C ) @ B2 ) )
= ( ring_1_of_int @ ( word @ C ) @ ( bit_se5824344872417868541ns_and @ int @ ( numeral_numeral @ int @ A2 ) @ ( numeral_numeral @ int @ B2 ) ) ) ) ) ).
% word_no_log_defs(3)
thf(fact_3438_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_3439_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_3440_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_3441_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_3442_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_3443_word__bitwise__1__simps_I4_J,axiom,
! [D4: $tType] :
( ( type_len @ D4 )
=> ! [A2: num] :
( ( bit_se5824344872417868541ns_and @ ( word @ D4 ) @ ( numeral_numeral @ ( word @ D4 ) @ A2 ) @ ( one_one @ ( word @ D4 ) ) )
= ( ring_1_of_int @ ( word @ D4 ) @ ( bit_se5824344872417868541ns_and @ int @ ( numeral_numeral @ int @ A2 ) @ ( one_one @ int ) ) ) ) ) ).
% word_bitwise_1_simps(4)
thf(fact_3444_word__no__log__defs_I4_J,axiom,
! [D4: $tType] :
( ( type_len @ D4 )
=> ! [A2: num,B2: num] :
( ( bit_se5824344872417868541ns_and @ ( word @ D4 ) @ ( numeral_numeral @ ( word @ D4 ) @ A2 ) @ ( uminus_uminus @ ( word @ D4 ) @ ( numeral_numeral @ ( word @ D4 ) @ B2 ) ) )
= ( ring_1_of_int @ ( word @ D4 ) @ ( bit_se5824344872417868541ns_and @ int @ ( numeral_numeral @ int @ A2 ) @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ B2 ) ) ) ) ) ) ).
% word_no_log_defs(4)
thf(fact_3445_word__no__log__defs_I5_J,axiom,
! [E3: $tType] :
( ( type_len @ E3 )
=> ! [A2: num,B2: num] :
( ( bit_se5824344872417868541ns_and @ ( word @ E3 ) @ ( uminus_uminus @ ( word @ E3 ) @ ( numeral_numeral @ ( word @ E3 ) @ A2 ) ) @ ( numeral_numeral @ ( word @ E3 ) @ B2 ) )
= ( ring_1_of_int @ ( word @ E3 ) @ ( bit_se5824344872417868541ns_and @ int @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ A2 ) ) @ ( numeral_numeral @ int @ B2 ) ) ) ) ) ).
% word_no_log_defs(5)
thf(fact_3446_word__no__log__defs_I6_J,axiom,
! [F: $tType] :
( ( type_len @ F )
=> ! [A2: num,B2: num] :
( ( bit_se5824344872417868541ns_and @ ( word @ F ) @ ( uminus_uminus @ ( word @ F ) @ ( numeral_numeral @ ( word @ F ) @ A2 ) ) @ ( uminus_uminus @ ( word @ F ) @ ( numeral_numeral @ ( word @ F ) @ B2 ) ) )
= ( ring_1_of_int @ ( word @ F ) @ ( bit_se5824344872417868541ns_and @ int @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ A2 ) ) @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ B2 ) ) ) ) ) ) ).
% word_no_log_defs(6)
thf(fact_3447_Suc__0__and__eq,axiom,
! [N: nat] :
( ( bit_se5824344872417868541ns_and @ nat @ ( suc @ ( zero_zero @ nat ) ) @ N )
= ( modulo_modulo @ nat @ N @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ).
% Suc_0_and_eq
thf(fact_3448_and__Suc__0__eq,axiom,
! [N: nat] :
( ( bit_se5824344872417868541ns_and @ nat @ N @ ( suc @ ( zero_zero @ nat ) ) )
= ( modulo_modulo @ nat @ N @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ).
% and_Suc_0_eq
thf(fact_3449_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_3450_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_3451_word__log__esimps_I1_J,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [X: word @ A] :
( ( bit_se5824344872417868541ns_and @ ( word @ A ) @ X @ ( zero_zero @ ( word @ A ) ) )
= ( zero_zero @ ( word @ A ) ) ) ) ).
% word_log_esimps(1)
thf(fact_3452_word__log__esimps_I7_J,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [X: word @ A] :
( ( bit_se5824344872417868541ns_and @ ( word @ A ) @ ( zero_zero @ ( word @ A ) ) @ X )
= ( zero_zero @ ( word @ A ) ) ) ) ).
% word_log_esimps(7)
thf(fact_3453_vebt__pred_Osimps_I2_J,axiom,
! [A2: $o,Uw: $o] :
( ( A2
=> ( ( vEBT_vebt_pred @ ( vEBT_Leaf @ A2 @ Uw ) @ ( suc @ ( zero_zero @ nat ) ) )
= ( some @ nat @ ( zero_zero @ nat ) ) ) )
& ( ~ A2
=> ( ( vEBT_vebt_pred @ ( vEBT_Leaf @ A2 @ Uw ) @ ( suc @ ( zero_zero @ nat ) ) )
= ( none @ nat ) ) ) ) ).
% vebt_pred.simps(2)
thf(fact_3454_vebt__succ_Osimps_I1_J,axiom,
! [B2: $o,Uu: $o] :
( ( B2
=> ( ( vEBT_vebt_succ @ ( vEBT_Leaf @ Uu @ B2 ) @ ( zero_zero @ nat ) )
= ( some @ nat @ ( one_one @ nat ) ) ) )
& ( ~ B2
=> ( ( vEBT_vebt_succ @ ( vEBT_Leaf @ Uu @ B2 ) @ ( zero_zero @ nat ) )
= ( none @ nat ) ) ) ) ).
% vebt_succ.simps(1)
thf(fact_3455_vebt__pred_Osimps_I3_J,axiom,
! [B2: $o,A2: $o,Va2: nat] :
( ( B2
=> ( ( vEBT_vebt_pred @ ( vEBT_Leaf @ A2 @ B2 ) @ ( suc @ ( suc @ Va2 ) ) )
= ( some @ nat @ ( one_one @ nat ) ) ) )
& ( ~ B2
=> ( ( A2
=> ( ( vEBT_vebt_pred @ ( vEBT_Leaf @ A2 @ B2 ) @ ( suc @ ( suc @ Va2 ) ) )
= ( some @ nat @ ( zero_zero @ nat ) ) ) )
& ( ~ A2
=> ( ( vEBT_vebt_pred @ ( vEBT_Leaf @ A2 @ B2 ) @ ( suc @ ( suc @ Va2 ) ) )
= ( none @ nat ) ) ) ) ) ) ).
% vebt_pred.simps(3)
thf(fact_3456_word__unat__and__lt,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [X: word @ A,N: nat,Y: word @ A] :
( ( ( ord_less @ nat @ ( semiring_1_unsigned @ A @ nat @ X ) @ N )
| ( ord_less @ nat @ ( semiring_1_unsigned @ A @ nat @ Y ) @ N ) )
=> ( ord_less @ nat @ ( semiring_1_unsigned @ A @ nat @ ( bit_se5824344872417868541ns_and @ ( word @ A ) @ X @ Y ) ) @ N ) ) ) ).
% word_unat_and_lt
thf(fact_3457_VEBT__internal_Oreplicatei_Ocases,axiom,
! [A: $tType,X: product_prod @ nat @ ( heap_Time_Heap @ A )] :
( ! [X3: heap_Time_Heap @ A] :
( X
!= ( product_Pair @ nat @ ( heap_Time_Heap @ A ) @ ( zero_zero @ nat ) @ X3 ) )
=> ~ ! [N2: nat,X3: heap_Time_Heap @ A] :
( X
!= ( product_Pair @ nat @ ( heap_Time_Heap @ A ) @ ( suc @ N2 ) @ X3 ) ) ) ).
% VEBT_internal.replicatei.cases
thf(fact_3458_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_3459_VEBT__internal_Ovalid_H_Ocases,axiom,
! [X: product_prod @ vEBT_VEBT @ nat] :
( ! [Uu2: $o,Uv2: $o,D3: nat] :
( X
!= ( product_Pair @ vEBT_VEBT @ nat @ ( vEBT_Leaf @ Uu2 @ Uv2 ) @ 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_3460_vebt__succ_Osimps_I2_J,axiom,
! [Uv: $o,Uw: $o,N: nat] :
( ( vEBT_vebt_succ @ ( vEBT_Leaf @ Uv @ Uw ) @ ( suc @ N ) )
= ( none @ nat ) ) ).
% vebt_succ.simps(2)
thf(fact_3461_vebt__pred_Osimps_I1_J,axiom,
! [Uu: $o,Uv: $o] :
( ( vEBT_vebt_pred @ ( vEBT_Leaf @ Uu @ Uv ) @ ( zero_zero @ nat ) )
= ( none @ nat ) ) ).
% vebt_pred.simps(1)
thf(fact_3462_vebt__succ_Osimps_I3_J,axiom,
! [Ux: nat,Uy: list @ vEBT_VEBT,Uz: vEBT_VEBT,Va2: nat] :
( ( vEBT_vebt_succ @ ( vEBT_Node @ ( none @ ( product_prod @ nat @ nat ) ) @ Ux @ Uy @ Uz ) @ Va2 )
= ( none @ nat ) ) ).
% vebt_succ.simps(3)
thf(fact_3463_vebt__pred_Osimps_I4_J,axiom,
! [Uy: nat,Uz: list @ vEBT_VEBT,Va2: vEBT_VEBT,Vb: nat] :
( ( vEBT_vebt_pred @ ( vEBT_Node @ ( none @ ( product_prod @ nat @ nat ) ) @ Uy @ Uz @ Va2 ) @ Vb )
= ( none @ nat ) ) ).
% vebt_pred.simps(4)
thf(fact_3464_and__nat__def,axiom,
( ( bit_se5824344872417868541ns_and @ nat )
= ( ^ [M4: nat,N3: nat] : ( nat2 @ ( bit_se5824344872417868541ns_and @ int @ ( semiring_1_of_nat @ int @ M4 ) @ ( semiring_1_of_nat @ int @ N3 ) ) ) ) ) ).
% and_nat_def
thf(fact_3465_xor__num_Ocases,axiom,
! [X: product_prod @ num @ num] :
( ( X
!= ( product_Pair @ num @ num @ one2 @ one2 ) )
=> ( ! [N2: num] :
( X
!= ( product_Pair @ num @ num @ one2 @ ( bit0 @ N2 ) ) )
=> ( ! [N2: num] :
( X
!= ( product_Pair @ num @ num @ one2 @ ( bit1 @ N2 ) ) )
=> ( ! [M2: num] :
( X
!= ( product_Pair @ num @ num @ ( bit0 @ M2 ) @ one2 ) )
=> ( ! [M2: num,N2: num] :
( X
!= ( product_Pair @ num @ num @ ( bit0 @ M2 ) @ ( bit0 @ N2 ) ) )
=> ( ! [M2: num,N2: num] :
( X
!= ( product_Pair @ num @ num @ ( bit0 @ M2 ) @ ( bit1 @ N2 ) ) )
=> ( ! [M2: num] :
( X
!= ( product_Pair @ num @ num @ ( bit1 @ M2 ) @ one2 ) )
=> ( ! [M2: num,N2: num] :
( X
!= ( product_Pair @ num @ num @ ( bit1 @ M2 ) @ ( bit0 @ N2 ) ) )
=> ~ ! [M2: num,N2: num] :
( X
!= ( product_Pair @ num @ num @ ( bit1 @ M2 ) @ ( bit1 @ N2 ) ) ) ) ) ) ) ) ) ) ) ).
% xor_num.cases
thf(fact_3466_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_3467_vebt__succ_Osimps_I4_J,axiom,
! [V: product_prod @ nat @ nat,Vc2: list @ vEBT_VEBT,Vd: vEBT_VEBT,Ve: nat] :
( ( vEBT_vebt_succ @ ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ V ) @ ( zero_zero @ nat ) @ Vc2 @ Vd ) @ Ve )
= ( none @ nat ) ) ).
% vebt_succ.simps(4)
thf(fact_3468_VEBT__internal_Onaive__member_Ocases,axiom,
! [X: product_prod @ vEBT_VEBT @ nat] :
( ! [A6: $o,B3: $o,X3: nat] :
( X
!= ( product_Pair @ vEBT_VEBT @ nat @ ( vEBT_Leaf @ A6 @ B3 ) @ X3 ) )
=> ( ! [Uu2: option @ ( product_prod @ nat @ nat ),Uv2: list @ vEBT_VEBT,Uw2: vEBT_VEBT,Ux2: nat] :
( X
!= ( product_Pair @ vEBT_VEBT @ nat @ ( vEBT_Node @ Uu2 @ ( zero_zero @ nat ) @ Uv2 @ Uw2 ) @ Ux2 ) )
=> ~ ! [Uy2: option @ ( product_prod @ nat @ nat ),V4: nat,TreeList2: list @ vEBT_VEBT,S2: vEBT_VEBT,X3: nat] :
( X
!= ( product_Pair @ vEBT_VEBT @ nat @ ( vEBT_Node @ Uy2 @ ( suc @ V4 ) @ TreeList2 @ S2 ) @ X3 ) ) ) ) ).
% VEBT_internal.naive_member.cases
thf(fact_3469_even__word__iff,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ( ( even_word @ A )
= ( ^ [A4: word @ A] :
( ( bit_se5824344872417868541ns_and @ ( word @ A ) @ A4 @ ( one_one @ ( word @ A ) ) )
= ( zero_zero @ ( word @ A ) ) ) ) ) ) ).
% even_word_iff
thf(fact_3470_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_3471_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_3472_VEBT__internal_OT_092_060_094sub_062d_092_060_094sub_062e_092_060_094sub_062l_092_060_094sub_062e_092_060_094sub_062t_092_060_094sub_062e_H_Ocases,axiom,
! [X: product_prod @ vEBT_VEBT @ nat] :
( ! [A6: $o,B3: $o] :
( X
!= ( product_Pair @ vEBT_VEBT @ nat @ ( vEBT_Leaf @ A6 @ B3 ) @ ( zero_zero @ nat ) ) )
=> ( ! [A6: $o,B3: $o] :
( X
!= ( product_Pair @ vEBT_VEBT @ nat @ ( vEBT_Leaf @ A6 @ B3 ) @ ( suc @ ( zero_zero @ nat ) ) ) )
=> ( ! [A6: $o,B3: $o,N2: nat] :
( X
!= ( product_Pair @ vEBT_VEBT @ nat @ ( vEBT_Leaf @ A6 @ B3 ) @ ( suc @ ( suc @ N2 ) ) ) )
=> ( ! [Deg2: nat,TreeList2: list @ vEBT_VEBT,Summary2: vEBT_VEBT,Uu2: nat] :
( X
!= ( product_Pair @ vEBT_VEBT @ nat @ ( vEBT_Node @ ( none @ ( product_prod @ nat @ nat ) ) @ Deg2 @ TreeList2 @ Summary2 ) @ Uu2 ) )
=> ( ! [Mi: 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 @ Mi @ Ma2 ) ) @ ( zero_zero @ nat ) @ TreeList2 @ Summary2 ) @ X3 ) )
=> ( ! [Mi: 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 @ Mi @ Ma2 ) ) @ ( suc @ ( zero_zero @ nat ) ) @ TreeList2 @ Summary2 ) @ X3 ) )
=> ~ ! [Mi: nat,Ma2: nat,Va: nat,TreeList2: list @ vEBT_VEBT,Summary2: vEBT_VEBT,X3: nat] :
( X
!= ( product_Pair @ vEBT_VEBT @ nat @ ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi @ Ma2 ) ) @ ( suc @ ( suc @ Va ) ) @ TreeList2 @ Summary2 ) @ X3 ) ) ) ) ) ) ) ) ).
% VEBT_internal.T\<^sub>d\<^sub>e\<^sub>l\<^sub>e\<^sub>t\<^sub>e'.cases
thf(fact_3473_T_092_060_094sub_062p_092_060_094sub_062r_092_060_094sub_062e_092_060_094sub_062d_H_Ocases,axiom,
! [X: product_prod @ vEBT_VEBT @ nat] :
( ! [Uu2: $o,Uv2: $o] :
( X
!= ( product_Pair @ vEBT_VEBT @ nat @ ( vEBT_Leaf @ Uu2 @ Uv2 ) @ ( zero_zero @ nat ) ) )
=> ( ! [A6: $o,Uw2: $o] :
( X
!= ( product_Pair @ vEBT_VEBT @ nat @ ( vEBT_Leaf @ A6 @ Uw2 ) @ ( suc @ ( zero_zero @ nat ) ) ) )
=> ( ! [A6: $o,B3: $o,Va: nat] :
( X
!= ( product_Pair @ vEBT_VEBT @ nat @ ( vEBT_Leaf @ A6 @ B3 ) @ ( suc @ ( suc @ Va ) ) ) )
=> ( ! [Uy2: nat,Uz2: list @ vEBT_VEBT,Va3: vEBT_VEBT,Vb2: nat] :
( X
!= ( product_Pair @ vEBT_VEBT @ nat @ ( vEBT_Node @ ( none @ ( product_prod @ nat @ nat ) ) @ Uy2 @ Uz2 @ Va3 ) @ Vb2 ) )
=> ( ! [V4: product_prod @ nat @ nat,Vd2: list @ vEBT_VEBT,Ve2: vEBT_VEBT,Vf2: nat] :
( X
!= ( product_Pair @ vEBT_VEBT @ nat @ ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ V4 ) @ ( zero_zero @ nat ) @ Vd2 @ Ve2 ) @ Vf2 ) )
=> ( ! [V4: product_prod @ nat @ nat,Vh2: list @ vEBT_VEBT,Vi2: vEBT_VEBT,Vj2: nat] :
( X
!= ( product_Pair @ vEBT_VEBT @ nat @ ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ V4 ) @ ( suc @ ( zero_zero @ nat ) ) @ Vh2 @ Vi2 ) @ Vj2 ) )
=> ~ ! [Mi: nat,Ma2: nat,Va: nat,TreeList2: list @ vEBT_VEBT,Summary2: vEBT_VEBT,X3: nat] :
( X
!= ( product_Pair @ vEBT_VEBT @ nat @ ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi @ Ma2 ) ) @ ( suc @ ( suc @ Va ) ) @ TreeList2 @ Summary2 ) @ X3 ) ) ) ) ) ) ) ) ).
% T\<^sub>p\<^sub>r\<^sub>e\<^sub>d'.cases
thf(fact_3474_T_092_060_094sub_062s_092_060_094sub_062u_092_060_094sub_062c_092_060_094sub_062c_H_Ocases,axiom,
! [X: product_prod @ vEBT_VEBT @ nat] :
( ! [Uu2: $o,B3: $o] :
( X
!= ( product_Pair @ vEBT_VEBT @ nat @ ( vEBT_Leaf @ Uu2 @ B3 ) @ ( zero_zero @ nat ) ) )
=> ( ! [Uv2: $o,Uw2: $o,N2: nat] :
( X
!= ( product_Pair @ vEBT_VEBT @ nat @ ( vEBT_Leaf @ Uv2 @ Uw2 ) @ ( suc @ N2 ) ) )
=> ( ! [Ux2: nat,Uy2: list @ vEBT_VEBT,Uz2: vEBT_VEBT,Va3: nat] :
( X
!= ( product_Pair @ vEBT_VEBT @ nat @ ( vEBT_Node @ ( none @ ( product_prod @ nat @ nat ) ) @ Ux2 @ Uy2 @ Uz2 ) @ Va3 ) )
=> ( ! [V4: product_prod @ nat @ nat,Vc: list @ vEBT_VEBT,Vd2: vEBT_VEBT,Ve2: nat] :
( X
!= ( product_Pair @ vEBT_VEBT @ nat @ ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ V4 ) @ ( zero_zero @ nat ) @ Vc @ Vd2 ) @ Ve2 ) )
=> ( ! [V4: product_prod @ nat @ nat,Vg2: list @ vEBT_VEBT,Vh2: vEBT_VEBT,Vi2: nat] :
( X
!= ( product_Pair @ vEBT_VEBT @ nat @ ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ V4 ) @ ( suc @ ( zero_zero @ nat ) ) @ Vg2 @ Vh2 ) @ Vi2 ) )
=> ~ ! [Mi: nat,Ma2: nat,Va: nat,TreeList2: list @ vEBT_VEBT,Summary2: vEBT_VEBT,X3: nat] :
( X
!= ( product_Pair @ vEBT_VEBT @ nat @ ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi @ Ma2 ) ) @ ( suc @ ( suc @ Va ) ) @ TreeList2 @ Summary2 ) @ X3 ) ) ) ) ) ) ) ).
% T\<^sub>s\<^sub>u\<^sub>c\<^sub>c'.cases
thf(fact_3475_T_092_060_094sub_062i_092_060_094sub_062n_092_060_094sub_062s_092_060_094sub_062e_092_060_094sub_062r_092_060_094sub_062t_H_Ocases,axiom,
! [X: product_prod @ vEBT_VEBT @ nat] :
( ! [A6: $o,B3: $o,X3: nat] :
( X
!= ( product_Pair @ vEBT_VEBT @ nat @ ( vEBT_Leaf @ A6 @ 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 ) )
=> ( ! [V4: nat,TreeList2: list @ vEBT_VEBT,Summary2: vEBT_VEBT,X3: nat] :
( X
!= ( product_Pair @ vEBT_VEBT @ nat @ ( vEBT_Node @ ( none @ ( product_prod @ nat @ nat ) ) @ ( suc @ ( suc @ V4 ) ) @ TreeList2 @ Summary2 ) @ X3 ) )
=> ~ ! [Mi: nat,Ma2: nat,Va: nat,TreeList2: list @ vEBT_VEBT,Summary2: vEBT_VEBT,X3: nat] :
( X
!= ( product_Pair @ vEBT_VEBT @ nat @ ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi @ Ma2 ) ) @ ( suc @ ( suc @ Va ) ) @ TreeList2 @ Summary2 ) @ X3 ) ) ) ) ) ) ).
% T\<^sub>i\<^sub>n\<^sub>s\<^sub>e\<^sub>r\<^sub>t'.cases
thf(fact_3476_T_092_060_094sub_062m_092_060_094sub_062e_092_060_094sub_062m_092_060_094sub_062b_092_060_094sub_062e_092_060_094sub_062r_H_Ocases,axiom,
! [X: product_prod @ vEBT_VEBT @ nat] :
( ! [A6: $o,B3: $o,X3: nat] :
( X
!= ( product_Pair @ vEBT_VEBT @ nat @ ( vEBT_Leaf @ A6 @ B3 ) @ X3 ) )
=> ( ! [Uu2: nat,Uv2: list @ vEBT_VEBT,Uw2: vEBT_VEBT,X3: nat] :
( X
!= ( product_Pair @ vEBT_VEBT @ nat @ ( vEBT_Node @ ( none @ ( product_prod @ nat @ nat ) ) @ Uu2 @ Uv2 @ Uw2 ) @ X3 ) )
=> ( ! [V4: 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 ) @ V4 ) @ ( zero_zero @ nat ) @ Uy2 @ Uz2 ) @ X3 ) )
=> ( ! [V4: product_prod @ nat @ nat,Vb2: list @ vEBT_VEBT,Vc: vEBT_VEBT,X3: nat] :
( X
!= ( product_Pair @ vEBT_VEBT @ nat @ ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ V4 ) @ ( suc @ ( zero_zero @ nat ) ) @ Vb2 @ Vc ) @ X3 ) )
=> ~ ! [Mi: nat,Ma2: nat,Va: nat,TreeList2: list @ vEBT_VEBT,Summary2: vEBT_VEBT,X3: nat] :
( X
!= ( product_Pair @ vEBT_VEBT @ nat @ ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi @ Ma2 ) ) @ ( suc @ ( suc @ Va ) ) @ TreeList2 @ Summary2 ) @ X3 ) ) ) ) ) ) ).
% T\<^sub>m\<^sub>e\<^sub>m\<^sub>b\<^sub>e\<^sub>r'.cases
thf(fact_3477_VEBT__internal_Omembermima_Ocases,axiom,
! [X: product_prod @ vEBT_VEBT @ nat] :
( ! [Uu2: $o,Uv2: $o,Uw2: nat] :
( X
!= ( product_Pair @ vEBT_VEBT @ nat @ ( vEBT_Leaf @ Uu2 @ Uv2 ) @ Uw2 ) )
=> ( ! [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 ) )
=> ( ! [Mi: nat,Ma2: nat,Va3: list @ vEBT_VEBT,Vb2: vEBT_VEBT,X3: nat] :
( X
!= ( product_Pair @ vEBT_VEBT @ nat @ ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi @ Ma2 ) ) @ ( zero_zero @ nat ) @ Va3 @ Vb2 ) @ X3 ) )
=> ( ! [Mi: nat,Ma2: nat,V4: nat,TreeList2: list @ vEBT_VEBT,Vc: vEBT_VEBT,X3: nat] :
( X
!= ( product_Pair @ vEBT_VEBT @ nat @ ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi @ Ma2 ) ) @ ( suc @ V4 ) @ TreeList2 @ Vc ) @ X3 ) )
=> ~ ! [V4: nat,TreeList2: list @ vEBT_VEBT,Vd2: vEBT_VEBT,X3: nat] :
( X
!= ( product_Pair @ vEBT_VEBT @ nat @ ( vEBT_Node @ ( none @ ( product_prod @ nat @ nat ) ) @ ( suc @ V4 ) @ TreeList2 @ Vd2 ) @ X3 ) ) ) ) ) ) ).
% VEBT_internal.membermima.cases
thf(fact_3478_and__nat__unfold,axiom,
( ( bit_se5824344872417868541ns_and @ nat )
= ( ^ [M4: nat,N3: nat] :
( if @ nat
@ ( ( M4
= ( zero_zero @ nat ) )
| ( N3
= ( zero_zero @ nat ) ) )
@ ( zero_zero @ nat )
@ ( plus_plus @ nat @ ( times_times @ nat @ ( modulo_modulo @ nat @ M4 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ ( modulo_modulo @ nat @ N3 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( bit_se5824344872417868541ns_and @ nat @ ( divide_divide @ nat @ M4 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ ( divide_divide @ nat @ N3 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) ) ) ) ).
% and_nat_unfold
thf(fact_3479_and__nat__rec,axiom,
( ( bit_se5824344872417868541ns_and @ nat )
= ( ^ [M4: nat,N3: nat] :
( plus_plus @ nat
@ ( zero_neq_one_of_bool @ nat
@ ( ~ ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ M4 )
& ~ ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N3 ) ) )
@ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( bit_se5824344872417868541ns_and @ nat @ ( divide_divide @ nat @ M4 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ ( divide_divide @ nat @ N3 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) ) ) ).
% and_nat_rec
thf(fact_3480_minNull__delete__time__bound,axiom,
! [T: vEBT_VEBT,N: nat,X: nat] :
( ( vEBT_invar_vebt @ T @ N )
=> ( ( vEBT_VEBT_minNull @ ( vEBT_vebt_delete @ T @ X ) )
=> ( ord_less_eq @ nat @ ( vEBT_T_d_e_l_e_t_e @ T @ X ) @ ( numeral_numeral @ nat @ ( bit1 @ ( bit0 @ ( bit0 @ one2 ) ) ) ) ) ) ) ).
% minNull_delete_time_bound
thf(fact_3481_mintlistlength,axiom,
! [Mi3: nat,Ma: nat,Deg: nat,TreeList: list @ vEBT_VEBT,Summary: vEBT_VEBT,N: nat] :
( ( vEBT_invar_vebt @ ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi3 @ Ma ) ) @ Deg @ TreeList @ Summary ) @ N )
=> ( ( Mi3 != Ma )
=> ( ( ord_less @ nat @ Mi3 @ Ma )
& ? [M2: nat] :
( ( ( some @ nat @ M2 )
= ( vEBT_vebt_mint @ Summary ) )
& ( ord_less @ nat @ M2 @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( minus_minus @ nat @ N @ ( divide_divide @ nat @ N @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) ) ) ) ) ).
% mintlistlength
thf(fact_3482_vebt__delete_Osimps_I6_J,axiom,
! [Mi3: 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 @ Mi3 @ Ma ) ) @ ( suc @ ( zero_zero @ nat ) ) @ Tr @ Sm ) @ X )
= ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi3 @ Ma ) ) @ ( suc @ ( zero_zero @ nat ) ) @ Tr @ Sm ) ) ).
% vebt_delete.simps(6)
thf(fact_3483_vebt__delete_Osimps_I5_J,axiom,
! [Mi3: 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 @ Mi3 @ Ma ) ) @ ( zero_zero @ nat ) @ TrLst @ Smry ) @ X )
= ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi3 @ Ma ) ) @ ( zero_zero @ nat ) @ TrLst @ Smry ) ) ).
% vebt_delete.simps(5)
thf(fact_3484_not__min__Null__member,axiom,
! [T: vEBT_VEBT] :
( ~ ( vEBT_VEBT_minNull @ T )
=> ? [X_1: nat] : ( vEBT_V8194947554948674370ptions @ T @ X_1 ) ) ).
% not_min_Null_member
thf(fact_3485_min__Null__member,axiom,
! [T: vEBT_VEBT,X: nat] :
( ( vEBT_VEBT_minNull @ T )
=> ~ ( vEBT_vebt_member @ T @ X ) ) ).
% min_Null_member
thf(fact_3486_minNullmin,axiom,
! [T: vEBT_VEBT] :
( ( vEBT_VEBT_minNull @ T )
=> ( ( vEBT_vebt_mint @ T )
= ( none @ nat ) ) ) ).
% minNullmin
thf(fact_3487_minminNull,axiom,
! [T: vEBT_VEBT] :
( ( ( vEBT_vebt_mint @ T )
= ( none @ nat ) )
=> ( vEBT_VEBT_minNull @ T ) ) ).
% minminNull
thf(fact_3488_mint__member,axiom,
! [T: vEBT_VEBT,N: nat,Maxi: nat] :
( ( vEBT_invar_vebt @ T @ N )
=> ( ( ( vEBT_vebt_mint @ T )
= ( some @ nat @ Maxi ) )
=> ( vEBT_vebt_member @ T @ Maxi ) ) ) ).
% mint_member
thf(fact_3489_mint__corr__help,axiom,
! [T: vEBT_VEBT,N: nat,Mini: nat,X: nat] :
( ( vEBT_invar_vebt @ T @ N )
=> ( ( ( vEBT_vebt_mint @ T )
= ( some @ nat @ Mini ) )
=> ( ( vEBT_vebt_member @ T @ X )
=> ( ord_less_eq @ nat @ Mini @ X ) ) ) ) ).
% mint_corr_help
thf(fact_3490_mint__sound,axiom,
! [T: vEBT_VEBT,N: nat,X: nat] :
( ( vEBT_invar_vebt @ T @ N )
=> ( ( vEBT_VEBT_min_in_set @ ( vEBT_VEBT_set_vebt @ T ) @ X )
=> ( ( vEBT_vebt_mint @ T )
= ( some @ nat @ X ) ) ) ) ).
% mint_sound
thf(fact_3491_mint__corr,axiom,
! [T: vEBT_VEBT,N: nat,X: nat] :
( ( vEBT_invar_vebt @ T @ N )
=> ( ( ( vEBT_vebt_mint @ T )
= ( some @ nat @ X ) )
=> ( vEBT_VEBT_min_in_set @ ( vEBT_VEBT_set_vebt @ T ) @ X ) ) ) ).
% mint_corr
thf(fact_3492_minNull__delete__time__bound_H,axiom,
! [T: vEBT_VEBT,N: nat,X: nat] :
( ( vEBT_invar_vebt @ T @ N )
=> ( ( vEBT_VEBT_minNull @ ( vEBT_vebt_delete @ T @ X ) )
=> ( ord_less_eq @ nat @ ( vEBT_V1232361888498592333_e_t_e @ T @ X ) @ ( one_one @ nat ) ) ) ) ).
% minNull_delete_time_bound'
thf(fact_3493_misiz,axiom,
! [T: vEBT_VEBT,N: nat,M: nat] :
( ( vEBT_invar_vebt @ T @ N )
=> ( ( ( some @ nat @ M )
= ( vEBT_vebt_mint @ T ) )
=> ( ord_less @ nat @ M @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N ) ) ) ) ).
% misiz
thf(fact_3494_local_Opower__def,axiom,
( vEBT_VEBT_power
= ( vEBT_V2048590022279873568_shift @ nat @ ( power_power @ nat ) ) ) ).
% local.power_def
thf(fact_3495_VEBT__internal_OminNull_Osimps_I4_J,axiom,
! [Uw: nat,Ux: list @ vEBT_VEBT,Uy: vEBT_VEBT] : ( vEBT_VEBT_minNull @ ( vEBT_Node @ ( none @ ( product_prod @ nat @ nat ) ) @ Uw @ Ux @ Uy ) ) ).
% VEBT_internal.minNull.simps(4)
thf(fact_3496_VEBT__internal_OminNull_Oelims_I2_J,axiom,
! [X: vEBT_VEBT] :
( ( vEBT_VEBT_minNull @ X )
=> ( ( X
!= ( vEBT_Leaf @ $false @ $false ) )
=> ~ ! [Uw2: nat,Ux2: list @ vEBT_VEBT,Uy2: vEBT_VEBT] :
( X
!= ( vEBT_Node @ ( none @ ( product_prod @ nat @ nat ) ) @ Uw2 @ Ux2 @ Uy2 ) ) ) ) ).
% VEBT_internal.minNull.elims(2)
thf(fact_3497_VEBT__internal_OminNull_Oelims_I1_J,axiom,
! [X: vEBT_VEBT,Y: $o] :
( ( ( vEBT_VEBT_minNull @ X )
= Y )
=> ( ( ( X
= ( vEBT_Leaf @ $false @ $false ) )
=> ~ Y )
=> ( ( ? [Uv2: $o] :
( X
= ( vEBT_Leaf @ $true @ Uv2 ) )
=> Y )
=> ( ( ? [Uu2: $o] :
( X
= ( vEBT_Leaf @ Uu2 @ $true ) )
=> Y )
=> ( ( ? [Uw2: nat,Ux2: list @ vEBT_VEBT,Uy2: vEBT_VEBT] :
( X
= ( vEBT_Node @ ( none @ ( product_prod @ nat @ nat ) ) @ Uw2 @ Ux2 @ Uy2 ) )
=> ~ Y )
=> ~ ( ? [Uz2: product_prod @ nat @ nat,Va3: nat,Vb2: list @ vEBT_VEBT,Vc: vEBT_VEBT] :
( X
= ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ Uz2 ) @ Va3 @ Vb2 @ Vc ) )
=> Y ) ) ) ) ) ) ).
% VEBT_internal.minNull.elims(1)
thf(fact_3498_insertsimp,axiom,
! [T: vEBT_VEBT,N: nat,L: nat] :
( ( vEBT_invar_vebt @ T @ N )
=> ( ( vEBT_VEBT_minNull @ T )
=> ( ord_less_eq @ nat @ ( vEBT_T_i_n_s_e_r_t @ T @ L ) @ ( numeral_numeral @ nat @ ( bit1 @ one2 ) ) ) ) ) ).
% insertsimp
thf(fact_3499_vebt__delete_Osimps_I3_J,axiom,
! [A2: $o,B2: $o,N: nat] :
( ( vEBT_vebt_delete @ ( vEBT_Leaf @ A2 @ B2 ) @ ( suc @ ( suc @ N ) ) )
= ( vEBT_Leaf @ A2 @ B2 ) ) ).
% vebt_delete.simps(3)
thf(fact_3500_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_3501_vebt__delete_Osimps_I4_J,axiom,
! [Deg: nat,TreeList: list @ vEBT_VEBT,Summary: vEBT_VEBT,Uu: nat] :
( ( vEBT_vebt_delete @ ( vEBT_Node @ ( none @ ( product_prod @ nat @ nat ) ) @ Deg @ TreeList @ Summary ) @ Uu )
= ( vEBT_Node @ ( none @ ( product_prod @ nat @ nat ) ) @ Deg @ TreeList @ Summary ) ) ).
% vebt_delete.simps(4)
thf(fact_3502_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_3503_nested__mint,axiom,
! [Mi3: nat,Ma: nat,Deg: nat,TreeList: list @ vEBT_VEBT,Summary: vEBT_VEBT,N: nat,Va2: nat] :
( ( vEBT_invar_vebt @ ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi3 @ Ma ) ) @ Deg @ TreeList @ Summary ) @ N )
=> ( ( N
= ( suc @ ( suc @ Va2 ) ) )
=> ( ~ ( ord_less @ nat @ Ma @ Mi3 )
=> ( ( Ma != Mi3 )
=> ( ord_less @ nat @ ( vEBT_VEBT_high @ ( plus_plus @ nat @ ( times_times @ nat @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ Va2 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ ( suc @ ( divide_divide @ nat @ Va2 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( size_size @ ( list @ vEBT_VEBT ) @ TreeList ) ) ) ) ) ) ).
% nested_mint
thf(fact_3504_vebt__mint_Oelims,axiom,
! [X: vEBT_VEBT,Y: option @ nat] :
( ( ( vEBT_vebt_mint @ X )
= Y )
=> ( ! [A6: $o,B3: $o] :
( ( X
= ( vEBT_Leaf @ A6 @ B3 ) )
=> ~ ( ( A6
=> ( Y
= ( some @ nat @ ( zero_zero @ nat ) ) ) )
& ( ~ A6
=> ( ( B3
=> ( Y
= ( some @ nat @ ( one_one @ nat ) ) ) )
& ( ~ B3
=> ( Y
= ( none @ nat ) ) ) ) ) ) )
=> ( ( ? [Uu2: nat,Uv2: list @ vEBT_VEBT,Uw2: vEBT_VEBT] :
( X
= ( vEBT_Node @ ( none @ ( product_prod @ nat @ nat ) ) @ Uu2 @ Uv2 @ Uw2 ) )
=> ( Y
!= ( none @ nat ) ) )
=> ~ ! [Mi: nat] :
( ? [Ma2: nat,Ux2: nat,Uy2: list @ vEBT_VEBT,Uz2: vEBT_VEBT] :
( X
= ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi @ Ma2 ) ) @ Ux2 @ Uy2 @ Uz2 ) )
=> ( Y
!= ( some @ nat @ Mi ) ) ) ) ) ) ).
% vebt_mint.elims
thf(fact_3505_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_3506_greater__shift,axiom,
( ( ord_less @ nat )
= ( ^ [Y6: nat,X2: nat] : ( vEBT_VEBT_greater @ ( some @ nat @ X2 ) @ ( some @ nat @ Y6 ) ) ) ) ).
% greater_shift
thf(fact_3507_option_Ocollapse,axiom,
! [A: $tType,Option: option @ A] :
( ( Option
!= ( none @ A ) )
=> ( ( some @ A @ ( the2 @ A @ Option ) )
= Option ) ) ).
% option.collapse
thf(fact_3508_VEBT__internal_Ooption__shift_Osimps_I1_J,axiom,
! [A: $tType,Uu: A > A > A,Uv: option @ A] :
( ( vEBT_V2048590022279873568_shift @ A @ Uu @ ( none @ A ) @ Uv )
= ( none @ A ) ) ).
% VEBT_internal.option_shift.simps(1)
thf(fact_3509_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_3510_option_Oexhaust__sel,axiom,
! [A: $tType,Option: option @ A] :
( ( Option
!= ( none @ A ) )
=> ( Option
= ( some @ A @ ( the2 @ A @ Option ) ) ) ) ).
% option.exhaust_sel
thf(fact_3511_VEBT__internal_Ooption__shift_Osimps_I2_J,axiom,
! [A: $tType,Uw: A > A > A,V: A] :
( ( vEBT_V2048590022279873568_shift @ A @ Uw @ ( some @ A @ V ) @ ( none @ A ) )
= ( none @ A ) ) ).
% VEBT_internal.option_shift.simps(2)
thf(fact_3512_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 ) ) )
=> ( ( ? [V4: A] :
( Xa
= ( some @ A @ V4 ) )
=> ( ( Xb
= ( none @ A ) )
=> ( Y
!= ( none @ A ) ) ) )
=> ~ ! [A6: A] :
( ( Xa
= ( some @ A @ A6 ) )
=> ! [B3: A] :
( ( Xb
= ( some @ A @ B3 ) )
=> ( Y
!= ( some @ A @ ( X @ A6 @ B3 ) ) ) ) ) ) ) ) ).
% VEBT_internal.option_shift.elims
thf(fact_3513_VEBT__internal_Ooption__comp__shift_Ocases,axiom,
! [A: $tType,X: product_prod @ ( A > A > $o ) @ ( product_prod @ ( option @ A ) @ ( option @ A ) )] :
( ! [Uu2: A > A > $o,Uv2: option @ A] :
( X
!= ( product_Pair @ ( A > A > $o ) @ ( product_prod @ ( option @ A ) @ ( option @ A ) ) @ Uu2 @ ( product_Pair @ ( option @ A ) @ ( option @ A ) @ ( none @ A ) @ Uv2 ) ) )
=> ( ! [Uw2: A > A > $o,V4: A] :
( X
!= ( product_Pair @ ( A > A > $o ) @ ( product_prod @ ( option @ A ) @ ( option @ A ) ) @ Uw2 @ ( product_Pair @ ( option @ A ) @ ( option @ A ) @ ( some @ A @ V4 ) @ ( none @ A ) ) ) )
=> ~ ! [F5: A > A > $o,X3: A,Y3: A] :
( X
!= ( product_Pair @ ( A > A > $o ) @ ( product_prod @ ( option @ A ) @ ( option @ A ) ) @ F5 @ ( product_Pair @ ( option @ A ) @ ( option @ A ) @ ( some @ A @ X3 ) @ ( some @ A @ Y3 ) ) ) ) ) ) ).
% VEBT_internal.option_comp_shift.cases
thf(fact_3514_VEBT__internal_Ooption__shift_Ocases,axiom,
! [A: $tType,X: product_prod @ ( A > A > A ) @ ( product_prod @ ( option @ A ) @ ( option @ A ) )] :
( ! [Uu2: A > A > A,Uv2: option @ A] :
( X
!= ( product_Pair @ ( A > A > A ) @ ( product_prod @ ( option @ A ) @ ( option @ A ) ) @ Uu2 @ ( product_Pair @ ( option @ A ) @ ( option @ A ) @ ( none @ A ) @ Uv2 ) ) )
=> ( ! [Uw2: A > A > A,V4: A] :
( X
!= ( product_Pair @ ( A > A > A ) @ ( product_prod @ ( option @ A ) @ ( option @ A ) ) @ Uw2 @ ( product_Pair @ ( option @ A ) @ ( option @ A ) @ ( some @ A @ V4 ) @ ( none @ A ) ) ) )
=> ~ ! [F5: A > A > A,A6: A,B3: A] :
( X
!= ( product_Pair @ ( A > A > A ) @ ( product_prod @ ( option @ A ) @ ( option @ A ) ) @ F5 @ ( product_Pair @ ( option @ A ) @ ( option @ A ) @ ( some @ A @ A6 ) @ ( some @ A @ B3 ) ) ) ) ) ) ).
% VEBT_internal.option_shift.cases
thf(fact_3515_vebt__mint_Osimps_I2_J,axiom,
! [Uu: nat,Uv: list @ vEBT_VEBT,Uw: vEBT_VEBT] :
( ( vEBT_vebt_mint @ ( vEBT_Node @ ( none @ ( product_prod @ nat @ nat ) ) @ Uu @ Uv @ Uw ) )
= ( none @ nat ) ) ).
% vebt_mint.simps(2)
thf(fact_3516_summaxma,axiom,
! [Mi3: 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 @ Mi3 @ Ma ) ) @ Deg @ TreeList @ Summary ) @ Deg )
=> ( ( Mi3 != Ma )
=> ( ( the2 @ nat @ ( vEBT_vebt_maxt @ Summary ) )
= ( vEBT_VEBT_high @ Ma @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) ) ).
% summaxma
thf(fact_3517_mul__def,axiom,
( vEBT_VEBT_mul
= ( vEBT_V2048590022279873568_shift @ nat @ ( times_times @ nat ) ) ) ).
% mul_def
thf(fact_3518_add__def,axiom,
( vEBT_VEBT_add
= ( vEBT_V2048590022279873568_shift @ nat @ ( plus_plus @ nat ) ) ) ).
% add_def
thf(fact_3519_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_3520_maxbmo,axiom,
! [T: vEBT_VEBT,X: nat] :
( ( ( vEBT_vebt_maxt @ T )
= ( some @ nat @ X ) )
=> ( vEBT_V8194947554948674370ptions @ T @ X ) ) ).
% maxbmo
thf(fact_3521_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_3522_maxt__member,axiom,
! [T: vEBT_VEBT,N: nat,Maxi: nat] :
( ( vEBT_invar_vebt @ T @ N )
=> ( ( ( vEBT_vebt_maxt @ T )
= ( some @ nat @ Maxi ) )
=> ( vEBT_vebt_member @ T @ Maxi ) ) ) ).
% maxt_member
thf(fact_3523_list__update__overwrite,axiom,
! [A: $tType,Xs2: list @ A,I: nat,X: A,Y: A] :
( ( list_update @ A @ ( list_update @ A @ Xs2 @ I @ X ) @ I @ Y )
= ( list_update @ A @ Xs2 @ I @ Y ) ) ).
% list_update_overwrite
thf(fact_3524_maxt__corr__help,axiom,
! [T: vEBT_VEBT,N: nat,Maxi: nat,X: nat] :
( ( vEBT_invar_vebt @ T @ N )
=> ( ( ( vEBT_vebt_maxt @ T )
= ( some @ nat @ Maxi ) )
=> ( ( vEBT_vebt_member @ T @ X )
=> ( ord_less_eq @ nat @ X @ Maxi ) ) ) ) ).
% maxt_corr_help
thf(fact_3525_maxt__sound,axiom,
! [T: vEBT_VEBT,N: nat,X: nat] :
( ( vEBT_invar_vebt @ T @ N )
=> ( ( vEBT_VEBT_max_in_set @ ( vEBT_VEBT_set_vebt @ T ) @ X )
=> ( ( vEBT_vebt_maxt @ T )
= ( some @ nat @ X ) ) ) ) ).
% maxt_sound
thf(fact_3526_maxt__corr,axiom,
! [T: vEBT_VEBT,N: nat,X: nat] :
( ( vEBT_invar_vebt @ T @ N )
=> ( ( ( vEBT_vebt_maxt @ T )
= ( some @ nat @ X ) )
=> ( vEBT_VEBT_max_in_set @ ( vEBT_VEBT_set_vebt @ T ) @ X ) ) ) ).
% maxt_corr
thf(fact_3527_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_3528_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_3529_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_3530_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_3531_nth__update__invalid,axiom,
! [A: $tType,I: nat,L: list @ A,J: nat,X: A] :
( ~ ( ord_less @ nat @ I @ ( size_size @ ( list @ A ) @ L ) )
=> ( ( nth @ A @ ( list_update @ A @ L @ J @ X ) @ I )
= ( nth @ A @ L @ I ) ) ) ).
% nth_update_invalid
thf(fact_3532_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_3533_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_3534_del__x__not__mi__newnode__not__nil,axiom,
! [Mi3: nat,X: nat,Ma: nat,Deg: nat,H2: nat,L: nat,Newnode: vEBT_VEBT,TreeList: list @ vEBT_VEBT,Newlist: list @ vEBT_VEBT,Summary: vEBT_VEBT] :
( ( ( ord_less @ nat @ Mi3 @ X )
& ( ord_less_eq @ nat @ X @ Ma ) )
=> ( ( Mi3 != Ma )
=> ( ( ord_less_eq @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ Deg )
=> ( ( ( vEBT_VEBT_high @ X @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) )
= H2 )
=> ( ( ( vEBT_VEBT_low @ X @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) )
= L )
=> ( ( Newnode
= ( vEBT_vebt_delete @ ( nth @ vEBT_VEBT @ TreeList @ H2 ) @ L ) )
=> ( ~ ( vEBT_VEBT_minNull @ Newnode )
=> ( ( Newlist
= ( list_update @ vEBT_VEBT @ TreeList @ H2 @ Newnode ) )
=> ( ( ord_less @ nat @ ( vEBT_VEBT_high @ X @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( size_size @ ( list @ vEBT_VEBT ) @ TreeList ) )
=> ( ( vEBT_vebt_delete @ ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi3 @ Ma ) ) @ Deg @ TreeList @ Summary ) @ X )
= ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi3 @ ( 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_3535_del__x__mi__lets__in__not__minNull,axiom,
! [X: nat,Mi3: nat,Ma: nat,Deg: nat,Xn: nat,H2: nat,Summary: vEBT_VEBT,TreeList: list @ vEBT_VEBT,L: nat,Newnode: vEBT_VEBT,Newlist: list @ vEBT_VEBT] :
( ( ( X = Mi3 )
& ( ord_less @ nat @ X @ Ma ) )
=> ( ( Mi3 != Ma )
=> ( ( ord_less_eq @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ Deg )
=> ( ( ( vEBT_VEBT_high @ Xn @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) )
= H2 )
=> ( ( Xn
= ( plus_plus @ nat @ ( times_times @ nat @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) )
=> ( ( ( vEBT_VEBT_low @ Xn @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) )
= L )
=> ( ( ord_less @ nat @ ( vEBT_VEBT_high @ Xn @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( size_size @ ( list @ vEBT_VEBT ) @ TreeList ) )
=> ( ( Newnode
= ( vEBT_vebt_delete @ ( nth @ vEBT_VEBT @ TreeList @ H2 ) @ L ) )
=> ( ( Newlist
= ( list_update @ vEBT_VEBT @ TreeList @ H2 @ Newnode ) )
=> ( ~ ( vEBT_VEBT_minNull @ Newnode )
=> ( ( vEBT_vebt_delete @ ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi3 @ 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_3536_list__update__swap,axiom,
! [A: $tType,I: nat,I5: nat,Xs2: list @ A,X: A,X6: A] :
( ( I != I5 )
=> ( ( list_update @ A @ ( list_update @ A @ Xs2 @ I @ X ) @ I5 @ X6 )
= ( list_update @ A @ ( list_update @ A @ Xs2 @ I5 @ X6 ) @ I @ X ) ) ) ).
% list_update_swap
thf(fact_3537_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_3538_in__set__upd__eq,axiom,
! [A: $tType,I: nat,L: list @ A,X: A,Y: A] :
( ( ord_less @ nat @ I @ ( size_size @ ( list @ A ) @ L ) )
=> ( ( member @ A @ X @ ( set2 @ A @ ( list_update @ A @ L @ I @ Y ) ) )
= ( ( X = Y )
| ( ( member @ A @ X @ ( set2 @ A @ L ) )
& ! [Y6: A] : ( member @ A @ X @ ( set2 @ A @ ( list_update @ A @ L @ I @ Y6 ) ) ) ) ) ) ) ).
% in_set_upd_eq
thf(fact_3539_set__update__memI,axiom,
! [A: $tType,N: nat,Xs2: list @ A,X: A] :
( ( ord_less @ nat @ N @ ( size_size @ ( list @ A ) @ Xs2 ) )
=> ( member @ A @ X @ ( set2 @ A @ ( list_update @ A @ Xs2 @ N @ X ) ) ) ) ).
% set_update_memI
thf(fact_3540_in__set__upd__cases,axiom,
! [A: $tType,X: A,L: list @ A,I: nat,Y: A] :
( ( member @ A @ X @ ( set2 @ A @ ( list_update @ A @ L @ I @ Y ) ) )
=> ( ( ( ord_less @ nat @ I @ ( size_size @ ( list @ A ) @ L ) )
=> ( X != Y ) )
=> ( member @ A @ X @ ( set2 @ A @ L ) ) ) ) ).
% in_set_upd_cases
thf(fact_3541_in__set__upd__eq__aux,axiom,
! [A: $tType,I: nat,L: list @ A,X: A,Y: A] :
( ( ord_less @ nat @ I @ ( size_size @ ( list @ A ) @ L ) )
=> ( ( member @ A @ X @ ( set2 @ A @ ( list_update @ A @ L @ I @ Y ) ) )
= ( ( X = Y )
| ! [Y6: A] : ( member @ A @ X @ ( set2 @ A @ ( list_update @ A @ L @ I @ Y6 ) ) ) ) ) ) ).
% in_set_upd_eq_aux
thf(fact_3542_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_3543_nth__list__update_H,axiom,
! [A: $tType,I: nat,J: nat,L: list @ A,X: A] :
( ( ( ( I = J )
& ( ord_less @ nat @ I @ ( size_size @ ( list @ A ) @ L ) ) )
=> ( ( nth @ A @ ( list_update @ A @ L @ I @ X ) @ J )
= X ) )
& ( ~ ( ( I = J )
& ( ord_less @ nat @ I @ ( size_size @ ( list @ A ) @ L ) ) )
=> ( ( nth @ A @ ( list_update @ A @ L @ I @ X ) @ J )
= ( nth @ A @ L @ J ) ) ) ) ).
% nth_list_update'
thf(fact_3544_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_3545_vebt__maxt_Osimps_I2_J,axiom,
! [Uu: nat,Uv: list @ vEBT_VEBT,Uw: vEBT_VEBT] :
( ( vEBT_vebt_maxt @ ( vEBT_Node @ ( none @ ( product_prod @ nat @ nat ) ) @ Uu @ Uv @ Uw ) )
= ( none @ nat ) ) ).
% vebt_maxt.simps(2)
thf(fact_3546_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_3547_vebt__maxt_Oelims,axiom,
! [X: vEBT_VEBT,Y: option @ nat] :
( ( ( vEBT_vebt_maxt @ X )
= Y )
=> ( ! [A6: $o,B3: $o] :
( ( X
= ( vEBT_Leaf @ A6 @ B3 ) )
=> ~ ( ( B3
=> ( Y
= ( some @ nat @ ( one_one @ nat ) ) ) )
& ( ~ B3
=> ( ( A6
=> ( Y
= ( some @ nat @ ( zero_zero @ nat ) ) ) )
& ( ~ A6
=> ( Y
= ( none @ nat ) ) ) ) ) ) )
=> ( ( ? [Uu2: nat,Uv2: list @ vEBT_VEBT,Uw2: vEBT_VEBT] :
( X
= ( vEBT_Node @ ( none @ ( product_prod @ nat @ nat ) ) @ Uu2 @ Uv2 @ Uw2 ) )
=> ( Y
!= ( none @ nat ) ) )
=> ~ ! [Mi: nat,Ma2: nat] :
( ? [Ux2: nat,Uy2: list @ vEBT_VEBT,Uz2: vEBT_VEBT] :
( X
= ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi @ Ma2 ) ) @ Ux2 @ Uy2 @ Uz2 ) )
=> ( Y
!= ( some @ nat @ Ma2 ) ) ) ) ) ) ).
% vebt_maxt.elims
thf(fact_3548_insert__simp__norm,axiom,
! [X: nat,Deg: nat,TreeList: list @ vEBT_VEBT,Mi3: 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 @ Mi3 @ 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 @ Mi3 @ Ma ) ) @ Deg @ TreeList @ Summary ) @ X )
= ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi3 @ ( 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_3549_insert__simp__excp,axiom,
! [Mi3: nat,Deg: nat,TreeList: list @ vEBT_VEBT,X: nat,Ma: nat,Summary: vEBT_VEBT] :
( ( ord_less @ nat @ ( vEBT_VEBT_high @ Mi3 @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( size_size @ ( list @ vEBT_VEBT ) @ TreeList ) )
=> ( ( ord_less @ nat @ X @ Mi3 )
=> ( ( 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 @ Mi3 @ Ma ) ) @ Deg @ TreeList @ Summary ) @ X )
= ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ X @ ( ord_max @ nat @ Mi3 @ Ma ) ) ) @ Deg @ ( list_update @ vEBT_VEBT @ TreeList @ ( vEBT_VEBT_high @ Mi3 @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( vEBT_vebt_insert @ ( nth @ vEBT_VEBT @ TreeList @ ( vEBT_VEBT_high @ Mi3 @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( vEBT_VEBT_low @ Mi3 @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) @ ( if @ vEBT_VEBT @ ( vEBT_VEBT_minNull @ ( nth @ vEBT_VEBT @ TreeList @ ( vEBT_VEBT_high @ Mi3 @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) @ ( vEBT_vebt_insert @ Summary @ ( vEBT_VEBT_high @ Mi3 @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ Summary ) ) ) ) ) ) ) ).
% insert_simp_excp
thf(fact_3550_less__shift,axiom,
( ( ord_less @ nat )
= ( ^ [X2: nat,Y6: nat] : ( vEBT_VEBT_less @ ( some @ nat @ X2 ) @ ( some @ nat @ Y6 ) ) ) ) ).
% less_shift
thf(fact_3551_max__Suc__Suc,axiom,
! [M: nat,N: nat] :
( ( ord_max @ nat @ ( suc @ M ) @ ( suc @ N ) )
= ( suc @ ( ord_max @ nat @ M @ N ) ) ) ).
% max_Suc_Suc
thf(fact_3552_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_3553_max__nat_Oleft__neutral,axiom,
! [A2: nat] :
( ( ord_max @ nat @ ( zero_zero @ nat ) @ A2 )
= A2 ) ).
% max_nat.left_neutral
thf(fact_3554_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_3555_max__nat_Oright__neutral,axiom,
! [A2: nat] :
( ( ord_max @ nat @ A2 @ ( zero_zero @ nat ) )
= A2 ) ).
% max_nat.right_neutral
thf(fact_3556_max__0L,axiom,
! [N: nat] :
( ( ord_max @ nat @ ( zero_zero @ nat ) @ N )
= N ) ).
% max_0L
thf(fact_3557_max__0R,axiom,
! [N: nat] :
( ( ord_max @ nat @ N @ ( zero_zero @ nat ) )
= N ) ).
% max_0R
thf(fact_3558_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_3559_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_3560_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_3561_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_3562_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_3563_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_3564_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_3565_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_3566_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_3567_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_3568_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_3569_max__Suc__numeral,axiom,
! [N: nat,K: num] :
( ( ord_max @ nat @ ( suc @ N ) @ ( numeral_numeral @ nat @ K ) )
= ( suc @ ( ord_max @ nat @ N @ ( pred_numeral @ K ) ) ) ) ).
% max_Suc_numeral
thf(fact_3570_max__numeral__Suc,axiom,
! [K: num,N: nat] :
( ( ord_max @ nat @ ( numeral_numeral @ nat @ K ) @ ( suc @ N ) )
= ( suc @ ( ord_max @ nat @ ( pred_numeral @ K ) @ N ) ) ) ).
% max_numeral_Suc
thf(fact_3571_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_3572_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_3573_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_3574_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_3575_nat__mult__max__right,axiom,
! [M: nat,N: nat,Q3: nat] :
( ( times_times @ nat @ M @ ( ord_max @ nat @ N @ Q3 ) )
= ( ord_max @ nat @ ( times_times @ nat @ M @ N ) @ ( times_times @ nat @ M @ Q3 ) ) ) ).
% nat_mult_max_right
thf(fact_3576_nat__mult__max__left,axiom,
! [M: nat,N: nat,Q3: nat] :
( ( times_times @ nat @ ( ord_max @ nat @ M @ N ) @ Q3 )
= ( ord_max @ nat @ ( times_times @ nat @ M @ Q3 ) @ ( times_times @ nat @ N @ Q3 ) ) ) ).
% nat_mult_max_left
thf(fact_3577_nat__add__max__left,axiom,
! [M: nat,N: nat,Q3: nat] :
( ( plus_plus @ nat @ ( ord_max @ nat @ M @ N ) @ Q3 )
= ( ord_max @ nat @ ( plus_plus @ nat @ M @ Q3 ) @ ( plus_plus @ nat @ N @ Q3 ) ) ) ).
% nat_add_max_left
thf(fact_3578_nat__add__max__right,axiom,
! [M: nat,N: nat,Q3: nat] :
( ( plus_plus @ nat @ M @ ( ord_max @ nat @ N @ Q3 ) )
= ( ord_max @ nat @ ( plus_plus @ nat @ M @ N ) @ ( plus_plus @ nat @ M @ Q3 ) ) ) ).
% nat_add_max_right
thf(fact_3579_nat__minus__add__max,axiom,
! [N: nat,M: nat] :
( ( plus_plus @ nat @ ( minus_minus @ nat @ N @ M ) @ M )
= ( ord_max @ nat @ N @ M ) ) ).
% nat_minus_add_max
thf(fact_3580_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_3581_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_3582_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_3583_succ__greatereq__min,axiom,
! [Deg: nat,Mi3: 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 @ Mi3 @ X )
=> ( ( vEBT_vebt_succ @ ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi3 @ 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_3584_succ__less__length__list,axiom,
! [Deg: nat,Mi3: 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 @ Mi3 @ 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 @ Mi3 @ 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_3585_set__vebt_H__def,axiom,
( vEBT_VEBT_set_vebt
= ( ^ [T2: vEBT_VEBT] : ( collect @ nat @ ( vEBT_vebt_member @ T2 ) ) ) ) ).
% set_vebt'_def
thf(fact_3586_max__enat__simps_I2_J,axiom,
! [Q3: extended_enat] :
( ( ord_max @ extended_enat @ Q3 @ ( zero_zero @ extended_enat ) )
= Q3 ) ).
% max_enat_simps(2)
thf(fact_3587_max__enat__simps_I3_J,axiom,
! [Q3: extended_enat] :
( ( ord_max @ extended_enat @ ( zero_zero @ extended_enat ) @ Q3 )
= Q3 ) ).
% max_enat_simps(3)
thf(fact_3588_finite__interval__int4,axiom,
! [A2: int,B2: int] :
( finite_finite2 @ int
@ ( collect @ int
@ ^ [I2: int] :
( ( ord_less @ int @ A2 @ I2 )
& ( ord_less @ int @ I2 @ B2 ) ) ) ) ).
% finite_interval_int4
thf(fact_3589_finite__nth__roots,axiom,
! [N: nat,C2: complex] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ N )
=> ( finite_finite2 @ complex
@ ( collect @ complex
@ ^ [Z7: complex] :
( ( power_power @ complex @ Z7 @ N )
= C2 ) ) ) ) ).
% finite_nth_roots
thf(fact_3590_finite__interval__int3,axiom,
! [A2: int,B2: int] :
( finite_finite2 @ int
@ ( collect @ int
@ ^ [I2: int] :
( ( ord_less @ int @ A2 @ I2 )
& ( ord_less_eq @ int @ I2 @ B2 ) ) ) ) ).
% finite_interval_int3
thf(fact_3591_finite__interval__int2,axiom,
! [A2: int,B2: int] :
( finite_finite2 @ int
@ ( collect @ int
@ ^ [I2: int] :
( ( ord_less_eq @ int @ A2 @ I2 )
& ( ord_less @ int @ I2 @ B2 ) ) ) ) ).
% finite_interval_int2
thf(fact_3592_del__x__not__mi,axiom,
! [Mi3: nat,X: nat,Ma: nat,Deg: nat,H2: nat,L: nat,Newnode: vEBT_VEBT,TreeList: list @ vEBT_VEBT,Newlist: list @ vEBT_VEBT,Summary: vEBT_VEBT] :
( ( ( ord_less @ nat @ Mi3 @ X )
& ( ord_less_eq @ nat @ X @ Ma ) )
=> ( ( Mi3 != Ma )
=> ( ( ord_less_eq @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ Deg )
=> ( ( ( vEBT_VEBT_high @ X @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) )
= H2 )
=> ( ( ( vEBT_VEBT_low @ X @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) )
= L )
=> ( ( Newnode
= ( vEBT_vebt_delete @ ( nth @ vEBT_VEBT @ TreeList @ H2 ) @ L ) )
=> ( ( Newlist
= ( list_update @ vEBT_VEBT @ TreeList @ H2 @ Newnode ) )
=> ( ( ord_less @ nat @ ( vEBT_VEBT_high @ X @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( size_size @ ( list @ vEBT_VEBT ) @ TreeList ) )
=> ( ( ( vEBT_VEBT_minNull @ Newnode )
=> ( ( vEBT_vebt_delete @ ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi3 @ Ma ) ) @ Deg @ TreeList @ Summary ) @ X )
= ( vEBT_Node
@ ( some @ ( product_prod @ nat @ nat )
@ ( product_Pair @ nat @ nat @ Mi3
@ ( if @ nat @ ( X = Ma )
@ ( if @ nat
@ ( ( vEBT_vebt_maxt @ ( vEBT_vebt_delete @ Summary @ H2 ) )
= ( none @ nat ) )
@ Mi3
@ ( plus_plus @ nat @ ( times_times @ nat @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_maxt @ ( vEBT_vebt_delete @ Summary @ 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 @ Mi3 @ Ma ) ) @ Deg @ TreeList @ Summary ) @ X )
= ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi3 @ ( 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_3593_del__x__not__mi__new__node__nil,axiom,
! [Mi3: nat,X: nat,Ma: nat,Deg: nat,H2: nat,L: nat,Newnode: vEBT_VEBT,TreeList: list @ vEBT_VEBT,Sn: vEBT_VEBT,Summary: vEBT_VEBT,Newlist: list @ vEBT_VEBT] :
( ( ( ord_less @ nat @ Mi3 @ X )
& ( ord_less_eq @ nat @ X @ Ma ) )
=> ( ( Mi3 != Ma )
=> ( ( ord_less_eq @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ Deg )
=> ( ( ( vEBT_VEBT_high @ X @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) )
= H2 )
=> ( ( ( vEBT_VEBT_low @ X @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) )
= L )
=> ( ( Newnode
= ( vEBT_vebt_delete @ ( nth @ vEBT_VEBT @ TreeList @ H2 ) @ L ) )
=> ( ( vEBT_VEBT_minNull @ Newnode )
=> ( ( Sn
= ( vEBT_vebt_delete @ Summary @ H2 ) )
=> ( ( Newlist
= ( list_update @ vEBT_VEBT @ TreeList @ H2 @ Newnode ) )
=> ( ( ord_less @ nat @ ( vEBT_VEBT_high @ X @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( size_size @ ( list @ vEBT_VEBT ) @ TreeList ) )
=> ( ( vEBT_vebt_delete @ ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi3 @ Ma ) ) @ Deg @ TreeList @ Summary ) @ X )
= ( vEBT_Node
@ ( some @ ( product_prod @ nat @ nat )
@ ( product_Pair @ nat @ nat @ Mi3
@ ( if @ nat @ ( X = Ma )
@ ( if @ nat
@ ( ( vEBT_vebt_maxt @ Sn )
= ( none @ nat ) )
@ Mi3
@ ( plus_plus @ nat @ ( times_times @ nat @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_maxt @ 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_3594_del__x__not__mia,axiom,
! [Mi3: nat,X: nat,Ma: nat,Deg: nat,H2: nat,L: nat,TreeList: list @ vEBT_VEBT,Summary: vEBT_VEBT] :
( ( ( ord_less @ nat @ Mi3 @ X )
& ( ord_less_eq @ nat @ X @ Ma ) )
=> ( ( Mi3 != Ma )
=> ( ( ord_less_eq @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ Deg )
=> ( ( ( vEBT_VEBT_high @ X @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) )
= H2 )
=> ( ( ( vEBT_VEBT_low @ X @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) )
= L )
=> ( ( ord_less @ nat @ ( vEBT_VEBT_high @ X @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( size_size @ ( list @ vEBT_VEBT ) @ TreeList ) )
=> ( ( vEBT_vebt_delete @ ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi3 @ Ma ) ) @ Deg @ TreeList @ Summary ) @ X )
= ( if @ vEBT_VEBT @ ( vEBT_VEBT_minNull @ ( vEBT_vebt_delete @ ( nth @ vEBT_VEBT @ TreeList @ H2 ) @ L ) )
@ ( vEBT_Node
@ ( some @ ( product_prod @ nat @ nat )
@ ( product_Pair @ nat @ nat @ Mi3
@ ( if @ nat @ ( X = Ma )
@ ( if @ nat
@ ( ( vEBT_vebt_maxt @ ( vEBT_vebt_delete @ Summary @ H2 ) )
= ( none @ nat ) )
@ Mi3
@ ( plus_plus @ nat @ ( times_times @ nat @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_maxt @ ( vEBT_vebt_delete @ Summary @ H2 ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_maxt @ ( nth @ vEBT_VEBT @ ( list_update @ vEBT_VEBT @ TreeList @ H2 @ ( vEBT_vebt_delete @ ( nth @ vEBT_VEBT @ TreeList @ H2 ) @ L ) ) @ ( the2 @ nat @ ( vEBT_vebt_maxt @ ( vEBT_vebt_delete @ Summary @ H2 ) ) ) ) ) ) ) )
@ Ma ) ) )
@ Deg
@ ( list_update @ vEBT_VEBT @ TreeList @ H2 @ ( vEBT_vebt_delete @ ( nth @ vEBT_VEBT @ TreeList @ H2 ) @ L ) )
@ ( vEBT_vebt_delete @ Summary @ H2 ) )
@ ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi3 @ ( if @ nat @ ( X = Ma ) @ ( plus_plus @ nat @ ( times_times @ nat @ H2 @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_maxt @ ( nth @ vEBT_VEBT @ ( list_update @ vEBT_VEBT @ TreeList @ H2 @ ( vEBT_vebt_delete @ ( nth @ vEBT_VEBT @ TreeList @ H2 ) @ L ) ) @ H2 ) ) ) ) @ Ma ) ) ) @ Deg @ ( list_update @ vEBT_VEBT @ TreeList @ H2 @ ( vEBT_vebt_delete @ ( nth @ vEBT_VEBT @ TreeList @ H2 ) @ L ) ) @ Summary ) ) ) ) ) ) ) ) ) ).
% del_x_not_mia
thf(fact_3595_del__x__mia,axiom,
! [X: nat,Mi3: nat,Ma: nat,Deg: nat,TreeList: list @ vEBT_VEBT,Summary: vEBT_VEBT] :
( ( ( X = Mi3 )
& ( ord_less @ nat @ X @ Ma ) )
=> ( ( Mi3 != 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 @ Mi3 @ 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 @ Mi3 @ Ma ) ) @ Deg @ TreeList @ Summary ) ) ) ) ) ) ).
% del_x_mia
thf(fact_3596_del__x__mi__lets__in__minNull,axiom,
! [X: nat,Mi3: nat,Ma: nat,Deg: nat,Xn: nat,H2: nat,Summary: vEBT_VEBT,TreeList: list @ vEBT_VEBT,L: nat,Newnode: vEBT_VEBT,Newlist: list @ vEBT_VEBT,Sn: vEBT_VEBT] :
( ( ( X = Mi3 )
& ( ord_less @ nat @ X @ Ma ) )
=> ( ( Mi3 != Ma )
=> ( ( ord_less_eq @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ Deg )
=> ( ( ( vEBT_VEBT_high @ Xn @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) )
= H2 )
=> ( ( Xn
= ( plus_plus @ nat @ ( times_times @ nat @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) )
=> ( ( ( vEBT_VEBT_low @ Xn @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) )
= L )
=> ( ( ord_less @ nat @ ( vEBT_VEBT_high @ Xn @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( size_size @ ( list @ vEBT_VEBT ) @ TreeList ) )
=> ( ( Newnode
= ( vEBT_vebt_delete @ ( nth @ vEBT_VEBT @ TreeList @ H2 ) @ L ) )
=> ( ( Newlist
= ( list_update @ vEBT_VEBT @ TreeList @ H2 @ Newnode ) )
=> ( ( vEBT_VEBT_minNull @ Newnode )
=> ( ( Sn
= ( vEBT_vebt_delete @ Summary @ H2 ) )
=> ( ( vEBT_vebt_delete @ ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi3 @ 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_3597_del__x__mi__lets__in,axiom,
! [X: nat,Mi3: nat,Ma: nat,Deg: nat,Xn: nat,H2: nat,Summary: vEBT_VEBT,TreeList: list @ vEBT_VEBT,L: nat,Newnode: vEBT_VEBT,Newlist: list @ vEBT_VEBT] :
( ( ( X = Mi3 )
& ( ord_less @ nat @ X @ Ma ) )
=> ( ( Mi3 != Ma )
=> ( ( ord_less_eq @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ Deg )
=> ( ( ( vEBT_VEBT_high @ Xn @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) )
= H2 )
=> ( ( Xn
= ( plus_plus @ nat @ ( times_times @ nat @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) )
=> ( ( ( vEBT_VEBT_low @ Xn @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) )
= L )
=> ( ( ord_less @ nat @ ( vEBT_VEBT_high @ Xn @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( size_size @ ( list @ vEBT_VEBT ) @ TreeList ) )
=> ( ( Newnode
= ( vEBT_vebt_delete @ ( nth @ vEBT_VEBT @ TreeList @ H2 ) @ L ) )
=> ( ( Newlist
= ( list_update @ vEBT_VEBT @ TreeList @ H2 @ Newnode ) )
=> ( ( ( vEBT_VEBT_minNull @ Newnode )
=> ( ( vEBT_vebt_delete @ ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi3 @ 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 @ Mi3 @ 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_3598_del__x__mi,axiom,
! [X: nat,Mi3: nat,Ma: nat,Deg: nat,Xn: nat,H2: nat,Summary: vEBT_VEBT,TreeList: list @ vEBT_VEBT,L: nat] :
( ( ( X = Mi3 )
& ( ord_less @ nat @ X @ Ma ) )
=> ( ( Mi3 != Ma )
=> ( ( ord_less_eq @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ Deg )
=> ( ( ( vEBT_VEBT_high @ Xn @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) )
= H2 )
=> ( ( Xn
= ( plus_plus @ nat @ ( times_times @ nat @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) )
=> ( ( ( vEBT_VEBT_low @ Xn @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) )
= L )
=> ( ( ord_less @ nat @ ( vEBT_VEBT_high @ Xn @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( size_size @ ( list @ vEBT_VEBT ) @ TreeList ) )
=> ( ( vEBT_vebt_delete @ ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi3 @ Ma ) ) @ Deg @ TreeList @ Summary ) @ X )
= ( if @ vEBT_VEBT @ ( vEBT_VEBT_minNull @ ( vEBT_vebt_delete @ ( nth @ vEBT_VEBT @ TreeList @ H2 ) @ L ) )
@ ( vEBT_Node
@ ( some @ ( product_prod @ nat @ nat )
@ ( product_Pair @ nat @ nat @ Xn
@ ( if @ nat @ ( Xn = Ma )
@ ( if @ nat
@ ( ( vEBT_vebt_maxt @ ( vEBT_vebt_delete @ Summary @ H2 ) )
= ( none @ nat ) )
@ Xn
@ ( plus_plus @ nat @ ( times_times @ nat @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_maxt @ ( vEBT_vebt_delete @ Summary @ H2 ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_maxt @ ( nth @ vEBT_VEBT @ ( list_update @ vEBT_VEBT @ TreeList @ H2 @ ( vEBT_vebt_delete @ ( nth @ vEBT_VEBT @ TreeList @ H2 ) @ L ) ) @ ( the2 @ nat @ ( vEBT_vebt_maxt @ ( vEBT_vebt_delete @ Summary @ H2 ) ) ) ) ) ) ) )
@ Ma ) ) )
@ Deg
@ ( list_update @ vEBT_VEBT @ TreeList @ H2 @ ( vEBT_vebt_delete @ ( nth @ vEBT_VEBT @ TreeList @ H2 ) @ L ) )
@ ( vEBT_vebt_delete @ Summary @ H2 ) )
@ ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Xn @ ( if @ nat @ ( Xn = Ma ) @ ( plus_plus @ nat @ ( times_times @ nat @ H2 @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_maxt @ ( nth @ vEBT_VEBT @ ( list_update @ vEBT_VEBT @ TreeList @ H2 @ ( vEBT_vebt_delete @ ( nth @ vEBT_VEBT @ TreeList @ H2 ) @ L ) ) @ H2 ) ) ) ) @ Ma ) ) ) @ Deg @ ( list_update @ vEBT_VEBT @ TreeList @ H2 @ ( vEBT_vebt_delete @ ( nth @ vEBT_VEBT @ TreeList @ H2 ) @ L ) ) @ Summary ) ) ) ) ) ) ) ) ) ) ).
% del_x_mi
thf(fact_3599_del__in__range,axiom,
! [Mi3: nat,X: nat,Ma: nat,Deg: nat,TreeList: list @ vEBT_VEBT,Summary: vEBT_VEBT] :
( ( ( ord_less_eq @ nat @ Mi3 @ X )
& ( ord_less_eq @ nat @ X @ Ma ) )
=> ( ( Mi3 != 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 @ Mi3 @ Ma ) ) @ Deg @ TreeList @ Summary ) @ X )
= ( if @ vEBT_VEBT @ ( ord_less @ nat @ ( vEBT_VEBT_high @ ( if @ nat @ ( X = Mi3 ) @ ( plus_plus @ nat @ ( times_times @ nat @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ X ) @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( size_size @ ( list @ vEBT_VEBT ) @ TreeList ) )
@ ( if @ vEBT_VEBT @ ( vEBT_VEBT_minNull @ ( vEBT_vebt_delete @ ( nth @ vEBT_VEBT @ TreeList @ ( vEBT_VEBT_high @ ( if @ nat @ ( X = Mi3 ) @ ( plus_plus @ nat @ ( times_times @ nat @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ X ) @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( vEBT_VEBT_low @ ( if @ nat @ ( X = Mi3 ) @ ( plus_plus @ nat @ ( times_times @ nat @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ X ) @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) )
@ ( vEBT_Node
@ ( some @ ( product_prod @ nat @ nat )
@ ( product_Pair @ nat @ nat @ ( if @ nat @ ( X = Mi3 ) @ ( if @ nat @ ( X = Mi3 ) @ ( plus_plus @ nat @ ( times_times @ nat @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ X ) @ Mi3 )
@ ( if @ nat
@ ( ( ( X = Mi3 )
=> ( ( plus_plus @ nat @ ( times_times @ nat @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) )
= Ma ) )
& ( ( X != Mi3 )
=> ( X = Ma ) ) )
@ ( if @ nat
@ ( ( vEBT_vebt_maxt @ ( vEBT_vebt_delete @ Summary @ ( vEBT_VEBT_high @ ( if @ nat @ ( X = Mi3 ) @ ( plus_plus @ nat @ ( times_times @ nat @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ X ) @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) )
= ( none @ nat ) )
@ ( if @ nat @ ( X = Mi3 ) @ ( if @ nat @ ( X = Mi3 ) @ ( plus_plus @ nat @ ( times_times @ nat @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ X ) @ Mi3 )
@ ( plus_plus @ nat @ ( times_times @ nat @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_maxt @ ( vEBT_vebt_delete @ Summary @ ( vEBT_VEBT_high @ ( if @ nat @ ( X = Mi3 ) @ ( plus_plus @ nat @ ( times_times @ nat @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ X ) @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_maxt @ ( nth @ vEBT_VEBT @ ( list_update @ vEBT_VEBT @ TreeList @ ( vEBT_VEBT_high @ ( if @ nat @ ( X = Mi3 ) @ ( plus_plus @ nat @ ( times_times @ nat @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ X ) @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( vEBT_vebt_delete @ ( nth @ vEBT_VEBT @ TreeList @ ( vEBT_VEBT_high @ ( if @ nat @ ( X = Mi3 ) @ ( plus_plus @ nat @ ( times_times @ nat @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ X ) @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( vEBT_VEBT_low @ ( if @ nat @ ( X = Mi3 ) @ ( plus_plus @ nat @ ( times_times @ nat @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ X ) @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_maxt @ ( vEBT_vebt_delete @ Summary @ ( vEBT_VEBT_high @ ( if @ nat @ ( X = Mi3 ) @ ( plus_plus @ nat @ ( times_times @ nat @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ X ) @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) ) ) ) ) ) )
@ Ma ) ) )
@ Deg
@ ( list_update @ vEBT_VEBT @ TreeList @ ( vEBT_VEBT_high @ ( if @ nat @ ( X = Mi3 ) @ ( plus_plus @ nat @ ( times_times @ nat @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ X ) @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( vEBT_vebt_delete @ ( nth @ vEBT_VEBT @ TreeList @ ( vEBT_VEBT_high @ ( if @ nat @ ( X = Mi3 ) @ ( plus_plus @ nat @ ( times_times @ nat @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ X ) @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( vEBT_VEBT_low @ ( if @ nat @ ( X = Mi3 ) @ ( plus_plus @ nat @ ( times_times @ nat @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ X ) @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) )
@ ( vEBT_vebt_delete @ Summary @ ( vEBT_VEBT_high @ ( if @ nat @ ( X = Mi3 ) @ ( plus_plus @ nat @ ( times_times @ nat @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ X ) @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) )
@ ( vEBT_Node
@ ( some @ ( product_prod @ nat @ nat )
@ ( product_Pair @ nat @ nat @ ( if @ nat @ ( X = Mi3 ) @ ( if @ nat @ ( X = Mi3 ) @ ( plus_plus @ nat @ ( times_times @ nat @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ X ) @ Mi3 )
@ ( if @ nat
@ ( ( ( X = Mi3 )
=> ( ( plus_plus @ nat @ ( times_times @ nat @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) )
= Ma ) )
& ( ( X != Mi3 )
=> ( X = Ma ) ) )
@ ( plus_plus @ nat @ ( times_times @ nat @ ( vEBT_VEBT_high @ ( if @ nat @ ( X = Mi3 ) @ ( plus_plus @ nat @ ( times_times @ nat @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ X ) @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_maxt @ ( nth @ vEBT_VEBT @ ( list_update @ vEBT_VEBT @ TreeList @ ( vEBT_VEBT_high @ ( if @ nat @ ( X = Mi3 ) @ ( plus_plus @ nat @ ( times_times @ nat @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ X ) @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( vEBT_vebt_delete @ ( nth @ vEBT_VEBT @ TreeList @ ( vEBT_VEBT_high @ ( if @ nat @ ( X = Mi3 ) @ ( plus_plus @ nat @ ( times_times @ nat @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ X ) @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( vEBT_VEBT_low @ ( if @ nat @ ( X = Mi3 ) @ ( plus_plus @ nat @ ( times_times @ nat @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ X ) @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) @ ( vEBT_VEBT_high @ ( if @ nat @ ( X = Mi3 ) @ ( plus_plus @ nat @ ( times_times @ nat @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ X ) @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) ) )
@ Ma ) ) )
@ Deg
@ ( list_update @ vEBT_VEBT @ TreeList @ ( vEBT_VEBT_high @ ( if @ nat @ ( X = Mi3 ) @ ( plus_plus @ nat @ ( times_times @ nat @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ X ) @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( vEBT_vebt_delete @ ( nth @ vEBT_VEBT @ TreeList @ ( vEBT_VEBT_high @ ( if @ nat @ ( X = Mi3 ) @ ( plus_plus @ nat @ ( times_times @ nat @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ X ) @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( vEBT_VEBT_low @ ( if @ nat @ ( X = Mi3 ) @ ( plus_plus @ nat @ ( times_times @ nat @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ X ) @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) )
@ Summary ) )
@ ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi3 @ Ma ) ) @ Deg @ TreeList @ Summary ) ) ) ) ) ) ).
% del_in_range
thf(fact_3600_pred__lesseq__max,axiom,
! [Deg: nat,X: nat,Ma: nat,Mi3: 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 @ Mi3 @ 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 @ Mi3 @ X ) @ ( some @ nat @ Mi3 ) @ ( 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_3601_pred__less__length__list,axiom,
! [Deg: nat,X: nat,Ma: nat,TreeList: list @ vEBT_VEBT,Mi3: 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 @ Mi3 @ 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 @ Mi3 @ X ) @ ( some @ nat @ Mi3 ) @ ( 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_3602_max__def__raw,axiom,
! [A: $tType] :
( ( ord @ A )
=> ( ( ord_max @ A )
= ( ^ [A4: A,B5: A] : ( if @ A @ ( ord_less_eq @ A @ A4 @ B5 ) @ B5 @ A4 ) ) ) ) ).
% max_def_raw
thf(fact_3603_numeral__code_I2_J,axiom,
! [A: $tType] :
( ( numeral @ A )
=> ! [N: num] :
( ( numeral_numeral @ A @ ( bit0 @ N ) )
= ( plus_plus @ A @ ( numeral_numeral @ A @ N ) @ ( numeral_numeral @ A @ N ) ) ) ) ).
% numeral_code(2)
thf(fact_3604_set__vebt__def,axiom,
( vEBT_set_vebt
= ( ^ [T2: vEBT_VEBT] : ( collect @ nat @ ( vEBT_V8194947554948674370ptions @ T2 ) ) ) ) ).
% set_vebt_def
thf(fact_3605_strict__subset__divisors__dvd,axiom,
! [A: $tType] :
( ( comm_monoid_mult @ A )
=> ! [A2: A,B2: A] :
( ( ord_less @ ( set @ A )
@ ( collect @ A
@ ^ [C5: A] : ( dvd_dvd @ A @ C5 @ A2 ) )
@ ( collect @ A
@ ^ [C5: A] : ( dvd_dvd @ A @ C5 @ B2 ) ) )
= ( ( dvd_dvd @ A @ A2 @ B2 )
& ~ ( dvd_dvd @ A @ B2 @ A2 ) ) ) ) ).
% strict_subset_divisors_dvd
thf(fact_3606_lambda__one,axiom,
! [A: $tType] :
( ( monoid_mult @ A )
=> ( ( ^ [X2: A] : X2 )
= ( times_times @ A @ ( one_one @ A ) ) ) ) ).
% lambda_one
thf(fact_3607_lambda__zero,axiom,
! [A: $tType] :
( ( mult_zero @ A )
=> ( ( ^ [H: A] : ( zero_zero @ A ) )
= ( times_times @ A @ ( zero_zero @ A ) ) ) ) ).
% lambda_zero
thf(fact_3608_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_3609_subset__divisors__dvd,axiom,
! [A: $tType] :
( ( comm_monoid_mult @ A )
=> ! [A2: A,B2: A] :
( ( ord_less_eq @ ( set @ A )
@ ( collect @ A
@ ^ [C5: A] : ( dvd_dvd @ A @ C5 @ A2 ) )
@ ( collect @ A
@ ^ [C5: A] : ( dvd_dvd @ A @ C5 @ B2 ) ) )
= ( dvd_dvd @ A @ A2 @ B2 ) ) ) ).
% subset_divisors_dvd
thf(fact_3610_less__set__def,axiom,
! [A: $tType] :
( ( ord_less @ ( set @ A ) )
= ( ^ [A7: set @ A,B8: set @ A] :
( ord_less @ ( A > $o )
@ ^ [X2: A] : ( member @ A @ X2 @ A7 )
@ ^ [X2: A] : ( member @ A @ X2 @ B8 ) ) ) ) ).
% less_set_def
thf(fact_3611_finite__divisors__int,axiom,
! [I: int] :
( ( I
!= ( zero_zero @ int ) )
=> ( finite_finite2 @ int
@ ( collect @ int
@ ^ [D: int] : ( dvd_dvd @ int @ D @ I ) ) ) ) ).
% finite_divisors_int
thf(fact_3612_finite__M__bounded__by__nat,axiom,
! [P: nat > $o,I: nat] :
( finite_finite2 @ nat
@ ( collect @ nat
@ ^ [K3: nat] :
( ( P @ K3 )
& ( ord_less @ nat @ K3 @ I ) ) ) ) ).
% finite_M_bounded_by_nat
thf(fact_3613_nat__less__as__int,axiom,
( ( ord_less @ nat )
= ( ^ [A4: nat,B5: nat] : ( ord_less @ int @ ( semiring_1_of_nat @ int @ A4 ) @ ( semiring_1_of_nat @ int @ B5 ) ) ) ) ).
% nat_less_as_int
thf(fact_3614_nat__leq__as__int,axiom,
( ( ord_less_eq @ nat )
= ( ^ [A4: nat,B5: nat] : ( ord_less_eq @ int @ ( semiring_1_of_nat @ int @ A4 ) @ ( semiring_1_of_nat @ int @ B5 ) ) ) ) ).
% nat_leq_as_int
thf(fact_3615_numeral__code_I3_J,axiom,
! [A: $tType] :
( ( numeral @ A )
=> ! [N: num] :
( ( numeral_numeral @ A @ ( bit1 @ N ) )
= ( plus_plus @ A @ ( plus_plus @ A @ ( numeral_numeral @ A @ N ) @ ( numeral_numeral @ A @ N ) ) @ ( one_one @ A ) ) ) ) ).
% numeral_code(3)
thf(fact_3616_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_3617_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_3618_nat__plus__as__int,axiom,
( ( plus_plus @ nat )
= ( ^ [A4: nat,B5: nat] : ( nat2 @ ( plus_plus @ int @ ( semiring_1_of_nat @ int @ A4 ) @ ( semiring_1_of_nat @ int @ B5 ) ) ) ) ) ).
% nat_plus_as_int
thf(fact_3619_finite__divisors__nat,axiom,
! [M: nat] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ M )
=> ( finite_finite2 @ nat
@ ( collect @ nat
@ ^ [D: nat] : ( dvd_dvd @ nat @ D @ M ) ) ) ) ).
% finite_divisors_nat
thf(fact_3620_nat__times__as__int,axiom,
( ( times_times @ nat )
= ( ^ [A4: nat,B5: nat] : ( nat2 @ ( times_times @ int @ ( semiring_1_of_nat @ int @ A4 ) @ ( semiring_1_of_nat @ int @ B5 ) ) ) ) ) ).
% nat_times_as_int
thf(fact_3621_nat__minus__as__int,axiom,
( ( minus_minus @ nat )
= ( ^ [A4: nat,B5: nat] : ( nat2 @ ( minus_minus @ int @ ( semiring_1_of_nat @ int @ A4 ) @ ( semiring_1_of_nat @ int @ B5 ) ) ) ) ) ).
% nat_minus_as_int
thf(fact_3622_nat__div__as__int,axiom,
( ( divide_divide @ nat )
= ( ^ [A4: nat,B5: nat] : ( nat2 @ ( divide_divide @ int @ ( semiring_1_of_nat @ int @ A4 ) @ ( semiring_1_of_nat @ int @ B5 ) ) ) ) ) ).
% nat_div_as_int
thf(fact_3623_nat__mod__as__int,axiom,
( ( modulo_modulo @ nat )
= ( ^ [A4: nat,B5: nat] : ( nat2 @ ( modulo_modulo @ int @ ( semiring_1_of_nat @ int @ A4 ) @ ( semiring_1_of_nat @ int @ B5 ) ) ) ) ) ).
% nat_mod_as_int
thf(fact_3624_finite__roots__unity,axiom,
! [A: $tType] :
( ( ( real_V8999393235501362500lgebra @ A )
& ( idom @ A ) )
=> ! [N: nat] :
( ( ord_less_eq @ nat @ ( one_one @ nat ) @ N )
=> ( finite_finite2 @ A
@ ( collect @ A
@ ^ [Z7: A] :
( ( power_power @ A @ Z7 @ N )
= ( one_one @ A ) ) ) ) ) ) ).
% finite_roots_unity
thf(fact_3625_finite__lists__length__eq,axiom,
! [A: $tType,A3: set @ A,N: 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 )
= N ) ) ) ) ) ).
% finite_lists_length_eq
thf(fact_3626_diff__nat__eq__if,axiom,
! [Z5: int,Z: int] :
( ( ( ord_less @ int @ Z5 @ ( zero_zero @ int ) )
=> ( ( minus_minus @ nat @ ( nat2 @ Z ) @ ( nat2 @ Z5 ) )
= ( nat2 @ Z ) ) )
& ( ~ ( ord_less @ int @ Z5 @ ( zero_zero @ int ) )
=> ( ( minus_minus @ nat @ ( nat2 @ Z ) @ ( nat2 @ Z5 ) )
= ( if @ nat @ ( ord_less @ int @ ( minus_minus @ int @ Z @ Z5 ) @ ( zero_zero @ int ) ) @ ( zero_zero @ nat ) @ ( nat2 @ ( minus_minus @ int @ Z @ Z5 ) ) ) ) ) ) ).
% diff_nat_eq_if
thf(fact_3627_finite__lists__length__le,axiom,
! [A: $tType,A3: set @ A,N: 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 ) @ N ) ) ) ) ) ).
% finite_lists_length_le
thf(fact_3628_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_3629_set__decode__def,axiom,
( nat_set_decode
= ( ^ [X2: nat] :
( collect @ nat
@ ^ [N3: nat] :
~ ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ X2 @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N3 ) ) ) ) ) ) ).
% set_decode_def
thf(fact_3630_pochhammer__code,axiom,
! [A: $tType] :
( ( comm_semiring_1 @ A )
=> ( ( comm_s3205402744901411588hammer @ A )
= ( ^ [A4: A,N3: nat] :
( if @ A
@ ( N3
= ( zero_zero @ nat ) )
@ ( one_one @ A )
@ ( set_fo6178422350223883121st_nat @ A
@ ^ [O: nat] : ( times_times @ A @ ( plus_plus @ A @ A4 @ ( semiring_1_of_nat @ A @ O ) ) )
@ ( zero_zero @ nat )
@ ( minus_minus @ nat @ N3 @ ( one_one @ nat ) )
@ ( one_one @ A ) ) ) ) ) ) ).
% pochhammer_code
thf(fact_3631_vebt__buildup_Osimps_I3_J,axiom,
! [Va2: nat] :
( ( ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( suc @ ( suc @ Va2 ) ) )
=> ( ( vEBT_vebt_buildup @ ( suc @ ( suc @ Va2 ) ) )
= ( vEBT_Node @ ( none @ ( product_prod @ nat @ nat ) ) @ ( suc @ ( suc @ Va2 ) ) @ ( replicate @ vEBT_VEBT @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( vEBT_vebt_buildup @ ( divide_divide @ nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( vEBT_vebt_buildup @ ( divide_divide @ nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) )
& ( ~ ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( suc @ ( suc @ Va2 ) ) )
=> ( ( vEBT_vebt_buildup @ ( suc @ ( suc @ Va2 ) ) )
= ( vEBT_Node @ ( none @ ( product_prod @ nat @ nat ) ) @ ( suc @ ( suc @ Va2 ) ) @ ( replicate @ vEBT_VEBT @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( suc @ ( divide_divide @ nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( vEBT_vebt_buildup @ ( divide_divide @ nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( vEBT_vebt_buildup @ ( suc @ ( divide_divide @ nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) ) ) ) ).
% vebt_buildup.simps(3)
thf(fact_3632_VEBT__internal_OT_092_060_094sub_062b_092_060_094sub_062u_092_060_094sub_062i_092_060_094sub_062l_092_060_094sub_062d_092_060_094sub_062u_092_060_094sub_062p_Osimps_I3_J,axiom,
! [Va2: nat] :
( ( ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( suc @ ( suc @ Va2 ) ) )
=> ( ( vEBT_V8346862874174094_d_u_p @ ( suc @ ( suc @ Va2 ) ) )
= ( plus_plus @ nat @ ( one_one @ nat ) @ ( plus_plus @ nat @ ( plus_plus @ nat @ ( numeral_numeral @ nat @ ( bit1 @ ( bit0 @ ( bit0 @ one2 ) ) ) ) @ ( vEBT_V8346862874174094_d_u_p @ ( divide_divide @ nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( times_times @ nat @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( plus_plus @ nat @ ( vEBT_V8346862874174094_d_u_p @ ( divide_divide @ nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( one_one @ nat ) ) ) ) ) ) )
& ( ~ ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( suc @ ( suc @ Va2 ) ) )
=> ( ( vEBT_V8346862874174094_d_u_p @ ( suc @ ( suc @ Va2 ) ) )
= ( plus_plus @ nat @ ( plus_plus @ nat @ ( numeral_numeral @ nat @ ( bit1 @ ( bit1 @ ( bit0 @ one2 ) ) ) ) @ ( vEBT_V8346862874174094_d_u_p @ ( suc @ ( divide_divide @ nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) @ ( times_times @ nat @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( suc @ ( divide_divide @ nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( plus_plus @ nat @ ( vEBT_V8346862874174094_d_u_p @ ( divide_divide @ nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( one_one @ nat ) ) ) ) ) ) ) ).
% VEBT_internal.T\<^sub>b\<^sub>u\<^sub>i\<^sub>l\<^sub>d\<^sub>u\<^sub>p.simps(3)
thf(fact_3633_VEBT__internal_OT_092_060_094sub_062b_092_060_094sub_062u_092_060_094sub_062i_092_060_094sub_062l_092_060_094sub_062d_Osimps_I3_J,axiom,
! [Va2: nat] :
( ( ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( suc @ ( suc @ Va2 ) ) )
=> ( ( vEBT_V8646137997579335489_i_l_d @ ( suc @ ( suc @ Va2 ) ) )
= ( plus_plus @ nat @ ( one_one @ nat ) @ ( plus_plus @ nat @ ( plus_plus @ nat @ ( numeral_numeral @ nat @ ( bit0 @ ( bit1 @ ( bit0 @ one2 ) ) ) ) @ ( vEBT_V8646137997579335489_i_l_d @ ( divide_divide @ nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( times_times @ nat @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( vEBT_V8646137997579335489_i_l_d @ ( divide_divide @ nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) ) ) )
& ( ~ ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( suc @ ( suc @ Va2 ) ) )
=> ( ( vEBT_V8646137997579335489_i_l_d @ ( suc @ ( suc @ Va2 ) ) )
= ( plus_plus @ nat @ ( plus_plus @ nat @ ( numeral_numeral @ nat @ ( bit0 @ ( bit0 @ ( bit1 @ one2 ) ) ) ) @ ( vEBT_V8646137997579335489_i_l_d @ ( suc @ ( divide_divide @ nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) @ ( times_times @ nat @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( suc @ ( divide_divide @ nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( vEBT_V8646137997579335489_i_l_d @ ( divide_divide @ nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) ) ) ) ).
% VEBT_internal.T\<^sub>b\<^sub>u\<^sub>i\<^sub>l\<^sub>d.simps(3)
thf(fact_3634_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_3635_VEBT__internal_OT_092_060_094sub_062b_092_060_094sub_062u_092_060_094sub_062i_092_060_094sub_062l_092_060_094sub_062d_092_060_094sub_062u_092_060_094sub_062p_Oelims,axiom,
! [X: nat,Y: nat] :
( ( ( vEBT_V8346862874174094_d_u_p @ X )
= Y )
=> ( ( ( X
= ( zero_zero @ nat ) )
=> ( Y
!= ( numeral_numeral @ nat @ ( bit1 @ one2 ) ) ) )
=> ( ( ( X
= ( suc @ ( zero_zero @ nat ) ) )
=> ( Y
!= ( numeral_numeral @ nat @ ( bit1 @ one2 ) ) ) )
=> ~ ! [Va: nat] :
( ( X
= ( suc @ ( suc @ Va ) ) )
=> ~ ( ( ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( suc @ ( suc @ Va ) ) )
=> ( Y
= ( plus_plus @ nat @ ( one_one @ nat ) @ ( plus_plus @ nat @ ( plus_plus @ nat @ ( numeral_numeral @ nat @ ( bit1 @ ( bit0 @ ( bit0 @ one2 ) ) ) ) @ ( vEBT_V8346862874174094_d_u_p @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( times_times @ nat @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( plus_plus @ nat @ ( vEBT_V8346862874174094_d_u_p @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( one_one @ nat ) ) ) ) ) ) )
& ( ~ ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( suc @ ( suc @ Va ) ) )
=> ( Y
= ( plus_plus @ nat @ ( plus_plus @ nat @ ( numeral_numeral @ nat @ ( bit1 @ ( bit1 @ ( bit0 @ one2 ) ) ) ) @ ( vEBT_V8346862874174094_d_u_p @ ( suc @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) @ ( times_times @ nat @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( suc @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( plus_plus @ nat @ ( vEBT_V8346862874174094_d_u_p @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( one_one @ nat ) ) ) ) ) ) ) ) ) ) ) ).
% VEBT_internal.T\<^sub>b\<^sub>u\<^sub>i\<^sub>l\<^sub>d\<^sub>u\<^sub>p.elims
thf(fact_3636_VEBT__internal_OT_092_060_094sub_062b_092_060_094sub_062u_092_060_094sub_062i_092_060_094sub_062l_092_060_094sub_062d_Oelims,axiom,
! [X: nat,Y: nat] :
( ( ( vEBT_V8646137997579335489_i_l_d @ X )
= Y )
=> ( ( ( X
= ( zero_zero @ nat ) )
=> ( Y
!= ( numeral_numeral @ nat @ ( bit0 @ ( bit0 @ one2 ) ) ) ) )
=> ( ( ( X
= ( suc @ ( zero_zero @ nat ) ) )
=> ( Y
!= ( numeral_numeral @ nat @ ( bit0 @ ( bit0 @ one2 ) ) ) ) )
=> ~ ! [Va: nat] :
( ( X
= ( suc @ ( suc @ Va ) ) )
=> ~ ( ( ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( suc @ ( suc @ Va ) ) )
=> ( Y
= ( plus_plus @ nat @ ( one_one @ nat ) @ ( plus_plus @ nat @ ( plus_plus @ nat @ ( numeral_numeral @ nat @ ( bit0 @ ( bit1 @ ( bit0 @ one2 ) ) ) ) @ ( vEBT_V8646137997579335489_i_l_d @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( times_times @ nat @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( vEBT_V8646137997579335489_i_l_d @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) ) ) )
& ( ~ ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( suc @ ( suc @ Va ) ) )
=> ( Y
= ( plus_plus @ nat @ ( plus_plus @ nat @ ( numeral_numeral @ nat @ ( bit0 @ ( bit0 @ ( bit1 @ one2 ) ) ) ) @ ( vEBT_V8646137997579335489_i_l_d @ ( suc @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) @ ( times_times @ nat @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( suc @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( vEBT_V8646137997579335489_i_l_d @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) ) ) ) ) ) ) ) ).
% VEBT_internal.T\<^sub>b\<^sub>u\<^sub>i\<^sub>l\<^sub>d.elims
thf(fact_3637_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_3638_vebt__buildup_Oelims,axiom,
! [X: nat,Y: vEBT_VEBT] :
( ( ( vEBT_vebt_buildup @ X )
= Y )
=> ( ( ( X
= ( zero_zero @ nat ) )
=> ( Y
!= ( vEBT_Leaf @ $false @ $false ) ) )
=> ( ( ( X
= ( suc @ ( zero_zero @ nat ) ) )
=> ( Y
!= ( vEBT_Leaf @ $false @ $false ) ) )
=> ~ ! [Va: nat] :
( ( X
= ( suc @ ( suc @ Va ) ) )
=> ~ ( ( ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( suc @ ( suc @ Va ) ) )
=> ( Y
= ( vEBT_Node @ ( none @ ( product_prod @ nat @ nat ) ) @ ( suc @ ( suc @ Va ) ) @ ( replicate @ vEBT_VEBT @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( vEBT_vebt_buildup @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( vEBT_vebt_buildup @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) )
& ( ~ ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( suc @ ( suc @ Va ) ) )
=> ( Y
= ( vEBT_Node @ ( none @ ( product_prod @ nat @ nat ) ) @ ( suc @ ( suc @ Va ) ) @ ( replicate @ vEBT_VEBT @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( suc @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( vEBT_vebt_buildup @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( vEBT_vebt_buildup @ ( suc @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) ) ) ) ) ) ) ) ).
% vebt_buildup.elims
thf(fact_3639_vebt__member_Osimps_I5_J,axiom,
! [Mi3: nat,Ma: nat,Va2: nat,TreeList: list @ vEBT_VEBT,Summary: vEBT_VEBT,X: nat] :
( ( vEBT_vebt_member @ ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi3 @ Ma ) ) @ ( suc @ ( suc @ Va2 ) ) @ TreeList @ Summary ) @ X )
= ( ( X != Mi3 )
=> ( ( X != Ma )
=> ( ~ ( ord_less @ nat @ X @ Mi3 )
& ( ~ ( ord_less @ nat @ X @ Mi3 )
=> ( ~ ( ord_less @ nat @ Ma @ X )
& ( ~ ( ord_less @ nat @ Ma @ X )
=> ( ( ( ord_less @ nat @ ( vEBT_VEBT_high @ X @ ( divide_divide @ nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( size_size @ ( list @ vEBT_VEBT ) @ TreeList ) )
=> ( vEBT_vebt_member @ ( nth @ vEBT_VEBT @ TreeList @ ( vEBT_VEBT_high @ X @ ( divide_divide @ nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( vEBT_VEBT_low @ X @ ( divide_divide @ nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) )
& ( ord_less @ nat @ ( vEBT_VEBT_high @ X @ ( divide_divide @ nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( size_size @ ( list @ vEBT_VEBT ) @ TreeList ) ) ) ) ) ) ) ) ) ) ).
% vebt_member.simps(5)
thf(fact_3640_VEBT__internal_Omembermima_Osimps_I4_J,axiom,
! [Mi3: nat,Ma: nat,V: nat,TreeList: list @ vEBT_VEBT,Vc2: vEBT_VEBT,X: nat] :
( ( vEBT_VEBT_membermima @ ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi3 @ Ma ) ) @ ( suc @ V ) @ TreeList @ Vc2 ) @ X )
= ( ( X = Mi3 )
| ( 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_3641_VEBT__internal_Onaive__member_Oelims_I3_J,axiom,
! [X: vEBT_VEBT,Xa: nat] :
( ~ ( vEBT_V5719532721284313246member @ X @ Xa )
=> ( ! [A6: $o,B3: $o] :
( ( X
= ( vEBT_Leaf @ A6 @ B3 ) )
=> ( ( ( Xa
= ( zero_zero @ nat ) )
=> A6 )
& ( ( Xa
!= ( zero_zero @ nat ) )
=> ( ( ( Xa
= ( one_one @ nat ) )
=> B3 )
& ( Xa
= ( one_one @ nat ) ) ) ) ) )
=> ( ! [Uu2: option @ ( product_prod @ nat @ nat ),Uv2: list @ vEBT_VEBT,Uw2: vEBT_VEBT] :
( X
!= ( vEBT_Node @ Uu2 @ ( zero_zero @ nat ) @ Uv2 @ Uw2 ) )
=> ~ ! [Uy2: option @ ( product_prod @ nat @ nat ),V4: nat,TreeList2: list @ vEBT_VEBT] :
( ? [S2: vEBT_VEBT] :
( X
= ( vEBT_Node @ Uy2 @ ( suc @ V4 ) @ TreeList2 @ S2 ) )
=> ( ( ( ord_less @ nat @ ( vEBT_VEBT_high @ Xa @ ( divide_divide @ nat @ ( suc @ V4 ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( size_size @ ( list @ vEBT_VEBT ) @ TreeList2 ) )
=> ( vEBT_V5719532721284313246member @ ( nth @ vEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ Xa @ ( divide_divide @ nat @ ( suc @ V4 ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( vEBT_VEBT_low @ Xa @ ( divide_divide @ nat @ ( suc @ V4 ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) )
& ( ord_less @ nat @ ( vEBT_VEBT_high @ Xa @ ( divide_divide @ nat @ ( suc @ V4 ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( size_size @ ( list @ vEBT_VEBT ) @ TreeList2 ) ) ) ) ) ) ) ).
% VEBT_internal.naive_member.elims(3)
thf(fact_3642_VEBT__internal_Onaive__member_Oelims_I2_J,axiom,
! [X: vEBT_VEBT,Xa: nat] :
( ( vEBT_V5719532721284313246member @ X @ Xa )
=> ( ! [A6: $o,B3: $o] :
( ( X
= ( vEBT_Leaf @ A6 @ B3 ) )
=> ~ ( ( ( Xa
= ( zero_zero @ nat ) )
=> A6 )
& ( ( Xa
!= ( zero_zero @ nat ) )
=> ( ( ( Xa
= ( one_one @ nat ) )
=> B3 )
& ( Xa
= ( one_one @ nat ) ) ) ) ) )
=> ~ ! [Uy2: option @ ( product_prod @ nat @ nat ),V4: nat,TreeList2: list @ vEBT_VEBT] :
( ? [S2: vEBT_VEBT] :
( X
= ( vEBT_Node @ Uy2 @ ( suc @ V4 ) @ TreeList2 @ S2 ) )
=> ~ ( ( ( ord_less @ nat @ ( vEBT_VEBT_high @ Xa @ ( divide_divide @ nat @ ( suc @ V4 ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( size_size @ ( list @ vEBT_VEBT ) @ TreeList2 ) )
=> ( vEBT_V5719532721284313246member @ ( nth @ vEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ Xa @ ( divide_divide @ nat @ ( suc @ V4 ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( vEBT_VEBT_low @ Xa @ ( divide_divide @ nat @ ( suc @ V4 ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) )
& ( ord_less @ nat @ ( vEBT_VEBT_high @ Xa @ ( divide_divide @ nat @ ( suc @ V4 ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( size_size @ ( list @ vEBT_VEBT ) @ TreeList2 ) ) ) ) ) ) ).
% VEBT_internal.naive_member.elims(2)
thf(fact_3643_VEBT__internal_Onaive__member_Oelims_I1_J,axiom,
! [X: vEBT_VEBT,Xa: nat,Y: $o] :
( ( ( vEBT_V5719532721284313246member @ X @ Xa )
= Y )
=> ( ! [A6: $o,B3: $o] :
( ( X
= ( vEBT_Leaf @ A6 @ B3 ) )
=> ( Y
= ( ~ ( ( ( Xa
= ( zero_zero @ nat ) )
=> A6 )
& ( ( Xa
!= ( zero_zero @ nat ) )
=> ( ( ( Xa
= ( one_one @ nat ) )
=> B3 )
& ( Xa
= ( one_one @ nat ) ) ) ) ) ) ) )
=> ( ( ? [Uu2: option @ ( product_prod @ nat @ nat ),Uv2: list @ vEBT_VEBT,Uw2: vEBT_VEBT] :
( X
= ( vEBT_Node @ Uu2 @ ( zero_zero @ nat ) @ Uv2 @ Uw2 ) )
=> Y )
=> ~ ! [Uy2: option @ ( product_prod @ nat @ nat ),V4: nat,TreeList2: list @ vEBT_VEBT] :
( ? [S2: vEBT_VEBT] :
( X
= ( vEBT_Node @ Uy2 @ ( suc @ V4 ) @ TreeList2 @ S2 ) )
=> ( Y
= ( ~ ( ( ( ord_less @ nat @ ( vEBT_VEBT_high @ Xa @ ( divide_divide @ nat @ ( suc @ V4 ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( size_size @ ( list @ vEBT_VEBT ) @ TreeList2 ) )
=> ( vEBT_V5719532721284313246member @ ( nth @ vEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ Xa @ ( divide_divide @ nat @ ( suc @ V4 ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( vEBT_VEBT_low @ Xa @ ( divide_divide @ nat @ ( suc @ V4 ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) )
& ( ord_less @ nat @ ( vEBT_VEBT_high @ Xa @ ( divide_divide @ nat @ ( suc @ V4 ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( size_size @ ( list @ vEBT_VEBT ) @ TreeList2 ) ) ) ) ) ) ) ) ) ).
% VEBT_internal.naive_member.elims(1)
thf(fact_3644_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_3645_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_3646_max_Ostrict__order__iff,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ( ( ord_less @ A )
= ( ^ [B5: A,A4: A] :
( ( A4
= ( ord_max @ A @ A4 @ B5 ) )
& ( A4 != B5 ) ) ) ) ) ).
% max.strict_order_iff
thf(fact_3647_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_3648_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_3649_VEBT__internal_Omembermima_Oelims_I2_J,axiom,
! [X: vEBT_VEBT,Xa: nat] :
( ( vEBT_VEBT_membermima @ X @ Xa )
=> ( ! [Mi: nat,Ma2: nat] :
( ? [Va3: list @ vEBT_VEBT,Vb2: vEBT_VEBT] :
( X
= ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi @ Ma2 ) ) @ ( zero_zero @ nat ) @ Va3 @ Vb2 ) )
=> ~ ( ( Xa = Mi )
| ( Xa = Ma2 ) ) )
=> ( ! [Mi: nat,Ma2: nat,V4: nat,TreeList2: list @ vEBT_VEBT] :
( ? [Vc: vEBT_VEBT] :
( X
= ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi @ Ma2 ) ) @ ( suc @ V4 ) @ TreeList2 @ Vc ) )
=> ~ ( ( Xa = Mi )
| ( Xa = Ma2 )
| ( ( ( ord_less @ nat @ ( vEBT_VEBT_high @ Xa @ ( divide_divide @ nat @ ( suc @ V4 ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( size_size @ ( list @ vEBT_VEBT ) @ TreeList2 ) )
=> ( vEBT_VEBT_membermima @ ( nth @ vEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ Xa @ ( divide_divide @ nat @ ( suc @ V4 ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( vEBT_VEBT_low @ Xa @ ( divide_divide @ nat @ ( suc @ V4 ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) )
& ( ord_less @ nat @ ( vEBT_VEBT_high @ Xa @ ( divide_divide @ nat @ ( suc @ V4 ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( size_size @ ( list @ vEBT_VEBT ) @ TreeList2 ) ) ) ) )
=> ~ ! [V4: nat,TreeList2: list @ vEBT_VEBT] :
( ? [Vd2: vEBT_VEBT] :
( X
= ( vEBT_Node @ ( none @ ( product_prod @ nat @ nat ) ) @ ( suc @ V4 ) @ TreeList2 @ Vd2 ) )
=> ~ ( ( ( ord_less @ nat @ ( vEBT_VEBT_high @ Xa @ ( divide_divide @ nat @ ( suc @ V4 ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( size_size @ ( list @ vEBT_VEBT ) @ TreeList2 ) )
=> ( vEBT_VEBT_membermima @ ( nth @ vEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ Xa @ ( divide_divide @ nat @ ( suc @ V4 ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( vEBT_VEBT_low @ Xa @ ( divide_divide @ nat @ ( suc @ V4 ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) )
& ( ord_less @ nat @ ( vEBT_VEBT_high @ Xa @ ( divide_divide @ nat @ ( suc @ V4 ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( size_size @ ( list @ vEBT_VEBT ) @ TreeList2 ) ) ) ) ) ) ) ).
% VEBT_internal.membermima.elims(2)
thf(fact_3650_vebt__member_Oelims_I2_J,axiom,
! [X: vEBT_VEBT,Xa: nat] :
( ( vEBT_vebt_member @ X @ Xa )
=> ( ! [A6: $o,B3: $o] :
( ( X
= ( vEBT_Leaf @ A6 @ B3 ) )
=> ~ ( ( ( Xa
= ( zero_zero @ nat ) )
=> A6 )
& ( ( Xa
!= ( zero_zero @ nat ) )
=> ( ( ( Xa
= ( one_one @ nat ) )
=> B3 )
& ( Xa
= ( one_one @ nat ) ) ) ) ) )
=> ~ ! [Mi: nat,Ma2: nat,Va: nat,TreeList2: list @ vEBT_VEBT] :
( ? [Summary2: vEBT_VEBT] :
( X
= ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi @ Ma2 ) ) @ ( suc @ ( suc @ Va ) ) @ TreeList2 @ Summary2 ) )
=> ~ ( ( Xa != Mi )
=> ( ( Xa != Ma2 )
=> ( ~ ( ord_less @ nat @ Xa @ Mi )
& ( ~ ( ord_less @ nat @ Xa @ Mi )
=> ( ~ ( ord_less @ nat @ Ma2 @ Xa )
& ( ~ ( ord_less @ nat @ Ma2 @ Xa )
=> ( ( ( ord_less @ nat @ ( vEBT_VEBT_high @ Xa @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( size_size @ ( list @ vEBT_VEBT ) @ TreeList2 ) )
=> ( vEBT_vebt_member @ ( nth @ vEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ Xa @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( vEBT_VEBT_low @ Xa @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) )
& ( ord_less @ nat @ ( vEBT_VEBT_high @ Xa @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( size_size @ ( list @ vEBT_VEBT ) @ TreeList2 ) ) ) ) ) ) ) ) ) ) ) ) ).
% vebt_member.elims(2)
thf(fact_3651_VEBT__internal_Omembermima_Oelims_I1_J,axiom,
! [X: vEBT_VEBT,Xa: nat,Y: $o] :
( ( ( vEBT_VEBT_membermima @ X @ Xa )
= Y )
=> ( ( ? [Uu2: $o,Uv2: $o] :
( X
= ( vEBT_Leaf @ Uu2 @ Uv2 ) )
=> Y )
=> ( ( ? [Ux2: list @ vEBT_VEBT,Uy2: vEBT_VEBT] :
( X
= ( vEBT_Node @ ( none @ ( product_prod @ nat @ nat ) ) @ ( zero_zero @ nat ) @ Ux2 @ Uy2 ) )
=> Y )
=> ( ! [Mi: nat,Ma2: nat] :
( ? [Va3: list @ vEBT_VEBT,Vb2: vEBT_VEBT] :
( X
= ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi @ Ma2 ) ) @ ( zero_zero @ nat ) @ Va3 @ Vb2 ) )
=> ( Y
= ( ~ ( ( Xa = Mi )
| ( Xa = Ma2 ) ) ) ) )
=> ( ! [Mi: nat,Ma2: nat,V4: nat,TreeList2: list @ vEBT_VEBT] :
( ? [Vc: vEBT_VEBT] :
( X
= ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi @ Ma2 ) ) @ ( suc @ V4 ) @ TreeList2 @ Vc ) )
=> ( Y
= ( ~ ( ( Xa = Mi )
| ( Xa = Ma2 )
| ( ( ( ord_less @ nat @ ( vEBT_VEBT_high @ Xa @ ( divide_divide @ nat @ ( suc @ V4 ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( size_size @ ( list @ vEBT_VEBT ) @ TreeList2 ) )
=> ( vEBT_VEBT_membermima @ ( nth @ vEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ Xa @ ( divide_divide @ nat @ ( suc @ V4 ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( vEBT_VEBT_low @ Xa @ ( divide_divide @ nat @ ( suc @ V4 ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) )
& ( ord_less @ nat @ ( vEBT_VEBT_high @ Xa @ ( divide_divide @ nat @ ( suc @ V4 ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( size_size @ ( list @ vEBT_VEBT ) @ TreeList2 ) ) ) ) ) ) )
=> ~ ! [V4: nat,TreeList2: list @ vEBT_VEBT] :
( ? [Vd2: vEBT_VEBT] :
( X
= ( vEBT_Node @ ( none @ ( product_prod @ nat @ nat ) ) @ ( suc @ V4 ) @ TreeList2 @ Vd2 ) )
=> ( Y
= ( ~ ( ( ( ord_less @ nat @ ( vEBT_VEBT_high @ Xa @ ( divide_divide @ nat @ ( suc @ V4 ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( size_size @ ( list @ vEBT_VEBT ) @ TreeList2 ) )
=> ( vEBT_VEBT_membermima @ ( nth @ vEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ Xa @ ( divide_divide @ nat @ ( suc @ V4 ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( vEBT_VEBT_low @ Xa @ ( divide_divide @ nat @ ( suc @ V4 ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) )
& ( ord_less @ nat @ ( vEBT_VEBT_high @ Xa @ ( divide_divide @ nat @ ( suc @ V4 ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( size_size @ ( list @ vEBT_VEBT ) @ TreeList2 ) ) ) ) ) ) ) ) ) ) ) ).
% VEBT_internal.membermima.elims(1)
thf(fact_3652_VEBT__internal_Omembermima_Oelims_I3_J,axiom,
! [X: vEBT_VEBT,Xa: nat] :
( ~ ( vEBT_VEBT_membermima @ X @ Xa )
=> ( ! [Uu2: $o,Uv2: $o] :
( X
!= ( vEBT_Leaf @ Uu2 @ Uv2 ) )
=> ( ! [Ux2: list @ vEBT_VEBT,Uy2: vEBT_VEBT] :
( X
!= ( vEBT_Node @ ( none @ ( product_prod @ nat @ nat ) ) @ ( zero_zero @ nat ) @ Ux2 @ Uy2 ) )
=> ( ! [Mi: nat,Ma2: nat] :
( ? [Va3: list @ vEBT_VEBT,Vb2: vEBT_VEBT] :
( X
= ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi @ Ma2 ) ) @ ( zero_zero @ nat ) @ Va3 @ Vb2 ) )
=> ( ( Xa = Mi )
| ( Xa = Ma2 ) ) )
=> ( ! [Mi: nat,Ma2: nat,V4: nat,TreeList2: list @ vEBT_VEBT] :
( ? [Vc: vEBT_VEBT] :
( X
= ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi @ Ma2 ) ) @ ( suc @ V4 ) @ TreeList2 @ Vc ) )
=> ( ( Xa = Mi )
| ( Xa = Ma2 )
| ( ( ( ord_less @ nat @ ( vEBT_VEBT_high @ Xa @ ( divide_divide @ nat @ ( suc @ V4 ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( size_size @ ( list @ vEBT_VEBT ) @ TreeList2 ) )
=> ( vEBT_VEBT_membermima @ ( nth @ vEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ Xa @ ( divide_divide @ nat @ ( suc @ V4 ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( vEBT_VEBT_low @ Xa @ ( divide_divide @ nat @ ( suc @ V4 ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) )
& ( ord_less @ nat @ ( vEBT_VEBT_high @ Xa @ ( divide_divide @ nat @ ( suc @ V4 ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( size_size @ ( list @ vEBT_VEBT ) @ TreeList2 ) ) ) ) )
=> ~ ! [V4: nat,TreeList2: list @ vEBT_VEBT] :
( ? [Vd2: vEBT_VEBT] :
( X
= ( vEBT_Node @ ( none @ ( product_prod @ nat @ nat ) ) @ ( suc @ V4 ) @ TreeList2 @ Vd2 ) )
=> ( ( ( ord_less @ nat @ ( vEBT_VEBT_high @ Xa @ ( divide_divide @ nat @ ( suc @ V4 ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( size_size @ ( list @ vEBT_VEBT ) @ TreeList2 ) )
=> ( vEBT_VEBT_membermima @ ( nth @ vEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ Xa @ ( divide_divide @ nat @ ( suc @ V4 ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( vEBT_VEBT_low @ Xa @ ( divide_divide @ nat @ ( suc @ V4 ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) )
& ( ord_less @ nat @ ( vEBT_VEBT_high @ Xa @ ( divide_divide @ nat @ ( suc @ V4 ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( size_size @ ( list @ vEBT_VEBT ) @ TreeList2 ) ) ) ) ) ) ) ) ) ).
% VEBT_internal.membermima.elims(3)
thf(fact_3653_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,
! [Mi3: nat,Ma: nat,Va2: nat,TreeList: list @ vEBT_VEBT,Summary: vEBT_VEBT,X: nat] :
( ( vEBT_T_m_e_m_b_e_r @ ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi3 @ Ma ) ) @ ( suc @ ( suc @ Va2 ) ) @ TreeList @ Summary ) @ X )
= ( plus_plus @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( if @ nat @ ( X = Mi3 ) @ ( 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 @ Mi3 ) @ ( one_one @ nat ) @ ( plus_plus @ nat @ ( one_one @ nat ) @ ( if @ nat @ ( ord_less @ nat @ Ma @ X ) @ ( one_one @ nat ) @ ( plus_plus @ nat @ ( numeral_numeral @ nat @ ( bit1 @ ( bit0 @ ( bit0 @ one2 ) ) ) ) @ ( if @ nat @ ( ord_less @ nat @ ( vEBT_VEBT_high @ X @ ( divide_divide @ nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( size_size @ ( list @ vEBT_VEBT ) @ TreeList ) ) @ ( plus_plus @ nat @ ( one_one @ nat ) @ ( vEBT_T_m_e_m_b_e_r @ ( nth @ vEBT_VEBT @ TreeList @ ( vEBT_VEBT_high @ X @ ( divide_divide @ nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( vEBT_VEBT_low @ X @ ( divide_divide @ nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) @ ( one_one @ nat ) ) ) ) ) ) ) ) ) ) ) ) ).
% T\<^sub>m\<^sub>e\<^sub>m\<^sub>b\<^sub>e\<^sub>r.simps(5)
thf(fact_3654_vebt__insert_Osimps_I5_J,axiom,
! [Mi3: nat,Ma: nat,Va2: nat,TreeList: list @ vEBT_VEBT,Summary: vEBT_VEBT,X: nat] :
( ( vEBT_vebt_insert @ ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi3 @ Ma ) ) @ ( suc @ ( suc @ Va2 ) ) @ TreeList @ Summary ) @ X )
= ( if @ vEBT_VEBT
@ ( ( ord_less @ nat @ ( vEBT_VEBT_high @ ( if @ nat @ ( ord_less @ nat @ X @ Mi3 ) @ Mi3 @ X ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( size_size @ ( list @ vEBT_VEBT ) @ TreeList ) )
& ~ ( ( X = Mi3 )
| ( X = Ma ) ) )
@ ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ ( if @ nat @ ( ord_less @ nat @ X @ Mi3 ) @ X @ Mi3 ) @ ( ord_max @ nat @ ( if @ nat @ ( ord_less @ nat @ X @ Mi3 ) @ Mi3 @ X ) @ Ma ) ) ) @ ( suc @ ( suc @ Va2 ) ) @ ( list_update @ vEBT_VEBT @ TreeList @ ( vEBT_VEBT_high @ ( if @ nat @ ( ord_less @ nat @ X @ Mi3 ) @ Mi3 @ X ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( vEBT_vebt_insert @ ( nth @ vEBT_VEBT @ TreeList @ ( vEBT_VEBT_high @ ( if @ nat @ ( ord_less @ nat @ X @ Mi3 ) @ Mi3 @ X ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( vEBT_VEBT_low @ ( if @ nat @ ( ord_less @ nat @ X @ Mi3 ) @ Mi3 @ X ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) @ ( if @ vEBT_VEBT @ ( vEBT_VEBT_minNull @ ( nth @ vEBT_VEBT @ TreeList @ ( vEBT_VEBT_high @ ( if @ nat @ ( ord_less @ nat @ X @ Mi3 ) @ Mi3 @ X ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) @ ( vEBT_vebt_insert @ Summary @ ( vEBT_VEBT_high @ ( if @ nat @ ( ord_less @ nat @ X @ Mi3 ) @ Mi3 @ X ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ Summary ) )
@ ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi3 @ Ma ) ) @ ( suc @ ( suc @ Va2 ) ) @ TreeList @ Summary ) ) ) ).
% vebt_insert.simps(5)
thf(fact_3655_vebt__member_Oelims_I1_J,axiom,
! [X: vEBT_VEBT,Xa: nat,Y: $o] :
( ( ( vEBT_vebt_member @ X @ Xa )
= Y )
=> ( ! [A6: $o,B3: $o] :
( ( X
= ( vEBT_Leaf @ A6 @ B3 ) )
=> ( Y
= ( ~ ( ( ( Xa
= ( zero_zero @ nat ) )
=> A6 )
& ( ( Xa
!= ( zero_zero @ nat ) )
=> ( ( ( Xa
= ( one_one @ nat ) )
=> B3 )
& ( Xa
= ( one_one @ nat ) ) ) ) ) ) ) )
=> ( ( ? [Uu2: nat,Uv2: list @ vEBT_VEBT,Uw2: vEBT_VEBT] :
( X
= ( vEBT_Node @ ( none @ ( product_prod @ nat @ nat ) ) @ Uu2 @ Uv2 @ Uw2 ) )
=> Y )
=> ( ( ? [V4: product_prod @ nat @ nat,Uy2: list @ vEBT_VEBT,Uz2: vEBT_VEBT] :
( X
= ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ V4 ) @ ( zero_zero @ nat ) @ Uy2 @ Uz2 ) )
=> Y )
=> ( ( ? [V4: product_prod @ nat @ nat,Vb2: list @ vEBT_VEBT,Vc: vEBT_VEBT] :
( X
= ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ V4 ) @ ( suc @ ( zero_zero @ nat ) ) @ Vb2 @ Vc ) )
=> Y )
=> ~ ! [Mi: nat,Ma2: nat,Va: nat,TreeList2: list @ vEBT_VEBT] :
( ? [Summary2: vEBT_VEBT] :
( X
= ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi @ Ma2 ) ) @ ( suc @ ( suc @ Va ) ) @ TreeList2 @ Summary2 ) )
=> ( Y
= ( ~ ( ( Xa != Mi )
=> ( ( Xa != Ma2 )
=> ( ~ ( ord_less @ nat @ Xa @ Mi )
& ( ~ ( ord_less @ nat @ Xa @ Mi )
=> ( ~ ( ord_less @ nat @ Ma2 @ Xa )
& ( ~ ( ord_less @ nat @ Ma2 @ Xa )
=> ( ( ( ord_less @ nat @ ( vEBT_VEBT_high @ Xa @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( size_size @ ( list @ vEBT_VEBT ) @ TreeList2 ) )
=> ( vEBT_vebt_member @ ( nth @ vEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ Xa @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( vEBT_VEBT_low @ Xa @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) )
& ( ord_less @ nat @ ( vEBT_VEBT_high @ Xa @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( size_size @ ( list @ vEBT_VEBT ) @ TreeList2 ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).
% vebt_member.elims(1)
thf(fact_3656_vebt__member_Oelims_I3_J,axiom,
! [X: vEBT_VEBT,Xa: nat] :
( ~ ( vEBT_vebt_member @ X @ Xa )
=> ( ! [A6: $o,B3: $o] :
( ( X
= ( vEBT_Leaf @ A6 @ B3 ) )
=> ( ( ( Xa
= ( zero_zero @ nat ) )
=> A6 )
& ( ( Xa
!= ( zero_zero @ nat ) )
=> ( ( ( Xa
= ( one_one @ nat ) )
=> B3 )
& ( Xa
= ( one_one @ nat ) ) ) ) ) )
=> ( ! [Uu2: nat,Uv2: list @ vEBT_VEBT,Uw2: vEBT_VEBT] :
( X
!= ( vEBT_Node @ ( none @ ( product_prod @ nat @ nat ) ) @ Uu2 @ Uv2 @ Uw2 ) )
=> ( ! [V4: product_prod @ nat @ nat,Uy2: list @ vEBT_VEBT,Uz2: vEBT_VEBT] :
( X
!= ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ V4 ) @ ( zero_zero @ nat ) @ Uy2 @ Uz2 ) )
=> ( ! [V4: product_prod @ nat @ nat,Vb2: list @ vEBT_VEBT,Vc: vEBT_VEBT] :
( X
!= ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ V4 ) @ ( suc @ ( zero_zero @ nat ) ) @ Vb2 @ Vc ) )
=> ~ ! [Mi: nat,Ma2: nat,Va: nat,TreeList2: list @ vEBT_VEBT] :
( ? [Summary2: vEBT_VEBT] :
( X
= ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi @ Ma2 ) ) @ ( suc @ ( suc @ Va ) ) @ TreeList2 @ Summary2 ) )
=> ( ( Xa != Mi )
=> ( ( Xa != Ma2 )
=> ( ~ ( ord_less @ nat @ Xa @ Mi )
& ( ~ ( ord_less @ nat @ Xa @ Mi )
=> ( ~ ( ord_less @ nat @ Ma2 @ Xa )
& ( ~ ( ord_less @ nat @ Ma2 @ Xa )
=> ( ( ( ord_less @ nat @ ( vEBT_VEBT_high @ Xa @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( size_size @ ( list @ vEBT_VEBT ) @ TreeList2 ) )
=> ( vEBT_vebt_member @ ( nth @ vEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ Xa @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( vEBT_VEBT_low @ Xa @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) )
& ( ord_less @ nat @ ( vEBT_VEBT_high @ Xa @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( size_size @ ( list @ vEBT_VEBT ) @ TreeList2 ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).
% vebt_member.elims(3)
thf(fact_3657_T_092_060_094sub_062m_092_060_094sub_062e_092_060_094sub_062m_092_060_094sub_062b_092_060_094sub_062e_092_060_094sub_062r_Oelims,axiom,
! [X: vEBT_VEBT,Xa: nat,Y: nat] :
( ( ( vEBT_T_m_e_m_b_e_r @ X @ Xa )
= Y )
=> ( ( ? [A6: $o,B3: $o] :
( X
= ( vEBT_Leaf @ A6 @ 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 ) ) ) ) ) )
=> ( ( ? [Uu2: nat,Uv2: list @ vEBT_VEBT,Uw2: vEBT_VEBT] :
( X
= ( vEBT_Node @ ( none @ ( product_prod @ nat @ nat ) ) @ Uu2 @ Uv2 @ Uw2 ) )
=> ( Y
!= ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) )
=> ( ( ? [V4: product_prod @ nat @ nat,Uy2: list @ vEBT_VEBT,Uz2: vEBT_VEBT] :
( X
= ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ V4 ) @ ( zero_zero @ nat ) @ Uy2 @ Uz2 ) )
=> ( Y
!= ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) )
=> ( ( ? [V4: product_prod @ nat @ nat,Vb2: list @ vEBT_VEBT,Vc: vEBT_VEBT] :
( X
= ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ V4 ) @ ( suc @ ( zero_zero @ nat ) ) @ Vb2 @ Vc ) )
=> ( Y
!= ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) )
=> ~ ! [Mi: nat,Ma2: nat,Va: nat,TreeList2: list @ vEBT_VEBT] :
( ? [Summary2: vEBT_VEBT] :
( X
= ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi @ Ma2 ) ) @ ( suc @ ( suc @ Va ) ) @ TreeList2 @ Summary2 ) )
=> ( Y
!= ( plus_plus @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( if @ nat @ ( Xa = Mi ) @ ( 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 @ Mi ) @ ( one_one @ nat ) @ ( plus_plus @ nat @ ( one_one @ nat ) @ ( if @ nat @ ( ord_less @ nat @ Ma2 @ Xa ) @ ( one_one @ nat ) @ ( plus_plus @ nat @ ( numeral_numeral @ nat @ ( bit1 @ ( bit0 @ ( bit0 @ one2 ) ) ) ) @ ( if @ nat @ ( ord_less @ nat @ ( vEBT_VEBT_high @ Xa @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( size_size @ ( list @ vEBT_VEBT ) @ TreeList2 ) ) @ ( plus_plus @ nat @ ( one_one @ nat ) @ ( vEBT_T_m_e_m_b_e_r @ ( nth @ vEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ Xa @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( vEBT_VEBT_low @ Xa @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) @ ( one_one @ nat ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).
% T\<^sub>m\<^sub>e\<^sub>m\<^sub>b\<^sub>e\<^sub>r.elims
thf(fact_3658_vebt__insert_Oelims,axiom,
! [X: vEBT_VEBT,Xa: nat,Y: vEBT_VEBT] :
( ( ( vEBT_vebt_insert @ X @ Xa )
= Y )
=> ( ! [A6: $o,B3: $o] :
( ( X
= ( vEBT_Leaf @ A6 @ B3 ) )
=> ~ ( ( ( Xa
= ( zero_zero @ nat ) )
=> ( Y
= ( vEBT_Leaf @ $true @ B3 ) ) )
& ( ( Xa
!= ( zero_zero @ nat ) )
=> ( ( ( Xa
= ( one_one @ nat ) )
=> ( Y
= ( vEBT_Leaf @ A6 @ $true ) ) )
& ( ( Xa
!= ( one_one @ nat ) )
=> ( Y
= ( vEBT_Leaf @ A6 @ 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 ) ) )
=> ( ! [V4: nat,TreeList2: list @ vEBT_VEBT,Summary2: vEBT_VEBT] :
( ( X
= ( vEBT_Node @ ( none @ ( product_prod @ nat @ nat ) ) @ ( suc @ ( suc @ V4 ) ) @ TreeList2 @ Summary2 ) )
=> ( Y
!= ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Xa @ Xa ) ) @ ( suc @ ( suc @ V4 ) ) @ TreeList2 @ Summary2 ) ) )
=> ~ ! [Mi: nat,Ma2: nat,Va: nat,TreeList2: list @ vEBT_VEBT,Summary2: vEBT_VEBT] :
( ( X
= ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi @ Ma2 ) ) @ ( suc @ ( suc @ Va ) ) @ TreeList2 @ Summary2 ) )
=> ( Y
!= ( if @ vEBT_VEBT
@ ( ( ord_less @ nat @ ( vEBT_VEBT_high @ ( if @ nat @ ( ord_less @ nat @ Xa @ Mi ) @ Mi @ Xa ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( size_size @ ( list @ vEBT_VEBT ) @ TreeList2 ) )
& ~ ( ( Xa = Mi )
| ( Xa = Ma2 ) ) )
@ ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ ( if @ nat @ ( ord_less @ nat @ Xa @ Mi ) @ Xa @ Mi ) @ ( ord_max @ nat @ ( if @ nat @ ( ord_less @ nat @ Xa @ Mi ) @ Mi @ Xa ) @ Ma2 ) ) ) @ ( suc @ ( suc @ Va ) ) @ ( list_update @ vEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ ( if @ nat @ ( ord_less @ nat @ Xa @ Mi ) @ Mi @ Xa ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( vEBT_vebt_insert @ ( nth @ vEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ ( if @ nat @ ( ord_less @ nat @ Xa @ Mi ) @ Mi @ Xa ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( vEBT_VEBT_low @ ( if @ nat @ ( ord_less @ nat @ Xa @ Mi ) @ Mi @ Xa ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) @ ( if @ vEBT_VEBT @ ( vEBT_VEBT_minNull @ ( nth @ vEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ ( if @ nat @ ( ord_less @ nat @ Xa @ Mi ) @ Mi @ Xa ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) @ ( vEBT_vebt_insert @ Summary2 @ ( vEBT_VEBT_high @ ( if @ nat @ ( ord_less @ nat @ Xa @ Mi ) @ Mi @ Xa ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ Summary2 ) )
@ ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi @ Ma2 ) ) @ ( suc @ ( suc @ Va ) ) @ TreeList2 @ Summary2 ) ) ) ) ) ) ) ) ) ).
% vebt_insert.elims
thf(fact_3659_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,Mi3: nat,Va2: nat,TreeList: list @ vEBT_VEBT,Summary: vEBT_VEBT] :
( ( ( ord_less @ nat @ Ma @ X )
=> ( ( vEBT_T_p_r_e_d2 @ ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi3 @ Ma ) ) @ ( suc @ ( suc @ Va2 ) ) @ TreeList @ Summary ) @ X )
= ( one_one @ nat ) ) )
& ( ~ ( ord_less @ nat @ Ma @ X )
=> ( ( vEBT_T_p_r_e_d2 @ ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi3 @ Ma ) ) @ ( suc @ ( suc @ Va2 ) ) @ TreeList @ Summary ) @ X )
= ( if @ nat @ ( ord_less @ nat @ ( vEBT_VEBT_high @ X @ ( divide_divide @ nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( size_size @ ( list @ vEBT_VEBT ) @ TreeList ) )
@ ( if @ nat
@ ( ( ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList @ ( vEBT_VEBT_high @ X @ ( divide_divide @ nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) )
!= ( none @ nat ) )
& ( vEBT_VEBT_greater @ ( some @ nat @ ( vEBT_VEBT_low @ X @ ( divide_divide @ nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList @ ( vEBT_VEBT_high @ X @ ( divide_divide @ nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) ) )
@ ( plus_plus @ nat @ ( one_one @ nat ) @ ( vEBT_T_p_r_e_d2 @ ( nth @ vEBT_VEBT @ TreeList @ ( vEBT_VEBT_high @ X @ ( divide_divide @ nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( vEBT_VEBT_low @ X @ ( divide_divide @ nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) )
@ ( plus_plus @ nat @ ( vEBT_T_p_r_e_d2 @ Summary @ ( vEBT_VEBT_high @ X @ ( divide_divide @ nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( one_one @ nat ) ) )
@ ( one_one @ nat ) ) ) ) ) ).
% T\<^sub>p\<^sub>r\<^sub>e\<^sub>d'.simps(7)
thf(fact_3660_T_092_060_094sub_062s_092_060_094sub_062u_092_060_094sub_062c_092_060_094sub_062c_H_Osimps_I6_J,axiom,
! [X: nat,Mi3: nat,Ma: nat,Va2: nat,TreeList: list @ vEBT_VEBT,Summary: vEBT_VEBT] :
( ( ( ord_less @ nat @ X @ Mi3 )
=> ( ( vEBT_T_s_u_c_c2 @ ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi3 @ Ma ) ) @ ( suc @ ( suc @ Va2 ) ) @ TreeList @ Summary ) @ X )
= ( one_one @ nat ) ) )
& ( ~ ( ord_less @ nat @ X @ Mi3 )
=> ( ( vEBT_T_s_u_c_c2 @ ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi3 @ Ma ) ) @ ( suc @ ( suc @ Va2 ) ) @ TreeList @ Summary ) @ X )
= ( if @ nat @ ( ord_less @ nat @ ( vEBT_VEBT_high @ X @ ( divide_divide @ nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( size_size @ ( list @ vEBT_VEBT ) @ TreeList ) )
@ ( if @ nat
@ ( ( ( vEBT_vebt_maxt @ ( nth @ vEBT_VEBT @ TreeList @ ( vEBT_VEBT_high @ X @ ( divide_divide @ nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) )
!= ( none @ nat ) )
& ( vEBT_VEBT_less @ ( some @ nat @ ( vEBT_VEBT_low @ X @ ( divide_divide @ nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( vEBT_vebt_maxt @ ( nth @ vEBT_VEBT @ TreeList @ ( vEBT_VEBT_high @ X @ ( divide_divide @ nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) ) )
@ ( plus_plus @ nat @ ( one_one @ nat ) @ ( vEBT_T_s_u_c_c2 @ ( nth @ vEBT_VEBT @ TreeList @ ( vEBT_VEBT_high @ X @ ( divide_divide @ nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( vEBT_VEBT_low @ X @ ( divide_divide @ nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) )
@ ( plus_plus @ nat @ ( vEBT_T_s_u_c_c2 @ Summary @ ( vEBT_VEBT_high @ X @ ( divide_divide @ nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( one_one @ nat ) ) )
@ ( one_one @ nat ) ) ) ) ) ).
% T\<^sub>s\<^sub>u\<^sub>c\<^sub>c'.simps(6)
thf(fact_3661_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,Mi3: nat,Ma: nat,Va2: nat,TreeList: list @ vEBT_VEBT,Summary: vEBT_VEBT] :
( ( ( ( ord_less @ nat @ X @ Mi3 )
| ( ord_less @ nat @ Ma @ X ) )
=> ( ( vEBT_V1232361888498592333_e_t_e @ ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi3 @ Ma ) ) @ ( suc @ ( suc @ Va2 ) ) @ TreeList @ Summary ) @ X )
= ( one_one @ nat ) ) )
& ( ~ ( ( ord_less @ nat @ X @ Mi3 )
| ( ord_less @ nat @ Ma @ X ) )
=> ( ( ( ( X = Mi3 )
& ( X = Ma ) )
=> ( ( vEBT_V1232361888498592333_e_t_e @ ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi3 @ Ma ) ) @ ( suc @ ( suc @ Va2 ) ) @ TreeList @ Summary ) @ X )
= ( one_one @ nat ) ) )
& ( ~ ( ( X = Mi3 )
& ( X = Ma ) )
=> ( ( vEBT_V1232361888498592333_e_t_e @ ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi3 @ Ma ) ) @ ( suc @ ( suc @ Va2 ) ) @ TreeList @ Summary ) @ X )
= ( if @ nat @ ( ord_less @ nat @ ( vEBT_VEBT_high @ ( if @ nat @ ( X = Mi3 ) @ ( plus_plus @ nat @ ( times_times @ nat @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ X ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( size_size @ ( list @ vEBT_VEBT ) @ TreeList ) ) @ ( plus_plus @ nat @ ( vEBT_V1232361888498592333_e_t_e @ ( nth @ vEBT_VEBT @ TreeList @ ( vEBT_VEBT_high @ ( if @ nat @ ( X = Mi3 ) @ ( plus_plus @ nat @ ( times_times @ nat @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ X ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( vEBT_VEBT_low @ ( if @ nat @ ( X = Mi3 ) @ ( plus_plus @ nat @ ( times_times @ nat @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ X ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( if @ nat @ ( vEBT_VEBT_minNull @ ( vEBT_vebt_delete @ ( nth @ vEBT_VEBT @ TreeList @ ( vEBT_VEBT_high @ ( if @ nat @ ( X = Mi3 ) @ ( plus_plus @ nat @ ( times_times @ nat @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ X ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( vEBT_VEBT_low @ ( if @ nat @ ( X = Mi3 ) @ ( plus_plus @ nat @ ( times_times @ nat @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ X ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) @ ( vEBT_V1232361888498592333_e_t_e @ Summary @ ( vEBT_VEBT_high @ ( if @ nat @ ( X = Mi3 ) @ ( plus_plus @ nat @ ( times_times @ nat @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ X ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( one_one @ nat ) ) ) @ ( one_one @ nat ) ) ) ) ) ) ) ).
% VEBT_internal.T\<^sub>d\<^sub>e\<^sub>l\<^sub>e\<^sub>t\<^sub>e'.simps(7)
thf(fact_3662_vebt__pred_Osimps_I7_J,axiom,
! [Ma: nat,X: nat,Mi3: nat,Va2: nat,TreeList: list @ vEBT_VEBT,Summary: vEBT_VEBT] :
( ( ( ord_less @ nat @ Ma @ X )
=> ( ( vEBT_vebt_pred @ ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi3 @ Ma ) ) @ ( suc @ ( suc @ Va2 ) ) @ TreeList @ Summary ) @ X )
= ( some @ nat @ Ma ) ) )
& ( ~ ( ord_less @ nat @ Ma @ X )
=> ( ( vEBT_vebt_pred @ ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi3 @ Ma ) ) @ ( suc @ ( suc @ Va2 ) ) @ TreeList @ Summary ) @ X )
= ( if @ ( option @ nat ) @ ( ord_less @ nat @ ( vEBT_VEBT_high @ X @ ( divide_divide @ nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( size_size @ ( list @ vEBT_VEBT ) @ TreeList ) )
@ ( if @ ( option @ nat )
@ ( ( ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList @ ( vEBT_VEBT_high @ X @ ( divide_divide @ nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) )
!= ( none @ nat ) )
& ( vEBT_VEBT_greater @ ( some @ nat @ ( vEBT_VEBT_low @ X @ ( divide_divide @ nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList @ ( vEBT_VEBT_high @ X @ ( divide_divide @ nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) ) )
@ ( vEBT_VEBT_add @ ( vEBT_VEBT_mul @ ( some @ nat @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( some @ nat @ ( vEBT_VEBT_high @ X @ ( divide_divide @ nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) @ ( vEBT_vebt_pred @ ( nth @ vEBT_VEBT @ TreeList @ ( vEBT_VEBT_high @ X @ ( divide_divide @ nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( vEBT_VEBT_low @ X @ ( divide_divide @ nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) )
@ ( if @ ( option @ nat )
@ ( ( vEBT_vebt_pred @ Summary @ ( vEBT_VEBT_high @ X @ ( divide_divide @ nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) )
= ( none @ nat ) )
@ ( if @ ( option @ nat ) @ ( ord_less @ nat @ Mi3 @ X ) @ ( some @ nat @ Mi3 ) @ ( none @ nat ) )
@ ( vEBT_VEBT_add @ ( vEBT_VEBT_mul @ ( some @ nat @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( vEBT_vebt_pred @ Summary @ ( vEBT_VEBT_high @ X @ ( divide_divide @ nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) @ ( vEBT_vebt_maxt @ ( nth @ vEBT_VEBT @ TreeList @ ( the2 @ nat @ ( vEBT_vebt_pred @ Summary @ ( vEBT_VEBT_high @ X @ ( divide_divide @ nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) ) ) ) ) )
@ ( none @ nat ) ) ) ) ) ).
% vebt_pred.simps(7)
thf(fact_3663_vebt__succ_Osimps_I6_J,axiom,
! [X: nat,Mi3: nat,Ma: nat,Va2: nat,TreeList: list @ vEBT_VEBT,Summary: vEBT_VEBT] :
( ( ( ord_less @ nat @ X @ Mi3 )
=> ( ( vEBT_vebt_succ @ ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi3 @ Ma ) ) @ ( suc @ ( suc @ Va2 ) ) @ TreeList @ Summary ) @ X )
= ( some @ nat @ Mi3 ) ) )
& ( ~ ( ord_less @ nat @ X @ Mi3 )
=> ( ( vEBT_vebt_succ @ ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi3 @ Ma ) ) @ ( suc @ ( suc @ Va2 ) ) @ TreeList @ Summary ) @ X )
= ( if @ ( option @ nat ) @ ( ord_less @ nat @ ( vEBT_VEBT_high @ X @ ( divide_divide @ nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( size_size @ ( list @ vEBT_VEBT ) @ TreeList ) )
@ ( if @ ( option @ nat )
@ ( ( ( vEBT_vebt_maxt @ ( nth @ vEBT_VEBT @ TreeList @ ( vEBT_VEBT_high @ X @ ( divide_divide @ nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) )
!= ( none @ nat ) )
& ( vEBT_VEBT_less @ ( some @ nat @ ( vEBT_VEBT_low @ X @ ( divide_divide @ nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( vEBT_vebt_maxt @ ( nth @ vEBT_VEBT @ TreeList @ ( vEBT_VEBT_high @ X @ ( divide_divide @ nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) ) )
@ ( vEBT_VEBT_add @ ( vEBT_VEBT_mul @ ( some @ nat @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( some @ nat @ ( vEBT_VEBT_high @ X @ ( divide_divide @ nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) @ ( vEBT_vebt_succ @ ( nth @ vEBT_VEBT @ TreeList @ ( vEBT_VEBT_high @ X @ ( divide_divide @ nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( vEBT_VEBT_low @ X @ ( divide_divide @ nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) )
@ ( if @ ( option @ nat )
@ ( ( vEBT_vebt_succ @ Summary @ ( vEBT_VEBT_high @ X @ ( divide_divide @ nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) )
= ( none @ nat ) )
@ ( none @ nat )
@ ( vEBT_VEBT_add @ ( vEBT_VEBT_mul @ ( some @ nat @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( vEBT_vebt_succ @ Summary @ ( vEBT_VEBT_high @ X @ ( divide_divide @ nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) @ ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList @ ( the2 @ nat @ ( vEBT_vebt_succ @ Summary @ ( vEBT_VEBT_high @ X @ ( divide_divide @ nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) ) ) ) ) )
@ ( none @ nat ) ) ) ) ) ).
% vebt_succ.simps(6)
thf(fact_3664_vebt__delete_Osimps_I7_J,axiom,
! [X: nat,Mi3: nat,Ma: nat,Va2: nat,TreeList: list @ vEBT_VEBT,Summary: vEBT_VEBT] :
( ( ( ( ord_less @ nat @ X @ Mi3 )
| ( ord_less @ nat @ Ma @ X ) )
=> ( ( vEBT_vebt_delete @ ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi3 @ Ma ) ) @ ( suc @ ( suc @ Va2 ) ) @ TreeList @ Summary ) @ X )
= ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi3 @ Ma ) ) @ ( suc @ ( suc @ Va2 ) ) @ TreeList @ Summary ) ) )
& ( ~ ( ( ord_less @ nat @ X @ Mi3 )
| ( ord_less @ nat @ Ma @ X ) )
=> ( ( ( ( X = Mi3 )
& ( X = Ma ) )
=> ( ( vEBT_vebt_delete @ ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi3 @ Ma ) ) @ ( suc @ ( suc @ Va2 ) ) @ TreeList @ Summary ) @ X )
= ( vEBT_Node @ ( none @ ( product_prod @ nat @ nat ) ) @ ( suc @ ( suc @ Va2 ) ) @ TreeList @ Summary ) ) )
& ( ~ ( ( X = Mi3 )
& ( X = Ma ) )
=> ( ( vEBT_vebt_delete @ ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi3 @ Ma ) ) @ ( suc @ ( suc @ Va2 ) ) @ TreeList @ Summary ) @ X )
= ( if @ vEBT_VEBT @ ( ord_less @ nat @ ( vEBT_VEBT_high @ ( if @ nat @ ( X = Mi3 ) @ ( plus_plus @ nat @ ( times_times @ nat @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ X ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( size_size @ ( list @ vEBT_VEBT ) @ TreeList ) )
@ ( if @ vEBT_VEBT @ ( vEBT_VEBT_minNull @ ( vEBT_vebt_delete @ ( nth @ vEBT_VEBT @ TreeList @ ( vEBT_VEBT_high @ ( if @ nat @ ( X = Mi3 ) @ ( plus_plus @ nat @ ( times_times @ nat @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ X ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( vEBT_VEBT_low @ ( if @ nat @ ( X = Mi3 ) @ ( plus_plus @ nat @ ( times_times @ nat @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ X ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) )
@ ( vEBT_Node
@ ( some @ ( product_prod @ nat @ nat )
@ ( product_Pair @ nat @ nat @ ( if @ nat @ ( X = Mi3 ) @ ( if @ nat @ ( X = Mi3 ) @ ( plus_plus @ nat @ ( times_times @ nat @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ X ) @ Mi3 )
@ ( if @ nat
@ ( ( ( X = Mi3 )
=> ( ( plus_plus @ nat @ ( times_times @ nat @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) )
= Ma ) )
& ( ( X != Mi3 )
=> ( X = Ma ) ) )
@ ( if @ nat
@ ( ( vEBT_vebt_maxt @ ( vEBT_vebt_delete @ Summary @ ( vEBT_VEBT_high @ ( if @ nat @ ( X = Mi3 ) @ ( plus_plus @ nat @ ( times_times @ nat @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ X ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) )
= ( none @ nat ) )
@ ( if @ nat @ ( X = Mi3 ) @ ( if @ nat @ ( X = Mi3 ) @ ( plus_plus @ nat @ ( times_times @ nat @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ X ) @ Mi3 )
@ ( plus_plus @ nat @ ( times_times @ nat @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_maxt @ ( vEBT_vebt_delete @ Summary @ ( vEBT_VEBT_high @ ( if @ nat @ ( X = Mi3 ) @ ( plus_plus @ nat @ ( times_times @ nat @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ X ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_maxt @ ( nth @ vEBT_VEBT @ ( list_update @ vEBT_VEBT @ TreeList @ ( vEBT_VEBT_high @ ( if @ nat @ ( X = Mi3 ) @ ( plus_plus @ nat @ ( times_times @ nat @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ X ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( vEBT_vebt_delete @ ( nth @ vEBT_VEBT @ TreeList @ ( vEBT_VEBT_high @ ( if @ nat @ ( X = Mi3 ) @ ( plus_plus @ nat @ ( times_times @ nat @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ X ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( vEBT_VEBT_low @ ( if @ nat @ ( X = Mi3 ) @ ( plus_plus @ nat @ ( times_times @ nat @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ X ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_maxt @ ( vEBT_vebt_delete @ Summary @ ( vEBT_VEBT_high @ ( if @ nat @ ( X = Mi3 ) @ ( plus_plus @ nat @ ( times_times @ nat @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ X ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) ) ) ) ) ) )
@ Ma ) ) )
@ ( suc @ ( suc @ Va2 ) )
@ ( list_update @ vEBT_VEBT @ TreeList @ ( vEBT_VEBT_high @ ( if @ nat @ ( X = Mi3 ) @ ( plus_plus @ nat @ ( times_times @ nat @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ X ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( vEBT_vebt_delete @ ( nth @ vEBT_VEBT @ TreeList @ ( vEBT_VEBT_high @ ( if @ nat @ ( X = Mi3 ) @ ( plus_plus @ nat @ ( times_times @ nat @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ X ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( vEBT_VEBT_low @ ( if @ nat @ ( X = Mi3 ) @ ( plus_plus @ nat @ ( times_times @ nat @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ X ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) )
@ ( vEBT_vebt_delete @ Summary @ ( vEBT_VEBT_high @ ( if @ nat @ ( X = Mi3 ) @ ( plus_plus @ nat @ ( times_times @ nat @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ X ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) )
@ ( vEBT_Node
@ ( some @ ( product_prod @ nat @ nat )
@ ( product_Pair @ nat @ nat @ ( if @ nat @ ( X = Mi3 ) @ ( if @ nat @ ( X = Mi3 ) @ ( plus_plus @ nat @ ( times_times @ nat @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ X ) @ Mi3 )
@ ( if @ nat
@ ( ( ( X = Mi3 )
=> ( ( plus_plus @ nat @ ( times_times @ nat @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) )
= Ma ) )
& ( ( X != Mi3 )
=> ( X = Ma ) ) )
@ ( plus_plus @ nat @ ( times_times @ nat @ ( vEBT_VEBT_high @ ( if @ nat @ ( X = Mi3 ) @ ( plus_plus @ nat @ ( times_times @ nat @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ X ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_maxt @ ( nth @ vEBT_VEBT @ ( list_update @ vEBT_VEBT @ TreeList @ ( vEBT_VEBT_high @ ( if @ nat @ ( X = Mi3 ) @ ( plus_plus @ nat @ ( times_times @ nat @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ X ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( vEBT_vebt_delete @ ( nth @ vEBT_VEBT @ TreeList @ ( vEBT_VEBT_high @ ( if @ nat @ ( X = Mi3 ) @ ( plus_plus @ nat @ ( times_times @ nat @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ X ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( vEBT_VEBT_low @ ( if @ nat @ ( X = Mi3 ) @ ( plus_plus @ nat @ ( times_times @ nat @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ X ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) @ ( vEBT_VEBT_high @ ( if @ nat @ ( X = Mi3 ) @ ( plus_plus @ nat @ ( times_times @ nat @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ X ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) ) )
@ Ma ) ) )
@ ( suc @ ( suc @ Va2 ) )
@ ( list_update @ vEBT_VEBT @ TreeList @ ( vEBT_VEBT_high @ ( if @ nat @ ( X = Mi3 ) @ ( plus_plus @ nat @ ( times_times @ nat @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ X ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( vEBT_vebt_delete @ ( nth @ vEBT_VEBT @ TreeList @ ( vEBT_VEBT_high @ ( if @ nat @ ( X = Mi3 ) @ ( plus_plus @ nat @ ( times_times @ nat @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ X ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( vEBT_VEBT_low @ ( if @ nat @ ( X = Mi3 ) @ ( plus_plus @ nat @ ( times_times @ nat @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ X ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) )
@ Summary ) )
@ ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi3 @ Ma ) ) @ ( suc @ ( suc @ Va2 ) ) @ TreeList @ Summary ) ) ) ) ) ) ) ).
% vebt_delete.simps(7)
thf(fact_3665_T_092_060_094sub_062p_092_060_094sub_062r_092_060_094sub_062e_092_060_094sub_062d_H_Oelims,axiom,
! [X: vEBT_VEBT,Xa: nat,Y: nat] :
( ( ( vEBT_T_p_r_e_d2 @ X @ Xa )
= Y )
=> ( ( ? [Uu2: $o,Uv2: $o] :
( X
= ( vEBT_Leaf @ Uu2 @ Uv2 ) )
=> ( ( Xa
= ( zero_zero @ nat ) )
=> ( Y
!= ( one_one @ nat ) ) ) )
=> ( ( ? [A6: $o,Uw2: $o] :
( X
= ( vEBT_Leaf @ A6 @ Uw2 ) )
=> ( ( Xa
= ( suc @ ( zero_zero @ nat ) ) )
=> ( Y
!= ( one_one @ nat ) ) ) )
=> ( ( ? [A6: $o,B3: $o] :
( X
= ( vEBT_Leaf @ A6 @ B3 ) )
=> ( ? [Va: nat] :
( Xa
= ( suc @ ( suc @ Va ) ) )
=> ( Y
!= ( one_one @ nat ) ) ) )
=> ( ( ? [Uy2: nat,Uz2: list @ vEBT_VEBT,Va3: vEBT_VEBT] :
( X
= ( vEBT_Node @ ( none @ ( product_prod @ nat @ nat ) ) @ Uy2 @ Uz2 @ Va3 ) )
=> ( Y
!= ( one_one @ nat ) ) )
=> ( ( ? [V4: product_prod @ nat @ nat,Vd2: list @ vEBT_VEBT,Ve2: vEBT_VEBT] :
( X
= ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ V4 ) @ ( zero_zero @ nat ) @ Vd2 @ Ve2 ) )
=> ( Y
!= ( one_one @ nat ) ) )
=> ( ( ? [V4: product_prod @ nat @ nat,Vh2: list @ vEBT_VEBT,Vi2: vEBT_VEBT] :
( X
= ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ V4 ) @ ( suc @ ( zero_zero @ nat ) ) @ Vh2 @ Vi2 ) )
=> ( Y
!= ( one_one @ nat ) ) )
=> ~ ! [Mi: nat,Ma2: nat,Va: nat,TreeList2: list @ vEBT_VEBT,Summary2: vEBT_VEBT] :
( ( X
= ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi @ Ma2 ) ) @ ( suc @ ( suc @ Va ) ) @ TreeList2 @ Summary2 ) )
=> ~ ( ( ( ord_less @ nat @ Ma2 @ Xa )
=> ( Y
= ( one_one @ nat ) ) )
& ( ~ ( ord_less @ nat @ Ma2 @ Xa )
=> ( Y
= ( if @ nat @ ( ord_less @ nat @ ( vEBT_VEBT_high @ Xa @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( size_size @ ( list @ vEBT_VEBT ) @ TreeList2 ) )
@ ( if @ nat
@ ( ( ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ Xa @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) )
!= ( none @ nat ) )
& ( vEBT_VEBT_greater @ ( some @ nat @ ( vEBT_VEBT_low @ Xa @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ Xa @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) ) )
@ ( plus_plus @ nat @ ( one_one @ nat ) @ ( vEBT_T_p_r_e_d2 @ ( nth @ vEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ Xa @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( vEBT_VEBT_low @ Xa @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) )
@ ( plus_plus @ nat @ ( vEBT_T_p_r_e_d2 @ Summary2 @ ( vEBT_VEBT_high @ Xa @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( one_one @ nat ) ) )
@ ( one_one @ nat ) ) ) ) ) ) ) ) ) ) ) ) ) ).
% T\<^sub>p\<^sub>r\<^sub>e\<^sub>d'.elims
thf(fact_3666_T_092_060_094sub_062s_092_060_094sub_062u_092_060_094sub_062c_092_060_094sub_062c_H_Oelims,axiom,
! [X: vEBT_VEBT,Xa: nat,Y: nat] :
( ( ( vEBT_T_s_u_c_c2 @ X @ Xa )
= Y )
=> ( ( ? [Uu2: $o,B3: $o] :
( X
= ( vEBT_Leaf @ Uu2 @ B3 ) )
=> ( ( Xa
= ( zero_zero @ nat ) )
=> ( Y
!= ( one_one @ nat ) ) ) )
=> ( ( ? [Uv2: $o,Uw2: $o] :
( X
= ( vEBT_Leaf @ Uv2 @ Uw2 ) )
=> ( ? [N2: nat] :
( Xa
= ( suc @ N2 ) )
=> ( 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 ) ) )
=> ( ( ? [V4: product_prod @ nat @ nat,Vc: list @ vEBT_VEBT,Vd2: vEBT_VEBT] :
( X
= ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ V4 ) @ ( zero_zero @ nat ) @ Vc @ Vd2 ) )
=> ( Y
!= ( one_one @ nat ) ) )
=> ( ( ? [V4: product_prod @ nat @ nat,Vg2: list @ vEBT_VEBT,Vh2: vEBT_VEBT] :
( X
= ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ V4 ) @ ( suc @ ( zero_zero @ nat ) ) @ Vg2 @ Vh2 ) )
=> ( Y
!= ( one_one @ nat ) ) )
=> ~ ! [Mi: nat,Ma2: nat,Va: nat,TreeList2: list @ vEBT_VEBT,Summary2: vEBT_VEBT] :
( ( X
= ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi @ Ma2 ) ) @ ( suc @ ( suc @ Va ) ) @ TreeList2 @ Summary2 ) )
=> ~ ( ( ( ord_less @ nat @ Xa @ Mi )
=> ( Y
= ( one_one @ nat ) ) )
& ( ~ ( ord_less @ nat @ Xa @ Mi )
=> ( Y
= ( if @ nat @ ( ord_less @ nat @ ( vEBT_VEBT_high @ Xa @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( size_size @ ( list @ vEBT_VEBT ) @ TreeList2 ) )
@ ( if @ nat
@ ( ( ( vEBT_vebt_maxt @ ( nth @ vEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ Xa @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) )
!= ( none @ nat ) )
& ( vEBT_VEBT_less @ ( some @ nat @ ( vEBT_VEBT_low @ Xa @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( vEBT_vebt_maxt @ ( nth @ vEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ Xa @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) ) )
@ ( plus_plus @ nat @ ( one_one @ nat ) @ ( vEBT_T_s_u_c_c2 @ ( nth @ vEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ Xa @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( vEBT_VEBT_low @ Xa @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) )
@ ( plus_plus @ nat @ ( vEBT_T_s_u_c_c2 @ Summary2 @ ( vEBT_VEBT_high @ Xa @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( one_one @ nat ) ) )
@ ( one_one @ nat ) ) ) ) ) ) ) ) ) ) ) ) ).
% T\<^sub>s\<^sub>u\<^sub>c\<^sub>c'.elims
thf(fact_3667_vebt__delete_Oelims,axiom,
! [X: vEBT_VEBT,Xa: nat,Y: vEBT_VEBT] :
( ( ( vEBT_vebt_delete @ X @ Xa )
= Y )
=> ( ! [A6: $o,B3: $o] :
( ( X
= ( vEBT_Leaf @ A6 @ B3 ) )
=> ( ( Xa
= ( zero_zero @ nat ) )
=> ( Y
!= ( vEBT_Leaf @ $false @ B3 ) ) ) )
=> ( ! [A6: $o] :
( ? [B3: $o] :
( X
= ( vEBT_Leaf @ A6 @ B3 ) )
=> ( ( Xa
= ( suc @ ( zero_zero @ nat ) ) )
=> ( Y
!= ( vEBT_Leaf @ A6 @ $false ) ) ) )
=> ( ! [A6: $o,B3: $o] :
( ( X
= ( vEBT_Leaf @ A6 @ B3 ) )
=> ( ? [N2: nat] :
( Xa
= ( suc @ ( suc @ N2 ) ) )
=> ( Y
!= ( vEBT_Leaf @ A6 @ 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 ) ) )
=> ( ! [Mi: nat,Ma2: nat,TrLst2: list @ vEBT_VEBT,Smry2: vEBT_VEBT] :
( ( X
= ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi @ Ma2 ) ) @ ( zero_zero @ nat ) @ TrLst2 @ Smry2 ) )
=> ( Y
!= ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi @ Ma2 ) ) @ ( zero_zero @ nat ) @ TrLst2 @ Smry2 ) ) )
=> ( ! [Mi: nat,Ma2: nat,Tr2: list @ vEBT_VEBT,Sm2: vEBT_VEBT] :
( ( X
= ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi @ Ma2 ) ) @ ( suc @ ( zero_zero @ nat ) ) @ Tr2 @ Sm2 ) )
=> ( Y
!= ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi @ Ma2 ) ) @ ( suc @ ( zero_zero @ nat ) ) @ Tr2 @ Sm2 ) ) )
=> ~ ! [Mi: nat,Ma2: nat,Va: nat,TreeList2: list @ vEBT_VEBT,Summary2: vEBT_VEBT] :
( ( X
= ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi @ Ma2 ) ) @ ( suc @ ( suc @ Va ) ) @ TreeList2 @ Summary2 ) )
=> ~ ( ( ( ( ord_less @ nat @ Xa @ Mi )
| ( ord_less @ nat @ Ma2 @ Xa ) )
=> ( Y
= ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi @ Ma2 ) ) @ ( suc @ ( suc @ Va ) ) @ TreeList2 @ Summary2 ) ) )
& ( ~ ( ( ord_less @ nat @ Xa @ Mi )
| ( ord_less @ nat @ Ma2 @ Xa ) )
=> ( ( ( ( Xa = Mi )
& ( Xa = Ma2 ) )
=> ( Y
= ( vEBT_Node @ ( none @ ( product_prod @ nat @ nat ) ) @ ( suc @ ( suc @ Va ) ) @ TreeList2 @ Summary2 ) ) )
& ( ~ ( ( Xa = Mi )
& ( Xa = Ma2 ) )
=> ( Y
= ( if @ vEBT_VEBT @ ( ord_less @ nat @ ( vEBT_VEBT_high @ ( if @ nat @ ( Xa = Mi ) @ ( plus_plus @ nat @ ( times_times @ nat @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary2 ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList2 @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary2 ) ) ) ) ) ) @ Xa ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( size_size @ ( list @ vEBT_VEBT ) @ TreeList2 ) )
@ ( if @ vEBT_VEBT @ ( vEBT_VEBT_minNull @ ( vEBT_vebt_delete @ ( nth @ vEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ ( if @ nat @ ( Xa = Mi ) @ ( plus_plus @ nat @ ( times_times @ nat @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary2 ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList2 @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary2 ) ) ) ) ) ) @ Xa ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( vEBT_VEBT_low @ ( if @ nat @ ( Xa = Mi ) @ ( plus_plus @ nat @ ( times_times @ nat @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary2 ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList2 @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary2 ) ) ) ) ) ) @ Xa ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) )
@ ( vEBT_Node
@ ( some @ ( product_prod @ nat @ nat )
@ ( product_Pair @ nat @ nat @ ( if @ nat @ ( Xa = Mi ) @ ( if @ nat @ ( Xa = Mi ) @ ( plus_plus @ nat @ ( times_times @ nat @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary2 ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList2 @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary2 ) ) ) ) ) ) @ Xa ) @ Mi )
@ ( if @ nat
@ ( ( ( Xa = Mi )
=> ( ( plus_plus @ nat @ ( times_times @ nat @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary2 ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList2 @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary2 ) ) ) ) ) )
= Ma2 ) )
& ( ( Xa != Mi )
=> ( Xa = Ma2 ) ) )
@ ( if @ nat
@ ( ( vEBT_vebt_maxt @ ( vEBT_vebt_delete @ Summary2 @ ( vEBT_VEBT_high @ ( if @ nat @ ( Xa = Mi ) @ ( plus_plus @ nat @ ( times_times @ nat @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary2 ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList2 @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary2 ) ) ) ) ) ) @ Xa ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) )
= ( none @ nat ) )
@ ( if @ nat @ ( Xa = Mi ) @ ( if @ nat @ ( Xa = Mi ) @ ( plus_plus @ nat @ ( times_times @ nat @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary2 ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList2 @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary2 ) ) ) ) ) ) @ Xa ) @ 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 @ Summary2 @ ( vEBT_VEBT_high @ ( if @ nat @ ( Xa = Mi ) @ ( plus_plus @ nat @ ( times_times @ nat @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary2 ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList2 @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary2 ) ) ) ) ) ) @ Xa ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_maxt @ ( nth @ vEBT_VEBT @ ( list_update @ vEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ ( if @ nat @ ( Xa = Mi ) @ ( plus_plus @ nat @ ( times_times @ nat @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary2 ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList2 @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary2 ) ) ) ) ) ) @ Xa ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( vEBT_vebt_delete @ ( nth @ vEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ ( if @ nat @ ( Xa = Mi ) @ ( plus_plus @ nat @ ( times_times @ nat @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary2 ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList2 @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary2 ) ) ) ) ) ) @ Xa ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( vEBT_VEBT_low @ ( if @ nat @ ( Xa = Mi ) @ ( plus_plus @ nat @ ( times_times @ nat @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary2 ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList2 @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary2 ) ) ) ) ) ) @ Xa ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_maxt @ ( vEBT_vebt_delete @ Summary2 @ ( vEBT_VEBT_high @ ( if @ nat @ ( Xa = Mi ) @ ( plus_plus @ nat @ ( times_times @ nat @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary2 ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList2 @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary2 ) ) ) ) ) ) @ Xa ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) ) ) ) ) ) )
@ Ma2 ) ) )
@ ( suc @ ( suc @ Va ) )
@ ( list_update @ vEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ ( if @ nat @ ( Xa = Mi ) @ ( plus_plus @ nat @ ( times_times @ nat @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary2 ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList2 @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary2 ) ) ) ) ) ) @ Xa ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( vEBT_vebt_delete @ ( nth @ vEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ ( if @ nat @ ( Xa = Mi ) @ ( plus_plus @ nat @ ( times_times @ nat @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary2 ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList2 @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary2 ) ) ) ) ) ) @ Xa ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( vEBT_VEBT_low @ ( if @ nat @ ( Xa = Mi ) @ ( plus_plus @ nat @ ( times_times @ nat @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary2 ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList2 @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary2 ) ) ) ) ) ) @ Xa ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) )
@ ( vEBT_vebt_delete @ Summary2 @ ( vEBT_VEBT_high @ ( if @ nat @ ( Xa = Mi ) @ ( plus_plus @ nat @ ( times_times @ nat @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary2 ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList2 @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary2 ) ) ) ) ) ) @ Xa ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) )
@ ( vEBT_Node
@ ( some @ ( product_prod @ nat @ nat )
@ ( product_Pair @ nat @ nat @ ( if @ nat @ ( Xa = Mi ) @ ( if @ nat @ ( Xa = Mi ) @ ( plus_plus @ nat @ ( times_times @ nat @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary2 ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList2 @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary2 ) ) ) ) ) ) @ Xa ) @ Mi )
@ ( if @ nat
@ ( ( ( Xa = Mi )
=> ( ( plus_plus @ nat @ ( times_times @ nat @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary2 ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList2 @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary2 ) ) ) ) ) )
= Ma2 ) )
& ( ( Xa != Mi )
=> ( Xa = Ma2 ) ) )
@ ( plus_plus @ nat @ ( times_times @ nat @ ( vEBT_VEBT_high @ ( if @ nat @ ( Xa = Mi ) @ ( plus_plus @ nat @ ( times_times @ nat @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary2 ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList2 @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary2 ) ) ) ) ) ) @ Xa ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_maxt @ ( nth @ vEBT_VEBT @ ( list_update @ vEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ ( if @ nat @ ( Xa = Mi ) @ ( plus_plus @ nat @ ( times_times @ nat @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary2 ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList2 @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary2 ) ) ) ) ) ) @ Xa ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( vEBT_vebt_delete @ ( nth @ vEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ ( if @ nat @ ( Xa = Mi ) @ ( plus_plus @ nat @ ( times_times @ nat @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary2 ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList2 @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary2 ) ) ) ) ) ) @ Xa ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( vEBT_VEBT_low @ ( if @ nat @ ( Xa = Mi ) @ ( plus_plus @ nat @ ( times_times @ nat @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary2 ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList2 @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary2 ) ) ) ) ) ) @ Xa ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) @ ( vEBT_VEBT_high @ ( if @ nat @ ( Xa = Mi ) @ ( plus_plus @ nat @ ( times_times @ nat @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary2 ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList2 @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary2 ) ) ) ) ) ) @ Xa ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) ) )
@ Ma2 ) ) )
@ ( suc @ ( suc @ Va ) )
@ ( list_update @ vEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ ( if @ nat @ ( Xa = Mi ) @ ( plus_plus @ nat @ ( times_times @ nat @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary2 ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList2 @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary2 ) ) ) ) ) ) @ Xa ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( vEBT_vebt_delete @ ( nth @ vEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ ( if @ nat @ ( Xa = Mi ) @ ( plus_plus @ nat @ ( times_times @ nat @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary2 ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList2 @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary2 ) ) ) ) ) ) @ Xa ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( vEBT_VEBT_low @ ( if @ nat @ ( Xa = Mi ) @ ( plus_plus @ nat @ ( times_times @ nat @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary2 ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList2 @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary2 ) ) ) ) ) ) @ Xa ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) )
@ Summary2 ) )
@ ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi @ Ma2 ) ) @ ( suc @ ( suc @ Va ) ) @ TreeList2 @ Summary2 ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).
% vebt_delete.elims
thf(fact_3668_VEBT__internal_OT_092_060_094sub_062d_092_060_094sub_062e_092_060_094sub_062l_092_060_094sub_062e_092_060_094sub_062t_092_060_094sub_062e_H_Oelims,axiom,
! [X: vEBT_VEBT,Xa: nat,Y: nat] :
( ( ( vEBT_V1232361888498592333_e_t_e @ X @ Xa )
= Y )
=> ( ( ? [A6: $o,B3: $o] :
( X
= ( vEBT_Leaf @ A6 @ B3 ) )
=> ( ( Xa
= ( zero_zero @ nat ) )
=> ( Y
!= ( one_one @ nat ) ) ) )
=> ( ( ? [A6: $o,B3: $o] :
( X
= ( vEBT_Leaf @ A6 @ B3 ) )
=> ( ( Xa
= ( suc @ ( zero_zero @ nat ) ) )
=> ( Y
!= ( one_one @ nat ) ) ) )
=> ( ( ? [A6: $o,B3: $o] :
( X
= ( vEBT_Leaf @ A6 @ B3 ) )
=> ( ? [N2: nat] :
( Xa
= ( suc @ ( suc @ N2 ) ) )
=> ( Y
!= ( one_one @ nat ) ) ) )
=> ( ( ? [Deg2: nat,TreeList2: list @ vEBT_VEBT,Summary2: vEBT_VEBT] :
( X
= ( vEBT_Node @ ( none @ ( product_prod @ nat @ nat ) ) @ Deg2 @ TreeList2 @ Summary2 ) )
=> ( Y
!= ( one_one @ nat ) ) )
=> ( ( ? [Mi: nat,Ma2: nat,TreeList2: list @ vEBT_VEBT,Summary2: vEBT_VEBT] :
( X
= ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi @ Ma2 ) ) @ ( zero_zero @ nat ) @ TreeList2 @ Summary2 ) )
=> ( Y
!= ( one_one @ nat ) ) )
=> ( ( ? [Mi: nat,Ma2: nat,TreeList2: list @ vEBT_VEBT,Summary2: vEBT_VEBT] :
( X
= ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi @ Ma2 ) ) @ ( suc @ ( zero_zero @ nat ) ) @ TreeList2 @ Summary2 ) )
=> ( Y
!= ( one_one @ nat ) ) )
=> ~ ! [Mi: nat,Ma2: nat,Va: nat,TreeList2: list @ vEBT_VEBT,Summary2: vEBT_VEBT] :
( ( X
= ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi @ Ma2 ) ) @ ( suc @ ( suc @ Va ) ) @ TreeList2 @ Summary2 ) )
=> ~ ( ( ( ( ord_less @ nat @ Xa @ Mi )
| ( ord_less @ nat @ Ma2 @ Xa ) )
=> ( Y
= ( one_one @ nat ) ) )
& ( ~ ( ( ord_less @ nat @ Xa @ Mi )
| ( ord_less @ nat @ Ma2 @ Xa ) )
=> ( ( ( ( Xa = Mi )
& ( Xa = Ma2 ) )
=> ( Y
= ( one_one @ nat ) ) )
& ( ~ ( ( Xa = Mi )
& ( Xa = Ma2 ) )
=> ( Y
= ( if @ nat @ ( ord_less @ nat @ ( vEBT_VEBT_high @ ( if @ nat @ ( Xa = Mi ) @ ( plus_plus @ nat @ ( times_times @ nat @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary2 ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList2 @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary2 ) ) ) ) ) ) @ Xa ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( size_size @ ( list @ vEBT_VEBT ) @ TreeList2 ) ) @ ( plus_plus @ nat @ ( vEBT_V1232361888498592333_e_t_e @ ( nth @ vEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ ( if @ nat @ ( Xa = Mi ) @ ( plus_plus @ nat @ ( times_times @ nat @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary2 ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList2 @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary2 ) ) ) ) ) ) @ Xa ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( vEBT_VEBT_low @ ( if @ nat @ ( Xa = Mi ) @ ( plus_plus @ nat @ ( times_times @ nat @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary2 ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList2 @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary2 ) ) ) ) ) ) @ Xa ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( if @ nat @ ( vEBT_VEBT_minNull @ ( vEBT_vebt_delete @ ( nth @ vEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ ( if @ nat @ ( Xa = Mi ) @ ( plus_plus @ nat @ ( times_times @ nat @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary2 ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList2 @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary2 ) ) ) ) ) ) @ Xa ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( vEBT_VEBT_low @ ( if @ nat @ ( Xa = Mi ) @ ( plus_plus @ nat @ ( times_times @ nat @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary2 ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList2 @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary2 ) ) ) ) ) ) @ Xa ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) @ ( vEBT_V1232361888498592333_e_t_e @ Summary2 @ ( vEBT_VEBT_high @ ( if @ nat @ ( Xa = Mi ) @ ( plus_plus @ nat @ ( times_times @ nat @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary2 ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList2 @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary2 ) ) ) ) ) ) @ Xa ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( one_one @ nat ) ) ) @ ( one_one @ nat ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).
% VEBT_internal.T\<^sub>d\<^sub>e\<^sub>l\<^sub>e\<^sub>t\<^sub>e'.elims
thf(fact_3669_vebt__pred_Oelims,axiom,
! [X: vEBT_VEBT,Xa: nat,Y: option @ nat] :
( ( ( vEBT_vebt_pred @ X @ Xa )
= Y )
=> ( ( ? [Uu2: $o,Uv2: $o] :
( X
= ( vEBT_Leaf @ Uu2 @ Uv2 ) )
=> ( ( Xa
= ( zero_zero @ nat ) )
=> ( Y
!= ( none @ nat ) ) ) )
=> ( ! [A6: $o] :
( ? [Uw2: $o] :
( X
= ( vEBT_Leaf @ A6 @ Uw2 ) )
=> ( ( Xa
= ( suc @ ( zero_zero @ nat ) ) )
=> ~ ( ( A6
=> ( Y
= ( some @ nat @ ( zero_zero @ nat ) ) ) )
& ( ~ A6
=> ( Y
= ( none @ nat ) ) ) ) ) )
=> ( ! [A6: $o,B3: $o] :
( ( X
= ( vEBT_Leaf @ A6 @ B3 ) )
=> ( ? [Va: nat] :
( Xa
= ( suc @ ( suc @ Va ) ) )
=> ~ ( ( B3
=> ( Y
= ( some @ nat @ ( one_one @ nat ) ) ) )
& ( ~ B3
=> ( ( A6
=> ( Y
= ( some @ nat @ ( zero_zero @ nat ) ) ) )
& ( ~ A6
=> ( Y
= ( none @ nat ) ) ) ) ) ) ) )
=> ( ( ? [Uy2: nat,Uz2: list @ vEBT_VEBT,Va3: vEBT_VEBT] :
( X
= ( vEBT_Node @ ( none @ ( product_prod @ nat @ nat ) ) @ Uy2 @ Uz2 @ Va3 ) )
=> ( Y
!= ( none @ nat ) ) )
=> ( ( ? [V4: product_prod @ nat @ nat,Vd2: list @ vEBT_VEBT,Ve2: vEBT_VEBT] :
( X
= ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ V4 ) @ ( zero_zero @ nat ) @ Vd2 @ Ve2 ) )
=> ( Y
!= ( none @ nat ) ) )
=> ( ( ? [V4: product_prod @ nat @ nat,Vh2: list @ vEBT_VEBT,Vi2: vEBT_VEBT] :
( X
= ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ V4 ) @ ( suc @ ( zero_zero @ nat ) ) @ Vh2 @ Vi2 ) )
=> ( Y
!= ( none @ nat ) ) )
=> ~ ! [Mi: nat,Ma2: nat,Va: nat,TreeList2: list @ vEBT_VEBT,Summary2: vEBT_VEBT] :
( ( X
= ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi @ Ma2 ) ) @ ( suc @ ( suc @ Va ) ) @ TreeList2 @ Summary2 ) )
=> ~ ( ( ( ord_less @ nat @ Ma2 @ Xa )
=> ( Y
= ( some @ nat @ Ma2 ) ) )
& ( ~ ( ord_less @ nat @ Ma2 @ Xa )
=> ( Y
= ( if @ ( option @ nat ) @ ( ord_less @ nat @ ( vEBT_VEBT_high @ Xa @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( size_size @ ( list @ vEBT_VEBT ) @ TreeList2 ) )
@ ( if @ ( option @ nat )
@ ( ( ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ Xa @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) )
!= ( none @ nat ) )
& ( vEBT_VEBT_greater @ ( some @ nat @ ( vEBT_VEBT_low @ Xa @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ Xa @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) ) )
@ ( vEBT_VEBT_add @ ( vEBT_VEBT_mul @ ( some @ nat @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( some @ nat @ ( vEBT_VEBT_high @ Xa @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) @ ( vEBT_vebt_pred @ ( nth @ vEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ Xa @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( vEBT_VEBT_low @ Xa @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) )
@ ( if @ ( option @ nat )
@ ( ( vEBT_vebt_pred @ Summary2 @ ( vEBT_VEBT_high @ Xa @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) )
= ( none @ nat ) )
@ ( if @ ( option @ nat ) @ ( ord_less @ nat @ Mi @ Xa ) @ ( 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 @ Summary2 @ ( vEBT_VEBT_high @ Xa @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) @ ( vEBT_vebt_maxt @ ( nth @ vEBT_VEBT @ TreeList2 @ ( the2 @ nat @ ( vEBT_vebt_pred @ Summary2 @ ( vEBT_VEBT_high @ Xa @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) ) ) ) ) )
@ ( none @ nat ) ) ) ) ) ) ) ) ) ) ) ) ) ).
% vebt_pred.elims
thf(fact_3670_vebt__succ_Oelims,axiom,
! [X: vEBT_VEBT,Xa: nat,Y: option @ nat] :
( ( ( vEBT_vebt_succ @ X @ Xa )
= Y )
=> ( ! [Uu2: $o,B3: $o] :
( ( X
= ( vEBT_Leaf @ Uu2 @ B3 ) )
=> ( ( Xa
= ( zero_zero @ nat ) )
=> ~ ( ( B3
=> ( Y
= ( some @ nat @ ( one_one @ nat ) ) ) )
& ( ~ B3
=> ( Y
= ( none @ nat ) ) ) ) ) )
=> ( ( ? [Uv2: $o,Uw2: $o] :
( X
= ( vEBT_Leaf @ Uv2 @ Uw2 ) )
=> ( ? [N2: nat] :
( Xa
= ( suc @ N2 ) )
=> ( 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 ) ) )
=> ( ( ? [V4: product_prod @ nat @ nat,Vc: list @ vEBT_VEBT,Vd2: vEBT_VEBT] :
( X
= ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ V4 ) @ ( zero_zero @ nat ) @ Vc @ Vd2 ) )
=> ( Y
!= ( none @ nat ) ) )
=> ( ( ? [V4: product_prod @ nat @ nat,Vg2: list @ vEBT_VEBT,Vh2: vEBT_VEBT] :
( X
= ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ V4 ) @ ( suc @ ( zero_zero @ nat ) ) @ Vg2 @ Vh2 ) )
=> ( Y
!= ( none @ nat ) ) )
=> ~ ! [Mi: nat,Ma2: nat,Va: nat,TreeList2: list @ vEBT_VEBT,Summary2: vEBT_VEBT] :
( ( X
= ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi @ Ma2 ) ) @ ( suc @ ( suc @ Va ) ) @ TreeList2 @ Summary2 ) )
=> ~ ( ( ( ord_less @ nat @ Xa @ Mi )
=> ( Y
= ( some @ nat @ Mi ) ) )
& ( ~ ( ord_less @ nat @ Xa @ Mi )
=> ( Y
= ( if @ ( option @ nat ) @ ( ord_less @ nat @ ( vEBT_VEBT_high @ Xa @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( size_size @ ( list @ vEBT_VEBT ) @ TreeList2 ) )
@ ( if @ ( option @ nat )
@ ( ( ( vEBT_vebt_maxt @ ( nth @ vEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ Xa @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) )
!= ( none @ nat ) )
& ( vEBT_VEBT_less @ ( some @ nat @ ( vEBT_VEBT_low @ Xa @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( vEBT_vebt_maxt @ ( nth @ vEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ Xa @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) ) )
@ ( vEBT_VEBT_add @ ( vEBT_VEBT_mul @ ( some @ nat @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( some @ nat @ ( vEBT_VEBT_high @ Xa @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) @ ( vEBT_vebt_succ @ ( nth @ vEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ Xa @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( vEBT_VEBT_low @ Xa @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) )
@ ( if @ ( option @ nat )
@ ( ( vEBT_vebt_succ @ Summary2 @ ( vEBT_VEBT_high @ Xa @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) )
= ( none @ nat ) )
@ ( none @ nat )
@ ( vEBT_VEBT_add @ ( vEBT_VEBT_mul @ ( some @ nat @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( vEBT_vebt_succ @ Summary2 @ ( vEBT_VEBT_high @ Xa @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) @ ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList2 @ ( the2 @ nat @ ( vEBT_vebt_succ @ Summary2 @ ( vEBT_VEBT_high @ Xa @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) ) ) ) ) )
@ ( none @ nat ) ) ) ) ) ) ) ) ) ) ) ) ).
% vebt_succ.elims
thf(fact_3671_of__int__code__if,axiom,
! [A: $tType] :
( ( ring_1 @ A )
=> ( ( ring_1_of_int @ A )
= ( ^ [K3: int] :
( if @ A
@ ( K3
= ( zero_zero @ int ) )
@ ( zero_zero @ A )
@ ( if @ A @ ( ord_less @ int @ K3 @ ( zero_zero @ int ) ) @ ( uminus_uminus @ A @ ( ring_1_of_int @ A @ ( uminus_uminus @ int @ K3 ) ) )
@ ( if @ A
@ ( ( modulo_modulo @ int @ K3 @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) )
= ( zero_zero @ int ) )
@ ( times_times @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ ( ring_1_of_int @ A @ ( divide_divide @ int @ K3 @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) ) ) )
@ ( plus_plus @ A @ ( times_times @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ ( ring_1_of_int @ A @ ( divide_divide @ int @ K3 @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) ) ) ) @ ( one_one @ A ) ) ) ) ) ) ) ) ).
% of_int_code_if
thf(fact_3672_lowi__hT,axiom,
! [X: nat,N: nat] :
( time_htt @ nat @ ( one_one @ assn ) @ ( vEBT_VEBT_lowi @ X @ N )
@ ^ [R5: nat] :
( pure_assn
@ ( R5
= ( vEBT_VEBT_low @ X @ N ) ) )
@ ( one_one @ nat ) ) ).
% lowi_hT
thf(fact_3673_highi__hT,axiom,
! [X: nat,N: nat] :
( time_htt @ nat @ ( one_one @ assn ) @ ( vEBT_VEBT_highi @ X @ N )
@ ^ [R5: nat] :
( pure_assn
@ ( R5
= ( vEBT_VEBT_high @ X @ N ) ) )
@ ( one_one @ nat ) ) ).
% highi_hT
thf(fact_3674_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_3675_T_092_060_094sub_062i_092_060_094sub_062n_092_060_094sub_062s_092_060_094sub_062e_092_060_094sub_062r_092_060_094sub_062t_Oelims,axiom,
! [X: vEBT_VEBT,Xa: nat,Y: nat] :
( ( ( vEBT_T_i_n_s_e_r_t @ X @ Xa )
= Y )
=> ( ( ? [A6: $o,B3: $o] :
( X
= ( vEBT_Leaf @ A6 @ 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 ) ) )
=> ( ( ? [V4: nat,TreeList2: list @ vEBT_VEBT,Summary2: vEBT_VEBT] :
( X
= ( vEBT_Node @ ( none @ ( product_prod @ nat @ nat ) ) @ ( suc @ ( suc @ V4 ) ) @ TreeList2 @ Summary2 ) )
=> ( Y
!= ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) )
=> ~ ! [Mi: nat,Ma2: nat,Va: nat,TreeList2: list @ vEBT_VEBT,Summary2: vEBT_VEBT] :
( ( X
= ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi @ Ma2 ) ) @ ( suc @ ( suc @ Va ) ) @ TreeList2 @ Summary2 ) )
=> ( Y
!= ( plus_plus @ nat @ ( numeral_numeral @ nat @ ( bit1 @ ( bit1 @ ( bit0 @ ( bit0 @ one2 ) ) ) ) )
@ ( if @ nat
@ ( ( ord_less @ nat @ ( vEBT_VEBT_high @ ( if @ nat @ ( ord_less @ nat @ Xa @ Mi ) @ Mi @ Xa ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( size_size @ ( list @ vEBT_VEBT ) @ TreeList2 ) )
& ~ ( ( Xa = Mi )
| ( 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 @ Mi ) @ Mi @ Xa ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( vEBT_VEBT_low @ ( if @ nat @ ( ord_less @ nat @ Xa @ Mi ) @ Mi @ Xa ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( vEBT_T_m_i_n_N_u_l_l @ ( nth @ vEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ ( if @ nat @ ( ord_less @ nat @ Xa @ Mi ) @ Mi @ Xa ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) ) @ ( if @ nat @ ( vEBT_VEBT_minNull @ ( nth @ vEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ ( if @ nat @ ( ord_less @ nat @ Xa @ Mi ) @ Mi @ Xa ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) @ ( vEBT_T_i_n_s_e_r_t @ Summary2 @ ( vEBT_VEBT_high @ ( if @ nat @ ( ord_less @ nat @ Xa @ Mi ) @ Mi @ Xa ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( one_one @ nat ) ) )
@ ( one_one @ nat ) ) ) ) ) ) ) ) ) ) ).
% T\<^sub>i\<^sub>n\<^sub>s\<^sub>e\<^sub>r\<^sub>t.elims
thf(fact_3676_T_092_060_094sub_062m_092_060_094sub_062e_092_060_094sub_062m_092_060_094sub_062b_092_060_094sub_062e_092_060_094sub_062r_H_Oelims,axiom,
! [X: vEBT_VEBT,Xa: nat,Y: nat] :
( ( ( vEBT_T_m_e_m_b_e_r2 @ X @ Xa )
= Y )
=> ( ( ? [A6: $o,B3: $o] :
( X
= ( vEBT_Leaf @ A6 @ B3 ) )
=> ( Y
!= ( one_one @ nat ) ) )
=> ( ( ? [Uu2: nat,Uv2: list @ vEBT_VEBT,Uw2: vEBT_VEBT] :
( X
= ( vEBT_Node @ ( none @ ( product_prod @ nat @ nat ) ) @ Uu2 @ Uv2 @ Uw2 ) )
=> ( Y
!= ( one_one @ nat ) ) )
=> ( ( ? [V4: product_prod @ nat @ nat,Uy2: list @ vEBT_VEBT,Uz2: vEBT_VEBT] :
( X
= ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ V4 ) @ ( zero_zero @ nat ) @ Uy2 @ Uz2 ) )
=> ( Y
!= ( one_one @ nat ) ) )
=> ( ( ? [V4: product_prod @ nat @ nat,Vb2: list @ vEBT_VEBT,Vc: vEBT_VEBT] :
( X
= ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ V4 ) @ ( suc @ ( zero_zero @ nat ) ) @ Vb2 @ Vc ) )
=> ( Y
!= ( one_one @ nat ) ) )
=> ~ ! [Mi: nat,Ma2: nat,Va: nat,TreeList2: list @ vEBT_VEBT] :
( ? [Summary2: vEBT_VEBT] :
( X
= ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi @ Ma2 ) ) @ ( suc @ ( suc @ Va ) ) @ TreeList2 @ Summary2 ) )
=> ( Y
!= ( plus_plus @ nat @ ( one_one @ nat )
@ ( if @ nat @ ( Xa = Mi ) @ ( zero_zero @ nat )
@ ( if @ nat @ ( Xa = Ma2 ) @ ( zero_zero @ nat )
@ ( if @ nat @ ( ord_less @ nat @ Xa @ Mi ) @ ( zero_zero @ nat )
@ ( if @ nat @ ( ord_less @ nat @ Ma2 @ Xa ) @ ( zero_zero @ nat )
@ ( if @ nat
@ ( ( ord_less @ nat @ Mi @ Xa )
& ( ord_less @ nat @ Xa @ Ma2 ) )
@ ( if @ nat @ ( ord_less @ nat @ ( vEBT_VEBT_high @ Xa @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( size_size @ ( list @ vEBT_VEBT ) @ TreeList2 ) ) @ ( vEBT_T_m_e_m_b_e_r2 @ ( nth @ vEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ Xa @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( vEBT_VEBT_low @ Xa @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( zero_zero @ nat ) )
@ ( zero_zero @ nat ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).
% T\<^sub>m\<^sub>e\<^sub>m\<^sub>b\<^sub>e\<^sub>r'.elims
thf(fact_3677_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,
! [Mi3: nat,Ma: nat,Va2: nat,TreeList: list @ vEBT_VEBT,Summary: vEBT_VEBT,X: nat] :
( ( vEBT_T_i_n_s_e_r_t @ ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi3 @ Ma ) ) @ ( suc @ ( suc @ Va2 ) ) @ TreeList @ Summary ) @ X )
= ( plus_plus @ nat @ ( numeral_numeral @ nat @ ( bit1 @ ( bit1 @ ( bit0 @ ( bit0 @ one2 ) ) ) ) )
@ ( if @ nat
@ ( ( ord_less @ nat @ ( vEBT_VEBT_high @ ( if @ nat @ ( ord_less @ nat @ X @ Mi3 ) @ Mi3 @ X ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( size_size @ ( list @ vEBT_VEBT ) @ TreeList ) )
& ~ ( ( X = Mi3 )
| ( 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 @ Mi3 ) @ Mi3 @ X ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( vEBT_VEBT_low @ ( if @ nat @ ( ord_less @ nat @ X @ Mi3 ) @ Mi3 @ X ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( vEBT_T_m_i_n_N_u_l_l @ ( nth @ vEBT_VEBT @ TreeList @ ( vEBT_VEBT_high @ ( if @ nat @ ( ord_less @ nat @ X @ Mi3 ) @ Mi3 @ X ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) ) @ ( if @ nat @ ( vEBT_VEBT_minNull @ ( nth @ vEBT_VEBT @ TreeList @ ( vEBT_VEBT_high @ ( if @ nat @ ( ord_less @ nat @ X @ Mi3 ) @ Mi3 @ X ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) @ ( vEBT_T_i_n_s_e_r_t @ Summary @ ( vEBT_VEBT_high @ ( if @ nat @ ( ord_less @ nat @ X @ Mi3 ) @ Mi3 @ X ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( one_one @ nat ) ) )
@ ( one_one @ nat ) ) ) ) ).
% T\<^sub>i\<^sub>n\<^sub>s\<^sub>e\<^sub>r\<^sub>t.simps(5)
thf(fact_3678_T_092_060_094sub_062i_092_060_094sub_062n_092_060_094sub_062s_092_060_094sub_062e_092_060_094sub_062r_092_060_094sub_062t_H_Oelims,axiom,
! [X: vEBT_VEBT,Xa: nat,Y: nat] :
( ( ( vEBT_T_i_n_s_e_r_t2 @ X @ Xa )
= Y )
=> ( ( ? [A6: $o,B3: $o] :
( X
= ( vEBT_Leaf @ A6 @ 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 ) ) )
=> ( ( ? [V4: nat,TreeList2: list @ vEBT_VEBT,Summary2: vEBT_VEBT] :
( X
= ( vEBT_Node @ ( none @ ( product_prod @ nat @ nat ) ) @ ( suc @ ( suc @ V4 ) ) @ TreeList2 @ Summary2 ) )
=> ( Y
!= ( one_one @ nat ) ) )
=> ~ ! [Mi: nat,Ma2: nat,Va: nat,TreeList2: list @ vEBT_VEBT,Summary2: vEBT_VEBT] :
( ( X
= ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi @ Ma2 ) ) @ ( suc @ ( suc @ Va ) ) @ TreeList2 @ Summary2 ) )
=> ( Y
!= ( if @ nat
@ ( ( ord_less @ nat @ ( vEBT_VEBT_high @ ( if @ nat @ ( ord_less @ nat @ Xa @ Mi ) @ Mi @ Xa ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( size_size @ ( list @ vEBT_VEBT ) @ TreeList2 ) )
& ~ ( ( Xa = Mi )
| ( 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 @ Mi ) @ Mi @ Xa ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( vEBT_VEBT_low @ ( if @ nat @ ( ord_less @ nat @ Xa @ Mi ) @ Mi @ Xa ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( if @ nat @ ( vEBT_VEBT_minNull @ ( nth @ vEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ ( if @ nat @ ( ord_less @ nat @ Xa @ Mi ) @ Mi @ Xa ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) @ ( vEBT_T_i_n_s_e_r_t2 @ Summary2 @ ( vEBT_VEBT_high @ ( if @ nat @ ( ord_less @ nat @ Xa @ Mi ) @ Mi @ Xa ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( one_one @ nat ) ) )
@ ( one_one @ nat ) ) ) ) ) ) ) ) ) ).
% T\<^sub>i\<^sub>n\<^sub>s\<^sub>e\<^sub>r\<^sub>t'.elims
thf(fact_3679_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_3680_minNull__bound,axiom,
! [T: vEBT_VEBT] : ( ord_less_eq @ nat @ ( vEBT_T_m_i_n_N_u_l_l @ T ) @ ( one_one @ nat ) ) ).
% minNull_bound
thf(fact_3681_T_092_060_094sub_062m_092_060_094sub_062i_092_060_094sub_062n_092_060_094sub_062N_092_060_094sub_062u_092_060_094sub_062l_092_060_094sub_062l_Osimps_I3_J,axiom,
! [Uu: $o] :
( ( vEBT_T_m_i_n_N_u_l_l @ ( vEBT_Leaf @ Uu @ $true ) )
= ( one_one @ nat ) ) ).
% T\<^sub>m\<^sub>i\<^sub>n\<^sub>N\<^sub>u\<^sub>l\<^sub>l.simps(3)
thf(fact_3682_T_092_060_094sub_062m_092_060_094sub_062i_092_060_094sub_062n_092_060_094sub_062N_092_060_094sub_062u_092_060_094sub_062l_092_060_094sub_062l_Osimps_I2_J,axiom,
! [Uv: $o] :
( ( vEBT_T_m_i_n_N_u_l_l @ ( vEBT_Leaf @ $true @ Uv ) )
= ( one_one @ nat ) ) ).
% T\<^sub>m\<^sub>i\<^sub>n\<^sub>N\<^sub>u\<^sub>l\<^sub>l.simps(2)
thf(fact_3683_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_3684_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_3685_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_3686_T_092_060_094sub_062m_092_060_094sub_062i_092_060_094sub_062n_092_060_094sub_062N_092_060_094sub_062u_092_060_094sub_062l_092_060_094sub_062l_Osimps_I5_J,axiom,
! [Uz: product_prod @ nat @ nat,Va2: nat,Vb: list @ vEBT_VEBT,Vc2: vEBT_VEBT] :
( ( vEBT_T_m_i_n_N_u_l_l @ ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ Uz ) @ Va2 @ Vb @ Vc2 ) )
= ( one_one @ nat ) ) ).
% T\<^sub>m\<^sub>i\<^sub>n\<^sub>N\<^sub>u\<^sub>l\<^sub>l.simps(5)
thf(fact_3687_T_092_060_094sub_062m_092_060_094sub_062i_092_060_094sub_062n_092_060_094sub_062N_092_060_094sub_062u_092_060_094sub_062l_092_060_094sub_062l_Osimps_I4_J,axiom,
! [Uw: nat,Ux: list @ vEBT_VEBT,Uy: vEBT_VEBT] :
( ( vEBT_T_m_i_n_N_u_l_l @ ( vEBT_Node @ ( none @ ( product_prod @ nat @ nat ) ) @ Uw @ Ux @ Uy ) )
= ( one_one @ nat ) ) ).
% T\<^sub>m\<^sub>i\<^sub>n\<^sub>N\<^sub>u\<^sub>l\<^sub>l.simps(4)
thf(fact_3688_T_092_060_094sub_062m_092_060_094sub_062e_092_060_094sub_062m_092_060_094sub_062b_092_060_094sub_062e_092_060_094sub_062r_H_Osimps_I2_J,axiom,
! [Uu: nat,Uv: list @ vEBT_VEBT,Uw: vEBT_VEBT,X: nat] :
( ( vEBT_T_m_e_m_b_e_r2 @ ( vEBT_Node @ ( none @ ( product_prod @ nat @ nat ) ) @ Uu @ Uv @ Uw ) @ X )
= ( one_one @ nat ) ) ).
% T\<^sub>m\<^sub>e\<^sub>m\<^sub>b\<^sub>e\<^sub>r'.simps(2)
thf(fact_3689_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_3690_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_3691_insersimp_H,axiom,
! [T: vEBT_VEBT,N: nat,Y: nat] :
( ( vEBT_invar_vebt @ T @ N )
=> ( ~ ? [X_1: nat] : ( vEBT_V8194947554948674370ptions @ T @ X_1 )
=> ( ord_less_eq @ nat @ ( vEBT_T_i_n_s_e_r_t2 @ T @ Y ) @ ( one_one @ nat ) ) ) ) ).
% insersimp'
thf(fact_3692_insertsimp_H,axiom,
! [T: vEBT_VEBT,N: nat,L: nat] :
( ( vEBT_invar_vebt @ T @ N )
=> ( ( vEBT_VEBT_minNull @ T )
=> ( ord_less_eq @ nat @ ( vEBT_T_i_n_s_e_r_t2 @ T @ L ) @ ( one_one @ nat ) ) ) ) ).
% insertsimp'
thf(fact_3693_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_3694_insert_H__bound__height,axiom,
! [T: vEBT_VEBT,N: nat,X: nat] :
( ( vEBT_invar_vebt @ T @ N )
=> ( ord_less_eq @ nat @ ( vEBT_T_i_n_s_e_r_t2 @ T @ X ) @ ( plus_plus @ nat @ ( one_one @ nat ) @ ( vEBT_VEBT_height @ T ) ) ) ) ).
% insert'_bound_height
thf(fact_3695_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 ) ) )
=> ( ( ? [Uv2: $o] :
( X
= ( vEBT_Leaf @ $true @ Uv2 ) )
=> ( Y
!= ( one_one @ nat ) ) )
=> ( ( ? [Uu2: $o] :
( X
= ( vEBT_Leaf @ Uu2 @ $true ) )
=> ( Y
!= ( one_one @ nat ) ) )
=> ( ( ? [Uw2: nat,Ux2: list @ vEBT_VEBT,Uy2: vEBT_VEBT] :
( X
= ( vEBT_Node @ ( none @ ( product_prod @ nat @ nat ) ) @ Uw2 @ Ux2 @ Uy2 ) )
=> ( Y
!= ( one_one @ nat ) ) )
=> ~ ( ? [Uz2: product_prod @ nat @ nat,Va3: nat,Vb2: list @ vEBT_VEBT,Vc: vEBT_VEBT] :
( X
= ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ Uz2 ) @ Va3 @ Vb2 @ Vc ) )
=> ( Y
!= ( one_one @ nat ) ) ) ) ) ) ) ) ).
% T\<^sub>m\<^sub>i\<^sub>n\<^sub>N\<^sub>u\<^sub>l\<^sub>l.elims
thf(fact_3696_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,Vc2: 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 @ Vc2 ) @ X )
= ( one_one @ nat ) ) ).
% T\<^sub>m\<^sub>e\<^sub>m\<^sub>b\<^sub>e\<^sub>r'.simps(4)
thf(fact_3697_member__bound__height_H,axiom,
! [T: vEBT_VEBT,N: nat,X: nat] :
( ( vEBT_invar_vebt @ T @ N )
=> ( ord_less_eq @ nat @ ( vEBT_T_m_e_m_b_e_r2 @ T @ X ) @ ( plus_plus @ nat @ ( one_one @ nat ) @ ( vEBT_VEBT_height @ T ) ) ) ) ).
% member_bound_height'
thf(fact_3698_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,
! [Mi3: nat,Ma: nat,Va2: nat,TreeList: list @ vEBT_VEBT,Summary: vEBT_VEBT,X: nat] :
( ( vEBT_T_i_n_s_e_r_t2 @ ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi3 @ Ma ) ) @ ( suc @ ( suc @ Va2 ) ) @ TreeList @ Summary ) @ X )
= ( if @ nat
@ ( ( ord_less @ nat @ ( vEBT_VEBT_high @ ( if @ nat @ ( ord_less @ nat @ X @ Mi3 ) @ Mi3 @ X ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( size_size @ ( list @ vEBT_VEBT ) @ TreeList ) )
& ~ ( ( X = Mi3 )
| ( 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 @ Mi3 ) @ Mi3 @ X ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( vEBT_VEBT_low @ ( if @ nat @ ( ord_less @ nat @ X @ Mi3 ) @ Mi3 @ X ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( if @ nat @ ( vEBT_VEBT_minNull @ ( nth @ vEBT_VEBT @ TreeList @ ( vEBT_VEBT_high @ ( if @ nat @ ( ord_less @ nat @ X @ Mi3 ) @ Mi3 @ X ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) @ ( vEBT_T_i_n_s_e_r_t2 @ Summary @ ( vEBT_VEBT_high @ ( if @ nat @ ( ord_less @ nat @ X @ Mi3 ) @ Mi3 @ X ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( one_one @ nat ) ) )
@ ( one_one @ nat ) ) ) ).
% T\<^sub>i\<^sub>n\<^sub>s\<^sub>e\<^sub>r\<^sub>t'.simps(5)
thf(fact_3699_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,
! [Mi3: nat,Ma: nat,Va2: nat,TreeList: list @ vEBT_VEBT,Summary: vEBT_VEBT,X: nat] :
( ( vEBT_T_m_e_m_b_e_r2 @ ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi3 @ Ma ) ) @ ( suc @ ( suc @ Va2 ) ) @ TreeList @ Summary ) @ X )
= ( plus_plus @ nat @ ( one_one @ nat )
@ ( if @ nat @ ( X = Mi3 ) @ ( zero_zero @ nat )
@ ( if @ nat @ ( X = Ma ) @ ( zero_zero @ nat )
@ ( if @ nat @ ( ord_less @ nat @ X @ Mi3 ) @ ( zero_zero @ nat )
@ ( if @ nat @ ( ord_less @ nat @ Ma @ X ) @ ( zero_zero @ nat )
@ ( if @ nat
@ ( ( ord_less @ nat @ Mi3 @ X )
& ( ord_less @ nat @ X @ Ma ) )
@ ( if @ nat @ ( ord_less @ nat @ ( vEBT_VEBT_high @ X @ ( divide_divide @ nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( size_size @ ( list @ vEBT_VEBT ) @ TreeList ) ) @ ( vEBT_T_m_e_m_b_e_r2 @ ( nth @ vEBT_VEBT @ TreeList @ ( vEBT_VEBT_high @ X @ ( divide_divide @ nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( vEBT_VEBT_low @ X @ ( divide_divide @ nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( zero_zero @ nat ) )
@ ( zero_zero @ nat ) ) ) ) ) ) ) ) ).
% T\<^sub>m\<^sub>e\<^sub>m\<^sub>b\<^sub>e\<^sub>r'.simps(5)
thf(fact_3700_finite__Collect__less__nat,axiom,
! [K: nat] :
( finite_finite2 @ nat
@ ( collect @ nat
@ ^ [N3: nat] : ( ord_less @ nat @ N3 @ K ) ) ) ).
% finite_Collect_less_nat
thf(fact_3701_T_092_060_094sub_062d_092_060_094sub_062e_092_060_094sub_062l_092_060_094sub_062e_092_060_094sub_062t_092_060_094sub_062e_Oelims,axiom,
! [X: vEBT_VEBT,Xa: nat,Y: nat] :
( ( ( vEBT_T_d_e_l_e_t_e @ X @ Xa )
= Y )
=> ( ( ? [A6: $o,B3: $o] :
( X
= ( vEBT_Leaf @ A6 @ B3 ) )
=> ( ( Xa
= ( zero_zero @ nat ) )
=> ( Y
!= ( one_one @ nat ) ) ) )
=> ( ( ? [A6: $o,B3: $o] :
( X
= ( vEBT_Leaf @ A6 @ B3 ) )
=> ( ( Xa
= ( suc @ ( zero_zero @ nat ) ) )
=> ( Y
!= ( one_one @ nat ) ) ) )
=> ( ( ? [A6: $o,B3: $o] :
( X
= ( vEBT_Leaf @ A6 @ B3 ) )
=> ( ? [N2: nat] :
( Xa
= ( suc @ ( suc @ N2 ) ) )
=> ( Y
!= ( one_one @ nat ) ) ) )
=> ( ( ? [Deg2: nat,TreeList2: list @ vEBT_VEBT,Summary2: vEBT_VEBT] :
( X
= ( vEBT_Node @ ( none @ ( product_prod @ nat @ nat ) ) @ Deg2 @ TreeList2 @ Summary2 ) )
=> ( Y
!= ( one_one @ nat ) ) )
=> ( ( ? [Mi: nat,Ma2: nat,TreeList2: list @ vEBT_VEBT,Summary2: vEBT_VEBT] :
( X
= ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi @ Ma2 ) ) @ ( zero_zero @ nat ) @ TreeList2 @ Summary2 ) )
=> ( Y
!= ( one_one @ nat ) ) )
=> ( ( ? [Mi: nat,Ma2: nat,TreeList2: list @ vEBT_VEBT,Summary2: vEBT_VEBT] :
( X
= ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi @ Ma2 ) ) @ ( suc @ ( zero_zero @ nat ) ) @ TreeList2 @ Summary2 ) )
=> ( Y
!= ( one_one @ nat ) ) )
=> ~ ! [Mi: nat,Ma2: nat,Va: nat,TreeList2: list @ vEBT_VEBT,Summary2: vEBT_VEBT] :
( ( X
= ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi @ Ma2 ) ) @ ( suc @ ( suc @ Va ) ) @ TreeList2 @ Summary2 ) )
=> ( Y
!= ( plus_plus @ nat @ ( numeral_numeral @ nat @ ( bit1 @ one2 ) )
@ ( if @ nat
@ ( ( ord_less @ nat @ Xa @ Mi )
| ( ord_less @ nat @ Ma2 @ Xa ) )
@ ( one_one @ nat )
@ ( plus_plus @ nat @ ( numeral_numeral @ nat @ ( bit1 @ one2 ) )
@ ( if @ nat
@ ( ( Xa = Mi )
& ( 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 = Mi ) @ ( 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 = Mi ) @ ( plus_plus @ nat @ ( times_times @ nat @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary2 ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList2 @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary2 ) ) ) ) ) ) @ Xa ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( size_size @ ( list @ vEBT_VEBT ) @ TreeList2 ) )
@ ( plus_plus @ nat @ ( plus_plus @ nat @ ( numeral_numeral @ nat @ ( bit0 @ ( bit0 @ one2 ) ) ) @ ( vEBT_T_d_e_l_e_t_e @ ( nth @ vEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ ( if @ nat @ ( Xa = Mi ) @ ( plus_plus @ nat @ ( times_times @ nat @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary2 ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList2 @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary2 ) ) ) ) ) ) @ Xa ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( vEBT_VEBT_low @ ( if @ nat @ ( Xa = Mi ) @ ( plus_plus @ nat @ ( times_times @ nat @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary2 ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList2 @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary2 ) ) ) ) ) ) @ Xa ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) )
@ ( plus_plus @ nat @ ( plus_plus @ nat @ ( one_one @ nat ) @ ( vEBT_T_m_i_n_N_u_l_l @ ( vEBT_vebt_delete @ ( nth @ vEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ ( if @ nat @ ( Xa = Mi ) @ ( plus_plus @ nat @ ( times_times @ nat @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary2 ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList2 @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary2 ) ) ) ) ) ) @ Xa ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( vEBT_VEBT_low @ ( if @ nat @ ( Xa = Mi ) @ ( plus_plus @ nat @ ( times_times @ nat @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary2 ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList2 @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary2 ) ) ) ) ) ) @ Xa ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) )
@ ( if @ nat @ ( vEBT_VEBT_minNull @ ( vEBT_vebt_delete @ ( nth @ vEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ ( if @ nat @ ( Xa = Mi ) @ ( plus_plus @ nat @ ( times_times @ nat @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary2 ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList2 @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary2 ) ) ) ) ) ) @ Xa ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( vEBT_VEBT_low @ ( if @ nat @ ( Xa = Mi ) @ ( plus_plus @ nat @ ( times_times @ nat @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary2 ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList2 @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary2 ) ) ) ) ) ) @ Xa ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) )
@ ( plus_plus @ nat @ ( plus_plus @ nat @ ( one_one @ nat ) @ ( vEBT_T_d_e_l_e_t_e @ Summary2 @ ( vEBT_VEBT_high @ ( if @ nat @ ( Xa = Mi ) @ ( plus_plus @ nat @ ( times_times @ nat @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary2 ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList2 @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary2 ) ) ) ) ) ) @ Xa ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) )
@ ( plus_plus @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) )
@ ( if @ nat
@ ( ( ( Xa = Mi )
=> ( ( plus_plus @ nat @ ( times_times @ nat @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary2 ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList2 @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary2 ) ) ) ) ) )
= Ma2 ) )
& ( ( Xa != Mi )
=> ( 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 = Mi ) @ ( plus_plus @ nat @ ( times_times @ nat @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary2 ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList2 @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary2 ) ) ) ) ) ) @ Xa ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) )
@ ( plus_plus @ nat @ ( one_one @ nat )
@ ( if @ nat
@ ( ( vEBT_vebt_maxt @ ( vEBT_vebt_delete @ Summary2 @ ( vEBT_VEBT_high @ ( if @ nat @ ( Xa = Mi ) @ ( plus_plus @ nat @ ( times_times @ nat @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary2 ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList2 @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary2 ) ) ) ) ) ) @ Xa ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) )
= ( none @ nat ) )
@ ( one_one @ nat )
@ ( plus_plus @ nat @ ( numeral_numeral @ nat @ ( bit0 @ ( bit0 @ ( bit0 @ one2 ) ) ) ) @ ( vEBT_T_m_a_x_t @ ( nth @ vEBT_VEBT @ ( list_update @ vEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ ( if @ nat @ ( Xa = Mi ) @ ( plus_plus @ nat @ ( times_times @ nat @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary2 ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList2 @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary2 ) ) ) ) ) ) @ Xa ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( vEBT_vebt_delete @ ( nth @ vEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ ( if @ nat @ ( Xa = Mi ) @ ( plus_plus @ nat @ ( times_times @ nat @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary2 ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList2 @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary2 ) ) ) ) ) ) @ Xa ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( vEBT_VEBT_low @ ( if @ nat @ ( Xa = Mi ) @ ( plus_plus @ nat @ ( times_times @ nat @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary2 ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList2 @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary2 ) ) ) ) ) ) @ Xa ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_maxt @ ( vEBT_vebt_delete @ Summary2 @ ( vEBT_VEBT_high @ ( if @ nat @ ( Xa = Mi ) @ ( plus_plus @ nat @ ( times_times @ nat @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary2 ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList2 @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary2 ) ) ) ) ) ) @ Xa ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) ) ) ) ) ) ) )
@ ( one_one @ nat ) ) ) )
@ ( plus_plus @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) )
@ ( if @ nat
@ ( ( ( Xa = Mi )
=> ( ( plus_plus @ nat @ ( times_times @ nat @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary2 ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList2 @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary2 ) ) ) ) ) )
= Ma2 ) )
& ( ( Xa != Mi )
=> ( 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 = Mi ) @ ( plus_plus @ nat @ ( times_times @ nat @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary2 ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList2 @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary2 ) ) ) ) ) ) @ Xa ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( vEBT_vebt_delete @ ( nth @ vEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ ( if @ nat @ ( Xa = Mi ) @ ( plus_plus @ nat @ ( times_times @ nat @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary2 ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList2 @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary2 ) ) ) ) ) ) @ Xa ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( vEBT_VEBT_low @ ( if @ nat @ ( Xa = Mi ) @ ( plus_plus @ nat @ ( times_times @ nat @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary2 ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList2 @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary2 ) ) ) ) ) ) @ Xa ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) @ ( vEBT_VEBT_high @ ( if @ nat @ ( Xa = Mi ) @ ( plus_plus @ nat @ ( times_times @ nat @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary2 ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList2 @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary2 ) ) ) ) ) ) @ Xa ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) )
@ ( one_one @ nat ) ) ) ) ) )
@ ( one_one @ nat ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).
% T\<^sub>d\<^sub>e\<^sub>l\<^sub>e\<^sub>t\<^sub>e.elims
thf(fact_3702_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,
! [Mi3: nat,Ma: nat,Va2: nat,TreeList: list @ vEBT_VEBT,Summary: vEBT_VEBT,X: nat] :
( ( vEBT_T_d_e_l_e_t_e @ ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi3 @ Ma ) ) @ ( suc @ ( suc @ Va2 ) ) @ TreeList @ Summary ) @ X )
= ( plus_plus @ nat @ ( numeral_numeral @ nat @ ( bit1 @ one2 ) )
@ ( if @ nat
@ ( ( ord_less @ nat @ X @ Mi3 )
| ( ord_less @ nat @ Ma @ X ) )
@ ( one_one @ nat )
@ ( plus_plus @ nat @ ( numeral_numeral @ nat @ ( bit1 @ one2 ) )
@ ( if @ nat
@ ( ( X = Mi3 )
& ( 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 = Mi3 ) @ ( 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 = Mi3 ) @ ( plus_plus @ nat @ ( times_times @ nat @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ X ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( size_size @ ( list @ vEBT_VEBT ) @ TreeList ) )
@ ( plus_plus @ nat @ ( plus_plus @ nat @ ( numeral_numeral @ nat @ ( bit0 @ ( bit0 @ one2 ) ) ) @ ( vEBT_T_d_e_l_e_t_e @ ( nth @ vEBT_VEBT @ TreeList @ ( vEBT_VEBT_high @ ( if @ nat @ ( X = Mi3 ) @ ( plus_plus @ nat @ ( times_times @ nat @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ X ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( vEBT_VEBT_low @ ( if @ nat @ ( X = Mi3 ) @ ( plus_plus @ nat @ ( times_times @ nat @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ X ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) )
@ ( plus_plus @ nat @ ( plus_plus @ nat @ ( one_one @ nat ) @ ( vEBT_T_m_i_n_N_u_l_l @ ( vEBT_vebt_delete @ ( nth @ vEBT_VEBT @ TreeList @ ( vEBT_VEBT_high @ ( if @ nat @ ( X = Mi3 ) @ ( plus_plus @ nat @ ( times_times @ nat @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ X ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( vEBT_VEBT_low @ ( if @ nat @ ( X = Mi3 ) @ ( plus_plus @ nat @ ( times_times @ nat @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ X ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) )
@ ( if @ nat @ ( vEBT_VEBT_minNull @ ( vEBT_vebt_delete @ ( nth @ vEBT_VEBT @ TreeList @ ( vEBT_VEBT_high @ ( if @ nat @ ( X = Mi3 ) @ ( plus_plus @ nat @ ( times_times @ nat @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ X ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( vEBT_VEBT_low @ ( if @ nat @ ( X = Mi3 ) @ ( plus_plus @ nat @ ( times_times @ nat @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ X ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) )
@ ( plus_plus @ nat @ ( plus_plus @ nat @ ( one_one @ nat ) @ ( vEBT_T_d_e_l_e_t_e @ Summary @ ( vEBT_VEBT_high @ ( if @ nat @ ( X = Mi3 ) @ ( plus_plus @ nat @ ( times_times @ nat @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ X ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) )
@ ( plus_plus @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) )
@ ( if @ nat
@ ( ( ( X = Mi3 )
=> ( ( plus_plus @ nat @ ( times_times @ nat @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) )
= Ma ) )
& ( ( X != Mi3 )
=> ( 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 = Mi3 ) @ ( plus_plus @ nat @ ( times_times @ nat @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ X ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) )
@ ( plus_plus @ nat @ ( one_one @ nat )
@ ( if @ nat
@ ( ( vEBT_vebt_maxt @ ( vEBT_vebt_delete @ Summary @ ( vEBT_VEBT_high @ ( if @ nat @ ( X = Mi3 ) @ ( plus_plus @ nat @ ( times_times @ nat @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ X ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) )
= ( none @ nat ) )
@ ( one_one @ nat )
@ ( plus_plus @ nat @ ( numeral_numeral @ nat @ ( bit0 @ ( bit0 @ ( bit0 @ one2 ) ) ) ) @ ( vEBT_T_m_a_x_t @ ( nth @ vEBT_VEBT @ ( list_update @ vEBT_VEBT @ TreeList @ ( vEBT_VEBT_high @ ( if @ nat @ ( X = Mi3 ) @ ( plus_plus @ nat @ ( times_times @ nat @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ X ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( vEBT_vebt_delete @ ( nth @ vEBT_VEBT @ TreeList @ ( vEBT_VEBT_high @ ( if @ nat @ ( X = Mi3 ) @ ( plus_plus @ nat @ ( times_times @ nat @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ X ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( vEBT_VEBT_low @ ( if @ nat @ ( X = Mi3 ) @ ( plus_plus @ nat @ ( times_times @ nat @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ X ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_maxt @ ( vEBT_vebt_delete @ Summary @ ( vEBT_VEBT_high @ ( if @ nat @ ( X = Mi3 ) @ ( plus_plus @ nat @ ( times_times @ nat @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ X ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) ) ) ) ) ) ) )
@ ( one_one @ nat ) ) ) )
@ ( plus_plus @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) )
@ ( if @ nat
@ ( ( ( X = Mi3 )
=> ( ( plus_plus @ nat @ ( times_times @ nat @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) )
= Ma ) )
& ( ( X != Mi3 )
=> ( 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 = Mi3 ) @ ( plus_plus @ nat @ ( times_times @ nat @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ X ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( vEBT_vebt_delete @ ( nth @ vEBT_VEBT @ TreeList @ ( vEBT_VEBT_high @ ( if @ nat @ ( X = Mi3 ) @ ( plus_plus @ nat @ ( times_times @ nat @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ X ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( vEBT_VEBT_low @ ( if @ nat @ ( X = Mi3 ) @ ( plus_plus @ nat @ ( times_times @ nat @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ X ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) @ ( vEBT_VEBT_high @ ( if @ nat @ ( X = Mi3 ) @ ( plus_plus @ nat @ ( times_times @ nat @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ X ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) )
@ ( one_one @ nat ) ) ) ) ) )
@ ( one_one @ nat ) ) ) ) ) ) ) ) ).
% T\<^sub>d\<^sub>e\<^sub>l\<^sub>e\<^sub>t\<^sub>e.simps(7)
thf(fact_3703_T_092_060_094sub_062s_092_060_094sub_062u_092_060_094sub_062c_092_060_094sub_062c_Oelims,axiom,
! [X: vEBT_VEBT,Xa: nat,Y: nat] :
( ( ( vEBT_T_s_u_c_c @ X @ Xa )
= Y )
=> ( ( ? [Uu2: $o,B3: $o] :
( X
= ( vEBT_Leaf @ Uu2 @ B3 ) )
=> ( ( Xa
= ( zero_zero @ nat ) )
=> ( Y
!= ( plus_plus @ nat @ ( one_one @ nat ) @ ( one_one @ nat ) ) ) ) )
=> ( ( ? [Uv2: $o,Uw2: $o] :
( X
= ( vEBT_Leaf @ Uv2 @ Uw2 ) )
=> ( ? [N2: nat] :
( Xa
= ( suc @ N2 ) )
=> ( 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 ) ) )
=> ( ( ? [V4: product_prod @ nat @ nat,Vc: list @ vEBT_VEBT,Vd2: vEBT_VEBT] :
( X
= ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ V4 ) @ ( zero_zero @ nat ) @ Vc @ Vd2 ) )
=> ( Y
!= ( one_one @ nat ) ) )
=> ( ( ? [V4: product_prod @ nat @ nat,Vg2: list @ vEBT_VEBT,Vh2: vEBT_VEBT] :
( X
= ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ V4 ) @ ( suc @ ( zero_zero @ nat ) ) @ Vg2 @ Vh2 ) )
=> ( Y
!= ( one_one @ nat ) ) )
=> ~ ! [Mi: nat,Ma2: nat,Va: nat,TreeList2: list @ vEBT_VEBT,Summary2: vEBT_VEBT] :
( ( X
= ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi @ Ma2 ) ) @ ( suc @ ( suc @ Va ) ) @ TreeList2 @ Summary2 ) )
=> ( Y
!= ( plus_plus @ nat @ ( one_one @ nat )
@ ( if @ nat @ ( ord_less @ nat @ Xa @ Mi ) @ ( one_one @ nat )
@ ( plus_plus @ nat @ ( numeral_numeral @ nat @ ( bit0 @ ( bit1 @ ( bit0 @ one2 ) ) ) )
@ ( if @ nat @ ( ord_less @ nat @ ( vEBT_VEBT_high @ Xa @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( size_size @ ( list @ vEBT_VEBT ) @ TreeList2 ) )
@ ( plus_plus @ nat @ ( plus_plus @ nat @ ( one_one @ nat ) @ ( vEBT_T_m_a_x_t @ ( nth @ vEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ Xa @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) )
@ ( plus_plus @ nat @ ( numeral_numeral @ nat @ ( bit1 @ one2 ) )
@ ( if @ nat
@ ( ( ( vEBT_vebt_maxt @ ( nth @ vEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ Xa @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) )
!= ( none @ nat ) )
& ( vEBT_VEBT_less @ ( some @ nat @ ( vEBT_VEBT_low @ Xa @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( vEBT_vebt_maxt @ ( nth @ vEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ Xa @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) ) )
@ ( plus_plus @ nat @ ( numeral_numeral @ nat @ ( bit0 @ ( bit0 @ one2 ) ) ) @ ( vEBT_T_s_u_c_c @ ( nth @ vEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ Xa @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( vEBT_VEBT_low @ Xa @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) )
@ ( plus_plus @ nat @ ( plus_plus @ nat @ ( plus_plus @ nat @ ( one_one @ nat ) @ ( vEBT_T_s_u_c_c @ Summary2 @ ( vEBT_VEBT_high @ Xa @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) @ ( one_one @ nat ) )
@ ( if @ nat
@ ( ( vEBT_vebt_succ @ Summary2 @ ( vEBT_VEBT_high @ Xa @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) )
= ( none @ nat ) )
@ ( one_one @ nat )
@ ( plus_plus @ nat @ ( numeral_numeral @ nat @ ( bit0 @ ( bit0 @ one2 ) ) ) @ ( vEBT_T_m_i_n_t @ ( nth @ vEBT_VEBT @ TreeList2 @ ( the2 @ nat @ ( vEBT_vebt_succ @ Summary2 @ ( vEBT_VEBT_high @ Xa @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) ) ) ) ) ) ) ) )
@ ( one_one @ nat ) ) ) ) ) ) ) ) ) ) ) ) ) ).
% T\<^sub>s\<^sub>u\<^sub>c\<^sub>c.elims
thf(fact_3704_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_3705_T_092_060_094sub_062m_092_060_094sub_062a_092_060_094sub_062x_092_060_094sub_062t_Osimps_I2_J,axiom,
! [Uu: nat,Uv: list @ vEBT_VEBT,Uw: vEBT_VEBT] :
( ( vEBT_T_m_a_x_t @ ( vEBT_Node @ ( none @ ( product_prod @ nat @ nat ) ) @ Uu @ Uv @ Uw ) )
= ( one_one @ nat ) ) ).
% T\<^sub>m\<^sub>a\<^sub>x\<^sub>t.simps(2)
thf(fact_3706_T_092_060_094sub_062m_092_060_094sub_062i_092_060_094sub_062n_092_060_094sub_062t_Osimps_I2_J,axiom,
! [Uu: nat,Uv: list @ vEBT_VEBT,Uw: vEBT_VEBT] :
( ( vEBT_T_m_i_n_t @ ( vEBT_Node @ ( none @ ( product_prod @ nat @ nat ) ) @ Uu @ Uv @ Uw ) )
= ( one_one @ nat ) ) ).
% T\<^sub>m\<^sub>i\<^sub>n\<^sub>t.simps(2)
thf(fact_3707_maxt__bound,axiom,
! [T: vEBT_VEBT] : ( ord_less_eq @ nat @ ( vEBT_T_m_a_x_t @ T ) @ ( numeral_numeral @ nat @ ( bit1 @ one2 ) ) ) ).
% maxt_bound
thf(fact_3708_T_092_060_094sub_062m_092_060_094sub_062a_092_060_094sub_062x_092_060_094sub_062t_Osimps_I3_J,axiom,
! [Mi3: 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 @ Mi3 @ Ma ) ) @ Ux @ Uy @ Uz ) )
= ( one_one @ nat ) ) ).
% T\<^sub>m\<^sub>a\<^sub>x\<^sub>t.simps(3)
thf(fact_3709_mint__bound,axiom,
! [T: vEBT_VEBT] : ( ord_less_eq @ nat @ ( vEBT_T_m_i_n_t @ T ) @ ( numeral_numeral @ nat @ ( bit1 @ one2 ) ) ) ).
% mint_bound
thf(fact_3710_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_3711_T_092_060_094sub_062m_092_060_094sub_062i_092_060_094sub_062n_092_060_094sub_062t_Osimps_I3_J,axiom,
! [Mi3: 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 @ Mi3 @ Ma ) ) @ Ux @ Uy @ Uz ) )
= ( one_one @ nat ) ) ).
% T\<^sub>m\<^sub>i\<^sub>n\<^sub>t.simps(3)
thf(fact_3712_finite__psubset__induct,axiom,
! [A: $tType,A3: set @ A,P: ( set @ A ) > $o] :
( ( finite_finite2 @ A @ A3 )
=> ( ! [A9: set @ A] :
( ( finite_finite2 @ A @ A9 )
=> ( ! [B9: set @ A] :
( ( ord_less @ ( set @ A ) @ B9 @ A9 )
=> ( P @ B9 ) )
=> ( P @ A9 ) ) )
=> ( P @ A3 ) ) ) ).
% finite_psubset_induct
thf(fact_3713_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 )
=> ( ! [A6: $o,B3: $o] :
( ( X
= ( vEBT_Leaf @ A6 @ B3 ) )
=> ( Y
!= ( plus_plus @ nat @ ( one_one @ nat ) @ ( if @ nat @ B3 @ ( one_one @ nat ) @ ( plus_plus @ nat @ ( one_one @ nat ) @ ( one_one @ nat ) ) ) ) ) )
=> ( ( ? [Uu2: nat,Uv2: list @ vEBT_VEBT,Uw2: vEBT_VEBT] :
( X
= ( vEBT_Node @ ( none @ ( product_prod @ nat @ nat ) ) @ Uu2 @ Uv2 @ Uw2 ) )
=> ( Y
!= ( one_one @ nat ) ) )
=> ~ ( ? [Mi: nat,Ma2: nat,Ux2: nat,Uy2: list @ vEBT_VEBT,Uz2: vEBT_VEBT] :
( X
= ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi @ Ma2 ) ) @ Ux2 @ Uy2 @ Uz2 ) )
=> ( Y
!= ( one_one @ nat ) ) ) ) ) ) ).
% T\<^sub>m\<^sub>a\<^sub>x\<^sub>t.elims
thf(fact_3714_T_092_060_094sub_062m_092_060_094sub_062i_092_060_094sub_062n_092_060_094sub_062t_Oelims,axiom,
! [X: vEBT_VEBT,Y: nat] :
( ( ( vEBT_T_m_i_n_t @ X )
= Y )
=> ( ! [A6: $o] :
( ? [B3: $o] :
( X
= ( vEBT_Leaf @ A6 @ B3 ) )
=> ( Y
!= ( plus_plus @ nat @ ( one_one @ nat ) @ ( if @ nat @ A6 @ ( zero_zero @ nat ) @ ( plus_plus @ nat @ ( one_one @ nat ) @ ( one_one @ nat ) ) ) ) ) )
=> ( ( ? [Uu2: nat,Uv2: list @ vEBT_VEBT,Uw2: vEBT_VEBT] :
( X
= ( vEBT_Node @ ( none @ ( product_prod @ nat @ nat ) ) @ Uu2 @ Uv2 @ Uw2 ) )
=> ( Y
!= ( one_one @ nat ) ) )
=> ~ ( ? [Mi: nat,Ma2: nat,Ux2: nat,Uy2: list @ vEBT_VEBT,Uz2: vEBT_VEBT] :
( X
= ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi @ Ma2 ) ) @ Ux2 @ Uy2 @ Uz2 ) )
=> ( Y
!= ( one_one @ nat ) ) ) ) ) ) ).
% T\<^sub>m\<^sub>i\<^sub>n\<^sub>t.elims
thf(fact_3715_T_092_060_094sub_062p_092_060_094sub_062r_092_060_094sub_062e_092_060_094sub_062d_Osimps_I7_J,axiom,
! [Mi3: nat,Ma: nat,Va2: nat,TreeList: list @ vEBT_VEBT,Summary: vEBT_VEBT,X: nat] :
( ( vEBT_T_p_r_e_d @ ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi3 @ Ma ) ) @ ( suc @ ( suc @ Va2 ) ) @ TreeList @ Summary ) @ X )
= ( plus_plus @ nat @ ( one_one @ nat )
@ ( if @ nat @ ( ord_less @ nat @ Ma @ X ) @ ( one_one @ nat )
@ ( plus_plus @ nat @ ( plus_plus @ nat @ ( numeral_numeral @ nat @ ( bit0 @ ( bit1 @ ( bit0 @ one2 ) ) ) ) @ ( one_one @ nat ) )
@ ( if @ nat @ ( ord_less @ nat @ ( vEBT_VEBT_high @ X @ ( divide_divide @ nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( size_size @ ( list @ vEBT_VEBT ) @ TreeList ) )
@ ( plus_plus @ nat @ ( plus_plus @ nat @ ( plus_plus @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( vEBT_T_m_i_n_t @ ( nth @ vEBT_VEBT @ TreeList @ ( vEBT_VEBT_high @ X @ ( divide_divide @ nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) ) @ ( numeral_numeral @ nat @ ( bit1 @ one2 ) ) )
@ ( if @ nat
@ ( ( ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList @ ( vEBT_VEBT_high @ X @ ( divide_divide @ nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) )
!= ( none @ nat ) )
& ( vEBT_VEBT_greater @ ( some @ nat @ ( vEBT_VEBT_low @ X @ ( divide_divide @ nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList @ ( vEBT_VEBT_high @ X @ ( divide_divide @ nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) ) )
@ ( plus_plus @ nat @ ( numeral_numeral @ nat @ ( bit0 @ ( bit0 @ one2 ) ) ) @ ( vEBT_T_p_r_e_d @ ( nth @ vEBT_VEBT @ TreeList @ ( vEBT_VEBT_high @ X @ ( divide_divide @ nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( vEBT_VEBT_low @ X @ ( divide_divide @ nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) )
@ ( plus_plus @ nat @ ( plus_plus @ nat @ ( plus_plus @ nat @ ( one_one @ nat ) @ ( vEBT_T_p_r_e_d @ Summary @ ( vEBT_VEBT_high @ X @ ( divide_divide @ nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) @ ( one_one @ nat ) )
@ ( if @ nat
@ ( ( vEBT_vebt_pred @ Summary @ ( vEBT_VEBT_high @ X @ ( divide_divide @ nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) )
= ( none @ nat ) )
@ ( plus_plus @ nat @ ( one_one @ nat ) @ ( one_one @ nat ) )
@ ( plus_plus @ nat @ ( numeral_numeral @ nat @ ( bit0 @ ( bit0 @ one2 ) ) ) @ ( vEBT_T_m_a_x_t @ ( nth @ vEBT_VEBT @ TreeList @ ( the2 @ nat @ ( vEBT_vebt_pred @ Summary @ ( vEBT_VEBT_high @ X @ ( divide_divide @ nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) ) ) ) ) ) ) )
@ ( one_one @ nat ) ) ) ) ) ) ).
% T\<^sub>p\<^sub>r\<^sub>e\<^sub>d.simps(7)
thf(fact_3716_T_092_060_094sub_062s_092_060_094sub_062u_092_060_094sub_062c_092_060_094sub_062c_Osimps_I6_J,axiom,
! [Mi3: nat,Ma: nat,Va2: nat,TreeList: list @ vEBT_VEBT,Summary: vEBT_VEBT,X: nat] :
( ( vEBT_T_s_u_c_c @ ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi3 @ Ma ) ) @ ( suc @ ( suc @ Va2 ) ) @ TreeList @ Summary ) @ X )
= ( plus_plus @ nat @ ( one_one @ nat )
@ ( if @ nat @ ( ord_less @ nat @ X @ Mi3 ) @ ( one_one @ nat )
@ ( plus_plus @ nat @ ( numeral_numeral @ nat @ ( bit0 @ ( bit1 @ ( bit0 @ one2 ) ) ) )
@ ( if @ nat @ ( ord_less @ nat @ ( vEBT_VEBT_high @ X @ ( divide_divide @ nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( size_size @ ( list @ vEBT_VEBT ) @ TreeList ) )
@ ( plus_plus @ nat @ ( plus_plus @ nat @ ( one_one @ nat ) @ ( vEBT_T_m_a_x_t @ ( nth @ vEBT_VEBT @ TreeList @ ( vEBT_VEBT_high @ X @ ( divide_divide @ nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) )
@ ( plus_plus @ nat @ ( numeral_numeral @ nat @ ( bit1 @ one2 ) )
@ ( if @ nat
@ ( ( ( vEBT_vebt_maxt @ ( nth @ vEBT_VEBT @ TreeList @ ( vEBT_VEBT_high @ X @ ( divide_divide @ nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) )
!= ( none @ nat ) )
& ( vEBT_VEBT_less @ ( some @ nat @ ( vEBT_VEBT_low @ X @ ( divide_divide @ nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( vEBT_vebt_maxt @ ( nth @ vEBT_VEBT @ TreeList @ ( vEBT_VEBT_high @ X @ ( divide_divide @ nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) ) )
@ ( plus_plus @ nat @ ( numeral_numeral @ nat @ ( bit0 @ ( bit0 @ one2 ) ) ) @ ( vEBT_T_s_u_c_c @ ( nth @ vEBT_VEBT @ TreeList @ ( vEBT_VEBT_high @ X @ ( divide_divide @ nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( vEBT_VEBT_low @ X @ ( divide_divide @ nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) )
@ ( plus_plus @ nat @ ( plus_plus @ nat @ ( plus_plus @ nat @ ( one_one @ nat ) @ ( vEBT_T_s_u_c_c @ Summary @ ( vEBT_VEBT_high @ X @ ( divide_divide @ nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) @ ( one_one @ nat ) )
@ ( if @ nat
@ ( ( vEBT_vebt_succ @ Summary @ ( vEBT_VEBT_high @ X @ ( divide_divide @ nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) )
= ( none @ nat ) )
@ ( one_one @ nat )
@ ( plus_plus @ nat @ ( numeral_numeral @ nat @ ( bit0 @ ( bit0 @ one2 ) ) ) @ ( vEBT_T_m_i_n_t @ ( nth @ vEBT_VEBT @ TreeList @ ( the2 @ nat @ ( vEBT_vebt_succ @ Summary @ ( vEBT_VEBT_high @ X @ ( divide_divide @ nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) ) ) ) ) ) ) ) )
@ ( one_one @ nat ) ) ) ) ) ) ).
% T\<^sub>s\<^sub>u\<^sub>c\<^sub>c.simps(6)
thf(fact_3717_T_092_060_094sub_062p_092_060_094sub_062r_092_060_094sub_062e_092_060_094sub_062d_Oelims,axiom,
! [X: vEBT_VEBT,Xa: nat,Y: nat] :
( ( ( vEBT_T_p_r_e_d @ X @ Xa )
= Y )
=> ( ( ? [Uu2: $o,Uv2: $o] :
( X
= ( vEBT_Leaf @ Uu2 @ Uv2 ) )
=> ( ( Xa
= ( zero_zero @ nat ) )
=> ( Y
!= ( one_one @ nat ) ) ) )
=> ( ( ? [A6: $o,Uw2: $o] :
( X
= ( vEBT_Leaf @ A6 @ Uw2 ) )
=> ( ( Xa
= ( suc @ ( zero_zero @ nat ) ) )
=> ( Y
!= ( plus_plus @ nat @ ( one_one @ nat ) @ ( one_one @ nat ) ) ) ) )
=> ( ! [A6: $o,B3: $o] :
( ( X
= ( vEBT_Leaf @ A6 @ B3 ) )
=> ( ? [Va: nat] :
( Xa
= ( suc @ ( suc @ Va ) ) )
=> ( 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,Va3: vEBT_VEBT] :
( X
= ( vEBT_Node @ ( none @ ( product_prod @ nat @ nat ) ) @ Uy2 @ Uz2 @ Va3 ) )
=> ( Y
!= ( one_one @ nat ) ) )
=> ( ( ? [V4: product_prod @ nat @ nat,Vd2: list @ vEBT_VEBT,Ve2: vEBT_VEBT] :
( X
= ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ V4 ) @ ( zero_zero @ nat ) @ Vd2 @ Ve2 ) )
=> ( Y
!= ( one_one @ nat ) ) )
=> ( ( ? [V4: product_prod @ nat @ nat,Vh2: list @ vEBT_VEBT,Vi2: vEBT_VEBT] :
( X
= ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ V4 ) @ ( suc @ ( zero_zero @ nat ) ) @ Vh2 @ Vi2 ) )
=> ( Y
!= ( one_one @ nat ) ) )
=> ~ ! [Mi: nat,Ma2: nat,Va: nat,TreeList2: list @ vEBT_VEBT,Summary2: vEBT_VEBT] :
( ( X
= ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi @ Ma2 ) ) @ ( suc @ ( suc @ Va ) ) @ TreeList2 @ Summary2 ) )
=> ( Y
!= ( plus_plus @ nat @ ( one_one @ nat )
@ ( if @ nat @ ( ord_less @ nat @ Ma2 @ Xa ) @ ( one_one @ nat )
@ ( plus_plus @ nat @ ( plus_plus @ nat @ ( numeral_numeral @ nat @ ( bit0 @ ( bit1 @ ( bit0 @ one2 ) ) ) ) @ ( one_one @ nat ) )
@ ( if @ nat @ ( ord_less @ nat @ ( vEBT_VEBT_high @ Xa @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( size_size @ ( list @ vEBT_VEBT ) @ TreeList2 ) )
@ ( plus_plus @ nat @ ( plus_plus @ nat @ ( plus_plus @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( vEBT_T_m_i_n_t @ ( nth @ vEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ Xa @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) ) @ ( numeral_numeral @ nat @ ( bit1 @ one2 ) ) )
@ ( if @ nat
@ ( ( ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ Xa @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) )
!= ( none @ nat ) )
& ( vEBT_VEBT_greater @ ( some @ nat @ ( vEBT_VEBT_low @ Xa @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ Xa @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) ) )
@ ( plus_plus @ nat @ ( numeral_numeral @ nat @ ( bit0 @ ( bit0 @ one2 ) ) ) @ ( vEBT_T_p_r_e_d @ ( nth @ vEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ Xa @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( vEBT_VEBT_low @ Xa @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) )
@ ( plus_plus @ nat @ ( plus_plus @ nat @ ( plus_plus @ nat @ ( one_one @ nat ) @ ( vEBT_T_p_r_e_d @ Summary2 @ ( vEBT_VEBT_high @ Xa @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) @ ( one_one @ nat ) )
@ ( if @ nat
@ ( ( vEBT_vebt_pred @ Summary2 @ ( vEBT_VEBT_high @ Xa @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) )
= ( none @ nat ) )
@ ( plus_plus @ nat @ ( one_one @ nat ) @ ( one_one @ nat ) )
@ ( plus_plus @ nat @ ( numeral_numeral @ nat @ ( bit0 @ ( bit0 @ one2 ) ) ) @ ( vEBT_T_m_a_x_t @ ( nth @ vEBT_VEBT @ TreeList2 @ ( the2 @ nat @ ( vEBT_vebt_pred @ Summary2 @ ( vEBT_VEBT_high @ Xa @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) ) ) ) ) ) ) )
@ ( one_one @ nat ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).
% T\<^sub>p\<^sub>r\<^sub>e\<^sub>d.elims
thf(fact_3718_T_092_060_094sub_062d_092_060_094sub_062e_092_060_094sub_062l_092_060_094sub_062e_092_060_094sub_062t_092_060_094sub_062e_Opelims,axiom,
! [X: vEBT_VEBT,Xa: nat,Y: nat] :
( ( ( vEBT_T_d_e_l_e_t_e @ X @ Xa )
= Y )
=> ( ( accp @ ( product_prod @ vEBT_VEBT @ nat ) @ vEBT_T8441311223069195367_e_rel @ ( product_Pair @ vEBT_VEBT @ nat @ X @ Xa ) )
=> ( ! [A6: $o,B3: $o] :
( ( X
= ( vEBT_Leaf @ A6 @ 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 @ A6 @ B3 ) @ ( zero_zero @ nat ) ) ) ) ) )
=> ( ! [A6: $o,B3: $o] :
( ( X
= ( vEBT_Leaf @ A6 @ 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 @ A6 @ B3 ) @ ( suc @ ( zero_zero @ nat ) ) ) ) ) ) )
=> ( ! [A6: $o,B3: $o] :
( ( X
= ( vEBT_Leaf @ A6 @ B3 ) )
=> ! [N2: nat] :
( ( Xa
= ( suc @ ( suc @ N2 ) ) )
=> ( ( Y
= ( one_one @ nat ) )
=> ~ ( accp @ ( product_prod @ vEBT_VEBT @ nat ) @ vEBT_T8441311223069195367_e_rel @ ( product_Pair @ vEBT_VEBT @ nat @ ( vEBT_Leaf @ A6 @ B3 ) @ ( suc @ ( suc @ N2 ) ) ) ) ) ) )
=> ( ! [Deg2: nat,TreeList2: list @ vEBT_VEBT,Summary2: vEBT_VEBT] :
( ( X
= ( vEBT_Node @ ( none @ ( product_prod @ nat @ nat ) ) @ Deg2 @ TreeList2 @ Summary2 ) )
=> ( ( Y
= ( one_one @ nat ) )
=> ~ ( accp @ ( product_prod @ vEBT_VEBT @ nat ) @ vEBT_T8441311223069195367_e_rel @ ( product_Pair @ vEBT_VEBT @ nat @ ( vEBT_Node @ ( none @ ( product_prod @ nat @ nat ) ) @ Deg2 @ TreeList2 @ Summary2 ) @ Xa ) ) ) )
=> ( ! [Mi: nat,Ma2: nat,TreeList2: list @ vEBT_VEBT,Summary2: vEBT_VEBT] :
( ( X
= ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi @ 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 @ Mi @ Ma2 ) ) @ ( zero_zero @ nat ) @ TreeList2 @ Summary2 ) @ Xa ) ) ) )
=> ( ! [Mi: nat,Ma2: nat,TreeList2: list @ vEBT_VEBT,Summary2: vEBT_VEBT] :
( ( X
= ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi @ 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 @ Mi @ Ma2 ) ) @ ( suc @ ( zero_zero @ nat ) ) @ TreeList2 @ Summary2 ) @ Xa ) ) ) )
=> ~ ! [Mi: nat,Ma2: nat,Va: nat,TreeList2: list @ vEBT_VEBT,Summary2: vEBT_VEBT] :
( ( X
= ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi @ Ma2 ) ) @ ( suc @ ( suc @ Va ) ) @ TreeList2 @ Summary2 ) )
=> ( ( Y
= ( plus_plus @ nat @ ( numeral_numeral @ nat @ ( bit1 @ one2 ) )
@ ( if @ nat
@ ( ( ord_less @ nat @ Xa @ Mi )
| ( ord_less @ nat @ Ma2 @ Xa ) )
@ ( one_one @ nat )
@ ( plus_plus @ nat @ ( numeral_numeral @ nat @ ( bit1 @ one2 ) )
@ ( if @ nat
@ ( ( Xa = Mi )
& ( 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 = Mi ) @ ( 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 = Mi ) @ ( plus_plus @ nat @ ( times_times @ nat @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary2 ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList2 @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary2 ) ) ) ) ) ) @ Xa ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( size_size @ ( list @ vEBT_VEBT ) @ TreeList2 ) )
@ ( plus_plus @ nat @ ( plus_plus @ nat @ ( numeral_numeral @ nat @ ( bit0 @ ( bit0 @ one2 ) ) ) @ ( vEBT_T_d_e_l_e_t_e @ ( nth @ vEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ ( if @ nat @ ( Xa = Mi ) @ ( plus_plus @ nat @ ( times_times @ nat @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary2 ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList2 @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary2 ) ) ) ) ) ) @ Xa ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( vEBT_VEBT_low @ ( if @ nat @ ( Xa = Mi ) @ ( plus_plus @ nat @ ( times_times @ nat @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary2 ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList2 @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary2 ) ) ) ) ) ) @ Xa ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) )
@ ( plus_plus @ nat @ ( plus_plus @ nat @ ( one_one @ nat ) @ ( vEBT_T_m_i_n_N_u_l_l @ ( vEBT_vebt_delete @ ( nth @ vEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ ( if @ nat @ ( Xa = Mi ) @ ( plus_plus @ nat @ ( times_times @ nat @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary2 ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList2 @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary2 ) ) ) ) ) ) @ Xa ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( vEBT_VEBT_low @ ( if @ nat @ ( Xa = Mi ) @ ( plus_plus @ nat @ ( times_times @ nat @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary2 ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList2 @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary2 ) ) ) ) ) ) @ Xa ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) )
@ ( if @ nat @ ( vEBT_VEBT_minNull @ ( vEBT_vebt_delete @ ( nth @ vEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ ( if @ nat @ ( Xa = Mi ) @ ( plus_plus @ nat @ ( times_times @ nat @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary2 ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList2 @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary2 ) ) ) ) ) ) @ Xa ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( vEBT_VEBT_low @ ( if @ nat @ ( Xa = Mi ) @ ( plus_plus @ nat @ ( times_times @ nat @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary2 ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList2 @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary2 ) ) ) ) ) ) @ Xa ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) )
@ ( plus_plus @ nat @ ( plus_plus @ nat @ ( one_one @ nat ) @ ( vEBT_T_d_e_l_e_t_e @ Summary2 @ ( vEBT_VEBT_high @ ( if @ nat @ ( Xa = Mi ) @ ( plus_plus @ nat @ ( times_times @ nat @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary2 ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList2 @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary2 ) ) ) ) ) ) @ Xa ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) )
@ ( plus_plus @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) )
@ ( if @ nat
@ ( ( ( Xa = Mi )
=> ( ( plus_plus @ nat @ ( times_times @ nat @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary2 ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList2 @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary2 ) ) ) ) ) )
= Ma2 ) )
& ( ( Xa != Mi )
=> ( 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 = Mi ) @ ( plus_plus @ nat @ ( times_times @ nat @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary2 ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList2 @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary2 ) ) ) ) ) ) @ Xa ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) )
@ ( plus_plus @ nat @ ( one_one @ nat )
@ ( if @ nat
@ ( ( vEBT_vebt_maxt @ ( vEBT_vebt_delete @ Summary2 @ ( vEBT_VEBT_high @ ( if @ nat @ ( Xa = Mi ) @ ( plus_plus @ nat @ ( times_times @ nat @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary2 ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList2 @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary2 ) ) ) ) ) ) @ Xa ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) )
= ( none @ nat ) )
@ ( one_one @ nat )
@ ( plus_plus @ nat @ ( numeral_numeral @ nat @ ( bit0 @ ( bit0 @ ( bit0 @ one2 ) ) ) ) @ ( vEBT_T_m_a_x_t @ ( nth @ vEBT_VEBT @ ( list_update @ vEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ ( if @ nat @ ( Xa = Mi ) @ ( plus_plus @ nat @ ( times_times @ nat @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary2 ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList2 @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary2 ) ) ) ) ) ) @ Xa ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( vEBT_vebt_delete @ ( nth @ vEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ ( if @ nat @ ( Xa = Mi ) @ ( plus_plus @ nat @ ( times_times @ nat @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary2 ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList2 @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary2 ) ) ) ) ) ) @ Xa ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( vEBT_VEBT_low @ ( if @ nat @ ( Xa = Mi ) @ ( plus_plus @ nat @ ( times_times @ nat @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary2 ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList2 @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary2 ) ) ) ) ) ) @ Xa ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_maxt @ ( vEBT_vebt_delete @ Summary2 @ ( vEBT_VEBT_high @ ( if @ nat @ ( Xa = Mi ) @ ( plus_plus @ nat @ ( times_times @ nat @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary2 ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList2 @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary2 ) ) ) ) ) ) @ Xa ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) ) ) ) ) ) ) )
@ ( one_one @ nat ) ) ) )
@ ( plus_plus @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) )
@ ( if @ nat
@ ( ( ( Xa = Mi )
=> ( ( plus_plus @ nat @ ( times_times @ nat @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary2 ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList2 @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary2 ) ) ) ) ) )
= Ma2 ) )
& ( ( Xa != Mi )
=> ( 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 = Mi ) @ ( plus_plus @ nat @ ( times_times @ nat @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary2 ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList2 @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary2 ) ) ) ) ) ) @ Xa ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( vEBT_vebt_delete @ ( nth @ vEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ ( if @ nat @ ( Xa = Mi ) @ ( plus_plus @ nat @ ( times_times @ nat @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary2 ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList2 @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary2 ) ) ) ) ) ) @ Xa ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( vEBT_VEBT_low @ ( if @ nat @ ( Xa = Mi ) @ ( plus_plus @ nat @ ( times_times @ nat @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary2 ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList2 @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary2 ) ) ) ) ) ) @ Xa ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) @ ( vEBT_VEBT_high @ ( if @ nat @ ( Xa = Mi ) @ ( plus_plus @ nat @ ( times_times @ nat @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary2 ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList2 @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary2 ) ) ) ) ) ) @ Xa ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) )
@ ( one_one @ nat ) ) ) ) ) )
@ ( one_one @ nat ) ) ) ) ) ) ) )
=> ~ ( accp @ ( product_prod @ vEBT_VEBT @ nat ) @ vEBT_T8441311223069195367_e_rel @ ( product_Pair @ vEBT_VEBT @ nat @ ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi @ Ma2 ) ) @ ( suc @ ( suc @ Va ) ) @ TreeList2 @ Summary2 ) @ Xa ) ) ) ) ) ) ) ) ) ) ) ) ).
% T\<^sub>d\<^sub>e\<^sub>l\<^sub>e\<^sub>t\<^sub>e.pelims
thf(fact_3719_T_092_060_094sub_062s_092_060_094sub_062u_092_060_094sub_062c_092_060_094sub_062c_Opelims,axiom,
! [X: vEBT_VEBT,Xa: nat,Y: nat] :
( ( ( vEBT_T_s_u_c_c @ X @ Xa )
= Y )
=> ( ( accp @ ( product_prod @ vEBT_VEBT @ nat ) @ vEBT_T_s_u_c_c_rel2 @ ( product_Pair @ vEBT_VEBT @ nat @ X @ Xa ) )
=> ( ! [Uu2: $o,B3: $o] :
( ( X
= ( vEBT_Leaf @ Uu2 @ 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 @ Uu2 @ B3 ) @ ( zero_zero @ nat ) ) ) ) ) )
=> ( ! [Uv2: $o,Uw2: $o] :
( ( X
= ( vEBT_Leaf @ Uv2 @ Uw2 ) )
=> ! [N2: nat] :
( ( Xa
= ( suc @ N2 ) )
=> ( ( Y
= ( one_one @ nat ) )
=> ~ ( accp @ ( product_prod @ vEBT_VEBT @ nat ) @ vEBT_T_s_u_c_c_rel2 @ ( product_Pair @ vEBT_VEBT @ nat @ ( vEBT_Leaf @ Uv2 @ Uw2 ) @ ( suc @ N2 ) ) ) ) ) )
=> ( ! [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 ) ) ) )
=> ( ! [V4: product_prod @ nat @ nat,Vc: list @ vEBT_VEBT,Vd2: vEBT_VEBT] :
( ( X
= ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ V4 ) @ ( zero_zero @ nat ) @ Vc @ Vd2 ) )
=> ( ( Y
= ( one_one @ nat ) )
=> ~ ( accp @ ( product_prod @ vEBT_VEBT @ nat ) @ vEBT_T_s_u_c_c_rel2 @ ( product_Pair @ vEBT_VEBT @ nat @ ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ V4 ) @ ( zero_zero @ nat ) @ Vc @ Vd2 ) @ Xa ) ) ) )
=> ( ! [V4: product_prod @ nat @ nat,Vg2: list @ vEBT_VEBT,Vh2: vEBT_VEBT] :
( ( X
= ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ V4 ) @ ( suc @ ( zero_zero @ nat ) ) @ Vg2 @ Vh2 ) )
=> ( ( Y
= ( one_one @ nat ) )
=> ~ ( accp @ ( product_prod @ vEBT_VEBT @ nat ) @ vEBT_T_s_u_c_c_rel2 @ ( product_Pair @ vEBT_VEBT @ nat @ ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ V4 ) @ ( suc @ ( zero_zero @ nat ) ) @ Vg2 @ Vh2 ) @ Xa ) ) ) )
=> ~ ! [Mi: nat,Ma2: nat,Va: nat,TreeList2: list @ vEBT_VEBT,Summary2: vEBT_VEBT] :
( ( X
= ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi @ Ma2 ) ) @ ( suc @ ( suc @ Va ) ) @ TreeList2 @ Summary2 ) )
=> ( ( Y
= ( plus_plus @ nat @ ( one_one @ nat )
@ ( if @ nat @ ( ord_less @ nat @ Xa @ Mi ) @ ( one_one @ nat )
@ ( plus_plus @ nat @ ( numeral_numeral @ nat @ ( bit0 @ ( bit1 @ ( bit0 @ one2 ) ) ) )
@ ( if @ nat @ ( ord_less @ nat @ ( vEBT_VEBT_high @ Xa @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( size_size @ ( list @ vEBT_VEBT ) @ TreeList2 ) )
@ ( plus_plus @ nat @ ( plus_plus @ nat @ ( one_one @ nat ) @ ( vEBT_T_m_a_x_t @ ( nth @ vEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ Xa @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) )
@ ( plus_plus @ nat @ ( numeral_numeral @ nat @ ( bit1 @ one2 ) )
@ ( if @ nat
@ ( ( ( vEBT_vebt_maxt @ ( nth @ vEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ Xa @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) )
!= ( none @ nat ) )
& ( vEBT_VEBT_less @ ( some @ nat @ ( vEBT_VEBT_low @ Xa @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( vEBT_vebt_maxt @ ( nth @ vEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ Xa @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) ) )
@ ( plus_plus @ nat @ ( numeral_numeral @ nat @ ( bit0 @ ( bit0 @ one2 ) ) ) @ ( vEBT_T_s_u_c_c @ ( nth @ vEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ Xa @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( vEBT_VEBT_low @ Xa @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) )
@ ( plus_plus @ nat @ ( plus_plus @ nat @ ( plus_plus @ nat @ ( one_one @ nat ) @ ( vEBT_T_s_u_c_c @ Summary2 @ ( vEBT_VEBT_high @ Xa @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) @ ( one_one @ nat ) )
@ ( if @ nat
@ ( ( vEBT_vebt_succ @ Summary2 @ ( vEBT_VEBT_high @ Xa @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) )
= ( none @ nat ) )
@ ( one_one @ nat )
@ ( plus_plus @ nat @ ( numeral_numeral @ nat @ ( bit0 @ ( bit0 @ one2 ) ) ) @ ( vEBT_T_m_i_n_t @ ( nth @ vEBT_VEBT @ TreeList2 @ ( the2 @ nat @ ( vEBT_vebt_succ @ Summary2 @ ( vEBT_VEBT_high @ Xa @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) ) ) ) ) ) ) ) )
@ ( one_one @ nat ) ) ) ) ) )
=> ~ ( accp @ ( product_prod @ vEBT_VEBT @ nat ) @ vEBT_T_s_u_c_c_rel2 @ ( product_Pair @ vEBT_VEBT @ nat @ ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi @ Ma2 ) ) @ ( suc @ ( suc @ Va ) ) @ TreeList2 @ Summary2 ) @ Xa ) ) ) ) ) ) ) ) ) ) ) ).
% T\<^sub>s\<^sub>u\<^sub>c\<^sub>c.pelims
thf(fact_3720_T_092_060_094sub_062p_092_060_094sub_062r_092_060_094sub_062e_092_060_094sub_062d_Opelims,axiom,
! [X: vEBT_VEBT,Xa: nat,Y: nat] :
( ( ( vEBT_T_p_r_e_d @ X @ Xa )
= Y )
=> ( ( accp @ ( product_prod @ vEBT_VEBT @ nat ) @ vEBT_T_p_r_e_d_rel2 @ ( product_Pair @ vEBT_VEBT @ nat @ X @ Xa ) )
=> ( ! [Uu2: $o,Uv2: $o] :
( ( X
= ( vEBT_Leaf @ Uu2 @ Uv2 ) )
=> ( ( Xa
= ( zero_zero @ nat ) )
=> ( ( Y
= ( one_one @ nat ) )
=> ~ ( accp @ ( product_prod @ vEBT_VEBT @ nat ) @ vEBT_T_p_r_e_d_rel2 @ ( product_Pair @ vEBT_VEBT @ nat @ ( vEBT_Leaf @ Uu2 @ Uv2 ) @ ( zero_zero @ nat ) ) ) ) ) )
=> ( ! [A6: $o,Uw2: $o] :
( ( X
= ( vEBT_Leaf @ A6 @ Uw2 ) )
=> ( ( Xa
= ( suc @ ( zero_zero @ nat ) ) )
=> ( ( Y
= ( plus_plus @ nat @ ( one_one @ nat ) @ ( one_one @ nat ) ) )
=> ~ ( accp @ ( product_prod @ vEBT_VEBT @ nat ) @ vEBT_T_p_r_e_d_rel2 @ ( product_Pair @ vEBT_VEBT @ nat @ ( vEBT_Leaf @ A6 @ Uw2 ) @ ( suc @ ( zero_zero @ nat ) ) ) ) ) ) )
=> ( ! [A6: $o,B3: $o] :
( ( X
= ( vEBT_Leaf @ A6 @ B3 ) )
=> ! [Va: nat] :
( ( Xa
= ( suc @ ( suc @ Va ) ) )
=> ( ( 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 @ A6 @ B3 ) @ ( suc @ ( suc @ Va ) ) ) ) ) ) )
=> ( ! [Uy2: nat,Uz2: list @ vEBT_VEBT,Va3: vEBT_VEBT] :
( ( X
= ( vEBT_Node @ ( none @ ( product_prod @ nat @ nat ) ) @ Uy2 @ Uz2 @ Va3 ) )
=> ( ( Y
= ( one_one @ nat ) )
=> ~ ( accp @ ( product_prod @ vEBT_VEBT @ nat ) @ vEBT_T_p_r_e_d_rel2 @ ( product_Pair @ vEBT_VEBT @ nat @ ( vEBT_Node @ ( none @ ( product_prod @ nat @ nat ) ) @ Uy2 @ Uz2 @ Va3 ) @ Xa ) ) ) )
=> ( ! [V4: product_prod @ nat @ nat,Vd2: list @ vEBT_VEBT,Ve2: vEBT_VEBT] :
( ( X
= ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ V4 ) @ ( zero_zero @ nat ) @ Vd2 @ Ve2 ) )
=> ( ( Y
= ( one_one @ nat ) )
=> ~ ( accp @ ( product_prod @ vEBT_VEBT @ nat ) @ vEBT_T_p_r_e_d_rel2 @ ( product_Pair @ vEBT_VEBT @ nat @ ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ V4 ) @ ( zero_zero @ nat ) @ Vd2 @ Ve2 ) @ Xa ) ) ) )
=> ( ! [V4: product_prod @ nat @ nat,Vh2: list @ vEBT_VEBT,Vi2: vEBT_VEBT] :
( ( X
= ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ V4 ) @ ( suc @ ( zero_zero @ nat ) ) @ Vh2 @ Vi2 ) )
=> ( ( Y
= ( one_one @ nat ) )
=> ~ ( accp @ ( product_prod @ vEBT_VEBT @ nat ) @ vEBT_T_p_r_e_d_rel2 @ ( product_Pair @ vEBT_VEBT @ nat @ ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ V4 ) @ ( suc @ ( zero_zero @ nat ) ) @ Vh2 @ Vi2 ) @ Xa ) ) ) )
=> ~ ! [Mi: nat,Ma2: nat,Va: nat,TreeList2: list @ vEBT_VEBT,Summary2: vEBT_VEBT] :
( ( X
= ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi @ Ma2 ) ) @ ( suc @ ( suc @ Va ) ) @ TreeList2 @ Summary2 ) )
=> ( ( Y
= ( plus_plus @ nat @ ( one_one @ nat )
@ ( if @ nat @ ( ord_less @ nat @ Ma2 @ Xa ) @ ( one_one @ nat )
@ ( plus_plus @ nat @ ( plus_plus @ nat @ ( numeral_numeral @ nat @ ( bit0 @ ( bit1 @ ( bit0 @ one2 ) ) ) ) @ ( one_one @ nat ) )
@ ( if @ nat @ ( ord_less @ nat @ ( vEBT_VEBT_high @ Xa @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( size_size @ ( list @ vEBT_VEBT ) @ TreeList2 ) )
@ ( plus_plus @ nat @ ( plus_plus @ nat @ ( plus_plus @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( vEBT_T_m_i_n_t @ ( nth @ vEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ Xa @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) ) @ ( numeral_numeral @ nat @ ( bit1 @ one2 ) ) )
@ ( if @ nat
@ ( ( ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ Xa @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) )
!= ( none @ nat ) )
& ( vEBT_VEBT_greater @ ( some @ nat @ ( vEBT_VEBT_low @ Xa @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ Xa @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) ) )
@ ( plus_plus @ nat @ ( numeral_numeral @ nat @ ( bit0 @ ( bit0 @ one2 ) ) ) @ ( vEBT_T_p_r_e_d @ ( nth @ vEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ Xa @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( vEBT_VEBT_low @ Xa @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) )
@ ( plus_plus @ nat @ ( plus_plus @ nat @ ( plus_plus @ nat @ ( one_one @ nat ) @ ( vEBT_T_p_r_e_d @ Summary2 @ ( vEBT_VEBT_high @ Xa @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) @ ( one_one @ nat ) )
@ ( if @ nat
@ ( ( vEBT_vebt_pred @ Summary2 @ ( vEBT_VEBT_high @ Xa @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) )
= ( none @ nat ) )
@ ( plus_plus @ nat @ ( one_one @ nat ) @ ( one_one @ nat ) )
@ ( plus_plus @ nat @ ( numeral_numeral @ nat @ ( bit0 @ ( bit0 @ one2 ) ) ) @ ( vEBT_T_m_a_x_t @ ( nth @ vEBT_VEBT @ TreeList2 @ ( the2 @ nat @ ( vEBT_vebt_pred @ Summary2 @ ( vEBT_VEBT_high @ Xa @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) ) ) ) ) ) ) )
@ ( one_one @ nat ) ) ) ) ) )
=> ~ ( accp @ ( product_prod @ vEBT_VEBT @ nat ) @ vEBT_T_p_r_e_d_rel2 @ ( product_Pair @ vEBT_VEBT @ nat @ ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi @ Ma2 ) ) @ ( suc @ ( suc @ Va ) ) @ TreeList2 @ Summary2 ) @ Xa ) ) ) ) ) ) ) ) ) ) ) ) ).
% T\<^sub>p\<^sub>r\<^sub>e\<^sub>d.pelims
thf(fact_3721_pred__empty,axiom,
! [T: vEBT_VEBT,N: nat,X: nat] :
( ( vEBT_invar_vebt @ T @ N )
=> ( ( ( vEBT_vebt_pred @ T @ X )
= ( none @ nat ) )
= ( ( collect @ nat
@ ^ [Y6: nat] :
( ( vEBT_vebt_member @ T @ Y6 )
& ( ord_less @ nat @ Y6 @ X ) ) )
= ( bot_bot @ ( set @ nat ) ) ) ) ) ).
% pred_empty
thf(fact_3722_buildup__gives__empty,axiom,
! [N: nat] :
( ( vEBT_VEBT_set_vebt @ ( vEBT_vebt_buildup @ N ) )
= ( bot_bot @ ( set @ nat ) ) ) ).
% buildup_gives_empty
thf(fact_3723_mint__corr__help__empty,axiom,
! [T: vEBT_VEBT,N: nat] :
( ( vEBT_invar_vebt @ T @ N )
=> ( ( ( vEBT_vebt_mint @ T )
= ( none @ nat ) )
=> ( ( vEBT_VEBT_set_vebt @ T )
= ( bot_bot @ ( set @ nat ) ) ) ) ) ).
% mint_corr_help_empty
thf(fact_3724_maxt__corr__help__empty,axiom,
! [T: vEBT_VEBT,N: nat] :
( ( vEBT_invar_vebt @ T @ N )
=> ( ( ( vEBT_vebt_maxt @ T )
= ( none @ nat ) )
=> ( ( vEBT_VEBT_set_vebt @ T )
= ( bot_bot @ ( set @ nat ) ) ) ) ) ).
% maxt_corr_help_empty
thf(fact_3725_succ__empty,axiom,
! [T: vEBT_VEBT,N: nat,X: nat] :
( ( vEBT_invar_vebt @ T @ N )
=> ( ( ( vEBT_vebt_succ @ T @ X )
= ( none @ nat ) )
= ( ( collect @ nat
@ ^ [Y6: nat] :
( ( vEBT_vebt_member @ T @ Y6 )
& ( ord_less @ nat @ X @ Y6 ) ) )
= ( bot_bot @ ( set @ nat ) ) ) ) ) ).
% succ_empty
thf(fact_3726_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_3727_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_3728_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_3729_mod__h__bot__iff_I4_J,axiom,
! [B: $tType] :
( ( heap @ B )
=> ! [Q3: array @ B,Y: list @ B,H2: heap_ext @ product_unit] :
~ ( rep_assn @ ( snga_assn @ B @ Q3 @ Y ) @ ( product_Pair @ ( heap_ext @ product_unit ) @ ( set @ nat ) @ H2 @ ( bot_bot @ ( set @ nat ) ) ) ) ) ).
% mod_h_bot_iff(4)
thf(fact_3730_set__decode__zero,axiom,
( ( nat_set_decode @ ( zero_zero @ nat ) )
= ( bot_bot @ ( set @ nat ) ) ) ).
% set_decode_zero
thf(fact_3731_set__encode__empty,axiom,
( ( nat_set_encode @ ( bot_bot @ ( set @ nat ) ) )
= ( zero_zero @ nat ) ) ).
% set_encode_empty
thf(fact_3732_mod__h__bot__indep,axiom,
! [P: assn,H2: heap_ext @ product_unit,H4: 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 ) @ H4 @ ( bot_bot @ ( set @ nat ) ) ) ) ) ).
% mod_h_bot_indep
thf(fact_3733_not__psubset__empty,axiom,
! [A: $tType,A3: set @ A] :
~ ( ord_less @ ( set @ A ) @ A3 @ ( bot_bot @ ( set @ A ) ) ) ).
% not_psubset_empty
thf(fact_3734_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_3735_infinite__growing,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [X7: set @ A] :
( ( X7
!= ( bot_bot @ ( set @ A ) ) )
=> ( ! [X3: A] :
( ( member @ A @ X3 @ X7 )
=> ? [Xa2: A] :
( ( member @ A @ Xa2 @ X7 )
& ( ord_less @ A @ X3 @ Xa2 ) ) )
=> ~ ( finite_finite2 @ A @ X7 ) ) ) ) ).
% infinite_growing
thf(fact_3736_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_3737_vebt__succ_Opelims,axiom,
! [X: vEBT_VEBT,Xa: nat,Y: option @ nat] :
( ( ( vEBT_vebt_succ @ X @ Xa )
= Y )
=> ( ( accp @ ( product_prod @ vEBT_VEBT @ nat ) @ vEBT_vebt_succ_rel @ ( product_Pair @ vEBT_VEBT @ nat @ X @ Xa ) )
=> ( ! [Uu2: $o,B3: $o] :
( ( X
= ( vEBT_Leaf @ Uu2 @ 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 @ Uu2 @ B3 ) @ ( zero_zero @ nat ) ) ) ) ) )
=> ( ! [Uv2: $o,Uw2: $o] :
( ( X
= ( vEBT_Leaf @ Uv2 @ Uw2 ) )
=> ! [N2: nat] :
( ( Xa
= ( suc @ N2 ) )
=> ( ( Y
= ( none @ nat ) )
=> ~ ( accp @ ( product_prod @ vEBT_VEBT @ nat ) @ vEBT_vebt_succ_rel @ ( product_Pair @ vEBT_VEBT @ nat @ ( vEBT_Leaf @ Uv2 @ Uw2 ) @ ( suc @ N2 ) ) ) ) ) )
=> ( ! [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 ) ) ) )
=> ( ! [V4: product_prod @ nat @ nat,Vc: list @ vEBT_VEBT,Vd2: vEBT_VEBT] :
( ( X
= ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ V4 ) @ ( zero_zero @ nat ) @ Vc @ Vd2 ) )
=> ( ( Y
= ( none @ nat ) )
=> ~ ( accp @ ( product_prod @ vEBT_VEBT @ nat ) @ vEBT_vebt_succ_rel @ ( product_Pair @ vEBT_VEBT @ nat @ ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ V4 ) @ ( zero_zero @ nat ) @ Vc @ Vd2 ) @ Xa ) ) ) )
=> ( ! [V4: product_prod @ nat @ nat,Vg2: list @ vEBT_VEBT,Vh2: vEBT_VEBT] :
( ( X
= ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ V4 ) @ ( suc @ ( zero_zero @ nat ) ) @ Vg2 @ Vh2 ) )
=> ( ( Y
= ( none @ nat ) )
=> ~ ( accp @ ( product_prod @ vEBT_VEBT @ nat ) @ vEBT_vebt_succ_rel @ ( product_Pair @ vEBT_VEBT @ nat @ ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ V4 ) @ ( suc @ ( zero_zero @ nat ) ) @ Vg2 @ Vh2 ) @ Xa ) ) ) )
=> ~ ! [Mi: nat,Ma2: nat,Va: nat,TreeList2: list @ vEBT_VEBT,Summary2: vEBT_VEBT] :
( ( X
= ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi @ Ma2 ) ) @ ( suc @ ( suc @ Va ) ) @ TreeList2 @ Summary2 ) )
=> ( ( ( ( ord_less @ nat @ Xa @ Mi )
=> ( Y
= ( some @ nat @ Mi ) ) )
& ( ~ ( ord_less @ nat @ Xa @ Mi )
=> ( Y
= ( if @ ( option @ nat ) @ ( ord_less @ nat @ ( vEBT_VEBT_high @ Xa @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( size_size @ ( list @ vEBT_VEBT ) @ TreeList2 ) )
@ ( if @ ( option @ nat )
@ ( ( ( vEBT_vebt_maxt @ ( nth @ vEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ Xa @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) )
!= ( none @ nat ) )
& ( vEBT_VEBT_less @ ( some @ nat @ ( vEBT_VEBT_low @ Xa @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( vEBT_vebt_maxt @ ( nth @ vEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ Xa @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) ) )
@ ( vEBT_VEBT_add @ ( vEBT_VEBT_mul @ ( some @ nat @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( some @ nat @ ( vEBT_VEBT_high @ Xa @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) @ ( vEBT_vebt_succ @ ( nth @ vEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ Xa @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( vEBT_VEBT_low @ Xa @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) )
@ ( if @ ( option @ nat )
@ ( ( vEBT_vebt_succ @ Summary2 @ ( vEBT_VEBT_high @ Xa @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) )
= ( none @ nat ) )
@ ( none @ nat )
@ ( vEBT_VEBT_add @ ( vEBT_VEBT_mul @ ( some @ nat @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( vEBT_vebt_succ @ Summary2 @ ( vEBT_VEBT_high @ Xa @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) @ ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList2 @ ( the2 @ nat @ ( vEBT_vebt_succ @ Summary2 @ ( vEBT_VEBT_high @ Xa @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) ) ) ) ) )
@ ( none @ nat ) ) ) ) )
=> ~ ( accp @ ( product_prod @ vEBT_VEBT @ nat ) @ vEBT_vebt_succ_rel @ ( product_Pair @ vEBT_VEBT @ nat @ ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi @ Ma2 ) ) @ ( suc @ ( suc @ Va ) ) @ TreeList2 @ Summary2 ) @ Xa ) ) ) ) ) ) ) ) ) ) ) ).
% vebt_succ.pelims
thf(fact_3738_vebt__pred_Opelims,axiom,
! [X: vEBT_VEBT,Xa: nat,Y: option @ nat] :
( ( ( vEBT_vebt_pred @ X @ Xa )
= Y )
=> ( ( accp @ ( product_prod @ vEBT_VEBT @ nat ) @ vEBT_vebt_pred_rel @ ( product_Pair @ vEBT_VEBT @ nat @ X @ Xa ) )
=> ( ! [Uu2: $o,Uv2: $o] :
( ( X
= ( vEBT_Leaf @ Uu2 @ Uv2 ) )
=> ( ( Xa
= ( zero_zero @ nat ) )
=> ( ( Y
= ( none @ nat ) )
=> ~ ( accp @ ( product_prod @ vEBT_VEBT @ nat ) @ vEBT_vebt_pred_rel @ ( product_Pair @ vEBT_VEBT @ nat @ ( vEBT_Leaf @ Uu2 @ Uv2 ) @ ( zero_zero @ nat ) ) ) ) ) )
=> ( ! [A6: $o,Uw2: $o] :
( ( X
= ( vEBT_Leaf @ A6 @ Uw2 ) )
=> ( ( Xa
= ( suc @ ( zero_zero @ nat ) ) )
=> ( ( ( A6
=> ( Y
= ( some @ nat @ ( zero_zero @ nat ) ) ) )
& ( ~ A6
=> ( Y
= ( none @ nat ) ) ) )
=> ~ ( accp @ ( product_prod @ vEBT_VEBT @ nat ) @ vEBT_vebt_pred_rel @ ( product_Pair @ vEBT_VEBT @ nat @ ( vEBT_Leaf @ A6 @ Uw2 ) @ ( suc @ ( zero_zero @ nat ) ) ) ) ) ) )
=> ( ! [A6: $o,B3: $o] :
( ( X
= ( vEBT_Leaf @ A6 @ B3 ) )
=> ! [Va: nat] :
( ( Xa
= ( suc @ ( suc @ Va ) ) )
=> ( ( ( B3
=> ( Y
= ( some @ nat @ ( one_one @ nat ) ) ) )
& ( ~ B3
=> ( ( A6
=> ( Y
= ( some @ nat @ ( zero_zero @ nat ) ) ) )
& ( ~ A6
=> ( Y
= ( none @ nat ) ) ) ) ) )
=> ~ ( accp @ ( product_prod @ vEBT_VEBT @ nat ) @ vEBT_vebt_pred_rel @ ( product_Pair @ vEBT_VEBT @ nat @ ( vEBT_Leaf @ A6 @ B3 ) @ ( suc @ ( suc @ Va ) ) ) ) ) ) )
=> ( ! [Uy2: nat,Uz2: list @ vEBT_VEBT,Va3: vEBT_VEBT] :
( ( X
= ( vEBT_Node @ ( none @ ( product_prod @ nat @ nat ) ) @ Uy2 @ Uz2 @ Va3 ) )
=> ( ( Y
= ( none @ nat ) )
=> ~ ( accp @ ( product_prod @ vEBT_VEBT @ nat ) @ vEBT_vebt_pred_rel @ ( product_Pair @ vEBT_VEBT @ nat @ ( vEBT_Node @ ( none @ ( product_prod @ nat @ nat ) ) @ Uy2 @ Uz2 @ Va3 ) @ Xa ) ) ) )
=> ( ! [V4: product_prod @ nat @ nat,Vd2: list @ vEBT_VEBT,Ve2: vEBT_VEBT] :
( ( X
= ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ V4 ) @ ( zero_zero @ nat ) @ Vd2 @ Ve2 ) )
=> ( ( Y
= ( none @ nat ) )
=> ~ ( accp @ ( product_prod @ vEBT_VEBT @ nat ) @ vEBT_vebt_pred_rel @ ( product_Pair @ vEBT_VEBT @ nat @ ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ V4 ) @ ( zero_zero @ nat ) @ Vd2 @ Ve2 ) @ Xa ) ) ) )
=> ( ! [V4: product_prod @ nat @ nat,Vh2: list @ vEBT_VEBT,Vi2: vEBT_VEBT] :
( ( X
= ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ V4 ) @ ( suc @ ( zero_zero @ nat ) ) @ Vh2 @ Vi2 ) )
=> ( ( Y
= ( none @ nat ) )
=> ~ ( accp @ ( product_prod @ vEBT_VEBT @ nat ) @ vEBT_vebt_pred_rel @ ( product_Pair @ vEBT_VEBT @ nat @ ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ V4 ) @ ( suc @ ( zero_zero @ nat ) ) @ Vh2 @ Vi2 ) @ Xa ) ) ) )
=> ~ ! [Mi: nat,Ma2: nat,Va: nat,TreeList2: list @ vEBT_VEBT,Summary2: vEBT_VEBT] :
( ( X
= ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi @ Ma2 ) ) @ ( suc @ ( suc @ Va ) ) @ TreeList2 @ Summary2 ) )
=> ( ( ( ( ord_less @ nat @ Ma2 @ Xa )
=> ( Y
= ( some @ nat @ Ma2 ) ) )
& ( ~ ( ord_less @ nat @ Ma2 @ Xa )
=> ( Y
= ( if @ ( option @ nat ) @ ( ord_less @ nat @ ( vEBT_VEBT_high @ Xa @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( size_size @ ( list @ vEBT_VEBT ) @ TreeList2 ) )
@ ( if @ ( option @ nat )
@ ( ( ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ Xa @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) )
!= ( none @ nat ) )
& ( vEBT_VEBT_greater @ ( some @ nat @ ( vEBT_VEBT_low @ Xa @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ Xa @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) ) )
@ ( vEBT_VEBT_add @ ( vEBT_VEBT_mul @ ( some @ nat @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( some @ nat @ ( vEBT_VEBT_high @ Xa @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) @ ( vEBT_vebt_pred @ ( nth @ vEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ Xa @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( vEBT_VEBT_low @ Xa @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) )
@ ( if @ ( option @ nat )
@ ( ( vEBT_vebt_pred @ Summary2 @ ( vEBT_VEBT_high @ Xa @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) )
= ( none @ nat ) )
@ ( if @ ( option @ nat ) @ ( ord_less @ nat @ Mi @ Xa ) @ ( 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 @ Summary2 @ ( vEBT_VEBT_high @ Xa @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) @ ( vEBT_vebt_maxt @ ( nth @ vEBT_VEBT @ TreeList2 @ ( the2 @ nat @ ( vEBT_vebt_pred @ Summary2 @ ( vEBT_VEBT_high @ Xa @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) ) ) ) ) )
@ ( none @ nat ) ) ) ) )
=> ~ ( accp @ ( product_prod @ vEBT_VEBT @ nat ) @ vEBT_vebt_pred_rel @ ( product_Pair @ vEBT_VEBT @ nat @ ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi @ Ma2 ) ) @ ( suc @ ( suc @ Va ) ) @ TreeList2 @ Summary2 ) @ Xa ) ) ) ) ) ) ) ) ) ) ) ) ).
% vebt_pred.pelims
thf(fact_3739_VEBT__internal_OT_092_060_094sub_062d_092_060_094sub_062e_092_060_094sub_062l_092_060_094sub_062e_092_060_094sub_062t_092_060_094sub_062e_H_Opelims,axiom,
! [X: vEBT_VEBT,Xa: nat,Y: nat] :
( ( ( vEBT_V1232361888498592333_e_t_e @ X @ Xa )
= Y )
=> ( ( accp @ ( product_prod @ vEBT_VEBT @ nat ) @ vEBT_V6368547301243506412_e_rel @ ( product_Pair @ vEBT_VEBT @ nat @ X @ Xa ) )
=> ( ! [A6: $o,B3: $o] :
( ( X
= ( vEBT_Leaf @ A6 @ 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 @ A6 @ B3 ) @ ( zero_zero @ nat ) ) ) ) ) )
=> ( ! [A6: $o,B3: $o] :
( ( X
= ( vEBT_Leaf @ A6 @ 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 @ A6 @ B3 ) @ ( suc @ ( zero_zero @ nat ) ) ) ) ) ) )
=> ( ! [A6: $o,B3: $o] :
( ( X
= ( vEBT_Leaf @ A6 @ B3 ) )
=> ! [N2: nat] :
( ( Xa
= ( suc @ ( suc @ N2 ) ) )
=> ( ( Y
= ( one_one @ nat ) )
=> ~ ( accp @ ( product_prod @ vEBT_VEBT @ nat ) @ vEBT_V6368547301243506412_e_rel @ ( product_Pair @ vEBT_VEBT @ nat @ ( vEBT_Leaf @ A6 @ B3 ) @ ( suc @ ( suc @ N2 ) ) ) ) ) ) )
=> ( ! [Deg2: nat,TreeList2: list @ vEBT_VEBT,Summary2: vEBT_VEBT] :
( ( X
= ( vEBT_Node @ ( none @ ( product_prod @ nat @ nat ) ) @ Deg2 @ TreeList2 @ Summary2 ) )
=> ( ( Y
= ( one_one @ nat ) )
=> ~ ( accp @ ( product_prod @ vEBT_VEBT @ nat ) @ vEBT_V6368547301243506412_e_rel @ ( product_Pair @ vEBT_VEBT @ nat @ ( vEBT_Node @ ( none @ ( product_prod @ nat @ nat ) ) @ Deg2 @ TreeList2 @ Summary2 ) @ Xa ) ) ) )
=> ( ! [Mi: nat,Ma2: nat,TreeList2: list @ vEBT_VEBT,Summary2: vEBT_VEBT] :
( ( X
= ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi @ 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 @ Mi @ Ma2 ) ) @ ( zero_zero @ nat ) @ TreeList2 @ Summary2 ) @ Xa ) ) ) )
=> ( ! [Mi: nat,Ma2: nat,TreeList2: list @ vEBT_VEBT,Summary2: vEBT_VEBT] :
( ( X
= ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi @ 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 @ Mi @ Ma2 ) ) @ ( suc @ ( zero_zero @ nat ) ) @ TreeList2 @ Summary2 ) @ Xa ) ) ) )
=> ~ ! [Mi: nat,Ma2: nat,Va: nat,TreeList2: list @ vEBT_VEBT,Summary2: vEBT_VEBT] :
( ( X
= ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi @ Ma2 ) ) @ ( suc @ ( suc @ Va ) ) @ TreeList2 @ Summary2 ) )
=> ( ( ( ( ( ord_less @ nat @ Xa @ Mi )
| ( ord_less @ nat @ Ma2 @ Xa ) )
=> ( Y
= ( one_one @ nat ) ) )
& ( ~ ( ( ord_less @ nat @ Xa @ Mi )
| ( ord_less @ nat @ Ma2 @ Xa ) )
=> ( ( ( ( Xa = Mi )
& ( Xa = Ma2 ) )
=> ( Y
= ( one_one @ nat ) ) )
& ( ~ ( ( Xa = Mi )
& ( Xa = Ma2 ) )
=> ( Y
= ( if @ nat @ ( ord_less @ nat @ ( vEBT_VEBT_high @ ( if @ nat @ ( Xa = Mi ) @ ( plus_plus @ nat @ ( times_times @ nat @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary2 ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList2 @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary2 ) ) ) ) ) ) @ Xa ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( size_size @ ( list @ vEBT_VEBT ) @ TreeList2 ) ) @ ( plus_plus @ nat @ ( vEBT_V1232361888498592333_e_t_e @ ( nth @ vEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ ( if @ nat @ ( Xa = Mi ) @ ( plus_plus @ nat @ ( times_times @ nat @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary2 ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList2 @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary2 ) ) ) ) ) ) @ Xa ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( vEBT_VEBT_low @ ( if @ nat @ ( Xa = Mi ) @ ( plus_plus @ nat @ ( times_times @ nat @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary2 ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList2 @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary2 ) ) ) ) ) ) @ Xa ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( if @ nat @ ( vEBT_VEBT_minNull @ ( vEBT_vebt_delete @ ( nth @ vEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ ( if @ nat @ ( Xa = Mi ) @ ( plus_plus @ nat @ ( times_times @ nat @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary2 ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList2 @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary2 ) ) ) ) ) ) @ Xa ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( vEBT_VEBT_low @ ( if @ nat @ ( Xa = Mi ) @ ( plus_plus @ nat @ ( times_times @ nat @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary2 ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList2 @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary2 ) ) ) ) ) ) @ Xa ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) @ ( vEBT_V1232361888498592333_e_t_e @ Summary2 @ ( vEBT_VEBT_high @ ( if @ nat @ ( Xa = Mi ) @ ( plus_plus @ nat @ ( times_times @ nat @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary2 ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList2 @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary2 ) ) ) ) ) ) @ Xa ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( one_one @ nat ) ) ) @ ( one_one @ nat ) ) ) ) ) ) )
=> ~ ( accp @ ( product_prod @ vEBT_VEBT @ nat ) @ vEBT_V6368547301243506412_e_rel @ ( product_Pair @ vEBT_VEBT @ nat @ ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi @ Ma2 ) ) @ ( suc @ ( suc @ Va ) ) @ TreeList2 @ Summary2 ) @ Xa ) ) ) ) ) ) ) ) ) ) ) ) ).
% VEBT_internal.T\<^sub>d\<^sub>e\<^sub>l\<^sub>e\<^sub>t\<^sub>e'.pelims
thf(fact_3740_vebt__delete_Opelims,axiom,
! [X: vEBT_VEBT,Xa: nat,Y: vEBT_VEBT] :
( ( ( vEBT_vebt_delete @ X @ Xa )
= Y )
=> ( ( accp @ ( product_prod @ vEBT_VEBT @ nat ) @ vEBT_vebt_delete_rel @ ( product_Pair @ vEBT_VEBT @ nat @ X @ Xa ) )
=> ( ! [A6: $o,B3: $o] :
( ( X
= ( vEBT_Leaf @ A6 @ 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 @ A6 @ B3 ) @ ( zero_zero @ nat ) ) ) ) ) )
=> ( ! [A6: $o,B3: $o] :
( ( X
= ( vEBT_Leaf @ A6 @ B3 ) )
=> ( ( Xa
= ( suc @ ( zero_zero @ nat ) ) )
=> ( ( Y
= ( vEBT_Leaf @ A6 @ $false ) )
=> ~ ( accp @ ( product_prod @ vEBT_VEBT @ nat ) @ vEBT_vebt_delete_rel @ ( product_Pair @ vEBT_VEBT @ nat @ ( vEBT_Leaf @ A6 @ B3 ) @ ( suc @ ( zero_zero @ nat ) ) ) ) ) ) )
=> ( ! [A6: $o,B3: $o] :
( ( X
= ( vEBT_Leaf @ A6 @ B3 ) )
=> ! [N2: nat] :
( ( Xa
= ( suc @ ( suc @ N2 ) ) )
=> ( ( Y
= ( vEBT_Leaf @ A6 @ B3 ) )
=> ~ ( accp @ ( product_prod @ vEBT_VEBT @ nat ) @ vEBT_vebt_delete_rel @ ( product_Pair @ vEBT_VEBT @ nat @ ( vEBT_Leaf @ A6 @ B3 ) @ ( suc @ ( suc @ N2 ) ) ) ) ) ) )
=> ( ! [Deg2: nat,TreeList2: list @ vEBT_VEBT,Summary2: vEBT_VEBT] :
( ( X
= ( vEBT_Node @ ( none @ ( product_prod @ nat @ nat ) ) @ Deg2 @ TreeList2 @ Summary2 ) )
=> ( ( Y
= ( vEBT_Node @ ( none @ ( product_prod @ nat @ nat ) ) @ Deg2 @ TreeList2 @ Summary2 ) )
=> ~ ( accp @ ( product_prod @ vEBT_VEBT @ nat ) @ vEBT_vebt_delete_rel @ ( product_Pair @ vEBT_VEBT @ nat @ ( vEBT_Node @ ( none @ ( product_prod @ nat @ nat ) ) @ Deg2 @ TreeList2 @ Summary2 ) @ Xa ) ) ) )
=> ( ! [Mi: nat,Ma2: nat,TrLst2: list @ vEBT_VEBT,Smry2: vEBT_VEBT] :
( ( X
= ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi @ Ma2 ) ) @ ( zero_zero @ nat ) @ TrLst2 @ Smry2 ) )
=> ( ( Y
= ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi @ 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 @ Mi @ Ma2 ) ) @ ( zero_zero @ nat ) @ TrLst2 @ Smry2 ) @ Xa ) ) ) )
=> ( ! [Mi: nat,Ma2: nat,Tr2: list @ vEBT_VEBT,Sm2: vEBT_VEBT] :
( ( X
= ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi @ Ma2 ) ) @ ( suc @ ( zero_zero @ nat ) ) @ Tr2 @ Sm2 ) )
=> ( ( Y
= ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi @ 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 @ Mi @ Ma2 ) ) @ ( suc @ ( zero_zero @ nat ) ) @ Tr2 @ Sm2 ) @ Xa ) ) ) )
=> ~ ! [Mi: nat,Ma2: nat,Va: nat,TreeList2: list @ vEBT_VEBT,Summary2: vEBT_VEBT] :
( ( X
= ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi @ Ma2 ) ) @ ( suc @ ( suc @ Va ) ) @ TreeList2 @ Summary2 ) )
=> ( ( ( ( ( ord_less @ nat @ Xa @ Mi )
| ( ord_less @ nat @ Ma2 @ Xa ) )
=> ( Y
= ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi @ Ma2 ) ) @ ( suc @ ( suc @ Va ) ) @ TreeList2 @ Summary2 ) ) )
& ( ~ ( ( ord_less @ nat @ Xa @ Mi )
| ( ord_less @ nat @ Ma2 @ Xa ) )
=> ( ( ( ( Xa = Mi )
& ( Xa = Ma2 ) )
=> ( Y
= ( vEBT_Node @ ( none @ ( product_prod @ nat @ nat ) ) @ ( suc @ ( suc @ Va ) ) @ TreeList2 @ Summary2 ) ) )
& ( ~ ( ( Xa = Mi )
& ( Xa = Ma2 ) )
=> ( Y
= ( if @ vEBT_VEBT @ ( ord_less @ nat @ ( vEBT_VEBT_high @ ( if @ nat @ ( Xa = Mi ) @ ( plus_plus @ nat @ ( times_times @ nat @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary2 ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList2 @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary2 ) ) ) ) ) ) @ Xa ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( size_size @ ( list @ vEBT_VEBT ) @ TreeList2 ) )
@ ( if @ vEBT_VEBT @ ( vEBT_VEBT_minNull @ ( vEBT_vebt_delete @ ( nth @ vEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ ( if @ nat @ ( Xa = Mi ) @ ( plus_plus @ nat @ ( times_times @ nat @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary2 ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList2 @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary2 ) ) ) ) ) ) @ Xa ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( vEBT_VEBT_low @ ( if @ nat @ ( Xa = Mi ) @ ( plus_plus @ nat @ ( times_times @ nat @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary2 ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList2 @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary2 ) ) ) ) ) ) @ Xa ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) )
@ ( vEBT_Node
@ ( some @ ( product_prod @ nat @ nat )
@ ( product_Pair @ nat @ nat @ ( if @ nat @ ( Xa = Mi ) @ ( if @ nat @ ( Xa = Mi ) @ ( plus_plus @ nat @ ( times_times @ nat @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary2 ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList2 @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary2 ) ) ) ) ) ) @ Xa ) @ Mi )
@ ( if @ nat
@ ( ( ( Xa = Mi )
=> ( ( plus_plus @ nat @ ( times_times @ nat @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary2 ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList2 @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary2 ) ) ) ) ) )
= Ma2 ) )
& ( ( Xa != Mi )
=> ( Xa = Ma2 ) ) )
@ ( if @ nat
@ ( ( vEBT_vebt_maxt @ ( vEBT_vebt_delete @ Summary2 @ ( vEBT_VEBT_high @ ( if @ nat @ ( Xa = Mi ) @ ( plus_plus @ nat @ ( times_times @ nat @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary2 ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList2 @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary2 ) ) ) ) ) ) @ Xa ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) )
= ( none @ nat ) )
@ ( if @ nat @ ( Xa = Mi ) @ ( if @ nat @ ( Xa = Mi ) @ ( plus_plus @ nat @ ( times_times @ nat @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary2 ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList2 @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary2 ) ) ) ) ) ) @ Xa ) @ 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 @ Summary2 @ ( vEBT_VEBT_high @ ( if @ nat @ ( Xa = Mi ) @ ( plus_plus @ nat @ ( times_times @ nat @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary2 ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList2 @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary2 ) ) ) ) ) ) @ Xa ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_maxt @ ( nth @ vEBT_VEBT @ ( list_update @ vEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ ( if @ nat @ ( Xa = Mi ) @ ( plus_plus @ nat @ ( times_times @ nat @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary2 ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList2 @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary2 ) ) ) ) ) ) @ Xa ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( vEBT_vebt_delete @ ( nth @ vEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ ( if @ nat @ ( Xa = Mi ) @ ( plus_plus @ nat @ ( times_times @ nat @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary2 ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList2 @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary2 ) ) ) ) ) ) @ Xa ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( vEBT_VEBT_low @ ( if @ nat @ ( Xa = Mi ) @ ( plus_plus @ nat @ ( times_times @ nat @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary2 ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList2 @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary2 ) ) ) ) ) ) @ Xa ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_maxt @ ( vEBT_vebt_delete @ Summary2 @ ( vEBT_VEBT_high @ ( if @ nat @ ( Xa = Mi ) @ ( plus_plus @ nat @ ( times_times @ nat @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary2 ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList2 @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary2 ) ) ) ) ) ) @ Xa ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) ) ) ) ) ) )
@ Ma2 ) ) )
@ ( suc @ ( suc @ Va ) )
@ ( list_update @ vEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ ( if @ nat @ ( Xa = Mi ) @ ( plus_plus @ nat @ ( times_times @ nat @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary2 ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList2 @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary2 ) ) ) ) ) ) @ Xa ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( vEBT_vebt_delete @ ( nth @ vEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ ( if @ nat @ ( Xa = Mi ) @ ( plus_plus @ nat @ ( times_times @ nat @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary2 ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList2 @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary2 ) ) ) ) ) ) @ Xa ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( vEBT_VEBT_low @ ( if @ nat @ ( Xa = Mi ) @ ( plus_plus @ nat @ ( times_times @ nat @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary2 ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList2 @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary2 ) ) ) ) ) ) @ Xa ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) )
@ ( vEBT_vebt_delete @ Summary2 @ ( vEBT_VEBT_high @ ( if @ nat @ ( Xa = Mi ) @ ( plus_plus @ nat @ ( times_times @ nat @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary2 ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList2 @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary2 ) ) ) ) ) ) @ Xa ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) )
@ ( vEBT_Node
@ ( some @ ( product_prod @ nat @ nat )
@ ( product_Pair @ nat @ nat @ ( if @ nat @ ( Xa = Mi ) @ ( if @ nat @ ( Xa = Mi ) @ ( plus_plus @ nat @ ( times_times @ nat @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary2 ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList2 @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary2 ) ) ) ) ) ) @ Xa ) @ Mi )
@ ( if @ nat
@ ( ( ( Xa = Mi )
=> ( ( plus_plus @ nat @ ( times_times @ nat @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary2 ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList2 @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary2 ) ) ) ) ) )
= Ma2 ) )
& ( ( Xa != Mi )
=> ( Xa = Ma2 ) ) )
@ ( plus_plus @ nat @ ( times_times @ nat @ ( vEBT_VEBT_high @ ( if @ nat @ ( Xa = Mi ) @ ( plus_plus @ nat @ ( times_times @ nat @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary2 ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList2 @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary2 ) ) ) ) ) ) @ Xa ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_maxt @ ( nth @ vEBT_VEBT @ ( list_update @ vEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ ( if @ nat @ ( Xa = Mi ) @ ( plus_plus @ nat @ ( times_times @ nat @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary2 ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList2 @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary2 ) ) ) ) ) ) @ Xa ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( vEBT_vebt_delete @ ( nth @ vEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ ( if @ nat @ ( Xa = Mi ) @ ( plus_plus @ nat @ ( times_times @ nat @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary2 ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList2 @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary2 ) ) ) ) ) ) @ Xa ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( vEBT_VEBT_low @ ( if @ nat @ ( Xa = Mi ) @ ( plus_plus @ nat @ ( times_times @ nat @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary2 ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList2 @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary2 ) ) ) ) ) ) @ Xa ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) @ ( vEBT_VEBT_high @ ( if @ nat @ ( Xa = Mi ) @ ( plus_plus @ nat @ ( times_times @ nat @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary2 ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList2 @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary2 ) ) ) ) ) ) @ Xa ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) ) )
@ Ma2 ) ) )
@ ( suc @ ( suc @ Va ) )
@ ( list_update @ vEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ ( if @ nat @ ( Xa = Mi ) @ ( plus_plus @ nat @ ( times_times @ nat @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary2 ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList2 @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary2 ) ) ) ) ) ) @ Xa ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( vEBT_vebt_delete @ ( nth @ vEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ ( if @ nat @ ( Xa = Mi ) @ ( plus_plus @ nat @ ( times_times @ nat @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary2 ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList2 @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary2 ) ) ) ) ) ) @ Xa ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( vEBT_VEBT_low @ ( if @ nat @ ( Xa = Mi ) @ ( plus_plus @ nat @ ( times_times @ nat @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary2 ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList2 @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary2 ) ) ) ) ) ) @ Xa ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) )
@ Summary2 ) )
@ ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi @ Ma2 ) ) @ ( suc @ ( suc @ Va ) ) @ TreeList2 @ Summary2 ) ) ) ) ) ) )
=> ~ ( accp @ ( product_prod @ vEBT_VEBT @ nat ) @ vEBT_vebt_delete_rel @ ( product_Pair @ vEBT_VEBT @ nat @ ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi @ Ma2 ) ) @ ( suc @ ( suc @ Va ) ) @ TreeList2 @ Summary2 ) @ Xa ) ) ) ) ) ) ) ) ) ) ) ) ).
% vebt_delete.pelims
thf(fact_3741_one__assn__raw_Ocases,axiom,
! [X: product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat )] :
~ ! [H3: heap_ext @ product_unit,As: set @ nat] :
( X
!= ( product_Pair @ ( heap_ext @ product_unit ) @ ( set @ nat ) @ H3 @ As ) ) ).
% one_assn_raw.cases
thf(fact_3742_pure__assn__raw_Ocases,axiom,
! [A: $tType,B: $tType,X: product_prod @ $o @ ( product_prod @ A @ ( set @ B ) )] :
~ ! [B3: $o,H3: A,As: set @ B] :
( X
!= ( product_Pair @ $o @ ( product_prod @ A @ ( set @ B ) ) @ B3 @ ( product_Pair @ A @ ( set @ B ) @ H3 @ As ) ) ) ).
% pure_assn_raw.cases
thf(fact_3743_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,H3: 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 ) @ H3 @ As ) ) ) ) ) ).
% snga_assn_raw.cases
thf(fact_3744_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,H3: 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 ) @ H3 @ As ) ) ) ) ) ).
% sngr_assn_raw.cases
thf(fact_3745_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 ) ) )] :
~ ! [P7: ( product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ) ) > $o,Q8: ( product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ) ) > $o,H3: 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 ) ) ) @ P7 @ ( 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 ) @ H3 @ As ) ) ) ) ).
% times_assn_raw.cases
thf(fact_3746_T_092_060_094sub_062s_092_060_094sub_062u_092_060_094sub_062c_092_060_094sub_062c_H_Opelims,axiom,
! [X: vEBT_VEBT,Xa: nat,Y: nat] :
( ( ( vEBT_T_s_u_c_c2 @ X @ Xa )
= Y )
=> ( ( accp @ ( product_prod @ vEBT_VEBT @ nat ) @ vEBT_T_s_u_c_c_rel @ ( product_Pair @ vEBT_VEBT @ nat @ X @ Xa ) )
=> ( ! [Uu2: $o,B3: $o] :
( ( X
= ( vEBT_Leaf @ Uu2 @ 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 @ Uu2 @ B3 ) @ ( zero_zero @ nat ) ) ) ) ) )
=> ( ! [Uv2: $o,Uw2: $o] :
( ( X
= ( vEBT_Leaf @ Uv2 @ Uw2 ) )
=> ! [N2: nat] :
( ( Xa
= ( suc @ N2 ) )
=> ( ( Y
= ( one_one @ nat ) )
=> ~ ( accp @ ( product_prod @ vEBT_VEBT @ nat ) @ vEBT_T_s_u_c_c_rel @ ( product_Pair @ vEBT_VEBT @ nat @ ( vEBT_Leaf @ Uv2 @ Uw2 ) @ ( suc @ N2 ) ) ) ) ) )
=> ( ! [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 ) ) ) )
=> ( ! [V4: product_prod @ nat @ nat,Vc: list @ vEBT_VEBT,Vd2: vEBT_VEBT] :
( ( X
= ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ V4 ) @ ( zero_zero @ nat ) @ Vc @ Vd2 ) )
=> ( ( Y
= ( one_one @ nat ) )
=> ~ ( accp @ ( product_prod @ vEBT_VEBT @ nat ) @ vEBT_T_s_u_c_c_rel @ ( product_Pair @ vEBT_VEBT @ nat @ ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ V4 ) @ ( zero_zero @ nat ) @ Vc @ Vd2 ) @ Xa ) ) ) )
=> ( ! [V4: product_prod @ nat @ nat,Vg2: list @ vEBT_VEBT,Vh2: vEBT_VEBT] :
( ( X
= ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ V4 ) @ ( suc @ ( zero_zero @ nat ) ) @ Vg2 @ Vh2 ) )
=> ( ( Y
= ( one_one @ nat ) )
=> ~ ( accp @ ( product_prod @ vEBT_VEBT @ nat ) @ vEBT_T_s_u_c_c_rel @ ( product_Pair @ vEBT_VEBT @ nat @ ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ V4 ) @ ( suc @ ( zero_zero @ nat ) ) @ Vg2 @ Vh2 ) @ Xa ) ) ) )
=> ~ ! [Mi: nat,Ma2: nat,Va: nat,TreeList2: list @ vEBT_VEBT,Summary2: vEBT_VEBT] :
( ( X
= ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi @ Ma2 ) ) @ ( suc @ ( suc @ Va ) ) @ TreeList2 @ Summary2 ) )
=> ( ( ( ( ord_less @ nat @ Xa @ Mi )
=> ( Y
= ( one_one @ nat ) ) )
& ( ~ ( ord_less @ nat @ Xa @ Mi )
=> ( Y
= ( if @ nat @ ( ord_less @ nat @ ( vEBT_VEBT_high @ Xa @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( size_size @ ( list @ vEBT_VEBT ) @ TreeList2 ) )
@ ( if @ nat
@ ( ( ( vEBT_vebt_maxt @ ( nth @ vEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ Xa @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) )
!= ( none @ nat ) )
& ( vEBT_VEBT_less @ ( some @ nat @ ( vEBT_VEBT_low @ Xa @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( vEBT_vebt_maxt @ ( nth @ vEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ Xa @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) ) )
@ ( plus_plus @ nat @ ( one_one @ nat ) @ ( vEBT_T_s_u_c_c2 @ ( nth @ vEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ Xa @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( vEBT_VEBT_low @ Xa @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) )
@ ( plus_plus @ nat @ ( vEBT_T_s_u_c_c2 @ Summary2 @ ( vEBT_VEBT_high @ Xa @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( one_one @ nat ) ) )
@ ( one_one @ nat ) ) ) ) )
=> ~ ( accp @ ( product_prod @ vEBT_VEBT @ nat ) @ vEBT_T_s_u_c_c_rel @ ( product_Pair @ vEBT_VEBT @ nat @ ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi @ Ma2 ) ) @ ( suc @ ( suc @ Va ) ) @ TreeList2 @ Summary2 ) @ Xa ) ) ) ) ) ) ) ) ) ) ) ).
% T\<^sub>s\<^sub>u\<^sub>c\<^sub>c'.pelims
thf(fact_3747_T_092_060_094sub_062i_092_060_094sub_062n_092_060_094sub_062s_092_060_094sub_062e_092_060_094sub_062r_092_060_094sub_062t_Opelims,axiom,
! [X: vEBT_VEBT,Xa: nat,Y: nat] :
( ( ( vEBT_T_i_n_s_e_r_t @ X @ Xa )
= Y )
=> ( ( accp @ ( product_prod @ vEBT_VEBT @ nat ) @ vEBT_T9217963907923527482_t_rel @ ( product_Pair @ vEBT_VEBT @ nat @ X @ Xa ) )
=> ( ! [A6: $o,B3: $o] :
( ( X
= ( vEBT_Leaf @ A6 @ 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 @ A6 @ 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 ) ) ) )
=> ( ! [V4: nat,TreeList2: list @ vEBT_VEBT,Summary2: vEBT_VEBT] :
( ( X
= ( vEBT_Node @ ( none @ ( product_prod @ nat @ nat ) ) @ ( suc @ ( suc @ V4 ) ) @ TreeList2 @ Summary2 ) )
=> ( ( Y
= ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) )
=> ~ ( accp @ ( product_prod @ vEBT_VEBT @ nat ) @ vEBT_T9217963907923527482_t_rel @ ( product_Pair @ vEBT_VEBT @ nat @ ( vEBT_Node @ ( none @ ( product_prod @ nat @ nat ) ) @ ( suc @ ( suc @ V4 ) ) @ TreeList2 @ Summary2 ) @ Xa ) ) ) )
=> ~ ! [Mi: nat,Ma2: nat,Va: nat,TreeList2: list @ vEBT_VEBT,Summary2: vEBT_VEBT] :
( ( X
= ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi @ Ma2 ) ) @ ( suc @ ( suc @ Va ) ) @ TreeList2 @ Summary2 ) )
=> ( ( Y
= ( plus_plus @ nat @ ( numeral_numeral @ nat @ ( bit1 @ ( bit1 @ ( bit0 @ ( bit0 @ one2 ) ) ) ) )
@ ( if @ nat
@ ( ( ord_less @ nat @ ( vEBT_VEBT_high @ ( if @ nat @ ( ord_less @ nat @ Xa @ Mi ) @ Mi @ Xa ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( size_size @ ( list @ vEBT_VEBT ) @ TreeList2 ) )
& ~ ( ( Xa = Mi )
| ( 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 @ Mi ) @ Mi @ Xa ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( vEBT_VEBT_low @ ( if @ nat @ ( ord_less @ nat @ Xa @ Mi ) @ Mi @ Xa ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( vEBT_T_m_i_n_N_u_l_l @ ( nth @ vEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ ( if @ nat @ ( ord_less @ nat @ Xa @ Mi ) @ Mi @ Xa ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) ) @ ( if @ nat @ ( vEBT_VEBT_minNull @ ( nth @ vEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ ( if @ nat @ ( ord_less @ nat @ Xa @ Mi ) @ Mi @ Xa ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) @ ( vEBT_T_i_n_s_e_r_t @ Summary2 @ ( vEBT_VEBT_high @ ( if @ nat @ ( ord_less @ nat @ Xa @ Mi ) @ Mi @ Xa ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( one_one @ nat ) ) )
@ ( one_one @ nat ) ) ) )
=> ~ ( accp @ ( product_prod @ vEBT_VEBT @ nat ) @ vEBT_T9217963907923527482_t_rel @ ( product_Pair @ vEBT_VEBT @ nat @ ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi @ Ma2 ) ) @ ( suc @ ( suc @ Va ) ) @ TreeList2 @ Summary2 ) @ Xa ) ) ) ) ) ) ) ) ) ) ).
% T\<^sub>i\<^sub>n\<^sub>s\<^sub>e\<^sub>r\<^sub>t.pelims
thf(fact_3748_T_092_060_094sub_062p_092_060_094sub_062r_092_060_094sub_062e_092_060_094sub_062d_H_Opelims,axiom,
! [X: vEBT_VEBT,Xa: nat,Y: nat] :
( ( ( vEBT_T_p_r_e_d2 @ X @ Xa )
= Y )
=> ( ( accp @ ( product_prod @ vEBT_VEBT @ nat ) @ vEBT_T_p_r_e_d_rel @ ( product_Pair @ vEBT_VEBT @ nat @ X @ Xa ) )
=> ( ! [Uu2: $o,Uv2: $o] :
( ( X
= ( vEBT_Leaf @ Uu2 @ Uv2 ) )
=> ( ( Xa
= ( zero_zero @ nat ) )
=> ( ( Y
= ( one_one @ nat ) )
=> ~ ( accp @ ( product_prod @ vEBT_VEBT @ nat ) @ vEBT_T_p_r_e_d_rel @ ( product_Pair @ vEBT_VEBT @ nat @ ( vEBT_Leaf @ Uu2 @ Uv2 ) @ ( zero_zero @ nat ) ) ) ) ) )
=> ( ! [A6: $o,Uw2: $o] :
( ( X
= ( vEBT_Leaf @ A6 @ Uw2 ) )
=> ( ( Xa
= ( suc @ ( zero_zero @ nat ) ) )
=> ( ( Y
= ( one_one @ nat ) )
=> ~ ( accp @ ( product_prod @ vEBT_VEBT @ nat ) @ vEBT_T_p_r_e_d_rel @ ( product_Pair @ vEBT_VEBT @ nat @ ( vEBT_Leaf @ A6 @ Uw2 ) @ ( suc @ ( zero_zero @ nat ) ) ) ) ) ) )
=> ( ! [A6: $o,B3: $o] :
( ( X
= ( vEBT_Leaf @ A6 @ B3 ) )
=> ! [Va: nat] :
( ( Xa
= ( suc @ ( suc @ Va ) ) )
=> ( ( Y
= ( one_one @ nat ) )
=> ~ ( accp @ ( product_prod @ vEBT_VEBT @ nat ) @ vEBT_T_p_r_e_d_rel @ ( product_Pair @ vEBT_VEBT @ nat @ ( vEBT_Leaf @ A6 @ B3 ) @ ( suc @ ( suc @ Va ) ) ) ) ) ) )
=> ( ! [Uy2: nat,Uz2: list @ vEBT_VEBT,Va3: vEBT_VEBT] :
( ( X
= ( vEBT_Node @ ( none @ ( product_prod @ nat @ nat ) ) @ Uy2 @ Uz2 @ Va3 ) )
=> ( ( Y
= ( one_one @ nat ) )
=> ~ ( accp @ ( product_prod @ vEBT_VEBT @ nat ) @ vEBT_T_p_r_e_d_rel @ ( product_Pair @ vEBT_VEBT @ nat @ ( vEBT_Node @ ( none @ ( product_prod @ nat @ nat ) ) @ Uy2 @ Uz2 @ Va3 ) @ Xa ) ) ) )
=> ( ! [V4: product_prod @ nat @ nat,Vd2: list @ vEBT_VEBT,Ve2: vEBT_VEBT] :
( ( X
= ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ V4 ) @ ( zero_zero @ nat ) @ Vd2 @ Ve2 ) )
=> ( ( Y
= ( one_one @ nat ) )
=> ~ ( accp @ ( product_prod @ vEBT_VEBT @ nat ) @ vEBT_T_p_r_e_d_rel @ ( product_Pair @ vEBT_VEBT @ nat @ ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ V4 ) @ ( zero_zero @ nat ) @ Vd2 @ Ve2 ) @ Xa ) ) ) )
=> ( ! [V4: product_prod @ nat @ nat,Vh2: list @ vEBT_VEBT,Vi2: vEBT_VEBT] :
( ( X
= ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ V4 ) @ ( suc @ ( zero_zero @ nat ) ) @ Vh2 @ Vi2 ) )
=> ( ( Y
= ( one_one @ nat ) )
=> ~ ( accp @ ( product_prod @ vEBT_VEBT @ nat ) @ vEBT_T_p_r_e_d_rel @ ( product_Pair @ vEBT_VEBT @ nat @ ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ V4 ) @ ( suc @ ( zero_zero @ nat ) ) @ Vh2 @ Vi2 ) @ Xa ) ) ) )
=> ~ ! [Mi: nat,Ma2: nat,Va: nat,TreeList2: list @ vEBT_VEBT,Summary2: vEBT_VEBT] :
( ( X
= ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi @ Ma2 ) ) @ ( suc @ ( suc @ Va ) ) @ TreeList2 @ Summary2 ) )
=> ( ( ( ( ord_less @ nat @ Ma2 @ Xa )
=> ( Y
= ( one_one @ nat ) ) )
& ( ~ ( ord_less @ nat @ Ma2 @ Xa )
=> ( Y
= ( if @ nat @ ( ord_less @ nat @ ( vEBT_VEBT_high @ Xa @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( size_size @ ( list @ vEBT_VEBT ) @ TreeList2 ) )
@ ( if @ nat
@ ( ( ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ Xa @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) )
!= ( none @ nat ) )
& ( vEBT_VEBT_greater @ ( some @ nat @ ( vEBT_VEBT_low @ Xa @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ Xa @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) ) )
@ ( plus_plus @ nat @ ( one_one @ nat ) @ ( vEBT_T_p_r_e_d2 @ ( nth @ vEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ Xa @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( vEBT_VEBT_low @ Xa @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) )
@ ( plus_plus @ nat @ ( vEBT_T_p_r_e_d2 @ Summary2 @ ( vEBT_VEBT_high @ Xa @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( one_one @ nat ) ) )
@ ( one_one @ nat ) ) ) ) )
=> ~ ( accp @ ( product_prod @ vEBT_VEBT @ nat ) @ vEBT_T_p_r_e_d_rel @ ( product_Pair @ vEBT_VEBT @ nat @ ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi @ Ma2 ) ) @ ( suc @ ( suc @ Va ) ) @ TreeList2 @ Summary2 ) @ Xa ) ) ) ) ) ) ) ) ) ) ) ) ).
% T\<^sub>p\<^sub>r\<^sub>e\<^sub>d'.pelims
thf(fact_3749_vebt__insert_Opelims,axiom,
! [X: vEBT_VEBT,Xa: nat,Y: vEBT_VEBT] :
( ( ( vEBT_vebt_insert @ X @ Xa )
= Y )
=> ( ( accp @ ( product_prod @ vEBT_VEBT @ nat ) @ vEBT_vebt_insert_rel @ ( product_Pair @ vEBT_VEBT @ nat @ X @ Xa ) )
=> ( ! [A6: $o,B3: $o] :
( ( X
= ( vEBT_Leaf @ A6 @ B3 ) )
=> ( ( ( ( Xa
= ( zero_zero @ nat ) )
=> ( Y
= ( vEBT_Leaf @ $true @ B3 ) ) )
& ( ( Xa
!= ( zero_zero @ nat ) )
=> ( ( ( Xa
= ( one_one @ nat ) )
=> ( Y
= ( vEBT_Leaf @ A6 @ $true ) ) )
& ( ( Xa
!= ( one_one @ nat ) )
=> ( Y
= ( vEBT_Leaf @ A6 @ B3 ) ) ) ) ) )
=> ~ ( accp @ ( product_prod @ vEBT_VEBT @ nat ) @ vEBT_vebt_insert_rel @ ( product_Pair @ vEBT_VEBT @ nat @ ( vEBT_Leaf @ A6 @ 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 ) ) ) )
=> ( ! [V4: nat,TreeList2: list @ vEBT_VEBT,Summary2: vEBT_VEBT] :
( ( X
= ( vEBT_Node @ ( none @ ( product_prod @ nat @ nat ) ) @ ( suc @ ( suc @ V4 ) ) @ TreeList2 @ Summary2 ) )
=> ( ( Y
= ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Xa @ Xa ) ) @ ( suc @ ( suc @ V4 ) ) @ TreeList2 @ Summary2 ) )
=> ~ ( accp @ ( product_prod @ vEBT_VEBT @ nat ) @ vEBT_vebt_insert_rel @ ( product_Pair @ vEBT_VEBT @ nat @ ( vEBT_Node @ ( none @ ( product_prod @ nat @ nat ) ) @ ( suc @ ( suc @ V4 ) ) @ TreeList2 @ Summary2 ) @ Xa ) ) ) )
=> ~ ! [Mi: nat,Ma2: nat,Va: nat,TreeList2: list @ vEBT_VEBT,Summary2: vEBT_VEBT] :
( ( X
= ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi @ Ma2 ) ) @ ( suc @ ( suc @ Va ) ) @ TreeList2 @ Summary2 ) )
=> ( ( Y
= ( if @ vEBT_VEBT
@ ( ( ord_less @ nat @ ( vEBT_VEBT_high @ ( if @ nat @ ( ord_less @ nat @ Xa @ Mi ) @ Mi @ Xa ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( size_size @ ( list @ vEBT_VEBT ) @ TreeList2 ) )
& ~ ( ( Xa = Mi )
| ( Xa = Ma2 ) ) )
@ ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ ( if @ nat @ ( ord_less @ nat @ Xa @ Mi ) @ Xa @ Mi ) @ ( ord_max @ nat @ ( if @ nat @ ( ord_less @ nat @ Xa @ Mi ) @ Mi @ Xa ) @ Ma2 ) ) ) @ ( suc @ ( suc @ Va ) ) @ ( list_update @ vEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ ( if @ nat @ ( ord_less @ nat @ Xa @ Mi ) @ Mi @ Xa ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( vEBT_vebt_insert @ ( nth @ vEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ ( if @ nat @ ( ord_less @ nat @ Xa @ Mi ) @ Mi @ Xa ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( vEBT_VEBT_low @ ( if @ nat @ ( ord_less @ nat @ Xa @ Mi ) @ Mi @ Xa ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) @ ( if @ vEBT_VEBT @ ( vEBT_VEBT_minNull @ ( nth @ vEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ ( if @ nat @ ( ord_less @ nat @ Xa @ Mi ) @ Mi @ Xa ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) @ ( vEBT_vebt_insert @ Summary2 @ ( vEBT_VEBT_high @ ( if @ nat @ ( ord_less @ nat @ Xa @ Mi ) @ Mi @ Xa ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ Summary2 ) )
@ ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi @ Ma2 ) ) @ ( suc @ ( suc @ Va ) ) @ TreeList2 @ Summary2 ) ) )
=> ~ ( accp @ ( product_prod @ vEBT_VEBT @ nat ) @ vEBT_vebt_insert_rel @ ( product_Pair @ vEBT_VEBT @ nat @ ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi @ Ma2 ) ) @ ( suc @ ( suc @ Va ) ) @ TreeList2 @ Summary2 ) @ Xa ) ) ) ) ) ) ) ) ) ) ).
% vebt_insert.pelims
thf(fact_3750_T_092_060_094sub_062i_092_060_094sub_062n_092_060_094sub_062s_092_060_094sub_062e_092_060_094sub_062r_092_060_094sub_062t_H_Opelims,axiom,
! [X: vEBT_VEBT,Xa: nat,Y: nat] :
( ( ( vEBT_T_i_n_s_e_r_t2 @ X @ Xa )
= Y )
=> ( ( accp @ ( product_prod @ vEBT_VEBT @ nat ) @ vEBT_T5076183648494686801_t_rel @ ( product_Pair @ vEBT_VEBT @ nat @ X @ Xa ) )
=> ( ! [A6: $o,B3: $o] :
( ( X
= ( vEBT_Leaf @ A6 @ B3 ) )
=> ( ( Y
= ( one_one @ nat ) )
=> ~ ( accp @ ( product_prod @ vEBT_VEBT @ nat ) @ vEBT_T5076183648494686801_t_rel @ ( product_Pair @ vEBT_VEBT @ nat @ ( vEBT_Leaf @ A6 @ 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 ) ) ) )
=> ( ! [V4: nat,TreeList2: list @ vEBT_VEBT,Summary2: vEBT_VEBT] :
( ( X
= ( vEBT_Node @ ( none @ ( product_prod @ nat @ nat ) ) @ ( suc @ ( suc @ V4 ) ) @ TreeList2 @ Summary2 ) )
=> ( ( Y
= ( one_one @ nat ) )
=> ~ ( accp @ ( product_prod @ vEBT_VEBT @ nat ) @ vEBT_T5076183648494686801_t_rel @ ( product_Pair @ vEBT_VEBT @ nat @ ( vEBT_Node @ ( none @ ( product_prod @ nat @ nat ) ) @ ( suc @ ( suc @ V4 ) ) @ TreeList2 @ Summary2 ) @ Xa ) ) ) )
=> ~ ! [Mi: nat,Ma2: nat,Va: nat,TreeList2: list @ vEBT_VEBT,Summary2: vEBT_VEBT] :
( ( X
= ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi @ Ma2 ) ) @ ( suc @ ( suc @ Va ) ) @ TreeList2 @ Summary2 ) )
=> ( ( Y
= ( if @ nat
@ ( ( ord_less @ nat @ ( vEBT_VEBT_high @ ( if @ nat @ ( ord_less @ nat @ Xa @ Mi ) @ Mi @ Xa ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( size_size @ ( list @ vEBT_VEBT ) @ TreeList2 ) )
& ~ ( ( Xa = Mi )
| ( 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 @ Mi ) @ Mi @ Xa ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( vEBT_VEBT_low @ ( if @ nat @ ( ord_less @ nat @ Xa @ Mi ) @ Mi @ Xa ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( if @ nat @ ( vEBT_VEBT_minNull @ ( nth @ vEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ ( if @ nat @ ( ord_less @ nat @ Xa @ Mi ) @ Mi @ Xa ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) @ ( vEBT_T_i_n_s_e_r_t2 @ Summary2 @ ( vEBT_VEBT_high @ ( if @ nat @ ( ord_less @ nat @ Xa @ Mi ) @ Mi @ Xa ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( one_one @ nat ) ) )
@ ( one_one @ nat ) ) )
=> ~ ( accp @ ( product_prod @ vEBT_VEBT @ nat ) @ vEBT_T5076183648494686801_t_rel @ ( product_Pair @ vEBT_VEBT @ nat @ ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi @ Ma2 ) ) @ ( suc @ ( suc @ Va ) ) @ TreeList2 @ Summary2 ) @ Xa ) ) ) ) ) ) ) ) ) ) ).
% T\<^sub>i\<^sub>n\<^sub>s\<^sub>e\<^sub>r\<^sub>t'.pelims
thf(fact_3751_T_092_060_094sub_062m_092_060_094sub_062e_092_060_094sub_062m_092_060_094sub_062b_092_060_094sub_062e_092_060_094sub_062r_Opelims,axiom,
! [X: vEBT_VEBT,Xa: nat,Y: nat] :
( ( ( vEBT_T_m_e_m_b_e_r @ X @ Xa )
= Y )
=> ( ( accp @ ( product_prod @ vEBT_VEBT @ nat ) @ vEBT_T5837161174952499735_r_rel @ ( product_Pair @ vEBT_VEBT @ nat @ X @ Xa ) )
=> ( ! [A6: $o,B3: $o] :
( ( X
= ( vEBT_Leaf @ A6 @ 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 @ A6 @ B3 ) @ Xa ) ) ) )
=> ( ! [Uu2: nat,Uv2: list @ vEBT_VEBT,Uw2: vEBT_VEBT] :
( ( X
= ( vEBT_Node @ ( none @ ( product_prod @ nat @ nat ) ) @ Uu2 @ Uv2 @ Uw2 ) )
=> ( ( Y
= ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) )
=> ~ ( accp @ ( product_prod @ vEBT_VEBT @ nat ) @ vEBT_T5837161174952499735_r_rel @ ( product_Pair @ vEBT_VEBT @ nat @ ( vEBT_Node @ ( none @ ( product_prod @ nat @ nat ) ) @ Uu2 @ Uv2 @ Uw2 ) @ Xa ) ) ) )
=> ( ! [V4: product_prod @ nat @ nat,Uy2: list @ vEBT_VEBT,Uz2: vEBT_VEBT] :
( ( X
= ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ V4 ) @ ( 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 ) @ V4 ) @ ( zero_zero @ nat ) @ Uy2 @ Uz2 ) @ Xa ) ) ) )
=> ( ! [V4: product_prod @ nat @ nat,Vb2: list @ vEBT_VEBT,Vc: vEBT_VEBT] :
( ( X
= ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ V4 ) @ ( suc @ ( zero_zero @ nat ) ) @ Vb2 @ Vc ) )
=> ( ( Y
= ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) )
=> ~ ( accp @ ( product_prod @ vEBT_VEBT @ nat ) @ vEBT_T5837161174952499735_r_rel @ ( product_Pair @ vEBT_VEBT @ nat @ ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ V4 ) @ ( suc @ ( zero_zero @ nat ) ) @ Vb2 @ Vc ) @ Xa ) ) ) )
=> ~ ! [Mi: nat,Ma2: nat,Va: nat,TreeList2: list @ vEBT_VEBT,Summary2: vEBT_VEBT] :
( ( X
= ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi @ Ma2 ) ) @ ( suc @ ( suc @ Va ) ) @ TreeList2 @ Summary2 ) )
=> ( ( Y
= ( plus_plus @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( if @ nat @ ( Xa = Mi ) @ ( 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 @ Mi ) @ ( one_one @ nat ) @ ( plus_plus @ nat @ ( one_one @ nat ) @ ( if @ nat @ ( ord_less @ nat @ Ma2 @ Xa ) @ ( one_one @ nat ) @ ( plus_plus @ nat @ ( numeral_numeral @ nat @ ( bit1 @ ( bit0 @ ( bit0 @ one2 ) ) ) ) @ ( if @ nat @ ( ord_less @ nat @ ( vEBT_VEBT_high @ Xa @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( size_size @ ( list @ vEBT_VEBT ) @ TreeList2 ) ) @ ( plus_plus @ nat @ ( one_one @ nat ) @ ( vEBT_T_m_e_m_b_e_r @ ( nth @ vEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ Xa @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( vEBT_VEBT_low @ Xa @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) @ ( one_one @ nat ) ) ) ) ) ) ) ) ) ) ) )
=> ~ ( accp @ ( product_prod @ vEBT_VEBT @ nat ) @ vEBT_T5837161174952499735_r_rel @ ( product_Pair @ vEBT_VEBT @ nat @ ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi @ Ma2 ) ) @ ( suc @ ( suc @ Va ) ) @ TreeList2 @ Summary2 ) @ Xa ) ) ) ) ) ) ) ) ) ) ).
% T\<^sub>m\<^sub>e\<^sub>m\<^sub>b\<^sub>e\<^sub>r.pelims
thf(fact_3752_T_092_060_094sub_062m_092_060_094sub_062e_092_060_094sub_062m_092_060_094sub_062b_092_060_094sub_062e_092_060_094sub_062r_H_Opelims,axiom,
! [X: vEBT_VEBT,Xa: nat,Y: nat] :
( ( ( vEBT_T_m_e_m_b_e_r2 @ X @ Xa )
= Y )
=> ( ( accp @ ( product_prod @ vEBT_VEBT @ nat ) @ vEBT_T8099345112685741742_r_rel @ ( product_Pair @ vEBT_VEBT @ nat @ X @ Xa ) )
=> ( ! [A6: $o,B3: $o] :
( ( X
= ( vEBT_Leaf @ A6 @ B3 ) )
=> ( ( Y
= ( one_one @ nat ) )
=> ~ ( accp @ ( product_prod @ vEBT_VEBT @ nat ) @ vEBT_T8099345112685741742_r_rel @ ( product_Pair @ vEBT_VEBT @ nat @ ( vEBT_Leaf @ A6 @ B3 ) @ Xa ) ) ) )
=> ( ! [Uu2: nat,Uv2: list @ vEBT_VEBT,Uw2: vEBT_VEBT] :
( ( X
= ( vEBT_Node @ ( none @ ( product_prod @ nat @ nat ) ) @ Uu2 @ Uv2 @ Uw2 ) )
=> ( ( Y
= ( one_one @ nat ) )
=> ~ ( accp @ ( product_prod @ vEBT_VEBT @ nat ) @ vEBT_T8099345112685741742_r_rel @ ( product_Pair @ vEBT_VEBT @ nat @ ( vEBT_Node @ ( none @ ( product_prod @ nat @ nat ) ) @ Uu2 @ Uv2 @ Uw2 ) @ Xa ) ) ) )
=> ( ! [V4: product_prod @ nat @ nat,Uy2: list @ vEBT_VEBT,Uz2: vEBT_VEBT] :
( ( X
= ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ V4 ) @ ( 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 ) @ V4 ) @ ( zero_zero @ nat ) @ Uy2 @ Uz2 ) @ Xa ) ) ) )
=> ( ! [V4: product_prod @ nat @ nat,Vb2: list @ vEBT_VEBT,Vc: vEBT_VEBT] :
( ( X
= ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ V4 ) @ ( suc @ ( zero_zero @ nat ) ) @ Vb2 @ Vc ) )
=> ( ( Y
= ( one_one @ nat ) )
=> ~ ( accp @ ( product_prod @ vEBT_VEBT @ nat ) @ vEBT_T8099345112685741742_r_rel @ ( product_Pair @ vEBT_VEBT @ nat @ ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ V4 ) @ ( suc @ ( zero_zero @ nat ) ) @ Vb2 @ Vc ) @ Xa ) ) ) )
=> ~ ! [Mi: nat,Ma2: nat,Va: nat,TreeList2: list @ vEBT_VEBT,Summary2: vEBT_VEBT] :
( ( X
= ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi @ Ma2 ) ) @ ( suc @ ( suc @ Va ) ) @ TreeList2 @ Summary2 ) )
=> ( ( Y
= ( plus_plus @ nat @ ( one_one @ nat )
@ ( if @ nat @ ( Xa = Mi ) @ ( zero_zero @ nat )
@ ( if @ nat @ ( Xa = Ma2 ) @ ( zero_zero @ nat )
@ ( if @ nat @ ( ord_less @ nat @ Xa @ Mi ) @ ( zero_zero @ nat )
@ ( if @ nat @ ( ord_less @ nat @ Ma2 @ Xa ) @ ( zero_zero @ nat )
@ ( if @ nat
@ ( ( ord_less @ nat @ Mi @ Xa )
& ( ord_less @ nat @ Xa @ Ma2 ) )
@ ( if @ nat @ ( ord_less @ nat @ ( vEBT_VEBT_high @ Xa @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( size_size @ ( list @ vEBT_VEBT ) @ TreeList2 ) ) @ ( vEBT_T_m_e_m_b_e_r2 @ ( nth @ vEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ Xa @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( vEBT_VEBT_low @ Xa @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( zero_zero @ nat ) )
@ ( zero_zero @ nat ) ) ) ) ) ) ) )
=> ~ ( accp @ ( product_prod @ vEBT_VEBT @ nat ) @ vEBT_T8099345112685741742_r_rel @ ( product_Pair @ vEBT_VEBT @ nat @ ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi @ Ma2 ) ) @ ( suc @ ( suc @ Va ) ) @ TreeList2 @ Summary2 ) @ Xa ) ) ) ) ) ) ) ) ) ) ).
% T\<^sub>m\<^sub>e\<^sub>m\<^sub>b\<^sub>e\<^sub>r'.pelims
thf(fact_3753_vebt__member_Opelims_I3_J,axiom,
! [X: vEBT_VEBT,Xa: nat] :
( ~ ( vEBT_vebt_member @ X @ Xa )
=> ( ( accp @ ( product_prod @ vEBT_VEBT @ nat ) @ vEBT_vebt_member_rel @ ( product_Pair @ vEBT_VEBT @ nat @ X @ Xa ) )
=> ( ! [A6: $o,B3: $o] :
( ( X
= ( vEBT_Leaf @ A6 @ B3 ) )
=> ( ( accp @ ( product_prod @ vEBT_VEBT @ nat ) @ vEBT_vebt_member_rel @ ( product_Pair @ vEBT_VEBT @ nat @ ( vEBT_Leaf @ A6 @ B3 ) @ Xa ) )
=> ( ( ( Xa
= ( zero_zero @ nat ) )
=> A6 )
& ( ( Xa
!= ( zero_zero @ nat ) )
=> ( ( ( Xa
= ( one_one @ nat ) )
=> B3 )
& ( Xa
= ( one_one @ nat ) ) ) ) ) ) )
=> ( ! [Uu2: nat,Uv2: list @ vEBT_VEBT,Uw2: vEBT_VEBT] :
( ( X
= ( vEBT_Node @ ( none @ ( product_prod @ nat @ nat ) ) @ Uu2 @ Uv2 @ Uw2 ) )
=> ~ ( accp @ ( product_prod @ vEBT_VEBT @ nat ) @ vEBT_vebt_member_rel @ ( product_Pair @ vEBT_VEBT @ nat @ ( vEBT_Node @ ( none @ ( product_prod @ nat @ nat ) ) @ Uu2 @ Uv2 @ Uw2 ) @ Xa ) ) )
=> ( ! [V4: product_prod @ nat @ nat,Uy2: list @ vEBT_VEBT,Uz2: vEBT_VEBT] :
( ( X
= ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ V4 ) @ ( zero_zero @ nat ) @ Uy2 @ Uz2 ) )
=> ~ ( accp @ ( product_prod @ vEBT_VEBT @ nat ) @ vEBT_vebt_member_rel @ ( product_Pair @ vEBT_VEBT @ nat @ ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ V4 ) @ ( zero_zero @ nat ) @ Uy2 @ Uz2 ) @ Xa ) ) )
=> ( ! [V4: product_prod @ nat @ nat,Vb2: list @ vEBT_VEBT,Vc: vEBT_VEBT] :
( ( X
= ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ V4 ) @ ( suc @ ( zero_zero @ nat ) ) @ Vb2 @ Vc ) )
=> ~ ( accp @ ( product_prod @ vEBT_VEBT @ nat ) @ vEBT_vebt_member_rel @ ( product_Pair @ vEBT_VEBT @ nat @ ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ V4 ) @ ( suc @ ( zero_zero @ nat ) ) @ Vb2 @ Vc ) @ Xa ) ) )
=> ~ ! [Mi: nat,Ma2: nat,Va: nat,TreeList2: list @ vEBT_VEBT,Summary2: vEBT_VEBT] :
( ( X
= ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi @ Ma2 ) ) @ ( suc @ ( suc @ Va ) ) @ TreeList2 @ Summary2 ) )
=> ( ( accp @ ( product_prod @ vEBT_VEBT @ nat ) @ vEBT_vebt_member_rel @ ( product_Pair @ vEBT_VEBT @ nat @ ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi @ Ma2 ) ) @ ( suc @ ( suc @ Va ) ) @ TreeList2 @ Summary2 ) @ Xa ) )
=> ( ( Xa != Mi )
=> ( ( Xa != Ma2 )
=> ( ~ ( ord_less @ nat @ Xa @ Mi )
& ( ~ ( ord_less @ nat @ Xa @ Mi )
=> ( ~ ( ord_less @ nat @ Ma2 @ Xa )
& ( ~ ( ord_less @ nat @ Ma2 @ Xa )
=> ( ( ( ord_less @ nat @ ( vEBT_VEBT_high @ Xa @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( size_size @ ( list @ vEBT_VEBT ) @ TreeList2 ) )
=> ( vEBT_vebt_member @ ( nth @ vEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ Xa @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( vEBT_VEBT_low @ Xa @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) )
& ( ord_less @ nat @ ( vEBT_VEBT_high @ Xa @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( size_size @ ( list @ vEBT_VEBT ) @ TreeList2 ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).
% vebt_member.pelims(3)
thf(fact_3754_vebt__member_Opelims_I2_J,axiom,
! [X: vEBT_VEBT,Xa: nat] :
( ( vEBT_vebt_member @ X @ Xa )
=> ( ( accp @ ( product_prod @ vEBT_VEBT @ nat ) @ vEBT_vebt_member_rel @ ( product_Pair @ vEBT_VEBT @ nat @ X @ Xa ) )
=> ( ! [A6: $o,B3: $o] :
( ( X
= ( vEBT_Leaf @ A6 @ B3 ) )
=> ( ( accp @ ( product_prod @ vEBT_VEBT @ nat ) @ vEBT_vebt_member_rel @ ( product_Pair @ vEBT_VEBT @ nat @ ( vEBT_Leaf @ A6 @ B3 ) @ Xa ) )
=> ~ ( ( ( Xa
= ( zero_zero @ nat ) )
=> A6 )
& ( ( Xa
!= ( zero_zero @ nat ) )
=> ( ( ( Xa
= ( one_one @ nat ) )
=> B3 )
& ( Xa
= ( one_one @ nat ) ) ) ) ) ) )
=> ~ ! [Mi: nat,Ma2: nat,Va: nat,TreeList2: list @ vEBT_VEBT,Summary2: vEBT_VEBT] :
( ( X
= ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi @ Ma2 ) ) @ ( suc @ ( suc @ Va ) ) @ TreeList2 @ Summary2 ) )
=> ( ( accp @ ( product_prod @ vEBT_VEBT @ nat ) @ vEBT_vebt_member_rel @ ( product_Pair @ vEBT_VEBT @ nat @ ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi @ Ma2 ) ) @ ( suc @ ( suc @ Va ) ) @ TreeList2 @ Summary2 ) @ Xa ) )
=> ~ ( ( Xa != Mi )
=> ( ( Xa != Ma2 )
=> ( ~ ( ord_less @ nat @ Xa @ Mi )
& ( ~ ( ord_less @ nat @ Xa @ Mi )
=> ( ~ ( ord_less @ nat @ Ma2 @ Xa )
& ( ~ ( ord_less @ nat @ Ma2 @ Xa )
=> ( ( ( ord_less @ nat @ ( vEBT_VEBT_high @ Xa @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( size_size @ ( list @ vEBT_VEBT ) @ TreeList2 ) )
=> ( vEBT_vebt_member @ ( nth @ vEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ Xa @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( vEBT_VEBT_low @ Xa @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) )
& ( ord_less @ nat @ ( vEBT_VEBT_high @ Xa @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( size_size @ ( list @ vEBT_VEBT ) @ TreeList2 ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).
% vebt_member.pelims(2)
thf(fact_3755_vebt__member_Opelims_I1_J,axiom,
! [X: vEBT_VEBT,Xa: nat,Y: $o] :
( ( ( vEBT_vebt_member @ X @ Xa )
= Y )
=> ( ( accp @ ( product_prod @ vEBT_VEBT @ nat ) @ vEBT_vebt_member_rel @ ( product_Pair @ vEBT_VEBT @ nat @ X @ Xa ) )
=> ( ! [A6: $o,B3: $o] :
( ( X
= ( vEBT_Leaf @ A6 @ B3 ) )
=> ( ( Y
= ( ( ( Xa
= ( zero_zero @ nat ) )
=> A6 )
& ( ( 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 @ A6 @ B3 ) @ Xa ) ) ) )
=> ( ! [Uu2: nat,Uv2: list @ vEBT_VEBT,Uw2: vEBT_VEBT] :
( ( X
= ( vEBT_Node @ ( none @ ( product_prod @ nat @ nat ) ) @ Uu2 @ Uv2 @ Uw2 ) )
=> ( ~ Y
=> ~ ( accp @ ( product_prod @ vEBT_VEBT @ nat ) @ vEBT_vebt_member_rel @ ( product_Pair @ vEBT_VEBT @ nat @ ( vEBT_Node @ ( none @ ( product_prod @ nat @ nat ) ) @ Uu2 @ Uv2 @ Uw2 ) @ Xa ) ) ) )
=> ( ! [V4: product_prod @ nat @ nat,Uy2: list @ vEBT_VEBT,Uz2: vEBT_VEBT] :
( ( X
= ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ V4 ) @ ( 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 ) @ V4 ) @ ( zero_zero @ nat ) @ Uy2 @ Uz2 ) @ Xa ) ) ) )
=> ( ! [V4: product_prod @ nat @ nat,Vb2: list @ vEBT_VEBT,Vc: vEBT_VEBT] :
( ( X
= ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ V4 ) @ ( suc @ ( zero_zero @ nat ) ) @ Vb2 @ Vc ) )
=> ( ~ Y
=> ~ ( accp @ ( product_prod @ vEBT_VEBT @ nat ) @ vEBT_vebt_member_rel @ ( product_Pair @ vEBT_VEBT @ nat @ ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ V4 ) @ ( suc @ ( zero_zero @ nat ) ) @ Vb2 @ Vc ) @ Xa ) ) ) )
=> ~ ! [Mi: nat,Ma2: nat,Va: nat,TreeList2: list @ vEBT_VEBT,Summary2: vEBT_VEBT] :
( ( X
= ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi @ Ma2 ) ) @ ( suc @ ( suc @ Va ) ) @ TreeList2 @ Summary2 ) )
=> ( ( Y
= ( ( Xa != Mi )
=> ( ( Xa != Ma2 )
=> ( ~ ( ord_less @ nat @ Xa @ Mi )
& ( ~ ( ord_less @ nat @ Xa @ Mi )
=> ( ~ ( ord_less @ nat @ Ma2 @ Xa )
& ( ~ ( ord_less @ nat @ Ma2 @ Xa )
=> ( ( ( ord_less @ nat @ ( vEBT_VEBT_high @ Xa @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( size_size @ ( list @ vEBT_VEBT ) @ TreeList2 ) )
=> ( vEBT_vebt_member @ ( nth @ vEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ Xa @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( vEBT_VEBT_low @ Xa @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) )
& ( ord_less @ nat @ ( vEBT_VEBT_high @ Xa @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( size_size @ ( list @ vEBT_VEBT ) @ TreeList2 ) ) ) ) ) ) ) ) ) )
=> ~ ( accp @ ( product_prod @ vEBT_VEBT @ nat ) @ vEBT_vebt_member_rel @ ( product_Pair @ vEBT_VEBT @ nat @ ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi @ Ma2 ) ) @ ( suc @ ( suc @ Va ) ) @ TreeList2 @ Summary2 ) @ Xa ) ) ) ) ) ) ) ) ) ) ).
% vebt_member.pelims(1)
thf(fact_3756_VEBT__internal_Onaive__member_Opelims_I3_J,axiom,
! [X: vEBT_VEBT,Xa: nat] :
( ~ ( vEBT_V5719532721284313246member @ X @ Xa )
=> ( ( accp @ ( product_prod @ vEBT_VEBT @ nat ) @ vEBT_V5765760719290551771er_rel @ ( product_Pair @ vEBT_VEBT @ nat @ X @ Xa ) )
=> ( ! [A6: $o,B3: $o] :
( ( X
= ( vEBT_Leaf @ A6 @ B3 ) )
=> ( ( accp @ ( product_prod @ vEBT_VEBT @ nat ) @ vEBT_V5765760719290551771er_rel @ ( product_Pair @ vEBT_VEBT @ nat @ ( vEBT_Leaf @ A6 @ B3 ) @ Xa ) )
=> ( ( ( Xa
= ( zero_zero @ nat ) )
=> A6 )
& ( ( Xa
!= ( zero_zero @ nat ) )
=> ( ( ( Xa
= ( one_one @ nat ) )
=> B3 )
& ( Xa
= ( one_one @ nat ) ) ) ) ) ) )
=> ( ! [Uu2: option @ ( product_prod @ nat @ nat ),Uv2: list @ vEBT_VEBT,Uw2: vEBT_VEBT] :
( ( X
= ( vEBT_Node @ Uu2 @ ( zero_zero @ nat ) @ Uv2 @ Uw2 ) )
=> ~ ( accp @ ( product_prod @ vEBT_VEBT @ nat ) @ vEBT_V5765760719290551771er_rel @ ( product_Pair @ vEBT_VEBT @ nat @ ( vEBT_Node @ Uu2 @ ( zero_zero @ nat ) @ Uv2 @ Uw2 ) @ Xa ) ) )
=> ~ ! [Uy2: option @ ( product_prod @ nat @ nat ),V4: nat,TreeList2: list @ vEBT_VEBT,S2: vEBT_VEBT] :
( ( X
= ( vEBT_Node @ Uy2 @ ( suc @ V4 ) @ TreeList2 @ S2 ) )
=> ( ( accp @ ( product_prod @ vEBT_VEBT @ nat ) @ vEBT_V5765760719290551771er_rel @ ( product_Pair @ vEBT_VEBT @ nat @ ( vEBT_Node @ Uy2 @ ( suc @ V4 ) @ TreeList2 @ S2 ) @ Xa ) )
=> ( ( ( ord_less @ nat @ ( vEBT_VEBT_high @ Xa @ ( divide_divide @ nat @ ( suc @ V4 ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( size_size @ ( list @ vEBT_VEBT ) @ TreeList2 ) )
=> ( vEBT_V5719532721284313246member @ ( nth @ vEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ Xa @ ( divide_divide @ nat @ ( suc @ V4 ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( vEBT_VEBT_low @ Xa @ ( divide_divide @ nat @ ( suc @ V4 ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) )
& ( ord_less @ nat @ ( vEBT_VEBT_high @ Xa @ ( divide_divide @ nat @ ( suc @ V4 ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( size_size @ ( list @ vEBT_VEBT ) @ TreeList2 ) ) ) ) ) ) ) ) ) ).
% VEBT_internal.naive_member.pelims(3)
thf(fact_3757_VEBT__internal_Onaive__member_Opelims_I2_J,axiom,
! [X: vEBT_VEBT,Xa: nat] :
( ( vEBT_V5719532721284313246member @ X @ Xa )
=> ( ( accp @ ( product_prod @ vEBT_VEBT @ nat ) @ vEBT_V5765760719290551771er_rel @ ( product_Pair @ vEBT_VEBT @ nat @ X @ Xa ) )
=> ( ! [A6: $o,B3: $o] :
( ( X
= ( vEBT_Leaf @ A6 @ B3 ) )
=> ( ( accp @ ( product_prod @ vEBT_VEBT @ nat ) @ vEBT_V5765760719290551771er_rel @ ( product_Pair @ vEBT_VEBT @ nat @ ( vEBT_Leaf @ A6 @ B3 ) @ Xa ) )
=> ~ ( ( ( Xa
= ( zero_zero @ nat ) )
=> A6 )
& ( ( Xa
!= ( zero_zero @ nat ) )
=> ( ( ( Xa
= ( one_one @ nat ) )
=> B3 )
& ( Xa
= ( one_one @ nat ) ) ) ) ) ) )
=> ~ ! [Uy2: option @ ( product_prod @ nat @ nat ),V4: nat,TreeList2: list @ vEBT_VEBT,S2: vEBT_VEBT] :
( ( X
= ( vEBT_Node @ Uy2 @ ( suc @ V4 ) @ TreeList2 @ S2 ) )
=> ( ( accp @ ( product_prod @ vEBT_VEBT @ nat ) @ vEBT_V5765760719290551771er_rel @ ( product_Pair @ vEBT_VEBT @ nat @ ( vEBT_Node @ Uy2 @ ( suc @ V4 ) @ TreeList2 @ S2 ) @ Xa ) )
=> ~ ( ( ( ord_less @ nat @ ( vEBT_VEBT_high @ Xa @ ( divide_divide @ nat @ ( suc @ V4 ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( size_size @ ( list @ vEBT_VEBT ) @ TreeList2 ) )
=> ( vEBT_V5719532721284313246member @ ( nth @ vEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ Xa @ ( divide_divide @ nat @ ( suc @ V4 ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( vEBT_VEBT_low @ Xa @ ( divide_divide @ nat @ ( suc @ V4 ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) )
& ( ord_less @ nat @ ( vEBT_VEBT_high @ Xa @ ( divide_divide @ nat @ ( suc @ V4 ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( size_size @ ( list @ vEBT_VEBT ) @ TreeList2 ) ) ) ) ) ) ) ) ).
% VEBT_internal.naive_member.pelims(2)
thf(fact_3758_VEBT__internal_Onaive__member_Opelims_I1_J,axiom,
! [X: vEBT_VEBT,Xa: nat,Y: $o] :
( ( ( vEBT_V5719532721284313246member @ X @ Xa )
= Y )
=> ( ( accp @ ( product_prod @ vEBT_VEBT @ nat ) @ vEBT_V5765760719290551771er_rel @ ( product_Pair @ vEBT_VEBT @ nat @ X @ Xa ) )
=> ( ! [A6: $o,B3: $o] :
( ( X
= ( vEBT_Leaf @ A6 @ B3 ) )
=> ( ( Y
= ( ( ( Xa
= ( zero_zero @ nat ) )
=> A6 )
& ( ( 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 @ A6 @ B3 ) @ Xa ) ) ) )
=> ( ! [Uu2: option @ ( product_prod @ nat @ nat ),Uv2: list @ vEBT_VEBT,Uw2: vEBT_VEBT] :
( ( X
= ( vEBT_Node @ Uu2 @ ( zero_zero @ nat ) @ Uv2 @ Uw2 ) )
=> ( ~ Y
=> ~ ( accp @ ( product_prod @ vEBT_VEBT @ nat ) @ vEBT_V5765760719290551771er_rel @ ( product_Pair @ vEBT_VEBT @ nat @ ( vEBT_Node @ Uu2 @ ( zero_zero @ nat ) @ Uv2 @ Uw2 ) @ Xa ) ) ) )
=> ~ ! [Uy2: option @ ( product_prod @ nat @ nat ),V4: nat,TreeList2: list @ vEBT_VEBT,S2: vEBT_VEBT] :
( ( X
= ( vEBT_Node @ Uy2 @ ( suc @ V4 ) @ TreeList2 @ S2 ) )
=> ( ( Y
= ( ( ( ord_less @ nat @ ( vEBT_VEBT_high @ Xa @ ( divide_divide @ nat @ ( suc @ V4 ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( size_size @ ( list @ vEBT_VEBT ) @ TreeList2 ) )
=> ( vEBT_V5719532721284313246member @ ( nth @ vEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ Xa @ ( divide_divide @ nat @ ( suc @ V4 ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( vEBT_VEBT_low @ Xa @ ( divide_divide @ nat @ ( suc @ V4 ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) )
& ( ord_less @ nat @ ( vEBT_VEBT_high @ Xa @ ( divide_divide @ nat @ ( suc @ V4 ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( size_size @ ( list @ vEBT_VEBT ) @ TreeList2 ) ) ) )
=> ~ ( accp @ ( product_prod @ vEBT_VEBT @ nat ) @ vEBT_V5765760719290551771er_rel @ ( product_Pair @ vEBT_VEBT @ nat @ ( vEBT_Node @ Uy2 @ ( suc @ V4 ) @ TreeList2 @ S2 ) @ Xa ) ) ) ) ) ) ) ) ).
% VEBT_internal.naive_member.pelims(1)
thf(fact_3759_VEBT__internal_Omembermima_Opelims_I3_J,axiom,
! [X: vEBT_VEBT,Xa: nat] :
( ~ ( vEBT_VEBT_membermima @ X @ Xa )
=> ( ( accp @ ( product_prod @ vEBT_VEBT @ nat ) @ vEBT_V4351362008482014158ma_rel @ ( product_Pair @ vEBT_VEBT @ nat @ X @ Xa ) )
=> ( ! [Uu2: $o,Uv2: $o] :
( ( X
= ( vEBT_Leaf @ Uu2 @ Uv2 ) )
=> ~ ( accp @ ( product_prod @ vEBT_VEBT @ nat ) @ vEBT_V4351362008482014158ma_rel @ ( product_Pair @ vEBT_VEBT @ nat @ ( vEBT_Leaf @ Uu2 @ Uv2 ) @ Xa ) ) )
=> ( ! [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 ) ) )
=> ( ! [Mi: nat,Ma2: nat,Va3: list @ vEBT_VEBT,Vb2: vEBT_VEBT] :
( ( X
= ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi @ Ma2 ) ) @ ( zero_zero @ nat ) @ Va3 @ Vb2 ) )
=> ( ( accp @ ( product_prod @ vEBT_VEBT @ nat ) @ vEBT_V4351362008482014158ma_rel @ ( product_Pair @ vEBT_VEBT @ nat @ ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi @ Ma2 ) ) @ ( zero_zero @ nat ) @ Va3 @ Vb2 ) @ Xa ) )
=> ( ( Xa = Mi )
| ( Xa = Ma2 ) ) ) )
=> ( ! [Mi: nat,Ma2: nat,V4: nat,TreeList2: list @ vEBT_VEBT,Vc: vEBT_VEBT] :
( ( X
= ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi @ Ma2 ) ) @ ( suc @ V4 ) @ TreeList2 @ Vc ) )
=> ( ( 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 @ Mi @ Ma2 ) ) @ ( suc @ V4 ) @ TreeList2 @ Vc ) @ Xa ) )
=> ( ( Xa = Mi )
| ( Xa = Ma2 )
| ( ( ( ord_less @ nat @ ( vEBT_VEBT_high @ Xa @ ( divide_divide @ nat @ ( suc @ V4 ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( size_size @ ( list @ vEBT_VEBT ) @ TreeList2 ) )
=> ( vEBT_VEBT_membermima @ ( nth @ vEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ Xa @ ( divide_divide @ nat @ ( suc @ V4 ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( vEBT_VEBT_low @ Xa @ ( divide_divide @ nat @ ( suc @ V4 ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) )
& ( ord_less @ nat @ ( vEBT_VEBT_high @ Xa @ ( divide_divide @ nat @ ( suc @ V4 ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( size_size @ ( list @ vEBT_VEBT ) @ TreeList2 ) ) ) ) ) )
=> ~ ! [V4: nat,TreeList2: list @ vEBT_VEBT,Vd2: vEBT_VEBT] :
( ( X
= ( vEBT_Node @ ( none @ ( product_prod @ nat @ nat ) ) @ ( suc @ V4 ) @ TreeList2 @ Vd2 ) )
=> ( ( accp @ ( product_prod @ vEBT_VEBT @ nat ) @ vEBT_V4351362008482014158ma_rel @ ( product_Pair @ vEBT_VEBT @ nat @ ( vEBT_Node @ ( none @ ( product_prod @ nat @ nat ) ) @ ( suc @ V4 ) @ TreeList2 @ Vd2 ) @ Xa ) )
=> ( ( ( ord_less @ nat @ ( vEBT_VEBT_high @ Xa @ ( divide_divide @ nat @ ( suc @ V4 ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( size_size @ ( list @ vEBT_VEBT ) @ TreeList2 ) )
=> ( vEBT_VEBT_membermima @ ( nth @ vEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ Xa @ ( divide_divide @ nat @ ( suc @ V4 ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( vEBT_VEBT_low @ Xa @ ( divide_divide @ nat @ ( suc @ V4 ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) )
& ( ord_less @ nat @ ( vEBT_VEBT_high @ Xa @ ( divide_divide @ nat @ ( suc @ V4 ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( size_size @ ( list @ vEBT_VEBT ) @ TreeList2 ) ) ) ) ) ) ) ) ) ) ) ).
% VEBT_internal.membermima.pelims(3)
thf(fact_3760_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 ) )
=> ( ! [Mi: nat,Ma2: nat,Va3: list @ vEBT_VEBT,Vb2: vEBT_VEBT] :
( ( X
= ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi @ Ma2 ) ) @ ( zero_zero @ nat ) @ Va3 @ Vb2 ) )
=> ( ( accp @ ( product_prod @ vEBT_VEBT @ nat ) @ vEBT_V4351362008482014158ma_rel @ ( product_Pair @ vEBT_VEBT @ nat @ ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi @ Ma2 ) ) @ ( zero_zero @ nat ) @ Va3 @ Vb2 ) @ Xa ) )
=> ~ ( ( Xa = Mi )
| ( Xa = Ma2 ) ) ) )
=> ( ! [Mi: nat,Ma2: nat,V4: nat,TreeList2: list @ vEBT_VEBT,Vc: vEBT_VEBT] :
( ( X
= ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi @ Ma2 ) ) @ ( suc @ V4 ) @ TreeList2 @ Vc ) )
=> ( ( 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 @ Mi @ Ma2 ) ) @ ( suc @ V4 ) @ TreeList2 @ Vc ) @ Xa ) )
=> ~ ( ( Xa = Mi )
| ( Xa = Ma2 )
| ( ( ( ord_less @ nat @ ( vEBT_VEBT_high @ Xa @ ( divide_divide @ nat @ ( suc @ V4 ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( size_size @ ( list @ vEBT_VEBT ) @ TreeList2 ) )
=> ( vEBT_VEBT_membermima @ ( nth @ vEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ Xa @ ( divide_divide @ nat @ ( suc @ V4 ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( vEBT_VEBT_low @ Xa @ ( divide_divide @ nat @ ( suc @ V4 ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) )
& ( ord_less @ nat @ ( vEBT_VEBT_high @ Xa @ ( divide_divide @ nat @ ( suc @ V4 ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( size_size @ ( list @ vEBT_VEBT ) @ TreeList2 ) ) ) ) ) )
=> ~ ! [V4: nat,TreeList2: list @ vEBT_VEBT,Vd2: vEBT_VEBT] :
( ( X
= ( vEBT_Node @ ( none @ ( product_prod @ nat @ nat ) ) @ ( suc @ V4 ) @ TreeList2 @ Vd2 ) )
=> ( ( accp @ ( product_prod @ vEBT_VEBT @ nat ) @ vEBT_V4351362008482014158ma_rel @ ( product_Pair @ vEBT_VEBT @ nat @ ( vEBT_Node @ ( none @ ( product_prod @ nat @ nat ) ) @ ( suc @ V4 ) @ TreeList2 @ Vd2 ) @ Xa ) )
=> ~ ( ( ( ord_less @ nat @ ( vEBT_VEBT_high @ Xa @ ( divide_divide @ nat @ ( suc @ V4 ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( size_size @ ( list @ vEBT_VEBT ) @ TreeList2 ) )
=> ( vEBT_VEBT_membermima @ ( nth @ vEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ Xa @ ( divide_divide @ nat @ ( suc @ V4 ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( vEBT_VEBT_low @ Xa @ ( divide_divide @ nat @ ( suc @ V4 ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) )
& ( ord_less @ nat @ ( vEBT_VEBT_high @ Xa @ ( divide_divide @ nat @ ( suc @ V4 ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( size_size @ ( list @ vEBT_VEBT ) @ TreeList2 ) ) ) ) ) ) ) ) ) ).
% VEBT_internal.membermima.pelims(2)
thf(fact_3761_VEBT__internal_Omembermima_Opelims_I1_J,axiom,
! [X: vEBT_VEBT,Xa: nat,Y: $o] :
( ( ( vEBT_VEBT_membermima @ X @ Xa )
= Y )
=> ( ( accp @ ( product_prod @ vEBT_VEBT @ nat ) @ vEBT_V4351362008482014158ma_rel @ ( product_Pair @ vEBT_VEBT @ nat @ X @ Xa ) )
=> ( ! [Uu2: $o,Uv2: $o] :
( ( X
= ( vEBT_Leaf @ Uu2 @ Uv2 ) )
=> ( ~ Y
=> ~ ( accp @ ( product_prod @ vEBT_VEBT @ nat ) @ vEBT_V4351362008482014158ma_rel @ ( product_Pair @ vEBT_VEBT @ nat @ ( vEBT_Leaf @ Uu2 @ Uv2 ) @ Xa ) ) ) )
=> ( ! [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 ) ) ) )
=> ( ! [Mi: nat,Ma2: nat,Va3: list @ vEBT_VEBT,Vb2: vEBT_VEBT] :
( ( X
= ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi @ Ma2 ) ) @ ( zero_zero @ nat ) @ Va3 @ Vb2 ) )
=> ( ( Y
= ( ( Xa = Mi )
| ( 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 @ Mi @ Ma2 ) ) @ ( zero_zero @ nat ) @ Va3 @ Vb2 ) @ Xa ) ) ) )
=> ( ! [Mi: nat,Ma2: nat,V4: nat,TreeList2: list @ vEBT_VEBT,Vc: vEBT_VEBT] :
( ( X
= ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi @ Ma2 ) ) @ ( suc @ V4 ) @ TreeList2 @ Vc ) )
=> ( ( Y
= ( ( Xa = Mi )
| ( Xa = Ma2 )
| ( ( ( ord_less @ nat @ ( vEBT_VEBT_high @ Xa @ ( divide_divide @ nat @ ( suc @ V4 ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( size_size @ ( list @ vEBT_VEBT ) @ TreeList2 ) )
=> ( vEBT_VEBT_membermima @ ( nth @ vEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ Xa @ ( divide_divide @ nat @ ( suc @ V4 ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( vEBT_VEBT_low @ Xa @ ( divide_divide @ nat @ ( suc @ V4 ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) )
& ( ord_less @ nat @ ( vEBT_VEBT_high @ Xa @ ( divide_divide @ nat @ ( suc @ V4 ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( size_size @ ( list @ vEBT_VEBT ) @ TreeList2 ) ) ) ) )
=> ~ ( accp @ ( product_prod @ vEBT_VEBT @ nat ) @ vEBT_V4351362008482014158ma_rel @ ( product_Pair @ vEBT_VEBT @ nat @ ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi @ Ma2 ) ) @ ( suc @ V4 ) @ TreeList2 @ Vc ) @ Xa ) ) ) )
=> ~ ! [V4: nat,TreeList2: list @ vEBT_VEBT,Vd2: vEBT_VEBT] :
( ( X
= ( vEBT_Node @ ( none @ ( product_prod @ nat @ nat ) ) @ ( suc @ V4 ) @ TreeList2 @ Vd2 ) )
=> ( ( Y
= ( ( ( ord_less @ nat @ ( vEBT_VEBT_high @ Xa @ ( divide_divide @ nat @ ( suc @ V4 ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( size_size @ ( list @ vEBT_VEBT ) @ TreeList2 ) )
=> ( vEBT_VEBT_membermima @ ( nth @ vEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ Xa @ ( divide_divide @ nat @ ( suc @ V4 ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( vEBT_VEBT_low @ Xa @ ( divide_divide @ nat @ ( suc @ V4 ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) )
& ( ord_less @ nat @ ( vEBT_VEBT_high @ Xa @ ( divide_divide @ nat @ ( suc @ V4 ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( size_size @ ( list @ vEBT_VEBT ) @ TreeList2 ) ) ) )
=> ~ ( accp @ ( product_prod @ vEBT_VEBT @ nat ) @ vEBT_V4351362008482014158ma_rel @ ( product_Pair @ vEBT_VEBT @ nat @ ( vEBT_Node @ ( none @ ( product_prod @ nat @ nat ) ) @ ( suc @ V4 ) @ TreeList2 @ Vd2 ) @ Xa ) ) ) ) ) ) ) ) ) ) ).
% VEBT_internal.membermima.pelims(1)
thf(fact_3762_monoseq__arctan__series,axiom,
! [X: real] :
( ( ord_less_eq @ real @ ( abs_abs @ real @ X ) @ ( one_one @ real ) )
=> ( topological_monoseq @ real
@ ^ [N3: nat] : ( times_times @ real @ ( divide_divide @ real @ ( one_one @ real ) @ ( semiring_1_of_nat @ real @ ( plus_plus @ nat @ ( times_times @ nat @ N3 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ ( one_one @ nat ) ) ) ) @ ( power_power @ real @ X @ ( plus_plus @ nat @ ( times_times @ nat @ N3 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ ( one_one @ nat ) ) ) ) ) ) ).
% monoseq_arctan_series
thf(fact_3763_delete__correct,axiom,
! [T: vEBT_VEBT,N: nat,X: nat] :
( ( vEBT_invar_vebt @ T @ N )
=> ( ( vEBT_VEBT_set_vebt @ ( vEBT_vebt_delete @ T @ X ) )
= ( minus_minus @ ( set @ nat ) @ ( vEBT_set_vebt @ T ) @ ( insert @ nat @ X @ ( bot_bot @ ( set @ nat ) ) ) ) ) ) ).
% delete_correct
thf(fact_3764_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_3765_foldr__zero,axiom,
! [Xs2: list @ nat,D2: nat] :
( ! [I3: nat] :
( ( ord_less @ nat @ I3 @ ( size_size @ ( list @ nat ) @ Xs2 ) )
=> ( ord_less @ nat @ ( zero_zero @ nat ) @ ( nth @ nat @ Xs2 @ I3 ) ) )
=> ( ord_less_eq @ nat @ ( size_size @ ( list @ nat ) @ Xs2 ) @ ( minus_minus @ nat @ ( foldr @ nat @ nat @ ( plus_plus @ nat ) @ Xs2 @ D2 ) @ D2 ) ) ) ).
% foldr_zero
thf(fact_3766_foldr__one,axiom,
! [D2: nat,Ys: list @ nat] : ( ord_less_eq @ nat @ D2 @ ( foldr @ nat @ nat @ ( plus_plus @ nat ) @ Ys @ D2 ) ) ).
% foldr_one
thf(fact_3767_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_3768_delete__correct_H,axiom,
! [T: vEBT_VEBT,N: nat,X: nat] :
( ( vEBT_invar_vebt @ T @ N )
=> ( ( vEBT_VEBT_set_vebt @ ( vEBT_vebt_delete @ T @ X ) )
= ( minus_minus @ ( set @ nat ) @ ( vEBT_VEBT_set_vebt @ T ) @ ( insert @ nat @ X @ ( bot_bot @ ( set @ nat ) ) ) ) ) ) ).
% delete_correct'
thf(fact_3769_foldr__mono,axiom,
! [Xs2: list @ nat,Ys: list @ nat,C2: nat,D2: nat] :
( ( ( size_size @ ( list @ nat ) @ Xs2 )
= ( size_size @ ( list @ nat ) @ Ys ) )
=> ( ! [I3: nat] :
( ( ord_less @ nat @ I3 @ ( size_size @ ( list @ nat ) @ Xs2 ) )
=> ( ord_less @ nat @ ( nth @ nat @ Xs2 @ I3 ) @ ( nth @ nat @ Ys @ I3 ) ) )
=> ( ( ord_less_eq @ nat @ C2 @ D2 )
=> ( 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 @ D2 ) ) ) ) ) ).
% foldr_mono
thf(fact_3770_foldr__length,axiom,
! [A: $tType,L: list @ A] :
( ( foldr @ A @ nat
@ ^ [X2: A] : suc
@ L
@ ( zero_zero @ nat ) )
= ( size_size @ ( list @ A ) @ L ) ) ).
% foldr_length
thf(fact_3771_set__replicate,axiom,
! [A: $tType,N: nat,X: A] :
( ( N
!= ( zero_zero @ nat ) )
=> ( ( set2 @ A @ ( replicate @ A @ N @ X ) )
= ( insert @ A @ X @ ( bot_bot @ ( set @ A ) ) ) ) ) ).
% set_replicate
thf(fact_3772_set__encode__insert,axiom,
! [A3: set @ nat,N: nat] :
( ( finite_finite2 @ nat @ A3 )
=> ( ~ ( member @ nat @ N @ A3 )
=> ( ( nat_set_encode @ ( insert @ nat @ N @ A3 ) )
= ( plus_plus @ nat @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N ) @ ( nat_set_encode @ A3 ) ) ) ) ) ).
% set_encode_insert
thf(fact_3773_foldr__cong,axiom,
! [B: $tType,A: $tType,A2: A,B2: A,L: list @ B,K: list @ B,F2: B > A > A,G: B > A > A] :
( ( A2 = B2 )
=> ( ( L = K )
=> ( ! [A6: A,X3: B] :
( ( member @ B @ X3 @ ( set2 @ B @ L ) )
=> ( ( F2 @ X3 @ A6 )
= ( G @ X3 @ A6 ) ) )
=> ( ( foldr @ B @ A @ F2 @ L @ A2 )
= ( foldr @ B @ A @ G @ K @ B2 ) ) ) ) ) ).
% foldr_cong
thf(fact_3774_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,A9: set @ A] :
( ( finite_finite2 @ A @ A9 )
=> ( ! [X5: A] :
( ( member @ A @ X5 @ A9 )
=> ( ord_less @ A @ B3 @ X5 ) )
=> ( ( P @ A9 )
=> ( P @ ( insert @ A @ B3 @ A9 ) ) ) ) )
=> ( P @ A3 ) ) ) ) ) ).
% finite_linorder_min_induct
thf(fact_3775_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,A9: set @ A] :
( ( finite_finite2 @ A @ A9 )
=> ( ! [X5: A] :
( ( member @ A @ X5 @ A9 )
=> ( ord_less @ A @ X5 @ B3 ) )
=> ( ( P @ A9 )
=> ( P @ ( insert @ A @ B3 @ A9 ) ) ) ) )
=> ( P @ A3 ) ) ) ) ) ).
% finite_linorder_max_induct
thf(fact_3776_atLeast0__atMost__Suc,axiom,
! [N: nat] :
( ( set_or1337092689740270186AtMost @ nat @ ( zero_zero @ nat ) @ ( suc @ N ) )
= ( insert @ nat @ ( suc @ N ) @ ( set_or1337092689740270186AtMost @ nat @ ( zero_zero @ nat ) @ N ) ) ) ).
% atLeast0_atMost_Suc
thf(fact_3777_atLeastAtMost__insertL,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq @ nat @ M @ N )
=> ( ( insert @ nat @ M @ ( set_or1337092689740270186AtMost @ nat @ ( suc @ M ) @ N ) )
= ( set_or1337092689740270186AtMost @ nat @ M @ N ) ) ) ).
% atLeastAtMost_insertL
thf(fact_3778_atLeastAtMostSuc__conv,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq @ nat @ M @ ( suc @ N ) )
=> ( ( set_or1337092689740270186AtMost @ nat @ M @ ( suc @ N ) )
= ( insert @ nat @ ( suc @ N ) @ ( set_or1337092689740270186AtMost @ nat @ M @ N ) ) ) ) ).
% atLeastAtMostSuc_conv
thf(fact_3779_Icc__eq__insert__lb__nat,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq @ nat @ M @ N )
=> ( ( set_or1337092689740270186AtMost @ nat @ M @ N )
= ( insert @ nat @ M @ ( set_or1337092689740270186AtMost @ nat @ ( suc @ M ) @ N ) ) ) ) ).
% Icc_eq_insert_lb_nat
thf(fact_3780_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_3781_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_3782_foldr__length__aux,axiom,
! [A: $tType,L: list @ A,A2: nat] :
( ( foldr @ A @ nat
@ ^ [X2: A] : suc
@ L
@ A2 )
= ( plus_plus @ nat @ A2 @ ( size_size @ ( list @ A ) @ L ) ) ) ).
% foldr_length_aux
thf(fact_3783_finite__induct__select,axiom,
! [A: $tType,S3: set @ A,P: ( set @ A ) > $o] :
( ( finite_finite2 @ A @ S3 )
=> ( ( P @ ( bot_bot @ ( set @ A ) ) )
=> ( ! [T3: set @ A] :
( ( ord_less @ ( set @ A ) @ T3 @ S3 )
=> ( ( P @ T3 )
=> ? [X5: A] :
( ( member @ A @ X5 @ ( minus_minus @ ( set @ A ) @ S3 @ T3 ) )
& ( P @ ( insert @ A @ X5 @ T3 ) ) ) ) )
=> ( P @ S3 ) ) ) ) ).
% finite_induct_select
thf(fact_3784_set__replicate__Suc,axiom,
! [A: $tType,N: nat,X: A] :
( ( set2 @ A @ ( replicate @ A @ ( suc @ N ) @ X ) )
= ( insert @ A @ X @ ( bot_bot @ ( set @ A ) ) ) ) ).
% set_replicate_Suc
thf(fact_3785_psubset__insert__iff,axiom,
! [A: $tType,A3: set @ A,X: A,B4: set @ A] :
( ( ord_less @ ( set @ A ) @ A3 @ ( insert @ A @ X @ B4 ) )
= ( ( ( member @ A @ X @ B4 )
=> ( ord_less @ ( set @ A ) @ A3 @ B4 ) )
& ( ~ ( member @ A @ X @ B4 )
=> ( ( ( member @ A @ X @ A3 )
=> ( ord_less @ ( set @ A ) @ ( minus_minus @ ( set @ A ) @ A3 @ ( insert @ A @ X @ ( bot_bot @ ( set @ A ) ) ) ) @ B4 ) )
& ( ~ ( member @ A @ X @ A3 )
=> ( ord_less_eq @ ( set @ A ) @ A3 @ B4 ) ) ) ) ) ) ).
% psubset_insert_iff
thf(fact_3786_set__replicate__conv__if,axiom,
! [A: $tType,N: nat,X: A] :
( ( ( N
= ( zero_zero @ nat ) )
=> ( ( set2 @ A @ ( replicate @ A @ N @ X ) )
= ( bot_bot @ ( set @ A ) ) ) )
& ( ( N
!= ( zero_zero @ nat ) )
=> ( ( set2 @ A @ ( replicate @ A @ N @ X ) )
= ( insert @ A @ X @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ).
% set_replicate_conv_if
thf(fact_3787_insert__swap__set__eq,axiom,
! [A: $tType,I: nat,L: list @ A,X: A] :
( ( ord_less @ nat @ I @ ( size_size @ ( list @ A ) @ L ) )
=> ( ( insert @ A @ ( nth @ A @ L @ I ) @ ( set2 @ A @ ( list_update @ A @ L @ I @ X ) ) )
= ( insert @ A @ X @ ( set2 @ A @ L ) ) ) ) ).
% insert_swap_set_eq
thf(fact_3788_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_3789_set__decode__plus__power__2,axiom,
! [N: nat,Z: nat] :
( ~ ( member @ nat @ N @ ( nat_set_decode @ Z ) )
=> ( ( nat_set_decode @ ( plus_plus @ nat @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N ) @ Z ) )
= ( insert @ nat @ N @ ( nat_set_decode @ Z ) ) ) ) ).
% set_decode_plus_power_2
thf(fact_3790_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_3791_diff__preserves__multiset,axiom,
! [A: $tType,M7: A > nat,N4: A > nat] :
( ( finite_finite2 @ A
@ ( collect @ A
@ ^ [X2: A] : ( ord_less @ nat @ ( zero_zero @ nat ) @ ( M7 @ X2 ) ) ) )
=> ( finite_finite2 @ A
@ ( collect @ A
@ ^ [X2: A] : ( ord_less @ nat @ ( zero_zero @ nat ) @ ( minus_minus @ nat @ ( M7 @ X2 ) @ ( N4 @ X2 ) ) ) ) ) ) ).
% diff_preserves_multiset
thf(fact_3792_add__mset__in__multiset,axiom,
! [A: $tType,M7: A > nat,A2: A] :
( ( finite_finite2 @ A
@ ( collect @ A
@ ^ [X2: A] : ( ord_less @ nat @ ( zero_zero @ nat ) @ ( M7 @ X2 ) ) ) )
=> ( finite_finite2 @ A
@ ( collect @ A
@ ^ [X2: A] : ( ord_less @ nat @ ( zero_zero @ nat ) @ ( if @ nat @ ( X2 = A2 ) @ ( suc @ ( M7 @ X2 ) ) @ ( M7 @ X2 ) ) ) ) ) ) ).
% add_mset_in_multiset
thf(fact_3793_pochhammer__times__pochhammer__half,axiom,
! [A: $tType] :
( ( field_char_0 @ A )
=> ! [Z: A,N: nat] :
( ( times_times @ A @ ( comm_s3205402744901411588hammer @ A @ Z @ ( suc @ N ) ) @ ( comm_s3205402744901411588hammer @ A @ ( plus_plus @ A @ Z @ ( divide_divide @ A @ ( one_one @ A ) @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) ) @ ( suc @ N ) ) )
= ( groups7121269368397514597t_prod @ nat @ A
@ ^ [K3: nat] : ( plus_plus @ A @ Z @ ( divide_divide @ A @ ( semiring_1_of_nat @ A @ K3 ) @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) )
@ ( set_or1337092689740270186AtMost @ nat @ ( zero_zero @ nat ) @ ( plus_plus @ nat @ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N ) @ ( one_one @ nat ) ) ) ) ) ) ).
% pochhammer_times_pochhammer_half
thf(fact_3794_foldr0,axiom,
! [Xs2: list @ real,C2: real,D2: real] :
( ( foldr @ real @ real @ ( plus_plus @ real ) @ Xs2 @ ( plus_plus @ real @ C2 @ D2 ) )
= ( plus_plus @ real @ ( foldr @ real @ real @ ( plus_plus @ real ) @ Xs2 @ D2 ) @ C2 ) ) ).
% foldr0
thf(fact_3795_size__eq__0__iff__empty,axiom,
! [A: $tType,M7: multiset @ A] :
( ( ( size_size @ ( multiset @ A ) @ M7 )
= ( zero_zero @ nat ) )
= ( M7
= ( zero_zero @ ( multiset @ A ) ) ) ) ).
% size_eq_0_iff_empty
thf(fact_3796_size__empty,axiom,
! [A: $tType] :
( ( size_size @ ( multiset @ A ) @ ( zero_zero @ ( multiset @ A ) ) )
= ( zero_zero @ nat ) ) ).
% size_empty
thf(fact_3797_size__union,axiom,
! [A: $tType,M7: multiset @ A,N4: multiset @ A] :
( ( size_size @ ( multiset @ A ) @ ( plus_plus @ ( multiset @ A ) @ M7 @ N4 ) )
= ( plus_plus @ nat @ ( size_size @ ( multiset @ A ) @ M7 ) @ ( size_size @ ( multiset @ A ) @ N4 ) ) ) ).
% size_union
thf(fact_3798_prod_Ocl__ivl__Suc,axiom,
! [A: $tType] :
( ( comm_monoid_mult @ A )
=> ! [N: nat,M: nat,G: nat > A] :
( ( ( ord_less @ nat @ ( suc @ N ) @ M )
=> ( ( groups7121269368397514597t_prod @ nat @ A @ G @ ( set_or1337092689740270186AtMost @ nat @ M @ ( suc @ N ) ) )
= ( one_one @ A ) ) )
& ( ~ ( ord_less @ nat @ ( suc @ N ) @ M )
=> ( ( groups7121269368397514597t_prod @ nat @ A @ G @ ( set_or1337092689740270186AtMost @ nat @ M @ ( suc @ N ) ) )
= ( times_times @ A @ ( groups7121269368397514597t_prod @ nat @ A @ G @ ( set_or1337092689740270186AtMost @ nat @ M @ N ) ) @ ( G @ ( suc @ N ) ) ) ) ) ) ) ).
% prod.cl_ivl_Suc
thf(fact_3799_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
@ ^ [I2: B] : ( modulo_modulo @ A @ ( F2 @ I2 ) @ A2 )
@ A3 )
@ A2 )
= ( modulo_modulo @ A @ ( groups7121269368397514597t_prod @ B @ A @ F2 @ A3 ) @ A2 ) ) ) ).
% mod_prod_eq
thf(fact_3800_prod_Oshift__bounds__cl__Suc__ivl,axiom,
! [A: $tType] :
( ( comm_monoid_mult @ A )
=> ! [G: nat > A,M: nat,N: nat] :
( ( groups7121269368397514597t_prod @ nat @ A @ G @ ( set_or1337092689740270186AtMost @ nat @ ( suc @ M ) @ ( suc @ N ) ) )
= ( groups7121269368397514597t_prod @ nat @ A
@ ^ [I2: nat] : ( G @ ( suc @ I2 ) )
@ ( set_or1337092689740270186AtMost @ nat @ M @ N ) ) ) ) ).
% prod.shift_bounds_cl_Suc_ivl
thf(fact_3801_prod_Oshift__bounds__cl__nat__ivl,axiom,
! [A: $tType] :
( ( comm_monoid_mult @ A )
=> ! [G: nat > A,M: nat,K: nat,N: nat] :
( ( groups7121269368397514597t_prod @ nat @ A @ G @ ( set_or1337092689740270186AtMost @ nat @ ( plus_plus @ nat @ M @ K ) @ ( plus_plus @ nat @ N @ K ) ) )
= ( groups7121269368397514597t_prod @ nat @ A
@ ^ [I2: nat] : ( G @ ( plus_plus @ nat @ I2 @ K ) )
@ ( set_or1337092689740270186AtMost @ nat @ M @ N ) ) ) ) ).
% prod.shift_bounds_cl_nat_ivl
thf(fact_3802_prod_OatLeastAtMost__rev,axiom,
! [A: $tType] :
( ( comm_monoid_mult @ A )
=> ! [G: nat > A,N: nat,M: nat] :
( ( groups7121269368397514597t_prod @ nat @ A @ G @ ( set_or1337092689740270186AtMost @ nat @ N @ M ) )
= ( groups7121269368397514597t_prod @ nat @ A
@ ^ [I2: nat] : ( G @ ( minus_minus @ nat @ ( plus_plus @ nat @ M @ N ) @ I2 ) )
@ ( set_or1337092689740270186AtMost @ nat @ N @ M ) ) ) ) ).
% prod.atLeastAtMost_rev
thf(fact_3803_prod_OatLeast0__atMost__Suc,axiom,
! [A: $tType] :
( ( comm_monoid_mult @ A )
=> ! [G: nat > A,N: nat] :
( ( groups7121269368397514597t_prod @ nat @ A @ G @ ( set_or1337092689740270186AtMost @ nat @ ( zero_zero @ nat ) @ ( suc @ N ) ) )
= ( times_times @ A @ ( groups7121269368397514597t_prod @ nat @ A @ G @ ( set_or1337092689740270186AtMost @ nat @ ( zero_zero @ nat ) @ N ) ) @ ( G @ ( suc @ N ) ) ) ) ) ).
% prod.atLeast0_atMost_Suc
thf(fact_3804_prod_Onat__ivl__Suc_H,axiom,
! [A: $tType] :
( ( comm_monoid_mult @ A )
=> ! [M: nat,N: nat,G: nat > A] :
( ( ord_less_eq @ nat @ M @ ( suc @ N ) )
=> ( ( groups7121269368397514597t_prod @ nat @ A @ G @ ( set_or1337092689740270186AtMost @ nat @ M @ ( suc @ N ) ) )
= ( times_times @ A @ ( G @ ( suc @ N ) ) @ ( groups7121269368397514597t_prod @ nat @ A @ G @ ( set_or1337092689740270186AtMost @ nat @ M @ N ) ) ) ) ) ) ).
% prod.nat_ivl_Suc'
thf(fact_3805_prod_OatLeast__Suc__atMost,axiom,
! [A: $tType] :
( ( comm_monoid_mult @ A )
=> ! [M: nat,N: nat,G: nat > A] :
( ( ord_less_eq @ nat @ M @ N )
=> ( ( groups7121269368397514597t_prod @ nat @ A @ G @ ( set_or1337092689740270186AtMost @ nat @ M @ N ) )
= ( times_times @ A @ ( G @ M ) @ ( groups7121269368397514597t_prod @ nat @ A @ G @ ( set_or1337092689740270186AtMost @ nat @ ( suc @ M ) @ N ) ) ) ) ) ) ).
% prod.atLeast_Suc_atMost
thf(fact_3806_prod_OSuc__reindex__ivl,axiom,
! [A: $tType] :
( ( comm_monoid_mult @ A )
=> ! [M: nat,N: nat,G: nat > A] :
( ( ord_less_eq @ nat @ M @ N )
=> ( ( times_times @ A @ ( groups7121269368397514597t_prod @ nat @ A @ G @ ( set_or1337092689740270186AtMost @ nat @ M @ N ) ) @ ( G @ ( suc @ N ) ) )
= ( times_times @ A @ ( G @ M )
@ ( groups7121269368397514597t_prod @ nat @ A
@ ^ [I2: nat] : ( G @ ( suc @ I2 ) )
@ ( set_or1337092689740270186AtMost @ nat @ M @ N ) ) ) ) ) ) ).
% prod.Suc_reindex_ivl
thf(fact_3807_fact__prod,axiom,
! [A: $tType] :
( ( semiring_char_0 @ A )
=> ( ( semiring_char_0_fact @ A )
= ( ^ [N3: nat] :
( semiring_1_of_nat @ A
@ ( groups7121269368397514597t_prod @ nat @ nat
@ ^ [X2: nat] : X2
@ ( set_or1337092689740270186AtMost @ nat @ ( one_one @ nat ) @ N3 ) ) ) ) ) ) ).
% fact_prod
thf(fact_3808_atLeastAtMostPlus1__int__conv,axiom,
! [M: int,N: int] :
( ( ord_less_eq @ int @ M @ ( plus_plus @ int @ ( one_one @ int ) @ N ) )
=> ( ( set_or1337092689740270186AtMost @ int @ M @ ( plus_plus @ int @ ( one_one @ int ) @ N ) )
= ( insert @ int @ ( plus_plus @ int @ ( one_one @ int ) @ N ) @ ( set_or1337092689740270186AtMost @ int @ M @ N ) ) ) ) ).
% atLeastAtMostPlus1_int_conv
thf(fact_3809_simp__from__to,axiom,
( ( set_or1337092689740270186AtMost @ int )
= ( ^ [I2: int,J2: int] : ( if @ ( set @ int ) @ ( ord_less @ int @ J2 @ I2 ) @ ( bot_bot @ ( set @ int ) ) @ ( insert @ int @ I2 @ ( set_or1337092689740270186AtMost @ int @ ( plus_plus @ int @ I2 @ ( one_one @ int ) ) @ J2 ) ) ) ) ) ).
% simp_from_to
thf(fact_3810_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
@ ^ [A4: nat] : ( times_times @ A @ ( F2 @ A4 ) )
@ A2
@ B2
@ ( one_one @ A ) ) ) ) ).
% prod_atLeastAtMost_code
thf(fact_3811_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_3812_prod_Oub__add__nat,axiom,
! [A: $tType] :
( ( comm_monoid_mult @ A )
=> ! [M: nat,N: nat,G: nat > A,P4: nat] :
( ( ord_less_eq @ nat @ M @ ( plus_plus @ nat @ N @ ( one_one @ nat ) ) )
=> ( ( groups7121269368397514597t_prod @ nat @ A @ G @ ( set_or1337092689740270186AtMost @ nat @ M @ ( plus_plus @ nat @ N @ P4 ) ) )
= ( times_times @ A @ ( groups7121269368397514597t_prod @ nat @ A @ G @ ( set_or1337092689740270186AtMost @ nat @ M @ N ) ) @ ( groups7121269368397514597t_prod @ nat @ A @ G @ ( set_or1337092689740270186AtMost @ nat @ ( plus_plus @ nat @ N @ ( one_one @ nat ) ) @ ( plus_plus @ nat @ N @ P4 ) ) ) ) ) ) ) ).
% prod.ub_add_nat
thf(fact_3813_fact__eq__fact__times,axiom,
! [N: nat,M: nat] :
( ( ord_less_eq @ nat @ N @ M )
=> ( ( semiring_char_0_fact @ nat @ M )
= ( times_times @ nat @ ( semiring_char_0_fact @ nat @ N )
@ ( groups7121269368397514597t_prod @ nat @ nat
@ ^ [X2: nat] : X2
@ ( set_or1337092689740270186AtMost @ nat @ ( suc @ N ) @ M ) ) ) ) ) ).
% fact_eq_fact_times
thf(fact_3814_pochhammer__Suc__prod,axiom,
! [A: $tType] :
( ( comm_semiring_1 @ A )
=> ! [A2: A,N: nat] :
( ( comm_s3205402744901411588hammer @ A @ A2 @ ( suc @ N ) )
= ( groups7121269368397514597t_prod @ nat @ A
@ ^ [I2: nat] : ( plus_plus @ A @ A2 @ ( semiring_1_of_nat @ A @ I2 ) )
@ ( set_or1337092689740270186AtMost @ nat @ ( zero_zero @ nat ) @ N ) ) ) ) ).
% pochhammer_Suc_prod
thf(fact_3815_pochhammer__prod__rev,axiom,
! [A: $tType] :
( ( comm_semiring_1 @ A )
=> ( ( comm_s3205402744901411588hammer @ A )
= ( ^ [A4: A,N3: nat] :
( groups7121269368397514597t_prod @ nat @ A
@ ^ [I2: nat] : ( plus_plus @ A @ A4 @ ( semiring_1_of_nat @ A @ ( minus_minus @ nat @ N3 @ I2 ) ) )
@ ( set_or1337092689740270186AtMost @ nat @ ( one_one @ nat ) @ N3 ) ) ) ) ) ).
% pochhammer_prod_rev
thf(fact_3816_fact__div__fact,axiom,
! [N: nat,M: nat] :
( ( ord_less_eq @ nat @ N @ M )
=> ( ( divide_divide @ nat @ ( semiring_char_0_fact @ nat @ M ) @ ( semiring_char_0_fact @ nat @ N ) )
= ( groups7121269368397514597t_prod @ nat @ nat
@ ^ [X2: nat] : X2
@ ( set_or1337092689740270186AtMost @ nat @ ( plus_plus @ nat @ N @ ( one_one @ nat ) ) @ M ) ) ) ) ).
% fact_div_fact
thf(fact_3817_prod_Oin__pairs,axiom,
! [A: $tType] :
( ( comm_monoid_mult @ A )
=> ! [G: nat > A,M: nat,N: nat] :
( ( groups7121269368397514597t_prod @ nat @ A @ G @ ( set_or1337092689740270186AtMost @ nat @ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ M ) @ ( suc @ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N ) ) ) )
= ( groups7121269368397514597t_prod @ nat @ A
@ ^ [I2: nat] : ( times_times @ A @ ( G @ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ I2 ) ) @ ( G @ ( suc @ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ I2 ) ) ) )
@ ( set_or1337092689740270186AtMost @ nat @ M @ N ) ) ) ) ).
% prod.in_pairs
thf(fact_3818_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_3819_pochhammer__Suc__prod__rev,axiom,
! [A: $tType] :
( ( comm_semiring_1 @ A )
=> ! [A2: A,N: nat] :
( ( comm_s3205402744901411588hammer @ A @ A2 @ ( suc @ N ) )
= ( groups7121269368397514597t_prod @ nat @ A
@ ^ [I2: nat] : ( plus_plus @ A @ A2 @ ( semiring_1_of_nat @ A @ ( minus_minus @ nat @ N @ I2 ) ) )
@ ( set_or1337092689740270186AtMost @ nat @ ( zero_zero @ nat ) @ N ) ) ) ) ).
% pochhammer_Suc_prod_rev
thf(fact_3820_filter__preserves__multiset,axiom,
! [A: $tType,M7: A > nat,P: A > $o] :
( ( finite_finite2 @ A
@ ( collect @ A
@ ^ [X2: A] : ( ord_less @ nat @ ( zero_zero @ nat ) @ ( M7 @ X2 ) ) ) )
=> ( finite_finite2 @ A
@ ( collect @ A
@ ^ [X2: A] : ( ord_less @ nat @ ( zero_zero @ nat ) @ ( if @ nat @ ( P @ X2 ) @ ( M7 @ X2 ) @ ( zero_zero @ nat ) ) ) ) ) ) ).
% filter_preserves_multiset
thf(fact_3821_and__int_Osimps,axiom,
( ( bit_se5824344872417868541ns_and @ int )
= ( ^ [K3: int,L2: int] :
( if @ int
@ ( ( member @ int @ K3 @ ( insert @ int @ ( zero_zero @ int ) @ ( insert @ int @ ( uminus_uminus @ int @ ( one_one @ int ) ) @ ( bot_bot @ ( set @ int ) ) ) ) )
& ( member @ int @ L2 @ ( insert @ int @ ( zero_zero @ int ) @ ( insert @ int @ ( uminus_uminus @ int @ ( one_one @ int ) ) @ ( bot_bot @ ( set @ int ) ) ) ) ) )
@ ( uminus_uminus @ int
@ ( zero_neq_one_of_bool @ int
@ ( ~ ( dvd_dvd @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ K3 )
& ~ ( dvd_dvd @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ L2 ) ) ) )
@ ( plus_plus @ int
@ ( zero_neq_one_of_bool @ int
@ ( ~ ( dvd_dvd @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ K3 )
& ~ ( dvd_dvd @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ L2 ) ) )
@ ( times_times @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ ( bit_se5824344872417868541ns_and @ int @ ( divide_divide @ int @ K3 @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) ) @ ( divide_divide @ int @ L2 @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) ) ) ) ) ) ) ) ).
% and_int.simps
thf(fact_3822_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_3823_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_3824_prod_Odelta,axiom,
! [B: $tType,A: $tType] :
( ( comm_monoid_mult @ A )
=> ! [S3: set @ B,A2: B,B2: B > A] :
( ( finite_finite2 @ B @ S3 )
=> ( ( ( member @ B @ A2 @ S3 )
=> ( ( groups7121269368397514597t_prod @ B @ A
@ ^ [K3: B] : ( if @ A @ ( K3 = A2 ) @ ( B2 @ K3 ) @ ( one_one @ A ) )
@ S3 )
= ( B2 @ A2 ) ) )
& ( ~ ( member @ B @ A2 @ S3 )
=> ( ( groups7121269368397514597t_prod @ B @ A
@ ^ [K3: B] : ( if @ A @ ( K3 = A2 ) @ ( B2 @ K3 ) @ ( one_one @ A ) )
@ S3 )
= ( one_one @ A ) ) ) ) ) ) ).
% prod.delta
thf(fact_3825_prod_Odelta_H,axiom,
! [B: $tType,A: $tType] :
( ( comm_monoid_mult @ A )
=> ! [S3: set @ B,A2: B,B2: B > A] :
( ( finite_finite2 @ B @ S3 )
=> ( ( ( member @ B @ A2 @ S3 )
=> ( ( groups7121269368397514597t_prod @ B @ A
@ ^ [K3: B] : ( if @ A @ ( A2 = K3 ) @ ( B2 @ K3 ) @ ( one_one @ A ) )
@ S3 )
= ( B2 @ A2 ) ) )
& ( ~ ( member @ B @ A2 @ S3 )
=> ( ( groups7121269368397514597t_prod @ B @ A
@ ^ [K3: B] : ( if @ A @ ( A2 = K3 ) @ ( B2 @ K3 ) @ ( one_one @ A ) )
@ S3 )
= ( one_one @ A ) ) ) ) ) ) ).
% prod.delta'
thf(fact_3826_dvd__prodI,axiom,
! [A: $tType,B: $tType] :
( ( comm_semiring_1 @ A )
=> ! [A3: set @ B,A2: B,F2: B > A] :
( ( finite_finite2 @ B @ A3 )
=> ( ( member @ B @ A2 @ A3 )
=> ( dvd_dvd @ A @ ( F2 @ A2 ) @ ( groups7121269368397514597t_prod @ B @ A @ F2 @ A3 ) ) ) ) ) ).
% dvd_prodI
thf(fact_3827_subset__mset_Oadd__eq__0__iff__both__eq__0,axiom,
! [A: $tType,X: multiset @ A,Y: multiset @ A] :
( ( ( plus_plus @ ( multiset @ A ) @ X @ Y )
= ( zero_zero @ ( multiset @ A ) ) )
= ( ( X
= ( zero_zero @ ( multiset @ A ) ) )
& ( Y
= ( zero_zero @ ( multiset @ A ) ) ) ) ) ).
% subset_mset.add_eq_0_iff_both_eq_0
thf(fact_3828_subset__mset_Ozero__eq__add__iff__both__eq__0,axiom,
! [A: $tType,X: multiset @ A,Y: multiset @ A] :
( ( ( zero_zero @ ( multiset @ A ) )
= ( plus_plus @ ( multiset @ A ) @ X @ Y ) )
= ( ( X
= ( zero_zero @ ( multiset @ A ) ) )
& ( Y
= ( zero_zero @ ( multiset @ A ) ) ) ) ) ).
% subset_mset.zero_eq_add_iff_both_eq_0
thf(fact_3829_empty__eq__union,axiom,
! [A: $tType,M7: multiset @ A,N4: multiset @ A] :
( ( ( zero_zero @ ( multiset @ A ) )
= ( plus_plus @ ( multiset @ A ) @ M7 @ N4 ) )
= ( ( M7
= ( zero_zero @ ( multiset @ A ) ) )
& ( N4
= ( zero_zero @ ( multiset @ A ) ) ) ) ) ).
% empty_eq_union
thf(fact_3830_union__eq__empty,axiom,
! [A: $tType,M7: multiset @ A,N4: multiset @ A] :
( ( ( plus_plus @ ( multiset @ A ) @ M7 @ N4 )
= ( zero_zero @ ( multiset @ A ) ) )
= ( ( M7
= ( zero_zero @ ( multiset @ A ) ) )
& ( N4
= ( zero_zero @ ( multiset @ A ) ) ) ) ) ).
% union_eq_empty
thf(fact_3831_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_3832_prod__zero__iff,axiom,
! [A: $tType,B: $tType] :
( ( semidom @ A )
=> ! [A3: set @ B,F2: B > A] :
( ( finite_finite2 @ B @ A3 )
=> ( ( ( groups7121269368397514597t_prod @ B @ A @ F2 @ A3 )
= ( zero_zero @ A ) )
= ( ? [X2: B] :
( ( member @ B @ X2 @ A3 )
& ( ( F2 @ X2 )
= ( zero_zero @ A ) ) ) ) ) ) ) ).
% prod_zero_iff
thf(fact_3833_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_3834_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_3835_dvd__prod__eqI,axiom,
! [A: $tType,B: $tType] :
( ( comm_semiring_1 @ A )
=> ! [A3: set @ B,A2: B,B2: A,F2: B > A] :
( ( finite_finite2 @ B @ A3 )
=> ( ( member @ B @ A2 @ A3 )
=> ( ( B2
= ( F2 @ A2 ) )
=> ( dvd_dvd @ A @ B2 @ ( groups7121269368397514597t_prod @ B @ A @ F2 @ A3 ) ) ) ) ) ) ).
% dvd_prod_eqI
thf(fact_3836_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_3837_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_3838_Multiset_Odiff__cancel,axiom,
! [A: $tType,A3: multiset @ A] :
( ( minus_minus @ ( multiset @ A ) @ A3 @ A3 )
= ( zero_zero @ ( multiset @ A ) ) ) ).
% Multiset.diff_cancel
thf(fact_3839_diff__empty,axiom,
! [A: $tType,M7: multiset @ A] :
( ( ( minus_minus @ ( multiset @ A ) @ M7 @ ( zero_zero @ ( multiset @ A ) ) )
= M7 )
& ( ( minus_minus @ ( multiset @ A ) @ ( zero_zero @ ( multiset @ A ) ) @ M7 )
= ( zero_zero @ ( multiset @ A ) ) ) ) ).
% diff_empty
thf(fact_3840_union__less__mono,axiom,
! [A: $tType] :
( ( preorder @ A )
=> ! [A3: multiset @ A,C6: multiset @ A,B4: multiset @ A,D5: multiset @ A] :
( ( ord_less @ ( multiset @ A ) @ A3 @ C6 )
=> ( ( ord_less @ ( multiset @ A ) @ B4 @ D5 )
=> ( ord_less @ ( multiset @ A ) @ ( plus_plus @ ( multiset @ A ) @ A3 @ B4 ) @ ( plus_plus @ ( multiset @ A ) @ C6 @ D5 ) ) ) ) ) ).
% union_less_mono
thf(fact_3841_union__le__mono2,axiom,
! [A: $tType] :
( ( preorder @ A )
=> ! [B4: multiset @ A,D5: multiset @ A,C6: multiset @ A] :
( ( ord_less @ ( multiset @ A ) @ B4 @ D5 )
=> ( ord_less @ ( multiset @ A ) @ ( plus_plus @ ( multiset @ A ) @ C6 @ B4 ) @ ( plus_plus @ ( multiset @ A ) @ C6 @ D5 ) ) ) ) ).
% union_le_mono2
thf(fact_3842_union__le__mono1,axiom,
! [A: $tType] :
( ( preorder @ A )
=> ! [B4: multiset @ A,D5: multiset @ A,C6: multiset @ A] :
( ( ord_less @ ( multiset @ A ) @ B4 @ D5 )
=> ( ord_less @ ( multiset @ A ) @ ( plus_plus @ ( multiset @ A ) @ B4 @ C6 ) @ ( plus_plus @ ( multiset @ A ) @ D5 @ C6 ) ) ) ) ).
% union_le_mono1
thf(fact_3843_empty__neutral_I2_J,axiom,
! [A: $tType,X: multiset @ A] :
( ( plus_plus @ ( multiset @ A ) @ X @ ( zero_zero @ ( multiset @ A ) ) )
= X ) ).
% empty_neutral(2)
thf(fact_3844_empty__neutral_I1_J,axiom,
! [A: $tType,X: multiset @ A] :
( ( plus_plus @ ( multiset @ A ) @ ( zero_zero @ ( multiset @ A ) ) @ X )
= X ) ).
% empty_neutral(1)
thf(fact_3845_union__diff__assoc,axiom,
! [A: $tType,C6: multiset @ A,B4: multiset @ A,A3: multiset @ A] :
( ( ( minus_minus @ ( multiset @ A ) @ C6 @ B4 )
= ( zero_zero @ ( multiset @ A ) ) )
=> ( ( minus_minus @ ( multiset @ A ) @ ( plus_plus @ ( multiset @ A ) @ A3 @ B4 ) @ C6 )
= ( plus_plus @ ( multiset @ A ) @ A3 @ ( minus_minus @ ( multiset @ A ) @ B4 @ C6 ) ) ) ) ).
% union_diff_assoc
thf(fact_3846_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_3847_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_3848_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 ) )
=> ~ ! [A6: B] :
( ( member @ B @ A6 @ A3 )
=> ( ( G @ A6 )
= ( one_one @ A ) ) ) ) ) ).
% prod.not_neutral_contains_not_neutral
thf(fact_3849_prod__dvd__prod,axiom,
! [A: $tType,B: $tType] :
( ( comm_monoid_mult @ A )
=> ! [A3: set @ B,F2: B > A,G: B > A] :
( ! [A6: B] :
( ( member @ B @ A6 @ A3 )
=> ( dvd_dvd @ A @ ( F2 @ A6 ) @ ( G @ A6 ) ) )
=> ( dvd_dvd @ A @ ( groups7121269368397514597t_prod @ B @ A @ F2 @ A3 ) @ ( groups7121269368397514597t_prod @ B @ A @ G @ A3 ) ) ) ) ).
% prod_dvd_prod
thf(fact_3850_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_3851_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_3852_prod__power__distrib,axiom,
! [B: $tType,A: $tType] :
( ( comm_semiring_1 @ B )
=> ! [F2: A > B,A3: set @ A,N: nat] :
( ( power_power @ B @ ( groups7121269368397514597t_prod @ A @ B @ F2 @ A3 ) @ N )
= ( groups7121269368397514597t_prod @ A @ B
@ ^ [X2: A] : ( power_power @ B @ ( F2 @ X2 ) @ N )
@ A3 ) ) ) ).
% prod_power_distrib
thf(fact_3853_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_3854_prod__mono,axiom,
! [A: $tType,B: $tType] :
( ( linordered_semidom @ A )
=> ! [A3: set @ B,F2: B > A,G: B > A] :
( ! [I3: B] :
( ( member @ B @ I3 @ A3 )
=> ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( F2 @ I3 ) )
& ( ord_less_eq @ A @ ( F2 @ I3 ) @ ( G @ I3 ) ) ) )
=> ( ord_less_eq @ A @ ( groups7121269368397514597t_prod @ B @ A @ F2 @ A3 ) @ ( groups7121269368397514597t_prod @ B @ A @ G @ A3 ) ) ) ) ).
% prod_mono
thf(fact_3855_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_3856_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_3857_prod__zero,axiom,
! [B: $tType,A: $tType] :
( ( comm_semiring_1 @ A )
=> ! [A3: set @ B,F2: B > A] :
( ( finite_finite2 @ B @ A3 )
=> ( ? [X5: B] :
( ( member @ B @ X5 @ A3 )
& ( ( F2 @ X5 )
= ( zero_zero @ A ) ) )
=> ( ( groups7121269368397514597t_prod @ B @ A @ F2 @ A3 )
= ( zero_zero @ A ) ) ) ) ) ).
% prod_zero
thf(fact_3858_sum_Ofinite__Collect__op,axiom,
! [A: $tType,B: $tType] :
( ( comm_monoid_add @ A )
=> ! [I6: set @ B,X: B > A,Y: B > A] :
( ( finite_finite2 @ B
@ ( collect @ B
@ ^ [I2: B] :
( ( member @ B @ I2 @ I6 )
& ( ( X @ I2 )
!= ( zero_zero @ A ) ) ) ) )
=> ( ( finite_finite2 @ B
@ ( collect @ B
@ ^ [I2: B] :
( ( member @ B @ I2 @ I6 )
& ( ( Y @ I2 )
!= ( zero_zero @ A ) ) ) ) )
=> ( finite_finite2 @ B
@ ( collect @ B
@ ^ [I2: B] :
( ( member @ B @ I2 @ I6 )
& ( ( plus_plus @ A @ ( X @ I2 ) @ ( Y @ I2 ) )
!= ( zero_zero @ A ) ) ) ) ) ) ) ) ).
% sum.finite_Collect_op
thf(fact_3859_prod_Ofinite__Collect__op,axiom,
! [A: $tType,B: $tType] :
( ( comm_monoid_mult @ A )
=> ! [I6: set @ B,X: B > A,Y: B > A] :
( ( finite_finite2 @ B
@ ( collect @ B
@ ^ [I2: B] :
( ( member @ B @ I2 @ I6 )
& ( ( X @ I2 )
!= ( one_one @ A ) ) ) ) )
=> ( ( finite_finite2 @ B
@ ( collect @ B
@ ^ [I2: B] :
( ( member @ B @ I2 @ I6 )
& ( ( Y @ I2 )
!= ( one_one @ A ) ) ) ) )
=> ( finite_finite2 @ B
@ ( collect @ B
@ ^ [I2: B] :
( ( member @ B @ I2 @ I6 )
& ( ( times_times @ A @ ( X @ I2 ) @ ( Y @ I2 ) )
!= ( one_one @ A ) ) ) ) ) ) ) ) ).
% prod.finite_Collect_op
thf(fact_3860_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_3861_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_3862_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_3863_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_3864_prod__dvd__prod__subset2,axiom,
! [A: $tType,B: $tType] :
( ( comm_semiring_1 @ A )
=> ! [B4: set @ B,A3: set @ B,F2: B > A,G: B > A] :
( ( finite_finite2 @ B @ B4 )
=> ( ( ord_less_eq @ ( set @ B ) @ A3 @ B4 )
=> ( ! [A6: B] :
( ( member @ B @ A6 @ A3 )
=> ( dvd_dvd @ A @ ( F2 @ A6 ) @ ( G @ A6 ) ) )
=> ( dvd_dvd @ A @ ( groups7121269368397514597t_prod @ B @ A @ F2 @ A3 ) @ ( groups7121269368397514597t_prod @ B @ A @ G @ B4 ) ) ) ) ) ) ).
% prod_dvd_prod_subset2
thf(fact_3865_prod__dvd__prod__subset,axiom,
! [A: $tType,B: $tType] :
( ( comm_monoid_mult @ A )
=> ! [B4: set @ B,A3: set @ B,F2: B > A] :
( ( finite_finite2 @ B @ B4 )
=> ( ( ord_less_eq @ ( set @ B ) @ A3 @ B4 )
=> ( dvd_dvd @ A @ ( groups7121269368397514597t_prod @ B @ A @ F2 @ A3 ) @ ( groups7121269368397514597t_prod @ B @ A @ F2 @ B4 ) ) ) ) ) ).
% prod_dvd_prod_subset
thf(fact_3866_prod_Oreindex__bij__witness__not__neutral,axiom,
! [B: $tType,A: $tType,C: $tType] :
( ( comm_monoid_mult @ A )
=> ! [S4: set @ B,T4: set @ C,S3: set @ B,I: C > B,J: B > C,T5: set @ C,G: B > A,H2: C > A] :
( ( finite_finite2 @ B @ S4 )
=> ( ( finite_finite2 @ C @ T4 )
=> ( ! [A6: B] :
( ( member @ B @ A6 @ ( minus_minus @ ( set @ B ) @ S3 @ S4 ) )
=> ( ( I @ ( J @ A6 ) )
= A6 ) )
=> ( ! [A6: B] :
( ( member @ B @ A6 @ ( minus_minus @ ( set @ B ) @ S3 @ S4 ) )
=> ( member @ C @ ( J @ A6 ) @ ( minus_minus @ ( set @ C ) @ T5 @ T4 ) ) )
=> ( ! [B3: C] :
( ( member @ C @ B3 @ ( minus_minus @ ( set @ C ) @ T5 @ T4 ) )
=> ( ( J @ ( I @ B3 ) )
= B3 ) )
=> ( ! [B3: C] :
( ( member @ C @ B3 @ ( minus_minus @ ( set @ C ) @ T5 @ T4 ) )
=> ( member @ B @ ( I @ B3 ) @ ( minus_minus @ ( set @ B ) @ S3 @ S4 ) ) )
=> ( ! [A6: B] :
( ( member @ B @ A6 @ S4 )
=> ( ( G @ A6 )
= ( one_one @ A ) ) )
=> ( ! [B3: C] :
( ( member @ C @ B3 @ T4 )
=> ( ( H2 @ B3 )
= ( one_one @ A ) ) )
=> ( ! [A6: B] :
( ( member @ B @ A6 @ S3 )
=> ( ( H2 @ ( J @ A6 ) )
= ( G @ A6 ) ) )
=> ( ( groups7121269368397514597t_prod @ B @ A @ G @ S3 )
= ( groups7121269368397514597t_prod @ C @ A @ H2 @ T5 ) ) ) ) ) ) ) ) ) ) ) ) ).
% prod.reindex_bij_witness_not_neutral
thf(fact_3867_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_3868_less__1__prod2,axiom,
! [B: $tType,A: $tType] :
( ( linordered_idom @ B )
=> ! [I6: set @ A,I: A,F2: A > B] :
( ( finite_finite2 @ A @ I6 )
=> ( ( member @ A @ I @ I6 )
=> ( ( ord_less @ B @ ( one_one @ B ) @ ( F2 @ I ) )
=> ( ! [I3: A] :
( ( member @ A @ I3 @ I6 )
=> ( ord_less_eq @ B @ ( one_one @ B ) @ ( F2 @ I3 ) ) )
=> ( ord_less @ B @ ( one_one @ B ) @ ( groups7121269368397514597t_prod @ A @ B @ F2 @ I6 ) ) ) ) ) ) ) ).
% less_1_prod2
thf(fact_3869_less__1__prod,axiom,
! [B: $tType,A: $tType] :
( ( linordered_idom @ B )
=> ! [I6: set @ A,F2: A > B] :
( ( finite_finite2 @ A @ I6 )
=> ( ( I6
!= ( bot_bot @ ( set @ A ) ) )
=> ( ! [I3: A] :
( ( member @ A @ I3 @ I6 )
=> ( ord_less @ B @ ( one_one @ B ) @ ( F2 @ I3 ) ) )
=> ( ord_less @ B @ ( one_one @ B ) @ ( groups7121269368397514597t_prod @ A @ B @ F2 @ I6 ) ) ) ) ) ) ).
% less_1_prod
thf(fact_3870_prod_Osame__carrier,axiom,
! [A: $tType,B: $tType] :
( ( comm_monoid_mult @ A )
=> ! [C6: set @ B,A3: set @ B,B4: set @ B,G: B > A,H2: B > A] :
( ( finite_finite2 @ B @ C6 )
=> ( ( ord_less_eq @ ( set @ B ) @ A3 @ C6 )
=> ( ( ord_less_eq @ ( set @ B ) @ B4 @ C6 )
=> ( ! [A6: B] :
( ( member @ B @ A6 @ ( minus_minus @ ( set @ B ) @ C6 @ A3 ) )
=> ( ( G @ A6 )
= ( one_one @ A ) ) )
=> ( ! [B3: B] :
( ( member @ B @ B3 @ ( minus_minus @ ( set @ B ) @ C6 @ B4 ) )
=> ( ( H2 @ B3 )
= ( one_one @ A ) ) )
=> ( ( ( groups7121269368397514597t_prod @ B @ A @ G @ A3 )
= ( groups7121269368397514597t_prod @ B @ A @ H2 @ B4 ) )
= ( ( groups7121269368397514597t_prod @ B @ A @ G @ C6 )
= ( groups7121269368397514597t_prod @ B @ A @ H2 @ C6 ) ) ) ) ) ) ) ) ) ).
% prod.same_carrier
thf(fact_3871_prod_Osame__carrierI,axiom,
! [A: $tType,B: $tType] :
( ( comm_monoid_mult @ A )
=> ! [C6: set @ B,A3: set @ B,B4: set @ B,G: B > A,H2: B > A] :
( ( finite_finite2 @ B @ C6 )
=> ( ( ord_less_eq @ ( set @ B ) @ A3 @ C6 )
=> ( ( ord_less_eq @ ( set @ B ) @ B4 @ C6 )
=> ( ! [A6: B] :
( ( member @ B @ A6 @ ( minus_minus @ ( set @ B ) @ C6 @ A3 ) )
=> ( ( G @ A6 )
= ( one_one @ A ) ) )
=> ( ! [B3: B] :
( ( member @ B @ B3 @ ( minus_minus @ ( set @ B ) @ C6 @ B4 ) )
=> ( ( H2 @ B3 )
= ( one_one @ A ) ) )
=> ( ( ( groups7121269368397514597t_prod @ B @ A @ G @ C6 )
= ( groups7121269368397514597t_prod @ B @ A @ H2 @ C6 ) )
=> ( ( groups7121269368397514597t_prod @ B @ A @ G @ A3 )
= ( groups7121269368397514597t_prod @ B @ A @ H2 @ B4 ) ) ) ) ) ) ) ) ) ).
% prod.same_carrierI
thf(fact_3872_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_3873_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_3874_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_3875_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_3876_prod_Osubset__diff,axiom,
! [A: $tType,B: $tType] :
( ( comm_monoid_mult @ A )
=> ! [B4: set @ B,A3: set @ B,G: B > A] :
( ( ord_less_eq @ ( set @ B ) @ B4 @ A3 )
=> ( ( finite_finite2 @ B @ A3 )
=> ( ( groups7121269368397514597t_prod @ B @ A @ G @ A3 )
= ( times_times @ A @ ( groups7121269368397514597t_prod @ B @ A @ G @ ( minus_minus @ ( set @ B ) @ A3 @ B4 ) ) @ ( groups7121269368397514597t_prod @ B @ A @ G @ B4 ) ) ) ) ) ) ).
% prod.subset_diff
thf(fact_3877_prod__mono__strict,axiom,
! [A: $tType,B: $tType] :
( ( linordered_semidom @ A )
=> ! [A3: set @ B,F2: B > A,G: B > A] :
( ( finite_finite2 @ B @ A3 )
=> ( ! [I3: B] :
( ( member @ B @ I3 @ A3 )
=> ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( F2 @ I3 ) )
& ( ord_less @ A @ ( F2 @ I3 ) @ ( G @ I3 ) ) ) )
=> ( ( 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_3878_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_3879_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_3880_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
@ ^ [K3: B] : ( if @ A @ ( K3 = A2 ) @ ( B2 @ K3 ) @ ( C2 @ K3 ) )
@ 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
@ ^ [K3: B] : ( if @ A @ ( K3 = A2 ) @ ( B2 @ K3 ) @ ( C2 @ K3 ) )
@ S3 )
= ( groups7121269368397514597t_prod @ B @ A @ C2 @ ( minus_minus @ ( set @ B ) @ S3 @ ( insert @ B @ A2 @ ( bot_bot @ ( set @ B ) ) ) ) ) ) ) ) ) ) ).
% prod.delta_remove
thf(fact_3881_prod__mono2,axiom,
! [B: $tType,A: $tType] :
( ( linordered_idom @ B )
=> ! [B4: set @ A,A3: set @ A,F2: A > B] :
( ( finite_finite2 @ A @ B4 )
=> ( ( ord_less_eq @ ( set @ A ) @ A3 @ B4 )
=> ( ! [B3: A] :
( ( member @ A @ B3 @ ( minus_minus @ ( set @ A ) @ B4 @ A3 ) )
=> ( ord_less_eq @ B @ ( one_one @ B ) @ ( F2 @ B3 ) ) )
=> ( ! [A6: A] :
( ( member @ A @ A6 @ A3 )
=> ( ord_less_eq @ B @ ( zero_zero @ B ) @ ( F2 @ A6 ) ) )
=> ( ord_less_eq @ B @ ( groups7121269368397514597t_prod @ A @ B @ F2 @ A3 ) @ ( groups7121269368397514597t_prod @ A @ B @ F2 @ B4 ) ) ) ) ) ) ) ).
% prod_mono2
thf(fact_3882_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_3883_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_3884_ln__series,axiom,
! [X: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ X )
=> ( ( ord_less @ real @ X @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) )
=> ( ( ln_ln @ real @ X )
= ( suminf @ real
@ ^ [N3: nat] : ( times_times @ real @ ( times_times @ real @ ( power_power @ real @ ( uminus_uminus @ real @ ( one_one @ real ) ) @ N3 ) @ ( divide_divide @ real @ ( one_one @ real ) @ ( semiring_1_of_nat @ real @ ( plus_plus @ nat @ N3 @ ( one_one @ nat ) ) ) ) ) @ ( power_power @ real @ ( minus_minus @ real @ X @ ( one_one @ real ) ) @ ( suc @ N3 ) ) ) ) ) ) ) ).
% ln_series
thf(fact_3885_f__g__map__foldr__bound,axiom,
! [A: $tType,Xs2: list @ A,F2: A > real,C2: real,G: A > real,D2: 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 ) @ D2 ) @ ( plus_plus @ real @ ( times_times @ real @ C2 @ ( foldr @ real @ real @ ( plus_plus @ real ) @ ( map @ A @ real @ G @ Xs2 ) @ ( zero_zero @ real ) ) ) @ D2 ) ) ) ).
% f_g_map_foldr_bound
thf(fact_3886_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_3887_map__ident,axiom,
! [A: $tType] :
( ( map @ A @ A
@ ^ [X2: A] : X2 )
= ( ^ [Xs: list @ A] : Xs ) ) ).
% map_ident
thf(fact_3888_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_3889_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_3890_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_3891_map__replicate,axiom,
! [A: $tType,B: $tType,F2: B > A,N: nat,X: B] :
( ( map @ B @ A @ F2 @ ( replicate @ B @ N @ X ) )
= ( replicate @ A @ N @ ( F2 @ X ) ) ) ).
% map_replicate
thf(fact_3892_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_3893_nth__map,axiom,
! [B: $tType,A: $tType,N: nat,Xs2: list @ A,F2: A > B] :
( ( ord_less @ nat @ N @ ( size_size @ ( list @ A ) @ Xs2 ) )
=> ( ( nth @ B @ ( map @ A @ B @ F2 @ Xs2 ) @ N )
= ( F2 @ ( nth @ A @ Xs2 @ N ) ) ) ) ).
% nth_map
thf(fact_3894_powser__zero,axiom,
! [A: $tType] :
( ( real_V2822296259951069270ebra_1 @ A )
=> ! [F2: nat > A] :
( ( suminf @ A
@ ^ [N3: nat] : ( times_times @ A @ ( F2 @ N3 ) @ ( power_power @ A @ ( zero_zero @ A ) @ N3 ) ) )
= ( F2 @ ( zero_zero @ nat ) ) ) ) ).
% powser_zero
thf(fact_3895_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_3896_map__update,axiom,
! [A: $tType,B: $tType,F2: B > A,Xs2: list @ B,K: nat,Y: B] :
( ( map @ B @ A @ F2 @ ( list_update @ B @ Xs2 @ K @ Y ) )
= ( list_update @ A @ ( map @ B @ A @ F2 @ Xs2 ) @ K @ ( F2 @ Y ) ) ) ).
% map_update
thf(fact_3897_list_Omap__ident,axiom,
! [A: $tType,T: list @ A] :
( ( map @ A @ A
@ ^ [X2: A] : X2
@ T )
= T ) ).
% list.map_ident
thf(fact_3898_less__eq__multiset__def,axiom,
! [A: $tType] :
( ( preorder @ A )
=> ( ( ord_less_eq @ ( multiset @ A ) )
= ( ^ [M8: multiset @ A,N8: multiset @ A] :
( ( ord_less @ ( multiset @ A ) @ M8 @ N8 )
| ( M8 = N8 ) ) ) ) ) ).
% less_eq_multiset_def
thf(fact_3899_mset__le__asym,axiom,
! [A: $tType] :
( ( preorder @ A )
=> ! [M7: multiset @ A,N4: multiset @ A] :
( ( ord_less @ ( multiset @ A ) @ M7 @ N4 )
=> ~ ( ord_less @ ( multiset @ A ) @ N4 @ M7 ) ) ) ).
% mset_le_asym
thf(fact_3900_mset__le__trans,axiom,
! [A: $tType] :
( ( preorder @ A )
=> ! [K6: multiset @ A,M7: multiset @ A,N4: multiset @ A] :
( ( ord_less @ ( multiset @ A ) @ K6 @ M7 )
=> ( ( ord_less @ ( multiset @ A ) @ M7 @ N4 )
=> ( ord_less @ ( multiset @ A ) @ K6 @ N4 ) ) ) ) ).
% mset_le_trans
thf(fact_3901_mset__le__irrefl,axiom,
! [A: $tType] :
( ( preorder @ A )
=> ! [M7: multiset @ A] :
~ ( ord_less @ ( multiset @ A ) @ M7 @ M7 ) ) ).
% mset_le_irrefl
thf(fact_3902_mset__le__not__sym,axiom,
! [A: $tType] :
( ( preorder @ A )
=> ! [M7: multiset @ A,N4: multiset @ A] :
( ( ord_less @ ( multiset @ A ) @ M7 @ N4 )
=> ~ ( ord_less @ ( multiset @ A ) @ N4 @ M7 ) ) ) ).
% mset_le_not_sym
thf(fact_3903_mset__le__not__refl,axiom,
! [A: $tType] :
( ( preorder @ A )
=> ! [M7: multiset @ A] :
~ ( ord_less @ ( multiset @ A ) @ M7 @ M7 ) ) ).
% mset_le_not_refl
thf(fact_3904_list_Omap__cong,axiom,
! [B: $tType,A: $tType,X: list @ A,Ya: list @ A,F2: A > B,G: A > B] :
( ( X = Ya )
=> ( ! [Z3: A] :
( ( member @ A @ Z3 @ ( set2 @ A @ Ya ) )
=> ( ( F2 @ Z3 )
= ( G @ Z3 ) ) )
=> ( ( map @ A @ B @ F2 @ X )
= ( map @ A @ B @ G @ Ya ) ) ) ) ).
% list.map_cong
thf(fact_3905_list_Omap__cong0,axiom,
! [B: $tType,A: $tType,X: list @ A,F2: A > B,G: A > B] :
( ! [Z3: A] :
( ( member @ A @ Z3 @ ( set2 @ A @ X ) )
=> ( ( F2 @ Z3 )
= ( G @ Z3 ) ) )
=> ( ( map @ A @ B @ F2 @ X )
= ( map @ A @ B @ G @ X ) ) ) ).
% list.map_cong0
thf(fact_3906_list_Oinj__map__strong,axiom,
! [B: $tType,A: $tType,X: list @ A,Xa: list @ A,F2: A > B,Fa: A > B] :
( ! [Z3: A,Za: A] :
( ( member @ A @ Z3 @ ( set2 @ A @ X ) )
=> ( ( member @ A @ Za @ ( set2 @ A @ Xa ) )
=> ( ( ( F2 @ Z3 )
= ( Fa @ Za ) )
=> ( Z3 = Za ) ) ) )
=> ( ( ( map @ A @ B @ F2 @ X )
= ( map @ A @ B @ Fa @ Xa ) )
=> ( X = Xa ) ) ) ).
% list.inj_map_strong
thf(fact_3907_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_3908_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_3909_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_3910_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 ) )
=> ? [Y6: A] :
( X2
= ( F2 @ Y6 ) ) ) ) ) ).
% ex_map_conv
thf(fact_3911_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_3912_map__upd__eq,axiom,
! [B: $tType,A: $tType,I: nat,L: list @ A,F2: A > B,X: A] :
( ( ( ord_less @ nat @ I @ ( size_size @ ( list @ A ) @ L ) )
=> ( ( F2 @ ( nth @ A @ L @ I ) )
= ( F2 @ X ) ) )
=> ( ( map @ A @ B @ F2 @ ( list_update @ A @ L @ I @ X ) )
= ( map @ A @ B @ F2 @ L ) ) ) ).
% map_upd_eq
thf(fact_3913_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_3914_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_3915_VEBT__internal_Ocnt_H_Oelims,axiom,
! [X: vEBT_VEBT,Y: nat] :
( ( ( vEBT_VEBT_cnt2 @ X )
= Y )
=> ( ( ? [A6: $o,B3: $o] :
( X
= ( vEBT_Leaf @ A6 @ 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_3916_VEBT__internal_Ocnt_Oelims,axiom,
! [X: vEBT_VEBT,Y: real] :
( ( ( vEBT_VEBT_cnt @ X )
= Y )
=> ( ( ? [A6: $o,B3: $o] :
( X
= ( vEBT_Leaf @ A6 @ 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_3917_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_3918_VEBT__internal_Ospace_H_Oelims,axiom,
! [X: vEBT_VEBT,Y: nat] :
( ( ( vEBT_VEBT_space2 @ X )
= Y )
=> ( ( ? [A6: $o,B3: $o] :
( X
= ( vEBT_Leaf @ A6 @ 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_3919_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_3920_VEBT__internal_Ospace_Oelims,axiom,
! [X: vEBT_VEBT,Y: nat] :
( ( ( vEBT_VEBT_space @ X )
= Y )
=> ( ( ? [A6: $o,B3: $o] :
( X
= ( vEBT_Leaf @ A6 @ 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_3921_arctan__series,axiom,
! [X: real] :
( ( ord_less_eq @ real @ ( abs_abs @ real @ X ) @ ( one_one @ real ) )
=> ( ( arctan @ X )
= ( suminf @ real
@ ^ [K3: nat] : ( times_times @ real @ ( power_power @ real @ ( uminus_uminus @ real @ ( one_one @ real ) ) @ K3 ) @ ( times_times @ real @ ( divide_divide @ real @ ( one_one @ real ) @ ( semiring_1_of_nat @ real @ ( plus_plus @ nat @ ( times_times @ nat @ K3 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ ( one_one @ nat ) ) ) ) @ ( power_power @ real @ X @ ( plus_plus @ nat @ ( times_times @ nat @ K3 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ ( one_one @ nat ) ) ) ) ) ) ) ) ).
% arctan_series
thf(fact_3922_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_3923_suminf__zero,axiom,
! [A: $tType] :
( ( ( comm_monoid_add @ A )
& ( topological_t2_space @ A ) )
=> ( ( suminf @ A
@ ^ [N3: nat] : ( zero_zero @ A ) )
= ( zero_zero @ A ) ) ) ).
% suminf_zero
thf(fact_3924_pi__series,axiom,
( ( divide_divide @ real @ pi @ ( numeral_numeral @ real @ ( bit0 @ ( bit0 @ one2 ) ) ) )
= ( suminf @ real
@ ^ [K3: nat] : ( divide_divide @ real @ ( times_times @ real @ ( power_power @ real @ ( uminus_uminus @ real @ ( one_one @ real ) ) @ K3 ) @ ( one_one @ real ) ) @ ( semiring_1_of_nat @ real @ ( plus_plus @ nat @ ( times_times @ nat @ K3 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ ( one_one @ nat ) ) ) ) ) ) ).
% pi_series
thf(fact_3925_big__assn__simp,axiom,
! [H2: nat,TreeList: list @ vEBT_VEBT,L: nat,X: vEBT_VEBTi,Xaa: option @ nat,X13: array @ vEBT_VEBTi,Tree_is: list @ vEBT_VEBTi,Summary: vEBT_VEBT,X14: vEBT_VEBTi] :
( ( ord_less @ nat @ H2 @ ( size_size @ ( list @ vEBT_VEBT ) @ TreeList ) )
=> ( entails
@ ( times_times @ assn
@ ( times_times @ assn @ ( vEBT_vebt_assn_raw @ ( vEBT_vebt_delete @ ( nth @ vEBT_VEBT @ TreeList @ H2 ) @ L ) @ X )
@ ( pure_assn
@ ( Xaa
= ( vEBT_vebt_mint @ ( vEBT_vebt_delete @ ( nth @ vEBT_VEBT @ TreeList @ H2 ) @ L ) ) ) ) )
@ ( times_times @ assn @ ( times_times @ assn @ ( snga_assn @ vEBT_VEBTi @ X13 @ ( list_update @ vEBT_VEBTi @ Tree_is @ H2 @ X ) ) @ ( vEBT_vebt_assn_raw @ Summary @ X14 ) ) @ ( vEBT_List_listI_assn @ vEBT_VEBT @ vEBT_VEBTi @ ( minus_minus @ ( set @ nat ) @ ( set_or7035219750837199246ssThan @ nat @ ( zero_zero @ nat ) @ ( size_size @ ( list @ vEBT_VEBT ) @ TreeList ) ) @ ( insert @ nat @ H2 @ ( bot_bot @ ( set @ nat ) ) ) ) @ vEBT_vebt_assn_raw @ TreeList @ Tree_is ) ) )
@ ( times_times @ assn
@ ( times_times @ assn @ ( times_times @ assn @ ( snga_assn @ vEBT_VEBTi @ X13 @ ( list_update @ vEBT_VEBTi @ Tree_is @ H2 @ X ) ) @ ( vEBT_vebt_assn_raw @ Summary @ X14 ) )
@ ( pure_assn
@ ( Xaa
= ( vEBT_vebt_mint @ ( vEBT_vebt_delete @ ( nth @ vEBT_VEBT @ TreeList @ H2 ) @ L ) ) ) ) )
@ ( vEBT_List_list_assn @ vEBT_VEBT @ vEBT_VEBTi @ vEBT_vebt_assn_raw @ ( list_update @ vEBT_VEBT @ TreeList @ H2 @ ( vEBT_vebt_delete @ ( nth @ vEBT_VEBT @ TreeList @ H2 ) @ L ) ) @ ( list_update @ vEBT_VEBTi @ Tree_is @ H2 @ X ) ) ) ) ) ).
% big_assn_simp
thf(fact_3926_atLeastLessThan__iff,axiom,
! [A: $tType] :
( ( ord @ A )
=> ! [I: A,L: A,U: A] :
( ( member @ A @ I @ ( set_or7035219750837199246ssThan @ A @ L @ U ) )
= ( ( ord_less_eq @ A @ L @ I )
& ( ord_less @ A @ I @ U ) ) ) ) ).
% atLeastLessThan_iff
thf(fact_3927_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_3928_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_3929_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_3930_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_3931_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_3932_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_3933_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_3934_atLeastLessThan__singleton,axiom,
! [M: nat] :
( ( set_or7035219750837199246ssThan @ nat @ M @ ( suc @ M ) )
= ( insert @ nat @ M @ ( bot_bot @ ( set @ nat ) ) ) ) ).
% atLeastLessThan_singleton
thf(fact_3935_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_3936_big__assn__simp_H,axiom,
! [H2: nat,TreeList: list @ vEBT_VEBT,Xaa: vEBT_VEBT,L: nat,X: vEBT_VEBTi,Xb: option @ nat,X13: array @ vEBT_VEBTi,Tree_is: list @ vEBT_VEBTi,Summary: vEBT_VEBT,X14: vEBT_VEBTi] :
( ( ord_less @ nat @ H2 @ ( size_size @ ( list @ vEBT_VEBT ) @ TreeList ) )
=> ( ( Xaa
= ( vEBT_vebt_delete @ ( nth @ vEBT_VEBT @ TreeList @ H2 ) @ L ) )
=> ( entails
@ ( times_times @ assn
@ ( times_times @ assn @ ( vEBT_vebt_assn_raw @ Xaa @ X )
@ ( pure_assn
@ ( Xb
= ( vEBT_vebt_mint @ Xaa ) ) ) )
@ ( times_times @ assn @ ( times_times @ assn @ ( snga_assn @ vEBT_VEBTi @ X13 @ ( list_update @ vEBT_VEBTi @ Tree_is @ H2 @ X ) ) @ ( vEBT_vebt_assn_raw @ Summary @ X14 ) ) @ ( vEBT_List_listI_assn @ vEBT_VEBT @ vEBT_VEBTi @ ( minus_minus @ ( set @ nat ) @ ( set_or7035219750837199246ssThan @ nat @ ( zero_zero @ nat ) @ ( size_size @ ( list @ vEBT_VEBT ) @ TreeList ) ) @ ( insert @ nat @ H2 @ ( bot_bot @ ( set @ nat ) ) ) ) @ vEBT_vebt_assn_raw @ TreeList @ Tree_is ) ) )
@ ( times_times @ assn
@ ( times_times @ assn @ ( times_times @ assn @ ( snga_assn @ vEBT_VEBTi @ X13 @ ( list_update @ vEBT_VEBTi @ Tree_is @ H2 @ X ) ) @ ( vEBT_vebt_assn_raw @ Summary @ X14 ) )
@ ( pure_assn
@ ( Xb
= ( vEBT_vebt_mint @ Xaa ) ) ) )
@ ( vEBT_List_list_assn @ vEBT_VEBT @ vEBT_VEBTi @ vEBT_vebt_assn_raw @ ( list_update @ vEBT_VEBT @ TreeList @ H2 @ Xaa ) @ ( list_update @ vEBT_VEBTi @ Tree_is @ H2 @ X ) ) ) ) ) ) ).
% big_assn_simp'
thf(fact_3937_prod_Oop__ivl__Suc,axiom,
! [A: $tType] :
( ( comm_monoid_mult @ A )
=> ! [N: nat,M: nat,G: nat > A] :
( ( ( ord_less @ nat @ N @ M )
=> ( ( groups7121269368397514597t_prod @ nat @ A @ G @ ( set_or7035219750837199246ssThan @ nat @ M @ ( suc @ N ) ) )
= ( one_one @ A ) ) )
& ( ~ ( ord_less @ nat @ N @ M )
=> ( ( groups7121269368397514597t_prod @ nat @ A @ G @ ( set_or7035219750837199246ssThan @ nat @ M @ ( suc @ N ) ) )
= ( times_times @ A @ ( groups7121269368397514597t_prod @ nat @ A @ G @ ( set_or7035219750837199246ssThan @ nat @ M @ N ) ) @ ( G @ N ) ) ) ) ) ) ).
% prod.op_ivl_Suc
thf(fact_3938_atLeastLessThan__eq__iff,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A2: A,B2: A,C2: A,D2: A] :
( ( ord_less @ A @ A2 @ B2 )
=> ( ( ord_less @ A @ C2 @ D2 )
=> ( ( ( set_or7035219750837199246ssThan @ A @ A2 @ B2 )
= ( set_or7035219750837199246ssThan @ A @ C2 @ D2 ) )
= ( ( A2 = C2 )
& ( B2 = D2 ) ) ) ) ) ) ).
% atLeastLessThan_eq_iff
thf(fact_3939_atLeastLessThan__inj_I1_J,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A2: A,B2: A,C2: A,D2: A] :
( ( ( set_or7035219750837199246ssThan @ A @ A2 @ B2 )
= ( set_or7035219750837199246ssThan @ A @ C2 @ D2 ) )
=> ( ( ord_less @ A @ A2 @ B2 )
=> ( ( ord_less @ A @ C2 @ D2 )
=> ( A2 = C2 ) ) ) ) ) ).
% atLeastLessThan_inj(1)
thf(fact_3940_atLeastLessThan__inj_I2_J,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A2: A,B2: A,C2: A,D2: A] :
( ( ( set_or7035219750837199246ssThan @ A @ A2 @ B2 )
= ( set_or7035219750837199246ssThan @ A @ C2 @ D2 ) )
=> ( ( ord_less @ A @ A2 @ B2 )
=> ( ( ord_less @ A @ C2 @ D2 )
=> ( B2 = D2 ) ) ) ) ) ).
% atLeastLessThan_inj(2)
thf(fact_3941_pi__neq__zero,axiom,
( pi
!= ( zero_zero @ real ) ) ).
% pi_neq_zero
thf(fact_3942_listI__assn__conv,axiom,
! [A: $tType,B: $tType,N: nat,Xs2: list @ A,A3: A > B > assn,Xsi: list @ B] :
( ( N
= ( size_size @ ( list @ A ) @ Xs2 ) )
=> ( ( vEBT_List_listI_assn @ A @ B @ ( set_or7035219750837199246ssThan @ nat @ ( zero_zero @ nat ) @ N ) @ A3 @ Xs2 @ Xsi )
= ( vEBT_List_list_assn @ A @ B @ A3 @ Xs2 @ Xsi ) ) ) ).
% listI_assn_conv
thf(fact_3943_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_3944_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_3945_all__nat__less__eq,axiom,
! [N: nat,P: nat > $o] :
( ( ! [M4: nat] :
( ( ord_less @ nat @ M4 @ N )
=> ( P @ M4 ) ) )
= ( ! [X2: nat] :
( ( member @ nat @ X2 @ ( set_or7035219750837199246ssThan @ nat @ ( zero_zero @ nat ) @ N ) )
=> ( P @ X2 ) ) ) ) ).
% all_nat_less_eq
thf(fact_3946_ex__nat__less__eq,axiom,
! [N: nat,P: nat > $o] :
( ( ? [M4: nat] :
( ( ord_less @ nat @ M4 @ N )
& ( P @ M4 ) ) )
= ( ? [X2: nat] :
( ( member @ nat @ X2 @ ( set_or7035219750837199246ssThan @ nat @ ( zero_zero @ nat ) @ N ) )
& ( P @ X2 ) ) ) ) ).
% ex_nat_less_eq
thf(fact_3947_atLeastLessThanSuc__atLeastAtMost,axiom,
! [L: nat,U: nat] :
( ( set_or7035219750837199246ssThan @ nat @ L @ ( suc @ U ) )
= ( set_or1337092689740270186AtMost @ nat @ L @ U ) ) ).
% atLeastLessThanSuc_atLeastAtMost
thf(fact_3948_atLeastLessThan0,axiom,
! [M: nat] :
( ( set_or7035219750837199246ssThan @ nat @ M @ ( zero_zero @ nat ) )
= ( bot_bot @ ( set @ nat ) ) ) ).
% atLeastLessThan0
thf(fact_3949_pi__not__less__zero,axiom,
~ ( ord_less @ real @ pi @ ( zero_zero @ real ) ) ).
% pi_not_less_zero
thf(fact_3950_pi__gt__zero,axiom,
ord_less @ real @ ( zero_zero @ real ) @ pi ).
% pi_gt_zero
thf(fact_3951_pi__ge__zero,axiom,
ord_less_eq @ real @ ( zero_zero @ real ) @ pi ).
% pi_ge_zero
thf(fact_3952_listI__assn__conv_H,axiom,
! [B: $tType,A: $tType,N: nat,Xs2: list @ A,A3: A > B > assn,Xsi: list @ B,F3: assn] :
( ( N
= ( size_size @ ( list @ A ) @ Xs2 ) )
=> ( ( times_times @ assn @ ( vEBT_List_listI_assn @ A @ B @ ( set_or7035219750837199246ssThan @ nat @ ( zero_zero @ nat ) @ N ) @ A3 @ Xs2 @ Xsi ) @ F3 )
= ( times_times @ assn @ ( vEBT_List_list_assn @ A @ B @ A3 @ Xs2 @ Xsi ) @ F3 ) ) ) ).
% listI_assn_conv'
thf(fact_3953_prod_Oshift__bounds__Suc__ivl,axiom,
! [A: $tType] :
( ( comm_monoid_mult @ A )
=> ! [G: nat > A,M: nat,N: nat] :
( ( groups7121269368397514597t_prod @ nat @ A @ G @ ( set_or7035219750837199246ssThan @ nat @ ( suc @ M ) @ ( suc @ N ) ) )
= ( groups7121269368397514597t_prod @ nat @ A
@ ^ [I2: nat] : ( G @ ( suc @ I2 ) )
@ ( set_or7035219750837199246ssThan @ nat @ M @ N ) ) ) ) ).
% prod.shift_bounds_Suc_ivl
thf(fact_3954_prod_Oshift__bounds__nat__ivl,axiom,
! [A: $tType] :
( ( comm_monoid_mult @ A )
=> ! [G: nat > A,M: nat,K: nat,N: nat] :
( ( groups7121269368397514597t_prod @ nat @ A @ G @ ( set_or7035219750837199246ssThan @ nat @ ( plus_plus @ nat @ M @ K ) @ ( plus_plus @ nat @ N @ K ) ) )
= ( groups7121269368397514597t_prod @ nat @ A
@ ^ [I2: nat] : ( G @ ( plus_plus @ nat @ I2 @ K ) )
@ ( set_or7035219750837199246ssThan @ nat @ M @ N ) ) ) ) ).
% prod.shift_bounds_nat_ivl
thf(fact_3955_prod_Oivl__cong,axiom,
! [A: $tType,B: $tType] :
( ( ( ord @ B )
& ( comm_monoid_mult @ A ) )
=> ! [A2: B,C2: B,B2: B,D2: B,G: B > A,H2: B > A] :
( ( A2 = C2 )
=> ( ( B2 = D2 )
=> ( ! [X3: B] :
( ( ord_less_eq @ B @ C2 @ X3 )
=> ( ( ord_less @ B @ X3 @ D2 )
=> ( ( G @ X3 )
= ( H2 @ X3 ) ) ) )
=> ( ( groups7121269368397514597t_prod @ B @ A @ G @ ( set_or7035219750837199246ssThan @ B @ A2 @ B2 ) )
= ( groups7121269368397514597t_prod @ B @ A @ H2 @ ( set_or7035219750837199246ssThan @ B @ C2 @ D2 ) ) ) ) ) ) ) ).
% prod.ivl_cong
thf(fact_3956_prod_OatLeastLessThan__concat,axiom,
! [A: $tType] :
( ( comm_monoid_mult @ A )
=> ! [M: nat,N: nat,P4: nat,G: nat > A] :
( ( ord_less_eq @ nat @ M @ N )
=> ( ( ord_less_eq @ nat @ N @ P4 )
=> ( ( times_times @ A @ ( groups7121269368397514597t_prod @ nat @ A @ G @ ( set_or7035219750837199246ssThan @ nat @ M @ N ) ) @ ( groups7121269368397514597t_prod @ nat @ A @ G @ ( set_or7035219750837199246ssThan @ nat @ N @ P4 ) ) )
= ( groups7121269368397514597t_prod @ nat @ A @ G @ ( set_or7035219750837199246ssThan @ nat @ M @ P4 ) ) ) ) ) ) ).
% prod.atLeastLessThan_concat
thf(fact_3957_atLeast0__lessThan__Suc,axiom,
! [N: nat] :
( ( set_or7035219750837199246ssThan @ nat @ ( zero_zero @ nat ) @ ( suc @ N ) )
= ( insert @ nat @ N @ ( set_or7035219750837199246ssThan @ nat @ ( zero_zero @ nat ) @ N ) ) ) ).
% atLeast0_lessThan_Suc
thf(fact_3958_subset__eq__atLeast0__lessThan__finite,axiom,
! [N4: set @ nat,N: nat] :
( ( ord_less_eq @ ( set @ nat ) @ N4 @ ( set_or7035219750837199246ssThan @ nat @ ( zero_zero @ nat ) @ N ) )
=> ( finite_finite2 @ nat @ N4 ) ) ).
% subset_eq_atLeast0_lessThan_finite
thf(fact_3959_listI__assn__cong,axiom,
! [A: $tType,B: $tType,I6: set @ nat,I7: set @ nat,Xs2: list @ A,Xs4: list @ A,Xsi: list @ B,Xsi2: list @ B,A3: A > B > assn,A8: A > B > assn] :
( ( I6 = I7 )
=> ( ( ( size_size @ ( list @ A ) @ Xs2 )
= ( size_size @ ( list @ A ) @ Xs4 ) )
=> ( ( ( size_size @ ( list @ B ) @ Xsi )
= ( size_size @ ( list @ B ) @ Xsi2 ) )
=> ( ! [I3: nat] :
( ( member @ nat @ I3 @ I6 )
=> ( ( ord_less @ nat @ I3 @ ( size_size @ ( list @ A ) @ Xs2 ) )
=> ( ( ( size_size @ ( list @ A ) @ Xs2 )
= ( size_size @ ( list @ B ) @ Xsi ) )
=> ( ( ( nth @ A @ Xs2 @ I3 )
= ( nth @ A @ Xs4 @ I3 ) )
& ( ( nth @ B @ Xsi @ I3 )
= ( nth @ B @ Xsi2 @ I3 ) )
& ( ( A3 @ ( nth @ A @ Xs2 @ I3 ) @ ( nth @ B @ Xsi @ I3 ) )
= ( A8 @ ( nth @ A @ Xs4 @ I3 ) @ ( nth @ B @ Xsi2 @ I3 ) ) ) ) ) ) )
=> ( ( vEBT_List_listI_assn @ A @ B @ I6 @ A3 @ Xs2 @ Xsi )
= ( vEBT_List_listI_assn @ A @ B @ I7 @ A8 @ Xs4 @ Xsi2 ) ) ) ) ) ) ).
% listI_assn_cong
thf(fact_3960_listI__assn__weak__cong,axiom,
! [A: $tType,B: $tType,I6: set @ nat,I7: set @ nat,A3: A > B > assn,A8: A > B > assn,Xs2: list @ A,Xs4: list @ A,Xsi: list @ B,Xsi2: list @ B] :
( ( I6 = I7 )
=> ( ( A3 = A8 )
=> ( ( ( size_size @ ( list @ A ) @ Xs2 )
= ( size_size @ ( list @ A ) @ Xs4 ) )
=> ( ( ( size_size @ ( list @ B ) @ Xsi )
= ( size_size @ ( list @ B ) @ Xsi2 ) )
=> ( ! [I3: nat] :
( ( member @ nat @ I3 @ I6 )
=> ( ( ord_less @ nat @ I3 @ ( size_size @ ( list @ A ) @ Xs2 ) )
=> ( ( ( size_size @ ( list @ A ) @ Xs2 )
= ( size_size @ ( list @ B ) @ Xsi ) )
=> ( ( ( nth @ A @ Xs2 @ I3 )
= ( nth @ A @ Xs4 @ I3 ) )
& ( ( nth @ B @ Xsi @ I3 )
= ( nth @ B @ Xsi2 @ I3 ) ) ) ) ) )
=> ( ( vEBT_List_listI_assn @ A @ B @ I6 @ A3 @ Xs2 @ Xsi )
= ( vEBT_List_listI_assn @ A @ B @ I7 @ A8 @ Xs4 @ Xsi2 ) ) ) ) ) ) ) ).
% listI_assn_weak_cong
thf(fact_3961_atLeastLessThan__subseteq__atLeastAtMost__iff,axiom,
! [A: $tType] :
( ( dense_linorder @ A )
=> ! [A2: A,B2: A,C2: A,D2: A] :
( ( ord_less_eq @ ( set @ A ) @ ( set_or7035219750837199246ssThan @ A @ A2 @ B2 ) @ ( set_or1337092689740270186AtMost @ A @ C2 @ D2 ) )
= ( ( ord_less @ A @ A2 @ B2 )
=> ( ( ord_less_eq @ A @ C2 @ A2 )
& ( ord_less_eq @ A @ B2 @ D2 ) ) ) ) ) ).
% atLeastLessThan_subseteq_atLeastAtMost_iff
thf(fact_3962_atLeastAtMost__subseteq__atLeastLessThan__iff,axiom,
! [A: $tType] :
( ( preorder @ A )
=> ! [A2: A,B2: A,C2: A,D2: A] :
( ( ord_less_eq @ ( set @ A ) @ ( set_or1337092689740270186AtMost @ A @ A2 @ B2 ) @ ( set_or7035219750837199246ssThan @ A @ C2 @ D2 ) )
= ( ( ord_less_eq @ A @ A2 @ B2 )
=> ( ( ord_less_eq @ A @ C2 @ A2 )
& ( ord_less @ A @ B2 @ D2 ) ) ) ) ) ).
% atLeastAtMost_subseteq_atLeastLessThan_iff
thf(fact_3963_subst__not__in,axiom,
! [A: $tType,B: $tType,I: nat,I6: set @ nat,Xs2: list @ A,A3: A > B > assn,X15: A,Xsi: list @ B,X22: B] :
( ~ ( member @ nat @ I @ I6 )
=> ( ( ord_less @ nat @ I @ ( size_size @ ( list @ A ) @ Xs2 ) )
=> ( ( vEBT_List_listI_assn @ A @ B @ I6 @ A3 @ ( list_update @ A @ Xs2 @ I @ X15 ) @ ( list_update @ B @ Xsi @ I @ X22 ) )
= ( vEBT_List_listI_assn @ A @ B @ I6 @ A3 @ Xs2 @ Xsi ) ) ) ) ).
% subst_not_in
thf(fact_3964_prod_OatLeast0__lessThan__Suc,axiom,
! [A: $tType] :
( ( comm_monoid_mult @ A )
=> ! [G: nat > A,N: nat] :
( ( groups7121269368397514597t_prod @ nat @ A @ G @ ( set_or7035219750837199246ssThan @ nat @ ( zero_zero @ nat ) @ ( suc @ N ) ) )
= ( times_times @ A @ ( groups7121269368397514597t_prod @ nat @ A @ G @ ( set_or7035219750837199246ssThan @ nat @ ( zero_zero @ nat ) @ N ) ) @ ( G @ N ) ) ) ) ).
% prod.atLeast0_lessThan_Suc
thf(fact_3965_prod_OatLeast__Suc__lessThan,axiom,
! [A: $tType] :
( ( comm_monoid_mult @ A )
=> ! [M: nat,N: nat,G: nat > A] :
( ( ord_less @ nat @ M @ N )
=> ( ( groups7121269368397514597t_prod @ nat @ A @ G @ ( set_or7035219750837199246ssThan @ nat @ M @ N ) )
= ( times_times @ A @ ( G @ M ) @ ( groups7121269368397514597t_prod @ nat @ A @ G @ ( set_or7035219750837199246ssThan @ nat @ ( suc @ M ) @ N ) ) ) ) ) ) ).
% prod.atLeast_Suc_lessThan
thf(fact_3966_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_3967_prod_Olast__plus,axiom,
! [A: $tType] :
( ( comm_monoid_mult @ A )
=> ! [M: nat,N: nat,G: nat > A] :
( ( ord_less_eq @ nat @ M @ N )
=> ( ( groups7121269368397514597t_prod @ nat @ A @ G @ ( set_or1337092689740270186AtMost @ nat @ M @ N ) )
= ( times_times @ A @ ( G @ N ) @ ( groups7121269368397514597t_prod @ nat @ A @ G @ ( set_or7035219750837199246ssThan @ nat @ M @ N ) ) ) ) ) ) ).
% prod.last_plus
thf(fact_3968_pi__less__4,axiom,
ord_less @ real @ pi @ ( numeral_numeral @ real @ ( bit0 @ ( bit0 @ one2 ) ) ) ).
% pi_less_4
thf(fact_3969_pi__ge__two,axiom,
ord_less_eq @ real @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) @ pi ).
% pi_ge_two
thf(fact_3970_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_3971_atLeastLessThanSuc,axiom,
! [M: nat,N: nat] :
( ( ( ord_less_eq @ nat @ M @ N )
=> ( ( set_or7035219750837199246ssThan @ nat @ M @ ( suc @ N ) )
= ( insert @ nat @ N @ ( set_or7035219750837199246ssThan @ nat @ M @ N ) ) ) )
& ( ~ ( ord_less_eq @ nat @ M @ N )
=> ( ( set_or7035219750837199246ssThan @ nat @ M @ ( suc @ N ) )
= ( bot_bot @ ( set @ nat ) ) ) ) ) ).
% atLeastLessThanSuc
thf(fact_3972_prod_OatLeastLessThan__rev,axiom,
! [A: $tType] :
( ( comm_monoid_mult @ A )
=> ! [G: nat > A,N: nat,M: nat] :
( ( groups7121269368397514597t_prod @ nat @ A @ G @ ( set_or7035219750837199246ssThan @ nat @ N @ M ) )
= ( groups7121269368397514597t_prod @ nat @ A
@ ^ [I2: nat] : ( G @ ( minus_minus @ nat @ ( plus_plus @ nat @ M @ N ) @ ( suc @ I2 ) ) )
@ ( set_or7035219750837199246ssThan @ nat @ N @ M ) ) ) ) ).
% prod.atLeastLessThan_rev
thf(fact_3973_prod_Onested__swap,axiom,
! [A: $tType] :
( ( comm_monoid_mult @ A )
=> ! [A2: nat > nat > A,N: nat] :
( ( groups7121269368397514597t_prod @ nat @ A
@ ^ [I2: nat] : ( groups7121269368397514597t_prod @ nat @ A @ ( A2 @ I2 ) @ ( set_or7035219750837199246ssThan @ nat @ ( zero_zero @ nat ) @ I2 ) )
@ ( set_or1337092689740270186AtMost @ nat @ ( zero_zero @ nat ) @ N ) )
= ( groups7121269368397514597t_prod @ nat @ A
@ ^ [J2: nat] :
( groups7121269368397514597t_prod @ nat @ A
@ ^ [I2: nat] : ( A2 @ I2 @ J2 )
@ ( set_or1337092689740270186AtMost @ nat @ ( suc @ J2 ) @ N ) )
@ ( set_or7035219750837199246ssThan @ nat @ ( zero_zero @ nat ) @ N ) ) ) ) ).
% prod.nested_swap
thf(fact_3974_prod__Suc__Suc__fact,axiom,
! [N: nat] :
( ( groups7121269368397514597t_prod @ nat @ nat @ suc @ ( set_or7035219750837199246ssThan @ nat @ ( suc @ ( zero_zero @ nat ) ) @ N ) )
= ( semiring_char_0_fact @ nat @ N ) ) ).
% prod_Suc_Suc_fact
thf(fact_3975_prod__Suc__fact,axiom,
! [N: nat] :
( ( groups7121269368397514597t_prod @ nat @ nat @ suc @ ( set_or7035219750837199246ssThan @ nat @ ( zero_zero @ nat ) @ N ) )
= ( semiring_char_0_fact @ nat @ N ) ) ).
% prod_Suc_fact
thf(fact_3976_prod_Ohead__if,axiom,
! [A: $tType] :
( ( comm_monoid_mult @ A )
=> ! [N: nat,M: nat,G: nat > A] :
( ( ( ord_less @ nat @ N @ M )
=> ( ( groups7121269368397514597t_prod @ nat @ A @ G @ ( set_or1337092689740270186AtMost @ nat @ M @ N ) )
= ( one_one @ A ) ) )
& ( ~ ( ord_less @ nat @ N @ M )
=> ( ( groups7121269368397514597t_prod @ nat @ A @ G @ ( set_or1337092689740270186AtMost @ nat @ M @ N ) )
= ( times_times @ A @ ( groups7121269368397514597t_prod @ nat @ A @ G @ ( set_or7035219750837199246ssThan @ nat @ M @ N ) ) @ ( G @ N ) ) ) ) ) ) ).
% prod.head_if
thf(fact_3977_pi__half__neq__zero,axiom,
( ( divide_divide @ real @ pi @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) )
!= ( zero_zero @ real ) ) ).
% pi_half_neq_zero
thf(fact_3978_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_3979_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_3980_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_3981_fact__prod__Suc,axiom,
! [A: $tType] :
( ( semiring_char_0 @ A )
=> ( ( semiring_char_0_fact @ A )
= ( ^ [N3: nat] : ( semiring_1_of_nat @ A @ ( groups7121269368397514597t_prod @ nat @ nat @ suc @ ( set_or7035219750837199246ssThan @ nat @ ( zero_zero @ nat ) @ N3 ) ) ) ) ) ) ).
% fact_prod_Suc
thf(fact_3982_prod_OatLeastLessThan__rev__at__least__Suc__atMost,axiom,
! [A: $tType] :
( ( comm_monoid_mult @ A )
=> ! [G: nat > A,N: nat,M: nat] :
( ( groups7121269368397514597t_prod @ nat @ A @ G @ ( set_or7035219750837199246ssThan @ nat @ N @ M ) )
= ( groups7121269368397514597t_prod @ nat @ A
@ ^ [I2: nat] : ( G @ ( minus_minus @ nat @ ( plus_plus @ nat @ M @ N ) @ I2 ) )
@ ( set_or1337092689740270186AtMost @ nat @ ( suc @ N ) @ M ) ) ) ) ).
% prod.atLeastLessThan_rev_at_least_Suc_atMost
thf(fact_3983_pochhammer__prod,axiom,
! [A: $tType] :
( ( comm_semiring_1 @ A )
=> ( ( comm_s3205402744901411588hammer @ A )
= ( ^ [A4: A,N3: nat] :
( groups7121269368397514597t_prod @ nat @ A
@ ^ [I2: nat] : ( plus_plus @ A @ A4 @ ( semiring_1_of_nat @ A @ I2 ) )
@ ( set_or7035219750837199246ssThan @ nat @ ( zero_zero @ nat ) @ N3 ) ) ) ) ) ).
% pochhammer_prod
thf(fact_3984_fact__prod__rev,axiom,
! [A: $tType] :
( ( semiring_char_0 @ A )
=> ( ( semiring_char_0_fact @ A )
= ( ^ [N3: nat] : ( semiring_1_of_nat @ A @ ( groups7121269368397514597t_prod @ nat @ nat @ ( minus_minus @ nat @ N3 ) @ ( set_or7035219750837199246ssThan @ nat @ ( zero_zero @ nat ) @ N3 ) ) ) ) ) ) ).
% fact_prod_rev
thf(fact_3985_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_3986_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_3987_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_3988_arctan__ubound,axiom,
! [Y: real] : ( ord_less @ real @ ( arctan @ Y ) @ ( divide_divide @ real @ pi @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) ) ).
% arctan_ubound
thf(fact_3989_arctan__one,axiom,
( ( arctan @ ( one_one @ real ) )
= ( divide_divide @ real @ pi @ ( numeral_numeral @ real @ ( bit0 @ ( bit0 @ one2 ) ) ) ) ) ).
% arctan_one
thf(fact_3990_listI__assn__insert,axiom,
! [A: $tType,B: $tType,I: nat,I6: set @ nat,Xs2: list @ A,A3: A > B > assn,Xsi: list @ B] :
( ~ ( member @ nat @ I @ I6 )
=> ( ( ord_less @ nat @ I @ ( size_size @ ( list @ A ) @ Xs2 ) )
=> ( ( vEBT_List_listI_assn @ A @ B @ ( insert @ nat @ I @ I6 ) @ A3 @ Xs2 @ Xsi )
= ( times_times @ assn @ ( A3 @ ( nth @ A @ Xs2 @ I ) @ ( nth @ B @ Xsi @ I ) ) @ ( vEBT_List_listI_assn @ A @ B @ I6 @ A3 @ Xs2 @ Xsi ) ) ) ) ) ).
% listI_assn_insert
thf(fact_3991_listI__assn__subst,axiom,
! [A: $tType,B: $tType,I: nat,I6: set @ nat,Xs2: list @ A,A3: A > B > assn,X15: A,Xsi: list @ B,X22: B] :
( ~ ( member @ nat @ I @ I6 )
=> ( ( ord_less @ nat @ I @ ( size_size @ ( list @ A ) @ Xs2 ) )
=> ( ( vEBT_List_listI_assn @ A @ B @ ( insert @ nat @ I @ I6 ) @ 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 @ I6 @ A3 @ Xs2 @ Xsi ) ) ) ) ) ).
% listI_assn_subst
thf(fact_3992_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_3993_fact__split,axiom,
! [A: $tType] :
( ( semiring_char_0 @ A )
=> ! [K: nat,N: nat] :
( ( ord_less_eq @ nat @ K @ N )
=> ( ( semiring_char_0_fact @ A @ N )
= ( times_times @ A @ ( semiring_1_of_nat @ A @ ( groups7121269368397514597t_prod @ nat @ nat @ suc @ ( set_or7035219750837199246ssThan @ nat @ ( minus_minus @ nat @ N @ K ) @ N ) ) ) @ ( semiring_char_0_fact @ A @ ( minus_minus @ nat @ N @ K ) ) ) ) ) ) ).
% fact_split
thf(fact_3994_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_3995_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_3996_binomial__altdef__of__nat,axiom,
! [A: $tType] :
( ( field_char_0 @ A )
=> ! [K: nat,N: nat] :
( ( ord_less_eq @ nat @ K @ N )
=> ( ( semiring_1_of_nat @ A @ ( binomial @ N @ K ) )
= ( groups7121269368397514597t_prod @ nat @ A
@ ^ [I2: nat] : ( divide_divide @ A @ ( semiring_1_of_nat @ A @ ( minus_minus @ nat @ N @ I2 ) ) @ ( semiring_1_of_nat @ A @ ( minus_minus @ nat @ K @ I2 ) ) )
@ ( set_or7035219750837199246ssThan @ nat @ ( zero_zero @ nat ) @ K ) ) ) ) ) ).
% binomial_altdef_of_nat
thf(fact_3997_listI__assn__extract,axiom,
! [A: $tType,B: $tType,I: nat,I6: set @ nat,Xs2: list @ A,A3: A > B > assn,Xsi: list @ B] :
( ( member @ nat @ I @ I6 )
=> ( ( ord_less @ nat @ I @ ( size_size @ ( list @ A ) @ Xs2 ) )
=> ( ( vEBT_List_listI_assn @ A @ B @ I6 @ 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 ) @ I6 @ ( insert @ nat @ I @ ( bot_bot @ ( set @ nat ) ) ) ) @ A3 @ Xs2 @ Xsi ) ) ) ) ) ).
% listI_assn_extract
thf(fact_3998_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_3999_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_4000_listI__assn__reinsert,axiom,
! [B: $tType,A: $tType,P: assn,A3: A > B > assn,Xs2: list @ A,I: nat,Xsi: list @ B,I6: set @ nat,F3: 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 ) @ I6 @ ( insert @ nat @ I @ ( bot_bot @ ( set @ nat ) ) ) ) @ A3 @ Xs2 @ Xsi ) ) @ F3 ) )
=> ( ( ord_less @ nat @ I @ ( size_size @ ( list @ A ) @ Xs2 ) )
=> ( ( member @ nat @ I @ I6 )
=> ( ( entails @ ( times_times @ assn @ ( vEBT_List_listI_assn @ A @ B @ I6 @ A3 @ Xs2 @ Xsi ) @ F3 ) @ Q )
=> ( entails @ P @ Q ) ) ) ) ) ).
% listI_assn_reinsert
thf(fact_4001_listI__assn__reinsert__upd,axiom,
! [D4: $tType,C: $tType,P: assn,A3: C > D4 > assn,X: C,Xi2: D4,I6: set @ nat,I: nat,Xs2: list @ C,Xsi: list @ D4,F3: assn,Q: assn] :
( ( entails @ P @ ( times_times @ assn @ ( times_times @ assn @ ( A3 @ X @ Xi2 ) @ ( vEBT_List_listI_assn @ C @ D4 @ ( minus_minus @ ( set @ nat ) @ I6 @ ( insert @ nat @ I @ ( bot_bot @ ( set @ nat ) ) ) ) @ A3 @ Xs2 @ Xsi ) ) @ F3 ) )
=> ( ( ord_less @ nat @ I @ ( size_size @ ( list @ C ) @ Xs2 ) )
=> ( ( member @ nat @ I @ I6 )
=> ( ( entails @ ( times_times @ assn @ ( vEBT_List_listI_assn @ C @ D4 @ I6 @ A3 @ ( list_update @ C @ Xs2 @ I @ X ) @ ( list_update @ D4 @ Xsi @ I @ Xi2 ) ) @ F3 ) @ Q )
=> ( entails @ P @ Q ) ) ) ) ) ).
% listI_assn_reinsert_upd
thf(fact_4002_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_4003_sin__cos__npi,axiom,
! [N: nat] :
( ( sin @ real @ ( divide_divide @ real @ ( times_times @ real @ ( semiring_1_of_nat @ real @ ( suc @ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N ) ) ) @ pi ) @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) )
= ( power_power @ real @ ( uminus_uminus @ real @ ( one_one @ real ) ) @ N ) ) ).
% sin_cos_npi
thf(fact_4004_divmod__algorithm__code_I8_J,axiom,
! [A: $tType] :
( ( unique1627219031080169319umeral @ A )
=> ! [M: num,N: num] :
( ( ( ord_less @ num @ M @ N )
=> ( ( unique8689654367752047608divmod @ A @ ( bit1 @ M ) @ ( bit1 @ N ) )
= ( product_Pair @ A @ A @ ( zero_zero @ A ) @ ( numeral_numeral @ A @ ( bit1 @ M ) ) ) ) )
& ( ~ ( ord_less @ num @ M @ N )
=> ( ( unique8689654367752047608divmod @ A @ ( bit1 @ M ) @ ( bit1 @ N ) )
= ( unique1321980374590559556d_step @ A @ ( bit1 @ N ) @ ( unique8689654367752047608divmod @ A @ ( bit1 @ M ) @ ( bit0 @ ( bit1 @ N ) ) ) ) ) ) ) ) ).
% divmod_algorithm_code(8)
thf(fact_4005_divmod__algorithm__code_I7_J,axiom,
! [A: $tType] :
( ( unique1627219031080169319umeral @ A )
=> ! [M: num,N: num] :
( ( ( ord_less_eq @ num @ M @ N )
=> ( ( unique8689654367752047608divmod @ A @ ( bit0 @ M ) @ ( bit1 @ N ) )
= ( product_Pair @ A @ A @ ( zero_zero @ A ) @ ( numeral_numeral @ A @ ( bit0 @ M ) ) ) ) )
& ( ~ ( ord_less_eq @ num @ M @ N )
=> ( ( unique8689654367752047608divmod @ A @ ( bit0 @ M ) @ ( bit1 @ N ) )
= ( unique1321980374590559556d_step @ A @ ( bit1 @ N ) @ ( unique8689654367752047608divmod @ A @ ( bit0 @ M ) @ ( bit0 @ ( bit1 @ N ) ) ) ) ) ) ) ) ).
% divmod_algorithm_code(7)
thf(fact_4006_divides__aux__eq,axiom,
! [A: $tType] :
( ( unique1627219031080169319umeral @ A )
=> ! [Q3: A,R2: A] :
( ( unique5940410009612947441es_aux @ A @ ( product_Pair @ A @ A @ Q3 @ R2 ) )
= ( R2
= ( zero_zero @ A ) ) ) ) ).
% divides_aux_eq
thf(fact_4007_sin__zero,axiom,
! [A: $tType] :
( ( ( real_Vector_banach @ A )
& ( real_V2822296259951069270ebra_1 @ A ) )
=> ( ( sin @ A @ ( zero_zero @ A ) )
= ( zero_zero @ A ) ) ) ).
% sin_zero
thf(fact_4008_sin__pi,axiom,
( ( sin @ real @ pi )
= ( zero_zero @ real ) ) ).
% sin_pi
thf(fact_4009_sin__of__real__pi,axiom,
! [A: $tType] :
( ( ( real_Vector_banach @ A )
& ( real_V2822296259951069270ebra_1 @ A ) )
=> ( ( sin @ A @ ( real_Vector_of_real @ A @ pi ) )
= ( zero_zero @ A ) ) ) ).
% sin_of_real_pi
thf(fact_4010_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_4011_dvd__numeral__simp,axiom,
! [A: $tType] :
( ( unique1627219031080169319umeral @ A )
=> ! [M: num,N: num] :
( ( dvd_dvd @ A @ ( numeral_numeral @ A @ M ) @ ( numeral_numeral @ A @ N ) )
= ( unique5940410009612947441es_aux @ A @ ( unique8689654367752047608divmod @ A @ N @ M ) ) ) ) ).
% dvd_numeral_simp
thf(fact_4012_sin__npi,axiom,
! [N: nat] :
( ( sin @ real @ ( times_times @ real @ ( semiring_1_of_nat @ real @ N ) @ pi ) )
= ( zero_zero @ real ) ) ).
% sin_npi
thf(fact_4013_sin__npi2,axiom,
! [N: nat] :
( ( sin @ real @ ( times_times @ real @ pi @ ( semiring_1_of_nat @ real @ N ) ) )
= ( zero_zero @ real ) ) ).
% sin_npi2
thf(fact_4014_sin__npi__int,axiom,
! [N: int] :
( ( sin @ real @ ( times_times @ real @ pi @ ( ring_1_of_int @ real @ N ) ) )
= ( zero_zero @ real ) ) ).
% sin_npi_int
thf(fact_4015_divmod__algorithm__code_I3_J,axiom,
! [A: $tType] :
( ( unique1627219031080169319umeral @ A )
=> ! [N: num] :
( ( unique8689654367752047608divmod @ A @ one2 @ ( bit0 @ N ) )
= ( product_Pair @ A @ A @ ( zero_zero @ A ) @ ( numeral_numeral @ A @ one2 ) ) ) ) ).
% divmod_algorithm_code(3)
thf(fact_4016_divmod__algorithm__code_I4_J,axiom,
! [A: $tType] :
( ( unique1627219031080169319umeral @ A )
=> ! [N: num] :
( ( unique8689654367752047608divmod @ A @ one2 @ ( bit1 @ N ) )
= ( product_Pair @ A @ A @ ( zero_zero @ A ) @ ( numeral_numeral @ A @ one2 ) ) ) ) ).
% divmod_algorithm_code(4)
thf(fact_4017_sin__two__pi,axiom,
( ( sin @ real @ ( times_times @ real @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) @ pi ) )
= ( zero_zero @ real ) ) ).
% sin_two_pi
thf(fact_4018_sin__pi__half,axiom,
( ( sin @ real @ ( divide_divide @ real @ pi @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) )
= ( one_one @ real ) ) ).
% sin_pi_half
thf(fact_4019_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_4020_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_4021_sin__2npi,axiom,
! [N: nat] :
( ( sin @ real @ ( times_times @ real @ ( times_times @ real @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) @ ( semiring_1_of_nat @ real @ N ) ) @ pi ) )
= ( zero_zero @ real ) ) ).
% sin_2npi
thf(fact_4022_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_4023_sin__int__2pin,axiom,
! [N: int] :
( ( sin @ real @ ( times_times @ real @ ( times_times @ real @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) @ pi ) @ ( ring_1_of_int @ real @ N ) ) )
= ( zero_zero @ real ) ) ).
% sin_int_2pin
thf(fact_4024_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_4025_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_4026_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_4027_sin__le__one,axiom,
! [X: real] : ( ord_less_eq @ real @ ( sin @ real @ X ) @ ( one_one @ real ) ) ).
% sin_le_one
thf(fact_4028_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_4029_finite__atLeastZeroLessThan__int,axiom,
! [U: int] : ( finite_finite2 @ int @ ( set_or7035219750837199246ssThan @ int @ ( zero_zero @ int ) @ U ) ) ).
% finite_atLeastZeroLessThan_int
thf(fact_4030_atLeastLessThanPlusOne__atLeastAtMost__integer,axiom,
! [L: code_integer,U: code_integer] :
( ( set_or7035219750837199246ssThan @ code_integer @ L @ ( plus_plus @ code_integer @ U @ ( one_one @ code_integer ) ) )
= ( set_or1337092689740270186AtMost @ code_integer @ L @ U ) ) ).
% atLeastLessThanPlusOne_atLeastAtMost_integer
thf(fact_4031_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_4032_sin__gt__zero,axiom,
! [X: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ X )
=> ( ( ord_less @ real @ X @ pi )
=> ( ord_less @ real @ ( zero_zero @ real ) @ ( sin @ real @ X ) ) ) ) ).
% sin_gt_zero
thf(fact_4033_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_4034_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_4035_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_4036_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_4037_atLeastLessThanPlusOne__atLeastAtMost__int,axiom,
! [L: int,U: int] :
( ( set_or7035219750837199246ssThan @ int @ L @ ( plus_plus @ int @ U @ ( one_one @ int ) ) )
= ( set_or1337092689740270186AtMost @ int @ L @ U ) ) ).
% atLeastLessThanPlusOne_atLeastAtMost_int
thf(fact_4038_sin__eq__0__pi,axiom,
! [X: real] :
( ( ord_less @ real @ ( uminus_uminus @ real @ pi ) @ X )
=> ( ( ord_less @ real @ X @ pi )
=> ( ( ( sin @ real @ X )
= ( zero_zero @ real ) )
=> ( X
= ( zero_zero @ real ) ) ) ) ) ).
% sin_eq_0_pi
thf(fact_4039_divmod__int__def,axiom,
( ( unique8689654367752047608divmod @ int )
= ( ^ [M4: num,N3: num] : ( product_Pair @ int @ int @ ( divide_divide @ int @ ( numeral_numeral @ int @ M4 ) @ ( numeral_numeral @ int @ N3 ) ) @ ( modulo_modulo @ int @ ( numeral_numeral @ int @ M4 ) @ ( numeral_numeral @ int @ N3 ) ) ) ) ) ).
% divmod_int_def
thf(fact_4040_sin__zero__pi__iff,axiom,
! [X: real] :
( ( ord_less @ real @ ( abs_abs @ real @ X ) @ pi )
=> ( ( ( sin @ real @ X )
= ( zero_zero @ real ) )
= ( X
= ( zero_zero @ real ) ) ) ) ).
% sin_zero_pi_iff
thf(fact_4041_sin__zero__iff__int2,axiom,
! [X: real] :
( ( ( sin @ real @ X )
= ( zero_zero @ real ) )
= ( ? [I2: int] :
( X
= ( times_times @ real @ ( ring_1_of_int @ real @ I2 ) @ pi ) ) ) ) ).
% sin_zero_iff_int2
thf(fact_4042_word__atLeastLessThan__Suc__atLeastAtMost,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [U: word @ A,L: word @ A] :
( ( U
!= ( uminus_uminus @ ( word @ A ) @ ( one_one @ ( word @ A ) ) ) )
=> ( ( set_or7035219750837199246ssThan @ ( word @ A ) @ L @ ( plus_plus @ ( word @ A ) @ U @ ( one_one @ ( word @ A ) ) ) )
= ( set_or1337092689740270186AtMost @ ( word @ A ) @ L @ U ) ) ) ) ).
% word_atLeastLessThan_Suc_atLeastAtMost
thf(fact_4043_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_4044_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_4045_divmod__def,axiom,
! [A: $tType] :
( ( unique1627219031080169319umeral @ A )
=> ( ( unique8689654367752047608divmod @ A )
= ( ^ [M4: num,N3: num] : ( product_Pair @ A @ A @ ( divide_divide @ A @ ( numeral_numeral @ A @ M4 ) @ ( numeral_numeral @ A @ N3 ) ) @ ( modulo_modulo @ A @ ( numeral_numeral @ A @ M4 ) @ ( numeral_numeral @ A @ N3 ) ) ) ) ) ) ).
% divmod_def
thf(fact_4046_divmod_H__nat__def,axiom,
( ( unique8689654367752047608divmod @ nat )
= ( ^ [M4: num,N3: num] : ( product_Pair @ nat @ nat @ ( divide_divide @ nat @ ( numeral_numeral @ nat @ M4 ) @ ( numeral_numeral @ nat @ N3 ) ) @ ( modulo_modulo @ nat @ ( numeral_numeral @ nat @ M4 ) @ ( numeral_numeral @ nat @ N3 ) ) ) ) ) ).
% divmod'_nat_def
thf(fact_4047_sin__pi__divide__n__ge__0,axiom,
! [N: nat] :
( ( N
!= ( zero_zero @ nat ) )
=> ( ord_less_eq @ real @ ( zero_zero @ real ) @ ( sin @ real @ ( divide_divide @ real @ pi @ ( semiring_1_of_nat @ real @ N ) ) ) ) ) ).
% sin_pi_divide_n_ge_0
thf(fact_4048_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_4049_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_4050_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_4051_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_4052_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_4053_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_4054_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_4055_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_4056_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_4057_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_4058_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_4059_sin__pi__divide__n__gt__0,axiom,
! [N: nat] :
( ( ord_less_eq @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N )
=> ( ord_less @ real @ ( zero_zero @ real ) @ ( sin @ real @ ( divide_divide @ real @ pi @ ( semiring_1_of_nat @ real @ N ) ) ) ) ) ).
% sin_pi_divide_n_gt_0
thf(fact_4060_divmod__divmod__step,axiom,
! [A: $tType] :
( ( unique1627219031080169319umeral @ A )
=> ( ( unique8689654367752047608divmod @ A )
= ( ^ [M4: num,N3: num] : ( if @ ( product_prod @ A @ A ) @ ( ord_less @ num @ M4 @ N3 ) @ ( product_Pair @ A @ A @ ( zero_zero @ A ) @ ( numeral_numeral @ A @ M4 ) ) @ ( unique1321980374590559556d_step @ A @ N3 @ ( unique8689654367752047608divmod @ A @ M4 @ ( bit0 @ N3 ) ) ) ) ) ) ) ).
% divmod_divmod_step
thf(fact_4061_sin__zero__iff__int,axiom,
! [X: real] :
( ( ( sin @ real @ X )
= ( zero_zero @ real ) )
= ( ? [I2: int] :
( ( dvd_dvd @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ I2 )
& ( X
= ( times_times @ real @ ( ring_1_of_int @ real @ I2 ) @ ( divide_divide @ real @ pi @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) ) ) ) ) ) ).
% sin_zero_iff_int
thf(fact_4062_sin__zero__lemma,axiom,
! [X: real] :
( ( ord_less_eq @ real @ ( zero_zero @ real ) @ X )
=> ( ( ( sin @ real @ X )
= ( zero_zero @ real ) )
=> ? [N2: nat] :
( ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N2 )
& ( X
= ( times_times @ real @ ( semiring_1_of_nat @ real @ N2 ) @ ( divide_divide @ real @ pi @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) ) ) ) ) ) ).
% sin_zero_lemma
thf(fact_4063_sin__zero__iff,axiom,
! [X: real] :
( ( ( sin @ real @ X )
= ( zero_zero @ real ) )
= ( ? [N3: nat] :
( ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N3 )
& ( X
= ( times_times @ real @ ( semiring_1_of_nat @ real @ N3 ) @ ( divide_divide @ real @ pi @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) ) ) )
| ? [N3: nat] :
( ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N3 )
& ( X
= ( uminus_uminus @ real @ ( times_times @ real @ ( semiring_1_of_nat @ real @ N3 ) @ ( divide_divide @ real @ pi @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) ) ) ) ) ) ) ).
% sin_zero_iff
thf(fact_4064_one__div__minus__numeral,axiom,
! [N: num] :
( ( divide_divide @ int @ ( one_one @ int ) @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ N ) ) )
= ( uminus_uminus @ int @ ( adjust_div @ ( unique8689654367752047608divmod @ int @ one2 @ N ) ) ) ) ).
% one_div_minus_numeral
thf(fact_4065_minus__one__div__numeral,axiom,
! [N: num] :
( ( divide_divide @ int @ ( uminus_uminus @ int @ ( one_one @ int ) ) @ ( numeral_numeral @ int @ N ) )
= ( uminus_uminus @ int @ ( adjust_div @ ( unique8689654367752047608divmod @ int @ one2 @ N ) ) ) ) ).
% minus_one_div_numeral
thf(fact_4066_htt__cons__rule,axiom,
! [A: $tType,P3: assn,C2: heap_Time_Heap @ A,Q4: A > assn,T6: nat,P: assn,Q: A > assn,T: nat] :
( ( time_htt @ A @ P3 @ C2 @ Q4 @ T6 )
=> ( ( entails @ P @ P3 )
=> ( ! [X3: A] : ( entails @ ( Q4 @ X3 ) @ ( Q @ X3 ) )
=> ( ( ord_less_eq @ nat @ T6 @ T )
=> ( time_htt @ A @ P @ C2 @ Q @ T ) ) ) ) ) ).
% htt_cons_rule
thf(fact_4067_summable__arctan__series,axiom,
! [X: real] :
( ( ord_less_eq @ real @ ( abs_abs @ real @ X ) @ ( one_one @ real ) )
=> ( summable @ real
@ ^ [K3: nat] : ( times_times @ real @ ( power_power @ real @ ( uminus_uminus @ real @ ( one_one @ real ) ) @ K3 ) @ ( times_times @ real @ ( divide_divide @ real @ ( one_one @ real ) @ ( semiring_1_of_nat @ real @ ( plus_plus @ nat @ ( times_times @ nat @ K3 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ ( one_one @ nat ) ) ) ) @ ( power_power @ real @ X @ ( plus_plus @ nat @ ( times_times @ nat @ K3 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ ( one_one @ nat ) ) ) ) ) ) ) ).
% summable_arctan_series
thf(fact_4068_summable__single,axiom,
! [A: $tType] :
( ( ( comm_monoid_add @ A )
& ( topolo4958980785337419405_space @ A ) )
=> ! [I: nat,F2: nat > A] :
( summable @ A
@ ^ [R5: nat] : ( if @ A @ ( R5 = I ) @ ( F2 @ R5 ) @ ( zero_zero @ A ) ) ) ) ).
% summable_single
thf(fact_4069_summable__zero,axiom,
! [A: $tType] :
( ( ( comm_monoid_add @ A )
& ( topolo4958980785337419405_space @ A ) )
=> ( summable @ A
@ ^ [N3: nat] : ( zero_zero @ A ) ) ) ).
% summable_zero
thf(fact_4070_summable__iff__shift,axiom,
! [A: $tType] :
( ( real_V822414075346904944vector @ A )
=> ! [F2: nat > A,K: nat] :
( ( summable @ A
@ ^ [N3: nat] : ( F2 @ ( plus_plus @ nat @ N3 @ K ) ) )
= ( summable @ A @ F2 ) ) ) ).
% summable_iff_shift
thf(fact_4071_summable__cmult__iff,axiom,
! [A: $tType] :
( ( real_V3459762299906320749_field @ A )
=> ! [C2: A,F2: nat > A] :
( ( summable @ A
@ ^ [N3: nat] : ( times_times @ A @ C2 @ ( F2 @ N3 ) ) )
= ( ( C2
= ( zero_zero @ A ) )
| ( summable @ A @ F2 ) ) ) ) ).
% summable_cmult_iff
thf(fact_4072_summable__divide__iff,axiom,
! [A: $tType] :
( ( real_V3459762299906320749_field @ A )
=> ! [F2: nat > A,C2: A] :
( ( summable @ A
@ ^ [N3: nat] : ( divide_divide @ A @ ( F2 @ N3 ) @ C2 ) )
= ( ( C2
= ( zero_zero @ A ) )
| ( summable @ A @ F2 ) ) ) ) ).
% summable_divide_iff
thf(fact_4073_summable__If__finite__set,axiom,
! [A: $tType] :
( ( ( comm_monoid_add @ A )
& ( topolo4958980785337419405_space @ A ) )
=> ! [A3: set @ nat,F2: nat > A] :
( ( finite_finite2 @ nat @ A3 )
=> ( summable @ A
@ ^ [R5: nat] : ( if @ A @ ( member @ nat @ R5 @ A3 ) @ ( F2 @ R5 ) @ ( zero_zero @ A ) ) ) ) ) ).
% summable_If_finite_set
thf(fact_4074_summable__If__finite,axiom,
! [A: $tType] :
( ( ( comm_monoid_add @ A )
& ( topolo4958980785337419405_space @ A ) )
=> ! [P: nat > $o,F2: nat > A] :
( ( finite_finite2 @ nat @ ( collect @ nat @ P ) )
=> ( summable @ A
@ ^ [R5: nat] : ( if @ A @ ( P @ R5 ) @ ( F2 @ R5 ) @ ( zero_zero @ A ) ) ) ) ) ).
% summable_If_finite
thf(fact_4075_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_4076_Divides_Oadjust__div__eq,axiom,
! [Q3: int,R2: int] :
( ( adjust_div @ ( product_Pair @ int @ int @ Q3 @ R2 ) )
= ( plus_plus @ int @ Q3
@ ( zero_neq_one_of_bool @ int
@ ( R2
!= ( zero_zero @ int ) ) ) ) ) ).
% Divides.adjust_div_eq
thf(fact_4077_minus__numeral__div__numeral,axiom,
! [M: num,N: num] :
( ( divide_divide @ int @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ M ) ) @ ( numeral_numeral @ int @ N ) )
= ( uminus_uminus @ int @ ( adjust_div @ ( unique8689654367752047608divmod @ int @ M @ N ) ) ) ) ).
% minus_numeral_div_numeral
thf(fact_4078_numeral__div__minus__numeral,axiom,
! [M: num,N: num] :
( ( divide_divide @ int @ ( numeral_numeral @ int @ M ) @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ N ) ) )
= ( uminus_uminus @ int @ ( adjust_div @ ( unique8689654367752047608divmod @ int @ M @ N ) ) ) ) ).
% numeral_div_minus_numeral
thf(fact_4079_summable__const__iff,axiom,
! [A: $tType] :
( ( real_V822414075346904944vector @ A )
=> ! [C2: A] :
( ( summable @ A
@ ^ [Uu3: nat] : C2 )
= ( C2
= ( zero_zero @ A ) ) ) ) ).
% summable_const_iff
thf(fact_4080_summable__add,axiom,
! [A: $tType] :
( ( ( topolo5987344860129210374id_add @ A )
& ( topological_t2_space @ A ) )
=> ! [F2: nat > A,G: nat > A] :
( ( summable @ A @ F2 )
=> ( ( summable @ A @ G )
=> ( summable @ A
@ ^ [N3: nat] : ( plus_plus @ A @ ( F2 @ N3 ) @ ( G @ N3 ) ) ) ) ) ) ).
% summable_add
thf(fact_4081_summable__ignore__initial__segment,axiom,
! [A: $tType] :
( ( real_V822414075346904944vector @ A )
=> ! [F2: nat > A,K: nat] :
( ( summable @ A @ F2 )
=> ( summable @ A
@ ^ [N3: nat] : ( F2 @ ( plus_plus @ nat @ N3 @ K ) ) ) ) ) ).
% summable_ignore_initial_segment
thf(fact_4082_summable__mult2,axiom,
! [A: $tType] :
( ( real_V4412858255891104859lgebra @ A )
=> ! [F2: nat > A,C2: A] :
( ( summable @ A @ F2 )
=> ( summable @ A
@ ^ [N3: nat] : ( times_times @ A @ ( F2 @ N3 ) @ C2 ) ) ) ) ).
% summable_mult2
thf(fact_4083_summable__mult,axiom,
! [A: $tType] :
( ( real_V4412858255891104859lgebra @ A )
=> ! [F2: nat > A,C2: A] :
( ( summable @ A @ F2 )
=> ( summable @ A
@ ^ [N3: nat] : ( times_times @ A @ C2 @ ( F2 @ N3 ) ) ) ) ) ).
% summable_mult
thf(fact_4084_summable__Suc__iff,axiom,
! [A: $tType] :
( ( real_V822414075346904944vector @ A )
=> ! [F2: nat > A] :
( ( summable @ A
@ ^ [N3: nat] : ( F2 @ ( suc @ N3 ) ) )
= ( summable @ A @ F2 ) ) ) ).
% summable_Suc_iff
thf(fact_4085_summable__divide,axiom,
! [A: $tType] :
( ( real_V3459762299906320749_field @ A )
=> ! [F2: nat > A,C2: A] :
( ( summable @ A @ F2 )
=> ( summable @ A
@ ^ [N3: nat] : ( divide_divide @ A @ ( F2 @ N3 ) @ C2 ) ) ) ) ).
% summable_divide
thf(fact_4086_powser__insidea,axiom,
! [A: $tType] :
( ( real_V8999393235501362500lgebra @ A )
=> ! [F2: nat > A,X: A,Z: A] :
( ( summable @ A
@ ^ [N3: nat] : ( times_times @ A @ ( F2 @ N3 ) @ ( power_power @ A @ X @ N3 ) ) )
=> ( ( ord_less @ real @ ( real_V7770717601297561774m_norm @ A @ Z ) @ ( real_V7770717601297561774m_norm @ A @ X ) )
=> ( summable @ real
@ ^ [N3: nat] : ( real_V7770717601297561774m_norm @ A @ ( times_times @ A @ ( F2 @ N3 ) @ ( power_power @ A @ Z @ N3 ) ) ) ) ) ) ) ).
% powser_insidea
thf(fact_4087_summable__finite,axiom,
! [A: $tType] :
( ( ( comm_monoid_add @ A )
& ( topolo4958980785337419405_space @ A ) )
=> ! [N4: set @ nat,F2: nat > A] :
( ( finite_finite2 @ nat @ N4 )
=> ( ! [N2: nat] :
( ~ ( member @ nat @ N2 @ N4 )
=> ( ( F2 @ N2 )
= ( zero_zero @ A ) ) )
=> ( summable @ A @ F2 ) ) ) ) ).
% summable_finite
thf(fact_4088_summable__mult__D,axiom,
! [A: $tType] :
( ( real_V3459762299906320749_field @ A )
=> ! [C2: A,F2: nat > A] :
( ( summable @ A
@ ^ [N3: nat] : ( times_times @ A @ C2 @ ( F2 @ N3 ) ) )
=> ( ( C2
!= ( zero_zero @ A ) )
=> ( summable @ A @ F2 ) ) ) ) ).
% summable_mult_D
thf(fact_4089_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_4090_suminf__add,axiom,
! [A: $tType] :
( ( ( topolo5987344860129210374id_add @ A )
& ( topological_t2_space @ A ) )
=> ! [F2: nat > A,G: nat > A] :
( ( summable @ A @ F2 )
=> ( ( summable @ A @ G )
=> ( ( plus_plus @ A @ ( suminf @ A @ F2 ) @ ( suminf @ A @ G ) )
= ( suminf @ A
@ ^ [N3: nat] : ( plus_plus @ A @ ( F2 @ N3 ) @ ( G @ N3 ) ) ) ) ) ) ) ).
% suminf_add
thf(fact_4091_suminf__mult,axiom,
! [A: $tType] :
( ( real_V4412858255891104859lgebra @ A )
=> ! [F2: nat > A,C2: A] :
( ( summable @ A @ F2 )
=> ( ( suminf @ A
@ ^ [N3: nat] : ( times_times @ A @ C2 @ ( F2 @ N3 ) ) )
= ( times_times @ A @ C2 @ ( suminf @ A @ F2 ) ) ) ) ) ).
% suminf_mult
thf(fact_4092_suminf__mult2,axiom,
! [A: $tType] :
( ( real_V4412858255891104859lgebra @ A )
=> ! [F2: nat > A,C2: A] :
( ( summable @ A @ F2 )
=> ( ( times_times @ A @ ( suminf @ A @ F2 ) @ C2 )
= ( suminf @ A
@ ^ [N3: nat] : ( times_times @ A @ ( F2 @ N3 ) @ C2 ) ) ) ) ) ).
% suminf_mult2
thf(fact_4093_suminf__divide,axiom,
! [A: $tType] :
( ( real_V3459762299906320749_field @ A )
=> ! [F2: nat > A,C2: A] :
( ( summable @ A @ F2 )
=> ( ( suminf @ A
@ ^ [N3: nat] : ( divide_divide @ A @ ( F2 @ N3 ) @ C2 ) )
= ( divide_divide @ A @ ( suminf @ A @ F2 ) @ C2 ) ) ) ) ).
% suminf_divide
thf(fact_4094_suminf__nonneg,axiom,
! [A: $tType] :
( ( ( ordere6911136660526730532id_add @ A )
& ( topolo1944317154257567458pology @ A ) )
=> ! [F2: nat > A] :
( ( summable @ A @ F2 )
=> ( ! [N2: nat] : ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( F2 @ N2 ) )
=> ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( suminf @ A @ F2 ) ) ) ) ) ).
% suminf_nonneg
thf(fact_4095_suminf__eq__zero__iff,axiom,
! [A: $tType] :
( ( ( ordere6911136660526730532id_add @ A )
& ( topolo1944317154257567458pology @ A ) )
=> ! [F2: nat > A] :
( ( summable @ A @ F2 )
=> ( ! [N2: nat] : ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( F2 @ N2 ) )
=> ( ( ( suminf @ A @ F2 )
= ( zero_zero @ A ) )
= ( ! [N3: nat] :
( ( F2 @ N3 )
= ( zero_zero @ A ) ) ) ) ) ) ) ).
% suminf_eq_zero_iff
thf(fact_4096_suminf__pos,axiom,
! [A: $tType] :
( ( ( ordere6911136660526730532id_add @ A )
& ( topolo1944317154257567458pology @ A ) )
=> ! [F2: nat > A] :
( ( summable @ A @ F2 )
=> ( ! [N2: nat] : ( ord_less @ A @ ( zero_zero @ A ) @ ( F2 @ N2 ) )
=> ( ord_less @ A @ ( zero_zero @ A ) @ ( suminf @ A @ F2 ) ) ) ) ) ).
% suminf_pos
thf(fact_4097_summable__zero__power_H,axiom,
! [A: $tType] :
( ( ( ring_1 @ A )
& ( topolo4958980785337419405_space @ A ) )
=> ! [F2: nat > A] :
( summable @ A
@ ^ [N3: nat] : ( times_times @ A @ ( F2 @ N3 ) @ ( power_power @ A @ ( zero_zero @ A ) @ N3 ) ) ) ) ).
% summable_zero_power'
thf(fact_4098_summable__0__powser,axiom,
! [A: $tType] :
( ( real_V8999393235501362500lgebra @ A )
=> ! [F2: nat > A] :
( summable @ A
@ ^ [N3: nat] : ( times_times @ A @ ( F2 @ N3 ) @ ( power_power @ A @ ( zero_zero @ A ) @ N3 ) ) ) ) ).
% summable_0_powser
thf(fact_4099_summable__powser__split__head,axiom,
! [A: $tType] :
( ( real_V8999393235501362500lgebra @ A )
=> ! [F2: nat > A,Z: A] :
( ( summable @ A
@ ^ [N3: nat] : ( times_times @ A @ ( F2 @ ( suc @ N3 ) ) @ ( power_power @ A @ Z @ N3 ) ) )
= ( summable @ A
@ ^ [N3: nat] : ( times_times @ A @ ( F2 @ N3 ) @ ( power_power @ A @ Z @ N3 ) ) ) ) ) ).
% summable_powser_split_head
thf(fact_4100_powser__split__head_I3_J,axiom,
! [A: $tType] :
( ( ( real_Vector_banach @ A )
& ( real_V8999393235501362500lgebra @ A ) )
=> ! [F2: nat > A,Z: A] :
( ( summable @ A
@ ^ [N3: nat] : ( times_times @ A @ ( F2 @ N3 ) @ ( power_power @ A @ Z @ N3 ) ) )
=> ( summable @ A
@ ^ [N3: nat] : ( times_times @ A @ ( F2 @ ( suc @ N3 ) ) @ ( power_power @ A @ Z @ N3 ) ) ) ) ) ).
% powser_split_head(3)
thf(fact_4101_summable__powser__ignore__initial__segment,axiom,
! [A: $tType] :
( ( real_V8999393235501362500lgebra @ A )
=> ! [F2: nat > A,M: nat,Z: A] :
( ( summable @ A
@ ^ [N3: nat] : ( times_times @ A @ ( F2 @ ( plus_plus @ nat @ N3 @ M ) ) @ ( power_power @ A @ Z @ N3 ) ) )
= ( summable @ A
@ ^ [N3: nat] : ( times_times @ A @ ( F2 @ N3 ) @ ( power_power @ A @ Z @ N3 ) ) ) ) ) ).
% summable_powser_ignore_initial_segment
thf(fact_4102_suminf__pos2,axiom,
! [A: $tType] :
( ( ( ordere6911136660526730532id_add @ A )
& ( topolo1944317154257567458pology @ A ) )
=> ! [F2: nat > A,I: nat] :
( ( summable @ A @ F2 )
=> ( ! [N2: nat] : ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( F2 @ N2 ) )
=> ( ( ord_less @ A @ ( zero_zero @ A ) @ ( F2 @ I ) )
=> ( ord_less @ A @ ( zero_zero @ A ) @ ( suminf @ A @ F2 ) ) ) ) ) ) ).
% suminf_pos2
thf(fact_4103_suminf__pos__iff,axiom,
! [A: $tType] :
( ( ( ordere6911136660526730532id_add @ A )
& ( topolo1944317154257567458pology @ A ) )
=> ! [F2: nat > A] :
( ( summable @ A @ F2 )
=> ( ! [N2: nat] : ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( F2 @ N2 ) )
=> ( ( ord_less @ A @ ( zero_zero @ A ) @ ( suminf @ A @ F2 ) )
= ( ? [I2: nat] : ( ord_less @ A @ ( zero_zero @ A ) @ ( F2 @ I2 ) ) ) ) ) ) ) ).
% suminf_pos_iff
thf(fact_4104_powser__inside,axiom,
! [A: $tType] :
( ( ( real_Vector_banach @ A )
& ( real_V8999393235501362500lgebra @ A ) )
=> ! [F2: nat > A,X: A,Z: A] :
( ( summable @ A
@ ^ [N3: nat] : ( times_times @ A @ ( F2 @ N3 ) @ ( power_power @ A @ X @ N3 ) ) )
=> ( ( ord_less @ real @ ( real_V7770717601297561774m_norm @ A @ Z ) @ ( real_V7770717601297561774m_norm @ A @ X ) )
=> ( summable @ A
@ ^ [N3: nat] : ( times_times @ A @ ( F2 @ N3 ) @ ( power_power @ A @ Z @ N3 ) ) ) ) ) ) ).
% powser_inside
thf(fact_4105_suminf__split__head,axiom,
! [A: $tType] :
( ( real_V822414075346904944vector @ A )
=> ! [F2: nat > A] :
( ( summable @ A @ F2 )
=> ( ( suminf @ A
@ ^ [N3: nat] : ( F2 @ ( suc @ N3 ) ) )
= ( minus_minus @ A @ ( suminf @ A @ F2 ) @ ( F2 @ ( zero_zero @ nat ) ) ) ) ) ) ).
% suminf_split_head
thf(fact_4106_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_4107_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_4108_powser__split__head_I1_J,axiom,
! [A: $tType] :
( ( ( real_Vector_banach @ A )
& ( real_V8999393235501362500lgebra @ A ) )
=> ! [F2: nat > A,Z: A] :
( ( summable @ A
@ ^ [N3: nat] : ( times_times @ A @ ( F2 @ N3 ) @ ( power_power @ A @ Z @ N3 ) ) )
=> ( ( suminf @ A
@ ^ [N3: nat] : ( times_times @ A @ ( F2 @ N3 ) @ ( power_power @ A @ Z @ N3 ) ) )
= ( plus_plus @ A @ ( F2 @ ( zero_zero @ nat ) )
@ ( times_times @ A
@ ( suminf @ A
@ ^ [N3: nat] : ( times_times @ A @ ( F2 @ ( suc @ N3 ) ) @ ( power_power @ A @ Z @ N3 ) ) )
@ Z ) ) ) ) ) ).
% powser_split_head(1)
thf(fact_4109_powser__split__head_I2_J,axiom,
! [A: $tType] :
( ( ( real_Vector_banach @ A )
& ( real_V8999393235501362500lgebra @ A ) )
=> ! [F2: nat > A,Z: A] :
( ( summable @ A
@ ^ [N3: nat] : ( times_times @ A @ ( F2 @ N3 ) @ ( power_power @ A @ Z @ N3 ) ) )
=> ( ( times_times @ A
@ ( suminf @ A
@ ^ [N3: nat] : ( times_times @ A @ ( F2 @ ( suc @ N3 ) ) @ ( power_power @ A @ Z @ N3 ) ) )
@ Z )
= ( minus_minus @ A
@ ( suminf @ A
@ ^ [N3: nat] : ( times_times @ A @ ( F2 @ N3 ) @ ( power_power @ A @ Z @ N3 ) ) )
@ ( F2 @ ( zero_zero @ nat ) ) ) ) ) ) ).
% powser_split_head(2)
thf(fact_4110_suminf__exist__split,axiom,
! [A: $tType] :
( ( real_V822414075346904944vector @ A )
=> ! [R2: real,F2: nat > A] :
( ( ord_less @ real @ ( zero_zero @ real ) @ R2 )
=> ( ( summable @ A @ F2 )
=> ? [N9: nat] :
! [N10: nat] :
( ( ord_less_eq @ nat @ N9 @ N10 )
=> ( ord_less @ real
@ ( real_V7770717601297561774m_norm @ A
@ ( suminf @ A
@ ^ [I2: nat] : ( F2 @ ( plus_plus @ nat @ I2 @ N10 ) ) ) )
@ R2 ) ) ) ) ) ).
% suminf_exist_split
thf(fact_4111_summable__power__series,axiom,
! [F2: nat > real,Z: real] :
( ! [I3: nat] : ( ord_less_eq @ real @ ( F2 @ I3 ) @ ( one_one @ real ) )
=> ( ! [I3: nat] : ( ord_less_eq @ real @ ( zero_zero @ real ) @ ( F2 @ I3 ) )
=> ( ( ord_less_eq @ real @ ( zero_zero @ real ) @ Z )
=> ( ( ord_less @ real @ Z @ ( one_one @ real ) )
=> ( summable @ real
@ ^ [I2: nat] : ( times_times @ real @ ( F2 @ I2 ) @ ( power_power @ real @ Z @ I2 ) ) ) ) ) ) ) ).
% summable_power_series
thf(fact_4112_Abel__lemma,axiom,
! [A: $tType] :
( ( real_V822414075346904944vector @ A )
=> ! [R2: real,R0: real,A2: nat > A,M7: real] :
( ( ord_less_eq @ real @ ( zero_zero @ real ) @ R2 )
=> ( ( ord_less @ real @ R2 @ R0 )
=> ( ! [N2: nat] : ( ord_less_eq @ real @ ( times_times @ real @ ( real_V7770717601297561774m_norm @ A @ ( A2 @ N2 ) ) @ ( power_power @ real @ R0 @ N2 ) ) @ M7 )
=> ( summable @ real
@ ^ [N3: nat] : ( times_times @ real @ ( real_V7770717601297561774m_norm @ A @ ( A2 @ N3 ) ) @ ( power_power @ real @ R2 @ N3 ) ) ) ) ) ) ) ).
% Abel_lemma
thf(fact_4113_summable__ratio__test,axiom,
! [A: $tType] :
( ( real_Vector_banach @ A )
=> ! [C2: real,N4: nat,F2: nat > A] :
( ( ord_less @ real @ C2 @ ( one_one @ real ) )
=> ( ! [N2: nat] :
( ( ord_less_eq @ nat @ N4 @ N2 )
=> ( ord_less_eq @ real @ ( real_V7770717601297561774m_norm @ A @ ( F2 @ ( suc @ N2 ) ) ) @ ( times_times @ real @ C2 @ ( real_V7770717601297561774m_norm @ A @ ( F2 @ N2 ) ) ) ) )
=> ( summable @ A @ F2 ) ) ) ) ).
% summable_ratio_test
thf(fact_4114_norm__pre__pure__iff__htt,axiom,
! [A: $tType,P: assn,B2: $o,F2: heap_Time_Heap @ A,Q: A > assn,T: nat] :
( ( time_htt @ A @ ( times_times @ assn @ P @ ( pure_assn @ B2 ) ) @ F2 @ Q @ T )
= ( B2
=> ( time_htt @ A @ P @ F2 @ Q @ T ) ) ) ).
% norm_pre_pure_iff_htt
thf(fact_4115_norm__pre__pure__iff__htt_H,axiom,
! [A: $tType,B2: $o,P: assn,F2: heap_Time_Heap @ A,Q: A > assn,T: nat] :
( ( time_htt @ A @ ( times_times @ assn @ ( pure_assn @ B2 ) @ P ) @ F2 @ Q @ T )
= ( B2
=> ( time_htt @ A @ P @ F2 @ Q @ T ) ) ) ).
% norm_pre_pure_iff_htt'
thf(fact_4116_neg__eucl__rel__int__mult__2,axiom,
! [B2: int,A2: int,Q3: int,R2: int] :
( ( ord_less_eq @ int @ B2 @ ( zero_zero @ int ) )
=> ( ( eucl_rel_int @ ( plus_plus @ int @ A2 @ ( one_one @ int ) ) @ B2 @ ( product_Pair @ int @ int @ Q3 @ R2 ) )
=> ( eucl_rel_int @ ( plus_plus @ int @ ( one_one @ int ) @ ( times_times @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ A2 ) ) @ ( times_times @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ B2 ) @ ( product_Pair @ int @ int @ Q3 @ ( minus_minus @ int @ ( times_times @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ R2 ) @ ( one_one @ int ) ) ) ) ) ) ).
% neg_eucl_rel_int_mult_2
thf(fact_4117_pos__eucl__rel__int__mult__2,axiom,
! [B2: int,A2: int,Q3: int,R2: int] :
( ( ord_less_eq @ int @ ( zero_zero @ int ) @ B2 )
=> ( ( eucl_rel_int @ A2 @ B2 @ ( product_Pair @ int @ int @ Q3 @ R2 ) )
=> ( eucl_rel_int @ ( plus_plus @ int @ ( one_one @ int ) @ ( times_times @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ A2 ) ) @ ( times_times @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ B2 ) @ ( product_Pair @ int @ int @ Q3 @ ( plus_plus @ int @ ( one_one @ int ) @ ( times_times @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ R2 ) ) ) ) ) ) ).
% pos_eucl_rel_int_mult_2
thf(fact_4118_product__nth,axiom,
! [A: $tType,B: $tType,N: nat,Xs2: list @ A,Ys: list @ B] :
( ( ord_less @ nat @ N @ ( times_times @ nat @ ( size_size @ ( list @ A ) @ Xs2 ) @ ( size_size @ ( list @ B ) @ Ys ) ) )
=> ( ( nth @ ( product_prod @ A @ B ) @ ( product @ A @ B @ Xs2 @ Ys ) @ N )
= ( product_Pair @ A @ B @ ( nth @ A @ Xs2 @ ( divide_divide @ nat @ N @ ( size_size @ ( list @ B ) @ Ys ) ) ) @ ( nth @ B @ Ys @ ( modulo_modulo @ nat @ N @ ( size_size @ ( list @ B ) @ Ys ) ) ) ) ) ) ).
% product_nth
thf(fact_4119_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_4120_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_4121_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_4122_cos__pi,axiom,
( ( cos @ real @ pi )
= ( uminus_uminus @ real @ ( one_one @ real ) ) ) ).
% cos_pi
thf(fact_4123_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_4124_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_4125_cos__pi__half,axiom,
( ( cos @ real @ ( divide_divide @ real @ pi @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) )
= ( zero_zero @ real ) ) ).
% cos_pi_half
thf(fact_4126_cos__two__pi,axiom,
( ( cos @ real @ ( times_times @ real @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) @ pi ) )
= ( one_one @ real ) ) ).
% cos_two_pi
thf(fact_4127_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_4128_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_4129_cos__npi,axiom,
! [N: nat] :
( ( cos @ real @ ( times_times @ real @ ( semiring_1_of_nat @ real @ N ) @ pi ) )
= ( power_power @ real @ ( uminus_uminus @ real @ ( one_one @ real ) ) @ N ) ) ).
% cos_npi
thf(fact_4130_cos__npi2,axiom,
! [N: nat] :
( ( cos @ real @ ( times_times @ real @ pi @ ( semiring_1_of_nat @ real @ N ) ) )
= ( power_power @ real @ ( uminus_uminus @ real @ ( one_one @ real ) ) @ N ) ) ).
% cos_npi2
thf(fact_4131_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_4132_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_4133_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_4134_cos__2npi,axiom,
! [N: nat] :
( ( cos @ real @ ( times_times @ real @ ( times_times @ real @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) @ ( semiring_1_of_nat @ real @ N ) ) @ pi ) )
= ( one_one @ real ) ) ).
% cos_2npi
thf(fact_4135_cos__int__2pin,axiom,
! [N: int] :
( ( cos @ real @ ( times_times @ real @ ( times_times @ real @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) @ pi ) @ ( ring_1_of_int @ real @ N ) ) )
= ( one_one @ real ) ) ).
% cos_int_2pin
thf(fact_4136_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_4137_cos__npi__int,axiom,
! [N: int] :
( ( ( dvd_dvd @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ N )
=> ( ( cos @ real @ ( times_times @ real @ pi @ ( ring_1_of_int @ real @ N ) ) )
= ( one_one @ real ) ) )
& ( ~ ( dvd_dvd @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ N )
=> ( ( cos @ real @ ( times_times @ real @ pi @ ( ring_1_of_int @ real @ N ) ) )
= ( uminus_uminus @ real @ ( one_one @ real ) ) ) ) ) ).
% cos_npi_int
thf(fact_4138_unique__remainder,axiom,
! [A2: int,B2: int,Q3: int,R2: int,Q7: int,R4: int] :
( ( eucl_rel_int @ A2 @ B2 @ ( product_Pair @ int @ int @ Q3 @ R2 ) )
=> ( ( eucl_rel_int @ A2 @ B2 @ ( product_Pair @ int @ int @ Q7 @ R4 ) )
=> ( R2 = R4 ) ) ) ).
% unique_remainder
thf(fact_4139_unique__quotient,axiom,
! [A2: int,B2: int,Q3: int,R2: int,Q7: int,R4: int] :
( ( eucl_rel_int @ A2 @ B2 @ ( product_Pair @ int @ int @ Q3 @ R2 ) )
=> ( ( eucl_rel_int @ A2 @ B2 @ ( product_Pair @ int @ int @ Q7 @ R4 ) )
=> ( Q3 = Q7 ) ) ) ).
% unique_quotient
thf(fact_4140_cos__le__one,axiom,
! [X: real] : ( ord_less_eq @ real @ ( cos @ real @ X ) @ ( one_one @ real ) ) ).
% cos_le_one
thf(fact_4141_polar__Ex,axiom,
! [X: real,Y: real] :
? [R3: real,A6: real] :
( ( X
= ( times_times @ real @ R3 @ ( cos @ real @ A6 ) ) )
& ( Y
= ( times_times @ real @ R3 @ ( sin @ real @ A6 ) ) ) ) ).
% polar_Ex
thf(fact_4142_cos__arctan__not__zero,axiom,
! [X: real] :
( ( cos @ real @ ( arctan @ X ) )
!= ( zero_zero @ real ) ) ).
% cos_arctan_not_zero
thf(fact_4143_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_4144_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_4145_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_4146_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_4147_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_4148_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_4149_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_4150_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_4151_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_4152_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_4153_mod__int__unique,axiom,
! [K: int,L: int,Q3: int,R2: int] :
( ( eucl_rel_int @ K @ L @ ( product_Pair @ int @ int @ Q3 @ R2 ) )
=> ( ( modulo_modulo @ int @ K @ L )
= R2 ) ) ).
% mod_int_unique
thf(fact_4154_div__int__unique,axiom,
! [K: int,L: int,Q3: int,R2: int] :
( ( eucl_rel_int @ K @ L @ ( product_Pair @ int @ int @ Q3 @ R2 ) )
=> ( ( divide_divide @ int @ K @ L )
= Q3 ) ) ).
% div_int_unique
thf(fact_4155_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_4156_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_4157_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_4158_cos__two__neq__zero,axiom,
( ( cos @ real @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) )
!= ( zero_zero @ real ) ) ).
% cos_two_neq_zero
thf(fact_4159_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_4160_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_4161_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_4162_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_4163_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_4164_eucl__rel__int__dividesI,axiom,
! [L: int,K: int,Q3: int] :
( ( L
!= ( zero_zero @ int ) )
=> ( ( K
= ( times_times @ int @ Q3 @ L ) )
=> ( eucl_rel_int @ K @ L @ ( product_Pair @ int @ int @ Q3 @ ( zero_zero @ int ) ) ) ) ) ).
% eucl_rel_int_dividesI
thf(fact_4165_eucl__rel__int,axiom,
! [K: int,L: int] : ( eucl_rel_int @ K @ L @ ( product_Pair @ int @ int @ ( divide_divide @ int @ K @ L ) @ ( modulo_modulo @ int @ K @ L ) ) ) ).
% eucl_rel_int
thf(fact_4166_cos__two__less__zero,axiom,
ord_less @ real @ ( cos @ real @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) @ ( zero_zero @ real ) ).
% cos_two_less_zero
thf(fact_4167_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_4168_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_4169_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_4170_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_4171_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_4172_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_4173_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_4174_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_4175_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_4176_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_4177_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_4178_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_4179_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_4180_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_4181_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_4182_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_4183_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_4184_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_4185_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_4186_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_4187_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_4188_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_4189_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_4190_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_4191_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_4192_zminus1__lemma,axiom,
! [A2: int,B2: int,Q3: int,R2: int] :
( ( eucl_rel_int @ A2 @ B2 @ ( product_Pair @ int @ int @ Q3 @ R2 ) )
=> ( ( B2
!= ( zero_zero @ int ) )
=> ( eucl_rel_int @ ( uminus_uminus @ int @ A2 ) @ B2
@ ( product_Pair @ int @ int
@ ( if @ int
@ ( R2
= ( zero_zero @ int ) )
@ ( uminus_uminus @ int @ Q3 )
@ ( minus_minus @ int @ ( uminus_uminus @ int @ Q3 ) @ ( one_one @ int ) ) )
@ ( if @ int
@ ( R2
= ( zero_zero @ int ) )
@ ( zero_zero @ int )
@ ( minus_minus @ int @ B2 @ R2 ) ) ) ) ) ) ).
% zminus1_lemma
thf(fact_4193_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_4194_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_4195_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_4196_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_4197_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_4198_eucl__rel__int__iff,axiom,
! [K: int,L: int,Q3: int,R2: int] :
( ( eucl_rel_int @ K @ L @ ( product_Pair @ int @ int @ Q3 @ R2 ) )
= ( ( K
= ( plus_plus @ int @ ( times_times @ int @ L @ Q3 ) @ R2 ) )
& ( ( ord_less @ int @ ( zero_zero @ int ) @ L )
=> ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ R2 )
& ( ord_less @ int @ R2 @ L ) ) )
& ( ~ ( ord_less @ int @ ( zero_zero @ int ) @ L )
=> ( ( ( ord_less @ int @ L @ ( zero_zero @ int ) )
=> ( ( ord_less @ int @ L @ R2 )
& ( ord_less_eq @ int @ R2 @ ( zero_zero @ int ) ) ) )
& ( ~ ( ord_less @ int @ L @ ( zero_zero @ int ) )
=> ( Q3
= ( zero_zero @ int ) ) ) ) ) ) ) ).
% eucl_rel_int_iff
thf(fact_4199_eucl__rel__int__remainderI,axiom,
! [R2: int,L: int,K: int,Q3: int] :
( ( ( sgn_sgn @ int @ R2 )
= ( sgn_sgn @ int @ L ) )
=> ( ( ord_less @ int @ ( abs_abs @ int @ R2 ) @ ( abs_abs @ int @ L ) )
=> ( ( K
= ( plus_plus @ int @ ( times_times @ int @ Q3 @ L ) @ R2 ) )
=> ( eucl_rel_int @ K @ L @ ( product_Pair @ int @ int @ Q3 @ R2 ) ) ) ) ) ).
% eucl_rel_int_remainderI
thf(fact_4200_sincos__total__pi,axiom,
! [Y: real,X: real] :
( ( ord_less_eq @ real @ ( zero_zero @ real ) @ Y )
=> ( ( ( plus_plus @ real @ ( power_power @ real @ X @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ ( power_power @ real @ Y @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) )
= ( one_one @ real ) )
=> ? [T7: real] :
( ( ord_less_eq @ real @ ( zero_zero @ real ) @ T7 )
& ( ord_less_eq @ real @ T7 @ pi )
& ( X
= ( cos @ real @ T7 ) )
& ( Y
= ( sin @ real @ T7 ) ) ) ) ) ).
% sincos_total_pi
thf(fact_4201_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_4202_cos__zero__iff__int,axiom,
! [X: real] :
( ( ( cos @ real @ X )
= ( zero_zero @ real ) )
= ( ? [I2: int] :
( ~ ( dvd_dvd @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ I2 )
& ( X
= ( times_times @ real @ ( ring_1_of_int @ real @ I2 ) @ ( divide_divide @ real @ pi @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) ) ) ) ) ) ).
% cos_zero_iff_int
thf(fact_4203_eucl__rel__int_Osimps,axiom,
( eucl_rel_int
= ( ^ [A12: int,A23: int,A32: product_prod @ int @ int] :
( ? [K3: int] :
( ( A12 = K3 )
& ( A23
= ( zero_zero @ int ) )
& ( A32
= ( product_Pair @ int @ int @ ( zero_zero @ int ) @ K3 ) ) )
| ? [L2: int,K3: int,Q6: int] :
( ( A12 = K3 )
& ( A23 = L2 )
& ( A32
= ( product_Pair @ int @ int @ Q6 @ ( zero_zero @ int ) ) )
& ( L2
!= ( zero_zero @ int ) )
& ( K3
= ( times_times @ int @ Q6 @ L2 ) ) )
| ? [R5: int,L2: int,K3: int,Q6: int] :
( ( A12 = K3 )
& ( A23 = L2 )
& ( A32
= ( product_Pair @ int @ int @ Q6 @ R5 ) )
& ( ( sgn_sgn @ int @ R5 )
= ( sgn_sgn @ int @ L2 ) )
& ( ord_less @ int @ ( abs_abs @ int @ R5 ) @ ( abs_abs @ int @ L2 ) )
& ( K3
= ( plus_plus @ int @ ( times_times @ int @ Q6 @ L2 ) @ R5 ) ) ) ) ) ) ).
% eucl_rel_int.simps
thf(fact_4204_eucl__rel__int_Ocases,axiom,
! [A1: int,A22: int,A33: product_prod @ int @ int] :
( ( eucl_rel_int @ A1 @ A22 @ A33 )
=> ( ( ( A22
= ( zero_zero @ int ) )
=> ( A33
!= ( product_Pair @ int @ int @ ( zero_zero @ int ) @ A1 ) ) )
=> ( ! [Q5: int] :
( ( A33
= ( product_Pair @ int @ int @ Q5 @ ( zero_zero @ int ) ) )
=> ( ( A22
!= ( zero_zero @ int ) )
=> ( A1
!= ( times_times @ int @ Q5 @ A22 ) ) ) )
=> ~ ! [R3: int,Q5: int] :
( ( A33
= ( product_Pair @ int @ int @ Q5 @ R3 ) )
=> ( ( ( sgn_sgn @ int @ R3 )
= ( sgn_sgn @ int @ A22 ) )
=> ( ( ord_less @ int @ ( abs_abs @ int @ R3 ) @ ( abs_abs @ int @ A22 ) )
=> ( A1
!= ( plus_plus @ int @ ( times_times @ int @ Q5 @ A22 ) @ R3 ) ) ) ) ) ) ) ) ).
% eucl_rel_int.cases
thf(fact_4205_cos__zero__lemma,axiom,
! [X: real] :
( ( ord_less_eq @ real @ ( zero_zero @ real ) @ X )
=> ( ( ( cos @ real @ X )
= ( zero_zero @ real ) )
=> ? [N2: nat] :
( ~ ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N2 )
& ( X
= ( times_times @ real @ ( semiring_1_of_nat @ real @ N2 ) @ ( divide_divide @ real @ pi @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) ) ) ) ) ) ).
% cos_zero_lemma
thf(fact_4206_cos__zero__iff,axiom,
! [X: real] :
( ( ( cos @ real @ X )
= ( zero_zero @ real ) )
= ( ? [N3: nat] :
( ~ ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N3 )
& ( X
= ( times_times @ real @ ( semiring_1_of_nat @ real @ N3 ) @ ( divide_divide @ real @ pi @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) ) ) )
| ? [N3: nat] :
( ~ ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N3 )
& ( X
= ( uminus_uminus @ real @ ( times_times @ real @ ( semiring_1_of_nat @ real @ N3 ) @ ( divide_divide @ real @ pi @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) ) ) ) ) ) ) ).
% cos_zero_iff
thf(fact_4207_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_4208_sincos__total__pi__half,axiom,
! [X: real,Y: real] :
( ( ord_less_eq @ real @ ( zero_zero @ real ) @ X )
=> ( ( ord_less_eq @ real @ ( zero_zero @ real ) @ Y )
=> ( ( ( plus_plus @ real @ ( power_power @ real @ X @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ ( power_power @ real @ Y @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) )
= ( one_one @ real ) )
=> ? [T7: real] :
( ( ord_less_eq @ real @ ( zero_zero @ real ) @ T7 )
& ( ord_less_eq @ real @ T7 @ ( divide_divide @ real @ pi @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) )
& ( X
= ( cos @ real @ T7 ) )
& ( Y
= ( sin @ real @ T7 ) ) ) ) ) ) ).
% sincos_total_pi_half
thf(fact_4209_sincos__total__2pi__le,axiom,
! [X: real,Y: real] :
( ( ( plus_plus @ real @ ( power_power @ real @ X @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ ( power_power @ real @ Y @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) )
= ( one_one @ real ) )
=> ? [T7: real] :
( ( ord_less_eq @ real @ ( zero_zero @ real ) @ T7 )
& ( ord_less_eq @ real @ T7 @ ( times_times @ real @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) @ pi ) )
& ( X
= ( cos @ real @ T7 ) )
& ( Y
= ( sin @ real @ T7 ) ) ) ) ).
% sincos_total_2pi_le
thf(fact_4210_sincos__total__2pi,axiom,
! [X: real,Y: real] :
( ( ( plus_plus @ real @ ( power_power @ real @ X @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ ( power_power @ real @ Y @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) )
= ( one_one @ real ) )
=> ~ ! [T7: real] :
( ( ord_less_eq @ real @ ( zero_zero @ real ) @ T7 )
=> ( ( ord_less @ real @ T7 @ ( times_times @ real @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) @ pi ) )
=> ( ( X
= ( cos @ real @ T7 ) )
=> ( Y
!= ( sin @ real @ T7 ) ) ) ) ) ) ).
% sincos_total_2pi
thf(fact_4211_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_4212__C11_C,axiom,
! [Xa: array @ vEBT_VEBTi,X: list @ vEBT_VEBTi] :
( hoare_hoare_triple @ vEBT_VEBTi @ ( times_times @ assn @ ( snga_assn @ vEBT_VEBTi @ Xa @ X ) @ ( vEBT_List_list_assn @ vEBT_VEBT @ vEBT_VEBTi @ vEBT_vebt_assn_raw @ ( replicate @ vEBT_VEBT @ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ ( bit0 @ one2 ) ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ na @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( vEBT_vebt_buildup @ ( suc @ ( divide_divide @ nat @ na @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) @ X ) ) @ ( vEBT_V739175172307565963ildupi @ ( suc @ ( divide_divide @ nat @ ( suc @ ( suc @ na ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) )
@ ^ [R5: vEBT_VEBTi] : ( times_times @ assn @ ( times_times @ assn @ ( snga_assn @ vEBT_VEBTi @ Xa @ X ) @ ( vEBT_List_list_assn @ vEBT_VEBT @ vEBT_VEBTi @ vEBT_vebt_assn_raw @ ( replicate @ vEBT_VEBT @ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ ( bit0 @ one2 ) ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ na @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( vEBT_vebt_buildup @ ( suc @ ( divide_divide @ nat @ na @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) @ X ) ) @ ( vEBT_vebt_assn_raw @ ( vEBT_vebt_buildup @ ( suc @ ( divide_divide @ nat @ ( suc @ ( suc @ na ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ R5 ) ) ) ).
% "11"
thf(fact_4213_sgn__integer__code,axiom,
( ( sgn_sgn @ code_integer )
= ( ^ [K3: code_integer] :
( if @ code_integer
@ ( K3
= ( zero_zero @ code_integer ) )
@ ( zero_zero @ code_integer )
@ ( if @ code_integer @ ( ord_less @ code_integer @ K3 @ ( zero_zero @ code_integer ) ) @ ( uminus_uminus @ code_integer @ ( one_one @ code_integer ) ) @ ( one_one @ code_integer ) ) ) ) ) ).
% sgn_integer_code
thf(fact_4214_infinite__int__iff__unbounded,axiom,
! [S3: set @ int] :
( ( ~ ( finite_finite2 @ int @ S3 ) )
= ( ! [M4: int] :
? [N3: int] :
( ( ord_less @ int @ M4 @ ( abs_abs @ int @ N3 ) )
& ( member @ int @ N3 @ S3 ) ) ) ) ).
% infinite_int_iff_unbounded
thf(fact_4215_tan__pi,axiom,
( ( tan @ real @ pi )
= ( zero_zero @ real ) ) ).
% tan_pi
thf(fact_4216_tan__zero,axiom,
! [A: $tType] :
( ( ( real_Vector_banach @ A )
& ( real_V3459762299906320749_field @ A ) )
=> ( ( tan @ A @ ( zero_zero @ A ) )
= ( zero_zero @ A ) ) ) ).
% tan_zero
thf(fact_4217__C3_OIH_C_I4_J,axiom,
! [X: nat] :
( ~ ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( suc @ ( suc @ na ) ) )
=> ( ( X
= ( divide_divide @ nat @ ( suc @ ( suc @ na ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) )
=> ( hoare_hoare_triple @ vEBT_VEBTi @ ( one_one @ assn ) @ ( vEBT_V739175172307565963ildupi @ ( suc @ X ) ) @ ( vEBT_vebt_assn_raw @ ( vEBT_vebt_buildup @ ( suc @ X ) ) ) ) ) ) ).
% "3.IH"(4)
thf(fact_4218__C3_OIH_C_I3_J,axiom,
! [X: nat] :
( ~ ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( suc @ ( suc @ na ) ) )
=> ( ( X
= ( divide_divide @ nat @ ( suc @ ( suc @ na ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) )
=> ( hoare_hoare_triple @ vEBT_VEBTi @ ( one_one @ assn ) @ ( vEBT_V739175172307565963ildupi @ X ) @ ( vEBT_vebt_assn_raw @ ( vEBT_vebt_buildup @ X ) ) ) ) ) ).
% "3.IH"(3)
thf(fact_4219__C3_OIH_C_I1_J,axiom,
! [X: nat] :
( ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( suc @ ( suc @ na ) ) )
=> ( ( X
= ( divide_divide @ nat @ ( suc @ ( suc @ na ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) )
=> ( hoare_hoare_triple @ vEBT_VEBTi @ ( one_one @ assn ) @ ( vEBT_V739175172307565963ildupi @ X ) @ ( vEBT_vebt_assn_raw @ ( vEBT_vebt_buildup @ X ) ) ) ) ) ).
% "3.IH"(1)
thf(fact_4220_tan__npi,axiom,
! [N: nat] :
( ( tan @ real @ ( times_times @ real @ ( semiring_1_of_nat @ real @ N ) @ pi ) )
= ( zero_zero @ real ) ) ).
% tan_npi
thf(fact_4221_tan__periodic__n,axiom,
! [X: real,N: num] :
( ( tan @ real @ ( plus_plus @ real @ X @ ( times_times @ real @ ( numeral_numeral @ real @ N ) @ pi ) ) )
= ( tan @ real @ X ) ) ).
% tan_periodic_n
thf(fact_4222_tan__periodic__nat,axiom,
! [X: real,N: nat] :
( ( tan @ real @ ( plus_plus @ real @ X @ ( times_times @ real @ ( semiring_1_of_nat @ real @ N ) @ pi ) ) )
= ( tan @ real @ X ) ) ).
% tan_periodic_nat
thf(fact_4223_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_4224_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_4225_frame__rule__left,axiom,
! [A: $tType,P: assn,C2: heap_Time_Heap @ A,Q: A > assn,R: assn] :
( ( hoare_hoare_triple @ A @ P @ C2 @ Q )
=> ( hoare_hoare_triple @ A @ ( times_times @ assn @ R @ P ) @ C2
@ ^ [X2: A] : ( times_times @ assn @ R @ ( Q @ X2 ) ) ) ) ).
% frame_rule_left
thf(fact_4226_fi__rule,axiom,
! [A: $tType,P: assn,C2: heap_Time_Heap @ A,Q: A > assn,Ps: assn,F3: assn] :
( ( hoare_hoare_triple @ A @ P @ C2 @ Q )
=> ( ( entails @ Ps @ ( times_times @ assn @ P @ F3 ) )
=> ( hoare_hoare_triple @ A @ Ps @ C2
@ ^ [X2: A] : ( times_times @ assn @ ( Q @ X2 ) @ F3 ) ) ) ) ).
% fi_rule
thf(fact_4227_return__cons__rule,axiom,
! [A: $tType,P: assn,Q: A > assn,X: A] :
( ( entails @ P @ ( Q @ X ) )
=> ( hoare_hoare_triple @ A @ P @ ( heap_Time_return @ A @ X ) @ Q ) ) ).
% return_cons_rule
thf(fact_4228_zero__natural_Orsp,axiom,
( ( zero_zero @ nat )
= ( zero_zero @ nat ) ) ).
% zero_natural.rsp
thf(fact_4229_one__natural_Orsp,axiom,
( ( one_one @ nat )
= ( one_one @ nat ) ) ).
% one_natural.rsp
thf(fact_4230_zero__integer_Orsp,axiom,
( ( zero_zero @ int )
= ( zero_zero @ int ) ) ).
% zero_integer.rsp
thf(fact_4231_one__integer_Orsp,axiom,
( ( one_one @ int )
= ( one_one @ int ) ) ).
% one_integer.rsp
thf(fact_4232_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_4233_uminus__integer__code_I1_J,axiom,
( ( uminus_uminus @ code_integer @ ( zero_zero @ code_integer ) )
= ( zero_zero @ code_integer ) ) ).
% uminus_integer_code(1)
thf(fact_4234_times__integer__code_I2_J,axiom,
! [L: code_integer] :
( ( times_times @ code_integer @ ( zero_zero @ code_integer ) @ L )
= ( zero_zero @ code_integer ) ) ).
% times_integer_code(2)
thf(fact_4235_times__integer__code_I1_J,axiom,
! [K: code_integer] :
( ( times_times @ code_integer @ K @ ( zero_zero @ code_integer ) )
= ( zero_zero @ code_integer ) ) ).
% times_integer_code(1)
thf(fact_4236_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_4237_tan__45,axiom,
( ( tan @ real @ ( divide_divide @ real @ pi @ ( numeral_numeral @ real @ ( bit0 @ ( bit0 @ one2 ) ) ) ) )
= ( one_one @ real ) ) ).
% tan_45
thf(fact_4238_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_4239_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_4240_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_4241_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_4242_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_4243_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_4244_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_4245_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_4246_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_4247_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_4248_infinite__nat__iff__unbounded,axiom,
! [S3: set @ nat] :
( ( ~ ( finite_finite2 @ nat @ S3 ) )
= ( ! [M4: nat] :
? [N3: nat] :
( ( ord_less @ nat @ M4 @ N3 )
& ( member @ nat @ N3 @ S3 ) ) ) ) ).
% infinite_nat_iff_unbounded
thf(fact_4249_unbounded__k__infinite,axiom,
! [K: nat,S3: set @ nat] :
( ! [M2: nat] :
( ( ord_less @ nat @ K @ M2 )
=> ? [N10: nat] :
( ( ord_less @ nat @ M2 @ N10 )
& ( member @ nat @ N10 @ S3 ) ) )
=> ~ ( finite_finite2 @ nat @ S3 ) ) ).
% unbounded_k_infinite
thf(fact_4250_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_4251_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_4252_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_4253_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_4254_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_4255_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_4256_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_4257_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_4258_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_4259_abs__integer__code,axiom,
( ( abs_abs @ code_integer )
= ( ^ [K3: code_integer] : ( if @ code_integer @ ( ord_less @ code_integer @ K3 @ ( zero_zero @ code_integer ) ) @ ( uminus_uminus @ code_integer @ K3 ) @ K3 ) ) ) ).
% abs_integer_code
thf(fact_4260_less__integer__code_I1_J,axiom,
~ ( ord_less @ code_integer @ ( zero_zero @ code_integer ) @ ( zero_zero @ code_integer ) ) ).
% less_integer_code(1)
thf(fact_4261_plus__integer__code_I1_J,axiom,
! [K: code_integer] :
( ( plus_plus @ code_integer @ K @ ( zero_zero @ code_integer ) )
= K ) ).
% plus_integer_code(1)
thf(fact_4262_plus__integer__code_I2_J,axiom,
! [L: code_integer] :
( ( plus_plus @ code_integer @ ( zero_zero @ code_integer ) @ L )
= L ) ).
% plus_integer_code(2)
thf(fact_4263_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_4264_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_4265_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_4266_minus__integer__code_I2_J,axiom,
! [L: code_integer] :
( ( minus_minus @ code_integer @ ( zero_zero @ code_integer ) @ L )
= ( uminus_uminus @ code_integer @ L ) ) ).
% minus_integer_code(2)
thf(fact_4267_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_4268_tan__total__pi4,axiom,
! [X: real] :
( ( ord_less @ real @ ( abs_abs @ real @ X ) @ ( one_one @ real ) )
=> ? [Z3: real] :
( ( ord_less @ real @ ( uminus_uminus @ real @ ( divide_divide @ real @ pi @ ( numeral_numeral @ real @ ( bit0 @ ( bit0 @ one2 ) ) ) ) ) @ Z3 )
& ( ord_less @ real @ Z3 @ ( divide_divide @ real @ pi @ ( numeral_numeral @ real @ ( bit0 @ ( bit0 @ one2 ) ) ) ) )
& ( ( tan @ real @ Z3 )
= X ) ) ) ).
% tan_total_pi4
thf(fact_4269_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,I6: set @ nat,F3: 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 ) @ I6 @ ( insert @ nat @ I @ ( bot_bot @ ( set @ nat ) ) ) ) @ A3 @ Xs2 @ Xsi ) ) @ F3 ) )
=> ( ( ord_less @ nat @ I @ ( size_size @ ( list @ A ) @ Xs2 ) )
=> ( ( member @ nat @ I @ I6 )
=> ( ( hoare_hoare_triple @ C @ ( times_times @ assn @ ( vEBT_List_listI_assn @ A @ B @ I6 @ A3 @ Xs2 @ Xsi ) @ F3 ) @ C2 @ Q )
=> ( hoare_hoare_triple @ C @ P @ C2 @ Q ) ) ) ) ) ).
% listI_assn_reinsert'
thf(fact_4270_listI__assn__reinsert__upd_H,axiom,
! [C: $tType,D4: $tType,E3: $tType,P: assn,A3: C > D4 > assn,X: C,Xi2: D4,I6: set @ nat,I: nat,Xs2: list @ C,Xsi: list @ D4,F3: 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 @ D4 @ ( minus_minus @ ( set @ nat ) @ I6 @ ( insert @ nat @ I @ ( bot_bot @ ( set @ nat ) ) ) ) @ A3 @ Xs2 @ Xsi ) ) @ F3 ) )
=> ( ( ord_less @ nat @ I @ ( size_size @ ( list @ C ) @ Xs2 ) )
=> ( ( member @ nat @ I @ I6 )
=> ( ( hoare_hoare_triple @ E3 @ ( times_times @ assn @ ( vEBT_List_listI_assn @ C @ D4 @ I6 @ A3 @ ( list_update @ C @ Xs2 @ I @ X ) @ ( list_update @ D4 @ Xsi @ I @ Xi2 ) ) @ F3 ) @ C2 @ Q )
=> ( hoare_hoare_triple @ E3 @ P @ C2 @ Q ) ) ) ) ) ).
% listI_assn_reinsert_upd'
thf(fact_4271_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_4272_rule__at__index,axiom,
! [A: $tType,B: $tType,C: $tType,P: assn,A3: A > B > assn,Xs2: list @ A,Xsi: list @ B,F3: assn,I: nat,C2: heap_Time_Heap @ C,Q4: C > assn,F6: C > assn] :
( ( entails @ P @ ( times_times @ assn @ ( vEBT_List_list_assn @ A @ B @ A3 @ Xs2 @ Xsi ) @ F3 ) )
=> ( ( 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 ) ) @ F3 ) @ C2 @ Q4 )
=> ( ! [R3: C] : ( entails @ ( Q4 @ 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 ) ) @ ( F6 @ R3 ) ) )
=> ( hoare_hoare_triple @ C @ P @ C2
@ ^ [R5: C] : ( times_times @ assn @ ( vEBT_List_list_assn @ A @ B @ A3 @ Xs2 @ Xsi ) @ ( F6 @ R5 ) ) ) ) ) ) ) ).
% rule_at_index
thf(fact_4273_repli__emp,axiom,
! [A: $tType,B: $tType,X: heap_Time_Heap @ A,A3: B > A > assn,Y: B,N: nat] :
( ( hoare_hoare_triple @ A @ ( one_one @ assn ) @ X @ ( A3 @ Y ) )
=> ( hoare_hoare_triple @ ( list @ A ) @ ( one_one @ assn ) @ ( vEBT_VEBT_replicatei @ A @ N @ X ) @ ( vEBT_List_list_assn @ B @ A @ A3 @ ( replicate @ B @ N @ Y ) ) ) ) ).
% repli_emp
thf(fact_4274_repli__cons__repl,axiom,
! [B: $tType,A: $tType,Q: assn,X: heap_Time_Heap @ A,A3: B > A > assn,Y: B,N: 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 @ N @ X )
@ ^ [R5: list @ A] : ( times_times @ assn @ Q @ ( vEBT_List_list_assn @ B @ A @ A3 @ ( replicate @ B @ N @ Y ) @ R5 ) ) ) ) ).
% repli_cons_repl
thf(fact_4275_T__vebt__buildupi,axiom,
! [N: nat,H2: heap_ext @ product_unit] : ( ord_less_eq @ nat @ ( time_time @ vEBT_VEBTi @ ( vEBT_V739175172307565963ildupi @ N ) @ H2 ) @ ( vEBT_V441764108873111860ildupi @ N ) ) ).
% T_vebt_buildupi
thf(fact_4276_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_4277_highi__h,axiom,
! [X: nat,N: nat] :
( hoare_hoare_triple @ nat @ ( one_one @ assn ) @ ( vEBT_VEBT_highi @ X @ N )
@ ^ [R5: nat] :
( pure_assn
@ ( R5
= ( vEBT_VEBT_high @ X @ N ) ) ) ) ).
% highi_h
thf(fact_4278_lowi__h,axiom,
! [X: nat,N: nat] :
( hoare_hoare_triple @ nat @ ( one_one @ assn ) @ ( vEBT_VEBT_lowi @ X @ N )
@ ^ [R5: nat] :
( pure_assn
@ ( R5
= ( vEBT_VEBT_low @ X @ N ) ) ) ) ).
% lowi_h
thf(fact_4279_time__replicate,axiom,
! [A: $tType,X: heap_Time_Heap @ A,C2: nat,N: nat,H2: heap_ext @ product_unit] :
( ! [H3: heap_ext @ product_unit] : ( ord_less_eq @ nat @ ( time_time @ A @ X @ H3 ) @ C2 )
=> ( ord_less_eq @ nat @ ( time_time @ ( list @ A ) @ ( vEBT_VEBT_replicatei @ A @ N @ X ) @ H2 ) @ ( plus_plus @ nat @ ( one_one @ nat ) @ ( times_times @ nat @ ( plus_plus @ nat @ ( one_one @ nat ) @ C2 ) @ N ) ) ) ) ).
% time_replicate
thf(fact_4280_norm__pre__pure__iff,axiom,
! [A: $tType,P: assn,B2: $o,F2: heap_Time_Heap @ A,Q: A > assn] :
( ( hoare_hoare_triple @ A @ ( times_times @ assn @ P @ ( pure_assn @ B2 ) ) @ F2 @ Q )
= ( B2
=> ( hoare_hoare_triple @ A @ P @ F2 @ Q ) ) ) ).
% norm_pre_pure_iff
thf(fact_4281_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_4282_cons__rule,axiom,
! [A: $tType,P: assn,P3: assn,Q: A > assn,Q4: A > assn,C2: heap_Time_Heap @ A] :
( ( entails @ P @ P3 )
=> ( ! [X3: A] : ( entails @ ( Q @ X3 ) @ ( Q4 @ X3 ) )
=> ( ( hoare_hoare_triple @ A @ P3 @ C2 @ Q )
=> ( hoare_hoare_triple @ A @ P @ C2 @ Q4 ) ) ) ) ).
% cons_rule
thf(fact_4283_cons__post__rule,axiom,
! [A: $tType,P: assn,C2: heap_Time_Heap @ A,Q: A > assn,Q4: A > assn] :
( ( hoare_hoare_triple @ A @ P @ C2 @ Q )
=> ( ! [X3: A] : ( entails @ ( Q @ X3 ) @ ( Q4 @ X3 ) )
=> ( hoare_hoare_triple @ A @ P @ C2 @ Q4 ) ) ) ).
% cons_post_rule
thf(fact_4284_return__wp__rule,axiom,
! [A: $tType,Q: A > assn,X: A] : ( hoare_hoare_triple @ A @ ( Q @ X ) @ ( heap_Time_return @ A @ X ) @ Q ) ).
% return_wp_rule
thf(fact_4285_frame__rule,axiom,
! [A: $tType,P: assn,C2: heap_Time_Heap @ A,Q: A > assn,R: assn] :
( ( hoare_hoare_triple @ A @ P @ C2 @ Q )
=> ( hoare_hoare_triple @ A @ ( times_times @ assn @ P @ R ) @ C2
@ ^ [X2: A] : ( times_times @ assn @ ( Q @ X2 ) @ R ) ) ) ).
% frame_rule
thf(fact_4286_cons__pre__rule,axiom,
! [A: $tType,P: assn,P3: assn,C2: heap_Time_Heap @ A,Q: A > assn] :
( ( entails @ P @ P3 )
=> ( ( hoare_hoare_triple @ A @ P3 @ C2 @ Q )
=> ( hoare_hoare_triple @ A @ P @ C2 @ Q ) ) ) ).
% cons_pre_rule
thf(fact_4287_norm__pre__pure__rule1,axiom,
! [A: $tType,B2: $o,P: assn,F2: heap_Time_Heap @ A,Q: A > assn] :
( ( B2
=> ( hoare_hoare_triple @ A @ P @ F2 @ Q ) )
=> ( hoare_hoare_triple @ A @ ( times_times @ assn @ P @ ( pure_assn @ B2 ) ) @ F2 @ Q ) ) ).
% norm_pre_pure_rule1
thf(fact_4288_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_4289_return__sp__rule,axiom,
! [A: $tType,P: assn,X: A] :
( hoare_hoare_triple @ A @ P @ ( heap_Time_return @ A @ X )
@ ^ [R5: A] : ( times_times @ assn @ P @ ( pure_assn @ ( R5 = X ) ) ) ) ).
% return_sp_rule
thf(fact_4290_integer__of__int__code,axiom,
( code_integer_of_int
= ( ^ [K3: int] :
( if @ code_integer @ ( ord_less @ int @ K3 @ ( zero_zero @ int ) ) @ ( uminus_uminus @ code_integer @ ( code_integer_of_int @ ( uminus_uminus @ int @ K3 ) ) )
@ ( if @ code_integer
@ ( K3
= ( zero_zero @ int ) )
@ ( zero_zero @ code_integer )
@ ( if @ code_integer
@ ( ( modulo_modulo @ int @ K3 @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) )
= ( zero_zero @ int ) )
@ ( times_times @ code_integer @ ( numeral_numeral @ code_integer @ ( bit0 @ one2 ) ) @ ( code_integer_of_int @ ( divide_divide @ int @ K3 @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) ) ) )
@ ( plus_plus @ code_integer @ ( times_times @ code_integer @ ( numeral_numeral @ code_integer @ ( bit0 @ one2 ) ) @ ( code_integer_of_int @ ( divide_divide @ int @ K3 @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) ) ) ) @ ( one_one @ code_integer ) ) ) ) ) ) ) ).
% integer_of_int_code
thf(fact_4291_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_4292_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_4293_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_4294_real__sqrt__eq__iff,axiom,
! [X: real,Y: real] :
( ( ( sqrt @ X )
= ( sqrt @ Y ) )
= ( X = Y ) ) ).
% real_sqrt_eq_iff
thf(fact_4295_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_4296_real__sqrt__zero,axiom,
( ( sqrt @ ( zero_zero @ real ) )
= ( zero_zero @ real ) ) ).
% real_sqrt_zero
thf(fact_4297_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_4298_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_4299_real__sqrt__one,axiom,
( ( sqrt @ ( one_one @ real ) )
= ( one_one @ real ) ) ).
% real_sqrt_one
thf(fact_4300_real__sqrt__eq__1__iff,axiom,
! [X: real] :
( ( ( sqrt @ X )
= ( one_one @ real ) )
= ( X
= ( one_one @ real ) ) ) ).
% real_sqrt_eq_1_iff
thf(fact_4301_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_4302_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_4303_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_4304_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_4305_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_4306_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_4307_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_4308_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_4309_real__sqrt__mult__self,axiom,
! [A2: real] :
( ( times_times @ real @ ( sqrt @ A2 ) @ ( sqrt @ A2 ) )
= ( abs_abs @ real @ A2 ) ) ).
% real_sqrt_mult_self
thf(fact_4310_real__sqrt__abs2,axiom,
! [X: real] :
( ( sqrt @ ( times_times @ real @ X @ X ) )
= ( abs_abs @ real @ X ) ) ).
% real_sqrt_abs2
thf(fact_4311_real__sqrt__four,axiom,
( ( sqrt @ ( numeral_numeral @ real @ ( bit0 @ ( bit0 @ one2 ) ) ) )
= ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) ).
% real_sqrt_four
thf(fact_4312_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_4313_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_4314_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_4315_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_4316_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_4317_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_4318_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_4319_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_4320_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_4321_divmod__integer_H__def,axiom,
( ( unique8689654367752047608divmod @ code_integer )
= ( ^ [M4: num,N3: num] : ( product_Pair @ code_integer @ code_integer @ ( divide_divide @ code_integer @ ( numeral_numeral @ code_integer @ M4 ) @ ( numeral_numeral @ code_integer @ N3 ) ) @ ( modulo_modulo @ code_integer @ ( numeral_numeral @ code_integer @ M4 ) @ ( numeral_numeral @ code_integer @ N3 ) ) ) ) ) ).
% divmod_integer'_def
thf(fact_4322_real__sqrt__minus,axiom,
! [X: real] :
( ( sqrt @ ( uminus_uminus @ real @ X ) )
= ( uminus_uminus @ real @ ( sqrt @ X ) ) ) ).
% real_sqrt_minus
thf(fact_4323_Code__Numeral_Oset__bit__integer_Oabs__eq,axiom,
! [Xa: nat,X: int] :
( ( bit_se5668285175392031749et_bit @ code_integer @ Xa @ ( code_integer_of_int @ X ) )
= ( code_integer_of_int @ ( bit_se5668285175392031749et_bit @ int @ Xa @ X ) ) ) ).
% Code_Numeral.set_bit_integer.abs_eq
thf(fact_4324_divide__integer_Oabs__eq,axiom,
! [Xa: int,X: int] :
( ( divide_divide @ code_integer @ ( code_integer_of_int @ Xa ) @ ( code_integer_of_int @ X ) )
= ( code_integer_of_int @ ( divide_divide @ int @ Xa @ X ) ) ) ).
% divide_integer.abs_eq
thf(fact_4325_modulo__integer_Oabs__eq,axiom,
! [Xa: int,X: int] :
( ( modulo_modulo @ code_integer @ ( code_integer_of_int @ Xa ) @ ( code_integer_of_int @ X ) )
= ( code_integer_of_int @ ( modulo_modulo @ int @ Xa @ X ) ) ) ).
% modulo_integer.abs_eq
thf(fact_4326_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_4327_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_4328_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_4329_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_4330_zero__integer__def,axiom,
( ( zero_zero @ code_integer )
= ( code_integer_of_int @ ( zero_zero @ int ) ) ) ).
% zero_integer_def
thf(fact_4331_less__integer_Oabs__eq,axiom,
! [Xa: int,X: int] :
( ( ord_less @ code_integer @ ( code_integer_of_int @ Xa ) @ ( code_integer_of_int @ X ) )
= ( ord_less @ int @ Xa @ X ) ) ).
% less_integer.abs_eq
thf(fact_4332_one__integer__def,axiom,
( ( one_one @ code_integer )
= ( code_integer_of_int @ ( one_one @ int ) ) ) ).
% one_integer_def
thf(fact_4333_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_4334_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_4335_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_4336_sqrt2__less__2,axiom,
ord_less @ real @ ( sqrt @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ).
% sqrt2_less_2
thf(fact_4337_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_4338_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_4339_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_4340_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_4341_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_4342_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_4343_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_4344_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_4345_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_4346_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_4347_real__sqrt__sum__squares__triangle__ineq,axiom,
! [A2: real,C2: real,B2: real,D2: 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 @ D2 ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( plus_plus @ real @ ( sqrt @ ( plus_plus @ real @ ( power_power @ real @ A2 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ ( power_power @ real @ B2 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( sqrt @ ( plus_plus @ real @ ( power_power @ real @ C2 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ ( power_power @ real @ D2 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) ) ).
% real_sqrt_sum_squares_triangle_ineq
thf(fact_4348_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_4349_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_4350_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_4351_tan__60,axiom,
( ( tan @ real @ ( divide_divide @ real @ pi @ ( numeral_numeral @ real @ ( bit1 @ one2 ) ) ) )
= ( sqrt @ ( numeral_numeral @ real @ ( bit1 @ one2 ) ) ) ) ).
% tan_60
thf(fact_4352_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_4353_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_4354_sqrt__even__pow2,axiom,
! [N: nat] :
( ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N )
=> ( ( sqrt @ ( power_power @ real @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) @ N ) )
= ( power_power @ real @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ N @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) ).
% sqrt_even_pow2
thf(fact_4355_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_4356_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_4357_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_4358_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_4359_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_4360_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_4361_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_4362_real__sqrt__power__even,axiom,
! [N: nat,X: real] :
( ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N )
=> ( ( ord_less_eq @ real @ ( zero_zero @ real ) @ X )
=> ( ( power_power @ real @ ( sqrt @ X ) @ N )
= ( power_power @ real @ X @ ( divide_divide @ nat @ N @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) ) ).
% real_sqrt_power_even
thf(fact_4363_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_4364_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_4365_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_4366_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_4367_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_4368_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_4369_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_4370_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_4371_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_4372_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_4373_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_4374_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_4375_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_4376_sin__paired,axiom,
! [X: real] :
( sums @ real
@ ^ [N3: nat] : ( times_times @ real @ ( divide_divide @ real @ ( power_power @ real @ ( uminus_uminus @ real @ ( one_one @ real ) ) @ N3 ) @ ( semiring_char_0_fact @ real @ ( plus_plus @ nat @ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N3 ) @ ( one_one @ nat ) ) ) ) @ ( power_power @ real @ X @ ( plus_plus @ nat @ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N3 ) @ ( one_one @ nat ) ) ) )
@ ( sin @ real @ X ) ) ).
% sin_paired
thf(fact_4377_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_4378_and__int_Opsimps,axiom,
! [K: int,L: int] :
( ( accp @ ( product_prod @ int @ int ) @ bit_and_int_rel @ ( product_Pair @ int @ int @ K @ L ) )
=> ( ( ( ( member @ int @ K @ ( insert @ int @ ( zero_zero @ int ) @ ( insert @ int @ ( uminus_uminus @ int @ ( one_one @ int ) ) @ ( bot_bot @ ( set @ int ) ) ) ) )
& ( member @ int @ L @ ( insert @ int @ ( zero_zero @ int ) @ ( insert @ int @ ( uminus_uminus @ int @ ( one_one @ int ) ) @ ( bot_bot @ ( set @ int ) ) ) ) ) )
=> ( ( bit_se5824344872417868541ns_and @ int @ K @ L )
= ( uminus_uminus @ int
@ ( zero_neq_one_of_bool @ int
@ ( ~ ( dvd_dvd @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ K )
& ~ ( dvd_dvd @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ L ) ) ) ) ) )
& ( ~ ( ( member @ int @ K @ ( insert @ int @ ( zero_zero @ int ) @ ( insert @ int @ ( uminus_uminus @ int @ ( one_one @ int ) ) @ ( bot_bot @ ( set @ int ) ) ) ) )
& ( member @ int @ L @ ( insert @ int @ ( zero_zero @ int ) @ ( insert @ int @ ( uminus_uminus @ int @ ( one_one @ int ) ) @ ( bot_bot @ ( set @ int ) ) ) ) ) )
=> ( ( bit_se5824344872417868541ns_and @ int @ K @ L )
= ( plus_plus @ int
@ ( zero_neq_one_of_bool @ int
@ ( ~ ( dvd_dvd @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ K )
& ~ ( dvd_dvd @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ L ) ) )
@ ( times_times @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ ( bit_se5824344872417868541ns_and @ int @ ( divide_divide @ int @ K @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) ) @ ( divide_divide @ int @ L @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) ) ) ) ) ) ) ) ) ).
% and_int.psimps
thf(fact_4379_freeze__rule,axiom,
! [A: $tType] :
( ( heap @ A )
=> ! [A2: array @ A,Xs2: list @ A] :
( hoare_hoare_triple @ ( list @ A ) @ ( snga_assn @ A @ A2 @ Xs2 ) @ ( array_freeze @ A @ A2 )
@ ^ [R5: list @ A] : ( times_times @ assn @ ( snga_assn @ A @ A2 @ Xs2 ) @ ( pure_assn @ ( R5 = Xs2 ) ) ) ) ) ).
% freeze_rule
thf(fact_4380_sums__zero,axiom,
! [A: $tType] :
( ( ( comm_monoid_add @ A )
& ( topolo4958980785337419405_space @ A ) )
=> ( sums @ A
@ ^ [N3: nat] : ( zero_zero @ A )
@ ( zero_zero @ A ) ) ) ).
% sums_zero
thf(fact_4381_powser__sums__zero__iff,axiom,
! [A: $tType] :
( ( real_V8999393235501362500lgebra @ A )
=> ! [A2: nat > A,X: A] :
( ( sums @ A
@ ^ [N3: nat] : ( times_times @ A @ ( A2 @ N3 ) @ ( power_power @ A @ ( zero_zero @ A ) @ N3 ) )
@ X )
= ( ( A2 @ ( zero_zero @ nat ) )
= X ) ) ) ).
% powser_sums_zero_iff
thf(fact_4382_sums__divide,axiom,
! [A: $tType] :
( ( real_V3459762299906320749_field @ A )
=> ! [F2: nat > A,A2: A,C2: A] :
( ( sums @ A @ F2 @ A2 )
=> ( sums @ A
@ ^ [N3: nat] : ( divide_divide @ A @ ( F2 @ N3 ) @ C2 )
@ ( divide_divide @ A @ A2 @ C2 ) ) ) ) ).
% sums_divide
thf(fact_4383_sums__add,axiom,
! [A: $tType] :
( ( ( topolo5987344860129210374id_add @ A )
& ( topological_t2_space @ A ) )
=> ! [F2: nat > A,A2: A,G: nat > A,B2: A] :
( ( sums @ A @ F2 @ A2 )
=> ( ( sums @ A @ G @ B2 )
=> ( sums @ A
@ ^ [N3: nat] : ( plus_plus @ A @ ( F2 @ N3 ) @ ( G @ N3 ) )
@ ( plus_plus @ A @ A2 @ B2 ) ) ) ) ) ).
% sums_add
thf(fact_4384_sums__single,axiom,
! [A: $tType] :
( ( ( comm_monoid_add @ A )
& ( topolo4958980785337419405_space @ A ) )
=> ! [I: nat,F2: nat > A] :
( sums @ A
@ ^ [R5: nat] : ( if @ A @ ( R5 = I ) @ ( F2 @ R5 ) @ ( zero_zero @ A ) )
@ ( F2 @ I ) ) ) ).
% sums_single
thf(fact_4385_sums__0,axiom,
! [A: $tType] :
( ( ( comm_monoid_add @ A )
& ( topolo4958980785337419405_space @ A ) )
=> ! [F2: nat > A] :
( ! [N2: nat] :
( ( F2 @ N2 )
= ( zero_zero @ A ) )
=> ( sums @ A @ F2 @ ( zero_zero @ A ) ) ) ) ).
% sums_0
thf(fact_4386_sums__mult2,axiom,
! [A: $tType] :
( ( real_V4412858255891104859lgebra @ A )
=> ! [F2: nat > A,A2: A,C2: A] :
( ( sums @ A @ F2 @ A2 )
=> ( sums @ A
@ ^ [N3: nat] : ( times_times @ A @ ( F2 @ N3 ) @ C2 )
@ ( times_times @ A @ A2 @ C2 ) ) ) ) ).
% sums_mult2
thf(fact_4387_sums__mult,axiom,
! [A: $tType] :
( ( real_V4412858255891104859lgebra @ A )
=> ! [F2: nat > A,A2: A,C2: A] :
( ( sums @ A @ F2 @ A2 )
=> ( sums @ A
@ ^ [N3: nat] : ( times_times @ A @ C2 @ ( F2 @ N3 ) )
@ ( times_times @ A @ C2 @ A2 ) ) ) ) ).
% sums_mult
thf(fact_4388_sums__mult2__iff,axiom,
! [A: $tType] :
( ( ( field @ A )
& ( real_V4412858255891104859lgebra @ A ) )
=> ! [C2: A,F2: nat > A,D2: A] :
( ( C2
!= ( zero_zero @ A ) )
=> ( ( sums @ A
@ ^ [N3: nat] : ( times_times @ A @ ( F2 @ N3 ) @ C2 )
@ ( times_times @ A @ D2 @ C2 ) )
= ( sums @ A @ F2 @ D2 ) ) ) ) ).
% sums_mult2_iff
thf(fact_4389_sums__mult__iff,axiom,
! [A: $tType] :
( ( ( field @ A )
& ( real_V4412858255891104859lgebra @ A ) )
=> ! [C2: A,F2: nat > A,D2: A] :
( ( C2
!= ( zero_zero @ A ) )
=> ( ( sums @ A
@ ^ [N3: nat] : ( times_times @ A @ C2 @ ( F2 @ N3 ) )
@ ( times_times @ A @ C2 @ D2 ) )
= ( sums @ A @ F2 @ D2 ) ) ) ) ).
% sums_mult_iff
thf(fact_4390_sums__mult__D,axiom,
! [A: $tType] :
( ( real_V3459762299906320749_field @ A )
=> ! [C2: A,F2: nat > A,A2: A] :
( ( sums @ A
@ ^ [N3: nat] : ( times_times @ A @ C2 @ ( F2 @ N3 ) )
@ A2 )
=> ( ( C2
!= ( zero_zero @ A ) )
=> ( sums @ A @ F2 @ ( divide_divide @ A @ A2 @ C2 ) ) ) ) ) ).
% sums_mult_D
thf(fact_4391_sums__Suc__imp,axiom,
! [A: $tType] :
( ( real_V822414075346904944vector @ A )
=> ! [F2: nat > A,S: A] :
( ( ( F2 @ ( zero_zero @ nat ) )
= ( zero_zero @ A ) )
=> ( ( sums @ A
@ ^ [N3: nat] : ( F2 @ ( suc @ N3 ) )
@ S )
=> ( sums @ A @ F2 @ S ) ) ) ) ).
% sums_Suc_imp
thf(fact_4392_sums__Suc__iff,axiom,
! [A: $tType] :
( ( real_V822414075346904944vector @ A )
=> ! [F2: nat > A,S: A] :
( ( sums @ A
@ ^ [N3: nat] : ( F2 @ ( suc @ N3 ) )
@ S )
= ( sums @ A @ F2 @ ( plus_plus @ A @ S @ ( F2 @ ( zero_zero @ nat ) ) ) ) ) ) ).
% sums_Suc_iff
thf(fact_4393_sums__Suc,axiom,
! [A: $tType] :
( ( ( topolo5987344860129210374id_add @ A )
& ( topological_t2_space @ A ) )
=> ! [F2: nat > A,L: A] :
( ( sums @ A
@ ^ [N3: nat] : ( F2 @ ( suc @ N3 ) )
@ L )
=> ( sums @ A @ F2 @ ( plus_plus @ A @ L @ ( F2 @ ( zero_zero @ nat ) ) ) ) ) ) ).
% sums_Suc
thf(fact_4394_sums__zero__iff__shift,axiom,
! [A: $tType] :
( ( real_V822414075346904944vector @ A )
=> ! [N: nat,F2: nat > A,S: A] :
( ! [I3: nat] :
( ( ord_less @ nat @ I3 @ N )
=> ( ( F2 @ I3 )
= ( zero_zero @ A ) ) )
=> ( ( sums @ A
@ ^ [I2: nat] : ( F2 @ ( plus_plus @ nat @ I2 @ N ) )
@ S )
= ( sums @ A @ F2 @ S ) ) ) ) ).
% sums_zero_iff_shift
thf(fact_4395_powser__sums__if,axiom,
! [A: $tType] :
( ( ( ring_1 @ A )
& ( topolo4958980785337419405_space @ A ) )
=> ! [M: nat,Z: A] :
( sums @ A
@ ^ [N3: nat] : ( times_times @ A @ ( if @ A @ ( N3 = M ) @ ( one_one @ A ) @ ( zero_zero @ A ) ) @ ( power_power @ A @ Z @ N3 ) )
@ ( power_power @ A @ Z @ M ) ) ) ).
% powser_sums_if
thf(fact_4396_powser__sums__zero,axiom,
! [A: $tType] :
( ( real_V8999393235501362500lgebra @ A )
=> ! [A2: nat > A] :
( sums @ A
@ ^ [N3: nat] : ( times_times @ A @ ( A2 @ N3 ) @ ( power_power @ A @ ( zero_zero @ A ) @ N3 ) )
@ ( A2 @ ( zero_zero @ nat ) ) ) ) ).
% powser_sums_zero
thf(fact_4397_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_4398_power__half__series,axiom,
( sums @ real
@ ^ [N3: nat] : ( power_power @ real @ ( divide_divide @ real @ ( one_one @ real ) @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) @ ( suc @ N3 ) )
@ ( one_one @ real ) ) ).
% power_half_series
thf(fact_4399_sums__if_H,axiom,
! [G: nat > real,X: real] :
( ( sums @ real @ G @ X )
=> ( sums @ real
@ ^ [N3: nat] : ( if @ real @ ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N3 ) @ ( zero_zero @ real ) @ ( G @ ( divide_divide @ nat @ ( minus_minus @ nat @ N3 @ ( one_one @ nat ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) )
@ X ) ) ).
% sums_if'
thf(fact_4400_sums__if,axiom,
! [G: nat > real,X: real,F2: nat > real,Y: real] :
( ( sums @ real @ G @ X )
=> ( ( sums @ real @ F2 @ Y )
=> ( sums @ real
@ ^ [N3: nat] : ( if @ real @ ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N3 ) @ ( F2 @ ( divide_divide @ nat @ N3 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( G @ ( divide_divide @ nat @ ( minus_minus @ nat @ N3 @ ( one_one @ nat ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) )
@ ( plus_plus @ real @ X @ Y ) ) ) ) ).
% sums_if
thf(fact_4401_and__int_Opinduct,axiom,
! [A0: int,A1: int,P: int > int > $o] :
( ( accp @ ( product_prod @ int @ int ) @ bit_and_int_rel @ ( product_Pair @ int @ int @ A0 @ A1 ) )
=> ( ! [K2: int,L4: int] :
( ( accp @ ( product_prod @ int @ int ) @ bit_and_int_rel @ ( product_Pair @ int @ int @ K2 @ L4 ) )
=> ( ( ~ ( ( member @ int @ K2 @ ( insert @ int @ ( zero_zero @ int ) @ ( insert @ int @ ( uminus_uminus @ int @ ( one_one @ int ) ) @ ( bot_bot @ ( set @ int ) ) ) ) )
& ( member @ int @ L4 @ ( insert @ int @ ( zero_zero @ int ) @ ( insert @ int @ ( uminus_uminus @ int @ ( one_one @ int ) ) @ ( bot_bot @ ( set @ int ) ) ) ) ) )
=> ( P @ ( divide_divide @ int @ K2 @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) ) @ ( divide_divide @ int @ L4 @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) ) ) )
=> ( P @ K2 @ L4 ) ) )
=> ( P @ A0 @ A1 ) ) ) ).
% and_int.pinduct
thf(fact_4402_cos__paired,axiom,
! [X: real] :
( sums @ real
@ ^ [N3: nat] : ( times_times @ real @ ( divide_divide @ real @ ( power_power @ real @ ( uminus_uminus @ real @ ( one_one @ real ) ) @ N3 ) @ ( semiring_char_0_fact @ real @ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N3 ) ) ) @ ( power_power @ real @ X @ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N3 ) ) )
@ ( cos @ real @ X ) ) ).
% cos_paired
thf(fact_4403_geometric__deriv__sums,axiom,
! [A: $tType] :
( ( ( real_Vector_banach @ A )
& ( real_V3459762299906320749_field @ A ) )
=> ! [Z: A] :
( ( ord_less @ real @ ( real_V7770717601297561774m_norm @ A @ Z ) @ ( one_one @ real ) )
=> ( sums @ A
@ ^ [N3: nat] : ( times_times @ A @ ( semiring_1_of_nat @ A @ ( suc @ N3 ) ) @ ( power_power @ A @ Z @ N3 ) )
@ ( divide_divide @ A @ ( one_one @ A ) @ ( power_power @ A @ ( minus_minus @ A @ ( one_one @ A ) @ Z ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) ) ).
% geometric_deriv_sums
thf(fact_4404_upto_Opinduct,axiom,
! [A0: int,A1: int,P: int > int > $o] :
( ( accp @ ( product_prod @ int @ int ) @ upto_rel @ ( product_Pair @ int @ int @ A0 @ A1 ) )
=> ( ! [I3: int,J3: int] :
( ( accp @ ( product_prod @ int @ int ) @ upto_rel @ ( product_Pair @ int @ int @ I3 @ J3 ) )
=> ( ( ( ord_less_eq @ int @ I3 @ J3 )
=> ( P @ ( plus_plus @ int @ I3 @ ( one_one @ int ) ) @ J3 ) )
=> ( P @ I3 @ J3 ) ) )
=> ( P @ A0 @ A1 ) ) ) ).
% upto.pinduct
thf(fact_4405_diffs__equiv,axiom,
! [A: $tType] :
( ( ( real_V822414075346904944vector @ A )
& ( ring_1 @ A ) )
=> ! [C2: nat > A,X: A] :
( ( summable @ A
@ ^ [N3: nat] : ( times_times @ A @ ( diffs @ A @ C2 @ N3 ) @ ( power_power @ A @ X @ N3 ) ) )
=> ( sums @ A
@ ^ [N3: nat] : ( times_times @ A @ ( times_times @ A @ ( semiring_1_of_nat @ A @ N3 ) @ ( C2 @ N3 ) ) @ ( power_power @ A @ X @ ( minus_minus @ nat @ N3 @ ( suc @ ( zero_zero @ nat ) ) ) ) )
@ ( suminf @ A
@ ^ [N3: nat] : ( times_times @ A @ ( diffs @ A @ C2 @ N3 ) @ ( power_power @ A @ X @ N3 ) ) ) ) ) ) ).
% diffs_equiv
thf(fact_4406_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_4407_sum__gp,axiom,
! [A: $tType] :
( ( ( division_ring @ A )
& ( comm_ring @ A ) )
=> ! [N: nat,M: nat,X: A] :
( ( ( ord_less @ nat @ N @ M )
=> ( ( groups7311177749621191930dd_sum @ nat @ A @ ( power_power @ A @ X ) @ ( set_or1337092689740270186AtMost @ nat @ M @ N ) )
= ( zero_zero @ A ) ) )
& ( ~ ( ord_less @ nat @ N @ M )
=> ( ( ( X
= ( one_one @ A ) )
=> ( ( groups7311177749621191930dd_sum @ nat @ A @ ( power_power @ A @ X ) @ ( set_or1337092689740270186AtMost @ nat @ M @ N ) )
= ( semiring_1_of_nat @ A @ ( minus_minus @ nat @ ( plus_plus @ nat @ N @ ( one_one @ nat ) ) @ M ) ) ) )
& ( ( X
!= ( one_one @ A ) )
=> ( ( groups7311177749621191930dd_sum @ nat @ A @ ( power_power @ A @ X ) @ ( set_or1337092689740270186AtMost @ nat @ M @ N ) )
= ( divide_divide @ A @ ( minus_minus @ A @ ( power_power @ A @ X @ M ) @ ( power_power @ A @ X @ ( suc @ N ) ) ) @ ( minus_minus @ A @ ( one_one @ A ) @ X ) ) ) ) ) ) ) ) ).
% sum_gp
thf(fact_4408_arcsin__0,axiom,
( ( arcsin @ ( zero_zero @ real ) )
= ( zero_zero @ real ) ) ).
% arcsin_0
thf(fact_4409_sum_Oneutral__const,axiom,
! [B: $tType,A: $tType] :
( ( comm_monoid_add @ A )
=> ! [A3: set @ B] :
( ( groups7311177749621191930dd_sum @ B @ A
@ ^ [Uu3: B] : ( zero_zero @ A )
@ A3 )
= ( zero_zero @ A ) ) ) ).
% sum.neutral_const
thf(fact_4410_sum_Oempty,axiom,
! [B: $tType,A: $tType] :
( ( comm_monoid_add @ A )
=> ! [G: B > A] :
( ( groups7311177749621191930dd_sum @ B @ A @ G @ ( bot_bot @ ( set @ B ) ) )
= ( zero_zero @ A ) ) ) ).
% sum.empty
thf(fact_4411_sum__eq__0__iff,axiom,
! [A: $tType,B: $tType] :
( ( canoni5634975068530333245id_add @ A )
=> ! [F3: set @ B,F2: B > A] :
( ( finite_finite2 @ B @ F3 )
=> ( ( ( groups7311177749621191930dd_sum @ B @ A @ F2 @ F3 )
= ( zero_zero @ A ) )
= ( ! [X2: B] :
( ( member @ B @ X2 @ F3 )
=> ( ( F2 @ X2 )
= ( zero_zero @ A ) ) ) ) ) ) ) ).
% sum_eq_0_iff
thf(fact_4412_sum_Oinfinite,axiom,
! [B: $tType,A: $tType] :
( ( comm_monoid_add @ A )
=> ! [A3: set @ B,G: B > A] :
( ~ ( finite_finite2 @ B @ A3 )
=> ( ( groups7311177749621191930dd_sum @ B @ A @ G @ A3 )
= ( zero_zero @ A ) ) ) ) ).
% sum.infinite
thf(fact_4413_sum_Odelta_H,axiom,
! [B: $tType,A: $tType] :
( ( comm_monoid_add @ A )
=> ! [S3: set @ B,A2: B,B2: B > A] :
( ( finite_finite2 @ B @ S3 )
=> ( ( ( member @ B @ A2 @ S3 )
=> ( ( groups7311177749621191930dd_sum @ B @ A
@ ^ [K3: B] : ( if @ A @ ( A2 = K3 ) @ ( B2 @ K3 ) @ ( zero_zero @ A ) )
@ S3 )
= ( B2 @ A2 ) ) )
& ( ~ ( member @ B @ A2 @ S3 )
=> ( ( groups7311177749621191930dd_sum @ B @ A
@ ^ [K3: B] : ( if @ A @ ( A2 = K3 ) @ ( B2 @ K3 ) @ ( zero_zero @ A ) )
@ S3 )
= ( zero_zero @ A ) ) ) ) ) ) ).
% sum.delta'
thf(fact_4414_sum_Odelta,axiom,
! [B: $tType,A: $tType] :
( ( comm_monoid_add @ A )
=> ! [S3: set @ B,A2: B,B2: B > A] :
( ( finite_finite2 @ B @ S3 )
=> ( ( ( member @ B @ A2 @ S3 )
=> ( ( groups7311177749621191930dd_sum @ B @ A
@ ^ [K3: B] : ( if @ A @ ( K3 = A2 ) @ ( B2 @ K3 ) @ ( zero_zero @ A ) )
@ S3 )
= ( B2 @ A2 ) ) )
& ( ~ ( member @ B @ A2 @ S3 )
=> ( ( groups7311177749621191930dd_sum @ B @ A
@ ^ [K3: B] : ( if @ A @ ( K3 = A2 ) @ ( B2 @ K3 ) @ ( zero_zero @ A ) )
@ S3 )
= ( zero_zero @ A ) ) ) ) ) ) ).
% sum.delta
thf(fact_4415_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_4416_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
@ ^ [I2: A] : ( abs_abs @ B @ ( F2 @ I2 ) )
@ A3 ) ) ) ).
% sum_abs_ge_zero
thf(fact_4417_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_4418_sum_Oop__ivl__Suc,axiom,
! [A: $tType] :
( ( comm_monoid_add @ A )
=> ! [N: nat,M: nat,G: nat > A] :
( ( ( ord_less @ nat @ N @ M )
=> ( ( groups7311177749621191930dd_sum @ nat @ A @ G @ ( set_or7035219750837199246ssThan @ nat @ M @ ( suc @ N ) ) )
= ( zero_zero @ A ) ) )
& ( ~ ( ord_less @ nat @ N @ M )
=> ( ( groups7311177749621191930dd_sum @ nat @ A @ G @ ( set_or7035219750837199246ssThan @ nat @ M @ ( suc @ N ) ) )
= ( plus_plus @ A @ ( groups7311177749621191930dd_sum @ nat @ A @ G @ ( set_or7035219750837199246ssThan @ nat @ M @ N ) ) @ ( G @ N ) ) ) ) ) ) ).
% sum.op_ivl_Suc
thf(fact_4419_sum_Ocl__ivl__Suc,axiom,
! [A: $tType] :
( ( comm_monoid_add @ A )
=> ! [N: nat,M: nat,G: nat > A] :
( ( ( ord_less @ nat @ ( suc @ N ) @ M )
=> ( ( groups7311177749621191930dd_sum @ nat @ A @ G @ ( set_or1337092689740270186AtMost @ nat @ M @ ( suc @ N ) ) )
= ( zero_zero @ A ) ) )
& ( ~ ( ord_less @ nat @ ( suc @ N ) @ M )
=> ( ( groups7311177749621191930dd_sum @ nat @ A @ G @ ( set_or1337092689740270186AtMost @ nat @ M @ ( suc @ N ) ) )
= ( plus_plus @ A @ ( groups7311177749621191930dd_sum @ nat @ A @ G @ ( set_or1337092689740270186AtMost @ nat @ M @ N ) ) @ ( G @ ( suc @ N ) ) ) ) ) ) ) ).
% sum.cl_ivl_Suc
thf(fact_4420_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
@ ^ [I2: nat] : ( times_times @ A @ ( C2 @ I2 ) @ ( power_power @ A @ ( zero_zero @ A ) @ I2 ) )
@ A3 )
= ( C2 @ ( zero_zero @ nat ) ) ) )
& ( ~ ( ( finite_finite2 @ nat @ A3 )
& ( member @ nat @ ( zero_zero @ nat ) @ A3 ) )
=> ( ( groups7311177749621191930dd_sum @ nat @ A
@ ^ [I2: nat] : ( times_times @ A @ ( C2 @ I2 ) @ ( power_power @ A @ ( zero_zero @ A ) @ I2 ) )
@ A3 )
= ( zero_zero @ A ) ) ) ) ) ).
% sum_zero_power
thf(fact_4421_arcsin__1,axiom,
( ( arcsin @ ( one_one @ real ) )
= ( divide_divide @ real @ pi @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) ) ).
% arcsin_1
thf(fact_4422_sum__zero__power_H,axiom,
! [A: $tType] :
( ( field @ A )
=> ! [A3: set @ nat,C2: nat > A,D2: nat > A] :
( ( ( ( finite_finite2 @ nat @ A3 )
& ( member @ nat @ ( zero_zero @ nat ) @ A3 ) )
=> ( ( groups7311177749621191930dd_sum @ nat @ A
@ ^ [I2: nat] : ( divide_divide @ A @ ( times_times @ A @ ( C2 @ I2 ) @ ( power_power @ A @ ( zero_zero @ A ) @ I2 ) ) @ ( D2 @ I2 ) )
@ A3 )
= ( divide_divide @ A @ ( C2 @ ( zero_zero @ nat ) ) @ ( D2 @ ( zero_zero @ nat ) ) ) ) )
& ( ~ ( ( finite_finite2 @ nat @ A3 )
& ( member @ nat @ ( zero_zero @ nat ) @ A3 ) )
=> ( ( groups7311177749621191930dd_sum @ nat @ A
@ ^ [I2: nat] : ( divide_divide @ A @ ( times_times @ A @ ( C2 @ I2 ) @ ( power_power @ A @ ( zero_zero @ A ) @ I2 ) ) @ ( D2 @ I2 ) )
@ A3 )
= ( zero_zero @ A ) ) ) ) ) ).
% sum_zero_power'
thf(fact_4423_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_4424_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
@ ^ [N3: B] : ( divide_divide @ A @ ( F2 @ N3 ) @ R2 )
@ A3 ) ) ) ).
% sum_divide_distrib
thf(fact_4425_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
@ ^ [I2: B] : ( modulo_modulo @ A @ ( F2 @ I2 ) @ A2 )
@ A3 )
@ A2 )
= ( modulo_modulo @ A @ ( groups7311177749621191930dd_sum @ B @ A @ F2 @ A3 ) @ A2 ) ) ) ).
% mod_sum_eq
thf(fact_4426_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
@ ^ [N3: B] : ( times_times @ A @ R2 @ ( F2 @ N3 ) )
@ A3 ) ) ) ).
% sum_distrib_left
thf(fact_4427_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
@ ^ [N3: B] : ( times_times @ A @ ( F2 @ N3 ) @ R2 )
@ A3 ) ) ) ).
% sum_distrib_right
thf(fact_4428_sum__product,axiom,
! [B: $tType,C: $tType,A: $tType] :
( ( semiring_0 @ B )
=> ! [F2: A > B,A3: set @ A,G: C > B,B4: set @ C] :
( ( times_times @ B @ ( groups7311177749621191930dd_sum @ A @ B @ F2 @ A3 ) @ ( groups7311177749621191930dd_sum @ C @ B @ G @ B4 ) )
= ( groups7311177749621191930dd_sum @ A @ B
@ ^ [I2: A] :
( groups7311177749621191930dd_sum @ C @ B
@ ^ [J2: C] : ( times_times @ B @ ( F2 @ I2 ) @ ( G @ J2 ) )
@ B4 )
@ A3 ) ) ) ).
% sum_product
thf(fact_4429_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_4430_dvd__sum,axiom,
! [A: $tType,B: $tType] :
( ( comm_semiring_1 @ A )
=> ! [A3: set @ B,D2: A,F2: B > A] :
( ! [A6: B] :
( ( member @ B @ A6 @ A3 )
=> ( dvd_dvd @ A @ D2 @ ( F2 @ A6 ) ) )
=> ( dvd_dvd @ A @ D2 @ ( groups7311177749621191930dd_sum @ B @ A @ F2 @ A3 ) ) ) ) ).
% dvd_sum
thf(fact_4431_sum_Onot__neutral__contains__not__neutral,axiom,
! [B: $tType,A: $tType] :
( ( comm_monoid_add @ A )
=> ! [G: B > A,A3: set @ B] :
( ( ( groups7311177749621191930dd_sum @ B @ A @ G @ A3 )
!= ( zero_zero @ A ) )
=> ~ ! [A6: B] :
( ( member @ B @ A6 @ A3 )
=> ( ( G @ A6 )
= ( zero_zero @ A ) ) ) ) ) ).
% sum.not_neutral_contains_not_neutral
thf(fact_4432_sum_Oneutral,axiom,
! [B: $tType,A: $tType] :
( ( comm_monoid_add @ A )
=> ! [A3: set @ B,G: B > A] :
( ! [X3: B] :
( ( member @ B @ X3 @ A3 )
=> ( ( G @ X3 )
= ( zero_zero @ A ) ) )
=> ( ( groups7311177749621191930dd_sum @ B @ A @ G @ A3 )
= ( zero_zero @ A ) ) ) ) ).
% sum.neutral
thf(fact_4433_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_4434_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_4435_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_4436_sum_Ointer__filter,axiom,
! [A: $tType,B: $tType] :
( ( comm_monoid_add @ A )
=> ! [A3: set @ B,G: B > A,P: B > $o] :
( ( finite_finite2 @ B @ A3 )
=> ( ( groups7311177749621191930dd_sum @ B @ A @ G
@ ( collect @ B
@ ^ [X2: B] :
( ( member @ B @ X2 @ A3 )
& ( P @ X2 ) ) ) )
= ( groups7311177749621191930dd_sum @ B @ A
@ ^ [X2: B] : ( if @ A @ ( P @ X2 ) @ ( G @ X2 ) @ ( zero_zero @ A ) )
@ A3 ) ) ) ) ).
% sum.inter_filter
thf(fact_4437_sum_Oshift__bounds__Suc__ivl,axiom,
! [A: $tType] :
( ( comm_monoid_add @ A )
=> ! [G: nat > A,M: nat,N: nat] :
( ( groups7311177749621191930dd_sum @ nat @ A @ G @ ( set_or7035219750837199246ssThan @ nat @ ( suc @ M ) @ ( suc @ N ) ) )
= ( groups7311177749621191930dd_sum @ nat @ A
@ ^ [I2: nat] : ( G @ ( suc @ I2 ) )
@ ( set_or7035219750837199246ssThan @ nat @ M @ N ) ) ) ) ).
% sum.shift_bounds_Suc_ivl
thf(fact_4438_sum_Oshift__bounds__cl__Suc__ivl,axiom,
! [A: $tType] :
( ( comm_monoid_add @ A )
=> ! [G: nat > A,M: nat,N: nat] :
( ( groups7311177749621191930dd_sum @ nat @ A @ G @ ( set_or1337092689740270186AtMost @ nat @ ( suc @ M ) @ ( suc @ N ) ) )
= ( groups7311177749621191930dd_sum @ nat @ A
@ ^ [I2: nat] : ( G @ ( suc @ I2 ) )
@ ( set_or1337092689740270186AtMost @ nat @ M @ N ) ) ) ) ).
% sum.shift_bounds_cl_Suc_ivl
thf(fact_4439_sum_Oshift__bounds__nat__ivl,axiom,
! [A: $tType] :
( ( comm_monoid_add @ A )
=> ! [G: nat > A,M: nat,K: nat,N: nat] :
( ( groups7311177749621191930dd_sum @ nat @ A @ G @ ( set_or7035219750837199246ssThan @ nat @ ( plus_plus @ nat @ M @ K ) @ ( plus_plus @ nat @ N @ K ) ) )
= ( groups7311177749621191930dd_sum @ nat @ A
@ ^ [I2: nat] : ( G @ ( plus_plus @ nat @ I2 @ K ) )
@ ( set_or7035219750837199246ssThan @ nat @ M @ N ) ) ) ) ).
% sum.shift_bounds_nat_ivl
thf(fact_4440_sum_Oshift__bounds__cl__nat__ivl,axiom,
! [A: $tType] :
( ( comm_monoid_add @ A )
=> ! [G: nat > A,M: nat,K: nat,N: nat] :
( ( groups7311177749621191930dd_sum @ nat @ A @ G @ ( set_or1337092689740270186AtMost @ nat @ ( plus_plus @ nat @ M @ K ) @ ( plus_plus @ nat @ N @ K ) ) )
= ( groups7311177749621191930dd_sum @ nat @ A
@ ^ [I2: nat] : ( G @ ( plus_plus @ nat @ I2 @ K ) )
@ ( set_or1337092689740270186AtMost @ nat @ M @ N ) ) ) ) ).
% sum.shift_bounds_cl_nat_ivl
thf(fact_4441_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_4442_sum__le__included,axiom,
! [B: $tType,A: $tType,C: $tType] :
( ( ordere6911136660526730532id_add @ A )
=> ! [S: set @ B,T: set @ C,G: C > A,I: C > B,F2: B > A] :
( ( finite_finite2 @ B @ S )
=> ( ( finite_finite2 @ C @ T )
=> ( ! [X3: C] :
( ( member @ C @ X3 @ T )
=> ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( G @ X3 ) ) )
=> ( ! [X3: B] :
( ( member @ B @ X3 @ S )
=> ? [Xa2: C] :
( ( member @ C @ Xa2 @ T )
& ( ( 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 @ T ) ) ) ) ) ) ) ).
% sum_le_included
thf(fact_4443_sum__strict__mono__ex1,axiom,
! [A: $tType,I8: $tType] :
( ( ordere8940638589300402666id_add @ A )
=> ! [A3: set @ I8,F2: I8 > A,G: I8 > A] :
( ( finite_finite2 @ I8 @ A3 )
=> ( ! [X3: I8] :
( ( member @ I8 @ X3 @ A3 )
=> ( ord_less_eq @ A @ ( F2 @ X3 ) @ ( G @ X3 ) ) )
=> ( ? [X5: I8] :
( ( member @ I8 @ X5 @ A3 )
& ( ord_less @ A @ ( F2 @ X5 ) @ ( G @ X5 ) ) )
=> ( ord_less @ A @ ( groups7311177749621191930dd_sum @ I8 @ A @ F2 @ A3 ) @ ( groups7311177749621191930dd_sum @ I8 @ A @ G @ A3 ) ) ) ) ) ) ).
% sum_strict_mono_ex1
thf(fact_4444_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_4445_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_4446_sum_Oivl__cong,axiom,
! [A: $tType,B: $tType] :
( ( ( ord @ B )
& ( comm_monoid_add @ A ) )
=> ! [A2: B,C2: B,B2: B,D2: B,G: B > A,H2: B > A] :
( ( A2 = C2 )
=> ( ( B2 = D2 )
=> ( ! [X3: B] :
( ( ord_less_eq @ B @ C2 @ X3 )
=> ( ( ord_less @ B @ X3 @ D2 )
=> ( ( G @ X3 )
= ( H2 @ X3 ) ) ) )
=> ( ( groups7311177749621191930dd_sum @ B @ A @ G @ ( set_or7035219750837199246ssThan @ B @ A2 @ B2 ) )
= ( groups7311177749621191930dd_sum @ B @ A @ H2 @ ( set_or7035219750837199246ssThan @ B @ C2 @ D2 ) ) ) ) ) ) ) ).
% sum.ivl_cong
thf(fact_4447_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_4448_sum_Oreindex__bij__witness__not__neutral,axiom,
! [B: $tType,A: $tType,C: $tType] :
( ( comm_monoid_add @ A )
=> ! [S4: set @ B,T4: set @ C,S3: set @ B,I: C > B,J: B > C,T5: set @ C,G: B > A,H2: C > A] :
( ( finite_finite2 @ B @ S4 )
=> ( ( finite_finite2 @ C @ T4 )
=> ( ! [A6: B] :
( ( member @ B @ A6 @ ( minus_minus @ ( set @ B ) @ S3 @ S4 ) )
=> ( ( I @ ( J @ A6 ) )
= A6 ) )
=> ( ! [A6: B] :
( ( member @ B @ A6 @ ( minus_minus @ ( set @ B ) @ S3 @ S4 ) )
=> ( member @ C @ ( J @ A6 ) @ ( minus_minus @ ( set @ C ) @ T5 @ T4 ) ) )
=> ( ! [B3: C] :
( ( member @ C @ B3 @ ( minus_minus @ ( set @ C ) @ T5 @ T4 ) )
=> ( ( J @ ( I @ B3 ) )
= B3 ) )
=> ( ! [B3: C] :
( ( member @ C @ B3 @ ( minus_minus @ ( set @ C ) @ T5 @ T4 ) )
=> ( member @ B @ ( I @ B3 ) @ ( minus_minus @ ( set @ B ) @ S3 @ S4 ) ) )
=> ( ! [A6: B] :
( ( member @ B @ A6 @ S4 )
=> ( ( G @ A6 )
= ( zero_zero @ A ) ) )
=> ( ! [B3: C] :
( ( member @ C @ B3 @ T4 )
=> ( ( H2 @ B3 )
= ( zero_zero @ A ) ) )
=> ( ! [A6: B] :
( ( member @ B @ A6 @ S3 )
=> ( ( H2 @ ( J @ A6 ) )
= ( G @ A6 ) ) )
=> ( ( groups7311177749621191930dd_sum @ B @ A @ G @ S3 )
= ( groups7311177749621191930dd_sum @ C @ A @ H2 @ T5 ) ) ) ) ) ) ) ) ) ) ) ) ).
% sum.reindex_bij_witness_not_neutral
thf(fact_4449_sum_OatLeastLessThan__concat,axiom,
! [A: $tType] :
( ( comm_monoid_add @ A )
=> ! [M: nat,N: nat,P4: nat,G: nat > A] :
( ( ord_less_eq @ nat @ M @ N )
=> ( ( ord_less_eq @ nat @ N @ P4 )
=> ( ( plus_plus @ A @ ( groups7311177749621191930dd_sum @ nat @ A @ G @ ( set_or7035219750837199246ssThan @ nat @ M @ N ) ) @ ( groups7311177749621191930dd_sum @ nat @ A @ G @ ( set_or7035219750837199246ssThan @ nat @ N @ P4 ) ) )
= ( groups7311177749621191930dd_sum @ nat @ A @ G @ ( set_or7035219750837199246ssThan @ nat @ M @ P4 ) ) ) ) ) ) ).
% sum.atLeastLessThan_concat
thf(fact_4450_sum__nonneg__leq__bound,axiom,
! [B: $tType,A: $tType] :
( ( ordere6911136660526730532id_add @ A )
=> ! [S: set @ B,F2: B > A,B4: A,I: B] :
( ( finite_finite2 @ B @ S )
=> ( ! [I3: B] :
( ( member @ B @ I3 @ S )
=> ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( F2 @ I3 ) ) )
=> ( ( ( groups7311177749621191930dd_sum @ B @ A @ F2 @ S )
= B4 )
=> ( ( member @ B @ I @ S )
=> ( ord_less_eq @ A @ ( F2 @ I ) @ B4 ) ) ) ) ) ) ).
% sum_nonneg_leq_bound
thf(fact_4451_sum__nonneg__0,axiom,
! [B: $tType,A: $tType] :
( ( ordere6911136660526730532id_add @ A )
=> ! [S: set @ B,F2: B > A,I: B] :
( ( finite_finite2 @ B @ S )
=> ( ! [I3: B] :
( ( member @ B @ I3 @ S )
=> ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( F2 @ I3 ) ) )
=> ( ( ( groups7311177749621191930dd_sum @ B @ A @ F2 @ S )
= ( zero_zero @ A ) )
=> ( ( member @ B @ I @ S )
=> ( ( F2 @ I )
= ( zero_zero @ A ) ) ) ) ) ) ) ).
% sum_nonneg_0
thf(fact_4452_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_4453_sum__power__add,axiom,
! [A: $tType] :
( ( ( monoid_mult @ A )
& ( comm_ring @ A ) )
=> ! [X: A,M: nat,I6: set @ nat] :
( ( groups7311177749621191930dd_sum @ nat @ A
@ ^ [I2: nat] : ( power_power @ A @ X @ ( plus_plus @ nat @ M @ I2 ) )
@ I6 )
= ( times_times @ A @ ( power_power @ A @ X @ M ) @ ( groups7311177749621191930dd_sum @ nat @ A @ ( power_power @ A @ X ) @ I6 ) ) ) ) ).
% sum_power_add
thf(fact_4454_sum_OatLeastAtMost__rev,axiom,
! [A: $tType] :
( ( comm_monoid_add @ A )
=> ! [G: nat > A,N: nat,M: nat] :
( ( groups7311177749621191930dd_sum @ nat @ A @ G @ ( set_or1337092689740270186AtMost @ nat @ N @ M ) )
= ( groups7311177749621191930dd_sum @ nat @ A
@ ^ [I2: nat] : ( G @ ( minus_minus @ nat @ ( plus_plus @ nat @ M @ N ) @ I2 ) )
@ ( set_or1337092689740270186AtMost @ nat @ N @ M ) ) ) ) ).
% sum.atLeastAtMost_rev
thf(fact_4455_sums__If__finite__set,axiom,
! [A: $tType] :
( ( ( comm_monoid_add @ A )
& ( topolo4958980785337419405_space @ A ) )
=> ! [A3: set @ nat,F2: nat > A] :
( ( finite_finite2 @ nat @ A3 )
=> ( sums @ A
@ ^ [R5: nat] : ( if @ A @ ( member @ nat @ R5 @ A3 ) @ ( F2 @ R5 ) @ ( zero_zero @ A ) )
@ ( groups7311177749621191930dd_sum @ nat @ A @ F2 @ A3 ) ) ) ) ).
% sums_If_finite_set
thf(fact_4456_sums__If__finite,axiom,
! [A: $tType] :
( ( ( comm_monoid_add @ A )
& ( topolo4958980785337419405_space @ A ) )
=> ! [P: nat > $o,F2: nat > A] :
( ( finite_finite2 @ nat @ ( collect @ nat @ P ) )
=> ( sums @ A
@ ^ [R5: nat] : ( if @ A @ ( P @ R5 ) @ ( F2 @ R5 ) @ ( zero_zero @ A ) )
@ ( groups7311177749621191930dd_sum @ nat @ A @ F2 @ ( collect @ nat @ P ) ) ) ) ) ).
% sums_If_finite
thf(fact_4457_sums__finite,axiom,
! [A: $tType] :
( ( ( comm_monoid_add @ A )
& ( topolo4958980785337419405_space @ A ) )
=> ! [N4: set @ nat,F2: nat > A] :
( ( finite_finite2 @ nat @ N4 )
=> ( ! [N2: nat] :
( ~ ( member @ nat @ N2 @ N4 )
=> ( ( F2 @ N2 )
= ( zero_zero @ A ) ) )
=> ( sums @ A @ F2 @ ( groups7311177749621191930dd_sum @ nat @ A @ F2 @ N4 ) ) ) ) ) ).
% sums_finite
thf(fact_4458_suminf__finite,axiom,
! [A: $tType] :
( ( ( comm_monoid_add @ A )
& ( topological_t2_space @ A ) )
=> ! [N4: set @ nat,F2: nat > A] :
( ( finite_finite2 @ nat @ N4 )
=> ( ! [N2: nat] :
( ~ ( member @ nat @ N2 @ N4 )
=> ( ( F2 @ N2 )
= ( zero_zero @ A ) ) )
=> ( ( suminf @ A @ F2 )
= ( groups7311177749621191930dd_sum @ nat @ A @ F2 @ N4 ) ) ) ) ) ).
% suminf_finite
thf(fact_4459_sum__pos2,axiom,
! [A: $tType,B: $tType] :
( ( ordere6911136660526730532id_add @ A )
=> ! [I6: set @ B,I: B,F2: B > A] :
( ( finite_finite2 @ B @ I6 )
=> ( ( member @ B @ I @ I6 )
=> ( ( ord_less @ A @ ( zero_zero @ A ) @ ( F2 @ I ) )
=> ( ! [I3: B] :
( ( member @ B @ I3 @ I6 )
=> ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( F2 @ I3 ) ) )
=> ( ord_less @ A @ ( zero_zero @ A ) @ ( groups7311177749621191930dd_sum @ B @ A @ F2 @ I6 ) ) ) ) ) ) ) ).
% sum_pos2
thf(fact_4460_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_4461_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_4462_sum__pos,axiom,
! [A: $tType,B: $tType] :
( ( ordere6911136660526730532id_add @ A )
=> ! [I6: set @ B,F2: B > A] :
( ( finite_finite2 @ B @ I6 )
=> ( ( I6
!= ( bot_bot @ ( set @ B ) ) )
=> ( ! [I3: B] :
( ( member @ B @ I3 @ I6 )
=> ( ord_less @ A @ ( zero_zero @ A ) @ ( F2 @ I3 ) ) )
=> ( ord_less @ A @ ( zero_zero @ A ) @ ( groups7311177749621191930dd_sum @ B @ A @ F2 @ I6 ) ) ) ) ) ) ).
% sum_pos
thf(fact_4463_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_4464_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_4465_sum_Osame__carrier,axiom,
! [A: $tType,B: $tType] :
( ( comm_monoid_add @ A )
=> ! [C6: set @ B,A3: set @ B,B4: set @ B,G: B > A,H2: B > A] :
( ( finite_finite2 @ B @ C6 )
=> ( ( ord_less_eq @ ( set @ B ) @ A3 @ C6 )
=> ( ( ord_less_eq @ ( set @ B ) @ B4 @ C6 )
=> ( ! [A6: B] :
( ( member @ B @ A6 @ ( minus_minus @ ( set @ B ) @ C6 @ A3 ) )
=> ( ( G @ A6 )
= ( zero_zero @ A ) ) )
=> ( ! [B3: B] :
( ( member @ B @ B3 @ ( minus_minus @ ( set @ B ) @ C6 @ B4 ) )
=> ( ( H2 @ B3 )
= ( zero_zero @ A ) ) )
=> ( ( ( groups7311177749621191930dd_sum @ B @ A @ G @ A3 )
= ( groups7311177749621191930dd_sum @ B @ A @ H2 @ B4 ) )
= ( ( groups7311177749621191930dd_sum @ B @ A @ G @ C6 )
= ( groups7311177749621191930dd_sum @ B @ A @ H2 @ C6 ) ) ) ) ) ) ) ) ) ).
% sum.same_carrier
thf(fact_4466_sum_Osame__carrierI,axiom,
! [A: $tType,B: $tType] :
( ( comm_monoid_add @ A )
=> ! [C6: set @ B,A3: set @ B,B4: set @ B,G: B > A,H2: B > A] :
( ( finite_finite2 @ B @ C6 )
=> ( ( ord_less_eq @ ( set @ B ) @ A3 @ C6 )
=> ( ( ord_less_eq @ ( set @ B ) @ B4 @ C6 )
=> ( ! [A6: B] :
( ( member @ B @ A6 @ ( minus_minus @ ( set @ B ) @ C6 @ A3 ) )
=> ( ( G @ A6 )
= ( zero_zero @ A ) ) )
=> ( ! [B3: B] :
( ( member @ B @ B3 @ ( minus_minus @ ( set @ B ) @ C6 @ B4 ) )
=> ( ( H2 @ B3 )
= ( zero_zero @ A ) ) )
=> ( ( ( groups7311177749621191930dd_sum @ B @ A @ G @ C6 )
= ( groups7311177749621191930dd_sum @ B @ A @ H2 @ C6 ) )
=> ( ( groups7311177749621191930dd_sum @ B @ A @ G @ A3 )
= ( groups7311177749621191930dd_sum @ B @ A @ H2 @ B4 ) ) ) ) ) ) ) ) ) ).
% sum.same_carrierI
thf(fact_4467_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_4468_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_4469_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_4470_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_4471_sum_Osubset__diff,axiom,
! [A: $tType,B: $tType] :
( ( comm_monoid_add @ A )
=> ! [B4: set @ B,A3: set @ B,G: B > A] :
( ( ord_less_eq @ ( set @ B ) @ B4 @ A3 )
=> ( ( finite_finite2 @ B @ A3 )
=> ( ( groups7311177749621191930dd_sum @ B @ A @ G @ A3 )
= ( plus_plus @ A @ ( groups7311177749621191930dd_sum @ B @ A @ G @ ( minus_minus @ ( set @ B ) @ A3 @ B4 ) ) @ ( groups7311177749621191930dd_sum @ B @ A @ G @ B4 ) ) ) ) ) ) ).
% sum.subset_diff
thf(fact_4472_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_4473_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_4474_sum_OatLeast0__lessThan__Suc,axiom,
! [A: $tType] :
( ( comm_monoid_add @ A )
=> ! [G: nat > A,N: nat] :
( ( groups7311177749621191930dd_sum @ nat @ A @ G @ ( set_or7035219750837199246ssThan @ nat @ ( zero_zero @ nat ) @ ( suc @ N ) ) )
= ( plus_plus @ A @ ( groups7311177749621191930dd_sum @ nat @ A @ G @ ( set_or7035219750837199246ssThan @ nat @ ( zero_zero @ nat ) @ N ) ) @ ( G @ N ) ) ) ) ).
% sum.atLeast0_lessThan_Suc
thf(fact_4475_sum_OatLeast__Suc__lessThan,axiom,
! [A: $tType] :
( ( comm_monoid_add @ A )
=> ! [M: nat,N: nat,G: nat > A] :
( ( ord_less @ nat @ M @ N )
=> ( ( groups7311177749621191930dd_sum @ nat @ A @ G @ ( set_or7035219750837199246ssThan @ nat @ M @ N ) )
= ( plus_plus @ A @ ( G @ M ) @ ( groups7311177749621191930dd_sum @ nat @ A @ G @ ( set_or7035219750837199246ssThan @ nat @ ( suc @ M ) @ N ) ) ) ) ) ) ).
% sum.atLeast_Suc_lessThan
thf(fact_4476_sum_OatLeast0__atMost__Suc,axiom,
! [A: $tType] :
( ( comm_monoid_add @ A )
=> ! [G: nat > A,N: nat] :
( ( groups7311177749621191930dd_sum @ nat @ A @ G @ ( set_or1337092689740270186AtMost @ nat @ ( zero_zero @ nat ) @ ( suc @ N ) ) )
= ( plus_plus @ A @ ( groups7311177749621191930dd_sum @ nat @ A @ G @ ( set_or1337092689740270186AtMost @ nat @ ( zero_zero @ nat ) @ N ) ) @ ( G @ ( suc @ N ) ) ) ) ) ).
% sum.atLeast0_atMost_Suc
thf(fact_4477_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_4478_sum_OatLeast__Suc__atMost,axiom,
! [A: $tType] :
( ( comm_monoid_add @ A )
=> ! [M: nat,N: nat,G: nat > A] :
( ( ord_less_eq @ nat @ M @ N )
=> ( ( groups7311177749621191930dd_sum @ nat @ A @ G @ ( set_or1337092689740270186AtMost @ nat @ M @ N ) )
= ( plus_plus @ A @ ( G @ M ) @ ( groups7311177749621191930dd_sum @ nat @ A @ G @ ( set_or1337092689740270186AtMost @ nat @ ( suc @ M ) @ N ) ) ) ) ) ) ).
% sum.atLeast_Suc_atMost
thf(fact_4479_sum_Onat__ivl__Suc_H,axiom,
! [A: $tType] :
( ( comm_monoid_add @ A )
=> ! [M: nat,N: nat,G: nat > A] :
( ( ord_less_eq @ nat @ M @ ( suc @ N ) )
=> ( ( groups7311177749621191930dd_sum @ nat @ A @ G @ ( set_or1337092689740270186AtMost @ nat @ M @ ( suc @ N ) ) )
= ( plus_plus @ A @ ( G @ ( suc @ N ) ) @ ( groups7311177749621191930dd_sum @ nat @ A @ G @ ( set_or1337092689740270186AtMost @ nat @ M @ N ) ) ) ) ) ) ).
% sum.nat_ivl_Suc'
thf(fact_4480_sum_Olast__plus,axiom,
! [A: $tType] :
( ( comm_monoid_add @ A )
=> ! [M: nat,N: nat,G: nat > A] :
( ( ord_less_eq @ nat @ M @ N )
=> ( ( groups7311177749621191930dd_sum @ nat @ A @ G @ ( set_or1337092689740270186AtMost @ nat @ M @ N ) )
= ( plus_plus @ A @ ( G @ N ) @ ( groups7311177749621191930dd_sum @ nat @ A @ G @ ( set_or7035219750837199246ssThan @ nat @ M @ N ) ) ) ) ) ) ).
% sum.last_plus
thf(fact_4481_diffs__def,axiom,
! [A: $tType] :
( ( ring_1 @ A )
=> ( ( diffs @ A )
= ( ^ [C5: nat > A,N3: nat] : ( times_times @ A @ ( semiring_1_of_nat @ A @ ( suc @ N3 ) ) @ ( C5 @ ( suc @ N3 ) ) ) ) ) ) ).
% diffs_def
thf(fact_4482_sum_OSuc__reindex__ivl,axiom,
! [A: $tType] :
( ( comm_monoid_add @ A )
=> ! [M: nat,N: nat,G: nat > A] :
( ( ord_less_eq @ nat @ M @ N )
=> ( ( plus_plus @ A @ ( groups7311177749621191930dd_sum @ nat @ A @ G @ ( set_or1337092689740270186AtMost @ nat @ M @ N ) ) @ ( G @ ( suc @ N ) ) )
= ( plus_plus @ A @ ( G @ M )
@ ( groups7311177749621191930dd_sum @ nat @ A
@ ^ [I2: nat] : ( G @ ( suc @ I2 ) )
@ ( set_or1337092689740270186AtMost @ nat @ M @ N ) ) ) ) ) ) ).
% sum.Suc_reindex_ivl
thf(fact_4483_sum__Suc__diff_H,axiom,
! [A: $tType] :
( ( ab_group_add @ A )
=> ! [M: nat,N: nat,F2: nat > A] :
( ( ord_less_eq @ nat @ M @ N )
=> ( ( groups7311177749621191930dd_sum @ nat @ A
@ ^ [I2: nat] : ( minus_minus @ A @ ( F2 @ ( suc @ I2 ) ) @ ( F2 @ I2 ) )
@ ( set_or7035219750837199246ssThan @ nat @ M @ N ) )
= ( minus_minus @ A @ ( F2 @ N ) @ ( F2 @ M ) ) ) ) ) ).
% sum_Suc_diff'
thf(fact_4484_sum__Suc__diff,axiom,
! [A: $tType] :
( ( ab_group_add @ A )
=> ! [M: nat,N: nat,F2: nat > A] :
( ( ord_less_eq @ nat @ M @ ( suc @ N ) )
=> ( ( groups7311177749621191930dd_sum @ nat @ A
@ ^ [I2: nat] : ( minus_minus @ A @ ( F2 @ ( suc @ I2 ) ) @ ( F2 @ I2 ) )
@ ( set_or1337092689740270186AtMost @ nat @ M @ N ) )
= ( minus_minus @ A @ ( F2 @ ( suc @ N ) ) @ ( F2 @ M ) ) ) ) ) ).
% sum_Suc_diff
thf(fact_4485_sum_OatLeastLessThan__rev,axiom,
! [A: $tType] :
( ( comm_monoid_add @ A )
=> ! [G: nat > A,N: nat,M: nat] :
( ( groups7311177749621191930dd_sum @ nat @ A @ G @ ( set_or7035219750837199246ssThan @ nat @ N @ M ) )
= ( groups7311177749621191930dd_sum @ nat @ A
@ ^ [I2: nat] : ( G @ ( minus_minus @ nat @ ( plus_plus @ nat @ M @ N ) @ ( suc @ I2 ) ) )
@ ( set_or7035219750837199246ssThan @ nat @ N @ M ) ) ) ) ).
% sum.atLeastLessThan_rev
thf(fact_4486_sum_Onested__swap,axiom,
! [A: $tType] :
( ( comm_monoid_add @ A )
=> ! [A2: nat > nat > A,N: nat] :
( ( groups7311177749621191930dd_sum @ nat @ A
@ ^ [I2: nat] : ( groups7311177749621191930dd_sum @ nat @ A @ ( A2 @ I2 ) @ ( set_or7035219750837199246ssThan @ nat @ ( zero_zero @ nat ) @ I2 ) )
@ ( set_or1337092689740270186AtMost @ nat @ ( zero_zero @ nat ) @ N ) )
= ( groups7311177749621191930dd_sum @ nat @ A
@ ^ [J2: nat] :
( groups7311177749621191930dd_sum @ nat @ A
@ ^ [I2: nat] : ( A2 @ I2 @ J2 )
@ ( set_or1337092689740270186AtMost @ nat @ ( suc @ J2 ) @ N ) )
@ ( set_or7035219750837199246ssThan @ nat @ ( zero_zero @ nat ) @ N ) ) ) ) ).
% sum.nested_swap
thf(fact_4487_sums__If__finite__set_H,axiom,
! [A: $tType] :
( ( ( topolo1287966508704411220up_add @ A )
& ( topological_t2_space @ A ) )
=> ! [G: nat > A,S3: A,A3: set @ nat,S4: A,F2: nat > A] :
( ( sums @ A @ G @ S3 )
=> ( ( finite_finite2 @ nat @ A3 )
=> ( ( S4
= ( plus_plus @ A @ S3
@ ( groups7311177749621191930dd_sum @ nat @ A
@ ^ [N3: nat] : ( minus_minus @ A @ ( F2 @ N3 ) @ ( G @ N3 ) )
@ A3 ) ) )
=> ( sums @ A
@ ^ [N3: nat] : ( if @ A @ ( member @ nat @ N3 @ A3 ) @ ( F2 @ N3 ) @ ( G @ N3 ) )
@ S4 ) ) ) ) ) ).
% sums_If_finite_set'
thf(fact_4488_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
@ ^ [A4: nat] : ( plus_plus @ A @ ( F2 @ A4 ) )
@ A2
@ B2
@ ( zero_zero @ A ) ) ) ) ).
% sum_atLeastAtMost_code
thf(fact_4489_sum__mono2,axiom,
! [A: $tType,B: $tType] :
( ( ordere6911136660526730532id_add @ A )
=> ! [B4: set @ B,A3: set @ B,F2: B > A] :
( ( finite_finite2 @ B @ B4 )
=> ( ( ord_less_eq @ ( set @ B ) @ A3 @ B4 )
=> ( ! [B3: B] :
( ( member @ B @ B3 @ ( minus_minus @ ( set @ B ) @ B4 @ 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 @ B4 ) ) ) ) ) ) ).
% sum_mono2
thf(fact_4490_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_4491_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_4492_termdiff__converges__all,axiom,
! [A: $tType] :
( ( ( real_Vector_banach @ A )
& ( real_V3459762299906320749_field @ A ) )
=> ! [C2: nat > A,X: A] :
( ! [X3: A] :
( summable @ A
@ ^ [N3: nat] : ( times_times @ A @ ( C2 @ N3 ) @ ( power_power @ A @ X3 @ N3 ) ) )
=> ( summable @ A
@ ^ [N3: nat] : ( times_times @ A @ ( diffs @ A @ C2 @ N3 ) @ ( power_power @ A @ X @ N3 ) ) ) ) ) ).
% termdiff_converges_all
thf(fact_4493_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_4494_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_4495_sum_Oub__add__nat,axiom,
! [A: $tType] :
( ( comm_monoid_add @ A )
=> ! [M: nat,N: nat,G: nat > A,P4: nat] :
( ( ord_less_eq @ nat @ M @ ( plus_plus @ nat @ N @ ( one_one @ nat ) ) )
=> ( ( groups7311177749621191930dd_sum @ nat @ A @ G @ ( set_or1337092689740270186AtMost @ nat @ M @ ( plus_plus @ nat @ N @ P4 ) ) )
= ( plus_plus @ A @ ( groups7311177749621191930dd_sum @ nat @ A @ G @ ( set_or1337092689740270186AtMost @ nat @ M @ N ) ) @ ( groups7311177749621191930dd_sum @ nat @ A @ G @ ( set_or1337092689740270186AtMost @ nat @ ( plus_plus @ nat @ N @ ( one_one @ nat ) ) @ ( plus_plus @ nat @ N @ P4 ) ) ) ) ) ) ) ).
% sum.ub_add_nat
thf(fact_4496_sum_Ohead__if,axiom,
! [A: $tType] :
( ( comm_monoid_add @ A )
=> ! [N: nat,M: nat,G: nat > A] :
( ( ( ord_less @ nat @ N @ M )
=> ( ( groups7311177749621191930dd_sum @ nat @ A @ G @ ( set_or1337092689740270186AtMost @ nat @ M @ N ) )
= ( zero_zero @ A ) ) )
& ( ~ ( ord_less @ nat @ N @ M )
=> ( ( groups7311177749621191930dd_sum @ nat @ A @ G @ ( set_or1337092689740270186AtMost @ nat @ M @ N ) )
= ( plus_plus @ A @ ( groups7311177749621191930dd_sum @ nat @ A @ G @ ( set_or7035219750837199246ssThan @ nat @ M @ N ) ) @ ( G @ N ) ) ) ) ) ) ).
% sum.head_if
thf(fact_4497_sum__le__suminf,axiom,
! [A: $tType] :
( ( ( ordere6911136660526730532id_add @ A )
& ( topolo1944317154257567458pology @ A ) )
=> ! [F2: nat > A,I6: set @ nat] :
( ( summable @ A @ F2 )
=> ( ( finite_finite2 @ nat @ I6 )
=> ( ! [N2: nat] :
( ( member @ nat @ N2 @ ( uminus_uminus @ ( set @ nat ) @ I6 ) )
=> ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( F2 @ N2 ) ) )
=> ( ord_less_eq @ A @ ( groups7311177749621191930dd_sum @ nat @ A @ F2 @ I6 ) @ ( suminf @ A @ F2 ) ) ) ) ) ) ).
% sum_le_suminf
thf(fact_4498_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
@ ^ [K3: B] : ( if @ A @ ( K3 = A2 ) @ ( B2 @ K3 ) @ ( C2 @ K3 ) )
@ 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
@ ^ [K3: B] : ( if @ A @ ( K3 = A2 ) @ ( B2 @ K3 ) @ ( C2 @ K3 ) )
@ S3 )
= ( groups7311177749621191930dd_sum @ B @ A @ C2 @ ( minus_minus @ ( set @ B ) @ S3 @ ( insert @ B @ A2 @ ( bot_bot @ ( set @ B ) ) ) ) ) ) ) ) ) ) ).
% sum.delta_remove
thf(fact_4499_set__encode__def,axiom,
( nat_set_encode
= ( groups7311177749621191930dd_sum @ nat @ nat @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ).
% set_encode_def
thf(fact_4500_sum_OatLeastLessThan__rev__at__least__Suc__atMost,axiom,
! [A: $tType] :
( ( comm_monoid_add @ A )
=> ! [G: nat > A,N: nat,M: nat] :
( ( groups7311177749621191930dd_sum @ nat @ A @ G @ ( set_or7035219750837199246ssThan @ nat @ N @ M ) )
= ( groups7311177749621191930dd_sum @ nat @ A
@ ^ [I2: nat] : ( G @ ( minus_minus @ nat @ ( plus_plus @ nat @ M @ N ) @ I2 ) )
@ ( set_or1337092689740270186AtMost @ nat @ ( suc @ N ) @ M ) ) ) ) ).
% sum.atLeastLessThan_rev_at_least_Suc_atMost
thf(fact_4501_sum__strict__mono2,axiom,
! [B: $tType,A: $tType] :
( ( ordere8940638589300402666id_add @ B )
=> ! [B4: set @ A,A3: set @ A,B2: A,F2: A > B] :
( ( finite_finite2 @ A @ B4 )
=> ( ( ord_less_eq @ ( set @ A ) @ A3 @ B4 )
=> ( ( member @ A @ B2 @ ( minus_minus @ ( set @ A ) @ B4 @ A3 ) )
=> ( ( ord_less @ B @ ( zero_zero @ B ) @ ( F2 @ B2 ) )
=> ( ! [X3: A] :
( ( member @ A @ X3 @ B4 )
=> ( ord_less_eq @ B @ ( zero_zero @ B ) @ ( F2 @ X3 ) ) )
=> ( ord_less @ B @ ( groups7311177749621191930dd_sum @ A @ B @ F2 @ A3 ) @ ( groups7311177749621191930dd_sum @ A @ B @ F2 @ B4 ) ) ) ) ) ) ) ) ).
% sum_strict_mono2
thf(fact_4502_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_4503_convex__sum__bound__le,axiom,
! [A: $tType,B: $tType] :
( ( linordered_idom @ B )
=> ! [I6: set @ A,X: A > B,A2: A > B,B2: B,Delta: B] :
( ! [I3: A] :
( ( member @ A @ I3 @ I6 )
=> ( ord_less_eq @ B @ ( zero_zero @ B ) @ ( X @ I3 ) ) )
=> ( ( ( groups7311177749621191930dd_sum @ A @ B @ X @ I6 )
= ( one_one @ B ) )
=> ( ! [I3: A] :
( ( member @ A @ I3 @ I6 )
=> ( ord_less_eq @ B @ ( abs_abs @ B @ ( minus_minus @ B @ ( A2 @ I3 ) @ B2 ) ) @ Delta ) )
=> ( ord_less_eq @ B
@ ( abs_abs @ B
@ ( minus_minus @ B
@ ( groups7311177749621191930dd_sum @ A @ B
@ ^ [I2: A] : ( times_times @ B @ ( A2 @ I2 ) @ ( X @ I2 ) )
@ I6 )
@ B2 ) )
@ Delta ) ) ) ) ) ).
% convex_sum_bound_le
thf(fact_4504_summable__Cauchy,axiom,
! [A: $tType] :
( ( real_Vector_banach @ A )
=> ( ( summable @ A )
= ( ^ [F4: nat > A] :
! [E4: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ E4 )
=> ? [N8: nat] :
! [M4: nat] :
( ( ord_less_eq @ nat @ N8 @ M4 )
=> ! [N3: nat] : ( ord_less @ real @ ( real_V7770717601297561774m_norm @ A @ ( groups7311177749621191930dd_sum @ nat @ A @ F4 @ ( set_or7035219750837199246ssThan @ nat @ M4 @ N3 ) ) ) @ E4 ) ) ) ) ) ) ).
% summable_Cauchy
thf(fact_4505_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_4506_sum__natinterval__diff,axiom,
! [A: $tType] :
( ( ab_group_add @ A )
=> ! [M: nat,N: nat,F2: nat > A] :
( ( ( ord_less_eq @ nat @ M @ N )
=> ( ( groups7311177749621191930dd_sum @ nat @ A
@ ^ [K3: nat] : ( minus_minus @ A @ ( F2 @ K3 ) @ ( F2 @ ( plus_plus @ nat @ K3 @ ( one_one @ nat ) ) ) )
@ ( set_or1337092689740270186AtMost @ nat @ M @ N ) )
= ( minus_minus @ A @ ( F2 @ M ) @ ( F2 @ ( plus_plus @ nat @ N @ ( one_one @ nat ) ) ) ) ) )
& ( ~ ( ord_less_eq @ nat @ M @ N )
=> ( ( groups7311177749621191930dd_sum @ nat @ A
@ ^ [K3: nat] : ( minus_minus @ A @ ( F2 @ K3 ) @ ( F2 @ ( plus_plus @ nat @ K3 @ ( one_one @ nat ) ) ) )
@ ( set_or1337092689740270186AtMost @ nat @ M @ N ) )
= ( zero_zero @ A ) ) ) ) ) ).
% sum_natinterval_diff
thf(fact_4507_sum__telescope_H_H,axiom,
! [A: $tType] :
( ( ab_group_add @ A )
=> ! [M: nat,N: nat,F2: nat > A] :
( ( ord_less_eq @ nat @ M @ N )
=> ( ( groups7311177749621191930dd_sum @ nat @ A
@ ^ [K3: nat] : ( minus_minus @ A @ ( F2 @ K3 ) @ ( F2 @ ( minus_minus @ nat @ K3 @ ( one_one @ nat ) ) ) )
@ ( set_or1337092689740270186AtMost @ nat @ ( suc @ M ) @ N ) )
= ( minus_minus @ A @ ( F2 @ N ) @ ( F2 @ M ) ) ) ) ) ).
% sum_telescope''
thf(fact_4508_sums__group,axiom,
! [A: $tType] :
( ( ( comm_monoid_add @ A )
& ( topolo4958980785337419405_space @ A ) )
=> ! [F2: nat > A,S: A,K: nat] :
( ( sums @ A @ F2 @ S )
=> ( ( ord_less @ nat @ ( zero_zero @ nat ) @ K )
=> ( sums @ A
@ ^ [N3: nat] : ( groups7311177749621191930dd_sum @ nat @ A @ F2 @ ( set_or7035219750837199246ssThan @ nat @ ( times_times @ nat @ N3 @ K ) @ ( plus_plus @ nat @ ( times_times @ nat @ N3 @ K ) @ K ) ) )
@ S ) ) ) ) ).
% sums_group
thf(fact_4509_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 )
=> ~ ! [N9: nat] :
~ ! [M3: nat] :
( ( ord_less_eq @ nat @ N9 @ M3 )
=> ! [N10: nat] : ( ord_less @ real @ ( real_V7770717601297561774m_norm @ A @ ( groups7311177749621191930dd_sum @ nat @ A @ F2 @ ( set_or1337092689740270186AtMost @ nat @ M3 @ N10 ) ) ) @ E ) ) ) ) ) ).
% summable_partial_sum_bound
thf(fact_4510_mask__eq__sum__exp,axiom,
! [A: $tType] :
( ( semiring_parity @ A )
=> ! [N: nat] :
( ( minus_minus @ A @ ( power_power @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ N ) @ ( one_one @ A ) )
= ( groups7311177749621191930dd_sum @ nat @ A @ ( power_power @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) )
@ ( collect @ nat
@ ^ [Q6: nat] : ( ord_less @ nat @ Q6 @ N ) ) ) ) ) ).
% mask_eq_sum_exp
thf(fact_4511_sum__gp__multiplied,axiom,
! [A: $tType] :
( ( ( monoid_mult @ A )
& ( comm_ring @ A ) )
=> ! [M: nat,N: nat,X: A] :
( ( ord_less_eq @ nat @ M @ N )
=> ( ( times_times @ A @ ( minus_minus @ A @ ( one_one @ A ) @ X ) @ ( groups7311177749621191930dd_sum @ nat @ A @ ( power_power @ A @ X ) @ ( set_or1337092689740270186AtMost @ nat @ M @ N ) ) )
= ( minus_minus @ A @ ( power_power @ A @ X @ M ) @ ( power_power @ A @ X @ ( suc @ N ) ) ) ) ) ) ).
% sum_gp_multiplied
thf(fact_4512_sum_Oin__pairs,axiom,
! [A: $tType] :
( ( comm_monoid_add @ A )
=> ! [G: nat > A,M: nat,N: nat] :
( ( groups7311177749621191930dd_sum @ nat @ A @ G @ ( set_or1337092689740270186AtMost @ nat @ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ M ) @ ( suc @ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N ) ) ) )
= ( groups7311177749621191930dd_sum @ nat @ A
@ ^ [I2: nat] : ( plus_plus @ A @ ( G @ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ I2 ) ) @ ( G @ ( suc @ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ I2 ) ) ) )
@ ( set_or1337092689740270186AtMost @ nat @ M @ N ) ) ) ) ).
% sum.in_pairs
thf(fact_4513_termdiff__converges,axiom,
! [A: $tType] :
( ( ( real_Vector_banach @ A )
& ( real_V3459762299906320749_field @ A ) )
=> ! [X: A,K6: real,C2: nat > A] :
( ( ord_less @ real @ ( real_V7770717601297561774m_norm @ A @ X ) @ K6 )
=> ( ! [X3: A] :
( ( ord_less @ real @ ( real_V7770717601297561774m_norm @ A @ X3 ) @ K6 )
=> ( summable @ A
@ ^ [N3: nat] : ( times_times @ A @ ( C2 @ N3 ) @ ( power_power @ A @ X3 @ N3 ) ) ) )
=> ( summable @ A
@ ^ [N3: nat] : ( times_times @ A @ ( diffs @ A @ C2 @ N3 ) @ ( power_power @ A @ X @ N3 ) ) ) ) ) ) ).
% termdiff_converges
thf(fact_4514_mask__eq__sum__exp__nat,axiom,
! [N: nat] :
( ( minus_minus @ nat @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N ) @ ( suc @ ( zero_zero @ nat ) ) )
= ( groups7311177749621191930dd_sum @ nat @ nat @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) )
@ ( collect @ nat
@ ^ [Q6: nat] : ( ord_less @ nat @ Q6 @ N ) ) ) ) ).
% mask_eq_sum_exp_nat
thf(fact_4515_gauss__sum__nat,axiom,
! [N: nat] :
( ( groups7311177749621191930dd_sum @ nat @ nat
@ ^ [X2: nat] : X2
@ ( set_or1337092689740270186AtMost @ nat @ ( zero_zero @ nat ) @ N ) )
= ( divide_divide @ nat @ ( times_times @ nat @ N @ ( suc @ N ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ).
% gauss_sum_nat
thf(fact_4516_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_4517_Sum__Ico__nat,axiom,
! [M: nat,N: nat] :
( ( groups7311177749621191930dd_sum @ nat @ nat
@ ^ [X2: nat] : X2
@ ( set_or7035219750837199246ssThan @ nat @ M @ N ) )
= ( divide_divide @ nat @ ( minus_minus @ nat @ ( times_times @ nat @ N @ ( minus_minus @ nat @ N @ ( one_one @ nat ) ) ) @ ( times_times @ nat @ M @ ( minus_minus @ nat @ M @ ( one_one @ nat ) ) ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ).
% Sum_Ico_nat
thf(fact_4518_double__gauss__sum,axiom,
! [A: $tType] :
( ( comm_semiring_1 @ A )
=> ! [N: nat] :
( ( times_times @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ ( groups7311177749621191930dd_sum @ nat @ A @ ( semiring_1_of_nat @ A ) @ ( set_or1337092689740270186AtMost @ nat @ ( zero_zero @ nat ) @ N ) ) )
= ( times_times @ A @ ( semiring_1_of_nat @ A @ N ) @ ( plus_plus @ A @ ( semiring_1_of_nat @ A @ N ) @ ( one_one @ A ) ) ) ) ) ).
% double_gauss_sum
thf(fact_4519_double__arith__series,axiom,
! [A: $tType] :
( ( comm_semiring_1 @ A )
=> ! [A2: A,D2: A,N: nat] :
( ( times_times @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) )
@ ( groups7311177749621191930dd_sum @ nat @ A
@ ^ [I2: nat] : ( plus_plus @ A @ A2 @ ( times_times @ A @ ( semiring_1_of_nat @ A @ I2 ) @ D2 ) )
@ ( set_or1337092689740270186AtMost @ nat @ ( zero_zero @ nat ) @ N ) ) )
= ( times_times @ A @ ( plus_plus @ A @ ( semiring_1_of_nat @ A @ N ) @ ( one_one @ A ) ) @ ( plus_plus @ A @ ( times_times @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ A2 ) @ ( times_times @ A @ ( semiring_1_of_nat @ A @ N ) @ D2 ) ) ) ) ) ).
% double_arith_series
thf(fact_4520_arith__series__nat,axiom,
! [A2: nat,D2: nat,N: nat] :
( ( groups7311177749621191930dd_sum @ nat @ nat
@ ^ [I2: nat] : ( plus_plus @ nat @ A2 @ ( times_times @ nat @ I2 @ D2 ) )
@ ( set_or1337092689740270186AtMost @ nat @ ( zero_zero @ nat ) @ N ) )
= ( divide_divide @ nat @ ( times_times @ nat @ ( suc @ N ) @ ( plus_plus @ nat @ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ A2 ) @ ( times_times @ nat @ N @ D2 ) ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ).
% arith_series_nat
thf(fact_4521_Sum__Icc__nat,axiom,
! [M: nat,N: nat] :
( ( groups7311177749621191930dd_sum @ nat @ nat
@ ^ [X2: nat] : X2
@ ( set_or1337092689740270186AtMost @ nat @ M @ N ) )
= ( divide_divide @ nat @ ( minus_minus @ nat @ ( times_times @ nat @ N @ ( plus_plus @ nat @ N @ ( one_one @ nat ) ) ) @ ( times_times @ nat @ M @ ( minus_minus @ nat @ M @ ( one_one @ nat ) ) ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ).
% Sum_Icc_nat
thf(fact_4522_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_4523_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_4524_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_4525_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_4526_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_4527_double__gauss__sum__from__Suc__0,axiom,
! [A: $tType] :
( ( comm_semiring_1 @ A )
=> ! [N: nat] :
( ( times_times @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ ( groups7311177749621191930dd_sum @ nat @ A @ ( semiring_1_of_nat @ A ) @ ( set_or1337092689740270186AtMost @ nat @ ( suc @ ( zero_zero @ nat ) ) @ N ) ) )
= ( times_times @ A @ ( semiring_1_of_nat @ A @ N ) @ ( plus_plus @ A @ ( semiring_1_of_nat @ A @ N ) @ ( one_one @ A ) ) ) ) ) ).
% double_gauss_sum_from_Suc_0
thf(fact_4528_gauss__sum,axiom,
! [A: $tType] :
( ( euclid5411537665997757685th_nat @ A )
=> ! [N: nat] :
( ( groups7311177749621191930dd_sum @ nat @ A @ ( semiring_1_of_nat @ A ) @ ( set_or1337092689740270186AtMost @ nat @ ( zero_zero @ nat ) @ N ) )
= ( divide_divide @ A @ ( times_times @ A @ ( semiring_1_of_nat @ A @ N ) @ ( plus_plus @ A @ ( semiring_1_of_nat @ A @ N ) @ ( one_one @ A ) ) ) @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) ) ) ).
% gauss_sum
thf(fact_4529_arith__series,axiom,
! [A: $tType] :
( ( euclid5411537665997757685th_nat @ A )
=> ! [A2: A,D2: A,N: nat] :
( ( groups7311177749621191930dd_sum @ nat @ A
@ ^ [I2: nat] : ( plus_plus @ A @ A2 @ ( times_times @ A @ ( semiring_1_of_nat @ A @ I2 ) @ D2 ) )
@ ( set_or1337092689740270186AtMost @ nat @ ( zero_zero @ nat ) @ N ) )
= ( divide_divide @ A @ ( times_times @ A @ ( plus_plus @ A @ ( semiring_1_of_nat @ A @ N ) @ ( one_one @ A ) ) @ ( plus_plus @ A @ ( times_times @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ A2 ) @ ( times_times @ A @ ( semiring_1_of_nat @ A @ N ) @ D2 ) ) ) @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) ) ) ).
% arith_series
thf(fact_4530_sum__gp__offset,axiom,
! [A: $tType] :
( ( ( division_ring @ A )
& ( comm_ring @ A ) )
=> ! [X: A,M: nat,N: nat] :
( ( ( X
= ( one_one @ A ) )
=> ( ( groups7311177749621191930dd_sum @ nat @ A @ ( power_power @ A @ X ) @ ( set_or1337092689740270186AtMost @ nat @ M @ ( plus_plus @ nat @ M @ N ) ) )
= ( plus_plus @ A @ ( semiring_1_of_nat @ A @ N ) @ ( one_one @ A ) ) ) )
& ( ( X
!= ( one_one @ A ) )
=> ( ( groups7311177749621191930dd_sum @ nat @ A @ ( power_power @ A @ X ) @ ( set_or1337092689740270186AtMost @ nat @ M @ ( plus_plus @ nat @ M @ N ) ) )
= ( divide_divide @ A @ ( times_times @ A @ ( power_power @ A @ X @ M ) @ ( minus_minus @ A @ ( one_one @ A ) @ ( power_power @ A @ X @ ( suc @ N ) ) ) ) @ ( minus_minus @ A @ ( one_one @ A ) @ X ) ) ) ) ) ) ).
% sum_gp_offset
thf(fact_4531_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_4532_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_4533_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_4534_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_4535_gauss__sum__from__Suc__0,axiom,
! [A: $tType] :
( ( euclid5411537665997757685th_nat @ A )
=> ! [N: nat] :
( ( groups7311177749621191930dd_sum @ nat @ A @ ( semiring_1_of_nat @ A ) @ ( set_or1337092689740270186AtMost @ nat @ ( suc @ ( zero_zero @ nat ) ) @ N ) )
= ( divide_divide @ A @ ( times_times @ A @ ( semiring_1_of_nat @ A @ N ) @ ( plus_plus @ A @ ( semiring_1_of_nat @ A @ N ) @ ( one_one @ A ) ) ) @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) ) ) ).
% gauss_sum_from_Suc_0
thf(fact_4536_Chebyshev__sum__upper,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [N: nat,A2: nat > A,B2: nat > A] :
( ! [I3: nat,J3: nat] :
( ( ord_less_eq @ nat @ I3 @ J3 )
=> ( ( ord_less @ nat @ J3 @ N )
=> ( ord_less_eq @ A @ ( A2 @ I3 ) @ ( A2 @ J3 ) ) ) )
=> ( ! [I3: nat,J3: nat] :
( ( ord_less_eq @ nat @ I3 @ J3 )
=> ( ( ord_less @ nat @ J3 @ N )
=> ( ord_less_eq @ A @ ( B2 @ J3 ) @ ( B2 @ I3 ) ) ) )
=> ( ord_less_eq @ A
@ ( times_times @ A @ ( semiring_1_of_nat @ A @ N )
@ ( groups7311177749621191930dd_sum @ nat @ A
@ ^ [K3: nat] : ( times_times @ A @ ( A2 @ K3 ) @ ( B2 @ K3 ) )
@ ( set_or7035219750837199246ssThan @ nat @ ( zero_zero @ nat ) @ N ) ) )
@ ( times_times @ A @ ( groups7311177749621191930dd_sum @ nat @ A @ A2 @ ( set_or7035219750837199246ssThan @ nat @ ( zero_zero @ nat ) @ N ) ) @ ( groups7311177749621191930dd_sum @ nat @ A @ B2 @ ( set_or7035219750837199246ssThan @ nat @ ( zero_zero @ nat ) @ N ) ) ) ) ) ) ) ).
% Chebyshev_sum_upper
thf(fact_4537_Chebyshev__sum__upper__nat,axiom,
! [N: nat,A2: nat > nat,B2: nat > nat] :
( ! [I3: nat,J3: nat] :
( ( ord_less_eq @ nat @ I3 @ J3 )
=> ( ( ord_less @ nat @ J3 @ N )
=> ( ord_less_eq @ nat @ ( A2 @ I3 ) @ ( A2 @ J3 ) ) ) )
=> ( ! [I3: nat,J3: nat] :
( ( ord_less_eq @ nat @ I3 @ J3 )
=> ( ( ord_less @ nat @ J3 @ N )
=> ( ord_less_eq @ nat @ ( B2 @ J3 ) @ ( B2 @ I3 ) ) ) )
=> ( ord_less_eq @ nat
@ ( times_times @ nat @ N
@ ( groups7311177749621191930dd_sum @ nat @ nat
@ ^ [I2: nat] : ( times_times @ nat @ ( A2 @ I2 ) @ ( B2 @ I2 ) )
@ ( set_or7035219750837199246ssThan @ nat @ ( zero_zero @ nat ) @ N ) ) )
@ ( times_times @ nat @ ( groups7311177749621191930dd_sum @ nat @ nat @ A2 @ ( set_or7035219750837199246ssThan @ nat @ ( zero_zero @ nat ) @ N ) ) @ ( groups7311177749621191930dd_sum @ nat @ nat @ B2 @ ( set_or7035219750837199246ssThan @ nat @ ( zero_zero @ nat ) @ N ) ) ) ) ) ) ).
% Chebyshev_sum_upper_nat
thf(fact_4538_Maclaurin__cos__expansion2,axiom,
! [X: real,N: nat] :
( ( ord_less @ real @ ( zero_zero @ real ) @ X )
=> ( ( ord_less @ nat @ ( zero_zero @ nat ) @ N )
=> ? [T7: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ T7 )
& ( ord_less @ real @ T7 @ X )
& ( ( cos @ real @ X )
= ( plus_plus @ real
@ ( groups7311177749621191930dd_sum @ nat @ real
@ ^ [M4: nat] : ( times_times @ real @ ( cos_coeff @ M4 ) @ ( power_power @ real @ X @ M4 ) )
@ ( set_ord_lessThan @ nat @ N ) )
@ ( times_times @ real @ ( divide_divide @ real @ ( cos @ real @ ( plus_plus @ real @ T7 @ ( times_times @ real @ ( times_times @ real @ ( divide_divide @ real @ ( one_one @ real ) @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) @ ( semiring_1_of_nat @ real @ N ) ) @ pi ) ) ) @ ( semiring_char_0_fact @ real @ N ) ) @ ( power_power @ real @ X @ N ) ) ) ) ) ) ) ).
% Maclaurin_cos_expansion2
thf(fact_4539_Maclaurin__minus__cos__expansion,axiom,
! [N: nat,X: real] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ N )
=> ( ( ord_less @ real @ X @ ( zero_zero @ real ) )
=> ? [T7: real] :
( ( ord_less @ real @ X @ T7 )
& ( ord_less @ real @ T7 @ ( zero_zero @ real ) )
& ( ( cos @ real @ X )
= ( plus_plus @ real
@ ( groups7311177749621191930dd_sum @ nat @ real
@ ^ [M4: nat] : ( times_times @ real @ ( cos_coeff @ M4 ) @ ( power_power @ real @ X @ M4 ) )
@ ( set_ord_lessThan @ nat @ N ) )
@ ( times_times @ real @ ( divide_divide @ real @ ( cos @ real @ ( plus_plus @ real @ T7 @ ( times_times @ real @ ( times_times @ real @ ( divide_divide @ real @ ( one_one @ real ) @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) @ ( semiring_1_of_nat @ real @ N ) ) @ pi ) ) ) @ ( semiring_char_0_fact @ real @ N ) ) @ ( power_power @ real @ X @ N ) ) ) ) ) ) ) ).
% Maclaurin_minus_cos_expansion
thf(fact_4540_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_4541_lessThan__0,axiom,
( ( set_ord_lessThan @ nat @ ( zero_zero @ nat ) )
= ( bot_bot @ ( set @ nat ) ) ) ).
% lessThan_0
thf(fact_4542_sum_OlessThan__Suc,axiom,
! [A: $tType] :
( ( comm_monoid_add @ A )
=> ! [G: nat > A,N: nat] :
( ( groups7311177749621191930dd_sum @ nat @ A @ G @ ( set_ord_lessThan @ nat @ ( suc @ N ) ) )
= ( plus_plus @ A @ ( groups7311177749621191930dd_sum @ nat @ A @ G @ ( set_ord_lessThan @ nat @ N ) ) @ ( G @ N ) ) ) ) ).
% sum.lessThan_Suc
thf(fact_4543_prod_OlessThan__Suc,axiom,
! [A: $tType] :
( ( comm_monoid_mult @ A )
=> ! [G: nat > A,N: nat] :
( ( groups7121269368397514597t_prod @ nat @ A @ G @ ( set_ord_lessThan @ nat @ ( suc @ N ) ) )
= ( times_times @ A @ ( groups7121269368397514597t_prod @ nat @ A @ G @ ( set_ord_lessThan @ nat @ N ) ) @ ( G @ N ) ) ) ) ).
% prod.lessThan_Suc
thf(fact_4544_sumr__cos__zero__one,axiom,
! [N: nat] :
( ( groups7311177749621191930dd_sum @ nat @ real
@ ^ [M4: nat] : ( times_times @ real @ ( cos_coeff @ M4 ) @ ( power_power @ real @ ( zero_zero @ real ) @ M4 ) )
@ ( set_ord_lessThan @ nat @ ( suc @ N ) ) )
= ( one_one @ real ) ) ).
% sumr_cos_zero_one
thf(fact_4545_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_4546_sum__eq__empty__iff,axiom,
! [B: $tType,A: $tType,A3: set @ A,F2: A > ( multiset @ B )] :
( ( finite_finite2 @ A @ A3 )
=> ( ( ( groups7311177749621191930dd_sum @ A @ ( multiset @ B ) @ F2 @ A3 )
= ( zero_zero @ ( multiset @ B ) ) )
= ( ! [X2: A] :
( ( member @ A @ X2 @ A3 )
=> ( ( F2 @ X2 )
= ( zero_zero @ ( multiset @ B ) ) ) ) ) ) ) ).
% sum_eq_empty_iff
thf(fact_4547_lessThan__atLeast0,axiom,
( ( set_ord_lessThan @ nat )
= ( set_or7035219750837199246ssThan @ nat @ ( zero_zero @ nat ) ) ) ).
% lessThan_atLeast0
thf(fact_4548_lessThan__strict__subset__iff,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [M: A,N: A] :
( ( ord_less @ ( set @ A ) @ ( set_ord_lessThan @ A @ M ) @ ( set_ord_lessThan @ A @ N ) )
= ( ord_less @ A @ M @ N ) ) ) ).
% lessThan_strict_subset_iff
thf(fact_4549_lessThan__empty__iff,axiom,
! [N: nat] :
( ( ( set_ord_lessThan @ nat @ N )
= ( bot_bot @ ( set @ nat ) ) )
= ( N
= ( zero_zero @ nat ) ) ) ).
% lessThan_empty_iff
thf(fact_4550_lessThan__Suc,axiom,
! [K: nat] :
( ( set_ord_lessThan @ nat @ ( suc @ K ) )
= ( insert @ nat @ K @ ( set_ord_lessThan @ nat @ K ) ) ) ).
% lessThan_Suc
thf(fact_4551_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
@ ^ [A4: B] : ( power_power @ A @ C2 @ ( F2 @ A4 ) )
@ A3 ) ) ) ).
% power_sum
thf(fact_4552_sum__SucD,axiom,
! [A: $tType,F2: A > nat,A3: set @ A,N: nat] :
( ( ( groups7311177749621191930dd_sum @ A @ nat @ F2 @ A3 )
= ( suc @ N ) )
=> ? [X3: A] :
( ( member @ A @ X3 @ A3 )
& ( ord_less @ nat @ ( zero_zero @ nat ) @ ( F2 @ X3 ) ) ) ) ).
% sum_SucD
thf(fact_4553_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 ) ) )
& ! [Y6: A] :
( ( member @ A @ Y6 @ A3 )
=> ( ( X2 != Y6 )
=> ( ( F2 @ Y6 )
= ( zero_zero @ nat ) ) ) ) ) ) ) ) ).
% sum_eq_Suc0_iff
thf(fact_4554_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 ) )
& ! [Y6: A] :
( ( member @ A @ Y6 @ A3 )
=> ( ( X2 != Y6 )
=> ( ( F2 @ Y6 )
= ( zero_zero @ nat ) ) ) ) ) ) ) ) ).
% sum_eq_1_iff
thf(fact_4555_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_4556_sum_Onat__diff__reindex,axiom,
! [A: $tType] :
( ( comm_monoid_add @ A )
=> ! [G: nat > A,N: nat] :
( ( groups7311177749621191930dd_sum @ nat @ A
@ ^ [I2: nat] : ( G @ ( minus_minus @ nat @ N @ ( suc @ I2 ) ) )
@ ( set_ord_lessThan @ nat @ N ) )
= ( groups7311177749621191930dd_sum @ nat @ A @ G @ ( set_ord_lessThan @ nat @ N ) ) ) ) ).
% sum.nat_diff_reindex
thf(fact_4557_prod_Onat__diff__reindex,axiom,
! [A: $tType] :
( ( comm_monoid_mult @ A )
=> ! [G: nat > A,N: nat] :
( ( groups7121269368397514597t_prod @ nat @ A
@ ^ [I2: nat] : ( G @ ( minus_minus @ nat @ N @ ( suc @ I2 ) ) )
@ ( set_ord_lessThan @ nat @ N ) )
= ( groups7121269368397514597t_prod @ nat @ A @ G @ ( set_ord_lessThan @ nat @ N ) ) ) ) ).
% prod.nat_diff_reindex
thf(fact_4558_sum__roots__unity,axiom,
! [N: nat] :
( ( ord_less @ nat @ ( one_one @ nat ) @ N )
=> ( ( groups7311177749621191930dd_sum @ complex @ complex
@ ^ [X2: complex] : X2
@ ( collect @ complex
@ ^ [Z7: complex] :
( ( power_power @ complex @ Z7 @ N )
= ( one_one @ complex ) ) ) )
= ( zero_zero @ complex ) ) ) ).
% sum_roots_unity
thf(fact_4559_sum__nth__roots,axiom,
! [N: nat,C2: complex] :
( ( ord_less @ nat @ ( one_one @ nat ) @ N )
=> ( ( groups7311177749621191930dd_sum @ complex @ complex
@ ^ [X2: complex] : X2
@ ( collect @ complex
@ ^ [Z7: complex] :
( ( power_power @ complex @ Z7 @ N )
= C2 ) ) )
= ( zero_zero @ complex ) ) ) ).
% sum_nth_roots
thf(fact_4560_ln__prod,axiom,
! [A: $tType,I6: set @ A,F2: A > real] :
( ( finite_finite2 @ A @ I6 )
=> ( ! [I3: A] :
( ( member @ A @ I3 @ I6 )
=> ( ord_less @ real @ ( zero_zero @ real ) @ ( F2 @ I3 ) ) )
=> ( ( ln_ln @ real @ ( groups7121269368397514597t_prod @ A @ real @ F2 @ I6 ) )
= ( groups7311177749621191930dd_sum @ A @ real
@ ^ [X2: A] : ( ln_ln @ real @ ( F2 @ X2 ) )
@ I6 ) ) ) ) ).
% ln_prod
thf(fact_4561_sum_OlessThan__Suc__shift,axiom,
! [A: $tType] :
( ( comm_monoid_add @ A )
=> ! [G: nat > A,N: nat] :
( ( groups7311177749621191930dd_sum @ nat @ A @ G @ ( set_ord_lessThan @ nat @ ( suc @ N ) ) )
= ( plus_plus @ A @ ( G @ ( zero_zero @ nat ) )
@ ( groups7311177749621191930dd_sum @ nat @ A
@ ^ [I2: nat] : ( G @ ( suc @ I2 ) )
@ ( set_ord_lessThan @ nat @ N ) ) ) ) ) ).
% sum.lessThan_Suc_shift
thf(fact_4562_sum__lessThan__telescope,axiom,
! [A: $tType] :
( ( ab_group_add @ A )
=> ! [F2: nat > A,M: nat] :
( ( groups7311177749621191930dd_sum @ nat @ A
@ ^ [N3: nat] : ( minus_minus @ A @ ( F2 @ ( suc @ N3 ) ) @ ( F2 @ N3 ) )
@ ( set_ord_lessThan @ nat @ M ) )
= ( minus_minus @ A @ ( F2 @ M ) @ ( F2 @ ( zero_zero @ nat ) ) ) ) ) ).
% sum_lessThan_telescope
thf(fact_4563_sum__lessThan__telescope_H,axiom,
! [A: $tType] :
( ( ab_group_add @ A )
=> ! [F2: nat > A,M: nat] :
( ( groups7311177749621191930dd_sum @ nat @ A
@ ^ [N3: nat] : ( minus_minus @ A @ ( F2 @ N3 ) @ ( F2 @ ( suc @ N3 ) ) )
@ ( set_ord_lessThan @ nat @ M ) )
= ( minus_minus @ A @ ( F2 @ ( zero_zero @ nat ) ) @ ( F2 @ M ) ) ) ) ).
% sum_lessThan_telescope'
thf(fact_4564_sumr__diff__mult__const2,axiom,
! [A: $tType] :
( ( ring_1 @ A )
=> ! [F2: nat > A,N: nat,R2: A] :
( ( minus_minus @ A @ ( groups7311177749621191930dd_sum @ nat @ A @ F2 @ ( set_ord_lessThan @ nat @ N ) ) @ ( times_times @ A @ ( semiring_1_of_nat @ A @ N ) @ R2 ) )
= ( groups7311177749621191930dd_sum @ nat @ A
@ ^ [I2: nat] : ( minus_minus @ A @ ( F2 @ I2 ) @ R2 )
@ ( set_ord_lessThan @ nat @ N ) ) ) ) ).
% sumr_diff_mult_const2
thf(fact_4565_prod_OlessThan__Suc__shift,axiom,
! [A: $tType] :
( ( comm_monoid_mult @ A )
=> ! [G: nat > A,N: nat] :
( ( groups7121269368397514597t_prod @ nat @ A @ G @ ( set_ord_lessThan @ nat @ ( suc @ N ) ) )
= ( times_times @ A @ ( G @ ( zero_zero @ nat ) )
@ ( groups7121269368397514597t_prod @ nat @ A
@ ^ [I2: nat] : ( G @ ( suc @ I2 ) )
@ ( set_ord_lessThan @ nat @ N ) ) ) ) ) ).
% prod.lessThan_Suc_shift
thf(fact_4566_summableI__nonneg__bounded,axiom,
! [A: $tType] :
( ( ( condit6923001295902523014norder @ A )
& ( ordere6911136660526730532id_add @ A )
& ( topolo1944317154257567458pology @ A ) )
=> ! [F2: nat > A,X: A] :
( ! [N2: nat] : ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( F2 @ N2 ) )
=> ( ! [N2: nat] : ( ord_less_eq @ A @ ( groups7311177749621191930dd_sum @ nat @ A @ F2 @ ( set_ord_lessThan @ nat @ N2 ) ) @ X )
=> ( summable @ A @ F2 ) ) ) ) ).
% summableI_nonneg_bounded
thf(fact_4567_sum_OatLeast1__atMost__eq,axiom,
! [A: $tType] :
( ( comm_monoid_add @ A )
=> ! [G: nat > A,N: nat] :
( ( groups7311177749621191930dd_sum @ nat @ A @ G @ ( set_or1337092689740270186AtMost @ nat @ ( suc @ ( zero_zero @ nat ) ) @ N ) )
= ( groups7311177749621191930dd_sum @ nat @ A
@ ^ [K3: nat] : ( G @ ( suc @ K3 ) )
@ ( set_ord_lessThan @ nat @ N ) ) ) ) ).
% sum.atLeast1_atMost_eq
thf(fact_4568_sum__bounds__lt__plus1,axiom,
! [A: $tType] :
( ( comm_monoid_add @ A )
=> ! [F2: nat > A,Mm: nat] :
( ( groups7311177749621191930dd_sum @ nat @ A
@ ^ [K3: nat] : ( F2 @ ( suc @ K3 ) )
@ ( set_ord_lessThan @ nat @ Mm ) )
= ( groups7311177749621191930dd_sum @ nat @ A @ F2 @ ( set_or1337092689740270186AtMost @ nat @ ( one_one @ nat ) @ Mm ) ) ) ) ).
% sum_bounds_lt_plus1
thf(fact_4569_sums__iff__shift,axiom,
! [A: $tType] :
( ( real_V822414075346904944vector @ A )
=> ! [F2: nat > A,N: nat,S: A] :
( ( sums @ A
@ ^ [I2: nat] : ( F2 @ ( plus_plus @ nat @ I2 @ N ) )
@ S )
= ( sums @ A @ F2 @ ( plus_plus @ A @ S @ ( groups7311177749621191930dd_sum @ nat @ A @ F2 @ ( set_ord_lessThan @ nat @ N ) ) ) ) ) ) ).
% sums_iff_shift
thf(fact_4570_sums__iff__shift_H,axiom,
! [A: $tType] :
( ( real_V822414075346904944vector @ A )
=> ! [F2: nat > A,N: nat,S: A] :
( ( sums @ A
@ ^ [I2: nat] : ( F2 @ ( plus_plus @ nat @ I2 @ N ) )
@ ( minus_minus @ A @ S @ ( groups7311177749621191930dd_sum @ nat @ A @ F2 @ ( set_ord_lessThan @ nat @ N ) ) ) )
= ( sums @ A @ F2 @ S ) ) ) ).
% sums_iff_shift'
thf(fact_4571_sums__split__initial__segment,axiom,
! [A: $tType] :
( ( real_V822414075346904944vector @ A )
=> ! [F2: nat > A,S: A,N: nat] :
( ( sums @ A @ F2 @ S )
=> ( sums @ A
@ ^ [I2: nat] : ( F2 @ ( plus_plus @ nat @ I2 @ N ) )
@ ( minus_minus @ A @ S @ ( groups7311177749621191930dd_sum @ nat @ A @ F2 @ ( set_ord_lessThan @ nat @ N ) ) ) ) ) ) ).
% sums_split_initial_segment
thf(fact_4572_prod_OatLeast1__atMost__eq,axiom,
! [A: $tType] :
( ( comm_monoid_mult @ A )
=> ! [G: nat > A,N: nat] :
( ( groups7121269368397514597t_prod @ nat @ A @ G @ ( set_or1337092689740270186AtMost @ nat @ ( suc @ ( zero_zero @ nat ) ) @ N ) )
= ( groups7121269368397514597t_prod @ nat @ A
@ ^ [K3: nat] : ( G @ ( suc @ K3 ) )
@ ( set_ord_lessThan @ nat @ N ) ) ) ) ).
% prod.atLeast1_atMost_eq
thf(fact_4573_sum_Onat__group,axiom,
! [A: $tType] :
( ( comm_monoid_add @ A )
=> ! [G: nat > A,K: nat,N: nat] :
( ( groups7311177749621191930dd_sum @ nat @ A
@ ^ [M4: nat] : ( groups7311177749621191930dd_sum @ nat @ A @ G @ ( set_or7035219750837199246ssThan @ nat @ ( times_times @ nat @ M4 @ K ) @ ( plus_plus @ nat @ ( times_times @ nat @ M4 @ K ) @ K ) ) )
@ ( set_ord_lessThan @ nat @ N ) )
= ( groups7311177749621191930dd_sum @ nat @ A @ G @ ( set_ord_lessThan @ nat @ ( times_times @ nat @ N @ K ) ) ) ) ) ).
% sum.nat_group
thf(fact_4574_prod_Onat__group,axiom,
! [A: $tType] :
( ( comm_monoid_mult @ A )
=> ! [G: nat > A,K: nat,N: nat] :
( ( groups7121269368397514597t_prod @ nat @ A
@ ^ [M4: nat] : ( groups7121269368397514597t_prod @ nat @ A @ G @ ( set_or7035219750837199246ssThan @ nat @ ( times_times @ nat @ M4 @ K ) @ ( plus_plus @ nat @ ( times_times @ nat @ M4 @ K ) @ K ) ) )
@ ( set_ord_lessThan @ nat @ N ) )
= ( groups7121269368397514597t_prod @ nat @ A @ G @ ( set_ord_lessThan @ nat @ ( times_times @ nat @ N @ K ) ) ) ) ) ).
% prod.nat_group
thf(fact_4575_norm__prod__diff,axiom,
! [A: $tType,I8: $tType] :
( ( ( comm_monoid_mult @ A )
& ( real_V2822296259951069270ebra_1 @ A ) )
=> ! [I6: set @ I8,Z: I8 > A,W: I8 > A] :
( ! [I3: I8] :
( ( member @ I8 @ I3 @ I6 )
=> ( ord_less_eq @ real @ ( real_V7770717601297561774m_norm @ A @ ( Z @ I3 ) ) @ ( one_one @ real ) ) )
=> ( ! [I3: I8] :
( ( member @ I8 @ I3 @ I6 )
=> ( ord_less_eq @ real @ ( real_V7770717601297561774m_norm @ A @ ( W @ I3 ) ) @ ( one_one @ real ) ) )
=> ( ord_less_eq @ real @ ( real_V7770717601297561774m_norm @ A @ ( minus_minus @ A @ ( groups7121269368397514597t_prod @ I8 @ A @ Z @ I6 ) @ ( groups7121269368397514597t_prod @ I8 @ A @ W @ I6 ) ) )
@ ( groups7311177749621191930dd_sum @ I8 @ real
@ ^ [I2: I8] : ( real_V7770717601297561774m_norm @ A @ ( minus_minus @ A @ ( Z @ I2 ) @ ( W @ I2 ) ) )
@ I6 ) ) ) ) ) ).
% norm_prod_diff
thf(fact_4576_one__diff__power__eq,axiom,
! [A: $tType] :
( ( ( monoid_mult @ A )
& ( comm_ring @ A ) )
=> ! [X: A,N: nat] :
( ( minus_minus @ A @ ( one_one @ A ) @ ( power_power @ A @ X @ N ) )
= ( times_times @ A @ ( minus_minus @ A @ ( one_one @ A ) @ X ) @ ( groups7311177749621191930dd_sum @ nat @ A @ ( power_power @ A @ X ) @ ( set_ord_lessThan @ nat @ N ) ) ) ) ) ).
% one_diff_power_eq
thf(fact_4577_power__diff__1__eq,axiom,
! [A: $tType] :
( ( ( monoid_mult @ A )
& ( comm_ring @ A ) )
=> ! [X: A,N: nat] :
( ( minus_minus @ A @ ( power_power @ A @ X @ N ) @ ( one_one @ A ) )
= ( times_times @ A @ ( minus_minus @ A @ X @ ( one_one @ A ) ) @ ( groups7311177749621191930dd_sum @ nat @ A @ ( power_power @ A @ X ) @ ( set_ord_lessThan @ nat @ N ) ) ) ) ) ).
% power_diff_1_eq
thf(fact_4578_geometric__sum,axiom,
! [A: $tType] :
( ( field @ A )
=> ! [X: A,N: nat] :
( ( X
!= ( one_one @ A ) )
=> ( ( groups7311177749621191930dd_sum @ nat @ A @ ( power_power @ A @ X ) @ ( set_ord_lessThan @ nat @ N ) )
= ( divide_divide @ A @ ( minus_minus @ A @ ( power_power @ A @ X @ N ) @ ( one_one @ A ) ) @ ( minus_minus @ A @ X @ ( one_one @ A ) ) ) ) ) ) ).
% geometric_sum
thf(fact_4579_suminf__split__initial__segment,axiom,
! [A: $tType] :
( ( real_V822414075346904944vector @ A )
=> ! [F2: nat > A,K: nat] :
( ( summable @ A @ F2 )
=> ( ( suminf @ A @ F2 )
= ( plus_plus @ A
@ ( suminf @ A
@ ^ [N3: nat] : ( F2 @ ( plus_plus @ nat @ N3 @ K ) ) )
@ ( groups7311177749621191930dd_sum @ nat @ A @ F2 @ ( set_ord_lessThan @ nat @ K ) ) ) ) ) ) ).
% suminf_split_initial_segment
thf(fact_4580_suminf__minus__initial__segment,axiom,
! [A: $tType] :
( ( real_V822414075346904944vector @ A )
=> ! [F2: nat > A,K: nat] :
( ( summable @ A @ F2 )
=> ( ( suminf @ A
@ ^ [N3: nat] : ( F2 @ ( plus_plus @ nat @ N3 @ K ) ) )
= ( minus_minus @ A @ ( suminf @ A @ F2 ) @ ( groups7311177749621191930dd_sum @ nat @ A @ F2 @ ( set_ord_lessThan @ nat @ K ) ) ) ) ) ) ).
% suminf_minus_initial_segment
thf(fact_4581_sum__less__suminf,axiom,
! [A: $tType] :
( ( ( ordere8940638589300402666id_add @ A )
& ( topolo1944317154257567458pology @ A ) )
=> ! [F2: nat > A,N: nat] :
( ( summable @ A @ F2 )
=> ( ! [M2: nat] :
( ( ord_less_eq @ nat @ N @ M2 )
=> ( ord_less @ A @ ( zero_zero @ A ) @ ( F2 @ M2 ) ) )
=> ( ord_less @ A @ ( groups7311177749621191930dd_sum @ nat @ A @ F2 @ ( set_ord_lessThan @ nat @ N ) ) @ ( suminf @ A @ F2 ) ) ) ) ) ).
% sum_less_suminf
thf(fact_4582_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_4583_sum__gp__strict,axiom,
! [A: $tType] :
( ( ( division_ring @ A )
& ( comm_ring @ A ) )
=> ! [X: A,N: nat] :
( ( ( X
= ( one_one @ A ) )
=> ( ( groups7311177749621191930dd_sum @ nat @ A @ ( power_power @ A @ X ) @ ( set_ord_lessThan @ nat @ N ) )
= ( semiring_1_of_nat @ A @ N ) ) )
& ( ( X
!= ( one_one @ A ) )
=> ( ( groups7311177749621191930dd_sum @ nat @ A @ ( power_power @ A @ X ) @ ( set_ord_lessThan @ nat @ N ) )
= ( divide_divide @ A @ ( minus_minus @ A @ ( one_one @ A ) @ ( power_power @ A @ X @ N ) ) @ ( minus_minus @ A @ ( one_one @ A ) @ X ) ) ) ) ) ) ).
% sum_gp_strict
thf(fact_4584_power__diff__sumr2,axiom,
! [A: $tType] :
( ( ( monoid_mult @ A )
& ( comm_ring @ A ) )
=> ! [X: A,N: nat,Y: A] :
( ( minus_minus @ A @ ( power_power @ A @ X @ N ) @ ( power_power @ A @ Y @ N ) )
= ( times_times @ A @ ( minus_minus @ A @ X @ Y )
@ ( groups7311177749621191930dd_sum @ nat @ A
@ ^ [I2: nat] : ( times_times @ A @ ( power_power @ A @ Y @ ( minus_minus @ nat @ N @ ( suc @ I2 ) ) ) @ ( power_power @ A @ X @ I2 ) )
@ ( set_ord_lessThan @ nat @ N ) ) ) ) ) ).
% power_diff_sumr2
thf(fact_4585_diff__power__eq__sum,axiom,
! [A: $tType] :
( ( ( monoid_mult @ A )
& ( comm_ring @ A ) )
=> ! [X: A,N: nat,Y: A] :
( ( minus_minus @ A @ ( power_power @ A @ X @ ( suc @ N ) ) @ ( power_power @ A @ Y @ ( suc @ N ) ) )
= ( times_times @ A @ ( minus_minus @ A @ X @ Y )
@ ( groups7311177749621191930dd_sum @ nat @ A
@ ^ [P5: nat] : ( times_times @ A @ ( power_power @ A @ X @ P5 ) @ ( power_power @ A @ Y @ ( minus_minus @ nat @ N @ P5 ) ) )
@ ( set_ord_lessThan @ nat @ ( suc @ N ) ) ) ) ) ) ).
% diff_power_eq_sum
thf(fact_4586_atLeast1__lessThan__eq__remove0,axiom,
! [N: nat] :
( ( set_or7035219750837199246ssThan @ nat @ ( suc @ ( zero_zero @ nat ) ) @ N )
= ( minus_minus @ ( set @ nat ) @ ( set_ord_lessThan @ nat @ N ) @ ( insert @ nat @ ( zero_zero @ nat ) @ ( bot_bot @ ( set @ nat ) ) ) ) ) ).
% atLeast1_lessThan_eq_remove0
thf(fact_4587_real__sum__nat__ivl__bounded2,axiom,
! [A: $tType] :
( ( linordered_semidom @ A )
=> ! [N: nat,F2: nat > A,K6: A,K: nat] :
( ! [P8: nat] :
( ( ord_less @ nat @ P8 @ N )
=> ( ord_less_eq @ A @ ( F2 @ P8 ) @ K6 ) )
=> ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ K6 )
=> ( ord_less_eq @ A @ ( groups7311177749621191930dd_sum @ nat @ A @ F2 @ ( set_ord_lessThan @ nat @ ( minus_minus @ nat @ N @ K ) ) ) @ ( times_times @ A @ ( semiring_1_of_nat @ A @ N ) @ K6 ) ) ) ) ) ).
% real_sum_nat_ivl_bounded2
thf(fact_4588_sum__less__suminf2,axiom,
! [A: $tType] :
( ( ( ordere8940638589300402666id_add @ A )
& ( topolo1944317154257567458pology @ A ) )
=> ! [F2: nat > A,N: nat,I: nat] :
( ( summable @ A @ F2 )
=> ( ! [M2: nat] :
( ( ord_less_eq @ nat @ N @ M2 )
=> ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( F2 @ M2 ) ) )
=> ( ( ord_less_eq @ nat @ N @ I )
=> ( ( ord_less @ A @ ( zero_zero @ A ) @ ( F2 @ I ) )
=> ( ord_less @ A @ ( groups7311177749621191930dd_sum @ nat @ A @ F2 @ ( set_ord_lessThan @ nat @ N ) ) @ ( suminf @ A @ F2 ) ) ) ) ) ) ) ).
% sum_less_suminf2
thf(fact_4589_one__diff__power__eq_H,axiom,
! [A: $tType] :
( ( ( monoid_mult @ A )
& ( comm_ring @ A ) )
=> ! [X: A,N: nat] :
( ( minus_minus @ A @ ( one_one @ A ) @ ( power_power @ A @ X @ N ) )
= ( times_times @ A @ ( minus_minus @ A @ ( one_one @ A ) @ X )
@ ( groups7311177749621191930dd_sum @ nat @ A
@ ^ [I2: nat] : ( power_power @ A @ X @ ( minus_minus @ nat @ N @ ( suc @ I2 ) ) )
@ ( set_ord_lessThan @ nat @ N ) ) ) ) ) ).
% one_diff_power_eq'
thf(fact_4590_Maclaurin__zero,axiom,
! [A: $tType] :
( ( zero @ A )
=> ! [X: real,N: nat,Diff: nat > A > real] :
( ( X
= ( zero_zero @ real ) )
=> ( ( N
!= ( zero_zero @ nat ) )
=> ( ( groups7311177749621191930dd_sum @ nat @ real
@ ^ [M4: nat] : ( times_times @ real @ ( divide_divide @ real @ ( Diff @ M4 @ ( zero_zero @ A ) ) @ ( semiring_char_0_fact @ real @ M4 ) ) @ ( power_power @ real @ X @ M4 ) )
@ ( set_ord_lessThan @ nat @ N ) )
= ( Diff @ ( zero_zero @ nat ) @ ( zero_zero @ A ) ) ) ) ) ) ).
% Maclaurin_zero
thf(fact_4591_Maclaurin__lemma,axiom,
! [H2: real,F2: real > real,J: nat > real,N: nat] :
( ( ord_less @ real @ ( zero_zero @ real ) @ H2 )
=> ? [B10: real] :
( ( F2 @ H2 )
= ( plus_plus @ real
@ ( groups7311177749621191930dd_sum @ nat @ real
@ ^ [M4: nat] : ( times_times @ real @ ( divide_divide @ real @ ( J @ M4 ) @ ( semiring_char_0_fact @ real @ M4 ) ) @ ( power_power @ real @ H2 @ M4 ) )
@ ( set_ord_lessThan @ nat @ N ) )
@ ( times_times @ real @ B10 @ ( divide_divide @ real @ ( power_power @ real @ H2 @ N ) @ ( semiring_char_0_fact @ real @ N ) ) ) ) ) ) ).
% Maclaurin_lemma
thf(fact_4592_sum__split__even__odd,axiom,
! [F2: nat > real,G: nat > real,N: nat] :
( ( groups7311177749621191930dd_sum @ nat @ real
@ ^ [I2: nat] : ( if @ real @ ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ I2 ) @ ( F2 @ I2 ) @ ( G @ I2 ) )
@ ( set_ord_lessThan @ nat @ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N ) ) )
= ( plus_plus @ real
@ ( groups7311177749621191930dd_sum @ nat @ real
@ ^ [I2: nat] : ( F2 @ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ I2 ) )
@ ( set_ord_lessThan @ nat @ N ) )
@ ( groups7311177749621191930dd_sum @ nat @ real
@ ^ [I2: nat] : ( G @ ( plus_plus @ nat @ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ I2 ) @ ( one_one @ nat ) ) )
@ ( set_ord_lessThan @ nat @ N ) ) ) ) ).
% sum_split_even_odd
thf(fact_4593_Maclaurin__exp__le,axiom,
! [X: real,N: nat] :
? [T7: real] :
( ( ord_less_eq @ real @ ( abs_abs @ real @ T7 ) @ ( abs_abs @ real @ X ) )
& ( ( exp @ real @ X )
= ( plus_plus @ real
@ ( groups7311177749621191930dd_sum @ nat @ real
@ ^ [M4: nat] : ( divide_divide @ real @ ( power_power @ real @ X @ M4 ) @ ( semiring_char_0_fact @ real @ M4 ) )
@ ( set_ord_lessThan @ nat @ N ) )
@ ( times_times @ real @ ( divide_divide @ real @ ( exp @ real @ T7 ) @ ( semiring_char_0_fact @ real @ N ) ) @ ( power_power @ real @ X @ N ) ) ) ) ) ).
% Maclaurin_exp_le
thf(fact_4594_Sum__Icc__int,axiom,
! [M: int,N: int] :
( ( ord_less_eq @ int @ M @ N )
=> ( ( groups7311177749621191930dd_sum @ int @ int
@ ^ [X2: int] : X2
@ ( set_or1337092689740270186AtMost @ int @ M @ N ) )
= ( divide_divide @ int @ ( minus_minus @ int @ ( times_times @ int @ N @ ( plus_plus @ int @ N @ ( one_one @ int ) ) ) @ ( times_times @ int @ M @ ( minus_minus @ int @ M @ ( one_one @ int ) ) ) ) @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) ) ) ) ).
% Sum_Icc_int
thf(fact_4595_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_4596_Maclaurin__exp__lt,axiom,
! [X: real,N: nat] :
( ( X
!= ( zero_zero @ real ) )
=> ( ( ord_less @ nat @ ( zero_zero @ nat ) @ N )
=> ? [T7: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ ( abs_abs @ real @ T7 ) )
& ( ord_less @ real @ ( abs_abs @ real @ T7 ) @ ( abs_abs @ real @ X ) )
& ( ( exp @ real @ X )
= ( plus_plus @ real
@ ( groups7311177749621191930dd_sum @ nat @ real
@ ^ [M4: nat] : ( divide_divide @ real @ ( power_power @ real @ X @ M4 ) @ ( semiring_char_0_fact @ real @ M4 ) )
@ ( set_ord_lessThan @ nat @ N ) )
@ ( times_times @ real @ ( divide_divide @ real @ ( exp @ real @ T7 ) @ ( semiring_char_0_fact @ real @ N ) ) @ ( power_power @ real @ X @ N ) ) ) ) ) ) ) ).
% Maclaurin_exp_lt
thf(fact_4597_lemma__termdiff2,axiom,
! [A: $tType] :
( ( field @ A )
=> ! [H2: A,Z: A,N: nat] :
( ( H2
!= ( zero_zero @ A ) )
=> ( ( minus_minus @ A @ ( divide_divide @ A @ ( minus_minus @ A @ ( power_power @ A @ ( plus_plus @ A @ Z @ H2 ) @ N ) @ ( power_power @ A @ Z @ N ) ) @ H2 ) @ ( times_times @ A @ ( semiring_1_of_nat @ A @ N ) @ ( power_power @ A @ Z @ ( minus_minus @ nat @ N @ ( suc @ ( zero_zero @ nat ) ) ) ) ) )
= ( times_times @ A @ H2
@ ( groups7311177749621191930dd_sum @ nat @ A
@ ^ [P5: nat] :
( groups7311177749621191930dd_sum @ nat @ A
@ ^ [Q6: nat] : ( times_times @ A @ ( power_power @ A @ ( plus_plus @ A @ Z @ H2 ) @ Q6 ) @ ( power_power @ A @ Z @ ( minus_minus @ nat @ ( minus_minus @ nat @ N @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ Q6 ) ) )
@ ( set_ord_lessThan @ nat @ ( minus_minus @ nat @ ( minus_minus @ nat @ N @ ( suc @ ( zero_zero @ nat ) ) ) @ P5 ) ) )
@ ( set_ord_lessThan @ nat @ ( minus_minus @ nat @ N @ ( suc @ ( zero_zero @ nat ) ) ) ) ) ) ) ) ) ).
% lemma_termdiff2
thf(fact_4598_Maclaurin__sin__expansion,axiom,
! [X: real,N: nat] :
? [T7: real] :
( ( sin @ real @ X )
= ( plus_plus @ real
@ ( groups7311177749621191930dd_sum @ nat @ real
@ ^ [M4: nat] : ( times_times @ real @ ( sin_coeff @ M4 ) @ ( power_power @ real @ X @ M4 ) )
@ ( set_ord_lessThan @ nat @ N ) )
@ ( times_times @ real @ ( divide_divide @ real @ ( sin @ real @ ( plus_plus @ real @ T7 @ ( times_times @ real @ ( times_times @ real @ ( divide_divide @ real @ ( one_one @ real ) @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) @ ( semiring_1_of_nat @ real @ N ) ) @ pi ) ) ) @ ( semiring_char_0_fact @ real @ N ) ) @ ( power_power @ real @ X @ N ) ) ) ) ).
% Maclaurin_sin_expansion
thf(fact_4599_Maclaurin__sin__expansion2,axiom,
! [X: real,N: nat] :
? [T7: real] :
( ( ord_less_eq @ real @ ( abs_abs @ real @ T7 ) @ ( abs_abs @ real @ X ) )
& ( ( sin @ real @ X )
= ( plus_plus @ real
@ ( groups7311177749621191930dd_sum @ nat @ real
@ ^ [M4: nat] : ( times_times @ real @ ( sin_coeff @ M4 ) @ ( power_power @ real @ X @ M4 ) )
@ ( set_ord_lessThan @ nat @ N ) )
@ ( times_times @ real @ ( divide_divide @ real @ ( sin @ real @ ( plus_plus @ real @ T7 @ ( times_times @ real @ ( times_times @ real @ ( divide_divide @ real @ ( one_one @ real ) @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) @ ( semiring_1_of_nat @ real @ N ) ) @ pi ) ) ) @ ( semiring_char_0_fact @ real @ N ) ) @ ( power_power @ real @ X @ N ) ) ) ) ) ).
% Maclaurin_sin_expansion2
thf(fact_4600_Maclaurin__cos__expansion,axiom,
! [X: real,N: nat] :
? [T7: real] :
( ( ord_less_eq @ real @ ( abs_abs @ real @ T7 ) @ ( abs_abs @ real @ X ) )
& ( ( cos @ real @ X )
= ( plus_plus @ real
@ ( groups7311177749621191930dd_sum @ nat @ real
@ ^ [M4: nat] : ( times_times @ real @ ( cos_coeff @ M4 ) @ ( power_power @ real @ X @ M4 ) )
@ ( set_ord_lessThan @ nat @ N ) )
@ ( times_times @ real @ ( divide_divide @ real @ ( cos @ real @ ( plus_plus @ real @ T7 @ ( times_times @ real @ ( times_times @ real @ ( divide_divide @ real @ ( one_one @ real ) @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) @ ( semiring_1_of_nat @ real @ N ) ) @ pi ) ) ) @ ( semiring_char_0_fact @ real @ N ) ) @ ( power_power @ real @ X @ N ) ) ) ) ) ).
% Maclaurin_cos_expansion
thf(fact_4601_Maclaurin__sin__expansion4,axiom,
! [X: real,N: nat] :
( ( ord_less @ real @ ( zero_zero @ real ) @ X )
=> ? [T7: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ T7 )
& ( ord_less_eq @ real @ T7 @ X )
& ( ( sin @ real @ X )
= ( plus_plus @ real
@ ( groups7311177749621191930dd_sum @ nat @ real
@ ^ [M4: nat] : ( times_times @ real @ ( sin_coeff @ M4 ) @ ( power_power @ real @ X @ M4 ) )
@ ( set_ord_lessThan @ nat @ N ) )
@ ( times_times @ real @ ( divide_divide @ real @ ( sin @ real @ ( plus_plus @ real @ T7 @ ( times_times @ real @ ( times_times @ real @ ( divide_divide @ real @ ( one_one @ real ) @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) @ ( semiring_1_of_nat @ real @ N ) ) @ pi ) ) ) @ ( semiring_char_0_fact @ real @ N ) ) @ ( power_power @ real @ X @ N ) ) ) ) ) ) ).
% Maclaurin_sin_expansion4
thf(fact_4602_Maclaurin__sin__expansion3,axiom,
! [N: nat,X: real] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ N )
=> ( ( ord_less @ real @ ( zero_zero @ real ) @ X )
=> ? [T7: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ T7 )
& ( ord_less @ real @ T7 @ X )
& ( ( sin @ real @ X )
= ( plus_plus @ real
@ ( groups7311177749621191930dd_sum @ nat @ real
@ ^ [M4: nat] : ( times_times @ real @ ( sin_coeff @ M4 ) @ ( power_power @ real @ X @ M4 ) )
@ ( set_ord_lessThan @ nat @ N ) )
@ ( times_times @ real @ ( divide_divide @ real @ ( sin @ real @ ( plus_plus @ real @ T7 @ ( times_times @ real @ ( times_times @ real @ ( divide_divide @ real @ ( one_one @ real ) @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) @ ( semiring_1_of_nat @ real @ N ) ) @ pi ) ) ) @ ( semiring_char_0_fact @ real @ N ) ) @ ( power_power @ real @ X @ N ) ) ) ) ) ) ) ).
% Maclaurin_sin_expansion3
thf(fact_4603_choose__odd__sum,axiom,
! [A: $tType] :
( ( comm_ring_1 @ A )
=> ! [N: nat] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ N )
=> ( ( times_times @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) )
@ ( groups7311177749621191930dd_sum @ nat @ A
@ ^ [I2: nat] :
( if @ A
@ ~ ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ I2 )
@ ( semiring_1_of_nat @ A @ ( binomial @ N @ I2 ) )
@ ( zero_zero @ A ) )
@ ( set_ord_atMost @ nat @ N ) ) )
= ( power_power @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ N ) ) ) ) ).
% choose_odd_sum
thf(fact_4604_choose__even__sum,axiom,
! [A: $tType] :
( ( comm_ring_1 @ A )
=> ! [N: nat] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ N )
=> ( ( times_times @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) )
@ ( groups7311177749621191930dd_sum @ nat @ A
@ ^ [I2: nat] : ( if @ A @ ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ I2 ) @ ( semiring_1_of_nat @ A @ ( binomial @ N @ I2 ) ) @ ( zero_zero @ A ) )
@ ( set_ord_atMost @ nat @ N ) ) )
= ( power_power @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ N ) ) ) ) ).
% choose_even_sum
thf(fact_4605_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_4606_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_4607_arccos__1,axiom,
( ( arccos @ ( one_one @ real ) )
= ( zero_zero @ real ) ) ).
% arccos_1
thf(fact_4608_sum_OatMost__Suc,axiom,
! [A: $tType] :
( ( comm_monoid_add @ A )
=> ! [G: nat > A,N: nat] :
( ( groups7311177749621191930dd_sum @ nat @ A @ G @ ( set_ord_atMost @ nat @ ( suc @ N ) ) )
= ( plus_plus @ A @ ( groups7311177749621191930dd_sum @ nat @ A @ G @ ( set_ord_atMost @ nat @ N ) ) @ ( G @ ( suc @ N ) ) ) ) ) ).
% sum.atMost_Suc
thf(fact_4609_prod_OatMost__Suc,axiom,
! [A: $tType] :
( ( comm_monoid_mult @ A )
=> ! [G: nat > A,N: nat] :
( ( groups7121269368397514597t_prod @ nat @ A @ G @ ( set_ord_atMost @ nat @ ( suc @ N ) ) )
= ( times_times @ A @ ( groups7121269368397514597t_prod @ nat @ A @ G @ ( set_ord_atMost @ nat @ N ) ) @ ( G @ ( suc @ N ) ) ) ) ) ).
% prod.atMost_Suc
thf(fact_4610_arccos__minus__1,axiom,
( ( arccos @ ( uminus_uminus @ real @ ( one_one @ real ) ) )
= pi ) ).
% arccos_minus_1
thf(fact_4611_atMost__0,axiom,
( ( set_ord_atMost @ nat @ ( zero_zero @ nat ) )
= ( insert @ nat @ ( zero_zero @ nat ) @ ( bot_bot @ ( set @ nat ) ) ) ) ).
% atMost_0
thf(fact_4612_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_4613_arccos__0,axiom,
( ( arccos @ ( zero_zero @ real ) )
= ( divide_divide @ real @ pi @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) ) ).
% arccos_0
thf(fact_4614_atMost__atLeast0,axiom,
( ( set_ord_atMost @ nat )
= ( set_or1337092689740270186AtMost @ nat @ ( zero_zero @ nat ) ) ) ).
% atMost_atLeast0
thf(fact_4615_lessThan__Suc__atMost,axiom,
! [K: nat] :
( ( set_ord_lessThan @ nat @ ( suc @ K ) )
= ( set_ord_atMost @ nat @ K ) ) ).
% lessThan_Suc_atMost
thf(fact_4616_atMost__Suc,axiom,
! [K: nat] :
( ( set_ord_atMost @ nat @ ( suc @ K ) )
= ( insert @ nat @ ( suc @ K ) @ ( set_ord_atMost @ nat @ K ) ) ) ).
% atMost_Suc
thf(fact_4617_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_4618_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_4619_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_4620_sum__choose__upper,axiom,
! [M: nat,N: nat] :
( ( groups7311177749621191930dd_sum @ nat @ nat
@ ^ [K3: nat] : ( binomial @ K3 @ M )
@ ( set_ord_atMost @ nat @ N ) )
= ( binomial @ ( suc @ N ) @ ( suc @ M ) ) ) ).
% sum_choose_upper
thf(fact_4621_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_4622_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_4623_sum_OatMost__Suc__shift,axiom,
! [A: $tType] :
( ( comm_monoid_add @ A )
=> ! [G: nat > A,N: nat] :
( ( groups7311177749621191930dd_sum @ nat @ A @ G @ ( set_ord_atMost @ nat @ ( suc @ N ) ) )
= ( plus_plus @ A @ ( G @ ( zero_zero @ nat ) )
@ ( groups7311177749621191930dd_sum @ nat @ A
@ ^ [I2: nat] : ( G @ ( suc @ I2 ) )
@ ( set_ord_atMost @ nat @ N ) ) ) ) ) ).
% sum.atMost_Suc_shift
thf(fact_4624_sum__telescope,axiom,
! [A: $tType] :
( ( ab_group_add @ A )
=> ! [F2: nat > A,I: nat] :
( ( groups7311177749621191930dd_sum @ nat @ A
@ ^ [I2: nat] : ( minus_minus @ A @ ( F2 @ I2 ) @ ( F2 @ ( suc @ I2 ) ) )
@ ( set_ord_atMost @ nat @ I ) )
= ( minus_minus @ A @ ( F2 @ ( zero_zero @ nat ) ) @ ( F2 @ ( suc @ I ) ) ) ) ) ).
% sum_telescope
thf(fact_4625_polyfun__eq__coeffs,axiom,
! [A: $tType] :
( ( ( real_V8999393235501362500lgebra @ A )
& ( idom @ A ) )
=> ! [C2: nat > A,N: nat,D2: nat > A] :
( ( ! [X2: A] :
( ( groups7311177749621191930dd_sum @ nat @ A
@ ^ [I2: nat] : ( times_times @ A @ ( C2 @ I2 ) @ ( power_power @ A @ X2 @ I2 ) )
@ ( set_ord_atMost @ nat @ N ) )
= ( groups7311177749621191930dd_sum @ nat @ A
@ ^ [I2: nat] : ( times_times @ A @ ( D2 @ I2 ) @ ( power_power @ A @ X2 @ I2 ) )
@ ( set_ord_atMost @ nat @ N ) ) ) )
= ( ! [I2: nat] :
( ( ord_less_eq @ nat @ I2 @ N )
=> ( ( C2 @ I2 )
= ( D2 @ I2 ) ) ) ) ) ) ).
% polyfun_eq_coeffs
thf(fact_4626_prod_OatMost__Suc__shift,axiom,
! [A: $tType] :
( ( comm_monoid_mult @ A )
=> ! [G: nat > A,N: nat] :
( ( groups7121269368397514597t_prod @ nat @ A @ G @ ( set_ord_atMost @ nat @ ( suc @ N ) ) )
= ( times_times @ A @ ( G @ ( zero_zero @ nat ) )
@ ( groups7121269368397514597t_prod @ nat @ A
@ ^ [I2: nat] : ( G @ ( suc @ I2 ) )
@ ( set_ord_atMost @ nat @ N ) ) ) ) ) ).
% prod.atMost_Suc_shift
thf(fact_4627_bounded__imp__summable,axiom,
! [A: $tType] :
( ( ( condit6923001295902523014norder @ A )
& ( linord2810124833399127020strict @ A )
& ( topolo1944317154257567458pology @ A ) )
=> ! [A2: nat > A,B4: A] :
( ! [N2: nat] : ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( A2 @ N2 ) )
=> ( ! [N2: nat] : ( ord_less_eq @ A @ ( groups7311177749621191930dd_sum @ nat @ A @ A2 @ ( set_ord_atMost @ nat @ N2 ) ) @ B4 )
=> ( summable @ A @ A2 ) ) ) ) ).
% bounded_imp_summable
thf(fact_4628_sum_Onested__swap_H,axiom,
! [A: $tType] :
( ( comm_monoid_add @ A )
=> ! [A2: nat > nat > A,N: nat] :
( ( groups7311177749621191930dd_sum @ nat @ A
@ ^ [I2: nat] : ( groups7311177749621191930dd_sum @ nat @ A @ ( A2 @ I2 ) @ ( set_ord_lessThan @ nat @ I2 ) )
@ ( set_ord_atMost @ nat @ N ) )
= ( groups7311177749621191930dd_sum @ nat @ A
@ ^ [J2: nat] :
( groups7311177749621191930dd_sum @ nat @ A
@ ^ [I2: nat] : ( A2 @ I2 @ J2 )
@ ( set_or1337092689740270186AtMost @ nat @ ( suc @ J2 ) @ N ) )
@ ( set_ord_lessThan @ nat @ N ) ) ) ) ).
% sum.nested_swap'
thf(fact_4629_prod_Onested__swap_H,axiom,
! [A: $tType] :
( ( comm_monoid_mult @ A )
=> ! [A2: nat > nat > A,N: nat] :
( ( groups7121269368397514597t_prod @ nat @ A
@ ^ [I2: nat] : ( groups7121269368397514597t_prod @ nat @ A @ ( A2 @ I2 ) @ ( set_ord_lessThan @ nat @ I2 ) )
@ ( set_ord_atMost @ nat @ N ) )
= ( groups7121269368397514597t_prod @ nat @ A
@ ^ [J2: nat] :
( groups7121269368397514597t_prod @ nat @ A
@ ^ [I2: nat] : ( A2 @ I2 @ J2 )
@ ( set_or1337092689740270186AtMost @ nat @ ( suc @ J2 ) @ N ) )
@ ( set_ord_lessThan @ nat @ N ) ) ) ) ).
% prod.nested_swap'
thf(fact_4630_sum__choose__lower,axiom,
! [R2: nat,N: nat] :
( ( groups7311177749621191930dd_sum @ nat @ nat
@ ^ [K3: nat] : ( binomial @ ( plus_plus @ nat @ R2 @ K3 ) @ K3 )
@ ( set_ord_atMost @ nat @ N ) )
= ( binomial @ ( suc @ ( plus_plus @ nat @ R2 @ N ) ) @ N ) ) ).
% sum_choose_lower
thf(fact_4631_choose__rising__sum_I2_J,axiom,
! [N: nat,M: nat] :
( ( groups7311177749621191930dd_sum @ nat @ nat
@ ^ [J2: nat] : ( binomial @ ( plus_plus @ nat @ N @ J2 ) @ N )
@ ( set_ord_atMost @ nat @ M ) )
= ( binomial @ ( plus_plus @ nat @ ( plus_plus @ nat @ N @ M ) @ ( one_one @ nat ) ) @ M ) ) ).
% choose_rising_sum(2)
thf(fact_4632_choose__rising__sum_I1_J,axiom,
! [N: nat,M: nat] :
( ( groups7311177749621191930dd_sum @ nat @ nat
@ ^ [J2: nat] : ( binomial @ ( plus_plus @ nat @ N @ J2 ) @ N )
@ ( set_ord_atMost @ nat @ M ) )
= ( binomial @ ( plus_plus @ nat @ ( plus_plus @ nat @ N @ M ) @ ( one_one @ nat ) ) @ ( plus_plus @ nat @ N @ ( one_one @ nat ) ) ) ) ).
% choose_rising_sum(1)
thf(fact_4633_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_4634_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_4635_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_4636_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_4637_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_4638_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_4639_polyfun__eq__0,axiom,
! [A: $tType] :
( ( ( real_V8999393235501362500lgebra @ A )
& ( idom @ A ) )
=> ! [C2: nat > A,N: nat] :
( ( ! [X2: A] :
( ( groups7311177749621191930dd_sum @ nat @ A
@ ^ [I2: nat] : ( times_times @ A @ ( C2 @ I2 ) @ ( power_power @ A @ X2 @ I2 ) )
@ ( set_ord_atMost @ nat @ N ) )
= ( zero_zero @ A ) ) )
= ( ! [I2: nat] :
( ( ord_less_eq @ nat @ I2 @ N )
=> ( ( C2 @ I2 )
= ( zero_zero @ A ) ) ) ) ) ) ).
% polyfun_eq_0
thf(fact_4640_zero__polynom__imp__zero__coeffs,axiom,
! [A: $tType] :
( ( ( ab_semigroup_mult @ A )
& ( real_V8999393235501362500lgebra @ A ) )
=> ! [C2: nat > A,N: nat,K: nat] :
( ! [W3: A] :
( ( groups7311177749621191930dd_sum @ nat @ A
@ ^ [I2: nat] : ( times_times @ A @ ( C2 @ I2 ) @ ( power_power @ A @ W3 @ I2 ) )
@ ( set_ord_atMost @ nat @ N ) )
= ( zero_zero @ A ) )
=> ( ( ord_less_eq @ nat @ K @ N )
=> ( ( C2 @ K )
= ( zero_zero @ A ) ) ) ) ) ).
% zero_polynom_imp_zero_coeffs
thf(fact_4641_sum_OatMost__shift,axiom,
! [A: $tType] :
( ( comm_monoid_add @ A )
=> ! [G: nat > A,N: nat] :
( ( groups7311177749621191930dd_sum @ nat @ A @ G @ ( set_ord_atMost @ nat @ N ) )
= ( plus_plus @ A @ ( G @ ( zero_zero @ nat ) )
@ ( groups7311177749621191930dd_sum @ nat @ A
@ ^ [I2: nat] : ( G @ ( suc @ I2 ) )
@ ( set_ord_lessThan @ nat @ N ) ) ) ) ) ).
% sum.atMost_shift
thf(fact_4642_sum__up__index__split,axiom,
! [A: $tType] :
( ( comm_monoid_add @ A )
=> ! [F2: nat > A,M: nat,N: nat] :
( ( groups7311177749621191930dd_sum @ nat @ A @ F2 @ ( set_ord_atMost @ nat @ ( plus_plus @ nat @ M @ N ) ) )
= ( plus_plus @ A @ ( groups7311177749621191930dd_sum @ nat @ A @ F2 @ ( set_ord_atMost @ nat @ M ) ) @ ( groups7311177749621191930dd_sum @ nat @ A @ F2 @ ( set_or1337092689740270186AtMost @ nat @ ( suc @ M ) @ ( plus_plus @ nat @ M @ N ) ) ) ) ) ) ).
% sum_up_index_split
thf(fact_4643_prod_OatMost__shift,axiom,
! [A: $tType] :
( ( comm_monoid_mult @ A )
=> ! [G: nat > A,N: nat] :
( ( groups7121269368397514597t_prod @ nat @ A @ G @ ( set_ord_atMost @ nat @ N ) )
= ( times_times @ A @ ( G @ ( zero_zero @ nat ) )
@ ( groups7121269368397514597t_prod @ nat @ A
@ ^ [I2: nat] : ( G @ ( suc @ I2 ) )
@ ( set_ord_lessThan @ nat @ N ) ) ) ) ) ).
% prod.atMost_shift
thf(fact_4644_sum__choose__diagonal,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq @ nat @ M @ N )
=> ( ( groups7311177749621191930dd_sum @ nat @ nat
@ ^ [K3: nat] : ( binomial @ ( minus_minus @ nat @ N @ K3 ) @ ( minus_minus @ nat @ M @ K3 ) )
@ ( set_ord_atMost @ nat @ M ) )
= ( binomial @ ( suc @ N ) @ M ) ) ) ).
% sum_choose_diagonal
thf(fact_4645_vandermonde,axiom,
! [M: nat,N: nat,R2: nat] :
( ( groups7311177749621191930dd_sum @ nat @ nat
@ ^ [K3: nat] : ( times_times @ nat @ ( binomial @ M @ K3 ) @ ( binomial @ N @ ( minus_minus @ nat @ R2 @ K3 ) ) )
@ ( set_ord_atMost @ nat @ R2 ) )
= ( binomial @ ( plus_plus @ nat @ M @ N ) @ R2 ) ) ).
% vandermonde
thf(fact_4646_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_4647_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_4648_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_4649_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_4650_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_4651_sum__gp__basic,axiom,
! [A: $tType] :
( ( ( monoid_mult @ A )
& ( comm_ring @ A ) )
=> ! [X: A,N: nat] :
( ( times_times @ A @ ( minus_minus @ A @ ( one_one @ A ) @ X ) @ ( groups7311177749621191930dd_sum @ nat @ A @ ( power_power @ A @ X ) @ ( set_ord_atMost @ nat @ N ) ) )
= ( minus_minus @ A @ ( one_one @ A ) @ ( power_power @ A @ X @ ( suc @ N ) ) ) ) ) ).
% sum_gp_basic
thf(fact_4652_polyfun__finite__roots,axiom,
! [A: $tType] :
( ( ( real_V8999393235501362500lgebra @ A )
& ( idom @ A ) )
=> ! [C2: nat > A,N: nat] :
( ( finite_finite2 @ A
@ ( collect @ A
@ ^ [X2: A] :
( ( groups7311177749621191930dd_sum @ nat @ A
@ ^ [I2: nat] : ( times_times @ A @ ( C2 @ I2 ) @ ( power_power @ A @ X2 @ I2 ) )
@ ( set_ord_atMost @ nat @ N ) )
= ( zero_zero @ A ) ) ) )
= ( ? [I2: nat] :
( ( ord_less_eq @ nat @ I2 @ N )
& ( ( C2 @ I2 )
!= ( zero_zero @ A ) ) ) ) ) ) ).
% polyfun_finite_roots
thf(fact_4653_polyfun__roots__finite,axiom,
! [A: $tType] :
( ( ( real_V8999393235501362500lgebra @ A )
& ( idom @ A ) )
=> ! [C2: nat > A,K: nat,N: nat] :
( ( ( C2 @ K )
!= ( zero_zero @ A ) )
=> ( ( ord_less_eq @ nat @ K @ N )
=> ( finite_finite2 @ A
@ ( collect @ A
@ ^ [Z7: A] :
( ( groups7311177749621191930dd_sum @ nat @ A
@ ^ [I2: nat] : ( times_times @ A @ ( C2 @ I2 ) @ ( power_power @ A @ Z7 @ I2 ) )
@ ( set_ord_atMost @ nat @ N ) )
= ( zero_zero @ A ) ) ) ) ) ) ) ).
% polyfun_roots_finite
thf(fact_4654_polyfun__linear__factor__root,axiom,
! [A: $tType] :
( ( idom @ A )
=> ! [C2: nat > A,A2: A,N: nat] :
( ( ( groups7311177749621191930dd_sum @ nat @ A
@ ^ [I2: nat] : ( times_times @ A @ ( C2 @ I2 ) @ ( power_power @ A @ A2 @ I2 ) )
@ ( set_ord_atMost @ nat @ N ) )
= ( zero_zero @ A ) )
=> ~ ! [B3: nat > A] :
~ ! [Z4: A] :
( ( groups7311177749621191930dd_sum @ nat @ A
@ ^ [I2: nat] : ( times_times @ A @ ( C2 @ I2 ) @ ( power_power @ A @ Z4 @ I2 ) )
@ ( set_ord_atMost @ nat @ N ) )
= ( times_times @ A @ ( minus_minus @ A @ Z4 @ A2 )
@ ( groups7311177749621191930dd_sum @ nat @ A
@ ^ [I2: nat] : ( times_times @ A @ ( B3 @ I2 ) @ ( power_power @ A @ Z4 @ I2 ) )
@ ( set_ord_lessThan @ nat @ N ) ) ) ) ) ) ).
% polyfun_linear_factor_root
thf(fact_4655_polyfun__linear__factor,axiom,
! [A: $tType] :
( ( idom @ A )
=> ! [C2: nat > A,N: nat,A2: A] :
? [B3: nat > A] :
! [Z4: A] :
( ( groups7311177749621191930dd_sum @ nat @ A
@ ^ [I2: nat] : ( times_times @ A @ ( C2 @ I2 ) @ ( power_power @ A @ Z4 @ I2 ) )
@ ( set_ord_atMost @ nat @ N ) )
= ( plus_plus @ A
@ ( times_times @ A @ ( minus_minus @ A @ Z4 @ A2 )
@ ( groups7311177749621191930dd_sum @ nat @ A
@ ^ [I2: nat] : ( times_times @ A @ ( B3 @ I2 ) @ ( power_power @ A @ Z4 @ I2 ) )
@ ( set_ord_lessThan @ nat @ N ) ) )
@ ( groups7311177749621191930dd_sum @ nat @ A
@ ^ [I2: nat] : ( times_times @ A @ ( C2 @ I2 ) @ ( power_power @ A @ A2 @ I2 ) )
@ ( set_ord_atMost @ nat @ N ) ) ) ) ) ).
% polyfun_linear_factor
thf(fact_4656_sum__power__shift,axiom,
! [A: $tType] :
( ( ( monoid_mult @ A )
& ( comm_ring @ A ) )
=> ! [M: nat,N: nat,X: A] :
( ( ord_less_eq @ nat @ M @ N )
=> ( ( groups7311177749621191930dd_sum @ nat @ A @ ( power_power @ A @ X ) @ ( set_or1337092689740270186AtMost @ nat @ M @ N ) )
= ( times_times @ A @ ( power_power @ A @ X @ M ) @ ( groups7311177749621191930dd_sum @ nat @ A @ ( power_power @ A @ X ) @ ( set_ord_atMost @ nat @ ( minus_minus @ nat @ N @ M ) ) ) ) ) ) ) ).
% sum_power_shift
thf(fact_4657_atLeast1__atMost__eq__remove0,axiom,
! [N: nat] :
( ( set_or1337092689740270186AtMost @ nat @ ( suc @ ( zero_zero @ nat ) ) @ N )
= ( minus_minus @ ( set @ nat ) @ ( set_ord_atMost @ nat @ N ) @ ( insert @ nat @ ( zero_zero @ nat ) @ ( bot_bot @ ( set @ nat ) ) ) ) ) ).
% atLeast1_atMost_eq_remove0
thf(fact_4658_choose__row__sum,axiom,
! [N: nat] :
( ( groups7311177749621191930dd_sum @ nat @ nat @ ( binomial @ N ) @ ( set_ord_atMost @ nat @ N ) )
= ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N ) ) ).
% choose_row_sum
thf(fact_4659_summable__Cauchy__product,axiom,
! [A: $tType] :
( ( ( real_Vector_banach @ A )
& ( real_V4412858255891104859lgebra @ A ) )
=> ! [A2: nat > A,B2: nat > A] :
( ( summable @ real
@ ^ [K3: nat] : ( real_V7770717601297561774m_norm @ A @ ( A2 @ K3 ) ) )
=> ( ( summable @ real
@ ^ [K3: nat] : ( real_V7770717601297561774m_norm @ A @ ( B2 @ K3 ) ) )
=> ( summable @ A
@ ^ [K3: nat] :
( groups7311177749621191930dd_sum @ nat @ A
@ ^ [I2: nat] : ( times_times @ A @ ( A2 @ I2 ) @ ( B2 @ ( minus_minus @ nat @ K3 @ I2 ) ) )
@ ( set_ord_atMost @ nat @ K3 ) ) ) ) ) ) ).
% summable_Cauchy_product
thf(fact_4660_Cauchy__product,axiom,
! [A: $tType] :
( ( ( real_Vector_banach @ A )
& ( real_V4412858255891104859lgebra @ A ) )
=> ! [A2: nat > A,B2: nat > A] :
( ( summable @ real
@ ^ [K3: nat] : ( real_V7770717601297561774m_norm @ A @ ( A2 @ K3 ) ) )
=> ( ( summable @ real
@ ^ [K3: nat] : ( real_V7770717601297561774m_norm @ A @ ( B2 @ K3 ) ) )
=> ( ( times_times @ A @ ( suminf @ A @ A2 ) @ ( suminf @ A @ B2 ) )
= ( suminf @ A
@ ^ [K3: nat] :
( groups7311177749621191930dd_sum @ nat @ A
@ ^ [I2: nat] : ( times_times @ A @ ( A2 @ I2 ) @ ( B2 @ ( minus_minus @ nat @ K3 @ I2 ) ) )
@ ( set_ord_atMost @ nat @ K3 ) ) ) ) ) ) ) ).
% Cauchy_product
thf(fact_4661_binomial,axiom,
! [A2: nat,B2: nat,N: nat] :
( ( power_power @ nat @ ( plus_plus @ nat @ A2 @ B2 ) @ N )
= ( groups7311177749621191930dd_sum @ nat @ nat
@ ^ [K3: nat] : ( times_times @ nat @ ( times_times @ nat @ ( semiring_1_of_nat @ nat @ ( binomial @ N @ K3 ) ) @ ( power_power @ nat @ A2 @ K3 ) ) @ ( power_power @ nat @ B2 @ ( minus_minus @ nat @ N @ K3 ) ) )
@ ( set_ord_atMost @ nat @ N ) ) ) ).
% binomial
thf(fact_4662_sum_Oin__pairs__0,axiom,
! [A: $tType] :
( ( comm_monoid_add @ A )
=> ! [G: nat > A,N: nat] :
( ( groups7311177749621191930dd_sum @ nat @ A @ G @ ( set_ord_atMost @ nat @ ( suc @ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N ) ) ) )
= ( groups7311177749621191930dd_sum @ nat @ A
@ ^ [I2: nat] : ( plus_plus @ A @ ( G @ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ I2 ) ) @ ( G @ ( suc @ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ I2 ) ) ) )
@ ( set_ord_atMost @ nat @ N ) ) ) ) ).
% sum.in_pairs_0
thf(fact_4663_polynomial__product,axiom,
! [A: $tType] :
( ( idom @ A )
=> ! [M: nat,A2: nat > A,N: nat,B2: nat > A,X: A] :
( ! [I3: nat] :
( ( ord_less @ nat @ M @ I3 )
=> ( ( A2 @ I3 )
= ( zero_zero @ A ) ) )
=> ( ! [J3: nat] :
( ( ord_less @ nat @ N @ J3 )
=> ( ( B2 @ J3 )
= ( zero_zero @ A ) ) )
=> ( ( times_times @ A
@ ( groups7311177749621191930dd_sum @ nat @ A
@ ^ [I2: nat] : ( times_times @ A @ ( A2 @ I2 ) @ ( power_power @ A @ X @ I2 ) )
@ ( set_ord_atMost @ nat @ M ) )
@ ( groups7311177749621191930dd_sum @ nat @ A
@ ^ [J2: nat] : ( times_times @ A @ ( B2 @ J2 ) @ ( power_power @ A @ X @ J2 ) )
@ ( set_ord_atMost @ nat @ N ) ) )
= ( groups7311177749621191930dd_sum @ nat @ A
@ ^ [R5: nat] :
( times_times @ A
@ ( groups7311177749621191930dd_sum @ nat @ A
@ ^ [K3: nat] : ( times_times @ A @ ( A2 @ K3 ) @ ( B2 @ ( minus_minus @ nat @ R5 @ K3 ) ) )
@ ( set_ord_atMost @ nat @ R5 ) )
@ ( power_power @ A @ X @ R5 ) )
@ ( set_ord_atMost @ nat @ ( plus_plus @ nat @ M @ N ) ) ) ) ) ) ) ).
% polynomial_product
thf(fact_4664_prod_Oin__pairs__0,axiom,
! [A: $tType] :
( ( comm_monoid_mult @ A )
=> ! [G: nat > A,N: nat] :
( ( groups7121269368397514597t_prod @ nat @ A @ G @ ( set_ord_atMost @ nat @ ( suc @ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N ) ) ) )
= ( groups7121269368397514597t_prod @ nat @ A
@ ^ [I2: nat] : ( times_times @ A @ ( G @ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ I2 ) ) @ ( G @ ( suc @ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ I2 ) ) ) )
@ ( set_ord_atMost @ nat @ N ) ) ) ) ).
% prod.in_pairs_0
thf(fact_4665_polyfun__eq__const,axiom,
! [A: $tType] :
( ( ( real_V8999393235501362500lgebra @ A )
& ( idom @ A ) )
=> ! [C2: nat > A,N: nat,K: A] :
( ( ! [X2: A] :
( ( groups7311177749621191930dd_sum @ nat @ A
@ ^ [I2: nat] : ( times_times @ A @ ( C2 @ I2 ) @ ( power_power @ A @ X2 @ I2 ) )
@ ( set_ord_atMost @ nat @ N ) )
= K ) )
= ( ( ( C2 @ ( zero_zero @ nat ) )
= K )
& ! [X2: nat] :
( ( member @ nat @ X2 @ ( set_or1337092689740270186AtMost @ nat @ ( one_one @ nat ) @ N ) )
=> ( ( C2 @ X2 )
= ( zero_zero @ A ) ) ) ) ) ) ).
% polyfun_eq_const
thf(fact_4666_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_4667_binomial__ring,axiom,
! [A: $tType] :
( ( comm_semiring_1 @ A )
=> ! [A2: A,B2: A,N: nat] :
( ( power_power @ A @ ( plus_plus @ A @ A2 @ B2 ) @ N )
= ( groups7311177749621191930dd_sum @ nat @ A
@ ^ [K3: nat] : ( times_times @ A @ ( times_times @ A @ ( semiring_1_of_nat @ A @ ( binomial @ N @ K3 ) ) @ ( power_power @ A @ A2 @ K3 ) ) @ ( power_power @ A @ B2 @ ( minus_minus @ nat @ N @ K3 ) ) )
@ ( set_ord_atMost @ nat @ N ) ) ) ) ).
% binomial_ring
thf(fact_4668_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_4669_pochhammer__binomial__sum,axiom,
! [A: $tType] :
( ( comm_ring_1 @ A )
=> ! [A2: A,B2: A,N: nat] :
( ( comm_s3205402744901411588hammer @ A @ ( plus_plus @ A @ A2 @ B2 ) @ N )
= ( groups7311177749621191930dd_sum @ nat @ A
@ ^ [K3: nat] : ( times_times @ A @ ( times_times @ A @ ( semiring_1_of_nat @ A @ ( binomial @ N @ K3 ) ) @ ( comm_s3205402744901411588hammer @ A @ A2 @ K3 ) ) @ ( comm_s3205402744901411588hammer @ A @ B2 @ ( minus_minus @ nat @ N @ K3 ) ) )
@ ( set_ord_atMost @ nat @ N ) ) ) ) ).
% pochhammer_binomial_sum
thf(fact_4670_polynomial__product__nat,axiom,
! [M: nat,A2: nat > nat,N: nat,B2: nat > nat,X: nat] :
( ! [I3: nat] :
( ( ord_less @ nat @ M @ I3 )
=> ( ( A2 @ I3 )
= ( zero_zero @ nat ) ) )
=> ( ! [J3: nat] :
( ( ord_less @ nat @ N @ J3 )
=> ( ( B2 @ J3 )
= ( zero_zero @ nat ) ) )
=> ( ( times_times @ nat
@ ( groups7311177749621191930dd_sum @ nat @ nat
@ ^ [I2: nat] : ( times_times @ nat @ ( A2 @ I2 ) @ ( power_power @ nat @ X @ I2 ) )
@ ( set_ord_atMost @ nat @ M ) )
@ ( groups7311177749621191930dd_sum @ nat @ nat
@ ^ [J2: nat] : ( times_times @ nat @ ( B2 @ J2 ) @ ( power_power @ nat @ X @ J2 ) )
@ ( set_ord_atMost @ nat @ N ) ) )
= ( groups7311177749621191930dd_sum @ nat @ nat
@ ^ [R5: nat] :
( times_times @ nat
@ ( groups7311177749621191930dd_sum @ nat @ nat
@ ^ [K3: nat] : ( times_times @ nat @ ( A2 @ K3 ) @ ( B2 @ ( minus_minus @ nat @ R5 @ K3 ) ) )
@ ( set_ord_atMost @ nat @ R5 ) )
@ ( power_power @ nat @ X @ R5 ) )
@ ( set_ord_atMost @ nat @ ( plus_plus @ nat @ M @ N ) ) ) ) ) ) ).
% polynomial_product_nat
thf(fact_4671_choose__square__sum,axiom,
! [N: nat] :
( ( groups7311177749621191930dd_sum @ nat @ nat
@ ^ [K3: nat] : ( power_power @ nat @ ( binomial @ N @ K3 ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) )
@ ( set_ord_atMost @ nat @ N ) )
= ( binomial @ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N ) @ N ) ) ).
% choose_square_sum
thf(fact_4672_Cauchy__product__sums,axiom,
! [A: $tType] :
( ( ( real_Vector_banach @ A )
& ( real_V4412858255891104859lgebra @ A ) )
=> ! [A2: nat > A,B2: nat > A] :
( ( summable @ real
@ ^ [K3: nat] : ( real_V7770717601297561774m_norm @ A @ ( A2 @ K3 ) ) )
=> ( ( summable @ real
@ ^ [K3: nat] : ( real_V7770717601297561774m_norm @ A @ ( B2 @ K3 ) ) )
=> ( sums @ A
@ ^ [K3: nat] :
( groups7311177749621191930dd_sum @ nat @ A
@ ^ [I2: nat] : ( times_times @ A @ ( A2 @ I2 ) @ ( B2 @ ( minus_minus @ nat @ K3 @ I2 ) ) )
@ ( set_ord_atMost @ nat @ K3 ) )
@ ( times_times @ A @ ( suminf @ A @ A2 ) @ ( suminf @ A @ B2 ) ) ) ) ) ) ).
% Cauchy_product_sums
thf(fact_4673_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
@ ^ [J2: nat] : ( if @ A @ ( ord_less @ nat @ J2 @ K ) @ ( G @ J2 ) @ ( if @ A @ ( J2 = K ) @ ( zero_zero @ A ) @ ( H2 @ ( minus_minus @ nat @ J2 @ ( suc @ ( zero_zero @ nat ) ) ) ) ) )
@ ( set_ord_atMost @ nat @ P4 ) )
= ( groups7311177749621191930dd_sum @ nat @ A
@ ^ [J2: nat] : ( if @ A @ ( ord_less @ nat @ J2 @ K ) @ ( G @ J2 ) @ ( H2 @ J2 ) )
@ ( set_ord_atMost @ nat @ ( minus_minus @ nat @ P4 @ ( suc @ ( zero_zero @ nat ) ) ) ) ) ) ) ) ) ).
% sum.zero_middle
thf(fact_4674_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
@ ^ [J2: nat] : ( if @ A @ ( ord_less @ nat @ J2 @ K ) @ ( G @ J2 ) @ ( if @ A @ ( J2 = K ) @ ( one_one @ A ) @ ( H2 @ ( minus_minus @ nat @ J2 @ ( suc @ ( zero_zero @ nat ) ) ) ) ) )
@ ( set_ord_atMost @ nat @ P4 ) )
= ( groups7121269368397514597t_prod @ nat @ A
@ ^ [J2: nat] : ( if @ A @ ( ord_less @ nat @ J2 @ K ) @ ( G @ J2 ) @ ( H2 @ J2 ) )
@ ( set_ord_atMost @ nat @ ( minus_minus @ nat @ P4 @ ( suc @ ( zero_zero @ nat ) ) ) ) ) ) ) ) ) ).
% prod.zero_middle
thf(fact_4675_root__polyfun,axiom,
! [A: $tType] :
( ( idom @ A )
=> ! [N: nat,Z: A,A2: A] :
( ( ord_less_eq @ nat @ ( one_one @ nat ) @ N )
=> ( ( ( power_power @ A @ Z @ N )
= A2 )
= ( ( groups7311177749621191930dd_sum @ nat @ A
@ ^ [I2: nat] :
( times_times @ A
@ ( if @ A
@ ( I2
= ( zero_zero @ nat ) )
@ ( uminus_uminus @ A @ A2 )
@ ( if @ A @ ( I2 = N ) @ ( one_one @ A ) @ ( zero_zero @ A ) ) )
@ ( power_power @ A @ Z @ I2 ) )
@ ( set_ord_atMost @ nat @ N ) )
= ( zero_zero @ A ) ) ) ) ) ).
% root_polyfun
thf(fact_4676_sum__gp0,axiom,
! [A: $tType] :
( ( ( division_ring @ A )
& ( comm_ring @ A ) )
=> ! [X: A,N: nat] :
( ( ( X
= ( one_one @ A ) )
=> ( ( groups7311177749621191930dd_sum @ nat @ A @ ( power_power @ A @ X ) @ ( set_ord_atMost @ nat @ N ) )
= ( semiring_1_of_nat @ A @ ( plus_plus @ nat @ N @ ( one_one @ nat ) ) ) ) )
& ( ( X
!= ( one_one @ A ) )
=> ( ( groups7311177749621191930dd_sum @ nat @ A @ ( power_power @ A @ X ) @ ( set_ord_atMost @ nat @ N ) )
= ( divide_divide @ A @ ( minus_minus @ A @ ( one_one @ A ) @ ( power_power @ A @ X @ ( suc @ N ) ) ) @ ( minus_minus @ A @ ( one_one @ A ) @ X ) ) ) ) ) ) ).
% sum_gp0
thf(fact_4677_choose__alternating__linear__sum,axiom,
! [A: $tType] :
( ( comm_ring_1 @ A )
=> ! [N: nat] :
( ( N
!= ( one_one @ nat ) )
=> ( ( groups7311177749621191930dd_sum @ nat @ A
@ ^ [I2: nat] : ( times_times @ A @ ( times_times @ A @ ( power_power @ A @ ( uminus_uminus @ A @ ( one_one @ A ) ) @ I2 ) @ ( semiring_1_of_nat @ A @ I2 ) ) @ ( semiring_1_of_nat @ A @ ( binomial @ N @ I2 ) ) )
@ ( set_ord_atMost @ nat @ N ) )
= ( zero_zero @ A ) ) ) ) ).
% choose_alternating_linear_sum
thf(fact_4678_polyfun__diff__alt,axiom,
! [A: $tType] :
( ( idom @ A )
=> ! [N: nat,A2: nat > A,X: A,Y: A] :
( ( ord_less_eq @ nat @ ( one_one @ nat ) @ N )
=> ( ( minus_minus @ A
@ ( groups7311177749621191930dd_sum @ nat @ A
@ ^ [I2: nat] : ( times_times @ A @ ( A2 @ I2 ) @ ( power_power @ A @ X @ I2 ) )
@ ( set_ord_atMost @ nat @ N ) )
@ ( groups7311177749621191930dd_sum @ nat @ A
@ ^ [I2: nat] : ( times_times @ A @ ( A2 @ I2 ) @ ( power_power @ A @ Y @ I2 ) )
@ ( set_ord_atMost @ nat @ N ) ) )
= ( times_times @ A @ ( minus_minus @ A @ X @ Y )
@ ( groups7311177749621191930dd_sum @ nat @ A
@ ^ [J2: nat] :
( groups7311177749621191930dd_sum @ nat @ A
@ ^ [K3: nat] : ( times_times @ A @ ( times_times @ A @ ( A2 @ ( plus_plus @ nat @ ( plus_plus @ nat @ J2 @ K3 ) @ ( one_one @ nat ) ) ) @ ( power_power @ A @ Y @ K3 ) ) @ ( power_power @ A @ X @ J2 ) )
@ ( set_ord_lessThan @ nat @ ( minus_minus @ nat @ N @ J2 ) ) )
@ ( set_ord_lessThan @ nat @ N ) ) ) ) ) ) ).
% polyfun_diff_alt
thf(fact_4679_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_4680_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_4681_choose__linear__sum,axiom,
! [N: nat] :
( ( groups7311177749621191930dd_sum @ nat @ nat
@ ^ [I2: nat] : ( times_times @ nat @ I2 @ ( binomial @ N @ I2 ) )
@ ( set_ord_atMost @ nat @ N ) )
= ( times_times @ nat @ N @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( minus_minus @ nat @ N @ ( one_one @ nat ) ) ) ) ) ).
% choose_linear_sum
thf(fact_4682_choose__alternating__sum,axiom,
! [A: $tType] :
( ( comm_ring_1 @ A )
=> ! [N: nat] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ N )
=> ( ( groups7311177749621191930dd_sum @ nat @ A
@ ^ [I2: nat] : ( times_times @ A @ ( power_power @ A @ ( uminus_uminus @ A @ ( one_one @ A ) ) @ I2 ) @ ( semiring_1_of_nat @ A @ ( binomial @ N @ I2 ) ) )
@ ( set_ord_atMost @ nat @ N ) )
= ( zero_zero @ A ) ) ) ) ).
% choose_alternating_sum
thf(fact_4683_polyfun__extremal__lemma,axiom,
! [A: $tType] :
( ( real_V8999393235501362500lgebra @ A )
=> ! [E: real,C2: nat > A,N: nat] :
( ( ord_less @ real @ ( zero_zero @ real ) @ E )
=> ? [M9: real] :
! [Z4: A] :
( ( ord_less_eq @ real @ M9 @ ( real_V7770717601297561774m_norm @ A @ Z4 ) )
=> ( ord_less_eq @ real
@ ( real_V7770717601297561774m_norm @ A
@ ( groups7311177749621191930dd_sum @ nat @ A
@ ^ [I2: nat] : ( times_times @ A @ ( C2 @ I2 ) @ ( power_power @ A @ Z4 @ I2 ) )
@ ( set_ord_atMost @ nat @ N ) ) )
@ ( times_times @ real @ E @ ( power_power @ real @ ( real_V7770717601297561774m_norm @ A @ Z4 ) @ ( suc @ N ) ) ) ) ) ) ) ).
% polyfun_extremal_lemma
thf(fact_4684_polyfun__diff,axiom,
! [A: $tType] :
( ( idom @ A )
=> ! [N: nat,A2: nat > A,X: A,Y: A] :
( ( ord_less_eq @ nat @ ( one_one @ nat ) @ N )
=> ( ( minus_minus @ A
@ ( groups7311177749621191930dd_sum @ nat @ A
@ ^ [I2: nat] : ( times_times @ A @ ( A2 @ I2 ) @ ( power_power @ A @ X @ I2 ) )
@ ( set_ord_atMost @ nat @ N ) )
@ ( groups7311177749621191930dd_sum @ nat @ A
@ ^ [I2: nat] : ( times_times @ A @ ( A2 @ I2 ) @ ( power_power @ A @ Y @ I2 ) )
@ ( set_ord_atMost @ nat @ N ) ) )
= ( times_times @ A @ ( minus_minus @ A @ X @ Y )
@ ( groups7311177749621191930dd_sum @ nat @ A
@ ^ [J2: nat] :
( times_times @ A
@ ( groups7311177749621191930dd_sum @ nat @ A
@ ^ [I2: nat] : ( times_times @ A @ ( A2 @ I2 ) @ ( power_power @ A @ Y @ ( minus_minus @ nat @ ( minus_minus @ nat @ I2 @ J2 ) @ ( one_one @ nat ) ) ) )
@ ( set_or1337092689740270186AtMost @ nat @ ( suc @ J2 ) @ N ) )
@ ( power_power @ A @ X @ J2 ) )
@ ( set_ord_lessThan @ nat @ N ) ) ) ) ) ) ).
% polyfun_diff
thf(fact_4685_arccos__cos__eq__abs__2pi,axiom,
! [Theta: real] :
~ ! [K2: int] :
( ( arccos @ ( cos @ real @ Theta ) )
!= ( abs_abs @ real @ ( minus_minus @ real @ Theta @ ( times_times @ real @ ( ring_1_of_int @ real @ K2 ) @ ( times_times @ real @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) @ pi ) ) ) ) ) ).
% arccos_cos_eq_abs_2pi
thf(fact_4686_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
@ ^ [N3: nat] :
( if @ A
@ ( ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ P5 )
& ~ ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N3 ) )
@ ( times_times @ A @ ( real_V8093663219630862766scaleR @ A @ ( uminus_uminus @ real @ ( divide_divide @ real @ ( ring_1_of_int @ real @ ( times_times @ int @ ( power_power @ int @ ( uminus_uminus @ int @ ( one_one @ int ) ) @ ( divide_divide @ nat @ P5 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( semiring_1_of_nat @ int @ ( binomial @ P5 @ N3 ) ) ) ) @ ( semiring_char_0_fact @ real @ P5 ) ) ) @ ( power_power @ A @ X @ N3 ) ) @ ( power_power @ A @ Y @ ( minus_minus @ nat @ P5 @ N3 ) ) )
@ ( zero_zero @ A ) )
@ ( set_ord_atMost @ nat @ P5 ) )
@ ( times_times @ A @ ( sin @ A @ X ) @ ( sin @ A @ Y ) ) ) ) ).
% sin_x_sin_y
thf(fact_4687_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
@ ^ [N3: nat] : ( if @ A @ ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ P5 ) @ ( times_times @ A @ ( real_V8093663219630862766scaleR @ A @ ( divide_divide @ real @ ( ring_1_of_int @ real @ ( times_times @ int @ ( power_power @ int @ ( uminus_uminus @ int @ ( one_one @ int ) ) @ ( divide_divide @ nat @ P5 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( semiring_1_of_nat @ int @ ( binomial @ P5 @ N3 ) ) ) ) @ ( semiring_char_0_fact @ real @ P5 ) ) @ ( power_power @ A @ X @ N3 ) ) @ ( power_power @ A @ Y @ ( minus_minus @ nat @ P5 @ N3 ) ) ) @ ( zero_zero @ A ) )
@ ( set_ord_atMost @ nat @ P5 ) )
@ ( cos @ A @ ( plus_plus @ A @ X @ Y ) ) ) ) ).
% sums_cos_x_plus_y
thf(fact_4688_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
@ ^ [N3: nat] :
( if @ A
@ ( ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ P5 )
& ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N3 ) )
@ ( times_times @ A @ ( real_V8093663219630862766scaleR @ A @ ( divide_divide @ real @ ( ring_1_of_int @ real @ ( times_times @ int @ ( power_power @ int @ ( uminus_uminus @ int @ ( one_one @ int ) ) @ ( divide_divide @ nat @ P5 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( semiring_1_of_nat @ int @ ( binomial @ P5 @ N3 ) ) ) ) @ ( semiring_char_0_fact @ real @ P5 ) ) @ ( power_power @ A @ X @ N3 ) ) @ ( power_power @ A @ Y @ ( minus_minus @ nat @ P5 @ N3 ) ) )
@ ( zero_zero @ A ) )
@ ( set_ord_atMost @ nat @ P5 ) )
@ ( times_times @ A @ ( cos @ A @ X ) @ ( cos @ A @ Y ) ) ) ) ).
% cos_x_cos_y
thf(fact_4689_of__nat__id,axiom,
( ( semiring_1_of_nat @ nat )
= ( ^ [N3: nat] : N3 ) ) ).
% of_nat_id
thf(fact_4690_scaleR__cancel__right,axiom,
! [A: $tType] :
( ( real_V4867850818363320053vector @ A )
=> ! [A2: real,X: A,B2: real] :
( ( ( real_V8093663219630862766scaleR @ A @ A2 @ X )
= ( real_V8093663219630862766scaleR @ A @ B2 @ X ) )
= ( ( A2 = B2 )
| ( X
= ( zero_zero @ A ) ) ) ) ) ).
% scaleR_cancel_right
thf(fact_4691_scaleR__zero__right,axiom,
! [A: $tType] :
( ( real_V4867850818363320053vector @ A )
=> ! [A2: real] :
( ( real_V8093663219630862766scaleR @ A @ A2 @ ( zero_zero @ A ) )
= ( zero_zero @ A ) ) ) ).
% scaleR_zero_right
thf(fact_4692_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_4693_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_4694_scaleR__cancel__left,axiom,
! [A: $tType] :
( ( real_V4867850818363320053vector @ A )
=> ! [A2: real,X: A,Y: A] :
( ( ( real_V8093663219630862766scaleR @ A @ A2 @ X )
= ( real_V8093663219630862766scaleR @ A @ A2 @ Y ) )
= ( ( X = Y )
| ( A2
= ( zero_zero @ real ) ) ) ) ) ).
% scaleR_cancel_left
thf(fact_4695_scaleR__one,axiom,
! [A: $tType] :
( ( real_V4867850818363320053vector @ A )
=> ! [X: A] :
( ( real_V8093663219630862766scaleR @ A @ ( one_one @ real ) @ X )
= X ) ) ).
% scaleR_one
thf(fact_4696_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_4697_scaleR__eq__0__iff,axiom,
! [A: $tType] :
( ( real_V4867850818363320053vector @ A )
=> ! [A2: real,X: A] :
( ( ( real_V8093663219630862766scaleR @ A @ A2 @ X )
= ( zero_zero @ A ) )
= ( ( A2
= ( zero_zero @ real ) )
| ( X
= ( zero_zero @ A ) ) ) ) ) ).
% scaleR_eq_0_iff
thf(fact_4698_scaleR__zero__left,axiom,
! [A: $tType] :
( ( real_V4867850818363320053vector @ A )
=> ! [X: A] :
( ( real_V8093663219630862766scaleR @ A @ ( zero_zero @ real ) @ X )
= ( zero_zero @ A ) ) ) ).
% scaleR_zero_left
thf(fact_4699_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_4700_scaleR__power,axiom,
! [A: $tType] :
( ( real_V2822296259951069270ebra_1 @ A )
=> ! [X: real,Y: A,N: nat] :
( ( power_power @ A @ ( real_V8093663219630862766scaleR @ A @ X @ Y ) @ N )
= ( real_V8093663219630862766scaleR @ A @ ( power_power @ real @ X @ N ) @ ( power_power @ A @ Y @ N ) ) ) ) ).
% scaleR_power
thf(fact_4701_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_4702_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_4703_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_4704_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_4705_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_4706_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_4707_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_4708_scaleR__right__imp__eq,axiom,
! [A: $tType] :
( ( real_V4867850818363320053vector @ A )
=> ! [X: A,A2: real,B2: real] :
( ( X
!= ( zero_zero @ A ) )
=> ( ( ( real_V8093663219630862766scaleR @ A @ A2 @ X )
= ( real_V8093663219630862766scaleR @ A @ B2 @ X ) )
=> ( A2 = B2 ) ) ) ) ).
% scaleR_right_imp_eq
thf(fact_4709_scaleR__left__imp__eq,axiom,
! [A: $tType] :
( ( real_V4867850818363320053vector @ A )
=> ! [A2: real,X: A,Y: A] :
( ( A2
!= ( zero_zero @ real ) )
=> ( ( ( real_V8093663219630862766scaleR @ A @ A2 @ X )
= ( real_V8093663219630862766scaleR @ A @ A2 @ Y ) )
=> ( X = Y ) ) ) ) ).
% scaleR_left_imp_eq
thf(fact_4710_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_4711_real__scaleR__def,axiom,
( ( real_V8093663219630862766scaleR @ real )
= ( times_times @ real ) ) ).
% real_scaleR_def
thf(fact_4712_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_4713_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_4714_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_4715_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_4716_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_4717_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_4718_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_4719_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_4720_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_4721_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_4722_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_4723_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_4724_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_4725_Real__Vector__Spaces_Ole__add__iff1,axiom,
! [A: $tType] :
( ( real_V5355595471888546746vector @ A )
=> ! [A2: real,E: A,C2: A,B2: real,D2: A] :
( ( ord_less_eq @ A @ ( plus_plus @ A @ ( real_V8093663219630862766scaleR @ A @ A2 @ E ) @ C2 ) @ ( plus_plus @ A @ ( real_V8093663219630862766scaleR @ A @ B2 @ E ) @ D2 ) )
= ( ord_less_eq @ A @ ( plus_plus @ A @ ( real_V8093663219630862766scaleR @ A @ ( minus_minus @ real @ A2 @ B2 ) @ E ) @ C2 ) @ D2 ) ) ) ).
% Real_Vector_Spaces.le_add_iff1
thf(fact_4726_Real__Vector__Spaces_Ole__add__iff2,axiom,
! [A: $tType] :
( ( real_V5355595471888546746vector @ A )
=> ! [A2: real,E: A,C2: A,B2: real,D2: A] :
( ( ord_less_eq @ A @ ( plus_plus @ A @ ( real_V8093663219630862766scaleR @ A @ A2 @ E ) @ C2 ) @ ( plus_plus @ A @ ( real_V8093663219630862766scaleR @ A @ B2 @ E ) @ D2 ) )
= ( ord_less_eq @ A @ C2 @ ( plus_plus @ A @ ( real_V8093663219630862766scaleR @ A @ ( minus_minus @ real @ B2 @ A2 ) @ E ) @ D2 ) ) ) ) ).
% Real_Vector_Spaces.le_add_iff2
thf(fact_4727_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_4728_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_4729_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_4730_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_4731_scaleR__mono_H,axiom,
! [A: $tType] :
( ( real_V5355595471888546746vector @ A )
=> ! [A2: real,B2: real,C2: A,D2: A] :
( ( ord_less_eq @ real @ A2 @ B2 )
=> ( ( ord_less_eq @ A @ C2 @ D2 )
=> ( ( 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 @ D2 ) ) ) ) ) ) ) ).
% scaleR_mono'
thf(fact_4732_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_4733_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_4734_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_4735_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_4736_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_4737_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_4738_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_4739_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_4740_sin__converges,axiom,
! [A: $tType] :
( ( ( real_Vector_banach @ A )
& ( real_V2822296259951069270ebra_1 @ A ) )
=> ! [X: A] :
( sums @ A
@ ^ [N3: nat] : ( real_V8093663219630862766scaleR @ A @ ( sin_coeff @ N3 ) @ ( power_power @ A @ X @ N3 ) )
@ ( sin @ A @ X ) ) ) ).
% sin_converges
thf(fact_4741_sin__def,axiom,
! [A: $tType] :
( ( ( real_Vector_banach @ A )
& ( real_V2822296259951069270ebra_1 @ A ) )
=> ( ( sin @ A )
= ( ^ [X2: A] :
( suminf @ A
@ ^ [N3: nat] : ( real_V8093663219630862766scaleR @ A @ ( sin_coeff @ N3 ) @ ( power_power @ A @ X2 @ N3 ) ) ) ) ) ) ).
% sin_def
thf(fact_4742_cos__converges,axiom,
! [A: $tType] :
( ( ( real_Vector_banach @ A )
& ( real_V2822296259951069270ebra_1 @ A ) )
=> ! [X: A] :
( sums @ A
@ ^ [N3: nat] : ( real_V8093663219630862766scaleR @ A @ ( cos_coeff @ N3 ) @ ( power_power @ A @ X @ N3 ) )
@ ( cos @ A @ X ) ) ) ).
% cos_converges
thf(fact_4743_cos__def,axiom,
! [A: $tType] :
( ( ( real_Vector_banach @ A )
& ( real_V2822296259951069270ebra_1 @ A ) )
=> ( ( cos @ A )
= ( ^ [X2: A] :
( suminf @ A
@ ^ [N3: nat] : ( real_V8093663219630862766scaleR @ A @ ( cos_coeff @ N3 ) @ ( power_power @ A @ X2 @ N3 ) ) ) ) ) ) ).
% cos_def
thf(fact_4744_summable__norm__sin,axiom,
! [A: $tType] :
( ( ( real_Vector_banach @ A )
& ( real_V2822296259951069270ebra_1 @ A ) )
=> ! [X: A] :
( summable @ real
@ ^ [N3: nat] : ( real_V7770717601297561774m_norm @ A @ ( real_V8093663219630862766scaleR @ A @ ( sin_coeff @ N3 ) @ ( power_power @ A @ X @ N3 ) ) ) ) ) ).
% summable_norm_sin
thf(fact_4745_summable__norm__cos,axiom,
! [A: $tType] :
( ( ( real_Vector_banach @ A )
& ( real_V2822296259951069270ebra_1 @ A ) )
=> ! [X: A] :
( summable @ real
@ ^ [N3: nat] : ( real_V7770717601297561774m_norm @ A @ ( real_V8093663219630862766scaleR @ A @ ( cos_coeff @ N3 ) @ ( power_power @ A @ X @ N3 ) ) ) ) ) ).
% summable_norm_cos
thf(fact_4746_sin__minus__converges,axiom,
! [A: $tType] :
( ( ( real_Vector_banach @ A )
& ( real_V2822296259951069270ebra_1 @ A ) )
=> ! [X: A] :
( sums @ A
@ ^ [N3: nat] : ( uminus_uminus @ A @ ( real_V8093663219630862766scaleR @ A @ ( sin_coeff @ N3 ) @ ( power_power @ A @ ( uminus_uminus @ A @ X ) @ N3 ) ) )
@ ( sin @ A @ X ) ) ) ).
% sin_minus_converges
thf(fact_4747_cos__minus__converges,axiom,
! [A: $tType] :
( ( ( real_Vector_banach @ A )
& ( real_V2822296259951069270ebra_1 @ A ) )
=> ! [X: A] :
( sums @ A
@ ^ [N3: nat] : ( real_V8093663219630862766scaleR @ A @ ( cos_coeff @ N3 ) @ ( power_power @ A @ ( uminus_uminus @ A @ X ) @ N3 ) )
@ ( cos @ A @ X ) ) ) ).
% cos_minus_converges
thf(fact_4748_gbinomial__partial__row__sum,axiom,
! [A: $tType] :
( ( field_char_0 @ A )
=> ! [A2: A,M: nat] :
( ( groups7311177749621191930dd_sum @ nat @ A
@ ^ [K3: nat] : ( times_times @ A @ ( gbinomial @ A @ A2 @ K3 ) @ ( minus_minus @ A @ ( divide_divide @ A @ A2 @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) @ ( semiring_1_of_nat @ A @ K3 ) ) )
@ ( set_ord_atMost @ nat @ M ) )
= ( times_times @ A @ ( divide_divide @ A @ ( plus_plus @ A @ ( semiring_1_of_nat @ A @ M ) @ ( one_one @ A ) ) @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) @ ( gbinomial @ A @ A2 @ ( plus_plus @ nat @ M @ ( one_one @ nat ) ) ) ) ) ) ).
% gbinomial_partial_row_sum
thf(fact_4749_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_4750_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_4751_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_4752_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_4753_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_4754_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_4755_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_4756_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_4757_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_4758_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_4759_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_4760_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_4761_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_4762_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_4763_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_4764_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_4765_gbinomial__parallel__sum,axiom,
! [A: $tType] :
( ( field_char_0 @ A )
=> ! [A2: A,N: nat] :
( ( groups7311177749621191930dd_sum @ nat @ A
@ ^ [K3: nat] : ( gbinomial @ A @ ( plus_plus @ A @ A2 @ ( semiring_1_of_nat @ A @ K3 ) ) @ K3 )
@ ( set_ord_atMost @ nat @ N ) )
= ( gbinomial @ A @ ( plus_plus @ A @ ( plus_plus @ A @ A2 @ ( semiring_1_of_nat @ A @ N ) ) @ ( one_one @ A ) ) @ N ) ) ) ).
% gbinomial_parallel_sum
thf(fact_4766_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_4767_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_4768_gbinomial__index__swap,axiom,
! [A: $tType] :
( ( field_char_0 @ A )
=> ! [K: nat,N: nat] :
( ( times_times @ A @ ( power_power @ A @ ( uminus_uminus @ A @ ( one_one @ A ) ) @ K ) @ ( gbinomial @ A @ ( minus_minus @ A @ ( uminus_uminus @ A @ ( semiring_1_of_nat @ A @ N ) ) @ ( one_one @ A ) ) @ K ) )
= ( times_times @ A @ ( power_power @ A @ ( uminus_uminus @ A @ ( one_one @ A ) ) @ N ) @ ( gbinomial @ A @ ( minus_minus @ A @ ( uminus_uminus @ A @ ( semiring_1_of_nat @ A @ K ) ) @ ( one_one @ A ) ) @ N ) ) ) ) ).
% gbinomial_index_swap
thf(fact_4769_gbinomial__negated__upper,axiom,
! [A: $tType] :
( ( field_char_0 @ A )
=> ( ( gbinomial @ A )
= ( ^ [A4: A,K3: nat] : ( times_times @ A @ ( power_power @ A @ ( uminus_uminus @ A @ ( one_one @ A ) ) @ K3 ) @ ( gbinomial @ A @ ( minus_minus @ A @ ( minus_minus @ A @ ( semiring_1_of_nat @ A @ K3 ) @ A4 ) @ ( one_one @ A ) ) @ K3 ) ) ) ) ) ).
% gbinomial_negated_upper
thf(fact_4770_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_4771_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_4772_gbinomial__pochhammer,axiom,
! [A: $tType] :
( ( field_char_0 @ A )
=> ( ( gbinomial @ A )
= ( ^ [A4: A,K3: nat] : ( divide_divide @ A @ ( times_times @ A @ ( power_power @ A @ ( uminus_uminus @ A @ ( one_one @ A ) ) @ K3 ) @ ( comm_s3205402744901411588hammer @ A @ ( uminus_uminus @ A @ A4 ) @ K3 ) ) @ ( semiring_char_0_fact @ A @ K3 ) ) ) ) ) ).
% gbinomial_pochhammer
thf(fact_4773_gbinomial__pochhammer_H,axiom,
! [A: $tType] :
( ( field_char_0 @ A )
=> ( ( gbinomial @ A )
= ( ^ [A4: A,K3: nat] : ( divide_divide @ A @ ( comm_s3205402744901411588hammer @ A @ ( plus_plus @ A @ ( minus_minus @ A @ A4 @ ( semiring_1_of_nat @ A @ K3 ) ) @ ( one_one @ A ) ) @ K3 ) @ ( semiring_char_0_fact @ A @ K3 ) ) ) ) ) ).
% gbinomial_pochhammer'
thf(fact_4774_gbinomial__sum__lower__neg,axiom,
! [A: $tType] :
( ( field_char_0 @ A )
=> ! [A2: A,M: nat] :
( ( groups7311177749621191930dd_sum @ nat @ A
@ ^ [K3: nat] : ( times_times @ A @ ( gbinomial @ A @ A2 @ K3 ) @ ( power_power @ A @ ( uminus_uminus @ A @ ( one_one @ A ) ) @ K3 ) )
@ ( set_ord_atMost @ nat @ M ) )
= ( times_times @ A @ ( power_power @ A @ ( uminus_uminus @ A @ ( one_one @ A ) ) @ M ) @ ( gbinomial @ A @ ( minus_minus @ A @ A2 @ ( one_one @ A ) ) @ M ) ) ) ) ).
% gbinomial_sum_lower_neg
thf(fact_4775_gbinomial__altdef__of__nat,axiom,
! [A: $tType] :
( ( field_char_0 @ A )
=> ( ( gbinomial @ A )
= ( ^ [A4: A,K3: nat] :
( groups7121269368397514597t_prod @ nat @ A
@ ^ [I2: nat] : ( divide_divide @ A @ ( minus_minus @ A @ A4 @ ( semiring_1_of_nat @ A @ I2 ) ) @ ( semiring_1_of_nat @ A @ ( minus_minus @ nat @ K3 @ I2 ) ) )
@ ( set_or7035219750837199246ssThan @ nat @ ( zero_zero @ nat ) @ K3 ) ) ) ) ) ).
% gbinomial_altdef_of_nat
thf(fact_4776_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
@ ^ [I2: nat] : ( minus_minus @ A @ A2 @ ( semiring_1_of_nat @ A @ I2 ) )
@ ( set_or7035219750837199246ssThan @ nat @ ( zero_zero @ nat ) @ K ) ) ) ) ).
% gbinomial_mult_fact
thf(fact_4777_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
@ ^ [I2: nat] : ( minus_minus @ A @ A2 @ ( semiring_1_of_nat @ A @ I2 ) )
@ ( set_or7035219750837199246ssThan @ nat @ ( zero_zero @ nat ) @ K ) ) ) ) ).
% gbinomial_mult_fact'
thf(fact_4778_gbinomial__prod__rev,axiom,
! [A: $tType] :
( ( ( semiring_char_0 @ A )
& ( semidom_divide @ A ) )
=> ( ( gbinomial @ A )
= ( ^ [A4: A,K3: nat] :
( divide_divide @ A
@ ( groups7121269368397514597t_prod @ nat @ A
@ ^ [I2: nat] : ( minus_minus @ A @ A4 @ ( semiring_1_of_nat @ A @ I2 ) )
@ ( set_or7035219750837199246ssThan @ nat @ ( zero_zero @ nat ) @ K3 ) )
@ ( semiring_char_0_fact @ A @ K3 ) ) ) ) ) ).
% gbinomial_prod_rev
thf(fact_4779_gbinomial__sum__up__index,axiom,
! [A: $tType] :
( ( field_char_0 @ A )
=> ! [K: nat,N: nat] :
( ( groups7311177749621191930dd_sum @ nat @ A
@ ^ [J2: nat] : ( gbinomial @ A @ ( semiring_1_of_nat @ A @ J2 ) @ K )
@ ( set_or1337092689740270186AtMost @ nat @ ( zero_zero @ nat ) @ N ) )
= ( gbinomial @ A @ ( plus_plus @ A @ ( semiring_1_of_nat @ A @ N ) @ ( one_one @ A ) ) @ ( plus_plus @ nat @ K @ ( one_one @ nat ) ) ) ) ) ).
% gbinomial_sum_up_index
thf(fact_4780_gbinomial__partial__sum__poly,axiom,
! [A: $tType] :
( ( field_char_0 @ A )
=> ! [M: nat,A2: A,X: A,Y: A] :
( ( groups7311177749621191930dd_sum @ nat @ A
@ ^ [K3: nat] : ( times_times @ A @ ( times_times @ A @ ( gbinomial @ A @ ( plus_plus @ A @ ( semiring_1_of_nat @ A @ M ) @ A2 ) @ K3 ) @ ( power_power @ A @ X @ K3 ) ) @ ( power_power @ A @ Y @ ( minus_minus @ nat @ M @ K3 ) ) )
@ ( set_ord_atMost @ nat @ M ) )
= ( groups7311177749621191930dd_sum @ nat @ A
@ ^ [K3: nat] : ( times_times @ A @ ( times_times @ A @ ( gbinomial @ A @ ( uminus_uminus @ A @ A2 ) @ K3 ) @ ( power_power @ A @ ( uminus_uminus @ A @ X ) @ K3 ) ) @ ( power_power @ A @ ( plus_plus @ A @ X @ Y ) @ ( minus_minus @ nat @ M @ K3 ) ) )
@ ( set_ord_atMost @ nat @ M ) ) ) ) ).
% gbinomial_partial_sum_poly
thf(fact_4781_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
@ ^ [I2: nat] : ( minus_minus @ A @ A2 @ ( semiring_1_of_nat @ A @ I2 ) )
@ ( set_or1337092689740270186AtMost @ nat @ ( zero_zero @ nat ) @ K ) )
@ ( semiring_char_0_fact @ A @ ( suc @ K ) ) ) ) ) ).
% gbinomial_Suc
thf(fact_4782_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_4783_gbinomial__sum__nat__pow2,axiom,
! [A: $tType] :
( ( field_char_0 @ A )
=> ! [M: nat] :
( ( groups7311177749621191930dd_sum @ nat @ A
@ ^ [K3: nat] : ( divide_divide @ A @ ( gbinomial @ A @ ( semiring_1_of_nat @ A @ ( plus_plus @ nat @ M @ K3 ) ) @ K3 ) @ ( power_power @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ K3 ) )
@ ( set_ord_atMost @ nat @ M ) )
= ( power_power @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ M ) ) ) ).
% gbinomial_sum_nat_pow2
thf(fact_4784_gbinomial__partial__sum__poly__xpos,axiom,
! [A: $tType] :
( ( field_char_0 @ A )
=> ! [M: nat,A2: A,X: A,Y: A] :
( ( groups7311177749621191930dd_sum @ nat @ A
@ ^ [K3: nat] : ( times_times @ A @ ( times_times @ A @ ( gbinomial @ A @ ( plus_plus @ A @ ( semiring_1_of_nat @ A @ M ) @ A2 ) @ K3 ) @ ( power_power @ A @ X @ K3 ) ) @ ( power_power @ A @ Y @ ( minus_minus @ nat @ M @ K3 ) ) )
@ ( set_ord_atMost @ nat @ M ) )
= ( groups7311177749621191930dd_sum @ nat @ A
@ ^ [K3: nat] : ( times_times @ A @ ( times_times @ A @ ( gbinomial @ A @ ( minus_minus @ A @ ( plus_plus @ A @ ( semiring_1_of_nat @ A @ K3 ) @ A2 ) @ ( one_one @ A ) ) @ K3 ) @ ( power_power @ A @ X @ K3 ) ) @ ( power_power @ A @ ( plus_plus @ A @ X @ Y ) @ ( minus_minus @ nat @ M @ K3 ) ) )
@ ( set_ord_atMost @ nat @ M ) ) ) ) ).
% gbinomial_partial_sum_poly_xpos
thf(fact_4785_gbinomial__code,axiom,
! [A: $tType] :
( ( field_char_0 @ A )
=> ( ( gbinomial @ A )
= ( ^ [A4: A,K3: nat] :
( if @ A
@ ( K3
= ( zero_zero @ nat ) )
@ ( one_one @ A )
@ ( divide_divide @ A
@ ( set_fo6178422350223883121st_nat @ A
@ ^ [L2: nat] : ( times_times @ A @ ( minus_minus @ A @ A4 @ ( semiring_1_of_nat @ A @ L2 ) ) )
@ ( zero_zero @ nat )
@ ( minus_minus @ nat @ K3 @ ( one_one @ nat ) )
@ ( one_one @ A ) )
@ ( semiring_char_0_fact @ A @ K3 ) ) ) ) ) ) ).
% gbinomial_code
thf(fact_4786_gchoose__row__sum__weighted,axiom,
! [A: $tType] :
( ( field_char_0 @ A )
=> ! [R2: A,M: nat] :
( ( groups7311177749621191930dd_sum @ nat @ A
@ ^ [K3: nat] : ( times_times @ A @ ( gbinomial @ A @ R2 @ K3 ) @ ( minus_minus @ A @ ( divide_divide @ A @ R2 @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) @ ( semiring_1_of_nat @ A @ K3 ) ) )
@ ( set_or1337092689740270186AtMost @ nat @ ( zero_zero @ nat ) @ M ) )
= ( times_times @ A @ ( divide_divide @ A @ ( semiring_1_of_nat @ A @ ( suc @ M ) ) @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) @ ( gbinomial @ A @ R2 @ ( suc @ M ) ) ) ) ) ).
% gchoose_row_sum_weighted
thf(fact_4787_monoseq__Suc,axiom,
! [A: $tType] :
( ( order @ A )
=> ( ( topological_monoseq @ A )
= ( ^ [X8: nat > A] :
( ! [N3: nat] : ( ord_less_eq @ A @ ( X8 @ N3 ) @ ( X8 @ ( suc @ N3 ) ) )
| ! [N3: nat] : ( ord_less_eq @ A @ ( X8 @ ( suc @ N3 ) ) @ ( X8 @ N3 ) ) ) ) ) ) ).
% monoseq_Suc
thf(fact_4788_mono__SucI2,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [X7: nat > A] :
( ! [N2: nat] : ( ord_less_eq @ A @ ( X7 @ ( suc @ N2 ) ) @ ( X7 @ N2 ) )
=> ( topological_monoseq @ A @ X7 ) ) ) ).
% mono_SucI2
thf(fact_4789_mono__SucI1,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [X7: nat > A] :
( ! [N2: nat] : ( ord_less_eq @ A @ ( X7 @ N2 ) @ ( X7 @ ( suc @ N2 ) ) )
=> ( topological_monoseq @ A @ X7 ) ) ) ).
% mono_SucI1
thf(fact_4790_exp__first__two__terms,axiom,
! [A: $tType] :
( ( ( real_Vector_banach @ A )
& ( real_V2822296259951069270ebra_1 @ A ) )
=> ( ( exp @ A )
= ( ^ [X2: A] :
( plus_plus @ A @ ( plus_plus @ A @ ( one_one @ A ) @ X2 )
@ ( suminf @ A
@ ^ [N3: nat] : ( real_V8093663219630862766scaleR @ A @ ( inverse_inverse @ real @ ( semiring_char_0_fact @ real @ ( plus_plus @ nat @ N3 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( power_power @ A @ X2 @ ( plus_plus @ nat @ N3 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) ) ) ) ) ).
% exp_first_two_terms
thf(fact_4791_of__nat__code,axiom,
! [A: $tType] :
( ( semiring_1 @ A )
=> ( ( semiring_1_of_nat @ A )
= ( ^ [N3: nat] :
( semiri8178284476397505188at_aux @ A
@ ^ [I2: A] : ( plus_plus @ A @ I2 @ ( one_one @ A ) )
@ N3
@ ( zero_zero @ A ) ) ) ) ) ).
% of_nat_code
thf(fact_4792_Maclaurin__sin__bound,axiom,
! [X: real,N: nat] :
( ord_less_eq @ real
@ ( abs_abs @ real
@ ( minus_minus @ real @ ( sin @ real @ X )
@ ( groups7311177749621191930dd_sum @ nat @ real
@ ^ [M4: nat] : ( times_times @ real @ ( sin_coeff @ M4 ) @ ( power_power @ real @ X @ M4 ) )
@ ( set_ord_lessThan @ nat @ N ) ) ) )
@ ( times_times @ real @ ( inverse_inverse @ real @ ( semiring_char_0_fact @ real @ N ) ) @ ( power_power @ real @ ( abs_abs @ real @ X ) @ N ) ) ) ).
% Maclaurin_sin_bound
thf(fact_4793_inverse__eq__iff__eq,axiom,
! [A: $tType] :
( ( division_ring @ A )
=> ! [A2: A,B2: A] :
( ( ( inverse_inverse @ A @ A2 )
= ( inverse_inverse @ A @ B2 ) )
= ( A2 = B2 ) ) ) ).
% inverse_eq_iff_eq
thf(fact_4794_inverse__inverse__eq,axiom,
! [A: $tType] :
( ( division_ring @ A )
=> ! [A2: A] :
( ( inverse_inverse @ A @ ( inverse_inverse @ A @ A2 ) )
= A2 ) ) ).
% inverse_inverse_eq
thf(fact_4795_inverse__zero,axiom,
! [A: $tType] :
( ( division_ring @ A )
=> ( ( inverse_inverse @ A @ ( zero_zero @ A ) )
= ( zero_zero @ A ) ) ) ).
% inverse_zero
thf(fact_4796_inverse__nonzero__iff__nonzero,axiom,
! [A: $tType] :
( ( division_ring @ A )
=> ! [A2: A] :
( ( ( inverse_inverse @ A @ A2 )
= ( zero_zero @ A ) )
= ( A2
= ( zero_zero @ A ) ) ) ) ).
% inverse_nonzero_iff_nonzero
thf(fact_4797_inverse__1,axiom,
! [A: $tType] :
( ( division_ring @ A )
=> ( ( inverse_inverse @ A @ ( one_one @ A ) )
= ( one_one @ A ) ) ) ).
% inverse_1
thf(fact_4798_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_4799_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_4800_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_4801_inverse__minus__eq,axiom,
! [A: $tType] :
( ( division_ring @ A )
=> ! [A2: A] :
( ( inverse_inverse @ A @ ( uminus_uminus @ A @ A2 ) )
= ( uminus_uminus @ A @ ( inverse_inverse @ A @ A2 ) ) ) ) ).
% inverse_minus_eq
thf(fact_4802_abs__inverse,axiom,
! [A: $tType] :
( ( field_abs_sgn @ A )
=> ! [A2: A] :
( ( abs_abs @ A @ ( inverse_inverse @ A @ A2 ) )
= ( inverse_inverse @ A @ ( abs_abs @ A @ A2 ) ) ) ) ).
% abs_inverse
thf(fact_4803_sgn__inverse,axiom,
! [A: $tType] :
( ( field_abs_sgn @ A )
=> ! [A2: A] :
( ( sgn_sgn @ A @ ( inverse_inverse @ A @ A2 ) )
= ( inverse_inverse @ A @ ( sgn_sgn @ A @ A2 ) ) ) ) ).
% sgn_inverse
thf(fact_4804_inverse__sgn,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [A2: A] :
( ( inverse_inverse @ A @ ( sgn_sgn @ A @ A2 ) )
= ( sgn_sgn @ A @ A2 ) ) ) ).
% inverse_sgn
thf(fact_4805_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_4806_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_4807_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_4808_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_4809_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_4810_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_4811_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_4812_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_4813_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_4814_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_4815_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_4816_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_4817_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_4818_inverse__eq__imp__eq,axiom,
! [A: $tType] :
( ( division_ring @ A )
=> ! [A2: A,B2: A] :
( ( ( inverse_inverse @ A @ A2 )
= ( inverse_inverse @ A @ B2 ) )
=> ( A2 = B2 ) ) ) ).
% inverse_eq_imp_eq
thf(fact_4819_power__inverse,axiom,
! [A: $tType] :
( ( division_ring @ A )
=> ! [A2: A,N: nat] :
( ( power_power @ A @ ( inverse_inverse @ A @ A2 ) @ N )
= ( inverse_inverse @ A @ ( power_power @ A @ A2 @ N ) ) ) ) ).
% power_inverse
thf(fact_4820_real__sqrt__inverse,axiom,
! [X: real] :
( ( sqrt @ ( inverse_inverse @ real @ X ) )
= ( inverse_inverse @ real @ ( sqrt @ X ) ) ) ).
% real_sqrt_inverse
thf(fact_4821_nonzero__of__real__inverse,axiom,
! [A: $tType] :
( ( real_V5047593784448816457lgebra @ A )
=> ! [X: real] :
( ( X
!= ( zero_zero @ real ) )
=> ( ( real_Vector_of_real @ A @ ( inverse_inverse @ real @ X ) )
= ( inverse_inverse @ A @ ( real_Vector_of_real @ A @ X ) ) ) ) ) ).
% nonzero_of_real_inverse
thf(fact_4822_field__class_Ofield__inverse__zero,axiom,
! [A: $tType] :
( ( field @ A )
=> ( ( inverse_inverse @ A @ ( zero_zero @ A ) )
= ( zero_zero @ A ) ) ) ).
% field_class.field_inverse_zero
thf(fact_4823_inverse__zero__imp__zero,axiom,
! [A: $tType] :
( ( division_ring @ A )
=> ! [A2: A] :
( ( ( inverse_inverse @ A @ A2 )
= ( zero_zero @ A ) )
=> ( A2
= ( zero_zero @ A ) ) ) ) ).
% inverse_zero_imp_zero
thf(fact_4824_nonzero__inverse__eq__imp__eq,axiom,
! [A: $tType] :
( ( division_ring @ A )
=> ! [A2: A,B2: A] :
( ( ( inverse_inverse @ A @ A2 )
= ( inverse_inverse @ A @ B2 ) )
=> ( ( A2
!= ( zero_zero @ A ) )
=> ( ( B2
!= ( zero_zero @ A ) )
=> ( A2 = B2 ) ) ) ) ) ).
% nonzero_inverse_eq_imp_eq
thf(fact_4825_nonzero__inverse__inverse__eq,axiom,
! [A: $tType] :
( ( division_ring @ A )
=> ! [A2: A] :
( ( A2
!= ( zero_zero @ A ) )
=> ( ( inverse_inverse @ A @ ( inverse_inverse @ A @ A2 ) )
= A2 ) ) ) ).
% nonzero_inverse_inverse_eq
thf(fact_4826_nonzero__imp__inverse__nonzero,axiom,
! [A: $tType] :
( ( division_ring @ A )
=> ! [A2: A] :
( ( A2
!= ( zero_zero @ A ) )
=> ( ( inverse_inverse @ A @ A2 )
!= ( zero_zero @ A ) ) ) ) ).
% nonzero_imp_inverse_nonzero
thf(fact_4827_nonzero__norm__inverse,axiom,
! [A: $tType] :
( ( real_V8999393235501362500lgebra @ A )
=> ! [A2: A] :
( ( A2
!= ( zero_zero @ A ) )
=> ( ( real_V7770717601297561774m_norm @ A @ ( inverse_inverse @ A @ A2 ) )
= ( inverse_inverse @ real @ ( real_V7770717601297561774m_norm @ A @ A2 ) ) ) ) ) ).
% nonzero_norm_inverse
thf(fact_4828_nonzero__inverse__scaleR__distrib,axiom,
! [A: $tType] :
( ( real_V5047593784448816457lgebra @ A )
=> ! [A2: real,X: A] :
( ( A2
!= ( zero_zero @ real ) )
=> ( ( X
!= ( zero_zero @ A ) )
=> ( ( inverse_inverse @ A @ ( real_V8093663219630862766scaleR @ A @ A2 @ X ) )
= ( real_V8093663219630862766scaleR @ A @ ( inverse_inverse @ real @ A2 ) @ ( inverse_inverse @ A @ X ) ) ) ) ) ) ).
% nonzero_inverse_scaleR_distrib
thf(fact_4829_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_4830_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_4831_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_4832_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_4833_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_4834_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_4835_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_4836_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_4837_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_4838_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_4839_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_4840_nonzero__inverse__minus__eq,axiom,
! [A: $tType] :
( ( division_ring @ A )
=> ! [A2: A] :
( ( A2
!= ( zero_zero @ A ) )
=> ( ( inverse_inverse @ A @ ( uminus_uminus @ A @ A2 ) )
= ( uminus_uminus @ A @ ( inverse_inverse @ A @ A2 ) ) ) ) ) ).
% nonzero_inverse_minus_eq
thf(fact_4841_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_4842_inverse__eq__divide,axiom,
! [A: $tType] :
( ( division_ring @ A )
=> ( ( inverse_inverse @ A )
= ( divide_divide @ A @ ( one_one @ A ) ) ) ) ).
% inverse_eq_divide
thf(fact_4843_field__class_Ofield__divide__inverse,axiom,
! [A: $tType] :
( ( field @ A )
=> ( ( divide_divide @ A )
= ( ^ [A4: A,B5: A] : ( times_times @ A @ A4 @ ( inverse_inverse @ A @ B5 ) ) ) ) ) ).
% field_class.field_divide_inverse
thf(fact_4844_divide__inverse,axiom,
! [A: $tType] :
( ( division_ring @ A )
=> ( ( divide_divide @ A )
= ( ^ [A4: A,B5: A] : ( times_times @ A @ A4 @ ( inverse_inverse @ A @ B5 ) ) ) ) ) ).
% divide_inverse
thf(fact_4845_divide__inverse__commute,axiom,
! [A: $tType] :
( ( field @ A )
=> ( ( divide_divide @ A )
= ( ^ [A4: A,B5: A] : ( times_times @ A @ ( inverse_inverse @ A @ B5 ) @ A4 ) ) ) ) ).
% divide_inverse_commute
thf(fact_4846_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_4847_power__mult__power__inverse__commute,axiom,
! [A: $tType] :
( ( division_ring @ A )
=> ! [X: A,M: nat,N: nat] :
( ( times_times @ A @ ( power_power @ A @ X @ M ) @ ( power_power @ A @ ( inverse_inverse @ A @ X ) @ N ) )
= ( times_times @ A @ ( power_power @ A @ ( inverse_inverse @ A @ X ) @ N ) @ ( power_power @ A @ X @ M ) ) ) ) ).
% power_mult_power_inverse_commute
thf(fact_4848_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_4849_nonzero__abs__inverse,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [A2: A] :
( ( A2
!= ( zero_zero @ A ) )
=> ( ( abs_abs @ A @ ( inverse_inverse @ A @ A2 ) )
= ( inverse_inverse @ A @ ( abs_abs @ A @ A2 ) ) ) ) ) ).
% nonzero_abs_inverse
thf(fact_4850_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_4851_divide__real__def,axiom,
( ( divide_divide @ real )
= ( ^ [X2: real,Y6: real] : ( times_times @ real @ X2 @ ( inverse_inverse @ real @ Y6 ) ) ) ) ).
% divide_real_def
thf(fact_4852_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_4853_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_4854_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_4855_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_4856_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_4857_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_4858_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_4859_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_4860_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_4861_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_4862_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_4863_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_4864_of__nat__aux_Osimps_I2_J,axiom,
! [A: $tType] :
( ( semiring_1 @ A )
=> ! [Inc: A > A,N: nat,I: A] :
( ( semiri8178284476397505188at_aux @ A @ Inc @ ( suc @ N ) @ I )
= ( semiri8178284476397505188at_aux @ A @ Inc @ N @ ( Inc @ I ) ) ) ) ).
% of_nat_aux.simps(2)
thf(fact_4865_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_4866_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_4867_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_4868_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_4869_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_4870_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_4871_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_4872_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_4873_reals__Archimedean,axiom,
! [A: $tType] :
( ( archim462609752435547400_field @ A )
=> ! [X: A] :
( ( ord_less @ A @ ( zero_zero @ A ) @ X )
=> ? [N2: nat] : ( ord_less @ A @ ( inverse_inverse @ A @ ( semiring_1_of_nat @ A @ ( suc @ N2 ) ) ) @ X ) ) ) ).
% reals_Archimedean
thf(fact_4874_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_4875_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_4876_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_4877_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_4878_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_4879_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_4880_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_4881_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_4882_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_4883_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_4884_forall__pos__mono__1,axiom,
! [P: real > $o,E: real] :
( ! [D3: real,E2: real] :
( ( ord_less @ real @ D3 @ E2 )
=> ( ( P @ D3 )
=> ( P @ E2 ) ) )
=> ( ! [N2: nat] : ( P @ ( inverse_inverse @ real @ ( semiring_1_of_nat @ real @ ( suc @ N2 ) ) ) )
=> ( ( ord_less @ real @ ( zero_zero @ real ) @ E )
=> ( P @ E ) ) ) ) ).
% forall_pos_mono_1
thf(fact_4885_forall__pos__mono,axiom,
! [P: real > $o,E: real] :
( ! [D3: real,E2: real] :
( ( ord_less @ real @ D3 @ E2 )
=> ( ( P @ D3 )
=> ( P @ E2 ) ) )
=> ( ! [N2: nat] :
( ( N2
!= ( zero_zero @ nat ) )
=> ( P @ ( inverse_inverse @ real @ ( semiring_1_of_nat @ real @ N2 ) ) ) )
=> ( ( ord_less @ real @ ( zero_zero @ real ) @ E )
=> ( P @ E ) ) ) ) ).
% forall_pos_mono
thf(fact_4886_real__arch__inverse,axiom,
! [E: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ E )
= ( ? [N3: nat] :
( ( N3
!= ( zero_zero @ nat ) )
& ( ord_less @ real @ ( zero_zero @ real ) @ ( inverse_inverse @ real @ ( semiring_1_of_nat @ real @ N3 ) ) )
& ( ord_less @ real @ ( inverse_inverse @ real @ ( semiring_1_of_nat @ real @ N3 ) ) @ E ) ) ) ) ).
% real_arch_inverse
thf(fact_4887_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_4888_ln__inverse,axiom,
! [X: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ X )
=> ( ( ln_ln @ real @ ( inverse_inverse @ real @ X ) )
= ( uminus_uminus @ real @ ( ln_ln @ real @ X ) ) ) ) ).
% ln_inverse
thf(fact_4889_summable__exp,axiom,
! [A: $tType] :
( ( ( real_Vector_banach @ A )
& ( real_V3459762299906320749_field @ A ) )
=> ! [X: A] :
( summable @ A
@ ^ [N3: nat] : ( times_times @ A @ ( inverse_inverse @ A @ ( semiring_char_0_fact @ A @ N3 ) ) @ ( power_power @ A @ X @ N3 ) ) ) ) ).
% summable_exp
thf(fact_4890_summable__exp__generic,axiom,
! [A: $tType] :
( ( ( real_Vector_banach @ A )
& ( real_V2822296259951069270ebra_1 @ A ) )
=> ! [X: A] :
( summable @ A
@ ^ [N3: nat] : ( real_V8093663219630862766scaleR @ A @ ( inverse_inverse @ real @ ( semiring_char_0_fact @ real @ N3 ) ) @ ( power_power @ A @ X @ N3 ) ) ) ) ).
% summable_exp_generic
thf(fact_4891_ex__inverse__of__nat__less,axiom,
! [A: $tType] :
( ( archim462609752435547400_field @ A )
=> ! [X: A] :
( ( ord_less @ A @ ( zero_zero @ A ) @ X )
=> ? [N2: nat] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ N2 )
& ( ord_less @ A @ ( inverse_inverse @ A @ ( semiring_1_of_nat @ A @ N2 ) ) @ X ) ) ) ) ).
% ex_inverse_of_nat_less
thf(fact_4892_power__diff__conv__inverse,axiom,
! [A: $tType] :
( ( division_ring @ A )
=> ! [X: A,M: nat,N: nat] :
( ( X
!= ( zero_zero @ A ) )
=> ( ( ord_less_eq @ nat @ M @ N )
=> ( ( power_power @ A @ X @ ( minus_minus @ nat @ N @ M ) )
= ( times_times @ A @ ( power_power @ A @ X @ N ) @ ( power_power @ A @ ( inverse_inverse @ A @ X ) @ M ) ) ) ) ) ) ).
% power_diff_conv_inverse
thf(fact_4893_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_4894_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_4895_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_4896_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_4897_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_4898_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_4899_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_4900_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_4901_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_4902_exp__converges,axiom,
! [A: $tType] :
( ( ( real_Vector_banach @ A )
& ( real_V2822296259951069270ebra_1 @ A ) )
=> ! [X: A] :
( sums @ A
@ ^ [N3: nat] : ( real_V8093663219630862766scaleR @ A @ ( inverse_inverse @ real @ ( semiring_char_0_fact @ real @ N3 ) ) @ ( power_power @ A @ X @ N3 ) )
@ ( exp @ A @ X ) ) ) ).
% exp_converges
thf(fact_4903_exp__def,axiom,
! [A: $tType] :
( ( ( real_Vector_banach @ A )
& ( real_V2822296259951069270ebra_1 @ A ) )
=> ( ( exp @ A )
= ( ^ [X2: A] :
( suminf @ A
@ ^ [N3: nat] : ( real_V8093663219630862766scaleR @ A @ ( inverse_inverse @ real @ ( semiring_char_0_fact @ real @ N3 ) ) @ ( power_power @ A @ X2 @ N3 ) ) ) ) ) ) ).
% exp_def
thf(fact_4904_summable__norm__exp,axiom,
! [A: $tType] :
( ( ( real_Vector_banach @ A )
& ( real_V2822296259951069270ebra_1 @ A ) )
=> ! [X: A] :
( summable @ real
@ ^ [N3: nat] : ( real_V7770717601297561774m_norm @ A @ ( real_V8093663219630862766scaleR @ A @ ( inverse_inverse @ real @ ( semiring_char_0_fact @ real @ N3 ) ) @ ( power_power @ A @ X @ N3 ) ) ) ) ) ).
% summable_norm_exp
thf(fact_4905_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_4906_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_4907_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_4908_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_4909_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_4910_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_4911_exp__series__add__commuting,axiom,
! [A: $tType] :
( ( ( real_Vector_banach @ A )
& ( real_V2822296259951069270ebra_1 @ A ) )
=> ! [X: A,Y: A,N: nat] :
( ( ( times_times @ A @ X @ Y )
= ( times_times @ A @ Y @ X ) )
=> ( ( real_V8093663219630862766scaleR @ A @ ( inverse_inverse @ real @ ( semiring_char_0_fact @ real @ N ) ) @ ( power_power @ A @ ( plus_plus @ A @ X @ Y ) @ N ) )
= ( groups7311177749621191930dd_sum @ nat @ A
@ ^ [I2: nat] : ( times_times @ A @ ( real_V8093663219630862766scaleR @ A @ ( inverse_inverse @ real @ ( semiring_char_0_fact @ real @ I2 ) ) @ ( power_power @ A @ X @ I2 ) ) @ ( real_V8093663219630862766scaleR @ A @ ( inverse_inverse @ real @ ( semiring_char_0_fact @ real @ ( minus_minus @ nat @ N @ I2 ) ) ) @ ( power_power @ A @ Y @ ( minus_minus @ nat @ N @ I2 ) ) ) )
@ ( set_ord_atMost @ nat @ N ) ) ) ) ) ).
% exp_series_add_commuting
thf(fact_4912_exp__first__term,axiom,
! [A: $tType] :
( ( ( real_Vector_banach @ A )
& ( real_V2822296259951069270ebra_1 @ A ) )
=> ( ( exp @ A )
= ( ^ [X2: A] :
( plus_plus @ A @ ( one_one @ A )
@ ( suminf @ A
@ ^ [N3: nat] : ( real_V8093663219630862766scaleR @ A @ ( inverse_inverse @ real @ ( semiring_char_0_fact @ real @ ( suc @ N3 ) ) ) @ ( power_power @ A @ X2 @ ( suc @ N3 ) ) ) ) ) ) ) ) ).
% exp_first_term
thf(fact_4913_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_4914_powr__real__of__int,axiom,
! [X: real,N: int] :
( ( ord_less @ real @ ( zero_zero @ real ) @ X )
=> ( ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ N )
=> ( ( powr @ real @ X @ ( ring_1_of_int @ real @ N ) )
= ( power_power @ real @ X @ ( nat2 @ N ) ) ) )
& ( ~ ( ord_less_eq @ int @ ( zero_zero @ int ) @ N )
=> ( ( powr @ real @ X @ ( ring_1_of_int @ real @ N ) )
= ( inverse_inverse @ real @ ( power_power @ real @ X @ ( nat2 @ ( uminus_uminus @ int @ N ) ) ) ) ) ) ) ) ).
% powr_real_of_int
thf(fact_4915_exp__first__terms,axiom,
! [A: $tType] :
( ( ( real_Vector_banach @ A )
& ( real_V2822296259951069270ebra_1 @ A ) )
=> ! [K: nat] :
( ( exp @ A )
= ( ^ [X2: A] :
( plus_plus @ A
@ ( groups7311177749621191930dd_sum @ nat @ A
@ ^ [N3: nat] : ( real_V8093663219630862766scaleR @ A @ ( inverse_inverse @ real @ ( semiring_char_0_fact @ real @ N3 ) ) @ ( power_power @ A @ X2 @ N3 ) )
@ ( set_ord_lessThan @ nat @ K ) )
@ ( suminf @ A
@ ^ [N3: nat] : ( real_V8093663219630862766scaleR @ A @ ( inverse_inverse @ real @ ( semiring_char_0_fact @ real @ ( plus_plus @ nat @ N3 @ K ) ) ) @ ( power_power @ A @ X2 @ ( plus_plus @ nat @ N3 @ K ) ) ) ) ) ) ) ) ).
% exp_first_terms
thf(fact_4916_sinh__converges,axiom,
! [A: $tType] :
( ( ( real_Vector_banach @ A )
& ( real_V2822296259951069270ebra_1 @ A ) )
=> ! [X: A] :
( sums @ A
@ ^ [N3: nat] : ( if @ A @ ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N3 ) @ ( zero_zero @ A ) @ ( real_V8093663219630862766scaleR @ A @ ( inverse_inverse @ real @ ( semiring_char_0_fact @ real @ N3 ) ) @ ( power_power @ A @ X @ N3 ) ) )
@ ( sinh @ A @ X ) ) ) ).
% sinh_converges
thf(fact_4917_cosh__converges,axiom,
! [A: $tType] :
( ( ( real_Vector_banach @ A )
& ( real_V2822296259951069270ebra_1 @ A ) )
=> ! [X: A] :
( sums @ A
@ ^ [N3: nat] : ( if @ A @ ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N3 ) @ ( real_V8093663219630862766scaleR @ A @ ( inverse_inverse @ real @ ( semiring_char_0_fact @ real @ N3 ) ) @ ( power_power @ A @ X @ N3 ) ) @ ( zero_zero @ A ) )
@ ( cosh @ A @ X ) ) ) ).
% cosh_converges
thf(fact_4918_complex__unimodular__polar,axiom,
! [Z: complex] :
( ( ( real_V7770717601297561774m_norm @ complex @ Z )
= ( one_one @ real ) )
=> ~ ! [T7: real] :
( ( ord_less_eq @ real @ ( zero_zero @ real ) @ T7 )
=> ( ( ord_less @ real @ T7 @ ( times_times @ real @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) @ pi ) )
=> ( Z
!= ( complex2 @ ( cos @ real @ T7 ) @ ( sin @ real @ T7 ) ) ) ) ) ) ).
% complex_unimodular_polar
thf(fact_4919_sinh__real__zero__iff,axiom,
! [X: real] :
( ( ( sinh @ real @ X )
= ( zero_zero @ real ) )
= ( X
= ( zero_zero @ real ) ) ) ).
% sinh_real_zero_iff
thf(fact_4920_sinh__real__less__iff,axiom,
! [X: real,Y: real] :
( ( ord_less @ real @ ( sinh @ real @ X ) @ ( sinh @ real @ Y ) )
= ( ord_less @ real @ X @ Y ) ) ).
% sinh_real_less_iff
thf(fact_4921_sinh__real__neg__iff,axiom,
! [X: real] :
( ( ord_less @ real @ ( sinh @ real @ X ) @ ( zero_zero @ real ) )
= ( ord_less @ real @ X @ ( zero_zero @ real ) ) ) ).
% sinh_real_neg_iff
thf(fact_4922_sinh__real__pos__iff,axiom,
! [X: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ ( sinh @ real @ X ) )
= ( ord_less @ real @ ( zero_zero @ real ) @ X ) ) ).
% sinh_real_pos_iff
thf(fact_4923_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_4924_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_4925_sinh__0,axiom,
! [A: $tType] :
( ( ( real_Vector_banach @ A )
& ( real_V2822296259951069270ebra_1 @ A ) )
=> ( ( sinh @ A @ ( zero_zero @ A ) )
= ( zero_zero @ A ) ) ) ).
% sinh_0
thf(fact_4926_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_4927_norm__cos__sin,axiom,
! [T: real] :
( ( real_V7770717601297561774m_norm @ complex @ ( complex2 @ ( cos @ real @ T ) @ ( sin @ real @ T ) ) )
= ( one_one @ real ) ) ).
% norm_cos_sin
thf(fact_4928_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_4929_cosh__real__nonzero,axiom,
! [X: real] :
( ( cosh @ real @ X )
!= ( zero_zero @ real ) ) ).
% cosh_real_nonzero
thf(fact_4930_complex__of__real__def,axiom,
( ( real_Vector_of_real @ complex )
= ( ^ [R5: real] : ( complex2 @ R5 @ ( zero_zero @ real ) ) ) ) ).
% complex_of_real_def
thf(fact_4931_complex__of__real__code,axiom,
( ( real_Vector_of_real @ complex )
= ( ^ [X2: real] : ( complex2 @ X2 @ ( zero_zero @ real ) ) ) ) ).
% complex_of_real_code
thf(fact_4932_complex__eq__cancel__iff2,axiom,
! [X: real,Y: real,Xa: real] :
( ( ( complex2 @ X @ Y )
= ( real_Vector_of_real @ complex @ Xa ) )
= ( ( X = Xa )
& ( Y
= ( zero_zero @ real ) ) ) ) ).
% complex_eq_cancel_iff2
thf(fact_4933_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_4934_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_4935_sinh__less__cosh__real,axiom,
! [X: real] : ( ord_less @ real @ ( sinh @ real @ X ) @ ( cosh @ real @ X ) ) ).
% sinh_less_cosh_real
thf(fact_4936_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_4937_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_4938_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_4939_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_4940_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_4941_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_4942_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_4943_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_4944_cosh__real__pos,axiom,
! [X: real] : ( ord_less @ real @ ( zero_zero @ real ) @ ( cosh @ real @ X ) ) ).
% cosh_real_pos
thf(fact_4945_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_4946_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_4947_cosh__real__nonneg,axiom,
! [X: real] : ( ord_less_eq @ real @ ( zero_zero @ real ) @ ( cosh @ real @ X ) ) ).
% cosh_real_nonneg
thf(fact_4948_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_4949_cosh__real__ge__1,axiom,
! [X: real] : ( ord_less_eq @ real @ ( one_one @ real ) @ ( cosh @ real @ X ) ) ).
% cosh_real_ge_1
thf(fact_4950_Complex__eq__0,axiom,
! [A2: real,B2: real] :
( ( ( complex2 @ A2 @ B2 )
= ( zero_zero @ complex ) )
= ( ( A2
= ( zero_zero @ real ) )
& ( B2
= ( zero_zero @ real ) ) ) ) ).
% Complex_eq_0
thf(fact_4951_zero__complex_Ocode,axiom,
( ( zero_zero @ complex )
= ( complex2 @ ( zero_zero @ real ) @ ( zero_zero @ real ) ) ) ).
% zero_complex.code
thf(fact_4952_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_4953_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_4954_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_4955_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_4956_complex__mult,axiom,
! [A2: real,B2: real,C2: real,D2: real] :
( ( times_times @ complex @ ( complex2 @ A2 @ B2 ) @ ( complex2 @ C2 @ D2 ) )
= ( complex2 @ ( minus_minus @ real @ ( times_times @ real @ A2 @ C2 ) @ ( times_times @ real @ B2 @ D2 ) ) @ ( plus_plus @ real @ ( times_times @ real @ A2 @ D2 ) @ ( times_times @ real @ B2 @ C2 ) ) ) ) ).
% complex_mult
thf(fact_4957_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_4958_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_4959_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_4960_one__complex_Ocode,axiom,
( ( one_one @ complex )
= ( complex2 @ ( one_one @ real ) @ ( zero_zero @ real ) ) ) ).
% one_complex.code
thf(fact_4961_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_4962_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_4963_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_4964_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_4965_Complex__sum_H,axiom,
! [A: $tType,F2: A > real,S: set @ A] :
( ( groups7311177749621191930dd_sum @ A @ complex
@ ^ [X2: A] : ( complex2 @ ( F2 @ X2 ) @ ( zero_zero @ real ) )
@ S )
= ( complex2 @ ( groups7311177749621191930dd_sum @ A @ real @ F2 @ S ) @ ( zero_zero @ real ) ) ) ).
% Complex_sum'
thf(fact_4966_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_4967_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_4968_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_4969_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_4970_cosh__field__def,axiom,
! [A: $tType] :
( ( ( real_Vector_banach @ A )
& ( real_V3459762299906320749_field @ A ) )
=> ( ( cosh @ A )
= ( ^ [Z7: A] : ( divide_divide @ A @ ( plus_plus @ A @ ( exp @ A @ Z7 ) @ ( exp @ A @ ( uminus_uminus @ A @ Z7 ) ) ) @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) ) ) ) ).
% cosh_field_def
thf(fact_4971_sinh__field__def,axiom,
! [A: $tType] :
( ( ( real_Vector_banach @ A )
& ( real_V3459762299906320749_field @ A ) )
=> ( ( sinh @ A )
= ( ^ [Z7: A] : ( divide_divide @ A @ ( minus_minus @ A @ ( exp @ A @ Z7 ) @ ( exp @ A @ ( uminus_uminus @ A @ Z7 ) ) ) @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) ) ) ) ).
% sinh_field_def
thf(fact_4972_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_4973_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_4974_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_4975_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_4976_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_4977_and__mask__arith_H,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [N: nat,W: word @ A] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ N )
=> ( ( bit_se5824344872417868541ns_and @ ( word @ A ) @ W @ ( bit_se2239418461657761734s_mask @ ( word @ A ) @ N ) )
= ( divide_divide @ ( word @ A ) @ ( times_times @ ( word @ A ) @ W @ ( power_power @ ( word @ A ) @ ( numeral_numeral @ ( word @ A ) @ ( bit0 @ one2 ) ) @ ( minus_minus @ nat @ ( size_size @ ( word @ A ) @ W ) @ N ) ) ) @ ( power_power @ ( word @ A ) @ ( numeral_numeral @ ( word @ A ) @ ( bit0 @ one2 ) ) @ ( minus_minus @ nat @ ( size_size @ ( word @ A ) @ W ) @ N ) ) ) ) ) ) ).
% and_mask_arith'
thf(fact_4978_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_4979_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_4980_cot__zero,axiom,
! [A: $tType] :
( ( ( real_Vector_banach @ A )
& ( real_V3459762299906320749_field @ A ) )
=> ( ( cot @ A @ ( zero_zero @ A ) )
= ( zero_zero @ A ) ) ) ).
% cot_zero
thf(fact_4981_mask__0,axiom,
! [A: $tType] :
( ( bit_se359711467146920520ations @ A )
=> ( ( bit_se2239418461657761734s_mask @ A @ ( zero_zero @ nat ) )
= ( zero_zero @ A ) ) ) ).
% mask_0
thf(fact_4982_mask__eq__0__iff,axiom,
! [A: $tType] :
( ( bit_se359711467146920520ations @ A )
=> ! [N: nat] :
( ( ( bit_se2239418461657761734s_mask @ A @ N )
= ( zero_zero @ A ) )
= ( N
= ( zero_zero @ nat ) ) ) ) ).
% mask_eq_0_iff
thf(fact_4983_cot__pi,axiom,
( ( cot @ real @ pi )
= ( zero_zero @ real ) ) ).
% cot_pi
thf(fact_4984_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_4985_take__bit__minus__one__eq__mask,axiom,
! [A: $tType] :
( ( bit_ri3973907225187159222ations @ A )
=> ! [N: nat] :
( ( bit_se2584673776208193580ke_bit @ A @ N @ ( uminus_uminus @ A @ ( one_one @ A ) ) )
= ( bit_se2239418461657761734s_mask @ A @ N ) ) ) ).
% take_bit_minus_one_eq_mask
thf(fact_4986_cot__npi,axiom,
! [N: nat] :
( ( cot @ real @ ( times_times @ real @ ( semiring_1_of_nat @ real @ N ) @ pi ) )
= ( zero_zero @ real ) ) ).
% cot_npi
thf(fact_4987_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_4988_of__nat__mask__eq,axiom,
! [A: $tType] :
( ( bit_se359711467146920520ations @ A )
=> ! [N: nat] :
( ( semiring_1_of_nat @ A @ ( bit_se2239418461657761734s_mask @ nat @ N ) )
= ( bit_se2239418461657761734s_mask @ A @ N ) ) ) ).
% of_nat_mask_eq
thf(fact_4989_of__int__mask__eq,axiom,
! [A: $tType] :
( ( bit_ri3973907225187159222ations @ A )
=> ! [N: nat] :
( ( ring_1_of_int @ A @ ( bit_se2239418461657761734s_mask @ int @ N ) )
= ( bit_se2239418461657761734s_mask @ A @ N ) ) ) ).
% of_int_mask_eq
thf(fact_4990_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_4991_mask__1,axiom,
! [A: $tType] :
( ( bit_se359711467146920520ations @ A )
=> ( ( bit_se2239418461657761734s_mask @ A @ ( one_one @ nat ) )
= ( one_one @ A ) ) ) ).
% mask_1
thf(fact_4992_take__bit__eq__mask,axiom,
! [A: $tType] :
( ( bit_se359711467146920520ations @ A )
=> ( ( bit_se2584673776208193580ke_bit @ A )
= ( ^ [N3: nat,A4: A] : ( bit_se5824344872417868541ns_and @ A @ A4 @ ( bit_se2239418461657761734s_mask @ A @ N3 ) ) ) ) ) ).
% take_bit_eq_mask
thf(fact_4993_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_4994_mask__bin,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ( ( bit_se2239418461657761734s_mask @ ( word @ A ) )
= ( ^ [N3: nat] : ( ring_1_of_int @ ( word @ A ) @ ( bit_se2584673776208193580ke_bit @ int @ N3 @ ( uminus_uminus @ int @ ( one_one @ int ) ) ) ) ) ) ) ).
% mask_bin
thf(fact_4995_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_4996_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_4997_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_4998_and__mask__no,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [I: num,N: nat] :
( ( bit_se5824344872417868541ns_and @ ( word @ A ) @ ( numeral_numeral @ ( word @ A ) @ I ) @ ( bit_se2239418461657761734s_mask @ ( word @ A ) @ N ) )
= ( ring_1_of_int @ ( word @ A ) @ ( bit_se2584673776208193580ke_bit @ int @ N @ ( numeral_numeral @ int @ I ) ) ) ) ) ).
% and_mask_no
thf(fact_4999_semiring__bit__operations__class_Oeven__mask__iff,axiom,
! [A: $tType] :
( ( bit_se359711467146920520ations @ A )
=> ! [N: nat] :
( ( dvd_dvd @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ ( bit_se2239418461657761734s_mask @ A @ N ) )
= ( N
= ( zero_zero @ nat ) ) ) ) ).
% semiring_bit_operations_class.even_mask_iff
thf(fact_5000_mask__plus__1,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [N: nat] :
( ( plus_plus @ ( word @ A ) @ ( bit_se2239418461657761734s_mask @ ( word @ A ) @ N ) @ ( one_one @ ( word @ A ) ) )
= ( power_power @ ( word @ A ) @ ( numeral_numeral @ ( word @ A ) @ ( bit0 @ one2 ) ) @ N ) ) ) ).
% mask_plus_1
thf(fact_5001_less__mask__eq,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [X: word @ A,N: nat] :
( ( ord_less @ ( word @ A ) @ X @ ( power_power @ ( word @ A ) @ ( numeral_numeral @ ( word @ A ) @ ( bit0 @ one2 ) ) @ N ) )
=> ( ( bit_se5824344872417868541ns_and @ ( word @ A ) @ X @ ( bit_se2239418461657761734s_mask @ ( word @ A ) @ N ) )
= X ) ) ) ).
% less_mask_eq
thf(fact_5002_mask__eq__decr__exp,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ( ( bit_se2239418461657761734s_mask @ ( word @ A ) )
= ( ^ [N3: nat] : ( minus_minus @ ( word @ A ) @ ( power_power @ ( word @ A ) @ ( numeral_numeral @ ( word @ A ) @ ( bit0 @ one2 ) ) @ N3 ) @ ( one_one @ ( word @ A ) ) ) ) ) ) ).
% mask_eq_decr_exp
thf(fact_5003_mask__eq__exp__minus__1,axiom,
! [A: $tType] :
( ( bit_se359711467146920520ations @ A )
=> ( ( bit_se2239418461657761734s_mask @ A )
= ( ^ [N3: nat] : ( minus_minus @ A @ ( power_power @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ N3 ) @ ( one_one @ A ) ) ) ) ) ).
% mask_eq_exp_minus_1
thf(fact_5004_mask__eq__iff,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [W: word @ A,N: nat] :
( ( ( bit_se5824344872417868541ns_and @ ( word @ A ) @ W @ ( bit_se2239418461657761734s_mask @ ( word @ A ) @ N ) )
= W )
= ( ord_less @ int @ ( semiring_1_unsigned @ A @ int @ W ) @ ( power_power @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ N ) ) ) ) ).
% mask_eq_iff
thf(fact_5005_and__mask__lt__2p,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [W: word @ A,N: nat] : ( ord_less @ int @ ( semiring_1_unsigned @ A @ int @ ( bit_se5824344872417868541ns_and @ ( word @ A ) @ W @ ( bit_se2239418461657761734s_mask @ ( word @ A ) @ N ) ) ) @ ( power_power @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ N ) ) ) ).
% and_mask_lt_2p
thf(fact_5006_mask__Suc__rec,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [N: nat] :
( ( bit_se2239418461657761734s_mask @ ( word @ A ) @ ( suc @ N ) )
= ( plus_plus @ ( word @ A ) @ ( times_times @ ( word @ A ) @ ( numeral_numeral @ ( word @ A ) @ ( bit0 @ one2 ) ) @ ( bit_se2239418461657761734s_mask @ ( word @ A ) @ N ) ) @ ( one_one @ ( word @ A ) ) ) ) ) ).
% mask_Suc_rec
thf(fact_5007_is__aligned__AND__less__0,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [U: word @ A,N: nat,V: word @ A] :
( ( ( bit_se5824344872417868541ns_and @ ( word @ A ) @ U @ ( bit_se2239418461657761734s_mask @ ( word @ A ) @ N ) )
= ( zero_zero @ ( word @ A ) ) )
=> ( ( ord_less @ ( word @ A ) @ V @ ( power_power @ ( word @ A ) @ ( numeral_numeral @ ( word @ A ) @ ( bit0 @ one2 ) ) @ N ) )
=> ( ( bit_se5824344872417868541ns_and @ ( word @ A ) @ U @ V )
= ( zero_zero @ ( word @ A ) ) ) ) ) ) ).
% is_aligned_AND_less_0
thf(fact_5008_add__mask__fold,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [X: word @ A,N: nat] :
( ( minus_minus @ ( word @ A ) @ ( plus_plus @ ( word @ A ) @ X @ ( power_power @ ( word @ A ) @ ( numeral_numeral @ ( word @ A ) @ ( bit0 @ one2 ) ) @ N ) ) @ ( one_one @ ( word @ A ) ) )
= ( plus_plus @ ( word @ A ) @ X @ ( bit_se2239418461657761734s_mask @ ( word @ A ) @ N ) ) ) ) ).
% add_mask_fold
thf(fact_5009_and__mask__dvd__nat,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [N: nat,W: word @ A] :
( ( dvd_dvd @ nat @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N ) @ ( semiring_1_unsigned @ A @ nat @ W ) )
= ( ( bit_se5824344872417868541ns_and @ ( word @ A ) @ W @ ( bit_se2239418461657761734s_mask @ ( word @ A ) @ N ) )
= ( zero_zero @ ( word @ A ) ) ) ) ) ).
% and_mask_dvd_nat
thf(fact_5010_and__mask__mod__2p,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [W: word @ A,N: nat] :
( ( bit_se5824344872417868541ns_and @ ( word @ A ) @ W @ ( bit_se2239418461657761734s_mask @ ( word @ A ) @ N ) )
= ( ring_1_of_int @ ( word @ A ) @ ( modulo_modulo @ int @ ( semiring_1_unsigned @ A @ int @ W ) @ ( power_power @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ N ) ) ) ) ) ).
% and_mask_mod_2p
thf(fact_5011_and__mask__dvd,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [N: nat,W: word @ A] :
( ( dvd_dvd @ int @ ( power_power @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ N ) @ ( semiring_1_unsigned @ A @ int @ W ) )
= ( ( bit_se5824344872417868541ns_and @ ( word @ A ) @ W @ ( bit_se2239418461657761734s_mask @ ( word @ A ) @ N ) )
= ( zero_zero @ ( word @ A ) ) ) ) ) ).
% and_mask_dvd
thf(fact_5012_mask__eq__iff__w2p,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [N: nat,W: word @ A] :
( ( ord_less @ nat @ N @ ( size_size @ ( word @ A ) @ W ) )
=> ( ( ( bit_se5824344872417868541ns_and @ ( word @ A ) @ W @ ( bit_se2239418461657761734s_mask @ ( word @ A ) @ N ) )
= W )
= ( ord_less @ ( word @ A ) @ W @ ( power_power @ ( word @ A ) @ ( numeral_numeral @ ( word @ A ) @ ( bit0 @ one2 ) ) @ N ) ) ) ) ) ).
% mask_eq_iff_w2p
thf(fact_5013_and__mask__less__size,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [N: nat,X: word @ A] :
( ( ord_less @ nat @ N @ ( size_size @ ( word @ A ) @ X ) )
=> ( ord_less @ ( word @ A ) @ ( bit_se5824344872417868541ns_and @ ( word @ A ) @ X @ ( bit_se2239418461657761734s_mask @ ( word @ A ) @ N ) ) @ ( power_power @ ( word @ A ) @ ( numeral_numeral @ ( word @ A ) @ ( bit0 @ one2 ) ) @ N ) ) ) ) ).
% and_mask_less_size
thf(fact_5014_word__and__mask__le__2pm1,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [W: word @ A,N: nat] : ( ord_less_eq @ ( word @ A ) @ ( bit_se5824344872417868541ns_and @ ( word @ A ) @ W @ ( bit_se2239418461657761734s_mask @ ( word @ A ) @ N ) ) @ ( minus_minus @ ( word @ A ) @ ( power_power @ ( word @ A ) @ ( numeral_numeral @ ( word @ A ) @ ( bit0 @ one2 ) ) @ N ) @ ( one_one @ ( word @ A ) ) ) ) ) ).
% word_and_mask_le_2pm1
thf(fact_5015_word__mod__2p__is__mask,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [N: nat,X: word @ A] :
( ( ord_less @ ( word @ A ) @ ( zero_zero @ ( word @ A ) ) @ ( power_power @ ( word @ A ) @ ( numeral_numeral @ ( word @ A ) @ ( bit0 @ one2 ) ) @ N ) )
=> ( ( modulo_modulo @ ( word @ A ) @ X @ ( power_power @ ( word @ A ) @ ( numeral_numeral @ ( word @ A ) @ ( bit0 @ one2 ) ) @ N ) )
= ( bit_se5824344872417868541ns_and @ ( word @ A ) @ X @ ( bit_se2239418461657761734s_mask @ ( word @ A ) @ N ) ) ) ) ) ).
% word_mod_2p_is_mask
thf(fact_5016_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_5017_and__mask__arith,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [W: word @ A,N: nat] :
( ( bit_se5824344872417868541ns_and @ ( word @ A ) @ W @ ( bit_se2239418461657761734s_mask @ ( word @ A ) @ N ) )
= ( divide_divide @ ( word @ A ) @ ( times_times @ ( word @ A ) @ W @ ( power_power @ ( word @ A ) @ ( numeral_numeral @ ( word @ A ) @ ( bit0 @ one2 ) ) @ ( minus_minus @ nat @ ( size_size @ ( word @ A ) @ W ) @ N ) ) ) @ ( power_power @ ( word @ A ) @ ( numeral_numeral @ ( word @ A ) @ ( bit0 @ one2 ) ) @ ( minus_minus @ nat @ ( size_size @ ( word @ A ) @ W ) @ N ) ) ) ) ) ).
% and_mask_arith
thf(fact_5018_mask__numeral,axiom,
! [A: $tType] :
( ( bit_se359711467146920520ations @ A )
=> ! [N: num] :
( ( bit_se2239418461657761734s_mask @ A @ ( numeral_numeral @ nat @ N ) )
= ( plus_plus @ A @ ( one_one @ A ) @ ( times_times @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ ( bit_se2239418461657761734s_mask @ A @ ( pred_numeral @ N ) ) ) ) ) ) ).
% mask_numeral
thf(fact_5019_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_5020_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_5021_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_5022_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_5023_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_5024_mask__nat__positive__iff,axiom,
! [N: nat] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ ( bit_se2239418461657761734s_mask @ nat @ N ) )
= ( ord_less @ nat @ ( zero_zero @ nat ) @ N ) ) ).
% mask_nat_positive_iff
thf(fact_5025_norm__ii,axiom,
( ( real_V7770717601297561774m_norm @ complex @ imaginary_unit )
= ( one_one @ real ) ) ).
% norm_ii
thf(fact_5026_i__squared,axiom,
( ( times_times @ complex @ imaginary_unit @ imaginary_unit )
= ( uminus_uminus @ complex @ ( one_one @ complex ) ) ) ).
% i_squared
thf(fact_5027_power2__i,axiom,
( ( power_power @ complex @ imaginary_unit @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) )
= ( uminus_uminus @ complex @ ( one_one @ complex ) ) ) ).
% power2_i
thf(fact_5028_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_5029_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_5030_i__even__power,axiom,
! [N: nat] :
( ( power_power @ complex @ imaginary_unit @ ( times_times @ nat @ N @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) )
= ( power_power @ complex @ ( uminus_uminus @ complex @ ( one_one @ complex ) ) @ N ) ) ).
% i_even_power
thf(fact_5031_less__eq__mask,axiom,
! [N: nat] : ( ord_less_eq @ nat @ N @ ( bit_se2239418461657761734s_mask @ nat @ N ) ) ).
% less_eq_mask
thf(fact_5032_nat__mask__eq,axiom,
! [N: nat] :
( ( nat2 @ ( bit_se2239418461657761734s_mask @ int @ N ) )
= ( bit_se2239418461657761734s_mask @ nat @ N ) ) ).
% nat_mask_eq
thf(fact_5033_complex__i__not__one,axiom,
( imaginary_unit
!= ( one_one @ complex ) ) ).
% complex_i_not_one
thf(fact_5034_complex__i__not__zero,axiom,
( imaginary_unit
!= ( zero_zero @ complex ) ) ).
% complex_i_not_zero
thf(fact_5035_mask__nonnegative__int,axiom,
! [N: nat] : ( ord_less_eq @ int @ ( zero_zero @ int ) @ ( bit_se2239418461657761734s_mask @ int @ N ) ) ).
% mask_nonnegative_int
thf(fact_5036_not__mask__negative__int,axiom,
! [N: nat] :
~ ( ord_less @ int @ ( bit_se2239418461657761734s_mask @ int @ N ) @ ( zero_zero @ int ) ) ).
% not_mask_negative_int
thf(fact_5037_less__mask,axiom,
! [N: nat] :
( ( ord_less @ nat @ ( suc @ ( zero_zero @ nat ) ) @ N )
=> ( ord_less @ nat @ N @ ( bit_se2239418461657761734s_mask @ nat @ N ) ) ) ).
% less_mask
thf(fact_5038_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_5039_imaginary__unit_Ocode,axiom,
( imaginary_unit
= ( complex2 @ ( zero_zero @ real ) @ ( one_one @ real ) ) ) ).
% imaginary_unit.code
thf(fact_5040_take__bit__eq__mask__iff,axiom,
! [N: nat,K: int] :
( ( ( bit_se2584673776208193580ke_bit @ int @ N @ K )
= ( bit_se2239418461657761734s_mask @ int @ N ) )
= ( ( bit_se2584673776208193580ke_bit @ int @ N @ ( plus_plus @ int @ K @ ( one_one @ int ) ) )
= ( zero_zero @ int ) ) ) ).
% take_bit_eq_mask_iff
thf(fact_5041_i__complex__of__real,axiom,
! [R2: real] :
( ( times_times @ complex @ imaginary_unit @ ( real_Vector_of_real @ complex @ R2 ) )
= ( complex2 @ ( zero_zero @ real ) @ R2 ) ) ).
% i_complex_of_real
thf(fact_5042_complex__of__real__i,axiom,
! [R2: real] :
( ( times_times @ complex @ ( real_Vector_of_real @ complex @ R2 ) @ imaginary_unit )
= ( complex2 @ ( zero_zero @ real ) @ R2 ) ) ).
% complex_of_real_i
thf(fact_5043_Suc__mask__eq__exp,axiom,
! [N: nat] :
( ( suc @ ( bit_se2239418461657761734s_mask @ nat @ N ) )
= ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N ) ) ).
% Suc_mask_eq_exp
thf(fact_5044_mask__nat__less__exp,axiom,
! [N: nat] : ( ord_less @ nat @ ( bit_se2239418461657761734s_mask @ nat @ N ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N ) ) ).
% mask_nat_less_exp
thf(fact_5045_mask__half__int,axiom,
! [N: nat] :
( ( divide_divide @ int @ ( bit_se2239418461657761734s_mask @ int @ N ) @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) )
= ( bit_se2239418461657761734s_mask @ int @ ( minus_minus @ nat @ N @ ( one_one @ nat ) ) ) ) ).
% mask_half_int
thf(fact_5046_mask__int__def,axiom,
( ( bit_se2239418461657761734s_mask @ int )
= ( ^ [N3: nat] : ( minus_minus @ int @ ( power_power @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ N3 ) @ ( one_one @ int ) ) ) ) ).
% mask_int_def
thf(fact_5047_mask__nat__def,axiom,
( ( bit_se2239418461657761734s_mask @ nat )
= ( ^ [N3: nat] : ( minus_minus @ nat @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N3 ) @ ( one_one @ nat ) ) ) ) ).
% mask_nat_def
thf(fact_5048_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_5049_take__bit__eq__mask__iff__exp__dvd,axiom,
! [N: nat,K: int] :
( ( ( bit_se2584673776208193580ke_bit @ int @ N @ K )
= ( bit_se2239418461657761734s_mask @ int @ N ) )
= ( dvd_dvd @ int @ ( power_power @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ N ) @ ( plus_plus @ int @ K @ ( one_one @ int ) ) ) ) ).
% take_bit_eq_mask_iff_exp_dvd
thf(fact_5050_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_5051_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_5052_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_5053_csqrt__eq__0,axiom,
! [Z: complex] :
( ( ( csqrt @ Z )
= ( zero_zero @ complex ) )
= ( Z
= ( zero_zero @ complex ) ) ) ).
% csqrt_eq_0
thf(fact_5054_csqrt__0,axiom,
( ( csqrt @ ( zero_zero @ complex ) )
= ( zero_zero @ complex ) ) ).
% csqrt_0
thf(fact_5055_csqrt__1,axiom,
( ( csqrt @ ( one_one @ complex ) )
= ( one_one @ complex ) ) ).
% csqrt_1
thf(fact_5056_csqrt__eq__1,axiom,
! [Z: complex] :
( ( ( csqrt @ Z )
= ( one_one @ complex ) )
= ( Z
= ( one_one @ complex ) ) ) ).
% csqrt_eq_1
thf(fact_5057_power2__csqrt,axiom,
! [Z: complex] :
( ( power_power @ complex @ ( csqrt @ Z ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) )
= Z ) ).
% power2_csqrt
thf(fact_5058_Arg__ii,axiom,
( ( arg @ imaginary_unit )
= ( divide_divide @ real @ pi @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) ) ).
% Arg_ii
thf(fact_5059_Arg__zero,axiom,
( ( arg @ ( zero_zero @ complex ) )
= ( zero_zero @ real ) ) ).
% Arg_zero
thf(fact_5060_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_5061_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_5062_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_5063_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_5064_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_5065_norm__cis,axiom,
! [A2: real] :
( ( real_V7770717601297561774m_norm @ complex @ ( cis @ A2 ) )
= ( one_one @ real ) ) ).
% norm_cis
thf(fact_5066_cis__zero,axiom,
( ( cis @ ( zero_zero @ real ) )
= ( one_one @ complex ) ) ).
% cis_zero
thf(fact_5067_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_5068_pred__numeral__simps_I2_J,axiom,
! [K: num] :
( ( pred_numeral @ ( bit0 @ K ) )
= ( numeral_numeral @ nat @ ( bitM @ K ) ) ) ).
% pred_numeral_simps(2)
thf(fact_5069_cis__pi,axiom,
( ( cis @ pi )
= ( uminus_uminus @ complex @ ( one_one @ complex ) ) ) ).
% cis_pi
thf(fact_5070_cis__pi__half,axiom,
( ( cis @ ( divide_divide @ real @ pi @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) )
= imaginary_unit ) ).
% cis_pi_half
thf(fact_5071_cis__2pi,axiom,
( ( cis @ ( times_times @ real @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) @ pi ) )
= ( one_one @ complex ) ) ).
% cis_2pi
thf(fact_5072_cis__neq__zero,axiom,
! [A2: real] :
( ( cis @ A2 )
!= ( zero_zero @ complex ) ) ).
% cis_neq_zero
thf(fact_5073_semiring__norm_I26_J,axiom,
( ( bitM @ one2 )
= one2 ) ).
% semiring_norm(26)
thf(fact_5074_semiring__norm_I27_J,axiom,
! [N: num] :
( ( bitM @ ( bit0 @ N ) )
= ( bit1 @ ( bitM @ N ) ) ) ).
% semiring_norm(27)
thf(fact_5075_semiring__norm_I28_J,axiom,
! [N: num] :
( ( bitM @ ( bit1 @ N ) )
= ( bit1 @ ( bit0 @ N ) ) ) ).
% semiring_norm(28)
thf(fact_5076_inc__BitM__eq,axiom,
! [N: num] :
( ( inc @ ( bitM @ N ) )
= ( bit0 @ N ) ) ).
% inc_BitM_eq
thf(fact_5077_BitM__inc__eq,axiom,
! [N: num] :
( ( bitM @ ( inc @ N ) )
= ( bit1 @ N ) ) ).
% BitM_inc_eq
thf(fact_5078_cis__Arg,axiom,
! [Z: complex] :
( ( Z
!= ( zero_zero @ complex ) )
=> ( ( cis @ ( arg @ Z ) )
= ( sgn_sgn @ complex @ Z ) ) ) ).
% cis_Arg
thf(fact_5079_eval__nat__numeral_I2_J,axiom,
! [N: num] :
( ( numeral_numeral @ nat @ ( bit0 @ N ) )
= ( suc @ ( numeral_numeral @ nat @ ( bitM @ N ) ) ) ) ).
% eval_nat_numeral(2)
thf(fact_5080_one__plus__BitM,axiom,
! [N: num] :
( ( plus_plus @ num @ one2 @ ( bitM @ N ) )
= ( bit0 @ N ) ) ).
% one_plus_BitM
thf(fact_5081_BitM__plus__one,axiom,
! [N: num] :
( ( plus_plus @ num @ ( bitM @ N ) @ one2 )
= ( bit0 @ N ) ) ).
% BitM_plus_one
thf(fact_5082_DeMoivre,axiom,
! [A2: real,N: nat] :
( ( power_power @ complex @ ( cis @ A2 ) @ N )
= ( cis @ ( times_times @ real @ ( semiring_1_of_nat @ real @ N ) @ A2 ) ) ) ).
% DeMoivre
thf(fact_5083_numeral__BitM,axiom,
! [A: $tType] :
( ( neg_numeral @ A )
=> ! [N: num] :
( ( numeral_numeral @ A @ ( bitM @ N ) )
= ( minus_minus @ A @ ( numeral_numeral @ A @ ( bit0 @ N ) ) @ ( one_one @ A ) ) ) ) ).
% numeral_BitM
thf(fact_5084_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_5085_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_5086_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_5087_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_5088_bij__betw__roots__unity,axiom,
! [N: nat] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ N )
=> ( bij_betw @ nat @ complex
@ ^ [K3: nat] : ( cis @ ( divide_divide @ real @ ( times_times @ real @ ( times_times @ real @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) @ pi ) @ ( semiring_1_of_nat @ real @ K3 ) ) @ ( semiring_1_of_nat @ real @ N ) ) )
@ ( set_ord_lessThan @ nat @ N )
@ ( collect @ complex
@ ^ [Z7: complex] :
( ( power_power @ complex @ Z7 @ N )
= ( one_one @ complex ) ) ) ) ) ).
% bij_betw_roots_unity
thf(fact_5089_divmod__algorithm__code_I6_J,axiom,
! [A: $tType] :
( ( unique1627219031080169319umeral @ A )
=> ! [M: num,N: num] :
( ( unique8689654367752047608divmod @ A @ ( bit1 @ M ) @ ( bit0 @ N ) )
= ( product_case_prod @ A @ A @ ( product_prod @ A @ A )
@ ^ [Q6: A,R5: A] : ( product_Pair @ A @ A @ Q6 @ ( plus_plus @ A @ ( times_times @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ R5 ) @ ( one_one @ A ) ) )
@ ( unique8689654367752047608divmod @ A @ M @ N ) ) ) ) ).
% divmod_algorithm_code(6)
thf(fact_5090_listI__assn__def,axiom,
! [A: $tType,B: $tType] :
( ( vEBT_List_listI_assn @ A @ B )
= ( ^ [I9: 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 ) @ I9 @ ( set_or7035219750837199246ssThan @ nat @ ( zero_zero @ nat ) @ ( size_size @ ( list @ A ) @ Xs ) ) ) ) )
@ ( finite_fold @ nat @ assn
@ ^ [I2: nat,A4: assn] : ( times_times @ assn @ A4 @ ( A7 @ ( nth @ A @ Xs @ I2 ) @ ( nth @ B @ Xsi3 @ I2 ) ) )
@ ( one_one @ assn )
@ I9 ) ) ) ) ).
% listI_assn_def
thf(fact_5091_divmod__algorithm__code_I5_J,axiom,
! [A: $tType] :
( ( unique1627219031080169319umeral @ A )
=> ! [M: num,N: num] :
( ( unique8689654367752047608divmod @ A @ ( bit0 @ M ) @ ( bit0 @ N ) )
= ( product_case_prod @ A @ A @ ( product_prod @ A @ A )
@ ^ [Q6: A,R5: A] : ( product_Pair @ A @ A @ Q6 @ ( times_times @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ R5 ) )
@ ( unique8689654367752047608divmod @ A @ M @ N ) ) ) ) ).
% divmod_algorithm_code(5)
thf(fact_5092_sum_Oreindex__bij__betw__not__neutral,axiom,
! [B: $tType,A: $tType,C: $tType] :
( ( comm_monoid_add @ A )
=> ! [S4: set @ B,T4: set @ C,H2: B > C,S3: set @ B,T5: set @ C,G: C > A] :
( ( finite_finite2 @ B @ S4 )
=> ( ( finite_finite2 @ C @ T4 )
=> ( ( bij_betw @ B @ C @ H2 @ ( minus_minus @ ( set @ B ) @ S3 @ S4 ) @ ( minus_minus @ ( set @ C ) @ T5 @ T4 ) )
=> ( ! [A6: B] :
( ( member @ B @ A6 @ S4 )
=> ( ( G @ ( H2 @ A6 ) )
= ( zero_zero @ A ) ) )
=> ( ! [B3: C] :
( ( member @ C @ B3 @ T4 )
=> ( ( 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_5093_prod_Oreindex__bij__betw__not__neutral,axiom,
! [B: $tType,A: $tType,C: $tType] :
( ( comm_monoid_mult @ A )
=> ! [S4: set @ B,T4: set @ C,H2: B > C,S3: set @ B,T5: set @ C,G: C > A] :
( ( finite_finite2 @ B @ S4 )
=> ( ( finite_finite2 @ C @ T4 )
=> ( ( bij_betw @ B @ C @ H2 @ ( minus_minus @ ( set @ B ) @ S3 @ S4 ) @ ( minus_minus @ ( set @ C ) @ T5 @ T4 ) )
=> ( ! [A6: B] :
( ( member @ B @ A6 @ S4 )
=> ( ( G @ ( H2 @ A6 ) )
= ( one_one @ A ) ) )
=> ( ! [B3: C] :
( ( member @ C @ B3 @ T4 )
=> ( ( 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_5094_divmod__step__nat__def,axiom,
( ( unique1321980374590559556d_step @ nat )
= ( ^ [L2: num] :
( product_case_prod @ nat @ nat @ ( product_prod @ nat @ nat )
@ ^ [Q6: nat,R5: nat] : ( if @ ( product_prod @ nat @ nat ) @ ( ord_less_eq @ nat @ ( numeral_numeral @ nat @ L2 ) @ R5 ) @ ( product_Pair @ nat @ nat @ ( plus_plus @ nat @ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ Q6 ) @ ( one_one @ nat ) ) @ ( minus_minus @ nat @ R5 @ ( numeral_numeral @ nat @ L2 ) ) ) @ ( product_Pair @ nat @ nat @ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ Q6 ) @ R5 ) ) ) ) ) ).
% divmod_step_nat_def
thf(fact_5095_divmod__step__int__def,axiom,
( ( unique1321980374590559556d_step @ int )
= ( ^ [L2: num] :
( product_case_prod @ int @ int @ ( product_prod @ int @ int )
@ ^ [Q6: int,R5: int] : ( if @ ( product_prod @ int @ int ) @ ( ord_less_eq @ int @ ( numeral_numeral @ int @ L2 ) @ R5 ) @ ( product_Pair @ int @ int @ ( plus_plus @ int @ ( times_times @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ Q6 ) @ ( one_one @ int ) ) @ ( minus_minus @ int @ R5 @ ( numeral_numeral @ int @ L2 ) ) ) @ ( product_Pair @ int @ int @ ( times_times @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ Q6 ) @ R5 ) ) ) ) ) ).
% divmod_step_int_def
thf(fact_5096_divmod__step__def,axiom,
! [A: $tType] :
( ( unique1627219031080169319umeral @ A )
=> ( ( unique1321980374590559556d_step @ A )
= ( ^ [L2: num] :
( product_case_prod @ A @ A @ ( product_prod @ A @ A )
@ ^ [Q6: A,R5: A] : ( if @ ( product_prod @ A @ A ) @ ( ord_less_eq @ A @ ( numeral_numeral @ A @ L2 ) @ R5 ) @ ( product_Pair @ A @ A @ ( plus_plus @ A @ ( times_times @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ Q6 ) @ ( one_one @ A ) ) @ ( minus_minus @ A @ R5 @ ( numeral_numeral @ A @ L2 ) ) ) @ ( product_Pair @ A @ A @ ( times_times @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ Q6 ) @ R5 ) ) ) ) ) ) ).
% divmod_step_def
thf(fact_5097_divmod__step__integer__def,axiom,
( ( unique1321980374590559556d_step @ code_integer )
= ( ^ [L2: num] :
( product_case_prod @ code_integer @ code_integer @ ( product_prod @ code_integer @ code_integer )
@ ^ [Q6: code_integer,R5: code_integer] : ( if @ ( product_prod @ code_integer @ code_integer ) @ ( ord_less_eq @ code_integer @ ( numeral_numeral @ code_integer @ L2 ) @ R5 ) @ ( product_Pair @ code_integer @ code_integer @ ( plus_plus @ code_integer @ ( times_times @ code_integer @ ( numeral_numeral @ code_integer @ ( bit0 @ one2 ) ) @ Q6 ) @ ( one_one @ code_integer ) ) @ ( minus_minus @ code_integer @ R5 @ ( numeral_numeral @ code_integer @ L2 ) ) ) @ ( product_Pair @ code_integer @ code_integer @ ( times_times @ code_integer @ ( numeral_numeral @ code_integer @ ( bit0 @ one2 ) ) @ Q6 ) @ R5 ) ) ) ) ) ).
% divmod_step_integer_def
thf(fact_5098_divmod__nat__if,axiom,
( divmod_nat
= ( ^ [M4: nat,N3: nat] :
( if @ ( product_prod @ nat @ nat )
@ ( ( N3
= ( zero_zero @ nat ) )
| ( ord_less @ nat @ M4 @ N3 ) )
@ ( product_Pair @ nat @ nat @ ( zero_zero @ nat ) @ M4 )
@ ( product_case_prod @ nat @ nat @ ( product_prod @ nat @ nat )
@ ^ [Q6: nat] : ( product_Pair @ nat @ nat @ ( suc @ Q6 ) )
@ ( divmod_nat @ ( minus_minus @ nat @ M4 @ N3 ) @ N3 ) ) ) ) ) ).
% divmod_nat_if
thf(fact_5099_Arg__def,axiom,
( arg
= ( ^ [Z7: complex] :
( if @ real
@ ( Z7
= ( zero_zero @ complex ) )
@ ( zero_zero @ real )
@ ( fChoice @ real
@ ^ [A4: real] :
( ( ( sgn_sgn @ complex @ Z7 )
= ( cis @ A4 ) )
& ( ord_less @ real @ ( uminus_uminus @ real @ pi ) @ A4 )
& ( ord_less_eq @ real @ A4 @ pi ) ) ) ) ) ) ).
% Arg_def
thf(fact_5100_arctan__def,axiom,
( arctan
= ( ^ [Y6: 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 )
= Y6 ) ) ) ) ) ).
% arctan_def
thf(fact_5101_verit__sko__ex_H,axiom,
! [A: $tType,P: A > $o,A3: $o] :
( ( ( P @ ( fChoice @ A @ P ) )
= A3 )
=> ( ( ? [X8: A] : ( P @ X8 ) )
= A3 ) ) ).
% verit_sko_ex'
thf(fact_5102_verit__sko__forall,axiom,
! [A: $tType] :
( ( ^ [P6: A > $o] :
! [X9: A] : ( P6 @ X9 ) )
= ( ^ [P2: A > $o] :
( P2
@ ( fChoice @ A
@ ^ [X2: A] :
~ ( P2 @ X2 ) ) ) ) ) ).
% verit_sko_forall
thf(fact_5103_verit__sko__forall_H,axiom,
! [A: $tType,P: A > $o,A3: $o] :
( ( ( P
@ ( fChoice @ A
@ ^ [X2: A] :
~ ( P @ X2 ) ) )
= A3 )
=> ( ( ! [X8: A] : ( P @ X8 ) )
= A3 ) ) ).
% verit_sko_forall'
thf(fact_5104_verit__sko__forall_H_H,axiom,
! [A: $tType,B4: A,A3: A,P: A > $o] :
( ( B4 = A3 )
=> ( ( ( fChoice @ A @ P )
= A3 )
= ( ( fChoice @ A @ P )
= B4 ) ) ) ).
% verit_sko_forall''
thf(fact_5105_verit__sko__ex__indirect,axiom,
! [A: $tType,X: A,P: A > $o] :
( ( X
= ( fChoice @ A @ P ) )
=> ( ( ? [X8: A] : ( P @ X8 ) )
= ( P @ X ) ) ) ).
% verit_sko_ex_indirect
thf(fact_5106_verit__sko__ex__indirect2,axiom,
! [A: $tType,X: A,P: A > $o,P3: A > $o] :
( ( X
= ( fChoice @ A @ P ) )
=> ( ! [X3: A] :
( ( P @ X3 )
= ( P3 @ X3 ) )
=> ( ( ? [X8: A] : ( P3 @ X8 ) )
= ( P @ X ) ) ) ) ).
% verit_sko_ex_indirect2
thf(fact_5107_verit__sko__forall__indirect,axiom,
! [A: $tType,X: A,P: A > $o] :
( ( X
= ( fChoice @ A
@ ^ [X2: A] :
~ ( P @ X2 ) ) )
=> ( ( ! [X8: A] : ( P @ X8 ) )
= ( P @ X ) ) ) ).
% verit_sko_forall_indirect
thf(fact_5108_verit__sko__forall__indirect2,axiom,
! [A: $tType,X: A,P: A > $o,P3: A > $o] :
( ( X
= ( fChoice @ A
@ ^ [X2: A] :
~ ( P @ X2 ) ) )
=> ( ! [X3: A] :
( ( P @ X3 )
= ( P3 @ X3 ) )
=> ( ( ! [X8: A] : ( P3 @ X8 ) )
= ( P @ X ) ) ) ) ).
% verit_sko_forall_indirect2
thf(fact_5109_sum_Otriangle__reindex__eq,axiom,
! [A: $tType] :
( ( comm_monoid_add @ A )
=> ! [G: nat > nat > A,N: nat] :
( ( groups7311177749621191930dd_sum @ ( product_prod @ nat @ nat ) @ A @ ( product_case_prod @ nat @ nat @ A @ G )
@ ( collect @ ( product_prod @ nat @ nat )
@ ( product_case_prod @ nat @ nat @ $o
@ ^ [I2: nat,J2: nat] : ( ord_less_eq @ nat @ ( plus_plus @ nat @ I2 @ J2 ) @ N ) ) ) )
= ( groups7311177749621191930dd_sum @ nat @ A
@ ^ [K3: nat] :
( groups7311177749621191930dd_sum @ nat @ A
@ ^ [I2: nat] : ( G @ I2 @ ( minus_minus @ nat @ K3 @ I2 ) )
@ ( set_ord_atMost @ nat @ K3 ) )
@ ( set_ord_atMost @ nat @ N ) ) ) ) ).
% sum.triangle_reindex_eq
thf(fact_5110_prod_Otriangle__reindex__eq,axiom,
! [A: $tType] :
( ( comm_monoid_mult @ A )
=> ! [G: nat > nat > A,N: nat] :
( ( groups7121269368397514597t_prod @ ( product_prod @ nat @ nat ) @ A @ ( product_case_prod @ nat @ nat @ A @ G )
@ ( collect @ ( product_prod @ nat @ nat )
@ ( product_case_prod @ nat @ nat @ $o
@ ^ [I2: nat,J2: nat] : ( ord_less_eq @ nat @ ( plus_plus @ nat @ I2 @ J2 ) @ N ) ) ) )
= ( groups7121269368397514597t_prod @ nat @ A
@ ^ [K3: nat] :
( groups7121269368397514597t_prod @ nat @ A
@ ^ [I2: nat] : ( G @ I2 @ ( minus_minus @ nat @ K3 @ I2 ) )
@ ( set_ord_atMost @ nat @ K3 ) )
@ ( set_ord_atMost @ nat @ N ) ) ) ) ).
% prod.triangle_reindex_eq
thf(fact_5111_sum_Otriangle__reindex,axiom,
! [A: $tType] :
( ( comm_monoid_add @ A )
=> ! [G: nat > nat > A,N: nat] :
( ( groups7311177749621191930dd_sum @ ( product_prod @ nat @ nat ) @ A @ ( product_case_prod @ nat @ nat @ A @ G )
@ ( collect @ ( product_prod @ nat @ nat )
@ ( product_case_prod @ nat @ nat @ $o
@ ^ [I2: nat,J2: nat] : ( ord_less @ nat @ ( plus_plus @ nat @ I2 @ J2 ) @ N ) ) ) )
= ( groups7311177749621191930dd_sum @ nat @ A
@ ^ [K3: nat] :
( groups7311177749621191930dd_sum @ nat @ A
@ ^ [I2: nat] : ( G @ I2 @ ( minus_minus @ nat @ K3 @ I2 ) )
@ ( set_ord_atMost @ nat @ K3 ) )
@ ( set_ord_lessThan @ nat @ N ) ) ) ) ).
% sum.triangle_reindex
thf(fact_5112_prod_Otriangle__reindex,axiom,
! [A: $tType] :
( ( comm_monoid_mult @ A )
=> ! [G: nat > nat > A,N: nat] :
( ( groups7121269368397514597t_prod @ ( product_prod @ nat @ nat ) @ A @ ( product_case_prod @ nat @ nat @ A @ G )
@ ( collect @ ( product_prod @ nat @ nat )
@ ( product_case_prod @ nat @ nat @ $o
@ ^ [I2: nat,J2: nat] : ( ord_less @ nat @ ( plus_plus @ nat @ I2 @ J2 ) @ N ) ) ) )
= ( groups7121269368397514597t_prod @ nat @ A
@ ^ [K3: nat] :
( groups7121269368397514597t_prod @ nat @ A
@ ^ [I2: nat] : ( G @ I2 @ ( minus_minus @ nat @ K3 @ I2 ) )
@ ( set_ord_atMost @ nat @ K3 ) )
@ ( set_ord_lessThan @ nat @ N ) ) ) ) ).
% prod.triangle_reindex
thf(fact_5113_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_5114_arccos__def,axiom,
( arccos
= ( ^ [Y6: real] :
( the @ real
@ ^ [X2: real] :
( ( ord_less_eq @ real @ ( zero_zero @ real ) @ X2 )
& ( ord_less_eq @ real @ X2 @ pi )
& ( ( cos @ real @ X2 )
= Y6 ) ) ) ) ) ).
% arccos_def
thf(fact_5115_divmod__nat__def,axiom,
( divmod_nat
= ( ^ [M4: nat,N3: nat] : ( product_Pair @ nat @ nat @ ( divide_divide @ nat @ M4 @ N3 ) @ ( modulo_modulo @ nat @ M4 @ N3 ) ) ) ) ).
% divmod_nat_def
thf(fact_5116_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_5117_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_5118_arcsin__def,axiom,
( arcsin
= ( ^ [Y6: 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 )
= Y6 ) ) ) ) ) ).
% arcsin_def
thf(fact_5119_of__nat__code__if,axiom,
! [A: $tType] :
( ( semiring_1 @ A )
=> ( ( semiring_1_of_nat @ A )
= ( ^ [N3: nat] :
( if @ A
@ ( N3
= ( zero_zero @ nat ) )
@ ( zero_zero @ A )
@ ( product_case_prod @ nat @ nat @ A
@ ^ [M4: nat,Q6: nat] :
( if @ A
@ ( Q6
= ( zero_zero @ nat ) )
@ ( times_times @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ ( semiring_1_of_nat @ A @ M4 ) )
@ ( plus_plus @ A @ ( times_times @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ ( semiring_1_of_nat @ A @ M4 ) ) @ ( one_one @ A ) ) )
@ ( divmod_nat @ N3 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) ) ) ).
% of_nat_code_if
thf(fact_5120_bij__betw__nth__root__unity,axiom,
! [C2: complex,N: nat] :
( ( C2
!= ( zero_zero @ complex ) )
=> ( ( ord_less @ nat @ ( zero_zero @ nat ) @ N )
=> ( bij_betw @ complex @ complex @ ( times_times @ complex @ ( times_times @ complex @ ( real_Vector_of_real @ complex @ ( root @ N @ ( real_V7770717601297561774m_norm @ complex @ C2 ) ) ) @ ( cis @ ( divide_divide @ real @ ( arg @ C2 ) @ ( semiring_1_of_nat @ real @ N ) ) ) ) )
@ ( collect @ complex
@ ^ [Z7: complex] :
( ( power_power @ complex @ Z7 @ N )
= ( one_one @ complex ) ) )
@ ( collect @ complex
@ ^ [Z7: complex] :
( ( power_power @ complex @ Z7 @ N )
= C2 ) ) ) ) ) ).
% bij_betw_nth_root_unity
thf(fact_5121_neg__mask__is__div_H,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [N: nat,W: word @ A] :
( ( ord_less @ nat @ N @ ( size_size @ ( word @ A ) @ W ) )
=> ( ( bit_se5824344872417868541ns_and @ ( word @ A ) @ W @ ( bit_ri4277139882892585799ns_not @ ( word @ A ) @ ( bit_se2239418461657761734s_mask @ ( word @ A ) @ N ) ) )
= ( times_times @ ( word @ A ) @ ( divide_divide @ ( word @ A ) @ W @ ( power_power @ ( word @ A ) @ ( numeral_numeral @ ( word @ A ) @ ( bit0 @ one2 ) ) @ N ) ) @ ( power_power @ ( word @ A ) @ ( numeral_numeral @ ( word @ A ) @ ( bit0 @ one2 ) ) @ N ) ) ) ) ) ).
% neg_mask_is_div'
thf(fact_5122_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_5123_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_5124_real__root__zero,axiom,
! [N: nat] :
( ( root @ N @ ( zero_zero @ real ) )
= ( zero_zero @ real ) ) ).
% real_root_zero
thf(fact_5125_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_5126_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_5127_real__root__Suc__0,axiom,
! [X: real] :
( ( root @ ( suc @ ( zero_zero @ nat ) ) @ X )
= X ) ).
% real_root_Suc_0
thf(fact_5128_real__root__eq__iff,axiom,
! [N: nat,X: real,Y: real] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ N )
=> ( ( ( root @ N @ X )
= ( root @ N @ Y ) )
= ( X = Y ) ) ) ).
% real_root_eq_iff
thf(fact_5129_root__0,axiom,
! [X: real] :
( ( root @ ( zero_zero @ nat ) @ X )
= ( zero_zero @ real ) ) ).
% root_0
thf(fact_5130_and__and__not,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [A2: word @ A,B2: word @ A] :
( ( bit_se5824344872417868541ns_and @ ( word @ A ) @ ( bit_se5824344872417868541ns_and @ ( word @ A ) @ A2 @ B2 ) @ ( bit_ri4277139882892585799ns_not @ ( word @ A ) @ B2 ) )
= ( zero_zero @ ( word @ A ) ) ) ) ).
% and_and_not
thf(fact_5131_word__and__not,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [X: word @ A] :
( ( bit_se5824344872417868541ns_and @ ( word @ A ) @ X @ ( bit_ri4277139882892585799ns_not @ ( word @ A ) @ X ) )
= ( zero_zero @ ( word @ A ) ) ) ) ).
% word_and_not
thf(fact_5132_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_5133_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_5134_real__root__eq__0__iff,axiom,
! [N: nat,X: real] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ N )
=> ( ( ( root @ N @ X )
= ( zero_zero @ real ) )
= ( X
= ( zero_zero @ real ) ) ) ) ).
% real_root_eq_0_iff
thf(fact_5135_real__root__less__iff,axiom,
! [N: nat,X: real,Y: real] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ N )
=> ( ( ord_less @ real @ ( root @ N @ X ) @ ( root @ N @ Y ) )
= ( ord_less @ real @ X @ Y ) ) ) ).
% real_root_less_iff
thf(fact_5136_NOT__mask__AND__mask,axiom,
! [A: $tType] :
( ( bit_ri3973907225187159222ations @ A )
=> ! [W: A,N: nat] :
( ( bit_se5824344872417868541ns_and @ A @ ( bit_se5824344872417868541ns_and @ A @ W @ ( bit_se2239418461657761734s_mask @ A @ N ) ) @ ( bit_ri4277139882892585799ns_not @ A @ ( bit_se2239418461657761734s_mask @ A @ N ) ) )
= ( zero_zero @ A ) ) ) ).
% NOT_mask_AND_mask
thf(fact_5137_real__root__le__iff,axiom,
! [N: nat,X: real,Y: real] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ N )
=> ( ( ord_less_eq @ real @ ( root @ N @ X ) @ ( root @ N @ Y ) )
= ( ord_less_eq @ real @ X @ Y ) ) ) ).
% real_root_le_iff
thf(fact_5138_real__root__one,axiom,
! [N: nat] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ N )
=> ( ( root @ N @ ( one_one @ real ) )
= ( one_one @ real ) ) ) ).
% real_root_one
thf(fact_5139_real__root__eq__1__iff,axiom,
! [N: nat,X: real] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ N )
=> ( ( ( root @ N @ X )
= ( one_one @ real ) )
= ( X
= ( one_one @ real ) ) ) ) ).
% real_root_eq_1_iff
thf(fact_5140_minus__not__numeral__eq,axiom,
! [A: $tType] :
( ( bit_ri3973907225187159222ations @ A )
=> ! [N: num] :
( ( uminus_uminus @ A @ ( bit_ri4277139882892585799ns_not @ A @ ( numeral_numeral @ A @ N ) ) )
= ( numeral_numeral @ A @ ( inc @ N ) ) ) ) ).
% minus_not_numeral_eq
thf(fact_5141_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_5142_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_5143_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_5144_real__root__gt__0__iff,axiom,
! [N: nat,Y: real] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ N )
=> ( ( ord_less @ real @ ( zero_zero @ real ) @ ( root @ N @ Y ) )
= ( ord_less @ real @ ( zero_zero @ real ) @ Y ) ) ) ).
% real_root_gt_0_iff
thf(fact_5145_real__root__lt__0__iff,axiom,
! [N: nat,X: real] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ N )
=> ( ( ord_less @ real @ ( root @ N @ X ) @ ( zero_zero @ real ) )
= ( ord_less @ real @ X @ ( zero_zero @ real ) ) ) ) ).
% real_root_lt_0_iff
thf(fact_5146_real__root__ge__0__iff,axiom,
! [N: nat,Y: real] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ N )
=> ( ( ord_less_eq @ real @ ( zero_zero @ real ) @ ( root @ N @ Y ) )
= ( ord_less_eq @ real @ ( zero_zero @ real ) @ Y ) ) ) ).
% real_root_ge_0_iff
thf(fact_5147_real__root__le__0__iff,axiom,
! [N: nat,X: real] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ N )
=> ( ( ord_less_eq @ real @ ( root @ N @ X ) @ ( zero_zero @ real ) )
= ( ord_less_eq @ real @ X @ ( zero_zero @ real ) ) ) ) ).
% real_root_le_0_iff
thf(fact_5148_real__root__gt__1__iff,axiom,
! [N: nat,Y: real] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ N )
=> ( ( ord_less @ real @ ( one_one @ real ) @ ( root @ N @ Y ) )
= ( ord_less @ real @ ( one_one @ real ) @ Y ) ) ) ).
% real_root_gt_1_iff
thf(fact_5149_real__root__lt__1__iff,axiom,
! [N: nat,X: real] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ N )
=> ( ( ord_less @ real @ ( root @ N @ X ) @ ( one_one @ real ) )
= ( ord_less @ real @ X @ ( one_one @ real ) ) ) ) ).
% real_root_lt_1_iff
thf(fact_5150_real__root__ge__1__iff,axiom,
! [N: nat,Y: real] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ N )
=> ( ( ord_less_eq @ real @ ( one_one @ real ) @ ( root @ N @ Y ) )
= ( ord_less_eq @ real @ ( one_one @ real ) @ Y ) ) ) ).
% real_root_ge_1_iff
thf(fact_5151_real__root__le__1__iff,axiom,
! [N: nat,X: real] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ N )
=> ( ( ord_less_eq @ real @ ( root @ N @ X ) @ ( one_one @ real ) )
= ( ord_less_eq @ real @ X @ ( one_one @ real ) ) ) ) ).
% real_root_le_1_iff
thf(fact_5152_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_5153_real__root__pow__pos2,axiom,
! [N: nat,X: real] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ N )
=> ( ( ord_less_eq @ real @ ( zero_zero @ real ) @ X )
=> ( ( power_power @ real @ ( root @ N @ X ) @ N )
= X ) ) ) ).
% real_root_pow_pos2
thf(fact_5154_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_5155_real__root__commute,axiom,
! [M: nat,N: nat,X: real] :
( ( root @ M @ ( root @ N @ X ) )
= ( root @ N @ ( root @ M @ X ) ) ) ).
% real_root_commute
thf(fact_5156_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_5157_take__bit__not__take__bit,axiom,
! [A: $tType] :
( ( bit_ri3973907225187159222ations @ A )
=> ! [N: nat,A2: A] :
( ( bit_se2584673776208193580ke_bit @ A @ N @ ( bit_ri4277139882892585799ns_not @ A @ ( bit_se2584673776208193580ke_bit @ A @ N @ A2 ) ) )
= ( bit_se2584673776208193580ke_bit @ A @ N @ ( bit_ri4277139882892585799ns_not @ A @ A2 ) ) ) ) ).
% take_bit_not_take_bit
thf(fact_5158_take__bit__not__iff,axiom,
! [A: $tType] :
( ( bit_ri3973907225187159222ations @ A )
=> ! [N: nat,A2: A,B2: A] :
( ( ( bit_se2584673776208193580ke_bit @ A @ N @ ( bit_ri4277139882892585799ns_not @ A @ A2 ) )
= ( bit_se2584673776208193580ke_bit @ A @ N @ ( bit_ri4277139882892585799ns_not @ A @ B2 ) ) )
= ( ( bit_se2584673776208193580ke_bit @ A @ N @ A2 )
= ( bit_se2584673776208193580ke_bit @ A @ N @ B2 ) ) ) ) ).
% take_bit_not_iff
thf(fact_5159_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_5160_real__root__mult__exp,axiom,
! [M: nat,N: nat,X: real] :
( ( root @ ( times_times @ nat @ M @ N ) @ X )
= ( root @ M @ ( root @ N @ X ) ) ) ).
% real_root_mult_exp
thf(fact_5161_real__root__mult,axiom,
! [N: nat,X: real,Y: real] :
( ( root @ N @ ( times_times @ real @ X @ Y ) )
= ( times_times @ real @ ( root @ N @ X ) @ ( root @ N @ Y ) ) ) ).
% real_root_mult
thf(fact_5162_real__root__minus,axiom,
! [N: nat,X: real] :
( ( root @ N @ ( uminus_uminus @ real @ X ) )
= ( uminus_uminus @ real @ ( root @ N @ X ) ) ) ).
% real_root_minus
thf(fact_5163_real__root__inverse,axiom,
! [N: nat,X: real] :
( ( root @ N @ ( inverse_inverse @ real @ X ) )
= ( inverse_inverse @ real @ ( root @ N @ X ) ) ) ).
% real_root_inverse
thf(fact_5164_real__root__divide,axiom,
! [N: nat,X: real,Y: real] :
( ( root @ N @ ( divide_divide @ real @ X @ Y ) )
= ( divide_divide @ real @ ( root @ N @ X ) @ ( root @ N @ Y ) ) ) ).
% real_root_divide
thf(fact_5165_real__root__pos__pos__le,axiom,
! [X: real,N: nat] :
( ( ord_less_eq @ real @ ( zero_zero @ real ) @ X )
=> ( ord_less_eq @ real @ ( zero_zero @ real ) @ ( root @ N @ X ) ) ) ).
% real_root_pos_pos_le
thf(fact_5166_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_5167_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_5168_mask__eq__0__eq__x,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [X: word @ A,W: word @ A] :
( ( ( bit_se5824344872417868541ns_and @ ( word @ A ) @ X @ W )
= ( zero_zero @ ( word @ A ) ) )
= ( ( bit_se5824344872417868541ns_and @ ( word @ A ) @ X @ ( bit_ri4277139882892585799ns_not @ ( word @ A ) @ W ) )
= X ) ) ) ).
% mask_eq_0_eq_x
thf(fact_5169_mask__eq__x__eq__0,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [X: word @ A,W: word @ A] :
( ( ( bit_se5824344872417868541ns_and @ ( word @ A ) @ X @ W )
= X )
= ( ( bit_se5824344872417868541ns_and @ ( word @ A ) @ X @ ( bit_ri4277139882892585799ns_not @ ( word @ A ) @ W ) )
= ( zero_zero @ ( word @ A ) ) ) ) ) ).
% mask_eq_x_eq_0
thf(fact_5170_real__root__less__mono,axiom,
! [N: nat,X: real,Y: real] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ N )
=> ( ( ord_less @ real @ X @ Y )
=> ( ord_less @ real @ ( root @ N @ X ) @ ( root @ N @ Y ) ) ) ) ).
% real_root_less_mono
thf(fact_5171_real__root__le__mono,axiom,
! [N: nat,X: real,Y: real] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ N )
=> ( ( ord_less_eq @ real @ X @ Y )
=> ( ord_less_eq @ real @ ( root @ N @ X ) @ ( root @ N @ Y ) ) ) ) ).
% real_root_le_mono
thf(fact_5172_real__root__power,axiom,
! [N: nat,X: real,K: nat] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ N )
=> ( ( root @ N @ ( power_power @ real @ X @ K ) )
= ( power_power @ real @ ( root @ N @ X ) @ K ) ) ) ).
% real_root_power
thf(fact_5173_minus__eq__not__plus__1,axiom,
! [A: $tType] :
( ( bit_ri3973907225187159222ations @ A )
=> ( ( uminus_uminus @ A )
= ( ^ [A4: A] : ( plus_plus @ A @ ( bit_ri4277139882892585799ns_not @ A @ A4 ) @ ( one_one @ A ) ) ) ) ) ).
% minus_eq_not_plus_1
thf(fact_5174_minus__eq__not__minus__1,axiom,
! [A: $tType] :
( ( bit_ri3973907225187159222ations @ A )
=> ( ( uminus_uminus @ A )
= ( ^ [A4: A] : ( bit_ri4277139882892585799ns_not @ A @ ( minus_minus @ A @ A4 @ ( one_one @ A ) ) ) ) ) ) ).
% minus_eq_not_minus_1
thf(fact_5175_not__eq__complement,axiom,
! [A: $tType] :
( ( bit_ri3973907225187159222ations @ A )
=> ( ( bit_ri4277139882892585799ns_not @ A )
= ( ^ [A4: A] : ( minus_minus @ A @ ( uminus_uminus @ A @ A4 ) @ ( one_one @ A ) ) ) ) ) ).
% not_eq_complement
thf(fact_5176_real__root__abs,axiom,
! [N: nat,X: real] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ N )
=> ( ( root @ N @ ( abs_abs @ real @ X ) )
= ( abs_abs @ real @ ( root @ N @ X ) ) ) ) ).
% real_root_abs
thf(fact_5177_take__bit__not__eq__mask__diff,axiom,
! [A: $tType] :
( ( bit_ri3973907225187159222ations @ A )
=> ! [N: nat,A2: A] :
( ( bit_se2584673776208193580ke_bit @ A @ N @ ( bit_ri4277139882892585799ns_not @ A @ A2 ) )
= ( minus_minus @ A @ ( bit_se2239418461657761734s_mask @ A @ N ) @ ( bit_se2584673776208193580ke_bit @ A @ N @ A2 ) ) ) ) ).
% take_bit_not_eq_mask_diff
thf(fact_5178_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_5179_mask__AND__NOT__mask,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [W: word @ A,N: nat] :
( ( bit_se5824344872417868541ns_and @ ( word @ A ) @ ( bit_se5824344872417868541ns_and @ ( word @ A ) @ W @ ( bit_ri4277139882892585799ns_not @ ( word @ A ) @ ( bit_se2239418461657761734s_mask @ ( word @ A ) @ N ) ) ) @ ( bit_se2239418461657761734s_mask @ ( word @ A ) @ N ) )
= ( zero_zero @ ( word @ A ) ) ) ) ).
% mask_AND_NOT_mask
thf(fact_5180_sgn__root,axiom,
! [N: nat,X: real] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ N )
=> ( ( sgn_sgn @ real @ ( root @ N @ X ) )
= ( sgn_sgn @ real @ X ) ) ) ).
% sgn_root
thf(fact_5181_minus__numeral__inc__eq,axiom,
! [A: $tType] :
( ( bit_ri3973907225187159222ations @ A )
=> ! [N: num] :
( ( uminus_uminus @ A @ ( numeral_numeral @ A @ ( inc @ N ) ) )
= ( bit_ri4277139882892585799ns_not @ A @ ( numeral_numeral @ A @ N ) ) ) ) ).
% minus_numeral_inc_eq
thf(fact_5182_real__root__gt__zero,axiom,
! [N: nat,X: real] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ N )
=> ( ( ord_less @ real @ ( zero_zero @ real ) @ X )
=> ( ord_less @ real @ ( zero_zero @ real ) @ ( root @ N @ X ) ) ) ) ).
% real_root_gt_zero
thf(fact_5183_real__root__strict__decreasing,axiom,
! [N: nat,N4: nat,X: real] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ N )
=> ( ( ord_less @ nat @ N @ N4 )
=> ( ( ord_less @ real @ ( one_one @ real ) @ X )
=> ( ord_less @ real @ ( root @ N4 @ X ) @ ( root @ N @ X ) ) ) ) ) ).
% real_root_strict_decreasing
thf(fact_5184_sqrt__def,axiom,
( sqrt
= ( root @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ).
% sqrt_def
thf(fact_5185_not__numeral__Bit0__eq,axiom,
! [A: $tType] :
( ( bit_ri3973907225187159222ations @ A )
=> ! [N: num] :
( ( bit_ri4277139882892585799ns_not @ A @ ( numeral_numeral @ A @ ( bit0 @ N ) ) )
= ( uminus_uminus @ A @ ( numeral_numeral @ A @ ( bit1 @ N ) ) ) ) ) ).
% not_numeral_Bit0_eq
thf(fact_5186_root__abs__power,axiom,
! [N: nat,Y: real] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ N )
=> ( ( abs_abs @ real @ ( root @ N @ ( power_power @ real @ Y @ N ) ) )
= ( abs_abs @ real @ Y ) ) ) ).
% root_abs_power
thf(fact_5187_take__bit__not__mask__eq__0,axiom,
! [A: $tType] :
( ( bit_ri3973907225187159222ations @ A )
=> ! [M: nat,N: nat] :
( ( ord_less_eq @ nat @ M @ N )
=> ( ( bit_se2584673776208193580ke_bit @ A @ M @ ( bit_ri4277139882892585799ns_not @ A @ ( bit_se2239418461657761734s_mask @ A @ N ) ) )
= ( zero_zero @ A ) ) ) ) ).
% take_bit_not_mask_eq_0
thf(fact_5188_not__numeral__BitM__eq,axiom,
! [A: $tType] :
( ( bit_ri3973907225187159222ations @ A )
=> ! [N: num] :
( ( bit_ri4277139882892585799ns_not @ A @ ( numeral_numeral @ A @ ( bitM @ N ) ) )
= ( uminus_uminus @ A @ ( numeral_numeral @ A @ ( bit0 @ N ) ) ) ) ) ).
% not_numeral_BitM_eq
thf(fact_5189_and__mask__0__iff__le__mask,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [W: word @ A,N: nat] :
( ( ( bit_se5824344872417868541ns_and @ ( word @ A ) @ W @ ( bit_ri4277139882892585799ns_not @ ( word @ A ) @ ( bit_se2239418461657761734s_mask @ ( word @ A ) @ N ) ) )
= ( zero_zero @ ( word @ A ) ) )
= ( ord_less_eq @ ( word @ A ) @ W @ ( bit_se2239418461657761734s_mask @ ( word @ A ) @ N ) ) ) ) ).
% and_mask_0_iff_le_mask
thf(fact_5190_real__root__pos__pos,axiom,
! [N: nat,X: real] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ N )
=> ( ( ord_less @ real @ ( zero_zero @ real ) @ X )
=> ( ord_less_eq @ real @ ( zero_zero @ real ) @ ( root @ N @ X ) ) ) ) ).
% real_root_pos_pos
thf(fact_5191_real__root__strict__increasing,axiom,
! [N: nat,N4: nat,X: real] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ N )
=> ( ( ord_less @ nat @ N @ N4 )
=> ( ( ord_less @ real @ ( zero_zero @ real ) @ X )
=> ( ( ord_less @ real @ X @ ( one_one @ real ) )
=> ( ord_less @ real @ ( root @ N @ X ) @ ( root @ N4 @ X ) ) ) ) ) ) ).
% real_root_strict_increasing
thf(fact_5192_real__root__decreasing,axiom,
! [N: nat,N4: nat,X: real] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ N )
=> ( ( ord_less_eq @ nat @ N @ N4 )
=> ( ( ord_less_eq @ real @ ( one_one @ real ) @ X )
=> ( ord_less_eq @ real @ ( root @ N4 @ X ) @ ( root @ N @ X ) ) ) ) ) ).
% real_root_decreasing
thf(fact_5193_odd__real__root__pow,axiom,
! [N: nat,X: real] :
( ~ ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N )
=> ( ( power_power @ real @ ( root @ N @ X ) @ N )
= X ) ) ).
% odd_real_root_pow
thf(fact_5194_odd__real__root__unique,axiom,
! [N: nat,Y: real,X: real] :
( ~ ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N )
=> ( ( ( power_power @ real @ Y @ N )
= X )
=> ( ( root @ N @ X )
= Y ) ) ) ).
% odd_real_root_unique
thf(fact_5195_odd__real__root__power__cancel,axiom,
! [N: nat,X: real] :
( ~ ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N )
=> ( ( root @ N @ ( power_power @ real @ X @ N ) )
= X ) ) ).
% odd_real_root_power_cancel
thf(fact_5196_real__root__pow__pos,axiom,
! [N: nat,X: real] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ N )
=> ( ( ord_less @ real @ ( zero_zero @ real ) @ X )
=> ( ( power_power @ real @ ( root @ N @ X ) @ N )
= X ) ) ) ).
% real_root_pow_pos
thf(fact_5197_real__root__power__cancel,axiom,
! [N: nat,X: real] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ N )
=> ( ( ord_less_eq @ real @ ( zero_zero @ real ) @ X )
=> ( ( root @ N @ ( power_power @ real @ X @ N ) )
= X ) ) ) ).
% real_root_power_cancel
thf(fact_5198_real__root__pos__unique,axiom,
! [N: nat,Y: real,X: real] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ N )
=> ( ( ord_less_eq @ real @ ( zero_zero @ real ) @ Y )
=> ( ( ( power_power @ real @ Y @ N )
= X )
=> ( ( root @ N @ X )
= Y ) ) ) ) ).
% real_root_pos_unique
thf(fact_5199_real__root__increasing,axiom,
! [N: nat,N4: nat,X: real] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ N )
=> ( ( ord_less_eq @ nat @ N @ N4 )
=> ( ( ord_less_eq @ real @ ( zero_zero @ real ) @ X )
=> ( ( ord_less_eq @ real @ X @ ( one_one @ real ) )
=> ( ord_less_eq @ real @ ( root @ N @ X ) @ ( root @ N4 @ X ) ) ) ) ) ) ).
% real_root_increasing
thf(fact_5200_minus__exp__eq__not__mask,axiom,
! [A: $tType] :
( ( bit_ri3973907225187159222ations @ A )
=> ! [N: nat] :
( ( uminus_uminus @ A @ ( power_power @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ N ) )
= ( bit_ri4277139882892585799ns_not @ A @ ( bit_se2239418461657761734s_mask @ A @ N ) ) ) ) ).
% minus_exp_eq_not_mask
thf(fact_5201_NOT__mask,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [N: nat] :
( ( bit_ri4277139882892585799ns_not @ ( word @ A ) @ ( bit_se2239418461657761734s_mask @ ( word @ A ) @ N ) )
= ( uminus_uminus @ ( word @ A ) @ ( power_power @ ( word @ A ) @ ( numeral_numeral @ ( word @ A ) @ ( bit0 @ one2 ) ) @ N ) ) ) ) ).
% NOT_mask
thf(fact_5202_root__sgn__power,axiom,
! [N: nat,Y: real] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ N )
=> ( ( root @ N @ ( times_times @ real @ ( sgn_sgn @ real @ Y ) @ ( power_power @ real @ ( abs_abs @ real @ Y ) @ N ) ) )
= Y ) ) ).
% root_sgn_power
thf(fact_5203_sgn__power__root,axiom,
! [N: nat,X: real] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ N )
=> ( ( times_times @ real @ ( sgn_sgn @ real @ ( root @ N @ X ) ) @ ( power_power @ real @ ( abs_abs @ real @ ( root @ N @ X ) ) @ N ) )
= X ) ) ).
% sgn_power_root
thf(fact_5204_log__root,axiom,
! [N: nat,A2: real,B2: real] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ N )
=> ( ( ord_less @ real @ ( zero_zero @ real ) @ A2 )
=> ( ( log @ B2 @ ( root @ N @ A2 ) )
= ( divide_divide @ real @ ( log @ B2 @ A2 ) @ ( semiring_1_of_nat @ real @ N ) ) ) ) ) ).
% log_root
thf(fact_5205_log__base__root,axiom,
! [N: nat,B2: real,X: real] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ N )
=> ( ( ord_less @ real @ ( zero_zero @ real ) @ B2 )
=> ( ( log @ ( root @ N @ B2 ) @ X )
= ( times_times @ real @ ( semiring_1_of_nat @ real @ N ) @ ( log @ B2 @ X ) ) ) ) ) ).
% log_base_root
thf(fact_5206_ln__root,axiom,
! [N: nat,B2: real] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ N )
=> ( ( ord_less @ real @ ( zero_zero @ real ) @ B2 )
=> ( ( ln_ln @ real @ ( root @ N @ B2 ) )
= ( divide_divide @ real @ ( ln_ln @ real @ B2 ) @ ( semiring_1_of_nat @ real @ N ) ) ) ) ) ).
% ln_root
thf(fact_5207_split__root,axiom,
! [P: real > $o,N: nat,X: real] :
( ( P @ ( root @ N @ X ) )
= ( ( ( N
= ( zero_zero @ nat ) )
=> ( P @ ( zero_zero @ real ) ) )
& ( ( ord_less @ nat @ ( zero_zero @ nat ) @ N )
=> ! [Y6: real] :
( ( ( times_times @ real @ ( sgn_sgn @ real @ Y6 ) @ ( power_power @ real @ ( abs_abs @ real @ Y6 ) @ N ) )
= X )
=> ( P @ Y6 ) ) ) ) ) ).
% split_root
thf(fact_5208_root__powr__inverse,axiom,
! [N: nat,X: real] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ N )
=> ( ( ord_less @ real @ ( zero_zero @ real ) @ X )
=> ( ( root @ N @ X )
= ( powr @ real @ X @ ( divide_divide @ real @ ( one_one @ real ) @ ( semiring_1_of_nat @ real @ N ) ) ) ) ) ) ).
% root_powr_inverse
thf(fact_5209_neg__mask__is__div,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [W: word @ A,N: nat] :
( ( bit_se5824344872417868541ns_and @ ( word @ A ) @ W @ ( bit_ri4277139882892585799ns_not @ ( word @ A ) @ ( bit_se2239418461657761734s_mask @ ( word @ A ) @ N ) ) )
= ( times_times @ ( word @ A ) @ ( divide_divide @ ( word @ A ) @ W @ ( power_power @ ( word @ A ) @ ( numeral_numeral @ ( word @ A ) @ ( bit0 @ one2 ) ) @ N ) ) @ ( power_power @ ( word @ A ) @ ( numeral_numeral @ ( word @ A ) @ ( bit0 @ one2 ) ) @ N ) ) ) ) ).
% neg_mask_is_div
thf(fact_5210_signed__take__bit__eq__take__bit__minus,axiom,
( ( bit_ri4674362597316999326ke_bit @ int )
= ( ^ [N3: nat,K3: int] : ( minus_minus @ int @ ( bit_se2584673776208193580ke_bit @ int @ ( suc @ N3 ) @ K3 ) @ ( times_times @ int @ ( power_power @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ ( suc @ N3 ) ) @ ( zero_neq_one_of_bool @ int @ ( bit_se5641148757651400278ts_bit @ int @ K3 @ N3 ) ) ) ) ) ) ).
% signed_take_bit_eq_take_bit_minus
thf(fact_5211_neg__mask__add__mask,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [X: word @ A,N: nat] :
( ( plus_plus @ ( word @ A ) @ ( bit_se5824344872417868541ns_and @ ( word @ A ) @ X @ ( bit_ri4277139882892585799ns_not @ ( word @ A ) @ ( bit_se2239418461657761734s_mask @ ( word @ A ) @ N ) ) ) @ ( minus_minus @ ( word @ A ) @ ( power_power @ ( word @ A ) @ ( numeral_numeral @ ( word @ A ) @ ( bit0 @ one2 ) ) @ N ) @ ( one_one @ ( word @ A ) ) ) )
= ( bit_se1065995026697491101ons_or @ ( word @ A ) @ X @ ( bit_se2239418461657761734s_mask @ ( word @ A ) @ N ) ) ) ) ).
% neg_mask_add_mask
thf(fact_5212_new__rule,axiom,
! [A: $tType] :
( ( heap @ A )
=> ! [N: nat,X: A] :
( hoare_hoare_triple @ ( array @ A ) @ ( one_one @ assn ) @ ( array_new @ A @ N @ X )
@ ^ [R5: array @ A] : ( snga_assn @ A @ R5 @ ( replicate @ A @ N @ X ) ) ) ) ).
% new_rule
thf(fact_5213_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_5214_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_5215_or_Oidem,axiom,
! [A: $tType] :
( ( bit_se359711467146920520ations @ A )
=> ! [A2: A] :
( ( bit_se1065995026697491101ons_or @ A @ A2 @ A2 )
= A2 ) ) ).
% or.idem
thf(fact_5216_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_5217_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_5218_take__bit__or,axiom,
! [A: $tType] :
( ( bit_se359711467146920520ations @ A )
=> ! [N: nat,A2: A,B2: A] :
( ( bit_se2584673776208193580ke_bit @ A @ N @ ( bit_se1065995026697491101ons_or @ A @ A2 @ B2 ) )
= ( bit_se1065995026697491101ons_or @ A @ ( bit_se2584673776208193580ke_bit @ A @ N @ A2 ) @ ( bit_se2584673776208193580ke_bit @ A @ N @ B2 ) ) ) ) ).
% take_bit_or
thf(fact_5219_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_5220_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_5221_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_5222_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_5223_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_5224_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_5225_bit__numeral__Bit0__Suc__iff,axiom,
! [A: $tType] :
( ( bit_un5681908812861735899ations @ A )
=> ! [M: num,N: nat] :
( ( bit_se5641148757651400278ts_bit @ A @ ( numeral_numeral @ A @ ( bit0 @ M ) ) @ ( suc @ N ) )
= ( bit_se5641148757651400278ts_bit @ A @ ( numeral_numeral @ A @ M ) @ N ) ) ) ).
% bit_numeral_Bit0_Suc_iff
thf(fact_5226_bit__numeral__Bit1__Suc__iff,axiom,
! [A: $tType] :
( ( bit_un5681908812861735899ations @ A )
=> ! [M: num,N: nat] :
( ( bit_se5641148757651400278ts_bit @ A @ ( numeral_numeral @ A @ ( bit1 @ M ) ) @ ( suc @ N ) )
= ( bit_se5641148757651400278ts_bit @ A @ ( numeral_numeral @ A @ M ) @ N ) ) ) ).
% bit_numeral_Bit1_Suc_iff
thf(fact_5227_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_5228_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_5229_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_5230_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_5231_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_5232_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_5233_signed__take__bit__nonnegative__iff,axiom,
! [N: nat,K: int] :
( ( ord_less_eq @ int @ ( zero_zero @ int ) @ ( bit_ri4674362597316999326ke_bit @ int @ N @ K ) )
= ( ~ ( bit_se5641148757651400278ts_bit @ int @ K @ N ) ) ) ).
% signed_take_bit_nonnegative_iff
thf(fact_5234_signed__take__bit__negative__iff,axiom,
! [N: nat,K: int] :
( ( ord_less @ int @ ( bit_ri4674362597316999326ke_bit @ int @ N @ K ) @ ( zero_zero @ int ) )
= ( bit_se5641148757651400278ts_bit @ int @ K @ N ) ) ).
% signed_take_bit_negative_iff
thf(fact_5235_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_5236_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_5237_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_5238_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_5239_bit__minus__numeral__Bit0__Suc__iff,axiom,
! [W: num,N: nat] :
( ( bit_se5641148757651400278ts_bit @ int @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ ( bit0 @ W ) ) ) @ ( suc @ N ) )
= ( bit_se5641148757651400278ts_bit @ int @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ W ) ) @ N ) ) ).
% bit_minus_numeral_Bit0_Suc_iff
thf(fact_5240_bit__numeral__simps_I2_J,axiom,
! [A: $tType] :
( ( bit_un5681908812861735899ations @ A )
=> ! [W: num,N: num] :
( ( bit_se5641148757651400278ts_bit @ A @ ( numeral_numeral @ A @ ( bit0 @ W ) ) @ ( numeral_numeral @ nat @ N ) )
= ( bit_se5641148757651400278ts_bit @ A @ ( numeral_numeral @ A @ W ) @ ( pred_numeral @ N ) ) ) ) ).
% bit_numeral_simps(2)
thf(fact_5241_bit__minus__numeral__Bit1__Suc__iff,axiom,
! [W: num,N: nat] :
( ( bit_se5641148757651400278ts_bit @ int @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ ( bit1 @ W ) ) ) @ ( suc @ N ) )
= ( ~ ( bit_se5641148757651400278ts_bit @ int @ ( numeral_numeral @ int @ W ) @ N ) ) ) ).
% bit_minus_numeral_Bit1_Suc_iff
thf(fact_5242_bit__numeral__simps_I3_J,axiom,
! [A: $tType] :
( ( bit_un5681908812861735899ations @ A )
=> ! [W: num,N: num] :
( ( bit_se5641148757651400278ts_bit @ A @ ( numeral_numeral @ A @ ( bit1 @ W ) ) @ ( numeral_numeral @ nat @ N ) )
= ( bit_se5641148757651400278ts_bit @ A @ ( numeral_numeral @ A @ W ) @ ( pred_numeral @ N ) ) ) ) ).
% bit_numeral_simps(3)
thf(fact_5243_word__no__log__defs_I1_J,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [A2: num] :
( ( bit_ri4277139882892585799ns_not @ ( word @ A ) @ ( numeral_numeral @ ( word @ A ) @ A2 ) )
= ( ring_1_of_int @ ( word @ A ) @ ( bit_ri4277139882892585799ns_not @ int @ ( numeral_numeral @ int @ A2 ) ) ) ) ) ).
% word_no_log_defs(1)
thf(fact_5244_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_5245_bit__minus__numeral__int_I1_J,axiom,
! [W: num,N: num] :
( ( bit_se5641148757651400278ts_bit @ int @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ ( bit0 @ W ) ) ) @ ( numeral_numeral @ nat @ N ) )
= ( bit_se5641148757651400278ts_bit @ int @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ W ) ) @ ( pred_numeral @ N ) ) ) ).
% bit_minus_numeral_int(1)
thf(fact_5246_bit__minus__numeral__int_I2_J,axiom,
! [W: num,N: num] :
( ( bit_se5641148757651400278ts_bit @ int @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ ( bit1 @ W ) ) ) @ ( numeral_numeral @ nat @ N ) )
= ( ~ ( bit_se5641148757651400278ts_bit @ int @ ( numeral_numeral @ int @ W ) @ ( pred_numeral @ N ) ) ) ) ).
% bit_minus_numeral_int(2)
thf(fact_5247_bit__mod__2__iff,axiom,
! [A: $tType] :
( ( bit_semiring_bits @ A )
=> ! [A2: A,N: nat] :
( ( bit_se5641148757651400278ts_bit @ A @ ( modulo_modulo @ A @ A2 @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) @ N )
= ( ( N
= ( zero_zero @ nat ) )
& ~ ( dvd_dvd @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ A2 ) ) ) ) ).
% bit_mod_2_iff
thf(fact_5248_bin__nth__minus__Bit0,axiom,
! [N: nat,W: num] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ N )
=> ( ( bit_se5641148757651400278ts_bit @ int @ ( numeral_numeral @ int @ ( bit0 @ W ) ) @ N )
= ( bit_se5641148757651400278ts_bit @ int @ ( numeral_numeral @ int @ W ) @ ( minus_minus @ nat @ N @ ( one_one @ nat ) ) ) ) ) ).
% bin_nth_minus_Bit0
thf(fact_5249_bin__nth__minus__Bit1,axiom,
! [N: nat,W: num] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ N )
=> ( ( bit_se5641148757651400278ts_bit @ int @ ( numeral_numeral @ int @ ( bit1 @ W ) ) @ N )
= ( bit_se5641148757651400278ts_bit @ int @ ( numeral_numeral @ int @ W ) @ ( minus_minus @ nat @ N @ ( one_one @ nat ) ) ) ) ) ).
% bin_nth_minus_Bit1
thf(fact_5250_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_5251_word__no__log__defs_I2_J,axiom,
! [B: $tType] :
( ( type_len @ B )
=> ! [A2: num] :
( ( bit_ri4277139882892585799ns_not @ ( word @ B ) @ ( uminus_uminus @ ( word @ B ) @ ( numeral_numeral @ ( word @ B ) @ A2 ) ) )
= ( ring_1_of_int @ ( word @ B ) @ ( bit_ri4277139882892585799ns_not @ int @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ A2 ) ) ) ) ) ) ).
% word_no_log_defs(2)
thf(fact_5252_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_5253_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_5254_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_5255_signed__take__bit__eq__if__negative,axiom,
! [A: $tType] :
( ( bit_ri3973907225187159222ations @ A )
=> ! [A2: A,N: nat] :
( ( bit_se5641148757651400278ts_bit @ A @ A2 @ N )
=> ( ( bit_ri4674362597316999326ke_bit @ A @ N @ A2 )
= ( bit_se1065995026697491101ons_or @ A @ ( bit_se2584673776208193580ke_bit @ A @ N @ A2 ) @ ( bit_ri4277139882892585799ns_not @ A @ ( bit_se2239418461657761734s_mask @ A @ N ) ) ) ) ) ) ).
% signed_take_bit_eq_if_negative
thf(fact_5256_bit__not__int__iff,axiom,
! [K: int,N: nat] :
( ( bit_se5641148757651400278ts_bit @ int @ ( bit_ri4277139882892585799ns_not @ int @ K ) @ N )
= ( ~ ( bit_se5641148757651400278ts_bit @ int @ K @ N ) ) ) ).
% bit_not_int_iff
thf(fact_5257_bit__minus__int__iff,axiom,
! [K: int,N: nat] :
( ( bit_se5641148757651400278ts_bit @ int @ ( uminus_uminus @ int @ K ) @ N )
= ( bit_se5641148757651400278ts_bit @ int @ ( bit_ri4277139882892585799ns_not @ int @ ( minus_minus @ int @ K @ ( one_one @ int ) ) ) @ N ) ) ).
% bit_minus_int_iff
thf(fact_5258_bit__numeral__simps_I1_J,axiom,
! [A: $tType] :
( ( bit_un5681908812861735899ations @ A )
=> ! [N: num] :
~ ( bit_se5641148757651400278ts_bit @ A @ ( one_one @ A ) @ ( numeral_numeral @ nat @ N ) ) ) ).
% bit_numeral_simps(1)
thf(fact_5259_not__bit__1__Suc,axiom,
! [A: $tType] :
( ( bit_se359711467146920520ations @ A )
=> ! [N: nat] :
~ ( bit_se5641148757651400278ts_bit @ A @ ( one_one @ A ) @ ( suc @ N ) ) ) ).
% not_bit_1_Suc
thf(fact_5260_disjunctive__add,axiom,
! [A: $tType] :
( ( bit_se359711467146920520ations @ A )
=> ! [A2: A,B2: A] :
( ! [N2: nat] :
( ~ ( bit_se5641148757651400278ts_bit @ A @ A2 @ N2 )
| ~ ( bit_se5641148757651400278ts_bit @ A @ B2 @ N2 ) )
=> ( ( plus_plus @ A @ A2 @ B2 )
= ( bit_se1065995026697491101ons_or @ A @ A2 @ B2 ) ) ) ) ).
% disjunctive_add
thf(fact_5261_bit__disjunctive__add__iff,axiom,
! [A: $tType] :
( ( bit_semiring_bits @ A )
=> ! [A2: A,B2: A,N: nat] :
( ! [N2: nat] :
( ~ ( bit_se5641148757651400278ts_bit @ A @ A2 @ N2 )
| ~ ( bit_se5641148757651400278ts_bit @ A @ B2 @ N2 ) )
=> ( ( bit_se5641148757651400278ts_bit @ A @ ( plus_plus @ A @ A2 @ B2 ) @ N )
= ( ( bit_se5641148757651400278ts_bit @ A @ A2 @ N )
| ( bit_se5641148757651400278ts_bit @ A @ B2 @ N ) ) ) ) ) ).
% bit_disjunctive_add_iff
thf(fact_5262_bit__numeral__iff,axiom,
! [A: $tType] :
( ( bit_un5681908812861735899ations @ A )
=> ! [M: num,N: nat] :
( ( bit_se5641148757651400278ts_bit @ A @ ( numeral_numeral @ A @ M ) @ N )
= ( bit_se5641148757651400278ts_bit @ nat @ ( numeral_numeral @ nat @ M ) @ N ) ) ) ).
% bit_numeral_iff
thf(fact_5263_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_5264_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_5265_word__log__esimps_I9_J,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [X: word @ A] :
( ( bit_se1065995026697491101ons_or @ ( word @ A ) @ ( zero_zero @ ( word @ A ) ) @ X )
= X ) ) ).
% word_log_esimps(9)
thf(fact_5266_word__log__esimps_I3_J,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [X: word @ A] :
( ( bit_se1065995026697491101ons_or @ ( word @ A ) @ X @ ( zero_zero @ ( word @ A ) ) )
= X ) ) ).
% word_log_esimps(3)
thf(fact_5267_word__or__zero,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [A2: word @ A,B2: word @ A] :
( ( ( bit_se1065995026697491101ons_or @ ( word @ A ) @ A2 @ B2 )
= ( zero_zero @ ( word @ A ) ) )
= ( ( A2
= ( zero_zero @ ( word @ A ) ) )
& ( B2
= ( zero_zero @ ( word @ A ) ) ) ) ) ) ).
% word_or_zero
thf(fact_5268_bit__unset__bit__iff,axiom,
! [A: $tType] :
( ( bit_se359711467146920520ations @ A )
=> ! [M: nat,A2: A,N: nat] :
( ( bit_se5641148757651400278ts_bit @ A @ ( bit_se2638667681897837118et_bit @ A @ M @ A2 ) @ N )
= ( ( bit_se5641148757651400278ts_bit @ A @ A2 @ N )
& ( M != N ) ) ) ) ).
% bit_unset_bit_iff
thf(fact_5269_of__int__or__eq,axiom,
! [A: $tType] :
( ( bit_ri3973907225187159222ations @ A )
=> ! [K: int,L: int] :
( ( ring_1_of_int @ A @ ( bit_se1065995026697491101ons_or @ int @ K @ L ) )
= ( bit_se1065995026697491101ons_or @ A @ ( ring_1_of_int @ A @ K ) @ ( ring_1_of_int @ A @ L ) ) ) ) ).
% of_int_or_eq
thf(fact_5270_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_5271_or_Ocommute,axiom,
! [A: $tType] :
( ( bit_se359711467146920520ations @ A )
=> ( ( bit_se1065995026697491101ons_or @ A )
= ( ^ [A4: A,B5: A] : ( bit_se1065995026697491101ons_or @ A @ B5 @ A4 ) ) ) ) ).
% or.commute
thf(fact_5272_bit__or__iff,axiom,
! [A: $tType] :
( ( bit_se359711467146920520ations @ A )
=> ! [A2: A,B2: A,N: nat] :
( ( bit_se5641148757651400278ts_bit @ A @ ( bit_se1065995026697491101ons_or @ A @ A2 @ B2 ) @ N )
= ( ( bit_se5641148757651400278ts_bit @ A @ A2 @ N )
| ( bit_se5641148757651400278ts_bit @ A @ B2 @ N ) ) ) ) ).
% bit_or_iff
thf(fact_5273_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_5274_bit__and__iff,axiom,
! [A: $tType] :
( ( bit_se359711467146920520ations @ A )
=> ! [A2: A,B2: A,N: nat] :
( ( bit_se5641148757651400278ts_bit @ A @ ( bit_se5824344872417868541ns_and @ A @ A2 @ B2 ) @ N )
= ( ( bit_se5641148757651400278ts_bit @ A @ A2 @ N )
& ( bit_se5641148757651400278ts_bit @ A @ B2 @ N ) ) ) ) ).
% bit_and_iff
thf(fact_5275_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_5276_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_5277_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_5278_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_5279_bit__and__int__iff,axiom,
! [K: int,L: int,N: nat] :
( ( bit_se5641148757651400278ts_bit @ int @ ( bit_se5824344872417868541ns_and @ int @ K @ L ) @ N )
= ( ( bit_se5641148757651400278ts_bit @ int @ K @ N )
& ( bit_se5641148757651400278ts_bit @ int @ L @ N ) ) ) ).
% bit_and_int_iff
thf(fact_5280_bit__of__nat__iff__bit,axiom,
! [A: $tType] :
( ( bit_un5681908812861735899ations @ A )
=> ! [M: nat,N: nat] :
( ( bit_se5641148757651400278ts_bit @ A @ ( semiring_1_of_nat @ A @ M ) @ N )
= ( bit_se5641148757651400278ts_bit @ nat @ M @ N ) ) ) ).
% bit_of_nat_iff_bit
thf(fact_5281_of__nat__or__eq,axiom,
! [A: $tType] :
( ( bit_se359711467146920520ations @ A )
=> ! [M: nat,N: nat] :
( ( semiring_1_of_nat @ A @ ( bit_se1065995026697491101ons_or @ nat @ M @ N ) )
= ( bit_se1065995026697491101ons_or @ A @ ( semiring_1_of_nat @ A @ M ) @ ( semiring_1_of_nat @ A @ N ) ) ) ) ).
% of_nat_or_eq
thf(fact_5282_bit__1__iff,axiom,
! [A: $tType] :
( ( bit_semiring_bits @ A )
=> ! [N: nat] :
( ( bit_se5641148757651400278ts_bit @ A @ ( one_one @ A ) @ N )
= ( N
= ( zero_zero @ nat ) ) ) ) ).
% bit_1_iff
thf(fact_5283_bit__take__bit__iff,axiom,
! [A: $tType] :
( ( bit_se359711467146920520ations @ A )
=> ! [M: nat,A2: A,N: nat] :
( ( bit_se5641148757651400278ts_bit @ A @ ( bit_se2584673776208193580ke_bit @ A @ M @ A2 ) @ N )
= ( ( ord_less @ nat @ N @ M )
& ( bit_se5641148757651400278ts_bit @ A @ A2 @ N ) ) ) ) ).
% bit_take_bit_iff
thf(fact_5284_bit__of__bool__iff,axiom,
! [A: $tType] :
( ( bit_semiring_bits @ A )
=> ! [B2: $o,N: nat] :
( ( bit_se5641148757651400278ts_bit @ A @ ( zero_neq_one_of_bool @ A @ B2 ) @ N )
= ( B2
& ( N
= ( zero_zero @ nat ) ) ) ) ) ).
% bit_of_bool_iff
thf(fact_5285_signed__take__bit__eq__if__positive,axiom,
! [A: $tType] :
( ( bit_ri3973907225187159222ations @ A )
=> ! [A2: A,N: nat] :
( ~ ( bit_se5641148757651400278ts_bit @ A @ A2 @ N )
=> ( ( bit_ri4674362597316999326ke_bit @ A @ N @ A2 )
= ( bit_se2584673776208193580ke_bit @ A @ N @ A2 ) ) ) ) ).
% signed_take_bit_eq_if_positive
thf(fact_5286_and__eq__not__not__or,axiom,
! [A: $tType] :
( ( bit_ri3973907225187159222ations @ A )
=> ( ( bit_se5824344872417868541ns_and @ A )
= ( ^ [A4: A,B5: A] : ( bit_ri4277139882892585799ns_not @ A @ ( bit_se1065995026697491101ons_or @ A @ ( bit_ri4277139882892585799ns_not @ A @ A4 ) @ ( bit_ri4277139882892585799ns_not @ A @ B5 ) ) ) ) ) ) ).
% and_eq_not_not_or
thf(fact_5287_or__eq__not__not__and,axiom,
! [A: $tType] :
( ( bit_ri3973907225187159222ations @ A )
=> ( ( bit_se1065995026697491101ons_or @ A )
= ( ^ [A4: A,B5: A] : ( bit_ri4277139882892585799ns_not @ A @ ( bit_se5824344872417868541ns_and @ A @ ( bit_ri4277139882892585799ns_not @ A @ A4 ) @ ( bit_ri4277139882892585799ns_not @ A @ B5 ) ) ) ) ) ) ).
% or_eq_not_not_and
thf(fact_5288_signed__take__bit__def,axiom,
! [A: $tType] :
( ( bit_ri3973907225187159222ations @ A )
=> ( ( bit_ri4674362597316999326ke_bit @ A )
= ( ^ [N3: nat,A4: A] : ( bit_se1065995026697491101ons_or @ A @ ( bit_se2584673776208193580ke_bit @ A @ N3 @ A4 ) @ ( times_times @ A @ ( zero_neq_one_of_bool @ A @ ( bit_se5641148757651400278ts_bit @ A @ A4 @ N3 ) ) @ ( bit_ri4277139882892585799ns_not @ A @ ( bit_se2239418461657761734s_mask @ A @ N3 ) ) ) ) ) ) ) ).
% signed_take_bit_def
thf(fact_5289_disjunctive__diff,axiom,
! [A: $tType] :
( ( bit_ri3973907225187159222ations @ A )
=> ! [B2: A,A2: A] :
( ! [N2: nat] :
( ( bit_se5641148757651400278ts_bit @ A @ B2 @ N2 )
=> ( bit_se5641148757651400278ts_bit @ A @ A2 @ N2 ) )
=> ( ( minus_minus @ A @ A2 @ B2 )
= ( bit_se5824344872417868541ns_and @ A @ A2 @ ( bit_ri4277139882892585799ns_not @ A @ B2 ) ) ) ) ) ).
% disjunctive_diff
thf(fact_5290_not__int__def,axiom,
( ( bit_ri4277139882892585799ns_not @ int )
= ( ^ [K3: int] : ( minus_minus @ int @ ( uminus_uminus @ int @ K3 ) @ ( one_one @ int ) ) ) ) ).
% not_int_def
thf(fact_5291_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_5292_word__plus__and__or__coroll,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [X: word @ A,Y: word @ A] :
( ( ( bit_se5824344872417868541ns_and @ ( word @ A ) @ X @ Y )
= ( zero_zero @ ( word @ A ) ) )
=> ( ( plus_plus @ ( word @ A ) @ X @ Y )
= ( bit_se1065995026697491101ons_or @ ( word @ A ) @ X @ Y ) ) ) ) ).
% word_plus_and_or_coroll
thf(fact_5293_bit__not__int__iff_H,axiom,
! [K: int,N: nat] :
( ( bit_se5641148757651400278ts_bit @ int @ ( minus_minus @ int @ ( uminus_uminus @ int @ K ) @ ( one_one @ int ) ) @ N )
= ( ~ ( bit_se5641148757651400278ts_bit @ int @ K @ N ) ) ) ).
% bit_not_int_iff'
thf(fact_5294_flip__bit__eq__if,axiom,
! [A: $tType] :
( ( bit_se359711467146920520ations @ A )
=> ( ( bit_se8732182000553998342ip_bit @ A )
= ( ^ [N3: nat,A4: A] : ( if @ ( nat > A > A ) @ ( bit_se5641148757651400278ts_bit @ A @ A4 @ N3 ) @ ( bit_se2638667681897837118et_bit @ A ) @ ( bit_se5668285175392031749et_bit @ A ) @ N3 @ A4 ) ) ) ) ).
% flip_bit_eq_if
thf(fact_5295_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_5296_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_5297_time__array__new,axiom,
! [A: $tType] :
( ( heap @ A )
=> ! [N: nat,X: A,H2: heap_ext @ product_unit] :
( ( time_time @ ( array @ A ) @ ( array_new @ A @ N @ X ) @ H2 )
= ( plus_plus @ nat @ N @ ( one_one @ nat ) ) ) ) ).
% time_array_new
thf(fact_5298_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_5299_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_5300_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_5301_and__not__numerals_I2_J,axiom,
! [N: num] :
( ( bit_se5824344872417868541ns_and @ int @ ( one_one @ int ) @ ( bit_ri4277139882892585799ns_not @ int @ ( numeral_numeral @ int @ ( bit0 @ N ) ) ) )
= ( one_one @ int ) ) ).
% and_not_numerals(2)
thf(fact_5302_bit__imp__take__bit__positive,axiom,
! [N: nat,M: nat,K: int] :
( ( ord_less @ nat @ N @ M )
=> ( ( bit_se5641148757651400278ts_bit @ int @ K @ N )
=> ( ord_less @ int @ ( zero_zero @ int ) @ ( bit_se2584673776208193580ke_bit @ int @ M @ K ) ) ) ) ).
% bit_imp_take_bit_positive
thf(fact_5303_exp__eq__0__imp__not__bit,axiom,
! [A: $tType] :
( ( bit_semiring_bits @ A )
=> ! [N: nat,A2: A] :
( ( ( power_power @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ N )
= ( zero_zero @ A ) )
=> ~ ( bit_se5641148757651400278ts_bit @ A @ A2 @ N ) ) ) ).
% exp_eq_0_imp_not_bit
thf(fact_5304_bit__Suc,axiom,
! [A: $tType] :
( ( bit_semiring_bits @ A )
=> ! [A2: A,N: nat] :
( ( bit_se5641148757651400278ts_bit @ A @ A2 @ ( suc @ N ) )
= ( bit_se5641148757651400278ts_bit @ A @ ( divide_divide @ A @ A2 @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) @ N ) ) ) ).
% bit_Suc
thf(fact_5305_stable__imp__bit__iff__odd,axiom,
! [A: $tType] :
( ( bit_semiring_bits @ A )
=> ! [A2: A,N: nat] :
( ( ( divide_divide @ A @ A2 @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) )
= A2 )
=> ( ( bit_se5641148757651400278ts_bit @ A @ A2 @ N )
= ( ~ ( dvd_dvd @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ A2 ) ) ) ) ) ).
% stable_imp_bit_iff_odd
thf(fact_5306_bit__iff__idd__imp__stable,axiom,
! [A: $tType] :
( ( bit_semiring_bits @ A )
=> ! [A2: A] :
( ! [N2: nat] :
( ( bit_se5641148757651400278ts_bit @ A @ A2 @ N2 )
= ( ~ ( dvd_dvd @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ A2 ) ) )
=> ( ( divide_divide @ A @ A2 @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) )
= A2 ) ) ) ).
% bit_iff_idd_imp_stable
thf(fact_5307_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_5308_int__bit__bound,axiom,
! [K: int] :
~ ! [N2: nat] :
( ! [M3: nat] :
( ( ord_less_eq @ nat @ N2 @ M3 )
=> ( ( bit_se5641148757651400278ts_bit @ int @ K @ M3 )
= ( bit_se5641148757651400278ts_bit @ int @ K @ N2 ) ) )
=> ~ ( ( ord_less @ nat @ ( zero_zero @ nat ) @ N2 )
=> ( ( bit_se5641148757651400278ts_bit @ int @ K @ ( minus_minus @ nat @ N2 @ ( one_one @ nat ) ) )
= ( ~ ( bit_se5641148757651400278ts_bit @ int @ K @ N2 ) ) ) ) ) ).
% int_bit_bound
thf(fact_5309_and__not__numerals_I5_J,axiom,
! [M: num,N: num] :
( ( bit_se5824344872417868541ns_and @ int @ ( numeral_numeral @ int @ ( bit0 @ M ) ) @ ( bit_ri4277139882892585799ns_not @ int @ ( numeral_numeral @ int @ ( bit0 @ N ) ) ) )
= ( times_times @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ ( bit_se5824344872417868541ns_and @ int @ ( numeral_numeral @ int @ M ) @ ( bit_ri4277139882892585799ns_not @ int @ ( numeral_numeral @ int @ N ) ) ) ) ) ).
% and_not_numerals(5)
thf(fact_5310_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_5311_and__not__numerals_I3_J,axiom,
! [N: num] :
( ( bit_se5824344872417868541ns_and @ int @ ( one_one @ int ) @ ( bit_ri4277139882892585799ns_not @ int @ ( numeral_numeral @ int @ ( bit1 @ N ) ) ) )
= ( zero_zero @ int ) ) ).
% and_not_numerals(3)
thf(fact_5312_bit__iff__odd,axiom,
! [A: $tType] :
( ( bit_semiring_bits @ A )
=> ( ( bit_se5641148757651400278ts_bit @ A )
= ( ^ [A4: A,N3: nat] :
~ ( dvd_dvd @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ ( divide_divide @ A @ A4 @ ( power_power @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ N3 ) ) ) ) ) ) ).
% bit_iff_odd
thf(fact_5313_and__exp__eq__0__iff__not__bit,axiom,
! [A: $tType] :
( ( bit_se359711467146920520ations @ A )
=> ! [A2: A,N: nat] :
( ( ( bit_se5824344872417868541ns_and @ A @ A2 @ ( power_power @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ N ) )
= ( zero_zero @ A ) )
= ( ~ ( bit_se5641148757651400278ts_bit @ A @ A2 @ N ) ) ) ) ).
% and_exp_eq_0_iff_not_bit
thf(fact_5314_bit__not__iff__eq,axiom,
! [A: $tType] :
( ( bit_ri3973907225187159222ations @ A )
=> ! [A2: A,N: nat] :
( ( bit_se5641148757651400278ts_bit @ A @ ( bit_ri4277139882892585799ns_not @ A @ A2 ) @ N )
= ( ( ( power_power @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ N )
!= ( zero_zero @ A ) )
& ~ ( bit_se5641148757651400278ts_bit @ A @ A2 @ N ) ) ) ) ).
% bit_not_iff_eq
thf(fact_5315_mask__Suc__exp,axiom,
! [A: $tType] :
( ( bit_se359711467146920520ations @ A )
=> ! [N: nat] :
( ( bit_se2239418461657761734s_mask @ A @ ( suc @ N ) )
= ( bit_se1065995026697491101ons_or @ A @ ( power_power @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ N ) @ ( bit_se2239418461657761734s_mask @ A @ N ) ) ) ) ).
% mask_Suc_exp
thf(fact_5316_and__not__numerals_I6_J,axiom,
! [M: num,N: num] :
( ( bit_se5824344872417868541ns_and @ int @ ( numeral_numeral @ int @ ( bit0 @ M ) ) @ ( bit_ri4277139882892585799ns_not @ int @ ( numeral_numeral @ int @ ( bit1 @ N ) ) ) )
= ( times_times @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ ( bit_se5824344872417868541ns_and @ int @ ( numeral_numeral @ int @ M ) @ ( bit_ri4277139882892585799ns_not @ int @ ( numeral_numeral @ int @ N ) ) ) ) ) ).
% and_not_numerals(6)
thf(fact_5317_and__not__numerals_I9_J,axiom,
! [M: num,N: num] :
( ( bit_se5824344872417868541ns_and @ int @ ( numeral_numeral @ int @ ( bit1 @ M ) ) @ ( bit_ri4277139882892585799ns_not @ int @ ( numeral_numeral @ int @ ( bit1 @ N ) ) ) )
= ( times_times @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ ( bit_se5824344872417868541ns_and @ int @ ( numeral_numeral @ int @ M ) @ ( bit_ri4277139882892585799ns_not @ int @ ( numeral_numeral @ int @ N ) ) ) ) ) ).
% and_not_numerals(9)
thf(fact_5318_bit__int__def,axiom,
( ( bit_se5641148757651400278ts_bit @ int )
= ( ^ [K3: int,N3: nat] :
~ ( dvd_dvd @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ ( divide_divide @ int @ K3 @ ( power_power @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ N3 ) ) ) ) ) ).
% bit_int_def
thf(fact_5319_even__bit__succ__iff,axiom,
! [A: $tType] :
( ( bit_semiring_bits @ A )
=> ! [A2: A,N: nat] :
( ( dvd_dvd @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ A2 )
=> ( ( bit_se5641148757651400278ts_bit @ A @ ( plus_plus @ A @ ( one_one @ A ) @ A2 ) @ N )
= ( ( bit_se5641148757651400278ts_bit @ A @ A2 @ N )
| ( N
= ( zero_zero @ nat ) ) ) ) ) ) ).
% even_bit_succ_iff
thf(fact_5320_odd__bit__iff__bit__pred,axiom,
! [A: $tType] :
( ( bit_semiring_bits @ A )
=> ! [A2: A,N: nat] :
( ~ ( dvd_dvd @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ A2 )
=> ( ( bit_se5641148757651400278ts_bit @ A @ A2 @ N )
= ( ( bit_se5641148757651400278ts_bit @ A @ ( minus_minus @ A @ A2 @ ( one_one @ A ) ) @ N )
| ( N
= ( zero_zero @ nat ) ) ) ) ) ) ).
% odd_bit_iff_bit_pred
thf(fact_5321_mask__Suc__double,axiom,
! [A: $tType] :
( ( bit_se359711467146920520ations @ A )
=> ! [N: nat] :
( ( bit_se2239418461657761734s_mask @ A @ ( suc @ N ) )
= ( bit_se1065995026697491101ons_or @ A @ ( one_one @ A ) @ ( times_times @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ ( bit_se2239418461657761734s_mask @ A @ N ) ) ) ) ) ).
% mask_Suc_double
thf(fact_5322_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_5323_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_5324_bit__sum__mult__2__cases,axiom,
! [A: $tType] :
( ( bit_se359711467146920520ations @ A )
=> ! [A2: A,B2: A,N: nat] :
( ! [J3: nat] :
~ ( bit_se5641148757651400278ts_bit @ A @ A2 @ ( suc @ J3 ) )
=> ( ( bit_se5641148757651400278ts_bit @ A @ ( plus_plus @ A @ A2 @ ( times_times @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ B2 ) ) @ N )
= ( ( ( N
= ( zero_zero @ nat ) )
=> ~ ( dvd_dvd @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ A2 ) )
& ( ( N
!= ( zero_zero @ nat ) )
=> ( bit_se5641148757651400278ts_bit @ A @ ( times_times @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ B2 ) @ N ) ) ) ) ) ) ).
% bit_sum_mult_2_cases
thf(fact_5325_bit__rec,axiom,
! [A: $tType] :
( ( bit_semiring_bits @ A )
=> ( ( bit_se5641148757651400278ts_bit @ A )
= ( ^ [A4: A,N3: nat] :
( ( ( N3
= ( zero_zero @ nat ) )
=> ~ ( dvd_dvd @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ A4 ) )
& ( ( N3
!= ( zero_zero @ nat ) )
=> ( bit_se5641148757651400278ts_bit @ A @ ( divide_divide @ A @ A4 @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) @ ( minus_minus @ nat @ N3 @ ( one_one @ nat ) ) ) ) ) ) ) ) ).
% bit_rec
thf(fact_5326_and__not__numerals_I8_J,axiom,
! [M: num,N: num] :
( ( bit_se5824344872417868541ns_and @ int @ ( numeral_numeral @ int @ ( bit1 @ M ) ) @ ( bit_ri4277139882892585799ns_not @ int @ ( numeral_numeral @ int @ ( bit0 @ N ) ) ) )
= ( plus_plus @ int @ ( one_one @ int ) @ ( times_times @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ ( bit_se5824344872417868541ns_and @ int @ ( numeral_numeral @ int @ M ) @ ( bit_ri4277139882892585799ns_not @ int @ ( numeral_numeral @ int @ N ) ) ) ) ) ) ).
% and_not_numerals(8)
thf(fact_5327_not__int__rec,axiom,
( ( bit_ri4277139882892585799ns_not @ int )
= ( ^ [K3: int] : ( plus_plus @ int @ ( zero_neq_one_of_bool @ int @ ( dvd_dvd @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ K3 ) ) @ ( times_times @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ ( bit_ri4277139882892585799ns_not @ int @ ( divide_divide @ int @ K3 @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) ) ) ) ) ) ) ).
% not_int_rec
thf(fact_5328_Bit__Operations_Oset__bit__eq,axiom,
( ( bit_se5668285175392031749et_bit @ int )
= ( ^ [N3: nat,K3: int] :
( plus_plus @ int @ K3
@ ( times_times @ int
@ ( zero_neq_one_of_bool @ int
@ ~ ( bit_se5641148757651400278ts_bit @ int @ K3 @ N3 ) )
@ ( power_power @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ N3 ) ) ) ) ) ).
% Bit_Operations.set_bit_eq
thf(fact_5329_unset__bit__eq,axiom,
( ( bit_se2638667681897837118et_bit @ int )
= ( ^ [N3: nat,K3: int] : ( minus_minus @ int @ K3 @ ( times_times @ int @ ( zero_neq_one_of_bool @ int @ ( bit_se5641148757651400278ts_bit @ int @ K3 @ N3 ) ) @ ( power_power @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ N3 ) ) ) ) ) ).
% unset_bit_eq
thf(fact_5330_take__bit__Suc__from__most,axiom,
! [N: nat,K: int] :
( ( bit_se2584673776208193580ke_bit @ int @ ( suc @ N ) @ K )
= ( plus_plus @ int @ ( times_times @ int @ ( power_power @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ N ) @ ( zero_neq_one_of_bool @ int @ ( bit_se5641148757651400278ts_bit @ int @ K @ N ) ) ) @ ( bit_se2584673776208193580ke_bit @ int @ N @ K ) ) ) ).
% take_bit_Suc_from_most
thf(fact_5331_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_5332_bitNOT__integer__code,axiom,
( ( bit_ri4277139882892585799ns_not @ code_integer )
= ( ^ [I2: code_integer] : ( minus_minus @ code_integer @ ( uminus_uminus @ code_integer @ I2 ) @ ( one_one @ code_integer ) ) ) ) ).
% bitNOT_integer_code
thf(fact_5333_xor__int__unfold,axiom,
( ( bit_se5824344971392196577ns_xor @ int )
= ( ^ [K3: int,L2: int] :
( if @ int
@ ( K3
= ( uminus_uminus @ int @ ( one_one @ int ) ) )
@ ( bit_ri4277139882892585799ns_not @ int @ L2 )
@ ( if @ int
@ ( L2
= ( uminus_uminus @ int @ ( one_one @ int ) ) )
@ ( bit_ri4277139882892585799ns_not @ int @ K3 )
@ ( if @ int
@ ( K3
= ( zero_zero @ int ) )
@ L2
@ ( if @ int
@ ( L2
= ( zero_zero @ int ) )
@ K3
@ ( plus_plus @ int @ ( abs_abs @ int @ ( minus_minus @ int @ ( modulo_modulo @ int @ K3 @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) ) @ ( modulo_modulo @ int @ L2 @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) ) ) ) @ ( times_times @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ ( bit_se5824344971392196577ns_xor @ int @ ( divide_divide @ int @ K3 @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) ) @ ( divide_divide @ int @ L2 @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) ) ) ) ) ) ) ) ) ) ) ).
% xor_int_unfold
thf(fact_5334_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_5335_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_5336_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_5337_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_5338_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_5339_test__bit__1,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [N: nat] :
( ( bit_se5641148757651400278ts_bit @ ( word @ A ) @ ( one_one @ ( word @ A ) ) @ N )
= ( N
= ( zero_zero @ nat ) ) ) ) ).
% test_bit_1
thf(fact_5340_take__bit__xor,axiom,
! [A: $tType] :
( ( bit_se359711467146920520ations @ A )
=> ! [N: nat,A2: A,B2: A] :
( ( bit_se2584673776208193580ke_bit @ A @ N @ ( bit_se5824344971392196577ns_xor @ A @ A2 @ B2 ) )
= ( bit_se5824344971392196577ns_xor @ A @ ( bit_se2584673776208193580ke_bit @ A @ N @ A2 ) @ ( bit_se2584673776208193580ke_bit @ A @ N @ B2 ) ) ) ) ).
% take_bit_xor
thf(fact_5341_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_5342_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_5343_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_5344_or__nonnegative__int__iff,axiom,
! [K: int,L: int] :
( ( ord_less_eq @ int @ ( zero_zero @ int ) @ ( bit_se1065995026697491101ons_or @ int @ K @ L ) )
= ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ K )
& ( ord_less_eq @ int @ ( zero_zero @ int ) @ L ) ) ) ).
% or_nonnegative_int_iff
thf(fact_5345_or__negative__int__iff,axiom,
! [K: int,L: int] :
( ( ord_less @ int @ ( bit_se1065995026697491101ons_or @ int @ K @ L ) @ ( zero_zero @ int ) )
= ( ( ord_less @ int @ K @ ( zero_zero @ int ) )
| ( ord_less @ int @ L @ ( zero_zero @ int ) ) ) ) ).
% or_negative_int_iff
thf(fact_5346_xor__nonnegative__int__iff,axiom,
! [K: int,L: int] :
( ( ord_less_eq @ int @ ( zero_zero @ int ) @ ( bit_se5824344971392196577ns_xor @ int @ K @ L ) )
= ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ K )
= ( ord_less_eq @ int @ ( zero_zero @ int ) @ L ) ) ) ).
% xor_nonnegative_int_iff
thf(fact_5347_xor__negative__int__iff,axiom,
! [K: int,L: int] :
( ( ord_less @ int @ ( bit_se5824344971392196577ns_xor @ int @ K @ L ) @ ( zero_zero @ int ) )
= ( ( ord_less @ int @ K @ ( zero_zero @ int ) )
!= ( ord_less @ int @ L @ ( zero_zero @ int ) ) ) ) ).
% xor_negative_int_iff
thf(fact_5348_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_5349_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_5350_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_5351_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_5352_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_5353_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_5354_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_5355_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_5356_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_5357_or__minus__numerals_I2_J,axiom,
! [N: num] :
( ( bit_se1065995026697491101ons_or @ int @ ( one_one @ int ) @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ ( bit1 @ N ) ) ) )
= ( uminus_uminus @ int @ ( numeral_numeral @ int @ ( bit1 @ N ) ) ) ) ).
% or_minus_numerals(2)
thf(fact_5358_or__minus__numerals_I6_J,axiom,
! [N: num] :
( ( bit_se1065995026697491101ons_or @ int @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ ( bit1 @ N ) ) ) @ ( one_one @ int ) )
= ( uminus_uminus @ int @ ( numeral_numeral @ int @ ( bit1 @ N ) ) ) ) ).
% or_minus_numerals(6)
thf(fact_5359_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_5360_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_5361_word__no__log__defs_I7_J,axiom,
! [G2: $tType] :
( ( type_len @ G2 )
=> ! [A2: num,B2: num] :
( ( bit_se1065995026697491101ons_or @ ( word @ G2 ) @ ( numeral_numeral @ ( word @ G2 ) @ A2 ) @ ( numeral_numeral @ ( word @ G2 ) @ B2 ) )
= ( ring_1_of_int @ ( word @ G2 ) @ ( bit_se1065995026697491101ons_or @ int @ ( numeral_numeral @ int @ A2 ) @ ( numeral_numeral @ int @ B2 ) ) ) ) ) ).
% word_no_log_defs(7)
thf(fact_5362_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_5363_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_5364_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_5365_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_5366_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_5367_word__no__log__defs_I8_J,axiom,
! [H5: $tType] :
( ( type_len @ H5 )
=> ! [A2: num,B2: num] :
( ( bit_se1065995026697491101ons_or @ ( word @ H5 ) @ ( numeral_numeral @ ( word @ H5 ) @ A2 ) @ ( uminus_uminus @ ( word @ H5 ) @ ( numeral_numeral @ ( word @ H5 ) @ B2 ) ) )
= ( ring_1_of_int @ ( word @ H5 ) @ ( bit_se1065995026697491101ons_or @ int @ ( numeral_numeral @ int @ A2 ) @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ B2 ) ) ) ) ) ) ).
% word_no_log_defs(8)
thf(fact_5368_word__no__log__defs_I9_J,axiom,
! [I8: $tType] :
( ( type_len @ I8 )
=> ! [A2: num,B2: num] :
( ( bit_se1065995026697491101ons_or @ ( word @ I8 ) @ ( uminus_uminus @ ( word @ I8 ) @ ( numeral_numeral @ ( word @ I8 ) @ A2 ) ) @ ( numeral_numeral @ ( word @ I8 ) @ B2 ) )
= ( ring_1_of_int @ ( word @ I8 ) @ ( bit_se1065995026697491101ons_or @ int @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ A2 ) ) @ ( numeral_numeral @ int @ B2 ) ) ) ) ) ).
% word_no_log_defs(9)
thf(fact_5369_word__no__log__defs_I10_J,axiom,
! [J4: $tType] :
( ( type_len @ J4 )
=> ! [A2: num,B2: num] :
( ( bit_se1065995026697491101ons_or @ ( word @ J4 ) @ ( uminus_uminus @ ( word @ J4 ) @ ( numeral_numeral @ ( word @ J4 ) @ A2 ) ) @ ( uminus_uminus @ ( word @ J4 ) @ ( numeral_numeral @ ( word @ J4 ) @ B2 ) ) )
= ( ring_1_of_int @ ( word @ J4 ) @ ( bit_se1065995026697491101ons_or @ int @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ A2 ) ) @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ B2 ) ) ) ) ) ) ).
% word_no_log_defs(10)
thf(fact_5370_or__minus__minus__numerals,axiom,
! [M: num,N: num] :
( ( bit_se1065995026697491101ons_or @ int @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ M ) ) @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ N ) ) )
= ( bit_ri4277139882892585799ns_not @ int @ ( bit_se5824344872417868541ns_and @ int @ ( minus_minus @ int @ ( numeral_numeral @ int @ M ) @ ( one_one @ int ) ) @ ( minus_minus @ int @ ( numeral_numeral @ int @ N ) @ ( one_one @ int ) ) ) ) ) ).
% or_minus_minus_numerals
thf(fact_5371_and__minus__minus__numerals,axiom,
! [M: num,N: num] :
( ( bit_se5824344872417868541ns_and @ int @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ M ) ) @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ N ) ) )
= ( bit_ri4277139882892585799ns_not @ int @ ( bit_se1065995026697491101ons_or @ int @ ( minus_minus @ int @ ( numeral_numeral @ int @ M ) @ ( one_one @ int ) ) @ ( minus_minus @ int @ ( numeral_numeral @ int @ N ) @ ( one_one @ int ) ) ) ) ) ).
% and_minus_minus_numerals
thf(fact_5372_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_5373_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_5374_word__bitwise__1__simps_I9_J,axiom,
! [I8: $tType] :
( ( type_len @ I8 )
=> ! [A2: num] :
( ( bit_se1065995026697491101ons_or @ ( word @ I8 ) @ ( uminus_uminus @ ( word @ I8 ) @ ( numeral_numeral @ ( word @ I8 ) @ A2 ) ) @ ( one_one @ ( word @ I8 ) ) )
= ( ring_1_of_int @ ( word @ I8 ) @ ( bit_se1065995026697491101ons_or @ int @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ A2 ) ) @ ( one_one @ int ) ) ) ) ) ).
% word_bitwise_1_simps(9)
thf(fact_5375_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_5376_xor__int__def,axiom,
( ( bit_se5824344971392196577ns_xor @ int )
= ( ^ [K3: int,L2: int] : ( bit_se1065995026697491101ons_or @ int @ ( bit_se5824344872417868541ns_and @ int @ K3 @ ( bit_ri4277139882892585799ns_not @ int @ L2 ) ) @ ( bit_se5824344872417868541ns_and @ int @ ( bit_ri4277139882892585799ns_not @ int @ K3 ) @ L2 ) ) ) ) ).
% xor_int_def
thf(fact_5377_int__or__code_I2_J,axiom,
! [I: int] :
( ( bit_se1065995026697491101ons_or @ int @ I @ ( zero_zero @ int ) )
= I ) ).
% int_or_code(2)
thf(fact_5378_int__or__code_I1_J,axiom,
! [J: int] :
( ( bit_se1065995026697491101ons_or @ int @ ( zero_zero @ int ) @ J )
= J ) ).
% int_or_code(1)
thf(fact_5379_nth__0,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [N: nat] :
~ ( bit_se5641148757651400278ts_bit @ ( word @ A ) @ ( zero_zero @ ( word @ A ) ) @ N ) ) ).
% nth_0
thf(fact_5380_word__exists__nth,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [W: word @ A] :
( ( W
!= ( zero_zero @ ( word @ A ) ) )
=> ? [X_1: nat] : ( bit_se5641148757651400278ts_bit @ ( word @ A ) @ W @ X_1 ) ) ) ).
% word_exists_nth
thf(fact_5381_bit__xor__int__iff,axiom,
! [K: int,L: int,N: nat] :
( ( bit_se5641148757651400278ts_bit @ int @ ( bit_se5824344971392196577ns_xor @ int @ K @ L ) @ N )
= ( ( bit_se5641148757651400278ts_bit @ int @ K @ N )
!= ( bit_se5641148757651400278ts_bit @ int @ L @ N ) ) ) ).
% bit_xor_int_iff
thf(fact_5382_bit__or__int__iff,axiom,
! [K: int,L: int,N: nat] :
( ( bit_se5641148757651400278ts_bit @ int @ ( bit_se1065995026697491101ons_or @ int @ K @ L ) @ N )
= ( ( bit_se5641148757651400278ts_bit @ int @ K @ N )
| ( bit_se5641148757651400278ts_bit @ int @ L @ N ) ) ) ).
% bit_or_int_iff
thf(fact_5383_bit__xor__iff,axiom,
! [A: $tType] :
( ( bit_se359711467146920520ations @ A )
=> ! [A2: A,B2: A,N: nat] :
( ( bit_se5641148757651400278ts_bit @ A @ ( bit_se5824344971392196577ns_xor @ A @ A2 @ B2 ) @ N )
= ( ( bit_se5641148757651400278ts_bit @ A @ A2 @ N )
!= ( bit_se5641148757651400278ts_bit @ A @ B2 @ N ) ) ) ) ).
% bit_xor_iff
thf(fact_5384_or__nat__def,axiom,
( ( bit_se1065995026697491101ons_or @ nat )
= ( ^ [M4: nat,N3: nat] : ( nat2 @ ( bit_se1065995026697491101ons_or @ int @ ( semiring_1_of_nat @ int @ M4 ) @ ( semiring_1_of_nat @ int @ N3 ) ) ) ) ) ).
% or_nat_def
thf(fact_5385_bit__Suc__0__iff,axiom,
! [N: nat] :
( ( bit_se5641148757651400278ts_bit @ nat @ ( suc @ ( zero_zero @ nat ) ) @ N )
= ( N
= ( zero_zero @ nat ) ) ) ).
% bit_Suc_0_iff
thf(fact_5386_not__bit__Suc__0__Suc,axiom,
! [N: nat] :
~ ( bit_se5641148757651400278ts_bit @ nat @ ( suc @ ( zero_zero @ nat ) ) @ ( suc @ N ) ) ).
% not_bit_Suc_0_Suc
thf(fact_5387_of__nat__xor__eq,axiom,
! [A: $tType] :
( ( bit_se359711467146920520ations @ A )
=> ! [M: nat,N: nat] :
( ( semiring_1_of_nat @ A @ ( bit_se5824344971392196577ns_xor @ nat @ M @ N ) )
= ( bit_se5824344971392196577ns_xor @ A @ ( semiring_1_of_nat @ A @ M ) @ ( semiring_1_of_nat @ A @ N ) ) ) ) ).
% of_nat_xor_eq
thf(fact_5388_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_5389_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_5390_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_5391_xor_Ocommute,axiom,
! [A: $tType] :
( ( bit_se359711467146920520ations @ A )
=> ( ( bit_se5824344971392196577ns_xor @ A )
= ( ^ [A4: A,B5: A] : ( bit_se5824344971392196577ns_xor @ A @ B5 @ A4 ) ) ) ) ).
% xor.commute
thf(fact_5392_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_5393_of__int__xor__eq,axiom,
! [A: $tType] :
( ( bit_ri3973907225187159222ations @ A )
=> ! [K: int,L: int] :
( ( ring_1_of_int @ A @ ( bit_se5824344971392196577ns_xor @ int @ K @ L ) )
= ( bit_se5824344971392196577ns_xor @ A @ ( ring_1_of_int @ A @ K ) @ ( ring_1_of_int @ A @ L ) ) ) ) ).
% of_int_xor_eq
thf(fact_5394_int__xor__code_I1_J,axiom,
! [J: int] :
( ( bit_se5824344971392196577ns_xor @ int @ ( zero_zero @ int ) @ J )
= J ) ).
% int_xor_code(1)
thf(fact_5395_int__xor__code_I2_J,axiom,
! [I: int] :
( ( bit_se5824344971392196577ns_xor @ int @ I @ ( zero_zero @ int ) )
= I ) ).
% int_xor_code(2)
thf(fact_5396_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_5397_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_5398_or__greater__eq,axiom,
! [L: int,K: int] :
( ( ord_less_eq @ int @ ( zero_zero @ int ) @ L )
=> ( ord_less_eq @ int @ K @ ( bit_se1065995026697491101ons_or @ int @ K @ L ) ) ) ).
% or_greater_eq
thf(fact_5399_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_5400_word__eqI,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [U: word @ A,V: word @ A] :
( ! [N2: nat] :
( ( ord_less @ nat @ N2 @ ( size_size @ ( word @ A ) @ U ) )
=> ( ( bit_se5641148757651400278ts_bit @ ( word @ A ) @ U @ N2 )
= ( bit_se5641148757651400278ts_bit @ ( word @ A ) @ V @ N2 ) ) )
=> ( U = V ) ) ) ).
% word_eqI
thf(fact_5401_test__bit__size,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [W: word @ A,N: nat] :
( ( bit_se5641148757651400278ts_bit @ ( word @ A ) @ W @ N )
=> ( ord_less @ nat @ N @ ( size_size @ ( word @ A ) @ W ) ) ) ) ).
% test_bit_size
thf(fact_5402_or__int__def,axiom,
( ( bit_se1065995026697491101ons_or @ int )
= ( ^ [K3: int,L2: int] : ( bit_ri4277139882892585799ns_not @ int @ ( bit_se5824344872417868541ns_and @ int @ ( bit_ri4277139882892585799ns_not @ int @ K3 ) @ ( bit_ri4277139882892585799ns_not @ int @ L2 ) ) ) ) ) ).
% or_int_def
thf(fact_5403_not__bit__Suc__0__numeral,axiom,
! [N: num] :
~ ( bit_se5641148757651400278ts_bit @ nat @ ( suc @ ( zero_zero @ nat ) ) @ ( numeral_numeral @ nat @ N ) ) ).
% not_bit_Suc_0_numeral
thf(fact_5404_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_5405_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_5406_word__leI,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [U: word @ A,V: word @ A] :
( ! [N2: nat] :
( ( ord_less @ nat @ N2 @ ( size_size @ ( word @ A ) @ U ) )
=> ( ( bit_se5641148757651400278ts_bit @ ( word @ A ) @ U @ N2 )
=> ( bit_se5641148757651400278ts_bit @ ( word @ A ) @ V @ N2 ) ) )
=> ( ord_less_eq @ ( word @ A ) @ U @ V ) ) ) ).
% word_leI
thf(fact_5407_nth__mask,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [N: nat,I: nat] :
( ( bit_se5641148757651400278ts_bit @ ( word @ A ) @ ( bit_se2239418461657761734s_mask @ ( word @ A ) @ N ) @ I )
= ( ( ord_less @ nat @ I @ N )
& ( ord_less @ nat @ I @ ( size_size @ ( word @ A ) @ ( bit_se2239418461657761734s_mask @ ( word @ A ) @ N ) ) ) ) ) ) ).
% nth_mask
thf(fact_5408_bit_Oxor__def,axiom,
! [A: $tType] :
( ( bit_ri3973907225187159222ations @ A )
=> ( ( bit_se5824344971392196577ns_xor @ A )
= ( ^ [X2: A,Y6: A] : ( bit_se1065995026697491101ons_or @ A @ ( bit_se5824344872417868541ns_and @ A @ X2 @ ( bit_ri4277139882892585799ns_not @ A @ Y6 ) ) @ ( bit_se5824344872417868541ns_and @ A @ ( bit_ri4277139882892585799ns_not @ A @ X2 ) @ Y6 ) ) ) ) ) ).
% bit.xor_def
thf(fact_5409_bit_Oxor__def2,axiom,
! [A: $tType] :
( ( bit_ri3973907225187159222ations @ A )
=> ( ( bit_se5824344971392196577ns_xor @ A )
= ( ^ [X2: A,Y6: A] : ( bit_se5824344872417868541ns_and @ A @ ( bit_se1065995026697491101ons_or @ A @ X2 @ Y6 ) @ ( bit_se1065995026697491101ons_or @ A @ ( bit_ri4277139882892585799ns_not @ A @ X2 ) @ ( bit_ri4277139882892585799ns_not @ A @ Y6 ) ) ) ) ) ) ).
% bit.xor_def2
thf(fact_5410_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_5411_or__not__numerals_I2_J,axiom,
! [N: num] :
( ( bit_se1065995026697491101ons_or @ int @ ( one_one @ int ) @ ( bit_ri4277139882892585799ns_not @ int @ ( numeral_numeral @ int @ ( bit0 @ N ) ) ) )
= ( bit_ri4277139882892585799ns_not @ int @ ( numeral_numeral @ int @ ( bit0 @ N ) ) ) ) ).
% or_not_numerals(2)
thf(fact_5412_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_5413_bit__nat__iff,axiom,
! [K: int,N: nat] :
( ( bit_se5641148757651400278ts_bit @ nat @ ( nat2 @ K ) @ N )
= ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ K )
& ( bit_se5641148757651400278ts_bit @ int @ K @ N ) ) ) ).
% bit_nat_iff
thf(fact_5414_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_5415_word__and__1,axiom,
! [B: $tType] :
( ( type_len @ B )
=> ! [N: word @ B] :
( ( ( bit_se5641148757651400278ts_bit @ ( word @ B ) @ N @ ( zero_zero @ nat ) )
=> ( ( bit_se5824344872417868541ns_and @ ( word @ B ) @ N @ ( one_one @ ( word @ B ) ) )
= ( one_one @ ( word @ B ) ) ) )
& ( ~ ( bit_se5641148757651400278ts_bit @ ( word @ B ) @ N @ ( zero_zero @ nat ) )
=> ( ( bit_se5824344872417868541ns_and @ ( word @ B ) @ N @ ( one_one @ ( word @ B ) ) )
= ( zero_zero @ ( word @ B ) ) ) ) ) ) ).
% word_and_1
thf(fact_5416_test__bit__bin_H,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ( ( bit_se5641148757651400278ts_bit @ ( word @ A ) )
= ( ^ [W2: word @ A,N3: nat] :
( ( ord_less @ nat @ N3 @ ( size_size @ ( word @ A ) @ W2 ) )
& ( bit_se5641148757651400278ts_bit @ int @ ( semiring_1_unsigned @ A @ int @ W2 ) @ N3 ) ) ) ) ) ).
% test_bit_bin'
thf(fact_5417_or__not__numerals_I3_J,axiom,
! [N: num] :
( ( bit_se1065995026697491101ons_or @ int @ ( one_one @ int ) @ ( bit_ri4277139882892585799ns_not @ int @ ( numeral_numeral @ int @ ( bit1 @ N ) ) ) )
= ( bit_ri4277139882892585799ns_not @ int @ ( numeral_numeral @ int @ ( bit0 @ N ) ) ) ) ).
% or_not_numerals(3)
thf(fact_5418_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_5419_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_5420_bit__nat__def,axiom,
( ( bit_se5641148757651400278ts_bit @ nat )
= ( ^ [M4: nat,N3: nat] :
~ ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ M4 @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N3 ) ) ) ) ) ).
% bit_nat_def
thf(fact_5421_or__not__numerals_I6_J,axiom,
! [M: num,N: num] :
( ( bit_se1065995026697491101ons_or @ int @ ( numeral_numeral @ int @ ( bit0 @ M ) ) @ ( bit_ri4277139882892585799ns_not @ int @ ( numeral_numeral @ int @ ( bit1 @ N ) ) ) )
= ( times_times @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ ( bit_se1065995026697491101ons_or @ int @ ( numeral_numeral @ int @ M ) @ ( bit_ri4277139882892585799ns_not @ int @ ( numeral_numeral @ int @ N ) ) ) ) ) ).
% or_not_numerals(6)
thf(fact_5422_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_5423_and__neq__0__is__nth,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [Y: word @ A,N: nat,X: word @ A] :
( ( Y
= ( power_power @ ( word @ A ) @ ( numeral_numeral @ ( word @ A ) @ ( bit0 @ one2 ) ) @ N ) )
=> ( ( ( bit_se5824344872417868541ns_and @ ( word @ A ) @ X @ Y )
!= ( zero_zero @ ( word @ A ) ) )
= ( bit_se5641148757651400278ts_bit @ ( word @ A ) @ X @ N ) ) ) ) ).
% and_neq_0_is_nth
thf(fact_5424_nth__is__and__neq__0,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ( ( bit_se5641148757651400278ts_bit @ ( word @ A ) )
= ( ^ [X2: word @ A,N3: nat] :
( ( bit_se5824344872417868541ns_and @ ( word @ A ) @ X2 @ ( power_power @ ( word @ A ) @ ( numeral_numeral @ ( word @ A ) @ ( bit0 @ one2 ) ) @ N3 ) )
!= ( zero_zero @ ( word @ A ) ) ) ) ) ) ).
% nth_is_and_neq_0
thf(fact_5425_XOR__upper,axiom,
! [X: int,N: nat,Y: int] :
( ( ord_less_eq @ int @ ( zero_zero @ int ) @ X )
=> ( ( ord_less @ int @ X @ ( power_power @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ N ) )
=> ( ( ord_less @ int @ Y @ ( power_power @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ N ) )
=> ( ord_less @ int @ ( bit_se5824344971392196577ns_xor @ int @ X @ Y ) @ ( power_power @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ N ) ) ) ) ) ).
% XOR_upper
thf(fact_5426_OR__upper,axiom,
! [X: int,N: nat,Y: int] :
( ( ord_less_eq @ int @ ( zero_zero @ int ) @ X )
=> ( ( ord_less @ int @ X @ ( power_power @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ N ) )
=> ( ( ord_less @ int @ Y @ ( power_power @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ N ) )
=> ( ord_less @ int @ ( bit_se1065995026697491101ons_or @ int @ X @ Y ) @ ( power_power @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ N ) ) ) ) ) ).
% OR_upper
thf(fact_5427_or__not__numerals_I5_J,axiom,
! [M: num,N: num] :
( ( bit_se1065995026697491101ons_or @ int @ ( numeral_numeral @ int @ ( bit0 @ M ) ) @ ( bit_ri4277139882892585799ns_not @ int @ ( numeral_numeral @ int @ ( bit0 @ N ) ) ) )
= ( plus_plus @ int @ ( one_one @ int ) @ ( times_times @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ ( bit_se1065995026697491101ons_or @ int @ ( numeral_numeral @ int @ M ) @ ( bit_ri4277139882892585799ns_not @ int @ ( numeral_numeral @ int @ N ) ) ) ) ) ) ).
% or_not_numerals(5)
thf(fact_5428_test__bit__int__code_I1_J,axiom,
! [N: nat] :
~ ( bit_se5641148757651400278ts_bit @ int @ ( zero_zero @ int ) @ N ) ).
% test_bit_int_code(1)
thf(fact_5429_int__and__code_I2_J,axiom,
! [I: int] :
( ( bit_se5824344872417868541ns_and @ int @ I @ ( zero_zero @ int ) )
= ( zero_zero @ int ) ) ).
% int_and_code(2)
thf(fact_5430_int__and__code_I1_J,axiom,
! [J: int] :
( ( bit_se5824344872417868541ns_and @ int @ ( zero_zero @ int ) @ J )
= ( zero_zero @ int ) ) ).
% int_and_code(1)
thf(fact_5431_Suc__0__or__eq,axiom,
! [N: nat] :
( ( bit_se1065995026697491101ons_or @ nat @ ( suc @ ( zero_zero @ nat ) ) @ N )
= ( plus_plus @ nat @ N @ ( zero_neq_one_of_bool @ nat @ ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N ) ) ) ) ).
% Suc_0_or_eq
thf(fact_5432_or__Suc__0__eq,axiom,
! [N: nat] :
( ( bit_se1065995026697491101ons_or @ nat @ N @ ( suc @ ( zero_zero @ nat ) ) )
= ( plus_plus @ nat @ N @ ( zero_neq_one_of_bool @ nat @ ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N ) ) ) ) ).
% or_Suc_0_eq
thf(fact_5433_or__nat__rec,axiom,
( ( bit_se1065995026697491101ons_or @ nat )
= ( ^ [M4: nat,N3: nat] :
( plus_plus @ nat
@ ( zero_neq_one_of_bool @ nat
@ ( ~ ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ M4 )
| ~ ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N3 ) ) )
@ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( bit_se1065995026697491101ons_or @ nat @ ( divide_divide @ nat @ M4 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ ( divide_divide @ nat @ N3 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) ) ) ).
% or_nat_rec
thf(fact_5434_or__not__numerals_I9_J,axiom,
! [M: num,N: num] :
( ( bit_se1065995026697491101ons_or @ int @ ( numeral_numeral @ int @ ( bit1 @ M ) ) @ ( bit_ri4277139882892585799ns_not @ int @ ( numeral_numeral @ int @ ( bit1 @ N ) ) ) )
= ( plus_plus @ int @ ( one_one @ int ) @ ( times_times @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ ( bit_se1065995026697491101ons_or @ int @ ( numeral_numeral @ int @ M ) @ ( bit_ri4277139882892585799ns_not @ int @ ( numeral_numeral @ int @ N ) ) ) ) ) ) ).
% or_not_numerals(9)
thf(fact_5435_or__not__numerals_I8_J,axiom,
! [M: num,N: num] :
( ( bit_se1065995026697491101ons_or @ int @ ( numeral_numeral @ int @ ( bit1 @ M ) ) @ ( bit_ri4277139882892585799ns_not @ int @ ( numeral_numeral @ int @ ( bit0 @ N ) ) ) )
= ( plus_plus @ int @ ( one_one @ int ) @ ( times_times @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ ( bit_se1065995026697491101ons_or @ int @ ( numeral_numeral @ int @ M ) @ ( bit_ri4277139882892585799ns_not @ int @ ( numeral_numeral @ int @ N ) ) ) ) ) ) ).
% or_not_numerals(8)
thf(fact_5436_xor__int__rec,axiom,
( ( bit_se5824344971392196577ns_xor @ int )
= ( ^ [K3: int,L2: int] :
( plus_plus @ int
@ ( zero_neq_one_of_bool @ int
@ ( ( ~ ( dvd_dvd @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ K3 ) )
!= ( ~ ( dvd_dvd @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ L2 ) ) ) )
@ ( times_times @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ ( bit_se5824344971392196577ns_xor @ int @ ( divide_divide @ int @ K3 @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) ) @ ( divide_divide @ int @ L2 @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) ) ) ) ) ) ) ).
% xor_int_rec
thf(fact_5437_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_5438_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_5439_or__int__rec,axiom,
( ( bit_se1065995026697491101ons_or @ int )
= ( ^ [K3: int,L2: int] :
( plus_plus @ int
@ ( zero_neq_one_of_bool @ int
@ ( ~ ( dvd_dvd @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ K3 )
| ~ ( dvd_dvd @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ L2 ) ) )
@ ( times_times @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ ( bit_se1065995026697491101ons_or @ int @ ( divide_divide @ int @ K3 @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) ) @ ( divide_divide @ int @ L2 @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) ) ) ) ) ) ) ).
% or_int_rec
thf(fact_5440_or__nat__unfold,axiom,
( ( bit_se1065995026697491101ons_or @ nat )
= ( ^ [M4: nat,N3: nat] :
( if @ nat
@ ( M4
= ( zero_zero @ nat ) )
@ N3
@ ( if @ nat
@ ( N3
= ( zero_zero @ nat ) )
@ M4
@ ( plus_plus @ nat @ ( ord_max @ nat @ ( modulo_modulo @ nat @ M4 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ ( modulo_modulo @ nat @ N3 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( bit_se1065995026697491101ons_or @ nat @ ( divide_divide @ nat @ M4 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ ( divide_divide @ nat @ N3 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) ) ) ) ) ).
% or_nat_unfold
thf(fact_5441_or__int__unfold,axiom,
( ( bit_se1065995026697491101ons_or @ int )
= ( ^ [K3: int,L2: int] :
( if @ int
@ ( ( K3
= ( uminus_uminus @ int @ ( one_one @ int ) ) )
| ( L2
= ( uminus_uminus @ int @ ( one_one @ int ) ) ) )
@ ( uminus_uminus @ int @ ( one_one @ int ) )
@ ( if @ int
@ ( K3
= ( zero_zero @ int ) )
@ L2
@ ( if @ int
@ ( L2
= ( zero_zero @ int ) )
@ K3
@ ( plus_plus @ int @ ( ord_max @ int @ ( modulo_modulo @ int @ K3 @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) ) @ ( modulo_modulo @ int @ L2 @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) ) ) @ ( times_times @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ ( bit_se1065995026697491101ons_or @ int @ ( divide_divide @ int @ K3 @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) ) @ ( divide_divide @ int @ L2 @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) ) ) ) ) ) ) ) ) ) ).
% or_int_unfold
thf(fact_5442_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_5443_or__minus__numerals_I1_J,axiom,
! [N: num] :
( ( bit_se1065995026697491101ons_or @ int @ ( one_one @ int ) @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ ( bit0 @ N ) ) ) )
= ( uminus_uminus @ int @ ( numeral_numeral @ int @ ( bit_or_not_num_neg @ one2 @ ( bitM @ N ) ) ) ) ) ).
% or_minus_numerals(1)
thf(fact_5444_or__minus__numerals_I5_J,axiom,
! [N: num] :
( ( bit_se1065995026697491101ons_or @ int @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ ( bit0 @ N ) ) ) @ ( one_one @ int ) )
= ( uminus_uminus @ int @ ( numeral_numeral @ int @ ( bit_or_not_num_neg @ one2 @ ( bitM @ N ) ) ) ) ) ).
% or_minus_numerals(5)
thf(fact_5445_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_5446_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_5447_word__no__log__defs_I11_J,axiom,
! [K8: $tType] :
( ( type_len @ K8 )
=> ! [A2: num,B2: num] :
( ( bit_se5824344971392196577ns_xor @ ( word @ K8 ) @ ( numeral_numeral @ ( word @ K8 ) @ A2 ) @ ( numeral_numeral @ ( word @ K8 ) @ B2 ) )
= ( ring_1_of_int @ ( word @ K8 ) @ ( bit_se5824344971392196577ns_xor @ int @ ( numeral_numeral @ int @ A2 ) @ ( numeral_numeral @ int @ B2 ) ) ) ) ) ).
% word_no_log_defs(11)
thf(fact_5448_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_5449_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_5450_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_5451_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_5452_or__minus__numerals_I4_J,axiom,
! [M: num,N: num] :
( ( bit_se1065995026697491101ons_or @ int @ ( numeral_numeral @ int @ M ) @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ ( bit1 @ N ) ) ) )
= ( uminus_uminus @ int @ ( numeral_numeral @ int @ ( bit_or_not_num_neg @ M @ ( bit0 @ N ) ) ) ) ) ).
% or_minus_numerals(4)
thf(fact_5453_or__minus__numerals_I8_J,axiom,
! [N: num,M: num] :
( ( bit_se1065995026697491101ons_or @ int @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ ( bit1 @ N ) ) ) @ ( numeral_numeral @ int @ M ) )
= ( uminus_uminus @ int @ ( numeral_numeral @ int @ ( bit_or_not_num_neg @ M @ ( bit0 @ N ) ) ) ) ) ).
% or_minus_numerals(8)
thf(fact_5454_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_5455_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_5456_word__no__log__defs_I14_J,axiom,
! [N11: $tType] :
( ( type_len @ N11 )
=> ! [A2: num,B2: num] :
( ( bit_se5824344971392196577ns_xor @ ( word @ N11 ) @ ( uminus_uminus @ ( word @ N11 ) @ ( numeral_numeral @ ( word @ N11 ) @ A2 ) ) @ ( uminus_uminus @ ( word @ N11 ) @ ( numeral_numeral @ ( word @ N11 ) @ B2 ) ) )
= ( ring_1_of_int @ ( word @ N11 ) @ ( bit_se5824344971392196577ns_xor @ int @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ A2 ) ) @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ B2 ) ) ) ) ) ) ).
% word_no_log_defs(14)
thf(fact_5457_word__no__log__defs_I13_J,axiom,
! [M10: $tType] :
( ( type_len @ M10 )
=> ! [A2: num,B2: num] :
( ( bit_se5824344971392196577ns_xor @ ( word @ M10 ) @ ( uminus_uminus @ ( word @ M10 ) @ ( numeral_numeral @ ( word @ M10 ) @ A2 ) ) @ ( numeral_numeral @ ( word @ M10 ) @ B2 ) )
= ( ring_1_of_int @ ( word @ M10 ) @ ( bit_se5824344971392196577ns_xor @ int @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ A2 ) ) @ ( numeral_numeral @ int @ B2 ) ) ) ) ) ).
% word_no_log_defs(13)
thf(fact_5458_word__no__log__defs_I12_J,axiom,
! [L5: $tType] :
( ( type_len @ L5 )
=> ! [A2: num,B2: num] :
( ( bit_se5824344971392196577ns_xor @ ( word @ L5 ) @ ( numeral_numeral @ ( word @ L5 ) @ A2 ) @ ( uminus_uminus @ ( word @ L5 ) @ ( numeral_numeral @ ( word @ L5 ) @ B2 ) ) )
= ( ring_1_of_int @ ( word @ L5 ) @ ( bit_se5824344971392196577ns_xor @ int @ ( numeral_numeral @ int @ A2 ) @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ B2 ) ) ) ) ) ) ).
% word_no_log_defs(12)
thf(fact_5459_or__minus__numerals_I7_J,axiom,
! [N: num,M: num] :
( ( bit_se1065995026697491101ons_or @ int @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ ( bit0 @ N ) ) ) @ ( numeral_numeral @ int @ M ) )
= ( uminus_uminus @ int @ ( numeral_numeral @ int @ ( bit_or_not_num_neg @ M @ ( bitM @ N ) ) ) ) ) ).
% or_minus_numerals(7)
thf(fact_5460_or__minus__numerals_I3_J,axiom,
! [M: num,N: num] :
( ( bit_se1065995026697491101ons_or @ int @ ( numeral_numeral @ int @ M ) @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ ( bit0 @ N ) ) ) )
= ( uminus_uminus @ int @ ( numeral_numeral @ int @ ( bit_or_not_num_neg @ M @ ( bitM @ N ) ) ) ) ) ).
% or_minus_numerals(3)
thf(fact_5461_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_5462_word__bitwise__1__simps_I13_J,axiom,
! [M10: $tType] :
( ( type_len @ M10 )
=> ! [A2: num] :
( ( bit_se5824344971392196577ns_xor @ ( word @ M10 ) @ ( uminus_uminus @ ( word @ M10 ) @ ( numeral_numeral @ ( word @ M10 ) @ A2 ) ) @ ( one_one @ ( word @ M10 ) ) )
= ( ring_1_of_int @ ( word @ M10 ) @ ( bit_se5824344971392196577ns_xor @ int @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ A2 ) ) @ ( one_one @ int ) ) ) ) ) ).
% word_bitwise_1_simps(13)
thf(fact_5463_word__log__esimps_I5_J,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [X: word @ A] :
( ( bit_se5824344971392196577ns_xor @ ( word @ A ) @ X @ ( zero_zero @ ( word @ A ) ) )
= X ) ) ).
% word_log_esimps(5)
thf(fact_5464_word__log__esimps_I11_J,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [X: word @ A] :
( ( bit_se5824344971392196577ns_xor @ ( word @ A ) @ ( zero_zero @ ( word @ A ) ) @ X )
= X ) ) ).
% word_log_esimps(11)
thf(fact_5465_word__bw__same_I3_J,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [X: word @ A] :
( ( bit_se5824344971392196577ns_xor @ ( word @ A ) @ X @ X )
= ( zero_zero @ ( word @ A ) ) ) ) ).
% word_bw_same(3)
thf(fact_5466_or__not__num__neg_Osimps_I1_J,axiom,
( ( bit_or_not_num_neg @ one2 @ one2 )
= one2 ) ).
% or_not_num_neg.simps(1)
thf(fact_5467_or__not__num__neg_Osimps_I4_J,axiom,
! [N: num] :
( ( bit_or_not_num_neg @ ( bit0 @ N ) @ one2 )
= ( bit0 @ one2 ) ) ).
% or_not_num_neg.simps(4)
thf(fact_5468_or__not__num__neg_Osimps_I6_J,axiom,
! [N: num,M: num] :
( ( bit_or_not_num_neg @ ( bit0 @ N ) @ ( bit1 @ M ) )
= ( bit0 @ ( bit_or_not_num_neg @ N @ M ) ) ) ).
% or_not_num_neg.simps(6)
thf(fact_5469_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_5470_or__not__num__neg_Osimps_I7_J,axiom,
! [N: num] :
( ( bit_or_not_num_neg @ ( bit1 @ N ) @ one2 )
= one2 ) ).
% or_not_num_neg.simps(7)
thf(fact_5471_or__not__num__neg_Osimps_I5_J,axiom,
! [N: num,M: num] :
( ( bit_or_not_num_neg @ ( bit0 @ N ) @ ( bit0 @ M ) )
= ( bitM @ ( bit_or_not_num_neg @ N @ M ) ) ) ).
% or_not_num_neg.simps(5)
thf(fact_5472_or__not__num__neg_Osimps_I9_J,axiom,
! [N: num,M: num] :
( ( bit_or_not_num_neg @ ( bit1 @ N ) @ ( bit1 @ M ) )
= ( bitM @ ( bit_or_not_num_neg @ N @ M ) ) ) ).
% or_not_num_neg.simps(9)
thf(fact_5473_xor__nat__def,axiom,
( ( bit_se5824344971392196577ns_xor @ nat )
= ( ^ [M4: nat,N3: nat] : ( nat2 @ ( bit_se5824344971392196577ns_xor @ int @ ( semiring_1_of_nat @ int @ M4 ) @ ( semiring_1_of_nat @ int @ N3 ) ) ) ) ) ).
% xor_nat_def
thf(fact_5474_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_5475_or__not__num__neg_Osimps_I8_J,axiom,
! [N: num,M: num] :
( ( bit_or_not_num_neg @ ( bit1 @ N ) @ ( bit0 @ M ) )
= ( bitM @ ( bit_or_not_num_neg @ N @ M ) ) ) ).
% or_not_num_neg.simps(8)
thf(fact_5476_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 )
=> ! [M2: num] :
( ( Xa
= ( bit0 @ M2 ) )
=> ( Y
!= ( bit1 @ M2 ) ) ) )
=> ( ( ( X = one2 )
=> ! [M2: num] :
( ( Xa
= ( bit1 @ M2 ) )
=> ( Y
!= ( bit1 @ M2 ) ) ) )
=> ( ( ? [N2: num] :
( X
= ( bit0 @ N2 ) )
=> ( ( Xa = one2 )
=> ( Y
!= ( bit0 @ one2 ) ) ) )
=> ( ! [N2: num] :
( ( X
= ( bit0 @ N2 ) )
=> ! [M2: num] :
( ( Xa
= ( bit0 @ M2 ) )
=> ( Y
!= ( bitM @ ( bit_or_not_num_neg @ N2 @ M2 ) ) ) ) )
=> ( ! [N2: num] :
( ( X
= ( bit0 @ N2 ) )
=> ! [M2: num] :
( ( Xa
= ( bit1 @ M2 ) )
=> ( Y
!= ( bit0 @ ( bit_or_not_num_neg @ N2 @ M2 ) ) ) ) )
=> ( ( ? [N2: num] :
( X
= ( bit1 @ N2 ) )
=> ( ( Xa = one2 )
=> ( Y != one2 ) ) )
=> ( ! [N2: num] :
( ( X
= ( bit1 @ N2 ) )
=> ! [M2: num] :
( ( Xa
= ( bit0 @ M2 ) )
=> ( Y
!= ( bitM @ ( bit_or_not_num_neg @ N2 @ M2 ) ) ) ) )
=> ~ ! [N2: num] :
( ( X
= ( bit1 @ N2 ) )
=> ! [M2: num] :
( ( Xa
= ( bit1 @ M2 ) )
=> ( Y
!= ( bitM @ ( bit_or_not_num_neg @ N2 @ M2 ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).
% or_not_num_neg.elims
thf(fact_5477_word__ops__nth__size,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [N: nat,X: word @ A,Y: word @ A] :
( ( ord_less @ nat @ N @ ( size_size @ ( word @ A ) @ X ) )
=> ( ( ( bit_se5641148757651400278ts_bit @ ( word @ A ) @ ( bit_se1065995026697491101ons_or @ ( word @ A ) @ X @ Y ) @ N )
= ( ( bit_se5641148757651400278ts_bit @ ( word @ A ) @ X @ N )
| ( bit_se5641148757651400278ts_bit @ ( word @ A ) @ Y @ N ) ) )
& ( ( bit_se5641148757651400278ts_bit @ ( word @ A ) @ ( bit_se5824344872417868541ns_and @ ( word @ A ) @ X @ Y ) @ N )
= ( ( bit_se5641148757651400278ts_bit @ ( word @ A ) @ X @ N )
& ( bit_se5641148757651400278ts_bit @ ( word @ A ) @ Y @ N ) ) )
& ( ( bit_se5641148757651400278ts_bit @ ( word @ A ) @ ( bit_se5824344971392196577ns_xor @ ( word @ A ) @ X @ Y ) @ N )
= ( ( bit_se5641148757651400278ts_bit @ ( word @ A ) @ X @ N )
!= ( bit_se5641148757651400278ts_bit @ ( word @ A ) @ Y @ N ) ) )
& ( ( bit_se5641148757651400278ts_bit @ ( word @ A ) @ ( bit_ri4277139882892585799ns_not @ ( word @ A ) @ X ) @ N )
= ( ~ ( bit_se5641148757651400278ts_bit @ ( word @ A ) @ X @ N ) ) ) ) ) ) ).
% word_ops_nth_size
thf(fact_5478_numeral__or__not__num__eq,axiom,
! [M: num,N: num] :
( ( numeral_numeral @ int @ ( bit_or_not_num_neg @ M @ N ) )
= ( uminus_uminus @ int @ ( bit_se1065995026697491101ons_or @ int @ ( numeral_numeral @ int @ M ) @ ( bit_ri4277139882892585799ns_not @ int @ ( numeral_numeral @ int @ N ) ) ) ) ) ).
% numeral_or_not_num_eq
thf(fact_5479_int__numeral__not__or__num__neg,axiom,
! [M: num,N: num] :
( ( bit_se1065995026697491101ons_or @ int @ ( bit_ri4277139882892585799ns_not @ int @ ( numeral_numeral @ int @ M ) ) @ ( numeral_numeral @ int @ N ) )
= ( uminus_uminus @ int @ ( numeral_numeral @ int @ ( bit_or_not_num_neg @ N @ M ) ) ) ) ).
% int_numeral_not_or_num_neg
thf(fact_5480_int__numeral__or__not__num__neg,axiom,
! [M: num,N: num] :
( ( bit_se1065995026697491101ons_or @ int @ ( numeral_numeral @ int @ M ) @ ( bit_ri4277139882892585799ns_not @ int @ ( numeral_numeral @ int @ N ) ) )
= ( uminus_uminus @ int @ ( numeral_numeral @ int @ ( bit_or_not_num_neg @ M @ N ) ) ) ) ).
% int_numeral_or_not_num_neg
thf(fact_5481_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_5482_xor__nat__unfold,axiom,
( ( bit_se5824344971392196577ns_xor @ nat )
= ( ^ [M4: nat,N3: nat] :
( if @ nat
@ ( M4
= ( zero_zero @ nat ) )
@ N3
@ ( if @ nat
@ ( N3
= ( zero_zero @ nat ) )
@ M4
@ ( plus_plus @ nat @ ( modulo_modulo @ nat @ ( plus_plus @ nat @ ( modulo_modulo @ nat @ M4 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ ( modulo_modulo @ nat @ N3 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( bit_se5824344971392196577ns_xor @ nat @ ( divide_divide @ nat @ M4 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ ( divide_divide @ nat @ N3 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) ) ) ) ) ).
% xor_nat_unfold
thf(fact_5483_xor__nat__rec,axiom,
( ( bit_se5824344971392196577ns_xor @ nat )
= ( ^ [M4: nat,N3: nat] :
( plus_plus @ nat
@ ( zero_neq_one_of_bool @ nat
@ ( ( ~ ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ M4 ) )
!= ( ~ ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N3 ) ) ) )
@ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( bit_se5824344971392196577ns_xor @ nat @ ( divide_divide @ nat @ M4 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ ( divide_divide @ nat @ N3 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) ) ) ).
% xor_nat_rec
thf(fact_5484_xor__Suc__0__eq,axiom,
! [N: nat] :
( ( bit_se5824344971392196577ns_xor @ nat @ N @ ( suc @ ( zero_zero @ nat ) ) )
= ( minus_minus @ nat @ ( plus_plus @ nat @ N @ ( zero_neq_one_of_bool @ nat @ ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N ) ) )
@ ( zero_neq_one_of_bool @ nat
@ ~ ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N ) ) ) ) ).
% xor_Suc_0_eq
thf(fact_5485_Suc__0__xor__eq,axiom,
! [N: nat] :
( ( bit_se5824344971392196577ns_xor @ nat @ ( suc @ ( zero_zero @ nat ) ) @ N )
= ( minus_minus @ nat @ ( plus_plus @ nat @ N @ ( zero_neq_one_of_bool @ nat @ ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N ) ) )
@ ( zero_neq_one_of_bool @ nat
@ ~ ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N ) ) ) ) ).
% Suc_0_xor_eq
thf(fact_5486_word__ops__lsb,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [X: word @ A,Y: word @ A] :
( ( ( bit_se5641148757651400278ts_bit @ ( word @ A ) @ ( bit_se1065995026697491101ons_or @ ( word @ A ) @ X @ Y ) @ ( zero_zero @ nat ) )
= ( ( bit_se5641148757651400278ts_bit @ ( word @ A ) @ X @ ( zero_zero @ nat ) )
| ( bit_se5641148757651400278ts_bit @ ( word @ A ) @ Y @ ( zero_zero @ nat ) ) ) )
& ( ( bit_se5641148757651400278ts_bit @ ( word @ A ) @ ( bit_se5824344872417868541ns_and @ ( word @ A ) @ X @ Y ) @ ( zero_zero @ nat ) )
= ( ( bit_se5641148757651400278ts_bit @ ( word @ A ) @ X @ ( zero_zero @ nat ) )
& ( bit_se5641148757651400278ts_bit @ ( word @ A ) @ Y @ ( zero_zero @ nat ) ) ) )
& ( ( bit_se5641148757651400278ts_bit @ ( word @ A ) @ ( bit_se5824344971392196577ns_xor @ ( word @ A ) @ X @ Y ) @ ( zero_zero @ nat ) )
= ( ( bit_se5641148757651400278ts_bit @ ( word @ A ) @ X @ ( zero_zero @ nat ) )
!= ( bit_se5641148757651400278ts_bit @ ( word @ A ) @ Y @ ( zero_zero @ nat ) ) ) )
& ( ( bit_se5641148757651400278ts_bit @ ( word @ A ) @ ( bit_ri4277139882892585799ns_not @ ( word @ A ) @ X ) @ ( zero_zero @ nat ) )
= ( ~ ( bit_se5641148757651400278ts_bit @ ( word @ A ) @ X @ ( zero_zero @ nat ) ) ) ) ) ) ).
% word_ops_lsb
thf(fact_5487_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_5488_aux,axiom,
! [B: $tType,A: $tType,P: A > B > assn,A2: A,As2: list @ A,C2: B,Cs: list @ B] :
( ( finite_fold @ nat @ assn
@ ^ [I2: nat,Aa2: assn] : ( times_times @ assn @ Aa2 @ ( P @ ( nth @ A @ ( cons @ A @ A2 @ As2 ) @ I2 ) @ ( nth @ B @ ( cons @ B @ C2 @ Cs ) @ I2 ) ) )
@ ( one_one @ assn )
@ ( set_or7035219750837199246ssThan @ nat @ ( zero_zero @ nat ) @ ( suc @ ( size_size @ ( list @ A ) @ As2 ) ) ) )
= ( times_times @ assn @ ( P @ A2 @ C2 )
@ ( finite_fold @ nat @ assn
@ ^ [I2: nat,Aa2: assn] : ( times_times @ assn @ Aa2 @ ( P @ ( nth @ A @ As2 @ I2 ) @ ( nth @ B @ Cs @ I2 ) ) )
@ ( one_one @ assn )
@ ( set_or7035219750837199246ssThan @ nat @ ( zero_zero @ nat ) @ ( size_size @ ( list @ A ) @ As2 ) ) ) ) ) ).
% aux
thf(fact_5489_list_Oinject,axiom,
! [A: $tType,X21: A,X222: list @ A,Y21: A,Y222: list @ A] :
( ( ( cons @ A @ X21 @ X222 )
= ( cons @ A @ Y21 @ Y222 ) )
= ( ( X21 = Y21 )
& ( X222 = Y222 ) ) ) ).
% list.inject
thf(fact_5490_length__nth__simps_I4_J,axiom,
! [B: $tType,X: B,Xs2: list @ B,N: nat] :
( ( nth @ B @ ( cons @ B @ X @ Xs2 ) @ ( suc @ N ) )
= ( nth @ B @ Xs2 @ N ) ) ).
% length_nth_simps(4)
thf(fact_5491_nth__Cons__Suc,axiom,
! [A: $tType,X: A,Xs2: list @ A,N: nat] :
( ( nth @ A @ ( cons @ A @ X @ Xs2 ) @ ( suc @ N ) )
= ( nth @ A @ Xs2 @ N ) ) ).
% nth_Cons_Suc
thf(fact_5492_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_5493_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_5494_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_5495_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_5496_nth__Cons__pos,axiom,
! [A: $tType,N: nat,X: A,Xs2: list @ A] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ N )
=> ( ( nth @ A @ ( cons @ A @ X @ Xs2 ) @ N )
= ( nth @ A @ Xs2 @ ( minus_minus @ nat @ N @ ( one_one @ nat ) ) ) ) ) ).
% nth_Cons_pos
thf(fact_5497_map__eq__Cons__conv,axiom,
! [A: $tType,B: $tType,F2: B > A,Xs2: list @ B,Y: A,Ys: list @ A] :
( ( ( map @ B @ A @ F2 @ Xs2 )
= ( cons @ A @ Y @ Ys ) )
= ( ? [Z7: B,Zs: list @ B] :
( ( Xs2
= ( cons @ B @ Z7 @ Zs ) )
& ( ( F2 @ Z7 )
= Y )
& ( ( map @ B @ A @ F2 @ Zs )
= Ys ) ) ) ) ).
% map_eq_Cons_conv
thf(fact_5498_Cons__eq__map__conv,axiom,
! [A: $tType,B: $tType,X: A,Xs2: list @ A,F2: B > A,Ys: list @ B] :
( ( ( cons @ A @ X @ Xs2 )
= ( map @ B @ A @ F2 @ Ys ) )
= ( ? [Z7: B,Zs: list @ B] :
( ( Ys
= ( cons @ B @ Z7 @ Zs ) )
& ( X
= ( F2 @ Z7 ) )
& ( Xs2
= ( map @ B @ A @ F2 @ Zs ) ) ) ) ) ).
% Cons_eq_map_conv
thf(fact_5499_map__eq__Cons__D,axiom,
! [B: $tType,A: $tType,F2: B > A,Xs2: list @ B,Y: A,Ys: list @ A] :
( ( ( map @ B @ A @ F2 @ Xs2 )
= ( cons @ A @ Y @ Ys ) )
=> ? [Z3: B,Zs2: list @ B] :
( ( Xs2
= ( cons @ B @ Z3 @ Zs2 ) )
& ( ( F2 @ Z3 )
= Y )
& ( ( map @ B @ A @ F2 @ Zs2 )
= Ys ) ) ) ).
% map_eq_Cons_D
thf(fact_5500_Cons__eq__map__D,axiom,
! [A: $tType,B: $tType,X: A,Xs2: list @ A,F2: B > A,Ys: list @ B] :
( ( ( cons @ A @ X @ Xs2 )
= ( map @ B @ A @ F2 @ Ys ) )
=> ? [Z3: B,Zs2: list @ B] :
( ( Ys
= ( cons @ B @ Z3 @ Zs2 ) )
& ( X
= ( F2 @ Z3 ) )
& ( Xs2
= ( map @ B @ A @ F2 @ Zs2 ) ) ) ) ).
% Cons_eq_map_D
thf(fact_5501_list_Osimps_I9_J,axiom,
! [B: $tType,A: $tType,F2: A > B,X21: A,X222: list @ A] :
( ( map @ A @ B @ F2 @ ( cons @ A @ X21 @ X222 ) )
= ( cons @ B @ ( F2 @ X21 ) @ ( map @ A @ B @ F2 @ X222 ) ) ) ).
% list.simps(9)
thf(fact_5502_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_5503_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_5504_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_5505_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_5506_not__Cons__self2,axiom,
! [A: $tType,X: A,Xs2: list @ A] :
( ( cons @ A @ X @ Xs2 )
!= Xs2 ) ).
% not_Cons_self2
thf(fact_5507_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_5508_Suc__length__conv,axiom,
! [A: $tType,N: nat,Xs2: list @ A] :
( ( ( suc @ N )
= ( size_size @ ( list @ A ) @ Xs2 ) )
= ( ? [Y6: A,Ys3: list @ A] :
( ( Xs2
= ( cons @ A @ Y6 @ Ys3 ) )
& ( ( size_size @ ( list @ A ) @ Ys3 )
= N ) ) ) ) ).
% Suc_length_conv
thf(fact_5509_length__Suc__conv,axiom,
! [A: $tType,Xs2: list @ A,N: nat] :
( ( ( size_size @ ( list @ A ) @ Xs2 )
= ( suc @ N ) )
= ( ? [Y6: A,Ys3: list @ A] :
( ( Xs2
= ( cons @ A @ Y6 @ Ys3 ) )
& ( ( size_size @ ( list @ A ) @ Ys3 )
= N ) ) ) ) ).
% length_Suc_conv
thf(fact_5510_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_5511_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_5512_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_5513_list__update__code_I2_J,axiom,
! [A: $tType,X: A,Xs2: list @ A,Y: A] :
( ( list_update @ A @ ( cons @ A @ X @ Xs2 ) @ ( zero_zero @ nat ) @ Y )
= ( cons @ A @ Y @ Xs2 ) ) ).
% list_update_code(2)
thf(fact_5514_replicate__Suc,axiom,
! [A: $tType,N: nat,X: A] :
( ( replicate @ A @ ( suc @ N ) @ X )
= ( cons @ A @ X @ ( replicate @ A @ N @ X ) ) ) ).
% replicate_Suc
thf(fact_5515_list__assn_Osimps_I2_J,axiom,
! [A: $tType,C: $tType,P: A > C > assn,A2: A,As2: list @ A,C2: C,Cs: list @ C] :
( ( vEBT_List_list_assn @ A @ C @ P @ ( cons @ A @ A2 @ As2 ) @ ( cons @ C @ C2 @ Cs ) )
= ( times_times @ assn @ ( P @ A2 @ C2 ) @ ( vEBT_List_list_assn @ A @ C @ P @ As2 @ Cs ) ) ) ).
% list_assn.simps(2)
thf(fact_5516_list__assn__simps_I2_J,axiom,
! [A: $tType,B: $tType,P: A > B > assn,A2: A,As2: list @ A,C2: B,Cs: list @ B] :
( ( vEBT_List_list_assn @ A @ B @ P @ ( cons @ A @ A2 @ As2 ) @ ( cons @ B @ C2 @ Cs ) )
= ( times_times @ assn @ ( P @ A2 @ C2 ) @ ( vEBT_List_list_assn @ A @ B @ P @ As2 @ Cs ) ) ) ).
% list_assn_simps(2)
thf(fact_5517_Suc__le__length__iff,axiom,
! [A: $tType,N: nat,Xs2: list @ A] :
( ( ord_less_eq @ nat @ ( suc @ N ) @ ( size_size @ ( list @ A ) @ Xs2 ) )
= ( ? [X2: A,Ys3: list @ A] :
( ( Xs2
= ( cons @ A @ X2 @ Ys3 ) )
& ( ord_less_eq @ nat @ N @ ( size_size @ ( list @ A ) @ Ys3 ) ) ) ) ) ).
% Suc_le_length_iff
thf(fact_5518_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_5519_nth__Cons_H,axiom,
! [A: $tType,N: nat,X: A,Xs2: list @ A] :
( ( ( N
= ( zero_zero @ nat ) )
=> ( ( nth @ A @ ( cons @ A @ X @ Xs2 ) @ N )
= X ) )
& ( ( N
!= ( zero_zero @ nat ) )
=> ( ( nth @ A @ ( cons @ A @ X @ Xs2 ) @ N )
= ( nth @ A @ Xs2 @ ( minus_minus @ nat @ N @ ( one_one @ nat ) ) ) ) ) ) ).
% nth_Cons'
thf(fact_5520_nth__non__equal__first__eq,axiom,
! [A: $tType,X: A,Y: A,Xs2: list @ A,N: nat] :
( ( X != Y )
=> ( ( ( nth @ A @ ( cons @ A @ X @ Xs2 ) @ N )
= Y )
= ( ( ( nth @ A @ Xs2 @ ( minus_minus @ nat @ N @ ( one_one @ nat ) ) )
= Y )
& ( ord_less @ nat @ ( zero_zero @ nat ) @ N ) ) ) ) ).
% nth_non_equal_first_eq
thf(fact_5521_nth__equal__first__eq,axiom,
! [A: $tType,X: A,Xs2: list @ A,N: nat] :
( ~ ( member @ A @ X @ ( set2 @ A @ Xs2 ) )
=> ( ( ord_less_eq @ nat @ N @ ( size_size @ ( list @ A ) @ Xs2 ) )
=> ( ( ( nth @ A @ ( cons @ A @ X @ Xs2 ) @ N )
= X )
= ( N
= ( zero_zero @ nat ) ) ) ) ) ).
% nth_equal_first_eq
thf(fact_5522_Cons__replicate__eq,axiom,
! [A: $tType,X: A,Xs2: list @ A,N: nat,Y: A] :
( ( ( cons @ A @ X @ Xs2 )
= ( replicate @ A @ N @ Y ) )
= ( ( X = Y )
& ( ord_less @ nat @ ( zero_zero @ nat ) @ N )
& ( Xs2
= ( replicate @ A @ ( minus_minus @ nat @ N @ ( one_one @ nat ) ) @ X ) ) ) ) ).
% Cons_replicate_eq
thf(fact_5523_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_5524_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_5525_lsb__odd,axiom,
! [A: $tType] :
( ( least_6119777620449941438nt_lsb @ A )
=> ( ( least_8051144512741203767sb_lsb @ A )
= ( ^ [A4: A] :
~ ( dvd_dvd @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ A4 ) ) ) ) ).
% lsb_odd
thf(fact_5526_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_5527_int__lsb__numeral_I1_J,axiom,
~ ( least_8051144512741203767sb_lsb @ int @ ( zero_zero @ int ) ) ).
% int_lsb_numeral(1)
thf(fact_5528_int__lsb__numeral_I2_J,axiom,
least_8051144512741203767sb_lsb @ int @ ( one_one @ int ) ).
% int_lsb_numeral(2)
thf(fact_5529_int__lsb__numeral_I6_J,axiom,
! [W: num] :
~ ( least_8051144512741203767sb_lsb @ int @ ( numeral_numeral @ int @ ( bit0 @ W ) ) ) ).
% int_lsb_numeral(6)
thf(fact_5530_int__lsb__numeral_I3_J,axiom,
least_8051144512741203767sb_lsb @ int @ ( numeral_numeral @ int @ one2 ) ).
% int_lsb_numeral(3)
thf(fact_5531_int__lsb__numeral_I7_J,axiom,
! [W: num] : ( least_8051144512741203767sb_lsb @ int @ ( numeral_numeral @ int @ ( bit1 @ W ) ) ) ).
% int_lsb_numeral(7)
thf(fact_5532_int__lsb__numeral_I4_J,axiom,
least_8051144512741203767sb_lsb @ int @ ( uminus_uminus @ int @ ( one_one @ int ) ) ).
% int_lsb_numeral(4)
thf(fact_5533_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_5534_int__lsb__numeral_I5_J,axiom,
least_8051144512741203767sb_lsb @ int @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ one2 ) ) ).
% int_lsb_numeral(5)
thf(fact_5535_int__lsb__numeral_I9_J,axiom,
! [W: num] : ( least_8051144512741203767sb_lsb @ int @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ ( bit1 @ W ) ) ) ) ).
% int_lsb_numeral(9)
thf(fact_5536_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_5537_lsb__integer__code,axiom,
( ( least_8051144512741203767sb_lsb @ code_integer )
= ( ^ [X2: code_integer] : ( bit_se5641148757651400278ts_bit @ code_integer @ X2 @ ( zero_zero @ nat ) ) ) ) ).
% lsb_integer_code
thf(fact_5538_lsb__int__def,axiom,
( ( least_8051144512741203767sb_lsb @ int )
= ( ^ [I2: int] : ( bit_se5641148757651400278ts_bit @ int @ I2 @ ( zero_zero @ nat ) ) ) ) ).
% lsb_int_def
thf(fact_5539_word__lsb__alt,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ( ( least_8051144512741203767sb_lsb @ ( word @ A ) )
= ( ^ [W2: word @ A] : ( bit_se5641148757651400278ts_bit @ ( word @ A ) @ W2 @ ( zero_zero @ nat ) ) ) ) ) ).
% word_lsb_alt
thf(fact_5540_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_5541_bin__last__conv__lsb,axiom,
( ( ^ [A4: int] :
~ ( dvd_dvd @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ A4 ) )
= ( least_8051144512741203767sb_lsb @ int ) ) ).
% bin_last_conv_lsb
thf(fact_5542_lsb__word__eq,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ( ( least_8051144512741203767sb_lsb @ ( word @ A ) )
= ( ^ [A4: word @ A] :
~ ( dvd_dvd @ ( word @ A ) @ ( numeral_numeral @ ( word @ A ) @ ( bit0 @ one2 ) ) @ A4 ) ) ) ) ).
% lsb_word_eq
thf(fact_5543_word__lsb__def,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ( ( least_8051144512741203767sb_lsb @ ( word @ A ) )
= ( ^ [A4: word @ A] :
~ ( dvd_dvd @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ ( semiring_1_unsigned @ A @ int @ A4 ) ) ) ) ) ).
% word_lsb_def
thf(fact_5544_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_5545_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_5546_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_5547_vebt__buildup_Opelims,axiom,
! [X: nat,Y: vEBT_VEBT] :
( ( ( vEBT_vebt_buildup @ X )
= Y )
=> ( ( accp @ nat @ vEBT_v4011308405150292612up_rel @ X )
=> ( ( ( X
= ( zero_zero @ nat ) )
=> ( ( Y
= ( vEBT_Leaf @ $false @ $false ) )
=> ~ ( accp @ nat @ vEBT_v4011308405150292612up_rel @ ( zero_zero @ nat ) ) ) )
=> ( ( ( X
= ( suc @ ( zero_zero @ nat ) ) )
=> ( ( Y
= ( vEBT_Leaf @ $false @ $false ) )
=> ~ ( accp @ nat @ vEBT_v4011308405150292612up_rel @ ( suc @ ( zero_zero @ nat ) ) ) ) )
=> ~ ! [Va: nat] :
( ( X
= ( suc @ ( suc @ Va ) ) )
=> ( ( ( ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( suc @ ( suc @ Va ) ) )
=> ( Y
= ( vEBT_Node @ ( none @ ( product_prod @ nat @ nat ) ) @ ( suc @ ( suc @ Va ) ) @ ( replicate @ vEBT_VEBT @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( vEBT_vebt_buildup @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( vEBT_vebt_buildup @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) )
& ( ~ ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( suc @ ( suc @ Va ) ) )
=> ( Y
= ( vEBT_Node @ ( none @ ( product_prod @ nat @ nat ) ) @ ( suc @ ( suc @ Va ) ) @ ( replicate @ vEBT_VEBT @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( suc @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( vEBT_vebt_buildup @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( vEBT_vebt_buildup @ ( suc @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) ) ) )
=> ~ ( accp @ nat @ vEBT_v4011308405150292612up_rel @ ( suc @ ( suc @ Va ) ) ) ) ) ) ) ) ) ).
% vebt_buildup.pelims
thf(fact_5548_sum__count__set,axiom,
! [A: $tType,Xs2: list @ A,X7: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ ( set2 @ A @ Xs2 ) @ X7 )
=> ( ( finite_finite2 @ A @ X7 )
=> ( ( groups7311177749621191930dd_sum @ A @ nat @ ( count_list @ A @ Xs2 ) @ X7 )
= ( size_size @ ( list @ A ) @ Xs2 ) ) ) ) ).
% sum_count_set
thf(fact_5549_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_5550_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_5551_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_5552_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_5553_VEBT__internal_OT_092_060_094sub_062b_092_060_094sub_062u_092_060_094sub_062i_092_060_094sub_062l_092_060_094sub_062d_Opelims,axiom,
! [X: nat,Y: nat] :
( ( ( vEBT_V8646137997579335489_i_l_d @ X )
= Y )
=> ( ( accp @ nat @ vEBT_V5144397997797733112_d_rel @ X )
=> ( ( ( X
= ( zero_zero @ nat ) )
=> ( ( Y
= ( numeral_numeral @ nat @ ( bit0 @ ( bit0 @ one2 ) ) ) )
=> ~ ( accp @ nat @ vEBT_V5144397997797733112_d_rel @ ( zero_zero @ nat ) ) ) )
=> ( ( ( X
= ( suc @ ( zero_zero @ nat ) ) )
=> ( ( Y
= ( numeral_numeral @ nat @ ( bit0 @ ( bit0 @ one2 ) ) ) )
=> ~ ( accp @ nat @ vEBT_V5144397997797733112_d_rel @ ( suc @ ( zero_zero @ nat ) ) ) ) )
=> ~ ! [Va: nat] :
( ( X
= ( suc @ ( suc @ Va ) ) )
=> ( ( ( ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( suc @ ( suc @ Va ) ) )
=> ( Y
= ( plus_plus @ nat @ ( one_one @ nat ) @ ( plus_plus @ nat @ ( plus_plus @ nat @ ( numeral_numeral @ nat @ ( bit0 @ ( bit1 @ ( bit0 @ one2 ) ) ) ) @ ( vEBT_V8646137997579335489_i_l_d @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( times_times @ nat @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( vEBT_V8646137997579335489_i_l_d @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) ) ) )
& ( ~ ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( suc @ ( suc @ Va ) ) )
=> ( Y
= ( plus_plus @ nat @ ( plus_plus @ nat @ ( numeral_numeral @ nat @ ( bit0 @ ( bit0 @ ( bit1 @ one2 ) ) ) ) @ ( vEBT_V8646137997579335489_i_l_d @ ( suc @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) @ ( times_times @ nat @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( suc @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( vEBT_V8646137997579335489_i_l_d @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) ) ) )
=> ~ ( accp @ nat @ vEBT_V5144397997797733112_d_rel @ ( suc @ ( suc @ Va ) ) ) ) ) ) ) ) ) ).
% VEBT_internal.T\<^sub>b\<^sub>u\<^sub>i\<^sub>l\<^sub>d.pelims
thf(fact_5554_VEBT__internal_OT_092_060_094sub_062b_092_060_094sub_062u_092_060_094sub_062i_092_060_094sub_062l_092_060_094sub_062d_092_060_094sub_062u_092_060_094sub_062p_Opelims,axiom,
! [X: nat,Y: nat] :
( ( ( vEBT_V8346862874174094_d_u_p @ X )
= Y )
=> ( ( accp @ nat @ vEBT_V1247956027447740395_p_rel @ X )
=> ( ( ( X
= ( zero_zero @ nat ) )
=> ( ( Y
= ( numeral_numeral @ nat @ ( bit1 @ one2 ) ) )
=> ~ ( accp @ nat @ vEBT_V1247956027447740395_p_rel @ ( zero_zero @ nat ) ) ) )
=> ( ( ( X
= ( suc @ ( zero_zero @ nat ) ) )
=> ( ( Y
= ( numeral_numeral @ nat @ ( bit1 @ one2 ) ) )
=> ~ ( accp @ nat @ vEBT_V1247956027447740395_p_rel @ ( suc @ ( zero_zero @ nat ) ) ) ) )
=> ~ ! [Va: nat] :
( ( X
= ( suc @ ( suc @ Va ) ) )
=> ( ( ( ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( suc @ ( suc @ Va ) ) )
=> ( Y
= ( plus_plus @ nat @ ( one_one @ nat ) @ ( plus_plus @ nat @ ( plus_plus @ nat @ ( numeral_numeral @ nat @ ( bit1 @ ( bit0 @ ( bit0 @ one2 ) ) ) ) @ ( vEBT_V8346862874174094_d_u_p @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( times_times @ nat @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( plus_plus @ nat @ ( vEBT_V8346862874174094_d_u_p @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( one_one @ nat ) ) ) ) ) ) )
& ( ~ ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( suc @ ( suc @ Va ) ) )
=> ( Y
= ( plus_plus @ nat @ ( plus_plus @ nat @ ( numeral_numeral @ nat @ ( bit1 @ ( bit1 @ ( bit0 @ one2 ) ) ) ) @ ( vEBT_V8346862874174094_d_u_p @ ( suc @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) @ ( times_times @ nat @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( suc @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( plus_plus @ nat @ ( vEBT_V8346862874174094_d_u_p @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( one_one @ nat ) ) ) ) ) ) )
=> ~ ( accp @ nat @ vEBT_V1247956027447740395_p_rel @ ( suc @ ( suc @ Va ) ) ) ) ) ) ) ) ) ).
% VEBT_internal.T\<^sub>b\<^sub>u\<^sub>i\<^sub>l\<^sub>d\<^sub>u\<^sub>p.pelims
thf(fact_5555_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_5556_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_5557_VEBT__internal_OT__vebt__buildupi_Opelims,axiom,
! [X: nat,Y: nat] :
( ( ( vEBT_V441764108873111860ildupi @ X )
= Y )
=> ( ( accp @ nat @ vEBT_V2957053500504383685pi_rel @ X )
=> ( ( ( X
= ( zero_zero @ nat ) )
=> ( ( Y
= ( suc @ ( zero_zero @ nat ) ) )
=> ~ ( accp @ nat @ vEBT_V2957053500504383685pi_rel @ ( zero_zero @ nat ) ) ) )
=> ( ( ( X
= ( suc @ ( zero_zero @ nat ) ) )
=> ( ( Y
= ( suc @ ( zero_zero @ nat ) ) )
=> ~ ( accp @ nat @ vEBT_V2957053500504383685pi_rel @ ( suc @ ( zero_zero @ nat ) ) ) ) )
=> ~ ! [N2: nat] :
( ( X
= ( suc @ ( suc @ N2 ) ) )
=> ( ( ( ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N2 )
=> ( Y
= ( suc @ ( suc @ ( suc @ ( plus_plus @ nat @ ( vEBT_V441764108873111860ildupi @ ( suc @ ( divide_divide @ nat @ N2 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( plus_plus @ nat @ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ ( bit0 @ one2 ) ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ N2 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( times_times @ nat @ ( vEBT_V441764108873111860ildupi @ ( suc @ ( divide_divide @ nat @ N2 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ N2 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) ) ) ) ) ) ) )
& ( ~ ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N2 )
=> ( Y
= ( suc @ ( suc @ ( suc @ ( plus_plus @ nat @ ( vEBT_V441764108873111860ildupi @ ( suc @ ( suc @ ( divide_divide @ nat @ N2 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) @ ( plus_plus @ nat @ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ ( bit0 @ ( bit0 @ one2 ) ) ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ N2 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ ( bit0 @ one2 ) ) ) @ ( times_times @ nat @ ( vEBT_V441764108873111860ildupi @ ( suc @ ( divide_divide @ nat @ N2 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ N2 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) ) ) ) ) ) ) ) )
=> ~ ( accp @ nat @ vEBT_V2957053500504383685pi_rel @ ( suc @ ( suc @ N2 ) ) ) ) ) ) ) ) ) ).
% VEBT_internal.T_vebt_buildupi.pelims
thf(fact_5558_dup__1,axiom,
( ( code_dup @ ( one_one @ code_integer ) )
= ( numeral_numeral @ code_integer @ ( bit0 @ one2 ) ) ) ).
% dup_1
thf(fact_5559_Code__Numeral_Odup__code_I1_J,axiom,
( ( code_dup @ ( zero_zero @ code_integer ) )
= ( zero_zero @ code_integer ) ) ).
% Code_Numeral.dup_code(1)
thf(fact_5560_vebt__assn__raw_Osimps_I2_J,axiom,
! [Mmo: option @ ( product_prod @ nat @ nat ),Deg: nat,Tree_list: list @ vEBT_VEBT,Summary: vEBT_VEBT,Mmoi: option @ ( product_prod @ nat @ nat ),Degi: nat,Tree_array: array @ vEBT_VEBTi,Summaryi: vEBT_VEBTi] :
( ( vEBT_vebt_assn_raw @ ( vEBT_Node @ Mmo @ Deg @ Tree_list @ Summary ) @ ( vEBT_Nodei @ Mmoi @ Degi @ Tree_array @ Summaryi ) )
= ( times_times @ assn
@ ( times_times @ assn
@ ( pure_assn
@ ( ( Mmoi = Mmo )
& ( Degi = Deg ) ) )
@ ( vEBT_vebt_assn_raw @ Summary @ Summaryi ) )
@ ( ex_assn @ ( list @ vEBT_VEBTi )
@ ^ [Tree_is2: list @ vEBT_VEBTi] : ( times_times @ assn @ ( snga_assn @ vEBT_VEBTi @ Tree_array @ Tree_is2 ) @ ( vEBT_List_list_assn @ vEBT_VEBT @ vEBT_VEBTi @ vEBT_vebt_assn_raw @ Tree_list @ Tree_is2 ) ) ) ) ) ).
% vebt_assn_raw.simps(2)
thf(fact_5561_cis__multiple__2pi,axiom,
! [N: real] :
( ( member @ real @ N @ ( ring_1_Ints @ real ) )
=> ( ( cis @ ( times_times @ real @ ( times_times @ real @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) @ pi ) @ N ) )
= ( one_one @ complex ) ) ) ).
% cis_multiple_2pi
thf(fact_5562_ran__nth__set__encoding__conv,axiom,
! [A: $tType,L: list @ A] :
( ( ran @ nat @ A
@ ^ [I2: nat] : ( if @ ( option @ A ) @ ( ord_less @ nat @ I2 @ ( size_size @ ( list @ A ) @ L ) ) @ ( some @ A @ ( nth @ A @ L @ I2 ) ) @ ( none @ A ) ) )
= ( set2 @ A @ L ) ) ).
% ran_nth_set_encoding_conv
thf(fact_5563_ex__assn__const,axiom,
! [A: $tType,C2: assn] :
( ( ex_assn @ A
@ ^ [X2: A] : C2 )
= C2 ) ).
% ex_assn_const
thf(fact_5564_norm__assertion__simps_I16_J,axiom,
! [A: $tType,Q: A > assn,R: assn] :
( ( times_times @ assn @ ( ex_assn @ A @ Q ) @ R )
= ( ex_assn @ A
@ ^ [X2: A] : ( times_times @ assn @ ( Q @ X2 ) @ R ) ) ) ).
% norm_assertion_simps(16)
thf(fact_5565_norm__assertion__simps_I17_J,axiom,
! [B: $tType,R: assn,Q: B > assn] :
( ( times_times @ assn @ R @ ( ex_assn @ B @ Q ) )
= ( ex_assn @ B
@ ^ [X2: B] : ( times_times @ assn @ R @ ( Q @ X2 ) ) ) ) ).
% norm_assertion_simps(17)
thf(fact_5566_triv__exI,axiom,
! [A: $tType,Q: A > assn,X: A] : ( entails @ ( Q @ X ) @ ( ex_assn @ A @ Q ) ) ).
% triv_exI
thf(fact_5567_mod__ex__dist,axiom,
! [A: $tType,P: A > assn,H2: product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat )] :
( ( rep_assn @ ( ex_assn @ A @ P ) @ H2 )
= ( ? [X2: A] : ( rep_assn @ ( P @ X2 ) @ H2 ) ) ) ).
% mod_ex_dist
thf(fact_5568_frac__eq__0__iff,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [X: A] :
( ( ( archimedean_frac @ A @ X )
= ( zero_zero @ A ) )
= ( member @ A @ X @ ( ring_1_Ints @ A ) ) ) ) ).
% frac_eq_0_iff
thf(fact_5569_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_5570_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_5571_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_5572_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_5573_Ints__numeral,axiom,
! [A: $tType] :
( ( ring_1 @ A )
=> ! [N: num] : ( member @ A @ ( numeral_numeral @ A @ N ) @ ( ring_1_Ints @ A ) ) ) ).
% Ints_numeral
thf(fact_5574_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_5575_ex__distrib__star,axiom,
! [A: $tType,P: A > assn,Q: assn] :
( ( ex_assn @ A
@ ^ [X2: A] : ( times_times @ assn @ ( P @ X2 ) @ Q ) )
= ( times_times @ assn @ ( ex_assn @ A @ P ) @ Q ) ) ).
% ex_distrib_star
thf(fact_5576_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_5577_Ints__1,axiom,
! [A: $tType] :
( ( ring_1 @ A )
=> ( member @ A @ ( one_one @ A ) @ ( ring_1_Ints @ A ) ) ) ).
% Ints_1
thf(fact_5578_Ints__0,axiom,
! [A: $tType] :
( ( ring_1 @ A )
=> ( member @ A @ ( zero_zero @ A ) @ ( ring_1_Ints @ A ) ) ) ).
% Ints_0
thf(fact_5579_Ints__power,axiom,
! [A: $tType] :
( ( ring_1 @ A )
=> ! [A2: A,N: nat] :
( ( member @ A @ A2 @ ( ring_1_Ints @ A ) )
=> ( member @ A @ ( power_power @ A @ A2 @ N ) @ ( ring_1_Ints @ A ) ) ) ) ).
% Ints_power
thf(fact_5580_ex__one__point__gen,axiom,
! [A: $tType,P: A > assn,V: A] :
( ! [H3: product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ),X3: A] :
( ( rep_assn @ ( P @ X3 ) @ H3 )
=> ( X3 = V ) )
=> ( ( ex_assn @ A @ P )
= ( P @ V ) ) ) ).
% ex_one_point_gen
thf(fact_5581_mod__exI,axiom,
! [A: $tType,P: A > assn,H2: product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat )] :
( ? [X5: A] : ( rep_assn @ ( P @ X5 ) @ H2 )
=> ( rep_assn @ ( ex_assn @ A @ P ) @ H2 ) ) ).
% mod_exI
thf(fact_5582_mod__exE,axiom,
! [A: $tType,P: A > assn,H2: product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat )] :
( ( rep_assn @ ( ex_assn @ A @ P ) @ H2 )
=> ~ ! [X3: A] :
~ ( rep_assn @ ( P @ X3 ) @ H2 ) ) ).
% mod_exE
thf(fact_5583_enorm__exI_H,axiom,
! [A: $tType,Z8: A > $o,P: assn,Q: A > assn] :
( ! [X3: A] :
( ( Z8 @ X3 )
=> ( entails @ P @ ( Q @ X3 ) ) )
=> ( ? [X_12: A] : ( Z8 @ X_12 )
=> ( entails @ P @ ( ex_assn @ A @ Q ) ) ) ) ).
% enorm_exI'
thf(fact_5584_ent__ex__preI,axiom,
! [A: $tType,P: A > assn,Q: assn] :
( ! [X3: A] : ( entails @ ( P @ X3 ) @ Q )
=> ( entails @ ( ex_assn @ A @ P ) @ Q ) ) ).
% ent_ex_preI
thf(fact_5585_ent__ex__postI,axiom,
! [A: $tType,P: assn,Q: A > assn,X: A] :
( ( entails @ P @ ( Q @ X ) )
=> ( entails @ P @ ( ex_assn @ A @ Q ) ) ) ).
% ent_ex_postI
thf(fact_5586_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_5587_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_5588_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_5589_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_5590_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_5591_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_5592_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_5593_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_5594_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_5595_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_5596_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_5597_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_5598_sin__integer__2pi,axiom,
! [N: real] :
( ( member @ real @ N @ ( ring_1_Ints @ real ) )
=> ( ( sin @ real @ ( times_times @ real @ ( times_times @ real @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) @ pi ) @ N ) )
= ( zero_zero @ real ) ) ) ).
% sin_integer_2pi
thf(fact_5599_cos__integer__2pi,axiom,
! [N: real] :
( ( member @ real @ N @ ( ring_1_Ints @ real ) )
=> ( ( cos @ real @ ( times_times @ real @ ( times_times @ real @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) @ pi ) @ N ) )
= ( one_one @ real ) ) ) ).
% cos_integer_2pi
thf(fact_5600_ran__empty,axiom,
! [B: $tType,A: $tType] :
( ( ran @ B @ A
@ ^ [X2: B] : ( none @ A ) )
= ( bot_bot @ ( set @ A ) ) ) ).
% ran_empty
thf(fact_5601_length__corresp,axiom,
! [B: $tType,A: $tType] :
( ( heap @ A )
=> ! [Tree_array: array @ A,Tree_is: list @ B] :
( ( ( ex_assn @ ( list @ A ) @ ( snga_assn @ A @ Tree_array ) )
= ( top_top @ assn ) )
=> ( ( heap_Time_return @ nat @ ( size_size @ ( list @ B ) @ Tree_is ) )
= ( array_len @ A @ Tree_array ) ) ) ) ).
% length_corresp
thf(fact_5602_VEBT__internal_Ospace_Opelims,axiom,
! [X: vEBT_VEBT,Y: nat] :
( ( ( vEBT_VEBT_space @ X )
= Y )
=> ( ( accp @ vEBT_VEBT @ vEBT_VEBT_space_rel2 @ X )
=> ( ! [A6: $o,B3: $o] :
( ( X
= ( vEBT_Leaf @ A6 @ B3 ) )
=> ( ( Y
= ( numeral_numeral @ nat @ ( bit1 @ one2 ) ) )
=> ~ ( accp @ vEBT_VEBT @ vEBT_VEBT_space_rel2 @ ( vEBT_Leaf @ A6 @ 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_5603_merge__true__star,axiom,
( ( times_times @ assn @ ( top_top @ assn ) @ ( top_top @ assn ) )
= ( top_top @ assn ) ) ).
% merge_true_star
thf(fact_5604_assn__basic__inequalities_I1_J,axiom,
( ( top_top @ assn )
!= ( one_one @ assn ) ) ).
% assn_basic_inequalities(1)
thf(fact_5605_ent__true,axiom,
! [P: assn] : ( entails @ P @ ( top_top @ assn ) ) ).
% ent_true
thf(fact_5606_merge__true__star__ctx,axiom,
! [P: assn] :
( ( times_times @ assn @ ( top_top @ assn ) @ ( times_times @ assn @ ( top_top @ assn ) @ P ) )
= ( times_times @ assn @ ( top_top @ assn ) @ P ) ) ).
% merge_true_star_ctx
thf(fact_5607_ent__true__drop_I2_J,axiom,
! [P: assn,Q: assn] :
( ( entails @ P @ Q )
=> ( entails @ P @ ( times_times @ assn @ Q @ ( top_top @ assn ) ) ) ) ).
% ent_true_drop(2)
thf(fact_5608_ent__true__drop_I1_J,axiom,
! [P: assn,Q: assn,R: assn] :
( ( entails @ P @ ( times_times @ assn @ Q @ ( top_top @ assn ) ) )
=> ( entails @ ( times_times @ assn @ P @ R ) @ ( times_times @ assn @ Q @ ( top_top @ assn ) ) ) ) ).
% ent_true_drop(1)
thf(fact_5609_ent__refl__true,axiom,
! [A3: assn] : ( entails @ A3 @ ( times_times @ assn @ A3 @ ( top_top @ assn ) ) ) ).
% ent_refl_true
thf(fact_5610_ent__star__mono__true,axiom,
! [A3: assn,A8: assn,B4: assn,B11: assn] :
( ( entails @ A3 @ ( times_times @ assn @ A8 @ ( top_top @ assn ) ) )
=> ( ( entails @ B4 @ ( times_times @ assn @ B11 @ ( top_top @ assn ) ) )
=> ( entails @ ( times_times @ assn @ ( times_times @ assn @ A3 @ B4 ) @ ( top_top @ assn ) ) @ ( times_times @ assn @ ( times_times @ assn @ A8 @ B11 ) @ ( top_top @ assn ) ) ) ) ) ).
% ent_star_mono_true
thf(fact_5611_mod__star__trueI,axiom,
! [P: assn,H2: product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat )] :
( ( rep_assn @ P @ H2 )
=> ( rep_assn @ ( times_times @ assn @ P @ ( top_top @ assn ) ) @ H2 ) ) ).
% mod_star_trueI
thf(fact_5612_mod__star__trueE,axiom,
! [P: assn,H2: product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat )] :
( ( rep_assn @ ( times_times @ assn @ P @ ( top_top @ assn ) ) @ H2 )
=> ~ ! [H6: product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat )] :
~ ( rep_assn @ P @ H6 ) ) ).
% mod_star_trueE
thf(fact_5613_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_5614_VEBT__internal_Ospace_H_Opelims,axiom,
! [X: vEBT_VEBT,Y: nat] :
( ( ( vEBT_VEBT_space2 @ X )
= Y )
=> ( ( accp @ vEBT_VEBT @ vEBT_VEBT_space_rel @ X )
=> ( ! [A6: $o,B3: $o] :
( ( X
= ( vEBT_Leaf @ A6 @ B3 ) )
=> ( ( Y
= ( numeral_numeral @ nat @ ( bit0 @ ( bit0 @ one2 ) ) ) )
=> ~ ( accp @ vEBT_VEBT @ vEBT_VEBT_space_rel @ ( vEBT_Leaf @ A6 @ 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_5615_VEBT__internal_Ocnt_Opelims,axiom,
! [X: vEBT_VEBT,Y: real] :
( ( ( vEBT_VEBT_cnt @ X )
= Y )
=> ( ( accp @ vEBT_VEBT @ vEBT_VEBT_cnt_rel2 @ X )
=> ( ! [A6: $o,B3: $o] :
( ( X
= ( vEBT_Leaf @ A6 @ B3 ) )
=> ( ( Y
= ( one_one @ real ) )
=> ~ ( accp @ vEBT_VEBT @ vEBT_VEBT_cnt_rel2 @ ( vEBT_Leaf @ A6 @ 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_5616_VEBT__internal_Ocnt_H_Opelims,axiom,
! [X: vEBT_VEBT,Y: nat] :
( ( ( vEBT_VEBT_cnt2 @ X )
= Y )
=> ( ( accp @ vEBT_VEBT @ vEBT_VEBT_cnt_rel @ X )
=> ( ! [A6: $o,B3: $o] :
( ( X
= ( vEBT_Leaf @ A6 @ B3 ) )
=> ( ( Y
= ( one_one @ nat ) )
=> ~ ( accp @ vEBT_VEBT @ vEBT_VEBT_cnt_rel @ ( vEBT_Leaf @ A6 @ 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_5617_vebt__maxt_Opelims,axiom,
! [X: vEBT_VEBT,Y: option @ nat] :
( ( ( vEBT_vebt_maxt @ X )
= Y )
=> ( ( accp @ vEBT_VEBT @ vEBT_vebt_maxt_rel @ X )
=> ( ! [A6: $o,B3: $o] :
( ( X
= ( vEBT_Leaf @ A6 @ B3 ) )
=> ( ( ( B3
=> ( Y
= ( some @ nat @ ( one_one @ nat ) ) ) )
& ( ~ B3
=> ( ( A6
=> ( Y
= ( some @ nat @ ( zero_zero @ nat ) ) ) )
& ( ~ A6
=> ( Y
= ( none @ nat ) ) ) ) ) )
=> ~ ( accp @ vEBT_VEBT @ vEBT_vebt_maxt_rel @ ( vEBT_Leaf @ A6 @ B3 ) ) ) )
=> ( ! [Uu2: nat,Uv2: list @ vEBT_VEBT,Uw2: vEBT_VEBT] :
( ( X
= ( vEBT_Node @ ( none @ ( product_prod @ nat @ nat ) ) @ Uu2 @ Uv2 @ Uw2 ) )
=> ( ( Y
= ( none @ nat ) )
=> ~ ( accp @ vEBT_VEBT @ vEBT_vebt_maxt_rel @ ( vEBT_Node @ ( none @ ( product_prod @ nat @ nat ) ) @ Uu2 @ Uv2 @ Uw2 ) ) ) )
=> ~ ! [Mi: nat,Ma2: nat,Ux2: nat,Uy2: list @ vEBT_VEBT,Uz2: vEBT_VEBT] :
( ( X
= ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi @ 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 @ Mi @ Ma2 ) ) @ Ux2 @ Uy2 @ Uz2 ) ) ) ) ) ) ) ) ).
% vebt_maxt.pelims
thf(fact_5618_vebt__mint_Opelims,axiom,
! [X: vEBT_VEBT,Y: option @ nat] :
( ( ( vEBT_vebt_mint @ X )
= Y )
=> ( ( accp @ vEBT_VEBT @ vEBT_vebt_mint_rel @ X )
=> ( ! [A6: $o,B3: $o] :
( ( X
= ( vEBT_Leaf @ A6 @ B3 ) )
=> ( ( ( A6
=> ( Y
= ( some @ nat @ ( zero_zero @ nat ) ) ) )
& ( ~ A6
=> ( ( B3
=> ( Y
= ( some @ nat @ ( one_one @ nat ) ) ) )
& ( ~ B3
=> ( Y
= ( none @ nat ) ) ) ) ) )
=> ~ ( accp @ vEBT_VEBT @ vEBT_vebt_mint_rel @ ( vEBT_Leaf @ A6 @ B3 ) ) ) )
=> ( ! [Uu2: nat,Uv2: list @ vEBT_VEBT,Uw2: vEBT_VEBT] :
( ( X
= ( vEBT_Node @ ( none @ ( product_prod @ nat @ nat ) ) @ Uu2 @ Uv2 @ Uw2 ) )
=> ( ( Y
= ( none @ nat ) )
=> ~ ( accp @ vEBT_VEBT @ vEBT_vebt_mint_rel @ ( vEBT_Node @ ( none @ ( product_prod @ nat @ nat ) ) @ Uu2 @ Uv2 @ Uw2 ) ) ) )
=> ~ ! [Mi: nat,Ma2: nat,Ux2: nat,Uy2: list @ vEBT_VEBT,Uz2: vEBT_VEBT] :
( ( X
= ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi @ Ma2 ) ) @ Ux2 @ Uy2 @ Uz2 ) )
=> ( ( Y
= ( some @ nat @ Mi ) )
=> ~ ( accp @ vEBT_VEBT @ vEBT_vebt_mint_rel @ ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi @ Ma2 ) ) @ Ux2 @ Uy2 @ Uz2 ) ) ) ) ) ) ) ) ).
% vebt_mint.pelims
thf(fact_5619_T_092_060_094sub_062m_092_060_094sub_062i_092_060_094sub_062n_092_060_094sub_062t_Opelims,axiom,
! [X: vEBT_VEBT,Y: nat] :
( ( ( vEBT_T_m_i_n_t @ X )
= Y )
=> ( ( accp @ vEBT_VEBT @ vEBT_T_m_i_n_t_rel @ X )
=> ( ! [A6: $o,B3: $o] :
( ( X
= ( vEBT_Leaf @ A6 @ B3 ) )
=> ( ( Y
= ( plus_plus @ nat @ ( one_one @ nat ) @ ( if @ nat @ A6 @ ( zero_zero @ nat ) @ ( plus_plus @ nat @ ( one_one @ nat ) @ ( one_one @ nat ) ) ) ) )
=> ~ ( accp @ vEBT_VEBT @ vEBT_T_m_i_n_t_rel @ ( vEBT_Leaf @ A6 @ B3 ) ) ) )
=> ( ! [Uu2: nat,Uv2: list @ vEBT_VEBT,Uw2: vEBT_VEBT] :
( ( X
= ( vEBT_Node @ ( none @ ( product_prod @ nat @ nat ) ) @ Uu2 @ Uv2 @ Uw2 ) )
=> ( ( Y
= ( one_one @ nat ) )
=> ~ ( accp @ vEBT_VEBT @ vEBT_T_m_i_n_t_rel @ ( vEBT_Node @ ( none @ ( product_prod @ nat @ nat ) ) @ Uu2 @ Uv2 @ Uw2 ) ) ) )
=> ~ ! [Mi: nat,Ma2: nat,Ux2: nat,Uy2: list @ vEBT_VEBT,Uz2: vEBT_VEBT] :
( ( X
= ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi @ 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 @ Mi @ Ma2 ) ) @ Ux2 @ Uy2 @ Uz2 ) ) ) ) ) ) ) ) ).
% T\<^sub>m\<^sub>i\<^sub>n\<^sub>t.pelims
thf(fact_5620_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 )
=> ( ! [A6: $o,B3: $o] :
( ( X
= ( vEBT_Leaf @ A6 @ 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 @ A6 @ B3 ) ) ) )
=> ( ! [Uu2: nat,Uv2: list @ vEBT_VEBT,Uw2: vEBT_VEBT] :
( ( X
= ( vEBT_Node @ ( none @ ( product_prod @ nat @ nat ) ) @ Uu2 @ Uv2 @ Uw2 ) )
=> ( ( Y
= ( one_one @ nat ) )
=> ~ ( accp @ vEBT_VEBT @ vEBT_T_m_a_x_t_rel @ ( vEBT_Node @ ( none @ ( product_prod @ nat @ nat ) ) @ Uu2 @ Uv2 @ Uw2 ) ) ) )
=> ~ ! [Mi: nat,Ma2: nat,Ux2: nat,Uy2: list @ vEBT_VEBT,Uz2: vEBT_VEBT] :
( ( X
= ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi @ 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 @ Mi @ Ma2 ) ) @ Ux2 @ Uy2 @ Uz2 ) ) ) ) ) ) ) ) ).
% T\<^sub>m\<^sub>a\<^sub>x\<^sub>t.pelims
thf(fact_5621_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 ) ) ) )
=> ( ! [Uv2: $o] :
( ( X
= ( vEBT_Leaf @ $true @ Uv2 ) )
=> ( ( Y
= ( one_one @ nat ) )
=> ~ ( accp @ vEBT_VEBT @ vEBT_T5462971552011256508_l_rel @ ( vEBT_Leaf @ $true @ Uv2 ) ) ) )
=> ( ! [Uu2: $o] :
( ( X
= ( vEBT_Leaf @ Uu2 @ $true ) )
=> ( ( Y
= ( one_one @ nat ) )
=> ~ ( accp @ vEBT_VEBT @ vEBT_T5462971552011256508_l_rel @ ( vEBT_Leaf @ Uu2 @ $true ) ) ) )
=> ( ! [Uw2: nat,Ux2: list @ vEBT_VEBT,Uy2: vEBT_VEBT] :
( ( X
= ( vEBT_Node @ ( none @ ( product_prod @ nat @ nat ) ) @ Uw2 @ Ux2 @ Uy2 ) )
=> ( ( Y
= ( one_one @ nat ) )
=> ~ ( accp @ vEBT_VEBT @ vEBT_T5462971552011256508_l_rel @ ( vEBT_Node @ ( none @ ( product_prod @ nat @ nat ) ) @ Uw2 @ Ux2 @ Uy2 ) ) ) )
=> ~ ! [Uz2: product_prod @ nat @ nat,Va3: nat,Vb2: list @ vEBT_VEBT,Vc: vEBT_VEBT] :
( ( X
= ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ Uz2 ) @ Va3 @ Vb2 @ Vc ) )
=> ( ( Y
= ( one_one @ nat ) )
=> ~ ( accp @ vEBT_VEBT @ vEBT_T5462971552011256508_l_rel @ ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ Uz2 ) @ Va3 @ Vb2 @ Vc ) ) ) ) ) ) ) ) ) ) ).
% T\<^sub>m\<^sub>i\<^sub>n\<^sub>N\<^sub>u\<^sub>l\<^sub>l.pelims
thf(fact_5622_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 ) ) ) )
=> ( ! [Uv2: $o] :
( ( X
= ( vEBT_Leaf @ $true @ Uv2 ) )
=> ( ~ Y
=> ~ ( accp @ vEBT_VEBT @ vEBT_V6963167321098673237ll_rel @ ( vEBT_Leaf @ $true @ Uv2 ) ) ) )
=> ( ! [Uu2: $o] :
( ( X
= ( vEBT_Leaf @ Uu2 @ $true ) )
=> ( ~ Y
=> ~ ( accp @ vEBT_VEBT @ vEBT_V6963167321098673237ll_rel @ ( vEBT_Leaf @ Uu2 @ $true ) ) ) )
=> ( ! [Uw2: nat,Ux2: list @ vEBT_VEBT,Uy2: vEBT_VEBT] :
( ( X
= ( vEBT_Node @ ( none @ ( product_prod @ nat @ nat ) ) @ Uw2 @ Ux2 @ Uy2 ) )
=> ( Y
=> ~ ( accp @ vEBT_VEBT @ vEBT_V6963167321098673237ll_rel @ ( vEBT_Node @ ( none @ ( product_prod @ nat @ nat ) ) @ Uw2 @ Ux2 @ Uy2 ) ) ) )
=> ~ ! [Uz2: product_prod @ nat @ nat,Va3: nat,Vb2: list @ vEBT_VEBT,Vc: vEBT_VEBT] :
( ( X
= ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ Uz2 ) @ Va3 @ Vb2 @ Vc ) )
=> ( ~ Y
=> ~ ( accp @ vEBT_VEBT @ vEBT_V6963167321098673237ll_rel @ ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ Uz2 ) @ Va3 @ Vb2 @ Vc ) ) ) ) ) ) ) ) ) ) ).
% VEBT_internal.minNull.pelims(1)
thf(fact_5623_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 ) ) )
=> ~ ! [Uw2: nat,Ux2: list @ vEBT_VEBT,Uy2: vEBT_VEBT] :
( ( X
= ( vEBT_Node @ ( none @ ( product_prod @ nat @ nat ) ) @ Uw2 @ Ux2 @ Uy2 ) )
=> ~ ( accp @ vEBT_VEBT @ vEBT_V6963167321098673237ll_rel @ ( vEBT_Node @ ( none @ ( product_prod @ nat @ nat ) ) @ Uw2 @ Ux2 @ Uy2 ) ) ) ) ) ) ).
% VEBT_internal.minNull.pelims(2)
thf(fact_5624_bin__last__integer__nbe,axiom,
( bits_b8758750999018896077nteger
= ( ^ [I2: code_integer] :
( ( modulo_modulo @ code_integer @ I2 @ ( numeral_numeral @ code_integer @ ( bit0 @ one2 ) ) )
!= ( zero_zero @ code_integer ) ) ) ) ).
% bin_last_integer_nbe
thf(fact_5625_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_5626_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_5627_drop__bit__of__0,axiom,
! [A: $tType] :
( ( bit_se359711467146920520ations @ A )
=> ! [N: nat] :
( ( bit_se4197421643247451524op_bit @ A @ N @ ( zero_zero @ A ) )
= ( zero_zero @ A ) ) ) ).
% drop_bit_of_0
thf(fact_5628_drop__bit__drop__bit,axiom,
! [A: $tType] :
( ( bit_se359711467146920520ations @ A )
=> ! [M: nat,N: nat,A2: A] :
( ( bit_se4197421643247451524op_bit @ A @ M @ ( bit_se4197421643247451524op_bit @ A @ N @ A2 ) )
= ( bit_se4197421643247451524op_bit @ A @ ( plus_plus @ nat @ M @ N ) @ A2 ) ) ) ).
% drop_bit_drop_bit
thf(fact_5629_drop__bit__and,axiom,
! [A: $tType] :
( ( bit_se359711467146920520ations @ A )
=> ! [N: nat,A2: A,B2: A] :
( ( bit_se4197421643247451524op_bit @ A @ N @ ( bit_se5824344872417868541ns_and @ A @ A2 @ B2 ) )
= ( bit_se5824344872417868541ns_and @ A @ ( bit_se4197421643247451524op_bit @ A @ N @ A2 ) @ ( bit_se4197421643247451524op_bit @ A @ N @ B2 ) ) ) ) ).
% drop_bit_and
thf(fact_5630_drop__bit__or,axiom,
! [A: $tType] :
( ( bit_se359711467146920520ations @ A )
=> ! [N: nat,A2: A,B2: A] :
( ( bit_se4197421643247451524op_bit @ A @ N @ ( bit_se1065995026697491101ons_or @ A @ A2 @ B2 ) )
= ( bit_se1065995026697491101ons_or @ A @ ( bit_se4197421643247451524op_bit @ A @ N @ A2 ) @ ( bit_se4197421643247451524op_bit @ A @ N @ B2 ) ) ) ) ).
% drop_bit_or
thf(fact_5631_drop__bit__xor,axiom,
! [A: $tType] :
( ( bit_se359711467146920520ations @ A )
=> ! [N: nat,A2: A,B2: A] :
( ( bit_se4197421643247451524op_bit @ A @ N @ ( bit_se5824344971392196577ns_xor @ A @ A2 @ B2 ) )
= ( bit_se5824344971392196577ns_xor @ A @ ( bit_se4197421643247451524op_bit @ A @ N @ A2 ) @ ( bit_se4197421643247451524op_bit @ A @ N @ B2 ) ) ) ) ).
% drop_bit_xor
thf(fact_5632_drop__bit__of__bool,axiom,
! [A: $tType] :
( ( bit_se359711467146920520ations @ A )
=> ! [N: nat,B2: $o] :
( ( bit_se4197421643247451524op_bit @ A @ N @ ( zero_neq_one_of_bool @ A @ B2 ) )
= ( zero_neq_one_of_bool @ A
@ ( ( N
= ( zero_zero @ nat ) )
& B2 ) ) ) ) ).
% drop_bit_of_bool
thf(fact_5633_drop__bit__Suc__bit0,axiom,
! [A: $tType] :
( ( bit_un5681908812861735899ations @ A )
=> ! [N: nat,K: num] :
( ( bit_se4197421643247451524op_bit @ A @ ( suc @ N ) @ ( numeral_numeral @ A @ ( bit0 @ K ) ) )
= ( bit_se4197421643247451524op_bit @ A @ N @ ( numeral_numeral @ A @ K ) ) ) ) ).
% drop_bit_Suc_bit0
thf(fact_5634_drop__bit__Suc__bit1,axiom,
! [A: $tType] :
( ( bit_un5681908812861735899ations @ A )
=> ! [N: nat,K: num] :
( ( bit_se4197421643247451524op_bit @ A @ ( suc @ N ) @ ( numeral_numeral @ A @ ( bit1 @ K ) ) )
= ( bit_se4197421643247451524op_bit @ A @ N @ ( numeral_numeral @ A @ K ) ) ) ) ).
% drop_bit_Suc_bit1
thf(fact_5635_drop__bit__of__1,axiom,
! [A: $tType] :
( ( bit_se359711467146920520ations @ A )
=> ! [N: nat] :
( ( bit_se4197421643247451524op_bit @ A @ N @ ( one_one @ A ) )
= ( zero_neq_one_of_bool @ A
@ ( N
= ( zero_zero @ nat ) ) ) ) ) ).
% drop_bit_of_1
thf(fact_5636_drop__bit__numeral__bit0,axiom,
! [A: $tType] :
( ( bit_un5681908812861735899ations @ A )
=> ! [L: num,K: num] :
( ( bit_se4197421643247451524op_bit @ A @ ( numeral_numeral @ nat @ L ) @ ( numeral_numeral @ A @ ( bit0 @ K ) ) )
= ( bit_se4197421643247451524op_bit @ A @ ( pred_numeral @ L ) @ ( numeral_numeral @ A @ K ) ) ) ) ).
% drop_bit_numeral_bit0
thf(fact_5637_drop__bit__numeral__bit1,axiom,
! [A: $tType] :
( ( bit_un5681908812861735899ations @ A )
=> ! [L: num,K: num] :
( ( bit_se4197421643247451524op_bit @ A @ ( numeral_numeral @ nat @ L ) @ ( numeral_numeral @ A @ ( bit1 @ K ) ) )
= ( bit_se4197421643247451524op_bit @ A @ ( pred_numeral @ L ) @ ( numeral_numeral @ A @ K ) ) ) ) ).
% drop_bit_numeral_bit1
thf(fact_5638_drop__bit__of__nat,axiom,
! [A: $tType] :
( ( bit_un5681908812861735899ations @ A )
=> ! [N: nat,M: nat] :
( ( bit_se4197421643247451524op_bit @ A @ N @ ( semiring_1_of_nat @ A @ M ) )
= ( semiring_1_of_nat @ A @ ( bit_se4197421643247451524op_bit @ nat @ N @ M ) ) ) ) ).
% drop_bit_of_nat
thf(fact_5639_of__nat__drop__bit,axiom,
! [A: $tType] :
( ( bit_un5681908812861735899ations @ A )
=> ! [M: nat,N: nat] :
( ( semiring_1_of_nat @ A @ ( bit_se4197421643247451524op_bit @ nat @ M @ N ) )
= ( bit_se4197421643247451524op_bit @ A @ M @ ( semiring_1_of_nat @ A @ N ) ) ) ) ).
% of_nat_drop_bit
thf(fact_5640_take__bit__eq__self__iff__drop__bit__eq__0,axiom,
! [A: $tType] :
( ( bit_se359711467146920520ations @ A )
=> ! [N: nat,A2: A] :
( ( ( bit_se2584673776208193580ke_bit @ A @ N @ A2 )
= A2 )
= ( ( bit_se4197421643247451524op_bit @ A @ N @ A2 )
= ( zero_zero @ A ) ) ) ) ).
% take_bit_eq_self_iff_drop_bit_eq_0
thf(fact_5641_take__bit__drop__bit,axiom,
! [A: $tType] :
( ( bit_se359711467146920520ations @ A )
=> ! [M: nat,N: nat,A2: A] :
( ( bit_se2584673776208193580ke_bit @ A @ M @ ( bit_se4197421643247451524op_bit @ A @ N @ A2 ) )
= ( bit_se4197421643247451524op_bit @ A @ N @ ( bit_se2584673776208193580ke_bit @ A @ ( plus_plus @ nat @ M @ N ) @ A2 ) ) ) ) ).
% take_bit_drop_bit
thf(fact_5642_drop__bit__take__bit,axiom,
! [A: $tType] :
( ( bit_se359711467146920520ations @ A )
=> ! [M: nat,N: nat,A2: A] :
( ( bit_se4197421643247451524op_bit @ A @ M @ ( bit_se2584673776208193580ke_bit @ A @ N @ A2 ) )
= ( bit_se2584673776208193580ke_bit @ A @ ( minus_minus @ nat @ N @ M ) @ ( bit_se4197421643247451524op_bit @ A @ M @ A2 ) ) ) ) ).
% drop_bit_take_bit
thf(fact_5643_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_5644_bit__word__iff__drop__bit__and,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ( ( bit_se5641148757651400278ts_bit @ ( word @ A ) )
= ( ^ [A4: word @ A,N3: nat] :
( ( bit_se5824344872417868541ns_and @ ( word @ A ) @ ( bit_se4197421643247451524op_bit @ ( word @ A ) @ N3 @ A4 ) @ ( one_one @ ( word @ A ) ) )
= ( one_one @ ( word @ A ) ) ) ) ) ) ).
% bit_word_iff_drop_bit_and
thf(fact_5645_bit__iff__and__drop__bit__eq__1,axiom,
! [A: $tType] :
( ( bit_se359711467146920520ations @ A )
=> ( ( bit_se5641148757651400278ts_bit @ A )
= ( ^ [A4: A,N3: nat] :
( ( bit_se5824344872417868541ns_and @ A @ ( bit_se4197421643247451524op_bit @ A @ N3 @ A4 ) @ ( one_one @ A ) )
= ( one_one @ A ) ) ) ) ) ).
% bit_iff_and_drop_bit_eq_1
thf(fact_5646_bin__last__integer__code,axiom,
( bits_b8758750999018896077nteger
= ( ^ [I2: code_integer] :
( ( bit_se5824344872417868541ns_and @ code_integer @ I2 @ ( one_one @ code_integer ) )
!= ( zero_zero @ code_integer ) ) ) ) ).
% bin_last_integer_code
thf(fact_5647_bitAND__integer__unfold,axiom,
( ( bit_se5824344872417868541ns_and @ code_integer )
= ( ^ [X2: code_integer,Y6: 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 ) ) )
@ Y6
@ ( bits_Bit_integer @ ( bit_se5824344872417868541ns_and @ code_integer @ ( bits_b2549910563261871055nteger @ X2 ) @ ( bits_b2549910563261871055nteger @ Y6 ) )
@ ( ( bits_b8758750999018896077nteger @ X2 )
& ( bits_b8758750999018896077nteger @ Y6 ) ) ) ) ) ) ) ).
% bitAND_integer_unfold
thf(fact_5648_bitOR__integer__unfold,axiom,
( ( bit_se1065995026697491101ons_or @ code_integer )
= ( ^ [X2: code_integer,Y6: code_integer] :
( if @ code_integer
@ ( X2
= ( zero_zero @ code_integer ) )
@ Y6
@ ( 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 @ Y6 ) )
@ ( ( bits_b8758750999018896077nteger @ X2 )
| ( bits_b8758750999018896077nteger @ Y6 ) ) ) ) ) ) ) ).
% bitOR_integer_unfold
thf(fact_5649_drop__bit__half,axiom,
! [A: $tType] :
( ( bit_se359711467146920520ations @ A )
=> ! [N: nat,A2: A] :
( ( bit_se4197421643247451524op_bit @ A @ N @ ( divide_divide @ A @ A2 @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) )
= ( divide_divide @ A @ ( bit_se4197421643247451524op_bit @ A @ N @ A2 ) @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) ) ) ).
% drop_bit_half
thf(fact_5650_stable__imp__drop__bit__eq,axiom,
! [A: $tType] :
( ( bit_se359711467146920520ations @ A )
=> ! [A2: A,N: nat] :
( ( ( divide_divide @ A @ A2 @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) )
= A2 )
=> ( ( bit_se4197421643247451524op_bit @ A @ N @ A2 )
= A2 ) ) ) ).
% stable_imp_drop_bit_eq
thf(fact_5651_drop__bit__Suc,axiom,
! [A: $tType] :
( ( bit_se359711467146920520ations @ A )
=> ! [N: nat,A2: A] :
( ( bit_se4197421643247451524op_bit @ A @ ( suc @ N ) @ A2 )
= ( bit_se4197421643247451524op_bit @ A @ N @ ( divide_divide @ A @ A2 @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) ) ) ) ).
% drop_bit_Suc
thf(fact_5652_drop__bit__eq__div,axiom,
! [A: $tType] :
( ( bit_se359711467146920520ations @ A )
=> ( ( bit_se4197421643247451524op_bit @ A )
= ( ^ [N3: nat,A4: A] : ( divide_divide @ A @ A4 @ ( power_power @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ N3 ) ) ) ) ) ).
% drop_bit_eq_div
thf(fact_5653_even__drop__bit__iff__not__bit,axiom,
! [A: $tType] :
( ( bit_se359711467146920520ations @ A )
=> ! [N: nat,A2: A] :
( ( dvd_dvd @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ ( bit_se4197421643247451524op_bit @ A @ N @ A2 ) )
= ( ~ ( bit_se5641148757651400278ts_bit @ A @ A2 @ N ) ) ) ) ).
% even_drop_bit_iff_not_bit
thf(fact_5654_bit__iff__odd__drop__bit,axiom,
! [A: $tType] :
( ( bit_se359711467146920520ations @ A )
=> ( ( bit_se5641148757651400278ts_bit @ A )
= ( ^ [A4: A,N3: nat] :
~ ( dvd_dvd @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ ( bit_se4197421643247451524op_bit @ A @ N3 @ A4 ) ) ) ) ) ).
% bit_iff_odd_drop_bit
thf(fact_5655_bitXOR__integer__unfold,axiom,
( ( bit_se5824344971392196577ns_xor @ code_integer )
= ( ^ [X2: code_integer,Y6: code_integer] :
( if @ code_integer
@ ( X2
= ( zero_zero @ code_integer ) )
@ Y6
@ ( if @ code_integer
@ ( X2
= ( uminus_uminus @ code_integer @ ( one_one @ code_integer ) ) )
@ ( bit_ri4277139882892585799ns_not @ code_integer @ Y6 )
@ ( bits_Bit_integer @ ( bit_se5824344971392196577ns_xor @ code_integer @ ( bits_b2549910563261871055nteger @ X2 ) @ ( bits_b2549910563261871055nteger @ Y6 ) )
@ ( ( ~ ( bits_b8758750999018896077nteger @ X2 ) )
= ( bits_b8758750999018896077nteger @ Y6 ) ) ) ) ) ) ) ).
% bitXOR_integer_unfold
thf(fact_5656_drop__bit__rec,axiom,
! [A: $tType] :
( ( bit_se359711467146920520ations @ A )
=> ( ( bit_se4197421643247451524op_bit @ A )
= ( ^ [N3: nat,A4: A] :
( if @ A
@ ( N3
= ( zero_zero @ nat ) )
@ A4
@ ( bit_se4197421643247451524op_bit @ A @ ( minus_minus @ nat @ N3 @ ( one_one @ nat ) ) @ ( divide_divide @ A @ A4 @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) ) ) ) ) ) ).
% drop_bit_rec
thf(fact_5657_bin__rest__integer__code,axiom,
( bits_b2549910563261871055nteger
= ( ^ [I2: code_integer] : ( divide_divide @ code_integer @ I2 @ ( numeral_numeral @ code_integer @ ( bit0 @ one2 ) ) ) ) ) ).
% bin_rest_integer_code
thf(fact_5658_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_5659_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_5660_upto__aux__rec,axiom,
( upto_aux
= ( ^ [I2: int,J2: int,Js: list @ int] : ( if @ ( list @ int ) @ ( ord_less @ int @ J2 @ I2 ) @ Js @ ( upto_aux @ I2 @ ( minus_minus @ int @ J2 @ ( one_one @ int ) ) @ ( cons @ int @ J2 @ Js ) ) ) ) ) ).
% upto_aux_rec
thf(fact_5661_concat__bit__Suc,axiom,
! [N: nat,K: int,L: int] :
( ( bit_concat_bit @ ( suc @ N ) @ K @ L )
= ( plus_plus @ int @ ( modulo_modulo @ int @ K @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) ) @ ( times_times @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ ( bit_concat_bit @ N @ ( divide_divide @ int @ K @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) ) @ L ) ) ) ) ).
% concat_bit_Suc
thf(fact_5662_push__bit__of__0,axiom,
! [A: $tType] :
( ( bit_se359711467146920520ations @ A )
=> ! [N: nat] :
( ( bit_se4730199178511100633sh_bit @ A @ N @ ( zero_zero @ A ) )
= ( zero_zero @ A ) ) ) ).
% push_bit_of_0
thf(fact_5663_push__bit__eq__0__iff,axiom,
! [A: $tType] :
( ( bit_un5681908812861735899ations @ A )
=> ! [N: nat,A2: A] :
( ( ( bit_se4730199178511100633sh_bit @ A @ N @ A2 )
= ( zero_zero @ A ) )
= ( A2
= ( zero_zero @ A ) ) ) ) ).
% push_bit_eq_0_iff
thf(fact_5664_drop__bit__nonnegative__int__iff,axiom,
! [N: nat,K: int] :
( ( ord_less_eq @ int @ ( zero_zero @ int ) @ ( bit_se4197421643247451524op_bit @ int @ N @ K ) )
= ( ord_less_eq @ int @ ( zero_zero @ int ) @ K ) ) ).
% drop_bit_nonnegative_int_iff
thf(fact_5665_drop__bit__negative__int__iff,axiom,
! [N: nat,K: int] :
( ( ord_less @ int @ ( bit_se4197421643247451524op_bit @ int @ N @ K ) @ ( zero_zero @ int ) )
= ( ord_less @ int @ K @ ( zero_zero @ int ) ) ) ).
% drop_bit_negative_int_iff
thf(fact_5666_drop__bit__minus__one,axiom,
! [N: nat] :
( ( bit_se4197421643247451524op_bit @ int @ N @ ( uminus_uminus @ int @ ( one_one @ int ) ) )
= ( uminus_uminus @ int @ ( one_one @ int ) ) ) ).
% drop_bit_minus_one
thf(fact_5667_push__bit__push__bit,axiom,
! [A: $tType] :
( ( bit_se359711467146920520ations @ A )
=> ! [M: nat,N: nat,A2: A] :
( ( bit_se4730199178511100633sh_bit @ A @ M @ ( bit_se4730199178511100633sh_bit @ A @ N @ A2 ) )
= ( bit_se4730199178511100633sh_bit @ A @ ( plus_plus @ nat @ M @ N ) @ A2 ) ) ) ).
% push_bit_push_bit
thf(fact_5668_push__bit__and,axiom,
! [A: $tType] :
( ( bit_se359711467146920520ations @ A )
=> ! [N: nat,A2: A,B2: A] :
( ( bit_se4730199178511100633sh_bit @ A @ N @ ( bit_se5824344872417868541ns_and @ A @ A2 @ B2 ) )
= ( bit_se5824344872417868541ns_and @ A @ ( bit_se4730199178511100633sh_bit @ A @ N @ A2 ) @ ( bit_se4730199178511100633sh_bit @ A @ N @ B2 ) ) ) ) ).
% push_bit_and
thf(fact_5669_push__bit__or,axiom,
! [A: $tType] :
( ( bit_se359711467146920520ations @ A )
=> ! [N: nat,A2: A,B2: A] :
( ( bit_se4730199178511100633sh_bit @ A @ N @ ( bit_se1065995026697491101ons_or @ A @ A2 @ B2 ) )
= ( bit_se1065995026697491101ons_or @ A @ ( bit_se4730199178511100633sh_bit @ A @ N @ A2 ) @ ( bit_se4730199178511100633sh_bit @ A @ N @ B2 ) ) ) ) ).
% push_bit_or
thf(fact_5670_push__bit__xor,axiom,
! [A: $tType] :
( ( bit_se359711467146920520ations @ A )
=> ! [N: nat,A2: A,B2: A] :
( ( bit_se4730199178511100633sh_bit @ A @ N @ ( bit_se5824344971392196577ns_xor @ A @ A2 @ B2 ) )
= ( bit_se5824344971392196577ns_xor @ A @ ( bit_se4730199178511100633sh_bit @ A @ N @ A2 ) @ ( bit_se4730199178511100633sh_bit @ A @ N @ B2 ) ) ) ) ).
% push_bit_xor
thf(fact_5671_concat__bit__0,axiom,
! [K: int,L: int] :
( ( bit_concat_bit @ ( zero_zero @ nat ) @ K @ L )
= L ) ).
% concat_bit_0
thf(fact_5672_concat__bit__of__zero__2,axiom,
! [N: nat,K: int] :
( ( bit_concat_bit @ N @ K @ ( zero_zero @ int ) )
= ( bit_se2584673776208193580ke_bit @ int @ N @ K ) ) ).
% concat_bit_of_zero_2
thf(fact_5673_concat__bit__nonnegative__iff,axiom,
! [N: nat,K: int,L: int] :
( ( ord_less_eq @ int @ ( zero_zero @ int ) @ ( bit_concat_bit @ N @ K @ L ) )
= ( ord_less_eq @ int @ ( zero_zero @ int ) @ L ) ) ).
% concat_bit_nonnegative_iff
thf(fact_5674_concat__bit__negative__iff,axiom,
! [N: nat,K: int,L: int] :
( ( ord_less @ int @ ( bit_concat_bit @ N @ K @ L ) @ ( zero_zero @ int ) )
= ( ord_less @ int @ L @ ( zero_zero @ int ) ) ) ).
% concat_bit_negative_iff
thf(fact_5675_push__bit__Suc__numeral,axiom,
! [A: $tType] :
( ( bit_se359711467146920520ations @ A )
=> ! [N: nat,K: num] :
( ( bit_se4730199178511100633sh_bit @ A @ ( suc @ N ) @ ( numeral_numeral @ A @ K ) )
= ( bit_se4730199178511100633sh_bit @ A @ N @ ( numeral_numeral @ A @ ( bit0 @ K ) ) ) ) ) ).
% push_bit_Suc_numeral
thf(fact_5676_drop__bit__Suc__minus__bit0,axiom,
! [N: nat,K: num] :
( ( bit_se4197421643247451524op_bit @ int @ ( suc @ N ) @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ ( bit0 @ K ) ) ) )
= ( bit_se4197421643247451524op_bit @ int @ N @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ K ) ) ) ) ).
% drop_bit_Suc_minus_bit0
thf(fact_5677_drop__bit__of__Suc__0,axiom,
! [N: nat] :
( ( bit_se4197421643247451524op_bit @ nat @ N @ ( suc @ ( zero_zero @ nat ) ) )
= ( zero_neq_one_of_bool @ nat
@ ( N
= ( zero_zero @ nat ) ) ) ) ).
% drop_bit_of_Suc_0
thf(fact_5678_push__bit__Suc__minus__numeral,axiom,
! [A: $tType] :
( ( bit_ri3973907225187159222ations @ A )
=> ! [N: nat,K: num] :
( ( bit_se4730199178511100633sh_bit @ A @ ( suc @ N ) @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ K ) ) )
= ( bit_se4730199178511100633sh_bit @ A @ N @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ ( bit0 @ K ) ) ) ) ) ) ).
% push_bit_Suc_minus_numeral
thf(fact_5679_push__bit__minus__one__eq__not__mask,axiom,
! [A: $tType] :
( ( bit_ri3973907225187159222ations @ A )
=> ! [N: nat] :
( ( bit_se4730199178511100633sh_bit @ A @ N @ ( uminus_uminus @ A @ ( one_one @ A ) ) )
= ( bit_ri4277139882892585799ns_not @ A @ ( bit_se2239418461657761734s_mask @ A @ N ) ) ) ) ).
% push_bit_minus_one_eq_not_mask
thf(fact_5680_push__bit__numeral,axiom,
! [A: $tType] :
( ( bit_se359711467146920520ations @ A )
=> ! [L: num,K: num] :
( ( bit_se4730199178511100633sh_bit @ A @ ( numeral_numeral @ nat @ L ) @ ( numeral_numeral @ A @ K ) )
= ( bit_se4730199178511100633sh_bit @ A @ ( pred_numeral @ L ) @ ( numeral_numeral @ A @ ( bit0 @ K ) ) ) ) ) ).
% push_bit_numeral
thf(fact_5681_drop__bit__numeral__minus__bit0,axiom,
! [L: num,K: num] :
( ( bit_se4197421643247451524op_bit @ int @ ( numeral_numeral @ nat @ L ) @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ ( bit0 @ K ) ) ) )
= ( bit_se4197421643247451524op_bit @ int @ ( pred_numeral @ L ) @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ K ) ) ) ) ).
% drop_bit_numeral_minus_bit0
thf(fact_5682_drop__bit__Suc__minus__bit1,axiom,
! [N: nat,K: num] :
( ( bit_se4197421643247451524op_bit @ int @ ( suc @ N ) @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ ( bit1 @ K ) ) ) )
= ( bit_se4197421643247451524op_bit @ int @ N @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ ( inc @ K ) ) ) ) ) ).
% drop_bit_Suc_minus_bit1
thf(fact_5683_push__bit__Suc,axiom,
! [A: $tType] :
( ( bit_se359711467146920520ations @ A )
=> ! [N: nat,A2: A] :
( ( bit_se4730199178511100633sh_bit @ A @ ( suc @ N ) @ A2 )
= ( bit_se4730199178511100633sh_bit @ A @ N @ ( times_times @ A @ A2 @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) ) ) ) ).
% push_bit_Suc
thf(fact_5684_push__bit__of__1,axiom,
! [A: $tType] :
( ( bit_se359711467146920520ations @ A )
=> ! [N: nat] :
( ( bit_se4730199178511100633sh_bit @ A @ N @ ( one_one @ A ) )
= ( power_power @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ N ) ) ) ).
% push_bit_of_1
thf(fact_5685_even__push__bit__iff,axiom,
! [A: $tType] :
( ( bit_se359711467146920520ations @ A )
=> ! [N: nat,A2: A] :
( ( dvd_dvd @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ ( bit_se4730199178511100633sh_bit @ A @ N @ A2 ) )
= ( ( N
!= ( zero_zero @ nat ) )
| ( dvd_dvd @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ A2 ) ) ) ) ).
% even_push_bit_iff
thf(fact_5686_push__bit__minus__numeral,axiom,
! [A: $tType] :
( ( bit_ri3973907225187159222ations @ A )
=> ! [L: num,K: num] :
( ( bit_se4730199178511100633sh_bit @ A @ ( numeral_numeral @ nat @ L ) @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ K ) ) )
= ( bit_se4730199178511100633sh_bit @ A @ ( pred_numeral @ L ) @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ ( bit0 @ K ) ) ) ) ) ) ).
% push_bit_minus_numeral
thf(fact_5687_drop__bit__numeral__minus__bit1,axiom,
! [L: num,K: num] :
( ( bit_se4197421643247451524op_bit @ int @ ( numeral_numeral @ nat @ L ) @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ ( bit1 @ K ) ) ) )
= ( bit_se4197421643247451524op_bit @ int @ ( pred_numeral @ L ) @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ ( inc @ K ) ) ) ) ) ).
% drop_bit_numeral_minus_bit1
thf(fact_5688_push__bit__numeral__minus__1,axiom,
! [A: $tType] :
( ( bit_ri3973907225187159222ations @ A )
=> ! [N: num] :
( ( bit_se4730199178511100633sh_bit @ A @ ( numeral_numeral @ nat @ N ) @ ( uminus_uminus @ A @ ( one_one @ A ) ) )
= ( uminus_uminus @ A @ ( power_power @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ ( numeral_numeral @ nat @ N ) ) ) ) ) ).
% push_bit_numeral_minus_1
thf(fact_5689_drop__bit__nat__eq,axiom,
! [N: nat,K: int] :
( ( bit_se4197421643247451524op_bit @ nat @ N @ ( nat2 @ K ) )
= ( nat2 @ ( bit_se4197421643247451524op_bit @ int @ N @ K ) ) ) ).
% drop_bit_nat_eq
thf(fact_5690_drop__bit__int__code_I1_J,axiom,
! [I: int] :
( ( bit_se4197421643247451524op_bit @ int @ ( zero_zero @ nat ) @ I )
= I ) ).
% drop_bit_int_code(1)
thf(fact_5691_drop__bit__int__code_I2_J,axiom,
! [N: nat] :
( ( bit_se4197421643247451524op_bit @ int @ ( suc @ N ) @ ( zero_zero @ int ) )
= ( zero_zero @ int ) ) ).
% drop_bit_int_code(2)
thf(fact_5692_push__bit__take__bit,axiom,
! [A: $tType] :
( ( bit_se359711467146920520ations @ A )
=> ! [M: nat,N: nat,A2: A] :
( ( bit_se4730199178511100633sh_bit @ A @ M @ ( bit_se2584673776208193580ke_bit @ A @ N @ A2 ) )
= ( bit_se2584673776208193580ke_bit @ A @ ( plus_plus @ nat @ M @ N ) @ ( bit_se4730199178511100633sh_bit @ A @ M @ A2 ) ) ) ) ).
% push_bit_take_bit
thf(fact_5693_concat__bit__assoc,axiom,
! [N: nat,K: int,M: nat,L: int,R2: int] :
( ( bit_concat_bit @ N @ K @ ( bit_concat_bit @ M @ L @ R2 ) )
= ( bit_concat_bit @ ( plus_plus @ nat @ M @ N ) @ ( bit_concat_bit @ N @ K @ L ) @ R2 ) ) ).
% concat_bit_assoc
thf(fact_5694_push__bit__add,axiom,
! [A: $tType] :
( ( bit_se359711467146920520ations @ A )
=> ! [N: nat,A2: A,B2: A] :
( ( bit_se4730199178511100633sh_bit @ A @ N @ ( plus_plus @ A @ A2 @ B2 ) )
= ( plus_plus @ A @ ( bit_se4730199178511100633sh_bit @ A @ N @ A2 ) @ ( bit_se4730199178511100633sh_bit @ A @ N @ B2 ) ) ) ) ).
% push_bit_add
thf(fact_5695_push__bit__minus,axiom,
! [A: $tType] :
( ( bit_ri3973907225187159222ations @ A )
=> ! [N: nat,A2: A] :
( ( bit_se4730199178511100633sh_bit @ A @ N @ ( uminus_uminus @ A @ A2 ) )
= ( uminus_uminus @ A @ ( bit_se4730199178511100633sh_bit @ A @ N @ A2 ) ) ) ) ).
% push_bit_minus
thf(fact_5696_push__bit__of__int,axiom,
! [A: $tType] :
( ( bit_ri3973907225187159222ations @ A )
=> ! [N: nat,K: int] :
( ( bit_se4730199178511100633sh_bit @ A @ N @ ( ring_1_of_int @ A @ K ) )
= ( ring_1_of_int @ A @ ( bit_se4730199178511100633sh_bit @ int @ N @ K ) ) ) ) ).
% push_bit_of_int
thf(fact_5697_concat__bit__take__bit__eq,axiom,
! [N: nat,B2: int] :
( ( bit_concat_bit @ N @ ( bit_se2584673776208193580ke_bit @ int @ N @ B2 ) )
= ( bit_concat_bit @ N @ B2 ) ) ).
% concat_bit_take_bit_eq
thf(fact_5698_concat__bit__eq__iff,axiom,
! [N: nat,K: int,L: int,R2: int,S: int] :
( ( ( bit_concat_bit @ N @ K @ L )
= ( bit_concat_bit @ N @ R2 @ S ) )
= ( ( ( bit_se2584673776208193580ke_bit @ int @ N @ K )
= ( bit_se2584673776208193580ke_bit @ int @ N @ R2 ) )
& ( L = S ) ) ) ).
% concat_bit_eq_iff
thf(fact_5699_push__bit__of__nat,axiom,
! [A: $tType] :
( ( bit_se359711467146920520ations @ A )
=> ! [N: nat,M: nat] :
( ( bit_se4730199178511100633sh_bit @ A @ N @ ( semiring_1_of_nat @ A @ M ) )
= ( semiring_1_of_nat @ A @ ( bit_se4730199178511100633sh_bit @ nat @ N @ M ) ) ) ) ).
% push_bit_of_nat
thf(fact_5700_of__nat__push__bit,axiom,
! [A: $tType] :
( ( bit_se359711467146920520ations @ A )
=> ! [M: nat,N: nat] :
( ( semiring_1_of_nat @ A @ ( bit_se4730199178511100633sh_bit @ nat @ M @ N ) )
= ( bit_se4730199178511100633sh_bit @ A @ M @ ( semiring_1_of_nat @ A @ N ) ) ) ) ).
% of_nat_push_bit
thf(fact_5701_take__bit__push__bit,axiom,
! [A: $tType] :
( ( bit_se359711467146920520ations @ A )
=> ! [M: nat,N: nat,A2: A] :
( ( bit_se2584673776208193580ke_bit @ A @ M @ ( bit_se4730199178511100633sh_bit @ A @ N @ A2 ) )
= ( bit_se4730199178511100633sh_bit @ A @ N @ ( bit_se2584673776208193580ke_bit @ A @ ( minus_minus @ nat @ M @ N ) @ A2 ) ) ) ) ).
% take_bit_push_bit
thf(fact_5702_div__push__bit__of__1__eq__drop__bit,axiom,
! [A: $tType] :
( ( bit_se359711467146920520ations @ A )
=> ! [A2: A,N: nat] :
( ( divide_divide @ A @ A2 @ ( bit_se4730199178511100633sh_bit @ A @ N @ ( one_one @ A ) ) )
= ( bit_se4197421643247451524op_bit @ A @ N @ A2 ) ) ) ).
% div_push_bit_of_1_eq_drop_bit
thf(fact_5703_bits__ident,axiom,
! [A: $tType] :
( ( bit_se359711467146920520ations @ A )
=> ! [N: nat,A2: A] :
( ( plus_plus @ A @ ( bit_se4730199178511100633sh_bit @ A @ N @ ( bit_se4197421643247451524op_bit @ A @ N @ A2 ) ) @ ( bit_se2584673776208193580ke_bit @ A @ N @ A2 ) )
= A2 ) ) ).
% bits_ident
thf(fact_5704_set__bit__eq__or,axiom,
! [A: $tType] :
( ( bit_se359711467146920520ations @ A )
=> ( ( bit_se5668285175392031749et_bit @ A )
= ( ^ [N3: nat,A4: A] : ( bit_se1065995026697491101ons_or @ A @ A4 @ ( bit_se4730199178511100633sh_bit @ A @ N3 @ ( one_one @ A ) ) ) ) ) ) ).
% set_bit_eq_or
thf(fact_5705_flip__bit__eq__xor,axiom,
! [A: $tType] :
( ( bit_se359711467146920520ations @ A )
=> ( ( bit_se8732182000553998342ip_bit @ A )
= ( ^ [N3: nat,A4: A] : ( bit_se5824344971392196577ns_xor @ A @ A4 @ ( bit_se4730199178511100633sh_bit @ A @ N3 @ ( one_one @ A ) ) ) ) ) ) ).
% flip_bit_eq_xor
thf(fact_5706_push__bit__double,axiom,
! [A: $tType] :
( ( bit_se359711467146920520ations @ A )
=> ! [N: nat,A2: A] :
( ( bit_se4730199178511100633sh_bit @ A @ N @ ( times_times @ A @ A2 @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) )
= ( times_times @ A @ ( bit_se4730199178511100633sh_bit @ A @ N @ A2 ) @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) ) ) ).
% push_bit_double
thf(fact_5707_bit__iff__and__push__bit__not__eq__0,axiom,
! [A: $tType] :
( ( bit_se359711467146920520ations @ A )
=> ( ( bit_se5641148757651400278ts_bit @ A )
= ( ^ [A4: A,N3: nat] :
( ( bit_se5824344872417868541ns_and @ A @ A4 @ ( bit_se4730199178511100633sh_bit @ A @ N3 @ ( one_one @ A ) ) )
!= ( zero_zero @ A ) ) ) ) ) ).
% bit_iff_and_push_bit_not_eq_0
thf(fact_5708_push__bit__mask__eq,axiom,
! [A: $tType] :
( ( bit_ri3973907225187159222ations @ A )
=> ! [M: nat,N: nat] :
( ( bit_se4730199178511100633sh_bit @ A @ M @ ( bit_se2239418461657761734s_mask @ A @ N ) )
= ( bit_se5824344872417868541ns_and @ A @ ( bit_se2239418461657761734s_mask @ A @ ( plus_plus @ nat @ N @ M ) ) @ ( bit_ri4277139882892585799ns_not @ A @ ( bit_se2239418461657761734s_mask @ A @ M ) ) ) ) ) ).
% push_bit_mask_eq
thf(fact_5709_unset__bit__eq__and__not,axiom,
! [A: $tType] :
( ( bit_ri3973907225187159222ations @ A )
=> ( ( bit_se2638667681897837118et_bit @ A )
= ( ^ [N3: nat,A4: A] : ( bit_se5824344872417868541ns_and @ A @ A4 @ ( bit_ri4277139882892585799ns_not @ A @ ( bit_se4730199178511100633sh_bit @ A @ N3 @ ( one_one @ A ) ) ) ) ) ) ) ).
% unset_bit_eq_and_not
thf(fact_5710_bit__concat__bit__iff,axiom,
! [M: nat,K: int,L: int,N: nat] :
( ( bit_se5641148757651400278ts_bit @ int @ ( bit_concat_bit @ M @ K @ L ) @ N )
= ( ( ( ord_less @ nat @ N @ M )
& ( bit_se5641148757651400278ts_bit @ int @ K @ N ) )
| ( ( ord_less_eq @ nat @ M @ N )
& ( bit_se5641148757651400278ts_bit @ int @ L @ ( minus_minus @ nat @ N @ M ) ) ) ) ) ).
% bit_concat_bit_iff
thf(fact_5711_shiftr__integer__conv__div__pow2,axiom,
( ( bit_se4197421643247451524op_bit @ code_integer )
= ( ^ [N3: nat,X2: code_integer] : ( divide_divide @ code_integer @ X2 @ ( power_power @ code_integer @ ( numeral_numeral @ code_integer @ ( bit0 @ one2 ) ) @ N3 ) ) ) ) ).
% shiftr_integer_conv_div_pow2
thf(fact_5712_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_5713_drop__bit__int__def,axiom,
( ( bit_se4197421643247451524op_bit @ int )
= ( ^ [N3: nat,K3: int] : ( divide_divide @ int @ K3 @ ( power_power @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ N3 ) ) ) ) ).
% drop_bit_int_def
thf(fact_5714_signed__take__bit__eq__concat__bit,axiom,
( ( bit_ri4674362597316999326ke_bit @ int )
= ( ^ [N3: nat,K3: int] : ( bit_concat_bit @ N3 @ K3 @ ( uminus_uminus @ int @ ( zero_neq_one_of_bool @ int @ ( bit_se5641148757651400278ts_bit @ int @ K3 @ N3 ) ) ) ) ) ) ).
% signed_take_bit_eq_concat_bit
thf(fact_5715_push__bit__eq__mult,axiom,
! [A: $tType] :
( ( bit_se359711467146920520ations @ A )
=> ( ( bit_se4730199178511100633sh_bit @ A )
= ( ^ [N3: nat,A4: A] : ( times_times @ A @ A4 @ ( power_power @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ N3 ) ) ) ) ) ).
% push_bit_eq_mult
thf(fact_5716_exp__dvdE,axiom,
! [A: $tType] :
( ( bit_se359711467146920520ations @ A )
=> ! [N: nat,A2: A] :
( ( dvd_dvd @ A @ ( power_power @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ N ) @ A2 )
=> ~ ! [B3: A] :
( A2
!= ( bit_se4730199178511100633sh_bit @ A @ N @ B3 ) ) ) ) ).
% exp_dvdE
thf(fact_5717_drop__bit__nat__def,axiom,
( ( bit_se4197421643247451524op_bit @ nat )
= ( ^ [N3: nat,M4: nat] : ( divide_divide @ nat @ M4 @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N3 ) ) ) ) ).
% drop_bit_nat_def
thf(fact_5718_slice__eq__mask,axiom,
! [A: $tType] :
( ( bit_ri3973907225187159222ations @ A )
=> ! [N: nat,M: nat,A2: A] :
( ( bit_se4730199178511100633sh_bit @ A @ N @ ( bit_se2584673776208193580ke_bit @ A @ M @ ( bit_se4197421643247451524op_bit @ A @ N @ A2 ) ) )
= ( bit_se5824344872417868541ns_and @ A @ A2 @ ( bit_se5824344872417868541ns_and @ A @ ( bit_se2239418461657761734s_mask @ A @ ( plus_plus @ nat @ M @ N ) ) @ ( bit_ri4277139882892585799ns_not @ A @ ( bit_se2239418461657761734s_mask @ A @ N ) ) ) ) ) ) ).
% slice_eq_mask
thf(fact_5719_take__bit__sum,axiom,
! [A: $tType] :
( ( bit_un5681908812861735899ations @ A )
=> ( ( bit_se2584673776208193580ke_bit @ A )
= ( ^ [N3: nat,A4: A] :
( groups7311177749621191930dd_sum @ nat @ A
@ ^ [K3: nat] : ( bit_se4730199178511100633sh_bit @ A @ K3 @ ( zero_neq_one_of_bool @ A @ ( bit_se5641148757651400278ts_bit @ A @ A4 @ K3 ) ) )
@ ( set_or7035219750837199246ssThan @ nat @ ( zero_zero @ nat ) @ N3 ) ) ) ) ) ).
% take_bit_sum
thf(fact_5720_word__and__mask__or__conv__and__mask,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [N: word @ A,Index: nat] :
( ( bit_se5641148757651400278ts_bit @ ( word @ A ) @ N @ Index )
=> ( ( bit_se1065995026697491101ons_or @ ( word @ A ) @ ( bit_se5824344872417868541ns_and @ ( word @ A ) @ N @ ( bit_se2239418461657761734s_mask @ ( word @ A ) @ Index ) ) @ ( bit_se4730199178511100633sh_bit @ ( word @ A ) @ Index @ ( one_one @ ( word @ A ) ) ) )
= ( bit_se5824344872417868541ns_and @ ( word @ A ) @ N @ ( bit_se2239418461657761734s_mask @ ( word @ A ) @ ( plus_plus @ nat @ Index @ ( one_one @ nat ) ) ) ) ) ) ) ).
% word_and_mask_or_conv_and_mask
thf(fact_5721_signed__take__bit__code,axiom,
! [A: $tType] :
( ( bit_ri3973907225187159222ations @ A )
=> ( ( bit_ri4674362597316999326ke_bit @ A )
= ( ^ [N3: nat,A4: A] : ( if @ A @ ( bit_se5641148757651400278ts_bit @ A @ ( bit_se2584673776208193580ke_bit @ A @ ( suc @ N3 ) @ A4 ) @ N3 ) @ ( plus_plus @ A @ ( bit_se2584673776208193580ke_bit @ A @ ( suc @ N3 ) @ A4 ) @ ( bit_se4730199178511100633sh_bit @ A @ ( suc @ N3 ) @ ( uminus_uminus @ A @ ( one_one @ A ) ) ) ) @ ( bit_se2584673776208193580ke_bit @ A @ ( suc @ N3 ) @ A4 ) ) ) ) ) ).
% signed_take_bit_code
thf(fact_5722_word__type__copy__bits__axioms_Ointro,axiom,
! [B: $tType,A: $tType] :
( ( ( cl_HOL_Oequal @ A )
& ( linorder @ A )
& ( comm_ring_1 @ A )
& ( semiring_modulo @ A )
& ( type_len @ B ) )
=> ! [Word_of: A > ( word @ B ),Signed_drop_bit: nat > A > A] :
( ! [P8: A] :
( ( bit_se5641148757651400278ts_bit @ A @ P8 )
= ( bit_se5641148757651400278ts_bit @ ( word @ B ) @ ( Word_of @ P8 ) ) )
=> ( ! [P8: A] :
( ( Word_of @ ( bit_ri4277139882892585799ns_not @ A @ P8 ) )
= ( bit_ri4277139882892585799ns_not @ ( word @ B ) @ ( Word_of @ P8 ) ) )
=> ( ! [P8: A,Q5: A] :
( ( Word_of @ ( bit_se5824344872417868541ns_and @ A @ P8 @ Q5 ) )
= ( bit_se5824344872417868541ns_and @ ( word @ B ) @ ( Word_of @ P8 ) @ ( Word_of @ Q5 ) ) )
=> ( ! [P8: A,Q5: A] :
( ( Word_of @ ( bit_se1065995026697491101ons_or @ A @ P8 @ Q5 ) )
= ( bit_se1065995026697491101ons_or @ ( word @ B ) @ ( Word_of @ P8 ) @ ( Word_of @ Q5 ) ) )
=> ( ! [P8: A,Q5: A] :
( ( Word_of @ ( bit_se5824344971392196577ns_xor @ A @ P8 @ Q5 ) )
= ( bit_se5824344971392196577ns_xor @ ( word @ B ) @ ( Word_of @ P8 ) @ ( Word_of @ Q5 ) ) )
=> ( ! [N2: nat] :
( ( Word_of @ ( bit_se2239418461657761734s_mask @ A @ N2 ) )
= ( bit_se2239418461657761734s_mask @ ( word @ B ) @ N2 ) )
=> ( ! [N2: nat,P8: A] :
( ( Word_of @ ( bit_se4730199178511100633sh_bit @ A @ N2 @ P8 ) )
= ( bit_se4730199178511100633sh_bit @ ( word @ B ) @ N2 @ ( Word_of @ P8 ) ) )
=> ( ! [N2: nat,P8: A] :
( ( Word_of @ ( bit_se4197421643247451524op_bit @ A @ N2 @ P8 ) )
= ( bit_se4197421643247451524op_bit @ ( word @ B ) @ N2 @ ( Word_of @ P8 ) ) )
=> ( ! [N2: nat,P8: A] :
( ( Word_of @ ( Signed_drop_bit @ N2 @ P8 ) )
= ( signed_drop_bit @ B @ N2 @ ( Word_of @ P8 ) ) )
=> ( ! [N2: nat,P8: A] :
( ( Word_of @ ( bit_se2584673776208193580ke_bit @ A @ N2 @ P8 ) )
= ( bit_se2584673776208193580ke_bit @ ( word @ B ) @ N2 @ ( Word_of @ P8 ) ) )
=> ( ! [N2: nat,P8: A] :
( ( Word_of @ ( bit_se5668285175392031749et_bit @ A @ N2 @ P8 ) )
= ( bit_se5668285175392031749et_bit @ ( word @ B ) @ N2 @ ( Word_of @ P8 ) ) )
=> ( ! [N2: nat,P8: A] :
( ( Word_of @ ( bit_se2638667681897837118et_bit @ A @ N2 @ P8 ) )
= ( bit_se2638667681897837118et_bit @ ( word @ B ) @ N2 @ ( Word_of @ P8 ) ) )
=> ( ! [N2: nat,P8: A] :
( ( Word_of @ ( bit_se8732182000553998342ip_bit @ A @ N2 @ P8 ) )
= ( bit_se8732182000553998342ip_bit @ ( word @ B ) @ N2 @ ( Word_of @ P8 ) ) )
=> ( word_T1405001541757447525axioms @ A @ B @ Word_of @ Signed_drop_bit ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).
% word_type_copy_bits_axioms.intro
thf(fact_5723_word__type__copy__bits__axioms__def,axiom,
! [B: $tType,A: $tType] :
( ( ( cl_HOL_Oequal @ A )
& ( linorder @ A )
& ( comm_ring_1 @ A )
& ( semiring_modulo @ A )
& ( type_len @ B ) )
=> ( ( word_T1405001541757447525axioms @ A @ B )
= ( ^ [Word_of2: A > ( word @ B ),Signed_drop_bit2: nat > A > A] :
( ! [P5: A] :
( ( bit_se5641148757651400278ts_bit @ A @ P5 )
= ( bit_se5641148757651400278ts_bit @ ( word @ B ) @ ( Word_of2 @ P5 ) ) )
& ! [P5: A] :
( ( Word_of2 @ ( bit_ri4277139882892585799ns_not @ A @ P5 ) )
= ( bit_ri4277139882892585799ns_not @ ( word @ B ) @ ( Word_of2 @ P5 ) ) )
& ! [P5: A,Q6: A] :
( ( Word_of2 @ ( bit_se5824344872417868541ns_and @ A @ P5 @ Q6 ) )
= ( bit_se5824344872417868541ns_and @ ( word @ B ) @ ( Word_of2 @ P5 ) @ ( Word_of2 @ Q6 ) ) )
& ! [P5: A,Q6: A] :
( ( Word_of2 @ ( bit_se1065995026697491101ons_or @ A @ P5 @ Q6 ) )
= ( bit_se1065995026697491101ons_or @ ( word @ B ) @ ( Word_of2 @ P5 ) @ ( Word_of2 @ Q6 ) ) )
& ! [P5: A,Q6: A] :
( ( Word_of2 @ ( bit_se5824344971392196577ns_xor @ A @ P5 @ Q6 ) )
= ( bit_se5824344971392196577ns_xor @ ( word @ B ) @ ( Word_of2 @ P5 ) @ ( Word_of2 @ Q6 ) ) )
& ! [N3: nat] :
( ( Word_of2 @ ( bit_se2239418461657761734s_mask @ A @ N3 ) )
= ( bit_se2239418461657761734s_mask @ ( word @ B ) @ N3 ) )
& ! [N3: nat,P5: A] :
( ( Word_of2 @ ( bit_se4730199178511100633sh_bit @ A @ N3 @ P5 ) )
= ( bit_se4730199178511100633sh_bit @ ( word @ B ) @ N3 @ ( Word_of2 @ P5 ) ) )
& ! [N3: nat,P5: A] :
( ( Word_of2 @ ( bit_se4197421643247451524op_bit @ A @ N3 @ P5 ) )
= ( bit_se4197421643247451524op_bit @ ( word @ B ) @ N3 @ ( Word_of2 @ P5 ) ) )
& ! [N3: nat,P5: A] :
( ( Word_of2 @ ( Signed_drop_bit2 @ N3 @ P5 ) )
= ( signed_drop_bit @ B @ N3 @ ( Word_of2 @ P5 ) ) )
& ! [N3: nat,P5: A] :
( ( Word_of2 @ ( bit_se2584673776208193580ke_bit @ A @ N3 @ P5 ) )
= ( bit_se2584673776208193580ke_bit @ ( word @ B ) @ N3 @ ( Word_of2 @ P5 ) ) )
& ! [N3: nat,P5: A] :
( ( Word_of2 @ ( bit_se5668285175392031749et_bit @ A @ N3 @ P5 ) )
= ( bit_se5668285175392031749et_bit @ ( word @ B ) @ N3 @ ( Word_of2 @ P5 ) ) )
& ! [N3: nat,P5: A] :
( ( Word_of2 @ ( bit_se2638667681897837118et_bit @ A @ N3 @ P5 ) )
= ( bit_se2638667681897837118et_bit @ ( word @ B ) @ N3 @ ( Word_of2 @ P5 ) ) )
& ! [N3: nat,P5: A] :
( ( Word_of2 @ ( bit_se8732182000553998342ip_bit @ A @ N3 @ P5 ) )
= ( bit_se8732182000553998342ip_bit @ ( word @ B ) @ N3 @ ( Word_of2 @ P5 ) ) ) ) ) ) ) ).
% word_type_copy_bits_axioms_def
thf(fact_5724_set__bits__aux__Suc,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [F2: nat > $o,N: nat,W: word @ A] :
( ( code_T2661198915054445665ts_aux @ A @ F2 @ ( suc @ N ) @ W )
= ( code_T2661198915054445665ts_aux @ A @ F2 @ N @ ( bit_se1065995026697491101ons_or @ ( word @ A ) @ ( bit_se4730199178511100633sh_bit @ ( word @ A ) @ ( one_one @ nat ) @ W ) @ ( if @ ( word @ A ) @ ( F2 @ N ) @ ( one_one @ ( word @ A ) ) @ ( zero_zero @ ( word @ A ) ) ) ) ) ) ) ).
% set_bits_aux_Suc
thf(fact_5725_signed__drop__bit__0,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [W: word @ A] :
( ( signed_drop_bit @ A @ ( zero_zero @ nat ) @ W )
= W ) ) ).
% signed_drop_bit_0
thf(fact_5726_signed__drop__bit__signed__drop__bit,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [M: nat,N: nat,W: word @ A] :
( ( signed_drop_bit @ A @ M @ ( signed_drop_bit @ A @ N @ W ) )
= ( signed_drop_bit @ A @ ( plus_plus @ nat @ M @ N ) @ W ) ) ) ).
% signed_drop_bit_signed_drop_bit
thf(fact_5727_signed__drop__bit__of__0,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [N: nat] :
( ( signed_drop_bit @ A @ N @ ( zero_zero @ ( word @ A ) ) )
= ( zero_zero @ ( word @ A ) ) ) ) ).
% signed_drop_bit_of_0
thf(fact_5728_set__bits__aux__0,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [F2: nat > $o,W: word @ A] :
( ( code_T2661198915054445665ts_aux @ A @ F2 @ ( zero_zero @ nat ) @ W )
= W ) ) ).
% set_bits_aux_0
thf(fact_5729_push__bit__nonnegative__int__iff,axiom,
! [N: nat,K: int] :
( ( ord_less_eq @ int @ ( zero_zero @ int ) @ ( bit_se4730199178511100633sh_bit @ int @ N @ K ) )
= ( ord_less_eq @ int @ ( zero_zero @ int ) @ K ) ) ).
% push_bit_nonnegative_int_iff
thf(fact_5730_push__bit__negative__int__iff,axiom,
! [N: nat,K: int] :
( ( ord_less @ int @ ( bit_se4730199178511100633sh_bit @ int @ N @ K ) @ ( zero_zero @ int ) )
= ( ord_less @ int @ K @ ( zero_zero @ int ) ) ) ).
% push_bit_negative_int_iff
thf(fact_5731_signed__drop__bit__of__minus__1,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [N: nat] :
( ( signed_drop_bit @ A @ N @ ( uminus_uminus @ ( word @ A ) @ ( one_one @ ( word @ A ) ) ) )
= ( uminus_uminus @ ( word @ A ) @ ( one_one @ ( word @ A ) ) ) ) ) ).
% signed_drop_bit_of_minus_1
thf(fact_5732_concat__bit__of__zero__1,axiom,
! [N: nat,L: int] :
( ( bit_concat_bit @ N @ ( zero_zero @ int ) @ L )
= ( bit_se4730199178511100633sh_bit @ int @ N @ L ) ) ).
% concat_bit_of_zero_1
thf(fact_5733_push__bit__of__Suc__0,axiom,
! [N: nat] :
( ( bit_se4730199178511100633sh_bit @ nat @ N @ ( suc @ ( zero_zero @ nat ) ) )
= ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N ) ) ).
% push_bit_of_Suc_0
thf(fact_5734_drop__bit__push__bit__int,axiom,
! [M: nat,N: nat,K: int] :
( ( bit_se4197421643247451524op_bit @ int @ M @ ( bit_se4730199178511100633sh_bit @ int @ N @ K ) )
= ( bit_se4197421643247451524op_bit @ int @ ( minus_minus @ nat @ M @ N ) @ ( bit_se4730199178511100633sh_bit @ int @ ( minus_minus @ nat @ N @ M ) @ K ) ) ) ).
% drop_bit_push_bit_int
thf(fact_5735_push__bit__int__code_I1_J,axiom,
! [I: int] :
( ( bit_se4730199178511100633sh_bit @ int @ ( zero_zero @ nat ) @ I )
= I ) ).
% push_bit_int_code(1)
thf(fact_5736_push__bit__nat__eq,axiom,
! [N: nat,K: int] :
( ( bit_se4730199178511100633sh_bit @ nat @ N @ ( nat2 @ K ) )
= ( nat2 @ ( bit_se4730199178511100633sh_bit @ int @ N @ K ) ) ) ).
% push_bit_nat_eq
thf(fact_5737_set__bit__nat__def,axiom,
( ( bit_se5668285175392031749et_bit @ nat )
= ( ^ [M4: nat,N3: nat] : ( bit_se1065995026697491101ons_or @ nat @ N3 @ ( bit_se4730199178511100633sh_bit @ nat @ M4 @ ( one_one @ nat ) ) ) ) ) ).
% set_bit_nat_def
thf(fact_5738_flip__bit__nat__def,axiom,
( ( bit_se8732182000553998342ip_bit @ nat )
= ( ^ [M4: nat,N3: nat] : ( bit_se5824344971392196577ns_xor @ nat @ N3 @ ( bit_se4730199178511100633sh_bit @ nat @ M4 @ ( one_one @ nat ) ) ) ) ) ).
% flip_bit_nat_def
thf(fact_5739_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_5740_bit__push__bit__iff__int,axiom,
! [M: nat,K: int,N: nat] :
( ( bit_se5641148757651400278ts_bit @ int @ ( bit_se4730199178511100633sh_bit @ int @ M @ K ) @ N )
= ( ( ord_less_eq @ nat @ M @ N )
& ( bit_se5641148757651400278ts_bit @ int @ K @ ( minus_minus @ nat @ N @ M ) ) ) ) ).
% bit_push_bit_iff_int
thf(fact_5741_concat__bit__eq,axiom,
( bit_concat_bit
= ( ^ [N3: nat,K3: int,L2: int] : ( plus_plus @ int @ ( bit_se2584673776208193580ke_bit @ int @ N3 @ K3 ) @ ( bit_se4730199178511100633sh_bit @ int @ N3 @ L2 ) ) ) ) ).
% concat_bit_eq
thf(fact_5742_concat__bit__def,axiom,
( bit_concat_bit
= ( ^ [N3: nat,K3: int,L2: int] : ( bit_se1065995026697491101ons_or @ int @ ( bit_se2584673776208193580ke_bit @ int @ N3 @ K3 ) @ ( bit_se4730199178511100633sh_bit @ int @ N3 @ L2 ) ) ) ) ).
% concat_bit_def
thf(fact_5743_Bit__Operations_Oset__bit__int__def,axiom,
( ( bit_se5668285175392031749et_bit @ int )
= ( ^ [N3: nat,K3: int] : ( bit_se1065995026697491101ons_or @ int @ K3 @ ( bit_se4730199178511100633sh_bit @ int @ N3 @ ( one_one @ int ) ) ) ) ) ).
% Bit_Operations.set_bit_int_def
thf(fact_5744_bit__push__bit__iff__nat,axiom,
! [M: nat,Q3: nat,N: nat] :
( ( bit_se5641148757651400278ts_bit @ nat @ ( bit_se4730199178511100633sh_bit @ nat @ M @ Q3 ) @ N )
= ( ( ord_less_eq @ nat @ M @ N )
& ( bit_se5641148757651400278ts_bit @ nat @ Q3 @ ( minus_minus @ nat @ N @ M ) ) ) ) ).
% bit_push_bit_iff_nat
thf(fact_5745_flip__bit__int__def,axiom,
( ( bit_se8732182000553998342ip_bit @ int )
= ( ^ [N3: nat,K3: int] : ( bit_se5824344971392196577ns_xor @ int @ K3 @ ( bit_se4730199178511100633sh_bit @ int @ N3 @ ( one_one @ int ) ) ) ) ) ).
% flip_bit_int_def
thf(fact_5746_shiftl__integer__conv__mult__pow2,axiom,
( ( bit_se4730199178511100633sh_bit @ code_integer )
= ( ^ [N3: nat,X2: code_integer] : ( times_times @ code_integer @ X2 @ ( power_power @ code_integer @ ( numeral_numeral @ code_integer @ ( bit0 @ one2 ) ) @ N3 ) ) ) ) ).
% shiftl_integer_conv_mult_pow2
thf(fact_5747_unset__bit__int__def,axiom,
( ( bit_se2638667681897837118et_bit @ int )
= ( ^ [N3: nat,K3: int] : ( bit_se5824344872417868541ns_and @ int @ K3 @ ( bit_ri4277139882892585799ns_not @ int @ ( bit_se4730199178511100633sh_bit @ int @ N3 @ ( one_one @ int ) ) ) ) ) ) ).
% unset_bit_int_def
thf(fact_5748_push__bit__int__def,axiom,
( ( bit_se4730199178511100633sh_bit @ int )
= ( ^ [N3: nat,K3: int] : ( times_times @ int @ K3 @ ( power_power @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ N3 ) ) ) ) ).
% push_bit_int_def
thf(fact_5749_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_5750_push__bit__nat__def,axiom,
( ( bit_se4730199178511100633sh_bit @ nat )
= ( ^ [N3: nat,M4: nat] : ( times_times @ nat @ M4 @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N3 ) ) ) ) ).
% push_bit_nat_def
thf(fact_5751_push__bit__minus__one,axiom,
! [N: nat] :
( ( bit_se4730199178511100633sh_bit @ int @ N @ ( uminus_uminus @ int @ ( one_one @ int ) ) )
= ( uminus_uminus @ int @ ( power_power @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ N ) ) ) ).
% push_bit_minus_one
thf(fact_5752_set__bits__aux__rec,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ( ( code_T2661198915054445665ts_aux @ A )
= ( ^ [F4: nat > $o,N3: nat,W2: word @ A] :
( if @ ( word @ A )
@ ( N3
= ( zero_zero @ nat ) )
@ W2
@ ( code_T2661198915054445665ts_aux @ A @ F4 @ ( minus_minus @ nat @ N3 @ ( one_one @ nat ) ) @ ( bit_se1065995026697491101ons_or @ ( word @ A ) @ ( bit_se4730199178511100633sh_bit @ ( word @ A ) @ ( one_one @ nat ) @ W2 ) @ ( if @ ( word @ A ) @ ( F4 @ ( minus_minus @ nat @ N3 @ ( one_one @ nat ) ) ) @ ( one_one @ ( word @ A ) ) @ ( zero_zero @ ( word @ A ) ) ) ) ) ) ) ) ) ).
% set_bits_aux_rec
thf(fact_5753_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 ) )
=> ( ! [N10: nat] :
( ( bit_se5641148757651400278ts_bit @ ( word @ B ) @ B2 @ N10 )
= ( ( ord_less @ nat @ N10 @ ( size_size @ ( word @ B ) @ B2 ) )
& ( bit_se5641148757651400278ts_bit @ ( word @ C ) @ C2 @ N10 ) ) )
& ! [M3: nat] :
( ( bit_se5641148757651400278ts_bit @ ( word @ A ) @ A2 @ M3 )
= ( ( ord_less @ nat @ M3 @ ( size_size @ ( word @ A ) @ A2 ) )
& ( bit_se5641148757651400278ts_bit @ ( word @ C ) @ C2 @ ( plus_plus @ nat @ M3 @ ( size_size @ ( word @ B ) @ B2 ) ) ) ) ) ) ) ) ).
% test_bit_split
thf(fact_5754_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 ) )
=> ! [N10: nat,M3: nat] :
( ( ( bit_se5641148757651400278ts_bit @ ( word @ B ) @ B2 @ N10 )
= ( ( ord_less @ nat @ N10 @ ( size_size @ ( word @ B ) @ B2 ) )
& ( bit_se5641148757651400278ts_bit @ ( word @ C ) @ C2 @ N10 ) ) )
& ( ( bit_se5641148757651400278ts_bit @ ( word @ A ) @ A2 @ M3 )
= ( ( ord_less @ nat @ M3 @ ( size_size @ ( word @ A ) @ A2 ) )
& ( bit_se5641148757651400278ts_bit @ ( word @ C ) @ C2 @ ( plus_plus @ nat @ M3 @ ( size_size @ ( word @ B ) @ B2 ) ) ) ) ) ) ) ) ).
% test_bit_split'
thf(fact_5755_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 ) )
= ( ! [N3: nat] :
( ( bit_se5641148757651400278ts_bit @ ( word @ B ) @ B2 @ N3 )
= ( ( ord_less @ nat @ N3 @ ( size_size @ ( word @ B ) @ B2 ) )
& ( bit_se5641148757651400278ts_bit @ ( word @ C ) @ C2 @ N3 ) ) )
& ! [M4: nat] :
( ( bit_se5641148757651400278ts_bit @ ( word @ A ) @ A2 @ M4 )
= ( ( ord_less @ nat @ M4 @ ( size_size @ ( word @ A ) @ A2 ) )
& ( bit_se5641148757651400278ts_bit @ ( word @ C ) @ C2 @ ( plus_plus @ nat @ M4 @ ( size_size @ ( word @ B ) @ B2 ) ) ) ) ) ) ) ) ).
% test_bit_split_eq
thf(fact_5756_set__bit__integer__conv__masks,axiom,
( ( generi7602027413899671122et_bit @ code_integer )
= ( ^ [X2: code_integer,I2: nat,B5: $o] : ( if @ code_integer @ B5 @ ( bit_se1065995026697491101ons_or @ code_integer @ X2 @ ( bit_se4730199178511100633sh_bit @ code_integer @ I2 @ ( one_one @ code_integer ) ) ) @ ( bit_se5824344872417868541ns_and @ code_integer @ X2 @ ( bit_ri4277139882892585799ns_not @ code_integer @ ( bit_se4730199178511100633sh_bit @ code_integer @ I2 @ ( one_one @ code_integer ) ) ) ) ) ) ) ).
% set_bit_integer_conv_masks
thf(fact_5757_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,N: 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 )
=> ( ( ( N
= ( zero_zero @ nat ) )
=> ( ( Set_bits_aux @ F2 @ N @ W )
= W ) )
& ( ( N
!= ( zero_zero @ nat ) )
=> ( ( Set_bits_aux @ F2 @ N @ W )
= ( Set_bits_aux @ F2 @ ( minus_minus @ nat @ N @ ( one_one @ nat ) ) @ ( bit_se1065995026697491101ons_or @ A @ ( bit_se4730199178511100633sh_bit @ A @ ( one_one @ nat ) @ W ) @ ( if @ A @ ( F2 @ ( minus_minus @ nat @ N @ ( one_one @ nat ) ) ) @ ( one_one @ A ) @ ( zero_zero @ A ) ) ) ) ) ) ) ) ) ).
% word_type_copy_misc.set_bits_aux_code
thf(fact_5758_Uint32__code,axiom,
( uint322
= ( ^ [I2: code_integer] : ( if @ uint32 @ ( bit_se5641148757651400278ts_bit @ code_integer @ ( bit_se5824344872417868541ns_and @ code_integer @ I2 @ ( numeral_numeral @ code_integer @ ( 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 @ ( bit1 @ one2 ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) @ ( numeral_numeral @ nat @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ one2 ) ) ) ) ) ) @ ( uint32_signed @ ( minus_minus @ code_integer @ ( bit_se5824344872417868541ns_and @ code_integer @ I2 @ ( numeral_numeral @ code_integer @ ( 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 @ ( 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 @ ( bit0 @ one2 ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) @ ( uint32_signed @ ( bit_se5824344872417868541ns_and @ code_integer @ I2 @ ( numeral_numeral @ code_integer @ ( 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 @ ( bit1 @ one2 ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).
% Uint32_code
thf(fact_5759_bit__set__bit__iff__2n,axiom,
! [A: $tType] :
( ( generic_set_set_bit @ A )
=> ! [A2: A,M: nat,B2: $o,N: nat] :
( ( bit_se5641148757651400278ts_bit @ A @ ( generi7602027413899671122et_bit @ A @ A2 @ M @ B2 ) @ N )
= ( ( ( M = N )
=> B2 )
& ( ( M != N )
=> ( bit_se5641148757651400278ts_bit @ A @ A2 @ N ) )
& ( ( power_power @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ N )
!= ( zero_zero @ A ) ) ) ) ) ).
% bit_set_bit_iff_2n
thf(fact_5760_Generic__set__bit_Oset__bit__eq,axiom,
! [A: $tType] :
( ( ( bit_ri3973907225187159222ations @ A )
& ( generic_set_set_bit @ A ) )
=> ( ( generi7602027413899671122et_bit @ A )
= ( ^ [A4: A,N3: nat,B5: $o] : ( if @ ( nat > A > A ) @ B5 @ ( bit_se5668285175392031749et_bit @ A ) @ ( bit_se2638667681897837118et_bit @ A ) @ N3 @ A4 ) ) ) ) ).
% Generic_set_bit.set_bit_eq
thf(fact_5761_Uint32__signed__def,axiom,
( uint32_signed
= ( ^ [I2: code_integer] :
( if @ uint32
@ ( ( ord_less @ code_integer @ I2 @ ( uminus_uminus @ code_integer @ ( 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 ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) )
| ( ord_less_eq @ code_integer @ ( 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 ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) @ I2 ) )
@ ( undefined @ ( ( code_integer > uint32 ) > code_integer > uint32 ) @ uint322 @ I2 )
@ ( uint322 @ I2 ) ) ) ) ).
% Uint32_signed_def
thf(fact_5762_set__bit__unfold,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ( ( generi7602027413899671122et_bit @ ( word @ A ) )
= ( ^ [W2: word @ A,N3: nat,B5: $o] : ( if @ ( word @ A ) @ B5 @ ( bit_se5668285175392031749et_bit @ ( word @ A ) @ N3 @ W2 ) @ ( bit_se2638667681897837118et_bit @ ( word @ A ) @ N3 @ W2 ) ) ) ) ) ).
% set_bit_unfold
thf(fact_5763_Generic__set__bit_Oset__bit__int__def,axiom,
( ( generi7602027413899671122et_bit @ int )
= ( ^ [I2: int,N3: nat,B5: $o] : ( if @ ( nat > int > int ) @ B5 @ ( bit_se5668285175392031749et_bit @ int ) @ ( bit_se2638667681897837118et_bit @ int ) @ N3 @ I2 ) ) ) ).
% Generic_set_bit.set_bit_int_def
thf(fact_5764_test__bit__set,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [W: word @ A,N: nat,X: $o] :
( ( bit_se5641148757651400278ts_bit @ ( word @ A ) @ ( generi7602027413899671122et_bit @ ( word @ A ) @ W @ N @ X ) @ N )
= ( ( ord_less @ nat @ N @ ( size_size @ ( word @ A ) @ W ) )
& X ) ) ) ).
% test_bit_set
thf(fact_5765_test__bit__set__gen,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [W: word @ A,N: nat,X: $o,M: nat] :
( ( bit_se5641148757651400278ts_bit @ ( word @ A ) @ ( generi7602027413899671122et_bit @ ( word @ A ) @ W @ N @ X ) @ M )
= ( ( ( M = N )
=> ( ( ord_less @ nat @ N @ ( size_size @ ( word @ A ) @ W ) )
& X ) )
& ( ( M != N )
=> ( bit_se5641148757651400278ts_bit @ ( word @ A ) @ W @ M ) ) ) ) ) ).
% test_bit_set_gen
thf(fact_5766_one__bit__pow,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [N: nat] :
( ( generi7602027413899671122et_bit @ ( word @ A ) @ ( zero_zero @ ( word @ A ) ) @ N @ $true )
= ( power_power @ ( word @ A ) @ ( numeral_numeral @ ( word @ A ) @ ( bit0 @ one2 ) ) @ N ) ) ) ).
% one_bit_pow
thf(fact_5767_int__set__bit__True__conv__OR,axiom,
! [I: int,N: nat] :
( ( generi7602027413899671122et_bit @ int @ I @ N @ $true )
= ( bit_se1065995026697491101ons_or @ int @ I @ ( bit_se4730199178511100633sh_bit @ int @ N @ ( one_one @ int ) ) ) ) ).
% int_set_bit_True_conv_OR
thf(fact_5768_int__set__bit__False__conv__NAND,axiom,
! [I: int,N: nat] :
( ( generi7602027413899671122et_bit @ int @ I @ N @ $false )
= ( bit_se5824344872417868541ns_and @ int @ I @ ( bit_ri4277139882892585799ns_not @ int @ ( bit_se4730199178511100633sh_bit @ int @ N @ ( one_one @ int ) ) ) ) ) ).
% int_set_bit_False_conv_NAND
thf(fact_5769_int__set__bit__conv__ops,axiom,
( ( generi7602027413899671122et_bit @ int )
= ( ^ [I2: int,N3: nat,B5: $o] : ( if @ int @ B5 @ ( bit_se1065995026697491101ons_or @ int @ I2 @ ( bit_se4730199178511100633sh_bit @ int @ N3 @ ( one_one @ int ) ) ) @ ( bit_se5824344872417868541ns_and @ int @ I2 @ ( bit_ri4277139882892585799ns_not @ int @ ( bit_se4730199178511100633sh_bit @ int @ N3 @ ( one_one @ int ) ) ) ) ) ) ) ).
% int_set_bit_conv_ops
thf(fact_5770_mod__h__bot__normalize,axiom,
! [A: $tType,H2: heap_ext @ product_unit,P: assn] :
( ( syntax7388354845996824322omatch @ A @ ( heap_ext @ product_unit ) @ ( undefined @ A ) @ H2 )
=> ( ( 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 ) @ ( undefined @ ( heap_ext @ product_unit ) ) @ ( bot_bot @ ( set @ nat ) ) ) ) ) ) ).
% mod_h_bot_normalize
thf(fact_5771_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_5772_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_5773_test__bit__cat,axiom,
! [A: $tType,B: $tType,C: $tType] :
( ( ( type_len @ C )
& ( type_len @ B )
& ( type_len @ A ) )
=> ! [A2: word @ B,B2: word @ C,N: nat] :
( ( bit_se5641148757651400278ts_bit @ ( word @ A ) @ ( word_cat @ B @ C @ A @ A2 @ B2 ) @ N )
= ( ( ord_less @ nat @ N @ ( size_size @ ( word @ A ) @ ( word_cat @ B @ C @ A @ A2 @ B2 ) ) )
& ( ( ord_less @ nat @ N @ ( size_size @ ( word @ C ) @ B2 ) )
=> ( bit_se5641148757651400278ts_bit @ ( word @ C ) @ B2 @ N ) )
& ( ~ ( ord_less @ nat @ N @ ( size_size @ ( word @ C ) @ B2 ) )
=> ( bit_se5641148757651400278ts_bit @ ( word @ B ) @ A2 @ ( minus_minus @ nat @ N @ ( size_size @ ( word @ C ) @ B2 ) ) ) ) ) ) ) ).
% test_bit_cat
thf(fact_5774_word__cat__split__size,axiom,
! [B: $tType,A: $tType,C: $tType] :
( ( ( type_len @ C )
& ( type_len @ A )
& ( type_len @ B ) )
=> ! [T: word @ A,U: word @ B,V: word @ C] :
( ( ord_less_eq @ nat @ ( size_size @ ( word @ A ) @ T ) @ ( 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 @ T ) )
=> ( T
= ( word_cat @ B @ C @ A @ U @ V ) ) ) ) ) ).
% word_cat_split_size
thf(fact_5775_cat__slices,axiom,
! [C: $tType,A: $tType,B: $tType] :
( ( ( type_len @ B )
& ( type_len @ A )
& ( type_len @ C ) )
=> ! [A2: word @ A,N: nat,C2: word @ B,B2: word @ C] :
( ( A2
= ( slice2 @ B @ A @ N @ C2 ) )
=> ( ( B2
= ( slice2 @ B @ C @ ( zero_zero @ nat ) @ C2 ) )
=> ( ( N
= ( 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_5776_mult_Osafe__commute,axiom,
! [A: $tType] :
( ( ab_semigroup_mult @ A )
=> ! [X: A,Y: A,A2: A,B2: A] :
( ( syntax7388354845996824322omatch @ A @ A @ ( times_times @ A @ X @ Y ) @ A2 )
=> ( ( times_times @ A @ A2 @ B2 )
= ( times_times @ A @ B2 @ A2 ) ) ) ) ).
% mult.safe_commute
thf(fact_5777_add_Osafe__commute,axiom,
! [A: $tType] :
( ( ab_semigroup_add @ A )
=> ! [X: A,Y: A,A2: A,B2: A] :
( ( syntax7388354845996824322omatch @ A @ A @ ( plus_plus @ A @ X @ Y ) @ A2 )
=> ( ( plus_plus @ A @ A2 @ B2 )
= ( plus_plus @ A @ B2 @ A2 ) ) ) ) ).
% add.safe_commute
thf(fact_5778_slice__0,axiom,
! [B: $tType,A: $tType] :
( ( ( type_len @ A )
& ( type_len @ B ) )
=> ! [N: nat] :
( ( slice2 @ B @ A @ N @ ( zero_zero @ ( word @ B ) ) )
= ( zero_zero @ ( word @ A ) ) ) ) ).
% slice_0
thf(fact_5779_slice__id,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [T: word @ A] :
( ( slice2 @ A @ A @ ( zero_zero @ nat ) @ T )
= T ) ) ).
% slice_id
thf(fact_5780_assn__aci_I11_J,axiom,
! [X: assn,Y: assn,A2: assn,B2: assn] :
( ( syntax7388354845996824322omatch @ assn @ assn @ ( times_times @ assn @ X @ Y ) @ A2 )
=> ( ( times_times @ assn @ A2 @ B2 )
= ( times_times @ assn @ B2 @ A2 ) ) ) ).
% assn_aci(11)
thf(fact_5781_add_Oright__assoc,axiom,
! [A: $tType] :
( ( ab_semigroup_add @ A )
=> ! [A2: A,B2: A,C2: A] :
( ( plus_plus @ A @ ( plus_plus @ A @ A2 @ B2 ) @ C2 )
= ( plus_plus @ A @ A2 @ ( plus_plus @ A @ B2 @ C2 ) ) ) ) ).
% add.right_assoc
thf(fact_5782_add_Oright__commute,axiom,
! [A: $tType] :
( ( ab_semigroup_add @ A )
=> ! [A2: A,B2: A,C2: A] :
( ( plus_plus @ A @ ( plus_plus @ A @ A2 @ B2 ) @ C2 )
= ( plus_plus @ A @ ( plus_plus @ A @ A2 @ C2 ) @ B2 ) ) ) ).
% add.right_commute
thf(fact_5783_mult_Oright__commute,axiom,
! [A: $tType] :
( ( ab_semigroup_mult @ A )
=> ! [A2: A,B2: A,C2: A] :
( ( times_times @ A @ ( times_times @ A @ A2 @ B2 ) @ C2 )
= ( times_times @ A @ ( times_times @ A @ A2 @ C2 ) @ B2 ) ) ) ).
% mult.right_commute
thf(fact_5784_mult_Oright__assoc,axiom,
! [A: $tType] :
( ( ab_semigroup_mult @ A )
=> ! [A2: A,B2: A,C2: A] :
( ( times_times @ A @ ( times_times @ A @ A2 @ B2 ) @ C2 )
= ( times_times @ A @ A2 @ ( times_times @ A @ B2 @ C2 ) ) ) ) ).
% mult.right_assoc
thf(fact_5785_ucast__slice,axiom,
! [B: $tType,A: $tType] :
( ( ( type_len @ A )
& ( type_len @ B ) )
=> ( ( semiring_1_unsigned @ B @ ( word @ A ) )
= ( slice2 @ B @ A @ ( zero_zero @ nat ) ) ) ) ).
% ucast_slice
thf(fact_5786_slice__cat2,axiom,
! [B: $tType,A: $tType] :
( ( ( type_len @ A )
& ( type_len @ B ) )
=> ! [A2: word @ B,T: word @ A] :
( ( slice2 @ A @ A @ ( zero_zero @ nat ) @ ( word_cat @ B @ A @ A @ A2 @ T ) )
= T ) ) ).
% slice_cat2
thf(fact_5787_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_5788_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_5789_sum__diff1_H__aux,axiom,
! [B: $tType,A: $tType] :
( ( ab_group_add @ B )
=> ! [F3: set @ A,I6: set @ A,F2: A > B,I: A] :
( ( finite_finite2 @ A @ F3 )
=> ( ( ord_less_eq @ ( set @ A )
@ ( collect @ A
@ ^ [I2: A] :
( ( member @ A @ I2 @ I6 )
& ( ( F2 @ I2 )
!= ( zero_zero @ B ) ) ) )
@ F3 )
=> ( ( ( member @ A @ I @ I6 )
=> ( ( groups1027152243600224163dd_sum @ A @ B @ F2 @ ( minus_minus @ ( set @ A ) @ I6 @ ( insert @ A @ I @ ( bot_bot @ ( set @ A ) ) ) ) )
= ( minus_minus @ B @ ( groups1027152243600224163dd_sum @ A @ B @ F2 @ I6 ) @ ( F2 @ I ) ) ) )
& ( ~ ( member @ A @ I @ I6 )
=> ( ( groups1027152243600224163dd_sum @ A @ B @ F2 @ ( minus_minus @ ( set @ A ) @ I6 @ ( insert @ A @ I @ ( bot_bot @ ( set @ A ) ) ) ) )
= ( groups1027152243600224163dd_sum @ A @ B @ F2 @ I6 ) ) ) ) ) ) ) ).
% sum_diff1'_aux
thf(fact_5790_bit__horner__sum__bit__iff,axiom,
! [A: $tType] :
( ( bit_un5681908812861735899ations @ A )
=> ! [Bs: list @ $o,N: nat] :
( ( bit_se5641148757651400278ts_bit @ A @ ( groups4207007520872428315er_sum @ $o @ A @ ( zero_neq_one_of_bool @ A ) @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ Bs ) @ N )
= ( ( ord_less @ nat @ N @ ( size_size @ ( list @ $o ) @ Bs ) )
& ( nth @ $o @ Bs @ N ) ) ) ) ).
% bit_horner_sum_bit_iff
thf(fact_5791_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_5792_sum_Oempty_H,axiom,
! [B: $tType,A: $tType] :
( ( comm_monoid_add @ A )
=> ! [P4: B > A] :
( ( groups1027152243600224163dd_sum @ B @ A @ P4 @ ( bot_bot @ ( set @ B ) ) )
= ( zero_zero @ A ) ) ) ).
% sum.empty'
thf(fact_5793_sum_Oinsert_H,axiom,
! [A: $tType,B: $tType] :
( ( comm_monoid_add @ A )
=> ! [I6: set @ B,P4: B > A,I: B] :
( ( finite_finite2 @ B
@ ( collect @ B
@ ^ [X2: B] :
( ( member @ B @ X2 @ I6 )
& ( ( P4 @ X2 )
!= ( zero_zero @ A ) ) ) ) )
=> ( ( ( member @ B @ I @ I6 )
=> ( ( groups1027152243600224163dd_sum @ B @ A @ P4 @ ( insert @ B @ I @ I6 ) )
= ( groups1027152243600224163dd_sum @ B @ A @ P4 @ I6 ) ) )
& ( ~ ( member @ B @ I @ I6 )
=> ( ( groups1027152243600224163dd_sum @ B @ A @ P4 @ ( insert @ B @ I @ I6 ) )
= ( plus_plus @ A @ ( P4 @ I ) @ ( groups1027152243600224163dd_sum @ B @ A @ P4 @ I6 ) ) ) ) ) ) ) ).
% sum.insert'
thf(fact_5794_sum_Onon__neutral_H,axiom,
! [A: $tType,B: $tType] :
( ( comm_monoid_add @ A )
=> ! [G: B > A,I6: set @ B] :
( ( groups1027152243600224163dd_sum @ B @ A @ G
@ ( collect @ B
@ ^ [X2: B] :
( ( member @ B @ X2 @ I6 )
& ( ( G @ X2 )
!= ( zero_zero @ A ) ) ) ) )
= ( groups1027152243600224163dd_sum @ B @ A @ G @ I6 ) ) ) ).
% sum.non_neutral'
thf(fact_5795_sum_Odistrib__triv_H,axiom,
! [A: $tType,B: $tType] :
( ( comm_monoid_add @ A )
=> ! [I6: set @ B,G: B > A,H2: B > A] :
( ( finite_finite2 @ B @ I6 )
=> ( ( groups1027152243600224163dd_sum @ B @ A
@ ^ [I2: B] : ( plus_plus @ A @ ( G @ I2 ) @ ( H2 @ I2 ) )
@ I6 )
= ( plus_plus @ A @ ( groups1027152243600224163dd_sum @ B @ A @ G @ I6 ) @ ( groups1027152243600224163dd_sum @ B @ A @ H2 @ I6 ) ) ) ) ) ).
% sum.distrib_triv'
thf(fact_5796_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_5797_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_5798_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 )
=> ( ! [I3: B] :
( ( member @ B @ I3 @ ( minus_minus @ ( set @ B ) @ T5 @ S3 ) )
=> ( ( H2 @ I3 )
= ( 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_5799_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_5800_sum_Odistrib_H,axiom,
! [A: $tType,B: $tType] :
( ( comm_monoid_add @ A )
=> ! [I6: set @ B,G: B > A,H2: B > A] :
( ( finite_finite2 @ B
@ ( collect @ B
@ ^ [X2: B] :
( ( member @ B @ X2 @ I6 )
& ( ( G @ X2 )
!= ( zero_zero @ A ) ) ) ) )
=> ( ( finite_finite2 @ B
@ ( collect @ B
@ ^ [X2: B] :
( ( member @ B @ X2 @ I6 )
& ( ( H2 @ X2 )
!= ( zero_zero @ A ) ) ) ) )
=> ( ( groups1027152243600224163dd_sum @ B @ A
@ ^ [I2: B] : ( plus_plus @ A @ ( G @ I2 ) @ ( H2 @ I2 ) )
@ I6 )
= ( plus_plus @ A @ ( groups1027152243600224163dd_sum @ B @ A @ G @ I6 ) @ ( groups1027152243600224163dd_sum @ B @ A @ H2 @ I6 ) ) ) ) ) ) ).
% sum.distrib'
thf(fact_5801_sum_OG__def,axiom,
! [A: $tType,B: $tType] :
( ( comm_monoid_add @ A )
=> ( ( groups1027152243600224163dd_sum @ B @ A )
= ( ^ [P5: B > A,I9: set @ B] :
( if @ A
@ ( finite_finite2 @ B
@ ( collect @ B
@ ^ [X2: B] :
( ( member @ B @ X2 @ I9 )
& ( ( P5 @ X2 )
!= ( zero_zero @ A ) ) ) ) )
@ ( groups7311177749621191930dd_sum @ B @ A @ P5
@ ( collect @ B
@ ^ [X2: B] :
( ( member @ B @ X2 @ I9 )
& ( ( P5 @ X2 )
!= ( zero_zero @ A ) ) ) ) )
@ ( zero_zero @ A ) ) ) ) ) ).
% sum.G_def
thf(fact_5802_sum__diff1_H,axiom,
! [B: $tType,A: $tType] :
( ( ab_group_add @ B )
=> ! [I6: set @ A,F2: A > B,I: A] :
( ( finite_finite2 @ A
@ ( collect @ A
@ ^ [I2: A] :
( ( member @ A @ I2 @ I6 )
& ( ( F2 @ I2 )
!= ( zero_zero @ B ) ) ) ) )
=> ( ( ( member @ A @ I @ I6 )
=> ( ( groups1027152243600224163dd_sum @ A @ B @ F2 @ ( minus_minus @ ( set @ A ) @ I6 @ ( insert @ A @ I @ ( bot_bot @ ( set @ A ) ) ) ) )
= ( minus_minus @ B @ ( groups1027152243600224163dd_sum @ A @ B @ F2 @ I6 ) @ ( F2 @ I ) ) ) )
& ( ~ ( member @ A @ I @ I6 )
=> ( ( groups1027152243600224163dd_sum @ A @ B @ F2 @ ( minus_minus @ ( set @ A ) @ I6 @ ( insert @ A @ I @ ( bot_bot @ ( set @ A ) ) ) ) )
= ( groups1027152243600224163dd_sum @ A @ B @ F2 @ I6 ) ) ) ) ) ) ).
% sum_diff1'
thf(fact_5803_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_5804_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_5805_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_5806_horner__sum__eq__sum,axiom,
! [A: $tType,B: $tType] :
( ( comm_semiring_1 @ A )
=> ( ( groups4207007520872428315er_sum @ B @ A )
= ( ^ [F4: B > A,A4: A,Xs: list @ B] :
( groups7311177749621191930dd_sum @ nat @ A
@ ^ [N3: nat] : ( times_times @ A @ ( F4 @ ( nth @ B @ Xs @ N3 ) ) @ ( power_power @ A @ A4 @ N3 ) )
@ ( set_or7035219750837199246ssThan @ nat @ ( zero_zero @ nat ) @ ( size_size @ ( list @ B ) @ Xs ) ) ) ) ) ) ).
% horner_sum_eq_sum
thf(fact_5807_horner__sum__foldr,axiom,
! [A: $tType,B: $tType] :
( ( comm_semiring_0 @ A )
=> ( ( groups4207007520872428315er_sum @ B @ A )
= ( ^ [F4: B > A,A4: A,Xs: list @ B] :
( foldr @ B @ A
@ ^ [X2: B,B5: A] : ( plus_plus @ A @ ( F4 @ X2 ) @ ( times_times @ A @ A4 @ B5 ) )
@ Xs
@ ( zero_zero @ A ) ) ) ) ) ).
% horner_sum_foldr
thf(fact_5808_horner__sum__bit__eq__take__bit,axiom,
! [A: $tType] :
( ( bit_se359711467146920520ations @ A )
=> ! [A2: A,N: nat] :
( ( groups4207007520872428315er_sum @ $o @ A @ ( zero_neq_one_of_bool @ A ) @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ ( map @ nat @ $o @ ( bit_se5641148757651400278ts_bit @ A @ A2 ) @ ( upt @ ( zero_zero @ nat ) @ N ) ) )
= ( bit_se2584673776208193580ke_bit @ A @ N @ A2 ) ) ) ).
% horner_sum_bit_eq_take_bit
thf(fact_5809_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_5810_Cauchy__iff2,axiom,
( ( topolo3814608138187158403Cauchy @ real )
= ( ^ [X8: nat > real] :
! [J2: nat] :
? [M8: nat] :
! [M4: nat] :
( ( ord_less_eq @ nat @ M8 @ M4 )
=> ! [N3: nat] :
( ( ord_less_eq @ nat @ M8 @ N3 )
=> ( ord_less @ real @ ( abs_abs @ real @ ( minus_minus @ real @ ( X8 @ M4 ) @ ( X8 @ N3 ) ) ) @ ( inverse_inverse @ real @ ( semiring_1_of_nat @ real @ ( suc @ J2 ) ) ) ) ) ) ) ) ).
% Cauchy_iff2
thf(fact_5811_int__sdiv__simps_I2_J,axiom,
! [A2: int] :
( ( signed7115095781618012415divide @ int @ A2 @ ( zero_zero @ int ) )
= ( zero_zero @ int ) ) ).
% int_sdiv_simps(2)
thf(fact_5812_sdiv__int__0__div,axiom,
! [X: int] :
( ( signed7115095781618012415divide @ int @ ( zero_zero @ int ) @ X )
= ( zero_zero @ int ) ) ).
% sdiv_int_0_div
thf(fact_5813_sdiv__int__div__0,axiom,
! [X: int] :
( ( signed7115095781618012415divide @ int @ X @ ( zero_zero @ int ) )
= ( zero_zero @ int ) ) ).
% sdiv_int_div_0
thf(fact_5814_int__sdiv__simps_I1_J,axiom,
! [A2: int] :
( ( signed7115095781618012415divide @ int @ A2 @ ( one_one @ int ) )
= A2 ) ).
% int_sdiv_simps(1)
thf(fact_5815_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_5816_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_5817_sdiv__int__numeral__numeral,axiom,
! [M: num,N: num] :
( ( signed7115095781618012415divide @ int @ ( numeral_numeral @ int @ M ) @ ( numeral_numeral @ int @ N ) )
= ( divide_divide @ int @ ( numeral_numeral @ int @ M ) @ ( numeral_numeral @ int @ N ) ) ) ).
% sdiv_int_numeral_numeral
thf(fact_5818_length__upt,axiom,
! [I: nat,J: nat] :
( ( size_size @ ( list @ nat ) @ ( upt @ I @ J ) )
= ( minus_minus @ nat @ J @ I ) ) ).
% length_upt
thf(fact_5819_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_5820_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_5821_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_5822_upt__conv__Cons__Cons,axiom,
! [M: nat,N: nat,Ns: list @ nat,Q3: nat] :
( ( ( cons @ nat @ M @ ( cons @ nat @ N @ Ns ) )
= ( upt @ M @ Q3 ) )
= ( ( cons @ nat @ N @ Ns )
= ( upt @ ( suc @ M ) @ Q3 ) ) ) ).
% upt_conv_Cons_Cons
thf(fact_5823_map__Suc__upt,axiom,
! [M: nat,N: nat] :
( ( map @ nat @ nat @ suc @ ( upt @ M @ N ) )
= ( upt @ ( suc @ M ) @ ( suc @ N ) ) ) ).
% map_Suc_upt
thf(fact_5824_map__add__upt_H,axiom,
! [Ofs: nat,A2: nat,B2: nat] :
( ( map @ nat @ nat
@ ^ [I2: nat] : ( plus_plus @ nat @ I2 @ Ofs )
@ ( upt @ A2 @ B2 ) )
= ( upt @ ( plus_plus @ nat @ A2 @ Ofs ) @ ( plus_plus @ nat @ B2 @ Ofs ) ) ) ).
% map_add_upt'
thf(fact_5825_map__add__upt,axiom,
! [N: nat,M: nat] :
( ( map @ nat @ nat
@ ^ [I2: nat] : ( plus_plus @ nat @ I2 @ N )
@ ( upt @ ( zero_zero @ nat ) @ M ) )
= ( upt @ N @ ( plus_plus @ nat @ M @ N ) ) ) ).
% map_add_upt
thf(fact_5826_atLeastLessThan__upt,axiom,
( ( set_or7035219750837199246ssThan @ nat )
= ( ^ [I2: nat,J2: nat] : ( set2 @ nat @ ( upt @ I2 @ J2 ) ) ) ) ).
% atLeastLessThan_upt
thf(fact_5827_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_5828_map__decr__upt,axiom,
! [M: nat,N: nat] :
( ( map @ nat @ nat
@ ^ [N3: nat] : ( minus_minus @ nat @ N3 @ ( suc @ ( zero_zero @ nat ) ) )
@ ( upt @ ( suc @ M ) @ ( suc @ N ) ) )
= ( upt @ M @ N ) ) ).
% map_decr_upt
thf(fact_5829_atLeastAtMost__upt,axiom,
( ( set_or1337092689740270186AtMost @ nat )
= ( ^ [N3: nat,M4: nat] : ( set2 @ nat @ ( upt @ N3 @ ( suc @ M4 ) ) ) ) ) ).
% atLeastAtMost_upt
thf(fact_5830_atLeast__upt,axiom,
( ( set_ord_lessThan @ nat )
= ( ^ [N3: nat] : ( set2 @ nat @ ( upt @ ( zero_zero @ nat ) @ N3 ) ) ) ) ).
% atLeast_upt
thf(fact_5831_map__replicate__trivial,axiom,
! [A: $tType,X: A,I: nat] :
( ( map @ nat @ A
@ ^ [I2: nat] : X
@ ( upt @ ( zero_zero @ nat ) @ I ) )
= ( replicate @ A @ I @ X ) ) ).
% map_replicate_trivial
thf(fact_5832_sgn__sdiv__eq__sgn__mult,axiom,
! [A2: int,B2: int] :
( ( ( signed7115095781618012415divide @ int @ A2 @ B2 )
!= ( zero_zero @ int ) )
=> ( ( sgn_sgn @ int @ ( signed7115095781618012415divide @ int @ A2 @ B2 ) )
= ( sgn_sgn @ int @ ( times_times @ int @ A2 @ B2 ) ) ) ) ).
% sgn_sdiv_eq_sgn_mult
thf(fact_5833_atMost__upto,axiom,
( ( set_ord_atMost @ nat )
= ( ^ [N3: nat] : ( set2 @ nat @ ( upt @ ( zero_zero @ nat ) @ ( suc @ N3 ) ) ) ) ) ).
% atMost_upto
thf(fact_5834_map__upt__Suc,axiom,
! [A: $tType,F2: nat > A,N: nat] :
( ( map @ nat @ A @ F2 @ ( upt @ ( zero_zero @ nat ) @ ( suc @ N ) ) )
= ( cons @ A @ ( F2 @ ( zero_zero @ nat ) )
@ ( map @ nat @ A
@ ^ [I2: nat] : ( F2 @ ( suc @ I2 ) )
@ ( upt @ ( zero_zero @ nat ) @ N ) ) ) ) ).
% map_upt_Suc
thf(fact_5835_map__bit__range__eq__if__take__bit__eq,axiom,
! [N: nat,K: int,L: int] :
( ( ( bit_se2584673776208193580ke_bit @ int @ N @ K )
= ( bit_se2584673776208193580ke_bit @ int @ N @ L ) )
=> ( ( map @ nat @ $o @ ( bit_se5641148757651400278ts_bit @ int @ K ) @ ( upt @ ( zero_zero @ nat ) @ N ) )
= ( map @ nat @ $o @ ( bit_se5641148757651400278ts_bit @ int @ L ) @ ( upt @ ( zero_zero @ nat ) @ N ) ) ) ) ).
% map_bit_range_eq_if_take_bit_eq
thf(fact_5836_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_5837_nth__map__upt,axiom,
! [A: $tType,I: nat,N: nat,M: nat,F2: nat > A] :
( ( ord_less @ nat @ I @ ( minus_minus @ nat @ N @ M ) )
=> ( ( nth @ A @ ( map @ nat @ A @ F2 @ ( upt @ M @ N ) ) @ I )
= ( F2 @ ( plus_plus @ nat @ M @ I ) ) ) ) ).
% nth_map_upt
thf(fact_5838_signed__divide__int__def,axiom,
( ( signed7115095781618012415divide @ int )
= ( ^ [K3: int,L2: int] : ( times_times @ int @ ( times_times @ int @ ( sgn_sgn @ int @ K3 ) @ ( sgn_sgn @ int @ L2 ) ) @ ( divide_divide @ int @ ( abs_abs @ int @ K3 ) @ ( abs_abs @ int @ L2 ) ) ) ) ) ).
% signed_divide_int_def
thf(fact_5839_map__upt__eqI,axiom,
! [A: $tType,Xs2: list @ A,N: nat,M: nat,F2: nat > A] :
( ( ( size_size @ ( list @ A ) @ Xs2 )
= ( minus_minus @ nat @ N @ M ) )
=> ( ! [I3: nat] :
( ( ord_less @ nat @ I3 @ ( size_size @ ( list @ A ) @ Xs2 ) )
=> ( ( nth @ A @ Xs2 @ I3 )
= ( F2 @ ( plus_plus @ nat @ M @ I3 ) ) ) )
=> ( ( map @ nat @ A @ F2 @ ( upt @ M @ N ) )
= Xs2 ) ) ) ).
% map_upt_eqI
thf(fact_5840_CauchyD,axiom,
! [A: $tType] :
( ( real_V822414075346904944vector @ A )
=> ! [X7: nat > A,E: real] :
( ( topolo3814608138187158403Cauchy @ A @ X7 )
=> ( ( ord_less @ real @ ( zero_zero @ real ) @ E )
=> ? [M9: nat] :
! [M3: nat] :
( ( ord_less_eq @ nat @ M9 @ M3 )
=> ! [N10: nat] :
( ( ord_less_eq @ nat @ M9 @ N10 )
=> ( ord_less @ real @ ( real_V7770717601297561774m_norm @ A @ ( minus_minus @ A @ ( X7 @ M3 ) @ ( X7 @ N10 ) ) ) @ E ) ) ) ) ) ) ).
% CauchyD
thf(fact_5841_CauchyI,axiom,
! [A: $tType] :
( ( real_V822414075346904944vector @ A )
=> ! [X7: nat > A] :
( ! [E2: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ E2 )
=> ? [M11: nat] :
! [M2: nat] :
( ( ord_less_eq @ nat @ M11 @ M2 )
=> ! [N2: nat] :
( ( ord_less_eq @ nat @ M11 @ N2 )
=> ( ord_less @ real @ ( real_V7770717601297561774m_norm @ A @ ( minus_minus @ A @ ( X7 @ M2 ) @ ( X7 @ N2 ) ) ) @ E2 ) ) ) )
=> ( topolo3814608138187158403Cauchy @ A @ X7 ) ) ) ).
% CauchyI
thf(fact_5842_Cauchy__iff,axiom,
! [A: $tType] :
( ( real_V822414075346904944vector @ A )
=> ( ( topolo3814608138187158403Cauchy @ A )
= ( ^ [X8: nat > A] :
! [E4: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ E4 )
=> ? [M8: nat] :
! [M4: nat] :
( ( ord_less_eq @ nat @ M8 @ M4 )
=> ! [N3: nat] :
( ( ord_less_eq @ nat @ M8 @ N3 )
=> ( ord_less @ real @ ( real_V7770717601297561774m_norm @ A @ ( minus_minus @ A @ ( X8 @ M4 ) @ ( X8 @ N3 ) ) ) @ E4 ) ) ) ) ) ) ) ).
% Cauchy_iff
thf(fact_5843_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_5844_make__rule,axiom,
! [A: $tType] :
( ( heap @ A )
=> ! [N: nat,F2: nat > A] :
( hoare_hoare_triple @ ( array @ A ) @ ( one_one @ assn ) @ ( array_make @ A @ N @ F2 )
@ ^ [R5: array @ A] : ( snga_assn @ A @ R5 @ ( map @ nat @ A @ F2 @ ( upt @ ( zero_zero @ nat ) @ N ) ) ) ) ) ).
% make_rule
thf(fact_5845_entails__solve__init_I1_J,axiom,
! [P: assn,Q: assn] :
( ( fI_QUERY @ P @ Q @ ( top_top @ assn ) )
=> ( entails @ P @ ( times_times @ assn @ Q @ ( top_top @ assn ) ) ) ) ).
% entails_solve_init(1)
thf(fact_5846_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_5847_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_5848_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_5849_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_5850_frame__inference__init,axiom,
! [P: assn,Q: assn,F3: assn] :
( ( fI_QUERY @ P @ Q @ F3 )
=> ( entails @ P @ ( times_times @ assn @ Q @ F3 ) ) ) ).
% frame_inference_init
thf(fact_5851_FI__QUERY__def,axiom,
( fI_QUERY
= ( ^ [P2: assn,Q2: assn,F7: assn] : ( entails @ P2 @ ( times_times @ assn @ Q2 @ F7 ) ) ) ) ).
% FI_QUERY_def
thf(fact_5852_entails__solve__init_I2_J,axiom,
! [P: assn,Q: assn] :
( ( fI_QUERY @ P @ Q @ ( one_one @ assn ) )
=> ( entails @ P @ Q ) ) ).
% entails_solve_init(2)
thf(fact_5853_time__array__make,axiom,
! [A: $tType] :
( ( heap @ A )
=> ! [N: nat,F2: nat > A,H2: heap_ext @ product_unit] :
( ( time_time @ ( array @ A ) @ ( array_make @ A @ N @ F2 ) @ H2 )
= ( plus_plus @ nat @ N @ ( one_one @ nat ) ) ) ) ).
% time_array_make
thf(fact_5854_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_5855_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_5856_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_5857_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_5858_smod__int__0__mod,axiom,
! [X: int] :
( ( signed6721504322012087516modulo @ int @ ( zero_zero @ int ) @ X )
= ( zero_zero @ int ) ) ).
% smod_int_0_mod
thf(fact_5859_smod__int__mod__0,axiom,
! [X: int] :
( ( signed6721504322012087516modulo @ int @ X @ ( zero_zero @ int ) )
= X ) ).
% smod_int_mod_0
thf(fact_5860_smod__int__numeral__numeral,axiom,
! [M: num,N: num] :
( ( signed6721504322012087516modulo @ int @ ( numeral_numeral @ int @ M ) @ ( numeral_numeral @ int @ N ) )
= ( modulo_modulo @ int @ ( numeral_numeral @ int @ M ) @ ( numeral_numeral @ int @ N ) ) ) ).
% smod_int_numeral_numeral
thf(fact_5861_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_5862_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_5863_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_5864_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_5865_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_5866_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_5867_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_5868_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_5869_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_5870_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_5871_smod__int__alt__def,axiom,
( ( signed6721504322012087516modulo @ int )
= ( ^ [A4: int,B5: int] : ( times_times @ int @ ( sgn_sgn @ int @ A4 ) @ ( modulo_modulo @ int @ ( abs_abs @ int @ A4 ) @ ( abs_abs @ int @ B5 ) ) ) ) ) ).
% smod_int_alt_def
thf(fact_5872_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_5873_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_5874_length__mul__elem,axiom,
! [A: $tType,Xs2: list @ ( list @ A ),N: nat] :
( ! [X3: list @ A] :
( ( member @ ( list @ A ) @ X3 @ ( set2 @ ( list @ A ) @ Xs2 ) )
=> ( ( size_size @ ( list @ A ) @ X3 )
= N ) )
=> ( ( size_size @ ( list @ A ) @ ( concat @ A @ Xs2 ) )
= ( times_times @ nat @ ( size_size @ ( list @ ( list @ A ) ) @ Xs2 ) @ N ) ) ) ).
% length_mul_elem
thf(fact_5875_valid__eq2,axiom,
! [T: vEBT_VEBT,D2: nat] :
( ( vEBT_VEBT_valid @ T @ D2 )
=> ( vEBT_invar_vebt @ T @ D2 ) ) ).
% valid_eq2
thf(fact_5876_valid__eq,axiom,
vEBT_VEBT_valid = vEBT_invar_vebt ).
% valid_eq
thf(fact_5877_valid__eq1,axiom,
! [T: vEBT_VEBT,D2: nat] :
( ( vEBT_invar_vebt @ T @ D2 )
=> ( vEBT_VEBT_valid @ T @ D2 ) ) ).
% valid_eq1
thf(fact_5878_map__concat,axiom,
! [A: $tType,B: $tType,F2: B > A,Xs2: list @ ( list @ B )] :
( ( map @ B @ A @ F2 @ ( concat @ B @ Xs2 ) )
= ( concat @ A @ ( map @ ( list @ B ) @ ( list @ A ) @ ( map @ B @ A @ F2 ) @ Xs2 ) ) ) ).
% map_concat
thf(fact_5879_subseqs__refl,axiom,
! [A: $tType,Xs2: list @ A] : ( member @ ( list @ A ) @ Xs2 @ ( set2 @ ( list @ A ) @ ( subseqs @ A @ Xs2 ) ) ) ).
% subseqs_refl
thf(fact_5880_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_5881_VEBT__internal_Ovalid_H_Osimps_I1_J,axiom,
! [Uu: $o,Uv: $o,D2: nat] :
( ( vEBT_VEBT_valid @ ( vEBT_Leaf @ Uu @ Uv ) @ D2 )
= ( D2
= ( one_one @ nat ) ) ) ).
% VEBT_internal.valid'.simps(1)
thf(fact_5882_product__lists_Osimps_I2_J,axiom,
! [A: $tType,Xs2: list @ A,Xss: list @ ( list @ A )] :
( ( product_lists @ A @ ( cons @ ( list @ A ) @ Xs2 @ Xss ) )
= ( concat @ ( list @ A )
@ ( map @ A @ ( list @ ( list @ A ) )
@ ^ [X2: A] : ( map @ ( list @ A ) @ ( list @ A ) @ ( cons @ A @ X2 ) @ ( product_lists @ A @ Xss ) )
@ Xs2 ) ) ) ).
% product_lists.simps(2)
thf(fact_5883_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_5884_set__n__lists,axiom,
! [A: $tType,N: nat,Xs2: list @ A] :
( ( set2 @ ( list @ A ) @ ( n_lists @ A @ N @ Xs2 ) )
= ( collect @ ( list @ A )
@ ^ [Ys3: list @ A] :
( ( ( size_size @ ( list @ A ) @ Ys3 )
= N )
& ( ord_less_eq @ ( set @ A ) @ ( set2 @ A @ Ys3 ) @ ( set2 @ A @ Xs2 ) ) ) ) ) ).
% set_n_lists
thf(fact_5885_Multiset_Ois__empty__def,axiom,
! [A: $tType] :
( ( is_empty @ A )
= ( ^ [A7: multiset @ A] :
( A7
= ( zero_zero @ ( multiset @ A ) ) ) ) ) ).
% Multiset.is_empty_def
thf(fact_5886_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_5887_drop__bit__word__beyond,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [N: nat,W: word @ A] :
( ( ord_less_eq @ nat @ ( type_len0_len_of @ A @ ( type2 @ A ) ) @ N )
=> ( ( bit_se4197421643247451524op_bit @ ( word @ A ) @ N @ W )
= ( zero_zero @ ( word @ A ) ) ) ) ) ).
% drop_bit_word_beyond
thf(fact_5888_push__bit__word__beyond,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [N: nat,W: word @ A] :
( ( ord_less_eq @ nat @ ( type_len0_len_of @ A @ ( type2 @ A ) ) @ N )
=> ( ( bit_se4730199178511100633sh_bit @ ( word @ A ) @ N @ W )
= ( zero_zero @ ( word @ A ) ) ) ) ) ).
% push_bit_word_beyond
thf(fact_5889_uint__bintrunc,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [Bin: num] :
( ( semiring_1_unsigned @ A @ int @ ( numeral_numeral @ ( word @ A ) @ Bin ) )
= ( bit_se2584673776208193580ke_bit @ int @ ( type_len0_len_of @ A @ ( type2 @ A ) ) @ ( numeral_numeral @ int @ Bin ) ) ) ) ).
% uint_bintrunc
thf(fact_5890_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_5891_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_5892_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_5893_less__word__numeral__numeral,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [A2: num,B2: num] :
( ( ord_less @ ( word @ A ) @ ( numeral_numeral @ ( word @ A ) @ A2 ) @ ( numeral_numeral @ ( word @ A ) @ B2 ) )
= ( ord_less @ int @ ( bit_se2584673776208193580ke_bit @ int @ ( type_len0_len_of @ A @ ( type2 @ A ) ) @ ( numeral_numeral @ int @ A2 ) ) @ ( bit_se2584673776208193580ke_bit @ int @ ( type_len0_len_of @ A @ ( type2 @ A ) ) @ ( numeral_numeral @ int @ B2 ) ) ) ) ) ).
% less_word_numeral_numeral
thf(fact_5894_unat__bintrunc,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [Bin: num] :
( ( semiring_1_unsigned @ A @ nat @ ( numeral_numeral @ ( word @ A ) @ Bin ) )
= ( nat2 @ ( bit_se2584673776208193580ke_bit @ int @ ( type_len0_len_of @ A @ ( type2 @ A ) ) @ ( numeral_numeral @ int @ Bin ) ) ) ) ) ).
% unat_bintrunc
thf(fact_5895_bit__numeral__word__iff,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [W: num,N: nat] :
( ( bit_se5641148757651400278ts_bit @ ( word @ A ) @ ( numeral_numeral @ ( word @ A ) @ W ) @ N )
= ( ( ord_less @ nat @ N @ ( type_len0_len_of @ A @ ( type2 @ A ) ) )
& ( bit_se5641148757651400278ts_bit @ int @ ( numeral_numeral @ int @ W ) @ N ) ) ) ) ).
% bit_numeral_word_iff
thf(fact_5896_ucast__bintr,axiom,
! [B: $tType,A: $tType] :
( ( ( type_len @ A )
& ( type_len @ B ) )
=> ! [W: num] :
( ( semiring_1_unsigned @ A @ ( word @ B ) @ ( numeral_numeral @ ( word @ A ) @ W ) )
= ( ring_1_of_int @ ( word @ B ) @ ( bit_se2584673776208193580ke_bit @ int @ ( type_len0_len_of @ A @ ( type2 @ A ) ) @ ( numeral_numeral @ int @ W ) ) ) ) ) ).
% ucast_bintr
thf(fact_5897_unsigned__numeral,axiom,
! [A: $tType,B: $tType] :
( ( ( type_len @ B )
& ( semiring_1 @ A ) )
=> ! [N: num] :
( ( semiring_1_unsigned @ B @ A @ ( numeral_numeral @ ( word @ B ) @ N ) )
= ( semiring_1_of_nat @ A @ ( bit_se2584673776208193580ke_bit @ nat @ ( type_len0_len_of @ B @ ( type2 @ B ) ) @ ( numeral_numeral @ nat @ N ) ) ) ) ) ).
% unsigned_numeral
thf(fact_5898_signed__drop__bit__of__1,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [N: nat] :
( ( signed_drop_bit @ A @ N @ ( one_one @ ( word @ A ) ) )
= ( zero_neq_one_of_bool @ ( word @ A )
@ ( ( ( type_len0_len_of @ A @ ( type2 @ A ) )
= ( one_one @ nat ) )
| ( N
= ( zero_zero @ nat ) ) ) ) ) ) ).
% signed_drop_bit_of_1
thf(fact_5899_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_5900_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_5901_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_5902_exp__eq__zero__iff,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [N: nat] :
( ( ( power_power @ ( word @ A ) @ ( numeral_numeral @ ( word @ A ) @ ( bit0 @ one2 ) ) @ N )
= ( zero_zero @ ( word @ A ) ) )
= ( ord_less_eq @ nat @ ( type_len0_len_of @ A @ ( type2 @ A ) ) @ N ) ) ) ).
% exp_eq_zero_iff
thf(fact_5903_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_5904_signed__take__bit__word__Suc__numeral,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [N: nat,K: num] :
( ( bit_ri4674362597316999326ke_bit @ ( word @ A ) @ ( suc @ N ) @ ( numeral_numeral @ ( word @ A ) @ K ) )
= ( ring_1_of_int @ ( word @ A ) @ ( bit_ri4674362597316999326ke_bit @ int @ ( suc @ N ) @ ( bit_se2584673776208193580ke_bit @ int @ ( type_len0_len_of @ A @ ( type2 @ A ) ) @ ( numeral_numeral @ int @ K ) ) ) ) ) ) ).
% signed_take_bit_word_Suc_numeral
thf(fact_5905_signed__take__bit__word__numeral,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [N: num,K: num] :
( ( bit_ri4674362597316999326ke_bit @ ( word @ A ) @ ( numeral_numeral @ nat @ N ) @ ( numeral_numeral @ ( word @ A ) @ K ) )
= ( ring_1_of_int @ ( word @ A ) @ ( bit_ri4674362597316999326ke_bit @ int @ ( numeral_numeral @ nat @ N ) @ ( bit_se2584673776208193580ke_bit @ int @ ( type_len0_len_of @ A @ ( type2 @ A ) ) @ ( numeral_numeral @ int @ K ) ) ) ) ) ) ).
% signed_take_bit_word_numeral
thf(fact_5906_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_5907_uint__bintrunc__neg,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [Bin: num] :
( ( semiring_1_unsigned @ A @ int @ ( uminus_uminus @ ( word @ A ) @ ( numeral_numeral @ ( word @ A ) @ Bin ) ) )
= ( bit_se2584673776208193580ke_bit @ int @ ( type_len0_len_of @ A @ ( type2 @ A ) ) @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ Bin ) ) ) ) ) ).
% uint_bintrunc_neg
thf(fact_5908_div__word__numeral__numeral,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [A2: num,B2: num] :
( ( divide_divide @ ( word @ A ) @ ( numeral_numeral @ ( word @ A ) @ A2 ) @ ( numeral_numeral @ ( word @ A ) @ B2 ) )
= ( ring_1_of_int @ ( word @ A ) @ ( divide_divide @ 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 ) ) ) ) ) ) ).
% div_word_numeral_numeral
thf(fact_5909_mod__word__numeral__numeral,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [A2: num,B2: num] :
( ( modulo_modulo @ ( word @ A ) @ ( numeral_numeral @ ( word @ A ) @ A2 ) @ ( numeral_numeral @ ( word @ A ) @ B2 ) )
= ( ring_1_of_int @ ( word @ A ) @ ( modulo_modulo @ 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 ) ) ) ) ) ) ).
% mod_word_numeral_numeral
thf(fact_5910_signed__numeral,axiom,
! [A: $tType,B: $tType] :
( ( ( type_len @ B )
& ( ring_1 @ A ) )
=> ! [N: num] :
( ( ring_1_signed @ B @ A @ ( numeral_numeral @ ( word @ B ) @ N ) )
= ( ring_1_of_int @ A @ ( bit_ri4674362597316999326ke_bit @ int @ ( minus_minus @ nat @ ( type_len0_len_of @ B @ ( type2 @ B ) ) @ ( one_one @ nat ) ) @ ( numeral_numeral @ int @ N ) ) ) ) ) ).
% signed_numeral
thf(fact_5911_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_5912_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_5913_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_5914_less__word__minus__numeral__minus__numeral,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [A2: num,B2: num] :
( ( ord_less @ ( word @ A ) @ ( uminus_uminus @ ( word @ A ) @ ( numeral_numeral @ ( word @ A ) @ A2 ) ) @ ( uminus_uminus @ ( word @ A ) @ ( numeral_numeral @ ( word @ A ) @ B2 ) ) )
= ( ord_less @ int @ ( bit_se2584673776208193580ke_bit @ int @ ( type_len0_len_of @ A @ ( type2 @ A ) ) @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ A2 ) ) ) @ ( bit_se2584673776208193580ke_bit @ int @ ( type_len0_len_of @ A @ ( type2 @ A ) ) @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ B2 ) ) ) ) ) ) ).
% less_word_minus_numeral_minus_numeral
thf(fact_5915_less__word__numeral__minus__numeral,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [A2: num,B2: num] :
( ( ord_less @ ( word @ A ) @ ( numeral_numeral @ ( word @ A ) @ A2 ) @ ( uminus_uminus @ ( word @ A ) @ ( numeral_numeral @ ( word @ A ) @ B2 ) ) )
= ( ord_less @ int @ ( bit_se2584673776208193580ke_bit @ int @ ( type_len0_len_of @ A @ ( type2 @ A ) ) @ ( numeral_numeral @ int @ A2 ) ) @ ( bit_se2584673776208193580ke_bit @ int @ ( type_len0_len_of @ A @ ( type2 @ A ) ) @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ B2 ) ) ) ) ) ) ).
% less_word_numeral_minus_numeral
thf(fact_5916_less__word__minus__numeral__numeral,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [A2: num,B2: num] :
( ( ord_less @ ( word @ A ) @ ( uminus_uminus @ ( word @ A ) @ ( numeral_numeral @ ( word @ A ) @ A2 ) ) @ ( numeral_numeral @ ( word @ A ) @ B2 ) )
= ( ord_less @ int @ ( bit_se2584673776208193580ke_bit @ int @ ( type_len0_len_of @ A @ ( type2 @ A ) ) @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ A2 ) ) ) @ ( bit_se2584673776208193580ke_bit @ int @ ( type_len0_len_of @ A @ ( type2 @ A ) ) @ ( numeral_numeral @ int @ B2 ) ) ) ) ) ).
% less_word_minus_numeral_numeral
thf(fact_5917_unat__bintrunc__neg,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [Bin: num] :
( ( semiring_1_unsigned @ A @ nat @ ( uminus_uminus @ ( word @ A ) @ ( numeral_numeral @ ( word @ A ) @ Bin ) ) )
= ( nat2 @ ( bit_se2584673776208193580ke_bit @ int @ ( type_len0_len_of @ A @ ( type2 @ A ) ) @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ Bin ) ) ) ) ) ) ).
% unat_bintrunc_neg
thf(fact_5918_bit__neg__numeral__word__iff,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [W: num,N: nat] :
( ( bit_se5641148757651400278ts_bit @ ( word @ A ) @ ( uminus_uminus @ ( word @ A ) @ ( numeral_numeral @ ( word @ A ) @ W ) ) @ N )
= ( ( ord_less @ nat @ N @ ( type_len0_len_of @ A @ ( type2 @ A ) ) )
& ( bit_se5641148757651400278ts_bit @ int @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ W ) ) @ N ) ) ) ) ).
% bit_neg_numeral_word_iff
thf(fact_5919_drop__bit__word__Suc__numeral,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [N: nat,K: num] :
( ( bit_se4197421643247451524op_bit @ ( word @ A ) @ ( suc @ N ) @ ( numeral_numeral @ ( word @ A ) @ K ) )
= ( ring_1_of_int @ ( word @ A ) @ ( bit_se4197421643247451524op_bit @ int @ ( suc @ N ) @ ( bit_se2584673776208193580ke_bit @ int @ ( type_len0_len_of @ A @ ( type2 @ A ) ) @ ( numeral_numeral @ int @ K ) ) ) ) ) ) ).
% drop_bit_word_Suc_numeral
thf(fact_5920_drop__bit__word__numeral,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [N: num,K: num] :
( ( bit_se4197421643247451524op_bit @ ( word @ A ) @ ( numeral_numeral @ nat @ N ) @ ( numeral_numeral @ ( word @ A ) @ K ) )
= ( ring_1_of_int @ ( word @ A ) @ ( bit_se4197421643247451524op_bit @ int @ ( numeral_numeral @ nat @ N ) @ ( bit_se2584673776208193580ke_bit @ int @ ( type_len0_len_of @ A @ ( type2 @ A ) ) @ ( numeral_numeral @ int @ K ) ) ) ) ) ) ).
% drop_bit_word_numeral
thf(fact_5921_unat__power__lower,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [N: nat] :
( ( ord_less @ nat @ N @ ( type_len0_len_of @ A @ ( type2 @ A ) ) )
=> ( ( semiring_1_unsigned @ A @ nat @ ( power_power @ ( word @ A ) @ ( numeral_numeral @ ( word @ A ) @ ( bit0 @ one2 ) ) @ N ) )
= ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N ) ) ) ) ).
% unat_power_lower
thf(fact_5922_unsigned__neg__numeral,axiom,
! [A: $tType,B: $tType] :
( ( ( type_len @ B )
& ( semiring_1 @ A ) )
=> ! [N: num] :
( ( semiring_1_unsigned @ B @ A @ ( uminus_uminus @ ( word @ B ) @ ( numeral_numeral @ ( word @ B ) @ N ) ) )
= ( semiring_1_of_nat @ A @ ( nat2 @ ( bit_se2584673776208193580ke_bit @ int @ ( type_len0_len_of @ B @ ( type2 @ B ) ) @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ N ) ) ) ) ) ) ) ).
% unsigned_neg_numeral
thf(fact_5923_signed__take__bit__word__Suc__minus__numeral,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [N: nat,K: num] :
( ( bit_ri4674362597316999326ke_bit @ ( word @ A ) @ ( suc @ N ) @ ( uminus_uminus @ ( word @ A ) @ ( numeral_numeral @ ( word @ A ) @ K ) ) )
= ( ring_1_of_int @ ( word @ A ) @ ( bit_ri4674362597316999326ke_bit @ int @ ( suc @ N ) @ ( bit_se2584673776208193580ke_bit @ int @ ( type_len0_len_of @ A @ ( type2 @ A ) ) @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ K ) ) ) ) ) ) ) ).
% signed_take_bit_word_Suc_minus_numeral
thf(fact_5924_signed__take__bit__word__minus__numeral,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [N: num,K: num] :
( ( bit_ri4674362597316999326ke_bit @ ( word @ A ) @ ( numeral_numeral @ nat @ N ) @ ( uminus_uminus @ ( word @ A ) @ ( numeral_numeral @ ( word @ A ) @ K ) ) )
= ( ring_1_of_int @ ( word @ A ) @ ( bit_ri4674362597316999326ke_bit @ int @ ( numeral_numeral @ nat @ N ) @ ( bit_se2584673776208193580ke_bit @ int @ ( type_len0_len_of @ A @ ( type2 @ A ) ) @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ K ) ) ) ) ) ) ) ).
% signed_take_bit_word_minus_numeral
thf(fact_5925_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_5926_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_5927_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_5928_div__word__minus__numeral__minus__numeral,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [A2: num,B2: num] :
( ( divide_divide @ ( 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 ) @ ( divide_divide @ 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 ) ) ) ) ) ) ) ).
% div_word_minus_numeral_minus_numeral
thf(fact_5929_div__word__numeral__minus__numeral,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [A2: num,B2: num] :
( ( divide_divide @ ( word @ A ) @ ( numeral_numeral @ ( word @ A ) @ A2 ) @ ( uminus_uminus @ ( word @ A ) @ ( numeral_numeral @ ( word @ A ) @ B2 ) ) )
= ( ring_1_of_int @ ( word @ A ) @ ( divide_divide @ 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 ) ) ) ) ) ) ) ).
% div_word_numeral_minus_numeral
thf(fact_5930_div__word__minus__numeral__numeral,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [A2: num,B2: num] :
( ( divide_divide @ ( word @ A ) @ ( uminus_uminus @ ( word @ A ) @ ( numeral_numeral @ ( word @ A ) @ A2 ) ) @ ( numeral_numeral @ ( word @ A ) @ B2 ) )
= ( ring_1_of_int @ ( word @ A ) @ ( divide_divide @ 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 ) ) ) ) ) ) ).
% div_word_minus_numeral_numeral
thf(fact_5931_mod__word__minus__numeral__minus__numeral,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [A2: num,B2: num] :
( ( modulo_modulo @ ( 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 ) @ ( modulo_modulo @ 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 ) ) ) ) ) ) ) ).
% mod_word_minus_numeral_minus_numeral
thf(fact_5932_mod__word__numeral__minus__numeral,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [A2: num,B2: num] :
( ( modulo_modulo @ ( word @ A ) @ ( numeral_numeral @ ( word @ A ) @ A2 ) @ ( uminus_uminus @ ( word @ A ) @ ( numeral_numeral @ ( word @ A ) @ B2 ) ) )
= ( ring_1_of_int @ ( word @ A ) @ ( modulo_modulo @ 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 ) ) ) ) ) ) ) ).
% mod_word_numeral_minus_numeral
thf(fact_5933_mod__word__minus__numeral__numeral,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [A2: num,B2: num] :
( ( modulo_modulo @ ( word @ A ) @ ( uminus_uminus @ ( word @ A ) @ ( numeral_numeral @ ( word @ A ) @ A2 ) ) @ ( numeral_numeral @ ( word @ A ) @ B2 ) )
= ( ring_1_of_int @ ( word @ A ) @ ( modulo_modulo @ 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 ) ) ) ) ) ) ).
% mod_word_minus_numeral_numeral
thf(fact_5934_word__less__sub__le,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [N: nat,X: word @ A] :
( ( ord_less @ nat @ N @ ( type_len0_len_of @ A @ ( type2 @ A ) ) )
=> ( ( ord_less_eq @ ( word @ A ) @ X @ ( minus_minus @ ( word @ A ) @ ( power_power @ ( word @ A ) @ ( numeral_numeral @ ( word @ A ) @ ( bit0 @ one2 ) ) @ N ) @ ( one_one @ ( word @ A ) ) ) )
= ( ord_less @ ( word @ A ) @ X @ ( power_power @ ( word @ A ) @ ( numeral_numeral @ ( word @ A ) @ ( bit0 @ one2 ) ) @ N ) ) ) ) ) ).
% word_less_sub_le
thf(fact_5935_signed__neg__numeral,axiom,
! [A: $tType,B: $tType] :
( ( ( type_len @ B )
& ( ring_1 @ A ) )
=> ! [N: num] :
( ( ring_1_signed @ B @ A @ ( uminus_uminus @ ( word @ B ) @ ( numeral_numeral @ ( word @ B ) @ N ) ) )
= ( ring_1_of_int @ A @ ( bit_ri4674362597316999326ke_bit @ int @ ( minus_minus @ nat @ ( type_len0_len_of @ B @ ( type2 @ B ) ) @ ( one_one @ nat ) ) @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ N ) ) ) ) ) ) ).
% signed_neg_numeral
thf(fact_5936_signed__drop__bit__word__Suc__numeral,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [N: nat,K: num] :
( ( signed_drop_bit @ A @ ( suc @ N ) @ ( numeral_numeral @ ( word @ A ) @ K ) )
= ( ring_1_of_int @ ( word @ A ) @ ( bit_se4197421643247451524op_bit @ int @ ( suc @ N ) @ ( bit_ri4674362597316999326ke_bit @ int @ ( minus_minus @ nat @ ( type_len0_len_of @ A @ ( type2 @ A ) ) @ ( one_one @ nat ) ) @ ( numeral_numeral @ int @ K ) ) ) ) ) ) ).
% signed_drop_bit_word_Suc_numeral
thf(fact_5937_signed__drop__bit__word__numeral,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [N: num,K: num] :
( ( signed_drop_bit @ A @ ( numeral_numeral @ nat @ N ) @ ( numeral_numeral @ ( word @ A ) @ K ) )
= ( ring_1_of_int @ ( word @ A ) @ ( bit_se4197421643247451524op_bit @ int @ ( numeral_numeral @ nat @ N ) @ ( bit_ri4674362597316999326ke_bit @ int @ ( minus_minus @ nat @ ( type_len0_len_of @ A @ ( type2 @ A ) ) @ ( one_one @ nat ) ) @ ( numeral_numeral @ int @ K ) ) ) ) ) ) ).
% signed_drop_bit_word_numeral
thf(fact_5938_drop__bit__word__Suc__minus__numeral,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [N: nat,K: num] :
( ( bit_se4197421643247451524op_bit @ ( word @ A ) @ ( suc @ N ) @ ( uminus_uminus @ ( word @ A ) @ ( numeral_numeral @ ( word @ A ) @ K ) ) )
= ( ring_1_of_int @ ( word @ A ) @ ( bit_se4197421643247451524op_bit @ int @ ( suc @ N ) @ ( bit_se2584673776208193580ke_bit @ int @ ( type_len0_len_of @ A @ ( type2 @ A ) ) @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ K ) ) ) ) ) ) ) ).
% drop_bit_word_Suc_minus_numeral
thf(fact_5939_drop__bit__word__minus__numeral,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [N: num,K: num] :
( ( bit_se4197421643247451524op_bit @ ( word @ A ) @ ( numeral_numeral @ nat @ N ) @ ( uminus_uminus @ ( word @ A ) @ ( numeral_numeral @ ( word @ A ) @ K ) ) )
= ( ring_1_of_int @ ( word @ A ) @ ( bit_se4197421643247451524op_bit @ int @ ( numeral_numeral @ nat @ N ) @ ( bit_se2584673776208193580ke_bit @ int @ ( type_len0_len_of @ A @ ( type2 @ A ) ) @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ K ) ) ) ) ) ) ) ).
% drop_bit_word_minus_numeral
thf(fact_5940_signed__drop__bit__word__Suc__minus__numeral,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [N: nat,K: num] :
( ( signed_drop_bit @ A @ ( suc @ N ) @ ( uminus_uminus @ ( word @ A ) @ ( numeral_numeral @ ( word @ A ) @ K ) ) )
= ( ring_1_of_int @ ( word @ A ) @ ( bit_se4197421643247451524op_bit @ int @ ( suc @ N ) @ ( bit_ri4674362597316999326ke_bit @ int @ ( minus_minus @ nat @ ( type_len0_len_of @ A @ ( type2 @ A ) ) @ ( one_one @ nat ) ) @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ K ) ) ) ) ) ) ) ).
% signed_drop_bit_word_Suc_minus_numeral
thf(fact_5941_signed__drop__bit__word__minus__numeral,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [N: num,K: num] :
( ( signed_drop_bit @ A @ ( numeral_numeral @ nat @ N ) @ ( uminus_uminus @ ( word @ A ) @ ( numeral_numeral @ ( word @ A ) @ K ) ) )
= ( ring_1_of_int @ ( word @ A ) @ ( bit_se4197421643247451524op_bit @ int @ ( numeral_numeral @ nat @ N ) @ ( bit_ri4674362597316999326ke_bit @ int @ ( minus_minus @ nat @ ( type_len0_len_of @ A @ ( type2 @ A ) ) @ ( one_one @ nat ) ) @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ K ) ) ) ) ) ) ) ).
% signed_drop_bit_word_minus_numeral
thf(fact_5942_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_5943_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_5944_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_5945_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_5946_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_5947_nth__slice,axiom,
! [B: $tType,A: $tType] :
( ( ( type_len @ A )
& ( type_len @ B ) )
=> ! [N: nat,W: word @ B,M: nat] :
( ( bit_se5641148757651400278ts_bit @ ( word @ A ) @ ( slice2 @ B @ A @ N @ W ) @ M )
= ( ( bit_se5641148757651400278ts_bit @ ( word @ B ) @ W @ ( plus_plus @ nat @ M @ N ) )
& ( ord_less @ nat @ M @ ( type_len0_len_of @ A @ ( type2 @ A ) ) ) ) ) ) ).
% nth_slice
thf(fact_5948_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_5949_test__bit__wi,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [X: int,N: nat] :
( ( bit_se5641148757651400278ts_bit @ ( word @ A ) @ ( ring_1_of_int @ ( word @ A ) @ X ) @ N )
= ( ( ord_less @ nat @ N @ ( type_len0_len_of @ A @ ( type2 @ A ) ) )
& ( bit_se5641148757651400278ts_bit @ int @ X @ N ) ) ) ) ).
% test_bit_wi
thf(fact_5950_bit__word__of__int__iff,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [K: int,N: nat] :
( ( bit_se5641148757651400278ts_bit @ ( word @ A ) @ ( ring_1_of_int @ ( word @ A ) @ K ) @ N )
= ( ( ord_less @ nat @ N @ ( type_len0_len_of @ A @ ( type2 @ A ) ) )
& ( bit_se5641148757651400278ts_bit @ int @ K @ N ) ) ) ) ).
% bit_word_of_int_iff
thf(fact_5951_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_5952_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_5953_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_5954_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_5955_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_5956_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_5957_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_5958_zero__word_Orsp,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ( ( bit_se2584673776208193580ke_bit @ int @ ( type_len0_len_of @ A @ ( type2 @ A ) ) @ ( zero_zero @ int ) )
= ( bit_se2584673776208193580ke_bit @ int @ ( type_len0_len_of @ A @ ( type2 @ A ) ) @ ( zero_zero @ int ) ) ) ) ).
% zero_word.rsp
thf(fact_5959_size__0__same,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [W: word @ A,V: word @ A] :
( ( ( type_len0_len_of @ A @ ( type2 @ A ) )
= ( zero_zero @ nat ) )
=> ( W = V ) ) ) ).
% size_0_same
thf(fact_5960_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_5961_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_5962_uint__word__arith__bintrs_I7_J,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ( ( semiring_1_unsigned @ A @ int @ ( zero_zero @ ( word @ A ) ) )
= ( bit_se2584673776208193580ke_bit @ int @ ( type_len0_len_of @ A @ ( type2 @ A ) ) @ ( zero_zero @ int ) ) ) ) ).
% uint_word_arith_bintrs(7)
thf(fact_5963_test__bit__1_H,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [N: nat] :
( ( bit_se5641148757651400278ts_bit @ ( word @ A ) @ ( one_one @ ( word @ A ) ) @ N )
= ( ( ord_less @ nat @ ( zero_zero @ nat ) @ ( type_len0_len_of @ A @ ( type2 @ A ) ) )
& ( N
= ( zero_zero @ nat ) ) ) ) ) ).
% test_bit_1'
thf(fact_5964_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_5965_num__of__bintr_H,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [A2: num,B2: num] :
( ( ( bit_se2584673776208193580ke_bit @ int @ ( type_len0_len_of @ A @ ( type2 @ A ) ) @ ( numeral_numeral @ int @ A2 ) )
= ( numeral_numeral @ int @ B2 ) )
=> ( ( numeral_numeral @ ( word @ A ) @ A2 )
= ( numeral_numeral @ ( word @ A ) @ B2 ) ) ) ) ).
% num_of_bintr'
thf(fact_5966_bit__imp__le__length,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [W: word @ A,N: nat] :
( ( bit_se5641148757651400278ts_bit @ ( word @ A ) @ W @ N )
=> ( ord_less @ nat @ N @ ( type_len0_len_of @ A @ ( type2 @ A ) ) ) ) ) ).
% bit_imp_le_length
thf(fact_5967_bit__word__eqI,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [A2: word @ A,B2: word @ A] :
( ! [N2: nat] :
( ( ord_less @ nat @ N2 @ ( type_len0_len_of @ A @ ( type2 @ A ) ) )
=> ( ( bit_se5641148757651400278ts_bit @ ( word @ A ) @ A2 @ N2 )
= ( bit_se5641148757651400278ts_bit @ ( word @ A ) @ B2 @ N2 ) ) )
=> ( A2 = B2 ) ) ) ).
% bit_word_eqI
thf(fact_5968_word__eq__iff,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ( ( ^ [Y2: word @ A,Z2: word @ A] : Y2 = Z2 )
= ( ^ [X2: word @ A,Y6: word @ A] :
! [N3: nat] :
( ( ord_less @ nat @ N3 @ ( type_len0_len_of @ A @ ( type2 @ A ) ) )
=> ( ( bit_se5641148757651400278ts_bit @ ( word @ A ) @ X2 @ N3 )
= ( bit_se5641148757651400278ts_bit @ ( word @ A ) @ Y6 @ N3 ) ) ) ) ) ) ).
% word_eq_iff
thf(fact_5969_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_5970_nth__ucast,axiom,
! [B: $tType,A: $tType] :
( ( ( type_len @ A )
& ( type_len @ B ) )
=> ! [W: word @ B,N: nat] :
( ( bit_se5641148757651400278ts_bit @ ( word @ A ) @ ( semiring_1_unsigned @ B @ ( word @ A ) @ W ) @ N )
= ( ( bit_se5641148757651400278ts_bit @ ( word @ B ) @ W @ N )
& ( ord_less @ nat @ N @ ( type_len0_len_of @ A @ ( type2 @ A ) ) ) ) ) ) ).
% nth_ucast
thf(fact_5971_bit__ucast__iff,axiom,
! [A: $tType,B: $tType] :
( ( ( type_len @ B )
& ( type_len @ A ) )
=> ! [A2: word @ B,N: nat] :
( ( bit_se5641148757651400278ts_bit @ ( word @ A ) @ ( semiring_1_unsigned @ B @ ( word @ A ) @ A2 ) @ N )
= ( ( ord_less @ nat @ N @ ( type_len0_len_of @ A @ ( type2 @ A ) ) )
& ( bit_se5641148757651400278ts_bit @ ( word @ B ) @ A2 @ N ) ) ) ) ).
% bit_ucast_iff
thf(fact_5972_bin__nth__uint__imp,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [W: word @ A,N: nat] :
( ( bit_se5641148757651400278ts_bit @ int @ ( semiring_1_unsigned @ A @ int @ W ) @ N )
=> ( ord_less @ nat @ N @ ( type_len0_len_of @ A @ ( type2 @ A ) ) ) ) ) ).
% bin_nth_uint_imp
thf(fact_5973_bit__word__ucast__iff,axiom,
! [B: $tType,A: $tType] :
( ( ( type_len @ A )
& ( type_len @ B ) )
=> ! [W: word @ A,N: nat] :
( ( bit_se5641148757651400278ts_bit @ ( word @ B ) @ ( semiring_1_unsigned @ A @ ( word @ B ) @ W ) @ N )
= ( ( ord_less @ nat @ N @ ( type_len0_len_of @ A @ ( type2 @ A ) ) )
& ( ord_less @ nat @ N @ ( type_len0_len_of @ B @ ( type2 @ B ) ) )
& ( bit_se5641148757651400278ts_bit @ ( word @ A ) @ W @ N ) ) ) ) ).
% bit_word_ucast_iff
thf(fact_5974_neg__test__bit,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [X: word @ A,N: nat] :
( ( bit_se5641148757651400278ts_bit @ ( word @ A ) @ ( bit_ri4277139882892585799ns_not @ ( word @ A ) @ X ) @ N )
= ( ~ ( bit_se5641148757651400278ts_bit @ ( word @ A ) @ X @ N )
& ( ord_less @ nat @ N @ ( type_len0_len_of @ A @ ( type2 @ A ) ) ) ) ) ) ).
% neg_test_bit
thf(fact_5975_num__abs__bintr,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ( ( numeral_numeral @ ( word @ A ) )
= ( ^ [X2: num] : ( ring_1_of_int @ ( word @ A ) @ ( bit_se2584673776208193580ke_bit @ int @ ( type_len0_len_of @ A @ ( type2 @ A ) ) @ ( numeral_numeral @ int @ X2 ) ) ) ) ) ) ).
% num_abs_bintr
thf(fact_5976_mask__over__length,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [N: nat] :
( ( ord_less_eq @ nat @ ( type_len0_len_of @ A @ ( type2 @ A ) ) @ N )
=> ( ( bit_se2239418461657761734s_mask @ ( word @ A ) @ N )
= ( uminus_uminus @ ( word @ A ) @ ( one_one @ ( word @ A ) ) ) ) ) ) ).
% mask_over_length
thf(fact_5977_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_5978_word__of__int__less__iff,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [K: int,L: int] :
( ( ord_less @ ( word @ A ) @ ( ring_1_of_int @ ( word @ A ) @ K ) @ ( ring_1_of_int @ ( word @ A ) @ L ) )
= ( ord_less @ int @ ( bit_se2584673776208193580ke_bit @ int @ ( type_len0_len_of @ A @ ( type2 @ A ) ) @ K ) @ ( bit_se2584673776208193580ke_bit @ int @ ( type_len0_len_of @ A @ ( type2 @ A ) ) @ L ) ) ) ) ).
% word_of_int_less_iff
thf(fact_5979_max__test__bit,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [N: nat] :
( ( bit_se5641148757651400278ts_bit @ ( word @ A ) @ ( uminus_uminus @ ( word @ A ) @ ( one_one @ ( word @ A ) ) ) @ N )
= ( ord_less @ nat @ N @ ( type_len0_len_of @ A @ ( type2 @ A ) ) ) ) ) ).
% max_test_bit
thf(fact_5980_bit__uint__iff,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [W: word @ A,N: nat] :
( ( bit_se5641148757651400278ts_bit @ int @ ( semiring_1_unsigned @ A @ int @ W ) @ N )
= ( ( ord_less @ nat @ N @ ( type_len0_len_of @ A @ ( type2 @ A ) ) )
& ( bit_se5641148757651400278ts_bit @ ( word @ A ) @ W @ N ) ) ) ) ).
% bit_uint_iff
thf(fact_5981_test__bit__bin,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ( ( bit_se5641148757651400278ts_bit @ ( word @ A ) )
= ( ^ [W2: word @ A,N3: nat] :
( ( ord_less @ nat @ N3 @ ( type_len0_len_of @ A @ ( type2 @ A ) ) )
& ( bit_se5641148757651400278ts_bit @ int @ ( semiring_1_unsigned @ A @ int @ W2 ) @ N3 ) ) ) ) ) ).
% test_bit_bin
thf(fact_5982_signed__take__bit__eq,axiom,
! [A: $tType,B: $tType] :
( ( ( type_len @ B )
& ( bit_ri3973907225187159222ations @ A ) )
=> ! [N: nat,W: word @ B] :
( ( ( ord_less @ nat @ N @ ( type_len0_len_of @ B @ ( type2 @ B ) ) )
=> ( ( ring_1_signed @ B @ A @ ( bit_se2584673776208193580ke_bit @ ( word @ B ) @ N @ W ) )
= ( bit_se2584673776208193580ke_bit @ A @ N @ ( ring_1_signed @ B @ A @ W ) ) ) )
& ( ~ ( ord_less @ nat @ N @ ( type_len0_len_of @ B @ ( type2 @ B ) ) )
=> ( ( ring_1_signed @ B @ A @ ( bit_se2584673776208193580ke_bit @ ( word @ B ) @ N @ W ) )
= ( ring_1_signed @ B @ A @ W ) ) ) ) ) ).
% signed_take_bit_eq
thf(fact_5983_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,D2: word @ B] :
( ( ord_less_eq @ nat @ ( plus_plus @ nat @ ( type_len0_len_of @ A @ ( type2 @ A ) ) @ ( type_len0_len_of @ B @ ( type2 @ B ) ) ) @ ( type_len0_len_of @ C @ ( type2 @ C ) ) )
=> ( ( ( word_cat @ A @ B @ C @ A2 @ B2 )
= ( word_cat @ A @ B @ C @ C2 @ D2 ) )
= ( ( A2 = C2 )
& ( B2 = D2 ) ) ) ) ) ).
% word_cat_inj
thf(fact_5984_bit__set__bit__word__iff,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [W: word @ A,M: nat,B2: $o,N: nat] :
( ( bit_se5641148757651400278ts_bit @ ( word @ A ) @ ( generi7602027413899671122et_bit @ ( word @ A ) @ W @ M @ B2 ) @ N )
= ( ( ( M = N )
=> ( ( ord_less @ nat @ N @ ( type_len0_len_of @ A @ ( type2 @ A ) ) )
& B2 ) )
& ( ( M != N )
=> ( bit_se5641148757651400278ts_bit @ ( word @ A ) @ W @ N ) ) ) ) ) ).
% bit_set_bit_word_iff
thf(fact_5985_word__of__nat__less__iff,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [M: nat,N: nat] :
( ( ord_less @ ( word @ A ) @ ( semiring_1_of_nat @ ( word @ A ) @ M ) @ ( semiring_1_of_nat @ ( word @ A ) @ N ) )
= ( ord_less @ nat @ ( bit_se2584673776208193580ke_bit @ nat @ ( type_len0_len_of @ A @ ( type2 @ A ) ) @ M ) @ ( bit_se2584673776208193580ke_bit @ nat @ ( type_len0_len_of @ A @ ( type2 @ A ) ) @ N ) ) ) ) ).
% word_of_nat_less_iff
thf(fact_5986_bit__set__bit__aux,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [F2: nat > $o,N: nat,W: word @ A,M: nat] :
( ( bit_se5641148757651400278ts_bit @ ( word @ A ) @ ( code_T2661198915054445665ts_aux @ A @ F2 @ N @ W ) @ M )
= ( ( ord_less @ nat @ M @ ( type_len0_len_of @ A @ ( type2 @ A ) ) )
& ( ( ord_less @ nat @ M @ N )
=> ( F2 @ M ) )
& ( ~ ( ord_less @ nat @ M @ N )
=> ( bit_se5641148757651400278ts_bit @ ( word @ A ) @ W @ ( minus_minus @ nat @ M @ N ) ) ) ) ) ) ).
% bit_set_bit_aux
thf(fact_5987_bit__word__cat__iff,axiom,
! [C: $tType,A: $tType,B: $tType] :
( ( ( type_len @ B )
& ( type_len @ A )
& ( type_len @ C ) )
=> ! [V: word @ A,W: word @ B,N: nat] :
( ( bit_se5641148757651400278ts_bit @ ( word @ C ) @ ( word_cat @ A @ B @ C @ V @ W ) @ N )
= ( ( ord_less @ nat @ N @ ( type_len0_len_of @ C @ ( type2 @ C ) ) )
& ( ( ord_less @ nat @ N @ ( type_len0_len_of @ B @ ( type2 @ B ) ) )
=> ( bit_se5641148757651400278ts_bit @ ( word @ B ) @ W @ N ) )
& ( ~ ( ord_less @ nat @ N @ ( type_len0_len_of @ B @ ( type2 @ B ) ) )
=> ( bit_se5641148757651400278ts_bit @ ( word @ A ) @ V @ ( minus_minus @ nat @ N @ ( type_len0_len_of @ B @ ( type2 @ B ) ) ) ) ) ) ) ) ).
% bit_word_cat_iff
thf(fact_5988_two__power__eq,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [N: nat,M: nat] :
( ( ord_less @ nat @ N @ ( type_len0_len_of @ A @ ( type2 @ A ) ) )
=> ( ( ord_less @ nat @ M @ ( type_len0_len_of @ A @ ( type2 @ A ) ) )
=> ( ( ( power_power @ ( word @ A ) @ ( numeral_numeral @ ( word @ A ) @ ( bit0 @ one2 ) ) @ N )
= ( power_power @ ( word @ A ) @ ( numeral_numeral @ ( word @ A ) @ ( bit0 @ one2 ) ) @ M ) )
= ( N = M ) ) ) ) ) ).
% two_power_eq
thf(fact_5989_signed__take__bit__decr__length__iff,axiom,
! [A: $tType,B: $tType] :
( ( ( bit_ri3973907225187159222ations @ B )
& ( type_len @ A ) )
=> ! [K: B,L: B] :
( ( ( bit_ri4674362597316999326ke_bit @ B @ ( minus_minus @ nat @ ( type_len0_len_of @ A @ ( type2 @ A ) ) @ ( suc @ ( zero_zero @ nat ) ) ) @ K )
= ( bit_ri4674362597316999326ke_bit @ B @ ( minus_minus @ nat @ ( type_len0_len_of @ A @ ( type2 @ A ) ) @ ( suc @ ( zero_zero @ nat ) ) ) @ L ) )
= ( ( bit_se2584673776208193580ke_bit @ B @ ( type_len0_len_of @ A @ ( type2 @ A ) ) @ K )
= ( bit_se2584673776208193580ke_bit @ B @ ( type_len0_len_of @ A @ ( type2 @ A ) ) @ L ) ) ) ) ).
% signed_take_bit_decr_length_iff
thf(fact_5990_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_5991_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_5992_bin__nth__sint,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [N: nat,W: word @ A] :
( ( ord_less_eq @ nat @ ( type_len0_len_of @ A @ ( type2 @ A ) ) @ N )
=> ( ( bit_se5641148757651400278ts_bit @ int @ ( ring_1_signed @ A @ int @ W ) @ N )
= ( bit_se5641148757651400278ts_bit @ int @ ( ring_1_signed @ A @ int @ W ) @ ( minus_minus @ nat @ ( type_len0_len_of @ A @ ( type2 @ A ) ) @ ( one_one @ nat ) ) ) ) ) ) ).
% bin_nth_sint
thf(fact_5993_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_5994_neg__mask__test__bit,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [N: nat,M: nat] :
( ( bit_se5641148757651400278ts_bit @ ( word @ A ) @ ( bit_ri4277139882892585799ns_not @ ( word @ A ) @ ( bit_se2239418461657761734s_mask @ ( word @ A ) @ N ) ) @ M )
= ( ( ord_less_eq @ nat @ N @ M )
& ( ord_less @ nat @ M @ ( type_len0_len_of @ A @ ( type2 @ A ) ) ) ) ) ) ).
% neg_mask_test_bit
thf(fact_5995_word__of__int__bin__cat__eq__iff,axiom,
! [C: $tType,B: $tType,A: $tType] :
( ( ( type_len @ A )
& ( type_len @ B )
& ( type_len @ C ) )
=> ! [B2: word @ B,A2: word @ A,D2: word @ B,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 @ D2 ) @ ( semiring_1_unsigned @ A @ int @ C2 ) ) ) )
= ( ( B2 = D2 )
& ( A2 = C2 ) ) ) ) ) ).
% word_of_int_bin_cat_eq_iff
thf(fact_5996_mask__exceed,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [N: nat,X: word @ A] :
( ( ord_less_eq @ nat @ ( type_len0_len_of @ A @ ( type2 @ A ) ) @ N )
=> ( ( bit_se5824344872417868541ns_and @ ( word @ A ) @ X @ ( bit_ri4277139882892585799ns_not @ ( word @ A ) @ ( bit_se2239418461657761734s_mask @ ( word @ A ) @ N ) ) )
= ( zero_zero @ ( word @ A ) ) ) ) ) ).
% mask_exceed
thf(fact_5997_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_5998_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_5999_word__nchotomy,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [W4: word @ A] :
? [N2: nat] :
( ( W4
= ( semiring_1_of_nat @ ( word @ A ) @ N2 ) )
& ( ord_less @ nat @ N2 @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( type_len0_len_of @ A @ ( type2 @ A ) ) ) ) ) ) ).
% word_nchotomy
thf(fact_6000_word__nat__cases,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [X: word @ A] :
~ ! [N2: nat] :
( ( X
= ( semiring_1_of_nat @ ( word @ A ) @ N2 ) )
=> ~ ( ord_less @ nat @ N2 @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( type_len0_len_of @ A @ ( type2 @ A ) ) ) ) ) ) ).
% word_nat_cases
thf(fact_6001_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_6002_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_6003_More__Word_Opower__not__zero,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [N: nat] :
( ( ord_less @ nat @ N @ ( type_len0_len_of @ A @ ( type2 @ A ) ) )
=> ( ( power_power @ ( word @ A ) @ ( numeral_numeral @ ( word @ A ) @ ( bit0 @ one2 ) ) @ N )
!= ( zero_zero @ ( word @ A ) ) ) ) ) ).
% More_Word.power_not_zero
thf(fact_6004_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_6005_power__overflow,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [N: nat] :
( ( ord_less_eq @ nat @ ( type_len0_len_of @ A @ ( type2 @ A ) ) @ N )
=> ( ( power_power @ ( word @ A ) @ ( numeral_numeral @ ( word @ A ) @ ( bit0 @ one2 ) ) @ N )
= ( zero_zero @ ( word @ A ) ) ) ) ) ).
% power_overflow
thf(fact_6006_nth__w2p__same,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [N: nat] :
( ( bit_se5641148757651400278ts_bit @ ( word @ A ) @ ( power_power @ ( word @ A ) @ ( numeral_numeral @ ( word @ A ) @ ( bit0 @ one2 ) ) @ N ) @ N )
= ( ord_less @ nat @ N @ ( type_len0_len_of @ A @ ( type2 @ A ) ) ) ) ) ).
% nth_w2p_same
thf(fact_6007_nth__w2p,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [N: nat,M: nat] :
( ( bit_se5641148757651400278ts_bit @ ( word @ A ) @ ( power_power @ ( word @ A ) @ ( numeral_numeral @ ( word @ A ) @ ( bit0 @ one2 ) ) @ N ) @ M )
= ( ( M = N )
& ( ord_less @ nat @ M @ ( type_len0_len_of @ A @ ( type2 @ A ) ) ) ) ) ) ).
% nth_w2p
thf(fact_6008_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_6009_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_6010_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_6011_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_6012_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_6013_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_6014_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_6015_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_6016_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_6017_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_6018_nth__sint,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [W: word @ A,N: nat] :
( ( bit_se5641148757651400278ts_bit @ int @ ( ring_1_signed @ A @ int @ W ) @ N )
= ( ( ( ord_less @ nat @ N @ ( minus_minus @ nat @ ( type_len0_len_of @ A @ ( type2 @ A ) ) @ ( one_one @ nat ) ) )
=> ( bit_se5641148757651400278ts_bit @ ( word @ A ) @ W @ N ) )
& ( ~ ( ord_less @ nat @ N @ ( minus_minus @ nat @ ( type_len0_len_of @ A @ ( type2 @ A ) ) @ ( one_one @ nat ) ) )
=> ( bit_se5641148757651400278ts_bit @ ( word @ A ) @ W @ ( minus_minus @ nat @ ( type_len0_len_of @ A @ ( type2 @ A ) ) @ ( one_one @ nat ) ) ) ) ) ) ) ).
% nth_sint
thf(fact_6019_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_6020_bit__sint__iff,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [W: word @ A,N: nat] :
( ( bit_se5641148757651400278ts_bit @ int @ ( ring_1_signed @ A @ int @ W ) @ N )
= ( ( ( ord_less_eq @ nat @ ( type_len0_len_of @ A @ ( type2 @ A ) ) @ N )
& ( bit_se5641148757651400278ts_bit @ ( word @ A ) @ W @ ( minus_minus @ nat @ ( type_len0_len_of @ A @ ( type2 @ A ) ) @ ( one_one @ nat ) ) ) )
| ( bit_se5641148757651400278ts_bit @ ( word @ A ) @ W @ N ) ) ) ) ).
% bit_sint_iff
thf(fact_6021_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_6022_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_6023_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_6024_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_6025_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_6026_less__Suc__unat__less__bound,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [N: nat,X: word @ A] :
( ( ord_less @ nat @ N @ ( suc @ ( semiring_1_unsigned @ A @ nat @ X ) ) )
=> ( ord_less @ nat @ N @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( type_len0_len_of @ A @ ( type2 @ A ) ) ) ) ) ) ).
% less_Suc_unat_less_bound
thf(fact_6027_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_6028_lt2p__lem,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [N: nat,W: word @ A] :
( ( ord_less_eq @ nat @ ( type_len0_len_of @ A @ ( type2 @ A ) ) @ N )
=> ( ord_less @ int @ ( semiring_1_unsigned @ A @ int @ W ) @ ( power_power @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ N ) ) ) ) ).
% lt2p_lem
thf(fact_6029_power__le__mono,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [N: nat,M: nat] :
( ( ord_less_eq @ ( word @ A ) @ ( power_power @ ( word @ A ) @ ( numeral_numeral @ ( word @ A ) @ ( bit0 @ one2 ) ) @ N ) @ ( power_power @ ( word @ A ) @ ( numeral_numeral @ ( word @ A ) @ ( bit0 @ one2 ) ) @ M ) )
=> ( ( ord_less @ nat @ N @ ( type_len0_len_of @ A @ ( type2 @ A ) ) )
=> ( ( ord_less @ nat @ M @ ( type_len0_len_of @ A @ ( type2 @ A ) ) )
=> ( ord_less_eq @ nat @ N @ M ) ) ) ) ) ).
% power_le_mono
thf(fact_6030_two__power__increasing,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [N: nat,M: nat] :
( ( ord_less_eq @ nat @ N @ M )
=> ( ( ord_less @ nat @ M @ ( type_len0_len_of @ A @ ( type2 @ A ) ) )
=> ( ord_less_eq @ ( word @ A ) @ ( power_power @ ( word @ A ) @ ( numeral_numeral @ ( word @ A ) @ ( bit0 @ one2 ) ) @ N ) @ ( power_power @ ( word @ A ) @ ( numeral_numeral @ ( word @ A ) @ ( bit0 @ one2 ) ) @ M ) ) ) ) ) ).
% two_power_increasing
thf(fact_6031_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_6032_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_6033_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_6034_unat__split,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [P: nat > $o,X: word @ A] :
( ( P @ ( semiring_1_unsigned @ A @ nat @ X ) )
= ( ! [N3: nat] :
( ( ( ( semiring_1_of_nat @ ( word @ A ) @ N3 )
= X )
& ( ord_less @ nat @ N3 @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( type_len0_len_of @ A @ ( type2 @ A ) ) ) ) )
=> ( P @ N3 ) ) ) ) ) ).
% unat_split
thf(fact_6035_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_6036_unat__split__asm,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [P: nat > $o,X: word @ A] :
( ( P @ ( semiring_1_unsigned @ A @ nat @ X ) )
= ( ~ ? [N3: nat] :
( ( ( semiring_1_of_nat @ ( word @ A ) @ N3 )
= X )
& ( ord_less @ nat @ N3 @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( type_len0_len_of @ A @ ( type2 @ A ) ) ) )
& ~ ( P @ N3 ) ) ) ) ) ).
% unat_split_asm
thf(fact_6037_unat__eq__of__nat,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [N: nat,X: word @ A] :
( ( ord_less @ nat @ N @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( type_len0_len_of @ A @ ( type2 @ A ) ) ) )
=> ( ( ( semiring_1_unsigned @ A @ nat @ X )
= N )
= ( X
= ( semiring_1_of_nat @ ( word @ A ) @ N ) ) ) ) ) ).
% unat_eq_of_nat
thf(fact_6038_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_6039_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_6040_test__bit__2p,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [N: nat,M: nat] :
( ( bit_se5641148757651400278ts_bit @ ( word @ A ) @ ( ring_1_of_int @ ( word @ A ) @ ( power_power @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ N ) ) @ M )
= ( ( M = N )
& ( ord_less @ nat @ M @ ( type_len0_len_of @ A @ ( type2 @ A ) ) ) ) ) ) ).
% test_bit_2p
thf(fact_6041_word__1__le__power,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [N: nat] :
( ( ord_less @ nat @ N @ ( type_len0_len_of @ A @ ( type2 @ A ) ) )
=> ( ord_less_eq @ ( word @ A ) @ ( one_one @ ( word @ A ) ) @ ( power_power @ ( word @ A ) @ ( numeral_numeral @ ( word @ A ) @ ( bit0 @ one2 ) ) @ N ) ) ) ) ).
% word_1_le_power
thf(fact_6042_Word_Oof__nat__0,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [M: nat] :
( ( ( semiring_1_of_nat @ ( word @ A ) @ M )
= ( zero_zero @ ( word @ A ) ) )
= ( ? [Q6: nat] :
( M
= ( times_times @ nat @ Q6 @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( type_len0_len_of @ A @ ( type2 @ A ) ) ) ) ) ) ) ) ).
% Word.of_nat_0
thf(fact_6043_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_6044_p2__gt__0,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [N: nat] :
( ( ord_less @ ( word @ A ) @ ( zero_zero @ ( word @ A ) ) @ ( power_power @ ( word @ A ) @ ( numeral_numeral @ ( word @ A ) @ ( bit0 @ one2 ) ) @ N ) )
= ( ord_less @ nat @ N @ ( type_len0_len_of @ A @ ( type2 @ A ) ) ) ) ) ).
% p2_gt_0
thf(fact_6045_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_6046_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_6047_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_6048_word__of__nat__eq__0__iff,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [N: nat] :
( ( ( semiring_1_of_nat @ ( word @ A ) @ N )
= ( zero_zero @ ( word @ A ) ) )
= ( dvd_dvd @ nat @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( type_len0_len_of @ A @ ( type2 @ A ) ) ) @ N ) ) ) ).
% word_of_nat_eq_0_iff
thf(fact_6049_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_6050_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_6051_mask__lt__2pn,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [N: nat] :
( ( ord_less @ nat @ N @ ( type_len0_len_of @ A @ ( type2 @ A ) ) )
=> ( ord_less @ ( word @ A ) @ ( bit_se2239418461657761734s_mask @ ( word @ A ) @ N ) @ ( power_power @ ( word @ A ) @ ( numeral_numeral @ ( word @ A ) @ ( bit0 @ one2 ) ) @ N ) ) ) ) ).
% mask_lt_2pn
thf(fact_6052_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_6053_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_6054_signed__of__int,axiom,
! [A: $tType,B: $tType] :
( ( ( type_len @ B )
& ( ring_1 @ A ) )
=> ! [N: int] :
( ( ring_1_signed @ B @ A @ ( ring_1_of_int @ ( word @ B ) @ N ) )
= ( ring_1_of_int @ A @ ( bit_ri4674362597316999326ke_bit @ int @ ( minus_minus @ nat @ ( type_len0_len_of @ B @ ( type2 @ B ) ) @ ( suc @ ( zero_zero @ nat ) ) ) @ N ) ) ) ) ).
% signed_of_int
thf(fact_6055_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_6056_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_6057_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_6058_of__nat__n__less__equal__power__2,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [N: nat] :
( ( ord_less @ nat @ N @ ( type_len0_len_of @ A @ ( type2 @ A ) ) )
=> ( ord_less @ ( word @ A ) @ ( semiring_1_of_nat @ ( word @ A ) @ N ) @ ( power_power @ ( word @ A ) @ ( numeral_numeral @ ( word @ A ) @ ( bit0 @ one2 ) ) @ N ) ) ) ) ).
% of_nat_n_less_equal_power_2
thf(fact_6059_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_6060_complement__nth__w2p,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [N5: nat,N: nat] :
( ( ord_less @ nat @ N5 @ ( type_len0_len_of @ A @ ( type2 @ A ) ) )
=> ( ( bit_se5641148757651400278ts_bit @ ( word @ A ) @ ( bit_ri4277139882892585799ns_not @ ( word @ A ) @ ( power_power @ ( word @ A ) @ ( numeral_numeral @ ( word @ A ) @ ( bit0 @ one2 ) ) @ N ) ) @ N5 )
= ( N5 != N ) ) ) ) ).
% complement_nth_w2p
thf(fact_6061_bit__word__scast__iff,axiom,
! [B: $tType,A: $tType] :
( ( ( type_len @ A )
& ( type_len @ B ) )
=> ! [W: word @ A,N: nat] :
( ( bit_se5641148757651400278ts_bit @ ( word @ B ) @ ( ring_1_signed @ A @ ( word @ B ) @ W ) @ N )
= ( ( ord_less @ nat @ N @ ( type_len0_len_of @ B @ ( type2 @ B ) ) )
& ( ( bit_se5641148757651400278ts_bit @ ( word @ A ) @ W @ N )
| ( ( ord_less_eq @ nat @ ( type_len0_len_of @ A @ ( type2 @ A ) ) @ N )
& ( bit_se5641148757651400278ts_bit @ ( word @ A ) @ W @ ( minus_minus @ nat @ ( type_len0_len_of @ A @ ( type2 @ A ) ) @ ( suc @ ( zero_zero @ nat ) ) ) ) ) ) ) ) ) ).
% bit_word_scast_iff
thf(fact_6062_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_6063_word__power__less__diff,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [N: nat,Q3: word @ A,M: nat] :
( ( ord_less @ ( word @ A ) @ ( times_times @ ( word @ A ) @ ( power_power @ ( word @ A ) @ ( numeral_numeral @ ( word @ A ) @ ( bit0 @ one2 ) ) @ N ) @ Q3 ) @ ( power_power @ ( word @ A ) @ ( numeral_numeral @ ( word @ A ) @ ( bit0 @ one2 ) ) @ M ) )
=> ( ( ord_less @ ( word @ A ) @ Q3 @ ( power_power @ ( word @ A ) @ ( numeral_numeral @ ( word @ A ) @ ( bit0 @ one2 ) ) @ ( minus_minus @ nat @ ( type_len0_len_of @ A @ ( type2 @ A ) ) @ N ) ) )
=> ( ord_less @ ( word @ A ) @ Q3 @ ( power_power @ ( word @ A ) @ ( numeral_numeral @ ( word @ A ) @ ( bit0 @ one2 ) ) @ ( minus_minus @ nat @ M @ N ) ) ) ) ) ) ).
% word_power_less_diff
thf(fact_6064_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_6065_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_6066_bit__signed__drop__bit__iff,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [M: nat,W: word @ A,N: nat] :
( ( bit_se5641148757651400278ts_bit @ ( word @ A ) @ ( signed_drop_bit @ A @ M @ W ) @ N )
= ( bit_se5641148757651400278ts_bit @ ( word @ A ) @ W
@ ( if @ nat
@ ( ( ord_less_eq @ nat @ ( minus_minus @ nat @ ( type_len0_len_of @ A @ ( type2 @ A ) ) @ M ) @ N )
& ( ord_less @ nat @ N @ ( type_len0_len_of @ A @ ( type2 @ A ) ) ) )
@ ( minus_minus @ nat @ ( type_len0_len_of @ A @ ( type2 @ A ) ) @ ( one_one @ nat ) )
@ ( plus_plus @ nat @ M @ N ) ) ) ) ) ).
% bit_signed_drop_bit_iff
thf(fact_6067_take__bit__word__eq__self__iff,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [N: nat,W: word @ A] :
( ( ( bit_se2584673776208193580ke_bit @ ( word @ A ) @ N @ W )
= W )
= ( ( ord_less_eq @ nat @ ( type_len0_len_of @ A @ ( type2 @ A ) ) @ N )
| ( ord_less @ ( word @ A ) @ W @ ( power_power @ ( word @ A ) @ ( numeral_numeral @ ( word @ A ) @ ( bit0 @ one2 ) ) @ N ) ) ) ) ) ).
% take_bit_word_eq_self_iff
thf(fact_6068_signed__push__bit__eq,axiom,
! [A: $tType,B: $tType] :
( ( ( type_len @ B )
& ( bit_ri3973907225187159222ations @ A ) )
=> ! [N: nat,W: word @ B] :
( ( ring_1_signed @ B @ A @ ( bit_se4730199178511100633sh_bit @ ( word @ B ) @ N @ W ) )
= ( bit_ri4674362597316999326ke_bit @ A @ ( minus_minus @ nat @ ( type_len0_len_of @ B @ ( type2 @ B ) ) @ ( suc @ ( zero_zero @ nat ) ) ) @ ( bit_se4730199178511100633sh_bit @ A @ N @ ( ring_1_signed @ B @ A @ W ) ) ) ) ) ).
% signed_push_bit_eq
thf(fact_6069_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_6070_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_6071_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_6072_More__Word_Oof__nat__0,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [N: nat] :
( ( ( semiring_1_of_nat @ ( word @ A ) @ N )
= ( zero_zero @ ( word @ A ) ) )
=> ( ( ord_less @ nat @ N @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( type_len0_len_of @ A @ ( type2 @ A ) ) ) )
=> ( N
= ( zero_zero @ nat ) ) ) ) ) ).
% More_Word.of_nat_0
thf(fact_6073_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_6074_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_6075_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_6076_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_6077_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_6078_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_6079_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_6080_word__int__cases,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [X: word @ A] :
~ ! [N2: int] :
( ( X
= ( ring_1_of_int @ ( word @ A ) @ N2 ) )
=> ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ N2 )
=> ~ ( ord_less @ int @ N2 @ ( power_power @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ ( type_len0_len_of @ A @ ( type2 @ A ) ) ) ) ) ) ) ).
% word_int_cases
thf(fact_6081_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_6082_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_6083_of__nat__eq,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [N: nat,W: word @ A] :
( ( ( semiring_1_of_nat @ ( word @ A ) @ N )
= W )
= ( ? [Q6: nat] :
( N
= ( plus_plus @ nat @ ( semiring_1_unsigned @ A @ nat @ W ) @ ( times_times @ nat @ Q6 @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( type_len0_len_of @ A @ ( type2 @ A ) ) ) ) ) ) ) ) ) ).
% of_nat_eq
thf(fact_6084_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_6085_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_6086_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_6087_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_6088_unat__ucast__less__no__overflow,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [N: nat,F2: word @ A] :
( ( ord_less @ nat @ N @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( type_len0_len_of @ A @ ( type2 @ A ) ) ) )
=> ( ( ord_less @ nat @ ( semiring_1_unsigned @ A @ nat @ F2 ) @ N )
=> ( ord_less @ ( word @ A ) @ F2 @ ( semiring_1_of_nat @ ( word @ A ) @ N ) ) ) ) ) ).
% unat_ucast_less_no_overflow
thf(fact_6089_unat__ucast__less__no__overflow__simp,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [N: nat,F2: word @ A] :
( ( ord_less @ nat @ N @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( type_len0_len_of @ A @ ( type2 @ A ) ) ) )
=> ( ( ord_less @ nat @ ( semiring_1_unsigned @ A @ nat @ F2 ) @ N )
= ( ord_less @ ( word @ A ) @ F2 @ ( semiring_1_of_nat @ ( word @ A ) @ N ) ) ) ) ) ).
% unat_ucast_less_no_overflow_simp
thf(fact_6090_less__2p__is__upper__bits__unset,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [P4: word @ A,N: nat] :
( ( ord_less @ ( word @ A ) @ P4 @ ( power_power @ ( word @ A ) @ ( numeral_numeral @ ( word @ A ) @ ( bit0 @ one2 ) ) @ N ) )
= ( ( ord_less @ nat @ N @ ( type_len0_len_of @ A @ ( type2 @ A ) ) )
& ! [N12: nat] :
( ( ord_less_eq @ nat @ N @ N12 )
=> ( ( ord_less @ nat @ N12 @ ( type_len0_len_of @ A @ ( type2 @ A ) ) )
=> ~ ( bit_se5641148757651400278ts_bit @ ( word @ A ) @ P4 @ N12 ) ) ) ) ) ) ).
% less_2p_is_upper_bits_unset
thf(fact_6091_upper__bits__unset__is__l2p,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [N: nat,P4: word @ A] :
( ( ord_less @ nat @ N @ ( type_len0_len_of @ A @ ( type2 @ A ) ) )
=> ( ( ! [N12: nat] :
( ( ord_less_eq @ nat @ N @ N12 )
=> ( ( ord_less @ nat @ N12 @ ( type_len0_len_of @ A @ ( type2 @ A ) ) )
=> ~ ( bit_se5641148757651400278ts_bit @ ( word @ A ) @ P4 @ N12 ) ) ) )
= ( ord_less @ ( word @ A ) @ P4 @ ( power_power @ ( word @ A ) @ ( numeral_numeral @ ( word @ A ) @ ( bit0 @ one2 ) ) @ N ) ) ) ) ) ).
% upper_bits_unset_is_l2p
thf(fact_6092_nth__bounded,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [X: word @ A,N: nat,M: nat] :
( ( bit_se5641148757651400278ts_bit @ ( word @ A ) @ X @ N )
=> ( ( ord_less @ ( word @ A ) @ X @ ( power_power @ ( word @ A ) @ ( numeral_numeral @ ( word @ A ) @ ( bit0 @ one2 ) ) @ M ) )
=> ( ( ord_less_eq @ nat @ M @ ( type_len0_len_of @ A @ ( type2 @ A ) ) )
=> ( ord_less @ nat @ N @ M ) ) ) ) ) ).
% nth_bounded
thf(fact_6093_uint__power__lower,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [N: nat] :
( ( ord_less @ nat @ N @ ( type_len0_len_of @ A @ ( type2 @ A ) ) )
=> ( ( semiring_1_unsigned @ A @ int @ ( power_power @ ( word @ A ) @ ( numeral_numeral @ ( word @ A ) @ ( bit0 @ one2 ) ) @ N ) )
= ( power_power @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ N ) ) ) ) ).
% uint_power_lower
thf(fact_6094_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_6095_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_6096_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_6097_wi__le,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [N: int,M: int] :
( ( ord_less_eq @ ( word @ A ) @ ( ring_1_of_int @ ( word @ A ) @ N ) @ ( ring_1_of_int @ ( word @ A ) @ M ) )
= ( ord_less_eq @ int @ ( modulo_modulo @ int @ N @ ( power_power @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ ( type_len0_len_of @ A @ ( type2 @ A ) ) ) ) @ ( modulo_modulo @ int @ M @ ( power_power @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ ( type_len0_len_of @ A @ ( type2 @ A ) ) ) ) ) ) ) ).
% wi_le
thf(fact_6098_word__2p__mult__inc,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [N: nat,M: nat] :
( ( ord_less @ ( word @ A ) @ ( times_times @ ( word @ A ) @ ( numeral_numeral @ ( word @ A ) @ ( bit0 @ one2 ) ) @ ( power_power @ ( word @ A ) @ ( numeral_numeral @ ( word @ A ) @ ( bit0 @ one2 ) ) @ N ) ) @ ( times_times @ ( word @ A ) @ ( numeral_numeral @ ( word @ A ) @ ( bit0 @ one2 ) ) @ ( power_power @ ( word @ A ) @ ( numeral_numeral @ ( word @ A ) @ ( bit0 @ one2 ) ) @ M ) ) )
=> ( ( ord_less @ nat @ ( suc @ N ) @ ( type_len0_len_of @ A @ ( type2 @ A ) ) )
=> ( ord_less @ ( word @ A ) @ ( power_power @ ( word @ A ) @ ( numeral_numeral @ ( word @ A ) @ ( bit0 @ one2 ) ) @ N ) @ ( power_power @ ( word @ A ) @ ( numeral_numeral @ ( word @ A ) @ ( bit0 @ one2 ) ) @ M ) ) ) ) ) ).
% word_2p_mult_inc
thf(fact_6099_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_6100_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_6101_power__2__ge__iff,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [N: nat] :
( ( ord_less @ ( word @ A ) @ ( minus_minus @ ( word @ A ) @ ( power_power @ ( word @ A ) @ ( numeral_numeral @ ( word @ A ) @ ( bit0 @ one2 ) ) @ N ) @ ( one_one @ ( word @ A ) ) ) @ ( power_power @ ( word @ A ) @ ( numeral_numeral @ ( word @ A ) @ ( bit0 @ one2 ) ) @ N ) )
= ( ord_less @ nat @ N @ ( type_len0_len_of @ A @ ( type2 @ A ) ) ) ) ) ).
% power_2_ge_iff
thf(fact_6102_wi__less,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [N: int,M: int] :
( ( ord_less @ ( word @ A ) @ ( ring_1_of_int @ ( word @ A ) @ N ) @ ( ring_1_of_int @ ( word @ A ) @ M ) )
= ( ord_less @ int @ ( modulo_modulo @ int @ N @ ( power_power @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ ( type_len0_len_of @ A @ ( type2 @ A ) ) ) ) @ ( modulo_modulo @ int @ M @ ( power_power @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ ( type_len0_len_of @ A @ ( type2 @ A ) ) ) ) ) ) ) ).
% wi_less
thf(fact_6103_le__mask__iff__lt__2n,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [N: nat,W: word @ A] :
( ( ord_less @ nat @ N @ ( type_len0_len_of @ A @ ( type2 @ A ) ) )
= ( ( ord_less_eq @ ( word @ A ) @ W @ ( bit_se2239418461657761734s_mask @ ( word @ A ) @ N ) )
= ( ord_less @ ( word @ A ) @ W @ ( power_power @ ( word @ A ) @ ( numeral_numeral @ ( word @ A ) @ ( bit0 @ one2 ) ) @ N ) ) ) ) ) ).
% le_mask_iff_lt_2n
thf(fact_6104_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_6105_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_6106_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_6107_and__mask__less_H,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [N: nat,W: word @ A] :
( ( ord_less @ nat @ N @ ( type_len0_len_of @ A @ ( type2 @ A ) ) )
=> ( ord_less @ ( word @ A ) @ ( bit_se5824344872417868541ns_and @ ( word @ A ) @ W @ ( bit_se2239418461657761734s_mask @ ( word @ A ) @ N ) ) @ ( power_power @ ( word @ A ) @ ( numeral_numeral @ ( word @ A ) @ ( bit0 @ one2 ) ) @ N ) ) ) ) ).
% and_mask_less'
thf(fact_6108_eq__mask__less,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [W: word @ A,N: nat] :
( ( W
= ( bit_se5824344872417868541ns_and @ ( word @ A ) @ W @ ( bit_se2239418461657761734s_mask @ ( word @ A ) @ N ) ) )
=> ( ( ord_less @ nat @ N @ ( type_len0_len_of @ A @ ( type2 @ A ) ) )
=> ( ord_less @ ( word @ A ) @ W @ ( power_power @ ( word @ A ) @ ( numeral_numeral @ ( word @ A ) @ ( bit0 @ one2 ) ) @ N ) ) ) ) ) ).
% eq_mask_less
thf(fact_6109_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_6110_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_6111_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_6112_signed__of__nat,axiom,
! [A: $tType,B: $tType] :
( ( ( type_len @ B )
& ( ring_1 @ A ) )
=> ! [N: nat] :
( ( ring_1_signed @ B @ A @ ( semiring_1_of_nat @ ( word @ B ) @ N ) )
= ( ring_1_of_int @ A @ ( bit_ri4674362597316999326ke_bit @ int @ ( minus_minus @ nat @ ( type_len0_len_of @ B @ ( type2 @ B ) ) @ ( suc @ ( zero_zero @ nat ) ) ) @ ( semiring_1_of_nat @ int @ N ) ) ) ) ) ).
% signed_of_nat
thf(fact_6113_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_6114_word__power__mod__div,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [N: nat,M: nat,X: word @ A] :
( ( ord_less @ nat @ N @ ( type_len0_len_of @ A @ ( type2 @ A ) ) )
=> ( ( ord_less @ nat @ M @ ( type_len0_len_of @ A @ ( type2 @ A ) ) )
=> ( ( divide_divide @ ( word @ A ) @ ( modulo_modulo @ ( word @ A ) @ X @ ( power_power @ ( word @ A ) @ ( numeral_numeral @ ( word @ A ) @ ( bit0 @ one2 ) ) @ N ) ) @ ( power_power @ ( word @ A ) @ ( numeral_numeral @ ( word @ A ) @ ( bit0 @ one2 ) ) @ M ) )
= ( modulo_modulo @ ( word @ A ) @ ( divide_divide @ ( word @ A ) @ X @ ( power_power @ ( word @ A ) @ ( numeral_numeral @ ( word @ A ) @ ( bit0 @ one2 ) ) @ M ) ) @ ( power_power @ ( word @ A ) @ ( numeral_numeral @ ( word @ A ) @ ( bit0 @ one2 ) ) @ ( minus_minus @ nat @ N @ M ) ) ) ) ) ) ) ).
% word_power_mod_div
thf(fact_6115_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_6116_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_6117_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_6118_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_6119_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_6120_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_6121_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_6122_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_6123_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_6124_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_6125_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_6126_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_6127_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_6128_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_6129_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_6130_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_6131_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_6132_uint__split,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [P: int > $o,X: word @ A] :
( ( P @ ( semiring_1_unsigned @ A @ int @ X ) )
= ( ! [I2: int] :
( ( ( ( ring_1_of_int @ ( word @ A ) @ I2 )
= X )
& ( ord_less_eq @ int @ ( zero_zero @ int ) @ I2 )
& ( ord_less @ int @ I2 @ ( power_power @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ ( type_len0_len_of @ A @ ( type2 @ A ) ) ) ) )
=> ( P @ I2 ) ) ) ) ) ).
% uint_split
thf(fact_6133_uint__split__asm,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [P: int > $o,X: word @ A] :
( ( P @ ( semiring_1_unsigned @ A @ int @ X ) )
= ( ~ ? [I2: int] :
( ( ( ring_1_of_int @ ( word @ A ) @ I2 )
= X )
& ( ord_less_eq @ int @ ( zero_zero @ int ) @ I2 )
& ( ord_less @ int @ I2 @ ( power_power @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ ( type_len0_len_of @ A @ ( type2 @ A ) ) ) )
& ~ ( P @ I2 ) ) ) ) ) ).
% uint_split_asm
thf(fact_6134_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_6135_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_6136_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_6137_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_6138_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_6139_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_6140_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_6141_word__le__exists_H,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [X: word @ A,Y: word @ A] :
( ( ord_less_eq @ ( word @ A ) @ X @ Y )
=> ? [Z3: word @ A] :
( ( Y
= ( plus_plus @ ( word @ A ) @ X @ Z3 ) )
& ( ord_less @ int @ ( plus_plus @ int @ ( semiring_1_unsigned @ A @ int @ X ) @ ( semiring_1_unsigned @ A @ int @ Z3 ) ) @ ( power_power @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ ( type_len0_len_of @ A @ ( type2 @ A ) ) ) ) ) ) ) ).
% word_le_exists'
thf(fact_6142_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_6143_length__n__lists__elem,axiom,
! [A: $tType,Ys: list @ A,N: nat,Xs2: list @ A] :
( ( member @ ( list @ A ) @ Ys @ ( set2 @ ( list @ A ) @ ( n_lists @ A @ N @ Xs2 ) ) )
=> ( ( size_size @ ( list @ A ) @ Ys )
= N ) ) ).
% length_n_lists_elem
thf(fact_6144_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_6145_word__less__power__trans2,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [N: word @ A,M: nat,K: nat] :
( ( ord_less @ ( word @ A ) @ N @ ( power_power @ ( word @ A ) @ ( numeral_numeral @ ( word @ A ) @ ( bit0 @ one2 ) ) @ ( minus_minus @ nat @ M @ K ) ) )
=> ( ( ord_less_eq @ nat @ K @ M )
=> ( ( ord_less @ nat @ M @ ( type_len0_len_of @ A @ ( type2 @ A ) ) )
=> ( ord_less @ ( word @ A ) @ ( times_times @ ( word @ A ) @ N @ ( power_power @ ( word @ A ) @ ( numeral_numeral @ ( word @ A ) @ ( bit0 @ one2 ) ) @ K ) ) @ ( power_power @ ( word @ A ) @ ( numeral_numeral @ ( word @ A ) @ ( bit0 @ one2 ) ) @ M ) ) ) ) ) ) ).
% word_less_power_trans2
thf(fact_6146_word__less__power__trans,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [N: word @ A,M: nat,K: nat] :
( ( ord_less @ ( word @ A ) @ N @ ( power_power @ ( word @ A ) @ ( numeral_numeral @ ( word @ A ) @ ( bit0 @ one2 ) ) @ ( minus_minus @ nat @ M @ K ) ) )
=> ( ( ord_less_eq @ nat @ K @ M )
=> ( ( ord_less @ nat @ M @ ( type_len0_len_of @ A @ ( type2 @ A ) ) )
=> ( ord_less @ ( word @ A ) @ ( times_times @ ( word @ A ) @ ( power_power @ ( word @ A ) @ ( numeral_numeral @ ( word @ A ) @ ( bit0 @ one2 ) ) @ K ) @ N ) @ ( power_power @ ( word @ A ) @ ( numeral_numeral @ ( word @ A ) @ ( bit0 @ one2 ) ) @ M ) ) ) ) ) ) ).
% word_less_power_trans
thf(fact_6147_word__less__two__pow__divI,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [X: word @ A,N: nat,M: nat] :
( ( ord_less @ ( word @ A ) @ X @ ( power_power @ ( word @ A ) @ ( numeral_numeral @ ( word @ A ) @ ( bit0 @ one2 ) ) @ ( minus_minus @ nat @ N @ M ) ) )
=> ( ( ord_less_eq @ nat @ M @ N )
=> ( ( ord_less @ nat @ N @ ( type_len0_len_of @ A @ ( type2 @ A ) ) )
=> ( ord_less @ ( word @ A ) @ X @ ( divide_divide @ ( word @ A ) @ ( power_power @ ( word @ A ) @ ( numeral_numeral @ ( word @ A ) @ ( bit0 @ one2 ) ) @ N ) @ ( power_power @ ( word @ A ) @ ( numeral_numeral @ ( word @ A ) @ ( bit0 @ one2 ) ) @ M ) ) ) ) ) ) ) ).
% word_less_two_pow_divI
thf(fact_6148_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_6149_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_6150_word__power__nonzero,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [X: word @ A,N: nat] :
( ( ord_less @ ( word @ A ) @ X @ ( power_power @ ( word @ A ) @ ( numeral_numeral @ ( word @ A ) @ ( bit0 @ one2 ) ) @ ( minus_minus @ nat @ ( type_len0_len_of @ A @ ( type2 @ A ) ) @ N ) ) )
=> ( ( ord_less @ nat @ N @ ( type_len0_len_of @ A @ ( type2 @ A ) ) )
=> ( ( X
!= ( zero_zero @ ( word @ A ) ) )
=> ( ( times_times @ ( word @ A ) @ X @ ( power_power @ ( word @ A ) @ ( numeral_numeral @ ( word @ A ) @ ( bit0 @ one2 ) ) @ N ) )
!= ( zero_zero @ ( word @ A ) ) ) ) ) ) ) ).
% word_power_nonzero
thf(fact_6151_mult__pow2__inj,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [M: nat,N: nat,X: word @ A,Y: word @ A] :
( ( ord_less_eq @ nat @ ( plus_plus @ nat @ M @ N ) @ ( type_len0_len_of @ A @ ( type2 @ A ) ) )
=> ( ( ord_less_eq @ ( word @ A ) @ X @ ( bit_se2239418461657761734s_mask @ ( word @ A ) @ M ) )
=> ( ( ord_less_eq @ ( word @ A ) @ Y @ ( bit_se2239418461657761734s_mask @ ( word @ A ) @ M ) )
=> ( ( ( times_times @ ( word @ A ) @ X @ ( power_power @ ( word @ A ) @ ( numeral_numeral @ ( word @ A ) @ ( bit0 @ one2 ) ) @ N ) )
= ( times_times @ ( word @ A ) @ Y @ ( power_power @ ( word @ A ) @ ( numeral_numeral @ ( word @ A ) @ ( bit0 @ one2 ) ) @ N ) ) )
=> ( X = Y ) ) ) ) ) ) ).
% mult_pow2_inj
thf(fact_6152_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_6153_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_6154_push__bit__word__eq__nonzero,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [W: word @ A,M: nat,N: nat] :
( ( ord_less @ ( word @ A ) @ W @ ( power_power @ ( word @ A ) @ ( numeral_numeral @ ( word @ A ) @ ( bit0 @ one2 ) ) @ M ) )
=> ( ( ord_less @ nat @ ( plus_plus @ nat @ M @ N ) @ ( type_len0_len_of @ A @ ( type2 @ A ) ) )
=> ( ( W
!= ( zero_zero @ ( word @ A ) ) )
=> ( ( bit_se4730199178511100633sh_bit @ ( word @ A ) @ N @ W )
!= ( zero_zero @ ( word @ A ) ) ) ) ) ) ) ).
% push_bit_word_eq_nonzero
thf(fact_6155_signed__drop__bit__beyond,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [N: nat,W: word @ A] :
( ( ord_less_eq @ nat @ ( type_len0_len_of @ A @ ( type2 @ A ) ) @ N )
=> ( ( ( bit_se5641148757651400278ts_bit @ ( word @ A ) @ W @ ( minus_minus @ nat @ ( type_len0_len_of @ A @ ( type2 @ A ) ) @ ( suc @ ( zero_zero @ nat ) ) ) )
=> ( ( signed_drop_bit @ A @ N @ W )
= ( uminus_uminus @ ( word @ A ) @ ( one_one @ ( word @ A ) ) ) ) )
& ( ~ ( bit_se5641148757651400278ts_bit @ ( word @ A ) @ W @ ( minus_minus @ nat @ ( type_len0_len_of @ A @ ( type2 @ A ) ) @ ( suc @ ( zero_zero @ nat ) ) ) )
=> ( ( signed_drop_bit @ A @ N @ W )
= ( zero_zero @ ( word @ A ) ) ) ) ) ) ) ).
% signed_drop_bit_beyond
thf(fact_6156_n__lists_Osimps_I2_J,axiom,
! [A: $tType,N: nat,Xs2: list @ A] :
( ( n_lists @ A @ ( suc @ N ) @ Xs2 )
= ( concat @ ( list @ A )
@ ( map @ ( list @ A ) @ ( list @ ( list @ A ) )
@ ^ [Ys3: list @ A] :
( map @ A @ ( list @ A )
@ ^ [Y6: A] : ( cons @ A @ Y6 @ Ys3 )
@ Xs2 )
@ ( n_lists @ A @ N @ Xs2 ) ) ) ) ).
% n_lists.simps(2)
thf(fact_6157_uint__and__mask__or__full,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [N: word @ A,Mask1: word @ A,Mask2: int] :
( ( bit_se5641148757651400278ts_bit @ ( word @ A ) @ N @ ( minus_minus @ nat @ ( type_len0_len_of @ A @ ( type2 @ A ) ) @ ( one_one @ nat ) ) )
=> ( ( Mask1
= ( bit_se2239418461657761734s_mask @ ( word @ A ) @ ( minus_minus @ nat @ ( type_len0_len_of @ A @ ( type2 @ A ) ) @ ( one_one @ nat ) ) ) )
=> ( ( Mask2
= ( bit_se4730199178511100633sh_bit @ int @ ( minus_minus @ nat @ ( type_len0_len_of @ A @ ( type2 @ A ) ) @ ( one_one @ nat ) ) @ ( one_one @ int ) ) )
=> ( ( bit_se1065995026697491101ons_or @ int @ ( semiring_1_unsigned @ A @ int @ ( bit_se5824344872417868541ns_and @ ( word @ A ) @ N @ Mask1 ) ) @ Mask2 )
= ( semiring_1_unsigned @ A @ int @ N ) ) ) ) ) ) ).
% uint_and_mask_or_full
thf(fact_6158_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_6159_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_6160_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_6161_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_6162_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_6163_le__2p__upper__bits,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [P4: word @ A,N: nat] :
( ( ord_less_eq @ ( word @ A ) @ P4 @ ( minus_minus @ ( word @ A ) @ ( power_power @ ( word @ A ) @ ( numeral_numeral @ ( word @ A ) @ ( bit0 @ one2 ) ) @ N ) @ ( one_one @ ( word @ A ) ) ) )
=> ( ( ord_less @ nat @ N @ ( type_len0_len_of @ A @ ( type2 @ A ) ) )
=> ! [N6: nat] :
( ( ord_less_eq @ nat @ N @ N6 )
=> ( ( ord_less @ nat @ N6 @ ( type_len0_len_of @ A @ ( type2 @ A ) ) )
=> ~ ( bit_se5641148757651400278ts_bit @ ( word @ A ) @ P4 @ N6 ) ) ) ) ) ) ).
% le_2p_upper_bits
thf(fact_6164_le2p__bits__unset,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [P4: word @ A,N: nat] :
( ( ord_less_eq @ ( word @ A ) @ P4 @ ( minus_minus @ ( word @ A ) @ ( power_power @ ( word @ A ) @ ( numeral_numeral @ ( word @ A ) @ ( bit0 @ one2 ) ) @ N ) @ ( one_one @ ( word @ A ) ) ) )
=> ! [N6: nat] :
( ( ord_less_eq @ nat @ N @ N6 )
=> ( ( ord_less @ nat @ N6 @ ( type_len0_len_of @ A @ ( type2 @ A ) ) )
=> ~ ( bit_se5641148757651400278ts_bit @ ( word @ A ) @ P4 @ N6 ) ) ) ) ) ).
% le2p_bits_unset
thf(fact_6165_word__add__offset__less,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [Y: word @ A,N: nat,X: word @ A,M: nat,Sz: nat] :
( ( ord_less @ ( word @ A ) @ Y @ ( power_power @ ( word @ A ) @ ( numeral_numeral @ ( word @ A ) @ ( bit0 @ one2 ) ) @ N ) )
=> ( ( ord_less @ ( word @ A ) @ X @ ( power_power @ ( word @ A ) @ ( numeral_numeral @ ( word @ A ) @ ( bit0 @ one2 ) ) @ M ) )
=> ( ( ord_less @ nat @ Sz @ ( type_len0_len_of @ A @ ( type2 @ A ) ) )
=> ( ( ord_less @ ( word @ A ) @ X @ ( power_power @ ( word @ A ) @ ( numeral_numeral @ ( word @ A ) @ ( bit0 @ one2 ) ) @ ( minus_minus @ nat @ ( type_len0_len_of @ A @ ( type2 @ A ) ) @ N ) ) )
=> ( ( Sz
= ( plus_plus @ nat @ M @ N ) )
=> ( ord_less @ ( word @ A ) @ ( plus_plus @ ( word @ A ) @ ( times_times @ ( word @ A ) @ X @ ( power_power @ ( word @ A ) @ ( numeral_numeral @ ( word @ A ) @ ( bit0 @ one2 ) ) @ N ) ) @ Y ) @ ( power_power @ ( word @ A ) @ ( numeral_numeral @ ( word @ A ) @ ( bit0 @ one2 ) ) @ Sz ) ) ) ) ) ) ) ) ).
% word_add_offset_less
thf(fact_6166_bit__horner__sum__uint__exp__iff,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [Ws: list @ ( word @ A ),N: nat] :
( ( bit_se5641148757651400278ts_bit @ int @ ( groups4207007520872428315er_sum @ ( word @ A ) @ int @ ( semiring_1_unsigned @ A @ int ) @ ( power_power @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ ( type_len0_len_of @ A @ ( type2 @ A ) ) ) @ Ws ) @ N )
= ( ( ord_less @ nat @ ( divide_divide @ nat @ N @ ( type_len0_len_of @ A @ ( type2 @ A ) ) ) @ ( size_size @ ( list @ ( word @ A ) ) @ Ws ) )
& ( bit_se5641148757651400278ts_bit @ ( word @ A ) @ ( nth @ ( word @ A ) @ Ws @ ( divide_divide @ nat @ N @ ( type_len0_len_of @ A @ ( type2 @ A ) ) ) ) @ ( modulo_modulo @ nat @ N @ ( type_len0_len_of @ A @ ( type2 @ A ) ) ) ) ) ) ) ).
% bit_horner_sum_uint_exp_iff
thf(fact_6167_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_6168_even__mult__exp__div__word__iff,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [A2: word @ A,M: nat,N: nat] :
( ( dvd_dvd @ ( word @ A ) @ ( numeral_numeral @ ( word @ A ) @ ( bit0 @ one2 ) ) @ ( divide_divide @ ( word @ A ) @ ( times_times @ ( word @ A ) @ A2 @ ( power_power @ ( word @ A ) @ ( numeral_numeral @ ( word @ A ) @ ( bit0 @ one2 ) ) @ M ) ) @ ( power_power @ ( word @ A ) @ ( numeral_numeral @ ( word @ A ) @ ( bit0 @ one2 ) ) @ N ) ) )
= ( ~ ( ( ord_less_eq @ nat @ M @ N )
& ( ord_less @ nat @ N @ ( type_len0_len_of @ A @ ( type2 @ A ) ) )
& ~ ( dvd_dvd @ ( word @ A ) @ ( numeral_numeral @ ( word @ A ) @ ( bit0 @ one2 ) ) @ ( divide_divide @ ( word @ A ) @ A2 @ ( power_power @ ( word @ A ) @ ( numeral_numeral @ ( word @ A ) @ ( bit0 @ one2 ) ) @ ( minus_minus @ nat @ N @ M ) ) ) ) ) ) ) ) ).
% even_mult_exp_div_word_iff
thf(fact_6169_Suc__2p__unat__mask,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [N: nat,K: nat] :
( ( ord_less @ nat @ N @ ( type_len0_len_of @ A @ ( type2 @ A ) ) )
=> ( ( suc @ ( plus_plus @ nat @ ( times_times @ nat @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N ) @ K ) @ ( semiring_1_unsigned @ A @ nat @ ( bit_se2239418461657761734s_mask @ ( word @ A ) @ N ) ) ) )
= ( times_times @ nat @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N ) @ ( plus_plus @ nat @ K @ ( one_one @ nat ) ) ) ) ) ) ).
% Suc_2p_unat_mask
thf(fact_6170_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_6171_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_6172_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_6173_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_6174_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_6175_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_6176_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_6177_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_6178_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_6179_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_6180_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_6181_word__bit__induct,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [P: ( word @ A ) > $o,A2: word @ A] :
( ( P @ ( zero_zero @ ( word @ A ) ) )
=> ( ! [A6: word @ A] :
( ( P @ A6 )
=> ( ( ord_less @ ( word @ A ) @ ( zero_zero @ ( word @ A ) ) @ A6 )
=> ( ( ord_less @ ( word @ A ) @ A6 @ ( 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 ) ) @ A6 ) ) ) ) )
=> ( ! [A6: word @ A] :
( ( P @ A6 )
=> ( ( ord_less @ ( word @ A ) @ A6 @ ( 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 ) ) @ A6 ) ) ) ) )
=> ( P @ A2 ) ) ) ) ) ).
% word_bit_induct
thf(fact_6182_word__less__power__trans__ofnat,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [N: nat,M: nat,K: nat] :
( ( ord_less @ nat @ N @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( minus_minus @ nat @ M @ K ) ) )
=> ( ( ord_less_eq @ nat @ K @ M )
=> ( ( ord_less @ nat @ M @ ( type_len0_len_of @ A @ ( type2 @ A ) ) )
=> ( ord_less @ ( word @ A ) @ ( times_times @ ( word @ A ) @ ( semiring_1_of_nat @ ( word @ A ) @ N ) @ ( power_power @ ( word @ A ) @ ( numeral_numeral @ ( word @ A ) @ ( bit0 @ one2 ) ) @ K ) ) @ ( power_power @ ( word @ A ) @ ( numeral_numeral @ ( word @ A ) @ ( bit0 @ one2 ) ) @ M ) ) ) ) ) ) ).
% word_less_power_trans_ofnat
thf(fact_6183_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_6184_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_6185_length__n__lists,axiom,
! [A: $tType,N: nat,Xs2: list @ A] :
( ( size_size @ ( list @ ( list @ A ) ) @ ( n_lists @ A @ N @ Xs2 ) )
= ( power_power @ nat @ ( size_size @ ( list @ A ) @ Xs2 ) @ N ) ) ).
% length_n_lists
thf(fact_6186_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_6187_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_6188_alignUp__div__helper,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [K: nat,N: nat,X: word @ A,A2: word @ A] :
( ( ord_less @ nat @ K @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( minus_minus @ nat @ ( type_len0_len_of @ A @ ( type2 @ A ) ) @ N ) ) )
=> ( ( X
= ( times_times @ ( word @ A ) @ ( power_power @ ( word @ A ) @ ( numeral_numeral @ ( word @ A ) @ ( bit0 @ one2 ) ) @ N ) @ ( semiring_1_of_nat @ ( word @ A ) @ K ) ) )
=> ( ( ord_less_eq @ ( word @ A ) @ A2 @ X )
=> ( ( ord_less @ nat @ N @ ( type_len0_len_of @ A @ ( type2 @ A ) ) )
=> ( ( ( modulo_modulo @ ( word @ A ) @ A2 @ ( power_power @ ( word @ A ) @ ( numeral_numeral @ ( word @ A ) @ ( bit0 @ one2 ) ) @ N ) )
!= ( zero_zero @ ( word @ A ) ) )
=> ( ord_less @ ( word @ A ) @ ( divide_divide @ ( word @ A ) @ A2 @ ( power_power @ ( word @ A ) @ ( numeral_numeral @ ( word @ A ) @ ( bit0 @ one2 ) ) @ N ) ) @ ( semiring_1_of_nat @ ( word @ A ) @ K ) ) ) ) ) ) ) ) ).
% alignUp_div_helper
thf(fact_6189_less__eq__decr__length__iff,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [N: nat] :
( ( ord_less_eq @ nat @ N @ ( minus_minus @ nat @ ( type_len0_len_of @ A @ ( type2 @ A ) ) @ ( suc @ ( zero_zero @ nat ) ) ) )
= ( ord_less @ nat @ N @ ( type_len0_len_of @ A @ ( type2 @ A ) ) ) ) ) ).
% less_eq_decr_length_iff
thf(fact_6190_decr__length__less__iff,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [N: nat] :
( ( ord_less @ nat @ ( minus_minus @ nat @ ( type_len0_len_of @ A @ ( type2 @ A ) ) @ ( suc @ ( zero_zero @ nat ) ) ) @ N )
= ( ord_less_eq @ nat @ ( type_len0_len_of @ A @ ( type2 @ A ) ) @ N ) ) ) ).
% decr_length_less_iff
thf(fact_6191_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_6192_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_6193_len__num0,axiom,
( ( type_len0_len_of @ numeral_num0 )
= ( ^ [Uu4: itself @ numeral_num0] : ( zero_zero @ nat ) ) ) ).
% len_num0
thf(fact_6194_len__num1,axiom,
( ( type_len0_len_of @ numeral_num1 )
= ( ^ [Uu4: itself @ numeral_num1] : ( one_one @ nat ) ) ) ).
% len_num1
thf(fact_6195_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_6196_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_6197_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_6198_len__not__eq__0,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ( ( type_len0_len_of @ A @ ( type2 @ A ) )
!= ( zero_zero @ nat ) ) ) ).
% len_not_eq_0
thf(fact_6199_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_6200_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_6201_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_6202_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_6203_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_6204_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_6205_word__sdiv__div0,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [A2: word @ A] :
( ( signed7115095781618012415divide @ ( word @ A ) @ A2 @ ( zero_zero @ ( word @ A ) ) )
= ( zero_zero @ ( word @ A ) ) ) ) ).
% word_sdiv_div0
thf(fact_6206_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_6207_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_6208_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_6209_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_6210_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_6211_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_6212_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_6213_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_6214_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_6215_word__sdiv__numerals_I5_J,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ( ( signed7115095781618012415divide @ ( word @ A ) @ ( zero_zero @ ( word @ A ) ) @ ( zero_zero @ ( word @ A ) ) )
= ( ring_1_of_int @ ( word @ A ) @ ( signed7115095781618012415divide @ int @ ( ring_1_signed @ A @ int @ ( zero_zero @ ( word @ A ) ) ) @ ( ring_1_signed @ A @ int @ ( zero_zero @ ( word @ A ) ) ) ) ) ) ) ).
% word_sdiv_numerals(5)
thf(fact_6216_word__sdiv__numerals__lhs_I2_J,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [W: word @ A] :
( ( signed7115095781618012415divide @ ( word @ A ) @ ( zero_zero @ ( word @ A ) ) @ W )
= ( ring_1_of_int @ ( word @ A ) @ ( signed7115095781618012415divide @ int @ ( ring_1_signed @ A @ int @ ( zero_zero @ ( word @ A ) ) ) @ ( ring_1_signed @ A @ int @ W ) ) ) ) ) ).
% word_sdiv_numerals_lhs(2)
thf(fact_6217_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_6218_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_6219_word__sdiv__numerals_I2_J,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [Y: num] :
( ( signed7115095781618012415divide @ ( word @ A ) @ ( zero_zero @ ( word @ A ) ) @ ( numeral_numeral @ ( word @ A ) @ Y ) )
= ( ring_1_of_int @ ( word @ A ) @ ( signed7115095781618012415divide @ int @ ( ring_1_signed @ A @ int @ ( zero_zero @ ( word @ A ) ) ) @ ( ring_1_signed @ A @ int @ ( numeral_numeral @ ( word @ A ) @ Y ) ) ) ) ) ) ).
% word_sdiv_numerals(2)
thf(fact_6220_word__sdiv__numerals_I4_J,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [X: num] :
( ( signed7115095781618012415divide @ ( word @ A ) @ ( numeral_numeral @ ( word @ A ) @ X ) @ ( zero_zero @ ( 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 @ ( zero_zero @ ( word @ A ) ) ) ) ) ) ) ).
% word_sdiv_numerals(4)
thf(fact_6221_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_6222_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_6223_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_6224_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 ) )
= ( ! [I2: int] :
( ( ( X
= ( ring_1_of_int @ ( word @ B ) @ I2 ) )
& ( ord_less_eq @ int @ ( zero_zero @ int ) @ I2 )
& ( ord_less @ int @ I2 @ ( power_power @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ ( type_len0_len_of @ B @ ( type2 @ B ) ) ) ) )
=> ( P @ ( F2 @ I2 ) ) ) ) ) ) ).
% word_int_split
thf(fact_6225_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 ) )
= ( ~ ? [N3: int] :
( ( X
= ( ring_1_of_int @ ( word @ B ) @ N3 ) )
& ( ord_less_eq @ int @ ( zero_zero @ int ) @ N3 )
& ( ord_less @ int @ N3 @ ( power_power @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ ( type_len0_len_of @ B @ ( type2 @ B ) ) ) )
& ~ ( P @ ( F2 @ N3 ) ) ) ) ) ) ).
% word_int_split_asm
thf(fact_6226_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_6227_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_6228_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_6229_smod__word__mod__0,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [X: word @ A] :
( ( signed6721504322012087516modulo @ ( word @ A ) @ X @ ( zero_zero @ ( word @ A ) ) )
= X ) ) ).
% smod_word_mod_0
thf(fact_6230_smod__word__0__mod,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [X: word @ A] :
( ( signed6721504322012087516modulo @ ( word @ A ) @ ( zero_zero @ ( word @ A ) ) @ X )
= ( zero_zero @ ( word @ A ) ) ) ) ).
% smod_word_0_mod
thf(fact_6231_smod__word__zero,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [W: word @ A] :
( ( signed6721504322012087516modulo @ ( word @ A ) @ W @ ( zero_zero @ ( word @ A ) ) )
= W ) ) ).
% smod_word_zero
thf(fact_6232_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_6233_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_6234_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_6235_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_6236_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_6237_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_6238_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_6239_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_6240_word__smod__numerals__lhs_I2_J,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [W: word @ A] :
( ( signed6721504322012087516modulo @ ( word @ A ) @ ( zero_zero @ ( word @ A ) ) @ W )
= ( ring_1_of_int @ ( word @ A ) @ ( signed6721504322012087516modulo @ int @ ( ring_1_signed @ A @ int @ ( zero_zero @ ( word @ A ) ) ) @ ( ring_1_signed @ A @ int @ W ) ) ) ) ) ).
% word_smod_numerals_lhs(2)
thf(fact_6241_word__smod__numerals_I5_J,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ( ( signed6721504322012087516modulo @ ( word @ A ) @ ( zero_zero @ ( word @ A ) ) @ ( zero_zero @ ( word @ A ) ) )
= ( ring_1_of_int @ ( word @ A ) @ ( signed6721504322012087516modulo @ int @ ( ring_1_signed @ A @ int @ ( zero_zero @ ( word @ A ) ) ) @ ( ring_1_signed @ A @ int @ ( zero_zero @ ( word @ A ) ) ) ) ) ) ) ).
% word_smod_numerals(5)
thf(fact_6242_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_6243_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_6244_word__smod__numerals_I2_J,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [Y: num] :
( ( signed6721504322012087516modulo @ ( word @ A ) @ ( zero_zero @ ( word @ A ) ) @ ( numeral_numeral @ ( word @ A ) @ Y ) )
= ( ring_1_of_int @ ( word @ A ) @ ( signed6721504322012087516modulo @ int @ ( ring_1_signed @ A @ int @ ( zero_zero @ ( word @ A ) ) ) @ ( ring_1_signed @ A @ int @ ( numeral_numeral @ ( word @ A ) @ Y ) ) ) ) ) ) ).
% word_smod_numerals(2)
thf(fact_6245_word__smod__numerals_I4_J,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [X: num] :
( ( signed6721504322012087516modulo @ ( word @ A ) @ ( numeral_numeral @ ( word @ A ) @ X ) @ ( zero_zero @ ( 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 @ ( zero_zero @ ( word @ A ) ) ) ) ) ) ) ).
% word_smod_numerals(4)
thf(fact_6246_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_6247_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_6248_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_6249_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_6250_slice1__def,axiom,
! [A: $tType,B: $tType] :
( ( ( type_len @ B )
& ( type_len @ A ) )
=> ( ( slice1 @ A @ B )
= ( ^ [N3: nat,W2: word @ A] : ( if @ ( word @ B ) @ ( ord_less @ nat @ N3 @ ( 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 ) ) @ N3 ) @ W2 ) ) @ ( bit_se4730199178511100633sh_bit @ ( word @ B ) @ ( minus_minus @ nat @ N3 @ ( type_len0_len_of @ A @ ( type2 @ A ) ) ) @ ( semiring_1_unsigned @ A @ ( word @ B ) @ W2 ) ) ) ) ) ) ).
% slice1_def
thf(fact_6251_slice1__0,axiom,
! [B: $tType,A: $tType] :
( ( ( type_len @ A )
& ( type_len @ B ) )
=> ! [N: nat] :
( ( slice1 @ B @ A @ N @ ( zero_zero @ ( word @ B ) ) )
= ( zero_zero @ ( word @ A ) ) ) ) ).
% slice1_0
thf(fact_6252_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_6253_succ__pred__no_I4_J,axiom,
! [D4: $tType] :
( ( type_len @ D4 )
=> ! [W: num] :
( ( word_pred @ D4 @ ( uminus_uminus @ ( word @ D4 ) @ ( numeral_numeral @ ( word @ D4 ) @ W ) ) )
= ( minus_minus @ ( word @ D4 ) @ ( uminus_uminus @ ( word @ D4 ) @ ( numeral_numeral @ ( word @ D4 ) @ W ) ) @ ( one_one @ ( word @ D4 ) ) ) ) ) ).
% succ_pred_no(4)
thf(fact_6254_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_6255_word__not__simps_I2_J,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [Y: word @ A] :
~ ( ord_less @ ( word @ A ) @ ( word_pred @ A @ ( zero_zero @ ( word @ A ) ) ) @ Y ) ) ).
% word_not_simps(2)
thf(fact_6256_word__pred__m1,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ( ( word_pred @ A )
= ( ^ [A4: word @ A] : ( minus_minus @ ( word @ A ) @ A4 @ ( one_one @ ( word @ A ) ) ) ) ) ) ).
% word_pred_m1
thf(fact_6257_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_6258_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_6259_word__pred__alt,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ( ( word_pred @ A )
= ( ^ [A4: word @ A] : ( ring_1_of_int @ ( word @ A ) @ ( minus_minus @ int @ ( semiring_1_unsigned @ A @ int @ A4 ) @ ( one_one @ int ) ) ) ) ) ) ).
% word_pred_alt
thf(fact_6260_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_6261_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_6262_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_6263_of__nat__less__two__pow__div__set,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [N: nat,M: nat] :
( ( ord_less @ nat @ N @ ( type_len0_len_of @ A @ ( type2 @ A ) ) )
=> ( ( collect @ ( word @ A )
@ ^ [X2: word @ A] : ( ord_less @ ( word @ A ) @ X2 @ ( divide_divide @ ( word @ A ) @ ( power_power @ ( word @ A ) @ ( numeral_numeral @ ( word @ A ) @ ( bit0 @ one2 ) ) @ N ) @ ( power_power @ ( word @ A ) @ ( numeral_numeral @ ( word @ A ) @ ( bit0 @ one2 ) ) @ M ) ) ) )
= ( image @ nat @ ( word @ A ) @ ( semiring_1_of_nat @ ( word @ A ) )
@ ( collect @ nat
@ ^ [K3: nat] : ( ord_less @ nat @ K3 @ ( divide_divide @ nat @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ M ) ) ) ) ) ) ) ) ).
% of_nat_less_two_pow_div_set
thf(fact_6264_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_6265_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_6266_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_6267_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_6268_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_6269_image__add__atLeastAtMost_H,axiom,
! [A: $tType] :
( ( linordered_semidom @ A )
=> ! [K: A,I: A,J: A] :
( ( image @ A @ A
@ ^ [N3: A] : ( plus_plus @ A @ N3 @ K )
@ ( set_or1337092689740270186AtMost @ A @ I @ J ) )
= ( set_or1337092689740270186AtMost @ A @ ( plus_plus @ A @ I @ K ) @ ( plus_plus @ A @ J @ K ) ) ) ) ).
% image_add_atLeastAtMost'
thf(fact_6270_image__add__atLeastLessThan_H,axiom,
! [A: $tType] :
( ( linordered_semidom @ A )
=> ! [K: A,I: A,J: A] :
( ( image @ A @ A
@ ^ [N3: A] : ( plus_plus @ A @ N3 @ K )
@ ( set_or7035219750837199246ssThan @ A @ I @ J ) )
= ( set_or7035219750837199246ssThan @ A @ ( plus_plus @ A @ I @ K ) @ ( plus_plus @ A @ J @ K ) ) ) ) ).
% image_add_atLeastLessThan'
thf(fact_6271_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_6272_image__mult__atLeastAtMost,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [D2: A,A2: A,B2: A] :
( ( ord_less @ A @ ( zero_zero @ A ) @ D2 )
=> ( ( image @ A @ A @ ( times_times @ A @ D2 ) @ ( set_or1337092689740270186AtMost @ A @ A2 @ B2 ) )
= ( set_or1337092689740270186AtMost @ A @ ( times_times @ A @ D2 @ A2 ) @ ( times_times @ A @ D2 @ B2 ) ) ) ) ) ).
% image_mult_atLeastAtMost
thf(fact_6273_image__divide__atLeastAtMost,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [D2: A,A2: A,B2: A] :
( ( ord_less @ A @ ( zero_zero @ A ) @ D2 )
=> ( ( image @ A @ A
@ ^ [C5: A] : ( divide_divide @ A @ C5 @ D2 )
@ ( set_or1337092689740270186AtMost @ A @ A2 @ B2 ) )
= ( set_or1337092689740270186AtMost @ A @ ( divide_divide @ A @ A2 @ D2 ) @ ( divide_divide @ A @ B2 @ D2 ) ) ) ) ) ).
% image_divide_atLeastAtMost
thf(fact_6274_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_6275_nth__image__indices,axiom,
! [A: $tType,L: list @ A] :
( ( image @ nat @ A @ ( nth @ A @ L ) @ ( set_or7035219750837199246ssThan @ nat @ ( zero_zero @ nat ) @ ( size_size @ ( list @ A ) @ L ) ) )
= ( set2 @ A @ L ) ) ).
% nth_image_indices
thf(fact_6276_nat__seg__image__imp__finite,axiom,
! [A: $tType,A3: set @ A,F2: nat > A,N: nat] :
( ( A3
= ( image @ nat @ A @ F2
@ ( collect @ nat
@ ^ [I2: nat] : ( ord_less @ nat @ I2 @ N ) ) ) )
=> ( finite_finite2 @ A @ A3 ) ) ).
% nat_seg_image_imp_finite
thf(fact_6277_finite__conv__nat__seg__image,axiom,
! [A: $tType] :
( ( finite_finite2 @ A )
= ( ^ [A7: set @ A] :
? [N3: nat,F4: nat > A] :
( A7
= ( image @ nat @ A @ F4
@ ( collect @ nat
@ ^ [I2: nat] : ( ord_less @ nat @ I2 @ N3 ) ) ) ) ) ) ).
% finite_conv_nat_seg_image
thf(fact_6278_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_6279_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_6280_word__succ__p1,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ( ( word_succ @ A )
= ( ^ [A4: word @ A] : ( plus_plus @ ( word @ A ) @ A4 @ ( one_one @ ( word @ A ) ) ) ) ) ) ).
% word_succ_p1
thf(fact_6281_set__image__eq__pointwiseI,axiom,
! [B: $tType,A: $tType,L: list @ A,L3: list @ A,F2: A > B] :
( ( ( size_size @ ( list @ A ) @ L )
= ( size_size @ ( list @ A ) @ L3 ) )
=> ( ! [I3: nat] :
( ( ord_less @ nat @ I3 @ ( size_size @ ( list @ A ) @ L ) )
=> ( ( F2 @ ( nth @ A @ L @ I3 ) )
= ( F2 @ ( nth @ A @ L3 @ I3 ) ) ) )
=> ( ( image @ A @ B @ F2 @ ( set2 @ A @ L ) )
= ( image @ A @ B @ F2 @ ( set2 @ A @ L3 ) ) ) ) ) ).
% set_image_eq_pointwiseI
thf(fact_6282_in__set__image__conv__nth,axiom,
! [B: $tType,A: $tType,F2: B > A,X: B,L: list @ B] :
( ( member @ A @ ( F2 @ X ) @ ( image @ B @ A @ F2 @ ( set2 @ B @ L ) ) )
= ( ? [I2: nat] :
( ( ord_less @ nat @ I2 @ ( size_size @ ( list @ B ) @ L ) )
& ( ( F2 @ ( nth @ B @ L @ I2 ) )
= ( F2 @ X ) ) ) ) ) ).
% in_set_image_conv_nth
thf(fact_6283_scaleR__image__atLeastAtMost,axiom,
! [A: $tType] :
( ( real_V5355595471888546746vector @ A )
=> ! [C2: real,X: A,Y: A] :
( ( ord_less @ real @ ( zero_zero @ real ) @ C2 )
=> ( ( image @ A @ A @ ( real_V8093663219630862766scaleR @ A @ C2 ) @ ( set_or1337092689740270186AtMost @ A @ X @ Y ) )
= ( set_or1337092689740270186AtMost @ A @ ( real_V8093663219630862766scaleR @ A @ C2 @ X ) @ ( real_V8093663219630862766scaleR @ A @ C2 @ Y ) ) ) ) ) ).
% scaleR_image_atLeastAtMost
thf(fact_6284_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_6285_word__arith__nat__Suc,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ( ( word_succ @ A )
= ( ^ [A4: word @ A] : ( semiring_1_of_nat @ ( word @ A ) @ ( suc @ ( semiring_1_unsigned @ A @ nat @ A4 ) ) ) ) ) ) ).
% word_arith_nat_Suc
thf(fact_6286_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_6287_word__succ__alt,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ( ( word_succ @ A )
= ( ^ [A4: word @ A] : ( ring_1_of_int @ ( word @ A ) @ ( plus_plus @ int @ ( semiring_1_unsigned @ A @ int @ A4 ) @ ( one_one @ int ) ) ) ) ) ) ).
% word_succ_alt
thf(fact_6288_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_6289_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_6290_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_6291_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_6292_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_6293_word__sp__01,axiom,
! [C: $tType,A: $tType,B: $tType,D4: $tType] :
( ( ( type_len @ D4 )
& ( 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 @ D4 @ ( one_one @ ( word @ D4 ) ) )
= ( zero_zero @ ( word @ D4 ) ) ) ) ) ).
% word_sp_01
thf(fact_6294_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_6295_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_6296_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_6297_bij__betw__add,axiom,
! [A: $tType] :
( ( cancel_semigroup_add @ A )
=> ! [A2: A,A3: set @ A,B4: set @ A] :
( ( bij_betw @ A @ A @ ( plus_plus @ A @ A2 ) @ A3 @ B4 )
= ( ( image @ A @ A @ ( plus_plus @ A @ A2 ) @ A3 )
= B4 ) ) ) ).
% bij_betw_add
thf(fact_6298_bit__word__roti__iff,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [K: int,W: word @ A,N: nat] :
( ( bit_se5641148757651400278ts_bit @ ( word @ A ) @ ( word_roti @ A @ K @ W ) @ N )
= ( ( ord_less @ nat @ N @ ( type_len0_len_of @ A @ ( type2 @ A ) ) )
& ( bit_se5641148757651400278ts_bit @ ( word @ A ) @ W @ ( nat2 @ ( modulo_modulo @ int @ ( plus_plus @ int @ ( semiring_1_of_nat @ int @ N ) @ K ) @ ( semiring_1_of_nat @ int @ ( type_len0_len_of @ A @ ( type2 @ A ) ) ) ) ) ) ) ) ) ).
% bit_word_roti_iff
thf(fact_6299_bij__betw__Suc,axiom,
! [M7: set @ nat,N4: set @ nat] :
( ( bij_betw @ nat @ nat @ suc @ M7 @ N4 )
= ( ( image @ nat @ nat @ suc @ M7 )
= N4 ) ) ).
% bij_betw_Suc
thf(fact_6300_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_6301_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_6302_word__roti__0,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [W: word @ A] :
( ( word_roti @ A @ ( zero_zero @ int ) @ W )
= W ) ) ).
% word_roti_0
thf(fact_6303_word__roti__0_H,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [N: int] :
( ( word_roti @ A @ N @ ( zero_zero @ ( word @ A ) ) )
= ( zero_zero @ ( word @ A ) ) ) ) ).
% word_roti_0'
thf(fact_6304_zero__notin__Suc__image,axiom,
! [A3: set @ nat] :
~ ( member @ nat @ ( zero_zero @ nat ) @ ( image @ nat @ nat @ suc @ A3 ) ) ).
% zero_notin_Suc_image
thf(fact_6305_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_6306_image__Suc__lessThan,axiom,
! [N: nat] :
( ( image @ nat @ nat @ suc @ ( set_ord_lessThan @ nat @ N ) )
= ( set_or1337092689740270186AtMost @ nat @ ( one_one @ nat ) @ N ) ) ).
% image_Suc_lessThan
thf(fact_6307_image__Suc__atMost,axiom,
! [N: nat] :
( ( image @ nat @ nat @ suc @ ( set_ord_atMost @ nat @ N ) )
= ( set_or1337092689740270186AtMost @ nat @ ( one_one @ nat ) @ ( suc @ N ) ) ) ).
% image_Suc_atMost
thf(fact_6308_atLeast0__lessThan__Suc__eq__insert__0,axiom,
! [N: nat] :
( ( set_or7035219750837199246ssThan @ nat @ ( zero_zero @ nat ) @ ( suc @ N ) )
= ( insert @ nat @ ( zero_zero @ nat ) @ ( image @ nat @ nat @ suc @ ( set_or7035219750837199246ssThan @ nat @ ( zero_zero @ nat ) @ N ) ) ) ) ).
% atLeast0_lessThan_Suc_eq_insert_0
thf(fact_6309_atLeast0__atMost__Suc__eq__insert__0,axiom,
! [N: nat] :
( ( set_or1337092689740270186AtMost @ nat @ ( zero_zero @ nat ) @ ( suc @ N ) )
= ( insert @ nat @ ( zero_zero @ nat ) @ ( image @ nat @ nat @ suc @ ( set_or1337092689740270186AtMost @ nat @ ( zero_zero @ nat ) @ N ) ) ) ) ).
% atLeast0_atMost_Suc_eq_insert_0
thf(fact_6310_lessThan__Suc__eq__insert__0,axiom,
! [N: nat] :
( ( set_ord_lessThan @ nat @ ( suc @ N ) )
= ( insert @ nat @ ( zero_zero @ nat ) @ ( image @ nat @ nat @ suc @ ( set_ord_lessThan @ nat @ N ) ) ) ) ).
% lessThan_Suc_eq_insert_0
thf(fact_6311_atMost__Suc__eq__insert__0,axiom,
! [N: nat] :
( ( set_ord_atMost @ nat @ ( suc @ N ) )
= ( insert @ nat @ ( zero_zero @ nat ) @ ( image @ nat @ nat @ suc @ ( set_ord_atMost @ nat @ N ) ) ) ) ).
% atMost_Suc_eq_insert_0
thf(fact_6312_subset__subseqs,axiom,
! [A: $tType,X7: set @ A,Xs2: list @ A] :
( ( ord_less_eq @ ( set @ A ) @ X7 @ ( set2 @ A @ Xs2 ) )
=> ( member @ ( set @ A ) @ X7 @ ( image @ ( list @ A ) @ ( set @ A ) @ ( set2 @ A ) @ ( set2 @ ( list @ A ) @ ( subseqs @ A @ Xs2 ) ) ) ) ) ).
% subset_subseqs
thf(fact_6313_image__add__int__atLeastLessThan,axiom,
! [L: int,U: int] :
( ( image @ int @ int
@ ^ [X2: int] : ( plus_plus @ int @ X2 @ L )
@ ( set_or7035219750837199246ssThan @ int @ ( zero_zero @ int ) @ ( minus_minus @ int @ U @ L ) ) )
= ( set_or7035219750837199246ssThan @ int @ L @ U ) ) ).
% image_add_int_atLeastLessThan
thf(fact_6314_word__roti__conv__mod_H,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ( ( word_roti @ A )
= ( ^ [N3: int,W2: word @ A] : ( word_roti @ A @ ( modulo_modulo @ int @ N3 @ ( semiring_1_of_nat @ int @ ( size_size @ ( word @ A ) @ W2 ) ) ) @ W2 ) ) ) ) ).
% word_roti_conv_mod'
thf(fact_6315_image__add__integer__atLeastLessThan,axiom,
! [L: code_integer,U: code_integer] :
( ( image @ code_integer @ code_integer
@ ^ [X2: code_integer] : ( plus_plus @ code_integer @ X2 @ L )
@ ( set_or7035219750837199246ssThan @ code_integer @ ( zero_zero @ code_integer ) @ ( minus_minus @ code_integer @ U @ L ) ) )
= ( set_or7035219750837199246ssThan @ code_integer @ L @ U ) ) ).
% image_add_integer_atLeastLessThan
thf(fact_6316_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_6317_image__minus__const__atLeastLessThan__nat,axiom,
! [C2: nat,Y: nat,X: nat] :
( ( ( ord_less @ nat @ C2 @ Y )
=> ( ( image @ nat @ nat
@ ^ [I2: nat] : ( minus_minus @ nat @ I2 @ 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
@ ^ [I2: nat] : ( minus_minus @ nat @ I2 @ C2 )
@ ( set_or7035219750837199246ssThan @ nat @ X @ Y ) )
= ( insert @ nat @ ( zero_zero @ nat ) @ ( bot_bot @ ( set @ nat ) ) ) ) )
& ( ~ ( ord_less @ nat @ X @ Y )
=> ( ( image @ nat @ nat
@ ^ [I2: nat] : ( minus_minus @ nat @ I2 @ C2 )
@ ( set_or7035219750837199246ssThan @ nat @ X @ Y ) )
= ( bot_bot @ ( set @ nat ) ) ) ) ) ) ) ).
% image_minus_const_atLeastLessThan_nat
thf(fact_6318_word__roti__conv__mod,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ( ( word_roti @ A )
= ( ^ [N3: int] : ( word_roti @ A @ ( modulo_modulo @ int @ N3 @ ( semiring_1_of_nat @ int @ ( type_len0_len_of @ A @ ( type2 @ A ) ) ) ) ) ) ) ) ).
% word_roti_conv_mod
thf(fact_6319_translation__diff,axiom,
! [A: $tType] :
( ( ab_group_add @ A )
=> ! [A2: A,S: set @ A,T: set @ A] :
( ( image @ A @ A @ ( plus_plus @ A @ A2 ) @ ( minus_minus @ ( set @ A ) @ S @ T ) )
= ( minus_minus @ ( set @ A ) @ ( image @ A @ A @ ( plus_plus @ A @ A2 ) @ S ) @ ( image @ A @ A @ ( plus_plus @ A @ A2 ) @ T ) ) ) ) ).
% translation_diff
thf(fact_6320_translation__Compl,axiom,
! [A: $tType] :
( ( ab_group_add @ A )
=> ! [A2: A,T: set @ A] :
( ( image @ A @ A @ ( plus_plus @ A @ A2 ) @ ( uminus_uminus @ ( set @ A ) @ T ) )
= ( uminus_uminus @ ( set @ A ) @ ( image @ A @ A @ ( plus_plus @ A @ A2 ) @ T ) ) ) ) ).
% translation_Compl
thf(fact_6321_setceilmax,axiom,
! [S: vEBT_VEBT,M: nat,Listy: list @ vEBT_VEBT,N: nat] :
( ( vEBT_invar_vebt @ S @ M )
=> ( ! [X3: vEBT_VEBT] :
( ( member @ vEBT_VEBT @ X3 @ ( set2 @ vEBT_VEBT @ Listy ) )
=> ( vEBT_invar_vebt @ X3 @ N ) )
=> ( ( M
= ( suc @ N ) )
=> ( ! [X3: vEBT_VEBT] :
( ( member @ vEBT_VEBT @ X3 @ ( set2 @ vEBT_VEBT @ Listy ) )
=> ( ( semiring_1_of_nat @ int @ ( vEBT_VEBT_height @ X3 ) )
= ( archimedean_ceiling @ real @ ( log @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) @ ( semiring_1_of_nat @ real @ N ) ) ) ) )
=> ( ( ( semiring_1_of_nat @ int @ ( vEBT_VEBT_height @ S ) )
= ( archimedean_ceiling @ real @ ( log @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) @ ( semiring_1_of_nat @ real @ M ) ) ) )
=> ( ( semiring_1_of_nat @ int @ ( lattic643756798349783984er_Max @ nat @ ( image @ vEBT_VEBT @ nat @ vEBT_VEBT_height @ ( insert @ vEBT_VEBT @ S @ ( set2 @ vEBT_VEBT @ Listy ) ) ) ) )
= ( archimedean_ceiling @ real @ ( log @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) @ ( semiring_1_of_nat @ real @ M ) ) ) ) ) ) ) ) ) ).
% setceilmax
thf(fact_6322_max__idx__list,axiom,
! [I: nat,X13: list @ vEBT_VEBT,N: nat,X14: vEBT_VEBT] :
( ( ord_less @ nat @ I @ ( size_size @ ( list @ vEBT_VEBT ) @ X13 ) )
=> ( ord_less_eq @ nat @ ( times_times @ nat @ N @ ( vEBT_VEBT_height @ ( nth @ vEBT_VEBT @ X13 @ I ) ) ) @ ( suc @ ( suc @ ( times_times @ nat @ N @ ( ord_max @ nat @ ( vEBT_VEBT_height @ X14 ) @ ( lattic643756798349783984er_Max @ nat @ ( image @ vEBT_VEBT @ nat @ vEBT_VEBT_height @ ( set2 @ vEBT_VEBT @ X13 ) ) ) ) ) ) ) ) ) ).
% max_idx_list
thf(fact_6323_height__compose__list,axiom,
! [T: vEBT_VEBT,TreeList: list @ vEBT_VEBT,Summary: vEBT_VEBT] :
( ( member @ vEBT_VEBT @ T @ ( set2 @ vEBT_VEBT @ TreeList ) )
=> ( ord_less_eq @ nat @ ( vEBT_VEBT_height @ T ) @ ( lattic643756798349783984er_Max @ nat @ ( image @ vEBT_VEBT @ nat @ vEBT_VEBT_height @ ( insert @ vEBT_VEBT @ Summary @ ( set2 @ vEBT_VEBT @ TreeList ) ) ) ) ) ) ).
% height_compose_list
thf(fact_6324_max__ins__scaled,axiom,
! [N: nat,X14: vEBT_VEBT,M: nat,X13: list @ vEBT_VEBT] : ( ord_less_eq @ nat @ ( times_times @ nat @ N @ ( vEBT_VEBT_height @ X14 ) ) @ ( plus_plus @ nat @ M @ ( times_times @ nat @ N @ ( lattic643756798349783984er_Max @ nat @ ( insert @ nat @ ( vEBT_VEBT_height @ X14 ) @ ( image @ vEBT_VEBT @ nat @ vEBT_VEBT_height @ ( set2 @ vEBT_VEBT @ X13 ) ) ) ) ) ) ) ).
% max_ins_scaled
thf(fact_6325_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_6326_Max__divisors__self__nat,axiom,
! [N: nat] :
( ( N
!= ( zero_zero @ nat ) )
=> ( ( lattic643756798349783984er_Max @ nat
@ ( collect @ nat
@ ^ [D: nat] : ( dvd_dvd @ nat @ D @ N ) ) )
= N ) ) ).
% Max_divisors_self_nat
thf(fact_6327_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_6328_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_6329_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_6330_divide__nat__def,axiom,
( ( divide_divide @ nat )
= ( ^ [M4: nat,N3: nat] :
( if @ nat
@ ( N3
= ( zero_zero @ nat ) )
@ ( zero_zero @ nat )
@ ( lattic643756798349783984er_Max @ nat
@ ( collect @ nat
@ ^ [K3: nat] : ( ord_less_eq @ nat @ ( times_times @ nat @ K3 @ N3 ) @ M4 ) ) ) ) ) ) ).
% divide_nat_def
thf(fact_6331_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_6332_VEBT__internal_Oheight_Osimps_I2_J,axiom,
! [Uu: option @ ( product_prod @ nat @ nat ),Deg: nat,TreeList: list @ vEBT_VEBT,Summary: vEBT_VEBT] :
( ( vEBT_VEBT_height @ ( vEBT_Node @ Uu @ Deg @ TreeList @ Summary ) )
= ( plus_plus @ nat @ ( one_one @ nat ) @ ( lattic643756798349783984er_Max @ nat @ ( image @ vEBT_VEBT @ nat @ vEBT_VEBT_height @ ( insert @ vEBT_VEBT @ Summary @ ( set2 @ vEBT_VEBT @ TreeList ) ) ) ) ) ) ).
% VEBT_internal.height.simps(2)
thf(fact_6333_VEBT__internal_Oheight_Oelims,axiom,
! [X: vEBT_VEBT,Y: nat] :
( ( ( vEBT_VEBT_height @ X )
= Y )
=> ( ( ? [A6: $o,B3: $o] :
( X
= ( vEBT_Leaf @ A6 @ B3 ) )
=> ( Y
!= ( zero_zero @ nat ) ) )
=> ~ ! [Uu2: option @ ( product_prod @ nat @ nat ),Deg2: nat,TreeList2: list @ vEBT_VEBT,Summary2: vEBT_VEBT] :
( ( X
= ( vEBT_Node @ Uu2 @ Deg2 @ TreeList2 @ Summary2 ) )
=> ( Y
!= ( plus_plus @ nat @ ( one_one @ nat ) @ ( lattic643756798349783984er_Max @ nat @ ( image @ vEBT_VEBT @ nat @ vEBT_VEBT_height @ ( insert @ vEBT_VEBT @ Summary2 @ ( set2 @ vEBT_VEBT @ TreeList2 ) ) ) ) ) ) ) ) ) ).
% VEBT_internal.height.elims
thf(fact_6334_VEBT__internal_Oheight_Opelims,axiom,
! [X: vEBT_VEBT,Y: nat] :
( ( ( vEBT_VEBT_height @ X )
= Y )
=> ( ( accp @ vEBT_VEBT @ vEBT_VEBT_height_rel @ X )
=> ( ! [A6: $o,B3: $o] :
( ( X
= ( vEBT_Leaf @ A6 @ B3 ) )
=> ( ( Y
= ( zero_zero @ nat ) )
=> ~ ( accp @ vEBT_VEBT @ vEBT_VEBT_height_rel @ ( vEBT_Leaf @ A6 @ B3 ) ) ) )
=> ~ ! [Uu2: option @ ( product_prod @ nat @ nat ),Deg2: nat,TreeList2: list @ vEBT_VEBT,Summary2: vEBT_VEBT] :
( ( X
= ( vEBT_Node @ Uu2 @ Deg2 @ TreeList2 @ Summary2 ) )
=> ( ( Y
= ( plus_plus @ nat @ ( one_one @ nat ) @ ( lattic643756798349783984er_Max @ nat @ ( image @ vEBT_VEBT @ nat @ vEBT_VEBT_height @ ( insert @ vEBT_VEBT @ Summary2 @ ( set2 @ vEBT_VEBT @ TreeList2 ) ) ) ) ) )
=> ~ ( accp @ vEBT_VEBT @ vEBT_VEBT_height_rel @ ( vEBT_Node @ Uu2 @ Deg2 @ TreeList2 @ Summary2 ) ) ) ) ) ) ) ).
% VEBT_internal.height.pelims
thf(fact_6335_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_6336_Max__divisors__self__int,axiom,
! [N: int] :
( ( N
!= ( zero_zero @ int ) )
=> ( ( lattic643756798349783984er_Max @ int
@ ( collect @ int
@ ^ [D: int] : ( dvd_dvd @ int @ D @ N ) ) )
= ( abs_abs @ int @ N ) ) ) ).
% Max_divisors_self_int
thf(fact_6337_extra__sle__sless__unfolds_I4_J,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [N: num] :
( ( word_sle @ A @ ( one_one @ ( word @ A ) ) @ ( numeral_numeral @ ( word @ A ) @ N ) )
= ( ord_less_eq @ int @ ( ring_1_signed @ A @ int @ ( one_one @ ( word @ A ) ) ) @ ( ring_1_signed @ A @ int @ ( numeral_numeral @ ( word @ A ) @ N ) ) ) ) ) ).
% extra_sle_sless_unfolds(4)
thf(fact_6338_extra__sle__sless__unfolds_I6_J,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [N: num] :
( ( word_sle @ A @ ( numeral_numeral @ ( word @ A ) @ N ) @ ( one_one @ ( word @ A ) ) )
= ( ord_less_eq @ int @ ( ring_1_signed @ A @ int @ ( numeral_numeral @ ( word @ A ) @ N ) ) @ ( ring_1_signed @ A @ int @ ( one_one @ ( word @ A ) ) ) ) ) ) ).
% extra_sle_sless_unfolds(6)
thf(fact_6339_extra__sle__sless__unfolds_I5_J,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [N: num] :
( ( word_sle @ A @ ( numeral_numeral @ ( word @ A ) @ N ) @ ( zero_zero @ ( word @ A ) ) )
= ( ord_less_eq @ int @ ( ring_1_signed @ A @ int @ ( numeral_numeral @ ( word @ A ) @ N ) ) @ ( ring_1_signed @ A @ int @ ( zero_zero @ ( word @ A ) ) ) ) ) ) ).
% extra_sle_sless_unfolds(5)
thf(fact_6340_extra__sle__sless__unfolds_I2_J,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [N: num] :
( ( word_sle @ A @ ( zero_zero @ ( word @ A ) ) @ ( numeral_numeral @ ( word @ A ) @ N ) )
= ( ord_less_eq @ int @ ( ring_1_signed @ A @ int @ ( zero_zero @ ( word @ A ) ) ) @ ( ring_1_signed @ A @ int @ ( numeral_numeral @ ( word @ A ) @ N ) ) ) ) ) ).
% extra_sle_sless_unfolds(2)
thf(fact_6341_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_6342_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_6343_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_6344_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_6345_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_6346_signed_Olift__Suc__mono__le,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [F2: nat > ( word @ A ),N: nat,N5: nat] :
( ! [N2: nat] : ( word_sle @ A @ ( F2 @ N2 ) @ ( F2 @ ( suc @ N2 ) ) )
=> ( ( ord_less_eq @ nat @ N @ N5 )
=> ( word_sle @ A @ ( F2 @ N ) @ ( F2 @ N5 ) ) ) ) ) ).
% signed.lift_Suc_mono_le
thf(fact_6347_signed_Olift__Suc__antimono__le,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [F2: nat > ( word @ A ),N: nat,N5: nat] :
( ! [N2: nat] : ( word_sle @ A @ ( F2 @ ( suc @ N2 ) ) @ ( F2 @ N2 ) )
=> ( ( ord_less_eq @ nat @ N @ N5 )
=> ( word_sle @ A @ ( F2 @ N5 ) @ ( F2 @ N ) ) ) ) ) ).
% signed.lift_Suc_antimono_le
thf(fact_6348_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_6349_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_6350_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_6351_extra__sle__sless__unfolds_I12_J,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [N: num] :
( ( word_sless @ A @ ( numeral_numeral @ ( word @ A ) @ N ) @ ( one_one @ ( word @ A ) ) )
= ( ord_less @ int @ ( ring_1_signed @ A @ int @ ( numeral_numeral @ ( word @ A ) @ N ) ) @ ( ring_1_signed @ A @ int @ ( one_one @ ( word @ A ) ) ) ) ) ) ).
% extra_sle_sless_unfolds(12)
thf(fact_6352_extra__sle__sless__unfolds_I10_J,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [N: num] :
( ( word_sless @ A @ ( one_one @ ( word @ A ) ) @ ( numeral_numeral @ ( word @ A ) @ N ) )
= ( ord_less @ int @ ( ring_1_signed @ A @ int @ ( one_one @ ( word @ A ) ) ) @ ( ring_1_signed @ A @ int @ ( numeral_numeral @ ( word @ A ) @ N ) ) ) ) ) ).
% extra_sle_sless_unfolds(10)
thf(fact_6353_extra__sle__sless__unfolds_I11_J,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [N: num] :
( ( word_sless @ A @ ( numeral_numeral @ ( word @ A ) @ N ) @ ( zero_zero @ ( word @ A ) ) )
= ( ord_less @ int @ ( ring_1_signed @ A @ int @ ( numeral_numeral @ ( word @ A ) @ N ) ) @ ( ring_1_signed @ A @ int @ ( zero_zero @ ( word @ A ) ) ) ) ) ) ).
% extra_sle_sless_unfolds(11)
thf(fact_6354_extra__sle__sless__unfolds_I8_J,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [N: num] :
( ( word_sless @ A @ ( zero_zero @ ( word @ A ) ) @ ( numeral_numeral @ ( word @ A ) @ N ) )
= ( ord_less @ int @ ( ring_1_signed @ A @ int @ ( zero_zero @ ( word @ A ) ) ) @ ( ring_1_signed @ A @ int @ ( numeral_numeral @ ( word @ A ) @ N ) ) ) ) ) ).
% extra_sle_sless_unfolds(8)
thf(fact_6355_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_6356_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_6357_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_6358_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_6359_signed_Olift__Suc__mono__less,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [F2: nat > ( word @ A ),N: nat,N5: nat] :
( ! [N2: nat] : ( word_sless @ A @ ( F2 @ N2 ) @ ( F2 @ ( suc @ N2 ) ) )
=> ( ( ord_less @ nat @ N @ N5 )
=> ( word_sless @ A @ ( F2 @ N ) @ ( F2 @ N5 ) ) ) ) ) ).
% signed.lift_Suc_mono_less
thf(fact_6360_signed_Olift__Suc__mono__less__iff,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [F2: nat > ( word @ A ),N: nat,M: nat] :
( ! [N2: nat] : ( word_sless @ A @ ( F2 @ N2 ) @ ( F2 @ ( suc @ N2 ) ) )
=> ( ( word_sless @ A @ ( F2 @ N ) @ ( F2 @ M ) )
= ( ord_less @ nat @ N @ M ) ) ) ) ).
% signed.lift_Suc_mono_less_iff
thf(fact_6361_word__sless__alt,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ( ( word_sless @ A )
= ( ^ [A4: word @ A,B5: word @ A] : ( ord_less @ int @ ( ring_1_signed @ A @ int @ A4 ) @ ( ring_1_signed @ A @ int @ B5 ) ) ) ) ) ).
% word_sless_alt
thf(fact_6362_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_6363_take__bit__word__minus__numeral,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [N: num,K: num] :
( ( bit_se2584673776208193580ke_bit @ ( word @ A ) @ ( numeral_numeral @ nat @ N ) @ ( uminus_uminus @ ( word @ A ) @ ( numeral_numeral @ ( word @ A ) @ K ) ) )
= ( ring_1_of_int @ ( word @ A ) @ ( bit_se2584673776208193580ke_bit @ int @ ( ord_min @ nat @ ( type_len0_len_of @ A @ ( type2 @ A ) ) @ ( numeral_numeral @ nat @ N ) ) @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ K ) ) ) ) ) ) ).
% take_bit_word_minus_numeral
thf(fact_6364_take__bit__word__Suc__minus__numeral,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [N: nat,K: num] :
( ( bit_se2584673776208193580ke_bit @ ( word @ A ) @ ( suc @ N ) @ ( uminus_uminus @ ( word @ A ) @ ( numeral_numeral @ ( word @ A ) @ K ) ) )
= ( ring_1_of_int @ ( word @ A ) @ ( bit_se2584673776208193580ke_bit @ int @ ( ord_min @ nat @ ( type_len0_len_of @ A @ ( type2 @ A ) ) @ ( suc @ N ) ) @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ K ) ) ) ) ) ) ).
% take_bit_word_Suc_minus_numeral
thf(fact_6365_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_6366_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_6367_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_6368_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_6369_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_6370_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_6371_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_6372_min__arg__not__ge_I2_J,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [M: A,N: A] :
( ( ~ ( ord_less @ A @ ( ord_min @ A @ M @ N ) @ N ) )
= ( ( ord_min @ A @ M @ N )
= N ) ) ) ).
% min_arg_not_ge(2)
thf(fact_6373_min__arg__not__ge_I1_J,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [M: A,N: A] :
( ( ~ ( ord_less @ A @ ( ord_min @ A @ M @ N ) @ M ) )
= ( ( ord_min @ A @ M @ N )
= M ) ) ) ).
% min_arg_not_ge(1)
thf(fact_6374_min__Suc__Suc,axiom,
! [M: nat,N: nat] :
( ( ord_min @ nat @ ( suc @ M ) @ ( suc @ N ) )
= ( suc @ ( ord_min @ nat @ M @ N ) ) ) ).
% min_Suc_Suc
thf(fact_6375_min__0R,axiom,
! [N: nat] :
( ( ord_min @ nat @ N @ ( zero_zero @ nat ) )
= ( zero_zero @ nat ) ) ).
% min_0R
thf(fact_6376_min__0L,axiom,
! [N: nat] :
( ( ord_min @ nat @ ( zero_zero @ nat ) @ N )
= ( zero_zero @ nat ) ) ).
% min_0L
thf(fact_6377_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_6378_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_6379_take__bit__take__bit,axiom,
! [A: $tType] :
( ( bit_se359711467146920520ations @ A )
=> ! [M: nat,N: nat,A2: A] :
( ( bit_se2584673776208193580ke_bit @ A @ M @ ( bit_se2584673776208193580ke_bit @ A @ N @ A2 ) )
= ( bit_se2584673776208193580ke_bit @ A @ ( ord_min @ nat @ M @ N ) @ A2 ) ) ) ).
% take_bit_take_bit
thf(fact_6380_signed__take__bit__signed__take__bit,axiom,
! [A: $tType] :
( ( bit_ri3973907225187159222ations @ A )
=> ! [M: nat,N: nat,A2: A] :
( ( bit_ri4674362597316999326ke_bit @ A @ M @ ( bit_ri4674362597316999326ke_bit @ A @ N @ A2 ) )
= ( bit_ri4674362597316999326ke_bit @ A @ ( ord_min @ nat @ M @ N ) @ A2 ) ) ) ).
% signed_take_bit_signed_take_bit
thf(fact_6381_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_6382_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_6383_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_6384_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_6385_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_6386_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_6387_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_6388_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_6389_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_6390_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_6391_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_6392_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_6393_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_6394_take__bit__of__mask,axiom,
! [A: $tType] :
( ( bit_se359711467146920520ations @ A )
=> ! [M: nat,N: nat] :
( ( bit_se2584673776208193580ke_bit @ A @ M @ ( bit_se2239418461657761734s_mask @ A @ N ) )
= ( bit_se2239418461657761734s_mask @ A @ ( ord_min @ nat @ M @ N ) ) ) ) ).
% take_bit_of_mask
thf(fact_6395_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_6396_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_6397_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_6398_min__numeral__Suc,axiom,
! [K: num,N: nat] :
( ( ord_min @ nat @ ( numeral_numeral @ nat @ K ) @ ( suc @ N ) )
= ( suc @ ( ord_min @ nat @ ( pred_numeral @ K ) @ N ) ) ) ).
% min_numeral_Suc
thf(fact_6399_min__Suc__numeral,axiom,
! [N: nat,K: num] :
( ( ord_min @ nat @ ( suc @ N ) @ ( numeral_numeral @ nat @ K ) )
= ( suc @ ( ord_min @ nat @ N @ ( pred_numeral @ K ) ) ) ) ).
% min_Suc_numeral
thf(fact_6400_take__bit__word__Suc__numeral,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [N: nat,K: num] :
( ( bit_se2584673776208193580ke_bit @ ( word @ A ) @ ( suc @ N ) @ ( numeral_numeral @ ( word @ A ) @ K ) )
= ( ring_1_of_int @ ( word @ A ) @ ( bit_se2584673776208193580ke_bit @ int @ ( ord_min @ nat @ ( type_len0_len_of @ A @ ( type2 @ A ) ) @ ( suc @ N ) ) @ ( numeral_numeral @ int @ K ) ) ) ) ) ).
% take_bit_word_Suc_numeral
thf(fact_6401_take__bit__word__numeral,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [N: num,K: num] :
( ( bit_se2584673776208193580ke_bit @ ( word @ A ) @ ( numeral_numeral @ nat @ N ) @ ( numeral_numeral @ ( word @ A ) @ K ) )
= ( ring_1_of_int @ ( word @ A ) @ ( bit_se2584673776208193580ke_bit @ int @ ( ord_min @ nat @ ( type_len0_len_of @ A @ ( type2 @ A ) ) @ ( numeral_numeral @ nat @ N ) ) @ ( numeral_numeral @ int @ K ) ) ) ) ) ).
% take_bit_word_numeral
thf(fact_6402_minus__max__eq__min,axiom,
! [A: $tType] :
( ( linord5086331880401160121up_add @ A )
=> ! [X: A,Y: A] :
( ( uminus_uminus @ A @ ( ord_max @ A @ X @ Y ) )
= ( ord_min @ A @ ( uminus_uminus @ A @ X ) @ ( uminus_uminus @ A @ Y ) ) ) ) ).
% minus_max_eq_min
thf(fact_6403_minus__min__eq__max,axiom,
! [A: $tType] :
( ( linord5086331880401160121up_add @ A )
=> ! [X: A,Y: A] :
( ( uminus_uminus @ A @ ( ord_min @ A @ X @ Y ) )
= ( ord_max @ A @ ( uminus_uminus @ A @ X ) @ ( uminus_uminus @ A @ Y ) ) ) ) ).
% minus_min_eq_max
thf(fact_6404_concat__bit__assoc__sym,axiom,
! [M: nat,N: nat,K: int,L: int,R2: int] :
( ( bit_concat_bit @ M @ ( bit_concat_bit @ N @ K @ L ) @ R2 )
= ( bit_concat_bit @ ( ord_min @ nat @ M @ N ) @ K @ ( bit_concat_bit @ ( minus_minus @ nat @ M @ N ) @ L @ R2 ) ) ) ).
% concat_bit_assoc_sym
thf(fact_6405_nat__mult__min__right,axiom,
! [M: nat,N: nat,Q3: nat] :
( ( times_times @ nat @ M @ ( ord_min @ nat @ N @ Q3 ) )
= ( ord_min @ nat @ ( times_times @ nat @ M @ N ) @ ( times_times @ nat @ M @ Q3 ) ) ) ).
% nat_mult_min_right
thf(fact_6406_nat__mult__min__left,axiom,
! [M: nat,N: nat,Q3: nat] :
( ( times_times @ nat @ ( ord_min @ nat @ M @ N ) @ Q3 )
= ( ord_min @ nat @ ( times_times @ nat @ M @ Q3 ) @ ( times_times @ nat @ N @ Q3 ) ) ) ).
% nat_mult_min_left
thf(fact_6407_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_6408_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_6409_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_6410_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_6411_min_Ostrict__order__iff,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ( ( ord_less @ A )
= ( ^ [A4: A,B5: A] :
( ( A4
= ( ord_min @ A @ A4 @ B5 ) )
& ( A4 != B5 ) ) ) ) ) ).
% min.strict_order_iff
thf(fact_6412_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_6413_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_6414_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_6415_min__diff,axiom,
! [M: nat,I: nat,N: nat] :
( ( ord_min @ nat @ ( minus_minus @ nat @ M @ I ) @ ( minus_minus @ nat @ N @ I ) )
= ( minus_minus @ nat @ ( ord_min @ nat @ M @ N ) @ I ) ) ).
% min_diff
thf(fact_6416_min__def__raw,axiom,
! [A: $tType] :
( ( ord @ A )
=> ( ( ord_min @ A )
= ( ^ [A4: A,B5: A] : ( if @ A @ ( ord_less_eq @ A @ A4 @ B5 ) @ A4 @ B5 ) ) ) ) ).
% min_def_raw
thf(fact_6417_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_6418_take__bit__concat__bit__eq,axiom,
! [M: nat,N: nat,K: int,L: int] :
( ( bit_se2584673776208193580ke_bit @ int @ M @ ( bit_concat_bit @ N @ K @ L ) )
= ( bit_concat_bit @ ( ord_min @ nat @ M @ N ) @ K @ ( bit_se2584673776208193580ke_bit @ int @ ( minus_minus @ nat @ M @ N ) @ L ) ) ) ).
% take_bit_concat_bit_eq
thf(fact_6419_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_6420_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_6421_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_6422_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_6423_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_6424_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_6425_mod__exp__eq,axiom,
! [A: $tType] :
( ( bit_semiring_bits @ A )
=> ! [A2: A,M: nat,N: nat] :
( ( modulo_modulo @ A @ ( modulo_modulo @ A @ A2 @ ( power_power @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ M ) ) @ ( power_power @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ N ) )
= ( modulo_modulo @ A @ A2 @ ( power_power @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ ( ord_min @ nat @ M @ N ) ) ) ) ) ).
% mod_exp_eq
thf(fact_6426_mod__mod__power,axiom,
! [K: nat,M: nat,N: nat] :
( ( modulo_modulo @ nat @ ( modulo_modulo @ nat @ K @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ M ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N ) )
= ( modulo_modulo @ nat @ K @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( ord_min @ nat @ M @ N ) ) ) ) ).
% mod_mod_power
thf(fact_6427_Word_Obit__mask__iff,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [M: nat,N: nat] :
( ( bit_se5641148757651400278ts_bit @ ( word @ A ) @ ( bit_se2239418461657761734s_mask @ ( word @ A ) @ M ) @ N )
= ( ord_less @ nat @ N @ ( ord_min @ nat @ ( type_len0_len_of @ A @ ( type2 @ A ) ) @ M ) ) ) ) ).
% Word.bit_mask_iff
thf(fact_6428_mask__mod__exp,axiom,
! [A: $tType] :
( ( euclid5411537665997757685th_nat @ A )
=> ! [N: nat,M: nat] :
( ( modulo_modulo @ A @ ( minus_minus @ A @ ( power_power @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ N ) @ ( one_one @ A ) ) @ ( power_power @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ M ) )
= ( minus_minus @ A @ ( power_power @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ ( ord_min @ nat @ M @ N ) ) @ ( one_one @ A ) ) ) ) ).
% mask_mod_exp
thf(fact_6429_bit__slice__iff,axiom,
! [B: $tType,A: $tType] :
( ( ( type_len @ A )
& ( type_len @ B ) )
=> ! [M: nat,W: word @ A,N: nat] :
( ( bit_se5641148757651400278ts_bit @ ( word @ B ) @ ( slice2 @ A @ B @ M @ W ) @ N )
= ( ( ord_less @ nat @ N @ ( ord_min @ nat @ ( type_len0_len_of @ B @ ( type2 @ B ) ) @ ( minus_minus @ nat @ ( type_len0_len_of @ A @ ( type2 @ A ) ) @ M ) ) )
& ( bit_se5641148757651400278ts_bit @ ( word @ A ) @ W @ ( minus_minus @ nat @ ( plus_plus @ nat @ N @ ( type_len0_len_of @ A @ ( type2 @ A ) ) ) @ ( minus_minus @ nat @ ( type_len0_len_of @ A @ ( type2 @ A ) ) @ M ) ) ) ) ) ) ).
% bit_slice_iff
thf(fact_6430_bit__slice1__iff,axiom,
! [B: $tType,A: $tType] :
( ( ( type_len @ A )
& ( type_len @ B ) )
=> ! [M: nat,W: word @ A,N: nat] :
( ( bit_se5641148757651400278ts_bit @ ( word @ B ) @ ( slice1 @ A @ B @ M @ W ) @ N )
= ( ( ord_less_eq @ nat @ ( minus_minus @ nat @ M @ ( type_len0_len_of @ A @ ( type2 @ A ) ) ) @ N )
& ( ord_less @ nat @ N @ ( ord_min @ nat @ ( type_len0_len_of @ B @ ( type2 @ B ) ) @ M ) )
& ( bit_se5641148757651400278ts_bit @ ( word @ A ) @ W @ ( minus_minus @ nat @ ( plus_plus @ nat @ N @ ( minus_minus @ nat @ ( type_len0_len_of @ A @ ( type2 @ A ) ) @ M ) ) @ ( minus_minus @ nat @ M @ ( type_len0_len_of @ A @ ( type2 @ A ) ) ) ) ) ) ) ) ).
% bit_slice1_iff
thf(fact_6431_unat__mask,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [N: nat] :
( ( semiring_1_unsigned @ A @ nat @ ( bit_se2239418461657761734s_mask @ ( word @ A ) @ N ) )
= ( minus_minus @ nat @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( ord_min @ nat @ N @ ( type_len0_len_of @ A @ ( type2 @ A ) ) ) ) @ ( one_one @ nat ) ) ) ) ).
% unat_mask
thf(fact_6432_bit__horner__sum__bit__word__iff,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [Bs: list @ $o,N: nat] :
( ( bit_se5641148757651400278ts_bit @ ( word @ A ) @ ( groups4207007520872428315er_sum @ $o @ ( word @ A ) @ ( zero_neq_one_of_bool @ ( word @ A ) ) @ ( numeral_numeral @ ( word @ A ) @ ( bit0 @ one2 ) ) @ Bs ) @ N )
= ( ( ord_less @ nat @ N @ ( ord_min @ nat @ ( type_len0_len_of @ A @ ( type2 @ A ) ) @ ( size_size @ ( list @ $o ) @ Bs ) ) )
& ( nth @ $o @ Bs @ N ) ) ) ) ).
% bit_horner_sum_bit_word_iff
thf(fact_6433_drop__bit__exp__eq,axiom,
! [A: $tType] :
( ( bit_se359711467146920520ations @ A )
=> ! [M: nat,N: nat] :
( ( bit_se4197421643247451524op_bit @ A @ M @ ( power_power @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ N ) )
= ( times_times @ A
@ ( zero_neq_one_of_bool @ A
@ ( ( ord_less_eq @ nat @ M @ N )
& ( bit_se6407376104438227557le_bit @ A @ ( type2 @ A ) @ N ) ) )
@ ( power_power @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ ( minus_minus @ nat @ N @ M ) ) ) ) ) ).
% drop_bit_exp_eq
thf(fact_6434_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_6435_min__enat__simps_I2_J,axiom,
! [Q3: extended_enat] :
( ( ord_min @ extended_enat @ Q3 @ ( zero_zero @ extended_enat ) )
= ( zero_zero @ extended_enat ) ) ).
% min_enat_simps(2)
thf(fact_6436_min__enat__simps_I3_J,axiom,
! [Q3: extended_enat] :
( ( ord_min @ extended_enat @ ( zero_zero @ extended_enat ) @ Q3 )
= ( zero_zero @ extended_enat ) ) ).
% min_enat_simps(3)
thf(fact_6437_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_6438_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_6439_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_6440_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_6441_bit__minus__1__iff,axiom,
! [A: $tType] :
( ( bit_ri3973907225187159222ations @ A )
=> ! [N: nat] :
( ( bit_se5641148757651400278ts_bit @ A @ ( uminus_uminus @ A @ ( one_one @ A ) ) @ N )
= ( bit_se6407376104438227557le_bit @ A @ ( type2 @ A ) @ N ) ) ) ).
% bit_minus_1_iff
thf(fact_6442_bit__minus__2__iff,axiom,
! [A: $tType] :
( ( bit_ri3973907225187159222ations @ A )
=> ! [N: nat] :
( ( bit_se5641148757651400278ts_bit @ A @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) @ N )
= ( ( bit_se6407376104438227557le_bit @ A @ ( type2 @ A ) @ N )
& ( ord_less @ nat @ ( zero_zero @ nat ) @ N ) ) ) ) ).
% bit_minus_2_iff
thf(fact_6443_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_6444_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_6445_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_6446_bit__eqI,axiom,
! [A: $tType] :
( ( bit_semiring_bits @ A )
=> ! [A2: A,B2: A] :
( ! [N2: nat] :
( ( bit_se6407376104438227557le_bit @ A @ ( type2 @ A ) @ N2 )
=> ( ( bit_se5641148757651400278ts_bit @ A @ A2 @ N2 )
= ( bit_se5641148757651400278ts_bit @ A @ B2 @ N2 ) ) )
=> ( A2 = B2 ) ) ) ).
% bit_eqI
thf(fact_6447_bit__eq__iff,axiom,
! [A: $tType] :
( ( bit_semiring_bits @ A )
=> ( ( ^ [Y2: A,Z2: A] : Y2 = Z2 )
= ( ^ [A4: A,B5: A] :
! [N3: nat] :
( ( bit_se6407376104438227557le_bit @ A @ ( type2 @ A ) @ N3 )
=> ( ( bit_se5641148757651400278ts_bit @ A @ A4 @ N3 )
= ( bit_se5641148757651400278ts_bit @ A @ B5 @ N3 ) ) ) ) ) ) ).
% bit_eq_iff
thf(fact_6448_impossible__bit,axiom,
! [A: $tType] :
( ( bit_semiring_bits @ A )
=> ! [N: nat,A2: A] :
( ~ ( bit_se6407376104438227557le_bit @ A @ ( type2 @ A ) @ N )
=> ~ ( bit_se5641148757651400278ts_bit @ A @ A2 @ N ) ) ) ).
% impossible_bit
thf(fact_6449_bit__imp__possible__bit,axiom,
! [A: $tType] :
( ( bit_semiring_bits @ A )
=> ! [A2: A,N: nat] :
( ( bit_se5641148757651400278ts_bit @ A @ A2 @ N )
=> ( bit_se6407376104438227557le_bit @ A @ ( type2 @ A ) @ N ) ) ) ).
% bit_imp_possible_bit
thf(fact_6450_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_6451_bit__not__iff,axiom,
! [A: $tType] :
( ( bit_ri3973907225187159222ations @ A )
=> ! [A2: A,N: nat] :
( ( bit_se5641148757651400278ts_bit @ A @ ( bit_ri4277139882892585799ns_not @ A @ A2 ) @ N )
= ( ~ ~ ( bit_se6407376104438227557le_bit @ A @ ( type2 @ A ) @ N )
& ~ ( bit_se5641148757651400278ts_bit @ A @ A2 @ N ) ) ) ) ).
% bit_not_iff
thf(fact_6452_semiring__bit__operations__class_Obit__set__bit__iff,axiom,
! [A: $tType] :
( ( bit_se359711467146920520ations @ A )
=> ! [M: nat,A2: A,N: nat] :
( ( bit_se5641148757651400278ts_bit @ A @ ( bit_se5668285175392031749et_bit @ A @ M @ A2 ) @ N )
= ( ( bit_se5641148757651400278ts_bit @ A @ A2 @ N )
| ( ( M = N )
& ( bit_se6407376104438227557le_bit @ A @ ( type2 @ A ) @ N ) ) ) ) ) ).
% semiring_bit_operations_class.bit_set_bit_iff
thf(fact_6453_bit__flip__bit__iff,axiom,
! [A: $tType] :
( ( bit_se359711467146920520ations @ A )
=> ! [M: nat,A2: A,N: nat] :
( ( bit_se5641148757651400278ts_bit @ A @ ( bit_se8732182000553998342ip_bit @ A @ M @ A2 ) @ N )
= ( ( ( M = N )
= ( ~ ( bit_se5641148757651400278ts_bit @ A @ A2 @ N ) ) )
& ( bit_se6407376104438227557le_bit @ A @ ( type2 @ A ) @ N ) ) ) ) ).
% bit_flip_bit_iff
thf(fact_6454_bit__mask__iff,axiom,
! [A: $tType] :
( ( bit_se359711467146920520ations @ A )
=> ! [M: nat,N: nat] :
( ( bit_se5641148757651400278ts_bit @ A @ ( bit_se2239418461657761734s_mask @ A @ M ) @ N )
= ( ( bit_se6407376104438227557le_bit @ A @ ( type2 @ A ) @ N )
& ( ord_less @ nat @ N @ M ) ) ) ) ).
% bit_mask_iff
thf(fact_6455_bit__of__int__iff,axiom,
! [A: $tType] :
( ( bit_ri3973907225187159222ations @ A )
=> ! [K: int,N: nat] :
( ( bit_se5641148757651400278ts_bit @ A @ ( ring_1_of_int @ A @ K ) @ N )
= ( ( bit_se6407376104438227557le_bit @ A @ ( type2 @ A ) @ N )
& ( bit_se5641148757651400278ts_bit @ int @ K @ N ) ) ) ) ).
% bit_of_int_iff
thf(fact_6456_bit__signed__take__bit__iff,axiom,
! [A: $tType] :
( ( bit_ri3973907225187159222ations @ A )
=> ! [M: nat,A2: A,N: nat] :
( ( bit_se5641148757651400278ts_bit @ A @ ( bit_ri4674362597316999326ke_bit @ A @ M @ A2 ) @ N )
= ( ( bit_se6407376104438227557le_bit @ A @ ( type2 @ A ) @ N )
& ( bit_se5641148757651400278ts_bit @ A @ A2 @ ( ord_min @ nat @ M @ N ) ) ) ) ) ).
% bit_signed_take_bit_iff
thf(fact_6457_bit__of__nat__iff,axiom,
! [A: $tType] :
( ( bit_semiring_bits @ A )
=> ! [M: nat,N: nat] :
( ( bit_se5641148757651400278ts_bit @ A @ ( semiring_1_of_nat @ A @ M ) @ N )
= ( ( bit_se6407376104438227557le_bit @ A @ ( type2 @ A ) @ N )
& ( bit_se5641148757651400278ts_bit @ nat @ M @ N ) ) ) ) ).
% bit_of_nat_iff
thf(fact_6458_possible__bit__def,axiom,
! [A: $tType] :
( ( bit_semiring_bits @ A )
=> ( ( bit_se6407376104438227557le_bit @ A )
= ( ^ [Tyrep2: itself @ A,N3: nat] :
( ( power_power @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ N3 )
!= ( zero_zero @ A ) ) ) ) ) ).
% possible_bit_def
thf(fact_6459_bit__minus__iff,axiom,
! [A: $tType] :
( ( bit_ri3973907225187159222ations @ A )
=> ! [A2: A,N: nat] :
( ( bit_se5641148757651400278ts_bit @ A @ ( uminus_uminus @ A @ A2 ) @ N )
= ( ( bit_se6407376104438227557le_bit @ A @ ( type2 @ A ) @ N )
& ~ ( bit_se5641148757651400278ts_bit @ A @ ( minus_minus @ A @ A2 @ ( one_one @ A ) ) @ N ) ) ) ) ).
% bit_minus_iff
thf(fact_6460_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_6461_bit__push__bit__iff,axiom,
! [A: $tType] :
( ( bit_se359711467146920520ations @ A )
=> ! [M: nat,A2: A,N: nat] :
( ( bit_se5641148757651400278ts_bit @ A @ ( bit_se4730199178511100633sh_bit @ A @ M @ A2 ) @ N )
= ( ( ord_less_eq @ nat @ M @ N )
& ( bit_se6407376104438227557le_bit @ A @ ( type2 @ A ) @ N )
& ( bit_se5641148757651400278ts_bit @ A @ A2 @ ( minus_minus @ nat @ N @ M ) ) ) ) ) ).
% bit_push_bit_iff
thf(fact_6462_fold__possible__bit,axiom,
! [A: $tType] :
( ( bit_semiring_bits @ A )
=> ! [N: nat] :
( ( ( power_power @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ N )
= ( zero_zero @ A ) )
= ( ~ ( bit_se6407376104438227557le_bit @ A @ ( type2 @ A ) @ N ) ) ) ) ).
% fold_possible_bit
thf(fact_6463_bit__exp__iff,axiom,
! [A: $tType] :
( ( bit_semiring_bits @ A )
=> ! [M: nat,N: nat] :
( ( bit_se5641148757651400278ts_bit @ A @ ( power_power @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ M ) @ N )
= ( ( bit_se6407376104438227557le_bit @ A @ ( type2 @ A ) @ N )
& ( M = N ) ) ) ) ).
% bit_exp_iff
thf(fact_6464_bit__2__iff,axiom,
! [A: $tType] :
( ( bit_semiring_bits @ A )
=> ! [N: nat] :
( ( bit_se5641148757651400278ts_bit @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ N )
= ( ( bit_se6407376104438227557le_bit @ A @ ( type2 @ A ) @ ( one_one @ nat ) )
& ( N
= ( one_one @ nat ) ) ) ) ) ).
% bit_2_iff
thf(fact_6465_bit__not__exp__iff,axiom,
! [A: $tType] :
( ( bit_ri3973907225187159222ations @ A )
=> ! [M: nat,N: nat] :
( ( bit_se5641148757651400278ts_bit @ A @ ( bit_ri4277139882892585799ns_not @ A @ ( power_power @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ M ) ) @ N )
= ( ( bit_se6407376104438227557le_bit @ A @ ( type2 @ A ) @ N )
& ( N != M ) ) ) ) ).
% bit_not_exp_iff
thf(fact_6466_bit__minus__exp__iff,axiom,
! [A: $tType] :
( ( bit_ri3973907225187159222ations @ A )
=> ! [M: nat,N: nat] :
( ( bit_se5641148757651400278ts_bit @ A @ ( uminus_uminus @ A @ ( power_power @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ M ) ) @ N )
= ( ( bit_se6407376104438227557le_bit @ A @ ( type2 @ A ) @ N )
& ( ord_less_eq @ nat @ M @ N ) ) ) ) ).
% bit_minus_exp_iff
thf(fact_6467_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_6468_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_6469_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_6470_bit__mask__sub__iff,axiom,
! [A: $tType] :
( ( bit_semiring_bits @ A )
=> ! [M: nat,N: nat] :
( ( bit_se5641148757651400278ts_bit @ A @ ( minus_minus @ A @ ( power_power @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ M ) @ ( one_one @ A ) ) @ N )
= ( ( bit_se6407376104438227557le_bit @ A @ ( type2 @ A ) @ N )
& ( ord_less @ nat @ N @ M ) ) ) ) ).
% bit_mask_sub_iff
thf(fact_6471_bit__double__iff,axiom,
! [A: $tType] :
( ( bit_semiring_bits @ A )
=> ! [A2: A,N: nat] :
( ( bit_se5641148757651400278ts_bit @ A @ ( times_times @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ A2 ) @ N )
= ( ( bit_se5641148757651400278ts_bit @ A @ A2 @ ( minus_minus @ nat @ N @ ( one_one @ nat ) ) )
& ( N
!= ( zero_zero @ nat ) )
& ( bit_se6407376104438227557le_bit @ A @ ( type2 @ A ) @ N ) ) ) ) ).
% bit_double_iff
thf(fact_6472_bit__signed__iff,axiom,
! [A: $tType,B: $tType] :
( ( ( type_len @ B )
& ( bit_ri3973907225187159222ations @ A ) )
=> ! [W: word @ B,N: nat] :
( ( bit_se5641148757651400278ts_bit @ A @ ( ring_1_signed @ B @ A @ W ) @ N )
= ( ( bit_se6407376104438227557le_bit @ A @ ( type2 @ A ) @ N )
& ( bit_se5641148757651400278ts_bit @ ( word @ B ) @ W @ ( ord_min @ nat @ ( minus_minus @ nat @ ( type_len0_len_of @ B @ ( type2 @ B ) ) @ ( suc @ ( zero_zero @ nat ) ) ) @ N ) ) ) ) ) ).
% bit_signed_iff
thf(fact_6473_word__set__bits__def,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ( ( bit_bi4170147762399595738t_bits @ ( word @ A ) )
= ( ^ [P2: nat > $o] : ( groups4207007520872428315er_sum @ $o @ ( word @ A ) @ ( zero_neq_one_of_bool @ ( word @ A ) ) @ ( numeral_numeral @ ( word @ A ) @ ( bit0 @ one2 ) ) @ ( map @ nat @ $o @ P2 @ ( upt @ ( zero_zero @ nat ) @ ( type_len0_len_of @ A @ ( type2 @ A ) ) ) ) ) ) ) ) ).
% word_set_bits_def
thf(fact_6474_uint__word__rotr__eq,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [N: nat,W: word @ A] :
( ( semiring_1_unsigned @ A @ int @ ( word_rotr @ A @ N @ W ) )
= ( bit_concat_bit @ ( minus_minus @ nat @ ( type_len0_len_of @ A @ ( type2 @ A ) ) @ ( modulo_modulo @ nat @ N @ ( type_len0_len_of @ A @ ( type2 @ A ) ) ) ) @ ( bit_se4197421643247451524op_bit @ int @ ( modulo_modulo @ nat @ N @ ( type_len0_len_of @ A @ ( type2 @ A ) ) ) @ ( semiring_1_unsigned @ A @ int @ W ) ) @ ( semiring_1_unsigned @ A @ int @ ( bit_se2584673776208193580ke_bit @ ( word @ A ) @ ( modulo_modulo @ nat @ N @ ( type_len0_len_of @ A @ ( type2 @ A ) ) ) @ W ) ) ) ) ) ).
% uint_word_rotr_eq
thf(fact_6475_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_6476_set__bits__False__eq,axiom,
! [A: $tType] :
( ( bit_bi6583157726757044596ension @ A )
=> ( ( bit_bi4170147762399595738t_bits @ A
@ ^ [Uu3: nat] : $false )
= ( zero_zero @ A ) ) ) ).
% set_bits_False_eq
thf(fact_6477_set__bits__K__False,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ( ( bit_bi4170147762399595738t_bits @ ( word @ A )
@ ^ [Uu3: nat] : $false )
= ( zero_zero @ ( word @ A ) ) ) ) ).
% set_bits_K_False
thf(fact_6478_infinite__UNIV__listI,axiom,
! [A: $tType] :
~ ( finite_finite2 @ ( list @ A ) @ ( top_top @ ( set @ ( list @ A ) ) ) ) ).
% infinite_UNIV_listI
thf(fact_6479_word__rotr__word__rotr__eq,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [M: nat,N: nat,W: word @ A] :
( ( word_rotr @ A @ M @ ( word_rotr @ A @ N @ W ) )
= ( word_rotr @ A @ ( plus_plus @ nat @ M @ N ) @ W ) ) ) ).
% word_rotr_word_rotr_eq
thf(fact_6480_bit__set__bits__word__iff,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [P: nat > $o,N: nat] :
( ( bit_se5641148757651400278ts_bit @ ( word @ A ) @ ( bit_bi4170147762399595738t_bits @ ( word @ A ) @ P ) @ N )
= ( ( ord_less @ nat @ N @ ( type_len0_len_of @ A @ ( type2 @ A ) ) )
& ( P @ N ) ) ) ) ).
% bit_set_bits_word_iff
thf(fact_6481_word__type__copy__misc_Oset__bits__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,P: nat > $o] :
( ( 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 )
=> ( ( bit_bi4170147762399595738t_bits @ A @ P )
= ( Set_bits_aux @ P @ Size @ ( zero_zero @ A ) ) ) ) ) ).
% word_type_copy_misc.set_bits_code
thf(fact_6482_range__mod,axiom,
! [N: nat] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ N )
=> ( ( image @ nat @ nat
@ ^ [M4: nat] : ( modulo_modulo @ nat @ M4 @ N )
@ ( top_top @ ( set @ nat ) ) )
= ( set_or7035219750837199246ssThan @ nat @ ( zero_zero @ nat ) @ N ) ) ) ).
% range_mod
thf(fact_6483_set__bits__conv__set__bits__aux,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ( ( bit_bi4170147762399595738t_bits @ ( word @ A ) )
= ( ^ [F4: nat > $o] : ( code_T2661198915054445665ts_aux @ A @ F4 @ ( type_len0_len_of @ A @ ( type2 @ A ) ) @ ( zero_zero @ ( word @ A ) ) ) ) ) ) ).
% set_bits_conv_set_bits_aux
thf(fact_6484_word__test__bit__set__bits,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [F2: nat > $o,N: nat] :
( ( bit_se5641148757651400278ts_bit @ ( word @ A ) @ ( bit_bi4170147762399595738t_bits @ ( word @ A ) @ F2 ) @ N )
= ( ( ord_less @ nat @ N @ ( type_len0_len_of @ A @ ( type2 @ A ) ) )
& ( F2 @ N ) ) ) ) ).
% word_test_bit_set_bits
thf(fact_6485_bit__word__rotr__iff,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [M: nat,W: word @ A,N: nat] :
( ( bit_se5641148757651400278ts_bit @ ( word @ A ) @ ( word_rotr @ A @ M @ W ) @ N )
= ( ( ord_less @ nat @ N @ ( type_len0_len_of @ A @ ( type2 @ A ) ) )
& ( bit_se5641148757651400278ts_bit @ ( word @ A ) @ W @ ( modulo_modulo @ nat @ ( plus_plus @ nat @ N @ M ) @ ( type_len0_len_of @ A @ ( type2 @ A ) ) ) ) ) ) ) ).
% bit_word_rotr_iff
thf(fact_6486_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_6487_word__roti__eq__word__rotr__word__rotl,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ( ( word_roti @ A )
= ( ^ [I2: int,W2: word @ A] : ( if @ ( word @ A ) @ ( ord_less_eq @ int @ ( zero_zero @ int ) @ I2 ) @ ( word_rotr @ A @ ( nat2 @ I2 ) @ W2 ) @ ( word_rotl @ A @ ( nat2 @ ( uminus_uminus @ int @ I2 ) ) @ W2 ) ) ) ) ) ).
% word_roti_eq_word_rotr_word_rotl
thf(fact_6488_int__set__bits__K__False,axiom,
( ( bit_bi4170147762399595738t_bits @ int
@ ^ [Uu3: nat] : $false )
= ( zero_zero @ int ) ) ).
% int_set_bits_K_False
thf(fact_6489_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_6490_word__rotx__0,axiom,
! [A: $tType,B: $tType] :
( ( ( type_len @ B )
& ( type_len @ A ) )
=> ! [I: nat] :
( ( ( word_rotr @ A @ I @ ( zero_zero @ ( word @ A ) ) )
= ( zero_zero @ ( word @ A ) ) )
& ( ( word_rotl @ B @ I @ ( zero_zero @ ( word @ B ) ) )
= ( zero_zero @ ( word @ B ) ) ) ) ) ).
% word_rotx_0
thf(fact_6491_word__rotl__eq__word__rotr,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ( ( word_rotl @ A )
= ( ^ [N3: nat] : ( word_rotr @ A @ ( minus_minus @ nat @ ( type_len0_len_of @ A @ ( type2 @ A ) ) @ ( modulo_modulo @ nat @ N3 @ ( type_len0_len_of @ A @ ( type2 @ A ) ) ) ) ) ) ) ) ).
% word_rotl_eq_word_rotr
thf(fact_6492_bit__word__rotl__iff,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [M: nat,W: word @ A,N: nat] :
( ( bit_se5641148757651400278ts_bit @ ( word @ A ) @ ( word_rotl @ A @ M @ W ) @ N )
= ( ( ord_less @ nat @ N @ ( type_len0_len_of @ A @ ( type2 @ A ) ) )
& ( bit_se5641148757651400278ts_bit @ ( word @ A ) @ W @ ( modulo_modulo @ nat @ ( plus_plus @ nat @ N @ ( minus_minus @ nat @ ( type_len0_len_of @ A @ ( type2 @ A ) ) @ ( modulo_modulo @ nat @ M @ ( type_len0_len_of @ A @ ( type2 @ A ) ) ) ) ) @ ( type_len0_len_of @ A @ ( type2 @ A ) ) ) ) ) ) ) ).
% bit_word_rotl_iff
thf(fact_6493_root__def,axiom,
( root
= ( ^ [N3: nat,X2: real] :
( if @ real
@ ( N3
= ( zero_zero @ nat ) )
@ ( zero_zero @ real )
@ ( the_inv_into @ real @ real @ ( top_top @ ( set @ real ) )
@ ^ [Y6: real] : ( times_times @ real @ ( sgn_sgn @ real @ Y6 ) @ ( power_power @ real @ ( abs_abs @ real @ Y6 ) @ N3 ) )
@ X2 ) ) ) ) ).
% root_def
thf(fact_6494_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_6495_one__word__def,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ( ( one_one @ ( word @ A ) )
= ( word2 @ A @ ( one_one @ int ) ) ) ) ).
% one_word_def
thf(fact_6496_zero__word__def,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ( ( zero_zero @ ( word @ A ) )
= ( word2 @ A @ ( zero_zero @ int ) ) ) ) ).
% zero_word_def
thf(fact_6497_set__bit__word_Oabs__eq,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [Xa: nat,X: int] :
( ( bit_se5668285175392031749et_bit @ ( word @ A ) @ Xa @ ( word2 @ A @ X ) )
= ( word2 @ A @ ( bit_se5668285175392031749et_bit @ int @ Xa @ X ) ) ) ) ).
% set_bit_word.abs_eq
thf(fact_6498_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_6499_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_6500_less__word_Oabs__eq,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [Xa: int,X: int] :
( ( ord_less @ ( word @ A ) @ ( word2 @ A @ Xa ) @ ( word2 @ A @ X ) )
= ( ord_less @ 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_word.abs_eq
thf(fact_6501_bit__word_Oabs__eq,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [X: int] :
( ( bit_se5641148757651400278ts_bit @ ( word @ A ) @ ( word2 @ A @ X ) )
= ( ^ [N3: nat] :
( ( ord_less @ nat @ N3 @ ( type_len0_len_of @ A @ ( type2 @ A ) ) )
& ( bit_se5641148757651400278ts_bit @ int @ X @ N3 ) ) ) ) ) ).
% bit_word.abs_eq
thf(fact_6502_divide__word_Oabs__eq,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [Xa: int,X: int] :
( ( divide_divide @ ( word @ A ) @ ( word2 @ A @ Xa ) @ ( word2 @ A @ X ) )
= ( word2 @ A @ ( divide_divide @ 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 ) ) ) ) ) ).
% divide_word.abs_eq
thf(fact_6503_modulo__word_Oabs__eq,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [Xa: int,X: int] :
( ( modulo_modulo @ ( word @ A ) @ ( word2 @ A @ Xa ) @ ( word2 @ A @ X ) )
= ( word2 @ A @ ( modulo_modulo @ 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 ) ) ) ) ) ).
% modulo_word.abs_eq
thf(fact_6504_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_6505_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_6506_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_6507_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_6508_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_6509_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_6510_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_6511_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_6512_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_6513_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_6514_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_6515_CHAR__eq0__iff,axiom,
! [A: $tType] :
( ( semiring_1 @ A )
=> ( ( ( semiri4206861660011772517g_char @ A @ ( type2 @ A ) )
= ( zero_zero @ nat ) )
= ( ! [N3: nat] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ N3 )
=> ( ( semiring_1_of_nat @ A @ N3 )
!= ( zero_zero @ A ) ) ) ) ) ) ).
% CHAR_eq0_iff
thf(fact_6516_wf__set__bits__int__Suc,axiom,
! [F2: nat > $o] :
( ( bit_wf_set_bits_int
@ ^ [N3: nat] : ( F2 @ ( suc @ N3 ) ) )
= ( bit_wf_set_bits_int @ F2 ) ) ).
% wf_set_bits_int_Suc
thf(fact_6517_CHAR__eq__0,axiom,
! [A: $tType] :
( ( semiring_char_0 @ A )
=> ( ( semiri4206861660011772517g_char @ A @ ( type2 @ A ) )
= ( zero_zero @ nat ) ) ) ).
% CHAR_eq_0
thf(fact_6518_of__nat__CHAR,axiom,
! [A: $tType] :
( ( semiring_1 @ A )
=> ( ( semiring_1_of_nat @ A @ ( semiri4206861660011772517g_char @ A @ ( type2 @ A ) ) )
= ( zero_zero @ A ) ) ) ).
% of_nat_CHAR
thf(fact_6519_CHAR__eqI,axiom,
! [A: $tType] :
( ( semiring_1 @ A )
=> ! [C2: nat] :
( ( ( semiring_1_of_nat @ A @ C2 )
= ( zero_zero @ A ) )
=> ( ! [X3: nat] :
( ( ( semiring_1_of_nat @ A @ X3 )
= ( zero_zero @ A ) )
=> ( dvd_dvd @ nat @ C2 @ X3 ) )
=> ( ( semiri4206861660011772517g_char @ A @ ( type2 @ A ) )
= C2 ) ) ) ) ).
% CHAR_eqI
thf(fact_6520_of__nat__eq__0__iff__char__dvd,axiom,
! [A: $tType] :
( ( semiring_1 @ A )
=> ! [N: nat] :
( ( ( semiring_1_of_nat @ A @ N )
= ( zero_zero @ A ) )
= ( dvd_dvd @ nat @ ( semiri4206861660011772517g_char @ A @ ( type2 @ A ) ) @ N ) ) ) ).
% of_nat_eq_0_iff_char_dvd
thf(fact_6521_CHAR__pos__iff,axiom,
! [A: $tType] :
( ( semiring_1 @ A )
=> ( ( ord_less @ nat @ ( zero_zero @ nat ) @ ( semiri4206861660011772517g_char @ A @ ( type2 @ A ) ) )
= ( ? [N3: nat] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ N3 )
& ( ( semiring_1_of_nat @ A @ N3 )
= ( zero_zero @ A ) ) ) ) ) ) ).
% CHAR_pos_iff
thf(fact_6522_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_6523_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_6524_bit__sshiftr__iff,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [W: word @ A,M: nat,N: nat] :
( ( bit_se5641148757651400278ts_bit @ ( word @ A ) @ ( bit_Sh8784991116023147202shiftr @ A @ W @ M ) @ N )
= ( bit_se5641148757651400278ts_bit @ ( word @ A ) @ W
@ ( if @ nat
@ ( ( ord_less_eq @ nat @ ( minus_minus @ nat @ ( type_len0_len_of @ A @ ( type2 @ A ) ) @ M ) @ N )
& ( ord_less @ nat @ N @ ( type_len0_len_of @ A @ ( type2 @ A ) ) ) )
@ ( minus_minus @ nat @ ( type_len0_len_of @ A @ ( type2 @ A ) ) @ ( one_one @ nat ) )
@ ( plus_plus @ nat @ M @ N ) ) ) ) ) ).
% bit_sshiftr_iff
thf(fact_6525_map__comp__map,axiom,
! [B: $tType,C: $tType,A: $tType,F2: C > B,G: A > C] :
( ( comp @ ( list @ C ) @ ( list @ B ) @ ( list @ A ) @ ( map @ C @ B @ F2 ) @ ( map @ A @ C @ G ) )
= ( map @ A @ B @ ( comp @ C @ B @ A @ F2 @ G ) ) ) ).
% map_comp_map
thf(fact_6526_map__map,axiom,
! [B: $tType,A: $tType,C: $tType,F2: B > A,G: C > B,Xs2: list @ C] :
( ( map @ B @ A @ F2 @ ( map @ C @ B @ G @ Xs2 ) )
= ( map @ C @ A @ ( comp @ B @ A @ C @ F2 @ G ) @ Xs2 ) ) ).
% map_map
thf(fact_6527_List_Omap_Ocompositionality,axiom,
! [B: $tType,C: $tType,A: $tType,F2: B > C,G: A > B,List: list @ A] :
( ( map @ B @ C @ F2 @ ( map @ A @ B @ G @ List ) )
= ( map @ A @ C @ ( comp @ B @ C @ A @ F2 @ G ) @ List ) ) ).
% List.map.compositionality
thf(fact_6528_list_Omap__comp,axiom,
! [B: $tType,C: $tType,A: $tType,G: B > C,F2: A > B,V: list @ A] :
( ( map @ B @ C @ G @ ( map @ A @ B @ F2 @ V ) )
= ( map @ A @ C @ ( comp @ B @ C @ A @ G @ F2 ) @ V ) ) ).
% list.map_comp
thf(fact_6529_List_Omap_Ocomp,axiom,
! [C: $tType,B: $tType,A: $tType,F2: B > C,G: A > B] :
( ( comp @ ( list @ B ) @ ( list @ C ) @ ( list @ A ) @ ( map @ B @ C @ F2 ) @ ( map @ A @ B @ G ) )
= ( map @ A @ C @ ( comp @ B @ C @ A @ F2 @ G ) ) ) ).
% List.map.comp
thf(fact_6530_sshiftr__of__0,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [W: word @ A] :
( ( bit_Sh8784991116023147202shiftr @ A @ W @ ( zero_zero @ nat ) )
= W ) ) ).
% sshiftr_of_0
thf(fact_6531_sshiftr__0,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [N: nat] :
( ( bit_Sh8784991116023147202shiftr @ A @ ( zero_zero @ ( word @ A ) ) @ N )
= ( zero_zero @ ( word @ A ) ) ) ) ).
% sshiftr_0
thf(fact_6532_sshiftr__numeral__Suc,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [M: num,N: nat] :
( ( bit_Sh8784991116023147202shiftr @ A @ ( numeral_numeral @ ( word @ A ) @ M ) @ ( suc @ N ) )
= ( signed_drop_bit @ A @ ( suc @ N ) @ ( numeral_numeral @ ( word @ A ) @ M ) ) ) ) ).
% sshiftr_numeral_Suc
thf(fact_6533_sshiftr__numeral__numeral,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [M: num,N: num] :
( ( bit_Sh8784991116023147202shiftr @ A @ ( numeral_numeral @ ( word @ A ) @ M ) @ ( numeral_numeral @ nat @ N ) )
= ( signed_drop_bit @ A @ ( numeral_numeral @ nat @ N ) @ ( numeral_numeral @ ( word @ A ) @ M ) ) ) ) ).
% sshiftr_numeral_numeral
thf(fact_6534_sshiftr__minus__numeral__Suc,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [M: num,N: nat] :
( ( bit_Sh8784991116023147202shiftr @ A @ ( uminus_uminus @ ( word @ A ) @ ( numeral_numeral @ ( word @ A ) @ M ) ) @ ( suc @ N ) )
= ( signed_drop_bit @ A @ ( suc @ N ) @ ( uminus_uminus @ ( word @ A ) @ ( numeral_numeral @ ( word @ A ) @ M ) ) ) ) ) ).
% sshiftr_minus_numeral_Suc
thf(fact_6535_sshiftr__minus__numeral__numeral,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [M: num,N: num] :
( ( bit_Sh8784991116023147202shiftr @ A @ ( uminus_uminus @ ( word @ A ) @ ( numeral_numeral @ ( word @ A ) @ M ) ) @ ( numeral_numeral @ nat @ N ) )
= ( signed_drop_bit @ A @ ( numeral_numeral @ nat @ N ) @ ( uminus_uminus @ ( word @ A ) @ ( numeral_numeral @ ( word @ A ) @ M ) ) ) ) ) ).
% sshiftr_minus_numeral_numeral
thf(fact_6536_sshiftr__1,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [N: nat] :
( ( bit_Sh8784991116023147202shiftr @ A @ ( one_one @ ( word @ A ) ) @ N )
= ( zero_neq_one_of_bool @ ( word @ A )
@ ( ( ( type_len0_len_of @ A @ ( type2 @ A ) )
= ( one_one @ nat ) )
| ( N
= ( zero_zero @ nat ) ) ) ) ) ) ).
% sshiftr_1
thf(fact_6537_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_6538_foldr_Ofoldr__Cons,axiom,
! [B: $tType,A: $tType,F2: A > B > B,X: A,Xs2: list @ A] :
( ( foldr @ A @ B @ F2 @ ( cons @ A @ X @ Xs2 ) )
= ( comp @ B @ B @ B @ ( F2 @ X ) @ ( foldr @ A @ B @ F2 @ Xs2 ) ) ) ).
% foldr.foldr_Cons
thf(fact_6539_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_6540_sum_OatLeast__Suc__lessThan__Suc__shift,axiom,
! [A: $tType] :
( ( comm_monoid_add @ A )
=> ! [G: nat > A,M: nat,N: nat] :
( ( groups7311177749621191930dd_sum @ nat @ A @ G @ ( set_or7035219750837199246ssThan @ nat @ ( suc @ M ) @ ( suc @ N ) ) )
= ( groups7311177749621191930dd_sum @ nat @ A @ ( comp @ nat @ A @ nat @ G @ suc ) @ ( set_or7035219750837199246ssThan @ nat @ M @ N ) ) ) ) ).
% sum.atLeast_Suc_lessThan_Suc_shift
thf(fact_6541_sum_OatLeast__Suc__atMost__Suc__shift,axiom,
! [A: $tType] :
( ( comm_monoid_add @ A )
=> ! [G: nat > A,M: nat,N: nat] :
( ( groups7311177749621191930dd_sum @ nat @ A @ G @ ( set_or1337092689740270186AtMost @ nat @ ( suc @ M ) @ ( suc @ N ) ) )
= ( groups7311177749621191930dd_sum @ nat @ A @ ( comp @ nat @ A @ nat @ G @ suc ) @ ( set_or1337092689740270186AtMost @ nat @ M @ N ) ) ) ) ).
% sum.atLeast_Suc_atMost_Suc_shift
thf(fact_6542_bit__drop__bit__eq,axiom,
! [A: $tType] :
( ( bit_se359711467146920520ations @ A )
=> ! [N: nat,A2: A] :
( ( bit_se5641148757651400278ts_bit @ A @ ( bit_se4197421643247451524op_bit @ A @ N @ A2 ) )
= ( comp @ nat @ $o @ nat @ ( bit_se5641148757651400278ts_bit @ A @ A2 ) @ ( plus_plus @ nat @ N ) ) ) ) ).
% bit_drop_bit_eq
thf(fact_6543_sum_OatLeastLessThan__shift__bounds,axiom,
! [A: $tType] :
( ( comm_monoid_add @ A )
=> ! [G: nat > A,M: nat,K: nat,N: nat] :
( ( groups7311177749621191930dd_sum @ nat @ A @ G @ ( set_or7035219750837199246ssThan @ nat @ ( plus_plus @ nat @ M @ K ) @ ( plus_plus @ nat @ N @ K ) ) )
= ( groups7311177749621191930dd_sum @ nat @ A @ ( comp @ nat @ A @ nat @ G @ ( plus_plus @ nat @ K ) ) @ ( set_or7035219750837199246ssThan @ nat @ M @ N ) ) ) ) ).
% sum.atLeastLessThan_shift_bounds
thf(fact_6544_sum_OatLeastAtMost__shift__bounds,axiom,
! [A: $tType] :
( ( comm_monoid_add @ A )
=> ! [G: nat > A,M: nat,K: nat,N: nat] :
( ( groups7311177749621191930dd_sum @ nat @ A @ G @ ( set_or1337092689740270186AtMost @ nat @ ( plus_plus @ nat @ M @ K ) @ ( plus_plus @ nat @ N @ K ) ) )
= ( groups7311177749621191930dd_sum @ nat @ A @ ( comp @ nat @ A @ nat @ G @ ( plus_plus @ nat @ K ) ) @ ( set_or1337092689740270186AtMost @ nat @ M @ N ) ) ) ) ).
% sum.atLeastAtMost_shift_bounds
thf(fact_6545_prod_OatLeast__Suc__lessThan__Suc__shift,axiom,
! [A: $tType] :
( ( comm_monoid_mult @ A )
=> ! [G: nat > A,M: nat,N: nat] :
( ( groups7121269368397514597t_prod @ nat @ A @ G @ ( set_or7035219750837199246ssThan @ nat @ ( suc @ M ) @ ( suc @ N ) ) )
= ( groups7121269368397514597t_prod @ nat @ A @ ( comp @ nat @ A @ nat @ G @ suc ) @ ( set_or7035219750837199246ssThan @ nat @ M @ N ) ) ) ) ).
% prod.atLeast_Suc_lessThan_Suc_shift
thf(fact_6546_prod_OatLeast__Suc__atMost__Suc__shift,axiom,
! [A: $tType] :
( ( comm_monoid_mult @ A )
=> ! [G: nat > A,M: nat,N: nat] :
( ( groups7121269368397514597t_prod @ nat @ A @ G @ ( set_or1337092689740270186AtMost @ nat @ ( suc @ M ) @ ( suc @ N ) ) )
= ( groups7121269368397514597t_prod @ nat @ A @ ( comp @ nat @ A @ nat @ G @ suc ) @ ( set_or1337092689740270186AtMost @ nat @ M @ N ) ) ) ) ).
% prod.atLeast_Suc_atMost_Suc_shift
thf(fact_6547_prod_OatLeastLessThan__shift__bounds,axiom,
! [A: $tType] :
( ( comm_monoid_mult @ A )
=> ! [G: nat > A,M: nat,K: nat,N: nat] :
( ( groups7121269368397514597t_prod @ nat @ A @ G @ ( set_or7035219750837199246ssThan @ nat @ ( plus_plus @ nat @ M @ K ) @ ( plus_plus @ nat @ N @ K ) ) )
= ( groups7121269368397514597t_prod @ nat @ A @ ( comp @ nat @ A @ nat @ G @ ( plus_plus @ nat @ K ) ) @ ( set_or7035219750837199246ssThan @ nat @ M @ N ) ) ) ) ).
% prod.atLeastLessThan_shift_bounds
thf(fact_6548_prod_OatLeastAtMost__shift__bounds,axiom,
! [A: $tType] :
( ( comm_monoid_mult @ A )
=> ! [G: nat > A,M: nat,K: nat,N: nat] :
( ( groups7121269368397514597t_prod @ nat @ A @ G @ ( set_or1337092689740270186AtMost @ nat @ ( plus_plus @ nat @ M @ K ) @ ( plus_plus @ nat @ N @ K ) ) )
= ( groups7121269368397514597t_prod @ nat @ A @ ( comp @ nat @ A @ nat @ G @ ( plus_plus @ nat @ K ) ) @ ( set_or1337092689740270186AtMost @ nat @ M @ N ) ) ) ) ).
% prod.atLeastAtMost_shift_bounds
thf(fact_6549_summable__inverse__divide,axiom,
! [A: $tType] :
( ( real_V3459762299906320749_field @ A )
=> ! [F2: nat > A,C2: A] :
( ( summable @ A @ ( comp @ A @ A @ nat @ ( inverse_inverse @ A ) @ F2 ) )
=> ( summable @ A
@ ^ [N3: nat] : ( divide_divide @ A @ C2 @ ( F2 @ N3 ) ) ) ) ) ).
% summable_inverse_divide
thf(fact_6550_sum_Oreindex__nontrivial,axiom,
! [C: $tType,A: $tType,B: $tType] :
( ( comm_monoid_add @ A )
=> ! [A3: set @ B,H2: B > C,G: C > A] :
( ( finite_finite2 @ B @ A3 )
=> ( ! [X3: B,Y3: B] :
( ( member @ B @ X3 @ A3 )
=> ( ( member @ B @ Y3 @ A3 )
=> ( ( X3 != Y3 )
=> ( ( ( H2 @ X3 )
= ( H2 @ Y3 ) )
=> ( ( G @ ( H2 @ X3 ) )
= ( zero_zero @ A ) ) ) ) ) )
=> ( ( groups7311177749621191930dd_sum @ C @ A @ G @ ( image @ B @ C @ H2 @ A3 ) )
= ( groups7311177749621191930dd_sum @ B @ A @ ( comp @ C @ A @ B @ G @ H2 ) @ A3 ) ) ) ) ) ).
% sum.reindex_nontrivial
thf(fact_6551_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_6552_sum__image__le,axiom,
! [A: $tType,B: $tType,C: $tType] :
( ( ordere6911136660526730532id_add @ B )
=> ! [I6: set @ C,G: A > B,F2: C > A] :
( ( finite_finite2 @ C @ I6 )
=> ( ! [I3: C] :
( ( member @ C @ I3 @ I6 )
=> ( ord_less_eq @ B @ ( zero_zero @ B ) @ ( G @ ( F2 @ I3 ) ) ) )
=> ( ord_less_eq @ B @ ( groups7311177749621191930dd_sum @ A @ B @ G @ ( image @ C @ A @ F2 @ I6 ) ) @ ( groups7311177749621191930dd_sum @ C @ B @ ( comp @ A @ B @ C @ G @ F2 ) @ I6 ) ) ) ) ) ).
% sum_image_le
thf(fact_6553_sum_OatLeast0__lessThan__Suc__shift,axiom,
! [A: $tType] :
( ( comm_monoid_add @ A )
=> ! [G: nat > A,N: nat] :
( ( groups7311177749621191930dd_sum @ nat @ A @ G @ ( set_or7035219750837199246ssThan @ nat @ ( zero_zero @ nat ) @ ( suc @ N ) ) )
= ( plus_plus @ A @ ( G @ ( zero_zero @ nat ) ) @ ( groups7311177749621191930dd_sum @ nat @ A @ ( comp @ nat @ A @ nat @ G @ suc ) @ ( set_or7035219750837199246ssThan @ nat @ ( zero_zero @ nat ) @ N ) ) ) ) ) ).
% sum.atLeast0_lessThan_Suc_shift
thf(fact_6554_sum_OatLeast0__atMost__Suc__shift,axiom,
! [A: $tType] :
( ( comm_monoid_add @ A )
=> ! [G: nat > A,N: nat] :
( ( groups7311177749621191930dd_sum @ nat @ A @ G @ ( set_or1337092689740270186AtMost @ nat @ ( zero_zero @ nat ) @ ( suc @ N ) ) )
= ( plus_plus @ A @ ( G @ ( zero_zero @ nat ) ) @ ( groups7311177749621191930dd_sum @ nat @ A @ ( comp @ nat @ A @ nat @ G @ suc ) @ ( set_or1337092689740270186AtMost @ nat @ ( zero_zero @ nat ) @ N ) ) ) ) ) ).
% sum.atLeast0_atMost_Suc_shift
thf(fact_6555_prod_OatLeast0__lessThan__Suc__shift,axiom,
! [A: $tType] :
( ( comm_monoid_mult @ A )
=> ! [G: nat > A,N: nat] :
( ( groups7121269368397514597t_prod @ nat @ A @ G @ ( set_or7035219750837199246ssThan @ nat @ ( zero_zero @ nat ) @ ( suc @ N ) ) )
= ( times_times @ A @ ( G @ ( zero_zero @ nat ) ) @ ( groups7121269368397514597t_prod @ nat @ A @ ( comp @ nat @ A @ nat @ G @ suc ) @ ( set_or7035219750837199246ssThan @ nat @ ( zero_zero @ nat ) @ N ) ) ) ) ) ).
% prod.atLeast0_lessThan_Suc_shift
thf(fact_6556_prod_OatLeast0__atMost__Suc__shift,axiom,
! [A: $tType] :
( ( comm_monoid_mult @ A )
=> ! [G: nat > A,N: nat] :
( ( groups7121269368397514597t_prod @ nat @ A @ G @ ( set_or1337092689740270186AtMost @ nat @ ( zero_zero @ nat ) @ ( suc @ N ) ) )
= ( times_times @ A @ ( G @ ( zero_zero @ nat ) ) @ ( groups7121269368397514597t_prod @ nat @ A @ ( comp @ nat @ A @ nat @ G @ suc ) @ ( set_or1337092689740270186AtMost @ nat @ ( zero_zero @ nat ) @ N ) ) ) ) ) ).
% prod.atLeast0_atMost_Suc_shift
thf(fact_6557_sum_OatLeastLessThan__shift__0,axiom,
! [A: $tType] :
( ( comm_monoid_add @ A )
=> ! [G: nat > A,M: nat,N: nat] :
( ( groups7311177749621191930dd_sum @ nat @ A @ G @ ( set_or7035219750837199246ssThan @ nat @ M @ N ) )
= ( groups7311177749621191930dd_sum @ nat @ A @ ( comp @ nat @ A @ nat @ G @ ( plus_plus @ nat @ M ) ) @ ( set_or7035219750837199246ssThan @ nat @ ( zero_zero @ nat ) @ ( minus_minus @ nat @ N @ M ) ) ) ) ) ).
% sum.atLeastLessThan_shift_0
thf(fact_6558_prod_OatLeastLessThan__shift__0,axiom,
! [A: $tType] :
( ( comm_monoid_mult @ A )
=> ! [G: nat > A,M: nat,N: nat] :
( ( groups7121269368397514597t_prod @ nat @ A @ G @ ( set_or7035219750837199246ssThan @ nat @ M @ N ) )
= ( groups7121269368397514597t_prod @ nat @ A @ ( comp @ nat @ A @ nat @ G @ ( plus_plus @ nat @ M ) ) @ ( set_or7035219750837199246ssThan @ nat @ ( zero_zero @ nat ) @ ( minus_minus @ nat @ N @ M ) ) ) ) ) ).
% prod.atLeastLessThan_shift_0
thf(fact_6559_sum_OatLeast__lessThan__pred__shift,axiom,
! [A: $tType] :
( ( comm_monoid_add @ A )
=> ! [G: nat > A,M: nat,N: nat] :
( ( groups7311177749621191930dd_sum @ nat @ A
@ ( comp @ nat @ A @ nat @ G
@ ^ [N3: nat] : ( minus_minus @ nat @ N3 @ ( suc @ ( zero_zero @ nat ) ) ) )
@ ( set_or7035219750837199246ssThan @ nat @ ( suc @ M ) @ ( suc @ N ) ) )
= ( groups7311177749621191930dd_sum @ nat @ A @ G @ ( set_or7035219750837199246ssThan @ nat @ M @ N ) ) ) ) ).
% sum.atLeast_lessThan_pred_shift
thf(fact_6560_sum_OatLeast__atMost__pred__shift,axiom,
! [A: $tType] :
( ( comm_monoid_add @ A )
=> ! [G: nat > A,M: nat,N: nat] :
( ( groups7311177749621191930dd_sum @ nat @ A
@ ( comp @ nat @ A @ nat @ G
@ ^ [N3: nat] : ( minus_minus @ nat @ N3 @ ( suc @ ( zero_zero @ nat ) ) ) )
@ ( set_or1337092689740270186AtMost @ nat @ ( suc @ M ) @ ( suc @ N ) ) )
= ( groups7311177749621191930dd_sum @ nat @ A @ G @ ( set_or1337092689740270186AtMost @ nat @ M @ N ) ) ) ) ).
% sum.atLeast_atMost_pred_shift
thf(fact_6561_prod_OatLeast__lessThan__pred__shift,axiom,
! [A: $tType] :
( ( comm_monoid_mult @ A )
=> ! [G: nat > A,M: nat,N: nat] :
( ( groups7121269368397514597t_prod @ nat @ A
@ ( comp @ nat @ A @ nat @ G
@ ^ [N3: nat] : ( minus_minus @ nat @ N3 @ ( suc @ ( zero_zero @ nat ) ) ) )
@ ( set_or7035219750837199246ssThan @ nat @ ( suc @ M ) @ ( suc @ N ) ) )
= ( groups7121269368397514597t_prod @ nat @ A @ G @ ( set_or7035219750837199246ssThan @ nat @ M @ N ) ) ) ) ).
% prod.atLeast_lessThan_pred_shift
thf(fact_6562_prod_OatLeast__atMost__pred__shift,axiom,
! [A: $tType] :
( ( comm_monoid_mult @ A )
=> ! [G: nat > A,M: nat,N: nat] :
( ( groups7121269368397514597t_prod @ nat @ A
@ ( comp @ nat @ A @ nat @ G
@ ^ [N3: nat] : ( minus_minus @ nat @ N3 @ ( suc @ ( zero_zero @ nat ) ) ) )
@ ( set_or1337092689740270186AtMost @ nat @ ( suc @ M ) @ ( suc @ N ) ) )
= ( groups7121269368397514597t_prod @ nat @ A @ G @ ( set_or1337092689740270186AtMost @ nat @ M @ N ) ) ) ) ).
% prod.atLeast_atMost_pred_shift
thf(fact_6563_sum_OatLeastAtMost__shift__0,axiom,
! [A: $tType] :
( ( comm_monoid_add @ A )
=> ! [M: nat,N: nat,G: nat > A] :
( ( ord_less_eq @ nat @ M @ N )
=> ( ( groups7311177749621191930dd_sum @ nat @ A @ G @ ( set_or1337092689740270186AtMost @ nat @ M @ N ) )
= ( groups7311177749621191930dd_sum @ nat @ A @ ( comp @ nat @ A @ nat @ G @ ( plus_plus @ nat @ M ) ) @ ( set_or1337092689740270186AtMost @ nat @ ( zero_zero @ nat ) @ ( minus_minus @ nat @ N @ M ) ) ) ) ) ) ).
% sum.atLeastAtMost_shift_0
thf(fact_6564_prod_OatLeastAtMost__shift__0,axiom,
! [A: $tType] :
( ( comm_monoid_mult @ A )
=> ! [M: nat,N: nat,G: nat > A] :
( ( ord_less_eq @ nat @ M @ N )
=> ( ( groups7121269368397514597t_prod @ nat @ A @ G @ ( set_or1337092689740270186AtMost @ nat @ M @ N ) )
= ( groups7121269368397514597t_prod @ nat @ A @ ( comp @ nat @ A @ nat @ G @ ( plus_plus @ nat @ M ) ) @ ( set_or1337092689740270186AtMost @ nat @ ( zero_zero @ nat ) @ ( minus_minus @ nat @ N @ M ) ) ) ) ) ) ).
% prod.atLeastAtMost_shift_0
thf(fact_6565_shiftl__Suc__0,axiom,
! [N: nat] :
( ( bit_Sh4282982442137083160shiftl @ nat @ ( suc @ ( zero_zero @ nat ) ) @ N )
= ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N ) ) ).
% shiftl_Suc_0
thf(fact_6566_shiftl__minus__1__numeral,axiom,
! [A: $tType] :
( ( bit_ri3973907225187159222ations @ A )
=> ! [N: num] :
( ( bit_Sh4282982442137083160shiftl @ A @ ( uminus_uminus @ A @ ( one_one @ A ) ) @ ( numeral_numeral @ nat @ N ) )
= ( bit_ri4277139882892585799ns_not @ A @ ( bit_se2239418461657761734s_mask @ A @ ( numeral_numeral @ nat @ N ) ) ) ) ) ).
% shiftl_minus_1_numeral
thf(fact_6567_shiftl__0,axiom,
! [A: $tType] :
( ( bit_se359711467146920520ations @ A )
=> ! [N: nat] :
( ( bit_Sh4282982442137083160shiftl @ A @ ( zero_zero @ A ) @ N )
= ( zero_zero @ A ) ) ) ).
% shiftl_0
thf(fact_6568_shiftl__of__0,axiom,
! [A: $tType] :
( ( bit_se359711467146920520ations @ A )
=> ! [A2: A] :
( ( bit_Sh4282982442137083160shiftl @ A @ A2 @ ( zero_zero @ nat ) )
= A2 ) ) ).
% shiftl_of_0
thf(fact_6569_size__list__map,axiom,
! [A: $tType,B: $tType,F2: A > nat,G: B > A,Xs2: list @ B] :
( ( size_list @ A @ F2 @ ( map @ B @ A @ G @ Xs2 ) )
= ( size_list @ B @ ( comp @ A @ nat @ B @ F2 @ G ) @ Xs2 ) ) ).
% size_list_map
thf(fact_6570_shiftl__numeral__Suc,axiom,
! [A: $tType] :
( ( bit_se359711467146920520ations @ A )
=> ! [M: num,N: nat] :
( ( bit_Sh4282982442137083160shiftl @ A @ ( numeral_numeral @ A @ M ) @ ( suc @ N ) )
= ( bit_se4730199178511100633sh_bit @ A @ ( suc @ N ) @ ( numeral_numeral @ A @ M ) ) ) ) ).
% shiftl_numeral_Suc
thf(fact_6571_shiftl__numeral__numeral,axiom,
! [A: $tType] :
( ( bit_se359711467146920520ations @ A )
=> ! [M: num,N: num] :
( ( bit_Sh4282982442137083160shiftl @ A @ ( numeral_numeral @ A @ M ) @ ( numeral_numeral @ nat @ N ) )
= ( bit_se4730199178511100633sh_bit @ A @ ( numeral_numeral @ nat @ N ) @ ( numeral_numeral @ A @ M ) ) ) ) ).
% shiftl_numeral_numeral
thf(fact_6572_shiftl__minus__numeral__Suc,axiom,
! [A: $tType] :
( ( bit_ri3973907225187159222ations @ A )
=> ! [M: num,N: nat] :
( ( bit_Sh4282982442137083160shiftl @ A @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ M ) ) @ ( suc @ N ) )
= ( bit_se4730199178511100633sh_bit @ A @ ( suc @ N ) @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ M ) ) ) ) ) ).
% shiftl_minus_numeral_Suc
thf(fact_6573_shiftl__minus__numeral__numeral,axiom,
! [A: $tType] :
( ( bit_ri3973907225187159222ations @ A )
=> ! [M: num,N: num] :
( ( bit_Sh4282982442137083160shiftl @ A @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ M ) ) @ ( numeral_numeral @ nat @ N ) )
= ( bit_se4730199178511100633sh_bit @ A @ ( numeral_numeral @ nat @ N ) @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ M ) ) ) ) ) ).
% shiftl_minus_numeral_numeral
thf(fact_6574_shiftl__of__Suc,axiom,
! [A: $tType] :
( ( bit_se359711467146920520ations @ A )
=> ! [A2: A,N: nat] :
( ( bit_Sh4282982442137083160shiftl @ A @ A2 @ ( suc @ N ) )
= ( bit_Sh4282982442137083160shiftl @ A @ ( times_times @ A @ A2 @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) @ N ) ) ) ).
% shiftl_of_Suc
thf(fact_6575_shiftl__1,axiom,
! [A: $tType] :
( ( bit_se359711467146920520ations @ A )
=> ! [N: nat] :
( ( bit_Sh4282982442137083160shiftl @ A @ ( one_one @ A ) @ N )
= ( power_power @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ N ) ) ) ).
% shiftl_1
thf(fact_6576_foldr__map,axiom,
! [C: $tType,B: $tType,A: $tType,G: B > A > A,F2: C > B,Xs2: list @ C,A2: A] :
( ( foldr @ B @ A @ G @ ( map @ C @ B @ F2 @ Xs2 ) @ A2 )
= ( foldr @ C @ A @ ( comp @ B @ ( A > A ) @ C @ G @ F2 ) @ Xs2 @ A2 ) ) ).
% foldr_map
thf(fact_6577_list_Osize__gen__o__map,axiom,
! [B: $tType,A: $tType,F2: B > nat,G: A > B] :
( ( comp @ ( list @ B ) @ nat @ ( list @ A ) @ ( size_list @ B @ F2 ) @ ( map @ A @ B @ G ) )
= ( size_list @ A @ ( comp @ B @ nat @ A @ F2 @ G ) ) ) ).
% list.size_gen_o_map
thf(fact_6578_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_6579_sum_Oeq__fold,axiom,
! [A: $tType,B: $tType] :
( ( comm_monoid_add @ A )
=> ( ( groups7311177749621191930dd_sum @ B @ A )
= ( ^ [G3: B > A] : ( finite_fold @ B @ A @ ( comp @ A @ ( A > A ) @ B @ ( plus_plus @ A ) @ G3 ) @ ( zero_zero @ A ) ) ) ) ) ).
% sum.eq_fold
thf(fact_6580_prod_Oeq__fold,axiom,
! [A: $tType,B: $tType] :
( ( comm_monoid_mult @ A )
=> ( ( groups7121269368397514597t_prod @ B @ A )
= ( ^ [G3: B > A] : ( finite_fold @ B @ A @ ( comp @ A @ ( A > A ) @ B @ ( times_times @ A ) @ G3 ) @ ( one_one @ A ) ) ) ) ) ).
% prod.eq_fold
thf(fact_6581_one__bit__shiftl,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [N: nat] :
( ( generi7602027413899671122et_bit @ ( word @ A ) @ ( zero_zero @ ( word @ A ) ) @ N @ $true )
= ( bit_Sh4282982442137083160shiftl @ ( word @ A ) @ ( one_one @ ( word @ A ) ) @ N ) ) ) ).
% one_bit_shiftl
thf(fact_6582_shiftl__eq__mult,axiom,
! [A: $tType] :
( ( bit_se359711467146920520ations @ A )
=> ( ( bit_Sh4282982442137083160shiftl @ A )
= ( ^ [X2: A,N3: nat] : ( times_times @ A @ X2 @ ( power_power @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ N3 ) ) ) ) ) ).
% shiftl_eq_mult
thf(fact_6583_Code__Target__Int_Onegative__def,axiom,
( code_Target_negative
= ( comp @ int @ int @ num @ ( uminus_uminus @ int ) @ ( numeral_numeral @ int ) ) ) ).
% Code_Target_Int.negative_def
thf(fact_6584_shiftr__minus__numeral__numeral,axiom,
! [A: $tType] :
( ( bit_ri3973907225187159222ations @ A )
=> ! [M: num,N: num] :
( ( bit_Sh4282982442137083166shiftr @ A @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ M ) ) @ ( numeral_numeral @ nat @ N ) )
= ( bit_se4197421643247451524op_bit @ A @ ( numeral_numeral @ nat @ N ) @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ M ) ) ) ) ) ).
% shiftr_minus_numeral_numeral
thf(fact_6585_shiftr__0,axiom,
! [A: $tType] :
( ( bit_se359711467146920520ations @ A )
=> ! [N: nat] :
( ( bit_Sh4282982442137083166shiftr @ A @ ( zero_zero @ A ) @ N )
= ( zero_zero @ A ) ) ) ).
% shiftr_0
thf(fact_6586_shiftr__of__0,axiom,
! [A: $tType] :
( ( bit_se359711467146920520ations @ A )
=> ! [A2: A] :
( ( bit_Sh4282982442137083166shiftr @ A @ A2 @ ( zero_zero @ nat ) )
= A2 ) ) ).
% shiftr_of_0
thf(fact_6587_shiftr__Suc__0,axiom,
! [N: nat] :
( ( bit_Sh4282982442137083166shiftr @ nat @ ( suc @ ( zero_zero @ nat ) ) @ N )
= ( zero_neq_one_of_bool @ nat
@ ( N
= ( zero_zero @ nat ) ) ) ) ).
% shiftr_Suc_0
thf(fact_6588_shiftr__numeral__Suc,axiom,
! [A: $tType] :
( ( bit_se359711467146920520ations @ A )
=> ! [M: num,N: nat] :
( ( bit_Sh4282982442137083166shiftr @ A @ ( numeral_numeral @ A @ M ) @ ( suc @ N ) )
= ( bit_se4197421643247451524op_bit @ A @ ( suc @ N ) @ ( numeral_numeral @ A @ M ) ) ) ) ).
% shiftr_numeral_Suc
thf(fact_6589_shiftr__numeral__numeral,axiom,
! [A: $tType] :
( ( bit_se359711467146920520ations @ A )
=> ! [M: num,N: num] :
( ( bit_Sh4282982442137083166shiftr @ A @ ( numeral_numeral @ A @ M ) @ ( numeral_numeral @ nat @ N ) )
= ( bit_se4197421643247451524op_bit @ A @ ( numeral_numeral @ nat @ N ) @ ( numeral_numeral @ A @ M ) ) ) ) ).
% shiftr_numeral_numeral
thf(fact_6590_shiftr__1,axiom,
! [A: $tType] :
( ( bit_se359711467146920520ations @ A )
=> ! [N: nat] :
( ( bit_Sh4282982442137083166shiftr @ A @ ( one_one @ A ) @ N )
= ( zero_neq_one_of_bool @ A
@ ( N
= ( zero_zero @ nat ) ) ) ) ) ).
% shiftr_1
thf(fact_6591_shiftr__minus__numeral__Suc,axiom,
! [A: $tType] :
( ( bit_ri3973907225187159222ations @ A )
=> ! [M: num,N: nat] :
( ( bit_Sh4282982442137083166shiftr @ A @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ M ) ) @ ( suc @ N ) )
= ( bit_se4197421643247451524op_bit @ A @ ( suc @ N ) @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ M ) ) ) ) ) ).
% shiftr_minus_numeral_Suc
thf(fact_6592_bit__shiftr__eq,axiom,
! [A: $tType] :
( ( bit_se359711467146920520ations @ A )
=> ! [A2: A,N: nat] :
( ( bit_se5641148757651400278ts_bit @ A @ ( bit_Sh4282982442137083166shiftr @ A @ A2 @ N ) )
= ( comp @ nat @ $o @ nat @ ( bit_se5641148757651400278ts_bit @ A @ A2 ) @ ( plus_plus @ nat @ N ) ) ) ) ).
% bit_shiftr_eq
thf(fact_6593_shiftr__eq__div,axiom,
! [A: $tType] :
( ( bit_se359711467146920520ations @ A )
=> ( ( bit_Sh4282982442137083166shiftr @ A )
= ( ^ [X2: A,N3: nat] : ( divide_divide @ A @ X2 @ ( power_power @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ N3 ) ) ) ) ) ).
% shiftr_eq_div
thf(fact_6594_bit__revcast__iff,axiom,
! [B: $tType,A: $tType] :
( ( ( type_len @ A )
& ( type_len @ B ) )
=> ! [W: word @ A,N: nat] :
( ( bit_se5641148757651400278ts_bit @ ( word @ B ) @ ( revcast @ A @ B @ W ) @ N )
= ( ( ord_less_eq @ nat @ ( minus_minus @ nat @ ( type_len0_len_of @ B @ ( type2 @ B ) ) @ ( type_len0_len_of @ A @ ( type2 @ A ) ) ) @ N )
& ( ord_less @ nat @ N @ ( type_len0_len_of @ B @ ( type2 @ B ) ) )
& ( bit_se5641148757651400278ts_bit @ ( word @ A ) @ W @ ( minus_minus @ nat @ ( plus_plus @ nat @ N @ ( minus_minus @ nat @ ( type_len0_len_of @ A @ ( type2 @ A ) ) @ ( type_len0_len_of @ B @ ( type2 @ B ) ) ) ) @ ( minus_minus @ nat @ ( type_len0_len_of @ B @ ( type2 @ B ) ) @ ( type_len0_len_of @ A @ ( type2 @ A ) ) ) ) ) ) ) ) ).
% bit_revcast_iff
thf(fact_6595_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_6596_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_6597_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_6598_word__sless__msb__less,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ( ( word_sless @ A )
= ( ^ [X2: word @ A,Y6: word @ A] :
( ( ( most_s684356279273892711sb_msb @ ( word @ A ) @ Y6 )
=> ( most_s684356279273892711sb_msb @ ( word @ A ) @ X2 ) )
& ( ( ( most_s684356279273892711sb_msb @ ( word @ A ) @ X2 )
& ~ ( most_s684356279273892711sb_msb @ ( word @ A ) @ Y6 ) )
| ( ord_less @ ( word @ A ) @ X2 @ Y6 ) ) ) ) ) ) ).
% word_sless_msb_less
thf(fact_6599_msb__word__iff__sless__0,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ( ( most_s684356279273892711sb_msb @ ( word @ A ) )
= ( ^ [W2: word @ A] : ( word_sless @ A @ W2 @ ( zero_zero @ ( word @ A ) ) ) ) ) ) ).
% msb_word_iff_sless_0
thf(fact_6600_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_6601_word__msb__0,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ~ ( most_s684356279273892711sb_msb @ ( word @ A ) @ ( zero_zero @ ( word @ A ) ) ) ) ).
% word_msb_0
thf(fact_6602_word__msb__sint,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ( ( most_s684356279273892711sb_msb @ ( word @ A ) )
= ( ^ [W2: word @ A] : ( ord_less @ int @ ( ring_1_signed @ A @ int @ W2 ) @ ( zero_zero @ int ) ) ) ) ) ).
% word_msb_sint
thf(fact_6603_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_6604_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_6605_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_6606_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_6607_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_6608_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_6609_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_6610_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_6611_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_6612_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_6613_inj__map__eq__map,axiom,
! [B: $tType,A: $tType,F2: A > B,Xs2: list @ A,Ys: list @ A] :
( ( inj_on @ A @ B @ F2 @ ( top_top @ ( set @ A ) ) )
=> ( ( ( map @ A @ B @ F2 @ Xs2 )
= ( map @ A @ B @ F2 @ Ys ) )
= ( Xs2 = Ys ) ) ) ).
% inj_map_eq_map
thf(fact_6614_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_6615_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_6616_inj__mapI,axiom,
! [B: $tType,A: $tType,F2: A > B] :
( ( inj_on @ A @ B @ F2 @ ( top_top @ ( set @ A ) ) )
=> ( inj_on @ ( list @ A ) @ ( list @ B ) @ ( map @ A @ B @ F2 ) @ ( top_top @ ( set @ ( list @ A ) ) ) ) ) ).
% inj_mapI
thf(fact_6617_inj__map,axiom,
! [B: $tType,A: $tType,F2: A > B] :
( ( inj_on @ ( list @ A ) @ ( list @ B ) @ ( map @ A @ B @ F2 ) @ ( top_top @ ( set @ ( list @ A ) ) ) )
= ( inj_on @ A @ B @ F2 @ ( top_top @ ( set @ A ) ) ) ) ).
% inj_map
thf(fact_6618_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_6619_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_6620_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_6621_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_6622_inj__on__strict__subset,axiom,
! [B: $tType,A: $tType,F2: A > B,B4: set @ A,A3: set @ A] :
( ( inj_on @ A @ B @ F2 @ B4 )
=> ( ( ord_less @ ( set @ A ) @ A3 @ B4 )
=> ( ord_less @ ( set @ B ) @ ( image @ A @ B @ F2 @ A3 ) @ ( image @ A @ B @ F2 @ B4 ) ) ) ) ).
% inj_on_strict_subset
thf(fact_6623_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_6624_sorted__list__of__set_Oinj__on,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ( inj_on @ A @ A
@ ^ [X2: A] : X2
@ ( top_top @ ( set @ A ) ) ) ) ).
% sorted_list_of_set.inj_on
thf(fact_6625_map__injective,axiom,
! [A: $tType,B: $tType,F2: B > A,Xs2: list @ B,Ys: list @ B] :
( ( ( map @ B @ A @ F2 @ Xs2 )
= ( map @ B @ A @ F2 @ Ys ) )
=> ( ( inj_on @ B @ A @ F2 @ ( top_top @ ( set @ B ) ) )
=> ( Xs2 = Ys ) ) ) ).
% map_injective
thf(fact_6626_inj__mapD,axiom,
! [B: $tType,A: $tType,F2: A > B] :
( ( inj_on @ ( list @ A ) @ ( list @ B ) @ ( map @ A @ B @ F2 ) @ ( top_top @ ( set @ ( list @ A ) ) ) )
=> ( inj_on @ A @ B @ F2 @ ( top_top @ ( set @ A ) ) ) ) ).
% inj_mapD
thf(fact_6627_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_6628_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_6629_injective__scaleR,axiom,
! [A: $tType] :
( ( real_V4867850818363320053vector @ A )
=> ! [C2: real] :
( ( C2
!= ( zero_zero @ real ) )
=> ( inj_on @ A @ A @ ( real_V8093663219630862766scaleR @ A @ C2 ) @ ( top_top @ ( set @ A ) ) ) ) ) ).
% injective_scaleR
thf(fact_6630_finite__imp__nat__seg__image__inj__on,axiom,
! [A: $tType,A3: set @ A] :
( ( finite_finite2 @ A @ A3 )
=> ? [N2: nat,F5: nat > A] :
( ( A3
= ( image @ nat @ A @ F5
@ ( collect @ nat
@ ^ [I2: nat] : ( ord_less @ nat @ I2 @ N2 ) ) ) )
& ( inj_on @ nat @ A @ F5
@ ( collect @ nat
@ ^ [I2: nat] : ( ord_less @ nat @ I2 @ N2 ) ) ) ) ) ).
% finite_imp_nat_seg_image_inj_on
thf(fact_6631_DERIV__real__root__generic,axiom,
! [N: nat,X: real,D5: real] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ N )
=> ( ( X
!= ( zero_zero @ real ) )
=> ( ( ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N )
=> ( ( ord_less @ real @ ( zero_zero @ real ) @ X )
=> ( D5
= ( inverse_inverse @ real @ ( times_times @ real @ ( semiring_1_of_nat @ real @ N ) @ ( power_power @ real @ ( root @ N @ X ) @ ( minus_minus @ nat @ N @ ( suc @ ( zero_zero @ nat ) ) ) ) ) ) ) ) )
=> ( ( ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N )
=> ( ( ord_less @ real @ X @ ( zero_zero @ real ) )
=> ( D5
= ( uminus_uminus @ real @ ( inverse_inverse @ real @ ( times_times @ real @ ( semiring_1_of_nat @ real @ N ) @ ( power_power @ real @ ( root @ N @ X ) @ ( minus_minus @ nat @ N @ ( suc @ ( zero_zero @ nat ) ) ) ) ) ) ) ) ) )
=> ( ( ~ ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N )
=> ( D5
= ( inverse_inverse @ real @ ( times_times @ real @ ( semiring_1_of_nat @ real @ N ) @ ( power_power @ real @ ( root @ N @ X ) @ ( minus_minus @ nat @ N @ ( suc @ ( zero_zero @ nat ) ) ) ) ) ) ) )
=> ( has_field_derivative @ real @ ( root @ N ) @ D5 @ ( topolo174197925503356063within @ real @ X @ ( top_top @ ( set @ real ) ) ) ) ) ) ) ) ) ).
% DERIV_real_root_generic
thf(fact_6632_DERIV__even__real__root,axiom,
! [N: nat,X: real] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ N )
=> ( ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N )
=> ( ( ord_less @ real @ X @ ( zero_zero @ real ) )
=> ( has_field_derivative @ real @ ( root @ N ) @ ( inverse_inverse @ real @ ( times_times @ real @ ( uminus_uminus @ real @ ( semiring_1_of_nat @ real @ N ) ) @ ( power_power @ real @ ( root @ N @ X ) @ ( minus_minus @ nat @ N @ ( suc @ ( zero_zero @ nat ) ) ) ) ) ) @ ( topolo174197925503356063within @ real @ X @ ( top_top @ ( set @ real ) ) ) ) ) ) ) ).
% DERIV_even_real_root
thf(fact_6633_inj__on__Cons1,axiom,
! [A: $tType,X: A,A3: set @ ( list @ A )] : ( inj_on @ ( list @ A ) @ ( list @ A ) @ ( cons @ A @ X ) @ A3 ) ).
% inj_on_Cons1
thf(fact_6634_inj__on__diff__nat,axiom,
! [N4: set @ nat,K: nat] :
( ! [N2: nat] :
( ( member @ nat @ N2 @ N4 )
=> ( ord_less_eq @ nat @ K @ N2 ) )
=> ( inj_on @ nat @ nat
@ ^ [N3: nat] : ( minus_minus @ nat @ N3 @ K )
@ N4 ) ) ).
% inj_on_diff_nat
thf(fact_6635_inj__on__set__encode,axiom,
inj_on @ ( set @ nat ) @ nat @ nat_set_encode @ ( collect @ ( set @ nat ) @ ( finite_finite2 @ nat ) ) ).
% inj_on_set_encode
thf(fact_6636_inj__Suc,axiom,
! [N4: set @ nat] : ( inj_on @ nat @ nat @ suc @ N4 ) ).
% inj_Suc
thf(fact_6637_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_6638_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_6639_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_6640_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_6641_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_6642_DERIV__chain__s,axiom,
! [A: $tType] :
( ( real_V3459762299906320749_field @ A )
=> ! [S: set @ A,G: A > A,G4: A > A,F2: A > A,F8: A,X: A] :
( ! [X3: A] :
( ( member @ A @ X3 @ S )
=> ( has_field_derivative @ A @ G @ ( G4 @ X3 ) @ ( topolo174197925503356063within @ A @ X3 @ ( top_top @ ( set @ A ) ) ) ) )
=> ( ( has_field_derivative @ A @ F2 @ F8 @ ( topolo174197925503356063within @ A @ X @ ( top_top @ ( set @ A ) ) ) )
=> ( ( member @ A @ ( F2 @ X ) @ S )
=> ( has_field_derivative @ A
@ ^ [X2: A] : ( G @ ( F2 @ X2 ) )
@ ( times_times @ A @ F8 @ ( G4 @ ( F2 @ X ) ) )
@ ( topolo174197925503356063within @ A @ X @ ( top_top @ ( set @ A ) ) ) ) ) ) ) ) ).
% DERIV_chain_s
thf(fact_6643_DERIV__chain3,axiom,
! [A: $tType] :
( ( real_V3459762299906320749_field @ A )
=> ! [G: A > A,G4: A > A,F2: A > A,F8: A,X: A] :
( ! [X3: A] : ( has_field_derivative @ A @ G @ ( G4 @ X3 ) @ ( topolo174197925503356063within @ A @ X3 @ ( top_top @ ( set @ A ) ) ) )
=> ( ( has_field_derivative @ A @ F2 @ F8 @ ( topolo174197925503356063within @ A @ X @ ( top_top @ ( set @ A ) ) ) )
=> ( has_field_derivative @ A
@ ^ [X2: A] : ( G @ ( F2 @ X2 ) )
@ ( times_times @ A @ F8 @ ( G4 @ ( F2 @ X ) ) )
@ ( topolo174197925503356063within @ A @ X @ ( top_top @ ( set @ A ) ) ) ) ) ) ) ).
% DERIV_chain3
thf(fact_6644_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_6645_DERIV__chain_H,axiom,
! [A: $tType] :
( ( real_V3459762299906320749_field @ A )
=> ! [F2: A > A,D5: A,X: A,S: set @ A,G: A > A,E5: A] :
( ( has_field_derivative @ A @ F2 @ D5 @ ( 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 @ D5 )
@ ( topolo174197925503356063within @ A @ X @ S ) ) ) ) ) ).
% DERIV_chain'
thf(fact_6646_MVT2,axiom,
! [A2: real,B2: real,F2: real > real,F8: real > real] :
( ( ord_less @ real @ A2 @ B2 )
=> ( ! [X3: real] :
( ( ord_less_eq @ real @ A2 @ X3 )
=> ( ( ord_less_eq @ real @ X3 @ B2 )
=> ( has_field_derivative @ real @ F2 @ ( F8 @ X3 ) @ ( topolo174197925503356063within @ real @ X3 @ ( top_top @ ( set @ real ) ) ) ) ) )
=> ? [Z3: real] :
( ( ord_less @ real @ A2 @ Z3 )
& ( ord_less @ real @ Z3 @ B2 )
& ( ( minus_minus @ real @ ( F2 @ B2 ) @ ( F2 @ A2 ) )
= ( times_times @ real @ ( minus_minus @ real @ B2 @ A2 ) @ ( F8 @ Z3 ) ) ) ) ) ) ).
% MVT2
thf(fact_6647_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_6648_DERIV__pos__inc__right,axiom,
! [F2: real > real,L: real,X: real] :
( ( has_field_derivative @ real @ F2 @ L @ ( topolo174197925503356063within @ real @ X @ ( top_top @ ( set @ real ) ) ) )
=> ( ( ord_less @ real @ ( zero_zero @ real ) @ L )
=> ? [D3: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ D3 )
& ! [H7: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ H7 )
=> ( ( ord_less @ real @ H7 @ D3 )
=> ( ord_less @ real @ ( F2 @ X ) @ ( F2 @ ( plus_plus @ real @ X @ H7 ) ) ) ) ) ) ) ) ).
% DERIV_pos_inc_right
thf(fact_6649_DERIV__neg__dec__right,axiom,
! [F2: real > real,L: real,X: real] :
( ( has_field_derivative @ real @ F2 @ L @ ( topolo174197925503356063within @ real @ X @ ( top_top @ ( set @ real ) ) ) )
=> ( ( ord_less @ real @ L @ ( zero_zero @ real ) )
=> ? [D3: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ D3 )
& ! [H7: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ H7 )
=> ( ( ord_less @ real @ H7 @ D3 )
=> ( ord_less @ real @ ( F2 @ ( plus_plus @ real @ X @ H7 ) ) @ ( F2 @ X ) ) ) ) ) ) ) ).
% DERIV_neg_dec_right
thf(fact_6650_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_6651_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_6652_DERIV__ln,axiom,
! [X: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ X )
=> ( has_field_derivative @ real @ ( ln_ln @ real ) @ ( inverse_inverse @ real @ X ) @ ( topolo174197925503356063within @ real @ X @ ( top_top @ ( set @ real ) ) ) ) ) ).
% DERIV_ln
thf(fact_6653_deriv__nonneg__imp__mono,axiom,
! [A2: real,B2: real,G: real > real,G4: real > real] :
( ! [X3: real] :
( ( member @ real @ X3 @ ( set_or1337092689740270186AtMost @ real @ A2 @ B2 ) )
=> ( has_field_derivative @ real @ G @ ( G4 @ X3 ) @ ( topolo174197925503356063within @ real @ X3 @ ( top_top @ ( set @ real ) ) ) ) )
=> ( ! [X3: real] :
( ( member @ real @ X3 @ ( set_or1337092689740270186AtMost @ real @ A2 @ B2 ) )
=> ( ord_less_eq @ real @ ( zero_zero @ real ) @ ( G4 @ X3 ) ) )
=> ( ( ord_less_eq @ real @ A2 @ B2 )
=> ( ord_less_eq @ real @ ( G @ A2 ) @ ( G @ B2 ) ) ) ) ) ).
% deriv_nonneg_imp_mono
thf(fact_6654_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_6655_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_6656_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_6657_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_6658_DERIV__local__const,axiom,
! [F2: real > real,L: real,X: real,D2: real] :
( ( has_field_derivative @ real @ F2 @ L @ ( topolo174197925503356063within @ real @ X @ ( top_top @ ( set @ real ) ) ) )
=> ( ( ord_less @ real @ ( zero_zero @ real ) @ D2 )
=> ( ! [Y3: real] :
( ( ord_less @ real @ ( abs_abs @ real @ ( minus_minus @ real @ X @ Y3 ) ) @ D2 )
=> ( ( F2 @ X )
= ( F2 @ Y3 ) ) )
=> ( L
= ( zero_zero @ real ) ) ) ) ) ).
% DERIV_local_const
thf(fact_6659_DERIV__pos__inc__left,axiom,
! [F2: real > real,L: real,X: real] :
( ( has_field_derivative @ real @ F2 @ L @ ( topolo174197925503356063within @ real @ X @ ( top_top @ ( set @ real ) ) ) )
=> ( ( ord_less @ real @ ( zero_zero @ real ) @ L )
=> ? [D3: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ D3 )
& ! [H7: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ H7 )
=> ( ( ord_less @ real @ H7 @ D3 )
=> ( ord_less @ real @ ( F2 @ ( minus_minus @ real @ X @ H7 ) ) @ ( F2 @ X ) ) ) ) ) ) ) ).
% DERIV_pos_inc_left
thf(fact_6660_DERIV__neg__dec__left,axiom,
! [F2: real > real,L: real,X: real] :
( ( has_field_derivative @ real @ F2 @ L @ ( topolo174197925503356063within @ real @ X @ ( top_top @ ( set @ real ) ) ) )
=> ( ( ord_less @ real @ L @ ( zero_zero @ real ) )
=> ? [D3: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ D3 )
& ! [H7: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ H7 )
=> ( ( ord_less @ real @ H7 @ D3 )
=> ( ord_less @ real @ ( F2 @ X ) @ ( F2 @ ( minus_minus @ real @ X @ H7 ) ) ) ) ) ) ) ) ).
% DERIV_neg_dec_left
thf(fact_6661_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_6662_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_6663_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_6664_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_6665_DERIV__inverse_H,axiom,
! [A: $tType] :
( ( real_V3459762299906320749_field @ A )
=> ! [F2: A > A,D5: A,X: A,S: set @ A] :
( ( has_field_derivative @ A @ F2 @ D5 @ ( 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 ) ) @ D5 ) @ ( inverse_inverse @ A @ ( F2 @ X ) ) ) )
@ ( topolo174197925503356063within @ A @ X @ S ) ) ) ) ) ).
% DERIV_inverse'
thf(fact_6666_DERIV__divide,axiom,
! [A: $tType] :
( ( real_V3459762299906320749_field @ A )
=> ! [F2: A > A,D5: A,X: A,S: set @ A,G: A > A,E5: A] :
( ( has_field_derivative @ A @ F2 @ D5 @ ( 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 @ D5 @ ( G @ X ) ) @ ( times_times @ A @ ( F2 @ X ) @ E5 ) ) @ ( times_times @ A @ ( G @ X ) @ ( G @ X ) ) )
@ ( topolo174197925503356063within @ A @ X @ S ) ) ) ) ) ) ).
% DERIV_divide
thf(fact_6667_DERIV__cmult__right,axiom,
! [A: $tType] :
( ( real_V3459762299906320749_field @ A )
=> ! [F2: A > A,D5: A,X: A,S: set @ A,C2: A] :
( ( has_field_derivative @ A @ F2 @ D5 @ ( topolo174197925503356063within @ A @ X @ S ) )
=> ( has_field_derivative @ A
@ ^ [X2: A] : ( times_times @ A @ ( F2 @ X2 ) @ C2 )
@ ( times_times @ A @ D5 @ C2 )
@ ( topolo174197925503356063within @ A @ X @ S ) ) ) ) ).
% DERIV_cmult_right
thf(fact_6668_DERIV__cmult,axiom,
! [A: $tType] :
( ( real_V3459762299906320749_field @ A )
=> ! [F2: A > A,D5: A,X: A,S: set @ A,C2: A] :
( ( has_field_derivative @ A @ F2 @ D5 @ ( topolo174197925503356063within @ A @ X @ S ) )
=> ( has_field_derivative @ A
@ ^ [X2: A] : ( times_times @ A @ C2 @ ( F2 @ X2 ) )
@ ( times_times @ A @ C2 @ D5 )
@ ( topolo174197925503356063within @ A @ X @ S ) ) ) ) ).
% DERIV_cmult
thf(fact_6669_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_6670_has__field__derivative__cosh,axiom,
! [A10: $tType] :
( ( ( real_Vector_banach @ A10 )
& ( real_V3459762299906320749_field @ A10 ) )
=> ! [G: A10 > A10,Db: A10,X: A10,S: set @ A10] :
( ( has_field_derivative @ A10 @ G @ Db @ ( topolo174197925503356063within @ A10 @ X @ S ) )
=> ( has_field_derivative @ A10
@ ^ [X2: A10] : ( cosh @ A10 @ ( G @ X2 ) )
@ ( times_times @ A10 @ ( sinh @ A10 @ ( G @ X ) ) @ Db )
@ ( topolo174197925503356063within @ A10 @ X @ S ) ) ) ) ).
% has_field_derivative_cosh
thf(fact_6671_has__field__derivative__sinh,axiom,
! [A10: $tType] :
( ( ( real_Vector_banach @ A10 )
& ( real_V3459762299906320749_field @ A10 ) )
=> ! [G: A10 > A10,Db: A10,X: A10,S: set @ A10] :
( ( has_field_derivative @ A10 @ G @ Db @ ( topolo174197925503356063within @ A10 @ X @ S ) )
=> ( has_field_derivative @ A10
@ ^ [X2: A10] : ( sinh @ A10 @ ( G @ X2 ) )
@ ( times_times @ A10 @ ( cosh @ A10 @ ( G @ X ) ) @ Db )
@ ( topolo174197925503356063within @ A10 @ X @ S ) ) ) ) ).
% has_field_derivative_sinh
thf(fact_6672_DERIV__mult_H,axiom,
! [A: $tType] :
( ( real_V3459762299906320749_field @ A )
=> ! [F2: A > A,D5: A,X: A,S: set @ A,G: A > A,E5: A] :
( ( has_field_derivative @ A @ F2 @ D5 @ ( 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 @ D5 @ ( G @ X ) ) )
@ ( topolo174197925503356063within @ A @ X @ S ) ) ) ) ) ).
% DERIV_mult'
thf(fact_6673_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_6674_has__real__derivative__neg__dec__right,axiom,
! [F2: real > real,L: real,X: real,S3: set @ real] :
( ( has_field_derivative @ real @ F2 @ L @ ( topolo174197925503356063within @ real @ X @ S3 ) )
=> ( ( ord_less @ real @ L @ ( zero_zero @ real ) )
=> ? [D3: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ D3 )
& ! [H7: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ H7 )
=> ( ( member @ real @ ( plus_plus @ real @ X @ H7 ) @ S3 )
=> ( ( ord_less @ real @ H7 @ D3 )
=> ( ord_less @ real @ ( F2 @ ( plus_plus @ real @ X @ H7 ) ) @ ( F2 @ X ) ) ) ) ) ) ) ) ).
% has_real_derivative_neg_dec_right
thf(fact_6675_has__real__derivative__pos__inc__right,axiom,
! [F2: real > real,L: real,X: real,S3: set @ real] :
( ( has_field_derivative @ real @ F2 @ L @ ( topolo174197925503356063within @ real @ X @ S3 ) )
=> ( ( ord_less @ real @ ( zero_zero @ real ) @ L )
=> ? [D3: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ D3 )
& ! [H7: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ H7 )
=> ( ( member @ real @ ( plus_plus @ real @ X @ H7 ) @ S3 )
=> ( ( ord_less @ real @ H7 @ D3 )
=> ( ord_less @ real @ ( F2 @ X ) @ ( F2 @ ( plus_plus @ real @ X @ H7 ) ) ) ) ) ) ) ) ) ).
% has_real_derivative_pos_inc_right
thf(fact_6676_field__differentiable__add,axiom,
! [A: $tType] :
( ( real_V3459762299906320749_field @ A )
=> ! [F2: A > A,F8: A,F3: filter @ A,G: A > A,G4: A] :
( ( has_field_derivative @ A @ F2 @ F8 @ F3 )
=> ( ( has_field_derivative @ A @ G @ G4 @ F3 )
=> ( has_field_derivative @ A
@ ^ [Z7: A] : ( plus_plus @ A @ ( F2 @ Z7 ) @ ( G @ Z7 ) )
@ ( plus_plus @ A @ F8 @ G4 )
@ F3 ) ) ) ) ).
% field_differentiable_add
thf(fact_6677_DERIV__add,axiom,
! [A: $tType] :
( ( real_V3459762299906320749_field @ A )
=> ! [F2: A > A,D5: A,X: A,S: set @ A,G: A > A,E5: A] :
( ( has_field_derivative @ A @ F2 @ D5 @ ( 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 @ D5 @ E5 )
@ ( topolo174197925503356063within @ A @ X @ S ) ) ) ) ) ).
% DERIV_add
thf(fact_6678_DERIV__cdivide,axiom,
! [A: $tType] :
( ( real_V3459762299906320749_field @ A )
=> ! [F2: A > A,D5: A,X: A,S: set @ A,C2: A] :
( ( has_field_derivative @ A @ F2 @ D5 @ ( topolo174197925503356063within @ A @ X @ S ) )
=> ( has_field_derivative @ A
@ ^ [X2: A] : ( divide_divide @ A @ ( F2 @ X2 ) @ C2 )
@ ( divide_divide @ A @ D5 @ C2 )
@ ( topolo174197925503356063within @ A @ X @ S ) ) ) ) ).
% DERIV_cdivide
thf(fact_6679_has__real__derivative__pos__inc__left,axiom,
! [F2: real > real,L: real,X: real,S3: set @ real] :
( ( has_field_derivative @ real @ F2 @ L @ ( topolo174197925503356063within @ real @ X @ S3 ) )
=> ( ( ord_less @ real @ ( zero_zero @ real ) @ L )
=> ? [D3: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ D3 )
& ! [H7: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ H7 )
=> ( ( member @ real @ ( minus_minus @ real @ X @ H7 ) @ S3 )
=> ( ( ord_less @ real @ H7 @ D3 )
=> ( ord_less @ real @ ( F2 @ ( minus_minus @ real @ X @ H7 ) ) @ ( F2 @ X ) ) ) ) ) ) ) ) ).
% has_real_derivative_pos_inc_left
thf(fact_6680_has__real__derivative__neg__dec__left,axiom,
! [F2: real > real,L: real,X: real,S3: set @ real] :
( ( has_field_derivative @ real @ F2 @ L @ ( topolo174197925503356063within @ real @ X @ S3 ) )
=> ( ( ord_less @ real @ L @ ( zero_zero @ real ) )
=> ? [D3: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ D3 )
& ! [H7: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ H7 )
=> ( ( member @ real @ ( minus_minus @ real @ X @ H7 ) @ S3 )
=> ( ( ord_less @ real @ H7 @ D3 )
=> ( ord_less @ real @ ( F2 @ X ) @ ( F2 @ ( minus_minus @ real @ X @ H7 ) ) ) ) ) ) ) ) ) ).
% has_real_derivative_neg_dec_left
thf(fact_6681_DERIV__ident,axiom,
! [A: $tType] :
( ( real_V3459762299906320749_field @ A )
=> ! [F3: filter @ A] :
( has_field_derivative @ A
@ ^ [X2: A] : X2
@ ( one_one @ A )
@ F3 ) ) ).
% DERIV_ident
thf(fact_6682_DERIV__const,axiom,
! [A: $tType] :
( ( real_V3459762299906320749_field @ A )
=> ! [K: A,F3: filter @ A] :
( has_field_derivative @ A
@ ^ [X2: A] : K
@ ( zero_zero @ A )
@ F3 ) ) ).
% DERIV_const
thf(fact_6683_DERIV__power__Suc,axiom,
! [A: $tType] :
( ( real_V3459762299906320749_field @ A )
=> ! [F2: A > A,D5: A,X: A,S: set @ A,N: nat] :
( ( has_field_derivative @ A @ F2 @ D5 @ ( topolo174197925503356063within @ A @ X @ S ) )
=> ( has_field_derivative @ A
@ ^ [X2: A] : ( power_power @ A @ ( F2 @ X2 ) @ ( suc @ N ) )
@ ( times_times @ A @ ( plus_plus @ A @ ( one_one @ A ) @ ( semiring_1_of_nat @ A @ N ) ) @ ( times_times @ A @ D5 @ ( power_power @ A @ ( F2 @ X ) @ N ) ) )
@ ( topolo174197925503356063within @ A @ X @ S ) ) ) ) ).
% DERIV_power_Suc
thf(fact_6684_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_6685_DERIV__power,axiom,
! [A: $tType] :
( ( real_V3459762299906320749_field @ A )
=> ! [F2: A > A,D5: A,X: A,S: set @ A,N: nat] :
( ( has_field_derivative @ A @ F2 @ D5 @ ( topolo174197925503356063within @ A @ X @ S ) )
=> ( has_field_derivative @ A
@ ^ [X2: A] : ( power_power @ A @ ( F2 @ X2 ) @ N )
@ ( times_times @ A @ ( semiring_1_of_nat @ A @ N ) @ ( times_times @ A @ D5 @ ( power_power @ A @ ( F2 @ X ) @ ( minus_minus @ nat @ N @ ( suc @ ( zero_zero @ nat ) ) ) ) ) )
@ ( topolo174197925503356063within @ A @ X @ S ) ) ) ) ).
% DERIV_power
thf(fact_6686_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_6687_DERIV__local__min,axiom,
! [F2: real > real,L: real,X: real,D2: real] :
( ( has_field_derivative @ real @ F2 @ L @ ( topolo174197925503356063within @ real @ X @ ( top_top @ ( set @ real ) ) ) )
=> ( ( ord_less @ real @ ( zero_zero @ real ) @ D2 )
=> ( ! [Y3: real] :
( ( ord_less @ real @ ( abs_abs @ real @ ( minus_minus @ real @ X @ Y3 ) ) @ D2 )
=> ( ord_less_eq @ real @ ( F2 @ X ) @ ( F2 @ Y3 ) ) )
=> ( L
= ( zero_zero @ real ) ) ) ) ) ).
% DERIV_local_min
thf(fact_6688_DERIV__local__max,axiom,
! [F2: real > real,L: real,X: real,D2: real] :
( ( has_field_derivative @ real @ F2 @ L @ ( topolo174197925503356063within @ real @ X @ ( top_top @ ( set @ real ) ) ) )
=> ( ( ord_less @ real @ ( zero_zero @ real ) @ D2 )
=> ( ! [Y3: real] :
( ( ord_less @ real @ ( abs_abs @ real @ ( minus_minus @ real @ X @ Y3 ) ) @ D2 )
=> ( ord_less_eq @ real @ ( F2 @ Y3 ) @ ( F2 @ X ) ) )
=> ( L
= ( zero_zero @ real ) ) ) ) ) ).
% DERIV_local_max
thf(fact_6689_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_6690_DERIV__pow,axiom,
! [N: nat,X: real,S: set @ real] :
( has_field_derivative @ real
@ ^ [X2: real] : ( power_power @ real @ X2 @ N )
@ ( times_times @ real @ ( semiring_1_of_nat @ real @ N ) @ ( power_power @ real @ X @ ( minus_minus @ nat @ N @ ( suc @ ( zero_zero @ nat ) ) ) ) )
@ ( topolo174197925503356063within @ real @ X @ S ) ) ).
% DERIV_pow
thf(fact_6691_termdiffs__strong__converges__everywhere,axiom,
! [A: $tType] :
( ( ( real_Vector_banach @ A )
& ( real_V3459762299906320749_field @ A ) )
=> ! [C2: nat > A,X: A] :
( ! [Y3: A] :
( summable @ A
@ ^ [N3: nat] : ( times_times @ A @ ( C2 @ N3 ) @ ( power_power @ A @ Y3 @ N3 ) ) )
=> ( has_field_derivative @ A
@ ^ [X2: A] :
( suminf @ A
@ ^ [N3: nat] : ( times_times @ A @ ( C2 @ N3 ) @ ( power_power @ A @ X2 @ N3 ) ) )
@ ( suminf @ A
@ ^ [N3: nat] : ( times_times @ A @ ( diffs @ A @ C2 @ N3 ) @ ( power_power @ A @ X @ N3 ) ) )
@ ( topolo174197925503356063within @ A @ X @ ( top_top @ ( set @ A ) ) ) ) ) ) ).
% termdiffs_strong_converges_everywhere
thf(fact_6692_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_6693_DERIV__fun__pow,axiom,
! [G: real > real,M: real,X: real,N: nat] :
( ( has_field_derivative @ real @ G @ M @ ( topolo174197925503356063within @ real @ X @ ( top_top @ ( set @ real ) ) ) )
=> ( has_field_derivative @ real
@ ^ [X2: real] : ( power_power @ real @ ( G @ X2 ) @ N )
@ ( times_times @ real @ ( times_times @ real @ ( semiring_1_of_nat @ real @ N ) @ ( power_power @ real @ ( G @ X ) @ ( minus_minus @ nat @ N @ ( one_one @ nat ) ) ) ) @ M )
@ ( topolo174197925503356063within @ real @ X @ ( top_top @ ( set @ real ) ) ) ) ) ).
% DERIV_fun_pow
thf(fact_6694_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_6695_DERIV__quotient,axiom,
! [A: $tType] :
( ( real_V3459762299906320749_field @ A )
=> ! [F2: A > A,D2: A,X: A,S: set @ A,G: A > A,E: A] :
( ( has_field_derivative @ A @ F2 @ D2 @ ( topolo174197925503356063within @ A @ X @ S ) )
=> ( ( has_field_derivative @ A @ G @ E @ ( topolo174197925503356063within @ A @ X @ S ) )
=> ( ( ( G @ X )
!= ( zero_zero @ A ) )
=> ( has_field_derivative @ A
@ ^ [Y6: A] : ( divide_divide @ A @ ( F2 @ Y6 ) @ ( G @ Y6 ) )
@ ( divide_divide @ A @ ( minus_minus @ A @ ( times_times @ A @ D2 @ ( 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_6696_DERIV__inverse__fun,axiom,
! [A: $tType] :
( ( real_V3459762299906320749_field @ A )
=> ! [F2: A > A,D2: A,X: A,S: set @ A] :
( ( has_field_derivative @ A @ F2 @ D2 @ ( 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 @ D2 @ ( inverse_inverse @ A @ ( power_power @ A @ ( F2 @ X ) @ ( suc @ ( suc @ ( zero_zero @ nat ) ) ) ) ) ) )
@ ( topolo174197925503356063within @ A @ X @ S ) ) ) ) ) ).
% DERIV_inverse_fun
thf(fact_6697_finite__imp__inj__to__nat__seg,axiom,
! [A: $tType,A3: set @ A] :
( ( finite_finite2 @ A @ A3 )
=> ? [F5: A > nat,N2: nat] :
( ( ( image @ A @ nat @ F5 @ A3 )
= ( collect @ nat
@ ^ [I2: nat] : ( ord_less @ nat @ I2 @ N2 ) ) )
& ( inj_on @ A @ nat @ F5 @ A3 ) ) ) ).
% finite_imp_inj_to_nat_seg
thf(fact_6698_finite__imp__inj__to__nat__seg_H,axiom,
! [A: $tType,A3: set @ A] :
( ( finite_finite2 @ A @ A3 )
=> ~ ! [F5: A > nat] :
( ? [N2: nat] :
( ( image @ A @ nat @ F5 @ A3 )
= ( collect @ nat
@ ^ [I2: nat] : ( ord_less @ nat @ I2 @ N2 ) ) )
=> ~ ( inj_on @ A @ nat @ F5 @ A3 ) ) ) ).
% finite_imp_inj_to_nat_seg'
thf(fact_6699_termdiffs__sums__strong,axiom,
! [A: $tType] :
( ( ( real_Vector_banach @ A )
& ( real_V3459762299906320749_field @ A ) )
=> ! [K6: real,C2: nat > A,F2: A > A,F8: A,Z: A] :
( ! [Z3: A] :
( ( ord_less @ real @ ( real_V7770717601297561774m_norm @ A @ Z3 ) @ K6 )
=> ( sums @ A
@ ^ [N3: nat] : ( times_times @ A @ ( C2 @ N3 ) @ ( power_power @ A @ Z3 @ N3 ) )
@ ( F2 @ Z3 ) ) )
=> ( ( has_field_derivative @ A @ F2 @ F8 @ ( topolo174197925503356063within @ A @ Z @ ( top_top @ ( set @ A ) ) ) )
=> ( ( ord_less @ real @ ( real_V7770717601297561774m_norm @ A @ Z ) @ K6 )
=> ( sums @ A
@ ^ [N3: nat] : ( times_times @ A @ ( diffs @ A @ C2 @ N3 ) @ ( power_power @ A @ Z @ N3 ) )
@ F8 ) ) ) ) ) ).
% termdiffs_sums_strong
thf(fact_6700_has__real__derivative__powr,axiom,
! [Z: real,R2: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ Z )
=> ( has_field_derivative @ real
@ ^ [Z7: real] : ( powr @ real @ Z7 @ 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_6701_termdiffs,axiom,
! [A: $tType] :
( ( ( real_Vector_banach @ A )
& ( real_V3459762299906320749_field @ A ) )
=> ! [C2: nat > A,K6: A,X: A] :
( ( summable @ A
@ ^ [N3: nat] : ( times_times @ A @ ( C2 @ N3 ) @ ( power_power @ A @ K6 @ N3 ) ) )
=> ( ( summable @ A
@ ^ [N3: nat] : ( times_times @ A @ ( diffs @ A @ C2 @ N3 ) @ ( power_power @ A @ K6 @ N3 ) ) )
=> ( ( summable @ A
@ ^ [N3: nat] : ( times_times @ A @ ( diffs @ A @ ( diffs @ A @ C2 ) @ N3 ) @ ( power_power @ A @ K6 @ N3 ) ) )
=> ( ( ord_less @ real @ ( real_V7770717601297561774m_norm @ A @ X ) @ ( real_V7770717601297561774m_norm @ A @ K6 ) )
=> ( has_field_derivative @ A
@ ^ [X2: A] :
( suminf @ A
@ ^ [N3: nat] : ( times_times @ A @ ( C2 @ N3 ) @ ( power_power @ A @ X2 @ N3 ) ) )
@ ( suminf @ A
@ ^ [N3: nat] : ( times_times @ A @ ( diffs @ A @ C2 @ N3 ) @ ( power_power @ A @ X @ N3 ) ) )
@ ( topolo174197925503356063within @ A @ X @ ( top_top @ ( set @ A ) ) ) ) ) ) ) ) ) ).
% termdiffs
thf(fact_6702_termdiffs__strong,axiom,
! [A: $tType] :
( ( ( real_Vector_banach @ A )
& ( real_V3459762299906320749_field @ A ) )
=> ! [C2: nat > A,K6: A,X: A] :
( ( summable @ A
@ ^ [N3: nat] : ( times_times @ A @ ( C2 @ N3 ) @ ( power_power @ A @ K6 @ N3 ) ) )
=> ( ( ord_less @ real @ ( real_V7770717601297561774m_norm @ A @ X ) @ ( real_V7770717601297561774m_norm @ A @ K6 ) )
=> ( has_field_derivative @ A
@ ^ [X2: A] :
( suminf @ A
@ ^ [N3: nat] : ( times_times @ A @ ( C2 @ N3 ) @ ( power_power @ A @ X2 @ N3 ) ) )
@ ( suminf @ A
@ ^ [N3: nat] : ( times_times @ A @ ( diffs @ A @ C2 @ N3 ) @ ( power_power @ A @ X @ N3 ) ) )
@ ( topolo174197925503356063within @ A @ X @ ( top_top @ ( set @ A ) ) ) ) ) ) ) ).
% termdiffs_strong
thf(fact_6703_termdiffs__strong_H,axiom,
! [A: $tType] :
( ( ( real_Vector_banach @ A )
& ( real_V3459762299906320749_field @ A ) )
=> ! [K6: real,C2: nat > A,Z: A] :
( ! [Z3: A] :
( ( ord_less @ real @ ( real_V7770717601297561774m_norm @ A @ Z3 ) @ K6 )
=> ( summable @ A
@ ^ [N3: nat] : ( times_times @ A @ ( C2 @ N3 ) @ ( power_power @ A @ Z3 @ N3 ) ) ) )
=> ( ( ord_less @ real @ ( real_V7770717601297561774m_norm @ A @ Z ) @ K6 )
=> ( has_field_derivative @ A
@ ^ [Z7: A] :
( suminf @ A
@ ^ [N3: nat] : ( times_times @ A @ ( C2 @ N3 ) @ ( power_power @ A @ Z7 @ N3 ) ) )
@ ( suminf @ A
@ ^ [N3: nat] : ( times_times @ A @ ( diffs @ A @ C2 @ N3 ) @ ( power_power @ A @ Z @ N3 ) ) )
@ ( topolo174197925503356063within @ A @ Z @ ( top_top @ ( set @ A ) ) ) ) ) ) ) ).
% termdiffs_strong'
thf(fact_6704_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_6705_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_6706_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_6707_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_6708_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_6709_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_6710_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_6711_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_6712_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_6713_has__field__derivative__tanh,axiom,
! [A10: $tType] :
( ( ( real_Vector_banach @ A10 )
& ( real_V3459762299906320749_field @ A10 ) )
=> ! [G: A10 > A10,X: A10,Db: A10,S: set @ A10] :
( ( ( cosh @ A10 @ ( G @ X ) )
!= ( zero_zero @ A10 ) )
=> ( ( has_field_derivative @ A10 @ G @ Db @ ( topolo174197925503356063within @ A10 @ X @ S ) )
=> ( has_field_derivative @ A10
@ ^ [X2: A10] : ( tanh @ A10 @ ( G @ X2 ) )
@ ( times_times @ A10 @ ( minus_minus @ A10 @ ( one_one @ A10 ) @ ( power_power @ A10 @ ( tanh @ A10 @ ( G @ X ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ Db )
@ ( topolo174197925503356063within @ A10 @ X @ S ) ) ) ) ) ).
% has_field_derivative_tanh
thf(fact_6714_DERIV__real__sqrt__generic,axiom,
! [X: real,D5: real] :
( ( X
!= ( zero_zero @ real ) )
=> ( ( ( ord_less @ real @ ( zero_zero @ real ) @ X )
=> ( D5
= ( divide_divide @ real @ ( inverse_inverse @ real @ ( sqrt @ X ) ) @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) ) )
=> ( ( ( ord_less @ real @ X @ ( zero_zero @ real ) )
=> ( D5
= ( divide_divide @ real @ ( uminus_uminus @ real @ ( inverse_inverse @ real @ ( sqrt @ X ) ) ) @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) ) )
=> ( has_field_derivative @ real @ sqrt @ D5 @ ( topolo174197925503356063within @ real @ X @ ( top_top @ ( set @ real ) ) ) ) ) ) ) ).
% DERIV_real_sqrt_generic
thf(fact_6715_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_6716_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_6717_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_6718_DERIV__real__root,axiom,
! [N: nat,X: real] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ N )
=> ( ( ord_less @ real @ ( zero_zero @ real ) @ X )
=> ( has_field_derivative @ real @ ( root @ N ) @ ( inverse_inverse @ real @ ( times_times @ real @ ( semiring_1_of_nat @ real @ N ) @ ( power_power @ real @ ( root @ N @ X ) @ ( minus_minus @ nat @ N @ ( suc @ ( zero_zero @ nat ) ) ) ) ) ) @ ( topolo174197925503356063within @ real @ X @ ( top_top @ ( set @ real ) ) ) ) ) ) ).
% DERIV_real_root
thf(fact_6719_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_6720_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_6721_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_6722_Maclaurin__all__le,axiom,
! [Diff: nat > real > real,F2: real > real,X: real,N: nat] :
( ( ( Diff @ ( zero_zero @ nat ) )
= F2 )
=> ( ! [M2: nat,X3: real] : ( has_field_derivative @ real @ ( Diff @ M2 ) @ ( Diff @ ( suc @ M2 ) @ X3 ) @ ( topolo174197925503356063within @ real @ X3 @ ( top_top @ ( set @ real ) ) ) )
=> ? [T7: real] :
( ( ord_less_eq @ real @ ( abs_abs @ real @ T7 ) @ ( abs_abs @ real @ X ) )
& ( ( F2 @ X )
= ( plus_plus @ real
@ ( groups7311177749621191930dd_sum @ nat @ real
@ ^ [M4: nat] : ( times_times @ real @ ( divide_divide @ real @ ( Diff @ M4 @ ( zero_zero @ real ) ) @ ( semiring_char_0_fact @ real @ M4 ) ) @ ( power_power @ real @ X @ M4 ) )
@ ( set_ord_lessThan @ nat @ N ) )
@ ( times_times @ real @ ( divide_divide @ real @ ( Diff @ N @ T7 ) @ ( semiring_char_0_fact @ real @ N ) ) @ ( power_power @ real @ X @ N ) ) ) ) ) ) ) ).
% Maclaurin_all_le
thf(fact_6723_Maclaurin__all__le__objl,axiom,
! [Diff: nat > real > real,F2: real > real,X: real,N: nat] :
( ( ( ( Diff @ ( zero_zero @ nat ) )
= F2 )
& ! [M2: nat,X3: real] : ( has_field_derivative @ real @ ( Diff @ M2 ) @ ( Diff @ ( suc @ M2 ) @ X3 ) @ ( topolo174197925503356063within @ real @ X3 @ ( top_top @ ( set @ real ) ) ) ) )
=> ? [T7: real] :
( ( ord_less_eq @ real @ ( abs_abs @ real @ T7 ) @ ( abs_abs @ real @ X ) )
& ( ( F2 @ X )
= ( plus_plus @ real
@ ( groups7311177749621191930dd_sum @ nat @ real
@ ^ [M4: nat] : ( times_times @ real @ ( divide_divide @ real @ ( Diff @ M4 @ ( zero_zero @ real ) ) @ ( semiring_char_0_fact @ real @ M4 ) ) @ ( power_power @ real @ X @ M4 ) )
@ ( set_ord_lessThan @ nat @ N ) )
@ ( times_times @ real @ ( divide_divide @ real @ ( Diff @ N @ T7 ) @ ( semiring_char_0_fact @ real @ N ) ) @ ( power_power @ real @ X @ N ) ) ) ) ) ) ).
% Maclaurin_all_le_objl
thf(fact_6724_DERIV__odd__real__root,axiom,
! [N: nat,X: real] :
( ~ ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N )
=> ( ( X
!= ( zero_zero @ real ) )
=> ( has_field_derivative @ real @ ( root @ N ) @ ( inverse_inverse @ real @ ( times_times @ real @ ( semiring_1_of_nat @ real @ N ) @ ( power_power @ real @ ( root @ N @ X ) @ ( minus_minus @ nat @ N @ ( suc @ ( zero_zero @ nat ) ) ) ) ) ) @ ( topolo174197925503356063within @ real @ X @ ( top_top @ ( set @ real ) ) ) ) ) ) ).
% DERIV_odd_real_root
thf(fact_6725_Maclaurin,axiom,
! [H2: real,N: nat,Diff: nat > real > real,F2: real > real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ H2 )
=> ( ( ord_less @ nat @ ( zero_zero @ nat ) @ N )
=> ( ( ( Diff @ ( zero_zero @ nat ) )
= F2 )
=> ( ! [M2: nat,T7: real] :
( ( ( ord_less @ nat @ M2 @ N )
& ( ord_less_eq @ real @ ( zero_zero @ real ) @ T7 )
& ( ord_less_eq @ real @ T7 @ H2 ) )
=> ( has_field_derivative @ real @ ( Diff @ M2 ) @ ( Diff @ ( suc @ M2 ) @ T7 ) @ ( topolo174197925503356063within @ real @ T7 @ ( top_top @ ( set @ real ) ) ) ) )
=> ? [T7: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ T7 )
& ( ord_less @ real @ T7 @ H2 )
& ( ( F2 @ H2 )
= ( plus_plus @ real
@ ( groups7311177749621191930dd_sum @ nat @ real
@ ^ [M4: nat] : ( times_times @ real @ ( divide_divide @ real @ ( Diff @ M4 @ ( zero_zero @ real ) ) @ ( semiring_char_0_fact @ real @ M4 ) ) @ ( power_power @ real @ H2 @ M4 ) )
@ ( set_ord_lessThan @ nat @ N ) )
@ ( times_times @ real @ ( divide_divide @ real @ ( Diff @ N @ T7 ) @ ( semiring_char_0_fact @ real @ N ) ) @ ( power_power @ real @ H2 @ N ) ) ) ) ) ) ) ) ) ).
% Maclaurin
thf(fact_6726_Maclaurin2,axiom,
! [H2: real,Diff: nat > real > real,F2: real > real,N: nat] :
( ( ord_less @ real @ ( zero_zero @ real ) @ H2 )
=> ( ( ( Diff @ ( zero_zero @ nat ) )
= F2 )
=> ( ! [M2: nat,T7: real] :
( ( ( ord_less @ nat @ M2 @ N )
& ( ord_less_eq @ real @ ( zero_zero @ real ) @ T7 )
& ( ord_less_eq @ real @ T7 @ H2 ) )
=> ( has_field_derivative @ real @ ( Diff @ M2 ) @ ( Diff @ ( suc @ M2 ) @ T7 ) @ ( topolo174197925503356063within @ real @ T7 @ ( top_top @ ( set @ real ) ) ) ) )
=> ? [T7: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ T7 )
& ( ord_less_eq @ real @ T7 @ H2 )
& ( ( F2 @ H2 )
= ( plus_plus @ real
@ ( groups7311177749621191930dd_sum @ nat @ real
@ ^ [M4: nat] : ( times_times @ real @ ( divide_divide @ real @ ( Diff @ M4 @ ( zero_zero @ real ) ) @ ( semiring_char_0_fact @ real @ M4 ) ) @ ( power_power @ real @ H2 @ M4 ) )
@ ( set_ord_lessThan @ nat @ N ) )
@ ( times_times @ real @ ( divide_divide @ real @ ( Diff @ N @ T7 ) @ ( semiring_char_0_fact @ real @ N ) ) @ ( power_power @ real @ H2 @ N ) ) ) ) ) ) ) ) ).
% Maclaurin2
thf(fact_6727_Maclaurin__minus,axiom,
! [H2: real,N: nat,Diff: nat > real > real,F2: real > real] :
( ( ord_less @ real @ H2 @ ( zero_zero @ real ) )
=> ( ( ord_less @ nat @ ( zero_zero @ nat ) @ N )
=> ( ( ( Diff @ ( zero_zero @ nat ) )
= F2 )
=> ( ! [M2: nat,T7: real] :
( ( ( ord_less @ nat @ M2 @ N )
& ( ord_less_eq @ real @ H2 @ T7 )
& ( ord_less_eq @ real @ T7 @ ( zero_zero @ real ) ) )
=> ( has_field_derivative @ real @ ( Diff @ M2 ) @ ( Diff @ ( suc @ M2 ) @ T7 ) @ ( topolo174197925503356063within @ real @ T7 @ ( top_top @ ( set @ real ) ) ) ) )
=> ? [T7: real] :
( ( ord_less @ real @ H2 @ T7 )
& ( ord_less @ real @ T7 @ ( zero_zero @ real ) )
& ( ( F2 @ H2 )
= ( plus_plus @ real
@ ( groups7311177749621191930dd_sum @ nat @ real
@ ^ [M4: nat] : ( times_times @ real @ ( divide_divide @ real @ ( Diff @ M4 @ ( zero_zero @ real ) ) @ ( semiring_char_0_fact @ real @ M4 ) ) @ ( power_power @ real @ H2 @ M4 ) )
@ ( set_ord_lessThan @ nat @ N ) )
@ ( times_times @ real @ ( divide_divide @ real @ ( Diff @ N @ T7 ) @ ( semiring_char_0_fact @ real @ N ) ) @ ( power_power @ real @ H2 @ N ) ) ) ) ) ) ) ) ) ).
% Maclaurin_minus
thf(fact_6728_Maclaurin__all__lt,axiom,
! [Diff: nat > real > real,F2: real > real,N: nat,X: real] :
( ( ( Diff @ ( zero_zero @ nat ) )
= F2 )
=> ( ( ord_less @ nat @ ( zero_zero @ nat ) @ N )
=> ( ( X
!= ( zero_zero @ real ) )
=> ( ! [M2: nat,X3: real] : ( has_field_derivative @ real @ ( Diff @ M2 ) @ ( Diff @ ( suc @ M2 ) @ X3 ) @ ( topolo174197925503356063within @ real @ X3 @ ( top_top @ ( set @ real ) ) ) )
=> ? [T7: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ ( abs_abs @ real @ T7 ) )
& ( ord_less @ real @ ( abs_abs @ real @ T7 ) @ ( abs_abs @ real @ X ) )
& ( ( F2 @ X )
= ( plus_plus @ real
@ ( groups7311177749621191930dd_sum @ nat @ real
@ ^ [M4: nat] : ( times_times @ real @ ( divide_divide @ real @ ( Diff @ M4 @ ( zero_zero @ real ) ) @ ( semiring_char_0_fact @ real @ M4 ) ) @ ( power_power @ real @ X @ M4 ) )
@ ( set_ord_lessThan @ nat @ N ) )
@ ( times_times @ real @ ( divide_divide @ real @ ( Diff @ N @ T7 ) @ ( semiring_char_0_fact @ real @ N ) ) @ ( power_power @ real @ X @ N ) ) ) ) ) ) ) ) ) ).
% Maclaurin_all_lt
thf(fact_6729_Maclaurin__bi__le,axiom,
! [Diff: nat > real > real,F2: real > real,N: nat,X: real] :
( ( ( Diff @ ( zero_zero @ nat ) )
= F2 )
=> ( ! [M2: nat,T7: real] :
( ( ( ord_less @ nat @ M2 @ N )
& ( ord_less_eq @ real @ ( abs_abs @ real @ T7 ) @ ( abs_abs @ real @ X ) ) )
=> ( has_field_derivative @ real @ ( Diff @ M2 ) @ ( Diff @ ( suc @ M2 ) @ T7 ) @ ( topolo174197925503356063within @ real @ T7 @ ( top_top @ ( set @ real ) ) ) ) )
=> ? [T7: real] :
( ( ord_less_eq @ real @ ( abs_abs @ real @ T7 ) @ ( abs_abs @ real @ X ) )
& ( ( F2 @ X )
= ( plus_plus @ real
@ ( groups7311177749621191930dd_sum @ nat @ real
@ ^ [M4: nat] : ( times_times @ real @ ( divide_divide @ real @ ( Diff @ M4 @ ( zero_zero @ real ) ) @ ( semiring_char_0_fact @ real @ M4 ) ) @ ( power_power @ real @ X @ M4 ) )
@ ( set_ord_lessThan @ nat @ N ) )
@ ( times_times @ real @ ( divide_divide @ real @ ( Diff @ N @ T7 ) @ ( semiring_char_0_fact @ real @ N ) ) @ ( power_power @ real @ X @ N ) ) ) ) ) ) ) ).
% Maclaurin_bi_le
thf(fact_6730_Taylor__down,axiom,
! [N: nat,Diff: nat > real > real,F2: real > real,A2: real,B2: real,C2: real] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ N )
=> ( ( ( Diff @ ( zero_zero @ nat ) )
= F2 )
=> ( ! [M2: nat,T7: real] :
( ( ( ord_less @ nat @ M2 @ N )
& ( ord_less_eq @ real @ A2 @ T7 )
& ( ord_less_eq @ real @ T7 @ B2 ) )
=> ( has_field_derivative @ real @ ( Diff @ M2 ) @ ( Diff @ ( suc @ M2 ) @ T7 ) @ ( topolo174197925503356063within @ real @ T7 @ ( top_top @ ( set @ real ) ) ) ) )
=> ( ( ord_less @ real @ A2 @ C2 )
=> ( ( ord_less_eq @ real @ C2 @ B2 )
=> ? [T7: real] :
( ( ord_less @ real @ A2 @ T7 )
& ( ord_less @ real @ T7 @ C2 )
& ( ( F2 @ A2 )
= ( plus_plus @ real
@ ( groups7311177749621191930dd_sum @ nat @ real
@ ^ [M4: nat] : ( times_times @ real @ ( divide_divide @ real @ ( Diff @ M4 @ C2 ) @ ( semiring_char_0_fact @ real @ M4 ) ) @ ( power_power @ real @ ( minus_minus @ real @ A2 @ C2 ) @ M4 ) )
@ ( set_ord_lessThan @ nat @ N ) )
@ ( times_times @ real @ ( divide_divide @ real @ ( Diff @ N @ T7 ) @ ( semiring_char_0_fact @ real @ N ) ) @ ( power_power @ real @ ( minus_minus @ real @ A2 @ C2 ) @ N ) ) ) ) ) ) ) ) ) ) ).
% Taylor_down
thf(fact_6731_Taylor__up,axiom,
! [N: nat,Diff: nat > real > real,F2: real > real,A2: real,B2: real,C2: real] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ N )
=> ( ( ( Diff @ ( zero_zero @ nat ) )
= F2 )
=> ( ! [M2: nat,T7: real] :
( ( ( ord_less @ nat @ M2 @ N )
& ( ord_less_eq @ real @ A2 @ T7 )
& ( ord_less_eq @ real @ T7 @ B2 ) )
=> ( has_field_derivative @ real @ ( Diff @ M2 ) @ ( Diff @ ( suc @ M2 ) @ T7 ) @ ( topolo174197925503356063within @ real @ T7 @ ( top_top @ ( set @ real ) ) ) ) )
=> ( ( ord_less_eq @ real @ A2 @ C2 )
=> ( ( ord_less @ real @ C2 @ B2 )
=> ? [T7: real] :
( ( ord_less @ real @ C2 @ T7 )
& ( ord_less @ real @ T7 @ B2 )
& ( ( F2 @ B2 )
= ( plus_plus @ real
@ ( groups7311177749621191930dd_sum @ nat @ real
@ ^ [M4: nat] : ( times_times @ real @ ( divide_divide @ real @ ( Diff @ M4 @ C2 ) @ ( semiring_char_0_fact @ real @ M4 ) ) @ ( power_power @ real @ ( minus_minus @ real @ B2 @ C2 ) @ M4 ) )
@ ( set_ord_lessThan @ nat @ N ) )
@ ( times_times @ real @ ( divide_divide @ real @ ( Diff @ N @ T7 ) @ ( semiring_char_0_fact @ real @ N ) ) @ ( power_power @ real @ ( minus_minus @ real @ B2 @ C2 ) @ N ) ) ) ) ) ) ) ) ) ) ).
% Taylor_up
thf(fact_6732_Taylor,axiom,
! [N: nat,Diff: nat > real > real,F2: real > real,A2: real,B2: real,C2: real,X: real] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ N )
=> ( ( ( Diff @ ( zero_zero @ nat ) )
= F2 )
=> ( ! [M2: nat,T7: real] :
( ( ( ord_less @ nat @ M2 @ N )
& ( ord_less_eq @ real @ A2 @ T7 )
& ( ord_less_eq @ real @ T7 @ B2 ) )
=> ( has_field_derivative @ real @ ( Diff @ M2 ) @ ( Diff @ ( suc @ M2 ) @ T7 ) @ ( topolo174197925503356063within @ real @ T7 @ ( 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 )
=> ? [T7: real] :
( ( ( ord_less @ real @ X @ C2 )
=> ( ( ord_less @ real @ X @ T7 )
& ( ord_less @ real @ T7 @ C2 ) ) )
& ( ~ ( ord_less @ real @ X @ C2 )
=> ( ( ord_less @ real @ C2 @ T7 )
& ( ord_less @ real @ T7 @ X ) ) )
& ( ( F2 @ X )
= ( plus_plus @ real
@ ( groups7311177749621191930dd_sum @ nat @ real
@ ^ [M4: nat] : ( times_times @ real @ ( divide_divide @ real @ ( Diff @ M4 @ C2 ) @ ( semiring_char_0_fact @ real @ M4 ) ) @ ( power_power @ real @ ( minus_minus @ real @ X @ C2 ) @ M4 ) )
@ ( set_ord_lessThan @ nat @ N ) )
@ ( times_times @ real @ ( divide_divide @ real @ ( Diff @ N @ T7 ) @ ( semiring_char_0_fact @ real @ N ) ) @ ( power_power @ real @ ( minus_minus @ real @ X @ C2 ) @ N ) ) ) ) ) ) ) ) ) ) ) ) ) ).
% Taylor
thf(fact_6733_inj__sgn__power,axiom,
! [N: nat] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ N )
=> ( inj_on @ real @ real
@ ^ [Y6: real] : ( times_times @ real @ ( sgn_sgn @ real @ Y6 ) @ ( power_power @ real @ ( abs_abs @ real @ Y6 ) @ N ) )
@ ( top_top @ ( set @ real ) ) ) ) ).
% inj_sgn_power
thf(fact_6734_Maclaurin__lemma2,axiom,
! [N: nat,H2: real,Diff: nat > real > real,K: nat,B4: real] :
( ! [M2: nat,T7: real] :
( ( ( ord_less @ nat @ M2 @ N )
& ( ord_less_eq @ real @ ( zero_zero @ real ) @ T7 )
& ( ord_less_eq @ real @ T7 @ H2 ) )
=> ( has_field_derivative @ real @ ( Diff @ M2 ) @ ( Diff @ ( suc @ M2 ) @ T7 ) @ ( topolo174197925503356063within @ real @ T7 @ ( top_top @ ( set @ real ) ) ) ) )
=> ( ( N
= ( suc @ K ) )
=> ! [M3: nat,T8: real] :
( ( ( ord_less @ nat @ M3 @ N )
& ( ord_less_eq @ real @ ( zero_zero @ real ) @ T8 )
& ( ord_less_eq @ real @ T8 @ H2 ) )
=> ( has_field_derivative @ real
@ ^ [U2: real] :
( minus_minus @ real @ ( Diff @ M3 @ U2 )
@ ( plus_plus @ real
@ ( groups7311177749621191930dd_sum @ nat @ real
@ ^ [P5: nat] : ( times_times @ real @ ( divide_divide @ real @ ( Diff @ ( plus_plus @ nat @ M3 @ P5 ) @ ( zero_zero @ real ) ) @ ( semiring_char_0_fact @ real @ P5 ) ) @ ( power_power @ real @ U2 @ P5 ) )
@ ( set_ord_lessThan @ nat @ ( minus_minus @ nat @ N @ M3 ) ) )
@ ( times_times @ real @ B4 @ ( divide_divide @ real @ ( power_power @ real @ U2 @ ( minus_minus @ nat @ N @ M3 ) ) @ ( semiring_char_0_fact @ real @ ( minus_minus @ nat @ N @ M3 ) ) ) ) ) )
@ ( minus_minus @ real @ ( Diff @ ( suc @ M3 ) @ T8 )
@ ( plus_plus @ real
@ ( groups7311177749621191930dd_sum @ nat @ real
@ ^ [P5: nat] : ( times_times @ real @ ( divide_divide @ real @ ( Diff @ ( plus_plus @ nat @ ( suc @ M3 ) @ P5 ) @ ( zero_zero @ real ) ) @ ( semiring_char_0_fact @ real @ P5 ) ) @ ( power_power @ real @ T8 @ P5 ) )
@ ( set_ord_lessThan @ nat @ ( minus_minus @ nat @ N @ ( suc @ M3 ) ) ) )
@ ( times_times @ real @ B4 @ ( divide_divide @ real @ ( power_power @ real @ T8 @ ( minus_minus @ nat @ N @ ( suc @ M3 ) ) ) @ ( semiring_char_0_fact @ real @ ( minus_minus @ nat @ N @ ( suc @ M3 ) ) ) ) ) ) )
@ ( topolo174197925503356063within @ real @ T8 @ ( top_top @ ( set @ real ) ) ) ) ) ) ) ).
% Maclaurin_lemma2
thf(fact_6735_DERIV__arctan__series,axiom,
! [X: real] :
( ( ord_less @ real @ ( abs_abs @ real @ X ) @ ( one_one @ real ) )
=> ( has_field_derivative @ real
@ ^ [X10: real] :
( suminf @ real
@ ^ [K3: nat] : ( times_times @ real @ ( power_power @ real @ ( uminus_uminus @ real @ ( one_one @ real ) ) @ K3 ) @ ( times_times @ real @ ( divide_divide @ real @ ( one_one @ real ) @ ( semiring_1_of_nat @ real @ ( plus_plus @ nat @ ( times_times @ nat @ K3 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ ( one_one @ nat ) ) ) ) @ ( power_power @ real @ X10 @ ( plus_plus @ nat @ ( times_times @ nat @ K3 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ ( one_one @ nat ) ) ) ) ) )
@ ( suminf @ real
@ ^ [K3: nat] : ( times_times @ real @ ( power_power @ real @ ( uminus_uminus @ real @ ( one_one @ real ) ) @ K3 ) @ ( power_power @ real @ X @ ( times_times @ nat @ K3 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) )
@ ( topolo174197925503356063within @ real @ X @ ( top_top @ ( set @ real ) ) ) ) ) ).
% DERIV_arctan_series
thf(fact_6736_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
@ ^ [N3: nat] : ( times_times @ real @ ( times_times @ real @ ( F2 @ N3 ) @ ( semiring_1_of_nat @ real @ ( suc @ N3 ) ) ) @ ( power_power @ real @ X3 @ N3 ) ) ) )
=> ( ( member @ real @ X0 @ ( set_or5935395276787703475ssThan @ real @ ( uminus_uminus @ real @ R ) @ R ) )
=> ( ( ord_less @ real @ ( zero_zero @ real ) @ R )
=> ( has_field_derivative @ real
@ ^ [X2: real] :
( suminf @ real
@ ^ [N3: nat] : ( times_times @ real @ ( F2 @ N3 ) @ ( power_power @ real @ X2 @ ( suc @ N3 ) ) ) )
@ ( suminf @ real
@ ^ [N3: nat] : ( times_times @ real @ ( times_times @ real @ ( F2 @ N3 ) @ ( semiring_1_of_nat @ real @ ( suc @ N3 ) ) ) @ ( power_power @ real @ X0 @ N3 ) ) )
@ ( topolo174197925503356063within @ real @ X0 @ ( top_top @ ( set @ real ) ) ) ) ) ) ) ).
% DERIV_power_series'
thf(fact_6737_has__derivative__arcsin,axiom,
! [A: $tType] :
( ( real_V822414075346904944vector @ A )
=> ! [G: A > real,X: A,G4: A > real,S: set @ A] :
( ( ord_less @ real @ ( uminus_uminus @ real @ ( one_one @ real ) ) @ ( G @ X ) )
=> ( ( ord_less @ real @ ( G @ X ) @ ( one_one @ real ) )
=> ( ( has_derivative @ A @ real @ G @ G4 @ ( topolo174197925503356063within @ A @ X @ S ) )
=> ( has_derivative @ A @ real
@ ^ [X2: A] : ( arcsin @ ( G @ X2 ) )
@ ^ [X2: A] : ( times_times @ real @ ( G4 @ X2 ) @ ( inverse_inverse @ real @ ( sqrt @ ( minus_minus @ real @ ( one_one @ real ) @ ( power_power @ real @ ( G @ X ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) )
@ ( topolo174197925503356063within @ A @ X @ S ) ) ) ) ) ) ).
% has_derivative_arcsin
thf(fact_6738_greaterThanLessThan__iff,axiom,
! [A: $tType] :
( ( ord @ A )
=> ! [I: A,L: A,U: A] :
( ( member @ A @ I @ ( set_or5935395276787703475ssThan @ A @ L @ U ) )
= ( ( ord_less @ A @ L @ I )
& ( ord_less @ A @ I @ U ) ) ) ) ).
% greaterThanLessThan_iff
thf(fact_6739_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_6740_has__field__derivative__def,axiom,
! [A: $tType] :
( ( real_V3459762299906320749_field @ A )
=> ( ( has_field_derivative @ A )
= ( ^ [F4: A > A,D7: A] : ( has_derivative @ A @ A @ F4 @ ( times_times @ A @ D7 ) ) ) ) ) ).
% has_field_derivative_def
thf(fact_6741_has__derivative__imp__has__field__derivative,axiom,
! [A: $tType] :
( ( real_V3459762299906320749_field @ A )
=> ! [F2: A > A,D5: A > A,F3: filter @ A,D8: A] :
( ( has_derivative @ A @ A @ F2 @ D5 @ F3 )
=> ( ! [X3: A] :
( ( times_times @ A @ X3 @ D8 )
= ( D5 @ X3 ) )
=> ( has_field_derivative @ A @ F2 @ D8 @ F3 ) ) ) ) ).
% has_derivative_imp_has_field_derivative
thf(fact_6742_has__field__derivative__imp__has__derivative,axiom,
! [A: $tType] :
( ( real_V3459762299906320749_field @ A )
=> ! [F2: A > A,D5: A,F3: filter @ A] :
( ( has_field_derivative @ A @ F2 @ D5 @ F3 )
=> ( has_derivative @ A @ A @ F2 @ ( times_times @ A @ D5 ) @ F3 ) ) ) ).
% has_field_derivative_imp_has_derivative
thf(fact_6743_has__derivative__scaleR,axiom,
! [C: $tType,D4: $tType] :
( ( ( real_V822414075346904944vector @ D4 )
& ( real_V822414075346904944vector @ C ) )
=> ! [F2: D4 > real,F8: D4 > real,X: D4,S: set @ D4,G: D4 > C,G4: D4 > C] :
( ( has_derivative @ D4 @ real @ F2 @ F8 @ ( topolo174197925503356063within @ D4 @ X @ S ) )
=> ( ( has_derivative @ D4 @ C @ G @ G4 @ ( topolo174197925503356063within @ D4 @ X @ S ) )
=> ( has_derivative @ D4 @ C
@ ^ [X2: D4] : ( real_V8093663219630862766scaleR @ C @ ( F2 @ X2 ) @ ( G @ X2 ) )
@ ^ [H: D4] : ( plus_plus @ C @ ( real_V8093663219630862766scaleR @ C @ ( F2 @ X ) @ ( G4 @ H ) ) @ ( real_V8093663219630862766scaleR @ C @ ( F8 @ H ) @ ( G @ X ) ) )
@ ( topolo174197925503356063within @ D4 @ X @ S ) ) ) ) ) ).
% has_derivative_scaleR
thf(fact_6744_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_6745_has__derivative__const,axiom,
! [B: $tType,A: $tType] :
( ( ( real_V822414075346904944vector @ A )
& ( real_V822414075346904944vector @ B ) )
=> ! [C2: B,F3: filter @ A] :
( has_derivative @ A @ B
@ ^ [X2: A] : C2
@ ^ [X2: A] : ( zero_zero @ B )
@ F3 ) ) ).
% has_derivative_const
thf(fact_6746_has__derivative__add,axiom,
! [B: $tType,A: $tType] :
( ( ( real_V822414075346904944vector @ A )
& ( real_V822414075346904944vector @ B ) )
=> ! [F2: A > B,F8: A > B,F3: filter @ A,G: A > B,G4: A > B] :
( ( has_derivative @ A @ B @ F2 @ F8 @ F3 )
=> ( ( has_derivative @ A @ B @ G @ G4 @ F3 )
=> ( has_derivative @ A @ B
@ ^ [X2: A] : ( plus_plus @ B @ ( F2 @ X2 ) @ ( G @ X2 ) )
@ ^ [X2: A] : ( plus_plus @ B @ ( F8 @ X2 ) @ ( G4 @ X2 ) )
@ F3 ) ) ) ) ).
% has_derivative_add
thf(fact_6747_has__derivative__mult__left,axiom,
! [A: $tType,C: $tType] :
( ( ( real_V822414075346904944vector @ C )
& ( real_V4412858255891104859lgebra @ A ) )
=> ! [G: C > A,G4: C > A,F3: filter @ C,Y: A] :
( ( has_derivative @ C @ A @ G @ G4 @ F3 )
=> ( has_derivative @ C @ A
@ ^ [X2: C] : ( times_times @ A @ ( G @ X2 ) @ Y )
@ ^ [X2: C] : ( times_times @ A @ ( G4 @ X2 ) @ Y )
@ F3 ) ) ) ).
% has_derivative_mult_left
thf(fact_6748_has__derivative__mult__right,axiom,
! [A: $tType,C: $tType] :
( ( ( real_V822414075346904944vector @ C )
& ( real_V4412858255891104859lgebra @ A ) )
=> ! [G: C > A,G4: C > A,F3: filter @ C,X: A] :
( ( has_derivative @ C @ A @ G @ G4 @ F3 )
=> ( has_derivative @ C @ A
@ ^ [X2: C] : ( times_times @ A @ X @ ( G @ X2 ) )
@ ^ [X2: C] : ( times_times @ A @ X @ ( G4 @ X2 ) )
@ F3 ) ) ) ).
% has_derivative_mult_right
thf(fact_6749_has__derivative__mult,axiom,
! [A: $tType,D4: $tType] :
( ( ( real_V822414075346904944vector @ D4 )
& ( real_V4412858255891104859lgebra @ A ) )
=> ! [F2: D4 > A,F8: D4 > A,X: D4,S: set @ D4,G: D4 > A,G4: D4 > A] :
( ( has_derivative @ D4 @ A @ F2 @ F8 @ ( topolo174197925503356063within @ D4 @ X @ S ) )
=> ( ( has_derivative @ D4 @ A @ G @ G4 @ ( topolo174197925503356063within @ D4 @ X @ S ) )
=> ( has_derivative @ D4 @ A
@ ^ [X2: D4] : ( times_times @ A @ ( F2 @ X2 ) @ ( G @ X2 ) )
@ ^ [H: D4] : ( plus_plus @ A @ ( times_times @ A @ ( F2 @ X ) @ ( G4 @ H ) ) @ ( times_times @ A @ ( F8 @ H ) @ ( G @ X ) ) )
@ ( topolo174197925503356063within @ D4 @ X @ S ) ) ) ) ) ).
% has_derivative_mult
thf(fact_6750_has__derivative__zero__unique,axiom,
! [B: $tType,A: $tType] :
( ( ( real_V822414075346904944vector @ A )
& ( real_V822414075346904944vector @ B ) )
=> ! [F3: A > B,X: A] :
( ( has_derivative @ A @ B
@ ^ [X2: A] : ( zero_zero @ B )
@ F3
@ ( topolo174197925503356063within @ A @ X @ ( top_top @ ( set @ A ) ) ) )
=> ( F3
= ( ^ [H: A] : ( zero_zero @ B ) ) ) ) ) ).
% has_derivative_zero_unique
thf(fact_6751_has__derivative__exp,axiom,
! [A: $tType] :
( ( real_V822414075346904944vector @ A )
=> ! [G: A > real,G4: A > real,X: A,S: set @ A] :
( ( has_derivative @ A @ real @ G @ G4 @ ( topolo174197925503356063within @ A @ X @ S ) )
=> ( has_derivative @ A @ real
@ ^ [X2: A] : ( exp @ real @ ( G @ X2 ) )
@ ^ [X2: A] : ( times_times @ real @ ( G4 @ X2 ) @ ( exp @ real @ ( G @ X ) ) )
@ ( topolo174197925503356063within @ A @ X @ S ) ) ) ) ).
% has_derivative_exp
thf(fact_6752_greaterThanLessThan__subseteq__greaterThanLessThan,axiom,
! [A: $tType] :
( ( dense_linorder @ A )
=> ! [A2: A,B2: A,C2: A,D2: A] :
( ( ord_less_eq @ ( set @ A ) @ ( set_or5935395276787703475ssThan @ A @ A2 @ B2 ) @ ( set_or5935395276787703475ssThan @ A @ C2 @ D2 ) )
= ( ( ord_less @ A @ A2 @ B2 )
=> ( ( ord_less_eq @ A @ C2 @ A2 )
& ( ord_less_eq @ A @ B2 @ D2 ) ) ) ) ) ).
% greaterThanLessThan_subseteq_greaterThanLessThan
thf(fact_6753_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_6754_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_6755_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_6756_has__derivative__sin,axiom,
! [A: $tType] :
( ( real_V822414075346904944vector @ A )
=> ! [G: A > real,G4: A > real,X: A,S: set @ A] :
( ( has_derivative @ A @ real @ G @ G4 @ ( topolo174197925503356063within @ A @ X @ S ) )
=> ( has_derivative @ A @ real
@ ^ [X2: A] : ( sin @ real @ ( G @ X2 ) )
@ ^ [X2: A] : ( times_times @ real @ ( G4 @ X2 ) @ ( cos @ real @ ( G @ X ) ) )
@ ( topolo174197925503356063within @ A @ X @ S ) ) ) ) ).
% has_derivative_sin
thf(fact_6757_greaterThanLessThan__subseteq__atLeastAtMost__iff,axiom,
! [A: $tType] :
( ( dense_linorder @ A )
=> ! [A2: A,B2: A,C2: A,D2: A] :
( ( ord_less_eq @ ( set @ A ) @ ( set_or5935395276787703475ssThan @ A @ A2 @ B2 ) @ ( set_or1337092689740270186AtMost @ A @ C2 @ D2 ) )
= ( ( ord_less @ A @ A2 @ B2 )
=> ( ( ord_less_eq @ A @ C2 @ A2 )
& ( ord_less_eq @ A @ B2 @ D2 ) ) ) ) ) ).
% greaterThanLessThan_subseteq_atLeastAtMost_iff
thf(fact_6758_greaterThanLessThan__subseteq__atLeastLessThan__iff,axiom,
! [A: $tType] :
( ( dense_linorder @ A )
=> ! [A2: A,B2: A,C2: A,D2: A] :
( ( ord_less_eq @ ( set @ A ) @ ( set_or5935395276787703475ssThan @ A @ A2 @ B2 ) @ ( set_or7035219750837199246ssThan @ A @ C2 @ D2 ) )
= ( ( ord_less @ A @ A2 @ B2 )
=> ( ( ord_less_eq @ A @ C2 @ A2 )
& ( ord_less_eq @ A @ B2 @ D2 ) ) ) ) ) ).
% greaterThanLessThan_subseteq_atLeastLessThan_iff
thf(fact_6759_has__derivative__divide_H,axiom,
! [A: $tType,C: $tType] :
( ( ( real_V822414075346904944vector @ C )
& ( real_V3459762299906320749_field @ A ) )
=> ! [F2: C > A,F8: C > A,X: C,S3: set @ C,G: C > A,G4: C > A] :
( ( has_derivative @ C @ A @ F2 @ F8 @ ( topolo174197925503356063within @ C @ X @ S3 ) )
=> ( ( has_derivative @ C @ A @ G @ G4 @ ( 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 @ ( F8 @ H ) @ ( G @ X ) ) @ ( times_times @ A @ ( F2 @ X ) @ ( G4 @ H ) ) ) @ ( times_times @ A @ ( G @ X ) @ ( G @ X ) ) )
@ ( topolo174197925503356063within @ C @ X @ S3 ) ) ) ) ) ) ).
% has_derivative_divide'
thf(fact_6760_has__derivative__inverse,axiom,
! [A: $tType,C: $tType] :
( ( ( real_V822414075346904944vector @ C )
& ( real_V8999393235501362500lgebra @ A ) )
=> ! [F2: C > A,X: C,F8: C > A,S3: set @ C] :
( ( ( F2 @ X )
!= ( zero_zero @ A ) )
=> ( ( has_derivative @ C @ A @ F2 @ F8 @ ( 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 ) ) @ ( F8 @ H ) ) @ ( inverse_inverse @ A @ ( F2 @ X ) ) ) )
@ ( topolo174197925503356063within @ C @ X @ S3 ) ) ) ) ) ).
% has_derivative_inverse
thf(fact_6761_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_6762_DERIV__compose__FDERIV,axiom,
! [A: $tType] :
( ( real_V822414075346904944vector @ A )
=> ! [F2: real > real,F8: real,G: A > real,X: A,G4: A > real,S: set @ A] :
( ( has_field_derivative @ real @ F2 @ F8 @ ( topolo174197925503356063within @ real @ ( G @ X ) @ ( top_top @ ( set @ real ) ) ) )
=> ( ( has_derivative @ A @ real @ G @ G4 @ ( topolo174197925503356063within @ A @ X @ S ) )
=> ( has_derivative @ A @ real
@ ^ [X2: A] : ( F2 @ ( G @ X2 ) )
@ ^ [X2: A] : ( times_times @ real @ ( G4 @ X2 ) @ F8 )
@ ( topolo174197925503356063within @ A @ X @ S ) ) ) ) ) ).
% DERIV_compose_FDERIV
thf(fact_6763_has__derivative__cos,axiom,
! [A: $tType] :
( ( real_V822414075346904944vector @ A )
=> ! [G: A > real,G4: A > real,X: A,S: set @ A] :
( ( has_derivative @ A @ real @ G @ G4 @ ( topolo174197925503356063within @ A @ X @ S ) )
=> ( has_derivative @ A @ real
@ ^ [X2: A] : ( cos @ real @ ( G @ X2 ) )
@ ^ [X2: A] : ( times_times @ real @ ( G4 @ X2 ) @ ( uminus_uminus @ real @ ( sin @ real @ ( G @ X ) ) ) )
@ ( topolo174197925503356063within @ A @ X @ S ) ) ) ) ).
% has_derivative_cos
thf(fact_6764_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_6765_has__derivative__power,axiom,
! [B: $tType,A: $tType] :
( ( ( real_V822414075346904944vector @ A )
& ( real_V3459762299906320749_field @ B ) )
=> ! [F2: A > B,F8: A > B,X: A,S3: set @ A,N: nat] :
( ( has_derivative @ A @ B @ F2 @ F8 @ ( topolo174197925503356063within @ A @ X @ S3 ) )
=> ( has_derivative @ A @ B
@ ^ [X2: A] : ( power_power @ B @ ( F2 @ X2 ) @ N )
@ ^ [Y6: A] : ( times_times @ B @ ( times_times @ B @ ( semiring_1_of_nat @ B @ N ) @ ( F8 @ Y6 ) ) @ ( power_power @ B @ ( F2 @ X ) @ ( minus_minus @ nat @ N @ ( one_one @ nat ) ) ) )
@ ( topolo174197925503356063within @ A @ X @ S3 ) ) ) ) ).
% has_derivative_power
thf(fact_6766_has__derivative__ln,axiom,
! [A: $tType] :
( ( real_V822414075346904944vector @ A )
=> ! [G: A > real,X: A,G4: A > real,S: set @ A] :
( ( ord_less @ real @ ( zero_zero @ real ) @ ( G @ X ) )
=> ( ( has_derivative @ A @ real @ G @ G4 @ ( topolo174197925503356063within @ A @ X @ S ) )
=> ( has_derivative @ A @ real
@ ^ [X2: A] : ( ln_ln @ real @ ( G @ X2 ) )
@ ^ [X2: A] : ( times_times @ real @ ( G4 @ X2 ) @ ( inverse_inverse @ real @ ( G @ X ) ) )
@ ( topolo174197925503356063within @ A @ X @ S ) ) ) ) ) ).
% has_derivative_ln
thf(fact_6767_has__derivative__divide,axiom,
! [A: $tType,C: $tType] :
( ( ( real_V822414075346904944vector @ C )
& ( real_V8999393235501362500lgebra @ A ) )
=> ! [F2: C > A,F8: C > A,X: C,S3: set @ C,G: C > A,G4: C > A] :
( ( has_derivative @ C @ A @ F2 @ F8 @ ( topolo174197925503356063within @ C @ X @ S3 ) )
=> ( ( has_derivative @ C @ A @ G @ G4 @ ( 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 ) ) @ ( G4 @ H ) ) @ ( inverse_inverse @ A @ ( G @ X ) ) ) ) @ ( divide_divide @ A @ ( F8 @ H ) @ ( G @ X ) ) )
@ ( topolo174197925503356063within @ C @ X @ S3 ) ) ) ) ) ) ).
% has_derivative_divide
thf(fact_6768_has__derivative__prod,axiom,
! [B: $tType,I8: $tType,A: $tType] :
( ( ( real_V822414075346904944vector @ A )
& ( real_V3459762299906320749_field @ B ) )
=> ! [I6: set @ I8,F2: I8 > A > B,F8: I8 > A > B,X: A,S3: set @ A] :
( ! [I3: I8] :
( ( member @ I8 @ I3 @ I6 )
=> ( has_derivative @ A @ B @ ( F2 @ I3 ) @ ( F8 @ I3 ) @ ( topolo174197925503356063within @ A @ X @ S3 ) ) )
=> ( has_derivative @ A @ B
@ ^ [X2: A] :
( groups7121269368397514597t_prod @ I8 @ B
@ ^ [I2: I8] : ( F2 @ I2 @ X2 )
@ I6 )
@ ^ [Y6: A] :
( groups7311177749621191930dd_sum @ I8 @ B
@ ^ [I2: I8] :
( times_times @ B @ ( F8 @ I2 @ Y6 )
@ ( groups7121269368397514597t_prod @ I8 @ B
@ ^ [J2: I8] : ( F2 @ J2 @ X )
@ ( minus_minus @ ( set @ I8 ) @ I6 @ ( insert @ I8 @ I2 @ ( bot_bot @ ( set @ I8 ) ) ) ) ) )
@ I6 )
@ ( topolo174197925503356063within @ A @ X @ S3 ) ) ) ) ).
% has_derivative_prod
thf(fact_6769_has__derivative__powr,axiom,
! [A: $tType] :
( ( real_V822414075346904944vector @ A )
=> ! [G: A > real,G4: A > real,X: A,X7: set @ A,F2: A > real,F8: A > real] :
( ( has_derivative @ A @ real @ G @ G4 @ ( topolo174197925503356063within @ A @ X @ X7 ) )
=> ( ( has_derivative @ A @ real @ F2 @ F8 @ ( topolo174197925503356063within @ A @ X @ X7 ) )
=> ( ( ord_less @ real @ ( zero_zero @ real ) @ ( G @ X ) )
=> ( ( member @ A @ X @ X7 )
=> ( 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 @ ( F8 @ H ) @ ( ln_ln @ real @ ( G @ X ) ) ) @ ( divide_divide @ real @ ( times_times @ real @ ( G4 @ H ) @ ( F2 @ X ) ) @ ( G @ X ) ) ) )
@ ( topolo174197925503356063within @ A @ X @ X7 ) ) ) ) ) ) ) ).
% has_derivative_powr
thf(fact_6770_has__derivative__real__sqrt,axiom,
! [A: $tType] :
( ( real_V822414075346904944vector @ A )
=> ! [G: A > real,X: A,G4: A > real,S: set @ A] :
( ( ord_less @ real @ ( zero_zero @ real ) @ ( G @ X ) )
=> ( ( has_derivative @ A @ real @ G @ G4 @ ( topolo174197925503356063within @ A @ X @ S ) )
=> ( has_derivative @ A @ real
@ ^ [X2: A] : ( sqrt @ ( G @ X2 ) )
@ ^ [X2: A] : ( times_times @ real @ ( G4 @ X2 ) @ ( divide_divide @ real @ ( inverse_inverse @ real @ ( sqrt @ ( G @ X ) ) ) @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) )
@ ( topolo174197925503356063within @ A @ X @ S ) ) ) ) ) ).
% has_derivative_real_sqrt
thf(fact_6771_has__derivative__arctan,axiom,
! [A: $tType] :
( ( real_V822414075346904944vector @ A )
=> ! [G: A > real,G4: A > real,X: A,S: set @ A] :
( ( has_derivative @ A @ real @ G @ G4 @ ( topolo174197925503356063within @ A @ X @ S ) )
=> ( has_derivative @ A @ real
@ ^ [X2: A] : ( arctan @ ( G @ X2 ) )
@ ^ [X2: A] : ( times_times @ real @ ( G4 @ X2 ) @ ( inverse_inverse @ real @ ( plus_plus @ real @ ( one_one @ real ) @ ( power_power @ real @ ( G @ X ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) )
@ ( topolo174197925503356063within @ A @ X @ S ) ) ) ) ).
% has_derivative_arctan
thf(fact_6772_has__derivative__tan,axiom,
! [A: $tType] :
( ( real_V822414075346904944vector @ A )
=> ! [G: A > real,X: A,G4: A > real,S: set @ A] :
( ( ( cos @ real @ ( G @ X ) )
!= ( zero_zero @ real ) )
=> ( ( has_derivative @ A @ real @ G @ G4 @ ( topolo174197925503356063within @ A @ X @ S ) )
=> ( has_derivative @ A @ real
@ ^ [X2: A] : ( tan @ real @ ( G @ X2 ) )
@ ^ [X2: A] : ( times_times @ real @ ( G4 @ X2 ) @ ( inverse_inverse @ real @ ( power_power @ real @ ( cos @ real @ ( G @ X ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) )
@ ( topolo174197925503356063within @ A @ X @ S ) ) ) ) ) ).
% has_derivative_tan
thf(fact_6773_DERIV__series_H,axiom,
! [F2: real > nat > real,F8: real > nat > real,X0: real,A2: real,B2: real,L6: nat > real] :
( ! [N2: nat] :
( has_field_derivative @ real
@ ^ [X2: real] : ( F2 @ X2 @ N2 )
@ ( F8 @ X0 @ N2 )
@ ( topolo174197925503356063within @ real @ X0 @ ( top_top @ ( set @ real ) ) ) )
=> ( ! [X3: real] :
( ( member @ real @ X3 @ ( set_or5935395276787703475ssThan @ real @ A2 @ B2 ) )
=> ( summable @ real @ ( F2 @ X3 ) ) )
=> ( ( member @ real @ X0 @ ( set_or5935395276787703475ssThan @ real @ A2 @ B2 ) )
=> ( ( summable @ real @ ( F8 @ X0 ) )
=> ( ( summable @ real @ L6 )
=> ( ! [N2: nat,X3: real,Y3: real] :
( ( member @ real @ X3 @ ( set_or5935395276787703475ssThan @ real @ A2 @ B2 ) )
=> ( ( member @ real @ Y3 @ ( set_or5935395276787703475ssThan @ real @ A2 @ B2 ) )
=> ( ord_less_eq @ real @ ( abs_abs @ real @ ( minus_minus @ real @ ( F2 @ X3 @ N2 ) @ ( F2 @ Y3 @ N2 ) ) ) @ ( times_times @ real @ ( L6 @ N2 ) @ ( abs_abs @ real @ ( minus_minus @ real @ X3 @ Y3 ) ) ) ) ) )
=> ( has_field_derivative @ real
@ ^ [X2: real] : ( suminf @ real @ ( F2 @ X2 ) )
@ ( suminf @ real @ ( F8 @ X0 ) )
@ ( topolo174197925503356063within @ real @ X0 @ ( top_top @ ( set @ real ) ) ) ) ) ) ) ) ) ) ).
% DERIV_series'
thf(fact_6774_has__derivative__arccos,axiom,
! [A: $tType] :
( ( real_V822414075346904944vector @ A )
=> ! [G: A > real,X: A,G4: A > real,S: set @ A] :
( ( ord_less @ real @ ( uminus_uminus @ real @ ( one_one @ real ) ) @ ( G @ X ) )
=> ( ( ord_less @ real @ ( G @ X ) @ ( one_one @ real ) )
=> ( ( has_derivative @ A @ real @ G @ G4 @ ( topolo174197925503356063within @ A @ X @ S ) )
=> ( has_derivative @ A @ real
@ ^ [X2: A] : ( arccos @ ( G @ X2 ) )
@ ^ [X2: A] : ( times_times @ real @ ( G4 @ X2 ) @ ( inverse_inverse @ real @ ( uminus_uminus @ real @ ( sqrt @ ( minus_minus @ real @ ( one_one @ real ) @ ( power_power @ real @ ( G @ X ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) ) )
@ ( topolo174197925503356063within @ A @ X @ S ) ) ) ) ) ) ).
% has_derivative_arccos
thf(fact_6775_has__derivative__floor,axiom,
! [Aa: $tType,A: $tType] :
( ( ( real_V822414075346904944vector @ A )
& ( archim2362893244070406136eiling @ Aa )
& ( topolo2564578578187576103pology @ Aa ) )
=> ! [G: A > real,X: A,F2: real > Aa,G4: A > real,S: set @ A] :
( ( topolo3448309680560233919inuous @ real @ Aa @ ( topolo174197925503356063within @ real @ ( G @ X ) @ ( top_top @ ( set @ real ) ) ) @ F2 )
=> ( ~ ( member @ Aa @ ( F2 @ ( G @ X ) ) @ ( ring_1_Ints @ Aa ) )
=> ( ( has_derivative @ A @ real @ G @ G4 @ ( topolo174197925503356063within @ A @ X @ S ) )
=> ( has_derivative @ A @ real
@ ^ [X2: A] : ( ring_1_of_int @ real @ ( archim6421214686448440834_floor @ Aa @ ( F2 @ ( G @ X2 ) ) ) )
@ ^ [X2: A] : ( times_times @ real @ ( G4 @ X2 ) @ ( zero_zero @ real ) )
@ ( topolo174197925503356063within @ A @ X @ S ) ) ) ) ) ) ).
% has_derivative_floor
thf(fact_6776_termdiffs__aux,axiom,
! [A: $tType] :
( ( ( real_Vector_banach @ A )
& ( real_V3459762299906320749_field @ A ) )
=> ! [C2: nat > A,K6: A,X: A] :
( ( summable @ A
@ ^ [N3: nat] : ( times_times @ A @ ( diffs @ A @ ( diffs @ A @ C2 ) @ N3 ) @ ( power_power @ A @ K6 @ N3 ) ) )
=> ( ( ord_less @ real @ ( real_V7770717601297561774m_norm @ A @ X ) @ ( real_V7770717601297561774m_norm @ A @ K6 ) )
=> ( filterlim @ A @ A
@ ^ [H: A] :
( suminf @ A
@ ^ [N3: nat] : ( times_times @ A @ ( C2 @ N3 ) @ ( minus_minus @ A @ ( divide_divide @ A @ ( minus_minus @ A @ ( power_power @ A @ ( plus_plus @ A @ X @ H ) @ N3 ) @ ( power_power @ A @ X @ N3 ) ) @ H ) @ ( times_times @ A @ ( semiring_1_of_nat @ A @ N3 ) @ ( power_power @ A @ X @ ( minus_minus @ nat @ N3 @ ( suc @ ( zero_zero @ nat ) ) ) ) ) ) ) )
@ ( topolo7230453075368039082e_nhds @ A @ ( zero_zero @ A ) )
@ ( topolo174197925503356063within @ A @ ( zero_zero @ A ) @ ( top_top @ ( set @ A ) ) ) ) ) ) ) ).
% termdiffs_aux
thf(fact_6777_tendsto__mult__right__iff,axiom,
! [A: $tType,B: $tType] :
( ( ( field @ A )
& ( topolo4211221413907600880p_mult @ A ) )
=> ! [C2: A,F2: B > A,L: A,F3: filter @ B] :
( ( C2
!= ( zero_zero @ A ) )
=> ( ( filterlim @ B @ A
@ ^ [X2: B] : ( times_times @ A @ ( F2 @ X2 ) @ C2 )
@ ( topolo7230453075368039082e_nhds @ A @ ( times_times @ A @ L @ C2 ) )
@ F3 )
= ( filterlim @ B @ A @ F2 @ ( topolo7230453075368039082e_nhds @ A @ L ) @ F3 ) ) ) ) ).
% tendsto_mult_right_iff
thf(fact_6778_tendsto__mult__left__iff,axiom,
! [A: $tType,B: $tType] :
( ( ( field @ A )
& ( topolo4211221413907600880p_mult @ A ) )
=> ! [C2: A,F2: B > A,L: A,F3: filter @ B] :
( ( C2
!= ( zero_zero @ A ) )
=> ( ( filterlim @ B @ A
@ ^ [X2: B] : ( times_times @ A @ C2 @ ( F2 @ X2 ) )
@ ( topolo7230453075368039082e_nhds @ A @ ( times_times @ A @ C2 @ L ) )
@ F3 )
= ( filterlim @ B @ A @ F2 @ ( topolo7230453075368039082e_nhds @ A @ L ) @ F3 ) ) ) ) ).
% tendsto_mult_left_iff
thf(fact_6779_power__tendsto__0__iff,axiom,
! [A: $tType,N: nat,F2: A > real,F3: filter @ A] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ N )
=> ( ( filterlim @ A @ real
@ ^ [X2: A] : ( power_power @ real @ ( F2 @ X2 ) @ N )
@ ( topolo7230453075368039082e_nhds @ real @ ( zero_zero @ real ) )
@ F3 )
= ( filterlim @ A @ real @ F2 @ ( topolo7230453075368039082e_nhds @ real @ ( zero_zero @ real ) ) @ F3 ) ) ) ).
% power_tendsto_0_iff
thf(fact_6780_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_6781_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_6782_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_6783_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_6784_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_6785_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_6786_LIM__not__zero,axiom,
! [Aa: $tType,A: $tType] :
( ( ( topolo8386298272705272623_space @ A )
& ( zero @ Aa )
& ( topological_t2_space @ Aa ) )
=> ! [K: Aa,A2: A] :
( ( K
!= ( zero_zero @ Aa ) )
=> ~ ( filterlim @ A @ Aa
@ ^ [X2: A] : K
@ ( topolo7230453075368039082e_nhds @ Aa @ ( zero_zero @ Aa ) )
@ ( topolo174197925503356063within @ A @ A2 @ ( top_top @ ( set @ A ) ) ) ) ) ) ).
% LIM_not_zero
thf(fact_6787_isCont__LIM__compose2,axiom,
! [C: $tType,B: $tType,A: $tType] :
( ( ( real_V822414075346904944vector @ A )
& ( topolo4958980785337419405_space @ B )
& ( topolo4958980785337419405_space @ C ) )
=> ! [A2: A,F2: A > B,G: B > C,L: C] :
( ( topolo3448309680560233919inuous @ A @ B @ ( topolo174197925503356063within @ A @ A2 @ ( top_top @ ( set @ A ) ) ) @ F2 )
=> ( ( filterlim @ B @ C @ G @ ( topolo7230453075368039082e_nhds @ C @ L ) @ ( topolo174197925503356063within @ B @ ( F2 @ A2 ) @ ( top_top @ ( set @ B ) ) ) )
=> ( ? [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 @ L )
@ ( topolo174197925503356063within @ A @ A2 @ ( top_top @ ( set @ A ) ) ) ) ) ) ) ) ).
% isCont_LIM_compose2
thf(fact_6788_has__field__derivative__iff,axiom,
! [A: $tType] :
( ( real_V3459762299906320749_field @ A )
=> ! [F2: A > A,D5: A,X: A,S3: set @ A] :
( ( has_field_derivative @ A @ F2 @ D5 @ ( topolo174197925503356063within @ A @ X @ S3 ) )
= ( filterlim @ A @ A
@ ^ [Y6: A] : ( divide_divide @ A @ ( minus_minus @ A @ ( F2 @ Y6 ) @ ( F2 @ X ) ) @ ( minus_minus @ A @ Y6 @ X ) )
@ ( topolo7230453075368039082e_nhds @ A @ D5 )
@ ( topolo174197925503356063within @ A @ X @ S3 ) ) ) ) ).
% has_field_derivative_iff
thf(fact_6789_has__field__derivativeD,axiom,
! [A: $tType] :
( ( real_V3459762299906320749_field @ A )
=> ! [F2: A > A,D5: A,X: A,S3: set @ A] :
( ( has_field_derivative @ A @ F2 @ D5 @ ( topolo174197925503356063within @ A @ X @ S3 ) )
=> ( filterlim @ A @ A
@ ^ [Y6: A] : ( divide_divide @ A @ ( minus_minus @ A @ ( F2 @ Y6 ) @ ( F2 @ X ) ) @ ( minus_minus @ A @ Y6 @ X ) )
@ ( topolo7230453075368039082e_nhds @ A @ D5 )
@ ( topolo174197925503356063within @ A @ X @ S3 ) ) ) ) ).
% has_field_derivativeD
thf(fact_6790_atLeastSucLessThan__greaterThanLessThan,axiom,
! [L: nat,U: nat] :
( ( set_or7035219750837199246ssThan @ nat @ ( suc @ L ) @ U )
= ( set_or5935395276787703475ssThan @ nat @ L @ U ) ) ).
% atLeastSucLessThan_greaterThanLessThan
thf(fact_6791_tendsto__artanh,axiom,
! [A: $tType,F2: A > real,A2: real,F3: filter @ A] :
( ( filterlim @ A @ real @ F2 @ ( topolo7230453075368039082e_nhds @ real @ A2 ) @ F3 )
=> ( ( 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 ) )
@ F3 ) ) ) ) ).
% tendsto_artanh
thf(fact_6792_tendsto__power,axiom,
! [B: $tType,A: $tType] :
( ( ( power @ B )
& ( real_V4412858255891104859lgebra @ B ) )
=> ! [F2: A > B,A2: B,F3: filter @ A,N: nat] :
( ( filterlim @ A @ B @ F2 @ ( topolo7230453075368039082e_nhds @ B @ A2 ) @ F3 )
=> ( filterlim @ A @ B
@ ^ [X2: A] : ( power_power @ B @ ( F2 @ X2 ) @ N )
@ ( topolo7230453075368039082e_nhds @ B @ ( power_power @ B @ A2 @ N ) )
@ F3 ) ) ) ).
% tendsto_power
thf(fact_6793_continuous__power_H,axiom,
! [B: $tType,C: $tType] :
( ( ( topological_t2_space @ C )
& ( topolo1898628316856586783d_mult @ B ) )
=> ! [F3: filter @ C,F2: C > B,G: C > nat] :
( ( topolo3448309680560233919inuous @ C @ B @ F3 @ F2 )
=> ( ( topolo3448309680560233919inuous @ C @ nat @ F3 @ G )
=> ( topolo3448309680560233919inuous @ C @ B @ F3
@ ^ [X2: C] : ( power_power @ B @ ( F2 @ X2 ) @ ( G @ X2 ) ) ) ) ) ) ).
% continuous_power'
thf(fact_6794_tendsto__power__strong,axiom,
! [B: $tType,C: $tType] :
( ( topolo1898628316856586783d_mult @ B )
=> ! [F2: C > B,A2: B,F3: filter @ C,G: C > nat,B2: nat] :
( ( filterlim @ C @ B @ F2 @ ( topolo7230453075368039082e_nhds @ B @ A2 ) @ F3 )
=> ( ( filterlim @ C @ nat @ G @ ( topolo7230453075368039082e_nhds @ nat @ B2 ) @ F3 )
=> ( filterlim @ C @ B
@ ^ [X2: C] : ( power_power @ B @ ( F2 @ X2 ) @ ( G @ X2 ) )
@ ( topolo7230453075368039082e_nhds @ B @ ( power_power @ B @ A2 @ B2 ) )
@ F3 ) ) ) ) ).
% tendsto_power_strong
thf(fact_6795_continuous__power,axiom,
! [B: $tType,A: $tType] :
( ( ( topological_t2_space @ A )
& ( power @ B )
& ( real_V4412858255891104859lgebra @ B ) )
=> ! [F3: filter @ A,F2: A > B,N: nat] :
( ( topolo3448309680560233919inuous @ A @ B @ F3 @ F2 )
=> ( topolo3448309680560233919inuous @ A @ B @ F3
@ ^ [X2: A] : ( power_power @ B @ ( F2 @ X2 ) @ N ) ) ) ) ).
% continuous_power
thf(fact_6796_tendsto__null__power,axiom,
! [B: $tType,A: $tType] :
( ( real_V2822296259951069270ebra_1 @ B )
=> ! [F2: A > B,F3: filter @ A,N: nat] :
( ( filterlim @ A @ B @ F2 @ ( topolo7230453075368039082e_nhds @ B @ ( zero_zero @ B ) ) @ F3 )
=> ( ( ord_less @ nat @ ( zero_zero @ nat ) @ N )
=> ( filterlim @ A @ B
@ ^ [X2: A] : ( power_power @ B @ ( F2 @ X2 ) @ N )
@ ( topolo7230453075368039082e_nhds @ B @ ( zero_zero @ B ) )
@ F3 ) ) ) ) ).
% tendsto_null_power
thf(fact_6797_tendsto__log,axiom,
! [A: $tType,F2: A > real,A2: real,F3: filter @ A,G: A > real,B2: real] :
( ( filterlim @ A @ real @ F2 @ ( topolo7230453075368039082e_nhds @ real @ A2 ) @ F3 )
=> ( ( filterlim @ A @ real @ G @ ( topolo7230453075368039082e_nhds @ real @ B2 ) @ F3 )
=> ( ( 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 ) )
@ F3 ) ) ) ) ) ) ).
% tendsto_log
thf(fact_6798_tendsto__mult__right__zero,axiom,
! [A: $tType,D4: $tType] :
( ( real_V4412858255891104859lgebra @ A )
=> ! [F2: D4 > A,F3: filter @ D4,C2: A] :
( ( filterlim @ D4 @ A @ F2 @ ( topolo7230453075368039082e_nhds @ A @ ( zero_zero @ A ) ) @ F3 )
=> ( filterlim @ D4 @ A
@ ^ [X2: D4] : ( times_times @ A @ C2 @ ( F2 @ X2 ) )
@ ( topolo7230453075368039082e_nhds @ A @ ( zero_zero @ A ) )
@ F3 ) ) ) ).
% tendsto_mult_right_zero
thf(fact_6799_tendsto__mult__left__zero,axiom,
! [A: $tType,D4: $tType] :
( ( real_V4412858255891104859lgebra @ A )
=> ! [F2: D4 > A,F3: filter @ D4,C2: A] :
( ( filterlim @ D4 @ A @ F2 @ ( topolo7230453075368039082e_nhds @ A @ ( zero_zero @ A ) ) @ F3 )
=> ( filterlim @ D4 @ A
@ ^ [X2: D4] : ( times_times @ A @ ( F2 @ X2 ) @ C2 )
@ ( topolo7230453075368039082e_nhds @ A @ ( zero_zero @ A ) )
@ F3 ) ) ) ).
% tendsto_mult_left_zero
thf(fact_6800_tendsto__mult__zero,axiom,
! [A: $tType,D4: $tType] :
( ( real_V4412858255891104859lgebra @ A )
=> ! [F2: D4 > A,F3: filter @ D4,G: D4 > A] :
( ( filterlim @ D4 @ A @ F2 @ ( topolo7230453075368039082e_nhds @ A @ ( zero_zero @ A ) ) @ F3 )
=> ( ( filterlim @ D4 @ A @ G @ ( topolo7230453075368039082e_nhds @ A @ ( zero_zero @ A ) ) @ F3 )
=> ( filterlim @ D4 @ A
@ ^ [X2: D4] : ( times_times @ A @ ( F2 @ X2 ) @ ( G @ X2 ) )
@ ( topolo7230453075368039082e_nhds @ A @ ( zero_zero @ A ) )
@ F3 ) ) ) ) ).
% tendsto_mult_zero
thf(fact_6801_tendsto__mult__one,axiom,
! [B: $tType,D4: $tType] :
( ( topolo1898628316856586783d_mult @ B )
=> ! [F2: D4 > B,F3: filter @ D4,G: D4 > B] :
( ( filterlim @ D4 @ B @ F2 @ ( topolo7230453075368039082e_nhds @ B @ ( one_one @ B ) ) @ F3 )
=> ( ( filterlim @ D4 @ B @ G @ ( topolo7230453075368039082e_nhds @ B @ ( one_one @ B ) ) @ F3 )
=> ( filterlim @ D4 @ B
@ ^ [X2: D4] : ( times_times @ B @ ( F2 @ X2 ) @ ( G @ X2 ) )
@ ( topolo7230453075368039082e_nhds @ B @ ( one_one @ B ) )
@ F3 ) ) ) ) ).
% tendsto_mult_one
thf(fact_6802_tendsto__mult__right,axiom,
! [A: $tType,B: $tType] :
( ( topolo4211221413907600880p_mult @ A )
=> ! [F2: B > A,L: A,F3: filter @ B,C2: A] :
( ( filterlim @ B @ A @ F2 @ ( topolo7230453075368039082e_nhds @ A @ L ) @ F3 )
=> ( filterlim @ B @ A
@ ^ [X2: B] : ( times_times @ A @ ( F2 @ X2 ) @ C2 )
@ ( topolo7230453075368039082e_nhds @ A @ ( times_times @ A @ L @ C2 ) )
@ F3 ) ) ) ).
% tendsto_mult_right
thf(fact_6803_tendsto__mult__left,axiom,
! [A: $tType,B: $tType] :
( ( topolo4211221413907600880p_mult @ A )
=> ! [F2: B > A,L: A,F3: filter @ B,C2: A] :
( ( filterlim @ B @ A @ F2 @ ( topolo7230453075368039082e_nhds @ A @ L ) @ F3 )
=> ( filterlim @ B @ A
@ ^ [X2: B] : ( times_times @ A @ C2 @ ( F2 @ X2 ) )
@ ( topolo7230453075368039082e_nhds @ A @ ( times_times @ A @ C2 @ L ) )
@ F3 ) ) ) ).
% tendsto_mult_left
thf(fact_6804_tendsto__mult,axiom,
! [A: $tType,B: $tType] :
( ( topolo4211221413907600880p_mult @ A )
=> ! [F2: B > A,A2: A,F3: filter @ B,G: B > A,B2: A] :
( ( filterlim @ B @ A @ F2 @ ( topolo7230453075368039082e_nhds @ A @ A2 ) @ F3 )
=> ( ( filterlim @ B @ A @ G @ ( topolo7230453075368039082e_nhds @ A @ B2 ) @ F3 )
=> ( filterlim @ B @ A
@ ^ [X2: B] : ( times_times @ A @ ( F2 @ X2 ) @ ( G @ X2 ) )
@ ( topolo7230453075368039082e_nhds @ A @ ( times_times @ A @ A2 @ B2 ) )
@ F3 ) ) ) ) ).
% tendsto_mult
thf(fact_6805_continuous__mult__right,axiom,
! [A: $tType,B: $tType] :
( ( ( topological_t2_space @ B )
& ( real_V4412858255891104859lgebra @ A ) )
=> ! [F3: filter @ B,F2: B > A,C2: A] :
( ( topolo3448309680560233919inuous @ B @ A @ F3 @ F2 )
=> ( topolo3448309680560233919inuous @ B @ A @ F3
@ ^ [X2: B] : ( times_times @ A @ ( F2 @ X2 ) @ C2 ) ) ) ) ).
% continuous_mult_right
thf(fact_6806_continuous__mult__left,axiom,
! [A: $tType,B: $tType] :
( ( ( topological_t2_space @ B )
& ( real_V4412858255891104859lgebra @ A ) )
=> ! [F3: filter @ B,F2: B > A,C2: A] :
( ( topolo3448309680560233919inuous @ B @ A @ F3 @ F2 )
=> ( topolo3448309680560233919inuous @ B @ A @ F3
@ ^ [X2: B] : ( times_times @ A @ C2 @ ( F2 @ X2 ) ) ) ) ) ).
% continuous_mult_left
thf(fact_6807_continuous__mult_H,axiom,
! [B: $tType,D4: $tType] :
( ( ( topological_t2_space @ D4 )
& ( topolo4211221413907600880p_mult @ B ) )
=> ! [F3: filter @ D4,F2: D4 > B,G: D4 > B] :
( ( topolo3448309680560233919inuous @ D4 @ B @ F3 @ F2 )
=> ( ( topolo3448309680560233919inuous @ D4 @ B @ F3 @ G )
=> ( topolo3448309680560233919inuous @ D4 @ B @ F3
@ ^ [X2: D4] : ( times_times @ B @ ( F2 @ X2 ) @ ( G @ X2 ) ) ) ) ) ) ).
% continuous_mult'
thf(fact_6808_continuous__mult,axiom,
! [A: $tType,D4: $tType] :
( ( ( topological_t2_space @ D4 )
& ( real_V4412858255891104859lgebra @ A ) )
=> ! [F3: filter @ D4,F2: D4 > A,G: D4 > A] :
( ( topolo3448309680560233919inuous @ D4 @ A @ F3 @ F2 )
=> ( ( topolo3448309680560233919inuous @ D4 @ A @ F3 @ G )
=> ( topolo3448309680560233919inuous @ D4 @ A @ F3
@ ^ [X2: D4] : ( times_times @ A @ ( F2 @ X2 ) @ ( G @ X2 ) ) ) ) ) ) ).
% continuous_mult
thf(fact_6809_tendsto__add__zero,axiom,
! [B: $tType,D4: $tType] :
( ( topolo6943815403480290642id_add @ B )
=> ! [F2: D4 > B,F3: filter @ D4,G: D4 > B] :
( ( filterlim @ D4 @ B @ F2 @ ( topolo7230453075368039082e_nhds @ B @ ( zero_zero @ B ) ) @ F3 )
=> ( ( filterlim @ D4 @ B @ G @ ( topolo7230453075368039082e_nhds @ B @ ( zero_zero @ B ) ) @ F3 )
=> ( filterlim @ D4 @ B
@ ^ [X2: D4] : ( plus_plus @ B @ ( F2 @ X2 ) @ ( G @ X2 ) )
@ ( topolo7230453075368039082e_nhds @ B @ ( zero_zero @ B ) )
@ F3 ) ) ) ) ).
% tendsto_add_zero
thf(fact_6810_continuous__add,axiom,
! [B: $tType,D4: $tType] :
( ( ( topological_t2_space @ D4 )
& ( topolo6943815403480290642id_add @ B ) )
=> ! [F3: filter @ D4,F2: D4 > B,G: D4 > B] :
( ( topolo3448309680560233919inuous @ D4 @ B @ F3 @ F2 )
=> ( ( topolo3448309680560233919inuous @ D4 @ B @ F3 @ G )
=> ( topolo3448309680560233919inuous @ D4 @ B @ F3
@ ^ [X2: D4] : ( plus_plus @ B @ ( F2 @ X2 ) @ ( G @ X2 ) ) ) ) ) ) ).
% continuous_add
thf(fact_6811_tendsto__add__const__iff,axiom,
! [A: $tType,B: $tType] :
( ( topolo1633459387980952147up_add @ A )
=> ! [C2: A,F2: B > A,D2: A,F3: filter @ B] :
( ( filterlim @ B @ A
@ ^ [X2: B] : ( plus_plus @ A @ C2 @ ( F2 @ X2 ) )
@ ( topolo7230453075368039082e_nhds @ A @ ( plus_plus @ A @ C2 @ D2 ) )
@ F3 )
= ( filterlim @ B @ A @ F2 @ ( topolo7230453075368039082e_nhds @ A @ D2 ) @ F3 ) ) ) ).
% tendsto_add_const_iff
thf(fact_6812_tendsto__add,axiom,
! [A: $tType,B: $tType] :
( ( topolo6943815403480290642id_add @ A )
=> ! [F2: B > A,A2: A,F3: filter @ B,G: B > A,B2: A] :
( ( filterlim @ B @ A @ F2 @ ( topolo7230453075368039082e_nhds @ A @ A2 ) @ F3 )
=> ( ( filterlim @ B @ A @ G @ ( topolo7230453075368039082e_nhds @ A @ B2 ) @ F3 )
=> ( filterlim @ B @ A
@ ^ [X2: B] : ( plus_plus @ A @ ( F2 @ X2 ) @ ( G @ X2 ) )
@ ( topolo7230453075368039082e_nhds @ A @ ( plus_plus @ A @ A2 @ B2 ) )
@ F3 ) ) ) ) ).
% tendsto_add
thf(fact_6813_tendsto__real__root,axiom,
! [A: $tType,F2: A > real,X: real,F3: filter @ A,N: nat] :
( ( filterlim @ A @ real @ F2 @ ( topolo7230453075368039082e_nhds @ real @ X ) @ F3 )
=> ( filterlim @ A @ real
@ ^ [X2: A] : ( root @ N @ ( F2 @ X2 ) )
@ ( topolo7230453075368039082e_nhds @ real @ ( root @ N @ X ) )
@ F3 ) ) ).
% tendsto_real_root
thf(fact_6814_tendsto__one__prod_H,axiom,
! [C: $tType,B: $tType,A: $tType] :
( ( topolo4987421752381908075d_mult @ C )
=> ! [I6: set @ B,F2: A > B > C,F3: filter @ A] :
( ! [I3: B] :
( ( member @ B @ I3 @ I6 )
=> ( filterlim @ A @ C
@ ^ [X2: A] : ( F2 @ X2 @ I3 )
@ ( topolo7230453075368039082e_nhds @ C @ ( one_one @ C ) )
@ F3 ) )
=> ( filterlim @ A @ C
@ ^ [I2: A] : ( groups7121269368397514597t_prod @ B @ C @ ( F2 @ I2 ) @ I6 )
@ ( topolo7230453075368039082e_nhds @ C @ ( one_one @ C ) )
@ F3 ) ) ) ).
% tendsto_one_prod'
thf(fact_6815_tendsto__null__sum,axiom,
! [C: $tType,B: $tType,A: $tType] :
( ( topolo5987344860129210374id_add @ C )
=> ! [I6: set @ B,F2: A > B > C,F3: filter @ A] :
( ! [I3: B] :
( ( member @ B @ I3 @ I6 )
=> ( filterlim @ A @ C
@ ^ [X2: A] : ( F2 @ X2 @ I3 )
@ ( topolo7230453075368039082e_nhds @ C @ ( zero_zero @ C ) )
@ F3 ) )
=> ( filterlim @ A @ C
@ ^ [I2: A] : ( groups7311177749621191930dd_sum @ B @ C @ ( F2 @ I2 ) @ I6 )
@ ( topolo7230453075368039082e_nhds @ C @ ( zero_zero @ C ) )
@ F3 ) ) ) ).
% tendsto_null_sum
thf(fact_6816_tendsto__sgn,axiom,
! [A: $tType,B: $tType] :
( ( real_V822414075346904944vector @ A )
=> ! [F2: B > A,L: A,F3: filter @ B] :
( ( filterlim @ B @ A @ F2 @ ( topolo7230453075368039082e_nhds @ A @ L ) @ F3 )
=> ( ( L
!= ( zero_zero @ A ) )
=> ( filterlim @ B @ A
@ ^ [X2: B] : ( sgn_sgn @ A @ ( F2 @ X2 ) )
@ ( topolo7230453075368039082e_nhds @ A @ ( sgn_sgn @ A @ L ) )
@ F3 ) ) ) ) ).
% tendsto_sgn
thf(fact_6817_tendsto__divide__zero,axiom,
! [A: $tType,B: $tType] :
( ( real_V3459762299906320749_field @ A )
=> ! [F2: B > A,F3: filter @ B,C2: A] :
( ( filterlim @ B @ A @ F2 @ ( topolo7230453075368039082e_nhds @ A @ ( zero_zero @ A ) ) @ F3 )
=> ( filterlim @ B @ A
@ ^ [X2: B] : ( divide_divide @ A @ ( F2 @ X2 ) @ C2 )
@ ( topolo7230453075368039082e_nhds @ A @ ( zero_zero @ A ) )
@ F3 ) ) ) ).
% tendsto_divide_zero
thf(fact_6818_tendsto__divide,axiom,
! [A: $tType,B: $tType] :
( ( real_V3459762299906320749_field @ A )
=> ! [F2: B > A,A2: A,F3: filter @ B,G: B > A,B2: A] :
( ( filterlim @ B @ A @ F2 @ ( topolo7230453075368039082e_nhds @ A @ A2 ) @ F3 )
=> ( ( filterlim @ B @ A @ G @ ( topolo7230453075368039082e_nhds @ A @ B2 ) @ F3 )
=> ( ( B2
!= ( zero_zero @ A ) )
=> ( filterlim @ B @ A
@ ^ [X2: B] : ( divide_divide @ A @ ( F2 @ X2 ) @ ( G @ X2 ) )
@ ( topolo7230453075368039082e_nhds @ A @ ( divide_divide @ A @ A2 @ B2 ) )
@ F3 ) ) ) ) ) ).
% tendsto_divide
thf(fact_6819_tendsto__inverse,axiom,
! [A: $tType,B: $tType] :
( ( real_V8999393235501362500lgebra @ A )
=> ! [F2: B > A,A2: A,F3: filter @ B] :
( ( filterlim @ B @ A @ F2 @ ( topolo7230453075368039082e_nhds @ A @ A2 ) @ F3 )
=> ( ( A2
!= ( zero_zero @ A ) )
=> ( filterlim @ B @ A
@ ^ [X2: B] : ( inverse_inverse @ A @ ( F2 @ X2 ) )
@ ( topolo7230453075368039082e_nhds @ A @ ( inverse_inverse @ A @ A2 ) )
@ F3 ) ) ) ) ).
% tendsto_inverse
thf(fact_6820_tendsto__tanh,axiom,
! [A: $tType,C: $tType] :
( ( ( real_Vector_banach @ A )
& ( real_V3459762299906320749_field @ A ) )
=> ! [F2: C > A,A2: A,F3: filter @ C] :
( ( filterlim @ C @ A @ F2 @ ( topolo7230453075368039082e_nhds @ A @ A2 ) @ F3 )
=> ( ( ( cosh @ A @ A2 )
!= ( zero_zero @ A ) )
=> ( filterlim @ C @ A
@ ^ [X2: C] : ( tanh @ A @ ( F2 @ X2 ) )
@ ( topolo7230453075368039082e_nhds @ A @ ( tanh @ A @ A2 ) )
@ F3 ) ) ) ) ).
% tendsto_tanh
thf(fact_6821_tendsto__tan,axiom,
! [A: $tType] :
( ( ( real_Vector_banach @ A )
& ( real_V3459762299906320749_field @ A ) )
=> ! [F2: A > A,A2: A,F3: filter @ A] :
( ( filterlim @ A @ A @ F2 @ ( topolo7230453075368039082e_nhds @ A @ A2 ) @ F3 )
=> ( ( ( cos @ A @ A2 )
!= ( zero_zero @ A ) )
=> ( filterlim @ A @ A
@ ^ [X2: A] : ( tan @ A @ ( F2 @ X2 ) )
@ ( topolo7230453075368039082e_nhds @ A @ ( tan @ A @ A2 ) )
@ F3 ) ) ) ) ).
% tendsto_tan
thf(fact_6822_tendsto__rabs__zero,axiom,
! [A: $tType,F2: A > real,F3: filter @ A] :
( ( filterlim @ A @ real @ F2 @ ( topolo7230453075368039082e_nhds @ real @ ( zero_zero @ real ) ) @ F3 )
=> ( filterlim @ A @ real
@ ^ [X2: A] : ( abs_abs @ real @ ( F2 @ X2 ) )
@ ( topolo7230453075368039082e_nhds @ real @ ( zero_zero @ real ) )
@ F3 ) ) ).
% tendsto_rabs_zero
thf(fact_6823_tendsto__rabs__zero__iff,axiom,
! [A: $tType,F2: A > real,F3: filter @ A] :
( ( filterlim @ A @ real
@ ^ [X2: A] : ( abs_abs @ real @ ( F2 @ X2 ) )
@ ( topolo7230453075368039082e_nhds @ real @ ( zero_zero @ real ) )
@ F3 )
= ( filterlim @ A @ real @ F2 @ ( topolo7230453075368039082e_nhds @ real @ ( zero_zero @ real ) ) @ F3 ) ) ).
% tendsto_rabs_zero_iff
thf(fact_6824_tendsto__rabs__zero__cancel,axiom,
! [A: $tType,F2: A > real,F3: filter @ A] :
( ( filterlim @ A @ real
@ ^ [X2: A] : ( abs_abs @ real @ ( F2 @ X2 ) )
@ ( topolo7230453075368039082e_nhds @ real @ ( zero_zero @ real ) )
@ F3 )
=> ( filterlim @ A @ real @ F2 @ ( topolo7230453075368039082e_nhds @ real @ ( zero_zero @ real ) ) @ F3 ) ) ).
% tendsto_rabs_zero_cancel
thf(fact_6825_Lim__transform__eq,axiom,
! [A: $tType,B: $tType] :
( ( real_V822414075346904944vector @ A )
=> ! [F2: B > A,G: B > A,F3: filter @ B,A2: A] :
( ( filterlim @ B @ A
@ ^ [X2: B] : ( minus_minus @ A @ ( F2 @ X2 ) @ ( G @ X2 ) )
@ ( topolo7230453075368039082e_nhds @ A @ ( zero_zero @ A ) )
@ F3 )
=> ( ( filterlim @ B @ A @ F2 @ ( topolo7230453075368039082e_nhds @ A @ A2 ) @ F3 )
= ( filterlim @ B @ A @ G @ ( topolo7230453075368039082e_nhds @ A @ A2 ) @ F3 ) ) ) ) ).
% Lim_transform_eq
thf(fact_6826_LIM__zero__cancel,axiom,
! [B: $tType,A: $tType] :
( ( real_V822414075346904944vector @ B )
=> ! [F2: A > B,L: B,F3: filter @ A] :
( ( filterlim @ A @ B
@ ^ [X2: A] : ( minus_minus @ B @ ( F2 @ X2 ) @ L )
@ ( topolo7230453075368039082e_nhds @ B @ ( zero_zero @ B ) )
@ F3 )
=> ( filterlim @ A @ B @ F2 @ ( topolo7230453075368039082e_nhds @ B @ L ) @ F3 ) ) ) ).
% LIM_zero_cancel
thf(fact_6827_Lim__transform2,axiom,
! [A: $tType,B: $tType] :
( ( real_V822414075346904944vector @ A )
=> ! [F2: B > A,A2: A,F3: filter @ B,G: B > A] :
( ( filterlim @ B @ A @ F2 @ ( topolo7230453075368039082e_nhds @ A @ A2 ) @ F3 )
=> ( ( filterlim @ B @ A
@ ^ [X2: B] : ( minus_minus @ A @ ( F2 @ X2 ) @ ( G @ X2 ) )
@ ( topolo7230453075368039082e_nhds @ A @ ( zero_zero @ A ) )
@ F3 )
=> ( filterlim @ B @ A @ G @ ( topolo7230453075368039082e_nhds @ A @ A2 ) @ F3 ) ) ) ) ).
% Lim_transform2
thf(fact_6828_Lim__transform,axiom,
! [A: $tType,B: $tType] :
( ( real_V822414075346904944vector @ A )
=> ! [G: B > A,A2: A,F3: filter @ B,F2: B > A] :
( ( filterlim @ B @ A @ G @ ( topolo7230453075368039082e_nhds @ A @ A2 ) @ F3 )
=> ( ( filterlim @ B @ A
@ ^ [X2: B] : ( minus_minus @ A @ ( F2 @ X2 ) @ ( G @ X2 ) )
@ ( topolo7230453075368039082e_nhds @ A @ ( zero_zero @ A ) )
@ F3 )
=> ( filterlim @ B @ A @ F2 @ ( topolo7230453075368039082e_nhds @ A @ A2 ) @ F3 ) ) ) ) ).
% Lim_transform
thf(fact_6829_LIM__zero__iff,axiom,
! [B: $tType,A: $tType] :
( ( real_V822414075346904944vector @ B )
=> ! [F2: A > B,L: B,F3: filter @ A] :
( ( filterlim @ A @ B
@ ^ [X2: A] : ( minus_minus @ B @ ( F2 @ X2 ) @ L )
@ ( topolo7230453075368039082e_nhds @ B @ ( zero_zero @ B ) )
@ F3 )
= ( filterlim @ A @ B @ F2 @ ( topolo7230453075368039082e_nhds @ B @ L ) @ F3 ) ) ) ).
% LIM_zero_iff
thf(fact_6830_LIM__zero,axiom,
! [B: $tType,A: $tType] :
( ( real_V822414075346904944vector @ B )
=> ! [F2: A > B,L: B,F3: filter @ A] :
( ( filterlim @ A @ B @ F2 @ ( topolo7230453075368039082e_nhds @ B @ L ) @ F3 )
=> ( filterlim @ A @ B
@ ^ [X2: A] : ( minus_minus @ B @ ( F2 @ X2 ) @ L )
@ ( topolo7230453075368039082e_nhds @ B @ ( zero_zero @ B ) )
@ F3 ) ) ) ).
% LIM_zero
thf(fact_6831_tendsto__powr,axiom,
! [A: $tType,F2: A > real,A2: real,F3: filter @ A,G: A > real,B2: real] :
( ( filterlim @ A @ real @ F2 @ ( topolo7230453075368039082e_nhds @ real @ A2 ) @ F3 )
=> ( ( filterlim @ A @ real @ G @ ( topolo7230453075368039082e_nhds @ real @ B2 ) @ F3 )
=> ( ( A2
!= ( zero_zero @ real ) )
=> ( filterlim @ A @ real
@ ^ [X2: A] : ( powr @ real @ ( F2 @ X2 ) @ ( G @ X2 ) )
@ ( topolo7230453075368039082e_nhds @ real @ ( powr @ real @ A2 @ B2 ) )
@ F3 ) ) ) ) ).
% tendsto_powr
thf(fact_6832_tendsto__ln,axiom,
! [A: $tType,F2: A > real,A2: real,F3: filter @ A] :
( ( filterlim @ A @ real @ F2 @ ( topolo7230453075368039082e_nhds @ real @ A2 ) @ F3 )
=> ( ( A2
!= ( zero_zero @ real ) )
=> ( filterlim @ A @ real
@ ^ [X2: A] : ( ln_ln @ real @ ( F2 @ X2 ) )
@ ( topolo7230453075368039082e_nhds @ real @ ( ln_ln @ real @ A2 ) )
@ F3 ) ) ) ).
% tendsto_ln
thf(fact_6833_tendsto__real__sqrt,axiom,
! [A: $tType,F2: A > real,X: real,F3: filter @ A] :
( ( filterlim @ A @ real @ F2 @ ( topolo7230453075368039082e_nhds @ real @ X ) @ F3 )
=> ( filterlim @ A @ real
@ ^ [X2: A] : ( sqrt @ ( F2 @ X2 ) )
@ ( topolo7230453075368039082e_nhds @ real @ ( sqrt @ X ) )
@ F3 ) ) ).
% tendsto_real_sqrt
thf(fact_6834_tendsto__arcosh,axiom,
! [B: $tType,F2: B > real,A2: real,F3: filter @ B] :
( ( filterlim @ B @ real @ F2 @ ( topolo7230453075368039082e_nhds @ real @ A2 ) @ F3 )
=> ( ( ord_less @ real @ ( one_one @ real ) @ A2 )
=> ( filterlim @ B @ real
@ ^ [X2: B] : ( arcosh @ real @ ( F2 @ X2 ) )
@ ( topolo7230453075368039082e_nhds @ real @ ( arcosh @ real @ A2 ) )
@ F3 ) ) ) ).
% tendsto_arcosh
thf(fact_6835_tendsto__norm__zero,axiom,
! [B: $tType,A: $tType] :
( ( real_V822414075346904944vector @ B )
=> ! [F2: A > B,F3: filter @ A] :
( ( filterlim @ A @ B @ F2 @ ( topolo7230453075368039082e_nhds @ B @ ( zero_zero @ B ) ) @ F3 )
=> ( filterlim @ A @ real
@ ^ [X2: A] : ( real_V7770717601297561774m_norm @ B @ ( F2 @ X2 ) )
@ ( topolo7230453075368039082e_nhds @ real @ ( zero_zero @ real ) )
@ F3 ) ) ) ).
% tendsto_norm_zero
thf(fact_6836_tendsto__norm__zero__iff,axiom,
! [B: $tType,A: $tType] :
( ( real_V822414075346904944vector @ B )
=> ! [F2: A > B,F3: filter @ A] :
( ( filterlim @ A @ real
@ ^ [X2: A] : ( real_V7770717601297561774m_norm @ B @ ( F2 @ X2 ) )
@ ( topolo7230453075368039082e_nhds @ real @ ( zero_zero @ real ) )
@ F3 )
= ( filterlim @ A @ B @ F2 @ ( topolo7230453075368039082e_nhds @ B @ ( zero_zero @ B ) ) @ F3 ) ) ) ).
% tendsto_norm_zero_iff
thf(fact_6837_tendsto__norm__zero__cancel,axiom,
! [B: $tType,A: $tType] :
( ( real_V822414075346904944vector @ B )
=> ! [F2: A > B,F3: filter @ A] :
( ( filterlim @ A @ real
@ ^ [X2: A] : ( real_V7770717601297561774m_norm @ B @ ( F2 @ X2 ) )
@ ( topolo7230453075368039082e_nhds @ real @ ( zero_zero @ real ) )
@ F3 )
=> ( filterlim @ A @ B @ F2 @ ( topolo7230453075368039082e_nhds @ B @ ( zero_zero @ B ) ) @ F3 ) ) ) ).
% tendsto_norm_zero_cancel
thf(fact_6838_tendsto__cot,axiom,
! [A: $tType] :
( ( ( real_Vector_banach @ A )
& ( real_V3459762299906320749_field @ A ) )
=> ! [F2: A > A,A2: A,F3: filter @ A] :
( ( filterlim @ A @ A @ F2 @ ( topolo7230453075368039082e_nhds @ A @ A2 ) @ F3 )
=> ( ( ( sin @ A @ A2 )
!= ( zero_zero @ A ) )
=> ( filterlim @ A @ A
@ ^ [X2: A] : ( cot @ A @ ( F2 @ X2 ) )
@ ( topolo7230453075368039082e_nhds @ A @ ( cot @ A @ A2 ) )
@ F3 ) ) ) ) ).
% tendsto_cot
thf(fact_6839_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_6840_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 )
=> ? [S5: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ S5 )
& ! [X3: A] :
( ( ( X3 != A2 )
& ( ord_less @ real @ ( real_V7770717601297561774m_norm @ A @ ( minus_minus @ A @ X3 @ A2 ) ) @ S5 ) )
=> ( 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_6841_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 )
=> ? [S6: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ S6 )
& ! [X2: A] :
( ( ( X2 != A2 )
& ( ord_less @ real @ ( real_V7770717601297561774m_norm @ A @ ( minus_minus @ A @ X2 @ A2 ) ) @ S6 ) )
=> ( ord_less @ real @ ( real_V7770717601297561774m_norm @ B @ ( minus_minus @ B @ ( F2 @ X2 ) @ L6 ) ) @ R5 ) ) ) ) ) ) ) ).
% LIM_eq
thf(fact_6842_LIM__equal2,axiom,
! [B: $tType,A: $tType] :
( ( ( real_V822414075346904944vector @ A )
& ( topolo4958980785337419405_space @ B ) )
=> ! [R: real,A2: A,F2: A > B,G: A > B,L: 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 @ L ) @ ( topolo174197925503356063within @ A @ A2 @ ( top_top @ ( set @ A ) ) ) )
=> ( filterlim @ A @ B @ F2 @ ( topolo7230453075368039082e_nhds @ B @ L ) @ ( topolo174197925503356063within @ A @ A2 @ ( top_top @ ( set @ A ) ) ) ) ) ) ) ) ).
% LIM_equal2
thf(fact_6843_DERIV__LIM__iff,axiom,
! [A: $tType] :
( ( ( inverse @ A )
& ( real_V822414075346904944vector @ A ) )
=> ! [F2: A > A,A2: A,D5: A] :
( ( filterlim @ A @ A
@ ^ [H: A] : ( divide_divide @ A @ ( minus_minus @ A @ ( F2 @ ( plus_plus @ A @ A2 @ H ) ) @ ( F2 @ A2 ) ) @ H )
@ ( topolo7230453075368039082e_nhds @ A @ D5 )
@ ( 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 @ D5 )
@ ( topolo174197925503356063within @ A @ A2 @ ( top_top @ ( set @ A ) ) ) ) ) ) ).
% DERIV_LIM_iff
thf(fact_6844_LIM__fun__less__zero,axiom,
! [F2: real > real,L: real,C2: real] :
( ( filterlim @ real @ real @ F2 @ ( topolo7230453075368039082e_nhds @ real @ L ) @ ( topolo174197925503356063within @ real @ C2 @ ( top_top @ ( set @ real ) ) ) )
=> ( ( ord_less @ real @ L @ ( zero_zero @ real ) )
=> ? [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_6845_LIM__fun__not__zero,axiom,
! [F2: real > real,L: real,C2: real] :
( ( filterlim @ real @ real @ F2 @ ( topolo7230453075368039082e_nhds @ real @ L ) @ ( topolo174197925503356063within @ real @ C2 @ ( top_top @ ( set @ real ) ) ) )
=> ( ( L
!= ( zero_zero @ real ) )
=> ? [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_6846_LIM__fun__gt__zero,axiom,
! [F2: real > real,L: real,C2: real] :
( ( filterlim @ real @ real @ F2 @ ( topolo7230453075368039082e_nhds @ real @ L ) @ ( topolo174197925503356063within @ real @ C2 @ ( top_top @ ( set @ real ) ) ) )
=> ( ( ord_less @ real @ ( zero_zero @ real ) @ L )
=> ? [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_6847_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_6848_isCont__real__sqrt,axiom,
! [X: real] : ( topolo3448309680560233919inuous @ real @ real @ ( topolo174197925503356063within @ real @ X @ ( top_top @ ( set @ real ) ) ) @ sqrt ) ).
% isCont_real_sqrt
thf(fact_6849_isCont__real__root,axiom,
! [X: real,N: nat] : ( topolo3448309680560233919inuous @ real @ real @ ( topolo174197925503356063within @ real @ X @ ( top_top @ ( set @ real ) ) ) @ ( root @ N ) ) ).
% isCont_real_root
thf(fact_6850_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_6851_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_6852_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_6853_isCont__power,axiom,
! [B: $tType,A: $tType] :
( ( ( topological_t2_space @ A )
& ( power @ B )
& ( real_V4412858255891104859lgebra @ B ) )
=> ! [A2: A,F2: A > B,N: nat] :
( ( topolo3448309680560233919inuous @ A @ B @ ( topolo174197925503356063within @ A @ A2 @ ( top_top @ ( set @ A ) ) ) @ F2 )
=> ( topolo3448309680560233919inuous @ A @ B @ ( topolo174197925503356063within @ A @ A2 @ ( top_top @ ( set @ A ) ) )
@ ^ [X2: A] : ( power_power @ B @ ( F2 @ X2 ) @ N ) ) ) ) ).
% isCont_power
thf(fact_6854_continuous__at__within__inverse,axiom,
! [B: $tType,A: $tType] :
( ( ( topological_t2_space @ A )
& ( real_V8999393235501362500lgebra @ B ) )
=> ! [A2: A,S: set @ A,F2: A > B] :
( ( topolo3448309680560233919inuous @ A @ B @ ( topolo174197925503356063within @ A @ A2 @ S ) @ F2 )
=> ( ( ( F2 @ A2 )
!= ( zero_zero @ B ) )
=> ( topolo3448309680560233919inuous @ A @ B @ ( topolo174197925503356063within @ A @ A2 @ S )
@ ^ [X2: A] : ( inverse_inverse @ B @ ( F2 @ X2 ) ) ) ) ) ) ).
% continuous_at_within_inverse
thf(fact_6855_continuous__at__within__sgn,axiom,
! [B: $tType,A: $tType] :
( ( ( topological_t2_space @ A )
& ( real_V822414075346904944vector @ B ) )
=> ! [A2: A,S: set @ A,F2: A > B] :
( ( topolo3448309680560233919inuous @ A @ B @ ( topolo174197925503356063within @ A @ A2 @ S ) @ F2 )
=> ( ( ( F2 @ A2 )
!= ( zero_zero @ B ) )
=> ( topolo3448309680560233919inuous @ A @ B @ ( topolo174197925503356063within @ A @ A2 @ S )
@ ^ [X2: A] : ( sgn_sgn @ B @ ( F2 @ X2 ) ) ) ) ) ) ).
% continuous_at_within_sgn
thf(fact_6856_DERIV__D,axiom,
! [A: $tType] :
( ( real_V3459762299906320749_field @ A )
=> ! [F2: A > A,D5: A,X: A] :
( ( has_field_derivative @ A @ F2 @ D5 @ ( 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 @ D5 )
@ ( topolo174197925503356063within @ A @ ( zero_zero @ A ) @ ( top_top @ ( set @ A ) ) ) ) ) ) ).
% DERIV_D
thf(fact_6857_DERIV__def,axiom,
! [A: $tType] :
( ( real_V3459762299906320749_field @ A )
=> ! [F2: A > A,D5: A,X: A] :
( ( has_field_derivative @ A @ F2 @ D5 @ ( 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 @ D5 )
@ ( topolo174197925503356063within @ A @ ( zero_zero @ A ) @ ( top_top @ ( set @ A ) ) ) ) ) ) ).
% DERIV_def
thf(fact_6858_lim__exp__minus__1,axiom,
! [A: $tType] :
( ( ( real_Vector_banach @ A )
& ( real_V3459762299906320749_field @ A ) )
=> ( filterlim @ A @ A
@ ^ [Z7: A] : ( divide_divide @ A @ ( minus_minus @ A @ ( exp @ A @ Z7 ) @ ( one_one @ A ) ) @ Z7 )
@ ( topolo7230453075368039082e_nhds @ A @ ( one_one @ A ) )
@ ( topolo174197925503356063within @ A @ ( zero_zero @ A ) @ ( top_top @ ( set @ A ) ) ) ) ) ).
% lim_exp_minus_1
thf(fact_6859_atLeastPlusOneLessThan__greaterThanLessThan__int,axiom,
! [L: int,U: int] :
( ( set_or7035219750837199246ssThan @ int @ ( plus_plus @ int @ L @ ( one_one @ int ) ) @ U )
= ( set_or5935395276787703475ssThan @ int @ L @ U ) ) ).
% atLeastPlusOneLessThan_greaterThanLessThan_int
thf(fact_6860_greaterThanLessThan__upt,axiom,
( ( set_or5935395276787703475ssThan @ nat )
= ( ^ [N3: nat,M4: nat] : ( set2 @ nat @ ( upt @ ( suc @ N3 ) @ M4 ) ) ) ) ).
% greaterThanLessThan_upt
thf(fact_6861_lemma__termdiff4,axiom,
! [B: $tType,A: $tType] :
( ( ( real_V822414075346904944vector @ A )
& ( real_V822414075346904944vector @ B ) )
=> ! [K: real,F2: A > B,K6: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ K )
=> ( ! [H3: A] :
( ( H3
!= ( zero_zero @ A ) )
=> ( ( ord_less @ real @ ( real_V7770717601297561774m_norm @ A @ H3 ) @ K )
=> ( ord_less_eq @ real @ ( real_V7770717601297561774m_norm @ B @ ( F2 @ H3 ) ) @ ( times_times @ real @ K6 @ ( real_V7770717601297561774m_norm @ A @ H3 ) ) ) ) )
=> ( filterlim @ A @ B @ F2 @ ( topolo7230453075368039082e_nhds @ B @ ( zero_zero @ B ) ) @ ( topolo174197925503356063within @ A @ ( zero_zero @ A ) @ ( top_top @ ( set @ A ) ) ) ) ) ) ) ).
% lemma_termdiff4
thf(fact_6862_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 )
=> ( ! [Z3: real] :
( ( ord_less_eq @ real @ A2 @ Z3 )
=> ( ( ord_less_eq @ real @ Z3 @ B2 )
=> ( ( G @ ( F2 @ Z3 ) )
= Z3 ) ) )
=> ( ! [Z3: real] :
( ( ord_less_eq @ real @ A2 @ Z3 )
=> ( ( ord_less_eq @ real @ Z3 @ B2 )
=> ( topolo3448309680560233919inuous @ real @ real @ ( topolo174197925503356063within @ real @ Z3 @ ( top_top @ ( set @ real ) ) ) @ F2 ) ) )
=> ( topolo3448309680560233919inuous @ real @ real @ ( topolo174197925503356063within @ real @ ( F2 @ X ) @ ( top_top @ ( set @ real ) ) ) @ G ) ) ) ) ) ).
% isCont_inverse_function2
thf(fact_6863_field__has__derivative__at,axiom,
! [A: $tType] :
( ( real_V3459762299906320749_field @ A )
=> ! [F2: A > A,D5: A,X: A] :
( ( has_derivative @ A @ A @ F2 @ ( times_times @ A @ D5 ) @ ( 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 @ D5 )
@ ( topolo174197925503356063within @ A @ ( zero_zero @ A ) @ ( top_top @ ( set @ A ) ) ) ) ) ) ).
% field_has_derivative_at
thf(fact_6864_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_6865_isCont__ln,axiom,
! [X: real] :
( ( X
!= ( zero_zero @ real ) )
=> ( topolo3448309680560233919inuous @ real @ real @ ( topolo174197925503356063within @ real @ X @ ( top_top @ ( set @ real ) ) ) @ ( ln_ln @ real ) ) ) ).
% isCont_ln
thf(fact_6866_isCont__sgn,axiom,
! [B: $tType,A: $tType] :
( ( ( topological_t2_space @ A )
& ( real_V822414075346904944vector @ B ) )
=> ! [A2: A,F2: A > B] :
( ( topolo3448309680560233919inuous @ A @ B @ ( topolo174197925503356063within @ A @ A2 @ ( top_top @ ( set @ A ) ) ) @ F2 )
=> ( ( ( F2 @ A2 )
!= ( zero_zero @ B ) )
=> ( topolo3448309680560233919inuous @ A @ B @ ( topolo174197925503356063within @ A @ A2 @ ( top_top @ ( set @ A ) ) )
@ ^ [X2: A] : ( sgn_sgn @ B @ ( F2 @ X2 ) ) ) ) ) ) ).
% isCont_sgn
thf(fact_6867_filterlim__at__to__0,axiom,
! [B: $tType,A: $tType] :
( ( real_V822414075346904944vector @ A )
=> ! [F2: A > B,F3: filter @ B,A2: A] :
( ( filterlim @ A @ B @ F2 @ F3 @ ( topolo174197925503356063within @ A @ A2 @ ( top_top @ ( set @ A ) ) ) )
= ( filterlim @ A @ B
@ ^ [X2: A] : ( F2 @ ( plus_plus @ A @ X2 @ A2 ) )
@ F3
@ ( topolo174197925503356063within @ A @ ( zero_zero @ A ) @ ( top_top @ ( set @ A ) ) ) ) ) ) ).
% filterlim_at_to_0
thf(fact_6868_continuous__within__tan,axiom,
! [A: $tType] :
( ( ( real_Vector_banach @ A )
& ( real_V3459762299906320749_field @ A ) )
=> ! [X: A,S: set @ A,F2: A > A] :
( ( topolo3448309680560233919inuous @ A @ A @ ( topolo174197925503356063within @ A @ X @ S ) @ F2 )
=> ( ( ( cos @ A @ ( F2 @ X ) )
!= ( zero_zero @ A ) )
=> ( topolo3448309680560233919inuous @ A @ A @ ( topolo174197925503356063within @ A @ X @ S )
@ ^ [X2: A] : ( tan @ A @ ( F2 @ X2 ) ) ) ) ) ) ).
% continuous_within_tan
thf(fact_6869_atLeastPlusOneLessThan__greaterThanLessThan__integer,axiom,
! [L: code_integer,U: code_integer] :
( ( set_or7035219750837199246ssThan @ code_integer @ ( plus_plus @ code_integer @ L @ ( one_one @ code_integer ) ) @ U )
= ( set_or5935395276787703475ssThan @ code_integer @ L @ U ) ) ).
% atLeastPlusOneLessThan_greaterThanLessThan_integer
thf(fact_6870_continuous__within__cot,axiom,
! [A: $tType] :
( ( ( real_Vector_banach @ A )
& ( real_V3459762299906320749_field @ A ) )
=> ! [X: A,S: set @ A,F2: A > A] :
( ( topolo3448309680560233919inuous @ A @ A @ ( topolo174197925503356063within @ A @ X @ S ) @ F2 )
=> ( ( ( sin @ A @ ( F2 @ X ) )
!= ( zero_zero @ A ) )
=> ( topolo3448309680560233919inuous @ A @ A @ ( topolo174197925503356063within @ A @ X @ S )
@ ^ [X2: A] : ( cot @ A @ ( F2 @ X2 ) ) ) ) ) ) ).
% continuous_within_cot
thf(fact_6871_continuous__at__within__tanh,axiom,
! [A: $tType,C: $tType] :
( ( ( topological_t2_space @ C )
& ( real_Vector_banach @ A )
& ( real_V3459762299906320749_field @ A ) )
=> ! [X: C,A3: set @ C,F2: C > A] :
( ( topolo3448309680560233919inuous @ C @ A @ ( topolo174197925503356063within @ C @ X @ A3 ) @ F2 )
=> ( ( ( cosh @ A @ ( F2 @ X ) )
!= ( zero_zero @ A ) )
=> ( topolo3448309680560233919inuous @ C @ A @ ( topolo174197925503356063within @ C @ X @ A3 )
@ ^ [X2: C] : ( tanh @ A @ ( F2 @ X2 ) ) ) ) ) ) ).
% continuous_at_within_tanh
thf(fact_6872_CARAT__DERIV,axiom,
! [A: $tType] :
( ( real_V3459762299906320749_field @ A )
=> ! [F2: A > A,L: A,X: A] :
( ( has_field_derivative @ A @ F2 @ L @ ( topolo174197925503356063within @ A @ X @ ( top_top @ ( set @ A ) ) ) )
= ( ? [G3: A > A] :
( ! [Z7: A] :
( ( minus_minus @ A @ ( F2 @ Z7 ) @ ( F2 @ X ) )
= ( times_times @ A @ ( G3 @ Z7 ) @ ( minus_minus @ A @ Z7 @ X ) ) )
& ( topolo3448309680560233919inuous @ A @ A @ ( topolo174197925503356063within @ A @ X @ ( top_top @ ( set @ A ) ) ) @ G3 )
& ( ( G3 @ X )
= L ) ) ) ) ) ).
% CARAT_DERIV
thf(fact_6873_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 ) )
=> ? [M9: A] :
( ! [X5: real] :
( ( ( ord_less_eq @ real @ A2 @ X5 )
& ( ord_less_eq @ real @ X5 @ B2 ) )
=> ( ord_less_eq @ A @ ( F2 @ X5 ) @ M9 ) )
& ! [N13: A] :
( ( ord_less @ A @ N13 @ M9 )
=> ? [X3: real] :
( ( ord_less_eq @ real @ A2 @ X3 )
& ( ord_less_eq @ real @ X3 @ B2 )
& ( ord_less @ A @ N13 @ ( F2 @ X3 ) ) ) ) ) ) ) ) ).
% isCont_has_Ub
thf(fact_6874_filterlim__shift,axiom,
! [B: $tType,A: $tType] :
( ( real_V822414075346904944vector @ A )
=> ! [F2: A > B,F3: filter @ B,A2: A,D2: A] :
( ( filterlim @ A @ B @ F2 @ F3 @ ( topolo174197925503356063within @ A @ A2 @ ( top_top @ ( set @ A ) ) ) )
=> ( filterlim @ A @ B @ ( comp @ A @ B @ A @ F2 @ ( plus_plus @ A @ D2 ) ) @ F3 @ ( topolo174197925503356063within @ A @ ( minus_minus @ A @ A2 @ D2 ) @ ( top_top @ ( set @ A ) ) ) ) ) ) ).
% filterlim_shift
thf(fact_6875_filterlim__shift__iff,axiom,
! [B: $tType,A: $tType] :
( ( real_V822414075346904944vector @ A )
=> ! [F2: A > B,D2: A,F3: filter @ B,A2: A] :
( ( filterlim @ A @ B @ ( comp @ A @ B @ A @ F2 @ ( plus_plus @ A @ D2 ) ) @ F3 @ ( topolo174197925503356063within @ A @ ( minus_minus @ A @ A2 @ D2 ) @ ( top_top @ ( set @ A ) ) ) )
= ( filterlim @ A @ B @ F2 @ F3 @ ( topolo174197925503356063within @ A @ A2 @ ( top_top @ ( set @ A ) ) ) ) ) ) ).
% filterlim_shift_iff
thf(fact_6876_isCont__tan,axiom,
! [A: $tType] :
( ( ( real_Vector_banach @ A )
& ( real_V3459762299906320749_field @ A ) )
=> ! [X: A] :
( ( ( cos @ A @ X )
!= ( zero_zero @ A ) )
=> ( topolo3448309680560233919inuous @ A @ A @ ( topolo174197925503356063within @ A @ X @ ( top_top @ ( set @ A ) ) ) @ ( tan @ A ) ) ) ) ).
% isCont_tan
thf(fact_6877_isCont__cot,axiom,
! [A: $tType] :
( ( ( real_Vector_banach @ A )
& ( real_V3459762299906320749_field @ A ) )
=> ! [X: A] :
( ( ( sin @ A @ X )
!= ( zero_zero @ A ) )
=> ( topolo3448309680560233919inuous @ A @ A @ ( topolo174197925503356063within @ A @ X @ ( top_top @ ( set @ A ) ) ) @ ( cot @ A ) ) ) ) ).
% isCont_cot
thf(fact_6878_isCont__tanh,axiom,
! [A: $tType] :
( ( ( real_Vector_banach @ A )
& ( real_V3459762299906320749_field @ A ) )
=> ! [X: A] :
( ( ( cosh @ A @ X )
!= ( zero_zero @ A ) )
=> ( topolo3448309680560233919inuous @ A @ A @ ( topolo174197925503356063within @ A @ X @ ( top_top @ ( set @ A ) ) ) @ ( tanh @ A ) ) ) ) ).
% isCont_tanh
thf(fact_6879_powser__limit__0__strong,axiom,
! [A: $tType] :
( ( ( real_Vector_banach @ A )
& ( real_V3459762299906320749_field @ A ) )
=> ! [S: real,A2: nat > A,F2: A > A] :
( ( ord_less @ real @ ( zero_zero @ real ) @ S )
=> ( ! [X3: A] :
( ( X3
!= ( zero_zero @ A ) )
=> ( ( ord_less @ real @ ( real_V7770717601297561774m_norm @ A @ X3 ) @ S )
=> ( sums @ A
@ ^ [N3: nat] : ( times_times @ A @ ( A2 @ N3 ) @ ( power_power @ A @ X3 @ N3 ) )
@ ( F2 @ X3 ) ) ) )
=> ( filterlim @ A @ A @ F2 @ ( topolo7230453075368039082e_nhds @ A @ ( A2 @ ( zero_zero @ nat ) ) ) @ ( topolo174197925503356063within @ A @ ( zero_zero @ A ) @ ( top_top @ ( set @ A ) ) ) ) ) ) ) ).
% powser_limit_0_strong
thf(fact_6880_powser__limit__0,axiom,
! [A: $tType] :
( ( ( real_Vector_banach @ A )
& ( real_V3459762299906320749_field @ A ) )
=> ! [S: real,A2: nat > A,F2: A > A] :
( ( ord_less @ real @ ( zero_zero @ real ) @ S )
=> ( ! [X3: A] :
( ( ord_less @ real @ ( real_V7770717601297561774m_norm @ A @ X3 ) @ S )
=> ( sums @ A
@ ^ [N3: nat] : ( times_times @ A @ ( A2 @ N3 ) @ ( power_power @ A @ X3 @ N3 ) )
@ ( F2 @ X3 ) ) )
=> ( filterlim @ A @ A @ F2 @ ( topolo7230453075368039082e_nhds @ A @ ( A2 @ ( zero_zero @ nat ) ) ) @ ( topolo174197925503356063within @ A @ ( zero_zero @ A ) @ ( top_top @ ( set @ A ) ) ) ) ) ) ) ).
% powser_limit_0
thf(fact_6881_lemma__termdiff5,axiom,
! [B: $tType,A: $tType] :
( ( ( real_V822414075346904944vector @ A )
& ( real_Vector_banach @ B ) )
=> ! [K: real,F2: nat > real,G: A > nat > B] :
( ( ord_less @ real @ ( zero_zero @ real ) @ K )
=> ( ( summable @ real @ F2 )
=> ( ! [H3: A,N2: nat] :
( ( H3
!= ( zero_zero @ A ) )
=> ( ( ord_less @ real @ ( real_V7770717601297561774m_norm @ A @ H3 ) @ K )
=> ( ord_less_eq @ real @ ( real_V7770717601297561774m_norm @ B @ ( G @ H3 @ N2 ) ) @ ( times_times @ real @ ( F2 @ N2 ) @ ( real_V7770717601297561774m_norm @ A @ H3 ) ) ) ) )
=> ( filterlim @ A @ B
@ ^ [H: A] : ( suminf @ B @ ( G @ H ) )
@ ( topolo7230453075368039082e_nhds @ B @ ( zero_zero @ B ) )
@ ( topolo174197925503356063within @ A @ ( zero_zero @ A ) @ ( top_top @ ( set @ A ) ) ) ) ) ) ) ) ).
% lemma_termdiff5
thf(fact_6882_isCont__tan_H,axiom,
! [A: $tType] :
( ( ( real_Vector_banach @ A )
& ( real_V3459762299906320749_field @ A ) )
=> ! [A2: A,F2: A > A] :
( ( topolo3448309680560233919inuous @ A @ A @ ( topolo174197925503356063within @ A @ A2 @ ( top_top @ ( set @ A ) ) ) @ F2 )
=> ( ( ( cos @ A @ ( F2 @ A2 ) )
!= ( zero_zero @ A ) )
=> ( topolo3448309680560233919inuous @ A @ A @ ( topolo174197925503356063within @ A @ A2 @ ( top_top @ ( set @ A ) ) )
@ ^ [X2: A] : ( tan @ A @ ( F2 @ X2 ) ) ) ) ) ) ).
% isCont_tan'
thf(fact_6883_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_6884_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_6885_isCont__cot_H,axiom,
! [A: $tType] :
( ( ( real_Vector_banach @ A )
& ( real_V3459762299906320749_field @ A ) )
=> ! [A2: A,F2: A > A] :
( ( topolo3448309680560233919inuous @ A @ A @ ( topolo174197925503356063within @ A @ A2 @ ( top_top @ ( set @ A ) ) ) @ F2 )
=> ( ( ( sin @ A @ ( F2 @ A2 ) )
!= ( zero_zero @ A ) )
=> ( topolo3448309680560233919inuous @ A @ A @ ( topolo174197925503356063within @ A @ A2 @ ( top_top @ ( set @ A ) ) )
@ ^ [X2: A] : ( cot @ A @ ( F2 @ X2 ) ) ) ) ) ) ).
% isCont_cot'
thf(fact_6886_DERIV__inverse__function,axiom,
! [F2: real > real,D5: real,G: real > real,X: real,A2: real,B2: real] :
( ( has_field_derivative @ real @ F2 @ D5 @ ( topolo174197925503356063within @ real @ ( G @ X ) @ ( top_top @ ( set @ real ) ) ) )
=> ( ( D5
!= ( 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 @ D5 ) @ ( topolo174197925503356063within @ real @ X @ ( top_top @ ( set @ real ) ) ) ) ) ) ) ) ) ) ).
% DERIV_inverse_function
thf(fact_6887_isCont__polynom,axiom,
! [A: $tType] :
( ( real_V8999393235501362500lgebra @ A )
=> ! [A2: A,C2: nat > A,N: nat] :
( topolo3448309680560233919inuous @ A @ A @ ( topolo174197925503356063within @ A @ A2 @ ( top_top @ ( set @ A ) ) )
@ ^ [W2: A] :
( groups7311177749621191930dd_sum @ nat @ A
@ ^ [I2: nat] : ( times_times @ A @ ( C2 @ I2 ) @ ( power_power @ A @ W2 @ I2 ) )
@ ( set_ord_atMost @ nat @ N ) ) ) ) ).
% isCont_polynom
thf(fact_6888_isCont__powser__converges__everywhere,axiom,
! [A: $tType] :
( ( ( real_Vector_banach @ A )
& ( real_V3459762299906320749_field @ A ) )
=> ! [C2: nat > A,X: A] :
( ! [Y3: A] :
( summable @ A
@ ^ [N3: nat] : ( times_times @ A @ ( C2 @ N3 ) @ ( power_power @ A @ Y3 @ N3 ) ) )
=> ( topolo3448309680560233919inuous @ A @ A @ ( topolo174197925503356063within @ A @ X @ ( top_top @ ( set @ A ) ) )
@ ^ [X2: A] :
( suminf @ A
@ ^ [N3: nat] : ( times_times @ A @ ( C2 @ N3 ) @ ( power_power @ A @ X2 @ N3 ) ) ) ) ) ) ).
% isCont_powser_converges_everywhere
thf(fact_6889_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_6890_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_6891_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_6892_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_6893_isCont__inverse__function,axiom,
! [D2: real,X: real,G: real > real,F2: real > real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ D2 )
=> ( ! [Z3: real] :
( ( ord_less_eq @ real @ ( abs_abs @ real @ ( minus_minus @ real @ Z3 @ X ) ) @ D2 )
=> ( ( G @ ( F2 @ Z3 ) )
= Z3 ) )
=> ( ! [Z3: real] :
( ( ord_less_eq @ real @ ( abs_abs @ real @ ( minus_minus @ real @ Z3 @ X ) ) @ D2 )
=> ( topolo3448309680560233919inuous @ real @ real @ ( topolo174197925503356063within @ real @ Z3 @ ( top_top @ ( set @ real ) ) ) @ F2 ) )
=> ( topolo3448309680560233919inuous @ real @ real @ ( topolo174197925503356063within @ real @ ( F2 @ X ) @ ( top_top @ ( set @ real ) ) ) @ G ) ) ) ) ).
% isCont_inverse_function
thf(fact_6894_GMVT_H,axiom,
! [A2: real,B2: real,F2: real > real,G: real > real,G4: real > real,F8: real > real] :
( ( ord_less @ real @ A2 @ B2 )
=> ( ! [Z3: real] :
( ( ord_less_eq @ real @ A2 @ Z3 )
=> ( ( ord_less_eq @ real @ Z3 @ B2 )
=> ( topolo3448309680560233919inuous @ real @ real @ ( topolo174197925503356063within @ real @ Z3 @ ( top_top @ ( set @ real ) ) ) @ F2 ) ) )
=> ( ! [Z3: real] :
( ( ord_less_eq @ real @ A2 @ Z3 )
=> ( ( ord_less_eq @ real @ Z3 @ B2 )
=> ( topolo3448309680560233919inuous @ real @ real @ ( topolo174197925503356063within @ real @ Z3 @ ( top_top @ ( set @ real ) ) ) @ G ) ) )
=> ( ! [Z3: real] :
( ( ord_less @ real @ A2 @ Z3 )
=> ( ( ord_less @ real @ Z3 @ B2 )
=> ( has_field_derivative @ real @ G @ ( G4 @ Z3 ) @ ( topolo174197925503356063within @ real @ Z3 @ ( top_top @ ( set @ real ) ) ) ) ) )
=> ( ! [Z3: real] :
( ( ord_less @ real @ A2 @ Z3 )
=> ( ( ord_less @ real @ Z3 @ B2 )
=> ( has_field_derivative @ real @ F2 @ ( F8 @ Z3 ) @ ( topolo174197925503356063within @ real @ Z3 @ ( 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 ) ) @ ( G4 @ C4 ) )
= ( times_times @ real @ ( minus_minus @ real @ ( G @ B2 ) @ ( G @ A2 ) ) @ ( F8 @ C4 ) ) ) ) ) ) ) ) ) ).
% GMVT'
thf(fact_6895_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_6896_isCont__powser,axiom,
! [A: $tType] :
( ( ( real_Vector_banach @ A )
& ( real_V3459762299906320749_field @ A ) )
=> ! [C2: nat > A,K6: A,X: A] :
( ( summable @ A
@ ^ [N3: nat] : ( times_times @ A @ ( C2 @ N3 ) @ ( power_power @ A @ K6 @ N3 ) ) )
=> ( ( ord_less @ real @ ( real_V7770717601297561774m_norm @ A @ X ) @ ( real_V7770717601297561774m_norm @ A @ K6 ) )
=> ( topolo3448309680560233919inuous @ A @ A @ ( topolo174197925503356063within @ A @ X @ ( top_top @ ( set @ A ) ) )
@ ^ [X2: A] :
( suminf @ A
@ ^ [N3: nat] : ( times_times @ A @ ( C2 @ N3 ) @ ( power_power @ A @ X2 @ N3 ) ) ) ) ) ) ) ).
% isCont_powser
thf(fact_6897_isCont__powser_H,axiom,
! [Aa: $tType,A: $tType] :
( ( ( topological_t2_space @ A )
& ( real_Vector_banach @ Aa )
& ( real_V3459762299906320749_field @ Aa ) )
=> ! [A2: A,F2: A > Aa,C2: nat > Aa,K6: Aa] :
( ( topolo3448309680560233919inuous @ A @ Aa @ ( topolo174197925503356063within @ A @ A2 @ ( top_top @ ( set @ A ) ) ) @ F2 )
=> ( ( summable @ Aa
@ ^ [N3: nat] : ( times_times @ Aa @ ( C2 @ N3 ) @ ( power_power @ Aa @ K6 @ N3 ) ) )
=> ( ( ord_less @ real @ ( real_V7770717601297561774m_norm @ Aa @ ( F2 @ A2 ) ) @ ( real_V7770717601297561774m_norm @ Aa @ K6 ) )
=> ( topolo3448309680560233919inuous @ A @ Aa @ ( topolo174197925503356063within @ A @ A2 @ ( top_top @ ( set @ A ) ) )
@ ^ [X2: A] :
( suminf @ Aa
@ ^ [N3: nat] : ( times_times @ Aa @ ( C2 @ N3 ) @ ( power_power @ Aa @ ( F2 @ X2 ) @ N3 ) ) ) ) ) ) ) ) ).
% isCont_powser'
thf(fact_6898_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 ) ) )
=> ! [N10: nat] :
( member @ real
@ ( suminf @ real
@ ^ [I2: nat] : ( times_times @ real @ ( power_power @ real @ ( uminus_uminus @ real @ ( one_one @ real ) ) @ I2 ) @ ( A2 @ I2 ) ) )
@ ( set_or1337092689740270186AtMost @ real
@ ( groups7311177749621191930dd_sum @ nat @ real
@ ^ [I2: nat] : ( times_times @ real @ ( power_power @ real @ ( uminus_uminus @ real @ ( one_one @ real ) ) @ I2 ) @ ( A2 @ I2 ) )
@ ( set_ord_lessThan @ nat @ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N10 ) ) )
@ ( groups7311177749621191930dd_sum @ nat @ real
@ ^ [I2: nat] : ( times_times @ real @ ( power_power @ real @ ( uminus_uminus @ real @ ( one_one @ real ) ) @ I2 ) @ ( A2 @ I2 ) )
@ ( set_ord_lessThan @ nat @ ( plus_plus @ nat @ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N10 ) @ ( one_one @ nat ) ) ) ) ) ) ) ) ) ).
% summable_Leibniz(2)
thf(fact_6899_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 ) )
=> ! [N10: nat] :
( member @ real
@ ( suminf @ real
@ ^ [I2: nat] : ( times_times @ real @ ( power_power @ real @ ( uminus_uminus @ real @ ( one_one @ real ) ) @ I2 ) @ ( A2 @ I2 ) ) )
@ ( set_or1337092689740270186AtMost @ real
@ ( groups7311177749621191930dd_sum @ nat @ real
@ ^ [I2: nat] : ( times_times @ real @ ( power_power @ real @ ( uminus_uminus @ real @ ( one_one @ real ) ) @ I2 ) @ ( A2 @ I2 ) )
@ ( set_ord_lessThan @ nat @ ( plus_plus @ nat @ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N10 ) @ ( one_one @ nat ) ) ) )
@ ( groups7311177749621191930dd_sum @ nat @ real
@ ^ [I2: nat] : ( times_times @ real @ ( power_power @ real @ ( uminus_uminus @ real @ ( one_one @ real ) ) @ I2 ) @ ( A2 @ I2 ) )
@ ( set_ord_lessThan @ nat @ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N10 ) ) ) ) ) ) ) ) ).
% summable_Leibniz(3)
thf(fact_6900_tendsto__zero__mult__left__iff,axiom,
! [A: $tType] :
( ( ( field @ A )
& ( topolo4211221413907600880p_mult @ A ) )
=> ! [C2: A,A2: nat > A] :
( ( C2
!= ( zero_zero @ A ) )
=> ( ( filterlim @ nat @ A
@ ^ [N3: nat] : ( times_times @ A @ C2 @ ( A2 @ N3 ) )
@ ( topolo7230453075368039082e_nhds @ A @ ( zero_zero @ A ) )
@ ( at_top @ nat ) )
= ( filterlim @ nat @ A @ A2 @ ( topolo7230453075368039082e_nhds @ A @ ( zero_zero @ A ) ) @ ( at_top @ nat ) ) ) ) ) ).
% tendsto_zero_mult_left_iff
thf(fact_6901_tendsto__zero__mult__right__iff,axiom,
! [A: $tType] :
( ( ( field @ A )
& ( topolo4211221413907600880p_mult @ A ) )
=> ! [C2: A,A2: nat > A] :
( ( C2
!= ( zero_zero @ A ) )
=> ( ( filterlim @ nat @ A
@ ^ [N3: nat] : ( times_times @ A @ ( A2 @ N3 ) @ C2 )
@ ( topolo7230453075368039082e_nhds @ A @ ( zero_zero @ A ) )
@ ( at_top @ nat ) )
= ( filterlim @ nat @ A @ A2 @ ( topolo7230453075368039082e_nhds @ A @ ( zero_zero @ A ) ) @ ( at_top @ nat ) ) ) ) ) ).
% tendsto_zero_mult_right_iff
thf(fact_6902_tendsto__zero__divide__iff,axiom,
! [A: $tType] :
( ( ( field @ A )
& ( topolo4211221413907600880p_mult @ A ) )
=> ! [C2: A,A2: nat > A] :
( ( C2
!= ( zero_zero @ A ) )
=> ( ( filterlim @ nat @ A
@ ^ [N3: nat] : ( divide_divide @ A @ ( A2 @ N3 ) @ C2 )
@ ( topolo7230453075368039082e_nhds @ A @ ( zero_zero @ A ) )
@ ( at_top @ nat ) )
= ( filterlim @ nat @ A @ A2 @ ( topolo7230453075368039082e_nhds @ A @ ( zero_zero @ A ) ) @ ( at_top @ nat ) ) ) ) ) ).
% tendsto_zero_divide_iff
thf(fact_6903_LIMSEQ__ignore__initial__segment,axiom,
! [A: $tType] :
( ( topolo4958980785337419405_space @ A )
=> ! [F2: nat > A,A2: A,K: nat] :
( ( filterlim @ nat @ A @ F2 @ ( topolo7230453075368039082e_nhds @ A @ A2 ) @ ( at_top @ nat ) )
=> ( filterlim @ nat @ A
@ ^ [N3: nat] : ( F2 @ ( plus_plus @ nat @ N3 @ K ) )
@ ( topolo7230453075368039082e_nhds @ A @ A2 )
@ ( at_top @ nat ) ) ) ) ).
% LIMSEQ_ignore_initial_segment
thf(fact_6904_LIMSEQ__offset,axiom,
! [A: $tType] :
( ( topolo4958980785337419405_space @ A )
=> ! [F2: nat > A,K: nat,A2: A] :
( ( filterlim @ nat @ A
@ ^ [N3: nat] : ( F2 @ ( plus_plus @ nat @ N3 @ K ) )
@ ( topolo7230453075368039082e_nhds @ A @ A2 )
@ ( at_top @ nat ) )
=> ( filterlim @ nat @ A @ F2 @ ( topolo7230453075368039082e_nhds @ A @ A2 ) @ ( at_top @ nat ) ) ) ) ).
% LIMSEQ_offset
thf(fact_6905_continuous__real__root,axiom,
! [A: $tType] :
( ( topological_t2_space @ A )
=> ! [F3: filter @ A,F2: A > real,N: nat] :
( ( topolo3448309680560233919inuous @ A @ real @ F3 @ F2 )
=> ( topolo3448309680560233919inuous @ A @ real @ F3
@ ^ [X2: A] : ( root @ N @ ( F2 @ X2 ) ) ) ) ) ).
% continuous_real_root
thf(fact_6906_continuous__real__sqrt,axiom,
! [A: $tType] :
( ( topological_t2_space @ A )
=> ! [F3: filter @ A,F2: A > real] :
( ( topolo3448309680560233919inuous @ A @ real @ F3 @ F2 )
=> ( topolo3448309680560233919inuous @ A @ real @ F3
@ ^ [X2: A] : ( sqrt @ ( F2 @ X2 ) ) ) ) ) ).
% continuous_real_sqrt
thf(fact_6907_approx__from__below__dense__linorder,axiom,
! [A: $tType] :
( ( ( dense_linorder @ A )
& ( topolo3112930676232923870pology @ A )
& ( topolo1944317154257567458pology @ A ) )
=> ! [Y: A,X: A] :
( ( ord_less @ A @ Y @ X )
=> ? [U3: nat > A] :
( ! [N10: nat] : ( ord_less @ A @ ( U3 @ N10 ) @ X )
& ( filterlim @ nat @ A @ U3 @ ( topolo7230453075368039082e_nhds @ A @ X ) @ ( at_top @ nat ) ) ) ) ) ).
% approx_from_below_dense_linorder
thf(fact_6908_approx__from__above__dense__linorder,axiom,
! [A: $tType] :
( ( ( dense_linorder @ A )
& ( topolo3112930676232923870pology @ A )
& ( topolo1944317154257567458pology @ A ) )
=> ! [X: A,Y: A] :
( ( ord_less @ A @ X @ Y )
=> ? [U3: nat > A] :
( ! [N10: nat] : ( ord_less @ A @ X @ ( U3 @ N10 ) )
& ( filterlim @ nat @ A @ U3 @ ( topolo7230453075368039082e_nhds @ A @ X ) @ ( at_top @ nat ) ) ) ) ) ).
% approx_from_above_dense_linorder
thf(fact_6909_LIMSEQ__Suc,axiom,
! [A: $tType] :
( ( topolo4958980785337419405_space @ A )
=> ! [F2: nat > A,L: A] :
( ( filterlim @ nat @ A @ F2 @ ( topolo7230453075368039082e_nhds @ A @ L ) @ ( at_top @ nat ) )
=> ( filterlim @ nat @ A
@ ^ [N3: nat] : ( F2 @ ( suc @ N3 ) )
@ ( topolo7230453075368039082e_nhds @ A @ L )
@ ( at_top @ nat ) ) ) ) ).
% LIMSEQ_Suc
thf(fact_6910_LIMSEQ__imp__Suc,axiom,
! [A: $tType] :
( ( topolo4958980785337419405_space @ A )
=> ! [F2: nat > A,L: A] :
( ( filterlim @ nat @ A
@ ^ [N3: nat] : ( F2 @ ( suc @ N3 ) )
@ ( topolo7230453075368039082e_nhds @ A @ L )
@ ( at_top @ nat ) )
=> ( filterlim @ nat @ A @ F2 @ ( topolo7230453075368039082e_nhds @ A @ L ) @ ( at_top @ nat ) ) ) ) ).
% LIMSEQ_imp_Suc
thf(fact_6911_summable__LIMSEQ__zero,axiom,
! [A: $tType] :
( ( real_V822414075346904944vector @ A )
=> ! [F2: nat > A] :
( ( summable @ A @ F2 )
=> ( filterlim @ nat @ A @ F2 @ ( topolo7230453075368039082e_nhds @ A @ ( zero_zero @ A ) ) @ ( at_top @ nat ) ) ) ) ).
% summable_LIMSEQ_zero
thf(fact_6912_continuous__at__within__powr,axiom,
! [C: $tType] :
( ( topological_t2_space @ C )
=> ! [A2: C,S: set @ C,F2: C > real,G: C > real] :
( ( topolo3448309680560233919inuous @ C @ real @ ( topolo174197925503356063within @ C @ A2 @ S ) @ F2 )
=> ( ( topolo3448309680560233919inuous @ C @ real @ ( topolo174197925503356063within @ C @ A2 @ S ) @ G )
=> ( ( ( F2 @ A2 )
!= ( zero_zero @ real ) )
=> ( topolo3448309680560233919inuous @ C @ real @ ( topolo174197925503356063within @ C @ A2 @ S )
@ ^ [X2: C] : ( powr @ real @ ( F2 @ X2 ) @ ( G @ X2 ) ) ) ) ) ) ) ).
% continuous_at_within_powr
thf(fact_6913_continuous__within__ln,axiom,
! [A: $tType] :
( ( topological_t2_space @ A )
=> ! [X: A,S: set @ A,F2: A > real] :
( ( topolo3448309680560233919inuous @ A @ real @ ( topolo174197925503356063within @ A @ X @ S ) @ F2 )
=> ( ( ( F2 @ X )
!= ( zero_zero @ real ) )
=> ( topolo3448309680560233919inuous @ A @ real @ ( topolo174197925503356063within @ A @ X @ S )
@ ^ [X2: A] : ( ln_ln @ real @ ( F2 @ X2 ) ) ) ) ) ) ).
% continuous_within_ln
thf(fact_6914_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_6915_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_6916_LIMSEQ__root,axiom,
( filterlim @ nat @ real
@ ^ [N3: nat] : ( root @ N3 @ ( semiring_1_of_nat @ real @ N3 ) )
@ ( topolo7230453075368039082e_nhds @ real @ ( one_one @ real ) )
@ ( at_top @ nat ) ) ).
% LIMSEQ_root
thf(fact_6917_isCont__powr,axiom,
! [C: $tType] :
( ( topological_t2_space @ C )
=> ! [A2: C,F2: C > real,G: C > real] :
( ( topolo3448309680560233919inuous @ C @ real @ ( topolo174197925503356063within @ C @ A2 @ ( top_top @ ( set @ C ) ) ) @ F2 )
=> ( ( topolo3448309680560233919inuous @ C @ real @ ( topolo174197925503356063within @ C @ A2 @ ( top_top @ ( set @ C ) ) ) @ G )
=> ( ( ( F2 @ A2 )
!= ( zero_zero @ real ) )
=> ( topolo3448309680560233919inuous @ C @ real @ ( topolo174197925503356063within @ C @ A2 @ ( top_top @ ( set @ C ) ) )
@ ^ [X2: C] : ( powr @ real @ ( F2 @ X2 ) @ ( G @ X2 ) ) ) ) ) ) ) ).
% isCont_powr
thf(fact_6918_isCont__ln_H,axiom,
! [A: $tType] :
( ( topological_t2_space @ A )
=> ! [X: A,F2: A > real] :
( ( topolo3448309680560233919inuous @ A @ real @ ( topolo174197925503356063within @ A @ X @ ( top_top @ ( set @ A ) ) ) @ F2 )
=> ( ( ( F2 @ X )
!= ( zero_zero @ real ) )
=> ( topolo3448309680560233919inuous @ A @ real @ ( topolo174197925503356063within @ A @ X @ ( top_top @ ( set @ A ) ) )
@ ^ [X2: A] : ( ln_ln @ real @ ( F2 @ X2 ) ) ) ) ) ) ).
% isCont_ln'
thf(fact_6919_lim__const__over__n,axiom,
! [A: $tType] :
( ( real_V3459762299906320749_field @ A )
=> ! [A2: A] :
( filterlim @ nat @ A
@ ^ [N3: nat] : ( divide_divide @ A @ A2 @ ( semiring_1_of_nat @ A @ N3 ) )
@ ( topolo7230453075368039082e_nhds @ A @ ( zero_zero @ A ) )
@ ( at_top @ nat ) ) ) ).
% lim_const_over_n
thf(fact_6920_LIMSEQ__linear,axiom,
! [A: $tType] :
( ( topolo4958980785337419405_space @ A )
=> ! [X7: nat > A,X: A,L: nat] :
( ( filterlim @ nat @ A @ X7 @ ( topolo7230453075368039082e_nhds @ A @ X ) @ ( at_top @ nat ) )
=> ( ( ord_less @ nat @ ( zero_zero @ nat ) @ L )
=> ( filterlim @ nat @ A
@ ^ [N3: nat] : ( X7 @ ( times_times @ nat @ N3 @ L ) )
@ ( topolo7230453075368039082e_nhds @ A @ X )
@ ( at_top @ nat ) ) ) ) ) ).
% LIMSEQ_linear
thf(fact_6921_lim__inverse__n,axiom,
! [A: $tType] :
( ( real_V3459762299906320749_field @ A )
=> ( filterlim @ nat @ A
@ ^ [N3: nat] : ( inverse_inverse @ A @ ( semiring_1_of_nat @ A @ N3 ) )
@ ( topolo7230453075368039082e_nhds @ A @ ( zero_zero @ A ) )
@ ( at_top @ nat ) ) ) ).
% lim_inverse_n
thf(fact_6922_telescope__summable,axiom,
! [A: $tType] :
( ( real_V822414075346904944vector @ A )
=> ! [F2: nat > A,C2: A] :
( ( filterlim @ nat @ A @ F2 @ ( topolo7230453075368039082e_nhds @ A @ C2 ) @ ( at_top @ nat ) )
=> ( summable @ A
@ ^ [N3: nat] : ( minus_minus @ A @ ( F2 @ ( suc @ N3 ) ) @ ( F2 @ N3 ) ) ) ) ) ).
% telescope_summable
thf(fact_6923_telescope__summable_H,axiom,
! [A: $tType] :
( ( real_V822414075346904944vector @ A )
=> ! [F2: nat > A,C2: A] :
( ( filterlim @ nat @ A @ F2 @ ( topolo7230453075368039082e_nhds @ A @ C2 ) @ ( at_top @ nat ) )
=> ( summable @ A
@ ^ [N3: nat] : ( minus_minus @ A @ ( F2 @ N3 ) @ ( F2 @ ( suc @ N3 ) ) ) ) ) ) ).
% telescope_summable'
thf(fact_6924_nested__sequence__unique,axiom,
! [F2: nat > real,G: nat > real] :
( ! [N2: nat] : ( ord_less_eq @ real @ ( F2 @ N2 ) @ ( F2 @ ( suc @ N2 ) ) )
=> ( ! [N2: nat] : ( ord_less_eq @ real @ ( G @ ( suc @ N2 ) ) @ ( G @ N2 ) )
=> ( ! [N2: nat] : ( ord_less_eq @ real @ ( F2 @ N2 ) @ ( G @ N2 ) )
=> ( ( filterlim @ nat @ real
@ ^ [N3: nat] : ( minus_minus @ real @ ( F2 @ N3 ) @ ( G @ N3 ) )
@ ( topolo7230453075368039082e_nhds @ real @ ( zero_zero @ real ) )
@ ( at_top @ nat ) )
=> ? [L4: real] :
( ! [N10: nat] : ( ord_less_eq @ real @ ( F2 @ N10 ) @ L4 )
& ( filterlim @ nat @ real @ F2 @ ( topolo7230453075368039082e_nhds @ real @ L4 ) @ ( at_top @ nat ) )
& ! [N10: nat] : ( ord_less_eq @ real @ L4 @ ( G @ N10 ) )
& ( filterlim @ nat @ real @ G @ ( topolo7230453075368039082e_nhds @ real @ L4 ) @ ( at_top @ nat ) ) ) ) ) ) ) ).
% nested_sequence_unique
thf(fact_6925_LIMSEQ__inverse__zero,axiom,
! [X7: nat > real] :
( ! [R3: real] :
? [N13: nat] :
! [N2: nat] :
( ( ord_less_eq @ nat @ N13 @ N2 )
=> ( ord_less @ real @ R3 @ ( X7 @ N2 ) ) )
=> ( filterlim @ nat @ real
@ ^ [N3: nat] : ( inverse_inverse @ real @ ( X7 @ N3 ) )
@ ( topolo7230453075368039082e_nhds @ real @ ( zero_zero @ real ) )
@ ( at_top @ nat ) ) ) ).
% LIMSEQ_inverse_zero
thf(fact_6926_lim__inverse__n_H,axiom,
( filterlim @ nat @ real
@ ^ [N3: nat] : ( divide_divide @ real @ ( one_one @ real ) @ ( semiring_1_of_nat @ real @ N3 ) )
@ ( topolo7230453075368039082e_nhds @ real @ ( zero_zero @ real ) )
@ ( at_top @ nat ) ) ).
% lim_inverse_n'
thf(fact_6927_LIMSEQ__inverse__real__of__nat,axiom,
( filterlim @ nat @ real
@ ^ [N3: nat] : ( inverse_inverse @ real @ ( semiring_1_of_nat @ real @ ( suc @ N3 ) ) )
@ ( topolo7230453075368039082e_nhds @ real @ ( zero_zero @ real ) )
@ ( at_top @ nat ) ) ).
% LIMSEQ_inverse_real_of_nat
thf(fact_6928_LIMSEQ__root__const,axiom,
! [C2: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ C2 )
=> ( filterlim @ nat @ real
@ ^ [N3: nat] : ( root @ N3 @ C2 )
@ ( topolo7230453075368039082e_nhds @ real @ ( one_one @ real ) )
@ ( at_top @ nat ) ) ) ).
% LIMSEQ_root_const
thf(fact_6929_LIMSEQ__inverse__real__of__nat__add,axiom,
! [R2: real] :
( filterlim @ nat @ real
@ ^ [N3: nat] : ( plus_plus @ real @ R2 @ ( inverse_inverse @ real @ ( semiring_1_of_nat @ real @ ( suc @ N3 ) ) ) )
@ ( topolo7230453075368039082e_nhds @ real @ R2 )
@ ( at_top @ nat ) ) ).
% LIMSEQ_inverse_real_of_nat_add
thf(fact_6930_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_6931_increasing__LIMSEQ,axiom,
! [F2: nat > real,L: real] :
( ! [N2: nat] : ( ord_less_eq @ real @ ( F2 @ N2 ) @ ( F2 @ ( suc @ N2 ) ) )
=> ( ! [N2: nat] : ( ord_less_eq @ real @ ( F2 @ N2 ) @ L )
=> ( ! [E2: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ E2 )
=> ? [N10: nat] : ( ord_less_eq @ real @ L @ ( plus_plus @ real @ ( F2 @ N10 ) @ E2 ) ) )
=> ( filterlim @ nat @ real @ F2 @ ( topolo7230453075368039082e_nhds @ real @ L ) @ ( at_top @ nat ) ) ) ) ) ).
% increasing_LIMSEQ
thf(fact_6932_lim__1__over__n,axiom,
! [A: $tType] :
( ( real_V3459762299906320749_field @ A )
=> ( filterlim @ nat @ A
@ ^ [N3: nat] : ( divide_divide @ A @ ( one_one @ A ) @ ( semiring_1_of_nat @ A @ N3 ) )
@ ( topolo7230453075368039082e_nhds @ A @ ( zero_zero @ A ) )
@ ( at_top @ nat ) ) ) ).
% lim_1_over_n
thf(fact_6933_LIMSEQ__Suc__n__over__n,axiom,
! [A: $tType] :
( ( real_V3459762299906320749_field @ A )
=> ( filterlim @ nat @ A
@ ^ [N3: nat] : ( divide_divide @ A @ ( semiring_1_of_nat @ A @ ( suc @ N3 ) ) @ ( semiring_1_of_nat @ A @ N3 ) )
@ ( topolo7230453075368039082e_nhds @ A @ ( one_one @ A ) )
@ ( at_top @ nat ) ) ) ).
% LIMSEQ_Suc_n_over_n
thf(fact_6934_LIMSEQ__n__over__Suc__n,axiom,
! [A: $tType] :
( ( real_V3459762299906320749_field @ A )
=> ( filterlim @ nat @ A
@ ^ [N3: nat] : ( divide_divide @ A @ ( semiring_1_of_nat @ A @ N3 ) @ ( semiring_1_of_nat @ A @ ( suc @ N3 ) ) )
@ ( topolo7230453075368039082e_nhds @ A @ ( one_one @ A ) )
@ ( at_top @ nat ) ) ) ).
% LIMSEQ_n_over_Suc_n
thf(fact_6935_telescope__sums_H,axiom,
! [A: $tType] :
( ( real_V822414075346904944vector @ A )
=> ! [F2: nat > A,C2: A] :
( ( filterlim @ nat @ A @ F2 @ ( topolo7230453075368039082e_nhds @ A @ C2 ) @ ( at_top @ nat ) )
=> ( sums @ A
@ ^ [N3: nat] : ( minus_minus @ A @ ( F2 @ N3 ) @ ( F2 @ ( suc @ N3 ) ) )
@ ( minus_minus @ A @ ( F2 @ ( zero_zero @ nat ) ) @ C2 ) ) ) ) ).
% telescope_sums'
thf(fact_6936_telescope__sums,axiom,
! [A: $tType] :
( ( real_V822414075346904944vector @ A )
=> ! [F2: nat > A,C2: A] :
( ( filterlim @ nat @ A @ F2 @ ( topolo7230453075368039082e_nhds @ A @ C2 ) @ ( at_top @ nat ) )
=> ( sums @ A
@ ^ [N3: nat] : ( minus_minus @ A @ ( F2 @ ( suc @ N3 ) ) @ ( F2 @ N3 ) )
@ ( minus_minus @ A @ C2 @ ( F2 @ ( zero_zero @ nat ) ) ) ) ) ) ).
% telescope_sums
thf(fact_6937_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_6938_LIMSEQ__divide__realpow__zero,axiom,
! [X: real,A2: real] :
( ( ord_less @ real @ ( one_one @ real ) @ X )
=> ( filterlim @ nat @ real
@ ^ [N3: nat] : ( divide_divide @ real @ A2 @ ( power_power @ real @ X @ N3 ) )
@ ( topolo7230453075368039082e_nhds @ real @ ( zero_zero @ real ) )
@ ( at_top @ nat ) ) ) ).
% LIMSEQ_divide_realpow_zero
thf(fact_6939_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_6940_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_6941_LIMSEQ__inverse__realpow__zero,axiom,
! [X: real] :
( ( ord_less @ real @ ( one_one @ real ) @ X )
=> ( filterlim @ nat @ real
@ ^ [N3: nat] : ( inverse_inverse @ real @ ( power_power @ real @ X @ N3 ) )
@ ( topolo7230453075368039082e_nhds @ real @ ( zero_zero @ real ) )
@ ( at_top @ nat ) ) ) ).
% LIMSEQ_inverse_realpow_zero
thf(fact_6942_sums__def_H,axiom,
! [A: $tType] :
( ( ( comm_monoid_add @ A )
& ( topolo4958980785337419405_space @ A ) )
=> ( ( sums @ A )
= ( ^ [F4: nat > A,S6: A] :
( filterlim @ nat @ A
@ ^ [N3: nat] : ( groups7311177749621191930dd_sum @ nat @ A @ F4 @ ( set_or1337092689740270186AtMost @ nat @ ( zero_zero @ nat ) @ N3 ) )
@ ( topolo7230453075368039082e_nhds @ A @ S6 )
@ ( at_top @ nat ) ) ) ) ) ).
% sums_def'
thf(fact_6943_LIMSEQ__inverse__real__of__nat__add__minus,axiom,
! [R2: real] :
( filterlim @ nat @ real
@ ^ [N3: nat] : ( plus_plus @ real @ R2 @ ( uminus_uminus @ real @ ( inverse_inverse @ real @ ( semiring_1_of_nat @ real @ ( suc @ N3 ) ) ) ) )
@ ( topolo7230453075368039082e_nhds @ real @ R2 )
@ ( at_top @ nat ) ) ).
% LIMSEQ_inverse_real_of_nat_add_minus
thf(fact_6944_root__test__convergence,axiom,
! [A: $tType] :
( ( real_Vector_banach @ A )
=> ! [F2: nat > A,X: real] :
( ( filterlim @ nat @ real
@ ^ [N3: nat] : ( root @ N3 @ ( real_V7770717601297561774m_norm @ A @ ( F2 @ N3 ) ) )
@ ( topolo7230453075368039082e_nhds @ real @ X )
@ ( at_top @ nat ) )
=> ( ( ord_less @ real @ X @ ( one_one @ real ) )
=> ( summable @ A @ F2 ) ) ) ) ).
% root_test_convergence
thf(fact_6945_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_6946_LIMSEQ__iff,axiom,
! [A: $tType] :
( ( real_V822414075346904944vector @ A )
=> ! [X7: nat > A,L6: A] :
( ( filterlim @ nat @ A @ X7 @ ( topolo7230453075368039082e_nhds @ A @ L6 ) @ ( at_top @ nat ) )
= ( ! [R5: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ R5 )
=> ? [No: nat] :
! [N3: nat] :
( ( ord_less_eq @ nat @ No @ N3 )
=> ( ord_less @ real @ ( real_V7770717601297561774m_norm @ A @ ( minus_minus @ A @ ( X7 @ N3 ) @ L6 ) ) @ R5 ) ) ) ) ) ) ).
% LIMSEQ_iff
thf(fact_6947_LIMSEQ__I,axiom,
! [A: $tType] :
( ( real_V822414075346904944vector @ A )
=> ! [X7: nat > A,L6: A] :
( ! [R3: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ R3 )
=> ? [No2: nat] :
! [N2: nat] :
( ( ord_less_eq @ nat @ No2 @ N2 )
=> ( ord_less @ real @ ( real_V7770717601297561774m_norm @ A @ ( minus_minus @ A @ ( X7 @ N2 ) @ L6 ) ) @ R3 ) ) )
=> ( filterlim @ nat @ A @ X7 @ ( topolo7230453075368039082e_nhds @ A @ L6 ) @ ( at_top @ nat ) ) ) ) ).
% LIMSEQ_I
thf(fact_6948_LIMSEQ__D,axiom,
! [A: $tType] :
( ( real_V822414075346904944vector @ A )
=> ! [X7: nat > A,L6: A,R2: real] :
( ( filterlim @ nat @ A @ X7 @ ( topolo7230453075368039082e_nhds @ A @ L6 ) @ ( at_top @ nat ) )
=> ( ( ord_less @ real @ ( zero_zero @ real ) @ R2 )
=> ? [No3: nat] :
! [N10: nat] :
( ( ord_less_eq @ nat @ No3 @ N10 )
=> ( ord_less @ real @ ( real_V7770717601297561774m_norm @ A @ ( minus_minus @ A @ ( X7 @ N10 ) @ L6 ) ) @ R2 ) ) ) ) ) ).
% LIMSEQ_D
thf(fact_6949_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_6950_tendsto__exp__limit__sequentially,axiom,
! [X: real] :
( filterlim @ nat @ real
@ ^ [N3: nat] : ( power_power @ real @ ( plus_plus @ real @ ( one_one @ real ) @ ( divide_divide @ real @ X @ ( semiring_1_of_nat @ real @ N3 ) ) ) @ N3 )
@ ( topolo7230453075368039082e_nhds @ real @ ( exp @ real @ X ) )
@ ( at_top @ nat ) ) ).
% tendsto_exp_limit_sequentially
thf(fact_6951_tendsto__power__zero,axiom,
! [A: $tType,B: $tType] :
( ( real_V2822296259951069270ebra_1 @ A )
=> ! [F2: B > nat,F3: filter @ B,X: A] :
( ( filterlim @ B @ nat @ F2 @ ( at_top @ nat ) @ F3 )
=> ( ( ord_less @ real @ ( real_V7770717601297561774m_norm @ A @ X ) @ ( one_one @ real ) )
=> ( filterlim @ B @ A
@ ^ [Y6: B] : ( power_power @ A @ X @ ( F2 @ Y6 ) )
@ ( topolo7230453075368039082e_nhds @ A @ ( zero_zero @ A ) )
@ F3 ) ) ) ) ).
% tendsto_power_zero
thf(fact_6952_LIMSEQ__inverse__real__of__nat__add__minus__mult,axiom,
! [R2: real] :
( filterlim @ nat @ real
@ ^ [N3: nat] : ( times_times @ real @ R2 @ ( plus_plus @ real @ ( one_one @ real ) @ ( uminus_uminus @ real @ ( inverse_inverse @ real @ ( semiring_1_of_nat @ real @ ( suc @ N3 ) ) ) ) ) )
@ ( topolo7230453075368039082e_nhds @ real @ R2 )
@ ( at_top @ nat ) ) ).
% LIMSEQ_inverse_real_of_nat_add_minus_mult
thf(fact_6953_LIMSEQ__norm__0,axiom,
! [A: $tType] :
( ( real_V822414075346904944vector @ A )
=> ! [F2: nat > A] :
( ! [N2: nat] : ( ord_less @ real @ ( real_V7770717601297561774m_norm @ A @ ( F2 @ N2 ) ) @ ( divide_divide @ real @ ( one_one @ real ) @ ( semiring_1_of_nat @ real @ ( suc @ N2 ) ) ) )
=> ( filterlim @ nat @ A @ F2 @ ( topolo7230453075368039082e_nhds @ A @ ( zero_zero @ A ) ) @ ( at_top @ nat ) ) ) ) ).
% LIMSEQ_norm_0
thf(fact_6954_summable__Leibniz_I1_J,axiom,
! [A2: nat > real] :
( ( filterlim @ nat @ real @ A2 @ ( topolo7230453075368039082e_nhds @ real @ ( zero_zero @ real ) ) @ ( at_top @ nat ) )
=> ( ( topological_monoseq @ real @ A2 )
=> ( summable @ real
@ ^ [N3: nat] : ( times_times @ real @ ( power_power @ real @ ( uminus_uminus @ real @ ( one_one @ real ) ) @ N3 ) @ ( A2 @ N3 ) ) ) ) ) ).
% summable_Leibniz(1)
thf(fact_6955_field__derivative__lim__unique,axiom,
! [A: $tType] :
( ( real_V3459762299906320749_field @ A )
=> ! [F2: A > A,Df: A,Z: A,S: nat > A,A2: A] :
( ( has_field_derivative @ A @ F2 @ Df @ ( topolo174197925503356063within @ A @ Z @ ( top_top @ ( set @ A ) ) ) )
=> ( ( filterlim @ nat @ A @ S @ ( topolo7230453075368039082e_nhds @ A @ ( zero_zero @ A ) ) @ ( at_top @ nat ) )
=> ( ! [N2: nat] :
( ( S @ N2 )
!= ( zero_zero @ A ) )
=> ( ( filterlim @ nat @ A
@ ^ [N3: nat] : ( divide_divide @ A @ ( minus_minus @ A @ ( F2 @ ( plus_plus @ A @ Z @ ( S @ N3 ) ) ) @ ( F2 @ Z ) ) @ ( S @ N3 ) )
@ ( topolo7230453075368039082e_nhds @ A @ A2 )
@ ( at_top @ nat ) )
=> ( Df = A2 ) ) ) ) ) ) ).
% field_derivative_lim_unique
thf(fact_6956_powser__times__n__limit__0,axiom,
! [A: $tType] :
( ( ( real_Vector_banach @ A )
& ( real_V8999393235501362500lgebra @ A ) )
=> ! [X: A] :
( ( ord_less @ real @ ( real_V7770717601297561774m_norm @ A @ X ) @ ( one_one @ real ) )
=> ( filterlim @ nat @ A
@ ^ [N3: nat] : ( times_times @ A @ ( semiring_1_of_nat @ A @ N3 ) @ ( power_power @ A @ X @ N3 ) )
@ ( topolo7230453075368039082e_nhds @ A @ ( zero_zero @ A ) )
@ ( at_top @ nat ) ) ) ) ).
% powser_times_n_limit_0
thf(fact_6957_lim__n__over__pown,axiom,
! [A: $tType] :
( ( ( real_Vector_banach @ A )
& ( real_V3459762299906320749_field @ A ) )
=> ! [X: A] :
( ( ord_less @ real @ ( one_one @ real ) @ ( real_V7770717601297561774m_norm @ A @ X ) )
=> ( filterlim @ nat @ A
@ ^ [N3: nat] : ( divide_divide @ A @ ( semiring_1_of_nat @ A @ N3 ) @ ( power_power @ A @ X @ N3 ) )
@ ( topolo7230453075368039082e_nhds @ A @ ( zero_zero @ A ) )
@ ( at_top @ nat ) ) ) ) ).
% lim_n_over_pown
thf(fact_6958_summable,axiom,
! [A2: nat > real] :
( ( filterlim @ nat @ real @ A2 @ ( topolo7230453075368039082e_nhds @ real @ ( zero_zero @ real ) ) @ ( at_top @ nat ) )
=> ( ! [N2: nat] : ( ord_less_eq @ real @ ( zero_zero @ real ) @ ( A2 @ N2 ) )
=> ( ! [N2: nat] : ( ord_less_eq @ real @ ( A2 @ ( suc @ N2 ) ) @ ( A2 @ N2 ) )
=> ( summable @ real
@ ^ [N3: nat] : ( times_times @ real @ ( power_power @ real @ ( uminus_uminus @ real @ ( one_one @ real ) ) @ N3 ) @ ( A2 @ N3 ) ) ) ) ) ) ).
% summable
thf(fact_6959_cos__diff__limit__1,axiom,
! [Theta: nat > real,Theta2: real] :
( ( filterlim @ nat @ real
@ ^ [J2: nat] : ( cos @ real @ ( minus_minus @ real @ ( Theta @ J2 ) @ Theta2 ) )
@ ( topolo7230453075368039082e_nhds @ real @ ( one_one @ real ) )
@ ( at_top @ nat ) )
=> ~ ! [K2: nat > int] :
~ ( filterlim @ nat @ real
@ ^ [J2: nat] : ( minus_minus @ real @ ( Theta @ J2 ) @ ( times_times @ real @ ( ring_1_of_int @ real @ ( K2 @ J2 ) ) @ ( times_times @ real @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) @ pi ) ) )
@ ( topolo7230453075368039082e_nhds @ real @ Theta2 )
@ ( at_top @ nat ) ) ) ).
% cos_diff_limit_1
thf(fact_6960_cos__limit__1,axiom,
! [Theta: nat > real] :
( ( filterlim @ nat @ real
@ ^ [J2: nat] : ( cos @ real @ ( Theta @ J2 ) )
@ ( topolo7230453075368039082e_nhds @ real @ ( one_one @ real ) )
@ ( at_top @ nat ) )
=> ? [K2: nat > int] :
( filterlim @ nat @ real
@ ^ [J2: nat] : ( minus_minus @ real @ ( Theta @ J2 ) @ ( times_times @ real @ ( ring_1_of_int @ real @ ( K2 @ J2 ) ) @ ( times_times @ real @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) @ pi ) ) )
@ ( topolo7230453075368039082e_nhds @ real @ ( zero_zero @ real ) )
@ ( at_top @ nat ) ) ) ).
% cos_limit_1
thf(fact_6961_summable__Leibniz_I4_J,axiom,
! [A2: nat > real] :
( ( filterlim @ nat @ real @ A2 @ ( topolo7230453075368039082e_nhds @ real @ ( zero_zero @ real ) ) @ ( at_top @ nat ) )
=> ( ( topological_monoseq @ real @ A2 )
=> ( filterlim @ nat @ real
@ ^ [N3: nat] :
( groups7311177749621191930dd_sum @ nat @ real
@ ^ [I2: nat] : ( times_times @ real @ ( power_power @ real @ ( uminus_uminus @ real @ ( one_one @ real ) ) @ I2 ) @ ( A2 @ I2 ) )
@ ( set_ord_lessThan @ nat @ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N3 ) ) )
@ ( topolo7230453075368039082e_nhds @ real
@ ( suminf @ real
@ ^ [I2: nat] : ( times_times @ real @ ( power_power @ real @ ( uminus_uminus @ real @ ( one_one @ real ) ) @ I2 ) @ ( A2 @ I2 ) ) ) )
@ ( at_top @ nat ) ) ) ) ).
% summable_Leibniz(4)
thf(fact_6962_zeroseq__arctan__series,axiom,
! [X: real] :
( ( ord_less_eq @ real @ ( abs_abs @ real @ X ) @ ( one_one @ real ) )
=> ( filterlim @ nat @ real
@ ^ [N3: nat] : ( times_times @ real @ ( divide_divide @ real @ ( one_one @ real ) @ ( semiring_1_of_nat @ real @ ( plus_plus @ nat @ ( times_times @ nat @ N3 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ ( one_one @ nat ) ) ) ) @ ( power_power @ real @ X @ ( plus_plus @ nat @ ( times_times @ nat @ N3 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ ( one_one @ nat ) ) ) )
@ ( topolo7230453075368039082e_nhds @ real @ ( zero_zero @ real ) )
@ ( at_top @ nat ) ) ) ).
% zeroseq_arctan_series
thf(fact_6963_summable__Leibniz_H_I2_J,axiom,
! [A2: nat > real,N: nat] :
( ( filterlim @ nat @ real @ A2 @ ( topolo7230453075368039082e_nhds @ real @ ( zero_zero @ real ) ) @ ( at_top @ nat ) )
=> ( ! [N2: nat] : ( ord_less_eq @ real @ ( zero_zero @ real ) @ ( A2 @ N2 ) )
=> ( ! [N2: nat] : ( ord_less_eq @ real @ ( A2 @ ( suc @ N2 ) ) @ ( A2 @ N2 ) )
=> ( ord_less_eq @ real
@ ( groups7311177749621191930dd_sum @ nat @ real
@ ^ [I2: nat] : ( times_times @ real @ ( power_power @ real @ ( uminus_uminus @ real @ ( one_one @ real ) ) @ I2 ) @ ( A2 @ I2 ) )
@ ( set_ord_lessThan @ nat @ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N ) ) )
@ ( suminf @ real
@ ^ [I2: nat] : ( times_times @ real @ ( power_power @ real @ ( uminus_uminus @ real @ ( one_one @ real ) ) @ I2 ) @ ( A2 @ I2 ) ) ) ) ) ) ) ).
% summable_Leibniz'(2)
thf(fact_6964_summable__Leibniz_H_I3_J,axiom,
! [A2: nat > real] :
( ( filterlim @ nat @ real @ A2 @ ( topolo7230453075368039082e_nhds @ real @ ( zero_zero @ real ) ) @ ( at_top @ nat ) )
=> ( ! [N2: nat] : ( ord_less_eq @ real @ ( zero_zero @ real ) @ ( A2 @ N2 ) )
=> ( ! [N2: nat] : ( ord_less_eq @ real @ ( A2 @ ( suc @ N2 ) ) @ ( A2 @ N2 ) )
=> ( filterlim @ nat @ real
@ ^ [N3: nat] :
( groups7311177749621191930dd_sum @ nat @ real
@ ^ [I2: nat] : ( times_times @ real @ ( power_power @ real @ ( uminus_uminus @ real @ ( one_one @ real ) ) @ I2 ) @ ( A2 @ I2 ) )
@ ( set_ord_lessThan @ nat @ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N3 ) ) )
@ ( topolo7230453075368039082e_nhds @ real
@ ( suminf @ real
@ ^ [I2: nat] : ( times_times @ real @ ( power_power @ real @ ( uminus_uminus @ real @ ( one_one @ real ) ) @ I2 ) @ ( A2 @ I2 ) ) ) )
@ ( at_top @ nat ) ) ) ) ) ).
% summable_Leibniz'(3)
thf(fact_6965_sums__alternating__upper__lower,axiom,
! [A2: nat > real] :
( ! [N2: nat] : ( ord_less_eq @ real @ ( A2 @ ( suc @ N2 ) ) @ ( A2 @ N2 ) )
=> ( ! [N2: nat] : ( ord_less_eq @ real @ ( zero_zero @ real ) @ ( A2 @ N2 ) )
=> ( ( filterlim @ nat @ real @ A2 @ ( topolo7230453075368039082e_nhds @ real @ ( zero_zero @ real ) ) @ ( at_top @ nat ) )
=> ? [L4: real] :
( ! [N10: nat] :
( ord_less_eq @ real
@ ( groups7311177749621191930dd_sum @ nat @ real
@ ^ [I2: nat] : ( times_times @ real @ ( power_power @ real @ ( uminus_uminus @ real @ ( one_one @ real ) ) @ I2 ) @ ( A2 @ I2 ) )
@ ( set_ord_lessThan @ nat @ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N10 ) ) )
@ L4 )
& ( filterlim @ nat @ real
@ ^ [N3: nat] :
( groups7311177749621191930dd_sum @ nat @ real
@ ^ [I2: nat] : ( times_times @ real @ ( power_power @ real @ ( uminus_uminus @ real @ ( one_one @ real ) ) @ I2 ) @ ( A2 @ I2 ) )
@ ( set_ord_lessThan @ nat @ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N3 ) ) )
@ ( topolo7230453075368039082e_nhds @ real @ L4 )
@ ( at_top @ nat ) )
& ! [N10: nat] :
( ord_less_eq @ real @ L4
@ ( groups7311177749621191930dd_sum @ nat @ real
@ ^ [I2: nat] : ( times_times @ real @ ( power_power @ real @ ( uminus_uminus @ real @ ( one_one @ real ) ) @ I2 ) @ ( A2 @ I2 ) )
@ ( set_ord_lessThan @ nat @ ( plus_plus @ nat @ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N10 ) @ ( one_one @ nat ) ) ) ) )
& ( filterlim @ nat @ real
@ ^ [N3: nat] :
( groups7311177749621191930dd_sum @ nat @ real
@ ^ [I2: nat] : ( times_times @ real @ ( power_power @ real @ ( uminus_uminus @ real @ ( one_one @ real ) ) @ I2 ) @ ( A2 @ I2 ) )
@ ( set_ord_lessThan @ nat @ ( plus_plus @ nat @ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N3 ) @ ( one_one @ nat ) ) ) )
@ ( topolo7230453075368039082e_nhds @ real @ L4 )
@ ( at_top @ nat ) ) ) ) ) ) ).
% sums_alternating_upper_lower
thf(fact_6966_summable__Leibniz_I5_J,axiom,
! [A2: nat > real] :
( ( filterlim @ nat @ real @ A2 @ ( topolo7230453075368039082e_nhds @ real @ ( zero_zero @ real ) ) @ ( at_top @ nat ) )
=> ( ( topological_monoseq @ real @ A2 )
=> ( filterlim @ nat @ real
@ ^ [N3: nat] :
( groups7311177749621191930dd_sum @ nat @ real
@ ^ [I2: nat] : ( times_times @ real @ ( power_power @ real @ ( uminus_uminus @ real @ ( one_one @ real ) ) @ I2 ) @ ( A2 @ I2 ) )
@ ( set_ord_lessThan @ nat @ ( plus_plus @ nat @ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N3 ) @ ( one_one @ nat ) ) ) )
@ ( topolo7230453075368039082e_nhds @ real
@ ( suminf @ real
@ ^ [I2: nat] : ( times_times @ real @ ( power_power @ real @ ( uminus_uminus @ real @ ( one_one @ real ) ) @ I2 ) @ ( A2 @ I2 ) ) ) )
@ ( at_top @ nat ) ) ) ) ).
% summable_Leibniz(5)
thf(fact_6967_summable__Leibniz_H_I5_J,axiom,
! [A2: nat > real] :
( ( filterlim @ nat @ real @ A2 @ ( topolo7230453075368039082e_nhds @ real @ ( zero_zero @ real ) ) @ ( at_top @ nat ) )
=> ( ! [N2: nat] : ( ord_less_eq @ real @ ( zero_zero @ real ) @ ( A2 @ N2 ) )
=> ( ! [N2: nat] : ( ord_less_eq @ real @ ( A2 @ ( suc @ N2 ) ) @ ( A2 @ N2 ) )
=> ( filterlim @ nat @ real
@ ^ [N3: nat] :
( groups7311177749621191930dd_sum @ nat @ real
@ ^ [I2: nat] : ( times_times @ real @ ( power_power @ real @ ( uminus_uminus @ real @ ( one_one @ real ) ) @ I2 ) @ ( A2 @ I2 ) )
@ ( set_ord_lessThan @ nat @ ( plus_plus @ nat @ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N3 ) @ ( one_one @ nat ) ) ) )
@ ( topolo7230453075368039082e_nhds @ real
@ ( suminf @ real
@ ^ [I2: nat] : ( times_times @ real @ ( power_power @ real @ ( uminus_uminus @ real @ ( one_one @ real ) ) @ I2 ) @ ( A2 @ I2 ) ) ) )
@ ( at_top @ nat ) ) ) ) ) ).
% summable_Leibniz'(5)
thf(fact_6968_summable__Leibniz_H_I4_J,axiom,
! [A2: nat > real,N: nat] :
( ( filterlim @ nat @ real @ A2 @ ( topolo7230453075368039082e_nhds @ real @ ( zero_zero @ real ) ) @ ( at_top @ nat ) )
=> ( ! [N2: nat] : ( ord_less_eq @ real @ ( zero_zero @ real ) @ ( A2 @ N2 ) )
=> ( ! [N2: nat] : ( ord_less_eq @ real @ ( A2 @ ( suc @ N2 ) ) @ ( A2 @ N2 ) )
=> ( ord_less_eq @ real
@ ( suminf @ real
@ ^ [I2: nat] : ( times_times @ real @ ( power_power @ real @ ( uminus_uminus @ real @ ( one_one @ real ) ) @ I2 ) @ ( A2 @ I2 ) ) )
@ ( groups7311177749621191930dd_sum @ nat @ real
@ ^ [I2: nat] : ( times_times @ real @ ( power_power @ real @ ( uminus_uminus @ real @ ( one_one @ real ) ) @ I2 ) @ ( A2 @ I2 ) )
@ ( set_ord_lessThan @ nat @ ( plus_plus @ nat @ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N ) @ ( one_one @ nat ) ) ) ) ) ) ) ) ).
% summable_Leibniz'(4)
thf(fact_6969_filterlim__sequentially__Suc,axiom,
! [A: $tType,F2: nat > A,F3: filter @ A] :
( ( filterlim @ nat @ A
@ ^ [X2: nat] : ( F2 @ ( suc @ X2 ) )
@ F3
@ ( at_top @ nat ) )
= ( filterlim @ nat @ A @ F2 @ F3 @ ( at_top @ nat ) ) ) ).
% filterlim_sequentially_Suc
thf(fact_6970_has__derivative__at2,axiom,
! [B: $tType,A: $tType] :
( ( ( real_V822414075346904944vector @ A )
& ( real_V822414075346904944vector @ B ) )
=> ! [F2: A > B,F8: A > B,X: A] :
( ( has_derivative @ A @ B @ F2 @ F8 @ ( topolo174197925503356063within @ A @ X @ ( top_top @ ( set @ A ) ) ) )
= ( ( real_V3181309239436604168linear @ A @ B @ F8 )
& ( filterlim @ A @ B
@ ^ [Y6: A] : ( real_V8093663219630862766scaleR @ B @ ( divide_divide @ real @ ( one_one @ real ) @ ( real_V7770717601297561774m_norm @ A @ ( minus_minus @ A @ Y6 @ X ) ) ) @ ( minus_minus @ B @ ( F2 @ Y6 ) @ ( plus_plus @ B @ ( F2 @ X ) @ ( F8 @ ( minus_minus @ A @ Y6 @ X ) ) ) ) )
@ ( topolo7230453075368039082e_nhds @ B @ ( zero_zero @ B ) )
@ ( topolo174197925503356063within @ A @ X @ ( top_top @ ( set @ A ) ) ) ) ) ) ) ).
% has_derivative_at2
thf(fact_6971_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_6972_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_6973_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_6974_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_6975_real__bounded__linear,axiom,
( ( real_V3181309239436604168linear @ real @ real )
= ( ^ [F4: real > real] :
? [C5: real] :
( F4
= ( ^ [X2: real] : ( times_times @ real @ X2 @ C5 ) ) ) ) ) ).
% real_bounded_linear
thf(fact_6976_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_6977_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_6978_bounded__linear__zero,axiom,
! [B: $tType,A: $tType] :
( ( ( real_V822414075346904944vector @ A )
& ( real_V822414075346904944vector @ B ) )
=> ( real_V3181309239436604168linear @ A @ B
@ ^ [X2: A] : ( zero_zero @ B ) ) ) ).
% bounded_linear_zero
thf(fact_6979_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_6980_bounded__linear_Otendsto__zero,axiom,
! [B: $tType,A: $tType,C: $tType] :
( ( ( real_V822414075346904944vector @ A )
& ( real_V822414075346904944vector @ B ) )
=> ! [F2: A > B,G: C > A,F3: filter @ C] :
( ( real_V3181309239436604168linear @ A @ B @ F2 )
=> ( ( filterlim @ C @ A @ G @ ( topolo7230453075368039082e_nhds @ A @ ( zero_zero @ A ) ) @ F3 )
=> ( filterlim @ C @ B
@ ^ [X2: C] : ( F2 @ ( G @ X2 ) )
@ ( topolo7230453075368039082e_nhds @ B @ ( zero_zero @ B ) )
@ F3 ) ) ) ) ).
% bounded_linear.tendsto_zero
thf(fact_6981_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_6982_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_6983_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_6984_filterlim__Suc,axiom,
filterlim @ nat @ nat @ suc @ ( at_top @ nat ) @ ( at_top @ nat ) ).
% filterlim_Suc
thf(fact_6985_has__derivative__iff__norm,axiom,
! [B: $tType,A: $tType] :
( ( ( real_V822414075346904944vector @ A )
& ( real_V822414075346904944vector @ B ) )
=> ! [F2: A > B,F8: A > B,X: A,S: set @ A] :
( ( has_derivative @ A @ B @ F2 @ F8 @ ( topolo174197925503356063within @ A @ X @ S ) )
= ( ( real_V3181309239436604168linear @ A @ B @ F8 )
& ( filterlim @ A @ real
@ ^ [Y6: A] : ( divide_divide @ real @ ( real_V7770717601297561774m_norm @ B @ ( minus_minus @ B @ ( minus_minus @ B @ ( F2 @ Y6 ) @ ( F2 @ X ) ) @ ( F8 @ ( minus_minus @ A @ Y6 @ X ) ) ) ) @ ( real_V7770717601297561774m_norm @ A @ ( minus_minus @ A @ Y6 @ X ) ) )
@ ( topolo7230453075368039082e_nhds @ real @ ( zero_zero @ real ) )
@ ( topolo174197925503356063within @ A @ X @ S ) ) ) ) ) ).
% has_derivative_iff_norm
thf(fact_6986_has__derivative__at__within,axiom,
! [B: $tType,A: $tType] :
( ( ( real_V822414075346904944vector @ A )
& ( real_V822414075346904944vector @ B ) )
=> ! [F2: A > B,F8: A > B,X: A,S: set @ A] :
( ( has_derivative @ A @ B @ F2 @ F8 @ ( topolo174197925503356063within @ A @ X @ S ) )
= ( ( real_V3181309239436604168linear @ A @ B @ F8 )
& ( filterlim @ A @ B
@ ^ [Y6: A] : ( real_V8093663219630862766scaleR @ B @ ( inverse_inverse @ real @ ( real_V7770717601297561774m_norm @ A @ ( minus_minus @ A @ Y6 @ X ) ) ) @ ( minus_minus @ B @ ( minus_minus @ B @ ( F2 @ Y6 ) @ ( F2 @ X ) ) @ ( F8 @ ( minus_minus @ A @ Y6 @ X ) ) ) )
@ ( topolo7230453075368039082e_nhds @ B @ ( zero_zero @ B ) )
@ ( topolo174197925503356063within @ A @ X @ S ) ) ) ) ) ).
% has_derivative_at_within
thf(fact_6987_has__derivativeI,axiom,
! [B: $tType,A: $tType] :
( ( ( real_V822414075346904944vector @ A )
& ( real_V822414075346904944vector @ B ) )
=> ! [F8: A > B,X: A,F2: A > B,S: set @ A] :
( ( real_V3181309239436604168linear @ A @ B @ F8 )
=> ( ( filterlim @ A @ B
@ ^ [Y6: A] : ( real_V8093663219630862766scaleR @ B @ ( inverse_inverse @ real @ ( real_V7770717601297561774m_norm @ A @ ( minus_minus @ A @ Y6 @ X ) ) ) @ ( minus_minus @ B @ ( minus_minus @ B @ ( F2 @ Y6 ) @ ( F2 @ X ) ) @ ( F8 @ ( minus_minus @ A @ Y6 @ X ) ) ) )
@ ( topolo7230453075368039082e_nhds @ B @ ( zero_zero @ B ) )
@ ( topolo174197925503356063within @ A @ X @ S ) )
=> ( has_derivative @ A @ B @ F2 @ F8 @ ( topolo174197925503356063within @ A @ X @ S ) ) ) ) ) ).
% has_derivativeI
thf(fact_6988_has__derivative__iff__Ex,axiom,
! [B: $tType,A: $tType] :
( ( ( real_V822414075346904944vector @ A )
& ( real_V822414075346904944vector @ B ) )
=> ! [F2: A > B,F8: A > B,X: A] :
( ( has_derivative @ A @ B @ F2 @ F8 @ ( topolo174197925503356063within @ A @ X @ ( top_top @ ( set @ A ) ) ) )
= ( ( real_V3181309239436604168linear @ A @ B @ F8 )
& ? [E4: A > B] :
( ! [H: A] :
( ( F2 @ ( plus_plus @ A @ X @ H ) )
= ( plus_plus @ B @ ( plus_plus @ B @ ( F2 @ X ) @ ( F8 @ 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_6989_has__derivative__within,axiom,
! [B: $tType,A: $tType] :
( ( ( real_V822414075346904944vector @ A )
& ( real_V822414075346904944vector @ B ) )
=> ! [F2: A > B,F8: A > B,X: A,S: set @ A] :
( ( has_derivative @ A @ B @ F2 @ F8 @ ( topolo174197925503356063within @ A @ X @ S ) )
= ( ( real_V3181309239436604168linear @ A @ B @ F8 )
& ( filterlim @ A @ B
@ ^ [Y6: A] : ( real_V8093663219630862766scaleR @ B @ ( divide_divide @ real @ ( one_one @ real ) @ ( real_V7770717601297561774m_norm @ A @ ( minus_minus @ A @ Y6 @ X ) ) ) @ ( minus_minus @ B @ ( F2 @ Y6 ) @ ( plus_plus @ B @ ( F2 @ X ) @ ( F8 @ ( minus_minus @ A @ Y6 @ X ) ) ) ) )
@ ( topolo7230453075368039082e_nhds @ B @ ( zero_zero @ B ) )
@ ( topolo174197925503356063within @ A @ X @ S ) ) ) ) ) ).
% has_derivative_within
thf(fact_6990_has__derivative__at,axiom,
! [B: $tType,A: $tType] :
( ( ( real_V822414075346904944vector @ A )
& ( real_V822414075346904944vector @ B ) )
=> ! [F2: A > B,D5: A > B,X: A] :
( ( has_derivative @ A @ B @ F2 @ D5 @ ( topolo174197925503356063within @ A @ X @ ( top_top @ ( set @ A ) ) ) )
= ( ( real_V3181309239436604168linear @ A @ B @ D5 )
& ( 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 ) ) @ ( D5 @ 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_6991_has__derivative__def,axiom,
! [B: $tType,A: $tType] :
( ( ( real_V822414075346904944vector @ A )
& ( real_V822414075346904944vector @ B ) )
=> ( ( has_derivative @ A @ B )
= ( ^ [F4: A > B,F9: A > B,F7: filter @ A] :
( ( real_V3181309239436604168linear @ A @ B @ F9 )
& ( filterlim @ A @ B
@ ^ [Y6: A] :
( real_V8093663219630862766scaleR @ B
@ ( inverse_inverse @ real
@ ( real_V7770717601297561774m_norm @ A
@ ( minus_minus @ A @ Y6
@ ( topolo3827282254853284352ce_Lim @ A @ A @ F7
@ ^ [X2: A] : X2 ) ) ) )
@ ( minus_minus @ B
@ ( minus_minus @ B @ ( F4 @ Y6 )
@ ( F4
@ ( topolo3827282254853284352ce_Lim @ A @ A @ F7
@ ^ [X2: A] : X2 ) ) )
@ ( F9
@ ( minus_minus @ A @ Y6
@ ( topolo3827282254853284352ce_Lim @ A @ A @ F7
@ ^ [X2: A] : X2 ) ) ) ) )
@ ( topolo7230453075368039082e_nhds @ B @ ( zero_zero @ B ) )
@ F7 ) ) ) ) ) ).
% has_derivative_def
thf(fact_6992_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,F8: A > B] :
( ( member @ A @ X @ S3 )
=> ( ( topolo1002775350975398744n_open @ A @ S3 )
=> ( ( has_derivative @ A @ B @ F2 @ F8 @ ( topolo174197925503356063within @ A @ X @ S3 ) )
= ( ( real_V3181309239436604168linear @ A @ B @ F8 )
& ? [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 ) @ ( F8 @ 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_6993_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_6994_continuous__divide,axiom,
! [B: $tType,A: $tType] :
( ( ( topological_t2_space @ A )
& ( real_V3459762299906320749_field @ B ) )
=> ! [F3: filter @ A,F2: A > B,G: A > B] :
( ( topolo3448309680560233919inuous @ A @ B @ F3 @ F2 )
=> ( ( topolo3448309680560233919inuous @ A @ B @ F3 @ G )
=> ( ( ( G
@ ( topolo3827282254853284352ce_Lim @ A @ A @ F3
@ ^ [X2: A] : X2 ) )
!= ( zero_zero @ B ) )
=> ( topolo3448309680560233919inuous @ A @ B @ F3
@ ^ [X2: A] : ( divide_divide @ B @ ( F2 @ X2 ) @ ( G @ X2 ) ) ) ) ) ) ) ).
% continuous_divide
thf(fact_6995_continuous__inverse,axiom,
! [B: $tType,A: $tType] :
( ( ( topological_t2_space @ A )
& ( real_V8999393235501362500lgebra @ B ) )
=> ! [F3: filter @ A,F2: A > B] :
( ( topolo3448309680560233919inuous @ A @ B @ F3 @ F2 )
=> ( ( ( F2
@ ( topolo3827282254853284352ce_Lim @ A @ A @ F3
@ ^ [X2: A] : X2 ) )
!= ( zero_zero @ B ) )
=> ( topolo3448309680560233919inuous @ A @ B @ F3
@ ^ [X2: A] : ( inverse_inverse @ B @ ( F2 @ X2 ) ) ) ) ) ) ).
% continuous_inverse
thf(fact_6996_continuous__sgn,axiom,
! [B: $tType,A: $tType] :
( ( ( topological_t2_space @ A )
& ( real_V822414075346904944vector @ B ) )
=> ! [F3: filter @ A,F2: A > B] :
( ( topolo3448309680560233919inuous @ A @ B @ F3 @ F2 )
=> ( ( ( F2
@ ( topolo3827282254853284352ce_Lim @ A @ A @ F3
@ ^ [X2: A] : X2 ) )
!= ( zero_zero @ B ) )
=> ( topolo3448309680560233919inuous @ A @ B @ F3
@ ^ [X2: A] : ( sgn_sgn @ B @ ( F2 @ X2 ) ) ) ) ) ) ).
% continuous_sgn
thf(fact_6997_continuous__powr,axiom,
! [A: $tType] :
( ( topological_t2_space @ A )
=> ! [F3: filter @ A,F2: A > real,G: A > real] :
( ( topolo3448309680560233919inuous @ A @ real @ F3 @ F2 )
=> ( ( topolo3448309680560233919inuous @ A @ real @ F3 @ G )
=> ( ( ( F2
@ ( topolo3827282254853284352ce_Lim @ A @ A @ F3
@ ^ [X2: A] : X2 ) )
!= ( zero_zero @ real ) )
=> ( topolo3448309680560233919inuous @ A @ real @ F3
@ ^ [X2: A] : ( powr @ real @ ( F2 @ X2 ) @ ( G @ X2 ) ) ) ) ) ) ) ).
% continuous_powr
thf(fact_6998_continuous__ln,axiom,
! [A: $tType] :
( ( topological_t2_space @ A )
=> ! [F3: filter @ A,F2: A > real] :
( ( topolo3448309680560233919inuous @ A @ real @ F3 @ F2 )
=> ( ( ( F2
@ ( topolo3827282254853284352ce_Lim @ A @ A @ F3
@ ^ [X2: A] : X2 ) )
!= ( zero_zero @ real ) )
=> ( topolo3448309680560233919inuous @ A @ real @ F3
@ ^ [X2: A] : ( ln_ln @ real @ ( F2 @ X2 ) ) ) ) ) ) ).
% continuous_ln
thf(fact_6999_continuous__tan,axiom,
! [A: $tType] :
( ( ( real_Vector_banach @ A )
& ( real_V3459762299906320749_field @ A ) )
=> ! [F3: filter @ A,F2: A > A] :
( ( topolo3448309680560233919inuous @ A @ A @ F3 @ F2 )
=> ( ( ( cos @ A
@ ( F2
@ ( topolo3827282254853284352ce_Lim @ A @ A @ F3
@ ^ [X2: A] : X2 ) ) )
!= ( zero_zero @ A ) )
=> ( topolo3448309680560233919inuous @ A @ A @ F3
@ ^ [X2: A] : ( tan @ A @ ( F2 @ X2 ) ) ) ) ) ) ).
% continuous_tan
thf(fact_7000_continuous__cot,axiom,
! [A: $tType] :
( ( ( real_Vector_banach @ A )
& ( real_V3459762299906320749_field @ A ) )
=> ! [F3: filter @ A,F2: A > A] :
( ( topolo3448309680560233919inuous @ A @ A @ F3 @ F2 )
=> ( ( ( sin @ A
@ ( F2
@ ( topolo3827282254853284352ce_Lim @ A @ A @ F3
@ ^ [X2: A] : X2 ) ) )
!= ( zero_zero @ A ) )
=> ( topolo3448309680560233919inuous @ A @ A @ F3
@ ^ [X2: A] : ( cot @ A @ ( F2 @ X2 ) ) ) ) ) ) ).
% continuous_cot
thf(fact_7001_continuous__tanh,axiom,
! [A: $tType,C: $tType] :
( ( ( topological_t2_space @ C )
& ( real_Vector_banach @ A )
& ( real_V3459762299906320749_field @ A ) )
=> ! [F3: filter @ C,F2: C > A] :
( ( topolo3448309680560233919inuous @ C @ A @ F3 @ F2 )
=> ( ( ( cosh @ A
@ ( F2
@ ( topolo3827282254853284352ce_Lim @ C @ C @ F3
@ ^ [X2: C] : X2 ) ) )
!= ( zero_zero @ A ) )
=> ( topolo3448309680560233919inuous @ C @ A @ F3
@ ^ [X2: C] : ( tanh @ A @ ( F2 @ X2 ) ) ) ) ) ) ).
% continuous_tanh
thf(fact_7002_continuous__arcosh,axiom,
! [A: $tType] :
( ( topological_t2_space @ A )
=> ! [F3: filter @ A,F2: A > real] :
( ( topolo3448309680560233919inuous @ A @ real @ F3 @ F2 )
=> ( ( ord_less @ real @ ( one_one @ real )
@ ( F2
@ ( topolo3827282254853284352ce_Lim @ A @ A @ F3
@ ^ [X2: A] : X2 ) ) )
=> ( topolo3448309680560233919inuous @ A @ real @ F3
@ ^ [X2: A] : ( arcosh @ real @ ( F2 @ X2 ) ) ) ) ) ) ).
% continuous_arcosh
thf(fact_7003_continuous__log,axiom,
! [A: $tType] :
( ( topological_t2_space @ A )
=> ! [F3: filter @ A,F2: A > real,G: A > real] :
( ( topolo3448309680560233919inuous @ A @ real @ F3 @ F2 )
=> ( ( topolo3448309680560233919inuous @ A @ real @ F3 @ G )
=> ( ( ord_less @ real @ ( zero_zero @ real )
@ ( F2
@ ( topolo3827282254853284352ce_Lim @ A @ A @ F3
@ ^ [X2: A] : X2 ) ) )
=> ( ( ( F2
@ ( topolo3827282254853284352ce_Lim @ A @ A @ F3
@ ^ [X2: A] : X2 ) )
!= ( one_one @ real ) )
=> ( ( ord_less @ real @ ( zero_zero @ real )
@ ( G
@ ( topolo3827282254853284352ce_Lim @ A @ A @ F3
@ ^ [X2: A] : X2 ) ) )
=> ( topolo3448309680560233919inuous @ A @ real @ F3
@ ^ [X2: A] : ( log @ ( F2 @ X2 ) @ ( G @ X2 ) ) ) ) ) ) ) ) ) ).
% continuous_log
thf(fact_7004_continuous__artanh,axiom,
! [A: $tType] :
( ( topological_t2_space @ A )
=> ! [F3: filter @ A,F2: A > real] :
( ( topolo3448309680560233919inuous @ A @ real @ F3 @ F2 )
=> ( ( member @ real
@ ( F2
@ ( topolo3827282254853284352ce_Lim @ A @ A @ F3
@ ^ [X2: A] : X2 ) )
@ ( set_or5935395276787703475ssThan @ real @ ( uminus_uminus @ real @ ( one_one @ real ) ) @ ( one_one @ real ) ) )
=> ( topolo3448309680560233919inuous @ A @ real @ F3
@ ^ [X2: A] : ( artanh @ real @ ( F2 @ X2 ) ) ) ) ) ) ).
% continuous_artanh
thf(fact_7005_tendsto__offset__zero__iff,axiom,
! [C: $tType,D4: $tType,A: $tType] :
( ( ( real_V822414075346904944vector @ A )
& ( topolo4958980785337419405_space @ D4 )
& ( zero @ C ) )
=> ! [A2: A,S3: set @ A,F2: A > D4,L6: D4] :
( ( nO_MATCH @ C @ A @ ( zero_zero @ C ) @ A2 )
=> ( ( member @ A @ A2 @ S3 )
=> ( ( topolo1002775350975398744n_open @ A @ S3 )
=> ( ( filterlim @ A @ D4 @ F2 @ ( topolo7230453075368039082e_nhds @ D4 @ L6 ) @ ( topolo174197925503356063within @ A @ A2 @ S3 ) )
= ( filterlim @ A @ D4
@ ^ [H: A] : ( F2 @ ( plus_plus @ A @ A2 @ H ) )
@ ( topolo7230453075368039082e_nhds @ D4 @ L6 )
@ ( topolo174197925503356063within @ A @ ( zero_zero @ A ) @ ( top_top @ ( set @ A ) ) ) ) ) ) ) ) ) ).
% tendsto_offset_zero_iff
thf(fact_7006_has__derivativeI__sandwich,axiom,
! [B: $tType,A: $tType] :
( ( ( real_V822414075346904944vector @ A )
& ( real_V822414075346904944vector @ B ) )
=> ! [E: real,F8: A > B,S: set @ A,X: A,F2: A > B,H8: A > real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ E )
=> ( ( real_V3181309239436604168linear @ A @ B @ F8 )
=> ( ! [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 ) ) @ ( F8 @ ( minus_minus @ A @ Y3 @ X ) ) ) ) @ ( real_V7770717601297561774m_norm @ A @ ( minus_minus @ A @ Y3 @ X ) ) ) @ ( H8 @ Y3 ) ) ) ) )
=> ( ( filterlim @ A @ real @ H8 @ ( topolo7230453075368039082e_nhds @ real @ ( zero_zero @ real ) ) @ ( topolo174197925503356063within @ A @ X @ S ) )
=> ( has_derivative @ A @ B @ F2 @ F8 @ ( topolo174197925503356063within @ A @ X @ S ) ) ) ) ) ) ) ).
% has_derivativeI_sandwich
thf(fact_7007_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_7008_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_7009_dist__eq__0__iff,axiom,
! [A: $tType] :
( ( real_V7819770556892013058_space @ A )
=> ! [X: A,Y: A] :
( ( ( real_V557655796197034286t_dist @ A @ X @ Y )
= ( zero_zero @ real ) )
= ( X = Y ) ) ) ).
% dist_eq_0_iff
thf(fact_7010_dist__self,axiom,
! [A: $tType] :
( ( real_V7819770556892013058_space @ A )
=> ! [X: A] :
( ( real_V557655796197034286t_dist @ A @ X @ X )
= ( zero_zero @ real ) ) ) ).
% dist_self
thf(fact_7011_dist__0__norm,axiom,
! [A: $tType] :
( ( real_V822414075346904944vector @ A )
=> ! [X: A] :
( ( real_V557655796197034286t_dist @ A @ ( zero_zero @ A ) @ X )
= ( real_V7770717601297561774m_norm @ A @ X ) ) ) ).
% dist_0_norm
thf(fact_7012_zero__less__dist__iff,axiom,
! [A: $tType] :
( ( real_V7819770556892013058_space @ A )
=> ! [X: A,Y: A] :
( ( ord_less @ real @ ( zero_zero @ real ) @ ( real_V557655796197034286t_dist @ A @ X @ Y ) )
= ( X != Y ) ) ) ).
% zero_less_dist_iff
thf(fact_7013_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_7014_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_7015_open__ball,axiom,
! [A: $tType] :
( ( real_V7819770556892013058_space @ A )
=> ! [X: A,D2: real] :
( topolo1002775350975398744n_open @ A
@ ( collect @ A
@ ^ [Y6: A] : ( ord_less @ real @ ( real_V557655796197034286t_dist @ A @ X @ Y6 ) @ D2 ) ) ) ) ).
% open_ball
thf(fact_7016_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_7017_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_7018_dist__commute__lessI,axiom,
! [A: $tType] :
( ( real_V7819770556892013058_space @ A )
=> ! [Y: A,X: A,E: real] :
( ( ord_less @ real @ ( real_V557655796197034286t_dist @ A @ Y @ X ) @ E )
=> ( ord_less @ real @ ( real_V557655796197034286t_dist @ A @ X @ Y ) @ E ) ) ) ).
% dist_commute_lessI
thf(fact_7019_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_7020_dist__not__less__zero,axiom,
! [A: $tType] :
( ( real_V7819770556892013058_space @ A )
=> ! [X: A,Y: A] :
~ ( ord_less @ real @ ( real_V557655796197034286t_dist @ A @ X @ Y ) @ ( zero_zero @ real ) ) ) ).
% dist_not_less_zero
thf(fact_7021_dist__pos__lt,axiom,
! [A: $tType] :
( ( real_V7819770556892013058_space @ A )
=> ! [X: A,Y: A] :
( ( X != Y )
=> ( ord_less @ real @ ( zero_zero @ real ) @ ( real_V557655796197034286t_dist @ A @ X @ Y ) ) ) ) ).
% dist_pos_lt
thf(fact_7022_norm__conv__dist,axiom,
! [A: $tType] :
( ( real_V822414075346904944vector @ A )
=> ( ( real_V7770717601297561774m_norm @ A )
= ( ^ [X2: A] : ( real_V557655796197034286t_dist @ A @ X2 @ ( zero_zero @ A ) ) ) ) ) ).
% norm_conv_dist
thf(fact_7023_open__dist,axiom,
! [A: $tType] :
( ( real_V7819770556892013058_space @ A )
=> ( ( topolo1002775350975398744n_open @ A )
= ( ^ [S7: set @ A] :
! [X2: A] :
( ( member @ A @ X2 @ S7 )
=> ? [E4: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ E4 )
& ! [Y6: A] :
( ( ord_less @ real @ ( real_V557655796197034286t_dist @ A @ Y6 @ X2 ) @ E4 )
=> ( member @ A @ Y6 @ S7 ) ) ) ) ) ) ) ).
% open_dist
thf(fact_7024_filterlim__transform__within,axiom,
! [B: $tType,A: $tType] :
( ( real_V7819770556892013058_space @ A )
=> ! [G: A > B,G5: filter @ B,X: A,S3: set @ A,F3: filter @ B,D2: real,F2: A > B] :
( ( filterlim @ A @ B @ G @ G5 @ ( topolo174197925503356063within @ A @ X @ S3 ) )
=> ( ( ord_less_eq @ ( filter @ B ) @ G5 @ F3 )
=> ( ( ord_less @ real @ ( zero_zero @ real ) @ D2 )
=> ( ! [X4: A] :
( ( member @ A @ X4 @ S3 )
=> ( ( ord_less @ real @ ( zero_zero @ real ) @ ( real_V557655796197034286t_dist @ A @ X4 @ X ) )
=> ( ( ord_less @ real @ ( real_V557655796197034286t_dist @ A @ X4 @ X ) @ D2 )
=> ( ( F2 @ X4 )
= ( G @ X4 ) ) ) ) )
=> ( filterlim @ A @ B @ F2 @ F3 @ ( topolo174197925503356063within @ A @ X @ S3 ) ) ) ) ) ) ) ).
% filterlim_transform_within
thf(fact_7025_has__field__derivative__transform__within,axiom,
! [A: $tType] :
( ( real_V3459762299906320749_field @ A )
=> ! [F2: A > A,F8: A,A2: A,S3: set @ A,D2: real,G: A > A] :
( ( has_field_derivative @ A @ F2 @ F8 @ ( topolo174197925503356063within @ A @ A2 @ S3 ) )
=> ( ( ord_less @ real @ ( zero_zero @ real ) @ D2 )
=> ( ( member @ A @ A2 @ S3 )
=> ( ! [X3: A] :
( ( member @ A @ X3 @ S3 )
=> ( ( ord_less @ real @ ( real_V557655796197034286t_dist @ A @ X3 @ A2 ) @ D2 )
=> ( ( F2 @ X3 )
= ( G @ X3 ) ) ) )
=> ( has_field_derivative @ A @ G @ F8 @ ( topolo174197925503356063within @ A @ A2 @ S3 ) ) ) ) ) ) ) ).
% has_field_derivative_transform_within
thf(fact_7026_has__derivative__transform__within,axiom,
! [B: $tType,A: $tType] :
( ( ( real_V822414075346904944vector @ A )
& ( real_V822414075346904944vector @ B ) )
=> ! [F2: A > B,F8: A > B,X: A,S: set @ A,D2: real,G: A > B] :
( ( has_derivative @ A @ B @ F2 @ F8 @ ( topolo174197925503356063within @ A @ X @ S ) )
=> ( ( ord_less @ real @ ( zero_zero @ real ) @ D2 )
=> ( ( member @ A @ X @ S )
=> ( ! [X4: A] :
( ( member @ A @ X4 @ S )
=> ( ( ord_less @ real @ ( real_V557655796197034286t_dist @ A @ X4 @ X ) @ D2 )
=> ( ( F2 @ X4 )
= ( G @ X4 ) ) ) )
=> ( has_derivative @ A @ B @ G @ F8 @ ( topolo174197925503356063within @ A @ X @ S ) ) ) ) ) ) ) ).
% has_derivative_transform_within
thf(fact_7027_metric__CauchyI,axiom,
! [A: $tType] :
( ( real_V7819770556892013058_space @ A )
=> ! [X7: nat > A] :
( ! [E2: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ E2 )
=> ? [M11: nat] :
! [M2: nat] :
( ( ord_less_eq @ nat @ M11 @ M2 )
=> ! [N2: nat] :
( ( ord_less_eq @ nat @ M11 @ N2 )
=> ( ord_less @ real @ ( real_V557655796197034286t_dist @ A @ ( X7 @ M2 ) @ ( X7 @ N2 ) ) @ E2 ) ) ) )
=> ( topolo3814608138187158403Cauchy @ A @ X7 ) ) ) ).
% metric_CauchyI
thf(fact_7028_metric__CauchyD,axiom,
! [A: $tType] :
( ( real_V7819770556892013058_space @ A )
=> ! [X7: nat > A,E: real] :
( ( topolo3814608138187158403Cauchy @ A @ X7 )
=> ( ( ord_less @ real @ ( zero_zero @ real ) @ E )
=> ? [M9: nat] :
! [M3: nat] :
( ( ord_less_eq @ nat @ M9 @ M3 )
=> ! [N10: nat] :
( ( ord_less_eq @ nat @ M9 @ N10 )
=> ( ord_less @ real @ ( real_V557655796197034286t_dist @ A @ ( X7 @ M3 ) @ ( X7 @ N10 ) ) @ E ) ) ) ) ) ) ).
% metric_CauchyD
thf(fact_7029_Cauchy__altdef2,axiom,
! [A: $tType] :
( ( real_V7819770556892013058_space @ A )
=> ( ( topolo3814608138187158403Cauchy @ A )
= ( ^ [S6: nat > A] :
! [E4: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ E4 )
=> ? [N8: nat] :
! [N3: nat] :
( ( ord_less_eq @ nat @ N8 @ N3 )
=> ( ord_less @ real @ ( real_V557655796197034286t_dist @ A @ ( S6 @ N3 ) @ ( S6 @ N8 ) ) @ E4 ) ) ) ) ) ) ).
% Cauchy_altdef2
thf(fact_7030_Cauchy__def,axiom,
! [A: $tType] :
( ( real_V7819770556892013058_space @ A )
=> ( ( topolo3814608138187158403Cauchy @ A )
= ( ^ [X8: nat > A] :
! [E4: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ E4 )
=> ? [M8: nat] :
! [M4: nat] :
( ( ord_less_eq @ nat @ M8 @ M4 )
=> ! [N3: nat] :
( ( ord_less_eq @ nat @ M8 @ N3 )
=> ( ord_less @ real @ ( real_V557655796197034286t_dist @ A @ ( X8 @ M4 ) @ ( X8 @ N3 ) ) @ E4 ) ) ) ) ) ) ) ).
% Cauchy_def
thf(fact_7031_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_7032_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_7033_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_7034_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_7035_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_7036_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_7037_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_7038_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_7039_dist__triangle__third,axiom,
! [A: $tType] :
( ( real_V7819770556892013058_space @ A )
=> ! [X15: A,X22: A,E: real,X32: 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 @ X32 ) @ ( divide_divide @ real @ E @ ( numeral_numeral @ real @ ( bit1 @ one2 ) ) ) )
=> ( ( ord_less @ real @ ( real_V557655796197034286t_dist @ A @ X32 @ 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_7040_Lim__transform__within,axiom,
! [B: $tType,A: $tType] :
( ( ( real_V7819770556892013058_space @ A )
& ( topolo4958980785337419405_space @ B ) )
=> ! [F2: A > B,L: B,X: A,S3: set @ A,D2: real,G: A > B] :
( ( filterlim @ A @ B @ F2 @ ( topolo7230453075368039082e_nhds @ B @ L ) @ ( topolo174197925503356063within @ A @ X @ S3 ) )
=> ( ( ord_less @ real @ ( zero_zero @ real ) @ D2 )
=> ( ! [X4: A] :
( ( member @ A @ X4 @ S3 )
=> ( ( ord_less @ real @ ( zero_zero @ real ) @ ( real_V557655796197034286t_dist @ A @ X4 @ X ) )
=> ( ( ord_less @ real @ ( real_V557655796197034286t_dist @ A @ X4 @ X ) @ D2 )
=> ( ( F2 @ X4 )
= ( G @ X4 ) ) ) ) )
=> ( filterlim @ A @ B @ G @ ( topolo7230453075368039082e_nhds @ B @ L ) @ ( topolo174197925503356063within @ A @ X @ S3 ) ) ) ) ) ) ).
% Lim_transform_within
thf(fact_7041_Cauchy__altdef,axiom,
! [A: $tType] :
( ( real_V7819770556892013058_space @ A )
=> ( ( topolo3814608138187158403Cauchy @ A )
= ( ^ [F4: nat > A] :
! [E4: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ E4 )
=> ? [M8: nat] :
! [M4: nat] :
( ( ord_less_eq @ nat @ M8 @ M4 )
=> ! [N3: nat] :
( ( ord_less @ nat @ M4 @ N3 )
=> ( ord_less @ real @ ( real_V557655796197034286t_dist @ A @ ( F4 @ M4 ) @ ( F4 @ N3 ) ) @ E4 ) ) ) ) ) ) ) ).
% Cauchy_altdef
thf(fact_7042_CauchyI_H,axiom,
! [A: $tType] :
( ( real_V7819770556892013058_space @ A )
=> ! [X7: nat > A] :
( ! [E2: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ E2 )
=> ? [M11: nat] :
! [M2: nat] :
( ( ord_less_eq @ nat @ M11 @ M2 )
=> ! [N2: nat] :
( ( ord_less @ nat @ M2 @ N2 )
=> ( ord_less @ real @ ( real_V557655796197034286t_dist @ A @ ( X7 @ M2 ) @ ( X7 @ N2 ) ) @ E2 ) ) ) )
=> ( topolo3814608138187158403Cauchy @ A @ X7 ) ) ) ).
% CauchyI'
thf(fact_7043_tendsto__dist__iff,axiom,
! [B: $tType,A: $tType] :
( ( real_V7819770556892013058_space @ B )
=> ! [F2: A > B,L: B,F3: filter @ A] :
( ( filterlim @ A @ B @ F2 @ ( topolo7230453075368039082e_nhds @ B @ L ) @ F3 )
= ( filterlim @ A @ real
@ ^ [X2: A] : ( real_V557655796197034286t_dist @ B @ ( F2 @ X2 ) @ L )
@ ( topolo7230453075368039082e_nhds @ real @ ( zero_zero @ real ) )
@ F3 ) ) ) ).
% tendsto_dist_iff
thf(fact_7044_LIM__def,axiom,
! [B: $tType,A: $tType] :
( ( ( real_V7819770556892013058_space @ A )
& ( real_V7819770556892013058_space @ B ) )
=> ! [F2: A > B,L6: B,A2: A] :
( ( filterlim @ A @ B @ F2 @ ( topolo7230453075368039082e_nhds @ B @ L6 ) @ ( topolo174197925503356063within @ A @ A2 @ ( top_top @ ( set @ A ) ) ) )
= ( ! [R5: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ R5 )
=> ? [S6: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ S6 )
& ! [X2: A] :
( ( ( X2 != A2 )
& ( ord_less @ real @ ( real_V557655796197034286t_dist @ A @ X2 @ A2 ) @ S6 ) )
=> ( ord_less @ real @ ( real_V557655796197034286t_dist @ B @ ( F2 @ X2 ) @ L6 ) @ R5 ) ) ) ) ) ) ) ).
% LIM_def
thf(fact_7045_metric__LIM__D,axiom,
! [B: $tType,A: $tType] :
( ( ( real_V7819770556892013058_space @ A )
& ( real_V7819770556892013058_space @ 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_V557655796197034286t_dist @ A @ X5 @ A2 ) @ S2 ) )
=> ( ord_less @ real @ ( real_V557655796197034286t_dist @ B @ ( F2 @ X5 ) @ L6 ) @ R2 ) ) ) ) ) ) ).
% metric_LIM_D
thf(fact_7046_metric__LIM__I,axiom,
! [B: $tType,A: $tType] :
( ( ( real_V7819770556892013058_space @ A )
& ( real_V7819770556892013058_space @ B ) )
=> ! [A2: A,F2: A > B,L6: B] :
( ! [R3: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ R3 )
=> ? [S5: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ S5 )
& ! [X3: A] :
( ( ( X3 != A2 )
& ( ord_less @ real @ ( real_V557655796197034286t_dist @ A @ X3 @ A2 ) @ S5 ) )
=> ( ord_less @ real @ ( real_V557655796197034286t_dist @ B @ ( F2 @ X3 ) @ L6 ) @ R3 ) ) ) )
=> ( filterlim @ A @ B @ F2 @ ( topolo7230453075368039082e_nhds @ B @ L6 ) @ ( topolo174197925503356063within @ A @ A2 @ ( top_top @ ( set @ A ) ) ) ) ) ) ).
% metric_LIM_I
thf(fact_7047_metric__LIM__equal2,axiom,
! [B: $tType,A: $tType] :
( ( ( real_V7819770556892013058_space @ A )
& ( topolo4958980785337419405_space @ B ) )
=> ! [G: A > B,L: B,A2: A,R: real,F2: A > B] :
( ( filterlim @ A @ B @ G @ ( topolo7230453075368039082e_nhds @ B @ L ) @ ( topolo174197925503356063within @ A @ A2 @ ( top_top @ ( set @ A ) ) ) )
=> ( ( ord_less @ real @ ( zero_zero @ real ) @ R )
=> ( ! [X3: A] :
( ( X3 != A2 )
=> ( ( ord_less @ real @ ( real_V557655796197034286t_dist @ A @ X3 @ A2 ) @ R )
=> ( ( F2 @ X3 )
= ( G @ X3 ) ) ) )
=> ( filterlim @ A @ B @ F2 @ ( topolo7230453075368039082e_nhds @ B @ L ) @ ( topolo174197925503356063within @ A @ A2 @ ( top_top @ ( set @ A ) ) ) ) ) ) ) ) ).
% metric_LIM_equal2
thf(fact_7048_lim__sequentially,axiom,
! [A: $tType] :
( ( real_V7819770556892013058_space @ A )
=> ! [X7: nat > A,L6: A] :
( ( filterlim @ nat @ A @ X7 @ ( topolo7230453075368039082e_nhds @ A @ L6 ) @ ( at_top @ nat ) )
= ( ! [R5: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ R5 )
=> ? [No: nat] :
! [N3: nat] :
( ( ord_less_eq @ nat @ No @ N3 )
=> ( ord_less @ real @ ( real_V557655796197034286t_dist @ A @ ( X7 @ N3 ) @ L6 ) @ R5 ) ) ) ) ) ) ).
% lim_sequentially
thf(fact_7049_metric__LIMSEQ__I,axiom,
! [A: $tType] :
( ( real_V7819770556892013058_space @ A )
=> ! [X7: nat > A,L6: A] :
( ! [R3: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ R3 )
=> ? [No2: nat] :
! [N2: nat] :
( ( ord_less_eq @ nat @ No2 @ N2 )
=> ( ord_less @ real @ ( real_V557655796197034286t_dist @ A @ ( X7 @ N2 ) @ L6 ) @ R3 ) ) )
=> ( filterlim @ nat @ A @ X7 @ ( topolo7230453075368039082e_nhds @ A @ L6 ) @ ( at_top @ nat ) ) ) ) ).
% metric_LIMSEQ_I
thf(fact_7050_metric__LIMSEQ__D,axiom,
! [A: $tType] :
( ( real_V7819770556892013058_space @ A )
=> ! [X7: nat > A,L6: A,R2: real] :
( ( filterlim @ nat @ A @ X7 @ ( topolo7230453075368039082e_nhds @ A @ L6 ) @ ( at_top @ nat ) )
=> ( ( ord_less @ real @ ( zero_zero @ real ) @ R2 )
=> ? [No3: nat] :
! [N10: nat] :
( ( ord_less_eq @ nat @ No3 @ N10 )
=> ( ord_less @ real @ ( real_V557655796197034286t_dist @ A @ ( X7 @ N10 ) @ L6 ) @ R2 ) ) ) ) ) ).
% metric_LIMSEQ_D
thf(fact_7051_power__minus_H,axiom,
! [A: $tType] :
( ( ring_1 @ A )
=> ! [X: A,N: nat] :
( ( nO_MATCH @ A @ A @ ( one_one @ A ) @ X )
=> ( ( power_power @ A @ ( uminus_uminus @ A @ X ) @ N )
= ( times_times @ A @ ( power_power @ A @ ( uminus_uminus @ A @ ( one_one @ A ) ) @ N ) @ ( power_power @ A @ X @ N ) ) ) ) ) ).
% power_minus'
thf(fact_7052_metric__Cauchy__iff2,axiom,
! [A: $tType] :
( ( real_V7819770556892013058_space @ A )
=> ( ( topolo3814608138187158403Cauchy @ A )
= ( ^ [X8: nat > A] :
! [J2: nat] :
? [M8: nat] :
! [M4: nat] :
( ( ord_less_eq @ nat @ M8 @ M4 )
=> ! [N3: nat] :
( ( ord_less_eq @ nat @ M8 @ N3 )
=> ( ord_less @ real @ ( real_V557655796197034286t_dist @ A @ ( X8 @ M4 ) @ ( X8 @ N3 ) ) @ ( inverse_inverse @ real @ ( semiring_1_of_nat @ real @ ( suc @ J2 ) ) ) ) ) ) ) ) ) ).
% metric_Cauchy_iff2
thf(fact_7053_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_7054_metric__LIM__compose2,axiom,
! [C: $tType,B: $tType,A: $tType] :
( ( ( real_V7819770556892013058_space @ A )
& ( topolo4958980785337419405_space @ B )
& ( topolo4958980785337419405_space @ C ) )
=> ! [F2: A > B,B2: B,A2: A,G: B > C,C2: C] :
( ( filterlim @ A @ B @ F2 @ ( topolo7230453075368039082e_nhds @ B @ B2 ) @ ( topolo174197925503356063within @ A @ A2 @ ( top_top @ ( set @ A ) ) ) )
=> ( ( filterlim @ B @ C @ G @ ( topolo7230453075368039082e_nhds @ C @ C2 ) @ ( topolo174197925503356063within @ B @ B2 @ ( top_top @ ( set @ B ) ) ) )
=> ( ? [D6: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ D6 )
& ! [X3: A] :
( ( ( X3 != A2 )
& ( ord_less @ real @ ( real_V557655796197034286t_dist @ 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 ) ) ) ) ) ) ) ) ).
% metric_LIM_compose2
thf(fact_7055_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_7056_metric__isCont__LIM__compose2,axiom,
! [D4: $tType,C: $tType,A: $tType] :
( ( ( real_V7819770556892013058_space @ A )
& ( topolo4958980785337419405_space @ C )
& ( topolo4958980785337419405_space @ D4 ) )
=> ! [A2: A,F2: A > C,G: C > D4,L: D4] :
( ( topolo3448309680560233919inuous @ A @ C @ ( topolo174197925503356063within @ A @ A2 @ ( top_top @ ( set @ A ) ) ) @ F2 )
=> ( ( filterlim @ C @ D4 @ G @ ( topolo7230453075368039082e_nhds @ D4 @ L ) @ ( topolo174197925503356063within @ C @ ( F2 @ A2 ) @ ( top_top @ ( set @ C ) ) ) )
=> ( ? [D6: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ D6 )
& ! [X3: A] :
( ( ( X3 != A2 )
& ( ord_less @ real @ ( real_V557655796197034286t_dist @ A @ X3 @ A2 ) @ D6 ) )
=> ( ( F2 @ X3 )
!= ( F2 @ A2 ) ) ) )
=> ( filterlim @ A @ D4
@ ^ [X2: A] : ( G @ ( F2 @ X2 ) )
@ ( topolo7230453075368039082e_nhds @ D4 @ L )
@ ( topolo174197925503356063within @ A @ A2 @ ( top_top @ ( set @ A ) ) ) ) ) ) ) ) ).
% metric_isCont_LIM_compose2
thf(fact_7057_LIMSEQ__iff__nz,axiom,
! [A: $tType] :
( ( real_V7819770556892013058_space @ A )
=> ! [X7: nat > A,L6: A] :
( ( filterlim @ nat @ A @ X7 @ ( 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 )
& ! [N3: nat] :
( ( ord_less_eq @ nat @ No @ N3 )
=> ( ord_less @ real @ ( real_V557655796197034286t_dist @ A @ ( X7 @ N3 ) @ L6 ) @ R5 ) ) ) ) ) ) ) ).
% LIMSEQ_iff_nz
thf(fact_7058_LIM__offset__zero__iff,axiom,
! [C: $tType,D4: $tType,A: $tType] :
( ( ( real_V822414075346904944vector @ A )
& ( topolo4958980785337419405_space @ D4 )
& ( zero @ C ) )
=> ! [A2: A,F2: A > D4,L6: D4] :
( ( nO_MATCH @ C @ A @ ( zero_zero @ C ) @ A2 )
=> ( ( filterlim @ A @ D4 @ F2 @ ( topolo7230453075368039082e_nhds @ D4 @ L6 ) @ ( topolo174197925503356063within @ A @ A2 @ ( top_top @ ( set @ A ) ) ) )
= ( filterlim @ A @ D4
@ ^ [H: A] : ( F2 @ ( plus_plus @ A @ A2 @ H ) )
@ ( topolo7230453075368039082e_nhds @ D4 @ L6 )
@ ( topolo174197925503356063within @ A @ ( zero_zero @ A ) @ ( top_top @ ( set @ A ) ) ) ) ) ) ) ).
% LIM_offset_zero_iff
thf(fact_7059_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_7060_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_7061_less__filter__def,axiom,
! [A: $tType] :
( ( ord_less @ ( filter @ A ) )
= ( ^ [F7: filter @ A,F10: filter @ A] :
( ( ord_less_eq @ ( filter @ A ) @ F7 @ F10 )
& ~ ( ord_less_eq @ ( filter @ A ) @ F10 @ F7 ) ) ) ) ).
% less_filter_def
thf(fact_7062_tendsto__exp__limit__at__right,axiom,
! [X: real] :
( filterlim @ real @ real
@ ^ [Y6: real] : ( powr @ real @ ( plus_plus @ real @ ( one_one @ real ) @ ( times_times @ real @ X @ Y6 ) ) @ ( divide_divide @ real @ ( one_one @ real ) @ Y6 ) )
@ ( 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_7063_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_7064_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_7065_greaterThan__def,axiom,
! [A: $tType] :
( ( ord @ A )
=> ( ( set_ord_greaterThan @ A )
= ( ^ [L2: A] : ( collect @ A @ ( ord_less @ A @ L2 ) ) ) ) ) ).
% greaterThan_def
thf(fact_7066_ln__at__0,axiom,
filterlim @ real @ real @ ( ln_ln @ real ) @ ( at_bot @ real ) @ ( topolo174197925503356063within @ real @ ( zero_zero @ real ) @ ( set_ord_greaterThan @ real @ ( zero_zero @ real ) ) ) ).
% ln_at_0
thf(fact_7067_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_7068_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_7069_filterlim__at__right__to__0,axiom,
! [A: $tType,F2: real > A,F3: filter @ A,A2: real] :
( ( filterlim @ real @ A @ F2 @ F3 @ ( topolo174197925503356063within @ real @ A2 @ ( set_ord_greaterThan @ real @ A2 ) ) )
= ( filterlim @ real @ A
@ ^ [X2: real] : ( F2 @ ( plus_plus @ real @ X2 @ A2 ) )
@ F3
@ ( topolo174197925503356063within @ real @ ( zero_zero @ real ) @ ( set_ord_greaterThan @ real @ ( zero_zero @ real ) ) ) ) ) ).
% filterlim_at_right_to_0
thf(fact_7070_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_7071_exp__at__bot,axiom,
filterlim @ real @ real @ ( exp @ real ) @ ( topolo7230453075368039082e_nhds @ real @ ( zero_zero @ real ) ) @ ( at_bot @ real ) ).
% exp_at_bot
thf(fact_7072_filterlim__inverse__at__bot__neg,axiom,
filterlim @ real @ real @ ( inverse_inverse @ real ) @ ( at_bot @ real ) @ ( topolo174197925503356063within @ real @ ( zero_zero @ real ) @ ( set_ord_lessThan @ real @ ( zero_zero @ real ) ) ) ).
% filterlim_inverse_at_bot_neg
thf(fact_7073_filterlim__times__pos,axiom,
! [A: $tType,B: $tType] :
( ( ( linordered_field @ A )
& ( topolo1944317154257567458pology @ A ) )
=> ! [F2: B > A,P4: A,F1: filter @ B,C2: A,L: A] :
( ( filterlim @ B @ A @ F2 @ ( topolo174197925503356063within @ A @ P4 @ ( set_ord_greaterThan @ A @ P4 ) ) @ F1 )
=> ( ( ord_less @ A @ ( zero_zero @ A ) @ C2 )
=> ( ( L
= ( times_times @ A @ C2 @ P4 ) )
=> ( filterlim @ B @ A
@ ^ [X2: B] : ( times_times @ A @ C2 @ ( F2 @ X2 ) )
@ ( topolo174197925503356063within @ A @ L @ ( set_ord_greaterThan @ A @ L ) )
@ F1 ) ) ) ) ) ).
% filterlim_times_pos
thf(fact_7074_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_7075_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_7076_filterlim__tendsto__pos__mult__at__bot,axiom,
! [A: $tType,F2: A > real,C2: real,F3: filter @ A,G: A > real] :
( ( filterlim @ A @ real @ F2 @ ( topolo7230453075368039082e_nhds @ real @ C2 ) @ F3 )
=> ( ( ord_less @ real @ ( zero_zero @ real ) @ C2 )
=> ( ( filterlim @ A @ real @ G @ ( at_bot @ real ) @ F3 )
=> ( filterlim @ A @ real
@ ^ [X2: A] : ( times_times @ real @ ( F2 @ X2 ) @ ( G @ X2 ) )
@ ( at_bot @ real )
@ F3 ) ) ) ) ).
% filterlim_tendsto_pos_mult_at_bot
thf(fact_7077_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_7078_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_7079_filterlim__pow__at__bot__odd,axiom,
! [N: nat,F2: real > real,F3: filter @ real] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ N )
=> ( ( filterlim @ real @ real @ F2 @ ( at_bot @ real ) @ F3 )
=> ( ~ ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N )
=> ( filterlim @ real @ real
@ ^ [X2: real] : ( power_power @ real @ ( F2 @ X2 ) @ N )
@ ( at_bot @ real )
@ F3 ) ) ) ) ).
% filterlim_pow_at_bot_odd
thf(fact_7080_filterlim__pow__at__bot__even,axiom,
! [N: nat,F2: real > real,F3: filter @ real] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ N )
=> ( ( filterlim @ real @ real @ F2 @ ( at_bot @ real ) @ F3 )
=> ( ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N )
=> ( filterlim @ real @ real
@ ^ [X2: real] : ( power_power @ real @ ( F2 @ X2 ) @ N )
@ ( at_top @ real )
@ F3 ) ) ) ) ).
% filterlim_pow_at_bot_even
thf(fact_7081_lim__zero__infinity,axiom,
! [A: $tType] :
( ( real_V3459762299906320749_field @ A )
=> ! [F2: A > A,L: A] :
( ( filterlim @ A @ A
@ ^ [X2: A] : ( F2 @ ( divide_divide @ A @ ( one_one @ A ) @ X2 ) )
@ ( topolo7230453075368039082e_nhds @ A @ L )
@ ( topolo174197925503356063within @ A @ ( zero_zero @ A ) @ ( top_top @ ( set @ A ) ) ) )
=> ( filterlim @ A @ A @ F2 @ ( topolo7230453075368039082e_nhds @ A @ L ) @ ( at_infinity @ A ) ) ) ) ).
% lim_zero_infinity
thf(fact_7082_filterlim__at__top__mult__at__top,axiom,
! [A: $tType,F2: A > real,F3: filter @ A,G: A > real] :
( ( filterlim @ A @ real @ F2 @ ( at_top @ real ) @ F3 )
=> ( ( filterlim @ A @ real @ G @ ( at_top @ real ) @ F3 )
=> ( filterlim @ A @ real
@ ^ [X2: A] : ( times_times @ real @ ( F2 @ X2 ) @ ( G @ X2 ) )
@ ( at_top @ real )
@ F3 ) ) ) ).
% filterlim_at_top_mult_at_top
thf(fact_7083_sqrt__at__top,axiom,
filterlim @ real @ real @ sqrt @ ( at_top @ real ) @ ( at_top @ real ) ).
% sqrt_at_top
thf(fact_7084_greaterThan__0,axiom,
( ( set_ord_greaterThan @ nat @ ( zero_zero @ nat ) )
= ( image @ nat @ nat @ suc @ ( top_top @ ( set @ nat ) ) ) ) ).
% greaterThan_0
thf(fact_7085_filterlim__pow__at__top,axiom,
! [A: $tType,N: nat,F2: A > real,F3: filter @ A] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ N )
=> ( ( filterlim @ A @ real @ F2 @ ( at_top @ real ) @ F3 )
=> ( filterlim @ A @ real
@ ^ [X2: A] : ( power_power @ real @ ( F2 @ X2 ) @ N )
@ ( at_top @ real )
@ F3 ) ) ) ).
% filterlim_pow_at_top
thf(fact_7086_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_7087_tendsto__add__filterlim__at__infinity,axiom,
! [B: $tType,A: $tType] :
( ( real_V822414075346904944vector @ B )
=> ! [F2: A > B,C2: B,F3: filter @ A,G: A > B] :
( ( filterlim @ A @ B @ F2 @ ( topolo7230453075368039082e_nhds @ B @ C2 ) @ F3 )
=> ( ( filterlim @ A @ B @ G @ ( at_infinity @ B ) @ F3 )
=> ( filterlim @ A @ B
@ ^ [X2: A] : ( plus_plus @ B @ ( F2 @ X2 ) @ ( G @ X2 ) )
@ ( at_infinity @ B )
@ F3 ) ) ) ) ).
% tendsto_add_filterlim_at_infinity
thf(fact_7088_tendsto__add__filterlim__at__infinity_H,axiom,
! [B: $tType,A: $tType] :
( ( real_V822414075346904944vector @ B )
=> ! [F2: A > B,F3: filter @ A,G: A > B,C2: B] :
( ( filterlim @ A @ B @ F2 @ ( at_infinity @ B ) @ F3 )
=> ( ( filterlim @ A @ B @ G @ ( topolo7230453075368039082e_nhds @ B @ C2 ) @ F3 )
=> ( filterlim @ A @ B
@ ^ [X2: A] : ( plus_plus @ B @ ( F2 @ X2 ) @ ( G @ X2 ) )
@ ( at_infinity @ B )
@ F3 ) ) ) ) ).
% tendsto_add_filterlim_at_infinity'
thf(fact_7089_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_7090_real__tendsto__divide__at__top,axiom,
! [A: $tType,F2: A > real,C2: real,F3: filter @ A,G: A > real] :
( ( filterlim @ A @ real @ F2 @ ( topolo7230453075368039082e_nhds @ real @ C2 ) @ F3 )
=> ( ( filterlim @ A @ real @ G @ ( at_top @ real ) @ F3 )
=> ( filterlim @ A @ real
@ ^ [X2: A] : ( divide_divide @ real @ ( F2 @ X2 ) @ ( G @ X2 ) )
@ ( topolo7230453075368039082e_nhds @ real @ ( zero_zero @ real ) )
@ F3 ) ) ) ).
% real_tendsto_divide_at_top
thf(fact_7091_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_7092_tendsto__inverse__0__at__top,axiom,
! [A: $tType,F2: A > real,F3: filter @ A] :
( ( filterlim @ A @ real @ F2 @ ( at_top @ real ) @ F3 )
=> ( filterlim @ A @ real
@ ^ [X2: A] : ( inverse_inverse @ real @ ( F2 @ X2 ) )
@ ( topolo7230453075368039082e_nhds @ real @ ( zero_zero @ real ) )
@ F3 ) ) ).
% tendsto_inverse_0_at_top
thf(fact_7093_tendsto__inverse__0,axiom,
! [A: $tType] :
( ( real_V8999393235501362500lgebra @ A )
=> ( filterlim @ A @ A @ ( inverse_inverse @ A ) @ ( topolo7230453075368039082e_nhds @ A @ ( zero_zero @ A ) ) @ ( at_infinity @ A ) ) ) ).
% tendsto_inverse_0
thf(fact_7094_filterlim__tendsto__pos__mult__at__top,axiom,
! [A: $tType,F2: A > real,C2: real,F3: filter @ A,G: A > real] :
( ( filterlim @ A @ real @ F2 @ ( topolo7230453075368039082e_nhds @ real @ C2 ) @ F3 )
=> ( ( ord_less @ real @ ( zero_zero @ real ) @ C2 )
=> ( ( filterlim @ A @ real @ G @ ( at_top @ real ) @ F3 )
=> ( filterlim @ A @ real
@ ^ [X2: A] : ( times_times @ real @ ( F2 @ X2 ) @ ( G @ X2 ) )
@ ( at_top @ real )
@ F3 ) ) ) ) ).
% filterlim_tendsto_pos_mult_at_top
thf(fact_7095_filterlim__at__top__mult__tendsto__pos,axiom,
! [A: $tType,F2: A > real,C2: real,F3: filter @ A,G: A > real] :
( ( filterlim @ A @ real @ F2 @ ( topolo7230453075368039082e_nhds @ real @ C2 ) @ F3 )
=> ( ( ord_less @ real @ ( zero_zero @ real ) @ C2 )
=> ( ( filterlim @ A @ real @ G @ ( at_top @ real ) @ F3 )
=> ( filterlim @ A @ real
@ ^ [X2: A] : ( times_times @ real @ ( G @ X2 ) @ ( F2 @ X2 ) )
@ ( at_top @ real )
@ F3 ) ) ) ) ).
% filterlim_at_top_mult_tendsto_pos
thf(fact_7096_tendsto__neg__powr,axiom,
! [A: $tType,S: real,F2: A > real,F3: filter @ A] :
( ( ord_less @ real @ S @ ( zero_zero @ real ) )
=> ( ( filterlim @ A @ real @ F2 @ ( at_top @ real ) @ F3 )
=> ( filterlim @ A @ real
@ ^ [X2: A] : ( powr @ real @ ( F2 @ X2 ) @ S )
@ ( topolo7230453075368039082e_nhds @ real @ ( zero_zero @ real ) )
@ F3 ) ) ) ).
% tendsto_neg_powr
thf(fact_7097_tendsto__mult__filterlim__at__infinity,axiom,
! [A: $tType,B: $tType] :
( ( real_V3459762299906320749_field @ A )
=> ! [F2: B > A,C2: A,F3: filter @ B,G: B > A] :
( ( filterlim @ B @ A @ F2 @ ( topolo7230453075368039082e_nhds @ A @ C2 ) @ F3 )
=> ( ( C2
!= ( zero_zero @ A ) )
=> ( ( filterlim @ B @ A @ G @ ( at_infinity @ A ) @ F3 )
=> ( filterlim @ B @ A
@ ^ [X2: B] : ( times_times @ A @ ( F2 @ X2 ) @ ( G @ X2 ) )
@ ( at_infinity @ A )
@ F3 ) ) ) ) ) ).
% tendsto_mult_filterlim_at_infinity
thf(fact_7098_tendsto__divide__0,axiom,
! [A: $tType,C: $tType] :
( ( real_V8999393235501362500lgebra @ A )
=> ! [F2: C > A,C2: A,F3: filter @ C,G: C > A] :
( ( filterlim @ C @ A @ F2 @ ( topolo7230453075368039082e_nhds @ A @ C2 ) @ F3 )
=> ( ( filterlim @ C @ A @ G @ ( at_infinity @ A ) @ F3 )
=> ( filterlim @ C @ A
@ ^ [X2: C] : ( divide_divide @ A @ ( F2 @ X2 ) @ ( G @ X2 ) )
@ ( topolo7230453075368039082e_nhds @ A @ ( zero_zero @ A ) )
@ F3 ) ) ) ) ).
% tendsto_divide_0
thf(fact_7099_filterlim__power__at__infinity,axiom,
! [B: $tType,A: $tType] :
( ( real_V8999393235501362500lgebra @ B )
=> ! [F2: A > B,F3: filter @ A,N: nat] :
( ( filterlim @ A @ B @ F2 @ ( at_infinity @ B ) @ F3 )
=> ( ( ord_less @ nat @ ( zero_zero @ nat ) @ N )
=> ( filterlim @ A @ B
@ ^ [X2: A] : ( power_power @ B @ ( F2 @ X2 ) @ N )
@ ( at_infinity @ B )
@ F3 ) ) ) ) ).
% filterlim_power_at_infinity
thf(fact_7100_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_7101_filterlim__at__top__to__right,axiom,
! [A: $tType,F2: real > A,F3: filter @ A] :
( ( filterlim @ real @ A @ F2 @ F3 @ ( at_top @ real ) )
= ( filterlim @ real @ A
@ ^ [X2: real] : ( F2 @ ( inverse_inverse @ real @ X2 ) )
@ F3
@ ( topolo174197925503356063within @ real @ ( zero_zero @ real ) @ ( set_ord_greaterThan @ real @ ( zero_zero @ real ) ) ) ) ) ).
% filterlim_at_top_to_right
thf(fact_7102_filterlim__at__right__to__top,axiom,
! [A: $tType,F2: real > A,F3: filter @ A] :
( ( filterlim @ real @ A @ F2 @ F3 @ ( topolo174197925503356063within @ real @ ( zero_zero @ real ) @ ( set_ord_greaterThan @ real @ ( zero_zero @ real ) ) ) )
= ( filterlim @ real @ A
@ ^ [X2: real] : ( F2 @ ( inverse_inverse @ real @ X2 ) )
@ F3
@ ( at_top @ real ) ) ) ).
% filterlim_at_right_to_top
thf(fact_7103_filterlim__inverse__at__top__right,axiom,
filterlim @ real @ real @ ( inverse_inverse @ real ) @ ( at_top @ real ) @ ( topolo174197925503356063within @ real @ ( zero_zero @ real ) @ ( set_ord_greaterThan @ real @ ( zero_zero @ real ) ) ) ).
% filterlim_inverse_at_top_right
thf(fact_7104_filterlim__inverse__at__right__top,axiom,
filterlim @ real @ real @ ( inverse_inverse @ real ) @ ( topolo174197925503356063within @ real @ ( zero_zero @ real ) @ ( set_ord_greaterThan @ real @ ( zero_zero @ real ) ) ) @ ( at_top @ real ) ).
% filterlim_inverse_at_right_top
thf(fact_7105_filterlim__inverse__at__infinity,axiom,
! [A: $tType] :
( ( real_V8999393235501362500lgebra @ A )
=> ( filterlim @ A @ A @ ( inverse_inverse @ A ) @ ( at_infinity @ A ) @ ( topolo174197925503356063within @ A @ ( zero_zero @ A ) @ ( top_top @ ( set @ A ) ) ) ) ) ).
% filterlim_inverse_at_infinity
thf(fact_7106_filterlim__tendsto__neg__mult__at__bot,axiom,
! [A: $tType,F2: A > real,C2: real,F3: filter @ A,G: A > real] :
( ( filterlim @ A @ real @ F2 @ ( topolo7230453075368039082e_nhds @ real @ C2 ) @ F3 )
=> ( ( ord_less @ real @ C2 @ ( zero_zero @ real ) )
=> ( ( filterlim @ A @ real @ G @ ( at_top @ real ) @ F3 )
=> ( filterlim @ A @ real
@ ^ [X2: A] : ( times_times @ real @ ( F2 @ X2 ) @ ( G @ X2 ) )
@ ( at_bot @ real )
@ F3 ) ) ) ) ).
% filterlim_tendsto_neg_mult_at_bot
thf(fact_7107_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_7108_filterlim__inverse__at__iff,axiom,
! [B: $tType,A: $tType] :
( ( real_V8999393235501362500lgebra @ B )
=> ! [G: A > B,F3: filter @ A] :
( ( filterlim @ A @ B
@ ^ [X2: A] : ( inverse_inverse @ B @ ( G @ X2 ) )
@ ( topolo174197925503356063within @ B @ ( zero_zero @ B ) @ ( top_top @ ( set @ B ) ) )
@ F3 )
= ( filterlim @ A @ B @ G @ ( at_infinity @ B ) @ F3 ) ) ) ).
% filterlim_inverse_at_iff
thf(fact_7109_tendsto__exp__limit__at__top,axiom,
! [X: real] :
( filterlim @ real @ real
@ ^ [Y6: real] : ( powr @ real @ ( plus_plus @ real @ ( one_one @ real ) @ ( divide_divide @ real @ X @ Y6 ) ) @ Y6 )
@ ( topolo7230453075368039082e_nhds @ real @ ( exp @ real @ X ) )
@ ( at_top @ real ) ) ).
% tendsto_exp_limit_at_top
thf(fact_7110_filterlim__divide__at__infinity,axiom,
! [A: $tType] :
( ( real_V3459762299906320749_field @ A )
=> ! [F2: A > A,C2: A,F3: filter @ A,G: A > A] :
( ( filterlim @ A @ A @ F2 @ ( topolo7230453075368039082e_nhds @ A @ C2 ) @ F3 )
=> ( ( filterlim @ A @ A @ G @ ( topolo174197925503356063within @ A @ ( zero_zero @ A ) @ ( top_top @ ( set @ A ) ) ) @ F3 )
=> ( ( C2
!= ( zero_zero @ A ) )
=> ( filterlim @ A @ A
@ ^ [X2: A] : ( divide_divide @ A @ ( F2 @ X2 ) @ ( G @ X2 ) )
@ ( at_infinity @ A )
@ F3 ) ) ) ) ) ).
% filterlim_divide_at_infinity
thf(fact_7111_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_7112_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_7113_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_7114_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_7115_lim__at__infinity__0,axiom,
! [A: $tType] :
( ( real_V3459762299906320749_field @ A )
=> ! [F2: A > A,L: A] :
( ( filterlim @ A @ A @ F2 @ ( topolo7230453075368039082e_nhds @ A @ L ) @ ( at_infinity @ A ) )
= ( filterlim @ A @ A @ ( comp @ A @ A @ A @ F2 @ ( inverse_inverse @ A ) ) @ ( topolo7230453075368039082e_nhds @ A @ L ) @ ( topolo174197925503356063within @ A @ ( zero_zero @ A ) @ ( top_top @ ( set @ A ) ) ) ) ) ) ).
% lim_at_infinity_0
thf(fact_7116_polyfun__extremal,axiom,
! [A: $tType] :
( ( real_V8999393235501362500lgebra @ A )
=> ! [C2: nat > A,K: nat,N: nat,B4: real] :
( ( ( C2 @ K )
!= ( zero_zero @ A ) )
=> ( ( ord_less_eq @ nat @ ( one_one @ nat ) @ K )
=> ( ( ord_less_eq @ nat @ K @ N )
=> ( eventually @ A
@ ^ [Z7: A] :
( ord_less_eq @ real @ B4
@ ( real_V7770717601297561774m_norm @ A
@ ( groups7311177749621191930dd_sum @ nat @ A
@ ^ [I2: nat] : ( times_times @ A @ ( C2 @ I2 ) @ ( power_power @ A @ Z7 @ I2 ) )
@ ( set_ord_atMost @ nat @ N ) ) ) )
@ ( at_infinity @ A ) ) ) ) ) ) ).
% polyfun_extremal
thf(fact_7117_lhopital__left__at__top,axiom,
! [G: real > real,X: real,G4: real > real,F2: real > real,F8: real > real,Y: real] :
( ( filterlim @ real @ real @ G @ ( at_top @ real ) @ ( topolo174197925503356063within @ real @ X @ ( set_ord_lessThan @ real @ X ) ) )
=> ( ( eventually @ real
@ ^ [X2: real] :
( ( G4 @ X2 )
!= ( zero_zero @ real ) )
@ ( topolo174197925503356063within @ real @ X @ ( set_ord_lessThan @ real @ X ) ) )
=> ( ( eventually @ real
@ ^ [X2: real] : ( has_field_derivative @ real @ F2 @ ( F8 @ X2 ) @ ( topolo174197925503356063within @ real @ X2 @ ( top_top @ ( set @ real ) ) ) )
@ ( topolo174197925503356063within @ real @ X @ ( set_ord_lessThan @ real @ X ) ) )
=> ( ( eventually @ real
@ ^ [X2: real] : ( has_field_derivative @ real @ G @ ( G4 @ X2 ) @ ( topolo174197925503356063within @ real @ X2 @ ( top_top @ ( set @ real ) ) ) )
@ ( topolo174197925503356063within @ real @ X @ ( set_ord_lessThan @ real @ X ) ) )
=> ( ( filterlim @ real @ real
@ ^ [X2: real] : ( divide_divide @ real @ ( F8 @ X2 ) @ ( G4 @ X2 ) )
@ ( topolo7230453075368039082e_nhds @ real @ Y )
@ ( topolo174197925503356063within @ real @ X @ ( set_ord_lessThan @ real @ X ) ) )
=> ( filterlim @ real @ real
@ ^ [X2: real] : ( divide_divide @ real @ ( F2 @ X2 ) @ ( G @ X2 ) )
@ ( topolo7230453075368039082e_nhds @ real @ Y )
@ ( topolo174197925503356063within @ real @ X @ ( set_ord_lessThan @ real @ X ) ) ) ) ) ) ) ) ).
% lhopital_left_at_top
thf(fact_7118_eventually__sequentially__Suc,axiom,
! [P: nat > $o] :
( ( eventually @ nat
@ ^ [I2: nat] : ( P @ ( suc @ I2 ) )
@ ( at_top @ nat ) )
= ( eventually @ nat @ P @ ( at_top @ nat ) ) ) ).
% eventually_sequentially_Suc
thf(fact_7119_eventually__sequentially__seg,axiom,
! [P: nat > $o,K: nat] :
( ( eventually @ nat
@ ^ [N3: nat] : ( P @ ( plus_plus @ nat @ N3 @ K ) )
@ ( at_top @ nat ) )
= ( eventually @ nat @ P @ ( at_top @ nat ) ) ) ).
% eventually_sequentially_seg
thf(fact_7120_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 )
& ! [Y6: A] :
( ( ord_less @ A @ X @ Y6 )
=> ( ( ord_less @ A @ Y6 @ B5 )
=> ( P @ Y6 ) ) ) ) ) ) ) ).
% eventually_at_right_field
thf(fact_7121_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 )
& ! [Y6: A] :
( ( ord_less @ A @ X @ Y6 )
=> ( ( ord_less @ A @ Y6 @ B5 )
=> ( P @ Y6 ) ) ) ) ) ) ) ) ).
% eventually_at_right
thf(fact_7122_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_7123_eventually__at__top__dense,axiom,
! [A: $tType] :
( ( ( linorder @ A )
& ( no_top @ A ) )
=> ! [P: A > $o] :
( ( eventually @ A @ P @ ( at_top @ A ) )
= ( ? [N8: A] :
! [N3: A] :
( ( ord_less @ A @ N8 @ N3 )
=> ( P @ N3 ) ) ) ) ) ).
% eventually_at_top_dense
thf(fact_7124_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 )
& ! [Y6: A] :
( ( ord_less @ A @ B5 @ Y6 )
=> ( ( ord_less @ A @ Y6 @ X )
=> ( P @ Y6 ) ) ) ) ) ) ) ).
% eventually_at_left_field
thf(fact_7125_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 )
& ! [Y6: A] :
( ( ord_less @ A @ B5 @ Y6 )
=> ( ( ord_less @ A @ Y6 @ X )
=> ( P @ Y6 ) ) ) ) ) ) ) ) ).
% eventually_at_left
thf(fact_7126_sequentially__offset,axiom,
! [P: nat > $o,K: nat] :
( ( eventually @ nat @ P @ ( at_top @ nat ) )
=> ( eventually @ nat
@ ^ [I2: nat] : ( P @ ( plus_plus @ nat @ I2 @ K ) )
@ ( at_top @ nat ) ) ) ).
% sequentially_offset
thf(fact_7127_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_7128_eventually__at__bot__dense,axiom,
! [A: $tType] :
( ( ( linorder @ A )
& ( no_bot @ A ) )
=> ! [P: A > $o] :
( ( eventually @ A @ P @ ( at_bot @ A ) )
= ( ? [N8: A] :
! [N3: A] :
( ( ord_less @ A @ N3 @ N8 )
=> ( P @ N3 ) ) ) ) ) ).
% eventually_at_bot_dense
thf(fact_7129_eventually__nhds__top,axiom,
! [A: $tType] :
( ( ( order_top @ A )
& ( topolo1944317154257567458pology @ A ) )
=> ! [B2: A,P: A > $o] :
( ( ord_less @ A @ B2 @ ( top_top @ A ) )
=> ( ( eventually @ A @ P @ ( topolo7230453075368039082e_nhds @ A @ ( top_top @ A ) ) )
= ( ? [B5: A] :
( ( ord_less @ A @ B5 @ ( top_top @ A ) )
& ! [Z7: A] :
( ( ord_less @ A @ B5 @ Z7 )
=> ( P @ Z7 ) ) ) ) ) ) ) ).
% eventually_nhds_top
thf(fact_7130_order__tendstoD_I2_J,axiom,
! [A: $tType,B: $tType] :
( ( topolo2564578578187576103pology @ A )
=> ! [F2: B > A,Y: A,F3: filter @ B,A2: A] :
( ( filterlim @ B @ A @ F2 @ ( topolo7230453075368039082e_nhds @ A @ Y ) @ F3 )
=> ( ( ord_less @ A @ Y @ A2 )
=> ( eventually @ B
@ ^ [X2: B] : ( ord_less @ A @ ( F2 @ X2 ) @ A2 )
@ F3 ) ) ) ) ).
% order_tendstoD(2)
thf(fact_7131_order__tendstoD_I1_J,axiom,
! [A: $tType,B: $tType] :
( ( topolo2564578578187576103pology @ A )
=> ! [F2: B > A,Y: A,F3: filter @ B,A2: A] :
( ( filterlim @ B @ A @ F2 @ ( topolo7230453075368039082e_nhds @ A @ Y ) @ F3 )
=> ( ( ord_less @ A @ A2 @ Y )
=> ( eventually @ B
@ ^ [X2: B] : ( ord_less @ A @ A2 @ ( F2 @ X2 ) )
@ F3 ) ) ) ) ).
% order_tendstoD(1)
thf(fact_7132_order__tendstoI,axiom,
! [A: $tType,B: $tType] :
( ( topolo2564578578187576103pology @ A )
=> ! [Y: A,F2: B > A,F3: filter @ B] :
( ! [A6: A] :
( ( ord_less @ A @ A6 @ Y )
=> ( eventually @ B
@ ^ [X2: B] : ( ord_less @ A @ A6 @ ( F2 @ X2 ) )
@ F3 ) )
=> ( ! [A6: A] :
( ( ord_less @ A @ Y @ A6 )
=> ( eventually @ B
@ ^ [X2: B] : ( ord_less @ A @ ( F2 @ X2 ) @ A6 )
@ F3 ) )
=> ( filterlim @ B @ A @ F2 @ ( topolo7230453075368039082e_nhds @ A @ Y ) @ F3 ) ) ) ) ).
% order_tendstoI
thf(fact_7133_order__tendsto__iff,axiom,
! [B: $tType,A: $tType] :
( ( topolo2564578578187576103pology @ A )
=> ! [F2: B > A,X: A,F3: filter @ B] :
( ( filterlim @ B @ A @ F2 @ ( topolo7230453075368039082e_nhds @ A @ X ) @ F3 )
= ( ! [L2: A] :
( ( ord_less @ A @ L2 @ X )
=> ( eventually @ B
@ ^ [X2: B] : ( ord_less @ A @ L2 @ ( F2 @ X2 ) )
@ F3 ) )
& ! [U2: A] :
( ( ord_less @ A @ X @ U2 )
=> ( eventually @ B
@ ^ [X2: B] : ( ord_less @ A @ ( F2 @ X2 ) @ U2 )
@ F3 ) ) ) ) ) ).
% order_tendsto_iff
thf(fact_7134_filterlim__at__top__dense,axiom,
! [A: $tType,B: $tType] :
( ( unboun7993243217541854897norder @ B )
=> ! [F2: A > B,F3: filter @ A] :
( ( filterlim @ A @ B @ F2 @ ( at_top @ B ) @ F3 )
= ( ! [Z9: B] :
( eventually @ A
@ ^ [X2: A] : ( ord_less @ B @ Z9 @ ( F2 @ X2 ) )
@ F3 ) ) ) ) ).
% filterlim_at_top_dense
thf(fact_7135_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_7136_filterlim__at__bot__dense,axiom,
! [A: $tType,B: $tType] :
( ( ( dense_linorder @ B )
& ( no_bot @ B ) )
=> ! [F2: A > B,F3: filter @ A] :
( ( filterlim @ A @ B @ F2 @ ( at_bot @ B ) @ F3 )
= ( ! [Z9: B] :
( eventually @ A
@ ^ [X2: A] : ( ord_less @ B @ ( F2 @ X2 ) @ Z9 )
@ F3 ) ) ) ) ).
% filterlim_at_bot_dense
thf(fact_7137_eventually__at,axiom,
! [A: $tType] :
( ( real_V7819770556892013058_space @ A )
=> ! [P: A > $o,A2: A,S3: set @ A] :
( ( eventually @ A @ P @ ( topolo174197925503356063within @ A @ A2 @ S3 ) )
= ( ? [D: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ D )
& ! [X2: A] :
( ( member @ A @ X2 @ S3 )
=> ( ( ( X2 != A2 )
& ( ord_less @ real @ ( real_V557655796197034286t_dist @ A @ X2 @ A2 ) @ D ) )
=> ( P @ X2 ) ) ) ) ) ) ) ).
% eventually_at
thf(fact_7138_eventually__nhds__metric,axiom,
! [A: $tType] :
( ( real_V7819770556892013058_space @ A )
=> ! [P: A > $o,A2: A] :
( ( eventually @ A @ P @ ( topolo7230453075368039082e_nhds @ A @ A2 ) )
= ( ? [D: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ D )
& ! [X2: A] :
( ( ord_less @ real @ ( real_V557655796197034286t_dist @ A @ X2 @ A2 ) @ D )
=> ( P @ X2 ) ) ) ) ) ) ).
% eventually_nhds_metric
thf(fact_7139_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_7140_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_7141_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_7142_decreasing__tendsto,axiom,
! [A: $tType,B: $tType] :
( ( topolo2564578578187576103pology @ A )
=> ! [L: A,F2: B > A,F3: filter @ B] :
( ( eventually @ B
@ ^ [N3: B] : ( ord_less_eq @ A @ L @ ( F2 @ N3 ) )
@ F3 )
=> ( ! [X3: A] :
( ( ord_less @ A @ L @ X3 )
=> ( eventually @ B
@ ^ [N3: B] : ( ord_less @ A @ ( F2 @ N3 ) @ X3 )
@ F3 ) )
=> ( filterlim @ B @ A @ F2 @ ( topolo7230453075368039082e_nhds @ A @ L ) @ F3 ) ) ) ) ).
% decreasing_tendsto
thf(fact_7143_increasing__tendsto,axiom,
! [A: $tType,B: $tType] :
( ( topolo2564578578187576103pology @ A )
=> ! [F2: B > A,L: A,F3: filter @ B] :
( ( eventually @ B
@ ^ [N3: B] : ( ord_less_eq @ A @ ( F2 @ N3 ) @ L )
@ F3 )
=> ( ! [X3: A] :
( ( ord_less @ A @ X3 @ L )
=> ( eventually @ B
@ ^ [N3: B] : ( ord_less @ A @ X3 @ ( F2 @ N3 ) )
@ F3 ) )
=> ( filterlim @ B @ A @ F2 @ ( topolo7230453075368039082e_nhds @ A @ L ) @ F3 ) ) ) ) ).
% increasing_tendsto
thf(fact_7144_filterlim__at__top__gt,axiom,
! [A: $tType,B: $tType] :
( ( unboun7993243217541854897norder @ B )
=> ! [F2: A > B,F3: filter @ A,C2: B] :
( ( filterlim @ A @ B @ F2 @ ( at_top @ B ) @ F3 )
= ( ! [Z9: B] :
( ( ord_less @ B @ C2 @ Z9 )
=> ( eventually @ A
@ ^ [X2: A] : ( ord_less_eq @ B @ Z9 @ ( F2 @ X2 ) )
@ F3 ) ) ) ) ) ).
% filterlim_at_top_gt
thf(fact_7145_filterlim__at__bot__lt,axiom,
! [A: $tType,B: $tType] :
( ( unboun7993243217541854897norder @ B )
=> ! [F2: A > B,F3: filter @ A,C2: B] :
( ( filterlim @ A @ B @ F2 @ ( at_bot @ B ) @ F3 )
= ( ! [Z9: B] :
( ( ord_less @ B @ Z9 @ C2 )
=> ( eventually @ A
@ ^ [X2: A] : ( ord_less_eq @ B @ ( F2 @ X2 ) @ Z9 )
@ F3 ) ) ) ) ) ).
% filterlim_at_bot_lt
thf(fact_7146_filterlim__at__infinity__imp__filterlim__at__top,axiom,
! [A: $tType,F2: A > real,F3: filter @ A] :
( ( filterlim @ A @ real @ F2 @ ( at_infinity @ real ) @ F3 )
=> ( ( eventually @ A
@ ^ [X2: A] : ( ord_less @ real @ ( zero_zero @ real ) @ ( F2 @ X2 ) )
@ F3 )
=> ( filterlim @ A @ real @ F2 @ ( at_top @ real ) @ F3 ) ) ) ).
% filterlim_at_infinity_imp_filterlim_at_top
thf(fact_7147_filterlim__at__infinity__imp__filterlim__at__bot,axiom,
! [A: $tType,F2: A > real,F3: filter @ A] :
( ( filterlim @ A @ real @ F2 @ ( at_infinity @ real ) @ F3 )
=> ( ( eventually @ A
@ ^ [X2: A] : ( ord_less @ real @ ( F2 @ X2 ) @ ( zero_zero @ real ) )
@ F3 )
=> ( filterlim @ A @ real @ F2 @ ( at_bot @ real ) @ F3 ) ) ) ).
% filterlim_at_infinity_imp_filterlim_at_bot
thf(fact_7148_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_7149_continuous__arcosh__strong,axiom,
! [A: $tType] :
( ( topological_t2_space @ A )
=> ! [F3: filter @ A,F2: A > real] :
( ( topolo3448309680560233919inuous @ A @ real @ F3 @ F2 )
=> ( ( eventually @ A
@ ^ [X2: A] : ( ord_less_eq @ real @ ( one_one @ real ) @ ( F2 @ X2 ) )
@ F3 )
=> ( topolo3448309680560233919inuous @ A @ real @ F3
@ ^ [X2: A] : ( arcosh @ real @ ( F2 @ X2 ) ) ) ) ) ) ).
% continuous_arcosh_strong
thf(fact_7150_eventually__at__right__real,axiom,
! [A2: real,B2: real] :
( ( ord_less @ real @ A2 @ B2 )
=> ( eventually @ real
@ ^ [X2: real] : ( member @ real @ X2 @ ( set_or5935395276787703475ssThan @ real @ A2 @ B2 ) )
@ ( topolo174197925503356063within @ real @ A2 @ ( set_ord_greaterThan @ real @ A2 ) ) ) ) ).
% eventually_at_right_real
thf(fact_7151_eventually__at__left__real,axiom,
! [B2: real,A2: real] :
( ( ord_less @ real @ B2 @ A2 )
=> ( eventually @ real
@ ^ [X2: real] : ( member @ real @ X2 @ ( set_or5935395276787703475ssThan @ real @ B2 @ A2 ) )
@ ( topolo174197925503356063within @ real @ A2 @ ( set_ord_lessThan @ real @ A2 ) ) ) ) ).
% eventually_at_left_real
thf(fact_7152_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 ) )
= ( ? [D: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ D )
& ! [X2: A] :
( ( member @ A @ X2 @ S3 )
=> ( ( ( X2 != A2 )
& ( ord_less_eq @ real @ ( real_V557655796197034286t_dist @ A @ X2 @ A2 ) @ D ) )
=> ( P @ X2 ) ) ) ) ) ) ) ).
% eventually_at_le
thf(fact_7153_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_7154_tendsto__imp__filterlim__at__left,axiom,
! [B: $tType,A: $tType] :
( ( topolo2564578578187576103pology @ B )
=> ! [F2: A > B,L6: B,F3: filter @ A] :
( ( filterlim @ A @ B @ F2 @ ( topolo7230453075368039082e_nhds @ B @ L6 ) @ F3 )
=> ( ( eventually @ A
@ ^ [X2: A] : ( ord_less @ B @ ( F2 @ X2 ) @ L6 )
@ F3 )
=> ( filterlim @ A @ B @ F2 @ ( topolo174197925503356063within @ B @ L6 @ ( set_ord_lessThan @ B @ L6 ) ) @ F3 ) ) ) ) ).
% tendsto_imp_filterlim_at_left
thf(fact_7155_tendsto__imp__filterlim__at__right,axiom,
! [B: $tType,A: $tType] :
( ( topolo2564578578187576103pology @ B )
=> ! [F2: A > B,L6: B,F3: filter @ A] :
( ( filterlim @ A @ B @ F2 @ ( topolo7230453075368039082e_nhds @ B @ L6 ) @ F3 )
=> ( ( eventually @ A
@ ^ [X2: A] : ( ord_less @ B @ L6 @ ( F2 @ X2 ) )
@ F3 )
=> ( filterlim @ A @ B @ F2 @ ( topolo174197925503356063within @ B @ L6 @ ( set_ord_greaterThan @ B @ L6 ) ) @ F3 ) ) ) ) ).
% tendsto_imp_filterlim_at_right
thf(fact_7156_tendstoD,axiom,
! [A: $tType,B: $tType] :
( ( real_V7819770556892013058_space @ A )
=> ! [F2: B > A,L: A,F3: filter @ B,E: real] :
( ( filterlim @ B @ A @ F2 @ ( topolo7230453075368039082e_nhds @ A @ L ) @ F3 )
=> ( ( ord_less @ real @ ( zero_zero @ real ) @ E )
=> ( eventually @ B
@ ^ [X2: B] : ( ord_less @ real @ ( real_V557655796197034286t_dist @ A @ ( F2 @ X2 ) @ L ) @ E )
@ F3 ) ) ) ) ).
% tendstoD
thf(fact_7157_tendstoI,axiom,
! [A: $tType,B: $tType] :
( ( real_V7819770556892013058_space @ A )
=> ! [F2: B > A,L: A,F3: filter @ B] :
( ! [E2: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ E2 )
=> ( eventually @ B
@ ^ [X2: B] : ( ord_less @ real @ ( real_V557655796197034286t_dist @ A @ ( F2 @ X2 ) @ L ) @ E2 )
@ F3 ) )
=> ( filterlim @ B @ A @ F2 @ ( topolo7230453075368039082e_nhds @ A @ L ) @ F3 ) ) ) ).
% tendstoI
thf(fact_7158_tendsto__iff,axiom,
! [B: $tType,A: $tType] :
( ( real_V7819770556892013058_space @ A )
=> ! [F2: B > A,L: A,F3: filter @ B] :
( ( filterlim @ B @ A @ F2 @ ( topolo7230453075368039082e_nhds @ A @ L ) @ F3 )
= ( ! [E4: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ E4 )
=> ( eventually @ B
@ ^ [X2: B] : ( ord_less @ real @ ( real_V557655796197034286t_dist @ A @ ( F2 @ X2 ) @ L ) @ E4 )
@ F3 ) ) ) ) ) ).
% tendsto_iff
thf(fact_7159_eventually__at__top__to__right,axiom,
! [P: real > $o] :
( ( eventually @ real @ P @ ( at_top @ real ) )
= ( eventually @ real
@ ^ [X2: real] : ( P @ ( inverse_inverse @ real @ X2 ) )
@ ( topolo174197925503356063within @ real @ ( zero_zero @ real ) @ ( set_ord_greaterThan @ real @ ( zero_zero @ real ) ) ) ) ) ).
% eventually_at_top_to_right
thf(fact_7160_eventually__at__right__to__top,axiom,
! [P: real > $o] :
( ( eventually @ real @ P @ ( topolo174197925503356063within @ real @ ( zero_zero @ real ) @ ( set_ord_greaterThan @ real @ ( zero_zero @ real ) ) ) )
= ( eventually @ real
@ ^ [X2: real] : ( P @ ( inverse_inverse @ real @ X2 ) )
@ ( at_top @ real ) ) ) ).
% eventually_at_right_to_top
thf(fact_7161_tendsto__arcosh__strong,axiom,
! [B: $tType,F2: B > real,A2: real,F3: filter @ B] :
( ( filterlim @ B @ real @ F2 @ ( topolo7230453075368039082e_nhds @ real @ A2 ) @ F3 )
=> ( ( ord_less_eq @ real @ ( one_one @ real ) @ A2 )
=> ( ( eventually @ B
@ ^ [X2: B] : ( ord_less_eq @ real @ ( one_one @ real ) @ ( F2 @ X2 ) )
@ F3 )
=> ( filterlim @ B @ real
@ ^ [X2: B] : ( arcosh @ real @ ( F2 @ X2 ) )
@ ( topolo7230453075368039082e_nhds @ real @ ( arcosh @ real @ A2 ) )
@ F3 ) ) ) ) ).
% tendsto_arcosh_strong
thf(fact_7162_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_7163_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_7164_tendsto__0__le,axiom,
! [B: $tType,C: $tType,A: $tType] :
( ( ( real_V822414075346904944vector @ C )
& ( real_V822414075346904944vector @ B ) )
=> ! [F2: A > B,F3: filter @ A,G: A > C,K6: real] :
( ( filterlim @ A @ B @ F2 @ ( topolo7230453075368039082e_nhds @ B @ ( zero_zero @ B ) ) @ F3 )
=> ( ( eventually @ A
@ ^ [X2: A] : ( ord_less_eq @ real @ ( real_V7770717601297561774m_norm @ C @ ( G @ X2 ) ) @ ( times_times @ real @ ( real_V7770717601297561774m_norm @ B @ ( F2 @ X2 ) ) @ K6 ) )
@ F3 )
=> ( filterlim @ A @ C @ G @ ( topolo7230453075368039082e_nhds @ C @ ( zero_zero @ C ) ) @ F3 ) ) ) ) ).
% tendsto_0_le
thf(fact_7165_eventually__floor__less,axiom,
! [B: $tType,A: $tType] :
( ( ( archim2362893244070406136eiling @ B )
& ( topolo2564578578187576103pology @ B ) )
=> ! [F2: A > B,L: B,F3: filter @ A] :
( ( filterlim @ A @ B @ F2 @ ( topolo7230453075368039082e_nhds @ B @ L ) @ F3 )
=> ( ~ ( member @ B @ L @ ( ring_1_Ints @ B ) )
=> ( eventually @ A
@ ^ [X2: A] : ( ord_less @ B @ ( ring_1_of_int @ B @ ( archim6421214686448440834_floor @ B @ L ) ) @ ( F2 @ X2 ) )
@ F3 ) ) ) ) ).
% eventually_floor_less
thf(fact_7166_filterlim__at__infinity,axiom,
! [C: $tType,A: $tType] :
( ( real_V822414075346904944vector @ A )
=> ! [C2: real,F2: C > A,F3: filter @ C] :
( ( ord_less_eq @ real @ ( zero_zero @ real ) @ C2 )
=> ( ( filterlim @ C @ A @ F2 @ ( at_infinity @ A ) @ F3 )
= ( ! [R5: real] :
( ( ord_less @ real @ C2 @ R5 )
=> ( eventually @ C
@ ^ [X2: C] : ( ord_less_eq @ real @ R5 @ ( real_V7770717601297561774m_norm @ A @ ( F2 @ X2 ) ) )
@ F3 ) ) ) ) ) ) ).
% filterlim_at_infinity
thf(fact_7167_eventually__less__ceiling,axiom,
! [B: $tType,A: $tType] :
( ( ( archim2362893244070406136eiling @ B )
& ( topolo2564578578187576103pology @ B ) )
=> ! [F2: A > B,L: B,F3: filter @ A] :
( ( filterlim @ A @ B @ F2 @ ( topolo7230453075368039082e_nhds @ B @ L ) @ F3 )
=> ( ~ ( member @ B @ L @ ( ring_1_Ints @ B ) )
=> ( eventually @ A
@ ^ [X2: A] : ( ord_less @ B @ ( F2 @ X2 ) @ ( ring_1_of_int @ B @ ( archimedean_ceiling @ B @ L ) ) )
@ F3 ) ) ) ) ).
% eventually_less_ceiling
thf(fact_7168_tendsto__powr_H,axiom,
! [A: $tType,F2: A > real,A2: real,F3: filter @ A,G: A > real,B2: real] :
( ( filterlim @ A @ real @ F2 @ ( topolo7230453075368039082e_nhds @ real @ A2 ) @ F3 )
=> ( ( filterlim @ A @ real @ G @ ( topolo7230453075368039082e_nhds @ real @ B2 ) @ F3 )
=> ( ( ( A2
!= ( zero_zero @ real ) )
| ( ( ord_less @ real @ ( zero_zero @ real ) @ B2 )
& ( eventually @ A
@ ^ [X2: A] : ( ord_less_eq @ real @ ( zero_zero @ real ) @ ( F2 @ X2 ) )
@ F3 ) ) )
=> ( filterlim @ A @ real
@ ^ [X2: A] : ( powr @ real @ ( F2 @ X2 ) @ ( G @ X2 ) )
@ ( topolo7230453075368039082e_nhds @ real @ ( powr @ real @ A2 @ B2 ) )
@ F3 ) ) ) ) ).
% tendsto_powr'
thf(fact_7169_tendsto__powr2,axiom,
! [A: $tType,F2: A > real,A2: real,F3: filter @ A,G: A > real,B2: real] :
( ( filterlim @ A @ real @ F2 @ ( topolo7230453075368039082e_nhds @ real @ A2 ) @ F3 )
=> ( ( filterlim @ A @ real @ G @ ( topolo7230453075368039082e_nhds @ real @ B2 ) @ F3 )
=> ( ( eventually @ A
@ ^ [X2: A] : ( ord_less_eq @ real @ ( zero_zero @ real ) @ ( F2 @ X2 ) )
@ F3 )
=> ( ( 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 ) )
@ F3 ) ) ) ) ) ).
% tendsto_powr2
thf(fact_7170_tendsto__zero__powrI,axiom,
! [A: $tType,F2: A > real,F3: filter @ A,G: A > real,B2: real] :
( ( filterlim @ A @ real @ F2 @ ( topolo7230453075368039082e_nhds @ real @ ( zero_zero @ real ) ) @ F3 )
=> ( ( filterlim @ A @ real @ G @ ( topolo7230453075368039082e_nhds @ real @ B2 ) @ F3 )
=> ( ( eventually @ A
@ ^ [X2: A] : ( ord_less_eq @ real @ ( zero_zero @ real ) @ ( F2 @ X2 ) )
@ F3 )
=> ( ( ord_less @ real @ ( zero_zero @ real ) @ B2 )
=> ( filterlim @ A @ real
@ ^ [X2: A] : ( powr @ real @ ( F2 @ X2 ) @ ( G @ X2 ) )
@ ( topolo7230453075368039082e_nhds @ real @ ( zero_zero @ real ) )
@ F3 ) ) ) ) ) ).
% tendsto_zero_powrI
thf(fact_7171_LIM__at__top__divide,axiom,
! [A: $tType,F2: A > real,A2: real,F3: filter @ A,G: A > real] :
( ( filterlim @ A @ real @ F2 @ ( topolo7230453075368039082e_nhds @ real @ A2 ) @ F3 )
=> ( ( ord_less @ real @ ( zero_zero @ real ) @ A2 )
=> ( ( filterlim @ A @ real @ G @ ( topolo7230453075368039082e_nhds @ real @ ( zero_zero @ real ) ) @ F3 )
=> ( ( eventually @ A
@ ^ [X2: A] : ( ord_less @ real @ ( zero_zero @ real ) @ ( G @ X2 ) )
@ F3 )
=> ( filterlim @ A @ real
@ ^ [X2: A] : ( divide_divide @ real @ ( F2 @ X2 ) @ ( G @ X2 ) )
@ ( at_top @ real )
@ F3 ) ) ) ) ) ).
% LIM_at_top_divide
thf(fact_7172_filterlim__at__top__iff__inverse__0,axiom,
! [A: $tType,F2: A > real,F3: filter @ A] :
( ( eventually @ A
@ ^ [X2: A] : ( ord_less @ real @ ( zero_zero @ real ) @ ( F2 @ X2 ) )
@ F3 )
=> ( ( filterlim @ A @ real @ F2 @ ( at_top @ real ) @ F3 )
= ( filterlim @ A @ real @ ( comp @ real @ real @ A @ ( inverse_inverse @ real ) @ F2 ) @ ( topolo7230453075368039082e_nhds @ real @ ( zero_zero @ real ) ) @ F3 ) ) ) ).
% filterlim_at_top_iff_inverse_0
thf(fact_7173_filterlim__inverse__at__top,axiom,
! [A: $tType,F2: A > real,F3: filter @ A] :
( ( filterlim @ A @ real @ F2 @ ( topolo7230453075368039082e_nhds @ real @ ( zero_zero @ real ) ) @ F3 )
=> ( ( eventually @ A
@ ^ [X2: A] : ( ord_less @ real @ ( zero_zero @ real ) @ ( F2 @ X2 ) )
@ F3 )
=> ( filterlim @ A @ real
@ ^ [X2: A] : ( inverse_inverse @ real @ ( F2 @ X2 ) )
@ ( at_top @ real )
@ F3 ) ) ) ).
% filterlim_inverse_at_top
thf(fact_7174_filterlim__inverse__at__top__iff,axiom,
! [A: $tType,F2: A > real,F3: filter @ A] :
( ( eventually @ A
@ ^ [X2: A] : ( ord_less @ real @ ( zero_zero @ real ) @ ( F2 @ X2 ) )
@ F3 )
=> ( ( filterlim @ A @ real
@ ^ [X2: A] : ( inverse_inverse @ real @ ( F2 @ X2 ) )
@ ( at_top @ real )
@ F3 )
= ( filterlim @ A @ real @ F2 @ ( topolo7230453075368039082e_nhds @ real @ ( zero_zero @ real ) ) @ F3 ) ) ) ).
% filterlim_inverse_at_top_iff
thf(fact_7175_filterlim__inverse__at__bot,axiom,
! [A: $tType,F2: A > real,F3: filter @ A] :
( ( filterlim @ A @ real @ F2 @ ( topolo7230453075368039082e_nhds @ real @ ( zero_zero @ real ) ) @ F3 )
=> ( ( eventually @ A
@ ^ [X2: A] : ( ord_less @ real @ ( F2 @ X2 ) @ ( zero_zero @ real ) )
@ F3 )
=> ( filterlim @ A @ real
@ ^ [X2: A] : ( inverse_inverse @ real @ ( F2 @ X2 ) )
@ ( at_bot @ real )
@ F3 ) ) ) ).
% filterlim_inverse_at_bot
thf(fact_7176_lhopital__at__top__at__top,axiom,
! [F2: real > real,A2: real,G: real > real,F8: real > real,G4: real > real] :
( ( filterlim @ real @ real @ F2 @ ( at_top @ real ) @ ( topolo174197925503356063within @ real @ A2 @ ( top_top @ ( set @ real ) ) ) )
=> ( ( filterlim @ real @ real @ G @ ( at_top @ real ) @ ( topolo174197925503356063within @ real @ A2 @ ( top_top @ ( set @ real ) ) ) )
=> ( ( eventually @ real
@ ^ [X2: real] : ( has_field_derivative @ real @ F2 @ ( F8 @ X2 ) @ ( topolo174197925503356063within @ real @ X2 @ ( top_top @ ( set @ real ) ) ) )
@ ( topolo174197925503356063within @ real @ A2 @ ( top_top @ ( set @ real ) ) ) )
=> ( ( eventually @ real
@ ^ [X2: real] : ( has_field_derivative @ real @ G @ ( G4 @ X2 ) @ ( topolo174197925503356063within @ real @ X2 @ ( top_top @ ( set @ real ) ) ) )
@ ( topolo174197925503356063within @ real @ A2 @ ( top_top @ ( set @ real ) ) ) )
=> ( ( filterlim @ real @ real
@ ^ [X2: real] : ( divide_divide @ real @ ( F8 @ X2 ) @ ( G4 @ X2 ) )
@ ( at_top @ real )
@ ( topolo174197925503356063within @ real @ A2 @ ( top_top @ ( set @ real ) ) ) )
=> ( filterlim @ real @ real
@ ^ [X2: real] : ( divide_divide @ real @ ( F2 @ X2 ) @ ( G @ X2 ) )
@ ( at_top @ real )
@ ( topolo174197925503356063within @ real @ A2 @ ( top_top @ ( set @ real ) ) ) ) ) ) ) ) ) ).
% lhopital_at_top_at_top
thf(fact_7177_lhopital,axiom,
! [F2: real > real,X: real,G: real > real,G4: real > real,F8: real > real,F3: filter @ real] :
( ( filterlim @ real @ real @ F2 @ ( topolo7230453075368039082e_nhds @ real @ ( zero_zero @ real ) ) @ ( topolo174197925503356063within @ real @ X @ ( top_top @ ( set @ real ) ) ) )
=> ( ( filterlim @ real @ real @ G @ ( topolo7230453075368039082e_nhds @ real @ ( zero_zero @ real ) ) @ ( topolo174197925503356063within @ real @ X @ ( top_top @ ( set @ real ) ) ) )
=> ( ( eventually @ real
@ ^ [X2: real] :
( ( G @ X2 )
!= ( zero_zero @ real ) )
@ ( topolo174197925503356063within @ real @ X @ ( top_top @ ( set @ real ) ) ) )
=> ( ( eventually @ real
@ ^ [X2: real] :
( ( G4 @ X2 )
!= ( zero_zero @ real ) )
@ ( topolo174197925503356063within @ real @ X @ ( top_top @ ( set @ real ) ) ) )
=> ( ( eventually @ real
@ ^ [X2: real] : ( has_field_derivative @ real @ F2 @ ( F8 @ X2 ) @ ( topolo174197925503356063within @ real @ X2 @ ( top_top @ ( set @ real ) ) ) )
@ ( topolo174197925503356063within @ real @ X @ ( top_top @ ( set @ real ) ) ) )
=> ( ( eventually @ real
@ ^ [X2: real] : ( has_field_derivative @ real @ G @ ( G4 @ X2 ) @ ( topolo174197925503356063within @ real @ X2 @ ( top_top @ ( set @ real ) ) ) )
@ ( topolo174197925503356063within @ real @ X @ ( top_top @ ( set @ real ) ) ) )
=> ( ( filterlim @ real @ real
@ ^ [X2: real] : ( divide_divide @ real @ ( F8 @ X2 ) @ ( G4 @ X2 ) )
@ F3
@ ( topolo174197925503356063within @ real @ X @ ( top_top @ ( set @ real ) ) ) )
=> ( filterlim @ real @ real
@ ^ [X2: real] : ( divide_divide @ real @ ( F2 @ X2 ) @ ( G @ X2 ) )
@ F3
@ ( topolo174197925503356063within @ real @ X @ ( top_top @ ( set @ real ) ) ) ) ) ) ) ) ) ) ) ).
% lhopital
thf(fact_7178_lhopital__right__at__top__at__top,axiom,
! [F2: real > real,A2: real,G: real > real,F8: real > real,G4: real > real] :
( ( filterlim @ real @ real @ F2 @ ( at_top @ real ) @ ( topolo174197925503356063within @ real @ A2 @ ( set_ord_greaterThan @ real @ A2 ) ) )
=> ( ( filterlim @ real @ real @ G @ ( at_top @ real ) @ ( topolo174197925503356063within @ real @ A2 @ ( set_ord_greaterThan @ real @ A2 ) ) )
=> ( ( eventually @ real
@ ^ [X2: real] : ( has_field_derivative @ real @ F2 @ ( F8 @ X2 ) @ ( topolo174197925503356063within @ real @ X2 @ ( top_top @ ( set @ real ) ) ) )
@ ( topolo174197925503356063within @ real @ A2 @ ( set_ord_greaterThan @ real @ A2 ) ) )
=> ( ( eventually @ real
@ ^ [X2: real] : ( has_field_derivative @ real @ G @ ( G4 @ X2 ) @ ( topolo174197925503356063within @ real @ X2 @ ( top_top @ ( set @ real ) ) ) )
@ ( topolo174197925503356063within @ real @ A2 @ ( set_ord_greaterThan @ real @ A2 ) ) )
=> ( ( filterlim @ real @ real
@ ^ [X2: real] : ( divide_divide @ real @ ( F8 @ X2 ) @ ( G4 @ X2 ) )
@ ( at_top @ real )
@ ( topolo174197925503356063within @ real @ A2 @ ( set_ord_greaterThan @ real @ A2 ) ) )
=> ( filterlim @ real @ real
@ ^ [X2: real] : ( divide_divide @ real @ ( F2 @ X2 ) @ ( G @ X2 ) )
@ ( at_top @ real )
@ ( topolo174197925503356063within @ real @ A2 @ ( set_ord_greaterThan @ real @ A2 ) ) ) ) ) ) ) ) ).
% lhopital_right_at_top_at_top
thf(fact_7179_lhopital__at__top__at__bot,axiom,
! [F2: real > real,A2: real,G: real > real,F8: real > real,G4: real > real] :
( ( filterlim @ real @ real @ F2 @ ( at_top @ real ) @ ( topolo174197925503356063within @ real @ A2 @ ( top_top @ ( set @ real ) ) ) )
=> ( ( filterlim @ real @ real @ G @ ( at_bot @ real ) @ ( topolo174197925503356063within @ real @ A2 @ ( top_top @ ( set @ real ) ) ) )
=> ( ( eventually @ real
@ ^ [X2: real] : ( has_field_derivative @ real @ F2 @ ( F8 @ X2 ) @ ( topolo174197925503356063within @ real @ X2 @ ( top_top @ ( set @ real ) ) ) )
@ ( topolo174197925503356063within @ real @ A2 @ ( top_top @ ( set @ real ) ) ) )
=> ( ( eventually @ real
@ ^ [X2: real] : ( has_field_derivative @ real @ G @ ( G4 @ X2 ) @ ( topolo174197925503356063within @ real @ X2 @ ( top_top @ ( set @ real ) ) ) )
@ ( topolo174197925503356063within @ real @ A2 @ ( top_top @ ( set @ real ) ) ) )
=> ( ( filterlim @ real @ real
@ ^ [X2: real] : ( divide_divide @ real @ ( F8 @ X2 ) @ ( G4 @ X2 ) )
@ ( at_bot @ real )
@ ( topolo174197925503356063within @ real @ A2 @ ( top_top @ ( set @ real ) ) ) )
=> ( filterlim @ real @ real
@ ^ [X2: real] : ( divide_divide @ real @ ( F2 @ X2 ) @ ( G @ X2 ) )
@ ( at_bot @ real )
@ ( topolo174197925503356063within @ real @ A2 @ ( top_top @ ( set @ real ) ) ) ) ) ) ) ) ) ).
% lhopital_at_top_at_bot
thf(fact_7180_lhopital__left__at__top__at__top,axiom,
! [F2: real > real,A2: real,G: real > real,F8: real > real,G4: real > real] :
( ( filterlim @ real @ real @ F2 @ ( at_top @ real ) @ ( topolo174197925503356063within @ real @ A2 @ ( set_ord_lessThan @ real @ A2 ) ) )
=> ( ( filterlim @ real @ real @ G @ ( at_top @ real ) @ ( topolo174197925503356063within @ real @ A2 @ ( set_ord_lessThan @ real @ A2 ) ) )
=> ( ( eventually @ real
@ ^ [X2: real] : ( has_field_derivative @ real @ F2 @ ( F8 @ X2 ) @ ( topolo174197925503356063within @ real @ X2 @ ( top_top @ ( set @ real ) ) ) )
@ ( topolo174197925503356063within @ real @ A2 @ ( set_ord_lessThan @ real @ A2 ) ) )
=> ( ( eventually @ real
@ ^ [X2: real] : ( has_field_derivative @ real @ G @ ( G4 @ X2 ) @ ( topolo174197925503356063within @ real @ X2 @ ( top_top @ ( set @ real ) ) ) )
@ ( topolo174197925503356063within @ real @ A2 @ ( set_ord_lessThan @ real @ A2 ) ) )
=> ( ( filterlim @ real @ real
@ ^ [X2: real] : ( divide_divide @ real @ ( F8 @ X2 ) @ ( G4 @ X2 ) )
@ ( at_top @ real )
@ ( topolo174197925503356063within @ real @ A2 @ ( set_ord_lessThan @ real @ A2 ) ) )
=> ( filterlim @ real @ real
@ ^ [X2: real] : ( divide_divide @ real @ ( F2 @ X2 ) @ ( G @ X2 ) )
@ ( at_top @ real )
@ ( topolo174197925503356063within @ real @ A2 @ ( set_ord_lessThan @ real @ A2 ) ) ) ) ) ) ) ) ).
% lhopital_left_at_top_at_top
thf(fact_7181_lhopital__at__top,axiom,
! [G: real > real,X: real,G4: real > real,F2: real > real,F8: real > real,Y: real] :
( ( filterlim @ real @ real @ G @ ( at_top @ real ) @ ( topolo174197925503356063within @ real @ X @ ( top_top @ ( set @ real ) ) ) )
=> ( ( eventually @ real
@ ^ [X2: real] :
( ( G4 @ X2 )
!= ( zero_zero @ real ) )
@ ( topolo174197925503356063within @ real @ X @ ( top_top @ ( set @ real ) ) ) )
=> ( ( eventually @ real
@ ^ [X2: real] : ( has_field_derivative @ real @ F2 @ ( F8 @ X2 ) @ ( topolo174197925503356063within @ real @ X2 @ ( top_top @ ( set @ real ) ) ) )
@ ( topolo174197925503356063within @ real @ X @ ( top_top @ ( set @ real ) ) ) )
=> ( ( eventually @ real
@ ^ [X2: real] : ( has_field_derivative @ real @ G @ ( G4 @ X2 ) @ ( topolo174197925503356063within @ real @ X2 @ ( top_top @ ( set @ real ) ) ) )
@ ( topolo174197925503356063within @ real @ X @ ( top_top @ ( set @ real ) ) ) )
=> ( ( filterlim @ real @ real
@ ^ [X2: real] : ( divide_divide @ real @ ( F8 @ X2 ) @ ( G4 @ X2 ) )
@ ( topolo7230453075368039082e_nhds @ real @ Y )
@ ( topolo174197925503356063within @ real @ X @ ( top_top @ ( set @ real ) ) ) )
=> ( filterlim @ real @ real
@ ^ [X2: real] : ( divide_divide @ real @ ( F2 @ X2 ) @ ( G @ X2 ) )
@ ( topolo7230453075368039082e_nhds @ real @ Y )
@ ( topolo174197925503356063within @ real @ X @ ( top_top @ ( set @ real ) ) ) ) ) ) ) ) ) ).
% lhopital_at_top
thf(fact_7182_lhospital__at__top__at__top,axiom,
! [G: real > real,G4: real > real,F2: real > real,F8: real > real,X: real] :
( ( filterlim @ real @ real @ G @ ( at_top @ real ) @ ( at_top @ real ) )
=> ( ( eventually @ real
@ ^ [X2: real] :
( ( G4 @ X2 )
!= ( zero_zero @ real ) )
@ ( at_top @ real ) )
=> ( ( eventually @ real
@ ^ [X2: real] : ( has_field_derivative @ real @ F2 @ ( F8 @ X2 ) @ ( topolo174197925503356063within @ real @ X2 @ ( top_top @ ( set @ real ) ) ) )
@ ( at_top @ real ) )
=> ( ( eventually @ real
@ ^ [X2: real] : ( has_field_derivative @ real @ G @ ( G4 @ X2 ) @ ( topolo174197925503356063within @ real @ X2 @ ( top_top @ ( set @ real ) ) ) )
@ ( at_top @ real ) )
=> ( ( filterlim @ real @ real
@ ^ [X2: real] : ( divide_divide @ real @ ( F8 @ X2 ) @ ( G4 @ X2 ) )
@ ( topolo7230453075368039082e_nhds @ real @ X )
@ ( at_top @ real ) )
=> ( filterlim @ real @ real
@ ^ [X2: real] : ( divide_divide @ real @ ( F2 @ X2 ) @ ( G @ X2 ) )
@ ( topolo7230453075368039082e_nhds @ real @ X )
@ ( at_top @ real ) ) ) ) ) ) ) ).
% lhospital_at_top_at_top
thf(fact_7183_lhopital__right,axiom,
! [F2: real > real,X: real,G: real > real,G4: real > real,F8: real > real,F3: filter @ real] :
( ( filterlim @ real @ real @ F2 @ ( topolo7230453075368039082e_nhds @ real @ ( zero_zero @ real ) ) @ ( topolo174197925503356063within @ real @ X @ ( set_ord_greaterThan @ real @ X ) ) )
=> ( ( filterlim @ real @ real @ G @ ( topolo7230453075368039082e_nhds @ real @ ( zero_zero @ real ) ) @ ( topolo174197925503356063within @ real @ X @ ( set_ord_greaterThan @ real @ X ) ) )
=> ( ( eventually @ real
@ ^ [X2: real] :
( ( G @ X2 )
!= ( zero_zero @ real ) )
@ ( topolo174197925503356063within @ real @ X @ ( set_ord_greaterThan @ real @ X ) ) )
=> ( ( eventually @ real
@ ^ [X2: real] :
( ( G4 @ X2 )
!= ( zero_zero @ real ) )
@ ( topolo174197925503356063within @ real @ X @ ( set_ord_greaterThan @ real @ X ) ) )
=> ( ( eventually @ real
@ ^ [X2: real] : ( has_field_derivative @ real @ F2 @ ( F8 @ X2 ) @ ( topolo174197925503356063within @ real @ X2 @ ( top_top @ ( set @ real ) ) ) )
@ ( topolo174197925503356063within @ real @ X @ ( set_ord_greaterThan @ real @ X ) ) )
=> ( ( eventually @ real
@ ^ [X2: real] : ( has_field_derivative @ real @ G @ ( G4 @ X2 ) @ ( topolo174197925503356063within @ real @ X2 @ ( top_top @ ( set @ real ) ) ) )
@ ( topolo174197925503356063within @ real @ X @ ( set_ord_greaterThan @ real @ X ) ) )
=> ( ( filterlim @ real @ real
@ ^ [X2: real] : ( divide_divide @ real @ ( F8 @ X2 ) @ ( G4 @ X2 ) )
@ F3
@ ( topolo174197925503356063within @ real @ X @ ( set_ord_greaterThan @ real @ X ) ) )
=> ( filterlim @ real @ real
@ ^ [X2: real] : ( divide_divide @ real @ ( F2 @ X2 ) @ ( G @ X2 ) )
@ F3
@ ( topolo174197925503356063within @ real @ X @ ( set_ord_greaterThan @ real @ X ) ) ) ) ) ) ) ) ) ) ).
% lhopital_right
thf(fact_7184_lhopital__right__0,axiom,
! [F0: real > real,G0: real > real,G4: real > real,F8: real > real,F3: filter @ real] :
( ( filterlim @ real @ real @ F0 @ ( topolo7230453075368039082e_nhds @ real @ ( zero_zero @ real ) ) @ ( topolo174197925503356063within @ real @ ( zero_zero @ real ) @ ( set_ord_greaterThan @ real @ ( zero_zero @ real ) ) ) )
=> ( ( filterlim @ real @ real @ G0 @ ( topolo7230453075368039082e_nhds @ real @ ( zero_zero @ real ) ) @ ( topolo174197925503356063within @ real @ ( zero_zero @ real ) @ ( set_ord_greaterThan @ real @ ( zero_zero @ real ) ) ) )
=> ( ( eventually @ real
@ ^ [X2: real] :
( ( G0 @ X2 )
!= ( zero_zero @ real ) )
@ ( topolo174197925503356063within @ real @ ( zero_zero @ real ) @ ( set_ord_greaterThan @ real @ ( zero_zero @ real ) ) ) )
=> ( ( eventually @ real
@ ^ [X2: real] :
( ( G4 @ X2 )
!= ( zero_zero @ real ) )
@ ( topolo174197925503356063within @ real @ ( zero_zero @ real ) @ ( set_ord_greaterThan @ real @ ( zero_zero @ real ) ) ) )
=> ( ( eventually @ real
@ ^ [X2: real] : ( has_field_derivative @ real @ F0 @ ( F8 @ X2 ) @ ( topolo174197925503356063within @ real @ X2 @ ( top_top @ ( set @ real ) ) ) )
@ ( topolo174197925503356063within @ real @ ( zero_zero @ real ) @ ( set_ord_greaterThan @ real @ ( zero_zero @ real ) ) ) )
=> ( ( eventually @ real
@ ^ [X2: real] : ( has_field_derivative @ real @ G0 @ ( G4 @ X2 ) @ ( topolo174197925503356063within @ real @ X2 @ ( top_top @ ( set @ real ) ) ) )
@ ( topolo174197925503356063within @ real @ ( zero_zero @ real ) @ ( set_ord_greaterThan @ real @ ( zero_zero @ real ) ) ) )
=> ( ( filterlim @ real @ real
@ ^ [X2: real] : ( divide_divide @ real @ ( F8 @ X2 ) @ ( G4 @ X2 ) )
@ F3
@ ( topolo174197925503356063within @ real @ ( zero_zero @ real ) @ ( set_ord_greaterThan @ real @ ( zero_zero @ real ) ) ) )
=> ( filterlim @ real @ real
@ ^ [X2: real] : ( divide_divide @ real @ ( F0 @ X2 ) @ ( G0 @ X2 ) )
@ F3
@ ( topolo174197925503356063within @ real @ ( zero_zero @ real ) @ ( set_ord_greaterThan @ real @ ( zero_zero @ real ) ) ) ) ) ) ) ) ) ) ) ).
% lhopital_right_0
thf(fact_7185_lhopital__left,axiom,
! [F2: real > real,X: real,G: real > real,G4: real > real,F8: real > real,F3: filter @ real] :
( ( filterlim @ real @ real @ F2 @ ( topolo7230453075368039082e_nhds @ real @ ( zero_zero @ real ) ) @ ( topolo174197925503356063within @ real @ X @ ( set_ord_lessThan @ real @ X ) ) )
=> ( ( filterlim @ real @ real @ G @ ( topolo7230453075368039082e_nhds @ real @ ( zero_zero @ real ) ) @ ( topolo174197925503356063within @ real @ X @ ( set_ord_lessThan @ real @ X ) ) )
=> ( ( eventually @ real
@ ^ [X2: real] :
( ( G @ X2 )
!= ( zero_zero @ real ) )
@ ( topolo174197925503356063within @ real @ X @ ( set_ord_lessThan @ real @ X ) ) )
=> ( ( eventually @ real
@ ^ [X2: real] :
( ( G4 @ X2 )
!= ( zero_zero @ real ) )
@ ( topolo174197925503356063within @ real @ X @ ( set_ord_lessThan @ real @ X ) ) )
=> ( ( eventually @ real
@ ^ [X2: real] : ( has_field_derivative @ real @ F2 @ ( F8 @ X2 ) @ ( topolo174197925503356063within @ real @ X2 @ ( top_top @ ( set @ real ) ) ) )
@ ( topolo174197925503356063within @ real @ X @ ( set_ord_lessThan @ real @ X ) ) )
=> ( ( eventually @ real
@ ^ [X2: real] : ( has_field_derivative @ real @ G @ ( G4 @ X2 ) @ ( topolo174197925503356063within @ real @ X2 @ ( top_top @ ( set @ real ) ) ) )
@ ( topolo174197925503356063within @ real @ X @ ( set_ord_lessThan @ real @ X ) ) )
=> ( ( filterlim @ real @ real
@ ^ [X2: real] : ( divide_divide @ real @ ( F8 @ X2 ) @ ( G4 @ X2 ) )
@ F3
@ ( topolo174197925503356063within @ real @ X @ ( set_ord_lessThan @ real @ X ) ) )
=> ( filterlim @ real @ real
@ ^ [X2: real] : ( divide_divide @ real @ ( F2 @ X2 ) @ ( G @ X2 ) )
@ F3
@ ( topolo174197925503356063within @ real @ X @ ( set_ord_lessThan @ real @ X ) ) ) ) ) ) ) ) ) ) ).
% lhopital_left
thf(fact_7186_lhopital__right__at__top__at__bot,axiom,
! [F2: real > real,A2: real,G: real > real,F8: real > real,G4: real > real] :
( ( filterlim @ real @ real @ F2 @ ( at_top @ real ) @ ( topolo174197925503356063within @ real @ A2 @ ( set_ord_greaterThan @ real @ A2 ) ) )
=> ( ( filterlim @ real @ real @ G @ ( at_bot @ real ) @ ( topolo174197925503356063within @ real @ A2 @ ( set_ord_greaterThan @ real @ A2 ) ) )
=> ( ( eventually @ real
@ ^ [X2: real] : ( has_field_derivative @ real @ F2 @ ( F8 @ X2 ) @ ( topolo174197925503356063within @ real @ X2 @ ( top_top @ ( set @ real ) ) ) )
@ ( topolo174197925503356063within @ real @ A2 @ ( set_ord_greaterThan @ real @ A2 ) ) )
=> ( ( eventually @ real
@ ^ [X2: real] : ( has_field_derivative @ real @ G @ ( G4 @ X2 ) @ ( topolo174197925503356063within @ real @ X2 @ ( top_top @ ( set @ real ) ) ) )
@ ( topolo174197925503356063within @ real @ A2 @ ( set_ord_greaterThan @ real @ A2 ) ) )
=> ( ( filterlim @ real @ real
@ ^ [X2: real] : ( divide_divide @ real @ ( F8 @ X2 ) @ ( G4 @ X2 ) )
@ ( at_bot @ real )
@ ( topolo174197925503356063within @ real @ A2 @ ( set_ord_greaterThan @ real @ A2 ) ) )
=> ( filterlim @ real @ real
@ ^ [X2: real] : ( divide_divide @ real @ ( F2 @ X2 ) @ ( G @ X2 ) )
@ ( at_bot @ real )
@ ( topolo174197925503356063within @ real @ A2 @ ( set_ord_greaterThan @ real @ A2 ) ) ) ) ) ) ) ) ).
% lhopital_right_at_top_at_bot
thf(fact_7187_lhopital__left__at__top__at__bot,axiom,
! [F2: real > real,A2: real,G: real > real,F8: real > real,G4: real > real] :
( ( filterlim @ real @ real @ F2 @ ( at_top @ real ) @ ( topolo174197925503356063within @ real @ A2 @ ( set_ord_lessThan @ real @ A2 ) ) )
=> ( ( filterlim @ real @ real @ G @ ( at_bot @ real ) @ ( topolo174197925503356063within @ real @ A2 @ ( set_ord_lessThan @ real @ A2 ) ) )
=> ( ( eventually @ real
@ ^ [X2: real] : ( has_field_derivative @ real @ F2 @ ( F8 @ X2 ) @ ( topolo174197925503356063within @ real @ X2 @ ( top_top @ ( set @ real ) ) ) )
@ ( topolo174197925503356063within @ real @ A2 @ ( set_ord_lessThan @ real @ A2 ) ) )
=> ( ( eventually @ real
@ ^ [X2: real] : ( has_field_derivative @ real @ G @ ( G4 @ X2 ) @ ( topolo174197925503356063within @ real @ X2 @ ( top_top @ ( set @ real ) ) ) )
@ ( topolo174197925503356063within @ real @ A2 @ ( set_ord_lessThan @ real @ A2 ) ) )
=> ( ( filterlim @ real @ real
@ ^ [X2: real] : ( divide_divide @ real @ ( F8 @ X2 ) @ ( G4 @ X2 ) )
@ ( at_bot @ real )
@ ( topolo174197925503356063within @ real @ A2 @ ( set_ord_lessThan @ real @ A2 ) ) )
=> ( filterlim @ real @ real
@ ^ [X2: real] : ( divide_divide @ real @ ( F2 @ X2 ) @ ( G @ X2 ) )
@ ( at_bot @ real )
@ ( topolo174197925503356063within @ real @ A2 @ ( set_ord_lessThan @ real @ A2 ) ) ) ) ) ) ) ) ).
% lhopital_left_at_top_at_bot
thf(fact_7188_lhopital__right__at__top,axiom,
! [G: real > real,X: real,G4: real > real,F2: real > real,F8: real > real,Y: real] :
( ( filterlim @ real @ real @ G @ ( at_top @ real ) @ ( topolo174197925503356063within @ real @ X @ ( set_ord_greaterThan @ real @ X ) ) )
=> ( ( eventually @ real
@ ^ [X2: real] :
( ( G4 @ X2 )
!= ( zero_zero @ real ) )
@ ( topolo174197925503356063within @ real @ X @ ( set_ord_greaterThan @ real @ X ) ) )
=> ( ( eventually @ real
@ ^ [X2: real] : ( has_field_derivative @ real @ F2 @ ( F8 @ X2 ) @ ( topolo174197925503356063within @ real @ X2 @ ( top_top @ ( set @ real ) ) ) )
@ ( topolo174197925503356063within @ real @ X @ ( set_ord_greaterThan @ real @ X ) ) )
=> ( ( eventually @ real
@ ^ [X2: real] : ( has_field_derivative @ real @ G @ ( G4 @ X2 ) @ ( topolo174197925503356063within @ real @ X2 @ ( top_top @ ( set @ real ) ) ) )
@ ( topolo174197925503356063within @ real @ X @ ( set_ord_greaterThan @ real @ X ) ) )
=> ( ( filterlim @ real @ real
@ ^ [X2: real] : ( divide_divide @ real @ ( F8 @ X2 ) @ ( G4 @ X2 ) )
@ ( topolo7230453075368039082e_nhds @ real @ Y )
@ ( topolo174197925503356063within @ real @ X @ ( set_ord_greaterThan @ real @ X ) ) )
=> ( filterlim @ real @ real
@ ^ [X2: real] : ( divide_divide @ real @ ( F2 @ X2 ) @ ( G @ X2 ) )
@ ( topolo7230453075368039082e_nhds @ real @ Y )
@ ( topolo174197925503356063within @ real @ X @ ( set_ord_greaterThan @ real @ X ) ) ) ) ) ) ) ) ).
% lhopital_right_at_top
thf(fact_7189_lhopital__right__0__at__top,axiom,
! [G: real > real,G4: real > real,F2: real > real,F8: real > real,X: real] :
( ( filterlim @ real @ real @ G @ ( at_top @ real ) @ ( topolo174197925503356063within @ real @ ( zero_zero @ real ) @ ( set_ord_greaterThan @ real @ ( zero_zero @ real ) ) ) )
=> ( ( eventually @ real
@ ^ [X2: real] :
( ( G4 @ X2 )
!= ( zero_zero @ real ) )
@ ( topolo174197925503356063within @ real @ ( zero_zero @ real ) @ ( set_ord_greaterThan @ real @ ( zero_zero @ real ) ) ) )
=> ( ( eventually @ real
@ ^ [X2: real] : ( has_field_derivative @ real @ F2 @ ( F8 @ X2 ) @ ( topolo174197925503356063within @ real @ X2 @ ( top_top @ ( set @ real ) ) ) )
@ ( topolo174197925503356063within @ real @ ( zero_zero @ real ) @ ( set_ord_greaterThan @ real @ ( zero_zero @ real ) ) ) )
=> ( ( eventually @ real
@ ^ [X2: real] : ( has_field_derivative @ real @ G @ ( G4 @ X2 ) @ ( topolo174197925503356063within @ real @ X2 @ ( top_top @ ( set @ real ) ) ) )
@ ( topolo174197925503356063within @ real @ ( zero_zero @ real ) @ ( set_ord_greaterThan @ real @ ( zero_zero @ real ) ) ) )
=> ( ( filterlim @ real @ real
@ ^ [X2: real] : ( divide_divide @ real @ ( F8 @ X2 ) @ ( G4 @ X2 ) )
@ ( topolo7230453075368039082e_nhds @ real @ X )
@ ( topolo174197925503356063within @ real @ ( zero_zero @ real ) @ ( set_ord_greaterThan @ real @ ( zero_zero @ real ) ) ) )
=> ( filterlim @ real @ real
@ ^ [X2: real] : ( divide_divide @ real @ ( F2 @ X2 ) @ ( G @ X2 ) )
@ ( topolo7230453075368039082e_nhds @ real @ X )
@ ( topolo174197925503356063within @ real @ ( zero_zero @ real ) @ ( set_ord_greaterThan @ real @ ( zero_zero @ real ) ) ) ) ) ) ) ) ) ).
% lhopital_right_0_at_top
thf(fact_7190_summable__Cauchy_H,axiom,
! [A: $tType] :
( ( real_Vector_banach @ A )
=> ! [F2: nat > A,G: nat > real] :
( ( eventually @ nat
@ ^ [M4: nat] :
! [N3: nat] :
( ( ord_less_eq @ nat @ M4 @ N3 )
=> ( ord_less_eq @ real @ ( real_V7770717601297561774m_norm @ A @ ( groups7311177749621191930dd_sum @ nat @ A @ F2 @ ( set_or7035219750837199246ssThan @ nat @ M4 @ N3 ) ) ) @ ( G @ M4 ) ) )
@ ( at_top @ nat ) )
=> ( ( filterlim @ nat @ real @ G @ ( topolo7230453075368039082e_nhds @ real @ ( zero_zero @ real ) ) @ ( at_top @ nat ) )
=> ( summable @ A @ F2 ) ) ) ) ).
% summable_Cauchy'
thf(fact_7191_Bfun__metric__def,axiom,
! [B: $tType,A: $tType] :
( ( real_V7819770556892013058_space @ B )
=> ( ( bfun @ A @ B )
= ( ^ [F4: A > B,F7: filter @ A] :
? [Y6: B,K7: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ K7 )
& ( eventually @ A
@ ^ [X2: A] : ( ord_less_eq @ real @ ( real_V557655796197034286t_dist @ B @ ( F4 @ X2 ) @ Y6 ) @ K7 )
@ F7 ) ) ) ) ) ).
% Bfun_metric_def
thf(fact_7192_Bseq__Suc__iff,axiom,
! [A: $tType] :
( ( real_V822414075346904944vector @ A )
=> ! [F2: nat > A] :
( ( bfun @ nat @ A
@ ^ [N3: nat] : ( F2 @ ( suc @ N3 ) )
@ ( at_top @ nat ) )
= ( bfun @ nat @ A @ F2 @ ( at_top @ nat ) ) ) ) ).
% Bseq_Suc_iff
thf(fact_7193_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_7194_Bseq__offset,axiom,
! [A: $tType] :
( ( real_V7819770556892013058_space @ A )
=> ! [X7: nat > A,K: nat] :
( ( bfun @ nat @ A
@ ^ [N3: nat] : ( X7 @ ( plus_plus @ nat @ N3 @ K ) )
@ ( at_top @ nat ) )
=> ( bfun @ nat @ A @ X7 @ ( at_top @ nat ) ) ) ) ).
% Bseq_offset
thf(fact_7195_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_7196_Bseq__ignore__initial__segment,axiom,
! [A: $tType] :
( ( real_V7819770556892013058_space @ A )
=> ! [X7: nat > A,K: nat] :
( ( bfun @ nat @ A @ X7 @ ( at_top @ nat ) )
=> ( bfun @ nat @ A
@ ^ [N3: nat] : ( X7 @ ( plus_plus @ nat @ N3 @ K ) )
@ ( at_top @ nat ) ) ) ) ).
% Bseq_ignore_initial_segment
thf(fact_7197_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_7198_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_7199_Bseq__def,axiom,
! [A: $tType] :
( ( real_V822414075346904944vector @ A )
=> ! [X7: nat > A] :
( ( bfun @ nat @ A @ X7 @ ( at_top @ nat ) )
= ( ? [K7: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ K7 )
& ! [N3: nat] : ( ord_less_eq @ real @ ( real_V7770717601297561774m_norm @ A @ ( X7 @ N3 ) ) @ K7 ) ) ) ) ) ).
% Bseq_def
thf(fact_7200_BseqI,axiom,
! [A: $tType] :
( ( real_V822414075346904944vector @ A )
=> ! [K6: real,X7: nat > A] :
( ( ord_less @ real @ ( zero_zero @ real ) @ K6 )
=> ( ! [N2: nat] : ( ord_less_eq @ real @ ( real_V7770717601297561774m_norm @ A @ ( X7 @ N2 ) ) @ K6 )
=> ( bfun @ nat @ A @ X7 @ ( at_top @ nat ) ) ) ) ) ).
% BseqI
thf(fact_7201_BseqE,axiom,
! [A: $tType] :
( ( real_V822414075346904944vector @ A )
=> ! [X7: nat > A] :
( ( bfun @ nat @ A @ X7 @ ( at_top @ nat ) )
=> ~ ! [K9: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ K9 )
=> ~ ! [N10: nat] : ( ord_less_eq @ real @ ( real_V7770717601297561774m_norm @ A @ ( X7 @ N10 ) ) @ K9 ) ) ) ) ).
% BseqE
thf(fact_7202_BseqD,axiom,
! [A: $tType] :
( ( real_V822414075346904944vector @ A )
=> ! [X7: nat > A] :
( ( bfun @ nat @ A @ X7 @ ( at_top @ nat ) )
=> ? [K9: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ K9 )
& ! [N10: nat] : ( ord_less_eq @ real @ ( real_V7770717601297561774m_norm @ A @ ( X7 @ N10 ) ) @ K9 ) ) ) ) ).
% BseqD
thf(fact_7203_Bseq__iff1a,axiom,
! [A: $tType] :
( ( real_V822414075346904944vector @ A )
=> ! [X7: nat > A] :
( ( bfun @ nat @ A @ X7 @ ( at_top @ nat ) )
= ( ? [N8: nat] :
! [N3: nat] : ( ord_less @ real @ ( real_V7770717601297561774m_norm @ A @ ( X7 @ N3 ) ) @ ( semiring_1_of_nat @ real @ ( suc @ N8 ) ) ) ) ) ) ).
% Bseq_iff1a
thf(fact_7204_Bseq__iff,axiom,
! [A: $tType] :
( ( real_V822414075346904944vector @ A )
=> ! [X7: nat > A] :
( ( bfun @ nat @ A @ X7 @ ( at_top @ nat ) )
= ( ? [N8: nat] :
! [N3: nat] : ( ord_less_eq @ real @ ( real_V7770717601297561774m_norm @ A @ ( X7 @ N3 ) ) @ ( semiring_1_of_nat @ real @ ( suc @ N8 ) ) ) ) ) ) ).
% Bseq_iff
thf(fact_7205_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_7206_Bseq__iff2,axiom,
! [A: $tType] :
( ( real_V822414075346904944vector @ A )
=> ! [X7: nat > A] :
( ( bfun @ nat @ A @ X7 @ ( at_top @ nat ) )
= ( ? [K3: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ K3 )
& ? [X2: A] :
! [N3: nat] : ( ord_less_eq @ real @ ( real_V7770717601297561774m_norm @ A @ ( plus_plus @ A @ ( X7 @ N3 ) @ ( uminus_uminus @ A @ X2 ) ) ) @ K3 ) ) ) ) ) ).
% Bseq_iff2
thf(fact_7207_Bseq__iff3,axiom,
! [A: $tType] :
( ( real_V822414075346904944vector @ A )
=> ! [X7: nat > A] :
( ( bfun @ nat @ A @ X7 @ ( at_top @ nat ) )
= ( ? [K3: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ K3 )
& ? [N8: nat] :
! [N3: nat] : ( ord_less_eq @ real @ ( real_V7770717601297561774m_norm @ A @ ( plus_plus @ A @ ( X7 @ N3 ) @ ( uminus_uminus @ A @ ( X7 @ N8 ) ) ) ) @ K3 ) ) ) ) ) ).
% Bseq_iff3
thf(fact_7208_Bfun__inverse,axiom,
! [A: $tType,B: $tType] :
( ( real_V8999393235501362500lgebra @ A )
=> ! [F2: B > A,A2: A,F3: filter @ B] :
( ( filterlim @ B @ A @ F2 @ ( topolo7230453075368039082e_nhds @ A @ A2 ) @ F3 )
=> ( ( A2
!= ( zero_zero @ A ) )
=> ( bfun @ B @ A
@ ^ [X2: B] : ( inverse_inverse @ A @ ( F2 @ X2 ) )
@ F3 ) ) ) ) ).
% Bfun_inverse
thf(fact_7209_BfunE,axiom,
! [B: $tType,A: $tType] :
( ( real_V822414075346904944vector @ B )
=> ! [F2: A > B,F3: filter @ A] :
( ( bfun @ A @ B @ F2 @ F3 )
=> ~ ! [B10: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ B10 )
=> ~ ( eventually @ A
@ ^ [X2: A] : ( ord_less_eq @ real @ ( real_V7770717601297561774m_norm @ B @ ( F2 @ X2 ) ) @ B10 )
@ F3 ) ) ) ) ).
% BfunE
thf(fact_7210_Bfun__def,axiom,
! [B: $tType,A: $tType] :
( ( real_V822414075346904944vector @ B )
=> ( ( bfun @ A @ B )
= ( ^ [F4: A > B,F7: filter @ A] :
? [K7: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ K7 )
& ( eventually @ A
@ ^ [X2: A] : ( ord_less_eq @ real @ ( real_V7770717601297561774m_norm @ B @ ( F4 @ X2 ) ) @ K7 )
@ F7 ) ) ) ) ) ).
% Bfun_def
thf(fact_7211_summable__bounded__partials,axiom,
! [A: $tType] :
( ( ( real_V8037385150606011577_space @ A )
& ( real_V822414075346904944vector @ A ) )
=> ! [F2: nat > A,G: nat > real] :
( ( eventually @ nat
@ ^ [X02: nat] :
! [A4: nat] :
( ( ord_less_eq @ nat @ X02 @ A4 )
=> ! [B5: nat] :
( ( ord_less @ nat @ A4 @ B5 )
=> ( ord_less_eq @ real @ ( real_V7770717601297561774m_norm @ A @ ( groups7311177749621191930dd_sum @ nat @ A @ F2 @ ( set_or3652927894154168847AtMost @ nat @ A4 @ B5 ) ) ) @ ( G @ A4 ) ) ) )
@ ( at_top @ nat ) )
=> ( ( filterlim @ nat @ real @ G @ ( topolo7230453075368039082e_nhds @ real @ ( zero_zero @ real ) ) @ ( at_top @ nat ) )
=> ( summable @ A @ F2 ) ) ) ) ).
% summable_bounded_partials
thf(fact_7212_sequentially__imp__eventually__at__right,axiom,
! [A: $tType] :
( ( ( topolo3112930676232923870pology @ A )
& ( topolo1944317154257567458pology @ A ) )
=> ! [A2: A,B2: A,P: A > $o] :
( ( ord_less @ A @ A2 @ B2 )
=> ( ! [F5: nat > A] :
( ! [N10: nat] : ( ord_less @ A @ A2 @ ( F5 @ N10 ) )
=> ( ! [N10: nat] : ( ord_less @ A @ ( F5 @ N10 ) @ B2 )
=> ( ( order_antimono @ nat @ A @ F5 )
=> ( ( filterlim @ nat @ A @ F5 @ ( topolo7230453075368039082e_nhds @ A @ A2 ) @ ( at_top @ nat ) )
=> ( eventually @ nat
@ ^ [N3: nat] : ( P @ ( F5 @ N3 ) )
@ ( at_top @ nat ) ) ) ) ) )
=> ( eventually @ A @ P @ ( topolo174197925503356063within @ A @ A2 @ ( set_ord_greaterThan @ A @ A2 ) ) ) ) ) ) ).
% sequentially_imp_eventually_at_right
thf(fact_7213_greaterThanAtMost__iff,axiom,
! [A: $tType] :
( ( ord @ A )
=> ! [I: A,L: A,U: A] :
( ( member @ A @ I @ ( set_or3652927894154168847AtMost @ A @ L @ U ) )
= ( ( ord_less @ A @ L @ I )
& ( ord_less_eq @ A @ I @ U ) ) ) ) ).
% greaterThanAtMost_iff
thf(fact_7214_greaterThanAtMost__empty__iff2,axiom,
! [A: $tType] :
( ( preorder @ A )
=> ! [K: A,L: A] :
( ( ( bot_bot @ ( set @ A ) )
= ( set_or3652927894154168847AtMost @ A @ K @ L ) )
= ( ~ ( ord_less @ A @ K @ L ) ) ) ) ).
% greaterThanAtMost_empty_iff2
thf(fact_7215_greaterThanAtMost__empty__iff,axiom,
! [A: $tType] :
( ( preorder @ A )
=> ! [K: A,L: A] :
( ( ( set_or3652927894154168847AtMost @ A @ K @ L )
= ( bot_bot @ ( set @ A ) ) )
= ( ~ ( ord_less @ A @ K @ L ) ) ) ) ).
% greaterThanAtMost_empty_iff
thf(fact_7216_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_7217_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_7218_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_7219_decseq__SucI,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [X7: nat > A] :
( ! [N2: nat] : ( ord_less_eq @ A @ ( X7 @ ( suc @ N2 ) ) @ ( X7 @ N2 ) )
=> ( order_antimono @ nat @ A @ X7 ) ) ) ).
% decseq_SucI
thf(fact_7220_decseq__Suc__iff,axiom,
! [A: $tType] :
( ( order @ A )
=> ( ( order_antimono @ nat @ A )
= ( ^ [F4: nat > A] :
! [N3: nat] : ( ord_less_eq @ A @ ( F4 @ ( suc @ N3 ) ) @ ( F4 @ N3 ) ) ) ) ) ).
% decseq_Suc_iff
thf(fact_7221_atLeastSucAtMost__greaterThanAtMost,axiom,
! [L: nat,U: nat] :
( ( set_or1337092689740270186AtMost @ nat @ ( suc @ L ) @ U )
= ( set_or3652927894154168847AtMost @ nat @ L @ U ) ) ).
% atLeastSucAtMost_greaterThanAtMost
thf(fact_7222_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_7223_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_7224_greaterThanAtMost__upt,axiom,
( ( set_or3652927894154168847AtMost @ nat )
= ( ^ [N3: nat,M4: nat] : ( set2 @ nat @ ( upt @ ( suc @ N3 ) @ ( suc @ M4 ) ) ) ) ) ).
% greaterThanAtMost_upt
thf(fact_7225_sum_Ohead,axiom,
! [A: $tType] :
( ( comm_monoid_add @ A )
=> ! [M: nat,N: nat,G: nat > A] :
( ( ord_less_eq @ nat @ M @ N )
=> ( ( groups7311177749621191930dd_sum @ nat @ A @ G @ ( set_or1337092689740270186AtMost @ nat @ M @ N ) )
= ( plus_plus @ A @ ( G @ M ) @ ( groups7311177749621191930dd_sum @ nat @ A @ G @ ( set_or3652927894154168847AtMost @ nat @ M @ N ) ) ) ) ) ) ).
% sum.head
thf(fact_7226_prod_Ohead,axiom,
! [A: $tType] :
( ( comm_monoid_mult @ A )
=> ! [M: nat,N: nat,G: nat > A] :
( ( ord_less_eq @ nat @ M @ N )
=> ( ( groups7121269368397514597t_prod @ nat @ A @ G @ ( set_or1337092689740270186AtMost @ nat @ M @ N ) )
= ( times_times @ A @ ( G @ M ) @ ( groups7121269368397514597t_prod @ nat @ A @ G @ ( set_or3652927894154168847AtMost @ nat @ M @ N ) ) ) ) ) ) ).
% prod.head
thf(fact_7227_greaterThanAtMost__subseteq__atLeastAtMost__iff,axiom,
! [A: $tType] :
( ( dense_linorder @ A )
=> ! [A2: A,B2: A,C2: A,D2: A] :
( ( ord_less_eq @ ( set @ A ) @ ( set_or3652927894154168847AtMost @ A @ A2 @ B2 ) @ ( set_or1337092689740270186AtMost @ A @ C2 @ D2 ) )
= ( ( ord_less @ A @ A2 @ B2 )
=> ( ( ord_less_eq @ A @ C2 @ A2 )
& ( ord_less_eq @ A @ B2 @ D2 ) ) ) ) ) ).
% greaterThanAtMost_subseteq_atLeastAtMost_iff
thf(fact_7228_greaterThanAtMost__subseteq__atLeastLessThan__iff,axiom,
! [A: $tType] :
( ( dense_linorder @ A )
=> ! [A2: A,B2: A,C2: A,D2: A] :
( ( ord_less_eq @ ( set @ A ) @ ( set_or3652927894154168847AtMost @ A @ A2 @ B2 ) @ ( set_or7035219750837199246ssThan @ A @ C2 @ D2 ) )
= ( ( ord_less @ A @ A2 @ B2 )
=> ( ( ord_less_eq @ A @ C2 @ A2 )
& ( ord_less @ A @ B2 @ D2 ) ) ) ) ) ).
% greaterThanAtMost_subseteq_atLeastLessThan_iff
thf(fact_7229_greaterThanLessThan__subseteq__greaterThanAtMost__iff,axiom,
! [A: $tType] :
( ( dense_linorder @ A )
=> ! [A2: A,B2: A,C2: A,D2: A] :
( ( ord_less_eq @ ( set @ A ) @ ( set_or5935395276787703475ssThan @ A @ A2 @ B2 ) @ ( set_or3652927894154168847AtMost @ A @ C2 @ D2 ) )
= ( ( ord_less @ A @ A2 @ B2 )
=> ( ( ord_less_eq @ A @ C2 @ A2 )
& ( ord_less_eq @ A @ B2 @ D2 ) ) ) ) ) ).
% greaterThanLessThan_subseteq_greaterThanAtMost_iff
thf(fact_7230_tendsto__at__right__sequentially,axiom,
! [C: $tType,B: $tType] :
( ( ( topolo3112930676232923870pology @ B )
& ( topolo1944317154257567458pology @ B )
& ( topolo4958980785337419405_space @ C ) )
=> ! [A2: B,B2: B,X7: B > C,L6: C] :
( ( ord_less @ B @ A2 @ B2 )
=> ( ! [S8: nat > B] :
( ! [N10: nat] : ( ord_less @ B @ A2 @ ( S8 @ N10 ) )
=> ( ! [N10: nat] : ( ord_less @ B @ ( S8 @ N10 ) @ B2 )
=> ( ( order_antimono @ nat @ B @ S8 )
=> ( ( filterlim @ nat @ B @ S8 @ ( topolo7230453075368039082e_nhds @ B @ A2 ) @ ( at_top @ nat ) )
=> ( filterlim @ nat @ C
@ ^ [N3: nat] : ( X7 @ ( S8 @ N3 ) )
@ ( topolo7230453075368039082e_nhds @ C @ L6 )
@ ( at_top @ nat ) ) ) ) ) )
=> ( filterlim @ B @ C @ X7 @ ( topolo7230453075368039082e_nhds @ C @ L6 ) @ ( topolo174197925503356063within @ B @ A2 @ ( set_ord_greaterThan @ B @ A2 ) ) ) ) ) ) ).
% tendsto_at_right_sequentially
thf(fact_7231_cauchy__filter__metric,axiom,
! [A: $tType] :
( ( ( real_V768167426530841204y_dist @ A )
& ( topolo7287701948861334536_space @ A ) )
=> ( ( topolo6773858410816713723filter @ A )
= ( ^ [F7: filter @ A] :
! [E4: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ E4 )
=> ? [P2: A > $o] :
( ( eventually @ A @ P2 @ F7 )
& ! [X2: A,Y6: A] :
( ( ( P2 @ X2 )
& ( P2 @ Y6 ) )
=> ( ord_less @ real @ ( real_V557655796197034286t_dist @ A @ X2 @ Y6 ) @ E4 ) ) ) ) ) ) ) ).
% cauchy_filter_metric
thf(fact_7232_rat__inverse__code,axiom,
! [P4: rat] :
( ( quotient_of @ ( inverse_inverse @ rat @ P4 ) )
= ( product_case_prod @ int @ int @ ( product_prod @ int @ int )
@ ^ [A4: int,B5: int] :
( if @ ( product_prod @ int @ int )
@ ( A4
= ( zero_zero @ int ) )
@ ( product_Pair @ int @ int @ ( zero_zero @ int ) @ ( one_one @ int ) )
@ ( product_Pair @ int @ int @ ( times_times @ int @ ( sgn_sgn @ int @ A4 ) @ B5 ) @ ( abs_abs @ int @ A4 ) ) )
@ ( quotient_of @ P4 ) ) ) ).
% rat_inverse_code
thf(fact_7233_rat__one__code,axiom,
( ( quotient_of @ ( one_one @ rat ) )
= ( product_Pair @ int @ int @ ( one_one @ int ) @ ( one_one @ int ) ) ) ).
% rat_one_code
thf(fact_7234_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_7235_rat__zero__code,axiom,
( ( quotient_of @ ( zero_zero @ rat ) )
= ( product_Pair @ int @ int @ ( zero_zero @ int ) @ ( one_one @ int ) ) ) ).
% rat_zero_code
thf(fact_7236_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_7237_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_7238_atLeastPlusOneAtMost__greaterThanAtMost__int,axiom,
! [L: int,U: int] :
( ( set_or1337092689740270186AtMost @ int @ ( plus_plus @ int @ L @ ( one_one @ int ) ) @ U )
= ( set_or3652927894154168847AtMost @ int @ L @ U ) ) ).
% atLeastPlusOneAtMost_greaterThanAtMost_int
thf(fact_7239_quotient__of__denom__pos,axiom,
! [R2: rat,P4: int,Q3: int] :
( ( ( quotient_of @ R2 )
= ( product_Pair @ int @ int @ P4 @ Q3 ) )
=> ( ord_less @ int @ ( zero_zero @ int ) @ Q3 ) ) ).
% quotient_of_denom_pos
thf(fact_7240_atLeastPlusOneAtMost__greaterThanAtMost__integer,axiom,
! [L: code_integer,U: code_integer] :
( ( set_or1337092689740270186AtMost @ code_integer @ ( plus_plus @ code_integer @ L @ ( one_one @ code_integer ) ) @ U )
= ( set_or3652927894154168847AtMost @ code_integer @ L @ U ) ) ).
% atLeastPlusOneAtMost_greaterThanAtMost_integer
thf(fact_7241_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_7242_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_7243_quotient__of__int,axiom,
! [A2: int] :
( ( quotient_of @ ( of_int @ A2 ) )
= ( product_Pair @ int @ int @ A2 @ ( one_one @ int ) ) ) ).
% quotient_of_int
thf(fact_7244_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_7245_differentiable__cmult__right__iff,axiom,
! [A: $tType,B: $tType] :
( ( ( real_V822414075346904944vector @ B )
& ( real_V3459762299906320749_field @ A ) )
=> ! [Q3: B > A,C2: A,T: B] :
( ( differentiable @ B @ A
@ ^ [T2: B] : ( times_times @ A @ ( Q3 @ T2 ) @ C2 )
@ ( topolo174197925503356063within @ B @ T @ ( top_top @ ( set @ B ) ) ) )
= ( ( C2
= ( zero_zero @ A ) )
| ( differentiable @ B @ A @ Q3 @ ( topolo174197925503356063within @ B @ T @ ( top_top @ ( set @ B ) ) ) ) ) ) ) ).
% differentiable_cmult_right_iff
thf(fact_7246_differentiable__cmult__left__iff,axiom,
! [A: $tType,B: $tType] :
( ( ( real_V822414075346904944vector @ B )
& ( real_V3459762299906320749_field @ A ) )
=> ! [C2: A,Q3: B > A,T: B] :
( ( differentiable @ B @ A
@ ^ [T2: B] : ( times_times @ A @ C2 @ ( Q3 @ T2 ) )
@ ( topolo174197925503356063within @ B @ T @ ( top_top @ ( set @ B ) ) ) )
= ( ( C2
= ( zero_zero @ A ) )
| ( differentiable @ B @ A @ Q3 @ ( topolo174197925503356063within @ B @ T @ ( top_top @ ( set @ B ) ) ) ) ) ) ) ).
% differentiable_cmult_left_iff
thf(fact_7247_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_7248_differentiable__add,axiom,
! [B: $tType,A: $tType] :
( ( ( real_V822414075346904944vector @ A )
& ( real_V822414075346904944vector @ B ) )
=> ! [F2: A > B,F3: filter @ A,G: A > B] :
( ( differentiable @ A @ B @ F2 @ F3 )
=> ( ( differentiable @ A @ B @ G @ F3 )
=> ( differentiable @ A @ B
@ ^ [X2: A] : ( plus_plus @ B @ ( F2 @ X2 ) @ ( G @ X2 ) )
@ F3 ) ) ) ) ).
% differentiable_add
thf(fact_7249_differentiable__power,axiom,
! [B: $tType,A: $tType] :
( ( ( real_V822414075346904944vector @ A )
& ( real_V3459762299906320749_field @ B ) )
=> ! [F2: A > B,X: A,S: set @ A,N: nat] :
( ( differentiable @ A @ B @ F2 @ ( topolo174197925503356063within @ A @ X @ S ) )
=> ( differentiable @ A @ B
@ ^ [X2: A] : ( power_power @ B @ ( F2 @ X2 ) @ N )
@ ( topolo174197925503356063within @ A @ X @ S ) ) ) ) ).
% differentiable_power
thf(fact_7250_obtain__pos__sum,axiom,
! [R2: rat] :
( ( ord_less @ rat @ ( zero_zero @ rat ) @ R2 )
=> ~ ! [S2: rat] :
( ( ord_less @ rat @ ( zero_zero @ rat ) @ S2 )
=> ! [T7: rat] :
( ( ord_less @ rat @ ( zero_zero @ rat ) @ T7 )
=> ( R2
!= ( plus_plus @ rat @ S2 @ T7 ) ) ) ) ) ).
% obtain_pos_sum
thf(fact_7251_abs__rat__def,axiom,
( ( abs_abs @ rat )
= ( ^ [A4: rat] : ( if @ rat @ ( ord_less @ rat @ A4 @ ( zero_zero @ rat ) ) @ ( uminus_uminus @ rat @ A4 ) @ A4 ) ) ) ).
% abs_rat_def
thf(fact_7252_sgn__rat__def,axiom,
( ( sgn_sgn @ rat )
= ( ^ [A4: rat] :
( if @ rat
@ ( A4
= ( zero_zero @ rat ) )
@ ( zero_zero @ rat )
@ ( if @ rat @ ( ord_less @ rat @ ( zero_zero @ rat ) @ A4 ) @ ( one_one @ rat ) @ ( uminus_uminus @ rat @ ( one_one @ rat ) ) ) ) ) ) ).
% sgn_rat_def
thf(fact_7253_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_7254_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_7255_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_7256_normalize__negative,axiom,
! [Q3: int,P4: int] :
( ( ord_less @ int @ Q3 @ ( zero_zero @ int ) )
=> ( ( normalize @ ( product_Pair @ int @ int @ P4 @ Q3 ) )
= ( normalize @ ( product_Pair @ int @ int @ ( uminus_uminus @ int @ P4 ) @ ( uminus_uminus @ int @ Q3 ) ) ) ) ) ).
% normalize_negative
thf(fact_7257_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_7258_less__eq__rat__def,axiom,
( ( ord_less_eq @ rat )
= ( ^ [X2: rat,Y6: rat] :
( ( ord_less @ rat @ X2 @ Y6 )
| ( X2 = Y6 ) ) ) ) ).
% less_eq_rat_def
thf(fact_7259_normalize__denom__pos,axiom,
! [R2: product_prod @ int @ int,P4: int,Q3: int] :
( ( ( normalize @ R2 )
= ( product_Pair @ int @ int @ P4 @ Q3 ) )
=> ( ord_less @ int @ ( zero_zero @ int ) @ Q3 ) ) ).
% normalize_denom_pos
thf(fact_7260_normalize__crossproduct,axiom,
! [Q3: int,S: int,P4: int,R2: int] :
( ( Q3
!= ( zero_zero @ int ) )
=> ( ( S
!= ( zero_zero @ int ) )
=> ( ( ( normalize @ ( product_Pair @ int @ int @ P4 @ Q3 ) )
= ( normalize @ ( product_Pair @ int @ int @ R2 @ S ) ) )
=> ( ( times_times @ int @ P4 @ S )
= ( times_times @ int @ R2 @ Q3 ) ) ) ) ) ).
% normalize_crossproduct
thf(fact_7261_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_7262_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_7263_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_7264_Frct__code__post_I6_J,axiom,
! [K: num,L: num] :
( ( frct @ ( product_Pair @ int @ int @ ( numeral_numeral @ int @ K ) @ ( numeral_numeral @ int @ L ) ) )
= ( divide_divide @ rat @ ( numeral_numeral @ rat @ K ) @ ( numeral_numeral @ rat @ L ) ) ) ).
% Frct_code_post(6)
thf(fact_7265_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_7266_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 ) ) ) ) ) )
=> ? [L4: real,Z3: real] :
( ( ord_less @ real @ A2 @ Z3 )
& ( ord_less @ real @ Z3 @ B2 )
& ( has_field_derivative @ real @ F2 @ L4 @ ( topolo174197925503356063within @ real @ Z3 @ ( top_top @ ( set @ real ) ) ) )
& ( ( minus_minus @ real @ ( F2 @ B2 ) @ ( F2 @ A2 ) )
= ( times_times @ real @ ( minus_minus @ real @ B2 @ A2 ) @ L4 ) ) ) ) ) ) ).
% MVT
thf(fact_7267_set__bits__int__def,axiom,
( ( bit_bi4170147762399595738t_bits @ int )
= ( ^ [F4: nat > $o] :
( if @ int
@ ? [N3: nat] :
! [M4: nat] :
( ( ord_less_eq @ nat @ N3 @ M4 )
=> ( ( F4 @ M4 )
= ( F4 @ N3 ) ) )
@ ( bit_ri4674362597316999326ke_bit @ int
@ ( ord_Least @ nat
@ ^ [N3: nat] :
! [M4: nat] :
( ( ord_less_eq @ nat @ N3 @ M4 )
=> ( ( F4 @ M4 )
= ( F4 @ N3 ) ) ) )
@ ( groups4207007520872428315er_sum @ $o @ int @ ( zero_neq_one_of_bool @ int ) @ ( numeral_numeral @ int @ ( bit0 @ one2 ) )
@ ( map @ nat @ $o @ F4
@ ( upt @ ( zero_zero @ nat )
@ ( suc
@ ( ord_Least @ nat
@ ^ [N3: nat] :
! [M4: nat] :
( ( ord_less_eq @ nat @ N3 @ M4 )
=> ( ( F4 @ M4 )
= ( F4 @ N3 ) ) ) ) ) ) ) ) )
@ ( zero_zero @ int ) ) ) ) ).
% set_bits_int_def
thf(fact_7268_Least__eq__0,axiom,
! [P: nat > $o] :
( ( P @ ( zero_zero @ nat ) )
=> ( ( ord_Least @ nat @ P )
= ( zero_zero @ nat ) ) ) ).
% Least_eq_0
thf(fact_7269_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_7270_continuous__on__sgn,axiom,
! [B: $tType,A: $tType] :
( ( ( topolo4958980785337419405_space @ A )
& ( real_V822414075346904944vector @ B ) )
=> ! [S: set @ A,F2: A > B] :
( ( topolo81223032696312382ous_on @ A @ B @ S @ F2 )
=> ( ! [X3: A] :
( ( member @ A @ X3 @ S )
=> ( ( F2 @ X3 )
!= ( zero_zero @ B ) ) )
=> ( topolo81223032696312382ous_on @ A @ B @ S
@ ^ [X2: A] : ( sgn_sgn @ B @ ( F2 @ X2 ) ) ) ) ) ) ).
% continuous_on_sgn
thf(fact_7271_open__subdiagonal,axiom,
! [A: $tType] :
( ( topolo1944317154257567458pology @ A )
=> ( topolo1002775350975398744n_open @ ( product_prod @ A @ A )
@ ( collect @ ( product_prod @ A @ A )
@ ^ [Uu3: product_prod @ A @ A] :
? [X2: A,Y6: A] :
( ( Uu3
= ( product_Pair @ A @ A @ X2 @ Y6 ) )
& ( ord_less @ A @ X2 @ Y6 ) ) ) ) ) ).
% open_subdiagonal
thf(fact_7272_open__superdiagonal,axiom,
! [A: $tType] :
( ( topolo1944317154257567458pology @ A )
=> ( topolo1002775350975398744n_open @ ( product_prod @ A @ A )
@ ( collect @ ( product_prod @ A @ A )
@ ^ [Uu3: product_prod @ A @ A] :
? [X2: A,Y6: A] :
( ( Uu3
= ( product_Pair @ A @ A @ X2 @ Y6 ) )
& ( ord_less @ A @ Y6 @ X2 ) ) ) ) ) ).
% open_superdiagonal
thf(fact_7273_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_7274_continuous__on__add,axiom,
! [B: $tType,D4: $tType] :
( ( ( topolo4958980785337419405_space @ D4 )
& ( topolo6943815403480290642id_add @ B ) )
=> ! [S: set @ D4,F2: D4 > B,G: D4 > B] :
( ( topolo81223032696312382ous_on @ D4 @ B @ S @ F2 )
=> ( ( topolo81223032696312382ous_on @ D4 @ B @ S @ G )
=> ( topolo81223032696312382ous_on @ D4 @ B @ S
@ ^ [X2: D4] : ( plus_plus @ B @ ( F2 @ X2 ) @ ( G @ X2 ) ) ) ) ) ) ).
% continuous_on_add
thf(fact_7275_continuous__on__power,axiom,
! [B: $tType,C: $tType] :
( ( ( topolo4958980785337419405_space @ C )
& ( power @ B )
& ( real_V4412858255891104859lgebra @ B ) )
=> ! [S: set @ C,F2: C > B,N: nat] :
( ( topolo81223032696312382ous_on @ C @ B @ S @ F2 )
=> ( topolo81223032696312382ous_on @ C @ B @ S
@ ^ [X2: C] : ( power_power @ B @ ( F2 @ X2 ) @ N ) ) ) ) ).
% continuous_on_power
thf(fact_7276_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_7277_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_7278_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_7279_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_7280_continuous__on__mult_H,axiom,
! [B: $tType,D4: $tType] :
( ( ( topolo4958980785337419405_space @ D4 )
& ( topolo4211221413907600880p_mult @ B ) )
=> ! [A3: set @ D4,F2: D4 > B,G: D4 > B] :
( ( topolo81223032696312382ous_on @ D4 @ B @ A3 @ F2 )
=> ( ( topolo81223032696312382ous_on @ D4 @ B @ A3 @ G )
=> ( topolo81223032696312382ous_on @ D4 @ B @ A3
@ ^ [X2: D4] : ( times_times @ B @ ( F2 @ X2 ) @ ( G @ X2 ) ) ) ) ) ) ).
% continuous_on_mult'
thf(fact_7281_continuous__on__mult,axiom,
! [A: $tType,D4: $tType] :
( ( ( topolo4958980785337419405_space @ D4 )
& ( real_V4412858255891104859lgebra @ A ) )
=> ! [S: set @ D4,F2: D4 > A,G: D4 > A] :
( ( topolo81223032696312382ous_on @ D4 @ A @ S @ F2 )
=> ( ( topolo81223032696312382ous_on @ D4 @ A @ S @ G )
=> ( topolo81223032696312382ous_on @ D4 @ A @ S
@ ^ [X2: D4] : ( times_times @ A @ ( F2 @ X2 ) @ ( G @ X2 ) ) ) ) ) ) ).
% continuous_on_mult
thf(fact_7282_continuous__on__real__root,axiom,
! [A: $tType] :
( ( topolo4958980785337419405_space @ A )
=> ! [S: set @ A,F2: A > real,N: nat] :
( ( topolo81223032696312382ous_on @ A @ real @ S @ F2 )
=> ( topolo81223032696312382ous_on @ A @ real @ S
@ ^ [X2: A] : ( root @ N @ ( F2 @ X2 ) ) ) ) ) ).
% continuous_on_real_root
thf(fact_7283_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_7284_Least__Suc,axiom,
! [P: nat > $o,N: nat] :
( ( P @ N )
=> ( ~ ( P @ ( zero_zero @ nat ) )
=> ( ( ord_Least @ nat @ P )
= ( suc
@ ( ord_Least @ nat
@ ^ [M4: nat] : ( P @ ( suc @ M4 ) ) ) ) ) ) ) ).
% Least_Suc
thf(fact_7285_continuous__on__powr,axiom,
! [C: $tType] :
( ( topolo4958980785337419405_space @ C )
=> ! [S: set @ C,F2: C > real,G: C > real] :
( ( topolo81223032696312382ous_on @ C @ real @ S @ F2 )
=> ( ( topolo81223032696312382ous_on @ C @ real @ S @ G )
=> ( ! [X3: C] :
( ( member @ C @ X3 @ S )
=> ( ( F2 @ X3 )
!= ( zero_zero @ real ) ) )
=> ( topolo81223032696312382ous_on @ C @ real @ S
@ ^ [X2: C] : ( powr @ real @ ( F2 @ X2 ) @ ( G @ X2 ) ) ) ) ) ) ) ).
% continuous_on_powr
thf(fact_7286_continuous__on__ln,axiom,
! [A: $tType] :
( ( topolo4958980785337419405_space @ A )
=> ! [S: set @ A,F2: A > real] :
( ( topolo81223032696312382ous_on @ A @ real @ S @ F2 )
=> ( ! [X3: A] :
( ( member @ A @ X3 @ S )
=> ( ( F2 @ X3 )
!= ( zero_zero @ real ) ) )
=> ( topolo81223032696312382ous_on @ A @ real @ S
@ ^ [X2: A] : ( ln_ln @ real @ ( F2 @ X2 ) ) ) ) ) ) ).
% continuous_on_ln
thf(fact_7287_Least__Suc2,axiom,
! [P: nat > $o,N: nat,Q: nat > $o,M: nat] :
( ( P @ N )
=> ( ( Q @ M )
=> ( ~ ( P @ ( zero_zero @ nat ) )
=> ( ! [K2: nat] :
( ( P @ ( suc @ K2 ) )
= ( Q @ K2 ) )
=> ( ( ord_Least @ nat @ P )
= ( suc @ ( ord_Least @ nat @ Q ) ) ) ) ) ) ) ).
% Least_Suc2
thf(fact_7288_continuous__on__inverse,axiom,
! [B: $tType,A: $tType] :
( ( ( topolo4958980785337419405_space @ A )
& ( real_V8999393235501362500lgebra @ B ) )
=> ! [S: set @ A,F2: A > B] :
( ( topolo81223032696312382ous_on @ A @ B @ S @ F2 )
=> ( ! [X3: A] :
( ( member @ A @ X3 @ S )
=> ( ( F2 @ X3 )
!= ( zero_zero @ B ) ) )
=> ( topolo81223032696312382ous_on @ A @ B @ S
@ ^ [X2: A] : ( inverse_inverse @ B @ ( F2 @ X2 ) ) ) ) ) ) ).
% continuous_on_inverse
thf(fact_7289_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_7290_continuous__on__tan,axiom,
! [A: $tType] :
( ( ( real_Vector_banach @ A )
& ( real_V3459762299906320749_field @ A ) )
=> ! [S: set @ A,F2: A > A] :
( ( topolo81223032696312382ous_on @ A @ A @ S @ F2 )
=> ( ! [X3: A] :
( ( member @ A @ X3 @ S )
=> ( ( cos @ A @ ( F2 @ X3 ) )
!= ( zero_zero @ A ) ) )
=> ( topolo81223032696312382ous_on @ A @ A @ S
@ ^ [X2: A] : ( tan @ A @ ( F2 @ X2 ) ) ) ) ) ) ).
% continuous_on_tan
thf(fact_7291_continuous__on__cot,axiom,
! [A: $tType] :
( ( ( real_Vector_banach @ A )
& ( real_V3459762299906320749_field @ A ) )
=> ! [S: set @ A,F2: A > A] :
( ( topolo81223032696312382ous_on @ A @ A @ S @ F2 )
=> ( ! [X3: A] :
( ( member @ A @ X3 @ S )
=> ( ( sin @ A @ ( F2 @ X3 ) )
!= ( zero_zero @ A ) ) )
=> ( topolo81223032696312382ous_on @ A @ A @ S
@ ^ [X2: A] : ( cot @ A @ ( F2 @ X2 ) ) ) ) ) ) ).
% continuous_on_cot
thf(fact_7292_continuous__on__tanh,axiom,
! [A: $tType,C: $tType] :
( ( ( topolo4958980785337419405_space @ C )
& ( real_Vector_banach @ A )
& ( real_V3459762299906320749_field @ A ) )
=> ! [A3: set @ C,F2: C > A] :
( ( topolo81223032696312382ous_on @ C @ A @ A3 @ F2 )
=> ( ! [X3: C] :
( ( member @ C @ X3 @ A3 )
=> ( ( cosh @ A @ ( F2 @ X3 ) )
!= ( zero_zero @ A ) ) )
=> ( topolo81223032696312382ous_on @ C @ A @ A3
@ ^ [X2: C] : ( tanh @ A @ ( F2 @ X2 ) ) ) ) ) ) ).
% continuous_on_tanh
thf(fact_7293_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_7294_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_7295_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_7296_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_7297_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_7298_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_7299_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_7300_set__conv__nth,axiom,
! [A: $tType] :
( ( set2 @ A )
= ( ^ [Xs: list @ A] :
( collect @ A
@ ^ [Uu3: A] :
? [I2: nat] :
( ( Uu3
= ( nth @ A @ Xs @ I2 ) )
& ( ord_less @ nat @ I2 @ ( size_size @ ( list @ A ) @ Xs ) ) ) ) ) ) ).
% set_conv_nth
thf(fact_7301_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_7302_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_7303_Rolle__deriv,axiom,
! [A2: real,B2: real,F2: real > real,F8: 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 @ ( F8 @ X3 ) @ ( topolo174197925503356063within @ real @ X3 @ ( top_top @ ( set @ real ) ) ) ) ) )
=> ? [Z3: real] :
( ( ord_less @ real @ A2 @ Z3 )
& ( ord_less @ real @ Z3 @ B2 )
& ( ( F8 @ Z3 )
= ( ^ [V5: real] : ( zero_zero @ real ) ) ) ) ) ) ) ) ).
% Rolle_deriv
thf(fact_7304_mvt,axiom,
! [A2: real,B2: real,F2: real > real,F8: 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 @ ( F8 @ 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 ) )
!= ( F8 @ Xi3 @ ( minus_minus @ real @ B2 @ A2 ) ) ) ) ) ) ) ) ).
% mvt
thf(fact_7305_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_7306_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_7307_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_7308_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_7309_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_7310_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_7311_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_7312_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 ) ) ) ) ) )
=> ? [Z3: real] :
( ( ord_less @ real @ A2 @ Z3 )
& ( ord_less @ real @ Z3 @ B2 )
& ( has_field_derivative @ real @ F2 @ ( zero_zero @ real ) @ ( topolo174197925503356063within @ real @ Z3 @ ( top_top @ ( set @ real ) ) ) ) ) ) ) ) ) ).
% Rolle
thf(fact_7313_VEBT__internal_Ovalid_H_Oelims_I1_J,axiom,
! [X: vEBT_VEBT,Xa: nat,Y: $o] :
( ( ( vEBT_VEBT_valid @ X @ Xa )
= Y )
=> ( ( ? [Uu2: $o,Uv2: $o] :
( X
= ( vEBT_Leaf @ Uu2 @ Uv2 ) )
=> ( Y
= ( Xa
!= ( one_one @ nat ) ) ) )
=> ~ ! [Mima: option @ ( product_prod @ nat @ nat ),Deg2: nat,TreeList2: list @ vEBT_VEBT,Summary2: vEBT_VEBT] :
( ( X
= ( vEBT_Node @ Mima @ Deg2 @ TreeList2 @ Summary2 ) )
=> ( Y
= ( ~ ( ( Deg2 = Xa )
& ! [X2: vEBT_VEBT] :
( ( member @ vEBT_VEBT @ X2 @ ( set2 @ vEBT_VEBT @ TreeList2 ) )
=> ( vEBT_VEBT_valid @ X2 @ ( divide_divide @ nat @ Deg2 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) )
& ( vEBT_VEBT_valid @ Summary2 @ ( minus_minus @ nat @ Deg2 @ ( divide_divide @ nat @ Deg2 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) )
& ( ( size_size @ ( list @ vEBT_VEBT ) @ TreeList2 )
= ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( minus_minus @ nat @ Deg2 @ ( divide_divide @ nat @ Deg2 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) )
& ( case_option @ $o @ ( product_prod @ nat @ nat )
@ ( ~ ? [X8: nat] : ( vEBT_V8194947554948674370ptions @ Summary2 @ X8 )
& ! [X2: vEBT_VEBT] :
( ( member @ vEBT_VEBT @ X2 @ ( set2 @ vEBT_VEBT @ TreeList2 ) )
=> ~ ? [X8: nat] : ( vEBT_V8194947554948674370ptions @ X2 @ X8 ) ) )
@ ( product_case_prod @ nat @ nat @ $o
@ ^ [Mi2: nat,Ma3: nat] :
( ( ord_less_eq @ nat @ Mi2 @ Ma3 )
& ( ord_less @ nat @ Ma3 @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ Deg2 ) )
& ! [I2: nat] :
( ( ord_less @ nat @ I2 @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( minus_minus @ nat @ Deg2 @ ( divide_divide @ nat @ Deg2 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) )
=> ( ( ? [X8: nat] : ( vEBT_V8194947554948674370ptions @ ( nth @ vEBT_VEBT @ TreeList2 @ I2 ) @ X8 ) )
= ( vEBT_V8194947554948674370ptions @ Summary2 @ I2 ) ) )
& ( ( Mi2 = Ma3 )
=> ! [X2: vEBT_VEBT] :
( ( member @ vEBT_VEBT @ X2 @ ( set2 @ vEBT_VEBT @ TreeList2 ) )
=> ~ ? [X8: nat] : ( vEBT_V8194947554948674370ptions @ X2 @ X8 ) ) )
& ( ( Mi2 != Ma3 )
=> ( ( vEBT_V5917875025757280293ildren @ ( divide_divide @ nat @ Deg2 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ TreeList2 @ Ma3 )
& ! [X2: nat] :
( ( ord_less @ nat @ X2 @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ Deg2 ) )
=> ( ( vEBT_V5917875025757280293ildren @ ( divide_divide @ nat @ Deg2 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ TreeList2 @ X2 )
=> ( ( ord_less @ nat @ Mi2 @ X2 )
& ( ord_less_eq @ nat @ X2 @ Ma3 ) ) ) ) ) ) ) )
@ Mima ) ) ) ) ) ) ) ).
% VEBT_internal.valid'.elims(1)
thf(fact_7314_VEBT__internal_Ovalid_H_Oelims_I2_J,axiom,
! [X: vEBT_VEBT,Xa: nat] :
( ( vEBT_VEBT_valid @ X @ Xa )
=> ( ( ? [Uu2: $o,Uv2: $o] :
( X
= ( vEBT_Leaf @ Uu2 @ Uv2 ) )
=> ( Xa
!= ( one_one @ nat ) ) )
=> ~ ! [Mima: option @ ( product_prod @ nat @ nat ),Deg2: nat,TreeList2: list @ vEBT_VEBT,Summary2: vEBT_VEBT] :
( ( X
= ( vEBT_Node @ Mima @ Deg2 @ TreeList2 @ Summary2 ) )
=> ~ ( ( Deg2 = Xa )
& ! [X5: vEBT_VEBT] :
( ( member @ vEBT_VEBT @ X5 @ ( set2 @ vEBT_VEBT @ TreeList2 ) )
=> ( vEBT_VEBT_valid @ X5 @ ( divide_divide @ nat @ Deg2 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) )
& ( vEBT_VEBT_valid @ Summary2 @ ( minus_minus @ nat @ Deg2 @ ( divide_divide @ nat @ Deg2 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) )
& ( ( size_size @ ( list @ vEBT_VEBT ) @ TreeList2 )
= ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( minus_minus @ nat @ Deg2 @ ( divide_divide @ nat @ Deg2 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) )
& ( case_option @ $o @ ( product_prod @ nat @ nat )
@ ( ~ ? [X8: nat] : ( vEBT_V8194947554948674370ptions @ Summary2 @ X8 )
& ! [X2: vEBT_VEBT] :
( ( member @ vEBT_VEBT @ X2 @ ( set2 @ vEBT_VEBT @ TreeList2 ) )
=> ~ ? [X8: nat] : ( vEBT_V8194947554948674370ptions @ X2 @ X8 ) ) )
@ ( product_case_prod @ nat @ nat @ $o
@ ^ [Mi2: nat,Ma3: nat] :
( ( ord_less_eq @ nat @ Mi2 @ Ma3 )
& ( ord_less @ nat @ Ma3 @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ Deg2 ) )
& ! [I2: nat] :
( ( ord_less @ nat @ I2 @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( minus_minus @ nat @ Deg2 @ ( divide_divide @ nat @ Deg2 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) )
=> ( ( ? [X8: nat] : ( vEBT_V8194947554948674370ptions @ ( nth @ vEBT_VEBT @ TreeList2 @ I2 ) @ X8 ) )
= ( vEBT_V8194947554948674370ptions @ Summary2 @ I2 ) ) )
& ( ( Mi2 = Ma3 )
=> ! [X2: vEBT_VEBT] :
( ( member @ vEBT_VEBT @ X2 @ ( set2 @ vEBT_VEBT @ TreeList2 ) )
=> ~ ? [X8: nat] : ( vEBT_V8194947554948674370ptions @ X2 @ X8 ) ) )
& ( ( Mi2 != Ma3 )
=> ( ( vEBT_V5917875025757280293ildren @ ( divide_divide @ nat @ Deg2 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ TreeList2 @ Ma3 )
& ! [X2: nat] :
( ( ord_less @ nat @ X2 @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ Deg2 ) )
=> ( ( vEBT_V5917875025757280293ildren @ ( divide_divide @ nat @ Deg2 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ TreeList2 @ X2 )
=> ( ( ord_less @ nat @ Mi2 @ X2 )
& ( ord_less_eq @ nat @ X2 @ Ma3 ) ) ) ) ) ) ) )
@ Mima ) ) ) ) ) ).
% VEBT_internal.valid'.elims(2)
thf(fact_7315_option_Odisc__eq__case_I1_J,axiom,
! [A: $tType,Option: option @ A] :
( ( Option
= ( none @ A ) )
= ( case_option @ $o @ A @ $true
@ ^ [Uu3: A] : $false
@ Option ) ) ).
% option.disc_eq_case(1)
thf(fact_7316_option_Odisc__eq__case_I2_J,axiom,
! [A: $tType,Option: option @ A] :
( ( Option
!= ( none @ A ) )
= ( case_option @ $o @ A @ $false
@ ^ [Uu3: A] : $true
@ Option ) ) ).
% option.disc_eq_case(2)
thf(fact_7317_option_Osimps_I4_J,axiom,
! [A: $tType,B: $tType,F12: B,F22: A > B] :
( ( case_option @ B @ A @ F12 @ F22 @ ( none @ A ) )
= F12 ) ).
% option.simps(4)
thf(fact_7318_case__optionE,axiom,
! [A: $tType,P: $o,Q: A > $o,X: option @ A] :
( ( case_option @ $o @ A @ P @ Q @ X )
=> ( ( ( X
= ( none @ A ) )
=> ~ P )
=> ~ ! [Y3: A] :
( ( X
= ( some @ A @ Y3 ) )
=> ~ ( Q @ Y3 ) ) ) ) ).
% case_optionE
thf(fact_7319_Ball__def__raw,axiom,
! [A: $tType] :
( ( ball @ A )
= ( ^ [A7: set @ A,P2: A > $o] :
! [X2: A] :
( ( member @ A @ X2 @ A7 )
=> ( P2 @ X2 ) ) ) ) ).
% Ball_def_raw
thf(fact_7320_option_Ocase__eq__if,axiom,
! [A: $tType,B: $tType] :
( ( case_option @ B @ A )
= ( ^ [F13: B,F23: A > B,Option3: option @ A] :
( if @ B
@ ( Option3
= ( none @ A ) )
@ F13
@ ( F23 @ ( the2 @ A @ Option3 ) ) ) ) ) ).
% option.case_eq_if
thf(fact_7321_less__option__def,axiom,
! [A: $tType] :
( ( preorder @ A )
=> ( ( ord_less @ ( option @ A ) )
= ( ^ [X2: option @ A] :
( case_option @ $o @ A @ $false
@ ^ [Y6: A] :
( case_option @ $o @ A @ $true
@ ^ [Z7: A] : ( ord_less @ A @ Z7 @ Y6 )
@ X2 ) ) ) ) ) ).
% less_option_def
thf(fact_7322_option_Osplit__sel__asm,axiom,
! [B: $tType,A: $tType,P: B > $o,F12: B,F22: A > B,Option: option @ A] :
( ( P @ ( case_option @ B @ A @ F12 @ F22 @ Option ) )
= ( ~ ( ( ( Option
= ( none @ A ) )
& ~ ( P @ F12 ) )
| ( ( Option
= ( some @ A @ ( the2 @ A @ Option ) ) )
& ~ ( P @ ( F22 @ ( the2 @ A @ Option ) ) ) ) ) ) ) ).
% option.split_sel_asm
thf(fact_7323_option_Osplit__sel,axiom,
! [B: $tType,A: $tType,P: B > $o,F12: B,F22: A > B,Option: option @ A] :
( ( P @ ( case_option @ B @ A @ F12 @ F22 @ Option ) )
= ( ( ( Option
= ( none @ A ) )
=> ( P @ F12 ) )
& ( ( Option
= ( some @ A @ ( the2 @ A @ Option ) ) )
=> ( P @ ( F22 @ ( the2 @ A @ Option ) ) ) ) ) ) ).
% option.split_sel
thf(fact_7324_Max_Oeq__fold_H,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ( ( lattic643756798349783984er_Max @ A )
= ( ^ [A7: set @ A] :
( the2 @ A
@ ( finite_fold @ A @ ( option @ A )
@ ^ [X2: A,Y6: option @ A] : ( some @ A @ ( case_option @ A @ A @ X2 @ ( ord_max @ A @ X2 ) @ Y6 ) )
@ ( none @ A )
@ A7 ) ) ) ) ) ).
% Max.eq_fold'
thf(fact_7325_VEBT__internal_Ovalid_H_Osimps_I2_J,axiom,
! [Mima2: option @ ( product_prod @ nat @ nat ),Deg: nat,TreeList: list @ vEBT_VEBT,Summary: vEBT_VEBT,Deg4: nat] :
( ( vEBT_VEBT_valid @ ( vEBT_Node @ Mima2 @ Deg @ TreeList @ Summary ) @ Deg4 )
= ( ( Deg = Deg4 )
& ! [X2: vEBT_VEBT] :
( ( member @ vEBT_VEBT @ X2 @ ( set2 @ vEBT_VEBT @ TreeList ) )
=> ( vEBT_VEBT_valid @ X2 @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) )
& ( vEBT_VEBT_valid @ Summary @ ( minus_minus @ nat @ Deg @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) )
& ( ( size_size @ ( list @ vEBT_VEBT ) @ TreeList )
= ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( minus_minus @ nat @ Deg @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) )
& ( case_option @ $o @ ( product_prod @ nat @ nat )
@ ( ~ ? [X8: nat] : ( vEBT_V8194947554948674370ptions @ Summary @ X8 )
& ! [X2: vEBT_VEBT] :
( ( member @ vEBT_VEBT @ X2 @ ( set2 @ vEBT_VEBT @ TreeList ) )
=> ~ ? [X8: nat] : ( vEBT_V8194947554948674370ptions @ X2 @ X8 ) ) )
@ ( product_case_prod @ nat @ nat @ $o
@ ^ [Mi2: nat,Ma3: nat] :
( ( ord_less_eq @ nat @ Mi2 @ Ma3 )
& ( ord_less @ nat @ Ma3 @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ Deg ) )
& ! [I2: nat] :
( ( ord_less @ nat @ I2 @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( minus_minus @ nat @ Deg @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) )
=> ( ( ? [X8: nat] : ( vEBT_V8194947554948674370ptions @ ( nth @ vEBT_VEBT @ TreeList @ I2 ) @ X8 ) )
= ( vEBT_V8194947554948674370ptions @ Summary @ I2 ) ) )
& ( ( Mi2 = Ma3 )
=> ! [X2: vEBT_VEBT] :
( ( member @ vEBT_VEBT @ X2 @ ( set2 @ vEBT_VEBT @ TreeList ) )
=> ~ ? [X8: nat] : ( vEBT_V8194947554948674370ptions @ X2 @ X8 ) ) )
& ( ( Mi2 != Ma3 )
=> ( ( vEBT_V5917875025757280293ildren @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ TreeList @ Ma3 )
& ! [X2: nat] :
( ( ord_less @ nat @ X2 @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ Deg ) )
=> ( ( vEBT_V5917875025757280293ildren @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ TreeList @ X2 )
=> ( ( ord_less @ nat @ Mi2 @ X2 )
& ( ord_less_eq @ nat @ X2 @ Ma3 ) ) ) ) ) ) ) )
@ Mima2 ) ) ) ).
% VEBT_internal.valid'.simps(2)
thf(fact_7326_VEBT__internal_Ovalid_H_Oelims_I3_J,axiom,
! [X: vEBT_VEBT,Xa: nat] :
( ~ ( vEBT_VEBT_valid @ X @ Xa )
=> ( ( ? [Uu2: $o,Uv2: $o] :
( X
= ( vEBT_Leaf @ Uu2 @ Uv2 ) )
=> ( Xa
= ( one_one @ nat ) ) )
=> ~ ! [Mima: option @ ( product_prod @ nat @ nat ),Deg2: nat,TreeList2: list @ vEBT_VEBT,Summary2: vEBT_VEBT] :
( ( X
= ( vEBT_Node @ Mima @ Deg2 @ TreeList2 @ Summary2 ) )
=> ( ( Deg2 = Xa )
& ! [X3: vEBT_VEBT] :
( ( member @ vEBT_VEBT @ X3 @ ( set2 @ vEBT_VEBT @ TreeList2 ) )
=> ( vEBT_VEBT_valid @ X3 @ ( divide_divide @ nat @ Deg2 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) )
& ( vEBT_VEBT_valid @ Summary2 @ ( minus_minus @ nat @ Deg2 @ ( divide_divide @ nat @ Deg2 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) )
& ( ( size_size @ ( list @ vEBT_VEBT ) @ TreeList2 )
= ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( minus_minus @ nat @ Deg2 @ ( divide_divide @ nat @ Deg2 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) )
& ( case_option @ $o @ ( product_prod @ nat @ nat )
@ ( ~ ? [X8: nat] : ( vEBT_V8194947554948674370ptions @ Summary2 @ X8 )
& ! [X2: vEBT_VEBT] :
( ( member @ vEBT_VEBT @ X2 @ ( set2 @ vEBT_VEBT @ TreeList2 ) )
=> ~ ? [X8: nat] : ( vEBT_V8194947554948674370ptions @ X2 @ X8 ) ) )
@ ( product_case_prod @ nat @ nat @ $o
@ ^ [Mi2: nat,Ma3: nat] :
( ( ord_less_eq @ nat @ Mi2 @ Ma3 )
& ( ord_less @ nat @ Ma3 @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ Deg2 ) )
& ! [I2: nat] :
( ( ord_less @ nat @ I2 @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( minus_minus @ nat @ Deg2 @ ( divide_divide @ nat @ Deg2 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) )
=> ( ( ? [X8: nat] : ( vEBT_V8194947554948674370ptions @ ( nth @ vEBT_VEBT @ TreeList2 @ I2 ) @ X8 ) )
= ( vEBT_V8194947554948674370ptions @ Summary2 @ I2 ) ) )
& ( ( Mi2 = Ma3 )
=> ! [X2: vEBT_VEBT] :
( ( member @ vEBT_VEBT @ X2 @ ( set2 @ vEBT_VEBT @ TreeList2 ) )
=> ~ ? [X8: nat] : ( vEBT_V8194947554948674370ptions @ X2 @ X8 ) ) )
& ( ( Mi2 != Ma3 )
=> ( ( vEBT_V5917875025757280293ildren @ ( divide_divide @ nat @ Deg2 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ TreeList2 @ Ma3 )
& ! [X2: nat] :
( ( ord_less @ nat @ X2 @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ Deg2 ) )
=> ( ( vEBT_V5917875025757280293ildren @ ( divide_divide @ nat @ Deg2 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ TreeList2 @ X2 )
=> ( ( ord_less @ nat @ Mi2 @ X2 )
& ( ord_less_eq @ nat @ X2 @ Ma3 ) ) ) ) ) ) ) )
@ Mima ) ) ) ) ) ).
% VEBT_internal.valid'.elims(3)
thf(fact_7327_VEBT__internal_Ovalid_H_Opelims_I1_J,axiom,
! [X: vEBT_VEBT,Xa: nat,Y: $o] :
( ( ( vEBT_VEBT_valid @ X @ Xa )
= Y )
=> ( ( accp @ ( product_prod @ vEBT_VEBT @ nat ) @ vEBT_VEBT_valid_rel @ ( product_Pair @ vEBT_VEBT @ nat @ X @ Xa ) )
=> ( ! [Uu2: $o,Uv2: $o] :
( ( X
= ( vEBT_Leaf @ Uu2 @ Uv2 ) )
=> ( ( Y
= ( Xa
= ( one_one @ nat ) ) )
=> ~ ( accp @ ( product_prod @ vEBT_VEBT @ nat ) @ vEBT_VEBT_valid_rel @ ( product_Pair @ vEBT_VEBT @ nat @ ( vEBT_Leaf @ Uu2 @ Uv2 ) @ Xa ) ) ) )
=> ~ ! [Mima: option @ ( product_prod @ nat @ nat ),Deg2: nat,TreeList2: list @ vEBT_VEBT,Summary2: vEBT_VEBT] :
( ( X
= ( vEBT_Node @ Mima @ Deg2 @ TreeList2 @ Summary2 ) )
=> ( ( Y
= ( ( Deg2 = Xa )
& ! [X2: vEBT_VEBT] :
( ( member @ vEBT_VEBT @ X2 @ ( set2 @ vEBT_VEBT @ TreeList2 ) )
=> ( vEBT_VEBT_valid @ X2 @ ( divide_divide @ nat @ Deg2 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) )
& ( vEBT_VEBT_valid @ Summary2 @ ( minus_minus @ nat @ Deg2 @ ( divide_divide @ nat @ Deg2 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) )
& ( ( size_size @ ( list @ vEBT_VEBT ) @ TreeList2 )
= ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( minus_minus @ nat @ Deg2 @ ( divide_divide @ nat @ Deg2 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) )
& ( case_option @ $o @ ( product_prod @ nat @ nat )
@ ( ~ ? [X8: nat] : ( vEBT_V8194947554948674370ptions @ Summary2 @ X8 )
& ! [X2: vEBT_VEBT] :
( ( member @ vEBT_VEBT @ X2 @ ( set2 @ vEBT_VEBT @ TreeList2 ) )
=> ~ ? [X8: nat] : ( vEBT_V8194947554948674370ptions @ X2 @ X8 ) ) )
@ ( product_case_prod @ nat @ nat @ $o
@ ^ [Mi2: nat,Ma3: nat] :
( ( ord_less_eq @ nat @ Mi2 @ Ma3 )
& ( ord_less @ nat @ Ma3 @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ Deg2 ) )
& ! [I2: nat] :
( ( ord_less @ nat @ I2 @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( minus_minus @ nat @ Deg2 @ ( divide_divide @ nat @ Deg2 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) )
=> ( ( ? [X8: nat] : ( vEBT_V8194947554948674370ptions @ ( nth @ vEBT_VEBT @ TreeList2 @ I2 ) @ X8 ) )
= ( vEBT_V8194947554948674370ptions @ Summary2 @ I2 ) ) )
& ( ( Mi2 = Ma3 )
=> ! [X2: vEBT_VEBT] :
( ( member @ vEBT_VEBT @ X2 @ ( set2 @ vEBT_VEBT @ TreeList2 ) )
=> ~ ? [X8: nat] : ( vEBT_V8194947554948674370ptions @ X2 @ X8 ) ) )
& ( ( Mi2 != Ma3 )
=> ( ( vEBT_V5917875025757280293ildren @ ( divide_divide @ nat @ Deg2 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ TreeList2 @ Ma3 )
& ! [X2: nat] :
( ( ord_less @ nat @ X2 @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ Deg2 ) )
=> ( ( vEBT_V5917875025757280293ildren @ ( divide_divide @ nat @ Deg2 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ TreeList2 @ X2 )
=> ( ( ord_less @ nat @ Mi2 @ X2 )
& ( ord_less_eq @ nat @ X2 @ Ma3 ) ) ) ) ) ) ) )
@ Mima ) ) )
=> ~ ( accp @ ( product_prod @ vEBT_VEBT @ nat ) @ vEBT_VEBT_valid_rel @ ( product_Pair @ vEBT_VEBT @ nat @ ( vEBT_Node @ Mima @ Deg2 @ TreeList2 @ Summary2 ) @ Xa ) ) ) ) ) ) ) ).
% VEBT_internal.valid'.pelims(1)
thf(fact_7328_VEBT__internal_Ovalid_H_Opelims_I2_J,axiom,
! [X: vEBT_VEBT,Xa: nat] :
( ( vEBT_VEBT_valid @ X @ Xa )
=> ( ( accp @ ( product_prod @ vEBT_VEBT @ nat ) @ vEBT_VEBT_valid_rel @ ( product_Pair @ vEBT_VEBT @ nat @ X @ Xa ) )
=> ( ! [Uu2: $o,Uv2: $o] :
( ( X
= ( vEBT_Leaf @ Uu2 @ Uv2 ) )
=> ( ( accp @ ( product_prod @ vEBT_VEBT @ nat ) @ vEBT_VEBT_valid_rel @ ( product_Pair @ vEBT_VEBT @ nat @ ( vEBT_Leaf @ Uu2 @ Uv2 ) @ Xa ) )
=> ( Xa
!= ( one_one @ nat ) ) ) )
=> ~ ! [Mima: option @ ( product_prod @ nat @ nat ),Deg2: nat,TreeList2: list @ vEBT_VEBT,Summary2: vEBT_VEBT] :
( ( X
= ( vEBT_Node @ Mima @ Deg2 @ TreeList2 @ Summary2 ) )
=> ( ( accp @ ( product_prod @ vEBT_VEBT @ nat ) @ vEBT_VEBT_valid_rel @ ( product_Pair @ vEBT_VEBT @ nat @ ( vEBT_Node @ Mima @ Deg2 @ TreeList2 @ Summary2 ) @ Xa ) )
=> ~ ( ( Deg2 = Xa )
& ! [X5: vEBT_VEBT] :
( ( member @ vEBT_VEBT @ X5 @ ( set2 @ vEBT_VEBT @ TreeList2 ) )
=> ( vEBT_VEBT_valid @ X5 @ ( divide_divide @ nat @ Deg2 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) )
& ( vEBT_VEBT_valid @ Summary2 @ ( minus_minus @ nat @ Deg2 @ ( divide_divide @ nat @ Deg2 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) )
& ( ( size_size @ ( list @ vEBT_VEBT ) @ TreeList2 )
= ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( minus_minus @ nat @ Deg2 @ ( divide_divide @ nat @ Deg2 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) )
& ( case_option @ $o @ ( product_prod @ nat @ nat )
@ ( ~ ? [X8: nat] : ( vEBT_V8194947554948674370ptions @ Summary2 @ X8 )
& ! [X2: vEBT_VEBT] :
( ( member @ vEBT_VEBT @ X2 @ ( set2 @ vEBT_VEBT @ TreeList2 ) )
=> ~ ? [X8: nat] : ( vEBT_V8194947554948674370ptions @ X2 @ X8 ) ) )
@ ( product_case_prod @ nat @ nat @ $o
@ ^ [Mi2: nat,Ma3: nat] :
( ( ord_less_eq @ nat @ Mi2 @ Ma3 )
& ( ord_less @ nat @ Ma3 @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ Deg2 ) )
& ! [I2: nat] :
( ( ord_less @ nat @ I2 @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( minus_minus @ nat @ Deg2 @ ( divide_divide @ nat @ Deg2 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) )
=> ( ( ? [X8: nat] : ( vEBT_V8194947554948674370ptions @ ( nth @ vEBT_VEBT @ TreeList2 @ I2 ) @ X8 ) )
= ( vEBT_V8194947554948674370ptions @ Summary2 @ I2 ) ) )
& ( ( Mi2 = Ma3 )
=> ! [X2: vEBT_VEBT] :
( ( member @ vEBT_VEBT @ X2 @ ( set2 @ vEBT_VEBT @ TreeList2 ) )
=> ~ ? [X8: nat] : ( vEBT_V8194947554948674370ptions @ X2 @ X8 ) ) )
& ( ( Mi2 != Ma3 )
=> ( ( vEBT_V5917875025757280293ildren @ ( divide_divide @ nat @ Deg2 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ TreeList2 @ Ma3 )
& ! [X2: nat] :
( ( ord_less @ nat @ X2 @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ Deg2 ) )
=> ( ( vEBT_V5917875025757280293ildren @ ( divide_divide @ nat @ Deg2 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ TreeList2 @ X2 )
=> ( ( ord_less @ nat @ Mi2 @ X2 )
& ( ord_less_eq @ nat @ X2 @ Ma3 ) ) ) ) ) ) ) )
@ Mima ) ) ) ) ) ) ) ).
% VEBT_internal.valid'.pelims(2)
thf(fact_7329_case__option__rule,axiom,
! [A: $tType,B: $tType,V: option @ A,P: assn,Fn: heap_Time_Heap @ B,Q: B > assn,Fs: A > ( heap_Time_Heap @ B )] :
( ( ( V
= ( none @ A ) )
=> ( hoare_hoare_triple @ B @ P @ Fn @ Q ) )
=> ( ! [X3: A] :
( ( V
= ( some @ A @ X3 ) )
=> ( hoare_hoare_triple @ B @ P @ ( Fs @ X3 ) @ Q ) )
=> ( hoare_hoare_triple @ B @ P @ ( case_option @ ( heap_Time_Heap @ B ) @ A @ Fn @ Fs @ V ) @ Q ) ) ) ).
% case_option_rule
thf(fact_7330_VEBT__internal_Ovalid_H_Opelims_I3_J,axiom,
! [X: vEBT_VEBT,Xa: nat] :
( ~ ( vEBT_VEBT_valid @ X @ Xa )
=> ( ( accp @ ( product_prod @ vEBT_VEBT @ nat ) @ vEBT_VEBT_valid_rel @ ( product_Pair @ vEBT_VEBT @ nat @ X @ Xa ) )
=> ( ! [Uu2: $o,Uv2: $o] :
( ( X
= ( vEBT_Leaf @ Uu2 @ Uv2 ) )
=> ( ( accp @ ( product_prod @ vEBT_VEBT @ nat ) @ vEBT_VEBT_valid_rel @ ( product_Pair @ vEBT_VEBT @ nat @ ( vEBT_Leaf @ Uu2 @ Uv2 ) @ Xa ) )
=> ( Xa
= ( one_one @ nat ) ) ) )
=> ~ ! [Mima: option @ ( product_prod @ nat @ nat ),Deg2: nat,TreeList2: list @ vEBT_VEBT,Summary2: vEBT_VEBT] :
( ( X
= ( vEBT_Node @ Mima @ Deg2 @ TreeList2 @ Summary2 ) )
=> ( ( accp @ ( product_prod @ vEBT_VEBT @ nat ) @ vEBT_VEBT_valid_rel @ ( product_Pair @ vEBT_VEBT @ nat @ ( vEBT_Node @ Mima @ Deg2 @ TreeList2 @ Summary2 ) @ Xa ) )
=> ( ( Deg2 = Xa )
& ! [X3: vEBT_VEBT] :
( ( member @ vEBT_VEBT @ X3 @ ( set2 @ vEBT_VEBT @ TreeList2 ) )
=> ( vEBT_VEBT_valid @ X3 @ ( divide_divide @ nat @ Deg2 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) )
& ( vEBT_VEBT_valid @ Summary2 @ ( minus_minus @ nat @ Deg2 @ ( divide_divide @ nat @ Deg2 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) )
& ( ( size_size @ ( list @ vEBT_VEBT ) @ TreeList2 )
= ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( minus_minus @ nat @ Deg2 @ ( divide_divide @ nat @ Deg2 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) )
& ( case_option @ $o @ ( product_prod @ nat @ nat )
@ ( ~ ? [X8: nat] : ( vEBT_V8194947554948674370ptions @ Summary2 @ X8 )
& ! [X2: vEBT_VEBT] :
( ( member @ vEBT_VEBT @ X2 @ ( set2 @ vEBT_VEBT @ TreeList2 ) )
=> ~ ? [X8: nat] : ( vEBT_V8194947554948674370ptions @ X2 @ X8 ) ) )
@ ( product_case_prod @ nat @ nat @ $o
@ ^ [Mi2: nat,Ma3: nat] :
( ( ord_less_eq @ nat @ Mi2 @ Ma3 )
& ( ord_less @ nat @ Ma3 @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ Deg2 ) )
& ! [I2: nat] :
( ( ord_less @ nat @ I2 @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( minus_minus @ nat @ Deg2 @ ( divide_divide @ nat @ Deg2 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) )
=> ( ( ? [X8: nat] : ( vEBT_V8194947554948674370ptions @ ( nth @ vEBT_VEBT @ TreeList2 @ I2 ) @ X8 ) )
= ( vEBT_V8194947554948674370ptions @ Summary2 @ I2 ) ) )
& ( ( Mi2 = Ma3 )
=> ! [X2: vEBT_VEBT] :
( ( member @ vEBT_VEBT @ X2 @ ( set2 @ vEBT_VEBT @ TreeList2 ) )
=> ~ ? [X8: nat] : ( vEBT_V8194947554948674370ptions @ X2 @ X8 ) ) )
& ( ( Mi2 != Ma3 )
=> ( ( vEBT_V5917875025757280293ildren @ ( divide_divide @ nat @ Deg2 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ TreeList2 @ Ma3 )
& ! [X2: nat] :
( ( ord_less @ nat @ X2 @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ Deg2 ) )
=> ( ( vEBT_V5917875025757280293ildren @ ( divide_divide @ nat @ Deg2 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ TreeList2 @ X2 )
=> ( ( ord_less @ nat @ Mi2 @ X2 )
& ( ord_less_eq @ nat @ X2 @ Ma3 ) ) ) ) ) ) ) )
@ Mima ) ) ) ) ) ) ) ).
% VEBT_internal.valid'.pelims(3)
thf(fact_7331_take__bit__numeral__minus__numeral__word,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [M: num,N: num] :
( ( bit_se2584673776208193580ke_bit @ ( word @ A ) @ ( numeral_numeral @ nat @ M ) @ ( uminus_uminus @ ( word @ A ) @ ( numeral_numeral @ ( word @ A ) @ N ) ) )
= ( case_option @ ( word @ A ) @ num @ ( zero_zero @ ( word @ A ) )
@ ^ [Q6: num] : ( bit_se2584673776208193580ke_bit @ ( word @ A ) @ ( numeral_numeral @ nat @ M ) @ ( minus_minus @ ( word @ A ) @ ( power_power @ ( word @ A ) @ ( numeral_numeral @ ( word @ A ) @ ( bit0 @ one2 ) ) @ ( numeral_numeral @ nat @ M ) ) @ ( numeral_numeral @ ( word @ A ) @ Q6 ) ) )
@ ( bit_take_bit_num @ ( numeral_numeral @ nat @ M ) @ N ) ) ) ) ).
% take_bit_numeral_minus_numeral_word
thf(fact_7332_take__bit__numeral__minus__numeral__int,axiom,
! [M: num,N: num] :
( ( bit_se2584673776208193580ke_bit @ int @ ( numeral_numeral @ nat @ M ) @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ N ) ) )
= ( case_option @ int @ num @ ( zero_zero @ int )
@ ^ [Q6: num] : ( bit_se2584673776208193580ke_bit @ int @ ( numeral_numeral @ nat @ M ) @ ( minus_minus @ int @ ( power_power @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ ( numeral_numeral @ nat @ M ) ) @ ( numeral_numeral @ int @ Q6 ) ) )
@ ( bit_take_bit_num @ ( numeral_numeral @ nat @ M ) @ N ) ) ) ).
% take_bit_numeral_minus_numeral_int
thf(fact_7333_take__bit__num__simps_I1_J,axiom,
! [M: num] :
( ( bit_take_bit_num @ ( zero_zero @ nat ) @ M )
= ( none @ num ) ) ).
% take_bit_num_simps(1)
thf(fact_7334_take__bit__num__simps_I2_J,axiom,
! [N: nat] :
( ( bit_take_bit_num @ ( suc @ N ) @ one2 )
= ( some @ num @ one2 ) ) ).
% take_bit_num_simps(2)
thf(fact_7335_take__bit__num__simps_I5_J,axiom,
! [R2: num] :
( ( bit_take_bit_num @ ( numeral_numeral @ nat @ R2 ) @ one2 )
= ( some @ num @ one2 ) ) ).
% take_bit_num_simps(5)
thf(fact_7336_take__bit__num__simps_I3_J,axiom,
! [N: nat,M: num] :
( ( bit_take_bit_num @ ( suc @ N ) @ ( bit0 @ M ) )
= ( case_option @ ( option @ num ) @ num @ ( none @ num )
@ ^ [Q6: num] : ( some @ num @ ( bit0 @ Q6 ) )
@ ( bit_take_bit_num @ N @ M ) ) ) ).
% take_bit_num_simps(3)
thf(fact_7337_take__bit__num__simps_I4_J,axiom,
! [N: nat,M: num] :
( ( bit_take_bit_num @ ( suc @ N ) @ ( bit1 @ M ) )
= ( some @ num @ ( case_option @ num @ num @ one2 @ bit1 @ ( bit_take_bit_num @ N @ M ) ) ) ) ).
% take_bit_num_simps(4)
thf(fact_7338_take__bit__num__simps_I6_J,axiom,
! [R2: num,M: num] :
( ( bit_take_bit_num @ ( numeral_numeral @ nat @ R2 ) @ ( bit0 @ M ) )
= ( case_option @ ( option @ num ) @ num @ ( none @ num )
@ ^ [Q6: num] : ( some @ num @ ( bit0 @ Q6 ) )
@ ( bit_take_bit_num @ ( pred_numeral @ R2 ) @ M ) ) ) ).
% take_bit_num_simps(6)
thf(fact_7339_take__bit__num__simps_I7_J,axiom,
! [R2: num,M: num] :
( ( bit_take_bit_num @ ( numeral_numeral @ nat @ R2 ) @ ( bit1 @ M ) )
= ( some @ num @ ( case_option @ num @ num @ one2 @ bit1 @ ( bit_take_bit_num @ ( pred_numeral @ R2 ) @ M ) ) ) ) ).
% take_bit_num_simps(7)
thf(fact_7340_take__bit__numeral__numeral,axiom,
! [A: $tType] :
( ( bit_se359711467146920520ations @ A )
=> ! [M: num,N: num] :
( ( bit_se2584673776208193580ke_bit @ A @ ( numeral_numeral @ nat @ M ) @ ( numeral_numeral @ A @ N ) )
= ( case_option @ A @ num @ ( zero_zero @ A ) @ ( numeral_numeral @ A ) @ ( bit_take_bit_num @ ( numeral_numeral @ nat @ M ) @ N ) ) ) ) ).
% take_bit_numeral_numeral
thf(fact_7341_take__bit__num__eq__Some__imp,axiom,
! [A: $tType] :
( ( bit_se359711467146920520ations @ A )
=> ! [M: nat,N: num,Q3: num] :
( ( ( bit_take_bit_num @ M @ N )
= ( some @ num @ Q3 ) )
=> ( ( bit_se2584673776208193580ke_bit @ A @ M @ ( numeral_numeral @ A @ N ) )
= ( numeral_numeral @ A @ Q3 ) ) ) ) ).
% take_bit_num_eq_Some_imp
thf(fact_7342_take__bit__num__eq__None__imp,axiom,
! [A: $tType] :
( ( bit_se359711467146920520ations @ A )
=> ! [M: nat,N: num] :
( ( ( bit_take_bit_num @ M @ N )
= ( none @ num ) )
=> ( ( bit_se2584673776208193580ke_bit @ A @ M @ ( numeral_numeral @ A @ N ) )
= ( zero_zero @ A ) ) ) ) ).
% take_bit_num_eq_None_imp
thf(fact_7343_and__minus__numerals_I7_J,axiom,
! [N: num,M: num] :
( ( bit_se5824344872417868541ns_and @ int @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ ( bit0 @ N ) ) ) @ ( numeral_numeral @ int @ M ) )
= ( case_option @ int @ num @ ( zero_zero @ int ) @ ( numeral_numeral @ int ) @ ( bit_and_not_num @ M @ ( bitM @ N ) ) ) ) ).
% and_minus_numerals(7)
thf(fact_7344_and__minus__numerals_I3_J,axiom,
! [M: num,N: num] :
( ( bit_se5824344872417868541ns_and @ int @ ( numeral_numeral @ int @ M ) @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ ( bit0 @ N ) ) ) )
= ( case_option @ int @ num @ ( zero_zero @ int ) @ ( numeral_numeral @ int ) @ ( bit_and_not_num @ M @ ( bitM @ N ) ) ) ) ).
% and_minus_numerals(3)
thf(fact_7345_and__minus__numerals_I8_J,axiom,
! [N: num,M: num] :
( ( bit_se5824344872417868541ns_and @ int @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ ( bit1 @ N ) ) ) @ ( numeral_numeral @ int @ M ) )
= ( case_option @ int @ num @ ( zero_zero @ int ) @ ( numeral_numeral @ int ) @ ( bit_and_not_num @ M @ ( bit0 @ N ) ) ) ) ).
% and_minus_numerals(8)
thf(fact_7346_and__minus__numerals_I4_J,axiom,
! [M: num,N: num] :
( ( bit_se5824344872417868541ns_and @ int @ ( numeral_numeral @ int @ M ) @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ ( bit1 @ N ) ) ) )
= ( case_option @ int @ num @ ( zero_zero @ int ) @ ( numeral_numeral @ int ) @ ( bit_and_not_num @ M @ ( bit0 @ N ) ) ) ) ).
% and_minus_numerals(4)
thf(fact_7347_and__not__num_Osimps_I1_J,axiom,
( ( bit_and_not_num @ one2 @ one2 )
= ( none @ num ) ) ).
% and_not_num.simps(1)
thf(fact_7348_and__not__num_Osimps_I4_J,axiom,
! [M: num] :
( ( bit_and_not_num @ ( bit0 @ M ) @ one2 )
= ( some @ num @ ( bit0 @ M ) ) ) ).
% and_not_num.simps(4)
thf(fact_7349_and__not__num_Osimps_I2_J,axiom,
! [N: num] :
( ( bit_and_not_num @ one2 @ ( bit0 @ N ) )
= ( some @ num @ one2 ) ) ).
% and_not_num.simps(2)
thf(fact_7350_and__not__num_Osimps_I3_J,axiom,
! [N: num] :
( ( bit_and_not_num @ one2 @ ( bit1 @ N ) )
= ( none @ num ) ) ).
% and_not_num.simps(3)
thf(fact_7351_and__not__num_Osimps_I7_J,axiom,
! [M: num] :
( ( bit_and_not_num @ ( bit1 @ M ) @ one2 )
= ( some @ num @ ( bit0 @ M ) ) ) ).
% and_not_num.simps(7)
thf(fact_7352_and__not__num__eq__Some__iff,axiom,
! [M: num,N: num,Q3: num] :
( ( ( bit_and_not_num @ M @ N )
= ( some @ num @ Q3 ) )
= ( ( bit_se5824344872417868541ns_and @ int @ ( numeral_numeral @ int @ M ) @ ( bit_ri4277139882892585799ns_not @ int @ ( numeral_numeral @ int @ N ) ) )
= ( numeral_numeral @ int @ Q3 ) ) ) ).
% and_not_num_eq_Some_iff
thf(fact_7353_and__not__num_Osimps_I8_J,axiom,
! [M: num,N: num] :
( ( bit_and_not_num @ ( bit1 @ M ) @ ( bit0 @ N ) )
= ( case_option @ ( option @ num ) @ num @ ( some @ num @ one2 )
@ ^ [N12: num] : ( some @ num @ ( bit1 @ N12 ) )
@ ( bit_and_not_num @ M @ N ) ) ) ).
% and_not_num.simps(8)
thf(fact_7354_and__not__num__eq__None__iff,axiom,
! [M: num,N: num] :
( ( ( bit_and_not_num @ M @ N )
= ( none @ num ) )
= ( ( bit_se5824344872417868541ns_and @ int @ ( numeral_numeral @ int @ M ) @ ( bit_ri4277139882892585799ns_not @ int @ ( numeral_numeral @ int @ N ) ) )
= ( zero_zero @ int ) ) ) ).
% and_not_num_eq_None_iff
thf(fact_7355_int__numeral__and__not__num,axiom,
! [M: num,N: num] :
( ( bit_se5824344872417868541ns_and @ int @ ( numeral_numeral @ int @ M ) @ ( bit_ri4277139882892585799ns_not @ int @ ( numeral_numeral @ int @ N ) ) )
= ( case_option @ int @ num @ ( zero_zero @ int ) @ ( numeral_numeral @ int ) @ ( bit_and_not_num @ M @ N ) ) ) ).
% int_numeral_and_not_num
thf(fact_7356_int__numeral__not__and__num,axiom,
! [M: num,N: num] :
( ( bit_se5824344872417868541ns_and @ int @ ( bit_ri4277139882892585799ns_not @ int @ ( numeral_numeral @ int @ M ) ) @ ( numeral_numeral @ int @ N ) )
= ( case_option @ int @ num @ ( zero_zero @ int ) @ ( numeral_numeral @ int ) @ ( bit_and_not_num @ N @ M ) ) ) ).
% int_numeral_not_and_num
thf(fact_7357_Code__Abstract__Nat_Otake__bit__num__code_I3_J,axiom,
! [N: nat,M: num] :
( ( bit_take_bit_num @ N @ ( bit1 @ M ) )
= ( case_nat @ ( option @ num ) @ ( none @ num )
@ ^ [N3: nat] : ( some @ num @ ( case_option @ num @ num @ one2 @ bit1 @ ( bit_take_bit_num @ N3 @ M ) ) )
@ N ) ) ).
% Code_Abstract_Nat.take_bit_num_code(3)
thf(fact_7358_Code__Abstract__Nat_Otake__bit__num__code_I2_J,axiom,
! [N: nat,M: num] :
( ( bit_take_bit_num @ N @ ( bit0 @ M ) )
= ( case_nat @ ( option @ num ) @ ( none @ num )
@ ^ [N3: nat] :
( case_option @ ( option @ num ) @ num @ ( none @ num )
@ ^ [Q6: num] : ( some @ num @ ( bit0 @ Q6 ) )
@ ( bit_take_bit_num @ N3 @ M ) )
@ N ) ) ).
% Code_Abstract_Nat.take_bit_num_code(2)
thf(fact_7359_case__nat__numeral,axiom,
! [A: $tType,A2: A,F2: nat > A,V: num] :
( ( case_nat @ A @ A2 @ F2 @ ( numeral_numeral @ nat @ V ) )
= ( F2 @ ( pred_numeral @ V ) ) ) ).
% case_nat_numeral
thf(fact_7360_case__nat__add__eq__if,axiom,
! [A: $tType,A2: A,F2: nat > A,V: num,N: nat] :
( ( case_nat @ A @ A2 @ F2 @ ( plus_plus @ nat @ ( numeral_numeral @ nat @ V ) @ N ) )
= ( F2 @ ( plus_plus @ nat @ ( pred_numeral @ V ) @ N ) ) ) ).
% case_nat_add_eq_if
thf(fact_7361_nat_Ocase__distrib,axiom,
! [B: $tType,A: $tType,H2: A > B,F12: A,F22: nat > A,Nat: nat] :
( ( H2 @ ( case_nat @ A @ F12 @ F22 @ Nat ) )
= ( case_nat @ B @ ( H2 @ F12 )
@ ^ [X2: nat] : ( H2 @ ( F22 @ X2 ) )
@ Nat ) ) ).
% nat.case_distrib
thf(fact_7362_old_Onat_Osimps_I4_J,axiom,
! [A: $tType,F12: A,F22: nat > A] :
( ( case_nat @ A @ F12 @ F22 @ ( zero_zero @ nat ) )
= F12 ) ).
% old.nat.simps(4)
thf(fact_7363_old_Onat_Osimps_I5_J,axiom,
! [A: $tType,F12: A,F22: nat > A,X22: nat] :
( ( case_nat @ A @ F12 @ F22 @ ( suc @ X22 ) )
= ( F22 @ X22 ) ) ).
% old.nat.simps(5)
thf(fact_7364_nth__Cons,axiom,
! [A: $tType,X: A,Xs2: list @ A,N: nat] :
( ( nth @ A @ ( cons @ A @ X @ Xs2 ) @ N )
= ( case_nat @ A @ X @ ( nth @ A @ Xs2 ) @ N ) ) ).
% nth_Cons
thf(fact_7365_Nitpick_Ocase__nat__unfold,axiom,
! [A: $tType] :
( ( case_nat @ A )
= ( ^ [X2: A,F4: nat > A,N3: nat] :
( if @ A
@ ( N3
= ( zero_zero @ nat ) )
@ X2
@ ( F4 @ ( minus_minus @ nat @ N3 @ ( one_one @ nat ) ) ) ) ) ) ).
% Nitpick.case_nat_unfold
thf(fact_7366_Code__Abstract__Nat_Otake__bit__num__code_I1_J,axiom,
! [N: nat] :
( ( bit_take_bit_num @ N @ one2 )
= ( case_nat @ ( option @ num ) @ ( none @ num )
@ ^ [N3: nat] : ( some @ num @ one2 )
@ N ) ) ).
% Code_Abstract_Nat.take_bit_num_code(1)
thf(fact_7367_take__bit__num__def,axiom,
( bit_take_bit_num
= ( ^ [N3: nat,M4: num] :
( if @ ( option @ num )
@ ( ( bit_se2584673776208193580ke_bit @ nat @ N3 @ ( numeral_numeral @ nat @ M4 ) )
= ( zero_zero @ nat ) )
@ ( none @ num )
@ ( some @ num @ ( num_of_nat @ ( bit_se2584673776208193580ke_bit @ nat @ N3 @ ( numeral_numeral @ nat @ M4 ) ) ) ) ) ) ) ).
% take_bit_num_def
thf(fact_7368_not__less__Least,axiom,
! [A: $tType] :
( ( wellorder @ A )
=> ! [K: A,P: A > $o] :
( ( ord_less @ A @ K @ ( ord_Least @ A @ P ) )
=> ~ ( P @ K ) ) ) ).
% not_less_Least
thf(fact_7369_num__of__nat__numeral__eq,axiom,
! [Q3: num] :
( ( num_of_nat @ ( numeral_numeral @ nat @ Q3 ) )
= Q3 ) ).
% num_of_nat_numeral_eq
thf(fact_7370_nat_Odisc__eq__case_I2_J,axiom,
! [Nat: nat] :
( ( Nat
!= ( zero_zero @ nat ) )
= ( case_nat @ $o @ $false
@ ^ [Uu3: nat] : $true
@ Nat ) ) ).
% nat.disc_eq_case(2)
thf(fact_7371_nat_Odisc__eq__case_I1_J,axiom,
! [Nat: nat] :
( ( Nat
= ( zero_zero @ nat ) )
= ( case_nat @ $o @ $true
@ ^ [Uu3: nat] : $false
@ Nat ) ) ).
% nat.disc_eq_case(1)
thf(fact_7372_num__of__nat_Osimps_I1_J,axiom,
( ( num_of_nat @ ( zero_zero @ nat ) )
= one2 ) ).
% num_of_nat.simps(1)
thf(fact_7373_lt__ex,axiom,
! [A: $tType] :
( ( no_bot @ A )
=> ! [X: A] :
? [Y3: A] : ( ord_less @ A @ Y3 @ X ) ) ).
% lt_ex
thf(fact_7374_gt__ex,axiom,
! [A: $tType] :
( ( no_top @ A )
=> ! [X: A] :
? [X_1: A] : ( ord_less @ A @ X @ X_1 ) ) ).
% gt_ex
thf(fact_7375_dense,axiom,
! [A: $tType] :
( ( dense_order @ A )
=> ! [X: A,Y: A] :
( ( ord_less @ A @ X @ Y )
=> ? [Z3: A] :
( ( ord_less @ A @ X @ Z3 )
& ( ord_less @ A @ Z3 @ Y ) ) ) ) ).
% dense
thf(fact_7376_less__imp__neq,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [X: A,Y: A] :
( ( ord_less @ A @ X @ Y )
=> ( X != Y ) ) ) ).
% less_imp_neq
thf(fact_7377_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_7378_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_7379_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_7380_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_7381_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_7382_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_7383_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_7384_dual__order_Oirrefl,axiom,
! [A: $tType] :
( ( preorder @ A )
=> ! [A2: A] :
~ ( ord_less @ A @ A2 @ A2 ) ) ).
% dual_order.irrefl
thf(fact_7385_exists__least__iff,axiom,
! [A: $tType] :
( ( wellorder @ A )
=> ( ( ^ [P6: A > $o] :
? [X9: A] : ( P6 @ X9 ) )
= ( ^ [P2: A > $o] :
? [N3: A] :
( ( P2 @ N3 )
& ! [M4: A] :
( ( ord_less @ A @ M4 @ N3 )
=> ~ ( P2 @ M4 ) ) ) ) ) ) ).
% exists_least_iff
thf(fact_7386_linorder__less__wlog,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [P: A > A > $o,A2: A,B2: A] :
( ! [A6: A,B3: A] :
( ( ord_less @ A @ A6 @ B3 )
=> ( P @ A6 @ B3 ) )
=> ( ! [A6: A] : ( P @ A6 @ A6 )
=> ( ! [A6: A,B3: A] :
( ( P @ B3 @ A6 )
=> ( P @ A6 @ B3 ) )
=> ( P @ A2 @ B2 ) ) ) ) ) ).
% linorder_less_wlog
thf(fact_7387_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_7388_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_7389_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_7390_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_7391_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_7392_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_7393_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_7394_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_7395_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_7396_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_7397_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_7398_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_7399_order__less__irrefl,axiom,
! [A: $tType] :
( ( preorder @ A )
=> ! [X: A] :
~ ( ord_less @ A @ X @ X ) ) ).
% order_less_irrefl
thf(fact_7400_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_7401_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_7402_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_7403_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_7404_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_7405_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_7406_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_7407_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_7408_less__eq__nat_Osimps_I2_J,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq @ nat @ ( suc @ M ) @ N )
= ( case_nat @ $o @ $false @ ( ord_less_eq @ nat @ M ) @ N ) ) ).
% less_eq_nat.simps(2)
thf(fact_7409_max__Suc1,axiom,
! [N: nat,M: nat] :
( ( ord_max @ nat @ ( suc @ N ) @ M )
= ( case_nat @ nat @ ( suc @ N )
@ ^ [M5: nat] : ( suc @ ( ord_max @ nat @ N @ M5 ) )
@ M ) ) ).
% max_Suc1
thf(fact_7410_max__Suc2,axiom,
! [M: nat,N: nat] :
( ( ord_max @ nat @ M @ ( suc @ N ) )
= ( case_nat @ nat @ ( suc @ N )
@ ^ [M5: nat] : ( suc @ ( ord_max @ nat @ M5 @ N ) )
@ M ) ) ).
% max_Suc2
thf(fact_7411_list__update_Osimps_I2_J,axiom,
! [A: $tType,X: A,Xs2: list @ A,I: nat,V: A] :
( ( list_update @ A @ ( cons @ A @ X @ Xs2 ) @ I @ V )
= ( case_nat @ ( list @ A ) @ ( cons @ A @ V @ Xs2 )
@ ^ [J2: nat] : ( cons @ A @ X @ ( list_update @ A @ Xs2 @ J2 @ V ) )
@ I ) ) ).
% list_update.simps(2)
thf(fact_7412_numeral__num__of__nat,axiom,
! [N: nat] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ N )
=> ( ( numeral_numeral @ nat @ ( num_of_nat @ N ) )
= N ) ) ).
% numeral_num_of_nat
thf(fact_7413_num__of__nat__One,axiom,
! [N: nat] :
( ( ord_less_eq @ nat @ N @ ( one_one @ nat ) )
=> ( ( num_of_nat @ N )
= one2 ) ) ).
% num_of_nat_One
thf(fact_7414_diff__Suc,axiom,
! [M: nat,N: nat] :
( ( minus_minus @ nat @ M @ ( suc @ N ) )
= ( case_nat @ nat @ ( zero_zero @ nat )
@ ^ [K3: nat] : K3
@ ( minus_minus @ nat @ M @ N ) ) ) ).
% diff_Suc
thf(fact_7415_bit__numeral__rec_I1_J,axiom,
! [A: $tType] :
( ( bit_un5681908812861735899ations @ A )
=> ! [W: num,N: nat] :
( ( bit_se5641148757651400278ts_bit @ A @ ( numeral_numeral @ A @ ( bit0 @ W ) ) @ N )
= ( case_nat @ $o @ $false @ ( bit_se5641148757651400278ts_bit @ A @ ( numeral_numeral @ A @ W ) ) @ N ) ) ) ).
% bit_numeral_rec(1)
thf(fact_7416_bit__numeral__rec_I2_J,axiom,
! [A: $tType] :
( ( bit_un5681908812861735899ations @ A )
=> ! [W: num,N: nat] :
( ( bit_se5641148757651400278ts_bit @ A @ ( numeral_numeral @ A @ ( bit1 @ W ) ) @ N )
= ( case_nat @ $o @ $true @ ( bit_se5641148757651400278ts_bit @ A @ ( numeral_numeral @ A @ W ) ) @ N ) ) ) ).
% bit_numeral_rec(2)
thf(fact_7417_min__Suc1,axiom,
! [N: nat,M: nat] :
( ( ord_min @ nat @ ( suc @ N ) @ M )
= ( case_nat @ nat @ ( zero_zero @ nat )
@ ^ [M5: nat] : ( suc @ ( ord_min @ nat @ N @ M5 ) )
@ M ) ) ).
% min_Suc1
thf(fact_7418_min__Suc2,axiom,
! [M: nat,N: nat] :
( ( ord_min @ nat @ M @ ( suc @ N ) )
= ( case_nat @ nat @ ( zero_zero @ nat )
@ ^ [M5: nat] : ( suc @ ( ord_min @ nat @ M5 @ N ) )
@ M ) ) ).
% min_Suc2
thf(fact_7419_numeral__num__of__nat__unfold,axiom,
! [A: $tType] :
( ( semiring_1 @ A )
=> ! [N: nat] :
( ( ( N
= ( zero_zero @ nat ) )
=> ( ( numeral_numeral @ A @ ( num_of_nat @ N ) )
= ( one_one @ A ) ) )
& ( ( N
!= ( zero_zero @ nat ) )
=> ( ( numeral_numeral @ A @ ( num_of_nat @ N ) )
= ( semiring_1_of_nat @ A @ N ) ) ) ) ) ).
% numeral_num_of_nat_unfold
thf(fact_7420_num__of__nat__double,axiom,
! [N: nat] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ N )
=> ( ( num_of_nat @ ( plus_plus @ nat @ N @ N ) )
= ( bit0 @ ( num_of_nat @ N ) ) ) ) ).
% num_of_nat_double
thf(fact_7421_num__of__nat__plus__distrib,axiom,
! [M: nat,N: nat] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ M )
=> ( ( ord_less @ nat @ ( zero_zero @ nat ) @ N )
=> ( ( num_of_nat @ ( plus_plus @ nat @ M @ N ) )
= ( plus_plus @ num @ ( num_of_nat @ M ) @ ( num_of_nat @ N ) ) ) ) ) ).
% num_of_nat_plus_distrib
thf(fact_7422_less__fun__def,axiom,
! [B: $tType,A: $tType] :
( ( ord @ B )
=> ( ( ord_less @ ( A > B ) )
= ( ^ [F4: A > B,G3: A > B] :
( ( ord_less_eq @ ( A > B ) @ F4 @ G3 )
& ~ ( ord_less_eq @ ( A > B ) @ G3 @ F4 ) ) ) ) ) ).
% less_fun_def
thf(fact_7423_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_7424_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_7425_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_7426_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_7427_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_7428_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_7429_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_7430_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_7431_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_7432_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_7433_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_7434_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_7435_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_7436_order__less__le,axiom,
! [A: $tType] :
( ( order @ A )
=> ( ( ord_less @ A )
= ( ^ [X2: A,Y6: A] :
( ( ord_less_eq @ A @ X2 @ Y6 )
& ( X2 != Y6 ) ) ) ) ) ).
% order_less_le
thf(fact_7437_order__le__less,axiom,
! [A: $tType] :
( ( order @ A )
=> ( ( ord_less_eq @ A )
= ( ^ [X2: A,Y6: A] :
( ( ord_less @ A @ X2 @ Y6 )
| ( X2 = Y6 ) ) ) ) ) ).
% order_le_less
thf(fact_7438_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_7439_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_7440_dual__order_Ostrict__iff__not,axiom,
! [A: $tType] :
( ( preorder @ A )
=> ( ( ord_less @ A )
= ( ^ [B5: A,A4: A] :
( ( ord_less_eq @ A @ B5 @ A4 )
& ~ ( ord_less_eq @ A @ A4 @ B5 ) ) ) ) ) ).
% dual_order.strict_iff_not
thf(fact_7441_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_7442_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_7443_dual__order_Ostrict__iff__order,axiom,
! [A: $tType] :
( ( order @ A )
=> ( ( ord_less @ A )
= ( ^ [B5: A,A4: A] :
( ( ord_less_eq @ A @ B5 @ A4 )
& ( A4 != B5 ) ) ) ) ) ).
% dual_order.strict_iff_order
thf(fact_7444_dual__order_Oorder__iff__strict,axiom,
! [A: $tType] :
( ( order @ A )
=> ( ( ord_less_eq @ A )
= ( ^ [B5: A,A4: A] :
( ( ord_less @ A @ B5 @ A4 )
| ( A4 = B5 ) ) ) ) ) ).
% dual_order.order_iff_strict
thf(fact_7445_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_7446_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_7447_order_Ostrict__iff__not,axiom,
! [A: $tType] :
( ( preorder @ A )
=> ( ( ord_less @ A )
= ( ^ [A4: A,B5: A] :
( ( ord_less_eq @ A @ A4 @ B5 )
& ~ ( ord_less_eq @ A @ B5 @ A4 ) ) ) ) ) ).
% order.strict_iff_not
thf(fact_7448_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_7449_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_7450_order_Ostrict__iff__order,axiom,
! [A: $tType] :
( ( order @ A )
=> ( ( ord_less @ A )
= ( ^ [A4: A,B5: A] :
( ( ord_less_eq @ A @ A4 @ B5 )
& ( A4 != B5 ) ) ) ) ) ).
% order.strict_iff_order
thf(fact_7451_order_Oorder__iff__strict,axiom,
! [A: $tType] :
( ( order @ A )
=> ( ( ord_less_eq @ A )
= ( ^ [A4: A,B5: A] :
( ( ord_less @ A @ A4 @ B5 )
| ( A4 = B5 ) ) ) ) ) ).
% order.order_iff_strict
thf(fact_7452_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_7453_less__le__not__le,axiom,
! [A: $tType] :
( ( preorder @ A )
=> ( ( ord_less @ A )
= ( ^ [X2: A,Y6: A] :
( ( ord_less_eq @ A @ X2 @ Y6 )
& ~ ( ord_less_eq @ A @ Y6 @ X2 ) ) ) ) ) ).
% less_le_not_le
thf(fact_7454_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_7455_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_7456_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_7457_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_7458_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_7459_leI,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [X: A,Y: A] :
( ~ ( ord_less @ A @ X @ Y )
=> ( ord_less_eq @ A @ Y @ X ) ) ) ).
% leI
thf(fact_7460_leD,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [Y: A,X: A] :
( ( ord_less_eq @ A @ Y @ X )
=> ~ ( ord_less @ A @ X @ Y ) ) ) ).
% leD
thf(fact_7461_bot_Oextremum__strict,axiom,
! [A: $tType] :
( ( order_bot @ A )
=> ! [A2: A] :
~ ( ord_less @ A @ A2 @ ( bot_bot @ A ) ) ) ).
% bot.extremum_strict
thf(fact_7462_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_7463_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_7464_top_Oextremum__strict,axiom,
! [A: $tType] :
( ( order_top @ A )
=> ! [A2: A] :
~ ( ord_less @ A @ ( top_top @ A ) @ A2 ) ) ).
% top.extremum_strict
thf(fact_7465_num__of__nat_Osimps_I2_J,axiom,
! [N: nat] :
( ( ( ord_less @ nat @ ( zero_zero @ nat ) @ N )
=> ( ( num_of_nat @ ( suc @ N ) )
= ( inc @ ( num_of_nat @ N ) ) ) )
& ( ~ ( ord_less @ nat @ ( zero_zero @ nat ) @ N )
=> ( ( num_of_nat @ ( suc @ N ) )
= one2 ) ) ) ).
% num_of_nat.simps(2)
thf(fact_7466_set__bits__int__unfold_H,axiom,
( ( bit_bi4170147762399595738t_bits @ int )
= ( ^ [F4: nat > $o] :
( if @ int
@ ? [N3: nat] :
! [N12: nat] :
( ( ord_less_eq @ nat @ N3 @ N12 )
=> ~ ( F4 @ N12 ) )
@ ( groups4207007520872428315er_sum @ $o @ int @ ( zero_neq_one_of_bool @ int ) @ ( numeral_numeral @ int @ ( bit0 @ one2 ) )
@ ( map @ nat @ $o @ F4
@ ( upt @ ( zero_zero @ nat )
@ ( ord_Least @ nat
@ ^ [N3: nat] :
! [N12: nat] :
( ( ord_less_eq @ nat @ N3 @ N12 )
=> ~ ( F4 @ N12 ) ) ) ) ) )
@ ( if @ int
@ ? [N3: nat] :
! [N12: nat] :
( ( ord_less_eq @ nat @ N3 @ N12 )
=> ( F4 @ N12 ) )
@ ( bit_ri4674362597316999326ke_bit @ int
@ ( ord_Least @ nat
@ ^ [N3: nat] :
! [N12: nat] :
( ( ord_less_eq @ nat @ N3 @ N12 )
=> ( F4 @ N12 ) ) )
@ ( groups4207007520872428315er_sum @ $o @ int @ ( zero_neq_one_of_bool @ int ) @ ( numeral_numeral @ int @ ( bit0 @ one2 ) )
@ ( append @ $o
@ ( map @ nat @ $o @ F4
@ ( upt @ ( zero_zero @ nat )
@ ( ord_Least @ nat
@ ^ [N3: nat] :
! [N12: nat] :
( ( ord_less_eq @ nat @ N3 @ N12 )
=> ( F4 @ N12 ) ) ) ) )
@ ( cons @ $o @ $true @ ( nil @ $o ) ) ) ) )
@ ( zero_zero @ int ) ) ) ) ) ).
% set_bits_int_unfold'
thf(fact_7467_Eps__Opt__def,axiom,
! [A: $tType] :
( ( eps_Opt @ A )
= ( ^ [P2: A > $o] :
( if @ ( option @ A )
@ ? [X8: A] : ( P2 @ X8 )
@ ( some @ A @ ( fChoice @ A @ P2 ) )
@ ( none @ A ) ) ) ) ).
% Eps_Opt_def
thf(fact_7468_append_Oassoc,axiom,
! [A: $tType,A2: list @ A,B2: list @ A,C2: list @ A] :
( ( append @ A @ ( append @ A @ A2 @ B2 ) @ C2 )
= ( append @ A @ A2 @ ( append @ A @ B2 @ C2 ) ) ) ).
% append.assoc
thf(fact_7469_append__assoc,axiom,
! [A: $tType,Xs2: list @ A,Ys: list @ A,Zs3: list @ A] :
( ( append @ A @ ( append @ A @ Xs2 @ Ys ) @ Zs3 )
= ( append @ A @ Xs2 @ ( append @ A @ Ys @ Zs3 ) ) ) ).
% append_assoc
thf(fact_7470_append__same__eq,axiom,
! [A: $tType,Ys: list @ A,Xs2: list @ A,Zs3: list @ A] :
( ( ( append @ A @ Ys @ Xs2 )
= ( append @ A @ Zs3 @ Xs2 ) )
= ( Ys = Zs3 ) ) ).
% append_same_eq
thf(fact_7471_same__append__eq,axiom,
! [A: $tType,Xs2: list @ A,Ys: list @ A,Zs3: list @ A] :
( ( ( append @ A @ Xs2 @ Ys )
= ( append @ A @ Xs2 @ Zs3 ) )
= ( Ys = Zs3 ) ) ).
% same_append_eq
thf(fact_7472_star__false__left,axiom,
! [P: assn] :
( ( times_times @ assn @ ( bot_bot @ assn ) @ P )
= ( bot_bot @ assn ) ) ).
% star_false_left
thf(fact_7473_star__false__right,axiom,
! [P: assn] :
( ( times_times @ assn @ P @ ( bot_bot @ assn ) )
= ( bot_bot @ assn ) ) ).
% star_false_right
thf(fact_7474_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_7475_pure__false,axiom,
( ( pure_assn @ $false )
= ( bot_bot @ assn ) ) ).
% pure_false
thf(fact_7476_pure__assn__eq__false__iff,axiom,
! [P: $o] :
( ( ( pure_assn @ P )
= ( bot_bot @ assn ) )
= ~ P ) ).
% pure_assn_eq_false_iff
thf(fact_7477_assn__basic__inequalities_I3_J,axiom,
( ( bot_bot @ assn )
!= ( one_one @ assn ) ) ).
% assn_basic_inequalities(3)
thf(fact_7478_assn__basic__inequalities_I5_J,axiom,
( ( top_top @ assn )
!= ( bot_bot @ assn ) ) ).
% assn_basic_inequalities(5)
thf(fact_7479_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_7480_list_Omap__disc__iff,axiom,
! [B: $tType,A: $tType,F2: A > B,A2: list @ A] :
( ( ( map @ A @ B @ F2 @ A2 )
= ( nil @ B ) )
= ( A2
= ( nil @ A ) ) ) ).
% list.map_disc_iff
thf(fact_7481_Nil__is__map__conv,axiom,
! [A: $tType,B: $tType,F2: B > A,Xs2: list @ B] :
( ( ( nil @ A )
= ( map @ B @ A @ F2 @ Xs2 ) )
= ( Xs2
= ( nil @ B ) ) ) ).
% Nil_is_map_conv
thf(fact_7482_map__is__Nil__conv,axiom,
! [A: $tType,B: $tType,F2: B > A,Xs2: list @ B] :
( ( ( map @ B @ A @ F2 @ Xs2 )
= ( nil @ A ) )
= ( Xs2
= ( nil @ B ) ) ) ).
% map_is_Nil_conv
thf(fact_7483_append_Oright__neutral,axiom,
! [A: $tType,A2: list @ A] :
( ( append @ A @ A2 @ ( nil @ A ) )
= A2 ) ).
% append.right_neutral
thf(fact_7484_append__Nil2,axiom,
! [A: $tType,Xs2: list @ A] :
( ( append @ A @ Xs2 @ ( nil @ A ) )
= Xs2 ) ).
% append_Nil2
thf(fact_7485_append__self__conv,axiom,
! [A: $tType,Xs2: list @ A,Ys: list @ A] :
( ( ( append @ A @ Xs2 @ Ys )
= Xs2 )
= ( Ys
= ( nil @ A ) ) ) ).
% append_self_conv
thf(fact_7486_self__append__conv,axiom,
! [A: $tType,Y: list @ A,Ys: list @ A] :
( ( Y
= ( append @ A @ Y @ Ys ) )
= ( Ys
= ( nil @ A ) ) ) ).
% self_append_conv
thf(fact_7487_append__self__conv2,axiom,
! [A: $tType,Xs2: list @ A,Ys: list @ A] :
( ( ( append @ A @ Xs2 @ Ys )
= Ys )
= ( Xs2
= ( nil @ A ) ) ) ).
% append_self_conv2
thf(fact_7488_self__append__conv2,axiom,
! [A: $tType,Y: list @ A,Xs2: list @ A] :
( ( Y
= ( append @ A @ Xs2 @ Y ) )
= ( Xs2
= ( nil @ A ) ) ) ).
% self_append_conv2
thf(fact_7489_Nil__is__append__conv,axiom,
! [A: $tType,Xs2: list @ A,Ys: list @ A] :
( ( ( nil @ A )
= ( append @ A @ Xs2 @ Ys ) )
= ( ( Xs2
= ( nil @ A ) )
& ( Ys
= ( nil @ A ) ) ) ) ).
% Nil_is_append_conv
thf(fact_7490_append__is__Nil__conv,axiom,
! [A: $tType,Xs2: list @ A,Ys: list @ A] :
( ( ( append @ A @ Xs2 @ Ys )
= ( nil @ A ) )
= ( ( Xs2
= ( nil @ A ) )
& ( Ys
= ( nil @ A ) ) ) ) ).
% append_is_Nil_conv
thf(fact_7491_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_7492_map__append,axiom,
! [A: $tType,B: $tType,F2: B > A,Xs2: list @ B,Ys: list @ B] :
( ( map @ B @ A @ F2 @ ( append @ B @ Xs2 @ Ys ) )
= ( append @ A @ ( map @ B @ A @ F2 @ Xs2 ) @ ( map @ B @ A @ F2 @ Ys ) ) ) ).
% map_append
thf(fact_7493_list__update__nonempty,axiom,
! [A: $tType,Xs2: list @ A,K: nat,X: A] :
( ( ( list_update @ A @ Xs2 @ K @ X )
= ( nil @ A ) )
= ( Xs2
= ( nil @ A ) ) ) ).
% list_update_nonempty
thf(fact_7494_concat__replicate__trivial,axiom,
! [A: $tType,I: nat] :
( ( concat @ A @ ( replicate @ ( list @ A ) @ I @ ( nil @ A ) ) )
= ( nil @ A ) ) ).
% concat_replicate_trivial
thf(fact_7495_foldr__append,axiom,
! [B: $tType,A: $tType,F2: B > A > A,Xs2: list @ B,Ys: list @ B,A2: A] :
( ( foldr @ B @ A @ F2 @ ( append @ B @ Xs2 @ Ys ) @ A2 )
= ( foldr @ B @ A @ F2 @ Xs2 @ ( foldr @ B @ A @ F2 @ Ys @ A2 ) ) ) ).
% foldr_append
thf(fact_7496_concat__append,axiom,
! [A: $tType,Xs2: list @ ( list @ A ),Ys: list @ ( list @ A )] :
( ( concat @ A @ ( append @ ( list @ A ) @ Xs2 @ Ys ) )
= ( append @ A @ ( concat @ A @ Xs2 ) @ ( concat @ A @ Ys ) ) ) ).
% concat_append
thf(fact_7497_ent__false__iff,axiom,
! [P: assn] :
( ( entails @ P @ ( bot_bot @ assn ) )
= ( ! [H: product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat )] :
~ ( rep_assn @ P @ H ) ) ) ).
% ent_false_iff
thf(fact_7498_snga__same__false,axiom,
! [A: $tType] :
( ( heap @ A )
=> ! [P4: array @ A,X: list @ A,Y: list @ A] :
( ( times_times @ assn @ ( snga_assn @ A @ P4 @ X ) @ ( snga_assn @ A @ P4 @ Y ) )
= ( bot_bot @ assn ) ) ) ).
% snga_same_false
thf(fact_7499_listI__assn__finite,axiom,
! [B: $tType,A: $tType,I6: set @ nat,A3: A > B > assn,Xs2: list @ A,Xsi: list @ B] :
( ~ ( finite_finite2 @ nat @ I6 )
=> ( ( vEBT_List_listI_assn @ A @ B @ I6 @ A3 @ Xs2 @ Xsi )
= ( bot_bot @ assn ) ) ) ).
% listI_assn_finite
thf(fact_7500_some__opt__false__trivial,axiom,
! [A: $tType] :
( ( eps_Opt @ A
@ ^ [Uu3: A] : $false )
= ( none @ A ) ) ).
% some_opt_false_trivial
thf(fact_7501_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_7502_set__empty2,axiom,
! [A: $tType,Xs2: list @ A] :
( ( ( bot_bot @ ( set @ A ) )
= ( set2 @ A @ Xs2 ) )
= ( Xs2
= ( nil @ A ) ) ) ).
% set_empty2
thf(fact_7503_set__empty,axiom,
! [A: $tType,Xs2: list @ A] :
( ( ( set2 @ A @ Xs2 )
= ( bot_bot @ ( set @ A ) ) )
= ( Xs2
= ( nil @ A ) ) ) ).
% set_empty
thf(fact_7504_append1__eq__conv,axiom,
! [A: $tType,Xs2: list @ A,X: A,Ys: list @ A,Y: A] :
( ( ( append @ A @ Xs2 @ ( cons @ A @ X @ ( nil @ A ) ) )
= ( append @ A @ Ys @ ( cons @ A @ Y @ ( nil @ A ) ) ) )
= ( ( Xs2 = Ys )
& ( X = Y ) ) ) ).
% append1_eq_conv
thf(fact_7505_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_7506_replicate__empty,axiom,
! [A: $tType,N: nat,X: A] :
( ( ( replicate @ A @ N @ X )
= ( nil @ A ) )
= ( N
= ( zero_zero @ nat ) ) ) ).
% replicate_empty
thf(fact_7507_empty__replicate,axiom,
! [A: $tType,N: nat,X: A] :
( ( ( nil @ A )
= ( replicate @ A @ N @ X ) )
= ( N
= ( zero_zero @ nat ) ) ) ).
% empty_replicate
thf(fact_7508_horner__sum__simps_I1_J,axiom,
! [B: $tType,A: $tType] :
( ( comm_semiring_0 @ A )
=> ! [F2: B > A,A2: A] :
( ( groups4207007520872428315er_sum @ B @ A @ F2 @ A2 @ ( nil @ B ) )
= ( zero_zero @ A ) ) ) ).
% horner_sum_simps(1)
thf(fact_7509_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_7510_list__assn__aux__simps_I2_J,axiom,
! [B: $tType,A: $tType,P: A > B > assn,L: list @ A] :
( ( vEBT_List_list_assn @ A @ B @ P @ L @ ( nil @ B ) )
= ( pure_assn
@ ( L
= ( nil @ A ) ) ) ) ).
% list_assn_aux_simps(2)
thf(fact_7511_list__assn__aux__simps_I1_J,axiom,
! [A: $tType,B: $tType,P: A > B > assn,L3: list @ B] :
( ( vEBT_List_list_assn @ A @ B @ P @ ( nil @ A ) @ L3 )
= ( pure_assn
@ ( L3
= ( nil @ B ) ) ) ) ).
% list_assn_aux_simps(1)
thf(fact_7512_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_7513_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_7514_Eps__Opt__eq__None,axiom,
! [A: $tType,P: A > $o] :
( ( ( eps_Opt @ A @ P )
= ( none @ A ) )
= ( ~ ? [X8: A] : ( P @ X8 ) ) ) ).
% Eps_Opt_eq_None
thf(fact_7515_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_7516_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_7517_nth__append__first,axiom,
! [A: $tType,I: nat,L: list @ A,L3: list @ A] :
( ( ord_less @ nat @ I @ ( size_size @ ( list @ A ) @ L ) )
=> ( ( nth @ A @ ( append @ A @ L @ L3 ) @ I )
= ( nth @ A @ L @ I ) ) ) ).
% nth_append_first
thf(fact_7518_nth__append__length__plus,axiom,
! [A: $tType,Xs2: list @ A,Ys: list @ A,N: nat] :
( ( nth @ A @ ( append @ A @ Xs2 @ Ys ) @ ( plus_plus @ nat @ ( size_size @ ( list @ A ) @ Xs2 ) @ N ) )
= ( nth @ A @ Ys @ N ) ) ).
% nth_append_length_plus
thf(fact_7519_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_7520_list__assn__aux__append,axiom,
! [A: $tType,B: $tType,L1: list @ A,L12: list @ B,P: A > B > assn,L22: list @ A,L23: list @ B] :
( ( ( size_size @ ( list @ A ) @ L1 )
= ( size_size @ ( list @ B ) @ L12 ) )
=> ( ( vEBT_List_list_assn @ A @ B @ P @ ( append @ A @ L1 @ L22 ) @ ( append @ B @ L12 @ L23 ) )
= ( times_times @ assn @ ( vEBT_List_list_assn @ A @ B @ P @ L1 @ L12 ) @ ( vEBT_List_list_assn @ A @ B @ P @ L22 @ L23 ) ) ) ) ).
% list_assn_aux_append
thf(fact_7521_list__assn__aux__append2,axiom,
! [A: $tType,B: $tType,L22: list @ A,L23: list @ B,P: A > B > assn,L1: list @ A,L12: list @ B] :
( ( ( size_size @ ( list @ A ) @ L22 )
= ( size_size @ ( list @ B ) @ L23 ) )
=> ( ( vEBT_List_list_assn @ A @ B @ P @ ( append @ A @ L1 @ L22 ) @ ( append @ B @ L12 @ L23 ) )
= ( times_times @ assn @ ( vEBT_List_list_assn @ A @ B @ P @ L1 @ L12 ) @ ( vEBT_List_list_assn @ A @ B @ P @ L22 @ L23 ) ) ) ) ).
% list_assn_aux_append2
thf(fact_7522_n__lists__Nil,axiom,
! [A: $tType,N: nat] :
( ( ( N
= ( zero_zero @ nat ) )
=> ( ( n_lists @ A @ N @ ( nil @ A ) )
= ( cons @ ( list @ A ) @ ( nil @ A ) @ ( nil @ ( list @ A ) ) ) ) )
& ( ( N
!= ( zero_zero @ nat ) )
=> ( ( n_lists @ A @ N @ ( nil @ A ) )
= ( nil @ ( list @ A ) ) ) ) ) ).
% n_lists_Nil
thf(fact_7523_length__ge__1__conv,axiom,
! [A: $tType,L: list @ A] :
( ( ord_less_eq @ nat @ ( suc @ ( zero_zero @ nat ) ) @ ( size_size @ ( list @ A ) @ L ) )
= ( L
!= ( nil @ A ) ) ) ).
% length_ge_1_conv
thf(fact_7524_concat__map__singleton,axiom,
! [A: $tType,B: $tType,F2: B > A,Xs2: list @ B] :
( ( concat @ A
@ ( map @ B @ ( list @ A )
@ ^ [X2: B] : ( cons @ A @ ( F2 @ X2 ) @ ( nil @ A ) )
@ Xs2 ) )
= ( map @ B @ A @ F2 @ Xs2 ) ) ).
% concat_map_singleton
thf(fact_7525_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_7526_bot__option__def,axiom,
! [A: $tType] :
( ( order @ A )
=> ( ( bot_bot @ ( option @ A ) )
= ( none @ A ) ) ) ).
% bot_option_def
thf(fact_7527_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_7528_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_7529_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 ) ) ) )
=> ( ! [A6: A,As: list @ A] :
( ( Xa
= ( cons @ A @ A6 @ As ) )
=> ! [C4: C,Cs2: list @ C] :
( ( Xb
= ( cons @ C @ C4 @ Cs2 ) )
=> ( Y
!= ( times_times @ assn @ ( X @ A6 @ C4 ) @ ( vEBT_List_list_assn @ A @ C @ X @ As @ Cs2 ) ) ) ) )
=> ( ( ? [V4: A,Va: list @ A] :
( Xa
= ( cons @ A @ V4 @ Va ) )
=> ( ( Xb
= ( nil @ C ) )
=> ( Y
!= ( bot_bot @ assn ) ) ) )
=> ~ ( ( Xa
= ( nil @ A ) )
=> ( ? [V4: C,Va: list @ C] :
( Xb
= ( cons @ C @ V4 @ Va ) )
=> ( Y
!= ( bot_bot @ assn ) ) ) ) ) ) ) ) ).
% list_assn.elims
thf(fact_7530_bot__nat__def,axiom,
( ( bot_bot @ nat )
= ( zero_zero @ nat ) ) ).
% bot_nat_def
thf(fact_7531_bot__enat__def,axiom,
( ( bot_bot @ extended_enat )
= ( zero_zero @ extended_enat ) ) ).
% bot_enat_def
thf(fact_7532_VEBT__internal_Oreplicatei_Osimps_I1_J,axiom,
! [A: $tType,X: heap_Time_Heap @ A] :
( ( vEBT_VEBT_replicatei @ A @ ( zero_zero @ nat ) @ X )
= ( heap_Time_return @ ( list @ A ) @ ( nil @ A ) ) ) ).
% VEBT_internal.replicatei.simps(1)
thf(fact_7533_list_Osize__gen_I1_J,axiom,
! [A: $tType,X: A > nat] :
( ( size_list @ A @ X @ ( nil @ A ) )
= ( zero_zero @ nat ) ) ).
% list.size_gen(1)
thf(fact_7534_replicate__0,axiom,
! [A: $tType,X: A] :
( ( replicate @ A @ ( zero_zero @ nat ) @ X )
= ( nil @ A ) ) ).
% replicate_0
thf(fact_7535_length__nth__simps_I1_J,axiom,
! [A: $tType] :
( ( size_size @ ( list @ A ) @ ( nil @ A ) )
= ( zero_zero @ nat ) ) ).
% length_nth_simps(1)
thf(fact_7536_n__lists_Osimps_I1_J,axiom,
! [A: $tType,Xs2: list @ A] :
( ( n_lists @ A @ ( zero_zero @ nat ) @ Xs2 )
= ( cons @ ( list @ A ) @ ( nil @ A ) @ ( nil @ ( list @ A ) ) ) ) ).
% n_lists.simps(1)
thf(fact_7537_count__list_Osimps_I1_J,axiom,
! [A: $tType,Y: A] :
( ( count_list @ A @ ( nil @ A ) @ Y )
= ( zero_zero @ nat ) ) ).
% count_list.simps(1)
thf(fact_7538_subseqs_Osimps_I1_J,axiom,
! [A: $tType] :
( ( subseqs @ A @ ( nil @ A ) )
= ( cons @ ( list @ A ) @ ( nil @ A ) @ ( nil @ ( list @ A ) ) ) ) ).
% subseqs.simps(1)
thf(fact_7539_concat__eq__append__conv,axiom,
! [A: $tType,Xss: list @ ( list @ A ),Ys: list @ A,Zs3: list @ A] :
( ( ( concat @ A @ Xss )
= ( append @ A @ Ys @ Zs3 ) )
= ( ( ( Xss
= ( nil @ ( list @ A ) ) )
=> ( ( Ys
= ( nil @ A ) )
& ( Zs3
= ( nil @ A ) ) ) )
& ( ( Xss
!= ( nil @ ( list @ A ) ) )
=> ? [Xss1: list @ ( list @ A ),Xs: list @ A,Xs5: list @ A,Xss2: list @ ( list @ A )] :
( ( Xss
= ( append @ ( list @ A ) @ Xss1 @ ( cons @ ( list @ A ) @ ( append @ A @ Xs @ Xs5 ) @ Xss2 ) ) )
& ( Ys
= ( append @ A @ ( concat @ A @ Xss1 ) @ Xs ) )
& ( Zs3
= ( append @ A @ Xs5 @ ( concat @ A @ Xss2 ) ) ) ) ) ) ) ).
% concat_eq_append_conv
thf(fact_7540_list__induct4,axiom,
! [C: $tType,A: $tType,B: $tType,D4: $tType,Xs2: list @ A,Ys: list @ B,Zs3: list @ C,Ws: list @ D4,P: ( list @ A ) > ( list @ B ) > ( list @ C ) > ( list @ D4 ) > $o] :
( ( ( size_size @ ( list @ A ) @ Xs2 )
= ( size_size @ ( list @ B ) @ Ys ) )
=> ( ( ( size_size @ ( list @ B ) @ Ys )
= ( size_size @ ( list @ C ) @ Zs3 ) )
=> ( ( ( size_size @ ( list @ C ) @ Zs3 )
= ( size_size @ ( list @ D4 ) @ Ws ) )
=> ( ( P @ ( nil @ A ) @ ( nil @ B ) @ ( nil @ C ) @ ( nil @ D4 ) )
=> ( ! [X3: A,Xs3: list @ A,Y3: B,Ys4: list @ B,Z3: C,Zs2: list @ C,W3: D4,Ws2: list @ D4] :
( ( ( 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 @ D4 ) @ Ws2 ) )
=> ( ( P @ Xs3 @ Ys4 @ Zs2 @ Ws2 )
=> ( P @ ( cons @ A @ X3 @ Xs3 ) @ ( cons @ B @ Y3 @ Ys4 ) @ ( cons @ C @ Z3 @ Zs2 ) @ ( cons @ D4 @ W3 @ Ws2 ) ) ) ) ) )
=> ( P @ Xs2 @ Ys @ Zs3 @ Ws ) ) ) ) ) ) ).
% list_induct4
thf(fact_7541_list__induct3,axiom,
! [B: $tType,A: $tType,C: $tType,Xs2: list @ A,Ys: list @ B,Zs3: 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 ) @ Zs3 ) )
=> ( ( P @ ( nil @ A ) @ ( nil @ B ) @ ( nil @ C ) )
=> ( ! [X3: A,Xs3: list @ A,Y3: B,Ys4: list @ B,Z3: 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 @ Z3 @ Zs2 ) ) ) ) )
=> ( P @ Xs2 @ Ys @ Zs3 ) ) ) ) ) ).
% list_induct3
thf(fact_7542_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_7543_replicate__app__Cons__same,axiom,
! [A: $tType,N: nat,X: A,Xs2: list @ A] :
( ( append @ A @ ( replicate @ A @ N @ X ) @ ( cons @ A @ X @ Xs2 ) )
= ( cons @ A @ X @ ( append @ A @ ( replicate @ A @ N @ X ) @ Xs2 ) ) ) ).
% replicate_app_Cons_same
thf(fact_7544_list__nonempty__induct,axiom,
! [A: $tType,Xs2: list @ A,P: ( list @ A ) > $o] :
( ( Xs2
!= ( nil @ A ) )
=> ( ! [X3: A] : ( P @ ( cons @ A @ X3 @ ( nil @ A ) ) )
=> ( ! [X3: A,Xs3: list @ A] :
( ( Xs3
!= ( nil @ A ) )
=> ( ( P @ Xs3 )
=> ( P @ ( cons @ A @ X3 @ Xs3 ) ) ) )
=> ( P @ Xs2 ) ) ) ) ).
% list_nonempty_induct
thf(fact_7545_rev__nonempty__induct,axiom,
! [A: $tType,Xs2: list @ A,P: ( list @ A ) > $o] :
( ( Xs2
!= ( nil @ A ) )
=> ( ! [X3: A] : ( P @ ( cons @ A @ X3 @ ( nil @ A ) ) )
=> ( ! [X3: A,Xs3: list @ A] :
( ( Xs3
!= ( nil @ A ) )
=> ( ( P @ Xs3 )
=> ( P @ ( append @ A @ Xs3 @ ( cons @ A @ X3 @ ( nil @ A ) ) ) ) ) )
=> ( P @ Xs2 ) ) ) ) ).
% rev_nonempty_induct
thf(fact_7546_append__eq__Cons__conv,axiom,
! [A: $tType,Ys: list @ A,Zs3: list @ A,X: A,Xs2: list @ A] :
( ( ( append @ A @ Ys @ Zs3 )
= ( cons @ A @ X @ Xs2 ) )
= ( ( ( Ys
= ( nil @ A ) )
& ( Zs3
= ( cons @ A @ X @ Xs2 ) ) )
| ? [Ys5: list @ A] :
( ( Ys
= ( cons @ A @ X @ Ys5 ) )
& ( ( append @ A @ Ys5 @ Zs3 )
= Xs2 ) ) ) ) ).
% append_eq_Cons_conv
thf(fact_7547_Cons__eq__append__conv,axiom,
! [A: $tType,X: A,Xs2: list @ A,Ys: list @ A,Zs3: list @ A] :
( ( ( cons @ A @ X @ Xs2 )
= ( append @ A @ Ys @ Zs3 ) )
= ( ( ( Ys
= ( nil @ A ) )
& ( ( cons @ A @ X @ Xs2 )
= Zs3 ) )
| ? [Ys5: list @ A] :
( ( ( cons @ A @ X @ Ys5 )
= Ys )
& ( Xs2
= ( append @ A @ Ys5 @ Zs3 ) ) ) ) ) ).
% Cons_eq_append_conv
thf(fact_7548_Cons__eq__appendI,axiom,
! [A: $tType,X: A,Xs1: list @ A,Ys: list @ A,Xs2: list @ A,Zs3: list @ A] :
( ( ( cons @ A @ X @ Xs1 )
= Ys )
=> ( ( Xs2
= ( append @ A @ Xs1 @ Zs3 ) )
=> ( ( cons @ A @ X @ Xs2 )
= ( append @ A @ Ys @ Zs3 ) ) ) ) ).
% Cons_eq_appendI
thf(fact_7549_list__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 @ ( cons @ A @ X3 @ Xs3 ) @ ( nil @ B ) )
=> ( ! [Y3: B,Ys4: list @ B] : ( P @ ( nil @ A ) @ ( cons @ B @ Y3 @ Ys4 ) )
=> ( ! [X3: A,Xs3: list @ A,Y3: B,Ys4: list @ B] :
( ( P @ Xs3 @ Ys4 )
=> ( P @ ( cons @ A @ X3 @ Xs3 ) @ ( cons @ B @ Y3 @ Ys4 ) ) )
=> ( P @ Xs2 @ Ys ) ) ) ) ) ).
% list_induct2'
thf(fact_7550_neq__Nil__conv,axiom,
! [A: $tType,Xs2: list @ A] :
( ( Xs2
!= ( nil @ A ) )
= ( ? [Y6: A,Ys3: list @ A] :
( Xs2
= ( cons @ A @ Y6 @ Ys3 ) ) ) ) ).
% neq_Nil_conv
thf(fact_7551_rev__exhaust,axiom,
! [A: $tType,Xs2: list @ A] :
( ( Xs2
!= ( nil @ A ) )
=> ~ ! [Ys4: list @ A,Y3: A] :
( Xs2
!= ( append @ A @ Ys4 @ ( cons @ A @ Y3 @ ( nil @ A ) ) ) ) ) ).
% rev_exhaust
thf(fact_7552_rev__induct,axiom,
! [A: $tType,P: ( list @ A ) > $o,Xs2: list @ A] :
( ( P @ ( nil @ A ) )
=> ( ! [X3: A,Xs3: list @ A] :
( ( P @ Xs3 )
=> ( P @ ( append @ A @ Xs3 @ ( cons @ A @ X3 @ ( nil @ A ) ) ) ) )
=> ( P @ Xs2 ) ) ) ).
% rev_induct
thf(fact_7553_append__Cons,axiom,
! [A: $tType,X: A,Xs2: list @ A,Ys: list @ A] :
( ( append @ A @ ( cons @ A @ X @ Xs2 ) @ Ys )
= ( cons @ A @ X @ ( append @ A @ Xs2 @ Ys ) ) ) ).
% append_Cons
thf(fact_7554_remdups__adj_Ocases,axiom,
! [A: $tType,X: list @ A] :
( ( X
!= ( nil @ A ) )
=> ( ! [X3: A] :
( X
!= ( cons @ A @ X3 @ ( nil @ A ) ) )
=> ~ ! [X3: A,Y3: A,Xs3: list @ A] :
( X
!= ( cons @ A @ X3 @ ( cons @ A @ Y3 @ Xs3 ) ) ) ) ) ).
% remdups_adj.cases
thf(fact_7555_transpose_Ocases,axiom,
! [A: $tType,X: list @ ( list @ A )] :
( ( X
!= ( nil @ ( list @ A ) ) )
=> ( ! [Xss3: list @ ( list @ A )] :
( X
!= ( cons @ ( list @ A ) @ ( nil @ A ) @ Xss3 ) )
=> ~ ! [X3: A,Xs3: list @ A,Xss3: list @ ( list @ A )] :
( X
!= ( cons @ ( list @ A ) @ ( cons @ A @ X3 @ Xs3 ) @ Xss3 ) ) ) ) ).
% transpose.cases
thf(fact_7556_min__list_Ocases,axiom,
! [A: $tType] :
( ( ord @ A )
=> ! [X: list @ A] :
( ! [X3: A,Xs3: list @ A] :
( X
!= ( cons @ A @ X3 @ Xs3 ) )
=> ( X
= ( nil @ A ) ) ) ) ).
% min_list.cases
thf(fact_7557_list_Oexhaust,axiom,
! [A: $tType,Y: list @ A] :
( ( Y
!= ( nil @ A ) )
=> ~ ! [X212: A,X223: list @ A] :
( Y
!= ( cons @ A @ X212 @ X223 ) ) ) ).
% list.exhaust
thf(fact_7558_list_OdiscI,axiom,
! [A: $tType,List: list @ A,X21: A,X222: list @ A] :
( ( List
= ( cons @ A @ X21 @ X222 ) )
=> ( List
!= ( nil @ A ) ) ) ).
% list.discI
thf(fact_7559_list_Odistinct_I1_J,axiom,
! [A: $tType,X21: A,X222: list @ A] :
( ( nil @ A )
!= ( cons @ A @ X21 @ X222 ) ) ).
% list.distinct(1)
thf(fact_7560_list__assn_Ocases,axiom,
! [A: $tType,C: $tType,X: product_prod @ ( A > C > assn ) @ ( product_prod @ ( list @ A ) @ ( list @ C ) )] :
( ! [P7: A > C > assn] :
( X
!= ( product_Pair @ ( A > C > assn ) @ ( product_prod @ ( list @ A ) @ ( list @ C ) ) @ P7 @ ( product_Pair @ ( list @ A ) @ ( list @ C ) @ ( nil @ A ) @ ( nil @ C ) ) ) )
=> ( ! [P7: A > C > assn,A6: A,As: list @ A,C4: C,Cs2: list @ C] :
( X
!= ( product_Pair @ ( A > C > assn ) @ ( product_prod @ ( list @ A ) @ ( list @ C ) ) @ P7 @ ( product_Pair @ ( list @ A ) @ ( list @ C ) @ ( cons @ A @ A6 @ As ) @ ( cons @ C @ C4 @ Cs2 ) ) ) )
=> ( ! [Uu2: A > C > assn,V4: A,Va: list @ A] :
( X
!= ( product_Pair @ ( A > C > assn ) @ ( product_prod @ ( list @ A ) @ ( list @ C ) ) @ Uu2 @ ( product_Pair @ ( list @ A ) @ ( list @ C ) @ ( cons @ A @ V4 @ Va ) @ ( nil @ C ) ) ) )
=> ~ ! [Uu2: A > C > assn,V4: C,Va: list @ C] :
( X
!= ( product_Pair @ ( A > C > assn ) @ ( product_prod @ ( list @ A ) @ ( list @ C ) ) @ Uu2 @ ( product_Pair @ ( list @ A ) @ ( list @ C ) @ ( nil @ A ) @ ( cons @ C @ V4 @ Va ) ) ) ) ) ) ) ).
% list_assn.cases
thf(fact_7561_map__tailrec__rev_Ocases,axiom,
! [A: $tType,B: $tType,X: product_prod @ ( A > B ) @ ( product_prod @ ( list @ A ) @ ( list @ B ) )] :
( ! [F5: A > B,Bs2: list @ B] :
( X
!= ( product_Pair @ ( A > B ) @ ( product_prod @ ( list @ A ) @ ( list @ B ) ) @ F5 @ ( product_Pair @ ( list @ A ) @ ( list @ B ) @ ( nil @ A ) @ Bs2 ) ) )
=> ~ ! [F5: A > B,A6: A,As: list @ A,Bs2: list @ B] :
( X
!= ( product_Pair @ ( A > B ) @ ( product_prod @ ( list @ A ) @ ( list @ B ) ) @ F5 @ ( product_Pair @ ( list @ A ) @ ( list @ B ) @ ( cons @ A @ A6 @ As ) @ Bs2 ) ) ) ) ).
% map_tailrec_rev.cases
thf(fact_7562_successively_Ocases,axiom,
! [A: $tType,X: product_prod @ ( A > A > $o ) @ ( list @ A )] :
( ! [P7: A > A > $o] :
( X
!= ( product_Pair @ ( A > A > $o ) @ ( list @ A ) @ P7 @ ( nil @ A ) ) )
=> ( ! [P7: A > A > $o,X3: A] :
( X
!= ( product_Pair @ ( A > A > $o ) @ ( list @ A ) @ P7 @ ( cons @ A @ X3 @ ( nil @ A ) ) ) )
=> ~ ! [P7: A > A > $o,X3: A,Y3: A,Xs3: list @ A] :
( X
!= ( product_Pair @ ( A > A > $o ) @ ( list @ A ) @ P7 @ ( cons @ A @ X3 @ ( cons @ A @ Y3 @ Xs3 ) ) ) ) ) ) ).
% successively.cases
thf(fact_7563_arg__min__list_Ocases,axiom,
! [B: $tType,A: $tType] :
( ( linorder @ B )
=> ! [X: product_prod @ ( A > B ) @ ( list @ A )] :
( ! [F5: A > B,X3: A] :
( X
!= ( product_Pair @ ( A > B ) @ ( list @ A ) @ F5 @ ( cons @ A @ X3 @ ( nil @ A ) ) ) )
=> ( ! [F5: A > B,X3: A,Y3: A,Zs2: list @ A] :
( X
!= ( product_Pair @ ( A > B ) @ ( list @ A ) @ F5 @ ( cons @ A @ X3 @ ( cons @ A @ Y3 @ Zs2 ) ) ) )
=> ~ ! [A6: A > B] :
( X
!= ( product_Pair @ ( A > B ) @ ( list @ A ) @ A6 @ ( nil @ A ) ) ) ) ) ) ).
% arg_min_list.cases
thf(fact_7564_sorted__wrt_Ocases,axiom,
! [A: $tType,X: product_prod @ ( A > A > $o ) @ ( list @ A )] :
( ! [P7: A > A > $o] :
( X
!= ( product_Pair @ ( A > A > $o ) @ ( list @ A ) @ P7 @ ( nil @ A ) ) )
=> ~ ! [P7: A > A > $o,X3: A,Ys4: list @ A] :
( X
!= ( product_Pair @ ( A > A > $o ) @ ( list @ A ) @ P7 @ ( cons @ A @ X3 @ Ys4 ) ) ) ) ).
% sorted_wrt.cases
thf(fact_7565_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_7566_replicate__Suc__conv__snoc,axiom,
! [A: $tType,N: nat,X: A] :
( ( replicate @ A @ ( suc @ N ) @ X )
= ( append @ A @ ( replicate @ A @ N @ X ) @ ( cons @ A @ X @ ( nil @ A ) ) ) ) ).
% replicate_Suc_conv_snoc
thf(fact_7567_replicate__append__same,axiom,
! [A: $tType,I: nat,X: A] :
( ( append @ A @ ( replicate @ A @ I @ X ) @ ( cons @ A @ X @ ( nil @ A ) ) )
= ( cons @ A @ X @ ( replicate @ A @ I @ X ) ) ) ).
% replicate_append_same
thf(fact_7568_product__lists_Osimps_I1_J,axiom,
! [A: $tType] :
( ( product_lists @ A @ ( nil @ ( list @ A ) ) )
= ( cons @ ( list @ A ) @ ( nil @ A ) @ ( nil @ ( list @ A ) ) ) ) ).
% product_lists.simps(1)
thf(fact_7569_List_Omap__eq__append__conv,axiom,
! [A: $tType,B: $tType,F2: B > A,Xs2: list @ B,Ys: list @ A,Zs3: list @ A] :
( ( ( map @ B @ A @ F2 @ Xs2 )
= ( append @ A @ Ys @ Zs3 ) )
= ( ? [Us2: list @ B,Vs2: list @ B] :
( ( Xs2
= ( append @ B @ Us2 @ Vs2 ) )
& ( Ys
= ( map @ B @ A @ F2 @ Us2 ) )
& ( Zs3
= ( map @ B @ A @ F2 @ Vs2 ) ) ) ) ) ).
% List.map_eq_append_conv
thf(fact_7570_List_Oappend__eq__map__conv,axiom,
! [A: $tType,B: $tType,Ys: list @ A,Zs3: list @ A,F2: B > A,Xs2: list @ B] :
( ( ( append @ A @ Ys @ Zs3 )
= ( map @ B @ A @ F2 @ Xs2 ) )
= ( ? [Us2: list @ B,Vs2: list @ B] :
( ( Xs2
= ( append @ B @ Us2 @ Vs2 ) )
& ( Ys
= ( map @ B @ A @ F2 @ Us2 ) )
& ( Zs3
= ( map @ B @ A @ F2 @ Vs2 ) ) ) ) ) ).
% List.append_eq_map_conv
thf(fact_7571_list_Osimps_I8_J,axiom,
! [A: $tType,B: $tType,F2: A > B] :
( ( map @ A @ B @ F2 @ ( nil @ A ) )
= ( nil @ B ) ) ).
% list.simps(8)
thf(fact_7572_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_7573_length__Suc__rev__conv,axiom,
! [A: $tType,Xs2: list @ A,N: nat] :
( ( ( size_size @ ( list @ A ) @ Xs2 )
= ( suc @ N ) )
= ( ? [Ys3: list @ A,Y6: A] :
( ( Xs2
= ( append @ A @ Ys3 @ ( cons @ A @ Y6 @ ( nil @ A ) ) ) )
& ( ( size_size @ ( list @ A ) @ Ys3 )
= N ) ) ) ) ).
% length_Suc_rev_conv
thf(fact_7574_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,Xs6: list @ A,Y3: A,Ys6: list @ A] :
( ( X3 != Y3 )
& ( Xs2
= ( append @ A @ Pre @ ( append @ A @ ( cons @ A @ X3 @ ( nil @ A ) ) @ Xs6 ) ) )
& ( Ys
= ( append @ A @ Pre @ ( append @ A @ ( cons @ A @ Y3 @ ( nil @ A ) ) @ Ys6 ) ) ) ) ) ) ).
% same_length_different
thf(fact_7575_length__Suc__conv__rev,axiom,
! [A: $tType,Xs2: list @ A,N: nat] :
( ( ( size_size @ ( list @ A ) @ Xs2 )
= ( suc @ N ) )
= ( ? [Y6: A,Ys3: list @ A] :
( ( Xs2
= ( append @ A @ Ys3 @ ( cons @ A @ Y6 @ ( nil @ A ) ) ) )
& ( ( size_size @ ( list @ A ) @ Ys3 )
= N ) ) ) ) ).
% length_Suc_conv_rev
thf(fact_7576_concat_Osimps_I2_J,axiom,
! [A: $tType,X: list @ A,Xs2: list @ ( list @ A )] :
( ( concat @ A @ ( cons @ ( list @ A ) @ X @ Xs2 ) )
= ( append @ A @ X @ ( concat @ A @ Xs2 ) ) ) ).
% concat.simps(2)
thf(fact_7577_append__replicate__commute,axiom,
! [A: $tType,N: nat,X: A,K: nat] :
( ( append @ A @ ( replicate @ A @ N @ X ) @ ( replicate @ A @ K @ X ) )
= ( append @ A @ ( replicate @ A @ K @ X ) @ ( replicate @ A @ N @ X ) ) ) ).
% append_replicate_commute
thf(fact_7578_product_Osimps_I1_J,axiom,
! [B: $tType,A: $tType,Uu: list @ B] :
( ( product @ A @ B @ ( nil @ A ) @ Uu )
= ( nil @ ( product_prod @ A @ B ) ) ) ).
% product.simps(1)
thf(fact_7579_append__Nil,axiom,
! [A: $tType,Ys: list @ A] :
( ( append @ A @ ( nil @ A ) @ Ys )
= Ys ) ).
% append_Nil
thf(fact_7580_append_Oleft__neutral,axiom,
! [A: $tType,A2: list @ A] :
( ( append @ A @ ( nil @ A ) @ A2 )
= A2 ) ).
% append.left_neutral
thf(fact_7581_eq__Nil__appendI,axiom,
! [A: $tType,Xs2: list @ A,Ys: list @ A] :
( ( Xs2 = Ys )
=> ( Xs2
= ( append @ A @ ( nil @ A ) @ Ys ) ) ) ).
% eq_Nil_appendI
thf(fact_7582_append__eq__appendI,axiom,
! [A: $tType,Xs2: list @ A,Xs1: list @ A,Zs3: list @ A,Ys: list @ A,Us: list @ A] :
( ( ( append @ A @ Xs2 @ Xs1 )
= Zs3 )
=> ( ( Ys
= ( append @ A @ Xs1 @ Us ) )
=> ( ( append @ A @ Xs2 @ Ys )
= ( append @ A @ Zs3 @ Us ) ) ) ) ).
% append_eq_appendI
thf(fact_7583_append__eq__append__conv2,axiom,
! [A: $tType,Xs2: list @ A,Ys: list @ A,Zs3: list @ A,Ts: list @ A] :
( ( ( append @ A @ Xs2 @ Ys )
= ( append @ A @ Zs3 @ Ts ) )
= ( ? [Us2: list @ A] :
( ( ( Xs2
= ( append @ A @ Zs3 @ Us2 ) )
& ( ( append @ A @ Us2 @ Ys )
= Ts ) )
| ( ( ( append @ A @ Xs2 @ Us2 )
= Zs3 )
& ( Ys
= ( append @ A @ Us2 @ Ts ) ) ) ) ) ) ).
% append_eq_append_conv2
thf(fact_7584_list__update__code_I1_J,axiom,
! [A: $tType,I: nat,Y: A] :
( ( list_update @ A @ ( nil @ A ) @ I @ Y )
= ( nil @ A ) ) ).
% list_update_code(1)
thf(fact_7585_list__update_Osimps_I1_J,axiom,
! [A: $tType,I: nat,V: A] :
( ( list_update @ A @ ( nil @ A ) @ I @ V )
= ( nil @ A ) ) ).
% list_update.simps(1)
thf(fact_7586_comm__append__are__replicate,axiom,
! [A: $tType,Xs2: list @ A,Ys: list @ A] :
( ( ( append @ A @ Xs2 @ Ys )
= ( append @ A @ Ys @ Xs2 ) )
=> ? [M2: nat,N2: nat,Zs2: list @ A] :
( ( ( concat @ A @ ( replicate @ ( list @ A ) @ M2 @ Zs2 ) )
= Xs2 )
& ( ( concat @ A @ ( replicate @ ( list @ A ) @ N2 @ Zs2 ) )
= Ys ) ) ) ).
% comm_append_are_replicate
thf(fact_7587_concat__conv__foldr,axiom,
! [A: $tType] :
( ( concat @ A )
= ( ^ [Xss4: list @ ( list @ A )] : ( foldr @ ( list @ A ) @ ( list @ A ) @ ( append @ A ) @ Xss4 @ ( nil @ A ) ) ) ) ).
% concat_conv_foldr
thf(fact_7588_concat_Osimps_I1_J,axiom,
! [A: $tType] :
( ( concat @ A @ ( nil @ ( list @ A ) ) )
= ( nil @ A ) ) ).
% concat.simps(1)
thf(fact_7589_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_7590_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_7591_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_7592_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_7593_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_7594_append__Cons__eq__iff,axiom,
! [A: $tType,X: A,Xs2: list @ A,Ys: list @ A,Xs4: list @ A,Ys7: 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 @ Ys7 ) ) )
= ( ( Xs2 = Xs4 )
& ( Ys = Ys7 ) ) ) ) ) ).
% append_Cons_eq_iff
thf(fact_7595_in__set__conv__decomp,axiom,
! [A: $tType,X: A,Xs2: list @ A] :
( ( member @ A @ X @ ( set2 @ A @ Xs2 ) )
= ( ? [Ys3: list @ A,Zs: list @ A] :
( Xs2
= ( append @ A @ Ys3 @ ( cons @ A @ X @ Zs ) ) ) ) ) ).
% in_set_conv_decomp
thf(fact_7596_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_7597_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_7598_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_7599_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_7600_in__set__conv__decomp__last,axiom,
! [A: $tType,X: A,Xs2: list @ A] :
( ( member @ A @ X @ ( set2 @ A @ Xs2 ) )
= ( ? [Ys3: list @ A,Zs: list @ A] :
( ( Xs2
= ( append @ A @ Ys3 @ ( cons @ A @ X @ Zs ) ) )
& ~ ( member @ A @ X @ ( set2 @ A @ Zs ) ) ) ) ) ).
% in_set_conv_decomp_last
thf(fact_7601_in__set__conv__decomp__first,axiom,
! [A: $tType,X: A,Xs2: list @ A] :
( ( member @ A @ X @ ( set2 @ A @ Xs2 ) )
= ( ? [Ys3: list @ A,Zs: list @ A] :
( ( Xs2
= ( append @ A @ Ys3 @ ( cons @ A @ X @ Zs ) ) )
& ~ ( member @ A @ X @ ( set2 @ A @ Ys3 ) ) ) ) ) ).
% in_set_conv_decomp_first
thf(fact_7602_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,Zs: list @ A] :
( ( Xs2
= ( append @ A @ Ys3 @ ( cons @ A @ X2 @ Zs ) ) )
& ( P @ X2 )
& ! [Y6: A] :
( ( member @ A @ Y6 @ ( set2 @ A @ Zs ) )
=> ~ ( P @ Y6 ) ) ) ) ) ).
% split_list_last_prop_iff
thf(fact_7603_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] :
( ? [Zs: list @ A] :
( Xs2
= ( append @ A @ Ys3 @ ( cons @ A @ X2 @ Zs ) ) )
& ( P @ X2 )
& ! [Y6: A] :
( ( member @ A @ Y6 @ ( set2 @ A @ Ys3 ) )
=> ~ ( P @ Y6 ) ) ) ) ) ).
% split_list_first_prop_iff
thf(fact_7604_empty__set,axiom,
! [A: $tType] :
( ( bot_bot @ ( set @ A ) )
= ( set2 @ A @ ( nil @ A ) ) ) ).
% empty_set
thf(fact_7605_ent__false,axiom,
! [P: assn] : ( entails @ ( bot_bot @ assn ) @ P ) ).
% ent_false
thf(fact_7606_mod__false,axiom,
! [H2: product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat )] :
~ ( rep_assn @ ( bot_bot @ assn ) @ H2 ) ).
% mod_false
thf(fact_7607_list__assn__aux__ineq__len,axiom,
! [B: $tType,A: $tType,L: list @ A,Li2: list @ B,A3: A > B > assn] :
( ( ( size_size @ ( list @ A ) @ L )
!= ( size_size @ ( list @ B ) @ Li2 ) )
=> ( ( vEBT_List_list_assn @ A @ B @ A3 @ L @ Li2 )
= ( bot_bot @ assn ) ) ) ).
% list_assn_aux_ineq_len
thf(fact_7608_list__assn__simps_I3_J,axiom,
! [B: $tType,A: $tType,P: A > B > assn,A2: A,As2: list @ A] :
( ( vEBT_List_list_assn @ A @ B @ P @ ( cons @ A @ A2 @ As2 ) @ ( nil @ B ) )
= ( bot_bot @ assn ) ) ).
% list_assn_simps(3)
thf(fact_7609_list__assn__simps_I4_J,axiom,
! [B: $tType,A: $tType,P: A > B > assn,C2: B,Cs: list @ B] :
( ( vEBT_List_list_assn @ A @ B @ P @ ( nil @ A ) @ ( cons @ B @ C2 @ Cs ) )
= ( bot_bot @ assn ) ) ).
% list_assn_simps(4)
thf(fact_7610_list__assn_Osimps_I3_J,axiom,
! [C: $tType,A: $tType,Uu: A > C > assn,V: A,Va2: list @ A] :
( ( vEBT_List_list_assn @ A @ C @ Uu @ ( cons @ A @ V @ Va2 ) @ ( nil @ C ) )
= ( bot_bot @ assn ) ) ).
% list_assn.simps(3)
thf(fact_7611_list__assn_Osimps_I4_J,axiom,
! [C: $tType,A: $tType,Uu: A > C > assn,V: C,Va2: list @ C] :
( ( vEBT_List_list_assn @ A @ C @ Uu @ ( nil @ A ) @ ( cons @ C @ V @ Va2 ) )
= ( bot_bot @ assn ) ) ).
% list_assn.simps(4)
thf(fact_7612_replicate__add,axiom,
! [A: $tType,N: nat,M: nat,X: A] :
( ( replicate @ A @ ( plus_plus @ nat @ N @ M ) @ X )
= ( append @ A @ ( replicate @ A @ N @ X ) @ ( replicate @ A @ M @ X ) ) ) ).
% replicate_add
thf(fact_7613_len__greater__imp__nonempty,axiom,
! [A: $tType,X: nat,L: list @ A] :
( ( ord_less @ nat @ X @ ( size_size @ ( list @ A ) @ L ) )
=> ( L
!= ( nil @ A ) ) ) ).
% len_greater_imp_nonempty
thf(fact_7614_comm__append__is__replicate,axiom,
! [A: $tType,Xs2: list @ A,Ys: list @ A] :
( ( Xs2
!= ( nil @ A ) )
=> ( ( Ys
!= ( nil @ A ) )
=> ( ( ( append @ A @ Xs2 @ Ys )
= ( append @ A @ Ys @ Xs2 ) )
=> ? [N2: nat,Zs2: list @ A] :
( ( ord_less @ nat @ ( one_one @ nat ) @ N2 )
& ( ( concat @ A @ ( replicate @ ( list @ A ) @ N2 @ Zs2 ) )
= ( append @ A @ Xs2 @ Ys ) ) ) ) ) ) ).
% comm_append_is_replicate
thf(fact_7615_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_7616_list__decomp__1,axiom,
! [A: $tType,L: list @ A] :
( ( ( size_size @ ( list @ A ) @ L )
= ( one_one @ nat ) )
=> ? [A6: A] :
( L
= ( cons @ A @ A6 @ ( nil @ A ) ) ) ) ).
% list_decomp_1
thf(fact_7617_vebt__assn__raw_Osimps_I4_J,axiom,
! [Vd: $o,Ve: $o,V: option @ ( product_prod @ nat @ nat ),Va2: nat,Vb: array @ vEBT_VEBTi,Vc2: vEBT_VEBTi] :
( ( vEBT_vebt_assn_raw @ ( vEBT_Leaf @ Vd @ Ve ) @ ( vEBT_Nodei @ V @ Va2 @ Vb @ Vc2 ) )
= ( bot_bot @ assn ) ) ).
% vebt_assn_raw.simps(4)
thf(fact_7618_nth__append,axiom,
! [A: $tType,N: nat,Xs2: list @ A,Ys: list @ A] :
( ( ( ord_less @ nat @ N @ ( size_size @ ( list @ A ) @ Xs2 ) )
=> ( ( nth @ A @ ( append @ A @ Xs2 @ Ys ) @ N )
= ( nth @ A @ Xs2 @ N ) ) )
& ( ~ ( ord_less @ nat @ N @ ( size_size @ ( list @ A ) @ Xs2 ) )
=> ( ( nth @ A @ ( append @ A @ Xs2 @ Ys ) @ N )
= ( nth @ A @ Ys @ ( minus_minus @ nat @ N @ ( size_size @ ( list @ A ) @ Xs2 ) ) ) ) ) ) ).
% nth_append
thf(fact_7619_list__update__append,axiom,
! [A: $tType,N: nat,Xs2: list @ A,Ys: list @ A,X: A] :
( ( ( ord_less @ nat @ N @ ( size_size @ ( list @ A ) @ Xs2 ) )
=> ( ( list_update @ A @ ( append @ A @ Xs2 @ Ys ) @ N @ X )
= ( append @ A @ ( list_update @ A @ Xs2 @ N @ X ) @ Ys ) ) )
& ( ~ ( ord_less @ nat @ N @ ( size_size @ ( list @ A ) @ Xs2 ) )
=> ( ( list_update @ A @ ( append @ A @ Xs2 @ Ys ) @ N @ X )
= ( append @ A @ Xs2 @ ( list_update @ A @ Ys @ ( minus_minus @ nat @ N @ ( size_size @ ( list @ A ) @ Xs2 ) ) @ X ) ) ) ) ) ).
% list_update_append
thf(fact_7620_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_7621_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_7622_list__decomp__2,axiom,
! [A: $tType,L: list @ A] :
( ( ( size_size @ ( list @ A ) @ L )
= ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) )
=> ? [A6: A,B3: A] :
( L
= ( cons @ A @ A6 @ ( cons @ A @ B3 @ ( nil @ A ) ) ) ) ) ).
% list_decomp_2
thf(fact_7623_vebt__assn__raw_Oelims,axiom,
! [X: vEBT_VEBT,Xa: vEBT_VEBTi,Y: assn] :
( ( ( vEBT_vebt_assn_raw @ X @ Xa )
= Y )
=> ( ! [A6: $o,B3: $o] :
( ( X
= ( vEBT_Leaf @ A6 @ B3 ) )
=> ! [Ai: $o,Bi: $o] :
( ( Xa
= ( vEBT_Leafi @ Ai @ Bi ) )
=> ( Y
!= ( pure_assn
@ ( ( Ai = A6 )
& ( Bi = B3 ) ) ) ) ) )
=> ( ! [Mmo2: option @ ( product_prod @ nat @ nat ),Deg2: nat,Tree_list2: list @ vEBT_VEBT,Summary2: vEBT_VEBT] :
( ( X
= ( vEBT_Node @ Mmo2 @ Deg2 @ Tree_list2 @ Summary2 ) )
=> ! [Mmoi2: option @ ( product_prod @ nat @ nat ),Degi2: nat,Tree_array2: array @ vEBT_VEBTi,Summaryi2: vEBT_VEBTi] :
( ( Xa
= ( vEBT_Nodei @ Mmoi2 @ Degi2 @ Tree_array2 @ Summaryi2 ) )
=> ( Y
!= ( times_times @ assn
@ ( times_times @ assn
@ ( pure_assn
@ ( ( Mmoi2 = Mmo2 )
& ( Degi2 = Deg2 ) ) )
@ ( vEBT_vebt_assn_raw @ Summary2 @ Summaryi2 ) )
@ ( ex_assn @ ( list @ vEBT_VEBTi )
@ ^ [Tree_is2: list @ vEBT_VEBTi] : ( times_times @ assn @ ( snga_assn @ vEBT_VEBTi @ Tree_array2 @ Tree_is2 ) @ ( vEBT_List_list_assn @ vEBT_VEBT @ vEBT_VEBTi @ vEBT_vebt_assn_raw @ Tree_list2 @ Tree_is2 ) ) ) ) ) ) )
=> ( ( ? [V4: option @ ( product_prod @ nat @ nat ),Va: nat,Vb3: list @ vEBT_VEBT,Vc3: vEBT_VEBT] :
( X
= ( vEBT_Node @ V4 @ Va @ Vb3 @ Vc3 ) )
=> ( ? [Vd3: $o,Ve3: $o] :
( Xa
= ( vEBT_Leafi @ Vd3 @ Ve3 ) )
=> ( Y
!= ( bot_bot @ assn ) ) ) )
=> ~ ( ? [Vd3: $o,Ve3: $o] :
( X
= ( vEBT_Leaf @ Vd3 @ Ve3 ) )
=> ( ? [V4: option @ ( product_prod @ nat @ nat ),Va: nat,Vb3: array @ vEBT_VEBTi,Vc3: vEBT_VEBTi] :
( Xa
= ( vEBT_Nodei @ V4 @ Va @ Vb3 @ Vc3 ) )
=> ( Y
!= ( bot_bot @ assn ) ) ) ) ) ) ) ) ).
% vebt_assn_raw.elims
thf(fact_7624_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,C7: $o] :
( ( ord_less_eq @ char @ ( char2 @ B0 @ B1 @ B22 @ B32 @ B42 @ B52 @ B62 @ B72 ) @ ( char2 @ C0 @ C1 @ C22 @ C32 @ C42 @ C52 @ C62 @ C7 ) )
= ( ord_less_eq @ nat
@ ( foldr @ $o @ nat
@ ^ [B5: $o,K3: nat] : ( plus_plus @ nat @ ( zero_neq_one_of_bool @ nat @ B5 ) @ ( times_times @ nat @ K3 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) )
@ ( cons @ $o @ B0 @ ( cons @ $o @ B1 @ ( cons @ $o @ B22 @ ( cons @ $o @ B32 @ ( cons @ $o @ B42 @ ( cons @ $o @ B52 @ ( cons @ $o @ B62 @ ( cons @ $o @ B72 @ ( nil @ $o ) ) ) ) ) ) ) ) )
@ ( zero_zero @ nat ) )
@ ( foldr @ $o @ nat
@ ^ [B5: $o,K3: nat] : ( plus_plus @ nat @ ( zero_neq_one_of_bool @ nat @ B5 ) @ ( times_times @ nat @ K3 @ ( 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 @ C7 @ ( nil @ $o ) ) ) ) ) ) ) ) )
@ ( zero_zero @ nat ) ) ) ) ).
% less_eq_char_simp
thf(fact_7625_VEBTi_Oinject_I2_J,axiom,
! [X21: $o,X222: $o,Y21: $o,Y222: $o] :
( ( ( vEBT_Leafi @ X21 @ X222 )
= ( vEBT_Leafi @ Y21 @ Y222 ) )
= ( ( X21 = Y21 )
& ( X222 = Y222 ) ) ) ).
% VEBTi.inject(2)
thf(fact_7626_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_7627_upt__conv__Nil,axiom,
! [J: nat,I: nat] :
( ( ord_less_eq @ nat @ J @ I )
=> ( ( upt @ I @ J )
= ( nil @ nat ) ) ) ).
% upt_conv_Nil
thf(fact_7628_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_7629_upt__rec__numeral,axiom,
! [M: num,N: num] :
( ( ( ord_less @ nat @ ( numeral_numeral @ nat @ M ) @ ( numeral_numeral @ nat @ N ) )
=> ( ( upt @ ( numeral_numeral @ nat @ M ) @ ( numeral_numeral @ nat @ N ) )
= ( cons @ nat @ ( numeral_numeral @ nat @ M ) @ ( upt @ ( suc @ ( numeral_numeral @ nat @ M ) ) @ ( numeral_numeral @ nat @ N ) ) ) ) )
& ( ~ ( ord_less @ nat @ ( numeral_numeral @ nat @ M ) @ ( numeral_numeral @ nat @ N ) )
=> ( ( upt @ ( numeral_numeral @ nat @ M ) @ ( numeral_numeral @ nat @ N ) )
= ( nil @ nat ) ) ) ) ).
% upt_rec_numeral
thf(fact_7630_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,C7: $o] :
( ( ord_less @ char @ ( char2 @ B0 @ B1 @ B22 @ B32 @ B42 @ B52 @ B62 @ B72 ) @ ( char2 @ C0 @ C1 @ C22 @ C32 @ C42 @ C52 @ C62 @ C7 ) )
= ( ord_less @ nat
@ ( foldr @ $o @ nat
@ ^ [B5: $o,K3: nat] : ( plus_plus @ nat @ ( zero_neq_one_of_bool @ nat @ B5 ) @ ( times_times @ nat @ K3 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) )
@ ( cons @ $o @ B0 @ ( cons @ $o @ B1 @ ( cons @ $o @ B22 @ ( cons @ $o @ B32 @ ( cons @ $o @ B42 @ ( cons @ $o @ B52 @ ( cons @ $o @ B62 @ ( cons @ $o @ B72 @ ( nil @ $o ) ) ) ) ) ) ) ) )
@ ( zero_zero @ nat ) )
@ ( foldr @ $o @ nat
@ ^ [B5: $o,K3: nat] : ( plus_plus @ nat @ ( zero_neq_one_of_bool @ nat @ B5 ) @ ( times_times @ nat @ K3 @ ( 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 @ C7 @ ( nil @ $o ) ) ) ) ) ) ) ) )
@ ( zero_zero @ nat ) ) ) ) ).
% less_char_simp
thf(fact_7631_upt__0,axiom,
! [I: nat] :
( ( upt @ I @ ( zero_zero @ nat ) )
= ( nil @ nat ) ) ).
% upt_0
thf(fact_7632_list__encode_Ocases,axiom,
! [X: list @ nat] :
( ( X
!= ( nil @ nat ) )
=> ~ ! [X3: nat,Xs3: list @ nat] :
( X
!= ( cons @ nat @ X3 @ Xs3 ) ) ) ).
% list_encode.cases
thf(fact_7633_vebt__assn__raw_Osimps_I3_J,axiom,
! [V: option @ ( product_prod @ nat @ nat ),Va2: nat,Vb: list @ vEBT_VEBT,Vc2: vEBT_VEBT,Vd: $o,Ve: $o] :
( ( vEBT_vebt_assn_raw @ ( vEBT_Node @ V @ Va2 @ Vb @ Vc2 ) @ ( vEBT_Leafi @ Vd @ Ve ) )
= ( bot_bot @ assn ) ) ).
% vebt_assn_raw.simps(3)
thf(fact_7634_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_7635_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_7636_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_7637_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_7638_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_7639_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_7640_VEBT__internal_Ovebt__buildupi_H_Osimps_I1_J,axiom,
( ( vEBT_V739175172307565963ildupi @ ( zero_zero @ nat ) )
= ( heap_Time_return @ vEBT_VEBTi @ ( vEBT_Leafi @ $false @ $false ) ) ) ).
% VEBT_internal.vebt_buildupi'.simps(1)
thf(fact_7641_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_7642_VEBTi_Osize__gen_I2_J,axiom,
! [X21: $o,X222: $o] :
( ( vEBT_size_VEBTi @ ( vEBT_Leafi @ X21 @ X222 ) )
= ( zero_zero @ nat ) ) ).
% VEBTi.size_gen(2)
thf(fact_7643_concat__eq__appendD,axiom,
! [A: $tType,Xss: list @ ( list @ A ),Ys: list @ A,Zs3: list @ A] :
( ( ( concat @ A @ Xss )
= ( append @ A @ Ys @ Zs3 ) )
=> ( ( Xss
!= ( nil @ ( list @ A ) ) )
=> ? [Xss12: list @ ( list @ A ),Xs3: list @ A,Xs6: list @ A,Xss22: list @ ( list @ A )] :
( ( Xss
= ( append @ ( list @ A ) @ Xss12 @ ( cons @ ( list @ A ) @ ( append @ A @ Xs3 @ Xs6 ) @ Xss22 ) ) )
& ( Ys
= ( append @ A @ ( concat @ A @ Xss12 ) @ Xs3 ) )
& ( Zs3
= ( append @ A @ Xs6 @ ( concat @ A @ Xss22 ) ) ) ) ) ) ).
% concat_eq_appendD
thf(fact_7644_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_7645_vebt__assn__raw_Ocases,axiom,
! [X: product_prod @ vEBT_VEBT @ vEBT_VEBTi] :
( ! [A6: $o,B3: $o,Ai: $o,Bi: $o] :
( X
!= ( product_Pair @ vEBT_VEBT @ vEBT_VEBTi @ ( vEBT_Leaf @ A6 @ B3 ) @ ( vEBT_Leafi @ Ai @ Bi ) ) )
=> ( ! [Mmo2: option @ ( product_prod @ nat @ nat ),Deg2: nat,Tree_list2: list @ vEBT_VEBT,Summary2: vEBT_VEBT,Mmoi2: option @ ( product_prod @ nat @ nat ),Degi2: nat,Tree_array2: array @ vEBT_VEBTi,Summaryi2: vEBT_VEBTi] :
( X
!= ( product_Pair @ vEBT_VEBT @ vEBT_VEBTi @ ( vEBT_Node @ Mmo2 @ Deg2 @ Tree_list2 @ Summary2 ) @ ( vEBT_Nodei @ Mmoi2 @ Degi2 @ Tree_array2 @ Summaryi2 ) ) )
=> ( ! [V4: option @ ( product_prod @ nat @ nat ),Va: nat,Vb3: list @ vEBT_VEBT,Vc3: vEBT_VEBT,Vd3: $o,Ve3: $o] :
( X
!= ( product_Pair @ vEBT_VEBT @ vEBT_VEBTi @ ( vEBT_Node @ V4 @ Va @ Vb3 @ Vc3 ) @ ( vEBT_Leafi @ Vd3 @ Ve3 ) ) )
=> ~ ! [Vd3: $o,Ve3: $o,V4: option @ ( product_prod @ nat @ nat ),Va: nat,Vb3: array @ vEBT_VEBTi,Vc3: vEBT_VEBTi] :
( X
!= ( product_Pair @ vEBT_VEBT @ vEBT_VEBTi @ ( vEBT_Leaf @ Vd3 @ Ve3 ) @ ( vEBT_Nodei @ V4 @ Va @ Vb3 @ Vc3 ) ) ) ) ) ) ).
% vebt_assn_raw.cases
thf(fact_7646_subseqs_Osimps_I2_J,axiom,
! [A: $tType,X: A,Xs2: list @ A] :
( ( subseqs @ A @ ( cons @ A @ X @ Xs2 ) )
= ( append @ ( list @ A ) @ ( map @ ( list @ A ) @ ( list @ A ) @ ( cons @ A @ X ) @ ( subseqs @ A @ Xs2 ) ) @ ( subseqs @ A @ Xs2 ) ) ) ).
% subseqs.simps(2)
thf(fact_7647_vebt__assn__raw_Osimps_I1_J,axiom,
! [A2: $o,B2: $o,Ai2: $o,Bi2: $o] :
( ( vEBT_vebt_assn_raw @ ( vEBT_Leaf @ A2 @ B2 ) @ ( vEBT_Leafi @ Ai2 @ Bi2 ) )
= ( pure_assn
@ ( ( Ai2 = A2 )
& ( Bi2 = B2 ) ) ) ) ).
% vebt_assn_raw.simps(1)
thf(fact_7648_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_7649_upt__rec,axiom,
( upt
= ( ^ [I2: nat,J2: nat] : ( if @ ( list @ nat ) @ ( ord_less @ nat @ I2 @ J2 ) @ ( cons @ nat @ I2 @ ( upt @ ( suc @ I2 ) @ J2 ) ) @ ( nil @ nat ) ) ) ) ).
% upt_rec
thf(fact_7650_upt__eq__lel__conv,axiom,
! [L: nat,H2: nat,Is1: list @ nat,I: nat,Is2: list @ nat] :
( ( ( upt @ L @ H2 )
= ( append @ nat @ Is1 @ ( cons @ nat @ I @ Is2 ) ) )
= ( ( Is1
= ( upt @ L @ I ) )
& ( Is2
= ( upt @ ( suc @ I ) @ H2 ) )
& ( ord_less_eq @ nat @ L @ I )
& ( ord_less @ nat @ I @ H2 ) ) ) ).
% upt_eq_lel_conv
thf(fact_7651_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_7652_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_7653_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_7654_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_7655_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_7656_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_7657_char_Osize_I2_J,axiom,
! [X15: $o,X22: $o,X32: $o,X42: $o,X52: $o,X62: $o,X72: $o,X82: $o] :
( ( size_size @ char @ ( char2 @ X15 @ X22 @ X32 @ X42 @ X52 @ X62 @ X72 @ X82 ) )
= ( zero_zero @ nat ) ) ).
% char.size(2)
thf(fact_7658_char__of__def,axiom,
! [A: $tType] :
( ( bit_un5681908812861735899ations @ A )
=> ( ( unique5772411509450598832har_of @ A )
= ( ^ [N3: A] :
( char2
@ ~ ( dvd_dvd @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ N3 )
@ ( bit_se5641148757651400278ts_bit @ A @ N3 @ ( one_one @ nat ) )
@ ( bit_se5641148757651400278ts_bit @ A @ N3 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) )
@ ( bit_se5641148757651400278ts_bit @ A @ N3 @ ( numeral_numeral @ nat @ ( bit1 @ one2 ) ) )
@ ( bit_se5641148757651400278ts_bit @ A @ N3 @ ( numeral_numeral @ nat @ ( bit0 @ ( bit0 @ one2 ) ) ) )
@ ( bit_se5641148757651400278ts_bit @ A @ N3 @ ( numeral_numeral @ nat @ ( bit1 @ ( bit0 @ one2 ) ) ) )
@ ( bit_se5641148757651400278ts_bit @ A @ N3 @ ( numeral_numeral @ nat @ ( bit0 @ ( bit1 @ one2 ) ) ) )
@ ( bit_se5641148757651400278ts_bit @ A @ N3 @ ( numeral_numeral @ nat @ ( bit1 @ ( bit1 @ one2 ) ) ) ) ) ) ) ) ).
% char_of_def
thf(fact_7659_char_Osize__gen,axiom,
! [X15: $o,X22: $o,X32: $o,X42: $o,X52: $o,X62: $o,X72: $o,X82: $o] :
( ( size_char @ ( char2 @ X15 @ X22 @ X32 @ X42 @ X52 @ X62 @ X72 @ X82 ) )
= ( zero_zero @ nat ) ) ).
% char.size_gen
thf(fact_7660_char__of__quasi__inj,axiom,
! [A: $tType] :
( ( bit_un5681908812861735899ations @ A )
=> ! [M: A,N: A] :
( ( ( unique5772411509450598832har_of @ A @ M )
= ( unique5772411509450598832har_of @ A @ N ) )
= ( ( modulo_modulo @ A @ M @ ( numeral_numeral @ A @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ one2 ) ) ) ) ) ) ) ) ) )
= ( modulo_modulo @ A @ N @ ( numeral_numeral @ A @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ one2 ) ) ) ) ) ) ) ) ) ) ) ) ) ).
% char_of_quasi_inj
thf(fact_7661_char__of__mod__256,axiom,
! [A: $tType] :
( ( bit_un5681908812861735899ations @ A )
=> ! [N: A] :
( ( unique5772411509450598832har_of @ A @ ( modulo_modulo @ A @ N @ ( numeral_numeral @ A @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ one2 ) ) ) ) ) ) ) ) ) ) )
= ( unique5772411509450598832har_of @ A @ N ) ) ) ).
% char_of_mod_256
thf(fact_7662_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_7663_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_7664_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_7665_char__of__take__bit__eq,axiom,
! [A: $tType] :
( ( bit_un5681908812861735899ations @ A )
=> ! [N: nat,M: A] :
( ( ord_less_eq @ nat @ ( numeral_numeral @ nat @ ( bit0 @ ( bit0 @ ( bit0 @ one2 ) ) ) ) @ N )
=> ( ( unique5772411509450598832har_of @ A @ ( bit_se2584673776208193580ke_bit @ A @ N @ M ) )
= ( unique5772411509450598832har_of @ A @ M ) ) ) ) ).
% char_of_take_bit_eq
thf(fact_7666_char__of__eq__iff,axiom,
! [A: $tType] :
( ( bit_un5681908812861735899ations @ A )
=> ! [N: A,C2: char] :
( ( ( unique5772411509450598832har_of @ A @ N )
= C2 )
= ( ( bit_se2584673776208193580ke_bit @ A @ ( numeral_numeral @ nat @ ( bit0 @ ( bit0 @ ( bit0 @ one2 ) ) ) ) @ N )
= ( comm_s6883823935334413003f_char @ A @ C2 ) ) ) ) ).
% char_of_eq_iff
thf(fact_7667_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 ) )
=> ( ! [A6: $o,B3: $o] :
( ( X
= ( vEBT_Leaf @ A6 @ B3 ) )
=> ! [Ai: $o,Bi: $o] :
( ( Xa
= ( vEBT_Leafi @ Ai @ Bi ) )
=> ( ( Y
= ( pure_assn
@ ( ( Ai = A6 )
& ( Bi = B3 ) ) ) )
=> ~ ( accp @ ( product_prod @ vEBT_VEBT @ vEBT_VEBTi ) @ vEBT_v8524038756793281170aw_rel @ ( product_Pair @ vEBT_VEBT @ vEBT_VEBTi @ ( vEBT_Leaf @ A6 @ B3 ) @ ( vEBT_Leafi @ Ai @ Bi ) ) ) ) ) )
=> ( ! [Mmo2: option @ ( product_prod @ nat @ nat ),Deg2: nat,Tree_list2: list @ vEBT_VEBT,Summary2: vEBT_VEBT] :
( ( X
= ( vEBT_Node @ Mmo2 @ Deg2 @ Tree_list2 @ Summary2 ) )
=> ! [Mmoi2: option @ ( product_prod @ nat @ nat ),Degi2: nat,Tree_array2: array @ vEBT_VEBTi,Summaryi2: vEBT_VEBTi] :
( ( Xa
= ( vEBT_Nodei @ Mmoi2 @ Degi2 @ Tree_array2 @ Summaryi2 ) )
=> ( ( Y
= ( times_times @ assn
@ ( times_times @ assn
@ ( pure_assn
@ ( ( Mmoi2 = Mmo2 )
& ( Degi2 = Deg2 ) ) )
@ ( vEBT_vebt_assn_raw @ Summary2 @ Summaryi2 ) )
@ ( ex_assn @ ( list @ vEBT_VEBTi )
@ ^ [Tree_is2: list @ vEBT_VEBTi] : ( times_times @ assn @ ( snga_assn @ vEBT_VEBTi @ Tree_array2 @ Tree_is2 ) @ ( vEBT_List_list_assn @ vEBT_VEBT @ vEBT_VEBTi @ vEBT_vebt_assn_raw @ Tree_list2 @ Tree_is2 ) ) ) ) )
=> ~ ( accp @ ( product_prod @ vEBT_VEBT @ vEBT_VEBTi ) @ vEBT_v8524038756793281170aw_rel @ ( product_Pair @ vEBT_VEBT @ vEBT_VEBTi @ ( vEBT_Node @ Mmo2 @ Deg2 @ Tree_list2 @ Summary2 ) @ ( vEBT_Nodei @ Mmoi2 @ Degi2 @ Tree_array2 @ Summaryi2 ) ) ) ) ) )
=> ( ! [V4: option @ ( product_prod @ nat @ nat ),Va: nat,Vb3: list @ vEBT_VEBT,Vc3: vEBT_VEBT] :
( ( X
= ( vEBT_Node @ V4 @ Va @ 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 @ V4 @ Va @ Vb3 @ Vc3 ) @ ( vEBT_Leafi @ Vd3 @ Ve3 ) ) ) ) ) )
=> ~ ! [Vd3: $o,Ve3: $o] :
( ( X
= ( vEBT_Leaf @ Vd3 @ Ve3 ) )
=> ! [V4: option @ ( product_prod @ nat @ nat ),Va: nat,Vb3: array @ vEBT_VEBTi,Vc3: vEBT_VEBTi] :
( ( Xa
= ( vEBT_Nodei @ V4 @ Va @ Vb3 @ Vc3 ) )
=> ( ( Y
= ( bot_bot @ assn ) )
=> ~ ( accp @ ( product_prod @ vEBT_VEBT @ vEBT_VEBTi ) @ vEBT_v8524038756793281170aw_rel @ ( product_Pair @ vEBT_VEBT @ vEBT_VEBTi @ ( vEBT_Leaf @ Vd3 @ Ve3 ) @ ( vEBT_Nodei @ V4 @ Va @ Vb3 @ Vc3 ) ) ) ) ) ) ) ) ) ) ) ).
% vebt_assn_raw.pelims
thf(fact_7668_transpose__rectangle,axiom,
! [A: $tType,Xs2: list @ ( list @ A ),N: nat] :
( ( ( Xs2
= ( nil @ ( list @ A ) ) )
=> ( N
= ( zero_zero @ nat ) ) )
=> ( ! [I3: nat] :
( ( ord_less @ nat @ I3 @ ( size_size @ ( list @ ( list @ A ) ) @ Xs2 ) )
=> ( ( size_size @ ( list @ A ) @ ( nth @ ( list @ A ) @ Xs2 @ I3 ) )
= N ) )
=> ( ( transpose @ A @ Xs2 )
= ( map @ nat @ ( list @ A )
@ ^ [I2: nat] :
( map @ nat @ A
@ ^ [J2: nat] : ( nth @ A @ ( nth @ ( list @ A ) @ Xs2 @ J2 ) @ I2 )
@ ( upt @ ( zero_zero @ nat ) @ ( size_size @ ( list @ ( list @ A ) ) @ Xs2 ) ) )
@ ( upt @ ( zero_zero @ nat ) @ N ) ) ) ) ) ).
% transpose_rectangle
thf(fact_7669_transpose__map__map,axiom,
! [A: $tType,B: $tType,F2: B > A,Xs2: list @ ( list @ B )] :
( ( transpose @ A @ ( map @ ( list @ B ) @ ( list @ A ) @ ( map @ B @ A @ F2 ) @ Xs2 ) )
= ( map @ ( list @ B ) @ ( list @ A ) @ ( map @ B @ A @ F2 ) @ ( transpose @ B @ Xs2 ) ) ) ).
% transpose_map_map
thf(fact_7670_transpose_Osimps_I1_J,axiom,
! [A: $tType] :
( ( transpose @ A @ ( nil @ ( list @ A ) ) )
= ( nil @ ( list @ A ) ) ) ).
% transpose.simps(1)
thf(fact_7671_transpose_Osimps_I2_J,axiom,
! [A: $tType,Xss: list @ ( list @ A )] :
( ( transpose @ A @ ( cons @ ( list @ A ) @ ( nil @ A ) @ Xss ) )
= ( transpose @ A @ Xss ) ) ).
% transpose.simps(2)
thf(fact_7672_transpose__empty,axiom,
! [A: $tType,Xs2: list @ ( list @ A )] :
( ( ( transpose @ A @ Xs2 )
= ( nil @ ( list @ A ) ) )
= ( ! [X2: list @ A] :
( ( member @ ( list @ A ) @ X2 @ ( set2 @ ( list @ A ) @ Xs2 ) )
=> ( X2
= ( nil @ A ) ) ) ) ) ).
% transpose_empty
thf(fact_7673_length__transpose,axiom,
! [A: $tType,Xs2: list @ ( list @ A )] :
( ( size_size @ ( list @ ( list @ A ) ) @ ( transpose @ A @ Xs2 ) )
= ( foldr @ ( list @ A ) @ nat
@ ^ [Xs: list @ A] : ( ord_max @ nat @ ( size_size @ ( list @ A ) @ Xs ) )
@ Xs2
@ ( zero_zero @ nat ) ) ) ).
% length_transpose
thf(fact_7674_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_7675_upto_Opelims,axiom,
! [X: int,Xa: int,Y: list @ int] :
( ( ( upto @ X @ Xa )
= Y )
=> ( ( accp @ ( product_prod @ int @ int ) @ upto_rel @ ( product_Pair @ int @ int @ X @ Xa ) )
=> ~ ( ( ( ( ord_less_eq @ int @ X @ Xa )
=> ( Y
= ( cons @ int @ X @ ( upto @ ( plus_plus @ int @ X @ ( one_one @ int ) ) @ Xa ) ) ) )
& ( ~ ( ord_less_eq @ int @ X @ Xa )
=> ( Y
= ( nil @ int ) ) ) )
=> ~ ( accp @ ( product_prod @ int @ int ) @ upto_rel @ ( product_Pair @ int @ int @ X @ Xa ) ) ) ) ) ).
% upto.pelims
thf(fact_7676_upto__Nil,axiom,
! [I: int,J: int] :
( ( ( upto @ I @ J )
= ( nil @ int ) )
= ( ord_less @ int @ J @ I ) ) ).
% upto_Nil
thf(fact_7677_upto__Nil2,axiom,
! [I: int,J: int] :
( ( ( nil @ int )
= ( upto @ I @ J ) )
= ( ord_less @ int @ J @ I ) ) ).
% upto_Nil2
thf(fact_7678_upto__empty,axiom,
! [J: int,I: int] :
( ( ord_less @ int @ J @ I )
=> ( ( upto @ I @ J )
= ( nil @ int ) ) ) ).
% upto_empty
thf(fact_7679_upto__single,axiom,
! [I: int] :
( ( upto @ I @ I )
= ( cons @ int @ I @ ( nil @ int ) ) ) ).
% upto_single
thf(fact_7680_nth__upto,axiom,
! [I: int,K: nat,J: int] :
( ( ord_less_eq @ int @ ( plus_plus @ int @ I @ ( semiring_1_of_nat @ int @ K ) ) @ J )
=> ( ( nth @ int @ ( upto @ I @ J ) @ K )
= ( plus_plus @ int @ I @ ( semiring_1_of_nat @ int @ K ) ) ) ) ).
% nth_upto
thf(fact_7681_length__upto,axiom,
! [I: int,J: int] :
( ( size_size @ ( list @ int ) @ ( upto @ I @ J ) )
= ( nat2 @ ( plus_plus @ int @ ( minus_minus @ int @ J @ I ) @ ( one_one @ int ) ) ) ) ).
% length_upto
thf(fact_7682_upto__rec__numeral_I1_J,axiom,
! [M: num,N: num] :
( ( ( ord_less_eq @ int @ ( numeral_numeral @ int @ M ) @ ( numeral_numeral @ int @ N ) )
=> ( ( upto @ ( numeral_numeral @ int @ M ) @ ( numeral_numeral @ int @ N ) )
= ( cons @ int @ ( numeral_numeral @ int @ M ) @ ( upto @ ( plus_plus @ int @ ( numeral_numeral @ int @ M ) @ ( one_one @ int ) ) @ ( numeral_numeral @ int @ N ) ) ) ) )
& ( ~ ( ord_less_eq @ int @ ( numeral_numeral @ int @ M ) @ ( numeral_numeral @ int @ N ) )
=> ( ( upto @ ( numeral_numeral @ int @ M ) @ ( numeral_numeral @ int @ N ) )
= ( nil @ int ) ) ) ) ).
% upto_rec_numeral(1)
thf(fact_7683_upto__rec__numeral_I2_J,axiom,
! [M: num,N: num] :
( ( ( ord_less_eq @ int @ ( numeral_numeral @ int @ M ) @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ N ) ) )
=> ( ( upto @ ( numeral_numeral @ int @ M ) @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ N ) ) )
= ( cons @ int @ ( numeral_numeral @ int @ M ) @ ( upto @ ( plus_plus @ int @ ( numeral_numeral @ int @ M ) @ ( one_one @ int ) ) @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ N ) ) ) ) ) )
& ( ~ ( ord_less_eq @ int @ ( numeral_numeral @ int @ M ) @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ N ) ) )
=> ( ( upto @ ( numeral_numeral @ int @ M ) @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ N ) ) )
= ( nil @ int ) ) ) ) ).
% upto_rec_numeral(2)
thf(fact_7684_upto__rec__numeral_I3_J,axiom,
! [M: num,N: num] :
( ( ( ord_less_eq @ int @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ M ) ) @ ( numeral_numeral @ int @ N ) )
=> ( ( upto @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ M ) ) @ ( numeral_numeral @ int @ N ) )
= ( cons @ int @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ M ) ) @ ( upto @ ( plus_plus @ int @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ M ) ) @ ( one_one @ int ) ) @ ( numeral_numeral @ int @ N ) ) ) ) )
& ( ~ ( ord_less_eq @ int @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ M ) ) @ ( numeral_numeral @ int @ N ) )
=> ( ( upto @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ M ) ) @ ( numeral_numeral @ int @ N ) )
= ( nil @ int ) ) ) ) ).
% upto_rec_numeral(3)
thf(fact_7685_upto__rec__numeral_I4_J,axiom,
! [M: num,N: num] :
( ( ( ord_less_eq @ int @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ M ) ) @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ N ) ) )
=> ( ( upto @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ M ) ) @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ N ) ) )
= ( cons @ int @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ M ) ) @ ( upto @ ( plus_plus @ int @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ M ) ) @ ( one_one @ int ) ) @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ N ) ) ) ) ) )
& ( ~ ( ord_less_eq @ int @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ M ) ) @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ N ) ) )
=> ( ( upto @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ M ) ) @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ N ) ) )
= ( nil @ int ) ) ) ) ).
% upto_rec_numeral(4)
thf(fact_7686_upto__aux__def,axiom,
( upto_aux
= ( ^ [I2: int,J2: int] : ( append @ int @ ( upto @ I2 @ J2 ) ) ) ) ).
% upto_aux_def
thf(fact_7687_upto__code,axiom,
( upto
= ( ^ [I2: int,J2: int] : ( upto_aux @ I2 @ J2 @ ( nil @ int ) ) ) ) ).
% upto_code
thf(fact_7688_atLeastAtMost__upto,axiom,
( ( set_or1337092689740270186AtMost @ int )
= ( ^ [I2: int,J2: int] : ( set2 @ int @ ( upto @ I2 @ J2 ) ) ) ) ).
% atLeastAtMost_upto
thf(fact_7689_upto__split2,axiom,
! [I: int,J: int,K: int] :
( ( ord_less_eq @ int @ I @ J )
=> ( ( ord_less_eq @ int @ J @ K )
=> ( ( upto @ I @ K )
= ( append @ int @ ( upto @ I @ J ) @ ( upto @ ( plus_plus @ int @ J @ ( one_one @ int ) ) @ K ) ) ) ) ) ).
% upto_split2
thf(fact_7690_upto__split1,axiom,
! [I: int,J: int,K: int] :
( ( ord_less_eq @ int @ I @ J )
=> ( ( ord_less_eq @ int @ J @ K )
=> ( ( upto @ I @ K )
= ( append @ int @ ( upto @ I @ ( minus_minus @ int @ J @ ( one_one @ int ) ) ) @ ( upto @ J @ K ) ) ) ) ) ).
% upto_split1
thf(fact_7691_atLeastLessThan__upto,axiom,
( ( set_or7035219750837199246ssThan @ int )
= ( ^ [I2: int,J2: int] : ( set2 @ int @ ( upto @ I2 @ ( minus_minus @ int @ J2 @ ( one_one @ int ) ) ) ) ) ) ).
% atLeastLessThan_upto
thf(fact_7692_greaterThanAtMost__upto,axiom,
( ( set_or3652927894154168847AtMost @ int )
= ( ^ [I2: int,J2: int] : ( set2 @ int @ ( upto @ ( plus_plus @ int @ I2 @ ( one_one @ int ) ) @ J2 ) ) ) ) ).
% greaterThanAtMost_upto
thf(fact_7693_upto_Osimps,axiom,
( upto
= ( ^ [I2: int,J2: int] : ( if @ ( list @ int ) @ ( ord_less_eq @ int @ I2 @ J2 ) @ ( cons @ int @ I2 @ ( upto @ ( plus_plus @ int @ I2 @ ( one_one @ int ) ) @ J2 ) ) @ ( nil @ int ) ) ) ) ).
% upto.simps
thf(fact_7694_upto_Oelims,axiom,
! [X: int,Xa: int,Y: list @ int] :
( ( ( upto @ X @ Xa )
= Y )
=> ( ( ( ord_less_eq @ int @ X @ Xa )
=> ( Y
= ( cons @ int @ X @ ( upto @ ( plus_plus @ int @ X @ ( one_one @ int ) ) @ Xa ) ) ) )
& ( ~ ( ord_less_eq @ int @ X @ Xa )
=> ( Y
= ( nil @ int ) ) ) ) ) ).
% upto.elims
thf(fact_7695_upto__rec1,axiom,
! [I: int,J: int] :
( ( ord_less_eq @ int @ I @ J )
=> ( ( upto @ I @ J )
= ( cons @ int @ I @ ( upto @ ( plus_plus @ int @ I @ ( one_one @ int ) ) @ J ) ) ) ) ).
% upto_rec1
thf(fact_7696_upto__rec2,axiom,
! [I: int,J: int] :
( ( ord_less_eq @ int @ I @ J )
=> ( ( upto @ I @ J )
= ( append @ int @ ( upto @ I @ ( minus_minus @ int @ J @ ( one_one @ int ) ) ) @ ( cons @ int @ J @ ( nil @ int ) ) ) ) ) ).
% upto_rec2
thf(fact_7697_greaterThanLessThan__upto,axiom,
( ( set_or5935395276787703475ssThan @ int )
= ( ^ [I2: int,J2: int] : ( set2 @ int @ ( upto @ ( plus_plus @ int @ I2 @ ( one_one @ int ) ) @ ( minus_minus @ int @ J2 @ ( one_one @ int ) ) ) ) ) ) ).
% greaterThanLessThan_upto
thf(fact_7698_upto__split3,axiom,
! [I: int,J: int,K: int] :
( ( ord_less_eq @ int @ I @ J )
=> ( ( ord_less_eq @ int @ J @ K )
=> ( ( upto @ I @ K )
= ( append @ int @ ( upto @ I @ ( minus_minus @ int @ J @ ( one_one @ int ) ) ) @ ( cons @ int @ J @ ( upto @ ( plus_plus @ int @ J @ ( one_one @ int ) ) @ K ) ) ) ) ) ) ).
% upto_split3
thf(fact_7699_upto_Opsimps,axiom,
! [I: int,J: int] :
( ( accp @ ( product_prod @ int @ int ) @ upto_rel @ ( product_Pair @ int @ int @ I @ J ) )
=> ( ( ( ord_less_eq @ int @ I @ J )
=> ( ( upto @ I @ J )
= ( cons @ int @ I @ ( upto @ ( plus_plus @ int @ I @ ( one_one @ int ) ) @ J ) ) ) )
& ( ~ ( ord_less_eq @ int @ I @ J )
=> ( ( upto @ I @ J )
= ( nil @ int ) ) ) ) ) ).
% upto.psimps
thf(fact_7700_set__Cons__sing__Nil,axiom,
! [A: $tType,A3: set @ A] :
( ( set_Cons @ A @ A3 @ ( insert @ ( list @ A ) @ ( nil @ A ) @ ( bot_bot @ ( set @ ( list @ A ) ) ) ) )
= ( image @ A @ ( list @ A )
@ ^ [X2: A] : ( cons @ A @ X2 @ ( nil @ A ) )
@ A3 ) ) ).
% set_Cons_sing_Nil
thf(fact_7701_sorted__list__of__set__atMost__Suc,axiom,
! [K: nat] :
( ( linord4507533701916653071of_set @ nat @ ( set_ord_atMost @ nat @ ( suc @ K ) ) )
= ( append @ nat @ ( linord4507533701916653071of_set @ nat @ ( set_ord_atMost @ nat @ K ) ) @ ( cons @ nat @ ( suc @ K ) @ ( nil @ nat ) ) ) ) ).
% sorted_list_of_set_atMost_Suc
thf(fact_7702_sorted__list__of__set_Osorted__key__list__of__set__empty,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ( ( linord4507533701916653071of_set @ A @ ( bot_bot @ ( set @ A ) ) )
= ( nil @ A ) ) ) ).
% sorted_list_of_set.sorted_key_list_of_set_empty
thf(fact_7703_sorted__list__of__set_Ofold__insort__key_Oinfinite,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A3: set @ A] :
( ~ ( finite_finite2 @ A @ A3 )
=> ( ( linord4507533701916653071of_set @ A @ A3 )
= ( nil @ A ) ) ) ) ).
% sorted_list_of_set.fold_insort_key.infinite
thf(fact_7704_sorted__list__of__set_Oset__sorted__key__list__of__set,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A3: set @ A] :
( ( finite_finite2 @ A @ A3 )
=> ( ( set2 @ A @ ( linord4507533701916653071of_set @ A @ A3 ) )
= A3 ) ) ) ).
% sorted_list_of_set.set_sorted_key_list_of_set
thf(fact_7705_sorted__list__of__set__range,axiom,
! [M: nat,N: nat] :
( ( linord4507533701916653071of_set @ nat @ ( set_or7035219750837199246ssThan @ nat @ M @ N ) )
= ( upt @ M @ N ) ) ).
% sorted_list_of_set_range
thf(fact_7706_sorted__list__of__set_Osorted__key__list__of__set__eq__Nil__iff,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A3: set @ A] :
( ( finite_finite2 @ A @ A3 )
=> ( ( ( linord4507533701916653071of_set @ A @ A3 )
= ( nil @ A ) )
= ( A3
= ( bot_bot @ ( set @ A ) ) ) ) ) ) ).
% sorted_list_of_set.sorted_key_list_of_set_eq_Nil_iff
thf(fact_7707_sorted__list__of__set__lessThan__Suc,axiom,
! [K: nat] :
( ( linord4507533701916653071of_set @ nat @ ( set_ord_lessThan @ nat @ ( suc @ K ) ) )
= ( append @ nat @ ( linord4507533701916653071of_set @ nat @ ( set_ord_lessThan @ nat @ K ) ) @ ( cons @ nat @ K @ ( nil @ nat ) ) ) ) ).
% sorted_list_of_set_lessThan_Suc
thf(fact_7708_sorted__list__of__set_Osorted__key__list__of__set__inject,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A3: set @ A,B4: set @ A] :
( ( ( linord4507533701916653071of_set @ A @ A3 )
= ( linord4507533701916653071of_set @ A @ B4 ) )
=> ( ( finite_finite2 @ A @ A3 )
=> ( ( finite_finite2 @ A @ B4 )
=> ( A3 = B4 ) ) ) ) ) ).
% sorted_list_of_set.sorted_key_list_of_set_inject
thf(fact_7709_sorted__list__of__set__greaterThanAtMost,axiom,
! [I: nat,J: nat] :
( ( ord_less_eq @ nat @ ( suc @ I ) @ J )
=> ( ( linord4507533701916653071of_set @ nat @ ( set_or3652927894154168847AtMost @ nat @ I @ J ) )
= ( cons @ nat @ ( suc @ I ) @ ( linord4507533701916653071of_set @ nat @ ( set_or3652927894154168847AtMost @ nat @ ( suc @ I ) @ J ) ) ) ) ) ).
% sorted_list_of_set_greaterThanAtMost
thf(fact_7710_sorted__list__of__set__greaterThanLessThan,axiom,
! [I: nat,J: nat] :
( ( ord_less @ nat @ ( suc @ I ) @ J )
=> ( ( linord4507533701916653071of_set @ nat @ ( set_or5935395276787703475ssThan @ nat @ I @ J ) )
= ( cons @ nat @ ( suc @ I ) @ ( linord4507533701916653071of_set @ nat @ ( set_or5935395276787703475ssThan @ nat @ ( suc @ I ) @ J ) ) ) ) ) ).
% sorted_list_of_set_greaterThanLessThan
thf(fact_7711_set__Cons__def,axiom,
! [A: $tType] :
( ( set_Cons @ A )
= ( ^ [A7: set @ A,XS: set @ ( list @ A )] :
( collect @ ( list @ A )
@ ^ [Z7: list @ A] :
? [X2: A,Xs: list @ A] :
( ( Z7
= ( cons @ A @ X2 @ Xs ) )
& ( member @ A @ X2 @ A7 )
& ( member @ ( list @ A ) @ Xs @ XS ) ) ) ) ) ).
% set_Cons_def
thf(fact_7712_nth__sorted__list__of__set__greaterThanAtMost,axiom,
! [N: nat,J: nat,I: nat] :
( ( ord_less @ nat @ N @ ( minus_minus @ nat @ J @ I ) )
=> ( ( nth @ nat @ ( linord4507533701916653071of_set @ nat @ ( set_or3652927894154168847AtMost @ nat @ I @ J ) ) @ N )
= ( suc @ ( plus_plus @ nat @ I @ N ) ) ) ) ).
% nth_sorted_list_of_set_greaterThanAtMost
thf(fact_7713_nth__sorted__list__of__set__greaterThanLessThan,axiom,
! [N: nat,J: nat,I: nat] :
( ( ord_less @ nat @ N @ ( minus_minus @ nat @ J @ ( suc @ I ) ) )
=> ( ( nth @ nat @ ( linord4507533701916653071of_set @ nat @ ( set_or5935395276787703475ssThan @ nat @ I @ J ) ) @ N )
= ( suc @ ( plus_plus @ nat @ I @ N ) ) ) ) ).
% nth_sorted_list_of_set_greaterThanLessThan
thf(fact_7714_listset_Osimps_I1_J,axiom,
! [A: $tType] :
( ( listset @ A @ ( nil @ ( set @ A ) ) )
= ( insert @ ( list @ A ) @ ( nil @ A ) @ ( bot_bot @ ( set @ ( list @ A ) ) ) ) ) ).
% listset.simps(1)
thf(fact_7715_list_Osimps_I5_J,axiom,
! [B: $tType,A: $tType,F12: B,F22: A > ( list @ A ) > B,X21: A,X222: list @ A] :
( ( case_list @ B @ A @ F12 @ F22 @ ( cons @ A @ X21 @ X222 ) )
= ( F22 @ X21 @ X222 ) ) ).
% list.simps(5)
thf(fact_7716_list_Ocase__distrib,axiom,
! [B: $tType,C: $tType,A: $tType,H2: B > C,F12: B,F22: A > ( list @ A ) > B,List: list @ A] :
( ( H2 @ ( case_list @ B @ A @ F12 @ F22 @ List ) )
= ( case_list @ C @ A @ ( H2 @ F12 )
@ ^ [X16: A,X24: list @ A] : ( H2 @ ( F22 @ X16 @ X24 ) )
@ List ) ) ).
% list.case_distrib
thf(fact_7717_list_Osimps_I4_J,axiom,
! [A: $tType,B: $tType,F12: B,F22: A > ( list @ A ) > B] :
( ( case_list @ B @ A @ F12 @ F22 @ ( nil @ A ) )
= F12 ) ).
% list.simps(4)
thf(fact_7718_listset_Osimps_I2_J,axiom,
! [A: $tType,A3: set @ A,As3: list @ ( set @ A )] :
( ( listset @ A @ ( cons @ ( set @ A ) @ A3 @ As3 ) )
= ( set_Cons @ A @ A3 @ ( listset @ A @ As3 ) ) ) ).
% listset.simps(2)
thf(fact_7719_transpose_Oelims,axiom,
! [A: $tType,X: list @ ( list @ A ),Y: list @ ( list @ A )] :
( ( ( transpose @ A @ X )
= Y )
=> ( ( ( X
= ( nil @ ( list @ A ) ) )
=> ( Y
!= ( nil @ ( list @ A ) ) ) )
=> ( ! [Xss3: list @ ( list @ A )] :
( ( X
= ( cons @ ( list @ A ) @ ( nil @ A ) @ Xss3 ) )
=> ( Y
!= ( transpose @ A @ Xss3 ) ) )
=> ~ ! [X3: A,Xs3: list @ A,Xss3: list @ ( list @ A )] :
( ( X
= ( cons @ ( list @ A ) @ ( cons @ A @ X3 @ Xs3 ) @ Xss3 ) )
=> ( Y
!= ( cons @ ( list @ A )
@ ( cons @ A @ X3
@ ( concat @ A
@ ( map @ ( list @ A ) @ ( list @ A )
@ ( case_list @ ( list @ A ) @ A @ ( nil @ A )
@ ^ [H: A,T2: list @ A] : ( cons @ A @ H @ ( nil @ A ) ) )
@ Xss3 ) ) )
@ ( transpose @ A
@ ( cons @ ( list @ A ) @ Xs3
@ ( concat @ ( list @ A )
@ ( map @ ( list @ A ) @ ( list @ ( list @ A ) )
@ ( case_list @ ( list @ ( list @ A ) ) @ A @ ( nil @ ( list @ A ) )
@ ^ [H: A,T2: list @ A] : ( cons @ ( list @ A ) @ T2 @ ( nil @ ( list @ A ) ) ) )
@ Xss3 ) ) ) ) ) ) ) ) ) ) ).
% transpose.elims
thf(fact_7720_transpose_Osimps_I3_J,axiom,
! [A: $tType,X: A,Xs2: list @ A,Xss: list @ ( list @ A )] :
( ( transpose @ A @ ( cons @ ( list @ A ) @ ( cons @ A @ X @ Xs2 ) @ Xss ) )
= ( cons @ ( list @ A )
@ ( cons @ A @ X
@ ( concat @ A
@ ( map @ ( list @ A ) @ ( list @ A )
@ ( case_list @ ( list @ A ) @ A @ ( nil @ A )
@ ^ [H: A,T2: list @ A] : ( cons @ A @ H @ ( nil @ A ) ) )
@ Xss ) ) )
@ ( transpose @ A
@ ( cons @ ( list @ A ) @ Xs2
@ ( concat @ ( list @ A )
@ ( map @ ( list @ A ) @ ( list @ ( list @ A ) )
@ ( case_list @ ( list @ ( list @ A ) ) @ A @ ( nil @ ( list @ A ) )
@ ^ [H: A,T2: list @ A] : ( cons @ ( list @ A ) @ T2 @ ( nil @ ( list @ A ) ) ) )
@ Xss ) ) ) ) ) ) ).
% transpose.simps(3)
thf(fact_7721_min__list_Oelims,axiom,
! [A: $tType] :
( ( ord @ A )
=> ! [X: list @ A,Y: A] :
( ( ( min_list @ A @ X )
= Y )
=> ( ! [X3: A,Xs3: list @ A] :
( ( X
= ( cons @ A @ X3 @ Xs3 ) )
=> ( Y
!= ( case_list @ A @ A @ X3
@ ^ [A4: A,List2: list @ A] : ( ord_min @ A @ X3 @ ( min_list @ A @ Xs3 ) )
@ Xs3 ) ) )
=> ~ ( ( X
= ( nil @ A ) )
=> ( Y
!= ( undefined @ A ) ) ) ) ) ) ).
% min_list.elims
thf(fact_7722_transpose_Opsimps_I3_J,axiom,
! [A: $tType,X: A,Xs2: list @ A,Xss: list @ ( list @ A )] :
( ( accp @ ( list @ ( list @ A ) ) @ ( transpose_rel @ A ) @ ( cons @ ( list @ A ) @ ( cons @ A @ X @ Xs2 ) @ Xss ) )
=> ( ( transpose @ A @ ( cons @ ( list @ A ) @ ( cons @ A @ X @ Xs2 ) @ Xss ) )
= ( cons @ ( list @ A )
@ ( cons @ A @ X
@ ( concat @ A
@ ( map @ ( list @ A ) @ ( list @ A )
@ ( case_list @ ( list @ A ) @ A @ ( nil @ A )
@ ^ [H: A,T2: list @ A] : ( cons @ A @ H @ ( nil @ A ) ) )
@ Xss ) ) )
@ ( transpose @ A
@ ( cons @ ( list @ A ) @ Xs2
@ ( concat @ ( list @ A )
@ ( map @ ( list @ A ) @ ( list @ ( list @ A ) )
@ ( case_list @ ( list @ ( list @ A ) ) @ A @ ( nil @ ( list @ A ) )
@ ^ [H: A,T2: list @ A] : ( cons @ ( list @ A ) @ T2 @ ( nil @ ( list @ A ) ) ) )
@ Xss ) ) ) ) ) ) ) ).
% transpose.psimps(3)
thf(fact_7723_list_Odisc__eq__case_I1_J,axiom,
! [A: $tType,List: list @ A] :
( ( List
= ( nil @ A ) )
= ( case_list @ $o @ A @ $true
@ ^ [Uu3: A,Uv3: list @ A] : $false
@ List ) ) ).
% list.disc_eq_case(1)
thf(fact_7724_list_Odisc__eq__case_I2_J,axiom,
! [A: $tType,List: list @ A] :
( ( List
!= ( nil @ A ) )
= ( case_list @ $o @ A @ $false
@ ^ [Uu3: A,Uv3: list @ A] : $true
@ List ) ) ).
% list.disc_eq_case(2)
thf(fact_7725_transpose_Opsimps_I2_J,axiom,
! [A: $tType,Xss: list @ ( list @ A )] :
( ( accp @ ( list @ ( list @ A ) ) @ ( transpose_rel @ A ) @ ( cons @ ( list @ A ) @ ( nil @ A ) @ Xss ) )
=> ( ( transpose @ A @ ( cons @ ( list @ A ) @ ( nil @ A ) @ Xss ) )
= ( transpose @ A @ Xss ) ) ) ).
% transpose.psimps(2)
thf(fact_7726_transpose_Opsimps_I1_J,axiom,
! [A: $tType] :
( ( accp @ ( list @ ( list @ A ) ) @ ( transpose_rel @ A ) @ ( nil @ ( list @ A ) ) )
=> ( ( transpose @ A @ ( nil @ ( list @ A ) ) )
= ( nil @ ( list @ A ) ) ) ) ).
% transpose.psimps(1)
thf(fact_7727_transpose_Opinduct,axiom,
! [A: $tType,A0: list @ ( list @ A ),P: ( list @ ( list @ A ) ) > $o] :
( ( accp @ ( list @ ( list @ A ) ) @ ( transpose_rel @ A ) @ A0 )
=> ( ( ( accp @ ( list @ ( list @ A ) ) @ ( transpose_rel @ A ) @ ( nil @ ( list @ A ) ) )
=> ( P @ ( nil @ ( list @ A ) ) ) )
=> ( ! [Xss3: list @ ( list @ A )] :
( ( accp @ ( list @ ( list @ A ) ) @ ( transpose_rel @ A ) @ ( cons @ ( list @ A ) @ ( nil @ A ) @ Xss3 ) )
=> ( ( P @ Xss3 )
=> ( P @ ( cons @ ( list @ A ) @ ( nil @ A ) @ Xss3 ) ) ) )
=> ( ! [X3: A,Xs3: list @ A,Xss3: list @ ( list @ A )] :
( ( accp @ ( list @ ( list @ A ) ) @ ( transpose_rel @ A ) @ ( cons @ ( list @ A ) @ ( cons @ A @ X3 @ Xs3 ) @ Xss3 ) )
=> ( ( P
@ ( cons @ ( list @ A ) @ Xs3
@ ( concat @ ( list @ A )
@ ( map @ ( list @ A ) @ ( list @ ( list @ A ) )
@ ( case_list @ ( list @ ( list @ A ) ) @ A @ ( nil @ ( list @ A ) )
@ ^ [H: A,T2: list @ A] : ( cons @ ( list @ A ) @ T2 @ ( nil @ ( list @ A ) ) ) )
@ Xss3 ) ) ) )
=> ( P @ ( cons @ ( list @ A ) @ ( cons @ A @ X3 @ Xs3 ) @ Xss3 ) ) ) )
=> ( P @ A0 ) ) ) ) ) ).
% transpose.pinduct
thf(fact_7728_min__list_Osimps,axiom,
! [A: $tType] :
( ( ord @ A )
=> ! [X: A,Xs2: list @ A] :
( ( min_list @ A @ ( cons @ A @ X @ Xs2 ) )
= ( case_list @ A @ A @ X
@ ^ [A4: A,List2: list @ A] : ( ord_min @ A @ X @ ( min_list @ A @ Xs2 ) )
@ Xs2 ) ) ) ).
% min_list.simps
thf(fact_7729_transpose_Opelims,axiom,
! [A: $tType,X: list @ ( list @ A ),Y: list @ ( list @ A )] :
( ( ( transpose @ A @ X )
= Y )
=> ( ( accp @ ( list @ ( list @ A ) ) @ ( transpose_rel @ A ) @ X )
=> ( ( ( X
= ( nil @ ( list @ A ) ) )
=> ( ( Y
= ( nil @ ( list @ A ) ) )
=> ~ ( accp @ ( list @ ( list @ A ) ) @ ( transpose_rel @ A ) @ ( nil @ ( list @ A ) ) ) ) )
=> ( ! [Xss3: list @ ( list @ A )] :
( ( X
= ( cons @ ( list @ A ) @ ( nil @ A ) @ Xss3 ) )
=> ( ( Y
= ( transpose @ A @ Xss3 ) )
=> ~ ( accp @ ( list @ ( list @ A ) ) @ ( transpose_rel @ A ) @ ( cons @ ( list @ A ) @ ( nil @ A ) @ Xss3 ) ) ) )
=> ~ ! [X3: A,Xs3: list @ A,Xss3: list @ ( list @ A )] :
( ( X
= ( cons @ ( list @ A ) @ ( cons @ A @ X3 @ Xs3 ) @ Xss3 ) )
=> ( ( Y
= ( cons @ ( list @ A )
@ ( cons @ A @ X3
@ ( concat @ A
@ ( map @ ( list @ A ) @ ( list @ A )
@ ( case_list @ ( list @ A ) @ A @ ( nil @ A )
@ ^ [H: A,T2: list @ A] : ( cons @ A @ H @ ( nil @ A ) ) )
@ Xss3 ) ) )
@ ( transpose @ A
@ ( cons @ ( list @ A ) @ Xs3
@ ( concat @ ( list @ A )
@ ( map @ ( list @ A ) @ ( list @ ( list @ A ) )
@ ( case_list @ ( list @ ( list @ A ) ) @ A @ ( nil @ ( list @ A ) )
@ ^ [H: A,T2: list @ A] : ( cons @ ( list @ A ) @ T2 @ ( nil @ ( list @ A ) ) ) )
@ Xss3 ) ) ) ) ) )
=> ~ ( accp @ ( list @ ( list @ A ) ) @ ( transpose_rel @ A ) @ ( cons @ ( list @ A ) @ ( cons @ A @ X3 @ Xs3 ) @ Xss3 ) ) ) ) ) ) ) ) ).
% transpose.pelims
thf(fact_7730_min__list_Opelims,axiom,
! [A: $tType] :
( ( ord @ A )
=> ! [X: list @ A,Y: A] :
( ( ( min_list @ A @ X )
= Y )
=> ( ( accp @ ( list @ A ) @ ( min_list_rel @ A ) @ X )
=> ( ! [X3: A,Xs3: list @ A] :
( ( X
= ( cons @ A @ X3 @ Xs3 ) )
=> ( ( Y
= ( case_list @ A @ A @ X3
@ ^ [A4: A,List2: list @ A] : ( ord_min @ A @ X3 @ ( min_list @ A @ Xs3 ) )
@ Xs3 ) )
=> ~ ( accp @ ( list @ A ) @ ( min_list_rel @ A ) @ ( cons @ A @ X3 @ Xs3 ) ) ) )
=> ~ ( ( X
= ( nil @ A ) )
=> ( ( Y
= ( undefined @ A ) )
=> ~ ( accp @ ( list @ A ) @ ( min_list_rel @ A ) @ ( nil @ A ) ) ) ) ) ) ) ) ).
% min_list.pelims
thf(fact_7731_filter__upt__last,axiom,
! [A: $tType,P: A > $o,L: list @ A,Js2: list @ nat,J: nat,I: nat] :
( ( ( filter2 @ nat
@ ^ [K3: nat] : ( P @ ( nth @ A @ L @ K3 ) )
@ ( upt @ ( zero_zero @ nat ) @ ( size_size @ ( list @ A ) @ L ) ) )
= ( append @ nat @ Js2 @ ( cons @ nat @ J @ ( nil @ nat ) ) ) )
=> ( ( ord_less @ nat @ J @ I )
=> ( ( ord_less @ nat @ I @ ( size_size @ ( list @ A ) @ L ) )
=> ~ ( P @ ( nth @ A @ L @ I ) ) ) ) ) ).
% filter_upt_last
thf(fact_7732_filter__filter,axiom,
! [A: $tType,P: A > $o,Q: A > $o,Xs2: list @ A] :
( ( filter2 @ A @ P @ ( filter2 @ A @ Q @ Xs2 ) )
= ( filter2 @ A
@ ^ [X2: A] :
( ( Q @ X2 )
& ( P @ X2 ) )
@ Xs2 ) ) ).
% filter_filter
thf(fact_7733_filter__True,axiom,
! [A: $tType,Xs2: list @ A,P: A > $o] :
( ! [X3: A] :
( ( member @ A @ X3 @ ( set2 @ A @ Xs2 ) )
=> ( P @ X3 ) )
=> ( ( filter2 @ A @ P @ Xs2 )
= Xs2 ) ) ).
% filter_True
thf(fact_7734_filter__append,axiom,
! [A: $tType,P: A > $o,Xs2: list @ A,Ys: list @ A] :
( ( filter2 @ A @ P @ ( append @ A @ Xs2 @ Ys ) )
= ( append @ A @ ( filter2 @ A @ P @ Xs2 ) @ ( filter2 @ A @ P @ Ys ) ) ) ).
% filter_append
thf(fact_7735_set__filter,axiom,
! [A: $tType,P: A > $o,Xs2: list @ A] :
( ( set2 @ A @ ( filter2 @ A @ P @ Xs2 ) )
= ( collect @ A
@ ^ [X2: A] :
( ( member @ A @ X2 @ ( set2 @ A @ Xs2 ) )
& ( P @ X2 ) ) ) ) ).
% set_filter
thf(fact_7736_filter__False,axiom,
! [A: $tType,Xs2: list @ A,P: A > $o] :
( ! [X3: A] :
( ( member @ A @ X3 @ ( set2 @ A @ Xs2 ) )
=> ~ ( P @ X3 ) )
=> ( ( filter2 @ A @ P @ Xs2 )
= ( nil @ A ) ) ) ).
% filter_False
thf(fact_7737_length__filter__map,axiom,
! [A: $tType,B: $tType,P: A > $o,F2: B > A,Xs2: list @ B] :
( ( size_size @ ( list @ A ) @ ( filter2 @ A @ P @ ( map @ B @ A @ F2 @ Xs2 ) ) )
= ( size_size @ ( list @ B ) @ ( filter2 @ B @ ( comp @ A @ $o @ B @ P @ F2 ) @ Xs2 ) ) ) ).
% length_filter_map
thf(fact_7738_filter_Osimps_I2_J,axiom,
! [A: $tType,P: A > $o,X: A,Xs2: list @ A] :
( ( ( P @ X )
=> ( ( filter2 @ A @ P @ ( cons @ A @ X @ Xs2 ) )
= ( cons @ A @ X @ ( filter2 @ A @ P @ Xs2 ) ) ) )
& ( ~ ( P @ X )
=> ( ( filter2 @ A @ P @ ( cons @ A @ X @ Xs2 ) )
= ( filter2 @ A @ P @ Xs2 ) ) ) ) ).
% filter.simps(2)
thf(fact_7739_Cons__eq__filterD,axiom,
! [A: $tType,X: A,Xs2: list @ A,P: A > $o,Ys: list @ A] :
( ( ( cons @ A @ X @ Xs2 )
= ( filter2 @ A @ P @ Ys ) )
=> ? [Us3: list @ A,Vs3: list @ A] :
( ( Ys
= ( append @ A @ Us3 @ ( cons @ A @ X @ Vs3 ) ) )
& ! [X5: A] :
( ( member @ A @ X5 @ ( set2 @ A @ Us3 ) )
=> ~ ( P @ X5 ) )
& ( P @ X )
& ( Xs2
= ( filter2 @ A @ P @ Vs3 ) ) ) ) ).
% Cons_eq_filterD
thf(fact_7740_filter__eq__ConsD,axiom,
! [A: $tType,P: A > $o,Ys: list @ A,X: A,Xs2: list @ A] :
( ( ( filter2 @ A @ P @ Ys )
= ( cons @ A @ X @ Xs2 ) )
=> ? [Us3: list @ A,Vs3: list @ A] :
( ( Ys
= ( append @ A @ Us3 @ ( cons @ A @ X @ Vs3 ) ) )
& ! [X5: A] :
( ( member @ A @ X5 @ ( set2 @ A @ Us3 ) )
=> ~ ( P @ X5 ) )
& ( P @ X )
& ( Xs2
= ( filter2 @ A @ P @ Vs3 ) ) ) ) ).
% filter_eq_ConsD
thf(fact_7741_Cons__eq__filter__iff,axiom,
! [A: $tType,X: A,Xs2: list @ A,P: A > $o,Ys: list @ A] :
( ( ( cons @ A @ X @ Xs2 )
= ( filter2 @ A @ P @ Ys ) )
= ( ? [Us2: list @ A,Vs2: list @ A] :
( ( Ys
= ( append @ A @ Us2 @ ( cons @ A @ X @ Vs2 ) ) )
& ! [X2: A] :
( ( member @ A @ X2 @ ( set2 @ A @ Us2 ) )
=> ~ ( P @ X2 ) )
& ( P @ X )
& ( Xs2
= ( filter2 @ A @ P @ Vs2 ) ) ) ) ) ).
% Cons_eq_filter_iff
thf(fact_7742_filter__eq__Cons__iff,axiom,
! [A: $tType,P: A > $o,Ys: list @ A,X: A,Xs2: list @ A] :
( ( ( filter2 @ A @ P @ Ys )
= ( cons @ A @ X @ Xs2 ) )
= ( ? [Us2: list @ A,Vs2: list @ A] :
( ( Ys
= ( append @ A @ Us2 @ ( cons @ A @ X @ Vs2 ) ) )
& ! [X2: A] :
( ( member @ A @ X2 @ ( set2 @ A @ Us2 ) )
=> ~ ( P @ X2 ) )
& ( P @ X )
& ( Xs2
= ( filter2 @ A @ P @ Vs2 ) ) ) ) ) ).
% filter_eq_Cons_iff
thf(fact_7743_length__filter__less,axiom,
! [A: $tType,X: A,Xs2: list @ A,P: A > $o] :
( ( member @ A @ X @ ( set2 @ A @ Xs2 ) )
=> ( ~ ( P @ X )
=> ( ord_less @ nat @ ( size_size @ ( list @ A ) @ ( filter2 @ A @ P @ Xs2 ) ) @ ( size_size @ ( list @ A ) @ Xs2 ) ) ) ) ).
% length_filter_less
thf(fact_7744_sum__length__filter__compl,axiom,
! [A: $tType,P: A > $o,Xs2: list @ A] :
( ( plus_plus @ nat @ ( size_size @ ( list @ A ) @ ( filter2 @ A @ P @ Xs2 ) )
@ ( size_size @ ( list @ A )
@ ( filter2 @ A
@ ^ [X2: A] :
~ ( P @ X2 )
@ Xs2 ) ) )
= ( size_size @ ( list @ A ) @ Xs2 ) ) ).
% sum_length_filter_compl
thf(fact_7745_replicate__length__filter,axiom,
! [A: $tType,X: A,Xs2: list @ A] :
( ( replicate @ A
@ ( size_size @ ( list @ A )
@ ( filter2 @ A
@ ( ^ [Y2: A,Z2: A] : Y2 = Z2
@ X )
@ Xs2 ) )
@ X )
= ( filter2 @ A
@ ( ^ [Y2: A,Z2: A] : Y2 = Z2
@ X )
@ Xs2 ) ) ).
% replicate_length_filter
thf(fact_7746_length__filter__le,axiom,
! [A: $tType,P: A > $o,Xs2: list @ A] : ( ord_less_eq @ nat @ ( size_size @ ( list @ A ) @ ( filter2 @ A @ P @ Xs2 ) ) @ ( size_size @ ( list @ A ) @ Xs2 ) ) ).
% length_filter_le
thf(fact_7747_filter__cong,axiom,
! [A: $tType,Xs2: list @ A,Ys: list @ A,P: A > $o,Q: A > $o] :
( ( Xs2 = Ys )
=> ( ! [X3: A] :
( ( member @ A @ X3 @ ( set2 @ A @ Ys ) )
=> ( ( P @ X3 )
= ( Q @ X3 ) ) )
=> ( ( filter2 @ A @ P @ Xs2 )
= ( filter2 @ A @ Q @ Ys ) ) ) ) ).
% filter_cong
thf(fact_7748_filter__id__conv,axiom,
! [A: $tType,P: A > $o,Xs2: list @ A] :
( ( ( filter2 @ A @ P @ Xs2 )
= Xs2 )
= ( ! [X2: A] :
( ( member @ A @ X2 @ ( set2 @ A @ Xs2 ) )
=> ( P @ X2 ) ) ) ) ).
% filter_id_conv
thf(fact_7749_filter__is__subset,axiom,
! [A: $tType,P: A > $o,Xs2: list @ A] : ( ord_less_eq @ ( set @ A ) @ ( set2 @ A @ ( filter2 @ A @ P @ Xs2 ) ) @ ( set2 @ A @ Xs2 ) ) ).
% filter_is_subset
thf(fact_7750_inj__on__filter__key__eq,axiom,
! [B: $tType,A: $tType,F2: A > B,Y: A,Xs2: list @ A] :
( ( inj_on @ A @ B @ F2 @ ( insert @ A @ Y @ ( set2 @ A @ Xs2 ) ) )
=> ( ( filter2 @ A
@ ^ [X2: A] :
( ( F2 @ Y )
= ( F2 @ X2 ) )
@ Xs2 )
= ( filter2 @ A
@ ( ^ [Y2: A,Z2: A] : Y2 = Z2
@ Y )
@ Xs2 ) ) ) ).
% inj_on_filter_key_eq
thf(fact_7751_filter__concat,axiom,
! [A: $tType,P4: A > $o,Xs2: list @ ( list @ A )] :
( ( filter2 @ A @ P4 @ ( concat @ A @ Xs2 ) )
= ( concat @ A @ ( map @ ( list @ A ) @ ( list @ A ) @ ( filter2 @ A @ P4 ) @ Xs2 ) ) ) ).
% filter_concat
thf(fact_7752_filter__map,axiom,
! [A: $tType,B: $tType,P: A > $o,F2: B > A,Xs2: list @ B] :
( ( filter2 @ A @ P @ ( map @ B @ A @ F2 @ Xs2 ) )
= ( map @ B @ A @ F2 @ ( filter2 @ B @ ( comp @ A @ $o @ B @ P @ F2 ) @ Xs2 ) ) ) ).
% filter_map
thf(fact_7753_filter__replicate,axiom,
! [A: $tType,P: A > $o,X: A,N: nat] :
( ( ( P @ X )
=> ( ( filter2 @ A @ P @ ( replicate @ A @ N @ X ) )
= ( replicate @ A @ N @ X ) ) )
& ( ~ ( P @ X )
=> ( ( filter2 @ A @ P @ ( replicate @ A @ N @ X ) )
= ( nil @ A ) ) ) ) ).
% filter_replicate
thf(fact_7754_filter_Osimps_I1_J,axiom,
! [A: $tType,P: A > $o] :
( ( filter2 @ A @ P @ ( nil @ A ) )
= ( nil @ A ) ) ).
% filter.simps(1)
thf(fact_7755_empty__filter__conv,axiom,
! [A: $tType,P: A > $o,Xs2: list @ A] :
( ( ( nil @ A )
= ( filter2 @ A @ P @ Xs2 ) )
= ( ! [X2: A] :
( ( member @ A @ X2 @ ( set2 @ A @ Xs2 ) )
=> ~ ( P @ X2 ) ) ) ) ).
% empty_filter_conv
thf(fact_7756_filter__empty__conv,axiom,
! [A: $tType,P: A > $o,Xs2: list @ A] :
( ( ( filter2 @ A @ P @ Xs2 )
= ( nil @ A ) )
= ( ! [X2: A] :
( ( member @ A @ X2 @ ( set2 @ A @ Xs2 ) )
=> ~ ( P @ X2 ) ) ) ) ).
% filter_empty_conv
thf(fact_7757_set__minus__filter__out,axiom,
! [A: $tType,Xs2: list @ A,Y: A] :
( ( minus_minus @ ( set @ A ) @ ( set2 @ A @ Xs2 ) @ ( insert @ A @ Y @ ( bot_bot @ ( set @ A ) ) ) )
= ( set2 @ A
@ ( filter2 @ A
@ ^ [X2: A] : X2 != Y
@ Xs2 ) ) ) ).
% set_minus_filter_out
thf(fact_7758_upt__filter__extend,axiom,
! [U: nat,U4: nat,P: nat > $o] :
( ( ord_less_eq @ nat @ U @ U4 )
=> ( ! [I3: nat] :
( ( ( ord_less_eq @ nat @ U @ I3 )
& ( ord_less @ nat @ I3 @ U4 ) )
=> ~ ( P @ I3 ) )
=> ( ( filter2 @ nat @ P @ ( upt @ ( zero_zero @ nat ) @ U ) )
= ( filter2 @ nat @ P @ ( upt @ ( zero_zero @ nat ) @ U4 ) ) ) ) ) ).
% upt_filter_extend
thf(fact_7759_sort__by__quicksort,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [Xs2: list @ A] :
( ( linorder_sort_key @ A @ A
@ ^ [X2: A] : X2
@ Xs2 )
= ( append @ A
@ ( linorder_sort_key @ A @ A
@ ^ [X2: A] : X2
@ ( filter2 @ A
@ ^ [X2: A] : ( ord_less @ A @ X2 @ ( nth @ A @ Xs2 @ ( divide_divide @ nat @ ( size_size @ ( list @ A ) @ Xs2 ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) )
@ Xs2 ) )
@ ( append @ A
@ ( filter2 @ A
@ ^ [X2: A] :
( X2
= ( nth @ A @ Xs2 @ ( divide_divide @ nat @ ( size_size @ ( list @ A ) @ Xs2 ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) )
@ Xs2 )
@ ( linorder_sort_key @ A @ A
@ ^ [X2: A] : X2
@ ( filter2 @ A @ ( ord_less @ A @ ( nth @ A @ Xs2 @ ( divide_divide @ nat @ ( size_size @ ( list @ A ) @ Xs2 ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ Xs2 ) ) ) ) ) ) ).
% sort_by_quicksort
thf(fact_7760_sort__key__by__quicksort,axiom,
! [A: $tType,B: $tType] :
( ( linorder @ A )
=> ( ( linorder_sort_key @ B @ A )
= ( ^ [F4: B > A,Xs: list @ B] :
( append @ B
@ ( linorder_sort_key @ B @ A @ F4
@ ( filter2 @ B
@ ^ [X2: B] : ( ord_less @ A @ ( F4 @ X2 ) @ ( F4 @ ( nth @ B @ Xs @ ( divide_divide @ nat @ ( size_size @ ( list @ B ) @ Xs ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) )
@ Xs ) )
@ ( append @ B
@ ( filter2 @ B
@ ^ [X2: B] :
( ( F4 @ X2 )
= ( F4 @ ( nth @ B @ Xs @ ( divide_divide @ nat @ ( size_size @ ( list @ B ) @ Xs ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) )
@ Xs )
@ ( linorder_sort_key @ B @ A @ F4
@ ( filter2 @ B
@ ^ [X2: B] : ( ord_less @ A @ ( F4 @ ( nth @ B @ Xs @ ( divide_divide @ nat @ ( size_size @ ( list @ B ) @ Xs ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( F4 @ X2 ) )
@ Xs ) ) ) ) ) ) ) ).
% sort_key_by_quicksort
thf(fact_7761_sort__upt,axiom,
! [M: nat,N: nat] :
( ( linorder_sort_key @ nat @ nat
@ ^ [X2: nat] : X2
@ ( upt @ M @ N ) )
= ( upt @ M @ N ) ) ).
% sort_upt
thf(fact_7762_sort__upto,axiom,
! [I: int,J: int] :
( ( linorder_sort_key @ int @ int
@ ^ [X2: int] : X2
@ ( upto @ I @ J ) )
= ( upto @ I @ J ) ) ).
% sort_upto
thf(fact_7763_sort__key__simps_I1_J,axiom,
! [A: $tType,B: $tType] :
( ( linorder @ A )
=> ! [F2: B > A] :
( ( linorder_sort_key @ B @ A @ F2 @ ( nil @ B ) )
= ( nil @ B ) ) ) ).
% sort_key_simps(1)
thf(fact_7764_length__sort,axiom,
! [A: $tType,B: $tType] :
( ( linorder @ A )
=> ! [F2: B > A,Xs2: list @ B] :
( ( size_size @ ( list @ B ) @ ( linorder_sort_key @ B @ A @ F2 @ Xs2 ) )
= ( size_size @ ( list @ B ) @ Xs2 ) ) ) ).
% length_sort
thf(fact_7765_set__sort,axiom,
! [A: $tType,B: $tType] :
( ( linorder @ A )
=> ! [F2: B > A,Xs2: list @ B] :
( ( set2 @ B @ ( linorder_sort_key @ B @ A @ F2 @ Xs2 ) )
= ( set2 @ B @ Xs2 ) ) ) ).
% set_sort
thf(fact_7766_filter__sort,axiom,
! [A: $tType,B: $tType] :
( ( linorder @ A )
=> ! [P: B > $o,F2: B > A,Xs2: list @ B] :
( ( filter2 @ B @ P @ ( linorder_sort_key @ B @ A @ F2 @ Xs2 ) )
= ( linorder_sort_key @ B @ A @ F2 @ ( filter2 @ B @ P @ Xs2 ) ) ) ) ).
% filter_sort
thf(fact_7767_sort__key__stable,axiom,
! [B: $tType,A: $tType] :
( ( linorder @ B )
=> ! [F2: A > B,K: B,Xs2: list @ A] :
( ( filter2 @ A
@ ^ [Y6: A] :
( ( F2 @ Y6 )
= K )
@ ( linorder_sort_key @ A @ B @ F2 @ Xs2 ) )
= ( filter2 @ A
@ ^ [Y6: A] :
( ( F2 @ Y6 )
= K )
@ Xs2 ) ) ) ).
% sort_key_stable
thf(fact_7768_sort__key__const,axiom,
! [B: $tType,A: $tType] :
( ( linorder @ B )
=> ! [C2: B,Xs2: list @ A] :
( ( linorder_sort_key @ A @ B
@ ^ [X2: A] : C2
@ Xs2 )
= Xs2 ) ) ).
% sort_key_const
thf(fact_7769_transpose__aux__max,axiom,
! [A: $tType,B: $tType,Xs2: list @ A,Xss: list @ ( list @ B )] :
( ( ord_max @ nat @ ( suc @ ( size_size @ ( list @ A ) @ Xs2 ) )
@ ( foldr @ ( list @ B ) @ nat
@ ^ [Xs: list @ B] : ( ord_max @ nat @ ( size_size @ ( list @ B ) @ Xs ) )
@ Xss
@ ( zero_zero @ nat ) ) )
= ( suc
@ ( ord_max @ nat @ ( size_size @ ( list @ A ) @ Xs2 )
@ ( foldr @ ( list @ B ) @ nat
@ ^ [X2: list @ B] : ( ord_max @ nat @ ( minus_minus @ nat @ ( size_size @ ( list @ B ) @ X2 ) @ ( suc @ ( zero_zero @ nat ) ) ) )
@ ( filter2 @ ( list @ B )
@ ^ [Ys3: list @ B] :
( Ys3
!= ( nil @ B ) )
@ Xss )
@ ( zero_zero @ nat ) ) ) ) ) ).
% transpose_aux_max
thf(fact_7770_nth__transpose,axiom,
! [A: $tType,I: nat,Xs2: list @ ( list @ A )] :
( ( ord_less @ nat @ I @ ( size_size @ ( list @ ( list @ A ) ) @ ( transpose @ A @ Xs2 ) ) )
=> ( ( nth @ ( list @ A ) @ ( transpose @ A @ Xs2 ) @ I )
= ( map @ ( list @ A ) @ A
@ ^ [Xs: list @ A] : ( nth @ A @ Xs @ I )
@ ( filter2 @ ( list @ A )
@ ^ [Ys3: list @ A] : ( ord_less @ nat @ I @ ( size_size @ ( list @ A ) @ Ys3 ) )
@ Xs2 ) ) ) ) ).
% nth_transpose
thf(fact_7771_transpose__max__length,axiom,
! [A: $tType,Xs2: list @ ( list @ A )] :
( ( foldr @ ( list @ A ) @ nat
@ ^ [Xs: list @ A] : ( ord_max @ nat @ ( size_size @ ( list @ A ) @ Xs ) )
@ ( transpose @ A @ Xs2 )
@ ( zero_zero @ nat ) )
= ( size_size @ ( list @ ( list @ A ) )
@ ( filter2 @ ( list @ A )
@ ^ [X2: list @ A] :
( X2
!= ( nil @ A ) )
@ Xs2 ) ) ) ).
% transpose_max_length
thf(fact_7772_map__filter__def,axiom,
! [B: $tType,A: $tType] :
( ( map_filter @ A @ B )
= ( ^ [F4: A > ( option @ B ),Xs: list @ A] :
( map @ A @ B @ ( comp @ ( option @ B ) @ B @ A @ ( the2 @ B ) @ F4 )
@ ( filter2 @ A
@ ^ [X2: A] :
( ( F4 @ X2 )
!= ( none @ B ) )
@ Xs ) ) ) ) ).
% map_filter_def
thf(fact_7773_Bleast__code,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [Xs2: list @ A,P: A > $o] :
( ( bleast @ A @ ( set2 @ A @ Xs2 ) @ P )
= ( case_list @ A @ A @ ( abort_Bleast @ A @ ( set2 @ A @ Xs2 ) @ P )
@ ^ [X2: A,Xs: list @ A] : X2
@ ( filter2 @ A @ P
@ ( linorder_sort_key @ A @ A
@ ^ [X2: A] : X2
@ Xs2 ) ) ) ) ) ).
% Bleast_code
thf(fact_7774_map__filter__simps_I1_J,axiom,
! [A: $tType,B: $tType,F2: B > ( option @ A ),X: B,Xs2: list @ B] :
( ( map_filter @ B @ A @ F2 @ ( cons @ B @ X @ Xs2 ) )
= ( case_option @ ( list @ A ) @ A @ ( map_filter @ B @ A @ F2 @ Xs2 )
@ ^ [Y6: A] : ( cons @ A @ Y6 @ ( map_filter @ B @ A @ F2 @ Xs2 ) )
@ ( F2 @ X ) ) ) ).
% map_filter_simps(1)
thf(fact_7775_map__filter__simps_I2_J,axiom,
! [B: $tType,A: $tType,F2: B > ( option @ A )] :
( ( map_filter @ B @ A @ F2 @ ( nil @ B ) )
= ( nil @ A ) ) ).
% map_filter_simps(2)
thf(fact_7776_Bleast__def,axiom,
! [A: $tType] :
( ( ord @ A )
=> ( ( bleast @ A )
= ( ^ [S7: set @ A,P2: A > $o] :
( ord_Least @ A
@ ^ [X2: A] :
( ( member @ A @ X2 @ S7 )
& ( P2 @ X2 ) ) ) ) ) ) ).
% Bleast_def
thf(fact_7777_abort__Bleast__def,axiom,
! [A: $tType] :
( ( ord @ A )
=> ( ( abort_Bleast @ A )
= ( ^ [S7: set @ A,P2: A > $o] :
( ord_Least @ A
@ ^ [X2: A] :
( ( member @ A @ X2 @ S7 )
& ( P2 @ X2 ) ) ) ) ) ) ).
% abort_Bleast_def
thf(fact_7778_map__filter__map__filter,axiom,
! [A: $tType,B: $tType,F2: B > A,P: B > $o,Xs2: list @ B] :
( ( map @ B @ A @ F2 @ ( filter2 @ B @ P @ Xs2 ) )
= ( map_filter @ B @ A
@ ^ [X2: B] : ( if @ ( option @ A ) @ ( P @ X2 ) @ ( some @ A @ ( F2 @ X2 ) ) @ ( none @ A ) )
@ Xs2 ) ) ).
% map_filter_map_filter
thf(fact_7779_transpose__aux__filter__tail,axiom,
! [A: $tType,Xss: list @ ( list @ A )] :
( ( concat @ ( list @ A )
@ ( map @ ( list @ A ) @ ( list @ ( list @ A ) )
@ ( case_list @ ( list @ ( list @ A ) ) @ A @ ( nil @ ( list @ A ) )
@ ^ [H: A,T2: list @ A] : ( cons @ ( list @ A ) @ T2 @ ( nil @ ( list @ A ) ) ) )
@ Xss ) )
= ( map @ ( list @ A ) @ ( list @ A ) @ ( tl @ A )
@ ( filter2 @ ( list @ A )
@ ^ [Ys3: list @ A] :
( Ys3
!= ( nil @ A ) )
@ Xss ) ) ) ).
% transpose_aux_filter_tail
thf(fact_7780_transpose__aux__filter__head,axiom,
! [A: $tType,Xss: list @ ( list @ A )] :
( ( concat @ A
@ ( map @ ( list @ A ) @ ( list @ A )
@ ( case_list @ ( list @ A ) @ A @ ( nil @ A )
@ ^ [H: A,T2: list @ A] : ( cons @ A @ H @ ( nil @ A ) ) )
@ Xss ) )
= ( map @ ( list @ A ) @ A @ ( hd @ A )
@ ( filter2 @ ( list @ A )
@ ^ [Ys3: list @ A] :
( Ys3
!= ( nil @ A ) )
@ Xss ) ) ) ).
% transpose_aux_filter_head
thf(fact_7781_tl__upt,axiom,
! [M: nat,N: nat] :
( ( tl @ nat @ ( upt @ M @ N ) )
= ( upt @ ( suc @ M ) @ N ) ) ).
% tl_upt
thf(fact_7782_hd__upt,axiom,
! [I: nat,J: nat] :
( ( ord_less @ nat @ I @ J )
=> ( ( hd @ nat @ ( upt @ I @ J ) )
= I ) ) ).
% hd_upt
thf(fact_7783_hd__append2,axiom,
! [A: $tType,Xs2: list @ A,Ys: list @ A] :
( ( Xs2
!= ( nil @ A ) )
=> ( ( hd @ A @ ( append @ A @ Xs2 @ Ys ) )
= ( hd @ A @ Xs2 ) ) ) ).
% hd_append2
thf(fact_7784_tl__append2,axiom,
! [A: $tType,Xs2: list @ A,Ys: list @ A] :
( ( Xs2
!= ( nil @ A ) )
=> ( ( tl @ A @ ( append @ A @ Xs2 @ Ys ) )
= ( append @ A @ ( tl @ A @ Xs2 ) @ Ys ) ) ) ).
% tl_append2
thf(fact_7785_hd__replicate,axiom,
! [A: $tType,N: nat,X: A] :
( ( N
!= ( zero_zero @ nat ) )
=> ( ( hd @ A @ ( replicate @ A @ N @ X ) )
= X ) ) ).
% hd_replicate
thf(fact_7786_length__tl,axiom,
! [A: $tType,Xs2: list @ A] :
( ( size_size @ ( list @ A ) @ ( tl @ A @ Xs2 ) )
= ( minus_minus @ nat @ ( size_size @ ( list @ A ) @ Xs2 ) @ ( one_one @ nat ) ) ) ).
% length_tl
thf(fact_7787_hd__Cons__tl,axiom,
! [A: $tType,Xs2: list @ A] :
( ( Xs2
!= ( nil @ A ) )
=> ( ( cons @ A @ ( hd @ A @ Xs2 ) @ ( tl @ A @ Xs2 ) )
= Xs2 ) ) ).
% hd_Cons_tl
thf(fact_7788_list_Ocollapse,axiom,
! [A: $tType,List: list @ A] :
( ( List
!= ( nil @ A ) )
=> ( ( cons @ A @ ( hd @ A @ List ) @ ( tl @ A @ List ) )
= List ) ) ).
% list.collapse
thf(fact_7789_tl__replicate,axiom,
! [A: $tType,N: nat,X: A] :
( ( tl @ A @ ( replicate @ A @ N @ X ) )
= ( replicate @ A @ ( minus_minus @ nat @ N @ ( one_one @ nat ) ) @ X ) ) ).
% tl_replicate
thf(fact_7790_map__tl,axiom,
! [A: $tType,B: $tType,F2: B > A,Xs2: list @ B] :
( ( map @ B @ A @ F2 @ ( tl @ B @ Xs2 ) )
= ( tl @ A @ ( map @ B @ A @ F2 @ Xs2 ) ) ) ).
% map_tl
thf(fact_7791_list_Omap__sel_I2_J,axiom,
! [B: $tType,A: $tType,A2: list @ A,F2: A > B] :
( ( A2
!= ( nil @ A ) )
=> ( ( tl @ B @ ( map @ A @ B @ F2 @ A2 ) )
= ( map @ A @ B @ F2 @ ( tl @ A @ A2 ) ) ) ) ).
% list.map_sel(2)
thf(fact_7792_list_Omap__sel_I1_J,axiom,
! [B: $tType,A: $tType,A2: list @ A,F2: A > B] :
( ( A2
!= ( nil @ A ) )
=> ( ( hd @ B @ ( map @ A @ B @ F2 @ A2 ) )
= ( F2 @ ( hd @ A @ A2 ) ) ) ) ).
% list.map_sel(1)
thf(fact_7793_hd__map,axiom,
! [B: $tType,A: $tType,Xs2: list @ A,F2: A > B] :
( ( Xs2
!= ( nil @ A ) )
=> ( ( hd @ B @ ( map @ A @ B @ F2 @ Xs2 ) )
= ( F2 @ ( hd @ A @ Xs2 ) ) ) ) ).
% hd_map
thf(fact_7794_tl__append,axiom,
! [A: $tType,Xs2: list @ A,Ys: list @ A] :
( ( tl @ A @ ( append @ A @ Xs2 @ Ys ) )
= ( case_list @ ( list @ A ) @ A @ ( tl @ A @ Ys )
@ ^ [Z7: A,Zs: list @ A] : ( append @ A @ Zs @ Ys )
@ Xs2 ) ) ).
% tl_append
thf(fact_7795_hd__def,axiom,
! [A: $tType] :
( ( hd @ A )
= ( case_list @ A @ A @ ( undefined @ A )
@ ^ [X213: A,X224: list @ A] : X213 ) ) ).
% hd_def
thf(fact_7796_list_Ocase__eq__if,axiom,
! [A: $tType,B: $tType] :
( ( case_list @ B @ A )
= ( ^ [F13: B,F23: A > ( list @ A ) > B,List2: list @ A] :
( if @ B
@ ( List2
= ( nil @ A ) )
@ F13
@ ( F23 @ ( hd @ A @ List2 ) @ ( tl @ A @ List2 ) ) ) ) ) ).
% list.case_eq_if
thf(fact_7797_list_Osplit__sel,axiom,
! [B: $tType,A: $tType,P: B > $o,F12: B,F22: A > ( list @ A ) > B,List: list @ A] :
( ( P @ ( case_list @ B @ A @ F12 @ F22 @ List ) )
= ( ( ( List
= ( nil @ A ) )
=> ( P @ F12 ) )
& ( ( List
= ( cons @ A @ ( hd @ A @ List ) @ ( tl @ A @ List ) ) )
=> ( P @ ( F22 @ ( hd @ A @ List ) @ ( tl @ A @ List ) ) ) ) ) ) ).
% list.split_sel
thf(fact_7798_list_Osplit__sel__asm,axiom,
! [B: $tType,A: $tType,P: B > $o,F12: B,F22: A > ( list @ A ) > B,List: list @ A] :
( ( P @ ( case_list @ B @ A @ F12 @ F22 @ List ) )
= ( ~ ( ( ( List
= ( nil @ A ) )
& ~ ( P @ F12 ) )
| ( ( List
= ( cons @ A @ ( hd @ A @ List ) @ ( tl @ A @ List ) ) )
& ~ ( P @ ( F22 @ ( hd @ A @ List ) @ ( tl @ A @ List ) ) ) ) ) ) ) ).
% list.split_sel_asm
thf(fact_7799_tl__def,axiom,
! [A: $tType] :
( ( tl @ A )
= ( case_list @ ( list @ A ) @ A @ ( nil @ A )
@ ^ [X213: A,X224: list @ A] : X224 ) ) ).
% tl_def
thf(fact_7800_Misc_Onth__tl,axiom,
! [A: $tType,Xs2: list @ A,N: nat] :
( ( Xs2
!= ( nil @ A ) )
=> ( ( nth @ A @ ( tl @ A @ Xs2 ) @ N )
= ( nth @ A @ Xs2 @ ( suc @ N ) ) ) ) ).
% Misc.nth_tl
thf(fact_7801_hd__conv__nth,axiom,
! [A: $tType,Xs2: list @ A] :
( ( Xs2
!= ( nil @ A ) )
=> ( ( hd @ A @ Xs2 )
= ( nth @ A @ Xs2 @ ( zero_zero @ nat ) ) ) ) ).
% hd_conv_nth
thf(fact_7802_hd__append,axiom,
! [A: $tType,Xs2: list @ A,Ys: list @ A] :
( ( ( Xs2
= ( nil @ A ) )
=> ( ( hd @ A @ ( append @ A @ Xs2 @ Ys ) )
= ( hd @ A @ Ys ) ) )
& ( ( Xs2
!= ( nil @ A ) )
=> ( ( hd @ A @ ( append @ A @ Xs2 @ Ys ) )
= ( hd @ A @ Xs2 ) ) ) ) ).
% hd_append
thf(fact_7803_longest__common__prefix,axiom,
! [A: $tType,Xs2: list @ A,Ys: list @ A] :
? [Ps2: list @ A,Xs6: list @ A,Ys6: list @ A] :
( ( Xs2
= ( append @ A @ Ps2 @ Xs6 ) )
& ( Ys
= ( append @ A @ Ps2 @ Ys6 ) )
& ( ( Xs6
= ( nil @ A ) )
| ( Ys6
= ( nil @ A ) )
| ( ( hd @ A @ Xs6 )
!= ( hd @ A @ Ys6 ) ) ) ) ).
% longest_common_prefix
thf(fact_7804_hd__in__set,axiom,
! [A: $tType,Xs2: list @ A] :
( ( Xs2
!= ( nil @ A ) )
=> ( member @ A @ ( hd @ A @ Xs2 ) @ ( set2 @ A @ Xs2 ) ) ) ).
% hd_in_set
thf(fact_7805_list_Oset__sel_I1_J,axiom,
! [A: $tType,A2: list @ A] :
( ( A2
!= ( nil @ A ) )
=> ( member @ A @ ( hd @ A @ A2 ) @ ( set2 @ A @ A2 ) ) ) ).
% list.set_sel(1)
thf(fact_7806_list_Oset__sel_I2_J,axiom,
! [A: $tType,A2: list @ A,X: A] :
( ( A2
!= ( nil @ A ) )
=> ( ( member @ A @ X @ ( set2 @ A @ ( tl @ A @ A2 ) ) )
=> ( member @ A @ X @ ( set2 @ A @ A2 ) ) ) ) ).
% list.set_sel(2)
thf(fact_7807_Nitpick_Osize__list__simp_I1_J,axiom,
! [A: $tType] :
( ( size_list @ A )
= ( ^ [F4: A > nat,Xs: list @ A] :
( if @ nat
@ ( Xs
= ( nil @ A ) )
@ ( zero_zero @ nat )
@ ( suc @ ( plus_plus @ nat @ ( F4 @ ( hd @ A @ Xs ) ) @ ( size_list @ A @ F4 @ ( tl @ A @ Xs ) ) ) ) ) ) ) ).
% Nitpick.size_list_simp(1)
thf(fact_7808_hd__concat,axiom,
! [A: $tType,Xs2: list @ ( list @ A )] :
( ( Xs2
!= ( nil @ ( list @ A ) ) )
=> ( ( ( hd @ ( list @ A ) @ Xs2 )
!= ( nil @ A ) )
=> ( ( hd @ A @ ( concat @ A @ Xs2 ) )
= ( hd @ A @ ( hd @ ( list @ A ) @ Xs2 ) ) ) ) ) ).
% hd_concat
thf(fact_7809_list_Oexpand,axiom,
! [A: $tType,List: list @ A,List3: list @ A] :
( ( ( List
= ( nil @ A ) )
= ( List3
= ( nil @ A ) ) )
=> ( ( ( List
!= ( nil @ A ) )
=> ( ( List3
!= ( nil @ A ) )
=> ( ( ( hd @ A @ List )
= ( hd @ A @ List3 ) )
& ( ( tl @ A @ List )
= ( tl @ A @ List3 ) ) ) ) )
=> ( List = List3 ) ) ) ).
% list.expand
thf(fact_7810_list_Osel_I2_J,axiom,
! [A: $tType] :
( ( tl @ A @ ( nil @ A ) )
= ( nil @ A ) ) ).
% list.sel(2)
thf(fact_7811_list__take__induct__tl2,axiom,
! [B: $tType,A: $tType,Xs2: list @ A,Ys: list @ B,P: B > A > $o] :
( ( ( size_size @ ( list @ A ) @ Xs2 )
= ( size_size @ ( list @ B ) @ Ys ) )
=> ( ! [N2: nat] :
( ( ord_less @ nat @ N2 @ ( size_size @ ( list @ A ) @ Xs2 ) )
=> ( P @ ( nth @ B @ Ys @ N2 ) @ ( nth @ A @ Xs2 @ N2 ) ) )
=> ! [N10: nat] :
( ( ord_less @ nat @ N10 @ ( size_size @ ( list @ A ) @ ( tl @ A @ Xs2 ) ) )
=> ( P @ ( nth @ B @ ( tl @ B @ Ys ) @ N10 ) @ ( nth @ A @ ( tl @ A @ Xs2 ) @ N10 ) ) ) ) ) ).
% list_take_induct_tl2
thf(fact_7812_list_Osel_I1_J,axiom,
! [A: $tType,X21: A,X222: list @ A] :
( ( hd @ A @ ( cons @ A @ X21 @ X222 ) )
= X21 ) ).
% list.sel(1)
thf(fact_7813_list_Osel_I3_J,axiom,
! [A: $tType,X21: A,X222: list @ A] :
( ( tl @ A @ ( cons @ A @ X21 @ X222 ) )
= X222 ) ).
% list.sel(3)
thf(fact_7814_tl__Nil,axiom,
! [A: $tType,Xs2: list @ A] :
( ( ( tl @ A @ Xs2 )
= ( nil @ A ) )
= ( ( Xs2
= ( nil @ A ) )
| ? [X2: A] :
( Xs2
= ( cons @ A @ X2 @ ( nil @ A ) ) ) ) ) ).
% tl_Nil
thf(fact_7815_Nil__tl,axiom,
! [A: $tType,Xs2: list @ A] :
( ( ( nil @ A )
= ( tl @ A @ Xs2 ) )
= ( ( Xs2
= ( nil @ A ) )
| ? [X2: A] :
( Xs2
= ( cons @ A @ X2 @ ( nil @ A ) ) ) ) ) ).
% Nil_tl
thf(fact_7816_list_Oexhaust__sel,axiom,
! [A: $tType,List: list @ A] :
( ( List
!= ( nil @ A ) )
=> ( List
= ( cons @ A @ ( hd @ A @ List ) @ ( tl @ A @ List ) ) ) ) ).
% list.exhaust_sel
thf(fact_7817_Nitpick_Osize__list__simp_I2_J,axiom,
! [A: $tType] :
( ( size_size @ ( list @ A ) )
= ( ^ [Xs: list @ A] :
( if @ nat
@ ( Xs
= ( nil @ A ) )
@ ( zero_zero @ nat )
@ ( suc @ ( size_size @ ( list @ A ) @ ( tl @ A @ Xs ) ) ) ) ) ) ).
% Nitpick.size_list_simp(2)
thf(fact_7818_List_Onth__tl,axiom,
! [A: $tType,N: nat,Xs2: list @ A] :
( ( ord_less @ nat @ N @ ( size_size @ ( list @ A ) @ ( tl @ A @ Xs2 ) ) )
=> ( ( nth @ A @ ( tl @ A @ Xs2 ) @ N )
= ( nth @ A @ Xs2 @ ( suc @ N ) ) ) ) ).
% List.nth_tl
thf(fact_7819_slice__head,axiom,
! [A: $tType,From: nat,To: nat,Xs2: list @ A] :
( ( ord_less @ nat @ From @ To )
=> ( ( ord_less_eq @ nat @ To @ ( size_size @ ( list @ A ) @ Xs2 ) )
=> ( ( hd @ A @ ( slice @ A @ From @ To @ Xs2 ) )
= ( nth @ A @ Xs2 @ From ) ) ) ) ).
% slice_head
thf(fact_7820_part__def,axiom,
! [A: $tType,B: $tType] :
( ( linorder @ A )
=> ( ( linorder_part @ B @ A )
= ( ^ [F4: B > A,Pivot: A,Xs: list @ B] :
( product_Pair @ ( list @ B ) @ ( product_prod @ ( list @ B ) @ ( list @ B ) )
@ ( filter2 @ B
@ ^ [X2: B] : ( ord_less @ A @ ( F4 @ X2 ) @ Pivot )
@ Xs )
@ ( product_Pair @ ( list @ B ) @ ( list @ B )
@ ( filter2 @ B
@ ^ [X2: B] :
( ( F4 @ X2 )
= Pivot )
@ Xs )
@ ( filter2 @ B
@ ^ [X2: B] : ( ord_less @ A @ Pivot @ ( F4 @ X2 ) )
@ Xs ) ) ) ) ) ) ).
% part_def
thf(fact_7821_csqrt_Osimps_I1_J,axiom,
! [Z: complex] :
( ( re @ ( csqrt @ Z ) )
= ( sqrt @ ( divide_divide @ real @ ( plus_plus @ real @ ( real_V7770717601297561774m_norm @ complex @ Z ) @ ( re @ Z ) ) @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) ) ) ).
% csqrt.simps(1)
thf(fact_7822_complex__Re__numeral,axiom,
! [V: num] :
( ( re @ ( numeral_numeral @ complex @ V ) )
= ( numeral_numeral @ real @ V ) ) ).
% complex_Re_numeral
thf(fact_7823_Re__divide__of__nat,axiom,
! [Z: complex,N: nat] :
( ( re @ ( divide_divide @ complex @ Z @ ( semiring_1_of_nat @ complex @ N ) ) )
= ( divide_divide @ real @ ( re @ Z ) @ ( semiring_1_of_nat @ real @ N ) ) ) ).
% Re_divide_of_nat
thf(fact_7824_Re__divide__of__real,axiom,
! [Z: complex,R2: real] :
( ( re @ ( divide_divide @ complex @ Z @ ( real_Vector_of_real @ complex @ R2 ) ) )
= ( divide_divide @ real @ ( re @ Z ) @ R2 ) ) ).
% Re_divide_of_real
thf(fact_7825_Re__sgn,axiom,
! [Z: complex] :
( ( re @ ( sgn_sgn @ complex @ Z ) )
= ( divide_divide @ real @ ( re @ Z ) @ ( real_V7770717601297561774m_norm @ complex @ Z ) ) ) ).
% Re_sgn
thf(fact_7826_Re__divide__numeral,axiom,
! [Z: complex,W: num] :
( ( re @ ( divide_divide @ complex @ Z @ ( numeral_numeral @ complex @ W ) ) )
= ( divide_divide @ real @ ( re @ Z ) @ ( numeral_numeral @ real @ W ) ) ) ).
% Re_divide_numeral
thf(fact_7827_cos__Arg__i__mult__zero,axiom,
! [Y: complex] :
( ( Y
!= ( zero_zero @ complex ) )
=> ( ( ( re @ Y )
= ( zero_zero @ real ) )
=> ( ( cos @ real @ ( arg @ Y ) )
= ( zero_zero @ real ) ) ) ) ).
% cos_Arg_i_mult_zero
thf(fact_7828_scaleR__complex_Osimps_I1_J,axiom,
! [R2: real,X: complex] :
( ( re @ ( real_V8093663219630862766scaleR @ complex @ R2 @ X ) )
= ( times_times @ real @ R2 @ ( re @ X ) ) ) ).
% scaleR_complex.simps(1)
thf(fact_7829_one__complex_Osimps_I1_J,axiom,
( ( re @ ( one_one @ complex ) )
= ( one_one @ real ) ) ).
% one_complex.simps(1)
thf(fact_7830_imaginary__unit_Osimps_I1_J,axiom,
( ( re @ imaginary_unit )
= ( zero_zero @ real ) ) ).
% imaginary_unit.simps(1)
thf(fact_7831_zero__complex_Osimps_I1_J,axiom,
( ( re @ ( zero_zero @ complex ) )
= ( zero_zero @ real ) ) ).
% zero_complex.simps(1)
thf(fact_7832_Re__csqrt,axiom,
! [Z: complex] : ( ord_less_eq @ real @ ( zero_zero @ real ) @ ( re @ ( csqrt @ Z ) ) ) ).
% Re_csqrt
thf(fact_7833_cmod__plus__Re__le__0__iff,axiom,
! [Z: complex] :
( ( ord_less_eq @ real @ ( plus_plus @ real @ ( real_V7770717601297561774m_norm @ complex @ Z ) @ ( re @ Z ) ) @ ( zero_zero @ real ) )
= ( ( re @ Z )
= ( uminus_uminus @ real @ ( real_V7770717601297561774m_norm @ complex @ Z ) ) ) ) ).
% cmod_plus_Re_le_0_iff
thf(fact_7834_cos__n__Re__cis__pow__n,axiom,
! [N: nat,A2: real] :
( ( cos @ real @ ( times_times @ real @ ( semiring_1_of_nat @ real @ N ) @ A2 ) )
= ( re @ ( power_power @ complex @ ( cis @ A2 ) @ N ) ) ) ).
% cos_n_Re_cis_pow_n
thf(fact_7835_part__code_I2_J,axiom,
! [A: $tType,B: $tType] :
( ( linorder @ A )
=> ! [F2: B > A,Pivot2: A,X: B,Xs2: list @ B] :
( ( linorder_part @ B @ A @ F2 @ Pivot2 @ ( cons @ B @ X @ Xs2 ) )
= ( product_case_prod @ ( list @ B ) @ ( product_prod @ ( list @ B ) @ ( list @ B ) ) @ ( product_prod @ ( list @ B ) @ ( product_prod @ ( list @ B ) @ ( list @ B ) ) )
@ ^ [Lts: list @ B] :
( product_case_prod @ ( list @ B ) @ ( list @ B ) @ ( product_prod @ ( list @ B ) @ ( product_prod @ ( list @ B ) @ ( list @ B ) ) )
@ ^ [Eqs: list @ B,Gts: list @ B] : ( if @ ( product_prod @ ( list @ B ) @ ( product_prod @ ( list @ B ) @ ( list @ B ) ) ) @ ( ord_less @ A @ ( F2 @ X ) @ Pivot2 ) @ ( product_Pair @ ( list @ B ) @ ( product_prod @ ( list @ B ) @ ( list @ B ) ) @ ( cons @ B @ X @ Lts ) @ ( product_Pair @ ( list @ B ) @ ( list @ B ) @ Eqs @ Gts ) ) @ ( if @ ( product_prod @ ( list @ B ) @ ( product_prod @ ( list @ B ) @ ( list @ B ) ) ) @ ( ord_less @ A @ Pivot2 @ ( F2 @ X ) ) @ ( product_Pair @ ( list @ B ) @ ( product_prod @ ( list @ B ) @ ( list @ B ) ) @ Lts @ ( product_Pair @ ( list @ B ) @ ( list @ B ) @ Eqs @ ( cons @ B @ X @ Gts ) ) ) @ ( product_Pair @ ( list @ B ) @ ( product_prod @ ( list @ B ) @ ( list @ B ) ) @ Lts @ ( product_Pair @ ( list @ B ) @ ( list @ B ) @ ( cons @ B @ X @ Eqs ) @ Gts ) ) ) ) )
@ ( linorder_part @ B @ A @ F2 @ Pivot2 @ Xs2 ) ) ) ) ).
% part_code(2)
thf(fact_7836_csqrt_Ocode,axiom,
( csqrt
= ( ^ [Z7: complex] :
( complex2 @ ( sqrt @ ( divide_divide @ real @ ( plus_plus @ real @ ( real_V7770717601297561774m_norm @ complex @ Z7 ) @ ( re @ Z7 ) ) @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) )
@ ( times_times @ real
@ ( if @ real
@ ( ( im @ Z7 )
= ( zero_zero @ real ) )
@ ( one_one @ real )
@ ( sgn_sgn @ real @ ( im @ Z7 ) ) )
@ ( sqrt @ ( divide_divide @ real @ ( minus_minus @ real @ ( real_V7770717601297561774m_norm @ complex @ Z7 ) @ ( re @ Z7 ) ) @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) ) ) ) ) ) ).
% csqrt.code
thf(fact_7837_csqrt_Osimps_I2_J,axiom,
! [Z: complex] :
( ( im @ ( csqrt @ Z ) )
= ( times_times @ real
@ ( if @ real
@ ( ( im @ Z )
= ( zero_zero @ real ) )
@ ( one_one @ real )
@ ( sgn_sgn @ real @ ( im @ Z ) ) )
@ ( sqrt @ ( divide_divide @ real @ ( minus_minus @ real @ ( real_V7770717601297561774m_norm @ complex @ Z ) @ ( re @ Z ) ) @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) ) ) ) ).
% csqrt.simps(2)
thf(fact_7838_complex__Im__fact,axiom,
! [N: nat] :
( ( im @ ( semiring_char_0_fact @ complex @ N ) )
= ( zero_zero @ real ) ) ).
% complex_Im_fact
thf(fact_7839_complex__Im__of__int,axiom,
! [Z: int] :
( ( im @ ( ring_1_of_int @ complex @ Z ) )
= ( zero_zero @ real ) ) ).
% complex_Im_of_int
thf(fact_7840_Im__complex__of__real,axiom,
! [Z: real] :
( ( im @ ( real_Vector_of_real @ complex @ Z ) )
= ( zero_zero @ real ) ) ).
% Im_complex_of_real
thf(fact_7841_Im__power__real,axiom,
! [X: complex,N: nat] :
( ( ( im @ X )
= ( zero_zero @ real ) )
=> ( ( im @ ( power_power @ complex @ X @ N ) )
= ( zero_zero @ real ) ) ) ).
% Im_power_real
thf(fact_7842_complex__Im__numeral,axiom,
! [V: num] :
( ( im @ ( numeral_numeral @ complex @ V ) )
= ( zero_zero @ real ) ) ).
% complex_Im_numeral
thf(fact_7843_complex__Im__of__nat,axiom,
! [N: nat] :
( ( im @ ( semiring_1_of_nat @ complex @ N ) )
= ( zero_zero @ real ) ) ).
% complex_Im_of_nat
thf(fact_7844_Im__divide__of__real,axiom,
! [Z: complex,R2: real] :
( ( im @ ( divide_divide @ complex @ Z @ ( real_Vector_of_real @ complex @ R2 ) ) )
= ( divide_divide @ real @ ( im @ Z ) @ R2 ) ) ).
% Im_divide_of_real
thf(fact_7845_Im__sgn,axiom,
! [Z: complex] :
( ( im @ ( sgn_sgn @ complex @ Z ) )
= ( divide_divide @ real @ ( im @ Z ) @ ( real_V7770717601297561774m_norm @ complex @ Z ) ) ) ).
% Im_sgn
thf(fact_7846_Re__power__real,axiom,
! [X: complex,N: nat] :
( ( ( im @ X )
= ( zero_zero @ real ) )
=> ( ( re @ ( power_power @ complex @ X @ N ) )
= ( power_power @ real @ ( re @ X ) @ N ) ) ) ).
% Re_power_real
thf(fact_7847_Im__divide__numeral,axiom,
! [Z: complex,W: num] :
( ( im @ ( divide_divide @ complex @ Z @ ( numeral_numeral @ complex @ W ) ) )
= ( divide_divide @ real @ ( im @ Z ) @ ( numeral_numeral @ real @ W ) ) ) ).
% Im_divide_numeral
thf(fact_7848_Im__divide__of__nat,axiom,
! [Z: complex,N: nat] :
( ( im @ ( divide_divide @ complex @ Z @ ( semiring_1_of_nat @ complex @ N ) ) )
= ( divide_divide @ real @ ( im @ Z ) @ ( semiring_1_of_nat @ real @ N ) ) ) ).
% Im_divide_of_nat
thf(fact_7849_csqrt__of__real__nonneg,axiom,
! [X: complex] :
( ( ( im @ X )
= ( zero_zero @ real ) )
=> ( ( ord_less_eq @ real @ ( zero_zero @ real ) @ ( re @ X ) )
=> ( ( csqrt @ X )
= ( real_Vector_of_real @ complex @ ( sqrt @ ( re @ X ) ) ) ) ) ) ).
% csqrt_of_real_nonneg
thf(fact_7850_csqrt__minus,axiom,
! [X: complex] :
( ( ( ord_less @ real @ ( im @ X ) @ ( zero_zero @ real ) )
| ( ( ( im @ X )
= ( zero_zero @ real ) )
& ( ord_less_eq @ real @ ( zero_zero @ real ) @ ( re @ X ) ) ) )
=> ( ( csqrt @ ( uminus_uminus @ complex @ X ) )
= ( times_times @ complex @ imaginary_unit @ ( csqrt @ X ) ) ) ) ).
% csqrt_minus
thf(fact_7851_csqrt__of__real__nonpos,axiom,
! [X: complex] :
( ( ( im @ X )
= ( zero_zero @ real ) )
=> ( ( ord_less_eq @ real @ ( re @ X ) @ ( zero_zero @ real ) )
=> ( ( csqrt @ X )
= ( times_times @ complex @ imaginary_unit @ ( real_Vector_of_real @ complex @ ( sqrt @ ( abs_abs @ real @ ( re @ X ) ) ) ) ) ) ) ) ).
% csqrt_of_real_nonpos
thf(fact_7852_complex__is__Int__iff,axiom,
! [Z: complex] :
( ( member @ complex @ Z @ ( ring_1_Ints @ complex ) )
= ( ( ( im @ Z )
= ( zero_zero @ real ) )
& ? [I2: int] :
( ( re @ Z )
= ( ring_1_of_int @ real @ I2 ) ) ) ) ).
% complex_is_Int_iff
thf(fact_7853_imaginary__unit_Osimps_I2_J,axiom,
( ( im @ imaginary_unit )
= ( one_one @ real ) ) ).
% imaginary_unit.simps(2)
thf(fact_7854_one__complex_Osimps_I2_J,axiom,
( ( im @ ( one_one @ complex ) )
= ( zero_zero @ real ) ) ).
% one_complex.simps(2)
thf(fact_7855_zero__complex_Osimps_I2_J,axiom,
( ( im @ ( zero_zero @ complex ) )
= ( zero_zero @ real ) ) ).
% zero_complex.simps(2)
thf(fact_7856_scaleR__complex_Osimps_I2_J,axiom,
! [R2: real,X: complex] :
( ( im @ ( real_V8093663219630862766scaleR @ complex @ R2 @ X ) )
= ( times_times @ real @ R2 @ ( im @ X ) ) ) ).
% scaleR_complex.simps(2)
thf(fact_7857_times__complex_Osimps_I2_J,axiom,
! [X: complex,Y: complex] :
( ( im @ ( times_times @ complex @ X @ Y ) )
= ( plus_plus @ real @ ( times_times @ real @ ( re @ X ) @ ( im @ Y ) ) @ ( times_times @ real @ ( im @ X ) @ ( re @ Y ) ) ) ) ).
% times_complex.simps(2)
thf(fact_7858_cmod__eq__Re,axiom,
! [Z: complex] :
( ( ( im @ Z )
= ( zero_zero @ real ) )
=> ( ( real_V7770717601297561774m_norm @ complex @ Z )
= ( abs_abs @ real @ ( re @ Z ) ) ) ) ).
% cmod_eq_Re
thf(fact_7859_cmod__eq__Im,axiom,
! [Z: complex] :
( ( ( re @ Z )
= ( zero_zero @ real ) )
=> ( ( real_V7770717601297561774m_norm @ complex @ Z )
= ( abs_abs @ real @ ( im @ Z ) ) ) ) ).
% cmod_eq_Im
thf(fact_7860_Im__eq__0,axiom,
! [Z: complex] :
( ( ( abs_abs @ real @ ( re @ Z ) )
= ( real_V7770717601297561774m_norm @ complex @ Z ) )
=> ( ( im @ Z )
= ( zero_zero @ real ) ) ) ).
% Im_eq_0
thf(fact_7861_times__complex_Osimps_I1_J,axiom,
! [X: complex,Y: complex] :
( ( re @ ( times_times @ complex @ X @ Y ) )
= ( minus_minus @ real @ ( times_times @ real @ ( re @ X ) @ ( re @ Y ) ) @ ( times_times @ real @ ( im @ X ) @ ( im @ Y ) ) ) ) ).
% times_complex.simps(1)
thf(fact_7862_scaleR__complex_Ocode,axiom,
( ( real_V8093663219630862766scaleR @ complex )
= ( ^ [R5: real,X2: complex] : ( complex2 @ ( times_times @ real @ R5 @ ( re @ X2 ) ) @ ( times_times @ real @ R5 @ ( im @ X2 ) ) ) ) ) ).
% scaleR_complex.code
thf(fact_7863_csqrt__principal,axiom,
! [Z: complex] :
( ( ord_less @ real @ ( zero_zero @ real ) @ ( re @ ( csqrt @ Z ) ) )
| ( ( ( re @ ( csqrt @ Z ) )
= ( zero_zero @ real ) )
& ( ord_less_eq @ real @ ( zero_zero @ real ) @ ( im @ ( csqrt @ Z ) ) ) ) ) ).
% csqrt_principal
thf(fact_7864_sin__n__Im__cis__pow__n,axiom,
! [N: nat,A2: real] :
( ( sin @ real @ ( times_times @ real @ ( semiring_1_of_nat @ real @ N ) @ A2 ) )
= ( im @ ( power_power @ complex @ ( cis @ A2 ) @ N ) ) ) ).
% sin_n_Im_cis_pow_n
thf(fact_7865_Re__exp,axiom,
! [Z: complex] :
( ( re @ ( exp @ complex @ Z ) )
= ( times_times @ real @ ( exp @ real @ ( re @ Z ) ) @ ( cos @ real @ ( im @ Z ) ) ) ) ).
% Re_exp
thf(fact_7866_Im__exp,axiom,
! [Z: complex] :
( ( im @ ( exp @ complex @ Z ) )
= ( times_times @ real @ ( exp @ real @ ( re @ Z ) ) @ ( sin @ real @ ( im @ Z ) ) ) ) ).
% Im_exp
thf(fact_7867_times__complex_Ocode,axiom,
( ( times_times @ complex )
= ( ^ [X2: complex,Y6: complex] : ( complex2 @ ( minus_minus @ real @ ( times_times @ real @ ( re @ X2 ) @ ( re @ Y6 ) ) @ ( times_times @ real @ ( im @ X2 ) @ ( im @ Y6 ) ) ) @ ( plus_plus @ real @ ( times_times @ real @ ( re @ X2 ) @ ( im @ Y6 ) ) @ ( times_times @ real @ ( im @ X2 ) @ ( re @ Y6 ) ) ) ) ) ) ).
% times_complex.code
thf(fact_7868_cmod__power2,axiom,
! [Z: complex] :
( ( power_power @ real @ ( real_V7770717601297561774m_norm @ complex @ Z ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) )
= ( plus_plus @ real @ ( power_power @ real @ ( re @ Z ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ ( power_power @ real @ ( im @ Z ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ).
% cmod_power2
thf(fact_7869_Im__power2,axiom,
! [X: complex] :
( ( im @ ( power_power @ complex @ X @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) )
= ( times_times @ real @ ( times_times @ real @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) @ ( re @ X ) ) @ ( im @ X ) ) ) ).
% Im_power2
thf(fact_7870_Re__power2,axiom,
! [X: complex] :
( ( re @ ( power_power @ complex @ X @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) )
= ( minus_minus @ real @ ( power_power @ real @ ( re @ X ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ ( power_power @ real @ ( im @ X ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ).
% Re_power2
thf(fact_7871_complex__eq__0,axiom,
! [Z: complex] :
( ( Z
= ( zero_zero @ complex ) )
= ( ( plus_plus @ real @ ( power_power @ real @ ( re @ Z ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ ( power_power @ real @ ( im @ Z ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) )
= ( zero_zero @ real ) ) ) ).
% complex_eq_0
thf(fact_7872_norm__complex__def,axiom,
( ( real_V7770717601297561774m_norm @ complex )
= ( ^ [Z7: complex] : ( sqrt @ ( plus_plus @ real @ ( power_power @ real @ ( re @ Z7 ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ ( power_power @ real @ ( im @ Z7 ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) ) ).
% norm_complex_def
thf(fact_7873_inverse__complex_Osimps_I1_J,axiom,
! [X: complex] :
( ( re @ ( inverse_inverse @ complex @ X ) )
= ( divide_divide @ real @ ( re @ X ) @ ( plus_plus @ real @ ( power_power @ real @ ( re @ X ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ ( power_power @ real @ ( im @ X ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) ).
% inverse_complex.simps(1)
thf(fact_7874_complex__neq__0,axiom,
! [Z: complex] :
( ( Z
!= ( zero_zero @ complex ) )
= ( ord_less @ real @ ( zero_zero @ real ) @ ( plus_plus @ real @ ( power_power @ real @ ( re @ Z ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ ( power_power @ real @ ( im @ Z ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) ).
% complex_neq_0
thf(fact_7875_Re__divide,axiom,
! [X: complex,Y: complex] :
( ( re @ ( divide_divide @ complex @ X @ Y ) )
= ( divide_divide @ real @ ( plus_plus @ real @ ( times_times @ real @ ( re @ X ) @ ( re @ Y ) ) @ ( times_times @ real @ ( im @ X ) @ ( im @ Y ) ) ) @ ( plus_plus @ real @ ( power_power @ real @ ( re @ Y ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ ( power_power @ real @ ( im @ Y ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) ).
% Re_divide
thf(fact_7876_csqrt__square,axiom,
! [B2: complex] :
( ( ( ord_less @ real @ ( zero_zero @ real ) @ ( re @ B2 ) )
| ( ( ( re @ B2 )
= ( zero_zero @ real ) )
& ( ord_less_eq @ real @ ( zero_zero @ real ) @ ( im @ B2 ) ) ) )
=> ( ( csqrt @ ( power_power @ complex @ B2 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) )
= B2 ) ) ).
% csqrt_square
thf(fact_7877_csqrt__unique,axiom,
! [W: complex,Z: complex] :
( ( ( power_power @ complex @ W @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) )
= Z )
=> ( ( ( ord_less @ real @ ( zero_zero @ real ) @ ( re @ W ) )
| ( ( ( re @ W )
= ( zero_zero @ real ) )
& ( ord_less_eq @ real @ ( zero_zero @ real ) @ ( im @ W ) ) ) )
=> ( ( csqrt @ Z )
= W ) ) ) ).
% csqrt_unique
thf(fact_7878_inverse__complex_Osimps_I2_J,axiom,
! [X: complex] :
( ( im @ ( inverse_inverse @ complex @ X ) )
= ( divide_divide @ real @ ( uminus_uminus @ real @ ( im @ X ) ) @ ( plus_plus @ real @ ( power_power @ real @ ( re @ X ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ ( power_power @ real @ ( im @ X ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) ).
% inverse_complex.simps(2)
thf(fact_7879_Im__divide,axiom,
! [X: complex,Y: complex] :
( ( im @ ( divide_divide @ complex @ X @ Y ) )
= ( divide_divide @ real @ ( minus_minus @ real @ ( times_times @ real @ ( im @ X ) @ ( re @ Y ) ) @ ( times_times @ real @ ( re @ X ) @ ( im @ Y ) ) ) @ ( plus_plus @ real @ ( power_power @ real @ ( re @ Y ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ ( power_power @ real @ ( im @ Y ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) ).
% Im_divide
thf(fact_7880_complex__abs__le__norm,axiom,
! [Z: complex] : ( ord_less_eq @ real @ ( plus_plus @ real @ ( abs_abs @ real @ ( re @ Z ) ) @ ( abs_abs @ real @ ( im @ Z ) ) ) @ ( times_times @ real @ ( sqrt @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) @ ( real_V7770717601297561774m_norm @ complex @ Z ) ) ) ).
% complex_abs_le_norm
thf(fact_7881_complex__unit__circle,axiom,
! [Z: complex] :
( ( Z
!= ( zero_zero @ complex ) )
=> ( ( plus_plus @ real @ ( power_power @ real @ ( divide_divide @ real @ ( re @ Z ) @ ( real_V7770717601297561774m_norm @ complex @ Z ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ ( power_power @ real @ ( divide_divide @ real @ ( im @ Z ) @ ( real_V7770717601297561774m_norm @ complex @ Z ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) )
= ( one_one @ real ) ) ) ).
% complex_unit_circle
thf(fact_7882_inverse__complex_Ocode,axiom,
( ( inverse_inverse @ complex )
= ( ^ [X2: complex] : ( complex2 @ ( divide_divide @ real @ ( re @ X2 ) @ ( plus_plus @ real @ ( power_power @ real @ ( re @ X2 ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ ( power_power @ real @ ( im @ X2 ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( divide_divide @ real @ ( uminus_uminus @ real @ ( im @ X2 ) ) @ ( plus_plus @ real @ ( power_power @ real @ ( re @ X2 ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ ( power_power @ real @ ( im @ X2 ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) ) ) ).
% inverse_complex.code
thf(fact_7883_Complex__divide,axiom,
( ( divide_divide @ complex )
= ( ^ [X2: complex,Y6: complex] : ( complex2 @ ( divide_divide @ real @ ( plus_plus @ real @ ( times_times @ real @ ( re @ X2 ) @ ( re @ Y6 ) ) @ ( times_times @ real @ ( im @ X2 ) @ ( im @ Y6 ) ) ) @ ( plus_plus @ real @ ( power_power @ real @ ( re @ Y6 ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ ( power_power @ real @ ( im @ Y6 ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( divide_divide @ real @ ( minus_minus @ real @ ( times_times @ real @ ( im @ X2 ) @ ( re @ Y6 ) ) @ ( times_times @ real @ ( re @ X2 ) @ ( im @ Y6 ) ) ) @ ( plus_plus @ real @ ( power_power @ real @ ( re @ Y6 ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ ( power_power @ real @ ( im @ Y6 ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) ) ) ).
% Complex_divide
thf(fact_7884_Im__Reals__divide,axiom,
! [R2: complex,Z: complex] :
( ( member @ complex @ R2 @ ( real_Vector_Reals @ complex ) )
=> ( ( im @ ( divide_divide @ complex @ R2 @ Z ) )
= ( divide_divide @ real @ ( times_times @ real @ ( uminus_uminus @ real @ ( re @ R2 ) ) @ ( im @ Z ) ) @ ( power_power @ real @ ( real_V7770717601297561774m_norm @ complex @ Z ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) ).
% Im_Reals_divide
thf(fact_7885_Re__Reals__divide,axiom,
! [R2: complex,Z: complex] :
( ( member @ complex @ R2 @ ( real_Vector_Reals @ complex ) )
=> ( ( re @ ( divide_divide @ complex @ R2 @ Z ) )
= ( divide_divide @ real @ ( times_times @ real @ ( re @ R2 ) @ ( re @ Z ) ) @ ( power_power @ real @ ( real_V7770717601297561774m_norm @ complex @ Z ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) ).
% Re_Reals_divide
thf(fact_7886_Re__divide__Reals,axiom,
! [R2: complex,Z: complex] :
( ( member @ complex @ R2 @ ( real_Vector_Reals @ complex ) )
=> ( ( re @ ( divide_divide @ complex @ Z @ R2 ) )
= ( divide_divide @ real @ ( re @ Z ) @ ( re @ R2 ) ) ) ) ).
% Re_divide_Reals
thf(fact_7887_real__eq__imaginary__iff,axiom,
! [Y: complex,X: complex] :
( ( member @ complex @ Y @ ( real_Vector_Reals @ complex ) )
=> ( ( member @ complex @ X @ ( real_Vector_Reals @ complex ) )
=> ( ( X
= ( times_times @ complex @ imaginary_unit @ Y ) )
= ( ( X
= ( zero_zero @ complex ) )
& ( Y
= ( zero_zero @ complex ) ) ) ) ) ) ).
% real_eq_imaginary_iff
thf(fact_7888_imaginary__eq__real__iff,axiom,
! [Y: complex,X: complex] :
( ( member @ complex @ Y @ ( real_Vector_Reals @ complex ) )
=> ( ( member @ complex @ X @ ( real_Vector_Reals @ complex ) )
=> ( ( ( times_times @ complex @ imaginary_unit @ Y )
= X )
= ( ( X
= ( zero_zero @ complex ) )
& ( Y
= ( zero_zero @ complex ) ) ) ) ) ) ).
% imaginary_eq_real_iff
thf(fact_7889_Im__divide__Reals,axiom,
! [R2: complex,Z: complex] :
( ( member @ complex @ R2 @ ( real_Vector_Reals @ complex ) )
=> ( ( im @ ( divide_divide @ complex @ Z @ R2 ) )
= ( divide_divide @ real @ ( im @ Z ) @ ( re @ R2 ) ) ) ) ).
% Im_divide_Reals
thf(fact_7890_complex__is__Real__iff,axiom,
! [Z: complex] :
( ( member @ complex @ Z @ ( real_Vector_Reals @ complex ) )
= ( ( im @ Z )
= ( zero_zero @ real ) ) ) ).
% complex_is_Real_iff
thf(fact_7891_nonzero__Reals__divide,axiom,
! [A: $tType] :
( ( real_V7773925162809079976_field @ A )
=> ! [A2: A,B2: A] :
( ( member @ A @ A2 @ ( real_Vector_Reals @ A ) )
=> ( ( member @ A @ B2 @ ( real_Vector_Reals @ A ) )
=> ( ( B2
!= ( zero_zero @ A ) )
=> ( member @ A @ ( divide_divide @ A @ A2 @ B2 ) @ ( real_Vector_Reals @ A ) ) ) ) ) ) ).
% nonzero_Reals_divide
thf(fact_7892_Reals__divide,axiom,
! [A: $tType] :
( ( real_V7773925162809079976_field @ A )
=> ! [A2: A,B2: A] :
( ( member @ A @ A2 @ ( real_Vector_Reals @ A ) )
=> ( ( member @ A @ B2 @ ( real_Vector_Reals @ A ) )
=> ( member @ A @ ( divide_divide @ A @ A2 @ B2 ) @ ( real_Vector_Reals @ A ) ) ) ) ) ).
% Reals_divide
thf(fact_7893_Reals__add,axiom,
! [A: $tType] :
( ( real_V2191834092415804123ebra_1 @ A )
=> ! [A2: A,B2: A] :
( ( member @ A @ A2 @ ( real_Vector_Reals @ A ) )
=> ( ( member @ A @ B2 @ ( real_Vector_Reals @ A ) )
=> ( member @ A @ ( plus_plus @ A @ A2 @ B2 ) @ ( real_Vector_Reals @ A ) ) ) ) ) ).
% Reals_add
thf(fact_7894_Reals__power,axiom,
! [A: $tType] :
( ( real_V2191834092415804123ebra_1 @ A )
=> ! [A2: A,N: nat] :
( ( member @ A @ A2 @ ( real_Vector_Reals @ A ) )
=> ( member @ A @ ( power_power @ A @ A2 @ N ) @ ( real_Vector_Reals @ A ) ) ) ) ).
% Reals_power
thf(fact_7895_Reals__mult,axiom,
! [A: $tType] :
( ( real_V2191834092415804123ebra_1 @ A )
=> ! [A2: A,B2: A] :
( ( member @ A @ A2 @ ( real_Vector_Reals @ A ) )
=> ( ( member @ A @ B2 @ ( real_Vector_Reals @ A ) )
=> ( member @ A @ ( times_times @ A @ A2 @ B2 ) @ ( real_Vector_Reals @ A ) ) ) ) ) ).
% Reals_mult
thf(fact_7896_Complex__in__Reals,axiom,
! [X: real] : ( member @ complex @ ( complex2 @ X @ ( zero_zero @ real ) ) @ ( real_Vector_Reals @ complex ) ) ).
% Complex_in_Reals
thf(fact_7897_Reals__0,axiom,
! [A: $tType] :
( ( real_V2191834092415804123ebra_1 @ A )
=> ( member @ A @ ( zero_zero @ A ) @ ( real_Vector_Reals @ A ) ) ) ).
% Reals_0
thf(fact_7898_Reals__1,axiom,
! [B: $tType] :
( ( real_V2191834092415804123ebra_1 @ B )
=> ( member @ B @ ( one_one @ B ) @ ( real_Vector_Reals @ B ) ) ) ).
% Reals_1
thf(fact_7899_Reals__numeral,axiom,
! [A: $tType] :
( ( real_V2191834092415804123ebra_1 @ A )
=> ! [W: num] : ( member @ A @ ( numeral_numeral @ A @ W ) @ ( real_Vector_Reals @ A ) ) ) ).
% Reals_numeral
thf(fact_7900_nonzero__Reals__inverse,axiom,
! [A: $tType] :
( ( real_V5047593784448816457lgebra @ A )
=> ! [A2: A] :
( ( member @ A @ A2 @ ( real_Vector_Reals @ A ) )
=> ( ( A2
!= ( zero_zero @ A ) )
=> ( member @ A @ ( inverse_inverse @ A @ A2 ) @ ( real_Vector_Reals @ A ) ) ) ) ) ).
% nonzero_Reals_inverse
thf(fact_7901_series__comparison__complex,axiom,
! [A: $tType] :
( ( real_Vector_banach @ A )
=> ! [G: nat > complex,N4: nat,F2: nat > A] :
( ( summable @ complex @ G )
=> ( ! [N2: nat] : ( member @ complex @ ( G @ N2 ) @ ( real_Vector_Reals @ complex ) )
=> ( ! [N2: nat] : ( ord_less_eq @ real @ ( zero_zero @ real ) @ ( re @ ( G @ N2 ) ) )
=> ( ! [N2: nat] :
( ( ord_less_eq @ nat @ N4 @ N2 )
=> ( ord_less_eq @ real @ ( real_V7770717601297561774m_norm @ A @ ( F2 @ N2 ) ) @ ( real_V7770717601297561774m_norm @ complex @ ( G @ N2 ) ) ) )
=> ( summable @ A @ F2 ) ) ) ) ) ) ).
% series_comparison_complex
thf(fact_7902_extract__SomeE,axiom,
! [A: $tType,P: A > $o,Xs2: list @ A,Ys: list @ A,Y: A,Zs3: list @ A] :
( ( ( extract @ A @ P @ Xs2 )
= ( some @ ( product_prod @ ( list @ A ) @ ( product_prod @ A @ ( list @ A ) ) ) @ ( product_Pair @ ( list @ A ) @ ( product_prod @ A @ ( list @ A ) ) @ Ys @ ( product_Pair @ A @ ( list @ A ) @ Y @ Zs3 ) ) ) )
=> ( ( Xs2
= ( append @ A @ Ys @ ( cons @ A @ Y @ Zs3 ) ) )
& ( P @ Y )
& ~ ? [X5: A] :
( ( member @ A @ X5 @ ( set2 @ A @ Ys ) )
& ( P @ X5 ) ) ) ) ).
% extract_SomeE
thf(fact_7903_extract__Some__iff,axiom,
! [A: $tType,P: A > $o,Xs2: list @ A,Ys: list @ A,Y: A,Zs3: list @ A] :
( ( ( extract @ A @ P @ Xs2 )
= ( some @ ( product_prod @ ( list @ A ) @ ( product_prod @ A @ ( list @ A ) ) ) @ ( product_Pair @ ( list @ A ) @ ( product_prod @ A @ ( list @ A ) ) @ Ys @ ( product_Pair @ A @ ( list @ A ) @ Y @ Zs3 ) ) ) )
= ( ( Xs2
= ( append @ A @ Ys @ ( cons @ A @ Y @ Zs3 ) ) )
& ( P @ Y )
& ~ ? [X2: A] :
( ( member @ A @ X2 @ ( set2 @ A @ Ys ) )
& ( P @ X2 ) ) ) ) ).
% extract_Some_iff
thf(fact_7904_extract__Nil__code,axiom,
! [A: $tType,P: A > $o] :
( ( extract @ A @ P @ ( nil @ A ) )
= ( none @ ( product_prod @ ( list @ A ) @ ( product_prod @ A @ ( list @ A ) ) ) ) ) ).
% extract_Nil_code
thf(fact_7905_extract__None__iff,axiom,
! [A: $tType,P: A > $o,Xs2: list @ A] :
( ( ( extract @ A @ P @ Xs2 )
= ( none @ ( product_prod @ ( list @ A ) @ ( product_prod @ A @ ( list @ A ) ) ) ) )
= ( ~ ? [X2: A] :
( ( member @ A @ X2 @ ( set2 @ A @ Xs2 ) )
& ( P @ X2 ) ) ) ) ).
% extract_None_iff
thf(fact_7906_complex__mult__cnj,axiom,
! [Z: complex] :
( ( times_times @ complex @ Z @ ( cnj @ Z ) )
= ( real_Vector_of_real @ complex @ ( plus_plus @ real @ ( power_power @ real @ ( re @ Z ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ ( power_power @ real @ ( im @ Z ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) ).
% complex_mult_cnj
thf(fact_7907_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
@ ^ [A4: B] :
( ( member @ B @ A4 @ A3 )
& ~ ( dvd_dvd @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ ( F2 @ A4 ) ) ) ) ) ) ) ) ) ).
% even_sum_iff
thf(fact_7908_card__UNIV__unit,axiom,
( ( finite_card @ product_unit @ ( top_top @ ( set @ product_unit ) ) )
= ( one_one @ nat ) ) ).
% card_UNIV_unit
thf(fact_7909_card__Collect__less__nat,axiom,
! [N: nat] :
( ( finite_card @ nat
@ ( collect @ nat
@ ^ [I2: nat] : ( ord_less @ nat @ I2 @ N ) ) )
= N ) ).
% card_Collect_less_nat
thf(fact_7910_card__atMost,axiom,
! [U: nat] :
( ( finite_card @ nat @ ( set_ord_atMost @ nat @ U ) )
= ( suc @ U ) ) ).
% card_atMost
thf(fact_7911_complex__cnj__zero,axiom,
( ( cnj @ ( zero_zero @ complex ) )
= ( zero_zero @ complex ) ) ).
% complex_cnj_zero
thf(fact_7912_complex__cnj__zero__iff,axiom,
! [Z: complex] :
( ( ( cnj @ Z )
= ( zero_zero @ complex ) )
= ( Z
= ( zero_zero @ complex ) ) ) ).
% complex_cnj_zero_iff
thf(fact_7913_complex__cnj__one__iff,axiom,
! [Z: complex] :
( ( ( cnj @ Z )
= ( one_one @ complex ) )
= ( Z
= ( one_one @ complex ) ) ) ).
% complex_cnj_one_iff
thf(fact_7914_complex__cnj__one,axiom,
( ( cnj @ ( one_one @ complex ) )
= ( one_one @ complex ) ) ).
% complex_cnj_one
thf(fact_7915_card__Collect__le__nat,axiom,
! [N: nat] :
( ( finite_card @ nat
@ ( collect @ nat
@ ^ [I2: nat] : ( ord_less_eq @ nat @ I2 @ N ) ) )
= ( suc @ N ) ) ).
% card_Collect_le_nat
thf(fact_7916_card__UNIV__bool,axiom,
( ( finite_card @ $o @ ( top_top @ ( set @ $o ) ) )
= ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ).
% card_UNIV_bool
thf(fact_7917_card_Oempty,axiom,
! [A: $tType] :
( ( finite_card @ A @ ( bot_bot @ ( set @ A ) ) )
= ( zero_zero @ nat ) ) ).
% card.empty
thf(fact_7918_card_Oinfinite,axiom,
! [A: $tType,A3: set @ A] :
( ~ ( finite_finite2 @ A @ A3 )
=> ( ( finite_card @ A @ A3 )
= ( zero_zero @ nat ) ) ) ).
% card.infinite
thf(fact_7919_card__atLeastAtMost,axiom,
! [L: nat,U: nat] :
( ( finite_card @ nat @ ( set_or1337092689740270186AtMost @ nat @ L @ U ) )
= ( minus_minus @ nat @ ( suc @ U ) @ L ) ) ).
% card_atLeastAtMost
thf(fact_7920_sorted__list__of__set_Olength__sorted__key__list__of__set,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A3: set @ A] :
( ( size_size @ ( list @ A ) @ ( linord4507533701916653071of_set @ A @ A3 ) )
= ( finite_card @ A @ A3 ) ) ) ).
% sorted_list_of_set.length_sorted_key_list_of_set
thf(fact_7921_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_7922_card__greaterThanLessThan,axiom,
! [L: nat,U: nat] :
( ( finite_card @ nat @ ( set_or5935395276787703475ssThan @ nat @ L @ U ) )
= ( minus_minus @ nat @ U @ ( suc @ L ) ) ) ).
% card_greaterThanLessThan
thf(fact_7923_card__0__eq,axiom,
! [A: $tType,A3: set @ A] :
( ( finite_finite2 @ A @ A3 )
=> ( ( ( finite_card @ A @ A3 )
= ( zero_zero @ nat ) )
= ( A3
= ( bot_bot @ ( set @ A ) ) ) ) ) ).
% card_0_eq
thf(fact_7924_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_7925_complex__In__mult__cnj__zero,axiom,
! [Z: complex] :
( ( im @ ( times_times @ complex @ Z @ ( cnj @ Z ) ) )
= ( zero_zero @ real ) ) ).
% complex_In_mult_cnj_zero
thf(fact_7926_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_7927_card__Diff__insert,axiom,
! [A: $tType,A2: A,A3: set @ A,B4: set @ A] :
( ( member @ A @ A2 @ A3 )
=> ( ~ ( member @ A @ A2 @ B4 )
=> ( ( finite_card @ A @ ( minus_minus @ ( set @ A ) @ A3 @ ( insert @ A @ A2 @ B4 ) ) )
= ( minus_minus @ nat @ ( finite_card @ A @ ( minus_minus @ ( set @ A ) @ A3 @ B4 ) ) @ ( one_one @ nat ) ) ) ) ) ).
% card_Diff_insert
thf(fact_7928_card__atLeastAtMost__int,axiom,
! [L: int,U: int] :
( ( finite_card @ int @ ( set_or1337092689740270186AtMost @ int @ L @ U ) )
= ( nat2 @ ( plus_plus @ int @ ( minus_minus @ int @ U @ L ) @ ( one_one @ int ) ) ) ) ).
% card_atLeastAtMost_int
thf(fact_7929_card__greaterThanLessThan__int,axiom,
! [L: int,U: int] :
( ( finite_card @ int @ ( set_or5935395276787703475ssThan @ int @ L @ U ) )
= ( nat2 @ ( minus_minus @ int @ U @ ( plus_plus @ int @ L @ ( one_one @ int ) ) ) ) ) ).
% card_greaterThanLessThan_int
thf(fact_7930_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_7931_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_7932_card__map__elide,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [N: nat] :
( ( ord_less_eq @ nat @ N @ ( finite_card @ ( word @ A ) @ ( top_top @ ( set @ ( word @ A ) ) ) ) )
=> ( ( finite_card @ ( word @ A ) @ ( image @ nat @ ( word @ A ) @ ( semiring_1_of_nat @ ( word @ A ) ) @ ( set_or7035219750837199246ssThan @ nat @ ( zero_zero @ nat ) @ N ) ) )
= ( finite_card @ nat @ ( set_or7035219750837199246ssThan @ nat @ ( zero_zero @ nat ) @ N ) ) ) ) ) ).
% card_map_elide
thf(fact_7933_card__less__Suc2,axiom,
! [M7: set @ nat,I: nat] :
( ~ ( member @ nat @ ( zero_zero @ nat ) @ M7 )
=> ( ( finite_card @ nat
@ ( collect @ nat
@ ^ [K3: nat] :
( ( member @ nat @ ( suc @ K3 ) @ M7 )
& ( ord_less @ nat @ K3 @ I ) ) ) )
= ( finite_card @ nat
@ ( collect @ nat
@ ^ [K3: nat] :
( ( member @ nat @ K3 @ M7 )
& ( ord_less @ nat @ K3 @ ( suc @ I ) ) ) ) ) ) ) ).
% card_less_Suc2
thf(fact_7934_card__less__Suc,axiom,
! [M7: set @ nat,I: nat] :
( ( member @ nat @ ( zero_zero @ nat ) @ M7 )
=> ( ( suc
@ ( finite_card @ nat
@ ( collect @ nat
@ ^ [K3: nat] :
( ( member @ nat @ ( suc @ K3 ) @ M7 )
& ( ord_less @ nat @ K3 @ I ) ) ) ) )
= ( finite_card @ nat
@ ( collect @ nat
@ ^ [K3: nat] :
( ( member @ nat @ K3 @ M7 )
& ( ord_less @ nat @ K3 @ ( suc @ I ) ) ) ) ) ) ) ).
% card_less_Suc
thf(fact_7935_card__less,axiom,
! [M7: set @ nat,I: nat] :
( ( member @ nat @ ( zero_zero @ nat ) @ M7 )
=> ( ( finite_card @ nat
@ ( collect @ nat
@ ^ [K3: nat] :
( ( member @ nat @ K3 @ M7 )
& ( ord_less @ nat @ K3 @ ( suc @ I ) ) ) ) )
!= ( zero_zero @ nat ) ) ) ).
% card_less
thf(fact_7936_card__lists__length__eq,axiom,
! [A: $tType,A3: set @ A,N: 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 )
= N ) ) ) )
= ( power_power @ nat @ ( finite_card @ A @ A3 ) @ N ) ) ) ).
% card_lists_length_eq
thf(fact_7937_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_7938_card__eq__0__iff,axiom,
! [A: $tType,A3: set @ A] :
( ( ( finite_card @ A @ A3 )
= ( zero_zero @ nat ) )
= ( ( A3
= ( bot_bot @ ( set @ A ) ) )
| ~ ( finite_finite2 @ A @ A3 ) ) ) ).
% card_eq_0_iff
thf(fact_7939_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_7940_card__Suc__eq__finite,axiom,
! [A: $tType,A3: set @ A,K: nat] :
( ( ( finite_card @ A @ A3 )
= ( suc @ K ) )
= ( ? [B5: A,B8: set @ A] :
( ( A3
= ( insert @ A @ B5 @ B8 ) )
& ~ ( member @ A @ B5 @ B8 )
& ( ( finite_card @ A @ B8 )
= K )
& ( finite_finite2 @ A @ B8 ) ) ) ) ).
% card_Suc_eq_finite
thf(fact_7941_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_7942_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_7943_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_7944_card__less__sym__Diff,axiom,
! [A: $tType,A3: set @ A,B4: set @ A] :
( ( finite_finite2 @ A @ A3 )
=> ( ( finite_finite2 @ A @ B4 )
=> ( ( ord_less @ nat @ ( finite_card @ A @ A3 ) @ ( finite_card @ A @ B4 ) )
=> ( ord_less @ nat @ ( finite_card @ A @ ( minus_minus @ ( set @ A ) @ A3 @ B4 ) ) @ ( finite_card @ A @ ( minus_minus @ ( set @ A ) @ B4 @ A3 ) ) ) ) ) ) ).
% card_less_sym_Diff
thf(fact_7945_psubset__card__mono,axiom,
! [A: $tType,B4: set @ A,A3: set @ A] :
( ( finite_finite2 @ A @ B4 )
=> ( ( ord_less @ ( set @ A ) @ A3 @ B4 )
=> ( ord_less @ nat @ ( finite_card @ A @ A3 ) @ ( finite_card @ A @ B4 ) ) ) ) ).
% psubset_card_mono
thf(fact_7946_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 )
& ? [Y6: A] :
( ( member @ A @ Y6 @ S3 )
& ( X2 != Y6 )
& ! [Z7: A] :
( ( member @ A @ Z7 @ S3 )
=> ( ( Z7 = X2 )
| ( Z7 = Y6 ) ) ) ) ) ) ) ).
% card_2_iff'
thf(fact_7947_card_Oeq__fold,axiom,
! [A: $tType] :
( ( finite_card @ A )
= ( finite_fold @ A @ nat
@ ^ [Uu3: A] : suc
@ ( zero_zero @ nat ) ) ) ).
% card.eq_fold
thf(fact_7948_card__eq__sum,axiom,
! [A: $tType] :
( ( finite_card @ A )
= ( groups7311177749621191930dd_sum @ A @ nat
@ ^ [X2: A] : ( one_one @ nat ) ) ) ).
% card_eq_sum
thf(fact_7949_card__atLeastZeroLessThan__int,axiom,
! [U: int] :
( ( finite_card @ int @ ( set_or7035219750837199246ssThan @ int @ ( zero_zero @ int ) @ U ) )
= ( nat2 @ U ) ) ).
% card_atLeastZeroLessThan_int
thf(fact_7950_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_7951_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
@ ^ [I2: A] :
( ( member @ A @ I2 @ S3 )
& ( R @ I2 @ X3 ) ) ) )
= K ) )
=> ( ( groups7311177749621191930dd_sum @ A @ nat
@ ^ [I2: A] :
( finite_card @ B
@ ( collect @ B
@ ^ [J2: B] :
( ( member @ B @ J2 @ T5 )
& ( R @ I2 @ J2 ) ) ) )
@ S3 )
= ( times_times @ nat @ K @ ( finite_card @ B @ T5 ) ) ) ) ) ) ).
% sum_multicount
thf(fact_7952_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_7953_sum__bounded__above,axiom,
! [B: $tType,A: $tType] :
( ( ( ordere6911136660526730532id_add @ A )
& ( semiring_1 @ A ) )
=> ! [A3: set @ B,F2: B > A,K6: A] :
( ! [I3: B] :
( ( member @ B @ I3 @ A3 )
=> ( ord_less_eq @ A @ ( F2 @ I3 ) @ 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_7954_sum__bounded__below,axiom,
! [A: $tType,B: $tType] :
( ( ( ordere6911136660526730532id_add @ A )
& ( semiring_1 @ A ) )
=> ! [A3: set @ B,K6: A,F2: B > A] :
( ! [I3: B] :
( ( member @ B @ I3 @ A3 )
=> ( ord_less_eq @ A @ K6 @ ( F2 @ I3 ) ) )
=> ( 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_7955_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_7956_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_7957_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 )
=> ! [Y6: A] :
( ( member @ A @ Y6 @ A3 )
=> ( X2 = Y6 ) ) ) ) ) ) ).
% card_le_Suc0_iff_eq
thf(fact_7958_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_7959_card__eq__SucD,axiom,
! [A: $tType,A3: set @ A,K: nat] :
( ( ( finite_card @ A @ A3 )
= ( suc @ K ) )
=> ? [B3: A,B10: set @ A] :
( ( A3
= ( insert @ A @ B3 @ B10 ) )
& ~ ( member @ A @ B3 @ B10 )
& ( ( finite_card @ A @ B10 )
= K )
& ( ( K
= ( zero_zero @ nat ) )
=> ( B10
= ( bot_bot @ ( set @ A ) ) ) ) ) ) ).
% card_eq_SucD
thf(fact_7960_card__Suc__eq,axiom,
! [A: $tType,A3: set @ A,K: nat] :
( ( ( finite_card @ A @ A3 )
= ( suc @ K ) )
= ( ? [B5: A,B8: set @ A] :
( ( A3
= ( insert @ A @ B5 @ B8 ) )
& ~ ( member @ A @ B5 @ B8 )
& ( ( finite_card @ A @ B8 )
= K )
& ( ( K
= ( zero_zero @ nat ) )
=> ( B8
= ( bot_bot @ ( set @ A ) ) ) ) ) ) ) ).
% card_Suc_eq
thf(fact_7961_card__le__Suc__iff,axiom,
! [A: $tType,N: nat,A3: set @ A] :
( ( ord_less_eq @ nat @ ( suc @ N ) @ ( finite_card @ A @ A3 ) )
= ( ? [A4: A,B8: set @ A] :
( ( A3
= ( insert @ A @ A4 @ B8 ) )
& ~ ( member @ A @ A4 @ B8 )
& ( ord_less_eq @ nat @ N @ ( finite_card @ A @ B8 ) )
& ( finite_finite2 @ A @ B8 ) ) ) ) ).
% card_le_Suc_iff
thf(fact_7962_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_7963_card__psubset,axiom,
! [A: $tType,B4: set @ A,A3: set @ A] :
( ( finite_finite2 @ A @ B4 )
=> ( ( ord_less_eq @ ( set @ A ) @ A3 @ B4 )
=> ( ( ord_less @ nat @ ( finite_card @ A @ A3 ) @ ( finite_card @ A @ B4 ) )
=> ( ord_less @ ( set @ A ) @ A3 @ B4 ) ) ) ) ).
% card_psubset
thf(fact_7964_card__lists__length__le,axiom,
! [A: $tType,A3: set @ A,N: 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 ) @ N ) ) ) )
= ( groups7311177749621191930dd_sum @ nat @ nat @ ( power_power @ nat @ ( finite_card @ A @ A3 ) ) @ ( set_ord_atMost @ nat @ N ) ) ) ) ).
% card_lists_length_le
thf(fact_7965_Re__complex__div__eq__0,axiom,
! [A2: complex,B2: complex] :
( ( ( re @ ( divide_divide @ complex @ A2 @ B2 ) )
= ( zero_zero @ real ) )
= ( ( re @ ( times_times @ complex @ A2 @ ( cnj @ B2 ) ) )
= ( zero_zero @ real ) ) ) ).
% Re_complex_div_eq_0
thf(fact_7966_Im__complex__div__eq__0,axiom,
! [A2: complex,B2: complex] :
( ( ( im @ ( divide_divide @ complex @ A2 @ B2 ) )
= ( zero_zero @ real ) )
= ( ( im @ ( times_times @ complex @ A2 @ ( cnj @ B2 ) ) )
= ( zero_zero @ real ) ) ) ).
% Im_complex_div_eq_0
thf(fact_7967_ex__bij__betw__nat__finite,axiom,
! [A: $tType,M7: set @ A] :
( ( finite_finite2 @ A @ M7 )
=> ? [H3: nat > A] : ( bij_betw @ nat @ A @ H3 @ ( set_or7035219750837199246ssThan @ nat @ ( zero_zero @ nat ) @ ( finite_card @ A @ M7 ) ) @ M7 ) ) ).
% ex_bij_betw_nat_finite
thf(fact_7968_ex__bij__betw__nat__finite__1,axiom,
! [A: $tType,M7: set @ A] :
( ( finite_finite2 @ A @ M7 )
=> ? [H3: nat > A] : ( bij_betw @ nat @ A @ H3 @ ( set_or1337092689740270186AtMost @ nat @ ( one_one @ nat ) @ ( finite_card @ A @ M7 ) ) @ M7 ) ) ).
% ex_bij_betw_nat_finite_1
thf(fact_7969_card__roots__unity,axiom,
! [A: $tType] :
( ( ( real_V8999393235501362500lgebra @ A )
& ( idom @ A ) )
=> ! [N: nat] :
( ( ord_less_eq @ nat @ ( one_one @ nat ) @ N )
=> ( ord_less_eq @ nat
@ ( finite_card @ A
@ ( collect @ A
@ ^ [Z7: A] :
( ( power_power @ A @ Z7 @ N )
= ( one_one @ A ) ) ) )
@ N ) ) ) ).
% card_roots_unity
thf(fact_7970_ex__bij__betw__finite__nat,axiom,
! [A: $tType,M7: set @ A] :
( ( finite_finite2 @ A @ M7 )
=> ? [H3: A > nat] : ( bij_betw @ A @ nat @ H3 @ M7 @ ( set_or7035219750837199246ssThan @ nat @ ( zero_zero @ nat ) @ ( finite_card @ A @ M7 ) ) ) ) ).
% ex_bij_betw_finite_nat
thf(fact_7971_subset__eq__atLeast0__lessThan__card,axiom,
! [N4: set @ nat,N: nat] :
( ( ord_less_eq @ ( set @ nat ) @ N4 @ ( set_or7035219750837199246ssThan @ nat @ ( zero_zero @ nat ) @ N ) )
=> ( ord_less_eq @ nat @ ( finite_card @ nat @ N4 ) @ N ) ) ).
% subset_eq_atLeast0_lessThan_card
thf(fact_7972_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_7973_length__filter__conv__card,axiom,
! [A: $tType,P4: A > $o,Xs2: list @ A] :
( ( size_size @ ( list @ A ) @ ( filter2 @ A @ P4 @ Xs2 ) )
= ( finite_card @ nat
@ ( collect @ nat
@ ^ [I2: nat] :
( ( ord_less @ nat @ I2 @ ( size_size @ ( list @ A ) @ Xs2 ) )
& ( P4 @ ( nth @ A @ Xs2 @ I2 ) ) ) ) ) ) ).
% length_filter_conv_card
thf(fact_7974_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_7975_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_7976_card__nth__roots,axiom,
! [C2: complex,N: nat] :
( ( C2
!= ( zero_zero @ complex ) )
=> ( ( ord_less @ nat @ ( zero_zero @ nat ) @ N )
=> ( ( finite_card @ complex
@ ( collect @ complex
@ ^ [Z7: complex] :
( ( power_power @ complex @ Z7 @ N )
= C2 ) ) )
= N ) ) ) ).
% card_nth_roots
thf(fact_7977_card__roots__unity__eq,axiom,
! [N: nat] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ N )
=> ( ( finite_card @ complex
@ ( collect @ complex
@ ^ [Z7: complex] :
( ( power_power @ complex @ Z7 @ N )
= ( one_one @ complex ) ) ) )
= N ) ) ).
% card_roots_unity_eq
thf(fact_7978_card__2__iff,axiom,
! [A: $tType,S3: set @ A] :
( ( ( finite_card @ A @ S3 )
= ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) )
= ( ? [X2: A,Y6: A] :
( ( S3
= ( insert @ A @ X2 @ ( insert @ A @ Y6 @ ( bot_bot @ ( set @ A ) ) ) ) )
& ( X2 != Y6 ) ) ) ) ).
% card_2_iff
thf(fact_7979_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_7980_card__3__iff,axiom,
! [A: $tType,S3: set @ A] :
( ( ( finite_card @ A @ S3 )
= ( numeral_numeral @ nat @ ( bit1 @ one2 ) ) )
= ( ? [X2: A,Y6: A,Z7: A] :
( ( S3
= ( insert @ A @ X2 @ ( insert @ A @ Y6 @ ( insert @ A @ Z7 @ ( bot_bot @ ( set @ A ) ) ) ) ) )
& ( X2 != Y6 )
& ( Y6 != Z7 )
& ( X2 != Z7 ) ) ) ) ).
% card_3_iff
thf(fact_7981_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_7982_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_7983_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_7984_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_7985_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_7986_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_7987_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_7988_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_7989_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_7990_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_7991_Re__complex__div__lt__0,axiom,
! [A2: complex,B2: complex] :
( ( ord_less @ real @ ( re @ ( divide_divide @ complex @ A2 @ B2 ) ) @ ( zero_zero @ real ) )
= ( ord_less @ real @ ( re @ ( times_times @ complex @ A2 @ ( cnj @ B2 ) ) ) @ ( zero_zero @ real ) ) ) ).
% Re_complex_div_lt_0
thf(fact_7992_Re__complex__div__gt__0,axiom,
! [A2: complex,B2: complex] :
( ( ord_less @ real @ ( zero_zero @ real ) @ ( re @ ( divide_divide @ complex @ A2 @ B2 ) ) )
= ( ord_less @ real @ ( zero_zero @ real ) @ ( re @ ( times_times @ complex @ A2 @ ( cnj @ B2 ) ) ) ) ) ).
% Re_complex_div_gt_0
thf(fact_7993_Re__complex__div__le__0,axiom,
! [A2: complex,B2: complex] :
( ( ord_less_eq @ real @ ( re @ ( divide_divide @ complex @ A2 @ B2 ) ) @ ( zero_zero @ real ) )
= ( ord_less_eq @ real @ ( re @ ( times_times @ complex @ A2 @ ( cnj @ B2 ) ) ) @ ( zero_zero @ real ) ) ) ).
% Re_complex_div_le_0
thf(fact_7994_Re__complex__div__ge__0,axiom,
! [A2: complex,B2: complex] :
( ( ord_less_eq @ real @ ( zero_zero @ real ) @ ( re @ ( divide_divide @ complex @ A2 @ B2 ) ) )
= ( ord_less_eq @ real @ ( zero_zero @ real ) @ ( re @ ( times_times @ complex @ A2 @ ( cnj @ B2 ) ) ) ) ) ).
% Re_complex_div_ge_0
thf(fact_7995_Im__complex__div__lt__0,axiom,
! [A2: complex,B2: complex] :
( ( ord_less @ real @ ( im @ ( divide_divide @ complex @ A2 @ B2 ) ) @ ( zero_zero @ real ) )
= ( ord_less @ real @ ( im @ ( times_times @ complex @ A2 @ ( cnj @ B2 ) ) ) @ ( zero_zero @ real ) ) ) ).
% Im_complex_div_lt_0
thf(fact_7996_Im__complex__div__gt__0,axiom,
! [A2: complex,B2: complex] :
( ( ord_less @ real @ ( zero_zero @ real ) @ ( im @ ( divide_divide @ complex @ A2 @ B2 ) ) )
= ( ord_less @ real @ ( zero_zero @ real ) @ ( im @ ( times_times @ complex @ A2 @ ( cnj @ B2 ) ) ) ) ) ).
% Im_complex_div_gt_0
thf(fact_7997_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_7998_Im__complex__div__le__0,axiom,
! [A2: complex,B2: complex] :
( ( ord_less_eq @ real @ ( im @ ( divide_divide @ complex @ A2 @ B2 ) ) @ ( zero_zero @ real ) )
= ( ord_less_eq @ real @ ( im @ ( times_times @ complex @ A2 @ ( cnj @ B2 ) ) ) @ ( zero_zero @ real ) ) ) ).
% Im_complex_div_le_0
thf(fact_7999_Im__complex__div__ge__0,axiom,
! [A2: complex,B2: complex] :
( ( ord_less_eq @ real @ ( zero_zero @ real ) @ ( im @ ( divide_divide @ complex @ A2 @ B2 ) ) )
= ( ord_less_eq @ real @ ( zero_zero @ real ) @ ( im @ ( times_times @ complex @ A2 @ ( cnj @ B2 ) ) ) ) ) ).
% Im_complex_div_ge_0
thf(fact_8000_prod__le__power,axiom,
! [B: $tType,A: $tType] :
( ( linordered_semidom @ A )
=> ! [A3: set @ B,F2: B > A,N: A,K: nat] :
( ! [I3: B] :
( ( member @ B @ I3 @ A3 )
=> ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( F2 @ I3 ) )
& ( ord_less_eq @ A @ ( F2 @ I3 ) @ N ) ) )
=> ( ( ord_less_eq @ nat @ ( finite_card @ B @ A3 ) @ K )
=> ( ( ord_less_eq @ A @ ( one_one @ A ) @ N )
=> ( ord_less_eq @ A @ ( groups7121269368397514597t_prod @ B @ A @ F2 @ A3 ) @ ( power_power @ A @ N @ K ) ) ) ) ) ) ).
% prod_le_power
thf(fact_8001_sum__bounded__above__strict,axiom,
! [B: $tType,A: $tType] :
( ( ( ordere8940638589300402666id_add @ A )
& ( semiring_1 @ A ) )
=> ! [A3: set @ B,F2: B > A,K6: A] :
( ! [I3: B] :
( ( member @ B @ I3 @ A3 )
=> ( ord_less @ A @ ( F2 @ I3 ) @ 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_8002_sum__bounded__above__divide,axiom,
! [B: $tType,A: $tType] :
( ( linordered_field @ A )
=> ! [A3: set @ B,F2: B > A,K6: A] :
( ! [I3: B] :
( ( member @ B @ I3 @ A3 )
=> ( ord_less_eq @ A @ ( F2 @ I3 ) @ ( 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_8003_complex__mod__mult__cnj,axiom,
! [Z: complex] :
( ( real_V7770717601297561774m_norm @ complex @ ( times_times @ complex @ Z @ ( cnj @ Z ) ) )
= ( power_power @ real @ ( real_V7770717601297561774m_norm @ complex @ Z ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ).
% complex_mod_mult_cnj
thf(fact_8004_card__insert__le__m1,axiom,
! [A: $tType,N: nat,Y: set @ A,X: A] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ N )
=> ( ( ord_less_eq @ nat @ ( finite_card @ A @ Y ) @ ( minus_minus @ nat @ N @ ( one_one @ nat ) ) )
=> ( ord_less_eq @ nat @ ( finite_card @ A @ ( insert @ A @ X @ Y ) ) @ N ) ) ) ).
% card_insert_le_m1
thf(fact_8005_complex__div__gt__0,axiom,
! [A2: complex,B2: complex] :
( ( ( ord_less @ real @ ( zero_zero @ real ) @ ( re @ ( divide_divide @ complex @ A2 @ B2 ) ) )
= ( ord_less @ real @ ( zero_zero @ real ) @ ( re @ ( times_times @ complex @ A2 @ ( cnj @ B2 ) ) ) ) )
& ( ( ord_less @ real @ ( zero_zero @ real ) @ ( im @ ( divide_divide @ complex @ A2 @ B2 ) ) )
= ( ord_less @ real @ ( zero_zero @ real ) @ ( im @ ( times_times @ complex @ A2 @ ( cnj @ B2 ) ) ) ) ) ) ).
% complex_div_gt_0
thf(fact_8006_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_8007_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
@ ^ [Y6: B] :
( times_times @ C
@ ( semiring_1_of_nat @ C
@ ( finite_card @ A
@ ( collect @ A
@ ^ [X2: A] :
( ( member @ A @ X2 @ S3 )
& ( ( G @ X2 )
= Y6 ) ) ) ) )
@ ( F2 @ Y6 ) )
@ R ) ) ) ) ) ) ).
% sum_fun_comp
thf(fact_8008_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
@ ^ [K3: B] : ( if @ A @ ( K3 = A2 ) @ ( B2 @ K3 ) @ 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
@ ^ [K3: B] : ( if @ A @ ( K3 = A2 ) @ ( B2 @ K3 ) @ C2 )
@ S3 )
= ( power_power @ A @ C2 @ ( finite_card @ B @ S3 ) ) ) ) ) ) ) ).
% prod_gen_delta
thf(fact_8009_polyfun__roots__card,axiom,
! [A: $tType] :
( ( ( real_V8999393235501362500lgebra @ A )
& ( idom @ A ) )
=> ! [C2: nat > A,K: nat,N: nat] :
( ( ( C2 @ K )
!= ( zero_zero @ A ) )
=> ( ( ord_less_eq @ nat @ K @ N )
=> ( ord_less_eq @ nat
@ ( finite_card @ A
@ ( collect @ A
@ ^ [Z7: A] :
( ( groups7311177749621191930dd_sum @ nat @ A
@ ^ [I2: nat] : ( times_times @ A @ ( C2 @ I2 ) @ ( power_power @ A @ Z7 @ I2 ) )
@ ( set_ord_atMost @ nat @ N ) )
= ( zero_zero @ A ) ) ) )
@ N ) ) ) ) ).
% polyfun_roots_card
thf(fact_8010_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_8011_complex__norm__square,axiom,
! [Z: complex] :
( ( real_Vector_of_real @ complex @ ( power_power @ real @ ( real_V7770717601297561774m_norm @ complex @ Z ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) )
= ( times_times @ complex @ Z @ ( cnj @ Z ) ) ) ).
% complex_norm_square
thf(fact_8012_card__map__elide2,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [N: nat] :
( ( ord_less_eq @ nat @ N @ ( finite_card @ ( word @ A ) @ ( top_top @ ( set @ ( word @ A ) ) ) ) )
=> ( ( finite_card @ ( word @ A ) @ ( image @ nat @ ( word @ A ) @ ( semiring_1_of_nat @ ( word @ A ) ) @ ( set_or7035219750837199246ssThan @ nat @ ( zero_zero @ nat ) @ N ) ) )
= N ) ) ) ).
% card_map_elide2
thf(fact_8013_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_8014_complex__add__cnj,axiom,
! [Z: complex] :
( ( plus_plus @ complex @ Z @ ( cnj @ Z ) )
= ( real_Vector_of_real @ complex @ ( times_times @ real @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) @ ( re @ Z ) ) ) ) ).
% complex_add_cnj
thf(fact_8015_polyfun__rootbound,axiom,
! [A: $tType] :
( ( ( real_V8999393235501362500lgebra @ A )
& ( idom @ A ) )
=> ! [C2: nat > A,K: nat,N: nat] :
( ( ( C2 @ K )
!= ( zero_zero @ A ) )
=> ( ( ord_less_eq @ nat @ K @ N )
=> ( ( finite_finite2 @ A
@ ( collect @ A
@ ^ [Z7: A] :
( ( groups7311177749621191930dd_sum @ nat @ A
@ ^ [I2: nat] : ( times_times @ A @ ( C2 @ I2 ) @ ( power_power @ A @ Z7 @ I2 ) )
@ ( set_ord_atMost @ nat @ N ) )
= ( zero_zero @ A ) ) ) )
& ( ord_less_eq @ nat
@ ( finite_card @ A
@ ( collect @ A
@ ^ [Z7: A] :
( ( groups7311177749621191930dd_sum @ nat @ A
@ ^ [I2: nat] : ( times_times @ A @ ( C2 @ I2 ) @ ( power_power @ A @ Z7 @ I2 ) )
@ ( set_ord_atMost @ nat @ N ) )
= ( zero_zero @ A ) ) ) )
@ N ) ) ) ) ) ).
% polyfun_rootbound
thf(fact_8016_complex__diff__cnj,axiom,
! [Z: complex] :
( ( minus_minus @ complex @ Z @ ( cnj @ Z ) )
= ( times_times @ complex @ ( real_Vector_of_real @ complex @ ( times_times @ real @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) @ ( im @ Z ) ) ) @ imaginary_unit ) ) ).
% complex_diff_cnj
thf(fact_8017_complex__div__cnj,axiom,
( ( divide_divide @ complex )
= ( ^ [A4: complex,B5: complex] : ( divide_divide @ complex @ ( times_times @ complex @ A4 @ ( cnj @ B5 ) ) @ ( real_Vector_of_real @ complex @ ( power_power @ real @ ( real_V7770717601297561774m_norm @ complex @ B5 ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) ) ).
% complex_div_cnj
thf(fact_8018_cnj__add__mult__eq__Re,axiom,
! [Z: complex,W: complex] :
( ( plus_plus @ complex @ ( times_times @ complex @ Z @ ( cnj @ W ) ) @ ( times_times @ complex @ ( cnj @ Z ) @ W ) )
= ( real_Vector_of_real @ complex @ ( times_times @ real @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) @ ( re @ ( times_times @ complex @ Z @ ( cnj @ W ) ) ) ) ) ) ).
% cnj_add_mult_eq_Re
thf(fact_8019_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_8020_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_8021_card__num0,axiom,
( ( finite_card @ numeral_num0 @ ( top_top @ ( set @ numeral_num0 ) ) )
= ( zero_zero @ nat ) ) ).
% card_num0
thf(fact_8022_card__num1,axiom,
( ( finite_card @ numeral_num1 @ ( top_top @ ( set @ numeral_num1 ) ) )
= ( one_one @ nat ) ) ).
% card_num1
thf(fact_8023_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_8024_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_8025_card__nat,axiom,
( ( finite_card @ nat @ ( top_top @ ( set @ nat ) ) )
= ( zero_zero @ nat ) ) ).
% card_nat
thf(fact_8026_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_8027_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_8028_card__literal,axiom,
( ( finite_card @ literal @ ( top_top @ ( set @ literal ) ) )
= ( zero_zero @ nat ) ) ).
% card_literal
thf(fact_8029_CARD__1,axiom,
! [A: $tType] :
( ( cARD_1 @ A )
=> ( ( finite_card @ A @ ( top_top @ ( set @ A ) ) )
= ( one_one @ nat ) ) ) ).
% CARD_1
thf(fact_8030_bit1_Osize0,axiom,
! [A: $tType] :
( ( finite_finite @ A )
=> ( ord_less @ int @ ( zero_zero @ int ) @ ( semiring_1_of_nat @ int @ ( finite_card @ ( numeral_bit1 @ A ) @ ( top_top @ ( set @ ( numeral_bit1 @ A ) ) ) ) ) ) ) ).
% bit1.size0
thf(fact_8031_bit0_Osize0,axiom,
! [A: $tType] :
( ( finite_finite @ A )
=> ( ord_less @ int @ ( zero_zero @ int ) @ ( semiring_1_of_nat @ int @ ( finite_card @ ( numeral_bit0 @ A ) @ ( top_top @ ( set @ ( numeral_bit0 @ A ) ) ) ) ) ) ) ).
% bit0.size0
thf(fact_8032_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_8033_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_8034_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_8035_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_8036_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_8037_bit0__cases,axiom,
! [A: $tType] :
( ( finite_finite @ A )
=> ! [X: numeral_bit0 @ A] :
~ ! [Z3: int] :
( ( X
= ( ring_1_of_int @ ( numeral_bit0 @ A ) @ Z3 ) )
=> ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ Z3 )
=> ~ ( ord_less @ int @ Z3 @ ( semiring_1_of_nat @ int @ ( finite_card @ ( numeral_bit0 @ A ) @ ( top_top @ ( set @ ( numeral_bit0 @ A ) ) ) ) ) ) ) ) ) ).
% bit0_cases
thf(fact_8038_bit1__cases,axiom,
! [A: $tType] :
( ( finite_finite @ A )
=> ! [X: numeral_bit1 @ A] :
~ ! [Z3: int] :
( ( X
= ( ring_1_of_int @ ( numeral_bit1 @ A ) @ Z3 ) )
=> ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ Z3 )
=> ~ ( ord_less @ int @ Z3 @ ( semiring_1_of_nat @ int @ ( finite_card @ ( numeral_bit1 @ A ) @ ( top_top @ ( set @ ( numeral_bit1 @ A ) ) ) ) ) ) ) ) ) ).
% bit1_cases
thf(fact_8039_bit0__induct,axiom,
! [A: $tType] :
( ( finite_finite @ A )
=> ! [P: ( numeral_bit0 @ A ) > $o,X: numeral_bit0 @ A] :
( ! [Z3: int] :
( ( ord_less_eq @ int @ ( zero_zero @ int ) @ Z3 )
=> ( ( ord_less @ int @ Z3 @ ( semiring_1_of_nat @ int @ ( finite_card @ ( numeral_bit0 @ A ) @ ( top_top @ ( set @ ( numeral_bit0 @ A ) ) ) ) ) )
=> ( P @ ( ring_1_of_int @ ( numeral_bit0 @ A ) @ Z3 ) ) ) )
=> ( P @ X ) ) ) ).
% bit0_induct
thf(fact_8040_bit1__induct,axiom,
! [A: $tType] :
( ( finite_finite @ A )
=> ! [P: ( numeral_bit1 @ A ) > $o,X: numeral_bit1 @ A] :
( ! [Z3: int] :
( ( ord_less_eq @ int @ ( zero_zero @ int ) @ Z3 )
=> ( ( ord_less @ int @ Z3 @ ( semiring_1_of_nat @ int @ ( finite_card @ ( numeral_bit1 @ A ) @ ( top_top @ ( set @ ( numeral_bit1 @ A ) ) ) ) ) )
=> ( P @ ( ring_1_of_int @ ( numeral_bit1 @ A ) @ Z3 ) ) ) )
=> ( P @ X ) ) ) ).
% bit1_induct
thf(fact_8041_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_8042_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_8043_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_8044_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_8045_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_8046_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_8047_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_8048_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_8049_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_8050_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_8051_zero__bit1__def,axiom,
! [A: $tType] :
( ( finite_finite @ A )
=> ( ( zero_zero @ ( numeral_bit1 @ A ) )
= ( numeral_Abs_bit12 @ A @ ( zero_zero @ int ) ) ) ) ).
% zero_bit1_def
thf(fact_8052_bit1_Oof__int__eq,axiom,
! [A: $tType] :
( ( finite_finite @ A )
=> ( ( ring_1_of_int @ ( numeral_bit1 @ A ) )
= ( ^ [Z7: int] : ( numeral_Abs_bit12 @ A @ ( modulo_modulo @ int @ Z7 @ ( semiring_1_of_nat @ int @ ( finite_card @ ( numeral_bit1 @ A ) @ ( top_top @ ( set @ ( numeral_bit1 @ A ) ) ) ) ) ) ) ) ) ) ).
% bit1.of_int_eq
thf(fact_8053_bit1_Oof__nat__eq,axiom,
! [A: $tType] :
( ( finite_finite @ A )
=> ( ( semiring_1_of_nat @ ( numeral_bit1 @ A ) )
= ( ^ [K3: nat] : ( numeral_Abs_bit12 @ A @ ( modulo_modulo @ int @ ( semiring_1_of_nat @ int @ K3 ) @ ( semiring_1_of_nat @ int @ ( finite_card @ ( numeral_bit1 @ A ) @ ( top_top @ ( set @ ( numeral_bit1 @ A ) ) ) ) ) ) ) ) ) ) ).
% bit1.of_nat_eq
thf(fact_8054_dependent__nat__choice,axiom,
! [A: $tType,P: nat > A > $o,Q: nat > A > A > $o] :
( ? [X_12: A] : ( P @ ( zero_zero @ nat ) @ X_12 )
=> ( ! [X3: A,N2: nat] :
( ( P @ N2 @ X3 )
=> ? [Y4: A] :
( ( P @ ( suc @ N2 ) @ Y4 )
& ( Q @ N2 @ X3 @ Y4 ) ) )
=> ? [F5: nat > A] :
! [N10: nat] :
( ( P @ N10 @ ( F5 @ N10 ) )
& ( Q @ N10 @ ( F5 @ N10 ) @ ( F5 @ ( suc @ N10 ) ) ) ) ) ) ).
% dependent_nat_choice
thf(fact_8055_bit1_OUNIV__eq,axiom,
! [A: $tType] :
( ( finite_finite @ A )
=> ( ( top_top @ ( set @ ( numeral_bit1 @ A ) ) )
= ( image @ int @ ( numeral_bit1 @ A ) @ ( numeral_Abs_bit12 @ A ) @ ( set_or7035219750837199246ssThan @ int @ ( zero_zero @ int ) @ ( semiring_1_of_nat @ int @ ( finite_card @ ( numeral_bit1 @ A ) @ ( top_top @ ( set @ ( numeral_bit1 @ A ) ) ) ) ) ) ) ) ) ).
% bit1.UNIV_eq
thf(fact_8056_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_8057_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_8058_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_8059_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_8060_Abs__bit1_H__def,axiom,
! [A: $tType] :
( ( finite_finite @ A )
=> ( ( numeral_Abs_bit1 @ A )
= ( ^ [X2: int] : ( numeral_Abs_bit12 @ A @ ( modulo_modulo @ int @ X2 @ ( semiring_1_of_nat @ int @ ( finite_card @ ( numeral_bit1 @ A ) @ ( top_top @ ( set @ ( numeral_bit1 @ A ) ) ) ) ) ) ) ) ) ) ).
% Abs_bit1'_def
thf(fact_8061_Abs__bit1_H__code,axiom,
! [A: $tType] :
( ( finite_finite @ A )
=> ! [X: int] :
( ( numeral_Rep_bit1 @ A @ ( numeral_Abs_bit1 @ A @ X ) )
= ( modulo_modulo @ int @ X @ ( semiring_1_of_nat @ int @ ( finite_card @ ( numeral_bit1 @ A ) @ ( top_top @ ( set @ ( numeral_bit1 @ A ) ) ) ) ) ) ) ) ).
% Abs_bit1'_code
thf(fact_8062_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_8063_bit1_ORep__Abs__0,axiom,
! [A: $tType] :
( ( finite_finite @ A )
=> ( ( numeral_Rep_bit1 @ A @ ( numeral_Abs_bit12 @ A @ ( zero_zero @ int ) ) )
= ( zero_zero @ int ) ) ) ).
% bit1.Rep_Abs_0
thf(fact_8064_less__bit1__def,axiom,
! [A: $tType] :
( ( finite_finite @ A )
=> ( ( ord_less @ ( numeral_bit1 @ A ) )
= ( ^ [A4: numeral_bit1 @ A,B5: numeral_bit1 @ A] : ( ord_less @ int @ ( numeral_Rep_bit1 @ A @ A4 ) @ ( numeral_Rep_bit1 @ A @ B5 ) ) ) ) ) ).
% less_bit1_def
thf(fact_8065_bit1_ORep__0,axiom,
! [A: $tType] :
( ( finite_finite @ A )
=> ( ( numeral_Rep_bit1 @ A @ ( zero_zero @ ( numeral_bit1 @ A ) ) )
= ( zero_zero @ int ) ) ) ).
% bit1.Rep_0
thf(fact_8066_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_8067_bit1_ORep__less__n,axiom,
! [A: $tType] :
( ( finite_finite @ A )
=> ! [X: numeral_bit1 @ A] : ( ord_less @ int @ ( numeral_Rep_bit1 @ A @ X ) @ ( semiring_1_of_nat @ int @ ( finite_card @ ( numeral_bit1 @ A ) @ ( top_top @ ( set @ ( numeral_bit1 @ A ) ) ) ) ) ) ) ).
% bit1.Rep_less_n
thf(fact_8068_bit1_ORep__mod,axiom,
! [A: $tType] :
( ( finite_finite @ A )
=> ! [X: numeral_bit1 @ A] :
( ( modulo_modulo @ int @ ( numeral_Rep_bit1 @ A @ X ) @ ( semiring_1_of_nat @ int @ ( finite_card @ ( numeral_bit1 @ A ) @ ( top_top @ ( set @ ( numeral_bit1 @ A ) ) ) ) ) )
= ( numeral_Rep_bit1 @ A @ X ) ) ) ).
% bit1.Rep_mod
thf(fact_8069_bit1_ORep__numeral,axiom,
! [A: $tType] :
( ( finite_finite @ A )
=> ! [W: num] :
( ( numeral_Rep_bit1 @ A @ ( numeral_numeral @ ( numeral_bit1 @ A ) @ W ) )
= ( modulo_modulo @ int @ ( numeral_numeral @ int @ W ) @ ( semiring_1_of_nat @ int @ ( finite_card @ ( numeral_bit1 @ A ) @ ( top_top @ ( set @ ( numeral_bit1 @ A ) ) ) ) ) ) ) ) ).
% bit1.Rep_numeral
thf(fact_8070_bit1_ORep__Abs__mod,axiom,
! [A: $tType] :
( ( finite_finite @ A )
=> ! [M: int] :
( ( numeral_Rep_bit1 @ A @ ( numeral_Abs_bit12 @ A @ ( modulo_modulo @ int @ M @ ( semiring_1_of_nat @ int @ ( finite_card @ ( numeral_bit1 @ A ) @ ( top_top @ ( set @ ( numeral_bit1 @ A ) ) ) ) ) ) ) )
= ( modulo_modulo @ int @ M @ ( semiring_1_of_nat @ int @ ( finite_card @ ( numeral_bit1 @ A ) @ ( top_top @ ( set @ ( numeral_bit1 @ A ) ) ) ) ) ) ) ) ).
% bit1.Rep_Abs_mod
thf(fact_8071_bit1_Ominus__def,axiom,
! [A: $tType] :
( ( finite_finite @ A )
=> ( ( uminus_uminus @ ( numeral_bit1 @ A ) )
= ( ^ [X2: numeral_bit1 @ A] : ( numeral_Abs_bit12 @ A @ ( modulo_modulo @ int @ ( uminus_uminus @ int @ ( numeral_Rep_bit1 @ A @ X2 ) ) @ ( semiring_1_of_nat @ int @ ( finite_card @ ( numeral_bit1 @ A ) @ ( top_top @ ( set @ ( numeral_bit1 @ A ) ) ) ) ) ) ) ) ) ) ).
% bit1.minus_def
thf(fact_8072_bit1_Oadd__def,axiom,
! [A: $tType] :
( ( finite_finite @ A )
=> ( ( plus_plus @ ( numeral_bit1 @ A ) )
= ( ^ [X2: numeral_bit1 @ A,Y6: numeral_bit1 @ A] : ( numeral_Abs_bit12 @ A @ ( modulo_modulo @ int @ ( plus_plus @ int @ ( numeral_Rep_bit1 @ A @ X2 ) @ ( numeral_Rep_bit1 @ A @ Y6 ) ) @ ( semiring_1_of_nat @ int @ ( finite_card @ ( numeral_bit1 @ A ) @ ( top_top @ ( set @ ( numeral_bit1 @ A ) ) ) ) ) ) ) ) ) ) ).
% bit1.add_def
thf(fact_8073_bit1_Omult__def,axiom,
! [A: $tType] :
( ( finite_finite @ A )
=> ( ( times_times @ ( numeral_bit1 @ A ) )
= ( ^ [X2: numeral_bit1 @ A,Y6: numeral_bit1 @ A] : ( numeral_Abs_bit12 @ A @ ( modulo_modulo @ int @ ( times_times @ int @ ( numeral_Rep_bit1 @ A @ X2 ) @ ( numeral_Rep_bit1 @ A @ Y6 ) ) @ ( semiring_1_of_nat @ int @ ( finite_card @ ( numeral_bit1 @ A ) @ ( top_top @ ( set @ ( numeral_bit1 @ A ) ) ) ) ) ) ) ) ) ) ).
% bit1.mult_def
thf(fact_8074_bit1_Odiff__def,axiom,
! [A: $tType] :
( ( finite_finite @ A )
=> ( ( minus_minus @ ( numeral_bit1 @ A ) )
= ( ^ [X2: numeral_bit1 @ A,Y6: numeral_bit1 @ A] : ( numeral_Abs_bit12 @ A @ ( modulo_modulo @ int @ ( minus_minus @ int @ ( numeral_Rep_bit1 @ A @ X2 ) @ ( numeral_Rep_bit1 @ A @ Y6 ) ) @ ( semiring_1_of_nat @ int @ ( finite_card @ ( numeral_bit1 @ A ) @ ( top_top @ ( set @ ( numeral_bit1 @ A ) ) ) ) ) ) ) ) ) ) ).
% bit1.diff_def
thf(fact_8075_bit1_OAbs__inverse,axiom,
! [A: $tType] :
( ( finite_finite @ A )
=> ! [M: int] :
( ( member @ int @ M @ ( set_or7035219750837199246ssThan @ int @ ( zero_zero @ int ) @ ( semiring_1_of_nat @ int @ ( finite_card @ ( numeral_bit1 @ A ) @ ( top_top @ ( set @ ( numeral_bit1 @ A ) ) ) ) ) ) )
=> ( ( numeral_Rep_bit1 @ A @ ( numeral_Abs_bit12 @ A @ M ) )
= M ) ) ) ).
% bit1.Abs_inverse
thf(fact_8076_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_8077_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_8078_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_8079_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_8080_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_8081_distinct__sort,axiom,
! [A: $tType,B: $tType] :
( ( linorder @ A )
=> ! [F2: B > A,Xs2: list @ B] :
( ( distinct @ B @ ( linorder_sort_key @ B @ A @ F2 @ Xs2 ) )
= ( distinct @ B @ Xs2 ) ) ) ).
% distinct_sort
thf(fact_8082_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_8083_finite__lists__distinct__length__eq,axiom,
! [A: $tType,A3: set @ A,N: nat] :
( ( finite_finite2 @ A @ A3 )
=> ( finite_finite2 @ ( list @ A )
@ ( collect @ ( list @ A )
@ ^ [Xs: list @ A] :
( ( ( size_size @ ( list @ A ) @ Xs )
= N )
& ( distinct @ A @ Xs )
& ( ord_less_eq @ ( set @ A ) @ ( set2 @ A @ Xs ) @ A3 ) ) ) ) ) ).
% finite_lists_distinct_length_eq
thf(fact_8084_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_8085_not__distinct__decomp,axiom,
! [A: $tType,Ws: list @ A] :
( ~ ( distinct @ A @ Ws )
=> ? [Xs3: list @ A,Ys4: list @ A,Zs2: list @ A,Y3: A] :
( Ws
= ( append @ A @ Xs3 @ ( append @ A @ ( cons @ A @ Y3 @ ( nil @ A ) ) @ ( append @ A @ Ys4 @ ( append @ A @ ( cons @ A @ Y3 @ ( nil @ A ) ) @ Zs2 ) ) ) ) ) ) ).
% not_distinct_decomp
thf(fact_8086_distinct__singleton,axiom,
! [A: $tType,X: A] : ( distinct @ A @ ( cons @ A @ X @ ( nil @ A ) ) ) ).
% distinct_singleton
thf(fact_8087_distinct__length__2__or__more,axiom,
! [A: $tType,A2: A,B2: A,Xs2: list @ A] :
( ( distinct @ A @ ( cons @ A @ A2 @ ( cons @ A @ B2 @ Xs2 ) ) )
= ( ( A2 != B2 )
& ( distinct @ A @ ( cons @ A @ A2 @ Xs2 ) )
& ( distinct @ A @ ( cons @ A @ B2 @ Xs2 ) ) ) ) ).
% distinct_length_2_or_more
thf(fact_8088_not__distinct__conv__prefix,axiom,
! [A: $tType,As2: list @ A] :
( ( ~ ( distinct @ A @ As2 ) )
= ( ? [Xs: list @ A,Y6: A,Ys3: list @ A] :
( ( member @ A @ Y6 @ ( set2 @ A @ Xs ) )
& ( distinct @ A @ Xs )
& ( As2
= ( append @ A @ Xs @ ( cons @ A @ Y6 @ Ys3 ) ) ) ) ) ) ).
% not_distinct_conv_prefix
thf(fact_8089_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_8090_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_8091_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_8092_distinct__conv__nth,axiom,
! [A: $tType] :
( ( distinct @ A )
= ( ^ [Xs: list @ A] :
! [I2: nat] :
( ( ord_less @ nat @ I2 @ ( size_size @ ( list @ A ) @ Xs ) )
=> ! [J2: nat] :
( ( ord_less @ nat @ J2 @ ( size_size @ ( list @ A ) @ Xs ) )
=> ( ( I2 != J2 )
=> ( ( nth @ A @ Xs @ I2 )
!= ( nth @ A @ Xs @ J2 ) ) ) ) ) ) ) ).
% distinct_conv_nth
thf(fact_8093_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_8094_distinct__upt,axiom,
! [I: nat,J: nat] : ( distinct @ nat @ ( upt @ I @ J ) ) ).
% distinct_upt
thf(fact_8095_sorted__list__of__set_Odistinct__if__distinct__map,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [Xs2: list @ A] :
( ( distinct @ A @ Xs2 )
=> ( distinct @ A @ Xs2 ) ) ) ).
% sorted_list_of_set.distinct_if_distinct_map
thf(fact_8096_distinct__product,axiom,
! [A: $tType,B: $tType,Xs2: list @ A,Ys: list @ B] :
( ( distinct @ A @ Xs2 )
=> ( ( distinct @ B @ Ys )
=> ( distinct @ ( product_prod @ A @ B ) @ ( product @ A @ B @ Xs2 @ Ys ) ) ) ) ).
% distinct_product
thf(fact_8097_distinct__upto,axiom,
! [I: int,J: int] : ( distinct @ int @ ( upto @ I @ J ) ) ).
% distinct_upto
thf(fact_8098_distinct_Osimps_I1_J,axiom,
! [A: $tType] : ( distinct @ A @ ( nil @ A ) ) ).
% distinct.simps(1)
thf(fact_8099_sorted__list__of__set_Odistinct__sorted__key__list__of__set,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A3: set @ A] : ( distinct @ A @ ( linord4507533701916653071of_set @ A @ A3 ) ) ) ).
% sorted_list_of_set.distinct_sorted_key_list_of_set
thf(fact_8100_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_8101_distinct__filter,axiom,
! [A: $tType,Xs2: list @ A,P: A > $o] :
( ( distinct @ A @ Xs2 )
=> ( distinct @ A @ ( filter2 @ A @ P @ Xs2 ) ) ) ).
% distinct_filter
thf(fact_8102_distinct__map__filter,axiom,
! [A: $tType,B: $tType,F2: B > A,Xs2: list @ B,P: B > $o] :
( ( distinct @ A @ ( map @ B @ A @ F2 @ Xs2 ) )
=> ( distinct @ A @ ( map @ B @ A @ F2 @ ( filter2 @ B @ P @ Xs2 ) ) ) ) ).
% distinct_map_filter
thf(fact_8103_distinct__tl,axiom,
! [A: $tType,Xs2: list @ A] :
( ( distinct @ A @ Xs2 )
=> ( distinct @ A @ ( tl @ A @ Xs2 ) ) ) ).
% distinct_tl
thf(fact_8104_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_8105_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_8106_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_8107_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_8108_distinct__idx,axiom,
! [B: $tType,A: $tType,F2: B > A,L: list @ B,I: nat,J: nat] :
( ( distinct @ A @ ( map @ B @ A @ F2 @ L ) )
=> ( ( ord_less @ nat @ I @ ( size_size @ ( list @ B ) @ L ) )
=> ( ( ord_less @ nat @ J @ ( size_size @ ( list @ B ) @ L ) )
=> ( ( ( F2 @ ( nth @ B @ L @ I ) )
= ( F2 @ ( nth @ B @ L @ J ) ) )
=> ( I = J ) ) ) ) ) ).
% distinct_idx
thf(fact_8109_inj__on__nth,axiom,
! [A: $tType,Xs2: list @ A,I6: set @ nat] :
( ( distinct @ A @ Xs2 )
=> ( ! [X3: nat] :
( ( member @ nat @ X3 @ I6 )
=> ( ord_less @ nat @ X3 @ ( size_size @ ( list @ A ) @ Xs2 ) ) )
=> ( inj_on @ nat @ A @ ( nth @ A @ Xs2 ) @ I6 ) ) ) ).
% inj_on_nth
thf(fact_8110_bij__betw__nth,axiom,
! [A: $tType,Xs2: list @ A,A3: set @ nat,B4: set @ A] :
( ( distinct @ A @ Xs2 )
=> ( ( A3
= ( set_ord_lessThan @ nat @ ( size_size @ ( list @ A ) @ Xs2 ) ) )
=> ( ( B4
= ( set2 @ A @ Xs2 ) )
=> ( bij_betw @ nat @ A @ ( nth @ A @ Xs2 ) @ A3 @ B4 ) ) ) ) ).
% bij_betw_nth
thf(fact_8111_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_8112_set__update__distinct,axiom,
! [A: $tType,Xs2: list @ A,N: nat,X: A] :
( ( distinct @ A @ Xs2 )
=> ( ( ord_less @ nat @ N @ ( size_size @ ( list @ A ) @ Xs2 ) )
=> ( ( set2 @ A @ ( list_update @ A @ Xs2 @ N @ X ) )
= ( insert @ A @ X @ ( minus_minus @ ( set @ A ) @ ( set2 @ A @ Xs2 ) @ ( insert @ A @ ( nth @ A @ Xs2 @ N ) @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ) ) ).
% set_update_distinct
thf(fact_8113_bit1_Otype,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 ) @ ( semiring_1_of_nat @ int @ ( finite_card @ ( numeral_bit1 @ A ) @ ( top_top @ ( set @ ( numeral_bit1 @ A ) ) ) ) ) ) ) ) ).
% bit1.type
thf(fact_8114_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_8115_mod__type__def,axiom,
! [A: $tType] :
( ( ( minus @ A )
& ( one @ A )
& ( plus @ A )
& ( times @ A )
& ( uminus @ A )
& ( zero @ A ) )
=> ( ( numeral_mod_type @ A )
= ( ^ [N3: int,Rep: A > int,Abs: int > A] :
( ( type_definition @ A @ int @ Rep @ Abs @ ( set_or7035219750837199246ssThan @ int @ ( zero_zero @ int ) @ N3 ) )
& ( ord_less @ int @ ( one_one @ int ) @ N3 )
& ( ( zero_zero @ A )
= ( Abs @ ( zero_zero @ int ) ) )
& ( ( one_one @ A )
= ( Abs @ ( one_one @ int ) ) )
& ! [X2: A,Y6: A] :
( ( plus_plus @ A @ X2 @ Y6 )
= ( Abs @ ( modulo_modulo @ int @ ( plus_plus @ int @ ( Rep @ X2 ) @ ( Rep @ Y6 ) ) @ N3 ) ) )
& ! [X2: A,Y6: A] :
( ( times_times @ A @ X2 @ Y6 )
= ( Abs @ ( modulo_modulo @ int @ ( times_times @ int @ ( Rep @ X2 ) @ ( Rep @ Y6 ) ) @ N3 ) ) )
& ! [X2: A,Y6: A] :
( ( minus_minus @ A @ X2 @ Y6 )
= ( Abs @ ( modulo_modulo @ int @ ( minus_minus @ int @ ( Rep @ X2 ) @ ( Rep @ Y6 ) ) @ N3 ) ) )
& ! [X2: A] :
( ( uminus_uminus @ A @ X2 )
= ( Abs @ ( modulo_modulo @ int @ ( uminus_uminus @ int @ ( Rep @ X2 ) ) @ N3 ) ) ) ) ) ) ) ).
% mod_type_def
thf(fact_8116_mod__type_Ointro,axiom,
! [A: $tType] :
( ( ( minus @ A )
& ( one @ A )
& ( plus @ A )
& ( times @ A )
& ( uminus @ A )
& ( zero @ A ) )
=> ! [Rep2: A > int,Abs2: int > A,N: int] :
( ( type_definition @ A @ int @ Rep2 @ Abs2 @ ( set_or7035219750837199246ssThan @ int @ ( zero_zero @ int ) @ N ) )
=> ( ( ord_less @ int @ ( one_one @ int ) @ N )
=> ( ( ( 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 ) ) @ N ) ) )
=> ( ! [X3: A,Y3: A] :
( ( times_times @ A @ X3 @ Y3 )
= ( Abs2 @ ( modulo_modulo @ int @ ( times_times @ int @ ( Rep2 @ X3 ) @ ( Rep2 @ Y3 ) ) @ N ) ) )
=> ( ! [X3: A,Y3: A] :
( ( minus_minus @ A @ X3 @ Y3 )
= ( Abs2 @ ( modulo_modulo @ int @ ( minus_minus @ int @ ( Rep2 @ X3 ) @ ( Rep2 @ Y3 ) ) @ N ) ) )
=> ( ! [X3: A] :
( ( uminus_uminus @ A @ X3 )
= ( Abs2 @ ( modulo_modulo @ int @ ( uminus_uminus @ int @ ( Rep2 @ X3 ) ) @ N ) ) )
=> ( numeral_mod_type @ A @ N @ Rep2 @ Abs2 ) ) ) ) ) ) ) ) ) ) ).
% mod_type.intro
thf(fact_8117_mod__type_Osize0,axiom,
! [A: $tType] :
( ( ( minus @ A )
& ( one @ A )
& ( plus @ A )
& ( times @ A )
& ( uminus @ A )
& ( zero @ A ) )
=> ! [N: int,Rep2: A > int,Abs2: int > A] :
( ( numeral_mod_type @ A @ N @ Rep2 @ Abs2 )
=> ( ord_less @ int @ ( zero_zero @ int ) @ N ) ) ) ).
% mod_type.size0
thf(fact_8118_mod__type_Osize1,axiom,
! [A: $tType] :
( ( ( minus @ A )
& ( one @ A )
& ( plus @ A )
& ( times @ A )
& ( uminus @ A )
& ( zero @ A ) )
=> ! [N: int,Rep2: A > int,Abs2: int > A] :
( ( numeral_mod_type @ A @ N @ Rep2 @ Abs2 )
=> ( ord_less @ int @ ( one_one @ int ) @ N ) ) ) ).
% mod_type.size1
thf(fact_8119_mod__type_ORep__less__n,axiom,
! [A: $tType] :
( ( ( minus @ A )
& ( one @ A )
& ( plus @ A )
& ( times @ A )
& ( uminus @ A )
& ( zero @ A ) )
=> ! [N: int,Rep2: A > int,Abs2: int > A,X: A] :
( ( numeral_mod_type @ A @ N @ Rep2 @ Abs2 )
=> ( ord_less @ int @ ( Rep2 @ X ) @ N ) ) ) ).
% mod_type.Rep_less_n
thf(fact_8120_mod__type_ORep__mod,axiom,
! [A: $tType] :
( ( ( minus @ A )
& ( one @ A )
& ( plus @ A )
& ( times @ A )
& ( uminus @ A )
& ( zero @ A ) )
=> ! [N: int,Rep2: A > int,Abs2: int > A,X: A] :
( ( numeral_mod_type @ A @ N @ Rep2 @ Abs2 )
=> ( ( modulo_modulo @ int @ ( Rep2 @ X ) @ N )
= ( Rep2 @ X ) ) ) ) ).
% mod_type.Rep_mod
thf(fact_8121_mod__type_ORep__Abs__mod,axiom,
! [A: $tType] :
( ( ( minus @ A )
& ( one @ A )
& ( plus @ A )
& ( times @ A )
& ( uminus @ A )
& ( zero @ A ) )
=> ! [N: int,Rep2: A > int,Abs2: int > A,M: int] :
( ( numeral_mod_type @ A @ N @ Rep2 @ Abs2 )
=> ( ( Rep2 @ ( Abs2 @ ( modulo_modulo @ int @ M @ N ) ) )
= ( modulo_modulo @ int @ M @ N ) ) ) ) ).
% mod_type.Rep_Abs_mod
thf(fact_8122_mod__type_OAbs__inverse,axiom,
! [A: $tType] :
( ( ( minus @ A )
& ( one @ A )
& ( plus @ A )
& ( times @ A )
& ( uminus @ A )
& ( zero @ A ) )
=> ! [N: int,Rep2: A > int,Abs2: int > A,M: int] :
( ( numeral_mod_type @ A @ N @ Rep2 @ Abs2 )
=> ( ( member @ int @ M @ ( set_or7035219750837199246ssThan @ int @ ( zero_zero @ int ) @ N ) )
=> ( ( Rep2 @ ( Abs2 @ M ) )
= M ) ) ) ) ).
% mod_type.Abs_inverse
thf(fact_8123_mod__type_ORep__1,axiom,
! [A: $tType] :
( ( ( minus @ A )
& ( one @ A )
& ( plus @ A )
& ( times @ A )
& ( uminus @ A )
& ( zero @ A ) )
=> ! [N: int,Rep2: A > int,Abs2: int > A] :
( ( numeral_mod_type @ A @ N @ Rep2 @ Abs2 )
=> ( ( Rep2 @ ( one_one @ A ) )
= ( one_one @ int ) ) ) ) ).
% mod_type.Rep_1
thf(fact_8124_mod__type_Oone__def,axiom,
! [A: $tType] :
( ( ( minus @ A )
& ( one @ A )
& ( plus @ A )
& ( times @ A )
& ( uminus @ A )
& ( zero @ A ) )
=> ! [N: int,Rep2: A > int,Abs2: int > A] :
( ( numeral_mod_type @ A @ N @ Rep2 @ Abs2 )
=> ( ( one_one @ A )
= ( Abs2 @ ( one_one @ int ) ) ) ) ) ).
% mod_type.one_def
thf(fact_8125_mod__type_ORep__0,axiom,
! [A: $tType] :
( ( ( minus @ A )
& ( one @ A )
& ( plus @ A )
& ( times @ A )
& ( uminus @ A )
& ( zero @ A ) )
=> ! [N: int,Rep2: A > int,Abs2: int > A] :
( ( numeral_mod_type @ A @ N @ Rep2 @ Abs2 )
=> ( ( Rep2 @ ( zero_zero @ A ) )
= ( zero_zero @ int ) ) ) ) ).
% mod_type.Rep_0
thf(fact_8126_mod__type_Ozero__def,axiom,
! [A: $tType] :
( ( ( minus @ A )
& ( one @ A )
& ( plus @ A )
& ( times @ A )
& ( uminus @ A )
& ( zero @ A ) )
=> ! [N: int,Rep2: A > int,Abs2: int > A] :
( ( numeral_mod_type @ A @ N @ Rep2 @ Abs2 )
=> ( ( zero_zero @ A )
= ( Abs2 @ ( zero_zero @ int ) ) ) ) ) ).
% mod_type.zero_def
thf(fact_8127_mod__type_ORep__Abs__1,axiom,
! [A: $tType] :
( ( ( minus @ A )
& ( one @ A )
& ( plus @ A )
& ( times @ A )
& ( uminus @ A )
& ( zero @ A ) )
=> ! [N: int,Rep2: A > int,Abs2: int > A] :
( ( numeral_mod_type @ A @ N @ Rep2 @ Abs2 )
=> ( ( Rep2 @ ( Abs2 @ ( one_one @ int ) ) )
= ( one_one @ int ) ) ) ) ).
% mod_type.Rep_Abs_1
thf(fact_8128_mod__type_ORep__Abs__0,axiom,
! [A: $tType] :
( ( ( minus @ A )
& ( one @ A )
& ( plus @ A )
& ( times @ A )
& ( uminus @ A )
& ( zero @ A ) )
=> ! [N: int,Rep2: A > int,Abs2: int > A] :
( ( numeral_mod_type @ A @ N @ Rep2 @ Abs2 )
=> ( ( Rep2 @ ( Abs2 @ ( zero_zero @ int ) ) )
= ( zero_zero @ int ) ) ) ) ).
% mod_type.Rep_Abs_0
thf(fact_8129_mod__type_Oadd__def,axiom,
! [A: $tType] :
( ( ( minus @ A )
& ( one @ A )
& ( plus @ A )
& ( times @ A )
& ( uminus @ A )
& ( zero @ A ) )
=> ! [N: int,Rep2: A > int,Abs2: int > A,X: A,Y: A] :
( ( numeral_mod_type @ A @ N @ Rep2 @ Abs2 )
=> ( ( plus_plus @ A @ X @ Y )
= ( Abs2 @ ( modulo_modulo @ int @ ( plus_plus @ int @ ( Rep2 @ X ) @ ( Rep2 @ Y ) ) @ N ) ) ) ) ) ).
% mod_type.add_def
thf(fact_8130_mod__type_Omult__def,axiom,
! [A: $tType] :
( ( ( minus @ A )
& ( one @ A )
& ( plus @ A )
& ( times @ A )
& ( uminus @ A )
& ( zero @ A ) )
=> ! [N: int,Rep2: A > int,Abs2: int > A,X: A,Y: A] :
( ( numeral_mod_type @ A @ N @ Rep2 @ Abs2 )
=> ( ( times_times @ A @ X @ Y )
= ( Abs2 @ ( modulo_modulo @ int @ ( times_times @ int @ ( Rep2 @ X ) @ ( Rep2 @ Y ) ) @ N ) ) ) ) ) ).
% mod_type.mult_def
thf(fact_8131_mod__type_Ominus__def,axiom,
! [A: $tType] :
( ( ( minus @ A )
& ( one @ A )
& ( plus @ A )
& ( times @ A )
& ( uminus @ A )
& ( zero @ A ) )
=> ! [N: int,Rep2: A > int,Abs2: int > A,X: A] :
( ( numeral_mod_type @ A @ N @ Rep2 @ Abs2 )
=> ( ( uminus_uminus @ A @ X )
= ( Abs2 @ ( modulo_modulo @ int @ ( uminus_uminus @ int @ ( Rep2 @ X ) ) @ N ) ) ) ) ) ).
% mod_type.minus_def
thf(fact_8132_mod__type_Odiff__def,axiom,
! [A: $tType] :
( ( ( minus @ A )
& ( one @ A )
& ( plus @ A )
& ( times @ A )
& ( uminus @ A )
& ( zero @ A ) )
=> ! [N: int,Rep2: A > int,Abs2: int > A,X: A,Y: A] :
( ( numeral_mod_type @ A @ N @ Rep2 @ Abs2 )
=> ( ( minus_minus @ A @ X @ Y )
= ( Abs2 @ ( modulo_modulo @ int @ ( minus_minus @ int @ ( Rep2 @ X ) @ ( Rep2 @ Y ) ) @ N ) ) ) ) ) ).
% mod_type.diff_def
thf(fact_8133_mod__type_Otype,axiom,
! [A: $tType] :
( ( ( minus @ A )
& ( one @ A )
& ( plus @ A )
& ( times @ A )
& ( uminus @ A )
& ( zero @ A ) )
=> ! [N: int,Rep2: A > int,Abs2: int > A] :
( ( numeral_mod_type @ A @ N @ Rep2 @ Abs2 )
=> ( type_definition @ A @ int @ Rep2 @ Abs2 @ ( set_or7035219750837199246ssThan @ int @ ( zero_zero @ int ) @ N ) ) ) ) ).
% mod_type.type
thf(fact_8134_bit1__iszero__numeral,axiom,
! [A: $tType] :
( ( finite_finite @ A )
=> ! [W: num] :
( ( ring_1_iszero @ ( numeral_bit1 @ A ) @ ( numeral_numeral @ ( numeral_bit1 @ A ) @ W ) )
= ( ( modulo_modulo @ int @ ( numeral_numeral @ int @ W ) @ ( semiring_1_of_nat @ int @ ( finite_card @ ( numeral_bit1 @ A ) @ ( top_top @ ( set @ ( numeral_bit1 @ A ) ) ) ) ) )
= ( zero_zero @ int ) ) ) ) ).
% bit1_iszero_numeral
thf(fact_8135_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_8136_iszero__neg__numeral,axiom,
! [A: $tType] :
( ( ring_1 @ A )
=> ! [W: num] :
( ( ring_1_iszero @ A @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ W ) ) )
= ( ring_1_iszero @ A @ ( numeral_numeral @ A @ W ) ) ) ) ).
% iszero_neg_numeral
thf(fact_8137_not__iszero__1,axiom,
! [A: $tType] :
( ( ring_1 @ A )
=> ~ ( ring_1_iszero @ A @ ( one_one @ A ) ) ) ).
% not_iszero_1
thf(fact_8138_iszero__def,axiom,
! [A: $tType] :
( ( ring_1 @ A )
=> ( ( ring_1_iszero @ A )
= ( ^ [Z7: A] :
( Z7
= ( zero_zero @ A ) ) ) ) ) ).
% iszero_def
thf(fact_8139_iszero__0,axiom,
! [A: $tType] :
( ( ring_1 @ A )
=> ( ring_1_iszero @ A @ ( zero_zero @ A ) ) ) ).
% iszero_0
thf(fact_8140_zero__bit0__def,axiom,
! [A: $tType] :
( ( finite_finite @ A )
=> ( ( zero_zero @ ( numeral_bit0 @ A ) )
= ( numeral_Abs_bit02 @ A @ ( zero_zero @ int ) ) ) ) ).
% zero_bit0_def
thf(fact_8141_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_8142_not__iszero__Numeral1,axiom,
! [A: $tType] :
( ( ring_1 @ A )
=> ~ ( ring_1_iszero @ A @ ( numeral_numeral @ A @ one2 ) ) ) ).
% not_iszero_Numeral1
thf(fact_8143_not__iszero__numeral,axiom,
! [A: $tType] :
( ( ring_char_0 @ A )
=> ! [W: num] :
~ ( ring_1_iszero @ A @ ( numeral_numeral @ A @ W ) ) ) ).
% not_iszero_numeral
thf(fact_8144_eq__iff__iszero__diff,axiom,
! [A: $tType] :
( ( ring_1 @ A )
=> ( ( ^ [Y2: A,Z2: A] : Y2 = Z2 )
= ( ^ [X2: A,Y6: A] : ( ring_1_iszero @ A @ ( minus_minus @ A @ X2 @ Y6 ) ) ) ) ) ).
% eq_iff_iszero_diff
thf(fact_8145_not__iszero__neg__1,axiom,
! [A: $tType] :
( ( ring_1 @ A )
=> ~ ( ring_1_iszero @ A @ ( uminus_uminus @ A @ ( one_one @ A ) ) ) ) ).
% not_iszero_neg_1
thf(fact_8146_eq__numeral__iff__iszero_I10_J,axiom,
! [A: $tType] :
( ( ring_1 @ A )
=> ! [Y: num] :
( ( ( zero_zero @ A )
= ( numeral_numeral @ A @ Y ) )
= ( ring_1_iszero @ A @ ( numeral_numeral @ A @ Y ) ) ) ) ).
% eq_numeral_iff_iszero(10)
thf(fact_8147_eq__numeral__iff__iszero_I9_J,axiom,
! [A: $tType] :
( ( ring_1 @ A )
=> ! [X: num] :
( ( ( numeral_numeral @ A @ X )
= ( zero_zero @ A ) )
= ( ring_1_iszero @ A @ ( numeral_numeral @ A @ X ) ) ) ) ).
% eq_numeral_iff_iszero(9)
thf(fact_8148_eq__numeral__iff__iszero_I12_J,axiom,
! [A: $tType] :
( ( ring_1 @ A )
=> ! [Y: num] :
( ( ( zero_zero @ A )
= ( uminus_uminus @ A @ ( numeral_numeral @ A @ Y ) ) )
= ( ring_1_iszero @ A @ ( numeral_numeral @ A @ Y ) ) ) ) ).
% eq_numeral_iff_iszero(12)
thf(fact_8149_eq__numeral__iff__iszero_I11_J,axiom,
! [A: $tType] :
( ( ring_1 @ A )
=> ! [X: num] :
( ( ( uminus_uminus @ A @ ( numeral_numeral @ A @ X ) )
= ( zero_zero @ A ) )
= ( ring_1_iszero @ A @ ( numeral_numeral @ A @ X ) ) ) ) ).
% eq_numeral_iff_iszero(11)
thf(fact_8150_not__iszero__neg__Numeral1,axiom,
! [A: $tType] :
( ( ring_1 @ A )
=> ~ ( ring_1_iszero @ A @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ one2 ) ) ) ) ).
% not_iszero_neg_Numeral1
thf(fact_8151_eq__numeral__iff__iszero_I2_J,axiom,
! [A: $tType] :
( ( ring_1 @ A )
=> ! [X: num,Y: num] :
( ( ( numeral_numeral @ A @ X )
= ( uminus_uminus @ A @ ( numeral_numeral @ A @ Y ) ) )
= ( ring_1_iszero @ A @ ( numeral_numeral @ A @ ( plus_plus @ num @ X @ Y ) ) ) ) ) ).
% eq_numeral_iff_iszero(2)
thf(fact_8152_eq__numeral__iff__iszero_I3_J,axiom,
! [A: $tType] :
( ( ring_1 @ A )
=> ! [X: num,Y: num] :
( ( ( uminus_uminus @ A @ ( numeral_numeral @ A @ X ) )
= ( numeral_numeral @ A @ Y ) )
= ( ring_1_iszero @ A @ ( numeral_numeral @ A @ ( plus_plus @ num @ X @ Y ) ) ) ) ) ).
% eq_numeral_iff_iszero(3)
thf(fact_8153_bit0_Oof__nat__eq,axiom,
! [A: $tType] :
( ( finite_finite @ A )
=> ( ( semiring_1_of_nat @ ( numeral_bit0 @ A ) )
= ( ^ [K3: nat] : ( numeral_Abs_bit02 @ A @ ( modulo_modulo @ int @ ( semiring_1_of_nat @ int @ K3 ) @ ( semiring_1_of_nat @ int @ ( finite_card @ ( numeral_bit0 @ A ) @ ( top_top @ ( set @ ( numeral_bit0 @ A ) ) ) ) ) ) ) ) ) ) ).
% bit0.of_nat_eq
thf(fact_8154_eq__numeral__iff__iszero_I8_J,axiom,
! [A: $tType] :
( ( ring_1 @ A )
=> ! [Y: num] :
( ( ( one_one @ A )
= ( uminus_uminus @ A @ ( numeral_numeral @ A @ Y ) ) )
= ( ring_1_iszero @ A @ ( numeral_numeral @ A @ ( plus_plus @ num @ one2 @ Y ) ) ) ) ) ).
% eq_numeral_iff_iszero(8)
thf(fact_8155_eq__numeral__iff__iszero_I7_J,axiom,
! [A: $tType] :
( ( ring_1 @ A )
=> ! [X: num] :
( ( ( uminus_uminus @ A @ ( numeral_numeral @ A @ X ) )
= ( one_one @ A ) )
= ( ring_1_iszero @ A @ ( numeral_numeral @ A @ ( plus_plus @ num @ X @ one2 ) ) ) ) ) ).
% eq_numeral_iff_iszero(7)
thf(fact_8156_bit0_OUNIV__eq,axiom,
! [A: $tType] :
( ( finite_finite @ A )
=> ( ( top_top @ ( set @ ( numeral_bit0 @ A ) ) )
= ( image @ int @ ( numeral_bit0 @ A ) @ ( numeral_Abs_bit02 @ A ) @ ( set_or7035219750837199246ssThan @ int @ ( zero_zero @ int ) @ ( semiring_1_of_nat @ int @ ( finite_card @ ( numeral_bit0 @ A ) @ ( top_top @ ( set @ ( numeral_bit0 @ A ) ) ) ) ) ) ) ) ) ).
% bit0.UNIV_eq
thf(fact_8157_bit0_Oof__int__eq,axiom,
! [A: $tType] :
( ( finite_finite @ A )
=> ( ( ring_1_of_int @ ( numeral_bit0 @ A ) )
= ( ^ [Z7: int] : ( numeral_Abs_bit02 @ A @ ( modulo_modulo @ int @ Z7 @ ( semiring_1_of_nat @ int @ ( finite_card @ ( numeral_bit0 @ A ) @ ( top_top @ ( set @ ( numeral_bit0 @ A ) ) ) ) ) ) ) ) ) ) ).
% bit0.of_int_eq
thf(fact_8158_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_8159_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_8160_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_8161_Abs__bit0_H__def,axiom,
! [A: $tType] :
( ( finite_finite @ A )
=> ( ( numeral_Abs_bit0 @ A )
= ( ^ [X2: int] : ( numeral_Abs_bit02 @ A @ ( modulo_modulo @ int @ X2 @ ( semiring_1_of_nat @ int @ ( finite_card @ ( numeral_bit0 @ A ) @ ( top_top @ ( set @ ( numeral_bit0 @ A ) ) ) ) ) ) ) ) ) ) ).
% Abs_bit0'_def
thf(fact_8162_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_8163_word__eq__numeral__iff__iszero_I10_J,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [Y: num] :
( ( ( zero_zero @ ( word @ A ) )
= ( numeral_numeral @ ( word @ A ) @ Y ) )
= ( ring_1_iszero @ ( word @ A ) @ ( numeral_numeral @ ( word @ A ) @ Y ) ) ) ) ).
% word_eq_numeral_iff_iszero(10)
thf(fact_8164_word__eq__numeral__iff__iszero_I9_J,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [X: num] :
( ( ( numeral_numeral @ ( word @ A ) @ X )
= ( zero_zero @ ( word @ A ) ) )
= ( ring_1_iszero @ ( word @ A ) @ ( numeral_numeral @ ( word @ A ) @ X ) ) ) ) ).
% word_eq_numeral_iff_iszero(9)
thf(fact_8165_word__eq__numeral__iff__iszero_I12_J,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [Y: num] :
( ( ( zero_zero @ ( word @ A ) )
= ( uminus_uminus @ ( word @ A ) @ ( numeral_numeral @ ( word @ A ) @ Y ) ) )
= ( ring_1_iszero @ ( word @ A ) @ ( numeral_numeral @ ( word @ A ) @ Y ) ) ) ) ).
% word_eq_numeral_iff_iszero(12)
thf(fact_8166_word__eq__numeral__iff__iszero_I11_J,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [X: num] :
( ( ( uminus_uminus @ ( word @ A ) @ ( numeral_numeral @ ( word @ A ) @ X ) )
= ( zero_zero @ ( word @ A ) ) )
= ( ring_1_iszero @ ( word @ A ) @ ( numeral_numeral @ ( word @ A ) @ X ) ) ) ) ).
% word_eq_numeral_iff_iszero(11)
thf(fact_8167_word__eq__numeral__iff__iszero_I3_J,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [X: num,Y: num] :
( ( ( uminus_uminus @ ( word @ A ) @ ( numeral_numeral @ ( word @ A ) @ X ) )
= ( numeral_numeral @ ( word @ A ) @ Y ) )
= ( ring_1_iszero @ ( word @ A ) @ ( numeral_numeral @ ( word @ A ) @ ( plus_plus @ num @ X @ Y ) ) ) ) ) ).
% word_eq_numeral_iff_iszero(3)
thf(fact_8168_word__eq__numeral__iff__iszero_I2_J,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [X: num,Y: num] :
( ( ( numeral_numeral @ ( word @ A ) @ X )
= ( uminus_uminus @ ( word @ A ) @ ( numeral_numeral @ ( word @ A ) @ Y ) ) )
= ( ring_1_iszero @ ( word @ A ) @ ( numeral_numeral @ ( word @ A ) @ ( plus_plus @ num @ X @ Y ) ) ) ) ) ).
% word_eq_numeral_iff_iszero(2)
thf(fact_8169_iszero__word__no,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [Bin: num] :
( ( ring_1_iszero @ ( word @ A ) @ ( numeral_numeral @ ( word @ A ) @ Bin ) )
= ( ring_1_iszero @ int @ ( bit_se2584673776208193580ke_bit @ int @ ( type_len0_len_of @ A @ ( type2 @ A ) ) @ ( numeral_numeral @ int @ Bin ) ) ) ) ) ).
% iszero_word_no
thf(fact_8170_word__eq__numeral__iff__iszero_I8_J,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [Y: num] :
( ( ( one_one @ ( word @ A ) )
= ( uminus_uminus @ ( word @ A ) @ ( numeral_numeral @ ( word @ A ) @ Y ) ) )
= ( ring_1_iszero @ ( word @ A ) @ ( numeral_numeral @ ( word @ A ) @ ( plus_plus @ num @ one2 @ Y ) ) ) ) ) ).
% word_eq_numeral_iff_iszero(8)
thf(fact_8171_word__eq__numeral__iff__iszero_I7_J,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [X: num] :
( ( ( uminus_uminus @ ( word @ A ) @ ( numeral_numeral @ ( word @ A ) @ X ) )
= ( one_one @ ( word @ A ) ) )
= ( ring_1_iszero @ ( word @ A ) @ ( numeral_numeral @ ( word @ A ) @ ( plus_plus @ num @ X @ one2 ) ) ) ) ) ).
% word_eq_numeral_iff_iszero(7)
thf(fact_8172_bit0__iszero__numeral,axiom,
! [A: $tType] :
( ( finite_finite @ A )
=> ! [W: num] :
( ( ring_1_iszero @ ( numeral_bit0 @ A ) @ ( numeral_numeral @ ( numeral_bit0 @ A ) @ W ) )
= ( ( modulo_modulo @ int @ ( numeral_numeral @ int @ W ) @ ( semiring_1_of_nat @ int @ ( finite_card @ ( numeral_bit0 @ A ) @ ( top_top @ ( set @ ( numeral_bit0 @ A ) ) ) ) ) )
= ( zero_zero @ int ) ) ) ) ).
% bit0_iszero_numeral
thf(fact_8173_bit0_ORep__Abs__0,axiom,
! [A: $tType] :
( ( finite_finite @ A )
=> ( ( numeral_Rep_bit0 @ A @ ( numeral_Abs_bit02 @ A @ ( zero_zero @ int ) ) )
= ( zero_zero @ int ) ) ) ).
% bit0.Rep_Abs_0
thf(fact_8174_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_8175_less__bit0__def,axiom,
! [A: $tType] :
( ( finite_finite @ A )
=> ( ( ord_less @ ( numeral_bit0 @ A ) )
= ( ^ [A4: numeral_bit0 @ A,B5: numeral_bit0 @ A] : ( ord_less @ int @ ( numeral_Rep_bit0 @ A @ A4 ) @ ( numeral_Rep_bit0 @ A @ B5 ) ) ) ) ) ).
% less_bit0_def
thf(fact_8176_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_8177_bit0_ORep__0,axiom,
! [A: $tType] :
( ( finite_finite @ A )
=> ( ( numeral_Rep_bit0 @ A @ ( zero_zero @ ( numeral_bit0 @ A ) ) )
= ( zero_zero @ int ) ) ) ).
% bit0.Rep_0
thf(fact_8178_Abs__bit0_H__code,axiom,
! [A: $tType] :
( ( finite_finite @ A )
=> ! [X: int] :
( ( numeral_Rep_bit0 @ A @ ( numeral_Abs_bit0 @ A @ X ) )
= ( modulo_modulo @ int @ X @ ( semiring_1_of_nat @ int @ ( finite_card @ ( numeral_bit0 @ A ) @ ( top_top @ ( set @ ( numeral_bit0 @ A ) ) ) ) ) ) ) ) ).
% Abs_bit0'_code
% Type constructors (1297)
thf(tcon_Product__Type_Ounit___Enum_Oenum,axiom,
enum @ product_unit ).
thf(tcon_Product__Type_Ounit___Complete__Lattices_Ocomplete__linorder,axiom,
comple5582772986160207858norder @ product_unit ).
thf(tcon_Product__Type_Oprod___Enum_Oenum_1,axiom,
! [A10: $tType,A11: $tType] :
( ( ( enum @ A10 )
& ( enum @ A11 ) )
=> ( enum @ ( product_prod @ A10 @ A11 ) ) ) ).
thf(tcon_Numeral__Type_Onum1___Enum_Oenum_2,axiom,
enum @ numeral_num1 ).
thf(tcon_Numeral__Type_Obit1___Enum_Oenum_3,axiom,
! [A10: $tType] :
( ( finite_finite @ A10 )
=> ( enum @ ( numeral_bit1 @ A10 ) ) ) ).
thf(tcon_Numeral__Type_Obit0___Enum_Oenum_4,axiom,
! [A10: $tType] :
( ( finite_finite @ A10 )
=> ( enum @ ( numeral_bit0 @ A10 ) ) ) ).
thf(tcon_Extended__Nat_Oenat___Complete__Lattices_Ocomplete__linorder_5,axiom,
comple5582772986160207858norder @ extended_enat ).
thf(tcon_Option_Ooption___Enum_Oenum_6,axiom,
! [A10: $tType] :
( ( enum @ A10 )
=> ( enum @ ( option @ A10 ) ) ) ).
thf(tcon_Option_Ooption___Complete__Lattices_Ocomplete__linorder_7,axiom,
! [A10: $tType] :
( ( comple5582772986160207858norder @ A10 )
=> ( comple5582772986160207858norder @ ( option @ A10 ) ) ) ).
thf(tcon_Enum_Ofinite__3___Enum_Oenum_8,axiom,
enum @ finite_3 ).
thf(tcon_Enum_Ofinite__3___Complete__Lattices_Ocomplete__linorder_9,axiom,
comple5582772986160207858norder @ finite_3 ).
thf(tcon_Enum_Ofinite__2___Enum_Oenum_10,axiom,
enum @ finite_2 ).
thf(tcon_Enum_Ofinite__2___Complete__Lattices_Ocomplete__linorder_11,axiom,
comple5582772986160207858norder @ finite_2 ).
thf(tcon_Enum_Ofinite__1___Enum_Oenum_12,axiom,
enum @ finite_1 ).
thf(tcon_Enum_Ofinite__1___Complete__Lattices_Ocomplete__linorder_13,axiom,
comple5582772986160207858norder @ finite_1 ).
thf(tcon_Sum__Type_Osum___Enum_Oenum_14,axiom,
! [A10: $tType,A11: $tType] :
( ( ( enum @ A10 )
& ( enum @ A11 ) )
=> ( enum @ ( sum_sum @ A10 @ A11 ) ) ) ).
thf(tcon_String_Ochar___Enum_Oenum_15,axiom,
enum @ char ).
thf(tcon_Word_Oword___Enum_Oenum_16,axiom,
! [A10: $tType] :
( ( type_len @ A10 )
=> ( enum @ ( word @ A10 ) ) ) ).
thf(tcon_HOL_Obool___Enum_Oenum_17,axiom,
enum @ $o ).
thf(tcon_Set_Oset___Enum_Oenum_18,axiom,
! [A10: $tType] :
( ( enum @ A10 )
=> ( enum @ ( set @ A10 ) ) ) ).
thf(tcon_fun___Enum_Oenum_19,axiom,
! [A10: $tType,A11: $tType] :
( ( ( enum @ A10 )
& ( enum @ A11 ) )
=> ( enum @ ( A10 > A11 ) ) ) ).
thf(tcon_fun___Boolean__Algebras_Oboolean__algebra,axiom,
! [A10: $tType,A11: $tType] :
( ( boolea8198339166811842893lgebra @ A11 )
=> ( boolea8198339166811842893lgebra @ ( A10 > A11 ) ) ) ).
thf(tcon_fun___Orderings_Oorder__top,axiom,
! [A10: $tType,A11: $tType] :
( ( order_top @ A11 )
=> ( order_top @ ( A10 > A11 ) ) ) ).
thf(tcon_fun___Orderings_Oorder__bot,axiom,
! [A10: $tType,A11: $tType] :
( ( order_bot @ A11 )
=> ( order_bot @ ( A10 > A11 ) ) ) ).
thf(tcon_fun___Orderings_Opreorder,axiom,
! [A10: $tType,A11: $tType] :
( ( preorder @ A11 )
=> ( preorder @ ( A10 > A11 ) ) ) ).
thf(tcon_fun___Finite__Set_Ofinite,axiom,
! [A10: $tType,A11: $tType] :
( ( ( finite_finite @ A10 )
& ( finite_finite @ A11 ) )
=> ( finite_finite @ ( A10 > A11 ) ) ) ).
thf(tcon_fun___Orderings_Oorder,axiom,
! [A10: $tType,A11: $tType] :
( ( order @ A11 )
=> ( order @ ( A10 > A11 ) ) ) ).
thf(tcon_fun___Orderings_Oord,axiom,
! [A10: $tType,A11: $tType] :
( ( ord @ A11 )
=> ( ord @ ( A10 > A11 ) ) ) ).
thf(tcon_fun___Groups_Ouminus,axiom,
! [A10: $tType,A11: $tType] :
( ( uminus @ A11 )
=> ( uminus @ ( A10 > A11 ) ) ) ).
thf(tcon_fun___Groups_Ominus,axiom,
! [A10: $tType,A11: $tType] :
( ( minus @ A11 )
=> ( minus @ ( A10 > A11 ) ) ) ).
thf(tcon_fun___HOL_Oequal,axiom,
! [A10: $tType,A11: $tType] :
( ( ( enum @ A10 )
& ( cl_HOL_Oequal @ A11 ) )
=> ( cl_HOL_Oequal @ ( A10 > A11 ) ) ) ).
thf(tcon_Int_Oint___Conditionally__Complete__Lattices_Oconditionally__complete__linorder,axiom,
condit6923001295902523014norder @ 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___Limits_Otopological__comm__monoid__mult,axiom,
topolo4987421752381908075d_mult @ int ).
thf(tcon_Int_Oint___Bit__Comprehension_Obit__comprehension,axiom,
bit_bi6583157726757044596ension @ int ).
thf(tcon_Int_Oint___Rings_Olinordered__semiring__1__strict,axiom,
linord715952674999750819strict @ int ).
thf(tcon_Int_Oint___Limits_Otopological__comm__monoid__add,axiom,
topolo5987344860129210374id_add @ int ).
thf(tcon_Int_Oint___Groups_Olinordered__ab__semigroup__add,axiom,
linord4140545234300271783up_add @ int ).
thf(tcon_Int_Oint___Bit__Operations_Oring__bit__operations,axiom,
bit_ri3973907225187159222ations @ int ).
thf(tcon_Int_Oint___Topological__Spaces_Oorder__topology,axiom,
topolo2564578578187576103pology @ int ).
thf(tcon_Int_Oint___Rings_Osemiring__1__no__zero__divisors,axiom,
semiri2026040879449505780visors @ int ).
thf(tcon_Int_Oint___Rings_Olinordered__nonzero__semiring,axiom,
linord181362715937106298miring @ int ).
thf(tcon_Int_Oint___Limits_Otopological__semigroup__mult,axiom,
topolo4211221413907600880p_mult @ int ).
thf(tcon_Int_Oint___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___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_20,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_21,axiom,
order @ int ).
thf(tcon_Int_Oint___Num_Oneg__numeral,axiom,
neg_numeral @ int ).
thf(tcon_Int_Oint___Nat_Oring__char__0,axiom,
ring_char_0 @ int ).
thf(tcon_Int_Oint___Rings_Osemiring,axiom,
semiring @ int ).
thf(tcon_Int_Oint___Rings_Osemidom,axiom,
semidom @ int ).
thf(tcon_Int_Oint___Orderings_Oord_22,axiom,
ord @ int ).
thf(tcon_Int_Oint___Groups_Ouminus_23,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_24,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_25,axiom,
cl_HOL_Oequal @ int ).
thf(tcon_Nat_Onat___Conditionally__Complete__Lattices_Oconditionally__complete__linorder_26,axiom,
condit6923001295902523014norder @ nat ).
thf(tcon_Nat_Onat___Bit__Operations_Ounique__euclidean__semiring__with__bit__operations_27,axiom,
bit_un5681908812861735899ations @ nat ).
thf(tcon_Nat_Onat___Semiring__Normalization_Ocomm__semiring__1__cancel__crossproduct_28,axiom,
semiri1453513574482234551roduct @ nat ).
thf(tcon_Nat_Onat___Euclidean__Division_Ounique__euclidean__semiring__with__nat_29,axiom,
euclid5411537665997757685th_nat @ nat ).
thf(tcon_Nat_Onat___Groups_Oordered__ab__semigroup__monoid__add__imp__le_30,axiom,
ordere1937475149494474687imp_le @ nat ).
thf(tcon_Nat_Onat___Euclidean__Division_Ounique__euclidean__semiring_31,axiom,
euclid3128863361964157862miring @ nat ).
thf(tcon_Nat_Onat___Euclidean__Division_Oeuclidean__semiring__cancel_32,axiom,
euclid4440199948858584721cancel @ nat ).
thf(tcon_Nat_Onat___Divides_Ounique__euclidean__semiring__numeral_33,axiom,
unique1627219031080169319umeral @ nat ).
thf(tcon_Nat_Onat___Rings_Osemiring__no__zero__divisors__cancel_34,axiom,
semiri6575147826004484403cancel @ nat ).
thf(tcon_Nat_Onat___Groups_Ostrict__ordered__ab__semigroup__add_35,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_36,axiom,
ordere580206878836729694up_add @ nat ).
thf(tcon_Nat_Onat___Groups_Oordered__ab__semigroup__add__imp__le_37,axiom,
ordere2412721322843649153imp_le @ nat ).
thf(tcon_Nat_Onat___Bit__Operations_Osemiring__bit__operations_38,axiom,
bit_se359711467146920520ations @ nat ).
thf(tcon_Nat_Onat___Rings_Olinordered__comm__semiring__strict_39,axiom,
linord2810124833399127020strict @ nat ).
thf(tcon_Nat_Onat___Groups_Ostrict__ordered__comm__monoid__add_40,axiom,
strict7427464778891057005id_add @ nat ).
thf(tcon_Nat_Onat___Groups_Oordered__cancel__comm__monoid__add_41,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_42,axiom,
euclid3725896446679973847miring @ nat ).
thf(tcon_Nat_Onat___Topological__Spaces_Otopological__space_43,axiom,
topolo4958980785337419405_space @ nat ).
thf(tcon_Nat_Onat___Topological__Spaces_Olinorder__topology_44,axiom,
topolo1944317154257567458pology @ nat ).
thf(tcon_Nat_Onat___Limits_Otopological__comm__monoid__mult_45,axiom,
topolo4987421752381908075d_mult @ nat ).
thf(tcon_Nat_Onat___Limits_Otopological__comm__monoid__add_46,axiom,
topolo5987344860129210374id_add @ nat ).
thf(tcon_Nat_Onat___Groups_Olinordered__ab__semigroup__add_47,axiom,
linord4140545234300271783up_add @ nat ).
thf(tcon_Nat_Onat___Topological__Spaces_Oorder__topology_48,axiom,
topolo2564578578187576103pology @ nat ).
thf(tcon_Nat_Onat___Rings_Osemiring__1__no__zero__divisors_49,axiom,
semiri2026040879449505780visors @ nat ).
thf(tcon_Nat_Onat___Rings_Olinordered__nonzero__semiring_50,axiom,
linord181362715937106298miring @ nat ).
thf(tcon_Nat_Onat___Limits_Otopological__semigroup__mult_51,axiom,
topolo4211221413907600880p_mult @ nat ).
thf(tcon_Nat_Onat___Rings_Olinordered__semiring__strict_52,axiom,
linord8928482502909563296strict @ nat ).
thf(tcon_Nat_Onat___Rings_Osemiring__no__zero__divisors_53,axiom,
semiri3467727345109120633visors @ nat ).
thf(tcon_Nat_Onat___Groups_Oordered__ab__semigroup__add_54,axiom,
ordere6658533253407199908up_add @ nat ).
thf(tcon_Nat_Onat___Limits_Otopological__monoid__mult_55,axiom,
topolo1898628316856586783d_mult @ nat ).
thf(tcon_Nat_Onat___Groups_Oordered__comm__monoid__add_56,axiom,
ordere6911136660526730532id_add @ nat ).
thf(tcon_Nat_Onat___Groups_Ocancel__ab__semigroup__add_57,axiom,
cancel2418104881723323429up_add @ nat ).
thf(tcon_Nat_Onat___Limits_Otopological__monoid__add_58,axiom,
topolo6943815403480290642id_add @ nat ).
thf(tcon_Nat_Onat___Groups_Ocancel__comm__monoid__add_59,axiom,
cancel1802427076303600483id_add @ nat ).
thf(tcon_Nat_Onat___Rings_Ocomm__semiring__1__cancel_60,axiom,
comm_s4317794764714335236cancel @ nat ).
thf(tcon_Nat_Onat___Bit__Operations_Osemiring__bits_61,axiom,
bit_semiring_bits @ nat ).
thf(tcon_Nat_Onat___Topological__Spaces_Ot2__space_62,axiom,
topological_t2_space @ nat ).
thf(tcon_Nat_Onat___Rings_Oordered__comm__semiring_63,axiom,
ordere2520102378445227354miring @ nat ).
thf(tcon_Nat_Onat___Groups_Ocancel__semigroup__add_64,axiom,
cancel_semigroup_add @ nat ).
thf(tcon_Nat_Onat___Rings_Olinordered__semiring_65,axiom,
linordered_semiring @ nat ).
thf(tcon_Nat_Onat___Rings_Oordered__semiring__0_66,axiom,
ordered_semiring_0 @ nat ).
thf(tcon_Nat_Onat___Rings_Olinordered__semidom_67,axiom,
linordered_semidom @ nat ).
thf(tcon_Nat_Onat___Groups_Oab__semigroup__mult_68,axiom,
ab_semigroup_mult @ nat ).
thf(tcon_Nat_Onat___Rings_Osemiring__1__cancel_69,axiom,
semiring_1_cancel @ nat ).
thf(tcon_Nat_Onat___Rings_Oalgebraic__semidom_70,axiom,
algebraic_semidom @ nat ).
thf(tcon_Nat_Onat___Groups_Ocomm__monoid__mult_71,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_72,axiom,
ab_semigroup_add @ nat ).
thf(tcon_Nat_Onat___Rings_Oordered__semiring_73,axiom,
ordered_semiring @ nat ).
thf(tcon_Nat_Onat___Parity_Osemiring__parity_74,axiom,
semiring_parity @ nat ).
thf(tcon_Nat_Onat___Groups_Ocomm__monoid__add_75,axiom,
comm_monoid_add @ nat ).
thf(tcon_Nat_Onat___Rings_Osemiring__modulo_76,axiom,
semiring_modulo @ nat ).
thf(tcon_Nat_Onat___Rings_Ocomm__semiring__1_77,axiom,
comm_semiring_1 @ nat ).
thf(tcon_Nat_Onat___Rings_Ocomm__semiring__0_78,axiom,
comm_semiring_0 @ nat ).
thf(tcon_Nat_Onat___Groups_Osemigroup__mult_79,axiom,
semigroup_mult @ nat ).
thf(tcon_Nat_Onat___Rings_Osemidom__modulo_80,axiom,
semidom_modulo @ nat ).
thf(tcon_Nat_Onat___Rings_Osemidom__divide_81,axiom,
semidom_divide @ nat ).
thf(tcon_Nat_Onat___Num_Osemiring__numeral_82,axiom,
semiring_numeral @ nat ).
thf(tcon_Nat_Onat___Groups_Osemigroup__add_83,axiom,
semigroup_add @ nat ).
thf(tcon_Nat_Onat___Rings_Ozero__less__one_84,axiom,
zero_less_one @ nat ).
thf(tcon_Nat_Onat___Rings_Ocomm__semiring_85,axiom,
comm_semiring @ nat ).
thf(tcon_Nat_Onat___Orderings_Owellorder,axiom,
wellorder @ nat ).
thf(tcon_Nat_Onat___Orderings_Oorder__bot_86,axiom,
order_bot @ nat ).
thf(tcon_Nat_Onat___Nat_Osemiring__char__0_87,axiom,
semiring_char_0 @ nat ).
thf(tcon_Nat_Onat___Rings_Ozero__neq__one_88,axiom,
zero_neq_one @ nat ).
thf(tcon_Nat_Onat___Orderings_Opreorder_89,axiom,
preorder @ nat ).
thf(tcon_Nat_Onat___Orderings_Olinorder_90,axiom,
linorder @ nat ).
thf(tcon_Nat_Onat___Groups_Omonoid__mult_91,axiom,
monoid_mult @ nat ).
thf(tcon_Nat_Onat___Groups_Omonoid__add_92,axiom,
monoid_add @ nat ).
thf(tcon_Nat_Onat___Rings_Osemiring__1_93,axiom,
semiring_1 @ nat ).
thf(tcon_Nat_Onat___Rings_Osemiring__0_94,axiom,
semiring_0 @ nat ).
thf(tcon_Nat_Onat___Orderings_Ono__top_95,axiom,
no_top @ nat ).
thf(tcon_Nat_Onat___GCD_Osemiring__gcd_96,axiom,
semiring_gcd @ nat ).
thf(tcon_Nat_Onat___Rings_Omult__zero_97,axiom,
mult_zero @ nat ).
thf(tcon_Nat_Onat___Orderings_Oorder_98,axiom,
order @ nat ).
thf(tcon_Nat_Onat___Rings_Osemiring_99,axiom,
semiring @ nat ).
thf(tcon_Nat_Onat___Rings_Osemidom_100,axiom,
semidom @ nat ).
thf(tcon_Nat_Onat___Orderings_Oord_101,axiom,
ord @ nat ).
thf(tcon_Nat_Onat___Groups_Otimes_102,axiom,
times @ nat ).
thf(tcon_Nat_Onat___Groups_Ominus_103,axiom,
minus @ nat ).
thf(tcon_Nat_Onat___Power_Opower_104,axiom,
power @ nat ).
thf(tcon_Nat_Onat___Num_Onumeral_105,axiom,
numeral @ nat ).
thf(tcon_Nat_Onat___Groups_Ozero_106,axiom,
zero @ nat ).
thf(tcon_Nat_Onat___Groups_Oplus_107,axiom,
plus @ nat ).
thf(tcon_Nat_Onat___Groups_Oone_108,axiom,
one @ nat ).
thf(tcon_Nat_Onat___Rings_Odvd_109,axiom,
dvd @ nat ).
thf(tcon_Nat_Onat___Heap_Oheap_110,axiom,
heap @ nat ).
thf(tcon_Nat_Onat___HOL_Oequal_111,axiom,
cl_HOL_Oequal @ nat ).
thf(tcon_Nat_Onat___Nat_Osize,axiom,
size @ nat ).
thf(tcon_Num_Onum___Orderings_Opreorder_112,axiom,
preorder @ num ).
thf(tcon_Num_Onum___Orderings_Olinorder_113,axiom,
linorder @ num ).
thf(tcon_Num_Onum___Orderings_Oorder_114,axiom,
order @ num ).
thf(tcon_Num_Onum___Orderings_Oord_115,axiom,
ord @ num ).
thf(tcon_Num_Onum___Groups_Otimes_116,axiom,
times @ num ).
thf(tcon_Num_Onum___Groups_Oplus_117,axiom,
plus @ num ).
thf(tcon_Num_Onum___HOL_Oequal_118,axiom,
cl_HOL_Oequal @ num ).
thf(tcon_Num_Onum___Nat_Osize_119,axiom,
size @ num ).
thf(tcon_Rat_Orat___Semiring__Normalization_Ocomm__semiring__1__cancel__crossproduct_120,axiom,
semiri1453513574482234551roduct @ rat ).
thf(tcon_Rat_Orat___Groups_Oordered__ab__semigroup__monoid__add__imp__le_121,axiom,
ordere1937475149494474687imp_le @ rat ).
thf(tcon_Rat_Orat___Rings_Osemiring__no__zero__divisors__cancel_122,axiom,
semiri6575147826004484403cancel @ rat ).
thf(tcon_Rat_Orat___Groups_Ostrict__ordered__ab__semigroup__add_123,axiom,
strict9044650504122735259up_add @ rat ).
thf(tcon_Rat_Orat___Groups_Oordered__cancel__ab__semigroup__add_124,axiom,
ordere580206878836729694up_add @ rat ).
thf(tcon_Rat_Orat___Groups_Oordered__ab__semigroup__add__imp__le_125,axiom,
ordere2412721322843649153imp_le @ rat ).
thf(tcon_Rat_Orat___Rings_Olinordered__comm__semiring__strict_126,axiom,
linord2810124833399127020strict @ rat ).
thf(tcon_Rat_Orat___Groups_Ostrict__ordered__comm__monoid__add_127,axiom,
strict7427464778891057005id_add @ rat ).
thf(tcon_Rat_Orat___Groups_Oordered__cancel__comm__monoid__add_128,axiom,
ordere8940638589300402666id_add @ rat ).
thf(tcon_Rat_Orat___Archimedean__Field_Oarchimedean__field,axiom,
archim462609752435547400_field @ rat ).
thf(tcon_Rat_Orat___Rings_Olinordered__semiring__1__strict_129,axiom,
linord715952674999750819strict @ rat ).
thf(tcon_Rat_Orat___Orderings_Ounbounded__dense__linorder,axiom,
unboun7993243217541854897norder @ rat ).
thf(tcon_Rat_Orat___Groups_Olinordered__ab__semigroup__add_130,axiom,
linord4140545234300271783up_add @ rat ).
thf(tcon_Rat_Orat___Rings_Osemiring__1__no__zero__divisors_131,axiom,
semiri2026040879449505780visors @ rat ).
thf(tcon_Rat_Orat___Rings_Olinordered__nonzero__semiring_132,axiom,
linord181362715937106298miring @ rat ).
thf(tcon_Rat_Orat___Rings_Olinordered__semiring__strict_133,axiom,
linord8928482502909563296strict @ rat ).
thf(tcon_Rat_Orat___Rings_Osemiring__no__zero__divisors_134,axiom,
semiri3467727345109120633visors @ rat ).
thf(tcon_Rat_Orat___Groups_Oordered__ab__semigroup__add_135,axiom,
ordere6658533253407199908up_add @ rat ).
thf(tcon_Rat_Orat___Groups_Oordered__ab__group__add__abs_136,axiom,
ordere166539214618696060dd_abs @ rat ).
thf(tcon_Rat_Orat___Archimedean__Field_Ofloor__ceiling,axiom,
archim2362893244070406136eiling @ rat ).
thf(tcon_Rat_Orat___Groups_Oordered__comm__monoid__add_137,axiom,
ordere6911136660526730532id_add @ rat ).
thf(tcon_Rat_Orat___Groups_Olinordered__ab__group__add_138,axiom,
linord5086331880401160121up_add @ rat ).
thf(tcon_Rat_Orat___Groups_Ocancel__ab__semigroup__add_139,axiom,
cancel2418104881723323429up_add @ rat ).
thf(tcon_Rat_Orat___Rings_Oring__1__no__zero__divisors_140,axiom,
ring_15535105094025558882visors @ rat ).
thf(tcon_Rat_Orat___Groups_Ocancel__comm__monoid__add_141,axiom,
cancel1802427076303600483id_add @ rat ).
thf(tcon_Rat_Orat___Rings_Olinordered__ring__strict_142,axiom,
linord4710134922213307826strict @ rat ).
thf(tcon_Rat_Orat___Rings_Ocomm__semiring__1__cancel_143,axiom,
comm_s4317794764714335236cancel @ rat ).
thf(tcon_Rat_Orat___Rings_Oordered__comm__semiring_144,axiom,
ordere2520102378445227354miring @ rat ).
thf(tcon_Rat_Orat___Rings_Olinordered__semiring__1_145,axiom,
linord6961819062388156250ring_1 @ rat ).
thf(tcon_Rat_Orat___Groups_Oordered__ab__group__add_146,axiom,
ordered_ab_group_add @ rat ).
thf(tcon_Rat_Orat___Groups_Ocancel__semigroup__add_147,axiom,
cancel_semigroup_add @ rat ).
thf(tcon_Rat_Orat___Rings_Olinordered__semiring_148,axiom,
linordered_semiring @ rat ).
thf(tcon_Rat_Orat___Rings_Oordered__semiring__0_149,axiom,
ordered_semiring_0 @ rat ).
thf(tcon_Rat_Orat___Rings_Olinordered__semidom_150,axiom,
linordered_semidom @ rat ).
thf(tcon_Rat_Orat___Orderings_Odense__linorder,axiom,
dense_linorder @ rat ).
thf(tcon_Rat_Orat___Groups_Oab__semigroup__mult_151,axiom,
ab_semigroup_mult @ rat ).
thf(tcon_Rat_Orat___Rings_Osemiring__1__cancel_152,axiom,
semiring_1_cancel @ rat ).
thf(tcon_Rat_Orat___Groups_Ocomm__monoid__mult_153,axiom,
comm_monoid_mult @ rat ).
thf(tcon_Rat_Orat___Groups_Oab__semigroup__add_154,axiom,
ab_semigroup_add @ rat ).
thf(tcon_Rat_Orat___Fields_Olinordered__field,axiom,
linordered_field @ rat ).
thf(tcon_Rat_Orat___Rings_Oordered__semiring_155,axiom,
ordered_semiring @ rat ).
thf(tcon_Rat_Orat___Rings_Oordered__ring__abs_156,axiom,
ordered_ring_abs @ rat ).
thf(tcon_Rat_Orat___Groups_Ocomm__monoid__add_157,axiom,
comm_monoid_add @ rat ).
thf(tcon_Rat_Orat___Rings_Olinordered__ring_158,axiom,
linordered_ring @ rat ).
thf(tcon_Rat_Orat___Rings_Olinordered__idom_159,axiom,
linordered_idom @ rat ).
thf(tcon_Rat_Orat___Rings_Ocomm__semiring__1_160,axiom,
comm_semiring_1 @ rat ).
thf(tcon_Rat_Orat___Rings_Ocomm__semiring__0_161,axiom,
comm_semiring_0 @ rat ).
thf(tcon_Rat_Orat___Orderings_Odense__order,axiom,
dense_order @ rat ).
thf(tcon_Rat_Orat___Groups_Osemigroup__mult_162,axiom,
semigroup_mult @ rat ).
thf(tcon_Rat_Orat___Rings_Osemidom__divide_163,axiom,
semidom_divide @ rat ).
thf(tcon_Rat_Orat___Num_Osemiring__numeral_164,axiom,
semiring_numeral @ rat ).
thf(tcon_Rat_Orat___Groups_Osemigroup__add_165,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_166,axiom,
zero_less_one @ rat ).
thf(tcon_Rat_Orat___Rings_Ocomm__semiring_167,axiom,
comm_semiring @ rat ).
thf(tcon_Rat_Orat___Nat_Osemiring__char__0_168,axiom,
semiring_char_0 @ rat ).
thf(tcon_Rat_Orat___Groups_Oab__group__add_169,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_170,axiom,
zero_neq_one @ rat ).
thf(tcon_Rat_Orat___Rings_Oordered__ring_171,axiom,
ordered_ring @ rat ).
thf(tcon_Rat_Orat___Rings_Oidom__abs__sgn_172,axiom,
idom_abs_sgn @ rat ).
thf(tcon_Rat_Orat___Orderings_Opreorder_173,axiom,
preorder @ rat ).
thf(tcon_Rat_Orat___Orderings_Olinorder_174,axiom,
linorder @ rat ).
thf(tcon_Rat_Orat___Groups_Omonoid__mult_175,axiom,
monoid_mult @ rat ).
thf(tcon_Rat_Orat___Rings_Oidom__divide_176,axiom,
idom_divide @ rat ).
thf(tcon_Rat_Orat___Rings_Ocomm__ring__1_177,axiom,
comm_ring_1 @ rat ).
thf(tcon_Rat_Orat___Groups_Omonoid__add_178,axiom,
monoid_add @ rat ).
thf(tcon_Rat_Orat___Rings_Osemiring__1_179,axiom,
semiring_1 @ rat ).
thf(tcon_Rat_Orat___Rings_Osemiring__0_180,axiom,
semiring_0 @ rat ).
thf(tcon_Rat_Orat___Orderings_Ono__top_181,axiom,
no_top @ rat ).
thf(tcon_Rat_Orat___Orderings_Ono__bot_182,axiom,
no_bot @ rat ).
thf(tcon_Rat_Orat___Groups_Ogroup__add_183,axiom,
group_add @ rat ).
thf(tcon_Rat_Orat___Rings_Omult__zero_184,axiom,
mult_zero @ rat ).
thf(tcon_Rat_Orat___Rings_Ocomm__ring_185,axiom,
comm_ring @ rat ).
thf(tcon_Rat_Orat___Orderings_Oorder_186,axiom,
order @ rat ).
thf(tcon_Rat_Orat___Num_Oneg__numeral_187,axiom,
neg_numeral @ rat ).
thf(tcon_Rat_Orat___Nat_Oring__char__0_188,axiom,
ring_char_0 @ rat ).
thf(tcon_Rat_Orat___Rings_Osemiring_189,axiom,
semiring @ rat ).
thf(tcon_Rat_Orat___Fields_Oinverse,axiom,
inverse @ rat ).
thf(tcon_Rat_Orat___Rings_Osemidom_190,axiom,
semidom @ rat ).
thf(tcon_Rat_Orat___Orderings_Oord_191,axiom,
ord @ rat ).
thf(tcon_Rat_Orat___Groups_Ouminus_192,axiom,
uminus @ rat ).
thf(tcon_Rat_Orat___Rings_Oring__1_193,axiom,
ring_1 @ rat ).
thf(tcon_Rat_Orat___Rings_Oabs__if_194,axiom,
abs_if @ rat ).
thf(tcon_Rat_Orat___Groups_Otimes_195,axiom,
times @ rat ).
thf(tcon_Rat_Orat___Groups_Ominus_196,axiom,
minus @ rat ).
thf(tcon_Rat_Orat___Fields_Ofield,axiom,
field @ rat ).
thf(tcon_Rat_Orat___Power_Opower_197,axiom,
power @ rat ).
thf(tcon_Rat_Orat___Num_Onumeral_198,axiom,
numeral @ rat ).
thf(tcon_Rat_Orat___Groups_Ozero_199,axiom,
zero @ rat ).
thf(tcon_Rat_Orat___Groups_Oplus_200,axiom,
plus @ rat ).
thf(tcon_Rat_Orat___Rings_Oring_201,axiom,
ring @ rat ).
thf(tcon_Rat_Orat___Rings_Oidom_202,axiom,
idom @ rat ).
thf(tcon_Rat_Orat___Groups_Oone_203,axiom,
one @ rat ).
thf(tcon_Rat_Orat___Rings_Odvd_204,axiom,
dvd @ rat ).
thf(tcon_Rat_Orat___HOL_Oequal_205,axiom,
cl_HOL_Oequal @ rat ).
thf(tcon_Set_Oset___Boolean__Algebras_Oboolean__algebra_206,axiom,
! [A10: $tType] : ( boolea8198339166811842893lgebra @ ( set @ A10 ) ) ).
thf(tcon_Set_Oset___Orderings_Oorder__top_207,axiom,
! [A10: $tType] : ( order_top @ ( set @ A10 ) ) ).
thf(tcon_Set_Oset___Orderings_Oorder__bot_208,axiom,
! [A10: $tType] : ( order_bot @ ( set @ A10 ) ) ).
thf(tcon_Set_Oset___Orderings_Opreorder_209,axiom,
! [A10: $tType] : ( preorder @ ( set @ A10 ) ) ).
thf(tcon_Set_Oset___Finite__Set_Ofinite_210,axiom,
! [A10: $tType] :
( ( finite_finite @ A10 )
=> ( finite_finite @ ( set @ A10 ) ) ) ).
thf(tcon_Set_Oset___Orderings_Oorder_211,axiom,
! [A10: $tType] : ( order @ ( set @ A10 ) ) ).
thf(tcon_Set_Oset___Orderings_Oord_212,axiom,
! [A10: $tType] : ( ord @ ( set @ A10 ) ) ).
thf(tcon_Set_Oset___Groups_Ouminus_213,axiom,
! [A10: $tType] : ( uminus @ ( set @ A10 ) ) ).
thf(tcon_Set_Oset___Groups_Ominus_214,axiom,
! [A10: $tType] : ( minus @ ( set @ A10 ) ) ).
thf(tcon_Set_Oset___HOL_Oequal_215,axiom,
! [A10: $tType] :
( ( cl_HOL_Oequal @ A10 )
=> ( cl_HOL_Oequal @ ( set @ A10 ) ) ) ).
thf(tcon_HOL_Obool___Topological__Spaces_Otopological__space_216,axiom,
topolo4958980785337419405_space @ $o ).
thf(tcon_HOL_Obool___Topological__Spaces_Olinorder__topology_217,axiom,
topolo1944317154257567458pology @ $o ).
thf(tcon_HOL_Obool___Topological__Spaces_Oorder__topology_218,axiom,
topolo2564578578187576103pology @ $o ).
thf(tcon_HOL_Obool___Boolean__Algebras_Oboolean__algebra_219,axiom,
boolea8198339166811842893lgebra @ $o ).
thf(tcon_HOL_Obool___Topological__Spaces_Ot2__space_220,axiom,
topological_t2_space @ $o ).
thf(tcon_HOL_Obool___Orderings_Oorder__top_221,axiom,
order_top @ $o ).
thf(tcon_HOL_Obool___Orderings_Oorder__bot_222,axiom,
order_bot @ $o ).
thf(tcon_HOL_Obool___Orderings_Opreorder_223,axiom,
preorder @ $o ).
thf(tcon_HOL_Obool___Orderings_Olinorder_224,axiom,
linorder @ $o ).
thf(tcon_HOL_Obool___Finite__Set_Ofinite_225,axiom,
finite_finite @ $o ).
thf(tcon_HOL_Obool___Orderings_Oorder_226,axiom,
order @ $o ).
thf(tcon_HOL_Obool___Orderings_Oord_227,axiom,
ord @ $o ).
thf(tcon_HOL_Obool___Groups_Ouminus_228,axiom,
uminus @ $o ).
thf(tcon_HOL_Obool___Groups_Ominus_229,axiom,
minus @ $o ).
thf(tcon_HOL_Obool___Heap_Oheap_230,axiom,
heap @ $o ).
thf(tcon_HOL_Obool___HOL_Oequal_231,axiom,
cl_HOL_Oequal @ $o ).
thf(tcon_Heap_Oref___Heap_Oheap_232,axiom,
! [A10: $tType] : ( heap @ ( ref @ A10 ) ) ).
thf(tcon_Heap_Oref___HOL_Oequal_233,axiom,
! [A10: $tType] : ( cl_HOL_Oequal @ ( ref @ A10 ) ) ).
thf(tcon_Heap_Oref___Nat_Osize_234,axiom,
! [A10: $tType] : ( size @ ( ref @ A10 ) ) ).
thf(tcon_List_Olist___Heap_Oheap_235,axiom,
! [A10: $tType] :
( ( heap @ A10 )
=> ( heap @ ( list @ A10 ) ) ) ).
thf(tcon_List_Olist___HOL_Oequal_236,axiom,
! [A10: $tType] : ( cl_HOL_Oequal @ ( list @ A10 ) ) ).
thf(tcon_List_Olist___Nat_Osize_237,axiom,
! [A10: $tType] : ( size @ ( list @ A10 ) ) ).
thf(tcon_Real_Oreal___Conditionally__Complete__Lattices_Oconditionally__complete__linorder_238,axiom,
condit6923001295902523014norder @ real ).
thf(tcon_Real_Oreal___Semiring__Normalization_Ocomm__semiring__1__cancel__crossproduct_239,axiom,
semiri1453513574482234551roduct @ real ).
thf(tcon_Real_Oreal___Groups_Oordered__ab__semigroup__monoid__add__imp__le_240,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_241,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_242,axiom,
strict9044650504122735259up_add @ real ).
thf(tcon_Real_Oreal___Groups_Oordered__cancel__ab__semigroup__add_243,axiom,
ordere580206878836729694up_add @ real ).
thf(tcon_Real_Oreal___Groups_Oordered__ab__semigroup__add__imp__le_244,axiom,
ordere2412721322843649153imp_le @ real ).
thf(tcon_Real_Oreal___Rings_Olinordered__comm__semiring__strict_245,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_246,axiom,
strict7427464778891057005id_add @ real ).
thf(tcon_Real_Oreal___Groups_Oordered__cancel__comm__monoid__add_247,axiom,
ordere8940638589300402666id_add @ real ).
thf(tcon_Real_Oreal___Topological__Spaces_Otopological__space_248,axiom,
topolo4958980785337419405_space @ real ).
thf(tcon_Real_Oreal___Topological__Spaces_Olinorder__topology_249,axiom,
topolo1944317154257567458pology @ real ).
thf(tcon_Real_Oreal___Real__Vector__Spaces_Oreal__normed__field,axiom,
real_V3459762299906320749_field @ real ).
thf(tcon_Real_Oreal___Real__Vector__Spaces_Oreal__div__algebra,axiom,
real_V5047593784448816457lgebra @ real ).
thf(tcon_Real_Oreal___Archimedean__Field_Oarchimedean__field_250,axiom,
archim462609752435547400_field @ real ).
thf(tcon_Real_Oreal___Rings_Olinordered__semiring__1__strict_251,axiom,
linord715952674999750819strict @ real ).
thf(tcon_Real_Oreal___Real__Vector__Spaces_Ouniformity__dist,axiom,
real_V768167426530841204y_dist @ real ).
thf(tcon_Real_Oreal___Orderings_Ounbounded__dense__linorder_252,axiom,
unboun7993243217541854897norder @ real ).
thf(tcon_Real_Oreal___Limits_Otopological__comm__monoid__add_253,axiom,
topolo5987344860129210374id_add @ real ).
thf(tcon_Real_Oreal___Groups_Olinordered__ab__semigroup__add_254,axiom,
linord4140545234300271783up_add @ real ).
thf(tcon_Real_Oreal___Topological__Spaces_Oorder__topology_255,axiom,
topolo2564578578187576103pology @ real ).
thf(tcon_Real_Oreal___Rings_Osemiring__1__no__zero__divisors_256,axiom,
semiri2026040879449505780visors @ real ).
thf(tcon_Real_Oreal___Rings_Olinordered__nonzero__semiring_257,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_258,axiom,
topolo4211221413907600880p_mult @ real ).
thf(tcon_Real_Oreal___Topological__Spaces_Ouniform__space,axiom,
topolo7287701948861334536_space @ real ).
thf(tcon_Real_Oreal___Topological__Spaces_Operfect__space,axiom,
topolo8386298272705272623_space @ real ).
thf(tcon_Real_Oreal___Rings_Olinordered__semiring__strict_259,axiom,
linord8928482502909563296strict @ real ).
thf(tcon_Real_Oreal___Rings_Osemiring__no__zero__divisors_260,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_261,axiom,
ordere6658533253407199908up_add @ real ).
thf(tcon_Real_Oreal___Groups_Oordered__ab__group__add__abs_262,axiom,
ordere166539214618696060dd_abs @ real ).
thf(tcon_Real_Oreal___Archimedean__Field_Ofloor__ceiling_263,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_264,axiom,
ordere6911136660526730532id_add @ real ).
thf(tcon_Real_Oreal___Groups_Olinordered__ab__group__add_265,axiom,
linord5086331880401160121up_add @ real ).
thf(tcon_Real_Oreal___Groups_Ocancel__ab__semigroup__add_266,axiom,
cancel2418104881723323429up_add @ real ).
thf(tcon_Real_Oreal___Rings_Oring__1__no__zero__divisors_267,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_268,axiom,
topolo6943815403480290642id_add @ real ).
thf(tcon_Real_Oreal___Groups_Ocancel__comm__monoid__add_269,axiom,
cancel1802427076303600483id_add @ real ).
thf(tcon_Real_Oreal___Rings_Olinordered__ring__strict_270,axiom,
linord4710134922213307826strict @ real ).
thf(tcon_Real_Oreal___Rings_Ocomm__semiring__1__cancel_271,axiom,
comm_s4317794764714335236cancel @ real ).
thf(tcon_Real_Oreal___Limits_Otopological__group__add,axiom,
topolo1633459387980952147up_add @ real ).
thf(tcon_Real_Oreal___Topological__Spaces_Ot2__space_272,axiom,
topological_t2_space @ real ).
thf(tcon_Real_Oreal___Rings_Oordered__comm__semiring_273,axiom,
ordere2520102378445227354miring @ real ).
thf(tcon_Real_Oreal___Rings_Olinordered__semiring__1_274,axiom,
linord6961819062388156250ring_1 @ real ).
thf(tcon_Real_Oreal___Groups_Oordered__ab__group__add_275,axiom,
ordered_ab_group_add @ real ).
thf(tcon_Real_Oreal___Groups_Ocancel__semigroup__add_276,axiom,
cancel_semigroup_add @ real ).
thf(tcon_Real_Oreal___Rings_Olinordered__semiring_277,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_278,axiom,
ordered_semiring_0 @ real ).
thf(tcon_Real_Oreal___Rings_Olinordered__semidom_279,axiom,
linordered_semidom @ real ).
thf(tcon_Real_Oreal___Orderings_Odense__linorder_280,axiom,
dense_linorder @ real ).
thf(tcon_Real_Oreal___Groups_Oab__semigroup__mult_281,axiom,
ab_semigroup_mult @ real ).
thf(tcon_Real_Oreal___Rings_Osemiring__1__cancel_282,axiom,
semiring_1_cancel @ real ).
thf(tcon_Real_Oreal___Groups_Ocomm__monoid__mult_283,axiom,
comm_monoid_mult @ real ).
thf(tcon_Real_Oreal___Groups_Oab__semigroup__add_284,axiom,
ab_semigroup_add @ real ).
thf(tcon_Real_Oreal___Fields_Olinordered__field_285,axiom,
linordered_field @ real ).
thf(tcon_Real_Oreal___Rings_Oordered__semiring_286,axiom,
ordered_semiring @ real ).
thf(tcon_Real_Oreal___Rings_Oordered__ring__abs_287,axiom,
ordered_ring_abs @ real ).
thf(tcon_Real_Oreal___Groups_Ocomm__monoid__add_288,axiom,
comm_monoid_add @ real ).
thf(tcon_Real_Oreal___Rings_Olinordered__ring_289,axiom,
linordered_ring @ real ).
thf(tcon_Real_Oreal___Rings_Olinordered__idom_290,axiom,
linordered_idom @ real ).
thf(tcon_Real_Oreal___Rings_Ocomm__semiring__1_291,axiom,
comm_semiring_1 @ real ).
thf(tcon_Real_Oreal___Rings_Ocomm__semiring__0_292,axiom,
comm_semiring_0 @ real ).
thf(tcon_Real_Oreal___Orderings_Odense__order_293,axiom,
dense_order @ real ).
thf(tcon_Real_Oreal___Groups_Osemigroup__mult_294,axiom,
semigroup_mult @ real ).
thf(tcon_Real_Oreal___Rings_Osemidom__divide_295,axiom,
semidom_divide @ real ).
thf(tcon_Real_Oreal___Num_Osemiring__numeral_296,axiom,
semiring_numeral @ real ).
thf(tcon_Real_Oreal___Groups_Osemigroup__add_297,axiom,
semigroup_add @ real ).
thf(tcon_Real_Oreal___Fields_Ofield__abs__sgn_298,axiom,
field_abs_sgn @ real ).
thf(tcon_Real_Oreal___Fields_Odivision__ring_299,axiom,
division_ring @ real ).
thf(tcon_Real_Oreal___Rings_Ozero__less__one_300,axiom,
zero_less_one @ real ).
thf(tcon_Real_Oreal___Rings_Ocomm__semiring_301,axiom,
comm_semiring @ real ).
thf(tcon_Real_Oreal___Nat_Osemiring__char__0_302,axiom,
semiring_char_0 @ real ).
thf(tcon_Real_Oreal___Groups_Oab__group__add_303,axiom,
ab_group_add @ real ).
thf(tcon_Real_Oreal___Fields_Ofield__char__0_304,axiom,
field_char_0 @ real ).
thf(tcon_Real_Oreal___Rings_Ozero__neq__one_305,axiom,
zero_neq_one @ real ).
thf(tcon_Real_Oreal___Rings_Oordered__ring_306,axiom,
ordered_ring @ real ).
thf(tcon_Real_Oreal___Rings_Oidom__abs__sgn_307,axiom,
idom_abs_sgn @ real ).
thf(tcon_Real_Oreal___Orderings_Opreorder_308,axiom,
preorder @ real ).
thf(tcon_Real_Oreal___Orderings_Olinorder_309,axiom,
linorder @ real ).
thf(tcon_Real_Oreal___Groups_Omonoid__mult_310,axiom,
monoid_mult @ real ).
thf(tcon_Real_Oreal___Transcendental_Oln,axiom,
ln @ real ).
thf(tcon_Real_Oreal___Rings_Oidom__divide_311,axiom,
idom_divide @ real ).
thf(tcon_Real_Oreal___Rings_Ocomm__ring__1_312,axiom,
comm_ring_1 @ real ).
thf(tcon_Real_Oreal___Groups_Omonoid__add_313,axiom,
monoid_add @ real ).
thf(tcon_Real_Oreal___Rings_Osemiring__1_314,axiom,
semiring_1 @ real ).
thf(tcon_Real_Oreal___Rings_Osemiring__0_315,axiom,
semiring_0 @ real ).
thf(tcon_Real_Oreal___Orderings_Ono__top_316,axiom,
no_top @ real ).
thf(tcon_Real_Oreal___Orderings_Ono__bot_317,axiom,
no_bot @ real ).
thf(tcon_Real_Oreal___Groups_Ogroup__add_318,axiom,
group_add @ real ).
thf(tcon_Real_Oreal___Rings_Omult__zero_319,axiom,
mult_zero @ real ).
thf(tcon_Real_Oreal___Rings_Ocomm__ring_320,axiom,
comm_ring @ real ).
thf(tcon_Real_Oreal___Orderings_Oorder_321,axiom,
order @ real ).
thf(tcon_Real_Oreal___Num_Oneg__numeral_322,axiom,
neg_numeral @ real ).
thf(tcon_Real_Oreal___Nat_Oring__char__0_323,axiom,
ring_char_0 @ real ).
thf(tcon_Real_Oreal___Rings_Osemiring_324,axiom,
semiring @ real ).
thf(tcon_Real_Oreal___Fields_Oinverse_325,axiom,
inverse @ real ).
thf(tcon_Real_Oreal___Rings_Osemidom_326,axiom,
semidom @ real ).
thf(tcon_Real_Oreal___Orderings_Oord_327,axiom,
ord @ real ).
thf(tcon_Real_Oreal___Groups_Ouminus_328,axiom,
uminus @ real ).
thf(tcon_Real_Oreal___Rings_Oring__1_329,axiom,
ring_1 @ real ).
thf(tcon_Real_Oreal___Rings_Oabs__if_330,axiom,
abs_if @ real ).
thf(tcon_Real_Oreal___Groups_Otimes_331,axiom,
times @ real ).
thf(tcon_Real_Oreal___Groups_Ominus_332,axiom,
minus @ real ).
thf(tcon_Real_Oreal___Fields_Ofield_333,axiom,
field @ real ).
thf(tcon_Real_Oreal___Power_Opower_334,axiom,
power @ real ).
thf(tcon_Real_Oreal___Num_Onumeral_335,axiom,
numeral @ real ).
thf(tcon_Real_Oreal___Groups_Ozero_336,axiom,
zero @ real ).
thf(tcon_Real_Oreal___Groups_Oplus_337,axiom,
plus @ real ).
thf(tcon_Real_Oreal___Rings_Oring_338,axiom,
ring @ real ).
thf(tcon_Real_Oreal___Rings_Oidom_339,axiom,
idom @ real ).
thf(tcon_Real_Oreal___Groups_Oone_340,axiom,
one @ real ).
thf(tcon_Real_Oreal___Rings_Odvd_341,axiom,
dvd @ real ).
thf(tcon_Real_Oreal___HOL_Oequal_342,axiom,
cl_HOL_Oequal @ real ).
thf(tcon_Word_Oword___Bit__Operations_Osemiring__bit__operations_343,axiom,
! [A10: $tType] :
( ( type_len @ A10 )
=> ( bit_se359711467146920520ations @ ( word @ A10 ) ) ) ).
thf(tcon_Word_Oword___Bit__Comprehension_Obit__comprehension_344,axiom,
! [A10: $tType] :
( ( type_len @ A10 )
=> ( bit_bi6583157726757044596ension @ ( word @ A10 ) ) ) ).
thf(tcon_Word_Oword___Bit__Operations_Oring__bit__operations_345,axiom,
! [A10: $tType] :
( ( type_len @ A10 )
=> ( bit_ri3973907225187159222ations @ ( word @ A10 ) ) ) ).
thf(tcon_Word_Oword___Groups_Ocancel__ab__semigroup__add_346,axiom,
! [A10: $tType] :
( ( type_len @ A10 )
=> ( cancel2418104881723323429up_add @ ( word @ A10 ) ) ) ).
thf(tcon_Word_Oword___Groups_Ocancel__comm__monoid__add_347,axiom,
! [A10: $tType] :
( ( type_len @ A10 )
=> ( cancel1802427076303600483id_add @ ( word @ A10 ) ) ) ).
thf(tcon_Word_Oword___Rings_Ocomm__semiring__1__cancel_348,axiom,
! [A10: $tType] :
( ( type_len @ A10 )
=> ( comm_s4317794764714335236cancel @ ( word @ A10 ) ) ) ).
thf(tcon_Word_Oword___Bit__Operations_Osemiring__bits_349,axiom,
! [A10: $tType] :
( ( type_len @ A10 )
=> ( bit_semiring_bits @ ( word @ A10 ) ) ) ).
thf(tcon_Word_Oword___Groups_Ocancel__semigroup__add_350,axiom,
! [A10: $tType] :
( ( type_len @ A10 )
=> ( cancel_semigroup_add @ ( word @ A10 ) ) ) ).
thf(tcon_Word_Oword___Least__significant__bit_Olsb_351,axiom,
! [A10: $tType] :
( ( type_len @ A10 )
=> ( least_6119777620449941438nt_lsb @ ( word @ A10 ) ) ) ).
thf(tcon_Word_Oword___Groups_Oab__semigroup__mult_352,axiom,
! [A10: $tType] :
( ( type_len @ A10 )
=> ( ab_semigroup_mult @ ( word @ A10 ) ) ) ).
thf(tcon_Word_Oword___Rings_Osemiring__1__cancel_353,axiom,
! [A10: $tType] :
( ( type_len @ A10 )
=> ( semiring_1_cancel @ ( word @ A10 ) ) ) ).
thf(tcon_Word_Oword___Groups_Ocomm__monoid__mult_354,axiom,
! [A10: $tType] :
( ( type_len @ A10 )
=> ( comm_monoid_mult @ ( word @ A10 ) ) ) ).
thf(tcon_Word_Oword___Groups_Oab__semigroup__add_355,axiom,
! [A10: $tType] :
( ( type_len @ A10 )
=> ( ab_semigroup_add @ ( word @ A10 ) ) ) ).
thf(tcon_Word_Oword___Generic__set__bit_Oset__bit_356,axiom,
! [A10: $tType] :
( ( type_len @ A10 )
=> ( generic_set_set_bit @ ( word @ A10 ) ) ) ).
thf(tcon_Word_Oword___Parity_Osemiring__parity_357,axiom,
! [A10: $tType] :
( ( type_len @ A10 )
=> ( semiring_parity @ ( word @ A10 ) ) ) ).
thf(tcon_Word_Oword___Groups_Ocomm__monoid__add_358,axiom,
! [A10: $tType] :
( ( type_len @ A10 )
=> ( comm_monoid_add @ ( word @ A10 ) ) ) ).
thf(tcon_Word_Oword___Rings_Osemiring__modulo_359,axiom,
! [A10: $tType] :
( ( type_len @ A10 )
=> ( semiring_modulo @ ( word @ A10 ) ) ) ).
thf(tcon_Word_Oword___Rings_Ocomm__semiring__1_360,axiom,
! [A10: $tType] :
( ( type_len @ A10 )
=> ( comm_semiring_1 @ ( word @ A10 ) ) ) ).
thf(tcon_Word_Oword___Rings_Ocomm__semiring__0_361,axiom,
! [A10: $tType] :
( ( type_len @ A10 )
=> ( comm_semiring_0 @ ( word @ A10 ) ) ) ).
thf(tcon_Word_Oword___Groups_Osemigroup__mult_362,axiom,
! [A10: $tType] :
( ( type_len @ A10 )
=> ( semigroup_mult @ ( word @ A10 ) ) ) ).
thf(tcon_Word_Oword___Num_Osemiring__numeral_363,axiom,
! [A10: $tType] :
( ( type_len @ A10 )
=> ( semiring_numeral @ ( word @ A10 ) ) ) ).
thf(tcon_Word_Oword___Groups_Osemigroup__add_364,axiom,
! [A10: $tType] :
( ( type_len @ A10 )
=> ( semigroup_add @ ( word @ A10 ) ) ) ).
thf(tcon_Word_Oword___Rings_Ocomm__semiring_365,axiom,
! [A10: $tType] :
( ( type_len @ A10 )
=> ( comm_semiring @ ( word @ A10 ) ) ) ).
thf(tcon_Word_Oword___Orderings_Owellorder_366,axiom,
! [A10: $tType] :
( ( type_len @ A10 )
=> ( wellorder @ ( word @ A10 ) ) ) ).
thf(tcon_Word_Oword___Groups_Oab__group__add_367,axiom,
! [A10: $tType] :
( ( type_len @ A10 )
=> ( ab_group_add @ ( word @ A10 ) ) ) ).
thf(tcon_Word_Oword___Rings_Ozero__neq__one_368,axiom,
! [A10: $tType] :
( ( type_len @ A10 )
=> ( zero_neq_one @ ( word @ A10 ) ) ) ).
thf(tcon_Word_Oword___Parity_Oring__parity_369,axiom,
! [A10: $tType] :
( ( type_len @ A10 )
=> ( ring_parity @ ( word @ A10 ) ) ) ).
thf(tcon_Word_Oword___Orderings_Opreorder_370,axiom,
! [A10: $tType] :
( ( type_len @ A10 )
=> ( preorder @ ( word @ A10 ) ) ) ).
thf(tcon_Word_Oword___Orderings_Olinorder_371,axiom,
! [A10: $tType] :
( ( type_len @ A10 )
=> ( linorder @ ( word @ A10 ) ) ) ).
thf(tcon_Word_Oword___Groups_Omonoid__mult_372,axiom,
! [A10: $tType] :
( ( type_len @ A10 )
=> ( monoid_mult @ ( word @ A10 ) ) ) ).
thf(tcon_Word_Oword___Rings_Ocomm__ring__1_373,axiom,
! [A10: $tType] :
( ( type_len @ A10 )
=> ( comm_ring_1 @ ( word @ A10 ) ) ) ).
thf(tcon_Word_Oword___Groups_Omonoid__add_374,axiom,
! [A10: $tType] :
( ( type_len @ A10 )
=> ( monoid_add @ ( word @ A10 ) ) ) ).
thf(tcon_Word_Oword___Finite__Set_Ofinite_375,axiom,
! [A10: $tType] :
( ( type_len @ A10 )
=> ( finite_finite @ ( word @ A10 ) ) ) ).
thf(tcon_Word_Oword___Rings_Osemiring__1_376,axiom,
! [A10: $tType] :
( ( type_len @ A10 )
=> ( semiring_1 @ ( word @ A10 ) ) ) ).
thf(tcon_Word_Oword___Rings_Osemiring__0_377,axiom,
! [A10: $tType] :
( ( type_len @ A10 )
=> ( semiring_0 @ ( word @ A10 ) ) ) ).
thf(tcon_Word_Oword___Groups_Ogroup__add_378,axiom,
! [A10: $tType] :
( ( type_len @ A10 )
=> ( group_add @ ( word @ A10 ) ) ) ).
thf(tcon_Word_Oword___Rings_Omult__zero_379,axiom,
! [A10: $tType] :
( ( type_len @ A10 )
=> ( mult_zero @ ( word @ A10 ) ) ) ).
thf(tcon_Word_Oword___Rings_Ocomm__ring_380,axiom,
! [A10: $tType] :
( ( type_len @ A10 )
=> ( comm_ring @ ( word @ A10 ) ) ) ).
thf(tcon_Word_Oword___Orderings_Oorder_381,axiom,
! [A10: $tType] :
( ( type_len @ A10 )
=> ( order @ ( word @ A10 ) ) ) ).
thf(tcon_Word_Oword___Num_Oneg__numeral_382,axiom,
! [A10: $tType] :
( ( type_len @ A10 )
=> ( neg_numeral @ ( word @ A10 ) ) ) ).
thf(tcon_Word_Oword___Rings_Osemiring_383,axiom,
! [A10: $tType] :
( ( type_len @ A10 )
=> ( semiring @ ( word @ A10 ) ) ) ).
thf(tcon_Word_Oword___Orderings_Oord_384,axiom,
! [A10: $tType] :
( ( type_len @ A10 )
=> ( ord @ ( word @ A10 ) ) ) ).
thf(tcon_Word_Oword___Groups_Ouminus_385,axiom,
! [A10: $tType] :
( ( type_len @ A10 )
=> ( uminus @ ( word @ A10 ) ) ) ).
thf(tcon_Word_Oword___Rings_Oring__1_386,axiom,
! [A10: $tType] :
( ( type_len @ A10 )
=> ( ring_1 @ ( word @ A10 ) ) ) ).
thf(tcon_Word_Oword___Groups_Otimes_387,axiom,
! [A10: $tType] :
( ( type_len @ A10 )
=> ( times @ ( word @ A10 ) ) ) ).
thf(tcon_Word_Oword___Groups_Ominus_388,axiom,
! [A10: $tType] :
( ( type_len @ A10 )
=> ( minus @ ( word @ A10 ) ) ) ).
thf(tcon_Word_Oword___Power_Opower_389,axiom,
! [A10: $tType] :
( ( type_len @ A10 )
=> ( power @ ( word @ A10 ) ) ) ).
thf(tcon_Word_Oword___Num_Onumeral_390,axiom,
! [A10: $tType] :
( ( type_len @ A10 )
=> ( numeral @ ( word @ A10 ) ) ) ).
thf(tcon_Word_Oword___Groups_Ozero_391,axiom,
! [A10: $tType] :
( ( type_len @ A10 )
=> ( zero @ ( word @ A10 ) ) ) ).
thf(tcon_Word_Oword___Groups_Oplus_392,axiom,
! [A10: $tType] :
( ( type_len @ A10 )
=> ( plus @ ( word @ A10 ) ) ) ).
thf(tcon_Word_Oword___Rings_Oring_393,axiom,
! [A10: $tType] :
( ( type_len @ A10 )
=> ( ring @ ( word @ A10 ) ) ) ).
thf(tcon_Word_Oword___Groups_Oone_394,axiom,
! [A10: $tType] :
( ( type_len @ A10 )
=> ( one @ ( word @ A10 ) ) ) ).
thf(tcon_Word_Oword___Rings_Odvd_395,axiom,
! [A10: $tType] :
( ( type_len @ A10 )
=> ( dvd @ ( word @ A10 ) ) ) ).
thf(tcon_Word_Oword___HOL_Oequal_396,axiom,
! [A10: $tType] :
( ( type_len @ A10 )
=> ( cl_HOL_Oequal @ ( word @ A10 ) ) ) ).
thf(tcon_Word_Oword___Nat_Osize_397,axiom,
! [A10: $tType] :
( ( type_len @ A10 )
=> ( size @ ( word @ A10 ) ) ) ).
thf(tcon_Heap_Oarray___Heap_Oheap_398,axiom,
! [A10: $tType] : ( heap @ ( array @ A10 ) ) ).
thf(tcon_Heap_Oarray___HOL_Oequal_399,axiom,
! [A10: $tType] : ( cl_HOL_Oequal @ ( array @ A10 ) ) ).
thf(tcon_Heap_Oarray___Nat_Osize_400,axiom,
! [A10: $tType] : ( size @ ( array @ A10 ) ) ).
thf(tcon_String_Ochar___Orderings_Opreorder_401,axiom,
preorder @ char ).
thf(tcon_String_Ochar___Orderings_Olinorder_402,axiom,
linorder @ char ).
thf(tcon_String_Ochar___Finite__Set_Ofinite_403,axiom,
finite_finite @ char ).
thf(tcon_String_Ochar___Orderings_Oorder_404,axiom,
order @ char ).
thf(tcon_String_Ochar___Orderings_Oord_405,axiom,
ord @ char ).
thf(tcon_String_Ochar___Heap_Oheap_406,axiom,
heap @ char ).
thf(tcon_String_Ochar___HOL_Oequal_407,axiom,
cl_HOL_Oequal @ char ).
thf(tcon_String_Ochar___Nat_Osize_408,axiom,
size @ char ).
thf(tcon_Sum__Type_Osum___Finite__Set_Ofinite_409,axiom,
! [A10: $tType,A11: $tType] :
( ( ( finite_finite @ A10 )
& ( finite_finite @ A11 ) )
=> ( finite_finite @ ( sum_sum @ A10 @ A11 ) ) ) ).
thf(tcon_Sum__Type_Osum___Heap_Oheap_410,axiom,
! [A10: $tType,A11: $tType] :
( ( ( heap @ A10 )
& ( heap @ A11 ) )
=> ( heap @ ( sum_sum @ A10 @ A11 ) ) ) ).
thf(tcon_Sum__Type_Osum___HOL_Oequal_411,axiom,
! [A10: $tType,A11: $tType] : ( cl_HOL_Oequal @ ( sum_sum @ A10 @ A11 ) ) ).
thf(tcon_Sum__Type_Osum___Nat_Osize_412,axiom,
! [A10: $tType,A11: $tType] : ( size @ ( sum_sum @ A10 @ A11 ) ) ).
thf(tcon_Enum_Ofinite__1___Conditionally__Complete__Lattices_Oconditionally__complete__linorder_413,axiom,
condit6923001295902523014norder @ finite_1 ).
thf(tcon_Enum_Ofinite__1___Groups_Oordered__ab__semigroup__monoid__add__imp__le_414,axiom,
ordere1937475149494474687imp_le @ finite_1 ).
thf(tcon_Enum_Ofinite__1___Rings_Osemiring__no__zero__divisors__cancel_415,axiom,
semiri6575147826004484403cancel @ finite_1 ).
thf(tcon_Enum_Ofinite__1___Groups_Ostrict__ordered__ab__semigroup__add_416,axiom,
strict9044650504122735259up_add @ finite_1 ).
thf(tcon_Enum_Ofinite__1___Groups_Oordered__cancel__comm__monoid__diff_417,axiom,
ordere1170586879665033532d_diff @ finite_1 ).
thf(tcon_Enum_Ofinite__1___Groups_Oordered__cancel__ab__semigroup__add_418,axiom,
ordere580206878836729694up_add @ finite_1 ).
thf(tcon_Enum_Ofinite__1___Groups_Oordered__ab__semigroup__add__imp__le_419,axiom,
ordere2412721322843649153imp_le @ finite_1 ).
thf(tcon_Enum_Ofinite__1___Rings_Olinordered__comm__semiring__strict_420,axiom,
linord2810124833399127020strict @ finite_1 ).
thf(tcon_Enum_Ofinite__1___Groups_Ostrict__ordered__comm__monoid__add_421,axiom,
strict7427464778891057005id_add @ finite_1 ).
thf(tcon_Enum_Ofinite__1___Groups_Oordered__cancel__comm__monoid__add_422,axiom,
ordere8940638589300402666id_add @ finite_1 ).
thf(tcon_Enum_Ofinite__1___Groups_Ocanonically__ordered__monoid__add_423,axiom,
canoni5634975068530333245id_add @ finite_1 ).
thf(tcon_Enum_Ofinite__1___Groups_Olinordered__ab__semigroup__add_424,axiom,
linord4140545234300271783up_add @ finite_1 ).
thf(tcon_Enum_Ofinite__1___Rings_Olinordered__semiring__strict_425,axiom,
linord8928482502909563296strict @ finite_1 ).
thf(tcon_Enum_Ofinite__1___Boolean__Algebras_Oboolean__algebra_426,axiom,
boolea8198339166811842893lgebra @ finite_1 ).
thf(tcon_Enum_Ofinite__1___Rings_Osemiring__no__zero__divisors_427,axiom,
semiri3467727345109120633visors @ finite_1 ).
thf(tcon_Enum_Ofinite__1___Groups_Oordered__ab__semigroup__add_428,axiom,
ordere6658533253407199908up_add @ finite_1 ).
thf(tcon_Enum_Ofinite__1___Groups_Oordered__ab__group__add__abs_429,axiom,
ordere166539214618696060dd_abs @ finite_1 ).
thf(tcon_Enum_Ofinite__1___Groups_Oordered__comm__monoid__add_430,axiom,
ordere6911136660526730532id_add @ finite_1 ).
thf(tcon_Enum_Ofinite__1___Groups_Olinordered__ab__group__add_431,axiom,
linord5086331880401160121up_add @ finite_1 ).
thf(tcon_Enum_Ofinite__1___Groups_Ocancel__ab__semigroup__add_432,axiom,
cancel2418104881723323429up_add @ finite_1 ).
thf(tcon_Enum_Ofinite__1___Groups_Ocancel__comm__monoid__add_433,axiom,
cancel1802427076303600483id_add @ finite_1 ).
thf(tcon_Enum_Ofinite__1___Rings_Olinordered__ring__strict_434,axiom,
linord4710134922213307826strict @ finite_1 ).
thf(tcon_Enum_Ofinite__1___Rings_Oordered__comm__semiring_435,axiom,
ordere2520102378445227354miring @ finite_1 ).
thf(tcon_Enum_Ofinite__1___Groups_Oordered__ab__group__add_436,axiom,
ordered_ab_group_add @ finite_1 ).
thf(tcon_Enum_Ofinite__1___Groups_Ocancel__semigroup__add_437,axiom,
cancel_semigroup_add @ finite_1 ).
thf(tcon_Enum_Ofinite__1___Rings_Olinordered__semiring_438,axiom,
linordered_semiring @ finite_1 ).
thf(tcon_Enum_Ofinite__1___Rings_Oordered__semiring__0_439,axiom,
ordered_semiring_0 @ finite_1 ).
thf(tcon_Enum_Ofinite__1___Orderings_Odense__linorder_440,axiom,
dense_linorder @ finite_1 ).
thf(tcon_Enum_Ofinite__1___Groups_Oab__semigroup__mult_441,axiom,
ab_semigroup_mult @ finite_1 ).
thf(tcon_Enum_Ofinite__1___Groups_Ocomm__monoid__mult_442,axiom,
comm_monoid_mult @ finite_1 ).
thf(tcon_Enum_Ofinite__1___Groups_Ocomm__monoid__diff_443,axiom,
comm_monoid_diff @ finite_1 ).
thf(tcon_Enum_Ofinite__1___Groups_Oab__semigroup__add_444,axiom,
ab_semigroup_add @ finite_1 ).
thf(tcon_Enum_Ofinite__1___Rings_Oordered__semiring_445,axiom,
ordered_semiring @ finite_1 ).
thf(tcon_Enum_Ofinite__1___Rings_Oordered__ring__abs_446,axiom,
ordered_ring_abs @ finite_1 ).
thf(tcon_Enum_Ofinite__1___Groups_Ocomm__monoid__add_447,axiom,
comm_monoid_add @ finite_1 ).
thf(tcon_Enum_Ofinite__1___Rings_Olinordered__ring_448,axiom,
linordered_ring @ finite_1 ).
thf(tcon_Enum_Ofinite__1___Rings_Ocomm__semiring__0_449,axiom,
comm_semiring_0 @ finite_1 ).
thf(tcon_Enum_Ofinite__1___Orderings_Odense__order_450,axiom,
dense_order @ finite_1 ).
thf(tcon_Enum_Ofinite__1___Groups_Osemigroup__mult_451,axiom,
semigroup_mult @ finite_1 ).
thf(tcon_Enum_Ofinite__1___Groups_Osemigroup__add_452,axiom,
semigroup_add @ finite_1 ).
thf(tcon_Enum_Ofinite__1___Rings_Ocomm__semiring_453,axiom,
comm_semiring @ finite_1 ).
thf(tcon_Enum_Ofinite__1___Orderings_Owellorder_454,axiom,
wellorder @ finite_1 ).
thf(tcon_Enum_Ofinite__1___Orderings_Oorder__top_455,axiom,
order_top @ finite_1 ).
thf(tcon_Enum_Ofinite__1___Orderings_Oorder__bot_456,axiom,
order_bot @ finite_1 ).
thf(tcon_Enum_Ofinite__1___Groups_Oab__group__add_457,axiom,
ab_group_add @ finite_1 ).
thf(tcon_Enum_Ofinite__1___Rings_Oordered__ring_458,axiom,
ordered_ring @ finite_1 ).
thf(tcon_Enum_Ofinite__1___Orderings_Opreorder_459,axiom,
preorder @ finite_1 ).
thf(tcon_Enum_Ofinite__1___Orderings_Olinorder_460,axiom,
linorder @ finite_1 ).
thf(tcon_Enum_Ofinite__1___Groups_Omonoid__mult_461,axiom,
monoid_mult @ finite_1 ).
thf(tcon_Enum_Ofinite__1___Groups_Omonoid__add_462,axiom,
monoid_add @ finite_1 ).
thf(tcon_Enum_Ofinite__1___Finite__Set_Ofinite_463,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_464,axiom,
semiring_0 @ finite_1 ).
thf(tcon_Enum_Ofinite__1___Groups_Ogroup__add_465,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_466,axiom,
mult_zero @ finite_1 ).
thf(tcon_Enum_Ofinite__1___Rings_Ocomm__ring_467,axiom,
comm_ring @ finite_1 ).
thf(tcon_Enum_Ofinite__1___Orderings_Oorder_468,axiom,
order @ finite_1 ).
thf(tcon_Enum_Ofinite__1___Rings_Osemiring_469,axiom,
semiring @ finite_1 ).
thf(tcon_Enum_Ofinite__1___Fields_Oinverse_470,axiom,
inverse @ finite_1 ).
thf(tcon_Enum_Ofinite__1___Orderings_Oord_471,axiom,
ord @ finite_1 ).
thf(tcon_Enum_Ofinite__1___Groups_Ouminus_472,axiom,
uminus @ finite_1 ).
thf(tcon_Enum_Ofinite__1___Rings_Oabs__if_473,axiom,
abs_if @ finite_1 ).
thf(tcon_Enum_Ofinite__1___Groups_Otimes_474,axiom,
times @ finite_1 ).
thf(tcon_Enum_Ofinite__1___Groups_Ominus_475,axiom,
minus @ finite_1 ).
thf(tcon_Enum_Ofinite__1___Power_Opower_476,axiom,
power @ finite_1 ).
thf(tcon_Enum_Ofinite__1___Groups_Ozero_477,axiom,
zero @ finite_1 ).
thf(tcon_Enum_Ofinite__1___Groups_Oplus_478,axiom,
plus @ finite_1 ).
thf(tcon_Enum_Ofinite__1___Rings_Oring_479,axiom,
ring @ finite_1 ).
thf(tcon_Enum_Ofinite__1___Groups_Oone_480,axiom,
one @ finite_1 ).
thf(tcon_Enum_Ofinite__1___Rings_Odvd_481,axiom,
dvd @ finite_1 ).
thf(tcon_Enum_Ofinite__1___HOL_Oequal_482,axiom,
cl_HOL_Oequal @ finite_1 ).
thf(tcon_Enum_Ofinite__2___Conditionally__Complete__Lattices_Oconditionally__complete__linorder_483,axiom,
condit6923001295902523014norder @ finite_2 ).
thf(tcon_Enum_Ofinite__2___Semiring__Normalization_Ocomm__semiring__1__cancel__crossproduct_484,axiom,
semiri1453513574482234551roduct @ finite_2 ).
thf(tcon_Enum_Ofinite__2___Euclidean__Division_Ounique__euclidean__semiring_485,axiom,
euclid3128863361964157862miring @ finite_2 ).
thf(tcon_Enum_Ofinite__2___Euclidean__Division_Oeuclidean__semiring__cancel_486,axiom,
euclid4440199948858584721cancel @ finite_2 ).
thf(tcon_Enum_Ofinite__2___Rings_Osemiring__no__zero__divisors__cancel_487,axiom,
semiri6575147826004484403cancel @ finite_2 ).
thf(tcon_Enum_Ofinite__2___Euclidean__Division_Oeuclidean__semiring_488,axiom,
euclid3725896446679973847miring @ finite_2 ).
thf(tcon_Enum_Ofinite__2___Rings_Osemiring__1__no__zero__divisors_489,axiom,
semiri2026040879449505780visors @ finite_2 ).
thf(tcon_Enum_Ofinite__2___Rings_Osemiring__no__zero__divisors_490,axiom,
semiri3467727345109120633visors @ finite_2 ).
thf(tcon_Enum_Ofinite__2___Groups_Ocancel__ab__semigroup__add_491,axiom,
cancel2418104881723323429up_add @ finite_2 ).
thf(tcon_Enum_Ofinite__2___Rings_Oring__1__no__zero__divisors_492,axiom,
ring_15535105094025558882visors @ finite_2 ).
thf(tcon_Enum_Ofinite__2___Groups_Ocancel__comm__monoid__add_493,axiom,
cancel1802427076303600483id_add @ finite_2 ).
thf(tcon_Enum_Ofinite__2___Rings_Ocomm__semiring__1__cancel_494,axiom,
comm_s4317794764714335236cancel @ finite_2 ).
thf(tcon_Enum_Ofinite__2___Groups_Ocancel__semigroup__add_495,axiom,
cancel_semigroup_add @ finite_2 ).
thf(tcon_Enum_Ofinite__2___Groups_Oab__semigroup__mult_496,axiom,
ab_semigroup_mult @ finite_2 ).
thf(tcon_Enum_Ofinite__2___Rings_Osemiring__1__cancel_497,axiom,
semiring_1_cancel @ finite_2 ).
thf(tcon_Enum_Ofinite__2___Rings_Oalgebraic__semidom_498,axiom,
algebraic_semidom @ finite_2 ).
thf(tcon_Enum_Ofinite__2___Groups_Ocomm__monoid__mult_499,axiom,
comm_monoid_mult @ finite_2 ).
thf(tcon_Enum_Ofinite__2___Groups_Oab__semigroup__add_500,axiom,
ab_semigroup_add @ finite_2 ).
thf(tcon_Enum_Ofinite__2___Groups_Ocomm__monoid__add_501,axiom,
comm_monoid_add @ finite_2 ).
thf(tcon_Enum_Ofinite__2___Rings_Osemiring__modulo_502,axiom,
semiring_modulo @ finite_2 ).
thf(tcon_Enum_Ofinite__2___Rings_Ocomm__semiring__1_503,axiom,
comm_semiring_1 @ finite_2 ).
thf(tcon_Enum_Ofinite__2___Rings_Ocomm__semiring__0_504,axiom,
comm_semiring_0 @ finite_2 ).
thf(tcon_Enum_Ofinite__2___Groups_Osemigroup__mult_505,axiom,
semigroup_mult @ finite_2 ).
thf(tcon_Enum_Ofinite__2___Rings_Osemidom__modulo_506,axiom,
semidom_modulo @ finite_2 ).
thf(tcon_Enum_Ofinite__2___Rings_Osemidom__divide_507,axiom,
semidom_divide @ finite_2 ).
thf(tcon_Enum_Ofinite__2___Num_Osemiring__numeral_508,axiom,
semiring_numeral @ finite_2 ).
thf(tcon_Enum_Ofinite__2___Groups_Osemigroup__add_509,axiom,
semigroup_add @ finite_2 ).
thf(tcon_Enum_Ofinite__2___Fields_Odivision__ring_510,axiom,
division_ring @ finite_2 ).
thf(tcon_Enum_Ofinite__2___Rings_Ocomm__semiring_511,axiom,
comm_semiring @ finite_2 ).
thf(tcon_Enum_Ofinite__2___Orderings_Owellorder_512,axiom,
wellorder @ finite_2 ).
thf(tcon_Enum_Ofinite__2___Orderings_Oorder__top_513,axiom,
order_top @ finite_2 ).
thf(tcon_Enum_Ofinite__2___Orderings_Oorder__bot_514,axiom,
order_bot @ finite_2 ).
thf(tcon_Enum_Ofinite__2___Groups_Oab__group__add_515,axiom,
ab_group_add @ finite_2 ).
thf(tcon_Enum_Ofinite__2___Rings_Ozero__neq__one_516,axiom,
zero_neq_one @ finite_2 ).
thf(tcon_Enum_Ofinite__2___Rings_Oidom__abs__sgn_517,axiom,
idom_abs_sgn @ finite_2 ).
thf(tcon_Enum_Ofinite__2___Orderings_Opreorder_518,axiom,
preorder @ finite_2 ).
thf(tcon_Enum_Ofinite__2___Orderings_Olinorder_519,axiom,
linorder @ finite_2 ).
thf(tcon_Enum_Ofinite__2___Groups_Omonoid__mult_520,axiom,
monoid_mult @ finite_2 ).
thf(tcon_Enum_Ofinite__2___Rings_Oidom__modulo_521,axiom,
idom_modulo @ finite_2 ).
thf(tcon_Enum_Ofinite__2___Rings_Oidom__divide_522,axiom,
idom_divide @ finite_2 ).
thf(tcon_Enum_Ofinite__2___Rings_Ocomm__ring__1_523,axiom,
comm_ring_1 @ finite_2 ).
thf(tcon_Enum_Ofinite__2___Groups_Omonoid__add_524,axiom,
monoid_add @ finite_2 ).
thf(tcon_Enum_Ofinite__2___Finite__Set_Ofinite_525,axiom,
finite_finite @ finite_2 ).
thf(tcon_Enum_Ofinite__2___Type__Length_Olen0_526,axiom,
type_len0 @ finite_2 ).
thf(tcon_Enum_Ofinite__2___Rings_Osemiring__1_527,axiom,
semiring_1 @ finite_2 ).
thf(tcon_Enum_Ofinite__2___Rings_Osemiring__0_528,axiom,
semiring_0 @ finite_2 ).
thf(tcon_Enum_Ofinite__2___Groups_Ogroup__add_529,axiom,
group_add @ finite_2 ).
thf(tcon_Enum_Ofinite__2___Type__Length_Olen_530,axiom,
type_len @ finite_2 ).
thf(tcon_Enum_Ofinite__2___Rings_Omult__zero_531,axiom,
mult_zero @ finite_2 ).
thf(tcon_Enum_Ofinite__2___Rings_Ocomm__ring_532,axiom,
comm_ring @ finite_2 ).
thf(tcon_Enum_Ofinite__2___Orderings_Oorder_533,axiom,
order @ finite_2 ).
thf(tcon_Enum_Ofinite__2___Num_Oneg__numeral_534,axiom,
neg_numeral @ finite_2 ).
thf(tcon_Enum_Ofinite__2___Rings_Osemiring_535,axiom,
semiring @ finite_2 ).
thf(tcon_Enum_Ofinite__2___Fields_Oinverse_536,axiom,
inverse @ finite_2 ).
thf(tcon_Enum_Ofinite__2___Rings_Osemidom_537,axiom,
semidom @ finite_2 ).
thf(tcon_Enum_Ofinite__2___Orderings_Oord_538,axiom,
ord @ finite_2 ).
thf(tcon_Enum_Ofinite__2___Groups_Ouminus_539,axiom,
uminus @ finite_2 ).
thf(tcon_Enum_Ofinite__2___Rings_Oring__1_540,axiom,
ring_1 @ finite_2 ).
thf(tcon_Enum_Ofinite__2___Groups_Otimes_541,axiom,
times @ finite_2 ).
thf(tcon_Enum_Ofinite__2___Groups_Ominus_542,axiom,
minus @ finite_2 ).
thf(tcon_Enum_Ofinite__2___Fields_Ofield_543,axiom,
field @ finite_2 ).
thf(tcon_Enum_Ofinite__2___Power_Opower_544,axiom,
power @ finite_2 ).
thf(tcon_Enum_Ofinite__2___Num_Onumeral_545,axiom,
numeral @ finite_2 ).
thf(tcon_Enum_Ofinite__2___Groups_Ozero_546,axiom,
zero @ finite_2 ).
thf(tcon_Enum_Ofinite__2___Groups_Oplus_547,axiom,
plus @ finite_2 ).
thf(tcon_Enum_Ofinite__2___Rings_Oring_548,axiom,
ring @ finite_2 ).
thf(tcon_Enum_Ofinite__2___Rings_Oidom_549,axiom,
idom @ finite_2 ).
thf(tcon_Enum_Ofinite__2___Groups_Oone_550,axiom,
one @ finite_2 ).
thf(tcon_Enum_Ofinite__2___Rings_Odvd_551,axiom,
dvd @ finite_2 ).
thf(tcon_Enum_Ofinite__2___HOL_Oequal_552,axiom,
cl_HOL_Oequal @ finite_2 ).
thf(tcon_Enum_Ofinite__3___Conditionally__Complete__Lattices_Oconditionally__complete__linorder_553,axiom,
condit6923001295902523014norder @ finite_3 ).
thf(tcon_Enum_Ofinite__3___Semiring__Normalization_Ocomm__semiring__1__cancel__crossproduct_554,axiom,
semiri1453513574482234551roduct @ finite_3 ).
thf(tcon_Enum_Ofinite__3___Euclidean__Division_Ounique__euclidean__semiring_555,axiom,
euclid3128863361964157862miring @ finite_3 ).
thf(tcon_Enum_Ofinite__3___Euclidean__Division_Oeuclidean__semiring__cancel_556,axiom,
euclid4440199948858584721cancel @ finite_3 ).
thf(tcon_Enum_Ofinite__3___Rings_Osemiring__no__zero__divisors__cancel_557,axiom,
semiri6575147826004484403cancel @ finite_3 ).
thf(tcon_Enum_Ofinite__3___Euclidean__Division_Oeuclidean__semiring_558,axiom,
euclid3725896446679973847miring @ finite_3 ).
thf(tcon_Enum_Ofinite__3___Rings_Osemiring__1__no__zero__divisors_559,axiom,
semiri2026040879449505780visors @ finite_3 ).
thf(tcon_Enum_Ofinite__3___Rings_Osemiring__no__zero__divisors_560,axiom,
semiri3467727345109120633visors @ finite_3 ).
thf(tcon_Enum_Ofinite__3___Groups_Ocancel__ab__semigroup__add_561,axiom,
cancel2418104881723323429up_add @ finite_3 ).
thf(tcon_Enum_Ofinite__3___Rings_Oring__1__no__zero__divisors_562,axiom,
ring_15535105094025558882visors @ finite_3 ).
thf(tcon_Enum_Ofinite__3___Groups_Ocancel__comm__monoid__add_563,axiom,
cancel1802427076303600483id_add @ finite_3 ).
thf(tcon_Enum_Ofinite__3___Rings_Ocomm__semiring__1__cancel_564,axiom,
comm_s4317794764714335236cancel @ finite_3 ).
thf(tcon_Enum_Ofinite__3___Groups_Ocancel__semigroup__add_565,axiom,
cancel_semigroup_add @ finite_3 ).
thf(tcon_Enum_Ofinite__3___Groups_Oab__semigroup__mult_566,axiom,
ab_semigroup_mult @ finite_3 ).
thf(tcon_Enum_Ofinite__3___Rings_Osemiring__1__cancel_567,axiom,
semiring_1_cancel @ finite_3 ).
thf(tcon_Enum_Ofinite__3___Rings_Oalgebraic__semidom_568,axiom,
algebraic_semidom @ finite_3 ).
thf(tcon_Enum_Ofinite__3___Groups_Ocomm__monoid__mult_569,axiom,
comm_monoid_mult @ finite_3 ).
thf(tcon_Enum_Ofinite__3___Groups_Oab__semigroup__add_570,axiom,
ab_semigroup_add @ finite_3 ).
thf(tcon_Enum_Ofinite__3___Groups_Ocomm__monoid__add_571,axiom,
comm_monoid_add @ finite_3 ).
thf(tcon_Enum_Ofinite__3___Rings_Osemiring__modulo_572,axiom,
semiring_modulo @ finite_3 ).
thf(tcon_Enum_Ofinite__3___Rings_Ocomm__semiring__1_573,axiom,
comm_semiring_1 @ finite_3 ).
thf(tcon_Enum_Ofinite__3___Rings_Ocomm__semiring__0_574,axiom,
comm_semiring_0 @ finite_3 ).
thf(tcon_Enum_Ofinite__3___Groups_Osemigroup__mult_575,axiom,
semigroup_mult @ finite_3 ).
thf(tcon_Enum_Ofinite__3___Rings_Osemidom__modulo_576,axiom,
semidom_modulo @ finite_3 ).
thf(tcon_Enum_Ofinite__3___Rings_Osemidom__divide_577,axiom,
semidom_divide @ finite_3 ).
thf(tcon_Enum_Ofinite__3___Num_Osemiring__numeral_578,axiom,
semiring_numeral @ finite_3 ).
thf(tcon_Enum_Ofinite__3___Groups_Osemigroup__add_579,axiom,
semigroup_add @ finite_3 ).
thf(tcon_Enum_Ofinite__3___Fields_Odivision__ring_580,axiom,
division_ring @ finite_3 ).
thf(tcon_Enum_Ofinite__3___Rings_Ocomm__semiring_581,axiom,
comm_semiring @ finite_3 ).
thf(tcon_Enum_Ofinite__3___Orderings_Owellorder_582,axiom,
wellorder @ finite_3 ).
thf(tcon_Enum_Ofinite__3___Orderings_Oorder__top_583,axiom,
order_top @ finite_3 ).
thf(tcon_Enum_Ofinite__3___Orderings_Oorder__bot_584,axiom,
order_bot @ finite_3 ).
thf(tcon_Enum_Ofinite__3___Groups_Oab__group__add_585,axiom,
ab_group_add @ finite_3 ).
thf(tcon_Enum_Ofinite__3___Rings_Ozero__neq__one_586,axiom,
zero_neq_one @ finite_3 ).
thf(tcon_Enum_Ofinite__3___Rings_Oidom__abs__sgn_587,axiom,
idom_abs_sgn @ finite_3 ).
thf(tcon_Enum_Ofinite__3___Orderings_Opreorder_588,axiom,
preorder @ finite_3 ).
thf(tcon_Enum_Ofinite__3___Orderings_Olinorder_589,axiom,
linorder @ finite_3 ).
thf(tcon_Enum_Ofinite__3___Groups_Omonoid__mult_590,axiom,
monoid_mult @ finite_3 ).
thf(tcon_Enum_Ofinite__3___Rings_Oidom__modulo_591,axiom,
idom_modulo @ finite_3 ).
thf(tcon_Enum_Ofinite__3___Rings_Oidom__divide_592,axiom,
idom_divide @ finite_3 ).
thf(tcon_Enum_Ofinite__3___Rings_Ocomm__ring__1_593,axiom,
comm_ring_1 @ finite_3 ).
thf(tcon_Enum_Ofinite__3___Groups_Omonoid__add_594,axiom,
monoid_add @ finite_3 ).
thf(tcon_Enum_Ofinite__3___Finite__Set_Ofinite_595,axiom,
finite_finite @ finite_3 ).
thf(tcon_Enum_Ofinite__3___Type__Length_Olen0_596,axiom,
type_len0 @ finite_3 ).
thf(tcon_Enum_Ofinite__3___Rings_Osemiring__1_597,axiom,
semiring_1 @ finite_3 ).
thf(tcon_Enum_Ofinite__3___Rings_Osemiring__0_598,axiom,
semiring_0 @ finite_3 ).
thf(tcon_Enum_Ofinite__3___Groups_Ogroup__add_599,axiom,
group_add @ finite_3 ).
thf(tcon_Enum_Ofinite__3___Type__Length_Olen_600,axiom,
type_len @ finite_3 ).
thf(tcon_Enum_Ofinite__3___Rings_Omult__zero_601,axiom,
mult_zero @ finite_3 ).
thf(tcon_Enum_Ofinite__3___Rings_Ocomm__ring_602,axiom,
comm_ring @ finite_3 ).
thf(tcon_Enum_Ofinite__3___Orderings_Oorder_603,axiom,
order @ finite_3 ).
thf(tcon_Enum_Ofinite__3___Num_Oneg__numeral_604,axiom,
neg_numeral @ finite_3 ).
thf(tcon_Enum_Ofinite__3___Rings_Osemiring_605,axiom,
semiring @ finite_3 ).
thf(tcon_Enum_Ofinite__3___Fields_Oinverse_606,axiom,
inverse @ finite_3 ).
thf(tcon_Enum_Ofinite__3___Rings_Osemidom_607,axiom,
semidom @ finite_3 ).
thf(tcon_Enum_Ofinite__3___Orderings_Oord_608,axiom,
ord @ finite_3 ).
thf(tcon_Enum_Ofinite__3___Groups_Ouminus_609,axiom,
uminus @ finite_3 ).
thf(tcon_Enum_Ofinite__3___Rings_Oring__1_610,axiom,
ring_1 @ finite_3 ).
thf(tcon_Enum_Ofinite__3___Groups_Otimes_611,axiom,
times @ finite_3 ).
thf(tcon_Enum_Ofinite__3___Groups_Ominus_612,axiom,
minus @ finite_3 ).
thf(tcon_Enum_Ofinite__3___Fields_Ofield_613,axiom,
field @ finite_3 ).
thf(tcon_Enum_Ofinite__3___Power_Opower_614,axiom,
power @ finite_3 ).
thf(tcon_Enum_Ofinite__3___Num_Onumeral_615,axiom,
numeral @ finite_3 ).
thf(tcon_Enum_Ofinite__3___Groups_Ozero_616,axiom,
zero @ finite_3 ).
thf(tcon_Enum_Ofinite__3___Groups_Oplus_617,axiom,
plus @ finite_3 ).
thf(tcon_Enum_Ofinite__3___Rings_Oring_618,axiom,
ring @ finite_3 ).
thf(tcon_Enum_Ofinite__3___Rings_Oidom_619,axiom,
idom @ finite_3 ).
thf(tcon_Enum_Ofinite__3___Groups_Oone_620,axiom,
one @ finite_3 ).
thf(tcon_Enum_Ofinite__3___Rings_Odvd_621,axiom,
dvd @ finite_3 ).
thf(tcon_Enum_Ofinite__3___HOL_Oequal_622,axiom,
cl_HOL_Oequal @ finite_3 ).
thf(tcon_Filter_Ofilter___Orderings_Oorder__top_623,axiom,
! [A10: $tType] : ( order_top @ ( filter @ A10 ) ) ).
thf(tcon_Filter_Ofilter___Orderings_Oorder__bot_624,axiom,
! [A10: $tType] : ( order_bot @ ( filter @ A10 ) ) ).
thf(tcon_Filter_Ofilter___Orderings_Opreorder_625,axiom,
! [A10: $tType] : ( preorder @ ( filter @ A10 ) ) ).
thf(tcon_Filter_Ofilter___Orderings_Oorder_626,axiom,
! [A10: $tType] : ( order @ ( filter @ A10 ) ) ).
thf(tcon_Filter_Ofilter___Orderings_Oord_627,axiom,
! [A10: $tType] : ( ord @ ( filter @ A10 ) ) ).
thf(tcon_Filter_Ofilter___HOL_Oequal_628,axiom,
! [A10: $tType] :
( ( cl_HOL_Oequal @ A10 )
=> ( cl_HOL_Oequal @ ( filter @ A10 ) ) ) ).
thf(tcon_Option_Ooption___Conditionally__Complete__Lattices_Oconditionally__complete__linorder_629,axiom,
! [A10: $tType] :
( ( comple5582772986160207858norder @ A10 )
=> ( condit6923001295902523014norder @ ( option @ A10 ) ) ) ).
thf(tcon_Option_Ooption___Orderings_Owellorder_630,axiom,
! [A10: $tType] :
( ( wellorder @ A10 )
=> ( wellorder @ ( option @ A10 ) ) ) ).
thf(tcon_Option_Ooption___Orderings_Oorder__top_631,axiom,
! [A10: $tType] :
( ( order_top @ A10 )
=> ( order_top @ ( option @ A10 ) ) ) ).
thf(tcon_Option_Ooption___Orderings_Oorder__bot_632,axiom,
! [A10: $tType] :
( ( order @ A10 )
=> ( order_bot @ ( option @ A10 ) ) ) ).
thf(tcon_Option_Ooption___Orderings_Opreorder_633,axiom,
! [A10: $tType] :
( ( preorder @ A10 )
=> ( preorder @ ( option @ A10 ) ) ) ).
thf(tcon_Option_Ooption___Orderings_Olinorder_634,axiom,
! [A10: $tType] :
( ( linorder @ A10 )
=> ( linorder @ ( option @ A10 ) ) ) ).
thf(tcon_Option_Ooption___Finite__Set_Ofinite_635,axiom,
! [A10: $tType] :
( ( finite_finite @ A10 )
=> ( finite_finite @ ( option @ A10 ) ) ) ).
thf(tcon_Option_Ooption___Orderings_Oorder_636,axiom,
! [A10: $tType] :
( ( order @ A10 )
=> ( order @ ( option @ A10 ) ) ) ).
thf(tcon_Option_Ooption___Orderings_Oord_637,axiom,
! [A10: $tType] :
( ( preorder @ A10 )
=> ( ord @ ( option @ A10 ) ) ) ).
thf(tcon_Option_Ooption___Heap_Oheap_638,axiom,
! [A10: $tType] :
( ( heap @ A10 )
=> ( heap @ ( option @ A10 ) ) ) ).
thf(tcon_Option_Ooption___HOL_Oequal_639,axiom,
! [A10: $tType] : ( cl_HOL_Oequal @ ( option @ A10 ) ) ).
thf(tcon_Option_Ooption___Nat_Osize_640,axiom,
! [A10: $tType] : ( size @ ( option @ A10 ) ) ).
thf(tcon_Uint32_Ouint32___Bit__Operations_Osemiring__bit__operations_641,axiom,
bit_se359711467146920520ations @ uint32 ).
thf(tcon_Uint32_Ouint32___Bit__Comprehension_Obit__comprehension_642,axiom,
bit_bi6583157726757044596ension @ uint32 ).
thf(tcon_Uint32_Ouint32___Bit__Operations_Oring__bit__operations_643,axiom,
bit_ri3973907225187159222ations @ uint32 ).
thf(tcon_Uint32_Ouint32___Groups_Ocancel__ab__semigroup__add_644,axiom,
cancel2418104881723323429up_add @ uint32 ).
thf(tcon_Uint32_Ouint32___Groups_Ocancel__comm__monoid__add_645,axiom,
cancel1802427076303600483id_add @ uint32 ).
thf(tcon_Uint32_Ouint32___Rings_Ocomm__semiring__1__cancel_646,axiom,
comm_s4317794764714335236cancel @ uint32 ).
thf(tcon_Uint32_Ouint32___Bit__Operations_Osemiring__bits_647,axiom,
bit_semiring_bits @ uint32 ).
thf(tcon_Uint32_Ouint32___Groups_Ocancel__semigroup__add_648,axiom,
cancel_semigroup_add @ uint32 ).
thf(tcon_Uint32_Ouint32___Least__significant__bit_Olsb_649,axiom,
least_6119777620449941438nt_lsb @ uint32 ).
thf(tcon_Uint32_Ouint32___Groups_Oab__semigroup__mult_650,axiom,
ab_semigroup_mult @ uint32 ).
thf(tcon_Uint32_Ouint32___Rings_Osemiring__1__cancel_651,axiom,
semiring_1_cancel @ uint32 ).
thf(tcon_Uint32_Ouint32___Groups_Ocomm__monoid__mult_652,axiom,
comm_monoid_mult @ uint32 ).
thf(tcon_Uint32_Ouint32___Groups_Oab__semigroup__add_653,axiom,
ab_semigroup_add @ uint32 ).
thf(tcon_Uint32_Ouint32___Generic__set__bit_Oset__bit_654,axiom,
generic_set_set_bit @ uint32 ).
thf(tcon_Uint32_Ouint32___Parity_Osemiring__parity_655,axiom,
semiring_parity @ uint32 ).
thf(tcon_Uint32_Ouint32___Groups_Ocomm__monoid__add_656,axiom,
comm_monoid_add @ uint32 ).
thf(tcon_Uint32_Ouint32___Rings_Osemiring__modulo_657,axiom,
semiring_modulo @ uint32 ).
thf(tcon_Uint32_Ouint32___Rings_Ocomm__semiring__1_658,axiom,
comm_semiring_1 @ uint32 ).
thf(tcon_Uint32_Ouint32___Rings_Ocomm__semiring__0_659,axiom,
comm_semiring_0 @ uint32 ).
thf(tcon_Uint32_Ouint32___Groups_Osemigroup__mult_660,axiom,
semigroup_mult @ uint32 ).
thf(tcon_Uint32_Ouint32___Num_Osemiring__numeral_661,axiom,
semiring_numeral @ uint32 ).
thf(tcon_Uint32_Ouint32___Groups_Osemigroup__add_662,axiom,
semigroup_add @ uint32 ).
thf(tcon_Uint32_Ouint32___Rings_Ocomm__semiring_663,axiom,
comm_semiring @ uint32 ).
thf(tcon_Uint32_Ouint32___Groups_Oab__group__add_664,axiom,
ab_group_add @ uint32 ).
thf(tcon_Uint32_Ouint32___Rings_Ozero__neq__one_665,axiom,
zero_neq_one @ uint32 ).
thf(tcon_Uint32_Ouint32___Parity_Oring__parity_666,axiom,
ring_parity @ uint32 ).
thf(tcon_Uint32_Ouint32___Orderings_Opreorder_667,axiom,
preorder @ uint32 ).
thf(tcon_Uint32_Ouint32___Orderings_Olinorder_668,axiom,
linorder @ uint32 ).
thf(tcon_Uint32_Ouint32___Groups_Omonoid__mult_669,axiom,
monoid_mult @ uint32 ).
thf(tcon_Uint32_Ouint32___Rings_Ocomm__ring__1_670,axiom,
comm_ring_1 @ uint32 ).
thf(tcon_Uint32_Ouint32___Groups_Omonoid__add_671,axiom,
monoid_add @ uint32 ).
thf(tcon_Uint32_Ouint32___Rings_Osemiring__1_672,axiom,
semiring_1 @ uint32 ).
thf(tcon_Uint32_Ouint32___Rings_Osemiring__0_673,axiom,
semiring_0 @ uint32 ).
thf(tcon_Uint32_Ouint32___Groups_Ogroup__add_674,axiom,
group_add @ uint32 ).
thf(tcon_Uint32_Ouint32___Rings_Omult__zero_675,axiom,
mult_zero @ uint32 ).
thf(tcon_Uint32_Ouint32___Rings_Ocomm__ring_676,axiom,
comm_ring @ uint32 ).
thf(tcon_Uint32_Ouint32___Orderings_Oorder_677,axiom,
order @ uint32 ).
thf(tcon_Uint32_Ouint32___Num_Oneg__numeral_678,axiom,
neg_numeral @ uint32 ).
thf(tcon_Uint32_Ouint32___Rings_Osemiring_679,axiom,
semiring @ uint32 ).
thf(tcon_Uint32_Ouint32___Orderings_Oord_680,axiom,
ord @ uint32 ).
thf(tcon_Uint32_Ouint32___Groups_Ouminus_681,axiom,
uminus @ uint32 ).
thf(tcon_Uint32_Ouint32___Rings_Oring__1_682,axiom,
ring_1 @ uint32 ).
thf(tcon_Uint32_Ouint32___Groups_Otimes_683,axiom,
times @ uint32 ).
thf(tcon_Uint32_Ouint32___Groups_Ominus_684,axiom,
minus @ uint32 ).
thf(tcon_Uint32_Ouint32___Power_Opower_685,axiom,
power @ uint32 ).
thf(tcon_Uint32_Ouint32___Num_Onumeral_686,axiom,
numeral @ uint32 ).
thf(tcon_Uint32_Ouint32___Groups_Ozero_687,axiom,
zero @ uint32 ).
thf(tcon_Uint32_Ouint32___Groups_Oplus_688,axiom,
plus @ uint32 ).
thf(tcon_Uint32_Ouint32___Rings_Oring_689,axiom,
ring @ uint32 ).
thf(tcon_Uint32_Ouint32___Groups_Oone_690,axiom,
one @ uint32 ).
thf(tcon_Uint32_Ouint32___Rings_Odvd_691,axiom,
dvd @ uint32 ).
thf(tcon_Uint32_Ouint32___HOL_Oequal_692,axiom,
cl_HOL_Oequal @ uint32 ).
thf(tcon_Uint32_Ouint32___Nat_Osize_693,axiom,
size @ uint32 ).
thf(tcon_String_Oliteral___Groups_Osemigroup__add_694,axiom,
semigroup_add @ literal ).
thf(tcon_String_Oliteral___Orderings_Opreorder_695,axiom,
preorder @ literal ).
thf(tcon_String_Oliteral___Orderings_Olinorder_696,axiom,
linorder @ literal ).
thf(tcon_String_Oliteral___Groups_Omonoid__add_697,axiom,
monoid_add @ literal ).
thf(tcon_String_Oliteral___Orderings_Oorder_698,axiom,
order @ literal ).
thf(tcon_String_Oliteral___Orderings_Oord_699,axiom,
ord @ literal ).
thf(tcon_String_Oliteral___Groups_Ozero_700,axiom,
zero @ literal ).
thf(tcon_String_Oliteral___Groups_Oplus_701,axiom,
plus @ literal ).
thf(tcon_String_Oliteral___Heap_Oheap_702,axiom,
heap @ literal ).
thf(tcon_String_Oliteral___HOL_Oequal_703,axiom,
cl_HOL_Oequal @ literal ).
thf(tcon_String_Oliteral___Nat_Osize_704,axiom,
size @ literal ).
thf(tcon_Assertions_Oassn___Boolean__Algebras_Oboolean__algebra_705,axiom,
boolea8198339166811842893lgebra @ assn ).
thf(tcon_Assertions_Oassn___Groups_Oab__semigroup__mult_706,axiom,
ab_semigroup_mult @ assn ).
thf(tcon_Assertions_Oassn___Groups_Ocomm__monoid__mult_707,axiom,
comm_monoid_mult @ assn ).
thf(tcon_Assertions_Oassn___Groups_Osemigroup__mult_708,axiom,
semigroup_mult @ assn ).
thf(tcon_Assertions_Oassn___Orderings_Oorder__top_709,axiom,
order_top @ assn ).
thf(tcon_Assertions_Oassn___Orderings_Oorder__bot_710,axiom,
order_bot @ assn ).
thf(tcon_Assertions_Oassn___Orderings_Opreorder_711,axiom,
preorder @ assn ).
thf(tcon_Assertions_Oassn___Groups_Omonoid__mult_712,axiom,
monoid_mult @ assn ).
thf(tcon_Assertions_Oassn___Orderings_Oorder_713,axiom,
order @ assn ).
thf(tcon_Assertions_Oassn___Orderings_Oord_714,axiom,
ord @ assn ).
thf(tcon_Assertions_Oassn___Groups_Ouminus_715,axiom,
uminus @ assn ).
thf(tcon_Assertions_Oassn___Groups_Otimes_716,axiom,
times @ assn ).
thf(tcon_Assertions_Oassn___Groups_Ominus_717,axiom,
minus @ assn ).
thf(tcon_Assertions_Oassn___Power_Opower_718,axiom,
power @ assn ).
thf(tcon_Assertions_Oassn___Groups_Oone_719,axiom,
one @ assn ).
thf(tcon_Assertions_Oassn___Rings_Odvd_720,axiom,
dvd @ assn ).
thf(tcon_Complex_Ocomplex___Semiring__Normalization_Ocomm__semiring__1__cancel__crossproduct_721,axiom,
semiri1453513574482234551roduct @ complex ).
thf(tcon_Complex_Ocomplex___Topological__Spaces_Ofirst__countable__topology_722,axiom,
topolo3112930676232923870pology @ complex ).
thf(tcon_Complex_Ocomplex___Real__Vector__Spaces_Oreal__normed__div__algebra_723,axiom,
real_V8999393235501362500lgebra @ complex ).
thf(tcon_Complex_Ocomplex___Real__Vector__Spaces_Oreal__normed__algebra__1_724,axiom,
real_V2822296259951069270ebra_1 @ complex ).
thf(tcon_Complex_Ocomplex___Rings_Osemiring__no__zero__divisors__cancel_725,axiom,
semiri6575147826004484403cancel @ complex ).
thf(tcon_Complex_Ocomplex___Real__Vector__Spaces_Oreal__normed__algebra_726,axiom,
real_V4412858255891104859lgebra @ complex ).
thf(tcon_Complex_Ocomplex___Real__Vector__Spaces_Oreal__normed__vector_727,axiom,
real_V822414075346904944vector @ complex ).
thf(tcon_Complex_Ocomplex___Topological__Spaces_Otopological__space_728,axiom,
topolo4958980785337419405_space @ complex ).
thf(tcon_Complex_Ocomplex___Real__Vector__Spaces_Oreal__normed__field_729,axiom,
real_V3459762299906320749_field @ complex ).
thf(tcon_Complex_Ocomplex___Real__Vector__Spaces_Oreal__div__algebra_730,axiom,
real_V5047593784448816457lgebra @ complex ).
thf(tcon_Complex_Ocomplex___Real__Vector__Spaces_Ouniformity__dist_731,axiom,
real_V768167426530841204y_dist @ complex ).
thf(tcon_Complex_Ocomplex___Limits_Otopological__comm__monoid__add_732,axiom,
topolo5987344860129210374id_add @ complex ).
thf(tcon_Complex_Ocomplex___Rings_Osemiring__1__no__zero__divisors_733,axiom,
semiri2026040879449505780visors @ complex ).
thf(tcon_Complex_Ocomplex___Real__Vector__Spaces_Oreal__algebra__1_734,axiom,
real_V2191834092415804123ebra_1 @ complex ).
thf(tcon_Complex_Ocomplex___Real__Vector__Spaces_Ocomplete__space_735,axiom,
real_V8037385150606011577_space @ complex ).
thf(tcon_Complex_Ocomplex___Limits_Otopological__semigroup__mult_736,axiom,
topolo4211221413907600880p_mult @ complex ).
thf(tcon_Complex_Ocomplex___Topological__Spaces_Ouniform__space_737,axiom,
topolo7287701948861334536_space @ complex ).
thf(tcon_Complex_Ocomplex___Topological__Spaces_Operfect__space_738,axiom,
topolo8386298272705272623_space @ complex ).
thf(tcon_Complex_Ocomplex___Rings_Osemiring__no__zero__divisors_739,axiom,
semiri3467727345109120633visors @ complex ).
thf(tcon_Complex_Ocomplex___Real__Vector__Spaces_Oreal__algebra_740,axiom,
real_V6157519004096292374lgebra @ complex ).
thf(tcon_Complex_Ocomplex___Real__Vector__Spaces_Ometric__space_741,axiom,
real_V7819770556892013058_space @ complex ).
thf(tcon_Complex_Ocomplex___Limits_Otopological__ab__group__add_742,axiom,
topolo1287966508704411220up_add @ complex ).
thf(tcon_Complex_Ocomplex___Real__Vector__Spaces_Oreal__vector_743,axiom,
real_V4867850818363320053vector @ complex ).
thf(tcon_Complex_Ocomplex___Groups_Ocancel__ab__semigroup__add_744,axiom,
cancel2418104881723323429up_add @ complex ).
thf(tcon_Complex_Ocomplex___Rings_Oring__1__no__zero__divisors_745,axiom,
ring_15535105094025558882visors @ complex ).
thf(tcon_Complex_Ocomplex___Real__Vector__Spaces_Oreal__field_746,axiom,
real_V7773925162809079976_field @ complex ).
thf(tcon_Complex_Ocomplex___Limits_Otopological__monoid__add_747,axiom,
topolo6943815403480290642id_add @ complex ).
thf(tcon_Complex_Ocomplex___Groups_Ocancel__comm__monoid__add_748,axiom,
cancel1802427076303600483id_add @ complex ).
thf(tcon_Complex_Ocomplex___Rings_Ocomm__semiring__1__cancel_749,axiom,
comm_s4317794764714335236cancel @ complex ).
thf(tcon_Complex_Ocomplex___Limits_Otopological__group__add_750,axiom,
topolo1633459387980952147up_add @ complex ).
thf(tcon_Complex_Ocomplex___Topological__Spaces_Ot2__space_751,axiom,
topological_t2_space @ complex ).
thf(tcon_Complex_Ocomplex___Groups_Ocancel__semigroup__add_752,axiom,
cancel_semigroup_add @ complex ).
thf(tcon_Complex_Ocomplex___Real__Vector__Spaces_Obanach_753,axiom,
real_Vector_banach @ complex ).
thf(tcon_Complex_Ocomplex___Groups_Oab__semigroup__mult_754,axiom,
ab_semigroup_mult @ complex ).
thf(tcon_Complex_Ocomplex___Rings_Osemiring__1__cancel_755,axiom,
semiring_1_cancel @ complex ).
thf(tcon_Complex_Ocomplex___Groups_Ocomm__monoid__mult_756,axiom,
comm_monoid_mult @ complex ).
thf(tcon_Complex_Ocomplex___Groups_Oab__semigroup__add_757,axiom,
ab_semigroup_add @ complex ).
thf(tcon_Complex_Ocomplex___Groups_Ocomm__monoid__add_758,axiom,
comm_monoid_add @ complex ).
thf(tcon_Complex_Ocomplex___Rings_Ocomm__semiring__1_759,axiom,
comm_semiring_1 @ complex ).
thf(tcon_Complex_Ocomplex___Rings_Ocomm__semiring__0_760,axiom,
comm_semiring_0 @ complex ).
thf(tcon_Complex_Ocomplex___Groups_Osemigroup__mult_761,axiom,
semigroup_mult @ complex ).
thf(tcon_Complex_Ocomplex___Rings_Osemidom__divide_762,axiom,
semidom_divide @ complex ).
thf(tcon_Complex_Ocomplex___Num_Osemiring__numeral_763,axiom,
semiring_numeral @ complex ).
thf(tcon_Complex_Ocomplex___Groups_Osemigroup__add_764,axiom,
semigroup_add @ complex ).
thf(tcon_Complex_Ocomplex___Fields_Ofield__abs__sgn_765,axiom,
field_abs_sgn @ complex ).
thf(tcon_Complex_Ocomplex___Fields_Odivision__ring_766,axiom,
division_ring @ complex ).
thf(tcon_Complex_Ocomplex___Rings_Ocomm__semiring_767,axiom,
comm_semiring @ complex ).
thf(tcon_Complex_Ocomplex___Nat_Osemiring__char__0_768,axiom,
semiring_char_0 @ complex ).
thf(tcon_Complex_Ocomplex___Groups_Oab__group__add_769,axiom,
ab_group_add @ complex ).
thf(tcon_Complex_Ocomplex___Fields_Ofield__char__0_770,axiom,
field_char_0 @ complex ).
thf(tcon_Complex_Ocomplex___Rings_Ozero__neq__one_771,axiom,
zero_neq_one @ complex ).
thf(tcon_Complex_Ocomplex___Rings_Oidom__abs__sgn_772,axiom,
idom_abs_sgn @ complex ).
thf(tcon_Complex_Ocomplex___Groups_Omonoid__mult_773,axiom,
monoid_mult @ complex ).
thf(tcon_Complex_Ocomplex___Rings_Oidom__divide_774,axiom,
idom_divide @ complex ).
thf(tcon_Complex_Ocomplex___Rings_Ocomm__ring__1_775,axiom,
comm_ring_1 @ complex ).
thf(tcon_Complex_Ocomplex___Groups_Omonoid__add_776,axiom,
monoid_add @ complex ).
thf(tcon_Complex_Ocomplex___Rings_Osemiring__1_777,axiom,
semiring_1 @ complex ).
thf(tcon_Complex_Ocomplex___Rings_Osemiring__0_778,axiom,
semiring_0 @ complex ).
thf(tcon_Complex_Ocomplex___Groups_Ogroup__add_779,axiom,
group_add @ complex ).
thf(tcon_Complex_Ocomplex___Rings_Omult__zero_780,axiom,
mult_zero @ complex ).
thf(tcon_Complex_Ocomplex___Rings_Ocomm__ring_781,axiom,
comm_ring @ complex ).
thf(tcon_Complex_Ocomplex___Num_Oneg__numeral_782,axiom,
neg_numeral @ complex ).
thf(tcon_Complex_Ocomplex___Nat_Oring__char__0_783,axiom,
ring_char_0 @ complex ).
thf(tcon_Complex_Ocomplex___Rings_Osemiring_784,axiom,
semiring @ complex ).
thf(tcon_Complex_Ocomplex___Fields_Oinverse_785,axiom,
inverse @ complex ).
thf(tcon_Complex_Ocomplex___Rings_Osemidom_786,axiom,
semidom @ complex ).
thf(tcon_Complex_Ocomplex___Groups_Ouminus_787,axiom,
uminus @ complex ).
thf(tcon_Complex_Ocomplex___Rings_Oring__1_788,axiom,
ring_1 @ complex ).
thf(tcon_Complex_Ocomplex___Groups_Otimes_789,axiom,
times @ complex ).
thf(tcon_Complex_Ocomplex___Groups_Ominus_790,axiom,
minus @ complex ).
thf(tcon_Complex_Ocomplex___Fields_Ofield_791,axiom,
field @ complex ).
thf(tcon_Complex_Ocomplex___Power_Opower_792,axiom,
power @ complex ).
thf(tcon_Complex_Ocomplex___Num_Onumeral_793,axiom,
numeral @ complex ).
thf(tcon_Complex_Ocomplex___Groups_Ozero_794,axiom,
zero @ complex ).
thf(tcon_Complex_Ocomplex___Groups_Oplus_795,axiom,
plus @ complex ).
thf(tcon_Complex_Ocomplex___Rings_Oring_796,axiom,
ring @ complex ).
thf(tcon_Complex_Ocomplex___Rings_Oidom_797,axiom,
idom @ complex ).
thf(tcon_Complex_Ocomplex___Groups_Oone_798,axiom,
one @ complex ).
thf(tcon_Complex_Ocomplex___Rings_Odvd_799,axiom,
dvd @ complex ).
thf(tcon_Complex_Ocomplex___HOL_Oequal_800,axiom,
cl_HOL_Oequal @ complex ).
thf(tcon_Extended__Nat_Oenat___Conditionally__Complete__Lattices_Oconditionally__complete__linorder_801,axiom,
condit6923001295902523014norder @ extended_enat ).
thf(tcon_Extended__Nat_Oenat___Groups_Ostrict__ordered__ab__semigroup__add_802,axiom,
strict9044650504122735259up_add @ extended_enat ).
thf(tcon_Extended__Nat_Oenat___Groups_Ostrict__ordered__comm__monoid__add_803,axiom,
strict7427464778891057005id_add @ extended_enat ).
thf(tcon_Extended__Nat_Oenat___Groups_Ocanonically__ordered__monoid__add_804,axiom,
canoni5634975068530333245id_add @ extended_enat ).
thf(tcon_Extended__Nat_Oenat___Groups_Olinordered__ab__semigroup__add_805,axiom,
linord4140545234300271783up_add @ extended_enat ).
thf(tcon_Extended__Nat_Oenat___Rings_Olinordered__nonzero__semiring_806,axiom,
linord181362715937106298miring @ extended_enat ).
thf(tcon_Extended__Nat_Oenat___Rings_Osemiring__no__zero__divisors_807,axiom,
semiri3467727345109120633visors @ extended_enat ).
thf(tcon_Extended__Nat_Oenat___Groups_Oordered__ab__semigroup__add_808,axiom,
ordere6658533253407199908up_add @ extended_enat ).
thf(tcon_Extended__Nat_Oenat___Groups_Oordered__comm__monoid__add_809,axiom,
ordere6911136660526730532id_add @ extended_enat ).
thf(tcon_Extended__Nat_Oenat___Rings_Oordered__comm__semiring_810,axiom,
ordere2520102378445227354miring @ extended_enat ).
thf(tcon_Extended__Nat_Oenat___Groups_Oab__semigroup__mult_811,axiom,
ab_semigroup_mult @ extended_enat ).
thf(tcon_Extended__Nat_Oenat___Groups_Ocomm__monoid__mult_812,axiom,
comm_monoid_mult @ extended_enat ).
thf(tcon_Extended__Nat_Oenat___Groups_Oab__semigroup__add_813,axiom,
ab_semigroup_add @ extended_enat ).
thf(tcon_Extended__Nat_Oenat___Rings_Oordered__semiring_814,axiom,
ordered_semiring @ extended_enat ).
thf(tcon_Extended__Nat_Oenat___Groups_Ocomm__monoid__add_815,axiom,
comm_monoid_add @ extended_enat ).
thf(tcon_Extended__Nat_Oenat___Rings_Ocomm__semiring__1_816,axiom,
comm_semiring_1 @ extended_enat ).
thf(tcon_Extended__Nat_Oenat___Rings_Ocomm__semiring__0_817,axiom,
comm_semiring_0 @ extended_enat ).
thf(tcon_Extended__Nat_Oenat___Groups_Osemigroup__mult_818,axiom,
semigroup_mult @ extended_enat ).
thf(tcon_Extended__Nat_Oenat___Num_Osemiring__numeral_819,axiom,
semiring_numeral @ extended_enat ).
thf(tcon_Extended__Nat_Oenat___Groups_Osemigroup__add_820,axiom,
semigroup_add @ extended_enat ).
thf(tcon_Extended__Nat_Oenat___Rings_Ozero__less__one_821,axiom,
zero_less_one @ extended_enat ).
thf(tcon_Extended__Nat_Oenat___Rings_Ocomm__semiring_822,axiom,
comm_semiring @ extended_enat ).
thf(tcon_Extended__Nat_Oenat___Orderings_Owellorder_823,axiom,
wellorder @ extended_enat ).
thf(tcon_Extended__Nat_Oenat___Orderings_Oorder__top_824,axiom,
order_top @ extended_enat ).
thf(tcon_Extended__Nat_Oenat___Orderings_Oorder__bot_825,axiom,
order_bot @ extended_enat ).
thf(tcon_Extended__Nat_Oenat___Nat_Osemiring__char__0_826,axiom,
semiring_char_0 @ extended_enat ).
thf(tcon_Extended__Nat_Oenat___Rings_Ozero__neq__one_827,axiom,
zero_neq_one @ extended_enat ).
thf(tcon_Extended__Nat_Oenat___Orderings_Opreorder_828,axiom,
preorder @ extended_enat ).
thf(tcon_Extended__Nat_Oenat___Orderings_Olinorder_829,axiom,
linorder @ extended_enat ).
thf(tcon_Extended__Nat_Oenat___Groups_Omonoid__mult_830,axiom,
monoid_mult @ extended_enat ).
thf(tcon_Extended__Nat_Oenat___Groups_Omonoid__add_831,axiom,
monoid_add @ extended_enat ).
thf(tcon_Extended__Nat_Oenat___Rings_Osemiring__1_832,axiom,
semiring_1 @ extended_enat ).
thf(tcon_Extended__Nat_Oenat___Rings_Osemiring__0_833,axiom,
semiring_0 @ extended_enat ).
thf(tcon_Extended__Nat_Oenat___Rings_Omult__zero_834,axiom,
mult_zero @ extended_enat ).
thf(tcon_Extended__Nat_Oenat___Orderings_Oorder_835,axiom,
order @ extended_enat ).
thf(tcon_Extended__Nat_Oenat___Rings_Osemiring_836,axiom,
semiring @ extended_enat ).
thf(tcon_Extended__Nat_Oenat___Orderings_Oord_837,axiom,
ord @ extended_enat ).
thf(tcon_Extended__Nat_Oenat___Groups_Otimes_838,axiom,
times @ extended_enat ).
thf(tcon_Extended__Nat_Oenat___Groups_Ominus_839,axiom,
minus @ extended_enat ).
thf(tcon_Extended__Nat_Oenat___Power_Opower_840,axiom,
power @ extended_enat ).
thf(tcon_Extended__Nat_Oenat___Num_Onumeral_841,axiom,
numeral @ extended_enat ).
thf(tcon_Extended__Nat_Oenat___Groups_Ozero_842,axiom,
zero @ extended_enat ).
thf(tcon_Extended__Nat_Oenat___Groups_Oplus_843,axiom,
plus @ extended_enat ).
thf(tcon_Extended__Nat_Oenat___Groups_Oone_844,axiom,
one @ extended_enat ).
thf(tcon_Extended__Nat_Oenat___Rings_Odvd_845,axiom,
dvd @ extended_enat ).
thf(tcon_Extended__Nat_Oenat___HOL_Oequal_846,axiom,
cl_HOL_Oequal @ extended_enat ).
thf(tcon_Multiset_Omultiset___Groups_Oordered__ab__semigroup__add_847,axiom,
! [A10: $tType] :
( ( preorder @ A10 )
=> ( ordere6658533253407199908up_add @ ( multiset @ A10 ) ) ) ).
thf(tcon_Multiset_Omultiset___Groups_Ocancel__ab__semigroup__add_848,axiom,
! [A10: $tType] : ( cancel2418104881723323429up_add @ ( multiset @ A10 ) ) ).
thf(tcon_Multiset_Omultiset___Groups_Ocancel__comm__monoid__add_849,axiom,
! [A10: $tType] : ( cancel1802427076303600483id_add @ ( multiset @ A10 ) ) ).
thf(tcon_Multiset_Omultiset___Groups_Ocancel__semigroup__add_850,axiom,
! [A10: $tType] : ( cancel_semigroup_add @ ( multiset @ A10 ) ) ).
thf(tcon_Multiset_Omultiset___Groups_Ocomm__monoid__diff_851,axiom,
! [A10: $tType] : ( comm_monoid_diff @ ( multiset @ A10 ) ) ).
thf(tcon_Multiset_Omultiset___Groups_Oab__semigroup__add_852,axiom,
! [A10: $tType] : ( ab_semigroup_add @ ( multiset @ A10 ) ) ).
thf(tcon_Multiset_Omultiset___Groups_Ocomm__monoid__add_853,axiom,
! [A10: $tType] : ( comm_monoid_add @ ( multiset @ A10 ) ) ).
thf(tcon_Multiset_Omultiset___Groups_Osemigroup__add_854,axiom,
! [A10: $tType] : ( semigroup_add @ ( multiset @ A10 ) ) ).
thf(tcon_Multiset_Omultiset___Orderings_Opreorder_855,axiom,
! [A10: $tType] :
( ( preorder @ A10 )
=> ( preorder @ ( multiset @ A10 ) ) ) ).
thf(tcon_Multiset_Omultiset___Groups_Omonoid__add_856,axiom,
! [A10: $tType] : ( monoid_add @ ( multiset @ A10 ) ) ).
thf(tcon_Multiset_Omultiset___Orderings_Oorder_857,axiom,
! [A10: $tType] :
( ( preorder @ A10 )
=> ( order @ ( multiset @ A10 ) ) ) ).
thf(tcon_Multiset_Omultiset___Orderings_Oord_858,axiom,
! [A10: $tType] :
( ( preorder @ A10 )
=> ( ord @ ( multiset @ A10 ) ) ) ).
thf(tcon_Multiset_Omultiset___Groups_Ominus_859,axiom,
! [A10: $tType] : ( minus @ ( multiset @ A10 ) ) ).
thf(tcon_Multiset_Omultiset___Groups_Ozero_860,axiom,
! [A10: $tType] : ( zero @ ( multiset @ A10 ) ) ).
thf(tcon_Multiset_Omultiset___Groups_Oplus_861,axiom,
! [A10: $tType] : ( plus @ ( multiset @ A10 ) ) ).
thf(tcon_Multiset_Omultiset___HOL_Oequal_862,axiom,
! [A10: $tType] :
( ( cl_HOL_Oequal @ A10 )
=> ( cl_HOL_Oequal @ ( multiset @ A10 ) ) ) ).
thf(tcon_Multiset_Omultiset___Nat_Osize_863,axiom,
! [A10: $tType] : ( size @ ( multiset @ A10 ) ) ).
thf(tcon_Numeral__Type_Obit0___Groups_Ocancel__ab__semigroup__add_864,axiom,
! [A10: $tType] :
( ( finite_finite @ A10 )
=> ( cancel2418104881723323429up_add @ ( numeral_bit0 @ A10 ) ) ) ).
thf(tcon_Numeral__Type_Obit0___Groups_Ocancel__comm__monoid__add_865,axiom,
! [A10: $tType] :
( ( finite_finite @ A10 )
=> ( cancel1802427076303600483id_add @ ( numeral_bit0 @ A10 ) ) ) ).
thf(tcon_Numeral__Type_Obit0___Rings_Ocomm__semiring__1__cancel_866,axiom,
! [A10: $tType] :
( ( finite_finite @ A10 )
=> ( comm_s4317794764714335236cancel @ ( numeral_bit0 @ A10 ) ) ) ).
thf(tcon_Numeral__Type_Obit0___Groups_Ocancel__semigroup__add_867,axiom,
! [A10: $tType] :
( ( finite_finite @ A10 )
=> ( cancel_semigroup_add @ ( numeral_bit0 @ A10 ) ) ) ).
thf(tcon_Numeral__Type_Obit0___Groups_Oab__semigroup__mult_868,axiom,
! [A10: $tType] :
( ( finite_finite @ A10 )
=> ( ab_semigroup_mult @ ( numeral_bit0 @ A10 ) ) ) ).
thf(tcon_Numeral__Type_Obit0___Rings_Osemiring__1__cancel_869,axiom,
! [A10: $tType] :
( ( finite_finite @ A10 )
=> ( semiring_1_cancel @ ( numeral_bit0 @ A10 ) ) ) ).
thf(tcon_Numeral__Type_Obit0___Groups_Ocomm__monoid__mult_870,axiom,
! [A10: $tType] :
( ( finite_finite @ A10 )
=> ( comm_monoid_mult @ ( numeral_bit0 @ A10 ) ) ) ).
thf(tcon_Numeral__Type_Obit0___Groups_Oab__semigroup__add_871,axiom,
! [A10: $tType] :
( ( finite_finite @ A10 )
=> ( ab_semigroup_add @ ( numeral_bit0 @ A10 ) ) ) ).
thf(tcon_Numeral__Type_Obit0___Groups_Ocomm__monoid__add_872,axiom,
! [A10: $tType] :
( ( finite_finite @ A10 )
=> ( comm_monoid_add @ ( numeral_bit0 @ A10 ) ) ) ).
thf(tcon_Numeral__Type_Obit0___Rings_Ocomm__semiring__1_873,axiom,
! [A10: $tType] :
( ( finite_finite @ A10 )
=> ( comm_semiring_1 @ ( numeral_bit0 @ A10 ) ) ) ).
thf(tcon_Numeral__Type_Obit0___Rings_Ocomm__semiring__0_874,axiom,
! [A10: $tType] :
( ( finite_finite @ A10 )
=> ( comm_semiring_0 @ ( numeral_bit0 @ A10 ) ) ) ).
thf(tcon_Numeral__Type_Obit0___Groups_Osemigroup__mult_875,axiom,
! [A10: $tType] :
( ( finite_finite @ A10 )
=> ( semigroup_mult @ ( numeral_bit0 @ A10 ) ) ) ).
thf(tcon_Numeral__Type_Obit0___Num_Osemiring__numeral_876,axiom,
! [A10: $tType] :
( ( finite_finite @ A10 )
=> ( semiring_numeral @ ( numeral_bit0 @ A10 ) ) ) ).
thf(tcon_Numeral__Type_Obit0___Groups_Osemigroup__add_877,axiom,
! [A10: $tType] :
( ( finite_finite @ A10 )
=> ( semigroup_add @ ( numeral_bit0 @ A10 ) ) ) ).
thf(tcon_Numeral__Type_Obit0___Rings_Ocomm__semiring_878,axiom,
! [A10: $tType] :
( ( finite_finite @ A10 )
=> ( comm_semiring @ ( numeral_bit0 @ A10 ) ) ) ).
thf(tcon_Numeral__Type_Obit0___Orderings_Owellorder_879,axiom,
! [A10: $tType] :
( ( finite_finite @ A10 )
=> ( wellorder @ ( numeral_bit0 @ A10 ) ) ) ).
thf(tcon_Numeral__Type_Obit0___Groups_Oab__group__add_880,axiom,
! [A10: $tType] :
( ( finite_finite @ A10 )
=> ( ab_group_add @ ( numeral_bit0 @ A10 ) ) ) ).
thf(tcon_Numeral__Type_Obit0___Rings_Ozero__neq__one_881,axiom,
! [A10: $tType] :
( ( finite_finite @ A10 )
=> ( zero_neq_one @ ( numeral_bit0 @ A10 ) ) ) ).
thf(tcon_Numeral__Type_Obit0___Orderings_Opreorder_882,axiom,
! [A10: $tType] :
( ( finite_finite @ A10 )
=> ( preorder @ ( numeral_bit0 @ A10 ) ) ) ).
thf(tcon_Numeral__Type_Obit0___Orderings_Olinorder_883,axiom,
! [A10: $tType] :
( ( finite_finite @ A10 )
=> ( linorder @ ( numeral_bit0 @ A10 ) ) ) ).
thf(tcon_Numeral__Type_Obit0___Groups_Omonoid__mult_884,axiom,
! [A10: $tType] :
( ( finite_finite @ A10 )
=> ( monoid_mult @ ( numeral_bit0 @ A10 ) ) ) ).
thf(tcon_Numeral__Type_Obit0___Rings_Ocomm__ring__1_885,axiom,
! [A10: $tType] :
( ( finite_finite @ A10 )
=> ( comm_ring_1 @ ( numeral_bit0 @ A10 ) ) ) ).
thf(tcon_Numeral__Type_Obit0___Groups_Omonoid__add_886,axiom,
! [A10: $tType] :
( ( finite_finite @ A10 )
=> ( monoid_add @ ( numeral_bit0 @ A10 ) ) ) ).
thf(tcon_Numeral__Type_Obit0___Finite__Set_Ofinite_887,axiom,
! [A10: $tType] :
( ( finite_finite @ A10 )
=> ( finite_finite @ ( numeral_bit0 @ A10 ) ) ) ).
thf(tcon_Numeral__Type_Obit0___Cardinality_Ocard2,axiom,
! [A10: $tType] :
( ( finite_finite @ A10 )
=> ( card2 @ ( numeral_bit0 @ A10 ) ) ) ).
thf(tcon_Numeral__Type_Obit0___Type__Length_Olen0_888,axiom,
! [A10: $tType] :
( ( type_len0 @ A10 )
=> ( type_len0 @ ( numeral_bit0 @ A10 ) ) ) ).
thf(tcon_Numeral__Type_Obit0___Rings_Osemiring__1_889,axiom,
! [A10: $tType] :
( ( finite_finite @ A10 )
=> ( semiring_1 @ ( numeral_bit0 @ A10 ) ) ) ).
thf(tcon_Numeral__Type_Obit0___Rings_Osemiring__0_890,axiom,
! [A10: $tType] :
( ( finite_finite @ A10 )
=> ( semiring_0 @ ( numeral_bit0 @ A10 ) ) ) ).
thf(tcon_Numeral__Type_Obit0___Groups_Ogroup__add_891,axiom,
! [A10: $tType] :
( ( finite_finite @ A10 )
=> ( group_add @ ( numeral_bit0 @ A10 ) ) ) ).
thf(tcon_Numeral__Type_Obit0___Type__Length_Olen_892,axiom,
! [A10: $tType] :
( ( type_len @ A10 )
=> ( type_len @ ( numeral_bit0 @ A10 ) ) ) ).
thf(tcon_Numeral__Type_Obit0___Rings_Omult__zero_893,axiom,
! [A10: $tType] :
( ( finite_finite @ A10 )
=> ( mult_zero @ ( numeral_bit0 @ A10 ) ) ) ).
thf(tcon_Numeral__Type_Obit0___Rings_Ocomm__ring_894,axiom,
! [A10: $tType] :
( ( finite_finite @ A10 )
=> ( comm_ring @ ( numeral_bit0 @ A10 ) ) ) ).
thf(tcon_Numeral__Type_Obit0___Orderings_Oorder_895,axiom,
! [A10: $tType] :
( ( finite_finite @ A10 )
=> ( order @ ( numeral_bit0 @ A10 ) ) ) ).
thf(tcon_Numeral__Type_Obit0___Num_Oneg__numeral_896,axiom,
! [A10: $tType] :
( ( finite_finite @ A10 )
=> ( neg_numeral @ ( numeral_bit0 @ A10 ) ) ) ).
thf(tcon_Numeral__Type_Obit0___Rings_Osemiring_897,axiom,
! [A10: $tType] :
( ( finite_finite @ A10 )
=> ( semiring @ ( numeral_bit0 @ A10 ) ) ) ).
thf(tcon_Numeral__Type_Obit0___Orderings_Oord_898,axiom,
! [A10: $tType] :
( ( finite_finite @ A10 )
=> ( ord @ ( numeral_bit0 @ A10 ) ) ) ).
thf(tcon_Numeral__Type_Obit0___Groups_Ouminus_899,axiom,
! [A10: $tType] :
( ( finite_finite @ A10 )
=> ( uminus @ ( numeral_bit0 @ A10 ) ) ) ).
thf(tcon_Numeral__Type_Obit0___Rings_Oring__1_900,axiom,
! [A10: $tType] :
( ( finite_finite @ A10 )
=> ( ring_1 @ ( numeral_bit0 @ A10 ) ) ) ).
thf(tcon_Numeral__Type_Obit0___Groups_Otimes_901,axiom,
! [A10: $tType] :
( ( finite_finite @ A10 )
=> ( times @ ( numeral_bit0 @ A10 ) ) ) ).
thf(tcon_Numeral__Type_Obit0___Groups_Ominus_902,axiom,
! [A10: $tType] :
( ( finite_finite @ A10 )
=> ( minus @ ( numeral_bit0 @ A10 ) ) ) ).
thf(tcon_Numeral__Type_Obit0___Power_Opower_903,axiom,
! [A10: $tType] :
( ( finite_finite @ A10 )
=> ( power @ ( numeral_bit0 @ A10 ) ) ) ).
thf(tcon_Numeral__Type_Obit0___Num_Onumeral_904,axiom,
! [A10: $tType] :
( ( finite_finite @ A10 )
=> ( numeral @ ( numeral_bit0 @ A10 ) ) ) ).
thf(tcon_Numeral__Type_Obit0___Groups_Ozero_905,axiom,
! [A10: $tType] :
( ( finite_finite @ A10 )
=> ( zero @ ( numeral_bit0 @ A10 ) ) ) ).
thf(tcon_Numeral__Type_Obit0___Groups_Oplus_906,axiom,
! [A10: $tType] :
( ( finite_finite @ A10 )
=> ( plus @ ( numeral_bit0 @ A10 ) ) ) ).
thf(tcon_Numeral__Type_Obit0___Rings_Oring_907,axiom,
! [A10: $tType] :
( ( finite_finite @ A10 )
=> ( ring @ ( numeral_bit0 @ A10 ) ) ) ).
thf(tcon_Numeral__Type_Obit0___Groups_Oone_908,axiom,
! [A10: $tType] :
( ( finite_finite @ A10 )
=> ( one @ ( numeral_bit0 @ A10 ) ) ) ).
thf(tcon_Numeral__Type_Obit0___Rings_Odvd_909,axiom,
! [A10: $tType] :
( ( finite_finite @ A10 )
=> ( dvd @ ( numeral_bit0 @ A10 ) ) ) ).
thf(tcon_Numeral__Type_Obit0___HOL_Oequal_910,axiom,
! [A10: $tType] :
( ( finite_finite @ A10 )
=> ( cl_HOL_Oequal @ ( numeral_bit0 @ A10 ) ) ) ).
thf(tcon_Numeral__Type_Obit1___Groups_Ocancel__ab__semigroup__add_911,axiom,
! [A10: $tType] :
( ( finite_finite @ A10 )
=> ( cancel2418104881723323429up_add @ ( numeral_bit1 @ A10 ) ) ) ).
thf(tcon_Numeral__Type_Obit1___Groups_Ocancel__comm__monoid__add_912,axiom,
! [A10: $tType] :
( ( finite_finite @ A10 )
=> ( cancel1802427076303600483id_add @ ( numeral_bit1 @ A10 ) ) ) ).
thf(tcon_Numeral__Type_Obit1___Rings_Ocomm__semiring__1__cancel_913,axiom,
! [A10: $tType] :
( ( finite_finite @ A10 )
=> ( comm_s4317794764714335236cancel @ ( numeral_bit1 @ A10 ) ) ) ).
thf(tcon_Numeral__Type_Obit1___Groups_Ocancel__semigroup__add_914,axiom,
! [A10: $tType] :
( ( finite_finite @ A10 )
=> ( cancel_semigroup_add @ ( numeral_bit1 @ A10 ) ) ) ).
thf(tcon_Numeral__Type_Obit1___Groups_Oab__semigroup__mult_915,axiom,
! [A10: $tType] :
( ( finite_finite @ A10 )
=> ( ab_semigroup_mult @ ( numeral_bit1 @ A10 ) ) ) ).
thf(tcon_Numeral__Type_Obit1___Rings_Osemiring__1__cancel_916,axiom,
! [A10: $tType] :
( ( finite_finite @ A10 )
=> ( semiring_1_cancel @ ( numeral_bit1 @ A10 ) ) ) ).
thf(tcon_Numeral__Type_Obit1___Groups_Ocomm__monoid__mult_917,axiom,
! [A10: $tType] :
( ( finite_finite @ A10 )
=> ( comm_monoid_mult @ ( numeral_bit1 @ A10 ) ) ) ).
thf(tcon_Numeral__Type_Obit1___Groups_Oab__semigroup__add_918,axiom,
! [A10: $tType] :
( ( finite_finite @ A10 )
=> ( ab_semigroup_add @ ( numeral_bit1 @ A10 ) ) ) ).
thf(tcon_Numeral__Type_Obit1___Groups_Ocomm__monoid__add_919,axiom,
! [A10: $tType] :
( ( finite_finite @ A10 )
=> ( comm_monoid_add @ ( numeral_bit1 @ A10 ) ) ) ).
thf(tcon_Numeral__Type_Obit1___Rings_Ocomm__semiring__1_920,axiom,
! [A10: $tType] :
( ( finite_finite @ A10 )
=> ( comm_semiring_1 @ ( numeral_bit1 @ A10 ) ) ) ).
thf(tcon_Numeral__Type_Obit1___Rings_Ocomm__semiring__0_921,axiom,
! [A10: $tType] :
( ( finite_finite @ A10 )
=> ( comm_semiring_0 @ ( numeral_bit1 @ A10 ) ) ) ).
thf(tcon_Numeral__Type_Obit1___Groups_Osemigroup__mult_922,axiom,
! [A10: $tType] :
( ( finite_finite @ A10 )
=> ( semigroup_mult @ ( numeral_bit1 @ A10 ) ) ) ).
thf(tcon_Numeral__Type_Obit1___Num_Osemiring__numeral_923,axiom,
! [A10: $tType] :
( ( finite_finite @ A10 )
=> ( semiring_numeral @ ( numeral_bit1 @ A10 ) ) ) ).
thf(tcon_Numeral__Type_Obit1___Groups_Osemigroup__add_924,axiom,
! [A10: $tType] :
( ( finite_finite @ A10 )
=> ( semigroup_add @ ( numeral_bit1 @ A10 ) ) ) ).
thf(tcon_Numeral__Type_Obit1___Rings_Ocomm__semiring_925,axiom,
! [A10: $tType] :
( ( finite_finite @ A10 )
=> ( comm_semiring @ ( numeral_bit1 @ A10 ) ) ) ).
thf(tcon_Numeral__Type_Obit1___Orderings_Owellorder_926,axiom,
! [A10: $tType] :
( ( finite_finite @ A10 )
=> ( wellorder @ ( numeral_bit1 @ A10 ) ) ) ).
thf(tcon_Numeral__Type_Obit1___Groups_Oab__group__add_927,axiom,
! [A10: $tType] :
( ( finite_finite @ A10 )
=> ( ab_group_add @ ( numeral_bit1 @ A10 ) ) ) ).
thf(tcon_Numeral__Type_Obit1___Rings_Ozero__neq__one_928,axiom,
! [A10: $tType] :
( ( finite_finite @ A10 )
=> ( zero_neq_one @ ( numeral_bit1 @ A10 ) ) ) ).
thf(tcon_Numeral__Type_Obit1___Orderings_Opreorder_929,axiom,
! [A10: $tType] :
( ( finite_finite @ A10 )
=> ( preorder @ ( numeral_bit1 @ A10 ) ) ) ).
thf(tcon_Numeral__Type_Obit1___Orderings_Olinorder_930,axiom,
! [A10: $tType] :
( ( finite_finite @ A10 )
=> ( linorder @ ( numeral_bit1 @ A10 ) ) ) ).
thf(tcon_Numeral__Type_Obit1___Groups_Omonoid__mult_931,axiom,
! [A10: $tType] :
( ( finite_finite @ A10 )
=> ( monoid_mult @ ( numeral_bit1 @ A10 ) ) ) ).
thf(tcon_Numeral__Type_Obit1___Rings_Ocomm__ring__1_932,axiom,
! [A10: $tType] :
( ( finite_finite @ A10 )
=> ( comm_ring_1 @ ( numeral_bit1 @ A10 ) ) ) ).
thf(tcon_Numeral__Type_Obit1___Groups_Omonoid__add_933,axiom,
! [A10: $tType] :
( ( finite_finite @ A10 )
=> ( monoid_add @ ( numeral_bit1 @ A10 ) ) ) ).
thf(tcon_Numeral__Type_Obit1___Finite__Set_Ofinite_934,axiom,
! [A10: $tType] :
( ( finite_finite @ A10 )
=> ( finite_finite @ ( numeral_bit1 @ A10 ) ) ) ).
thf(tcon_Numeral__Type_Obit1___Cardinality_Ocard2_935,axiom,
! [A10: $tType] :
( ( finite_finite @ A10 )
=> ( card2 @ ( numeral_bit1 @ A10 ) ) ) ).
thf(tcon_Numeral__Type_Obit1___Type__Length_Olen0_936,axiom,
! [A10: $tType] :
( ( type_len0 @ A10 )
=> ( type_len0 @ ( numeral_bit1 @ A10 ) ) ) ).
thf(tcon_Numeral__Type_Obit1___Rings_Osemiring__1_937,axiom,
! [A10: $tType] :
( ( finite_finite @ A10 )
=> ( semiring_1 @ ( numeral_bit1 @ A10 ) ) ) ).
thf(tcon_Numeral__Type_Obit1___Rings_Osemiring__0_938,axiom,
! [A10: $tType] :
( ( finite_finite @ A10 )
=> ( semiring_0 @ ( numeral_bit1 @ A10 ) ) ) ).
thf(tcon_Numeral__Type_Obit1___Groups_Ogroup__add_939,axiom,
! [A10: $tType] :
( ( finite_finite @ A10 )
=> ( group_add @ ( numeral_bit1 @ A10 ) ) ) ).
thf(tcon_Numeral__Type_Obit1___Type__Length_Olen_940,axiom,
! [A10: $tType] :
( ( type_len0 @ A10 )
=> ( type_len @ ( numeral_bit1 @ A10 ) ) ) ).
thf(tcon_Numeral__Type_Obit1___Rings_Omult__zero_941,axiom,
! [A10: $tType] :
( ( finite_finite @ A10 )
=> ( mult_zero @ ( numeral_bit1 @ A10 ) ) ) ).
thf(tcon_Numeral__Type_Obit1___Rings_Ocomm__ring_942,axiom,
! [A10: $tType] :
( ( finite_finite @ A10 )
=> ( comm_ring @ ( numeral_bit1 @ A10 ) ) ) ).
thf(tcon_Numeral__Type_Obit1___Orderings_Oorder_943,axiom,
! [A10: $tType] :
( ( finite_finite @ A10 )
=> ( order @ ( numeral_bit1 @ A10 ) ) ) ).
thf(tcon_Numeral__Type_Obit1___Num_Oneg__numeral_944,axiom,
! [A10: $tType] :
( ( finite_finite @ A10 )
=> ( neg_numeral @ ( numeral_bit1 @ A10 ) ) ) ).
thf(tcon_Numeral__Type_Obit1___Rings_Osemiring_945,axiom,
! [A10: $tType] :
( ( finite_finite @ A10 )
=> ( semiring @ ( numeral_bit1 @ A10 ) ) ) ).
thf(tcon_Numeral__Type_Obit1___Orderings_Oord_946,axiom,
! [A10: $tType] :
( ( finite_finite @ A10 )
=> ( ord @ ( numeral_bit1 @ A10 ) ) ) ).
thf(tcon_Numeral__Type_Obit1___Groups_Ouminus_947,axiom,
! [A10: $tType] :
( ( finite_finite @ A10 )
=> ( uminus @ ( numeral_bit1 @ A10 ) ) ) ).
thf(tcon_Numeral__Type_Obit1___Rings_Oring__1_948,axiom,
! [A10: $tType] :
( ( finite_finite @ A10 )
=> ( ring_1 @ ( numeral_bit1 @ A10 ) ) ) ).
thf(tcon_Numeral__Type_Obit1___Groups_Otimes_949,axiom,
! [A10: $tType] :
( ( finite_finite @ A10 )
=> ( times @ ( numeral_bit1 @ A10 ) ) ) ).
thf(tcon_Numeral__Type_Obit1___Groups_Ominus_950,axiom,
! [A10: $tType] :
( ( finite_finite @ A10 )
=> ( minus @ ( numeral_bit1 @ A10 ) ) ) ).
thf(tcon_Numeral__Type_Obit1___Power_Opower_951,axiom,
! [A10: $tType] :
( ( finite_finite @ A10 )
=> ( power @ ( numeral_bit1 @ A10 ) ) ) ).
thf(tcon_Numeral__Type_Obit1___Num_Onumeral_952,axiom,
! [A10: $tType] :
( ( finite_finite @ A10 )
=> ( numeral @ ( numeral_bit1 @ A10 ) ) ) ).
thf(tcon_Numeral__Type_Obit1___Groups_Ozero_953,axiom,
! [A10: $tType] :
( ( finite_finite @ A10 )
=> ( zero @ ( numeral_bit1 @ A10 ) ) ) ).
thf(tcon_Numeral__Type_Obit1___Groups_Oplus_954,axiom,
! [A10: $tType] :
( ( finite_finite @ A10 )
=> ( plus @ ( numeral_bit1 @ A10 ) ) ) ).
thf(tcon_Numeral__Type_Obit1___Rings_Oring_955,axiom,
! [A10: $tType] :
( ( finite_finite @ A10 )
=> ( ring @ ( numeral_bit1 @ A10 ) ) ) ).
thf(tcon_Numeral__Type_Obit1___Groups_Oone_956,axiom,
! [A10: $tType] :
( ( finite_finite @ A10 )
=> ( one @ ( numeral_bit1 @ A10 ) ) ) ).
thf(tcon_Numeral__Type_Obit1___Rings_Odvd_957,axiom,
! [A10: $tType] :
( ( finite_finite @ A10 )
=> ( dvd @ ( numeral_bit1 @ A10 ) ) ) ).
thf(tcon_Numeral__Type_Obit1___HOL_Oequal_958,axiom,
! [A10: $tType] :
( ( finite_finite @ A10 )
=> ( cl_HOL_Oequal @ ( numeral_bit1 @ A10 ) ) ) ).
thf(tcon_Numeral__Type_Onum0___Type__Length_Olen0_959,axiom,
type_len0 @ numeral_num0 ).
thf(tcon_Numeral__Type_Onum0___HOL_Oequal_960,axiom,
cl_HOL_Oequal @ numeral_num0 ).
thf(tcon_Numeral__Type_Onum1___Groups_Ocancel__ab__semigroup__add_961,axiom,
cancel2418104881723323429up_add @ numeral_num1 ).
thf(tcon_Numeral__Type_Onum1___Groups_Ocancel__comm__monoid__add_962,axiom,
cancel1802427076303600483id_add @ numeral_num1 ).
thf(tcon_Numeral__Type_Onum1___Groups_Ocancel__semigroup__add_963,axiom,
cancel_semigroup_add @ numeral_num1 ).
thf(tcon_Numeral__Type_Onum1___Groups_Oab__semigroup__mult_964,axiom,
ab_semigroup_mult @ numeral_num1 ).
thf(tcon_Numeral__Type_Onum1___Groups_Ocomm__monoid__mult_965,axiom,
comm_monoid_mult @ numeral_num1 ).
thf(tcon_Numeral__Type_Onum1___Groups_Oab__semigroup__add_966,axiom,
ab_semigroup_add @ numeral_num1 ).
thf(tcon_Numeral__Type_Onum1___Groups_Ocomm__monoid__add_967,axiom,
comm_monoid_add @ numeral_num1 ).
thf(tcon_Numeral__Type_Onum1___Rings_Ocomm__semiring__0_968,axiom,
comm_semiring_0 @ numeral_num1 ).
thf(tcon_Numeral__Type_Onum1___Groups_Osemigroup__mult_969,axiom,
semigroup_mult @ numeral_num1 ).
thf(tcon_Numeral__Type_Onum1___Groups_Osemigroup__add_970,axiom,
semigroup_add @ numeral_num1 ).
thf(tcon_Numeral__Type_Onum1___Rings_Ocomm__semiring_971,axiom,
comm_semiring @ numeral_num1 ).
thf(tcon_Numeral__Type_Onum1___Orderings_Owellorder_972,axiom,
wellorder @ numeral_num1 ).
thf(tcon_Numeral__Type_Onum1___Groups_Oab__group__add_973,axiom,
ab_group_add @ numeral_num1 ).
thf(tcon_Numeral__Type_Onum1___Orderings_Opreorder_974,axiom,
preorder @ numeral_num1 ).
thf(tcon_Numeral__Type_Onum1___Orderings_Olinorder_975,axiom,
linorder @ numeral_num1 ).
thf(tcon_Numeral__Type_Onum1___Groups_Omonoid__mult_976,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_977,axiom,
monoid_add @ numeral_num1 ).
thf(tcon_Numeral__Type_Onum1___Finite__Set_Ofinite_978,axiom,
finite_finite @ numeral_num1 ).
thf(tcon_Numeral__Type_Onum1___Type__Length_Olen0_979,axiom,
type_len0 @ numeral_num1 ).
thf(tcon_Numeral__Type_Onum1___Rings_Osemiring__0_980,axiom,
semiring_0 @ numeral_num1 ).
thf(tcon_Numeral__Type_Onum1___Groups_Ogroup__add_981,axiom,
group_add @ numeral_num1 ).
thf(tcon_Numeral__Type_Onum1___Type__Length_Olen_982,axiom,
type_len @ numeral_num1 ).
thf(tcon_Numeral__Type_Onum1___Rings_Omult__zero_983,axiom,
mult_zero @ numeral_num1 ).
thf(tcon_Numeral__Type_Onum1___Rings_Ocomm__ring_984,axiom,
comm_ring @ numeral_num1 ).
thf(tcon_Numeral__Type_Onum1___Orderings_Oorder_985,axiom,
order @ numeral_num1 ).
thf(tcon_Numeral__Type_Onum1___Rings_Osemiring_986,axiom,
semiring @ numeral_num1 ).
thf(tcon_Numeral__Type_Onum1___Orderings_Oord_987,axiom,
ord @ numeral_num1 ).
thf(tcon_Numeral__Type_Onum1___Groups_Ouminus_988,axiom,
uminus @ numeral_num1 ).
thf(tcon_Numeral__Type_Onum1___Groups_Otimes_989,axiom,
times @ numeral_num1 ).
thf(tcon_Numeral__Type_Onum1___Groups_Ominus_990,axiom,
minus @ numeral_num1 ).
thf(tcon_Numeral__Type_Onum1___Power_Opower_991,axiom,
power @ numeral_num1 ).
thf(tcon_Numeral__Type_Onum1___Num_Onumeral_992,axiom,
numeral @ numeral_num1 ).
thf(tcon_Numeral__Type_Onum1___Groups_Ozero_993,axiom,
zero @ numeral_num1 ).
thf(tcon_Numeral__Type_Onum1___Groups_Oplus_994,axiom,
plus @ numeral_num1 ).
thf(tcon_Numeral__Type_Onum1___Rings_Oring_995,axiom,
ring @ numeral_num1 ).
thf(tcon_Numeral__Type_Onum1___Groups_Oone_996,axiom,
one @ numeral_num1 ).
thf(tcon_Numeral__Type_Onum1___Rings_Odvd_997,axiom,
dvd @ numeral_num1 ).
thf(tcon_Numeral__Type_Onum1___HOL_Oequal_998,axiom,
cl_HOL_Oequal @ numeral_num1 ).
thf(tcon_Product__Type_Oprod___Topological__Spaces_Otopological__space_999,axiom,
! [A10: $tType,A11: $tType] :
( ( ( topolo4958980785337419405_space @ A10 )
& ( topolo4958980785337419405_space @ A11 ) )
=> ( topolo4958980785337419405_space @ ( product_prod @ A10 @ A11 ) ) ) ).
thf(tcon_Product__Type_Oprod___Topological__Spaces_Ot2__space_1000,axiom,
! [A10: $tType,A11: $tType] :
( ( ( topological_t2_space @ A10 )
& ( topological_t2_space @ A11 ) )
=> ( topological_t2_space @ ( product_prod @ A10 @ A11 ) ) ) ).
thf(tcon_Product__Type_Oprod___Finite__Set_Ofinite_1001,axiom,
! [A10: $tType,A11: $tType] :
( ( ( finite_finite @ A10 )
& ( finite_finite @ A11 ) )
=> ( finite_finite @ ( product_prod @ A10 @ A11 ) ) ) ).
thf(tcon_Product__Type_Oprod___Heap_Oheap_1002,axiom,
! [A10: $tType,A11: $tType] :
( ( ( heap @ A10 )
& ( heap @ A11 ) )
=> ( heap @ ( product_prod @ A10 @ A11 ) ) ) ).
thf(tcon_Product__Type_Oprod___HOL_Oequal_1003,axiom,
! [A10: $tType,A11: $tType] : ( cl_HOL_Oequal @ ( product_prod @ A10 @ A11 ) ) ).
thf(tcon_Product__Type_Oprod___Nat_Osize_1004,axiom,
! [A10: $tType,A11: $tType] : ( size @ ( product_prod @ A10 @ A11 ) ) ).
thf(tcon_Product__Type_Ounit___Conditionally__Complete__Lattices_Oconditionally__complete__linorder_1005,axiom,
condit6923001295902523014norder @ product_unit ).
thf(tcon_Product__Type_Ounit___Boolean__Algebras_Oboolean__algebra_1006,axiom,
boolea8198339166811842893lgebra @ product_unit ).
thf(tcon_Product__Type_Ounit___Orderings_Owellorder_1007,axiom,
wellorder @ product_unit ).
thf(tcon_Product__Type_Ounit___Orderings_Oorder__top_1008,axiom,
order_top @ product_unit ).
thf(tcon_Product__Type_Ounit___Orderings_Oorder__bot_1009,axiom,
order_bot @ product_unit ).
thf(tcon_Product__Type_Ounit___Orderings_Opreorder_1010,axiom,
preorder @ product_unit ).
thf(tcon_Product__Type_Ounit___Orderings_Olinorder_1011,axiom,
linorder @ product_unit ).
thf(tcon_Product__Type_Ounit___Finite__Set_Ofinite_1012,axiom,
finite_finite @ product_unit ).
thf(tcon_Product__Type_Ounit___Orderings_Oorder_1013,axiom,
order @ product_unit ).
thf(tcon_Product__Type_Ounit___Orderings_Oord_1014,axiom,
ord @ product_unit ).
thf(tcon_Product__Type_Ounit___Groups_Ouminus_1015,axiom,
uminus @ product_unit ).
thf(tcon_Product__Type_Ounit___Groups_Ominus_1016,axiom,
minus @ product_unit ).
thf(tcon_Product__Type_Ounit___Heap_Oheap_1017,axiom,
heap @ product_unit ).
thf(tcon_Product__Type_Ounit___HOL_Oequal_1018,axiom,
cl_HOL_Oequal @ product_unit ).
thf(tcon_Heap_Oheap_Oheap__ext___HOL_Oequal_1019,axiom,
! [A10: $tType] : ( cl_HOL_Oequal @ ( heap_ext @ A10 ) ) ).
thf(tcon_Code__Numeral_Ointeger___Bit__Operations_Ounique__euclidean__semiring__with__bit__operations_1020,axiom,
bit_un5681908812861735899ations @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Semiring__Normalization_Ocomm__semiring__1__cancel__crossproduct_1021,axiom,
semiri1453513574482234551roduct @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Euclidean__Division_Ounique__euclidean__semiring__with__nat_1022,axiom,
euclid5411537665997757685th_nat @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Euclidean__Division_Ounique__euclidean__ring__with__nat_1023,axiom,
euclid8789492081693882211th_nat @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Groups_Oordered__ab__semigroup__monoid__add__imp__le_1024,axiom,
ordere1937475149494474687imp_le @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Euclidean__Division_Ounique__euclidean__semiring_1025,axiom,
euclid3128863361964157862miring @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Euclidean__Division_Oeuclidean__semiring__cancel_1026,axiom,
euclid4440199948858584721cancel @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Divides_Ounique__euclidean__semiring__numeral_1027,axiom,
unique1627219031080169319umeral @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Euclidean__Division_Oeuclidean__ring__cancel_1028,axiom,
euclid8851590272496341667cancel @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Rings_Osemiring__no__zero__divisors__cancel_1029,axiom,
semiri6575147826004484403cancel @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Groups_Ostrict__ordered__ab__semigroup__add_1030,axiom,
strict9044650504122735259up_add @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Groups_Oordered__cancel__ab__semigroup__add_1031,axiom,
ordere580206878836729694up_add @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Groups_Oordered__ab__semigroup__add__imp__le_1032,axiom,
ordere2412721322843649153imp_le @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Bit__Operations_Osemiring__bit__operations_1033,axiom,
bit_se359711467146920520ations @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Rings_Olinordered__comm__semiring__strict_1034,axiom,
linord2810124833399127020strict @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Groups_Ostrict__ordered__comm__monoid__add_1035,axiom,
strict7427464778891057005id_add @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Groups_Oordered__cancel__comm__monoid__add_1036,axiom,
ordere8940638589300402666id_add @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Euclidean__Division_Oeuclidean__semiring_1037,axiom,
euclid3725896446679973847miring @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Rings_Olinordered__semiring__1__strict_1038,axiom,
linord715952674999750819strict @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Groups_Olinordered__ab__semigroup__add_1039,axiom,
linord4140545234300271783up_add @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Bit__Operations_Oring__bit__operations_1040,axiom,
bit_ri3973907225187159222ations @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Rings_Osemiring__1__no__zero__divisors_1041,axiom,
semiri2026040879449505780visors @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Rings_Olinordered__nonzero__semiring_1042,axiom,
linord181362715937106298miring @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Euclidean__Division_Oeuclidean__ring_1043,axiom,
euclid5891614535332579305n_ring @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Rings_Olinordered__semiring__strict_1044,axiom,
linord8928482502909563296strict @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Rings_Osemiring__no__zero__divisors_1045,axiom,
semiri3467727345109120633visors @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Groups_Oordered__ab__semigroup__add_1046,axiom,
ordere6658533253407199908up_add @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Groups_Oordered__ab__group__add__abs_1047,axiom,
ordere166539214618696060dd_abs @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Groups_Oordered__comm__monoid__add_1048,axiom,
ordere6911136660526730532id_add @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Groups_Olinordered__ab__group__add_1049,axiom,
linord5086331880401160121up_add @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Groups_Ocancel__ab__semigroup__add_1050,axiom,
cancel2418104881723323429up_add @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Rings_Oring__1__no__zero__divisors_1051,axiom,
ring_15535105094025558882visors @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Groups_Ocancel__comm__monoid__add_1052,axiom,
cancel1802427076303600483id_add @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Rings_Olinordered__ring__strict_1053,axiom,
linord4710134922213307826strict @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Rings_Ocomm__semiring__1__cancel_1054,axiom,
comm_s4317794764714335236cancel @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Bit__Operations_Osemiring__bits_1055,axiom,
bit_semiring_bits @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Rings_Oordered__comm__semiring_1056,axiom,
ordere2520102378445227354miring @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Rings_Olinordered__semiring__1_1057,axiom,
linord6961819062388156250ring_1 @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Groups_Oordered__ab__group__add_1058,axiom,
ordered_ab_group_add @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Groups_Ocancel__semigroup__add_1059,axiom,
cancel_semigroup_add @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Rings_Olinordered__semiring_1060,axiom,
linordered_semiring @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Least__significant__bit_Olsb_1061,axiom,
least_6119777620449941438nt_lsb @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Rings_Oordered__semiring__0_1062,axiom,
ordered_semiring_0 @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Rings_Olinordered__semidom_1063,axiom,
linordered_semidom @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Groups_Oab__semigroup__mult_1064,axiom,
ab_semigroup_mult @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Rings_Osemiring__1__cancel_1065,axiom,
semiring_1_cancel @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Rings_Oalgebraic__semidom_1066,axiom,
algebraic_semidom @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Groups_Ocomm__monoid__mult_1067,axiom,
comm_monoid_mult @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Groups_Oab__semigroup__add_1068,axiom,
ab_semigroup_add @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Generic__set__bit_Oset__bit_1069,axiom,
generic_set_set_bit @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Rings_Oordered__semiring_1070,axiom,
ordered_semiring @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Rings_Oordered__ring__abs_1071,axiom,
ordered_ring_abs @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Parity_Osemiring__parity_1072,axiom,
semiring_parity @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Groups_Ocomm__monoid__add_1073,axiom,
comm_monoid_add @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Rings_Osemiring__modulo_1074,axiom,
semiring_modulo @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Rings_Olinordered__ring_1075,axiom,
linordered_ring @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Rings_Olinordered__idom_1076,axiom,
linordered_idom @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Rings_Ocomm__semiring__1_1077,axiom,
comm_semiring_1 @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Rings_Ocomm__semiring__0_1078,axiom,
comm_semiring_0 @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Groups_Osemigroup__mult_1079,axiom,
semigroup_mult @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Rings_Osemidom__modulo_1080,axiom,
semidom_modulo @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Rings_Osemidom__divide_1081,axiom,
semidom_divide @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Num_Osemiring__numeral_1082,axiom,
semiring_numeral @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Groups_Osemigroup__add_1083,axiom,
semigroup_add @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Rings_Ozero__less__one_1084,axiom,
zero_less_one @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Rings_Ocomm__semiring_1085,axiom,
comm_semiring @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Nat_Osemiring__char__0_1086,axiom,
semiring_char_0 @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Groups_Oab__group__add_1087,axiom,
ab_group_add @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Rings_Ozero__neq__one_1088,axiom,
zero_neq_one @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Rings_Oordered__ring_1089,axiom,
ordered_ring @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Rings_Oidom__abs__sgn_1090,axiom,
idom_abs_sgn @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Parity_Oring__parity_1091,axiom,
ring_parity @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Orderings_Opreorder_1092,axiom,
preorder @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Orderings_Olinorder_1093,axiom,
linorder @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Groups_Omonoid__mult_1094,axiom,
monoid_mult @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Rings_Oidom__modulo_1095,axiom,
idom_modulo @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Rings_Oidom__divide_1096,axiom,
idom_divide @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Rings_Ocomm__ring__1_1097,axiom,
comm_ring_1 @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Groups_Omonoid__add_1098,axiom,
monoid_add @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Rings_Osemiring__1_1099,axiom,
semiring_1 @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Rings_Osemiring__0_1100,axiom,
semiring_0 @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Groups_Ogroup__add_1101,axiom,
group_add @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Rings_Omult__zero_1102,axiom,
mult_zero @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Rings_Ocomm__ring_1103,axiom,
comm_ring @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Orderings_Oorder_1104,axiom,
order @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Num_Oneg__numeral_1105,axiom,
neg_numeral @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Nat_Oring__char__0_1106,axiom,
ring_char_0 @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Rings_Osemiring_1107,axiom,
semiring @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Rings_Osemidom_1108,axiom,
semidom @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Orderings_Oord_1109,axiom,
ord @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Groups_Ouminus_1110,axiom,
uminus @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Rings_Oring__1_1111,axiom,
ring_1 @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Rings_Oabs__if_1112,axiom,
abs_if @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Groups_Otimes_1113,axiom,
times @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Groups_Ominus_1114,axiom,
minus @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Power_Opower_1115,axiom,
power @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Num_Onumeral_1116,axiom,
numeral @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Groups_Ozero_1117,axiom,
zero @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Groups_Oplus_1118,axiom,
plus @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Rings_Oring_1119,axiom,
ring @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Rings_Oidom_1120,axiom,
idom @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Groups_Oone_1121,axiom,
one @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Rings_Odvd_1122,axiom,
dvd @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___HOL_Oequal_1123,axiom,
cl_HOL_Oequal @ code_integer ).
thf(tcon_Heap__Time__Monad_OHeap___HOL_Oequal_1124,axiom,
! [A10: $tType] : ( cl_HOL_Oequal @ ( heap_Time_Heap @ A10 ) ) ).
thf(tcon_Heap__Time__Monad_OHeap___Nat_Osize_1125,axiom,
! [A10: $tType] : ( size @ ( heap_Time_Heap @ A10 ) ) ).
thf(tcon_VEBT__Definitions_OVEBT___HOL_Oequal_1126,axiom,
cl_HOL_Oequal @ vEBT_VEBT ).
thf(tcon_VEBT__Definitions_OVEBT___Nat_Osize_1127,axiom,
size @ vEBT_VEBT ).
thf(tcon_VEBT__BuildupMemImp_OVEBTi___Heap_Oheap_1128,axiom,
heap @ vEBT_VEBTi ).
thf(tcon_VEBT__BuildupMemImp_OVEBTi___HOL_Oequal_1129,axiom,
cl_HOL_Oequal @ vEBT_VEBTi ).
thf(tcon_VEBT__BuildupMemImp_OVEBTi___Nat_Osize_1130,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 ) )
= ( ? [X8: A] : ( P @ X8 ) ) ) ).
% Conjectures (1)
thf(conj_0,conjecture,
( entails
@ ( times_times @ assn @ ( times_times @ assn @ ( times_times @ assn @ ( snga_assn @ vEBT_VEBTi @ xa @ x ) @ ( vEBT_List_list_assn @ vEBT_VEBT @ vEBT_VEBTi @ vEBT_vebt_assn_raw @ ( replicate @ vEBT_VEBT @ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ ( bit0 @ one2 ) ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ na @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( vEBT_vebt_buildup @ ( suc @ ( divide_divide @ nat @ na @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) @ x ) ) @ ( vEBT_vebt_assn_raw @ ( vEBT_vebt_buildup @ ( suc @ ( divide_divide @ nat @ ( suc @ ( suc @ na ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ xb ) )
@ ( pure_assn
@ ( xc
= ( vEBT_Nodei @ ( none @ ( product_prod @ nat @ nat ) ) @ ( suc @ ( suc @ na ) ) @ xa @ xb ) ) ) )
@ ( vEBT_vebt_assn_raw @ ( vEBT_vebt_buildup @ ( suc @ ( suc @ na ) ) ) @ xc ) ) ).
%------------------------------------------------------------------------------